{"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}
{"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}
{"title":"Properties and Generalizations of the Fibonacci Word Fractal","authors":"J. L. Ramírez, Gustavo N. Rubiano","doi":"10.3888/TMJ.16-2","DOIUrl":"https://doi.org/10.3888/TMJ.16-2","url":null,"abstract":"This article implements some combinatorial properties of the Fibonacci word and generalizations that can be generated from the iteration of a morphism between languages. Some graphic properties of the fractal curve are associated with these words; the curves can be generated from drawing rules similar to those used in L-systems. Simple changes to the programs generate other interesting curves. is certainly one of the most studied words in the field of combinatorics on words [1–4]. It is the archetype of a Sturmian word [5]. The word f can be associated with a fractal curve with combinatorial properties [6–7]. This article implements Mathematica programs to generate curves from f and a set of drawing rules. These rules are similar to those used in L-systems. The outline of this article is as follows. Section 2 recalls some definitions and ideas of combinatorics on words. Section 3 introduces the Fibonacci word, its fractal curve, and a family of words whose limit is the Fibonacci word fractal. Finally, Section 4 generalizes the Fibonacci word and its Fibonacci word fractal.","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":"69961801","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":"Using Boolean Computation to Solve Some Problems from Ramsey Theory","authors":"R. Cowen","doi":"10.3888/TMJ.15-10","DOIUrl":"https://doi.org/10.3888/TMJ.15-10","url":null,"abstract":"Mathematica’s industrial-strength Boolean computation capability is not used as often as it should be. There probably are several reasons for this lack of use, but it is our view that a primary reason is lack of experience in expressing mathematical problems in the form required for Boolean computation. We look at a typical problem that is susceptible to Boolean analysis and show how to translate it so that it can be tested for satisfiability with Mathematica’s built-in function SatisfiableQ. The problems we investigate come from an area of mathematics called Ramsey theory. Although Ramsey theory has been studied extensively for over 80 years and still provides many challenges, we neglect the theory (for the most part) and instead concentrate on translating the problems so that they are amenable to Boolean computation and then see what can be accomplished by computation alone. Those interested in learning a little more about Ramsey theory can consult [1]; for a standard reference, see [2].","PeriodicalId":91418,"journal":{"name":"The Mathematica journal","volume":"15 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2013-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"69961483","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":"Using the Logistic Map to Generate Scratching Sounds","authors":"Y. Miki","doi":"10.3888/tmj.15-5","DOIUrl":"https://doi.org/10.3888/tmj.15-5","url":null,"abstract":"This article presents a mathematical model for generating annoying scratching sounds. Such sounds are generated by frictional motion and have been attributed to the chaotic nature of the frequency spectrum thereby produced. The proposed model is based on the logistic map and is modified to have the stick-slip property of a frictional vibration. The resulting sound is similar to that generated by scratching a chalkboard or glass plate with the fingernails.","PeriodicalId":91418,"journal":{"name":"The Mathematica journal","volume":"14 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2013-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"69961583","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":"Negative Binomial Regression","authors":"Michael L. Zwilling","doi":"10.3888/TMJ.15-6","DOIUrl":"https://doi.org/10.3888/TMJ.15-6","url":null,"abstract":"where m > 0 is the mean of Y and a > 0 is the heterogeneity parameter. Hilbe [1] derives this parametrization as a Poisson-gamma mixture, or alternatively as the number of failures before the H1 e aLth success, though we will not require 1 e a to be an integer. The traditional negative binomial regression model, designated the NB2 model in [1], is (2) ln m = b0 + b1 x1 + b2 x2 +o⋯+ bp xp, where the predictor variables x1, x2, ..., xp are given, and the population regression coefficients b0, b1, b2, ..., bp are to be estimated. Given a random sample of n subjects, we observe for subject i the dependent variable yi and the predictor variables x1i, x2i, ..., xpi. Utilizing vector and matrix notation, we let b = H b0 b1 b2 o⋯ bp L¬, and we gather the predictor data into the design matrix X as follows:","PeriodicalId":91418,"journal":{"name":"The Mathematica journal","volume":"15 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2013-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"69961596","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":"Detecting Differential Gene Expression Using Affymetrix Microarrays","authors":"Todd Allen","doi":"10.3888/TMJ.15-11","DOIUrl":"https://doi.org/10.3888/TMJ.15-11","url":null,"abstract":"This article describes the development of a novel program to process Affymetrix microarray files, which are used in the biological sciences to establish differences in gene expression between two conditions (e.g., diseased tissue versus healthy tissue).","PeriodicalId":91418,"journal":{"name":"The Mathematica journal","volume":"15 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2013-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"69961493","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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