Showing papers in "The Mathematica journal in 2006"
Journal Article•
TL;DR: This algorithm combines a multilevel approach, which effectively overcomes local minimums, with the Barnes and Hut octree technique, which approximates short and long-range force efficiently.
Abstract: We propose a graph drawing algorithm that is both efficient and high quality. This algorithm combines a multilevel approach, which effectively overcomes local minimums, with the Barnes and Hut [1] octree technique, which approximates shortand long-range force efficiently. Our numerical results show that the algorithm is comparable in speed to Walshaw’s [2] highly efficient multilevel graph drawing algorithm, yet gives better results on some of the difficult problems. In addition, an adaptive cooling scheme for the force-directed algorithms and a general repulsive force model are proposed. The proposed graph drawing algorithm and others are included with Mathematica 5.1 and later versions in the package DiscreteMath‘GraphÑ Plot.
563 citations
Journal Article•
TL;DR: The number of ways of writing a positive integer n as a product of integer factors greater than one has been surveyed in this paper, and the speed of various approaches to generating these factorizations has been compared in practice.
Abstract: We study the number of ways of writing a positive integer n as a product of integer factors greater than one. We survey methods from the literature for enumerating and also generating lists of such factorizations for a given number n. In addition, we consider the same questions with respect to factorizations that satisfy constraints, such as having all factors distinct. We implement all these methods in Mathematica and compare the speeds of various approaches to generating these factorizations in practice.
20 citations
Journal Article•
TL;DR: This article discusses global optimization in multiextremal models and tools to handle such models in Mathematica, and introduces MathOptimizer, an application package for continuous nonlinear (convex and global) optimization.
Abstract: In engineering, economic, and scientific studies, decisions are frequently modeled by applying optimization concepts and techniques. This article discusses global optimization in multiextremal models and tools to handle such models in Mathematica. Since we assume that not all readers are familiar with optimization models and methods, a general modeling framework is presented. We also review several built-in Mathematica optimization functions and then introduce MathOptimizer, an application package for continuous nonlinear (convex and global) optimization. To illustrate the usage of MathOptimizer, several configuration analysis and design models are formulated and solved. We also provide some comparative notes related to current Mathematica optimization functionality (namely, the function NMinimize) and to the recently introduced MathOptimizer Professional package.
13 citations
Journal Article•
TL;DR: The accessible MathModelica internal form allows the user to extend the system with new functionality, as well as to perform queries on the model representation and write scripts for automatic model generation.
Abstract: MathModelica is an integrated interactive development environment for advanced object-oriented system modeling and simulation. The environment integrates Modelica-based modeling and simulation with ...
11 citations
Journal Article•
TL;DR: This article presents a compact and yet flexible implementation of SVMs in Mathematica, traditionally named MathSVM, designed to be easy to extend and modify, drawing on the powerful high-level language ofMathematica.
Abstract: Support vector machines (SVMs) are learning algorithms that have many applications in pattern recognition and nonlinear regression. Being very popular, SVM software is available in many versions. Still, existing implementations, usually in low-level languages such as C, are often difficult to understand and adapt to specific research tasks. In this article, we present a compact and yet flexible implementation of SVMs in Mathematica, traditionally named MathSVM. This software is designed to be easy to extend and modify, drawing on the powerful high-level language of Mathematica.
6 citations
TL;DR: In this article, the authors describe the electron wavefunctions whose evolution is governed by the Schrodinger equation and the Pauli exclusion principle, and they describe properties normally associated with real atoms such as quantised energy levels and shell structures.
Abstract: Quantum dots are artificially fabricated atoms, in which charge carriers are confined in all three dimensions just like electrons in real atoms. Consequently, they exhibit properties normally associated with real atoms such as quantised energy levels and shell structures. These properties are described by the electron wavefunctions whose evolution is governed by the Schrodinger equation and the Pauli exclusion principle.
3 citations
Journal Article•
TL;DR: This package introduces object-oriented paradigms into Mathematica and is used to develop a modeling environment built on a knowledge base where class and object properties and relations are maintained in a consistent, transparent, and extensible way.
Abstract: This article discusses an object-oriented approach to industrial software development using Mathematica. We present the package Elements for structured representation of physical, engineering, and mathematical objects. This package introduces object-oriented paradigms into Mathematica and is used to develop a modeling environment built on a knowledge base where class and object properties and relations are maintained in a consistent, transparent, and extensible way. We show how this tool can be applied to design models parametrized by structured objects instead of just simple values.
