Quasi-classical limit of BKP hierarchy andW-infinity symmetries

Previous results on quasi-classical limit of the KP and Toda hierarchies are now extended to the BKP hierarchy. Basic tools such as the Lax representation, the Baker-Akhiezer function and the tau function are reformulated so as to fit into the analysis of quasi-classical limit. Two subalgebrasW ₁ ^B ₊ andw ₁ ^B ₊ of theW-infinity algebrasW _{1 +} andw _{1 +} are introduced as fundamental Lie algebras of the BKP hierarchy and its quasi-classical limit, the dispersionless BKP hierarchy. The quantumW-infinity algebraW ₁ ^B ₊ emerges in symmetries of the BKP hierarchy. In quasi-classical limit, theseW ₁ ^B ₊ symmetries are shown to be contracted intow ₁ ^B ₊ symmetries of the dispersionless BKP hierarchy.     

On the completeness of the set of classicalW-algebras obtained from DS reductions-algebras obtained from DS reductions

We clarify the notion of the DS — generalized Drinfeld-Sokolov — reduction approach to classicalW-algebras. We first strengthen an earlier theorem which showed that ansl(2) embeddingL G can be associated to every DS reduction. We then use the fact that aW-algebra must have a quasi-primary basis to derive severe restrictions on the possible reductions corresponding to a givesl(2) embedding. In the known DS reductions found to data, for which theW-algebras are denoted byW _L ^G -algebras and are called canonical, the quasi-primary basis corresponds to the highest weights of thesl(2). Here we find some examples of noncanonical DS reductions leading toW-algebras which are direct products ofW _L ^G -algebras and free field algebras with conformal weights {0, 1/2, 1}. We also show that if the conformal weights of the generators of aW-algebra obtained from DS reduction are nonnegative 0 (which is the case for all DS reductions known to date), then the 3/2 subsectors of the weights are necessarily the same as in the correspondingW _L ^G -algebra. These results are consistent with an earlier result by Bowcock and Watts on the spectra ofW-algebras derived by different means. We are led to the conjecture that, up to free fields, the set ofW-algebras with nonnegative spectra >-0 that may be obtained from DS reduction is exhausted by the canonical ones.     

A theorem on the first heteroclinic tangency in two-dimensional maps. Orientation-preserving cases

Using the properties of the Jordan curve, the following theorem on the heteroclinic tangency in orientation-preserving two-dimensional maps is proved: LetT :R ² R ² be a one-parameter family ofC ¹ diffeomorphisms andJ=DetDT be such that 0<J1 or 1J<. LetW _u ⁿ be the unstable manifold of a hyperbolicn-cycle andW _s ^m the stable manifold of a hyperbolicm-cycle. Suppose that for< _c ,W _u ⁿ andW _s ^m have no common points, and that for> _c ,W _u ⁿ andW s/m have a transversal heteroclinic point. Then at= _c ,W _u ⁿ andW _s ^m are in the first asymptotic heteroclinic tangency except for the following three cases: (1)n=m; both cycles are without reflection. (2)m=2n; then- andm-cycles are with and without reflection, respectively; (3)n=2m; then- andm-cycles are without and with reflection, respectively.     

Conservation of Energy,Entropy Identity,and Local Stability for the Spatially Homogeneous Boltzmann Equation

For nonsoft potential collision kernels with angular cutoff, we prove that under the initial condition f ₀(v)(1+|v|²+|logf ₀(v)|)L ¹(R ³), the classical formal entropy identity holds for all nonnegative solutions of the spatially homogeneous Boltzmann equation in the class L ([0, ); L ¹ ₂(R ³))C ¹([0, ); L ¹(R ³)) [where L ¹ _s (R ³)={ff(v)(1+|v|²) ^s/2L ¹(R ³)}], and in this class, the nonincrease of energy always implies the conservation of energy and therefore the solutions obtained all conserve energy. Moreover, for hard potentials and the hard-sphere model, a local stability result for conservative solutions (i.e., satisfying the conservation of mass, momentum, and energy) is obtained. As an application of the local stability, a sufficient and necessary condition on the initial data f ₀ such that the conservative solutions f belong to L ¹ _loc([0, ); L ¹ ₂₊ (R ³)) is also given.     

Differential equations in the spectral parameter,Darboux transformations and a hierarchy of master symmetries for KdV

We study a certain family of Schrödinger operators whose eigenfunctions (, ) satisfy a differential equation in the spectral parameter of the formB(, )=(x). We show that the flows of a hierarchy of master symmetries for KdV are tangent to the manifolds that compose the strata of this class ofbispectral potentials. This extends and complements a result of Duistermaat and Grünbaum concerning a similar property for the Adler and Moser potentials and the flows of the KdV hierarchy.     

