{"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}
{"title":"Controlling Robots Built with the LEGO® MINDSTORMS® NXT Brick","authors":"D. Cousineau","doi":"10.3888/TMJ.15-3","DOIUrl":"https://doi.org/10.3888/TMJ.15-3","url":null,"abstract":"The NXT is a general-purpose processor that can be used to control motors and sensors; it is ideal for building autonomous robots. It can also communicate with more elaborate software located on a computer, using a Bluetooth communication port. In this article, we show how to communicate with the NXT by sending the correct bytes. We also introduce a package that manages all the exchanges through functions. These functions can be used in conjunction with dynamic cells to display the robotʼs status and control the robotʼs motor.","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":"69961541","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 General Method for Constructing Ramanujan-Type Formulas for Powers of 1/π","authors":"N. Bagis","doi":"10.3888/TMJ.15-8","DOIUrl":"https://doi.org/10.3888/TMJ.15-8","url":null,"abstract":"","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":"69961654","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":"Random Walks on the World Wide Web","authors":"T. Silvestri","doi":"10.3888/TMJ.15-9","DOIUrl":"https://doi.org/10.3888/TMJ.15-9","url":null,"abstract":"This article presents RandomWalkWeb, a package developed to perform random walks on the World Wide Web and to visualize the resulting data. Building upon the packageʼs functionality, we collected empirical network data consisting of 35,616 unique URLs (approximately 133,500 steps). An analysis was performed at the domain level and several properties of the web were measured. In particular, we estimated the power-law exponent g for the inand out-degree distributions, and obtained values of 2.10± 0.09 and 2.36± 0.1, respectively. These values were found to be in good agreement with previously published results.","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":"69961720","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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