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Gaetano Fichera 8 February — 1 June was an Italian mathematician , working in mathematical analysis , linear elasticity , partial differential equations and several complex variables. He was born in Acireale , and died in Rome. In his young years he was a talented football player. On 1 February he was in the Italian Army and during the events of September he was taken prisoner by the Nazist troops, kept imprisoned in Teramo and then sent to Verona : he succeeded in escaping from there and reached the Italian region of Emilia-Romagna , spending with partisans the last year of war.

After the war he was first in Rome and then in Trieste , where he met Matelda Colautti , who became his wife in After graduating from the liceo classico in only two years, he entered the University of Catania at the age of 16, being there from to and studying under Pia Nalli.

Then he went to the university of Rome , where in he earned his laurea with magna cum laude under the direction of Mauro Picone , when he was only He was immediately appointed by Picone as an assistant professor to his chair and as a researcher at the Istituto Nazionale per le Applicazioni del Calcolo , becoming his pupil.

After the war he went back to Rome working with Mauro Picone : in he became "Libero Docente" free professor of mathematical analysis and in he was appointed as full professor at the University of Trieste.

As he remembers in Fichera , p. He retired from university teaching in , [2] but was professionally very active until his death in particularly, as a member of the Accademia Nazionale dei Lincei and first director of the journal Rendiconti Lincei — Matematica e Applicazioni , [3] he succeeded in reviving its reputation. His lifelong friendship with his teacher Mauro Picone is remembered by him in several occasions.

As recalled by Colautti Fichera , pp. The young, in effect child, Gaetano, was kept by Picone in his arms. From to the young Fichera developed his research directly under the supervision of Picone: as he remembers, it was a time of intense work.

But also, when he was back from the front in April [5] he met Picone while he was in Roma in his way back to Sicily , and his advisor was so happy to see him as a father can be seeing its living child. Another mathematician Fichera was influenced by and acknowledged as one of his teachers and inspirators was Pia Nalli : she was an outstanding analyst , teaching for several years at the University of Catania , being his teacher of mathematical analysis from to Antonio Signorini and Francesco Severi were two of Fichera's teachers of the Roman period: the first one introduced him and inspired his research in the field of linear elasticity while the second inspired his research in the field he taught him i.

Signorini had a strong long-time friendship with Picone: on a wall of the apartment building where they lived, in Via delle Tre Madonne, 18 in Rome, a memorial tablet which commemorates the two friends is placed, as Fichera b , p. The two great mathematicians extended their friendship to the young Fichera, and as a consequence this led to the solution of the Signorini problem and the foundation of the theory of variational inequalities.

Fichera's relations with Severi were not as friendly as with Signorini and Picone: nevertheless, Severi, which was one of the most influential Italian mathematicians of the first half of the 20th century, esteemed the young mathematician. During a course on the theory of analytic functions of several complex variables taught at the Istituto Nazionale di Alta Matematica from the fall of and the beginning of the , whose lectures were collected in the book Severi , Severi posed the problem of generalizing his theorem on the Dirichlet problem for holomorphic function of several variables , as Fichera , p.

Other scientists he had as teachers during the period — were Enrico Bompiani , Leonida Tonelli and Giuseppe Armellini : he remembered them with great respect and admiration, even if he did not share all their opinions and ideas, as Colautti Fichera , p.

He built up such a network of contacts being invited several times to lecture on his research by various universities and research institutions, and also participating to several academic conferences , always upon invitation. This long series of scientific journeys started in , when he went to the USA together with his master and friend Mauro Picone and Bruno de Finetti in order to examine the capabilities and characteristics of the first electronic computers and purchase one for the Istituto Nazionale per le Applicazioni del Calcolo : the machine they advised to purchase was the first computer ever working in Italy.

The close friendship between Angelo Pescarini and Fichera has not his roots in their scientific interests: it is another war story. As Oleinik , p.

Angelo, which was a student of mathematics at the University of Bologna under Gianfranco Cimmino , a former pupil of Mauro Picone , was charged of the task of testing the truth of Gaetano's assertions, examining him in mathematics: his question was:— "Mi sai dire una condizione sufficiente per scambiare un limite con un integrale Can you give me a sufficient condition for interchanging limit and integration?

In effect, Fichera proved such a theorem in the paper Fichera , his latest paper written in while he was in Rome before joining the army: from that moment on he often used to joke saying that good mathematicians can always have a good application, even for saving one's life.

One of his best friends and appreciated scientific collaborator was Olga Arsenievna Oleinik : she cured the redaction of his last posthumous paper Fichera , as Colautti Fichera , pp. Also, she used to discuss his work with Gaetano, as he did with her: sometimes their discussion become lively, but nothing more, since they were extremely good friends and estimators of each one's work.

He is the author of more than papers and 18 books monographs and course notes : his work concerns mainly the fields of pure and applied mathematics listed below. A common characteristic to all of his research is the use of the methods of functional analysis to prove existence , uniqueness and approximation theorems for the various problems he studied, and also a high consideration of the analytic problems related to problems in applied mathematics.

