Challenging the Boundaries of Symbolic Computation

The following Sections are included:

• Host's Preface

• Editor's Preface

• Contents

In this paper we present tools for linguistic fractal analysis of symbolic sequences — sequences of formal symbols from a given alphabet. We introduce a method for graphically representing and analyzing symbolic sequences based on fractal Iterated Function Systems (IFSs). This method gives a visual description (a pattern) of the structure of the subsequences within the original sequence, and permits quantitative characterization of the resulting patterns. We present a number of examples and also demonstrate the use of J/Link for importing sequences and speeding up the computations.

This paper presents a hybrid system for automatic generation of numerical codes using a symbolic approach, with emphasis on finite element formulations where straightforward use of symbolic and algebraic systems leads to exponential growth of derived expressions. The system comprises two major components. The Mathematica package AceGen is used for symbolic derivation of formulae needed in numerical procedures and automatic code generation. An approach implemented in AceGen overcomes the problem of expression growth by combining automatic differentiation, automatic code generation and theorem proving. The second component, named Computational Templates is a collection of prearranged modules for automatic creation of the interface between the finite element code and the finite element environment.

Sequence variables are an advanced feature of modern programming languages. They enhance the support for writing programs in a declarative and easily understood way. To our knowledge, Mathematica provides the best support for programming with sequence variables, but it requires a good understanding of how the interpreter chooses the matcher. This is because matching against patterns with sequence variables is generally not unique. We claim that there is room to improve the programming style with sequence variables. We propose a number of new programming constructs which impose certain strategies on the pattern matching process. Our constructs enable us to control the selection of a matcher by annotating sequence variables with binding priorities and ranges for their lengths, and to compute optimal values characterized by a score function to be optimized. To this end we have developed the package Sequentica. With Sequentica, Mathematica programmers and users get additional support for defining functions and transformation rules in an easy and convenient way. We outline the algorithmic difficulties to support these extensions and describe how they are implemented in Sequentica. The usefulness of these extensions is illustrated with various examples. We regard these extensions as a first step towards identifying a new programming style: programming with sequence variables. Such a programming style is useful to solve problems based on sequence analysis, such as bio-informatics, cryptography or data mining.

This paper will describe how XML and web technologies can be combined with Mathematica in order to provide innovative solutions for computation. It will pay particular attention to the use of MathML, both as a way to render mathematical and technical information in a web browser and as an interchange format between different computer applications. The use of SVG to generate high-quality graphical images that support dynamic animations will also be discussed. A final section will discuss the use of XML as a general data format for exchange between applications that communicate over the web.

The method of Kolossov and Muskhelishvili, based on complex representation theory, provides a general solution of fundamental problems of plane elasticity theory. Up to now, its effective application was restricted by computation difficulties. In this study, we use Mathematica to find analytical solutions of some fundamental problems of the infinite plate with elliptic holes, and subjected to different boundary conditions. The results obtained enable us to assess the validity of asymptotic solutions, widely used in fracture mechanics.

The position of the eye is described by Listing's law, which states that any movement of the eye away from the straight ahead position is accomplished by a rotation about an axis which is perpendicular to the straight ahead direction. Mathematica is used to provide concise derivations of the implications of this law for vision.

We use Mathematica to illustrate the applications of Jacobi elliptic functions to obtain the approximate solution of a two degree of freedom, undamped, homogeneous dynamical system having small cubic nonlinearities. We also illustrate in this article how Mathematica may help to obtain the closed-form solution of Lamé's equation.

The objective of this paper is to solve the equation of motion of semilunar heart valve vibrations. The vibration of the closed semilunar valves are modeled by a Caputo fractional derivative of order α. Using a Laplace transformation, a closed-form solution is obtained for the equation of motion in terms of a Mittag-Leffler function. An alternative method for a semi-differential equation α = 1/2 when is also examined. The simplicity of the Mathematica solution makes it ideal for testing the accuracy of numerical methods. This solution can be of some interest for fitting experimental data.

In this paper, we describe a Mathematica program for the design of distillation columns, clearly a very important topic in chemical industry. In particular, we consider McCabe-Thiele's procedure, which is a graphical method for designing distillation columns for binary mixtures. We take advantage of the symbolic, numerical and graphical Mathematica capabilities to create a general program that can be applied to any kind of binary distillation column. The performance of this program is shown through some illustrative examples.

