# Search Results for simple

## Biographies

1. John Thompson (1932-)
• The reason was that suddenly progress began to be made on one of the main problems of finite group theory, namely the classification of finite simple groups.
• Every finite group can be viewed as built from a finite collection of finite simple groups.
• The finite simple groups are therefore the building blocks from which finite groups are built.
• To classify finite groups therefore reduces to two problems, namely the classification of finite simple groups and the solution of the extension problem, that is the problem of how to fit the building blocks together.
• Claude Chevalley showed in 1955 that the Lie groups have finite analogues which are finite simple groups.
• M Suzuki in 1960 discovered new infinite families of finite simple groups.
• Thompson, working with Walter Feit, proved in 1963 that all nonabelian finite simple groups were of even order.
• This result stunned the world of mathematics but it also led mathematicians to believe that a classification of finite simple groups might prove possible.
• Another major early step by Thompson towards the classification of finite simple groups was his classification of those finite simple groups in which every soluble subgroup has a soluble normaliser.
• Here, the authors proved a famous conjecture, to the effect that all non-cyclic finite simple groups have even order.
• In it he determined the minimal simple finite groups, this is to say, the simple groups whose proper subgroups are solvable.
• These results are the first substantial results achieved concerning simple groups.
• The nonabelian finite simple groups fall into a small number of infinite series and 26 sporadic groups.
• I like to say that I would like to see the solution of the problem of the finite simple groups and the part I expect Thompson's work to play in it.
• Thompson revolutionised the theory of finite groups by proving extraordinarily deep theorems that laid the foundation for the complete classification of finite simple groups, one of the greatest achievements of twentieth century mathematics.
• Simple groups are atoms from which all finite groups are built.
• In a major breakthrough, Feit and Thompson proved that every non-elementary simple group has an even number of elements.
• Later Thompson extended this result to establish a classification of an important kind of finite simple group called an N-group.
• Its almost incredible conclusion is that all finite simple groups belong to certain standard families, except for 26 sporadic groups.
• AMS (Classification of finite simple groups) .

2. 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).
• As usual, we denote by LF(2,p) the simple group of order p(p-1)(p+1)/2, (p > 3 a prime number).
• If all proper subgroups of a non-cyclic simple group G are of one of these four types, then G is isomorphic to a group LF(2,p).
• Then in 1960 he discovered a new class of finite simple groups, something which stunned the world of mathematics.
• A simple group is one which has no normal subgroups other than the identity subgroup and the group itself.
• Simple groups are the building blocks from which any group is made up.
• Cyclic groups of prime order are simple, but the interesting problem with simple groups is investigating non-abelian finite simple groups.
• Burnside had conjectured that there did not exist non-abelian finite simple groups of odd order.
• Up to the early 1960s, really nothing of real interest was known about general simple groups of finite order.
• In his paper The nonexistence of a certain type of simple groups of odd order he showed exactly what the title indicates.
• For those who know some group theory, we note that Suzuki proved that a finite group of odd order cannot be simple if the centraliser of every non-identity element is abelian.
• Suzuki's discovery of a new class of finite simple groups in 1960 shook mathematics.
• No new finite simple groups had been discovered since Emile Mathieu discovered five such groups in 1860 and 1873.
• In 1967 Suzuki discovered another new finite simple group.
• His work was a major factor in inspiring the remarkable combined effort by a large number of outstanding mathematicians on the classification of finite simple groups which many regard as the most important mathematical achievement of the 20th century.
• We believe that his work in the 1950's ignited work on the classification of finite simple groups, and in the 1960's and 70's he led its development.
• for his achievements in the domain of group theory, above all in recognition of his path-breaking works on the classification of finite simple groups as well as for his fundamental work on lattices of subgroups and his contributions to the theory of permutation groups.
• He continued to work on this final paper during his final days in Japan and the paper On the prime graph of a finite simple group - an application of the method of Feit-Thompson-Bender-Glauberman was published in [',' E Bannai, H Suzuki, H Yamaki and T Yoshida (eds.), Groups and combinatorics - in memory of Michio Suzuki (Mathematical Society of Japan, Tokyo, 2001).','1].

3. Rimhak Ree (1922-2005)
• In 1957 Ree published his first paper on finite simple groups, the topic for which he is best known today.
• In this paper On some simple groups defined by C Chevalley, Ree identifies many classes of finite simple groups defined by Claude Chevalley in his ground-breaking 1955 paper, with classes of classical simple groups.
• Ree's most famous papers were A family of simple groups associated with the simple Lie algebra of type G2 (1960) and A family of simple groups associated with the simple Lie algebra of type F4 (1961) in which he announced the construction of new families of finite simple groups.
• Joseph Gallian writes [',' J A Gallian, The Search for Finite Simple Groups, Mathematics Magazine 49 (4) (1976), 163-180.','3]:- .
• Michio Suzuki [in 'A new type of simple groups of finite order' (1960)], while in the process of classifying a certain type of doubly transitive permutation groups, discovered another new infinite family.
• This in turn led him to investigate two other similar situations and eventually discover his two families of simple groups.
• The Suzuki and Ree groups together with those of Chevalley and Steinberg are collectively referred to as the simple groups of Lie type.
• In fact these were the final two families of finite simple groups of Lie type to be discovered, bringing the total to sixteen classes.
• His achievements of research in some 29 simple groups including the two found in 1960s made him a gigantic figure in world mathematic circles and remain as important accomplishments in the world history of mathematics.

4. Daniel Gorenstein (1923-1992)
• His involvement in the classification of finite simple groups began in the year 1960-61 when he attended the Group Theory Year at the University of Chicago.
• My first foray into simple group theory dated from the famous 1960-61 group theory year at the University of Chicago, during which Walter Feit and John Thompson settled the solvability of groups of odd order.
• The classification of finite simple groups involved contributions by a host of mathematicians world wide.
• It is for the classification of finite simple groups that his name will always be remembered, certainly the mathematical achievement of the 20th century.
• Conway in England, and Fischer in Germany, each discovering three new sporadic groups, stimulated considerable additional interest, leading to an intensification of the search for further simple groups.
• But it was Aschbacher's entry into the field in the early 1970s that irrevocably altered the simple group landscape.
• Simultaneously with this burgeoning research effort, finite simple group theory was establishing a well-deserved reputation for inaccessibility because of the inordinate lengths of the papers pouring out.
• No mathematical theorem could require the number of pages these fellows were taking! Surely they were missing some geometric interpretation of the simple groups that would lead to a substantially shorter classification proof.
• Gorenstein's books on finite groups and the classification of finite simple groups are Finite groups (1968), Finite simple groups : an introduction to their classification (1982), The local structure of finite groups of characteristic 2 type (jointly written with Richard Lyons) (1983) and The classification of the finite simple groups (jointly written with Richard Lyons and Ronald Solomon) (1994).
• AMS (Classification of finite simple groups) .

5. Richard Brauer (1901-1977)
• This was the time when Brauer made his fundamental contribution to the algebraic theory of simple algebras.
• a theory of central division algebras over a given perfect field, and showed that the isomorphism classes of these algebras can be used to form a commutative group whose properties gave great insight into the structure of simple algebras.
• This he used to obtain results on finite groups, particularly finite simple groups, and the theory of blocks would play a big part in much of Brauer's later work.
• He began to formulate a method to classify all finite simple groups and his first step on this road was a group-theoretical characterisation of the simple groups PSL(2,q) in 1951 (although for a complicated number of reasons explained in [',' J A Green, Richard Dagobert Brauer, Bull.
• The paper was On groups of even order and it provided the key to the major breakthrough by Walter Feit and John Thompson when they proved that every finite simple group has even order.
• Brauer was to spend the rest of his life working on the problem of classifying the finite simple groups.
• (See the biography of Gorenstein for further details on the programme to classify finite simple groups.) Most important was Brauer's vital step in setting the direction for the whole classification programme in the paper On groups of even order where it is shown that there are only finitely many finite simple groups containing an involution whose centraliser is a given finite group.
• Brauer had announced these results and his programme for classifying finite simple groups at the International Congress of Mathematicians in Amsterdam in 1954.
• AMS (Classification of finite simple groups) .

6. John Conway (1937-)
• Knowing that he did not have the group theory skills necessary to prove his conjectures he tried to interest others, see [',' T M Thompson, From error-correcting codes through sphere packings to simple groups (Washington, 1983).
• A detailed description of this discovery is given in [',' T M Thompson, From error-correcting codes through sphere packings to simple groups (Washington, 1983).
• He showed that the symmetry group G of the Leech lattice, when factored by a central subgroup of order 2, was a previously undiscovered finite simple group of order 8,315,553,613,086,720,000.
• It had a great number of very interesting subgroups, including two more previously unknown simple groups, as well as groups having as homomorphic images almost all the finite sporadic simple groups known at that time.
• A finite sporadic simple group is a finite simple group which is not a member of one of the standard infinite families.
• I [EFR] cannot vouch for this change in confidence since I did not hear Conway lecture before his discovery of new simple groups.

7. Charles Sims (1937-2017)
• In 1967 Sims made a major breakthrough when, working with Donald Higman, he discovered the previously unknown sporadic simple group now known as the Higman-Sims group.
• They described the group in the paper A simple group of order 44n352n000 which appeared in Mathematische Zeitschrift in 1968.
• I think that Charles silently followed the good old saying that the proof of a pudding is in its eating and that similarly the proof of an algorithm is in its use and he did indeed prove the existence of three sporadic simple groups by constructing them as permutation groups using his 'Schreier-Sims' method.
• At the conference Computational methods for representations of groups and algebras at the University of Essen, Essen, 1-5 April 1997, Sims and George Havas gave a presentation for this Lyons finite simple sporadic group.
• We give a presentation of the Lyons simple group together with information on a complete computational proof that the presentation is correct.
• This fills a long-standing gap in the literature on the sporadic simple groups.
• In 1977, in collaboration with Jeffrey Leon, Sims proved the existence and uniqueness of a simple group generated by {3, 4}-transpositions.
• His name is attached to two sporadic simple groups; the one I know best, the Higman-Sims group, was found without any recourse to computation at all.

8. Wilhelm Killing (1847-1923)
• They discussed the simple Lie algebras which they knew about and Killing conjectured (wrongly) on 12 April 1886 that the only simple algebras were those related to the special linear group and orthogonal groups.
• By the time he wrote to Engel on 23 May Killing had discovered that his conjecture about simple algebras was wrong, for he had discovered G, and by 18 October he had discovered the complete list of simple algebras.
• The most remarkable part of this work is his discovery of the exceptional simple Lie algebras.
• The exceptional simple Lie algebras are the subject of the final Section 18 of Killing's paper.
• It was Cartan, in his doctoral thesis submitted in 1894, who found concrete representations of all the exceptional simple Lie algebras (although he did not work out all the details in his thesis).

9. Marshall Hall Jr (1910-1990)
• It was at this time that he was able to confirm the existence of a simple group of order 604,800 which had been predicted by Zvoninir Janko.
• He published his findings in the paper A search for simple groups of order less than one million presented to the conference 'Computational Problems in Abstract Algebra' held in Oxford 29 August to 2 September 1967.
• Very recently Z Janko announced that a simple group with certain properties would have order 604,800 and have a specific character table.
• The construction of a simple group of order 604,800 is given for the first time in this paper.
• The construction of the simple group of order 604,800 was carried out in August 1967 at the University of Warwick and at Cambridge University.
• The uniqueness of the simple group of this order was proved in Hall's paper, written jointly with David Wales, The simple group of order 604,800 (1968).

10. Maryam Mirzakhani (1977-2017)
• She also published A small non-4-choosable planar graph in 1996 and A simple proof of a theorem of Schur in 1998.
• Mirzakhani looked at what happens to the "prime number theorem for geodesics" when one considers only the closed geodesics that are simple, meaning that they do not intersect themselves.
• She was awarded her doctorate in 2004 for her 130-page thesis Simple Geodesics on Hyperbolic Surfaces and Volume of the Moduli Space of Curves.
• These results include a recursive formula for Weil-Petersson volumes of moduli spaces of Riemann surfaces, a determination of the asymptotics of the number of simple closed geodesics on a hyperbolic surface in terms of length, and a new proof of Witten's Conjecture (originally established by Kontsevich) establishing the KdV recursion for the intersection numbers on moduli space.
• These papers were: Weil-Petersson volumes and intersection theory on the moduli space of curves (2007); Simple geodesics and Weil-Petersson volumes of moduli spaces of bordered Riemann surfaces (2007); Random hyperbolic surfaces and measured laminations (2007); Growth of the number of simple closed geodesics on hyperbolic surfaces (2008); Ergodic theory of the earthquake flow (2008); and (with Elon Lindenstrauss) Ergodic theory of the space of measured laminations (2008).

11. Ottó Steinfeld (1924-1990)
• Other topics on which Steinfeld undertook research included the structure of simple artinian rings, for example in Some characterizations of semisimple rings with minimum condition on principal left ideals, and analogues in other algebraic systems.
• His article Uber die Verallgemeinerungen und Analoga der Wedderburn-Artinschen und Noetherschen Struktursatze Ⓣ (1967) discussed generalizations of the Noether and Wedderburn-Artin characterizations of the semi-simple and simple Artinian rings to F-rings, to the MHL-rings, for semi-simple linear compact rings, for semirings, for semi-simple near-rings, and for semi-groups which are unions of completely 0-simple.

12. Aleksei Kostrikin (1929-2000)
• This text, intended for 19- or 20-year old students, includes such topics as: general properties of mappings and of binary relations; some properties of simple groups; theory of representations; elements of the theory of finite fields; fields of algebraic numbers; as well as traditional subjects in a first course in algebra.
• In the 1960's, Kostrikin studied infinite-dimensional Lie algebras of Cartan type for finite characteristic and, with Shafarevich, formulated a conjecture describing all simple Lie p-algebras for characteristic p > 5.
• In the 1990's, Kostrikin discovered, with many young mathematicians, a theory of integral lattices in simple Lie algebras which are invariant under the Killing form.
• These had unexpectedly rich applications to other areas such as representations of certain finite simple groups.
• The book deals with certain "concrete" aspects of the representation theory of finite (almost) simple groups, namely with the realization of certain classes of these groups as automorphism groups of integral lattices and of related algebraic and combinatorial objects (root systems, symplectic spreads).
• The lattices are of a particular kind, embedded into a simple Lie algebra over C and endowed with the Killing form.

13. Edith Hirsch Luchins (1921-2002)
• In 1958 her fifth child was born and in the following year the two papers On radicals and continuity of homomorphisms into Banach algebras and On strictly semi-simple Banach algebras appeared, both in the Pacific Journal of Mathematics.
• A Banach algebra is said to be absolute if every homomorphism of a Banach algebra into it is continuous, and is said to be strictly semi-simple if its two-sided regular maximal right ideals have zero intersection.
• It is proved that an absolute Banach algebra contains no non-zero nilpotent elements, and that a strictly semi-simple Banach algebra is absolute.
• For certain special Banach algebras (including semi-simple annihilator algebras) it is proved that if B contains no non-zero nilpotent elements, then B is strictly semi-simple (and hence absolute).
• If the strict radical is zero, the algebra is called strictly semi-simple (sss).

14. Jacques Tits (1930-)
• This paper is an essay on how the development of group theory led to the discovery of various families of simple groups, and how these in turn led to the theory of buildings.
• Galois first used the term 'group' in the technical sense, and found the first simple groups.
• Killing came to such groups independently, and in 1888 found the classification of the simple Lie groups, using semisimple complex Lie algebras (families A through G).
• (2) Schemas en groupes a fibre generique simple sur les anneaux d'entiers Ⓣ; .
• The theory of buildings is a central unifying principle with an amazing range of applications, for example to the classification of algebraic and Lie groups as well as finite simple groups, to Kac-Moody groups (used by theoretical physicists), to combinatorial geometry (used in computer science), and to the study of rigidity phenomena in negatively curved spaces.

15. François Bruhat (1929-2007)
• This paper considered induced representations which Bruhat then applied in his next paper Representations induites des groupes de Lie semi-simples complexes Ⓣ (1954) to a complex semi-simple Lie group none of whose simple factor groups is one of the exceptional simple Lie groups.
• In the same year his paper Representations induites des groupes de Lie semi-simples reels Ⓣ appeared which dealt with the case of a real semi-simple Lie group.
• Together with Jacques Tits, Bruhat developed the theory of simple algebraic groups over local fields.

16. Michael Stifel (1487-1567)
• These productive years at Holzdorf ended when religious wars broke out in 1546 but these were far from simple Catholic versus Protestant affairs.
• At this time he produced a new edition of Rudolff's Coss (1552-1553) but this was certainly not a simple editing exercise but rather he more than doubled its length by adding much material of his own.
• But if one places this sign before a simple number which has not the root which the sign indicates, then from that simple number arises a surd number.
• In other words √4 is a simple number but √2 is a surd.

17. Robert May (1936-)
• What makes populations stabilise? What makes them fluctuate? Are populations in complex ecosystems more stable than populations in simple ecosystems? In 1973, Robert May addressed these questions in this classic book.
• He published highly significant papers such as Biological populations with nonoverlapping generations: stable points, stable cycles, and chaos (1974), Simple mathematical models with very complicated dynamics (1976), Bifurcations and dynamic complexity in simple ecological models (1976), and Thresholds and breakpoints in ecosystems with a multiplicity of stable states (1977).
• The 'Simple mathematical models' paper has been cited an incredible number of times.
• Such equations, even though simple and deterministic, can exhibit a surprising array of dynamical behaviour, from stable points, to bifurcating hierarchy of stable cycles, to apparently random fluctuations.

18. Alfred Hoblitzelle Clifford (1908-1992)
• as an assistant professor, a paper famous in the semigroup community about union of groups semigroups, that Clifford learned about Rees's Theorem determining the structure of completely 0-simple semigroups, generalizing the Wedderburn theory of rings.
• The impact was profound, first because [Clifford's first paper] was a special case (in fact of the Suschkiewitsch paper), second, its application to paper 'Semigroups admitting relative inverses' in hand, where Clifford proved S is a union of groups if and only if S is a semilattice of completely simple semigroups (to which Rees structure theorem applies), and finally because of its intrinsic beauty and importance.
• Clifford's 1953 paper 'A class of d-simple semigroups' on d-simple semigroups was his first reaction to Green and was really about inverse d-simple (one D class, and remember Green introduced D) semigroups.

19. Akos Seress (1958-2013)
• 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.
• They played an indispensable role in the proof of many deep results, including the construction and study of sporadic finite simple groups.
• This book describes the theory behind permutation group algorithms, up to the most recent developments based on the classification of finite simple groups.
• on Symbolic and Algebraic Manipulation) and was hailed as "a groundbreaking work" that "marks a turning point in Majorana Theory." His most recent work, with Harald Helfgott, under publication in the Annals of Mathematics, gives a long-sought bound on the diameter of the alternating and symmetric groups and represents a tour de force in the study of the geometry of finite simple groups.
• After some discussion, I suggested that Graham Higman's finitely presented simple group should give him the example he was seeking and pointed him towards a reference.

20. Élie Cartan (1869-1951)
• However, although Killing had shown that only certain exceptional simple algebras were possible, he had not proved that in fact these algebras exist.
• This was shown by Cartan in his thesis when he constructed each of the exceptional simple Lie algebras over the complex field.
• His first papers, published in 1893, were two notes stating his results on simple Lie groups.
• After the work of his thesis on finite continuous Lie groups, he later classified the semisimple Lie algebras over the real field and found all the irreducible linear representations of the simple Lie algebras.
• M Cartan points out that, in their most general mathematical form, spinors were discovered by him in 1913 in his work on linear representations of simple groups, and he emphasises their connection ..

21. Alan Day (1941-1990)
• One of the first was in the paper A simple solution to the word problem for lattices (1970) where he gave a simple solution to the word problem in free lattices.
• Alan once told me that he really liked elegant mathematics: simple ideas that give profound insights.
• It is a method which is simultaneously powerful and simple, with subtleties that go beyond the surface.
• His proofs involved a subtle manipulation of terms and, when asked by a colleague about how he found one particularly hard proof, he replied "Simple.

22. Walter Feit (1930-2004)
• Solomon, in [',' R Solomon, A brief history of the classification of the finite simple groups, Bull.
• It defined the monumental scale of the classification project for finite simple groups and threw down a gauntlet to other researchers in the field.
• It resolved a seemingly intractable case of the problem and offered entirely new and powerful ways of thinking about finite simple groups - ways of thinking that proved powerful enough to complete the entire project.
• He addressed the International Congresses of Mathematicians in Nice in 1970 on The Current Situation in the Theory of Finite Simple Groups.

23. Philip Hall (1904-1982)
• Hall offered parts of that book for examination in the Tripos and gave a proof that no group of order pn, n > 1, can be simple.
• In On non-strictly simple groups published in 1963 Hall established the existence of simple groups which were the infinite union of a chain of subgroups, each normal in the next.
• 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.

24. Ivor Etherington (1908-1994)
• Although he was working on general relativity, he published the following two papers in 1932: On errors in determinants and A simple method of finding sums of powers of the natural numbers.
• Combination of distributions by random mating is usually symbolised by the mathematical sign for multiplication; but this sign is not taken literally for the simple reason that the genetical laws connecting the distributions of progenitors and progeny are inconsistent with the laws governing multiplication in ordinary algebra.
• Here I propose to consider the symbolism more from the geneticist's point of view, applying it to some simple population problems, without going into the details of the mathematical background.
• I wish that this thesis may not be judged as a finished achievement in biological investigations but may be judged primarily as a contribution to algebra, suggested by biological problems, and perhaps having possibilities of applications beyond the simple ones so far demonstrated.

25. Graham Higman (1917-2008)
• Higman published further important papers in 1951 when he gave an example of a finitely presented group which is isomorphic to a proper factor of itself, and Higman's famous example of a finitely generated infinite simple group.
• After working on finitely generated nilpotent groups and infinite simple permutation groups, Higman, together with Philip Hall, produced another of his landmark papers in 1956 On the p-length of p-soluble groups and reduction theorems for Burnside's problem.
• Then in 1967 Higman became interested in the sporadic finite simple groups being discovered at this time and played an important role in constructing certain of these groups from a knowledge of their character tables.
• He published papers on the Higman-Sims simple group (named after D G Higman and not Graham Higman) and on Janko's group of order 50232960.

26. Wendelin Werner (1968-)
• In an interview after being awarded the Fields Medal, Werner was asked if he could explain, in simple terms, one of the problems he worked on.
• Understanding the behaviour of certain natural very long random curves in the plane is a seemingly simple question that has turned out to raise deep questions, some of which remain unsolved.
• have made predictions concerning the existence and values of critical exponents for various two-dimensional systems in statistical physics (such as self-avoiding walks, critical percolation, intersections of simple random walk) using considerations related to several branches of mathematics (probability theory, complex variables, representation theory of infinite-dimensional Lie algebras).
• One simple but important example is percolation ..

27. John Leech (1926-1992)
• Leech is, however, best known for the Leech lattice which gives rise to three sporadic simple groups.
• Knowing that he did not have the group theory skills necessary to prove his conjectures he tried to interest others, see [',' T M Thompson, From error-correcting codes through sphere packings to simple groups (Washington, 1983).','1]:- .
• A detailed description of this discovery is given in [',' T M Thompson, From error-correcting codes through sphere packings to simple groups (Washington, 1983).','1].
• Leech died almost exactly one month after Gorenstein who had overseen the classification of finite simple groups.

28. William Burnside (1852-1927)
• However it was in 1893 that he published his first paper on finite simple groups, showing that the alternating group A5 is the only finite simple group whose order is the product of four (not necessarily distinct) primes.
• They are concerned chiefly with the proof of certain tests that may be applied in particular cases to determine whether it is possible for a simple group of a given order to exist.
• For example he proved in a paper published in 1895 that if a group of even order has a cyclic Sylow 2-subgroup then the group cannot be simple.

29. Donald Higman (1928-2006)
• In 1967 Higman made a major breakthrough when, working with Charles Sims, he discovered the previously unknown sporadic simple group now known as the Higman-Sims group.
• At the conference Marshall Hall lectured on the construction of the Hall-Janko sporadic simple group as a rank-3 permutation group on 100 points and this prompted a discussion.
• Higman and Sims described the group in the paper A simple group of order 44,352,000 which appeared in Mathematische Zeitschrift in 1968.
• In June 1966 he was in Japan and, in that month, gave the lecture 'Remarks on Finite Permutation Groups' at the University of Tokyo in which he described the discovery of Janko's simple group and discussed Donald Livingstone's construction of it.

30. Emil Artin (1898-1962)
• He presented new insight into semi-simple algebras over the rationals.
• In 1955 he produced two important papers on finite simple groups, proving that the only coincidences in orders of the known (in 1955) finite simple groups were those given by Dickson in his Linear groups.
• This important piece of work is one of a number of results leading to the intense interest in finite simple groups which eventually led to their classification.

31. Marion Walter (1928-)
• This book has many mirror tricks in it that are simple enough for a 41/2 year old to enjoy.
• This is the most entertaining book ever! Simple yet dynamic.
• "Can you make this shape bigger? Can you make this shape smaller?" Simple and inexpensive.

32. Jakob Steiner (1796-1863)
• There exist a limited number of very simple fundamental relationships that together constitute the schema by means of which the remaining theorems can be developed logically and without difficulty.
• Here the main thing is neither the synthetic nor the analytic method, but the discovery of the mutual dependence of the figures and of the way in which their properties are carried over from the simple to the more complex ones.
• Starting from a few spatial properties Steiner attempted, by means of simple schema, to attain a comprehensive view of the multitude of geometric theorems that had been rent asunder.

33. Doris Schattschneider (1939-)
• for her 45-page thesis Restricted roots of a semi-simple algebraic group.
• A paper giving the main results of her thesis appeared in 1969 with the title On restricted roots of semi-simple algebraic groups.
• The second contribution was his deformation of Polya's simple motifs (bounded by line segments or arcs of circles) into recognizable birds, lizards, fishes, etc.

34. Claude Chevalley (1909-1984)
• Chevalley groups play a central role in the classification of finite simple groups.
• His name is also attached to Chevalley decompositions and to a Chevalley type of semi-simple algebraic group.
• The basic definitions are chosen deftly, and each topic is developed with simple directness.

35. Richard Borcherds (1959-)
• From the automorphism group of this lattice, Conway had discovered three previously unknown finite simple groups in 1968.
• In the classification of finite simple groups, one of the most mysterious objects found was the monster group.
• The concept was used extensively by Frenkel, Lepowski and Meurman in their work on the Monster simple group, and very recently it has been an essential tool in the progress made by E Frenkel on a conjecture of Drinfeld about the Langlands correspondence for representations of loop groups.

36. James McConnell (1915-1999)
• The next five chapters contain a simple, readable account on the general notions concerning semi-simple Lie algebras, their root diagrams, weight diagrams, and the reduction of product representations.
• The first half of this brochure is devoted to a detailed and elaborate discussion of the weight diagrams of representations of the simple Lie groups of rank two: A2, B2 and G2.

37. Christian Kramp (1760-1826)
• I use the very simple notation n! to designate the product of numbers decreasing from n to unity, i.e.
• Kramp went further than simple factorials, however, in his article Memoire sur les facultes numeriques Ⓣ published in Gergonne's Annales de Mathematiques in 1812.
• I note that any numerical faculty whatever is always reduced to the very simple form .

38. Jules Bienaymé (1796-1878)
• Bienayme published the Bienayme-Chebyshev inequality, which was used to give a very simple and precise demonstration of the generalised law of large numbers, in his important paper Considerations a l'appui de la decouverte de Laplace sur la loi de probabilite dans la methode des moindres carres Ⓣ (1853).
• Bienayme also worked on independent binomial trials and his most important contribution was his statement of the criticality theorem for simple branching processes which he gave in 1845 - eventual extinction of a family name has probability one if and only if the mean number of male children is one or less.
• He also gave a simple test for randomness of observations on a continuously varying quantity.

39. Abraham Wald (1902-1950)
• points out that the two major problems of statistical theory at that time, testing hypotheses and estimation, can both be regarded as simple special cases of a more general problem - known nowadays as a "statistical decision problem".
• The idea here is a simple one yet Wald was the first to build it into a statistical theory.
• He was a master at deriving complicated results in amazingly simple ways.

40. Thomas Graham (1905-1974)
• Tommy Graham wanted to go to the University of Cambridge to undertake research in mathematics for his doctorate but obtaining the necessary funding was not simple.
• Returning to Cambridge in 1932, Graham submitted his thesis On the Structure of Simple and Semi-Simple Groups and, after being examined, was awarded a Ph.D.

41. Ludwig Wittgenstein (1889-1951)
• This is essentially an atomic theory with the world built from simple objects.
• Why is philosophy so complicated? It ought to be entirely simple.
• Although the result of philosophy is simple, its method cannot be if it is to succeed.

42. Edward Lorenz (1917-2008)
• It was a simple event in 1961 which led Lorenz to results which brought him worldwide fame.
• Another account of aperiodic behaviour in ordinary differential equations, and difference equations, in which Lorenz describes how he arrived, starting from the description of convection in meteorology, at the Lorenz equations is contained in his paper On the prevalence of aperiodicity in simple systems delivered at the Biennial Seminar of the Canadian Mathematical Congress in Calgary, Canada, in 1978.
• The primary purpose of this study is to find out what the attractor set looks like for some simple atmospheric model.

43. Benoit Mandelbrot (1924-2010)
• Despite their pathological qualities, their extraordinary complexity, especially when viewed in greater and greater detail, they were often very simple to describe in the sense that the rules which generated them were absurdly simple to state.
• They occur in physics in the description of the extraordinarily complex behaviour of some simple physical systems like the forced pendulum and in the hugely complex behaviour of turbulence and phase transition.

44. Clarence Lewis (1883-1964)
• My mother was a vital young woman - nineteen when I was born - of simple faith and with the love of life.
• These studies are of so recent an origin that there has been till now no opportunity to consolidate into a single treatise anything but their most simple and primitive aspects.
• Accordingly the student, after leaving the almost childishly simple Boolean algebra as presented in the writings of Couturat and del Re, is immediately confronted with that forbidding monument of patience and research, the 'Principia Mathematica' of Whitehead and Russell.

45. Bill Boone (1920-1983)
• His thesis was entitled Several Simple, Unsolvable Problems of Group Theory Related to the Word Problem.
• He wrote up the results of his thesis in four parts under the title Certain simple, unsolvable problems of group theory; two parts appeared in 1954, and two more in 1955.
• It gives an algebraic characterisation of groups with soluble word problem connecting this property with embeddability in a simple group.

46. Atle Selberg (1917-2007)
• the basic ideas were rather simple always, and could be explained in rather simple terms ..
• His breakthroughs on long-standing problems were based on imaginative and novel ideas which, once digested, were appreciated as simple and decisive.

• The finite control was programmable with very simple instructions, and one could easily write a program that would read a string of A, T, C and G on the input tape and write the Watson-Crick complementary string on the output tape.
• But there was one important piece of information that made this similarity truly striking: Turing's toy computer had turned out to be universal - simple as it was, it could be programmed to compute anything that was computable at all.
• Theoretically, only two things were needed to build a computer capable of computing anything computable - a method by which information was stored and simple operations which acted on it.

48. Eduard Stiefel (1909-1978)
• Following earlier work of H Cartan and H Weyl, he introduced the so-called Stiefel Diagram for continuous groups, relating closed semi-simple groups and discontinuous reflection groups.
• Accordingly he was looking for a way to gain access to computing power beyond the level that could be performed by simple desktop calculators.
• However, we tried to use simple and transparent proofs, presented in detail.

• The rapid extension during recent years of the ideas of simple correlation has imposed their use upon many scientists not trained in the mathematical theory underlying them.
• One object of this bulletin is to present in simple un-technical language some explanation of the meaning and uses of the various correlation coefficients, simple partial and multiple.

50. Hans Samelson (1916-2005)
• My aim has been to follow as direct a path to these topics as I could, avoiding detours and side trips, and to keep all arguments as simple as possible.
• deals with the structure and representation theory of semi-simple Lie algebras and succeeds in covering a good deal of material.
• The purpose, as before, is to present a simple straightforward introduction, for the general mathematical reader, to the theory of Lie algebras, specifically to the structure and the (finite dimensional) representations of the semisimple Lie algebras.

51. Lyudmila Vsevolodovna Keldysh (1904-1976)
• It was in 1934 that her first papers were published: On the Homeomorphism of Canonical Elements of the 3rd Class; On Simple Functions of Class a; and On the Structure of B Measurable Functions of Class a.
• [She and her husband Petr Sergeevich Novikov] had a very simple lifestyle in the family, no excesses, simple furniture, only interesting pictures on the walls, and warm human relations.

52. Robert Boyle (1627-1691)
• He did follow Descartes in his overall belief that the world was basically a complex system governed by a small number of simple mathematical laws.
• Although others before him had applied mathematics to physics, Boyle was one of the first to extend the application of mathematics to chemistry which he tried to develop as a science whose complex appearance was merely the result on simple mathematical laws applied to simple fundamental particles.

53. Alexander Animalu (1938-)
• We list most of these in the following list: The spin-orbit interaction in metals and semiconductors (1966); (with F Bonsignori and V Bortolani) Electron-phonon contribution to the specific heat of alkalines (1966); (with F Bonsignori and V Bortolani) The phonon spectra of alkali metals and aluminium (1966); Optical conductivity of simple metals (1967); Self-consistent theory of optical transitions in simple metals (1967); The pressure dependence of the electrical resistivity thermopower and phonon dispersion in liquid mercury (1967); (with B Vasvari and V Heine) Electronic structure of Ca, Sr, and Ba under pressure (1967) [Note: Vasvari was a Hungarian who worked at Cambridge, England, while in receipt of a scholarship.
• Animalu writes, "This work was completed, supported in part by the Advanced Research Projects Agency through the Center for Materials Research, Stanford University, Stanford, California]; Many-electron effects in the optical conductivity of simple metals by Kubo formula (1970); General theory of magnetic-field-induced surface states (1970); Mass ratio of quarks (1971); Charge spectrum of four-component fields with O(4, 2) symmetry (1971); Bound states and mass spectra of hadrons in the quark model (1971); Scale symmetry (1972); Lepton and hadron currents in O(4, 2) current algebra (1972); High-field magnetoresistance of metals by Kubo-Mott formula (1972); Pseudopotential approach to magnetic energy bandstructure and magnetic breakdown in metals (1972); Josephson current in tunneling between coupled superconductors (1973); A relativistic model of quark-quark strong interactions (1973); Electronic structure of transition metals.

54. Émile Mathieu (1835-1890)
• Emile Mathieu is remembered especially for his discovery (in 1860 and 1873) of five sporadic simple groups named after him.
• By March 1859 he had been awarded his Docteur es Sciences by the Faculty of Science in Paris for his thesis Sur le nombre de valeurs que peut acquerir une fonction quand on y permute ses lettres de toutes les manieres possibles Ⓣ on transitive functions, the work which led to his initial discovery of sporadic simple groups.
• He lived in simple style dividing his time between his lectures and his mathematical researches.

55. Charles Fefferman (1949-)
• When I was a little boy I was interested in children's science: how rockets work and things like that, but I wasn't satisfied with simple explanations, so I checked a Physics textbook out of the public library and I couldn't understand anything.
• Fourier analysis is the study of how complicated vibrations break up into simple ones.
• A photograph is a two-dimensional image, also built up from simple pieces analogous to the fundamental note and overtones of a string.

56. Mark Aronovich Naimark (1909-1978)
• This is a fairly comprehensive account of the irreducible continuous unitary representations of the classical simple (complex) Lie groups, and of related aspects of harmonic analysis on such groups.
• (The title is somewhat misleading in that partial differential operators are not analysed.) Starting from simple facts about boundary-value problems, the author develops the theory of expansion by eigenfunctions, and the spectra of ordinary differential operators, including many of the results obtained recently by Russian mathematicians.
• The style is simple and lucid.

57. Øystein Ore (1899-1968)
• An effort has been made to present the subject matter in the book in as simple a form as possible.
• Many of these are quite simple; others are more in the nature of proposed research problems; these have been marked with an asterisk.

58. Johannes Robert Rydberg (1854-1919)
• Rydberg's most important work is on spectroscopy where he found a relatively simple expression relating the various lines in the spectra of the elements.
• 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.

59. Isaac Todhunter (1820-1884)
• His habits and tastes were singularly simple; and to a gentle kindly disposition he united a high sense of honour, a warm sympathy with all that was calculated to advance the cause of genuinely scientific study in the university, and considerable humour.
• From the majority of the papers in our few mathematical journals, one would almost be led to fancy that British mathematicians have too much pride to use a simple method, while an unnecessarily complex one be had.

60. 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.
• But it is impossible by words to convey any idea of the effect of his conversation, and of the impression made by so much philosophy, gaiety nd humour, accompanied by a manner at once so animated and simple.

61. David Mumford (1937-)
• Klein studied infinitely repeated reflections and was led to forms with multiple co-existing symmetries, each simple in itself, but whose interactions produce fractals on the edge of chaos.
• I will try to explain what this means, using simple examples and then go on show why it is proposed as a new tool in the diagnosis of medical conditions.

62. Jan Tinbergen (1903-1994)
• The ability of a planning expert to communicate with politicians and with citizens constitutes an important element in any type of democratic or semi-democratic planning and such communication can be enhanced by relatively simple models.
• In a neat, simple house, virtually indistinguishable from others on his block in the middle class neighbourhood of The Haviklaan, The Hague, lives one of the world's most distinguished economists, a co-winner of the first Nobel Prize in Economics in 1969, and a man who is known for his gentleness, his modesty, and his selfless dedication to the cause of human welfare.

