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Generalized Tribonacci Hyperbolic Spinors 广义 Tribonacci 双曲旋子
arXiv - MATH - General Mathematics Pub Date : 2024-05-21 DOI: arxiv-2405.13184
Zehra İşbilir, Bahar Doğan Yazıcı, Murat Tosun
{"title":"Generalized Tribonacci Hyperbolic Spinors","authors":"Zehra İşbilir, Bahar Doğan Yazıcı, Murat Tosun","doi":"arxiv-2405.13184","DOIUrl":"https://doi.org/arxiv-2405.13184","url":null,"abstract":"In this study, we introduce the generalized Tribonacci hyperbolic spinors and\u0000properties of this new special numbers system by the generalized Tribonacci\u0000numbers, which are one of the most general form of the third-order recurrence\u0000sequences, generalized Tribonacci quaternions, and hyperbolic spinors, which\u0000have quite an importance and framework from mathematics to physics. This study\u0000especially improves the relations between the hyperbolic spinors and\u0000generalized Tribonacci numbers with the help of the generalized Tribonacci\u0000split quaternions. Furthermore, we examine some special cases of them and\u0000construct both new equalities and fundamental properties such as recurrence\u0000relation, Binet formula, generating function, exponential generating function,\u0000Poisson generating function, summation formulas, special determinant\u0000properties, matrix formula, and special determinant equations. Also, we give\u0000some numerical algorithms with respect to the obtained materials. In addition\u0000to these, we give a brief introduction for further research: generalized\u0000Tribonacci polynomial hyperbolic spinor sequence.","PeriodicalId":501502,"journal":{"name":"arXiv - MATH - General Mathematics","volume":"128 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141146268","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
On the approximation of the Hardy $Z$-function via high-order sections 论通过高阶部分逼近哈代 Z$ 函数
arXiv - MATH - General Mathematics Pub Date : 2024-05-21 DOI: arxiv-2405.12557
Yochay Jerby
{"title":"On the approximation of the Hardy $Z$-function via high-order sections","authors":"Yochay Jerby","doi":"arxiv-2405.12557","DOIUrl":"https://doi.org/arxiv-2405.12557","url":null,"abstract":"Sections of the Hardy $Z$-function are given by $Z_N(t) := sum_{k=1}^{N}\u0000frac{cos(theta(t)-ln(k) t) }{sqrt{k}}$ for any $N in mathbb{N}$. Sections\u0000approximate the Hardy $Z$-function in two ways: (a) $2Z_{widetilde{N}(t)}(t)$\u0000is the Hardy-Littlewood approximate functional equation (AFE) approximation for\u0000$widetilde{N}(t) = left [ sqrt{frac{t}{2 pi}} right ]$. (b) $Z_{N(t)}(t)$\u0000is Spira's approximation for $N(t) = left [frac{t}{2} right ]$. Spira\u0000conjectured, based on experimental observations, that, contrary to the\u0000classical approximation $(a)$, approximation (b) satisfies the Riemann\u0000Hypothesis (RH) in the sense that all of its zeros are real. We present\u0000theoretical justification for Spira's conjecture, via new techniques of\u0000acceleration of series, showing that it is essentially equivalent to RH itself.","PeriodicalId":501502,"journal":{"name":"arXiv - MATH - General Mathematics","volume":"44 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141146243","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
On Edwards' Speculation and a New Variational Method for the Zeros of the $Z$-Function 论爱德华兹的推测和$Z$函数零点的新变量法
arXiv - MATH - General Mathematics Pub Date : 2024-05-21 DOI: arxiv-2405.12657
Yochay Jerby
{"title":"On Edwards' Speculation and a New Variational Method for the Zeros of the $Z$-Function","authors":"Yochay Jerby","doi":"arxiv-2405.12657","DOIUrl":"https://doi.org/arxiv-2405.12657","url":null,"abstract":"In his foundational book, Edwards introduced a unique \"speculation\" regarding\u0000the possible theoretical origins of the Riemann Hypothesis, based on the\u0000properties of the Riemann-Siegel formula. Essentially Edwards asks whether one\u0000can find a method to transition from zeros of $Z_0(t)=cos(theta(t))$, where\u0000$theta(t)$ is Riemann-Siegel theta function, to zeros of $Z(t)$, the Hardy\u0000$Z$-function. However, when applied directly to the classical Riemann-Siegel\u0000formula, it faces significant obstacles in forming a robust plausibility\u0000argument for the Riemann Hypothesis. In a recent work, we introduced an alternative to the Riemann-Siegel formula\u0000that utilizes series acceleration techniques. In this paper, we explore\u0000Edwards' speculation through the lens of our accelerated approach, which avoids\u0000many of the challenges encountered in the classical case. Our approach leads to\u0000the description of a novel variational framework for relating zeros of $Z_0(t)$\u0000to zeros of $Z(t)$ through paths in a high-dimensional parameter space\u0000$mathcal{Z}_N$, recasting the RH as a modern non-linear optimization problem.","PeriodicalId":501502,"journal":{"name":"arXiv - MATH - General Mathematics","volume":"67 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141146138","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Algebraic Curve Interpolation for Intervals via Symbolic-Numeric Computation 通过符号-数字计算进行区间代数曲线插值