On the correlation for Kac-like models in the convex case

The aim of this paper is to stu the behavior asm tends to of a family of measures exp[- ^(m)(x)]dx ^(m) on ^m , where ^(m) is a potential on ^m which is a perturbation in a suitable sense of the harmonic potential _j x _j ² .     

Possible generalization of Boltzmann-Gibbs statistics

With the use of a quantity normally scaled in multifractals, a generalized form is postulated for entropy, namelyS _q k [1 – _i=1 ^W p _i ^q ]/(q-1), whereq characterizes the generalization andp _i are the probabilities associated withW (microscopic) configurations (W). The main properties associated with this entropy are established, particularly those corresponding to the microcanonical and canonical ensembles. The Boltzmann-Gibbs statistics is recovered as theq1 limit.     

Traces,Dispersions of States and Hidden Variables

No Heading The interplay between the tracial property and minimality of dispersions of states on projections of von Neumann algebras and C^*-algebras is investigated. Let be a state on a C^*-algebra A with the projection structure P(A). The dispersion () is defined as () = sup{(p) – (p)² | p P(A)}. It is proved that () 2/9 whenever is a state on a real rank zero C^*-algebra with no nonzero abelian representation. New characterization of traces in terms of dispersions is proved: A state on a von Neumann algebra without abelian and Type I₂ direct summands is a trace if and only if has the minimal dispersion on all 3x3 matrix substructures. A similar characterization of semifinite normal traces on von Neumann algebras is obtained. The connection between unitary invariance of states and minimal dispersion property on C^*-algebras is studied. Besides providing a new characterization of trace in terms of physically relevant properties, the existing results on hidden variables in W^*- and C^*-formalism of quantum mechanics are strengthen.     

Geometrical interpretation of the bilinear equations for the KP hierarchy

The Grassmann manifold approach to the KP hierarchy, in the spirit of Segal and Wilson, is used to define the wavefunction _W and its adjoint _W. From the fact that _W and _W are the orthogonal, we derive the bilinear equations. The modified equations are treated at the same time.     

Muon catalyzed fusion: old and new aspects of energy production

Muon catalyzed fusion of deuterium and tritium (CF) yields the same energy gain per reaction as fusion with magnetic or inertial confinement (17.6 MeV). The crucial points of Cf are, however, very different, namely (a) the energy costW ₍₎ for production of one ^– and (b) the numbern of reactions a single muon can catalyze on the average. (b) is ultimately limited by theeffective sticking probability _f :n1/ _f. With standard methods one hasW ₍₎5 GeV, _f=0.5%. Hence a standard CF reactor can never reach a net energy gain. To solve this problem, ways discussed since about a decade are to increase the efficiency by both (i) energy multiplication using a fissionable blanket and (ii) breeding. A new way to increase the safety of fission devices mostly due to Yu. Petrov is outlined. On the other hand there is a hope to lowerW ₍₎ slightly and _f drastically, the latter by artificial reactivation. New theoretical results for beam cooling in an omegatron type driven integrated CF reactor, important forW ₍₎ and, in particular, _f, is presented.     

The transverse correlation length for randomly rough surfaces

It is shown by numerical simulations for a random, one-dimensional surface defined by the equationx ₃=(x ₁), where the surface profile function (x ₁) is a stationary, stochastic, Gaussian process, that the transverse correlation lengtha of the surface roughness is a good measure of the mean distance d between consecutive peaks and valleys on the surface. In the case that the surface height correlation function (x ₁)(x ₁)/²(x ₁)=W (|x ₁–x ₁|) has the Lorentzian formW(|x ₁|)=a ²/(x ₁ ² +a ²) we find that d=0.9080a; when it has the Gaussian formW(|x ₁|)=exp(–x ₁ ² /a ²), we find that d=1.2837a; and when it has the nonmonotonic formW(|x ₁|)=sin(x ₁/a)/(x ₁/a), we find that d=1.2883a. These results suggest that d is larger, the faster the surface structure factorg(|Q|) [the Fourier transform ofW(|x ₁|)] decays to zero with increasing |Q|. We also obtain the functionP(itx ₁), which is defined in such a way that, ifx ₁=0 is a zero of (x ₁),P(x ₁)dx ₁ is the probability that the nearest zero of (x ₁) for positivex ₁ lies betweenx ₁ andx ₁+dx ₁.     

Logically independent von Neumann lattices

Three definitions of logical independence of two von Neumann latticesP₁,P₂ of two sub-von Neumann algebras ₁, ₂ of a von Neumann algebra are given and the relations of the definitions clarified. It is shown that under weak assumptions the following notion, called logical independence is the strongest:A B 0 for any 0 A P₁, 0 B P₂. Propositions relating logical independence ofP₁,P₂ toC ^*-independence,W ^* independence, and strict locality of ₁, ₂ are presented.     

