We present new formulas for computing greatest common divisors (GCDs) and extracting the prime factors of semiprimes using only elementary arithmetic operations: addition, subtraction, multiplication, floored division, and exponentiation. Our GCD formula simplifies a result of Mazzanti, and is derived using Kronecker substitution techniques from our previous work. We utilize the GCD formula, along with recent developments on arithmetic terms for square roots and factorials, to derive explicit expressions for the prime factors of a semiprime $n=pq$.
{"title":"Elementary Formulas for Greatest Common Divisors and Semiprime Factors","authors":"Joseph M. Shunia","doi":"arxiv-2407.03357","DOIUrl":"https://doi.org/arxiv-2407.03357","url":null,"abstract":"We present new formulas for computing greatest common divisors (GCDs) and\u0000extracting the prime factors of semiprimes using only elementary arithmetic\u0000operations: addition, subtraction, multiplication, floored division, and\u0000exponentiation. Our GCD formula simplifies a result of Mazzanti, and is derived\u0000using Kronecker substitution techniques from our previous work. We utilize the\u0000GCD formula, along with recent developments on arithmetic terms for square\u0000roots and factorials, to derive explicit expressions for the prime factors of a\u0000semiprime $n=pq$.","PeriodicalId":501502,"journal":{"name":"arXiv - MATH - General Mathematics","volume":"156 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141570924","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}
This article aims to investigate the characteristics of (alpha, beta) Ricci Yamabe soliton (briefly (alpha, beta) (RYS)) and its spacetime. The inclusion of killing vector field and the Lorentzian metrics make the Ricci-Yamabe soliton richer and interesting. We study the cosmological and dust fluid model on (RYS) equipped with Lorentzian para Sasakian (LPS) spacetime. The cases of eta-parallel Ricci tensor and the Poisson structure have been studied on (RYS) equipped with (LPS) manifold. Gradient (RYS) equipped with (LPS) manifold also reveal. Finally, we establish an example of four-dimensional LP Sasakian manifold (LPS) that satisfy (alpha, beta) (RYS) and some results.
{"title":"Some Novel Results on (alpha, beta)-Ricci-Yamabe Soliton and its Spacetime","authors":"Pankaj Pandey, Kamakshi Sharma","doi":"arxiv-2407.05940","DOIUrl":"https://doi.org/arxiv-2407.05940","url":null,"abstract":"This article aims to investigate the characteristics of (alpha, beta) Ricci\u0000Yamabe soliton (briefly (alpha, beta) (RYS)) and its spacetime. The inclusion\u0000of killing vector field and the Lorentzian metrics make the Ricci-Yamabe\u0000soliton richer and interesting. We study the cosmological and dust fluid model\u0000on (RYS) equipped with Lorentzian para Sasakian (LPS) spacetime. The cases of\u0000eta-parallel Ricci tensor and the Poisson structure have been studied on (RYS)\u0000equipped with (LPS) manifold. Gradient (RYS) equipped with (LPS) manifold also\u0000reveal. Finally, we establish an example of four-dimensional LP Sasakian\u0000manifold (LPS) that satisfy (alpha, beta) (RYS) and some results.","PeriodicalId":501502,"journal":{"name":"arXiv - MATH - General Mathematics","volume":"13 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141570925","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}
Preferential equality is an equivalence relation on fuzzy subsets of finite sets and is a generalization of classical equality of subsets. In this paper we introduce a tightened version of the preferential equality on fuzzy subsets and derive some important combinatorial formulae for the number of such tight fuzzy subsets of an n-element set where n is a natural number. We also offer some asymptotic results
优先相等是有限集的模糊子集上的等价关系，是经典子集相等的一般化。在本文中，我们介绍了模糊子集优先相等关系的收紧版本，并给出了 n 元素集合（n 为自然数）中此类收紧模糊子集数量的一些重要组合公式。我们还提供了一些渐近结果
