The paper discusses the edge hyper-Zagreb index of a graph, which is calculated by replacing vertex degrees with edge degrees. The degree of an edge is determined by adding up the degrees of the end vertices of the edge and subtracting 2. We examine the edge hyper-Zagreb index of the Cartesian product and join of graphs, and also calculate it for organic linear Acenes molecules with the formula (C4n+2H2n+4). We establish a correlation between topological indices based on the number of rings and predict thermodynamic properties of Acenes family, such as electron affinity, bond, heat of formation and gap energy, using the Topological Indices Method (T IM).
{"title":"Exploring the Edge Hyper-Zagreb Index of Graphs: Applications and Predictions of Thermodynamic Properties for Organic Linear Acenes Molecules","authors":"Z. Aliannejadi, S. Shafiee Alamoti","doi":"arxiv-2406.16916","DOIUrl":"https://doi.org/arxiv-2406.16916","url":null,"abstract":"The paper discusses the edge hyper-Zagreb index of a graph, which is\u0000calculated by replacing vertex degrees with edge degrees. The degree of an edge\u0000is determined by adding up the degrees of the end vertices of the edge and\u0000subtracting 2. We examine the edge hyper-Zagreb index of the Cartesian product\u0000and join of graphs, and also calculate it for organic linear Acenes molecules\u0000with the formula (C4n+2H2n+4). We establish a correlation between topological\u0000indices based on the number of rings and predict thermodynamic properties of\u0000Acenes family, such as electron affinity, bond, heat of formation and gap\u0000energy, using the Topological Indices Method (T IM).","PeriodicalId":501502,"journal":{"name":"arXiv - MATH - General Mathematics","volume":"20 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141531665","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 main purpose and motivation of this article is to create a linear transformation on the polynomial ring of rational numbers. A matrix representation of this linear transformation based on standard fundamentals will be given. For some special cases of this matrix, matrix equations including inverse matrices, the Bell polynomials will be given. With the help of these equations, new formulas containing different polynomials, especially the Bernoulli polynomials, will be given. Finally, by applying the Laplace transform to the generating function for the Bernoulli polynomials, we derive some novel formulas involving the Hurwitz zeta function and infinite series.
{"title":"Formulas of special polynomials involving Bernoulli polynomials derived from matrix equations and Laplace transform","authors":"Ezgi Polat, Yilmaz Simsek","doi":"arxiv-2406.08503","DOIUrl":"https://doi.org/arxiv-2406.08503","url":null,"abstract":"The main purpose and motivation of this article is to create a linear\u0000transformation on the polynomial ring of rational numbers. A matrix\u0000representation of this linear transformation based on standard fundamentals\u0000will be given. For some special cases of this matrix, matrix equations\u0000including inverse matrices, the Bell polynomials will be given. With the help\u0000of these equations, new formulas containing different polynomials, especially\u0000the Bernoulli polynomials, will be given. Finally, by applying the Laplace\u0000transform to the generating function for the Bernoulli polynomials, we derive\u0000some novel formulas involving the Hurwitz zeta function and infinite series.","PeriodicalId":501502,"journal":{"name":"arXiv - MATH - General Mathematics","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141504853","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 following article summarizes research where theorems and their respective demonstrations are postulated based on quadratic equations with special properties given by the Pythagorean triplets and the Fibonacci sequence given the second order of equations where integer solutions are found an environment in number theory and its applications to calculus.
{"title":"Fibonacci sequence and Pythagorean triples in the composition of functions for integer solutions from certain operator","authors":"Pablo José Vega Esparza","doi":"arxiv-2405.21039","DOIUrl":"https://doi.org/arxiv-2405.21039","url":null,"abstract":"The following article summarizes research where theorems and their respective\u0000demonstrations are postulated based on quadratic equations with special\u0000properties given by the Pythagorean triplets and the Fibonacci sequence given\u0000the second order of equations where integer solutions are found an environment\u0000in number theory and its applications to calculus.","PeriodicalId":501502,"journal":{"name":"arXiv - MATH - General Mathematics","volume":"43 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141258757","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 present a range of difficult integration formulas involving Fibonacci and Lucas numbers and trigonometric functions. These formulas are often expressed in terms of special functions like the dilogarithm and Clausen's function. We also prove complements of integral identities of Dilcher (2000) and Stewart (2022). Many of our results are based on a fundamental lemma dealing with differentiation of complex-valued Fibonacci (Lucas) functions.
