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Clusters Produced by Placing Rhombic Triacontahedra at the Vertices of Polyhedra 在多面体的顶点放置菱形三面体所产生的簇
The Mathematica journal Pub Date : 2012-10-15 DOI: 10.3888/TMJ.14-14
S. Kabai, S. Bérczi, L. Szilassi
{"title":"Clusters Produced by Placing Rhombic Triacontahedra at the Vertices of Polyhedra","authors":"S. Kabai, S. Bérczi, L. Szilassi","doi":"10.3888/TMJ.14-14","DOIUrl":"https://doi.org/10.3888/TMJ.14-14","url":null,"abstract":"In this article we explore possible clusters of rhombic triacontahedra (RTs), usually by connecting them face to face, which happens when they are placed at the vertices of certain polyhedra. The edge length of such polyhedra is set to be twice the distance of a face of an RT from the origin (about 2.7527). The clusters thus produced can be used to build further clusters using an RT and a rhombic hexecontahedron (RH), the logo of Wolfram|Alpha. We briefly look at other kinds of connections and produce new clusters from old by using matching polyhedra instead of RTs.","PeriodicalId":91418,"journal":{"name":"The Mathematica journal","volume":"14 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2012-10-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"69961671","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}
引用次数: 0
Relativistic Motion of a Charged Particle and Pauli Algebra 带电粒子的相对论运动与泡利代数
The Mathematica journal Pub Date : 2012-05-09 DOI: 10.3888/tmj.14-10
J. Vrbik
{"title":"Relativistic Motion of a Charged Particle and Pauli Algebra","authors":"J. Vrbik","doi":"10.3888/tmj.14-10","DOIUrl":"https://doi.org/10.3888/tmj.14-10","url":null,"abstract":"An element of Pauli algebra consists of a complex scalar, say A, and a three-dimensional complex vector a, denoted A a HA, aL, which thus has eight real dimensions. In effect, this is a generalization of quaternion algebra, but with complex instead of real components. In this article, we call these elements “spinors.” A product of two spinors is a spinor defined by (1) HA, aLÄ⊗ HB, bL = HA B+ a ÿ b , A a+ B a+ i aäbL, where · and ä are the dot and cross products, respectively. Note that this multiplication is associative, implying that we do not need parentheses when multiplying three or more spinors. But multiplication is not commutative (the result depends on the order of factors). There are two important unary (single-argument) operations on spinors: the first is called a reflection (denoted A-), which changes the sign of the vector part, that is, Aa HA, -aL; the second takes the complex conjugate of A and of each component of a; it is denoted A*. Finally, just for convenience, we let A+ denote the combination of both of these, that is, HA-L* = HA*L-. Note that","PeriodicalId":91418,"journal":{"name":"The Mathematica journal","volume":"14 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2012-05-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"69961117","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}
引用次数: 2
Some Integrals Involving Symmetric-Top Eigenfunctions 涉及对称顶特征函数的若干积分
The Mathematica journal Pub Date : 2012-05-07 DOI: 10.3888/TMJ.14-12
R. Bonham
{"title":"Some Integrals Involving Symmetric-Top Eigenfunctions","authors":"R. Bonham","doi":"10.3888/TMJ.14-12","DOIUrl":"https://doi.org/10.3888/TMJ.14-12","url":null,"abstract":"Using closure over a complete set of rotational states for methane to evaluate the intensity for quasi-elastic electron scattering in the first Born approximation, a number of integrals were encountered that appear not to have been evaluated previously. Mathematica was employed to evaluate these and similar integrals, and it was discovered that in all the cases studied the results could be represented by simple formulas.","PeriodicalId":91418,"journal":{"name":"The Mathematica journal","volume":"14 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2012-05-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"69961609","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}
引用次数: 0
Perimetric Complexity of Binary Digital Images 二值数字图像的周长复杂度
The Mathematica journal Pub Date : 2012-03-12 DOI: 10.3888/TMJ.14-5
Andrew Watson
{"title":"Perimetric Complexity of Binary Digital Images","authors":"Andrew Watson","doi":"10.3888/TMJ.14-5","DOIUrl":"https://doi.org/10.3888/TMJ.14-5","url":null,"abstract":"","PeriodicalId":91418,"journal":{"name":"The Mathematica journal","volume":"14 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2012-03-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"69961375","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}
引用次数: 30
Gambler's Ruin and First Passage Time 赌徒的废墟和第一次穿越时间
The Mathematica journal Pub Date : 2012-03-12 DOI: 10.3888/TMJ.14-8
J. Vrbik
{"title":"Gambler's Ruin and First Passage Time","authors":"J. Vrbik","doi":"10.3888/TMJ.14-8","DOIUrl":"https://doi.org/10.3888/TMJ.14-8","url":null,"abstract":"Our first objective is to find the probability that our player wins the game, ending up with a+ b dollars. To do this, we have to imagine that the game has been going on for some time, and the player has reached the point of having exactly i dollars in his pocket, so that his opponent has a+ bi. Given that, we denote our player’s probability of winning the game by wi. If one more round is played, one can see that","PeriodicalId":91418,"journal":{"name":"The Mathematica journal","volume":"14 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2012-03-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"69961386","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}