{"title":"Extended Equivalence of Fuzzy Sets","authors":"Venkat Murali, Sithembele Nkonkobe","doi":"arxiv-2406.16951","DOIUrl":"https://doi.org/arxiv-2406.16951","url":null,"abstract":"Preferential equality is an equivalence relation on fuzzy subsets of finite\u0000sets and is a generalization of classical equality of subsets. In this paper we\u0000introduce a tightened version of the preferential equality on fuzzy subsets and\u0000derive some important combinatorial formulae for the number of such tight fuzzy\u0000subsets of an n-element set where n is a natural number. We also offer some\u0000asymptotic results","PeriodicalId":501502,"journal":{"name":"arXiv - MATH - General Mathematics","volume":"23 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141504907","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 paper, we introduce a couple of dynamical systems that are related to the Chaos Game. We begin by discussing different methods of generating the Sierpinski gasket. Then we show how the transition from random to uniform selection reduces the Sierpinski gasket to simple periodic orbits. Next, we provide a simple formula for the attractor of each of the introduced dynamical systems based only on the contraction ratio and the regular n-gon on which the game is played. Finally, we show how the basins of attraction of a particular dynamical system can generate some novel motifs that can tile the plane.
在本文中，我们将介绍几个与混沌博弈相关的动力系统。我们首先讨论了生成西尔平斯基垫圈的不同方法。然后，我们展示了从随机选择到均匀选择的过渡如何将西尔平斯基垫圈还原为简单的周期轨道。接下来，我们仅根据收缩比和游戏所处的正则 n 冈，就为每个引入的动力学系统的吸引子提供了一个简单的公式。最后，我们展示了特定动力学系统的吸引盆地如何产生一些可以铺满平面的新颖图案。
{"title":"The Chaos Game Versus Uniform Rotation: From Sierpinski Gaskets to Periodic Orbits","authors":"Abdulrahman Abdulaziz","doi":"arxiv-2407.02506","DOIUrl":"https://doi.org/arxiv-2407.02506","url":null,"abstract":"In this paper, we introduce a couple of dynamical systems that are related to\u0000the Chaos Game. We begin by discussing different methods of generating the\u0000Sierpinski gasket. Then we show how the transition from random to uniform\u0000selection reduces the Sierpinski gasket to simple periodic orbits. Next, we\u0000provide a simple formula for the attractor of each of the introduced dynamical\u0000systems based only on the contraction ratio and the regular n-gon on which the\u0000game is played. Finally, we show how the basins of attraction of a particular\u0000dynamical system can generate some novel motifs that can tile the plane.","PeriodicalId":501502,"journal":{"name":"arXiv - MATH - General Mathematics","volume":"35 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141550838","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 consider a system of three analytic functions, two of which are known to have all their zeros on the critical line $Re (s)=sigma=1/2$. We construct inequalities which constrain the third function, $xi(s)$, on $Im(s)=0$ to lie between the other two functions, in a sandwich structure. We investigate what can be said about the location of zeros and radius of convergence of expansions of $xi(s)$, with promising results.
{"title":"Sandwiching the Riemann hypothesis","authors":"R. C. McPhedran","doi":"arxiv-2407.00060","DOIUrl":"https://doi.org/arxiv-2407.00060","url":null,"abstract":"We consider a system of three analytic functions, two of which are known to\u0000have all their zeros on the critical line $Re (s)=sigma=1/2$. We construct\u0000inequalities which constrain the third function, $xi(s)$, on $Im(s)=0$ to lie\u0000between the other two functions, in a sandwich structure. We investigate what\u0000can be said about the location of zeros and radius of convergence of expansions\u0000of $xi(s)$, with promising results.","PeriodicalId":501502,"journal":{"name":"arXiv - MATH - General Mathematics","volume":"8 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141504908","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}
Zahra Belarbi, Benaoumeur Bayour, Delfim F. M. Torres
We consider a SICA model for HIV transmission on time scales. We prove permanence of solutions and we derive sufficient conditions for the existence and uniform asymptotic stability of a unique positive almost periodic solution of the system in terms of a Lyapunov function.