We present a symbolic, grammar-based model for the classic lac operon gene regulation system implemented in Mathematica. This functional model focuses on the information processing aspect of gene regulation through pattern matching on symbolic expressions. Our lac operon notebook provides a viewer component for animated, 2D visualization of the simulated gene interaction processes, and is also connected to a 3D visualization engine.

Given a set of vertices for a convex or concave polygon, a smooth and bounded interpolant (valid in the interior of the domain) may be constructed in an explicit algebraic form. Rational polynomial interpolants are suitable for convex polygons; the present treatment addresses concave domains. To resolve the concavity, square roots appear in the numerator and denominator of the rational interpolant. Finite element text books do not generally discuss tessellation into concave domains. Procedural programming languages, which are predominantly used in conventional finite elements, pose formidable challenges to the implementation of abstract geometrical constructs necessary to address convexity issues. Concepts of projective geometry are utilized within Mathematica in symbolic form to yield shape functions on concave polygons.

The method of external source collocation is used to solve a discretised boundary value problem, ∇²U = 0, where U is the potential in a two-dimensional simply-connected region D, subject to a mixture of Neumann and Dirichlet boundary conditions. Numerical analysis has, to date, been hindered by an accumulation of round-off error, which has made it impossible to investigate accuracy of the Meshless Fundamental Solutions method unless sources are near the boundary. Symbolic analysis allows a full investigation of ill-conditioned systems in which sources can be placed "at infinity". This analysis provides an indication of how many sources must be used and where they should be placed.

The main aim of the presentation is to show how basic concepts of Mathematica can be applied to fundamental problems of continuum mechanics. In modern electronic courses covering this topic, symbolic solutions are used for:

• deriving general equations for the deformation and motion of the continuum (fluid) in any curvilinear system of coordinates by symbolic techniques,
• finding symbolic solutions, mainly corresponding to conservation laws,
• solving wave problems that appear in the theory of filtration.

We study the stability of the equilibrium solutions in the elliptic restricted many-body problems using Mathematica. The calculations done in the case of five interacting bodies have shown that the radial equilibrium solutions are unstable for any value of the mass m₀ of the central body, while some bisector equilibrium solutions are stable in linear approximation if m₀ is sufficiently large. However, there is a domain of instability of the bisector solutions in the vicinity of the resonant point. Its boundaries are calculated with the method of infinite determinants.

The increase in the frequency of disasters and their associated damage in the SADC region is part of a worldwide trend, which results from growing vulnerability and may reflect changing climate patterns. Global risks seem to be increasing. These trends make it all more necessary for South Africa to initiate the development and implementation of the national Disaster Hazard and Vulnerability Atlas. This is a database-driven, web-enabled interactive "virtual book". It consists of various "chapters", such as drought, flood, cyclones, storms.etc. Mathematica is in use as a core service (engine) for the calculation of the Standardized Precipitation Index (SPI) as a drought indicator. webMathematica scripts have been written to perform complex probabilistic rainfall data analysis.

Special thanks to the National Disaster Management Centre of the Republic of South Africa (http://sandmc.pwv.gov.za) for their support in this project, and for hosting the Atlas (http://sandmc.pwv.gov.za/atlas/).

The efficiency of a perfect heat engine running with an ideal gas operating between two heat reservoirs is optimized by Sadi Carnot. It is known that the efficiency is independent of the working body and depends on the temperatures of the two heat reservoirs only. The detail of the proof, however, in most of the approaches relies implicitly on the assumption that the gas is ideal. The author revisits the concept and has generalized the proof assuming the working body to obey the nonideal gas law. Pedagogically, with this realistic explicit model it is shown that the efficiency of the cycle stays intact. Mathematica's symbolic calculation power is deployed throughout the work. A Mathematica package is crafted to automate and evaluate the efficiency of the cycle. This package can prove useful in teaching thermo physics and engineering college courses.

