Harvey K. Shepard

Measuring complexity using information fluctuation

Abstract A method for analyzing deterministic dynamical systems is presented. New measures of complexity are proposed, based on fluctuation in net information gain and its dependence on system size. These measures are applied to one-dimensional cellular automata and shown to be useful in selecting rules that support slow-moving gliders in quiescent backgrounds.

Solitons in the Camassa-Holm shallow water equation

Abstract We study the class of shallow water equations of Camassa and Hold derived from the Lagrangian, L = ∫ [ 1 2 ( ϕ x x x x − ϕ x ) ϕ t − 1 2 ( ϕ x ) 3 − 1 2 ϕ x ( ϕ x x ) 2 − 1 2 κ ϕ x 2 ] d x , using a variational approach. This class contains “peakons” for k=0, which are solitons whose peaks have a discontinuous first derivative. We derive approximate solitary wave solutions to this class of equations using trial variational functions of the form u(x,t)=ψx=A(t) expt[-β(t)|x-q(t)|2n] in a time-dependent variational calculation. For the case k=0 we obtain the exact answer. For k≠0 we obtain the optimal variational solution. For the variational solution having fixed conserved momentum P=∝1/2 (u2+ux2), dx, the solitons scaled amplitude, A/P1/2, and velocity, q ˙ / P 1 / 2 , depend only on the variable z = κ / P . We prove that these scaling relations are true for the exact soliton solutions to the Camassa-Holm equation.

SUSY-based variational method for the anharmonic oscillator

Using a newly suggested algorithm of Gozzi, Reuter and Thacker for calculating the excited states of one-dimensional systems, we determine approximately the eigenvalues and eigenfunctions of the anharmonic oscillator, described by the Hamiltonian H=12p2+gx4. We use ground state post-Gaussian trial wave functions of the form Ψ(x)=N exp(−b|x|2n), where n and b are continuous variational parameters. This algorithm is based on the hierarchy of Hamiltonians related by supersymmetry (SUSY) and the factorization method. We find that our two-parameter family of trial wave functions yields excellent energy eigenvalues and wave functions for the first few levels of the anharmonic oscillator.

Alteration of the ground state by external magnetic fields: The Abelian Higgs model

Abstract By fully exploiting the mathematical and physical analogy to the Ginzburg-Landau theory of superconductivity, we present a complete discussion of the ground state behavior of the four-dimensional Abelian Higgs model in the static tree level approximation. We show that a sufficiently strong external magnetic field can alter the ground state of the theory by restoring a spontaneously broken symmetry, or by creating a qualitatively different “vortex” state. The energetically favored ground state is explicitly determined as a function of the external field and the ratio between coupling constants of the theory.

Variational method for studying solitons in the Korteweg-de Vries equation

Abstract We use a variation method based on the principle of least action to obtain approximate time-dependent single soliton solutions to the KdV equation. A class of trial variational functions of the form u(x,t) = −A(t)exp[−β(t)|x−q(t)|2n], with n a continuous real variable, is used to parametrize time-dependent solutions. We find that this class of trial functions leads to soliton-like solutions for all n, moving with fixed shape and constant velocity, and with energy and mass conserved. Minimizing the energy of the soliton with respect to the parameter n, we obtain a variational solution that gives an extremely accurate approximation to the exact solution.

Optimized data extrapolation off the boundary of an analytic function

Abstract We discuss the problem of extrapolation of a function whose real and imaginary parts are not equally well known on the boundary of its analytic domain. Using an integral equation for the error correlation function, we examine the optimized extrapolated value of the function and the probable error in extrapolation. For several cases, specific solutions are obtained which illustrate general properties and are also useful in phenomenological applications. Comparing with ordinary dispersion relations, we note when the extrapolation error may be uncontrolled, or only weakly constrained, by the input data, and how a knowledge of other properties of the function; e.g. its threshold or high energy behavior, may be necessary in order to reduce the extrapolation error.

Determination of macroion diffusion coefficients using an analytical electrophoresis apparatus

Abstract Methods are presented to determine the effective macroion diffusion coefficient in the presence or absence of an electric field, using a unique analytical electrophoresis apparatus in which both electrophoretic mobility and steady-state electrophoresis may be studied. Approximate analytic solutions to the differential equations and boundary conditions describing diffusion are derived. These solutions are shown to be good approximations to numerial simulations of the differential equations, and provide a good phenomenological description of experimental data for the oligonucleotide p(dA)20⋅p(dT)20 in 100 mM KCl, 20 mM Tris-HCl, pH 8.0 buffer. Diffusion and electrophoresis measurements are made on a single sample without changing the buffer or the macroion concentration, thus the data are directly comparable.

Duality and the role of Daughters in KN→πΛ

In the reaction KN→πΛ at Ecm=1.54−2.15 GeV we observe semi-local averaging to zero of the t-channel helicity amplitudes at t≈tmin, as predicted by quark graph duality. The dominant mechanism for averaging is cancellation between Σγ(32−) and Σα(52+), and between Σβ(52−) and Σδ(72+). Daughter resonances appear to cancel semi-locally among themselves following the same pattern. The overall cancellation pattern is between consecutive towers of resonances.

Semiquantum chaos and the large N expansion

Abstract We consider the dynamical system consisting of a quantum degree of freedom A interacting with N quantum oscillators described by the Lagrangian L = 1 2 A 2 + ∑ i=1 N { 1 2 x i 2 − 1 2 (m 2 + e 2 A 2 )x i 2 } . In the limit N → ∞, with e2N fixed, the quantum fluctuations in A are of order 1 N . In this limit, the x oscillators behave as harmonic oscillators with a time dependent mass determined by the solution of a semiclassical equation for the expectation value 〈A(t)〉. This system can be described, when 〈x(t)〉 = 0, by a classical Hamiltonian for the variables G(t) = 〈x 2 (t), G (t) , A c (t) = 〈A(t)〉, and A c (t) . The dynamics of this latter system turns out to be chaotic. We propose to study the nature of this large N limit by considering both the exact quantum system as well as by studying an expansion in powers of 1 N for the equations of motion using the closed time path formalism of quantum dynamics.

Why the world is simple

1994, 495 pp. ,