1) Suppose you have to guess a 3 digit binary (i.e. 0's and 1's) code on a keypad. a) How many different codes are possible? b) Suppose that the door opens as soon as the 3 digit codes is entered. For example, if the code is 000, the door opens after 1000 is entered. Try to come up with the shortest binary sequence that is guaranteed to open the door. For example, if we had a 2 digit code, the sequence 00110 works. c)* Start exploring codes of length 4, length 5, etc. 2) Below are two directed graphs (A, B). (Note a vertex can have an edge to itself.) A Eulerian cycle is a path through ALL of the edges in a graph (using each only once) which starts and ends at the same vertex. For example, aedcb is an Eulerian cycle of graph A. a) Find all of the Eulerian cycles of graph A. Why is this not as hard as it seems? b) Find 3 different Eulerian cycles of graph B, all starting with a. Argue that there are at least 24 different Eulerian cycles of graph B. (In fact, there are exactly 24 different Eulerian cycles of graph B.)
{"title":"Combinatorics on Words","authors":"V. Keränen","doi":"10.3888/TMJ.11.3-4","DOIUrl":"https://doi.org/10.3888/TMJ.11.3-4","url":null,"abstract":"1) Suppose you have to guess a 3 digit binary (i.e. 0's and 1's) code on a keypad. a) How many different codes are possible? b) Suppose that the door opens as soon as the 3 digit codes is entered. For example, if the code is 000, the door opens after 1000 is entered. Try to come up with the shortest binary sequence that is guaranteed to open the door. For example, if we had a 2 digit code, the sequence 00110 works. c)* Start exploring codes of length 4, length 5, etc. 2) Below are two directed graphs (A, B). (Note a vertex can have an edge to itself.) A Eulerian cycle is a path through ALL of the edges in a graph (using each only once) which starts and ends at the same vertex. For example, aedcb is an Eulerian cycle of graph A. a) Find all of the Eulerian cycles of graph A. Why is this not as hard as it seems? b) Find 3 different Eulerian cycles of graph B, all starting with a. Argue that there are at least 24 different Eulerian cycles of graph B. (In fact, there are exactly 24 different Eulerian cycles of graph B.)","PeriodicalId":91418,"journal":{"name":"The Mathematica journal","volume":"11 1","pages":"358-375"},"PeriodicalIF":0.0,"publicationDate":"2010-02-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"69959379","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}
Optimization of parameters or ’systems’ in general plays an ever-increasing role in mathematics, economics, engineering, and life sciences. As a result, a wide variety of both traditional analytical, mathematical and non-traditional algorithmic approaches have been introduced to solve challenging and practically relevant optimization problems. Evolutionary optimization methods~namely, genetic algorithms, genetic programming, and evolution strategies~represent a category of non-traditional optimization algorithms drawing inspirations from the process of natural evolution. Particle swarm optimization represents another set of more recently developed algorithmic optimizers inspired by social behaviours of organisms such as birds [8] and social insects. These new evolutionary approaches in optimization are now entering the stage, and are thus far very successful in solving real-world optimization problems [12]. Although these evolutionary approaches share many concepts, each one has its strengths and weaknesses. The best way to understand these techniques is through practical experience, in particular on smaller-scale problems or on commonly accepted benchmark functions. In [11], we describe how evolution strategies and particle swarm optimizers compare on benchmarks prepared for a much more complex optimization task regarding a kinematic model of a soccer kick. The Mathematica notebooks that we created throughout these evaluation experiments and for the final design of the muscle control algorithms for the soccer kick are now also available through a webMathematica interface. The new Evolutionary & Swarm Optimization web site is integrated with the collection of notebooks from the EVOLVICA package, which covers evolution-based optimizers from genetic algorithms and evolution strategies to evolutionary programming and genetic programming. The EVOLVICA database of notebooks, along with the newly added swarm algorithms, provide a large experimentation and inquiry platform for introducing evolutionary and swarm-based optimization techniques to those who either wish to further their knowledge in the evolutionary computation domain or require a streamlined platform to build prototypical strategies to solve their optimization tasks. Making these notebooks available through a web Mathematica site means that anyone with an internet browser available will have instant access to a wide range of optimization algorithms.
