Absence of symmetry breaking for systems of rotors with random interactions

We prove that Gibbs states for the Hamiltonian , with thes _x varying on theN-dimensional unit sphere, obtained with nonrandom boundary conditions (in a suitable sense), are almost surely rotationally invariant if withJ _xy i.i.d. bounded random variables with zero average, 1 in one dimension, and 2 in two dimensions.     

SL(4, R)-invariant chiral fields

Using a method developed before a set of exact solutions of the chiral equations , wheregSL(4,R) are presented.Work supported in part by CONACYT, México.     

Superconductive superheating field for finiteκ

We have calculated analytically the superheating fieldH _sh for bulk superconductors, correct to second order in. We find , which agrees well with numerical computations for<0.5. The surface order parameter is , and the penetration depth is .     

Tunneling Across an Inhomogeneous Delta-Barrier

The authors deal with the tunneling of electrons across an inhomogeneous delta-barrier defined by the potential energy (where 0$$ " align="middle" border="0"> and 0$$ " align="middle" border="0"> are two constants). In particular, the perpendicular incidence of an electron with a given value of the wave vector is considered. The electron is forward-scattered into the region behind the barrier (region 2: 0$$ " align="middle" border="0"> ), i. e. the wave function is composed of plane waves with all wave vectors such that and \left. 0 \right)} $$ " align="middle" border="0"> ) (where ). Therefore, if 0$$ " align="middle" border="0"> , the wave function of the electron is represented as , where . An approximate formula is derived for the amplitude . The authors pay a special attention to the flow density and calculate this function in two cases: 1. for the plane and 2. for high values of is the diffraction angle). The authors discuss the relevance of their diffraction problem in a prospective quantum-mechanical theory of the tunneling of electrons across a randomly inhomogeneous Schottky barrier.     

Bound States in a Locally Deformed Waveguide: The Critical Case

We consider the Dirichlet Laplacian for astrip in with one straight boundary and a width , where $f$ is a smooth function of acompact support with a length 2b. We show that in the criticalcase, , the operator has nobound statesfor small .On the otherhand, a weakly bound state existsprovided . In thatcase, there are positive c ₁,c ₂ suchthat the corresponding eigenvalue satisfies for all sufficiently small.     

Observables and Statistical Maps

This article begins with a review of the framework of fuzzy probability theory. The basic structure is given by the -effect algebra of effects (fuzzy events) and the set of probability measures on a measurable space . An observable is defined, where is the value space of X. It is noted that there exists a one-to-one correspondence between states on and elements of and between observables and -morphisms from to . Various combinations of observables are discussed. These include compositions, products, direct products, and mixtures. Fuzzy stochastic processes are introduced and an application to quantum dynamics is considered. Quantum effects are characterized from among a more general class of effects. An alternative definition of a statistical map is given and it is shown that any statistical map has a unique extension to a statistical operator. Finally, various combinations of statistical maps are discussed and their relationships to the corresponding combinations of observables are derived.     

Long time behavior for the focusing nonlinear schroedinger equation with real spectral singularities

We consider the effect of real spectral singularities on the long time behavior of the solutions of the focusing Nonlinear Schroedinger equation. We find that for each spectral singularity , such an effect is limited to the region of the (x,t)-plane in which is close to the point of stationary phase (the phase here being defined in a standard way by, say, the evolution of the Jost functions). In that region, the solution performs decaying oscillations of the same form as in the other regions, but with different parameters. The order of decay is .We prove our result by using the Riemann-Hilbert factorization formulation of the inverse scattering problem. We recover our asymptotics by transforming our problem to one which is equivalent for large time, and which can be interpreted as the one corresponding to the genus 0 algebro-geometric solution of the equation.     

