There exists a range of explicit and approximate solutions to the cubic polynomial Rayleigh equation for the speed of surface waves across an elastic half-space. This article presents an alternative approach that uses Pade approximants to estimate the Rayleigh wave speed with five different approximations derived for two expansions about different points. Maximum relative absolute errors of between 0.34% and 0.00011% occur for the full range of the Poisson ratio from -1 to 0.5. Even smaller errors occur when the Poisson ratio is restricted within a range of 0 to 0.5. For higher-order approximants, the derived expressions for the Rayleigh wave speed are more accurate than previously published solutions, but incur a slight cost in extra arithmetic operations, depending on the desired accuracy.
{"title":"Use of Padé Approximants to Estimate the Rayleigh Wave Speed","authors":"A. Spathis","doi":"10.3888/TMJ.17-1","DOIUrl":"https://doi.org/10.3888/TMJ.17-1","url":null,"abstract":"There exists a range of explicit and approximate solutions to the cubic polynomial Rayleigh equation for the speed of surface waves across an elastic half-space. This article presents an alternative approach that uses Pade approximants to estimate the Rayleigh wave speed with five different approximations derived for two expansions about different points. Maximum relative absolute errors of between 0.34% and 0.00011% occur for the full range of the Poisson ratio from -1 to 0.5. Even smaller errors occur when the Poisson ratio is restricted within a range of 0 to 0.5. For higher-order approximants, the derived expressions for the Rayleigh wave speed are more accurate than previously published solutions, but incur a slight cost in extra arithmetic operations, depending on the desired accuracy.","PeriodicalId":91418,"journal":{"name":"The Mathematica journal","volume":"17 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2015-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"69962046","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}
When my son brought home a paper from school with a 7μ 8 grid of numbers on it, I was immediately interested. The goal: cover the puzzle with all the dominoes from the “bone pile,” making sure that each number of the puzzle is covered by the same number on a domino. Many similar puzzles can be found online and in puzzle collections: see [1, 2, 3, 4, 5] for several online resources, which are the source of some of the examples considered here.
{"title":"Three Ways to Solve Domino Grids","authors":"Ken Caviness","doi":"10.3888/TMJ.16-10","DOIUrl":"https://doi.org/10.3888/TMJ.16-10","url":null,"abstract":"When my son brought home a paper from school with a 7μ 8 grid of numbers on it, I was immediately interested. The goal: cover the puzzle with all the dominoes from the “bone pile,” making sure that each number of the puzzle is covered by the same number on a domino. Many similar puzzles can be found online and in puzzle collections: see [1, 2, 3, 4, 5] for several online resources, which are the source of some of the examples considered here.","PeriodicalId":91418,"journal":{"name":"The Mathematica journal","volume":"16 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2014-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"69961335","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}
Remarkable advances have been made in radiotherapy for cancer. Tumors inside the body that move with respiration can now be tracked during irradiation or can be irradiated at a specific phase of respiration [1]. Since radiation of these wavelengths is invisible to the human eye, it is not possible to see whether a tumor that is moving during respiration is being irradiated accurately. Therefore, simulations that use the actual respiration wave to show the tumor being irradiated are useful for training and to explain the procedure to patients.
{"title":"Development of Simulation Models of Respiratory Tracking and Synchronizing for Radiotherapy","authors":"H. Sekine, Yuki Otani","doi":"10.3888/TMJ.16-12","DOIUrl":"https://doi.org/10.3888/TMJ.16-12","url":null,"abstract":"Remarkable advances have been made in radiotherapy for cancer. Tumors inside the body that move with respiration can now be tracked during irradiation or can be irradiated at a specific phase of respiration [1]. Since radiation of these wavelengths is invisible to the human eye, it is not possible to see whether a tumor that is moving during respiration is being irradiated accurately. Therefore, simulations that use the actual respiration wave to show the tumor being irradiated are useful for training and to explain the procedure to patients.","PeriodicalId":91418,"journal":{"name":"The Mathematica journal","volume":"16 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2014-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"69961508","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 maximum likelihood estimator has been applied to find regression parameters of a straight line in case of different error models. Assuming Gaussian-type noise for the measurement errors, explicit results for the parameters can be given employing Mathematica. In the case of the ordinary least squares (OLSy), total least squares (TLS), and least geometric mean deviation (LGMD) approaches, as well as the error model of combining ordinary least squares (OLSx and OLSy) in the Pareto sense, simple formulas are given to compute the parameters via a reduced Gröbner basis. Numerical examples illustrate the methods, and the results are checked via direct global minimization of the residuals.
{"title":"Fitting Data with Different Error Models","authors":"B. Paláncz","doi":"10.3888/TMJ.16-4","DOIUrl":"https://doi.org/10.3888/TMJ.16-4","url":null,"abstract":"A maximum likelihood estimator has been applied to find regression parameters of a straight line in case of different error models. Assuming Gaussian-type noise for the measurement errors, explicit results for the parameters can be given employing Mathematica. In the case of the ordinary least squares (OLSy), total least squares (TLS), and least geometric mean deviation (LGMD) approaches, as well as the error model of combining ordinary least squares (OLSx and OLSy) in the Pareto sense, simple formulas are given to compute the parameters via a reduced Gröbner basis. Numerical examples illustrate the methods, and the results are checked via direct global minimization of the residuals.","PeriodicalId":91418,"journal":{"name":"The Mathematica journal","volume":"16 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2014-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"69961925","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}
Probabilistic programming is a programming language paradigm receiving both government support [1] and the attention of the popular technology press [2]. Probabilistic programming concerns writing programs with segments that can be interpreted as parameter and conditional distributions, yielding statistical findings through nonstandard execution. Mathematica not only has great support for statistics, but has another language feature particular to probabilistic language elements, namely memoization, which is the ability for functions to retain their value for particular function calls across parameters, creating random trials that retain their value. Recent research has found that reasoning about processes instead of given parameters has allowed Bayesian inference to undertake more flexible models that require computational support. This article explains this nonparametric Bayesian inference, shows how Mathematicaʼs capacity for memoization supports probabilistic programming features, and demonstrates this capability through two examples, learning systems of relations and learning arithmetic functions based on output.
