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

Influence of a strong longitudinal static electric field on free carrier faraday effect in an n-type InSb at room temperature at submillimeter wave frequencies

The expression for free carrier Faraday rotation and for ellipticity , as the function of the applied parallel static electric field and static magnetic field for a given value of wave angular frequency and electron concentration N₀, are obtained and theoretically analyzed with the aid of one-dimensional linearized wave theory and Kane's non-parabolic isotropic dispersion law. It is shown that the maximum Faraday rotation occurs near the cyclotron resonance condition, which can be expressed as , where , , and . Here m^* and e denote the effective mass and charge of electron, respectively. _g is the forbidden bandgap of semiconductor. v₀ is the carrier drift velocity, which is a non-linear function of E₀ in high field condition. A possibility of a simple way of determining the non-linear v₀ vs E₀ characteristics of semiconductors by the measurement of Faraday rotation is also discussed.     

Light Σ hypernuclei and Σ− atoms

The structures of ⁴ He and ⁷ Li are investigated within the frame of the [³H + ⁺] + + [³He + ^o] model and the microscopic cluster model +2N+, respectively. The different binding mechanism between ⁴ He and ⁷ Li is emphasized together with a brief discussion on the ⁴ Be hypernucleus. It is clarified that the ⁴ He hypernucleus is bound due to the cooperative contribution from the and terms of -N interaction, while the ⁷ Li hypernucleus is bound mainly due to the term of -N interaction. The combined analysis of ^– atoms and ⁴ He hypernucleus is shown to be useful to reveal the characteristic differences among the model D, model F and soft core versions of the Nijmegen OBE potentials.The authors would like to express their sincere thanks to J. ofka, O. Richter, M. Sotona, L. Majling, A. Cieply and other colleagues in Nuclear Physics Institute (e/Prague), and H. Band, Y. Yamamoto and T. Motoba, for collaboration and useful discussions. We dedicate this article to the memory of the excellent activities of the late Prof. Jan ofka.     

Higher spin BRS cohomology of supersymmetric chiral matter inD=4

We examine the BRS cohomology of chiral matter inN=1,D=4 supersymmetry to determine a general form of composite superfield operators which can suffer from supersymmetry anomalies. Composite superfield operators _{(a, b)} are products of the elementary chiral superfieldsS and and the derivative operatorsD , and . Such superfields _{(a, b)} can be chosen to have a symmetrized undotted indices _i and b symmetrized dotted indices . The result derived here is that each composite superfield _(a,b) is subject to potential supersymmetry anomalies ifa–b is an odd number, which means that _(a,b) is a fermionic superfield.     

An upper bound on the free energy for classical systems with Coulomb interactions in a varying external field

The thermodynamic limit is taken using a sequence of regions all the same shape as a given region of volume ||, with a specified distribution of normal field component on . We show that with magnetostatic interactions the limiting free energy density is bounded above by jhen where (,B) is the free energy density for a system of density in a uniform external fieldB and the inf is taken over all divergence-free fieldsB with given normal component on and all densities (x) compatible with particle number constraints of the form where _i is a sub-region of . A physical argument suggests that this upper bound is the true thermodynamic limit, and that it takes account demagnetization effects. Electrostatic interactions can be treated similarly.     

On the Leibniz Homology of Poisson Algebras

We study the Leibniz homology of the Poisson algebra of polynomial functions over (²ⁿ ,) where is the standard symplectic structure. We identify it with certain highest-weight vectors of some _2n ( )-modules and obtain some explicit result in low degree.     

Spectral and scattering theory for a Schrödinger operator with a class of momentum-dependent long-range potentials

Let be the selfadjoint operator for the static electromagnetic field where W _j for 0, 1, 2, ..., n is a sum of (i) a short-range potential and (ii) a smooth long-range potential decreasing at as |x|^- with in (0, 1]. Then for >1/2, asymptotic completeness holds for the scattering system (H, H ₀).     

Analyticity properties of eigenfunctions and scattering matrix

For potentialsV=V(x)=O(|x|^–2–) for |x|,x³ we prove that if theS-matrix of (–, –+V) has an analytic extension to a regionO in the lower half-plane, then the family of generalized eigenfunctions of –+V has an analytic extension toO such that for |Imk|<b. Consequently, the resolvent (–+V–z ²)^–1 has an analytic continuation from ⁺ to {kOImk|<b} as an operator from _b ={f=e ^–b|x| g|gL ₂(³)} to _–b . Based on this, we define for potentialsW=o(e ^–2b|x|) resonances of (–+V, –+V+W) as poles of and identify these resonances with poles of the analytically continuedS-matrix of (–+V, –+V+W).The author would like to thank the Institute for Advanced Study for its hospitality and the National Science Foundation for financial support under Grant No. DMS-8610730(1)     

Ensemble and trajectory statistics in a nonlinear partial differential equation

We have examined the influence of parametric noise on the solution behavioru(t, x) of a nonlinear initial value() problem arising in cell kinetics. In terms of ensemble statistics, the eventual limiting solution mean and variance are well-characterized functions of the noise statistics, and and depend on . When noise is continuously present along the trajectory, and are independent of the noise statistics and . However, in their evolution toward and , both _u (t, x) and _u ² (t, x) depend on the noise and.     

BV Structure of the Cohomology of Nilpotent Subalgebras and the Geometry of(W) Strings

Given a simple, simply laced, complex Lie algebra corresponding to the Lie group G, let be thesubalgebra generated by the positive roots. In this Letter we construct aBV algebra whose underlying graded commutative algebra is given by the cohomology, with respect to , of the algebra of regular functions on G with values in . We conjecture that describes the algebra of allphysical (i.e., BRST invariant) operators of the noncritical string. The conjecture is verified in the two explicitly known cases, ₂ (the Virasoro string) and ₃ (the string).     

