Distributions, which are the various ways of distributing a certain number of objects of different classes among a collection of targets, have been the subject of combinatorial investigations since MacMahonʼs 1917 monograph. In this paper we apply them to a simulation of superimposed random coding. Furthermore, asymptotic estimates are provided using logarithmic polynomials (related to the well-known Bell polynomials) for symbolic and numeric calculation.
{"title":"The Combinatorics of Distributions","authors":"B. Günther","doi":"10.3888/TMJ.13-10","DOIUrl":"https://doi.org/10.3888/TMJ.13-10","url":null,"abstract":"Distributions, which are the various ways of distributing a certain number of objects of different classes among a collection of targets, have been the subject of combinatorial investigations since MacMahonʼs 1917 monograph. In this paper we apply them to a simulation of superimposed random coding. Furthermore, asymptotic estimates are provided using logarithmic polynomials (related to the well-known Bell polynomials) for symbolic and numeric calculation.","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":"69959656","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 author has written a series of guided tours showing how to visualize and solve puzzles programmatically, creating animated visualizations, showcasing various programming tricks and algorithms, and using some good old-fashioned physics problemsolving strategies with an occasional foray into abstract mathematics. Here the “Free the Key” puzzle is solved and animated together with basically equivalent (read isomorphic) alternative representations.
{"title":"Maze for Free the Key Puzzle","authors":"Ken Caviness","doi":"10.3888/TMJ.13-1","DOIUrl":"https://doi.org/10.3888/TMJ.13-1","url":null,"abstract":"The author has written a series of guided tours showing how to visualize and solve puzzles programmatically, creating animated visualizations, showcasing various programming tricks and algorithms, and using some good old-fashioned physics problemsolving strategies with an occasional foray into abstract mathematics. Here the “Free the Key” puzzle is solved and animated together with basically equivalent (read isomorphic) alternative representations.","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":"69959641","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}
Consider a chain consisting of n+ 1 point-like particles of the same mass (equal to 1, by a choice of units), connected by n massless, perfectly flexible, inelastic links of equal length (also equal to 1). The chain is laid on a table top, straight and perpendicular to the edge. Then the first particle is pulled (together with the rest of the chain) gently over the edge of the table. This causes the chain to start sliding down, due to gravity (also of unit magnitude), in a frictionless manner [1]. Let us assume now that k particles have already left the table, and that their positions are defined by k angles j1, j2, ..., jk by which the first k links deviate from the vertical, and by s, the distance of the last particle to have left the table edge (jk is thus the angle of the hanging part of the corresponding link; the rest of it still lies flat on the table). Collectively, these k + 1 variables are known as generalized coordinates [2], as they fully specify the position of every particle. Now, using rectangular coordinates with the origin at the table’s edge, the x axis oriented vertically downward, and the y axis pointing horizontally, away from the table, we can compute the corresponding x and y coordinates of each particle by
{"title":"Simulating a Chain Sliding off a Desktop","authors":"J. Vrbik","doi":"10.3888/TMJ.13-3","DOIUrl":"https://doi.org/10.3888/TMJ.13-3","url":null,"abstract":"Consider a chain consisting of n+ 1 point-like particles of the same mass (equal to 1, by a choice of units), connected by n massless, perfectly flexible, inelastic links of equal length (also equal to 1). The chain is laid on a table top, straight and perpendicular to the edge. Then the first particle is pulled (together with the rest of the chain) gently over the edge of the table. This causes the chain to start sliding down, due to gravity (also of unit magnitude), in a frictionless manner [1]. Let us assume now that k particles have already left the table, and that their positions are defined by k angles j1, j2, ..., jk by which the first k links deviate from the vertical, and by s, the distance of the last particle to have left the table edge (jk is thus the angle of the hanging part of the corresponding link; the rest of it still lies flat on the table). Collectively, these k + 1 variables are known as generalized coordinates [2], as they fully specify the position of every particle. Now, using rectangular coordinates with the origin at the table’s edge, the x axis oriented vertically downward, and the y axis pointing horizontally, away from the table, we can compute the corresponding x and y coordinates of each particle by","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":"69960063","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 mixed, within-between subjects ANOVA (also called a splitplot ANOVA) is a statistical test of means commonly used in the behavioral sciences. One approach to computing this analysis is to use a corrected between-subjects ANOVA. A second approach uses the general linear model by partitioning the sum of squares and cross-product matrices. Both approaches are detailed in this article. Finally, a package called MixedDesignANOVA is introduced that runs mixed-design ANOVAs using the second approach and displays summary statistics as well as a mean plot.
