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Convex sets appear in various mathematical theories, and are used to define notions such as convex functions and hulls. As an abstraction from the usual definition of convex sets in vector spaces, we formalize in Coq an intrinsic…
The present paper is devoted to possible generalizations of the classic Lagrange Mean Value Theorem. We consider a real-valued function of several variables that is only assumed to be continuous. The main concept is to replace the notion of…
This article shows a very elementary and straightforward proof of the Implicit Function Theorem for differentiable maps $F(x,y)$ defined on a finite-dimensional Euclidean space. There are no hypothesis on the continuity of the partial…
The idea of using unfolding as a way of computing a program semantics has been applied successfully to logic programs and has shown itself a powerful tool that provides concrete, implementable results, as its outcome is actually source…
We study systems of nonlinear partial differential equations of parabolic type, in which the elliptic operator is replaced by the first order divergence operator acting on a flux function, which is related to the spatial gradient of the…
The purpose of this paper is to clarify the relationship between various conditions implying essential undecidability: our main result is that there exists a theory $T$ in which all partially recursive functions are representable, yet $T$…
We develop a novel formal theory of finite structures, based on a view of finite structures as a fundamental artifact of computing and programming, forming a common platform for computing both within particular finite structures, and in the…
In this article we will analyse how to compute the contribution of each input value to its aggregate output in some nonlinear models. Regression and classification applications, together with related algorithms for deep neural networks are…
We study adaptive approximation algorithms for general multivariate linear problems where the sets of input functions are non-convex cones. While it is known that adaptive algorithms perform essentially no better than non-adaptive…
In this paper we give simple extension and uniqueness theorems for restricted additive and logarithmic functional equations.
Linear models have shown great effectiveness and flexibility in many fields such as machine learning, signal processing and statistics. They can represent rich spaces of functions while preserving the convexity of the optimization problems…
Using techniques from the theory of foliations, we establish the cone theorem and the contraction theorem for lc generalized pairs in full generality, and meanwhile develop the minimal model program for $\mathbb Q$-factorial foliated dlt…
The first part of the paper provides new characterizations of the normal cone to the effective domain of the supremum of an arbitrary family of convex functions. These results are applied in the second part to give new formulas for the…
In this paper we provide a finite-sample and an infinite-sample representer theorem for the concatenation of (linear combinations of) kernel functions of reproducing kernel Hilbert spaces. These results serve as mathematical foundation for…
The incomplete version of the Macdonald function has various appellations in literature and earns a well-deserved reputation of being a computational challenge. This paper ties together the previously disjoint literature and presents the…
Sets of $d\times d$ matrices sharing a common invariant cone enjoy special properties, which are widely used in applications. However, finding this cone or even proving its existence/non-existence is hard. This problem is known to be…
We prove two versions of a global implicit function theorem, which involve no loss of derivative, for Keller's $ C_c^1 $-mappings between arbitrary Fr\'{e}chet spaces. Subsequently, within this framework, we apply these theorems to…
This is an exposition of some basic ideas in the realm of Global Inverse Function theorems. We address ourselves mainly to readers who are interested in the applications to Differential Equations. But we do not deal with those applications…
The purpose of this paper is to investigate coefficient matrices of functional equations of zeta functions associated with homogeneous cones, which are given explicitly in the previous paper, in detail. We prove that the coefficient matrix…
Prompted by an observation about the integral of exponential functions of the form $f(x)=\lambda e^{\alpha x}$, we investigate the possibility to exactly integrate families of functions generated from a given function by scaling or by…