We only consider vertex colorings, so a "colored graph" always means a vertex-colored graph. An n-coloring of a graph is a partition of the vertices into n disjoint subsets. We start with 2-colorings; call the colors red and blue. Two vertices are neighbors if they are connected by an edge. We say that two vertices of the same color are friends and two vertices of opposite colors are strangers. If more than half the neighbors of a colored vertex v are friends of v, we say that v lives in a friendly neighborhood; otherwise, v is said to live in an unfriendly neighborhood. If all the vertices of the graph have the same color, every vertex lives in a friendly neighborhood. Is there a 2-coloring such that every vertex lives in an unfriendly neighborhood? The surprising answer to this question is yes, as we shall show. A 2-coloring of a graph is unfriendly if each vertex lives in an unfriendly neighborhood, that is, at least half its neighbors are colored differently from itself. It is a theorem that every finite graph has an unfriendly coloring. (The situation is much more complicated for infinite graphs [1, 2]). The proof is clever, but not very long and we give it next. Define the mixing number of a colored graph to be the number of its edges whose vertices have different colors. Proceed by successively "flipping," that is, changing the color of those vertices that live in friendly neighborhoods. When a vertex is flipped, it may change the neighborhood status of other vertices; however, each flip increases the mixing number of the graph. Since the mixing number is bounded by the number of edges in the graph, this flipping process must eventually end with no more flippable vertices, that is, no more vertices living in friendly neighborhoods.
{"title":"Mixing Numbers and Unfriendly Colorings of Graphs","authors":"R. Cowen","doi":"10.3888/tmj.23-4","DOIUrl":"https://doi.org/10.3888/tmj.23-4","url":null,"abstract":"We only consider vertex colorings, so a \"colored graph\" always means a vertex-colored graph. An n-coloring of a graph is a partition of the vertices into n disjoint subsets. We start with 2-colorings; call the colors red and blue. Two vertices are neighbors if they are connected by an edge. We say that two vertices of the same color are friends and two vertices of opposite colors are strangers. If more than half the neighbors of a colored vertex v are friends of v, we say that v lives in a friendly neighborhood; otherwise, v is said to live in an unfriendly neighborhood. If all the vertices of the graph have the same color, every vertex lives in a friendly neighborhood. Is there a 2-coloring such that every vertex lives in an unfriendly neighborhood? The surprising answer to this question is yes, as we shall show. A 2-coloring of a graph is unfriendly if each vertex lives in an unfriendly neighborhood, that is, at least half its neighbors are colored differently from itself. It is a theorem that every finite graph has an unfriendly coloring. (The situation is much more complicated for infinite graphs [1, 2]). The proof is clever, but not very long and we give it next. Define the mixing number of a colored graph to be the number of its edges whose vertices have different colors. Proceed by successively \"flipping,\" that is, changing the color of those vertices that live in friendly neighborhoods. When a vertex is flipped, it may change the neighborhood status of other vertices; however, each flip increases the mixing number of the graph. Since the mixing number is bounded by the number of edges in the graph, this flipping process must eventually end with no more flippable vertices, that is, no more vertices living in friendly neighborhoods.","PeriodicalId":91418,"journal":{"name":"The Mathematica journal","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"69962312","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 is intended to help students understand the concept of a coverage probability involving confidence intervals. Mathematica is used as a language for describing an algorithm to compute the coverage probability for a simple confidence interval based on the binomial distribution. Then, higher-level functions are used to compute probabilities of expressions in order to obtain coverage probabilities. Several examples are presented: two confidence intervals for a population proportion based on the binomial distribution, an asymptotic confidence interval for the mean of the Poisson distribution, and an asymptotic confidence interval for a population proportion based on the negative binomial distribution.
{"title":"Coverage versus Confidence","authors":"Peyton Cook","doi":"10.3888/TMJ.23-1","DOIUrl":"https://doi.org/10.3888/TMJ.23-1","url":null,"abstract":"This article is intended to help students understand the concept of a coverage probability involving confidence intervals. Mathematica is used as a language for describing an algorithm to compute the coverage probability for a simple confidence interval based on the binomial distribution. Then, higher-level functions are used to compute probabilities of expressions in order to obtain coverage probabilities. Several examples are presented: two confidence intervals for a population proportion based on the binomial distribution, an asymptotic confidence interval for the mean of the Poisson distribution, and an asymptotic confidence interval for a population proportion based on the negative binomial distribution.","PeriodicalId":91418,"journal":{"name":"The Mathematica journal","volume":"23 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"69962758","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 shows how to use some of Mathematica’s built-in financial functions and define new functions useful for the practical analysis of real-world financial data. The main topics covered are linear programming and its application in bond portfolio management, conditional value-at-risk minimization, introductory time-series analysis, simulation, bootstrapping, robust equity portfolio optimization and artificial intelligence.
