On a regularity problem occurring in connection with Anosov diffeomorphisms

Let be aC -manifold and _s and _u be two Hölder foliations, transverse, and with uniformlyC leaves. If a functionf is uniformlyC along the leaves of the two foliations, then it isC on . The proof is elementary.     

Global stability of the rarefaction wave of a one-dimensional model system for compressible viscous gas

This paper is concerned with the asymptotic behavior toward the rarefaction wave of the solution of a one-dimensional barotropic model system for compressible viscous gas. We assume that the initial data tend to constant states atx=±, respectively, and the Riemann problem for the corresponding hyperbolic system admits a weak continuous rarefaction wave. If the adiabatic constant satisfies 12, then the solution is proved to tend to the rarefaction wave ast under no smallness conditions of both the difference of asymptotic values atx=± and the initial data. The proof is given by an elementaryL ²-energy method.     

Anomalies and curvature ofW manifolds

We study the holomorphic structure of certain complex manifolds associated withW algebras, namely, the flag manifoldsW /T andW ₁₊/T ₁₊, and the spacesW /SL(),R) andW ₁₊/GL(,R), whereT andT ₁₊ are the maximal tori inW andW ₁₊. We compute their Ricci curvature and show how the results are related to the anomaly-freedom conditions forW andW ₁₊. We discuss the relation of these manifolds with extensions of universal Teichmüller space.Supported in part by the U.S. Department of Energy, under grant DE-AS05-81ER40039Supported in part by the U.S. Department of Energy, under grant DE-FG03-84ER40168     

The number of bound states of singular oscillating potentials

For a spherically symmetric potential such that rVL ¹(a, ), a>0, and is such that, if we define W=– _r V(t) d(t), W belongs to L ¹ (0, ) and rW0 as r0, we show that the number of bound states in any partial-wave satisfies the bound n2 ₀ r W ² dr. It was shown in a previous paper [1] that this class of potentials is regular from the point of view of abstract scattering theory as well as from the time-independent theory and the Jost function approach. We show also that, for large values of the coupling constant, n(gV) has the asymptotic behaviour C _±g ₀ W(r) dr as g±.     

Solutions without phase-slip for the Ginsburg-Landau equation

We consider the Ginsburg-Landau equation for a complex scalar field in one dimension and consider initial data which have two different stationary solutions as their limits in space asx±. If these solutions are not very different, then we show that the initial data will evolve to a stationary solution by a phase melting process which avoids phase slips, i.e., which does not go through zero amplitude.     

Absence of asymptotic-time symmetry breaking in zero-bias spin-boson model with partial relaxation

The symmetric spin-boson model without external field is treated for any type of coupling to the boson bath and any initial bath density matrix. With initially fully aligned spin (_z (0)= =1), the proof is given that a partial relaxation (_z (+) t1<) implies that there is no asymptotic-time (up-and-down) symmetry breaking (i.e. that _z (+)=0). For the problem of a particle (interacting with free bosons) in a symmetric double well without spatial symmetry breaking before the infinite time limit, this means that att + the particle distribution becomes symmetric (irrespective of the full initial asymmetry) unless the particle fully remains (att + ) in Ihe starting well.     

Classification of all Poisson-Lie structures on an infinite-dimensional jet group

A local classification of all Poisson-Lie structures on an infinite-dimensional group G of formal power series is given. All Lie bialgebra structures on the Lie algebra G of G are also classified.     

Rigorous Spectral Analysis of the Metal–Insulator Transition in a Limit-Periodic Potential

We consider the limit-periodic Jacobi matrices associated with the real Julia sets of f (z)=z ²– for which [2, ) can be seen as the strength of the limit-periodic coefficients. The typical local spectral exponent of their spectral measures is shown to be a harmonic function in decreasing logarithmically from 1 to 0.     

Properties of theL p -solutions of linear impulsive equations in a banach space

Conditions for the existence ofL _p-solutions (1p) of linear impulsive equations in a Banach space are found.     

Calculation and optimization of parameters of radiating distributed feedback structures

A simple method is suggested for calculation of reflection, radiation and transmission coefficients for the distributed feedback structure in the second diffraction order. The method is based on a slight difference between coefficients of reflectionR and radiationI of the surface wave for = (where is the light wavelength corresponding to a precise resonance for the grating length I) and those for =_l (where _l is the light wavelength corresponding to the resonance for the finite grating length). The simplicity of the method makes it possible to use it for optimization of the distributed feedback structure by a number of parameters. The technique can be used in the case of thin-film and diffused waveguides for both TE and TM modes.     

Existence of a Constant for Finite System Extinction

We show the existence of a constant (0, ) such that if ⁿ is the extinction time for a supercritical contact process on [1, n] ^d starting from full occupancy, then {log(E[ ⁿ])}/n ^d tend to as n tends to infinity.     

