Einstein-Ōtsuki vacuum equations

The generalisation of the Einstein vacuum theory to tsuki geometry is considered. It is shown that the theory based on Lagrangian density -gR is consistent and leads to a theory that is classically indistinguishable from the Einstein theory.     

A Note on Static Solutions of a Lorentz Invariant Equation in Dimension 3

Cut-and-project sets with convex acceptance windows, based on irrationalities = (1+5), =1+2, =2+3 are models for experimentally observed quasicrystals – materials with diffraction patterns consisting of sharp Bragg peaks in crystallographically disallowed patterns. We show that for each of these three irrationalities there exists a unique binary operation of the type x _s y:=sx+(1–s)y, such that one-dimensional cut-and-project sets are precisely Delone sets closed under this operation.     

On the Eigenstates of the Elliptic Calogero–Moser Model

It is known that the trigonometric Calogero–Sutherland model is obtained by the trigonometric limit (–1) of the elliptic Calogero–Moser model, where (1, ) is a basic period of the elliptic function. We show that for all square-integrable eigenstates and eigenvalues of the Hamiltonian of the Calogero–Sutherland model, if exp(2–1) is small enough then there exist square-integrable eigenstates and eigenvalues of the Hamiltonian of the elliptic Calogero–Moser model which converge to the ones of the Calogero–Sutherland model for the 2-particle and the coupling constant l is positive integer cases and the 3-particle and l=1 case. In other words, we justify the regular perturbation with respect to the parameter exp(2–1). With some assumptions, we show analogous results for N-particle and l is positive integer cases.     

Large charge fluctuations in classical Coulomb systems

The typical fluctuation of the net electric chargeQ contained in a subregion of an infinitely extended equilibrium Coulomb system is expected to grow only as S, whereS is the surface area of. For some cases it has been previously shown thatQ/S has a Gaussian distribution as ¦¦. Here we study the probability law for larger charge fluctuations (large-deviation problem). We discuss the case when both ¦¦ andQ are large, but now withQ of an order larger than S. For a given value ofQ, the dominant microscopic configurations are assumed to be those associated with the formation of a double electrical layer along the surface of. The probability law forQ is then determined by the free energy of the double electrical layer. In the case of a one-component plasma, this free energy can be computed, for large enoughQ, by macroscopic electrostatics. There are solvable two-dimensional models for which exact microscopic calculations can be done, providing more complete results in these cases. A variety of behaviors of the probability law are exhibited.     

Analytic study of power spectra of the tent maps near band-splitting transitions

Successive band-splitting transitions occur in the one-dimensional map x_i+1=g(x_i),i=0, 1, 2,... withg(x)=x, (0 x 1/2) –x +, (1/2 <x 1) as the parameter is changed from 2 to 1. The transition point fromN (=2ⁿ) bands to 2Nbands is given by=(2)^1/N (n=0, 1,2,...). The time-correlation function _i=x_ix₀/(x₀)²,x_i x_i–x_i is studied in terms of the eigenvalues and eigenfunctions of the Frobenius-Perron operator of the map. It is shown that, near the transition point=2, _i–[(10–42)/17] _i,0-[(102-8)/51]_i,1 + [(7 + 42)/17](–1)ⁱe^–yi, where2(–2) is the damping constant and vanishes at=2, representing the critical slowing-down. This critical phenomenon is in strong contrast to the topologically invariant quantities, such as the Lyapunov exponent, which do not exhibit any anomaly at=2. The asymptotic expression for _i has been obtained by deriving an analytic form of _i for a sequence of which accumulates to 2 from the above. Near the transition point=(2)^1/N, the damping constant of _i fori N is given by _N=2(^N-2)/N. Numerical calculation is also carried out for arbitrary a and is shown to be consistent with the analytic results.     