{"title":"Uniform Stability of Dynamic SICA HIV Transmission Models on Time Scales","authors":"Zahra Belarbi, Benaoumeur Bayour, Delfim F. M. Torres","doi":"arxiv-2406.18596","DOIUrl":"https://doi.org/arxiv-2406.18596","url":null,"abstract":"We consider a SICA model for HIV transmission on time scales. We prove\u0000permanence of solutions and we derive sufficient conditions for the existence\u0000and uniform asymptotic stability of a unique positive almost periodic solution\u0000of the system in terms of a Lyapunov function.","PeriodicalId":501502,"journal":{"name":"arXiv - MATH - General Mathematics","volume":"57 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141531663","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 objective of this manuscript is to offer explicit expressions for diverse categories of infinite series incorporating the Fibonacci (Lucas) sequence and the Riemann zeta function. In demonstrating our findings, we will utilize conventional methodologies and integrate the Binet formulas pertinent to these sequences with generating functions that encompass the Riemann zeta function alongside established evaluations of certain series.
{"title":"Some Classes of series involving the Riemann zeta function, Fibonacci numbers and the Lucas numbers","authors":"Akerele Olofin Segun","doi":"arxiv-2406.16922","DOIUrl":"https://doi.org/arxiv-2406.16922","url":null,"abstract":"The objective of this manuscript is to offer explicit expressions for diverse\u0000categories of infinite series incorporating the Fibonacci (Lucas) sequence and\u0000the Riemann zeta function. In demonstrating our findings, we will utilize\u0000conventional methodologies and integrate the Binet formulas pertinent to these\u0000sequences with generating functions that encompass the Riemann zeta function\u0000alongside established evaluations of certain series.","PeriodicalId":501502,"journal":{"name":"arXiv - MATH - General Mathematics","volume":"12 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141531664","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}
J. F. Du Plessis, Zurab Janelidze, Bernardus A. Wessels
In point-free topology, one abstracts the poset of open subsets of a topological space, by replacing it with a frame (a complete lattice, where meet distributes over arbitrary join). In this paper we propose a similar abstraction of the posets of connected subsets in various space-like structures. The analogue of a frame is called a chainmail, which is defined as a poset admitting joins of its mails, i.e., subsets having a lower bound. The main result of the paper is an equivalence between a subcategory of the category of complete join-semilattices and the category of chainmails.
在无点拓扑学中，人们通过用一个框架（一个完整的网格，其中 meetdistributes over arbitrary join）来替代它，从而抽象出了拓扑空间的开放子集的正集（poset of open subsets of atopological space）。在本文中，我们提出了对各种空间结构中连通子集的正集进行类似抽象的方法。框架的类似物被称为链锁，它被定义为一个容许其邮件连接的集合，即具有下界的子集。本文的主要结果是完整连接半网格范畴的一个子范畴与链式邮件范畴之间的等价性。
{"title":"A Primer on Chainmails: Structures for Point-free Connectivity","authors":"J. F. Du Plessis, Zurab Janelidze, Bernardus A. Wessels","doi":"arxiv-2406.16923","DOIUrl":"https://doi.org/arxiv-2406.16923","url":null,"abstract":"In point-free topology, one abstracts the poset of open subsets of a\u0000topological space, by replacing it with a frame (a complete lattice, where meet\u0000distributes over arbitrary join). In this paper we propose a similar\u0000abstraction of the posets of connected subsets in various space-like\u0000structures. The analogue of a frame is called a chainmail, which is defined as\u0000a poset admitting joins of its mails, i.e., subsets having a lower bound. The\u0000main result of the paper is an equivalence between a subcategory of the\u0000category of complete join-semilattices and the category of chainmails.","PeriodicalId":501502,"journal":{"name":"arXiv - MATH - General Mathematics","volume":"133 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141521190","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}
Sanjar M. Abrarov, Rehan Siddiqui, Rajinder Kumar Jagpal, Brendan M. Quine
In this work, we consider the properties of the two-term Machin-like formula and develop an algorithm for computing digits of $pi$ by using its rational approximation. In this approximation, both terms are constructed by using a representation of $1/pi$ in the binary form. This approach provides the squared convergence in computing digits of $pi$ without any trigonometric functions and surd numbers. The Mathematica codes showing some examples are presented.