63. Raphael Robinson (1911-1995)
• The book gives an introductory account of the methods introduced by Tarski for establishing the undecidability of several fairly simple branches of mathematics (group theory, lattices, abstract projective geometry, closure algebras and others).
• In each of them he takes a problem, old or new, which can be stated in simple and intelligible terms, and either solves it, or at least adds much that is new.

64. Charles-Marie de La Condamine (1701-1774)
• There was never any discussion of constructing a fancy edifice, but rather a simple and durable monument appropriate for showing clearly the two endpoints of our base.
• widely known in every society, possessing the art of persuading the ignorant people to whom he had listened, bringing back singular observations to pique the frivolous curiosity of the people of the world, writing with enough charm to have people read his work, with enough neglect and too simple a tone to foster envy or threaten the self-esteem of others, interesting for his bravery and piquant for his faults.

65. Bronius Grigelionis (1935-2014)
• As an application, the problem of testing a simple hypothesis against a simple alternative in a Poisson process is considered.

66. Lazar Matveevich Gluskin (1922-1985)
• His other early papers included Homomorphisms of unilaterally simple semigroups on groups (Russian) (1955), Simple semigroups with zero (Russian) (1955), and Elementary generalized groups (Russian) (1957).

67. Pedro Abellanas (1914-1999)
• In particular, it is shown that the geometric definition of a simple point of a surface (in terms of the multiplicity of the intersection with two generic primes) is equivalent to the arithmetic definition [O Zariski, (1939)].
• According to the author's preface, this book is devoted to the study of the basic spaces (vectorial, affine, euclidean and projective) and the maps among them, especially the linear ones and also the differential functions in simple cases.

68. John Carr (1948-2016)
• He readily explained his insights in patient and simple terms both to students and to many others who approached him for help.
• He readily explained his insights in patient and simple terms both to students and to many others who approached him for help.

69. Wanda Szmielew (1918-1976)
• Why did Maria [Tarski's wife] put up with this? The simple answer is: She did not think she had much choice.
• Every movement seems simple, natural, the way it has to be.

70. Jean Richer (1630-1696)
• Again we do not know the date of his birth; in fact the year of 1630 that we give, although generally accepted by most historians, is simple a guess based on the fact he joined the Academie as a junior astronomer in 1666.
• The vibrations of the simple pendulum which was used were very short and remained quite perceptible up to 52 minutes, and were compared with those of an extremely good clock whose vibrations indicated seconds.

71. James Whitbread Lee Glaisher (1848-1928)
• Their fundamental principles are derived from observations so simple as to be almost axiomatic.
• There was no shred of pomposity in his bearing, which was frank and simple.

72. Bruce Kellogg (1930-2012)
• He summarises Some simple boundary value problems with corner singularities and boundary layers (2006) as follows:- .
• He co-authored several papers with Martin Stynes as, for example, Sharpened bounds for corner singularities and boundary layers in a simple convection-diffusion problem (2007).

73. Joseph Wedderburn (1882-1948)
• In this paper On hypercomplex numbers which appeared in the Proceedings of the London Mathematical Society, he showed that every semisimple algebra is a direct sum of simple algebras and that a simple algebra was a matrix algebra over a division ring.

74. Paul Painlevé (1863-1933)
• brought up in the simple democratic atmosphere of French skilled artisan family life.
• Painleve had a naturally simple and unaffected manner, and was possessed of a singular charm that few persons, even among his opponents, were able to resist.

75. Isaac Newton (1643-1727)
• Taking differentiation as the basic operation, Newton produced simple analytical methods that unified many separate techniques previously developed to solve apparently unrelated problems such as finding areas, tangents, the lengths of curves and the maxima and minima of functions.
• He had reached the conclusion during the two plague years that white light is not a simple entity.

76. Étienne Bézout (1730-1783)
• Of course on the face of it this does not help solve the equation but Bezout made the simplifying assumption that one of the two equations was of a particularly simple form.
• One has to understand the problems that faced Bezout for he did not have our simple suffix notation to denote the unknowns by x1, x2, x3, ..

77. Kathleen McNulty Antonelli (1921-2006)
• We did have desk calculators at that time, mechanical and driven with electric motors, that could do simple arithmetic.
• We were preparing a firing table for each gun, with maybe 1,800 simple trajectories.

78. Evangelista Torricelli (1608-1647)
• This theory allowed Cavalieri to find, in a simple and rapid way, the area and volume of various geometric figures.
• He was a skilled lens grinder, making excellent telescopes and small, short focus, simple microscopes, and he seems to have learnt these techniques during the time he lived with Galileo.

79. Karl Gräffe (1799-1873)
• The law by which the new equations are constructed is exceedingly simple.
• A simple notion, but effective and it is just what everyone does today.

80. Boris Yakovlevic Bukreev (1859-1962)
• The basic principles Bukreev adopted as a teacher included ensuring completeness of coverage of the topic under discussion, accuracy and clarity of presentation, and presenting his material in simple language.
• These books are written at a high scientific level, yet in simple and clear language.

81. George Greenhill (1847-1927)
• There, with his books around him, his tables covered in neat disorder with innumerable scraps of material and apparatus to be used as dynamical models, his walls festooned with every variety of pendulum, simple or compound, contrived from articles purchased below a prescribed limit of cost at the local stores, upon his floor the treasured roll of Turkish carpet from his room of long ago at St John's, and above the mantelpiece the portrait of his beloved teacher, Clerk Maxwell, smiling approval - with all these and the precious memories they recalled, the scholar was content.
• To the question of whether he would take tea or coffee his reply was a simple affirmative ..

82. Geoffrey Bennett (1868-1943)
• In the simple cases, when the modulus is a real number which is an odd prime, a power of an odd prime, or double the power of an odd prime, we know that there exist primitive roots of the modulus: that is, that there are numbers whose successive powers have for their rests the complete set of numbers less than, and prime to, the modulus.
• Many of these are authors giving their thanks to Bennett with words such as: "G T Bennett, of Emmanuel College, Cambridge, sent the author the following simple construction in August, 1902..

83. Georges Darmois (1888-1960)
• Distribution functions, mean values, characteristic functions with their various properties, and moments are presented in a very simple manner.
• Equally simple is the presentation of pairs of variables with the corresponding Laplace-Gauss law, convergence theorems, domains of attraction of the Laplace-Gauss law, with elements of the theory of errors and the influence of dependence on random quantities.

84. David Enskog (1884-1947)
• A simple theory does not predict this behaviour.
• He derived general formulas for the viscosity, conduction, and diffusion in simple and mixed gases, and also determined the pressure-tensor to a third approximation.

85. Ismael Boulliau (1605-1694)
• The simple suggestion I put to him should make him wiser and more reserved in making injurious statements against someone who has never made such cruel remarks, and who has never offended him.
• There is one aspect of Boulliau's philosophy which is well worth commenting on - namely the fact that he believed in simple explanations and moreover he wanted many different observed properties to result from a single cause.

86. Florence Nightingale David (1909-1993)
• the ideas are simple and easy to understand, but the manipulative procedures are discouraging in that they depend on mathematical tricks which have to be learnt.
• The book is written with care but without pedantry, and takes the reader from deceivingly simple problems to serious and sometimes quite involved mathematical theory.

87. Yitz Herstein (1923-1988)
• Before being appointed to Chicago he had published papers such as A proof of a conjecture of Vandiver (1950), On a conjecture on simple groups (1950), and Group-rings as *-algebras (1950).
• The book largely concerns Herstein's work on Lie and Jordan structure of simple associative rings which he published in various papers in the early 1950s.

• Columella was a Roman soldier and farmer who wrote extensively on agriculture and similar subjects, hoping to foster in people a love for farming and a liking for the simple life.
• They give methods of lifting heavy weights and describe simple mechanical machines.

89. Helmut Hasse (1898-1979)
• At Halle Hasse obtained fundamental results on the structure of central simple algebras over local fields.
• While in Marburg he began joint work with Brauer and Emmy Noether on simple algebras, culminating in the complete determination of what is today called the Brauer group of an algebraic number field.

90. Donald Knuth (1938-)
• There has been much recent interest in languages whose grammar is sufficiently simple that an efficient left-to-right parsing algorithm can be mechanically produced from the grammar.
• In this paper, we define LR(k) grammars, which are perhaps the most general ones of this type, and they provide the basis for understanding all of the special tricks which have been used in the construction of parsing algorithms for languages with simple structure, e.g., algebraic languages.

91. Giorgio Bidone (1781-1839)
• Simple and modest in his desires, he loved science for its own sake, and he never used it to pursue dreams of ambition and wealth.
• I entered into this clean apartment which was so simple and austere; exact and precisely ordered like a page of calculations.

92. Charles René Reyneau (1656-1728)
• Fortunately, in the last century, the lines and figures were expressed by the familiar characters of the alphabet, and these expressions were reduced to an easy calculus, which also expresses all the simple and compound relations which these lines and figures can have.
• He added to it the excellent method of employing indeterminate expressions, which, however simple they were, represented an infinity of magnitudes; and to determine from all of them the particular magnitudes which they may satisfy.

93. John Crank (1916-2006)
• In the 1940s such calculations were carried out on simple mechanical desk machines.
• Crank and Nicolson's method, which is numerically stable, requires the solution of a very simple system of linear equations (a tridiagonal system) at each time level.

94. R H Bing (1914-1986)
• Bing worked for his doctorate under R L Moore's supervision, undertaking research on simple plane webs.
• He worked on topological classification of the 2-sphere, the 3-sphere, pseudo arcs, simple closed curves and Hilbert space.

95. Lyman Spitzer (1914-1997)
• This approach is necessary in discussing containment by magnetic mirrors and the lack of it in a simple torus.
• This is joined with Maxwell's equations, and the simple limits of high and low magnetic fields are briefly considered.

96. Matyá Lerch (1860-1922)
• However, things are not quite so simple since his given name (the one that appears on his birth certificate) is Matěj.
• He attended courses on the theory of elliptic functions by Weierstrass, and courses on the theory of algebraic equations and on simple and multiple integrals by Kronecker.

97. Gaetano Fichera (1922-1996)
• In pure mathematics Gaetano Fichera achieved considerable results in the following fields: mixed boundary value problems of elliptic equations; generalized potential of a simple layer; second order elliptic-parabolic equations; well posed problems; weak solutions; semicontinuity of quasi-regular integrals of the calculus of variations; two-sided approximation of the eigenvalues of a certain type of positive operators and computation of their multiplicity; uniform approximation of a complex function f(z); extension and generalization of the theory for potentials of simple and double layer; specification of the necessary and sufficient conditions for the passage to the limit under integral sign for an arbitrary set; analytic functions of several complex variables; solution of the Dirichlet problem for a holomorphic function in a bounded domain with a connected boundary, without the strong conditions assumed by Francesco Severi in a former study; construction of a general abstract axiomatic theory of differential forms; convergence proof of an approximating method in numerical analysis and explicit bounds for the error.

98. Efim Zelmanov (1955-)
• The Schreier conjecture, that the outer automorphism groups of finite simple groups are soluble, was shown to be true as a consequence of the classification of finite simple groups.

99. William Edge (1904-1997)
• He also used geometrical configurations to investigate groups and, although his work was out of fashion at a time when group theorists were moving towards the classification of finite simple groups, his work did provide a deeper understanding of some of these groups, for example Conway's simple groups.

100. Wolfgang Gaschütz (1920-2016)
• The effects that this simple exchange of books had on the development of the mathematical seminar in Kiel can arguably still be perceived today.
• After completion of the classification of simple groups the main problem in the theory of finite groups remains the problem of mastering mechanisms of their interaction in arbitrary groups.

101. Jacob Levitzki (1904-1956)
• In it he showed that every semisimple algebra is a direct sum of simple algebras and that a simple algebra was a matrix algebra over a division ring.

102. Shiing-shen Chern (1911-2004)
• He knew all these papers on simple Lie groups, Lie algebras, all by heart.
• Through war and peace and through bad and good times we have shared a life for forty years, which is both simple and rich.

103. Hubert Wall (1902-1971)
• Over less than two hundred pages the reader travels from elementary number theory to simple graphs, from integrals and surfaces to linear spaces of simple graphs.

104. Leo Moser (1921-1970)
• a simple, courteous and soft spoken person full of anecdotes, humour and problems.
• He had an infinite stock of amusing stories, and a huge storehouse of simple puzzles of both a mathematical and a non-mathematical nature.

105. Oded Schramm (1961-2008)
• (The spheres must have disjoint interiors, but they don't have to be the same size.) It's a standard theorem in classical geometry, also related to important work in hyperbolic geometry and complex analysis, that you can realize any planar simple graph by kissing circles in R2, i.e., the circles are the vertices and the kissing pairs are the edges.
• The simple random walk on the square grid in the plane converges to Brownian motion under appropriate scaling.

106. Hans Zassenhaus (1912-1991)
• During this time he proved the Zassenhaus (butterfly) lemma, a beautiful result on subgroups which can be used to give a simple, and very beautiful, proof of the Jordan-Holder theorem.
• These groups play an important role in the classification of finite simple groups coordinated by Daniel Gorenstein.

107. Guido Grandi (1671-1742)
• The Accademia Arcadia was a literary academy which was founded in Rome in 1690 to promote a more natural, simple poetic style and around 1700 Grandi attended this Academy.
• However, he lived a simple retiring life among a small circle of friends.

108. Robert Edward Bowen (1947-1978)
• This three page paper gives simple bounds (using Euler's polyhedron formula) for the number of edges in the sets obtained when the vertices of a planar graph are partitioned into two sets.
• life was simple and unpretentious, punctuated by occasional parties full of noise and dancing.

109. Raymond Brink (1890-1973)
• 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.
• Its statement and proof are simple for convergence, but rather awkward for the divergence test.

110. Herman Hollerith (1860-1929)
• The punch was constructed in a similar way to a typewriter having a simple keyboard.
• By this time he had added a mechanism to feed the cards automatically and other automatic sorting procedures which added sophistication to the original simple mechanical counting process.

111. Leonid Vital'evich Kantorovich (1912-1986)
• The method of successive approximations is often applied to proving existence of solutions to various classes of functional equations; moreover, the proof of convergence of these approximations leans on the fact that the equation under study may be majorised by another equation of a simple kind.
• The basic principle of their effective use was the paralleling of similar calculations, which made it possible to introduce simple program changes on the plugboard (of course, by hand).

112. Leslie Valiant (1949-)
• His results here range from simple, but powerful and elegant, insights to reexamining the very foundations.
• An example of a simple insight is his parallel routing scheme, described in the paper "A scheme for fast parallel communication" (1982).

113. Jan A Schouten (1883-1971)
• For example in 1924 he published Uber die Geometrie der halb-symmetrischen Ubertragungen Ⓣ jointly with Alexander Friedmann, and in 1926 he published two papers written jointly with Elie Cartan: On Riemaniann geometries admitting an absolute parallelism, and On the Geometry of the Group-manifold of Simple and Semi-simple Groups.

• J-P Serre showed with simple examples that no such functorial theory exists for abstract algebraic varieties which reflects the usual (singular) integral cohomology of spaces.
• I will describe a simple model which is useful for the study of the relationship between the history of a population and its genetic properties.

115. Aleksei Alekseevich Dezin (1923-2008)
• The potential reader is transported from very simple systems of linear equations to concepts as complex as that of linear space, invertible operator, eigenvalue and eigenvector, norm, adjoint, unitary and selfadjoint operator.
• He shows a special gift for making complicated facts look simple.

116. John Charles Fields (1863-1932)
• This long period of study, which exercised a decisive influence on his life and outlook, was rendered possible by a modest private income, combined with simple living and abstemious habits.
• The machinery, which he had to invent for the purpose, is simple, and its parts are beautifully coordinated.

117. Olabisi Ugbebor (1951-)
• We did some simple ones.
• Then, I showed them why if we were using that simple method for a larger collection, we would be there all day.

118. Ernst Witt (1911-1991)
• Having seen a remarkably simple proof by Witt of Wedderburn's theorem that every finite skew field is commutative, Herglotz encouraged him to submit it for publication and it became Witt's first paper appearing in 1931.
• joined the SA, urged on by the simple wish ..

119. Józeph Petzval (1807-1891)
• Petzval produced an achromatic portrait lens that was vastly superior to the simple meniscus lens then in use.
• His attack was based on the fact that Doppler derived the principle in a few lines using only simple equations.

120. Hermann von Helmholtz (1821-1894)
• Theoretical natural science must, therefore, if it is not to rest content with a partial view of the nature of things, take a position in harmony with the present conception of the nature of simple forces and the consequences of this conception.
• Its task will be completed when the reduction of phenomena to simple forces is completed, and when it can at the same time be proved that the reduction given is the only one possible which the phenomena will permit.

121. Timothy Pedley (1942-)
• Section 2 outlines the simple one-dimensional theory of pulse propagation in distended vessels, based on a 'tube law' to describe the elastic properties, and points out that there are a number of features (involving (a) wave attenuation and (b) localized constraints) that the simple theory still cannot explain.

• While an undergraduate he had already published some important papers (all in Russian): Simple examples of unsolvable canonical calculi (1967), Simple examples of unsolvable associative calculi (1967), Arithmetic representations of powers (1968), A connection between systems of word and length equations and Hilbert's tenth problem (1968), and Two reductions of Hilbert's tenth problem (1968).

123. Bella Abramovna Subbotovskaya (1938-1982)
• "A problem cannot be uninteresting, it can only be simple or complicated" - this was her saying.
• Bella Abramovna's and her like-minded people's idea was humane and simple: attempt to at least partially restore fairness by offering students who were seriously interested in mathematics the possibility of receiving that fundamental mathematical education which the administrators of Mekh-Mat deprived them.

124. Charles Noble (1867-1962)
• On 14 August 1907 Noble was in Zurich when he submitted his paper Singular points of a simple kind of differential equation of the second order to the Bulletin of the American Mathematical Society.
• 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.

125. Herbert Pahlings (1939-2012)
• At that time the classification of finite simple groups had just been finished and the preparation of the 'Atlas of Finite Groups' was on the way.
• Character tables of simple and related groups are a dominant feature of the Atlas and programs such as the ones of CAS were welcome in particular for interactive handling the character tables of groups which were far too big to be worked with from their elements.

126. 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.
• He was, according to one of his many illustrious pupils, 'a charming teacher, so simple, unaffected and sincere in manner, so chaste in style, so clear in demonstration'.

127. Mineo Chini (1866-1933)
• Chini examined one of the writings of Eugenio Beltrami, Sulla flessione delle superfici rigate Ⓣ, in which the author studied the deformation of such surfaces; Chini was able to reduce to the minimum the number of possible elements that identifies the shape of a ruled surface, he researched the formulae - rather simple in this case - that gave all the bump-shaped surfaces and applied these formulae to treat some problems of the same type, but less simple, than those tackled by Beltrami in his essay.

128. Francesco Faà di Bruno (1825-1888)
• The subject is thoroughly and brilliantly set out, the exposition is simple, clear and, in several places, elegant.
• Faa di Bruno was tall and not always well dressed, but he was simple and good natured.

129. D'Arcy Thompson (1860-1948)
• It behoves us always to remember that in physics it has taken great men to discover simple things.
• When he meets with a simple geometrical construction, for instance in the honeycomb, he would fain refer it to psychical instinct, or to skill and ingenuity, rather than to the operation of physical forces or mathematical laws; when he sees in snail, or nautilus, or tiny foraminiferal or radiolarian shell a close approach to sphere or spiral, he is prone of old habit to believe that after all it is something more than a spiral or a sphere, and that in this 'something more' there lies what neither mathematics nor physics can explain.

130. Francesco Severi (1879-1961)
• His lectures on his own work were unforgettable, the style was beautifully simple ..
• Personal relationships with Severi, however complicated in appearance, were always reducible to two basically simple situations: either he had just taken offence or else he was in the process of giving it - and quite often genuinely unaware that he was doing so.

131. Ingrid Daubechies (1954-)
• But I also was interested in seeing how machinery worked, or in why certain mathematical things were true (like the fact that a number is divisible by nine if, when you add all its digits together, you get another number divisible by 9 - try it with 73512 and 8577, both multiples of 9; there is no rule that is quite as simple for divisibility by 7, say).
• The use of wavelets as an analytical tool is like Fourier analysis - simple and yet very powerful.

132. Ernst Jacobsthal (1882-1965)
• Indeed, he was so humanly simple and natural, that you had to be fond of him.
• He also showed that it is possible to find a solution p = x2 + y2 where x and y can be expressed with simple sums over Legendre symbols.

133. Jan Mikusiski (1913-1987)
• It introduces natural numbers through a new mathematical approach; replaces the Riemann integral with the more general Lebesgue integral; and rigorously develops the real number system from four simple axioms of natural numbers.
• Additional features include a wider range of problems than other texts - including simple and routine as well as problems requiring more in depth creativity, answers to common questions, a new approach to the concept of equivalence relation which simplifies the construction of real numbers, and a large number of computational applications.

134. Johannes de Groot (1914-1972)
• As is well known, any field can be obtained from its prime field by a succession of simple transcendental extensions followed by a succession of algebraic extensions.
• This trait was his strength, where the riches came from his grasp of simple ideas without much background knowledge, which made it possible for him to lead others to work together and to encourage them.

135. Philip Maini (1959-)
• Now, when kicking a football about, I dream of solving maths problems instead! I first saw the power and beauty of mathematics when, in the first year of A levels, my teacher wrote down the equation for simple harmonic motion for a swinging pendulum and I saw how this simple equation could describe everything about the motion of the pendulum.

• It, like the ideas of Pythagoras, tried to explain the multitude of complexity seen in the world as being the consequence of a small number of simple underlying properties.
• Although we no longer believe in Empedocles' four element theory, we do still look for simple mathematics which will explain the complex phenomena that surround us.

137. Sydney Chapman (1888-1970)
• There was a simple directness about his mode of expression, which often concealed deep thought.
• Chapman's mild manner veiled a strong will and great determination; his tastes and habits were simple.

138. Jean d'Alembert (1717-1783)
• Rational mechanics was a science based on simple necessary principles from which all particular phenomenon could be deduced by rigorous mathematical methods.
• In order to avoid delicate experiments or long tedious calculations, in order to substitute analytical methods which cost them less trouble, they often make hypotheses which have no place in nature; they pursue theories that are foreign to their object, whereas a little constancy in the execution of a perfectly simple method would have surely brought them to their goal.

139. Charles Augustin Coulomb (1736-1806)
• his simple, elegant solution to the problem of torsion in cylinders and his use of the torsion balance in physical applications were important to numerous physicists in succeeding years.
• Viewing fortresses as nothing more than immense permanent batteries designed to pour overwhelming fire on attacking armies, Montalembert simplified the intricate geometric designs of Vauban and relied on simple polygonal structures, often with detached peripheral forts instead of projecting bastions.

140. Christian Heinrich von Nagel (1803-1882)
• to expose the ideas laid down in nature: the simple pure utterances of the deity.
• This is constructed in a simple way.

141. Max Dehn (1878-1952)
• Written in 1914, not long after the discovery of the fundamental group of a topological space, it tackles a simple and beautiful problem: to confirm a property of the simplest knot which is suggested by five minutes of experimentation: that the right and left trefoil knots are not isotopic.
• before 1984 we really didn't have any simple tests for non-amphicheirality, so that Dehn's work (which at first glance looks like the use of a cannon to kill a sparrow) remained central to the subject for nearly 70 years.

142. Otto Hölder (1859-1937)
• He searched for finite simple groups and in the 1892 paper Die einfachen Gruppen im ersten und zweiten Hundert der Ordnungszahlen Ⓣ in Mathematische Annalen he showed that all simple groups up to order 200 are already known.

143. Ali Moustafa Mosharrafa (1898-1950)
• Mosharrafa's next paper, On the quantum theory of the simple Zeeman effect was submitted for publication on 1 September 1922 and published in February 1923.
• The aim of this paper is to put forward a theory of the simple Zeeman effect which possesses the same general features as those of the corresponding theory in the case of the Stark effect already developed by Epstein and Schwarzschild.

144. James Murray (1931-)
• For example A theoretical study of the effect of impulse on the human torso (1966), A simple method for obtaining approximate solutions for a class of diffusion-kinetic enzyme problems (Part I, 1968; Part II, 1968), and On the molecular mechanism of facilitated oxygen diffusion by haemoglobin and myoglobin (1971) are on mathematical biology while A simple method for determining asymptotic forms of Navier-Stokes solutions for a class of large Reynolds number flows (1967), Singular perturbations of a class of nonlinear hyperbolic and parabolic equations (1968) and On Burgers' model equations for turbulence (1973) are on fluid dynamics.

145. Herbert Richmond (1863-1948)
• It is true that the scope of these methods is restricted, but there is compensation in the fact that when geometry is successful in solving a problem the solution is almost invariably both simple and beautiful.
• A result already known is obtained in a simple manner.

146. Otto Szász (1884-1952)
• In fact Szasz worked on problems associated with both Riesz brothers, and he gave a very simple proof a theorem by Marcel Riesz on rational functions with given bounds on the unit circle.
• His life and energy were dedicated to the promotion of simple and beautiful problems of mathematics, in particular of the classical analysis.

147. Maurits Escher (1898-1972)
• After all his efforts, how far short of the originally so lucid and misleading simple idea did this result fall! .
• Circle Limit III was created using only simple drawing instruments and Escher's great intuition, but Coxeter proved that [',' D Schattschneider, Escher: A mathematician in spite of himself, in R K Guy and R E Woodrow (eds), The Lighter Side of Mathematics (Washington, 1994), 91-100.','8]:- .

• Zeno's challenge to simple pluralism is successful, in that he forces anti-Parmenideans to go beyond common sense.
• Zeno bases both the dichotomy paradox and the attack on simple pluralism on the fact that once a thing is divisible, then it is infinitely divisible.

149. Gregori Margulis (1946-)
• However, PSL2(R) was for a long time the only simple Lie group which was known to contain non-arithmetic discrete subgroups of finite covolume, and further examples discovered in 1965 by Makarov and Vinberg involved only few other Lie groups, thus adding credit to conjectures of Selberg and Pyatetski-Shapiro to the effect that "for most semisimple Lie groups" discrete subgroups of finite covolume are necessarily arithmetic.
• for his monumental contributions to algebra, in particular to the theory of lattices in semi-simple Lie groups, and striking applications of this to ergodic theory, representation theory, number theory, combinatorics and measure theory.

150. Raymond Wilder (1896-1982)
• The best known example of such a positional invariant is embodied in the Jordan curve theorem: A simple closed curve in the 2-sphere has precisely two complementary domains and is the boundary of each of them.
• A converse to the Jordan curve theorem, proved by Schonflies, states that a subset of the 2-sphere is a simple closed curve if it has two complementary domains, is the boundary of each of them, and is accessible from each of these domains.

151. Bertrand Russell (1872-1970)
• Thus, in its details, the theory admits of two versions, the "simple theory" and the "ramified theory".

152. Volker Strassen (1936-)
• It is an intricate yet simple algorithm that remains the method of choice for multiplying dense matrices of size 30 by 30 or more on machines today.

153. Wilhelm Ackermann (1896-1962)
• A remarkably simple axiomatization of a system of set theory is presented which the reviewer feels deserves serious consideration.

154. Bernhard Neumann (1909-2002)
• The present note, written in gratitude, affection and esteem, in Bernhard Neumann's honour, comprises some simple variations on the themes of that paper.

155. Paul Ehrenfest (1880-1933)
• Although he knew mathematics it was not simple for him.

156. Alfred Kempe (1849-1922)
• Hence many problems - such as, for example, the trisection of an angle - which can readily be effected by employing other simple means, are said to have no geometrical solution, since they cannot be achieved by straight lines and circles only.

157. Nikolai Luzin (1883-1950)
• His presentation was always very elegant and at first sight apparently unnecessarily simple - the result of his great pedagogic talent.

158. Kurt Mahler (1903-1988)
• His attitude to mathematics was like his attitude to life: he liked things as simple as possible and usually eschewed abstraction, but with his direct methods was often able to go surprisingly far.

159. Herbert Dingle (1890-1978)
• He wrote Relativity for All in 1922, in which he explained the subject (which was considered highly specialised at the time) in simple terms.

160. Charles Hermite (1822-1901)
• What radiates from the text is [Hermite's] humility, his Catholicism, his concern for his (very extended) family, his willingness to fight for colleagues whose merit he discerns, and his devotion to family, merit, and principle rather than simple influence.

161. John Colenso (1814-1883)
• "a simple-minded, but intelligent, native" asked him if he truly believed the story of Noah and a worldwide flood.

162. Agner Erlang (1878-1929)
• However, Magdalene and Hans Nielsen made a happy if simple home for their family making sure that they had sufficient food prepared which they prepared in the most hygienic manner possible.

163. Tom Whiteside (1932-2008)
• He held the most august and the most lowly colleagues to the same simple intellectual standards and judged - and treated - each only in terms of their intellectual integrity.

164. Paul Dirac (1902-1984)
• reflects Dirac's very characteristic approach: abstract but simple, always selecting the important points and arguing with unbeatable logic.

165. Pierre-Simon Laplace (1749-1827)
• The book continues with methods of finding probabilities of compound events when the probabilities of their simple components are known, then a discussion of the method of least squares, Buffon's needle problem, and inverse probability.

166. Andor Kertész (1929-1974)
• Kertesz defended his thesis entitled Operator modules and semi-simple rings in 1954.

167. Kollagunta Ramanathan (1920-1992)
• Simple proofs are given of a number of known theorems, such as the one that asserts that the product of 2sin π( n/m) taken over all n less than and prime to m has the value p or 1 according as m is or is not a power of the prime p.

168. Theodor Estermann (1902-1991)
• Until I read this book I would not have believed it possible to give so lucid and simple an account of the proofs of these three difficult and important theorems.

169. Tosio Kato (1917-1999)
• emphasizes clear and simple explanations of the fundamental notions of functional analysis.

170. Werner Rogosinski (1894-1964)
• The so-called Lindelof principle is nothing more than a transformation and systematic application of the simple Schwarz lemma.

171. Mikhail Yakovlevich Suslin (1894-1919)
• He had plenty of time to do this since he found his school work rather simple.

172. Anton Kazimirovich Suschkevich (1889-1961)
• In this Chapter there is an excellent contribution to the major structure theorem for algebraic semigroup theory which today is known as the Rees Theorem (named after David Rees) which classifies completely 0-simple semigroups.

173. John Stallings (1935-2008)
• This simple sounding theorem proves to be very powerful, implying (with a little work) the following two theorems: .

174. Gilles Pisier (1950-)
• Pisier's unique and clear way of presenting the material might even surprise researchers in the field: complicated results look very natural and simple in Pisier's presentation.

175. Sophus Lie (1842-1899)
• I have found it, it is quite simple! .

176. John Kemeny (1926-1992)
• Just as von Neumann realised that a computer that did only ordinary arithmetic operations could have extraordinary power, Kemeny realised that to make this power available to everyone, a programming language could and should be exceedingly simple.

177. Piers Bohl (1865-1921)
• There are many seemingly simple questions in this area which still seem to be open.

178. Francisco José Duarte (1883-1972)
• He published papers on the general solution of a diophantine equation of the third degree x3 + y3 + z3 - 3xyz = v3, simplified Kummer's criterion and gave a simple proof of the impossibility of solving the Fermat equation x3 + y3 + z3 = 0 in nonzero integers.

• Rogers in [',' Biography by C Ambrose Rogers, in Dictionary of National Biography (Oxford, 2004).','2] illustrates this with a simple example:- .

180. Raoul Bott (1923-2005)
• Usually he likes to discuss a simple key example that encapsulates the essence of the problem.

181. Dmitrii Matveevich Sintsov (1867-1946)
• in 1903) elementary simple proofs of its general real solutions.

182. Alexander Ostrowski (1893-1986)
• Life is often not that simple, however, since there was a quota on the number of Jewish students allowed into the university and entrance was decided by a lottery rather than on merit.

183. 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.

184. Paul Erds (1913-1996)
• He posed and solved problems that were beautiful, simple to understand, but notoriously difficult to solve.

185. Ernesto Pascal (1865-1940)
• The chief fault of the book, from our point of view, is that it sacrifices simple and natural discussion to the pursuit of the end so dear to Italian mathematicians, the greatest possible generality.

186. Julius Gysel (1851-1935)
• In his obituary, his parents' home is described as a place with 'simple living, hard work, and a lot of music' [',' Obituary: Julius Gysel (1881-1972), Verhandlungen der Schweizerischen Naturforschenden Gesellschaft 153, 1973, 264-265','9].

187. Étienne Bobillier (1798-1840)
• He first set up a problem in the form of an equation in a particular case, simple enough so that the analytic geometry of his time could deal with it.

188. Ivan Georgievich Petrovsky (1901-1973)
• In art, as in science, he values depth, simplicity and clarity; in painting he likes Rembrandt, Serov, Nesterov, in music, Bach, Vivaldi, in architecture, simple and severe forms.

189. Vaughan Jones (1952-)
• These included (in addition to knots and links) that part of statistical mechanics having to do with exactly solvable models, the very new area of quantum groups, and also Dynkin diagrams and the representation theory of simple Lie algebras.

190. Tobias Mayer (1723-1762)
• The celebrate Tobias Mayer contrived, however, a method to determine, at one reading, instead of the simple angle observed, a multiple of the same angle; and, by this means, the instrument became, in practice, capable of any degree of accuracy, as far as regards the above mentioned errors.

191. Gabriel Mouton (1618-1694)
• He conducted experiments which led him to the conclusion that a simple pendulum of length one virgula would oscillate 3959.2 times in 30 minutes.

192. Lois Griffiths (1899-1981)
• Even moderately elaborate theorems are resolved into simple elements; illustrative examples and particular cases are introduced to pave the way for the formal proofs which follow; and there is much restatement.

193. Irving John Good (1916-2009)
• He was awarded a Smith's Prize in 1940 for his essay on fractional dimensions of sets of simple continued fractions, and he received his doctorate in 1941 for his thesis The topological concept of partial dimension based on the ideas of Henri Lebesgue.

194. Max Born (1882-1970)
• He was widely known for his exposition of the ideas of physics to the layman, and he was held in affection by his many colleagues and pupils for the warmth and simple directness of his personality.

195. Francesco Cecioni (1884-1968)
• A master of life, nobody failed to take his precious advice; he helped everyone with detached disinterest; he was paternal and understanding, simple, patient, humble, generously charitable, particularly with the young.

196. Mikhael Leonidovich Gromov (1943-)
• His work is unique through the abundance and the force of the concepts he has created, as well as through the new techniques he has devised and applied to solve problems, often simple to state and to understand, and which seem, at first sight, inaccessible.

197. Daniel Pedoe (1910-1998)
• However, it is not that simple; for his father, who was a Polish Jew from the priestly tribe of Kohanim, changed his name to Cohen when he arrived in Britain in the 1890s.

198. Alan Mercer (1931-2014)
• Mercer published Some simple duration-dependent stochastic processes (1959) and A queueing problem in which the arrival times of the customers are scheduled (1960) giving his address in both as Birkbeck College, University of London and Atomic Weapons Research Establishment at Aldermaston.

199. François-Joseph Servois (1768-1847)
• I confess that I do not yet see in this notation anything but a geometric mask applied to analytic forms the direct use of which seems to me simple and more expeditious.

200. Onorato Nicoletti (1872-1929)
• He had great power of analysis; examining questions dealt with by others, even the most extreme, he often succeeded in overcoming large-scale hypotheses and unnecessary conditions, thus reducing his treatment to a very simple logical scheme, which allowed him, sometimes overcoming serious difficulties with uncommon abilities and deductive force, to obtain far more general results than those already known.

201. John von Neumann (1903-1957)
• After a talk with him one always came away with a feeling that the problem was really simple and transparent.

• that a scholion replaced, perhaps in a damaged copy, the first of four proofs by a simple reference to generally known theorems.

203. Fatma Moalla (1939-)
• And I hope that one day one will stop making such a fuss, largely for a simple chronological chance ..

204. Hector Macdonald (1865-1935)
• The problem was simple.

205. James Booth (1806-1878)
• I was then led to the discovery of a simple method and compact notation from the following considerations.

206. Lóránd Eötvös (1848-1919)
• He discovered Eotvos's law of surface tension which states that the temperature coefficient of the molecular surface energy of a liquid is independent of the nature of simple unassociated liquids.

207. Hendrik de Vries (1867-1954)
• He knew how to give fascinating talks about the origins of Analytical Geometry, the misunderstood 'Rough draft for an essay on the results of taking plane sections of a cone' of Desargues, the brilliant young man Blaise Pascal, and especially about Gaspard Monge, who as a student at the Ecole Militaire, using some simple constructions, solved an important problem ..

208. F F P Bisacre (1885-1954)
• In §1 a simple test, using polarized light, for the best setting of a diffraction grating is described.

209. Kenneth Appel (1932-2013)
• I cannot believe that most mathematicians could have accepted our announcement as utterly convincing, although later work in the classification of simple groups showed it to be correct.

210. Mary Somerville (1780-1872)
• Her conversation very simple and pleasing.

211. Bernard de Fontenelle (1657-1757)
• Simple, exact, unaffected, and as varied in their scientific content as the sixty-nine astronomers, chemists, physicists, anatomists and others whom they commemorated, the Eloges exemplify a new literary form, moulded and created by Fontenelle, peculiarly French and still neither easily nor very successfully imitated in other languages ..

