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We improve both dimension compression and expansion bounds for homeomorphisms with $p$-exponentially integrable distortion. To the first direction we also introduce estimates for the compression multifractal spectra, which will be used to…
We re-examine the notion of relative $(p,\eps)$-approximations, recently introduced in [CKMS06], and establish upper bounds on their size, in general range spaces of finite VC-dimension, using the sampling theory developed in [LLS01] and in…
We extend the definition of quasi-finite complexes by considering not necessarily countable complexes. We provide a characterization of quasi-finite complexes in terms of L-invertible maps and dimensional properties of compactifications.…
In this paper, we extend several approximation theorems, originally formulated in the context of the standard $L^p$ norm, to the more general framework of variable exponent spaces. Our study is motivated by applications in neural networks,…
We enhance the approximation capabilities of algebraic polynomials by composing them with homeomorphisms. This composition yields families of functions that remain dense in the space of continuous functions, while enabling more accurate…
We discuss finite local extensions of quantum field theories in low space time dimensions in connection with categorical structures and the question of modular invariants in conformal field theory, also touching upon purely mathematical…
In this short note we give counterexamples to several results related to extension theorems published recently.
We survey classical and recent results on exponents of Diophantine approximation. We give only a few proofs and highlight several open problems.
In this work, we propose a convenient framework for infinite-dimensional analysis (including both real and complex analysis in infinite dimensions), in which differentiation (in some weak sense) and integration operations can be easily…
This brief survey deals with multi-dimensional Diophantine approximations in sense of linear form and with simultaneous Diophantine approximations. We discuss the phenomenon of degenerate dimension of linear subspaces generated by the best…
The aim of this paper is to investigate, which infinite dimensional consequences follow from the main results of recently published paper of the authors (2009) (see Theorems 2 and 3). We show that the finite dimensional Theorem 3 implies…
The task of approximating a function of d variables from its evaluations at a given number of points is ubiquitous in numerical analysis and engineering applications. When d is large, this task is challenged by the so-called curse of…
Laplace approximations are commonly used to approximate high-dimensional integrals in statistical applications, but the quality of such approximations as the dimension of the integral grows is not well understood. In this paper, we prove a…
In this paper we develop a new machinery to study the capacity of artificial neural networks (ANNs) to approximate high-dimensional functions without suffering from the curse of dimensionality. Specifically, we introduce a concept which we…
We prove some extension results for holomorphic mappings with values in complex Hilbert manifolds
We prove some extension theorems involving uniformly continuous maps of the universal Urysohn space. We also prove reconstruction theorems for certain groups of autohomeomorphisms of this space and of its open subsets.
The paper is mostly a survey on recent results in Diophantine approximation, with emphasis on properties of exponents measuring various notions of Diophantine <approximation.
We consider the problem of domain approximation in finite element methods for Maxwell equations on curved domains, i.e., when affine or polynomial meshes fail to exactly cover the domain of interest. In such cases, one is forced to…
We propose the concepts of vicinal mappings and firmly vicinal mappings in metric spaces. We obtain fixed point and convergence theorems for these mappings in complete geodesic spaces with curvature bounded above by one and apply our…
In this paper, we introduce the concept of nearly convex set-valued mappings and investigate fundamental properties of these mappings. Additionally, we establish a geometric approach for generalized differentiation of nearly convex…