Fonction homogène | EPFL Graph Search

vignette|Exemple de fonction homogène de degré 1 En mathématiques, une fonction homogène est une fonction qui a un comportement d’échelle multiplicatif par rapport à son ou ses arguments : si l'argument (vectoriel au besoin) est multiplié par un scalaire, alors le résultat sera multiplié par ce scalaire porté à une certaine puissance. Définitions Soient E et F deux espaces vectoriels sur un même corps commutatif K. Une fonction f de E dans F est dite homogène de degré α si \forall t\in K\quad\forall x \in E\qquad f(tx)=t^\alpha f(x). Si K est un sous-corps des réels, on dit que f est positivement homogène de degré α si \forall t>0\quad\forall x\in E\qquad f(tx)=t^\alpha f(x). Si K est un sous-corps des complexes, on dit que f est absolument homogène de degré α si \forall t\in K\quad\forall x\in E\qquad f(tx)=|t|^\alpha f(x). Selon le contexte, « positivement homogène » peut signifier « positivement homogène de degré α

Unique decomposition of homogeneous languages and application to isothetic regions

Nicolas René Jean Ninin

A language is said to be homogeneous when all its words have the same length. Homogeneous languages thus form a monoid under concatenation. It becomes freely commutative under the simultaneous actions of every permutation group G(n) on the collection of homogeneous languages of length n is an element of N. One recovers the isothetic regions from (Haucourt 2017, to appear (online since October 2017)) by considering the alphabet of connected subsets of the space vertical bar G vertical bar, viz the geometric realization of a finite graph G. Factoring the geometric model of a conservative program amounts to parallelize it, and there exists an efficient factoring algorithm for isothetic regions. Yet, from the theoretical point of view, one wishes to go beyond the class of conservative programs, which implies relaxing the finiteness hypothesis on the graph G. Provided that the collections of n-dimensional isothetic regions over G (denoted by R-n vertical bar G vertical bar) are co -unital distributive lattices, the prime decomposition of isothetic regions is given by an algorithm which is, unfortunately, very inefficient. Nevertheless, if the collections R-n vertical bar G vertical bar satisfy the stronger property of being Boolean algebras, then the efficient factoring algorithm is available again. We relate the algebraic properties of the collections R-n vertical bar G vertical bar to the geometric properties of the space I GI. On the way, the algebraic structure R-n vertical bar G vertical bar is proven to be the universal tensor product, in the category of semilattices with zero, of n copies of the algebraic structure R-1 vertical bar G vertical bar.

2019

The role of perspective functions in convexity, polyconvexity, rank-one convexity and separate convexity

Bernard Dacorogna

Any finite, separately convex, positively homogeneous function on

\mathbb{R}^2

is convex. This was first established by the first author ["Direct methods in calculus of variations", Springer-Verlag (1989)]. Here we give a new and concise proof of this result, and we show that it fails in higher dimension. The key of the new proof is the notion of {\it perspective} of a convex function

f

, namely, the function

(x,y)\to yf(x/y)

y>0

. In recent works of the second author [Math. Programming 89A (2001) 505--516; J. Optimization Theory Appl. 126 (2005) 175--189 and 357--366], the perspective has been substantially generalized by considering functions of the form

(x,y) \to g(y)f(x/g(y))

, with suitable assumptions on

g

. Here, this {\it generalized perspective} is shown to be a powerful tool for the analysis of convexity properties of parametrized families of matrix functions.

2008

Eigenfunction expansions of ultradifferentiable functions and ultradistributions

Aparajita Dasgupta

In this paper we give a global characterisation of classes of ultradi_erentiable functions and corresponding ultradistributions on a compact manifold X. The characterisation is given in terms of the eigenfunction expansion of an elliptic operator on X. This extends the result for analytic functions on compact manifold by Seeley [See69], and the characterisation of Gevrey functions and Gevrey ultradistributions on compact Lie groups and homogeneous spaces by the authors.

2016