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Mixing Numbers and Unfriendly Colorings of Graphs 混合数字和不友好的图着色
Pub Date : 2021-01-01 DOI: 10.3888/tmj.23-4
R. Cowen
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.
我们只考虑顶点着色,所以“有色图”总是指顶点着色的图。图的n着色是将顶点划分为n个不相交的子集。我们从两种颜色开始;称这些颜色为红色和蓝色。如果两个顶点由一条边连接,它们就是邻居。我们说两个颜色相同的顶点是朋友,两个颜色相反的顶点是陌生人。如果一个有色顶点v的邻居中有一半以上是v的朋友,我们就说v在一个友好的邻居中;否则,据说v住在一个不友好的社区。如果图中所有顶点的颜色相同,则每个顶点都在友好邻域中。是否存在两种着色使得每个顶点都处于不友好邻域?这个问题令人惊讶的答案是肯定的,我们将会展示。一个图的2色是不友好的,如果每个顶点都在一个不友好的邻居中,也就是说,至少有一半的邻居的颜色与它自己不同。这是一个定理,每个有限图都有不友好着色。(无限图的情况要复杂得多[1,2])。这个证明很聪明,但不是很长,我们接下来给出它。定义彩色图形的混合数为其顶点具有不同颜色的边的数量。通过连续的“翻转”来进行,也就是说,改变那些生活在友好社区中的顶点的颜色。当一个顶点翻转时,可能会改变其他顶点的邻域状态;然而,每次翻转都会增加图的混合次数。由于混合数受图中边数的限制,这个翻转过程最终必须以没有可翻转的顶点结束,也就是说,没有更多的顶点生活在友好的邻居中。
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引用次数: 0
Coverage versus Confidence 覆盖率与信心
Pub Date : 2021-01-01 DOI: 10.3888/TMJ.23-1
Peyton Cook
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.
本文旨在帮助学生理解涉及置信区间的覆盖概率的概念。使用Mathematica作为描述基于二项分布计算简单置信区间的覆盖概率的算法的语言。然后,使用更高级的函数来计算表达式的概率,从而获得覆盖概率。给出了基于二项分布的总体比例的两个置信区间,泊松分布均值的一个渐近置信区间,以及基于负二项分布的总体比例的一个渐近置信区间。
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引用次数: 0
Selected Financial Applications 选定的金融应用
Pub Date : 2021-01-01 DOI: 10.3888/tmj.23-5
Ramesh Adhikari
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.
本文展示了如何使用Mathematica的一些内置财务函数,以及如何定义对实际金融数据分析有用的新函数。主要内容包括线性规划及其在债券投资组合管理中的应用、条件风险价值最小化、时间序列分析入门、模拟、自举、稳健股票投资组合优化和人工智能。
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引用次数: 1
Numerical Contour Integration 数值轮廓积分
Pub Date : 2021-01-01 DOI: 10.3888/tmj.23-3
Erickson Tjoa
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.
我们通过设置integration和nintegration选项,利用复杂分析的强大结果,提出了轮廓积分的简单实现。因此,本文可以看作是使用现有内置工具进行数值轮廓积分的文档。我们提供了一些例子,说明如何使用这种方法对一些常用的分布(如量子场论中的Wightman函数)进行解析和数值积分。当涉及时间顺序时,我们还提供了一种近似技术,这是量子场论中计算戴森级数展开和费曼传播子中的二阶项时经常遇到的情况。我们相信我们的实现将对涉及高级或迟钝格林函数、传播子、核等的更一般的计算有用。
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引用次数: 1
Unconditional Applicability of Lehmer’s Measure to the Two-Term Machin-like Formula for π Lehmer测度对π两项类机公式的无条件适用性
Pub Date : 2020-04-23 DOI: 10.3888/tmj.23-2
S. Abrarov, R. Siddiqui, R. Jagpal, B. Quine
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.
Lehmer根据类似Machin的π公式中使用的数字betai定义了一个测度。当betai是整数时,Lehmer测度可以用来确定给定的类Machin公式对pi的计算效率。然而,由于计算很复杂,当一个或多个beta_i是有理的时,尚不清楚Lehmer的测度是否适用。在本文中,我们为π的两项类Machin公式开发了一个新的算法,作为Lehmer测度无条件适用性的例子。这种方法不涉及任何无理数,并且可以通过切线函数的Newton-Raphson迭代方法快速计算pi。
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引用次数: 4
Sectional Curvature in Riemannian Manifolds 黎曼流形中的截面曲率
Pub Date : 2020-01-01 DOI: 10.3888/TMJ.22-1
B. Healy, Elliott Fairchild, Francis Owen
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.
黎曼流形或伪黎曼流形上的度量结构完全由它的度量张量决定,它在任何给定的图中都有矩阵表示。在这个度规中编码的是截面曲率,它经常引起数学物理学家、微分几何学者和几何群理论家的兴趣。在本文中,我们提供了一个函数来计算给定黎曼流形的度量张量的截面曲率。我们还定义了一个函数来获得里奇张量,一个密切相关的对象。
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引用次数: 4
Foundations of Computational Finance 计算金融基础
Pub Date : 2020-01-01 DOI: 10.3888/tmj.22-2
R. Adhikari
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引用次数: 2
Degree versus Dimension for Rational Parametric Curves 有理参数曲线的度与维
Pub Date : 2020-01-01 DOI: 10.3888/tmj.22-3
Barry H. Dayton
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引用次数: 0
Generating Minimally Unsatisfiable Conjunctive Normal Forms 生成最小不可满足的合取范式
Pub Date : 2020-01-01 DOI: 10.3888/tmj.22-4
R. Cowen
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引用次数: 0
Structural Equation Modeling 结构方程建模
Pub Date : 2020-01-01 DOI: 10.3888/tmj.22-5
R. Oldenburg
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引用次数: 1
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