By L. Auslander, R. Tolimieri

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Vito Volterra (1860-1940) was once some of the most well-known representatives of Italian technology in his day. Angelo Guerragio and Giovanni Paolini research Volterra's most vital contributions to arithmetic and their functions, in addition to his striking organizational achievements in medical coverage.

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None of this is yet present in Volterra. He had no need for an abstraction ‘pushed’ this far, and for him the motivations behind his studies remained essential for determining the level of abstraction. Volterra’s functions of lines – it is no coincidence that the terminology is also different – are definite correspondences to a ‘concrete’ set and are in any case specific, such as that constituted of functions that are continuous in an interval. There is no general definition of metric space. The definition of derivative itself still refers to the limit of a real parameter (and to a procedure that is familiar to all those who work with the calculus of variations): considering an initial function f 0 and the corresponding value of the functional U ðf 0 Þ, the incremented function is written as f 0 þ tg where t is a real parameter and g is a pre-established function that continues to give, as t tends to 0, the ‘direction’ of the increment.

To phrase it lightly, we can say that at the origin of the dispute, on the part of Peano, there was a cat. On the part of Volterra, there was a series of papers published in the winter and spring of 1895 in the Atti of Torino’s Accademia della Scienze. The problem addressed was the analysis of systems in which there are motions due to the actions of internal forces. The specific reference is to the earth. ) were already characterised by a certain consistency and tradition. Instead, in Volterra’s opinion, what had remained relatively unexamined where the cyclical motions that, even though present on the earth and within it, do not noticeably modify the form of its surface or the distribution of the masses on it.

At that time Volterra did not yet use term, which would be coined in 1903 by Jacques Hadamard (1865–1963), one of the most important mathematicians of the first half of the twentieth century, and another of Volterra’s close French friends. He preferred to use the terms that appear in the titles of his papers: ‘functions that depend on other functions’ and ‘functions that depend on lines’. 2 Scientific Work During the Period in Pisa 19 later generations, and would become the definitive one. What is a functional, or a function that depends on lines?