On the asymptotic behaviour of infinite gradient systems

We consider gradient systems of infinitely many particles in one-dimensional space interacting via a positive invariant pair potential with a hard core. The main assumption is that is strictly convex within the rangeR of (whereR is a fixed number ). Under some technical conditions we prove the following theorems: Let the initial distribution be given by a translation invariant point process onR ¹. Then there exists only one extreme equilibrium state with a given intensityI() satisfyingI()R ^–1, and all ergodic initial distributions with an intensityI()R ^–1 converge weakly ast to the extreme equilibrium state with the same intensity.     

Positive resonances and the limiting amplitude principle

We define the positive resonance points of self-adjoint operators without using the analytical continuation of corresponding resolvents and show that the limiting amplitude principle for the abstract wave equation does not take place in general, if ² = , where is the disturbing frequency and is the resonance point. The asymptotics of corresponding solutions as t are obtained, which imply the growth of the oscillation amplitude as t , 0<<1, or as ln t, t .     

Rate equations in the hopping transport

The usual kinetic equations for the site occupation probabilities in an external field are solved exactly in a simple one-dimensional periodic model with two kinds of atoms using a) free boundary conditions and order of limitsN, 0 needed for a proper treatment of the dc conductivity here b) boundary conditions with metallic contacts and order of limitsN, 0 and c) the same boundary conditions but reversed order of limiting processes 0,N typical of e.g. numerical and percolation treatments. (N and are the number of sites and frequency.) It is demonstrated that though the bulk dc conductivity is the same in all three cases, local bulk properties of the material are strongly dependent on the régime used. The role of the order of all three limiting processes 0,N+ andn+ (N–n+) for local shifts of the chemical potential _n in the dc limit is examined (n is the number of the relevant site calculated from a boundary of the chain). It is shown especially that the rate equation treatment (régime a) on the one hand and numerical or percolation treatments (régime c) on the other hand never yield the same bulk values of _r.     

Variance calculations and the Bessel kernel

In the Laguerre ensembleof n xN Hermitian matrices, it is of interest both theoretically and for applications to quantum transport problems to compute the variance of a linear statistic, denoted var_N f, asN . Furthermore, this statistic often contains an additional parameter a for which the limit is most interesting and most difficult to compute numerically. We derive exact expressions for both lim_N var_N f and lim , lim_N var_N f.     

Two results concerning asymptotic behavior of solutions of the Burgers equation with force

We consider the Burgers equation with an external force. For the case of the force periodic in space and time we prove the existence of a solution periodic in space and time which is the limit of a wide class of solutions ast . If the force is the product of a periodic function ofx and white noise in time, we prove the existence of an invariant distribution concentrated on the space of space-periodic functions which is the limit of a wide class of distributions ast .     

Quasifinite highest weight modules over the superW 1+∞algebra

We study quasifinite highest weight modules over the supersymmetric extension of theW ₁₊ algebra on the basis of the analysis by Kac and Radul. We find that the quasifiniteness of the modules is again characterized by polynomials, and obtain the differential equations for highest weights. The spectral flow, free field realization over the (B, C)-system, and the embedding into (|) are also presented.Address after April 1, 1994: Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606, JapanAddress after April 1, 1994: Uji Research Center, Yukawa Institute for Theoretical Physics, Kyoto University, Uji 611, Japan     

The Percolation Transition for the Zero-Temperature Stochastic Ising Model on the Hexagonal Lattice

On the planar hexagonal lattice , we analyze the Markov process whose state (t), in , updates each site v asynchronously in continuous time t0, so that _v (t) agrees with a majority of its (three) neighbors. The initial _v (0)'s are i.i.d. with P[ _v (0)=+1]=p[0,1]. We study, both rigorously and by Monte Carlo simulation, the existence and nature of the percolation transition as t and p1/2. Denoting by ⁺(t,p) the expected size of the plus cluster containing the origin, we (1) prove that ⁺(,1/2)= and (2) study numerically critical exponents associated with the divergence of ⁺(,p) as p1/2. A detailed finite-size scaling analysis suggests that the exponents and of this t= (dependent) percolation model have the same values, 4/3 and 43/18, as standard two-dimensional independent percolation. We also present numerical evidence that the rate at which (t)() as t is exponential.     

Notes on the classical and the nonrelativistic limits in quantum mechanics

A method is presented which can be used to discuss both the classical and also the nonrelativistic limit of quantum mechanics. A one-to-one correspondence may be established between the asymptotic convergence of the resolvent and that of the timedependent solution. In so far as the question of dynamics is concerned we investigate the relation between families of nonrelativistic Hamiltonians and the corresponding Dirac-Hamiltonians when c± or when c±0. The nonrelativistic free theory formally shows the same pattern when ±0 (the classical limit) or when ±. The investigation finally shows how the asymptotic convergence of the relativistic theory can take place under some fairly general conditions of the radiation field.     

The free energy of a class of Hopfield models

We show that in the limitp ,N 0,=p/N 0 the limit free energy of the Hopfield model equals in probability the Curie-Weiss free energy. We prove also that the free energy of the Hopfield model is self-averaging for any finite .