The number of bound states of singular oscillating potentials

For a spherically symmetric potential such that rVL ¹(a, ), a>0, and is such that, if we define W=– _r V(t) d(t), W belongs to L ¹ (0, ) and rW0 as r0, we show that the number of bound states in any partial-wave satisfies the bound n2 ₀ r W ² dr. It was shown in a previous paper [1] that this class of potentials is regular from the point of view of abstract scattering theory as well as from the time-independent theory and the Jost function approach. We show also that, for large values of the coupling constant, n(gV) has the asymptotic behaviour C _±g ₀ W(r) dr as g±.     

Decay of the Bethe–Salpeter Kernel and Bound States for Lattice Classical Ferromagnetic Spin Systems at High Temperature

We consider lattice classical ferromagnetic spin systems at high temperature (1) with nearest neighbor interactions and even single-spin distributions (ssd). Associated with each system is an imaginary time lattice quantum field theory. It is known that there is a particle of mass m–ln in the energy-momentum spectrum. If s ⁴–3s ²²<0, where s ^k is the kth moment of the ssd, and is sufficiently small, we show that in the two-particle subspace there is no mass spectrum up to 2m. For >0 we show that the only mass spectrum in (m, 2m) is a bound state of mass m _b=2m+ln(1–)+O(), where =(+2s ²²)^–1. A bound on the decay of the kernel of a Bethe–Salpeter equation is obtained and used to prove these results.     

Analyticity of scattering for the ϕ4 theory

We consider scattering for the equation (+m ²)+³=0 on four-dimensional Minkowski space. Form>0, one-to-one and onto wave operatorsW _± :HH are known to exist for all 0, whereH denotes the Hilbert space of finite-energy Cauchy data. We prove that the maps (,u)W _± (u) and (,u)(W _± )^–1 (u) are continuous from [0, )×H toH, and extend to real-analytic functions from an open neighborhood of {0}×H×{0}×H to the Hilbert spaceH _–1 of Cauchy data with Poincaré-invariant norm. Form=0, wave operatorsW _± are known to exist as diffeomorphisms ofH for all 0, where hereH denotes the Hilbert space of finite Einstein energy Cauchy data. In this case we prove that the maps (,u)(W _± ) (u) and (,u)(W _± )^–1 (u) extend to real-analytic functions from a neighborhood of [0, )×H×H toH.     

Operator representations of U q (sl2(ℝ))

Integrable (well-behaved) operator representations of the *-algebra U _q (sl₂()), |q|= 1, q ² 1, |q|=1, q² 1, in Hilbert space are defined and classified up to unitary equivalence.     

Random media and eigenvalues of the Laplacian

Let be a fixed number >1. We remove [m ]-balls of centersw ₁, ..., with the same radius /m from a bounded domain inR ³. We consider the asymptotic behaviour of thek ^th eigenvalue of the Laplacian in under the Dirichlet condition as a random variable on a probability space , whenm.     

Typical cluster size for two-dimensional percolation processes

The typical cluster size for two-dimensional percolation models is discussed. It is shown that, forW ₀={xZ ²0x}, [–lim _n(1/n) logP _p (W ₀=n)]^–1p–p _c ^– aspp _c , provided thatE _p (W ₀²)/E _p (W ₀)p–P _c ^– aspp _c . Furthermore, we introduce a new quantityf _s (p), which may be thought of as the singular part of the free energy, and show thatf _s (p)¦p–p _c ¦^2v provided that the correlation length ¦p–p _c ¦^–v aspp _c .     

Fabrication and low threshold current density CW operation of GaInAsP/InP multiple-reflector microcavity laser

We report CW operation of a GaInAsP/InP multiple-reflector microcavity (MRMC) laser operated at fairly low threshold current density. The threshold current density with broad contact (stripe widthW=240 m, cavity lengthL=60 m) under pulsed operation was 180 A cm^–2 (l _th=20 mA), and was 230 A cm^–2 under CW operation at room temperature operating at 1.52 m wavelength.     

Attractor States and Quantum Instabilities in de Sitter Space

The asymptotic behavior of the energy–momentum tensor for a free quantized scalar field with mass m and curvature coupling in de Sitter space is investigated. It is shown that for an arbitrary, homogeneous, and isotropic, fourth-order adiabatic state for which the two-point function is infrared finite, T _ab approaches the Bunch–Davies de Sitter invariant value at late times if m ² + R > 0. In the case m = = 0, the energy–momentum tensor approaches the de Sitter invariant Allen–Folacci value for such a state. For m ² + R = 0 but m and not separately zero, it is shown that at late times T _ab grows linearly in terms of cosmic time leading to an instability of de Sitter space. The asymptotic behavior is again independent of the state of the field. For m ² + R < 0, it is shown that, for most values of m and , T _ab grows exponentially in terms of cosmic time at late times in a state dependent manner.