Existence theory for nonclassical entropy solutions of scalar conservation laws

We establish a general existence theory for the Cauchy problem associated with a scalar conservation law in one-space dimension. The flux-function is assumed to be nonconvex and we consider nonclassical entropy solutions selected by a kinetic relation. To solve the Cauchy problem, we construct a sequence of approximate solutions using a wave-front tracking scheme. The main difficulty is deriving a uniform estimate on the total variation of the approximate solutions. This is achieved here by introducing a generalized total variation functional, which is decreasing in time and, additionally, reduces to the standard total variation functional when the solutions contain only classical shocks. This functional seems sufficiently robust to be useful for systems as well.Received: June 3, 2002; revised: November 12, 2002     

Explicit exact solutions for a new generalized Hamiltonian amplitude equation with nonlinear terms of any order

Making use of a proper transformation and a generalized ansatz, we consider a new generalized Hamiltonian amplitude equation with nonlinear terms of any order, iu_x + u_tt + (|u|^p + |u|^2p)u + u_xt = 0. As a result, many explicit exact solutions, which include kink-shaped soliton solutions, bell-shaped soliton solutions, periodic wave solutions, the combined formal solitary wave solutions and rational solutions, are obtained.Received: April 4, 2002     

Wave front shape for a constrained cylindrically anisotropic elastic solid

The two-dimensional wave front shape caused by a point impulse excitation in a cylindrically anisotropic elastic solid is considered. The elastic parameters of the solid are constrained such that E = G This constraint allows the parametric equations of the wave front to be expressed exactly in terms of elementary transcendental functions. The precise location of double and cusp points on the front is treated in detail. Time histories of several wave front patterns are presented and an interesting feature of the front is generalized to the unconstrained solid.     

I.I. Rabi: The Two Cultures and the Universal Culture of Science

I set forth and discuss I.I. Rabis views on the nature of science and society, focusing specifically on Rabis views on science and government, science and education, and science and religion. I also explore the influence of Rabi and C.P. Snow on each other. In the 1960s, Rabi set forth his mature and boldest positions, advocating science as the center of education and science as the replacement for religion. His positions culminate in science as the universal culture. I highlight Rabi not only as a scientist and public servant, but also as a public intellectual actively engaged with and connected to major issues of his time.Michael A. Day is Professor of Physics at Lebanon Valley College. He holds doctorates in both physics and philosophy.     

Hardness and phase analysis of IN 718 deformed at high strain rate

Specimens of the nickel base alloy IN 718 deformed at high strain rate (~10 s^–1), as realized in a screw press, have higher strength than parts forged conventionally in a hydraulic press. Microstructure analyses in light and transmission electron microscopes reveal the precipitation of very small Ni₃Nb particles (-phase) to be the reason for the increased hardness. Several processing routes are discussed and analyzed in relation to the TTT-diagram of IN 718.     

Holomorphic functions and quasiconformal mappings with smooth moduli

We discuss the implication , where f is a holomorphic function (resp., a quasiconformal mapping) on a domain (resp., ) and Λ_ω(G) is the Lipschitz space associated with a majorant ω.     

Algebraic independence of the values of Mahler functions associated with a certain continued fraction expansion

It is proved that the function , which can be expressed as a certain continued fraction, takes algebraically independent values at any distinct nonzero algebraic numbers inside the unit circle if the sequence {R_k}_k?0 is the generalized Fibonacci numbers.     

Global Solutions of an Obstacle-Problem-Like Equation with Two Phases

Concerning the obstacle-problem-like equation , where ₊ > 0 and _– > 0, we give a complete characterization of all global two-phase solutions with quadratic growth both at 0 and infinity.     

Unbounded Disjointness Preserving Linear Functionals

Let X be a locally compact Hausdorff space and C ₀(X) the Banach space of continuous functions on X vanishing at infinity. In this paper, we shall study unbounded disjointness preserving linear functionals on C ₀(X). They arise from prime ideals of C ₀(X), and we translate it into the cozero set ideal setting. In particular, every unbounded disjointness preserving linear functional of c ₀ can be constructed explicitly through an ultrafilter on complementary to a cozero set ideal. This ultrafilter method can be extended to produce many, but in general not all, such functionals on C ₀(X) for arbitrary X. We also make some remarks where C ₀(X) is replaced by a non-commutative C*-algebra.     

Hyperbolic spaces at large primes and a conjecture of Moore

A simply connected finite complex X is called elliptic if its rational homotopy Lie algebra is of finite dimension and hyperbolic otherwise. According to a conjecture of Moore, there exists an exponent for the p-torsion part of if and only if X is elliptic. In this note, it is shown that, provided the prime p is sufficiently large, a hyperbolic space with p-torsion free loop space homology has no exponent in the p-torsion of the homotopy groups. For a class of formal spaces, this result is obtained for every odd prime.