The trace formula for Schrödinger operators on the line

This paper discusses certain aspects of the spectral and inverse spectral problems for the Schrödinger operator , for q(x)C(), the space of bounded continuous functions. The trace formula of the title is the relation

Covariant (hh′)-Deformed Bosonic and Fermionic Algebras as Contraction Limits of q-Deformed Ones

GL_h(n) ×GL_h(m)-covariant (hh)-bosonic[or (hh)-fermionic] algebras are built in terms of thecorresponding R_h and -matrices by contracting theGL_q(n) × -covariant q-bosonic (or q-fermionic) algebras , = 1, 2.When using a basis of wherein theannihilation operators are contragredient to thecreation ones, this contraction procedure can be carried out for any n, m values. Whenemploying instead a basis wherein the annihilationoperators, like the creation ones, are irreducibletensor operators with respect to the dual quantumalgebra U_q(gl(n)) , a contraction limit only exists forn, m {1, 2, 4, 6, . . .}. For n = 2, m = 1, andn = m = 2, the resulting relations can be expressed interms of coupled (anti)commutators (as in the classical case), by usingU_h(sl(2)) [instead of s1(2)] Clebsch-Gordancoefficients. Some U_h(sl(2)) rank-1/2irreducible tensor operators recently constructed byAizawa are shown to provide a realization of (2, 1).     

On the Modified Korteweg–De Vries Equation

We consider the large time asymptotic behavior of solutions to the Cauchy problem for the modified Korteweg–de Vries equation , with initial data . We assume that the coefficient is real, bounded and slowly varying function, such that , where . We suppose that the initial data are real-valued and small enough, belonging to the weighted Sobolev space . In comparison with the previous paper (Internat. Res. Notices 8 (1999), 395–418), here we exclude the condition that the integral of the initial data u ₀ is zero. We prove the time decay estimates and for all , where . We also find the asymptotics for large time of the solution in the neighborhood of the self-similar solution.     

Distribution of roots of random real generalized polynomials

The average density of zeros for monic generalized polynomials, , with real holomorphic ,f _k and real Gaussian coefficients is expressed in terms of correlation functions of the values of the polynomial and its derivative. We obtain compact expressions for both the regular component (generated by the complex roots) and the singular one (real roots) of the average density of roots. The density of the regular component goes to zero in the vicinity of the real axis like |lmz|. We present the low- and high-disorder asymptotic behaviors. Then we particularize to the large-n limit of the average density of complex roots of monic algebraic polynomials of the form with real independent, identically distributed Gaussian coefficients having zero mean and dispersion . The average density tends to a simple,universal function of =2nlog|z| and in the domain coth(/2)n|sin arg(z)|, where nearly all the roots are located for largen.     

Logarithmic Sobolev inequality for lattice gases with mixing conditions

Let denote the grand canonical Gibbs measure of a lattice gas in a cube of sizeL with the chemical potential and a fixed boundary condition. Let be the corresponding canonical measure defined by conditioning on . Consider the lattice gas dynamics for which each particle performs random walk with rates depending on near-by particles. The rates are chosen such that, for everyn andL fixed, is a reversible measure. Suppose that the Dobrushin-Shlosman mixing conditions holds for forall chemical potentials . We prove that for any probability densityf with respect to ; here the constant is independent ofn orL andD denotes the Dirichlet form of the dynamics. The dependence onL is optimal.Research partially supported by U.S. National Science Foundations grant 9403462, Sloan Foundation Fellowship and David and Lucile Packard Foundation Fellowship.     

Uncertainty principle and the horizon size of our universe

Quantum uncertainties prevent simultaneous measurement of the expansion factor S(t) and its time derivative . Consequently the Hubble size has an inherent uncertainty in the quantum state that describes the semiclassical evolution of the universe. We show that the quantum uncertainty in the Hubble size of the universe is amplified to unacceptably large values in any inflationary process.This essay received an honorable mention from the Gravity Research Foundation, 1986-Ed.     