我们提出了一系列涉及斐波那契数、卢卡斯数和三角函数的困难积分公式。这些公式通常用稀释算术和克劳森函数等特殊函数表示。我们还证明了 Dilcher (2000) 和 Stewart (2022) 的积分等式的补全。我们的许多结果都是基于处理复值斐波那契（卢卡斯）函数微分的基本定理。
{"title":"Integration Formulas Involving Fibonacci and Lucas Numbers","authors":"Kunle Adegoke, Robert Frontczak","doi":"arxiv-2406.00064","DOIUrl":"https://doi.org/arxiv-2406.00064","url":null,"abstract":"We present a range of difficult integration formulas involving Fibonacci and\u0000Lucas numbers and trigonometric functions. These formulas are often expressed\u0000in terms of special functions like the dilogarithm and Clausen's function. We\u0000also prove complements of integral identities of Dilcher (2000) and Stewart\u0000(2022). Many of our results are based on a fundamental lemma dealing with\u0000differentiation of complex-valued Fibonacci (Lucas) functions.","PeriodicalId":501502,"journal":{"name":"arXiv - MATH - General Mathematics","volume":"31 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141259051","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 hierarchical structure of the butterfly fractal -- the Hofstader butterfly, is found to be described by an octonary tree. In this framework of building the butterfly graph, every iteration generates sextuplets of butterflies, each with a tail that is made up of an infinity of butterflies. Identifying {it butterfly with a tale} as the building block, the tree is constructed with eight generators represented by unimodular matrices with integer coefficients. This Diophantine description provides one to one mapping with the butterfly fractal, encoding the magnetic flux interval and the topological quantum numbers of every butterfly. The butterfly tree is a generalization of the ternary tree describing the set of primitive Pythagorean triplets.
{"title":"Building the Butterfly Fractal: The Eightfold Way","authors":"Indubala I Satija","doi":"arxiv-2406.00068","DOIUrl":"https://doi.org/arxiv-2406.00068","url":null,"abstract":"The hierarchical structure of the butterfly fractal -- the Hofstader\u0000butterfly, is found to be described by an octonary tree. In this framework of\u0000building the butterfly graph, every iteration generates sextuplets of\u0000butterflies, each with a tail that is made up of an infinity of butterflies.\u0000Identifying {it butterfly with a tale} as the building block, the tree is\u0000constructed with eight generators represented by unimodular matrices with\u0000integer coefficients. This Diophantine description provides one to one mapping\u0000with the butterfly fractal, encoding the magnetic flux interval and the\u0000topological quantum numbers of every butterfly. The butterfly tree is a\u0000generalization of the ternary tree describing the set of primitive Pythagorean\u0000triplets.","PeriodicalId":501502,"journal":{"name":"arXiv - MATH - General Mathematics","volume":"8 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141258748","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}
Understanding distance metrics in high-dimensional spaces is crucial for various fields such as data analysis, machine learning, and optimization. The Manhattan distance, a fundamental metric in multi-dimensional settings, measures the distance between two points by summing the absolute differences along each dimension. This study investigates the behavior of Manhattan distance as the dimensionality of the space increases, addressing the question: how does the Manhattan distance between two points change as the number of dimensions n increases?. We analyze the theoretical properties and statistical behavior of Manhattan distance through mathematical derivations and computational simulations using Python. By examining random points uniformly distributed in fixed intervals across dimensions, we explore the asymptotic behavior of Manhattan distance and validate theoretical expectations empirically. Our findings reveal that the mean and variance of Manhattan distance exhibit predictable trends as dimensionality increases, aligning closely with theoretical predictions. Visualizations of Manhattan distance distributions across varying dimensionalities offer intuitive insights into its behavior. This study contributes to the understanding of distance metrics in high-dimensional spaces, providing insights for applications requiring efficient navigation and analysis in multi-dimensional domains.
