Simulated annealing via Sobolev inequalities

We use Sobolev inequalities to study the simulated annealing algorithm. This approach takes advantage of the local time reversibility of the process and yields the optimal freezing schedule as well as quantitative information about the rate at which the process is tending to its ground state.Research supported in part by NSF Grant DMS-8609944Research supported in part by NSF Grant DMS-8611487 and ARO DAAL03-86-K-0171     

Strong electric field eigenvalue asymptotics for the Schrödinger operator with electromagnetic potential

We consider the Schrödinger operator with electric potential V, which decays at infinity, and magnetic potential A. We study the asymptotic behaviour for large values of the electric field coupling constant of the eigenvalues situated under the essential-spectrum lower bound. We concentrate on the cases of rapidly decaying V (e.g. V L ^m/2( ^m ) for m 3) and arbitrary A, or slowly decaying V (i.e. V(x |x|^– , (0,2), as |x| ) and magnetic potentials A corresponding to constant magnetic fields B = curl A.Partially supported by the Ministry of Culture, Science and Education under Grant No. 52.     

The non-relativistic limit ofP(ϕ)2 quantum field theories: Two-particle phenomena

It is proved that for two-particle phenomena theP()₂ quantum field theories with speed of lightc converge to non-relativistic quantum mechanics with a function potential in the limitc.Supported by NSF Grant No. PHY 7506746     

A new geometrical gravitational theory

A geometrical gravitational theory based on the connection ={ } + ln + ln –g ln is developed. The field equations for the new theory are uniquely determined apart from one unknown dimensionless parameter ₂. The geometry on which our theory is based is an extension of the Weyl geometry, and by the extension the gravitational coupling constant and the gravitational mass are made to be dynamical and geometrical. The fundamental geometrical objects in the theory are a metricg and two gauge scalars and. Physically the gravitational potential corresponds tog in the same way as in general relativity, the gravitational coupling constant to ^–2, and the gravitational mass tou(, ), which is a coscalar of power –1 algebraically made of and. The theory satisfies the weak equivalence principle, but breaks the strong one generally. We shall find outu(, )= on the assumption that the strong one keeps holding good at least for bosons of low spins. Thus we have the simple correspondence between the geometrical objects and the gravitational objects. Since the theory satisfies the weak one, the inertial mass is also dynamical and geometrical in the same way as is the gravitational mass. Moreover, the cosmological term in the theory is a coscalar of power –4 algebraically made of andu(, ), so it is dynamical, too. Finally we give spherically symmetric exact solutions. The permissible range of the unknown parameter ₂ is experimentally determined by applying the solutions to the solar system.     

The borel transform in Euclidean ϕ v 4 local existence for Rev < 4

We consider the ⁴ theory in Euclidean space of complex dimensionv and prove that, for Rev < 4 the renormalized Feynman amplitudes grow at worst exponentially in the number of vertices in the graph. This implies that the Borel transform of any Schwinger function may be defined in a neighborhood of the origin in the Borel plane.Research partially supported by National Science Foundation, Grant Number Phy 77-02277 A 01     

Anisotropy of the low temperature resistivity of hexagonal single crystalline spin glass systems

The impurity contribution to the resistivity in zero field (T) of dilute hexagonal single crystals of ZnMn, CdMn and MgMn has been studied in the mK range on samples cut parallel () and perpendicular () to thec-axis, using a SQUID technique for the measurements. Typical spin glass behavior is found in (T) as well as (T) for all alloys, with Kondo like logarithmic increases at higher temperatures and maxima atT _m at lower temperatures, indicating the influence of impurity interactions. The differences in the corresponding isotropic resistivity _poly(T) between the three systems can qualitatively be understood within the framework of a theoretical model by Larsen, describing (T) as a function of universal quantitiesT/T _K and _RKKY/T _K , where _RKKY is the RKKY-interaction strength andT _K the Kondo temperature. With respect to the two lattice directions studied, the behavior of (T and (T is anisotropic in the Kondo regime as well as in the range where ordering becomes important. While the anisotropy in the Kondo slope can be understood by an anisotropic unitarity limit, the understanding of the anisotropy in region where impurity interactions are important remains problematic.Dedicated to Prof. Dr. S. Methfessel on the occasion of his 60th birthday     

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     

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.     

