Search Results for Series


Biographies

  1. Nina Bari (1901-1961)
    • In fact, it was then that she first began to study the uniqueness problem for trigonometric series.
    • She undertook research on the theory of trigonometrical series and her major results were announced in her first paper Sur l'unicite du developpement trigonometrique Ⓣ published by the Academie des Sciences in Paris in 1923.
    • Already this first piece of work by Nina Bari testified to her great mathematical talent, since it included the solution of several very difficult problems in the theory of trigonometric series that had lately been engaging the attention of many outstanding mathematicians.
    • Her numerous papers exerted a great influence on the development of such fundamental branches of the theory of functions as the theory of trigonometric series, orthogonal series and bases, etc.
    • Several of her investigations are rightly regarded as classics, for example her papers on the theory of uniqueness for trigonometric series and on the superpositions of functions.
    • Let us mention in particular her paper The uniqueness problem of the representation of functions by trigonometric series (Russian) which she published in 1949.
    • an exhaustive review, in many cases accompanied by complete proofs, of the existing results in the theory of uniqueness of representation of functions by trigonometric series.
    • In [',' D E Cameron and J Spetich, Nina Karlovna Bari, in L S Grinstein and P J Campbell (eds.), Women of Mathematics (Greenwood, Westport, Conn., 1987), 6-12.','1] her final publication, a 936-page research monograph on trigonometric series, is described as follows:- .
    • It has become a standard reference for mathematicians specializing in the theory of functions and the theory of trigonometric series.
    • The fifteen chapters of the book are: Basic concepts and theorems; Fourier coefficients; Convergence of a Fourier series at a point; Fourier series of continuous functions; Convergence and divergence of a Fourier series on a set; "Correcting" a function on a set of small measure; Summability of Fourier series; Conjugate trigonometric series; Absolute convergence of Fourier series; Sine and cosine series with decreasing coefficients; Lacunary series; Convergence and divergence of general trigonometric series; Absolute convergence of general trigonometric series; The problem of uniqueness of the expansion of a function in a trigonometric series; and Representation of functions by trigonometric series.
    • Bari also wrote textbooks for use in teaching training institutes such as Higher Algebra (1932) and The Theory of Series (1936).

  2. Antoni Zygmund (1900-1992)
    • Aleksander Rajchman was interested in the theory of trigonometric series.
    • This gave Zygmund a life-long interest in the trigonometric series.
    • from the University of Warsaw in 1923 for a dissertation on the Riemannian theory of trigonometric series written under Aleksander Rajchman's supervision.
    • Among other topics, he worked on summability of numerical series, summability of general orthogonal series, trigonometric integrals, sets of uniqueness, summability of Fourier series, differentiability of functions, smooth functions, approximation theory, absolutely convergent Fourier series, multipliers and translation invariant operators, conjugate series and Taylor series, lacunary trigonometric series, series of independent random variables, random trigonometric series, the Littlewood-Paley, Luzin and Marcinkiewicz functions, boundary values of analytic and harmonic functions, singular integrals, partial differential equations and interpolation operators.
    • He studied topics such as Riemann summability, differentiability properties of trigonometric series and sets of uniqueness.
    • For example in 1926 he published six papers in Mathematische Zeitschrift in French: Contribution a l'unicite du developpement trigonometrique Ⓣ; Sur la theorie riemannienne des series trigonometriques Ⓣ; Sur la possibilite d'appliquer la methode de Riemann aux series trigonometriques sommables par le procede de Poisson Ⓣ; Sur les series trigonometriques sommables par le procede de Poisson Ⓣ; Sur un theoreme de la theorie de la sommabilite Ⓣ and Une remarque sur un theoreme de M Kaczmarz Ⓣ.
    • The joint Zygmund-Paley work played an important role in Zygmund's book Trigonometric Series (1935).
    • Each volume of the series [Monografie Mat.] published so far represents an important event in the development of mathematical research, and the present volume in this respect is second to none of its predecessors.
    • If one looks through the long list of books on Fourier series one cannot help feeling that even the bulkiest of them are far from giving an adequate picture of the present status of the field.
    • The non-existence of a monograph giving such a picture was very badly felt not only by beginners but also by specialists, and the failure of so many attempts to write a real book on Fourier series created an impression that the task was almost hopeless.
    • The author of the present monograph completely succeeded in dispelling this "inferiority complex" and produced a book which not only introduces the reader into the immense field the theory of Fourier series but at the same time almost imperceptibly brings him to the latest achievements, many of them being due author himself.
    • Surely, Antoni Zygmund's "Trigonometric series" has been, and continues to be, one of the most influential books in the history of mathematical analysis.
    • Generations of mathematicians from Hardy and Littlewood to recent classes of graduate students specializing in analysis have viewed "Trigonometric series" with enormous admiration and have profited greatly from reading it.
    • In light of the importance of Antoni Zygmund as a mathematician and of the impact of "Trigonometric series", it is only fitting that a brief discussion of his life and mathematics accompany the present volume, and this is what I have attempted to give here.
    • The general part of the book is followed by three chapters, one on entire and meromorphic functions, one on elliptic functions, and one on G(s), Z(s) and Dirichlet series.
    • The purpose of these notes is to present those aspects of trigonometric interpolation which resemble the theory of Fourier series.

  3. Mikhail Iosiphovich Kadets (1923-2011)
    • Baltaga, who gave the course on mathematical analysis, drew Kadets' attention to the question of infinite-dimensional extensions of the theorem of Steinitz on conditionally convergent series.
    • From 1965, Kadets worked at the Kharkov State Academy of Municipal Economy as well as at Kharkov University where he gave courses on Functional Analysis and, in particular, high powered courses on Banach spaces such as 'Series in Banach spaces', 'Biorthogonal systems and bases', 'Theory of renormings' and similar specialist topics related to his research interests.
    • The two Kadets, father and son, coauthored the book Rearrangements of series in Banach spaces (Russian) (1988).
    • The "sum-set" of a series will mean the set of all sums of convergent rearrangements of the series.
    • Steinitz's theorem says that if ∑xn is a convergent series in a finite-dimensional Banach space X, with sum s, then its sum-set is a linear subset of X ..
    • In particular, a result of V M Kadets states that every infinite-dimensional space contains series with nonconvex sum-set.
    • The book also includes some of the theory of unconditionally convergent series (for which, of course, rearrangement introduces no new points).
    • The two authors published Series in Banach spaces.
    • Series of scalars, vectors, or functions are among the fundamental objects of mathematical analysis.
    • When the arrangement of the terms is fixed, investigating a series amounts to investigating the sequence of its partial sums.
    • In this case the theory of series is a part of the theory of sequences, which deals with their convergence, asymptotic behaviour etc.
    • The specific character of the theory of series manifests itself when one considers rearrangements (permutations) of the terms of a series, which brings combinatorial considerations into the problem studied.
    • The phenomenon that a numerical series can change its sum when the order of its terms is changed is one of the most impressive facts encountered in a university analysis course.
    • The present book is devoted precisely to this aspect of the theory of series whose terms are elements of Banach (as well as other topological linear) spaces.
    • The first is to characterise those series in a given space that remain convergent (and have the same sum) for any rearrangement of their terms; such series are usually called unconditionally convergent.
    • The second problem is, when a series converges only for certain rearrangements of its terms (in other words, converges conditionally), to describe its sum range, i.e.
    • His interests are very similar to those of his father, namely Banach space theory, in particular the study of sequences and series, bases, vector-valued measures and integration, isomorphic and isometric structures of Banach spaces, and operators in Banach spaces.

  4. Madhava (1350-1425)
    • In [',' T Hayashi, T Kusuba and M Yano, The correction of the Madhava series for the circumference of a circle, Centaurus 33 (2-3) (1990), 149-174.','10] Rajagopal and Rangachari put his achievement into context when they write:- .
    • Madhava discovered the series equivalent to the Maclaurin expansions of sin x, cos x, and arctan x around 1400, which is over two hundred years before they were rediscovered in Europe.
    • This is discussed in detail in [',' D Gold and D Pingree, A hitherto unknown Sanskrit work concerning Madhava’s derivation of the power series for sine and cosine, Historia Sci.
    • In [',' R C Gupta, The Madhava-Gregory series, Math.
    • Jyesthadeva describes Madhava's series as follows:- .
    • This is a remarkable passage describing Madhava's series, but remember that even this passage by Jyesthadeva was written more than 100 years before James Gregory rediscovered this series expansion.
    • Perhaps we should write down in modern symbols exactly what the series is that Madhava has found.
    • Thus the series is .
    • which is equivalent to Gregory's series .
    • Now Madhava put q = π/4 into his series to obtain .
    • and he also put θ = π/6 into his series to obtain .
    • which can be obtained from the last of Madhava's series above by taking 21 terms.
    • Perhaps even more impressive is the fact that Madhava gave a remainder term for his series which improved the approximation.
    • He improved the approximation of the series for π/4 by adding a correction term Rn to obtain .
    • It is thought that the way that he found these highly accurate tables was to use the equivalent of the series expansions .
    • Jyesthadeva in Yukti-Bhasa gave an explanation of how Madhava found his series expansions around 1400 which are equivalent to these modern versions rediscovered by Newton around 1676.

  5. Charles Whish (1795-1833)
    • Whish discovered that certain Hindu texts contained approximations to π which had been found using series expansions.
    • For example Hyne wrote [',' U K V Sarma, V Bhat, V Pai and K Ramasubramanian, The discovery of Madhava series by Whish: an episode in historiography of science, Ganita Bharati 32 (1) (2010), 115-126.','14]:- .
    • the Hindus never invented the series; it was communicated with many others, by Europeans, to some learned Natives in modern times ..
    • Although Whish believed at first that these series had been found by the Hindus, his two colleagues made him change his opinion.
    • The following written by Warren shows this [',' U K V Sarma, V Bhat, V Pai and K Ramasubramanian, The discovery of Madhava series by Whish: an episode in historiography of science, Ganita Bharati 32 (1) (2010), 115-126.','14]:- .
    • in Mr Hyne's opinion the Hindus never invented the series referring in the Quadrature of the Circle which were found in their possession in various parts of India; and that Mr Whish, from whom he had obtained some of those which were communicated to the Madras Literary Society, after having first expressed a belief that they were indigenous, had subsequently reasons for thinking them entirely modern, and derived from the Europeans; observing that not one of the Jyautish Sastras who used these Rules, were capable of demonstrating them.
    • So at this stage Whish, Warren and Hyne believed that the series must have been communicated to the Hindus by Europeans because the Hindus who told them about the series were not able to prove that the results were correct.
    • Whish had been convinced by his older colleagues to change his view and believe that the series had been given to the Hindus by Europeans, but he reverted to his earlier opinion when he discovered proofs of the results in the Yuktibhasa.
    • He published a now famous paper On the Hindu Quadrature of the Circle, and the Infinite Series of the Proportion of the Circumference to the Diameter Exhibited in the Four Sastras, the Tantra Sangraham, Yucti Bhasha, Carana Padhati, and Sadratnamala which was published in the Transactions of the Royal Asiatic Society of Great Britain and Ireland in 1834.
    • Having thus submitted to the inspection of the curious eight different infinite series, extracted from Brahmanical works for the quadrature of the circle, it will be proper to explain by what steps the Hindu mathematician has been led to these forms, which have only been made known to Europeans, through the method of fluxions, the invention of the illustrious Newton.
    • it is a fact which I have ascertained beyond a doubt, that the invention of infinite series of these forms has originated in Malabar, and is not, even to this day, known to the eastward of the range of Ghats which divides that country, called in the earliest times Ceralam, from the countries of Madura, Coimbatore, Mysore, and those in succession, to that northward of these provinces.
    • A further account of the Yuktibhasa, the demonstrations of the rules for the quadrature of the circle by infinite series, with the series for the sines, cosines, and their demonstrations, will be given in a separate paper.
    • It seems that these findings first came to light in 1834 when Charles Whish published a paper in the 'Transactions of the Royal Asiatic Society' entitled "On the Hindu quadrature of the circle and the infinite series of the proportion of the circumference to the diameter exhibited in the four sastras, the Tantrasangraham, Yukti Bhasha, Caruna Paddhati, and Sadratnamala".
    • One example I can give you relates to the Indian Madhava's demonstration, in about 1400 A.D., of the infinite power series of trigonometrical functions using geometrical and algebraic arguments.
    • The matter resurfaced in the 1950s, and now we have the Sanskrit texts properly edited, and we understand the clever way that Madhava derived the series without the calculus; but many historians still find it impossible to conceive of the problem and its solution in terms of anything other than the calculus and proclaim that the calculus is what Madhava found.

  6. James Stirling (1692-1770)
    • The terms of the Snell Exhibitions is described in [',' I Tweddle, James Stirling: this about series and such things (Edinburgh, 1988).','3]:- .
    • Tweddle [',' I Tweddle, James Stirling: this about series and such things (Edinburgh, 1988).','3] notes that a student with the name 'James Stirling' matriculated at the University of Edinburgh on 24 March 1710, did not graduate, and has a signature which is similar to that of the mathematician.
    • This book is a treatise on infinite series, summation, interpolation and quadrature.
    • One of the main aims of the book was to consider methods of speeding up the convergence of series.
    • As an example of the problem he is trying to solve Stirling gives the example of the series ∑ 1/[',' Biography in Encyclopaedia Britannica.','2n(2n-1)] which had been studied by Brouncker in his work on the area under a hyperbola.
    • .if anyone would find an accurate value of this series to nine places ..
    • they would require one thousand million of terms; and this series converges much swifter than many others..
    • For example he defined the series Tn+1 = nTn with T1 = 1.
    • However, before doing so we will look at a correspondence that Stirling had with Euler since this relates to the work we have just discussed on series.
    • 10 (1957), 117-158.','7] or [',' I Tweddle, James Stirling: this about series and such things (Edinburgh, 1988).','3]):- .
    • the more I have learned from your excellent articles, which I have seen here and there in your Transactions, concerning the nature of series, a study in which I have indeed expended much effort, the more I have wished to become acquainted with you in order that I could receive more from you yourself and also submit my own deliberations to your judgement.
    • Now that I have read through it diligently, I am truly astonished at the great abundance of excellent methods contained in such a small volume, by means of which you show how to sum slowly converging series with ease and how to interpolate progressions which are very difficult to deal with.
    • But especially pleasing to me was proposition XIV of part 1 in which you give a method by which series, whose law of progression is not even established, may be summed with great ease using only the relation of the last terms, certainly this method extends very widely and is of the greatest use.
    • 10 (1957), 117-158.','7] or [',' I Tweddle, James Stirling: this about series and such things (Edinburgh, 1988).','3]):- .
    • He certainly did not give up mathematics when he took up the post in the mining company, and in [',' I Tweddle, James Stirling: this about series and such things (Edinburgh, 1988).','3] there is a discussion of unpublished mathematical work in notebooks of Stirling that were probably written between 1730 and 1745.
    • he surveyed the Clyde with a view to rendering it navigable by a series of locks, thus taking the first step towards making Glasgow the commercial capital of Scotland.
    • As Stirling's unpublished manuscripts show [',' I Tweddle, James Stirling: this about series and such things (Edinburgh, 1988).','3], he did go much further than the 1735 paper but probably the pressure of work at the mining company gave him too little time to polish the work.

  7. Brook Taylor (1685-1731)
    • The first of these books contains what is now known as the Taylor series, though it would only be known as this in 1785.
    • In particular they discussed infinite series and probability.
    • His life, however, suffered a series of personal tragedies beginning around 1721.
    • Taylor added to mathematics a new branch now called the "calculus of finite differences", invented integration by parts, and discovered the celebrated series known as Taylor's expansion.
    • It was, wrote Taylor, due to a comment that Machin made in Child's Coffeehouse when he had commented on using "Sir Isaac Newton's series" to solve Kepler's problem, and also using "Dr Halley's method of extracting roots" of polynomial equations.
    • Taylor initially derived the version which occurs as Proposition 11 as a generalisation of Halley's method of approximating roots of the Kepler equation, but soon discovered that it was a consequence of the Bernoulli series.
    • The second version occurs as Corollary 2 to Proposition 7 and was thought of as a method of expanding solutions of fluxional equations in infinite series.
    • and his methods are discussed in [',' A Malet, James Gregorie on tangents and the ’Taylor’ rule for series expansions, Arch.
    • The differences in Newton's ideas of Taylor series and those of Gregory are discussed in [',' S S Petrova and D A Romanovska, On the history of the discovery of Taylor series (Russian), Istor.-Mat.
    • The term "Taylor's series" seems to have used for the first time by Lhuilier in 1786.
    • Taylor series for sine .
    • Taylor series for cosine .
    • Taylor series .

  8. Dmitrii Menshov (1892-1988)
    • Menshov's first degree was awarded in 1916 for the thesis which he wrote on The Riemann theory of trigonometric series which was examined by Egorov and Luzin.
    • However, only three weeks after he graduated, Menshov discovered one of his most fundamental results on the uniqueness problem for trigonometric series.
    • Consider the trigonometric series .
    • Cantor had proved that if this series converges to 0 for all x in [0, 2π] - E, for a countable set E, then an = bn = 0 for all n.
    • Vallee Poussin had proved that if the above series converged to a finite Lebesgue integrable function f (x) then the given series is the Fourier series of f (x).
    • The remarkable, and unexpected, result that Menshov discovered in 1916 was that this was not so, for he constructed a trigonometric series which converges to 0 for all x in [0, 2π] - E, for a set E of measure zero, yet not all the coefficients of the trigonometric series are zero.
    • His scientific interests relate principally to the theory of trigonometric series, the theory of orthogonal series and the problem of monogenity of functions of a complex variable.
    • For his work on the representation of functions by trigonometric series, Menshov was awarded a State Prize in 1951.
    • In 1958 Menshov attended the International Congress of Mathematicians in Edinburgh and he was invited to address the Congress with his paper On the convergence of trigonometric series.

  9. Srinivasa Ramanujan (1887-1920)
    • He made substantial contributions to the analytical theory of numbers and worked on elliptic functions, continued fractions, and infinite series.
    • In 1900 he began to work on his own on mathematics summing geometric and arithmetic series.
    • He investigated the series ∑(1/n) and calculated Euler's constant to 15 decimal places.
    • He continued his mathematical work, however, and at this time he worked on hypergeometric series and investigated relations between integrals and series.
    • Continuing his mathematical work Ramanujan studied continued fractions and divergent series in 1908.
    • Hill replied in a fairly encouraging way but showed that he had failed to understand Ramanujan's results on divergent series.
    • The recommendation to Ramanujan that he read Bromwich's Theory of infinite series did not please Ramanujan much.
    • I have made a special investigation of divergent series in general and the results I get are termed by the local mathematicians as 'startling'.
    • Ramanujan worked out the Riemann series, the elliptic integrals, hypergeometric series and functional equations of the zeta function.
    • Ramanujan independently discovered results of Gauss, Kummer and others on hypergeometric series.
    • Ramanujan's own work on partial sums and products of hypergeometric series have led to major development in the topic.

  10. Raymond Brink (1890-1973)
    • He was awarded his doctorate in 1916 for his thesis Some Integral Tests for the Convergence and Divergence of Infinite Series.
    • He published the paper A New Integral Test for the Convergence and Divergence of Infinite Series (1918) which was based on his thesis.
    • A new sequence of integral tests for the convergence and divergence of infinite series has been developed by the author.
    • These he published in the paper A New Sequence of Integral Tests for the Convergence and Divergence of Infinite Series which appeared in the Annals of Mathematics in 1919.
    • Finally we give some examples of Brink's papers: A new integral test for the convergence and divergence of infinite series (1918); A new sequence of integral tests for the convergence and divergence of infinite series (1919); The May Meeting of the Minnesota Section (1927); Recent Publications: Reviews: Studies in the History of Statistical Method - With Special Reference to Certain Education Problems (1929); The May Meeting of the Minnesota Section (1930); A Simplified Integral Test for the Convergence of Infinite Series (1931); Recent Publications: Reviews: Differential Equations (1932); The Annual Meeting of the Minnesota Section (1937); College Mathematics During Reconstruction (1944), and A Course in Mathematics for the Purpose of General Education (1947).
    • The paper A Simplified Integral Test for the Convergence of Infinite Series (1931) is really a follow-up paper to his 1918 and 1919 papers.
    • In other papers the author has presented certain integral tests for the convergence and divergence of infinite series.
    • Such tests are interesting not only because they can be used for testing types of series which are very difficult to examine by other methods, but also because, through the natural connection between integration and summation, they offer a simple and attractive means of unifying and establishing many tests of other kinds.
    • Du Bois-Reymond gave the name "tests of the first kind" to series tests which make direct use of the general term of the series itself.
    • In a similar way, the familiar Maclaurin-Cauchy integral test, which requires the use of a function u(x) where u(n) is the general term of the series, may be called an "integral test of the first kind." And an "integral test of the second kind" is one in which an analogous role is played by a function r(x), where r(n) is the ratio of the nth term to the preceding term.

  11. Leonhard Euler (1707-1783)
    • This was to find a closed form for the sum of the infinite series ζ(2) = ∑ (1/n2), a problem which had defeated many of the top mathematicians including Jacob Bernoulli, Johann Bernoulli and Daniel Bernoulli.
    • In 1737 he proved the connection of the zeta function with the series of prime numbers giving the famous relation .
    • Other work done by Euler on infinite series included the introduction of his famous Euler's constant γ, in 1735, which he showed to be the limit of .
    • Euler also studied Fourier series and in 1744 he was the first to express an algebraic function by such a series when he gave the result .
    • Euler wrote to James Stirling on 8 June 1736 telling him about his results on summing reciprocals of powers, the harmonic series and Euler's constant and other results on series.
    • Concerning the summation of very slowly converging series, in the past year I have lectured to our Academy on a special method of which I have given the sums of very many series sufficiently accurately and with very little effort.
    • he has two theorems for summing series by means of derivatives of the terms, one of which is the self-same result that you sent me.
    • I have very little desire for anything to be detracted from the fame of the celebrated Mr Maclaurin since he probably came upon the same theorem for summing series before me, and consequently deserves to be named as its first discoverer.
    • He considered linear equations with constant coefficients, second order differential equations with variable coefficients, power series solutions of differential equations, a method of variation of constants, integrating factors, a method of approximating solutions, and many others.

  12. Nilakantha (1444-1544)
    • The series π/4 = 1 - 1/3 + 1/5 - 1/7 + ..
    • is a special case of the series representation for arctan, namely .
    • It is well known that one simply puts x = 1 to obtain the series for π/4.
    • The author of [',' R Roy, The discovery of the series formula for š by Leibniz, Gregory and Nilakantha, Math.
    • 63 (5) (1990), 291-306.','4] reports on the appearance of these series in the work of Leibniz and James Gregory from the 1670s.
    • The contributions of the two European mathematicians to this series are well known but in [',' R Roy, The discovery of the series formula for š by Leibniz, Gregory and Nilakantha, Math.
    • 63 (5) (1990), 291-306.','4] the results on this series in the work of Madhava nearly three hundred years earlier as presented by Nilakantha in the Tantrasamgraha is also discussed.
    • Nilakantha derived the series expansion .
    • An interesting feature of his work was his introduction of several additional series for π/4 that converged more rapidly than .
    • The author of [',' R Roy, The discovery of the series formula for š by Leibniz, Gregory and Nilakantha, Math.
    • 63 (5) (1990), 291-306.','4] provides a reconstruction of how he may have arrived at these results based on the assumption that he possessed a certain continued fraction representation for the tail series .

  13. Pietro Mengoli (1626-1686)
    • Mengoli used infinite series to good effect in Novae quadraturae arithmeticae, seu de additione fractionum Ⓣ published in Bologna in 1650, developing ideas which had first been investigated by Cataldi.
    • He begins with the summation of geometric series, then shows that the harmonic series does not converge.
    • In doing so he became the first person to prove that it was possible for a series whose terms tend to zero to be made larger than any given number.
    • He also investigated the harmonic series with alternating signs which he proved converges to log(2).
    • This series was also investigated by Nicolaus Mercator.
    • Other interesting results about series in Novae quadraturae arithmeticae Ⓣ include a study of the sum of reciprocals of the triangular numbers ½n(n+1).
    • He then argued that the difference 1 - n-1/n+1 could be made smaller than any given positive number by taking n large enough so the sum of the series was 1.
    • , 10 Mengoli showed that the series whose nth term is 1/n(n+r) converges having sum S where .
    • He also showed that the series whose nth term was 1/n(n+1)(n+2) converges with sum 1/4.
    • However, perhaps not surprisingly, he failed to be able to sum the series whose nth term is 1/n2.
    • He defines limits of positive variable quantities using ideas that he had used in looking at limits of series.

  14. Raphaël Salem (1898-1963)
    • Whenever he had free time in the evenings he worked on Fourier series, a topic which interested him throughout his life.
    • He became attracted to Fourier series, and the interest in the subject remained undiminished throughout his life.
    • Although he read some of the current literature on Fourier series, he apparently worked all alone ..
    • He did have some connections with Paris mathematicians, however, particularly with Denjoy who may have influenced him towards Fourier series.
    • He was in the right place to carry on with his interest in Fourier series, and he collaborated on this topic with Norbert Wiener and Zygmund (with whom he wrote joint papers).
    • Zygmund, reviewing [',' Oeuvres mathematique de Raphael Salem (Paris, 1967).','2], puts Salem's contributions to Fourier series into perspective:- .
    • For the last few decades two problems were central in the field: convergence almost everywhere of Fourier series and the nature of the sets of uniqueness for trigonometric series.
    • Another direction in which [Salem] did a lot was applications of the calculus of probability to Fourier series and, curiously enough, this has connection with problems of uniqueness.
    • Moreover, it seem that, far from being incidental, as it might have appeared some 30 or so years ago, the calculus of probability is becoming a standard method of attacking problems of trigonometric series.
    • After Salem died his wife established an international prize for outstanding contributions to Fourier series [',' J-P Kahane, Biography in Dictionary of Scientific Biography (New York 1970-1990).','1]:- .

  15. Henri Lebesgue (1875-1941)
    • Lebesgue wrote two monographs Lecons sur l'integration et la recherche des fonctions primitives Ⓣ (1904) and Lecons sur les series trigonometriques Ⓣ (1906) which arose from these two lecture courses and served to make his important ideas more widely known.
    • Fourier had assumed that for bounded functions term by term integration of an infinite series representing the function was possible.
    • From this he was able to prove that if a function was representable by a trigonometric series then this series is necessarily its Fourier series.
    • There is a problem here, namely that a function which is not Riemann integrable may be represented as a uniformly bounded series of Riemann integrable functions.
    • In 1905 Lebesgue gave a deep discussion of the various conditions Lipschitz and Jordan had used in order to ensure that a function f (x) is the sum of its Fourier series.
    • What Lebesgue was able to show was that term by term integration of a uniformly bounded series of Lebesgue integrable functions was always valid.
    • This now meant that Fourier's proof that if a function was representable by a trigonometric series then this series is necessarily its Fourier series became valid, since it could now be founded on a correct result regarding term by term integration of series.

  16. Anneli Lax (1922-1999)
    • The NML was intended as a series of monographs on various mathematical topics.
    • The monographs were written by individual mathematicians most of whom had not written for the high school level prior to their work in the series.
    • Instead, books are still being published in the NML series, though at a slower pace than during its height in the 1960s.
    • Lax was at the centre of the MAA's publication program for thirty-three years, overseeing the NML series.
    • The NML series was planned by Lax and the editorial board to [',' Anneli Lax, New Mathematical Library ','5]:- .
    • Lax was a skilled editor and strove to bring out the best work of the mathematicians who wrote for the series.
    • Around 1960 Anneli approached me about contributing a volume for an upcoming series called The New Mathematical Library that she was editing and that was designed to overcome this reluctance to write mathematical texts for students.
    • After much persuasion, I agreed and wrote 'The Lore of Large Numbers', Number 6 in the series currently still in print, but horribly out of date! .
    • There are now also more than 35 books in the NML series.
    • Many of Lax's admirers thought the NML should be re-titled 'ANML,' Anneli's New Mathematical Library, because of her care in developing and sustaining the series.
    • No other person in the history of the Association's book publishing effort has played a larger role in developing and nurturing a book series.

  17. Mary Cartwright (1900-1998)
    • Having received permission to attend these inspiring sessions which ([',' C Series, Obituary : Dame Mary Cartwright DBE (1900-1998) (9 April 1998, Guardian).','5] or [',' C Series, Obituary : Dame Mary Cartwright DBE (1900-1998), European Mathematical Society Newsletter 30 (1999), 21-23.','6]):- .
    • From this time her research flourished and ([',' C Series, Obituary : Dame Mary Cartwright DBE (1900-1998) (9 April 1998, Guardian).','5] or [',' C Series, Obituary : Dame Mary Cartwright DBE (1900-1998), European Mathematical Society Newsletter 30 (1999), 21-23.','6]):- .
    • in a long series of papers she continued to explore the theory of complex (especially entire) functions; particularly their strange behaviour where they "blow up" ..
    • As Mistress of Girton [',' C Series, Obituary : Dame Mary Cartwright DBE (1900-1998) (9 April 1998, Guardian).','5]:- .
    • Monthly 103 (10) (1996), 833-845.','3], [',' C Series, Obituary : Dame Mary Cartwright DBE (1900-1998) (9 April 1998, Guardian).','5], and [',' C Series, Obituary : Dame Mary Cartwright DBE (1900-1998), European Mathematical Society Newsletter 30 (1999), 21-23.','6] as "wry" and Caroline Series writes in [',' C Series, Obituary : Dame Mary Cartwright DBE (1900-1998), European Mathematical Society Newsletter 30 (1999), 21-23.','6]:- .
    • I [EFR] watched the TV documentary and fully agree with Caroline Series' comment.

  18. Gilles Pisier (1950-)
    • In October 1974 Pisier became an Attache de Recherche having already published a whole range of papers including: Bases suites lacunaires dans les espaces Lpd'apres Kadec et Pelczynski p spaces of Kadec and Pelczynski',5067)">Ⓣ (1973); (with Bernard Maurey) Un theoreme d'extrapolation et ses consequences Ⓣ (1973); "Type" des espaces normes Ⓣ (1973); Sur les espaces de Banach qui ne contiennent pas uniformement de l1n 1n',5070)">Ⓣ (1973) and (with Bernard Maurey) Characterisation d'une classe d'espaces de Banach par des proprietes de series aleatoires vectorielles Ⓣ (1973).
    • by J P Kahane ( "Random series of functions", 1968) based on the remarkable work of Paley, Salem and Zygmund and the work of his student P Billiard on random Fourier series.
    • Pisier's main contributions to random Fourier series are contained in two works, both with Michael Marcus.
    • First there is the book Random Fourier series with applications to harmonic analysis (1981).
    • The authors generalize to the framework of locally compact abelian groups some results which were first established for Rademacher series and later for more general random Fourier series and for Gaussian processes.
    • While the results are stated in this general setting, the intuition developed in the concrete case of random Fourier series on a finite interval of the line pervades the presentation.
    • uniform convergence of a random Fourier series and most other results are dependent on the existence (or not) of this property.
    • The second of his major works with Michael Marcus on random Fourier series is the paper Characterization of almost surely continuous p-stable random Fourier series and strongly stationary processes (1984).

  19. Marcel Riesz (1886-1969)
    • He studied at Budapest University and, influenced by Fejer, undertook research on problems from the theory of series.
    • In it he gave the correct generalisation of Cantor's uniqueness theorem for convergent trigonometric series to trigonometric series summable by the Cesaro method.
    • The first period of his work, from the beginning of his doctoral research up to around the beginning of World War I, concentrated on the theory of series, in particular the summability theory of power series, trigonometric series and Dirichlet series.
    • Another highlight from this period is his beautiful proof of Fatou's theorem which give conditions under which the power series of an analytic function converges to a point on its circle of convergence.
    • In a joint work with Hardy The general theory of Dirichlet's series, published by Cambridge University Press in 1915, he introduced Riesz means.
    • He gave an important series of lectures Clifford numbers and spinors at the University of Maryland between October 1957 and January 1958.

  20. Szolem Mandelbrojt (1899-1983)
    • He remained in the United States as a lecturer at the Rice Institute in Houston during 1926-27 and published Modern researches on the singularities of functions defined by Taylor's series in a Rice Institute Pamphlet (1927).
    • Mandelbrojt published an important book Series de Fourier et classes quasi-analytiques de fonctions Ⓣ in 1935.
    • The present lectures give an excellent account of the modern theory of classes of infinitely differentiable functions of a real variable and may be regarded as a second edition of the author's book "Series de Fourier et classes quasi-analytiques de fonctions" to include the work of the author and of Henri Cartan since that date.
    • Continuing to work at the Rice Institute during World War II, Mandelbrojt continued to publish important work In 1944 he published a series of lectures he had given at the Institute under the title Dirichlet series:- .
    • This monograph consists of a series of lectures delivered by the author and is not a complete treatment of general Dirichlet series.
    • He continued as association with the Rice Institute, howver, and continued to publish work in their Rice Institute Pamphlet series.
    • He published in that series General theorems of closure in 1951 which presented results concerning closure of translations and closure of linear combinations of derivatives of a function.
    • In the following year he published Series adherentes, regularisation des suites, applications Ⓣ in Paris, then in 1958 Composition theorems in the Rice Institute Series.
    • various density functions for sets of positive integers, adherent series, the Fourier-Carleman transform and the spectrum of a function defined by means of this transform.
    • In 1969 Mandelbrojt published Series de Dirichlet.

  21. Siobhan Vernon (1932-2002)
    • On 17 December 1956 she submitted her paper Note on an integrability theorem for sine series to the Quarterly Journal of Mathematics, Oxford, and it was published in the following year.
    • She was appointed to the new full-time post of Assistant in Mathematics at University College, Cork, in 1957 and submitted her second paper On a divergent trigonometrical series given by Steinhaus to the Proceedings of the American Mathematical Society on 28 May 1958.
    • Her third paper The absolute convergence of certain lacunary Fourier series was published in the Proceedings of the Royal Irish Academy in 1960-61.
    • She continued to be encouraged by Kennedy and acknowledged this in her next paper On power series with non-negative coefficients (1966).
    • O'Shea's next two publications were Note on power series of a certain type (1968) and A class of power series with no real zeros (1971).
    • Her publications were generally in the field of infinite series, in particular trigonometric series, which is mathematical analysis but in her teaching she had an inclination to algebra.
    • Under her married name of Siobhan Vernon, she published the three papers On power series with non-negative coefficients.

  22. Dunham Jackson (1888-1946)
    • Many a worker will be inspired by the pages devoted to Fourier and Legendre series and to Chebyshev-Hermite polynomials to try his hand in obtaining similar results for other classes of orthogonal functions, and in particular, for other classes of orthogonal Chebyshev polynomials.
    • As well as this monograph, Jackson published a student text Fourier Series and Orthogonal Polynomials in 1941 in the Carus Mathematical Monographs of the Mathematical Association of America.
    • Copson writes [',' E T Copson, Review: Fourier series and Orthogonal Polynomials by Dunham Jackson, The Mathematical Gazette 27 (273) (1943), 39-40.','2]:- .
    • Shohat [',' J Shohat, Review: Fourier series and Orthogonal Polynomials by Dunham Jackson, National Mathematics Magazine 16 (5) (1942), 267-268.','10] begins his review by explaining that the book:- .
    • Curtiss writes [',' J H Curtiss, Review: Fourier series and Orthogonal Polynomials by Dunham Jackson, Amer.
    • The Association is indeed fortunate to be able to add to the Carus series an exposition by one of the world's foremost authorities on orthogonal polynomials.
    • Charles Moore write in his review [',' C N Moore, Fourier series and Orthogonal Polynomials by Dunham Jackson, Science, New Series 96 (2486) (1942), 183-184.','6]:- .
    • Then he began to suffer a series of minor heart attacks which prevented him from carrying out his teaching duties.

  23. Paul Cohen (1934-2007)
    • Continuing to study at Chicago for his doctorate under the supervision of Antoni Zygmund he was awarded his PhD in 1958 for his doctoral thesis Topics in the Theory of Uniqueness of Trigonometric Series.
    • In this thesis, Cohen states that he [',' P J Cohen, Topics in the Theory of Uniqueness of Trigonometric Series, Ph.D.
    • He begins the Introduction by putting the topic of the thesis into context [',' P J Cohen, Topics in the Theory of Uniqueness of Trigonometric Series, Ph.D.
    • The theory of uniqueness of trigonometrical series can be regarded as arsing from the question of deciding in what sense the Fourier series of a function may be considered as the legitimate expansion of the function in an infinite trigonometrical series.
    • We know, of course, that if the series converges boundedly to the function, then indeed the coefficients of the series must be given by the Euler-Fourier Formulas.
    • However, in the absence of such a condition, we may ask ourselves whether two trigonometrical series may converge to the same function everywhere.

  24. Lipót Fejér (1880-1959)
    • In Berlin he attended courses by Georg Frobenius and Lazarus Fuchs but it was his discussions with Hermann Schwarz that led him look at the convergence of Fourier series and prove the highly significant "Fejer's theorem" published in Sur les fonctions bornees et integrables Ⓣ.
    • The Fourier series is summable (C, 1) to the value of the function at each point of continuity.
    • Henri Poincare, Paris, 1981), 67-84.','8], its discovery restored to Fourier series a fundamental role in analysis for at least fifty years.
    • We note that it was while he was in Berlin that he changed his name from Weiss to Fejer and, after he did so, Schwarz jokingly refused to talk to him! Fejer's fundamental summation theorem for Fourier series formed the basis of his doctoral thesis which he presented to the University of Budapest in 1902.
    • If the Fourier series of a function converges at a point of continuity of the function, its sum is the value of the function at that point; .
    • However there were problems regarding his appointment to the chair as is recounted in [',' K Tandori, The life and works of Lipot Fejer, in Functions, series, operators (2 Vols.), Colloq.
    • He worked on power series and on potential theory.
    • Much of his work is on Fourier series and their singularities but he also contributed to approximation theory.
    • One of Fejer's students, Agnes Berger, described his lecturing style in the following way (see [',' K Tandori, The life and works of Lipot Fejer, in Functions, series, operators (2 Vols.), Colloq.

  25. John Machin (1680-1751)
    • This series (among others for the same purpose, and drawn from the same principle) I received from the excellent analyst, and my much esteemed friend Mr John Machin; and by means thereof, van Ceulen's number, or that in Art.
    • No indication is given in Jones's work, however, as to how Machin discovered his series expansion for π so when de Moivre wrote to Johann Bernoulli on 8 July 1706 telling him about Machin's series for π he suggested that Johann Bernoulli might tell Jakob Hermann about Machin's unproved result.
    • He did so and Hermann quickly discovered a proof that Machin's series converges to π.
    • He produced techniques that show other similar series also converge rapidly to π and he wrote on 21 August 1706 to Leibniz giving details.
    • Two years later, on 6 July 1708, de Moivre wrote again to Johann Bernoulli about Machin's series, on this occasion giving two different proofs that it converged to π.
    • Machin had explained to Taylor in Child's Coffeehouse how to use Newton's series to solve Kepler's problem and also how Halley's method finds roots of polynomial equations.
    • Mr William Jones's Synopsis palmariorum matheseo which was published in the year 1706 is the only book in which as I believe the series of Mr Machin had ever made its appearance before the publication of my Dissertation on the Use of the Negative Sign in Algebra in the year 1758.
    • Machin's work on the series for has proved of lasting importance, but most of his other contributions are not of the same high standard.

  26. Hugo Steinhaus (1887-1972)
    • Steinhaus studied Lebesgue's two major books Lecons sur l'integration et la recherche des fonctions primitives Ⓣ (1904) and Lecons sur les series trigonmetriques Ⓣ (1906) around 1912 after completing his doctorate.
    • by Steinhaus and Banach, concentrated mainly on functional analysis and its diverse applications, the general theory of orthogonal series, and probability theory.
    • Another important publishing venture in which Steinhaus was involved, begun in 1931, was a new series of Mathematical Monographs.
    • The series was set up under the editorship of Steinhaus and Banach from Lvov and Knaster, Kuratowski, Mazurkiewicz, and Sierpiński from Warsaw.
    • An important contribution to the series was a volume written by Steinhaus jointly with Kaczmarz in 1937, The theory of orthogonal series.
    • Some of Steinhaus's early work was on trigonometric series.
    • He gave an example of a trigonometric series which diverged at every point, yet its coefficients tended to zero.
    • He also gave an example of a trigonometric series which converged in one interval but diverged in a second interval.
    • As we have noted above, other contributions by Steinhaus were on orthogonal series, probability theory, real functions and their applications.

  27. Lejeune Dirichlet (1805-1859)
    • These papers introduce Dirichlet series and determine, among other things, the formula for the class number for quadratic forms.
    • He turned to Laplace's problem of proving the stability of the solar system and produced an analysis which avoided the problem of using series expansion with quadratic and higher terms disregarded.
    • Dirichlet is also well known for his papers on conditions for the convergence of trigonometric series and the use of the series to represent arbitrary functions.
    • These series had been used previously by Fourier in solving differential equations.
    • Earlier work by Poisson on the convergence of Fourier series was shown to be non-rigorous by Cauchy.
    • Because of this work Dirichlet is considered the founder of the theory of Fourier series.
    • Riemann, who was a student of Dirichlet, wrote in the introduction to his habilitation thesis on Fourier series that it was Dirichlet [',' H Koch, Gustav Peter Lejeune Dirichlet, in Mathematics in Berlin (Berlin, 1998), 33-40.','11]:- .

  28. Stefan Kaczmarz (1895-1939)
    • Other papers he published, some coauthored with his colleagues at Lwow, are: Uber die Konvergenz der Reihen von Orthogonalfunktionen Ⓣ (1925), Uber die Summierbarkeit der Orthogonalreihen Ⓣ (1927), Uber Reihen von allgemeinen Orthogonalfunktionen Ⓣ (1927), (with L Nikliborc) Sur les suites de fonctions convergentes en moyenne Ⓣ (1928), Sur la convergence et la sommabilite des developpements orthogonaux Ⓣ (1929), Zur Theorie der Fourierschen Doppelreihe Ⓣ (1930), (With H Steinhaus) Le systeme orthogonal de M Rademacher Ⓣ (1930), Integrale vom Dini'schen Typus Ⓣ (1931), Une remarque sur les series Ⓣ (1931), Axioms for Arithmetic (1932), On Some Classes of Fourier Series (1933), Sur les multiplicateurs des series orthogonales Ⓣ (1933), Note on general transforms (1933), Notes on orthogonal series I (1934), Notes on orthogonal series II (1934), Notes on orthogonal series III (1936), (with J Marcinkiewicz) Sur les multiplicateurs des series orthogonales Ⓣ (1938), and Sur l'irrationalite des integrales indefinies Ⓣ (1939).
    • The present volume of the excellent Polish Series is devoted to the theory of general orthogonal functions of a single real variable.
    • This book gives a good account of recent researches in orthogonal series, and may be recommended not only to specialists in that subject but also to those interested in trigonometric series and in linear operations.
    • From 1923 to 1939 Kaczmarz taught many university level courses at Lwow such as: Analytical Geometry, Higher Analysis, Integral Equations, Algebraic Curves, Trigonometric Series, Non-Euclidean Geometry and the Theory of Groups, and Differential Geometry.

  29. Werner Rogosinski (1894-1964)
    • At this stage Landau was interested in number theory, in particular Dirichlet series that he had studied for his own doctorate, and it was in this area that Rogosinski worked for his thesis.
    • The particular problem Rogosinski studied involved the Dirichlet divisor problem, particularly studying a series expansion given by Georgy Voronoy.
    • Rogosinski was appointed to the University of Konigsberg as a Privatdocent in 1923 and began publishing papers which typified the contributions he made throughout his life to the theory of functions and to the theory of series.
    • This book on Fourier series was based on lectures that Rogosinski gave on the topic at the University of Konigsberg.
    • In 1944 Rogosinski's collaboration with Hardy led to the publication of their book Fourier Series.
    • E C Titchmarsh writes [',' E C Titchmarsh, Review: Fourier Series by G H Hardy and W W Rogosinski, The Mathematical Gazette 28 (281) (1944), 164.','5]:- .
    • Rogosinski lectured to the Edinburgh Mathematical Society in the summer of 1945 and, a consequence of this lecture became the book Volume and integral (1952) published in the Oliver and Boyd Series of University Mathematical Texts.
    • Distinguished for his contributions to Mathematical Analysis, especially in the theory of Fourier Series and Allied Subjects.

  30. Samuel Verblunsky (1906-1996)
    • in 1927 he continued to undertake research at Magdalene College with J E Littlewood as his thesis advisor and was awarded his doctorate in 1930 for his thesis Researches In The Theory Of Fourier Series.
    • The 1930 Mathematical Proceedings of the Cambridge Philosophical Society contains eight papers by Verblunsky, namely: A property of continuous arcs (submitted May 1929), The relation between Riemann's method of summation and Cesaro's (submitted May 1929), Note on the sum of an oscillating series (submitted October 1929), Note on the Gibbs Phenomenon (submitted November 1929), The convergence of singular integrals (submitted February 1930), Note on the modified Heine-Borel theorem (submitted April 1930), A property of continuous arcs II (submitted July 1930), and Note on the sum of an oscillating series II (submitted July 1930).
    • The papers On Summable Trigonometric Series (submitted September 1929) and The Generalized Third Derivative and its Application to the Theory of Trigonometric Series (submitted September 1929) were both published in the Proceedings of the London Mathematical Society in 1930.
    • Among the results that Verblunsky proved in his early papers was that a trigonometric series cannot be summable (C, 1) to 0 unless its coefficients are nul, thus confirming a 1911 conjecture of Marcel Riesz.
    • The Mathematical Proceedings of the Cambridge Philosophical Society of 1931 contains four Verblunsky papers, namely Note on Continuous Functionals (submitted June 1930), The symmetric derivative and its application to the theory of trigonometric series (submitted October 1930), On summable trigonometric integrals (submitted November 1930), and Note on the Gibbs Phenomenon II (submitted May 1931).
    • A contribution to the algebra of Fourier series while the second, which contains a proof of what today is called Verblunsky's Theorem, was On positive harmonic functions.
    • in 1953 for his thesis Some Studies in Trigonometrical Ratios and he published the paper Uniqueness theorems for a class of generalized trigonometrical series in 1954 in which he wrote:- .
    • students, Johnston Andrew Anderson and Gordon Harper Fullerton, wrote the joint paper On a class of Cauchy exponential series (1965) in which they write:- .

  31. Colin Maclaurin (1698-1746)
    • It is in the Treatise of fluxions that Maclaurin uses the special case of Taylor series now named after him and for which he is undoubtedly best remembered today.
    • The Maclaurin series was not an idea discovered independently of the more general result of Taylor for Maclaurin acknowledges Taylor's contribution.
    • Another important result given by Maclaurin, which has not been named after him or any other mathematician, is the important integral test for the convergence of an infinite series.
    • As mentioned above, Maclaurin is best known for the Maclaurin Series, which is a special case of the Taylor series.
    • Maclaurin series for sine .
    • Maclaurin series for cosine .
    • Maclaurin series .

  32. Georges Valiron (1884-1955)
    • He was sent to the University of Strasbourg to teach mathematics, presenting a course on 'Dirichlet series and factorial series' in March and April 1921.
    • In February and March 1922 Valiron delivered a series of lectures to honours students at the University College of Wales at Aberystwyth.
    • In the following year he published Theorie Generale des Series de Dirichlet Ⓣ (1926).
    • L L Smail writes in a review [',' L L Smail, Review: Theorie Generale des Series de Dirichlet by G Valiron, Bull.
    • The aim of the series to which this work belongs is to present in brief and compact form the most important results and outlines of methods in various mathematical subjects of current interest.
    • This particular number on Dirichlet series gives an excellent survey of this interesting field which is now growing so rapidly.
    • Among the papers that Valiron published in the years following World War I, we mention: Les theoremes generaux de M Borel dans la theorie des fonctions entieres Ⓣ (1920); Recherches sur le theoreme de M Picard Ⓣ (1921); Recherches sur le theoreme de M Picard dans la theorie des fonctions entieres Ⓣ (1922); Sur les fonctions entieres verifiant une classe d'equations differentielles Ⓣ (1923); Sur l'abscisse de convergence des series de Dirichlet Ⓣ (1924); Sur les surfaces qui admettent un plan tangent en chaque point Ⓣ (1926); and Sur la distribution des valeurs des fonctions meromorphes Ⓣ (1926).
    • Infinite series and products; III.
    • Functions of a single complex variable defined or represented by series or integrals; VII.
    • Trigonometric series and generalizations; VIII.

  33. Germund Dahlquist (1925-2005)
    • Three methods, old but not so well known, transform an infinite series into a complex integral over an infinite interval.
    • Applications are made to slowly convergent alternating and positive series, to Fourier series, to the numerical analytic continuation of power series outside the circle of convergence, and to ill-conditioned power series.
    • [The second part] is mainly concerned with the derivation, analysis and applications of a summation formula, due to Lindelof, for alternating series and complex power series, including ill-conditioned power series.

  34. Stephen Bosanquet (1903-1984)
    • Bosanquet's early 1930s papers were on series and integration, a 1930 paper being on fractional integration, a topic he would return to many times.
    • For example he was a visiting professor at the University of Utah during 1964-65 where he gave a major lecture series on The history and development of the theory of divergent series and integrals.
    • During 1969-70 he visited the University of Western Ontario and gave another major lecture series, this time on Matrix transformations and sequence spaces with applications to summability.
    • Bosanquet wrote many papers on the convergence and summability of Fourier series.
    • He also wrote on the convergence and summability of Dirichlet series and studied specific kinds of summability such as summability factors for Cesaro means.
    • He saw Hardy's great book 'Divergent Series' through the press during Hardy's last illness and he later edited the volume on 'Series' in Hardy's Collected Works; he was chief editor for the last two of the seven volumes.

  35. Lennart Carleson (1928-)
    • In 1966 Carleson solved one of the outstanding problems of mathematics in his paper On convergence and growth of partial sums of Fourier series.
    • Fourier, in 1807, had claimed that every function equals the sum of its Fourier series.
    • A major research area throughout the 19th century concerned the convergence of Fourier series, and continuous functions whose Fourier series diverges at a dense set of points were constructed by du Bois-Reymond.
    • In 1913 Luzin conjectured that if a function f is square Lebesgue integrable then the Fourier series of f converges pointwise to f almost everywhere.
    • Carleson lectured on his spectacular result at the International Congress of Mathematicians at Moscow in 1966 when he gave the address Convergence and summability of Fourier series.
    • The citation emphasizes not only Carleson's fundamental scientific contributions, the best known of which perhaps are the proof of Luzin's conjecture on the convergence of Fourier series, the solutions of the corona problem and the interpolation problem for bounded analytic functions, the solution of the extension problem for quasiconformal mappings in higher dimensions, and the proof of the existence of 'strange attractors' in the Henon family of planar maps, but also his outstanding role as scientific leader and advisor.
    • His research in analysis is a series of towering and fundamental discoveries.

  36. Julio Rey Pastor (1888-1962)
    • Between 1911 and 1916, La Junta Para Ampliacion de Estudios funded Rey Pastor to carry out a series of visits to Germany.
    • However he was not one to remain fixed in one place for a long time and went to Barcelona in 1915 to give a series of lectures at the Institut d'Estudia.
    • It dealt with the study of the method of summation of series.
    • This article by Rey Pastor is framed by a long series of works, begun at the beginning of the twentieth century, on problems of summing series, convergence algorithms, singular integrals and comparative studies of series and integrals.
    • He first presented his work in this area in 1926 in his course on "Series and Integrals" which he gave at Buenos Aires University.
    • He continued working on problems related to the theory of summation of divergent series throughout the 1930's and published much of his work in international journals.

  37. Nathan Fine (1916-1994)
    • who will present a series of at most three lectures accessible to a large fraction of those who teach college mathematics.
    • He is perhaps best known for his book Basic hypergeometric series and applications published in the Mathematical Surveys and Monographs Series of the American Mathematical Society.
    • Fine was at that time engaged in his own special development of q-hypergeometric series, and as the years passed he kept adding to his results and polishing his presentation.
    • We became somewhat diverted while looking at Fine's text Basic hypergeometric series and applications when we began to look at Andrews' Introduction.
    • For far too long, there has been a dearth of good references on basic hypergeometric series.
    • The present book and Basic hypergeometric series by G Gasper and M Rahman have appeared in the past two years to greatly rectify this situation.
    • This is a very personal book, a distillation of those results in basic hypergeometric series which hold the most appeal to its author.

  38. Johannes Robert Rydberg (1854-1919)
    • Notwithstanding the imperfect spectroscopic tables then at his disposal Rydberg discovered most of the important properties of series spectra, including the relation between corresponding series in the spectra of related elements, and foreshadowed discoveries which were made later, when experimental work has sufficiently advanced.
    • Some of the features noted by Rydberg were observed about the same time by Kayser and Runge, but his work had the special merit of connecting different series in the spectrum of the same element into one system, which could be represented by a set of simple formulae having but few adjustable constants.
    • He especially insisted that the hydrogen constant, now generally called the "Rydberg constant," should appear in all series and, apart from slight variations from element to element suggested by the theoretical work of Bohr, nearly all subsequent attempts to improve the representation series have involved this supposition, and have had Rydberg's formula as a basis.
    • the wave-lengths and wave numbers of corresponding lines, as well as the values of the three constants of the corresponding series of different elements, are periodical functions of the atomic weight.
    • As I have pointed out already in my general exposition of the constitution of line spectra, and have afterwards tried further to confirm, there can be no doubt that these series are really parts of a single group of lines with two variable integral parameters, the general formula of which can be written approximately.

  39. Hubert Wall (1902-1971)
    • in 1927 for his thesis On the Pade Approximants Associated with the Continued Fraction and Series of Stieltjes.
    • Over the following years he produced an excellent series of papers on continued fractions such as: On extended Stieltjes series (1929); On the Pade approximants associated with a positive definite power series (1931); Convergence criteria for continued fractions (1931); General theorems on the convergence of sequences of Pade approximants (1932); On the relationship among the diagonal files of a Pade table (1932); On the expansion of an integral of Stieltjes (1932) and On the continued fractions which represent meromorphic functions (1933).
    • This is the first volume to appear in "The University Series in Higher Mathematics" which is planned to be a collection of "advanced text and reference books in pure and applied mathematics." In order to make the book suitable as a text book the author gives detailed proofs and includes material which might be unfamiliar to "a student of rather modest preparation." Into this category fall such topics as: the Stieltjes-Vitali theorem, Schwarz's inequalities, matrix calculus, elementary properties of the Stieltjes integral, and basic concepts and formulae of the theory of continued fractions.
    • As part of the Mathematical Association of America Classroom Resource Materials series, the book is intended for just that: supplementary classroom material for students with an unusual approach for presenting mathematical ideas.
    • Pade approximants and continued fractions are closely related to many disciplines of pure and applied mathematics, including analytic function theory, the theory of moments, asymptotics and summability of divergent series.

  40. Wilhelm Lexis (1837-1914)
    • From 1876-79 Lexis studied data presented as a series over time.
    • He initiated the study of time series with his 1879 article On the theory of the stability of statistical series.
    • Lexis required a binomial dispersion for his series to be stable and this ruled out most interesting series.
    • After what is not much more than a three year period working on statistics in Freiburg, he began to produce a series of papers on economics.
    • In addition to studying economics, the theory of population, and statistics, he also worked on the production of an economic encyclopaedia, edited a series of works on education in general and university education in particular, and was the director of the first institute of actuarial science in Germany.

  41. Niels Norlund (1885-1981)
    • a long series of papers developing the theory of difference equations.
    • He studied the factorial series and interpolation series entering in their solutions, determining their region of convergence and by analytic prolongation and different summation methods he extended them in the complex plane, determining their singularities and their behaviour at infinity, also by use of their relations to continued fractions and asymptotic series.
    • In particular the little paper 'Sur une application des fonctions permutables' Ⓣ from 1919 should be mentioned: there he states some universal results on the summability of series based on a specific - but rather general - choice of the weights given to the elements; the method includes the better of the known summability methods, such as Cesaro's method, and is now known under the standard designation of Norlund-summation.
    • The Geodesic Institute could not continue with its work, but Norland used the opportunity to produce a series of atlases, detailing the history of the mapping of Denmark, the Faroe Islands and Iceland.
    • It is likely to become the standard memoir on the generalized hypergeometric series ..

  42. Christian Goldbach (1690-1764)
    • When Bernoulli started to discuss infinite series with Goldbach as they talked in Oxford, Goldbach confessed that he knew nothing about the topic.
    • Bernoulli gave him a loan of a book on the topic by his uncle Jacob Bernoulli but Goldbach found infinite series too difficult at that time, and gave up his attempts to understand Jacob Bernoulli's text.
    • We mentioned that Goldbach gave up his attempts to understand infinite series in 1712.
    • However in 1717 he read an article by Leibniz on computing the area of a circle and this led him to look again at the theory of infinite series.
    • We should, however, mention his another two of his papers on infinite series De transformatione serierum Ⓣ (1729) and De terminis generalibus serierum Ⓣ (1732).
    • The first of these introduced a method of transforming one series into another while the sum of the series remains fixed.

  43. Rudolf Wolf (1816-1893)
    • Wolf wrote on prime number theory and geometry, then later on probability and statistics - a series of papers discussed Buffon's needle experiment in which he estimated π by Monte Carlo methods.
    • I decided to carry out corresponding series of tests, hoping to obtain, not , but at least new proofs about the rules governing a finite number of trials.
    • On a plate of about one square foot I drew a series of parallels at a distance of 45 mm, and from a knitting needle I broke a piece of 36 mm length - thus getting as close as 1/100 to the ideal ratio according to the instruction above.
    • Thus I obtained in the first series of trials for 100 tosses a mean of 21.76 ± 0.64 tosses in which the needle intersected the parallels.
    • In the second series of trials I obtained 71.34 ± 1.25.
    • In the third series of trials I obtained 50.64 ± 0.83.
    • a number that lies within the error limits of the mean from the third series of trials.

  44. James Gregory (1638-1675)
    • He visited Flanders, Rome and Paris on his journey but spent most time at the University of Padua where he worked on using infinite convergent series to find the areas of the circle and hyperbola.
    • We can now be certain that during the summer of 1668 Gregory was completely familiar with the series expansions of sin, cos and tan.
    • Although he did not disclose his methods in the small treatise he discussed topics including various series expansions, the integral of the logarithmic function, and other related ideas.
    • In February 1671 he discovered Taylor series (not published by Taylor until 1715), and the theorem is contained in a letter sent to Collins on 15 February 1671.
    • For his reluctance to publish his "several universal methods in geometry and analysis" when he heard through Collins of Newton's own advances in calculus and infinite series, he postumously paid a heavy price ..
    • However, we now summarise these and other contributions in the hope that, despite his reluctance to publish his methods, his remarkable contributions might indeed be more widely understood: Gregory anticipated Newton in discovering both the interpolation formula and the general binomial theorem as early as 1670; he discovered Taylor series more than 40 years before Taylor; he solved Kepler's famous problem of how to divide a semicircle by a straight line through a given point of the diameter in a given ratio (his method was to apply Taylor series to the general cycloid); he gives one of the earliest examples of a comparison test for convergence, essentially giving Cauchy's ratio test, together with an understanding of the remainder; he gave a definition of the integral which is essentially as general as that given by Riemann; his understanding of all solutions to a differential equation, including singular solutions, is impressive; he appears to be the first to attempt to prove that π and e are not the solution of algebraic equations; he knew how to express the sum of the nth powers of the roots of an algebraic equation in terms of the coefficients; and a remark in his last letter to Collins suggests that he had begun to realise that algebraic equations of degree greater than four could not be solved by radicals.

  45. Frank Jackson (1870-1960)
    • Over the following couple of years he published two further papers in the same Proceedings, namely A certain linear differential equation (1896) and Certain expansions of xn in hypergeometric series (1897).
    • Bruce Berndt writes [',' B C Berndt, What is a q-series, University of Illinois at Urbana-Champaign.','1]:- .
    • Two English mathematicians, Frank H Jackson and L James Rogers, at the end of the 19th and beginning of the 20th centuries devoted most of their mathematical careers to further developing the theory of q-series, but their efforts were not appreciated by their contemporary researchers.
    • The Bible in the theory of basic hypergeometric series is the text 'Basic Hypergeometric Series' by G Gasper and M Rahman.
    • Jackson's most important papers on q-series are On q-definite integrals (1910), On basic double hypergeometric functions (1942) and Basic double hypergeometric functions (1944).
    • Cuthbert, Chester-le-Street, in the series Famous churches and abbeys.

  46. Elias Stein (1931-)
    • He held this position for two years during which time a whole series of his papers appeared in print: Interpolation of linear operators (1956), Functions of exponential type (1957), Interpolation in polynomial classes and Markoff's inequality (1957), Note on singular integrals (1957), (with G Weiss) On the inerpolation of analytic families of operators action on Hpspaces (1957), (with E H Ostrow) A generalization of lemmas of Marcinkiewicz and Fine with applications to singular integrals (1957), A maximal function with applications to Fourier series (1958), (with G Weiss) Fractional integrals on n-dimensional Euclidean space (1958), (with G Weiss) Interpolation of operators with change of measures (1958), Localization and summability of multiple Fourier series (1958), and On the functions of Littlewood-Paley, Luzin, Marcinkiewicz (1958).
    • Considerable portions of the theory of Fourier series and integrals are known to extend to various categories of topological groups.
    • In 1971 Analytic continuation of group representations appeared based on a series of James Whittemore lectures that Stein had given at Yale University in November 1967.
    • I look forward to reading and reviewing the next three books in this series (by the same authors).
    • The second in this three volume series Complex analysis was published in 2003 and the third volume Real analysis: Measure theory, integration, and Hilbert spaces in 2005.

  47. Gregory of Saint-Vincent (1584-1667)
    • In searching for ways to obtain a trisection, St Vincent came across the series .
    • This series according to St Vincent equals 2/3, which he called the terminus.
    • There are many topics covered in the book including a study of circles, triangles, geometric series, ellipses, parabolas and hyperbolas.
    • Book II looks at geometric series which Saint-Vincent is able to sum using the transformations he introduced in the first Book.
    • He applies his results to a number of interesting problems such as the trisection of an angle which he achieves through an infinite series of bisections.
    • He also applies his summation of series to the classical Greek problem of Zeno, namely Achilles and the tortoise.
    • Huygens recommended the section on geometric series to Leibniz who later came to make a thorough study of the entire work.

  48. Otto Hölder (1859-1937)
    • Otto Holder worked on the convergence of Fourier series and in 1884 he discovered the inequality now named after him.
    • He became interested in group theory through Kronecker and Klein and proved the uniqueness of the factor groups in a composition series.
    • His habilitation thesis examined the convergence of the Fourier series of a function that was not assumed to be either continuous or bounded.
    • At first at Gottingen he continued to work on the convergence of Fourier series.
    • Klein's lectures on Galois theory at Gottingen had interested Holder who began to study the Galois theory of equations and from there he was led to study composition series of groups.
    • Holder proved the uniqueness of the factor groups in a composition series, the theorem now called the Jordan-Holder theorem, and published the result in Mathematische Annalen in 1889 in the paper Zuruckfuhrung einer beliebigen algebraischen Gleichung auf eine Kette von Gleichungen Ⓣ.
    • A thorough study of the methods of ratiocination employed in mathematics, mechanics, and the exact natural sciences has led Professor Holder to the conviction that the deductive method there employed is made up of series of concatenated conclusions of quite characteristic form, and that consequently these sciences have a peculiar method and logic of their own.

  49. Abraham Plessner (1900-1961)
    • In 1921 Plessner went to the University of Gottingen where, between May and August, he took courses on Dirichlet series and Galois theory by Edmund Landau; algebraic number fields by Emmy Noether; and the calculus of variations by Richard Courant.
    • Plessner obtained his doctorate from Giessen in 1922 for a thesis on conjugate trigonometrical series entitled Zur Theorie der konjugierten trigonometrischen Reihen Ⓣ.
    • Plessner's theorem states that if a trigonometric series converges everywhere in a set E of positive measure, then its conjugate series converges almost everywhere in E.
    • In point of difficulty, it stands between these and de la Vallee Poussin's monograph in the Borel series.
    • The book is divided into six chapters, whose titles are self-explanatory: (1) The fundamental concepts of the theory of sets; (2) The measure of sets of points; (3) Functions of real variables; (4) The Lebesgue integral; (5) Functions of one and two variables; and (6) Fourier series.
    • The book provides, in a clear and concise manner, an overview of the theory of the Lebesgue integral and its application to the theory of Fourier series, so that it can be read without difficulty by a student in their middle semesters.

  50. Jakob Hermann (1678-1733)
    • He received his first degree in 1695 and defended a dissertation on infinite series for a Master's Degree in the following year.
    • Hermann was taught mathematics by Jacob Bernoulli who, at that time, was working on infinite series.
    • Jacob Bernoulli divided up his work on infinite series into a number of dissertations and these were defended by his students for their Master's Degrees.
    • Four of these students were Johann Jacob Fritz, who defended his dissertation on series in 1689, Hieronymus Beck who defended his thesis in 1692, Jakob Hermann who, as we noted above, defended his dissertation in 1696, and Nicolaus Harscher who defended his thesis on series in 1698.
    • Slightly later, Nicolaus Bernoulli, a nephew of Jacob Bernoulli, also studied with his uncle and defended a dissertation on infinite series in 1704.
    • Hermann discussed such topics as finding the radius of curvature and normals to plane curves; the division of an angle or an arc of a circumference into n parts, by the use of an infinite series; orthogonal trajectories for a given family of curves, by the use of differential equations; and the use of polar coordinates in the analysis of plane curves other than spirals.

  51. Jacob Bernoulli (1655-1705)
    • Jacob Bernoulli returned to Switzerland and taught mechanics at the University in Basel from 1683, giving a series of important lectures on the mechanics of solids and liquids.
    • By 1689 he had published important work on infinite series and published his law of large numbers in probability theory.
    • Jacob Bernoulli published five treatises on infinite series between 1682 and 1704.
    • Euler was the first to find the sum of this series in 1737.
    • Bernoulli also studied the exponential series which came out of examining compound interest.
    • The Bernoulli numbers appear in the book in a discussion of the exponential series.
    • Bernoulli greatly advanced algebra, the infinitesimal calculus, the calculus of variations, mechanics, the theory of series, and the theory of probability.

  52. John Farey (1766-1826)
    • The reason we have included him is that he made one mathematical observation and, from this, the Farey series of fractions has been named.
    • We shall discuss below Farey's contribution to mathematics and also look at others who contributed to Farey series.
    • In the second paragraph he defines the Farey series and states the "curious property".
    • The Farey series (really a sequence) is defined as follows.
    • Historical references to the Farey sequence have been examined by the authors of [',' M Bruckheimer and A Arcavi, Farey series and Pick’s area theorem, The Mathematical Intelligencer 17 (4) (1995), 64-67.','3].
    • The article [',' M Bruckheimer and A Arcavi, Farey series and Pick’s area theorem, The Mathematical Intelligencer 17 (4) (1995), 64-67.','3] contains other interesting information on Farey's sequence, its relation to Pick's area theorem, and the inaccurate historical comments made about the sequence over many years.

  53. Karl Weierstrass (1815-1897)
    • The transformation of his conception of an analytic function from a differentiable function to a function expansible into a convergent power series was made during this early period of Weierstrass's mathematical activity.
    • This paper did not give the full theory of inversion of hyperelliptic integrals that Weierstrass had developed but rather gave a preliminary description of his methods involving representing abelian functions as constantly converging power series.
    • The topics of his lectures included:- the application of Fourier series and integrals to mathematical physics (1856/57), an introduction to the theory of analytic functions (where he set out results he had obtained in 1841 but never published), the theory of elliptic functions (his main research topic), and applications to problems in geometry and mechanics.
    • Its contents were: numbers, the function concept with Weierstrass's power series approach, continuity and differentiability, analytic continuation, points of singularity, analytic functions of several variables, in particular Weierstrass's "preparation theorem", and contour integrals.
    • The standards of rigour that Weierstrass set, defining, for example, irrational numbers as limits of convergent series, strongly affected the future of mathematics.
    • Known as the father of modern analysis, Weierstrass devised tests for the convergence of series and contributed to the theory of periodic functions, functions of real variables, elliptic functions, Abelian functions, converging infinite products, and the calculus of variations.

  54. Paul du Bois-Reymond (1831-1889)
    • The standard technique to solve partial differential equations used Fourier series but Cauchy, Abel and Dirichlet had all pointed out problems associated with the convergence of the Fourier series of an arbitrary function.
    • In 1873 du Bois-Reymond was the first person to give an example of a continuous function whose Fourier series diverges at a point.
    • Perhaps what was even more surprising, the Fourier series of du Bois-Reymond function diverged at a dense set of points.
    • The conception of space as static and unchanging can never generate the notion of a sharply defined, uniform line from a series of points however dense, for, after all, points are devoid of size, and hence no matter how dense a series of points may be, it can never become an interval, which must always be regarded as the sum of intervals between points.

  55. Sixto Ríos (1913-2008)
    • He began publishing papers, both in French and in Spanish: Sobre una generalizacion del algoritmo de convergencia de Euler Ⓣ (1932); Sur l'ensemble singulier d'une classe des series potentielles de Taylor qui presentent des lacunes Ⓣ (1933); Estado actual de la teoria de la hiperconvergencia Ⓣ (1934); Sopra l'ultraconvergenza delle serie di Dirichlet Ⓣ (1934); Algunos resultados relativos a la hiperconvergencia en las series de Dirichlet Ⓣ (1934); and Sobre un teorema de M.
    • His thesis, which was a continuation of Alexander Markowich Ostrowski's works on potential series, was published in 1936.
    • In 1937 he was awarded the First Prize of the Spanish Royal Academy of Sciences for his paper Sobre el problema de hiperconvergencia de las series de Dirichlet cuyas sucesiones de exponentes poseen densidad maxima infinita Ⓣ (1940) and Conferencias sobre sucesiones de funciones analiticas y sus aplicaciones Ⓣ (1940).
    • Before taking up the chair of statistics, Rios had published a number of books on analysis such as Lectures on the theory of the integral (Spanish) (1942), Lectures on the theory of the analytic continuation of Dirichlet series (Spanish) (1943), La Prolongacion Analitica de la Integral de Dirichlet-Stieltjes Ⓣ (1944), Conferencias sobre la Representacion Analitica de Funciones Ⓣ (1945), and Conceptos de Integral Ⓣ (1946).
    • Even after taking up the Chair of Statistics he published the analysis book Introduccion a la Teoria de Series Trigonometricas Ⓣ (1949).
    • Series A (General) 118 (1) (1955), 110-111.','34]:- .

  56. Alexander Ivanovich Skopin (1927-2003)
    • The first significant result which Skopin produced was concerned with the upper central series of groups [',' A I Skopin, The factor groups of an upper central series of free groups (Russian), Doklady Akad.
    • From the 1970's, he became interested in problems concerning the structure of the lower central series of groups of the Burnside type, that is groups of prime-power exponent.
    • It was very natural to fix the exponent and the number of generators and to study the lower central series of a group in more detail by performing direct calculations of its factors.
    • At first, Skopin considered the case of metabelian groups with two generators with exponents 8, 9, or 27 ([',' A I Skopin, Factors of the nilpotent series of some metabelian groups of prime-power exponent (Russian), Modules and algebraic groups 2, Zap.
    • (LOMI) 132 (1983), 129-163.','18], [',' A I Skopin, Investigation on a BESM-6 computer of the structure of the nilpotent series of a metabelian 2-generator group of exponent 27 (Russian), Zap.

  57. Ferran Sunyer (1912-1967)
    • Sunyer lost contact with him at this time and it was only after Hadamard had spent a year in England and returned to Paris as soon as the war ended that he was able to resume contact sending him a memoir on lacunary Taylor series.
    • Two notes with results from the memoir on lacunary Taylor series were quickly published.
    • Sunyer continued to publish papers such as On a class of transformations of the algorithms for summation of analytic series (Spanish) (1948), On the exclusion of an exceptional function by a gap condition (Spanish) (1948), Une generalisation des fonctions presque-periodiques Ⓣ (1949), A new generalization of almost periodic functions (Catalan) (1949), and Properties of entire functions (of finite order) represented by lacunary Taylor series (Spanish) (1949).
    • The bulk of his work, published between 1939 and 1970, is on topics in the theory of entire and meromorphic functions, where his main tendency is that of generalizing important results, and is grouped in the volume under review into eight sections: (i) "miscellaneous papers", one of which contains generalizations of the theorems of Landau, Schottky, Picard, and of Montel's normality criterion, another on a new method for the summation of power series, but also a paper that falls entirely outside of the realm of analysis (to which all other papers are devoted), answering a question raised by Sierpinski in 1951 on order types; (ii) and (iii), spread over 220 pages, the study of exceptional values of an entire function as influenced by the lacunarity of the Taylor (or Dirichlet) series defining it, (ii) containing F Sunyer i Balaguer's 'Acta Mathematica' paper of 1952; (iv) work on the behaviour of an entire function and its derivatives and primitives along Borel-Valiron directions of maximal type; (v) generalizations of S Mandelbrojt's fundamental inequality (vi) topics in the theory of entire functions and overconvergence; (vii) generalizations of quasi-periodic and elliptic functions; (viii) differentiable functions of a real variable, containing a lovely result (the only one with a co-author, E Corominas) published in 1954) ..

  58. Camille Jordan (1838-1922)
    • Indeed Jordan introduced the concept of a composition series (a series of subgroups each normal in the preceding with the property that no further terms could be added to the series so that it retains that property).
    • The composition factors of a group G are the groups obtained by computing the factor groups of adjacent groups in the composition series.
    • Jordan proved the Jordan-Holder theorem, namely that although groups can have different composition series, the set of composition factors is an invariant of the group.
    • Among Jordan's many contributions to analysis we should also mention his generalisation of the criteria for the convergence of a Fourier series.

  59. Georg Cantor (1845-1918)
    • This was due to Heine, one of his senior colleagues at Halle, who challenged Cantor to prove the open problem on the uniqueness of representation of a function as a trigonometric series.
    • He published further papers between 1870 and 1872 dealing with trigonometric series and these all show the influence of Weierstrass's teaching.
    • Cantor published a paper on trigonometric series in 1872 in which he defined irrational numbers in terms of convergent sequences of rational numbers.
    • Between 1879 and 1884 Cantor published a series of six papers in Mathematische Annalen designed to provide a basic introduction to set theory.
    • Soon Cantor was publishing in Mittag-Leffler's journal Acta Mathematica but his important series of six papers in Mathematische Annalen also continued to appear.
    • The fifth paper in this series Grundlagen einer allgemeinen Mannigfaltigkeitslehre Ⓣ was also published as a separate monograph and was especially important for a number of reasons.

  60. Mikio Sato (1928-)
    • I worked out hyperfunction series and outlined the theory for several variables - though the complete theory was finished later, since it required a generalization of cohomology theory.
    • In 1968 he began to give a series of talks on algebraic analysis at Tokyo University which was attended by two students Takahiro Kawai and Masaki Kashiwara.
    • In 1978, with coauthors Tetsuji Miwa and Michio Jimbo, he began publishing a series of papers Holonomic quantum fields.
    • This is the first one of a series of papers on holonomic quantum fields.
    • In this series we deal with these objects: (1) Deformation theory for linear differential equations (Riemann-Hilbert problem and its generalization to higher dimensions), (2) Quantum fields with critical strength (2-dimensional Ising model, etc.) and (3) Theory of Clifford group.
    • By 1980 they had reached the 17th paper in this series.

  61. Pierre Laurent (1813-1854)
    • Cauchy reported on Laurent's entry Memoire sur le calcul des variations Ⓣ, which contains the Laurent series for a complex function, on 20 May 1843.
    • In modern terminology the theorem defined in an annulus (a ring bounded by two concentric circles) and on its boundary can be developed in a general, power series in increasing and decreasing powers of the variable.
    • There exist functions, such as certain Bessel functions, which cannot be expanded into Taylor series, and Laurent's theorem would be applicable to these functions.
    • Given Cauchy's attempt to claim the result of Laurent's paper it is not surprising the Academy of Sciences chose not to publish it but, despite this, the theorem and series are named for Laurent.
    • Let x designate a real or imaginary variable; a real or imaginary function of x will be developable in a convergent series ordered according to the ascending powers of this variable, while the modulus of the variable will preserve a value less than the smallest of the values for which the function or its derivative ceases to be finite or continuous.
    • Let x designate a real or imaginary variable; a real or imaginary function of x can be represented by the sum of two convergent series, one ordered according to the integral and ascending powers of x, and the other according to the integral and descending powers of x; and the modulus of x will take on a value in an interval within which the function or its derivative does not cease to be finite and continuous.

  62. Tom Apostol (1923-2016)
    • He studied Konrad Knopp's book Theory and Applications of Infinite Series along with one other student and they took turns to lecture each day on what they had learnt, with Zuckerman continually questioning their understanding.
    • The papers he published in 1950-51 are: Generalized Dedekind sums and transformation formulae of certain Lambert series (1950); Asymptotic series related to the partition function (1951); Identities involving the coefficients of certain Dirichlet series (1951); Remark on the Hurwitz zeta function (1951); and On the Lerch zeta function (1951).
    • The book began as a series of notes that Apostol produced, following the ideas they had already discussed, for delivering the first year course, and then the second year course.
    • The two volumes Introduction to analytic number theory and Modular functions and Dirichlet series in number theory were both published in 1976.

  63. Clement Durell (1882-1968)
    • He was promoted to senior mathematics master at Winchester College in 1910 and began publishing a series of articles in the Mathematical Gazette.
    • After the end of the war he returned to Winchester College and began publishing a series of articles in the Mathematical Gazette and a remarkable series of textbooks which would make him the best known writer of English school mathematics texts.
    • All chapters conclude with a series of exercises, with solutions at the end of the book.
    • Contents include the properties of the triangle and the quadrilateral; equations, sub-multiple angles, and inverse functions; hyperbolic, logarithmic, and exponential functions; and expansions in power-series.
    • Further topics encompass the special hyperbolic functions; projection and finite series; complex numbers; de Moivre's theorem and its applications; one- and many-valued functions of a complex variable; and roots of equations.

  64. Ralph James (1909-1979)
    • In certain problems (for example, in the theory of uniqueness of trigonometric series) it is desirable to define a second integral of a given function without defining the first integral.
    • The main result of the present paper is that the sum of an everywhere convergent trigonometric series is integrable in the proposed sense, and the series itself is the Fourier series of the sum.
    • Some later papers continuing this theme were Generalized nth primitives (1954), Integrals and summable trigonometric series (1955), and Summable trigonometric series (1956).

  65. Francis Murnaghan (1893-1976)
    • He also published The orthogonal and symplectic groups in 1958 which arose from a series of twenty lectures he gave in Dublin in 1957.
    • It covers topics such as: vectors and matrices; Fourier series; boundary value problems; Legendre and Bessel functions; integral equations; the calculus of variations and dynamics; and the operational calculus.
    • The first of these is a short book of less that 100 pages written for engineers and scientists, while the second consists of 19 lectures on such topics as: the Fourier integral; the Laplace integral transformation; the differential equations of Laguerre and Bessel; properties of special functions; asymptotic series for an error function, and for certain Bessel functions.
    • The 'converging factor' for an asymptotic series representing a function f(x) is that number by which the (n+1)st term of the series must be multiplied so that the result of adding this product to the sum of the first n terms will be f(x).
    • This report describes the determination to high precision of this factor for the asymptotic series representing the probability integral.

  66. Lev Arkad'evich Kaluznin (1914-1990)
    • He worked for the CNRS, published a series of papers on the structure of Sylow p-subgroups of symmetric groups, and in 1948 defended his doctoral thesis on the same topic.
    • His activities and achivements during the decade 1960-1970 includes: conducting research, teaching at Kiev State University and the Kiev Pedagogical Institute, consulting for the Department of Mathematical Linguistics, serving as a senior researcher at the Institute of Cybernetics of the Ukrainian Academy of Sciences, organising series of public lectures on mathematics, and serving as a member on editorial boards of several scientific journals.
    • Using his techniques, he was able to describe the characteristic subgroups of the Sylow p-subgroups, their derived series, their upper and lower central series, and more.
    • A particularly important result is the well-known theorem of Krasner and Kaluznin concerning the embeddings of a group with a subnormal series into the wreath product of the factors of the series.

  67. Jacob Bronowski (1908-1974)
    • He continued his research in geometry publishing a series of papers On triple planes and a paper The figure of six points in space of four dimensions.
    • [I am a] mathematician trained in physics, who was taken into the life sciences in middle age by a series of lucky chances.
    • He had presented a series for BBC television in the early 1960s called Insight in which he had looked at mathematical ideas such as probability, scientific ideas such as entropy and also the extent of human intelligence.
    • His last major project was to write and narrate the BBC television series The Ascent of Man which was filmed between July 1971 and December 1972.
    • The thirteen part series was broadcast in 1973 and also published in book form in that year.
    • I [EFR] remember watching this remarkable television series which, in a style much copied since, described in humanist terms the work of Pythagoras, Newton, Einstein, Galen, Versalius, Darwin, Mendel, Szilard, John von Neumann and many others showing their contributions as significant highlights in human development.

  68. Linda Goldway Keen (1940-)
    • In the 1980s she collaborated with Caroline Series on work on Riemann surfaces which returned to ideas that she had studied for her doctoral thesis:- .
    • Using powerful techniques developed by Thurston that involve hyperbolic three-manifolds, Keen and Series gave a geometric interpretation to Maskit's parameters.
    • Much of Keen's work has been done in collaborations with other mathematicians and after her work with Caroline Series she teamed up with Paul Blanchard, Robert Devaney, and Lisa Goldberg to produce important results on dynamical systems.
    • She teamed up again with Caroline Series for their joint article Pleating invariants for punctured torus groups (2004).
    • Keen, working with Nikola Lakic, wrote the book Hyperbolic geometry from a local viewpoint which was published in 2007 by Cambridge University Press in the London Mathematical Society Student Texts series.
    • The Emmy Noether Lectures are a prestigious series of lectures organised by the Association for Women in Mathematics.

  69. Li Shanlan (1811-1882)
    • Xu may have been a governor but he was also an excellent mathematician with an interest in infinite series.
    • He produced his own versions of logarithms, infinite series, and combinatorics but he did not follow the style of western mathematics but made his research naturally develop out of the foundations of Chinese mathematics.
    • The summation of series constitutes a branch of Chinese mathematics called Short Width [Chapter 4 of the Nine Chapters on the Mathematical Art.
    • The works of the great astronomer Guo Shoujing concerning the inequalities of the solar and lunar motion, Wang Lai's iterated sums, Dong Fangli's cyclotomical computations, and lastly the summation of series which appear in the algebra and the differential calculus of the Westerners constitute the major part of this chapter.
    • Zhu Shijie from the Yuan dynasty is the only one who has made use of the prescriptions relating to summation of series.
    • But his intention was only to expound the algebra and for that reason he presents the summation of series neither precisely nor methodically.

