Te-waves guided by arbitrary nonlinear dielectrics bounded by linear media

Considerable interest is currently being devoted to nonlinear propagation in dielectric slab waveguides for integrated optics and millimetric applications. Much of the current work is based on Kerr-like ( E ²) nonlinear media and numerically discussed. We present a exact solution of TE-waves for arbitrary nonlinear dielectric ( |E| ). We applicate this solution to the planar nonlinear optic waveguides and the dispersion relations are given. The results show that the propagation constants are a function of the field magnitude.Supported by deutscher akademischer austauschdienst (DAAD)He is now with Dept. Electrical Engineering, Duisburg University, 4100 Duisburg, Germany     

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.     

Solutions to the Boltzmann Equation in the Boussinesq Regime

We consider a gas in a horizontal slab in which the top and bottom walls are kept at different temperatures. The system is described by the Boltzmann equation (BE) with Maxwellian boundary conditions specifying the wall temperatures. We study the behavior of the system when the Knudsen number is small and the temperature difference between the walls as well as the velocity field is of order , while the gravitational force is of order ². We prove that there exists a solution to the BE for which is near a global Maxwellian, and whose moments are close, up to order ², to the density, velocity and temperature obtained from the smooth solution of the Oberbeck–Boussinesq equations assumed to exist for .     

Quantum-Logics-Valued Measure Convergence Theorem

In this paper, the following quantum-logic valued measure convergence theorem is proved: Let (L ₁, 0, 1) be a Boolean algebra, (L ₂, , , 0, 1) be a quantum logic and { _n : n N} be a sequence of s-bounded (L ₂, , , 0, 1)-valued measures which are defined on (L ₁, 0, 1). If for each a (L ₁, 0, 1), { _n (a)} _{n N} is an order topology Cauchy sequence, when {v(a)} convergent to 0, { _n (a)} is order topology convergent to 0 for each n N, where v is a nonnegative finite additive measure which is defined on (L ₁, 0, 1), then when {v(a)} convergent to 0, { _n (a)} are order topology convergent to 0 uniformly with respect to n N.     

Absolute Continuity of Vitali–Hahn–Saks Measure Convergence Theorems

In this paper, we prove the following improved Vitali–Hahn–Saks measure convergence theorem: Let (L, 0, 1) be a Boolean algebra with the sequential completeness property, (G, ) be an Abelian topological group, be a nonnegative finitely additive measure defined on L, {_n: n N} be a sequence of finitely additive s-bounded G-valued measures defined on L, too. If for each a L, {_n(a)}_{n N} is a -convergent sequence, for each nN, when { (a)} convergent to 0, {_n(a)} is -convergent, then when { (a)} convergent to 0, {_n(a)} are -convergent uniformly with respect to nN     

Long range scattering for nonlinear Schrödinger equations in one space dimension

We consider the scattering problem for the nonlinear Schrödinger equation in 1+1 dimensions: where = /x,R{0},R,p>3. We show that modified wave operators for (*) exist on a dense set of a neighborhood of zero in the Lebesgue spaceL ²(R) or in the Sobolev spaceH ¹(R)., The modified wave operators are introduced in order to control the long range nonlinearity |u|² u.Laboratoire associé au Centre National de la Recherche Scientifique     

Singular Perturbations of Self-Adjoint Operators

Singular finite rank perturbations of an unbounded self-adjoint operator A ₀ in a Hilbert space ₀ are defined formally as A ₍₎=A ₀+GG ^*, where G is an injective linear mapping from = ^d to the scale space _-k(A₀)k , kN, of generalized elements associated with the self-adjoint operator A ₀, and where is a self-adjoint operator in . The cases k=1 and k=2 have been studied extensively in the literature with applications to problems involving point interactions or zero range potentials. The scalar case with k=2n>1 has been considered recently by various authors from a mathematical point of view. In this paper, singular finite rank perturbations A ₍₎ in the general setting ran G – _k (A ₀), kN, are studied by means of a recent operator model induced by a class of matrix polynomials. As an application, singular perturbations of the Dirac operator are considered.     

The classification of differential structures on quantum 2-spheres

Exterior algebras of differential forms on quantum 2-spheresS _qc ² ,q[–1, 1]/{0},c[0, ] (c=0 forq=±1), are classified. In the definition of exterior algebras we assume the invariance w.r.t. the action of the quantumSU(2) group and dimensionality conditions (which imply that we deal with two-dimensional manifolds). The exterior algebras exist only forc=0 and are unique in that case. The corresponding generalized directional derivatives are provided.     

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.     

