Related papers: Combinatorial theorems relative to a random set

By presenting the proofs of a few sample results, we introduce the reader to the use of nonstandard analysis in aspects of combinatorics of numbers.

In recent literature there are an increasing number of papers where the forbidden sets of difference equations are computed. We review and complete different attempts to describe the forbidden set and propose new perspectives for further…

We survey the classical results of the Dirichlet Approximation Theorem.

We obtain new partial results supporting the spectral set conjecture in dimension 1.

In this paper some new ways of generalizing perfect numbers are investigated, numerical results are presented and some conjectures are established.

In this note, we give an alternate proof of the multinomial theorem using a probabilistic approach. Although the multinomial theorem is basically a combinatorial result, our proof may be simpler for a student familiar with only basic…

We present an overview of selected topics in random permutations and random partitions highlighting analogies with random matrix theory.

A combinatorial methods are used to investigate some properties of certain generalized Stirling numbers, including explicit formula and recurrence relations. Furthermore, an expression of these numbers with symmetric function is deduced.

We collect here various conjectures on congruences made by the author in a series of papers, some of which involve binary quadratic forms and other advanced theories. Part A consists of 100 unsolved conjectures of the author while…

Several new invariants for Lie algebroids have been discovered recently. We give an overview of these invariants and establish several relationships between them.

We study the combinatorics of hyperplane arrangements over arbitrary fields. Specifically, we determine in which situation an arrangement and its reduction modulo a prime number have isomorphic lattices via the use of minimal strong…

We propose a categorical setting for the study of the combinatorics of rational numbers. We find combinatorial interpretation for the Bernoulli and Euler numbers and polynomials.

Natural linear and coalgebra transformations of tensor algebras are studied. The representations of certain combinatorial groups are given. These representations are connected to natural transformations of tensor algebras and to the groups…

We call attention on the fact that recent unprecedented technological achievements, in particular in the field of quantum optics, seem to open the way to new experimental tests which might be relevant both for the foundational problems of…

New cases of the multiplicity conjecture are considered.

This paper surveys the recent attempts at leveraging machine learning to solve constrained optimization problems. It focuses on surveying the work on integrating combinatorial solvers and optimization methods with machine learning…

We prove limit theorems for sums of randomly chosen random variables conditioned on the summands. We consider several versions of the corner growth setting, including specific cases of dependence amongst the summands and summands with heavy…

We survey recent developments in the theory of achievement sets and present a substantial collection of open problems.

This is a detailed survey -- with rigorous and self-contained proofs -- of some of the basics of elementary combinatorics and algebra, including the properties of finite sums, binomial coefficients, permutations and determinants. It is…

This is a survey of old and new problems and results in additive number theory.