In this study, we introduce the generalized Tribonacci hyperbolic spinors and properties of this new special numbers system by the generalized Tribonacci numbers, which are one of the most general form of the third-order recurrence sequences, generalized Tribonacci quaternions, and hyperbolic spinors, which have quite an importance and framework from mathematics to physics. This study especially improves the relations between the hyperbolic spinors and generalized Tribonacci numbers with the help of the generalized Tribonacci split quaternions. Furthermore, we examine some special cases of them and construct both new equalities and fundamental properties such as recurrence relation, Binet formula, generating function, exponential generating function, Poisson generating function, summation formulas, special determinant properties, matrix formula, and special determinant equations. Also, we give some numerical algorithms with respect to the obtained materials. In addition to these, we give a brief introduction for further research: generalized Tribonacci polynomial hyperbolic spinor sequence.
{"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}
Sections of the Hardy $Z$-function are given by $Z_N(t) := sum_{k=1}^{N} frac{cos(theta(t)-ln(k) t) }{sqrt{k}}$ for any $N in mathbb{N}$. Sections approximate the Hardy $Z$-function in two ways: (a) $2Z_{widetilde{N}(t)}(t)$ is the Hardy-Littlewood approximate functional equation (AFE) approximation for $widetilde{N}(t) = left [ sqrt{frac{t}{2 pi}} right ]$. (b) $Z_{N(t)}(t)$ is Spira's approximation for $N(t) = left [frac{t}{2} right ]$. Spira conjectured, based on experimental observations, that, contrary to the classical approximation $(a)$, approximation (b) satisfies the Riemann Hypothesis (RH) in the sense that all of its zeros are real. We present theoretical justification for Spira's conjecture, via new techniques of acceleration of series, showing that it is essentially equivalent to RH itself.
哈代 Z 元函数的截面由 $Z_N(t) := sum_{k=1}^{N}frac{cos(theta(t)-ln(k) t) }{sqrt{k}}$ 给出，适用于 mathbb{N}$ 中的任意 $N。各节以两种方式近似哈代 Z 函数：（a）$2Z_{widetilde{N}(t)}(t)$ 是哈代-利特尔伍德近似函数方程（AFE）对$widetilde{N}(t) = left [ sqrtfrac{t}{2 pi}} right ]$ 的近似。(b) $Z_{N(t)}(t)$是斯派拉对 $N(t) = left [frac{t}{2} right ]$ 的近似值。斯派拉根据实验观察推测，与经典近似值 $(a)$ 相反，近似值 (b) 满足黎曼假设（RH），即它的所有零点都是实数。我们通过新的加速数列技术，从理论上证明了斯皮拉的猜想，表明它本质上等同于黎曼假设本身。
{"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}
In his foundational book, Edwards introduced a unique "speculation" regarding the possible theoretical origins of the Riemann Hypothesis, based on the properties of the Riemann-Siegel formula. Essentially Edwards asks whether one can find a method to transition from zeros of $Z_0(t)=cos(theta(t))$, where $theta(t)$ is Riemann-Siegel theta function, to zeros of $Z(t)$, the Hardy $Z$-function. However, when applied directly to the classical Riemann-Siegel formula, it faces significant obstacles in forming a robust plausibility argument for the Riemann Hypothesis. In a recent work, we introduced an alternative to the Riemann-Siegel formula that utilizes series acceleration techniques. In this paper, we explore Edwards' speculation through the lens of our accelerated approach, which avoids many of the challenges encountered in the classical case. Our approach leads to the description of a novel variational framework for relating zeros of $Z_0(t)$ to zeros of $Z(t)$ through paths in a high-dimensional parameter space $mathcal{Z}_N$, recasting the RH as a modern non-linear optimization problem.
{"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}
Algebraic curve interpolation is described by specifying the location of N points in the plane and constructing an algebraic curve of a function f that should pass through them. In this paper, we propose a novel approach to construct the algebraic curve that interpolates a set of data (points or neighborhoods). This approach aims to search the polynomial with the smallest degree interpolating the given data. Moreover, the paper also presents an efficient method to reconstruct the algebraic curve of integer coefficients with the smallest degree and the least monomials that interpolates the provided data. The problems are converted into optimization problems and are solved via Lagrange multipliers methods and symbolic computation. Various examples are presented to illustrate the proposed approaches.
