Evaluation of Gaussian Molecular Integrals

CURRENT ARTICLES: VOLUME 16

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

A maximum likelihood estimator has been applied to find regression parameters of a straight line in case of different error models. Assuming Gaussian-type noise for the measurement errors, explicit results for the parameters can be given employing Mathematica. In the case of the ordinary least squares (), total least squares (), and least geometric mean deviation () approaches, as well as the error model of combining ordinary least squares ( and ) in the Pareto sense, simple formulas are given to compute the parameters via a reduced Gröbner basis. Numerical examples illustrate the methods, and the results are checked via direct global minimization of the residuals. Read More »

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 »

Implementing Nonparametric Bayesian Inference

John Cassel
Published January 9, 2014

Probabilistic programming is a programming language paradigm receiving both government support [1] and the attention of the popular technology press [2]. Probabilistic programming concerns writing programs with segments that can be interpreted as parameter and conditional distributions, yielding statistical findings through nonstandard execution. Mathematica not only has great support for statistics, but has another language feature particular to probabilistic language elements, namely memoization, which is the ability for functions to retain their value for particular function calls across parameters, creating random trials that retain their value. Recent research has found that reasoning about processes instead of given parameters has allowed Bayesian inference to undertake more flexible models that require computational support. This article explains this nonparametric Bayesian inference, shows how Mathematica’s capacity for memoization supports probabilistic programming features, and demonstrates this capability through two examples, learning systems of relations and learning arithmetic functions based on output. Read More »