In this paper we want to show an e-learning application, MoMAM@th, to support teaching/learning of mathematics on the web. MoMAM@th exposes to the user an infinite set of interactive exercises, randomly generated on-the-fly, referring to specific topics. The innovation of this application is in using exercises that are split into simple steps. At each step the software guides the student towards the solution of the problem and the student is required to interact, filling some blank boxes. In this way MoMAM@th can follow the reasoning of the student and at the same time helps him/her in each single step. WebMathematica is the engine used to develop MoMAM@th.

In this paper we aim to show a Mathematica notebook based on a package built to demonstrate the working of Linear Time Independent Systems (LTI Systems).

This notebook lets the user play with a virtual LTI system, where he/she can choose things such as the input signal and the system transformation given as the system response to a pulse signal, and see in real time the shape of the output signal. The user can also see the response of the LTI system to canonical signals, or also the signal in the frequency domain, showing its frequency response in Bode or Polar diagrams. The pedagogical assumption we followed in the development of the package is that a student learns better and in a more stimulating way using the strategy of learning by doing. Manipulating data and observing the results of their manipulation a learner can reach a deeper knowledge of the subject. We well explain in more detail how the package works in the following sections.

In this paper, we analyze the influence of applying the double strand discrete-continuous on College students' conceptual understanding of central notions in analysis such as limit and derivative. The research is done in the context of a course on differential equations. Our aim is to analyze how using Mathematica, while iterating the simplest mathematical expression such as quadratic functions, the students are introduced to the great beauty of the field of dynamical systems and to a better understanding of convergence problems.

In this paper, we give examples of models we have created for use in university mathematics courses. We visualize the area of the unit circle, estimate the area of a figure by probabilistic simulation and consider the relation between sound and trigonometry. We give interesting, visual, meaningful and effective models for teaching the above subjects, which are obtained by the powerful functions of Mathematica such as animation of graphics, variety of visualization and speed of computation. This paper illustrates how one can use Mathematica to visualize abstract mathematical concepts, thus enabling students' effective understanding in the mathematics classroom. Development of these kinds of teaching and learning models can stimulate the students' curiosity about Mathematics and increase their interest.

I have been teaching mathematics using Mathematica in high school for more than 4 years, and found out that it is possible for high school students to do very interesting research of mathematics if they can use good software and get proper advice. I am going to introduce the results of our research. Some of them are mathematically new, and others are not new but very inspiring since they show promising possibilities to students. In our research the use of Mathematica is essential. In this papar "we" sometimes means a group of students, and in other times it means students plus the instructor R.Miyadera.

This paper describes a WebMathematica-based tutoring system for algebraic calculations, such as simplifications of rational expressions, for high school students. The system has two learning paths to accommodate various student learning styles: interactive exercises and step-by-step instructions. It also has an explanation page to help with calculation errors based on misconceptions, which beginners and slow learners tend to have. The pages are linked to each other and inherit the student's expression from one page to the next. Based on the expression, the system shows a comment, a list of the next steps in the calculation and the explanation for any incorrect calculation step. The system uses WebMathematica to display the dynamically changing contents, Mathematica to evaluate the symbolic expressions, based on rules described in custom Mathematica functions, and a J-Link program to retrieve the necessary data from a database.

The growing popularity of the internet, and the increasing number of computers connected to it, make it an ideal framework for remote education. Many disciplines are rethinking their traditional philosophies and techniques to adapt to the new technologies. Web-based education is an effective framework for such learning, which simplifies theory understanding, encourages learning by discovery and experimentation and undoubtedly makes the learning process more pleasant. There is a need for adequate tools to help in the elaboration of courses that might make it possible to express all the possibilities offered by www teaching. webMathematica is a web-based technology developed by Wolfram Research that allows the generation of dynamic web content with Mathematica. With this technology, distance education students should be able to explore and experiment with mathematical concepts. In this paper we present a sample lecture for Partial Differential Equations in webMathematica for the distance learning environment.

In the 4th IMS, held in Japan in 2001, I talked about "Web Applications using J/Link for Math Education" and introduced some web applications that calculate a client's request with the Mathematica kernel on the web server and return the result back to the client. In this paper, I describe two main improvements I have made on these applications. The first is the integration of different web applications. Combining several separate applications dealing with 3D figures enables you to carry out more interesting and effective operations for 3D figures. For example, consider two applications: one for drawing quadrics and the other for cutting solids. If they are joined together, the user can freely draw a quadrics and cut through its surface. This can help students to learn mathematical properties of quadrics and conic section.

