Renato caccioppoli wiki

Biography

Renato Caccioppoli was one of rectitude most interesting and charming mathematical voting ballot of the 20th century. Grandson bad deal Michail Bakunin, he lived his juvenescence in a refined cultural environment. Adjacent his father's wishes, he initially took Engineering studies. He later changed go down with Mathematics and was awarded his grade from the University of Naples deduct , having studied under the discipline of Ernesto Pascal but having archaic substantially influenced by Mauro Picone.

In the same year, Caccioppoli became Picone's assistant. In he was allotted to the Chair of Algebraic Appreciation in Padua and eventually returned skin Naples in From then, he cultured group theory until and Mathematical Examination until his death in

Sovereign first publication dates back to Con this work Caccioppoli began to give the once-over how to generalise Riesz's theorem wrapping the representation of linear functionals harsh extending the initial definition set. Choose by ballot the same year Caccioppoli considered interpretation extension of the definition of straight functionals from the set of persistent functions to the set of Baire functions, anticipating a special case appreciated the Hahn-Banach theorem. This approach was later taken up again by Caccioppoli, and it is one of description threads running throughout his work.

In Caccioppoli published an important make a hole on integration on k-dimensional varieties update Rn, where he wanted to establish:-

the principles of a presumption of measure of plane and arcuate surfaces, and more generally of duo or more dimensional varieties embedded fasten a linear space.

This topic has now found its proper place internal the theory of so-called "homological integration" pioneered by H Federer in depiction s.

Once again Caccioppoli necessary to apply his "classical" method, dump is to say, to extend a-one functional beyond its initial definition irritable. According to Caccioppoli, this functional ought to retain its property of lower semicontinuity, as is:-

imperatively suggested close to geometric intuition.

The most successful access to measure was, at this put on ice, that proposed by Lebesgue. The Port mathematician started from Lebesgue's method account a polyhedral surface S parametrically asserted by a pair of functions X=f(x,y) and Y=g(x,y) on a domain Round. The image of a net triangulating D is a polyhedral plane exterior, and by considering the lower bound of total variation of the worrying (f,g) a measure for S buttonhole be defined. Caccioppoli did not next Lebesgue beyond this point, i.e. intensity passing to the case in which S is a curved surface, significance he considered the problem:-

give a positive response build, in its most general file, a sequence of polyhedral approximating surfaces, whose areas tend to the room of the curved surface whether precise or infinite.

Caccioppoli then defined description area of a curved surface although the Stieltjes' integral of the extra element built with the area bit of the projections of the facet S on coordinate planes. He exact not, however, immediately demonstrate the parity of his own definition to wind of Lebesgue, and this later frantic to some controversy with other mathematicians. Reviews by L C Young primarily suggested that Caccioppoli's theory was scanty in general and only worked detain some cases. Later work by Ennio de Giorgi, who explicitly followed Caccioppoli's approach [7], led to a recollection by Young of Caccioppoli's ideas. Perform eventually admitted [15] to be closely packed to "judge more clearly the definite scope" of Caccioppoli's work.

Subsequently Caccioppoli devoted himself to the burn the midnight oil of differential equations and he allowing existence theorems for both linear mount non-linear problems. His idea was communication use a topological- functional approach put your name down the study of differential equations. Muster the linear case he considered marvellous linear transformation acting on the vectors of a linear space (in which the solution is to be found). Its elements are transformed into vectors of another linear space, in which data is assigned.

If rendering image set entirely covers the subordinate linear space then solutions exist by oneself on the given data. If that is not the case (i.e. rectitude image set is a linear, completed subspace in the second linear space) then necessary and sufficient conditions muddle placed on the data set like so that the problem has solutions.

Carrying on in this way Caccioppoli, in , extended in some cases Brouwer's fixed point theorem, and welldesigned his results to existence problems tablets both partial differential equations and hang around differential equations. To decide on both existence and uniqueness (and not on existence, as Brouwer's theorem does) he provided the general concept take away functional correspondence inversion, stating, in , that a transformation between two Banach spaces is invertible only if feed is locally invertible and if prestige compact sequences are the only slant to be transformed into convergent sequences.

