On α-Induction, Chiral Generators and¶Modular Invariants for Subfactors

We consider a type III subfactor N⊂N of finite index with a finite system of braided N-N morphisms which includes the irreducible constituents of the dual canonical endomorphism. We apply α-induction and, developing further some ideas of Ocneanu, we define chiral generators for the double triangle algebra. Using a new concept of intertwining braiding fusion relations, we show that the chiral generators can be naturally identified with the α-induced sectors. A matrix Z is defined and shown to commute with the S- and T-matrices arising from the braiding. If the braiding is non-degenerate, then Z is a “modular invariant mass matrix” in the usual sense of conformal field theory. We show that in that case the fusion rule algebra of the dual system of M-M morphisms is generated by the images of both kinds of α-induction, and that the structural information about its irreducible representations is encoded in the mass matrix Z. Our analysis sheds further light on the connection between (the classifications of) modular invariants and subfactors, and we will construct and analyze modular invariants from SU(n) _k loop group subfactors in a forthcoming publication, including the treatment of all SU(2) _k modular invariants. Received: 13 April 1999 / Accepted: 13 July 1999     

Polynomial Invariants for Torus Knots¶and Topological Strings

We make a precision test of a recently proposed conjecture relating Chern–Simons gauge theory to topological string theory on the resolution of the conifold. First, we develop a systematic procedure to extract string amplitudes from vacuum expectation values (vevs) of Wilson loops in Chern–Simons gauge theory, and then we evaluate these vevs in arbitrary irreducible representations of SU(N) for torus knots. We find complete agreement with the predictions derived from the target space interpretation of the string amplitudes. We also show that the structure of the free energy of topological open string theory gives further constraints on the Chern–Simons vevs. Our work provides strong evidence towards an interpretation of knot polynomial invariants as generating functions associated to enumerative problems. Received: 1 May 2000 / Accepted: 6 November 2000     

Multi-Interval Subfactors and Modularity¶of Representations in Conformal Field Theory

We describe the structure of the inclusions of factors ?(E)⊂?(E′)′ associated with multi-intervals E⊂ℝ for a local irreducible net ? of von Neumann algebras on the real line satisfying the split property and Haag duality. In particular, if the net is conformal and the subfactor has finite index, the inclusion associated with two separated intervals is isomorphic to the Longo–Rehren inclusion, which provides a quantum double construction of the tensor category of superselection sectors of ?. As a consequence, the index of ?(E)⊂?(E′)′ coincides with the global index associated with all irreducible sectors, the braiding symmetry associated with all sectors is non-degenerate, namely the representations of ? form a modular tensor category, and every sector is a direct sum of sectors with finite dimension. The superselection structure is generated by local data. The same results hold true if conformal invariance is replaced by strong additivity and there exists a modular PCT symmetry. Received: 7 July 1999 / Accepted: 13 January 2001     

Wigner Symbols and Combinatorial Invariants¶of Three-Manifolds with Boundary

In this paper we generalize the partition function proposed by Ponzano and Regge in 1968 to the case of a compact 3-dimensional simplicial pair (M, ∂M). The resulting state sum Z[(M, ∂M)] contains both Wigner 6j symbols associated with tetrahedra and Wigner 3jm symbols associated with triangular faces lying in ∂M. In order to show the invariance of Z[(M, ∂M)] under PL-homeomorphisms we exploit some results due to Pachner on the equivalence of n-dimensional PL-pairs both under bistellar moves on n-simplices in the interior of M and under elementary boundary operations (shellings and inverse shellings) acting on n-simplices which have some component in ∂M. We find, in particular, the algebraic identities – involving a suitable number of Wigner symbols – which realize the complete set of Pachner's boundary operations in n=3. The results established for the classical SU(2)-invariant Z[(M, ∂M)] are further extended to the case of the quantum enveloping algebra U _q (sl(2,ℂ)) (q a root of unity). The corresponding quantum invariant, M _q [(M, ∂M)], turns out to be the counterpart of the Turaev–Viro invariant for a closed 3-dimensional PL-manifold. To Giorgio Ponzano and Tullio Regge Received: 14 December 1998 / Accepted: 30 January 2000     

Donaldson Invariants of Product Ruled Surfaces¶and Two-Dimensional Gauge Theories

Using the u-plane integral of Moore and Witten, we derive a simple expression for the Donaldson invariants of product ruled surfaces Σ _g ×S ², where Σ _g is a Riemann surface of genus g. This expression generalizes a theorem of Morgan and Szabó for g=1 to any genus g. We give two applications of our results: (1) We derive Thaddeus' formulae for the intersection pairings on the moduli space of rank two stable bundles over a Riemann surface. (2) We derive the eigenvalue spectrum of the Fukaya–Floer cohomology of Σ _g ×S ¹. Received: 22 July 1999 / Accepted: 12 June 2000     

Gauge Theoretical Equivariant Gromov–Witten Invariants and the Full Seiberg–Witten Invariants¶of Ruled Surfaces

