相关论文: An abstract formulation of image partition regular…

We address partition regularity problems for homogeneous quadratic equations. A consequence of our main results is that, under natural conditions on the coefficients $a,b,c$, for any finite coloring of the positive integers, there exists a…

The number of plane partitions contained in a given box was shown by MacMahon to be given by a simple product formula. By a simple bijection, this formula also enumerates lozenge tilings of hexagons of side-lengths $a,b,c,a,b,c$ (in cyclic…

Normally, in mathematics and physics, only point particle systems, which are either finite or countable, are studied. We introduce new formal mathematical object called regular continuum system of point particles (with continuum number of…

We derive an asymptotic formula for $A(n,j,r)$ the number of integer partitions of $n$ into at most $j$ parts each part $\le r$. We assume $j$ and $r$ are near their mean values. We also investigate the second largest part, the number of…

A theorem of Andrews equates partitions in which no part is repeated more than 2k-1 times to partitions in which, if j appears at least k times, all parts less than j also do so. This paper proves the theorem bijectively, with some of the…

In this paper we analyse and compare two different notions of regularity for filters on complete Boolean algebras. We also announce two results from a forthcoming paper in preparation, which provide a characterization of Keisler's order in…

We study Rado functionals and the maximal condition (first introduced by J. M. Barret et al.) in terms of the partition regularity of mixed systems of linear equations and inequalities. By strengthening the maximal Rado condition, we…

In this note we present new examples of determinantal point processes with infinitely many particles. The particles live on the half-lattice {1,2,...} or on the open half-line (0,+\infty). The main result is the computation of the…

In this article, we first investigate the partitions whose parts are congruent to $a$ or $b$ modulo $k$ with the aid of separable integer partition classes with modulus $k$ introduced by Andrews. Then, we introduce the…

We present a conjecture about partitions, with a very elementary formulation.

We focus on explaining image classifiers, taking the work of Mothilal et al. [2021] (MMTS) as our point of departure. We observe that, although MMTS claim to be using the definition of explanation proposed by Halpern [2016], they do not…

Existence and uniqueness are established for a degenerate regularization of the well-known Perona-Malik equation proposed by the first author for non-smooth initial data. The results heavily rely on the choice of appropriate functional…

In 2019, Fox, Tidor and Zhao (arXiv:1911.03427) proved an induced arithmetic removal lemma for linear patterns of complexity 1 in vector spaces over a fixed finite field. With no further assumptions on the pattern, this induced removal…

We provide a novel perspective on "regularity" as a property of representations of the Weyl algebra. We first critique a proposal by Halvorson [2004, "Complementarity of representations in quantum mechanics", Studies in History and…

We give short elementary expositions of combinatorial proofs of some variants of Euler's partitition problem that were first addressed analytically by George Andrews, and later combinatorially by others. Our methods, based on ideas from a…

In a recent interesting work [15], W.Y. He established the important partial regularity theory and the almost optimal higher regularity theory for energy minimizing harmonic almost complex structures. Based on a new observation on the…

We provide a generalization of first-order necessary conditions of optimality for infinite-dimensional optimization problems with a finite number of inequality constraints and with a finite number of inequality and equality constraints. Our…

Implicit neural representations (INR) have been recently proposed as deep learning (DL) based solutions for image compression. An image can be compressed by training an INR model with fewer weights than the number of image pixels to map the…

In this article, we introduce the notion of almost consecutive partitions. A partition is almost consecutive if every term is consecutive, with the possible exception of the smallest one. We find formulas relating to the smallest parts of…

This survey synthesizes the current state of the art on the regularity theory for solutions to the optimal partition problem. Namely, we consider non-negative, vector-valued Sobolev functions whose components have mutually disjoint support,…