We introduce some key formulas of special relativity and apply them to the motion of a spinless, charged point particle of unit mass, subject to the Lorentz force due to an external electromagnetic field. Read More »
Differential equation models for population dynamics are now standard fare in single-variable calculus. Building on these ordinary differential equation (ODE) models provides the opportunity for a meaningful and intuitive introduction to partial differential equations (PDEs). This article illustrates PDE models for location-dependent carrying capacities, migrations, and the dispersion of a population. The PDE models themselves are built from the logistic equation with location-dependent parameters, the traveling wave equation, and the diffusion equation. The approach presented here is suitable for a single lecture, reading assignment, and exercise set in multivariable calculus. Interactive examples accompany the text and the article is designed for use as a CDF document in which some of the input can remain hidden. Read More »
We develop symbolic methods of asymptotic approximations for solutions of linear ordinary differential equations and use them to stabilize numerical calculations. Our method follows classical analysis for first-order systems and higher-order scalar equations where growth behavior is expressed in terms of elementary functions. We then recast our equations in modified form, thereby obtaining stability. Read More »
We investigate the classical problem of a gambler repeatedly betting $1 on the flip of a potentially biased coin until he either loses all his money or wins the money of his opponent. This is then extended to the case of his adversary (a casino) having practically unlimited resources and used to derive the inverse Gaussian distribution of the first passage time. Read More »
This article describes how to model diffusion using NDSolve, and then compares that to constructing your own methods using procedural, functional, rule-based, and modular programming. While based on the diffusion equation, these techniques can be applied to any partial differential equation. Read More »
Perimetric complexity is a measure of the complexity of binary pictures. It is defined as the sum of inside and outside perimeters of the foreground, squared, divided by the foreground area, divided by 
. Difficulties arise when this definition is applied to digital images composed of binary pixels. In this article we identify these problems and propose solutions. Perimetric complexity is often used as a measure of visual complexity, in which case it should take into account the limited resolution of the visual system. We propose a measure of visual perimetric complexity that meets this requirement. Read More »
Both approximate and exact solutions for the motion of a spinning top are constructed with the help of quaternions. Read More »
This article discusses the evaluation of molecular overlap integrals for Gaussian-type functions with arbitrary angular dependence. As an example, we calculate the overlap matrix for the water molecule in the STO-3G basis set. Read More »
Reflection and transmission (scattering) of plane waves at a planar boundary between two elastic half-spaces are important fundamental processes in seismology. Such plane waves may be compressional (P) or shear (S) in an elastic medium. In this article we apply Mathematica to computing the complex algebraic expressions describing the reflection and transmission amplitudes, phases, and angles of propagation. I also illustrate the energy flux quantities. Brewster’s angle, which represents where zero reflected energy occurs, is an important variable for SH waves, which are polarized out of the cross section, and is dependent on both the velocity and density ratio of the two half-spaces. For the P-SV system, the P and SV waves are polarized in the plane of the cross section. Poisson’s ratio (a function of the P and S velocities) affects the energy flux behavior in the case of incident SV but not incident P. Also for the P-SV system, there are four critical angles that affect the energy flux behavior. I also study the limiting P-SV case where the top medium becomes a vacuum, thus creating a “free surface.” For the P-SV system, energy flux of the four scattered waves for any incident wave can be partitioned into top and bottom layer contributions and into total P and total SV contributions, providing further insight into the nature of P-SV scattering. The single-boundary formulation can easily be extended to a stack of layers giving the amplitude and phase at receivers offset along the surface from the source. Read More »
We first solve the planar Kepler problem of an asteroid’s motion, perturbed by the gravitational pull of Jupiter. Analyzing the resulting differential equations for its orbital elements, we demonstrate the mechanism for creating a gap at the 2:1 resonance (the asteroid making two orbits for Jupiter’s one), and briefly mention the case of other resonances (3:2, 3:1, etc.). We also discuss reasons why the motion becomes chaotic at these resonances. Read More »
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