On the positivity of the effective action in a theory of random surfaces

It is shown that the functional , defined onC functions on the two-dimensional sphere, satisfies the inequalityS[]0 if is subject to the constraint . The minimumS[]=0 is attained at the solutions of the Euler-Lagrange equations. The proof is based on a sharper version of Moser-Trudinger's inequality (due to Aubin) which holds under the additional constraint ; this condition can always be satisfied by exploiting the invariance ofS[] under the conformal transformations ofS ². The result is relevant for a recently proposed formulation of a theory of random surfaces.On leave from: Istituto di Fisica dell'Università di Parma, Sezione di Fisica Teorica, Parma, Italy     

Long-Time Asymptotics of Solutions to the Cauchy Problem for the Defocusing Nonlinear Schrödinger Equation with Finite-Density Initial Data. II. Dark Solitons on Continua

For Lax-pair isospectral deformations whose associated spectrum, for given initial data, consists of the disjoint union of a finitely denumerable discrete spectrum (solitons) and a continuous spectrum (continuum), the matrix Riemann–Hilbert problem approach is used to derive the leading-order asymptotics as of solutions to the Cauchy problem for the defocusing nonlinear Schrödinger equation ( NLSE), , with finite-density initial data

Classification of three-dimensional covariant differential calculi on Podles' quantum spheres and on related spaces

Covariant differential calculi on the quantum space for the quantum group SL _q (2) are classified. Our main assumptions are thatq is not a root of unity and that the differentials de _j of the generators of form a free right module basis for the first-order forms. Our result says, in particular, that apart from the two casesc =c(3), there exists a unique differential calculus with the above properties on the space which corresponds to Podles' quantum sphereS _qc ^/2 .     

Separable coordinates for four-dimensional Riemannian spaces

We present a complete list of all separable coordinate systems for the equations and with special emphasis on nonorthogonal coordinates. Applications to general relativity theory are indicated.     

Gelation and Cluster Growth with Cluster-Wall Interactions

Metallic cluster growth within a reactive polymer matrix is modeled by augmenting coagulation equations to include the influence of side reactions of metal atoms with the polymer matrix: where > 0 and where c _k denotes the concentration of the kth cluster and p denotes the concentration of reactive sites available within the polymer matrix for reaction with metallic atoms. The initial conditions are required to be non-negative and satisfy and p(0) = p ₀. We assume that for 01, which encompasses both bond linking kernels (R _jk = j k ) and surface reaction kernels (R _jk = j + k ). Our analytical and numerical results indicate that the side reactions delay gelation in some cases and inhibit gelation in others. We provide numerical evidence that gelation occurs for the classical coagulation equations ( = 0) with the bond linking kernel (d ) for 1/2<1. We examine the relative fraction of metal atoms, which coagulate compared to those which interact with the polymer matrix, and demonstrate in particular a linear dependence on ^–1 in the limiting case R _{= jk} , p ₀=1.     

A nonlinear instability for 3×3 systems of conservation laws

The phenomenon of nonlinear resonance provides a mechanism for the unbounded amplification of small solutions of systems of conservation laws. We construct spatially 2-periodic solutionsu ^N C ([0,t _N ] × witht _N bounded, satisfying

Scaling characteristics of ac conductivity and dielectric function in fractal regime of the metal-insulator transition

An analysis of the ac conductivity _ac(), and the ac dielectric constant, (), of the metal-insulator percolation systems is presented in the critical regime near the transition threshold. It is argued that the polarization and relaxation of the finite fractal metallic clusters play dominant roles in controlling the dynamic response of the system on both sides of the threshold. The relaxation time constant of a fractal cluster is shown to scale with its size as withd _t = 4 – 2d +d _c + /, whered is tge Euclidean dimension, andd _c , , and are the scaling indices for the charging, the dc conductivity, and the correlation length respectively. The average time dependent response of the system is shown to scale with a new time scale , where is the correlation length and ₀ is a microscopic time constant. It is shown that at frequencies and with /d_t 1, in close agreement with experiments. The effects of the anomalous transport along the infinite cluster and the medium polarizability are also discussed.