{"title":"Exploratory Toolkit for Evolutionary and Swarm-Based Optimization","authors":"Namrata Khemka, C. Jacob","doi":"10.3888/TMJ.11.3-5","DOIUrl":"https://doi.org/10.3888/TMJ.11.3-5","url":null,"abstract":"Optimization of parameters or ’systems’ in general plays an ever-increasing role in mathematics, economics, engineering, and life sciences. As a result, a wide variety of both traditional analytical, mathematical and non-traditional algorithmic approaches have been introduced to solve challenging and practically relevant optimization problems. Evolutionary optimization methods~namely, genetic algorithms, genetic programming, and evolution strategies~represent a category of non-traditional optimization algorithms drawing inspirations from the process of natural evolution. Particle swarm optimization represents another set of more recently developed algorithmic optimizers inspired by social behaviours of organisms such as birds [8] and social insects. These new evolutionary approaches in optimization are now entering the stage, and are thus far very successful in solving real-world optimization problems [12]. Although these evolutionary approaches share many concepts, each one has its strengths and weaknesses. The best way to understand these techniques is through practical experience, in particular on smaller-scale problems or on commonly accepted benchmark functions. In [11], we describe how evolution strategies and particle swarm optimizers compare on benchmarks prepared for a much more complex optimization task regarding a kinematic model of a soccer kick. The Mathematica notebooks that we created throughout these evaluation experiments and for the final design of the muscle control algorithms for the soccer kick are now also available through a webMathematica interface. The new Evolutionary & Swarm Optimization web site is integrated with the collection of notebooks from the EVOLVICA package, which covers evolution-based optimizers from genetic algorithms and evolution strategies to evolutionary programming and genetic programming. The EVOLVICA database of notebooks, along with the newly added swarm algorithms, provide a large experimentation and inquiry platform for introducing evolutionary and swarm-based optimization techniques to those who either wish to further their knowledge in the evolutionary computation domain or require a streamlined platform to build prototypical strategies to solve their optimization tasks. Making these notebooks available through a web Mathematica site means that anyone with an internet browser available will have instant access to a wide range of optimization algorithms.","PeriodicalId":91418,"journal":{"name":"The Mathematica journal","volume":"11 1","pages":"376-391"},"PeriodicalIF":0.0,"publicationDate":"2010-02-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"69959395","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 graphs of complex-valued functions f : Ø or functions of the type f : 2 Ø 2 are in general two-dimensional manifolds in the space 4. The article presents a method for the visualization of such a graph. The graph is first projected to three-dimensional space with parallel projection and the image~the surface in three-dimensional space~is rendered on the screen in the usual way. The visualization can be improved in two ways: the graph can be rotated in four-dimensional space or the direction line of the projection can be changed, which means that the observer flies around the graph in four dimensions. The animation and manipulation capabilities of Mathematica are appropriate tools for the purpose.
{"title":"On the Visualization of Riemann Surfaces","authors":"Simo Kivelä","doi":"10.3888/TMJ.11.3-6","DOIUrl":"https://doi.org/10.3888/TMJ.11.3-6","url":null,"abstract":"The graphs of complex-valued functions f : Ø or functions of the type f : 2 Ø 2 are in general two-dimensional manifolds in the space 4. The article presents a method for the visualization of such a graph. The graph is first projected to three-dimensional space with parallel projection and the image~the surface in three-dimensional space~is rendered on the screen in the usual way. The visualization can be improved in two ways: the graph can be rotated in four-dimensional space or the direction line of the projection can be changed, which means that the observer flies around the graph in four dimensions. The animation and manipulation capabilities of Mathematica are appropriate tools for the purpose.","PeriodicalId":91418,"journal":{"name":"The Mathematica journal","volume":"11 1","pages":"392-403"},"PeriodicalIF":0.0,"publicationDate":"2010-02-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"69959439","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}
It is well known that the set of all natural numbers divisible by a fixed modulus m can be recognized by a finite state machine, assuming that the numbers are written in standard base-B representation. It is much harder to determine the state complexity of the minimal recognizer [1]. In this article we discuss the size of minimal recognizers for a variety of numeration systems, including reverse base-B representation and the Fibonacci system.
{"title":"Divisibility and State Complexity","authors":"Klaus Sutner","doi":"10.3888/TMJ.11.3-8","DOIUrl":"https://doi.org/10.3888/TMJ.11.3-8","url":null,"abstract":"It is well known that the set of all natural numbers divisible by a fixed modulus m can be recognized by a finite state machine, assuming that the numbers are written in standard base-B representation. It is much harder to determine the state complexity of the minimal recognizer [1]. In this article we discuss the size of minimal recognizers for a variety of numeration systems, including reverse base-B representation and the Fibonacci system.","PeriodicalId":91418,"journal":{"name":"The Mathematica journal","volume":"11 1","pages":"430-445"},"PeriodicalIF":0.0,"publicationDate":"2010-02-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"69959448","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}
In 1857 Sir William Rowan Hamilton invented the Icosian game [1]. In a world based on the dodecahedral graph, a traveler must visit 20 cities, without revisiting any of them. Today, when the trip makes a loop through all the vertices of the graph, it is called a Hamiltonian tour (or cycle). When the first and last vertices in a trip are not connected, it is called a Hamiltonian path (or trail). The first image shown is a tour; the second is a path.
