Kac-Moody symmetry of generalized non-linear Schrödinger equations

The classical non-linear Schrödinger equation associated with a symmetric Lie algebra =km is known to possess a class of conserved quantities which from a realization of the algebrak []. The construction is now extended to provide a realization of the Kac-Moody algebrak[, ^–1] (with central extension). One can then define auxiliary quantities to obtain the full algebra [, ^–1]. This leads to the formal linearization of the system.     

The asymmetric contact process on a finite set

The contact process onZ has one phase transition; let _c be the critical value at which the transition occurs. Let _N be the extinction time of the contact process on {0,...,N}. Durrett and Liu (1988), Durrett and Schonmann (1988), and Durrett, Schonmann, and Tanaka (1989) have respectively proved that the subcritical, supercritical, and critical phases can be characterized using a large finite system (instead ofZ) in the following way. There are constants ₁() and ₂() such that if < _c , lim _{N N} /logN = 1/₁(); if > _c , lim _N log _N /N = ₂(); if = _c , lim _{N N} /N= and lim _{N N} /N ⁴=0 in probability. In this paper we consider the asymmetric contact process onZ when it has two distinct critical values _c1< _c2. The arguments of Durrett and Liu and of Durrett and Schonmann hold for < _c1 and > _c2. We show that for [ _c1< _c2), lim _{N N} /N=-1/, (where _i is an edge speed) and for = _c2, lim _N log _N /logN=2 in probability.     

Kinetic equation in the kinetic region of the dilute and nonuniform electron plasma

Mori's scaling method is used to derive the kinetic equation for a dilute, nonuniform electron plasma in the kinetic region where the space-time cutoff (b, t _c) satisfies _Dbl _f , _D t _{c f} , with _D the Debye length, _D ^–1= _p the plasma frequency, andl _f and _f the mean free path and time, respectively. The kinetic equation takes account of the nonuniformity of the order ofl _f and _D for the single-and the two-particle distribution function, respectively. Thus the Vlasov term associated with the two-particle distribution function is retained. This kinetic equation is deduced from the kinetic equation in the coherent region obtained by Morita, Mori, and Tokuyama, where the space-time cutoff of the coherent region satisfies _Dbr ₀, _Dt _{c 0}, withr ₀ the Landau length and ₀ the corresponding time scale.     

C λ-extended oscillator algebra and parasupersymmetric quantum mechanics

The C -extended oscillator algebra is generated by {1, a, a , N, T}, where T is the generator of the cyclic group C of order . It can be realized as a generalized deformed oscillator algebra (GDOA). Its unirreps can thus be easily exhibited using the representation theory of GDOAs and their carrier spaces show a Z-grading structure. Within its infinite-dimensional Fock space representation, this algebra provides a bosonization of parasupersymmetric quantum mechanics of order p = – 1.     

Energy level crossings in quantum mechanics

From the eigenvalue equationH \ _n () =E _n ()\ _n () withH H ₀ +V one can derive an autonomous system of first order differential equations for the eigenvaluesE _n () and the matrix elementsV _mn () where is the independent variable. To solve the dynamical system we need the initial valuesE _n ( = 0) and \ _n ( = 0). Thus one finds the motion of the energy levelsE _n (). We discuss the question of energy level crossing. Furthermore we describe the connection with the stationary state perturbation theory. The dependence of the survival probability as well as some thermodynamic quantities on is derived. This means we calculate the differential equations which these quantities obey. Finally we derive the equations of motion for the extended caseH =H ₀ +V ₁ + ² V ₂ and give an application to a supersymmetric Hamiltonian.     

Perturbation about the mean field critical point

We consider two models that are small perturbations of Gaussian or mean field models: the first one is a double well /4⁴ — /2² perturbation of a massless Gaussian lattice field in the weak coupling limit (0, proportional to ). The other consists of a spin 1/2 Ising model with long-range Kac type interactions; the inverse range of the interaction, , is the small parameter. The second model is related to the first one via a sine-Gordon transformation. The lattice ^d has dimensiond3.In both cases we derive an asymptotic estimate to first order (in or ²) on the location of the critical point. Moreover, we prove bounds on the remainder of an expansion in or around the Gaussian or mean field critical points.The appendix, due to E. Speer, contains an extension of Weinberg's theorem on the divergence of Feynman graphs which is used in the proofs.Supported by NSF Grant # MCS 78-01885Supported by NSF Grant # PHY 78-15920     

Geometry and combinatorics of Julia sets of real quadratic maps

For real a correspondence is made between the Julia setB forz(z–)², in the hyperbolic case, and the set of-chains±(±(±..., with the aid of Cremer's theorem. It is shown how a number of features ofB can be understood in terms of-chains. The structure ofB is determined by certain equivalence classes of-chains, fixed by orders of visitation of certain real cycles; and the bifurcation history of a given cycle can be conveniently computed via the combinatorics of-chains. The functional equations obeyed by attractive cycles are investigated, and their relation to-chains is given. The first cascade of period-doubling bifurcations is described from the point of view of the associated Julia sets and-chains. Certain Julia sets associated with the Feigenbaum function and some theorems of Lanford are discussed.Supported by NSF grant No. MCS-8104862.Supported by NSF grant No. MCS-8203325.     