Multidimensional random walks in random environments with subclassical limiting behavior

In this paper we will describe and analyze a class of multidimensional random walks in random environments which contain the one dimensional nearest neighbor situation as a special case and have the pleasant feature that quite a lot can be said about them. Our results make rigorous a heuristic argument of Marinari et al. (1983), and show that in anyd< we can have (a)X _n is recurrent and (b)X _n (logn)².AMS Mid Career Fellowship 1984–1986. Research also partially supported by NSF Grant MCS 83-00836     

Learning withQ-state clock neurons

We study the optimal learning capacity for neural networks withQ-state clock neurons, i.e. the states arecomplex numbers with magnitude 1 and azimuthal anglesn·2/Q, withn=0, 1, ...,Q–1. Performing a phase space analysis, the learning capacity _c for given stability can be expressed by means of a double-integral with a simple geometrical interpretation, which for vanishing reduces to _c (Q) = 4Q/(3Q–4), forQ3. Then we define a training algorithm, which generalizes the well-known AdaTron algorithm fromQ=2 toQ3 and converges very fast to the network with optimal stability, if the numberp of random patterns to be learned is smaller than _c (Q). Finally, in the conclusions, we also give hints on applications for image recognition and in a note added in proof we generalize some results to Potts model networks.     

Decay of a Yang-Mills field coupled to a scalar field

Consider a gauge fieldF and a scalar field with a self-couplingV() as well as the standard coupling betweenF and . If 02V()·V(), there are no classical lumps. IfV()=||⁴ the system is conformally invariant and all the energy radiates out along the light cone.Research supported in part by NSF grants MCS 77-01340 and MCS 78-03567     

Limits on stability of positive molecular ions

It is shown that a positive diatomic molecule consisting of N electrons and two nuclei with charges Z ₁ and Z ₂ is unstable with respect to breakup into two atomic subsystems if the nuclear charges are sufficiently large. Bounds on the critical charge are obtained in the limit as N .Research supported in part by NSF grants DMS-8709805 and DMS-8808112.     

Phases of the number-theoretic spin chain

We present numerical and analytical evidence for a first-order phase transition of the ferromagnetic spin chain with partition functionZ()=(–1)/() at the inverse temperature _cr=2.     

Classical and quantum-mechanical systems of Toda lattice type. I

The structure of the commutant of Laplace operators in the enveloping and Poisson algebra of certain generalized ax +b groups leads (in this article) to a determination of classical and quantum mechanical first integrals to generalized periodic and non-periodic Toda lattices. Certain new Hamiltonian systems of Toda lattice type are also shown to fit in this framework. Finite dimensional Lax forms for the (periodic) Toda lattices are given generalizing results of Flaschke.Research partially supported by NSF grant MCS 79-03223Research partially supported by NSF grant MCS 79-03153     

Inertial effects on the escape rate of a particle driven by colored noise: An instanton approach

A recent calculation, in the weak-noise limit, of the rate of escape of a particle over a one-dimensional potential barrier is extended by including an inertial term in the Langevin equation. Specifically, we consider a system described by the Langevin equation , where is a Gaussian colored noise with mean zero and correlator (t)(t')=(D/)exp(–|t–t'|/). A pathintegral formulation is augmented by a steepest descent calculation valid in the weak-noise (D0) limit. This yields an escape rateexp(–S/D), where the actionS is the minimum, over paths characterizing escape over the barrier, of a generalized Onsager-Machlup functional, the extremal path being an instanton of the theory. The extremal actionS is calculated analytically for smallm and for general potentials, and numerical results forS are displayed for various ranges ofm and for the typical case of the quartic potentialV(x)=–x ₂/2+x ₄/4.     