  70. Sheila Edmonds (1916-2002)
    • Two papers appeared in 1942: On the multiplication of series which are infinite in both directions was published in the Journal of the London Mathematical Society while On the Parseval formulae for Fourier transforms appeared in the Proceedings of the Cambridge Philosophical Society.
    • In the first of these papers Edmonds looks at the doubly infinite series ∑ an where the sum is over both positive and negative integers.
    • The series is said to converge if the two series, one defined over the positive integers, the other defined over the negative integers, both converge.
    • In this case the original series is said to have sum A = A' + A'' where A' and A'' are the sums over the positive and negative integers respectively.
    • In a series of papers published over the following years Edmonds examined a whole variety of different conditions on the functions f and g which give the required equalities.

  71. Jyesthadeva (about 1500-about 1575)
    • In [',' R C Gupta, The Madhava-Gregory series, Math.
    • Jyesthadeva describes Madhava's series as follows:- .
    • This is a remarkable passage describing Madhava's series, but remember that even this passage by Jyesthadeva was written more than 100 years before James Gregory rediscovered this series expansion.
    • To see how this description of the series fits with Gregory's series for arctan(x) see the biography of Madhava.

  72. Adriaan Cornelis Zaanen (1913-2003)
    • As a student, he came into contact with the ideas of modern analysis via Zygmund's book on trigonometric series and Banach's book on linear transformations.
    • During the war years he published papers such as On some orthogonal systems of functions (1939), A theorem on a certain orthogonal series and its conjugate series (1940), On some orthogonal systems of functions (1940), Uber die Existenz der Eigenfunktionen eines symmetrisierbaren Kernes Ⓣ (1942), Uber vollstetige symmetrische und symmetrisierbare Operatoren Ⓣ (1943), Transformations in Hilbert space which depend upon one parameter (1944), and On the absolute convergence of Fourier series (1945).
    • The results appeared in 1946 and 1947 in a series of seven papers in the Proceedings of the Royal Academy of Sciences of The Netherlands.
    • Their series of sixteen notes (all published in the Nederl.

  73. Salvatore Pincherle (1853-1936)
    • Remaining faithful to the ideas of Weierstrass, he did not take the topological approach that later proved to be most successful, but tried to start from a series of powers of the D derivation symbol.
    • Although his efforts did not prove very fruitful, he was able to study in depth the Laplace transform, iteration problems, and series of generalised factors.
    • Indeed, the conception of a function as a series of coefficients was still dominant at this time ..
    • He also studied linear sets of numerical sequences and of Laurent series.
    • He judged the mood of the majority of mathematicians in the world correctly and through a series of hard negotiations, he found his way around roadblocks with patience yet with determination.

  74. Sheila Scott Macintyre (1910-1960)
    • The ideas of her thesis were entirely her own; starting from a paper by Wright on functional inequalities she pursued the problems involved into the field of interpolation series in which her husband was also interested.
    • Much of her work was on interpolation series associated with analytic functions.
    • In a joint paper with her husband, convergence properties of the Abel series of a function, which had been discussed by several writers in the case of an integral function, were investigated for a class of functions regular in an angle.
    • Mrs Macintyre later devised an integral transform in which the kernel was obtained from that of the Laplace transform by a process involving fractional differentiation, and applied it to extend the theory of the Gregory-Newton and Abel interpolation series.
    • A few years later, in 1952, the two Macintyres published a more conventional type of joint paper, namely the 2-author work Theorems on the convergence and asymptotic validity of Abel's series (mentioned by Cossar in the quote above) which was published in the Proceedings of the Royal Society of Edinburgh.

  75. Arthur Jules Morin (1795-1880)
    • These memoirs presented the results of a series of carefully executed experiments on friction which he began planning in 1829.
    • During 1853-56 Morin undertook a series of experiments on the resistance of building materials which he published in a series of papers.
    • a totalizing anemometer, one of a series which was to be set into chimneys in the new buildings of the Palais de Justice in order to monitor ventilation.
    • The revolutions were counted through a series of gear wheels, so as to indicate the volume of air passing through the shaft in a period of time.

  76. James Clunie (1926-2013)
    • The two papers referred to in this quote are The determination of an integral function of finite order by its Taylor series (1953) and On the determination of an integral function from its Taylor series (1955).
    • This paper contains a revised account, for integral functions of finite order, of Wiman's analysis of the determination of an integral function by its power series.
    • Other papers are: An extension of quasi-monotone series (1953); On Bose-Einstein functions (1954); Univalent regions of integral functions (1954); The asymptotic paths of integral functions of infinite order (1955); On a theorem of Collingwood and Valiron (1955); The asymptotic behaviour of integral functions (1955); Note on integral functions of infinite order (1955); Note on a theorem of Parthasarathy (1955); The maximum modulus of an integral function of an integral function (1955); and Series of positive terms (1955).

  77. Ernst Lindelöf (1870-1946)
    • Then he worked on analytic functions, applying results of Mittag-Leffler in a study of the asymptotic investigation of Taylor series.
    • He considered analogues of Fourier series and applied them to gamma functions.
    • Moreover he considers series analogous to Fourier summation formulas and applications to the gamma function and the Riemann function.
    • In addition, new results concerning the Stirling series and analytic continuation are presented.
    • The book concludes with an asymptotic investigation of series defined by Taylor's formula.

  78. Charles Fox (1897-1977)
    • This paper, A class of null series, was published in the Proceedings of the London Mathematical Society in 1926.
    • Various Schlomilch series, representing null functions, have been discovered by Nielsen [','','1899], who gave proofs of them by arguments similar to those by which the Riemann-Lebesgue lemma is proved.
    • He submitted two further papers in June 1925, The Expression of Hypergeometric Series in Terms of Similar Series, and Some Further Contributions to the Theory of Null Series and Their Connexion with Null Integrals to the same Proceedings; both were published in 1927 as was his next paper A Generalization of an Integral Equation Due to Bateman which he submitted in 1926.

  79. Arnaud Denjoy (1884-1974)
    • In 1934 he wrote that his greatest achievements had been the integration of derivatives, the computation of the coefficients of a converging trigonometric series, a theorem on quasi-analytic functions, and differential equations on a torus.
    • The second of these topics, computation of the coefficients of a converging trigonometric series, was the subject of a four volume work Lectures on the computation of coefficients in a trigonometric series which appeared between 1941 and 1949.
    • These four volumes were an expanded version of work which had appeared in a series of papers by Denjoy beginning in 1920.
    • However, Choquet describes the four volume work Lectures on the computation of coefficients in a trigonometric series which contains the famous Denjoy integral, as [',' G Choquet, Biography in Dictionary of Scientific Biography (New York 1970-1990).','1]:- .

  80. Friedrich Hirzebruch (1927-2012)
    • In 1962 Hirzebruch gave a series of seminars at Brandeis and Berkeley.
    • In 1981 a series of five lectures by Hirzebruch and eight lectures by Gerard van der Geer were combined into notes by the two authors entitled Lectures on Hilbert modular surfaces.
    • The first 30 Arbeitstagungen which he organised form the "First Series" with a "Second Series" beginning in 1993.
    • The first papers were stimulated by the series of lectures by A Grothendieck during the first Arbeitstagung 1957 generalizing Hirzebruch's Riemann-Roch theorem.

  81. Abraham Seidenberg (1916-1988)
    • He held a Visiting Professorship at the University of Milan and he gave several series of lectures there.
    • In fact he was in Milan in the middle of giving a lecture series at the time of his death.
    • His career included a Guggenheim Fellowship [awarded 1953], visiting Professorships at Harvard and at the University of Milan, and numerous invited addresses, including several series of lectures at the University of Milan, the National University of Mexico, and at the Accademia dei Lincei in Rome.
    • At the time of his death, he was in the midst of another series of lectures at the University of Milan.
    • Concepts such as plane curve, intersection multiplicity, branch, genus, and linear series are introduced in a concrete, computational way; the necessary abstract algebra is kept in a secondary position whenever possible.

  82. Émile Borel (1871-1956)
    • the theory of measure, Borel's theory of divergent series, his theory of non-analytic continuation and the theory of quasi-analytic functions all derive from ideas which make their first appearance in this paper.
    • In [',' M Frechet, La vie et l’oeuvre d’Emile Borel, Enseignement mathematique 11 (1965), 1-95.','8] Borel's mathematical work is divided into the following topics: Arithmetic; Numerical series; Set theory; Measure of sets; Rarefaction of a set of measure zero; Real functions of real variables; Complex functions of complex variables; Differential equations; Geometry; Calculus of probabilities; and Mathematical physics.
    • Borel, although not the first to define the sum of a divergent series, was the first to develop a systematic theory for a divergent series which he did in 1899.
    • In addition, between 1921 and 1927, Borel published a series of papers on game theory and became the first to define games of strategy.
    • He published many outstanding works over the years including Lecons sur la theorie des fonctions Ⓣ (1898), Lecons sur les series entieres Ⓣ (1900), Lecons sur les fonctions divergentes Ⓣ (1901), Lecons sur les fonctions de variables reelles et les developpements en series de polynomes Ⓣ (1905), Le Hasard Ⓣ (1913), Lecons sur les fonctions monogenes uniformes d'une variable complexe Ⓣ (1917), L'Espace et le temps Ⓣ (1921), Methodes et problemes de la theorie des fonctions Ⓣ (1922), Traite du calcul des probabilites et ses applications Ⓣ (1924-1934), and Principes et formules classiques du calcul des probabilites Ⓣ (1925).

  83. Henri Padé (1863-1953)
    • In his thesis Pade made the first systematic study of what we call today Pade approximants, which are rational approximations to functions given by their power series.
    • At around the same time Euler used Pade-type approximation to find the sum of a series.
    • The method continued to be used from time to time by various mathematicians, for example Kummer in 1837 used Pade approximants to sum series which only converged very slowly.
    • Pade established various properties of this table in his thesis and developed the ideas further in later papers, particularly in 1899 when he studies the exponential series and in 1901 when he considered (1+x)m, for m not an integer.
    • Although the theory of Pade approximants which he had developed in his thesis, and in many later papers, was not quick to be taken up by many other mathematicians, it did become well known after Borel presented Pade approximants in his 1901 book on divergent series.

  84. Paul Butzer (1928-)
    • Theory 160 (1-2) (2009), 3-18.','5] how he organised a remarkably successful series of conferences:- .
    • In all, these Oberwolfach symposia - with Bela Szokefalvi-Nagy as co-organizer from the fourth onwards - drew about 250 different experts from 24 countries including Hungary, Bulgaria, Poland, Romania and eventually Russia, the roster of participants almost representing a Who's Who in approximation theory and associated fields such as harmonic analysis, functional analysis and operator theory, integral transform theory, orthogonal polynomials, interpolation, special functions, divergent series.
    • Approximately half of this volume deals with the theories of Fourier series and of Fourier integrals from a transform point of view.
    • Special as this approach may seem, it not only embraces many of the topics of the classical theory but also leads to significant new results, e.g., on summation processes of Fourier series, conjugate functions, fractional integration and differentiation, limiting behaviour of solutions of partial differential equations, and saturation theory.
    • The book received outstanding reviews; for example James Rovnyak [',' J Rovnyak, Review: E B Christoffel: The Influence of His Work on Mathematics and the Physical Sciences edited by P L Butzer and F Feher, Science, New Series 217 (4555) (1982), 145.','14] writes:- .

  85. Boris Yakovlevic Bukreev (1859-1962)
    • During the 1890s Bukreev produced a series of high quality papers including: On the theory of gamma functions; On some formulas in the theory of elliptic functions of Weierstrass; On the distribution of the roots of a class of entire transcendental functions; and Theorems for elliptic functions of Weierstrass.
    • After 1900 he became interested in the theory of series, publishing papers such as Notes on the theory of series and he also worked on the Calculus of Variations.
    • He taught courses on analysis, differential and integral calculus and their applications to geometry, the theory of integration of differential equations, the theory of series, algebra, and other topics.
    • For example: Introduction to the theory of series; Elements of the theory of determinants; Course on definite integrals (1903); and Elements of algebraic analysis (1912).

  86. Luis Caffarelli (1948-)
    • dissertation and some other articles on summability and conjugation of series of special polynomials.
    • In 1971 Caffarelli submitted his thesis Sobre conjugacion y sumabilidad de series de Jacobi Ⓣ to the University of Buenos Aires and he was awarded a Ph.D.
    • He undertook joint research with his thesis advisor Calixto Pedro Calderon and they wrote two joint papers: Weak type estimates for the Hardy-Littlewood maximal functions (1974-74); and On Abel summability of multiple Jacobi series (1974).
    • Shortly after my arrival, I attended a fascinating series of lectures by Hans Lewy and became interested in nonlinear partial differential equations, variations inequalities and free-boundary problems.
    • In a series of pioneering papers, Caffarelli put forward a novel methodology which eventually leads, after several truly amazing technical estimates that step by step improve the regularity of the solutions and the boundary, to full regularity under very mild assumptions.

  87. Gaston Julia (1893-1978)
    • Volume 2, in three parts, consists of articles on (i) J points of functions of one variable, (ii) J points of functions of several variables, and (iii) Series of iterates.
    • Volume 3 contains four parts: (i) Functional equations and conformal mapping; (ii) Conformal mapping; (iii) General lectures; and (iv) Isolated works in analysis on Implicit function defined by the vanishing of an active function, and on certain series.
    • An admirable account of the theory of Fourier series (pp.
    • This book is the sixteenth of the well known series, 'Cahiers Scientifiques,' and is the first of a series which proposes to give the mathematical foundation of quantum mechanics.

  88. Matyá Lerch (1860-1922)
    • He also attended several lecture courses by Fuchs, namely (i) Introduction to the theory of infinite series, (ii) Integration of differential equations, (iii) The theory of linear differential equations, and (iv) Invariant theory.
    • This set the direction of his future research to be on special functions, infinite series and analytic functions.
    • Over the following fourteen years he wrote 31 papers on infinite series, geometry, special functions, and number theory.
    • In that topic he studied infinite series, and the gamma function as well as other special functions.
    • He also studied the principle of most rapid convergence of a series.

  89. Pia Nalli (1886-1964)
    • A function f can be expanded as a Fourier series and the coefficients calculated as integrals.
    • She also proved uniqueness theorems for the trigonometric series expansion for this class of functions.
    • Also during 1915-1917, she focused her attention on the problem of summation of series, with special reference to Dirichlet series.
    • She published her results in a series of four notes entitled Sulle equazioni integrali Ⓣ in 1919 and the memoir Generalizzazione di alcuni punti della teoria delle equazioni integrali di Fredholm Ⓣ in the same year.

  90. Paddy Kennedy (1929-1966)
    • He continued with a remarkable research output, with papers: Integrability theorems for power series (1955), Conformal mapping of bounded domains (1956), and A class of integral functions bounded on certain curves (1956).
    • His remarkable publication record continued with: A note on uniformly distributed sequences (1956), Fourier series with gaps (1956), General integrability theorems for power series (1957), Fourier series with gaps.
    • II (1957), Remark on a theorem of Zygmund (1958), On the coefficients in certain Fourier series (1958).

  91. Norman Levinson (1912-1975)
    • The turning point in Levinson's studies had come when he signed up for Wiener's graduate course on Fourier series and integrals in 1933-34.
    • In 1940 Levinson published Gap and density theorems in the American Mathematical Society Colloquium Publication Series.
    • It was a great tribute to the young mathematician that he had been invited to write a book in a series which was reserved for distinguished senior mathematicians.
    • Levinson wrote only two papers on time series, but these had a large impact.
    • Shortly before his death he wrote a series of important papers on the Riemann hypothesis arising from this fundamental number theory paper.

  92. Leopold Kronecker (1823-1891)
    • students to hear that Kronecker was questioned at his oral on a wide range of topics including the theory of probability as applied to astronomical observations, the theory of definite integrals, series and differential equations, as well as on Greek, and the history of philosophy.
    • In 1870 Heine published a paper On trigonometric series in Crelle's Journal, but Kronecker had tried to persuade Heine to withdraw the paper.
    • Even the concept of an infinite series, for example one which increases according to definite powers of variables, is in my opinion only permissible with the reservation that in every special case, on the basis of the arithmetic laws of constructing terms (or coefficients), ..
    • certain assumptions must be shown to hold which are applicable to the series like finite expressions, and which thus make the extension beyond the concept of a finite series really unnecessary.

  93. Ivan Ivanovich Privalov (1891-1941)
    • He was allowed to choose the topic for one of these lectures - Picard's Theorem - while the other - Summation of Trigonometrical Series - was on a topic set by the Faculty.
    • The titles of his lectures over the years 1922-25 were: A generalization of a theorem of Fatou (1922), Properties of the coefficients of a Taylor series (1923), The uniform convergence of sequences of analytic functions which give a 'schlicht' conformal mapping (1923), The uniqueness of an analytic function (1923), A generalization of Vitali's theorem on sequences of analytic functions (1924), The convergence of conjugate trigonometric series (1924), The convergence of sequences of analytic functions (1924), Concerning a condition of Blaschke (1925), A new definition of a harmonic function (1925), Harmonic functions (1925).
    • In this domain he has obtained profound results on the theory of trigonometric series, on integrals of Cauchy type, on boundary value problems, in the study of properties of analytic functions inside domains and on the theory of subharmonic functions.
    • Later textbook were: Fourier series (1930); Course of differential calculus (1934); Course of integral calculus (1934); Integral equations (1935); Foundation of the analysis of infinitesimals, textbook for self-education (1935); and Elements of the theory of elliptic functions (1939).

  94. Otto Szász (1884-1952)
    • Szasz's main work was in real analysis, particularly Fourier series.
    • His most important contributions are probably between 1915 and 1930 when he made a series of remarkable contributions to a number of different areas.
    • Other work by Szasz made major contributions to questions posed by Landau on the maximum modulus of the partial sums of a power series.
    • He also studied problems on power series related to work of Frigyes Riesz.
    • Some of Szasz's contributions to Fourier series related to results proved by Bernstein, Hardy, Littlewood and Fejer.

  95. John Knopfmacher (1937-1999)
    • This was followed by a series of articles on several aspects of algebra, including non-associative algebra, finite groups, Lie algebras, homology, and finite topological spaces, in journals such as the 'Bulletin' of the American Mathematical Society, and the 'Oxford Quarterly Journal of Mathematics'.
    • In a profound series of papers, starting with an announcement in the 'Bulletin' of the American Mathematical Society, and developed in six successive papers in the 'Journal fur die Reine und Angewandte Mathematik' from 1972 to 1975, he laid the foundations for the theory, and obtained many of its most significant results.
    • Just as we can regard the Cantor product as being a product analogue of the series of Sylvester, this new product is analogous to the classical Engel representation for real numbers.
    • The growth conditions satisfied by the digits in the product are likewise shown to correspond to those required for the Engel series.
    • When I left Graz, one week ago, I talked to him on his plans for his series of lectures here in Graz ..

  96. Takebe Katahiro (1664-1739)
    • Perhaps Takebe's greatest achievement was to devise a method to calculate a series expansion of a function.
    • Now we know that no such polynomial exists but Takebe found an infinite series expressing s in terms of k, namely .
    • In fact, looked at in modern terms, what Takebe was calculating was the Taylor series expansion of arcsin(√k)2 about k = 0.
    • The two approaches are often complementary, as he demonstrated by showing that an infinite series that he had obtained inductively could also be derived algebraically.
    • His procedure for calculating the infinite series played a key role in the development of analysis in Japan in the following decades.

  97. Brian Kuttner (1908-1992)
    • Fourier series, strong summability, Riesz means, Norland methods, and Tauberian theory.
    • Most of Kuttner's early work is on Fourier series and summability.
    • Hardy quotes some of these early results of Kuttner's in his treatise Divergent series (1949).
    • at the age of only 26, Kuttner proved a basic theorem in the general theory of trigonometric series, a result delightful for both the deceptive simplicity of its statement and the elegance of its proof.
    • Zygmund greatly admired this theorem of Kuttner, which now occupies an honoured place in Zygmund's monumental work on trigonometric series.

  98. Hugh MacColl (1837-1909)
    • For example The Calculus of Equivalent Statements was a series of eight papers published in the Proceedings of the London Mathematical Society between 1877 and 1898.
    • He published a series of nine papers entitled Symbolic Logic in The Athenaeum between 1903 and 1904, and a series of eight articles Symbolic Reasoning in Mind between 1880 and 1906.
    • This little volume may be regarded as the final concentrated outcome of a series of researches begun in 1872 and continued (though with some long breaks) until today.
    • Bertrand Russell reviewed the work in a five-page article [',' B Russell, Review: Symbolic Logic and Its Applications by Hugh MacColl, Mind, New Series 15 (58) (1906), 255-260.','12].

  99. C T Rajagopal (1903-1978)
    • Rajagopal studied sequences, series, summability.
    • He also studied functions of a complex variable giving an analogue of a theorem of Edmund Landau on partial sums of Fourier series.
    • In several papers he studied the relation between the growth of the mean values of an entire function and that of its Dirichlet series.
    • He showed that the series for tan-1x discovered by Gregory and those for sin x and cos x discovered by Newton were known to the Hindus 150 years earlier.
    • He identified the Hindu mathematician Madhava as the first discoverer of these series.

  100. Isaac Schoenberg (1903-1990)
    • He read many mathematics textbooks in French, including Jacques Hadamard's Lecons de Geometrie Elementaire Ⓣ and other mathematics books in the same series.
    • He attended Edmund Landau's courses on 'Entire Functions', 'Trigonometric Series', 'The Big Fermat Problem', and 'Analytic Number Theory'.
    • By the time he returned to Gottingen in 1924, Alexander Ostrowski was teaching there and Schoenberg attended his course on 'Overconvergence of Power Series' and also the seminar that Ostrowski ran.
    • Schoenberg made further outstanding contributions in a series of papers between 1950 and 1959 on the theory of Polya frequency functions.
    • He investigated their wide applications in approximation theory in a series of three papers between 1969 and 1973.

  101. Jozéf Hoëné Wronski (1778-1853)
    • He criticised Lagrange's use of infinite series and introduced his own ideas for series expansions of a function.
    • Out of this came his "universal Hoene-Wronski series" or "la serie universelle de Wronski".
    • This consisted of the development of a function as a series in terms of arbitrary functions.
    • The coefficients in this series are determinants now known as Wronskians (so named by Muir in 1882).

  102. Alfred Tauber (1866-1942)
    • He lectured in Vienna on the theory of series, trigonometric series, and potential theory.
    • Tauber's lack of success in being given a professorial position was certainly not due to any lack of mathematical ability, for he continued to publish a series of high quality papers.
    • He obtained important results on divergent series and the name 'Tauberian Theorems' was coined by Hardy and Littlewood.
    • The conditions which Tauber gave to allow him to prove the converse of Abel's limit theorem on power series are now known as 'Tauberian conditions' and appeared in Ein Satz aus der Theorie der unendlichen Reihen Ⓣ (1897).

  103. Maurits Escher (1898-1972)
    • He later tried working with the concept of similarities, using identical motifs of diminishing size, arranged in a series of concentric circles, but as with much of his work, he was unhappy about the final quality.
    • Escher used pictures to tell a story in his Metamorphosis series of designs.
    • These designs brought together many of Escher's skills and show the transformation from one distinct object to another, by means of a series of slight changes to a regular pattern in the plane.
    • An Italian coastline is transformed through a series of convex polygons into a regular pattern in the plane until finally a distinct, coloured, human motif emerges, signifying his change of perspective from landscape work to regular division of the plane.
    • Escher fell ill initially in 1964 whilst delivering a series of lectures in North America.

  104. Edward Van Vleck (1863-1943)
    • For example he published On the determination of a series of Sturm's functions by the calculation of a single determinant (1899), On linear criteria for the determination of the radius of convergence of a power series (1900), On the convergence of continued fractions with complex elements (1901), A determination of the number of real and imaginary roots of the hypergeometric series (1902), On an extension of the 1894 memoir of Stieltjes (1903), and On the extension of a theorem of Poincare for difference-equations (1912).
    • Van Vleck was American Mathematical Society Colloquium lecturer in 1903 giving six lectures on divergent series and continued fractions.
    • He published these lectures in the first volume of the series American Mathematical Society Colloquium Publications.

  105. Bernhard Riemann (1826-1866)
    • He spent thirty months working on his Habilitation dissertation which was on the representability of functions by trigonometric series.
    • While preceding papers have shown that if a function possesses such and such a property, then it can be represented by a Fourier series, we pose the reverse question: if a function can be represented by a trigonometric series, what can one say about its behaviour.
    • Riemann studied the convergence of the series representation of the zeta function and found a functional equation for the zeta function.

  106. Thomas Bromwich (1875-1929)
    • He worked on infinite series, particularly during his time in Galway.
    • In 1908 he published his only large treatise An introduction to the theory of infinite series which was based on lectures on analysis he had given at Galway.
    • In a series of papers he put Heaviside's calculus on a rigorous basis treating the operators as contour integrals.
    • Thomas Bromwich's Infinite Series .

  107. Walter Rudin (1921-2010)
    • Rudin was awarded his doctorate in 1949 for his thesis Uniqueness Theory for Laplace Series.
    • He presented the paper Uniqueness theory for Hermite series to the Congress on 1 September.
    • As we have seen, Rudin's early work was on trigonometric series and holomorphic functions of one complex variable.
    • At the NSF regional conference from 1-5 June 1970, held at the University of Missouri, St Louis, Rudin gave a series of lecture which were published as Lectures on the edge-of-the-wedge theorem (1971).

  108. Beppo Levi (1875-1961)
    • He founded the journal Mathematicae Notae, the series Publicaciones del Instituto de matematicas, and the series of books Monografias.
    • He published Sistemas de ecuaciones analiticas en terminos finitos, diferenciales y en derivadas partiales (Systems of Analytic Equations: Equations in Finite Terms, Ordinary and Partial Differential Equations) (1944) as the first volume in the Monografias series.
    • we should like to remark upon the elegance of mathematical expression which is possible in Spanish and to suggest that with the present volume the new series of monographs has made a most satisfactory beginning.

  109. Hilda Geiringer (1893-1973)
    • This was awarded in 1917 for a thesis on Fourier series in two variables.
    • Siegmund-Schultze writes in [',' R Siegmund-Schultze, Hilda Geiringer von Mises, Charlier Series, Ideology, and the human side of the emancipation of applied mathematics at the University of Berlin during the 1920s, Historia Mathematica 20 (1993), 364-381.','3]:- .
    • The debate over Geiringer's theses for Habilitation opens up a chapter of the history of mathematical statistics, namely, expansions of a discrete distribution with an infinite number of values in a series in successive derivatives of the Poisson distribution with respect to the parameter.
    • She wrote up her outstanding series of lectures on the geometrical foundations of mechanics and, although they were never properly published, these were widely used in the United States for many years.

  110. Kollagunta Ramanathan (1920-1992)
    • After a series of papers published in North American journals such as Identities and congruences of Ramanujan type (1950), The Theory of Units of Quadratic and Hermitian Forms (1951), Abelian quadratic forms (1952), and Units of quadratic forms (1952), he returned to publishing his research in Indian journals.
    • For several years, Professor Ramanathan had been actively interested in the study of published and unpublished work of Srinivasa Ramanujan, expounding, elucidating and extending Ramanujan's beautiful work on singular values of certain modular functions, Rogers-Ramanujan continued fractions and hypergeometric series.
    • An example of a paper motivated by his study of Ramanujan's work is Hypergeometric series and continued fractions (1987).
    • The author extends much of the work mentioned above to basic hypergeometric series.

  111. Maurice Auslander (1926-1994)
    • It may appear strange that his first paper is the third in a series of papers entitled On the dimension of modules and algebras but this is because it was one of a series of papers with this title in the Nagoya Mathematics Journal by a number of distinguished mathematicians.
    • In 1957 he published the sixth paper in the series On the dimension of modules and algebras.
    • Auslander's two papers in this series contain the now standard result that the global dimension of a ring can be computed from knowledge of the cyclic modules.

  112. Hermann Laurent (1841-1908)
    • However he was already working on writing mathematics texts, his first being Traite des series in 1862.
    • He wrote 30 books and a fair number of papers on infinite series, equations, differential equations and geometry.
    • N Ya Sanin and A V Letnikov published a series of papers in Matem.
    • Particularly given the topics on which Laurent worked, it is easy to assume that Laurent series must be named after him.
    • However this is not the case and Laurent series are in fact named after Pierre Laurent.

  113. Alfréd Haar (1885-1933)
    • Haar asked a series of fundamental questions about systems of orthonormal functions on the interval [0, 1].
    • Haar wrote: one wants to be able to determine sufficient conditions that a series of such functions is convergent; one wants examples of relatively sensible functions which do not converge in the pointwise or uniform sense; one wants to understand how summation methods may be used to overcome the problems of divergence; and one wants to know exactly when, if the series of partial sums of an orthogonal expansion of a function converges, its limit equals the original function.
    • He constructed what is now known as Haar's orthonormal basis to answer the question of divergence of continuous functions expanded as series of orthonormal systems of functions.

  114. Patrick Moran (1917-1988)
    • He attended a variety of courses in 1938: J A Todd on differential geometry, M H A Newman on analysis, A E Ingham on algebra, H Heilbronn on the theory of numbers, S Goldstein on electricity and hydrodynamics, E Cunningham on statics and dynamics, J C Burkill on theory of real functions and W W Rogosinski on Fourier series.
    • In 1939 he took courses: M H A Newman on topology, A S Besicovitch on integration, F Smithies on integral equations, J C Burkill on the theory of real functions, W W Rogosinski on Fourier series, and G U Yule on statistics.
    • Some of the first papers he published on statistics were Random associations on a lattice (1947), The random division of an interval (1947), Some theorems on time series (1947), On the method of paired comparisons (1947), and Rank correlation and permutation distributions (1948).
    • One of the first areas he worked on after taking up the statistics chair in Canberra led first to the paper A probability theory of dams and storage systems and, after a series of others on the topic, the book The Theory of Storage (1959).

  115. George Lidstone (1870-1952)
    • He read the paper Note on the Summation of a Trigonometrical Series to the Society at its meeting on Friday 10 March 1922.
    • This paper is composed of a series of commentaries, illustrated frequently with numerical examples, on the subject of interpolation.
    • that the corresponding Charlier type B series ..
    • The author shows by numerical examples that in many cases this series gives a better fit than Pearson's type III curves.

  116. Maurice Kendall (1907-1983)
    • Kendall continued a remarkable stream of research papers on topics such as the theory of k-statistics, time series, and rank correlation methods and a monograph Rank Correlation in 1948.
    • In 1963 he published (jointly with P A P Moran) Geometrical probability followed by Time series (1973) in which Kendall states his objectives to bridge the gap between "sophisticated theory and practical applications" in the field of time series and to "treat the subject in its entirety for the benefit of the practising statistician".
    • He also published A course in multivariate analysis and Cluster analysis as well a whole series of articles Studies in the history of probability and statistics.

  117. George Mackey (1916-2006)
    • He then produced a series of important papers on group representations including On induced representations of groups (1951), Induced representations of locally compact groups (1952), and Symmetric and anti symmetric Kronecker squares and intertwining numbers of induced representations of finite groups (1953).
    • During the academic year 1966-67, Mackey delivered a series of lectures on group representations and their applications at Oxford University in England where he was George Eastman visiting professor.
    • Mackey has written many beautiful survey articles and in 1992 the American Mathematical Society and the London Mathematical Society in their wonderful series 'History of Mathematics' published The scope and history of commutative and noncommutative harmonic analysis by Mackey.
    • 1968), and Caroline Series (Ph.D.

  118. Frantisek Wolf (1904-1989)
    • His thesis Contribution a la theorie des series trigonometriques generalisees et des series a fonctions orthogonales Ⓣ was supervised by Otakar Borvka, and he was awarded the degree Rerum Naturum Doctor in 1928.
    • The first was An extension of the Phragmen-Lindelof theorem while the second was On summable trigonometrical series: an extension of uniqueness theorems.
    • The first gives, under certain rather complex conditions, inversion formulae for trigonometric integrals and the second asserts that under the same conditions the difference between the given trigonometric integral and the trigonometric series of a certain function will be uniformly summable to zero throughout a given interval.
    • In Chapter VI he gives some results on the inversion of order of integration in a trigonometric integral equivalent to the integration of trigonometric series term by term.

  119. Gösta Mittag-Leffler (1846-1927)
    • In this paper Mittag-Leffler proposed a series of general topological notions on infinite point sets based on Cantor's new set theory [',' L Garding, Mittag-Leffler’s and Sonya Kovalevski’s mathematical papers, in Mathematics and Mathematicians : Mathematics in Sweden before 1950 (Providence, R.I., 1998), 85-96.','8]:- .
    • Between 1900 and 1905 Mittag-Leffler published a series of five papers which he called "Notes" on the summation of divergent series.
    • The aim of these notes was to construct the analytical continuation of a power series outside its circle of convergence.

  120. Ulisse Dini (1845-1918)
    • Dini looked at infinite series and generalised results such as a theorem of Kummer and one of Riemann, the ideas for which had first emerged in work of Dirichlet.
    • He discovered a condition, now known as the Dini condition, ensuring the convergence of a Fourier series in terms of the convergence of a definite integral.
    • As well as trigonometric series, Dini studied results on potential theory.
    • He published Foundations of the theory of functions of a real variable in 1878; a treatise on Fourier series in 1880; and a two volume work Lessons on infinitesimal analysis with the first volume appearing in 1907 and the second in 1915.

  121. Anastácio da Cunha (1744-1787)
    • De Pombal put through a series of major reforms, and around 1758-59 he moved against the Jesuits and the Society of Jesus.
    • Da Cunha develops a criterion for the convergence of a series which he uses to define the exponential function in a rather modern way, and from these develops the binomial series.
    • In Principios Matematicos da Cunha also gave a definition of the convergence of a series which is equivalent to Cauchy's convergence criterion.

  122. Grace Alele-Williams (1932-)
    • The African Mathematics Programme organized writing workshops in Africa that produced the Entebbe Modern Mathematics Series.
    • Her paper gives the following conclusion about the Entebbe Mathematics Series comprising 67 volumes of texts which cover the entire primary, training, secondary and sixth form levels:- .
    • The Entebbe Mathematics Series have sometimes been dubbed American but this is to ignore the valuable contribution of the African participants, who feel keenly the African origin of the series.

  123. Vojtch Jarník (1897-1970)
    • He studied the problem for the particular case of the ellipsoid in a series of papers.
    • During the decade 1939-49 he wrote a series of papers dealing with the geometry of numbers, in particular dealing with Minkowski's inequality for convex bodies.
    • He also wrote on rearrangement of infinite series, trigonometric series and other areas of analysis.

  124. Joseph Ritt (1893-1951)
    • One year before the publication of this work, Ritt had published Theory of Functions which provides an introduction to the theory of functions in a series of short and to the point lecture notes giving the student an account of the fundamental definitions and theorems of the subject.
    • He produced two books on the subject which contained the results from a long series of papers which he produced on the topic.
    • In a remarkable series of papers which appeared in the Annals of Mathematics he investigated a differential group of order n was he defined as a power series, with certain properties, in two sets of n indeterminates and their derivatives of various orders.

  125. Amédée Mannheim (1831-1906)
    • On the Rule will be seen, along its whole length, and close to the upper edge of the groove, a series of graduations, with an identically similar series along the upper edge of the Slide.
    • Another series of graduations will also be seen on the Rule along the lower edge of the groove, with a corresponding series on the Slide.

  126. Christoph Gudermann (1798-1852)
    • His own contributions tended to be a whole series of special cases (although this could not have been obvious at the time) which were forgotten later when the general results which included them were found.
    • Gudermann, at this time, was particularly interested in the theory of elliptic functions and in the expansion of functions by power series.
    • In particular his use of power series in the study of the hyperbolic functions is of importance.
    • The transformation of his conception of an analytic function from a differentiable function to a function expandable into a convergent power series was made during this early period of Weierstrass's mathematical activity.

  127. Ernesto Pascal (1865-1940)
    • Let us note, however, that in 1897 he began publication of his Repertorio di matematiche superiori Ⓣ series of texts.
    • This series was translated into several languages, including German, and reviews of some of the Italian texts and German translations can be read here: .
    • Other texts by Pascal appeared in the Manuale Hoepli series.
    • As an example of one of the many works in this series we quote from the review by Edwin Bailey Elliott of Pascal's I Gruppi Continui di Trasformazioni Ⓣ (1903):- .

  128. Paul Bachmann (1837-1920)
    • In view of the ambitious series of volumes by Bachmann; giving a comprehensive exposition of number theory, a series not yet completed, the appearance of a new volume on the elements of the subject, quite independent of the series mentioned, will doubtless cause some surprise.
    • The author believes that the present book, both in contents and in foundation, may well be considered as a supplementary volume to his former series.

  129. Pierre Bonnet (1819-1892)
    • One year before this, in 1843, Bonnet had written a paper on the convergence of series with positive terms.
    • Another paper on series in 1849 was to earn him an award from the Brussels Academy.
    • However between these two papers on series, Bonnet had begun his work on differential geometry in 1844.
    • Between 1844 and 1867 he published a series of papers on the differential geometry of surfaces.

  130. Abraham Gelbart (1911-1994)
    • in 1940 for his 26-page thesis On the Growth Properties of a Function of Two Complex Variables Given by its Power Series Expansion.
    • It is of particular value in the Weierstrass-Hadamard approach, i.e., in obtaining properties of a function from the coefficients of its power series expansion.
    • In this paper we shall obtain growth properties in terms of the coefficients of the power series expansion of a function f (z1, z2) of two complex variables analytic in special domains of the type mentioned above; first, with the aid of Bergman's integral formula, along the two-dimensional surfaces common to the bounding hypersurfaces, and then, along a class of two-dimensional surfaces lying in only one of the bounding hypersurfaces and having a line of contact with another bounding hypersurface.
    • The Distinguished Scientist Lecture Series at Bard College originated in 1979 when Nobel laureate physicist Paul Dirac accepted an invitation from Abe Gelbart and The Bard Center to deliver a lecture titled "The Discovery of Antimatter." The talk presented a view of science rarely seen by the general public - as a record of personal achievement as well as a body of facts and theories.

  131. Joseph Raabe (1801-1859)
    • Today Raabe's name is mostly remembered for Raabe's Test for convergence of series.
    • He had published another paper on series while still in Vienna, however, namely Uber Reihen, deren Differenzenreihen wiederkehren Ⓣ (1829).
    • Then the series a1 + a2 + a3 + ..
    • This test, which is an extension of d'Alembert's ratio test, often succeeds for series in which the terms contain factorials, where d'Alembert's simple ratio test is inconclusive.

  132. Pierre Cartier (1932-)
    • According to Bourbaki's method, we studied reports on the topics which were to be treated in the series.
    • I remember that we discussed many things, especially a text written by Laurent Schwartz on the foundations of Lie groups; it was one of the first drafts in the well-known series of Bourbaki on Lie groups .
    • In this paper, the distinguished author presents a series of philosophical musings based on the distinction between a "true finite" set and a "theoretical finite" set.
    • Through a series of illustrations the author suggests that the set-theoretic conception is a liberating notion that has important implications for mathematics.

  133. Yitz Herstein (1923-1988)
    • Noncommutative rings appeared in The Carus Mathematical Monographs series published by The Mathematical Association of America.
    • This colourful and informative book on noncommutative ring theory is based on a series of expository lectures given by the author in the summer of 1965 at Bowdoin College before an audience of teachers from colleges and small universities.
    • The spirit of the Carus Monograph series is clearly embodied in this moving and excellently written account of important aspects of classical and modern ring theory.
    • Topics in Ring Theory was based on lectures Herstein gave at the University of Chicago and first published in the University of Chicago Mathematics Lecture Notes series.

  134. Tiberiu Popoviciu (1905-1975)
    • Over the following years, he published his thesis and a series of papers on this topic beginning with Sur le prolongement des fonctions convexes d'ordre superieur Ⓣ (1934), Sur l'approximation des fonctions convexes d'ordre superieur Ⓣ (1934) and Notes sur les fontions convexes d'ordre superieur Ⓣ (1936).
    • It was, however, only in the early 1950s that he began to publish a whole series of important papers on the topic.
    • It was the first book in an intended series on Computing Theory, Numerical Analysis and Information Theory.
    • This is planned to be the first volume in a series on numerical analysis.

  135. Yakov Davydovich Tamarkin (1888-1945)
    • It was published in English in Mathematische Zeitschrift in 1928 as Some general problems of the theory of ordinary linear differential equations and expansion of an arbitrary function in series of fundamental functions.
    • Tamarkin published a series of papers written jointly with Hille between 1932 and 1934.
    • For example they published: On the summability of Fourier series (two papers), On a theorem of Hahn-Steinhaus, On a theorem of Paley and Wiener, On the theory of linear integral equations.
    • He published a major text The Problem of Moments written jointly with J A Shohat in the American Mathematical Society Mathematical Surveys Series (1943).

  136. Lionel Cooper (1915-1977)
    • The theory of the absolute summability of series has been treated by various authors and the results have been applied to Fourier series.
    • In this paper results for Fourier integrals parallel to those of Bosanquet for series are obtained.
    • After a series of medical tests, doctors discovered that he had a heart defect and said that surgery was the only way to solve the problem.

  137. Wadysaw Orlicz (1903-1990)
    • In 1928 he wrote his doctoral thesis Some problems in the theory of orthogonal series under the supervision of Eustachy Żyliński.
    • His book Linear Functional Analysis, (Peking 1963, 138 pp - in Chinese), based on a series of lectures delivered in German on selected topics of functional analysis at the Institute of Mathematics of the Academia Sinica in Beijing in 1958, was translated into English and published in 1992 by World Scientific, Singapore.
    • Orlicz's contribution is important in the following areas in mathematics: function spaces (mainly Orlicz spaces), orthogonal series, unconditional convergence in Banach spaces, summability, vector-valued functions, metric locally convex spaces, Saks spaces, real functions, measure theory and integration, polynomial operators and modular spaces.
    • For example, the Orlicz-Pettis theorem says that in Banach spaces the classes of weakly subseries convergent and norm unconditionally convergent series coincide.

  138. Alfonso Del Re (1859-1921)
    • With a later 1911 memoir, Alfonso Del Re illustrated a series of arguments on the independence of his series of postulates, in addition to those implicit in the reference to Huntington.
    • B A Bernstein in a paper read before the American Mathematical Society (in San Francisco) on 25 October 1913, after quoting the contributions of Charles Peirce, Edward Huntington and H M Sheffer, proposed a series of postulates of completion in terms of the operation "exception".
    • Bernstein himself deduced the sufficiency of these from the series of postulates indicated by A Del Re in his 'Logical Algebra', adding new evidence to those implicit in the reference to Huntington.

  139. Ernst Steinitz (1871-1928)
    • He also wrote several significant papers on conditionally convergent series such as Bedingt konvergente Reihen und konvexe Systeme Ⓣ (1913).
    • In fact Edmund Landau had suggested to Steinitz that he read Paul Levy's 1905 paper on conditionally convergent series but Steinitz found this unsatisfactory writing in his 1913 paper that Landau's suggestion:- .
    • However much we must deplore publications which are in such a deficient form that extensive commentaries are necessary if they are to be understood, we must admit that M Levy has largely proved the proposition on the indicated series with usually complex numbers in the quoted work.
    • In fact, perhaps surprisingly, it was in the context of his work on series that he set out what today is often called the 'Steinitz replacement theorem' for vector spaces.

  140. Douglas Northcott (1916-2005)
    • He then went on to study for Part III, taking courses on Fourier series and divergent series from G H Hardy which strongly influenced him.
    • He wrote up the results he had obtained while in hospital in India as the paper Abstract Tauberian theorems with applications to power series and Hilbert series which was published in 1947.

  141. Simon Newcomb (1835-1909)
    • He wrote a paper showing how the coordinates of a planet might be represented by trigonometric series.
    • Laplace had devised a method involving cosine series for computing the perturbing force on a planet caused by other planets.
    • The coefficients in the series were known as 'Laplace coefficients' but the drawback of the method was that it only worked for circular orbits.
    • Newcomb showed how to extend Laplace's series to give a perturbing function in the case of elliptical orbits by introducing differential operators which act on the Laplace coefficients.

  142. Karl Aubert (1924-1990)
    • The first was Summation of some series of binomial coefficients by means of Cauchy's integral formula (1944) followed by Remark on the middle binomial coefficient (Norwegian) (1944), Summation of some series of binomial coefficients on the basis of Cauchy's integral formula (Norwegian) (1945), A group-theoretical remark of E Hoff-Hansen concerning certain expressions in the quantification theory (1947), and On making precise and generalizing the concept of relation (Norwegian) (1948).
    • Aubert was also a driving force behind the meetings in the series "Ski and mathematics" at Gausdal Hotel, where teachers and students from universities and colleges from all over the country gathered during the first week of the new year for outdoor activities and inspiring lectures.
    • In the early 1980s, Aubert criticised Simon's irrelevant use of mathematics beginning a fascinating debate in which they responded to each other in a series of papers.

  143. Yudell Luke (1918-1983)
    • Not only did he use rational approximation, but Luke also developed series expansions as an approximation method.
    • For example he expanded hypergeometric functions in series of Laguerre and Hermite polynomials.
    • He gave a wonderful series of lectures on special functions, asymptotic analysis, and approximation theory.
    • While at MRI he gave an extensive series of lectures on the history of philosophy, focusing especially on Spinoza, whose work he believed, contains the most meaningful elements of those ethical and intellectual ideals which alone can provide a personal bedrock in an uncertain, frenetically changing world.

  144. Terence Tao (1975-)
    • The material starts at the very beginning - the construction of the number systems and set theory, and then goes on to the basics of analysis (limits, series, continuity, differentiation, Riemann integration), through to power series, several-variable calculus and Fourier analysis, and finally the Lebesgue integral.
    • In 2010 the next in Tao's series was published An epsilon of room, I: real analysis.
    • One can anticipate a long and fascinating series of books that will appear over the next years.

  145. Carl Friedrich Hindenburg (1741-1808)
    • His first papers on mathematics were published in 1776 when he studied series.
    • Hindenburg published a series of works on combinatorial mathematics, in particular probability, series and formulae for higher differentials.
    • Gudermann, best known as the teacher of Weierstrass, worked on the expansion of functions into power series and, as shown by Manning in [',' K R Manning, The emergence of the Weierstrassian approach to complex analysis, Arch.

  146. Edward Maitland Wright (1906-2005)
    • The resulting financial disaster caused Kate and Maitland to separate and Kate Wright, being an excellent musician and trained music teacher, took up a series of positions at boarding schools in the south of England.
    • Also among his early work was a series of three papers titled Asymptotic partition formulae.
    • The third in the series Asymptotic partition formulae, III.
    • These topics are: prime numbers; congruences and the quadratic reciprocity law; continued fractions; irrational, algebraic and transcendental numbers; quadratic fields; arithmetical functions, their order of magnitude and the Dirichlet or power series which generate them; partitions and representations of numbers as sums of squares, cubes and higher powers; Diophantine approximation; and the geometry of numbers.

  147. Eduard Kummer (1810-1893)
    • He published a paper on hypergeometric series in Crelle's Journal in 1836 and he sent a copy of the paper to Jacobi.
    • This resulted in a series of sympathetic revolutions against the governments of the German Confederation.
    • He extended Gauss's work on hypergeometric series, giving developments that are useful in the theory of differential equations.
    • He was the first to compute the monodromy groups of these series.

  148. Eduard Stiefel (1909-1978)
    • There followed a series of four papers by Stiefel and his two assistants Heinz Rutishauser and Ambros Speiser, Programmgesteuerte digitale Rechengerate (elektronische Rechenmaschinen) Ⓣ appearing in 1950 and 1951.
    • In this series of papers the authors discuss in very considerable detail a number of the important mathematical questions that naturally arise in the design of a digital computer.
    • The first of the series was in 1964 followed by meetings in 1967, 1969, 1972, 1975 and 1978 [',' V Szebehely, D Saari, J Waldvogal and U Kirchgraber, Eduard L Stiefel (1909-1978), in Proceedings of the Sixth Conference on Mathematical Methods in Celestial Mechanics, Math.
    • We develop the averaging method, based consistently on Lie series, and we deal in detail with the implications of this basic concept.

  149. George Pólya (1887-1985)
    • While the book was being worked on, Polya continued a remarkable series of publications, with a total of 31 papers appearing during the three years 1926-28.
    • For example, in 1918 he published papers on series, number theory, combinatorics and voting systems.
    • Polya's interest in complex analysis led him to investigate singularities of power series, gap theorems, power series with integral coefficients and those taking integral values at the positive integers, the Polya representation for entire functions of exponential type, and the location of zeros.

  150. Benjamin Osgood Peirce (1854-1914)
    • Arthur Gordon Webster, who was a student at Harvard at this time, writes [',' A G Webster, Benjamin Osgood Peirce, Science, New Series 39 (999) (1914), 274-277.','12]:- .
    • But the most notable course instituted by Peirce and shared by this pair of masterly teachers was that one in which Peirce treated the theory of the Newtonian potential function, and Byerly the theory of Fourier's series.
    • There are numerous auxiliary formulae, for example, those arising in trigonometry, the principal relations between the elliptic integrals (Jacobian notation), and series for frequently occurring functions.
    • Arthur Gordon Webster writes [',' A G Webster, Benjamin Osgood Peirce, Science, New Series 39 (999) (1914), 274-277.','12]:- .

  151. Akos Seress (1958-2013)
    • It was the first in a series of joint papers with Laszlo Babai on the complexity of permutation group algorithms, some others being On the diameter of Cayley graphs of the symmetric group (1988) and the three author papers Permutation groups (1987) and Fast management of permutation groups (1988) which had Seress, Babai and E M Luks as authors.
    • He continued his work in computational group theory with a series of important papers on the statistical theory of finite simple groups with Bill Kantor and others; this line of work contributed to a recent definitive result on the complexity of algorithms for matrix groups over finite fields by Seress and coauthors.
    • The writing of this book began in 1993, on the suggestion of Joachim Neubuser, who envisioned a series of books covering the major areas of computational group theory.
    • Seress, along with Bill Kantor, has been a chief organizer of a series of meetings that brought these two communities together.

  152. J A Green (1926-2014)
    • By that time M H A Newman's plan to use specially designed electronic computers to assist in the decipherment of the "Fish" series of coded messages was well advanced.
    • Let us mention Sandy's little book Sequences and series (1958) whose aim is stated in the Preface:- .
    • We also mention the lectures given by Sandy Green at Groups St Andrews 1989 when he was a main speaker giving a series of lectures on Schur algebras and general linear groups.
    • Green: Sequences and Series .

  153. John Hellins (1749-1827)
    • In our days he must be a slender mathematician who does not know that they are useful, not only in trigonometry, navigation, astronomy, the calculation of compound interest and annuities, but also in the finding of fluents, and the summation of infinite series.
    • Hellins published many papers; the following were all in the Philosophical Transactions of the Royal Society: A new method of finding the equal roots of an equation by division (1782); Dr Halley's method of computing the quadrature of the circle improved; being a transformation of his series for that purpose, to others which converge by the powers of 60 (1794); Mr Jones' computation of the hyperbolic logarithm of 10 compared (1796); A method of computing the value of a slowly converging series, of which all the terms are affirmative (1798); An improved solution of a problem in physical astronomy, by which swiftly converging series are obtained, which are useful in computing the perturbations of the motions of the Earth, Mars, and Venus, by their mutual attraction (1798); A second appendix to the improved solution of a problem in physical astronomy (1800); and On the rectification of the conic sections (1802).

  154. Edward Lorenz (1917-2008)
    • The generalized vorticity equation is satisfied by formal infinite series representing the density and wind fields.
    • The first few terms of a particular series solution are obtained explicitly.
    • The series appear to converge near the north pole, and determine a model of a polar air mass.
    • He used this series of three lectures as a basis for his famous text The essence of chaos (1993).

  155. Norrie Everitt (1924-2011)
    • This type of problem has been considered, for general-order equations, by many writers [see in particular G D Birkhoff, 'Boundary-value and expansion problems of ordinary differential equations' (1908) and J Tamarkin, 'Some general problems of the theory of ordinary linear differential equations and expansion of an arbitrary function in a series of fundamental functions' (1927)].
    • The first, held in Dundee from 28 to 31 March 1972, had Proceedings published in the Springer Lecture Notes in Mathematics series with Norrie Everitt and Brian Sleeman as editors.
    • The success of this conference led to it becoming the first in a series, and in 1978 the fifth 'Ordinary and Partial Differential Equations Conference' was held in Dundee with Everitt as the sole editor of the Proceedings.
    • The seventh conference in the series took place in Dundee in March/April 1982, and later that year Everitt left Dundee to take up an appointment at the University of Birmingham as Mason Chair and Head of the Department of Mathematics.

  156. John Shepherdson (1926-2015)
    • His main work turned towards set theory and he published the paper Well-ordered sub-series of general series (1951).
    • We are concerned here with the investigation of the structure of ordered series by means of their well-ordered sub-series.

  157. Nicolaas de Bruijn (1918-2012)
    • He continued to hold this position until June 1944 [',' F D Kamareddine, Editorial preface, in F D Kamareddine (ed.), Thirty-five years of automating mathematics, Applied Logic Series 28 (Kluwer Academic Publishers, Dordrecht, 2003).','1]:- .
    • Also in 1943, in addition to his doctoral thesis, he published On the absolute convergence of Dirichlet series, On the number of solutions of the system ..
    • He began publishing papers on combinatorics relevant to his work during this period such as The problem of optimum antenna current distribution (1946), A combinatorial problem (1946), On the zeros of a polynomial and of its derivative (1946), and A note on van der Pol's equation (1946) [',' F D Kamareddine, Editorial preface, in F D Kamareddine (ed.), Thirty-five years of automating mathematics, Applied Logic Series 28 (Kluwer Academic Publishers, Dordrecht, 2003).','1]:- .
    • The Preface [',' F D Kamareddine, Editorial preface, in F D Kamareddine (ed.), Thirty-five years of automating mathematics, Applied Logic Series 28 (Kluwer Academic Publishers, Dordrecht, 2003).','1] records:- .

  158. Solomon Grigoryevich Mikhlin (1908-1990)
    • The book [',' S G Mikhlin, Multidimensional singular integrals and integral equations, International Series of Monographs in Pure and Applied Mathematics 83 (Pergamon Press, Oxford-London-Edinburgh-New York-Paris-Frankfurt, 1965).','1] is dedicated to her memory.
    • The latter became Mikhlin's master thesis supervisor: the topic of the thesis, defended in 1929, was the convergence of double power series.
    • Mikhlin also proved a now classical theorem on multipliers of Fourier transform in the Lp-space, based on an analogous theorem of Jozef Marcinkiewicz on Fourier series.
    • A complete collection of his results in this field up to 1965, as well as contributions by Francesco Tricomi, Georges Giraud, Alberto Calderon and Antoni Zygmund, is contained in the monograph [',' S G Mikhlin, Multidimensional singular integrals and integral equations, International Series of Monographs in Pure and Applied Mathematics 83 (Pergamon Press, Oxford-London-Edinburgh-New York-Paris-Frankfurt, 1965).','1].

  159. Sheila Power Tinney (1918-2010)
    • Born wrote the first in a series of papers on this topic with the title On the stability of crystal lattices I which was published in the Mathematical Proceedings of the Cambridge Philosophical Society in 1940.
    • The second paper in this series was written by his student Rama Dhar Misra, the third by Born with Reinhold Furth, the fourth by Born with Rama Dhar Misra, while the fifth and sixth were written by Reinhold Furth.
    • The next in the series, published in the same journal in 1942, was by Power and it was entitled On the stability of crystal lattices.
    • The next paper in the series, also in the same journal in 1942, was On the stability of crystal lattices.

  160. Cyril Offord (1906-2000)
    • He began publishing papers in 1932 with On the summability of power series and On the summability of trigonometric series appearing in that year and the paper Fourier and Hankel transforms in the following year.
    • It was during these last three years at Cambridge that he worked with J E Littlewood on the topic for which he is best known today, and they published a series of important joint papers beginning with On the number of real roots of a random algebraic equation in 1938.
    • In it Offord and Littlewood considered the family of entire functions of finite order obtained by introducing independent plus or minus signs, each with probability 1/2, in the Taylor series of a given function.

  161. J W S Cassels (1922-2015)
    • His mathematical publications started in about 1947 with a series of papers on the geometry of numbers, in particular papers on theorems of Khinchin and of Davenport, and on a problem of Mahler.
    • After further papers on Diophantine equations and Diophantine approximation he wrote a series of five papers on Some metrical theorems in Diophantine approximation.
    • After further papers on subgroups of infinite abelian groups and normal numbers he wrote a series of eight papers on Arithmetic on curves of genus 1.
    • Ian Cassels has made many distinguished contributions to the theory of numbers; possibly his most important work is on the arithmetic of elliptic curves, published in a series of papers between 1959 and 1964.

  162. Goro Shimura (1930-2019)
    • He published Euler products and Eisenstein series (1997) which M Ram Murty reviewed - we give the first and last paragraphs of his interesting review:- .
    • This monograph focuses on three objectives: (i) the determination of local Euler factors on classical groups, in an explicit rational form; (ii) Euler products and Eisenstein series on a unitary group of an arbitrary signature; (iii) a class number formula for a totally definite Hermitian form.
    • It is notable that the collection of appendices, as well as the material on algebraic groups and their localizations, Eisenstein series and their analytic continuations, that was scattered in the research literature, sometimes without proof, and often relegated to the background as "well-known", is now gathered together in this volume.
    • In 2007 he published Elementary Dirichlet series and modular forms and in 2010 Arithmetic of quadratic forms.

  163. Édouard Benjamin Baillaud (1848-1934)
    • He also managed to improve the status of astronomy by re-launching the journal Annales de l'Observatoire de Toulouse in 1880, producing an initial volume of a new series, published in Paris.
    • The series continued through to 1968.
    • A congress, held in the Paris Observatory on 16 April 1887, had resolved: (i) to create a chart of all stars down to the fourteenth magnitude, the plates to be in duplicate (ii) to create a second series of photographs with shorter exposure, including stars to the eleventh magnitude, to be made concurrently to form a catalogue and to determine fundamental positions in the first series.

  164. Thomas Hakon Grönwall (1877-1932)
    • Gronwall's work contains classical analysis (Fourier series, Gibbs phenomenon, summability theory, Laplace and Legendre series), differential and integral equations, analytic number theory (transcendental numbers, divisor function, L-function of Dirichlet), complex function theory (Dirichlet L-series, conformal mappings, univalent functions), differential geometry, mathematical physics (problems of elasticity, ballistics, induction, potential theory, kinetic theory of gases, optics), nomography, atomic physics (wave mechanics of hydrogen and helium atom, lattice theory of crystals) and physical chemistry where he is especially known as a very important contributor.
    • The Gronwall summation method of series appeared in 1932 as a generalization of de la Vallee Poussin and Cesaro summation methods (cf.

  165. Alfréd Rényi (1921-1970)
    • started regularly to attend a lecture series I held to June of 1942.
    • from the University of Szeged, with Frigyes Riesz as his thesis advisor, for a thesis on Cauchy-Fourier series.
    • Results from his doctoral thesis appeared in the paper On the summability of Cauchy-Fourier series (1950).
    • The first result, regarding the representation of an even number, is an approximation to the unproved Goldbach conjecture and supersedes an earlier proof of the same proposition by Estermann (1932) which made use of an unproved generalized Riemann hypothesis for all Dirichlet L-series.

  166. Panagiotis Zervos (1878-1952)
    • thesis entitled On the series and the theorem of Descartes (Greek) for which he was awarded his doctorate by the University of Athens in 1901.
    • Since 1925 the Paris Academy of Sciences had been commissioning a series of monographs reviewing current mathematical problems, the series having the title Memorial des sciences mathematiques Ⓣ.
    • Zervos, as a recognised world leader on the Monge Problem was asked to write Le probleme de Monge Ⓣ which appeared as No 53 in the series in 1932, see [',' P Zervos, Le probleme de Monge, Memorial des sciences mathematiques 53 (1932), 62 pages.','2].

  167. Onorato Nicoletti (1872-1929)
    • The other works relate to ordinary differential equations, or to those with partial derivatives also of a higher order than the 2nd, and they all have particular importance for the completion of the results obtained with the method of successive approximations and with that of Riemann as well as for the studies that are taking place on the 2nd order equations of hyperbolic type, in which the Laplace series is finite.
    • He also jointly edited a series of mathematics textbooks for secondary schools along with Roberto Marcolongo which was published by Perrella in Naples.
    • His two books with Maroni and with Sansone appear in this series.
    • Other publications concern differential geometry, Taylor's multiple series, the theory of limits, that of analytic functions; two of these works were also translated abroad.

  168. Aleksandr Osipovich Gelfond (1906-1968)
    • (In 1966 Alan Baker proved Gelfond's Conjecture in general.) Gelfond's papers in 1933 and 1934, which include his remarkable achievement, are: Gram determinants for stationary series (written jointly with Khinchin) (1933); A necessary and sufficient criterion for the transcendence of a number (1933); Functions that take integer values at the points of a geometric progression (1933); On the seventh problem of D Hilbert (1934); and On the seventh problem of Hilbert (1934).
    • The chapter titles of this book are: Residues; Singular points and series representations of a function; Expansion of a function in a series and properties of the gamma function; Some functional identities and asymptotic estimates; and Laplace transformation and some problems which are solved by the use of residue theory.

  169. Carl Runge (1856-1927)
    • Runge then worked on a procedure for the numerical solution of algebraic equations in which the roots were expressed as infinite series of rational functions of the coefficients.
    • He succeeded in arranging the spectral lines of helium in two spectral series and, until 1897, this was thought to be evidence that hydrogen was a mixture of two elements.
    • In these spectra we found for the first time series systems of two different multiplicities.

  170. William Jones (1675-1749)
    • It included the differential calculus, infinite series, and is also famed since the symbol π is used in it with its modern meaning.
    • These included transcripts of Newton's manuscripts, letters and results obtained with the method of infinite series which Newton had discovered in about 1664.
    • With assistance from Newton himself, Jones produced Analysis per quantitatum series, fluxiones, ac differentia Ⓣ in 1711 although it should be noted that this first edition of 1711 did not record either Newton's name nor that of Jones.

  171. Howard Aiken (1900-1973)
    • Aiken's 1937 proposal for a calculating machine began with a series of paragraphs devoted to an account of the pioneers in machine calculation: Pascal, Moreland, Leibniz, and, above all, Babbage.
    • This same historical homage characterizes the series of articles [with Grace Hopper] in 'Electrical Engineering' in 1946.
    • He continued to work at Harvard on this series of machines, working next on the Mark III and finally the Mark IV up to 1952.

  172. Henry White (1861-1943)
    • Klein attended the Columbian Exposition and then went to Evanston to spend two weeks there at White's invitation to give a series of lectures on contemporary mathematical research.
    • Now, why would it not be possible to combine with this miscellaneous program (which ought by all means to be kept up) something more akin to university models? Would not a series of three or six lectures on nearly related topics, if well chosen, prove attractive and useful to larger numbers? .
    • White's proposal was indeed taken up by the American Mathematical Society and the result was the American Mathematical Society Colloquium lectures which are published as the American Mathematical Society Colloquium Publication series.

  173. Edmond Laguerre (1834-1886)
    • He found a divergent series, the first few terms of which gave a good approximation to the integral.
    • He went on to investigate properties of the polynomials, proving orthogonality relations and also showing that an arbitrary function could be expanded in a 'Fourier type' series in Laguerre polynomials.
    • That it was developed from a divergent series is especially remarkable.

  174. Victor Puiseux (1820-1883)
    • During this period Puiseux published a series of more than ten papers in Liouville's Journal.
    • He examined series expansions and looked at series with fractional powers.

  175. Guido Fubini (1879-1943)
    • A series of decrees removed Jews from positions of influence in government, banking and education.
    • His technical mastery often permitted him to discover simpler demonstrations of such theorems as those of Berstein and Pringsheim on the development of Taylor series.
    • This led him to solve a whole series of engineering problems which he was writing up as a textbook towards the end of his life.

  176. Yang Hui (about 1238-about 1298)
    • Yang also gave formulae for the sum of certain series, for example he found the sum of the squares of the natural numbers from m2 to (m+n)2 and showed that .
    • 8 (1) (1981), 61-66.','14] for a discussion of the geometrical ideas which lie behind Yang's approach to summing series.
    • The topics covered by Yang include multiplication, division, root-extraction, quadratic and simultaneous equations, series, computations of areas of a rectangle, a trapezium, a circle, and other figures.

  177. Al-Umawi (about 1400-1489)
    • After describing the very briefly the basic arithmetical operations of addition and multiplication, al-Umawi moves on to discuss the summation of series.
    • Among the series al-Umawi considers are arithmetic and geometric series.

  178. Alice T Schafer (1915-2009)
    • The curve is assumed to be analytic and thus can be represented in the neighbourhood of the point in question by power series.
    • Choice of the proper projective coordinate system permits the reduction of these power series to simple canonical forms.
    • With the aid of these canonical power series, [Schafer] derives numerous theorems concerning surfaces which osculate the given curve at the singular point, concerning the sections of the tangent developable in the neighbourhood of the singular point and concerning the projections of the given curve.

  179. David Enskog (1884-1947)
    • Is it now certain that all the terms that appear in your calculations have a real significance? Is it in other words certain that the function which is sought can be developed in a series of the type assumed? But however fundamentally important I may consider this question, and however much I may advise that everything be done to demonstrate the correctness of the assumptions made, I would nevertheless conclude by saying that I do not regard it as impossible to defend the study in question, even if the difficulty cannot be overcome.
    • There was still a problem in that Enskog had failed to prove that his series converged.
    • Enskog used Hilbert's methods to work out a series expansion of the velocity distribution function f and wrote this up for his doctoral dissertation at Uppsala in 1917.

  180. Christian Juel (1855-1935)
    • However, at first it was a journal aimed at school teachers of mathematics but in 1890, when Juel was editor, it was split into Series A containing school level mathematics, and Series B which was a scientific journal containing articles on advanced topics.
    • Series B allowed articles in foreign languages and contributions from foreign mathematicians began to appear in the journal beginning at this time.

  181. Mihailo Petrovi (1868-1943)
    • Since 1925 the Paris Academy of Sciences had been commissioning a series of monographs reviewing current mathematical problems, the series having the title Memorial des sciences mathematiques Ⓣ.
    • Petrović was asked to write Integration qualitative des equations differentielles Ⓣ which appeared as No 48 in the series in 1931.

  182. Emilio Baiada (1914-1984)
    • The author uses the results of Toeplitz and Caratheodory concerning the coefficients of the Fourier-Stieltjes series of a non-decreasing function and some theorems of Ghizzetti concerning the coefficients of a function between given bounds in order to characterize functions which are i) non-negative, ii) with values between 0 and 1, iii) positive and bounded away from zero.
    • We have mentioned some of Baiada's publications above but we note that his output totals 60 scientific publications on a wide range of different fields in analysis: ordinary and partial differential equations, Fourier series and the series expansion of orthonormal functions, topology, real analysis, functional analysis, calculus of variations, measure and integration, optimisation, and the theory of functions.

  183. Jacques-Louis Lions (1928-2001)
    • In Paris he began a weekly numerical analysis seminar series and, later, he set up a numerical analysis laboratory.
    • He was even able to teach this material in the classroom, which was quite a pedagogical challenge! In a series of works begun with Evariste Sanchez-Palencia in 1995, he also developed the theory of 'sensitive problems', particularly as they arise in the theory of elastic shells.
    • In a long series of notes published in the Comptes Rendus until 2001, Lions returned to numerical analysis, and in particular to parallel computation and domain decomposition methods.

  184. Michael Fekete (1886-1957)
    • There he worked with Edmund Landau and during his year in Gottingen wrote a number of further papers including: On necessary and sufficient conditions for the summability of power series (Hungarian) (1910), Sur les series de Dirichlet Ⓣ (1910), and Sur un theoreme de M Landau Ⓣ (1910).
    • He gave his lecture Transfinite diameter and Fourier series to the congress on Monday, 6 September, to Section IId with Jean Leray in the chair.
    • In 1916 the author showed that analytic continuation of a power series can be represented as a matrix-transformation of the partial sums by [a specific] upper triangular matrix ..

  185. Nicolaus Mercator (1620-1687)
    • Mercator discovered the well known series, sometimes called Mercator's series, .
    • This series was also investigated by Mengoli.

  186. Józef Marcinkiewicz (1910-1940)
    • Zygmund was undertaking research on trigonometric series and in 1931-32 he gave a course on this topic at Wilno for the first time.
    • The only visible trace of Schauder's influence is a very interesting paper of Marcinkiewicz on the multipliers of Fourier series, a paper which originated in connection with a problem proposed by Schauder ..
    • He suggested problems on general orthogonal systems to Marcinkiewicz and this resulted in a series of papers from him on this topic.

  187. Vladimir A Steklov (1864-1926)
    • He wrote General Theory of Fundamental Functions in which he examined expansions of functions as series in an infinite system of orthogonal eigenfunctions.
    • Steklov was not the first to examine series expansions in terms of infinite sets of orthogonal eigenfunctions, of course Fourier had examined a special case of this situation many years before.
    • He studied a generalisation of Parseval's equality for Fourier series to his general setting showing this to be a fundamental property.

  188. Raymond Paley (1907-1933)
    • Paley had already proved impressive results on Fourier series and had collaborated with Littlewood, his supervisor.
    • Zygmund's book Trigonometric Series published in 1935 owes a debt to the joint work that he carried out with Paley.
    • Soon after his arrival in America, however, certain studies of lacunary series which Paley had already begun suggested a new attack on the theory of interpolation and allied trigonometrical problems.

  189. Eduard Heine (1821-1881)
    • Before arriving at Halle, Heine published on partial differential equations and during his first few years teaching at Halle he wrote papers on the theory of heat, summation of series, continued fractions and elliptic functions.
    • At Halle, Heine taught a variety of courses such as: potential theory and its applications, number theory, Fourier series, trigonometric series, mechanics, and the theory of heat.

  190. Jean-Pierre Serre (1926-)
    • This is how I learned about derivatives, integrals, series and such (I did that in a purely formal manner - Euler's style so to speak: I did not like, and did not understand, epsilons and deltas.) .
    • Since a series of lectures for a year's course is about 20 hours, that's quite a lot.
    • of all kinds, from Giono to Boll to Kawabata, including fairy tales and the Harry Potter series.

  191. John Brinkley (1766-1835)
    • In Which Is Contained the General Doctrine of Reversion of Series, of Approximating to the Roots of Equations, and of the Solution of Fluxional Equations by Series (1800); General Demonstrations of the Theorems for the Sines and Cosines of Multiple Circular Arcs, and Also of the Theorems for Expressing the Powers of Sines and Cosines by the Sines and Cosines of Multiple Arcs; to Which Is Added a Theorem by Help Whereof the Same Method May Be Applied to Demonstrate the Properties of Multiple Hyperbolic Areas (1800); On Determining Innumerable Portions of a Sphere, the Solidities and Spherical Superficies of Which Portions Are at the Same Time Algebraically Assignable (1802); A Theorem for Finding the Surface of an Oblique Cylinder, with Its Geometrical Demonstration.
    • Also, an Appendix, Containing Some Observations on the Methods of Finding the Circumference of a very Excentric Ellipse; including a Geometrical Demonstration of the Remarkable Property of Elliptic Arcs Discovered by Count Fagnani (1803); An Investigation of the General Term of an Important Series in the Inverse Method of Finite Differences (1807); and Observations Relative to the Form of the Arbitrary Constant Quantities, That Occur in the Integration of Certain Differential Equations; and, Also, in the Integration of a Certain Equation of Finite Differences (1818).

  192. Winifred Sargent (1905-1979)
    • She did produce good results, despite any feelings that she may have had about them, for she published On Young's criteria for the convergence of Fourier series and their conjugates in the Proceedings of the Cambridge Philosophical Society based on this work which appeared in print in 1929.
    • Another paper published in the same year On the summability (C) of allied series and the existence of (CP) extends the conditions for the Cesaro summability of Fourier-Lebesgue series and of their conjugates given by Bosanquet in 1937 to the case of functions integrable in the Cesaro-Perron sense.

  193. John Whittaker (1905-1984)
    • He was interested over many years in expanding functions in a series of polynomials and Whittaker's constant is named after him.
    • Interpolatory function theory (1939, reprinted 1964), Series of Polynomials (1944), and Sur les Series de Base de Polynomes Quelconques (1949).
    • The second book is only 43 pages long and is the result of a series of lectures given by Whittaker at the Fouad I University, Egypt in 1943.

  194. Paul-André Meyer (1934-2003)
    • In 1967 the first of the series Seminaire de Probabilites Ⓣ appeared.
    • This series, published by Springer in their Lecture Notes in Mathematics series, contains remarkable work by Meyer.

  195. Thomas MacRobert (1884-1962)
    • The E-function was a generalisation of the generalised hypergeometric functions, and from 1938 onwards MacRobert produced a whole series of works on the properties of the E-function and integrals with E-functions.
    • Formulae for generalized hypergeometric functions as particular cases of more general formulae (1939) showed how certain known formulae for generalized hypergeometric functions can be derived as particular cases of formulae of more general type involving multiple series; Some formulae for the E-function (1941) showed how special cases of the formulae derived lead to interesting relations between Bessel functions, Legendre functions and confluent hypergeometric functions; and Proofs of some formulae for the hypergeometric function and the E-function (1943) gave alternative proofs for some known theorems on hypergeometric functions, then gives a formula for an integral involving the product of two E-functions.
    • He continued to produce papers on the E-function such as On an identity involving E-functions (1948), Integral of an E-function expressed as a sum of two E-functions (1953), An integral involving an E-function and an associated Legendre functions of the first kind (1953), Integrals involving E-functions (1958), Infinite series of E-functions (1959).

  196. Maxime Bôcher (1867-1918)
    • Bocher published around 100 papers on differential equations, series, and algebra.
    • In a seventy page article in 1906, Introduction to the theory of Fourier's series published in the Annals of Mathematics, he gave the first satisfactory treatment of the Gibbs phenomenon (he wrote another paper On Gibbs' phenomenon in 1914).
    • Bocher was honoured by the American Mathematical Society when he was chosen to give the first series of Colloquium lectures in 1896.

  197. Stefan Banach (1892-1945)
    • Another important publishing venture, begun in 1931, was a new series of Mathematical Monographs.
    • The first volume in the series Theorie des Operations lineaires Ⓣ was written by Banach and appeared in 1932.
    • In addition, he contributed to measure theory, integration, the theory of sets, and orthogonal series.

  198. J C Burkill (1900-1993)
    • He resumed his research in mathematics winning the Adams prize in 1948 for an essay on integrals and trigonometric series.
    • The book covers: sets and functions, metric spaces, continuous functions on metric spaces, real and complex limits and series, uniform convergence, Riemann-Stieltjes integration, multivariable differential and integral calculus, Fourier series, Cauchy's theorem, Laurent expansions, residue calculus, infinite products, the factor theorem of Weierstrass, asymptotic expansions, and applications to special functions in particular the gamma function.

  199. Robert Geary (1896-1983)
    • Maurice Kendall gave the Sixth Geary Lecture in 1973 and spoke about the man the series honoured:- .
    • My distinguished predecessor in contributing to this series of lectures, Jan Tinbergen, included in the title of his lecture the word 'interdisciplinary'.
    • Series A (General) 148 (2) (1985), 162.','9]:- .

  200. Guido Grandi (1671-1742)
    • In this he studied the logarithmic curve proposed by Christiaan Huygens, using generalised algebraic methods, series expansions and infinitesimal methods.
    • He used the series expansion .
    • However, he did not neglect mathematics, publishing an Italian version of Euclid's Elements in 1731 and a series of works on mechanics Instituzioni meccaniche Ⓣ (1739), arithmetic Instituzioni di aritmetica pratica Ⓣ (1740) and geometry Instituzioni geometriche Ⓣ (1741).

  201. Hans Rademacher (1892-1969)
    • He also wrote a number of textbooks such as Lectures on analytic number theory (1955), Lectures on elementary number theory (1964), Dedekind sums (1972), Topics in analytic number theory (1973), and Higher mathematics from an elementary point of view which was only published in 1983 but was based on a series of lectures he delivered at Stanford University in 1947.
    • In this remarkable series of lectures the author has taken a number of interesting mathematical threads and woven them into a colorful tapestry.
    • They appeared as Dedekind sums in the Carus Mathematical Monographs series in 1972.

  202. Gábor Szeg (1895-1985)
    • 1: Series, Integral Calculus, Theory of Functions (1972), by George Polya and Gabor Szego, American Scientist 61 (3) (1973), 376.','27]:- .
    • Dunham Jackson writes [',' D Jackson, Review: Orthogonal Polynomials, by Gabor Szego, Science, New Series 91 (2370) (1940), 526.','11]:- .
    • Mark Kac praises the book highly, writing [',' M Kac, Review: Toeplitz forms and their applications, by Ulf Grenander and Gabor Szego, Science, New Series 128 (3316) (1958), 137-138.','13]:- .

  203. Galileo Galilei (1564-1642)
    • Galileo spent three years holding this post at the university of Pisa and during this time he wrote De Motu a series of essays on the theory of motion which he never published.
    • From these reports, and using his own technical skills as a mathematician and as a craftsman, Galileo began to make a series of telescopes whose optical performance was much better than that of the Dutch instrument.
    • He made a long series of observations and was able to give accurate periods by 1612.

  204. Steve Rallis (1942-2012)
    • They were almost all long term collaborations resulting in series of papers.
    • The L-function is expressed as an integral involving Eisenstein series on a larger group.
    • The method, known as the descent map, uses certain Fourier coefficients (of Gelfand-Graev or Fourier-Jacobi type, depending on the context) of residual Eisenstein series on a bigger classical group induced from the automorphic representations on GLn.

  205. Ilya Iosifovich Piatetski-Shapiro (1929-2009)
    • In the following year the Moscow Mathematical Society awarded him their Young Mathematician Prize for his work On the problem of uniqueness of expansion of a function in a trigonometric series which he had undertaken while an undergraduate.
    • Among his main achievements are: the solution of Salem's problem about the uniqueness of the expansion of a function into a trigonometric series; the example of a non symmetric homogeneous domain in dimension 4 answering Cartan's question, and the complete classification (with E Vinberg and G Gindikin) of all bounded homogeneous domains; the solution of Torelli's problem for K3 surfaces (with I Shafarevich); a solution of a special case of Selberg's conjecture on unipotent elements, which paved the way for important advances in the theory of discrete groups, and many important results in the theory of automorphic functions, e.g., the extension of the theory to the general context of semi-simple Lie groups (with I Gelfand), the general theory of arithmetic groups operating on bounded symmetric domains, the first 'converse theorem' for GL(3), the construction of L-functions for automorphic representations for all the classical groups (with S Rallis) and the proof of the existence of non arithmetic lattices in hyperbolic spaces of arbitrary large dimension (with M Gromov).
    • He became a collaborator and friend and together they wrote the monograph The arithmetic and spectral analysis of Poincare series published in 1990.

  206. Jean Delsarte (1903-1968)
    • He published a series of papers on this topic in 1934-35: Les fonctions moyenne-periodiques Ⓣ (1934); Application de la theorie des fonctions moyenne-periodiques a la resolution de certaines equations integrales Ⓣ (1934); Application de la theorie des fonctions moyenne-periodiques a la resolution des equations de Fredholm-Norlund Ⓣ (1935); and Les fonctions moyenne-periodiques Ⓣ (1935).
    • Delsarte worked in analysis extending work on series expansions due to Whittaker and Watson.
    • These works had convinced him that a good understanding of the formal properties of [series expansions of functions] was necessary to a fruitful study of their domains of definition and their mode of convergence.

  207. Ernesto Cesŕro (1859-1906)
    • Sur diverses questions d'arithmetique Ⓣ was the first of a series which Cesaro wrote on the theory of numbers.
    • the number of common divisors of two numerals, determination of the values of the sum totals of their squares, the probability of incommensurability of three arbitrary numbers, and so on; to these he attempted to apply obtained results in the theory of Fourier series.
    • He also contributed to the study of divergent series, a topic which interested him early in his career, and we should note that in his work on mathematical physics he was a staunch follower of Maxwell.

  208. Robert Woodhouse (1773-1827)
    • He wrote an three papers in the Philosophical Transactions of the Royal Society in 1801, 1802 and an important book Principles of Analytic Calculation in 1803 which attempted to put the calculus on a rigorous algebraic foundation using a formal series expansions method similar to that developed by Lagrange [',' Biography by Harvey W Becher, in Dictionary of National Biography (Oxford, 2004).','2]:- .
    • In essence Woodhouse was dealing with Taylor series of a function, from which he could directly read off the first, second, third etc.
    • derivatives from the coefficients of the terms of the series without involving any limiting process.

  209. Brian Hartley (1939-1994)
    • thesis The stability group of a series of subgroups was submitted in 1964 but before that, in May 1963, he had submitted a paper jointly written with Philip Hall, with the same title as his thesis to the London Mathematical Society.
    • In fact Hartley's first published paper was The order-types of central series which appeared in 1965.
    • The Hartleys owned a series of tandems, but Mary [his wife] tended to fold her arms and sing as they went up hills, leaving Brian to do all the work.

  210. George Udny Yule (1871-1951)
    • In 1895 Yule was elected to the Royal Statistical Society and over the next few years, inspired by Pearson, he produced a series of important articles on the statistics of regression and correlation.
    • He wrote papers on time-correlation in which he introduced the correlogram and he did fundamental work on the theory of autoregressive series.
    • The last chapters discuss interpolation and graduation, index numbers, and time series.

  211. Charles Méray (1835-1911)
    • This work is a book Nouveau precis d'analyse infinitesimale Ⓣ which aims to present the theory of functions of a complex variable using power series.
    • It is another rigorous work and in fact between 1872 and 1894 Meray produced a series of papers which remove geometric considerations from analytic proofs.
    • Meray's work consistently follows Lagrange in basing the whole of analysis on the concept of functions written as Taylor series.

  212. Ralph Fox (1913-1973)
    • The quotient groups of the lower central series (1958); and Free differential calculus.
    • In the summer of 1951, Fox went to the Mathematics Institute of the Universidad Nacional Autonoma de Mexico in Mexico City where he gave a series of lectures, supported by a Guggenheim Fellowship.
    • In the spring of 1956 Fox was invited to give a series of lectures at Haverford College (an institution founded on Quaker values on a campus just outside Philadelphia) under the Philips Lecture Program.The Philips Grant consists of funds left by Haverford alumnus William Pyle Philips (who graduated in 1902) for (i) the purchase of rare books which the college would not otherwise buy and (ii) to invite distinguished scientists and statesmen to Haverford.

  213. John Lewis (1932-2004)
    • They also explored the structure of the series representation of atomic interactions in inverse powers of the interatomic distance and demonstrated that the series were unique and asymptotically divergent.
    • He became expert in the representation theory of groups and during the academic year 1966-67, attended a series of lectures on group representations and their applications at Oxford given by George Whitelaw Mackey who was George Eastman visiting professor.

