One criterion of semigroup product convergence

Let exp(-tA) and exp(-tB) be C ₀ contraction semigroups on both K and , where K is a Hilbert space and is a reflexive Banach space such that the linear space K is dense both in K and . Let ^* be a dual pair of Banach spaces. In this paper we study some properties of infinitesimal operators of these semigroups. We show that under suitable assumptions there is some connection between the form-sum A+B and a closure of A ₁+B₁, where -A ₁ is an infinitesimal operator of C ₀ contraction semigroup exp(-tA ₁) which is the extension by continuity on of C ₀ contraction semigroup exp(-tA) Kin . In particular we give some criterion of an m-accretive closability A ₁+B ₁ which may be applied for example to the Schrödinger operators acting in suitable L ^p-spaces. Also this criterion together with properties of semigroups under consideration results in the establishment of the Lie-Trotter formulae.     

On the schwarz problem for the [^(\textsu)]\text2 \widehat{{\text{su}}}_{\text{2}} Knizhnik-Zamolodchikov equation

We study the monodromy representations ^k,Iof the mapping class group ₄ acting on 4-point blocks satisfying the Knizhnik-Zamolodchikov equation for the levelk su₂ current algebra. We classify all irreducible ^k,Iwhich are realized by finite groups; we also display finite irreducible components for the reducible representations corresponding tok = 10.Supported by the Federal Ministry of Science and Research, Austria.     

The heat semigroup and integrability of Lie algebras: Lipschitz spaces and smoothness properties

We define and analyze Lipschitz spaces _,q associated with a representationxgV(x) of the Lie algebrag by closed operatorsV(x) on the Banach space together with a heat semigroupS. If the action ofS satisfies certain minimal smoothness hypotheses with respect to the differential structure of (,g,V) then the Lipschitz spaces support representations ofg for which productsV(x)V(y) are relatively bounded by the Laplacian generatingS. These regularity properties of the _,q can then be exploited to obtain improved smoothness properties ofS on . In particularC ₄-estimates on the action ofS automatically implyC -estimates. Finally we use these results to discuss integrability criteria for (,g,V).Dedicated to Res Jost and Arthur Wightman     

Flow Polynomials and Their Asymptotic Limits for Lattice Strip Graphs

We present exact calculations of flow polynomials F(G,q) for lattice strips of various fixed widths L _y 4 and arbitrarily great lengths L _x , with several different boundary conditions. Square, honeycomb, and triangular lattice strips are considered. We introduce the notion of flows per face fl in the infinite-length limit. We study the zeros of F(G,q) in the complex q plane and determine exactly the asymptotic accumulation sets of these zeros in the infinite-length limit for the various families of strips. The function fl is nonanalytic on this locus. The loci are found to be noncompact for many strip graphs with periodic (or twisted periodic) longitudinal boundary conditions, and compact for strips with free longitudinal boundary conditions. We also find the interesting feature that, aside from the trivial case L _y =1, the maximal point, q _cf , where crosses the real axis, is universal on cyclic and Möbius strips of the square lattice for all widths for which we have calculated it and is equal to the asymptotic value q _cf =3 for the infinite square lattice.     

Nonexistence of hidden variables in the algebraic approach

Given two unital C*-algebrasA, and their state spacesE _A , E respectively, (A,E _A ) is said to have (, E) as a hidden theory via a linear, positive, unit-preserving map L: A if, for all E _A , L* can be decomposed in E into states with pointwise strictly less dispersion than that of . Conditions onA and L are found that exclude (A,E _A ) from having a hidden theory via L. It is shown in particular that, ifA is simple, then no (, E) can be a hidden theory of (A,E _A ) via a Jordan homomorphism; it is proved furthermore that, ifA is a UHF algebra, it cannot be embedded into a larger C*-algebra such that (, E) is a hidden theory of (A,E _A ) via a conditional expectation from ontoA.     

Phenomenological determination of the beauty meson decay parameterf B and the CP-violating angle δ

We fit the CKM-matrix to all recent data with the following free parameters: three mixing angles, the CP-violating angle in the Maiani parametrisation, the top quark massm _t, and the productf _{B B0} ^1/2 , wheref _B is theB-meson decay parameter and _B ⁰ is the bag parameter. Our fits span a contiguous region in the (f _{B B0} ^1/2 , cos )-plane. The parametersf _{B B0} ^1/2 and cos are strongly positively correlated.     