了解高维空间中的距离度量对于数据分析、机器学习和优化等多个领域至关重要。曼哈顿距离是多维环境中的一个基本度量，它通过对每个维度上的绝对差值求和来测量两点之间的距离。本研究探讨了曼哈顿距离在空间维度增加时的行为，解决了 "当维度数 n 增加时，两点间的曼哈顿距离会如何变化 "这一问题。我们通过数学推导和使用 Python 进行计算模拟，分析了曼哈顿距离的理论性质和统计行为。通过研究均匀分布在各维度固定区间的随机点，我们探索了曼哈顿距离的渐近行为，并从经验上验证了理论预期。我们的研究结果表明，随着维度的增加，曼哈顿距离的均值和方差呈现出可预测的趋势，这与理论预测非常吻合。不同维度下曼哈顿距离分布的可视化提供了对其行为的直观见解。这项研究有助于理解高维空间中的距离度量，为需要在多维领域中进行高效导航和分析的应用提供启示。
{"title":"Asymptotic behavior of the Manhattan distance in $n$-dimensions: Estimating multidimensional scenarios in empirical experiments","authors":"Ergon Cugler de Moraes Silva","doi":"arxiv-2406.15441","DOIUrl":"https://doi.org/arxiv-2406.15441","url":null,"abstract":"Understanding distance metrics in high-dimensional spaces is crucial for\u0000various fields such as data analysis, machine learning, and optimization. The\u0000Manhattan distance, a fundamental metric in multi-dimensional settings,\u0000measures the distance between two points by summing the absolute differences\u0000along each dimension. This study investigates the behavior of Manhattan\u0000distance as the dimensionality of the space increases, addressing the question:\u0000how does the Manhattan distance between two points change as the number of\u0000dimensions n increases?. We analyze the theoretical properties and statistical\u0000behavior of Manhattan distance through mathematical derivations and\u0000computational simulations using Python. By examining random points uniformly\u0000distributed in fixed intervals across dimensions, we explore the asymptotic\u0000behavior of Manhattan distance and validate theoretical expectations\u0000empirically. Our findings reveal that the mean and variance of Manhattan\u0000distance exhibit predictable trends as dimensionality increases, aligning\u0000closely with theoretical predictions. Visualizations of Manhattan distance\u0000distributions across varying dimensionalities offer intuitive insights into its\u0000behavior. This study contributes to the understanding of distance metrics in\u0000high-dimensional spaces, providing insights for applications requiring\u0000efficient navigation and analysis in multi-dimensional domains.","PeriodicalId":501502,"journal":{"name":"arXiv - MATH - General Mathematics","volume":"26 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141531666","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 the direct method, we prove the generalised Hyers-Ulam stability of the following functional equation begin{equation} phi(x+y, z+w)+phi(x-y, z-w)-2 phi(x, z)-2 phi(x, w)=0 end{equation} in modular space satisfying the Fatou property or $Delta_2$-condition.
{"title":"Stability of additive-quadratic functional equation in modular space","authors":"Abderrahman Baza, Mohamed Rossafi, Choonkil Park","doi":"arxiv-2406.15436","DOIUrl":"https://doi.org/arxiv-2406.15436","url":null,"abstract":"Using the direct method, we prove the generalised Hyers-Ulam stability of the\u0000following functional equation begin{equation} phi(x+y, z+w)+phi(x-y, z-w)-2\u0000phi(x, z)-2 phi(x, w)=0 end{equation} in modular space satisfying the Fatou\u0000property or $Delta_2$-condition.","PeriodicalId":501502,"journal":{"name":"arXiv - MATH - General Mathematics","volume":"48 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141531667","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}
Special functions like the polygamma, Hurwitz zeta, and Lerch zeta functions have sporadically been connected with the nth derivatives of trigonometric functions. We show the polylogarithm $text{Li}_s(z)$, a function of complex argument and order $z$ and $s$, encodes the nth derivatives of the cotangent, tangent, cosecant and secant functions, and their hyperbolic equivalents, at negative integer orders $s = -n$. We then show how at the same orders, the polylogarithm represents the nth application of the operator $x frac{d}{dx}$ on the inverse trigonometric and hyperbolic functions. Finally, we construct a sum relating two polylogarithms of order $-n$ to a linear combination of polylogarithms of orders $s = 0, -1, -2, ..., -n$.