Eigenvalue Asymptotics for the Schrödinger Operator with a δ-Interaction on a Punctured Surface

Given n2, we put r=min . Let be a compact, C ^r -smooth surface in ⁿ which contains the origin. Let further be a family of measurable subsets of such that as . We derive an asymptotic expansion for the discrete spectrum of the Schrödinger operator in L ²( ⁿ ), where is a positive constant, as . An analogous result is given also for geometrically induced bound states due to a interaction supported by an infinite planar curve.     

Exact Large Deviation Functional of a Stationary Open Driven Diffusive System: The Asymmetric Exclusion Process

We consider the asymmetric exclusion process (ASEP) in one dimension on sites i=1,...,N, in contact at sites i=1 and i=N with infinite particle reservoirs at densities _a and _b . As _a and _b are varied, the typical macroscopic steady state density profile ¯(x), x[a,b], obtained in the limit N=L(b–a), exhibits shocks and phase transitions. Here we derive an exact asymptotic expression for the probability of observing an arbitrary macroscopic profile , so that is the large deviation functional, a quantity similar to the free energy of equilibrium systems. We find, as in the symmetric, purely diffusive case q=1 (treated in an earlier work), that is in general a non-local functional of (x). Unlike the symmetric case, however, the asymmetric case exhibits ranges of the parameters for which is not convex and others for which has discontinuities in its second derivatives at (x)=¯(x). In the latter ranges the fluctuations of order in the density profile near ¯(x) are then non-Gaussian and cannot be calculated from the large deviation function.     

Internal Lifschitz singularities for one dimensional Schrödinger operators

The integrated density of states of the periodic plus random one-dimensional Schrödinger operator ;f0,q _i ()0, has Lifschitz singularities at the edges of the gaps inSp(H ). We use Dirichlet-Neumann bracketing based on a specifically one-dimensional construction of bracketing operators without eigenvalues in a given gap of the periodic ones.     

Yet another formulation of the Dirac equation

The 2-by-2 Pauli matrix algebra is used to write the 1-by-4 Dirac field in anequivalent 2-by-2 matrix . The current 4-vectors and *_µ are then compared and the latter is shown to not be easily interpretable as a probability density, and also tocontain .     

On the tensor product of supersingleton representations ofosp(1, 2n)

The tensor product of two supersingleton representations _n of the Lie superalgebraosp (1, 2n) is studied forn2. The main results are as follows: (a) anticommutators and commutators of the odd generators in _{n n} form a skew-symmetric representation of the Lie algebrau(n, n); (b) simple explicit form of all irreducible components of _{n n}, which are labelled by a single parameterJ=0, 1, ..., has been found. Each of them is a^*-representation ofosp (1, 2n) for which assertion (a) is valid. The dimension of its vacuum subspace equals , i.e., the nondegenerate vacuum occurs for J=0 only. Basic property of this family of irreducible^*-representations of osp(1, 2n) are analogous to those of massless representations of osp(1, 4).Dedicated to Academician Václav Votruba on the occasion of his eightieth birthday.     

Conformally Invariant Quantization at Order Three

Let (M, g) be a pseudo-Riemannian manifold and the space of densities of degree on M. Denote the space of differential operators from to of order k and S _k with = – the corresponding space of symbols. We construct (the unique) conformally invariant quantization map . This result generalizes that of Duval and Ovsienko.     

On a Shock Front in Burgers Turbulence

We investigate how chaos propagates in the solution of Burgers equation _t u+u _x u=0 with initial condition u(,0) distributed as a white noise on and 0 on . We describe the evolution of the shock front that travels to the left. Asymptotics are given for both large and small time t.     

Layering transition in SOS model with external magnetic field

For the SOS model defined by the Hamiltonian , where _x , _x ,{1,2,...},h>0,x ^d ,d2 it is shown that in the low-temperature region an infinite sequence of first-order phase transitions takes place whenh»0 and the temperature is fixed.     

Conformal Quantum Effects and the Anisotropic Singularities of Scalar-Tensor Theories of Gravity

We show that the inclusion of a term C _abcd C ^abcd in the action can remove the recently described anisotropic singularity occurring on the hypersurface F () = 0 of scalar-tensor theories of gravity of the type

Influence of gravity on surface melting

A chemical network system for use in the study of reaction cascades is described by the nonlinear rate equation , in which (x) is derived from the –G/x of Taylor's expansion of the order parameterx of the thermodynamic potential, Gibbs' functionG(T,P,x), at about the critical pointC(T _C ,P _C ) of the control variables (parameters)T andP. The behavior of the system around the stable pointx=0 is analyzed only with the sign ofk ₁(T, P), becausek ₂(T, P) is always positive. The system is not closed and is affected by physical and chemical changes in the neighbor systems. WhenT andP fluctuate ( ) through the changes andk ₁ passes zero, the system in the steady state becomes instable andk ₁ jumps at the reaction threshold. Then, the products are formed at the number of moleculesx=(|k ₁|/k ₂)^1/2. To describe such a transition,k ₁ giving theG curvature atx=0 is represented phenomenologically by an approximate function, tanh(G/RT), for a metastable state with a relatively smallk ₁(>0). The reaction takes place in a cascade in accordance with a cubic state equation obtained from tanh(G/RT), which describes a jump of the reaction energy G at the reaction threshold.