Singular Dimensions¶of the N= 2 Superconformal Algebras II:¶The Twisted N= 2 Algebra

We introduce a suitable adapted ordering for the twisted N= 2 superconformal algebra (i.e. with mixed boundary conditions for the fermionic fields). We show that the ordering kernels for complete Verma modules have two elements and the ordering kernels for G-closed Verma modules just one. Therefore, spaces of singular vectors may be two-dimensional for complete Verma modules whilst for G-closed Verma modules they can only be one-dimensional. We give all singular vectors for the levels , 1, and for both complete Verma modules and G-closed Verma modules. We also give explicit examples of degenerate cases with two-dimensional singular vector spaces in complete Verma modules. General expressions are conjectured for the relevant terms of all (primitive) singular vectors, i.e. for the coefficients with respect to the ordering kernel. These expressions allow to identify all degenerate cases as well as all G-closed singular vectors. They also lead to the discovery of subsingular vectors for the twisted N= 2 superconformal algebra. Explicit examples of these subsingular vectors are given for the levels , 1, and . Finally, the multiplication rules for singular vector operators are derived using the ordering kernel coefficients. This sets the basis for the analysis of the twisted N= 2 embedding diagrams. Received: Received: 15 March 1999 / Accepted: 12 November 2000     

On the characterization of the stationary state of a class of dynamical semigroups

We consider a representation of the entropy production for a completely positive, trace-preserving dynamical semigroup satisfying detailed balance with respect to its faithful stationary state denned on aW*-algebra(): it is expressed as a positive Hermitian form on(), which is analogous to the quantum correlation functions used in the Kubo theory. By considering this Hermitian form as a variation function of a vector in(), an exact characterization of the stationary states of semigroups in a certain class is obtained. On this basis, the problem of characterizing the stationary states discussed by Spohn and Lebowitz for manyreservoir open systems is solved without the restriction to situations near thermal equilibrium.     

The uniqueness theorem for the universal R-matrix

Up to now, the universal R-matrix for quantized Kac-Moody algebras is believed to be uniquely determined (for some ansatz) by properties of a quasi-cocommutativity and a quasi-triangularity. We prove here that the universal R-matrix (for the same ansatz) is uniquely determined by the property of the quasi-cocommutativity only. Thus, the quasi-triangular property (and the Yang-Baxter equation!) for the universal R-matrix is a consequence of the linear equation of the quasi-cocommutativity. The proof is based on properties of singular vectors in the tensor product of the Verma modules and the structure of extremal projector for quantized algebras. Explicit expressions of the universal R-matrix for quantized algebras U _q (A _inf1 ^sup(1) ) and U _q (A _inf2 ^sup(2) ) are given.

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Symmetric states of composite systems

Størmer proved a theorem on the integral decomposition of symmetric states on a C ^*-algebra . Motivated by problems in statistical mechanics, we define summetric states on a composite algebra A ( )and extend Størmer's theorem to this situation. Applications to spin-boson models are sketched.     

Group duality and the Kubo-Martin-Schwinger condition

We consider clusteringG-invariant states of aC*-algebraU endowed with an action of a locally compact abelian groupG. Denoting as usual byF _AB,G _AB, the corresponding two-point functions, we give criteria for the fulfillment of the KMS condition (w.r.t. some one-parameter subgroup ofG) based upon the existence of a closable mapT such thatTF _AB =G _AB for allA,BU. Closability is either inL (G),B(G), orC (G), according to clustering assumptions. Our criteria originate from the combination of duality results for the groupG (phrased in terms of functions systems), with density results for the two-point functions.Supported in part by the National Science Foundation     

Two equivalent criteria for modularity of the lattice of all physical decision effects

This paper answers the open question 1 of [3] in the affirmative and, conditionally, the open question 2 of [3], too. Assuming irreducibility of the orthomodular latticeG of all physical decision effectsE, we shall prove in the first section that modularity ofG implies symmetry of the physical probability function . In the second section, we shall consider the filter algebra (B) being assumed to possess an involution * such thatT*T=0 impliesT=0. Then this will be proved: If every atomic filterT _P is a fixpoint of * and * is, in a restricted manner, norm-preserving on the minimal left ideal _P :=(B)T _P , thenG is modular.     

Möbius function for the set of acyclic directed backbone graphs

Consider a two-rooted graphG, the edges of which are directed in such a way that there are no cycles and every edge belongs to some self-avoiding walk from rootu to rootv which follows the direction of the edges. Au-v backbone ofG is a subgraph formed by taking the union of any subset of directed self-avoiding walks fromu tov. Let _uv be the set of all such backbones ofG partially ordered by set-inclusion. We prove the conjecture of Bhatti and Essam that the Möbius function of this set is given, for acyclicb,b _uv withbb, by(b,b)=(–1) ^c-c , wherec andc are the respective cycle ranks ofb andb. The significance of this result in percolation theory is reviewed together with previous results for other sets of subgraphs.     

Quantum deformations of singletons and of free zero-mass fields

We consider quantum deformations of the real symplectic (or anti-De Sitter) algebra sp(4), spin(3, 2) and of its singleton and (4-dimensional) zero-mass representations. For q a root of –1, these representations admit finite-dimensional unitary subrepresentations. It is pointed out that U_q (sp(4, )), unlike U_q (su(2, 2)), contains U_q (sl ₂ ) as a quantum subalgebra.To Asim Barut, with all our friendship.     

