## Every Clutter Is a Tree of Blobs »

Given a finite vertex set, one can construct every connected spanning hypergraph by first choosing a spanning hypertree, then choosing a blob on each of its edges. Read More »

Published December 4, 2017

Published October 20, 2017

We present a Mathematica implementation of an algorithm for computing new closed-form evaluations for classes of trig-logarithmic and hyperbolic-logarithmic definite integrals based on the substitution of logarithmic functions into the Maclaurin series expansions of trigonometric and hyperbolic functions. Using this algorithm, we offer new closed-form evaluations for a variety of trig-logarithmic integrals that state-of-the-art computer algebra systems cannot evaluate directly. We also show how this algorithm may be used to evaluate interesting infinite series and products. Read More »

Published September 26, 2017

Rice–Rampsberger–Kassel–Marcus (RRKM) theory calculates an energy-dependent microcanonical unimolecular rate constant for a chemical reaction from a sum and density of vibrational quantum states. This article demonstrates how to program the Beyer–Swinehart direct count of the sum and density of states for harmonic oscillators, as well as the Stein–Rabinovitch extension for anharmonic oscillators. Microcanonical rate constants are calculated for the decomposition of vinyl cyanide () into , and as an example. Read More »

Published August 24, 2017

This article continues the presentation of a variety of applications around the theme of inversion: quandles, inversion of one circle into another and inverting a pair of circles into congruent or concentric circles. Read More »

Published June 15, 2017

Input shaping is an established technique to generate prefilters so that flexible mechanical systems move with minimal residual vibration. Many examples of such systems are found in engineering—for example, space structures, robots, cranes and so on. The problem of vibration control is serious when precise motion is required in the presence of structural flexibility. In a wind turbine blade, untreated flapwise vibrations may reduce the life of the blade and unexpected vibrations can spread to the supporting structure. This article investigates one of the tools available to control vibrations within flexible mechanical systems using the input shaping technique. Read More »

Published May 17, 2017

This didactic synthesis compares three solution methods for polynomial approximation and systematically presents their common characteristics and their close interrelations:

1. Classical Gram–Schmidt orthonormalization and Fourier approximation in

2. Linear least-squares solution via QR factorization on an equally spaced grid in

3. Linear least-squares solution via the normal equations method in and on an equally

spaced grid in

The first two methods are linear least-squares systems with Vandermonde matrices ; the normal equations contain matrices of Hilbert type . The solutions on equally spaced grids in converge to the solutions in All solution characteristics and their relations are illustrated by symbolic or numeric examples and graphs. Read More »

Published March 28, 2017

The derivation of the scattering force and the gradient force on a spherical particle due to an electromagnetic wave often invokes the Clausius–Mossotti factor, based on an ad hoc physical model. In this article, we derive the expressions including the Clausius–Mossotti factor directly from the fundamental equations of classical electromagnetism. Starting from an analytic expression for the force on a spherical particle in a vacuum using the Maxwell stress tensor, as well as the Mie solution for the response of dielectric particles to an electromagnetic plane wave, we derive the scattering and gradient forces. In both cases, the Clausius–Mossotti factor arises rigorously from the derivation without any physical argumentation. The limits agree with expressions in the literature. Read More »