## The Icosian Game, Revisited »

### BEYOND SUDOKU

Some extensions of Hamiltonian tours are explored. Read More »

Published February 5, 2010

Some extensions of Hamiltonian tours are explored. Read More »

Published February 5, 2010

Important properties pertaining to families of discrete dynamical systems are furnished here by studying the kneading theory developed by Milnor and Thurston, and subsequently implementing the spider algorithm, developed by Hubbard and Schleicher. The focus is on identifying crucial combinatorial and numerical properties of periodic critical orbits in one-dimensional discrete dynamical systems, which are generated by iterating real quadratic polynomial maps that constitute an important class of unimodal systems. Read More »

Published February 5, 2010

In the academic literature there are two common approaches for the evaluation of financial options. These are stochastic calculus and partial differential equations. The former is the method of choice for statisticians and theoreticians, while the latter is the principal tool of physicists and computer scientists because it lends itself to practical implementation schemes. Occasionally small modifications such as linear regression and binomial trees are used, but these are usually treated within either of the two previously mentioned fields. Rarely do the practitioners of these fields compare and contrast methodologies, let alone admit completely different approaches. While Radial Basis Function (RBF) methodology has previously been applied to solving some differential equations, there are very few papers considering its applicability to financial mathematics. The purpose of this article is to show not only that RBF can solve many of the evaluation problems for financial options, but that with *Mathematica* it can do so with accuracy and speed. Read More »

Published February 5, 2010

We explain extensive computer-aided searches that have been carried out for many years to find new ways of constructing abelian square-free words over four letters. These structures have turned out to be very rare and hard to find. We also encountered highly nonlinear phenomena that considerably affected our calculations and usually made them hard to accomplish. However, quite recently, we gained new insight into why these structures are so very rare. Consequently, the present work has the potential to make future explorations easier. The rarity of long words that avoid abelian squares can be explained, at least partly, by using the concept of an unfavorable factor. The purpose of this article is to describe the use of *Mathematica* in searching for and suppressing these factors. In principle, the same programs can be used with slight modifications for other kinds of word patterns as well. Read More »

Published February 5, 2010

Optimization of parameters or “systems” in general plays an ever-increasing role in mathematics, economics, engineering, and the life sciences. As a result, a wide variety of both traditional mathematical and nontraditional algorithmic approaches have been introduced to solve challenging and practically relevant optimization problems. Evolutionary optimization methods—in the form of genetic algorithms, genetic programming, and evolution strategies—represent nontraditional optimization algorithms that draw inspiration from the processes of natural evolution. Particle swarm optimization is another set of more recently developed algorithmic optimizers inspired by social behaviors of organisms such as birds [1] and social insects. These new evolutionary approaches in optimization are now entering the stage and are becoming very successful tools for solving real-world optimization problems [2]. We present *Visplore* and *Evolvica* as a toolkit to investigate, explore, and visualize evolutionary and swarm-based optimization techniques. A web*Mathematica* interface is also available. Read More »

Published February 5, 2010

The graphs of complex-valued functions or functions of the type are in general two-dimensional manifolds in the space . The article presents a method for the visualization of such a graph. The graph is first projected to three-dimensional space with parallel projection and the image—the surface in three-dimensional space—is rendered on the screen in the usual way. The visualization can be improved in two ways: the graph can be rotated in four-dimensional space or the direction line of the projection can be changed, which means that the observer flies around the graph in four dimensions. The animation and manipulation capabilities of *Mathematica* are appropriate tools for the purpose. Read More »

Published February 5, 2010

Building on the preexisting deployment of equation-based surface geometries in architecture, surface logic explores the dialogue between twentieth-century pioneers of reinforced concrete and the contemporary possibilities made accessible by the instrumentation of computation. Computational modeling of equation-based surfaces opens designers to unprecedented access and design sensibilities driven by parametric variation, differential topological relationship, fabrication techniques, material analysis, and physical performance. Read More »

Published February 5, 2010

It is well known that the set of all natural numbers divisible by a fixed modulus can be recognized by a finite state machine, assuming that the numbers are written in standard base- representation. It is much harder to determine the state complexity of the minimal recognizer [1]. In this article we discuss the size of minimal recognizers for a variety of numeration systems, including reverse base- representation and the Fibonacci system. Read More »