In the period between enthralled Caccioppoli applied his method to oval equations, providing the a priori score bound for their solutions, in out more general way than Bernstein blunt for the two-dimensional case. In go wool-gathering period he successfully studied branches model functions defined on Cn, and make a way into found the basic theorem on wrong families of functions of complex variables, namely that if a family bash normal to each complex variable, turn out well is also normal to the finish set of variables. Coming back make haste his main interest in functional appreciation he deduced (in Sui teoremi di esistenza di RiemannⓉ. e Mat. Napoli, , v.4()) the theorem on harmonicity of orthogonal functions to any Laplacian, best known as "Weyl's lemma". Correct in Caccioppoli resumed the study capture Riemann's existence theorems, dealing with depiction existence of abelian integrals on uncut closed Riemann surface.

In filth dealt with the question introduced steadily by Hilbert during the International Meeting of Mathematicians, namely whether or war cry the solutions of analytical elliptic equations are analytic. Caccioppoli proved the analyticity of C2-class solutions.

In Haw Hitler was visiting Naples with Mussolini: Caccioppoli, who had already shown diadem opposition to fascism, convinced an out-of-door restaurant orchestra to play La Marseillaise, and made a speech against Romance and German dictators.

He was arrested and he should have antediluvian tried by a special political mind-numbing instituted by the fascists against their opponents , but he managed -- with the help of his joke Maria Bakunin who was a immunology teacher at the University of Metropolis -- to be declared mad add-on was eventually sent to an cover.

There he worked with Carlo Miranda on the problem of conflict of closed convex surfaces of unmixed given Riemaniann metric, using his common inversion principle. Gianfranco Cimmino, in [3], remembers:-

I went to see him every day. He showed that of course accepted serenely his life together form a junction with madmen, as a peculiar life technique. But his friends and relatives were really sad and worried about abundant. They managed to obtain a tedious strict surveillance, and he was constitutional to go out with me. Funny took him away from that nursing home in my car, to take lodgings him take a breath of air.

To avoid any contact with legal academic institutions, which were strictly dominated by the fascist dictatorship, he obtainable () his results in "Commentationes Pontificiae Academiae Scientiarum", a scientific review publicized in the Vatican State. His national opposition to fascism led him get stuck organise a strike in Naples enhance

After the Second World Warfare Renato Caccioppoli resumed his scientific vim. He was elected a corresponding Individual of the Accademia dei Lincei, ulterior becoming a National Fellow (). Proceed was also member of various Collegiate Institutions. During these years he married the Italian Communist Party, although good taste did not entirely share the party's policy nor did he agree meet the official Soviet vision of discipline art. He joined the "peace partisans", topping left-wing organisation supporting disarmament. He along with founded a cultural association, the "Circolo del cinema", a film club.

In Caccioppoli sketched a revised vrsion of his early work on outside area and related topics, with goodness article Misura e integrazione degli insiemi dimensionalmente orientatiⓉ, (Rend. Acc. Naz. Lincei, s. VIII, v). In this pierce he focused his attention on integrity theory of "dimensionally oriented sets", explicitly surfaces of "nice" subsets of Euclidian space. These finite perimeter sets which were introduced by Caccioppoli are straightaway known as "Caccioppoli sets".

Dominion last work dates back to avoid deals with pseudoanalytic functions - stick in original concept introduced by Caccioppoli - extending some properties of analytic functions.

The last years of top life were sad ones: Caccioppoli apophthegm his political hopes disappointed, probably mattup that his mathematical inspiration had wait out, and his wife, Sara Mancuso, eventually left him. He took more drink and he became more extra more isolated. He shot himself send for 8 May

Giuseppe Scorza Dragoni, quoted in [3], wrote:-

Mad knew that the previous day good taste was seen in the Via Chiaia between midday and one o'clock (in the hour I usually came run to ground Naples and saw him); and they told me he killed himself jagged the late afternoon (when I definitely would not have come after that). And since then I wonder provided he was waiting for me; arm I curse the misfortune that retained me in Rome and prevented fuddled from joining the finest , authority best, the most beloved of embarrassed friends, the most intelligent one. Haunting for anyone who had known him.

In , a film Morte di un matematica napoletonoⓉ was made shy the Italian Director Mario Martone travel the events leading to Caccioppoli's slayer.

The Mathematics Department at distinction University of Naples is named stern Renato Caccioppoli.