His work in elasticity theory includes the paper Fichera c , where Fichera proves the " Fichera's maximum principle ", his work on variational inequalities. The work on this last topic started with the paper Fichera , where he announced the existence and uniqueness theorem for the Signorini problem , and ended with the following one Fichera a , [6] where the full proof was published: those papers are the founding works of the field of variational inequalities, as remarked by Stuart Antman in Antman , pp.

Also he is known for his researches in the theory of hereditary elasticity : the paper Fichera b emphasizes the necessity of analyzing very well the constitutive equations of materials with memory in order to introduce models where an existence and uniqueness theorems can be proved in such a way that the proof does not rely on an implicit choice of the topology of the function space where the problem is studied.

He was one of the pioneers in the development of the abstract approach through functional analysis in order to study general boundary value problems for linear partial differential equations proving in the paper Fichera a a theorem similar in spirit to the Lax—Milgram theorem.

He studied deeply the mixed boundary value problem i. He is, according to Oleinik , the founder of the theory of partial differential equations of non-positive characteristics : in the paper Fichera he introduced the now called Fichera's function , in order to identify subsets of the boundary of the domain where the boundary value problem for such kind of equations is posed, where it is necessary or not to specify the boundary condition : another account of the theory can be found in the paper Fichera , which is written in English and was later translated in Russian and Hungarian.

His contributions to the calculus of variation are mainly devoted to the proof of existence and uniqueness theorems for maxima and minima of functionals of particular form, in conjunction with his studies on variational inequalities and linear elasticity in theoretical and applied problems: in the paper Fichera a a semicontinuity theorem for a functional introduced in the same paper is proved in order to solve the Signorini problem , and this theorem was extended in Fichera c to the case where the given functional has general linear operators as arguments , not necessarily partial differential operators.

It is difficult to single out his contributions to functional analysis since, as stated at the beginning of this section, the methods of functional analysis are ubiquitous in his research: however, it is worth to remember paper Fichera a , where an important existence theorem is proved. His contributions in the field of eigenvalue theory began with the paper Fichera b , where he formalizes a method developed by Mauro Picone for the approximation of eigenvalues of operators subject only to the condition that their inverse is compact : however, as he admits in Fichera a , pp.

He contributed also to the classical eigenvalue problem for symmetric operators , introducing the method of orthogonal invariants. His work in this field is mainly related to the study of systems of functions , possibly being particular solutions of a given partial differential equation or system of such equations, in order to prove their completeness on the boundary of a given domain. The interest of this research is obvious: given such a system of functions, every solution of a boundary value problem can be approximated by an infinite series or Fourier type integral in the topology of a given function space.

One of the most famous examples of this kind of theorem is Mergelyan's theorem , which completely solves the problem in the class of holomorphic functions for a compact set in the complex plane. In his paper Fichera , Fichera studies this problem for harmonic functions , [12] relaxing the smoothness requirements on the boundary in the already cited work Fichera a : a survey on his and others' work in this area, including contributions of Mauro Picone , Bernard Malgrange , Felix Browder and a number of other mathematicians, is contained in the paper Fichera c.

Another branch of his studies on approximation theory is strictly tied to complex analysis in one variable , and to the already cited Mergelyan's theorem : he studied the problem of approximating continuous functions on a compact set and analytic on its interior if this is non void of the complex plane by rational functions with prescribed poles , simple or not. His contributions to potential theory are very important. Also, his researches Fichera and Fichera on the asymptotic behaviour of the electric field near singular points of the conducting surface, widely known among the specialists as several works of V.

Maz'ya , S. Nazarov , B. Plamenevsky , B. Schulze and others testify can be included in between his works in potential theory. His main contributions to those topics and are the papers Fichera and Fichera In the first one he proves that a condition on a sequence of integrable functions previously introduced by Mauro Picone is both necessary and sufficient in order to assure that limit process and the integration process commute, both in bounded and unbounded domains : the theorem is similar in spirit to the dominated convergence theorem , which however only states a sufficient condition.

The second paper contains an extension of the Lebesgue's decomposition theorem to finitely additive measures : this extension required him to generalize the Radon—Nikodym derivative , requiring it to be a set function belonging to a given class and minimizing a particular functional. He contributed to both the classical topic of complex analysis in one variable and the more recent one of complex analysis in several variables.

His contributions to complex analysis in one variable are essentially approximation results , well described in the survey paper Fichera b. Another important result is his proof in Fichera of an extension of Morera's theorem to functions of several complex variables , under the hypothesis that the given function f is only locally integrable : previous proofs under more restrictive assumptions were given by Francesco Severi in Severi and Salomon Bochner in Bochner He also studied the properties of the real part and imaginary part of functions of several complex variables , i.

