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     

On the massive sine-Gordon equation in all regions of collapse

The ultraviolet stability for the cosine interaction in two dimensions and finite volume is rederived for values ² [4, 32/5[ and proven for the remaining ² [4, 8[ by using renormalization group methods developed in [G, GN1] to portray renormalized effective potentials arising from a multiscale decomposition.Supported by the Studienstiftung des Deutschen VolkesSupported in part by the Schweizerischer Nationalfonds     

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 Phase Transition in Statistical Models Defined on Farey Fractions

We consider several statistical models defined on the Farey fractions. Two of these models may be regarded as spin chains, with long-range interactions, while another arises in the study of multifractals associated with chaotic maps exhibiting intermittency. We prove that these models all have the same free energy. Their thermodynamic behavior is determined by the spectrum of the transfer operator (Ruelle–Perron–Frobenius operator), which is defined using the maps (presentation functions) generating the Farey tree. The spectrum of this operator was completely determined by Prellberg. It follows that these models have a second-order phase transition with a specific heat divergence of the form C [ ln² ]^–1. The spin chain models are also rigorously known to have a discontinuity in the magnetization at the phase transition.     

A complete proof of the Feigenbaum conjectures

The Feigenbaum phenomenon is studied by analyzing an extended renormalization group map . This map acts on functions that are jointly analytic in a position variable (t) and in the parameter () that controls the period doubling phenomenon. A fixed point ^* for this map is found. The usual renormalization group doubling operatorN acts on this function ^* simply by multiplication of with the universal Feigenbaum ratio ^*= 4.669201..., i.e., (N ^*(,t)= ^*( ^* ,t). Therefore, the one-parameter family of functions, ^* , ^* (t)=( ^*(,t), is invariant underN. In particular, the function ₀ ^* is the Feigenbaum fixed point ofN, while ^* represents the unstable manifold ofN. It is proven that this unstable manifold crosses the manifold of functions with superstable period two transversally.     

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.     

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.     

Wavelength distribution of X-rays in the focus of a monochromator and an estimate of the influence of this distribution on precision measurements of lattice constants

Wavelength distribution in the focus of a Johansson type monochromator is computed assuming the tube focus emissivityG(), reflection curveR() and wavelength distribution of the incident radiationJ(-₀) are known. It is shown e.g. that the centre of gravity may be shifted in accordance with the position of the crystal on the focal circle which may considerably influence precision measurements of lattice parameters.

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, G(,) R() J(-₀) . , , , .

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The author is grateful to Z. Hemanová for carrying out all the computations very carefully.     

Theory of fiber bundle with local metric of internal space and gravitation with torsion

In this paper we propose a new theory of a fiber bundle provided with a local metric of internal space. The fibers differ from usual fibers, having an enlarged factor. The enlargement may be procured by a differential mapping(x) from structure groupG to the fiberF _x atx M, and(x)R. The torsion presented stems from the local metric of internal space and the local metric stems from a induced mapping _*(x) of(x). From the theory we can get the Brans-Dicke theory with torsion. If we assume the spin density of the gauge field determines the enlarged factor of the fiberF _x, our theory is an extended Cartan theory.     

Periodic inverse problem for a new hierarchy of coupled evolution equations

We study the inverse problem with periodic boundary condition for a new class of integrable nonlinear evolution equations. The multiphase periodic solutions for the nonlinear fields (p, q, r) are expressed in terms of the Riemann theta function, which is obtained via the linearization of the flows of the set of auxiliary variables _j on a Riemann surface. An explicit case is evaluated to obtain the form of the algebraic curve on which the variables _j move.     

On the Haar measure of the quantumSU(N) group

We prove that the Haar state associated to the compact matrix quantum groupSU (N) is faithful for ]–1,1[,0, and anyN2.     

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.     

Raman effect of light on argon atoms

Scalar scattering of light on the 3p and 3s subshells of the argon atom (Stokes scattering) and anti-Stokes scattering on the excited 4p and 4s states of argon are examined in the Hartree-Fock approximation. The calculation is made in a velocity form and in a length form. It is shown that the value in the r form is 1.5–2 times greater than in the form.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 1, pp. 36–42, January, 1977.     

Some properties of topological geons

We investigate the Finkelstein-Misner geons for a non-simply-connected space-time manifold (M, g ₀). We use relations between different Lorentzian structures unequivalent tog ₀ and topological properties ofM given by the Morse theory. It implies that to some pieces of geons we have to associate Wheeler's worm-holes. Geons that correspond to time-orientable Lorentz structures are related tog ₀ by Morse functions that describe the attaching of a handle of index one. In the case of geons associated to time-nonorientable Lorentzian structures, appropriate handles are related to loops along which the notion of time reverses. If we assume electromagnetic properties of geons, then only four species, v, e, p, m, of different geons can exist and geon m has to decay according to mv+p+e.     

Self and mutual admittance of slot antennas on a dielectric half-space

In this paper, an efficient implementation of the spectral domain moment technique is presented for computing the self and mutual coupling between slot antennas on a dielectric half-space. It is demonstrated that by the proper selection of the weighting functions in the method of moments, the analytic evaluation or simplification of the transverse moment integrals is enabled. This results into a significant reduction of the required computational labor. The method is then utilized in order to provide design data for the self and mutual admittances between two slot antennas on a dielectric substrate lens in the case of fused quartz ( _r =3.80), crystal quartz ( _r =4.53), silicon ( _r =11.9) and GaAs ( _r =12.8). The presented technique and associated results are useful when designing twin slot quasi-optical receivers, imaging arrays, phased arrays or power-combining arrays of slot elements at millimeter-wave frequencies.     

Batalin-Vilkovisky algebras and two-dimensional topological field theories

By a Batalin-Vilkovisky algebra, we mean a graded commutative algebraA, together with an operator :A A ₊₁ such that ² = 0, and [,a]–a is a graded derivation ofA for allaA. In this article, we show that there is a natural structure of a Batalin-Vilkovisky algebra on the cohomology of a topological conformal field theory in two dimensions. We make use of a technique from algebraic topology: the theory of operads.The author is partially supported by a fellowship of the Sloan Foundation and a research grant of the NSF     

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.     

ИНТЕРФЕРЕНЦИЯ ДВУХ Э СТЕСТВЕННYХ ПУЧКОВ СВ ЕТА, ПРОШЕДШИХ ЧЕРЕЗ ОПТИ ЧЕСКИ АКТИВНУЮ) СРЕДУ

, .     

Nontrivial extensions of a representation of the Poincaré group with mass and helicity zero by its tensorial product

Let U(a, ) be a representation of the Poincaré group with mass and helicity zero, realized in the space of C -functions with compact support on ³, without the origin. Let U ⁽²⁾(a, ) denote the tensorial product of U(a, ) by itself. We explicitly determine the cocycles of extension of U(a, ) by U ⁽²⁾(a, ) and we prove that the nontrivial cohomology is indexed by (u(),),u()D ¹ _]0,3\,.