Takashi Yoshino
Published December 31, 2017
Gus Wiseman
Published December 4, 2017
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 »
John M. Campbell
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 »
Adam C. Mansell, David J. Kahle, Darrin J. Bellert
Published September 26, 2017
Part 3: Quandles, Inverting Triangles to Triangles and Inverting into Concentric Circles
Jaime Rangel-Mondragón
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 »
Desmond Adair, Martin Jaeger
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 »
Zachary H. Levine, J. J. Curry
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 »
NB
consists of 27 unit subcubes. Each face 
of 
determines a set 
of nine subcubes that have a face in the same plane as 
. The set 
can be rotated around the normal through the center of 
. Rubik’s 4-cube (or 4D hypercube) 
consists of 81 unit 4-subcubes, each containing eight 3D subcubes. Each 3-face 
of 
determines a set 
of 27 4-subcubes that have a cube in the same hyperplane as 
. The set 
can be rotated around the normal (a plane) through the center of 
. Projecting the whole 4D configuration to 3D exhibits Rubik’s 4-cube as a four-dimensional extension of Rubik’s cube. Starting from a random coloring of the 4-cube, the goal of the puzzle is to return to the initial coloring of the 3-faces. Read More »
) into 
, 
and 
as an example. 
approximation and systematically presents their common characteristics and their close interrelations:
and on an equally
; 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. 