On learning and energy-entropy dependence in recurrent and nonrecurrent signed networks

Learning of patterns by neural networks obeying general rules of sensory transduction and of converting membrane potentials to spiking frequencies is considered. Any finite number of cellsA can sample a pattern playing on any finite number of cells without causing irrevocable sampling bias ifA = orA =. Total energy transfer from inputs ofA to outputs of depends on the entropy of the input distribution. Pattern completion on recall trials can occur without destroying perfect memory even ifA = by choosing the signal thresholds sufficiently large. The mathematical results are global limit and oscillation theorems for a class of nonlinear functional-differential systems.The preparation of this work was supported in part by the National Science Foundation (GP 9003), the Office of Naval Research (N00014-67-A-024-OQ16), and the A.P. Sloan Foundation.     

On weak and monotone σ-closures ofC*-algebras

It is proved that the monotone -closure of the self-adjoint part of anyC*-algebraA is the self-adjoint part of aC*-algebra . IfA is of type I it is proved that is weakly -closed, i.e. is a*-algebra. The physical importance of*-algebras was explained in [1] and [7].     

Linking numbers of closed manifolds at random in ℝ n,inductances and contacts

We give here new results of topology and integral geometry concerning the Gauss linking number I of closed manifolds inn-dimensional space. The rigid manifolds have arbitrary shapes and dimensions, and are statistically at random positions in ⁿ . Generalizing Pohl's work, for two closed manifoldsC ₁ ^r ,C ₂ ^s , of respective dimensionsr ands, with 0rn–1, andr+s+1=n, we consider the kinematic linking integralI=<I ²(x,O)d ⁿ x>, of the square linking number I ofC ₁ ^r andC ₂ ^s , over the group of Euclidean motions of one manifold (translationsx, rotationsO). Introducing a new tensorial method, and using group theory, we show quite generally thatI=num. fact. , where is a length variable and whereA , (=1, 2) are characteristic functions associated with the manifoldC only. We study functionsA and of a manifoldC ^r , of dimensionr, in all cases 0rn–1.A always exists.A(0) givesC's area, whereas equals the interior volume of a hypersurfaceC. is found to exist and not to vanish only if 2 dimC+1=n andn=3+4q=3, 7, 11 ...A and are explicitly calculated for segments andr-spheresS ^r . As an application the topological excluded volume of a gas of nonlinked spheresS ^r moving in ^2r+1 is calculated. We generalize toN manifoldsC , =1, ...,N, linked successively to each other and forming a ring. The cyclic product of their linking numbers is integrated over the group of motions of the manifolds. It is shown to factorize completely in Fourier space, with special algebraic rules, over the set of 2N characteristic functionsA , , associated with theC 's. The same algebra of characteristic functions is shown to describe a larger class of topology and electromagnetism properties: a new theorem is given for a family of Euclidean group integrals involving the random linking numbers, mutual inductances and contact distributions ofN manifolds.     

q-Deformed orthogonal and pseudo-orthogonal algebras and their representations

Deformed orthogonal and pseudo-orthogonal Lie algebras are constructed which differ from deformations of Lie algebras in terms of Cartan subalgebra and root vectors and which make it possible to construct representations by operators acting according to Gel'fand-Tsetlin-type formulas. Unitary representations of the q-deformed algebras U _q (so _n,1) are found.     

Some remarkable degenerations of quantum groups

We show that an irreducible representation of a quantized enveloping algebraU at a ^th root of 1 has maximal dimension (= ^N ) if the corresponding symplectic leaf has maximal dimension (=2N). The method of the proof consists of a construction of a sequence of degenerations ofU , the last one being aq-commutative algebraU ^(2N) . This allows us to reduce many problems concerningU to that concerningU ^(2N) .To Armand Borel on his 70^th birthdaySupported in part by the NSF grant DMS-9103792     

Kinetics of phase separation in ramified polymer blends of arbitrary topology

We present here a theoretical study of kinetics of phase separation within a mixture made of two chemically incompatible ramified polymers. For simplicity, we assume that they have the same topology. We are interested in the variation of the relaxation rate, _q, versus the wave number q, in the vicinity of the spinodal temperature. The kinetics is governed by local (Rouse) and reptation motions (faster and slower modes). For qR_G 1 (R_G being the gyration radius), kinetics is entirely controlled by local motions where each chain moves inside its own tube, and we show that the corresponding characteristic frequency, ^{-1}_q, scales as ^{-1}_{q G}q⁶, where _G is a known topological factor. For qR_G 1, however, kinetics is rather dominated by long-wavelength (reptation) motions where unlike ramified polymers creep inside a long tube. For this case, we find that ^{-1}_q ( 0 )q² (_c - ), where ( 0 ) is another known topological factor that represents the total mobility of free monomers belonging to connected chains and reticulation points, and _c accounts for the critical value of the segregation parameter. Finally, the derived relaxation rate must be compared to that relative to a linear polymer mixture.