212. Richard Fuchs (1873-1944)
• During these years, busy with the release of the publications of his father, he promoted his life's work, the theory of linear differential equations in the complex domain, through its own investigations, and a wide readership benefited from his ability present a clear presentation and a simple argument.

213. Wim Cohen (1923-2000)
• It is an excellent introduction to the power of the regenerative-process approach to queueing theory, especially when it comes to providing simple, intuitively based arguments for well-known results in a general setting.

214. Leone Battista Alberti (1404-1472)
• Polyalphabetic substitution was introduced into diplomatic practice by Alberti, who also invented a simple mechanical device to speed up coding and decoding, consisting of a fixed and a movable ring.

• It appears both as al-Karaji and as al-Karkhi but this is not a simple matter of two different transliterations of the same Arabic name.

216. Julius Plücker (1801-1868)
• Things were not so simple, however, for the chair of mathematics in Berlin had just been filled by Jakob Steiner.

217. Tom Cowling (1906-1990)
• In the important chapter on the non-uniform state for a simple gas, use is made of Enskog's method of solving the integral equation and of Burnett's calculation of certain quantities A and B with the aid of Sonine's polynomials.

218. Eliakim Moore (1862-1932)
• He also studied infinite series of finite simple groups.

219. Irmgard Flügge-Lotz (1903-1974)
• The purpose of this book is to acquaint the reader with the problem of discontinuous control by presenting the essential phenomena in simple examples before guiding him to an understanding of systems of higher order.

220. Édouard Goursat (1858-1936)
• The Cauchy-Goursat theorem states the integral of a function round a simple closed contour is zero if the function is analytic inside the contour.

221. Carlos Benjamin de Lyra (1927-1974)
• From 1956 on he directed a weekly seminar on the subject covering the most important topics of the field as for instance: simple homotopy type, topological localization, homology of fibre spaces, cohomological operations, applications of cohomological operations, Postnikov systems, localization and applications.

222. Augustin Fresnel (1788-1827)
• It was a great chance for Fresnel to put his revolutionary work before the world and he was very confident of his theory since his mathematical deductions from the one simple hypothesis led to results which he had verified experimentally giving a highly accurate agreement between theory and experimental evidence.

223. Pierre-Louis Moreau de Maupertuis (1698-1759)
• These laws, so beautiful and so simple, are perhaps the only ones which the Creator and Organizer of things has established in matter in order to effect all the phenomena of the visible world ..

224. Endre Szemerédi (1940-)
• Of course quite often the proofs using only elementary methods are not simple because one may have to put together basic ingredients in extremely complicated and sophisticated ways.

225. Sigekatu Kuroda (1905-1972)
• Therefore, in formulating these systems, some special conditions to restrict the free application of logic are needed, for instance, simple or ramified type theory, introduced to logic first by Russell, or the restriction of the comprehension axiom of set theory.

226. Jacob Wolfowitz (1910-1981)
• It is also a handy introductory text because of its brief and simple formulations of problems and estimates.

227. Karl Reinhardt (1895-1941)
• The choice of material includes not only the integration of the ordinary simple functions, but contains also the derivation of the fundamental rules for manipulating integrals, such as the methods for introducing new variables and for integrating a product of two functions.

228. Emil Grosswald (1912-1989)
• He also wrote two papers which were published in 1950, the first being On a simple property of the derivatives of Legendre's polynomials while the second was Functions of bounded variation.

• The topics are: positive and negative numbers; zero; the unknown; surds; the kuttaka; indeterminate quadratic equations; simple equations; quadratic equations; equations with more than one unknown; quadratic equations with more than one unknown; operations with products of several unknowns; and the author and his work.

230. Emanuels Grinbergs (1911-1982)
• By 1954 he was allowed to lecture at the University of Latvia and in 1956 he defended a second thesis (to replace the one declared void by the authorities) Problems of analysis and synthesis of simple linear circuits.

231. Giuseppe Biancani (1566-1624)
• Before the work could be published, Biancani had to remove the description of Galileo's work on floating bodies, and replace it with a simple reference indicating where Galileo's theory could be found.

232. Willem 'sGravesande (1688-1742)
• the theory of matter, elementary mechanics, the five simple machines, Newton's laws of motion, gravity, central forces, hydrostatics, hydraulics, sound, and wave motion.

233. Hermann Grassmann (1809-1877)
• By this demonstration Grassmann also undermined the notion that language developed from an analytic to a synthetic structure through [combining simple words without changing their form to make new words].

234. Alicia Boole Stott (1860-1940)
• She then produced three-dimensional central cross-sections of all the six regular polytopes by purely Euclidean constructions and synthetic methods for the simple reason that she had never learned any analytic geometry.

235. Yves Rocard (1903-1992)
• 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.

236. Pierre Fatou (1878-1929)
• I had during those years a simple and modest pupil, who was a mathematical genius.

237. Oscar Zariski (1899-1986)
• His use of the notions of integral independence, valuation rings, and regular local rings, in algebraic geometry proved particularly fruitful and led him to such high points as the resolution of singularities for threefolds in characteristic 0 in 1944, the clarification of the notion of simple point in 1947, and the theory of holomorphic functions on algebraic varieties over arbitrary ground fields.

238. Guido Fubini (1879-1943)
• In addition to the areas of analysis detailed above, he worked on the calculus of variations where he studied reducing Weierstrass's integral to a Lebesgue integral and also he worked on the expression of surface integrals in terms of two simple integrations.

239. François Budan (1761-1840)
• He did not appeal to the theory of finite differences or to the calculus of these coefficients, preferring to give them "by means of simple additions and subtractions".

240. Yakov Grigorevich Sinai (1935-)
• Sinai's work centres round the grand aim of deriving the basic physical laws which describe the behaviour of many particle systems as a direct consequence of simple rules governing the interaction of individual particles.

241. John Polkinghorne (1930-)
• In particular, the author gives a simple motivation for the complicated definition of the transformation used by the reviewer.

242. Jean-Louis Koszul (1921-2018)
• The first was Sur le troisieme nombre de Betti des espaces de groupes de Lie compacts Ⓣ in which he completed the proof that the third Betti number of a simple compact Lie group is one by studying certain of the exceptional groups.

243. Mario Fiorentini (1918-)
• However [',' E Sernesi, A simple article, in Commutative algebra and algebraic geometry, Ferrara (Dekker, New York, 1999), x-xiii.

244. Jacopo Riccati (1676-1754)
• His way of life was a very simple one, and he travelled very little.

245. Dimitrei D Stancu (1927-2014)
• In some of them, only the cases of 2 or 3 variables are worked out, in order to keep notations reasonably simple.

246. Sheila Power Tinney (1918-2010)
• It is shown that among the cubic Bravais lattices contained in this group the face and body centred ones correspond to a minimum of potential energy, but the simple cubic lattice to a maximum.

247. Virgil Snyder (1869-1950)
• Simple, forceful language is employed throughout, the theorems are models of clear expression and, when a paragraph is completed, its connection with the rest of the subject is apparent.

248. André Weil (1906-1998)
• He didn't want to return to France to avoid being forced into the army, but it was not a simple matter to escape from the war in Europe at this time.

249. 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.

250. Fritz Ursell (1923-2012)
• A critical estimate is only possible when the meteorological conditions are sufficiently simple, but in one selected example it appears that the velocity of propagation is within 5% of the value prescribed by hydrodynamical theory.

• Vladimirov was assigned to assist Leonid Vitalevich Kantorovich calculating critical parameters of certain simple nuclear systems.

252. Edwin Spanier (1921-1996)
• No matter how complex the subject, at the end the reader feels the theorems are the right ones, the hypotheses natural, and the methods as simple as possible.

• For the present it is not known how to obtain simple and geometrically clear theorems on the distribution of the lattice points on the sphere by other methods.

254. Diederik Korteweg (1848-1941)
• The reason is simple, he did not have the necessary qualifications to enter a university even though he had already begun to publish papers while he worked as a school teacher.

255. Norman Ferrers (1829-1903)
• is so simple and instructive, that I am sure every logician will be delighted to meet with it here or elsewhere.

256. George Mackey (1916-2006)
• Of course, in order to avoid excessive pedantry I left many simple arguments to the imagination of the student - especially after the first few chapters.

257. Jean Dieudonné (1906-1992)
• Well the Bourbaki method is very simple-we cut the threads.

258. George Boole (1815-1864)
• Boole approached logic in a new way reducing it to a simple algebra, incorporating logic into mathematics.

259. André Bloch (1893-1948)
• Although there is a simple classification of Riemann surfaces (hyperbolic, elliptic, parabolic), any specific Riemann surface can be a nasty, brutish, intricate object.

260. Michelangelo Ricci (1619-1682)
• However, this is not quite as simple as it might at first appear since, when the Pope informed him that he would be made a cardinal, Ricci politely and humbly replied to the Pope in a long letter refusing to accept the position:- .

261. Jeremiah Horrocks (1618-1641)
• Horrocks purchased a simple telescope which he set up to project an image of the sun onto a graduated circle six inches in diameter.

262. Dimitri Fedorovich Egorov (1869-1931)
• In this paper, in addition to the independent, very elegant and simple solution of the problem proposed, the originality and logical rigour of the exposition of the basic general geometrical principles deserve special mention, as does also the very successful working out of many details.

263. Wolfgang Pauli (1900-1958)
• he had a genius of fastening on some one point which could be made simple, and so presented was seen at once to be important.

264. Solomon Grigoryevich Mikhlin (1908-1990)
• Its essence relies on the possibility of substituting the kernel of the integral operator by its variational-difference approximation, so that the resolvent of the new kernel can be expressed by simple recurrence formulae.

265. Edwin Olds (1898-1961)
• Finally a third test procedure is developed by using the Neyman-Pearson Lemma for testing simple hypotheses.

266. Maria Agnesi (1718-1799)
• She is a girl of about twenty years of age, neither ugly nor pretty, with a very simple and very sweet manner.

267. 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.

268. Tommaso Ceva (1648-1737)
• This academy was founded in Rome in 1690 to promote a more natural, simple poetic style.

269. David Spence (1926-2003)
• The second extension appeared in his paper Some simple results for two-dimensional jet-flap aerofoils which was also published in 1958.

270. Joseph Raabe (1801-1859)
• 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.

271. Simon Stevin (1548-1620)
• Before presenting the numerical tables, Stevin gave rules for simple and compound interest and also gave many examples of their use.

272. Stanisaw wierczkowski (1932-2015)
• Although the discipline was congenial, its methods were not! He writes [',' S Świerczkowski, Looking Astern autobiography.','1]: "Mathematically the work was simple.

273. Georges Buffon (1707-1788)
• Voltaire did not appreciate his style, and d'Alembert called him "the great phrasemonger." According to the writer J-F Marmontel, Buffon had to put up with snubs from the mathematicians, chemists, and astronomers, while the naturalists themselves gave him little support and some even reproached him for writing ostentatiously in a subject that required a simple and natural style.

274. Niels Abel (1802-1829)
• It was a monument resplendent in its simple lines - the main theorem from his Paris memoir, formulated in few words.

275. Wilhelm Lexis (1837-1914)
• It posed the question of whether an empirical index of dispersion is consistent with the assumption that sex is governed by a simple chance mechanism.

276. Oliver Byrne (1810-1880)
• He published A Treatise on Diophantine Algebra in 1831 but he referred to this work as "A Treatise on Algebra" in the 35-page pamphlet A Short Practical Treatise on Spherical Trigonometry: Containing a Few Simple Rules, by which the Great Difficulties to be Encountered by the Student in this Branch of Mathematics are Effectually Obviated which he published in 1835.

• There is no simple relation between the length of these two types of years hence they were to be calculated as exactly as possible.

278. Paul Cohen (1934-2007)
• He made mathematics look simple and unified.

279. Avicenna (980-1037)
• In his work Mi'yar al-'aqul ibn Sina defines simple machines and combinations of them which involve rollers, levers, windlasses, pulleys, and many others.

280. Brian Hartley (1939-1994)
• Although in a different area of group theory from Hartley, John Thompson was also at Chicago and had just gained world fame with his 1963 paper, written with Walter Feit, proving all nonabelian finite simple groups were of even order.

281. Archimedes (287 BC-212 BC)
• It is not possible to find in all geometry more difficult and intricate questions, or more simple and lucid explanations.

• The exercises are deliberately not "graded" - after all the problems we meet in mathematical "real life" do not come in order of difficulty; some of them are very simple illustrative examples; others are in the nature of tutorial problems for a conventional course, while others are quite difficult results.

283. Frank Cole (1861-1926)
• He published The linear functions of a complex variable in the Annals of Mathematics in 1890 then, between the years 1891 to 1893, he found the complete list of simple groups with orders between 200 and 600.

284. William Whewell (1794-1866)
• It secures me a comfortable establishment for life at least so long as my life is a simple one.

285. Agnes Mary Clerke (1842-1907)
• It has thus become practicable to describe in simple language the most essential parts of recent astronomical discoveries.

• Clearly this simple device was not understood at the time.

287. Evgeny Sergeevich Lyapin (1914-2005)
• His classes were always full of deep concepts and new ideas expressed in a simple, rigorous, and crisp form.

288. Henri Poincaré (1854-1912)
• remainder of the thesis is a little confused and shows that the author was still unable to express his ideas in a clear and simple manner.

289. Leone Burton (1936-2007)
• A simple grouping of these words and phrases characterizes Leone for us very effectively: (i) Leone was a sensitive and caring friend, good company, generous, kind, warm, honest, wise and supportive.

290. Aleksei Krylov (1863-1945)
• is to present simple methods of composition of the secular equation in the developed form, after which, its solution, i.e.

291. Robert Fricke (1861-1930)
• Fricke's long experience with the latter subject made it easy for him to give a simple authoritative exposition of those portions of it which suffice for the transcendental solutions of equations of low degrees.

292. Sergei Novikov (1938-)
• simple, elegant and natural.

293. Frank Harary (1921-2005)
• As a rule the theory is presented as a sequence of simple theorems, each with a clear and precise proof.

294. Bent Christiansen (1921-1996)
• Yet his great integrity led him to be intolerant of injustice, of those who were rude, self-seeking, inefficient and not disposed to think, and of those who peddled simple solutions to complex problems.

295. Giuseppe Veronese (1854-1917)
• He illustrated the fact that difficulties arose when a simple surface in high dimension was projected onto 3-space.

296. Thomas Hirst (1830-1892)
• Yet with all his aloofness of manner he could be very simple, very patient, and extremely kind.

297. Athanase Dupré (1808-1869)
• In 1866 Athanase and Paul Dupre jointly published the article On the law of the union of simple substances, and on attractions at small distances.

298. Witold Hurewicz (1904-1956)
• In this book it has been the aim of the authors to give a connected and simple account of the most essential parts of dimension theory.

299. Henry More (1614-1687)
• More does not deny this fact which any simple experiment will verify, but he claimed that the motion of the second ball is from an internal property of its own, awakened by the impact of the first ball.

300. Yozo Matsushima (1921-1983)
• Zassenhaus had conjectured that every semisimple Lie algebra L over a field of prime characteristic, with [L, L] = L, is the direct sum of simple ideal and Matsushima was able to construct a counterexample.

301. Charles-François Sturm (1803-1855)
• Sturm achieved fame with his paper which, using ideas of Fourier, gave a simple solution.

302. Leopold Löwenheim (1878-1957)
• Simple devices, some of which may be useful in investigating the intuitionistic validity of propositions.

303. Dmitry Aleksandrovich Grave (1863-1939)
• He could explain deep mathematical ideas in a remarkably clear and simple way, and this talent led to a large number of students attending his lectures.

304. Tommaso Boggio (1877-1963)
• He was a modest man, with simple ways and needs, yet he was strong and decent, friendly towards his colleagues and kind to his students.

305. George E Andrews (1938-)
• His list of publications continued to grow with the paper A simple proof of Jacobi's triple product identity (1965) appearing before three papers were published in the following year based on the work of his doctoral thesis on mock theta functions and partitions.

306. David Rittenhouse (1732-1796)
• He published A method of finding the sum of several powers of the sines in 1793 and in a paper of 12 August 1795 he gave an expansion of log10 n, where n a positive number, as a simple continued fraction and then computed log10 99 to nine decimal places.

307. Bernard Lamy (1640-1715)
• The fourth part examines style in a larger sense: imagination, memory, and judgment as the basis of good style; the three levels of style; the lofty, the simple, and the middle; and the differences between styles of an orator or preacher, a historian, and a poet.

308. Edmund Landau (1877-1938)
• Written with the greatest care, Landau's books are characterised by argumentation which is complete, and as simple as possible.

309. John Maynard Keynes (1883-1946)
• his axioms are good; they are simple and few and by the aid of the symbolism he deduces the whole subject from them by rigid reasoning.

310. Brooke Benjamin (1929-1995)
• a careful and thorough analysis of the flows over a simple harmonic wavy boundary which is either (a) rigid (b) a flexible solid or (c) completely mobile, as if it were the interface with a second fluid.

311. George Atwood (1745-1807)
• Atwood is best known for a work A Treatise on the Rectilinear Motion and Rotation of Bodies (published by Cambridge University Press in 1784) which is a textbook on Newtonian mechanics describing impact and simple harmonic motion.

312. Gaston Darboux (1842-1917)
• are as pure as they are simple and beautiful.

313. Eugenio Levi (1883-1917)
• The work of E E Levi almost always deal with issues of fundamental importance: he was not discouraged by the difficulties, even major ones, encountered by other mathematicians, and with a more profound analysis, often very simple and ingenious, was able to clarify and overcome them.

314. Carlo Bonferroni (1892-1960)
• To indicate the interest in this area we note that an generalisation of Bonferroni's inequalities by S Holm in the paper A simple sequentially rejective multiple test procedure published in the Scandinavian Journal of Statistics 6 (1979), 65-70, has received around 2000 citations.

315. Andrew Wiles (1953-)
• it was so indescribably beautiful, it was so simple and so elegant, and I just stared in disbelief for twenty minutes, then during the day I walked round the department.

316. Henry Scheffé (1907-1977)
• For models possessing this property, it turns out that both testing and estimation become particularly simple.

317. Walter Shewhart (1891-1967)
• About a third of that page was given over to a simple diagram which we would all recognize today as a schematic control chart.

318. Guido Stampacchia (1922-1978)
• The personality of Stampacchia was both strong and simple, open and helpful.

319. Gottfried Köthe (1905-1989)
• Linear functionals are of course studied intensively here; while the standard simple Banach spaces ..

320. Martin Gardner (1914-2010)
• I think of myself as like a person who loves classical music, but whose talents never advanced beyond playing simple tunes on a musical saw.

321. Olli Lehto (1925-)
• It led in due time to a simple solution of the geometric problem of moduli, and there are encouraging signs of a fruitful theory in several dimensions.

322. Paul Bachmann (1837-1920)
• The author is usually scrupulous in crediting even simple and commonly current results to their original publisher.

323. Udita Narayana Singh (1920-1989)
• We have given a simple overview of Udita Narayana Singh's career, but a detailed account is given in [',' B S Yadav, U N Singh: His life and Work, Indian Journal of Mathematics 33 (1991), i-xxiv.','1].

324. Gaspard de Prony (1755-1839)
• The present generation would never have witnessed the end of this monumental work if M de Prony had not had the fortunate idea of applying the powerful method of division of labour, conceiving methods to reduce the long and laborious part of the production of the tables to simple additions and subtractions..

325. Félix Savart (1791-1841)
• For example, he would use, in combination, wheels with numbers of teeth which bore a simple relationship to each other.

326. Gilbert Hunt (1916-2008)
• "Two maximal abelian subgroups of a compact connected Lie group G are conjugate within G." I present a simple metric proof.

327. Alonzo Church (1903-1995)
• He published A formulation of the simple theory of types in 1940 in which he attempted to give a system related to that of Whitehead and Russell's Principia Mathematica which was designed to avoid the paradoxes of naive set theory.

328. Victor Amédée Lebesgue (1791-1875)
• Very simple manners, of a character full of frankness, and independence, virtuous at every test, never seeking the opportunities of putting himself to the fore, Lebesgue lived a very solitary life, constantly occupied with his favourite studies.

329. Ernest Esclangon (1876-1954)
• The method, so simple in principle, was not made a practical success without several years of experimenting.

330. Robert Recorde (1510-1558)
• He therefore wrote all his books in English and, in addition, he tried to use clear and simple expressions.

331. Hans Wussing (1927-2011)
• the lively, clear, and simple style nicely conveys its main message: that mathematics is a human pursuit whose aims and motivations can be understood by everyone.

332. Euphemia Lofton Haynes (1890-1980)
• I give and devise unto my son, Joseph William Lofton, and unto my daughter, Martha Euphemia Haynes, in fee simple, as tenants in common, Lots Twenty Three (23) and Twenty Four (24), in Square One Hundred and ninety six (196), improved by premises No.

333. Hermann Weyl (1885-1955)
• There I attended his lectures on the Elie Cartan calculus of differential forms and their application to electromagnetism - eloquent, simple, full of insights.

334. Otto Schreier (1901-1929)
• His first paper in 1924 On the groups AaBb = 1 gave a simple algebraic proof of a theorem on knot groups, which generalised a theorem given by Max Dehn ten years earlier that the trefoil knot and its mirror image are not equivalent.

335. Beno Eckmann (1917-2008)
• Peter Hilton, who had been a personal friend of Eckmann's for many years spoke in detail of Eckmann's research in topology: continuous solutions of systems of linear equations, a group-theoretical proof of the Hurwitz-Radon theorem, complexes with operators, spaces with means, simple homotopy type.

• Difficult modern theories become quite clear and simple in his exposition.

337. R A Fisher (1890-1962)
• In fact the reasons for the feud were not nearly as simple as those usually given.

338. Donald Eperson (1904-2001)
• This was probably because they were a kind of word puzzle whose solution depended upon finding words with simple rhythms that fitted into a musical framework of pentameters and hexameters.

339. William Threlfall (1888-1949)
• In our conception of space a simple-minded idea of continuity comes before everything else.

340. Robert Carmichael (1879-1967)
• This simple and logical account will serve a useful purpose by showing what assumptions we are in the habit of making, and wherein these admit of modification without contradicting the evidence of our senses.

341. Attia Ashour (1924-2017)
• Simple explanations are suggested for some known features of micropulsations, and for some well-known phenomena of magnetic disturbance, including Sangster's rotating disturbance vector.

342. John Semple (1904-1985)
• Projective geometry is a subject that lends itself naturally to algebraic treatment, and we have had no hesitation in developing it in this way - both because to do so affords a simple means of giving mathematical precision to intuitive geometrical concepts and arguments, and also because the extent to which algebra is now used in almost all branches of mathematics makes it reasonable to assume that the reader already possesses a working knowledge of its methods.

343. André Lichnerowicz (1915-1998)
• Apart from its intrinsic merits, not the least of which is its simple and clear style, the book therefore provides a good introduction to the works of Cartan ..

344. Hans Hahn (1879-1934)
• He wrote papers on the theory of curves including one which gave a rigorous proof of the Jordan's theorem for simple closed polygons which he based on Veblen's geometrical axioms.

345. Ernst Öpik (1893-1985)
• He steered me towards planets and satellites, and taught me to use simple physical principles in place of more obscure mathematical approaches.

346. Edwin Pitman (1897-1993)
• Using the familiar fact that two simple linear combinations of two normally correlated variates are independently and normally distributed, an exact test is derived for the significance of the ratio of sample variances in samples from a normal bivariate population.

347. Bibhutibhushan Datta (1888-1958)
• He lived a simple itinerant life over the following years, drifting from place to place.

348. Nathan Jacobson (1910-1999)
• Florie did not give up mathematics for she was a joint author with her husband on their 1949 paper Classification and representation of semi-simple Jordan algebras.

349. Carl Neumann (1832-1925)
• with one and the same relatively simple mathematical expression." He noted with satisfaction that Neumann had required many hypotheses to reach a similar result.

350. Pierre Humbert (1891-1953)
• Moreover, he was unsatisfied with the simple juxtaposition of knowledge and religious faith.

351. Enzo Martinelli (1911-1999)
• For example, in order to be able to provide simple and interesting examples, I soon adopted cellular cross-links, while postponing the justification of their use until later.

352. Paul Guldin (1577-1643)
• A rotation is a simple and perfectly circular motion, around a fixed centre, or an unmoved axis, which is called the 'axis of rotation', turning around either a point, or a line, or a plane surface, which, almost as leaving a trace behind it, describes or generates a circular quantity, either a line, or a surface, or a body.

• he discussed for instance the difference between the broader meaning of the word element (in which any proposition leading to another may be said to be an element of it) and the stricter meaning of something simple and fundamental standing to consequences drawn from it in the relation of a principle, which is capable of being universally applied and enters into the proof of all manner of propositions.

354. Karl Sundman (1873-1949)
• Its adaptation for these other aims will be relatively simple to achieve.

355. Arthur Schönflies (1853-1928)
• He introduced the topological notions of accessible point, closed curve and simple closed curve.

356. Sharaf al-Din al-Tusi (about 1135-1213)
• a simple wooden rod with graduated markings but without sights.

357. William Berwick (1888-1944)
• Berwick was an algebraist who worked on the problem of computing an integral basis for the algebraic integers in a simple algebraic extension of the rationals.

358. James Clerk Maxwell (1831-1879)
• Maxwell showed that a few relatively simple mathematical equations could express the behaviour of electric and magnetic fields and their interrelation.

• He discovered many propositions himself, and instructed his successors in the principles underlying many others, his method of attacking problems had greater generality in some cases and was more in the nature of simple inspection and observation in other cases.

360. Rudolf Kalman (1930-)
• Randomness does, and to capture it better he proposes a new definition: random is not uniquely determined by simple classical rules.

361. Gerbert of Aurillac (946-1003)
• So astonishing was his skill, that the simple folk of his day, in sheer bewilderment, accepted without question the belief that his knowledge was universal ..

362. Heisuke Hironaka (1931-)
• Some fundamental theorems in the theory of several complex variables and of the geometry of complex manifolds are proved in a simple but rigorous form.

363. Lothar Collatz (1910-1990)
• The Collatz problem is simple to state.

364. Erland Bring (1736-1798)
• The coefficient of y2 is rather simple, m2 - mb - 2n + 3a, but the next coefficient has degree three and nine terms.

365. Johann Franz Encke (1791-1865)
• They bear strong and uniform testimony to his eminent frankness and truthfulness; his labours, they say, were incessant, his recreations few; he was simple in his manners, and in all his habits temperate.

366. Dorothea Beale (1831-1906)
• She also suggests that girls aged sixteen to eighteen should study advanced pure and applied mathematics, which is quite different to the simple arithmetic previously thought acceptable.

367. Arnold Sommerfeld (1868-1951)
• In the evenings, when the simple meal was cooked, the dishes were washed, the weather and snow properly discussed, the talk invariably turned to mathematical physics, and this was the occasion for the receptive students to learn the master's inner thoughts.

368. Benjamin Moiseiwitsch (1927-2016)
• It begins with a straightforward account, accompanied by simple examples of a variety of integral equations and the methods of their solution.

369. Shreeram Shankar Abhyankar (1930-2012)
• He broke this rule the first time when he used the classification of finite simple groups.

370. Jean-Baptiste Bélanger (1790-1874)
• Belanger provided a stepwise integration of this equation in the simple case of the horizontal aqueduct that had been built recently to bring the waters of the Ourcq River into Paris.

371. Urbain Le Verrier (1811-1877)
• One should have seen M Lescarbault, so small, so simple, so modest, and so timid, in order to understand the emotion with which he was seized, when Le Verrier, from his great height, and with that blunt intonation which he can command, thus addressed him: "It is then you, Sir, who pretend to have observed the intra-Mercurial planet, and who have committed the grave offence of keeping your observation secret for nine months.

372. Rajeev Motwani (1962-2009)
• His lectures were so perfectly crafted, from the progression of describing a simple approach providing the intuition to generalizing it, to doing an impeccable formal analysis, to the perfect board technique, that I left every lecture excited about a new powerful topic that I have just learned and understood.

373. Krystyna Kuperberg (1944-)
• The paper is an important contribution to the theory of dynamical systems, and it solves in a simple but elegant way the long-standing Seifert conjecture.

374. Leslie Woods (1922-2007)
• If F is harmonic or is a solution to Poisson's equation, it may have singular points in the field or on the boundary at which it (a) has finite values, but has infinite derivatives, (b) has logarithmic infinities, or (c) has simple discontinuities.

375. Albert Einstein (1879-1955)
• said hardly anything beyond presenting a very simple objection to the probability interpretation ..

376. John Aitchison (1926-2016)
• Recognition that the study of compositions must satisfy simple principles has led recently to the advocacy of new forms of analysis of compositional data.

377. Roberto Frucht (1906-1997)
• Our object in this note is to construct a new and simple operation on two graphs G1 and G2, called their corona, with the property that the group of the new graph is in general isomorphic with the wreath product of the groups of G1 and G2.

378. Sixto Ríos (1913-2008)
• One day Rios placed on the chalkboard, before Barinaga arrived, an easy and simple solution he had found to one of the difficult problems.

379. Hanno Rund (1925-1993)
• The theoretical physicist is shown how the theory of non-homogeneous single integral problems give rise to relativistic particle mechanics, in which the special invariant Hamiltonian function permits a particularly simple method of quantization, from which the relativistic wave equations (Dirac, Kemmer, etc.) may be obtained directly.

380. Gordon Whyburn (1904-1969)
• The theory was based on cyclic elements, that is a region C such that any two points of C are contained in a simple closed curve of C.

381. Josip Plemelj (1873-1967)
• Another contribution that we should mention was Plemelj's simple proof of the n = 5 case of Fermat's Last Theorem which he published in 1912.

382. Charles Chree (1860-1928)
• etc.; there are thus enormous obvious differences between the simple mathematical problems of my papers and the actual state of matters on the earth; and if there is any resemblance between the results in the papers and actualities, it may quite as likely be a pure accident as not.

383. Lejeune Dirichlet (1805-1859)
• His proofs characteristically started with surprisingly simple observations, followed by extremely sharp analysis of the remaining problem.

384. James Alexander (1888-1971)
• He returned to Princeton where he submitted his dissertation Functions which map the interior of the unit circle upon simple regions and, in 1915, was awarded his Ph.D.

385. Dimitrie Pompeiu (1873-1954)
• This simple remark has led to many interesting problems in analysis known as the problem of Pompeiu.

386. Chukwuka Okonjo (1928-)
• The concepts of growth, population and simple growth; III.

387. Hugh Dowker (1912-1982)
• This first step is here reduced to a simple algorithm suitable for computer use.

388. Thomas Flett (1923-1976)
• Mean value theorems of differential and integral calculus provide a relatively simple, but very powerful tool of mathematical analysis suitable for solving many diverse problems.

389. Alfred Goldie (1920-2005)
• In fact Goldie's first paper in this area Decompositions of semi-simple rings (1956) made an immediate impact since Jacobson included one of Goldie's theorems in his classic monograph Structure of Rings of 1956, acknowledging that it had been communicated by Goldie.

390. Pieter Hendrik Schoute (1846-1913)
• This paper may be regarded as a continuation of [On the Angles of the Regular Polytopes of Four-Dimensional Space]; it is concerned with polytopes of S4 characterized by the property of admitting one kind of vertex and one length of edge, which polytopes will be called "semiregular." These polytopes, corresponding to the Archimedian semiregular polyhedra of ordinary space, have been deduced from the regular ones by very simple geometrical operations called "expansions" and "contractions" in a masterly memoir of A Boole Stott; they will be indicated here by the symbols introduced in that memoir.

391. Guido Ascoli (1887-1957)
• Ascoli does not limit himself to dryly following the guidelines for university courses, but attempts to "discern in the admirable edifice of concepts and results that small bit that is essential in the very first study from what would otherwise be destined to remain a lifeless and inexpressive knowledge; to present to young people a simple and harmonic organism of fundamental ideas that contribute effectively towards their intellectual formation" (Preface).

392. Ron Book (1937-1997)
• [He] did not like large towns, he did not like the sophistication of the traditional elite, he liked simple people working day after day to grow crops.

393. Gustav Elfving (1908-1984)
• The results had a simple geometric interpretation and were computationally easy before computer technology was highly developed.

394. Roger Cotes (1682-1716)
• a new sort of construction in geometry which appear to me very easy, simple and general.

395. William Jones (1675-1749)
• An obvious question would be: why was his father not named 'Jones'? The answer is simple, he was named Jones since this is the English version of the Welsh Sion.

396. René Eugène Gateaux (1889-1914)
• Gateaux's body was buried near the St Anne Chapel in Rouvroy, a simple cross without inscription marking the place.

397. Charles Graves (1812-1899)
• This is a simple error arising from the fact that, as Bishop of Limerick, Graves would sign himself Charles Limerick or C Limerick.

398. Gilbert Bliss (1876-1951)
• The book starts with a typical simple problem, the non-parametric problem in 3-space with fixed end-points.

399. Andre-Louis Cholesky (1875-1918)
• The problem of adjusting the grid greatly worried officers in the Geographical Service, who were anxious to find a method which was simple, fast and precise.

400. Abigail Thompson (1958-)
• Surprisingly, any closed orientable 3-manifold can be split into two simple pieces, called handlebodies.

401. Heinz Prüfer (1896-1934)
• In it Prufer gives a very simple proof of an expansion theorem for a particular second order linear homogeneous differential equation coming from the oscillation and evolution theorem.

402. Rudolf Peierls (1907-1995)
• Surprises in theoretical physics are either theoretical results in disagreement with naive physical intuition, or simple solutions to apparently unmanageable problems.

403. Alfred Pringsheim (1850-1941)
• He gave a very simple proof of Cauchy's integral theorem.

404. Girolamo Cardano (1501-1576)
• In the same year, Cardan's first two mathematical books were published, the second The Practice of Arithmetic and Simple Mensuration was a sign of greater things to come.

405. Petre Sergescu (1893-1954)
• They were exceedingly hospitable to friends, and it was my privilege to enjoy their simple but gallant welcome when I visited them in their little home in the rue Daubenton no.

406. Grace Alele-Williams (1932-)
• Teaching the teachers mathematics is a relatively simple task but changing their attitude and practice is harder.

407. Judita Cofman (1936-2001)
• There she continued her work on finite geometries and published papers such as: On a characterization of finite desarguesian projective planes (1966); Double transitivity in finite affine planes (1967); Triple transitivity in finite Mobius planes (1967); Translations in finite Mobius planes (1968); On Baer involutions of finite projective planes (1970); and Simple groups and Mobius planes of even order.

408. al-Kashi (1390-1450)
• Al-Kashi can no longer be considered as the inventor of decimal fractions; it remains nonetheless, that in his exposition the mathematician, far from being a simple compiler, went one step beyond al-Samawal and represents an important dimension in the history of decimal fractions.

409. Samarendra Nath Roy (1906-1964)
• He was a man who practiced simple living and high thinking.

410. Mikhail Egorovich Vashchenko-Zakharchenko (1825-1912)
• That a man so well acquainted with modern investigations of the principles of the science of space as Mr Vashchenko-Zakharchenko (a bibliography of this subject is also appended to the volume) should prove such an ardent adherent of Euclid, pure and simple, for the schools, is a truly remarkable fact.

411. Lazarus Fuchs (1833-1902)
• They lived in a great number of different houses, forced to live a very simple and modest life-style since, especially in the first few years, they lived off the income that Fuchs made through giving private lessons.

412. Charles Weatherburn (1884-1974)
• I cannot agree with those who would make nature more akin to the complex than to the simple.

413. Charles Coulson (1910-1974)
• From this simple experimental fact has developed the whole science of electrostatics, that is the properties of electricity at rest.

414. Louis de Branges (1932-)
• The proof is now available in a form that can be verified by any experienced mathematician as analysis that is "hard" in the original aesthetic sense of Hardy - simple algebraic manipulations linked by difficult inequalities.

415. Karl Aubert (1924-1990)
• The most important aspect of Aubert's research was the basic ideas, the simple and general concepts, that he introduced.

416. Duilio Gigli (1878-1933)
• How many of the combinations of n integers from 1 to m have the same sum s, and how many have a sum ≤ L? With the help of a generating function, the author solves these problems in a simple way.

417. Wilhelm Meyer (1856-1934)
• gave lectures discussing the essential aspects of mathematical research in the spirit of Klein's Erlangen programme, and gave lectures discussing the essential aspects of mathematical research in the spirit of the time and emphasizing the importance of simple algebraic identities, the symmetries of group theory, and transformation principles as a source of geometric theorems.

418. William Wager Cooper (1914-2012)
• In sum, most of the features of linear programming are illuminated by a deceptively simple example.

419. Derrick Norman Lehmer (1867-1938)
• The following course is intended to give, in as simple a way as possible, the essentials of synthetic projective geometry.

420. Daniel Bernoulli (1700-1782)
• While in St Petersburg he made one of his most famous discoveries when he defined the simple nodes and the frequencies of oscillation of a system.

• Based on his observations of solstices and equinoxes, Ptolemy found the lengths of the seasons and, based on these, he proposed a simple model for the sun which was a circular motion of uniform angular velocity, but the earth was not at the centre of the circle but at a distance called the eccentricity from this centre.

422. John Backus (1924-2007)
• I remember doing relatively simple calculations to get a few points on a curve for an amplifier.