{"title":"Unifying trigonometric and hyperbolic function derivatives via negative integer order polylogarithms","authors":"Andrew Ducharme","doi":"arxiv-2405.19371","DOIUrl":"https://doi.org/arxiv-2405.19371","url":null,"abstract":"Special functions like the polygamma, Hurwitz zeta, and Lerch zeta functions\u0000have sporadically been connected with the nth derivatives of trigonometric\u0000functions. We show the polylogarithm $text{Li}_s(z)$, a function of complex\u0000argument and order $z$ and $s$, encodes the nth derivatives of the cotangent,\u0000tangent, cosecant and secant functions, and their hyperbolic equivalents, at\u0000negative integer orders $s = -n$. We then show how at the same orders, the\u0000polylogarithm represents the nth application of the operator $x frac{d}{dx}$\u0000on the inverse trigonometric and hyperbolic functions. Finally, we construct a\u0000sum relating two polylogarithms of order $-n$ to a linear combination of\u0000polylogarithms of orders $s = 0, -1, -2, ..., -n$.","PeriodicalId":501502,"journal":{"name":"arXiv - MATH - General Mathematics","volume":"122 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141197274","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 (efficient and parsimonious) decomposition of higher-order tensors is a fundamental problem with numerous applications in a variety of fields. Several methods have been proposed in the literature to that end, with the Tucker and PARAFAC decompositions being the most prominent ones. Inspired by the latter, in this work we propose a multi-resolution low-rank tensor decomposition to describe (approximate) a tensor in a hierarchical fashion. The central idea of the decomposition is to recast the tensor into emph{multiple} lower-dimensional tensors to exploit the structure at different levels of resolution. The method is first explained, an alternating least squares algorithm is discussed, and preliminary simulations illustrating the potential practical relevance are provided.
{"title":"A Multi-resolution Low-rank Tensor Decomposition","authors":"Sergio Rozada, Antonio G. Marques","doi":"arxiv-2406.18560","DOIUrl":"https://doi.org/arxiv-2406.18560","url":null,"abstract":"The (efficient and parsimonious) decomposition of higher-order tensors is a\u0000fundamental problem with numerous applications in a variety of fields. Several\u0000methods have been proposed in the literature to that end, with the Tucker and\u0000PARAFAC decompositions being the most prominent ones. Inspired by the latter,\u0000in this work we propose a multi-resolution low-rank tensor decomposition to\u0000describe (approximate) a tensor in a hierarchical fashion. The central idea of\u0000the decomposition is to recast the tensor into emph{multiple}\u0000lower-dimensional tensors to exploit the structure at different levels of\u0000resolution. The method is first explained, an alternating least squares\u0000algorithm is discussed, and preliminary simulations illustrating the potential\u0000practical relevance are provided.","PeriodicalId":501502,"journal":{"name":"arXiv - MATH - General Mathematics","volume":"20 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141531668","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}
José Sanabria, Adolfo Pimienta, Semiramis Zambrano
In this manuscript the idea of soft convex structures is given and some of their properties are investigated. Also, soft convex sets, soft concave sets and soft convex hull operator are defined and their properties are studied. Moreover, the concepts of soft convexly derived operator and soft convex base are studied and their relationship to convex structures are explored.
{"title":"Soft convex structures","authors":"José Sanabria, Adolfo Pimienta, Semiramis Zambrano","doi":"arxiv-2405.19367","DOIUrl":"https://doi.org/arxiv-2405.19367","url":null,"abstract":"In this manuscript the idea of soft convex structures is given and some of\u0000their properties are investigated. Also, soft convex sets, soft concave sets\u0000and soft convex hull operator are defined and their properties are studied.\u0000Moreover, the concepts of soft convexly derived operator and soft convex base\u0000are studied and their relationship to convex structures are explored.","PeriodicalId":501502,"journal":{"name":"arXiv - MATH - General Mathematics","volume":"9 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141197278","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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