  214. Leopold Gegenbauer (1849-1903)
    • Theses polynomials are obtained from the hypergeometric series in certain cases where the series is in fact finite.
    • However, the name of Gegenbauer occurs in many other places, such as Gegenbauer functions, Gegenbauer transforms, Gegenbauer series, Fourier-Gegenbauer sums, Gauss-Gegenbauer quadrature, Gegenbauer's integral inequalities, Gegenbauer's partial differential operators, the Gegenbauer equation, Gegenbauer approximation, Gegenbauer weight functions, the Gegenbauer oscillator, and the Gegenbauer addition theorem published in 1875.

  215. Harry Pitt (1914-2005)
    • He was tutored by J C Burkill and attended courses by world-leading mathematicians such as: functions of a complex variable from A E Ingham, almost periodic functions from A S Besicovitch, the theory of functions from J E Littlewood, and divergent series from G H Hardy.
    • Few research students can have had a more productive beginning to their careers for, after publishing A note on bilinear forms in 1936, and Theorems on Fourier series and powers series in 1937, he then published no fewer than eight papers in 1938.

  216. Otto Stolz (1842-1905)
    • He later dedicated an increasing part of his research to real analysis, in particular to convergence problems in the theory of series, including double series; to the discussion of the limits of indeterminate ratios; and to integration.
    • He published a series of books under the title Lecons nouvelles sur l'analyse infinitesimale et ses applications geometriques Ⓣ.

  217. Paul Davies (1946-)
    • Among Davies's better-known media productions were a series of 45 minute BBC Radio 3 science documentaries.
    • In early 2000 he devised and presented a three-part series for BBC Radio 4 on the origin of life, entitled 'The Genesis Factor'.
    • His television projects include two six-part Australian series 'The Big Questions' and 'More Big Questions' and a 2003 BBC documentary about his work in astrobiology entitled 'The Cradle of Life'.

  218. Stanisaw Saks (1897-1942)
    • One of the works for which Saks is most famous is their joint book Analytic functions which appeared in 1938 as volume eight in the Mathematical Monographs series.
    • This was not Saks' first monograph, however, for he had already published an important volume in the Mathematical Monographs series.
    • This earlier volume, the volume two in the series published in 1933, was his famous work Theory of the integral.

  219. John Tukey (1915-2000)
    • Tukey's first major contribution to statistics was his introduction of modern techniques for the estimation of spectra of time series.
    • And when I have pondered about why such techniques as the spectrum analysis of time series have proved so useful, it has become clear that their 'dealing with fluctuations' aspects are, in many circumstances, of lesser importance than the aspects that would already have been required to deal effectively with the simpler case of very extensive data where fluctuations would no longer be a problem.
    • Time series : 1949-1964 (Belmont, CA, 1984).','2]:- .

  220. Ambrose Rogers (1920-2005)
    • His thesis advisors at Birkbeck were Lancelot Stephen Bosanquet and Richard G Cooke and in his first paper, Linear transformations which apply to all convergent sequences and series, published in 1946 in the Journal of the London Mathematical Society, he thanked them writing:- .
    • in 1949 for his thesis The Transformation of Sequences by Matrices in which he studied divergent series.
    • Absolute convergence is equivalent to unconditional convergence of series of points of a Banach space if and only if the space is finite-dimensional.

  221. Johann Balmer (1825-1898)
    • However, despite being a mathematics teacher and lecturer all his life, Balmer is best remembered for his work on spectral series and his formula, given in 1885, for the wavelengths of the spectral lines of the hydrogen atom.
    • In his paper of 1885 Balmer suggested that giving n other small integer values would give the wavelengths of other series produced by the hydrogen atom.
    • Indeed this prediction turned out to be correct and these series of lines were later observed.

  222. Louis de Branges (1932-)
    • Louis de Branges of Purdue University has received the first Ostrowski Prize for developing powerful Hilbert space methods which led him to a surprising proof of the Bieberbach Conjecture on power series for conformal mappings.
    • This result is the culmination of a series of publications on canonical unitary linear systems whose state space is a Krein space.
    • They supplement a previous series on canonical unitary linear systems whose state space is a Hilbert space.

  223. Rimhak Ree (1922-2005)
    • In the Bulletin Ree found the paper Note on power series by Max Zorn in which Zorn solved a problem originally posed by Salomon Bochner about the convergence of certain power series with complex coefficients.
    • In the paper Zorn posed the question of whether the same result held for power series with real coefficients.

  224. Friedrich Engel (1861-1941)
    • We immediately got down to preliminary editorial work of a series of chapters which, according to Lie's plan, would be included in the work.
    • He has also, during this time, developed a series of important ideas of his own, but has in a most unselfish manner declined to describe them here in any great detail or continuity, satisfying himself with submitting short pieces to 'Mathematische Annalen' and, particularly, 'Leipziger Berichte'.
    • Under such circumstances the third volume of the series, but the first one to be printed, has now been put into our hands.

  225. Aleksandr Yakovlevich Khinchin (1894-1959)
    • This first paper began a series of publications by Khinchin on properties of functions which are retained after deleting a set of density zero at a given point.
    • It was first published in 1943 and the eight lectures it contains are: Continuum; Limits; Functions; Series; Derivative; Integral; Series expansions of functions; and Differential equations.

  226. Philip Hall (1904-1982)
    • In June 1939 Hall gave a series of lectures at a small meeting at the Mathematical Institute in Gottingen.
    • In August 1957 Hall gave a series of lectures at the Canadian Mathematical Congress Summer Seminar in Edmonton, Canada, on nilpotent groups which have had great influence ever since.
    • Besides containing a discussion of the possible order types of abelian series in simple groups, the paper also presents an extremely informative survey of the inter-relations that are known or conjectured to exist between the various classes of generalized soluble groups.

  227. Jesse Douglas (1897-1965)
    • In a series of papers from 1927 onwards Douglas worked towards the complete solution: Extremals and transversality of the general calculus of variations problem of the first order in space (1927), The general geometry of paths (1927-28), and A method of numerical solution of the problem of Plateau (1927-28).
    • In the 47 page text, Douglas also mentions Fourier series and transforms, Denjoy integrals and the double integrals of Riemann and of Lebesgue.
    • He also presented a series of papers On the basis theorem for finite abelian groups.

  228. Antoine Parseval (1755-1836)
    • The first was Memoire sur la resolution des equations aux differences partielle lineaires du second ordre Ⓣ dated 5 May 1798, the second was Memoire sur les series et sur l'integration complete d'une equation aux differences partielle lineaires du second ordre, a coefficiens constans Ⓣ dated 5 April 1799, the third was Ingegration generale et complete des equations de la propogation du son, l'air etant considere avec les trois dimensions Ⓣ dated 5 July 1801, the fourth was Ingegration generale et complete de deux equations importantes dans la mecanique des fluides Ⓣ dated 16 August 1803, and finally Methode generale pour sommer, par le moyen des integrales definies, la suite donnee par le theoreme de M Lagrange, au moyen de laquelle il trouve une valeur qui satisfait a une equation algebrique ou transcendente Ⓣ dated 7 May 1804.
    • Today this theorem is seen in the context of Fourier series, and often also in more abstract settings which are quite far removed from Parseval's original ideas.
    • The original theorem was concerned with summing infinite series.
    • The improved version, as given in 1801, states that if two series .

  229. Robert P Langlands (1936-)
    • Then, over the next couple of years, he produced deep results on Eisenstein series and went on to apply Eisenstein series to prove a number theory conjecture due to Weil.
    • He received the Cole Prize in Number Theory from the American Mathematical Society in 1982 for his pioneering work on automorphic forms Eisenstein series, and product formulae.

  230. Emma Castelnuovo (1913-2014)
    • Just after the Second World War, Emma with a university professor and a young colleague organized a successful series of talks held by mathematicians, physicists, philosophers, and educators.
    • Science and technology education, Document Series N.
    • Science and technology education, Document Series N.

  231. Atle Selberg (1917-2007)
    • In a hand-written copy that his father had of Carl Stormer's lecture notes he came across the series .
    • automorphic functions, Dirichlet series.
    • Secondly, his papers up to 1947, which appeared mostly in Norwegian series or journals of limited distribution and partly even during World War II, are now at last easily accessible.

  232. Aryeh Dvoretzky (1916-2008)
    • He was awarded his doctorate in 1941 for his thesis Studies on general Dirichlet series but before submitting this work he had already published a number papers in French, the first being Sur les singularites des fonctions analytiques (1938).
    • In this paper they proved that, for a series of points in a Banach space, absolute convergence is equivalent to unconditional convergence if and only if the Banach space is finite-dimensional.
    • In 2009, the Einstein Institute of Mathematics at the Hebrew University established an annual lecture series in memory of Dvoretzky.

  233. Harald Bohr (1887-1951)
    • Harald Bohr worked on Dirichlet series, and applied analysis to the theory of numbers.
    • Bohr's interest in which functions could be represented by a Dirichlet series led him to devise the theory of almost periodic functions.
    • The fundamental theorem for almost periodic functions is a generalisation of the Parseval identity for Fourier series.

  234. Lee Lorch (1915-2014)
    • in 1941 for his thesis Some Problems on the Borel Summability of Fourier Series.
    • The existence of a continuous function whose Fourier series diverges at a point follows from the unboundedness of the sequence of Lebesgue constants for many summability methods.
    • His research, however, went well and he published The Lebesgue constants for (E, 1) summation of Fourier series (1952), Asymptotic expressions for some integrals which include certain Lebesgue and Fejer constants (1953), Derivatives of infinite order (1953), The principal term in the asymptotic expansion of the Lebesgue constants (1954), and The limit of a certain integral containing a parameter (1955).

  235. Ettore Bortolotti (1866-1947)
    • Bortolotti studied topology at first but later went in the direction of analysis considering the calculus of finite differences, continued fractions, convergence of infinite algorithms, summation of series, the asymptotic behaviour of series and improper integrals.
    • In the 1940 paper on Babylonian mathematics, Bortolotti gives a summary of problems published by Neugebauer but argues that the fact that large series of examples for quadratic equations are made up from the same roots demonstrates that this pair of roots has an 'arcane mystic property'.

  236. George Box (1919-2013)
    • The main areas to which Box has contributed are: statistical inference, robustness, and modelling strategy; experimental design and response surface methodology; time series analysis and forecasting; distribution theory, transformation of variables, and nonlinear estimation; and applications of statistics.
    • Times series analysis.
    • It is basically a series of thirteen papers published by the authors and their co-authors, between 1962 and 1968, cobbled together with a minimum of re-writing.

  237. Hermann Schubert (1848-1911)
    • Schubert was editor of Sammlung Schubert, a series of textbooks.
    • He wrote the first in the series Arithmetik und Algebra Ⓣ and a later book in the series on analysis Niedere Analysis Ⓣ.

  238. Mark Aronovich Naimark (1909-1978)
    • He continued to publish joint papers with Gelfand, in particular in 1946-47 they published seven papers: On unitary representations of a complex unimodular group; Unitary representations of the Lorentz group; Unitary representations of the group of complex matrices of the second order (the Lorentz group); Unitary representations of the Lorentz group; Unitary representations of the group of linear transformations of a line; The fundamental series of the irreducible representations of a complex unimodular group; and Auxiliary and degenerate series of the representations of a unimodular group.
    • It supersedes the material published in a long series of earlier notes.

  239. Otto Haupt (1887-1988)
    • This was a severe test with a week long series of written and oral examinations.
    • Following Sommerfeld's suggestions, he was able to solve a problem on series expansions by eigenfunctions of a boundary value problem and published the result in the paper Uber die Entwieklung einer willkurlichen Funktion nach den Eigenfunktionen des Turbulenzproblems Ⓣ published by the Bavarian Academy of Sciences.
    • He served as an aide in the Flying Division but suffered a series of illnesses.

  240. Soraya Sherif (1934-)
    • SorayanSherif, A Tauberian relation between the Borel and the Lototsky transforms of series,nPacific J.
    • SorayanSherif, Absolute Tauberian constants for quasi-Hausdorff series-to-series transformations,nIndian J.

  241. Peter Hilton (1923-2010)
    • Hilton had also published Differential calculus, a 56-page text in the Library of Mathematics series.
    • I think then I would have to jump and say that a series of papers I did with Joseph Roitberg and later also with Guido Mislin on questions relating to failures of cancellation in homotopy theory are good papers.
    • We were neither of us pure category- theorists and I think that these series of papers we wrote on group-like structures in general categories, and on general homotopy theory and duality, were two very significant contributions to the applications of these categorical notions that suggested ideas and problems.

  242. Adolf Hurwitz (1859-1919)
    • The lectures contained Weierstrass's version of the arithmetisation of analysis including his "construction" of the real numbers, the ε, δ approach to analysis and his theory of complex functions based on power series.
    • He also wrote several papers on Fourier series.
    • Hurwitz informed E Landau about Kakeya's result (corrected); Landau needed the result in a proof of a theorem on infinite power series.

  243. Charles Fefferman (1949-)
    • He had published five papers in 1971, namely: On the divergence of multiple Fourier series; Characterizations of bounded mean oscillation; Some maximal inequalities; The multiplier problem for the ball; and On the convergence of multiple Fourier series.
    • This prize was established by the heirs of Raphael Salem, the French banker and famous mathematician, to be awarded to a young mathematician for outstanding work connected with the theory of Fourier series and related problems.

  244. Gertrude Blanch (1897-1996)
    • She published several papers, most jointly with Arnold N Lowan, such as: Tables of Planck's radiation and photon functions (1940), Errors in Hayashi's table of Bessel functions for complex arguments (1941) and On the inversion of the q-series associated with Jacobian elliptic functions (1942).
    • The author remarks, "there does not seem to appear in the literature any method for improving the accuracy of the characteristic values, except by cumbersome iteration." She then develops a method which corrects not only an approximate characteristic value, but also the coefficients in the series for the periodic solutions.
    • In 1967 Blanch retired from her job at Wright Patterson Air Force Base and was honoured with the publication of Blanch anniversary volume (1967), which contained a series of papers by her friends.

  245. Joseph Fourier (1768-1830)
    • The first objection, made by Lagrange and Laplace in 1808, was to Fourier's expansions of functions as trigonometrical series, what we now call Fourier series.
    • Fourier's work provided the impetus for later work on trigonometric series and the theory of functions of a real variable.

  246. G H Hardy (1877-1947)
    • wrote many papers on the convergence of series and integrals and allied topics.
    • Hardy's interests covered many topics of pure mathematics - Diophantine analysis, summation of divergent series, Fourier series, the Riemann zeta function, and the distribution of primes.

  247. Ambros Speiser (1922-2003)
    • There followed a series of four papers by Stiefel, Rutishauser and Speiser, Programmgesteuerte digitale Rechengerate (elektronische Rechenmaschinen) Ⓣ appearing in 1950 and 1951.
    • In this series of papers the authors discuss in very considerable detail a number of the important mathematical questions that naturally arise in the design of a digital computer.
    • This seemed to Speiser a good time to move on to another venture [',' Presidential series: Exclusive Interview with Professor Ambros Speiser, International Federation for Information Processing.

  248. Hans-Joachim Bremermann (1926-1996)
    • In fact his interest in the theory of computation went back to his days as a graduate student in Munster when he attended a series of lectures on Turing machines.
    • In 1978 he gave the "What Physicists Do" series of lectures at The Sonoma State University.
    • In this series he discussed the physical limitations to mathematical understanding of physical and biological systems.

  249. Heinz Rutishauser (1918-1970)
    • Also in 1951 the first of a series of four papers by Rutishauser, Eduard Stiefel and Ambros Speiser, Programmgesteuerte digitale Rechengerate (elektronische Rechenmaschinen) Ⓣ appearing in 1950 and 1951.
    • In this series of papers the authors discuss in very considerable detail a number of the important mathematical questions that naturally arise in the design of a digital computer.
    • The QD algorithm represents a number of computational schemes for doing a surprising number of jobs: e.g., getting all eigenvalues of a matrix from its Schwarz constants, getting the zeros of a polynomial from its coefficients, finding the poles of a function from its power series, obtaining partial fraction representations of functions, and so forth.

  250. Stefan Bergman (1895-1977)
    • He lectured first at the Massachusetts Institute of Technology in Cambridge, Massachusetts, where he gave a series of lectures on Theory of pseudo-conformal transformations and its connection with differential geometry during 1939-40.
    • Results in the theory of one complex variable on such topics as analytic continuation, the residue theorem, Hadamard's theorems on the connection between the coefficients of the power series development of an analytic function and the character and location of the singularities and on Abelian integrals are used to give information concerning domains of regularity, series expansion, singularities and integral relations for the solutions.

  251. Thomas Carlyle (1795-1881)
    • He gave a series of lectures beginning in May 1837 on the German influence on Britain, and another series in the following year on European literature.
    • Further lectures series were given in 1839 and 1840.

  252. Curtis McMullen (1958-)
    • He has also been invited to give special lectures such as (since 2000): Nevanlinna Colloquium, Helsinki (2000); American Mathematical Society Colloquium Lectures, Washington DC (2000); Distinguished Lecture Series, Brown University (2001); American Mathematical Society Ross Lecture, Boston (2002); Namboodiri Lectures, University of Chicago (2003); Alaoglu Lecture, California Institute of Technology, Pasadena (2003); Mathematische Arbeitstagung, Max-Planck-Institut, Bonn (2003); Bowen Lectures, University of California, Berkeley (2004); Kolchin Lecture, Columbia University, New York (2005); Hopf Lectures, Eidgenossische Technische Hochschule, Zurich (2005); Mathematische Arbeitstagung, Max-Planck-Institut, Bonn (2007); Ziwet Lectures, University of Michigan, Ann Arbor (2008) and Nielsen Lecture, CTQM, Aarhus, Denmark (2008).
    • As the title of the talk suggests, there were many different areas of mathematics touched on by McMullen, including: Fermat's Last Theorem, Zeno's Paradoxes, hyperbolic and spherical geometry, the harmonic series, and tiling.
    • Near the end of his talk, McMullen showed a path that a human could take to elude the lion and used results about infinite series to demonstrate the path's effectiveness.

  253. Andreas Speiser (1885-1970)
    • From 1911 to 1937 twenty-six of these volumes have appeared, of which twenty volumes belong to Series I which contains the mathematical works.
    • Series II contains the works on mechanics, technology and astronomy, while Series III is devoted to the works on physics and philosophy.

  254. Oded Schramm (1961-2008)
    • His work in a spectacular series of papers has led to major progress in probability theory, in the theory of percolation and of random walks, as well as in related topics of conformal field theory.
    • He was also invited to give the prestigious Coxeter Lecture Series at the Fields Institute in September 2005.
    • He gave the following abstract for his three lecture series on Scaling limits of two dimensional random systems:- .

  255. Josip Plemelj (1873-1967)
    • He was soon making his own mathematical discoveries, for example he discovered for himself the series expansion for sin x and for cos x.
    • The way he did this was to first find the series expansion for arcsin x and then invert the series to obtain that for cos x.

  256. Raymond Smullyan (1919-2017)
    • Smullyan's publications have been quite remarkable with the two outstanding books on retrograde analysis chess problems [',' S Smullyan, The Chess Mysteries of the Sherlock Holmes (New York, 1979).','2] and [',' S Smullyan, The Chess Mysteries of the Arabian Knights (New York, 1981).','3], a whole series of marvellous popular puzzle books such as [',' S Smullyan, What is the name of this book? (New York, 1978).','1] and [',' S Smullyan, Satan, Cantor, and Infinity and other mind-boggling puzzles (New York, 1992).','4], and some books on the foundations of mathematics and mathematical logic which are in many ways in a class of their own.
    • This book was the first of a series of texts which appeared in quick succession.
    • A third volume in the series Diagonalization and self-reference was published in 1994 and presents a very difficult topic in such a way as to make it both understandable and enjoyable.

  257. Bruno de Finetti (1906-1985)
    • In this regard, D V Lindley [',' D V Lindley, Obituary : Bruno de Finetti, 1906-1985, Journal of the Royal Statistical Society, Series A 149 (1986), 252.','7], [',' D V Lindley, De Finetti, Bruno, in Encyclopedia of Statistical Sciences (Supplement) (New York, 1989), 46-47.','8] reports that Bruno de Finetti was especially fond of the aphorism:- .
    • Although the idea of probability as a measure of the observer's belief that an event will happen had already been conceived by F P Ramsey in 1926, Bruno de Finetti was unaware of Ramsey's work and, moreover, his chief interest was for coherent probability assessments and not for rational decisions; see the obituary by D V Lindley [',' D V Lindley, Obituary : Bruno de Finetti, 1906-1985, Journal of the Royal Statistical Society, Series A 149 (1986), 252.','7] for more information.
    • A key tool for him was nomenclature: for example, as reported by D V Lindley [',' D V Lindley, Obituary : Bruno de Finetti, 1906-1985, Journal of the Royal Statistical Society, Series A 149 (1986), 252.','7], [',' D V Lindley, De Finetti, Bruno, in Encyclopedia of Statistical Sciences (Supplement) (New York, 1989), 46-47.','8], he insisted that "random variables" should more appropriately be called "random quantities", for "What varies?".

  258. Eustachio Manfredi (1674-1739)
    • With his assistants, he carried out three series of observations of the pole star, two series with mobile quadrants and the third with a two and a half metre wall semicircle.
    • He then carried out a series of observations on stars, attempting to measure parallax.

  259. Michael Faraday (1791-1867)
    • Faraday introduced a series of six Christmas lectures for children at the Royal Institution in 1826.
    • He published his first paper in what was to become a series on Experimental researches on electricity in 1831.
    • These two final series of lectures by Faraday were published and have become classics.

  260. Hector Macdonald (1865-1935)
    • the relations between convergent series and asymptotic expansions, the zeros and the addition theorem of the Bessel functions, various Bessel integrals, spherical harmonics and Fourier series.
    • Macdonald worked on electric waves and solved difficult problems regarding diffraction of these waves by summing series of Bessel functions.

  261. Oscar Schlömilch (1823-1901)
    • In 1847 he gave a general remainder formula for the remainder in Taylor series.
    • He discovered an important series expansion of an arbitrary function in terms of Bessel functions in 1857.
    • In a militant manner, Barfuss defended an obsolete point of view on the question of symbolic calculation with divergent series against Schlomilch.

  262. Rózsa Péter (1905-1977)
    • In a series of papers she became a founder of recursive function theory.
    • In a series of articles, beginning in 1934, Peter developed various deep theorems about primitive recursive functions, most of them with an explicit algorithmic content.
    • Beginning in 1932, Rosza Peter has published a series of papers, examining the relationship of various special forms of recursion, and showing the definability of new functions by successively higher types of recursion, which establish her as the leading contributor to the special theory of recursive functions.

  263. D R Kaprekar (1905-1986)
    • The possible digitadition series are separated into three types: type A has all is members coprime to 3; type B has all is members divisible by 3 but not by 9; C has all is members divisible by 9.
    • Kaprekar notes that if x and y are of the same type (that is, each prime to 3, or each divisible by 3 but not 9, or each divisible by 9) then their digitadition series coincide after a certain point.
    • He conjectured that a digitadition series cannot contain more than 4 consecutive primes.

  264. Cesare Burali-Forti (1861-1931)
    • In 1893-94 Burali-Forti gave an informal series of lectures on mathematical logic at the University of Turin.
    • He urged that Burali-Forti's contradiction could be simply resolved by denying the premise that the series of all ordinal numbers is well-ordered, "..
    • Edgar Odell Lovett (1871-1957) reviewed Burali-Forti's book Introduction a la geometrie differentielle suivant la methode de H Grassman (1897) in [',' E O Lovett, Review: Introduction a la geometrie differentielle suivant la methode de H Grassman, by C Burali-Forti, professeur a l’Academie militaire de Turin, Science, New Series 10 (253) (1899), 653-654.','22].

  265. William McFadden Orr (1866-1934)
    • Orr's first publications were on hypergeometric series, Fourier double integrals involving Bessel functions which he followed up with a similar paper involving Legendre functions.
    • Then he advances a series of objections that may be raised against various ways of representing the foundations of thermodynamics; especially noteworthy among these objections is that of Bertrand, i.e., that the pressure, temperature, and entropy are defined only for the case that at least sufficiently small parts of a system can be regarded as being in equilibrium; a similar objection is raised with respect to the heat supplied.
    • Series A, Mathematical and Physical Sciences 148 (863) (1935), 1-31.','6]:- .

  266. David Rees (1918-2013)
    • There he attended a series of lectures by Philip Hall on semigroups and these lectures inspired him to produce deep results in this area.
    • Over the next years, Rees continued to produce a whole series of important papers on local rings.
    • Reflecting its birth as a series of lectures ..

  267. Guido Castelnuovo (1865-1952)
    • In 1873 Alexander von Brill and Max Noether had published a joint work on properties of linear series.
    • In the second of these, which appeared in 1891, he gave the first systematic use of the characteristic series and of the adjoint system.
    • Castelnuovo produced a series of papers over a period of 20 years which, together with Enriques, finally produced a classification of algebraic surfaces.

  268. Joseph Serret (1819-1885)
    • At the College of France he gave several series of lectures, taking a different topic in each academic year.
    • Finally, the sixth chapter, which ends the textbook, is primarily devoted to developing trigonometric solutions based on the use of series; these solutions relate to different situations that arise frequently in Astronomy and in Geodesy, and for which general methods become insufficient.
    • Each chapter ends with a series of questions relating to the material treated; I urge the reader to seek the solution.

  269. Piero della Francesca (1420-1492)
    • Piero almost certainly wrote all three works in the vernacular (his native dialect was Tuscan), and all three are in the style associated with the tradition of 'practical mathematics', that is, they consist largely of series of worked examples, with rather little discursive text.
    • It deals with arithmetic, starting with the use of fractions, and works through series of standard problems, then it turns to algebra, and works through similarly standard problems, then it turns to geometry and works through rather more problems than is standard before (without warning) coming up with some entirely original three-dimensional problems involving two of the 'Archimedean polyhedra' (those now known as the truncated tetrahedron and the cuboctahedron).
    • He accordingly starts with a series of mathematical theorems, some taken from the optical work of Euclid (possibly through medieval sources) but some original to Piero himself.

  270. Samuel Beatty (1881-1970)
    • Another venture led by Beatty was organising a series of Mathematical Expositions to be produced by the Department.
    • He wrote the following as a Preface to an early volume which shows his thinking behind the series:- .
    • A series of books, published under the auspices of the University of Toronto and bearing the title 'Mathematical Expositions', represents an attempt to meet this need.

  271. Édouard Goursat (1858-1936)
    • He then produced an impressive series of papers which contributed to almost every area of analysis which was being studied at that time.
    • Goursat's papers on the theory of linear differential equations and their rational transformations, as well as his studies on hypergeometric series, Kummer's equation, and the reduction of abelian integrals form, in the words of Picard "a remarkable ensemble of works evolving naturally one from the other".
    • Volume 1 covers applications to geometry, expansion in series, definite integrals, and derivatives and differentials.
    • Despite working on the many new editions of Cours d'analyse mathematique Ⓣ, Goursat found time to write other texts such as Le probleme de Backlund Ⓣ (1925), and Lecons sur les series hypergeometriques et sur quelles fonctions qui s'y rattachent Ⓣ (1936).

  272. Leopold Schmetterer (1919-2004)
    • One advantage of working in the Henschel aircraft factory was the fact that he had to use Fourier series, and now back in the Mathematical Institute he began to study these series more deeply.
    • In 1949 he obtained the right to teach in universities after submitting his Habilitation thesis Uber die Approximation gewisser trigonometrischer Reihen Ⓣ on the theory of trigonometrical series.

  273. Marjorie Senechal (1939-)
    • in 1956 for his thesis Summation formulas associated with a class of Dirichlet Series.
    • Analytic number theory became my research topic because my advisor, an expert on summation formulas for divergent infinite series, had a grant from the Office of Naval Research that included support for a graduate student.
    • She published the paper based on her thesis A summation formula and an identity for a class of Dirichlet series in 1966.

  274. Thomas Bayes (1702-1761)
    • Another mathematical publication on asymptotic series appeared after his death where he showed that the series for log z! given by Stirling and de Moivre, was not valid since it diverged.
    • This notebook contains a considerable amount of mathematical work, including discussions of probability, trigonometry, geometry, solution of equations, series, and differential calculus.

  275. Edwin Beckenbach (1906-1982)
    • He wrote a series of books and acted as editor for several more texts.
    • In 1961 he edited a second series of Modern mathematics for the engineer which was divided into three parts as was the earlier volume.
    • Beckenbach contributed to mathematics with a series of texts for schools and colleges.

  276. Sydney Chapman (1888-1970)
    • His first research was on summable series and he wrote two papers on this topic, one of them a joint paper with Hardy.
    • During the war, between 1915 and 1917, he completed a series of important papers on thermal diffusion and the fundamentals of gas dynamics.
    • Between 1913 and 1919 he published another important series of papers, this time on terrestrial magnetism which we comment on below.

  277. Simion Stoilow (1887-1961)
    • His work was having a major international impact and he was invited to Paris where he gave a series of lectures on his work in February 1931.
    • Stoilow gave a series of six lectures on Riemann surfaces at the Istituto di Alta Matematica in Rome in April, 1957.
    • After a fairly standard introduction to the general theory, beginning with power series, he goes on, in volume 1, to look at topics such as entire and meromorphic function, doubly periodic functions, conformal mapping on the boundary of a Jordan region, multiple-valued functions, and applications of modular functions to the Picard circle of ideas.

  278. Aldo Ghizzetti (1908-1992)
    • He published a series of papers in 1942: Sui momenti di una funzione limitata Ⓣ; Ricerche sui momenti di una funzione limitata compresa fra limiti assegnati Ⓣ; (with Renato Caccioppoli) Ricerche asintotiche per una particolare equazione differenziale non lineare Ⓣ; (with Renato Caccioppoli) Ricerche asintotiche per una classe di sistemi di equazioni differenziali ordinarie non lineari Ⓣ; and most importantly Sui problemi di Dirichlet per la striscia e per lo strato Ⓣ.
    • Alessandro Faedo was appointed in 1946 and began a series of brilliant appointments building Pisa into a world leading mathematical centre.
    • A feature of this work involved extending the classical concept of orthogonal polynomials to s-orthogonal polynomials and studying their associated series expansions.

  279. Mei Juecheng (1681-1763)
    • This contained the infinite series expansion for sin(x) which was discovered by James Gregory and Isaac Newton.
    • In fact the Jesuit missionary Pierre Jartoux (1669-1720) (known in China as Du Demei) introduced the infinite series for the sine into China in 1701 and it was known there by the name 'formula of Master Du'.
    • using infinite series which he does not prove and refers to as "formulas from the Western scholar Du Demei." .

  280. Gian-Carlo Rota (1932-1999)
    • As we have indicated above, Rota worked on functional analysis for his doctorate and, up to about 1960, he wrote a series of papers on operator theory.
    • This paper was the first of a series of ten papers with this main title, all ten have subtitles (for example this first one was subtitled Theory of Mobius functions ) and all the remaining nine have between one and three additional co-authors.
    • He had been due to give a series of three lectures at Temple University, the Groswald Memorial Lectures, on the previous day and, when he failed to arrive in Philadelphia, a check was made at his home.

  281. Ernst Peschl (1906-1986)
    • The titles of the chapters are: Algebra and geometry of complex numbers; Fundamental topological concepts, sets, sequences of complex numbers and infinite series; Functions, real and complex differentiability and holomorphy; Integral theorems and their consequences; Winding number and curves homologous to zero; Taylor development of holomorphic functions; Elementary transcendental functions; Laurent series, isolated singularities and residue calculus; Holomorphic and meromorphic functions obtained by limiting processes; Analytic continuation; and Conformal mappings.

  282. Bill Ferrar (1893-1990)
    • Ferrar wrote many research papers which deal with the convergence of series, an interest which came from working with G N Watson at Cambridge for during a summer vacation while an undergraduate.
    • From about 1930 his interests turned towards number theory and he examined the convergence of series and the evaluation of singular integrals.

  283. Mikhail Fedorovich Subbotin (1893-1966)
    • Subbotin not only showed the possibility of improving the convergence of the trigonometric series by which the behaviour of perturbing forces is represented, but also gave an expression for determining Laplace coefficients and presented formulas for computing the coefficients of the necessary members of the trigonometric series.

  284. Benjamin Moiseiwitsch (1927-2016)
    • In addition, the integral equation approach leads naturally to the solution of the problem - under suitable conditions - in the form of an infinite series.
    • The second part examines Fourier series and Fourier and Laplace transforms, integral equations, wave motion, heat conduction, tensor analysis, special and general relativity, quantum theory, and variational principles.

  285. Angelo Genocchi (1817-1889)
    • After a series of defeats, Charles Albert's army withdrew from Milan.
    • The main research topics which Genocchi worked on were number theory, series and the integral calculus.

  286. Johann Bernoulli (1667-1748)
    • In 1694 he considered the function y = xx and he also investigated series using the method of integration by parts.
    • He summed series, and discovered addition theorems for trigonometric and hyperbolic functions using the differential equations they satisfy.

  287. William Milne (1881-1967)
    • However, Milne was also interested in mathematical education and published a series of papers and mathematical notes in the Mathematical Gazette.
    • These included: The geometrical meaning of the triad of points (1910); A property of the complete quadrangle (1911); The teaching of limits and convergence to scholarship candidates (1911); The teaching of limits and convergence to scholarship candidates (1912); The teaching of limits and convergence to scholarship candidates (1913); Another proof and generalisation of the theorem given in note 339 (1913); The teaching of modern analysis in secondary schools (1915); The graphical treatment of power series (1918); The uses and functions of a school mathematical library (1918); Mathematics and the pivotal industries (1919); The training of the mathematical teacher (1920); and Noether's canonical curves (1920).

  288. William Kruskal (1919-2005)
    • In 1954, prior to the era of modern high speed computers, the present authors published the first of a series of four landmark papers on measures of association for cross classifications.
    • This series of papers evolved over a twenty-year period.

  289. Alexander von Brill (1842-1935)
    • In 1874 he brought out a series of paper models of second order surfaces.
    • By 1890 he was selling 16 series of models, seven of which were the original ones constructed at the Technische Hochschule in Munich under the direction of Brill, Klein and von Dyck.

  290. Ernst Stueckelberg (1905-1984)
    • In fact, these meals turned into a series of tutorials on the quantum mechanics.
    • Stueckelberg and I had laid out a whole series of calculations we wanted to attempt.

  291. Serge Lang (1927-2005)
    • Lang's mathematical research ranged over a wide range of topics such as algebraic geometry, Diophantine geometry (a term Lang invented), transcendental number theory, Diophantine approximation, analytic number theory and its connections to representation theory, modular curves and their applications in number theory, L-series, hyperbolic geometry, Arakelov theory, and differential geometry.
    • Three public dialogues (1985), Introduction to complex hyperbolic spaces (1987), Introduction to Arakelov theory (1988), Topics in Nevanlinna theory (1990), Basic analysis of regularized series and products (1993), Fundamentals of differential geometry (1999), and Math talks for undergraduates (1999).

  292. Viggo Brun (1885-1978)
    • Now the first question one would naturally ask is "What is the approximate value of the sum of the reciprocals of twin primes"? Unfortunately there is no accepted standard for exactly what series is being summed.
    • However, as we can see from the title of Brun's paper, he took a slightly different start to his series .

  293. Charles Augustin Coulomb (1736-1806)
    • He began to feel threatened by his political opponents in 1775 and began a series of reforms.
    • From examination of many physical parameters, he developed a series of two-term equations, the first term a constant and the second term varying with time, normal force, velocity, or other parameters.

  294. Peter Ladislaw Hammer (1936-2006)
    • There followed a series of papers, typical of which are Linear Programming and Transportation (1960), Applications of Mathematics to Economics (1960), Optimization of the Development Plan of an Industry (1961), and A Method for Solving Transportation Problems (1961).
    • RUTCOR played a major role, with Hammer as its director, with regular series of seminars, workshops, and courses put on for the numerous graduate students.

  295. Mstislav Vsevolodovich Keldysh (1911-1978)
    • Keldysh worked on aerohydrodynamics, publishing a series of papers on the topic between 1934 and 1937.
    • Keldysh submitted his doctoral thesis Functions of Complex Variable and Harmonic Functions Representation with Polynomial Series to the Steklov Mathematical Institute in 1937 - the degree was awarded in the following year [',' I N Sneddon, Mstislav Vsevolodovich Keldysh, Hon.

  296. Pietro Cataldi (1548-1626)
    • Cataldi found square roots of numbers by use of an infinite series leading to an early investigation into continued fractions.
    • In this work the square root of a number is found through the use of infinite series and unlimited continued fractions.

  297. Thomas Flett (1923-1976)
    • 9 (1977), 330-339.','13] Flett's mathematical contributions are described in detail under the headings (i) Fourier series and power series (11 publications), (ii) Summability (10 publications), (iii) Function theoretic identities and inequalities (16 publications), (iv) Geometric analysis (7 publications), (v) Complex analysis, harmonic functions (7 publications), and (vi) Miscellaneous (10 publications).

  298. Lajos Dávid (1881-1962)
    • It is to his merit that the book series titled Kozlemenyek a debreceni Egyetem Matematikai Szeminariumabol Ⓣ (1927-1940), containing 15 doctoral theses, started to appear.
    • There were only a few to assist him at the Mathematics Seminar, so Lajos David himself held lectures on a very wide variety of topics: descriptive geometry, infinite series, infinitesimal calculus and geometry, analysis, the practical solution of equations, the theory of differential equations, surface theory, probability theory, and practical mathematics.

  299. Pafnuty Chebyshev (1821-1894)
    • Chebyshev submitted a paper on The calculation of roots of equations in which he solved the equation y = f (x) by using a series expansion for the inverse function of f.
    • This paper was on the convergence of Taylor series.

  300. Andrey Kolmogorov (1903-1987)
    • However the person who made the deepest impression on Kolmogorov at this time was Stepanov who lectured to him on trigonometric series.
    • This was published jointly with Khinchin and contains the 'three series' theorem as well as results on inequalities of partial sums of random variables which would become the basis for martingale inequalities and the stochastic calculus.

  301. Morris Kline (1908-1992)
    • Ivor Grattan-Guinness, a well-known historian of mathematics, writes [',' I Grattan-Guinness, Review: Mathematical Thought from Ancient to Modern Times by Morris Kline, Science, New Series 180 (4086) (1973), 627-628.','19]:- .
    • On the contrary he had written a whole series of articles on the "new maths" in the 1950s and 1960s in journals such as The Mathematics Teacher.

  302. William Brouncker (1620-1684)
    • The English Civil War broke out in 1642, the Scots joined the Parliament forces and Charles I suffered a series of defeats.
    • Brouncker's mathematical achievements includes work on continued fractions and calculating logarithms by infinite series.

  303. Paul Appell (1855-1930)
    • In 1880 Appell defined a series of functions satisfying the condition that the derivative of the nth function is n times the (n - 1)th function.
    • his scientific work consists of a series of brilliant solutions of particular problems, some of the greatest difficulty.

  304. Béla Kerékjártó (1898-1946)
    • The first of these courses was enlarged into a book Vorlesungen uber Topologie Ⓣ which appeared in the series Grundlehren der Mathematischen Wissenschaften in 1923.
    • His final work was intended to be a series of five books, only two of which were written before his death.

  305. Hans Reichardt (1908-1991)
    • Schmidt had held a temporary post in Gottingen in 1933 and, in the same year succeeded Richard Courant as editor of Springer-Verlag's famous "Yellow Series" of mathematical monographs when Courant was dismissed because he was Jewish.
    • Schmidt was a Roman Catholic, and not Jewish, but he was quickly out of favour with the Nazis when he refused to remove Richard Courant's name from the title page of the Springer series.

  306. Solomon Lefschetz (1884-1972)
    • During these years he wrote a series of important papers on topology despite being out the mainstream of mathematical research.
    • Lefschetz had many students working in this area and, between 1950 and 1960, a series of important publications Contributions to the theory of nonlinear oscillations appeared in the Annals of Mathematics Studies, published by Princeton University Press.

  307. Pedro Abellanas (1914-1999)
    • This book gave an introduction to real and complex numbers, matrices, vectors, euclidean geometry, sequences and series, real and complex functions, and differential and integral calculus.
    • In Part 2 the author discusses limits, derivatives, and series.

  308. Walter Shewhart (1891-1967)
    • In this classic volume, based on a series of ground-breaking lectures given to the Graduate School of the Department of Agriculture in 1938, Dr Shewhart illuminates the fundamental principles and techniques basic to the efficient use of statistical method in attaining statistical control, establishing tolerance limits, presenting data, and specifying accuracy and precision.
    • During 1944-46 he served on the National Research Council and for over 20 years he served as editor of the Mathematical Statistics Series of John Wiley and Sons.

  309. Konrad Knopp (1882-1957)
    • Chapter III: Sets, sequences and power series.
    • Volume 1 covers numbers, functions, limits, analytic geometry, algebra, set theory; volume 2 covers differential calculus, infinite series, elements of differential geometry and of function theory; and volume 3 covers integral calculus and its applications, function theory, differential equations.

  310. Richard Varga (1928-)
    • They are Matrix iterative analysis (1962), Functional analysis and approximation theory in numerical analysis (1971), Topics in polynomial and rational interpolation and approximation (1982), (with Albert Edrei and Edward B Saff) Zeros of sections of power series (1983), Scientific computation on mathematical problems and conjectures (1990), and Gerschgorin and his circles (2004).
    • Zeros of sections of power series is described by Walter Hengartner as a 'remarkable monograph' but perhaps the greatest praise is heaped on Scientific computation on mathematical problems and conjectures.

  311. Mikhail Krawtchouk (1892-1942)
    • Later in 1918 an independent Ukraine was again declared in Kiev but there followed a series of struggles between Ukrainian nationalist, White, and Red forces.
    • There was a series of public trials and in a massive terror campaign against the population as a whole.

  312. Fabian Franklin (1853-1939)
    • Francis Dominic Murnaghan, the author of [',' F D Murnaghan, Fabian Franklin, Science, New Series 89 (2309) (1939), 283.','5], was never taught by Franklin but he was on the staff at Johns Hopkins and came in contact with many who had been taught by Franklin.
    • However, Murnaghan believes that Franklin did not achieve the depth of research results that his talents would have merited [',' F D Murnaghan, Fabian Franklin, Science, New Series 89 (2309) (1939), 283.','5]:- .

  313. Richard Askey (1933-)
    • Twenty five years after he gave his series of ten lectures to the National Science Foundation Regional Conference, Askey published another major work on special functions.
    • He also gave two series of lectures in 1984, namely the University of Illinois Trjitzinsky Lectures and the Pennsylvania State University College of Science Lectures.

  314. Evelyn Boyd Granville (1924-)
    • She wrote a doctoral thesis On Laguerre Series in the Complex Domain and in 1949, together with Marjorie Lee Browne who graduated from the University of Michigan in the same year, she became the one of the first black American women to be awarded a Ph.D.
    • math must not be taught as a series of disconnected, meaningless technical procedures from dull and empty textbooks.

  315. Donald Higman (1928-2006)
    • for his thesis Focal Series in Finite Groups.
    • His first publication Lattice homomorphisms induced by group homomorphisms appeared in 1951 while he was still a research student and four further publications appeared during the two years at McGill University, namely: Focal series in finite groups (1953); Remarks on splitting extensions (1954); Modules with a group of operators (1954); and Indecomposable representations at characteristic p (1954).

  316. Harish-Chandra (1923-1983)
    • Some major contributions by Harish-Chandra's work may be singled out: the explicit determination of the Plancherel measure for semisimple groups, the determination of the discrete series representations, his results on Eisenstein series and in the theory of automorphic forms, his "philosophy of cusp forms", as he called it, as a guiding principle to have a common view of certain phenomena in the representation theory of reductive groups in a rather broad sense, including not only the real Lie groups, but p-adic groups or groups over adele rings.

  317. Einar Carl Hille (1894-1980)
    • Hille's main work was on integral equations, differential equations, special functions, Dirichlet series and Fourier series.

  318. Pierre Fatou (1878-1929)
    • He began publishing the results of his mathematical research in 1904 with the short paper Sur les series entieres a coefficients entiers Ⓣ appearing in that year.
    • In 1905 he published four short papers: Sur l'integrale de Poisson et les lignes singulieres des fonctions analytiques Ⓣ; Sur quelques theoremes de Riemann Ⓣ; Sur l'approximation des incommensurables et les series trigonometriques Ⓣ; and La serie de Fourier et la serie de Taylor sur son cercle de convergence Ⓣ.
    • Three more papers were published in 1906: Sur le developpement en serie trigonometrique des fonctions non integrables Ⓣ; Series trigonometriques et series de Taylor Ⓣ; and Sur l'application de l'analyse de Dirichlet aux formes quadratiques a coefficients et a indeterminees conjuguees Ⓣ.
    • In 1906 he submitted his thesis, Series trigonometriques et series de Taylor Ⓣ which was on integration theory and complex function theory.
    • His famous work involves Taylor series where he examined the convergence and the analytic extension of the series.

  319. Henri Cartan (1904-2008)
    • Cartan published Les transformations analytiques des domaines cercles les uns dans les autres Ⓣ in 1930 and, since this paper contained generalisations of results proved by Heinrich Behnke, he was invited by Behnke to visit Germany in May 1931 and give a series of lectures at Munster in Westphalen where Behnke taught.
    • A central figure in this development has been Henri Cartan, whose series of papers in this field starting in the 1920's dealt with fundamental questions relating to Nevanlinna theory, generalizations of the Mittag-Leffler and Weierstrass theorems to functions of several variables, problems concerned with biholomorphic mappings and the biholomorphic equivalence problem, domains of holomorphy and holomorphic convexity, etc.

  320. Cecilia Krieger (1894-1974)
    • Her doctoral dissertation was On the summability of trigonometric series with localized properties - on Fourier constants and convergence factors of double Fourier series.

  321. Alston Householder (1904-1993)
    • In a remarkable series of papers he effectively classified the algorithms for solving linear equations and computing eigensystems, showing that in many cases essentially the same algorithm had been presented in a large variety of superficially quite different algorithms.
    • At the 13th in the series of Householder Symposia held in Pontresina, Switzerland in 1996, Friedrich L Bauer spoke on Memories of Alston Householder.

  322. Wilhelm Wirtinger (1865-1945)
    • The mathematician Wilhelm Wirtinger, who established his scientific reputation over 10 years ago with the solution of a difficult problem (the general theta function), reported today in the Academy of Sciences on "the development of some mathematical concepts in modern times." The definition of these terms is not always easy, as Wirtinger shows with the example of an infinite set: "Even Galileo had mentioned that the infinite set of integers seems to be far greater than the set of square numbers, since the square numbers are becoming increasingly rare the further one progresses along the series of numbers, while, on the other hand, the size of both sets would have to be the same because each number corresponds to a perfect square." .
    • At the age of 71 he wrote the first of a series ground-breaking papers on higher dimensional spaces.

  323. Ludwig Schlesinger (1864-1933)
    • In 1926 Schlesinger published a book on Lebesgue integration and Fourier series in collaboration with Abraham Plessner.
    • The work studies trigonometric series and the boundary behaviour of analytic functions.

  324. Jean Frenet (1816-1900)
    • He published these observations in the Memoires de l'Academie imperiale de Lyon in a continuing series, first in 1853, next in 1856, and again in 1858.
    • A Drian took over publishing the series of meteorological observations.

  325. Aurel Angelescu (1886-1938)
    • Aurel Angelescu's main fields of interest were generating functions for polynomials, linear differential equations, functional analysis, trigonometric series and the general theory of algebraic equations.
    • Examples of his contributions are: Extrait sur les polynomes Ⓣ (1915); Sur le quardatures mechanique Ⓣ (1920); On the linear homogeneous recurrence relations with constant coefficients (Romanian) (1925); The definition of the functions of complex variable (Romanian) (1925); Sur certains developpements des fonctions holomorphes en serie de fractions rationnelles Ⓣ (1925); Sur une formule de M Pompeiu Ⓣ (1929); On some means (Romanian) (1932); Sur le principe de Legendre Ⓣ (1933); Sur une certaine extension des series entieres Ⓣ (1937); and Sur certains polynomes generalisant les polynomes de Laguerre Ⓣ (1938).
    • He also partly wrote and was the editor of Lectures on Mathematical Analysis: sets, limits, series, derivatives and differentials, extreme points, change of variables (Romanian) (1927).

  326. Karl Gruenberg (1928-2007)
    • Typical of this is his famous Some cohomological topics in group theory which appeared in the Queen Mary College Mathematics Notes series in 1967.
    • In 1976 he published Relation modules of finite groups which appeared in the Conference Board of the Mathematical Sciences Regional Conference Series in Mathematics of the American Mathematical Society.

  327. Cesare Arzelŕ (1847-1912)
    • He elaborated the concept of stepwise uniform convergence which gives a necessary and sufficient condition for a series of continuous functions to converge to a continuous function (1883), and he proved the term-wise integration theorem for a series of functions using the Riemann integral (1885).

  328. Tadeusz Banachiewicz (1882-1954)
    • He was then employed at the Engelhardt Observatory near Kazan where he spent five years (1910-1915) making a series of heliometric observations.
    • In 1936 Banachiewicz organised a series of expeditions to Japan, Greece and Siberia to make solar observations.

  329. Alec Aitken (1895-1967)
    • With Rutherford he was editor of a series of the University Mathematical Texts and he himself wrote for the series Determinants and matrices (1939) and Statistical Mathematics (1939).

  330. Roberto Marcolongo (1862-1943)
    • At the Fourth International Congress of Mathematicians held in Rome in April 1908 he delivered a paper on L'unificazione delle teorie vettoriali Ⓣ, a research program derived from the geometric calculus of Giuseppe Peano and developed in the following years in a series of articles and monographs written in collaboration with Cesare Burali-Forti (of whom he later published the obituary [',' R Marcolongo, Cesare Burali-Forti, Bollettino dell’Unione matematica italiana 10 (1931), 182-185.','7] in the Bollettino dell'Unione matematica italiana Ⓣ).
    • These, like the two historical monographs mentioned above, were published in the particularly important series of Hoepli Manuals.

  331. Wendelin Werner (1968-)
    • The title for his series of talks was Two-dimensional continuous random systems and again the abstract for these talks gives good insight into topics on which he was making a remarkable impact in collaboration with Greg Lawler and Oded Schramm:- .
    • For example, in 2008 he was elected to the French Academy of Sciences and, in January of the same year, he was the American Mathematical Society Colloquium Lecturer, giving his series of lectures on Random conformally invariant pictures at San Diego Convention Center.

  332. Robert Simson (1687-1768)
    • That Simson's work was not restricted to Greek geometry is illustrated by Tweddle's paper [',' I Tweddle, John Machin and Robert Simson on inverse-tangent series for š, Arch.
    • 42 (1) (1991), 1-14.','3], in which he discusses an early manuscript of Simson dealing with inverse tangent series and their use in calculating π.

  333. Józeph Petzval (1807-1891)
    • Johann, born on 4 July 1775 in Lodenice, taught in the evangelical school in Spisska Bela and [',' Petzval History Series: The Early Life of Joseph Petzval, Lomography Magazine (9 June 2015).','13]:- .
    • As a teacher, Petzval was well-liked by his students [',' Petzval History Series: Joseph Petzval’s Unconventional Career, Lomography Magazine (14 June 2015).','15]:- .

  334. Alfred Pringsheim (1850-1941)
    • He also has important results on the singularities of power series with positive coefficients.
    • In 1893 he proved that a function is analytic if it is infinitely differentiable on an open interval and the radius of convergence r(x) of the Taylor series centred at x is bounded away from 0.

  335. Ferdinand Rudio (1856-1929)
    • He edited two volumes himself, namely Leonhard Euleri Opera Omnia: Series Prima: Commentationes Arithmeticae - 1st Part Ⓣ (1915) and Leonhard Euleri Opera Omnia: Series Prima: Commentationes Arithmeticae - 2nd Part Ⓣ (1917).

  336. Herbert Seifert (1907-1996)
    • The book was accepted by Blaschke for the Hamburg monograph series but the two authors ran into problems with a Latin epigraph which they wished to put at the beginning.
    • Seifert, still able to do mathematical research, worked on differential equations and wrote a series of papers on the topic through the war years.

  337. Madan Lal Puri (1929-)
    • Time series and related topics.
    • Professor Puri's research has lead to his being considered one of the most versatile and prolific researchers in the world in mathematical statistics in the areas of nonparametric statistics, order statistics, limit theory under mixing, time series, splines, tests of normality, generalized inverses of matrices and related topics, stochastic processes, statistics of directional data, random sets, and fuzzy sets and fuzzy measures.

  338. Bernard Bolzano (1781-1848)
    • Erste Lieferung Ⓣ (1810), the first of an intended series on the foundations of mathematics.
    • Bolzano wrote the second of his series but did not publish it.

  339. Antonio Mario Lorgna (1735-1796)
    • We indicate below the titles of some of his works but let us record here that, among the pure mathematical topics he worked on, was geometry, convergence of series and algebraic equations.
    • One of his works was translated into English as Dissertation on the Summation of Infinite Converging Series with Algebraic Divisor, Exhibiting a Method Not Only Entirely New, But Much More General (1779).

  340. David Gawen Champernowne (1912-2000)
    • This paper represented the first serious attempt at the application to time-series analysis of the techniques of Thomas Bayes.
    • While at Oxford Champernowne pursued his pre-war interest in Frank Ramsey's theory of probability: this led to work on the application of Bayesian analysis to autoregressive series at a time when the Bayesian approach was intellectually unfashionable.

  341. Deane Montgomery (1909-1992)
    • His interests turned from point-set topology to transformation groups quite early in his career and he published a series of papers on the topic in collaboration with Leo Zippin.
    • In a long series of papers written in the late 1960s and early 1970s, [Montgomery and C T Yang] used the study of group actions on homotopy 7-spheres to showcase and test the growing new techniques of differential topology, especially index theory and surgery theory.

  342. William Osgood (1864-1943)
    • Some papers over the next few years included: Sufficient conditions in the calculus of variations (1900), On a fundamental property of a minimum in the calculus of variations and the proof of a theorem of Weierstrass's (1901), A Jordan curve of positive area (1903), On Cantor's theorem concerning the coefficients of a convergent trigonometric series, with generalizations (1909), On the gyroscope (1922), and On normal forms of differential equations (1925).
    • Other classic texts included Introduction to Infinite Series (1897), A First Course in the Differential and Integral Calculus (1909), Topics in the theory of functions of several complex variables published by the American Mathematical Society in 1914, Plane and Solid Analytic Geometry (with W C Graustein, 1921), Advanced Calculus (1925), and Mechanics (1937).

  343. Frederick Justin Almgren (1933-1997)
    • The reasons for this are partly indicated, for readers with only an advanced calculus background, in terms of examples, illustrated by a series of rather beautiful diagrams in colour.
    • Throughout his career, Almgren brought great geometric insight, technical power, and relentless determination to bear on a series of the most important and difficult problems in his field.

  344. Mary Fasenmyer (1906-1996)
    • There her doctoral studies were supervised by Earl Rainville who suggest that she examine some combinatorial problems related to hypergeometric series.
    • In her thesis she gave algorithms to find recurrence relations between sums of terms in hypergeometric series.

  345. Erwin Schrödinger (1887-1961)
    • Schrodinger published his revolutionary work relating to wave mechanics and the general theory of relativity in a series of six papers in 1926.
    • After giving a brilliant series of lectures in Madison he was offered a permanent professorship there but [',' W J Moore, A life of Erwin Schrodinger (Cambridge, 1994).','8]:- .

  346. Lester R Ford (1886-1967)
    • In addition to his work on point-wise discontinuous functions which we mentioned above, Ford is best known for an "absolutely marvellous geometric interpretation of the Farey series".
    • Still others have to do with Ford's former activity as editor of the MONTHLY, where he started the series of papers with the titles "What is ..

  347. Andrzej Alexiewicz (1917-1995)
    • Żyliński had supervised Władysław Orlicz's doctoral thesis on orthogonal series and it was Orlicz who became a friend, advisor and collaborator of Alexiewicz.
    • In fact, because of the disruption caused by the war, his habilitation thesis was published before his doctoral thesis, but he had already a number of earlier publications such as: (with W Orlicz) Remarque sur l'equation fonctionelle f (x + y) = f (x) + f (y) Ⓣ (1945); Linear operations among bounded measurable functions I and II (1946); On Hausdorff classes (1947); On multiplication of infinite series (1948); and Linear functionals on Denjoy-integrable functions (1948).

  348. Georg Frobenius (1849-1917)
    • On the development of analytic functions in series.
    • In a series of letters to Dedekind, the first on 12 April 1896, his ideas on group characters quickly developed.

  349. Earle Raymond Hedrick (1876-1943)
    • He was editor of the American Mathematical Monthly from 1913 to 1915, editor-in-chief of the Bulletin of the American Mathematical Society from 1921 to 1937, he was editor of 34 volumes in the Engineering Science Series and 35 volumes in the Series of Mathematical Texts.

  350. Georges Darmois (1888-1960)
    • Darmois was also an invited speaker at the Congress (although not a plenary speaker) and he gave the lecture Sur l'analyse et comparasion des series statistiques qui se developpent dans le temps Ⓣ.
    • G Darmois, professor at the University of Nancy, will give a series of four lectures on the following topic: Statistical laws, correlation and covariance with applications to heredity, to social and economic sciences.
    • Darmois discusses Fisher's work on estimation - he already knew of Fisher's work in time series analysis and population genetics - yet even so his note in the 'Comptes Rendus' ('Sur les lois de probabilite a estimation exhaustive', Ⓣ(1935) comes as a surprise for it contained an important contribution to Fisher's theory of estimation: Darmois (1935) considered distributions admitting a sufficient statistic, what are now called the exponential, or Koopman-Pitman-Darmois, family of distributions - B O Koopman and E J Pitman were based in the United States and Australia respectively and their works appeared somewhat later.

  351. Gotthold Eisenstein (1823-1852)
    • developed his own independent analytic theory of elliptic functions, based on the technique of summing certain conditionally convergent series.
    • Moreover, this case provides not merely an illuminating introduction to his theory, but also the simplest proofs for a series of results, originally discussed by Euler ..

  352. Thomas Lancaster Wren (1889-1972)
    • It is reported in [',' Scientific Notes and News, Science, New Series 38 (986) (1913), 736-740.','4] that:- .
    • G H Hardy, in his 'Divergent Series', asserts that Francis William Newman (1805-1897) [brother of John Henry Newman (1801-1890)] was for a time Professor of Mathematics at University College, London; this seems to be a mistake.

  353. Carl Střrmer (1874-1957)
    • It is unusual for anyone to publish a paper in the year they enter university as an undergraduate, but this is exactly what Størmer did with a paper on the summation of trigonometric series.
    • His output of mathematical papers continued with twelve papers on series, number theory, and the theory of functions between 1896 and 1902.

  354. George Waddel Snedecor (1881-1974)
    • The change appears to be due to Henry Wallace, an agricultural graduate of Iowa State College, who gave a series of ten Saturday afternoon lectures to agricultural and biological research workers in 1924.
    • The author takes the reader on a series of excursions of rather exciting discovery.

  355. Adolphe Quetelet (1796-1874)
    • Chance, that mysterious, much abused word, should be considered only a veil for our ignorance; it is a phantom which exercises the most absolute empire over the common mind, accustomed to consider events only as isolated, but which is reduced to naught before the philosopher, whose eye embraces a long series of events and whose penetration is not led astray by variations, which disappear when he gives himself sufficient perspective to seize the laws of nature.
    • It seems to me that that which relates to the human species, considered en masse, is of the order of physical facts: the greater the number of individuals, the more the influence of the individual will is effaced, being replaced by the series of general facts that depend on the general causes according to which society exists and maintains itself.

  356. Yves Rocard (1903-1992)
    • Reviewing the second edition, V A Johnson writes [',' V A Johnson, Review: Electricite, by Y Rocard, Science, New Series 125 (3236) (1957), 28-29.','9]:- .
    • It is assumed that the reader has some facility in mathematics and thus is familiar with the common vector operations, simple manipulations with complex variables, linear differential equations, and series expansions.

  357. George Lusztig (1946-)
    • He reported on this work in the lecture On the discrete series representations of the classical groups over a finite field which he gave to the Algebraic Groups section at the International Congress of Mathematicians held in Vancouver in August 1974.
    • In 1974 he published the monograph The discrete series of GLn over a finite field.

  358. Isaac Newton (1643-1727)
    • Newton made notes on Wallis's treatment of series but also devised his own proofs of the theorems writing:- .
    • The title page of Analysis per quantitatum series fluxiones (1711) .

  359. Kazimierz Kuratowski (1896-1980)
    • He was also one of the founders and an editor of the important Mathematical Monographs series.
    • He contributed the third volume in this series with his monograph on topology which we will mention again below.

  360. Peter Henrici (1923-1987)
    • The first paper he published in English was A Neumann series for the product of two Whittaker functions which appeared in the Proceedings of the American Mathematical Society in 1953.
    • The first volume Power series - integration - conformal mapping - location of zeros first appeared in 1974, the second volume Special functions - integral transforms - asymptotics - continued fractions first appeared in 1977, and the third and final volume Discrete Fourier analysis - Cauchy integrals - construction of conformal maps - univalent functions first appeared in 1986.

  361. Wilhelm Bessel (1784-1846)
    • His meeting with important English scientists, including Herschel, impressed him deeply and stimulated him to finish and publish, despite his weakened health, a series of works.
    • Bessel functions appear as coefficients in the series expansion of the indirect perturbation of a planet, that is the motion caused by the motion of the Sun caused by the perturbing body.

  362. Mineo Chini (1866-1933)
    • Chini treated, in a clear and ordered way, differential equations of the first and second order, integration by series of differential equations and partial derivatives of the first and second order.
    • In the 1919-20 academic year Chini gave an open course on this topic, consisting of a series of seminars at the Institute of Further Studies in Florence and the result of the experiment was pleasing because there was a high attendance and interest from the students; he then decided to publish the lectures that have been mentioned above.

  363. Yuri Vladimirovich Linnik (1915-1972)
    • From this he was able to produce a whole series of papers proving powerful arithmetical consequences, including a variant of the Goldbach Conjecture.
    • Also in 1967 Linnik published Lecons sur les problemes de statistique analytique which came about as the result of a series of lectures he had given in the previous year at the Institut de Statistique of the University of Paris.

  364. John Walsh (1786-1847)
    • However this may have been, Mr Walsh was for a series of years engaged in a constant endeavour to induce the principal learned societies of Europe to print his communications.
    • This merely contains a series of definitions and axioms, etc., beginning with the 'doctrine of ratio'.

  365. Vladimir Aleksandrovich Marchenko (1922-)
    • Continuing his studies at Kharkov he was awarded his candidates degree (equivalent to a Ph.D.), advised by Naum Samoilovich Landkof who had been a student of Mikhail Alekseevich Lavrent'ev, in January 1948 after defending his thesis Summation methods for generalized Fourier series.
    • He was awarded the Lenin Prize in 1962 for his series of papers on inverse problems of spectral analysis.

  366. Theodore Samuel Motzkin (1908-1970)
    • Motzkin's first publication, however, was not on linear programming but rather on power series.
    • Both linear programming and power series were themes which ran through Motzkin's research throughout his life but he was an extremely broad mathematician and there were many other themes.

  367. Hillel Furstenberg (1935-)
    • while still an undergraduate, I was exposed to a series of high-level lectures in advanced topics given by prominent professors who visited [Yeshiva College] from a number of institutions.
    • In Jerusalem, which was the centre of his academic activities, he continued producing a long series of monumental mathematical works.

  368. Leonid Vital'evich Kantorovich (1912-1986)
    • Kantorovich spoke on 25 June in the session on 'Theory of functions and theory of series' chaired by Dmitrii Evgenevich Menshov.
    • As Herbert E Scarf remarks [',' H E Scarf, The 1975 Nobel Prize in Economics: Resource Allocation, Science, New Series 190 (4215) (1975), 649; 710; 712.','71]:- .

  369. Aurel Wintner (1903-1958)
    • Hill's methods used infinite matrices and series expansions which he assumed were convergent but gave no proof.
    • These were Lectures on asymptotic distributions and infinite convolutions (1938), Analytical foundations of celestial mechanics (1941), Eratosthenian averages (1943), Theory of measure in arithmetical semigroups (1944), The Fourier transforms of probability distributions (1947), and An arithmetical approach to ordinary Fourier series (1945).

  370. Charles Merrifield (1827-1884)
    • He was editor of a series of scientific textbooks and he contributed to the series the volume Technical Arithmetic and Mensuration (1870).

  371. Rex Tims (1926-1971)
    • The usual topics-limits, infinite series, continuity, differentiability (or derivability as the authors call it), definite integral, logarithmic and exponential functions, integration techniques, improper integrals, Taylor's theorem and real power series-are included.

  372. Wilhelm Weber (1804-1891)
    • He published a series of papers on this topic between 1828 and 1830 in Annalen der Physik und Chemie [',' A E Woodruff, Biography in Dictionary of Scientific Biography (New York 1970-1990).','1]:- .
    • In 1848 a series of republican revolts against European monarchies spread through France, Germany, Italy, and the Austrian Empire.

  373. Mary Everest Boole (1832-1916)
    • series of talks to a group of London mothers who, finding their religion threatened by Darwin's new theories, sought Mrs Boole's philosophic wisdom.
    • At the age of 50, she began writing a series of books and articles about this, publishing them regularly until the time of her death in 1916 aged 84.

  374. Daniel O'Connell (1896-1982)
    • While in Australia O'Connell presented several radio talks, including a series of three talks with the title 'According to Hoyle' on the Australian Broadcasting Commission station 2BL-2NC, broadcast in March and April 1952.
    • His own research on eclipsing binaries and variable stars led to a long series of research papers.

  375. Pascual Jordan (1902-1980)
    • My father used to read popular books about natural science especially the series of 'Kosmos' books which was popular at the beginning of this century.
    • Each year several books came out in this series, written by people such as Eilhelm Bolsche who was well-known as a popular author.

  376. Jean-Claude Bouquet (1819-1885)
    • With Briot he worked from 1853 onwards on deep studies of Cauchy's ideas of analysis and produced many fundamental papers on series expansions of functions and on elliptic functions.
    • In 1853 they established conditions for a function to be expandable into an entire series in their important paper Note sur le developpement des fonctions en series convergentes, ordonnees suivant les puissances croissantes de la variable Ⓣ.

  377. Jacqueline Ferrand (1918-2014)
    • Volume II covered series, elementary functions of a complex variable, and elementary measure and integration.
    • Volume III covered multivariable integral calculus, further topics in functions of a complex variable, Fourier series and ordinary differential equations.

  378. Gerard Mercator (1512-1594)
    • His 'atlas' continued with a further series of maps of France, Germany and the Netherlands in 1585.
    • Although the project was never completed Mercator did publish a further series in 1589 including maps to the Balkans (then called Sclavonia) and Greece.

  379. Ron Book (1937-1997)
    • He was editor of three monograph series, in particular being editor-in-chief of the series Progress in Theoretical Computer Science.

  380. Francesco Severi (1879-1961)
    • His doctoral thesis, Sopra alcune singolarita delle curve di un iperspazio Ⓣ, together with a series of other papers which he published had published while an undergraduate, deal with enumerative geometry, a subject which had been started by Hermann Schubert.
    • In some special cases I suggested the definition of the characteristic series of a continuous system to Severi.

  381. Arthur Lee Dixon (1867-1955)
    • In 1908 Dixon began a series of publications on algebraic eliminants, carrying the subject forward from the point where Cayley had left it.
    • In the latter part of his career, Dixon published a series of around twelve joint papers with W L Ferrar on analytic number theory, summation formulae, Bessel functions and other topics in analysis.

  382. Roger Paman (about 1688-1748)
    • A "radical Quantity" is close to what today would be called a "variable", even though Paman implies that an expression can be composed of this variable only by taking powers of it, and by multiplicating by scalars, which means that Paman is thinking of polynomials or power series.
    • But we would not be able to define the derivative using Paman's terms since we consider more complicated functions than the polynomials or power series which Paman considered.

  383. Heinrich Weber (1842-1913)
    • Also, the existence of primes in arithmetic progressions, using L-series, is included.
    • Kleine Ausgabe in einem Bande, by Heinrich Weber, Science, New Series 38 (981) (1913), 550-551.','11]:- .

  384. André Weil (1906-1998)
    • He gave a talk to this seminar on domains of convergence of power series in several complex variables.
    • Weil made a major contribution through his books that include Arithmetique et geometrie sur les varietes algebriques Ⓣ (1935), Sur les espaces a structure uniforme et sur la topologie generale Ⓣ (1937), L'integration dans les groupes topologiques et ses applications Ⓣ (1940), Foundations of Algebraic Geometry (1946), Sur les courbes algebriques et les varietes qui s'en deduisent Ⓣ (1948), Varietes abeliennes et courbes algebriques Ⓣ (1948), Introduction a l'etude des varietes kahleriennes Ⓣ (1958), Discontinuous subgroups of classical groups (1958), Adeles and algebraic groups (1961), Basic number theory (1967), Dirichlet Series and Automorphic Forms (1971), Essais historiques sur la theorie des nombres Ⓣ (1975), Elliptic Functions According to Eisenstein and Kronecker (1976), (with Maxwell Rosenlicht) Number Theory for Beginners (1979), Adeles and Algebraic Groups (1982), Number Theory: An Approach Through History From Hammurapi to Legendre (1984), and Correspondance entre Henri Cartan et Andre Weil Ⓣ (1928-1991) (2011).

  385. A Adrian Albert (1905-1972)
    • In addition he was beset by a series of illnesses ..
    • These matrices arise in the theory of complex manifolds and Albert went on to write an important series of papers on these questions over the following years.

  386. René Eugčne Gateaux (1889-1914)
    • Volterra himself, invited by Borel and Hadamard, came to Paris to give a series of lectures on functional analysis, published in 1913 ([',' Vito Volterra, Lecons sur les fonctions de lignes (Gauthier-Villars, Paris, 1913).','23]) and whose redaction was precisely made by Peres.
    • Nobody could predict the development and the range that this new series of researches could have attained.

  387. Giorgio Bidone (1781-1839)
    • His research at this time was on the solution of transcendental equations and also on definite integrals with papers such as Sur diverses integrals definies Ⓣ (1813), in which he used the method of Mascheroni series to reduce various integrals to known cases, and Sur les transcendantes elliptiques Ⓣ (1818) in which he extended the work of Legendre on the numerical values of elliptic functions of the first and second kind.
    • Another example of his use of interplay with theory and experiment is in his memoir Sur cause des ricochets que font les pierres et les boulets de canon, Lances obliquement sur la surface de l'eau Ⓣ (1813), where, through a long series of experiments conducted and interpreted with ingenuity, skill and acumen, he displayed an "analytical theory" of the phenomenon which at that time intrigued both physicists and mathematicians.

  388. Issai Schur (1875-1941)
    • In a series of papers he introduced the concept now known as the 'Schur multiplier'.
    • Fifth, in divergent series; .

  389. John Thompson (1932-)
    • The nonabelian finite simple groups fall into a small number of infinite series and 26 sporadic groups.
    • He gave a series on lectures on Galois groups at that meeting.

  390. William Whyburn (1901-1972)
    • However, Whyburn had already published three papers before submitting his doctoral thesis, namely An extension of the definition of the Green's function in one dimension (1924), On the Green's function for systems of differential equations (1927) and On the polynomial convergents of power series (1927).
    • We should marvel that, with such a limited amount of research time over his four years at the University of Texas, not only did he write a thesis but also the series of major papers we mentioned above.

  391. Carl Siegel (1896-1981)
    • In this general area Siegel considered the theory of discontinuous groups and their fundamental domains, algebraic relations between modular functions and between modular forms, and Fourier series of modular forms.
    • He examined Birkhoff's work on perturbation theory solutions for analytical Hamiltonian differential equations near an equilibrium point using formal power series.

  392. Israil Gelfand (1913-2009)
    • He saw the importance of the work of Sobolev and Schwartz on the theory of generalised functions and distributions, and he developed this theory in a series of monographs.
    • Between 1968 and 1972 Gelfand produced a series of important papers on the cohomology of infinite dimensional Lie algebras.

  393. John Scott Russell (1808-1882)
    • These reports, in fact all Russell's own work, contain a remarkable series of observations, at sea, in rivers and canals, and in Russell's own wave tank constructed for the purpose.
    • In M Lakshmanan, Solitons, Springer Series in Nonlinear Dynamics, (New York, 1988) 150-281.','4] and [',' R K Bullough and P J Caudrey, Solitons and the Korteweg-de Vries Equation: Integrable Systems in 1834-1995.

  394. Abraham Wald (1902-1950)
    • During this period Wald published 10 papers on economics and econometrics, and he also published an important monograph in 1936 on seasonal movements in time series.
    • seasonal corrections to time series, approximate formulas for economic index numbers, indifference surfaces, the existence and uniqueness of solutions of extended forms of the Walrasian system of equations of production, the Cournot duopoly problem, and finally, in his much used work written with Mann (1943), stochastic difference equations.

  395. David Gregory (1659-1708)
    • Gregory himself published Exercitatio geometria de dimensione curvarum in 1684 while at Edinburgh which was an interesting work developing his uncle's work on infinite series.
    • Gregory sent Newton a copy of his paper on infinite series, taking care to offer extensive praise to Newton.

  396. Horatio Carslaw (1870-1954)
    • an enthusiast in original research, and having studied the mathematical papers and memoirs bearing on Fourier's series and their application in mathematical physics, purposes writing a book on the subject.
    • The second book was Introduction to the theory of Fourier's series and integrals and the mathematical theory of the conduction of heat.

  397. Nikoloz Muskhelishvili (1891-1976)
    • Muskhelishvili then pursued systematic investigations on this subject and published a series of articles devoted to various boundary problems of the plane theory of elasticity and to other problems of mathematical physics.
    • In a series of articles Mushkelishvili worked out a method of solving a variety of boundary problems for analytic functions of a single complex variable.

  398. Heinrich Jung (1876-1953)
    • The analytic approach, followed in the present book, uses as fundamental concept that of a place of the field K, a place being an isomorphism of K into the quotient field of the ring of convergent power series in two variables (the uniformizing variables at the place) with the requirement that distinct couples of values of the variables u, v in these power series sufficiently near to (0, 0) should lead to distinct values for some function of the field.

  399. Hitoshi Kumano-Go (1935-1982)
    • During these years Kumano-Go published a series of papers which studied the local and global uniqueness of the solutions of the Cauchy problem for partial differential equations.
    • In addition to his work on pseudo-differential operators, Kumano-Go published a series of papers on the product of Fourier integral operators.

  400. J E Littlewood (1885-1977)
    • For 35 years he collaborated with G H Hardy working on the theory of series, the Riemann zeta function, inequalities, and the theory of functions.
    • The collaboration led to a series of papers Partitio numerorum using the Hardy-Littlewood-Ramanujan analytical method.

  401. André Lichnerowicz (1915-1998)
    • He had suggested that what was needed was a series of books on mathematical physics.
    • The second surveys the theory of exterior differential forms, the general form of Stokes's theorem and its specialization to two and three dimensions, orthogonal series, Fourier integrals, bounded linear operators in Hilbert space, and the classical theory of integral equations for L2 integrable kernels.

  402. Sergei Chernikov (1912-1987)
    • In Chernikov's articles, therefore, a series of geometrically obvious properties of linear inequalities is given in analytic form that is more convenient for the use of machine techniques.
    • A series of papers by Chernikov in the 1960s studied polyhedrally closed systems, special types of infinite systems of linear inequalities [',' I I Eremin, D I Zaitsev, M I Kargapolov and V S Charin, Sergei Nikolaevich Chernikov (on his sixtieth birthday) (Russian), Uspekhi Mat.

  403. Martin Ohm (1792-1872)
    • One finds contradictions of the theory of "opposed magnitudes;" - another is merely disquieted by "imaginary quantities;" - a third finally meets difficulties in "infinite series," either because Euler and other distinguished mathematicians have applied them with success in a divergent form, while the complainant thinks himself convinced that their convergence is a fundamental condition, - or because in general investigations general series occur, which, precisely because they are general, can be neither accounted divergent nor convergent.

  404. Chris Zeeman (1925-2016)
    • He was Gresham Professor of Geometry from 1988 to 1994, delivering an annual series of public lectures.
    • Naming the award for Zeeman was particularly appropriate for a number of reasons: he was first mathematician to deliver the Royal Institution's Christmas Lectures; and his 'Mathematics into Pictures' series is recognised as significantly influencing to young mathematicians.

  405. Emil Artin (1898-1962)
    • He defined a new type of L-series, which generalised Dirichlet's L-series, yet was quite different in nature.

  406. Robert Boyle (1627-1691)
    • The apparatus had been designed by Hooke and using it Boyle had discovered a whole series of important facts.
    • both a penetrating critique of Pascal's work on hydrostatics, full of acute observations upon Pascal's experimental method, and a presentation of a series of important and ingenious experiments on fluid pressure.

  407. Vito Volterra (1860-1940)
    • In 1909 he gave a series of lectures at Clark University in Massachusetts, United States, and received an honorary degree from the university as part of its 20th anniversary celebrations.
    • The theory of functionals as a generalization of the idea of a function of several independent variables was developed by Volterra in a series of papers published since 1887 and was inspired by the problems of the calculus of variations.

  408. Bibhutibhushan Datta (1888-1958)
    • The second part, we are told, is devoted to algebra and the third part contains the history of geometry, trigonometry, calculus and various other topics such as magic squares, theory of series and permutations and combinations.
    • Beginning in 1980 Kripa Shankar Shukla revised the material of the third volume and published it as a series of nine papers between 1980 and 1993.

  409. Frederick Mosteller (1916-2006)
    • Series A (General) 119 (1) (1956), 87.','38]:- .
    • Series C (Applied Statistics) 28 (2) (1979), 177-178.','24]:- .

  410. Charles Eugčne Delaunay (1816-1872)
    • Delaunay found the longitude, latitude and parallax of the Moon as infinite series.
    • These gave results correct to 1 second of arc but were not too practical as the series converged slowly.

  411. Spiru Haret (1851-1912)
    • Taking also into account commensurabilities, and using generalized Fourier series (which generate quasiperiodic solutions), Poincare proved the divergence of these series, which means instability, confirming in this way Haret's result.

  412. Nicolai Vasilievich Bugaev (1837-1903)
    • He wrote a Master's thesis on convergence of infinite series at the University of Moscow which he submitted and successfully defended in 1863.
    • (6) (1963), 71-78.','8] is concerned with the subsequent development of ideas on general tests of convergence of infinite series contained in Bugaev's thesis.

  413. Ian Sneddon (1919-2000)
    • The first was Special functions of mathematical physics and chemistry published in the Oliver and Boyd series in 1956.
    • It was, as all the Oliver and Boyd series books, sold at a price a student could afford and it provided a straightforward account of the topic in a short but very clear style.

  414. Mikhail Mikhailovich Postnikov (1927-2004)
    • We have changed the chronological sequence of Postnikov's publications slightly to list finally the six textbooks which he wrote corresponding to six series of lectures on geometry given to undergraduate and graduate students at Moscow State University.
    • This volume is the sixth in Postnikov's series of lecture notes in differential geometry, and provides an advanced overview of various topics in Riemannian geometry.

  415. Dimitrei D Stancu (1927-2014)
    • Stancu published a series of papers related to the work of his doctoral thesis in 1956-57.
    • Some other papers in this series were On polynomial interpolation formulas for functions of several variables (Romanian) (1957), Generalisation of some interpolation formulas for functions of several variables and certain thoughts on the numerical integration formula of Gauss (Romanian) (1957), A generalisation of the Gauss-Christoffel quadrature formula (Romanian) (1957), Generalisation of certain interpolation formulas for functions of several variables (Romanian) (1957), Sur une classe de polinomes orthogonaux et sur des formules generales de quadrature a nombre minimum de termes Ⓣ (1957), Contributions to the numerical integration of functions of several variables (Romanian) (1957), and On the Hermite interpolation formula and on some of its applications (Romanian) (1957).

  416. Hjalmar Mellin (1854-1933)
    • He applied this technique systematically in a long series of papers to the study of the gamma function, hypergeometric functions, Dirichlet series, the Riemann zeta function and related number-theoretic functions.

  417. David van Dantzig (1900-1959)
    • His most important work at this time was in topological algebra and in addition to his doctoral thesis, he wrote a whole series of papers on this topic.
    • For example in the paper [',' J A Schouten and D van Dantzig, On Projective Connexions and their Application to the General Field-Theory, Annals of Mathematics, Second Series 34 (2) (1933), 271-312.','22] published in 1933, van Dantzig and Schouten write in the introduction:- .

  418. Hermann Hankel (1839-1873)
    • He also studied functions, now named Hankel functions or Bessel functions of the third kind, in a series of papers which appeared in Mathematische Annalen.
    • In the same way that he saw the importance of Grassmann's work, Hankel also must have considerable credit for seeing the importance of Bolzano's work on infinite series.

  419. John Kingman (1939-)
    • He wrote on the theory of Markov processes and published an important series of articles on Markov transition probabilities which we gave details of above.
    • In 1979 Kingman gave a series of lectures at Iowa State University on the contributions of mathematics to the study of genetic evolution.

  420. Jacques Hadamard (1865-1963)
    • Already at this stage he began to undertake research, investigating the problem of finding an estimate for the determinant generated by coefficients of a power series.
    • Hadamard obtained his doctorate in 1892 for a thesis on functions defined by Taylor series.

  421. George G Lorentz (1910-2006)
    • While in Kislovodsk he found Antoni Zygmund's Trigonometrical series in the library of a small nearby college - the book proved significant in his mathematical development.
    • I wrote some 20 papers: joint papers with Kamke and Knopp, papers related to differential equations, papers on summability, on Fourier series, and papers where rearrangements play a role.

  422. Masayoshi Nagata (1927-2008)
    • A series of papers in the late 1950s on algebraic geometry over Dedekind domains laid the foundation for later developments of algebraic geometry in terms of schemes.
    • The concept of the Henselization of rings, developed in a series of papers in the 1950s, turned out to be fundamental for algebraic spaces and etale topology.

  423. Thomas Stieltjes (1856-1894)
    • He received his doctorate of science in 1886 for a thesis on asymptotic series.
    • Also important is his work on divergent series and discontinuous functions.

  424. Erich Hecke (1887-1947)
    • Schoeneberg describes Hecke's contributions to a number of topics which he lists as follows: Hilbert modular functions, Dedekind zeta functions, arithmetical notions and methods, elliptic modular forms of level N, algebraic functions, Dirichlet series with functional equation, Hecke-operators Tn, and physics where he made contributions to the kinetic theory of gases.
    • He then introduced the new concept of "Grossencharakter" and the corresponding L-series, to which he extended the properties of analytic continuation he had proved for the zeta functions in 1917.

  425. Clifford Ambrose Truesdell III (1919-2000)
    • He develops a formal method for deriving both 'membrane' and 'bending' theory simultaneously from a common expansion, and he studies the singular state at the apex of a cone, relying mainly on the tools of Fourier expansion and formal manipulation of power series.
    • In its use of formal methods, its reliance on special kinematic hypotheses (membranes of revolution only) and its presentation of series solutions of a great many special problems, this work can be considered properly isolated from his other work.