423. Giacinto Morera (1856-1907)
• He developed the study of the harmonic functions, applying results due to Pizzetti, finding a simple expression for the inner and outer gravitational field of an ellipsoid, solving the Dirichlet problem.

424. Elizabeth Fennema (1928-)
• However, things were not so simple.

425. Pietro Cataldi (1548-1626)
• We note that he does not use what are called today 'simple continued fractions' and his method will always give a continued fraction for a square root which has period 1.

426. Carl Johannes Thomae (1840-1921)
• The rules of chess are arbitrary; the system of rules for arithmetic is such that by means of simple axioms the numbers can be related to intuitive manifolds, so that they are of essential service in the knowledge of nature.

427. Joan Sylvia Lyttle Birman (1927-)
• A third will be the unifying principles provided by representations of simple Lie algebras and their universal enveloping algebras.

428. Roger Godement (1921-2016)
• Resume de lecons Ⓣ (1959); Cours d'Algebre Ⓣ (1963); (with Herve Jacquet) Zeta functions of simple algebras Ⓣ (1972); Introduction a la theorie des groupes de Lie Ⓣ (1982); Analyse Mathematique I.

429. Georg Simon Ohm (1789-1854)
• This may have no simple explanation but rather be the result of a number of different contributary factors.

430. Paul Mansion (1844-1919)
• studied, by a particularly simple original method, the multiplication and transformation of elliptic functions.

431. James Wiegold (1934-2009)
• In the first of these two papers he proved that a group G has the property that every normal subgroup is a direct factor if and only if G is a restricted direct product of simple groups, while the second paper extended the results of his earlier paper Nilpotent products of groups with amalgamations.

432. Félix Savary (1797-1841)
• Although this might appear to be a fairly simple consequence of Newton's law of gravitation, nevertheless it was important for it was the first verification of the laws for objects outside the solar system.

433. John Bell (1928-1990)
• As a simple example, the state-vector above might apply to an ensemble of many systems, but in addition a hidden variable for each system might say what the actual value of sz might be.

434. Hyman Bass (1932-)
• The (J H C) Whitehead torsion, introduced in order to study the topological notion of simple homotopy type, leads to the groups K1.

435. Georgios Remoundos (1878-1928)
• He considered his teaching work, according to his own words, "as a sacred duty of utmost importance." Cyparissos Stephanos, who taught Remoundos said that, "Remoundos is so clear and simple in his teaching, and so effective that one can say that he opens the head of the student, puts the mathematics in, then locks the head and takes the key, and leaves reassured!" .

436. Anneli Lax (1922-1999)
• When I think of that day I carry in my heart many of the things I loved about Anneli - her quiet determination, her openness and acceptance of the weaknesses of others and her joy in simple pleasures.

437. Hideo Tanaka (1938-2012)
• Possibility data analysis offers not only the general methodology to analyze and model the uncertainty in operations research but also the common and simple way to solve the problems.

438. Heraclides (387 BC-312 BC)
• It is an unconvincing article and it seems to only repeat van der Waerden's earlier hypothesis without making any attempt to counter the rather simple and totally convincing argument by Neugebauer.

439. Cyrus Colton MacDuffee (1895-1961)
• He continued to publish on rings and algebras with papers such as A correspondence between matrices and quadratic ideals (1927), An introduction to the theory of ideals in linear associative algebras (1929), The discriminant matrix of a semi-simple algebra (1931), and Matrices with elements in a principal ideal ring (1933).

440. Oskar Bolza (1857-1942)
• Papers which appeared in the Transactions of the American Mathematical Society over the next few years were: New proof of a theorem of Osgood's in the calculus of variations (1901); Proof of the sufficiency of Jacobi's condition for a permanent sign of the second variation in the so-called isoperimetric problems (1902); Weierstrass' theorem and Kneser's theorem on transversals for the most general case of an extremum of a simple definite integral (1906); and Existence proof for a field of extremals tangent to a given curve (1907).

441. John Pople (1925-2004)
• is devoted to theoretical principles and experimental methods and the authors have achieved a comprehensive and yet simple account of what can be a difficult subject.

• In it he presented a very simple algebraic approach but was aware of its limitations writing:- .

443. Phyllis Nicolson (1917-1968)
• Crank and Nicolson's method, which is numerically stable, requires the solution of a very simple system of linear equations (a tridiagonal system) at each time level.

444. John Dougall (1867-1960)
• By regarding Q as a 4-sphere in complex Euclidean 5-space, and making some projections, he relates this to a simple theorem of plane geometry: .

445. Alan Turing (1912-1954)
• In one sense 'decidability' was a simple question, namely given a mathematical proposition could one find an algorithm which would decide if the proposition was true of false.

446. Harald Cramér (1893-1985)
• ','1] sums up Cramer's contribution with simple but effective words:- .

447. Klaus Fuchs (1911-1988)
• For the symmetric group on n symbols, there is a procedure for constructing the simple matrix representation corresponding to a given partition of n [cf.

448. Annibale Giordano (1769-1835)
• Mathematics being of this kind, is not in effect, having the development of simple ideas constituting the ideas of different magnitudes we know, and of some of our primitive conventions above them.

449. Kunihiko Kodaira (1915-1997)
• Thus, Cauchy's integral formula is first proved in a topologically simple case from which the author deduces the basic properties of holomorphic functions.

• It was not a simple modification that al-Farisi made.

451. Louis Antoine (1888-1971)
• Antoine was trying to prove a three-dimensional analogue of the Jordan-Schonflies theorem, which says that, given a simple closed curve in the plane, there exists a homeomorphism of the plane that takes the curve into the standard circle.

452. Thoralf Skolem (1887-1963)
• It characterizes the automorphisms of simple algebras and was later rediscovered by Emmy Noether.

453. Barnabé Brisson (1777-1828)
• [They] applied descriptive geometry to the actual geography of France [discovering] a way to find the lowest points in the watersheds between basins by a simple examination of existing topographical maps, which lacked contour lines and all but a few isolated points of altitude.

454. William Clifford (1845-1879)
• If you can say a few more words about my husband I would love it - how brilliant he was, how witty and what an adorable nature he had: he was so gay and simple and light-hearted, and had an indescribable charm.

455. Friedrich Engel (1861-1941)
• But, now, Lie decided to tackle a major piece on transformation groups, which was certainly intended to be much more than a simple introduction to the elements of the theory.

456. Reinhold Baer (1902-1979)
• Baer also had a very positive effect on the development of the Mathematics Department: in particular he was responsible for Michio Suzuki coming to Illinois - a crucial event that led to the Department becoming a centre of research in finite simple group theory.

457. Giovanni Poleni (1683-1761)
• Poleni actually built this machine which was reportedly very simple and easy to operate; but when he heard of another machine presented to the Emperor by the Viennese mechanician Brauer, he destroyed his own and never rebuilt it.

• Peter's appointment at Rutgers University was not a simple move of a professor from one institution to another.

459. Gerard Murphy (1948-2006)
• This association helps in the study of Lie ideals, and is especially useful for studying simple algebras.

460. Wilhelm Fiedler (1832-1912)
• Fiedler never mentions Cremona in his paper, but in a letter to Cremona, at the beginning of 1873, he praises his book and the simple way in which Cremona introduces the topics.

461. Phillip Griffiths (1938-)
• Some of the author's most enduring results come from simple such calculations which show how different higher-dimensional variations of cycles or Hodge structures are, often referred to as Griffiths transversality.

• The powerful, yet simple, Platonic theism and morality which shines out of the De consolatione philosophiae made it extremely popular during the Middle Ages and the Renaissance.

463. Fabian Franklin (1853-1939)
• His simple demeanour and dignity of person commanded the instant respect of his students, a respect which was never lost.

464. Stephen C Kleene (1909-1994)
• Difficult proofs are broken down into a large number of simple cases; some of these cases are usually left to the reader.

465. Olinthus Gregory (1774-1841)
• We do not deny that the scheme of revelation has its difficulties: for if the things of nature are often difficult to comprehend, it would be strange indeed if supernatural matters were so simple, and obvious, and suited to finite capacities, as never to startle or puzzle us at all.

466. Hermann Schubert (1848-1911)
• In all these essays, which are of a simple and popular character, and designed for the general public, Professor Schubert has incorporated much of his original research.

467. Pavel Tichy (1936-1994)
• He was awarded his doctorate in 1959 for his thesis An Exposition of Godel's Incompleteness Theorem in the Simple Theory of Types (Czech).

468. Luigi Fantappiè (1901-1956)
• For example in Deduzione autonoma dell'equazione generalizzata di Schrodinger, nella teoria di relativita finale Ⓣ (1955) Fantappie deduces the Klein-Gordan equation in quantum mechanics as a limit, as the radius of the universe tends to infinity, of a classical (non-quantized) equation in his extension of relativity based on a simple (pseudo-orthogonal) group having the Lorentz group as a type of limit.

469. Karl Menger (1902-1985)
• There is an important phase in the development of modern point set theoretical geometry which has been closely associated with the concept of dimensionality, - we refer to the attempt to create precise mathematical meaning for the simple geometric spaces of our intuition in terms of primitive non-arithmetical concepts.

470. Richard Tapia (1939-)
• It is also shown that this procedure can be applied to a class of two point boundary value problems containing the Euler-Lagrange equation for simple variational problems and most second order ordinary differential equations.

471. Jan Stampioen (1610-1690)
• If this sounds like a particularly modern approach to selling, then let us simple say that human nature does not seem to have changed much over the centuries! .

472. Georges de Rham (1903-1990)
• Formes, courants, formes harmoniques Ⓣ (1955); (with S Maumary and M A Kervaire) Torsion et type simple d'homotopie Ⓣ (1967); and Lectures on introduction to algebraic topology (1969).

473. Jerzy o (1920-1998)
• For example he published A simple proof of the existence of equilibrium in a von Neumann model and some of its consequences (1971), Extended von Neumann models and game theory (1976), and Mathematical theory of von Neumann economic models.

474. Hans Meinhardt (1938-2016)
• It will be shown that a relatively simple set of interactions can explain seemingly complex experimental observations in a quantitative manner.

475. Valentina Mikhailovna Borok (1931-2004)
• In the same period she obtained formulae that made it possible to compute in simple algebraic terms the numerical parameters that determine classes of uniqueness and well-posedness of the Cauchy problem for systems of linear partial differential equations with constant coefficients.

476. Felix Behrend (1911-1962)
• In the same year in Note on the compactification of separated uniform spaces he gave a simple method of obtaining, for any uniform space S, a uniform structure which is totally bounded and compatible with the topology of S.

477. Raymond Smullyan (1919-2017)
• Before the class began, he tried to warm up the group, tossing out some simple puzzles ..

478. Charles Fox (1897-1977)
• The methods of contour integration, however, give extremely simple proofs of these results, and also give rise to many interesting results which, I believe, are new.

479. Roger Apéry (1916-1994)
• It has been my good fortune to find a very simple version of the proof a few months after Apery's announcement.

480. Pascual Jordan (1902-1980)
• Things, however, are not quite as simple as they might appear and one must not think that because Jordan was a staunch and enthusiastic Nazi supporter, he believed in all the Nazi policies.

481. Alexis Clairaut (1713-1765)
• In order to avoid delicate experiments or long tedious calculations, in order to substitute analytical methods which cost them less trouble, they often make hypotheses which have no place in nature; they pursue theories that are foreign to their object, whereas a little constancy in the execution of a perfectly simple method would have surely brought them to their goal.

482. Piero della Francesca (1420-1492)
• Also it is clear that Piero's Italian writing lacked style, and was rather simple and elementary.

483. Gustav Kirchhoff (1824-1887)
• In all his work he strove for clarity and rigour in the quantitative statement of experience, using a direct and straightforward approach and simple ideas.

484. Giovanni Vailati (1863-1909)
• I saw him universally celebrated and requested from all intervening scholars; in the streets, in the pubs, in gatherings and meetings he was always in the middle of a group which he fascinated with his simple, whole-hearted, and nonetheless interesting, informative conversation.

485. Vincenzo Riccati (1707-1775)
• Vincenzo Riccati, somehow, put an end to this trend by showing that one could construct in a simple continuous way all transcendental curves from the differential equations that define them.

486. Gheorghe Calugrenu (1902-1976)
• As a lecturer, Calugăreănu gave simple, clear explanations.

487. Monteiro da Rocha (1734-1819)
• He gave a simple method of calculating a parabolic orbit given three observations which he presented to the Academy of Sciences of Lisbon in 1782.

488. Jorgen Gram (1850-1916)
• This work provided a simple and natural framework for invariant theory.

489. Willem de Sitter (1872-1934)
• This is a particularly simple solution of the field equations of general relativity for an expanding universe.

490. Charlotte Angas Scott (1858-1931)
• When retiring for study after an extremely simple 'tea' in the Common Room, they would pick up three things en route to their rooms ..

• Kuku's first three papers were (i) Some algebraic K-theoretic applications of the LF and NF functors (1973), (ii) Whitehead group of orders in p-adic semi-simple algebras (1973) and (iii) Some finiteness theorems in the K-theory of orders in p-adic algebras (1976).

492. Victor Puiseux (1820-1883)
• His kindness, his charity, and above all his simple, unaffected modesty overshadowed even his talents.

493. Pierre Rémond de Montmort (1678-1719)
• The simple fact that Remond de Montmort and Taylor felt compelled to debate the physics of universal gravitation was an important marker of the changing intellectual climate of the time, yet the appearance of their thoroughly exchange within a journal self-consciously devoted to promoting and provoking public, critical debate of philosophical matters was even more catalysing.

494. Thomas Allen (1540-1632)
• He had a great many mathematical instruments and glasses in his chamber, which did also confirm the ignorant in their opinion; and his servitor [servant] (to impose on Freshmen and simple people) would tell them that sometimes he should meet the spirits coming up his stairs like bees.

495. Alice T Schafer (1915-2009)
• Choice of the proper projective coordinate system permits the reduction of these power series to simple canonical forms.

496. Benjamin Bramer (1588-1652)
• One must not think that the fighting was a simple conflict between Protestant and Roman Catholic forces.

497. Gustav Roch (1839-1866)
• This was not a simple matter, for Roch did not have a particularly strong background in either mathematics or physics, so in order to prepare himself for advanced study he took courses at a private institute as well as at the Polytechnic Institute.

498. Karl Schwarzschild (1873-1916)
• I had not expected that one could formulate the exact solution of the problem in such a simple way.

499. Hans Schubert (1908-1987)
• In Uber die Potentiale der auf dem Mantel eines Kreiszylinders ausgebreiteten einfachen und doppelten Belegung Ⓣ (1952) he derives a Fourier integral representation containing Bessel functions for the axially symmetric potential induced by a simple and double layer on the surface of a circular cylinder.

500. August Crelle (1780-1855)
• The solution was simple, even if it required a change in policy, and that was to have a second journal for more practical mathematics and this he moved to a second journal which he started in 1829, the Journal fur die Baukunst.

501. Herbert Robbins (1915-2001)
• From a level approximately that of a sound high-school training, the development proceeds by direct paths to some of the best content of mathematics; and fundamental ideas are made strikingly clear by well-chosen, simple examples.

502. Gordon Preston (1925-2015)
• He published papers over the next years such as Chains of congruences on a completely 0-simple semigroup (1965), Matrix representations of inverse semigroups (1969) and Free inverse semigroups (1973).

503. Jean-Pierre Serre (1926-)
• The events started in bright sunshine in Oslo on Sunday, 1 June 2003, with a simple ceremony at the Abel Monument in Slottsparken.

504. Winifred Sargent (1905-1979)
• provided a simple and direct proof for a theorem which is fundamental in the development of the Cesaro-Perron scale of integration.

505. Gaston Julia (1893-1978)
• This book presents a continuation of the first volume of the author dealing with those aspects of the modern theory of functions of a complex variable which are derivable from simple geometrical principles.

506. Lipót Fejér (1880-1959)
• Fejer's theorem is a simple, beautiful theorem, and, in the opinion of Jean-Pierre Kahane [',' J-P Kahane, Leopold Fejer et l’analyse mathematique au debut du XXe siecle, Cahiers du Seminaire d’Histoire des Mathematiques 2 (Inst.

507. Ernest Wilczynski (1876-1932)
• in the midst of a lecture [in 1923] he finally realised that he could go no further and, with a simple statement to that effect, walked from his classroom never to return, leaving his students amazed by the classic self-restraint with which he accepted his tragic fate.

508. Robert Hooke (1635-1703)
• He failed to develop major theories from his inspired ideas for the simple reason that he did not really have the technical ability to develop such comprehensive theories as some of his contemporaries like Newton and Huygens.

509. Angelo Genocchi (1817-1889)
• His explanations were calm, with no repetitions, and he aimed at rigorously presenting the fundamental concepts and studying them so as to arrive at simple procedures and clear exposition.

510. Hans Lewy (1904-1988)
• His paper An example of a smooth linear partial differential equation without solution (1957) gave a simple partial differential equation which has no solution, a result which had a substantial impact on the area.

511. Paul Kelly (1915-1995)
• The book is easy to read, notation is kept simple, and the proofs are clear and complete.

512. Elemér Kiss (1929-2006)
• Their family, as many families at that time, led a very simple life, but they brought up their three children with great love, all of them graduating from university.

513. Ilya Iosifovich Piatetski-Shapiro (1929-2009)
• 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).

514. Marius Lacombe (1862-1938)
• He is not one of those who consider mathematics to be simple gymnastics or an adornment of the mind.

515. Louis Arbogast (1759-1803)
• Do the arbitrary functions introduced when differential equations are integrated belong to any curves or surfaces either algebraic, transcendental, or mechanical, either discontinuous or produced by a simple movement of the hand? Or should they legitimately be applied only to continuous curves susceptible of being expressed by algebraic or transcendental equations? .

516. Max Zorn (1906-1993)
• He studied the structure of semisimple alternative rings in 1932, proving that such a ring is a direct sum of simple alternative algebras which he classified.

517. Elliott Montroll (1916-1983)
• At the Third Berkeley Symposium on Mathematical Statistics and Probability 1954-1955, Montroll gave a paper Theory of the vibration of simple cubic lattices with nearest neighbor interactions in which described vibrations of a cubic lattice with 1, 2, 3, and n dimensions where n is large.

518. Henrietta Swan Leavitt (1868-1921)
• 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.

519. Max Newman (1897-1984)
• At a time when the study of manifolds was based on a number of different combinatory concepts, he established a simple combinatory system of simplicial complexes with an equivalence relation based on elementary moves.

520. David Rees (1918-2013)
• They contain the concept known today as a Rees matrix semigroup which Rees defines and uses to classify completely 0-simple semigroups.

521. Floyd Burton Jones (1910-1999)
• Although one can see how this might have led him to mathematics the route was not as simple as that for the topic which he decided to take up instead of law was chemistry.

522. Maurice d'Ocagne (1862-1938)
• The purpose of Nomography is to reduce to simple readings on graphical charts, constructed once for all, the computations which necessarily intervene in the practice of various technical arts.

• He gives a wide variety of applications including problems involving ratios, barter, simple interest, mixtures, purchase and sale, rates of travel, wages, and filling of cisterns.

524. Herbert Wilf (1931-2012)
• The remarkably simple idea of the work of Wilf and Zeilberger has already changed a part of mathematics for the experts, for the high-level users outside the area, and the area itself.

525. Michael Freedman (1951-)
• The simple nature of his results in the topological case must be contrasted with the extreme complications which are now known to occur in the study of differentiable and piecewise linear 4-manifolds.

526. George Stokes (1819-1903)
• Though he was never narrow in his faith and religious sympathies, he always held fast by the simple evangelical truths he learnt from his father..

527. David Gale (1921-2008)
• We mention one further paper which in many ways is typical of the delightfully simple yet deep questions that Gale often investigated.

528. Maria Winckelmann (1670-1720)
• Her argument was simple: her husband had been ill for some time and she had actually been doing the job herself for that period, and after his death she had continued to produce the Kirch calendars.

529. Crispin Nash-Williams (1932-2001)
• of developing nontrivial and fairly deep mathematics from a very simple initial concept.

530. Samuel Haughton (1821-1897)
• The personal charm of Dr Haughton's character was something which cannot be expressed in words, while to the outside public he was a brilliant speaker, a racy raconteur, a versatile genius, and a sagacious man of affairs; to the inner circle of his friends he was a wise and willing advisor, one ever ready to help and to guide, affectionate, sincere, and intensely sympathetic, a calm and simple Christian who in all his work ever kept clearly before him his responsibilities as a Christian teacher.

531. Mitchell Feigenbaum (1944-)
• When Feigenbaum first found 4.669 in August 1975, which he only found to three places due to the limit of the accuracy of his HP65, he spend some time trying to see if it was a simple combination of 'well-known' numbers.

• In "Elements" Legendre gave a simple proof that π is irrational, as well as the first proof that π2 is irrational, and conjectured that π is not the root of any algebraic equation of finite degree with rational coefficients.

533. Nathan Divinsky (1925-2012)
• For example, he published On commuting automorphisms of rings (1955), Commutative subdirectly irreducible rings (1957), On simple, semi-radical and radical algebras (1959) and General radicals that coincide with the classical radical on rings with D.C.C.

534. David Eisenbud (1947-)
• Thus this book introduces big ideas with seemingly simple, concrete examples, generalizes from them to an appropriate abstract formulation, and then applies the concept to interesting classical problems in a meaningful way.

535. Richard Delamain (1600-1644)
• In Delamain's we have two (or three) flat brass rings, of the same thickness, graduated and grooved on the edges, one moving within, and in contact with, the other: Oughtred's instrument consists of one round plate, divided into several concentric circles, on which are laid down the logarithms of numbers, sines and tangents, and all operations are performed by means of two indices, radiating from a pin at the centre, like the legs of a sector; this mode of operation, it must be obvious, is far more complex, more inconvenient, and more liable to derangement, than the simple movement first proposed by Delamain.

536. Edwin Hubble (1889-1953)
• The explanation is simple, but revolutionary: the Universe is expanding.

537. Demetrios Kappos (1904-1985)
• It is doubtful whether anyone else could have been such a support to the young Kappos who later said, "I learned how to work because of Caratheodory." The relationship between the two Greeks was more than a simple professor-student relationship for Kappos was a frequent guest at the Caratheodory home, and also a companion of the professor at the park in the area known as the English garden.

## History Topics

1. Word problems
• Notice that there is a simple connection between the Conjugacy Problem and the Word Problem.
• He published these results in 1927 and at the same time gave a simple rigorous proof of the solution of the word problem in a free group.
• These functions were built up from simple functions.
• (x) simple.
• We note that although given a finite group presentation we cannot recursively recognisable whether the group is simple, if we know that a given presentation defines a simple group then that group has soluble word problem.

2. Physical world
• If we deduce results about mechanics from these laws, are we discovering properties of the physical world, or are we simply proving results in an abstract mathematical system? Does a mathematical model, no matter how good, only predict behaviour of the physical world or does it give us insight into the nature of that world? Does the belief that the world functions through simple mathematical relationships tell us something about the world, or does it only tell us something about the way humans think.
• Music, perhaps strangely, was the motivating factor for the Pythagoreans realised that musical harmonies were related to simple ratios.
• Moreover the same simple ratios hold for vibrating strings and for vibrating columns of air.
• It was a belief that a simple mathematical relationship must be physically significant which led Kepler to discover his third law of planetary motion.
• He set up an axiom system consisting of hard particles which were at rest or in motion, obeying three simple laws concerning motion and forces, and a universal law of gravitation.
• As we have suggested there were problems with Newton's system despite the fact that it appeared to reduce the whole of nature to consequences of simple mathematical laws.

3. Gravitation
• the simple bodies such as earth, fire, air and water; for we say that these things and things of this sort are natural.
• As well as giving a simple mathematical model for planetary motion, these laws were highly significant since they stated for the first time that the motions of the heavenly bodies are not composed of circular motion.
• Comets, he showed, were subject to the same gravitational forces and these forces gave a simple explanation of tides.
• It was a devastating attack on Descartes' vortex theory of gravitation and put forward a brilliantly simple theory from which so much could be explained.
• There was no simple solution to the problems that the different theories posed.

4. African women 1
• Biographical Data: Abstract of the thesis: "This research aims to study the conceptions expressed by Moroccan students and teachers at the end of secondary school about the notion of (simple) continuity of a function.
• Accordingly, their physical occurrences and phenomena, and simple solutions are discussed in chapter one, with derivation of conservation conditions given therein in chapter two, a theoretical study of methods is considered and sufficient conditions for stability and convergence results are presented.
• Given the means (or the totals) of auxiliary variables positively correlated with the character of interest, we construct a multivariate estimator for the population mean/total to be used with simple random sampling (srs) or with any probability proportional to size (pps) design.
• She has published The two-dimensional stability of a viscous fluid between rotating cylinders (1996), Symmetric simple map for a single-null divertor tokamak (1997), The principle of exchange of stabilities for Couette flow (2003), Derivation of the dipole map (2004), and Symplectic mappings for divertor tokamaks (2005).
• After a review of the various models described in the literature, our study deals with the simple zero-equation model and the more complex two-equation model of the k-e kind.

5. Kepler's Laws
• (Moreover, the same principle is invoked in relation to planetary motion when Kepler based his investigation on what Aristotle had specified as the only two simple motions, circular and rectilinear, discussed in Section 9.) This principle has far-reaching ramifications, as we will demonstrate in connection with the complementary pairings that recur in Kepler's mature work in Epitome Ⓣ Book V (1621) - where the term 'complementary' is used in the everyday sense that the pair complete one another, and also with the mathematical connotation of being at right angles.
• He adopted the traditional mechanism of deferent, epicycle, and eccentric, being aware, as the Ancients had been, that motion in the circle of radius a centred on A, when combined with motion in the epicyclet of radius ZQ = AB = ae (whose centre Z lies on the deferent), together produce a motion of Q equivalent to a simple motion of Q round the eccentric circle centre B radius a.
• The mathematical treatment carried out in Planetary motion tackled kinematically demonstrates that this angle is the uniquely appropriate foundation for a structure which is simple because it depends on orthogonality and therefore is the only workable basis for Kepler's astronomy.
• In De Caelo Ⓣ I, 3, Aristotle had declared that there were only two simple motions, circular and linear.
• This is the process that was described (in Section 4) as idealization because it ensured an exact solution (of the one-body problem) which was uniquely simple.

6. African men 1
• He has published around 70 papers including A class of algebraically special perfect fluid space-times (1970), Geometric properties of neutrino fields in curved spacetime (1971), Some exact cosmological models with gravitational waves (1979), Power law singularities in orthogonal spatially homogeneous cosmologies (1984), Mathematical cosmology (1990), Introduction to dynamical systems (1994), Cosmological models from a dynamical systems perspective (2005), The dynamics of Lemaitre-Tolman cosmologies (2009), and Simple expressions for second order density perturbations in standard cosmology (2014).
• He has published over 150 papers on Finite Groups, Simple Groups and Sporadic Simple Groups, Representation Theory of Finite Groups, Character Tables of Extension Groups, Cliûord-Fischer Matrices, Presentations of Group Extensions, Application of Finite Groups to Combinatorial Designs and Finite Geometries.
• Here are a few examples of Moori's papers: On certain groups associated with the smallest Fischer group (1981); Subgroups of 3-transposition groups generated by four 3-transpositions (1994); (p, q, r)-Generations of the Smallest Conway Group Co_3 (1997); Codes, Designs and Graphs from the Janko Groups J_1 and J_2 (2002); Permutation decoding for the binary codes from triangular graphs (2004); Some designs and codes invariant under the simple group Co_2 (2007); Codes associated with triangular graphs and permutation decoding (2010); and A survey on Clifford-Fischer Theory (2015).

7. Greek astronomy
• On the other side there is an important idea in the Pythagorean philosophy which had a lasting impact, namely the idea that all complex phenomena must reduce to simple ones.
• Another important philosophical idea which had important consequences from the time of Pythagoras, and was emphasised by Plato, was that complex phenomena must be consequences of basic simple phenomena.
• 45">The changing aspects of the revolution of the planets is because, being fixed in their own circles or in their own shperes whose movements they follow, they are carried across the zodiac, just as Pythagoras had first understood it, by a regulated simple and equal revolution but which results by combination in a movement that appears variable and unequal.
• Eudoxus was the first to propose a model whereby the apparently complex motions of the heavenly bodies did indeed result from simple circular motion.

8. Abstract linear spaces
• He starts with undefined elements which he calls 'simple quantities' and generates more complex quantities using specified rules.
• I go further, since I call these not just quantities but simple quantities.
• There are other quantities which are themselves compounded quantities and whose characteristics are as distinct relative to each other as the characteristics of the different simple quantities are to each other.

9. African women I
• This research aims to study the conceptions expressed by Moroccan students and teachers at the end of secondary school about the notion of (simple) continuity of a function.
• Has published The two-dimensional stability of a viscous fluid between rotating cylinders (1996), Symmetric simple map for a single-null divertor tokamak (1997), The principle of exchange of stabilities for Couette flow (2003), Derivation of the dipole map (2004), and Symplectic mappings for divertor tokamaks (2005).
• After a review of the various models described in the literature, our study deals with the simple zero-equation model and the more complex two-equation model of the k-e kind.

10. Burnside problem
• There are finitely many finite simple groups of exponent n, .
• The outer automorphism group Out(G) = Aut(G)/Inn(G) is soluble for any finite simple group of exponent n.
• Now (moving ahead), the classification of finite simple groups in the 1980's shows that ii.

11. Weather forecasting
• However, as the article should provide only an overview of the mathematical methods used in current forecasting models, I have chosen to include only simple equations and explain some mathematical symbols in order to make understanding the methods easier.
• The approximations described above are very simple examples illustrating the general idea of finite differences.
• 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 .

12. Newton's bucket
• The experiment is quite simple and any reader of this article can try the experiment for themselves.
• What is the problem? Is this not precisely what we would expect to happen? Newton asked the simple question: why does the surface of the water become concave? One is inclined to reply to Newton: that is an easy question - the surface becomes concave since the water is spinning.
• Why should that be? Well in simple terms, in a universe with no matter there is no gravity.

13. Elliptic functions
• For example the period of a simple pendulum was found to be related to an integral which expressed arc length but no form could be found in terms of 'simple' functions.
• This is a particularly simple case of an elliptic integral.

14. Chandrasekhar Eddington
• He reinforced the established idea of a simple model for the evolution of the stars (in which they all eventually become white dwarfs).
• He brought the matter to Bohr's attention and reported back to Chandrasekhar that they were "absolutely unable to see any meaning in Eddington's statements" but that the question seemed to be "quite simple." In the hopes of getting them to settle the controversy, Chandrasekhar sent Eddington's manuscript to Rosenfeld and Bohr, who then in turn sent it to Wolfgang Pauli.
• His ideas involved the seven primitive constants of physics which Eddington sought to relate in simple numerical ways.

15. Special relativity
• The simultaneity of two events or the order of their succession, as well as the equality of two time intervals, must be defined in such a way that the statements of the natural laws be as simple as possible.
• The conception of an ether absolutely at rest is the most simple and the most natural - at least if the ether is conceived to be not a substance but merely space endowed with certain physical properties.
• While Lorentz must be considered as the first to have found the mathematical content of the relativity principle, Einstein succeeded in reducing it to a simple principle.

16. Mathematics and Architecture
• But the first gardener in history to lay out a perfect ellipse with three stakes and a length of string certainly held no degree in the theory of cones! Nor did Egyptian architects have anything more than simple devices -- "tricks", "knacks" and methods of an entirely empirical kind, no doubt discovered by trial and error -- for laying out their ground plans.
• He made an art out of structural purity, using simple geometric forms for aesthetic as well as functional purposes.

17. Ring Theory
• In 1908 Wedderburn had the important idea of splitting the study of a ring into two parts, one part he called the radical, the part which was left being called semi-simple.
• He used matrix rings to classify the semi-simple part.

18. Fair book
• Walker uses six figure logs to do the simple multiplications.
• This must be a simple miscopying.

19. Infinity
• His argument is a simple one.
• Why does defining a set make the actual infinite a reality? The answer is simple.

20. Topology history
• It is interesting to realise that this, really rather simple, formula seems to have been missed by Archimedes and Descartes although both wrote extensively on polyhedra.
Go directly to this paragraph
• He called a simple closed curve on a surface which does not intersect itself an irreducible circuit if it cannot be continuously transformed into a point.
Go directly to this paragraph

21. Black holes
• I had not expected that one could formulate the exact solution of the problem in such a simple way.
• Israel, using general relativity, showed that non-rotating black holes had to be very simple; they were perfectly spherical, their size depended upon their mass only, and any two such black holes with the same mass must be identical.

22. Indian mathematics
• The idea seems so simple nowadays that its significance and profound importance is no longer appreciated.
• The main topics of Jaina mathematics in around 150 BC were: the theory of numbers, arithmetical operations, geometry, operations with fractions, simple equations, cubic equations, quartic equations, and permutations and combinations.

23. Mental arithmetic
• This may be due to the simple fact that such calculating abilities require continual practice for many hours each day and education occupies too much time to allow this to continue.
• My aim has been to demonstrate, in these various rather simple examples, some part of the repertoire, the armoury of resource upon which a mental calculator may draw, and in regard to the choice of which he must make instantaneous decisions, and keep to them.

24. African men 2
• Thesis title: Analyse sur les algebres de Jordan simples reelles [Analysis on real simple Jordan algebras].
• Thesis title: 2-Generations of the Sporadic Simple Groups.

25. Indian numerals
• The idea seems so simple nowadays that its significance and profound importance is no longer appreciated.
• The second aspect of the Indian number system which we want to investigate here is the place value system which, as Laplace comments in the quote which we gave at the beginning of this article, seems "so simple that its significance and profound importance is no longer appreciated." We should also note the fact, which is important to both aspects, that the Indian number systems are almost exclusively base 10, as opposed to the Babylonian base 60 systems.

26. Newton poetry
• Clouded in dust, from motion's simple laws, .
• From laws sublimely simple, speak thy fame .

27. Chinese overview
• The method of calculation is very simple to explain but has wide application.
• After having understood how to make use of the golden section, I began to believe that the different geometrical methods could be understood and that neither the missionaries attitude of considering this simple technique as a divine gift, nor the Chinese attitude of rejecting it as heresy is correct.

28. Planetary motion
• Therefore it is clear that this expression for the radius vector of a circle with its origin at an eccentric point is much less simple than that for the radius vector of the ellipse with the same origin when that point is its focus, as set out in (5) just above.

29. Real numbers 2
• In order to complete the connection presented in this section of the domains of the quantities defined [his determinate limits] with the geometry of the straight line, one must add an axiom which simple says that every numerical quantity also has a determined point on the straight line whose coordinate is equal to that quantity, indeed, equal in the sense in which this is explained in this section.

30. Mathematical games
• Mathematical puzzles vary from the simple to deep problems which are still unsolved.

31. Fractal Geometry
• Equally, no simple shape from Euclidean geometry comes to mind when contemplating things such as the path of a river.

32. Bourbaki 2
• Clearly this powerful mathematical team did not see their task simple to push the last of the chapters through the publishing process.

33. Nine chapters
• Many of the problems seem simple an excuse to give the reader practice at handling difficult calculations with fractions.

34. Alcuin's book
• This is not a simple copying error in the manuscript since 32788 is multiplied by 8 to get the final number.

35. Jaina mathematics
• the theory of numbers, arithmetical operations, geometry, operations with fractions, simple equations, cubic equations, quartic equations, and permutations and combinations.

36. Real numbers 3
• A simple code will let us translate these into letters, 00 become a, 01 become b, ..

37. Squaring the circle
• It neither prevented the stream of publications claiming that π had some simple rational value, nor did it prevent the stream of publications of quite correct constructions to approximately square the circle with ruler and compass.

38. References for Egyptian mathematics
• M Guillemot, Les methodes de simple fausse position dans les mathematiques egyptiennes et les mathematiques arabes, in Histoire des mathematiques arabes, Vol.

39. Greek numbers
• We have omitted the symbol for 'one', a simple '|', which was an obvious notation not coming from the initial letter of a number.

40. Copernicus autograph
• There is no simple progression so the idea that first he used C, moving on to D, then E and finally F is just not born out by the way the quires are made up of the papers.

41. Group theory

42. Wave versus matrix
• All that Foucault had shown is that the simple corpuscular model is not an accurate model to predict all the properties of light.

43. Neptune and Pluto
• if a simple study of its physical appearance can replace the rigorous determination of the positions of all the stars, the search will proceed much more rapidly.

44. The four colour theorem
• If you retort with some very simple case which makes me out a stupid animal, I think I must do as the Sphynx did..

45. Egyptian Papyri
• This is discussed in detail in [',' M Guillemot, Les methodes de simple fausse position dans les mathematiques egyptiennes et les mathematiques arabes, in Histoire des mathematiques arabes, Vol.

46. Perfect numbers
• 44">Among simple even numbers, some are superabundant, others are deficient: these two classes are as two extremes opposed to one another; as for those that occupy the middle position between the two, they are said to be perfect.

47. Golden ratio
• Of course if AB has length 1 and AC = x where C divides AB in the golden ratio, then we can use simple algebra to find x.

48. Braids arithmetic
• Simple proportion.
• To reduce a compound fraction to an equivalent simple fraction.

49. Fair book insert
• Part of the difficulty is that the material in the Fair Book itself does not always advance in difficulty, and sometimes after quite hard problems, simple ones of the same type will appear.

50. Orbits
• Even if the Earth - Moon system were considered as a two body problem, theoretically solved in the Principia, the orbits would not be simple ellipses.

