Related papers: Regularity properties of random wavelet series
In probability theory, there is a tendency to treat one random variable with a given distribution as being just as good as any other. By and large this is fine because probability is (mostly) concerned with distributional properties of…
We know that tilesets that can tile the plane always admit a quasi-periodic tiling [4, 8], yet they hold many uncomputable properties [3, 11, 21, 25]. The quasi-periodicity function is one way to measure the regularity of a quasi-periodic…
We define twelve variants of a Reifenberg's affine approximation property, which are known to be connected with the singular sets of minimal surfaces. With this motivation we investigate the regularity of the sets possessing these. We…
An approach to modelling random sets with locally finite perimeter as random elements in the corresponding subspace of $L^1$ functions is suggested. A Crofton formula for flat sections of the perimeter is shown. Finally, random processes of…
We prove a.s. (almost sure) unisolvency of interpolation by continuous random sampling with respect to any given density, in spaces of multivariate a.e. (almost everywhere) analytic functions. Examples are given concerning polynomial and…
Deep neural networks trained in an end-to-end manner are proven to be efficient in a wide range of machine learning tasks. However, there is one drawback of end-to-end learning: The learned features and information are implicitly…
In this paper we show that bisymmetry, which is an algebraic property, has a regularity improving feature. More precisely, we prove that every bisymmetric, partially strictly monotonic, reflexive and symmetric function $F:I^2\to I$ is…
In this work, some regularity properties of mild solutions for a class of stochastic linear functional differential equations driven by infinite dimensional Wiener processes are considered. In terms of retarded fundamental solutions, we…
We construct deterministic solutions to the Helmholtz equation in $\mathbb{R}^m$ which behave accordingly to the Random Wave Model. We then find the number of their nodal domains, their nodal volume and the topologies and nesting trees of…
We compute accurately the golden critical invariant circles of several area-preserving twist maps of the cylinder. We define some functions related to the invariant circle and to the dynamics of the map restricted to the circle (for…
We introduce a wide family of stochastic processes that are obtained as sums of self-similar localized "waveforms" with multiplicative intensity in the spirit of the Richardson cascade picture of turbulence. We establish the convergence and…
We investigate regularizations of distributional sections of vector bundles by means of nets of smooth sections that preserve the main regularity properties of the original distributions (singular support, wavefront set, Sobolev…
We study the asymptotic behavior of wavelet coefficients of random processes with long memory. These processes may be stationary or not and are obtained as the output of non--linear filter with Gaussian input. The wavelet coefficients that…
We investigate explicit functions that can produce truly random numbers. We use the analytical properties of the explicit functions to show that certain class of autonomous dynamical systems can generate random dynamics. This dynamics…
We discuss the statistical properties of the volume of the nodal set of wave function for two paradigmatic model systems which we consider in arbitrary dimension $s\ge 2$: the cuboid as a paradigm for a regular shape with separable wave…
Discrete wavelet-based methods promise to emerge as an excellent framework for the non-perturbative analysis of quantum field theories. In this work, we investigate aspects of renormalization in theories analyzed using wavelet-based…
Let $A = \{a_{1},a_{2},\dots{}\}$ $(a_{1} < a_{2} < \dots{})$ be an infinite sequence of nonnegative integers, and let $R_{A,2}(n)$ denote the number of solutions of $a_{x}+a_{y}=n$ $(a_{x},a_{y}\in A)$. P. Erd\H{o}s, A. S\'ark\"ozy and V.…
We study the effects of rounding on the moments of random variables. Specifically, given a random variable $X$ and its rounded counterpart $\operatorname{rd}(X)$, we study $|\mathbb{E}[X^k] - \mathbb{E}[\operatorname{rd}(X)^{k}]|$ for…
We show that generic H\"older continuous functions are $\rho$-irregular. The property of $\rho$-irregularity has been first introduced by Catellier and Gubinelli (Stoc. Proc. Appl. 126, 2016) and plays a key role in the study of…
We present new types of regularity for nonlinear generalized functions, based on the notion of regular growth with respect to the regularizing parameter of Colombeau's simplified model. This generalizes the notion of G^{\infty }-regularity…