**Mauro Picone**'s parents were Alfonso Picone and Anna Bongiovanni. Alfonso was a mining engineer in Sicily, employed in sulphur mining industry, living in Lercara Friddi in the Province of Palermo about 45 km southeast of the city of Palermo, Sicily. There were three children in the family, Mauro having two sisters. The sulphur mining industry had an important role in Sicily, with most of the world's sulphur being mined there during the first part of the 19

^{th}century. However, the industry declined towards the end of the century with the discovery of rich sulphur mines in America and, as a result, Alfonso's family were reduced to poverty. He left mining in 1889 and the family moved to Arezzo in Tuscany where Alfonso had won a competition for a professorship at the Technical Institute. Mauro began his elementary schooling in Arezzo and, since he showed great artistic skills and a great love of the paintings of Piero della Francesca in the Basilica of San Francesco, his father sent him to a studio to study drawing. He certainly had great talent for drawing for he won a prize in a local exhibition.

One might have expected that Picone would excel at school, but this seems not to have been the case. Because his progress at elementary school was poor, particularly in arithmetic, he was given extra tuition to bring him up to standard. However, in [15] Picone says that he did not profit from this. After elementary school, he enrolled in the Technical Institute in Arezzo where his father was teaching. Although when he entered the Institute his main interest was still in drawing, he became steadily more interested in mathematics but also began to like the lessons in Italian and geography. Of course, his skill in drawing was particularly useful in his study of geography where he excelled in drawing maps. In mathematics he used the books by Richard Baltzer, *Elements of mathematics*, translated from German to Italian by Luigi Cremona and [15]:-

In his third year at the Institute he was taught mathematics by his father for a short while, but in his third and fourth year his main mathematics teacher was Michele de Franchis (1875-1946) who taught him algebra and analytic geometry, taking him to the second year university level. Before he completed his studies, however, his father moved to the Technical Institute in Parma and Picone moved there to obtain his final school certificate. One of the professors there [15]:-I became fond of their study and, since then, I deeply loved mathematics, this queen of the sciences.

In October 1903 he was successful in the competition for admission to the Scuola Normale in Pisa. However, he became ill and his parents hurried to Pisa to take care of him. It was January 1904 before he was well enough to begin his studies, taking courses given by Ulisse Dini and Luigi Bianchi. He was also [7]:-... Abd el Kader Salza, a gentle and wise man who took a liking to me, advised me to compete for a internal student place at the Scuola Normale Superiore of Pisa.

At first he enrolled to study for a degree in physics, but Bianchi advised him to change to a mathematics degree which he did in his third year of study. He graduated in 1907, but remained at the Scuola Normale where he became Dini's assistant. He married Maria Jole Agonigi, from a wealthy family of merchants from Pisa, on 30 October 1913 [15]:-... particularly fascinated by the depth of the genius of the mathematician Eugenio Elia Levi, just two years older than him, but already established as a researcher of exceptional calibre.

On the day he married, he left Pisa and moved to the Polytechnic of Turin as an assistant to Guido Fubini who held the chair of Rational Mechanics and Infinitesimal Analysis. In this period he developed research on ordinary differential equations and partial derivatives. This is described in more detail by Silvio Cinquini in a review of [9]. In this early period of his career, Picone studied:-My wife had a very beneficial influence on my life as a scholar. ... being pleased to give me constant material, moral and spiritual assistance.

Picone's quiet, hardworking, life in Turin was disrupted with the outbreak of World War I. Italy did not enter the war when it broke out in 1914 although at this stage most people in the country would have supported Germany but had a long-standing rivalry with Austria-Hungary. Only in April 1915 did Italy enter the war, persuaded to come in on the side of the Allies, Russia, France and Britain. One year later, in April 1916, Picone was called up and assigned to the 6... three different topics:(i)boundary value problems for second order linear ordinary differential equations, for which Picone developed his well-known "identity", and the subsequent extension of these results to second order linear partial differential equations of elliptic and parabolic types;(ii)partial differential equations of hyperbolic type(in two independent variables), for which Picone studied problems generalizing Goursat's problem;(iii)research on differential geometry in the direction set by L Bianchi, with particular attention to the characterization of theds^{2}of a ruling and to W congruences.

