Related papers: Non-local distance functions and geometric regular…

Riemann's non-differentiable function is a classic example of a continuous function which is almost nowhere differentiable, and many results concerning its analytic regularity have been shown so far. However, it can also be given a…

This work revolves around properties and applications of functions whose nonlocal gradient, or more precisely, finite-horizon fractional gradient, vanishes. Surprisingly, in contrast to the classical local theory, we show that this class…

Using a standard linearization technique and previously obtained microlocal properties for pseudodifferential operators with smooth coefficients, the authors state results of microlocal regularity in generalized Besov spaces for solutions…

We study sets of local dimensions for self-similar measures in $\mathbb{R}$ satisfying the finite neighbour condition, which is formally stronger than the weak separation condition but satisfied in all known examples. Under a mild technical…

We present a nonlinear dynamical approximation method for time-dependent Partial Differential Equations (PDEs). The approach makes use of parametrized decoder functions, and provides a general, and principled way of understanding and…

We construct a non-local Benamou-Brenier-type transport distance on the space of stationary point processes and analyse the induced geometry. We show that our metric is a specific variant of the transport distance recently constructed in…

Many classical geometric inequalities on functionals of convex bodies depend on the dimension of the ambient space. We show that this dimension dependence may often be replaced (totally or partially) by different symmetry measures of the…

A finite metric space is called here distance degree regular if its distance degree sequence is the same for every vertex. A notion of designs in such spaces is introduced that generalizes that of designs in $Q$-polynomial distance-regular…

We consider the family of non-local and non-convex functionals proposed and investigated by J. Bourgain, H. Brezis and H.-M. Nguyen in a series of papers of the last decade. It was known that this family of functionals Gamma-converges to a…

We establish sharp geometric Holder regularity estimates for Gradient for bounded solutions of a class of fully nonlinear elliptic equations with non-homogeneous degeneracy. Such regularity estimates simplify and generalize, to some extent,…

This paper is devoted to dimensional reductions via the norm resolvent convergence. We derive explicit bounds on the resolvent difference as well as spectral asymptotics. The efficiency of our abstract tool is demonstrated by its…

We study the surface regularity of compact sets $G \subset R^n$ which is equal to the supremum of numbers $s\ge 0$ such that the measure of the set $G_{\varepsilon}\setminus G$ does not exceed $C\varepsilon^{s}, \varepsilon > 0$, where…

We consider distributions on a closed compact manifold $M$ as maps on smoothing operators. Thus spaces of certain maps between $\Psi^{-\infty}(M)\to \mathcal{C}^{\infty}(M)$ are considered as generalized functions. For any collection of…

Motivated by well-known obstacles to quantum gravity, I look for the most general geometrodynamical symmetries compatible with a reduced physical configuration space for metric gravity. I argue that they lead either to a completely static…

We study the basic computational problem of detecting approximate stationary points for continuous piecewise affine (PA) functions. Our contributions span multiple aspects, including complexity, regularity, and algorithms. Specifically, we…

We prove a global fractional differentiability result via the fractional Caccioppoli-type estimate for solutions to nonlinear elliptic problems with measure data. This work is in fact inspired by the recent paper [B. Avelin, T. Kuusi, G.…

The existence of an equidimensional morphism f with etale local sections from a regular algebraic space X to a locally noetherian normal algebraic space S of characteristic zero with excellent local rings implies that S is regular and f…

Given a collection A of holomorphic functions, we consider how to describe all the holomorphic functions locally definable from A. The notion of local definability of holomorphic functions was introduced by Wilkie, who gave a complete…

We introduce meshfree finite difference methods for approximating nonlinear elliptic operators that depend on second directional derivatives or the eigenvalues of the Hessian. Approximations are defined on unstructured point clouds, which…

Let $\mathbb{F}_p$ be a prime field, and ${\mathcal E}$ a set in $\mathbb{F}_p^2$. Let $\Delta({\mathcal E})=\{||x-y||: x,y \in {\mathcal E} \}$, the distance set of ${\mathcal E}$. In this paper, we provide a quantitative connection…