History and Overview

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New submissions for Fri, 10 May 24

[1]  arXiv:2405.05267 [pdf, ps, other]

Title: $v$-Palindromes: An Analogy to the Palindromes

Authors: Chris Bispels, Muhammet Boran, Steven J. Miller, Eliel Sosis, Daniel Tsai

Comments: 22 pages, 2 figures, 1 table. arXiv admin note: substantial text overlap with arXiv:2111.10211

Subjects: History and Overview (math.HO); Number Theory (math.NT)

Around the year 2007, one of the authors, Tsai, accidentally discovered a property of the number $198$ he saw on the license plate of a car. Namely, if we take $198$ and its reversal $891$, which have prime factorizations $198 = 2\cdot 3^2\cdot 11$ and $891 = 3^4\cdot 11$ respectively, and sum the numbers appearing in each factorization getting $2+3+2+11 = 18$ and $3+4+11 = 18$, both sums are $18$. Such numbers were later named $v$-palindromes because they can be viewed as an analogy to the usual palindromes. In this article, we introduce the concept of a $v$-palindrome in base $b$ and prove their existence for infinitely many bases. We also exhibit infinite families of $v$-palindromes in bases $p+1$ and $p^2+1$, for each odd prime $p$. Finally, we collect some conjectures and problems involving $v$-palindromes.

[2]  arXiv:2405.05270 [pdf, ps, other]

Title: Algorithmic methods of finite discrete structures. The Four Color Theorem. Theory, methods, algorithms

Authors: Sergey Kurapov, Maxim Davidovsky

Comments: 123 pages, in Ukrainian language, 140 figures, a preprint of monography

Subjects: History and Overview (math.HO); Discrete Mathematics (cs.DM); Combinatorics (math.CO)

The Four color problem is closely related to other branches of mathematics and practical applications. More than 20 of its reformulations are known, which connect this problem with problems of algebra, statistical mechanics and planning. And this is also typical for mathematics: the solution to a problem studied out of pure curiosity turns out to be useful in representing real objects and processes that are completely different in nature. Despite the published machine methods for combinatorial proof of the Four color conjecture, there is still no clear description of the mechanism for coloring a planar graph with four colors, its natural essence and its connection with the phenomenon of graph planarity. It is necessary not only to prove (preferably by deductive methods) that any planar graph can be colored with four colors, but also to show how to color it. The paper considers an approach based on the possibility of reducing a maximally flat graph to a regular flat cubic graph with its further coloring. Based on the Tate-Volynsky theorem, the vertices of a maximally flat graph can be colored with four colors, if the edges of its dual cubic graph can be colored with three colors. Considering the properties of a colored cubic graph, it can be shown that the addition of colors obeys the transformation laws of the fourth order Klein group. Using this property, it is possible to create algorithms for coloring planar graphs.

[3]  arXiv:2405.05273 [pdf, ps, other]

Title: Lovasz' Conjecture and Other Applications of Topological Methods in Discrete Mathematics

Authors: Jingsi Hou, Guangyan Huang, Sammy Suliman, Haoran Yan

Comments: 17 pages, 9 figures

Subjects: History and Overview (math.HO); Combinatorics (math.CO)

In 20th century mathematics, the field of topology, which concerns the properties of geometric objects under continuous transformation, has proved surprisingly useful in application to the study of discrete mathematics, such as combinatorics, graph theory, and theoretical computer science. In this paper, we seek to provide an introduction to the relevant topological concepts to non-specialists, as well as a selection of some existing applications to theorems in discrete mathematics.

[4]  arXiv:2405.05278 [pdf, other]

Title: The Thousand Faces of Pythagoras (As Mil Faces de Pitágoras)

Authors: André L. G. Mandolesi

Comments: in Portuguese language

Journal-ref: Revista de Matem\'atica Hip\'atia, 1(2):27--36, 2024

Subjects: History and Overview (math.HO)

The Pythagorean Theorem is one of the oldest, more famous and more useful theorems of Mathematics, and possibly the one that has had the most impact in the evolution of this and other sciences. In this article, we look at it from different perpectives, some of them uncommon. We recall some of its history, some well known applications and generalizations, other less known ones, and show it still has many surprising facets which are usually ignored.
(O Teorema de Pit\'agoras (TP) \'e um dos mais antigos, famosos e \'uteis teoremas da Matem\'atica, e possivelmente o que maior impacto teve na evolu\c{c}\~ao desta e outras ci\^encias. Neste artigo, vamos olhar para este velho conhecido de diferentes perspectivas, algumas pouco usuais. Iremos lembrar um pouco da sua hist\'oria, algumas aplica\c{c}\~oes e generaliza\c{c}\~oes bem conhecidas, outras nem tanto, e ver que ele guarda muitas facetas surpreendentes e geralmente ignoradas.)

[5]  arXiv:2405.05379 [pdf, other]

Title: Reflecting on beauty: the aesthetics of mathematical discovery

Authors: Filip D. Jevtić, Jovana Kostić, Katarina Maksimović

Subjects: History and Overview (math.HO)

Mathematical research is often motivated by the desire to reach a beautiful result or to prove it in an elegant way. Mathematician's work is thus strongly influenced by his aesthetic judgments. However, the criteria these judgments are based on remain unclear. In this article, we focus on the concept of mathematical beauty, as one of the central aesthetic concepts in mathematics. We argue that beauty in mathematics reveals connections between apparently non-related problems or areas and allows a better and wider insight into mathematical reality as a whole. We also explain the close relationship between beauty and other important notions such as depth, elegance, simplicity, fruitfulness, and others.

[6]  arXiv:2405.05720 [pdf, ps, other]

Title: Prolegomena to the Bestiary

Authors: Yang-Hui He

Comments: 10 pages

Subjects: History and Overview (math.HO); High Energy Physics - Theory (hep-th); Algebraic Geometry (math.AG)

``Calabi-Yau Manifolds: a Bestiary for Physicists'' by Tristan Hubsch in 1992 was a classic that served to introduce algebraic geometry to physicists when the first string theory revolution of 1984 - 94 brought, inter alia, the subject of Calabi-Yau manifolds to the staple of high-energy theorists. We are fortunate that a substantially expanded and updated new edition of the Bestiary will shortly appear. This brief note will serve as an afterword to the much anticipated volume.

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