Conditional Entropy and the Rokhlin Metric on an Orthomodular Lattice with Bayessian State

The present paper deals with the study of conditional entropy and its properties in a quantum space (L,s), where L is an orthomodular lattice and s is a Bayessian state on L. First, we obtained a pseudo-metric on the family of all partitions of the couple (B,s), where B is a Boolean algebra and s is a state on B. This pseudo-metric turns out to be a metric (called the Rokhlin metric) by using a new notion of s-refinement and by identifying those partitions of (B,s) which are s-equivalent. The present theory has then been extended to the quantum space (L,s), where L is an orthomodular lattice and s is a Bayessian state on L. Applying the theory of commutators and Bell inequalities, it is shown that the couple (L,s) can be equivalently replaced by a couple (B,s ₀), where B is a Boolean algebra and s ₀ is a state on B.     

A Finitely Additive State Criterion for the Completeness of Inner Product Spaces

We show that an inner product space S (real, complex or quaternion) is complete if, and only if, the system of all orthogonally closed subspaces in S, denoted by F(S), admits at least one finitely additive state which is not vanishing on the set of all finite dimensional subspaces of S. Although it gives only a partial solution to the problem formulated by Pták on the existence of a finitely additive state on F(S) for incomplete S, this gives an important insight into the structure of the set of states on F(S). This criterion has no analogue whatsoever in E(S), the system of splitting subspaces of S.     

On realizations of exterior calculus with d N = 0

We study realizations of the q-exterior calculus with exterior differential d satisfying d ^N = 0, N > 2 on the free associative algebra with one generator and on the generalized Clifford algebras. Analogs of the notions of connection and curvature are discussed in the case of the q-exterior calculus on the generalized Clifford algebra. We show that the q-exterior calculus on the free associative algebra with one generator is related to q-calculus on the braided line.     

Instantons and Yang–Mills Flows on Coset Spaces

We consider the Yang–Mills flow equations on a reductive coset space G/H and the Yang–Mills equations on the manifold \mathbbR×G/H{\mathbb{R}\times G/H}. On non-symmetric coset spaces G/H one can introduce geometric fluxes identified with the torsion of the spin connection. The condition of G-equivariance imposed on the gauge fields reduces the Yang–Mills equations to f⁴{\phi^4}-kink equations on \mathbbR{\mathbb{R}}. Depending on the boundary conditions and torsion, we obtain solutions to the Yang–Mills equations describing instantons, chains of instanton–anti-instanton pairs or modifications of gauge bundles. For Lorentzian signature on \mathbbR×G/H{\mathbb{R}\times G/H}, dyon-type configurations are constructed as well. We also present explicit solutions to the Yang–Mills flow equations and compare them with the Yang–Mills solutions on \mathbbR×G/H{\mathbb{R}\times G/H}.     

Absolute Continuity in Noncommutative Measure Theory

Recent results on absolute continuity of Banach space valued operators and convergence theorems on operator algebras are deepened and summarized. It is shown that absolute continuity of an operator T on a von Neumann algebra M with respect to a positive normal functional ψ on M is not implied by the fact that the null projections of ψ are the null projections of T. However, it is proved that the implication above is true whenever M is finite or T is weak^*-continuous. Further it is shown that the absolute value preserves the Vitali-Hahn-Saks property if, and only if, the underlying algebra is finite. This result improves classical results on weak compactness of sets of noncommutative measures.     

A new strategy for solid phase synthesis of a secondary amide library using sulfonamide linker via radical traceless cleavage

A new strategy for solid phase synthesis of a secondary amide library using sulfonamide linker via radical traceless cleavage is reported. Polystyrylsulfonyl chloride (1) reacted with primary amines to afford polystyryl-supported N-alkyl sulfonamides (2), which were acylated with acid chlorides and followed by radical cleavage with TiCl₄/Zn to afford secondary amides. It was interestingly found that the products released from acyl alkanesulfonamide resins are closely dependent on the substituents of benzene rings of alkyl or acyl groups on the resins. When the substituent on benzene ring of N-benzyl group of sulfonamides is an electron rich MeO-group, the products released from sulfonamide resins are dependent on the substitution position on benzene ring: para-MeO- to yield 1,2-bis (p-methoxylphenyl)ethane and N-p-methoxylbenzyl benzamide (30:1); ortho-MeO- to give 1,2-bis (o-methoxylphenyl)ethane and N-o-methoxylbenzyl benzamide (1:15); and meta-MeO- only to release N-m-methoxylbenzyl benzamide. Neither N-benzoyl sulfonamide resins on benzene ring with electron-drawing para-O₂N-, nor the one with electron-donating para-H₂N- could release any amide product, while the N-benzoyl sulfonamide resins on benzene ring with para-acetamido group released para-acetamidobenzamides. The conjugation effect to stabilize the radical groups in the radical cleaving process was observed.     