  426. Hyman Bass (1932-)
    • Kaplansky gave an inspiring series of courses on homological methods in commutative algebra.
    • Bass produced a series of papers during his first years at Columbia, for example Finitistic dimension and a homological generalization of semiprimary rings (1960), Projective modules over algebras (1961), Injective dimension in noetherian rings (1962), and Torsion free and projective modules (1962).

  427. David Eugene Smith (1860-1944)
    • Five years later he wrote in [',' D E Smith, The David Eugene Smith Gift of Historical-Mathematical Instruments to Columbia University, Science, New Series 83 (2158) (1936), 79-80.','17]:- .
    • In 1936 he donated his collection of mathematical instruments to Columbia University [',' D E Smith, The David Eugene Smith Gift of Historical-Mathematical Instruments to Columbia University, Science, New Series 83 (2158) (1936), 79-80.','17]:- .

  428. Enrico Magenes (1923-2010)
    • At our invitation Lions came to Genoa in April 1958 to give a series of lectures on mixed problems in the sense of Hadamard ..
    • In a remarkable series of papers, followed and made complete in a three-volume book in cooperation with J L Lions (Nonhomogeneous Boundary Value Problems and Applications), he set the foundations for the modern treatment of partial differential equations, and in particular the ones mostly used in applications.

  429. Leopold Vietoris (1891-2002)
    • After the war he worked on a number of different areas including statistics, publishing a series of four papers entitled Vergleich unbekannter Mittelwerte auf Grund von Versuchsreihen Ⓣ between 1979 and 1982.
    • He continued to publish papers, his last paper being the third in a series entitled Uber das Vorzeichen gewisser trigonometrischer Summen Ⓣ(1994) which appeared when he was 103 years old.

  430. David Hayes (1937-2011)
    • The singular series associated with a given additive problem is interpreted as a certain Radon-Nikodym derivative.
    • We mentioned above that Hayes spent time at Harvard University in the autumn of 1999 and while he was there he delivered a series of lectures on the Stark conjectures.

  431. Alessandro Padoa (1868-1937)
    • Beginning in 1898 he gave a series of lectures at the Universities of Brussels (1898), Pavia (1899), Rome (1900, 1901, 1903), Padua (1905), Cagliari (1907) and Geneva (1911).
    • M Padoa - previously my distinguished student and now my colleague and my friend - has given to this subject, since 1898, a series of well-attended conferences in the Universities of Brussels, Pavia, Rome, Padua, Cagliari and Geneva, and has presented highly regarded papers to the Congresses of philosophy and mathematics in Paris, Livorno, Parma, Padua and Bologna.

  432. Yakov Grigorevich Sinai (1935-)
    • Sinai has also been invited to give many prestigious lectures or lecture courses including: Loeb Lecturer, Harvard University (1978); Plenary Speaker at the International Congress on Mathematical Physics in Berlin (1981); Plenary Speaker at the International Congress on Mathematical Physics in Marseilles (1986); Distinguished Lecturer, Israel (1989); Solomon Lefschetz Lectures, Mexico (1990); Plenary Speaker at the International Congress of Mathematicians, Kyoto (1990); Landau Lectures, Hebrew University of Jerusalem (1993); Plenary Speaker at the First Latin American Congress in Mathematics (2000); Plenary Speaker at the American Mathematical Society Meeting "Challenges in Mathematics" (2000); Andreevski Lectures, Berlin, Germany (2001); Bowen Lectures, University of California at Berkeley (2001); Leonidas Alaoglu Memorial Lecture, California Institute of Technology (2002); Joseph Fels Ritt Lectures, Columbia University (2004); Leonardo da Vinci Lecture, Milan, Italy (2006); Galileo Chair, Pisa, Italy (2006); John T Lewis Lecture Series, Dublin Institute for Advanced Studies and the Hamilton Mathematics Institute, Trinity College, Dublin, Ireland (2007); and Milton Brockett Porter Lecture Series, Rice University, Houston, Texas (2007).

  433. Daniel Rutherford (1906-1966)
    • Rutherford wrote several other texts in the Oliver and Boyd series which he set up with Aitken.
    • Outstanding research contributions led to Rutherford being elected a fellow of the Royal Society of Edinburgh in 1934 and he received the Keith Prize from the Society for an outstanding series of papers he published in 1951-53.

  434. Andrei Tikhonov (1906-1993)
    • After a series of fundamental papers introducing the topic, the work was carried on by his students.
    • In the 1960s Tikhonov began to produce an important series of papers on ill-posed problems.

  435. Ernest William Barnes (1874-1953)
    • Barnes' episcopate was marked by a series of controversies stemming from his outspoken views and, rather surprisingly for someone who held such high office in the Church, often unorthodox religious beliefs.
    • Barnes next turned his attention to the theory of integral functions, where, in a series of papers, he investigated their asymptotic structure.

  436. Julius Petersen (1839-1910)
    • He wrote a series of school and undergraduate texts which achieved international acclaim despite being too difficult for all but the ablest pupils.
    • He wrote a series of textbooks based on courses he had given at the College of Technology: one on plane geometry in 1877; one on statics in 1881; one on kinematics in 1884; and one on dynamics in 1887.

  437. Christoph Scheiner (1573-1650)
    • Scheiner had drawn a series of diagrams showing sunspots on various days, made by putting dark coloured glass in front of his telescope.
    • This halo, now known as Scheiner's halo, is thought to be caused by refraction of sunlight by certain ice crystals in the earth's atmosphere, see [',' E Whalley, Scheiner’s Halo: Evidence for Ice Ic in the Atmosphere, Science, New Series 211 (4480) (1981), 389-390.','21].

  438. Thomas Cherry (1898-1966)
    • Since the author uses the solutions of Chaplygin, in the form of an infinite series of hypergeometric functions, of the linear second order partial differential equation in the hodograph variables of the potential function, this series diverges for values of the velocity whose speeds exceed the speed at infinity.

  439. Edward Titchmarsh (1899-1963)
    • He studied Fourier series and Fourier integrals writing Introduction to the Theory of Fourier Integrals (1937).
    • From 1939 Titchmarsh concentrated on the theory of series expansions of eigenfunctions of differential equations, work which helped to resolve problems in quantum mechanics.

  440. Aleksei Krylov (1863-1945)
    • In a paper on forced vibrations of fixed-section pivots (1905), he presented an original development of Fourier's method for solving boundary value problems, pointing out its applicability to a series of important questions: for example, the theory of steam-driven machine indicators, the measurement of gas pressure in the conduit of an instrument, and the twisting vibrations of a roller with a flywheel on its end.
    • He studied the acceleration of convergence of Fourier series in a paper in 1912, and studied the approximate solutions to differential equations in a paper published in 1917.

  441. Robert Fricke (1861-1930)
    • The authors of [',' D Mumford, C Series and D Wright, Indra’s pearls: the vision of Felix Klein (Cambridge University Press, Cambridge, 2002).','2] write:- .
    • The present volume is the first of a series of three which Dr Fricke proposes to write on the elliptic functions and their applications.

  442. Alfredo Capelli (1855-1910)
    • It is a remarkable contribution which contains a proof of Sylow's theorems and of Jordan's theorem on composition series.
    • 11 (2) (1991), 25-54.','3] discuss why Capelli was unaware of Sylow's results and why Burnside and other early 20th century authors were unaware of Capelli's results about nilpotent groups and composition series.

  443. Charles Coulson (1910-1974)
    • A mathematical account of the common types of wave motion in the Oliver and Boyd series.
    • He had maintained his links with the St Andrews applied mathematicians and he published another text in the Oliver and Boyd series, this one first appearing in 1948.

  444. J Willard Gibbs (1839-1903)
    • A series of five papers by Gibbs on the electromagnetic theory of light were published between 1882 and 1889.
    • The American Mathematical Society named a lecture series in honour of Gibbs.

  445. Douglas Munn (1929-2008)
    • A series of papers between 1966 and 1973 exploring these ideas gave rise to results that are now regarded as classical.
    • His discovery of Passman's books on infinite group rings brought about a further change in the main thrust of his work, and in the eighties, while still writing the occasional paper on pure semigroup theory, he returned to the study of semigroup algebras, publishing a series of remarkable papers linking semigroup properties to ring-theoretic properties of their algebras.

  446. Ronald Mitchell (1921-2007)
    • A joint Mitchell/Fairweather paper, published in 1964, was the first in a series on high order alternating direction finite difference methods for elliptic PDE's.
    • Another conference was held in Dundee in 1969, and while in fact the first of what was to become a long series of biennial meetings associated with Dundee, the two earlier St Andrews meetings have quite properly been included and so have pushed it into third place.

  447. Harald Cramér (1893-1985)
    • Also influenced by G H Hardy, Cramer's research resulted in the award of a PhD in 1917 for his thesis On a class of Dirichlet series.
    • He began to produce a series of papers on analytic number theory, and he addressed the Scandinavian Congress of Mathematicians in 1922 on Contributions to the analytic theory of numbers detailing his work on the topic up to that time.

  448. John Herschel (1792-1871)
    • He studied algebras and published papers on trigonometrical series.
    • I am going under my father's directions, to take up the series of his observations where he has left them (for he has now pretty well given over regular observing) and continuing his scrutiny of the heavens with powerful telescopes ..

  449. Nikolai Luzin (1883-1950)
    • I don't know how it happened, but I cannot be satisfied any more with analytic functions and Taylor series ..
    • He returned to Moscow in 1914 and he completed his thesis The integral and trigonometric series which he submitted in 1915.

  450. Paul Bernays (1888-1977)
    • Between 1937 and 1954 Bernays wrote a whole series of articles in the Journal of Symbolic Logic which attempted to achieve this goal.
    • But it is not immediately obvious that this idea of the infinite number series can be realised; the intellectual experience of its successful realisation is then essential for developing a feeling of familiarity, even of obviousness, as acquired evidence.

  451. Henrietta Swan Leavitt (1868-1921)
    • In order to furnish material for determining their periods, a series of sixteen plates, having exposures from two to four hours, was taken with the Bruce Telescope the following autumn.
    • A straight line can readily be drawn among each of the two series of points corresponding to maxima and minima, thus showing that there is a simple relation between the brightness of the variables and their periods.

  452. Richard von Mises (1883-1953)
    • Ostrowski wrote in a 1965 lecture (see for example [',' R Siegmund-Schultze, Hilda Geiringer von Mises, Charlier Series, Ideology, and the human side of the emancipation of applied mathematics at the University of Berlin during the 1920s, Historia Mathematica 20 (1993), 364-381.','16]):- .
    • Ostrowski in the same lecture which we quoted from above wrote (see for example [',' R Siegmund-Schultze, Hilda Geiringer von Mises, Charlier Series, Ideology, and the human side of the emancipation of applied mathematics at the University of Berlin during the 1920s, Historia Mathematica 20 (1993), 364-381.','16]):- .

  453. John Airey (1868-1937)
    • London, Series A, 94 (661) (1918), 307-314.
    • London, Series A, 96 (674) (1919), 1-8.

  454. Nicolas Vilant (1737-1807)
    • Plagued by ill-health, he was unable to teach for much of this time, and employed a series of assistants.
    • As a private arrangement, he employed a series of assistants to teach in his stead; but he gave occasional lectures, supervised the work of his assistants and took an interest in the best students.

  455. Giuseppe Bagnera (1865-1927)
    • He obtained his laurea in civil engineering in 1890 but by this stage he already had two mathematics publications: Sopra i determinanti che si possono formare con gli stessi elementi Ⓣ (1887), and Sur une propriete des series simplement convergentes Ⓣ (1888).
    • Bagnera began his scientific work with a series of works marked by the elegant simplicity which was the essential character of all his work.
    • During these years Michele de Franchis, who was also a Sicilian, was professor of projective and descriptive geometry at the University of Parma, and Bagnera and de Franchis collaborated on an outstanding series of papers on the theory of hyperelliptic surfaces.

  456. Max Zorn (1906-1993)
    • Or we may prescribe a seemingly much more powerful condition, namely, that the function possesses a development into (abstract) power series about each point of the domain of definition.
    • For it turns out that only a very weak continuity property has to be added to the existence of the Gateaux differential in order to ensure the existence of the power series development called for by the second definition.

  457. Edvard Phragmén (1863-1937)
    • The deadline for submissions was 1 June 1888 and there were four questions, the first of which involved the stability of the solar system and asked for series which describe the motion of bodies in a generalised 3-body problem.
    • Here the problem is more special, lying within the theory of uniform convergence of trigonometric series.

  458. Athanase Dupré (1808-1869)
    • During the 1860s Dupre published a series of memoirs on the mechanical theory of heat in the Annales de Chimie et de Physique.
    • Since the end of the year 1859 I have presented to the Academy a series of memoirs on the 'Theorie mecanique de la chaleur' almost all of which were published in the 'Annales de Chimie et de Physique'.

  459. Zdzisaw Pawlak (1926-2006)
    • The aim of the present note is to show that a well known electronic element of digital computers, the flip-flop, may be used for generating a series of random binary digits with equal probabilities.
    • Based on Professor Pawlak's ideas, an Electronic Digital Machine and, later (after Pawlak's transfer to the Mathematical Institute of Polish Academy of Sciences), a prototype (1960) and five machines of the test series of Universal Digital Machine (UMC-1) were built.

  460. Witold Hurewicz (1904-1956)
    • To do this he had to produce new techniques and he began to publish his results in a series of papers.
    • He gave a series of lectures at Brown University in 1943 and these were published in mimeographed form by Brown University as Ordinary differential equations in the real domain with emphasis on geometric method.

  461. Michel Plancherel (1885-1967)
    • In a series of articles he generalized results in the classical Fourier theory to more general spaces (Hilbert spaces) by investigating various orthonormal systems of functions, their summability and the representation of functions in such systems by Fourier series and Fourier integrals and more general integral transformations.

  462. Louis Arbogast (1759-1803)
    • Essentially he realised that there was no rigorous methods to deal with the convergence of series.
    • The formal algebraic manipulation of series investigated by Lagrange and Laplace in the 1770s was put in the form of operator equalities by Arbogast in 1800 in Calcul des derivations.

  463. Sydney Goldstein (1903-1989)
    • During his career, Goldstein gave many lectures series, some of which were written up and published.
    • At the Summer Seminar at Boulder, Colorado in 1957 he gave a series of lectures which were written up as 300-page book.

  464. Gottfried Leibniz (1646-1716)
    • On Huygens' advice, Leibniz read Saint-Vincent's work on summing series and made some discoveries of his own in this area.
    • While explaining his results on series to Pell, he was told that these were to be found in a book by Mouton.

  465. Kunihiko Kodaira (1915-1997)
    • In 1979 he published the five volume Introduction to analysis in Japanese covering real numbers, functions, differentiation, integration, infinite series, functions of several variables, curves and surfaces, Fourier series, Fourier transforms, ordinary differential equations, and distributions.

  466. Francesco Brioschi (1824-1897)
    • Brioschi used the findings of a series of major projects or participated in the projects' development - for example, in the regulation of the Po and Tiber ..
    • He made fundamental contributions in 1857 to changing the Annali di Scienze Matematiche e Fisiche into a journal of international standing, in 1886 to the new series of the journal Politecnico, to an Italian edition of Euclid's Elements for secondary education, and he edited Leonardo da Vinci's Codice Atlantico Ⓣ which was a major contribution to the understanding of the history of science and technology.

  467. Fibonacci (1170-1250)
    • There are also problems involving perfect numbers, problems involving the Chinese remainder theorem and problems involving summing arithmetic and geometric series.
    • For unity is a square and from it is produced the first square, namely 1; adding 3 to this makes the second square, namely 4, whose root is 2; if to this sum is added a third odd number, namely 5, the third square will be produced, namely 9, whose root is 3; and so the sequence and series of square numbers always rise through the regular addition of odd numbers.

  468. John William Strutt (1842-1919)
    • Turning my attention to nitrogen, I made a series of determinations ..
    • Having obtained a series of concordant observations on gas thus prepared I was at first disposed to consider the work on nitrogen as finished.

  469. Paul Lévy (1886-1971)
    • He chose to attend the Ecole Polytechnique and he while still an undergraduate there published his first paper on semiconvergent series in 1905.
    • Not only did Levy contribute to probability and functional analysis but he also worked on partial differential equations and series.

  470. Iossif Vladimirovich Ostrovskii (1934-)
    • In the 2000s he wrote On the zeros of tails of power series (2000) and On the zero distribution of remainders of entire power series (2001).

  471. Herman Goldstine (1913-2004)
    • They produced a series of reports on the EDVAC (Electronic Differential Variable Computer) which changed the whole concept of computers.
    • He did not undertake research solely on computers and their applications, however, for he published a series of three papers on Hilbert space with non-associative scalars (1962, 1964, 1966).

  472. Tom Whiteside (1932-2008)
    • An abundance of references direct the reader either to the contemporary literature or to the previous volumes of this series where related problems have already been treated.
    • This, however, did not end up in a series of major publications as his work on Newton had, rather he published only a few scraps.

  473. Iain Adamson (1928-2010)
    • In a recent series of articles and books, Menger has raised grave doubts about the meaning, if any, which can be attached to the phrase "a function f (x) of the real variable x".
    • It is perhaps a hard-headed rather than an exciting presentation, but the author, an ardent Scotsman writing in the Edinburgh series, would presumably wish to have it that way.

  474. Karl Maruhn (1904-1976)
    • In a collaboration with the algebraist Heinrich Grell (1903-1974) and the differential geometer and topologist Willi Ludwig August Rinow (1907-1979), he edited a series of books aimed at teaching mathematics to Gymnasium pupils and college students.
    • By the time of Maruhn's death in 1976 around 80 books had appeared in the series and it continued to produce texts until 1990.

  475. Nikolai Evgrafovich Kochin (1901-1944)
    • Great interest was aroused by the series of works by Vilhelm Bjerknes and his son Jacob, expounding a new theory of cyclone origin and development, which brought back the idea of atmospheric fronts.
    • This article was the start of a major series of works by Kochin, resulting in the creation of the linear theory of cyclogenesis in the early 1930s.

  476. Charles Babbage (1791-1871)
    • He wrote papers on several different mathematical topics over the next few years but none are particularly important and some, such as his work on infinite series, are clearly incorrect.
    • The drawings of the Analytical Engine have been made entirely at my own cost: I instituted a long series of experiments for the purpose of reducing the expense of its construction to limits which might be within the means I could myself afford to supply.

  477. Giovanni Cassini (1625-1712)
    • With these instruments Cassini made a series of new discoveries.
    • He published detailed series of observations of the moons of Jupiter in 1668.

  478. Charles Loewner (1893-1968)
    • He wrote a series of papers on this topic, culminating in one where he proved a special case of the Bieberbach conjecture in 1923.
    • given by the series .

  479. John Playfair (1748-1819)
    • Playfair's simple and eloquent style consisted of a series of chapters clearly stating the Huttonian theory, giving the facts to support it, and the arguments given against it.
    • The second volume was entirely devoted to astronomy, while a third volume, which was intended to complete the series and cover the subjects of optics, electricity, and magnetism, was never completed.

  480. Yulian Vasilievich Sokhotsky (1842-1927)
    • His doctoral dissertation On definite integrals and functions with applications to expansion of series was an early investigation of the theory of singular integral equations.
    • His work is important in the development of the theory of functions, in particular having applications in the theory of hypergeometric series and differential equations.

  481. Herman Chernoff (1923-)
    • This was in fact Chernoff's third paper since he published A note on the inversion of power series in 1947 which:- .
    • [treats] the multiplication of power series and their inversion by means of a movable strip of paper on which the coefficients are written.

  482. Friedrich Karl Schmidt (1901-1977)
    • Also in 1933, F-K Schmidt succeeded Richard Courant as editor of Springer-Verlag's famous "Yellow Series" of mathematical monographs when Courant was dismissed because he was Jewish.
    • F-K Schmidt was a Roman Catholic, and not Jewish, but he was quickly out of favour with the Nazis when he refused to remove Richard Courant's name from the title page of the Springer series.

  483. Frigyes Riesz (1880-1956)
    • A satisfactory theory of series of orthonormal functions only became possible after the invention of the Lebesgue integral and this theory was largely the work of Riesz.
    • He also studied orthonormal series and topology.

  484. Johan Ludwig Jensen (1859-1925)
    • Johan Ludwig Jensen's father was the sort of person who undertook a whole series of different projects yet, despite his good education and cultured style, the projects tended to end up financial failures.
    • He also studied infinite series, the gamma function and inequalities for convex functions.

  485. Federico Cafiero (1914-1980)
    • Alessandro Faedo had been appointed to succeed Leonida Tonelli at Pisa in 1946 and he began a series of inspired appointments, building Pisa into one of the leading mathematical centres in the world.
    • Together with his colleague Antonio Zitarosa, he founded a series devoted to works in mathematical analysis published by Liguori of Naples.

  486. William Tutte (1917-2002)
    • As a young mathematician and codebreaker, he deciphered a series of German military encryption codes known as FISH.
    • Among his books are: Connectivity in graphs published in 1966; Introduction to the theory of matroids (1971), based on a series of lectures given by Tutte at the Rand Corporation in 1965; Graph Theory (1984); and Graph Theory as I Have Known It (1998) which gives a fascinating account of how he discovered his many fundamental results.

  487. Erwin Hiebert (1919-2012)
    • First one by Arthur H Compton [',' A H Compton, Review: The Impact of Atomic Energy, by Erwin Hiebert, Science, New Series 134 (3486) (1961), 1231-1233.','3]:- .
    • The Harvard University obituary for Hiebert [',' A H Compton, Review: The Impact of Atomic Energy, by Erwin Hiebert, Science, New Series 134 (3486) (1961), 1231-1233.','3] states:- .

  488. William McCrea (1904-1999)
    • He proved conclusively that this was right in a paper which gave perhaps the first qualitative correct model of the solar atmosphere, with the currently accepted abundance of about three-quarters hydrogen and one-quarter helium by mass, and led to a series of equally authoritative papers on the atmospheres of other stars.
    • This was in the Oliver and Boyd series and McCrea writes in the Preface:- .

  489. Leonard Rogers (1862-1933)
    • Such names as Rogers-Ramanujan identities, Rogers-Ramanujan continued fractions and Rogers transformations are known in the theory of partitions, combinatorics and hypergeometric series.
    • [',' G H Hardy, Ramanujan (New York, 1940).','2], [',' G E Andrews, q-series: their Development and Application in Analysis, Number Theory, Combinatorics, Physics, and Computer Algebra, (Providence, 1986).','4], [',' G E Andrews, L J Rogers and the Rogers-Ramanujan identities, Math.

  490. George Salmon (1819-1904)
    • The first of these was his sermon Prayer published in 1849, which was followed by a series of publications of his sermons, for example many are collected in Sermons preached in Trinity College Chapel (1861) and Cathedral and University Sermons (1900).
    • He published much in the area of theology with works such as The eternity of future punishment (1864), The reign of law (1873), Non-miraculous Christianity (1881), Introduction to the New Testament (1885), The infallibility of the Church (1888), Thoughts on the textual criticism of the New Testament (1897), and a series of articles between 1877 and 1887 on the history of the early Christian Church in the Dictionary of Christian Biography.

  491. Robert Gillespie (1903-1977)
    • He began to publish a series of important papers in the Cambridge Philosophical Society and in the Proceedings of the Edinburgh Mathematical Society.
    • In this RSE obituary Gillespie's books Integration (Edinburgh, 1939) and Partial Differentiation (Edinburgh, 1951), both published in Oliver & Boyd's series of university mathematical texts, are mentioned.

  492. Adrien-Marie Legendre (1752-1833)
    • He then introduced what we call today the Legendre functions and used these to determine, using power series, the attraction of an ellipsoid at any exterior point.
    • The 1785 paper on number theory contains a number of important results such as the law of quadratic reciprocity for residues and the results that every arithmetic series with the first term coprime to the common difference contains an infinite number of primes.

  493. Woolsey Johnson (1841-1927)
    • This treatise on differential equations is in continuation of the series of mathematical textbooks, by the same author, of which have already appeared the differential and integral calculus.
    • Cassius Jackson Keyser (1862-1947), reviewing (a late abridged edition of) An Elementary Treatise on the Differential Calculus founded on the Method of Rates many years after it was first written, does take a rather "American" approach [',' C J Keyser, Review: An Elementary Treatise on the Differential Calculus founded on the Method of Rates, by William Woolsey Johnson, Science, New Series 29 (755) (1909), 974-977.','5]:- .

  494. Jacob T Schwartz (1930-2009)
    • Jack's style has been to enter a new field, master quickly the existing research literature, add the stamp of his own forceful vision in a series of research contributions, and finally, leave behind an active research group that continues fruitful research for many years along the lines he has laid down.
    • His style has been to enter a new field, quickly master the existing research literature, add the stamp of his own forceful vision in a series of research contributions, and finally leave behind an active research group that continues fruitful research for many years, along the lines he has laid down.

  495. Shigeo Sasaki (1912-1987)
    • One of them is a series of three papers on the relations between the structure of spaces with normal conformal connections and their holonomy groups.
    • It was this last series of three papers which formed the basis of Sasaki's doctoral thesis which he presented in 1943, receiving his doctorate in July of that year.

  496. William Young (1863-1942)
    • He studied Fourier series and orthogonal series in general, the ideas which he put forward being further developed by Littlewood and Hardy.

  497. Friedrich Hartogs (1874-1943)
    • Hartogs is best known for his results on the representation of analytic functions of several variables by means of power series.
    • Likewise, his name lives on in the Hartogs figure, which represents the simplest example of a region which is not a domain of holomorphy, and in Hartogs domains (convergence ranges for certain series).

  498. Bartel van der Waerden (1903-1996)
    • He then began to publish a series of articles in Mathematische Annalen on algebraic geometry.
    • About ten years ago, van der Waerden, already eminent as an algebraist, began, in a series of papers in the Mathematische Annalen, to create rigorous foundations for algebraic geometry.

  499. Edgar Raymond Lorch (1907-1990)
    • In a series of remarkable memoirs, Liouville demonstrated the impossibility of evaluating certain indefinite integrals, and of solving certain differential equations, in terms of elementary functions.
    • He gave a series of lectures in Italian at the University of Rome which he then rewrote in English to become his famous book Spectral Theory (1962) described by a reviewer as "a model of economy and clarity." You can read some extracts from reviews of this text at THIS LINK.

  500. William Spottiswoode (1825-1883)
    • His series of memoirs on the contact of curves and surfaces, contributed to the 'Philosophical Transactions' of 1862 and subsequent years, mainly gave him his high rank as a mathematician.
    • The interesting series of communications on the contact of curves and surfaces which are contained in the Philosophical Transactions of 1862 and subsequent years would alone account for the high rank he obtained as a mathematician.

  501. Vincenzo Riccati (1707-1775)
    • It probably came too late, at the end of the period of construction of the curves, when geometry has given way to algebra, and when series became the tool of choice to represent the solutions of differential equations.
    • Riccati and Saladini also considered the principle of the substitution of infinitesimals in the 'Institutiones analyticae' Ⓣ, together with the application of the series of integral calculus and the rules of integration for certain classes of circular and hyperbolic functions.

  502. Alexander Grothendieck (1928-2014)
    • A general theory of duality for locally convex spaces had to be worked out: Schwartz and I had started its study for Frechet spaces and their direct limits, but we had met a series of problems we could not solve.

  503. Carl Neumann (1832-1925)
    • Several terms are easy to establish as being named after Carl Neumann, however, such as the Neumann-Poincare Operator, the Neumann boundary value problem, the Neumann boundary condition, the Neumann series, and the Neumann problem.

  504. Claude-Louis Navier (1785-1836)
    • He worked on applied mathematics topics such as engineering, elasticity and fluid mechanics and, in addition, he made contributions to Fourier series and their application to physical problems.

  505. Mary Ellen Rudin (1924-2013)
    • In August 1974 Rudin gave a series of lectures on set theoretic topology at the CBMS Regional Conference held at the University of Wyoming, Laramie.

  506. Francesco Gerbaldi (1858-1934)
    • This is the paper Le frazione continue di Halphen and it was one of a series of three papers which Gerbaldi wrote on the continued fractions of George-Henri Halphen.

  507. Enzo Martinelli (1911-1999)
    • The representation formulas, extending Cauchy's classical style, were proved by Martinelli in a series of works ranging from 1937 to 1955.

  508. Felix Hausdorff (1868-1942)
    • He introduced the concept of a partially ordered set and from 1901 to 1909 he proved a series of results on ordered sets.

  509. Paul Guldin (1577-1643)
    • Guldin was clearly very interested in mechanics because he had a whole series of books on this subject including ones on military equipment, fortifications, artillery and pyrotechnics.

  510. Karl Sundman (1873-1949)
    • The most famous contribution of Sundman was his solution of the three-body problem which he accomplished using analytic methods to prove the existence of an infinite series solution.

  511. René de Sluze (1622-1685)
    • De Sluze received information that Newton had devised a method to find tangents and on other topics that Newton had been working on such as infinite series.

  512. William Feller (1906-1970)
    • Mathematics I-IV (with Marije Kiseljak), Differential and Integral Calculus (with Vladimir Varicak), Infinite Series (with Stjepan Bohnicek), Number Theory 1,2 (with Stjepan Bohnicek), Theory of Real Functions (with Vladimir Vranic), Calculus of Variations (with Vladimir Varicak), and two mathematical seminars (with Vladimir Varicak), along with a panoply of experimental and theoretical physics, some chemistry, and a sample of courses in pedagogy and psychology.

  513. Christian Wiener (1826-1896)
    • For example, he used imaginary projection and developed a grid method that can be derived from the theory of cyclically projected point series.

  514. Alexis Bouvard (1767-1843)
    • He also had at his disposal two fine series of post-discovery observations, one by the Paris observatory, the other by the Greenwich observatory.

  515. Barry Johnson (1937-2002)
    • His mathematical publications started in 1964 with a series of papers on topological algebras, measure algebras and Banach algebras.

  516. Gregorio Ricci-Curbastro (1853-1925)
    • The first was a series of articles on Maxwell's theory of electrodynamics and the work of Clausius which Betti asked him to write.

  517. Humphrey Lloyd (1800-1881)
    • Between 1827 and 1833 Hamilton published a series of papers in the 'Transactions' of the Royal Irish Academy.

  518. Pierre-Louis Lions (1956-)
    • Another major contribution by Lions, in a long series of important papers, is to variational problems.

  519. Leon Simon (1945-)
    • In a series of papers over the past ten years, Simon has developed methods for analysing this structure.

  520. Harry Bateman (1882-1946)
    • He accumulated a vast store of information on all the familiar special functions and on his death the publication of his manuscripts was undertaken by Erdelyi and his associates in the form of the well-known series Higher Transcendental Functions and Tables of Integral Transforms.

  521. Henry Whitehead (1904-1960)
    • The long series of collaborative papers written between 1950 and 1960 reflects his eagerness to share his ideas and to interest himself in the results of others, which remained undiminished to the end of his life.

  522. Johannes Boersma (1937-2004)
    • He published Computation of Fresnel integrals (1960) which gave a table of coefficients for approximation of Fresnel integrals by finite power series.

  523. Élie Cartan (1869-1951)
    • In 1936-37 he delivered a series of lectures at the Sorbonne which covered his contributions to the topic.

  524. Giovanni Battista Riccioli (1598-1671)
    • During the period from 1629 to 1631 he conducted a series of experiments with falling bodies finding that the distance of fall per second increased according to the sequence 1, 3, 9, 27.

  525. Iacopo Barsotti (1921-1987)
    • While at Pisa, Barsotti published a series of seven papers entitled Metodi analitici per varieta abeliane in caratteristica positiva.

  526. Philippe de la Hire (1640-1718)
    • Bosse had published a series of works developing the geometric ideas that he had learnt from Desargues and had established his own school of art in 1661.

  527. Franc Mocnik (1814-1892)
    • The textbook of arithmetic and algebra for upper classes of secondary schools which he published in 1874 is quite advanced covering topics such as divergence, convergence, binomial series and interpolation.

  528. Geoffrey Taylor (1886-1975)
    • At the age of 11 he attended a series of children's Christmas lectures on The principles of the electric telegraph and these made a strong impression on him.

  529. Shreeram Shankar Abhyankar (1930-2012)
    • Throughout his mathematical life, he took polynomials and power series, and related concepts such as determinants and discriminants as the focus of his interest, only considering the most fundamental and important problems.

  530. Nathan Mendelsohn (1917-2006)
    • He wrote papers on a wide variety of combinatorial problems, for example: Symbolic solution of card matching problems (1946), Applications of combinatorial formulae to generalizations of Wilson's theorem (1949), Representations of positive real numbers by infinite sequences of integers (1952), A problem in combinatorial analysis (1953), The asymptotic series for a certain class of permutation problems (1956), and Some elementary properties of ill conditioned matrices and linear equations (1956).

  531. Paul Halmos (1916-2006)
    • Halmos is known for both his outstanding contributions to operator theory, ergodic theory, functional analysis, in particular Hilbert spaces, and for his series of exceptionally well written textbooks.

  532. Edward Kasner (1878-1955)
    • It is packed with information and deals with all the elementary ingredients of mathematics, number, shape, line, series, pattern, infinity, paradox, chance, change, with copious and entertaining visual aids: and the treatment is refreshing.

  533. Sergei Bernstein (1880-1968)
    • Many of the annotations differ considerably from the usual kind of footnote; they contain a series of valuable ideas and observations and are to be regarded as new scientific work.

  534. Eugčne Cosserat (1866-1931)
    • He then carried out a series of observations on the satellites of Jupiter and those of Saturn.

  535. Mauro Picone (1885-1977)
    • Some of his most important books which Picone published during his years in Rome are: Appunti di Analisi superiore Ⓣ (1940), which studies harmonic functions, Fourier, Laplace and Legendre series and the equations of mathematical physics; Lezioni di Analisi funzionale Ⓣ (1946), which concerns the calculus of variations; Teoria moderna dell'integrazione delle funzioni Ⓣ (1946), containing a detailed discussion of the r-dimensional Stieltjes integrals; (with Tullio Viola) Lezioni sulla teoria moderna dell'integrazione Ⓣ (1952), which is basically the previous work by Picone with three extra chapters by Viola; and (with Gaetano Fichera) Trattato di Analisi matematica Ⓣ (Vol 1, 1954, Vol 2, 1955), which puts into a treatise Picone's way of teaching calculus particularly slanted towards the applications studied at the Institute for Applied Calculus.

  536. Giuseppe Battaglini (1826-1894)
    • This series of articles consists of a summary, with additions, of some of the chapters in Jordan's treatise as well as of parts drawn from Joseph Serret's 'Cours d'algebre superieure' and Peter Lejeune Dirichlet's 'Vorlesungen uber Zahlentheorie'.

  537. Hannes Alfvén (1908-1995)
    • One result of these interests was a series of books that Hannes wrote, some together with Kerstin.

  538. Alexander Oppenheim (1903-1997)
    • Examples of his papers are Rational approximations to irrationals (1941), On the representation of real numbers by products of rational numbers (1953), On indefinite binary quadratic forms (1954), On the Diophantine equation x3+ y3+ z3= x + y + z (1966), The irrationality of certain infinite products (1968), Representations of real numbers by series of reciprocals of odd integers (1971) and The prisoner's walk: an exercise in number theory (1984).

  539. Sridhara (about 870-about 930)
    • There are sections of the book devoted to arithmetic and geometric progressions, including progressions with a fractional numbers of terms, and formulae for the sum of certain finite series are given.

  540. Nicolaus(I) Bernoulli (1687-1759)
    • In his letters to Euler (1742-43) he criticises Euler's indiscriminate use of divergent series.

  541. Arthur Cayley (1821-1895)
    • Here is G B Halsted's tribute [',' G B Halsted, Review: The Collected Mathematical Papers of Arthur Cayley, by A Cayley, Science, New Series 9 (211) (1899), 59-63.','23]:- .

  542. Robert Lee Moore (1882-1974)
    • This volume, published in the Colloquium Lectures Series of the American Mathematical Society, arose from the colloquium lectures which Moore gave in 1929 and is a self-contained introduction to the topic concentrating on Moore's own contributions to the subject.

  543. Robert Woodward (1849-1924)
    • Since his first association with the Survey, in 1884, he has not only supervised the computations made in connection with the triangulation and astronomic determinations, conducted the computation of a series of tables for the use of the Topographic Branch, and given aid to geologists having occasion to read their data by mathematical methods, but he also made important additions to geologic science by discussing and advancing, on several lines, the theories of terrestrial physics.

  544. Charles De la Vallée Poussin (1866-1962)
    • Most of the additional material appeared in small type and covered topics such as set theory, in particular the Schroder-Bernstein theorem, the Lebesgue integral, functions of bounded variation, the Jordan curve theorem, polynomial approximation, Parseval's theorem on trigonometric series, results of Fejer, etc.

  545. Samuel Dickstein (1851-1939)
    • In 1884 he was one of the two founders of a series of mathematics and physics textbooks which were written in Polish.

  546. Al-Biruni (973-1048)
    • These include: theoretical and practical arithmetic, summation of series, combinatorial analysis, the rule of three, irrational numbers, ratio theory, algebraic definitions, method of solving algebraic equations, geometry, Archimedes' theorems, trisection of the angle and other problems which cannot be solved with ruler and compass alone, conic sections, stereometry, stereographic projection, trigonometry, the sine theorem in the plane, and solving spherical triangles.

  547. George Stokes (1819-1903)
    • I too feel that I have been thinking too much of late, but in a different way, my head running on divergent series, the discontinuity of arbitrary constants, ..

  548. Kurt Gödel (1906-1978)
    • In 1934 Godel gave a series of lectures at Princeton entitled On undecidable propositions of formal mathematical systems.

  549. Samuel Haughton (1821-1897)
    • Haughton was appointed registrar of Trinity Medical School in 1864 and this put him in a position to carry out a series of reforms.

  550. Bartholomew Lloyd (1772-1837)
    • In the thirteen years between Lloyd leaving the chair of mathematics and MacCullagh filling the chair there had been two professors, namely James Wilson, who had published First Elements of the Theory of Series and Differences (1822), and Franc Sadleir (1775-1851).

  551. Srinivasa Varadhan (1940-)
    • In a series of joint papers with Monroe D Donsker exploring the hierarchy of large deviations in the context of Markov processes, Varadhan demonstrated the relevance and the power of this new approach.

  552. William Birnbaum (1903-2000)
    • After arriving in Gottingen, Edmund Landau became his advisor, and he attended several lecture courses: differential equations given by Courant; calculus of variations given by Courant; power series given by Landau; higher geometry given by Herglotz; probability calculus given by Bernays; analysis of infinitely many variables given by Wegner; and attended the mathematical seminar directed by Courant and Herglotz.

  553. Enrico Fermi (1901-1954)
    • In the summer of 1954 Fermi returned to Italy and gave a series of lectures in the Villa Monastero in Varenna on Lake Como.

  554. James Taylor (1851-1910)
    • When Mr Taylor took pen in hand, he showed that he possessed the gift of lucid expression, as is illustrated in the series of articles on "The Transit of Venus" in 1882, which he contributed to The Dollar Magazine of that year.

  555. David Eisenbud (1947-)
    • Together, they developed the theory of Limit Linear Series and used it to solve a number of classical problems about the moduli spaces of complex algebraic curves.

  556. Edward Ince (1891-1941)
    • The nucleus of an integral equation for one of the periodic Lame functions is expanded in series of products of the characteristic functions ..

  557. Philipp Furtwängler (1869-1940)
    • Furtwangler continued to attack Hilbert's Ninth Problem in a series of papers: Allgemeiner Existenzbeweis fur den Klassenkorper eines beliebigen algebraischen Zahlkorpers Ⓣ (1907); Reziprozitatsgesetze fur Potenzreste mit Primzahlexponenten in algebraischen Zahlkorpern I Ⓣ, (1909); Reziprozitatsgesetze fur Potenzreste mit Primzahlexponenten in algebraischen Zahlkorpern II Ⓣ (1912); and Reziprozitatsgesetze fur Potenzreste mit Primzahlexponenten in algebraischen Zahlkorpern III Ⓣ, (1913).

  558. Steven Orszag (1943-2011)
    • In the early 1970s, in a series of landmark papers, Steven Orszag showed that spectral methods, and the closely related pseudospectral methods, could be used to simulate incompressible turbulence with N3 Fourier modes at a cost of only O(N3 log N) operations per timestep with zero numerical dispersion and dissipation.

  559. Georg Vega (1754-1802)
    • These French defensive lines consisted of a series of fortifications built much earlier to protect Alsace from an attack from the east.

  560. Martin Kruskal (1925-2006)
    • Analysing asymptotic series also led Kruskal to become interested in surreal numbers, generalisations of real numbers introduced by John Conway.

  561. Urbain Le Verrier (1811-1877)
    • Victor Regnault (1810-1878) also applied for this position and he was a strong candidate for, in 1835, he had begun a series of important researches on the haloid and other derivatives of unsaturated hydrocarbons.

  562. Nevil Maskelyne (1732-1811)
    • After making these observations, he set up an observatory on the north side of the mountain and made a similar series of observations.

  563. Louis Puissant (1769-1843)
    • The map was produced with considerable detail, the projection used spherical trigonometry, truncated power series and differential geometry.

  564. Eduard Weyr (1852-1903)
    • Research by Weyr on analysis deals particularly with infinite series and products, and with elliptic functions.

  565. Mary Newson (1869-1959)
    • She attended lectures by Klein on 'Hypergeometric Series' and also lectures by Heinrich Weber.

  566. Louis Goodstein (1912-1985)
    • Educated at St Paul's School London, Louis won scholarships and a prize for an essay on divergent series.

  567. Leslie Woods (1922-2007)
    • He now published a whole series of papers - the next two were: A new relaxation treatment of flow with axial symmetry (1951), and The numerical solution of two-dimensional fluid motion in the neighbourhood of stagnation points and sharp corners (1952).

  568. Luigi Bianchi (1856-1928)
    • In 1879 appeared a paper on the centro-surface of a helicoid in the Giornale di Matematiche; and from that time onwards until shortly before his death a series of papers of first-rate merit appeared each year from his prolific pen in spite of the many distractions caused by domestic troubles, the duties of his professorship, and his labours as an editor.

  569. Marcel-Paul Schützenberger (1920-1996)
    • Later he published a series of results on variable-length codes all of them reported in our book with Jean Berstel (Theory of Codes, Academic Press, 1984).

  570. John Aitchison (1926-2016)
    • He served as Editor of the Royal Statistical Society, Series B from 1963 to 1965 and Associate Editor of Biometrika from 1966 to 1969.

  571. Cheryl Praeger (1948-)
    • A series of three papers on a similar topic On the Sylow subgroups of a doubly transitive permutation group appeared in 1974 and 1975.

  572. Hermann Bondi (1919-2005)
    • Also with co-authors, he wrote a series of papers Gravitational waves in general relativity.

  573. Roger Penrose (1931-)
    • Beginning in 1959, Penrose published a series of important papers on cosmology.

  574. Yurii Vasilevich Prokhorov (1929-2013)
    • This course covered the foundations of functional analysis, measure theory and the theory of orthogonal series.

  575. Giovanni Sansone (1888-1979)
    • Later he worked on differential geometry followed by work on series of orthogonal functions, and then moved to linear and nonlinear differential equations.

  576. James Mercer (1883-1932)
    • Mercer was a mathematical analyst of originality and skill; he made noteworthy advances in the theory of integral equations, and especially in the theory of the expansion of arbitrary functions in series of orthogonal functions.

  577. Leonard Dickson (1874-1954)
    • Dickson's search for a counterexample led him to consider non-associative algebras and in a series of papers he determined all three and four-dimensional (non-associative) division algebras over a field.

  578. Charles Chree (1860-1928)
    • Series A, Containing Papers of a Mathematical and Physical Character Vol.

  579. Vladimir Voevodsky (1966-2017)
    • To get a flavour of this work we give his abstract of a series of lectures which he gave at the University of Miami in January 2005 entitled Categories, Population Genetics and a Little of Quantum Physics:- .

  580. Robion Kirby (1938-)
    • In 1989 Kirby published The topology of 4-manifolds in the Springer Lecture Notes in Mathematics Series.

  581. Andrew Gleason (1921-2008)
    • Chapters I to VI cover elementary logic and set theory; Chapters VII to X deal with the various "number systems" from the natural integers to the complex numbers; Chapter XI briefly returns to set theory (countable sets, cardinal numbers and the axiom of choice); finally, the last four chapters deal, respectively, with limits of complex sequences, infinite series and products, metric spaces, and the elementary theory of holomorphic functions of one variable (Cauchy integral excluded, but the logarithmic function is defined and studied).

  582. Hermann Schwarz (1843-1921)
    • In answering the problem of when Gauss's hypergeometric series was an algebraic function Schwarz, as he had done so many times, developed a method which would lead to much more general results.

  583. Karl Heinrich Weise (1909-1990)
    • Also mentioned are existence theorems as well as solutions by iteration, power series, and numerical methods.

  584. Irving Stringham (1847-1909)
    • This Congress was the model for the series of International Congresses of Mathematicians which began in Zurich in 1897.

  585. Rudolf Lipschitz (1832-1903)
    • He carried out many important and fruitful investigations in number theory, in the theory of Bessel functions and of Fourier series, in ordinary and partial differential equations, and in analytical mechanics and potential theory.

  586. Gloria Olive (1923-2006)
    • She published papers such as Binomial functions and combinatorial mathematics (1979), A combinatorial approach to generalized powers (1980), Binomial functions with the Stirling property (1981), Some functions that count (1983), Taylor series revisited (1984), Catalan numbers revisited (1985), A special class of infinite matrices (1987), and The ballot problem revisited (1988).

  587. Stephano degli Angeli (1623-1697)
    • James Gregory studied with Angeli in Padua from 1664 to 1668 and learnt from him about series expansions of functions.

  588. Alfred Goldie (1920-2005)
    • During his retirement he published papers such as (with Gunter Krause) Associated series and regular elements of Noetherian rings (1987), Rings with an additive rank function (1990), and (with Gunter Krause) Embedding rings with Krull dimension in Artinian rings (1996).

  589. Georg Scheffers (1866-1945)
    • The articles on number have been remodeled according to Dedekind's theory and the proofs of the theorems on continuous functions and on the convergence of series have thus been given a real backbone.

  590. Nikolai Nikolaevich Bogolyubov (1909-1992)
    • Bogolyubov himself completed a series of brilliant papers on the theory of stability of a plasma in a magnetic field and on the theory and applications of the kinetic equations, and he began his construction of axiomatic quantum field theory.

  591. John Hammersley (1920-2004)
    • He had already begun publishing statistical papers with The "effective" number of independent observations in an autocorrelated time series, a joint publication with G V Bayley apprearing in the Journal of the Royal Statistical Society in 1946.

  592. Paul Epstein (1871-1939)
    • In the first part of Zur Theorie allgemeiner Zetafunctionen Ⓣ (1903) Epstein introduced a function belonging to a class of Dirichlet series generalising the Riemann Zeta-function depending on a given quadratic form.

  593. Thomas Kirkman (1806-1895)
    • This was followed by a series of papers.

  594. Guglielmo Righini (1908-1978)
    • During these years he published a series of papers on the physical conditions of the solar corona, the intensity distribution of its continuous spectrum, its colour index and temperature, again on the basis of observational and theoretical considerations.

  595. Moritz Abraham Stern (1807-1894)
    • His publications reflect his diverse mathematical interests, including work on number theory, the theory of continued fractions, the theory of series, the theory of Bernoulli's and Euler's numbers, and the theory of functions.

  596. William Hamilton (1788-1856)
    • Hamilton was one of the first in a series of British logicians to create the algebra of logic and introduced the 'quantification of the predicate'.

  597. Max Deuring (1907-1984)
    • On this second visit he gave a series of lectures which were published as Lectures on the theory of algebraic functions of one variable in 1973.

  598. Gottlob Frege (1848-1925)
    • a series of brilliant philosophical articles in which he elaborated his philosophy of logic.

  599. Robert Edward Bowen (1947-1978)
    • In 1975 Bowen published the book Equilibrium states and the ergodic theory of Anosov diffeomorphisms in the Springer Lecture Notes in Mathematics Series.

  600. Abigail Thompson (1958-)
    • As a final comment on her research, let us quote the Abstract of the lecture The stabilization problem for 3-manifolds which Thompson gave at the University of Texas Distinguished Women in Mathematics Lecture Series in the spring of 2009:- .

  601. Petre Sergescu (1893-1954)
    • He helped to arrange the first hall, devoted to the history of numbers, a series of lectures on the history of science, and special exhibitions devoted to Pascal and Leonardo da Vinci.

  602. Brian Haselgrove (1926-1964)
    • A great many terms of the Euler-Maclaurin or Riemann-Siegel series were used to calculate each entry.

  603. Antoine Arnauld (1612-1694)
    • Pascal wrote a series of 18 letters now known as Les Provinciales during the years 1656 and 1657 in defence of Arnauld.

  604. Claude-Louis Mathieu (1783-1875)
    • In the same year, together with Biot, he embarked on a series of measurements of the length of the seconds pendulum at different points on the meridian, in particular at Bordeaux and at Dunkirk.

  605. Joseph Kruskal (1928-2010)
    • In the 5 September 1969 issue of Science, John H Wilson Jr published the letter Better Written Journal Papers - Who Wants Them? Kruskal replied in [',' J B Kruskal, Nature’s Chief Masterpiece Is Writing Well, Science, New Series 166 (3904) (1969), 454-455.','4]:- .

  606. Samarendra Nath Roy (1906-1964)
    • In 1970 the book Essays in probability and statistics, edited by R C Bose, I M Chakravarti, P C Mahalanobis, C R Rao and K J C Smith, dedicated to his memory was published in the University of North Carolina Monograph Series in Probability and Statistics.

  607. Lazarus Fuchs (1833-1902)
    • In a series of papers (1880-81) Fuchs studied functions obtained by inverting the integrals of solutions to a second-order linear differential equation in a manner generalising Jacobi's inversion problem.

  608. Don Carlson (1938-2010)
    • The authors obtain power series expansions in terms of a parameter e for plane deformations of incompressible, isotropic homogeneous elastic bodies.

  609. Vladimir Maz'ya (1937-)
    • In addition the American Mathematical Society published Perspectives in Partial Differential Equations, Harmonic Analysis and Applications: A Volume in Honor of Vladimir G Maz'ya's 70th Birthday in their Proceedings of Symposia in Pure Mathematics series.

  610. Hans Grauert (1930-2011)
    • This text is an excellent introduction to the classical themes of modern several complex variables theory: domains of holomorphy, holomorphic complexity, pseudoconvexity, the ring of convergent power series, analytic subvarieties and the several variables version of the Mittag-Leffler and Weierstrass problems ..

  611. Erasmus Reinhold (1511-1553)
    • They were a series of astronomical tables that showed that the heliocentric model was applicable in practice.

  612. Kuo-Tsai Chen (1923-1987)
    • Kuo-Tsai Chen is best known to the mathematics community for his work on iterated integrals and power series connections in conjunction with his research on the cohomology of loop spaces.

  613. Dusa McDuff (1945-)
    • Her work includes fundamental theorems on the symplectic blowup construction, a theorem on the symplectic packing problem (joint with Leonid Polterovich), and a series of seminal joint papers with Francois Lalonde on the symplectic energy and the stability of Hamiltonian flows.

  614. Claude E Shannon (1916-2001)
    • Working with John Riordan, Shannon published a paper in 1942 on the number of two-terminal series-parallel networks.

  615. Gheorghe Mihoc (1906-1981)
    • We mention in particular the series of conferences on probability in 1955, 1962, 1968, 1971, 1974 and 1979.

  616. William Wager Cooper (1914-2012)
    • International Series in Operations Research & Management Science 147 (2011), 201-216.','15]:- .

  617. Leonida Tonelli (1885-1946)
    • The fourth, and final, volume Argomenti vari Ⓣ, published in 1963, contains papers on trigonometric series, ordinary differential equations and integral equations (all published in or after 1924-25), and some miscellaneous work (from 1909 onwards), including Tonelli's biography of Salvatore Pincherle.

  618. Derrick Norman Lehmer (1867-1938)
    • Under each tooth in this second series of gears is a small hole.

  619. Daniel Bernoulli (1700-1782)
    • Thus, in one stroke he derived the entire series of such curves as the velaria, lintearia, catenaria..

  620. Ernest Hobson (1856-1933)
    • His research concentrated on convergence, in particular convergence of series of orthogonal functions.

  621. Konstantin Alekseevich Andreev (1848-1921)
    • Gram determinants were introduced by J P Gram in 1879 but Andreev invented them independently in the context of problems of expansion of functions into orthogonal series and the best quadratic approximation to functions.

  622. Erhard Schmidt (1876-1959)
    • He also expanded functions related to the integral of the kernel function as an infinite series in a set of orthonormal eigenfunctions.

  623. Nikolai Fuss (1755-1826)
    • Most of Fuss's papers are solutions to problems posed by Euler on spherical geometry, trigonometry, series, differential geometry and differential equations.

  624. Michael Artin (1934-)
    • The point of the extension is that Artin's theorem on approximating formal power series solutions allows one to show that many moduli spaces are actually algebraic spaces and so can be studied by the methods of algebraic geometry.

  625. Charles Tinseau (1748-1822)
    • He published a series of anti-Revolution writings from 1792 onwards and tried to organise uprisings in France, as did Charles-Philippe who made an unsuccessful attempt to land in the Vendee to lead a royalist rising there.

  626. Basil Rennie (1920-1996)
    • In On dominated convergence he proves a converse of Lebesgue's theorem of dominated convergence and gives an application to Fourier series.

  627. James Hutton (1726-1797)
    • His simple and eloquent style consisted of a series of chapters clearly stating the Huttonian theory, giving the facts to support it, and the arguments given against it.

  628. Lucien Le Cam (1924-2000)
    • The result of the trial and the related immunology research have been published in a series of papers in medical journals.

  629. John Backus (1924-2007)
    • The first problem he worked on was to write a program in machine code for the Selective Sequence Electronic Calculator (SSEC) to calculate the position of the moon from a function given by a series expansion with about 1000 terms.

  630. Ludwig Bieberbach (1886-1982)
    • Considerable care has been devoted to the simplification of known proofs and to the detailed discussion of phenomena which usually are given but casual attention", Analytische Fortsetzung Ⓣ (1955) "No mathematician who is interested in the Taylor series, in interpolatory theory, or in the study of the singularities of analytic functions, can afford to be without it", and Einfuhrung in die Theorie der Differentialgleichungen im reellen Gebiet Ⓣ (1956).

  631. Carl Friedrich Gauss (1777-1855)
    • His publications during this time include Disquisitiones generales circa seriem infinitam Ⓣ, a rigorous treatment of series and an introduction of the hypergeometric function, Methodus nova integralium valores per approximationem inveniendi Ⓣ, a practical essay on approximate integration, Bestimmung der Genauigkeit der Beobachtungen Ⓣ, a discussion of statistical estimators, and Theoria attractionis corporum sphaeroidicorum ellipticorum homogeneorum methodus nova tractata Ⓣ.

  632. Théodore Olivier (1793-1853)
    • In 1857, four years after Olivier died, Harvard University purchased 24 of Olivier's models from Fabre de Lagrange and after the university received the order Benjamin Peirce gave a series of lectures on the mathematics which they illustrated.

  633. Johannes Kepler (1571-1630)
    • From the first, Kepler had sought a rule relating the sizes of the orbits to the periods, but there was no slow series of steps towards this law as there had been towards the other two.

  634. Hans Zassenhaus (1912-1991)
    • In a long series of papers he applied Lie algebras to problems of theoretical physics.

  635. Nikolai Vladimirovich Efimov (1910-1982)
    • Nikolai Vladimirovich Efimov is one of those rare geometers whose international reputation rests not so much on a series of separate results as on one really big discovery.

  636. Eduard Study (1862-1930)
    • In July 1893 he visited the United States where he attended the mathematical conference at the Columbian Exposition in Chicago in August 1893 and then went to Evanston to attend Klein's series of lectures on contemporary mathematical research held from 28 August to 9 September 1893.

  637. Salomon Bochner (1899-1982)
    • Bochner found that the Riemann Localisation Theorem was not valid for Fourier series of several variables (1935 - 1936), which led him indirectly to consider functions of several complex variables (1937).

  638. Nicholas Oresme (1323-1382)
    • Oresme also worked on infinite series and argued for an infinite void beyond the Earth.

  639. Christoff Rudolff (1499-1543)
    • Next come formulas for summing arithmetic and geometric series.

  640. Johann Friedrich Pfaff (1765-1825)
    • Pfaff did important work in analysis working on partial differential equations, special functions and the theory of series.

  641. Albert Ingham (1900-1967)
    • Ingham's work was on the Riemann zeta function, the theory of numbers, the theory of series and Tauberian theorems.

  642. Jean-Baptiste-Joseph Delambre (1749-1822)
    • presents each major chronological period in a series of discrete analyses of one treatise after another.

  643. Shaun Wylie (1913-2009)
    • Alexander at least twice announced a series of lectures and each time abandoned it fairly early on.

  644. Frank Smithies (1912-2002)
    • Smithies early work was on integral equations and in 1958 his text Integral equations was published by Cambridge University Press in their Cambridge Tracts in Mathematics and Mathematical Physics Series.

  645. Eduard ech (1893-1960)
    • He continued this interest after the war ended and used his experiences with the school teachers to organise a series of school mathematics textbooks.

  646. Jan Mikusiski (1913-1987)
    • The Professor delivered for them a series of lectures on operational calculus.

  647. Jean Bourgain (1954-2018)
    • He has also been invited to give many prestigious lecture series such as: the A Zygmund Lectures, University of Chicago (1989), the R de Francia Memorial Lectures, Autonoma University, Spain (1991), the American Mathematical Society Colloquium Lectures (1994), the A Ziwet Lectures, University of Michigan (1995), the IAS/Park City Lectures, Park City (1995), the Fields Lectures, Toronto, Canada (2004), the T Wolff Memorial Lectures, California Institute of Technology (2004), the Landau Lectures, Hebrew University, Jerusalem (2005), and the Trjitzinsky Memorial Lectures, University of Illinois (2005).

  648. Pavel Aleksandrov (1896-1982)
    • He laid the foundations of homology theory in a series of fundamental papers between 1925 and 1929.

  649. Lazar Matveevich Gluskin (1922-1985)
    • he published a series of brilliant results on semigroups of linear transformations.

  650. Charles S Peirce (1839-1914)
    • Peirce lectured on Pragmatism at Harvard in March to May of 1903 and published a series of essays explaining his ideas in The Monist in 1905.

  651. Karen Smith (1965-)
    • In January 1996 Smith delivered a twenty-hour lecture series at the University of Jyvaskyla, the university where her husband had studied before going to the United States.

  652. Alfred Young (1873-1940)
    • He wrote a series of papers On quantitative substitutional analysis which arose out of the classical theory of invariants and contained his results in this area.

  653. Arnold Walfisz (1892-1962)
    • The first of these was in the Georgian Academy of Sciences' popular science series.

  654. Laszlo Lovász (1948-)
    • I could not resist, however, working out a series of exercises on random walks on graphs, and their relations to eigenvalues, expansion properties, and electrical resistance (this area has classical roots but has grown explosively in the last few years).

  655. John Dougall (1867-1960)
    • This paper contains a new derivation of the coefficients in the expansion into a series of Legendre polynomials of the product of two Legendre polynomials.

  656. Oskar Perron (1880-1975)
    • A whole series of outstanding scientists, not just opportunists who thought they could get rid of their competition, rushed to be Hitler's mouthpiece, did not hesitate to speak of the good Aryan and the bad "foreign" Jewish physics and mathematics, and thus ridicule their German fatherland to all foreign countries.

  657. Lars Ahlfors (1907-1996)
    • One year he gave a series of talks on Teichmuller's papers, probably with a view to preparing himself for his work on quasiconformal mappings.

  658. Alice Bache Gould (1868-1953)
    • The 150-page book was published in Boston in The Beacon Biographies of Eminent Americans series in 1901.

  659. Erich Kähler (1906-2000)
    • At first he was inspired by books his mother bought him about Sven Hedin, a Swedish explorer who led a series of expeditions through Central Asia and as a consequence made important archaeological and geographical findings.

  660. Anne Bosworth (1868-1907)
    • In [',' G B Halsted, Supplementary Report on Non-Euclidean Geometry, Science, New Series 14 (358) (1901), 705-717.','7] George Bruce Halsted writes:- .

  661. Pao Lu Hsu (1910-1970)
    • During this period [at University College, London] Hsu wrote a remarkable series of papers on statistical inference which show the strong influence of the Neyman-Pearson point of view.

  662. Efim Zelmanov (1955-)
    • At the Groups-St Andrews conference at Galway, Ireland in 1993, of which I [EFR] was a joint organiser, Zelmanov was one of the main speakers and he gave a series of five lectures on Nil rings methods in the theory of nilpotent groups.

  663. Annibale Giordano (1769-1835)
    • Metaphysics, which seemed so far from this perfection, thanks to the combined efforts of great men saw the dissipation of that thick haze in which the schools had enveloped it: It can be reduced to two very extensive branches, the first of which having man as its object, includes the history of our ideas; and the second, which concerns other objects, is the series of those few corollaries, which derive from the most general physical laws, known under the name Cosmological law.

  664. Anna Johnson Wheeler (1883-1966)
    • The records show that Anna studied Algebra and Trigonometry in 1899-1900, Modern Geometry, the Theory of Equations, and Solid Analytical Geometry in 1900-1901, Calculus, Analytical mechanics and Plane Analytical Geometry in 1901-02, and the Theory of Substitutions and Potential, Partial Differential Equations and Fourier Series, and Differential Equations in 1902-03.

  665. Thomas Fantet de Lagny (1660-1734)
    • De Lagny is well known for his contributions to computational mathematics, calculating π to 120 places and also making useful comments on the convergence of the series he was using.

  666. Marshall Stone (1903-1989)
    • For example he published An unusual type of expansion problem (1924), A comparison of the series of Fourier and Birkhoff (1926), Developments in Legendre polynomials (1926), and Developments in Hermite polynomials (1927).

  667. Guglielmo Libri (1803-1869)
    • These two sales of books imported from France contain a magnificent series of manuscripts and books by Galileo, Copernicus, Kepler, Cardan, etc., many with long notes pointing out their significance, and we must not allow ourselves to be blinded to the showmanship and originality of Libri's catalogue by his unenviable reputation as a forger and a thief.

  668. Wenceslaus Johann Gustav Karsten (1732-1787)
    • He also introduced Karsten to Euler and the two exchanged a series of letters (38 in all, of which 23 were written by Karsten) between 1758 and 1765.

  669. Joseph Doob (1910-2004)
    • In fact he undertook the work of writing the book because he had become intellectually bored while undertaking war work in Washington and so was enthusiastic when, in 1945, Shewhart invited him to publish a volume in the Wiley series in statistics.

  670. Giordano Bruno (1548-1600)
    • Oxford seemed a place of learning that looked attractive to Bruno who visited there in the summer of 1583 and gave a series of lecturers on Copernicus's theory that the Earth rotated round the fixed Sun.

  671. Ludwig Berwald (1883-1942)
    • Berwald wrote a series of major papers On Finsler and Cartan geometries.

  672. Domenico Montesano (1863-1930)
    • An involution mapped on the general cubic variety of S4 is probably an example; two series of such were given at the Bologna Congress of 1928 ..

  673. Pavel Urysohn (1898-1924)
    • He published a series of short notes on this topic during 1922.

  674. Augusta Ada Byron (1815-1852)
    • She considered writing a long review, perhaps in the style of her Notes, of Ohm's work On galvanic series, mathematically determined but Babbage, who she looked to for encouragement, was becoming depressed at his own lack of success with financing the development of his computers and failed to give her the necessary support.

  675. Wilhelm Fiedler (1832-1912)
    • Not only did Fiedler do a great deal of self-study but with his friends, the geologist Adolf Knop and the chemist Alexander Muller, he led an organised series of scientific and literary lecture evenings in Chemnitz.

  676. Luigi Fantappič (1901-1956)
    • For these the author establishes the usual concepts of the differential calculus such as the derivative, with the customary properties and rules and the series expansion of such a function.

  677. Pappus (about 290-about 350)
    • It seems likely that this work was not originally written as a single treatise but rather was written as a series of books dealing with different topics.

  678. Carlo Miranda (1912-1982)
    • Proceedings of international Meeting dedicated to the memory of Professor Carlo Miranda (Naples, 1983).','1] and divides these contributions into the following areas: (a) Integral equations, series expansions, summation methods; (b) Harmonic mappings, potential theory, holomorphic functions; (c) Calculus of variations, differential forms, elliptic systems; (d) Numerical analysis; (e) Propagation problems; (f) Differential geometry in the large; (g) General theory for elliptic equations; and (h) Functional transformations.

  679. Richard Tapia (1939-)
    • In the same year he was honoured with the inauguration of the Blackwell-Tapia Lecture Series by Cornell University, Ithaca.

  680. Demetrios Kappos (1904-1985)
    • Also in 1958 Kappos began work on a series of textbooks which would be published between 1960 and 1967.

  681. Martha Betz Shapley (1890-1981)
    • She produced a series of papers dedicated to computing the elements and the variation of the periods of eclipsing systems that represented outstanding contributions for the time.

  682. Hans Meinhardt (1938-2016)
    • The review [',' L G Harrison, Review: Models of Biological Pattern Formation, by Hans Meinhardt, Science, New Series 219 (4586) (1983), 841.','3] gives additional useful information:- .

  683. Georg Zehfuss (1832-1901)
    • However, a series of influential texts at and after the turn of the century permanently associated Kronecker's name with the ⊗ product, and this terminology is nearly universal today.

  684. Valentina Mikhailovna Borok (1931-2004)
    • Starting in the late 1960s, Valentina began a series of papers that lay the foundations for the theory of local and non-local boundary value problems in infinite layers for systems of partial differential equations.

  685. Friedrich Schottky (1851-1935)
    • Schottky's paper Uber eine specielle Function, welche bei einer bestimmten linearen Transformation ihres Arguments unverandert bleibt Ⓣ (1887), advanced the theory of Poincare series considerably.

  686. Raphael Robinson (1911-1995)
    • In a series of papers Robinson showed that a number of mathematical theories are undecidable.

  687. Philippe Flajolet (1948-2011)
    • His seminar series in Versailles was legendary.

  688. Axel Thue (1863-1922)
    • Thue has shown in a series of original investigations that he has many of the qualifications needed to become an outstanding mathematician.

  689. Joseph Raphson (1668-1712)
    • Mr Halley related that Mr Raphson had Invented a method of Solving all sorts of Equations, and giving their Roots in Infinite Series, which Converge apace, and that he had desired of him an Equation of the fifth power to be proposed to him, to which he returned Answers true to Seven Figures in much less time than it could have been effected by the Known methods of Vieta.

  690. Tjalling Charles Koopmans (1910-1985)
    • Koopmans, who lived in Hamden, Connecticut, died at Yale-New Haven Hospital after suffering a series of cerebral strokes.

  691. Joseph Walsh (1895-1973)
    • He continued to publish a steady stream of papers with On the location of the roots of the derivative of a polynomial appearing in 1920 and then two papers A generalization of the Fourier cosine series and A theorem on cross-ratios in the geometry of inversion in 1921.

  692. Ward Cheney (1929-2016)
    • For example he published Two new algorithms for rational approximation (1961) with Henry L Loeb, Tchebycheff approximation in locally convex spaces (1962) with Allen A Goldstein, On rational Chebyshev Approximation (1962) with Henry L Loeb, and two papers with Ambikeshwar Sharma which both appeared in 1964, namely On a generalization of Bernstein polynomials, and Bernstein power series.

  693. Wilhelm Blaschke (1885-1962)
    • He went to Italy, where he gave a series of lectures, before going on to Greece where again gave several lectures before returning to Germany on 16 April.

  694. John Charles Fields (1863-1932)
    • The series of International Congresses of Mathematicians began in Zurich in 1897 but no congress was held during World War I (1914-18).

  695. Oleksandr Mikolaiovich Sharkovsky (1936-)
    • For a series of scientific papers "Theory of Dynamical Systems: Methods and Applications".

  696. Luca Valerio (1552-1618)
    • In fact, his work represents, at least for the Archimedean tradition, the apex of that intellectual movement, in the sense that with it a series of themes that had appeared throughout the mathematics of the Cinquecento were brought to maturity, a level to which subsequent research would necessarily have to raise the framework of Renaissance mathematics, which was oriented especially toward rediscovery and translation of, and commentary on, classical texts.

  697. Alfred Barnard Basset (1854-1930)
    • Every three of four years the titles and head-notes of all papers relating to each separate branch of science should be copied out and arranged in proper order, and a series of digests of each separate branch of science should be published.

  698. William Moser (1927-2009)
    • The full collection, corrected and improved, has been published by Mathematics Magazine in its spectrum series as "500 Mathematical Challenges" in 1995.

  699. Sarvadaman Chowla (1907-1995)
    • He wrote on additive number theory (lattice points, partitions, Waring's problem), analysis, Bernoulli numbers, class invariants, definite integrals, elliptic integrals, infinite series, the Weierstrass approximation theorem), analytic number theory (Dirichlet L-functions, primes, Riemann and Epstein zeta functions), binary quadratic forms and class numbers, combinatorial problems (block designs, difference sets, Latin squares), Diophantine equations and Diophantine approximation, elementary number theory (arithmetic functions, continued fractions, and Ramanujan's tau function), and exponential and character sums (Gauss sums, Kloosterman sums, trigonometric sums).

  700. Leonid Andreevich Pastur (1937-)
    • Both formal properties of the perturbation series and the applications to the evaluation of the spectrum in the presence of impurity centres are discussed.

  701. Gheorghe Calugrenu (1902-1976)
    • Returning to Weierstrass's point of view on analyticity and the definition of analytic functions via Taylor series elements, Calugăreănu created the theory of invariants and covariants of analytic continuation.

  702. Monteiro da Rocha (1734-1819)
    • He proposed a method to accelerate the convergence of numerical series, similar to the one that would be formulated by Lewis Richardson in the beginning of the twentieth century.

  703. Jorgen Gram (1850-1916)
    • He published a paper on these topics On series expansions determined by the methods of least squares and for this work he was awarded the degree of Doctor of Science in 1879.

  704. Abraham Sharp (1653-1742)
    • In fact Sharp used Gregory's series with x = √3 to calculate π to 72 places, a task he had carried out in 1699.

  705. Stefan Warschawski (1904-1989)
    • In 1955 he published two papers in Experiments in the computation of conformal maps published in the National Bureau of Standards Applied Mathematics Series.

  706. Steven Vajda (1901-1995)
    • Ledermann and Vajda jointly edited most of the core volumes in the Handbook of Applicable Mathematics series.

  707. Willem de Sitter (1872-1934)
    • He published a series of papers (1916-17) on the astronomical consequences of Einstein's general theory of relativity.

  708. William Prager (1903-1980)
    • In November and December 1954 Prager gave a series of lectures at the Polytechnic Institute in Zurich.

  709. Hubert Linfoot (1905-1982)
    • His book can be considered as a collection of a series of monographs in special fields of optics in which he is interested.

  710. Heinrich Burkhardt (1861-1914)
    • His main work was in analysis, particularly the theory of trigonometric series, and on the history of mathematics.

  711. Bernadette Perrin-Riou (1955-)
    • In 1979 she published the 110-page Plongement d'une extension diedrale dans une extension diedrale ou quaternionienne Ⓣ in the Mathematical Publications of Orsay Series.

  712. Pierre Rémond de Montmort (1678-1719)
    • After his death his paper on summing infinite series, De seriebus infinitis tractatus Ⓣ, was published in the Philosophical Transactions of the Royal Society.

  713. Isadore Singer (1924-)
    • Singer's series of five papers with Michael F Atiyah on the Index Theorem for elliptic operators (which appeared in 1968 - 71) and his three papers with Atiyah and V K Patodi on the Index Theorem for manifolds with boundary (which appeared in 1975 - 76) are among the great classics of global analysis.

  714. George Szekeres (1911-2005)
    • In 1965 he wrote a numerical analysis paper Some estimates of the coefficients in the Chebyshev series expansion of a function and a paper dealing with a combinatorial problem On a problem of Schutte and Erdős written jointly with his wife Esther.

  715. Ralph Jeffery (1889-1975)
    • His presidential address to the Society was Trigonometric series which he gave in 1953 and three years later it was published as a 39 page book by the University of Toronto Press.

  716. Kenjiro Shoda (1902-1977)
    • develops generalized theories of normal chains, composition series, of direct and subdirect products, and generalizations of the Jordan-Holder and the Remak-Schmidt-Ore theorems.

  717. Corrado Gini (1884-1965)
    • Series A 126 (3) (1963), 466-467.','8] as follows:- .

  718. Wilhelm Wien (1864-1928)
    • He conducted an unsuccessful series of experiments, using platinum foil, trying to establish a new unit of light [',' H Kangro, Biography in Dictionary of Scientific Biography (New York 1970-1990).','1]:- .

  719. Theodoros Varopoulos (1894-1957)
    • Varopoulos's paper Sur quelque theorems de M Remoundos Ⓣ (1920) refines theorems, postulated by Remoundos in a series of lectures at the University of Athens in 1918-1919, about the theory of functions.

  720. John Torrence Tate (1925-)
    • In his thesis, which has become a classic, he proved the functional equation for Hecke's L-series by a novel method involving Fourier analysis on idele groups.

  721. Émile Picard (1856-1941)
    • Picard's solution was represented in the form of a convergent series.

  722. Bernard Malgrange (1928-)
    • He has also been awarded the Prix Peccot-Vimont from the College de France in 1962 and delivered the prestigious series of lectures on his research there in that year, the Cours Peccot.

  723. Charles Hayes (1678-1760)
    • He published a supplement of this last mentioned work in 1747 Series of Kings of Argos and of Emperors of China from Fohi to Jesus Christ.

  724. Gheorghe Pic (1907-1984)
    • The fourteen sections are: semigroups; divisibility in commutative semigroups; groups; semigroups of fractions; equivalences induced by a subgroup; conjugate subsets; inner automorphisms; Sylow subgroups; extensions of groups; direct products of subgroups; normal series; nilpotent groups; solvable groups; free semigroups and free groups.

  725. Vera Nikolaevna Kublanovskaya (1920-2012)
    • the method is applied to particular problems such as, for instance, solution of systems of linear equations, determination of eigenvalues and eigenvectors of a matrix, integration of differential equations by series, solution of Dirichlet problem by finite differences, solution of integral equations, etc.

  726. Achille-Pierre Dionis du Séjour (1734-1794)
    • Over a period of almost 20 years from 1764 he wrote a series of memoirs on eclipses, occultations (when one astronomical body comes in front of another), calculating orbits, and other such topics, and these were brought together in a two volume work Traite analytique des mouvements apparents des corps celestes Ⓣ which he published, volume one in 1786 and volume two in 1789.

  727. Hendrik Lorentz (1853-1928)
    • In 1909 he published his "Theory of Electrons", based on a series of lectures at Columbia University, and in 1916 he published in French at Leipzig an account of statistical thermodynamic theories, based on lectures delivered at the College de France in 1912.

  728. Heinrich Maschke (1853-1908)
    • Among the papers he published while at Chicago are: On systems of six points lying in three ways in involution (1896), Note on the unilateral surface of Mobius (1900), A new method of determining the differential parameters and invariants of quadratic differential quantics (1900), On superosculating quadric surfaces (1902), A symbolic treatment of the theory of invariants of quadratic differential quantics of n variables (1903), Differential parameters of the first order (1906); The Kronecker-Gaussian curvature of hyperspace (1906) and A geometrical problem connected with the continuation of a power-series (1906).

  729. John Landen (1719-1790)
    • Landen wrote on dynamics, summation of series and an important transformation giving a relation between elliptic functions.

  730. Annie Jump Cannon (1863-1941)
    • For hundreds of years, Oxford University has been giving honorary degrees to leading men in the fields of science and art, but for the first time, a woman was so honoured when the degree of Doctor of Science was conferred on Miss Annie Jump Cannon, of Harvard College Observatory, in recognition of a long series of valuable contributions to astronomy, chief of which is the completion of a catalogue of 225,300 stars - "The Henry Draper Catalogue of Stellar Spectra." ..

  731. Isaac Todhunter (1820-1884)
    • Still less can I imagine how it came to pass that he published a whole series of excellent mathematical works.

  732. Jean-Baptiste Biot (1774-1862)
    • Later in 1808, together with Claude-Louis Mathieu, he embarked on a series of measurements of the length of the seconds pendulum at different points on the meridian, in particular at Bordeaux and at Dunkirk.

  733. Ugo Amaldi (1875-1957)
    • This was the first of a series of textbooks written by Amaldi and Enriques for use in secondary schools.

  734. Joseph Privat de Moličres (1677-1742)
    • He also published a series of Memoirs of the Academy and in several articles in the Journal de Trevoux.

  735. William Burnside (1852-1927)
    • This paper was the first of a series which Burnside described himself as follows (see for example [',' C W Curtis, Pioneers of representation theory : Frobenius, Burnside, Schur, and Brauer (Providence, RI, 1999).','3]):- .

  736. Albrecht Fröhlich (1916-2001)
    • In the present paper the fields of at most class two over the rational field are studied, the class of a field being defined as the length of the central series of the Galois group.

  737. Joăo Baptista Lavanha (1555-1624)
    • He began work on a map of Aragon in about 1611 by making a series of geodesic measurements [',' L de Alburquerque, Biography in Dictionary of Scientific Biography (New York 1970-1990).','1]:- .

  738. Norman Routledge (1928-2013)
    • At Cambridge he attended J E Littlewood's lectures which were a series of comments on one of his books.

  739. Jakob Rosanes (1842-1922)
    • He also wrote a series of papers on linearly dependent point systems in a plane and in space.

  740. Philip Stein (1890-1974)
    • Roseveare wrote a number of papers including: A chapter on algebra (1903), On convergence of series (1905), On 'Circular Measure' and the product forms of the sine and cosine (1905), Expansions of trigonometrical functions (1905), and Expansions of functions in general (1905).

  741. Victor Amazaspovich Ambartsumian (1908-1996)
    • This led to a series of research articles developing the topic further.

  742. Peter Ramus (1515-1572)
    • Using this approach Ramus worked on many topics and wrote a whole series of textbooks on logic and rhetoric, grammar, mathematics, astronomy, and optics.

  743. Eustachy yliski (1889-1954)
    • His health deteriorated and he suffered a series of strokes.

  744. William Niven (1842-1917)
    • He also published On Certain Definite Integrals Occurring in Spherical Harmonic Analysis, and on the Expansion in Series of the Potentials of the Ellipsoid and of the Ellipse (1879), On Ellipsoidal Harmonics (1891), On the Calculation of the Trajectories of Shot (1877), and The Calculation of Ellipsoidal Harmonics (1906) in the Philosophical Transactions or the Proceedings of the Royal Society.

  745. Jean Beaugrand (about 1590-1640)
    • There is little wonder that he was not in favour among the French mathematicians for he had attacked the work of Desargues and published a series of pamphlets attacking the work of Descartes.

  746. Ottó Varga (1909-1969)
    • This paper was the third in a series of papers on integral geometry, the first two being written by Blaschke and both being published in 1935.

  747. Magnus Wenninger (1919-2017)
    • This book is, however, more than instructions for a series of models: the underlying ideas are explained in enough detail to add greatly to one's satisfaction without taking control of what is intended as a practical book.

  748. Gaspard Monge (1746-1818)
    • Over the next few years he submitted a series of important papers to the Academie on partial differential equations which he studied from a geometrical point of view.

  749. Anders Lexell (1740-1784)
    • Lexell did work in analysis on topics other than differential equations, for example he suggested a classification of elliptic integrals and he worked on the Lagrange series.

  750. Hans Lewy (1904-1988)
    • he published a series of fundamental papers on partial differential equations and the calculus of variations.

  751. Wacaw Sierpiski (1882-1969)
    • His work on functions of a real variable include results on functional series, differentiability of functions and Baire's classification.

  752. Ivan Vidav (1918-2015)
    • Besides his scientific and pedagogical obligations he was an active member of the Society of Mathematicians, Physicists and Astronomers of Slovenia from the foundation of the Society, serving as its chairman from 1951 to 1954, its vice chairman from 1955 to 1959, for many years the organizer of the national competition in mathematics for grammar school pupils, the main editor of several book series published by the Society, and more.

  753. Tomás Rodríguez Bachiller (1899-1980)
    • When Einstein gave a series of three public lectures on relativity on 5, 6 and 8 March, Bachiller was told to attend them, take notes and prepare summaries.

  754. Torsten Carleman (1892-1949)
    • Carleman is now remembered for remarkable results in integral equations (1923), quasi-analytic functions (1926), harmonic analysis (1944), trigonometric series (1918-23), approximation of functions (1922-27) and Boltzmann's equation (1944).

  755. Elliott Montroll (1916-1983)
    • During his time at Maryland, Montroll published Topics in statistical mechanics of interacting particles which was 86 pages of mimeographed notes of a lecture series, written jointly with G F Newell.

  756. Werner Romberg (1909-2003)
    • is exactly the same operation which Huygens performs but he does it out of a different motivation, namely that of summing a geometric series.

  757. Max Newman (1897-1984)
    • A series of papers by Newman on this topic between 1926 and 1932 revolutionised the field.

  758. James McConnell (1915-1999)
    • In 1964-65 he gave a series of seminars on the applications of group theory to elementary particles and he wrote these up in the 111-page tract Introduction to the group theory of elementary particles (1965).

  759. Bernhard Neumann (1909-2002)
    • In 1955 when I first arrived in Manchester to work with B H Neumann he suggested that I read his paper 'Ascending derived series' which had only just been submitted for publication.

  760. Grace Hopper (1906-1992)
    • Compute the coefficients of the arctan series by next Thursday.

  761. Alfred Kempe (1849-1922)
    • Kempe worked on the topic and presented a series of lectures at the Royal Institution on How to draw a straight line: A lecture on linkages in 1877.

  762. Jacob Amsler (1823-1912)
    • adapted easily to the determination of static and inertial moments and to the coefficients of Fourier series: it proved especially useful to shipbuilders and railway engineers.

  763. Anatoly Malcev (1909-1967)
    • We mentioned some of the prizes he received above, such as the State Prize in 1946, but another important honour which he received in 1964 was a Lenin Prize for his series of papers on the applications of mathematical logic to algebra.

  764. Luigi Cremona (1830-1903)
    • Also while at Bologna Cremona developed the theory of birational transformations, later known as Cremona transformations, and wrote a series of papers on twisted cubic surfaces.

  765. Aryabhata (476-550)
    • It also contains continued fractions, quadratic equations, sums of power series and a table of sines.

  766. Cathleen Morawetz (1923-2017)
    • In a series of three significant papers in the late 1950s, Cathleen Morawetz used functional analysis coupled with ingenious new estimates for an equation of mixed type, i.e.