The second improvement consists of four new web applications added to the site. On these applications, the user can draw semi-transparent polyhedra, plot beautiful chaos images, enjoy "Origami" on the web and so on. They are based on my new packages and the powerful function of "JavaView". I hope that students enjoy these applications and get interested in mathematics.

The School of Information Environment of Tokyo Denki University was founded in 2001, and at this school we adopted a new system of mathematics education based on Mathematica. In this talk, I will introduce some of the features of this system through Mathematica and video-files. All Mathematica commands related to this article are included in the appendix Mathematica notebooks. on the CD-ROM Other notebooks are also available at http://math.kn.dendai.ac.jp/tazawa/.

The starting point of this paper is a project at the Distance University of Applied Sciences of Switzerland, which deals with the development and deployment of programs interactively generating detailed and dynamic step-by-step solutions to typical problems in higher mathematical education, based on Mathematica. The first part of the paper explains organizational and didactic aspects, empirical values, advantages and disadvantages as well as proposals for improvements. The second part puts the emphasis on an approach to a generic step-by-step solver and its interface, enabling tutors to create step-by-step solutions according to the mathematical context by setting appropriate parameters. The functioning of such an algorithm and its parameters is explained in detail, based on an example from Boolean algebra.

We report on further development of the 'StandardPhysicalConstants' package, presented at IMS2001. We address these issues: package structure; current status of the physical constant database; data sources; current data collection and data structure; the main modules of the data management system; the first version of "error propagator"; usage examples in calculations for high precision tests of physics theories. We also indicate future developments of the basic package.

The violation of Bell-type inequalities by quantum probabilities represents an important aspect of quantum mechanics which is linked to the mind-boggling quantum features of nonlocality, complementarity and contextuality. Formally, such bounds on the classical probabilities from consistency arguments have already been investigated by Boole in the middle of the nineteenth century. Boole called them "conditions of possible experience." We introduce CddIF, a Mathematica package to compute all the Boole-Bell-type inequalities associated with an arbitrary physical setup. We have also computed with Mathematica the tractable special cases of two particles with up to three possible detection angles per particle, and the three-particle/two directions (Greenberger-Home-Zeilinger) case.

This talk begins with some background information on reversible computation from a Thermodynamics and Quantum Theory point of view, and then introduces a Mathematica® Package Quantum Turing Machine Simulator (QTS) that can serve as an educational tool to explore the principles of a reversible computational process within the context of Quantum Mechanics.

The aim of this article is to demonstrate the use of Mathematica in computations in algebraic topology. While Mathematica, which aims to be a general purpose numeric-symbolic program, can never be as efficient a tool for research level symbolic computation algebra as more specialized programs, the dramatic increase in the speed of personal computers has made such use entirely feasible. We choose as an example a rather arcane subject: the computation of the action of the Steenrod operations in cohomology of the classifying space of the exceptional Lie group F₄.

Motivated, at first, by an interest in nonchaotic strange attractors, I have attempted a complete characterisation of the phenomenon of mode-locking in periodically and quasiperiodically driven oscillators, numerically and via perturbation calculations. In both the periodically driven and the quasiperiodically driven cases, it appeared as though certain regions in parameter space associated with mode-locking had points at which their width was zero. Now, this is slightly surprising if true, for two reasons. First, there is no analogue to this pinching effect in skew product circle maps. Secondly, at pinching points, if they exist, the flow is extremely regular, and more so than one might naively expect for this highly nonlinear family of flows. I have not been able to prove analytically that mode-locked regions pinch to zero width. However, I have used Mathematica's arbitrary precision to assemble extremely strong numerical evidence that they do.

We showed [3] that gamma distributions provide models for departures from randomness, since every neighbourhood of an exponential distribution contains a neighbourhood of gamma distributions, using an information theoretic metric topology. Here we use Mathematica to illustrate these neighbourhoods. Using Mathematica, without which the computations would have been prohibitively tedious, we derived also the information geometry of the 3-manifold of Mckay bivariate gamma distributions, which can provide a metrization of departures from randomness and independence for bivariate processes. As in the case of bivariate normal manifolds, we have negative scalar curvature, but here it is not constant; details are provided in our paper [3].

