Related papers: Borel selection of dominating hyperplanes
Gowers norms have been studied extensively both in the direct sense, starting with a function and understanding the associated norm, and in the inverse sense, starting with the norm and deducing properties of the function. Instead of…
We give a partial answer to an important open problem in descriptive set theory, the Decomposability Conjecture for Borel functions on an analytic subset of a Polish space to a separable metrizable space. Our techniques employ deep results…
Indicator functions mentioned in the title are constructed on an arbitrary nondiscrete locally compact Abelian group of finite dimension. Moreover, they can be obtained by small perturbation from any indicator function fixed beforehand. In…
We study the behaviour of the algebraic degree of vectorial Boolean functions when their inputs are restricted to an affine subspace of their domain. Functions which maintain their degree on all subspaces of as high a codimension as…
We study the lattice of all Borel clones on $2 = \{0,1\}$: classes of Borel functions $f : 2^n \to 2$, $n \le \omega$, which are closed under composition and include all projections. This is a natural extension to countable arities of…
A continuous selection of polynomial functions is a continuous function whose domain can be partitioned into finitely many pieces on which the function coincides with a polynomial. Given a set of finitely many polynomials, we show that…
Neural networks are versatile tools for computation, having the ability to approximate a broad range of functions. An important problem in the theory of deep neural networks is expressivity; that is, we want to understand the functions that…
For every natural number k we introduce the notion of k-th order convolution of functions on abelian groups. We study the group of convolution preserving automorphisms of function algebras in the limit. It turns out that such groups have…
The nonzero level sets of a homogeneous, logarithmically homogeneous, or translationally homogeneous function are affine spheres if and only if the Hessian determinant of the function is a multiple of a power or an exponential of the…
We show that for any free probability measure-preserving action of $\mathbb{C}^{d}$ on a standard probability space, there exists a Borel entire function $F$ such that the factor map $x \mapsto F_{x}$, where $F_{x}(z) = F(z \cdot x)$, is…
We describe a web of connections between the following topics: the mathematical theory of voting and social choice; the computational complexity of the Maximum Cut problem; the Gaussian Isoperimetric Inequality and Borell's generalization…
We consider a class of optimization problems that involve determining the maximum value that a function in a particular class can attain subject to a collection of difference constraints. We show that a particular linear programming…
We consider A-hypergeometric functions associated to normal sets in the plane. We give a classification of all point configurations for which there exists a parameter vector such that the associated hypergeometric function is algebraic. In…
We prove general results about separation and weak$^\#$-convergence of boundedly finite measures on separable metric spaces and Souslin spaces. More precisely, we consider an algebra of bounded real-valued, or more generally a $*$-algebra…
A supervised learning problem is to find a function in a hypothesis function space given values on isolated data points. Inspired by the frequency principle in neural networks, we propose a Fourier-domain variational formulation for…
We consider the problem of variable selection in high-dimensional sparse additive models. We focus on the case that the components belong to nonparametric classes of functions. The proposed method is motivated by geometric considerations in…
Let X be an affine real algebraic set . We investigate on the theory of algebraically constructible functions on X and the description of the semi-algebraic subsets of X when we replace the polynomial functions on X by some rational…
We introduce a new combinatorial structure: the superselector. We show that superselectors subsume several important combinatorial structures used in the past few years to solve problems in group testing, compressed sensing, multi-channel…
In this paper we study definable families of functions from an ordered abelian group into various naturally arising definable quotients. We show that for an ordered abelian group $G$ and definable family of convex subgroups…
We study non-autonomous conformal iterated function systems, with finite or countably infinite alphabet alike. These differ from the usual (autonomous) iterated function systems in that the contractions applied at each step in time are…