  767. Charles Hermite (1822-1901)
    • The letters he exchanged with Jacobi show that Hermite had discovered some differential equations satisfied by theta-functions and he was using Fourier series to study them.

  768. Agner Erlang (1878-1929)
    • Series A 114 (1) (1951), 103-104.','14]:- .

  769. Hans-Jürgen Hoehnke (1925-2007)
    • In this series of papers he investigates the structural relations between Brandt groupoids, Ehresmann groupoids, semigroups, Brandt semigroups, categories and groups.

  770. Alan Day (1941-1990)
    • He even managed to travel to Connecticut to deliver a series of lectures.

  771. Andrzej Mostowski (1913-1975)
    • He was the editor of the Mathematical, Astronomical and Physical series of the Bulletin of the Polish Academy of Sciences, on the editorial board of several journals including Fundamenta Mathematicae, Dissertationes Mathematicae, the Journal of Symbolic Logic and Studia Logica.

  772. Shanti Swarup Gupta (1925-2002)
    • The first of these is Multiple decision procedures: theory and methodology of selecting and ranking populations written jointly with S Panchapakesan and published in the Wiley Series in Probability and Mathematical Statistics in 1979.

  773. Arne Beurling (1905-1986)
    • Beurling worked on the theory of generalized functions, differential equation, harmonic analysis, Dirichlet series and potential theory.

  774. Philipp Frank (1884-1966)
    • In mathematics he worked on the calculus of variations, Fourier series, function spaces, Hamiltonian geometrical optics, Schrodinger wave mechanics, and relativity.

  775. Jean-Charles de Borda (1733-1799)
    • He also developed a series of trigonometric tables in conjunction with his surveying techniques.

  776. Luis Antonio Santaló (1911-2001)
    • At the end of 1934 Luis Santalo found his now famous proofs of the isoperimetric inequality in the plane and Blaschke himself found the fundamental kinematic formula and started a series of papers under the general title of "integral geometry".

  777. Loo-Keng Hua (1910-1985)
    • In addition, Chinese Television (CCTV) produced a mini-series telling the story of Hua's life, which has been shown at least twice since then.

  778. Bill Morton (1930-)
    • The second conference in this series was held at the University of Reading in April 1985.

  779. Eugčne Rouché (1832-1910)
    • The first of these theses was on analysis and was entitled Sur le developpement des fonctions en series ordonnees suivant les denominateurs des reduites d'une fraction continue Ⓣ while the second was on mechanics and was entitled Sur les integrales communes a plusieurs problemes de mecanique relatifs au mouvement d'un point sur une surface Ⓣ.
    • The magnificent discoveries of modern geometry have not penetrated into teaching; abandoned by the official syllabuses, they do not occupy the place in the series of mathematical studies due to them; we speak of it only a little and incidentally in Analytical Geometry, where it seems to be wrong to replace the admirable ideas created by Descartes by new ideas.

  780. Alan Baker (1939-2018)
    • In fact eight of his papers had appeared in print before he submitted his doctoral dissertation: Continued fractions of transcendental numbers (1962); On Mahler's classification of transcendental numbers (1964); Rational approximations to certain algebraic numbers (1964); On an analogue of Littlewood's Diophantine approximation problem (1964); Approximations to the logarithms of certain rational numbers (1964); Rational approximations to the cube root of 2 and other algebraic numbers (1964); Power series representing algebraic functions (1965); and On some Diophantine inequalities involving the exponential function (1965).

  781. Theodor Estermann (1902-1991)
    • He submitted the paper On certain functions represented by Dirichlet series to the Proceedings of the London Mathematical Society in November 1926.

  782. Johann Werner (1468-1522)
    • Perhaps of more interest is a series of twelve supplementary notes to this work.

  783. Tosio Kato (1917-1999)
    • The Department of Mathematics of the University of California, Berkeley, published Kato's notes Quadratic forms in Hilbert spaces and asymptotic perturbation series in 1955.

  784. Alexander Dinghas (1908-1974)
    • Dinghas produced a series of papers on isoperimetric problems in spaces of constant curvature.

  785. Leslie Fox (1918-1992)
    • He contributed in many ways to promoting numerical analysis, for example in running summer schools, in developing links to industry, forming links with schools through the Mathematical Association, and with the writing of a wonderful series of books on the subject.

  786. Alfred Brauer (1894-1985)
    • From the late 1950s Brauer published a series of papers on nonnegative matrices, a topic studied by Frobenius towards the end of his career.

  787. Irving Reiner (1924-1986)
    • The book has developed from a series of lectures for graduate students, and the author's intention has been to make the book - and the subject - easily accessible to a large variety of readers.

  788. Al-Khwarizmi (about 790-about 850)
    • This would not be worth mentioning if a series of conclusions about al-Khwarizmi's personality, occasionally even the origins of his knowledge, had not been drawn.

  789. John Stallings (1935-2008)
    • Stallings visited the Tata Institute of Fundamental Research in 1967 and gave a series of lectures which were written up for publication by G Ananda Swarup who writes [',' G A Swarup, John Robert Stallings (1935-2008) (Monday, 8 December 2008).','3]:- .

  790. Andrea Tacquet (1612-1660)
    • Tacquet introduced several ways of thinking which proved important in giving a foundation for future progress, for example in noting that one could pass from a finite progression to an infinite series.

  791. Karl Peterson (1828-1881)
    • Peterson's most important paper was 'On the ratios and relationships between curved surfaces' (1866), devoted to deformation of surfaces, which laid the foundation for a series of papers on the problem of bending on a principal basis, i.e., preserving the conjugacy of a certain net on the surface, the first example of which for deformation of surfaces of revolution on a surface of revolution was found by Minding ..

  792. James Thomson (1786-1849)
    • They inspired a series of rebellions in 1798 which were brutally repressed by the British.

  793. Ivar Bendixson (1861-1935)
    • In this area he first studied uniform convergence of series of real functions and took an important step towards giving precise conditions when the limit function of continuous functions is continuous.

  794. Richard Rado (1906-1989)
    • Some of his more minor work was in topics such as the convergence of sequences and series.

  795. Neil Trudinger (1942-)
    • This 1998 edition was reprinted in the "Classics in Mathematics" series by Springer-Verlag in 2001.

  796. Alexander Ostrowski (1893-1986)
    • One consequence of this association was his monograph Solution of equations and systems of equations which was published in 1960 and was the result of a series of lectures he had given at the National Bureau of Standards.

  797. Alexander Aleksandrovich Kirillov (1936-)
    • Kirillov published a series of important papers in Functional Analysis and Applications.

  798. Charles Hutton (1737-1823)
    • In 1776 he published A new and general method of finding simple and quickly converging series and two year later, in the same Transactions he published The force of fired gunpowder and the velocity of cannon balls.

  799. Maurice Fréchet (1878-1973)
    • The book Lecons sur les fonctions de variables reelles et les developpements en series de polynomes Ⓣ was published in 1905.
    • This campaign took the unusual form of a survey sent to colleagues all over the world as well as a series of papers, committee reports, and censure motions within the International Institute of Statistics.

  800. Charles Pisot (1910-1984)
    • In the summer of 1963, Pisot gave a series of lectures at the University of Montreal in Canada.

  801. Franciszek Szafraniec (1940-)
    • The first of these when written up for the proceedings became the fourth in a series of papers On normal extensions of unbounded operators some of which were written with Jan Stochel, one of his colleagues at the Jagiellonian University.

  802. Wang Yuan (1930-)
    • Wang Yuan fell in love with analytic number theory and gave a series of lectures to the graduate seminar based on Ingham's book The distribution of prime numbers.

  803. Lars Hörmander (1931-2012)
    • Hormander spent the summers 1960-61 at Stanford University as an invited professor, and took advantage of this time to honour the offer of the 'Springer Grundlehren series' of publishing a book about partial differential equations.

  804. Blagoj Popov (1923-2014)
    • "At the moment I am interested in the problem of linearization of the product of orthogonal functions and the summation of certain series," Dr Popov said.

  805. Ernst Straus (1922-1983)
    • In 1949 Straus collaborated with Richard Bellman publishing Continued fractions, algebraic functions and the Pade table in which they gave a method for obtaining the rational approximants of Frobenius-Pade for power series expansions of algebraic functions.

  806. Albert Girard (1595-1632)
    • The second thing which Albert Girard mentions, is a way of exhibiting a series of rational fractions, that converge to the square root of any number proposed, and that very fast.

  807. Ludwig Sylow (1832-1918)
    • G A Miller writes [',' G A Miller, Professor Ludvig Sylow, Science, New Series 49 (1256) (1919), 85.','10]:- .

  808. Lev Landau (1908-1968)
    • This was a series of examinations that Landau set for which students had to spend a long time preparing.

  809. Benjamin Franklin Finkel (1865-1947)
    • He retired from his professorship at Drury College in 1937 and, three years later, began publishing a series of articles entitled A History of American Mathematical Journals in the National Mathematics Magazine.

  810. Irving John Good (1916-2009)
    • His first publications appeared at this time: The approximate local monotony of measurable functions (1940), The fractional dimensional theory of continued fractions (1941), and Note on the summation of a classical divergent series (1941).

  811. Eugen Netto (1846-1919)
    • There he taught courses on advanced algebra, the calculus of variations, mechanics, Fourier series, and synthetic geometry.

  812. Kiyosi Ito (1915-2008)
    • In 1960 Ito visited the Tata Institute in Bombay, India, where he gave a series of lectures surveying his own work and that of other on Markov processes, Levy processes, Brownian motion and linear diffusion.

  813. Daniel Pedoe (1910-1998)
    • I was not asked to teach Dyson's class geometry, but just to continue with a set of problems on infinite series.

  814. Cora Sadosky (1940-2010)
    • (eds.), Harmonic Analysis, Partial Differential Equations, Complex Analysis, Banach Spaces, and Operator Theory 1, Association for Women in Mathematics Series 4 (Springer International Publishing, Switzerland, 2016), 3-24.','14]):- .

  815. Alexander Weinstein (1897-1979)
    • For example he solved Helmholtz's problem for jets, giving the first uniqueness and existence theorems for free jets in a series of papers from 1923 to 1929.

  816. Pierre Deligne (1944-)
    • The areas on which he has worked, in addition to algebraic geometry, are Hilbert's 21st problem, Hodge theory, theory of moduli, modular forms, Galois representations, L-series and the Langlands conjectures, and representations of algebraic groups.

  817. Zoltán Balogh (1953-2002)
    • What makes Zoli's research especially stand out are a series of solutions to several long-standing problems in the field, which he obtained at an amazing pace starting in the mid-1980s, continuing essentially until his death.

  818. Bill Boone (1920-1983)
    • The method is a refinement of that used by the author in the earlier papers of this series ..

  819. Brahmagupta (598-670)
    • Rules for summing series are also given.

  820. John von Neumann (1903-1957)
    • In the second half of the 1930's and the early 1940s von Neumann, working with his collaborator F J Murray, laid the foundations for the study of von Neumann algebras in a fundamental series of papers.

  821. Ernest Corominas i Vigneaux (1913-1992)
    • Is it possible to extend to the derivatives of higher order or differential quotients, the classical theorems on the formal derivation of series? This problem quickly led me to a more general one.

  822. William Horner (1786-1837)
    • Horner made other mathematical contributions, however, publishing a series of papers on transforming and solving algebraic equations, and he also applied similar techniques to functional equations.

  823. Archibald Macintyre (1908-1967)
    • A few years later, in 1952, the two Macintyres published a more conventional type of joint paper, namely the 2-author work Theorems on the convergence and asymptotic validity of Abel's series which was published in the Proceedings of the Royal Society of Edinburgh.

  824. Hendrik de Vries (1867-1954)
    • These culminated in a series of articles in the 'Nieuw Tijdschrift voor Wiskunde' (New Journal of Mathematics), which were later collected, together with some other items, in a three volume publication entitled 'Historische Studien' (1926).

  825. Kenneth Appel (1932-2013)
    • Eight days later, the two shared a canoe, talking about various subjects, including Carole's recent purchase of a season ticket to a classical music concert series.

  826. Allen Shields (1927-1989)
    • I probably fell in love with Allen the first time I heard him lecture (it was on power series, in the University of Michigan Math Club).

  827. Bernard de Fontenelle (1657-1757)
    • Nowhere is the meaning of science made more clear, and its value so appreciated, as in that series of "lives of the scientists" that we know as the "Eloges." .

  828. Eduard Helly (1884-1943)
    • He taught in a Gymnasium, gave private tuition, and wrote solution manuals for a series of standard textbooks.

  829. Ali Moustafa Mosharrafa (1898-1950)
    • in 1923 for his thesis The Quantum Theory of Spectral Series.

  830. William Metzler (1863-1943)
    • Furthermore, he was the one who first pointed out that one could have the transcendental functions of a square matrix simply by substituting it into the appropriate Taylor series.

  831. Cornelius Lanczos (1893-1974)
    • Lanczos was much influenced by Fejer; he learnt from him about Fourier series, orthogonal polynomials, and interpolation.

  832. Timothy Gowers (1963-)
    • Dr W T Gowers of University College, London, is awarded a Junior Whitehead Prize for his work in applying infinite combinatorics to resolve a series of longstanding questions in Banach space theory, some originating with Banach himself.

  833. Wim Cohen (1923-2000)
    • He worked on the problem of analysing congestion in telephone systems and from 1956 began publishing a series of papers.

  834. Nassim Taleb (1960-)
    • Counterintuitively, even a long series of wins can be the result of chance; it all depends on how many attempts you make.

  835. Paramesvara (about 1370-about 1460)
    • Paramesvara made a series of eclipse observations between 1393 and 1432 which we have referred to above.

  836. Narayana (about 1340-about 1400)
    • He used formulae and rules for the relations between magic squares and arithmetic series.

  837. Konrad Zuse (1910-1995)
    • In fact Zuse designed several computers other than those of his Z series.

  838. Tom Cowling (1906-1990)
    • During his time at Leeds the heavy load on a conscientious university professor with both departmental responsibilities and a commitment to scholarship took its toll; a series of health problems - a duodenal ulcer operation in 1954, a slipped disk in 1957 and a mild heart attack in 1960 - caused a slowing down of his activities well before his retirement.

  839. Eliakim Moore (1862-1932)
    • He also studied infinite series of finite simple groups.

  840. Louis Lefébure de Fourcy (1787-1869)
    • He began publishing a series of books in 1827 and over the next few years various texts appeared: Lecons de geometrie analytique Ⓣ (1827), Theories du plus grand commun diviseur algebrique et de l'elimination entre deux equations a deux inconnue Ⓣ (1827), Traite de geometrie descriptive Ⓣ (2 vols.

  841. Johannes de Groot (1914-1972)
    • In another 1942 paper, Bemerkung uber die analytische Fortsetzung in bewerteten Korpern Ⓣ, he gives a new proof of the theorem that it is impossible to obtain an analytic extension of a power series with coefficients in a p-adic field.

  842. Georg Faber (1877-1966)
    • He received his university teaching qualification from the University of Wurzburg in 1905, after submitting his Habilitationsschrift presenting work on power series in several variables.

  843. Pierre-Louis Moreau de Maupertuis (1698-1759)
    • This work, and other work by Maupertuis on heredity, proposed a series of conjectures which some see as an early version of the theory of evolution.

  844. Endre Szemerédi (1940-)
    • He won the Rolf Schock Prize in Mathematics decided by the Royal Swedish Academy of Sciences, 2008; the DeLong Lecture Series, University of Colorado, 2010; the Abel Prize, 2012 [',' Citation, Endre Szemeredi, The Abel Prize Laureate 2012, The Abel Prize (2012).','4]:- .

  845. Sigekatu Kuroda (1905-1972)
    • It has been published in a series of thirteen papers under the common title "An investigation of the logical structure of mathematics." .

  846. Vladimir Abramovich Rokhlin (1919-1984)
    • He planned a series of papers and books on popular and general scientific subjects, discussed problems of education, methodology, publication of literature, etc.

  847. Hans Petersson (1902-1984)
    • In 1982 Petersson published an important book Modulfunktionen und quadratische Formen Ⓣ in Springer-Verlag's Ergebnisse der Mathematik und ihrer Grenzgebiete series.

  848. Percy Daniell (1889-1946)
    • If ever a boy ventured to ask whether the work in hand was of any help for his scholarship examination, Levett's invariable reply would be: "It is not my business to win scholarships for you, I have to make you love beautiful series." .

  849. Wilbur Knorr (1945-1997)
    • Early in the 1990s he had begun a series of projects on medieval illuminated astronomical manuscripts, some of which he published.

  850. John T Graves (1806-1870)
    • From the recent researches of MM Poisson and Poinsot on angular section, and their discovery of error in trigonometrical formulae usually considered complete, my attention has been drawn to analogous incorrectness in logarithmic series.

  851. William Thomson (1824-1907)
    • This paper Fourier's expansions of functions in trigonometrical series was written to defend Fourier's mathematics against criticism from the professor of mathematics at the university of Edinburgh.

  852. Jules Bienaymé (1796-1878)
    • Series A (General) 142 (2) (1979), 259-260.','12] (reviewing C C Heyde and E Seneta's book [',' C C Heyde and E Seneta, I J Bienayme : Statistical theory anticipated (Springer-Verlag, New York-Heidelberg, 1977).','1]):- .

  853. Tullio Levi-Civita (1873-1941)
    • Levi-Civita's work was of extreme importance in the theory of relativity, and he produced a series of papers elegantly treating the problem of a static gravitational field.

  854. Georges Lemaître (1894-1966)
    • In 1933 Einstein and Lemaitre gave a series of lectures in California.

  855. Jules Lissajous (1822-1880)
    • At the conclusion of this beautiful series of experiments, which, thanks to the skill of those who performed them, were all successful, on the motion of Mr Faraday, the thanks of the meeting were unanimously voted to M M Lissajous and Duboscq and communicated to those gentlemen by his Grace the President, The Duke of Northumberland.

  856. Hermann Grassmann (1809-1877)
    • After writing a series of articles on constitutional law, Grassmann became increasingly at odds with the political direction the newspaper was going and withdrew form it.

  857. Alicia Boole Stott (1860-1940)
    • These are On certain series of sections of the regular four-dimensional hypersolids (1900) and Geometrical deduction of semiregular from regular polytopes and space fillings (1910).

  858. Gyula Maurer (1927-2012)
    • These are: Remark on multiplicative arithmetic functions; Groups of infinite permutations; Contribution to the study of groups from their quasi-centre; On the notion of power; and On the normal series of the group of generalized infinite permutations.

  859. Jürgen Moser (1928-1999)
    • First we mention Lectures on Hamiltonian systems (1968) which examines problems of the stability of solutions, the convergence of power series expansions, and integrals for Hamiltonian systems near a critical point.

  860. Philip Franklin (1898-1965)
    • However, he is best known for textbooks he published on calculus, differential equations, complex variable and Fourier series.

  861. Ivor Grattan-Guinness (1941-2014)
    • I was already amazed both that many techniques and definitions in mathematical analysis, and also set theory itself, had been stimulated by problems in Fourier series, and equally that no teacher or book on the subject ever mentioned the links (because, I soon realised, the authors themselves were ignorant of them).

  862. László Filep (1941-2004)
    • We must not give the impression that Filep's only research interest was in the history of mathematics for he also published a long series of papers on fuzzy groups, some written with his collaborator Gyula Iulius Maurer, beginning in 1987.

  863. William Whiston (1667-1752)
    • While he was being subjected to charges of heresy he was bold enough to set out his religious beliefs in a series of pamphlets Primitive Christianity Revived (1711-12).

  864. John Wallis (1616-1703)
    • About the beginning of my mathematical studies, as soon as the works of our celebrated countryman, Dr Wallis, fell into my hands, by considering the Series, by the Intercalation of which, he exhibits the Area of the Circle and the Hyperbola..

  865. Ivan Matveevich Vinogradov (1891-1983)
    • However it was Vinogradov who, in a series of papers in the 1930s, brought the method to its full potential.

  866. François Budan (1761-1840)
    • In total Budan published ten mathematical works and as one further example of his contributions we note that he submitted a paper on the summation of series to the Academy of Sciences in 1802 which was refereed by Biot and Lacroix.

  867. Lev Pontryagin (1908-1988)
    • He then produced a series of papers on differential games which extends his work on control theory.

  868. Ralph Fowler (1889-1944)
    • This work continued in a series of papers through the 1920s leading to the Adams Prize of the University of Cambridge in 1923-24 and was published in 1929 as the seminal volume, Statistical Mechanics, which had a second edition, minus the astrophysical applications, published in 1936.

  869. Freeman Dyson (1923-)
    • It gives us the opportunity to follow his own research about congruence properties of partitions, special series and infinite products, generating functions, and modular functions, ..

  870. Florence Nightingale David (1909-1993)
    • The first investigates the behaviour of the nonparametric test for randomness based on the number of alternations of an event and its complement in a series of trials.

  871. Jean-Louis Koszul (1921-2018)
    • In the autumn of 1958 he again held a seminar series in Sao Paulo, this time on symmetric spaces.

  872. Farkas Bolyai (1775-1856)
    • His study of the convergence of series includes a test equivalent to Raabe's test which he discovered independently and at about the same time as Raabe.

  873. Kurt Hensel (1861-1941)
    • In 1897 the Weierstrass method of power-series development for algebraic functions led him to the invention of the p-adic numbers.

  874. Apollonius (about 262 BC-about 190 BC)
    • Included in it are a series of propositions which, though worked out by the purest geometrical methods, actually lead immediately to the determination of the evolute of each of the three conics; that is to say, the Cartesian equations of the evolutes can be easily deduced from the results obtained by Apollonius.

  875. Lev Shnirelman (1905-1938)
    • L A Lyusternik became a friend and important collaborator with Shnirelman and together they made significant contributions to topological methods in the calculus of variations in a series of paper written jointly between 1927 and 1929.

  876. Blaise Pascal (1623-1662)
    • From about this time Pascal began a series of experiments on atmospheric pressure.

  877. Heinrich Scholz (1884-1956)
    • Scholz's connections with Bieberbach had led earlier to funds being provided for a series of monographs on mathematical logic which had started in 1937.

  878. Guillaume Bigourdan (1851-1932)
    • Bigourdan made the first of two series of observations between 18 April 1902 and 29 June 1902.

  879. Elena Moldovan Popoviciu (1924-2009)
    • In a long series of articles she studied approximation by interpolatory sets, convexity with respect to them, etc.

  880. Joseph Boussinesq (1842-1929)
    • Lagrange had already tried this route and written the resulting series of differential equations, but had found their integration to exceed the possibilities of contemporary analysis unless nonlinear terms were dropped.

  881. Tibor Radó (1895-1965)
    • He gave his series of talks on his major contributions on surface area.

  882. Wilhelm Magnus (1907-1990)
    • During this period Magnus introduced Lie ring methods to study the lower central series of free groups.

  883. Robert Thompson (1931-1995)
    • As a result he published a number of series of papers attacking particular problems.

  884. Oskar Anderson (1887-1960)
    • From 1907 to 1915 he was A A Chuprov's assistant and his dissertation was on variance-difference methods for analysing time series.

  885. Shiing-shen Chern (1911-2004)
    • In the spring of 1932 Blaschke visited Peking and gave a series on topological questions in differential geometry.

  886. Joseph Liouville (1809-1882)
    • Sturm and Liouville examined general linear second order differential equations and examined properties of their eigenvalues, the behaviour of the eigenfunctions and the series expansion of arbitrary functions in terms of these eigenfunctions.

  887. Wassily Hoeffding (1914-1991)
    • I performed series of random tossings and recorded their outcomes before I knew much about probability theory.

  888. Aleksandr Mikhailovich Lyapunov (1857-1918)
    • He returned to the problem that Chebyshev had placed before him and, in an extensive series of papers which continued until his death, developed the theory of figures of equilibrium of rotating heavy liquids.

  889. Luigi Menabrea (1809-1896)
    • In August 1840 Charles Babbage gave a series of lectures on his Analytical Engine at the Academy of Sciences in Turin.

  890. Vilhelm Bjerknes (1862-1951)
    • The next step forward in the mathematical approach was due to Richardson in 1922 when he reduced the complicated equations produced by Bjerknes's Bergen School to long series of simple arithmetic operations.

  891. Fritz Ursell (1923-2012)
    • After taking courses by G H Hardy on divergent series, J E Littlewood on complex analysis, A E Ingham on number theory, W V D Hodge on Riemann surfaces, and P A M Dirac on quantum mechanics, Ursell graduated with distinction in 1943.

  892. Shokichi Iyanaga (1906-2006)
    • This came about through talking to the Departmental Assistant T Shimizu who discussed with him questions about power series.

  893. Edwin Spanier (1921-1996)
    • Spanier began joint work with Henry Whitehead and in a series of papers they introduced the method of duality in homotopy theory.

  894. Stefan Mazurkiewicz (1888-1945)
    • Although he only managed to recreate part of the work it was completed eleven years after his death and published as number 32 in the Mathematical Monographs series.

  895. Ralph Sampson (1866-1939)
    • Sampson used a series of accurate observations from Harvard College Observatory to amend the existing theory of the satellite orbits, but the disagreement between theory and observation persisted.

  896. Paul Dubreil (1904-1994)
    • thesis, Dubreil published a series of papers: Recherches sur la valeur des exposants des composants primaires des ideaux de polynomes Ⓣ (1930); Sur quelques proprietes des systemes de points dans le plan et des courbes gauches algebriques Ⓣ (1933); Sur les intersections totales mixtes dans l'espace a trois dimensions Ⓣ (1933); Sur quelques proprietes des varietes algebriques Ⓣ (1934); and Quelques proprietes des varietes algebriques se rattachant aux theories de l'algebre moderne Ⓣ (1935).

  897. Anatolii Volodymyrovych Skorokhod (1930-2011)
    • Series A (General) 129 (3) (1966), 476.','13]:- .

  898. Paul Turán (1910-1976)
    • As regards the latter, Turan found new approaches to such topics as quasi-analytic classes, Fabry's gap theorem and the theory of lacunary series, amongst others.

  899. Diederik Korteweg (1848-1941)
    • Van der Waals was working on the phase separation of binary mixtures, and Korteweg supplied the necessary mathematical input with his work on folds on surfaces in a series of papers between 1891 and 1903, such as Over plooipunten en bijbehorende plooien in de nabijheid der randlijnen van het E'-vlak van Van der Waals Ⓣ (1902).

  900. Henri Andoyer (1862-1929)
    • In the ninth chapter a short discussion is found concerning the convergence of series used in astronomy.

  901. Sidney Luxton Loney (1860-1939)
    • He is best known to mathematicians as the author of a series of very popular books.

  902. Aleksei Alekseevich Dezin (1923-2008)
    • This was written in a period when he was publishing a particularly outstanding series of papers on invariant systems of first-order partial differential equations on smooth Riemannian manifolds.

  903. Jean Dieudonné (1906-1992)
    • He was one of the main contributors to the Bourbaki series of texts from the time that the group came into existence and in many ways he was the leading influence in a group whose whole object was to avoid anyone taking on this role.

  904. Alfred North Whitehead (1861-1947)
    • Science and the Modern World (1925), a series of lectures given in the United States, served as an introduction to his later metaphysics.

  905. John Coates (1945-)
    • John's father, J R Coates, taught French at a high school but a series of mental breakdowns led to him retiring from teaching and managing the farm his father owned near the Manning River.

  906. Richard Brauer (1901-1977)
    • In 1949 Brauer was awarded the Cole Prize from the American Mathematical Society for his paper On Artin's L-series with general group characters which he published in the Annals of Mathematics in 1947.

  907. Felix Klein (1849-1925)
    • This work led him to consider elliptic modular functions which he studied in a series of papers.

  908. John Turner (1871-)
    • At Easter Mr John Turner, Deputy Rector and Principal Teacher of Mathematics, retired after forty years' service, and the best memorial of his work is the unbroken series of success gained by his pupils in Mathematics, and their abiding gratitude.

  909. Evgeny Evgenievich Slutsky (1880-1948)
    • He also studied correlations of related series for a limited number of trials.

  910. Ernest de Jonquičres (1820-1901)
    • his results form a series of detailed supplements to the work of others and reflect Jonquieres's inventiveness in calculating rather than a more profound contribution to the advancement of the field.

  911. Samuel Molyneux (1689-1728)
    • Locke had written Some thought concerning education (1693) which was based on a series of letters he had written to Edward Clarke from Holland (where he had been in exile) advising him on how to bring up his son.

  912. Wolfgang Pauli (1900-1958)
    • This was in 1922, when he gave a series of guest lectures at Gottingen when he reported on his theoretical investigations on the periodic system of elements.

  913. László Rédei (1900-1980)
    • The classical approach to the study of p-groups consists in the investigation of their subgroups and central series.

  914. Fabio Conforto (1909-1954)
    • As well as a series of interesting papers on algebraic geometry, he also became interested in the history of the topic and published Il contributo italiano al progresso della geometria algebrica negli ultimi cento anni Ⓣ in 1939.

  915. Norbert Wiener (1894-1964)
    • Moreover, it led me very directly to the periodogram, and to the study of forms of harmonic analysis more general than the classical Fourier series and Fourier integral.

  916. Edwin Olds (1898-1961)
    • He also was able to turn the task of checking over to a clerk because now all that had to be done was a series of additions to get a definite check.

  917. Maria Agnesi (1718-1799)
    • In 1738 she published Propositiones Philosophicae a series of essays on philosophy and natural science.

  918. Jérôme Franel (1859-1939)
    • That the relationship between a series of fractions so simple can be connected to a mathematical hypothesis so profound with such economy is the mark of a teacher of mathematics of the very highest order.

  919. Josif Zakharovich Shtokalo (1897-1987)
    • After 1945 he became particularly interested in the qualitative and stability theory of solutions of systems of linear ordinary differential equations in the Lyapunov sense and in the 1940s and 1950 published a series of articles and three monographs in these areas.

  920. Beniamino Segre (1903-1977)
    • He gave a series of three lectures in London in 1950 which were published as Arithmetical questions on algebraic varieties in 1951.

  921. Lamberto Cesari (1910-1990)
    • Cesari also proved that the double Fourier series of any BVC function f (x, y) converges almost everywhere to f (x, y), a sharp result.

  922. Richard Schoen (1950-)
    • As the authors note in their introduction, the book under review was written for the lecture series given at Princeton University in 1983 and at the University of California, San Diego, in 1984 and 1985.

  923. John Ringrose (1932-)
    • He has written on operators of Volterra-type, compact linear operators, the Neumann series of integral operators, algebras of operators, automorphisms and derivations of operator algebras, and the cohomology of operator algebras.

  924. Poul Heegaard (1871-1948)
    • Heegaard had been interested in astronomy since he was a child and in 1901 he began publishing a series of popular article on the topic.

  925. Edith Hirsch Luchins (1921-2002)
    • Despite having to interrupt her doctoral studies, Luchins began to publish a series of papers with her husband including: Towards Intrinsic Methods in Testing (1946), A Structural Approach to the Teaching of the Concept of Area in Intuitive Geometry (1947), The Satiation Theory of Figural After-Effects and Gestalt Principles of Perception (1953), and Variables and Functions (1954).

  926. Robert Rankin (1915-2001)
    • With his usual fine sense of history, [Rankin] begins the discussion, not with Ramanujan himself, but rather with the older English mathematician J W L Glaisher (born in 1848), who initiated the study of multiplicative properties of the Fourier coefficients of modular forms in his series of papers, published in 1907, dealing with ..

  927. Simon Stevin (1548-1620)
    • With Prince Maurits now head of the army of the republic, and with Stevin as an advisor in his service, a series of military triumphs over the Spanish forces followed.

  928. Henry Jack (1917-1978)
    • New methods were all carefully kept in a series of notebooks for use in his class-work.

  929. Georges Buffon (1707-1788)
    • His main mathematical contribution of this period was the publication of his translation of Newton's Method of Fluxions and infinite series in 1740.

  930. Frederick Atkinson (1916-2002)
    • In fact much of his early research followed on from this beginning with papers such as A summation formula for p(n), the partition function (1939), The mean value of the zeta-function on the critical line (1941), A divisor problem (1941), The Abel summation of certain Dirichlet series (1948), A mean value property of the Riemann zeta-function (1948), The mean-values of arithmetical functions (1949), and The mean-value of the Riemann zeta function (1949).

  931. William Hopkins (1793-1866)
    • referred to a series of important experiments which he had instituted at Manchester with the advice of Sir William Thomson and the assistance of Messrs Joule and Fairbairn, to determine the temperature of melting substances under great pressure.

  932. Ernst Hellinger (1883-1950)
    • Hellinger's position at Evanston throughout the war was precarious with a series of one-year appointments but he acquired American citizenship in 1944 and worked at Evanston until 1949 when he retired.

  933. Niels Abel (1802-1829)
    • If you disregard the very simplest cases, there is in all of mathematics not a single infinite series whose sum had been rigorously determined.

  934. Emanuel Lasker (1868-1941)
    • While in England he gave a series of lectures on chess which he wrote up for publication as Common Sense in Chess.

  935. Joseph Gergonne (1771-1859)
    • Gergonne's first contributions to duality appear in a series of papers beginning in 1810.

  936. Tycho Brahe (1546-1601)
    • In fact, however, their construction can be traced in his logs and rationalized as several series of experiments which only produced his major instruments in the mid-1580's.

  937. Francis Edgeworth (1845-1926)
    • In 1892 Edgeworth examined correlation and methods of estimating correlation coefficients in a series of papers.

  938. Louis Karpinski (1878-1956)
    • For over two decades he had had a series of operations and injections, suffered from anemia and diabetes, and surprised his contemporaries and colleagues by his ability to continue active, physically and mentally, in spite of these physical handicaps, simply because he had the will and determination to do so.

  939. George Chrystal (1851-1911)
    • In [',' The Student (New Series) 4 (7) (3 December, 1890), 98.','26] he is described by the Edinburgh students of 1890:- .

  940. William Spence (1777-1815)
    • In 1808 Spence was again in London, and during the several months that he lived there, he published An Essay on the various Orders of logarithmic Transcendents; with an Inquiry into their Applications to the Integral Calculus, and the Summation of Series.

  941. James Pierpont (1866-1938)
    • Two series of lectures were given, one by Maxime Bocher on Linear Differential Equations, and their Application and the other by Pierpont on Galois's Theory of Equations.

  942. Corrado Segre (1863-1924)
    • It was in two parts, one on quadrics in higher dimensional spaces with the other part on the geometry of the right line and of its quadratic series.

  943. George Forsythe (1917-1972)
    • He did his best to remedy the problem through writing texts and also by acting as editor for Prentice-Hall's excellent Series in Automatic Computation.

  944. William Whewell (1794-1866)
    • Whewell entered Cambridge in October 1812, but by this time the family had suffered a series of tragedies with his mother dying in 1807 and three of his younger brothers dying before William began his university studies.

  945. Heinrich Bruns (1848-1919)
    • He worked on the three-body problem showing that the series solutions of the Lagrange equations can change between convergent to divergent for small perturbations of the constants on which the coefficients of the time depend.

  946. Igor Kluvánek (1931-1993)
    • Pavol recalled in [',' J R Higgins, Five short stories about the cardinal series, Bull.

  947. Ralph Boas (1912-1992)
    • Among the awards that Boas received we mention the 1970 Lester R Ford Award from the Mathematical Association of America for his paper Inequalities for the derivatives of polynomials (1969) and the 1978 Lester R Ford Award for his paper Partial sums of infinite series, and how they grow (1977).

  948. Zeno of Elea (about 490 BC-about 425 BC)
    • Although they have often been dismissed as logical nonsense, many attempts have also been made to dispose of them by means of mathematical theorems, such as the theory of convergent series or the theory of sets.

  949. Tadeusz Waewski (1896-1972)
    • His interest in that topic began around 1960 and he published a series of important papers on the topic through the 1960s.

  950. Joseph Plateau (1801-1883)
    • He then carried out a series of experiments repeating the original accident but also investigating the shape of the drops of oil when the mixture of water and alcohol is rotating.

  951. Igor Rostislavovich Shafarevich (1923-2017)
    • I published a book on socialism and a series of nonmathematical articles.

  952. Kenkichi Iwasawa (1917-1998)
    • Iwasawa himself produced a series of deep papers throughout the 1960s which pushed his ideas much further.

  953. Henri Poincaré (1854-1912)
    • He also showed that series expansions previously used in studying the 3-body problem were convergent, but not in general uniformly convergent, so putting in doubt the stability proofs of Lagrange and Laplace.

  954. Alexander Andreevich Samarskii (1919-2008)
    • In those years Samarskii, together with A N Tikhonov, wrote a series of papers on electrodynamics and the excitation of electromagnetic waves in waveguides.

  955. Hanna Neumann (1914-1971)
    • Within months of becoming a professor, she gave a series of courses to secondary schoolteachers and participated in discussions on new syllabuses for senior students.

  956. Giuseppe Basso (1842-1895)
    • The series of works on optical physics is the one that defines and assigns to Giuseppe Basso his true place in the ranks of our workers of science: a place, the importance of which will be evident to those who think of the deplorable neglect in which here as elsewhere, in schools as laboratories, is at present left the beautiful branch of physics that Basso favoured.

  957. Uriel Rothblum (1947-2012)
    • Rothblum served on the editorial board of several journals: Letters in Linear Algebra and Its Application (1980-81); SIAM Journal on Algebraic and Discrete Methods (1983-87); SIAM Journal on Matrix Analysis and Applications (1988-93); Operations Research (1996-99); Journal on Combinatorial Optimization (2005-2012); World Scientific Series on applied mathematics (2006-2012); Linear Algebra and Its Applications (1982-2012); and Mathematics of Operations Research (1979-2012).

  958. Hermann von Helmholtz (1821-1894)
    • In mathematical appendices he advocated the use of Fourier series.

  959. Samuel Wilks (1906-1964)
    • Series A (General) 127 (4) (1964), 597-599.','36]:- .

  960. Fritz John (1910-1994)
    • He wrote an important series of papers on numerical analysis, studying ill-posed problems.

  961. Charles Noble (1867-1962)
    • In a series of four memoirs in the 'Journal de Mathematiques', Poincare has, among other things, discussed the topology of curves defined by ordinary differential equations of a simple character.

  962. Frank Harary (1921-2005)
    • Harary gave a series of inspiring lectures on graph theory which had a major influence of me and soon after I introduced graph theory into my undergraduate teaching.

  963. Jacques Tits (1930-)
    • He followed this with three series of lectures on the following topics .

  964. Hans Hamburger (1889-1956)
    • The course of his work on each of the main problems that he tackled follows the same pattern; a series of shorter preparatory papers on different aspects of the problem, followed by a very detailed complete account, usually spread over two or three papers.

  965. Zyoiti Suetuna (1898-1970)
    • This book, based mainly on the Riemann zeta-functions and L-functions, is a unique exposition of the analytical theory of numbers in a modern sense as can be seen from the chapter headings: I) Riemann's zeta-functions; II) Hecke's L-functions; III) Dirichlet's L-functions; and IV) Artin's L-series.

  966. Charles Hughes Terrot (1790-1872)
    • 1853 On the Summation of a Compound Series, and its Application to a Problem in Probabilities .

  967. Federigo Enriques (1871-1946)
    • He produced a series of papers over a period of 20 years which, together with Castelnuovo, finally produced a classification of algebraic surfaces [',' D Babbitt and J Goodstein, Federigo Enriques’s Quest to Prove the ’Completeness Theorem’, Notices Amer.

  968. Joan Clarke (1917-1996)
    • Joan Murray's greatest achievement was to establish the sequence of gold unicorns and heavy groats of James III and James IV, an extremely complex series which caused great difficulty for previous students.

  969. Henry More (1614-1687)
    • He was a committed experimental scientist and he undertook a series of hydrostatic and pneumatic experiments to disprove Boyle's theory.

  970. Gomes Teixeira (1851-1933)
    • His performance had been outstanding and in 1871, while still an undergraduate, he wrote Desenvolvimento das funcoes em fraccao continua Ⓣ which showed how to develop functions as continued fractions and applied these techniques to approximate roots of equations using rapidly converging series.

  971. Yozo Matsushima (1921-1983)
    • His research in Osaka took a somewhat different direction and he wrote a series of papers on cohomology of locally symmetric spaces.

  972. Charles-François Sturm (1803-1855)
    • Papers of 1836-1837 by Sturm and Liouville on differential equations involved expansions of functions in series and is today well-known as the Sturm-Liouville problem, an eigenvalue problem in second order differential equations.

  973. Michio Suzuki (1926-1998)
    • During this period he published a series of excellent papers: The lattice of subgroups of a finite group (in Japanese) (1950); On the finite group with a complete partition (1950); On the lattice of subgroups of finite groups (1951); On the L-homomorphisms of finite groups (1951); and A characterization of simple groups LF(2,p) (1951).

  974. Anatoly Mykhailovych Samoilenko (1938-)
    • Samoilenko has written a series of monographs with N I Ronto.

  975. Leopold Löwenheim (1878-1957)
    • Despite war service in France, Hungary and Serbia between August 1915 and December 1916, he published a series of important papers on mathematical logic during the eleven years from 1908 to 1919, extending work by Charles Peirce, Schroder, and Whitehead.

  976. Jack Warga (1922-2011)
    • Over the next few years he published a whole series of papers on these topics.

  977. Horst Tietz (1921-2012)
    • Series expansions; 7.

  978. Jack Todd (1911-2007)
    • He became ill while giving a lecture series on group theory and quantum mechanics.

  979. Jack van Lint (1932-2004)
    • The Dutch had assumed that they would be able to return to the pre-war situation in Indonesia, but an Independence movement made this impossible and a series of risings marked the beginning of a revolution.

  980. Michele Cipolla (1880-1947)
    • He published some one hundred works in addition to a series of texts written for secondary school and a number of remarkable university-level treatises.

  981. Frank Adams (1930-1989)
    • He continued to produce work of outstanding depth and originality, and during his first few years at Manchester he wrote a series of papers On the groups J(X) which were highly influential in homotopy theory.

  982. Frank Bonsall (1920-2011)
    • discusses a variety of inequalities, mostly connected with Hilbert's double series theorem ..

  983. Saunders Mac Lane (1909-2005)
    • Osa said [',' M Ward, Review: Survey of Modern Algebra, by Garrett Birkhoff and Saunders Mac Lane, Science, New Series 95 (2467) (1942), 386-387.','55]:- .

  984. Peter Lax (1926-)
    • SIAM published Lax's Hyperbolic systems of conservation laws and the mathematical theory of shock waves in their Conference Series in Applied Mathematics in 1973.

  985. Bryce McLeod (1929-2014)
    • In the interview [',' J Ball, Interview: Bryce McLeod, Oxford Mathematical Institute video series.','1] McLeod gives some wonderful insights into his love of mathematics.

  986. Andrei Andreyevich Markov (1856-1922)
    • Markov's early work was mainly in number theory and analysis, algebraic continued fractions, limits of integrals, approximation theory and the convergence of series.

  987. George E Andrews (1938-)
    • The combinatorial and formal power series aspects of the subject have usually been treated in books on elementary number theory or combinatorial analysis.

  988. David Rittenhouse (1732-1796)
    • This work was not new in the sense that had he known of Taylor series published in 1717 he could have deduced his results easily.

  989. Chike Obi (1921-2008)
    • This was, in fact, the first of a series of ten papers he published on Analytical theory of non-linear oscillations.

  990. Michel Rolle (1652-1719)
    • It amplified the concepts of limits of roots of equations, provided the fundamentals from which Maclaurin derived his formula, began modern methods of series for determining roots, and discussed the relationship of imaginary roots in equations and their derivatives.

  991. Jean-Victor Poncelet (1788-1867)
    • The lectures he gave at Metz were first produced in lithographed form then, after a series of versions, were eventually published.

  992. Alexandru Lupas (1942-2007)
    • For example On Bernstein power series (1966) was reviewed by D E Wulbert who wrote:- .

  993. Dmitrii Konstantinovich Faddeev (1907-1989)
    • He produced a series of 'Problem' books aimed at pupils in various different years at high school.

  994. Giulio Vivanti (1859-1949)
    • To remember the works that are most pertinent to questions of calculus, we will limit ourselves to hinting at the interesting question proposed with the memoir: "On the series of powers whose coefficients depend on a variable".

  995. Carl Boyer (1906-1976)
    • Judith Victor Grabiner, reviewing A History of Mathematics (1968), writes [',' J V Grabiner, Review: A History of Mathematics, by Carl B Boyer, Science, New Series 163 (3863) (1969), 171.','10]:- .

  996. Bernard Lamy (1640-1715)
    • Next came Demonstration de la verite et de la saintete de la morale chretienne (1688), and Harmonia sive Concordia quatuor evangelistarum, in qua vera series actuum et sermonum Domini nostri (1689).

  997. Dmitrii Viktorovich Anosov (1936-2014)
    • Anosov published a series of deep yet readable books (like detective novels) for high-school and college students.

  998. John Colson (1680-1760)
    • In 1736, he published an English version of Newton's 'Method of Fluxions and Infinite Series' originally written in Latin.

  999. Edmund Landau (1877-1938)
    • He submitted this habilitation thesis in 1901, only two years after his doctorate, consisted of his work on Dirichlet series, a topic in analytic number theory.

  1000. Pietro Paoli (1759-1839)
    • Among Paoli's publications we mention Liburnensis Opuscula analytica Ⓣ (1780), Ricerche sulle serie Ⓣ (1788) which corrects an error in a 1779 paper by Laplace on series, Della integrazione dell'equazioni a differenze parziali finite ed infinitesime Ⓣ (1800), Sulle oscillazioni di un corpo pendente da un filo estendibile memoria Ⓣ(1815), and Sull'uso del calcolo delle differenze finite nella dottrina degl'integrali definiti memoria Ⓣ (1828).

  1001. Leon Mirsky (1918-1983)
    • He read widely on his own and he kept a record of the parts of the theory that pleased him by filling up a whole series of notebooks.

  1002. John Henry Michell (1863-1940)
    • II, by J H Michell and M H Belz, Science, New Series 119 (3095) (1954), 549-550.','10], writes:- .

  1003. David Mumford (1937-)
    • Let us mention the book Indra's Pearls: The Vision of Felix Klein which he published with Caroline Series and David Wright in 2002.

  1004. Stanisaw Leniewski (1886-1939)
    • From then until 1939 he published a series of twelve papers giving his theories of logic and mathematics.

  1005. Nikolaos Hatzidakis (1872-1942)
    • As part of his role as president, he also gave a series of lectures on a diverse range of material.

  1006. Bob Thomason (1952-1995)
    • During the six years he spent there he produced a series of outstanding papers solving, among others, problems arising from Grothendieck's work in his paper with Berthelot and Illusie Theorie des Intersections et Theoreme de Riemann-Roch (1971).

  1007. Alcuin (735-804)
    • These were a series of illuminated masterpieces written largely in gold, often on purple coloured vellum.

  1008. Armand Borel (1923-2003)
    • In the summer of 1951 he gave a series of lectures in Zurich on the Leray's ideas on the theory of homological invariants of locally compact spaces and of continuous mappings which was published as a 95 page book of mimeographed notes with the title Cohomologie des espaces localement compacts, d'apres J Leray Ⓣ.

  1009. Giovanni Magini (1555-1617)
    • But now he was unable to obtain engravers, owing to a series of misfortunes, and began a lengthy descriptive commentary which was to accompany the maps, but which was never published, and is for the most part lost.

  1010. Andrew Wiles (1953-)
    • He filled what he thought were the remaining few gaps and gave a series of lectures at the Isaac Newton Institute in Cambridge ending on 23 June 1993.

  1011. Helmut Wielandt (1910-2001)
    • Where Hall had started from arithmetical questions and product decompositions, my own work was triggered by a question of Robert Remak of a quite different type: is the group generated by two subgroups that occur in composition series always of the same kind? In my Habilitationsschrift I expanded the discovery that this question can be answered affirmatively to a detailed study of the normal structure of finite groups.

  1012. Albert Châtelet (1883-1960)
    • Notes written for a series of experiments undertaken in the spring of 1916 can help to convey an even clearer sense of the mathematicians' activities at Gavre.

  1013. John Carr (1948-2016)
    • These notes are based on a series of lectures given in the Lefschetz Center for Dynamical Systems in the Division of Applied Mathematics at Brown University during the academic year 1978-79.

  1014. Ismail Mohamed (1930-2013)
    • His thesis was entitled On Series of Subgroups Related to Groups of Automorphisms, and his thesis advisor had been Kurt Hirsch.

  1015. Siméon-Denis Poisson (1781-1840)
    • His approach to these problems was to use series expansions to derive approximate solutions.

  1016. Charles Sims (1937-2017)
    • It includes efforts to automate many of the techniques taught to high school students and college undergraduates, such as the manipulation of polynomials and rational functions, differentiation and integration in closed form, and expansion in Taylor series.

  1017. Colin Cherry (1914-1979)
    • He was honoured in 1987 when Imperial College inaugurated 'The Colin Cherry Memorial Lecture.' The information given about the Memorial Lecture series gives the following description of Cherry's contributions:- .

  1018. Augustin-Louis Cauchy (1789-1857)
    • Cauchy was the first to make a rigorous study of the conditions for convergence of infinite series in addition to his rigorous definition of an integral.

  1019. Herbert Federer (1920-2010)
    • After World War II ended in 1945, Tibor Rado was invited to be the American Mathematical Society Colloquium Lecturer, and he gave a series of talks on his major contributions on surface area.

  1020. Daniel Rudolph (1949-2010)
    • At the Institute for Advanced Studies of the Hebrew University he gave a lecture series on 'Nonequivalence' in the spring of 1976.

  1021. Persi Diaconis (1945-)
    • In 2001 he was a main speaker at Groups St Andrews 2001 in Oxford giving a series of lectures on Random walks on groups: characters and geometry.

  1022. Edward Collingwood (1900-1970)
    • he had great intellectual powers which enabled him to achieve excellence in diverse activities conducted in parallel and not in series.

  1023. Mario Pieri (1860-1913)
    • He also gave a series of lectures on polyhedra at the Scuola Normale Superiore.

  1024. Nikolay Sonin (1849-1915)
    • He obtained a Master's Degree with a thesis on the expansion of functions in infinite series submitted in 1871.

  1025. Émile Mathieu (1835-1890)
    • This generalised a discovery made by Cauchy in 1846 when he discovered the 3-transitive group of degree 5 of order 120 which one obtains as the p = 5 case of Mathieu's infinite series.

  1026. Kazimierz Bartel (1882-1941)
    • He lectured on this topic and his lecture series was published as Perspektywa malarska Ⓣ (1928).

  1027. Edward Copson (1901-1980)
    • Copson was honoured by election to the Royal Society of Edinburgh in 1924 and was awarded the Keith Prize of the Society in 1941 for an outstanding series of papers published in the Proceedings.

  1028. Fritz Zwicky (1898-1974)
    • Koenig writes [',' T Koenig, Fritz Zwicky: Novae Become Supernovae, in M Turatto, S Benetti, L Zampieri and W Shea (eds.), 1604-2004: Supernovae as Cosmological Lighthouses, ASP Conference Series 342, Proceedings of the conference held 15-19 June, 2004 in Padua, Italy (Astronomical Society of the Pacific, San Francisco, 2005), 53-60.','10]:- .

  1029. Udita Narayana Singh (1920-1989)
    • in 1949 for his thesis Strong Summability of Trigonometric Series.

  1030. Bhama Srinivasan (1935-)
    • Also in 1979 her book Representations of finite Chevalley groups appeared in the Springer-Verlag Lecture Notes in Mathematics Series.

  1031. Willard Van Quine (1908-2000)
    • And corresponds to terminals in series, or to those in parallel, so that if you simplify mathematical logical steps, you have simplified your wiring.

  1032. Richard Hamming (1915-1998)
    • He further developed methods introduced by Jacob D Tamarkin to investigate the characteristic numbers and to show that the series used converged uniformly.

  1033. Levi ben Gerson (1288-1344)
    • In this work he also looks at the summation of series, permutations and combinations, and basic algebraic identities.

  1034. Ernest Esclangon (1876-1954)
    • Esclangon broadcasts the time through a series of photoelectric cells, which activated 'pistes sonores' located on a rotating cylinder.

  1035. Hans Wussing (1927-2011)
    • This textbook is the first volume of a projected series intended as a basic introduction to the history of mathematics and designed for independent study, in particular for students and high school teachers.

  1036. Saharon Shelah (1945-)
    • The papers have to be written quickly, previous constructions are newly refreshed and modified, and so a labyrinthian network may result over a series of related papers.

  1037. Salvatore Cherubino (1885-1970)
    • Even the Cayley-Hamilton theorem is proved by a power series expansion.

  1038. Marcel Grossmann (1878-1936)
    • Finally we should mention the honour given to Marcel Grossmann by naming the series of conferences, the Marcel Grossmann Meetings (on Recent Developments in Theoretical and Experimental General Relativity, Gravitation, and Relativistic Field Theories).

  1039. Doris Hellman (1910-1973)
    • The article [',' C D Hellman, History of Science, Science, New Series 131 (3397) (1960), 362-364.','8] is Hellman's report on this International Congress which describes both the academic and social aspects of the meeting.

  1040. Richard Feynman (1918-1988)
    • Returning to their respective homes in the summer of 1936 the two exchanged a series of remarkable letters as they tried to develop a version of space-time where (quoted from one of the letters - see [',' J Gleick, Genius : The Life and Science of Richard Feynman (New York, 1992).','6]):- .

  1041. Alexander Merriles (1880-1950)
    • He was founder of the Gramophone Club, an institution which will likely be permanent and which enables a select band of kindred spirits to discover, study and exploit the musical treasures that are hidden away in a series of gramophone records.

  1042. Robert Dunbar (1889-1959)
    • She published the paper Absorption and scattering of X-rays and the characteristic radiations of the J series in collaboration with Charles Glover Barkla.

  1043. Johann Hudde (1628-1704)
    • In 1656 he gave the power series expansion of ln(1+x).

  1044. Otto Schreier (1901-1929)
    • Very pretty paper on Fourier series in Hahn's seminar ..

  1045. Maurice Bartlett (1910-2002)
    • By this time he was undertaking research on time series and stochastic processes and gave a lecture course on his research interests.

  1046. al-Khujandi (about 940-1000)
    • During the year 994 al-Khujandi used the very large instrument to observe a series of meridian transits of the sun near the solstices.

  1047. Eugene Lukacs (1906-1987)
    • Jointly with Z W Birnbaum, he was the founding editor of the Academic Press Series in Probability and Mathematical Statistics (1962-85).

  1048. Oswald Veblen (1880-1960)
    • He was the Colloquium Lecturer for the Society in 1916 when he gave a series of lectures on topology.

  1049. Ibn al-Banna (1256-1321)
    • Other interesting results on summing series are the results .

  1050. Pelageia Polubarinova Kochina (1899-1999)
    • In 1996 she was awarded the M V Keldysh Gold Medal for a series of studies into hydrodynamics and the theory of filtration.

  1051. William Threlfall (1888-1949)
    • The book was accepted by Wilhelm Blaschke for the Hamburg monograph series but, since Blaschke went along with the Nazi ideas, he objected to the Kepler quote on the grounds that it looked like a political statement - of course this is exactly what it was meant to be.

  1052. Emory McClintock (1840-1916)
    • A series of notebooks reflecting specific research into the Baskerville, Kemble, McClintock, and Wakeman families record the lines of descent through each ancestral surname.

  1053. William Edge (1904-1997)
    • These have interesting geometrical properties and Edge investigated them in a series of papers spanning 40 years.

  1054. Dudley Littlewood (1903-1979)
    • Another reason was certainly the work of Hilbert, but Littlewood tried to remedy the "tensor reason" in a series of papers on tensors and invariant theory.

  1055. John Semple (1904-1985)
    • The result was a series of fascinating papers and one further book Generalized Clifford parallelism (1971).

  1056. Alberto Calderón (1920-1998)
    • Zygmund posed Calderon a question and the puzzled Calderon replied that the answer was contained in Zygmund's own book Trigonometric Series.

  1057. Sofia Kovalevskaya (1850-1891)
    • The paper on the reduction of abelian integrals to simpler elliptic integrals is of less importance but it consisted of a skilled series of manipulations which showed her complete command of Weierstrass's theory.

  1058. Matteo Bottasso (1878-1918)
    • Following this series of appointments, he became a lecturer in algebra and analytic geometry at the University of Parvia (the city is 35 km south of Milan).

  1059. Edwin Pitman (1897-1993)
    • This was the first of a series of eight papers which he wrote during the next two years which included: Significance tests which may be applied to samples from any populations (1937), Significance test which may be applied to samples from any populations.

  1060. Mikhail Vasilevich Ostrogradski (1801-1862)
    • His important work on ordinary differential equations considered methods of solution of non-linear equations which involved power series expansions in a parameter alpha.

  1061. Alfred Foster (1904-1994)
    • is a continuation of a long series of articles by [Foster] and his students which investigates unique factorization in certain classes of abstract algebras.

  1062. Roy Kerr (1934-)
    • Roy's style as Head of Department was at once uncompromising and dashing; in a series of moves which affronted some of our colleagues in other departments, who had grown comfortable with the traditional Canterbury view that Mathematics should be a low-cost department devoted to service teaching, he contrived to reduce student-staff ratios, encourage research, and equip the department with a computer system at the sort of cost hitherto associated with spectrographs.

  1063. Stanislaw Knapowski (1931-1967)
    • This came about in September 1956 when Turan gave a series of lectures on a new analytic method in Lublin.


History Topics

  1. Weather forecasting
    • Furthermore, I have assumed that the reader is familiar with differential equations, differentiation of functions of several variables, Fourier series and Gaussian elimination.
    • One of the main causes for instability are truncation errors, which happen when a variable ψ is represented by a Taylor series, i.e.
    • Due to computational reasons, only the very first terms of the series, which are in fact the most important ones, can be used, but the higher-order terms influence the accuracy of the series.
    • Not only space, but also time has to be discretized, and time derivatives can also be represented as finite difference approximations, that is in terms of values at discrete time levels [','R W Riddaway, Numerical Methods, revised March 2001 (by M Hortal) (Meteorological Training Course Lecture Series, 2002) ','17, p.
    • The explicit scheme is much easier to solve than the implicit one, as it is possible to compute the new value of ψl at time n+1 for every grid point, provided the values of ψl are known for every grid point at the current time step n [','R W Riddaway, Numerical Methods, revised March 2001 (by M Hortal) (Meteorological Training Course Lecture Series, 2002) ','17, p.
    • The implicit scheme, on the other hand, is absolutely stable, but it results in a system of simultaneous equations, so is more difficult to solve [','R W Riddaway, Numerical Methods, revised March 2001 (by M Hortal) (Meteorological Training Course Lecture Series, 2002) ','17, p.
    • When this series is substituted into an equation of the form Lψ = f (x), where L is a differential operator, you get a so-called residual function: .
    • The residual function is zero when the solution of the equation above is exact, therefore the series coefficients an should be chosen such that the residual function is minimised, i.e.
    • In the majority of cases, polynomial approximations, such as Fourier series or Chebyshev polynomials, are the best choice; but when it comes to weather forecasting, the use of spherical coordinates demands that spherical harmonics are used as expansion functions.
    • A simple example that can be solved in terms of a Fourier series illustrates the idea of the spectral method: One of the processes described by the primitive equations is advection (which is the transport of for instance heat in the atmosphere), and the non-linear advection equation is given by .
    • Having chosen appropriate boundary conditions, the equation can be expanded in terms of a finite Fourier series: .
    • where the um are the complex expansion coefficients and M is the maximum wave number [','R W Riddaway, Numerical Methods, revised March 2001 (by M Hortal) (Meteorological Training Course Lecture Series, 2002) ','17, p.
    • where Fm is a series in terms of the um.
    • There are several methods which convert differential equations to discrete problems, for example the least-square method or the Galerkin method, and which can be used in order to choose the time derivative such that the residual function is as close to zero as possible [','R W Riddaway, Numerical Methods, revised March 2001 (by M Hortal) (Meteorological Training Course Lecture Series, 2002) ','17, p.
    • Most commonly, Fast Fourier Transforms are used, but in principle all transform methods make it possible to switch between a spectral representation and a grid-point representation [','R W Riddaway, Numerical Methods, revised March 2001 (by M Hortal) (Meteorological Training Course Lecture Series, 2002) ','17, p.
    • Furthermore, products with more than two components suffer from aliasing, meaning that waves that are too short to be resolved for a certain grid resolution falsely appear as longer waves [','R W Riddaway, Numerical Methods, revised March 2001 (by M Hortal) (Meteorological Training Course Lecture Series, 2002) ','17, p.
    • Spherical harmonics are two-dimensional, so they are much more difficult to solve than expansions in terms of Fourier series.
    • Moreover, a finite series expansion in terms of linearly independent functions approximates the variation of ψ within a specified element (e.g.
    • a set of grid points) [','R W Riddaway, Numerical Methods, revised March 2001 (by M Hortal) (Meteorological Training Course Lecture Series, 2002) ','17, p.

  2. Indian mathematics
    • An analysis of the weights discovered suggests that they belong to two series both being decimal in nature with each decimal number multiplied and divided by two, giving for the main series ratios of 0.05, 0.1, 0.2, 0.5, 1, 2, 5, 10, 20, 50, 100, 200, and 500.
    • For instance, the development of number theory, the theory of indeterminates infinite series expressions for sine, cosine and tangent, computational mathematics, etc.
    • The most remarkable contribution from this period, however, was by Madhava who invented Taylor series and rigorous mathematical analysis in some inspired contributions.
    • These include: a formula for the ecliptic; the Newton-Gauss interpolation formula; the formula for the sum of an infinite series; Lhuilier's formula for the circumradius of a cyclic quadrilateral.
    • Of particular interest is the approximation to the value of π which was the first to be made using a series.
    • Madhava's result which gave a series for π, translated into the language of modern mathematics, reads .
    • See for example [',' K M Marar and C T Rajagopal, Gregory’s series in the mathematical literature of Kerala, Math.
    • abound with fluxional forms and series to be found in no work of foreign countries.

  3. function concept
    • Euler allowed the algebraic operations in his analytic expressions to be used an infinite number of times, resulting in infinite series, infinite products, and infinite continued fractions.
    • He later suggests that a transcendental function should be studied by expanding it in a power series.
    • For example it had led him to define the gamma function and to solve the problem which had defeated mathematicians for some considerable time, namely summing the series .
    • Fourier showed that some discontinuous functions could be represented by what today we call a Fourier series.
    • Monthly 105 (3) (1998), 263-270.','18] that confusion regarding functions had been due to a lack of understanding of the distinction between a "function" and its "representation", for example as a series of sines and cosines.
    • Fourier's work would lead eventually to the clarification of the function concept when in 1829 Dirichlet proved results concerning the convergence of Fourier series, thus clarifying the distinction between a function and its representation.
    • However, despite this, when he begins to prove theorems about expressing an arbitrary function as a Fourier series, he uses the fact that his arbitrary function is continuous in the modern sense! .
    • It therefore has a Taylor series which converges everywhere but only equals the function at 0.
    • In 1876 Paul du Bois-Reymond made the distinction between a function and its representation even clearer when he constructed a continuous function whose Fourier series diverges at a point.

  4. Calculus history
    • This is the first known example of the summation of an infinite series.
    • Barrow was in some way to blame for this since the publisher of Barrow's work had gone bankrupt and publishers were, after this, wary of publishing mathematical works! Newton's work on Analysis with infinite series was written in 1669 and circulated in manuscript.
    • Similarly his Method of fluxions and infinite series was written in 1671 and published in English translation in 1736.
    • In these two works Newton calculated the series expansion for sin x and cos x and the expansion for what was actually the exponential function, although this function was not established until Euler introduced the present notation ex.
      Go directly to this paragraph
    • You can see graphs of the series expansions for sine at THIS LINK and for cosine at THIS LINK.
    • They are now called Taylor or Maclaurin series.
    • by the method of infinite series, .
    • Expansions of the Talor series of sin x .
    • Expansions of the Talor series of cosin x .

  5. African women 1
    • She has published around 20 papers including On p-nuclear and entropy quasinorms in Banach spaces (1979), On projection constant problems and the existence of metric projections in normed spaces (2001), On the projection constants of some topological spaces and some applications (2001), Interpolation methods to estimate eigenvalue distribution of some integral operators (2004), On The General Term of a Cauchy Product of Two Series of The Truncation Error for Some Restrictive Approximations for IBVP for Parabolic and Hyperbolic Equations (2004), Finite co-dimensional Banach spaces and some bounded recovery problems (2004), Generalization of Banach contraction principle in two directions (2007), Two population three-player prisoner's dilemma game (2016), The payoff matrix of repeated asymmetric 2×2 games (2016).
    • Thesis title: Polinoomreekse in meerveranderlike analise [Polynomial series in multivariate analysis].
    • She has published several papers including Generalised Laguerre series forms of Wishart distributions (1973), Some extensions of the Wishart moment generating function (1975), An extension of Geisser's discrimination model to proportional covariance matrices (1982), A general model for negative multinomial frequency counts (1989).
    • She had taken the five courses Theory of Statistics, Theory of Storage, Theory of Probability, Sequential Analysis, and Time Series in 1974-75.
    • Thesis title: Convergence of Pade approximants for some q-hypergeometric Series (Wynn's Power Series I, II and III).
    • Biographical Data: Here is the Abstract of her thesis: "The convergence of certain sequences of Pade approximates for three types of power series is investigated.
    • For a specified range of values of one of the parameters occurring in the power series, each of the functions has a natural boundary on its circle of convergence.

  6. Prime numbers
    • He was able to show that not only is the so-called Harmonic series ∑ (1/n) divergent, but the series .
    • The sum to n terms of the Harmonic series grows roughly like log(n), while the latter series diverges even more slowly like log[ log(n) ].
    • This means, for example, that summing the reciprocals of all the primes that have been listed, even by the most powerful computers, only gives a sum of about 4, but the series still diverges to ∞.

  7. Orbits
    • Although the motions of the planets were discussed by the Greeks they believed that the planets revolved round the Earth so are of little interest to us in this article although the method of epicycles is an early application of Fourier series.
      Go directly to this paragraph
    • He treated it as a restricted three body problem and used transformations to produce infinite series solutions for the longitude, latitude and parallax for the Moon.
      Go directly to this paragraph
    • The beginnings of his theory was published in 1847 and he had refined the theory until it was published in 2 volumes in 1860 and 1867 and was extremely accurate, its only drawback being the slow convergence of the infinite series.
      Go directly to this paragraph
    • He discussed convergence and uniform convergence of the series solutions discussed by earlier mathematicians and proved them not to be uniformly convergent.
      Go directly to this paragraph

  8. Bolzano's manuscripts
    • Before the first volume in the series appeared in 1969 there were a number of related publications.
    • The first volume in the new series Bernard Bolzano-Gesamtausgabe Ⓣ published by Friedrich Frommann Verlag and edited by Eduard Winter, Jan Berg, Friedrich Kambartel, Jaromir Louzil, and Bob van Rootselaar, contains a biography of Bolzano together with details of the topics on which he worked: mathematics, logic, theology, philosophy and aesthetics.
    • The second volume, which set the scene for the whole series, appeared in 1972.
    • Further details of this series of volumes is given at THIS LINK .

  9. Pi history
    • In Gregory's series, for example, to get 4 decimal places correct we require the error to be less than 0.00005 = 1/20000, and so we need about 10000 terms of the series.
    • from which the first series results if we put x = 1.
    • and then calculate the two series obtained by putting first 1/2 and the 1/3 into (3).

  10. African men 1
    • Thesis title: On Series of Subgroups Related to Groups of Automorphisms.
    • He has published around 50 papers including Series inequalities involving convex functions (1974), Variations of integral means and Hardy-Littlewood maximal functions (1982), Generalizations of Hardy's integral inequality (1987), On a general Ishikawa fixed point iteration process for continuous hemicontractive maps in Hilbert spaces (2001), New iteration methods for pseudocontractive and accretive operators in arbitrary Banach spaces (2003), and Some convergence results for the Jungck-Mann and the Jungck-Ishikawa iteration processes in the class of generalized Zamfirescu operators (2008).
    • Thesis title: Analyse des series stationnaires a temps discret; Application au cas d'un modele statistique du mouvement du pole.
    • He works on statistics, particularly on time series and stochastic processes.
    • His area of research is Homological Algebra, and he has published two graduate and research level books and over 60 research articles in this area including Balanced functors applied to modules (1985), Syzygies of resolvents over Gorenstein rings (1990), Resolutions by Gorenstein injective and projective modules and modules of finite injective dimension over Gorenstein rings (1995), Compact coGalois groups (2000), The existence of Gorenstein flat covers (2004), Closure under transfinite extensions (2007), and Submonoids of the formal power series (2017).

  11. African women I
    • Published around 20 papers including On p-nuclear and entropy quasinorms in Banach spaces (1979), On projection constant problems and the existence of metric projections in normed spaces (2001), On the projection constants of some topological spaces and some applications (2001), Interpolation methods to estimate eigenvalue distribution of some integral operators (2004), On The General Term of a Cauchy Product of Two Series of The Truncation Error for Some Restrictive Approximations for IBVP for Parabolic and Hyperbolic Equations (2004), Finite co-dimensional Banach spaces and some bounded recovery problems (2004), Generalization of Banach contraction principle in two directions (2007), Two population three-player prisoner's dilemma game (2016), The payoff matrix of repeated asymmetric 2×2 games (2016).
    • Thesis title: Polinoomreekse in meerveranderlike analise [Polynomial series in multivariate analysis].
    • Published papers Generalised Laguerre series forms of Wishart distributions (1973), Some extensions of the Wishart moment generating function (1975), An extension of Geisser's discrimination model to proportional covariance matrices (1982), A general model for negative multinomial frequency counts (1989).
    • Thesis title: Convergence of Pade Approximants for some q-Hypergeometric Series.

  12. Chinese overview
    • He described multiplication, division, root-extraction, quadratic and simultaneous equations, series, computations of areas of a rectangle, a trapezium, a circle, and other figures.
    • He also gave many results on sums of series.
    • He produced his own versions of logarithms, infinite series, and combinatorics which did not follow the style of western mathematics but his research naturally developed out of the foundations of Chinese mathematics.

  13. Fractal Geometry
    • sum of a convergent power series) would certainly produce such a curve.
    • Science, New Series 156 3775 (May 5, 1967): 636-638.','8], in which he linked the idea of previous mathematicians to the real world -- namely coastlines, which he claimed were "statistically self-similar".
    • Science, New Series 156 3775 (May 5, 1967): 636-638.','8] .

  14. Real numbers 2
    • This leads into the study of infinite series but without the necessary machinery to prove that these infinite series converged to a limit, he was never going to be able to progress much further in studying real numbers.
    • Among the forms of the completeness property he implicitly assumed are that a bounded monotone sequence converges to a limit and that the Cauchy criterion is a sufficient condition for the convergence of a series.

  15. Bourbaki 1
    • A large number of subcommittees were formed, given the size of the group, and these were to cover the following topics: algebra, analytic functions, integration theory, differential equations, existence theorems for differential equations, partial differential equations, differentials and differential forms, calculus of variations, special functions, geometry, Fourier series, and representations of functions.
    • He presented a series of theorems, all completely wrong, each attributed to a different fake mathematician.
    • Already in 1935 Bourbaki had taken the decision to produce a series of books which were linearly ordered in the sense that no reference could be made except to books earlier in the linear progression.

  16. Sundials
    • Series A, Mathematical and Physical Sciences, 1974.
    • Series A, Mathematical and Physical Sciences, 1974.
    • If Menaechmus or someone else marked this path with a series of dots on a given day, he would 'discover' a hyperbola.

  17. The number e
    • In 1668 Nicolaus Mercator published Logarithmotechnia which contains the series expansion of log(1+x).
    • The work involves the calculation of various exponential series and many results are achieved with term by term integration.

  18. Measurement
    • An analysis of the weights discovered in excavations suggests that they had two different series, both decimal in nature, with each decimal number multiplied and divided by two.
    • The main series has ratios of 0.05, 0.1, 0.2, 0.5, 1, 2, 5, 10, 20, 50, 100, 200, and 500.

  19. Ledermann interview
    • It was while he was at St Andrews that Walter suggested a cheap series of books for undergraduates.
    • His idea was taken up and the Oliver & Boyd series of mathematical texts was born.

  20. References for Pi history
    • R Roy, The discovery of the series formula for π by Leibniz, Gregory and Nilakantha, Math.
    • I Tweddle, John Machin and Robert Simson on inverse-tangent series for π, Archive for History of Exact Sciences 42 (1) (1991), 1-14.

  21. Topology history
    • The idea of connectivity was eventually put on a completely rigorous basis by Poincare in a series of papers Analysis situs Ⓣ in 1895.
      Go directly to this paragraph
    • Schmidt in 1907 examined the notion of convergence in sequence spaces, extending methods which Hilbert had used in his work on integral equations to generalise the idea of a Fourier series.
      Go directly to this paragraph

  22. Classical light
    • He used a rotating wheel with 720 teeth to break up a light beam into a series of pulses.

  23. Fermat's last theorem
    • Wiles gave a series of three lectures at the Isaac Newton Institute in Cambridge, England the first on Monday 21 June, the second on Tuesday 22 June.
      Go directly to this paragraph

  24. Indian numerals
    • The early use of such large numbers eventually led to the adoption of a series of names for successive powers of ten.

  25. Elliptic functions

  26. Kepler's Laws
    • Kepler carried out the reduction to heliocentricity, and further simplifying procedures, in a series of steps: .

  27. Egyptian mathematics
    • What is the quantity? Other problems involve geometric series such as Problem 64: divide 10 hekats of barley among 10 men so that each gets 1/8 of a hekat more than the one before.

  28. References for Sundials
    • Series A, Mathematical and Physical Sciences, 1974.

  29. African men 2
    • Thesis title: Modeles de series chronologiques a coefficients periodiques.
    • Thesis title: Some Power Series Transformations and Weighted Mean Transformations that Preserve Absolute Convergence.

  30. References for Golden ratio
    • R Archibald, The golden section - Fibonacci series, Amer.

  31. References for Fractal Geometry
    • Science, New Series 156 3775 (May 5, 1967): 636-638.

  32. References for Water-clocks
    • Series A, Mathematical and Physical Sciences, 1974.

  33. Real numbers 1
    • He makes a series of definitions.

  34. Coffee houses
    • These were not just impromptu lectures given in the course of discussion, but rather were properly advertised and usually not one off lectures but rather extended lecture series.

  35. Science in the 17th century
    • During the 17th century, Europe experienced a series of changes in thought, knowledge and beliefs that affected society, influenced politics and produced a cultural transformation.

  36. References for Indian mathematics
    • K M Marar and C T Rajagopal, Gregory's series in the mathematical literature of Kerala, Math.

  37. Poincaré - Inspector of mines
    • As one might imagine, Poincare's report is a remarkably carefully argued document where he details a whole series of possible causes and lists the evidence for and against each.

  38. References for function concept
    • G Ferraro, Some aspects of Euler's theory of series: inexplicable functions and the Euler-Maclaurin summation formula, Historia Math.

  39. References for Weather forecasting
    • R W Riddaway, Numerical Methods, revised March 2001 (by M Hortal) (Meteorological Training Course Lecture Series, 2002) .

  40. EMS History
    • The Training Colleges provide Summer Courses for teachers in Infant and Junior Schools, and give courses in Rural Gardening, Country Dancing and - possibly - Elocution, but as for a course say on the best way of teaching logarithms to those who know no algebra, or the best way of teaching the convergence and divergence of series to those who have merely reached the standard of the leaving certificate, these problems are never attempted; perhaps because there is no one in the Training Colleges competent to deal with them, or perhaps because if someone did attempt to deal with them then he would have no audience.

  41. Pell's equation
    • To find the infinite series of solutions take the powers of 170 + 39√19.

  42. Greek astronomy
    • Meton worked in Athens with another astronomer Euctemon, and they made a series of observations of the solstices (the points at which the sun is at greatest distance from the equator) in order to determine the length of the tropical year.

  43. Set theory


Societies etc

  1. Italian Society of Applied and Industrial Mathematics
    • The SEMA-SIMAI Book Series .
    • In 2013 the Italian Society of Applied and Industrial Mathematics was joined by the Spanish Society of Applied Mathematics in a series of texts published by Springer.
    • The following information about the 'SEMA-SIMAI Book Series' appears on the Springer page [',' SEMA-SIMAI Springer Series, Springer International Publishing (2018).','3]:- .
    • As of 2013, the SIMAI Springer Series opens to SEMA in order to publish a joint series aiming to publish advanced textbooks, research-level monographs and collected works that focus on applications of mathematics to social and industrial problems, including biology, medicine, engineering, environment and finance.
    • This series is meant to host selected contributions focusing on the relevance of mathematics in real life applications and to provide useful reference material to students, academic and industrial researchers at an international level.
    • Interdisciplinary contributions, showing a fruitful collaboration of mathematicians with researchers of other fields to address complex applications, are welcomed in this series.

  2. Spanish Society of Applied Mathematics
    • The SEMA-SIMAI Book Series .
    • In 2013 Spanish Society of Applied Mathematics joined the Societa Italiana di Matematica Applicata e Industriale (SIMAI) in their series of texts published by Springer.
    • The following information about the 'SEMA-SIMAI Book Series' appears on the Springer page [',' SEMA-SIMAI Springer Series, Springer International Publishing (2018).','4]:- .
    • As of 2013, the SIMAI Springer Series opens to SEMA in order to publish a joint series aiming to publish advanced textbooks, research-level monographs and collected works that focus on applications of mathematics to social and industrial problems, including biology, medicine, engineering, environment and finance.
    • This series is meant to host selected contributions focusing on the relevance of mathematics in real life applications and to provide useful reference material to students, academic and industrial researchers at an international level.
    • Interdisciplinary contributions, showing a fruitful collaboration of mathematicians with researchers of other fields to address complex applications, are welcomed in this series.

  3. Mathematical Society of Japan
    • The Mathematical Society of Japan publishes a number of journals and book series.
    • The journal is now in its third series which began in 2006.
    • This is a book series published by the Mathematical Society of Japan, the first volumes appearing in 1983.
    • The Mathematical Society of Japan has two series of memoirs.
    • One is a series intended for an international mathematical readership while the other is in Japanese and is intended for mathematicians based in Japan.
    • Both series are essentially textbooks on advanced topics, usually based on lecture notes of a course given by the author.
    • Both these series are aimed at graduate students or young researchers.

  4. Mexican Mathematical Society
    • Alfonso Napoles Gandara was the editor of this first series which continued publication until volume 11 in 1954.
    • The Bulletin was not published in 1955, then the second series began publication in 1956 with Jose Adem and Emilio Lluis Riera as editors.
    • This series of the Bulletin had four issues per year and 39 volumes were produced, the final issue of this series being in October 1995.
    • A third series of the Bulletin began publishing in 1995, with one volume consisting of two issues each year, and this series continues to be published.

  5. Danish Mathematical Society
    • The number of foreign speakers then grew and from 1921 an invitation was given every second year to a leading mathematician to give a series of lectures.
    • The first series was given in 1921 by Hilbert.
    • The outbreak of World War II caused this lecture series to stop.
    • Series A published work on elementary mathematics, while series B published advanced mathematical research.

  6. Hong Kong Mathematical Society
    • The Society has two book series, the Hong Kong Mathematical Society Texts in General Education and the Hong Kong Mathematical Society Undergraduate Textbook Series.
    • The Second in this series was held in conjunction with the First East Asian SIAM Symposium at the Honk Kong Baptist University, 12-16 December 2005.
    • Four further conferences in the Scientific Computing and Partial Differential Equations series were held, also in the Honk Kong Baptist University, the Third conference, 8-12 December 2008, the Fourth conference, 5-9 December 2011, the Fifth conference, 8-12 December 2015, and the Sixth conference, 5-8 June 2017.

  7. New York Academy of Sciences
    • In 1823 the Lyceum published the first volume of the Annals of the Lyceum of Natural History of New-York which was to become an extremely important series consisting of:- .
    • The Academy continued to support graph theory conferences with two one day meetings each year in May and November having the proceedings published in their Graph Theory Notes New York series.
    • The series continued, reaching "New York Graph Theory Day 40" which was held at the State University of New York, Purchase, New York, on 4 November 2000.
    • The conference held in Madison, Wisconsin from 26 June to 29 June 1991, the seventh in the series, was in honour of Mary Ellen Rudin.

  8. Athens Academy
    • There followed a series of attempts by Alexandros Rizos-Ranghavis (1809-1892).
    • c) The implementation of a series of scientific talks or seminars by the researchers of the Research Centre of Pure and Applied Mathematics, as well as, by other scientists.
    • The success of this series of conferences had led to plans to continue it during the next years.
    • The Research Centre for Astronomy and Applied Mathematics has also organized, with remarkable success, a series of talks for the general public in the Academy of Athens during 2009 in the spirit of the International Year of Astronomy.

  9. Malaysian Mathematical Society
    • When the Society changed its name in 1999 the Bulletin began a second series under a slightly different name, namely the Bulletin of the Malaysian Mathematical Sciences Society.
    • The last volume in this series was Volume 37 in 2014, with the current series beginning in 2015 with Volume 38.
    • This series is published by the Malaysian Mathematical Sciences Society, the Universiti Sains Malaysia and also Springer Singapore.

  10. European Women in Mathematics
    • The five were Bodil Branner, Caroline Series, Gudrun Kalmbach, Marie-Francoise Roy and Donna Strauss.
    • The third meeting was held at the University of Warwick, Coventry, England, in December 1988 was organised by Caroline Series.
    • Let us end with quoting Caroline Series.
    • She was interviewed in 2007 and asked about her involvement with the European Women in Mathematics [',' C Series, Interview, Mathematics Today (2007).','3]:- .

  11. Slovak Mathematicians and Physicists Union
    • Another conference series which they collaborate in organising is the Equadiff series of biannual conferences on mathematical analysis, numerical approximation and applications of differential equations held in rotation in the Czech Republic, Slovakia and Western Europe.
    • The ALGORITMY (Algorithms) series represents the oldest Central-European series of international high level scientific meetings devoted to applied mathematics and numerical methods in computational sciences and engineering.

  12. Allahabad Mathematical Society
    • He then went to Paris where he worked under Arnaud Denjoy and was awarded a Docteur es Science in 1932 for his thesis Contribution a l'etude de la series conjuguee d'une serie de Fourier.
    • The Society also publishes a Lecture Note Series [',' Allahabad Mathematical Society website.','1]:- .
    • With the objective of providing instructional courses on a wide range of basic topics in "mathematics today" to students in India, the Allahabad Mathematical Society initiated inviting authors to contribute to this series.
    • Four volumes of this series have appeared: Pramila Srivastava, Functions of Several Variables (1990); C J Mozzochi and S A Naimpally, Uniformity and Proximity (2009); John G Hocking and Somashekhar A Naimpally, Nearness - A Better Approach to Continuity and Limits (2009); and C J Mozzochi, The Fermat Proof.

