In this thesis possible characterizations of the category of primitive recursive functions and the category of recursive functions are studied. Closed cartesian categories, closed under the Peano-Lawvere axiom, whlch are called pre-recursive, are considered first. Representable functlons in such a category are introduced. Every primitive recursive function is representable in a pre-recursive category. In ⍕, the free pre-recursive category generated by the empty category Φ, every morphism T → N is a natural number, and every morphism N[n] → N[m] , n ∊ N, m ∊ N, represents a recursive function. Furthermore, a morphism representing a function which is not primitive recursive is found. A recursive function whlch is not representable in ⍕ is constructed. Following the above, structures of primitive recursive category and structures of recursive category are proposed, each structure generating a category whose class of representable functions is respectively the class of primitive recursive functions and the class of recursive functlons. / Pouvons-nous caractériser la catégorie des fonctions primitives récursives, la catégorie des fonctions récursives? Considérons les catégories cartésiennes fermées, fermées sous l'axiome de Peano-Lawvere, que nous appellerons pré-récursives et précisons ce qu'est une fonction représentable dans une telle catégorie. Toute fonction primitive récursive est représentable dans une catégorie pré-récursive. Dans, la catégorie pré-récursive libre engendrée par la catégorie vide, tout morphisme T->N représente un nombre naturel et tout morphisme N[n] -> N[m], n N, m N, représente une fonction récursive. De plus, on peut trouver un morphisme qui représente une fonction qui n'est pas primitive récursive et construire une fonction récursive non représentable dans $. A la suite de ces résultats, nous présentons des structures de catégorie primitive récursive et de catégorie récursive, chaque structure engendrantune catégorie dont la classe des fonctions représentables est respectivement la class des fonctions primitives récursives et celle des fonctions récursives. fr

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The principle content of this thesis could be divided roughly into three parts: a) to establish some of the more imprtant theorems of the convex and subharmonic functions; b) to give a solution fof the Dirichlet Problem for the circle which, as we will see, is essenstial in the development of the theory of subharmonic fuctions; c) to show the analogy existing between the theories of the convex and subharmonic functions. Historically the first elementary and systematic expostion of convex functions was written by J. Jensen in 1906. Of the many authors who, following Jensen, have developed the theory in its early stages we mention in particular F. Bernstein, G. Doetch, and L. Galvani [cv. Bernstein (1), *Bernstein and Doetsch (2), *and L. Galvanie, “Sulle funzioni convesse di una e duo vairabili definite in un aggregato qualunque” (Circele Mathematico di Palermo, Vol.41 1916, p.103-134)]. The theory of the subharmonic functions which was developed mainly by F. Riesz (in 1926) and Paul Montel (in 1928) appeared principally, as we will see, as a generalization of the convex functions of one variable to functions of two variables obeying the same fundamental properties [cf. F. Riesz (10) * and P. Montel (8)]*. Two other names must be mentioned in the connection of this theory, those of J.E. Littlewood and T. Rado who have further developed the general theory of Subharmonic functions [...]

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ix, 149 leaves / Title page, contents and abstract only. The complete thesis in print form is available from the University Library. / Thesis (Ph.D.) from the Dept. of Statistics, University of Adelaide, 1971

Thesis (Ph.D.) from the Dept. of Statistics, University of Adelaide, 1971.