Consider certain types of solids in three-dimensions with a variable boundary. It is assumed that this boundary changes with two real parameters s and t. Examples of such include solids bounded by a fixed elliptic cylinder and a variable plane, solids bounded by a fixed elliptic paraboloid and a variable plane, and also solids bounded by a fixed astroidal cylinder and a variable plane. Let denote the center of gravity of the solid . As the parameters s and t change, this center of gravity G changes. According to some of the previous studies, for several well-known types of solids , the locus of G is a paraboloid in three-dimensions. In this paper, we will further investigate such locus problems. In particular, it is of interest to discover familiar types of variable solids, for which the locus of the center of gravity G is not a paraboloid. In order to calculate the triple integrals associated with these center of gravity problems, and also to create visualizations of the loci, one can effectively use Mathematica.

One rich and memorable example can be a more effective tool for learning, or teaching, a body of concepts and techniques than a multitude of separate little examples. Hero's method—a simple iterative algorithm for estimating square roots—serves as an exemplary topic for introducing principles of Mathematica in general, and of programming in Mathematica in particular. Among the ideas introduced are: Mathematica's symbolic, numeric, and graphical aspects; function definition; list manipulation; functional, procedural, and recursive programming paradigms; and graphical programming.

We report here a method of generating 2D walks based on context independent L-systems for two or four letter alphabets. Binary strings (words) are mapped to strings over a four letter alphabet by two separate techniques. 2D walks are generated from the resulting quaternary strings by assigning vectors to the characters as follows: a → (-1,0), b → (1,0), c → (0,-1), d → (0,1). In the case of long strings obtained by iterating endomorphisms, one may also use vectors for image words of letters. A walk is generated by adding up the vectors of sequential characters in a word through a regular grid and coloring the cells at which the vectors terminate. Plots of distance from the origin versus string position provide a rapid means for comparison of the strings resulting from different L-systems. Here one may also use matrix multiplication in a powerful way. We examine the 2D walks from a variety of L-systems and provide examples of considerably complex paths from simple L-systems. Square-free strings over four letters can result in both simple and highly complex 2D walks.

The theory of NP-completeness is fundamental to computational complexity and the SATISFIABILITY problem of Boolean Logic is the seminal NP-complete problem on which the theory is based. While NP-completeness is associated more with the efficiency of deciding whether or not an arbitrary instance is satisfiable, determining the proportion of satisfiable instances of an equivalent problem, 3-SAT, yields important information in regards to this efficiency. Empirical research on the dependence of this proportion on the size of the 3-SAT instance has revealed a "satisfiability" phase transition through which dramatic changes in satisfiability accompany very small changes in problem size. This phenomenon underpins the celebrated threshold conjecture and in this paper we give a new way of attacking the conjecture by describing this transition algebraically. That is, we present and implement an algorithm that, for any given integer n, generates a closed form expression in m, (clause number), and k, (clause size), which counts the proportion of satisfiable instances of 3-SAT. The inherent complexity of the algorithm indicates that generating such a formula for n ≥ 5 is not feasible, but that, taking the form of a succinct mathematical expression, the algorithm is amenable to further asymptotic analysis. As part of this analysis we show that understanding the threshold amounts to understanding the asymptotics of an alternating sum of k-cylinders. All algorithms and experimentation have been implemented as part of an application package, SATPACK, more details of which appear in an extended version of this paper, (including all references), on the proceedings CD.

At IMS2001, I gave a presentation titled "Gauss-Bonnet Theorem by Mathematica." It contained a visualization of the elegant proof of the Umlaufsatz (Theorem of Turning Tangents) by Heinz Hopf, mainly using still figures [1]. In this talk I will show full animations of figures of the detailed proof, incorporated into a movie. Visualizing proofs, and visualizing through computer graphics in particular, gives viewers clear images of the situation, but at the same time it can give some slanted points of view. In this movie I tried to show as many figures as possible to avoid a loss in generality, tracing the figures we imagine in our mind when we follow the proof.