{"title":"Computing Mixed-Design (Split-Plot) ANOVA","authors":"S. Chartier, D. Cousineau","doi":"10.3888/TMJ.13-17","DOIUrl":"https://doi.org/10.3888/TMJ.13-17","url":null,"abstract":"The mixed, within-between subjects ANOVA (also called a splitplot ANOVA) is a statistical test of means commonly used in the behavioral sciences. One approach to computing this analysis is to use a corrected between-subjects ANOVA. A second approach uses the general linear model by partitioning the sum of squares and cross-product matrices. Both approaches are detailed in this article. Finally, a package called MixedDesignANOVA is introduced that runs mixed-design ANOVAs using the second approach and displays summary statistics as well as a mean plot.","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":"69959340","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 describes my investigation into several basic problems regarding random walks on graphs. On several occasions, I asked myself questions which my intuition failed to answer. I guessed at an answer, and spent some time in a fruitless attempt at proving that it was correct. Out of frustration I turned to computer simulations, only to discover that my guesses were faulty. Once I had the correct answer, I was able to supply the proofs. As every mathematician knows, it is much easier to solve a problem when you know the right answer ahead of time. This presentation is deliberately informal, as it represents the record of an actual investigation that took place, rather than a crafted paper. In fact, the notebook that I used to run my experiments has become the paper, with explanatory text added and unnecessary debris removed.
{"title":"Two Basic Results Concerning Random Walks on Graphs","authors":"Greg Markowsky","doi":"10.3888/TMJ.13-16","DOIUrl":"https://doi.org/10.3888/TMJ.13-16","url":null,"abstract":"This article describes my investigation into several basic problems regarding random walks on graphs. On several occasions, I asked myself questions which my intuition failed to answer. I guessed at an answer, and spent some time in a fruitless attempt at proving that it was correct. Out of frustration I turned to computer simulations, only to discover that my guesses were faulty. Once I had the correct answer, I was able to supply the proofs. As every mathematician knows, it is much easier to solve a problem when you know the right answer ahead of time. This presentation is deliberately informal, as it represents the record of an actual investigation that took place, rather than a crafted paper. In fact, the notebook that I used to run my experiments has become the paper, with explanatory text added and unnecessary debris removed.","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":"69959325","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 theoretical model of a kinematic chain impacting granular matter is studied. The force of the granular medium acting on the chain is a linear superposition of a static (depth-dependent) resistance force and a dynamic (velocity-dependent) frictional force. This resistance force is opposed to the direction of the velocity of the immersed chain. We present two methods (one using EventLocator and the other using FixedStep) for the problem. As examples, a single and a double pendulum are simulated using different initial impact velocity conditions. We analyze how rapidly the kinematic chain impacting the granular medium slows upon collision. For the analyzed cases the kinematic chain under high impact force (higher initial velocity) comes to rest faster in the granular matter than the same body under low impact force (lower initial velocity).
{"title":"Impact of a Planar Kinematic Chain with Granular Matter","authors":"D. Marghitu, Seunghun Lee","doi":"10.3888/TMJ.13-2","DOIUrl":"https://doi.org/10.3888/TMJ.13-2","url":null,"abstract":"The theoretical model of a kinematic chain impacting granular matter is studied. The force of the granular medium acting on the chain is a linear superposition of a static (depth-dependent) resistance force and a dynamic (velocity-dependent) frictional force. This resistance force is opposed to the direction of the velocity of the immersed chain. We present two methods (one using EventLocator and the other using FixedStep) for the problem. As examples, a single and a double pendulum are simulated using different initial impact velocity conditions. We analyze how rapidly the kinematic chain impacting the granular medium slows upon collision. For the analyzed cases the kinematic chain under high impact force (higher initial velocity) comes to rest faster in the granular matter than the same body under low impact force (lower initial velocity).","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":"69959962","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 demonstrate a symbolic elimination technique to solve a nine-parameter 3D affine transformation when only three known points in both systems are given. The system of nine equations is reduced to six by subtracting the equations and eliminating the translation parameters. From these six equations, five variables are eliminated using a Grobner basis to get a quadratic univariate polynomial, from which the solution can be expressed symbolically. The main advantage of this result is that we do not need to guess initial values of the nine parameters, which is necessary in the case of the traditional solution of the nonlinear system of equations. This result can be useful in geodesy, robotics, and photogrammetry when occasionally only three known points in both systems are given or when a Gauss‐ Jacobi combinatorial solution may be required for certain reasons, for example detecting outliers by using variancecovariance matrices.
