相关论文: On certain positive integer sequences

We use elementary arguments to prove results on the order of magnitude of certain sums concerning the gcd's and lcm's of $k$ positive integers, where $k\ge 2$ is fixed. We refine and generalize an asymptotic formula of Bordell\`{e}s (2007),…

We study pairs and m--tuples of compositions of a positive integer n with parts restricted to a subset P of positive integers. We obtain some exact enumeration results for the number of tuples of such compositions having the same number of…

This is a survey article on some recent developments in the arithmetic theory of linear algebraic groups over higher-dimensional fields, written for the Notices of the AMS.

The main purpose of this survey is to introduce an inexperienced reader to additive prime number theory and some related branches of analytic number theory. We state the main problems in the field, sketch their history and the basic…

To any finite ordered subset and any finite partition of a group a set of tuples of positive integers, named as configurations, is associated that describes the group's behavior. The present paper provides an exposition of this notion and…

The main results of this paper are generalizations some classical theorems about transversals for families of finite sets to some cases of families of infinite sets.

We survey current developments in the approximation theory of sequence modelling in machine learning. Particular emphasis is placed on classifying existing results for various model architectures through the lens of classical approximation…

We announce a number of conjectures associated with and arising from a study of primes and irrationals in $\mathbb{R}$. All are supported by numerical verification to the extent possible.

These notes derive a number of technical results on nonlinear contraction theory, a comparatively recent tool for system stability analysis. In particular, they provide new results on the preservation of contraction through system…

In this note, simple proofs of certain well-known results involving the positive square root of positive matrices are given.

An improved upper bound is obtained for the density of sequences of positive integers that contain no k-term geometric progression.

We consider the problem of sequencing a set of positive numbers. We try to find the optimal sequence to maximize the variance of its partial sums. The optimal sequence is shown to have a beautiful structure. It is interesting to note that…

This is a survey of results concerning the asymptotic equilibrium distribution of zeros of random holomorphic polynomials and holomorphic sections of high powers of a positive line bundle, as related to the authors' recent work. Our primary…

We examine the convergence properties of sequences of nonnegative real numbers that satisfy a particular class of recursive inequalities, from the perspective of proof theory and computability theory. We first establish a number of results…

This work delves into the {\it quotient of an affine semigroup by a positive integer}, exploring its intricate properties and broader implications. We unveil an {\it associated tree} that serves as a valuable tool for further analysis.…

Any power series with nonnegative coefficients has an associated family of probability distributions supported on the nonnegative integers. There is a close connection between the function theoretic properties of the power series and the…

These lecture notes provide an informal introduction to the theory of nonnegative polynomials and sums of squares. We highlight the history and some recent developments, especially the new connections with classical (complex) algebraic…

Prime number multiplet classifications and patterns are extended to negative integers. The extension from prime numbers to single prime powers is also studied. Prime number septets at equal distance are given. It is also shown that each…

The present paper studies structure of the ring of integer-valued entire functions. We characterize certain classes of prime and maximal ideals and investigate some of their properties.

The set of indices that correspond to the positive entries of a sequence of numbers is called its positivity set. In this paper, we study the density of the positivity set of a given linear recurrence sequence, that is the question of how…