相关论文: Integer Subsets with High Volume and Low Perimeter
Large-scale eigenvalue problems arise in various fields of science and engineering and demand computationally efficient solutions. In this study, we investigate the subspace approximation for parametric linear eigenvalue problems, aiming to…
In this monograph, we prove an asymptotic approximation for integrals of probability densities over sets in finite dimensional euclidean space, which are far away from the origin (asymptotic sets). We use this approximation to investigate…
In this paper we provide new existence results for isoperimetric sets of large volume in Riemannian manifolds with nonnegative Ricci curvature and Euclidean volume growth. We find sufficient conditions for their existence in terms of the…
Let $\{U_n\}_{n \geqslant 0}$ and $\{G_m\}_{m \geqslant 0}$ be two linear recurrence sequences defined over the integers. We establish an asymptotic formula for the number of integers $c$ in the range $[-x, x]$ which can be represented as…
We study the problem of generating interesting integer sequences with a combinatorial interpretation. For this we introduce a two-step approach. In the first step, we generate first-order logic sentences which define some combinatorial…
We study extremal problems about sets of integers that do not contain sumsets with summands of prescribed size. We analyse both finite sets and infinite sequences. We also study the connections of these problems with extremal problems of…
We define the parametric closure problem, in which the input is a partially ordered set whose elements have linearly varying weights and the goal is to compute the sequence of minimum-weight lower sets of the partial order as the weights…
We study some new isoperimetric inequalities on graphs. We etablish a relation between the volume entropy (or asymptotic volume), the systole and the first Betti number of weighted graphs. We also find bounds for the volume, associated to…
This paper deals with a variation of the classical isoperimetric problem in dimension $N\ge 2$ for a two-phase piecewise constant density whose discontinuity interface is a given hyperplane. We introduce a weighted perimeter functional with…
Let t[n] be a sequence that satisfies a first order homogeneous recurrence t[n] = Q[n]*t[n-1], where Q is a polynomial with integer coefficients. The asymptotic behavior of the p-adic valuation of t[n] is described under the assumption that…
We consider several problems about pseudoprimes. First, we look at the issue of their distribution in residue classes. There is a literature on this topic in the case that the residue class is coprime to the modulus. Here we provide some…
The study of the additive volume of sets can be reduced to the case of one-dimensional sets. The exact values of the volume of extremal sets are given as a conjecture.
We propose a multi-scale analysis method for studying arithmetic properties of integer sets, such as primality. Our approach organizes information through a hierarchy of nested sequences, where each level enables a hierarchical expression…
Given a sequence composed of a limit number of characters, we try to "read" it as a "text". This involves to segment the sequence into "words". The difficulty is to distinguish good segmentation from enormous number of random ones.Aiming at…
We consider the unique infinite connected component of supercritical bond percolation on the square lattice and study the geometric properties of isoperimetric sets, i.e., sets with minimal boundary for a given volume. For almost every…
The perimeter of a measurable subset of $\mathbb R^N$ is the total variation of its characteristic function. We generalize this notion to a subset $E$ of a closed Riemannian manifold. We show that the perimeter of $E$ is the limit of the…
Consider a set of integers $\mathscr A$ having finite diameter $X$, and a system of simultaneous polynomial equations to be solved over $\mathscr A$. In many circumstances, it is known that the number of solutions of this system is…
Most of the world's digital data is currently encoded in a sequential form, and compression methods for sequences have been studied extensively. However, there are many types of non-sequential data for which good compression techniques are…
We prove that a sumset of a TE subset of (\N) (these sets can be viewed as "aperiodic" sets) with a set of positive upper density intersects a set of values of any polynomial with integer coefficients., i.e. for any (A \subset \N ) a TE…
Computation of confidence sets is central to data science and machine learning, serving as the workhorse of A/B testing and underpinning the operation and analysis of reinforcement learning algorithms. This paper studies the geometry of the…