The π+ d → pp reaction at intermediate energies

Conclusion The joint efforts of the laboratories with intermediate energies accelerators at ANL, KEK, LANL, LNPI, Saclay, SIN and TRIUMF produced a rather definite picture of the ⁺pp reaction in the isobar region. However the problem of anomalies for the square-root branch point due to the production of isobar in the intermediate state have not been resolved so far. The poles obtained in different phenomenological analyses could be just the effective parametrization of these cuts (see e.g. [13]). Anyway, some singularities are definitely necessary to describe the data.The experimental data in regions s < 2·05 GeV and s < 2·25 GeV are clearly insufficient. Further accumulation of the data in these regions is necessary. Polarized deuteron data are badly needed in the whole kinematical range [13]. Besides as it was mentioned before it is necessary to measure accurately cross sections of other inelastic reactions which are important for the investigation (observation?) of dibaryonsNN N, NN andd N, NN [21]. It would be of interest also to measure cross sections of non quasi two-particle channels likeNN N, 2N andd N, 2N [22]. (In the case ofNN scattering quasi two-particle channels are known to dominate in a wide energy range.)Presented at the symposium Mesons and Light Nuclei, Bechyn, Czechoslovakia, May 27–June 1, 1985.I am grateful to L. A. Kondratyuk, M. G. Ryskin and M. I. Strikman for valuable discussions of the considered problems.     

Riccati-coupled similarity shock wave solutions for multispeed discrete Boltzmann models

We study nonstandard shock wave similarity solutions for three multispeed discrete Boltzmann models: (1) the square 8_i, model with speeds 1 and 2 with thex axis along one median, (2) the Cabannes cubic 14_i model with speeds 1 and 3 and thex axis perpendicular to one face, and (3) another 14_i, model with speeds 1 and 2. These models have five independent densities and two nonlinear Riccati-coupled equations. The standard similarity shock waves, solutions of scalar Riccati equations, are monotonic and the same behavior holds for the conservative macroscopic quantities. First, we determine exact similarity shock-wave solutions of coupled Riccati equations and we observe nonmonotonic behavior for one density and a smaller effect for one conservative macroscopic quantity when we allow a violation of the microreversibility. Second, we obtain new results on the Whitham weak shock wave propagation. Third, we solve numerically the corresponding dynamical system, with microreversibility satisfied or not, and we also observe the analogous nonmonotonic behavior.     

The Action of √−Δ on Weighted Sobolev Spaces

The action of – on distributions is examined within the context of weighted Sobolev spaces. The results obtained are as follows: (1) – is a continuous map of R ⁿ ), the space of rapidly decreasing functions, to L ^{2, s} (R ⁿ for any s < n/2 +1; (2) if k R and s > –n/2 – 1, then – is a continuous map from H ^{k, s} (R ⁿ ), the weighted Sobolev space, to H ^{k–1, t} (R ⁿ for some t. The results are optimal in a sense.     

On the extremality of the disordered state for the Ising model on the Bethe lattice

We give a simple proof that the limit Ising Gibbs measure with free boundary conditions on the Bethe lattice with the forward branching ratio k2 is extremal if and only if is less or equal to the spin glass transition value, given by tanh( _c ^SG = 1/k.The work was partially supported by the NSF grant DMS 9504513.     

Gauge and Space-Time Symmetry Unification

Unification ideas suggest an integral treatment of fermion and boson spin andgauge-group degrees of freedom. Hence, a generalized quantum field equation,based on Dirac's, is proposed and investigated which contains gauge and flavorsymmetries, determines vector gauge field and fermion solution representations,and fixes their mode of interaction. The simplest extension of the theory with a6-dimensional Clifford algebra has an SU(2) _L × U(1) symmetry, which isassociated with the isospin and the hypercharge, their vector carriers, two-flavorcharged and chargeless leptons, and scalar particles. A mass term producesbreaking of the symmetry to an electromagnetic U(1), and a Weinberg's angle_W with sin ²(_W) = 0.25. A more realistic 8D extension gives coupling constantsof the respective groups g = 1/2 .707 and g = 1/6 .408, with thesame _W.     

A remark on the KAM theorem applied to a four-vortex system

We consider a planar four-vortex system with unit intensities and apply the KAM theorem for two-dimensional tori with fixed frequency. We obtain a rigorous lower bound for the stochasticity threshold of the torus with rotation number=(5—1)/2 and compare our result with numerical experiments.     

On the dynamics of excitations in disordered systems

A new, time-local (TL) reduced equation of motion for the probability distribution of excitations in a disordered system is developed. ToO(k²) the TL equation results in a Gaussian spatial probability distribution, i.e, P(r, t) = [(2)^1/2]^–dexp(-r²/2²), where = (t) is a correlation length, andr = ¦r¦. The corresponding distribution derived from the Hahn-Zwanzig (HZ) equation is more complicated and assumes the asymptotic (r ) form: P(r, s)(s ^d )^–1exp(–r/) · (r/)^(1-d)/2 where = (s),d is the space dimensionality, ands is the Laplace transform variable conjugate tot. The HZ distribution generalizes the scaling form suggested by Alexanderet al. ford= 1. In the Markov limit (t)t, (s)1/s, and the two distributions are identical (ordinary diffusion).     