{"title":"The Icosian Game, Revisited","authors":"E. Pegg","doi":"10.3888/TMJ.11.3-1","DOIUrl":"https://doi.org/10.3888/TMJ.11.3-1","url":null,"abstract":"In 1857 Sir William Rowan Hamilton invented the Icosian game [1]. In a world based on the dodecahedral graph, a traveler must visit 20 cities, without revisiting any of them. Today, when the trip makes a loop through all the vertices of the graph, it is called a Hamiltonian tour (or cycle). When the first and last vertices in a trip are not connected, it is called a Hamiltonian path (or trail). The first image shown is a tour; the second is a path.","PeriodicalId":91418,"journal":{"name":"The Mathematica journal","volume":"11 1","pages":"310-314"},"PeriodicalIF":0.0,"publicationDate":"2010-02-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"69959301","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}
Important properties pertaining to families of discrete dynamical systems are furnished here by studying the kneading theory developed by Milnor and Thurston, and subsequently implementing the spider algorithm, developed by Hubbard and Schleicher. The focus is on identifying crucial combinatorial and numerical properties of periodic critical orbits in one-dimensional discrete dynamical systems, which are generated by iterating real quadratic polynomial maps that constitute an important class of unimodal systems.
{"title":"A Mathematica Implementation of Nonlinear Dynamical Systems Theory via the Spider Algorithm and Finding Critical Zeros of High-Degree Polynomials","authors":"T. Jonassen","doi":"10.3888/TMJ.11.3-2","DOIUrl":"https://doi.org/10.3888/TMJ.11.3-2","url":null,"abstract":"Important properties pertaining to families of discrete dynamical systems are furnished here by studying the kneading theory developed by Milnor and Thurston, and subsequently implementing the spider algorithm, developed by Hubbard and Schleicher. The focus is on identifying crucial combinatorial and numerical properties of periodic critical orbits in one-dimensional discrete dynamical systems, which are generated by iterating real quadratic polynomial maps that constitute an important class of unimodal systems.","PeriodicalId":91418,"journal":{"name":"The Mathematica journal","volume":"11 1","pages":"315-332"},"PeriodicalIF":0.0,"publicationDate":"2010-02-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"69959318","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}
In the academic literature there are two common approaches for the evaluation of financial options. These are stochastic calculus and partial differential equations. The former is the method of choice for statisticians and theoreticians, while the latter is the principal tool of physicists and computer scientists because it lends itself to practical implementation schemes. Occasionally small modifications such as linear regression and binomial trees are used, but these are usually treated within either of the two previously mentioned fields. Rarely do the practitioners of these fields compare and contrast methodologies, let alone admit completely different approaches. While Radial Basis Function (RBF) methodology has previously been applied to solving some differential equations, there are very few papers considering its applicability to financial mathematics. The purpose of this article is to show not only that RBF can solve many of the evaluation problems for financial options, but that with Mathematica it can do so with accuracy and speed.