  13. American Academy of Arts and Sciences
    • The original series had four volumes published between 1785 and 1821; a second series contained nineteen volumes published between 1833 and 1946; no volumes appeared between 1947 and 1956, then in 1957 one further volume, called Series 3, Volume 24 was published.

  14. Society for Industrial and Applied Mathematics
    • In addition a book publishing programme began in 1961 with the Series in Applied Mathematics.
    • It was followed by: Proceedings in Applied Mathematics (1969), Regional Conferences in Applied Mathematics (1972), Studies in Applied and Numerical Mathematics (1979), Frontiers of Applied Mathematics (1983), Classics in Applied Mathematics (1988), and eight further series since then.
    • The Society has inaugurated Prizes and established prestigious lecture series.

  15. Bulgarian Statistical Society
    • Before this there had been a series of Summer Schools on Probability and Statistics organised jointly by the Bulgarian Academy of Sciences and Sofia University.
    • A annual seminar series on Statistical Data Analysis began in 1982 before the founding of the Society, but since 1996 the Society has taken over the organisation of this seminar series.

  16. Max Planck Society for Advancement of Science
    • Max Planck Institute for Mathematics preprint series .
    • The Max Planck Institute for Mathematics preprint series was established in 1983 shortly after the institute itself.
    • For instance, Fourier's studies of the thermal conduction equations led to the development of the theory of Fourier series and in general to the creation of harmonic analysis.

  17. References for Spanish Applied
    • SEMA-SIMAI Springer Series, Springer International Publishing (2018).
    • http://www.springer.com/series/10532 .

  18. References for Italian Applied
    • SEMA-SIMAI Springer Series, Springer International Publishing (2018).
    • http://www.springer.com/series/10532 .

  19. South-Eastern Europe Mathematical Society
    • In accordance with the decision of the Executive Council of the Balkan Mathematical Union (taken in July 1984) in 1987, the National Committee for Mathematics of Bulgaria and the Bulgarian Academy of Sciences started the publication of Mathematica Balkanica - New Series, edited by Academician Blagovest Sendov.
    • At the MASSEE Congress it was decided to continue the publication of Mathematica Balkanica - New Series under the auspices of MASSEE.

  20. Ramanujan Mathematical Society
    • The Ramanujan Mathematical Society also has two series of memorial lectures.
    • It also has two series of Endowment Lectures.

  21. Southeast Asian Mathematical Society
    • The series came into existence due to a meeting between Yukiyoshi Kawada, Secretary of the International Congress on Mathematical Education from 1975 to 1978, and Lee Peng Yee, as the representative of the Southeast Asian Mathematical Society.
    • This conference series continued, one being held every three years in a Southeast Asian country: 1981 Kuala Lumpur, Malaysia; 1984 Hat Yai, Thailand; 1987 Singapore; 1990 Bandar Seri Begawan, Brunei; 1993 Surabaya, Indonesia; 1996 Hanoi, Vietnam; 1999 Manila, the Philippines; 2002 Singapore.

  22. Norwegian Mathematical Society
    • That problem found a temporary solution when Heegaard succeeded in obtaining funds for a series of pamphlets, Norsk matematisk forenings skrifter ..
    • For many years starting in 1922, Crown Prince Olav awarded a prize for the best solutions to a series of problems posed in the Journal.

  23. Armenian Academy of Sciences
    • Real Analysis: The main areas of research are trigonometric and general orthogonal series; bases in functional spaces; weighted functional spaces; differentiation of multidimensional integrals; representation and uniqueness for multiple Haar, Franklin, Walsh and trigonometric series; nonlinear approximation.

  24. Royal Statistical Society
    • Series B (Methodological):- .
    • Series A (General).

  25. Belgium Mathematical Society
    • In 1977 the Bulletin split into two series, with Hirsch remaining the sole editor of one of the two series until 1993.

  26. Mathematical Society of Philippines
    • As soon as it was founded the Mathematical Society of the Philippines set up a series of seminars held in various universities during 1973-74.
    • In the summer of 1974 they held a lecture series on Graph Theory followed by further seminars during 1974-75.

  27. Armenian Mathematical Union
    • Alexandr Andraniki Talalyan (born 22 September, 1928; died 9 August, 2016) studied for his Candidate's Degree at the Steklov Mathematical Institute and was awarded the degree in 1956 for his thesis On the convergence of orthogonal series.
    • He was awarded a doctorate (equivalent to a D.Sc.) from the Steklov Mathematical Institute in 1962 for his thesis Representation of measurable functions by series.

  28. French Statistical Society
    • The JES consists of a week of an increasingly deep study on a well defined theme, in an ideal setting, with a series of meetings and discussions.
    • 1984: Time series analysis.

  29. Israel Academy of Sciences
    • Among the active committees are those reviewing Israel's nuclear physics program, various life science topics, molecular medicine efforts, and one responsible for the publication of a series of volumes describing Israeli flora and fauna.

  30. Irish Mathematical Society
    • The Society planned a series of short instructional conferences.

  31. Warsaw Scientific Society
    • In 1884 he was one of the two founders of a series of mathematics and physics textbooks which were written in Polish.

  32. References for Trinity Cambridge
    • Theory Series B 25 (1978), 240-243.

  33. Egyptian Academy of Sciences
    • This has been accompanied by the development of a number of societies dealing with various branches of science, among which special mention may be made of the Societe Entomologique d'Egypte, with its fine series of Bulletins and Memoires dating from 1907.

  34. Argentina Mathematical Union
    • The second part of the first volume contains, among a number of papers, Hiperconvergencia de las series de Dirichlet cuyos exponentes forman una sucesion de densidad maxima infinita by Sixto Rios (Madrid) and Algunos complementos a la teoria de limites de las funciones reales en espacios abstractos by J Rey Pastor (Buenos Aires).

  35. German Academy of Scientists Leopoldina
    • The next President Emil Abderhalden reorganised the Sections of the Academy in 1932 and introduced the series Lebensdarstel-lungen deutscher Naturforscher (Biographies of German Natural Scientists).

  36. Serbian Academy of Sciences
    • In 1844 a series of laws were introduced concerning the administration and the education system in Serbia and the Society was able to recommence its work in August 1844.

  37. Mexican Academy of Sciences
    • This was not just a nominal shift; a series of meaningful changes accompanied it.

  38. South African Mathematical Society
    • There ensued a series of involved and somewhat delicate negotiations designed, on my part, to ensure that such a visit would take place under conditions which would satisfy my scruples.

  39. Brazilian Mathematical Society
    • In 1989 the Bulletin was relaunched with the English title Bulletin of the Brazilian Mathematical Society, beginning a new series and with an international editorial board.

  40. Chinese Mathematical Society
    • a professional society comprising thousands of members and publishes about ten mathematical journals and several book series.

  41. Association for Statistics and its Uses
    • 1984: Time series analysis.

  42. American Mathematical Society
    • Now, why would it not be possible to combine with this miscellaneous program (which ought by all means to be kept up), something more akin to university models? Would not a series of three to six lectures on nearly related topics, if well chosen, prove attractive and useful to larger numbers? .

  43. Heidelberg Academy of Sciences
    • The Academy organises scholarly and scientific symposia and public lecture series.

  44. Trinity Cambridge Mathematical Society
    • Theory Series B 25 (1978), 240-243.','2]).

  45. References for European Women
    • C Series, Interview, Mathematics Today (2007).

  46. Finnish Academy of Sciences
    • The Academy publishes the mathematics journal Annales Academiae Scientiarum Fennicae, Mathematica and also the monograph series Annales Academiae Scientiarum Fennicae, Mathematica Dissertationes which publishes doctoral theses.

  47. International Astronomical Union
    • The IAU Circulars were a series of postcards with information on recent discoveries that required prompt dissemination .

  48. Cyprus Mathematical Society
    • As an example of its contents, let us look at Volume 8 No 1 (2009) which was given over to publishing papers by plenary speakers from two previous conferences in the series 'Symposium on Elementary Maths Teaching' (SEMT).

  49. Serbian Society of Mathematicians and Physicists
    • From the year 1954 the title of the journal was Nastava matematike i fizike and in 1974, after 20 years of this title, a new series named Nastava matematike began having dropped the physics connection.

  50. Edinburgh Mathematical Society
    • A second colloquium was held in Edinburgh in the following year, prior to the outbreak of World War I, but the series had to be discontinued for the duration of the War.

  51. References for Lincei
    • M Sanchez Sorondo, The Pontifical Academy of Sciences : A Historical Profile, The Pontifical Academy of Sciences Extra Series 16 (Vatican City, 2003).

  52. Czech Mathematicians and Physicists Union
    • In the following year they began publishing a series of mathematics and physics textbooks.

  53. Spanish Mathematical Society
    • Rey Pastor had certainly played his part in supporting the journal, particularly in the period 1911-13, with a series of papers.

  54. International Mathematical Union
    • The series of International Congresses of Mathematicians had begun in Zurich in 1897 but no congress was held during World War I (1914-18).

  55. Lincei Accademia
    • It desired [',' M Sanchez Sorondo, The Pontifical Academy of Sciences : A Historical Profile, The Pontifical Academy of Sciences Extra Series 16 (Vatican City, 2003).','2]:- .

  56. Estonian Statistical Society
    • In order to achieve this aim, a series of national one- and two-days conferences has been organized.

  57. Egyptian Academy of Sciences
    • This has been accompanied by the development of a number of societies dealing with various branches of science, among which special mention may be made of the Societe Entomologique d'Egypte, with its fine series of Bulletins and Memoires dating from 1907.


Honours

  1. AMS Fulkerson Prize
    • for 'The matroids with the max-flow min-cut property', Journal of Combinatorial Theory Series B 23 (1977), 189-222.
    • for 'The width-length inequality and degenerate projective planes', W Cook and P D Seymour (eds.), Polyhedral Combinatorics, DIMACS Series in Discrete Mathematics and Theoretical Computer Science, volume 1, (American Mathematical Society, 1990) 101-105.
    • for 'Decomposition of balanced matrices', Journal of Combinatorial Theory, Series B, 77 (1999), no.
    • for 'The Excluded Minors for GF(4)-Representable Matroids', Journal of Combinatorial Theory Series B, 79 (2000), no.
    • for 'A characterization of weakly bipartite graphs', Journal of Combinatorial Theory Series B, 83 (2001), no.
    • for 'A combinatorial algorithm minimizing submodular functions in strongly polynomial time', Journal of Combinatorial Theory Series B 80 (2000), no.
    • Wagner's conjecture', Journal of Combinatorial Theory Series B 92 (2) 2004, 325-357.

  2. AMS Steele Prize
    • for his cumulative influence on the theory of Fourier series, real variables, and related areas of analysis.
    • for his papers "An interpolation problem for bounded analytic functions", "Interpolation by bounded analytic functions and the Corona problem", and "On convergence and growth of partial sums of Fourier series".
    • for his paper "On the existence and irreducibility of certain series of representations".
    • He later extended this work to a spectral theory for the automorphic Laplace operator, relying on the Radon transform on horospheres to avoid Eisenstein series.
    • Singer's series of five papers with Michael F Atiyah on the Index Theorem for elliptic operators (which appeared in 1968-71) and his three papers with Atiyah and V K Patodi on the Index Theorem for manifolds with boundary (which appeared in 1975-76) are among the great classics of global analysis.
    • for his contributions to low dimensional topology, and in particular for a series of highly original papers, starting with "Hyperbolic structures on 3-manifolds.

  3. Wilks Award of the ASS
    • in recognition of his many significant contributions to experimental design, robustness, Evolutionary Operations, Bayesian methods, and time series analysis, and for his leadership in relating theoretical results to practical problems.
    • for major contributions to our knowledge of time series and multivariate statistical analysis; and for pioneering in the advancement of statistics as researcher, teacher, author, editor, and adviser to the government and key national institutions, perpetuating in many ways the spirit in which Samuel S Wilks made his many contributions to statistics.
    • for outstanding research in Time Series Analysis, especially for his innovative introduction of reproducing kernel spaces, spectral analysis and spectrum smoothing; for pioneering contributions in quantile and density quantile functions and estimation; for unusually successful and influential textbooks in Probability and Stochastic Processes; for excellent and enthusiastic teaching and dissemination of statistical knowledge; and for a commitment to service on Society Councils, Government Advisory Committees, and Editorial Boards.
    • for maintaining the highest professional standards in research, teaching and service to the profession; for fundamental research into the mathematical basis of hypothesis tests and estimates; and for creating a series of textbooks that have inspired a generation of statisticians.
    • for significant and fundamental contributions to the theory and practice of statistics, particularly Bayesian inference, multiple time series modelling, intervention analysis, environmental statistics, seasonal adjustment, and forecasting; for leadership in research in business statistics, econometrics, finance, and atmospheric ozone; for being an outstanding mentor to Chinese statistical education and statisticians of many backgrounds; and for innovative service on government advisory committees and editorial boards.

  4. MAA Chauvenet Prize
    • The Convergence of Fourier Series, Amer.
    • Series of Orthogonal Polynomials, Annals of Mathematics 34 (1933), 527-545; .
    • Barcodes: The Persistent Topology of Data, Bulletin (New Series) of the American Mathematical Society, 45, no.

  5. AMS Cole Prize in Number Theory
    • for a series of three joint papers "Diophantine problems over local fields.
    • for pioneering work on automorphic forms, Eisenstein series and product formulas, particularly for his paper "Base change for GL(2)".
    • for their paper "Heegner points and derivatives of L-Series".

  6. Sylvester Medal
    • for his brilliant researches in the theories of aggregates and of sets of points of the arithmetic continuum, of transfinite numbers, and Fouriers series.
    • .for his major distinctive contributions to time series analysis, to optimisation theory, and to a wide range of topics in applied probability theory and the mathematics of operational research.
    • His most spectacular achievement was the proof of the convergence almost everywhere of the Fourier Series of square integrable and continuous functions.

  7. Royal Medal
    • for his investigations and discoveries contained in the series of experimental researches in electricity published in the Philosophical Transactions, and more particularly for the seventh series, relating to the definite nature of electrochemical action.
    • for his paper on the laws of the tides on the cost of Ireland, as inferred from an extensive series of observations made in connection with the Ordnance Survey of Ireland, published in the Philosophical Transactions for the present year.

  8. European Mathematical Society Prize
    • In his thesis and in the subsequent work with Braverman, Gaitsgory established fundamental properties of Eisenstein series in the geometric setting.
    • has created the method of dynamic diophantine approximation which has led to a series of remarkable results in complex geometry of algebraic varieties.
    • Very recently, he found they key to the problem of defining, in non-commutative Iwasawa theory, the analogue of the characteristic series of modules over Iwasawa algebras.

  9. The Moran Medal
    • Citation: Rob Hyndman has made major contributions to a wide range of fields, especially to forecasting, time-series, graphical methods and computational statistics.
    • His research in forecasting challenged the appropriateness of the most fundamental of Bayesian forecasting models for exponential-family time series and on state-space models for exponential smoothing.

  10. Galway Group Theory.html
    • For a brief history of this conference series see the article History of Groups in Galway.
    • Dave Johnson (University of Nottingham) Power series under substitution .

  11. International Congress Speaker
    • The series of International Congresses of Mathematicians began in Zurich in 1897 but no congress was held during World War I (1914-18) or World War II (1939-45).
    • Lennart Carleson, Convergence and Summability of Fourier Series.

  12. Copley Medal
    • For his Paper On the Summation of Series, whose general term is a determinate function of z the distance from the first term of the series.

  13. AMS Cole Prize in Algebra
    • for his paper "On Artin's L-series with general group characters".

  14. Groups St Andrews.html
    • For a brief history of this conference series see the article Twenty-Five Years of Groups St Andrews.

  15. Henry George Forder Lectures
    • 2003 Caroline Series .

  16. Gibbs Lectures.html
    • The lectures are given annually, but there have been a few years since the series began when no lecture was given.

  17. Ruth I Michler Memorial Prize
    • Appointed to the University of North Texas in Denton she began to earn an excellent international reputation with a series of outstanding papers.

  18. LMS Presidential Addresses
    • Considerations respecting the Translation of Series of Observations into Continuous Formulae.

  19. LMS President
    • 2118 - 2119 Caroline Series .

  20. IMU Leelavati Prize
    • In the book the author poses a series of problems in elementary arithmetic and algebra as challenges to a person named Leelavati, followed by indications of solutions.

  21. LMS Whitehead Prize
    • 1987 C M Series .

  22. MAA Hedrick Lecturer
    • who will present a series of at most three lectures accessible to a large fraction of those who teach college mathematics.

  23. New Zealand Mathematics Society Research Award
    • For an outstanding series of research articles on harmonic functions and potential theory, in which he has introduced new ideas and tools, and deep analyses, that have resulted in new and improved approaches to classical theorems and led to their generalisation to more abstract situations.

  24. Microsoft Research Prize in Algebra and Number Theory
    • Her book 'On the cohomology of certain noncompact Shimura varieties' published in the Annals of Mathematics Studies Series is described as a tour-de-force.


References

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  4. References for William Feller
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  6. References for Madhava
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    • Series A (General) 148 (2) (1985), 162.
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  8. References for Alexander Ivanovich Skopin
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    • Series A (General) 119 (1) (1956), 87.

  10. References for Dunham Jackson
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  11. References for Patrick Moran
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  13. References for James Gregory
    • R C Gupta, The Madhava-Gregory series, Math.
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  16. References for Dmitrii Menshov
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  17. References for Józeph Petzval
    • Petzval History Series: The Early Life of Joseph Petzval, Lomography Magazine (9 June 2015).
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  18. References for Cecilia Payne-Gaposchkin
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  19. References for Ernest Esclangon
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  20. References for Frank Harary
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  22. References for Gábor Szeg
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  25. References for Cora Sadosky
    • (eds.), Harmonic Analysis, Partial Differential Equations, Complex Analysis, Banach Spaces, and Operator Theory 1, Association for Women in Mathematics Series 4 (Springer International Publishing, Switzerland, 2016), E1.
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  27. References for Daniel Pedoe
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  36. References for Claude Berge
    • Series A (General) 126 (2) (1963), 322-323.
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  37. References for Lipót Fejér
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    • H Krakauer, Review: The Calculus of Selfishness, by Karl Sigmund, Science, New Series 328 (5981) (2010), 977-978.
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  42. References for John Tukey
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  43. References for Richard Askey
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  45. References for Evgenii Mikhailovich Lifshitz
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  46. References for Geoffrey Kneebone
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  47. References for Nicolas-Louis de Lacaille
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  48. References for Christoph Scheiner
    • J Casanovas, Early Observations of Sunspots: Scheiner and Galileo, in B Schmieder, J C del Toro Iniesta and M Vasquez (Eds.), 1st Advances in Solar Physics Euroconference, Advances in the Physics of Sunspots, ASP Conference Series 118 (1997), 3-20.
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  49. References for Joseph Fourier
    • S Bochner, Fourier series came first, Amer.
    • A C Bose, Fourier series and its influence on some of the developments of mathematical analysis, Bulletin of the Calcutta Mathematical Society 9 (1917-8), 71-84.

  50. References for Federigo Enriques
    • C D Broad, Review: Problems of Science by Federigo Enriques, Mind (New Series) 24 (93) (1915), 94-98.
    • C J Keyser, Review: Problems of Science by Federigo Enriques, Science (New Series) 40 (1027) (1914), 346-350.

  51. References for Lev Landau
    • Mechanics and Molecular Physics, by L D Landau, A I Akhiezer, and E M Lifshitz, Science, New Series 160 (3828) (1968), 667.
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  52. References for Gottfried Leibniz
    • P Costabel, Leibniz et les series numeriques, in Leibniz in Paris (1672-1676) Sympos.
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  53. References for Bryce McLeod
    • J Ball, Interview: Bryce McLeod, Oxford Mathematical Institute video series.
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  54. References for James Murray
    • Series D (The Statistician) 40 (3), Special Issue: Survey Design, Methodology and Analysis (2) (1991), 344-345.
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  55. References for Hans Meinhardt
    • E C Cox, Review: The Algorithmic Beauty of Sea Shells, by Hans Meinhardt, Science, New Series 270 (5233) (1995), 113-115.
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  56. References for Wilbur Knorr
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  57. References for Henry Scheffé
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  58. References for Vito Volterra
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  59. References for Madan Lal Puri
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  72. References for Louis Karpinski
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  95. References for Hilda Geiringer
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  101. References for Richard von Mises
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  112. References for al-Kashi
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  114. References for Charles Weatherburn
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  116. References for Walter Rouse Ball
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  117. References for William Wager Cooper
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    • Series A, Mathematical and Physical Sciences 148 (863) (1935), 1-31.

  120. References for Paul Butzer
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  121. References for Roger Godement
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  122. References for Nilakantha
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  123. References for William Whyburn
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  124. References for Lars Ahlfors
    • (eds.), Science, New Series 120 (3107) (1954), 100.

  125. References for Anne Bosworth
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  126. References for William Clifford
    • A Celebration of Two Lives: William Kingdon Clifford (1845-1979) and Lucy Clifford (1846-1929), Mind, New Series 104 (414) (1995), 447-448.

  127. References for John Farey
    • M Bruckheimer and A Arcavi, Farey series and Pick's area theorem, The Mathematical Intelligencer 17 (4) (1995), 64-67.

  128. References for Joseph Pérčs
    • G C Evans, Review: Theorie Generale des Fonctionelles, by Vito Volterra and Joseph Peres, Science, New Series 88 (2286) (1938), 380-381.

  129. References for Fabian Franklin
    • F D Murnaghan, Fabian Franklin, Science, New Series 89 (2309) (1939), 283.

  130. References for Richard Tapia
    • Cornell Honours David Blackwell and Richard Tapia with Lecture Series, FOCUS Newsletter of the Mathematical Association of America 20 (6) (August/September 2000), 4.

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    • J Witkowski, The life and work of Prof Dr Tadeusz Banachiewicz, Acta Astronomica Series C 5 (1955), 85-94.

  132. References for Alfred Barnard Basset
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  133. References for Giovanni Vailati
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  134. References for Isaac Newton
    • Biography Series (Moscow, 1987).

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  137. References for Agner Erlang
    • Series A 114 (1) (1951), 103-104.

  138. References for Werner Rogosinski
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  139. References for Jyesthadeva
    • R C Gupta, The Madhava-Gregory series, Math.

  140. References for Abraham Fraenkel
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  141. References for David Blackwell
    • T S Ferguson, L S Shapley and J B MacQueen (eds.), Statistics, probability and game theory, Papers in honor of David Blackwell, Institute of Mathematical Statistics Lecture Notes - Monograph Series 30 (Hayward, CA, 1996).

  142. References for Pafnuty Chebyshev
    • M G Cox, Piecewise Chebyshev series, Bull.

  143. References for William Brouncker
    • J Dutka, Wallis's product, Brouncker's continued fraction, and Leibniz's series, Arch.

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    • L Ehrenpreis, Review: Linear Partial Differential Operators, by Lars Hormander, Science, New Series 143 (3603) (1964), 234.

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    • Presidential series: Exclusive Interview with Professor Ambros Speiser, International Federation for Information Processing.

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  147. References for Eugen Netto
    • F Cajori, Review: Lehrbuch der Combinatorik, by Eugen Netto, Science, New Series 16 (403) (1902), 469-470.

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  150. References for Jules Bienaymé
    • Series A (General) 142 (2) (1979), 259-260.

  151. References for Georges Lemaître
    • M Heller, Lemaitre, big bang, and the quantum universe, Pachart History of Astronomy Series 10 (Pachart Publishing House, Tucson, AZ, 1996).

  152. References for Yves Rocard
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  153. References for Julio Rey Pastor
    • L Espanol Gonzalez and C Sanchez Fernandez, Julio Rey Pastor and the theory of summable divergent series (Spanish), LLULL 24 (49) (2001), 89-118.

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    • J Dutka, Wallis's product, Brouncker's continued fraction, and Leibniz's series, Arch.

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  157. References for Sheila Power Tinney
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    • I Tweddle, John Machin and Robert Simson on Inverse- tangent Series for p, Archive for History of Exact Sciences 42 (1) (1991), 1-14.

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    • Kleine Ausgabe in einem Bande, by Heinrich Weber, Science, New Series 38 (981) (1913), 550-551.

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    • I Tweddle, James Stirling: this about series and such things (Edinburgh, 1988).

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  173. References for Richard Delamain
    • Domestic series.


Additional material

  1. Bolzano publications
    • This is the first volume of the new series (which is still being produced) of Bernard Bolzano- Gesamtausgabe.
    • This volume in the series of posthumous writings of the Collected works of Bernard Bolzano contains transcriptions of some early manuscripts which present his views in the 1810's on the foundations of logic, mathematics and physics.
    • Contains his thoughts on Euclidean geometry, manipulations of series, functions and foundations of calculus, and topics in mechanics.
    • Band 11 Teil 1 (German), [Bernard Bolzano - collected works: Series I.
    • Band 11 Teil 3 (German), [Bernard Bolzano - collected works: Series I.
    • Band 12 Teil 1 (German), [Bernard Bolzano - collected works: Series I.
    • Band 11 Teil 2 (German), [Bernard Bolzano - collected works: Series I.
    • Band 12 Teil 2 (German), [Bernard Bolzano - collected works: Series I.
    • Band 12 Teil 3 (German), [Bernard Bolzano - collected works: Series I.
    • Band 18 (German), [Bernard Bolzano - Collected works: Series 1.
    • Collected works: Series II.
    • Collected works: Series I.
    • Band 16 Teil 1 (German), [Bernard Bolzano - Collected works: Series I.
    • Band 16 Teil 2 (German), [Bernard Bolzano - Collected works: Series I.
    • Band 13 Teil 2 (German), [Bernard Bolzano - Collected works: Series I.
    • Band 4 Teil 1 (German), [Bernard Bolzano - Collected works: Series II.
    • (German) [Bernard Bolzano - Collected works: Series II.
    • Band 13 Teil 3 (German), [Bernard Bolzano - Collected works: Series I.
    • (German) [Bernard Bolzano - Collected works: Series II.
    • Band 5 Teil 2 (German), [Bernard Bolzano - Collected works: Series II.
    • Band 14 Teil 1 (German), [Bernard Bolzano - collected works: Series I.
    • Band 6 Teil 1 (German), [Bernard Bolzano - collected works: Series II.
    • 1827-1840 (German), [Bernard Bolzano - Collected works: Series II.
    • Band 6 Teil 2 (German), [Bernard Bolzano - collected works: Series II.
    • Band 7 Teil 1 (German), [Bernard Bolzano - Collected works: Series II.
    • Band 7 Teil 2 (German), [Bernard Bolzano - Collected works: Series II.
    • Band 8 Teil 1 (German) [Bernard Bolzano - Collected works: Series II.
    • Band 8 Teil 2 (German), [Bernard Bolzano - Collected works: Series II.
    • Band 14 Teil 2 (German), [Bernard Bolzano - Collected works: Series I.
    • Band 9 Teil 1 (German), [Bernard Bolzano - Collected works: Series II.
    • He also jots down his ongoing ideas on trigonometric series, the binomial theorem, Taylor's theorem, the mean-value theorem, and convergence of infinite series.
    • Band 14 Teil 3 (German), [Bernard Bolzano - collected works: Series I.
    • Band 10 Teil 1, Grossenlehre IV (German), [Bernard Bolzano - Collected works: Series II.
    • Band 9 Teil 2 (German), [Bernard Bolzano - Collected works: Series II.
    • Band 10 Teil 1 (German), [Bernard Bolzano - Collected works: Series II.
    • 1841-1847 (German), [Bernard Bolzano - Collected works: Series II.
    • Band 10 Teil 2 (German), [Bernard Bolzano - Collected works: Series II.
    • He is collecting material for his future mathematical texts and also shows a particular interest in trigonometric series.
    • Band 11 Teil 1 (German), [Bernard Bolzano - Collected works: Series II.

  2. Thomas Bromwich: 'Infinite Series
    • Thomas Bromwich: Infinite Series .
    • In 1908 Thomas John Bromwich published An introduction to the theory of infinite series which was based on lectures on analysis he had given at Queen's College, Galway in each of the five sessions between 1902 and 1907.
    • INFINITE SERIES .
    • In the remainder of the book free use is made of the notation and principles of the Differential and Integral Calculus; I have for some time been convinced that beginners should not attempt to study Infinite Series in any detail until after they have mastered the differentiation and integration, of the simpler functions, and the geometrical meaning of these operations.
    • In Chapter V will be found an account of Pringsheim's theory of double series, which has not been easily accessible to English readers hitherto.
    • In obtaining the fundamental power-series and products constant reference is made to the principle of uniform convergence, and particularly to Tannery's theorems (Art.
    • 50, 51, 83) on the continuity of power-series, a theorem which, in spite of its importance, has usually not been adequately discussed in text-books.
    • Chapter XI contains a tolerably complete account of the recently developed theories of non-convergent and asymptotic series; the treatment has been confined to the arithmetic side, the applications to function-theory being outside the scope of the book.
    • xxi., 2nd ed.), Carslaw's Fourier Series and Integrals (ch.
    • xiv., xy).] I was therefore led to write out Appendix III, giving an introduction to the theory of integrals; here special attention is directed to the points of similarity and of difference between this theory and that of series.
    • 169, 171, 172) are called by the same names as in the case of series and the traditional form of the Second Theorem of Mean Value is replaced by inequalities (Art.
    • I hope that most double-limit problems, which present themselves naturally, in connexion with integration of series, differentiation of integrals, and so forth, can be settled without difficulty by using the results given here.
    • Chapter X of the first edition ("Complex Series and Products") has been broken up into two chapters, X and XI, the first of these containing the general theory of complex series and products, and the second dealing with special series and functions.
    • Chapter Xl of the first edition ("Non-Convergent and Asymptotic Series") now becomes Chapter XII.
    • Here the entire discussion of the theory of summable series, apart from the historical introduction, has been omitted, as Dr Bromwich felt that an adequate account of the subject with its later developments would require more space than could be given to it in the present volume.
    • The part of the chapter devoted to, asymptotic series has been enlarged, and contains, among other new matter, an exposition of the asymptotic expansions of the Bessel functions.
    • Room has also been found for a discussion of trigonometrical series, including Stokes's transformation and Gibbs's phenomenon.
    • Appendix III ("Infinite Integrals and Gama Functions") was originally written in connection with the discussion of summable series, and might therefore have been omitted.
    • https://www-history.mcs.st-andrews.ac.uk/Extras/Bromwich_Series.html .

  3. Konrad Knopp: Texts
    • The first is Infinite Sequences and Series which contains the following publisher's information:- .
    • He develops the theory of infinite sequences and series from its beginnings to a point where the reader will be in a position to investigate more advanced stages on his own.
    • In the treatment of sequences and series that follows, he covers arbitrary and null sequences; sequences and sets of numbers; convergence and divergence; Cauchy's limit theorem; main tests for sequences; and infinite series.
    • Chapter three deals with main tests for infinite series and operating with convergent series.
    • Chapters four and five explain power series and the development of the theory of convergence, while chapter six treats expansion of the elementary functions.
    • The book concludes with a discussion of numerical and closed evaluation of series.
    • 1.1 Preliminary remarks concerning sequences and series .
    • Sequences and Series .
    • 2.6 Infinite series .
    • The Main Tests for Infinite Series.
    • Operating with Convergent Series .
    • 3.1 Series of positive terms: The first main test and the comparison tests of the first and second kind .
    • 3.3 Series of positive, monotonically decreasing terms .
    • 3.6 Operating with convergent series .
    • Power Series .
    • 4.2 The functions represented by power series .
    • 4.3 Operating with power series.
    • 4.4 The inversion of a power series .
    • 5.6 Series transformations .
    • 5.7 Multiplication of series .
    • 6.5 The general power and the binomial series .
    • Numerical and Closed Evaluation of Series .
    • Series and the Expansion of Analytic Functions in Series .
    • Series with Variable Terms .
    • Uniformly Convergent Series of Analytic Functions .
    • The Expansion of Analytic Functions in Power Series .
    • Expansion and Identity Theorems for Power Series .
    • Continuation by Means of Power Series and Complete Definition of Analytic Functions .
    • Volume I contains more than 300 elementary problems dealing with fundamental concepts, infinite sequences and series, functions of a complex variable, conformal mapping, and more.
    • Infinite Sequences and Series .
    • Infinite Series with Constant Terms.
    • Convergence Properties of Power Series.
    • Expansion in Series .
    • Series with Variable Terms.
    • Expansion in Power Series.
    • Behaviour of Power Series on the Circle of Convergence.

  4. Hille publications
    • Note on Dirichlet's series with complex exponents, Ann.
    • Some remarks on Dirichlet series, Proc.
    • On Laguerre's series: First note, Proc.
    • On Laguerre's series: Second note, Proc.
    • On Laguerre's series: Third note, Proc.
    • Note on the behavior of certain power series on the circle of convergence with application to a problem of Carleman, Proc.
    • (With J D Tamarkin) On the summability of Fourier series, Proc.
    • Note on the preceding paper by Mr Peek: "Solution to a problem in diffusion employing a non-orthogonal sine series", Ann.
    • (With J D Tamarkin) On the summability of Fourier series.
    • Note on a power series considered by Hardy and Littlewood, J.
    • Note on Some Hypergeometric Series of Higher Order, J.
    • Review: Six Lectures on Recent Researches in the Theory of Fourier Series, by Ganesh Prasad, Amer.
    • (With J D Tamarkin) On the summability of Fourier series.
    • (With H F Bohnenblust) Sur la convergence absolue des series de Dirichlet, C.R.
    • (With H F Bohnenblust) On the absolute convergence of Dirichlet series, Ann.
    • (With J D Tamarkin) On the summability of Fourier series.
    • Review: Infinite Series, by Tomlinson Fort, Amer.
    • (With J D Tamarkin) On the summability of Fourier series.
    • Summation of Fourier series, Bull.
    • (With J D Tamarkin) On the summability of Fourier series.
    • (With J D Tamarkin) Addition to the paper "On the summability of Fourier series.
    • (With J D Tamarkin) On the summability of Fourier series.
    • (With J D Tamarkin) On the summability of Fourier series.
    • On the absolute convergence of polynomial series, Amer.
    • Contributions to the theory of Hermitian series, Duke Math.
    • Sur les series associees a une serie d'Hermite, C.
    • Contributions to the theory of Hermitian series.
    • Characteristic series of boundary value problems, Trans.
    • On the oscillation of differential transforms and the characteristic series of boundary-value problems, Univ.
    • (With George Klein) Riemann's localization theorem for Fourier series, Duke Math.
    • Review: Trigonometric Series Vols.
    • 1 and 2, by A Zygmund, Science New Series 130 (3376) (1959), 618.
    • Sur les fonctions analytiques definies par des series d'Hermite, J.
    • On a class of series expansions in the theory of Emden's equation, Proc.
    • Addison-Wesley Series in Mathematics (Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1972).
    • On a class of interpolation series, Proc.
    • Contributions to the theory of Hermitian series.

  5. D'Arcy Thompson on Greek irrationals
    • The following is a table of the side and diagonal numbers ([side and diagonal numbers] (literally [number of units in the sides and diagonals])) - Proclus (28, 10) gives the series as far as 12, 17, and adds - [and always thus (etc.)].
    • We begin, necessarily, with One, as the [ beginning] or origin of both series; for, as Theon says, Unity is the first principle of all configurations, and consequently there is in Unity a [ rule] both of diagonal and of side.
    • Firstly, the successive results are closer and closer approximations to that irrational number (viz., √2) which is the 'limit', the unattainable limit, of the series.
    • This property of the side- and- diagonal series, that not merely is the square of the one in alternate excess and defect as compared with twice the square on the other, but that this alternate excess and defect is in every case measured by one unit, is expressly stated by Theon and by Proclus.
    • [One], in short, is the word used both for that unit with which the series on either side begins, and for that unit which is at every successive stage the measure of excess or defect.
    • It is the beginning, the [start] of the whole series , then again, as the series proceeds, the 'One' has to be imported into each succeeding Dyad, where it defines ([defines]) the amount of excess or defect, and equates or equalizes ([equals]) the two incompatible quantities.
    • This point, this precise nature of the agency of the 'One', and the simple explanation which it involves of the precise meaning of [to define] or [to equal], both seem to me to be made clear by our study of the Greek side-and-diagonal series; but the point is lost as soon as we replace that formula by the continued fractions of our modem arithmetic.
    • Prof Taylor, so it seems to me, has treated the successive convergents of the continued fraction as identical with the successive fractions of the Greek series.
    • As he says, 'they never actually meet, since none of the "convergents" is ever the same as its successor, but by proceeding far enough with the series we can make the interval between two successive "convergents" less than any assigned difference, however small.' Precisely so; but all the while that 'monad' in which the excess or defect consists is never seen in the convergents of the continued fraction; and indeed it is so effectually concealed that Prof Taylor neither recognizes its importance nor even mentions it at all.
    • identical with the series of convergent fractions beginning 1/1 , 2/1 , 5/3 &c., which we may set forth as follows, in tabular form: .
    • which continued fraction is identical with the series of convergent fractions 1/5 , 10/51 , &c.
    • Whence we find the convergents to √3 to be 5/3 , 26/15 , 265/153 , 1351/780 , &c.; in short, we have the very series from which Archimedes may have drawn his examples, without omissions.
    • But he would very soon find that 32 was nearly the double of 22 ; searching for another such case, he would find that 72 was nearly the double of 52; and by the time he had found a third instance he would be on the brink of the rule which connects them all, and defines the series.
    • We have now seen that in the convergent series leading to √3, the 'One' is no longer the unique and indispensable 'equalizer'; and we shall soon see that it is by no means indispensable (though at first it seemed so) in the series of side and diagonal numbers which leads to √2.
    • This is the famous series, sometimes called the Fibonacci series, supposed to have been 'discovered' or first recorded by Leonardo of Pisa, nicknamed the Son of the Buffalo, or 'Fi Bonacci'.
    • It is the simplest of all additive series, for each number is merely the sum of its two predecessors.
    • It has no longer anything to do with sides or diagonals, and indeed we need no longer write it in columns, but in a single series, .
    • Here is another of the many curious properties of the series: .
    • Euclid himself is giving us a sort of algebraic geometry, or rather perhaps a geometrical algebra; and the series we are now speaking of arithmeticizes that geometry and that algebra.
    • It is surely much more than a coincidence that this series is closely related to Euclid II.
    • Our two series started alike, with 1 and 1 - the [beginning] of all arithmetic.
    • Of the two series which thus begin alike and then part company, the one leads to the square-root of 2 or the hypotenuse of an isosceles right-angled triangle, and the other leads to the Divine or Golden Section.
    • It is inconceivable that the Greeks should have been familiarly acquainted with the one and yet unacquainted with the other of these two series, so simple, so interesting and so important, so similar in their properties and so closely connected with one another.
    • Depend upon it, the series which has its limit in the Golden Mean was just as familiar to them as that other series whose limit is √2.
    • The Golden Mean series is a very curious one; and we have put it only in one and that the simplest of its many forms.
    • For the fact is, we may begin it as we please, with 1, 1, or 1, 2, or 1, 3, or any two numbers whatsoever, whole or fractional, and in the end it comes always to the same thing! For instance, we may have the series .
    • The side of the decagon, then, or the star-decagon, may be read off at once to any required degree of accuracy from our table of the Sectio Divina or Golden Mean, or in other words from our Fibonacci series.
    • All this is a beautifully simple illustration of a principle recognized in modern mathematics, that you may immensely extend the efficiency (so to speak) of the series of natural numbers if only you can add one other number to it.
    • carry us a long way; but if we add to this consecutive series either √2, or π, or the number we have now called τ, in each case an immense new field of operations is rendered possible.
    • We can neither represent it by a continued fraction nor by a series of side-and-diagonal numbers.

  6. Hardy Inaugural Lecture
    • In particular, the solution of the problem shows quite clearly that, if we are to attack these 'additive' problems by analytic methods, it is in the theory of integral power series .
    • In the latter theory the right weapon is generally not a power series, but what is called a Dirichlet's series, a series of the type .
    • The associated power series .
    • is easily transformed into the series .
    • called Lambert's series.
    • The corresponding Dirichlet's series is far more fundamental; it is in fact .
    • so that, in the theory of Dirichlet's series, the terms combine naturally with one another in a 'multiplicative' manner.
    • so that the multiplication of two terms of a power series involves an additive operation on their ranks.
    • It is thus that the Dirichlet's series rather than the power series proves to be the proper weapon in the theory of primes.
    • Here also we are led to a power series, or infinite product, convergent inside the unit circle; but there the resemblance ends.
    • It was conjectured by a very brilliant Hungarian mathematician, Mr G Polya, five or six years ago, that any function represented by a power series whose coefficients are integers, and which is convergent inside the unit circle, must behave, in this respect, like one or other of the two generating functions which we have considered.
    • The series is convergent when |x| < 1, and, by Cauchy's Theorem, we have .
    • This class of points is indeed an infinite class; but the infinity is, in Cantor's phrase, only an enumerable infinity; and the points can therefore be arranged in a simply infinite series, on the model of the series .
    • There is, therefore, at any rate, the hope that we may be able to isolate the contributions of each of these selected points, and obtain, by adding them together, a series which may give a genuine approximation to our coefficient.
    • We find that there is a certain series, which we call the singular series, which is plainly the key to the solution.
    • This series is .
    • The genesis of the series is this.
    • We associate with the rational point x = e2pπi/q an auxiliary power series .
    • We then add together all these auxiliary functions, and endeavour to approximate to the coefficient of our original series by summing the auxiliary coefficients over all values of p and q.
    • There are certain characteristics common to all these series.
    • The series for the cubes is easily shown to be positive; but we cannot deduce that r3,7 (n) is positive, and draw consequences as to the representation of numbers by 7 cubes, because in this case we cannot dispose satisfactorily of the error term O(nσ) in the general formula.
    • In the two cases relating to fourth powers which I have chosen, the discussion of the series itself is rather more delicate, for there is in each of them one term which can be negative and greater than 1.
    • It brings us for the first time into relation with the series on which the solution in the last resort depends, and tells us, approximately but truly, what the number of representations really is.

  7. Rios publications
    • Sixto Rios Garcia, Sur l'ensemble singulier d'une classe des series potentielles de Taylor qui presentent des lacunes, Comptes Rendus de l'Academie des Sciences de Paris 197 (1933), 1170-1173.
    • Sixto Rios Garcia, Sur l'ensemble singulier d'une classe des series potentielles de Taylor qui presentent des lacunes, Revista Matematica Hispano-Americana (2) 8 (1933), 221-224.
    • Sixto Rios Garcia, Algunos resultados relativos a la hiperconvergencia en las series de Dirichlet, Revista Matematica Hispano-Americana 9 (1934), 132-136 .
    • Sixto Rios Garcia, Las series de polinomios exponenciales y la hiperconvergencia de las series de Dirichlet, Bol.
    • Sixto Rios Garcia, Hiperconvergencia de las series de Dirichlet cuyos exponentes forman una sucesion de densidad maxima infinita, Revista de la Union Matematica Argentina 1 (2) (1937), 71-78.
    • Sixto Rios Garcia, Hiperconvergencia de las series de Dirichlet cuyos exponentes forman una sucesion de densidad maxima infinita, Bol.
    • Sixto Rios Garcia, Sobre las series de potencias desordenadas y la hiperconvergencia de una clase de series de Dirichlet, Revista de la Union Matematica Argentina 2 (4) (1938-1939), 27-33.
    • Sixto Rios Garcia, Las series de potencias desordenadas, Revista Matematica Hispano-Americana 15 (1940), 1-7.
    • Sixto Rios Garcia, Sobre el problema de hiperconvergencia de las series de Dirichlet cuyas sucesiones de exponentes poseen densidad maxima infinita, Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales 34 (2) (1940), 163-179.
    • Sixto Rios Garcia, On the analytic continuation of Dirichlet series with infinite maximum density (Spanish), Revista Union Matematica Argentina 7 (1941), 38-40.
    • Sixto Rios Garcia, Analytical continuation of functions defined by Dirichlet series (Spanish), Revista Acad.
    • Sixto Rios Garcia, Lectures on the theory of the analytic continuation of Dirichlet series (Spanish), Revista Acad.
    • Sixto Rios Garcia, On the rearrangement of series of functions and its applications (Spanish), Abh.
    • Sixto Rios Garcia, Note on the convergence of trigonometric series (Spanish), Mat.
    • Sixto Rios Garcia, On the rearrangement of series of functions and its applications (Spanish), Revista Mat.
    • Sixto Rios Garcia, The problem of the number of isomers in the homologous series of organic chemistry (Spanish), Investigacion y Progreso.
    • Sixto Rios Garcia, The problem of the number of isomers in the homologous series of organic chemistry (Spanish), Gaz.
    • Sixto Rios Garcia, On the rearrangement of series of functions.
    • Dirichlet series with real coefficients (Spanish), Revista Mat.
    • Sixto Rios Garcia, On the probability that a Taylor series admits an analytic continuation (Spanish), Revista Mat.
    • Sixto Rios Garcia, On the sets of continuable and noncontinuable Taylor series (Spanish), Publ.
    • Sixto Rios Garcia, Sur l'ultraconvergence des series d'interpolation, C.
    • Sixto Rios Garcia, Introduction to the theory of Fourier series (Spanish), Revista Acad.
    • Sixto Rios Garcia, Theory of the analytic continuation of Dirichlet series (Portuguese), Centro Estudos Mat.
    • Sixto Rios Garcia, Introduction to the theory of Fourier series I (Spanish), Revista Acad.
    • Sixto Rios Garcia, Introduction to the theory of Fourier series II (Spanish), Revista Acad.
    • Sixto Rios Garcia, Problems of the rearrangement of series (Spanish), Collectanea Math.
    • Sixto Rios Garcia, Introduccion a la Teoria de Series Trigonometricas (C.

  8. Sansone books
    • The improvements and additions of Sansone are very pleasing to the harmonious structure of the author; they contribute to making the work a reliable and easy guide for one who wants to know the most beautiful and the highest theories of the modern analysis of the real variable, as well as for a suitable introduction to the subject matter of which Sansone gives a concise and clear representation in the second volume: the theory of the developments in series of orthogonal functions, and especially in the series of Legendre, of Chebyshev- Laguerre, and from Chebyshev-Hermite polynomials.
    • Theory of series.
    • Differentiation of series.
    • Taylor's Series.
    • Completion of the theory of series.
    • The five chapters deal, respectively, with sets and transfinite numbers; measure of linear sets; measurable functions, functions of bounded variation; integration of measurable functions, integration of series; differentiation of integrals and of functions of bounded variation.
    • The topics discussed are, by chapters: (1) Developments in orthogonal functions, introduction to Hilbert space; (2) Fourier series; (3) Series of Legendre polynomials and of spherical harmonics; (4) Laguerre and Hermite series; (5) Approximation and interpolation; (6) Stieltjes integrals (including a discussion of Fourier transforms of distribution functions; this chapter has little connection with the rest of the book).
    • This is a translation of the first four chapters of the third edition (1952) of Sansone's second volume of Vitali's 'Moderna teoria delle funzioni di variabile reale', dealing with general theorems, Fourier series, series of Legendre polynomials and spherical harmonics, and series of Laguerre and Hermite functions.
    • Power series as holomorphic functions.
    • Expansion in Taylor series.
    • Dirichlet series, the zeta function of Riemann.
    • Summability of power series outside the circle of convergence.
    • Asymptotic series.
    • The first volume, on holomorphic functions, discusses Cauchy's theorem, singularities and residues, elliptic functions, the Riemann Zeta function, and summability of power series outside the circle of convergence.
    • Following Artin the theory of singular points is developed without the use of Laurent series.
    • Among the topics discussed are: power series, elementary functions, Cauchy integral theorem, residue theory, Weierstrass factor theorem, Mittag-Leffler theorem, elliptic functions, integral functions of finite order, Dirichlet series, Riemann zeta function, Laplace integral and asymptotic series.
    • Science, New Series 148 (3677) (1965), 1583.

  9. Green: 'Sequences and Series
    • Green: Sequences and Series .
    • A series of books called the Library of Mathematics were edited by W Ledermann.
    • One of the early texts in the series was Sequences and Series by J A Green.
    • Sequences and Series .
    • Its aims are, first, to present the fundamental mathematical ideas which underlie the notion of a convergent series, and secondly to develop, as far as the small space allows, a body of technique and a familiarity with particular examples sufficient to make the reader feel at home with such applications of infinite series as he is likely to meet in his scientific studies.
    • In particular the idea of convergence itself is directly involved in the practical problem of numerical calculation of the sum of a series, and I have devoted some space to this topic, traditionally neglected in elementary books on series.

  10. Groups St Andrews proceedings
    • The two volumes of the Proceedings of Groups - St Andrews 1989 are similar in style to 'Groups - St Andrews 1981' and 'Proceedings of Groups - St Andrews 1985' both published by Cambridge University Press in the London Mathematical Society Lecture Notes Series.
    • As mentioned above there was a series of lectures on CAYLEY and a CAYLEY workshop; there was also a lecture on GAP.
    • They invited T Hurley, S Tobin and J Ward to join them and continue the series in 1993 in Galway.
    • Also, as it transpired, one speaker was awarded a Fields Medal exactly one year later at the 1994 ICM in Zurich; the organisers have great pleasure in congratulating Professor Zelmanov most heartily - and hope perhaps that this may augur well for future speakers in the series! .
    • This was the fifth meeting of the four-yearly Groups St Andrews Conferences, and the series continues to flourish.
    • The shape of the conference was similar to the previous conferences in that the first week was dominated by five series of talks, each surveying an area of rapid contemporary development in group theory.
    • As the largest regular meeting on group theory in the world, this series has provided a continuing stimulus to research in group theory.
    • Groups St Andrews 2001 in Oxford was another highly successful conference in the continuing series.
    • The main speakers, who were invited to give a series of talks, were Marston D E Conder (Auckland), Persi Diaconis (Stanford), Peter P Palfy (Eotvos Lorand, Budapest), Marcus du Sautoy (Cambridge), and Michael R Vaughan-Lee (Oxford).
    • It is a measure of the success of this conference series and this subject that mathematical libraries around the world are collecting the series of St Andrews Conference Proceedings.
    • It is hoped that the next conference in this series will be held in 2005 and will be Groups St Andrews in St Andrews (revisiting the scene of the original crime), and it is also hoped that in 2009 we will return once more to Bath.
    • He attended the first Groups St Andrews conference in 1981 and some later ones in the series.
    • It is hoped that the next conference in this series will be held in 2009 and will be Groups St Andrews in Bath.
    • This was the eighth in the series of Groups St Andrews group theory conferences organised by Colin Campbell and Edmund Robertson of the University of St Andrews.
    • The shape of the conference was similar to the previous conferences (with the exception of Groups St Andrews 1981 and 2005) in that the first week was dominated by five series of talks, each surveying an area of rapid contemporary development in group theory and related areas.
    • The next conference in this series will be held in St Andrews in 2013.
    • This was the ninth in the series of Groups St Andrews group theory conferences organised by Colin Campbell and Edmund Robertson of the University of St Andrews.
    • This was the tenth in the series of Groups St Andrews group theory conferences.

  11. Vajda books
    • Series A (General) 119 (3) (1956), 340.
    • Series C (Applied Statistics) 6 (1) (1957), 79-80 .
    • Series A (General) 121 (4) (1958), 483-484.
    • Science, New Series 131 (3404) (1960), 916-917.
    • Series A (General) 123 (4) (1960), 493.
    • Science, New Series 132 (3436) (1960), 1306-1307.
    • The present closely related volume was developed from a series of lectures on the same subject matter and contains two main parts, one on linear programming and one on game theory.
    • Series A (General) 125 (3) (1962), 495-496.
    • Science, New Series 139 (3558) (1963), 898.
    • Series D (The Statistician) 14 (2) (1964), 176-177.
    • Series A (General) 126 (3) (1963), 470.
    • Series A (General) 131 (2) (1968), 233-234.
    • Series A (General) 131 (2) (1968), 233-234.
    • Series D (The Statistician) 25 (4) (1976), 307.
    • Series A (General) 139 (2) (1976), 273.
    • Series A (General) 142 (3) (1979), 384-385.
    • This book comprises a series of essays on a wide variety of topics, all related in some way to applied mathematics.
    • Series A (Statistics in Society) 160 (1) (1997),157.

  12. Broadbent's papers
    • An Early Method for Summation of Series.
    • The method given by Euler for the transformation of series was used by him to obtain "sums" for various divergent series, and recently Konrad Knopp [Math.
    • In Tract 8, "The Valuation of Infinite Series," written in 1780, Hutton gives an account of his method, claiming to have arrived at it independently of Euler's work, and employing it to approximate to the sums of series whose terms alternate in sign.
    • When the proposed series converges, we have an easy and rapid way of determining its sum, and in the case of a divergent series, a number is determined which would nowadays be recognized as a conventional sum.
    • 18, "On Interpolation and Summation"] refers to the "remarkable method" of Hutton, but he applies it only to convergent series, and makes no reference to its application to oscillating series.
    • This is noteworthy in view of the remarks made by De Morgan at the end of his Chapter 19 on "The Transformation of Divergent Developments," but it might be conjectured that even so acute a logician as De Morgan had not clearly perceived, at any rate at the time of writing his "Calculus," that the word "sum" applied to any infinite series is being used in a conventional sense.
    • In Tract 7 on "The Nature and Value of Infinite Series," he defines a convergent series as one in which the absolute magnitudes of the terms form a decreasing sequence and a divergent series as one in which they form an increasing sequence; a series of the type .
    • The theory given by Hutton turns on a fallacy connected with the use of this series.

  13. Gyula König Prize
    • I note that Szego's theorem, which he obtained in a roundabout way via the study of Toeplitz forms and the Fourier series of positive functions, can very easily be derived from a famous formula of Johan Jensen.
    • Another, extensive group of Szego's work belongs to the following sphere of ideas: from properties of the coefficients of power series, or from arithmetic properties of most of these coefficients, he deduces properties of the corresponding analytic functions.
    • More specifically: (i) theorems of Jacques Hadamard and Charles Fabry about lacunary power series; (ii) the theorem conjectured by Polya and proved by Fritz Carlson about power series with integer coefficients that are convergent inside the unit disk, which states that the function defined by such a power series either is rational or else cannot be extended beyond the unit disk; and (iii) the analogous theorem of Szego about power series having only finitely many different coefficients ("Uber Potenzreihen mit endlich vielen verschiedenen Koeffizienten," Sitzungsber.
    • In the same article, starting from the same principle, Szego also deduces a theorem of Alexander Markowich Ostrowski which leads us to a seemingly distant theorem of Robert Jentzsch (1890-1918), a young German mathematician who died in the war, about the distribution of zeros of the partial sums of power series.
    • Finally, I turn to the area belonging both to complex and real analysis to which Szego devoted the largest part of his work: the theory of orthogonal systems and the corresponding series expansions.
    • This contains, as a special case, both Legendre and power series.
    • These expansions, even in the general case, behave very much like power series, and provide a new and most natural solution to the following problem of Georg Faber: given a domain, find a system of polynomials in which every function that is holomorphic in this domain has an expansion.
    • In these two papers he examines the so-called "inner" asymptotics for orthogonal systems and the corresponding series expansions.
    • It is known, especially after Alfred Haar's dissertation, that with the help of these asymptotic expressions one can reduce questions of convergence and summability of series expansions to certain special cases, e.g., Fourier series.
    • Zeitschrift, Szego also shows that the same elementary method, without the use of asymptotic expressions of the polynomials, directly gives asymptotics for the partial sums [of orthogonal series] and in this way reduces convergence problems to analogous questions for Fourier series.

  14. Mathematicians and Music 3
    • In the eighteenth century when calculus had become a tool, there was a notable series of theoretical discussions of vibrating strings.
    • First in the series of theoretical discussions to which I have referred are those of Brook Taylor, who, according to his biographer, "possessed considerable ability as a musician and an artist." His discussions appeared in the Philosophical Transactions for 1713 and 1715 and in his book Methodus Incrementa Directa et Inversa, the first treatise dealing with finite differences, and the one which contains the celebrated theorem regarding expansions, now connected with Taylor's name.
    • He started with Taylor's particular solution and found, in effect, that the function for determining the position of the string after starting from rest could naturally be expressed in a form later called a Fourier series.
    • Thus were such series first introduced into mathematical physics.
    • In this way mathematicians were led to consideration of the famous problem of expanding an arbitrary function as a trigonometric series.
    • One of Euler's most notable papers connected with the history of Fourier's series did not appear in print till 1793, ten years after his death.
    • In such works, in the comparatively recent notable paper in this country by Harvey Davis, on vibrations of a rubbed string, and, of course, in other mathematical treatments of similar material, Fourier series must enter in a fundamental manner.
    • With specified conditions the series and its coefficients for a given tone or combination of tones may be determined.
    • Or, if we have a graph of the vibrations corresponding to such tones, the series may also be calculated, various terms in the series corresponding to simple elements compounded in the tone or tones.
    • In England, from 1905 to 1912, E H Barton and his associates published a series of papers illustrated by photographs of vibration curves particularly as issuing from the violin strings, bridge, and belly.
    • For the mathematician a great advantage of a photograph is that he can, after much labour, from it calculate the corresponding Fourier series.
    • By means of a Henrici machine, when the stylus of the instrument is moved along the curve of the photograph the numerical values of the coefficients in the corresponding Fourier series may be read off.
    • That is, a tone made up of 30 simple tones can be analyzed and the coefficients of the corresponding number of terms in the Fourier series written down.

  15. Bartlett reviews
    • Series C (Applied Statistics) 5 (1) (1956), 70.
    • Over recent years it has been developed in an ah hoc way by physicists studying statistical mechanics and the motion of fluids in turbulence; by mathematicians in games of chance; by electrical engineers in the communication of signals, the study of noise, and so on; and by statisticians studying population growth with births and deaths, sequential sampling, queues and renewals, and the correlation of time series.
    • S Geisser, Review: An Introduction to Stochastic Processes with Special Reference to Methods and Application by M S Bartlett, Science, New Series 122 (3166) (1955), 423-424.
    • Series A (General) 118 (4) (1955), 484-485.
    • Series A (General) 124 (2) (1961), 252.
    • All statisticians, together with the present and future generations of statistical students, will welcome the reappearance of Professor M S Bartlett's well-known book as a substantially bound paper-back in the Cambridge University Press series.
    • Series A (General) 130 (3) (1967), 429-430.
    • Series C (Applied Statistics) 10 (3) (1961), 189.
    • This monograph is one of a series on applied probability and statistics devoted to recent developments in these areas.
    • Mathematically-sophisticated population ecologists will welcome the appearance in the Methuen Monograph series of a book that is completely theirs.
    • Series A (General) 125 (3) (1962), 484-487 for a detailed review of these essays.
    • Series A (General) 140 (1) (1977), 105.
    • Series A (General) 140 (2) (1977), 248.
    • Series A (General) 133 (1) (1970), 101-102.

  16. Smith's Teaching Books
    • This book is the second of the Wentworth-Smith series.
    • This is the third of the Wentworth-Smith Series.
    • This text is a revised version of the old and widely known book of similar content of the "Wentworth series." With a few exceptions it consists of a partial rearrangement of a portion of the material of the older book, and users of the older book will have no difficulty in recognizing the text and many of the problems of the newer book.
    • This book differs from its predecessors in the series principally in the order of topics.
    • The wonder of the series really is the combination of authorship.
    • The basis of the series is the historic Wentworth series, than which no series has had a greater sale in the last quarter of a century.
    • There is something quite stimulating in seeing a series of arithmetics that is the latest evolution of a series that has been largely in use for many years.
    • mathematical induction, permutations, probability, determinants, theory of equations - followed by "Optional Special Topics" - partial fractions, interest and annuities, infinite series, and so on.
    • The exception is the treatment of infinite series.
    • It would not be easy to get together three men who would mean more in the mathematical world than do these authors of "Exercises and Tests in Algebra," and no one has prepared a more wholesome series of tests or measurements.
    • This compact collection of charming essays by Professor Smith is the first volume of a series announced by 'Scripta Mathematica' "designed to furnish, at a nominal price, material which will interest not only teachers of mathematics but all who recall their contact with the subject in their school or college days." ..

  17. Gibson History 5 - James Gregory
    • The two sequences (S) are called a "converging series," the corresponding pairs un , vn are called "converging terms," and the common limit is called "the termination of the series." It is from this beginning that the term "convergence" comes into use in connection with series.
    • The important point is that he seeks to prove that t, the termination of the converging series, can not be an algebraic function of any pair of the terms un , vn .
    • In a series of propositions he discusses the mensuration of the surface of paraboloids and hyperboloids of revolution and of spheroids and rectifies parabolic arcs.
    • Gregory's name is usually attached to a series for tan-1 x and except in this connection and in relation to the phrase "convergent series" it is rarely mentioned.
    • But the source from which the series issued sent forth many more theorems of great importance which seem to have been unnoticed.
    • (iii) general series for the sines and cosines of multiple angles, with a large variety of series for the mensuration of the circle.
    • Later, after seeing one of Newton's series, he developed many series and for the inspiration, though not for the methods, he was in these cases indebted, I think, to the simple statement (without explanations of any kind) of the Newtonian series.

  18. Grattan-Guinness books
    • as a convergent series of reciprocal powers of 2; he then finds the smallest zero of the function by taking the l.u.b.
    • Science, New Series 172 (3987) (1971), 1017.
    • Topics treated include definitions (for functions of a real variable) of limit, continuity, and the convergence of infinite series.
    • All the ingredients of a typical undergraduate course in Real Analysis are here: continuity, differentiability, limits, the convergence and uniform convergence of infinite series, integrability, Fourier series, etc.
    • In this book we see more clearly than before how Fourier came to his famous series.
    • From this first treatise Fourier developed all his subsequent writings on the theory of heat and on trigonometric series.
    • Submitted in December 1807 to the First Class of the French Institute (the revolutionary successor of the Academie des Sciences), it contained much that was original, both in its physics and in the range of novel mathematical techniques (including "Fourier series" and "Bessel functions") that it dis played.
    • Press History of Science series.
    • Mind, New Series 88 (352) (1979), 604-607.
    • The bulk of Grattan-Guinness's book is a series of extracts from this correspondence, in chronological order, linked by his own commentary.
    • And he exhibits that interest in a series of remarks explicitly directed to the textbook tradition .
    • Naturally, as he has done in other publications, Grattan-Guinness describes the unification of the relatively disjointed eighteenth-century differential and integral calculus techniques as limit theory and the algebra of inequalities gradually brought an understanding of the convergence and divergence of infinite series.

  19. Landau and Lifshitz reviews
    • Science, New Series 128 (3327) (1958), 767-768.
    • This volume is the second to appear in a projected series of nine volumes on theoretical physics by these authors.
    • 'Quantum mechanics' is the third volume of a series on theoretical physics by Landau and Lifshitz.
    • Landau and Lifshitz's multivolume series of texts on theoretical physics is justly acclaimed, especially for the unique, physically incisive presentation of the material.
    • The series of texts on theoretical physics by the authors has been announced for publication in English translation in 9 volumes.
    • This remarkable textbook is the latest translation of a nine-volume series on theoretical physics by one of the world's foremost and versatile physicists, Lev Landau, and his collaborator, E M Lifshitz.
    • Except for Volume 9, Physical Kinetics, all volumes of this series are now available in English, and their existence should exert considerable influence on the content of graduate courses in theoretical physics in English speaking countries because of their in comparable excellence, incredibly broad scope and up-to-date nature..
    • As in all previous volumes of this series, numerous interesting and important problems are worked out most ingeniously, albeit rather briefly, making these books so valuable.
    • This is the ninth volume of the famous series entitled 'A course of theoretical physics'.
    • Volume V of this series is regarded as part I of a course in statistical mechanics, and Volume IX (under review here) as part II.
    • This final volume in the celebrated Landau and Lifshitz series lives up to the standards of its predecessors.
    • This is the closing volume of a series of ten volumes, which actually represent a general course of theoretical physics as was planned by L.
    • Science, New Series 160 (3828) (1968), 667.

  20. De Montmort: 'Essai d'Analyse
    • Possibly Montmort had a contact with a pupil of Jacob - it is thought he did not meet Nicolaus Bernoulli until 1709 - and this contact inspired him to pursue the new calculus with its fascinating sidelines of the summation of infinite series and the manipulation of binomial coefficients.
    • The preceding solution furnishes a singular use of the figurate numbers (of which I shall speak later), for I find in examining the formula, that Pierre's chance is expressible by an infinite series of terms which have alternate + and - signs, and such that the numerator is the series of numbers which are found in the Table (i.e.
    • the Arithmetic Triangle) in the perpendicular column which corresponds to p, beginning with p, and the denominator the series of products p × p - 1 × p - 2 × p - 3 × p - 4 × p - 5; so that, cancelling out the common terms, we have for Pierre's chance the very simple .
    • whence by the method of inversion of series .
    • He gives a more general way of getting this series which he says he has obtained from a paper of Leibniz (Leipzig, 1693) in which is the problem Un logarithme etant donnee, trouver le nombre qui lui correspond.
    • One could make several interesting remarks about these series but that would take us outside the present subject and would lead us too far away.
    • It is clear that the Bernoullis helped considerably with this second edition, clarifying Montmort's ideas for him and contributing much in the way of summation of series.
    • Jacob was very good at summing series, so that this type of mathematical exercise was easy for Nicolaus and Johann.
    • This formula, the differences of zero series, had been reached by de Moivre in De Mensura Sortis in 1711.
    • The generalisations of the various topics discussed in the first edition are interesting, without adding anything particularly new to the probability calculus, although the various methods for the summation of series show the skill of the Bernoullis in that part of algebra.
    • Montmort obviously published the long series of letters because he wanted Nicolaus to have the credit of the results he had worked out.

  21. Charles Bossut on Leibniz and Newton
    • In the piece entitled De Analysi per Aequationes Numero Terminorum infinitas besides the method for resolving equations by approximation, which has nothing to do with us here, Newton teaches how to square curves, the ordinates of which are expressed by monomials or sums of monomials; and when the ordinates contain complex radicals, he reduces the question to the former case by evolving the ordinate into an infinite series of simple terms by means of the binomial theorem, which no one had done before.
    • While they agree that the evolution of radicals into series is a considerable step made by Newton, they immediately perceive, without the assistance of any subsequent and conjectural light, that the methods of Fermat, Wallis, and Barrow, might have been employed to find the results concerning quadratures which Newton contents himself with enunciating; since, after the evolution of radicals, if there be any, nothing more is necessary but to sum up the monomial quantities.
    • In one of his letters to Oldenburg, written even while he was in London, Leibniz says that having discovered a method of summing up certain series by means of their differences, this method was shown to him already published in a book by Mouton, canon of St Paul's at Lyon, On the Diameters of the Sun and Moon: that he then invented another method, which he explains, of forming the differences and thence deducing the sums of the series: that he is capable of summing up a series of fractions of which the numerators are unity and the denominators either the terms of the series of natural numbers, those of the series of triangular numbers, or those of the series of pyramidal numbers, etc.
    • He soon found the approximate quadrature of the circle by a series analogous to that which Mercator had given for the approximate quadrature of the hyperbola.
    • This series he communicated to Huygens by whom it was highly applauded; and to Oldenburg, who answered him that Newton had already invented similar things not only for the circle but for other curves of which he sent him sketches.
    • In fact the theory of series was already far advanced in England at that time; and though Leibniz had likewise penetrated deeply into it, he always acknowledged that the English, and Newton in particular, had preceded and surpassed him in that branch of analysis: but this is not the differential calculus, and the English have shown too evident a partiality in their endeavours to connect these two objects together, .
    • He relates that, on combining his old remarks on the differences of numbers with his recent meditations on geometry, he hit upon this calculus about the year 1676; that he made astonishing applications of it to geometry; that being obliged to return to Hanover about the same time he could not entirely follow the thread of his meditations; that endeavouring nevertheless to bring forward his new discovery, he went by the way of England and Holland; that he stayed some days in London where he became acquainted with Collins who showed him several letters from Gregory, Newton, and other mathematicians, which turned chiefly on series.

  22. Rios's books
    • Lectures on the theory of the analytic continuation of Dirichlet series (Spanish) (1943), by Sixto Rios Garcia.
    • This publication gives an account of some of the properties of Dirichlet series in the complex domain and their extension to Laplace-Stieltjes integrals.
    • Among the author's own contributions are studies on overconvergence, the determination of singularities, and, in the case of Dirichlet series, the question of analytic continuation by reordering.
    • This appears to be a revision of the author's lectures [Lectures on the theory of the analytic continuation of Dirichlet series (Spanish)].
    • In these lectures (written in collaboration with L Vigil) the author covers a wide variety of topics in the representation of analytic functions of a complex variable: Runge's theorems; analytic continuation by overconvergence and by rearrangement (he constructs, among other examples, a "universal" series of polynomials which can be rearranged to converge uniformly to any prescribed analytic function in any desired region); Mittag-Leffler, Borel and Painleve expansions; analytic continuation by summation of series; representation of functions by Laplace integrals and by Dirichlet, factorial, interpolation and Lambert series.
    • Introduccion a la Teoria de Series Trigonometricas (1949), by Sixto Rios Garcia.
    • The remaining chapters cover non-parametric analysis, a "theory of errors," analysis of variance, design of experiments, regression and correlation, sampling from finite populations, time series, and stochastic processes.
    • Time series.
    • Series A (General) 118 (1) (1955), 110-111.
    • But there are also chapters on decision functions, sequential analysis, non-parametric distributions, time series and stochastic processes.

  23. Archimedes on mechanical and geometric methods
    • If in two series of magnitudes those of the first series are, in order, proportional to those of the second series and further] the magnitudes [of the first series], either all or some of them, are in any ratio whatever [to those of a third series], and if the magnitudes of the second series are in the same ratio to the corresponding magnitudes [of a fourth series], then the sum of the magnitudes of the first series has to the sum of the selected magnitudes of the third series the same ratio which the sum of the magnitudes of the second series has to the sum of the (correspondingly) selected magnitudes of the fourth series.

  24. André Weil: 'Algebraic Geometry
    • My very cordial thanks go also to all those who read portions of the manuscript of this book, in preliminary or in final form, to whom many improvements are due, and to the Colloquium Committee and the staff of the American Mathematical Society, for doing me the honour of publishing this volume In their well-known series, and for the unfailing kindness and courtesy shown to me in all the arrangements connected- with this publication.
    • Thus for a time the indiscriminate use of divergent series threatened the whole of analysis; and who can say whether Abel and Cauchy acted more as "creative" or as "critical" mathematicians when they hurried to the rescue? One would be lacking in a sense of proportion, should one compare the present situation in algebraic geometry to that which these great men had to face; but there is no doubt that, in this field, the work of consolidation has so long been overdue that the delay is now seriously hampering progress in this and other branches of mathematics.
    • foundations of algebraic geometry may claim to be exhaustive unless it includes (among other topics) the definition and elementary properties of differential forms of the first and second kind, the so-called "principle of degeneration", and the method of formal power-series; but, concerning these subjects, nothing more than some cursory remarks in Chap.
    • How much the present book contributes to this, our readers, and future algebraic geometers, must judge; at any rate, as has been hinted above, and as will be shown in detail in a forthcoming series of papers, its language and its results have already been applied to the re-statement and extension of the theory of correspondences on algebraic curves, and of the geometry on Abelian varieties, and have successfully stood that test.
    • for instance, one will find here all that is needed for the proof of Bertini's theorems, for a detailed ideal-theoretic study (by geometric means) of the quotient-ring of a simple point, for the elementary part of the theory of linear series, and for a rigorous definition of the various concepts of equivalence.
    • As for my debt to my immediate predecessors, it will be obvious to any moderately well informed reader that I have greatly profited from van der Waerden's well-known series of papers [published in the Math.
    • formal power-series, and the representation of an ideal in a Noetherian ring as intersection of primary ideals) are used; the reader who is willing to take that theorem for granted, or successful in constructing a simpler proof of it, will not require, in all the rest of the book, any knowledge of these methods, or of anything beyond what has been mentioned above.
    • IX, it is possible to prove the same theorem, by means of Zariski's results on birational correspondences, without making any use of formal power-series; on the other hand, Chevalley, by giving [Trans.
    • 1-85], for some of the main results in the theory of intersections, alternative proofs which begin by establishing the corresponding theorems for algebroid varieties, has shown how the ring of formal power-series can be given the principal role, instead of the subordinate one which it plays in our treatment.
    • The contents of this chapter include all that is needed for the theory of the linear equivalence of divisors (and, in particular, of "virtual curves", i.e., in our language, of cycles of dimension 1, on surfaces), and consequently for the foundation of the theory of linear series on a variety.

  25. Ferrar: 'Textbook of Convergence
    • 166, (ii) a remark once made to me by Professor Hardy, and (iii) my own work on a special series.
    • The brief chapter on Fourier series will, I hope, prove useful in spite of its brevity and many omissions.
    • A First Course in the Theory of Sequences and Series .
    • Series Of Positive Terms .
    • Alternating Series .
    • The General Theory of Infinite Series .
    • The Product of Two Series .
    • Power Series .
    • Double Series .
    • Fourier Series .

  26. Gábor Szegö's books
    • These two volumes of examples in analysis are numbered XIX and XX respectively in the 'Grundlehren der mathematischen Wissenschaften' series, which is producing so many interesting works.
    • Science, New Series 91 (2370) (1940), 526.
    • Now a depth of critical understanding which scarcely went beyond the fundamental cases of Fourier and Legendre series has come to prevail with unifying authority over a wider range of generalization than had been even tentatively surveyed, and the diverse fields into which the applications extend derive clarification from a common body of coordinated knowledge.
    • However, these omissions are compensated by several interesting features: (a) an elaborate treatment of the asymptotic behaviour of orthogonal polynomials, by various methods, with applications, in particular, to the "classical" polynomials of Legendre, Jacobi, Laguerre and Hermite; (b) a detailed study of expansions in series of orthogonal polynomials, regarding convergence and summability; (c) a detailed study of orthogonal polynomials in the complex domain; (d) a study of the zeros of orthogonal polynomials, particularly of the classical ones, based upon an extension of Sturm's theorem for differential equations.
    • Science New Series 115 (2980) (1952), 155.
    • Science, New Series 128 (3316) (1958), 137-138.
    • The theory of Toeplitz forms has its roots in the work of Toeplitz, Fejer, Caratheodory, F Riesz on trigonometric series and harmonic functions.
    • Since their first publication in 1925 these books have served more than one generation of mathematicians as an introduction to important parts of Classical Analysis and number theory: the theory of infinite series; integral calculus, in particular inequalities and asymptotic expansions; functions of a complex variable, for instance geometric function theory and maximal principles; zeroes; polynomials; Determinants and Quadratic forms; number theory; and a little geometry.
    • The virtue of the book is that each section, after a brief introduction to the subject, consists of a series of problems so arranged as to make them of reasonable difficulty, as a whole, and an excellent means of thinking one's way into the subject, together with a very good crib for the occasions when the problems are too hard.
    • 1: Series, Integral Calculus, Theory of Functions (1972), by George Polya and Gabor Szego.

  27. The Dundee Numerical Analysis Conferences
    • The series of numerical analysis conferences held every two years in Dundee is part of the UK numerical analysis scene, if not of a wider picture.
    • John Morris had by now become the main organiser, and, following a lot of hard work negotiating with publishers, the Proceedings were published for the first time under his Editorship, by Springer Verlag in their Lecture Notes in Mathematics Series (Volume 109).
    • This was organised by John Morris, who edited the Proceedings which were published in the Springer Lecture Notes in Mathematics series.
    • I said in the Preface: "This was the 5th in a series of biennial conferences in numerical analysis, originating in St Andrews University, and held in Dundee since 1969".
    • So maybe this was the first explicit acknowledgement of the numbering system, and I assume I was interpreting the March, 1971, Conference as being the 4th in the series.
    • So the series was now well into its stride, and with a fairly well established pattern.
    • Following a change of policy by the Springer Editors, we moved to the Pitman Research Notes in Mathematics Series, published at that time by Longman.
    • The arrangement to publish the Proceedings in the Pitman series continued in a satisfactory way until 1999, although by this time the series had evolved into a Chapman and Hall/CRC series of Research Notes, published by CRC Press.

  28. Bahouri publications
    • Hajer Bahouri and Jean-Yves Chemin, Equations d'ondes quasi-lineaires et estimations de Strichartz, Notes aux Comptes-Rendus de l'Academie des Sciences de Paris (Series I) 325 (9) (1998), 803-806.
    • Hajer Bahouri and Jean-Yves Chemin, Equations d'ondes quasilineaires et effet dispersif, Notes aux Comptes-Rendus de l'Academie des Sciences de Paris (Series I) 329 (2) (1999), 117-120.
    • Hajer Bahouri, Isabelle Gallagher and Jean-Yves Chemin, Inegalites de Hardy precisees, Notes aux Comptes-Rendus de l'Academie des Sciences de Paris (Series I) 341 (2) (2005), 89-92.
    • Hajer Bahouri, Clotilde Fermanian-Kammerer and Isabelle Gallagher, Analyse de l'espace des phases et calcul pseudo-differential sur le groupe de Heisenberg, Notes aux Comptes- Rendus de l'Academie des Sciences de Paris (Series I) 347 (17-18) (2009), 1021-1024.
    • Hajer Bahour, Mohamed Majdoub and Nader Masmoudi, Lack of compactness in the 2D critical Sobolev embedding, the general case, Notes aux Comptes-Rendus de l'Academie des Sciences de Paris (Series I) 350 (3-4) (2012), 177-181.
    • Hajer Bahouri, Clotilde Fermanian-Kammerer and Isabelle Gallagher, Refined inequalities on Graded Lie groups, Notes aux Comptes-Rendus de l'Academie des Sciences de Paris (Series I) 350 (3-4) (2012), 393-397.
    • Hajer Bahouri, Jean-Yves Chemin and Isabelle Gallagher, Stability by rescaled weak convergence for the Navier-Stokes equations, Notes aux Comptes-Rendus de l'Academie des Sciences de Paris (Series I) 352 (4) (2014), 305-310.
    • Hajer Bahouri, Structure theorem for 2D linear and nonlinear Schrodinger equations, Notes aux Comptes-Rendus de l'Academie des Sciences de Paris (Series I) 353 (3) (2015), 235-240.
    • Hajer Bahouri, Jean-Yves Chemin and Isabelle Gallagher, On the role of anisotropy in weak stability for the Navier-Stokes system, London Mathematical Society Lecture Notes Series (Special volume on the honour of Abbas Bahri) (2019), 55 pages.

  29. Franklin's textbooks
    • deal with complex numbers, average values, and Fourier series.
    • Towards the end of the book [two chapters] deal with the theory of analytic functions both of real and complex variables, of their expansions in series, and of operations with these series, also with the theory of the convergence of Fourier series.
    • It is designed to show them how to operate with complex quantities and how to solve problems for their solution on the use of Fourier series and integrals, and Laplace transforms.
    • The author makes no attempt to lean on the mathematical rigour, but nevertheless the analysis is done in orderly fashion; the arguments are clear and plausible, and important facts, such as those concerning the convergence of Fourier series are plainly stated without claiming to be proved.
    • Apparently, this book was very carefully composed and its structure is such that the beginner can attain a solid knowledge of the subject by through a series of stages (81 articles in all) of fairly uniform and moderate difficulty.
    • The choice of title is accurate: the print is large and uncrowded on pages of medium size, the tersely phrased text is divided into sections of about one page, the total number of pages is small for the ground covered, which is differential and integral calculus of one variable, infinite series, and partial derivatives and multiple integrals ..
    • True, there is plenty of information here about Fourier series, Green's theorem, contour integration, the gamma function, and the rest; but these topics are set firmly in a more fundamental context, and much of the detail is relegated to the very full exercises at the ends of the chapters.

  30. Sikorski books
    • The third part of the book deals with applications of the theory of Lebesgue integration to orthogonal series, and to Fourier series and the Fourier integral.
    • The treatment of all this material is outstanding by its great clarity and in showing how the deeper results of set theory and the abstract theory of measure find applications in functional analysis in general, and the theory of orthogonal series.
    • The second volume of Sikorski's Real functions deals with function spaces, orthogonal series with special treatment of Fourier series and Fourier integrals.
    • There are four chapters: Function spaces; Hilbert space; Fourier series; Fourier integrals.
    • The appearance of this book in the well-known series "Ergebnisse der Mathematik" is timely.
    • But the coherent and smooth presentation of the whole body of material is, of course, more effective than a series of research papers.

  31. Samuel Wilks' books
    • In the contents of this book economists thus possess both a general theoretical framework and instructive examples, which will be of invaluable assistance when they set out to develop the statistical theory specifically adapted to the estimation problems concerning economic time series and the relations between them.
    • Wilks's book, in fact, is based on a series of lectures delivered at Princeton to graduate and advanced undergraduate students of mathematics.
    • Econometrica, New series 14 (55) (1947), 239-241.
    • The books of the other category, some of them excellent [Borel Series, Cramer, Kolmogorov, Uspensky], deal with mathematical theory of probability with only occasional glimpses on some particular questions pertaining to statistical theory.
    • Science, New Series 109 (2835) (1949), 450.
    • Series A (General) 126 (1) (1963), 128-129.
    • Series A (General) 129 (4) (1966), 593-594.
    • Series A (General) 136 (2) (1973), 262.

  32. Library of Mathematics
    • A series of books called the Library of Mathematics were edited by Walter Ledermann.
    • The title and editor of the series appear as: .
    • Ledermann describes the series as follows: .
    • This series of short text-books is primarily intended for readers who study mathematics as a tool rather than for its own sake.
    • 'These are all useful little books, and topics suitable for similar treatment are doubtless under consideration by the editor of the series.' .
    • Titles in the series are displayed on an early volume as follows: .
    • Sequences and Series J A Green .

  33. Boas books
    • Science, New Series 121 (3142) (1955), 390.
    • The objects of this monograph are to explain a general method, the method of kernel expansion, of expanding analytic functions in series ..
    • This little book differs from most items in the Ergebnisse series in that, rather than being a survey of a field, it is a semi-expository presentation of the authors' approach to a topic.
    • One thing that these questions show is that we are still short of a number of special series which will provide counterexamples.
    • As Boas observes early on, "complex analysis was originally developed for the sake of its applications." In addition to the usual applications to irrotational flows of incompressible, nonviscous fluids, there are discussions of the Plemelj formulas, Nyquist diagrams, and asymptotic series.
    • In addition to the standard topics based on the integral formulas and power series, there is an attractive collection of short discussions of additional topics including intuitive Riemann surfaces, non-Euclidean geometry, and Fourier series.
    • The book proper presents enough of a mix to delight readers of many stripes: reprints of a good number of Boas' expository mathematical papers (on Lion Hunting, Infinite Series, The Mean-Value Theorem and Indeterminate Forms, Complex Variables, Inverse Functions, Polynomials, The Teaching of Mathematics); several sections titled Recollections and Verse; Reminiscences by several close friends, students, and son Harold (also a mathematician); Reviews and other miscellany.