51. Arabic numerals
• The story of this transmission is not, however, a simple one.

52. Babylonian mathematics
• From the mathematical point of view these problems are comparatively simple ..

53. General relativity
• in all my life I have not laboured nearly so hard, and I have become imbued with great respect for mathematics, the subtler part of which I had in my simple-mindedness regarded as pure luxury until now.

54. Pell's equation
• The method relies on a simple observation, namely that, for any m, (1, m) satisfies the 'Pell type equation' .

55. Brachistochrone problem
• Even so, while the method is ingenious and rich, one must admit that it is not as simple as one might hope in a work of pure analysis ..

56. Classical light
• Newton carried out a very simple experiment.

57. Tartaglia versus Cardan
• Ferrari to Tartaglia: You have the infamy to say that Cardano is ignorant in mathematics, and you call him uncultured and simple-minded, a man of low standing and coarse talk and other similar offending words too tedious to repeat.

58. The Scottish Book
• As the reader will see, this general rule could not guarantee against an occasional question to which the answer was quite simple or even trivial.

59. Voting
• The simple system in which each voter gives a single vote to their favourite candidate can also lead to tactical voting.

60. Christianity and Mathematics
• The Creator is the great architect of all things; in the cognition of the mathematically simple structure of the universe man will become united with Him.

61. 20th century time
• The foundations on which the theory is based are remarkably simple.

62. Zero
• How could the brilliant mathematical advances of the Greeks not see them adopt a number system with all the advantages that the Babylonian place-value system possessed? The real answer to this question is more subtle than the simple answer that we are about to give, but basically the Greek mathematical achievements were based on geometry.

63. Greek sources I
• The truth, however, is not nearly so simple and we will illustrate the way that Greek mathematical texts have come down to us by looking first at perhaps the most famous example, namely Euclid's Elements.
Go directly to this paragraph

64. References for Egyptian Papyri
• M Guillemot, Les methodes de simple fausse position dans les mathematiques egyptiennes et les mathematiques arabes, in Histoire des mathematiques arabes, Vol.

## Societies etc

1. Trinity Cambridge Mathematical Society
• The second symbol, which is the official logo of the Society, is the unique smallest simple squared square.
• That it is simple means that no proper subset of the squares of size at least 2 forms a rectangle and smallest in that no square can be so divided with fewer squares.
• This squared square was discovered in March 1978 by A J W Duijvestijn using a computer search and published in his paper Simple perfect squared square of lowest order (1978) which contained the order 21 simple perfect squared square (see [',' A J W Duijvestijn, Simple perfect squared square of lowest order, J.
• However, these solutions were not simple in the sense defined above.
• Ten years later Tutte found a simple squared square with 69 different size squares in the dissection which he published in Squaring the square (1950), see [',' W T Tutte, Squaring the square, Canadian J.
• the logo of the Society shall be the (unique) smallest simple squared square, with the largest partitioning square in the top left corner and the largest of the squares adjacent to this to its right (rather than below it).

• "Everything is voluntary in our general Swiss society," it said, and the study of nature should be "for the greater part of its members their only love, everything else being incidental." Therefore, "only a very simple, unpretentious organization" should be established.
• The Academy began as a "very simple, unpretentious organization" but as it grew it became a complex, well-organised body.

3. References for Trinity Cambridge
• A J W Duijvestijn, Simple perfect squared square of lowest order, J.

4. Max Planck Society for Advancement of Science
• The changeover from the Kaiser Wilhelm Society, however, was not as simple as all that for the two Societies both continued to exist side by side for twelve years with the Kaiser Wilhelm Society only completing its dissolution following its last Annual General Meeting on 21 June 1960.

5. German Mathematical Society
• The potencies represent the simple and important generalisation of the finite cardinal numbers.

## Honours

1. Groups St Andrews.html
• Finite simple groups: a survey .
• Finite regularity of locally finite simple groups .
• Economical generating sets for finite simple groups .
• Width questions for finite simple groups .
• Finite simple groups and fusion systems .
• On characters and p-blocks of finite simple groups .
• Simple groups, generation and probabilistic methods .
• Representations and subgroup structure of simple algebraic groups .

2. Galway Group Theory.html
• T J Laffey (University College Dublin) On finite simple groups .
• M Liebeck (Cambridge) Some applications of the classification of finite simple groups to permutation group theory .
• B Hartley (Manchester) Simple locally finite groups .
• O Puglisi (Florence) Group algebras of locally finite simple groups .
• G Hiss (Aachen) Low dimensional representations of quasi-simple groups .
• Inna (Korchagina) Capdeboscq (University of Warwick) Finite simple groups with double life .
• Radu Stancu (Picardie - Jules Verne) Evaluations of simple biset functors .

3. AMS Steele Prize
• for his book "Finite Simple Groups, An Introduction to their Classification", and his two survey articles "The Classification of Finite Simple Groups" and "Classifying the Finite Simple Groups".
• for his construction of the "Monster" sporadic finite simple group.
• for their work, "The classification of finite simple groups: groups of characteristic 2 type" .

4. International Congress Speaker
• John Griggs Thompson, Characterizations of Finite Simple Groups.
• Walter Feit, The Current Situation in the Theory of Finite Simple Groups.
• Elias M Stein, Some Problems in Harmonic Analysis Suggested by Symmetric Spaces and Semi-Simple Groups.
• Daniel Gorenstein, The Classification of Finite Simple Groups.

5. AMS Cole Prize in Algebra
• for his groundbreaking research on representation theory, cohomology, and subgroup structure of finite quasi-simple groups, and the wide-ranging applications of this work to other areas of mathematics.

6. Wolf Prize
• for his monumental contributions to algebra, in particular to the theory of lattices in semi-simple Lie groups, and striking applications of this to ergodic theory, representation theory, number theory, combinatorics, and measure theory.

7. MAA Chauvenet Prize
• The Simple Continued Fraction Expansion of e, Amer.

• Odd characterisations of finite simple groups.

9. Gibbs Lectures.html
• December 1955; Houston, Texas; Joseph E Meyer; The structure of simple fields.

10. Sylvester Medal
• for his fundamental contributions leading to the complete classification of all finite simple groups.

11. AMS Conant Prize
• for his article "A Brief History of the Classification of the Finite Simple Groups".

12. Rolf Schock Prize
• for his fundamental contributions to one of the largest mathematical projects ever, the classification of finite simple groups, notably his contribution to the quasi-thin case.

## References

1. References for John Conway
• T M Thompson, From error-correcting codes through sphere packings to simple groups (Washington, 1983).

2. References for Georges de Rham
• J Milnor and O Burlet, Torsion et type simple d'homotopie, in A Haefliger and R Narasimhan (eds.), Essays on Topology and Related Topics : Memoires dedies a Georges de Rham (Springer, Berlin - Heidelberg - New York, 1970), 12-17.

3. References for Subrahmanyan Chandrasekhar
• N Panchapakesan, Seeing beauty in the simple and the complex : Chandrasekhar and general relativity, in Classical and quantum aspects of gravitation and cosmology, Madras, 1996 (Madras, 1998), 1-10.

4. References for Galileo Galilei
• R Naylor, Galileo's simple pendulum, Physis 16 (1974), 23-46.

5. References for Walter Feit
• R Solomon, A brief history of the classification of the finite simple groups, Bull.

6. References for Jules Bienaymé
• C C Heyde and E Seneta, The simple branching process, a turning point test and a fundamental inequality : A historical note on I-J Bienayme, Biometrica 59 (3) (1972), 680-683.

7. References for John Leech
• T M Thompson, From error-correcting codes through sphere packings to simple groups (Washington, 1983).

8. References for Mario Fiorentini
• E Sernesi, A simple article, in Commutative algebra and algebraic geometry, Ferrara (Dekker, New York, 1999), x-xiii.

9. References for Paul Dubreil
• C Hollings, Embedding semigroups in groups: not as simple as it might seem, Arch.

10. References for Rimhak Ree
• J A Gallian, The Search for Finite Simple Groups, Mathematics Magazine 49 (4) (1976), 163-180.

11. References for Émile Mathieu
• R Silvestri, Simple groups of finite order in the nineteenth century, Arch.

12. References for Levi ben Gerson
• R Glasner, Gersonides on simple and composite movements, Stud.

13. References for Fischer Black
• (1988c), A Simple Discounting Rule, Financial Management, 17(2), 7-11.

1. Hardy Inaugural Lecture
• There is nothing in the least popular about its methods, as to its votaries it is the most beautiful, so by common consent it is the most difficult of all branches of a difficult science; but many of the actual results are such as can be stated in a simple and striking form.
• There are various ways of solving this extremely simple problem.
• And even this problem, simple as it is, has sufficient content to bring out clearly certain principles of cardinal importance.
• It is easy to see this by considering a simple example.
• If s, the number of squares, is even and less than 10, the number of representations may be expressed in a very simple form by means of the divisors of n.
• When s is 3, 5, or 7, the number of representations can also be found in a simple form, though one of a very different character.
• Liouville's proof, which was first published in 1859, is quite simple and, as the simplest example of an important type of argument, is worth reproducing here.
• Thus G(4) ≥ 15; and Kempner, by a slight elaboration of this simple argument, has proved that G(4) ≥ 16.
• In the second row I have shown the best known lower bounds, which are given by the simple general formula [(3/2)k] + 2k - 2, in which [(3/2)k] denotes the integral part of (3/2)k.
• All this is simple enough; but the further study of the integral is very intricate and difficult, and I cannot attempt to do more than to give a rough idea of the obstacles that have to be surmounted.
• In the present case we have no such simple recourse; for every point of the unit circle is a singularity of an exceedingly complicated kind, and the circle as a whole is a barrier across which it is impossible to deform the contour.
• (sum is over n) which (a) is as simple and natural as we can make it, and (b) behaves perfectly regularly at all points of the unit circle except at the one point with which we are particularly concerned.
• The process is, at bottom, one of 'decomposition into simple elements', applied in an unusual way.
• It will be seen that these numbers conform to a simple law, and that is the third advantage of the method, that it is not a mere existence proof, but gives us a definite upper bound for G(k) for all values of k, viz.

2. Élie Cartan reviews
• This book begins with very simple and familiar ideas of vectors in Euclidean space in rectangular Cartesian coordinates and gradually arrives at the notion of a tensor and the algebraic and differential operations with tensors.
• After two chapters on the generalities of the theory there is a chapter on closed groups and one on closed simple groups.
• There follows a chapter on open groups, containing the theorem that the first Betti number of an open simple group is 0 or 1.
• The book concludes with a statement of all known theorems on the Betti numbers of closed simple groups - among others the results of L Pontrjagin and R Brauer, who have calculated them for the four main types of simple group.
• In the preface to the two volumes under review M Cartan points out that, in their most general mathematical form, spinors were discovered by him in 1913 in his work on linear representations of simple groups, and he emphasises their connection, shown in Vol.
• For example, he completed the work of Killing and Lie on the classification of simple Lie algebras.
• It was first published in 1966, when the work of Killing and Cartan on the classification of simple Lie groups was beginning to be applied in elementary particle physics.
• 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.
• Clearly, he is presenting a "vulgarization" of the general theory of semi-simple Lie algebras and groups, which he developed almost single-handedly (with the help of Hermann Weyl!) in the period 1893-1930.

3. German syllabus
• Simple equations of first degree with one unknown, in connection with operations with rational numbers.
• Equations of first degree with one or more unknowns; simple applications, especially from everyday life.
• Simple masses of shrubbery and borders of garden paths.
• Simple integral and rational functions.
• Simple equations, and systems of equations which c an be solved by quadratic equations - numerical and graphical treatment.
• Simple triangle calculations.
• Simple exercises in surveying and levelling.
• Simple representations by means of functions of a complex variable.
• Simple astronomical observations with measurements and calculations.

4. ELOGIUM OF EULER
• 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.
• This was done by searching for the sums or the expression of their general terms and to those of the roots or determinant equations, by which to obtain with a simple calculation the approximate value of the products or the indefinite sums of certain numbers.
• He abandoned his first ideas and submitted new ones to proof by experiments and enriched Dioptics with analytical formulas which were simple, useful, general and applicable for every instrument that could be built.
• At other times it would be a problem that appeared insurmountable that he resolved in an instant by a very simple method or an elementary problem with a very difficult solution that could only be overcome with the greatest efforts.
• At other times simple numbers, or a new series presented questions novel by their uniqueness which took him to unexpected proofs.
• Euler's work due to the telling of the very simple and unvaried events of his life.
• Euler's name, so highly regarded in the Sciences and the imposing way in which his insights reveal the most thorny and abstract ideas, reveals in these simple and easily readable Letters and unique charm and those who have not studied Mathematics, are astonished and flattered to be able to understand a work by Euler and are grateful that his message has been placed within their grasp.
• Euler's simple modesty felt his force and on more than on occasion used it to good purpose.
• For most of the Northern aristocracy to whom he was personally known, they had already provided him with marks of their esteem, or more like veneration that one can hardly deny when one sees the uniting of such simple virtues to such vast heightened genius.

5. Value of Mathematics
• Simple generalisation to others of a demand of their own allows us to see the impossibility of a desire that our blind selfishness imposed on us with a pressing imperative.
• It is, in short, to make use of the faculty which in Mathematics we call intuition (to look inside ourselves) and that should not be confused with the faculty called intuition by some psychologists and pedagogues that hardly differs from simple perception.
• If the student were accustomed to constantly project the data and results of the problems into the realm of reality, absurdities of this nature would be avoided, the pupil would become accustomed to keep in mind this simple and yet so often forgotten truth that all data translating a measure of the physical world is necessarily approximate, and that, therefore, the alleged accuracy in the results is not only a pure chimera but a grotesque falsification of reality.
• These simple laws serve, for example, to justify the implantation of the cyclical methods that establish the continuity in the study of the topic without breaking them up into separate areas; justify the introduction of intuitive methods in the first years of high school to fill the gap that existed between the empiricism of primary education and the rationalism of university education, and the progressive evolution of methods that without discontinuity or sudden jumps allow one to develop the psychological activities of the child gradually from early childhood to university.
• - Simple analysis, observation of the facts and points surrounding the child.
• But it is necessary in such a case that the technique of handling the book is adequate so that this management is not converted into simple memory exercises.
• How can we combine the two utilitarian and formative tendencies without reloading the programmes with the overwhelming and unbearable weight they suffer today? I propose a very simple formula.
• If the educational efficacy of mathematical teaching lies mainly in methods, respecting them, we will have the freedom to select the knowledge that will be most useful and thus arouse greatest interest, and thus the two utilitarian and educational points of view, which have so often been presented as opposed to one another, will be joined in a simple harmonizing formula: Teaching useful knowledge with educational methods.

6. A A Albert: 'Structure of Algebras
• It has been most fortunately possible at this time to give a new treatment of the early parts of our subject simplifying not only the proofs in the theory of normal simple algebras but even the exposition of the structure theorems of Wedderburn.
• Their exposition is begun in Chapter IV which contains the theory of the commutator subalgebra of a simple subalgebra of a normal simple algebra, the study of automorphisms of a simple algebra, splitting fields, and the index reduction factor theory.
• This is believed to be the first time the extension has been made in a really simple fashion.
• The theory of involutorial simple algebras arose in connection with the study of Riemann matrices but is now a separate branch of the theory of simple algebras with structure theorems on approximately the same level as those on arbitrary simple algebras.

7. Kelvin on the sun, Part 2
• The continually repeated blows upon any part of the walls or ceiling will in the aggregate be equivalent to a continuous pressure which will be in simple proportion to the average density of the crowd at the place.
• One very remarkable and important result which be finds is, that the density at the centre is about twenty times the mean density; and this, whether the mass be large or small, and whether of oxygen, nitrogen, or hydrogen, or other substance; provided only it be of one kind of gas throughout, and that the density in the central parts is not too great to allow the condensation to take place, according to the ordinary gaseous law of density, in simple proportion to pressure for the same temperatures.
• But when the compressing force is sufficiently increased, they all show greater resistance to condensation than according to the law of simple proportion, and it seems most probable that there is for every gas a limit beyond which the density cannot be increased by any pressure however great.
• Lane remarks that the density at the centre of the sun would be "nearly one-third greater than that of the metal platinum," if the gaseous law held up to so great a degree of condensation for the ingredients of the sun's mass; but he does not suggest this supposition as probable, and he no doubt agrees with the general opinion that in all probability the ingredients of the sun's mass, at the actual temperatures corresponding to their positions in his interior, obey the simple gaseous law through but a comparatively small space inwards from the surface; and that in the central regions they are much less condensed than according to that law.
• According to the simple gaseous law, the sun's central density would be thirty-one times that of water; we may assume that it is in all probability much less than this, though considerably greater than the mean density, 1.4.
• If we ask, How does the temperature of equi-dense portions of the sun vary from age to age? the answer certainly is that the matter of the sun of which the density has any stated value, for example, the ordinary density of our atmosphere, becomes always less and less hot, whatever be its place in the fluid, and whatever be the law of compression of the fluid, whether the simple gaseous law or anything from that to absolute incompressibility.
• But at a certain time in the history of a wholly fluid globe, primitively rare enough throughout to be gaseous, shrinking under the influence of its own gravitation and its radiation of heat outwards into cold surrounding space, when the central parts have become so much condensed as to resist further condensation greatly more than according to the gaseous law of simple proportions, it seems to me certain that the early process of becoming warmer, which has been demonstrated by Lane, and Newcomb, and Ball, must cease, and that the central temperature must begin to diminish on account of the cooling by radiation from the surface, and the mixing of the cooled fluid throughout the interior.
• If the substance were oxygen, or nitrogen, or other gas or mixture of gases simple or compound, of specific density equal to the specific density of our air, the central temperature would be 51,200° C, and the average translational velocity of the molecules 6.66 kilometres per second, being √(3/7) of 10.2, the velocity acquired by a heavy body falling unresisted from the outer boundary (of 40 times the radius of the earth's orbit) to the centre of the nebulous mass.

8. Max Planck: 'Quantum Theory
• This exceedingly simple relation is a complete and adequate expression of Wien's law of distribution of energy; for the dependence upon wave-length is always given immediately as well as the dependence upon energy by Wien's generally accepted law of displacements.
• Finally, the observations made by G Rubens and F Kurlbaum, with infra-red rays after transmission through fluorspar and rock salt, showed a totally different relation, which, under certain conditions, was still very simple.
• Thus, by direct experiment, two simple limits have been fixed for the function R, i.e.
• (It would be better to substitute temperature for energy here.) On this basis a comparatively simple combinatory method was derived for calculating the physical probability of a certain distribution of energy in a system of resonators.
• The interpretation of the second universal constant of the radiation formula was much less simple.
• 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.
• By greatly simplifying the assumptions regarding the nature of the oscillations, P Debye obtained a comparatively simple formula for the specific heat of a solid body.
• 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.

9. Jacobson: 'Structure of Rings
• These are: the structure theory of rings without finiteness assumptions, cohomology of algebras, and structure and representation theory of non-semi-simple rings (Frobenius algebras, quasi-Frobenius rings).
• Thus the present volume includes virtually all the results on semi-simple rings which can be found in the two books cited before.
• For example, the theory of centralizers of finite dimensional simple subalgebras of simple rings with minimum condition appears as a special case of the Galois theory of the complete ring of linear transformations of a vector space over a division ring.
• A semi-simple ring is one which has enough irreducible representations to distinguish elements.
• In the first part we consider the theory of semi-simple rings with minimum condition.
• In Chapter V we define Kronecker products of modules and algebras and we reduce the problems of determining the structure of Kronecker products of simple algebras to the case of division algebras and fields.

10. D'Arcy Thompson on Greek irrationals
• Aristotle gives us the following statement of Plato's concept of the 'genesis of number': [Number is derived from Unity and the indeterminate dyad]; but this apparently simple statement has never been satisfactorily explained.
• In short if we keep to this restricted definition of our problem, and if we then go a step or two farther in its interpretation than Prof Taylor has gone, we come to a very simple understanding of what [the one/unity] and the [infinite/indeterminate dyad] are; and of how, between them both, such a 'number' as √2 is generated.
• The continued fraction is an elegant arithmetical device, and the mathematician calls it a simplified expression; but it does not follow that it is simple to work with.
• 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.
• All this arithmetic is so simple that it can hardly have escaped the notice of any calculator who pondered over the elementary table with which we began.
• 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.
• 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.

11. Aitken: 'Statistical Mathematics
• What is the axiomatic basis of the science of statistics, and what are the facts upon which the inductive synthesis is based? The facts are certain regularities which have been observed in the proportionate frequency with which certain simple events happen or do not happen, when the circumstances under which they may occur are reconstructed again and again in repeated trials; and the axioms, and the structure of theorems founded upon them, constitute the subject called mathematical probability.
• The reader is recommended to experiment with simple repeated trials of this kind, and for future reference to record the results in sequence, in the order in which they occur.
• Simple ideas such as these suggest by generalization and abstraction the axioms of probability; but the choice of axioms may be made in various ways, which lead to different formulations of the theory of probability.
• As our simple illustrations of the coin and the die have suggested, the crude intuition of probability rests on the observation that when a given set of circumstances S, such as a symmetrical coin spun rapidly, has been present on numerous occasions in the past, it has been associated in a nearly constant proportion of those occasions with some event E, such as the fall of "heads." .
• To take a classical example, in the sequence defining a certain simple geometric series, .
• Let us examine more closely the system S, keeping some simple system such as a coin or die in mind.
• Now the question of assigning a measure to such aggregates has been deeply studied in modern pure mathematics, the guiding idea being that of extending as widely as possible the scope of a concept familiar in simple cases, namely the cardinal number of a finite set of objects, the length of a line, the area of a surface, the volume of a solid.

12. Maini papers
• A non-linear bifurcation analysis is presented for a simple version of the field equations: a non-standard element is required.
• A nonlinear bifurcation analysis is presented for a simple version of the governing field equations.
• Simple analytical solutions are obtained when the reaction rates and the initial substrate concentration satisfy a certain condition.
• Here we examine a simple one-dimensional caricature of their model which exhibits similar linear behaviour and present a nonlinear analysis which shows the possibility of superposition of modes subject to appropriate parameter values and initial conditions.
• We propose a simple partial differential equation model for a chemotactic system of two species, a population of cells and a chemo-attractant to which cells respond.
• By using mode selection from the linear analysis we produce simple pattern elements such as stripes and regular spots.
• More complex patterns evolve from these simple solutions as parameter values or domain shape change continuously.

13. Cheney books
• The usual questions from classical approximation theory can be posed for these approximating subspaces, such as (i) Do best approximations exist? (ii) Are best approximations unique? (iii) How are best approximations characterized? (iv) What algorithms can be devised for computing best approximations? (v) Do there exist simple procedures which provide "good" approximations, in contrast to "best" approximations? (vi) What are the projections of least norm on these subspaces? and (vii) what are the projection constants of these subspaces? This volume surveys only a part of this growing field of research.
• First we ask, "What subclasses of functions are suitable for approximating other functions?" Here interest focuses naturally on functions that are simple combinations of univariate functions.
• The important tensor-product subspaces play the principal role here because of their simple linear structure.
• In other cases, the reverse is true, and the students learn much from programming simple algorithms themselves and experimenting with them - although we offer a blanket admonition to use well-tested software.
• In style, we have tried to make the exposition as simple and clear as possible, electing to furnish proofs that are complete and relatively easy to read without the reader needing to resort to pencil and paper.
• To paraphrase Shaw: We have done our best to avoid conciseness! We have also made considerable efforts to find simple ways to introduce and explain each topic.

14. Truesdell's books
• With this tractate I aim to provide a simple logical structure for the classical thermodynamics of homogeneous fluid bodies.
• I think it is as simple and pretty as can be.
• That this tractate is a long one, results from its triple scope: (1) Conceptual: for those already expert in thermodynamics, to show how all the concepts of the traditional, elementary theory can be derived from simple and natural assumptions about heat engines, developed by simple and rigorous mathematics.
• For this reason I have included detailed proofs of propositions which to physicists and engineers may seem so obvious as to need no proof, to mathematicians so simple that anyone can prove them.
• (with R G Muncaster) Fundamentals of Maxwell's kinetic theory of a simple monatomic gas.

15. Raphson books
• Moreover, Mr Raphson explains his Method after the very same manner as he invented it, and to show the large Extent and Certainty thereof, he propounds a general Theorem, which he afterwards resolves more particularly in Two Propositions: Then he proceeds to illustrate his Method by Examples, in 32 Problems; wherein is exhibited the Resolution of Equations, of all manners of Dimensions, taken from the Resolution of a Simple Equations: Whereunto he adds Examples of Quadratic Equations, All his Operations are described at large; and to render the Practice more plain and obvious, the Author hath taken the Pains to compose certain Tables, which are inserted at the End; insomuch that if he continues to prosecute these Studies, as he hath begun, it is not to be doubted but that he will become one of the most skilful Mathematicians that are now living; since at the Age of 22 Years, he hath already attained to so great a Knowledge in those abstruse and difficult Sciences: Wherefore what improvement may we from not expect from the extraordinary Judgement of his riper Years? .
• First, that we admit of nothing as a first Principle, but what appears to be certain and most evidently true even to the meanest Capacity; such a Principle he reckons a Simple Idea to be, an Attribute Essential to the thing to which it belongs: That it may be certain tis requisite it should the first and undoubted Truth; and that it be evidently True, a clear and distinct Perception is Necessary.
• The other sort of Arguments for the Proof of real Space distinct from Matter, Mr Raphson in a Geometrical Way deduces from the Necessary and Natural Concatenation and Consequences of simple Ideas.
• Sixthly, That a Self-existent Being is in its own Nature a most simple Being.
• In the Conclusion of the first Part Mr Raphson observes, That as the Self-existent Being necessarily exists, so such Beings as are not self-existent owe their Being to something extraneous to themselves: That as the former is what it is of it self, so the later receive all that they are and have, from something else: That as the former is in its own Nature eternal, so the later are in the same manner temporal: As the one is infinite, the other are finite: As the one is necessarily and of it self one, the other owe their Unity not to themselves, but to what made them such: As the one is a most absolutely simple Being, so the other are either compounded, and so resolvable into the Principles of which they consist; or if they are in their Nature simple, as they were made, so they may be unmade; or, as far as they were produced, so far also are they capable of being destroyed by the Being that produced them: As the one is immutable, the other are mutable: And as the one is all that is or can be, in an absolute and infinitely perfect Sense, the other are of restrained and limited Essences, which is the Reason that there are many of them, for Finiteness is the natural Root of Plurality.

16. Moran reviews
• In fact, Chapter II of this monograph consists of an analysis of several simple inventory models.
• Simple ideas of linkage, cross-over, recombination fraction and so on must be familiar.
• For example, the untutored reader might imagine from the discussion here that the Canadian lynx data was an example of a simple sinusoidal regression with an added random error.
• Simple calculations are usually left to the reader and for difficult proofs the reader is often referred to the literature.
• Among stochastic processes considered, apart from simple Markov chains and processes, are less common models like Daniels's stiff chains and Hammersley's self-avoiding random walks.

17. Marion Walter's books
• This book will have great appeal, for it invites the young mathematician to explore, to discover, and to learn by doing as he uses the safe metal mirror that is provided with the book to answer the stimulating questions or to follow the simple directions.
• While having much fun, the child will grasp simple mathematical concepts.
• The young reader will have fun seeing, doing, thinking, and imagining as he uses the safe, metal mirror, which comes with the book, to follow the simple suggestions and to answer the intriguing questions.
• Simple mathematical concepts will be developed while the child is having fun.
• The puzzles move from simple to more challenging, along with some impossible puzzles.

18. Von Neumann: 'The Mathematician' Part 2
• This means that the criterion of success for such a theory is simply whether it can, by a simple and elegant classifying and correlating scheme, cover very many phenomena, which without this scheme would seem complicated and heterogeneous, and whether the scheme even covers phenomena which were not considered or even not known at the time when the scheme was evolved.
• One expects a mathematical theorem or a mathematical theory not only to describe and to classify in a simple and elegant way numerous and a priori disparate special cases.
• Also, if the deductions are lengthy or complicated, there should be some simple general principle involved, which "'explains" the complications and detours, reduces the apparent arbitrariness to a few simple guiding motivations, etc.
• These criteria are clearly those of any creative art, and the existence of some underlying empirical, worldly motif in the background - often in a very remote background - overgrown by aestheticizing developments and followed into a multitude of labyrinthine variants - all this is much more akin to the atmosphere of art pure and simple than to that of the empirical sciences.

19. Valiant Turing Award
• However, his model was so simple and compelling that it immediately captured the imagination of the community and led to widespread agreement that this was indeed the right approach.
• His results here range from simple, but powerful and elegant, insights to reexamining the very foundations.
• An example of a simple insight is his parallel routing scheme, described in the paper "A scheme for fast parallel communication" (SIAM J.
• Using simple greedy schemes to route the data can often lead to congestion - too many data paths may end up using the same link in the network at once.
• Valiant discovered a brilliant and simple randomized solution to the problem.

20. G H Hardy addresses the British Association in 1922
• They are seemingly simple questions, and it is not necessary to be anything of a mathematician to understand them; and I have chosen them for no better reason than that I happen to be interested in them myself.
• Is there, I am asking, any simple criterion by which such numbers can be distinguished? .
• Then, if it is of the form 4m + 1, it is a sum of squares, and in one way only, while if it is of the form 4m + 3 it is not so expressible; and this simple rule may readily be generalised so as to apply to numbers of any form.
• There is no case, except the simple case of squares, in which the solution is in any sense complete.
• There are no simple general tests by which the primality of a number chosen at random can be determined, and the amount of computation required in any particular case may be quite appalling.

21. Gordon Preston on semigroups
• What he gave was the structure, in terms of its maximal subgroups, of a completely 0-simple inverse semigroup.
• 1037-1049) showed that semigroups which are unions of groups are disjoint unions of completely simple semigroups, an important early structure theorem, emphasising also the importance of completely simple semigroups, introduced by Rees in his 1940 paper.
• It was said that semigroups were objects that were too simple to be interesting, and that useful mathematics would not stem from their study.
• It was an attempt at giving an integrated account of the Green relations, the Schutzenberger group and representations (we had just had what we called a "Schutzenberger week" at which Schutzenberger had lectured each day on his representations and associated matters), completely 0-simple semigroups, 0-minimal ideals, etc.

22. Mathematicians and Music 3
• The great feature of this work is the formulation and proof of the laws by which the ear bears musical sounds from one or more distinct sources; how the theory of combined musical sounds is reduced to the theory of combined simple sounds.
• From these laws we learn the nature of consonance and dissonance, knowledge so necessary for building up a system of harmony; we learn the principles which determined those degrees of musical sound selected by various nations at various times; we understand the reasons for the simple ratios of the lengths of strings producing consonant tones and the limitation of the numbers of these ratios; and we appreciate the value of temperaments for different instruments.
• 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.
• 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.
• In concluding references to activities of the past one hundred years, I should, however, take time to recall that when, in these latest days, there arose a question as to the manner in which our present musical notation for equal temperament scales could best be simplified, it was a former president of this Association who brought forward a scheme so beautifully simple that further advance in this regard cannot be imagined.

23. Hilbert reviews
• (3) the principle of not merely proving a proposition in the most simple way but indicating precisely what axioms are necessary and sufficient for the proof; .
• The many-mansioned discipline known as "symbolic logic" has for a long time ceased to be a simple affair, and introductions to it vary according to the special interests to which they cater.
• The paradoxes are presented, and Russell's simple theory of types is adopted for their avoidance.
• Six chapters (Simple curves and surfaces, regular systems of points, configurations, differential geometry, kinematics, topology) serve to lead the average reader to a number of vantage points from which large domains can be surveyed.
• Thus although the primary object is to present proofs of consistency (Widerspruchsfreiheit) of mathematical systems, no such proof appears in the first volume, whose 468 pages are occupied by preliminary considerations, such as the possibility of transforming formulae in the propositional calculus into a standard form, the extent to which logical quantifiers can be eliminated from an axiom system, the exact specification of the rules for the use of recursive definitions, and proofs of categoricity (Entscheidungsbarkeit) and completeness (Vollstandigkeit) for simple systems.

• The analogy is drawn with a simple model of frequency conversion.
• For a general class of two-mode, simple analytic expressions are derived for the evolution of the field quantum entropy in the bimodal field interacting with an effective two-level atom via the Raman transition, with an additional Kerr-like medium.
• Simple expressions for the atomic populations, the cavity photon statistics, and the reflection and transmission probabilities are given for any initial state of the atom-field system.
• We propose a generation of Bell-type states having a simple initial state preparation of the present system.
• We propose a method of generating Bell-type states from a simple initial state preparation of two different modes of electromagnetic field.

25. De Montmort: 'Essai d'Analyse
• The life of Pierre-Remond de Montmort, after a stormy start, was a simple, happy one.
• He discusses such simple games as Pharaoh, Bassette, Lansquenet and Treize, and then, not so fully or successfully, Ombre and Picquet.
• Having set down the rules, he solves simple cases in a method somewhat reminiscent of Huygens, and then takes a plunge into a general solution which appears to be correct but is not always demonstrably so.
• 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 .
• He doesn't bother with the rules (they must have been entirely established by this time) and he calculates several simple chances but remarks that in the majority of situations the solution cannot be found.

26. Slaught's books
• These exercises are for the most part very simple, but by bringing into play a large number of straight lines and circle arcs in a single problem, they are a valuable aid in the development of geometric imagination.
• The new Algebra contains numerous attractive features, all aiming to make the subject more simple and interesting and therefore more valuable to first year pupils.
• Among these features the most distinctive are perhaps the following four: (i) The presentation of the subject is as simple as it can be made.
• Like previous books by these authors there is great emphasis on simple presentation and easy gradation in each topic, and on the side of concrete applications.
• While recognizing the increased maturity of the pupils, the authors nevertheless maintain in this present text that simple and interesting form of presentation which characterizes the earlier book.

27. Craig books
• Perhaps the only division of the subject - omitting the simple case of perspective projection - that has ever been fully treated is that of projection by similarity of infinitely small areas.
• A few of the solutions of simple problems in the paper, it is believed by the author, are new and simpler than any he was able to find in the writings of others.
• With these few exceptions there is no claim of originality in what follows: the attempt having simply been made to present in as simple and natural form as possible what others have done.
• The object of that chapter is only to give in a simple manner some of the more important and elementary properties of the curves of the second order, so that convenient reference could be made in the subsequent part of the paper to the various formulas connected with these curves, and also simple means given for constructing them.

28. Halmos books 2
• The rules are simple and the advantages of following them to the conscientious and diligent reader are surely obvious and incalculable.
• What makes a problem interesting? Its statement should be simple, not requiring excessive explanations, and the solution should be readily understandable by the intended audience.
• Understanding simple things such as basic linear algebra does not seem to be an easy task.
• As to the form, the style is vivid and clear, using simple words, and free of long and complex technicalities.
• The text is rich in brief comments explaining the ideas behind the reasoning and calculations, and frequently refers to simple examples and to the basic notions of universal algebra.

29. Loney CUP
• We have been particularly struck with the manner in which the author combines perspicuity with brevity in his short chapter on "units and dimensions," a subject which, though apparently very simple, often presents considerable difficulties to a beginner.
• Mr Loney may be congratulated on the production of a most valuable text-book, at once simple and complete.
• We are glad to see that the method of the hodograph is used in treating of normal acceleration, and that cycloidal motion is considered as a case of simple harmonic motion.
• The author has succeeded in his purpose to produce "a fairly complete elementary text-book on Plane Trigonometry." The faithful student of this treatise "will have little to unlearn when he commences to read treatises of a more difficult character." The style is clear and simple; even when it is diffuse, the author never hides his thoughts with words either large or small.

30. D'Arcy Thompson on Plato and Planets
• Without recapitulating further details for Sun and Moon, we come to a very great difficulty in the case of the planets, namely to explain, by any simple imaginary mechanism, what are known as their stations and retrogradations.
• But the fourth sphere was not coaxial with the third, but was set somewhat obliquely to it (just as the first sphere was to the second), and thus introduced another component in the form of a simple harmonic motion, causing the planet to perform apparently a pendulum-like vibration in the plane of the ecliptic, while all the while it was being carried around that circle by the proper motion of the second sphere.
• The calculation, a very simple one, has been performed by Schiaparelli, who shows the angles to be as follows: .
• A chief source of obscurity in the whole passage is the simple circumstance that Plato was talking in riddles and in allegory, after the fashion of antiquity and the East, and therefore did not choose to tell us many things which he must have known.

31. Berge books
• In what are called simple graphs, the vertices are divided into two sets such that all arcs connect only members of one set with points of the other.
• Matching on a simple graph (assignment problem, Latin squares); 11.
• The Preface tells us that it "aims to demonstrate that a large part of these mathematical programming problems can be solved in a simple and elegant manner using ..
• Surprisingly, Vajda's excellent 'Mathematical Programming' (1961) [Steven Vajda (1901-1995)] is not mentioned, though in that book all the essential ideas of the present one are explained in simple numerical terms so that this one could almost have been subtitled "A Mathematical Gloss on a Text by Vajda".

32. Apostol Project
• Simple applications, Archimedes' discovery, computation, and extensions (lattices, random numbers, Buffon needle).
• This leads to elegant derivations of addition formulas, with applications to simple harmonic motion.
• Intrigue and drama are injected into the story when alternative theories are pro posed, for example, Did Eupalinos physically measure around the mountain or over the mountain? Site exploration, simple mathematics, and common sense sup ply the answer.
• After an introduction and a brief survey of mathematical events up to the seventeenth century, the units describe topics in or about numeration systems, number theory, the Pythagorean theorem, irrational numbers, pi, the evolution of trigonometry from astronomy, simple analytic geometry, and some fundamental calculus.

