Axioms

16 pages, 271 KiB

Open AccessArticle

On Ricci Solitons and Curvature Properties of Doubly Warped Products with QSMC

by Md Aquib, Vaishali Sah, Sarvesh Kumar Yadav and Jaya Upreti

Axioms 2025, 14(8), 548; https://doi.org/10.3390/axioms14080548 (registering DOI) - 22 Jul 2025

This paper explores the geometric interplay between the Levi–Civita connection and the quarter-symmetric metric connection on doubly warped product manifolds. We analyze the behavior of Ricci solitons on such manifolds, focusing on the influence of conformal and Killing vector fields within the framework of quarter-symmetric metric connections (QSMCs). Furthermore, we examine conditions under which the manifold exhibits Einstein properties, presenting new insights into Einstein-like structures in the context of doubly warped product manifolds endowed with a quarter-symmetric metric connection. Full article

(This article belongs to the Special Issue Recent Developments in Differential Geometry and Its Applications)

14 pages, 367 KiB

Open AccessArticle

A Cubic Spline Numerical Method for a Singularly Perturbed Two-Parameter Ordinary Differential Equation

by Hassan J. Al Salman, Fasika Wondimu Gelu and Ahmed A. Al Ghafli

Axioms 2025, 14(8), 547; https://doi.org/10.3390/axioms14080547 (registering DOI) - 22 Jul 2025

Abstract

This paper presents a uniformly convergent cubic spline numerical method for a singularly perturbed two-perturbation parameter ordinary differential equation. The considered differential equation is discretized using the cubic spline numerical method on a Bakhvalov-type mesh. The uniform convergence via the error analysis is established very well. The numerical findings indicate that the proposed method achieves second-order uniform convergence. Four test examples have been considered to perform numerical experimentations. Full article

(This article belongs to the Special Issue Numerical Methods and Approximation Theory)

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25 pages, 355 KiB

Open AccessArticle

Each ζ(n), 5 ≤ n ≤ 25, Is Not a Liouville Number

by Sidney A. Morris

Axioms 2025, 14(8), 546; https://doi.org/10.3390/axioms14080546 (registering DOI) - 22 Jul 2025

Abstract

We prove that for the odd integers

n \in {5, 7, 9, \dots, 25}

, the Riemann zeta value

ζ (n)

is not a Liouville number. Our method applies a general strategy pioneered by Wadim [...] Read more.

We prove that for the odd integers

n \in {5, 7, 9, \dots, 25}

, the Riemann zeta value

ζ (n)

is not a Liouville number. Our method applies a general strategy pioneered by Wadim Zudilin and D.V. Vasilyev. Specifically, we construct families of high-dimensional integrals that expand into rational linear combinations of odd zeta values, eliminate lower-order terms to isolate

ζ (n)

, and apply Nesterenko’s linear independence criterion. We verify the required asymptotic growth and decay conditions for each odd

n \leq 25

, establishing that

μ (ζ (n)) < \infty

, and thus that

ζ (n) \notin L

. This is the first unified proof covering all odd zeta values up to

ζ (25)

and highlights the structural barriers to extending the method beyond this point. We also give rigorous upper bounds on

μ (ζ (n))

for all odd integers

n \in {5, 7, \dots, 25}

, using multiple integral constructions due to Vasilyev and Zudilin, elimination of lower zeta terms, and the quantitative version of Nesterenko’s criterion. Full article

(This article belongs to the Section Algebra and Number Theory)

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Axioms, Volume 14, Issue 8 (August 2025) – 3 articles

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