Singular continuous spectrum on a cantor set of zero Lebesgue measure for the Fibonacci Hamiltonian

It is rigorously proven that the spectrum of the tight-binding Fibonacci Hamiltonian,H _mn= _{m, n+1}+ _{m, n–1}+ _{m, n} [(n+1)]–[n]) where =(5–1)/2 and [·] means integer part, is a Cantor set of zero Lebesgue measure for all real nonzero, and the spectral measures are purely singular continuous. This follows from a recent result by Kotani, coupled with the vanishing of the Lyapunov exponent in the spectrum.On leave from the Central Research Institute for Physics, Budapest, Hungary.     

Further consequences of Einstein locality

The results of a previous paper for the properties of the correlation functionP() at small angles following from Einstein locality are extended to the general case whereP(0)–1. Experimental investigations are suggested to check the validity of some theorems on the behavior of the correlation functionP() which are deduced directly from Einstein locality.     

Characters of Finite-Dimensional Irreducible q(n) -Modules

The solution of the Kac character problem for thequeer series of Lie superalgebras q(n) is announced. An explicit algorithm which computes the character of an arbitrary finite-dimensional irreducible q(n)-module is presented. As an illustration, the correction terms to the generic character formula of Penkov (Monatsh. Math.118 (1994), 419) are written down for all finite-dimensional irreducible representations of q(n), for n4, with nongeneric character.     

Quantum field theory and the coloring problem of graphs

The ^k theory is compared with the multilinear theory of scalar fields ₁, ₂, ..., _k having the same mass as that of . In particular, it is shown that Feynman integrals encountered in the ³ theory are not necessarily present also in the _{1 2 3} theory, but they are if they correspond to planar Feynman graphs having no tadpole part. Furthermore, a necessary and sufficient condition for the presence of a ³ Feynman integral in the _{1 2} ² theory is found. Those considerations are applications of graph theory, especially of the coloring problem of graphs, to Feynman graphs.     

On the invariant measures for the two-dimensional euler flow

It is proven that the canonical Gibbs measure associated with a gas of vortices of intensity ± converges, in the limitN, 0,Nconst, to a Gaussian measure, which is invariant for the two-dimensional Euler equation.On leave from Dipartimento di Matematica Università di Roma Tor Vergata Roma, Italy.On leave from Dipartimento di Matematica Università di Roma La Sapienza, Roma, Italy.     

Steady flux in a continuous-space Sinai chain

We examine the steady-state flux of particles diffusing in a one-dimensional finite chain with Sinai-type disorder, i.e., the system in which in addition to the thermal noise, particles are subject to a stationary random-correlated in space Gaussian force. For this model we calculate the disorder average (over configurations of the random force) flux exactly for arbitrary values of system's parameters, such as chain lengthN, strength of the force, and temperature. We prove that within the limitN1 the average flux decreases withN as J(N)=C/N and thus confirm our recent predictions that the flux in the discrete-space Sinai model is anomalous.     

Discontinuous behavior of effective transport coefficients in quasiperiodic media

We investigate the effective conductivity ^* of a quasiperiodic medium in ^d and the discontinuous dependence, found in ref. 1, of ^* on the wavelengths of the system. It was shown there, for example, that the effective conductivity ^*(k) for a layered medium with a one-dimensional local conductivity _k (x)=A+cosx+coskx, A>2, is discontinuous ink. An explicit class of higherdimensional examples which exhibit the discontinuity is constructed here. The conductivity ^*(k, L) of a sample of lengthL in one dimension asL is also analyzed and shown to have a plateau structure for any irrationalk well approximated by rationals.     

A Remark on the Use of Differential Forms for the Skyrme Problem

We introduce a formulation of the Skyrme problem using differential forms. By means of this formulation, we prove first that the homothetic map between the standard three-sphere of radius R, S³ _r R⁴, and S³ ₁ is the unique minimizer, modulo isometries, of the Skyrme energy in its homotopy class, for any R less than some critical value R₀ (3/2, 2]. We then establish a stability result for this Skyrme-form problem from which we can recover the result of M. Loss and N. S. Manton which states that this homothetic map is stable only up to R = 2.