Volume 18

This article illustrates how Mathematica can be employed to model stochastic processes via stochastic differential equations to compute trajectories and their statistical features. In addition, we discuss parameter estimation of the model via the maximum likelihood method with global optimization. We consider handling modeling error, system noise and measurement error, compare the stochastic and deterministic models and use the likelihood-ratio test to verify the possible improvement provided by stochastic modeling. The Mathematica code is simple and can be easily adapted to similar problems. Read More »

This series of articles showcases a variety of applications around the theme of inversion, which constitutes a strategic way of manipulating configurations of circles and lines in the plane. This article includes the Riemann sphere, rings of four tangent circles, and inverting the Sierpinski sieve. Read More »

We describe efficient algorithms for working with subgroups of . Operations discussed include join and meet, congruence testing, congruence closure, subgroup testing, cusp enumeration, supergroup lattice, generators and coset enumeration, and constructing a group from a list of generators. Read More »

We present an implementation of the Poisson-Influenced -Means Algorithm (PIKA), first developed to characterize the output of a superconducting transition edge sensor (TES) in the few-photon-counting regime. The algorithm seeks to group data into several clusters that minimize their distances from their means, as in classical -means clustering, but with the added knowledge that the cluster sizes should follow a Poisson distribution. Read More »

Picard’s iteration is used to find the analytical solutions of some Abel–Volterra equations. For many equations, the integrals involved in Picard’s iteration cannot be evaluated. The author approximates the solutions of those equations employing a semi-implicit product midpoint rule.The Aitken extrapolation is used to accelerate the convergence of both methods. A blow-up phenomena model, a wave propagation, and a superfluity equation are solved to show the practicality of the methods. Programs offered solve the general equations. A user needs only to enter the particular forcing and kernel functions, constants, and a step size for a particular problem. Read More »

Biological systems possess traits that are a product of coordinated gene expression, heavily influenced by environmental pressures. This is true of both desirable and undesirable traits (i.e. disease). In cases of unwanted traits, it would be tremendously useful to have a means of systematically lowering the expression of genes implicated in producing the undesirable traits. This article describes the implementation of a computational biology algorithm designed to compute small interfering RNA sequences, capable of systematically silencing the expression of specific genes. Read More »