Spreading of Lagrangian Regularity on Rational Invariant Tori

Let P _h be a self-adjoint semiclassical pseudodifferential operator on a manifold M such that the bicharacteristic flow of the principal symbol on T*M is completely integrable and the subprincipal symbol of P _h vanishes. Consider a semiclassical family of eigenfunctions, or, more generally, quasimodes u _h of P _h . We show that on a nondegenerate rational invariant torus, Lagrangian regularity of u _h (regularity under test operators characteristic on the torus) propagates both along bicharacteristics, and also in an additional “diffractive” manner. In particular, in addition to propagating along null bicharacteristics, regularity fills in the interiors of small annular tubes of bicharacteristics.     

Structure of the heavy electron compounds Ce(Cu x Al1-x )5 and Ce(Cu x Ga1-x )5, [0.6≦x≦0.8]

Neutron powder diffraction measurtements on the recently discovered heavy electron compounds Ce(Cu _x Al_1-x )₅ and Ce(Cu _x Ga_1-x )₅ show that Al or Ga atoms in these compounds locate exclusively on the 3g sites of space groupP6/mmm. Further Rietveld analysis shows that these atoms on the 3g sites are disordered.     

An explicit *-product on the cotangent bundle of a Lie group

We give explicit formulas for a ^*-product on the cotangent bundle T ^* G of a Lie group G; these formulas involve on the one hand the multiplicative structure of the universal enveloping algebra U(G) of the Lie algebra G of G and on the other hand bidifferential operators analogous to the ones used by Moyal to define a ^*-product on IR²ⁿ.Chargé de recherches au FNRS, on leave of absence from Université libre de Bruxelles.     

Solitons on Noncommutative Orbifold T 2/Z N

Following the construction of the projection operators on T ² presented by Gopakumar, Headrick and Spradlin, we construct a set of projection operators on the integral noncommutative orbifold T ²/G(G=Z _N , N=2, 3, 4, 6) which correspond to a set of solitons on T ²/Z _N in noncommutative field theory. In this way, we derive an explicit form of projector on T ²/Z ₆ as an example. We also construct a complete set of projectors on T ²/Z _N by series expansions for integral case.     

Smoothing and extending cosmic time functions

Any continuous time function on aC ^k space-timeV (i.e., a scalar onV that increases along any causal curve) can be approximated by smoothC ^k time functions. A time function defined on a (bounded) subset of a stably causalV can be extended to a time function on the whole ofV.     

In Situ Monitoring the Photolysis of Fluoranthene Adsorbed on Mangrove Leaves Using Fiber-Optic Fluorimetry

A newly established fiber-optic fluorimetry was used to determine the fluoranthene (Fla) adsorbed on the leaf surfaces of three mangrove species, such as Excoecarla agallocha (Ea), Kandelia candel (Kc) and Aegiceras corniculatum (Ac). With the method, the photolysis of Fla adsorbed on Ea, Kc and Ac leaf surfaces was studied. Under the laboratory conditions, using a high pressure mercury lamp as light source, photolysis was found to be the main transformation pathway for the Fla adsorbed on the three mangrove leaves, whereas disappearance of the adsorbed Fla as a result of volatilization and absorption could be negligible. The photolysis of Fla adsorbed on the three mangrove leaf surfaces followed first-order kinetics with photolysis rates of Fla in the order of Ac > Ea > Kc. The photolysis half lives of Fla adsorbed on the leaves were from 14.32 min to 69.31 min. The different initial concentrations of Fla adsorbed on the selected mangrove leaves did not affect the photolysis rates of Fla. The absorption characteristics of leaf-wax played an important role to influence the different photolysis rates of Fla adsorbed on these three mangrove species.     

