Classical and Metric Multidimensional Scaling
Published September 30, 2015
We describe the use of classical and metric multidimensional scaling methods for graphical representation of the proximity between collections of data consisting of cases characterized by multidimensional attributes. These methods can preserve metric differences between cases, while allowing for dimensional reduction and projection to two or three dimensions ideal for data exploration. We demonstrate these methods with three datasets for: (i) the immunological similarity of influenza proteins measured by a multidimensional assay; (ii) influenza protein sequence similarity; and (iii) reconstruction of airport-relative locations from paired proximity measurements. These examples highlight the use of proximity matrices, eigenvalues, eigenvectors, and linear and nonlinear mappings using numerical minimization methods. Some considerations and caveats for each method are also discussed, and compact Mathematica programs are provided. Read More »
Published April 26, 2015
Assume that a spacecraft is in a circular orbit and consider the problem of finding the largest possible circular orbit to which the spacecraft can be transferred with constant thrust during a set time, so that the variable parameter is the thrust-direction angle . Also assume that there is only one center of attraction at the common center of the two circular orbits. Finally, assume normalized values for all constants and variables.
This article is divided into five sections: the orbit transfer problem, equations of motion, the optimal control problem, necessary conditions for the Mayer problem, and a dynamic approach to the maximal orbit transfer problem using Mathematica’s built-in Manipulate function.
The Earth-Mars orbit transfer problem is timely, given the successful flight and smooth landing of the American Curiosity rover on Mars. Read More »
Published February 21, 2015
This article describes the implementation of RIFA, a computational biology algorithm designed to identify the genes most directly responsible for creating differences in phenotype between two biological states, for example, tumorous liver tissue versus cirrhotic liver tissue. Read More »
Published January 16, 2015
There exists a range of explicit and approximate solutions to the cubic polynomial Rayleigh equation for the speed of surface waves across an elastic half-space. This article presents an alternative approach that uses Padé approximants to estimate the Rayleigh wave speed with five different approximations derived for two expansions about different points. Maximum relative absolute errors of between 0.34% and 0.00011% occur for the full range of the Poisson ratio from to 0.5. Even smaller errors occur when the Poisson ratio is restricted within a range of 0 to 0.5. For higher-order approximants, the derived expressions for the Rayleigh wave speed are more accurate than previously published solutions, but incur a slight cost in extra arithmetic operations, depending on the desired accuracy. Read More »