{"title":"A rational approximation of the two-term Machin-like formula for $π$","authors":"Sanjar M. Abrarov, Rehan Siddiqui, Rajinder Kumar Jagpal, Brendan M. Quine","doi":"arxiv-2406.08510","DOIUrl":"https://doi.org/arxiv-2406.08510","url":null,"abstract":"In this work, we consider the properties of the two-term Machin-like formula\u0000and develop an algorithm for computing digits of $pi$ by using its rational\u0000approximation. In this approximation, both terms are constructed by using a\u0000representation of $1/pi$ in the binary form. This approach provides the\u0000squared convergence in computing digits of $pi$ without any trigonometric\u0000functions and surd numbers. The Mathematica codes showing some examples are\u0000presented.","PeriodicalId":501502,"journal":{"name":"arXiv - MATH - General Mathematics","volume":"42 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141504912","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}
This article contains a proof of the fact that, under certain mild technical conditions, the action of the automorphism group of a cyclic 3-manifold cover of the type SxR, where S is a compact surface, yields a compact quotient. This result is then immediately applied to extend a theorem on the fiberings over the circle of certain compact 3-manifolds which are torus sums. As a corollary, I prove the validity of the conditional main theorem in my article titled "A fibering theorem for 3-manifolds", which appeared in the Journal of Groups, Complexity, Cryptology in 2021 and its subsequent erratum. This paper also furnishes a proof of the irreducibility of the summands of compact 3-manifolds which are torus sums and irreducible.
本文证明了这样一个事实，即在某些温和的技术条件下，SxR 类型的环状 3-manifold 盖（其中 S 是一个紧凑曲面）的自变群作用会产生一个紧凑商。这一结果立即被应用于扩展关于某些紧凑 3-manifolds的圆上纤维的定理，这些 3-manifolds是环和。作为推论，我在 2021 年发表于《群、复杂性、密码学期刊》（Journal of Groups, Complexity, Cryptology）的题为《3-manifolds 的纤维化定理》（Afibering theorem for 3-manifolds）的文章中证明了条件主定理的有效性，并随后对其进行了勘误。这篇论文还证明了紧凑 3-manifolds的和的不可还原性，这些和是环和和不可还原的。
{"title":"On the cyclic 3-manifold covers of the type surface x R","authors":"Jordan A. Sahattchieve","doi":"arxiv-2406.15457","DOIUrl":"https://doi.org/arxiv-2406.15457","url":null,"abstract":"This article contains a proof of the fact that, under certain mild technical\u0000conditions, the action of the automorphism group of a cyclic 3-manifold cover\u0000of the type SxR, where S is a compact surface, yields a compact quotient. This\u0000result is then immediately applied to extend a theorem on the fiberings over\u0000the circle of certain compact 3-manifolds which are torus sums. As a corollary,\u0000I prove the validity of the conditional main theorem in my article titled \"A\u0000fibering theorem for 3-manifolds\", which appeared in the Journal of Groups,\u0000Complexity, Cryptology in 2021 and its subsequent erratum. This paper also\u0000furnishes a proof of the irreducibility of the summands of compact 3-manifolds\u0000which are torus sums and irreducible.","PeriodicalId":501502,"journal":{"name":"arXiv - MATH - General Mathematics","volume":"24 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141504852","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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