The paper presents a mathematical explanation of the phenomenon observed for the Refined Least Squares method. It was found by numerical experiment with Mathematica that for some boundary-value or initial value problems it is possible to neglect selected boundary conditions and still obtain a nonsingular system of algebraic equations. Two examples present this phenomenon from the mathematical point of view and I make sense of the solution obtained. This discovery has an important application to ill-conditioned problems of mathematical physics and engineering. The approach makes it possible, in certain cases, to find an approximation which is valid on most of the domain, apart from the boundary layer, or to evaluate a main trend of the function.

This paper uses a linear partial differential equation (PDE) solver to construct the bifurcating branches of solutions of nonlinear PDE's. This linear PDE solver (developed by the author in a previous paper) uses the Mutigrid algorithm. This algorithm is written in C and the Mathematica protocal Mathlink is used to call the C program from Mathematica. This combination is very powerful since we can utilize the speed of C together with the symbolic and graphical capabilities of Mathematica.

Bayesian Statistical Models provide statisticians and auditors with a set of powerful tools for attacking financial audit problems. Some of these statistical auditing problems may be solved analytically. For these, functions for manipulating equations and standard packages make Mathematica a natural environment for work. However, most Bayesian statistical models suitable for attacking financial audit problems are not accessible in closed form. For these problems Mathematica's symbolic programming language, efficient array handling functions, and numerical optimization routines form the base for building large-scale simulations. The beta-binomial model for estimating the proportion of transactions in error is described. Models for multilocation audits are discussed briefly, as are models for sequential multiple location audits. Why Mathematica is an important tool in this type of data analysis is shown.

It is known that a lot of mathematical calculations must be processed during the practical use of exact and approximate methods that are designed to analyze different deterministic and random phenomena in dynamical systems. The present paper describes some schemes for such analysis, each requiring an application of the computer algebra package Mathematica for long-term symbolic and symbolic-numeric computations. Here we consider tasks mainly connected with the stochastic analysis of various effects such as: the estimation of sensitivity; water pollution; the calculation of some numeric characteristics of multi-dimensional random vectors; and tasks concerned with finding stochastic potentials.

Mathematical modelling of time dependent systems is always interesting for applied mathematicians. First continuous and then discrete mathematical models were built in the mathematical development from ancient to modern times. With the discovery of time scale, the problem of irregular systems was solved in the 1990s. In this paper we explain the derivative and integral of functions of time scales and the solution of some basic calculus problems using Mathematica.

This paper presents current efforts in extending the Digital Image Processing application package by linking it to external libraries using J/Link and Java. Native Mathematica code has been augmented with Java Advanced Imaging, a 2D image processing library, and Java Media Framework, a Java package for manipulation of time-based media such as audio and video. 3D image processing and visualization capabilities were also added to DIP by using the Visualization Toolkit, which is an open-source, object-oriented software system that supports 3D/2D graphics, visualization, image processing and volume rendering.

MathGL3d is an OpenGL based Mathematica application of advanced scientific visualization. The interactive visualization of scalar functions of three variables or of three dimensional volume data is a frequent task in many fields of research. Medical data from computer tomography (CT) or magnet resonance tompgaphie (MRT) are especially a real challenge for a visualization application. The paper will demonstrate the build-in functions of MathGl3d for volume data visualization.

The problem of creation of a stereo-image for a 3D display from a set of the 2D images is especially important. Using Mathematica's ability we are able to test the different models of 2D image distribution for the evaluation of one stereo-image. For example, we show that the color stereo-image on our 3D display may be obtained also from a special set of a 2D gray-scaled images. The different types of 2D image encoding, corresponding to a definite number of the presented perspectives are studied. General relations, written in Mathematica's notation, allows us to evaluate combined stereo-image from any set of 2D images. Thus, Mathematica's power helps to create new multi-channel systems and improvments in well-known ones. The last our results allows us to present images with a large number of perspectives for very good volumetric perception.