33. Didactics of Mathematics
• Geometry consists of a set of definitions of geometric figures and calculation of perimeters of polygons, and areas of the same in simple cases.
• Geometry is practically reduced to applying formulas for the areas of simple polygons, of circles and some circular figures.
• The first course includes: natural number, fractions, decimal numbers, simple rule of three, elementary geometric figures, some geometric constructions and area calculation.
• It is possible that, in addition to the reasons just mentioned, there is a more powerful one that is really that it has reduced the teaching of mathematics, in its elementary phase, to a simple calculator, but with the aggravating aspect of automatism.

34. Piaggio Reviews
• The object of this book is to give an account of the central parts of the subject in as simple a form as possible, suitable for those with no previous knowledge of it, and yet at the same time to point out the different directions in which it may be developed.
• In devoting an early chapter to some simple partial differential equations, Prof Piaggio has put teachers and students alike under a debt which the latter cannot realise.
• Is it credible that some of us became acquainted with the equations of wave motion and with their simpler solutions surreptitiously in treatises on sound, because in pure mathematics a partial differential equation of the second order, however simple, was expected to yield precedence to the twenty-four solutions of the hypergeometric equation and to an abstract discrimination between general integrals, complete primitives, and the like? .
• "The object of this book is to give an account of the central parts of the subject in as simple a form as possible, suitable for those with no previous knowledge of it, and yet at the same time to point out the different directions in which it may be developed." The only previous knowledge assumed is that of the differential and integral calculus.

35. Orr Stability
• An explanation of the difficulty was given by showing that it is necessary to push Lord Rayleigh's investigations a step farther by resolving a disturbance into its constituent fundamental ones by quasi-Fourier analysis, and that, when this is done for disturbances of initially simple type in some of the most important and simplest cases of flow, it is found that the disturbance will, for suitable values of the constants, increase very much, so that the motion is practically unstable.
• As the fundamental modes of disturbance do, as is shown in Chapter II., possess stability of the simple exponential character, the "special" solution is, I believe, as a matter of fact, the solution for a given initial disturbance; if this be a simple trigonometrical function of the coordinates, the form of v is simple; but that of the "forced" disturbance in no case appears capable of being readily calculated.

36. Coulson: 'Electricity
• From this simple experimental fact has developed the whole science of electrostatics, that is, the properties of electricity at rest.
• First, the microscopic viewpoint throws light on the fundamental physical processes; this enables us to view our subject as one whole and means that we shall not have to introduce from time to time apparently unrelated physical assumptions, for we shall see how our macroscopic equations arise quite naturally from simple microscopic properties of the atom and the electron.
• The explanation is simple, for in these substances all the negative charges (or electrons) are firmly attached to corresponding positive charges.
• Thus the difference between substances which are or are not permanent magnets is not that they are made of essentially different material, but rather that with permanent magnets we have no way (or at any rate no simple way) of destroying the co-operative effect of the separate atoms, whereas with non-permanent magnets this co-operation is solely the result of forces exerted from outside, and automatically disappears when the force is removed.

37. Semple and Kneebone: 'Algebraic Projective Geometry
• Projective geometry is a subject that lends itself naturally to algebraic treatment, and we have had no hesitation in developing it in this way - both because to do so affords a simple means of giving mathematical precision to intuitive geometrical concepts and arguments, and also because the extent to which algebra is now used in almost all branches of mathematics makes it reasonable to assume that the reader already possesses a working knowledge of its methods.
• Our first rudimentary idea of number is arrived at by simple abstraction from the processes of counting and measuring ordinary objects, and this idea is adequate at the level of school arithmetic.
• In this book we shall study the structure of projective geometry which, as is well known, is closely associated with certain simple algebraic structures, and with linear algebra particularly.
• Finally, the essentials of euclidean geometry may be treated projectively by the simple artifice of introducing the line at infinity and the circular points.

38. H L F Helmholtz: 'Theory of Music' Introduction
• Later physics has extended the law of Pythagoras by passing from the lengths of strings to the number of vibrations, and thus making it applicable to the tones of all musical instruments, and the numerical relations 4 to 5 and 5 to 6 have been added to the above for the less perfect consonances of the major and minor Thirds, but I am not aware that any real step was ever made towards answering the question: What have musical consonances to do with the ratios of the first six numbers? Musicians, as well as philosophers and physicists, have generally contented themselves with saying in effect that human minds were in some unknown manner so constituted as to discover the numerical relations of musical vibrations, and to have a peculiar pleasure in contemplating, simple ratios which are readily comprehensible.
• E Hanslick, in his book On the Beautiful in Music (Ueber das musikalisch Schone), triumphantly attacked the false standpoint of exaggerated sentimentality, from which it was fashionable to theorise on music, and referred the critic to the simple elements of melodic movement.
• The question of how the ear is able to perceive these harmonic upper partial tones then leads to an hypothesis respecting the mode in which the auditory nerves are excited, which is well fitted to reduce all the facts and laws in this department to a relatively simple mechanical conception.
• 5a), but that I was unwilling to omit that hypothesis because it is so well suited to furnish an extremely simple connection between all the very various and very complicated phenomena which present themselves in the course of this investigation.

39. Max Planck and the quanta of energy
• The noteworthy result was found that this connection was in no way dependent upon the nature of the resonator, particularly its attenuation constants - a circumstance which I welcomed happily since the whole problem thus became simpler, for instead of the energy of radiation, the energy of the resonator could be taken and, thereby, a complex system, composed of many degrees of freedom, could be replaced by a simple system of one degree of freedom.
• This extremely simple relationship can be considered as the completely adequate expression of Wien's energy distribution law; for with the dependence upon the energy, the dependence upon the wavelength is always directly given through the general, well-established displacement law by Wien.
• Whilst for small values of the energy and for short waves, Wien's law was satisfactorily confirmed, noteworthy deviations for larger wavelengths were found, first by O Lummer and E Pringsheim, and finally by H Rubens and F Kurlbaum, whose measurements on the infrared residual rays of fluorite and rock salt revealed a totally different, though still extremely simple relationship, characterized by the fact that the quantity R is not proportional to the energy, but to the square of the energy, and in fact this holds with increasing accuracy for greater energies and wavelengths.
• So, through direct experiment, two simple limits were determined for the function R: for small energies, proportionality with the energy; for greater energies, proportionality with the square of the energy.

40. Teixeira on Rocha
• The solution thus obtained is more closely approximate than those given previously, but less simple.
• Analysis and geometry help each other, but there are questions in the domain of the latter science to which the mathematician, without consideration, throws himself on the wings of the first, and, by flying, seeks to find by complicated formulas results to which the latter leads by a simple path.
• Anastacio da Cunha gave two very simple geometric demonstrations of this formula, and he censured the Academy of Sciences not only for posing such a simple question, but also for having rewarded such a mediocre memoir.

41. A N Whitehead addresses the British Association in 1916, Part 2
• The theory of the interconnection between the truth-values of the general propositions arising from any such aggregate of propositional functions forms a simple and elegant chapter of mathematical logic.
• But it is objected that this process and its consequences are so simple that an elaborate science is out of place.
• What, then, is the use of an elaborate chemical analysis of sea-water? There is the general answer, that you cannot know too much of methods which you always employ; and there is the special answer, that logical forms and logical implications are not so very simple, and that the whole of mathematics is evidence to this effect.
• Another example of this law is the way physicists and chemists have dissolved the simple idea of an extended body, say of a chair, which a child understands, into a bewildering notion of a complex dance of molecules and atoms and electrons and waves of light.

42. Born on wave mechanics
• Frequency and wave-number, on the other hand, are properties of simple harmonic waves, whose definition implies that they extend indefinitely in time and space.
• Nevertheless, modern physics declares that the matter is not so simple as this, whenever we have to deal with the restless universe of atoms and electrons.
• But the position of a physicist who wishes to observe an electron in its path is not so simple.
• Physically there is no meaning in regarding this wave as a simple harmonic wave of infinite extent; we must, on the contrary, regard it as a wave packet consisting of a small group of indefinitely close wave-numbers, that is, of great extent in space.

43. Smith's Teaching Books
• It is believed that teachers will welcome the logical, and at the same time simple, presentation of subjects like evolution, factoring, the theory of indices, and the treatment of the quadratic as set forth in this work.
• The noteworthy features of the book are the early and simple introduction of graphs with a table of squares and cubes at the end of the book to facilitate computation, the large number of oral problems under each topic and the cumulative reviews at the end of the book.
• Every important magazine such as well informed persons read uses graphs, formulas, and simple equations.
• that the work should proceed from the simple to the complex" and that "..

44. Gyula König Prize
• Here he gives very simple numerical expressions for Lebesgue constants whose properties were previously studied by Lipot Fejer and Thomas Gronwall.
• The expansions studied by Szego have an interesting and very simple relationship with the conformal mapping of the finite and infinite domains bounded by the given curve onto the unit disk.
• In this area, where the first classical results are linked with the names of Laplace and Darboux, Szego not only obtains very general results, far overshadowing anything known previously, but he obtains these results exactly because he examines these questions, considered very difficult, using a simple, one can say elementary, method.
• The main point of his method is that he squeezes the weight function P(x) between two functions of a very simple structure that have the form √(1 - x2)/P(x) where P(x) is a polynomial.

45. Vector calculus problems
• The method of exposition adopted by Grassmann is exceedingly abstract and this fact has stood stubbornly in the way of the general adoption of the Ausdehnungslehre to such an extent that we use today the barycentric calculus, the theory of equipollences, quaternions, or the Cartesian geometry, for the resolution of geometric questions which are capable of much more simple resolution by the methods of Grassmann.
• In ordinary differential geometry simple properties most frequently yield themselves only after very complicated calculations.
• On the other hand the geometrical calculus makes no use whatever of coordinates; it operates directly on the geometric elements; each formula which it produces is an invariant, capable of a simple geometric interpretation and leading directly to the graphic representation of the elements considered.
• What advantage has this circuitous definition with its adventitious vectors which in no way affect the value of a × b - what advantage has this over the simple statement, let a × b = ab cos q? By the non-quaternionic approach, some definition must be given - why not choose the simplest? .

46. de Montessus publications
• Quelques statistiques reductibles et non reductibles a la loi de probabilite simple, Annales Societe sc.
• (with F J Duarte), Determination de la mode, ou ecart le plus probable dans les courbes de probabilite simple, Annales Societe sc.
• Les phenomenes de physique et la loi de probabilite simple.
• Determination rigoureuse de la frequence moyenne et de la mode dans les courbes de probabilite simple - Application a un exemple, La Meteorologie (1928), 241-250.

47. Basset prefaces
• I have devoted Chapter IX to the flexion and vibrations of naturally straight wires and rods; whilst an entirely new chapter has been added on the finite deformation of naturally straight and curved wires, in which I have discussed a variety of questions which admit of fairly simple mathematical treatment.
• I have accordingly included Plucker's equations, which determine the number and the species of the simple singularities of any algebraic curve; and have also considered all the compound singularities which a quartic curve can possess.
• I have therefore confined the discussion to the simple and compound singularities of curves of this degree, together with a few miscellaneous propositions; and in Chapter IX, I proceed to investigate the theory of bicircular quartics and cartesians, concluding with the general theory of circular cubics, which is better treated as a particular case of bicircular quartics than as a special case of cubic curves.

• Simple Consequences and Extensions .
• Simple Properties of the Elementary Functions.
• Simple Non-Linear Mapping Problems.

49. Hardy on the Tripos
• It seems to me also that, if I wish afterwards to be certain that they have understood me, the obviously sensible way of finding out is to ask them to reproduce the substance of what I said or to apply the theorems which I proved to simple examples.
• We have not to undertake a general defence of mathematics and the position which is at present allowed to it in education, or to repel the very formidable onslaught which might be directed against it by Philistinism pure and simple.
• We may ask in the first place, if it be granted that what I have said about the past is roughly true, how far have things improved? Is it not true already that the Tripos means a great deal less, and English mathematics appreciably more, than forty years ago, and is it not extremely likely that, even if there be no further radical changes, this process will continue? Then, if we are not content to answer this question by a simple affirmative and leave it there, we may ask what really are the fundamental faults of an examination on the Tripos model, and whether it is not possible to make less drastic suggestions model, and whether it is not possible to make less drastic suggestions for its improvement.

50. Edmund Landau: 'Foundations of Analysis' Prefaces
• The complex numbers, incidentally, are not needed by the student in his first semester, but their introduction, being quite simple, can be made without difficulty.
• The matter now looks so simple and the proof so similar to the other proofs in the first chapter, that not even the expert might have noticed this point had I not given above a detailed confession of crime and punishment.
• For x.y the same simple type of proof applies; however, ∑ xn and ∏ xn are possible only with the Dedekind procedure.

51. Hart books
• The application of the higher branches of mathematical analysis to the solution of mechanical problems has been so perfectly successful as to procure its universal adoption, not only in the treatment of abstruse and difficult questions, but also of the most simple and elementary parts of the science.
• In the hope of contributing to its removal, he has been induced to publish the following treatise, in which he has endeavoured, by means of simple geometrical constructions, to render the most important fundamental propositions easily understood by all classes of students.
• In this manner the fundamental propositions of Hydrostatics are demonstrated; but as this is the only part of the subject in which accurate results have been obtained, so it is the only part which appears capable of such simple and elementary demonstration.

52. James Jeans: 'Physics and Philosophy' I
• To take a simple illustration, the physicist finds that the spectrum of atomic hydrogen contains the line Ha which we have already mentioned, and also a very great number of other lines which are usually designated as Hb, Hg, Hd, etc.
• The wave-lengths of these lines can be measured, and are found to be related with one another in a very simple way which can be expressed by a quite simple mathematical formula.

53. Born Inaugural
• Relativity gave the first example in which the intrusion of the observer into the description of facts is not so simple, and leads to a new conception to conserve the idea of an objective world Einstein has acknowledged that his studies on this problem were deeply influenced by the ideas of Ernst Mach, a Viennese physicist who developed more and more into a philosopher.
• But if we take into account the simple quantitative law relating energy and frequency already discovered by Planck, the case becomes very serious.
• This is indeed the case, and the connecting law is extremely simple when all the particles of the beam have exactly the same velocity.

54. Cariolaro's papers
• In particular, we give generalizations of Vizing's Theorem, Shannon's Theorem and Vizing's Adjacency Lemma, and an extension to multigraphs of the simple graph version of Vizing's Theorem which is obtained by proving that the chromatic index of an arbitrary multigraph must assume one of only two possible values.
• Star multigraphs turn out to be useful tools in the study of the chromatic index of simple graphs.
• We show that the following fundamental edge-colouring problem can be solved in polynomial time for any given constant B: given a simple graph G, find an edge-colouring of G where each colour is assigned to at most B edges and which, subject to this condition, has the fewest number of colour classes.

55. Fejer descriptions
• It is due to such care spent on the elaboration of the solution that Fejer's papers are very clearly written, and easy to read and most of his proofs appear very clear and simple.
• Yet only the very naive may think that it is easy to write a paper that is easy to read, or that it is a simple thing to point out a significant problem that is capable of a simple solution.

56. Dahlin Extracts
• To begin with he defines a star as a simple, ethereal, shining and essentially spherical body, which, through an intrinsic divine force rotates around the axis mundi, completes its orbit in a fixed time and has been created by God for the sake of mankind.
• It was split into two classes: pure or simple and impure or mixed.
• Dahlin describes some of the calculations, which, except for a few trigonometric, mainly make use of simple numeric operations.

57. Wolfgang Pauli and the Exclusion Principle
• giving the lengths of the periods in the natural system of chemical elements, was zealously discussed in Munich, including the remark of the Swedish physicist, Rydberg, that these numbers are of the simple form 2n2 if n takes on all integer values.
• On the one hand, the anomalous type of splitting exhibited beautiful and simple laws and Lande had already succeeded to find the simpler splitting of the spectroscopic terms from the observed splitting of the lines.
• The fundamental idea can be stated in the following way: The complicated numbers of electrons in closed subgroups are reduced to the simple number one if the division of the groups by giving the values of the four quantum numbers of an electron is carried so far that every degeneracy is removed.

58. E W Hobson: 'Mathematical Education
• I do not of course contemplate the introduction into such a course of artificial problems on scales of notation; only the fundamental principles should be explained, with such quite simple illustrations as may be found necessary for their complete elucidation.
• I do not know to what extent some rudimentary and informal treatment of the properties of simple figures in three-dimensional space has at the present time become part of the normal instruction in Geometry in our schools.
• It is unnecessary to insist upon the importance of an endeavour to uproot ignorance of this kind, due as it is to lack of stimulation of the power of observing simple spatial properties.

59. Serre reviews
• In Chapter III, the author conjectures that every rational semi-simple abelian l-adic representation ..
• In particular, if these stabilizers are trivial, G is free so obtaining a simple proof of the theorem of Otto Schreier stating that any subgroup of a free group is free.
• After an overview of the main theorems on NX(p), the book offers simple, illustrative examples and discusses the Chebotarev density theorem, which is essential in studying frobenian functions and frobenian sets.

60. Menger on teaching
• (d) The symbol x, because of its equivocal use, is unfit to express in a simple way some of the most important properties of the identity function.
• But although these ideas can be presented in the form of deductive systems they deprive the student of the experience of developing such a theory from assumptions about really simple and purely geometric concepts such as points, lines, and incidence.
• The way to achieve these aims is to abandon the idea of teaching Euclidean geometry in its entirety and to present only a part or an aspect of Euclidean geometry - but that part or aspect with absolute rigour - as well as some simple related theories which, from the points of view of various students, are "relevant".

61. Max Planck: 'The Nature of Light
• What is this something which spreads through empty space and moves through the atmosphere at the enormous, speed of 300,000 kilometres per second? Isaac Newton, the founder of classical mechanics, made the most simple and obvious assumption that there are certain infinitesimally small corpuscles which are sent out in all directions with that velocity from a source of light, e.g.
• Instead of collecting as many as possible of the multifarious facts available, I shall simple examine one of them in detail.
• For we have long known that the chemical atom is not by any means the simple invariable element of which all matter is constituted, but rather that every single atom, particularly one of a heavy metal, must be considered as a world in itself, and the farther one penetrates, the richer and more varied the structure appears.

62. Newcomb Elements of Geometry
• The author has considered it more important to base the subject on natural and customary modes of thought than to adopt a system simple and rigorous, but not so based.
• The mode in which he has endeavoured to avoid the difficulty, and to render the natural system as rigorous and nearly as simple as the other, will be seen by an examination of the chapter on Proportion.
• From the fourth book onward a knowledge of simple equations is sometimes presupposed.

63. Ernest Hobson addresses the British Association in 1910, Part 3
• Here we have a case in which an enumeration, which appears to be not amenable to direct treatment, can actually be carried out in a simple manner when the underlying identity of the operation is recognised with that involved in certain operations due to differential operators, the calculus of which belongs superficially to a wholly different region of thought from that relating to Latin squares.
• One of the most general mathematical conceptions is that of functional relationship, or 'functionality.' Starting originally from simple cases such as a function represented by a power of a variable, this conception has, under the pressure of the needs of expanding mathematical theories, gradually attained the completeness of generality which, it possesses at the present time.
• But, in the schemes of purely deductive geometry, the systems of axioms and postulates are far from being of a very simple character; their real nature, and the necessity for many of them, can only be appreciated at a much later stage in mathematical education than the one of which I am speaking.

64. Mary Boole writing
• There seems to be evidence that in ancient times all people in good society were expected to know simple truths about geometric forms in the same way as we all know simple facts in natural history.
• The question asked by a parent should be, 'At what age would you recommend me to let my child begin learning such portions of Algebra (or Trigonometry) as can only be learned by the aid of complicated devices invented, centuries after the science itself was an actual working possession of our race, for the sake of projecting its action into fields which would be inaccessible to it if only natural and simple tools were used?' The answer should be, 'When the process of learning by the more direct means has become so familiar as to be performed sub-consciously.' .

65. Kingman autobiography
• After a couple of weeks, I graduated to some simple modelling the group was doing of the demand for telephone cable pairs to homes.
• Our models were very simple, and involved probability calculations little more advanced than those in Lindley's lectures.
• So I tried to solve simple queues with service in random order, and eventually found a formal solution for M/G/1 for this discipline.

66. Douglas Jones's books
• On the other hand, much attention is given to simple numerical processes, which provide excellent illustrations of the uses of analysis in a field of general importance.
• The concept of an antenna as a piece of wire or portion of dielectric which radiates electromagnetic energy is simple enough in principle, but the derivation of quantitative results of value for design purposes is fraught with difficulties.
• It would have been interesting to see an example of some simple problem being worked by both 'standard' and 'non-standard' processes, so that the reader may compare the two methods.

67. Marshall Hall books
• Here, the specific nature of the rule of combination did not matter as long as it satisfied certain simple properties (closure, associativity, possession of inverse), i.e., the collection with its rule of combination formed a group.
• If the specified rules are very simple, then the chief emphasis is on the enumeration of the number of ways in which the arrangement may be made.
• Combinatorial theory encompasses a wide variety of topics, from simple counting of permutations and use of the pigeonhole principle to partitions, map colourings, latin squares, rook polynomials, design of experiments and Ramsey theory.

68. Menger on the Calculus of Variations
• A simple but interesting example, due to the economist H Hotelling (Columbia University), is to find the most economic way of production in a mine.
• For example, we consider the two following extremely simple problems: two given points may be joined by all possible curves; which of them has the shortest length, and which of them has the greatest length? The first problem is soluble: The straight line segment joining the two points is the shortest line joining them.
• The most simple example of this theory, which calculates the number of minimizing and maximizing curves as well as of stationary curves, is the following "geographical" theorem quoted by Morse: If we add the number of peaks and the number of pits on the surface of the earth, and subtract the number of passes, then the result will be the number 2, whatever the shape of the mountains may be (highlands excluded).

69. Pólya on Fejér
• It is due to such care spent on the elaboration of the solution that Fejer's papers are very clearly written, and easy to read and most of his proofs appear very clear and simple.
• Yet only the very naive may think that it is easy to write a paper that is easy to read, or that it is a simple thing to point out a significant problem that is capable of a simple solution.

70. Geary's books
• It is nowhere clearly stated that each tableau is a representation of the original problem with the equations solved for the basic variables in terms of the non-basic variables, so that only a simple change of variable (Jordan elimination) is involved in progressing from one tableau to the next.
• Rather crucial in such an elementary work on linear programming is to explain the Simplex Method by means of a simple example.
• Part I, "Theory", begins by considering a simple example which is solved graphically and by a labored use of a simplex method.

71. R A Fisher: 'Statistical Methods' Introduction
• If an observation, such as a simple measurement, be repeated indefinitely, the aggregate of the results is a population of measurements.
• In all cases, perhaps, it is possible to reduce to a simple numerical form the main issues which the investigator has in view, in so far as the data are competent to throw light on such issues.
• Some simple examples of the application of the method of maximum likelihood, and other methods, to genetical problems are developed in the final chapter.

72. Segel books
• Part B (Some fundamental procedures illustrated on ordinary differential equations) contains chapters entitled: Simplification, dimensional analysis, and scaling; Regular perturbation theory; Illustration of techniques on a physiological flow problem; Introduction to singular perturbation theory; Singular perturbation theory applied to a problem in biochemical kinetics; Three techniques applied to the simple pendulum.
• Chapter 7: Regular Perturbation Theory; The series method applied to the simple pendulum; Projectile problem solved by perturbation theory; .
• Chapter 11: Three Techniques Applied to the Simple Pendulum; Stability of normal and inverted equilibrium of the pendulum; A multiple scale expansion; The phase plane; .

73. Mirsky books
• This work provides an elementary and easily readable account of linear algebra, in which the exposition is sufficiently simple to make it equally useful to readers whose principal interests lie in the fields of physics or technology.
• The book is intended mainly for students pursuing an honours course in mathematics, but I hope that the exposition is sufficiently simple to make it equally useful to readers whose principal interests lie in the fields of physics or technology.
• The proofs of these are generally very simple, but are widely scattered throughout the literature and are often not easily accessible.

74. Gender and Mathematics
• (See for example, [',' L O Adetula, Solution of simple word problems by Nigerian children: Language and schooling factors, Journal for Research in Mathematics Education 20 (1989), 489-497.
• Nothing appears to be simple and listing what I really know is difficult.
• But a closer examination reveals that nothing to do with gender is simple.

75. Newton by his contemporaries
• These indeed derive the causes of all things from the most simple principles possible; but then they assume nothing as a principle that is not proved by phenomena.
• From some select phenomena they deduce by analysis the forces of nature, and the more simple laws of forces; and from thence by synthesis show the constitution of the rest.
• Therefore that we may begin our reasoning from what is most simple and nearest to us, let us consider a little what is the nature of gravity with us on Earth, that we may proceed the more safely when we come to consider it in the heavenly bodies that lie at so vast a distance from us.

76. Biography of Mathematics
• For a very simple reason: because Geometry is a living subject and therefore changing.
• This simple fact produces an extraordinary simplification, because when forgetting the quantities, which were vectors, and that could not be multiplied as numbers, what required one to resort to introducing a multiplication of the tensor product, considering the symbol having empty content and operating with it as if it were a number, we immediately obtain a set, whose elements we now call polynomials, with which we operate as with integers.
• The new foundation of Mathematics is based on a very simple principle, which can be stated as follows: "In order to be able to demonstrate the propositions with rigour, it is necessary to empty the concepts of primitive content, limited to establishing the allowed logical mechanism and the fundamental relations between such concepts." .

77. Mathematicians and Music 2.2
• In the early part of the third period in the development of music, namely, the period of Harmonic or Modern Music, we have the first opera and the first oratorio, and, as I have already said, the discovery by Galileo that the simple ratios of the lengths of strings existed also for the pitch numbers of the tones they produced, an observation later generalized by Newton.
• when a string is plucked or struck, or, as we may add 'bowed' at any point in its length which is the node of any of its so-called harmonics those simple vibrational forms of the string which have a node in that point are not contained in the compound vibrational form.
• Hence if we attack at its middle point, all the simple vibrations due to the even numbered partials, each of which has a node at that point, will be absent.

78. Skolem: 'Abstract Set Theory
• However, the simple theory of types, Quine's theory and the ramified theory of types are treated to a certain extent.
• The simple infinite sequence.
• The simple theory of types .

79. Bratteli publications
• Ola Bratteli, George A Elliott, David E Evans and Akitaka Kishimoto, Homotopy of a pair of approximately commuting unitaries in a simple C*-algebra, J.
• Ola Bratteli, Palle E T Jorgensen, Ki Hang Kim and Fred Roush, Decidability of the isomorphism problem for stationary AF-algebras and the associated ordered simple dimension groups, Ergodic Theory Dynam.
• Ola Bratteli, Palle E T Jorgensen, Ki Hang Kim and Fred Roush, Corrigendum to the paper: "Decidability of the isomorphism problem for stationary AF-algebras and the associated ordered simple dimension groups'' [Ergodic Theory Dynam.

80. Poincaré on the future of mathematics
• An algebraical formula which gives us the solution of a type of numerical problems, if we finally replace the letters by numbers, is the simple example which occurs to one's mind at once.
• It is for the same reason that, when a somewhat lengthy calculation has conducted us to some simple and striking result, we are not satisfied until we have shown that we might have foreseen, if not the whole result, at least its most characteristic features.
• And since it enables us to foresee whether the solution of these problems will be simple, it shows us at least whether the calculation is worth undertaking.

81. Senechal on Delone
• But on the basis of his papers and through my friendship with Ravil V Galiulin (1940-2010), Nikolai P Dolbilin, and Mikhail I Shtogrin, I became and remain his disciple, trying always to emulate his clear and simple approach to crystallographic problems and his informal, lucid writing style.
• From these simple hardcore and homogeneity axioms they drew a surprising amount of information about the geometry of the distribution of these points in space.
• Not only is this approach simple and elegant, it has turned out to be useful.

82. Sigmund books
• What does one learn from this book? The biologists will learn of the mathematical unity of seemingly distinct biological problems, of the wealth of mathematical complexity hidden behind even relatively simple biological models, and of the mathematical rigour that can be usefully applied in the analysis of such models.
• He is the only mathematician about whom I dare make this assertion." This book wants to give a simple, intuitive and easily digestible introduction to Godel's life and work, meant for readers interested in the human and cultural aspects of science.
• Karl Sigmund, a pioneer in evolutionary game theory, uses simple and well-known game theory models to examine the foundations of collective action and the effects of reciprocity and reputation.

83. Rudio's Euler talk
• The humble rural conditions of Leonhard Euler's upbringing surely contributed to his simple, modest attitude, as well as his impartiality, which he managed to preserve up to old age.
• Since the natural phenomena are dependent on each other in the most varied ways, there are infinitely many mathematical functions -- but please do not believe that these dependencies and hence the corresponding functions are always as simple as in the examples I mentioned above.
• As an example, it would not be possible to describe the motions of Earth around the Sun in a more simple and comprehensive manner than by Kepler's laws.

84. Heath: Everyman's Library 'Euclid' Introduction
• The simple truth is that it was not written for schoolboys or schoolgirls, but for the grown man who would have the necessary knowledge and judgment to appreciate the highly contentious matters which have to be grappled with in any attempt to set out the essentials of Euclidean geometry as a strictly logical system, and, in particular, the difficulty of making the best selection of unproved postulates or axioms to form the foundation of the subject.
• Simson had, it is true, a "bee in his bonnet." The title-page of his first editions says that "in this edition the errors by which Theon or others have long ago vitiated these Books are corrected, and some of Euclid's Demonstrations are restored." Simson, however, was not in any real sense a competent textual critic; he acted on the simple but uncritical principle that whatever he found in the text which fell short of perfection, whether of form or content, must have been due to alterations made by Theon or "some unskilful editor.
• The simplest case of "application of areas," which is equivalent to the solution for x of the simple equation ax = S, can be read in this volume (Eucl.

85. Peter's books
• The illustrations are simple and interesting.
• Next follows fractions and how they do not fill the number line and a beautifully simple sketch of Cantor's proof that, contrary to appearances, there are as many positive integers as there are rational numbers (fractions).
• The objective of this book is principally to show that many of the definitional structures used in programming languages can be expressed formally as partial recursive functions and that these partial recursive functions can be implemented by straightforward techniques on a random access machine with a simple assembly language instruction set.

86. Feller Prefaces
• Some of them are simple exercises, but most of them serve as additional illustrative material to the text or contain various complements.
• In the resulting confusion closely related problems are not recognized as such and simple things are obscured by complicated methods.
• Some theorems which were considered strikingly original and deep now appear with simple proofs among more refined results.

87. Halmos popular papers
• A public lecture should be simple and elementary; it should not be complicated and technical.
• If you believe and can act on this injunction ("be simple"), you can stop reading here; the rest of what I have to say is, in comparison, just a matter of minor detail.
• My test for what makes a good teacher is very simple: it is the pragmatic one of judging the performance by the product.

88. Tverberg Bergen institute
• (It is quite interesting comparing the various proofs of Ramsey's theorem: Ramsey's original proof is an excellent instance of how one can refine the structure of a result so as to be able to prove it in many small steps; but Skolem's proof is simpler; and subsequent proofs of Erdos and Szekeres and Erdos and Rado illustrate how a simple change in strategy can effect a reduction in numerical bounds by several orders of magnitude.) But Skolem had an early, substantial interest in combinatorial problems per se, publishing a lengthy account Untersuchungen Ouber einige Klassen kombinatorischer Probleme in 1917; for example, Skolem includes a catalogue of connected graphs on up to 8 vertices, each of degree at most 3, clearly with an eye to what we would recognise as design-theoretic properties.
• Fundamental to Skolem's approach is the simple idea of partitioning the set of integers {1, 2, ..

89. E C Titchmarsh: 'Aftermath
• 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.
• But essentially their patterns are of the same sort as the simple ones of which we have given some examples here.

90. Edward Sang on his tables
• Accident brought it again before me, and this time, considering not the relations of the lines connected with it, but the relations of the areas concerned, an exceedingly simple solution was found.
• The mean anomaly of a planet may be deduced from its angle of position, or as it is generally called, its excentric anomaly, by simple additions and subtractions of these circular segments.

91. Godement's reviews
• Zeta functions of simple algebras (1972), by Roger Godement and Herve Jacquet.
• The aim of the authors is to define the Hecke zeta-functions for all simple algebras over algebraic number fields and to prove a functional equation for them.

92. Rudio's talk
• I do not even wish to remind you that this simple assumption, which forms the basis of our calculation today, has escaped a mathematically gifted people such as the Greeks.
• Children in public schools were not even taught simple calculations with digits.

93. Chrystal: 'Algebra' Preface
• I suppose that the student has gone in this way the length of, say, the solution of problems by means of simple or perhaps even quadratic equations, and that he is more or less familiar with the construction of literal formulae, such, for example, as that for the amount of a sum of money during a given term at simple interest.

94. Mercer's papers
• Some simple duration-dependent stochastic processes, J.
• Some simple wear-dependent renewal processes, J.

95. W Burnside: 'Theory of Groups of Finite Order
• Galois introduced into the theory the exceedingly important idea of a self-conjugate sub-group, and the corresponding division of groups into simple and composite.
• The last Chapter contains a series of results in connection with the classification of groups as simple, composite, or soluble.

96. Burton papers
• Conclusions: The key themes identified in the literature were: (i) the conceptualisation of management as masculine, to which we would add 'white'; (ii) discrimination in promotion; (iii) women's career patterns; (iv) domestic responsibilities, to which we would add responsibilities to their communities; (v) mentoring, which is clearly neither singular nor simple; and (vi) management styles.
• Although untangling the inter-relationships between these three is no simple matter, they make effective starting points in order to ask similar questions of mathematics to those asked by our colleagues in science.

97. Feller Reviews 1
• Starting with some rather simple problems of combinatorial analysis, the text then goes on to such advanced topics as Markov chains, recurrent events, random walks, waiting time, trunking problems, and time-dependent stochastic processes.
• To avoid advanced mathematical concepts (measure theory, etc.) and to make the work useful to beginners, the author limits it to questions which involve only a countable sample space; but about these simple questions it addresses the most advanced problems of probability theory, many of which have not until now been exposed in a book, so that the work is of the highest interest for specialists.

98. Loney reviews
• The author has succeeded in his purpose to produce "a fairly complete elementary text-book on Plane Trigonometry." The faithful student of this treatise "will have little to unlearn when he commences to read treatises of a more difficult character." The style is clear and simple; even when it is diffuse, the author never hides his thoughts with words either large or small.
• The sections dealing with curves of buoyancy and tensions of vessels are as simple as is necessary for ordinary students.

99. Wussing Reviews
• The virtues of this volume are simple.
• the lively, clear, and simple style nicely conveys its main message: that mathematics is a human pursuit whose aims and motivations can be understood by everyone.

100. Eddington: 'Mathematical Theory of Relativity' Introduction
• Or again, instead of cutting short the astronomical calculations when we reach the parallax, we might go on to take the cube of the result, and so obtain another manufactured quantity, a "cubic parallax." For some obscure reason we expect to see distance appearing plainly as a gulf in the true world-picture; parallax does not appear directly, though it can be exhibited as an angle by a comparatively simple construction; and cubic parallax is not in the picture at all The physicist would say that he finds a length, and manufactures a cubic parallax; but it is only because he has inherited a preconceived theory of the world that he makes the distinction.
• But to catalogue all the precautions and provisos in the operation of determining even so simple a thing as length, is a task which we shirk.

101. St Andrews Physics Examinations
• Define a simple pendulum and a compound pendulum.
• How may the length of a simple pendulum and its time of vibration be determined by observations made with a compound pendulum? (Kater's method.) .

102. Feller Reviews 2
• Most of the chapters, include numerous problems, ranging from simple exercises to applications and extensions of the text.
• He restricts himself to a discussion of enumerably infinite simple events.

103. Hamming's Reviews
• The algorithms are explained geometrically and often illustrated by a simple numerical example, sometimes showing the limitations of the algorithm.
• I have no hesitation in recommending the book as a simple introduction to computers and their uses.

104. Heinrich Weber's books
• Our admiration is no less excited by its pedagogic excellencies ; Weber's German is simple and concise, the demonstrations are clear and rigorous, and many of them are of extreme elegance.
• The theory of interest is based upon compound interest, in the sense that simple interest is looked upon as an annuity in perpetuity.

105. Pedoe's books
• The inclusion of unfamiliar, yet conceptually relatively simple, phenomena associated with circles furnishes examples that should be both interesting and challenging to students.
• It is, in essence, a pictorial essay requiring a persistent intense concentration on the simple and subtle beauty inherent in the plethora of diagrams.

106. Dehn on Aristotle
• And finally, it gives the mature mathematician great satisfaction to methodically examine large systems of propositions, to see the wonderful architecture of entire disciplines rise as a result of stringent combination from simple fundamentals to heights inaccessible to direct observation.
• Pre-Greek mathematics had primitive knowledge about integers and simple geometrical forms like points, straight lines, planes, etc.

107. Edinburgh Physics Examinations
• Define Simple Harmonic Motion; and show that the resultant of two S.H.M: of the same period, in one line, is another of the same period.
• Explain the action of a convex lens of short focus when employed as a simple microscope.