  34. Moran reviews
    • Series A (General) 123 (4) (1960), 485-486.
    • The approach will be familiar to those who have read J Gani's paper already published in Series B of this Journal.
    • It closes with two short chapters: one on Monte Carlo methods (of direct simulation type) and the other on the best or good Rules of release (of stock or water, as the case may be) by means of such examples as that where several dams in series, with electric generators of different efficiencies, have to supply a given amount of power with minimal expected loss due to overflow.
    • Series D (The Statistician) 14 (1) (1964), 77-78.
    • Series A (General) 128 (4) (1965), 596-597.
    • Sankhya: The Indian Journal of Statistics, Series A (1961-2002) 30 (1) (1968), 116-117.
    • Series A (General) 132 (1) (1969), 106.

  35. Champernowne reviews
    • Series D (The Statistician) 19 (3) (1970), 354-356.
    • Although part of a series aimed at "third-year undergraduates with an elementary knowledge of calculus and statistical methods," even the most advanced graduate students will find this volume very difficult - this, despite the fact that the book does not look very hard.
    • Sankhya: The Indian Journal of Statistics, Series B (1960-2002) 35 (1) (1973), 123.
    • [Review of Volume 1] This is the first of the three volume series by the author The author stresses the fact that most important economic decisions are made on the basis of imperfect information and in the face of considerable uncertainty about the probable effects of alternative possible decisions.
    • Series A (General) 138 (1) (1975), 111-112.
    • Economica, New Series 41 (161) (1974), 468-469.
    • Economica, New Series 67 (267) (2000), 461-462.

  36. Twenty-Five Years of Groups St Andrews Conferences
    • We realised that Groups 2005 would be the last of the series that we would organise before retiring so we made the decision to bring the conference 'home'.
    • Although attempting wide coverage of group theory topics, we made a conscious decision for the early conferences not to have a lecture series devoted to the classification.
    • At Groups St Andrews 2001 the lecture series were: Group actions on graphs, maps and surfaces with maximum symmetry; An introduction to random walks on finite groups - character theory and geometry; Groups and Lattices; Zeta functions of groups and counting p-groups; and Lie methods in group theory, and at Groups St Andrews 2005: Aspects of infinite permutation groups; On self-similarity and branching in group theory; Interactions between group theory and semigroup theory; and Graphs, automorphisms, and product action.
    • The influence of the series of conferences is, we believe, illustrated by the fact that 'Groups St Andrews' is mentioned in 285 reviews in MathSciNet and over 1500 papers in a beta version of the Google Scholar search.
    • The Proceedings of each of the first six conferences have been published by CUP as numbers 71, 121, 159 & 160, 211 & 212, 260 & 261, 304 & 305 in the London Mathematical Society Lecture Note Series.
    • The Proceedings of Groups St Andrews 2005 will again be published in two volumes in the same series.
    • We have enjoyed the twenty-five years of Groups St Andrews conferences and look forward to the continuation of the series.

  37. ELOGIUM OF EULER
    • At every turn in Euler's life, series analysis always occupied a special place.
    • His novel research into the series of indefinite products provided the necessary resources into solutions to a great many useful and curious questions.
    • It was above all by imagining the new series forms and by employing them not only to approximations, to which we are so often forced to take, but also into the discovery of absolute and rigorous proofs that M.
    • Taylor was made into an important branch of integral calculus by assigning a simple and workable notation which was found to apply successfully to the theory of series.
    • These rays are either the greatest or smallest of all those that belong to the series of curves formed in this way and that finally they always find themselves in planes perpendicular to one another.
    • At other times simple numbers, or a new series presented questions novel by their uniqueness which took him to unexpected proofs.
    • Once there were two of his students who had calculated a convergent series to the 17th term which was certainly complicated and needed to be written on paper but when the results were compared a discrepancy appeared by one number when the students asked the Master who was correct, Euler did the entire calculation in his head and his answer proved to be correct.

  38. Rydberg's application
    • Thus we find amongst his published works, besides a couple of essays on pure mathematics, several containing theoretical speculations on chemical atomic weight and further a whole series of investigations concerning the constitution of the spectra of the chemical elements.
    • The researches last mentioned, which perhaps ought to be considered the author's most important works, have their centre in his great memoir: "Recherches sur la constitution des spectres d'emission des elements chimiques." In this memoir, on the composition of which a considerable amount of work has been evidently expended, the author, similarly to what has been done by Kayser and Runge nearly at the same time, points out the possibility of arranging the lines of a spectrum in certain regular series, which, although at present without a theoretical foundation, will possibly be of importance for the search of the theory yet unknown of these extremely complicated phenomena.
    • To this belong the memoirs: "Uber den Bau der Linienspectren der chemischen Grundstoffe," "Recherches sur la constitution des spectres d'emission des elements chimiques," "Beitrage zur Kentniss der Linienspectren (I-IV)," "Die neuen Grundstoffe des Cleveitgases," "The New Series in the Spectrum of Hydrogen," "On Triplets with constant Differences in the Line Spectrum of Copper," "On the Constitution of the red Spectrum of Argon," and the memoirs handed in in manuscript: "Studien uber die Funktionsform der Spectralserien," and "Einige Liniengruppen mit constanten Schwingungsdifferenzen bei vierwerthigen Grundstoffen." The main problems of these memoirs have been to express the situation of the spectral lines of the elements through empirical formulae.
    • For example, I find admirable, that you discovered the second triplet series in the strontium spectrum based on Kayser's and my observations, one that we missed even though we had directed our attention especially to these series.
    • Also, regarding the new series in the spectrum of magnesium, which you announced, I have been convinced by later observations that you were completely correct, and Kayser and I were wrong when we disputed this series.

  39. Petrovic integration
    • Since 1925 the Paris Academy of Sciences had been commissioning a series of monographs reviewing current mathematical problems, the series having the title Memorial des sciences mathematiques.
    • Petrović was asked to write Integration qualitative des equations differentielles which appeared as No 48 in the series in 1931.
    • For this very reason it also facilitates the numerical calculation of the integral, because we already know convergent series which represent the integral in a region of the plane, and the main difficulty which presents itself as the extension of the numerical computation outside of this region, is to find a guide facilitating the passage from one region where the function is represented by a series, to another region where it is expressed by a different series.
    • Considerations of this kind have indeed guided Henri Poincare in these profound researches relating to the numerical computation of the integral of differential equations by means of series.

  40. More Smith History books
    • Having made an extensive collection of mathematical manuscripts, early printed works, and early instruments, and having brought together most of the European literature upon the subject and embodied it in a series of lectures for my classes in the history of mathematics, I welcomed the suggestion of Dr Carus that I join with Mr Mikami in the preparation of the present work.
    • Science, New Series 40 (1036) (1914), 675-676.
    • Science, New Series 58 (1502) (1923), 288-290.
    • It is needless to point out the difficulties that beset one who undertakes to write on the subject treated in this work (one of the series entitled Our Debt to Greece and Rome).
    • The editors of this series were fortunate in securing as the author of this book David Eugene Smith, whose History of Mathematics is the most complete and the most scientific work of its kind in our language.
    • it would seem only honest to state that while the book leaves a little to be desired in the matter of style and details, nevertheless, considering its brevity (necessitated by conforming with companion volumes of the series) it unquestionably represents a welcome addition to the literature of historical and expository mathematics.

  41. Einar Hille: 'Analytic Function Theory
    • The Cauchy integral is a much more pliable and versatile tool than the power series when it comes to doing things in function theory.
    • But before the student can really grasp integrals of analytic functions, he should have at his disposal a large number of such functions, and here the power series is invaluable as a source.
    • The power series also leads to important connections with real analysis, and it is indispensable for the problem of analytic continuation.
    • This is followed by a chapter on power series and one on the elementary transcendental functions.
    • Familiarity with abstract mathematical reasoning and some skill in manipulating identities, integrals, and series are the main prerequisites.
    • Finally, I wish to thank Ginn and Company for the honour they have shown me by letting my book inaugurate their new series "Introductions to Higher Mathematics" as well as for sympathetic consideration of an author's whims and wishes.

  42. Weil reviews
    • New Series 107 (2768) (1948), 75-76.
    • We find those fundamental facts without which, for example, a good treatment of the theory of linear series would be difficult.
    • This is the second of a series of papers with which the author promised to follow his book, Foundations of algebraic geometry, American Mathematical Society, 1946.
    • New Series 129 (3356) (1959), 1136-1137.
    • Its growth is made manifest by numerous conferences all over the world and is recorded, for example, in the "Lecture notes" series of Springer-Verlag.
    • New Series 226 (4682) (1984), 1412.

  43. Max Planck: 'Quantum Theory
    • The result of this long series of investigations was the establishment of a general relation between the energy of a resonator of given period and the radiant energy of the corresponding region of the spectrum in the surrounding field when the energy exchange is stationary.
    • Nothing can better illustrate the impetuous advance made in experimental methods in the last twenty years than the fact that since then, not one only, but a whole series of methods have been devised for measuring the mass of a single molecule with almost the same accuracy as that of a planet.
    • The first advance in this work was made by A Einstein, who proved, on the one hand, that the introduction of the energy quanta, required by the quantum of action, appeared suitable for deriving a simple explanation for a series of remarkable observations of light effects, such as Stokes's rule, emission of electrons, and ionization of gases.
    • The first brilliant result was Balmer's series for hydrogen and helium, including the reduction of the universal Rydberg constants to pure numbers, by which the small difference between hydrogen and helium was found to be due to the slower motion of the heavier atomic core.
    • This led immediately to the investigation of other series in the optical and Rontgen spectra by means of Ritz's useful combination principle, the fundamental meaning of which was now demonstrated for the first time.
    • Proceeding further along the same lines, P Epstein succeeded in giving a complete explanation of the Stark effect of the electrical separation of the spectral lines, and P Debye in giving a simple meaning to the K-series of the Rontgen spectrum, investigated by Manne Siegbahn.

  44. Cauchy's Calculus
    • Reasons for this latter approach, however widely they are accepted, above all in passing from convergent to divergent series and from real to imaginary quantities, can only be considered, it seems to me, as inductions, apt enough sometimes to set forth the truth, but ill founded according to the exactitude which is required in the mathematical sciences.
    • Cauchy's definition of convergence and sum of a series: .
    • If, for increasing values of n, the sum Sn approaches a certain limit S, the series will be called convergent and the limit in question will be called the sum of the series.
    • Conversely, when these many conditions are satisfied, the convergence of the series is assured.
    • He "proved" incorrectly that the limit of a convergent series of continuous functions is continuous.

  45. Bell books
    • Bell rightly disparages Pilate's undeservedly famous query, "What is truth?" The question means for him a series of ad hoc questions of the form, What results will follow from such and such an operation or set of operations? Of course, the question might also mean, What is absolute truth, or the secret of the universe? But this is next to nonsense.
    • Science, New Series 87 (2269) (1938), 578-579.
    • The reader of Men of Mathematics lays down the book after a first reading with a feeling of profound satisfaction that here is a fascinating set of short stories (it is more than a series of biographies), for frequent perusal in parts or as a whole, a book of reference for historical use, and a means of forming friend- ships with men already more or less well known.
    • Mr Bell has long been determined to write a history of mathematics and his recent books may be represented as a series of successive approximations to a genuine history.
    • Science, New Series 93 (2412) (1941), 281-283.
    • Science, New Series 113 (2938) (1951), 445-446.

  46. Douglas Jones's books
    • This book is an addition to the publishers' series, The Library of Mathematics.
    • It deals only with oscillations of systems with one degree of freedom, since more complicated systems are discussed elsewhere in the series.
    • In recent years several treatises on Electromagnetism have appeared; the volume under review is a worthy addition to the series.
    • These are two volumes of a series, edited by Professor Jones, entitled 'Introductory Mathematics for Scientists and Engineers'.
    • The book covers limits, functions, continuity, derivatives, maxima and minima, mean value theorem, sequences, series and (from both aspects) integrals.
    • Many personal computers are based on one of the 86 series of Intel microprocessors, namely the 8086, 80286, 80386, and the 80486, in order of increasing power.

  47. Frank Harary's books
    • Series A (General) 131 (1) (1968), 104-105.
    • Based on a series of lectures given in 1963 at the NATO Summer School on graph theory, this book concentrates on the relationship between graph theory and statistical mechanics and electric network theory, and in keeping with the editor's main interest six of the twelve lectures deal with enumerative problems.
    • Man, New Series 20 (4) (1985), 759-760.
    • American Anthropologist, New Series 87 (1) (1985), 171-172.
    • Man, New Series 27 (2) (1992), 425-426.
    • American Anthropologist, New Series 95 (2) (1993), 497-498.

  48. Hilbert reviews
    • Science, New Series 16 (399) (1902), 307-308.
    • Science, New Series 119 (3080) (1954), 75-76.
    • The text covers the following subjects: linear transformations and quadratic forms, development of arbitrary functions in series of orthogonal functions, linear integral equations, calculus of variations, eigenvalue and vibration problems, application of variational calculus to eigenvalue problems, and special functions, as in the German original.
    • Science, New Series 116 (3023) (1952), 643-644.
    • Science, New Series 99 (2573) (1944), 322.
    • Science, New Series 137 (3527) (1962), 334.

  49. Godement's reviews
    • The Artin zeta-functions (or L-series) are easy to define; those of Hecke are not.
    • They will be your students for the next two or three years, and your job is to lead them through calculus and into the beginnings of higher analysis - complex variables and Fourier series, for example.
    • The content is quite classical: sets and functions, convergence of sequences and series, continuous and differentiable functions, elementary functions.
    • The ones on set theory, real numbers, harmonic series, uniform convergence, Cauchy criterion, differentiable functions, logarithmic function, strange identities can be recommended.
    • Calcul differentiel et integral, series de Fourier, fonctions holomorphes (1998), by Roger Godement.
    • This is a review of the English translation Analysis II: Differential and Integral Calculus, Fourier Series, Holomorphic Functions (2005).
    • The volume under review, consisting of two chapters numbered XI and XII, is the fourth in a series.

  50. O'Brien Calculus
    • The method of Limits is generally allowed to be the best and most natural basis upon which to found the principles of the Differential Calculus; in the following pages this method is exclusively adopted, no use whatever being made of series in the demonstration of fundamental propositions.
    • IX contains the theory of Series, based upon one of the preceding Lemmas, without assuming that f (x + h) can be developed in the form .
    • and here I have endeavoured to show what the real nature of a series is, and to prove rigorously the principle of Indeterminate Coefficients.
    • XVII contains the general Theory of Contacts and Ultimate Intersections: no use is made of series in explaining the different orders of contact.
    • The assumption, that f(x + h) can be expanded in a series of the form A + Bha + Chb + &c.
    • seems to me to be a serious defect in the common method of establishing Taylor's Series, and thereupon the principles of the Differential Calculus.

  51. Gibson History 8 - James Stirling
    • Stirling made important contributions to mathematics in two different fields (i) in the theory of Higher Plane Curves, and (ii) in the theory of Series.
    • 1-13); (ii) Part I, Summation of Series (pp.
    • 15-84); (iii) Part II, Interpolation of Series (pp.
    • The developments of the Methodus Differentialis have an intimate relation to Gamma Functions and the Hypergeometric Series, but in the study of Stirling we appreciate the genius that enabled him to handle intractable series without the aids that the later developments put at our disposal.
    • It is of course not to be expected that the mathematical student can study at first hand even the majority of the older writers, but I do think that he should make a firsthand acquaintance with some of them; next to Newton I would place Stirling as the man whose work is specially valuable where series are in question.

  52. Charles Bossut on Leibniz and Newton Part 2
    • Some time before his death, Leibniz, wishing to feel the pulse of the English, as he expressed himself, caused the celebrated problem of orthogonal trajectories to be proposed to them, which consists in finding the curve that cuts a series of given curves at a constant angle, or at an angle varying according to a given law.
    • Johann Bernoulli, who had already made some attempts in this direction in the Memoirs of the Academy of Sciences for 1702, easily solved all these problems in the Leipzig Transactions for 1719; and from the results he obtained he formed a series of curious theorems, the development and demonstration of which were useful exercises for his son and nephew.
    • It does not properly belong to the new geometry, yet it contributed to it's progress by stimulating the spirit of combination in general, and by the extent which the author gave to the theory of series, a happy supplement to the imperfection of the rigorous methods in all branches of mathematics.
    • Three years afterwards de Moivre published a little treatise on the same subject entitled Mensura Sortis, chiefly remarkable for containing the elements of the theory of recurrent series and some very ingenious applications of it.
    • In this manner he has summed up some very curious series.
    • Nicole, however, a very distinguished French geometrician, was able to understand it: he very clearly unfolded the method for resolving finite differences and added several new series of his own invention.

  53. Apostol books
    • Science, New Series 127 (3293) (1958), 292.
    • Modular functions and Dirichlet series in number theory (1976), by Tom M Apostol.
    • Modular functions and Dirichlet series in number theory (Second edition) (1990), by Tom M Apostol.
    • For many years the author has been urged to develop a text on linear algebra based on material in the second edition of his two-volume Calculus, which presents calculus of functions of one or more variables, integrated with differential equations, infinite series, linear algebra, probability, and numerical analysis.
    • New horizons in geometry is a dense compilation of results from a series of papers published over the past 16 years.

  54. Smith's History Papers
    • Science, New Series 41 (1047) (1915),132-133.
    • Science, New Series 64 (1652) (1926), 204-206.
    • Professor Gregory D Walcott of Hamline University, St Paul, Minnesota, has for some time been planning a series of source books in the history of the sciences, covering the period 1500-1900.
    • To carry out this project the American Philosophical Society has recently secured from the Carnegie Foundation a grant sufficiently large to permit of a beginning in the publication of the series.
    • Science, New Series 81 (2107) (1935), 487.

  55. Edinburgh Mathematics Examinations
    • When is a series said to be convergent? .
    • Prove geometrically that the locus of the middle points of a series of parallel chords of a conic section is a straight line.
    • Deduce from the Exponential Series (or find independently) the series for loge(1 + z) (log to base e) in terms of x.
    • Shew that the locus of the middle points of a series of parallel chords of a conic section is a straight line.

  56. Three Sadleirian Professors
    • The later portions of this book were for many years the only place (with the exception of Chrystal's Algebra) where could be found an accurate account in English of complex numbers and of infinite series.
    • In 1907 the fame of his Trigonometry was eclipsed by that of his Treatise on the Functions of a Real Variable and the Theory of Fourier's Series.
    • Professor Hobson's smaller books are Squaring the Circle (1913) and The Domain of Natural Science (1923; a series of Gifford lectures delivered at Aberdeen).
    • Most of these deal with convergence of series or the analytic theory of numbers.
    • One has already been mentioned; the others are The Integration of Functions of a Single Variable (1905) and The General Theory of Dirichlet's Series (1915, in collaboration with M Riesz).

  57. Morton's reports
    • He is carrying out research there on Power Series.
    • B V Williams and J H C Whitehead, A Theorem on Linear Connections, Annals of Mathematics, Second Series 31 (1) (1930), 151-157.
    • V C Morton and M T Chapple, A point representation of a system of space cubic curves which pass through four given points and whose chords belong to a given tetrahedral complex, The Quarterly Journal of Mathematics, Oxford Series 19 (1948), 133-139.
    • D S Meyler, A point representation of a system of rational normal curves of order n through n+1 fixed points, The Quarterly Journal of Mathematics, Oxford Series (2) 1 (1950), 33-40.
    • D S Meyler, A point representation of a system of rational normal curves of order n through n+1 fixed points, The Quarterly Journal of Mathematics, Oxford Series (2) 1 (1950), 33-40.

  58. Harnack calculus book
    • To describe thoroughly the phenomena of motion is to assign every circumstance in numbers of concrete units: so that if the series of numbers is also to enable us to describe motion, it must contain a continuous series of quantities.
    • Thus the first problem of Analysis is: to develop the conception and the properties of the continuous series of numbers.
    • The natural series of numbers, which arises by adding on a thing to others in counting, advances always by unity; each number is defined by the preceding number and by unity.
    • This series of integers starting from unity can be continued on indefinitely.

  59. Zwicky books
    • Science, New Series 127 (3294) (1958), 343-344.
    • Management Science 14 (8), Application Series (1968), B539.
    • Science, New Series 163 (3873) (1969), 1317-1318.
    • Management Science 16 (4), Application Series (1969), B295-B296.
    • One of the difficulties is that the major expository device used by the author is the statement of a series of "examples" of morphological analysis.

  60. Von Neumann Silliman lectures
    • Each lecture course would be published in a series as a memorial to Mrs Silliman.
    • Although illness prevented him delivering the lectures, his unfinished manuscript was published as the 36th volume in the published series in May 1958 following von Neumann's death.
    • To give the Silliman Lectures, one of the oldest and most outstanding academic lecture series in the United States, is considered a privilege and an honour among scholars all over the world.
    • Traditionally the lecturer is asked to give a series of talks, over a period of about two weeks, and then to shape the manuscript of the lectures into a book to be published under the auspices of Yale University, the home and headquarters of the Silliman Lectures.
    • I should like to be permitted to express my deep gratitude to the Silliman Lecture Committee, to Yale University, and to the Yale University Press, all of which have been so helpful and kind during the last, sad years of Johnny's life and now honour his memory by admitting his unfinished and fragmentary manuscript to the series of the Silliman Lectures Publications.

  61. Basset papers
    • In the first series of experiments polarized light was reflected from the polished pole of an electromagnet, and it was found that when the circuit was closed, so that the reflecting surface became magnetized perpendicularly to itself, the reflected light exhibited certain peculiarities, which disappeared when the circuit was broken.
    • In the second series of experiments the reflector was a polished plate of soft iron laid upon the poles of a horse-shoe electromagnet, so that the direction of magnetization was parallel, or approximately so, to the reflecting surface; and it was found that the effect of the current was analogous to, though by no means identical with, the effect produced in the first series of experiments.
    • In both series of experiments it was found that the effects produced by magnetization materially varied with the angle of incidence.
    • The first error is that the three stresses R, S, T are accurately zero throughout the substance of the plate or shell; the second error is that it is not permissible to expand the various quantities involved, in a series of ascending powers of the distance of a point from the middle surface.

  62. Netto books
    • Science, New Series 16 (403) (1902), 469-470.
    • Under the leadership of C F Hindenburg it represents the culmination of an unfortunate tendency of eighteenth century mathematicians to develop analysis, particularly the subject of infinite series, with reference to form only, and to pay little or no attention to the actual contents of formulae.
    • One of the latest additions to the excellent series of monographs published in the Sammlung von Lehrbucher auf dem Gebiete der mathematischen Wissenschaften is from the pen of Professor Netto of the University of Giessen, whose name is familiar in connection with the substitution theory.
    • The scope of the treatise includes the theory of composition-series, chief-series, Sylow subgroups, and Abelian groups, but not the theory of commutants or groups of isomorphisms.

  63. G H Hardy: 'Integration of functions
    • The Cambridge Tracts in Mathematics and Mathematical Physics was a series of pamphlets published by Cambridge University Press.
    • When the series first began to be published the General Editors were J G Leathem and E T Whittaker.
    • The integration of functions of a single variable by G H Hardy was No 2 in the series and published in 1905.
    • I have borrowed largely from the Cours d'Analyse of Hermite and Goursat, but my greatest debt is to Liouville, who published in the years 1830-40 a series of remarkable memoirs on the general problem of integration which appear to have fallen into an oblivion which they certainly do not deserve.
    • It was Liouville who first gave rigid proofs of whole series of theorems of the most fundamental importance in analysis - that the exponential function is not algebraical, that the logarithmic function cannot be expressed by means of algebraical and exponential functions, and that the standard elliptic, integrals cannot be expressed by algebraical, exponential and logarithmic functions.

  64. Bradley letter
    • This points out to us the great Advantage of cultivating this as well as every other Branch of Natural Knowledge, by a regular Series of Observations and Experiments.
    • From that time to the present, I have continued to make Observations at Wansted, as Opportunity offered, with a View of discovering the Laws and Cause of this Phenomenon: For, by the Favour of my very kind and worthy Friend Matthew Wymondefold Esq, my Instrument has remained, where it was first erected; so that I have been able, without any Interruption, which the Removal of it to another Place would have occasioned, to proceed on with my intended Series of Observations, for the Space of twenty Years: a Term somewhat exceeding the whole Period of the Changes, that happen in this Phenomenon.
    • For these Reasons, we generally find, that the more exact the Instruments are, that we make use of, and the more regular the Series of Observations is, that we take; the sooner we are enabled to discover the Cause of any new Phenomenon.
    • And indeed it was on account of the Objections, which might have been raised against such a Postulate, that I thought it necessary, to continue my Series of Observations for so many Years, before I published the Conclusions, which I shall at present endeavour to draw from them.
    • I judged it proper to continue my Observations of the same Stars; hoping that, by a regular and longer Series of them, carried on through several succeeding Years, I might, at length, be enabled to discover the real Cause of such apparent Inconsistencies.

  65. Herivel's books
    • Science, New Series 152 (3724) (1966), 915.
    • Science, New Series 189 (4199) (1975), 279.
    • In addition, as one of a series of recent biographies of French exact scientists born in the eighteenth century, it enriches our knowledge of a remarkably fruitful period for French mathematics and physics.
    • In his introduction Herivel explicitly justifies his decision to forego attempting "that fully integrated biographico-scientific study of which historians of science sometimes dream" and to "make a clean division into two parts, Part I on Fourier the Man, and Part II on Fourier the Physicist." One of his reasons is that Fourier began his major work on heat only around 1804, after an exciting if not terribly significant series of careers as mathematics teacher, minor revolutionary, political prisoner, student of Egyptian culture, and civil administrator under both Napoleon and Louis XVIII.
    • When the name Joseph Fourier is mentioned, one is more likely to think of series, of transforms or of equations describing the propagation of heat in solids, than of a young man caught up in the complexities of a revolution, of a companion to Napoleon on an ill-fated excursion to Egypt, or of an excellent administrator serving the Emperor in a rural setting away from Paris.

  66. St Andrews Mathematics Examinations
    • Show how to find the sum of n terms of a geometrical series.
    • Explain what is meant by the sum of an infinite number of terms of such a series.
    • Sum to n terms the series- .
    • Find the sum of one of the following series (whose first three terms are given) to infinity, and of the other to n terms:- .
    • Show that the following series is convergent, and find its sum: .

  67. Ball papers
    • The following numbers, forming what I will call the series a, are expressible by one "4": 1, 2, 3, 4, 6, 9, 24, 265, 720, ..
    • From this series it follows that if m and n are two numbers such that n - m is less than 10, then every number between m and n is expressible by m or n and one "4." The numbers 1 to 13, 15 to 18, 20 to 28, 30, 33, 36, forming the series b, are expressible by two "4"s.
    • In an unpublished memorandum made some years later (cancelled, but believed to be correct in the part here quoted), he thus described his work of this time: "In the beginning of the year 1665 I found the method of approximating Series and the Rule for reducing any dignity of any Binomial into such a series.

  68. O'Brien tracts
    • In calculating the attractions of the Earth on any particle, he has arrived at the correct results, without considering diverging series as inadmissible; and this he conceives to be important, because there is evidently no good reason why a diverging series should not be as good a symbolical representative of a quantity as a converging series; or why there should be any occasion to enquire whether a series is diverging or converging, as long as we do not want to calculate its arithmetical value or determine its sign.
    • Instances, it is true, have been brought forward by Poisson in which the use of diverging series appears to lead to error; but if the reasoning employed in Chapter III of these Tracts be not incorrect, this error is due to quite a different cause; as will be immediately perceived on referring to Articles 33, 34, 35, and 37.

  69. Pedoe's books
    • There are 9 chapters: mathematical games, chance and choice, where does it end (i.e., transfinite numbers), automatic thinking (logic, algebra of classes, etc.), two-way stretch, rules of play (elementary algebra, groups, etc.), an accountant's nightmare (infinite series), double talk (antinomies), what is mathematics.
    • Science, New Series 131 (3396) (1960), 295-296.
    • The topics treated are mathematical games, chance and choice, infinity, sets and logic, topology, groups-rings-fields, series, and more logic.
    • Science, New Series 143 (3612) (1964), 1320.

  70. George William Hill's new theory of Jupiter and Saturn
    • The employment of the eccentric anomaly of the planet whose co-ordinates are sought as the independent variable undoubtedly augments the convergence of the series; but the adoption of this mode of proceeding would bring about the use of two independent variables, one of the co-ordinates of Jupiter, another for those of Saturn.
    • As the developments have to be pushed to terms of three dimensions with respect to disturbing forces the heaviest part of the labour consists in forming products of periodic series, one of which belongs to Jupiter, the other to Saturn; and as integration can not be performed unless these products are transformed so as to involve but one variable we should have an endless series of transformations to make.
    • In consequence, the final form adopted for all the periodic series is in terms of the mean anomalies, so that the time is always the independent variable.

  71. Gregory-Collins correspondence
    • The include integrating √(1 + 1/x2), the transcendence of the exponential function, various series expansions for sin x and a general form of the binomial theorem.
    • Notes which Gregory wrote on the blank spaces of Collins' 29 January 1671 letter shows that he was by this time quite expert in computing the Taylor series of functions.
    • This letter contains six examples of functions expanded using Taylor series many years before Brook Taylor was born.
    • Gregory assumes that Newton must be familiar with Taylor series.

  72. Loney reviews
    • In Part II the author first deduces the common exponential and logarithmic series, and then treats of complex quantities under their trigonometric form.
    • The author treats with marked clearness the exponential series for complex quantities, the trigonometric functions of complex angles, and the hyperbolic functions defined analytically.
    • Among the other subjects treated the most important are the value of π, summation of series, expansion in series, factoring of mathematical expressions, proportional parts, errors of observation, solution of cubic equations, and geometric representation of complex quantities.

  73. Obada publications
    • In the problem of propagation and scattering of light in crystals, the third-order correlation function already appears in the lowest order terms of the multiple scattering series.
    • A model is presented in the 3-level atom and 2-mode system, in which infinite series tend to closed forms, for thermal and coherent distributions for the modes.
    • F K Faramawy, M M Abu Sitta and A-S F Obada, Propagation in non-linear media: Refractive index in non-linear media, LAMP Series Report/91/7 International Centre for Theoretical Physics, Trieste (Italy) (August 1991), 22 pages.
    • A-S F Obada and Z M Omar, Dynamics of two three-level atoms interacting with two modes of radiation, LAMP Series Report/95/9, International Centre for Theoretical Physics, Trieste (Italy) (December 1995), 14 pages.

  74. Johnson pre1900 books
    • In Chapter IX, I have attempted to classify the principles employed in finding the equations of Geometrical Loci, and to explain and illustrate them fully by examples solved in the text, and a carefully graduated series of examples for practice.
    • The difficulties usually encountered on beginning the study of the Differential Calculus, when the fundamental idea employed is that of infinitesimals or that of limits, together with the objectionable use of infinite series involved in Lagrange's method of derived functions, have induced several writers on this subject to return to the employment of Newton's conception of rates or fluxions.
    • The expression "binomial equations" is applied in this work (in a sense introduced by Boole) to those linear equations which are included in the general form f1(ϑ )y + xs f2(ϑ )y = 0, and which constitute the class of equations best adapted to solution by development in series.
    • Chapter VIII is devoted to the general solution of the binomial equation in the notation of the hypergeometric series, and Chapter IX to Riccati's, Bessel's and Legendre's equations.

  75. Eulogy to Euler by Fuss
    • It is there that we find remarkably the full measure of the theory of curves: tautochrones, brachistochrone, trajectories and the very deep research in integral calculus, on the nature of numbers, concerning series, the motion of heavenly bodies, the attraction of spheroid-elliptical bodies and on an infinity of subjects of which one hundredth part would suffice in making the reputation of anyone else.
    • He gathered together everything that he found to be useful and interesting concerning the properties of infinite series and their summations; He opened a new road in which to treat exponential quantities and he deduced the way in which to furnish a more concise and fulsome way for logarithms and their usage.
    • He showed a new algorithm which he found for circular quantities, for which its introduction provided for an entire revolution in the science of calculations, and after having found the utility in the calculus of sine, for which he is truly the author, and the recurrent series, he provides for in the second part the general theory of curves with their divisions and sub-divisions and in a supplement the theory of solids and their surfaces while showing how their measurement leads to the equations with three variables and he ends finally this important work by developing the idea of curves with double curvature which provides for the consideration of the intersection of curved lined surfaces.
    • Euler had already envisaged the true principles in the systematic order in which he has exposed them and with the methodology which exists and the clarity with which he has shown the utility of the calculus in relation to the doctrine of series and to the theory of the maxima and the minima.

  76. R A Fisher: 'Statistical Methods' Introduction
    • In 1925 R A Fisher published Statistical Methods for Research Workers in the Biological Monographs and Manuals Series by the publisher Oliver and Boyd of Edinburgh in Scotland.
    • With an infinite population the frequency distribution specifies the fractions of the population assigned to the several classes; we may have (i) a finite number of fractions adding up to unity as in the Mendelian frequency distributions, or (ii) an infinite series of finite fractions adding up to unity, or (iii) a mathematical function expressing the fraction of the total in each of the infinitesimal elements in which the range of the variate may be divided.
    • Three of the distributions with which we shall be concerned, Bernoulli's binomial distribution, Laplace's normal distribution, and Poisson's series, were developed by writers on probability.
    • Examples of sufficient statistics are the arithmetic mean of samples from the normal distribution, or from the Poisson series; it is the fact of providing sufficient statistics for these two important types of distribution which gives to the arithmetic mean its theoretical importance.

  77. Berge books
    • 0ystein Ore, Graphs and their uses, forthcoming in the School Mathematics Study Group (SMSG) series.
    • Series A (General) 126 (2) (1963), 322-323.
    • Series A (General) 130 (1) (1967), 121-122.
    • For, although the two parts, respectively by Alain Ghouila-Houri and by Claude Berge, both deal with series of problems which cover much of the mathematical theory of the non-statistical part of operational research and both sets of problems are solved by the construction of algorithms of linear programming type (with proof of convergence), yet their mathematical nature is very different.

  78. George Gibson: 'Calculus
    • With respect to mathematical attainments, the reader is supposed to be familiar with Geometry, as represented by the parts of Euclid's Elements that are usually read., with Algebra up to the Binomial Theorem for positive integral indices, and with Plane Trigonometry as far as the Addition Theorem; but no use is made of Complex (imaginary) number, nor is a knowledge of Infinite Series presupposed.
    • As in some of the more recent text-books, the discussion of Taylor's Theorem has been postponed; the Mean Value Theorem is sufficient in the earlier stages, and the somewhat abstract theorems on Convergence and Continuity of Series are most profitably treated towards the end of the course.
    • The chapter on the Fourier Series will, I hope, be sufficient as an introduction to the subject; but the student can not be too earnestly recommended to read and to master the fascinating pages in which Fourier himself develops the process of representing an arbitrary function by means of a harmonic series.

  79. Harvey obituaries
    • How can that be assured of happening? The British mathematician, Maclaurin, who discovered mathematical series that are now taught in A-level mathematics, was a professor at the University of Aberdeen in the 1720s.
    • The university also teaches mathematical series more intensively than is normally done in most universities - just to raise and keep the tradition of series in the memory of Maclaurin.
    • Despite our very different childhoods, I recall a long, enjoyable evening with him in a shebeen [an unlicensed establishment or private house selling alcohol] in rural Zimbabwe talking about the various American and Japanese TV series we had both seen growing up (which I mentionednhere).

  80. L E Dickson: 'Linear algebras
    • The Cambridge Tracts in Mathematics and Mathematical Physics was a series of pamphlets published by Cambridge University Press.
    • When the series first began to be published the General Editors were J G Leathem and E T Whittaker.
    • However, G H Hardy took over from E T Whittaker and when Linear algebras by L E Dickson, which was No 16 in the series, was published in 1914 the General Editors were G H Hardy and J G Leathem.
    • My thanks are due to the editors for the opportunity to participate in this useful series of tracts.

  81. Walk Around Paris
    • Simeon Denis Poisson, whose main work was on integrals and Fourier Series.
    • Augustin Louis Cauchy, who worked mainly on analysis, introducing the criteria for convergence of series and holomorphic functions.
    • Joseph Fourier, who is most well known for having calculated the propagation of heat, by decomposing a function into a converging trigonometric series, which is known as a Fourier function, this method being called the Fourier Transform, which is the basis of signaling theory, digital imaging, data compression and systems like 3G and 4G .
    • His mathematical study includes projective geometry, calculus, and series of whole numbers, as well as working on Pascal's triangle, for binomial coefficients, in his book Traite du triangle arithmetique.

  82. Heath: Everyman's Library 'Euclid' Introduction
    • In 1905 the London publisher Joseph Malaby Dent had the idea of producing the Everyman's Library, a cheap series of reprints of classical texts.
    • Dent wanted the series:- .
    • (which had to be copied exactly) had to be similar to those of the other volumes in the series.
    • Todhunter, Senior Wrangler in 1848, was the author of a series of mathematical textbooks quite unrivalled in their day; and his notes to Euclid, admirably concise and to the point, fully deserve re-impression.

  83. Zariski and Samuel: 'Commutative Algebra
    • The algebro-geometric origin and motivation of the book will become more evident in the second volume (which will deal with valuation theory, polynomial and power series rings, and local algebra; more will be said of that volume in its preface) than they are in this first volume.
    • (Other variations of that theorem will be found in Volume II, in the chapter on polynomial and power series rings.) With Matusita we then define a Dedekind domain as an integral domain in which every ideal is a product of prime ideals and derive from that definition the usual characterization of Dedekind domains and their properties.
    • These topics are: (a) valuation theory; (b) theory of polynomial and power series rings (including generalizations to graded rings and modules); (c) local algebra.
    • The greater part of Chapter VII is devoted to classical properties of polynomial and power series rings (e.g., dimension theory) and their applications to algebraic geometry.

  84. Smith Major History books
    • Science, New Series 32 (812) (1910), 114-115.
    • The second volume of a series of books by David Eugene Smith is an outstanding contribution to the history of mathematics.
    • It is to be hoped that the success of the series will permit of a volume devoted to this important phase of the development of the science.
    • Series A (General) 123 (3) (1960), 340-341.

  85. Gender and Mathematics
    • The series of studies dealing with educational variables, reported and summarized in the book edited by Gilah Leder and me [',' E Fennema and G Leder (eds.), Mathematics and gender: Influences on teachers and students (Teachers College Press, New York, 1990).
    • In connection with this series of studies, Peterson and I proposed the Autonomous Learning Behaviours model, which suggested that because of societal influences (of which teachers and classrooms were main components) and personal belief systems (lowered confidence, attributional style, belief in usefulness), females do not participate in learning activities that enable them to become independent learners of mathematics [',' P L Peterson and E Fennema, Effective teaching, student engagement in classroom activities, and sex-related differences in learning mathematics, American Educational Research Journal 22 (3) (1985), 309-335.
    • For these studies, we did a series of meta-analyses of extant work on gender differences reported in the US, Australia, and Canada ([',' J S Hyde, E Fennema and S J Lamon, Gender differences in mathematics performance, Psychological Bulletin (1990), 139-155.','29], [',' J S Hyde, E Fennema, M Ryan and L A Frost, Gender differences in mathematics attitude and affect: A meta-analysis, Psychology of Women Quarterly 14 (1990), 299-324.','30]).
    • Jim Schuerich [',' J Schuerich, Methodological implications of feminist and poststructuralist views of science, The National Center for Science Teaching and Learning Monograph Series, 4 (1992).

  86. Segel books
    • Chapter 3: Random Processes and Partial Differential Equations; Random walk in one dimension; Langevin's equation; Asymptotic series, Laplace's method, gamma function, Stirling's formula; A difference equation and its limit; Further considerations pertinent to the relationship between probability and partial differential equations; .
    • Chapter 4: Superposition, Heat Flow, and Fourier Analysis; Conduction of heat; Fourier's theorem; On the nature of Fourier series; .
    • Chapter 7: Regular Perturbation Theory; The series method applied to the simple pendulum; Projectile problem solved by perturbation theory; .
    • This book, which is a volume in the Santa Fe Institute Studies in the Sciences of Complexity series, had its origins in a workshop of the same title held at the Santa Fe Institute in July 1999.

  87. Boyer's books
    • Mind, New Series 49 (194) (1940), 248-253.
    • Science, New Series 125 (3252) (1957), 823-824.
    • The greater part of the History of Analytic Geometry was first published as a series of articles in Scripta Mathematica, volumes 16 through 21 (1950-55).
    • Science, New Series 163 (3863) (1969), 171.

  88. Peres books
    • A final chapter contains suggestive remarks upon the connection between permutable functions and the summation of divergent series.
    • Science, New Series 88 (2286) (1938), 380-381.
    • The volume before us is the first in a series of three, the three together to give a resume of something over fifty years of the work of Volterra.
    • This volume and the other two in prospect, as well as the recent treatise, "Operations inflnitesimales lineaires," by Volterra and Hostinsky, form an amplification and modernization of the two volumes on functions of curves and integral equations published in the same Borel series on the theory of functions, some twenty-five years ago.

  89. H Weyl: 'Theory of groups and quantum mechanics' Introduction
    • The deduction of the Balmer series for the line spectrum of hydrogen and of the Rydberg number from universal atomic constants constituted its first convincing confirmation.
    • An equivalent original English book is that of Ruark and Urey, Atoms, Molecules and Quanta (New York, 1930), which appears in the International Series in Physics, edited by Richtmeyer.
    • The spectroscopic data, presented in accordance with the new quantum theory, together with complete references to the literature, are given in the following three volumes of the series Struktur der Materie, edited by Born and Franck:- .
    • The spectroscopic aspects of the subject are also discussed in Pauling and Goudsmit's recent The Structure of Line Spectra (1930), which also appears in the International Series in Physics.

  90. Eddington: 'Mathematical Theory of Relativity' Introduction
    • The parallax of a star is found by a well-known series of operations and calculations; the distance across the room is found by operations with a tape-measure.
    • The same series of operations will naturally manufacture the same result when world-conditions are the same, and different results when they are different.
    • A physical quantity is defined by the series of operations and calculations of which it is the result.
    • I should be puzzled to say off-hand what is the series of operations and calculations involved in measuring a length of 10-15 cm; nevertheless I shall refer to such a length when necessary as though it were a quantity of which the definition is obvious.

  91. Speiser books
    • From the title of the series it is clear that the aim is to deal with fundamental theories rather than to present details.
    • On the whole, the volume under review furnishes a very attractive introduction into some of the most modern developments of the theory of groups of finite order, with emphasis on its applications, and we can only wish it success along with the other volumes of the interesting series to which it belongs now being published under the general editorship of R Courant of the University of Gottingen.
    • The first edition of this famous book, in the "yellow" series of mathematical monographs of the Julius Springer Verlag, is 35 years old.
    • Speiser explains the cultural significance of mathematics in a series of 24 pieces from the literature.

  92. Analysis of Variance
    • For each treatment in each locality there is a mixing tank from which the fluid is pumped to all the tanks on this treatment, connected "in parallel:" We do not want a "series" connection, where the outflow from one tank is the inflow to another, because this would confound the effects of the varieties in these two tanks with the effects (if any) of order in the "series" connection.
    • Journal of the Royal Statistical Society, Series A (General) 123 (4) (1960), 482-483.
    • Economica, New Series 28 (112) (1961), 453-454.

  93. Murphy books
    • This subject is discussed in a series of propositions, the more clearly to impress the reader with the steps of the reasoning.
    • I have then given the theorems of both Sturm and Fourier relative to the discovery of the number of real and imaginary roots of an equation, the combination of which with the methods of approximation due to Newton and Lagrange conducts to the solution of all numerical equations of finite dimensions, except for imaginary roots, for the discovery of which I have employed a method deduced from recurring series.
    • The formation of literal equations being understood, I have explained the logarithmic method for obtaining with rapidity the series which analytically represent the different roots and their functions; and have then shown how to effect some general and useful transformations of equations, and explained the algebraical solutions of the equations of inferior degrees, and the analytical meaning of the different surd parts which constitute the roots.
    • After giving several useful analytical results springing from the employment of the methods before given, I have passed on to discuss recurring series, which have been used from an early date for the solution of equations.

  94. Datta's publications
    • Beginning in 1980 Kripa Shankar Shukla revised the material of the third volume and published it as a series of papers between 1980 and 1993.
    • On the stability of two rectilinear vortices of compressible fluid moving in an incompressible liquid, Philosophical Magazine (Series 6) 40 (1920), 138-148.
    • (with Avadhesh Narayan Singh) Use of series in India, Indian Journal of the History of Science 28 (2) (1993), 103-129.

  95. E C Titchmarsh: 'Aftermath
    • The book covers Counting (see this link), Arithmetic, Algebra, The use of numbers in geometry, Irrational numbers, Indices and logarithms, Infinite series and e, The square root of minus one, Trigonometry, Functions, The differential calculus, The integral calculus and Aftermath.
    • Algebra and geometry are to form the subjects of further volumes of this series, so that very little has been said about them here.
    • An analyst should be able to handle such things as integrals and infinite series just as well as if they were the simple expressions of elementary algebra.

  96. Turnbull lectures on Colin Maclaurin, Part 2
    • as a special case of the series for f(x + h) given by Brook Taylor in 1715.
    • An earlier passage contains the well-known integral test for the convergence or divergence of a monotonic series, where and are compared; a method which Cauchy rediscovered many years later.
    • This Maclaurin turned to practical account by developing a technique for computing the series from the integral, or vice versa, to a high order of accuracy, and gave him what is nowadays known as the Euler-Maclaurin summation formula, both mathematicians having discovered the method, as it would seem, independently and nearly contemporaneously.

  97. Otto Neugebauer - a biographical sketch
    • As part of his manifold activities Prof Neugebauer edits two important periodicals "Zentralblatt fur Mathematik und ihre Grenzgebiete" and the "Zentralblatt fur Mechanik"; in addition, he edits the two valuable series of monographs, the "Quellen und Studien zur Geschichte der Mathematik, Astronomie und Physik" and the "Ergebnisse der Mathematik und ihrer Grenzgebiete".
    • In 1932 appeared no less than six distinct contributions from his pen dealing with the history of ancient algebra, the sexagesimal system and Babylonian fractions, Apollonius, Babylonian series, square root approximations, and siege calculations.
    • Professor Neugebauer has announced a series of three volumes on the history of ancient astronomy and mathematics.

  98. Hodge's books
    • Science, New Series 95 (2457) (1942), 124-125.
    • Science, New Series 107 (2785) (1948), 511-512.
    • This second volume of the series enables one to see more clearly the plan of the whole work, although a final appreciation must wait until the appearance of the third and final volume.

  99. De Rham mountaineering
    • This was long and steep, a series of five stepped towers, with excellent rock almost too exciting for me.
    • Finally, we give a series of quotes from: Georges de Rham (1903-1990), Enseign.
    • This is the beginning of a long series of first ascents in the Alps.

  100. William Lowell Putnam Mathematical Competition
    • Again the winners were enthusiastic to continue the series and again the losers (this time Harvard) declined the challenge.
    • Prove that every positive rational number is the sum of a finite number of distinct terms of the series .
    • The second problem has particular historical interest since the ancient Egyptians knew how to do it! In fact every rational has infinitely many different such decompositions or, thought of another way, one can omit any finite number of terms of the series and a decomposition (even infinitely many) is still possible.

  101. M Bôcher: 'Integral equations
    • The Cambridge Tracts in Mathematics and Mathematical Physics was a series of pamphlets published by Cambridge University Press.
    • When the series first began to be published the General Editors were J G Leathem and E T Whittaker.
    • An introduction to the study of integral equations by Maxime Bocher was No 10 in the series and published in 1909.

  102. Feller Reviews 4
    • Journal of the Royal Statistical Society Series D (The Statistician) 17 (2) (1967), 197-198.
    • Journal of the Royal Statistical Society Series A (General) 130 (1) (1967), 109.
    • Journal of the Royal Statistical Society Series C (Applied Statistics) 16 (2) (1967), 177-179.

  103. Mordell reminiscences
    • It was really a good book, though not rigorous, and contained a great deal of material including the theory of equations, series, and a chapter on the theory of numbers.
    • He was the author of a book on college algebra and also one on operations with series.
    • He had just written his book on infinite series of which he was very proud.

  104. Gibson History 7 - Robert Simson
    • from this he deduced several series for π, among them the series known as Machin's Series.

  105. Mikou pioneer
    • It is in Rabat, at Lalla-Nezha high school, that she will get her baccalaureate (series D) in 1965.
    • After a series of dead ends, Noufissa Mikou, who is particularly interested in the applications of mathematics to computer and telecommunications networks, begins a thesis d'Etat which she will complete in 1981.
    • We were faced with a series of direct or indirect blockages from the administration and even from colleagues," she says.

  106. Kingman autobiography
    • After proving that I was not good at making the tea, I was given a handbook of common Fourier series which was known to be riddled with errors, and which I was told to revise.
    • This had the useful effect of giving me one Tripos question which I could answer at sight, since there was always a function whose Fourier series we had to compute, and I had all the easy examples by heart.
    • Finally, I strongly recommend Peter Whittle's lively history of the Cambridge Statistical Laboratory, which can be found on the Laboratory's website together with a fascinating series of group photographs of the members through the years.

  107. Truesdell's books
    • Leonhardi Euleri Opera Omnia, Series secunda (Opera mechanica et astronoca) (1960).
    • Though called an "introduction" to some of the corresponding Euler volumes in this series, this is really the first modern scholarly treatise on the theory of deformable solids: Todhunter and Pearson's and other older works dealing with this topic will henceforth have a place only in the historiography of the subject.
    • This book is based on a series of lectures given by the author at Syracuse University in 1965.

  108. Raphson books
    • Review of: Analysis Aequationem Universalis, &c An Universal Analysis of Equations; Or, A Short and General Method of resolving Algebraical Equations; deduced and demonstrated out of the New Doctrine of Infinite Series: By Joseph Raphson, Fellow of the Royal Society.
    • An Universal Analysis of Equations; Or, A General and Expeditious Method for resolving Algebraical Equations, Deduced and Demonstrated from the New Method of Infinite Series.
    • Edition the Second: To which is annexed an Appendix, concerning the Infinite Progress of Infinite Series for extracting the Roots of Algebraical Equations; As also a Mathematical-Metaphysical Essay concerning Real Space, or Infinite Being.

  109. L R Ford - Differential Equations
    • The first three chapters lead up to the later chapters by their discussions of direction fields, of solutions in series, of the Wronskian and linear dependence.
    • It is unusual to find Clairaut's equation and simple examples of solution in series in the first chapter of a text-book on differential equations, but the idea is a good one.
    • Subsequent chapters cover special methods for equations of first order, linear equations of any order with a brief account of the use of the Laplace transform, solution in series of the hypergeometric, Legendre's and Bessel's equations, approximate numerical solutions, and two chapters on partial differential equations.

  110. Mellin publications
    • Most of his publications are in mathematics, dealing mainly with Mellin transform, hypergeometric series and asymptotic expansions.
    • Hjalmar Mellin, On hypergeometric series of higher orders (Finnish), Acta Soc.
    • Hjalmar Mellin, A theorem on Dirichlet series (Finnish), Acta Soc.

  111. Teixeira books
    • The major part of these handsomely printed quartos [Volumes I and IV] is concerned with the developments of functions of various kinds in series.
    • For instance, the first hundred pages of Volume I comprise a series of papers on Taylor's Theorem in the case of functions of both real and complex variables, first in an elementary treatment and then by the methods of Cauchy, Riemann, Weierstrass and Mittag-Leffler.
    • The paper is completed by a discussion of the series of Burmann and Lagrange with a generalisation of the former.

  112. Smith's Obituaries and Biographies
    • Science, New Series 70 (1815) (1929), 347.
    • Science, New Series 72 (1864) (1930), 287-288.
    • Science, New Series 73 (1896) (1931), 468-469.

  113. Science at St Andrews
    • On returning to London in 1668 Gregory learnt from his friend Collins of a new method for expanding a logarithm by an infinite series that had just been published by Mercator.
    • How close Gregory and Newton were in mathematical thought may be judged from the fact that on one occasion independent statements of the same discovery - the infinite series for the inverse sine - crossed in the post.
    • He was physicist, astronomer, and engineer of eminent powers, and once carried out a long series of experiments, from 1876 to 1880, in Pitlochry and across the Firth of Clyde by means of arc lamps for directly measuring the velocity of light.

  114. Groups in Galway
    • At the inaugural meeting of the Irish Mathematical Society in Dublin in December 1977, I proposed that a series of instructional conferences be organised in various topics in Mathematics.
    • The Irish Mathematical Society intends to hold a series of short instructional conferences.
    • Thus the series of Groups in Galway was well launched.

  115. Ahlfors' Complex analysis
    • One of the most fundamental properties of analytic functions is that they can be represented through convergent power series.
    • Conversely, with trivial exceptions every convergent power series defines an analytic function.
    • Power series are very explicit analytic expressions and as such are extremely maniable.

  116. Mathematicians and Music 2.1
    • Reducing d an octave, a an octave, e two octaves, and b two octaves, we have the series .
    • To obtain the f missing in this series and to fill up the wide interval between e and g it appears that c as a fifth below the prime was raised an octave.
    • By beginning with different letters in the series thus determined, Euclid got the seven Pythagorean scales covering two octaves instead of one.

  117. Enciclopedia delle Matematiche
    • The twenty main headings of the present volume, with the number of pages devoted to each of these subjects and the authors of the articles, are as follows: Logic, (75), A Padoa; General arithmetic, (126), D Gigli; Practical arithmetic, (52), E Bortolotti and D Gigli; Theory of numbers and indeterminate analysis, (68), M Cipolla; Progressions, (17), A Finzi; Logarithms, (42), A Finzi; Mechanical calculus, (28), G Tacchella; Combinatory calculus, (9), L Berzolari; Elements of the theory of groups, (51), L Berzolari; Determinants, (30), L Berzolari; Linear equations, (13), L Berzolari; Linear substitutions and linear, bilinear and quadratic forms, (28), L Berzolari; Rational functions of one or more variables, (37), O Nicoletti; General properties of algebraic equations, (59), O Nicoletti; Equations of the second, third, and fourth degree, and other particular algebraic equations, systems of algebraic equations of elementary type, (57), E G Togliatti; Methods for the discussion of problems of the second degree and remarks on some of the third and fourth degree, (63), R Marcolongo, Limits, series, continued fractions, and infinite products, (45), G Vitali; Elements of infinitesimal analysis, (101), G Vivanti; Relations between the theory of aggregates and elementary mathematics, (11), G Vivanti; The analytic function from an elementary point of view, (29), S Pincherle.
    • It discusses only the traditional arithmetic, geometric, and harmonic series and some of the most elementary extensions of the first two to figurate numbers.
    • The thirteen main headings of this second part, with the number of pages devoted to each of these subjects and the authors of the articles, are as follows : combinatory calculus (9), L Berzolari , elements of the theory of groups (51), L Berzolari; determinants (30), L Berzolari; linear equations (13), L Berzolari; linear substitutions and linear, bilinear and quadratic forms (28), L Berzolari; rational functions of one or more variables (37), O Nicoletti; general properties of algebraic equations (59), O Niccletti; equations of the second, third and fourth degree and other particular algebraic equations, systems of algebraic equations of elementary type (57), E G Togliatti; methods for the discussion of problems of the second degree and remarks on some of the third and fourth degree (63), R Marcolongo, limits, series, continued fractions and infinite products (45), G Vitali: elements of infinitesimal analysis (101), G Vivanti, relations between the theory of aggregates and elementary mathematics (11), G Vivanti; the analytic functions from an elementary point of view (29), S Pincherle.

  118. Horace Lamb addresses the British Association in 1904
    • It was from him that many of us first learned that a great mathematical theory does not consist of a series of detached propositions carefully labelled and arranged like specimens on the shelves of a museum, but that it forms an organic whole, instinct with life, and with unlimited possibilities of future development.
    • So far as British universities are concerned, they have formed the starting point of a whole series of works conceived in a similar spirit, though naturally not always crowned by the same success.
    • It is now generally accepted that an analytical solution of a physical question, however elegant it may be made to appear by means of a judicious notation, is not complete so long as the results are given merely in terms of functions defined by infinite series or definite integrals, and cannot be exhibited in a numerical or graphical form.

  119. Ugbebor publications
    • Olabisi Oreofe Ugbebor, Probability Distribution and Elementary Limit Theorems (Ibadan External Studies Programme Series, University of Ibadan, Ibadan, 1991), 216 pages.
    • and N I Akinwande, Analytical Geometry and Mechanics (University Mathematics Series (1), 2000), 109 pages.
    • Olabisi Oreofe Ugbebor and U N Bassey, Mathematics Series (3) (Y-Books, Associated Book-Makers, Ibadan, Nigeria Ltd., 2003), 256 pages.

  120. Kline's books
    • Review by J Murray Barbour [Notes, Second Series 12 (1) (1954), 108-109.',4)">4]: .
    • Review by I Grattan-Guinness [Science, New Series 180 (4086) (1973), 627-628.',19)">19]: .
    • The book is too well constructed to permit the mere extraction of odd chapters, but not too much work would be needed to produce paperbacks on, say, 'Series', 'Differential equations' and 'The growth of abstract algebra' which would prove most valuable for use with undergraduate courses.

  121. George Salmon: from mathematics to theology
    • In the next year he produced his third and most controversial and important theological work, The Infallibility of the Church, which was a series of lectures in which he argues against the tradition of Papal Infallibly within the Roman Catholic Church.
    • Firstly, I will focus on his most famous theological work, his series of lectures on The Infallibility of the Church, to illustrate how his mathematical brilliance was a factor in his theological works, especially in his analytical approach and process orientated approach to find meaning and truth.
    • The argument continues to the conclusion that this creative process must have begun with a non-contingent being, and the inductive step is taken to assume that this is the 'being that which we call God.' Similarly, the ontological argument for the existence of God uses the principle of the necessity of an uncaused cause, The created world is like a chain of caused evolutionary events, and this process rather than being an infinite regression series had a starting point, an uncaused cause, again 'the being that which we call God.' .

  122. Mirsky books
    • I have also included a brief sketch of the theory of matrix power series, a topic of considerable interest and elegance not normally dealt with in elementary textbooks.
    • At the end of each chapter there is a series of miscellaneous problems arranged approximately in order of increasing difficulty.
    • However, a number of elementary inequalities involving complex matrices are given and there is a lengthy discussion of power series in a matrix.

  123. Aitken: 'Statistical Mathematics
    • The Oliver and Boyd series of mathematical texts were widely used by students throughout the 1940s to 1960s.
    • One of the books in the series was Statistical Mathematics by A C Aitken.
    • To take a classical example, in the sequence defining a certain simple geometric series, .

  124. Finkel's Solution Book
    • In cases where the formulae lead to series, as in the case of the circumference of the ellipse, the rule is given for a near approximation.
    • - Determinants; Elements of Mechanics; Quaternions with application to Geometry and Mechanics; Theory of Curves and Surfaces; Dynamics of a Rigid Body; Trigonometric Series.
    • - Hydrokinetics; Infinite Series and Products; The Theory of Functions; Algebra.

  125. MacRobert: 'Spherical Harmonics' Preface
    • Subsequently it was thought advantageous to include discussions on similar lines of Fourier Series and Bessel Functions, with corresponding applications.
    • The first chapter contains an elementary account of the theory of Fourier Series, while the second and third deal with the applications of Fourier Series to Conduction of Heat and Vibrations of Strings.

  126. Senechal reviews
    • Science, New Series 270 (5237) (1995), 839-840.
    • We also know Hardy from A Mathematician's Apology and classic books on special topics: Dirichlet series, divergent series, inequalities, the theory of numbers.

  127. Trakhtenbrot books
    • This is an English translation of the second edition of Trakhtenbrot's booklet and is one of a series of translations from the Russian series, "Popular lectures in mathematics," which are being prepared under a project at the University of Chicago.
    • This monograph is one in a series of scholarly studies on facets of work in the Soviet Union not well known or understood by the professional community or academia in the West.

  128. David Hilbert: 'Mathematical Problems
    • The same is true of the first problems of geometry, the problems bequeathed us by antiquity, such as the duplication of the cube, the squaring of the circle; also the oldest problems in the theory of the solution of numerical equations, in the theory of curves and the differential and integral calculus, in the calculus of variations, the theory of Fourier series and the theory of potential - to say nothing of the further abundance of problems properly belonging to mechanics, astronomy and physics.
    • Further, the proof that the power series permits the application of the four elementary arithmetical operations as well as the term by term differentiation and integration, and the recognition of the utility of the power series depending upon this proof contributed materially to the simplification of all analysis, particularly of the theory of elimination and the theory of differential equations, and also of the existence proofs demanded in those theories.

  129. Sneddon: 'Special functions
    • The Oliver and Boyd series of mathematical texts were widely used by students throughout the 1940s to 1960s.
    • One of the books in the series was Special functions of Mathematical Physics and Chemistry by Ian N Sneddon.
    • I should also like to thank Dr D E Rutherford, general editor of the series, for his advice and criticism throughout the preparation of the book.

  130. W H Young: 'Differential Calculus
    • The Cambridge Tracts in Mathematics and Mathematical Physics was a series of pamphlets published by Cambridge University Press.
    • When the series first began to be published the General Editors were J G Leathem and E T Whittaker.
    • The fundamental theorems of the differential calculus by W H Young was No 11 in the series and published in 1910.

  131. Geary's books
    • Series A (General) 128 (1) (1965), 148.
    • Economica, New Series 33 (132) (1966), 496.
    • Series A (General) 139 (1) (1976), 137-138.

  132. Feller Reviews 2
    • Journal of the Royal Statistical Society Series A (General) 121 (3) (1958), 354-355.
    • Journal of the Royal Statistical Society Series C (Applied Statistics) 7 (3) (1958), 203-204.
    • Science, New Series 129 (3354) (1959), 956.

  133. Collins and Gregory discuss Tschirnhaus
    • Upon the parting visit I received from him, in answer to the doubt I mentioned about that series, he said it was only fitted to the condition there proposed.
    • I am much obliged to you for yours of 20th August, those methods of Mr Tschirnhaus indeed are but very particular, though he asserts that he has general ones, and I told him at parting I should be more fully convinced he had if he but show me his series for finding the three roots of a cubic equation capable of so many that would express the same in cubic surds different from those of Cardan, which indeed (to my understanding) seems to imply that the said roots cannot be always (at least in that method, or any other I know of) be expressed by any manner of surds, to which he replied that his papers were packed up; otherwise he would impart the same, and there being present with him a Dane named George Mohr who lately published in low Dutch, two little books the one named Euclides Danicus where he pretends to perform all Euclid's problems with a pair of compasses only without ruler, and another entitled Euclides Curiosus, wherein with a ruler and a fork (or compasses at a fixed opening) he performs the same, he [Tschirnhaus] said he would speak to the said Mohr to impart the said series, which he promised to do but as yet has not performed.

  134. Landau and Lifshitz Prefaces
    • The present book is one of the series on 'Theoretical Physics', in which we endeavour to give an up-to-date account of various departments of that science.
    • the complete series will contain the following nine volumes: .
    • The present volume of the 'Theoretical physics' series is devoted to an exposition of statistical physics and thermodynamics.

  135. Who was who 1852
    • Dirichlet worked in number theory, in particular analytic number theory (Dirichlet's series commemorates this fact), functions of a real variable, Fourier series, potential theory etc.
    • The former was his paper on trigonometric series, the latter dealt with the basic hypotheses of geometry.

  136. Segel Asymptotic analysis
    • Another important aspect of the subject is regular perturbation theory, which gives rise to convergent series solutions to problems.
    • A convergent power series about a point is asymptotic as the independent variable approaches the point, but we wish to show the importance of "genuine" or non-convergent asymptotic series.

  137. Fraenkel books
    • This is the first of a series of monographs on modern mathematics, based on Fraenkel's talks in the Israel adult education programme; further volumes will deal with modem algebra and transfinite numbers.
    • Science, New Series 122 (3165) (1955), 380.
    • After a vivid historical introduction describing the paradoxes of set theory there follows an account of the axiomatic foundations of the Zermelo style system which Fraenkel used informally in his introductory volume in the same series, Abstract Set Theory.

  138. Serre reviews
    • Addison-Wesley has just reissued Serre's 1968 treatise on l-adic representations in their Advanced Book Classics series.
    • More importantly, it can be viewed as a toolbox which contains clear and concise explanations of fundamental facts about a series of related topics: abstract l-adic representations, Hodge-Tate decompositions, elliptic curves, L-functions, etc.
    • This book in the SUP series is based upon two courses of lectures given by the author in 1962 and 1964 at the Ecole Normale Superieure.

  139. Godement's preface
    • As for content, I did not hesitate to introduce, sometimes very early, subjects considered relatively advanced - multiple series and unconditional convergence, analytical functions, the definition and immediate properties of Radon measures and distributions, the integrals of semi-continuous functions, Weierstrass elliptic functions, etc.
    • Chapter VII develops, besides the classical theory of Fourier series and integrals, those classical properties of analytic functions or harmonics that can be proved without using the curvilinear Cauchy integral: the simplest results on Fourier series given there are sufficient and I have often taught this less common method; the remainder of the theory will come in Volume III.

  140. Perron books
    • A natural sequence of ideas leads to a discussion of the expressions of numbers as decimals and continued fractions as well as in the less familiar forms of Cantor's series and product and the series of Sylvester, Luroth and Engel.
    • The fourth chapter contains not only an excellent introduction to the theory of continued fractions, but also the representation of real numbers by the series of Cantor, Luroth, Engel and Sylvester.

  141. Kantorovich books
    • Science, New Series 134 (3487) (1961), 1358.
    • The first chapter deals with expansion in series, both orthogonal and nonorthogonal, with a section on the improvement of convergence.
    • Economica, New Series 44 (176) (1977), 427-428.

  142. Isaacs' Differential Games
    • An investigation, spurred by a series of headlined catastrophes, revealed an unexpected and elegant liaison with differential games.
    • Dr Issacs developed his theory in a series of Rand memoranda in 1954-55.
    • Series A (General) 129 (3) (1966), 474-475.

  143. Enriques' reviews
    • This is the first volume of a series "Per la Storia e la Filosofia delle Matematiche," published under the auspices of the Istituto Nazionale per la Storia delle Scienze Fisiche e Matematiche, and edited by Professor Enriques of the University of Bologna.
    • This is the third volume in the series "Per la Storia e la Filosofia delle Matematiche" edited by Professor Enriques.
    • The authors published their first account in Italian: 'Storia del Pensiero Scientifico', four years prior to the present French series.

  144. Élie Cartan reviews
    • In this sense, this is not a book from which to learn the skill of tensor formalism; just as a book in complex variables, if leaning in a geometric direction, need not be the appropriate source from which to learn the technique of power series manipulation.
    • In terms of contemporary Lie group theory, it deals with the B and D series of simple Lie algebras and the Lie groups which go along with them, i.e., the orthogonal matrix groups over the real and complex numbers and their simply connected covering groups.
    • An elucidation of the concepts that occur in these commentaries is presented in a series of three appendices by Hermann, entitled "The formalism of connection theory", "Cartan's method of the moving frame as a generalization of Klein's 'Erlanger Programm' ", "Excursions into the theory of Cartan connections".

  145. Poincaré on non-Euclidean geometry
    • From these hypotheses he deduces a series of theorems between which it is impossible to find any contradiction, and he constructs a geometry as impeccable in its logic as Euclidean geometry.
    • Let us consider a certain plane, which I shall call the fundamental plane, and let us construct a kind of dictionary by making a double series of terms written in two columns, and corresponding each to each, just as in ordinary dictionaries the words in two languages which have the same signification correspond to one another:- .
    • If the first point is conceded and the second rejected, we are led to a series of theorems even stranger than those of Lobachevsky and Riemann, but equally free from contradiction.

  146. Rios Honorary Degree
    • It can be said that it appears already in the Calculus of Probabilities when the Chevalier de Mere presented to Pascal frequencies of events in the game of dice whose probability was badly calculated by de Mere, but was to be correctly modelled by Pascal, and identified with the frequency in a long series of trials, led him to the first conscious and successful simulation of a random game, in short, to the creation of the Calculus of Probabilities.
    • Finally the results from the verified model are compared with those of the real system and if the degree of adjustment is unacceptable a new series of steps must be carried out that modify the first model in view of the new information acquired.
    • A series of classifications in which characteristics of the system (experimental, observable, designable), basic sciences (hard, such as Physics, Chemistry, ..

  147. Coulson: 'Electricity
    • The Oliver and Boyd series of mathematical texts were widely used by students throughout the 1940s to 1960s.
    • One of the books in the series was Electricity by Charles A Coulson.
    • But there is another entirely different way of measuring current; for when charges are flowing we discover a whole series of new phenomena, to which we give the name magnetism.

  148. Dingle books
    • Only one chapter is devoted to the General Theory, and the major part of the book is intended by the author to be an introduction to Professor McCrea's valuable little monograph in the same series, 'Relativity Physics'.
    • In this work Professor Dingle has collected a series of his essays on the history and philosophy of science.
    • Dingle's feeling of disillusion with the theory, on which he had previously written a textbook in the Methuen Monographs on Physical Subjects series, began in 1955 when, in the course of reading Sir George Thomson's book 'The Foreseeable Future', he encountered a speculation arising from the well known clock paradox of relativity, sometimes referred to as the paradox of the travelling twin.

  149. Ledermann: 'Complex Numbers
    • A series of books called the Library of Mathematics were edited by W Ledermann.
    • One of the early texts in the series was Complex Numbers by Ledermann himself.
    • I should like to thank my friend and colleague Dr J A Green for a number of valuable suggestions, especially in connection with the chapter on convergence, which is a sequel to his volume Sequences and Series in this Library.

  150. Gillespie: 'Integration
    • The Oliver and Boyd series of mathematical texts were widely used by students throughout the 1940s to 1960s.
    • One of the books in the series was Integration by R P Gillespie.
    • I should also like to thank the general editors of the series for their kindly advice throughout the preparation of the book.

  151. Reviews of Shafarevich's books
    • In the series, 'Popular lectures on mathematics'.
    • This book is the result of a series of lectures on linear algebra and the geometry of multidimensional spaces given in the 1950s through 1970s by Igor R.
    • Igor R Shafarevich gave a series of lectures on linear algebra and geometry at the Moscow State University from the 1950s through the 1970s.

  152. H L F Helmholtz: 'Theory of music' Prefaces
    • By this means that peculiar series of upper partial tones, on the existence of which the present theory of music is essentially founded, receives a new subjective value, entirely independent of external alterations in the quality of tone.
    • To illustrate the anatomical descriptions, I have been able to add a series of new woodcuts, principally from Henle's Manual of Anatomy, with the author's permission, for which I here take the opportunity of publicly thanking him.

  153. Sommerville: 'Geometry of n dimensions
    • [In the twenty-seven volumes of the new series of the Proceedings of the London Mathematical Society there are barely a dozen papers dealing with higher space.
    • There are two main ways in which we may arrive at an idea of higher dimensions: one geometrical, by extending in the upward direction the series of geometrical elements, point, line, surface, solid; the other by invoking algebra and giving extended geometrical interpretations to algebraic relationships.

  154. Durell and Robson: 'Advanced Trigonometry
    • Thus the methods for expanding functions in series focus attention on "remainders" and "limits"; the methods for factorizing functions turn on establishing possible forms and then using the fundamental factor-theorem; the discussion of complex numbers emphasises the fact that complex numbers are just as "real" as real numbers, etc.
    • The theory of Infinite Products has been left for this companion volume; it is not so easy to provide a satisfactory ab initio treatment for products as it is for series and the alternative of taking for granted everything that really matters is undesirable.

  155. Apostol Project
    • One of a series of motivational and instructional videotapes intended primarily for students in grades 7-9.
    • Animation demonstrates the Gibbs phenomenon of Fourier series.

  156. H W Turnbull: 'Scottish Contribution to the Calculus
    • Interpolation formulae involving successive order of finite differences as well as the power series, involving successive derivatives and found by Taylor and Maclaurin, were used over forty years earlier (1670-1671) by Gregory.
    • The notes provide evidence of work by Hudde at Amsterdam prior to 1660 on the logarithmic series, antedating Newton and Mercator.

  157. Horace Lamb addresses the British Association in 1904, Part 2
    • On this view the most refined geometrical demonstration can be resolved into a series of imagined experiments performed with such bodies, or rather with their conventional representations.
    • The not result of the preceding survey is that the systems of Geometry, of Mechanics, and even of Arithmetic, on which we base our study of Nature, are all contrivances of the same general kind: they consist of series of abstractions and conventions devised to represent, or rather to symbolise, what is most interesting and most accessible to us in the world of phenomena.

  158. Menger on teaching
    • Science, New Series 123 (3196) (1956), 547-548.
    • Science, New Series 127 (3310) (1958), 1320-1323.

  159. NAS Memoir of Chauvenet
    • Soon after leaving college he was selected by Professor Bache to assist in the series of magnetic observations undertaken at Girard College in Philadelphia.
    • The Gaussian equations, the finite variations and differentials of trigonometric expressions, the solution of the general spherical triangle, and the development of several functions into series of multiple angles, are instances most readily noted.

  160. Stringham address
    • When a series of elements operating upon each other in accordance with fixed laws produce only other elements belonging to the same series, they are said to constitute a group.

  161. Ball books
    • Science, New Series 5 (113) (1897), 352-353.
    • Several years ago the Editor of the 'College Monographs' conceived the idea of a series of volumes dealing separately with the Colleges of our two ancient Universities; since then many historical and illustrated works carrying out, in some degree, this idea have been published.

  162. Kelvin on the sun, Part 2
    • It will again fall inwards, and after a rapidly subsiding series of quicker and quicker oscillations it will subside, probably in the course of two or three years, into a globular star of about the same dimensions, heat, and brightness as our present sun, but differing from him in this, that it will have no rotation.
    • A diminishing series of out and in oscillations will follow, and the incandescent globe thus contracting and expanding alternately, in the course it may be of three or four hundred years, will settle to a radius of forty times the radius of the earth's orbit.

  163. Gattegno's books
    • I offer this book as the first in such a series; in it I restrict myself to elementary mathematics, and mainly to the algebra and theory of numbers.
    • This is the first in a series of books to be written by the author showing the possibility of making the study of education properly scientific.

  164. Heaton problems
    • Problem: Find the sum of the first n+1 terms of the series .
    • Problem: Sum the infinite series n2/(4n2 - 1)2 beginning with n = 1, n being always odd.

  165. James Gregory's manuscripts
    • His method depended on a series of eliminations, which he carried out fully for the cubic and biquadratic, but only partially for the quintic.
    • Actually Gregory did not set too great store by this general method: for, as he wrote in his very last letter to Collins, "I have a method of series going very far beyond it, and, to be ingenuous, I think one of its greatest uses is by drawing learned men from this contemplation which busied so many fine wits." .

  166. da Silva works
    • Propriedades geraes e resolucao directa das congruencias binomias: introduccao ao estudo da theoria dos numeros, Memorias da Academia das Ciencias (New series) I (1) (Imprensa Nacional, Lisbon, 1854), 1-16.
    • De varias formulas novas de geometria analytica relativa aos eixos coordenados obliquos, Memorias da Academia Real das Ciencias de Lisboa (New Series) V (1) (1872), 1-20.

  167. Whyburn's books
    • Economica, New Series 27 (108) (1960), 375-376.
    • Included, how ever, are some excellent materials not usually found in intermediate algebra books - the basic ratios and identities of trigonometry, the sine law, the cosine law, mathematical induction, series, a brief discussion of the number e, short introductions to algebra of matrices, and Diophantine equations, among others - materials which the inspired teacher will welcome.

  168. Skolem: 'Abstract Set Theory
    • The following pages contain a series of lectures on abstract set theory given at the University of Notre Dame during the Fall Semester 1957-58.
    • Almost 100 years ago the German mathematician Georg Cantor was studying the representation of functions of a real variable by trigonometric series.

  169. Ledermann: 'Finite Groups
    • The Oliver and Boyd series of mathematical texts were widely used by students throughout the 1940s to 1960s.
    • One of the books in the series was Introduction to the Theory of Finite Groups by Walter Ledermann.

  170. John Couch Adams' account of the discovery of Neptune
    • Meanwhile the Royal Academy of Sciences of Gottingen had proposed the theory of Uranus as the subject of their mathematical prize, and although the little time which I could spare from important duties in my college prevented me from attempting the complete examination of the theory which a competition for the prize would have required, yet this fact, together with the possession of such a valuable series of observations, induced me to undertake a new solution of the problem.
    • After obtaining several solutions differing little from each other, by gradually taking into account more and more terms of the series expressing the perturbations, I communicated to Professor Challis, in September 1845, the final values which I had obtained for the mass, heliocentric longitude, and elements of the orbit of the assumed planet.

  171. Weil on history
    • On the latter date, Euler, referring explicitly to Fagnano's work on the lemniscate, read to the Academy the first of a series of papers, eventually proving in full generality the addition and multiplication theorems for elliptic integrals.
    • ultimately upon the values, for suitable values of the arguments, of the simple series discussed above in our Chapter 7.

  172. R A Fisher: 'History of Statistics
    • In 1925 R A Fisher published Statistical Methods for Research Workers in the Biological Monographs and Manuals Series by the publisher Oliver and Boyd of Edinburgh in Scotland.
    • On the other hand, it is to him we owe the principle that the distribution of a quantity compounded of independent parts shows a whole series of features - the mean, variance, and other cumulants - which are simply the sums of like features of the distributions of the parts.

  173. G H Hardy addresses the British Association in 1922
    • One is to take refuge, as Professor Henry Smith, with visible reluctance, did then, in a series of general propositions to which mathematicians, physicists, and astronomers may all be, expected to return a polite assent.
    • The function of a mathematician, then, is simply to observe the facts about his own hard and intricate system of reality, that astonishingly beautiful complex of logical relations which forms the subject-matter of his science, as if he were an explorer looking at a distant range of mountains, and to record the results of his observations in a series of maps, each of which is a branch of pure mathematics.

  174. Gheorghe Mihoc's books
    • The second, entitled "Stochastic processes", deals with Markov chains, ergodic problems, distributions, limit theorems, time series, aleatory mechanics, and mixing processes.
    • The first part deals with elements of probability theory; the second is an analysis of statistical distributions, while the third one is devoted to the analysis of time series.

  175. Cotlar publications
    • Series, Belmont, CA, 1982), 258-269.
    • Series, Belmont, CA, 1982), 306-317.

  176. German syllabus
    • Arithmetic and geometric series.
    • Infinite geometric series.

  177. D'Arcy Thompson on Plato and Planets
    • We may be justified in including in our series the angle of the ecliptic for the Sun, that is to say about 23° 28', or rather, if we assume the epoch of Eudoxus (as Dr Copeland suggested to me long ago) about 23° 45': which angle, in the [ artificial sphere] of Eudoxus, is the angle which the axis of the Sun's second sphere makes with that of his first; and in including for the Moon an angle which, as Eudoxus tells us, is in her case somewhat greater than the Sun's, in fact about 5° more.
    • While writing on this subject let me add, in parenthesis, that the very ancient and very decorative 'Greek key-pattern' seems to me to be nothing more nor less than an archaic representation of a planet's apparent course, a series of simplified hippopedes.

  178. G H Hardy's schedule of lectures in the USA
    • Modern work in the theory of ordinary trigonometric series .
    • Fourier series and almost periodic functions from the standpoint of groups .

  179. Isaac Todhunter: 'Euclid' Preface
    • In 1905 the London publisher Joseph Malaby Dent had the idea of producing the Everyman's Library, a cheap series of reprints of classical texts.
    • Dent wanted the series:- .

  180. Mac Lane books
    • The chapter on special fields now includes power series fields and a treatment of the p-adic numbers.
    • The book originated in a series of lectures given by the author to postgraduate philosophy students specialising in the philosophy of mathematics.

  181. Poincaré on intuition in mathematics
    • Weierstrass leads everything back to the consideration of series and their analytic transformations; to express it better, he reduces analysis to a sort of prolongation of arithmetic; you may turn through all his books without finding a figure.
    • It is to perceive the inward reason which makes of this series of successive moves a sort of organized whole.

  182. Bützberger on Steiner
    • However, he did publish Uber bizentrische Polygone, Steinersche Kreis- und Kugelreihen und die Erfindung der Inversion (On Bicentric Polygons, Steiner Series of Circles and of Spheres, and the Invention of Inversion) in 1913, dedicating a separate section for each of the three topics.
    • In the second section in particular, he treats 'Steiner series of circles and of spheres; here [he] follows Geiser's view: "Einleitung in die synthetische Geometrie", last chapter "Das Prinzip der reziproken Radien" '.

  183. Carr Masterclasses
    • Each series of masterclasses consists of four consecutive Saturdays and four such series are held each year.

  184. Knorr's books
    • The Classical Review, New Series 39 (2) (1989), 364-365.
    • The Classical Review, New Series 41 (1) (1991), 210-212.

  185. Isaac Todhunter: 'Euclid' Introduction
    • In 1905 the London publisher Joseph Malaby Dent had the idea of producing the Everyman's Library, a cheap series of reprints of classical texts.
    • Dent wanted the series:- .

  186. Philip Jourdain and Georg Cantor
    • The Open Court Series of Classics of Science and Philosophy, No.
    • These memoirs are the final and logically purified statement of many of the most important results of the long series of memoirs begun by Cantor in 1870.

  187. The South-Troughton quarrel
    • The fault found with it was this: when the instrument was turned a little on its axis and then let go, a series of about a dozen short, quick vibrations followed, each lasting about 0.3 or 0.4 second.
    • (Drinkwater Bethune wrote lives of Galileo and Kepler in the Library of Useful Knowledge, and with Sir John Lubbock a little book On Probability, in the same series.) Maule at once insisted that Troughton & Simms should be allowed to finish their work according to the plan proposed by Sheepshanks, but only to be paid for if successful.

  188. Gottingen chairs
    • There are, however, a series of reservations in connection with you being called, and the question remains, to what extent I should submit to these reservations, and whether I should perhaps even say right away that Hilbert's coming here would be more suited to our needs in the end.
    • to his house and gave us a series of private lectures on some chapters of the theory of functions of complex variables, in particular on Mittag-Leffler's theorem, which I still consider as one of the most impressive experiences of my student life.

  189. Halmos Set Theory
    • Treated thus, the set theory appears as a series of facts about a reality outside us, rather like euclidian geometry, founded on axiomas, was conceived as a series of facts about the space surrounding us.

  190. Hardy on the Tripos
    • And as there is only one test of originality in mathematics, namely the accomplishment of original work, and as it is useless to ask a youth of twenty-two to perform original research under examination conditions, the examination necessarily degenerates into a kind of game, and instruction for it into initiation into a series of stunts and tricks.
    • Indeed I am afraid that my advice to reformers might sound like a series of stupid jokes.

  191. Smith Autograph Papers
    • In an earlier article in this series I have called attention to a little side-light thrown upon his life by a note from his collaborator Lalande.
    • Of all the mathematicians who added to the reputation of England in the closing years of Newton's life, no one arouses a more sympathetic interest than Abraham De Moivre, author of the well-known Doctrine of Chances, of the even more notable Miscellanea Analytica, of a work on annuities, and one on series, and of various monographs on geometry and the Newtonian calculus.