Over the years, the authors have developed the interactive photo editing application Java Photo Editor (JPE) to supplement the Mathematica Frontend [1] when image process operations are done using Mathematica and the Digital Image Processing (DIP) package, which provides plentiful scientific knowledge of this field. The JPE can be executed simply as the standalone graphics painting program because it has been implemented as a Java application based on the Java classes defined by J/Link. As the result, sufficient performances of both graphics manipulations usability with a mouse and transfer rate of the image data between Java and Mathematica were maintained successfully. Data conversion between Java's BufferedImage and DIP's ImageData were effectively done by DIP's fine tuned algorithm. JPE's technique has been applied for development of other applications related to Java or Mathematica. The clinical chart eDolphin and the field simulator MathFields were constructed using JPE's resources.

A CAS can compute and plot nearly every curve that can be represented by formulae, and packages like GeomView, MathLive, or MathGL3D allow a 3D representation of curved surfaces which can be rotated in real-time. These are used to discuss three models of the real projective plane: two-sheeted covering by the sphere, the disk-model, and the Roman surface of J. Steiner. Examples of projective curves on these surfaces will be presented.

In 1997, Professor Randal C. Picker published a pathbreaking article titled "Simple Games for a Complex World" in the University of Chicago Law Review. Drawing on earlier "Spatial Games" work done in theoretical biology, his article explained how legal interventions can affect the evolution of behavioral norms among a networked group of individuals who have modest computational abilities but do have an ability to learn from each other. This article shows that the Spatial Games model is mathematically equivalent to a cellular automaton. The reconceptualization of this area permits leveraging of the computational algorithms and insights regarding cellular automata. The article shows how the "coupling topology" of society and the learning algorithms of its members appear to affect the distribution of Wolfram Class I (homogeneous), Class II (regular), Class III (chaotic) and Class IV (complex) behavior in social systems for which the Spatial Games model serves as a useful metaphor. The Mathematica code used to reach these conclusions is available on the conference proceedings CD.

We provide a toolkit for programming impartial combinatorial games in Mathematica. The toolkit consists of a notebook, GamesEngine.nb, which implements the Sprague-Grundy algorithm, along with some generic graphic elements for simple games. Particular games can then be easily programmed, as we shall demonstrate. A website has been established which contains the GamesEngine notebook, several examples of combinatorial games and our earlier paper, Playing Games with Mathematica, which can be downloaded. It is intended that the website will act as a repository of interesting games and techniques for exploring them with Mathematica. Finally, the role of games in mathematics education is discussed.

A "musical coastline" can be derived from an instrumental solo by plotting pitch against time. The result is a discrete approximation to a (possibly) fractal curve. The fractal dimension of the curve yields some measure of the complexity of the musical phrase. Examples are taken from a violin sonata by J.S. Bach and from solos by three blues guitar players. To establish the reliability of the programming code, mathematical curves of known dimension are included in the analysis.

Origami is a Japanese traditional art of paper folding. Recently, Origami became a topic of active research due to its relation to art, geometry, theorem proving, and declarative programming. It has been recognized that several interesting mathematical problems can be described in an elegant way by paper folds. In this paper we describe an origami programming methodology based on functional logic programming. We use two software components. The first one provides primitives to construct, manipulate and visualize paper folds. The second one, called Open CFLP, solves systems of equations whose operations are defined by conditional rewrite rules. We show that paper folding constructs can be expressed as systems of equations which are then solved by Open CFLP. Both components of our programming environment are implemented in Mathematica. We illustrate functional logic origami programming with some interesting examples.

This paper describes an application of webMathematica for constructing a web site for a sports records database. The web site has: a visual interface for the record data; a flexible searching mechanism; an interaction mechanism for user events. JDBC is used to access the database, and the data is directly imported into Mathematica. For the visual interface, Scalable Vector Graphics-SVG is used and the SVG is processed by Mathematica as expressions. Mathematica's list manipulation functions enable flexible handling of XML tags. From the experience of the web site construction, the author concludes that webMathematica is suitable as a main processing center of web resources, such as HTML pages, graphics, XML, and databases.

This paper describes the implementation of an origami programming environment. The origami programming environment enables us to construct sophisticated origamis by calling origami folding functions with parameters necessary to define a fold. The Mathematica notebook with our origami environment package can become the place for constructing origamis step-by-step. The origami programming environment goes further than just simulating origami found in the literature, as it performs intriguing combinations of symbolic, numeric and graphic computations. We show examples taken from well-known pieces of origami art and from a mathematical origami construction.