108. Peacock Treatise
• If the first principles of Algebra had been consistent with themselves, or had led to no difficulties either in the reasoning immediately connected with them, or in their remote consequences, which did not admit of a simple and uniform explanation, we should very properly hesitate before we acceded to any innovations in those principles or in their exposition; for under such circumstances, the perfect union and attachment of the parts of the fabric would furnish the best evidence of the sufficiency of the foundations: but it is the admitted existence of difficulties in the consequences of the principles of Algebra, as they are commonly stated, both immediate and remote, which naturally, and indeed necessarily, induces us to suspect the existence likewise of imperfections or inaccuracies in the principles themselves: a suspicion which becomes confirmed when it appears, after the most careful examination of them, that the difficulties in question are not referable to their imperfect development.
• Algebra has always been considered as merely such a modification of Arithmetic as arose from the use of symbolic language, and the operations of one science have been transferred to the other without any statement of an extension of their meaning and application: thus symbols are assumed to be the general and unlimited representatives of every species of quantity: the operations of Addition and Subtraction in their simple arithmetical sense, are assumed to be denoted by the signs + and -, and to be used in connecting such symbols with each other: Multiplication and Division, two inverse operations in Arithmetic, are supposed to be equally applicable to all quantities which symbols may denote, without any necessary modification of their meaning: but at the same time that the primitive assumption of such signs and operations is thus carefully limited in the extent of their signification, there is no such limitation imposed upon the extent of their application: thus it is not considered necessary that the operations of Addition and Subtraction should be confined to quantities of the same kind, or that the quantities subtracted should be less than the quantities from which they are subtracted: and when the violation of this restriction, which would appear to be rendered necessary by the primitive meaning of those operation, has led to the independent existence of the signs + and -, as an assumption which is also necessary in order to preserve the assumed universality of the values of the symbols and of the possibility of the operations which they designate, it is not considered that by this additional usage of them, we have altogether abandoned the definitions of those operations in practice, though we have retained them in name: for the consequences of those operations, and the assumptions connected with them, must be determined by the fundamental rules for performing them, which are independent of each other, or whose necessary connection is dependent upon their assumed universality only: and the imposition of the names Addition and Subtraction upon such operations, and even their immediate derivation from a science in which their meaning and applications are perfectly understood and strictly limited, can exercise no influence upon the results of a science, which regards the combinations of signs and symbols only, according to determinate laws, which are altogether independent of the specific values of the symbols themselves.

109. Recollections of Mary Somerville
• A still smaller number of her own letters have been added, either as illustrating her opinions on events she witnessed, or else as affording some slight idea of her simple and loving disposition.
• Nor is her simple account of her early days without interest, when, as a lonely child, she wandered by the seashore, and on the links of Burntisland, collecting shells and flowers; or spent the clear cold nights at her window, watching the starlit heavens, whose mysteries she was destined one day to penetrate in all their profound and sublime laws, making clear to others that knowledge which she herself had acquired, at the cost of so hard a struggle.

110. Weyl on Hilbert
• His optimism, his spiritual passion, his unshakable faith in the supreme value of science, and his firm confidence in the power of reason to find simple and clear answers to simple and clear questions were irresistibly contagious.

111. Plucker Copley medal
• He has succeeded in obtaining the mathematical definition of these curved lines or surfaces, by a simple application of the known laws of electromagnetic action, regarding an element of the discharge as the element of an electric current.
• In a recent memoir, which has only just been published in the Philosophical Transactions, Professor Plucker has investigated the two totally different spectra frequently afforded by the same elementary substance according as it is submitted to the instantaneous discharge of a Leyden jar charged by an induction-coil, or rendered incandescent by the simple discharge of the coil, or else, in some cases, by ordinary flames.

112. James Jeans: 'Physics and Philosophy' II
• A simple specific example of this general argument will be found below.
• For suppose - to imagine a simple although not very likely possibility - that it had been found that the pattern of events could be fully explained by assuming that matter consisted of hard spherical atoms, and that each of these behaved like a minute billiard-ball.

113. Rhode Island College
• The work in algebra consists of a systematic drill in the fundamental operations, leading up to a study of the equation, both simple and quadratic, the theory of exponents, radicals, the progressions, the binomial formula, and the graphic representation of equations.
• It includes the differentiation of algebraic, trigonometric, anti-trigonometric, exponential and logarithmic functions, successive differentiation and the integration of simple forms, illustrated by applications to the rectification of plane curves, the areas of plane curves and the surface and volume of solids of revolution.

114. Whyburn's books
• Though the treatment is not difficult, it is likely to be more readily intelligible to those with some mathematical training who are seeking knowledge of economics than for economists without any initial facility in manipulating simple mathematics.
• The volume in question seems particularly interesting for Italian students of Economics; in fact, it constitutes a clear and elementary introduction to traditional mathematics, the introduction is presented in a very simple way but not without rigour, and therefore fills a gap in the range of books, because every argument put forward is always illustrated by examples with elementary mathematics drawn from economics.

115. W H Young addresses ICM 1928 Part 2
• And yet, these also deserve their place of honour, if only for services like that rendered by Sir Ronald Ross in utilising the simple idea that it is on the percentage of mosquitoes to the individual, not on their mere presence, or even their number, that depends the epidemic of malaria, thereby creating anew the science of tropical medicine.
• The necessity for the recognition of the region of validity may be illustrated by the following simple fable: .

116. Shepherdson Tribute
• The first I found, connected to Heilbronn, is on simple additive combinatorics (and was reviewed by Erdos).
• He used Tarski's work on real-closed fields, in a simple but brilliant way (and elsewhere alludes to doing things with open induction for exponentiation).

117. Reviews of Shafarevich's books
• Groups are illustrated by simple examples, the symmetry groups of polyhedra and various crystals, but also by more abstract cases such as the Brauer group.
• To help the reader grasp the material, simple problems are given to be solved.

118. Enciclopedia delle Matematiche
• The treatment is simple, interesting and in connection with topics treated reasonably comprehensive.
• But throughout the presentation is scholarly, the emphasis welt-placed , and the language simple, connected and interesting.

119. Magnus books
• The authors of the present volume have overcome this obstacle by leading the reader slowly from the concrete to the abstract, from the simple to the complex, employing effectively graphs or Cayley diagrams to help the student visualize some of the structural properties of groups.
• It may be appropriate to remark here that the theory of Hill's equation reveals the occurrence of a surprising phenomenon which can be described in rather simple terms.

120. H F Baker: 'A locus with 25920 linear self-transformations' Introduction
• The geometrical properties of this primal are very interesting; and they form a vivid and simple concrete representation of the group of the lines of a cubic surface, and its more important subgroups; and incidentally illustrate the elements of the theory of the substitutions of five and six objects.
• One remark should perhaps be added here to make the general statements of this introduction more precise: The group of the lines of a cubic surface is of order 24× 34× 40; this group has a subgroup of order 1/2 (24× 34× 40) or 23× 34× 40, which, as Jordan proved, is simple [it is PSp(4,3), the projective symplectic group of 4 × 4 matrices over the field of 3 elements].

121. André Weil: 'Algebraic Geometry
• 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.
• V deals with the intersections of an arbitrary variety and of a linear variety in an affine space, first (in § 1) when these varieties have complementary dimensions, then (in § 2) in general; § 3 contains some applications of these results to the theory of simple points.

122. Bell books
• Dr Bell, incidentally a successful novelist, has written in simple style, mathematically unconventional perhaps, but not flippant.
• [The reader] must on occasion be content to have some historical claim substantiated by simple repetition; he should remember his Undergraduate Society debating the eternal "Hen-or-Egg" Priority problem; but before all else he must be mature enough not to be muddled, as so many of us are, by irony and sarcasm - and he must certainly not be pedantic about detail.

123. Mordell reminiscences
• The reason is a very simple and natural one.
• So as I have said in the beginning of my talk, there was a very simple explanation of why I went to Cambridge.

124. Santalo honorary doctorate
• The first systematic work of pure geometry was the Euclid's Elements (3rd century BC) whose purity refers both to its constituent elements (points, lines, planes) that are simple and perfect, obtained by idealization of visual forms discernible by the senses, to the axiomatic construction, which served as a model for all subsequent mathematics, and also to the common notions with which the congruence of figures is introduced through the movements of the plane.
• We have tried to make it clear with a simple and very limited example, but surely an analogous evolution can be found in many other chapters of that science.

125. A N Whitehead: 'Autobiographical Notes
• Such Scripture lessons, on each Sunday afternoon and Monday morning, were popular, because the authors did not seem to know much more Greek than we did, and so kept their grammar simple.
• The only point on which I feel certain is that there is no widespread, simple solution.

126. Kepler's Planetary Laws
• (Moreover, the same principle is invoked in relation to planetary motion when Kepler based his investigation on what Aristotle had specified as the only two simple motions, circular and rectilinear, discussed in Section 9.) This principle has far-reaching ramifications, as we will demonstrate in connection with the complementary pairings that recur in Kepler's mature work in Epitome Book V (1621) - where the term 'complementary' is used in the everyday sense that the pair complete one another, and also with the mathematical connotation of being at right angles.
• He adopted the traditional mechanism of deferent, epicycle, and eccentric, being aware, as the Ancients had been, that motion in the circle of radius a centred on A, when combined with motion in the epicyclet of radius ZQ = AB = ae (whose centre Z lies on the deferent), together produce a motion of Q equivalent to a simple motion of Q round the eccentric circle centre B radius a.

127. David Hilbert: 'Mathematical Problems
• By the examples of the simple and double integral I will show briefly, at the close of my lecture, how this way leads at once to a surprising simplification of the calculus of variations.
• But what an important nerve, vital to mathematical science, would be cut by the extirpation of geometry and mathematical physics! On the contrary I think that wherever, from the side of the theory of knowledge or in geometry, or from the theories of natural or physical science, mathematical ideas come up, the problem arises for mathematical science to investigate the principles underlying these ideas and so to establish them upon a simple and complete system of axioms, that the exactness of the new ideas and their applicability to deduction shall be in no respect inferior to those of the old arithmetical concepts.

• But all this ingenuity, all this jovial character, would remain as a simple anecdote if it were not united to an impressive capacity to transmit ideas; and this ability he develops using the most diverse of strategies: "I wanted to keep this operation intact on the board - he tells his students one day, showing them the remains of a polynomial multiplication operation - but the janitor has come and erased almost all of it; I just arrived in time to stop him deleting the multiplicand and the product.
• Emphatic truth that, although simple enough, many teachers have not yet assimilated.

129. Gibson History 4 - John Napier
• But between Michael and Napier we can name no Scot whose interest lay specially in the domain of science, and the explanation is simple.
• It is quite obvious that the logarithm as thus defined is not so simple in actual work as the logarithm we now use.

130. Thomas Bromwich: 'Infinite Series
• 44 is not strictly historical, but is intended to emphasise the similarity between the tests for uniform convergence and for simple convergence (Arts.
• To illustrate the general theory, a short discussion of Dirichlet's integrals and of the Gamma integrals is given; it is hoped that these proofs will be found both simple and rigorous.

131. J A Schouten's Opening Address to ICM 1954
• The faculty of deduction belongs more to the conscious mind, the subconscious being in general only able to perform very simple and trivial deductions.
• In fact, there are machines, effecting a few simple logical deductions, and other machines, especially constructed for the investigation of big molecules, which are able to pass in a short time over say a million possible combinations of phases in order to single out some twenty five most suitable ones for a more detailed examination.

132. Frank Harary's books
• On the whole the book is clearly written with plenty of simple exercises, but Professor Harary's boundless enthusiasm makes rather extravagant calls on the reader's attention; he is so anxious that nothing be left out.
• Even at the simple level of drawing pictures and looking at them there is ample scope for experiment and investigation, since the number of distinct graphs increases so rapidly even with only a few vertices.

133. Carathéodory: 'Conformal representation
• In the proof of this theorem, which forms the foundation of the whole theory, he assumes as obvious that a certain problem in the calculus of variations possesses a solution, and this assumption, as Weierstrass (1815-1897) first pointed out, invalidates his proof Quite simple, analytic, and in every way regular problems in the calculus of variations axe now known which do not always possess solutions.
• During the present century the work of a number of mathematicians has created new methods which make possible a very simple treatment of our problem; it is the purpose of the following pages to give an account of these methods which, while as short as possible, shall yet be essentially complete.

134. Levitzki's papers
• A Galois theory in semi-simple rings, Bull.
• On the equivalence of the nilpotent elements of a semi simple ring, Compositio Math.

135. Bell papers
• The old assurances and arrogances are gone; the universe is not a book to be read in a cloister, nor is the solar system the simple parish it was in the middle ages.
• A typical simple specimen, which has passed unaltered into current usage, is his postulational definition of an ideal, and there are many others.

136. Charles Bossut on Leibniz and Newton Part 2
• At the same time Johann Bernoulli gave another method which, to the advantage of being incomparablely more simple, added that of embracing all the geometrical curves, all the mechanical curves completely similar, and lastly a great number of mechanical curves incompletely similar.
• Newton had determined the curve described by a projectile in a medium resisting in the ratio of the simple velocity: but had not touched on the case, at that time more difficult, where the resistance of the medium is as the square of the velocity.

137. L R Ford - Differential Equations
• 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.
• General solutions of simple types of partial differential equations are obtained before separation of variables is used to solve problems of vibration and the Laplace equation in two dimensions.

138. Samarskii's books
• The author gives a systematic exposition of the foundations of the theory of difference schemes and applications of this theory to the solution of simple typical problems of mathematical physics.
• The author has attempted to make his presentation understandable on the first reading, paying attention to the basic concepts of the theory of numerical methods and illustrating them by very simple examples.

139. Jordan algebras
• In a fundamental 1934 paper, Jordan, John von Neumann, and Eugene Wigner showed that every finite-dimensional formally real Jordan algebra is a direct sum of a finite number of simple ideals, and that there are only five basic types of simple building blocks ..

140. Byrne: Doctrine of Proportion
• Professor Young will not deny (for they are his own words) that "the term in reality denotes the quotient arising from the division of one magnitude or quantity by another of the same kind (or the multiple or submultiple which an antecedent is of its consequent); it is accurately assignable (in numbers) when the magnitudes are commensurable, but unassignable (in numbers) when they are incommensurable." When this simple fact is known, what is to be understood by the term cannot be misconstrued, although we do allow that in many cases the exact ratio of one magnitude to another of the same kind cannot be expressed by numbers; this may be a fault in our present system of notation, or in the plan adopted for finding a common measure, and not in our geometrical notion of that which is to be conveyed by the term.
• The student will readily perceive that the term ratio is not intended to convey a real and substantial essence, but merely a simple conception of the mind, which can be well defined, and not, as some writers would have it, an ill defined or unknown term.

141. M Bôcher: 'Integral equations
• Mathematicians have so far devoted their attention mainly to two peculiarly simple types of integral equations, - the linear equations of the first and second kinds, - and we shall not in this tract attempt to go beyond these cases.
• We shall also restrict ourselves to equations in which only simple (as distinguished from multiple) integrals occur.

142. Groups St Andrews proceedings
• Computational methods are surveyed in several articles in particular the major survey by Joachim Neubuser and find application in papers on Burnside groups and finite simple groups.
• The Theory of Groups continues to move forward on many fronts, and twenty years on from the announcement of the classification of the finite simple groups, it prospers perhaps surprisingly well (rather like Mark Twain).

143. E C Titchmarsh on Counting
• Number must have a meaning such that it is true that I have the same number of fingers on each hand, and the same number of buttons as buttonholes on my waistcoat (with coats the situation does not seem to be so simple).
• The conclusion of all this seems to be that we must do without a simple and direct answer to the question, "What is a number?" This will not prevent us from doing mathematics.

144. Rios Honorary Degree
• This has contributed to considering Monte Carlo simulation as more than a simple complement to modelling, because of the basic advantages of understanding, implementation, execution and memory requirements, etc.
• The extraordinary similarity of the structure of all parts of the human cortex to each other and of human cortex with the cortex of the most primitive mammals suggests that a relatively simple universal principle governs its operation, even in complex processes like language." .

145. Atiyah reviews
• Our philosophy has been to build up to the main theorems in a succession of simple steps and to omit routine verifications.
• Some of them are simple and others are rather difficult.

146. Jacobson: 'Theory of Rings
• That this has been possible in a book dealing with results of the significance of Wedderburn's theorems, the Albert-Brauer-Noether [A Adrain Albert, Richard Brauer, Emmy Noether] theory of simple algebras and the arithmetic ideal theory is another demonstration of one of the most remarkable characteristics of modern algebra, namely, the simplicity of its logical structure.
• In the first part of this chapter we consider the theory of simple algebras over a general field.

147. Michell Twisted Rings
• The problem is simple to state: "If a wire of isotropic section and naturally straight be twisted, and the ends joined so as to form a continuous curve, the circle will be a stable form of equilibrium for less than a certain amount of twist." In other words, consider an isotropic elastic rod (the rod has no preferred bending direction) that is stress-free when held straight.
• Apparently, Michell realized that when these frequencies become imaginary the equilibrium shape loses its stability and he applied this idea to derive a simple criterion for the instability of a twisted elastic ring.

148. Edinburgh Mathematics Examinations
• Show that their resultant may be treated as simple harmonic motion, in a direction which rotates slowly.
• (b) the other is forced to execute transverse simple harmonic motions of given period and range.

149. Rios's books
• Ten chapters take you in logical steps from "individual decisions in a probabilistic environment" up to "utility in a multistage environment" and "collective decisions." It includes also a reasonably full treatment of Bayesian methods and is illustrated with simple examples.
• This is an excellent simple introduction to decision theory.

150. James Jeans addresses the British Association in 1934, Part 2
• It may seem strange, and almost too good to be true, that nature should in the last resort consist of something we can really understand; but there is always the simple solution available that the external world is essentially of the same nature as mental ideas.
• Let me digress again to remind you of two simple instances of such conflicts and of the verdicts which observation has pronounced upon them.

151. Kurosh: 'The theory of groups' 1st edition
• Moreover, from the point of view of algebra itself - of which the theory of groups is a part - a situation could hardly be regarded as normal in which such very simple and important groups as, for example, the additive group of integers remained outside the limits of the theory.
• Furthermore, very often a problem that is simple and completely solved in the case of finite groups changes to a broad theory, yet far from complete, this happens, for example, in the theory of abelian groups, one of the most important parts of contemporary group theory.

152. Harold Jeffreys on Logic and Scientific Inference
• But the importance of simple laws in inference leads us to concentrate on those properties of sensations that actually satisfy simple laws as far as they have been tested.

153. Halsted Beltrami
• In the exordium of a memoir dated Pisa, 31 May 1866, Beltrami remarks that in treating of a map destined to serve for measurements of distance it would be most convenient to determine, that to the geodetics of the surface should correspond the straights of the plane, because, such a representation accomplished, the questions concerning geodetic triangles would be reduced to simple questions of plane trigonometry.
• At least it seems that to such attempts we owe a memoir where is studied with scrupulous care the surface generated by the rotation of the tractrix about its asymptote with the aim of deducing the elements by a construction simple and exact of the surface itself.

154. Bolyai house and grave
• Thanks to his tireless diligence and procurement, a campaign started in the capital city circles and the Mathematics and Physics Society marked the grave of the author of "Appendix" with a simple, pretty tacky gravestone.
• I was hoping, that even a simple gravestone would be given to him, rather to his grave, for the occasion of the marking of his father's gravestone, but the Committee was not willing to do that - the remaining 100 forints from the collected money they wished to use as a scholarship called "Bolyai-fund", whose interest would be distributed as a reward for the best mathematics students of the last 4 classes of secondary school.

155. Halmos books 1
• The idea is to emphasize the simple geometric notions common to many parts of mathematics and its applications, and to do so in a language that gives away the trade secrets and tells the student what is in the back of the minds of people proving theorems about integral equations and Hilbert spaces.
• Mathematical logic has been intertwined with algebra from its beginnings, through Boole's discovery that simple laws of logic can be expressed symbolically as algebraic equations.

156. Vanstone obituary
• With Ray even a simple conversation would almost always turn into a serious discussion of non-trivial matters.
• Their many acts of kindness to visitors and colleagues, and their families, went far beyond simple human civility.

157. Boas books
• The text is a valuable contribution to mathematical literature in that it sets forth in simple language and in short space the parts of the real variable theory that are essential to further study in the various fields of mathematics.
• His prose is simple and direct, and at the same time, elegant and witty ..

158. Centenary of John Leslie
• His wants, however, were simple, and he managed to support himself in comfort by his pen and by tutoring little Colin Maclaurin; and he combined these with travel and study.
• In character Leslie was simple, good-natured and straightforward, free from jealousy and ready in his appreciation of the work of others.

159. Ahrens book of quotes
• For a scientist, explaining is analogous to tracing something back to a handful of desirably simple fundamental laws, which cannot be overcome, but must simply be taken for granted, in order to exhaustively explain a phenomenon.
• During a conversation with the writer in the last weeks of his life, Sylvester remarked as curious that notwithstanding he had always considered the bent of his mind to be rather analytical than geometrical, he found in nearly every case that the solution of an analytical problem turned upon some quite simple geometrical notion, and that he was never satisfied until he could present the argument in geometrical language.

160. Primes abstract
• How can one predict when the next prime number will occur? Is there a formula which could generate primes? These apparently simple questions have confounded mathematicians ever since the Ancient Greeks.

161. Ramchundra Preface
• This is a method which appears extremely simple and easy, though it baffled all my endeavours for the space of three years.

162. Gregory's Astronomical Clock
• The movement of the clock is of simple construction and contains only three wheels with an ordinary tick-tack escapement driven by a weight, suspended on a slender chain, which requires to be pulled up every twenty-four hours.

163. NAS Memoir of Chauvenet
• At the time of its publication trigonometry in many of our colleges was restricted to the simple cases of plane and spherical triangles, by the trammelling geometric processes.

164. Schrödinger: 'Statistical Thermodynamics
• The object of this seminar is to develop briefly one simple, unified standard method, capable of dealing, without changing the fundamental attitude, with all cases (classical, quantum, Bose-Einstein, Fermi-Dirac, etc.) and with every new problem that may arise.

165. Netto books
• The simple fact that the able and fairly complete treatise now under review hardly mentions the work of Hindenburg shows that what are now considered the substantial parts of combinatoric have been developed outside of the German Combinatorial School.

• Be simple.

167. Practical Logical Beautiful
• Then there is percentages, and estimating various quantities with simple calculations (Chapter 4), some knowledge of graphs, some probability and statistics.

168. Kepler's 'Foundations of modern optics' Preface to a translation
• But Kepler's text is not only a Latin text of the late sixteenth century written by an author from the Germanic cultural sphere, it is a technical text in which numerous passages - notably Chapter IV - testify more to a pure and simple transcription of personal notes than to a patiently executed draft.

169. Hans Hahn: 'The crisis in intuition
• Again and again we have found that, even in simple and elementary geometric questions, intuition is a wholly unreliable guide.

170. Moiseiwitsch Variational Principles
• Variational principles have long played two major roles in mathematical physics; one as great unifying principles through which the different equations can be ex- pressed in elegantly simple form, and the other as remarkably useful computational tools for the accurate determination of discrete eigenvalues such as the vibration frequencies of classical systems and the bound state energies of quantum mechanical systems.

171. H F Baker: 'A locus with 25920 linear self-transformations' Preface
• This volume, is concerned with a locus - itself very interesting to explore geometrically - which exhibits in a simple way the structure of the group of the lines of a cubic surface in ordinary space, regarded as the group of the tritangent planes of the surface.

172. Feller Reviews 4
• And just as in volume one the author delights in giving many deceptively simple results which tease the probabilistic intuition or which would require sophisticated proof if viewed outside their natural probabilistic context.

173. System Reliability Theory
• For this purpose we treat simple situations first.

174. Gattegno reflections
• These words came to me in 1960 when I was interviewed for the Christchurch Daily (New Zealand), and was asked to characterize my work in a few simple words.

• It is the successful organisation of multitudinous sounds of words, associations of words, pictorial memories of diverse events and feelings ordinarily occurring in life, combined with a special narrative of great events: the whole so disposed to excite emotions which, as defined by Milton, are simple, sensuous, and passionate.

176. Kuku interview
• I came from a very simple background.

177. Newcomb School Algebra
• The First Course, which extends to Simple Equations, is intended to drill the student in all the fundamental processes by exercises which are, for the most part, of the simplest character.

178. Ernest Hobson addresses the British Association in 1910, Part 2
• Except in certain very simple cases no process of measurement, such as the determination of an area or a volume, can be carried out with exactitude by a finite number of applications of the operations of arithmetic.

179. Landau and Lifshitz Prefaces
• For similar reasons, the microscopic theory of superconductivity is described with the simple model od an isotropic Fermi gas with weak interaction, disregarding the complications due to the presence of the crystal lattice and the Coulomb interaction.

180. A I Khinchin on Information Theory
• On the whole, I follow the path indicated in the works of McMillan and Feinstein, deviating from them only in the comparatively few cases when I see a gap in their explanation, or when another explanation seems to me more complete and convincing (and sometimes, more simple).

181. Segel Asymptotic analysis
• To do this we shall briefly examine (i) certain basic theorems connected with asymptotic expansions, (ii) a differential equation in the neighbourhood of an essential singularity, (iii) a very simple singular perturbation problem, and (iv) the lack of genuine distinction between a large variable and a small one - all in order to emphasize the connection between asymptotic approximations and essential singularities.

182. Ahrens book reviews
• We do not find much of value in the story of the boys who convinced a simple old man that in their use of logarithm tables they were mastering the house numbers of Europe.

183. Closing Gap abstract
• Prime numbers have intrigued, inspired and infuriated mathematicians for millennia, and yet mathematicians' difficulty with answering simple questions about them reveals their depth and subtlety.

184. Laplace: 'Méchanique Céleste
• I hope, in consideration of the difficulty and importance of the subject, that mathematicians and astronomers will receive it with indulgence, and that they will find the results sufficiently simple to be used in their researches.

185. Noether's students
• Thesis title: Nichtgaloissche Zerfallungskorper einfacher Systeme (Non-Galois splitting fields of simple systems).

186. Karl Menger on Hans Hahn
• No one knew as well as he how to present those leading ideas in such a simple as well as thorough way, in such a logical as well as suggestive form." .

187. Galois Sainte Pelagie preface
• Long algebraic calculations were at first hardly necessary for progress in Mathematics; the very simple theorems hardly gained from being translated into the language of analysis.

188. Catalan manifesto
• If my name was not heard in the political assemblies, there is a very simple reason: in the oligarchic regime that you just reversed, I was not a voter, and I would never become a voter! But ask those whom I dare to call my illustrious friends: Dupont (de l'Eure), Arago, Ledru-Rollin, Louis Blanc, Garnier-Pages; ask the young people who left the Ecole Polytechnique the previous eight years; invoke memories of my old comrades of the eleventh legion: they will all tell you that I was always seen at the breach, and that my well known republican views prevented me from making the progress that I was legitimately due.

189. Champernowne reviews
• Those already familiar with the field will find Champernowne's book generally fascinating because of the many novel simple models developed in the book.

190. Kantorovich books
• The first chapter deals with simple cases of short-run planning of production.

191. Analysis of Variance
• An agricultural experiment of a relatively simple structure to which the analysis of variance would be applicable would be the following: In each of three localities four varieties of tomatoes are grown in tanks containing chemical solutions.

• I was fortunate enough to communicate with Nikolai Ivanovich many times - a refined intellectual, an unusually simple, benevolent person with a great sense of humour.

193. William and Grace Young: 'Sets of Points
• The present volume is an attempt at a simple presentation of one of the most recent branches of mathematical science.

194. Stringham books
• Starting with the theory of proportion as stated by Euclid, the author builds upon this the algebra of real quantities and establishes the laws of combination of such quantities by simple geometrical constructions.

195. Taleb reviews
• Non-technically written and built on episodes, stories and simple visualisations; the book is easy to read.

196. Dingle books
• "Into this simple unison of thought there broke with the Renaissance the discordant note of modern science," with its independent direct appeal to experience.

197. R A Fisher: 'History of Statistics
• Gauss, further, perfected the systematic fitting of regression formulae, simple and multiple, by the method of least squares, which, in the cases to which it is appropriate, is a particular example of the method of maximum likelihood.

198. Collected Papers of Paul Ehrenfest' Preface
• He has a great preference for the use of simple models that show the essential traits of a problem - and is a master at inventing them; this is a common feature of his lectures and his writings.

199. H L F Helmholtz: 'Theory of music' Prefaces
• Again, it appears that the peculiar articulation between the auditory ossicles called -hammer' and 'anvil' might easily cause within the ear itself the formation of harmonic upper partial tones for simple tones which are sounded loudly.

200. Klein Elementary Mathematics
• I shall endeavor to put before the teacher, as well as the maturing student, from the view-point of modern science, but in a manner as simple, stimulating, and convincing as possible, both the can lent and the foundations of the topics of instruction, with due regard for the current methods of teaching.

201. Kurosh: 'The theory of groups' 2nd edition
• Of course, even now the classification of extensions is far from having reached that degree of perfection which would allow the solving of any problem on extensions by a simple reference to this classification; but the whole position cannot be compared to what it was twelve years ago.

202. Knorr's books
• The implications of this simple conception struck me, as I was completing a paper on Apollonius' construction of the hyperbola (1980; published in Centaurus, 1982), for it served to unify a diverse range of geometric materials I had then been collecting for some five years.

203. Max Born's matrices
• A student occasionally goes to lectures about abstruse subjects just for fun and speedily forgets all about them This is what happened to me with a lecture on higher algebra, of which I recollected little more than the word "matrix" and a few simple theorems about these matrices.

204. Catalan retirement
• It would not be the same, I am convinced, that if I give my dear students, old and new, not a dissertation on the delights of mathematics (this would lead us too far afield), but a few simple thoughts relating to intellectual work.

205. Gibson History 7 - Robert Simson
• His method of teaching was simple and perspicuous, his elocution clear, and his manner easy and impressive.

206. J L Synge: 'Geometrical Optics
• A "perfect" scientific theory may be described as one which proceeds logically from a few simple hypotheses to conclusions which are in complete agreement with observation, to within the limits of accuracy of observation.

207. Henry Baker addresses the British Association in 1913
• And, alas! to deal only with one of the earliest problems of the subject, though the finally sufficient conditions for a minimum of a simple integral seemed settled long ago, and could be applied, for example, to Newton's celebrated problem of the solid of least resistance, it has since been shown to be a general fact that such a problem cannot have any definite solution at all.

208. J J Nassau - Practical Astronomy
• The most complex of them are remarkably clear as a result of the simple device of heavy-lining the principal parts so that they stand out from the background of reference circles.

209. L'Hôpital: 'Analyse des infiniment petits' Preface
• At the time, however, this method was not as simple as M Barrow has since made it, by having paid closer attention to the properties of polygons, which naturally suggest that one consider the small triangles each made up of a part of the curve cut off between two infinitely close ordinates, the difference between these ordinates and the difference between the corresponding abscissae.

210. Harold Jeffreys: 'Scientific Inference' Preface
• It is found to lead to an explanation and a justification of the high probabilities attached in practice to simple quantitative laws, and thereby to a recasting of the processes involved in description.

211. Halmos: creative art
• Even that can be done, and I could show you a perfectly simple method in one minute and convince you that it works in two more minutes.

212. George Temple's Inaugural Lecture I
• Many of these primers of natural philosophy give the impression that the various divisions of this great subject are mainly and essentially deductive systems, each solidly based on a few general principles, which themselves are almost immediate inferences from a few simple and unequivocal experiments or observations.

213. Sims computation
• While the author's presentation neatly embeds the Todd-Coxeter method into the general context of his automaton theoretic set up, it has to be said that the rather simple basic idea that the Todd-Coxeter method is trying to construct a transitive permutation representation by a backtrack method is not very easily understood coming in the guise of dealing with automata.

214. Comments by Charlotte Angas Scott
• In reviewing a book, one of the canons of fair criticism is to regard its adaptation to the readers for whom the author himself designs it; but as a preliminary to this notice, we must object to the selection implied in the preface, where Professor Smith describes his book as intended "to present in simple and intelligible form a body of geometric doctrine acquaintance with which may fairly be demanded of candidates for the Freshman class," and then points out that one year's study of geometry is about as much as can be expected in schools.

215. Finlay Freundlich's Inaugural Address, Part 2
• The spherical geometry is the most simple case of a non-Euclidean geometry.

216. Percy MacMahon addresses the British Association in 1901, Part 2
• 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.

217. Loney Prefaces
• In order to deal as fully as possible with the less elementary processes of Arithmetic, and at the same time to keep the book within a reasonable size, it is assumed that the student already knows the four "Simple" Rules and the "Compound" Rules.

218. Women mathematics teachers
• [','','8, page 72] The 1870 Education Act introduced a curriculum focused on the three Rs; reading, writing and arithmetic, thus women began to learn simple arithmetic with an aim to improve domestic skills.

219. Isaacs' Differential Games
• The theory is illustrated throughout by applications to deceptively simple pursuit games.

220. O'Brien Calculus
• XV contains a very simple method of tracing curves.

221. Bartlett reviews
• With a few notable exceptions, biologists have tended to avoid anything but the most trivial uses of mathematics, partly because of a lack of mathematical training, partly because of a feeling that the complexities of living organisms cannot be reduced to a few simple equations.

222. Ball papers
• Here I will assume that we allow the use of brackets and the symbols for square roots, decimals (simple and repeating), factorials, and subfactorials [this is n!(1 - 1/1! + 1/2! - 1/3! + ..

223. Dubreil Books
• This lucid and simple introduction to abstract algebra approaches the subject from an extremely general point of view.

224. Wall's Creative mathematics
• Thus in calculus the simple graph rather than the concept of variable is taken as fundamental.

225. Boyer's books
• One's envy that there are places of higher learning where such textbooks are widely wanted is somewhat modified when one sees the drudgery of a set of exercises at the end of each chapter; they contain, however, not only the usual essay-type revision questions, but also simple sums to test whether the plain mathematical content has been assimilated.

226. Santalo quotes
• Tycho Brahe could scarcely suspect that all his tables of observations could be condensed into the simple mathematical formula of Newton's universal law of attraction.

227. George William Hill's new theory of Jupiter and Saturn
• Later the terms factored by the simple power of the eccentricities were added by himself, Lalande, Lagrange, Bailly and Lambert.

228. Rydberg's application
• They can in the main be expressed through a few and simple propositions, but their importance becomes evident only, when we compare the state before and after the publication of his treatise and when we consider the labour it has cost to attain this, whereof the memoir of Docent Rydberg of 1890 bears sufficient witness.

• I was fortunate enough to communicate with Nikolai Ivanovich many times - a refined intellectual, an unusually simple, benevolent person with a great sense of humour.

230. Mary Boole Darwin
• But I cannot see how the belief that all organic beings including man have been genetically derived from some simple being, instead of having been separately created bears on your difficulties.

231. Heinrich Tietze on Numbers, Part 2
• Compare the time required to do a simple problem in addition in the decimal system with that required by the same problem using Roman numerals: .

232. University of Glasgow Examinations
• Investigate the relation between the length of a simple pendulum and the time of oscillation.

233. Atiyah on beauty
• But of course with a beautiful result it can be very simple to state yet proving it can be very complicated like Fermat's Last Theorem.

234. AMS war appeal.html
• In the cause of simple humanity and in the interests of the unborn generations for whom science can prepare benefits as yet but dimly descried, the American Mathematical Society now appeals to its sister-societies in every land - and most particularly to those in lands which are at war with one another - to exert all possible effort towards the conservation of the scientific resources of the world against the day when peace shall reign once more.

235. Burali-Forti Russell letter
• It will not have the importance that you would like and that I wish it would have; but it will have the sole merit of showing how both simple and precise is the notation of your great Hamilton, of whom I am ardent admirer.

236. Cofman books
• There are many collections of mathematical problems for various ages and levels of ability so why commend another? The answer is simple.

237. Encke Obituary
• They bear strong and uniform testimony to his eminent frankness and truthfulness; his labours, they say, were incessant, his recreations few; he was simple in his manners, and in all his habits temperate.

238. 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.

239. Zehfuss publications
• Georg Zehfuss, Deduction simple de l'Expression Γ(x) de Gauss, Nouvelles annales de mathematiques (1) 18 (1859), 356.

• And the reason is simple enough.

241. Ahlfors' Complex analysis
• Many situations which seem intuitively simple are logically quite involved and must be avoided.

242. Dehn on Mathematical abilities
• One-dimensional rhythm (simple music).

243. A I Khinchin: 'Statistical Mechanics' Introduction
• Darwin and Fowler also created a simple, convenient, and mathematically rigorous apparatus for the computation of asymptotic formulas.

244. Erdos document
• Number theory is full of incredibly simple, and seemingly almost unsolvable problems, Edmund Landau listed four of these in the international conference in Cambridge in 1912, whose solutions are "unreachable with our current scientific knowledge".

245. Von Neumann: 'The Mathematician
• Euclid's postulational treatment represents a great step away from empiricism, but it is not at all simple to defend the position that this was the decisive and final step, producing an absolute separation.

246. James Jeans addresses the British Association in 1934
• Yet a simple argument will show that he can never get beyond x, y and z.

247. Gender and Mathematics refs
• L O Adetula, Solution of simple word problems by Nigerian children: Language and schooling factors, Journal for Research in Mathematics Education 20 (1989), 489-497.

248. Valdivia Infinity
• The infinitesimal calculus, differential equations and even the calculus of probabilities and mathematical statistics are presented in a very simple way in non-standard analysis, but in return the meanings are not as clear as in classical mathematical analysis.