{"title":"Probabilistic Programming with Stochastic Memoization","authors":"I. Bayesian, John B. Cassel","doi":"10.3888/TMJ.16-1","DOIUrl":"https://doi.org/10.3888/TMJ.16-1","url":null,"abstract":"Probabilistic programming is a programming language paradigm receiving both government support [1] and the attention of the popular technology press [2]. Probabilistic programming concerns writing programs with segments that can be interpreted as parameter and conditional distributions, yielding statistical findings through nonstandard execution. Mathematica not only has great support for statistics, but has another language feature particular to probabilistic language elements, namely memoization, which is the ability for functions to retain their value for particular function calls across parameters, creating random trials that retain their value. Recent research has found that reasoning about processes instead of given parameters has allowed Bayesian inference to undertake more flexible models that require computational support. This article explains this nonparametric Bayesian inference, shows how Mathematicaʼs capacity for memoization supports probabilistic programming features, and demonstrates this capability through two examples, learning systems of relations and learning arithmetic functions based on output.","PeriodicalId":91418,"journal":{"name":"The Mathematica journal","volume":"16 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2014-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"69961776","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 introduces the notion of a representation of cellular automata rules based on a template. This enhances the standard representation based on a rule table, in that it refers to families of cellular automata, instead of a rule alone. The key for obtaining the templates is the role of the built-in equation-solving capabilities of Mathematica. Operations applicable to the templates are defined, and examples of their use are given in the context of finding representations for rule sets that share the properties of maximum internal symmetry or number conservation. The perspectives for using templates in further contexts are also discussed and current limitations are addressed.
{"title":"Representing Families of Cellular Automata Rules","authors":"P. D. Oliveira, Maurício Verardo","doi":"10.3888/TMJ.16-8","DOIUrl":"https://doi.org/10.3888/TMJ.16-8","url":null,"abstract":"This article introduces the notion of a representation of cellular automata rules based on a template. This enhances the standard representation based on a rule table, in that it refers to families of cellular automata, instead of a rule alone. The key for obtaining the templates is the role of the built-in equation-solving capabilities of Mathematica. Operations applicable to the templates are defined, and examples of their use are given in the context of finding representations for rule sets that share the properties of maximum internal symmetry or number conservation. The perspectives for using templates in further contexts are also discussed and current limitations are addressed.","PeriodicalId":91418,"journal":{"name":"The Mathematica journal","volume":"16 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2014-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"69962029","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 presents a compact analytic approximation to the solution of a nonlinear partial differential equation of the diffusion type by using Bürmannʼs theorem. Expanding an analytic function in powers of its derivative is shown to be a useful approach for solutions satisfying an integral relation, such as the error function and the heat integral for nonlinear heat transfer. Based on this approach, series expansions for solutions of nonlinear equations are constructed. The convergence of a Bürmann series can be enhanced by introducing basis functions depending on an additional parameter, which is determined by the boundary conditions. A nonlinear example, illustrating this enhancement, is embedded into a comprehensive presentation of Bürmannʼs theorem. Besides a recursive scheme for elementary cases, a fast algorithm for multivalued Bürmann expansions and inverse functions is developed using integer partitions. The present approach facilitates the search for expansions of analytic functions superior to commonly used Taylor series and shows how to apply these expansions to nonlinear PDEs of the diffusion type.
{"title":"On Bürmann's Theorem and Its Application to Problems of Linear and Nonlinear Heat Transfer and Diffusion","authors":"Harald M. Schöpf, P. Supancic","doi":"10.3888/TMJ.16-11","DOIUrl":"https://doi.org/10.3888/TMJ.16-11","url":null,"abstract":"This article presents a compact analytic approximation to the solution of a nonlinear partial differential equation of the diffusion type by using Bürmannʼs theorem. Expanding an analytic function in powers of its derivative is shown to be a useful approach for solutions satisfying an integral relation, such as the error function and the heat integral for nonlinear heat transfer. Based on this approach, series expansions for solutions of nonlinear equations are constructed. The convergence of a Bürmann series can be enhanced by introducing basis functions depending on an additional parameter, which is determined by the boundary conditions. A nonlinear example, illustrating this enhancement, is embedded into a comprehensive presentation of Bürmannʼs theorem. Besides a recursive scheme for elementary cases, a fast algorithm for multivalued Bürmann expansions and inverse functions is developed using integer partitions. The present approach facilitates the search for expansions of analytic functions superior to commonly used Taylor series and shows how to apply these expansions to nonlinear PDEs of the diffusion type.","PeriodicalId":91418,"journal":{"name":"The Mathematica journal","volume":"16 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2014-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"69961447","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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