Twisted Lie Group C*-Algebras as Strict Quantization

A nonzero 2-cocycle Z2(g, R) on the Lie algebra g of a compact Lie group G defines a twisted version of the Lie–Poisson structure on the dual Lie algebra g*, leading to a Poisson algebra C (g*()). Similarly, a multiplier c Z2(G, U(1)) on G which is smooth near the identity defines a twist in the convolution product on G, encoded by the twisted group C-algebra C*(G,c). Further to some superficial yet enlightening analogies between C (g*()) and C*(G,c), it is shown that the latter is a strict quantization of the former, where Plancks constant assumes values in (Z\{0})-1. This means that there exists a continuous field of C*-algebras, indexed by 0 (Z\{0})-1, for which A0= C0(g*) and A=C*(G,c) for 0, along with a cross-section of the field satisfying Diracs condition asymptotically relating the commutator in A to the Poisson bracket on C(g*()). Note that the quantization of does not occur for =0.     

Concrete quantum logics with covering properties

LetL be a concrete (=set-representable) quantum logic. Letn be a natural number (or, more generally, a cardinal). We say thatL admits intrinsic coverings of the ordern, and writeL C _n , if for any pairA, BL we can find a collection {C _i iI}, where cardI<n andC _i L for anyiI, such thatA B= _il C _i . Thus, in a certain sense, ifLC _n , then the rate of noncompatibility of an arbitrary pairA,BL is less than a given numbern. In this paper we first consider general and combinatorial properties of logics ofC _n and exhibit typical examples. In particular, for a givenn we construct examples ofL C _n+1\C _n . Further, we discuss the relation of the classesC _n to other classes of logics important within the quantum theories (e.g., we discover the interesting relation to the class of logics which have an abundance of Jauch-Piron states). We then consider conditions on which a class of concrete logics reduce to Boolean algebras. We conclude with some open questions.     

MV and Heyting Effect Algebras

We review the fact that an MV-algebra is the same thing as a lattice-ordered effect algebra in which disjoint elements are orthogonal. An HMV-algebra is an MV-effect algebra that is also a Heyting algebra and in which the Heyting center and the effect-algebra center coincide. We show that every effect algebra with the generalized comparability property is an HMV-algebra. We prove that, for an MV-effect algebra E, the following conditions are mutually equivalent: (i) E is HMV, (ii) E has a center valued pseudocomplementation, (iii) E admits a central cover mapping such that, for all p, qE, pq=0(p)q=0.     

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.     

Rigorous Spectral Analysis of the Metal–Insulator Transition in a Limit-Periodic Potential

We consider the limit-periodic Jacobi matrices associated with the real Julia sets of f (z)=z ²– for which [2, ) can be seen as the strength of the limit-periodic coefficients. The typical local spectral exponent of their spectral measures is shown to be a harmonic function in decreasing logarithmically from 1 to 0.     

Differential calculus on quantum spheres

Three-dimensional differential calculus on quantum spheres S _infc ^sup2 ,]–1, 1[{0}, c[0, ], is introduced and investigated. Spectra of generalized Laplacians are found. These operators are expressed by generalized directional derivatives. Classical limits of these objects are obtained and a simple approach to quantum mechanics on a quantum sphere is presented.     

ИЗМЕРЕНИЕ УПРУГОСТИ ПАРОВ НЕКОТОРЫХ ВЫСОКОМОЛ ЕКУЛЯРНЫХ ВЕЩЕСТВ РАДИОМЕТРИЧЕСКИМ МА НОМЕТРОМ С БОЛБШОЙ ЧУВСТВИТЕЛ ЬНОСТЬЮ

- — — - ; napc. - , - , , 9 . - . - 3. 10^–6–1. 10^–6 mmpm. cm.     

Conditional expectation and commutativity in von Neumann algebras

LetH be a Hilbert space,A the von Neumann algebra of all bounded operators onH,B a von Neumann subalgebra ofA, andw a bounded linear functional onA. The functionalw is said to commute withBA ifw(AB)=w(BA) for allAA. It is shown that the mapBw (BAB) is a complex measure on the orthocomplemented partially ordered set of all orthogonal projections inB for everyAA if and only ifw commutes with all members ofB. For anyAA, the conditional expectation ofA with respect toB andw is defined and it is shown that this expectation exists for an Abelian separableB ifw commutes with all members ofB. Using Gleason's theorem it is shown thatw commutes withB if and only if the density operator ofw commutes withB.     

The twisted SU(3) group. Irreducible *-representations of the C *-algebra C(SμU(3))

Irreducible ^*-representations of C(SU(3)) are constructed for ]0, 1[. It is proved that C(SU(3)) is a type-I C ^*-algebra.     

ПЛОТНОСТЬ ТОКА В КАТО ДНЫХ ПЯТНАХ В КОНДЕНСИРОВАННОМ Р АЗРЯДЕ

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О взаимодействии экс итонов с колебаниями кристаллической реш етки

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