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| viXra.org > Functions and Analysis > 2510 |
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[5] viXra:2510.0141 [pdf] submitted on 2025-10-28 11:41:01
Authors: Richard J. Mathar, Artur Jasinski
Comments: 21 pages including a Mathematica program
The manuscript contains one corrigendum to the evaluation of the generalized hypergeometric function 3F2(-1) in chapter 7.4.5 in volume 3 of the Prudnikov-Brychkov-Marichev book series, four corrigenda to the evaluation of 4F3(1) in chapter 7.5.3, sixteen corrigenda to the evaluation of 4F3(-1) in chapter 7.5.4, and six corrigenda to the evaluation of 5F4(1) in chapter 7.7.2.
Category: Functions and Analysis
[4] viXra:2510.0134 [pdf] submitted on 2025-10-27 06:01:48
Authors: Chunhua Jin, Yifu Wang
Comments: 38 Pages.
This paper is concerned with a predator-prey model in $N$-dimensional spaces ($N=1, 2, 3$), given bybegin{align*}left{begin{aligned}&frac{partial u}{partial t}=Delta u-chiablacdot(uabla v),&frac{partial v}{partial t}=Delta v+xiablacdot(vabla u),end{aligned}ight. end{align*}which describes random movement of both predator and prey species, as well as the spatial dynamics involving predators pursuing prey and prey attempting to evade predators. It is shown that any global strong solutions of the corresponding Cauchy problem converge to zero in the sense of $L^p$-norm for any $1<ple infty$, and also converge to the heat kernel with respect to $L^p$-norm for any $1le ple infty$. In particular, the decay rate thereof is optimal in the sense thatit is consistent with that of the heat equation in $mathbb R^N$ ($N=2, 3$).Undoubtedly, the global existence of solutions appears to be among the most challenging topic in the analysis of this model. Indeed even in the one-dimensional setting, only global weak solutions in a bounded domain have been successfully constructed by far. Nevertheless, to provide a comprehensive understanding of the main results, we append the conclusion on the global existence and asymptotic behavior of strong solutions, although certain smallness conditions on the initial data are required.
Category: Functions and Analysis
[3] viXra:2510.0126 [pdf] submitted on 2025-10-26 22:48:15
Authors: Amin Bagheri
Comments: 6 Pages. (Note by viXra Admin: Please cite listed scientific reference and submit article written with AI assistance to ai.viXra.org)
The number e, which is Euler’s number, has an important role in the fields of analysis, differential equations, and number theory. In 1873, Charles Hermite proved the transcendence of e , which was a fundamental achievement in the theory of transcendental numbers. In this note, we focus on the historical and mathematical context surrounding Hermite's original proof, followed by an outline of a modern, elegant, and straightforward approach proving e’s transcendence. This paper intends to present a clear, succinct, and self-explanatory assembly ideal for undergraduate students or early career researchers and shifts focus from severe technical overgeneralization to clarity of explanation and conceptual understanding.
Category: Functions and Analysis
[2] viXra:2510.0116 [pdf] replaced on 2026-04-20 04:16:16
Authors: Johan Aspegren
Comments: 13 Pages.
We give a proof for a sharp projection analogue of the slicing problem. Moreover, we show a geometric proof of the slicing problem.
Category: Functions and Analysis
[1] viXra:2510.0110 [pdf] submitted on 2025-10-23 21:00:25
Authors: Urs Frauenfelder, Joa Weber
Comments: 45 Pages. 3 figures
In this article we are considering the Hessian of the area functional in a non-Darboux chart. This does not seem to have been considered before and leads to an interesting new mathematical structure which we introduce in this article and refer to as almost extendable weak Hessian field.Our main result is a Fredholm theorem for Robbin-Salamon operators associated to non-continuous Hessians which we prove by taking advantage of this new structure.
Category: Functions and Analysis
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