{"title":"Selected Financial Applications","authors":"Ramesh Adhikari","doi":"10.3888/tmj.23-5","DOIUrl":"https://doi.org/10.3888/tmj.23-5","url":null,"abstract":"This article shows how to use some of Mathematica’s built-in financial functions and define new functions useful for the practical analysis of real-world financial data. The main topics covered are linear programming and its application in bond portfolio management, conditional value-at-risk minimization, introductory time-series analysis, simulation, bootstrapping, robust equity portfolio optimization and artificial intelligence.","PeriodicalId":91418,"journal":{"name":"The Mathematica journal","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"69962327","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 straightforward implementation of contour integration by setting options for Integrate and NIntegrate, taking advantage of powerful results in complex analysis. As such, this article can be viewed as documentation to perform numerical contour integration with the existing built-in tools. We provide examples of how this method can be used when integrating analytically and numerically some commonly used distributions, such as Wightman functions in quantum field theory. We also provide an approximating technique when time-ordering is involved, a commonly encountered scenario in quantum field theory for computing second-order terms in Dyson series expansion and Feynman propagators. We believe our implementation will be useful for more general calculations involving advanced or retarded Green’s functions, propagators, kernels and so on.
{"title":"Numerical Contour Integration","authors":"Erickson Tjoa","doi":"10.3888/tmj.23-3","DOIUrl":"https://doi.org/10.3888/tmj.23-3","url":null,"abstract":"We present a straightforward implementation of contour integration by setting options for Integrate and NIntegrate, taking advantage of powerful results in complex analysis. As such, this article can be viewed as documentation to perform numerical contour integration with the existing built-in tools. We provide examples of how this method can be used when integrating analytically and numerically some commonly used distributions, such as Wightman functions in quantum field theory. We also provide an approximating technique when time-ordering is involved, a commonly encountered scenario in quantum field theory for computing second-order terms in Dyson series expansion and Feynman propagators. We believe our implementation will be useful for more general calculations involving advanced or retarded Green’s functions, propagators, kernels and so on.","PeriodicalId":91418,"journal":{"name":"The Mathematica journal","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"69962771","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}
Lehmer defined a measure depending on numbers beta_i used in a Machin-like formula for pi. When the beta_i are integers, Lehmer's measure can be used to determine the computational efficiency of the given Machin-like formula for pi. However, because the computations are complicated, it is unclear if Lehmer's measure applies when one or more of the beta_i are rational. In this article, we develop a new algorithm for a two-term Machin-like formula for pi as an example of the unconditional applicability of Lehmer's measure. This approach does not involve any irrational numbers and may allow calculating pi rapidly by the Newton-Raphson iteration method for the tangent function.
{"title":"Unconditional Applicability of Lehmer’s Measure to the Two-Term Machin-like Formula for π","authors":"S. Abrarov, R. Siddiqui, R. Jagpal, B. Quine","doi":"10.3888/tmj.23-2","DOIUrl":"https://doi.org/10.3888/tmj.23-2","url":null,"abstract":"Lehmer defined a measure depending on numbers beta_i used in a Machin-like formula for pi. When the beta_i are integers, Lehmer's measure can be used to determine the computational efficiency of the given Machin-like formula for pi. However, because the computations are complicated, it is unclear if Lehmer's measure applies when one or more of the beta_i are rational. In this article, we develop a new algorithm for a two-term Machin-like formula for pi as an example of the unconditional applicability of Lehmer's measure. This approach does not involve any irrational numbers and may allow calculating pi rapidly by the Newton-Raphson iteration method for the tangent function.","PeriodicalId":91418,"journal":{"name":"The Mathematica journal","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-04-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41648509","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 metric structure on a Riemannian or pseudo-Riemannian manifold is entirely determined by its metric tensor, which has a matrix representation in any given chart. Encoded in this metric is the sectional curvature, which is often of interest to mathematical physicists, differential geometers and geometric group theorists alike. In this article, we provide a function to compute the sectional curvature for a Riemannian manifold given its metric tensor. We also define a function to obtain the Ricci tensor, a closely related object.
{"title":"Sectional Curvature in Riemannian Manifolds","authors":"B. Healy, Elliott Fairchild, Francis Owen","doi":"10.3888/TMJ.22-1","DOIUrl":"https://doi.org/10.3888/TMJ.22-1","url":null,"abstract":"The metric structure on a Riemannian or pseudo-Riemannian manifold is entirely determined by its metric tensor, which has a matrix representation in any given chart. Encoded in this metric is the sectional curvature, which is often of interest to mathematical physicists, differential geometers and geometric group theorists alike. In this article, we provide a function to compute the sectional curvature for a Riemannian manifold given its metric tensor. We also define a function to obtain the Ricci tensor, a closely related object.","PeriodicalId":91418,"journal":{"name":"The Mathematica journal","volume":"22 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"69962644","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}
{"title":"Degree versus Dimension for Rational Parametric Curves","authors":"Barry H. Dayton","doi":"10.3888/tmj.22-3","DOIUrl":"https://doi.org/10.3888/tmj.22-3","url":null,"abstract":"","PeriodicalId":91418,"journal":{"name":"The Mathematica journal","volume":"22 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"69962697","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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