How often do simple dynamical processes have infinitely many coexisting sinks?

This work concerns the nature of chaotic dynamical processes. Sheldon Newhouse wrote on dynamical processes (depending on a parameter )x _x+1=T(x _n ; ), wherex is in the plane, such as might arise when studying Poincaré return maps for autonomous differential equations in IR³. He proved that if the system is chaotic there will very often be existing parameter values for which there are infinitely many periodic attractors coexisting in a bounded region of the plane, and that such parameter values would be dense in some interval. The fact that infinitely many coexisting sinks can occur brings into question the very nature of the foundations of chaotic dynamical processes. We prove, for an apparently typical situation, that Newhouse's construction yields only a set of parameter values of measure zero.This research was supported in part by grants from the Air Force Office of Scientific Research AFOSR 81-0217, the Consiglio Nazionale delle Ricerche-Comitato per le Matematiche, and the National Science Foundation DMS 84-19110On leave from: Dipartimento di Matematica G. Castel nuovo Universita di Roma La Sapienza P. le Aldo Moro 5, I-00185 Rome, Italy     

Positron annihilation anomaly in the high-temperature superconductor YBa2Cu3O7−x

We have measured the ac susceptibility of a wire with a Nb core (1.27 mm diam.) and a Cu cladding (0.37 mm thickness) atT50 K andB0.1 mG. Due to its proximity to Nb, the Cu becomes fully superconducting. From the data we find a breakdown fieldH _b =1.2 (mG) and a coherence length =2.2T ^–1/2 (m) for the Cu, as well as a field penetration depth -34T ^1/2 (m) at the Cu/Nb interface.     

Exact results for the asymmetric simple exclusion process with a blockage

We present new results for the current as a function of transmission rate in the one-dimensional totally asymmetric simple exclusion process (TASEP) with a blockage that lowers the jump rate at one site from one tor<1. Exact finitevolume results serve to bound the allowed values for the current in the infinite system. This proves the existence of a nonequilibrium phase transition, corresponding to an immiscibility gap in the allowed values of the asymptotic densities which the infinite system can have in a stationary state. A series expansion inr, derived from the finite systems, is proven to be asymptotic for all sufficiently large systems. Padé approximants based on this series, which make specific assumptions about the nature of the singularity atr=1, match numerical data for the infinite system to 1 part in 10⁴.     

A nonequilibrium entropy for dynamical systems

It is proposed to define entropy for nonequilibrium ensembles using a method of coarse graining which partitions phase space into sets which typically have zero measure. These are chosen by considering the totality of future possibilities for observation on the system. It is shown that this entropy is necessarily a nondecreasing function of the timet. There is no contradiction with the reversibility of the laws of motion because this method of coarse graining is asymmetric under time reversal. Under suitable conditions (which are stated explicitly) this entropy approaches the equilibrium entropy ast+ and the fine-grained entropy ast–. In particular, the conditions can always be satisfied if the system is aK-system, as in the Sinai billiard models. Some theorems are given which give information about whether it is possible to generate the partition used here for coarse graining from time translates of a finite partition, and at the same time elucidate the connection between our concept of entropy and the entropy invariant of Kolmogorov and Sinai.Research supported in part by NSF grants PHY78-03816 and PHY78-15920.     

On the equivalence of different order parameters for Ising ferromagnets

In a recent note Barber showed, for a spin-1/2 Ising system with ferromagnetic pair interactions, that some critical exponents of the triplet order parameter _{i j k} are the same as those of the magnetization _i . Here we prove such results for all odd correlations and dispense with the requirement of pair interactions. We also prove that the critical temperatureT _c , defined as the temperature below which there is a spontaneous magnetization, is for fixed even spin interactionsJ _e independent of the way in which the odd interactionsJ _o approach zero from above. This is achieved by using only the simplest, Griffiths-Kelley-Sherman (GKS), inequalities, which apply to the most general many-spin, ferromagnetic interactions.Research supported in part by NSF Grant #MPS 75-20638.