### Expanding a Function in Powers of Its Derivative

H. M. Schöpf, P. H. Supancic

Published November 24, 2014

This article presents a compact analytic approximation to the solution of a nonlinear partial differential equation of the diffusion type by using Bürmann’s 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 Bürmann’s 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. Read More »

Kenneth E. Caviness

Published October 17, 2014

This article develops and compares three methods for solving domino grid puzzles: a “human-type” algorithm, a brute-force method, and a scheme using a generalized odometer. Read More »

### III. Nuclear-Electron Attraction Integrals

Minhhuy Hô, Julio Manuel Hernández-Pérez

Published September 15, 2014

This article carries out the evaluation of nuclear-electron attraction energy integrals using Gaussian-type functions with arbitrary Cartesian angular values. As an example, we calculate the corresponding matrix for the water molecule in the STO-3G basis set. Read More »

Pedro P. B. de Oliveira, Maurício Verardo

Published August 25, 2014

This article introduces the notion of a representation of cellular automata rules based on a template. This enhances the standard representation based on a rule table, in that it refers to families of cellular automata, instead of a rule alone. The key for obtaining the templates is the role of the built-in equation-solving capabilities of *Mathematica*. Operations applicable to the templates are defined, and examples of their use are given in the context of finding representations for rule sets that share the properties of maximum internal symmetry or number conservation. The perspectives for using templates in further contexts are also discussed and current limitations are addressed. Read More »

### A Graph Theory Approach

Todd Silvestri

Published July 16, 2014

We demonstrate a method of generating an exact analytical expression for the reliability of a complex system using a directed acyclic graph to represent the system’s reliability block diagram. Additionally, we show how statistical information stored in a reliability block diagram can be used to transform an analytical expression into a time-dependent function for system reliability. Read More »

Rodrigo Loureiro Malacarne

Published June 23, 2014

This article performs a canonical correlation analysis on financial data of country-specific Exchange Traded Funds (ETFs) to analyze the relationship between stock markets in developed and developing countries. We conclude, using Bartlett’s statistic, that there is a significant relationship between these two datasets. Read More »

### Incircle, Radical Circle, Radical Axis, Twins, Generalizations, and Proofs without Words

Jaime Rangel-Mondragón

Published May 20, 2014

This article systematically verifies a series of properties of an ancient figure called the arbelos. It includes some new discoveries and extensions contributed by the author. Read More »

Béla Paláncz

Published April 30, 2014

### Classroom Tools for Game Theory

Sérgio O. Parreiras

Published March 20, 2014

The Karush-Kuhn-Tucker equations (under suitable conditions) provide necessary and sufficient conditions for the solution of the problem of maximizing (minimizing) a concave (convex) function. This article corrects the program in [1], which computes the solution of Karush-Kuhn-Tucker equations. Our main goal, however, is to provide a program to compute the set of all Nash equilibria of a bimatrix game. The program works well for “small” games (i.e. 4×4 or smaller games); thus, in particular, it is suitable for constructing classroom examples and as an additional tool to empower students in classes using game theory. Read More »

### Exploring Fractal Curves

José L. Ramírez, Gustavo N. Rubiano

Published February 19, 2014

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. Read More »