249. Napier Tercentenary
• Mr H S Gay gave some simple and for practical purposes sufficiently accurate formula for determining the trigonometrical functions when the angle is given and conversely, without the use of elaborate tables.

250. Binet Papers
• Note sur le mouvement du pendule simple en ayant egard a l'influence de la rotation diurne de la Terre, Comptes Rendus des Seances de l'Academie des Sciences 32 (1851), 157-159; 160; 197-205.

251. Kuku Representation Theory
• Profinite K-theory of p-adic orders and semi-simple algebras .

252. Solve Applied Problems
• All the problems can be solved in closed analytical forms in terms of elementary functions or simple integrals.

253. Descartes' 'La Geometrie'
• Perceiving that in order to understand these relations I should sometimes have to consider them one by one, and sometimes only to bear them in mind, or embrace them in the aggregate, I thought that, in order the better to consider them individually, I should view them as subsisting between straight lines, than which I could find no objects more simple, or capable of being more distinctly represented to my imagination and senses; and on the other hand that in order to retain them in the memory, or embrace an aggregate of many, I should express them by certain characters, the briefest possible.

254. Mannheim publications
• A Mannheim, Determination simple et rapide d'une equation des surfaces du second ordre contenant six points donnes, Bulletin des Sciences Mathematiques et astronomiques (I) II (1871), 125-127.

255. Thomson on 'ether
• I feel that I have a right to drop the adjective luminiferous, because the medium, far above the earth's surface, through which we receive sun-heat (or light), and through which the planets move, was called ether 2000 years before chemists usurped the name for "sulphuric ether," "muriatic ether," and other compounds, fancifully supposed to be peculiarly ethereal; and I trust that chemists of the present day will not be angry with me if I use the word ether, pure and simple, to denote the medium whose undulatory motions constitute radiant heat (or light).

256. Ernesto Pascal's books
• The chief fault of the book, from our point of view, is that it sacrifices simple and natural discussion to the pursuit of the end so dear to Italian mathematicians, the greatest possible generality.

257. Etherington papers
• I M H Etherington, A simple method of finding sums of powers of the natural numbers, Edinburgh Math.

258. V Lebesgue books
• The work of Legendre is no longer sufficient in spite of its extent, and by this very fact the author has not been willing to confine himself to the simple role of translator.

259. Felix Klein on intuition
• [If a work like] Cours d'analyse of Camille Jordan is placed in the hands of a beginner a large part of the subject will remain unintelligible, and at a later stage, the student will not have gained the power of making use of the principles in the simple cases occurring in the applied sciences ..

260. Johnson pre1900 books
• The investigation thus initiated resulted in a satisfactory method of obtaining the differentials of the simple functions, which was embodied in a paper communicated to the American Academy of Arts and Sciences, January 14, 1873, by Professor J M Peirce, and published in the Proceedings of the Society.

261. Eulogy to Euler by Fuss
• His mood was always on an even-keel, a sweet, natural happiness, a good-natured sarcasm; a story-teller both innocent and simple made his conversation pleasant and coveted.

262. Descartes' Method
• The long chains of simple and easy reasonings by means of which geometers are accustomed to reach the conclusions of their most difficult demonstrations led me to imagine that all things, to the knowledge of which man is competent, are mutually connected in the same way, and that there is nothing so far removed from us as to be beyond our reach, or so hidden that we cannot discover it, provided only we abstain from accepting the false for the true, and always preserve in our thoughts the order necessary for the deduction of one truth from another.

263. Aitchison books
• It presents an elementary introduction to the ideas of statistical decision making, with little mathematical prerequisite or formal demand on the reader, and with the material firmly tied to an extensive framework of simple but realistic examples.

264. John Walsh's delusions
• He will find in it geometry more deep and subtle, and at the same time more simple and elegant, than it was ever contemplated human genius could invent.

265. Edwin Elliot: 'Algebra of Quantics
• In endeavouring to improve this second edition, which is the last I shall live to produce, and is probably definitive, I have continued to think mainly for him, and to picture him as one better prepared for being led on from the simple - it may be the crude -to the elaborate, than for first receiving, and then applying, comprehensive theory.

266. Franklin's textbooks
• He gives the most important results, with proofs where they are simple, and references to original papers where they are long and tedious; so that the brochure is very readable.

267. A N Whitehead: 'Mathematics in the History of Thought
• Suppose we project our imagination backwards through many thousands of years, and endeavour to realise the simple-mindedness of even the greatest intellects in those early societies.

268. Mac Lane books
• Throughout the book, the author's style is simple and direct, as one would expect.

269. L E Dickson: 'Linear algebras
• The remarkable properties of Cayley's algebra of eight units are here obtained for the first time in a simple manner, without computations.

270. Weatherburn books
• The book should serve as a simple introduction to these subjects treated by way of the vector methods, and for the purposes in view is admirably adapted to the student's needs.

271. Eperson contributions
• Ask a friend to choose a number with two digits, which you will endeavour to discover if he/she tells you the answer to some simple calculations.

272. McBride equal bisectors
• But, alas! a simple case of Euclid I.

273. R L Wilder: 'Cultural Basis of Mathematics III
• Without a symbolic apparatus to convey our ideas to one another, and to pass on our results to future generations, there wouldn't be any such thing as mathematics - indeed, there would be essentially no culture at all, since, with the possible exception of a few simple tools, culture is based on the use of symbols.

274. Fatou Fonctions Automorphes
• Fatou uses Montel's theory of normal families of functions to obtain a new and very simple proof of the theorem that a group of real linear transformations which has no infinitesimal transformation is properly discontinuous.

275. Cafaro's papers
• Very often, self-similarity of intermediate asymptotics can be derived from simple dimensional-analysis arguments.

276. Simplicius on astronomy and physics
• But he must go to the physicist for his first principles, namely, that the movements of the stars are simple, uniform, and ordered, and by means of these principles he will then prove that the rhythmic motion of all alike is in circles, some being turned in parallel circles, others in oblique circles.

277. Kelvin on the sun
• To advance another step, still through impracticable mechanism, towards the practical method by which the sun's heat is produced, let the thread of the screw be of uniformly decreasing steepness from the surface downwards, so that the velocity of the weight, as it is allowed to descend by the turning of the screw, shall be in simple proportion to distance from the sun's centre.

278. Gibson History 5 - James Gregory
• 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.

279. Mitchell Feigenbaum: the interviewer
• Briefly, he discovered a universal quantitative solution characterized by specific measurable constants that describes the crossover from simple to chaotic behaviors in many complex systems.

280. Bronowski and retrodigitisation
• Based on the fact that 7 is a divisor of 1001, it provides a simple way to compute remainders on division by 7.

• To illustrate by a very simple example, the function x2 has for its graph a parabola with its principal vertex at the origin of coordinates, and its principal diameter coincident with the y-axis.

282. Sikorski books
• There is no high points or climaxes, no broad over-view, no explanation of a simple illustration before meeting the full treatment.

283. Einstein NY Times
• Public interest in Albert Einstein's relativity theory had become so great in 1929 that, when he presented to the Prussian Academy his comprehensive theory fusing electromagnetism and gravitation in a single law, the New York Times urged him to prepare an explanation of his new work in terms as simple as the subject would allow.

284. Payne-Gaposchkin introduction
• Most of its concepts can be expressed in simple, everyday language.

285. Poincaré on non-Euclidean geometry
• Further, this interpretation is not unique, and several dictionaries may be constructed analogous to that above, which will enable us by a simple translation to convert Lobachevsky's theorems into the theorems of ordinary geometry.

286. Algebraic Triplets
• Mr Charles Graves stated that Sir Wm Hamilton had been the first to announce that if the real unit line, the factors, and the product line, be projected upon the symmetric axis, the projections will form a proportion in the simple sense of that term ..

287. Young Researchers
• Title: Decompositions of graphs: splitting huge structures into simple pieces .

288. Helmholtz on Thomas Young
• I include myself among the number; for I long toiled at the task, without getting any nearer my object, until I at last discovered that a wonderfully simple solution had been discovered at the beginning of this [nineteenth] century, and had been in print for any one to read who chose.

289. H Weyl: 'Theory of groups and quantum mechanics' Introduction
• In this chapter many details will be introduced with an eye to future use in the applications; it is to be hoped that in spite of this the simple thread of the argument has remained plainly visible.

290. What do mathematicians do?
• Nowadays quite simple proofs exist, but they use sophisticated new tools such as group theory and field theory.

291. Cardan: autobiography
• He confesses it without impudence and without feigned contrition, without even wishing to make himself an object of interest, but with the same simple and sincere love of fact which guided him in his scientific researches.

292. Kerr: 'Technical Education
• As confirmatory evidence of the volume and importance of the work done it is interesting to find in 1824 an eminent mechanical engineer, M Dupin, calling the attention of France to the Andersonian College, "a school for teaching the theory of the mechanical and chemical arts - intended not only for the directors of the workshops but particularly for the simple working man." He attributes the industrial supremacy of this country to the cultivation of science, and he calls upon Frenchmen "not to remain behind in this immense progress but to proceed on the same lines in order to outstrip, if possible, a people whom Nature has made our rival in every kind of glory." .

293. George Gibson: 'Calculus
• Simple exercises are attached to many of the sections; in the formal sets will be found several theorems and results for which room could not be made in the text, and which are yet of sufficient importance to be explicitly stated.

294. Plato on Mathematics
• 'Think a little,' I told him, 'and you will see that what has preceded will supply the answer; for if simple unity could be adequately perceived by the sight or by any other sense, then, there would be nothing to attract the mind towards reality any more than in the case of the finger we discussed.

295. Carol R Karp: 'Languages with expressions of infinite length
• My interest in infinitary logic dates back to a February day in 1956 when I remarked to my thesis supervisor, Professor Leon Henkin, that a particularly vexing problem would be so simple if only I could write a formula that would say x = 0 or x = 1 or x = 2 etc.

296. Ptolemy's hypotheses of astronomy
• The ground for this conviction which is readiest to hand, seeing that the earth has been proved to be spherical and situated in the middle of the universe, is this simple fact: in all parts of the earth without exception the tendencies and the motions of bodies which have weight - I mean their own proper motions - always and everywhere operate at right angles to the (tangent) plane drawn evenly through the point of contact where the object falls.

297. Turnbull lectures on Colin Maclaurin
• We may pause for a moment to consider what a boon this interval of unhampered leisure would bring to the lad upon the threshold of manhood: the simple life in the manse of Kilfinan upon the open easterly shore of Loch Fyne, but a few miles over the hill from Glendaruel, the home of his childhood: the opportunity for his thoughts upon geometry to ripen, the fruit of the teaching that he received from Robert Simson, his Professor at college.

298. Louis Auslander books
• In each chapter basic notions are introduced after illustrating and discussing simple cases of the objects, and almost all sections end with a set of exercises which facilitate the understanding of the subject.

299. Howie Thanksgiving Service
• Those of us who had the good fortune to enjoy John's friendship and hospitality, equally don't need reminding of his simple human qualities - devotion to family, enjoyment of good company and conversation, multitude of interests, including of course his love of music, and his service to the community.

300. Apostol books
• The presentation is simple and clear.

301. Malcev: 'Foundations of Linear Algebra' Introduction
• For example, the fundamental idea behind the solution of a system of linear equations in several unknowns is that of replacing such a system by a chain of these simple equations.

302. Lewis's papers
• I can only put forward certain statements intended to formulate attributes which are essential to mind; to point to phenomena of which we can say, "Whatever else is or is not comprehended under 'mind,' at least it is intended to include these." In particular, I shall wish to emphasize that whatever is called "content of consciousness" is so included, and to consider certain consequences of that simple fact.

303. Peres publications
• Joseph Peres, Action sur un obstacle d'un fluide visqueux; demonstration simple de formules de Faxen, Comptes Rendus Acad.

304. Arvesen publications
• Ole Peder Arvesen, Sur certaines surfaces algebriques, parmi lesquelles la surface de Steiner constitue le cas le plus simple, Norske Vid.

305. Kuratowski: 'Introduction to Topology
• and Frechet), and the space of continuous functions are metric spaces; at the same time, the very concept of a metric space is especially simple and geometrically clear.

306. A D Aleksandrov's view of Mathematics
• Of course, the rules here are very simple, but we should remember that in some period of antiquity they represented the most advanced mathematical achievements of the age.

307. William Herschel discoveries
• Herschel began to examine stellar spectra using a simple prism in 1798.

308. Gerard Murphy papers
• G J Murphy, Simple C*-algebras and subgroups of Q, Proc.

309. Weil on history
• ultimately upon the values, for suitable values of the arguments, of the simple series discussed above in our Chapter 7.

310. Perron books
• The style is simple and precise and presents no difficulties to a reader having a firm grasp of the fundamental principles of elementary analysis.

311. Green's students
• Thesis title: A Sporadic Simple Group of B Fischer of Order 64,561,751,654,400.

312. Mathematicians and Music 2.1
• Pythagoras proposed to find in the order of the universe, where whole numbers and simple ratios prevail, an answer to the question: Why is consonance (the beautiful in sound) determined by the ratio of small whole numbers? The correct numerical ratios existing between the seven tones of the diatonic scale corresponded, according to Pythagoras, to the sun, moon and five planets, and the distances of the celestial bodies from the central fire, etc.

313. Lorch books
• Chapter VI expounds the Gel_fand representation of semi-simple commutative Banach algebras.

314. Vajda books
• Linear programming computations consist of simple arithmetic processes, although the mathematical proofs and problems of formulation are sometimes quite deep and technical.

315. Leslie Origins Number
• These simple arrangements would, on their first application, carry the power of reckoning but a very little way.

316. De Rham books
• Torsion et type simple d'homotopie (1967), by G de Rham, S Maumary and M A Kervaire.

317. Douglas Jones publications
• D S Jones, The scattering of sound by a simple shear layer, Phil.

318. Survey of Modern Algebra
• Then the abstract definition appears simple, and the theoretical properties which are deduced from the definition exhibit the power of the concept.

319. Richard Courant: 'Differential and Integral calculus' English edition
• I felt that owing to the difference between the methods of teaching the calculus in Germany and in Britain and America a simple translation was out of the question, and that fundamental changes would be required in order to meet the needs of English-speaking students.

320. Krejci's book
• There is no simple and satisfactory explanation of this fact.

321. Groups in Galway
• Finite simple groups.

322. Ernest Hobson addresses the British Association in 1910
• These times must have been preceded by still earlier ages in which the mental evolution of man led him to the use of the tally, and of simple modes of measurement, long before the notions of number and of magnitude appeared in an explicit form.

323. Todd: 'Basic Numerical Mathematics
• However, most of the problems in Volume 1 can be dealt with using simple programmable hand calculators, but many of these in Volume 2 require the more sophisticated hand calculators (i.e., those with replaceable programs).

324. Cajori: 'A history of mathematics' Introduction
• After the pupils have learned how to bisect a given angle, surprise them by telling of the many futile attempts which have been made to solve, by elementary geometry, the apparently very simple problem of the trisection of an angle.

325. Gentry Berlin
• Berlin was chosen as place of residence for the first few weeks or months, as the case might be, for the simple reason that I knew of people here who would kindly take me in charge till I should have learned to take care of myself in a foreign land.

326. Hormander books
• A brief chapter on differential equations with no solutions is followed by chapters on operators of constant strength, operators with simple characteristics, the Cauchy problem, and a concluding chapter on elliptic boundary value problems.

327. Zwicky books
• In simple terms, morphological research is a method of ensuring "unbias" (a word coined by Zwicky for the purpose) by the systematic listing of all conceivable alternatives in a given complex situation.

328. Cheltenham exams
• This simple sum is made more difficult by the need to understand the definition of an improper fraction and the wordy nature of the question.

329. Borali-Forti preface
• Leibniz was the first to conceive the grand plan to create a universal writing system, by which every idea could be expressed by means composed of simple ideas, each represented by a special sign.

330. Horace Lamb addresses the British Association in 1904, Part 2
• The investigators of the classical school, as it may perhaps be styled, were animated by a simple and vigorous faith; they sought as a matter of course for a mechanical explanation of phenomena, and had no misgivings as to the trustiness of the analytical weapons which they wielded.

331. Einar Hille: 'Analytic Function Theory
• These general considerations have led to the following arrangement of the subject matter of Volume I: After a preliminary study of number systems, the geometry of the complex plane is developed, and simple functions such as linear fractions, powers, and roots are studied.

332. More Smith History books
• To cover the ground of whole numbers so completely in thirty-four pages is a masterpiece of condensation, more noticeable because the matter is given in simple words and explained as to a beginner.

333. Leslie works
• The first portion deals with the more simple properties of conics, while the higher part treats chiefly of the construction of conics to satisfy all sorts of conditions.

334. Peirce publications
• A simple device for measuring the deflections of a mirror galvanometer, Proc.

335. Galileo: 'Dialogue
• Salviatus: So that its motion should be compounded of two; from this it would follow that the stone would no longer describe that simple straight and perpendicular line but one transverse and perhaps not straight.

336. John Collins by Wood
• (4) 'The Doctrine of Decimal Arithmetic, simple Interest, etc.

337. Ahlfors' reviews
• The book contains several of the author's results; the style is clear; the proofs are simple; we find several examples and problems; but there is no index or references.

338. Einstein: 'Ether and Relativity
• The laws were clear and simple, the mechanical interpretations clumsy and contradictory.

339. Clifford's books
• It is therefore very refreshing to find someone who reminds us that in spite of all these learned trappings the basic rules of arithmetic and algebra are nothing more and nothing else but "common sense"; especially if this is done in as simple and as convincing a manner as in the book under review.

340. Heinrich Tietze on Numbers
• system is simple, compared to the strange and complicated duodecimal system.

341. Scholar and the World.html
• The point that I wish to make is a simple - even a trite - one.

342. Venice and statistics
• The towns, on the other hand, throughout the West must from very early times have treated production, which with them depended on industry and commerce, as exceedingly variable; but, even in the most flourishing times of the Hanseatic League, they never got beyond a simple commercial balance sheet.

343. Percy MacMahon addresses the British Association in 1901
• The gravitation formula has been recognised from the time of Newton as ruling the dynamics of the heavens, and the exact agreement of the facts derived from observation with the simple theory has established astronomy as the most exact of all the departments of applied science.

344. Coxeter and Moser: 'Generators and Relations
• Some of them play an essential role in the theory of simple Lie groups.

345. Weil reviews
• In the second part - Chapters III and IV - the author studies the zeta function of division algebras and central simple algebras and then uses the Poisson summation formula to calculate the Tamagawa numbers of "most" classical groups .

346. Twenty-Five Years of Groups St Andrews Conferences
• The twenty-five years since 1981 have been an important period in the development of group theory following the classification of finite simple groups.

347. Founding the Indian Mathematical Society
• I shall soon be submitting to members proposals for a simple constitution for the Society according to which the affairs of the Society will be managed by a committee consisting of a President, a few office bearers and some additional members.

348. Booth Analytic Method
• Among other applications of the method, I trust that to the theory of reciprocal polars will be found simple and satisfactory.

349. Keynes: 'Probability' Introduction Ch II
• All it can do is so to arrange the reasoning that the logical relations, which have to be perceived directly, are made explicit and are of a simple kind.

350. Gaschutz's My Path
• The effects that this simple exchange of books had on the development of the mathematical seminar in Kiel can arguably still be perceived today.

351. Catalan manifesto
• Si mon nom n'a pas retenti dans les Assemblees politiques, c'est par une raison bien simple: sous la regime oligarchique que vous venez de renverser, je n'etais pas electeur, et je ne serais jamais devenu electeur! Mais consultez ceux que j'ose appeler mes illustres amis: Dupont (de l'Eure), Arago, Ledru-Rollin, Louis Blanc, Garnier-Pages; interrogez les jeunes gens sortis de l'Ecole Polytechnique depuis huit ans; invoquez les souvenirs de mes anciens camarades de la XIe legion: tous vous diront qu'ils m'ont constamment vu sur la breche, et que mes opinions republicaines bien connues m'ont empeche d'obtenir l'avancement qui m'etait legitimement du.

352. Valdivia aspects of maths
• It can be said that Euclid was the systematiser of almost all the mathematical results known in his time, ordering them in a masterly way in a deductive system, demonstrating from a few simple geometric properties, self-evident and not requiring proof, according to the spirit of the time, all that follows as logical consequences of the former.

353. Science at St Andrews
• With a simple diagram Gregory explained how the light rays on passing along the cylindrical tube would strike a large parabolic mirror and be reflected through a focus to a small concave mirror standing on the central axis of the tube, from which by a second reflexion they resume their course and pass out at a central aperture in the first mirror to an eye-piece.

354. Andrew Forsyth addresses the British Association in 1905
• The simple laws of planetary motion were not formulated, for Kepler had them only in the making.

355. Mathematics at Aberdeen 4
• In Mathematics the minimum required was on the first six books of Euclid, plane trigonometry and algebra as far as simple and quadratic equations.

356. Alfred Tarski: 'Cardinal Algebras
• The derivations are not simple but, in general, are not more involved than direct proofs carried through by the method indicated above.

357. Kline's books
• His writing is clear and simple, though somewhat repetitious.

## Quotations

1. Quotations by Ampere
• By combining at random simple truths with each other, more complicated ones are deduced from them.
• [Ampere gives this example drawn from geometry to illustrate his meaning for  direct synthesis when deductions following from more simple, already-known theorems leads to a new discovery.] .
• There is analysis when from a complicated truth one deduces more simple truths.

2. Quotations by Bronowski
• Einstein was a man who could ask immensely simple questions.
• And what his work showed is that when the answers are simple too, then you can hear God thinking.

3. Quotations by Einstein
• Everything should be made as simple as possible, but not simpler.
• Most of the fundamental ideas of science are essentially simple, and may, as a rule, be expressed in a language comprehensible to everyone.

4. Quotations by Kepler
• My aim is to say that the machinery of the heavens is not like a divine animal but like a clock (and anyone who believes a clock has a soul gives the work the honour due to its maker) and that in it almost all the varety of motions is from one very simple magnetic force acting on bodies, as in the clock all motions are from a very simple weight.

5. Quotations by Brahe
• So Mathematical Truth prefers simple words since the language of Truth is itself simple.

6. Quotations by Weyl
• A modern mathematical proof is not very different from a modern machine, or a modern test setup: the simple fundamental principles are hidden and almost invisible under a mass of technical details.
• They are nothing but marks, and all that is in them we have put into them by the simple rule of straight succession.

7. Quotations by Hilbert
• Mathematics is a game played according to certain simple rules with meaningless marks on paper.

8. A quotation by Davenport
• A peculiarity of the higher arithmetic is the great difficulty which has often been experienced in proving simple general theorems which had been suggested quite naturally by numerical evidence.

9. A quotation by Maseres
• Negative numbers darken the very whole doctrines of the equations and make dark of the things which are in their nature excessively obvious and simple.

10. Quotations by Russell
• The point of philosophy is to start with something so simple as not to seem worth stating, and to end with something so paradoxical that no one will believe it.

11. Quotations by Laplace
• It is India that gave us the ingenious method of expressing all numbers by means of ten symbols, each symbol receiving a value of position as well as an absolute value; a profound and important idea which appears so simple to us now that we ignore its true merit.

12. Quotations by Sylvester
• During a conversation with the writer in the last weeks of his life, Sylvester remarked as curious that notwithstanding he had always considered the bent of his mind to be rather analytical than geometrical, he found in nearly every case that the solution of an analytical problem turned upon some quite simple geometrical notion, and that he was never satisfied until he could present the argument in geometrical language.

13. Quotations by Schrodinger
• The idea of the continuum seems simple to us.

14. Quotations by Born
• The problem of physics is how the actual phenomena, as observed with the help of our sense organs aided by instruments, can be reduced to simple notions which are suited for precise measurement and used of the formulation of quantitative laws.

15. Quotations by Von Neumann
• If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is.

16. A quotation by Chasles
• The doctrines of pure geometry often, and in many questions, give a simple and natural way to penetrate the origin of truths, to lay bare the mysterious chain which unites them, and to make them known individually, luminously and completely.

17. Quotations by Hawking
• My goal is simple.

18. Quotations by Condorcet
• uniformity of measures can only displease those lawyers who fear to see the number of lawsuits diminished, and those traders who fear a loss of profit from anything which renders commercial transactions easy and simple ..

19. A quotation by Lemoine
• A mathematical truth is neither simple nor complicated in itself, it is.

20. Quotations by Dantzig
• The two relatively simple problems -- the determination of the diagonal of a square and that of the circumference of a circle -- revealed the existence of new mathematical beings for which no place could be found within the rational domain.

21. A quotation by MacMahon
• During a conversation with the writer in the last weeks of his life, Sylvester remarked as curious that notwithstanding he had always considered the bent of his mind to be rather analytical than geometrical, he found in nearly every case that the solution of an analytical problem turned upon some quite simple geometrical notion, and that he was never satisfied until he could present the argument in geometrical language.

22. Quotations by Anaxagoras
• Anyhow we take in nourishment which is simple and homogeneous, such as bread or water, and by this are nourished hair, veins, arteries, flesh, sinews, bones and all the other parts of the body.

23. Quotations by Atiyah
• The most useful piece of advice I would give to a mathematics student is always to suspect an impressive sounding Theorem if it does not have a special case which is both simple and non-trivial.

24. Quotations by Dee
• A marveilous newtrality have these things mathematicall, and also a strange participation between things supernaturall, immortall, intellectuall, simple and indivisible, and things naturall, mortall, sensible, componded and divisible.

25. Quotations by Gauss
• A great part of its [higher arithmetic] theories derives an additional charm from the peculiarity that important propositions, with the impress of simplicity on them, are often easily discovered by induction, and yet are of so profound a character that we cannot find the demonstrations till after many vain attempts; and even then, when we do succeed, it is often by some tedious and artificial process, while the simple methods may long remain concealed.

26. Quotations by Descartes
• These long chains of perfectly simple and easy reasonings by means of which geometers are accustomed to carry out their most difficult demonstrations had led me to fancy that everything that can fall under human knowledge forms a similar sequence; and that so long as we avoid accepting as true what is not so, and always preserve the right order of deduction of one thing from another, there can be nothing too remote to be reached in the end, or to well hidden to be discovered.

27. Quotations by Turnbull
• A simple instance of failing in this is provided by the poll-man at Cambridge, who learned perfectly how to factorize a2 - b2 but was floored because the examiner unkindly asked for the factors of p2 - q2 .

28. A quotation by Mazur Barry
• In the history of the concept of number has been adjective (three cows, three monads) and noun (three, pure and simple), and now .

## Famous Curves

1. Folium
• There are three special forms of the folium, the simple folium, the double folium and the trifolium.
• The graph plotted above is the simple folium.
• The simple folium is the pedal curve of the tricuspoid where the pedal point is one of the cusps.

2. Double
• There are three special forms of the folium, the simple folium, the double folium and the trifolium.
• There are separate entries for the simple folium and the trifolium.

3. Trifolium
• There are three special forms of the folium, the simple folium, the double folium and the trifolium.
• There are separate entries for the simple folium and the double folium.

4. Cycloid
• The cycloid has the property that a particle P sliding on a cycloid will exhibit simple harmonic motion and the period will be independent of the starting point.

5. Tricuspoid
• The pedal of the tricuspoid, where the pedal point is the cusp, is a simple folium.

## Chronology

1. Mathematical Chronology
• The first symbols for numbers, simple straight lines, are used in Egypt.
• Maxwell publishes On Faraday's lines of force showing that a few relatively simple mathematical equations could express the behaviour of electric and magnetic fields and their interrelation.
• Cartan, in his doctoral dissertation, classifies all finite dimensional simple Lie algebras over the complex numbers.
• Russell discovers "Russell's paradox" which illustrates in a simple fashion the problems inherent in naive set theory.
• M Suzuki discovers new infinite families of finite simple groups.
• Edward Lorenz discovers a simple mathematical system with chaotic behaviour.
• John Thompson and Feit publish Solvability of Groups of Odd Order which proves that all nonabelian finite simple groups are of even order.
• Conway publishes details of his discovery of new sporadic finite simple groups.
• The classification of finite simple groups is complete.

2. Chronology for 1960 to 1970
• M Suzuki discovers new infinite families of finite simple groups.
• Edward Lorenz discovers a simple mathematical system with chaotic behaviour.
• John Thompson and Feit publish Solvability of Groups of Odd Order which proves that all nonabelian finite simple groups are of even order.
• Conway publishes details of his discovery of new sporadic finite simple groups.

3. Chronology for 1900 to 1910
• Russell discovers "Russell's paradox" which illustrates in a simple fashion the problems inherent in naive set theory.

4. Chronology for 1970 to 1980
• The classification of finite simple groups is complete.

5. Chronology for 1890 to 1900
• Cartan, in his doctoral dissertation, classifies all finite dimensional simple Lie algebras over the complex numbers.

6. Chronology for 1850 to 1860
• Maxwell publishes On Faraday's lines of force showing that a few relatively simple mathematical equations could express the behaviour of electric and magnetic fields and their interrelation.

7. Chronology for 1950 to 1960
• M Suzuki discovers new infinite families of finite simple groups.

8. Chronology for 30000BC to 500BC
• The first symbols for numbers, simple straight lines, are used in Egypt.

9. Chronology for 1980 to 1990
• The classification of finite simple groups is complete.

## EMS Archive

1. Edinburgh Mathematical Society Lecturers 1883-2016
• A simple method of finding any number of square numbers whose sum is a square .
• (Provincial Training College, Glasgow) On a simple theodolite suitable for use in schools .
• (Glasgow) Elementary methods for calculating first and second moments of simple configurations .
• (lived in Glasgow) A simple theodolite: A teaching appliance .
• (Edinburgh) A simple linkage for describing equal areas .
• (Glasgow) A simple link apparatus for the mechanical solution of quadratic equations .
• (Edinburgh) Exhibition of two simple nomograms .
• (Edinburgh) A simple form of integrometer .
• (Syracuse University, New York) A generalisation and simple proof of Kronecker's theorem concerning the minors of a compound determinant, {Communicated by David Gibb} .
• On the teaching of simple mathematical astronomy in schools; .
• (Cambridge) Some advances and retreats in the study of simple groups .
• (Warwick) Geometrical structures associated with simple groups of Lie type .
• (Cambridge) Finite simple groups - some special cases .
• (Oxford) Infinite simple groups .
• (University College, Cardiff) Problems about generating simple groups .
• (Leeds) Projective modules and simple rings .
• (Birmingham) Characterizations of simple groups .
• (Warwick) A simple partial differential equation with surprising behaviour .

2. EMS Proceedings papers
• A simple linkage for describing equal areas .
• A simple form of integrometer .

3. EMS 1930 Colloquium
• Simple rational curves in a plane: rational curves in space (cubics, quartics): general rational curves: conditions for a curve to be rational: simple rational surfaces (quadrics, cubics): general notions as to rational surfaces: conditions for a surface to be rational.

4. EMS Proceedings papers
• Elementary methods for calculating first and second moments of simple configurations .
• Common logarithms calculated by simple multiplication .

5. 1908-09 Jan meeting
• Miller, William: "A simple theodolite: A teaching appliance", [Title] .

6. 1883 Mar meeting
• A simple majority shall decide all questions not otherwise provided for in the Rules, the Chairman having in all cases both a deliberative and a casting vote .

7. 1905-06 Mar meeting
• Arneil, Loudon: "On a simple theodolite suitable for use in schools", [Title] .

8. 1906-07 Jun meeting
• Muirhead, Robert Franklin: "Elementary methods for calculating first and second moments of simple configurations" .

9. 1895-96 Feb meeting
• Martin, Artemas: "A simple method of finding any number of square numbers whose sum is a square" .

10. 1930-31 Rules meeting
• 16: A simple majority shall decide all questions not otherwise provided for in the Rules, the Chairman having in all cases a deliberative and a casting vote.

11. 1914-15 May meeting
• Whittaker, Edmund Taylor: "Exhibition of two simple nomograms", [Title] .

12. 1930-31 meeting
• 16: A simple majority shall decide all questions not otherwise provided for in the Rules, the Chairman having in all cases a deliberative and a casting vote.

13. EMS Proceedings papers
• A simple nomogram for the solution of quadratic equations .

14. EMS Proceedings papers
• A simple method of finding any number of square numbers whose sum is a square .

15. EMS 1938 Colloquium
• An analytic deformation of a function f(x) in the neighbourhood of x = infinity means replacing the independent variable by an analytic function of itself, x' = f(x), which has a simple pole at infinity.

16. EMS honours James Leslie
• His wants were simple, and he was able to travel widely.

17. EMS Freundlich
• Theoretical prediction and observational evidence as to the divergence from a simple Kepler orbit were compared and applied to test the accuracy of models of the constitution of stars.

18. EMS school mathematics
• (2) Encourage study on simple lines of the nature of mathematical reasoning.

19. 1923-24 May meeting
• Metzler, W H: "A generalisation and simple proof of Kronecker's theorem concerning the minors of a compound determinant", [Proceedings, session 43] {Communicated by David Gibb} .

20. 1914-15 Mar meeting
• Stokes, George D C: "A simple link apparatus for the mechanical solution of quadratic equations", [Title] .

21. 1915-16 Feb meeting
• Horsburgh, Ellice Martin: "A simple form of integrometer", [Title] .

22. 1912-13 Jun meeting
• Horsburgh, E M: "A simple linkage for describing equal areas" .

23. Napier Tercentenary
• Mr H S Gay gave some simple and for practical purposes sufficiently accurate formula for determining the trigonometrical functions when the angle is given and conversely, without the use of elaborate tables.

24. EMS 1980 Colloquium
• I had the privilege of attending these lectures which were extremely successful in meeting Nash-Williams' aim "of developing nontrivial and fairly deep mathematics from a very simple initial concept." .

25. Solution5.1.html
• By the way, by varying the coefficients in the recursion for the tn one can replace the above (E, P) by essentially any elliptic curve over Q and any rational point on it, so that in an elementary course in number theory one could develop (or at least introduce) the entire theory of elliptic curves just by starting with these simple recursions! .

## BMC Archive

1. BMC 2016
• Pyber, LHow to avoid the Classification Theorem of Finite Simple Groups in Asymptotic Group Theory .
• Malcolm, AThe involution width of a finite simple group .

2. BMC 1970
• Macdonald, I GRepresentation of semi-simple Lie groups .

3. BMC 1963
• Carter, R WSimple groups and simple Lie algebras .

4. BMC 1982
• Gorenstein, D Reworking the classification of finite simple groups .

5. BMC 2003
• Bavula, V Maximal commutative subalgebras of simple infinite-dimensional algebras .

6. BMC 1999
• Shalev, A Simple groups, Cayley groups and probability .

7. BMC 1976
• Aschbacher, MThin finite simple groups .

8. BMC 1977
• Collins, M JThe identification problem for finite simple groups .

9. BMC 1996
• Holland, M P Grothendieck groups of primitive factors of enveloping algebras of semi-simple Lie algebras .

10. BMC 1998
• Premet, A A Recent progress in the classification of finite-dimensonal simple Lie algebras in prime characteristic .

11. BMC 1987
• Wilson, R A Subgroups of simple groups .

12. BMC 2007
• Vassiliev, D Teleparallelism: difficult word but simple way of reinterpreting the Dirac equation .

13. BMC 1993
• Curtis, R T Symmetric generation of sporadic simple groups .

14. BMC 2018
• Grazian, VThe classification of simple fusion systems .

15. BMC 1969
• Thompson, J GFinite simple groups .

## Gazetteer of the British Isles

1. London individuals H-M
• First to observe that falling barometer indicates bad weather, to build a Gregorian telescope, to show that Mars and Jupiter rotated (being the first to observe the Great Red Spot of Jupiter), to observe that the tail of a comet was repelled by the sun, to invent the iris diaphragm, to note that movement where the restoring force is proportional to the displacement gives simple harmonic motion, to observe Chladni figures.
• However, he also produced a simple adding machine at the same time and Pepys disparaged it: 'very pretty, but not very useful' [The Diary of Samuel Pepys M.A., F.R.S; Clerk of the Acts and Secretary to the Admiralty Transcribed by the late Rev.

2. Oxford Institutions and Colleges
• 53-54',55)">Gunther] - I recall this is a rather simple adding device of 1666 which was rightly disparaged by Pepys, but Morland produced, at the same time, the first successful multiplying calculator).
• The Museum has the fine late 16C Flemish painting The Measurers depicting a mathematical instrument maker and numerous applications of simple measuring instruments.

3. London Museums
• Toward the left end are several diagrams of triangles which are area problems and at the left end are diagrams of pyramids where simple computations involving the slope are done.

4. Oxford individuals
• He has a floor slab in the middle of the north floor with the simple inscription 'Henricus Briggius' [Early Science in Oxford: Vol.

5. Exeter, Devon
• The Royal Albert Museum, Queen Street, Exeter, contains the Exeter Puzzle Jug, probably made in the Saintonge region of western France, c1300, perhaps the finest example of medieval pottery imported to England and the earliest extant example in England of a puzzle jug, though the puzzle aspect is quite simple.

6. Harpenden, Hertfordshire
• During this time he formally developed analysis of variance, he introduced the word variance and found the distribution of the correlation coefficient, the correct chi-squared distribution for contingency tables (Pearson failed to get the right number of degrees of freedom) and the distributions of the simple and multiple correlation coefficients.

7. Dorchester
• a simple adding machine (British Calculator, Model B, probably early 20C); .

## Astronomy section

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JOC/BS August 2001