arXiv - MATH - General Mathematics Pub Date : 2024-05-20 DOI: arxiv-2407.07095
Lydia Dehbi, Zhengfeng Yang, Chao Peng, Yaochen Xu, Zhenbing Zeng
{"title":"Algebraic Curve Interpolation for Intervals via Symbolic-Numeric Computation","authors":"Lydia Dehbi, Zhengfeng Yang, Chao Peng, Yaochen Xu, Zhenbing Zeng","doi":"arxiv-2407.07095","DOIUrl":"https://doi.org/arxiv-2407.07095","url":null,"abstract":"Algebraic curve interpolation is described by specifying the location of N\u0000points in the plane and constructing an algebraic curve of a function f that\u0000should pass through them. In this paper, we propose a novel approach to\u0000construct the algebraic curve that interpolates a set of data (points or\u0000neighborhoods). This approach aims to search the polynomial with the smallest\u0000degree interpolating the given data. Moreover, the paper also presents an\u0000efficient method to reconstruct the algebraic curve of integer coefficients\u0000with the smallest degree and the least monomials that interpolates the provided\u0000data. The problems are converted into optimization problems and are solved via\u0000Lagrange multipliers methods and symbolic computation. Various examples are\u0000presented to illustrate the proposed approaches.","PeriodicalId":501502,"journal":{"name":"arXiv - MATH - General Mathematics","volume":"68 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141584703","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Primitive Euler brick generator 原始欧拉砖生成器
arXiv - MATH - General Mathematics Pub Date : 2024-05-20 DOI: arxiv-2405.13061
Djamel Himane
{"title":"Primitive Euler brick generator","authors":"Djamel Himane","doi":"arxiv-2405.13061","DOIUrl":"https://doi.org/arxiv-2405.13061","url":null,"abstract":"The smallest Euler brick, discovered by Paul Halcke, has edges $(177, 44,\u0000240) $ and face diagonals $(125, 267, 244 ) $, generated by the primitive\u0000Pythagorean triple $ (3, 4, 5) $. Let $ (u,v,w) $ primitive Pythagorean triple,\u0000Sounderson made a generalization parameterization of the edges\u0000begin{equation*} a = vert u(4v^2 - w^2) vert, quad b = vert v(4u^2 -\u0000w^2)vert, quad c = vert 4uvw vert end{equation*} give face diagonals\u0000begin{equation*} {displaystyle d=w^{3},quad e=u(4v^{2}+w^{2}),quad\u0000f=v(4u^{2}+w^{2})} end{equation*} leads to an Euler brick. Finding other\u0000formulas that generate these primitive bricks, other than formula above, or\u0000making initial guesses that can be improved later, is the key to understanding\u0000how they are generated.","PeriodicalId":501502,"journal":{"name":"arXiv - MATH - General Mathematics","volume":"59 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141146260","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Series representations of positive integral powers of pi pi 的正积分幂的数列表示法
arXiv - MATH - General Mathematics Pub Date : 2024-05-19 DOI: arxiv-2405.12248
Mingzhou Xu
{"title":"Series representations of positive integral powers of pi","authors":"Mingzhou Xu","doi":"arxiv-2405.12248","DOIUrl":"https://doi.org/arxiv-2405.12248","url":null,"abstract":"Using a pointwise version of Fej'{e}r's theorem about Fourier series, we\u0000obtain two formulae related to the series representations of positive integral\u0000powers of $pi$. We also check the correctness of our formulae by the\u0000applications of the R software.","PeriodicalId":501502,"journal":{"name":"arXiv - MATH - General Mathematics","volume":"20 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141146264","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
A Family of New Formulas for the Euler-Mascheroni Constant 欧拉-马切洛尼常数的一系列新公式
arXiv - MATH - General Mathematics Pub Date : 2024-05-18 DOI: arxiv-2405.12246
Noah Ripke
{"title":"A Family of New Formulas for the Euler-Mascheroni Constant","authors":"Noah Ripke","doi":"arxiv-2405.12246","DOIUrl":"https://doi.org/arxiv-2405.12246","url":null,"abstract":"We introduce and prove several new formulas for the Euler-Mascheroni\u0000Constant. This is done through the introduction of the defined E-Harmonic\u0000function, whose properties, in this paper, lead to two novel formulas,\u0000alongside a family of formulas. While the paper does introduce many new\u0000approximations, it does not exhaust the possibilities of the E-Harmonic\u0000function but provides a strong first dive into its natural conclusions. We hope\u0000that the diversity of new formulas may provide stepping stones to a proof (or\u0000disproof) of the irrationality of the Euler-Mascheroni constant.","PeriodicalId":501502,"journal":{"name":"arXiv - MATH - General Mathematics","volume":"29 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141146133","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Linear canonical space-time transform and convolution theorems 线性时空变换和卷积定理
