The Mathematica Journal


Robert Cowen
Published July 30, 2018

An important problem in graph theory is to find the number of complete subgraphs of a given size in a graph. If the graph is very large, it is usually only possible to obtain upper bounds for these numbers based on the numbers of complete subgraphs of smaller sizes. The KruskalKatona bounds are often used for these calculations. We investigate these bounds in specific cases and study how they might be improved. Read More »

Part II: Root-Finding Methods

M. Irene Falcão, Fernando Miranda, Ricardo Severino, M. Joana Soares
Published July 1, 2018

This article explores the numerical mathematics and visualization capabilities of Mathematica in the framework of quaternion algebra. In this context, we discuss computational aspects of the recently introduced Newton and Weierstrass methods for finding the roots of a quaternionic polynomial. Read More »

Part I: Manipulating, Evaluating and Factoring

M. Irene Falcão, Fernando Miranda, Ricardo Severino, M. Joana Soares
Published May 4, 2018

This article discusses a recently developed Mathematica toola collection of functions for manipulating, evaluating and factoring quaternionic polynomials. relies on the package , which is available for download at Read More »

A Finitely Generated Group with Interesting Subgroups

Paul R. McCreary, Teri Jo Murphy, Christan Carter
Published March 26, 2018

The action of Möbius transformations with real coefficients preserves the hyperbolic metric in the upper half-plane model of the hyperbolic plane. The modular group is an interesting group of hyperbolic isometries generated by two Möbius transformations, namely, an order-two element and an element of infinite order . Viewing the action of the group elements on a model of the hyperbolic plane provides insight into the structure of hyperbolic 2-space. Animations provide dynamic illustrations of this action. Read More »

Célestin Wafo Soh
Published February 26, 2018

We propose and implement an algorithm for solving an overdetermined system of partial differential equations in one unknown. Our approach relies on the BourMayer method to determine compatibility conditions via JacobiMayer brackets. We solve compatible systems recursively by imitating what one would do with pen and paper: Solve one equation, substitute its solution into the remaining equations, and iterate the process until the equations of the system are exhausted. The method we employ for assessing the consistency of the underlying system differs from the traditional use of differential Gröbner bases, yet seems more efficient and straightforward to implement. Read More »

Robert Cowen
Published January 25, 2018

We simultaneously introduce effective techniques for solving Sudoku puzzles and explain how to implement them in Mathematica. The hardest puzzles require some guessing, and we include a simple backtracking technique that solves even the hardest puzzles. The programming skills required are kept at a minimum. Read More »