相关论文: Two Definite Integrals That Are Definitely (and Su…

In the process of studying a conjecture of Holly M. Green and Martin W. Liebeck, we obtain two interesting identities by elementary methods, one is a combinatorial identity, and the other is a number theoretic identity.

In this paper, we establish various inequalities for some mappings that are linked with the illustrious Hermite-Hadamard integral inequality for mappings whose absolute values belong to the class K?;s m;1 and K?;s m;2.

When searching for Calabi.Yau differential equations, often different formulas for the coefficients give the same differential equation. The coefficients are usually sums (simple, double or triple) of products of binomial coefficients. This…

This article presents an equivalent formulation of the implicit complementarity problem. We demonstrate that solution of the equivalent formulation is equivalent to the solution of the implicit complementarity problem. Moreover, we provide…

We investigate the special class of formulas made up of arbitrary but finite com- binations of addition, multiplication, and exponentiation gates. The inputs to these formulas are restricted to the integral unit 1. In connection with such…

We prove a combinatorial identity relating Catalan numbers to tangent numbers arising from the study of peak algebra that was conjectured by Aliniaeifard and Li. This identity leads to the discovery of the intriguing identity $$…

This article aims to find explicit congruences between Dirichlet characters and gives various results on how to find some effectively on a computer. It ends with concrete examples putting those ideas in application.

An approach to infinite dimensional integration which unifies the case of oscillatory integrals and the case of probabilistic type integrals is presented. It provides a truly infinite dimensional construction of integrals as linear…

In this paper, we prove the conjecture that if there is an odd perfect number, then there are infinitely many of them.

We prove that five ways to define entry A086377 in the On-Line Encyclopedia of Integer Sequences do lead to the same integer sequence.

Two infinite sequences A and B of non-negative integers are called additive complements, if their sum contains all sufficiently large integers. Let $A(x)$ and $B(x)$ be the counting functions of A and B. In this paper, we extend the results…

In this article, we study special values of the Dedekind zeta function over an imaginary quadratic field. The values of the Dedekind zeta function at any even integer over any totally real number field is quite well known in literature. In…

Two photon-pair creation processes can be arranged such that the paths of the emitted photons are identical. Thereby the path information is not erased but is never born in the first place. In addition to its implications for fundamental…

Among three natural numbers there is always one which is larger than or equal to the Nim sum of the remaining two numbers. This amazing fact has many applications.

We survey combinatorial interpretations of some dozen identities for the double factorial such as, for instance, (2n-2)!! + Sum_{k=2}^{n} (2n-1)!!(2k-4)!!/(2k-1)!! = (2n-1)!!. Our methods are mostly bijective.

An algebraic integer is said large if all its real or complex embeddings have absolute value larger than $1$. An integral ideal is said \emph{large} if it admits a large generator. We investigate the notion of largeness, relating it to some…

Let $a,b$ be elements in a unital C$^*$-algebra with $0\leq a,b\leq 1$. The element $a$ is absolutely compatible with $b$ if $$\vert a - b \vert + \vert 1 - a - b \vert = 1.$$ In this note, we describe a complete list of absolutely…

We find a $q$-analog of the following symmetrical identity involving binomial coefficients $\binom{n}{m}$ and Eulerian numbers $A_{n,m}$, due to Chung, Graham and Knuth [{\it J. Comb.}, {\bf 1} (2010), 29--38]: {equation*} \sum_{k\geq…

We present a class of one-to-one matching models with perfectly transferable utility. We discuss identification and inference in these separable models, and we show how their comparative statics are readily analyzed.

We present in this work a complete session in a Mathematica notebook. The aim of this notebook is to check identities in symmetric compositions. This notebook is a complement of our work [1] and it has all the explicit computations. We…