引用次数: 1
On the Integral of the Product of Three Bessel Functions over an Infinite Domain 无限域上三个贝塞尔函数积的积分
The Mathematica journal Pub Date : 2012-01-01 DOI: 10.3888/TMJ.14-15
S. Auluck
{"title":"On the Integral of the Product of Three Bessel Functions over an Infinite Domain","authors":"S. Auluck","doi":"10.3888/TMJ.14-15","DOIUrl":"https://doi.org/10.3888/TMJ.14-15","url":null,"abstract":"Fourier-space representation of the partial differential equations describing nonlinear dynamics of continuous media in cylindrical geometry can be achieved using Chandrasekhar–Kendall (C–K) functions defined over infinite domain as an orthogonal basis for solenoidal vector fields and their generating function and its gradient as orthogonal bases for scalar and irrotational vector fields, respectively. All differential and integral operations involved in translating the partial differential equations into transform space are then carried out on the basis functions, leaving a set of time evolution equations, which describe the rate of change of the spectral coefficient of an evolving mode in terms of an aggregate effect of pairs of interacting modes computed as an integral over a product of spectral coefficients of two physical quantities along with a kernel, which involves the following integral:","PeriodicalId":91418,"journal":{"name":"The Mathematica journal","volume":"14 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2012-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"69961685","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}
引用次数: 13
From Population Dynamics to Partial Differential Equations 从种群动力学到偏微分方程
The Mathematica journal Pub Date : 2012-01-01 DOI: 10.3888/TMJ.14-9
M. Kerckhove
{"title":"From Population Dynamics to Partial Differential Equations","authors":"M. Kerckhove","doi":"10.3888/TMJ.14-9","DOIUrl":"https://doi.org/10.3888/TMJ.14-9","url":null,"abstract":"Differential equation models for population dynamics are now standard fare in single-variable calculus. Building on these ordinary differential equation (ODE) models provides the opportunity for a meaningful and intuitive introduction to partial differential equations (PDEs). This article illustrates PDE models for location-dependent carrying capacities, migrations, and the dispersion of a population. The PDE models themselves are built from the logistic equation with location-dependent parameters, the traveling wave equation, and the diffusion equation. The approach presented here is suitable for a single lecture, reading assignment, and exercise set in multivariable calculus. Interactive examples accompany the text and the article is designed for use as a CDF document in which some of the input can remain hidden.","PeriodicalId":91418,"journal":{"name":"The Mathematica journal","volume":"14 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2012-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"69961436","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}
引用次数: 6
Exploring Reflection and Transmission Coefficients in Elastic Media 探索弹性介质中的反射和透射系数
The Mathematica journal Pub Date : 2012-01-01 DOI: 10.3888/TMJ.14-2
D. V. Seggern
{"title":"Exploring Reflection and Transmission Coefficients in Elastic Media","authors":"D. V. Seggern","doi":"10.3888/TMJ.14-2","DOIUrl":"https://doi.org/10.3888/TMJ.14-2","url":null,"abstract":"","PeriodicalId":91418,"journal":{"name":"The Mathematica journal","volume":"14 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2012-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"69961287","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}
引用次数: 1
Soft Landing on the Moon with Mathematica 用数学软件在月球上软着陆
The Mathematica journal Pub Date : 2012-01-01 DOI: 10.3888/TMJ.14-13
M. Muresan
{"title":"Soft Landing on the Moon with Mathematica","authors":"M. Muresan","doi":"10.3888/TMJ.14-13","DOIUrl":"https://doi.org/10.3888/TMJ.14-13","url":null,"abstract":"","PeriodicalId":91418,"journal":{"name":"The Mathematica journal","volume":"14 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2012-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"69961620","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}
引用次数: 3
Mathematical Exploration of Kirkwood Gaps 柯克伍德峡谷的数学探索
The Mathematica journal Pub Date : 2012-01-01 DOI: 10.3888/TMJ.14-1
J. Vrbik
{"title":"Mathematical Exploration of Kirkwood Gaps","authors":"J. Vrbik","doi":"10.3888/TMJ.14-1","DOIUrl":"https://doi.org/10.3888/TMJ.14-1","url":null,"abstract":"We first solve the planar Kepler problem of an asteroidʼs motion, perturbed by the gravitational pull of Jupiter. Analyzing the resulting differential equations for its orbital elements, we demonstrate the mechanism for creating a gap at the 2:1 resonance (the asteroid making two orbits for Jupiterʼs one), and briefly mention the case of other resonances (3:2, 3:1, etc.). We also discuss reasons why the motion becomes chaotic at these resonances.","PeriodicalId":91418,"journal":{"name":"The Mathematica journal","volume":"14 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2012-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"69960905","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}
引用次数: 0
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