Volume 13 (page 2)

This article considers both the creation of random-looking irregular surfaces and the triangular-facet visualization and interpolation of nonsmooth data arrays. This rough-surface simulation scheme is novel because cellular automata computations define the surface topology. The companion triangular-facet surface-plotting method produces a crisp visualization of nonsmooth data arrays. Read More »

Distributions, which are the various ways of distributing a certain number of objects of different classes among a collection of targets, have been the subject of combinatorial investigations since MacMahon’s 1917 monograph. In this paper we apply them to a simulation of superimposed random coding. Furthermore, asymptotic estimates are provided using logarithmic polynomials (related to the well-known Bell polynomials) for symbolic and numeric calculation. Read More »

We demonstrate a symbolic elimination technique to solve a nine-parameter 3D affine transformation when only three known points in both systems are given. The system of nine equations is reduced to six by subtracting the equations and eliminating the translation parameters. From these six equations, five variables are eliminated using a Gröbner basis to get a quadratic univariate polynomial, from which the solution can be expressed symbolically. The main advantage of this result is that we do not need to guess initial values of the nine parameters, which is necessary in the case of the traditional solution of the nonlinear system of equations. This result can be useful in geodesy, robotics, and photogrammetry when occasionally only three known points in both systems are given or when a Gauss-Jacobi combinatorial solution may be required for certain reasons, for example detecting outliers by using variance-covariance matrices. Read More »

In finite populations, evolutionary dynamics can no longer be described by deterministic differential equations, but require a stochastic formulation [1]. We show how Mathematica can be used to both simulate and visualize evolutionary processes in limited populations. The Moran process is introduced as the basic stochastic model of an evolutionary process in finite populations. This model is extended to mixed populations with relative fitness differences. We combine population ecology with game theoretic ideas, simulating evolutionary games in well-mixed and structured populations. Read More »

Determining whether Hamiltonian cycles exist in graphs is an NP-complete problem, so it is no wonder that the Combinatorica function HamiltonianCycle is slow for large graphs. Theorems by Dirac, Ore, Pósa, and Chvátal provide sufficient conditions that are easy to check for the existence of such cycles. This article provides Mathematica programs for those conditions, thus extending the capability of HamiltonianQ, which only tests the biconnectivity—a simple necessary condition—of a given graph. We also investigate experimentally the limiting behavior of whether the conditions are fulfilled for large random graphs. The phenomenon seen is proved as a theorem, closely related to earlier results by Karp and Pósa. Read More »

An enumeration of strings is developed, in which all strings of finite length of symbols from any alphabet appear, with no upper bounds for string length or alphabet size. A bijective indexing function and its inverse are found for the string enumeration, allowing iteration through the set of all strings, as well as identification of arbitrary strings by the associated index. The method is then extended to sequences of strings and to sequential substitution system (SSS) rulesets, providing a well-defined, relatively dense enumeration of all possible valid SSS rulesets for strings of arbitrary length and any number of symbols used in rulesets of any length, although in this case the indexing function is not one-to-one. Read More »

We show how a Monte Carlo procedure (based on random numbers) can generate a large sample of electron locations in any simple molecule. Based on this sampling, we can accurately estimate the molecule’s ground-state energy and other properties of interest. We demonstrate this using the LiH molecule. Read More »

In this article we demonstrate a method of constructing various types of graphics from polygonal arcs using replacement rules for vertices. The article is based on a presentation entitled “Constructing Graphics from Polygonal Arcs Using Replacement Rules for Vertices” given by the author at the Wolfram Technology Conference in Champaign, Illinois in October 2006. Read More »

The main purpose of this article is to develop an algorithm for simulating a chain sliding off a desktop and to design and demonstrate the corresponding program. Read More »

The theoretical model of a kinematic chain impacting granular matter is studied. The force of the granular medium acting on the chain is a linear superposition of a static (depth-dependent) resistance force and a dynamic (velocity-dependent) frictional force. This resistance force is opposed to the direction of the velocity of the immersed chain. We present two methods (one using EventLocator and the other using FixedStep) for the problem. As examples, a single and a double pendulum are simulated using different initial impact velocity conditions. We analyze how rapidly the kinematic chain impacting the granular medium slows upon collision. For the analyzed cases the kinematic chain under high impact force (higher initial velocity) comes to rest faster in the granular matter than the same body under low impact force (lower initial velocity). Read More »