{"title":"Motion of a Spinning Top","authors":"J. Vrbik","doi":"10.3888/TMJ.14-4","DOIUrl":"https://doi.org/10.3888/TMJ.14-4","url":null,"abstract":"A quaternion is a four-dimensional quantity consisting of a scalar, say A, and a threedimensional vector a, collectively denoted A a HA, aL. Addition of two quaternions is component-wise, (1) HA, aLÅ⊕ HB, bL = HA+ B, a+ bL, (we do not need to add quaternions in this article). Their multiplication follows the rule (2) HA, aLÄ⊗ HB, bL = HA Ba ÿ b , A b+ B aaäbL. It is important to note that such multiplication is associative (even though noncommutative). This can be verified by the following. 8A_, a_<Î8B_, b_< := 8A B a.b, A b + B a aäb< êê TrigReduce êê Simplify H8A, 8a1, a2, a3<<Î8B, 8b1, b2, b3<<LÎ8C, 8c1, c2, c3<< 8A, 8a1, a2, a3<<ÎH8B, 8b1, b2, b3<<Î8C, 8c1, c2, c3<<L êê Expand 80, 80, 0, 0<<","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":"69961360","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":"Evaluation of Gaussian Molecular Integrals","authors":"Minhhuy Hô, Julio M Hernández-Pérez","doi":"10.3888/TMJ.14-3","DOIUrl":"https://doi.org/10.3888/TMJ.14-3","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":"69961313","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":"The Walker Function","authors":"M. Mikhailov, A. Freire","doi":"10.3888/TMJ.14-11","DOIUrl":"https://doi.org/10.3888/TMJ.14-11","url":null,"abstract":"The quantitative description of turbulent flows is known to be severely hampered by the extremely rapid variations in the mean and higher-order statistics in the near-wall region. Some very early studies [1, 2, 3] showed that the basic structure of an attached turbulent boundary layer consists of a viscous wall layer, in which the turbulent and laminar stresses are of comparable magnitude, and a defect layer, in which the velocity profile may be expressed in terms of a small perturbation to the external flow solution [4]. Also, [1, 2, 3] showed that this structure naturally leads to a universal velocity solution that has logarithmic behavior and depends on the velocity and length scales based on the friction velocity.","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":"69961571","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":"A Toolbox for Quasirandom Simulation","authors":"M. Carter","doi":"10.3888/TMJ.13-21","DOIUrl":"https://doi.org/10.3888/TMJ.13-21","url":null,"abstract":"In the eighteenth century, Georges-Louis Leclerc, Comte de Buffon, proposed a novel method to estimate p—dropping a needle over and over again onto a wooden floor of parallel planks. The probability of a needle crossing a join in the floor is related to p. By counting the number of crosses, one can estimate this probability, and hence compute a value for p (see [1]). Buffon is said to have tried the method by tossing baguettes over his shoulder. A more direct way of estimating p is to throw darts randomly at a circular target inscribed in a square, and count the proportion that land inside the circle. These are simple examples of numerical integration by simulation.","PeriodicalId":91418,"journal":{"name":"The Mathematica journal","volume":"13 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2011-12-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"69960017","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":"Simulation of Evolutionary Dynamics in Finite Populations","authors":"B. Voelkl","doi":"10.3888/TMJ.13-8","DOIUrl":"https://doi.org/10.3888/TMJ.13-8","url":null,"abstract":"In finite populations, evolutionary dynamics can no longer be described by deterministic differential equations, but require a stochastic formulation [1]. We show how Mathematica can be used to both simulate and visualize evolutionary processes in limited populations. The Moran process is introduced as the basic stochastic model of an evolutionary process in finite populations. This model is extended to mixed populations with relative fitness differences. We combine population ecology with game theoretic ideas, simulating evolutionary games in wellmixed and structured populations.","PeriodicalId":91418,"journal":{"name":"The Mathematica journal","volume":"13 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2011-05-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"69959804","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":"Search for Hamiltonian Cycles","authors":"Csongor György Csehi, J. Tóth","doi":"10.3888/TMJ.13-7","DOIUrl":"https://doi.org/10.3888/TMJ.13-7","url":null,"abstract":"Determining whether Hamiltonian cycles exist in graphs is an NP-complete problem, so it is no wonder that the Combinatorica function HamiltonianCycle is slow for large graphs. Theorems by Dirac, Ore, Pósa, and Chvátal provide sufficient conditions that are easy to check for the existence of such cycles. This article provides Mathematica programs for those conditions, thus extending the capability of HamiltonianQ, which only tests the biconnectivity—a simple necessary condition—of a given graph. We also investigate experimentally the limiting behavior of whether the conditions are fulfilled for large random graphs. The phenomenon seen is proved as a theorem, closely related to earlier results by Karp and Pósa.","PeriodicalId":91418,"journal":{"name":"The Mathematica journal","volume":"13 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2011-05-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"69960234","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":"A New Method of Constructing Fractals and Other Graphics","authors":"J. Helmstedt","doi":"10.3888/TMJ.13-4","DOIUrl":"https://doi.org/10.3888/TMJ.13-4","url":null,"abstract":"The simplest type of Lindenmeyer or L-system can be used to construct graphics as follows. Two polygonal arcs A1 and A2 are chosen, such that the length of each line segment is an integral multiple of a fixed positive number, l, and if a line segment has length n l, then it is treated as a polygonal arc consisting of n line segments of equal length. Also, the angle between each pair of adjacent line segments is an integral multiple of a fixed angle, d. A1 is usually chosen as a single line segment or as the boundary of a regular polygon. Each line segment of A1 is replaced by a copy of A2, and then each line segment of the resulting polygonal arc is replaced by a copy of A2, and so on. The constructions are achieved by interpreting certain replacement rules for sequences as replacement rules for line segments [1].","PeriodicalId":91418,"journal":{"name":"The Mathematica journal","volume":"13 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2011-02-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"69960077","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":"Betting Two Patterns against Each Other","authors":"J. Vrbik","doi":"10.3888/TMJ.13-15","DOIUrl":"https://doi.org/10.3888/TMJ.13-15","url":null,"abstract":"We present a technique for computing the probability that a specific pattern of successes and failures is generated randomly before another such pattern, thus winning the corresponding game. The program we build for this purpose finds the mean and standard deviation of the number of trials needed to complete one round of such a game. It can be used to maximize the probability of winning a game by choosing the best possible pattern, and also by adjusting the probability of a success. Finally, we verify our theoretical results by a Monte Carlo simulation.","PeriodicalId":91418,"journal":{"name":"The Mathematica journal","volume":"13 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2011-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"69959774","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":"Sampling Distribution of ML Estimators: Cauchy Example","authors":"J. Vrbik","doi":"10.3888/TMJ.13-19","DOIUrl":"https://doi.org/10.3888/TMJ.13-19","url":null,"abstract":"p y 1 s2 + Hx mL2 , where x can have any real value. The distribution has two parameters m and s, which represent its median (the “location” parameter) and semi-interquartile deviation (the “scale” parameter), respectively. This rather unusual distribution has no mean and infinite standard deviation. The exact parameter values are usually not known, and need to be estimated by repeating the corresponding random experiment independently n times, and converting the information thus gathered into two respective estimates of m and s. The best way of doing this is by maximizing the corresponding likelihood function:","PeriodicalId":91418,"journal":{"name":"The Mathematica journal","volume":"13 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2011-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"69959950","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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