arXiv - MATH - General Mathematics Pub Date : 2024-05-16 DOI: arxiv-2405.10990
Yi-Qiao Xu, Bing-Zhao Li
{"title":"Linear canonical space-time transform and convolution theorems","authors":"Yi-Qiao Xu, Bing-Zhao Li","doi":"arxiv-2405.10990","DOIUrl":"https://doi.org/arxiv-2405.10990","url":null,"abstract":"Following the idea of the fractional space-time Fourier transform, a linear\u0000canonical space-time transform for 16-dimensional space-time\u0000$Cell_{3,1}$-valued signals is investigated in this paper. First, the\u0000definition of the proposed linear canonical space-time transform is given, and\u0000some related properties of this transform are obtained. Second, the convolution\u0000operator and the corresponding convolution theorem are proposed. Third, the\u0000convolution theorem associated with the two-sided linear canonical space-time\u0000transform is derived.","PeriodicalId":501502,"journal":{"name":"arXiv - MATH - General Mathematics","volume":"623 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141146261","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
A Categorical Development of Right Derived Functors 右衍生函数的分类发展
arXiv - MATH - General Mathematics Pub Date : 2024-05-12 DOI: arxiv-2405.10332
Skyler Marks
{"title":"A Categorical Development of Right Derived Functors","authors":"Skyler Marks","doi":"arxiv-2405.10332","DOIUrl":"https://doi.org/arxiv-2405.10332","url":null,"abstract":"Category theory is the language of homological algebra, allowing us to state\u0000broadly applicable theorems and results without needing to specify the details\u0000for every instance of analogous objects. However, authors often stray from the\u0000realm of pure abstract category theory in their development of the field,\u0000leveraging the Freyd-Mitchell embedding theorem or similar results, or\u0000otherwise using set-theoretic language to augment a general categorical\u0000discussion. This paper seeks to demonstrate that - while it is not necessary\u0000for most mathematicians' purposes - a development of homological concepts can\u0000be contrived from purely categorical notions. We begin by outlining the\u0000categories we will work within, namely Abelian categories (building off\u0000additive categories). We continue to develop cohomology groups of sequences,\u0000eventually culminating in a development of right derived functors. This paper\u0000is designed to be a minimalist construction, supplying no examples or\u0000motivation beyond what is necessary to develop the ideas presented.","PeriodicalId":501502,"journal":{"name":"arXiv - MATH - General Mathematics","volume":"26 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141146132","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Towards Point-Free Spacetimes 迈向无点时空
arXiv - MATH - General Mathematics Pub Date : 2024-05-10 DOI: arxiv-2406.15406
Nesta van der Schaaf
{"title":"Towards Point-Free Spacetimes","authors":"Nesta van der Schaaf","doi":"arxiv-2406.15406","DOIUrl":"https://doi.org/arxiv-2406.15406","url":null,"abstract":"In this thesis we propose and study a theory of ordered locales, a type of\u0000point-free space equipped with a preorder structure on its frame of opens. It\u0000is proved that the Stone-type duality between topological spaces and locales\u0000lifts to a new adjunction between a certain category of ordered topological\u0000spaces and the newly introduced category of ordered locales. As an application, we use these techniques to develop point-free analogues of\u0000some common aspects from the causality theory of Lorentzian manifolds. In\u0000particular, we show that so-called indecomposable past sets in a spacetime can\u0000be viewed as the points of the locale of futures. This builds towards a\u0000point-free causal boundary construction. Furthermore, we introduce a notion of\u0000causal coverage that leads naturally to a generalised notion of Grothendieck\u0000topology incorporating the order structure. From this naturally emerges a\u0000localic notion of domain of dependence.","PeriodicalId":501502,"journal":{"name":"arXiv - MATH - General Mathematics","volume":"237 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141521192","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
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