代数曲线插值是通过指定平面上 N 个点的位置，并构造一条应通过这些点的函数 f 的代数曲线来描述的。在本文中，我们提出了一种新方法来构建代数曲线，以插值一组数据（点或邻域）。这种方法旨在搜索与给定数据插值的最小度多项式。此外，本文还提出了一种高效的方法，用于重构整数系数最小、单项式最少的代数曲线，以对所给数据进行插值。这些问题被转化为优化问题，并通过拉格朗日乘法器方法和符号计算加以解决。本文列举了各种实例来说明所提出的方法。
{"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}
The smallest Euler brick, discovered by Paul Halcke, has edges $(177, 44, 240) $ and face diagonals $(125, 267, 244 ) $, generated by the primitive Pythagorean triple $ (3, 4, 5) $. Let $ (u,v,w) $ primitive Pythagorean triple, Sounderson made a generalization parameterization of the edges begin{equation*} a = vert u(4v^2 - w^2) vert, quad b = vert v(4u^2 - w^2)vert, quad c = vert 4uvw vert end{equation*} give face diagonals begin{equation*} {displaystyle d=w^{3},quad e=u(4v^{2}+w^{2}),quad f=v(4u^{2}+w^{2})} end{equation*} leads to an Euler brick. Finding other formulas that generate these primitive bricks, other than formula above, or making initial guesses that can be improved later, is the key to understanding how they are generated.
保罗-哈尔克（Paul Halcke）发现的最小欧拉砖的边长为 $(177,44,240)$，面对角线为 $(125, 267, 244 )$，由原始毕达哥拉斯三重 $ (3, 4, 5) $ 生成。让$(u,v,w)$原始勾股定理三重边，Sounderson 对边做了广义参数化：a = vert u(4v^2 - w^2) vert, quad b = vert v(4u^2 -w^2)vert, quad c = vert 4uvw vert end{equation*} 给出面对角线（begin{equation*}）。{displaystyle d=w^{3},quad e=u(4v^{2}+w^{2}),quadf=v(4u^{2}+w^{2})}end{equation*} 引出欧拉砖。除上述公式外，找到生成这些原始砖块的其他公式，或者做出可以在以后加以改进的初步猜测，是理解这些砖块是如何生成的关键。
{"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}
Using a pointwise version of Fej'{e}r's theorem about Fourier series, we obtain two formulae related to the series representations of positive integral powers of $pi$. We also check the correctness of our formulae by the applications of the R software.
利用关于傅里叶级数的 Fej'{e}r's theorem 的点式版本，我们得到了两个与 $pi$ 的正积分幂的级数表示有关的公式。我们还通过应用 R 软件检验了公式的正确性。
{"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}
We introduce and prove several new formulas for the Euler-Mascheroni Constant. This is done through the introduction of the defined E-Harmonic function, whose properties, in this paper, lead to two novel formulas, alongside a family of formulas. While the paper does introduce many new approximations, it does not exhaust the possibilities of the E-Harmonic function but provides a strong first dive into its natural conclusions. We hope that the diversity of new formulas may provide stepping stones to a proof (or disproof) of the irrationality of the Euler-Mascheroni constant.
{"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}
Following the idea of the fractional space-time Fourier transform, a linear canonical space-time transform for 16-dimensional space-time $Cell_{3,1}$-valued signals is investigated in this paper. First, the definition of the proposed linear canonical space-time transform is given, and some related properties of this transform are obtained. Second, the convolution operator and the corresponding convolution theorem are proposed. Third, the convolution theorem associated with the two-sided linear canonical space-time transform is derived.
{"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}
Category theory is the language of homological algebra, allowing us to state broadly applicable theorems and results without needing to specify the details for every instance of analogous objects. However, authors often stray from the realm of pure abstract category theory in their development of the field, leveraging the Freyd-Mitchell embedding theorem or similar results, or otherwise using set-theoretic language to augment a general categorical discussion. This paper seeks to demonstrate that - while it is not necessary for most mathematicians' purposes - a development of homological concepts can be contrived from purely categorical notions. We begin by outlining the categories we will work within, namely Abelian categories (building off additive categories). We continue to develop cohomology groups of sequences, eventually culminating in a development of right derived functors. This paper is designed to be a minimalist construction, supplying no examples or motivation beyond what is necessary to develop the ideas presented.
{"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}
In this thesis we propose and study a theory of ordered locales, a type of point-free space equipped with a preorder structure on its frame of opens. It is proved that the Stone-type duality between topological spaces and locales lifts to a new adjunction between a certain category of ordered topological spaces and the newly introduced category of ordered locales. As an application, we use these techniques to develop point-free analogues of some common aspects from the causality theory of Lorentzian manifolds. In particular, we show that so-called indecomposable past sets in a spacetime can be viewed as the points of the locale of futures. This builds towards a point-free causal boundary construction. Furthermore, we introduce a notion of causal coverage that leads naturally to a generalised notion of Grothendieck topology incorporating the order structure. From this naturally emerges a localic notion of domain of dependence.
{"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}
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