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.     

On the existence of invariant,absolutely continuous measures

Let (, , ) be a measure space with normalized measure,f: a nonsingular transformation. We prove: there exists anf-invariant normalized measure which is absolutely continuous with respect to if and only if there exist >0, and , 0<<1, such that (E)< implies (f ^–k(E))< for allk0.     

A Monomial Basis for the Virasoro Minimal Series M(p,p′) : The Case 1<p′/p<2

Quadratic relations are given explicitly in two cases of chiral conformal field theory, and monomial bases of the representation spaces are constructed by using the Fourier components of the intertwiners. The first case is the (2,1) primary fields for the (p,p)-minimal series M_r,s (1rp–1,1sp–1) for the Virasoro algebra where 1<p/p<2. We restrict ourselves to the case p3, for which the (2,1) primary field exists. The second case is the intertwiners corresponding to the two-dimensional representation for the level k integrable highest weight modules V() (0k) for the affine Lie algebra      

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.     

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.     

Dynamical bifurcation with noise

It was shown by A. Neishtadt that dynamical bifurcation, in which the control parameter is varied with a small but finite speed , is characterized by adelay in bifurcation, here denoted _j and depending on . Here we study dynamical bifurcation, in the framework and with the language of Landau theory of phase transitions, in the presence of a Gaussian noise of strength . By numerical experiments at fixed = ₀, we study the dependence of _j on a for order parameters of dimension 3; an exact scaling relation satisfied by the equations permits us to obtain for this the behavior for general . We find that in the smallnoise regime _j() a(^–b ), while in the strong-noise regime _j() – ce(^–d); we also measure the parameters in these formulas.     

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.     

On the invariant sets of a family of quadratic maps

The Julia setB for the mappingz (z–)² is considered, where is a complex parameter. For 2 a new upper bound for the Hausdorff dimension is given, and the monic polynomials orthogonal with respect to the equilibrium measure onB are introduced. A method for calculating all of the polynomials is provided, and certain identities which obtain among coefficients of the three-term recurrence relations are given. A unifying theme is the relationship betweenB and -chains ± (± (± ...), which is explored for –1/42 and for with ||1/4, with the aid of the Böttcher equation. ThenB is shown to be a Hölder continuous curve for ||<1/4.Supported by NSF Grant MCS-8104862Supported by NSF Grant MCS-8002731     

En">Characters and Composition Factor Multiplicities for the Lie Superalgebra $${\mathfrak{g}}{\mathfrak{l}}$$ ( m / n )

The multiplicities a of simple modules L in the composition series of Kac modules V lambda for the Lie superalgebra (m/n ) were described by Serganova, leading to her solution of the character problem for (m/n ). In Serganova's algorithm all with nonzero a are determined for a given this algorithm, turns out to be rather complicated. In this Letter, a simple rule is conjectured to find all nonzero a for any given weight . In particular, we claim that for an r-fold atypical weight there are 2r distinct weights such that a = 1, and a = 0 for all other weights . Some related properties on the multiplicities a are proved, and arguments in favour of our main conjecture are given. Finally, an extension of the conjecture describing the inverse of the matrix of Kazhdan–Lusztig polynomials is discussed.     

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.     

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.     

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.     

Approach to equilibrium in models of a system in contact with a heat bath

We investigate simple model systems in contact with an infinite heat bath. The former consists of a finite number of particles in a bounded region of ^d,d=1,2. The heat baths are infinite particle systems which can penetrate and interact with the system via elastic collisions. Outside the particles move freely and have a Gibbs probability measure prior to entering. We show that starting from almost any initial configuration, the system approaches, ast , the appropriate Gibbs distribution. The combined system plus bath is Bernoulli.Partially supported by NSF Grants PHY 8201708 and DMR 81-14726-02.     

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.     

The Two Dimensional Hubbard Model at Half-Filling. I. Convergent Contributions

We prove analyticity theorems in the coupling constant for the Hubbard model at half-filling. The model in a single renormalization group slice of index i is proved to be analytic in for ||c/i for some constant c, and the skeleton part of the model at temperature T (the sum of all graphs without two point insertions) is proved to be analytic in for ||c/|log T|². These theorems are necessary steps towards proving that the Hubbard model at half-filling is not a Fermi liquid (in the mathematically precise sense of Salmhofer).     

On the existence of rotating stars in general relativity

The Newtonian equations of motion, and Newton's law of gravitation can be obtained by a limit of Einstein's equations. For a sufficiently small constant the existence of a set of solutions (0) of Einstein's equations of a stationary, axisymmetric star is proven. This existence is proven in weighted Sobolev spaces with the implicit function theorem. Since the value of the causality constant depends only on the units used to measure the velocity, the existence of a solution for any small is physically interesting.     

Anderson localization for the almost Mathieu equation: A nonperturbative proof

We prove that for any diophantine rotation angle and a.e. phase the almost Mathieu operator (H()) _n = _n–1 + _n+1 +cos(2(+n)) _n has pure point spectrum with exponentially decaying eigenfunctions for 15. We also prove the existence of some pure point spectrum for any 5.4.     

Lyapunov Modes of Two-Dimensional Many-Body Systems; Soft Disks, Hard Disks, and Rotors

The dynamical instability of many-body systems can best be characterized through the local Lyapunov spectrum {}, its associated eigenvectors {}, and the time-averaged spectrum {}. Each local Lyapunov exponent describes the degree of instability associated with a well-defined direction—given by the associated unit vector —in the full many-body phase space. For a variety of hard-particle systems it is by now well-established that several of the vectors, all with relatively-small values of the time-averaged exponent , correspond to quite well-defined long-wavelength modes. We investigate soft particles from the same viewpoint here, and find no convincing evidence for corresponding modes. The situation is similar—no firm evidence for modes—in a simple two-dimensional lattice-rotor model. We believe that these differences are related to the form of the time-averaged Lyapunov spectrum near =0.     

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.     

Modified London model for the determination of λ and ζ in a high κ superconductor

We present a modified London model suggested by Brandt [1–3] which introduces a finite vortex core size appropriate for isotropic superconductors in which the average internal field is less than approximately (1/4)H _c2. TheSR lineshape resulting from this model possesses a distinctive shape due to the magnetic penetration depth and the vortex core diameter (approximately equal to twice the coherence length ). However, for a given lineshape, there is a large range of values of and which produce nearly the same lineshape. Lineshape smearing caused by disorder in the vortex lattice increases uncertainty in values for and . If well-determined values of either (T) or (T) are not available from another technique, both of them can be determined bySR measurements alone if runs in more than one applied field at the same temperature are fit with and as shared parameters. We also present our method of estimating the degree of disorder in the vortex lattice.