^{th}Artillery Regiment where he was given the rank of second lieutenant. In July he was sent to the front near Trento where the Italians were fighting the Austrians. On learning that Picone was a mathematician, his commander, Colonel Federico Baistrocchi, set him the task of computing new gunnery tables for the artillery. He adapted the old tables produced by Francesco Siacci to the particular geographic conditions they were fighting in around Trento. This work changed Picone's view of mathematics [15]:-

Picone was promoted to captain of artillery in 1917 and, in the following year, was awarded the Military Cross by Italy and the Croix de Guerre with silver star by France... you can imagine, after this success achieved with mathematics, the kind of new light in which it appeared to me. I thought: so mathematics is not only beautiful, it can be useful as well.

After the war ended and Picone was demobbed, he returned to his university teaching. In 1919 he was appointed as Professor of Analysis at the University of Catania, in 1920 he was briefly at the University of Cagliari, and then returned to Catania in 1921 as Head of Mathematics. While in Catania, he published two books: *Teoria introduttiva delle equazioni differenziali ordinarie e calcolo delle variazioni* Ⓣ (1922) and *Lezioni di Analisi infinitesimale* Ⓣ (1923). Then, after a short stay in Pisa in 1924-1925, he moved to the University of Naples. There Renato Caccioppoli, who graduated from Naples in 1925 having been a student of Ernesto Pascal, became Picone's assistant. Carlo Miranda entered the University of Naples in 1927 and studied with Picone, graduating in 1931. At this time Miranda also became an assistant of Picone who by this time had four assistants, the other two being Gianfranco Cimmino and Giuseppe Scorza Dragoni. These four assistants are referred to as "Mauro Picone's four musketeers" by Scorza Dragoni in his obituary of Miranda. They worked in the "Institute for Calculus" which had been founded by Picone in Naples in 1927 with financial assistance from the Banco di Napoli. This Institute was perhaps the earliest Institute for computing in the world.

Picone left Naples in 1932, together with Miranda, when he moved his Institute to Rome on being appointed to the chair of Higher Analysis there. Mario Salvadori explained in his autobiography *A Tangential Life* (see [18]) how this move came about:-

Picone was the director of the Institute for the Applications of the Calculus in Rome from its founding until 31 July 1960 when he retired from his chair at the university and was made professor emeritus. The early days of the Institute are described by Salvadori (see [18]):-Picone, a vivacious, bright and aggressive Sicilian had been the first mathematician in Italy to recognize the importance of numerical analysis ... When Picone became interested in numerical analysis the electronic computer was twenty-five years into the future the pure mathematicians at the University of Rome despised the subject, although the Germans had already given it high status in some of their universities. With admirable single mindedness and great political skill Picone succeeded in getting appointed to a chair of mathematical analysis in Rome, left Naples University together with his brilliant assistant Miranda and, besides teaching the kind of mathematics acceptable to his illustrious colleagues, obtained a small grant from the newly established Italian National Research Council to start in a small apartment in Rome in a new section of town the high-sounding "Institute for the Applications of the Calculus."

Salvadori also describes a visit by Mussolini to the Institute (see [18]):-The only calculators available to us then were mechanical and hand cranked. Later we were able to buy a few electrically powered mechanical calculators, which were a little faster and less noisy. Yet by1935Picone's vision and persistence had given the institute, by now the National Institute for the Applications of Calculus, the entire top floor of the new palace of the National Research Council and enough money to pay(miserly)a staff of thirty.

The work undertaken by the Institute included functional analysis, partial differentiation, integral equations, calculus of variations, special functions, probability theory, rational mechanics and mathematical physics. As well as much theoretical work, practical work was undertaken for various government ministries, particularly the Ministries of Aviation and War, as Italy prepared for war. After Italy entered World War II in June 1940, the Institute worked on military applications as part of the war effort, switching to work to aid Italy's economic reconstruction after the armistice agreement of September 1943. In the above quote, Salvadori pokes fun at Fascism but, in the years running up to World War II, Picone seems to have enthusiastically embraced its ideas. For example, when Mussolini joined the Fascist party in 1923 Picone wrote to him:-On the morning of the inauguration of the new headquarters we were requested to be at our desks at8a.m. When "il Duce" strutted along the corridor of the institute we cranked our machines by hand as fast as we could after setting all the levers of our calculators at the9positions, because in this configuration the calculators made the biggest racket. Il Duce could not miss the enormous significance to the future of the Fascist Empire of so many999,999,999s being so loudly multiplied by999,999,999. In a sense the bombast of these multiplifications was an honest representation of the empty racket of most Fascist activities. As I vigorously turned my crank I saw from the corner of my eye paunchy Mussolini in his black uniform followed by paunchy Picone in his black shirt. As soon as "he" left the racket stopped and we went back to our serious pioneering work.