Nucleon charge exchange on the deuteron: A critical review

The existing experimental data on the d(n, p)nn and d(p, n)pp cross-sections in the forward direction are reviewed in terms of the Dean sum rule. It is shown that the measurement of the ratio of the charge exchange on the deuteron to that on the proton might, if taken together with other experimental data, allow a direct construction of the np → np scattering amplitude in the backward direction with few ambiguities. Communicated by T. Hennino     

Loop and Path Spaces and¶Four-Dimensional BF Theories:¶Connections, Holonomies and Observables

We study the differential geometry of principal G-bundles whose base space is the space of free paths (loops) on a manifold M. In particular we consider connections defined in terms of pairs (A,B), where A is a connection for a fixed principal bundle P(M,G) and B is a 2-form on M. The relevant curvatures, parallel transports and holonomies are computed and their expressions in local coordinates are exhibited. When the 2-form B is given by the curvature of A, then the so-called non-abelian Stokes formula follows. For a generic 2-form B, we distinguish the cases when the parallel transport depends on the whole path of paths and when it depends only on the spanned surface. In particular we discuss generalizations of the non-abelian Stokes formula. We study also the invariance properties of the (trace of the) holonomy under suitable transformation groups acting on the pairs (A,B). In this way we are able to define observables for both topological and non-topological quantum field theories of the BF type. In the non-topological case, the surface terms may be relevant for the understanding of the quark-confinement problem. In the topological case the (perturbative) four-dimensional quantum BF-theory is expected to yield invariants of imbedded (or immersed) surfaces in a 4-manifold M. Received: 28 March 1998 / Accepted: 12 September 1998     

GLq(N)-covariant braided differential bialgebras

We study the possibility of defining the (braided) comultiplication for the GL _q (N)-covariant differential complexes on some quantum spaces. We discover suchdifferential bialgebras (and Hopf algebras) on the bosonic and fermionic quantum hyperplanes (with additive coproduct) and on the braided matrix algebra BM _q (N) with both multiplicative and additive coproducts. The latter case is related (forN = 2) to theq-Minkowski space andq-Poincaré algebra.     

Spanning Trees on the Sierpinski Gasket

We present the numbers of spanning trees on the Sierpinski gasket SG _d (n) at stage n with dimension d equal to two, three and four. The general expression for the number of spanning trees on SG _d (n) with arbitrary d is conjectured. The numbers of spanning trees on the generalized Sierpinski gasket SG _d,b (n) with d = 2 and b = 3,4 are also obtained.     

All Stable Characteristic Classes of Homological Vector Fields

An odd vector field Q on a supermanifold M is called homological, if Q ² = 0. The operator of Lie derivative L _Q makes the algebra of smooth tensor fields on M into a differential tensor algebra. In this paper, we give a complete classification of certain invariants of homological vector fields called characteristic classes. These take values in the cohomology of the operator L _Q and are represented by Q-invariant tensors made up of the homological vector field and a symmetric connection on M by means of the algebraic tensor operations and covariant differentiation.     

Idempotents as <Emphasis Type="Italic">J</Emphasis>-Projections

Let B(H) ^Id be the set of all bounded idempotents on a Hilbert space H. Fix p∈B(H) ^Id . The aim of the paper is to show a set of symmetries J on H for which p is a J-projection.     

Symplectic Structures of Moduli Space¶of Higgs Bundles over a Curve and Hilbert Scheme¶of Points on the Canonical Bundle

The moduli space of triples of the form (E,θ,s) are considered, where (E,θ) is a Higgs bundle on a fixed Riemann surface X, and s is a nonzero holomorphic section of E. Such a moduli space admits a natural map to the moduli space of Higgs bundles simply by forgetting s. If (Y,L) is the spectral data for the Higgs bundle (E,θ), then s defines a section of the line bundle L over Y. The divisor of this section gives a point of a Hilbert scheme, parametrizing 0-dimensional subschemes of the total space of the canonical bundle K _X , since Y is a curve on K _X . The main result says that the pullback of the symplectic form on the moduli space of Higgs bundles to the moduli space of triples coincides with the pullback of the natural symplectic form on the Hilbert scheme using the map that sends any triple (E,θ,s) to the divisor of the corresponding section of the line bundle on the spectral curve. Received: 15 January 2000 / Accepted: 25 March 2001     

The W k, p -Continuity of the Schrödinger Wave Operators on the Line

We prove that the wave operators for the Schr?dinger equation on the line are continuous on the Sobolev spaces W _{k, p} , 1 < p < ∞. Moreover, if the potential is exceptional and , where f ₁(x, 0) is a Jost solution at zero energy, the wave operators are continuous on W _{k ,1} and on W _{k ,∞}. Received: 21 April 1999 / Accepted: 15 July 1999