Hydromagnetic laminar flow of a conducting fluid in an annulus in presence of a radial magnetic field

The object of the present paper is to study the MHD effects on the laminar flow of a viscous, incompressible and conducting fluid in an annulus with arbitrary time-varying pressure gradient and arbitrary initial velocity in presence of a radial magnetic field. Using finite Hankel transform, solutions for both the unsteady and steady flows under different prescribed pressure gradients have been found out.Notation H a constant characterising the intensity of the magnetic field - p hydrostatic pressure - _e magnetic permeability - coefficient of viscosity - kinematic coefficient of voscosity - conductivity of the medium - density - a radius of the inner cylinder - b radius of the outer cylinder - parameter - s positive root - J (sr) Bessel's function of first kind of ordergl - Y (sr) Bessel's function of second kind of order     

Functional derivative for the distribution function of a Λ-nucleon pair in nuclear matter

An exact expression for the functional derivative of the distribution function of a -nucleon pair in nuclear matter is derived. An approximate expression is also derived by means of the Kirkwood superposition approximation. The latter expression is subsequently used to obtain the Euler equation for the correlation functionf(r1) of a -nucleon pair in nuclear matter.     

Mean-Field Critical Behavior for the Contact Process

The contact process is a model of spread of an infectious disease. Combining with the result of ref. 1, we prove that the critical exponents take on the mean-field values for sufficiently high dimensional nearest-neighbor models and for sufficiently spread-out models with d>4:()– _c as _c and ()( _c–)^–1 as _c, where () and () are the spread probability and the susceptibility of the infection respectively, and _c is the critical infection rate. Our results imply that the upper critical dimension for the contact process is at most 4.     

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.     

Singular measures without restrictive intervals

For the transformationT:[0,1][0,1] defined byT(x)=x(1–x) with 04, a is shown to exist for whichT has no restrictive intervals, hence is sensitive to initial conditions, but for which no finite absolutely continuous invariant measure exists forT.Supported by the U.S. Army Research Office through the Mathematical Sciences Institute of Cornell University     

Analysis of the order parameter for uniform nearest particle system

The uniform nearest particle system (UNPS) is studied, which is a continuoustime Markov process with state space . The rigorous upper bound ^(mf) = ( – 1)/ for the order parameter ₂, is given by the correlation identity and the FKG inequality. Then an improvement of this bound ^(mf) is shown in a similar fashion; C( – 1)/|log( – 1) for >1. Recently, Mountford proved that the critical value _c=1. Combining his result and our improved bound implies that if the critical exponent exists, it is strictly greater than the mean-field value 1 in the weak sense.     

Compact recording of information by the holographic method

Conclusions For holography using a convergent subject beam arrangement, the effect associated with increased extrafocal information density is most pronounced when the parameter 2ac/F is small. This relationship between the object parameters (a and c) and the characteristics of the holographic system (, F) is encountered most frequently in problems of microfilming (ac) and radar imaging (ac,a ²/F is small). It is only in the case 2acF99 with an accuracy of not worse than 1%, that we can assert that the sizes of the minimal Fourier holograms will coincide.Gorkii State University. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 22, No. 4, pp. 449–457, April, 1979.     

Energy relaxation in aϕ 4 with long range interactions

We investigate the influence of long range interactions on the relaxation behaviour of a lattice model with an on-site potential of ⁴-type and infinite range harmonic interactions. For finite number of particlesN, it is shown that the autocorrelation functions <E _n(t)E _n > of the fluctuations of the one-particle energiesE _n(t) decays exponentially. The corresponding relaxation time is proportional toN and is given by (T, N) =N₀(T). The temperature dependent time scale ₀ can explicitly be related to the dynamics of a one-particle correlator of the noninteracting system. The results are derived using Mori-Zwanzig projection formalism. The corresponding memory kernel is calculated within a mode coupling approximation and by a perturbative approach. Both results agree in leading order in 1/N. It is speculated that any interaction of range generates a timescale .     

On the thermodynamics of associative recall of structured patterns within a given context

Within a thermodynamic approach we study the associative recall of structured patterns, e.g. complex impressions composed of auditory and visual components, or words consisting of various letters. These words can be recognized either without context, i.e. letter by letter in an independent way, or they can be put into context by favouring a certain percentage of preferred letter combinations (meaningful words) in the Hamiltonian. Particular emphasis is put on the question, under which conditions the system can recognize the preferred words in an improved way without loosing the quality of retrieval of the remaining arbitrary letter combinations. For two-letter words we find a phase diagram depending on three parameters, i.e. the temperatureT, the ratio between the number of letters within the alphabet and the number of neurons, and the enhancement parameter for the preferred combinations. This phase diagram is very rich, e.g. for a given and 0<<1, below a critical temperatureT ₂() one can recognize both the preferred words and-with somewhat reduced error tolerance-also any arbitrary letter combination, whereas forT ₂()<T<T ₁() only the preferred words can be recognized. For <0 the role of preferred and non-preferred combinations is interchanged.     

The S-matrix for abstract scattering systems

Let S() be the S-matrix at energy for an abstract scattering system. We derive a bound, in terms of the interaction, on integrals of the form h () S()- _HS ² d, where denotes the Hilbert-Schmidt norm.Supported by the Swiss National Science Foundation.     

Fractional noise

Fractional noiseN(t),t 0, is a stochastic process for every , and is defined as the fractional derivative or fractional integral of white noise. For = 1 we recover Brownian motion and for = 1/2 we findf ^–1-noise. For 1/2 1, a superposition of fractional noise is related to the fractional diffusion equation.     

Renormalization Group study of a critical lattice model

The Renormalization Group is used to study the correlation functions of a nonlocal hierarchical model mimicking the ()⁴ model, dipole gas and the like. It is shown that the infrared behaviour of the correlations is that of the massless gaussian 1/2c()()².     

The ground state energy of Sdhrödinger operators

We studye()=inf spec(-+V) and examine whene()<0 for all 0. We prove thatc ²e()–d ² for suitableV and all small ||.Research partially funded under NSF grant number DMS-9101716.