2 citations
2 citations
Journal Article•
TL;DR: An improvement to the method of Sinha and Butcher is shown, able to calculate exact, rather then approximate, Picard iterates of high order.
Abstract: Linear ordinary differential equations (ODEs) with periodic coefficients appear in various interesting applications, such as determining the linear stability regions of systems of vertically driven multiple pendula. Sinha and Butcher [1, 2] have obtained very good approximations to the solutions of such equations by calculating approximate Picard iterates symbolically in the parameters on which the system depends. In this article we show an improvement to the method of Sinha and Butcher. We are able to calculate exact, rather then approximate, Picard iterates of high order. The key point in the programming is the necessity of introducing a user-defined function to carry out the integrations that appear in the definition of the Picard iterates. After introducing the concept of Picard iteration and explaining its fast implementation, we apply the method to determine the stability regions for linearized systems of vertically driven multiple pendula.
2 citations
TL;DR: A Trott constant is a number whose continued fraction representation (simple or not) is the same as the digits of its radix representation and this Corner implements a search method for such numbers and lists its results.
Abstract: A Trott constant is a number whose continued fraction representation (simple or not) is the same as the digits of its radix representation. For instance, in base 10, 0.1084101... = 0 + 1 / (1 + 1 / (0 + 1 / (8 + 1 / (4 + 1 / (1 + 1 / (0 + 1 / (1 ...))))))). This Corner implements a search method for such numbers and lists its results. The best number we found has more than 400 digits in common between its nonsimple, zero-free continued fraction representation and its radix representation.
2 citations
Journal Article•
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TL;DR: A pedagogic calculator is being developed using webMathematica that gives intermediate steps for calculations in a wide range of applications (integration, differentiation, algebra, equation solving, and so on).
Abstract: Xmath [1] and dMath [2] are projects supported by the European Commission through the Minerva Action (Xmath) [3] and the Leonardo da Vinci program (dMath) [4]. These programs seek to promote European cooperation in the fields of open and distance learning (Xmath) and vocational training (dMath). A pedagogic calculator [5] is being developed using webMathematica [6, 7] in connection with these programs. This calculator gives intermediate steps for calculations in a wide range of applications (integration, differentiation, algebra, equation solving, and so on). An expression is broken down and analyzed by Mathematica packages that have been developed to work with webMathematica. Rules that are familiar from hand calculation are given and so are the intermediate results. The user may scroll to look at a number of levels (the first level is the first hint) and then continue by hand. In the same way as an expression may be broken down into different levels, so can the calculation steps. This gives structured output, as a professor would do it on a blackboard, stating the rules at each level. The output in webMathematica may be written as MathMLForm, giving a nonimage format that can be rendered in Internet Explorer using proper style sheets. One of the main ideas of Xmath is implementing MathML [8] in general web pages containing mathematical expressions. The dMath project creates a database of mathematical modules using the SciLas system developed in the project. The Xmath calculator is connected to the database modules and will be further developed in dMath.
TL;DR: In fact, it is probably fair to say that almost nobody thinks I should be doing anything other than working very hard on the upcoming version of Mathematica, which I can assure you is going to be something extremely exciting as mentioned in this paper.
Abstract: It is a pleasure to be with you all today. My main excuse for not being there in person is that we really have to finish the next version of Mathematica. I think it is probably fair to say that almost nobody thinks I should be doing anything other than working very hard on the upcoming version of Mathematica, which I can assure you is going to be something extremely exciting, that we have been working on for a great many years.
TL;DR: A number of simulation models of the spread of organisms across a landscape using representative individual organisms to represent key features of the life cycle of the organism in question and one or more movement processes such as rain splash, dispersal by wind, human-mediated dispersal, or insect flight are constructed.
Abstract: We have constructed a number of simulation models of the spread of organisms across a landscape. These models are based on the spread of representative individual organisms. The models are quite simple but they are typically evaluated thousands or millions of times in a series of time-steps, often resulting in complex emergent behaviour. The models generally represent key features of the life cycle of the organism in question and one or more movement processes such as rain splash, dispersal by wind, human-mediated dispersal, or insect flight. In each spread event an individual moves along a path a distance that is chosen from a probability distribution. Both the path and the distribution of distances are functions of environmental or other parameters.