{"title":"A Symbolic Solution of a 3D Affine Transformation","authors":"B. Paláncz, Zaletnyik Piroska","doi":"10.3888/TMJ.13-9","DOIUrl":"https://doi.org/10.3888/TMJ.13-9","url":null,"abstract":"We demonstrate a symbolic elimination technique to solve a nine-parameter 3D affine transformation when only three known points in both systems are given. The system of nine equations is reduced to six by subtracting the equations and eliminating the translation parameters. From these six equations, five variables are eliminated using a Grobner basis to get a quadratic univariate polynomial, from which the solution can be expressed symbolically. The main advantage of this result is that we do not need to guess initial values of the nine parameters, which is necessary in the case of the traditional solution of the nonlinear system of equations. This result can be useful in geodesy, robotics, and photogrammetry when occasionally only three known points in both systems are given or when a Gauss‐ Jacobi combinatorial solution may be required for certain reasons, for example detecting outliers by using variancecovariance matrices.","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":"69959818","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}
Fisher first introduced the Fisher linear discriminant back in 1938. After the popularization of the support vector machine (SVM) and the kernel trick it became inevitable that the Fisher linear discriminant would be kernelized. Sebastian Mika accomplished this task as part of his Ph.D. in 2002 and the kernelized Fisher discriminant (KFD) now forms part of the largescale machine-learning tool Shogun. In this article we introduce the package MathKFD. We apply MathKFD to synthetic datasets to demonstrate nonlinear classification via kernels. We also test performance on datasets from the machine-learning literature. The construction of MathKFD follows closely in style the construction of MathSVM by Nilsson and colleagues. We hope these two packages and others of the same ilk will eventually be integrated to form a kernel-based machine-learning environment for Mathematica.
{"title":"Fisher Discrimination with Kernels","authors":"Hugh Murrell, K. Hashimoto, Daichi Takatori","doi":"10.3888/TMJ.13-13","DOIUrl":"https://doi.org/10.3888/TMJ.13-13","url":null,"abstract":"Fisher first introduced the Fisher linear discriminant back in 1938. After the popularization of the support vector machine (SVM) and the kernel trick it became inevitable that the Fisher linear discriminant would be kernelized. Sebastian Mika accomplished this task as part of his Ph.D. in 2002 and the kernelized Fisher discriminant (KFD) now forms part of the largescale machine-learning tool Shogun. In this article we introduce the package MathKFD. We apply MathKFD to synthetic datasets to demonstrate nonlinear classification via kernels. We also test performance on datasets from the machine-learning literature. The construction of MathKFD follows closely in style the construction of MathSVM by Nilsson and colleagues. We hope these two packages and others of the same ilk will eventually be integrated to form a kernel-based machine-learning environment for Mathematica.","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":"69959727","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}
A package for solving time-dependent partial differential equations (PDEs), MathPDE, is presented. It implements finite-difference methods. After making a sequence of symbolic transformations on the PDE and its initial and boundary conditions, MathPDE automatically generates a problem-specific set of Mathematica functions to solve the numerical problem, which is essentially a system of algebraic equations. MathPDE then internally calls MathCode, a Mathematica-to-C++ code generator, to generate a C++ program for solving the algebraic problem, and compiles it into an executable that can be run via MathLink. When the algebraic system is nonlinear, the Newton-Raphson method is used and SuperLU, a library for sparse systems, is used for matrix operations. This article discusses the wide range of PDEs that can be handled by MathPDE, the accuracy of the finite-difference schemes used, and importantly, the ability to handle both regular and irregular spatial domains. Since a standalone C++ program is generated to compute the numerical solution, the package offers portability.
{"title":"MathPDE: A Package to Solve PDEs by Finite Differences","authors":"K. Sheshadri, P. Fritzson","doi":"10.3888/TMJ.13-20","DOIUrl":"https://doi.org/10.3888/TMJ.13-20","url":null,"abstract":"A package for solving time-dependent partial differential equations (PDEs), MathPDE, is presented. It implements finite-difference methods. After making a sequence of symbolic transformations on the PDE and its initial and boundary conditions, MathPDE automatically generates a problem-specific set of Mathematica functions to solve the numerical problem, which is essentially a system of algebraic equations. MathPDE then internally calls MathCode, a Mathematica-to-C++ code generator, to generate a C++ program for solving the algebraic problem, and compiles it into an executable that can be run via MathLink. When the algebraic system is nonlinear, the Newton-Raphson method is used and SuperLU, a library for sparse systems, is used for matrix operations. This article discusses the wide range of PDEs that can be handled by MathPDE, the accuracy of the finite-difference schemes used, and importantly, the ability to handle both regular and irregular spatial domains. Since a standalone C++ program is generated to compute the numerical solution, the package offers portability.","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":"69959973","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 show how a Monte Carlo procedure (based on random numbers) can generate a large sample of electron locations in any simple molecule. Based on this sampling, we can accurately estimate the moleculeʼs ground-state energy and other properties of interest. We demonstrate this using the LiH molecule.
{"title":"Monte Carlo Simulation of Simple Molecules","authors":"J. Vrbik","doi":"10.3888/TMJ.13-5","DOIUrl":"https://doi.org/10.3888/TMJ.13-5","url":null,"abstract":"We show how a Monte Carlo procedure (based on random numbers) can generate a large sample of electron locations in any simple molecule. Based on this sampling, we can accurately estimate the moleculeʼs ground-state energy and other properties of interest. We demonstrate this using the LiH molecule.","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":"69960128","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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