However, after World War II ended, Picone made critical comments about Fascism, writing in 1948:-Illustrious Excellency, let me express to you my deepest innermost satisfaction for you giving your support to the Fascist Party to which I belong. ... Your major support of the Fascist Party ... will overcome the hesitation of many of my colleagues and will bring more new pure blood into the strong veins of the Party which is rebuilding and reorganising our country.

Let us be kind and suggest that by this time Picone had seen the error of his earlier Fascist beliefs.Unfortunately, ... university life was painfully interrupted for seven years, from1938to1945, because of those senseless racial measures which deprived Italy, in that long and difficult period, of the precious work of citizens of very high moral, spiritual and intellectual standing.

As to Picone's own mathematical contributions after setting up his Institutes, Silvio Cinquini writes in a review of [9]:-

Some of his most important books which Picone published during his years in Rome are:... one notes Picone's marked preference for the numerical direction of analysis which was more amply developed after the establishment of the Istituto Nazionale per le Applicazioni del Calcolo, since he knew that problems with an applied emphasis always give rise to new theoretical research. Resulting from this were Picone's results on a priori bounds for the solutions of ordinary differential equations, as well as for those of linear partial differential equations of elliptic type and parabolic type for which the bound is obtained by means of the boundary data and the known terms; these results are contained in his well-known 'Notes on higher analysis'(Italian)a volume published in1940and which was, for its time, "truly avant-garde". Gaetano Fichera highlights Picone's1936memoir which contains a characterization of a large class of linear partial differential equations whose solutions enjoy mean-value properties termed "integral properties" by Picone; using this theory Picone reconstructed M Nicolescu's theory of polyharmonic functions. However, the works which led to the broadest and most important research are those based on the translation of boundary value problems for linear partial differential equations into systems of Fischer-Riesz integral equations; this method, whose object is the numerical calculation of the solutions, is similar to that of subsequent authors, who considered weak solutions of the same problems.

*Appunti di Analisi superiore*Ⓣ (1940), which studies harmonic functions, Fourier, Laplace and Legendre series and the equations of mathematical physics;

*Lezioni di Analisi funzionale*Ⓣ (1946), which concerns the calculus of variations;

*Teoria moderna dell'integrazione delle funzioni*Ⓣ (1946), containing a detailed discussion of the r-dimensional Stieltjes integrals; (with Tullio Viola)

*Lezioni sulla teoria moderna dell'integrazione*Ⓣ (1952), which is basically the previous work by Picone with three extra chapters by Viola; and (with Gaetano Fichera)

*Trattato di Analisi matematica*Ⓣ (Vol 1, 1954, Vol 2, 1955), which puts into a treatise Picone's way of teaching calculus particularly slanted towards the applications studied at the Institute for Applied Calculus. After Picone retired in 1960, he was made professor emeritus. Over the last eight years in the chair he had concentrated his research on a classical approach to the integrals of the calculus of variations.

Picone received many honours for his remarkable contributions to mathematics. He was awarded honorary doctorates by the University of Sao Paulo in Brazil and the University of Bucharest. He was elected to, among many Societies and Academies: the Accademia dei Lincei; the National Academy of Sciences of Italy (the "Academy of Forty"); the National Society of Science, Letters and the Arts; the Turin Academy of Sciences; the Accademia Gioenia di Catania; the Modena Academy of Science, Letters and the Arts; the Palermo Academy of Sciences, Humanities and Fine Arts; the Buenos Aires Academy of Sciences; the Warsaw Society of Arts and Sciences; the Polish Academy of Sciences; the Royal Academy of Exact Sciences in Madrid; and the Academy of Sciences of the Socialist Republic of Romania. He received many prizes including: the Royal Prize for Mathematics from the Accademia dei Lincei (1938); the Tenore Prize of the Royal Society of Naples; the Severi Prize from the National Institute of Advanced Mathematics; the Gold Medal for merit of Culture and Art; the Gold Medal of the Faculty of Sciences of the University of Rome; the Gold Medal of the Italian National Research Council; the Gold Medal from the French Society for the Encouragement of Research and Invention; and the Fermat Medal of the Academy of Sciences of Toulouse.

**Article by:** *J J O'Connor* and *E F Robertson*