Related papers: Rectifiability and approximate differentiability o…

We identify a large class of constant (complex) coefficient, second order elliptic systems for which the Dirichlet problem in the upper-half space with data in $L^p$-based Sobolev spaces, $1<p<\infty$, of arbitrary smoothness $\ell$, is…

We consider mappings satisfying an upper bound for the distortion of families of curves. We establish lower bounds for the distortion of distances under such mappings. As applications, we obtain theorems on the discreteness of the limit…

For a field $\mathbb{F}$ and integers $d$ and $k$, a set ${\cal A} \subseteq \mathbb{F}^d$ is called $k$-nearly orthogonal if its members are non-self-orthogonal and every $k+1$ vectors of ${\cal A}$ include an orthogonal pair. We prove…

We develop a theory of higher order structures in compact abelian groups. In the frame of this theory we prove general inverse theorems and regularity lemmas for Gowers's uniformity norms. We put forward an algebraic interpretation of the…

The notion of pointwise differentials for distributions is a way to extract local information of distributions by rescaling the distribution at a point. In this paper, we study the pointwise differentials for distributions corresponding to…

This paper considers the arbitrary-proportional finite-set-partitioning problem which involves partitioning a finite set into multiple subsets with respect to arbitrary nonnegative proportions. This is the core art of many fundamental…

A comprehensive approach to Sobolev-type embeddings, involving arbitrary rearrangement- invariant norms on the entire Euclidean space R^n, is offered. In particular, the optimal target space in any such embedding is exhibited. Crucial in…

We derive a new bound on the effectiveness of the Petz map as a universal recovery channel in approximate quantum error correction using the second sandwiched R\'{e}nyi relative entropy $\tilde{D}_{2}$. For large Hilbert spaces, our bound…

In semidefinite programming (SDP), unlike in linear programming, Farkas' lemma may fail to prove infeasibility. Here we obtain an exact, short certificate of infeasibility in SDP by an elementary approach: we reformulate any semidefinite…

The purpose of this paper is to provide tools for analyzing the compactness of sequences in Sobolev spaces, in particular if the sequence gets mapped onto a compact set by some nonlinear operator. Here, our focus lies on a very general…

We give, for some Borel sets of a product of two Polish spaces, including the Borel sets with countable sections, a Hurewicz-like characterization of those which cannot become a transfinite difference of open sets by changing the two Polish…

We establish upper bounds for the size of two-distance sets in Euclidean space and spherical two-distance sets. The main recipe for obtaining upper bounds is the spectral method. We construct Seidel matrices to encode the distance relations…

We present a novel approach for high-order accurate numerical differentiation on unstructured meshes of quadrilateral elements. To differentiate a given function, an auxiliary function with greater smoothness properties is defined which…

In this paper, we study an unconventional but practically meaningful reversibility problem of commonly used image filters. We broadly define filters as operations to smooth images or to produce layers via global or local algorithms. And we…

Hereditary algebras are quasi-hereditary with respect to any adapted partial order on the indexing set of the isomorphism classes of their simple modules. For any adapted partial order on $\{1,\dots, n\}$, we compute the quiver and…

We demonstrate that the set $L^\infty(X, [-1,1])$ of all measurable functions over a Borel measure space $(X, \mathcal B, \mu )$ with values in the unit interval is typically non-polyhedric when interpreted as a subset of a dual space. Our…

For an algebraic number $\alpha$ we consider the orders of the reductions of $\alpha$ in finite fields. In the case where $\alpha$ is an integer, it is known by the work on Artin's primitive root conjecture that the order is "almost always…

We prove level-by-level upper and lower bounds on the strength of determinacy for finite differences of sets in the hyperarithmetical hierarchy in terms of subsystems of finite-and transfinite-order arithmetic, extending the…

We develop a new framework for generalizing approximation algorithms from the structural graph algorithm literature so that they apply to graphs somewhat close to that class (a scenario we expect is common when working with real-world…

We introduce novel mathematical and computational tools to develop a complete algorithm for computing the set of non-properness of polynomials maps in the plane. In particular, this set, which we call \emph{the Jelonek set}, is a subset of…