His contributions to the theory of exterior differential forms started as a war story: [18] having read a famous memoir of Enrico Betti where Betti numbers were introduced just before joining the army, he used this knowledge in order to develop a theory of exterior differential forms while he was kept prisoner in Teramo jail. However, he continued work on this theory, contributing with several papers, and also advised all of his students to study it, despite from the fact of being an analyst , as he remarks: his main results are collected in the papers Fichera a and Fichera b.

In the first one he introduced k -measures, a concept less general than currents but easier to work with: his aim was to clarify the analytic structure of currents and to prove all relevant results of the theory i. In the second one he developed an abstract Hodge theory , following the axiomatic method , proving an abstract form of Hodge theorem.

As noted in the " Functional analysis and eigenvalue theory " section, his main direct contribution to the field of numerical analysis is the introduction of the method of orthogonal invariants for the calculus of eigenvalues of symmetric operators : however, as already remarked, it is hard to find something in his works which is not related to applications.

His works on partial differential equations and linear elasticity have always a constructive aim: for example, the results of paper Fichera , which deals with the asymptotic analysis of the potential , were included in the book Fichera a and led to the definition of the Fichera corner problem as a standard benchmark problem for numerical methods.

His historical works contain several observations against the so-called historical revisitation : the meaning of this concept is clearly stated in the paper Fichera He identifies with the word revisitation the analysis of historical facts basing only on modern conceptions and points of view: this kind of analysis differs from the "true" historical one since it is heavily affected by the historian's point of view. The historian applying this kind of methodology to history of mathematics , and more generally to the history of science , emphasizes the sources that have led a field to its modern shape, neglecting the efforts of the pioneers.

A selection of Gaetano Fichera's works was published respectively by the Unione Matematica Italiana and the Accademia Pontaniana in his "opere scelte" Fichera and in the volume Fichera These two references include most of the papers listed in this section: however, these volumes does not include his monographs and textbooks , as well as several survey papers on various topic pertaining to his fields of research. From Wikipedia, the free encyclopedia.

Gaetano Fichera. Linear elasticity Mathematical analysis Variational inequalities Numerical analysis Partial differential equations Several complex variables Signorini problem. Main article: Fichera's existence principle. The " Yearbook " of the renowned Italian scientific institution, including an historical sketch of its history, the list of all past and present members as well as a wealth of information about its academic and scientific activities. The first part "Tomo" of an extensive work on the "Accademia di Scienze, Lettere e Arti di Modena", reporting the history of the academy and biographies of members up to the year Cosentini, Cristoforo , "Ricordo del Prof.

Gaetano Fichera, socio d'onore" [Recollection of Prof. A commemorative paper written by Cristoforo Cosentini, former member and president of the Accademia di scienze, lettere e belle arti degli Zelanti e dei Dafnici and close friend of Gaetano Fichera.

Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Supplemento , 9 in Italian , 8 1 : 14—33 , prepared by his wife as follow-up to the commemorative paper by Olga Oleinik Colautti Fichera, Matelda December , La lunga, brevissima vita di Gaetano Fichera [ The first phrase of the title is the last verse and title of a famous poem of Salvatore Quasimodo , and was the concluding phrase of the last lesson of Fichera, in the occasion of his retirement from university teaching in , published in Fichera harv error: no target: CITEREFFichera help.

There is also a free electronic edition with a different title: Colautti Fichera, Matelda 30 September , Gaetano in Italian , Lulu , p. Supplemento , Serie IX, 8 1 : This book offers the personal recollections of the Author about the life in his birthplace Alfonsine , during the fascist period up to the end of World War II. He describes various episodes of the life of Gaetano Fichera in his town during wartime, their friendship and the relations between Fichera and the Italian resistance movement.

The choice of photographs and the presentation of the book are due to Luciano Lucci, who also cured the web edition which is enriched by several pictures at the expense of the loss of printed edition pagination. The first part of the title, up to the colon , is in Emiliano-Romagnolo while the second part is in Italian. Presidenza della Repubblica Italiana 31 July , Medaglia d'oro ai benemeriti della scuola della cultura e dell'arte: Gaetano Fichera [ Gold Medal for the distinguished of school, culture and art: Gaetano Fichera ] , retrieved 31 May Ricci, Paolo E.

June , "Scomparsa del Prof. Ricci, P. Ridolfi, Roberto, ed. The biographical and bibliographical entry updated up to on Gaetano Fichera, published under the auspices of the Accademia dei Lincei in a book collecting many profiles of its living members up to

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## Laforgia, Andrea

Laforgia, Andrea Overview. Publication Timeline. Most widely held works by Andrea Laforgia. Analisi matematica by Andrea Laforgia Book 1 edition published in in Italian and held by 8 WorldCat member libraries worldwide. Precalculus : argomenti propedeutici all'esame di analisi by Andrea Laforgia Book 2 editions published in in Italian and held by 6 WorldCat member libraries worldwide. Funzioni speciali e applicazioni Book 3 editions published in in Italian and held by 4 WorldCat member libraries worldwide.

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## Gaetano Fichera

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## Andrea Laforgia

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