  192. Wolfgang Pauli and the Exclusion Principle
    • The series of whole numbers 2, 8, 18, 32 ..
    • This was in 1922, when he gave a series of guest lectures at Gottingen, in which he reported on his theoretical investigations on the Periodic System of Elements.

  193. Percy MacMahon addresses the British Association in 1901, Part 2
    • The results were important algebraically as throwing light on the theory of Algebraic series, but another large class of problems remained untouched, and was considered as being both outside the scope and beyond the power of the method.
    • In the case of simple unrestricted partition it gives directly the composition by rows of units which is in fact carried out by the Ferrers-Sylvester graphical representation, and led in the hands of the latter to important results connection with algebraical series which present themselves in elliptic functions and in other departments of mathematics.

  194. Henry Baker addresses the British Association in 1913
    • Though the recurrence of these inquiries is part of a wider consideration of functions of complex variables, it has been associated also with the theory of those series which Fourier used so boldly, and so wickedly, for the conduction of heat.
    • This problem has led to the precision of what is meant by a function of real variables, to the question of the uniform convergence of an infinite series, as you may see in early papers of Stokes, to new formulation of the conditions of integration and of the properties of multiple integrals, and so on.

  195. U N Singh
    • he undertook research under the guidance of Professor B N Prasad in the area of Fourier series.
    • His thesis examiner, none other than the late Professor E C Titchmarsh, F.R.S., of the University of Oxford had this to say while commenting on the work entitled "Strong Summability of Trigonometric Series": "The thesis displays considerable originality and the proofs of the theorems indicate that the candidate is an extremely talented mathematician".

  196. Rutherford: 'Fluid Dynamics
    • The Oliver and Boyd series of mathematical texts were widely used by students throughout the 1940s to 1960s.
    • One of the books in the series was Fluid Dynamics by D E Rutherford.

  197. Zwicky lecture
    • A review of the development of astronomy reveals a series of most entertaining adventures, errors and omissions in addition to great discoveries and achievements.
    • These successes are embodied in the construction and operation of a whole series of remarkable jet engines as well as in the integrated and extended knowledge which was acquired on the whole problem of propulsive power.

  198. Peter's books
    • A large part of the theory is made up of the author's own work which has appeared in a series of papers since 1932.
    • Beginning in 1932, Rosza Peter has published a series of papers, examining the relationship of various special forms of recursion, and showing the definability of new functions by successively higher types of recursion, which establish her as the leading contributor to the special theory of recursive functions.

  199. Hormander books
    • Science, New Series 143 (3603) (1964), 234.
    • The first (Distribution theory and Fourier analysis) and second (Differential operators with constant coefficients) volumes of this monograph [','','1983] can be regarded as an expansion and updating of my book Linear partial differential operators in the Grundlehren series in 1963.

  200. W H Young addresses ICM 1928
    • Yet when we try to express quality by a series of numbers, each number must itself have a Quality-factor, before the series of numbers individually and collectively can be intelligible.

  201. Feller Reviews 1
    • Journal of the Royal Statistical Society Series A (General) 114 (2) (1951), 249-250.
    • At the end of each chapter is a series of exercises, many of them very interesting in themselves.

  202. Solve Applied Problems
    • Although the subjects of Fourier series, Fourier and Laplace transforms, and integral equations, are not strictly applied mathematics, they are essential for the study of wave motions, including vibrating strings, sound waves and water waves, and for the study of heat conduction.
    • Thus the boxes labelled wave motion and heat conduction cannot be opened before the box containing Fourier series has been examined in some detail.

  203. Hamming's Reviews
    • Here and there one sees neat tricks of the trade and finds discussions of topics not usually found in texts - for example, the summation of series and the abovementioned band-limited functions.
    • Hamming's text is designed to introduce engineers and scientists to the basic ideas of numerical analysis through a brief survey of many (twelve) topics at an elementary level, including optimization, Fourier series, and random processes.

  204. Sigmund books
    • Science, New Series 264 (5156) (1994), 294-295.
    • Science, New Series 328 (5981) (2010), 977-978.

  205. Murray books
    • In the first there is a discussion of concepts and definitions of asymptotic expansions, sequences and series, and in the second Watson's lemma and Laplace's method for integrals are considered.
    • Series D (The Statistician) 40 (3), Special Issue: Survey Design, Methodology and Analysis (2) (1991), 344-345.

  206. Mary Boole writing
    • The questions so often put by parents, 'At what age do you think my child had better begin Algebra?' (or Trigonometry) and, 'Can you recommend me to a good teacher?' really mean something analogous to this:- 'I intend to keep my child ignorant of all experiences concerning fruit, and all processes connected with it , till he is old enough to begin receiving straight away, in one continuous series of lessons, information, conveyed by verbal explanations, about how to stand on one's legs, how to climb ladders, how to use sickles, how fruits taste, their hygienic and economic value, their botanical classification, and the best means of preserving them.
    • At what age do you consider this series of lessons should begin, and whom do you recommend me to employ to give it?' The only answer one could make to such a question would be that there is no age at which any such course should begin, and no person who ought to be asked to give it.

  207. Karpinski and Smith's numerals
    • The second is by Florian Cajori and appears in Science, New Series 35 (900) (1912), 501-504.
    • Science, New Series 35 (900) (1912), 501-504.

  208. Luca Valerio's scientific career
    • In fact, his work represents, at least for the Archimedean tradition, the apex of that intellectual movement, in the sense that with it a series of themes that had appeared throughout the mathematics of the Cinquecento were brought to maturity, a level to which subsequent research would necessarily have to raise the framework of Renaissance mathematics, which was oriented especially toward rediscovery and translation of, and commentary on, classical texts.
    • In this paper we present a series of unpublished materials that should help to shed light on some of the questions raised above, especially on the connections between Valerio and the Society of Jesus, on his academic career, on his relations with Margherita Sarrocchi and on some related matters.

  209. Charlotte Angas Scott's papers
    • The memoirs to which reference is made in this paper are those by Cayley, "On the Higher Singularities of a Plane Curve," 1866, and H J Smith, "On the Higher Singularities of Plane Curves," 1873-6, and the series of papers by Brill and Nother in the 'Mathematische Annalen', etc.
    • XIV of this Journal I gave an account of a geometrical method of analysing Higher Singularities, by means of which there may be found for any singularity a penultimate form involving a series of nodes with a certain number of evanescent loops.

  210. Gibson: 'History of Scottish Mathematics
    • In 1927 the Edinburgh Mathematical Society began to publish Series 2 of the Proceedings of the Edinburgh Mathematical Society.
    • The first paper in the new Series was by George Gibson and it was the first part of his two-part paper Sketch of the History of Mathematics in Scotland to the end of the 18th Century.

  211. Halmos books 1
    • Science, New Series 138 (3543) (1962), 886-887.
    • Science, New Series 144 (3618) (1964), 531-532.

  212. Haupt calculus textbooks
    • After the rules of differentiation are developed, the mean value theorem, Taylor series, what some texts call evaluation of indeterminates, the indefinite integral, and circular and hyperbolic functions are considered.
    • Some topics included are: continuity and differentiability, tangents, formal rules of differentiation, derivatives of the elementary functions, higher derivatives, the mean value theorem, Taylor's formula with remainder, Taylor's series, indeterminate forms, existence and uniqueness of primitive functions, elementary integration formulas, divided differences, Newton and Lagrange interpolation polynomials, derivatives (generalized derivatives) derivatives of higher order as limits of divided differences, convex and convexoid functions of nth order, osculating circle and curvature, limit sets of functions of one or more real variables, derivate sets and contingents, nowhere differentiable functions, differentiability properties of rectifiable arcs, contingents and generalized differentials of functions of several variables, differentials of higher order, partial derivatives of first and higher order, Taylor's formula with remainder for functions of several variables, implicit functions, functional determinants, maxima and minima of functions of several variables, multiplier rule, differentiable mappings, functional dependence and independence.

  213. Piaggio Reviews
    • If we travel along the purely analytical path, we are soon led to discuss Infinite Series, Existence Theorems and the Theory of Functions.
    • (4) solutions in series, .

  214. Brinkley Copley Medal
    • The volume for 1807 contains an important paper, on the General Term of a Series in the Inverse Method of finite Differences; in which, taking up a subject of investigation on which both Lagrange and Laplace had written, he has surmounted a difficulty which had remained even after the investigations of these illustrious geometers.
    • He has examined all Mr Pond's results, reasoning upon the law of the aberration of light, the effects of refraction and of differences of temperature, and has compared his own series of observations with those of other astronomers, and he seems entirely convinced of the accuracy of his general conclusions.

  215. Comments by Charlotte Angas Scott
    • No more damaging charge can be brought against any treatise laying claim to thoroughness than that of recklessness in the use of infinite series; and yet Mr Edwards has everywhere laid himself open to this charge.
    • One of the most difficult things to teach the beginner in mathematics is to give proper attention to the convergency of the series dealt with.

  216. Young Researchers
    • Speaker: Caroline Series, Warwick Mathematics Institute, University of Warwick, UK .
    • series .

  217. Aitchison books
    • Series A (General) 134 (2) (1971), 242-243.
    • Series C (Applied Statistics) 36 (3) (1987), 375.

  218. James Clerk Maxwell on the nature of Saturn's rings
    • The entire system of rings must therefore consist either of a series of many concentric rings, each moving with its own velocity, and having its own systems of waves, or else of a confused multitude of revolving particles, not arranged in rings, and continually coming into collision with each other.
    • These particles may be arranged in series of narrow rings, or they may move through each other irregularly.

  219. MacDuffee's books
    • Science, New Series 93 (2408) (1941), 185-186.
    • Science, New Series 119 (3099) (1954), 730.

  220. Gini Eugenics address
    • Regenerative Eugenics has the special purpose of studying, through series of successive generations, how new stocks rise, what circumstances determine their formation in the midst of the obscure mass of the population Ń a formation which can hardly be explained by the heredity of superior factors heretofore non-extant Ń and what importance may be ascribed in their formation to the influence of happy combinations arising from cross-breeding and favored by natural selection, such as the change of environment caused by emigration, or the selection of the original populations which occurs in emigration.
    • Not only has that Institute desired to be officially represented at this Congress, but it has also wished to make a worthy contribution to the annexed exhibition by sending a series of large colored diagrams showing density of population, birth-rate, and death-rate of the several Italian Communes, and two collections of graphs showing the variations in the Italian death-rate during the past forty years, and the composition of large Italian families, as well as many aspects of the marriage and death rates of their members.

  221. Temesvár letter from János to Farkas Bolyai
    • Tokelletes, ugy a mint meg-irta, hanem persze mar tudni kell elore a series .
    • proof, of course is perfect as you have written, but you need to know the form of the series, to be used in the proof, .

  222. Ahlfors' reviews
    • The emphasis throughout is on the geometric approach and power series are not introduced until the middle of the book.
    • The exponential and trigonometric functions are now defined by means of power series; the introduction to point set topology has been rewritten; normal families are more directly approached and the connection with compactness is emphasised; the Riemann mapping theorem has been combined with a section on the Schwarz-Christoffel transformation; a brief treatment of elliptic functions has been included; and exercise sections have been enlarged.

  223. Hardy and Veblen on Erdos
    • Who won the World Series? Widder doesn't know.

  224. Valdivia Infinity
    • Mathematicians, in addition to accepting the concept of dimensionless point, were somehow influenced by Zeno to rigorously establish the concept of boundary and introduce convergent series.

  225. Edward Sang on his tables
    • In this way the series of fundamental tables needed for the new system has been completed, so far as the limit of minutes goes.

  226. Rudio's talk
    • Published in the Sammlung gemeinverstandlicher wissenschaftlicher Vortrage, edited by R Virchow and W Wattenbach; new series, volume 6, issue 142, Verlagsanstalt und Druckerei A.G., Hamburg, 1892.

  227. Charles Tweedie on James Stirling
    • Gauss himself had most unwillingly to make use of Stirling's Series, though its lack of convergence was anathema to him.

  228. Iyahen tribute
    • The lecture which was delivered on 20th November 1980 was the 11th in the Inaugural Lecture series of the University of Benin.

  229. Rouche and de Comberousse
    • - Fundamental properties relating to two pencils which have a common homologous ray and an equal cross-ratio, or to two rectilinear series of four points which have an equal cross-ratio and a common homologous point.

  230. Survey of Modern Algebra
    • Science, New Series 95 (2467) (1942), 386-387.

  231. Jacques Hadamard's mathematician's mind
    • These analogies appeared when, in 1937, at the Centre de Synthese in Paris, a series of lectures was delivered on invention of various kinds, with the help of the great Genevese psychologist, Claparede.

  232. Finsler publications
    • P Finsler, Die Wahrscheinlichkeit seltener Erscheinungen, Annali di Matematica Pura ed Applicata (Series IV) 54 (1961), 311-323.

  233. Cotlar interview
    • At that time a lot of work was done, the Institute of Calculus was created and a series of publications edited by Cora Ratto de Sadosky was begun, also the Einstein Foundation whose mission was to facilitate the study of talented young people lacking resources.

  234. Mathematics in Edinburgh
    • A series of explanatory lectures will be given by the Class-Assistant before the examination.

  235. Bronowski and retrodigitisation
    • Although Jacob Bronowski's name is most remembered in association with the BBC television documentary series The Ascent of Man he made at the end of his life - it inspired discussion [',' A Orton and S M Flower, Analysis of an ancient tessellation, Math.

  236. W Burnside: 'Theory of Groups of Finite Order
    • The last Chapter contains a series of results in connection with the classification of groups as simple, composite, or soluble.

  237. Fatou Fonctions Automorphes
    • The theta series are studied in detail.

  238. Paul Halmos: the Moore method
    • That had two effects: it stopped the course from turning into an uninterrupted series of lectures by the best student, and it made for a fierce competitive attitude in the class - nobody wanted to stay at the bottom.

  239. University of Glasgow Examinations
    • From the series 1, 2, 3, 4, ..

  240. Montmort's Treize
    • Now if, in this whole series of cards, he never once turns over the card he is naming, he pays out what each other player has put up for the game, and the deal passes to the player sitting to his right.

  241. Sheppard Papers
    • "Summation of the coefficients of some terminating hypergeometric series." Proc.

  242. Harold Jeffreys: 'Scientific Inference' Preface
    • The present work had its beginnings in a series of papers published jointly some years ago by Dr Dorothy Wrinch and myself.

  243. Taleb reviews
    • This book proposes a systematic view of Nassim Nicholas Taleb's ideas, some of which have already been published in a series of articles and in a previous book.

  244. L R Ford - Automorphic Functions
    • In the fifth chapter existence theorems are established by means of the Poincare theta series, and some properties of the theta functions are proved.

  245. Somerville's Booklist
    • LacroixFinite differences and series .

  246. Mathematics at Aberdeen 3
    • To help and encourage beginners he published much expanded translations of two of Newton's tracts, on quadrature and series.

  247. Isaac Schoenberg: Mathematical time exposures
    • Others deal with themes as diverse as Fibonacci numbers, convex sets, spline functions, non-differentiable curves, iterative algorithms, the Kakeya problem (the smallest area within which a rod can be turned round in the plane) and geometrical porisms; and there are four each on various aspects of finite Fourier series and Konig-Szucs polygons, which are described as the paths of (weightless, infinitesimal, perfectly elastic) billiard balls confined within a cube.

  248. Kepler's 'Foundations of modern optics' Preface to a translation
    • Catherine Chevalley has first of all attempted to satisfy this series of requirements.

  249. Marshall Hall books
    • The length of the lower central series is an example.

  250. Sylow's 1913 Address
    • He says: When a function's growth is developed in power series according to the increment in the absolute variable, that term in the expansion, which contains the first power, is called the differential of the function.

  251. Wussing Reviews
    • This textbook is the first volume of a projected series intended as a basic introduction to the history of mathematics and designed for independent study, in particular for students and high school teachers.

  252. L R Ford: Monthly Editor
    • "Before the echoes of the Pearl Harbor catastrophe had died down, we were beset with a series of countless headaches which are concomitant with any such abrupt transition.

  253. Slaught's books
    • This book follows the plane geometry of the textbook series of the authors.

  254. Jacobson: 'Structure of Rings
    • We had planned originally to write a series of notes indicating individual contributions.

  255. Yung-Chow Wong
    • During the year 1940, four of my papers were accepted for publication by the Journal of the London Mathematics Society, the Quarterly Journal of Mathematics (Oxford series) and the Proceedings of the Edinburgh Mathematical Society.

  256. Heinrich Weber's books
    • Science, New Series 38 (981) (1913), 550-551.

  257. Green's students
    • Thesis title: On the Discrete Series Characters of Linear Groups.

  258. Clifford's 'Lectures and Essays'
    • I doubt if he studied historical works critically; it seems to me that he regarded history in a poetical rather than a scientific spirit, seeing events in a series of vivid pictures which had the force of present realities as each came in turn before the mind's eye.

  259. Madras College exams
    • This long series of examinations, embracing several hundred pupils and a vast variety of subjects, was as usual, one of the most interesting and popular of the proceedings of the occasion.

  260. Edinburgh Physics Examinations
    • What is meant by arranging electric conductors "in series," and "in multiple arc"? .

  261. Big Game Hunting
    • Hobson, The Theory of Functions of a Real Variable and the Theory of Fourier's Series, 1927, vol.

  262. Big Game Hunting
    • Hobson, The Theory of Functions of a Real Variable and the Theory of Fourier's Series, 1927, vol.

  263. Lorch books
    • Science, New Series 138 (3537) (1962), 132.

  264. Feller Reviews 5
    • Journal of the Royal Statistical Society Series A (General) 135 (3) (1972), 430.

  265. Surely you're joking Mr Feynman' Introduction
    • There may be no way to construct such a series of delightful stones about himself and his work: the challenge and frustration, the excitement that caps insight, the deep pleasure of scientific understanding that has been the wellspring of happiness in his life.

  266. Peacock Treatise
    • The assumption however of the independent existence of the signs + and - removes this limitation, and renders the performance of the operation denoted by - equally possible in all cases: and it is this assumption with effects the separation of arithmetical and symbolical Algebra, and which renders it necessary to establish the principles of this science upon a basis of their own: for the assumption in question can result from no process of reasoning from the principles or operations of Arithmetic, and if considered as a generalisation of them, it is not the last result in a series of propositions connected with them: it must be considered therefore as an independent principle, which is suggested as a means of ending a difficulty which results from the application of arithmetical operations to general symbols.

  267. Leslie Origins Number
    • The heap, so analysed by a series of partitions, might then be expressed with a very few low numbers, capable of being distinctly retained.

  268. Pappus Pandrosion
    • Pappus then proceeds through a series of problems related to him by students of Pandrosion.

  269. Dixmier reviews
    • The range of material and the book's particular emphasis can probably best be indicated by listing the ten chapter headings: Topological spaces, Limits and continuity, Constructions of spaces (subspaces, products and quotients), Compact spaces, Metric spaces, Limits and convergence of functions, Real-valued functions, Normed spaces, Infinite series, and Connected spaces.

  270. Todd: 'Basic Numerical Mathematics
    • We then take off with a study of "rate of convergence" and follow this with accounts of "acceleration process" and of "asymptotic series" - these permit illumination and consolidation of earlier concepts.

  271. Duran-Loriga's biography of Hermite
    • This article was translated into English by George Bruce Halsted and this translation was published in Science, New Series 13 (336) (1901), 883-885 and also in Amer.

  272. Cheltenham exams
    • Arithmetic and geometric progressions also feature heavily, question 13 in the 1875 paper asks, "deduce an expression for the sum of n terms of a geometric series." Factors are called common measures and question 8 of the same paper instructs the pupil to, "find the greatest common measure of 3a3 - 3a2b + ab2 - b3 and 4a2 - 5ab + b2 ".

  273. A comment about Napier
    • About half of the quantity he scattered around a newly seeded area and the other half was used as bait in a series of small paper cones which Napier had lined with "bird lime." Of course, the inebriated, birdbrained doves managed to stick their beaks into the wrong places and Napier collected several score and held them at ransom until he was paid the full value of all the seed he had lost.

  274. Brinkley obituary
    • In the course of this discussion he first made British Astronomers acquainted with the method of Minimum Squares, that powerful method of obtaining from a series of observations their most probable result.

  275. Percy MacMahon addresses the British Association in 1901
    • His death at a comparatively early age terminates the important series of discoveries which were proclaimed from his laboratory in the Johns Hopkins University at Baltimore.

  276. Hellman's books
    • Science, New Series 131 (3408) (1960), 1203.

  277. Bertrand Russell on Euclid
    • In spaces where the straight line is not a closed series, this follows from the axioms mentioned in connection with I.6 and I.7.

  278. W H Young addresses ICM 1928 Part 2
    • The combinatory properties of cardinal numbers, which comprise ultimately all formal analysis, provide a practically unlimited series of questions, as to the corresponding properties of the objects under consideration.

  279. R A Fisher: the life of a scientist' Preface
    • Prof G A Barnard arranged a valuable series of seminars on Fisher's statistical papers during our stay at Essex University in the academic year 1970-1971.

  280. James Jeans: 'Physics and Philosophy' I
    • Here they produce further changes, as the result of which - after a series of processes we do not in the least understand - his mind acquires perceptions - to use Hume's terminology - of the outer world.

  281. Johnson post1900 books
    • The expeditious symbolic methods of integration applicable to some forms of linear equations, and the subject of development of integrals in convergent series, have been treated as fully as space would allow.

  282. Slaught on Mathematics and Teaching
    • Science, New Series 55 (1415) (1922), 146-148.

  283. Peacock's Algebra
    • A student who masters these two volumes will have adequate preparation in trigonometry and algebra, including series, for the great body of elementary applications to physics.

  284. Vector calculus problems
    • The book was reviewed by Edgar Odell Lovett (1871-1957) and his review appeared in Science, New Series 10 (253) (1899), 653-654.

  285. Keynes: 'Probability' Introduction Ch II
    • Our logic is concerned with drawing conclusions by a series of steps of certain specified kinds from a limited body of premisses.

  286. Shepherdson Tribute
    • The first work to make him known internationally was the series of papers, beginning in 1951, on the minimal model for set theory, inspired by Godel's monograph from the late 1930's, and manifesting a deep understanding of that great work.

  287. Sansone publications
    • Translated from the Italian by Ainsley H Diamond, International Series of Monographs in Pure and Applied Mathematics 67 (The Macmillan Co., New York, 1964).

  288. Vailati Reviews
    • But it was too early for these ideas to catch on; moreover, after a series of subordinate posts, he abandoned university life, to teach in schools and technical institutes; and though he wrote a vast number of articles and reviews, he never attempted to write a book.

  289. Spanish.html
    • Rey Pastor had certainly played his part in supporting the journal, particularly in the period 1911-13, with a series of papers.

  290. Publications of Gino Fano
    • G Fano, A preface to a series of special lectures on Italian Geometry, and 2 general lectures.

  291. Magnus books
    • Volume I of that work contains the Preface and Foreword to the whole series, describing the history and the aims of the so-called Bateman Manuscript Project.

  292. H F Baker: 'A locus with 25920 linear self-transformations' Introduction
    • In a series of papers on hyperelliptic functions of two variables, in the Math.

  293. Writings of Charles S Peirce' Preface
    • Accordingly, it was necessary to depart occasionally from the strict chronological arrangement in order to present series of papers as uninterrupted units.

  294. Hardy and Veblen on Max Newman
    • Newman thinks that Cambridge Press should really represent us [for Princeton's Colloquium Lecture series], not B&B [Bowes and Bowes], as people think of them first always.

  295. MacRobert Professor
    • He collaborated with the late Professor Andrew Gray in the revision of Gray and Mathew's "Bessel Functions," and had a large share in the editing of the second edition of Bromwich's "Infinite Series." He has published numerous original papers in the Proceedings of the Edinburgh Mathematical Society, and of the Royal Society of Edinburgh.

  296. Cochran: 'Sampling Techniques' Preface
    • The tendency in sampling practice, where decisions must often be made quickly on inadequate knowledge, is to develop a series of working rules, each of which has some basis in theory.

  297. Marion Walter's books
    • This book is a series of problems involving the use of a mirror on one drawing to match other related drawings.

  298. Veblen's Opening Address to ICM 1950
    • The Organizing Committee of the present Congress has tried to meet this problem by means of a series of conferences, more informal than the regular program, but even in the conferences the problem of numbers will remain.

  299. A N Whitehead addresses the British Association in 1916
    • The nature of induction, its importance, and the rules of inductive logic have been considered by a long series of thinkers, especially English thinkers, Bacon, Herschel, J S Mill, Venn, Jevons, and others.

  300. Max Planck: 'The Nature of Light
    • The practical weakness of his position is that he is, consequently, compelled to renounce a series of important conclusions, immediately deduced from the theory of identity.

  301. Vailati writings
    • The best questions, for both purposes, are the ones that refer to the prediction of a specific fact, those where, after describing a given situation and a series of specific operations to the student, we ask what he would expect to find or to obtain if he were to perform them, or how he would act if he wanted to achieve a specific result given the circumstances.

  302. A I Khinchin on Information Theory
    • Unfortunately, there remains a whole series of significant difficulties.

  303. history of reliability
    • His task was to analyze the missile system, and he quickly derived the product probability law of series components.

  304. Taylor versus Continental mathematicians
    • In the summing of arithmetical series.

  305. Stringham books
    • Chapter I, consisting of a series of introductory lessons, is wholly new, and Chapter XIII is partly new and partly transferred from Chapter XXVIII of the second edition.

  306. Santalo honorary doctorate
    • Laplace (1749-1827) in his Analytical Theory of Probabilities (1812), considers the plane divided into congruent rectangles by two series of parallel lines and calculates the probability that a needle thrown at random on the plane does not cut any of those straight lines (problem of the Laplace needle).

  307. Hopper Aiken
    • He waved his hand and said: "That's a computing machine." I said, "Yes, Sir." What else could I say? He said he would like to have me compute the coefficients of the arc tangent series, for Thursday.

  308. George Temple's Inaugural Lecture I
    • The reader feels that he himself is assisting at a series of scientific discoveries, and that a world of new possibilities lies before him.

  309. Sims computation
    • In 1994 Charles Sims published Computation with finitely presented groups in the series Encyclopedia of Mathematics and Its Applications published by Cambridge University Press.

  310. Cheney books
    • What is prerequisite of the reader is a familiarity with such topics as sequences, vector spaces, series, uniform convergence, continuity, and the mean-value theorem - all of which are normally acquired in a good calculus course.

  311. Puig Adam publications
    • Pedro Puig Adam, Series divergentes cuyo termino general tiende a cero, Revista Matematica Hispano-Americana (1924).

  312. Kepler's Planetary Laws
    • Kepler carried out the reduction to heliocentricity, and further simplifying procedures, in a series of steps: .

  313. Hamburger Grimshaw
    • Science, New Series 115 (2990) (1952), 425.

  314. Sommerfeld: 'Atomic Structure
    • In the first half of the nineteenth century Electrodynamics consisted of a series of disconnected elementary laws.

  315. Teixeira on Rocha
    • Coelho da Maia obtained, by means of horrendous calculations, the extension of the formula of Fontaine we mentioned, full of series expansions lacking rigour, but added nothing remarkable about their convergence.

  316. The Tercentenary of the birth of James Gregory
    • Such a discovery was that of Nicolaus Mercator who had found the logarithmic series.

  317. Gibson History 9 - Colin Maclaurin
    • The range covered is very wide; many of the theorems, for example, respecting areas can be easily interpreted as theorems in integration: his test for the convergence of a series (pp.

  318. Craig Differential Equations
    • Goursat's Thesis on equations of the second order satisfied by the hypergeometric series.

  319. Edmund Landau: 'Foundations of Analysis' Prefaces
    • with the mysterious series of dots after the comma (called natural numbers in Chapter I), in the definition of the arithmetical operations with these numbers, and in the proofs of the associated theorems.

  320. Harold Jeffreys on Probability
    • Hence P determines a cut in the series of rational fractions.

  321. Brinkley Astronomy
    • I have also endeavoured to substitute new and simpler mathematical demonstrations in many cases where my experience led me to think that the old ones were too cumbrous, and have added a series of questions on the first thirteen chapters which I hope will he found useful to the Students preparing for examinations.

  322. Edinburgh's tribute to A C Aitken
    • He and D E Rutherford edited a series of texts on University Mathematics in which he wrote the first two volumes himself .

  323. Herschel Museum
    • Herschel set about promoting a rival series of concerts, using Mr Shaw a Pump-Room player and another arch enemy of Linley, at the Octagon Chapel where he was the organist.

  324. Halsted on Cayley
    • G B Halsted, Review: The Collected Mathematical Papers of Arthur Cayley, by A Cayley, Science, New Series 9 (211) (1899), 59-63.

  325. Knorr's papers
    • The present paper is one of a series developing out of my study of the chronological ordering of the Archimedean corpus ("Archimedes and the 'Elements'," to appear late this year).

  326. J A Schouten's Opening Address to ICM 1954
    • In fact, there is nowadays no big factory without its computing machines and no investigation involving series of experiments or observations is possible without an elaborate application of modern statistics.

  327. G A Miller - A letter to the editor
    • One paper of this type, which appeared in the American Mathematical Monthly, was recently reviewed by the following sentence: "A series of assertions are not correct." Zentralblatt fur Mathematik, vol.

  328. Smith Teaching Papers
    • Science, New Series 76 (1977) (1932), 468-471.

  329. Flett's books
    • This is followed by six chapters dealing with calculus of real-and complex-valued functions of one real variable (continuity, limits, derivatives, Riemann integrals, infinite series, and uniform convergence).

  330. Wall's Continued fractions
    • J-fraction expansions for power series .

  331. Gender and Mathematics refs
    • J Schuerich, Methodological implications of feminist and poststructuralist views of science, The National Center for Science Teaching and Learning Monograph Series, 4 (1992).

  332. EMS Rutherford
    • He played a decisive part in the development of the well-known series of University Mathematical Texts and, later, of University Mathematical Monographs.

  333. Rudio's Euler talk
    • Ladies and gentlemen, it must make you proud indeed to learn that among all the mathematicians alive then, the 34-year old Leonhard Euler from Basel was regarded as the most worthy one to head the series of many distinguished names that have since adorned this famous institute.

  334. Mathematics in France during World War II
    • We hoped that this would inaugurate a series of works on scientific humanism.

  335. Cofman teaching
    • The solutions to these problems had been sought for ages; the attempts led to a series of new discoveries and contributed to a further development of the entire science of mathematics.

  336. Marcolongo books
    • In view of the plan that the fourth international congress of mathematicians held at Rome in 1908 should discuss the notations of vector analysis and perhaps lend the weight of its recommendation to some particular system, Burali-Forti and Marcolongo awhile ago set themselves the laudable but somewhat thankless task of collecting and editing all the historical, critical, and scientific material which might be indispensable to a proper settlement of the question by the congress, and this material they published in a series of five notes beginning in the twenty-third volume (1907) of the Rendiconti of Palermo and running through several succeeding numbers and volumes.

  337. Gibson History 6 - More Gregorys
    • But he is specially good in the treatment of series.

  338. Combinatorial algorithms
    • Newton forms of a polynomial and the composition of power series are also discussed.

  339. Felix Klein on intuition
    • Note that we have here an example of a curve with indeterminate derivatives arising out of purely geometrical considerations, while it might be supposed from the usual treatment of such curves that they can only be defined by artificial analytical series ..

  340. E C Titchmarsh on Counting
    • The book covers Counting, Arithmetic, Algebra, The use of numbers in geometry, Irrational numbers, Indices and logarithms, Infinite series and e, The square root of minus one, Trigonometry, Functions, The differential calculus, The integral calculus and Aftermath (see this link).

  341. Weatherburn books
    • Economica, New Series 14 (55) (1947), 239-241.

  342. Arthur Eddington's 1927 Gifford Lectures
    • It makes all the difference in the world whether the paper before me is poised as it were on a swarm of flies and sustained in shuttlecock fashion by a series of tiny blows from the swarm underneath, or whether it is supported because there is substance below it, it being the intrinsic nature of substance to occupy space to the exclusion of other substance; all the difference in conception at least, but no difference to my practical task of writing on the paper.

  343. Thomas Muir: 'History of determinants
    • It was thus not until March 1900 that a second series of analytic abstracts began to appear in the Edinburgh Proceedings, and that the preparation of a third list of writings was methodically undertaken.

  344. Kelvin on the sun
    • Precisely the same series of events as we have been considering will take place in every one of the pits.

  345. Studies presented to Richard von Mises' Introduction
    • In fact, a long series of measurements is needed from which eventually "the value of the length" can be computed.

  346. American Mathematical Society Colloquium
    • Selected Topics in the Theory of Divergent Series and Continued Fractions.

  347. Carver sportsman
    • "I'll excuse the entire class from final examination if -" and then proceeds to challenge the group to a series of competitions in any kind of games or sport desired, Carver to compete against the picked representative in each event and guaranteeing to win at least seventy-five per cent of the matches; otherwise no examination.

  348. Jacobson: 'Theory of Rings
    • The only exposition of the subject in book form that has appeared to date is Deuring's Algebren published in the Ergebnisse series in 1935.

  349. Michell Twisted Rings
    • The connection with Zajac's work was only realized years later by Coleman, Tobias, Olson, and collaborators in a series of papers [B D Coleman, E H Dill, M Lembo, Z Lu and I Tobias, On the dynamics of rods in the theory of Kirchhoff and Clebsch (1993); I Tobias and W K Olson, The effect of intrinsic curvature on supercoiling - Predictions of elasticity theory (1993); Y Yang, I Tobias and W K Olson, Finite element analysis of DNA supercoiling (1993)].

  350. Herschel William papers
    • I was then engaged in a series of observations on the parallax of the fixed stars, which I hope soon to have the honour of laying before the Royal Society; and those observations requiring very high powers, I had ready at hand the several magnifiers of 227, 460, 932, 1536, 2010, &c.

  351. Carol R Karp: 'Languages with expressions of infinite length
    • Professor Tarski's interest in the area led to a series of new developments in set theory that grew out of William Hanf's work on models of infinitary languages, reported in 1960.

  352. Bradley works
    • I have to offer my thanks to them for their attention, as well as to the Rev W Lax, who upon this occasion, as upon every other for a series of years, has proved to me the warmth and constancy of his friendship.

  353. Mathematicians and Music
    • "Bound together?" Yes! in regularity of vibrations, in relations of tones to one another in melodies and harmonies, in tone-colour, in rhythm, in the many varieties of musical form, in Fourier's series arising in discussion of vibrating strings and development of arbitrary functions, and in modern discussions of acoustics.

  354. Everitt BVP
    • This work was published by the American Mathematical Society as volume 61 in their Mathematical Surveys and Monographs series.

  355. Louis Auslander books
    • One-variable text, treating numbers as approximations; hence power, Taylor series are shown to boost computational knowhow.

  356. H Weyl: 'Theory of groups and quantum mechanics'Preface to Second Edition
    • I may mention in this connection the derivation of the Clebsch-Gordan series, which is of fundamental importance for the whole of spectroscopy and for the applications of quantum theory to chemistry, the section on the Jordan-Holder theorem and its analogues, and above all the careful investigation of the connection between the algebra of symmetric transformations and the symmetric permutation group.

  357. Montmort Preface
    • Mr Bernoulli divided it into four Parts; in the first three he gave the solution to various Problems relating to Games of chance: one must find there many new things on infinite series, on combinations and permutations, with the solution of the five Problems proposed a long time ago to Mathematicians by Mr Huygens.

  358. Lewis's papers
    • Mind, New Series 21 (84) (1912), 522-531.

  359. William Herschel discoveries
    • I was then engaged in a series of observations on the parallax of the fixed stars, which I hope soon to have the honour of laying before the Royal Society; and those observations requiring very high powers, I had ready at hand the several magnifiers of 227, 460, 932, 1536, 2010, &c.

  360. T M MacRobert: 'Spherical Harmonics' Contents
    • Fourier series.

  361. Prufer's publications
    • Neue Herleitung der Sturm-Liouvilleschen Reihenentwicklung stetiger Funktionen (New derivation of the storm-Liouville series expansion of continuous functions) (1926).

  362. Douglas Jones publications
    • D S Jones, Asymptotic series and remainders, in B D Sleeman and R J Jarvis (eds), Ordinary and partial differential equations, vol.

  363. Burton reviews
    • In 'Girls into Maths Can Go', the editor has selected a series of eighteen articles, each of which emphasizes a different aspect of the effects of discriminatory practices in British schools on girls' mathematical learning.

  364. Statistics Journal in Estonia
    • In these Proceedings there were published papers on astronomy (Tartu has traditionally been a famous centre for astronomy), meteorology (in Tartu the time series of weather observations has been continuous since the middle of the 19th century), medical and anthropological sciences, etc.

  365. How likely abstract
    • John Haigh is an expert on probability who has written a popular book on the subject Taking chances: winning with probability and has given one of a series of popular lectures for the London Mathematical Society.

  366. Olunloyo interview
    • He told us that the publishers of the DO Fagunwa series had to borrow his personal copies so they could reproduce the books when they went out of print.

  367. Artzy books
    • That's not a term that seems to be commonly used these days, but Artzy is not alone in its use; another book, by Gruenberg and Weir, also has this title and is apparently still available from Springer-Verlag as an entry in their Graduate Texts in Mathematics series.

  368. A I Khinchin: 'Statistical Mechanics' Introduction
    • In places where we might have to use finite sums or series, we operate with integrals, continuous distributions of probability might be replaced by the discrete ones, for which completely analogous limit theorems hold true.

  369. James Jeans addresses the British Association in 1934
    • Physical science obtains its knowledge of the external world by a series of exact measurements, or, more precisely, by comparisons of measurements.

  370. Kuku Representation Theory
    • Aderemi Kuku's monograph Representation Theory and Higher Algebraic K-Theory was published by Chapman & Hall in their Pure and Applied mathematics Series in 2007.

  371. Feller Reviews 3
    • Management Science 14 (10, Application Series) (1968), B633-B634.

  372. Max Planck and the quanta of energy
    • The fruit of this long series of investigations, of which some, by comparison with existing observations, mainly the vapour measurements by Vilhelm Bjerknes, were susceptible to checking, and were thereby confirmed, was the establishment of the general connection between the energy of a resonator of specific natural period of vibration and the energy radiation of the corresponding spectral region in the surrounding field under conditions of stationary energy exchange.

  373. Laguardia Basic Sciences
    • The teaching of every subject that contains a novelty must be preceded by a series of reasons and living examples that show the student the convenience of mastering the new method.

  374. Papers about Lewis
    • S B Rosenthal, Logic and "Ontological Commitment": Lewis and Heidegger, The Journal of Speculative Philosophy, New Series 9 (4) (1995), 247-255.

  375. H L F Helmholtz: 'Theory of Music' Introduction
    • A series of qualities of tone are analysed in respect to their harmonic upper partial tones, and it results that these upper partial tones are not, as was hitherto thought, isolated phenomena of small importance, but that, with very few exceptions, they determine the qualities of tone of almost all instruments, and are of the greatest importance for those qualities of tone which are best adapted for musical purposes.

  376. Oskar Bolza: 'Calculus of Variations
    • To give a detailed account of this development was the object of a series of lectures which I delivered at the Colloquium held in connection with the summer meeting of the American Mathematical Society at Ithaca, N.

  377. Alfred Tarski: 'Cardinal Algebras
    • On the one hand, we have a series of very strong and general theorem.,-, which exhaust large portions of the arithmetic of cardinals, e.g., the theory of cardinal addition; these theorems have been established by .applying the so-called axiom of choice in its most general form and, in particular, the well-ordering principle.

  378. Tietze: 'Famous Problems of Mathematics
    • This is the aim in publishing this book, which is based on a series of lectures delivered for students of all faculties at the University of Munich.

  379. Kuratowski: 'Introduction to Set Theory
    • The stimulus to the investigations from which the theory of sets grew, was given by problems of analysis, the establishing of the foundations of the theory of irrational numbers, the theory of trigonometric series, etc, However, the further development of set theory went initially in an abstract direction, little connected with other branches of mathematics.

  380. Edwin Elliot: 'Algebra of Quantics
    • The reader will not, however, find that the present work is a compilation from others which have preceded it, great as has been the help which those others have afforded Constant recourse has been had to the original authorities, particularly of course to Cayley's series of memoirs, and to Sylvester's writings in the Cambridge and Dublin Mathematical Journal, the American Journal of Mathematics, and elsewhere.


Quotations

  1. Quotations by Euler
    • Notable enough, however, are the controversies over the series 1 - 1 + 1 - 1 + 1 - ..
    • Understanding of this question is to be sought in the word "sum"; this idea, if thus conceived -- namely, the sum of a series is said to be that quantity to which it is brought closer as more terms of the series are taken -- has relevance only for convergent series, and we should in general give up the idea of sum for divergent series.

  2. Quotations by Abel
    • If you disregard the very simplest cases, there is in all of mathematics not a single infinite series whose sum has been rigorously determined.
    • With the exception of the geometric series, there does not exist in all of mathematics a single infinite series whose sum has been determined rigorously.
    • The divergent series are the invention of the devil, and it is a shame to base on them any demonstration whatsoever.
    • By using them, one may draw any conclusion he pleases and that is why these series have produced so many fallacies and so many paradoxes ..

  3. Quotations by Bernoulli Jacob
    • The sum of an infinite series whose final term vanishes perhaps is infinite, perhaps finite.
    • Even as the finite encloses an infinite series .

  4. Quotations by Heaviside
    • This series is divergent, therefore we may be able to do something with it.


Famous Curves

No matches from this section


Chronology

  1. Mathematical Chronology
    • He uses zero and negative numbers, gives methods to solve quadratic equations, sum series, and compute square roots.
    • In these he solves quadratic equations, sums series, studies combinations, and gives methods of finding the areas of polygons.
    • His work on mathematics covers arithmetic, summation of series, combinatorial analysis, the rule of three, irrational numbers, ratio theory, algebraic definitions, method of solving algebraic equations, geometry, Archimedes' theorems, trisection of the angle and other problems which cannot be solved with ruler and compass alone, conic sections, stereometry, stereographic projection, trigonometry, the sine theorem in the plane, and solving spherical triangles.
    • He gives the well known series expansion of log(1 + x).
    • His series expansion for arctan(x) gives a series for π/4.
    • Mengoli publishes The Problem of Squaring the Circle which studies infinite series and gives an infinite product expansion for π/2.
    • It contains the Bernoulli numbers which appear in a discussion of the exponential series.
    • In the book he uses infinite series to find the definite integrals of functions.
    • Fourier discovers his method of representing continuous functions by the sum of a series of trigonometric functions and uses the method in his paper On the Propagation of Heat in Solid Bodies which he submits to the Paris Academy.
    • Cauchy gives power series expansions of analytic functions of a complex variable.
    • In his dissertation he studied the representability of functions by trigonometric series.
    • Delaunay solves the three-body problem by giving the longitude, latitude and parallax of the Moon as infinite series.
    • Meray publishes Nouveau precis d'analyse infinitesimale which aims to present the theory of functions of a complex variable using power series.
    • Lexis publishes On the theory of the stability of statistical series which begins the study of time series.
    • He also shows that series expansions previously used in studying the three-body problem, for example by Delaunay, were convergent, but not in general uniformly convergent.
    • Pearson publishes the first in a series of 18 papers, written over the next 18 years, which introduce a number of fundamental concepts to the study of statistics.
    • Fejer publishes a fundamental summation theorem for Fourier series.
    • Borel publishes the first in a series of papers on game theory and becomes the first to define games of strategy.
    • It results from a series of lectures given in the United States and serves as an introduction to his later metaphysics.
    • Quidong Wang finds infinite series solutions to the n-body problem (with minor exceptions).

  2. Chronology for 1650 to 1675
    • He gives the well known series expansion of log(1 + x).
    • His series expansion for arctan(x) gives a series for π/4.
    • Mengoli publishes The Problem of Squaring the Circle which studies infinite series and gives an infinite product expansion for π/2.

  3. Chronology for 1870 to 1880
    • Meray publishes Nouveau precis d'analyse infinitesimale which aims to present the theory of functions of a complex variable using power series.
    • Lexis publishes On the theory of the stability of statistical series which begins the study of time series.

  4. Chronology for 1890 to 1900
    • He also shows that series expansions previously used in studying the three-body problem, for example by Delaunay, were convergent, but not in general uniformly convergent.
    • Pearson publishes the first in a series of 18 papers, written over the next 18 years, which introduce a number of fundamental concepts to the study of statistics.
    • Fejer publishes a fundamental summation theorem for Fourier series.

  5. Chronology for 500 to 900
    • He uses zero and negative numbers, gives methods to solve quadratic equations, sum series, and compute square roots.
    • In these he solves quadratic equations, sums series, studies combinations, and gives methods of finding the areas of polygons.

  6. Chronology for 1850 to 1860
    • In his dissertation he studied the representability of functions by trigonometric series.
    • Delaunay solves the three-body problem by giving the longitude, latitude and parallax of the Moon as infinite series.

  7. Chronology for 900 to 1100
    • In these he solves quadratic equations, sums series, studies combinations, and gives methods of finding the areas of polygons.
    • His work on mathematics covers arithmetic, summation of series, combinatorial analysis, the rule of three, irrational numbers, ratio theory, algebraic definitions, method of solving algebraic equations, geometry, Archimedes' theorems, trisection of the angle and other problems which cannot be solved with ruler and compass alone, conic sections, stereometry, stereographic projection, trigonometry, the sine theorem in the plane, and solving spherical triangles.

  8. Chronology for 1920 to 1930
    • Borel publishes the first in a series of papers on game theory and becomes the first to define games of strategy.
    • It results from a series of lectures given in the United States and serves as an introduction to his later metaphysics.

  9. Chronology for 1800 to 1810
    • Fourier discovers his method of representing continuous functions by the sum of a series of trigonometric functions and uses the method in his paper On the Propagation of Heat in Solid Bodies which he submits to the Paris Academy.

  10. Chronology for 1900 to 1910
    • Fejer publishes a fundamental summation theorem for Fourier series.

  11. Chronology for 1830 to 1840
    • Cauchy gives power series expansions of analytic functions of a complex variable.

  12. Chronology for 1860 to 1870
    • Delaunay solves the three-body problem by giving the longitude, latitude and parallax of the Moon as infinite series.

  13. Chronology for 1700 to 1720
    • It contains the Bernoulli numbers which appear in a discussion of the exponential series.

  14. Chronology for 1720 to 1740
    • In the book he uses infinite series to find the definite integrals of functions.

  15. Chronology for 1990 to 2000
    • Quidong Wang finds infinite series solutions to the n-body problem (with minor exceptions).


EMS Archive

  1. Edinburgh Mathematical Society Lecturers 1883-2016
    • (Glasgow) On the history of the Fourier series .
    • (Glasgow) A proof of the uniform convergence of the Fourier series, with notes on the differentiation of the series .
    • (School House, Cowbridge) Certain expansions of xn in hypergeometric series .
    • (Glasgow) On the summation of a certain series .
    • (Glasgow) An extension of Abel's theorem on the continuity of a power series .
    • (Glasgow) Note on the use of Fourier's series in the problem of the transverse vibrations of strings .
    • (School House, Cowbridge) Generalised forms of the series of Bessel and Legendre .
    • (George Heriot's School, Edinburgh) A note on a theorem in double series .
    • (Victoria College, Wellington, New Zealand) On the fractional infinite series for cosec x, sec x, cot x and tan x .
    • (School House, Cowbridge) Certain series of generalised Bessel-functions .
    • (School House, Cowbridge) Certain series of basic Bessel co-efficients .
    • (Girls' High School, Glasgow) Theorems on the summation of series .
    • (Bhaonagar, Bombay) On series for calculating Euler's constant and the constant in Stirling's theorem .
    • An infinite series of triangles and conics with a common pole and polar .
    • (Victoria College, Wellington, New Zealand) On summable series and Bernoullian numbers .
    • (University College, London) The convergence of the series in Mathieu's functions, {Communicated by Professor Whittaker} .
    • New formulae about the theory of the series of alternate sign, {Communicated by A S Morrison} .
    • (Edinburgh) Some new expansions in series of polynomials: A formula for the solution of algebraic or transcendental equations .
    • On the arrangement of signs of the terms in a certain double series given by Arndt, {Communicated by W S Catto} .
    • The relation between series and elliptic function solutions .
    • (Scottish Widows' Fund in Edinburgh) A new form for the sum of a trigonometric series, {Communicated by George D C Stokes} .
    • (Edinburgh) The vibrations of a particle about a position of equilibrium, Part 3, The significance of the divergence of the series solution .
    • (Edinburgh) The vibrations of a particle about a position of equilibrium, Part 4, The convergence of the trigonometric series of dynamics .
    • (Glasgow) Expansions in series of spherical harmonics .
    • (Toronto) On the rearrangement of terms in a complex series, {Communicated by Bevan B Baker} .
    • (Edinburgh) Series formulae for the roots of equations .
    • (Edinburgh) A series for symmetric functions of roots of equations .
    • (Edinburgh) Note on Binet's inverse factorial series for μ(x) .
    • (Aberdeen) Note on a formula connected with Fourier series .
    • (Edinburgh) On the relation between inverse factorial series and binomial coefficient series .
    • An application of Abel's lemma to double series .
    • (Edinburgh) On the Poisson sum of a Fourier series .
    • (Edinburgh) The absolute summability (A) of Fourier series .
    • An apparatus for determining coefficients in power series, {Communicated by William Peddie and read by H S Ruse} .
    • (Hebrew University of Jerusalem) On the absolute summability (A) of infinite series .
    • Self-reciprocal functions in the form of series .
    • (Dundee) Some notes on divergent series .
    • (Glasgow) On the approach of a series to its Cesaro limit; .
    • Some sufficient conditions for the absolute Cesaro summability of series; .
    • (Liverpool) The representation of functions by series .
    • (University College, Dundee) Invariant matrices and the Gordan-Capelli series; .
    • (Birkbeck College, London) Series to series transformations and analytic continuation by matrix methods .
    • (Aberdeen) Integral functions with gap power series .
    • (St Andrews) On the abscissae of the summability of a Dirichlet series; .
    • (Liverpool) The summability of a power series on its circle of convergence .
    • (Aberdeen) Spherical summation of multiple Fourier series; .
    • (Oxford) On counting subgroups of finite index, and some rational Poincare series .
    • (Bristol) Divergent series: reaping Dingle's harvest .
    • Series, C.M.
    • (Nottingham) Divergent series and the role of exponentially small terms in differential equations .
    • (Konstanz) Automorphism groups of fields of generalised series .

  2. 1925-26 Jan meeting
    • I, Series 2] .
    • I, Series 2] {Title in minutes: "On the correlation of aggregates"} .
    • I, Series 2] {Read by title} .
    • Copson, Edward Thomas: "Note on the integral equations for the Lame functions", [Proceedings, Vol I, Series 2] {Read by title} .
    • I, Series 2] {Read by title} .
    • I, Series 2] {Title in minutes: "On the maxima and minima of functions of two variables".

  3. The EMS: the first 100 years (1883-1983) Part 2
    • By the middle 1920s it was felt that the size of page used was too small and a second series of the Proceedings was begun in 1927.
    • Each volume of the new series generally consisted of four parts, published over a period of two years.
    • Although it is not mentioned in the minutes, an important reason for embarking on a second series must have been the wish to raise the level of the papers published, which, although it had risen considerably over the years, remained somewhat uneven.
    • It was under Turnbull's editorship that the second series of the Proceedings was begun in 1927 and it was largely due to his efforts that the second series became a mathematical journal of repute.

  4. EMS 125th Anniversary booklet
    • He worked on the convergence of series.
    • Peter Comrie graduated from St Andrews and after a series of teaching posts became Rector of Leith Academy.
    • He worked on the convergence of series.
    • He graduated from University College Bangor and occupied a series of posts in Glasgow University, finishing as Professor of Applied Physics.
    • He returned to Glasgow to a series of posts culminating in the professorship.

  5. EMS 125th Anniversary booklet
    • Peter Comrie graduated from St Andrews and after a series of teaching posts became Rector of Leith Academy.
    • He graduated from University College Bangor and occupied a series of posts in Glasgow University, finishing as Professor of Applied Physics.
    • He returned to Glasgow to a series of posts culminating in the professorship.
    • He worked on the convergence of series.
    • He worked on the convergence of series.

  6. 1931-32 Jun meeting
    • III, Series 2] {Read by title} .
    • III, Series 2] {Read by title} .
    • III, Series 2] {Read by title} .
    • III, Series 2] {Read by title} .
    • Phillips, E G: "Self-reciprocal functions in the form of series", [Not printed in an EMS publication] {Read by title} .

  7. 1925-26 Jun meeting
    • I, Series 2] .
    • I, Series 2] .
    • I, Series 2.
    • I, Series 2] {Read by title.
    • I, Series 2] {Read by title} .

  8. 1931-32 May meeting
    • Mursi-Ahmed, M: "On the composition of simultaneous differential systems of the first order", [Proceedings, Vol.III, Series 2] {Read by title} .
    • III, Series 2] {Read by title} .
    • Fekete, M: "On the absolute summability (A) of infinite series", [Proceedings, Vol.
    • III, Series 2] {Read by title} .
    • III, Series 2] {Read by title} .

  9. 1928-29 Jun meeting
    • II, Series 2] {Title in minutes: "Systems of linear complexes associated with a quadric surface"} .
    • I, Series 2] {Title in minutes: "On Young's condition for the Stieltjes integral"} .
    • Whittaker, John Macnaughten: "The absolute summability of Fourier series", [Proceedings, Vol.
    • II, Series 2] {Title in minutes: "On the Poisson sum of a Fourier series"} .

  10. 1928-29 Mar meeting
    • II, Series 2] .
    • I, Series 2] .
    • Walsh, C E: "An application of Abel's lemma to double series", [Proceedings, Vol.
    • I, Series 2] .

  11. 1930-31 Nov meeting
    • II, Series 2] {Read by title} .
    • II, Series 2] {Read by title} .
    • II, Series 2] {Read by title} .
    • II, Series 2] {Read by title} .

  12. EMS Proceedings papers
    • On the history of the Fourier series .
    • A proof of the uniform convergence of the Fourier series, with notes on the differentiation of the series .
    • An extension of Abel's theorem on the continuity of a power series .

  13. 1925-26 Mar meeting
    • I, Series 2] {Read by title} .
    • I, Series 2, with subtitle "Read 5th February 1926".
    • I, Series 2] {Read by title} .
    • I., Series 2 (with subtitle "Read 4th February 1927")] .

  14. 1931-32 Feb meeting
    • Thomson, R W M: "An apparatus for determining coefficients in power series", [Mathematical Notes no 27, June 1932] {Communicated by William Peddie and read by H S Ruse} .
    • III, Series 2] .
    • III, Series 2] {Read by title} .
    • III, Series 2] {Read by title} .

  15. 1929-30 Nov meeting
    • II, Series 2] {Communicated by A C Aitken} .
    • II, Series 2] .
    • II, Series 2] {Read by title} .
    • Great interest was also shown in Professor Turnbull's successful attempt to expand a scalar function f(X+A) of two matrices A, X as a Taylor series.

  16. 1930-31 Jan meeting
    • II, Series 2] .
    • II, Series 2] {Read by title} .
    • II, Series 2] {Read by title} .
    • II, Series 2] {Read by title} .

  17. 1931-32 Nov meeting
    • III, Series 2] {Read by title} .
    • III, Series 2] {Read by title} .
    • III, Series 2] {Read by title} .

  18. 1931-32 Dec meeting
    • III, Series 2] {Read by title} .
    • III, Series 2] {Read by title} .
    • III, Series 2] {Read by title} .

  19. 1929-30 Jan meeting
    • II, Series 2] {Communicated by H W Turnbull} .
    • II, Series 2] {Communicated by H W Turnbull} .
    • II, Series 2] {Communicated by E T Copson} .

  20. 1929-30 Feb meeting
    • II, Series 2] {Communicated by G Timms} .
    • II, Series 2] .
    • II, Series 2] {Read by title} .

  21. EMS Ince
    • Solutions in the form of power series were discussed: sometimes very slow convergence of these series had troubled the lecturer in his work of numerical tabulation of Lame functions; but he had very recently found out that in such cases a vast improvement was obtained by using Fourier series solutions of Lame's equation in a trigonometric form.

  22. 1927-28 Nov meeting
    • I, Series 2] .
    • I, Series 2] {Read by title} .
    • I, Series 2] {Read by title} .

  23. 1929-30 May meeting
    • Prasad, B N: "The absolute summability (A) of Fourier series", [Proceedings, Vol.
    • II, Series 2] .
    • II, Series 2] {Communicated by E T Copson} .

  24. 1893-94 Feb meeting
    • Gibson,George Alexander: "A proof of the uniform convergence of the Fourier series, with notes on the differentiation of the series" .

  25. 1927-28 Jun meeting
    • I, Series 2] {Title in minutes: "Note on Numerical Integration"} .
    • I, Series 2] {Read by title} .

  26. 1928-29 Feb meeting
    • I, Series 2] {Title in minutes: "On some algebraical points connected with the four-nodal cubic surface"} .
    • I, Series 2] .

  27. 1922-23 Nov meeting
    • Baker, Bevan B: "The vibrations of a particle about a position of equilibrium, Part 3, The significance of the divergence of the series solution", {Part 3 of 4 (Part 1: March, session 39, part 2: January session 40, part 4: this meeting)} .
    • Baker, Bevan B: "The vibrations of a particle about a position of equilibrium, Part 4, The convergence of the trigonometric series of dynamics", {Part 4 of 4 (Part 1: March session 39, part 2: January session 40, Part 3: this meeting)} .

  28. 1925-26 Feb meeting
    • I, Series 2] {Read by title} .
    • I, Series 2] {Read by title} .

  29. 1924-25 Jun meeting
    • Ferrar, W L: "On the relation between inverse factorial series and binomial coefficient series", {Communicated by title} .

  30. 1930-31 May meeting
    • III, Series 2] .
    • II, Series 2] {Read by title} .

  31. EMS Proceedings papers
    • On the relation between inverse factorial series and binomial coefficient series .

  32. EMS 1988 Colloquium
    • The following gave a series of lectures: .
    • Principal speakers were Professor S Smale (Berkeley), Professor S J Patterson (Gottingen) and Dr J B MacLeod (Oxford); Professor A Baernstein (Washington University, St Louis) presented the second series of Copson Memorial Lectures.

  33. EMS 1972 Colloquium
    • The October 1971 minutes say that Professors Halmos, Harary and Vajda had accepted invitations to give lecture courses, and Professors Cohn and Eells had agreed to lead series of seminars.
    • Harary gave a series of inspiring lectures on graph theory which had a major influence of me and soon after I introduced graph theory into my undergraduate teaching.

  34. EMS 1976 Colloquium
    • The following gave a series of lectures: .
    • Firstly I remember his inspiring series of lectures on Lie Algebras in which he also spoke about locally compact groups and the solution to Hilbert's fifth problem.

  35. LMS Newsletter article
    • Indeed the authors of this note acknowledge the generous support and encouragement of the Edinburgh Mathematical Society for the Groups St Andrews series of conferences.
    • For many years from 1913 to 2003 there was a series of summer colloquia, the St Andrews Colloquia, which were held in St Andrews from 1926 at roughly four-yearly intervals.

  36. The EMS: the first hundred years
    • This suggestion, however, was not taken up and those present at the meeting turned their attention to a series of eight motions setting up the Edinburgh Mathematical Society, its aim being the mutual improvement of its members in the mathematical sciences, pure and applied.
    • Over the years the percentage of university members has, of course, increased; however, as late as 1926, just before the second series of the Proceedings was instituted, their percentage had only risen from 26% to 36%.

  37. Edinburgh Mathematical Notes
    • We should remark that The Second Series of the Proceedings started in 1927 with a larger page size.
    • The Mathematical Notes, which had a page size identical to the First Series of the Proceedings, was changed to keep a matching page size.

  38. EMS 1980 Colloquium
    • The following gave a series of lectures: .
    • Nash-Williams' lecture series formed the basis of the two-part survey A glance at graph theory which he published in the Bulletin of the London Mathematical Society.

  39. EMS 1955 Colloquium
    • The meeting opened with an address by Professor A C Aitken, Edinburgh, on "Numerical Analysis and Algebra." A series of five lectures is being given during the next week by Mr M F Atiyah, Cambridge, on "Topological Methods in Algebraic Geometry.
    • Professor P Hall, Cambridge, on "Symmetric Functions in the Theory of Groups." Professor W W Rogosinski, Newcastle, on "The Hahn-Banach Theorem and its Applications" and Professor J L Synge, on "Hamilton's Method in the Relativistic Theory of Waves, Particles and Photons." A series of three lectures is to be given by Professor E Stiefel, Zurich, on "Strategy in Relaxation and in the Numerical Computation of Eigen-Values," Single lectures are to be delivered by Professor A C Zanen, Delft, and Professor B van der Pol, Geneva.

  40. 1930-31 Dec meeting
    • II, Series 2] {Read by title} .
    • II, Series 2] {Read by title} .

  41. 1931-32 Jan meeting
    • III, Series 2] {Read by title} .

  42. 1921-22 Mar meeting
    • Lidstone, G J: "A new form for the sum of a trigonometric series", {Communicated by George D C Stokes} .

  43. 1901-02 Jan meeting
    • Carslaw, Horatio Scott: "Note on the use of Fourier's series in the problem of the transverse vibrations of strings" .

  44. 1894-95 Jun meeting
    • Chrystal, George: "A summary of the theory of the refraction of thin approximately axial pencils through a series of media bounded by coaxial spherical surfaces, with application to a photographic triplet, &c.", {Printed in Proceedings for Session 14.

  45. 1893-94 Dec meeting
    • XXXVII (5th series) pp.

  46. 1896-97 April meeting
    • Jackson, Frank Hilton: "Certain expansions of xn in hypergeometric series" .

  47. EMS 1914 Colloquium
    • In the second lecture of the series, Mr Cunningham reviewed the results of recent experiments which had for their aim the discovery of the detailed structure of matter.

  48. 1902-03 Mar meeting
    • Jackson, Frank Hilton: "Generalised forms of the series of Bessel and Legendre" .

  49. 1900-01 May meeting
    • Carslaw, Horatio Scott: "On the summation of a certain series", {Not mentioned in the Proceedings} .

  50. 1924-25 Nov meeting
    • Aitken, Alexander Craig: "A series for symmetric functions of roots of equations", [Title] .

  51. 1921-22 Jan meeting
    • The relation between series and elliptic function solutions" .

  52. 1927-28 Feb meeting
    • I, Series 2] {Communicated by Professor Turnbull} .

  53. 1887-88 Feb meeting
    • "He intimidated that the committee had met and drawn up a series of questions which it was proposed to submit for the opinion of members before a definite report should be drawn up.

  54. 1889-90 Dec meeting
    • Professor Steggall exhibited an exhaustive series of diagrams of the curves made by the tracing point.

  55. 1925-26 Nov meeting
    • I, Series 2] {Communicated by Herbert W Turnbull.

  56. 1903-04 Jan meeting
    • Picken, D K: "On the fractional infinite series for cosec x, sec x, cot x and tan x" .

  57. 1891-92 Feb meeting
    • John E A Steggall exhibited a series of magnetic curves traced by his method.

  58. 1900-01 Jun meeting
    • Gibson, George Alexander: "An extension of Abel's theorem on the continuity of a power series" .

  59. 1911-12 Jan meeting
    • Donaldson, J A: "An infinite series of triangles and conics with a common pole and polar" .

  60. 1909-10 Jan meeting
    • Sanjana, K J: "On series for calculating Euler's constant and the constant in Stirling's theorem" .

  61. 1930-31 Feb meeting
    • II, Series 2] {Read by title} .

  62. EMS 1996 Colloquium
    • The following each gave a series of five lectures: .

  63. 1923-24 Jun meeting
    • Aitken, Alexander Craig: "Series formulae for the roots of equations", [Title] .

  64. 1884-85 Jun meeting
    • Tait, Peter Guthrie: "Summation of certain series", [Abstract] {"On the Detection of amphicheiral knots, with special reference to the mathematical processes involved"} .

  65. 1883-84 Dec meeting
    • XIII (new series) pp.

  66. 1923-24 Mar meeting
    • Beatty, S: "On the rearrangement of terms in a complex series", [Proceedings, session 43] {Communicated by Bevan B Baker.} .

  67. 1927-28 Dec meeting
    • I, Series 2] {Title in minutes: "Cardinal function interpolation"} .

  68. EMS 1930 Colloquium
    • It has been very successful, over a hundred members attending the series of meetings in the University Residence Hall.

  69. 1917-18 Jun meeting
    • Whittaker, Edmund Taylor: "A formula for the solution of algebraic or transcendental equations", {Title in minutes: "Some new expansions in series of polynomials"} .

  70. 1929-30 Jun meeting
    • II, Series 2] .

  71. 1919-20 Mar meeting
    • Mitchell, J: "On the arrangement of signs of the terms in a certain double series given by Arndt", [Title] {Communicated by W S Catto} .

  72. 1914-15 Mar meeting
    • Tavani, F: "New formulae about the theory of the series of alternate sign", [Title] {Communicated by A S Morrison} .

  73. EMS Proceedings papers
    • A new form for the sum of a trigonometric series .

  74. EMS Proceedings papers
    • Expansions in series of spherical harmonics .

  75. Solution5.1.html
    • (The coefficient C±(n) in fact depends on the position of the point (wn, wn+1) on the above-mentioned curve (XY - 1)(5 - X - Y) = 6.) Expanding the function C±(n) numerically into a Fourier series, we discover that it is a Jacobi theta function, and since theta functions (or quotients of them) are elliptic functions, this leads quickly to elliptic curves and to the above curve E.

  76. EMS Proceedings papers
    • On the fractional infinite series for cosec x, sec x, cot x and tan x .

  77. EMS Proceedings papers
    • Note on geometric series .

  78. EMS Proceedings papers
    • The convergence of the series in Mathieu's functions .

  79. EMS Proceedings papers
    • Note on the use of Fourier's series in the problem of the transverse vibrations of strings .

  80. 1892-93 May meeting
    • Gibson, George Alexander: "On the history of the Fourier series" .

  81. EMS Proceedings papers
    • Note on a Theorem in Double Series .

  82. EMS Proceedings papers
    • A summary of the theory of the refraction of thin approximately axial pencils through a series of media bounded by coaxial spherical surfaces, with application to a photographic triplet, &c.

  83. EMS Proceedings papers
    • Summation of certain series [Abstract] .

  84. EMS Proceedings papers
    • Note on Binet's Inverse Factorial Series for μ(x) .

  85. EMS Proceedings papers
    • On the rearrangement of terms in a complex series, {Read in session 42} .

  86. EMS 1934 Colloquium
    • Of equal interest were the two lectures given by Professor B M Wilson (Dundee) on the notebooks of Ramanujan and the lecture by Professor J M Whittaker (Liverpool) on the representation of integral functions by series of polynomials.

  87. EMS 1938 Colloquium
    • Professor H W Turnbull is delivering a series of lectures on the work of James Gregory.

  88. EMS 1984 Colloquium
    • The following gave a series of seven lectures: .

  89. EMS 1913 Colloquium
    • After we had been taught that velocities did not compound according to the parallelogram law, it was a positive delight to find that the Fourier series remained ordinarily additive; and with this in possession we had no great difficulty in apprehending the possibility of a space devoid of parallel lines.

  90. 1924-25 Mar meeting
    • Copson, Edward Thomas: "Note on Binet's inverse factorial series for mu (x)" .

  91. EMS Mobius
    • Other papers of the programme were: Deformable quadrics and their circular sections, by Professor H W Turnbull, F.R.S., illustrated with a series of folding cardboard models; A note on the history of the fundamental theorem of the integral calculus, in which Professor Turnbull showed that an important theorem, commonly ascribed to Barrow in 1669, was actually contained in Gregory's Pars Universalis (1668), and should therefore be credited to the latter mathematician; On the intersection of certain quadrics, by L M Brown; and On the projective geometry of paths, by Dr J Haantjes.

  92. EMS
    • A second colloquium was held in Edinburgh in the following year, prior to the outbreak of World War I, but the series had to be discontinued for the duration of the War.

  93. 1893-94 Jun meeting
    • XXXVIII (5th series) pp.

  94. EMS 1926 Colloquium
    • However, after the war ended and life began to return to normal, the Society decided to start a new series of Colloquia to be held in St Andrews.

  95. 1930-31 Jun meeting
    • III, Series 2] {Read by title} .

  96. 1903-04 May meeting
    • Jackson, Frank Hilton: "Note on a theorem of Lommel", {Title in minutes: "Certain series of generalised Bessel-functions"} .

  97. 1905-06 Nov meeting
    • Jackson, Frank Hilton: "Certain series of basic Bessel co-efficients", [Title] .

  98. 1928-29 May meeting
    • I, Series 2] .

  99. 1913-14 Mar meeting
    • Picken, D K: "On summable series and Bernoullian numbers", [Title] .

  100. 1896-97 Apr meeting
    • Jackson, Frank Hilton: "Certain expansions of xn in hypergeometric series" .

  101. 1923-24 Dec meeting
    • MacRobert, Thomas M: "Expansions in series of spherical harmonics" .

  102. 1908-09 Mar meeting
    • Holm, Alexander: "Theorems on the summation of series" .

  103. 1883-84 Nov meeting
    • Tait, Peter Guthrie: "Listing's Topologie", [Title] [This address will be found in the Philosophical Magazine, Vol.XVII (fifth series) pp.

  104. 1924-25 May meeting
    • Macdonald, H M: "Note on a formula connected with Fourier series", [Proceedings, session 44 (25/26)] .

  105. 1925-26 Dec meeting
    • II, Series 2] {Read by title} .

  106. 1883-84 Jan meeting
    • Tait, Peter Guthrie: "Theorem relating to the sum of selected binomial-theorem cofficients", [Title] [This theorem will be found in the Messenger of Mathematics (new series) Vol.

  107. 1903-04 Nov meeting
    • Thomson, William L: "A note on a theorem in double series", [Title] .

  108. 1914-15 Nov meeting
    • Watson, G N: "The convergence of the series in Mathieu's functions", {Communicated by Professor Whittaker} .

  109. EMS 1992 Colloquium
    • The following gave a series of lectures: .


BMC Archive

  1. Minutes for 2005
    • Caroline Series (Warwick) - until 31 May 2006 .
    • Helen Robinson (Coventry), Caroline Series (Warwick), Rachel Camina (Cambridge) .
    • The BMC representatives at this meeting will be Edmund Robertson, Garth Dales, and Caroline Series and perhaps one of the Liverpool organizers.
    • Caroline Series (Warwick) - until 31 May 2006 cms@maths.warwick.ac.uk .

  2. Minutes for 1999
    • Series (Warwick), representative from Warwick for 54th Colloquium.
    • Series suggested that in the application form, people be encouraged to offer a talk in a Splinter Group, and possibly propose an additional area for a Splinter Group.
    • Series suggested that titles and abstracts of all talks be collected where possible, and put on the web.
    • Series remarked that one-day meetings should reinforce the BMC, but longer satellite events might compete, as people might not want to be away for long so close to Easter.

  3. Minutes for 2006
    • [Caroline Series, Iain Gordon] .
    • [Need new LMS representative to replace Caroline Series who is a member until 31 May 2006.
    • Apologies for absence of Iain Gordon (Glasgow), Michael White (Newcastle), Caroline Series (Warwick) were accepted.

  4. Minutes for January 2004
    • C Series (Warwick) - until 31 May 2006 .
    • (I) Welcome - especially Caroline Series (LMS) and Edmund Robertson (EMS); Nicholas Young and Zina Lykova (Newcastle) .
    • Garth Dales (Chairman), Rob Curtis (Birmingham), David Armitage and Martin Mathieu (Belfast), Peter Giblin and Hugh Morton (Liverpool), Zinaida Lykova and Nicholas Young (Newcastle), John Greenlees, Helen Robinson and Caroline Series (LMS), Graham Jameson and Edmund Robertson (EdMS), William Crawley-Boevey (elected at AGM) .

  5. BMC 2017
    • Bruinier, JGenerating series of special divisors on arithmetic ball quotients .
    • Cornelissen, GReconstructing global fields from L-series .

  6. Scientific Committee minutes 2004
    • Garth Dales (Chair); Norman Biggs, Helen Robinson and Caroline Series (LMS); Edmund Robertson and Rob Archbold (EdMS); Hugh Morton and Peter Giblin (Liverpool), joint secretaries for the meeting; Martin Mathieu (Belfast); Zinaida Lykova and Nicholas Young (Newcastle); Sandra Pott (elected at 2004 AGM); Francis Clarke and Niels Jacob (Swansea, by invitation from Item 3).
    • in the series which started at Manchester as the British Theoretical Mechanics Colloquium.

  7. Scientific Committee minutes 2004
    • Garth Dales (Chair); Norman Biggs, Helen Robinson and Caroline Series (LMS); Edmund Robertson and Rob Archbold (EdMS); Hugh Morton and Peter Giblin (Liverpool), joint secretaries for the meeting; Martin Mathieu (Belfast); Zinaida Lykova and Nicholas Young (Newcastle); Sandra Pott (elected at 2004 AGM); Francis Clarke and Niels Jacob (Swansea, by invitation from Item 3).
    • in the series which started at Manchester as the British Theoretical Mechanics Colloquium.

  8. Minutes for 1999
    • Professor C Series issued an invitation for the 54th BMC to be held at Warwick in 2002.
    • C Series (Warwick): .

  9. BMC 1992
    • Hawkes, J Series .
    • Huxley, M N Pixels, area and Fourier series .

  10. Minutes for 2000
    • C Series (Warwick) confirmed that there will be a combined meeting of the BMC and the BAMC in Warwick, 7-12 April 2002 (arrival on the Sunday evening).
    • Series suggested that the Scientific Committee should have a long-term chair, and perhaps meet more than once per year.

  11. Scientific Committee 2004
    • Present: Garth Dales (Chair), David Armitage and Martin Mathieu (Belfast), Peter Giblin and Hugh Morton (Liverpool), Zinaida Lykova and Nicholas Young (Newcastle), John Greenlees and Caroline Series (LMS), Graham Jameson and Edmund Robertson (EdMS), Sandra Pott (elected at AGM).
    • It was agreed to ask Peter Giblin, Hugh Morton, Caroline Series and Garth Dales to represent the BMC Scientific Committee at this meeting.

  12. Scientific Committee 2002
    • The meeting would be in parallel rather than in series (Warwick was in series).

  13. BMC 1978
    • Holland, FReflections on Hilbert's double series theorem .
    • Newns, W FBasic series .

  14. BMC 1994
    • Askey, R The q-series of L J Rogers as seen in 1894 and 1994 .
    • Series, C M Circles in limit sets of Kleinian groups .

  15. Minutes for 2005
    • LMS: Rachel Camina (Cambridge), Garth Dales (Leeds), Caroline Series (Warwick), .
    • Edmund Robertson reported on the meeting with BAMC representatives on 20 September, which Garth Dales, Caroline Series and Edmund Robertson attended.

  16. LMS report
    • I understand my replacement is Caroline Series.
    • The meeting at Liverpool will also be a joint one (with a somewhat different format to the Warwick one, parallel rather than series).

  17. BMC 2012
    • Iyudu, N The Anick conjecture on the minimal Hilbert series of quadratic algebras .

  18. BMC speakers
    • Series, C M : 1981, 1994, 2003, 2007 .

  19. BMC speakers
    • Series, C M : 1981, 1994, 2003, 2007 .

  20. BMC 1968
    • Kahane, J -P Perfect sets and trigonometric series .

  21. Minutes for 2003
    • Caroline Series (Warwick) reported that in Warwick they had argued for lower rates (and obtained them).

  22. Letter 1985
    • This was actually attempted a few years ago, and the files are full of a series of these entertainments at which every little detail was discussed by a committee of 10.

  23. BMC 2007
    • Series, CIndra's pearls: geometry and symmetry .

  24. Meeting 2003
    • The British Mathematical Colloquium (BMC) is the title accorded by the BMC Annual General Meeting to a series of independent conferences organised for the community of pure mathematicians based in the United Kingdom.

  25. BMC 2001
    • Gekeler, E -UObservations about Eisenstein series for the modular group .

  26. Minutes for 1994
    • To provide an element of continuity, and because we believe that the Scientific Committee will need to look ahead and consider the series of British Mathematical Colloquia, rather than just one meeting at a time, we propose that the normal period of service for the six nominated members should be three years (with staggered retirement dates).

  27. Minutes for 2000
    • C M Series (cms@maths.warwick.ac.uk), 54th BMC, Warwick, 2002 .

  28. BMC-BAMC meeting 2005
    • Caroline Series (Warwick) .

  29. Minutes for 1999
    • Garth Dales (BMC), Dugald McPherson (BMC), Bob Odoni (BMC), Chris Budd (BAMC), David Broomhead (BAMC), David Needham (BAMC), Alan Newell (Warwick), Andrew Stuart (Warwick), Caroline Series (Warwick), Colin Rourke (Warwick), Robert MacKay (Warwick) .

  30. BMC 2003
    • Series, C M Recent developments in hyperbolic geometry .

  31. Minutes for 2004
    • Caroline Series reported.

  32. Minutes for 1999
    • Prof C M Series (Warwick), representative from Warwick for 54th Colloquium .

  33. BMC 1973
    • Lusztig, GThe discrete series representations of the general linear groups over a finite field .

  34. BMC 1953
    • Pitt, H RConvergence of Fourier series .

  35. BMC Morning speakers
    • Series, C M : 1981, 1994, 2003 .

  36. BMC 1959
    • Noble, M ESome boundary properties of power series .

  37. Minutes for 2010
    • A novel aspect will be the replacement of special sessions by a series of 6 workshops on different areas of mathematics, each lasting two afternoons with expert organisers and small budgets.

  38. BMC Special Session speakers
    • Series, C M : 2007 .

  39. BMC 1981
    • Series, C M Infinite words and limit sets of Fuchsian groups .

  40. BMC special session speakers
    • Series, C M : 2007 .


Gazetteer of the British Isles

  1. References
    • First Series.
    • Fascination of London Series.
    • Medižval Towns Series, Dent, London, 1910.
    • The Medižval Towns Series, Dent, London, (1907), revised, 1926.
    • The Medižval Towns Series, Dent, London, (1899), revised, (1927), 2nd ptg, 1933.
    • Irish Heritage Series, No.
    • Kensington (The Fascination of London series).
    • College Histories series.
    • A series of some 40 leaflets, each of a few pages, once available from the Library on Richmond Green.
    • The Medižval Towns Series, Dent, London, 1909.
    • Irish Heritage Series, No.
    • (Spine says First Series and a note by a bookdealer on the flyleaf says 2 vol..
    • Lives of Great Men and Women Series IV.

  2. Oxford individuals
    • The Mediaeval Towns Series, Dent, London, (1907), revised, 1926.
    • The Mediaeval Towns Series, Dent, London, (1907), revised, 1926, pp.115-116',9)">Headlam] seems a bit dubious about the site.
    • Irish Heritage Series, No.
    • The Mediaeval Towns Series, Dent, London, (1907), revised, 1926.
    • The Mediaeval Towns Series, Dent, London, (1907), revised, 1926.
    • He introduced continued fractions in 1657-1658 and may have been the first to give a series for log x.
    • The Mediaeval Towns Series, Dent, London, (1907), revised, 1926.
    • The Mediaeval Towns Series, Dent, London, (1907), revised, 1926.
    • The Mediaeval Towns Series, Dent, London, (1907), revised, 1926.
    • The Mediaeval Towns Series, Dent, London, (1907), revised, 1926.
    • The Mediaeval Towns Series, Dent, London, (1907), revised, 1926.
    • The Mediaeval Towns Series, Dent, London, (1907), revised, 1926.

  3. Oxford Institutions and Colleges
    • The Mediaeval Towns Series, Dent, London, (1907), revised, 1926.
    • The Mediaeval Towns Series, Dent, London, (1907), revised, 1926.
    • The Mediaeval Towns Series, Dent, London, (1907), revised, 1926.
    • The Mediaeval Towns Series, Dent, London, (1907), revised, 1926.
    • The Mediaeval Towns Series, Dent, London, (1907), revised, 1926.
    • The Mediaeval Towns Series, Dent, London, (1907), revised, 1926.
    • Indeed, Sir Ray Lankester [Science From an Easy Chair, Second Series; (1913), 3rd ed., Eyre Methuen, 1920, p.313',36)">Lankester] says this was the first usage of the word 'museum' in its current sense.
    • The Mediaeval Towns Series, Dent, London, (1907), revised, 1926.

  4. Slough, Berkshire
    • First Series.
    • First Series.
    • First Series.

  5. London individuals S-Z
    • Fascination of London Series.
    • Fascination of London Series.
    • Fascination of London Series.

  6. Cambridge Individuals
    • In 1692-1693, he delivered the first series of eight Boyle Lectures, established by the will of Robert Boyle for proving the Christian Religion.
    • In Oct, he began the long series of stays in other nursing homes--his illness has generally been thought to be tuberculosis and/or stomach ulcer, but a recent theory is that it was amoebic dysentery--a disease which could easily have been cured even then.

  7. London individuals A-C
    • Irish Heritage Series, No.
    • He also found a (the first?) series for the logarithm and proposed a new musical scale with 17 notes.

  8. London individuals H-M
    • His paper "An improved solution of a problem in physical astronomy; by which swiftly converging series are obtained which are useful in computing the perturbations of the motions of the Earth, Mars and Venus ..
    • Thomas Robert Malthus (1766-1834) is known to have lived in a series of London garrets in the late 18C and early 19C.

  9. London Schools
    • College Histories series.

  10. Benvie, nr Dundee
    • William originated the ideas of time series charts and pie charts.

  11. London individuals N-R
    • 29',11)">Craig]), later called Orbell's Buildings, now the site of Newton Court, near the corner of Duke's Lane and Pitt Street, off Kensington Church Street [Kensington (The Fascination of London series).

  12. Woodstock, Oxfordshire
    • The Mediaeval Towns Series, Dent, London, (1907), revised, 1926.

  13. Other Institutions in central London
    • The stone parchment borne by the cherubs beside him once showed a 'Diagram' and a 'converging series', but these vanished when the monument was cleaned c1950.

  14. London Museums
    • The BL's one is safer, and for me is the most effective of this series that I have seen.) .

  15. Other London Institutions outside the centre
    • At the beginning of this room is material about the longitude problem: a copy of the Longitude Act; numerous proposals of methods; a copy of the first version of William Hogarth's The Madhouse from his series The Rake's Progress which shows a lunatic in Bedlam drawing a scheme proposed by William Whiston and Humphrey Ditton to determine longitude by firing bombs into the air; the first volumes of the Nautical Almanac; Huygens' attempt to make a sea-going clock; etc.

  16. Londonderry, Co. Londonderry
    • Irish Heritage Series, No.

  17. Manchester
    • In 1972-1974, they develop MU5 which ICL develops into their 2900 series.

  18. Mathematical Architecture and Mathematical Tiles
    • 68a Kelmscott Road, Wandsworth, SW11, is a modern building by Simon Humphreys, 1998, whose volume is defined by the Fibonacci series.

  19. Thomastown, Co. Kilkenny
    • Irish Heritage Series, No.


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