General Mathematics

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New submissions for Fri, 10 May 24

[1]  arXiv:2405.05264 [pdf, ps, other]

Title: Two integral representations for the logarithm of the Glaisher-Kinkelin constant

Authors: Jean-Christophe Pain

Subjects: General Mathematics (math.GM)

We present two integral representations of the logarithm of the Glaisher-Kinkelin constant. Both are based on a definite integral of $\ln[\Gamma(x + 1)]$, $\Gamma$ being the usual Gamma function. The first one relies on an integral representation of $\ln[\Gamma(x + 1)]$ due to Binet, and the second one results from the so-called Malmst\'en formula. The numerical evaluation is easier with the latter expression than with the former.

[2]  arXiv:2405.05268 [pdf, ps, other]

Title: Sums of powers of integers and the sequence A304330

Authors: José L. Cereceda

Comments: 20 pages

Subjects: General Mathematics (math.GM)

For integer $k \geq 1$, let $S_k(n)$ denote the sum of the $k$th powers of the first $n$ positive integers $1^k + 2^k + \cdots + n^k$. In this paper, we derive a new formula expressing $2^{2k}$ times $S_{2k}(n)$ as a sum of $k$ terms involving the numbers in the $k$th row of the integer sequence A304330, which is closely related to the central factorial numbers of the second kind with even indices. Furthermore, we provide an alternative proof of Knuth's formula for $S_{2k}(n)$ and show that it can equally be expressed in terms of A304330. We also deduce the corresponding formulas for the odd-indexed power sums $S_{2k-1}(n)$. Finally, we determine the Faulhaber form of $S_{2k}(n)$ and $S_{2k+1}(n)$ in terms of the sequence A304330 and the Legendre-Stirling numbers of the first kind.

[3]  arXiv:2405.05271 [pdf, ps, other]

Title: A mean value inequalities for the polygamma and zeta functions

Authors: Mohamed Bouali

Comments: No comments

Subjects: General Mathematics (math.GM)

A recently published result states inequalities of the harmonic mean of the digamma function. In this work, we prove among others results that for all positive real numbers $x\neq 1$, $$-\gamma<-\gamma H(x,1/x)<\frac{\gamma^2}{\psi\big(H(x,1/x)\big)}<\psi\Big(1/H(x,1/x)\Big)<H\Big(\psi(x), \psi(1/x)\Big),$$ $$H\Big(\zeta(x),\zeta(1/x)\Big)<-2,$$ and for all $x\in(0,1)$ $$\zeta(1/2)<H\Big(\zeta(x),\zeta(1-x)\Big)<-1,$$ $$\frac{\log 4}{1+\log 4}<H\Big(\eta(x),\eta(1-x)\Big)<(1-\sqrt 2)\zeta(1/2).$$ Here, $\psi=\Gamma'/\Gamma$ denotes the digamma function, $\gamma$ is Euler's constant, $\zeta$ is the Riemann's zeta function and $\eta$ is the Dirichlet's eta function.

[4]  arXiv:2405.05280 [pdf, ps, other]

Title: Monotonicity and inequalities for the ratios of two Bernoulli polynomials

Authors: Zhen-Hang Yang, Feng Qi

Comments: 22 pages

Subjects: General Mathematics (math.GM)

In the article, the authors establish the monotonicity of the ratios \begin{equation*} \frac{B_{2n-1}(t)}{B_{2n+1}(t)}, \quad \frac{B_{2n}(t)}{B_{2n+1}(t)},\quad \frac{B_{2m}(t)}{B_{2n}(t)},\quad \frac{B_{2n}(t)}{B_{2n-1}(t)} \end{equation*} and derive some known and new inequalities of the Bernoulli polynomials $B_n(t)$, the Bernoulli numbers $B_{2n}$, and their ratios such as $\frac{B_{2n+2}}{B_{2n}}$.

[5]  arXiv:2405.05281 [pdf, other]

Title: On Tournament Design

Authors: Leo Fried

Comments: 129 pages, advised by Professor Eric Maskin

Subjects: General Mathematics (math.GM)

A monograph on the theory of tournament design focusing on brackets and multibrackets in particular.

[6]  arXiv:2405.05283 [pdf, ps, other]

Title: The existence for the classical solution of the Navier-Stokes equations

Authors: Jianfeng Wang

Comments: arXiv admin note: substantial text overlap with arXiv:2303.16444

Subjects: General Mathematics (math.GM)

In this paper we will discuss the existence for the classical solution of the Navier-Stokes equations. First, we transform it into generalized integral equations. Next, we discuss the existence of the classical solution by Leray-Schauder degree and Sobolev space\ $H^{-m_{1}}(\Omega_{1})$.

[7]  arXiv:2405.05300 [pdf, ps, other]

Title: Sigma index in Trees with Given Degree Sequences

Authors: Jasem Hamoud, Artem Kurnosov

Comments: 3 pages, 5 figures, conference 66th All-Russia Scientific Conference MIPT

Subjects: General Mathematics (math.GM)

The sigma index in graph theory refers to a measure of the degree differences between vertices in a graph. The goal is to determine the graphs that have the maximum sigma index within certain classes of graphs. Abdo, Dimitrov, and Gutman characterized the graphs with the greatest sigma index among all connected graphs of a fixed order.

[8]  arXiv:2405.05810 [pdf, ps, other]

Title: Series involving rational, factorial and power functions

Authors: Robert Reynolds

Subjects: General Mathematics (math.GM)

This is an anthology of series involving rational, factorial, and power functions expressed in terms of special functions. New finite expansions involving quotient functions expressed in terms of the Hurwitz-Lerch zeta function are given. These results represent a new form of expressing this special function as a finite series where contour integration is required for derivation. Extended series previously known and derived are extended using differential equations and algebraic methods.

Replacements for Fri, 10 May 24

[9]  arXiv:2312.09552 (replaced) [pdf, other]

Title: The number of convex polyhedra edge-inscribed into one and edge-circumscribed around another convex polyhedron

Authors: Yagub N. Aliyev

Subjects: General Mathematics (math.GM)

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