{"title":"Evaluation of Financial Options Using Radial Basis Functions in Mathematica","authors":"M. Kelly","doi":"10.3888/TMJ.11.3-3","DOIUrl":"https://doi.org/10.3888/TMJ.11.3-3","url":null,"abstract":"In the academic literature there are two common approaches for the evaluation of financial options. These are stochastic calculus and partial differential equations. The former is the method of choice for statisticians and theoreticians, while the latter is the principal tool of physicists and computer scientists because it lends itself to practical implementation schemes. Occasionally small modifications such as linear regression and binomial trees are used, but these are usually treated within either of the two previously mentioned fields. Rarely do the practitioners of these fields compare and contrast methodologies, let alone admit completely different approaches. While Radial Basis Function (RBF) methodology has previously been applied to solving some differential equations, there are very few papers considering its applicability to financial mathematics. The purpose of this article is to show not only that RBF can solve many of the evaluation problems for financial options, but that with Mathematica it can do so with accuracy and speed.","PeriodicalId":91418,"journal":{"name":"The Mathematica journal","volume":"11 1","pages":"333-357"},"PeriodicalIF":0.0,"publicationDate":"2010-02-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"69959354","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}
DRIMA is a simple cellular model for the multi-agent setting in which reactive agents have their behavior changed by the behavior of others, as the outcome of their interactions; it is also the system that implements the model in Mathematica. It was conceived as a metaphor for the high-level issue of how agents “attract” others toward them, be it in the form of a change in any behavioral or conceptual orientation, habit, thinking, etc. This is modeled through a single behavior of the agents, which is their movement on a two-dimensional grid; as they move, they undergo interactions with each other that modify the way they move before and after the interaction. The focus of the model is on addressing issues related to the emergent dynamics of a particular setting, much in tune with an artificial life or complex systems perspective. DRIMA is purposefully meant to be simple and non-general, a minimal system for the kind of question it is designed to help address. The article is a presentation of the model and of key aspects of its implementation, not a discussion on its use to address any particular question.
{"title":"DRIMA: A Minimal System for Probing the Dynamics of Change in a Reactive Multi- Agent Setting","authors":"P. D. Oliveira","doi":"10.3888/TMJ.12-1","DOIUrl":"https://doi.org/10.3888/TMJ.12-1","url":null,"abstract":"DRIMA is a simple cellular model for the multi-agent setting in which reactive agents have their behavior changed by the behavior of others, as the outcome of their interactions; it is also the system that implements the model in Mathematica. It was conceived as a metaphor for the high-level issue of how agents “attract” others toward them, be it in the form of a change in any behavioral or conceptual orientation, habit, thinking, etc. This is modeled through a single behavior of the agents, which is their movement on a two-dimensional grid; as they move, they undergo interactions with each other that modify the way they move before and after the interaction. The focus of the model is on addressing issues related to the emergent dynamics of a particular setting, much in tune with an artificial life or complex systems perspective. DRIMA is purposefully meant to be simple and non-general, a minimal system for the kind of question it is designed to help address. The article is a presentation of the model and of key aspects of its implementation, not a discussion on its use to address any particular question.","PeriodicalId":91418,"journal":{"name":"The Mathematica journal","volume":"12 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2010-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"69959468","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}
Cross tabulations (also known as cross tabs, or contingency tables) often arise in data analysis, whenever data can be placed into two distinct sets of categories. In market research, for example, we might categorize purchases of a range of products made at selected locations; or in medical testing, we might record adverse drug reactions according to symptoms and whether the patient received the standard or placebo treatment.
{"title":"An Introduction to Correspondence Analysis","authors":"P. Yelland","doi":"10.3888/TMJ.12-4","DOIUrl":"https://doi.org/10.3888/TMJ.12-4","url":null,"abstract":"Cross tabulations (also known as cross tabs, or contingency tables) often arise in data analysis, whenever data can be placed into two distinct sets of categories. In market research, for example, we might categorize purchases of a range of products made at selected locations; or in medical testing, we might record adverse drug reactions according to symptoms and whether the patient received the standard or placebo treatment.","PeriodicalId":91418,"journal":{"name":"The Mathematica journal","volume":"12 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2010-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"69959596","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}
In this article, we introduce the package Moment Closure, which may be used to generate closure differential equations and closure approximations of the cumulants (moments) of a nonlinear stochastic compartmental model with Markov transitions. Specifically, this package defines the pair of functions MomentClosureSystem and MomentClosurePlots that achieves moment closure through the neglect of high-order cumulants. We demonstrate the application of these functions through the analysis of several test models. In select cases, the resulting cumulant approximations are compared across neglect levels and to exact answers.
{"title":"Achieving Moment Closure through Cumulant Neglect","authors":"T. Matis, I. Guardiola","doi":"10.3888/TMJ.12-2","DOIUrl":"https://doi.org/10.3888/TMJ.12-2","url":null,"abstract":"In this article, we introduce the package Moment Closure, which may be used to generate closure differential equations and closure approximations of the cumulants (moments) of a nonlinear stochastic compartmental model with Markov transitions. Specifically, this package defines the pair of functions MomentClosureSystem and MomentClosurePlots that achieves moment closure through the neglect of high-order cumulants. We demonstrate the application of these functions through the analysis of several test models. In select cases, the resulting cumulant approximations are compared across neglect levels and to exact answers.","PeriodicalId":91418,"journal":{"name":"The Mathematica journal","volume":"12 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2010-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"69959531","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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