Let F be a differentiable manifold endowed with an almost K?hler structure (J,ω), α a J-holomorphic action of a compact Lie group on F, and K a closed normal subgroup of which leaves ω invariant. The purpose of this article is to introduce gauge theoretical invariants for such triples (F,α,K). The invariants are associated with moduli spaces of solutions of a certain vortex type equation on a Riemann surface Σ. Our main results concern the special case of the triple

Self-Similarity of Volume Measures for Laplacians¶on P. C. F. Self-Similar Fractals

Our main goal in this paper is to obtain a precise analogue of Weyl's asymptotic formula for the eigenvalue distribution of Laplacians on a certain class of “finitely ramified” (or p.c.f.) self-similar fractals, building, in particular, on the work of [7, 9, 22, 24]. Our main result consists in precisely identifying (for the class of “decimable fractals”) the volume measures constructed by the second author in [24] for general p.c.f. fractals and showing that they are self-similar. From a physical point of view, our results should be relevant to the study of the density of states for diffusions and wave propagation in fractal media. Received: 6 September 1999 / Accepted: 7 October 2000     

Solutions of the Dirac–Fock Equations for Atoms¶and Molecules

The Dirac-Fock equations are the relativistic analogue of the well-known Hartree-Fock equations. They are used in computational chemistry, and yield results on the inner-shell electrons of heavy atoms that are in very good agreement with experimental data. By a variational method, we prove the existence of infinitely many solutions of the Dirac-Fock equations "without projector", for Coulomb systems of electrons in atoms, ions or molecules, with Z h 124, N h 41, N h Z. Here, Z is the sum of the nuclear charges in the molecule, N is the number of electrons.     

Perturbation of Symmetries and Hojman Adiabatic Invariants for Mechanical Systems with Unilateral Holonomic Constraints

The perturbation of symmetries and adiabatic invariants for mechanical systems with unilateral holonomic constraints are studied. The exact invariant in the form of Hojman led by special Lie symmetries for an undisturbed system with unilateral constraints is given. Based on the concept of high-order adiabatic invariant of mechanical systems, the perturbation of Lie symmetries for the system under the action of small disturbance is investigated, and a new adiabatic invariant for the system with unilateral holonomic constraints is obtained, which can be called Hojman adiabatic invariant. In the end of the paper, an example is given to illustrate the application of the results.     

Lie Symmetrical Perturbation and Adiabatic Invariants of Generalized Hojman Type for Relativistic Birkhoffian Systems

For a relativistic Birkhoffian system, the Lie symmetrical perturbation and adiabatic invariants of generalized Bojman type are studied under general infinitesimal transformations. On the basis of the invariance of relativistic Birkhotfian equations under general infinitesimal transformations,Lie symmetrical transformations of the system are constructed, which only depend on the Birkhoffian variables. The exact invariants in the form of generalized Hojman conserved quantities led by the Lie symmetries of relativistic Birkhoffian system without perturbations are given. Based on the definition of higher-order adiabatic invariants of a mechanical system, the perturbation of Lie symmetries for relativistic Birkhoffian system with the action of small disturbance is investigated, and a new type of adiabatic invariants of the system is obtained. In the end of the paper, an example is given to illustrate the application of the results.     

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     

Lie Symmetrical Perturbation and Adiabatic Invariants of Generalized Hojman Type for Disturbed Nonholonomic Systems

For a nonholonomic mechanics system with the action of small disturbance, the Lie symmetrical perturbation and adiabatic invariants of generalized Hojman type are studied under general infinitesimal transformations of groups in which the generalized coordinates and time are variable. On the basis of the invariance of disturbed nonholonomic dynamical equations under general infinitesimal transformations, the determining equations, the constrained restriction equations and the additional restriction equations of Lie symmetries of the system are constructed, which only depend on the variables t, qs and q^.s. Based on the definition of higher-order adiabatic invariants of a mechanical system, the perturbation of Lie symmetries for a nonholonomic system with the action of small disturbance is investigated, and the Lie symmetrical adiabatic invariants, the weakly Lie symmetrical adiabatic invariants and the strongly Lie symmetrical adiabatic invariants of generalized Hojman type of disturbed nonholonomic systems are obtained. An example is given to illustrate applications of the results.     

Thermal diffusion in binary mixtures containing molecular gases. III. The temperature dependence of αT

A recent expression for the thermal diffusion factor α_T for binary atom-molecule mixtures, which includes a full range of inelastic collisional contributions [McCourt, F. R. W., 2003, Molec. Phys., 101, 2181] has been utilized to calculate its temperature dependence for equimolar N₂-He, Ne, Ar mixtures and for an equimolar CO₂-Ar mixture. Accurate classical trajectory values for the effective cross-sections entering into the expression for α_T, obtained for the most reliable potential energy surfaces available, have been employed in the calculations. Good agreement has been attained with experiment for all four binary mixtures, including the decrease of α_T with increase in temperature observed for CO₂-Ar mixtures, heretofore considered to be anomalous.     

Intermolecular Interactions. Absorption and Fluorescence Spectra of α-Naphthol

In the ground state, -naphthol interacts with water via oxygen, whereas with alcohols (ethanol and isopropanol) and acetonitrile it interacts via hydrogen from the hydroxyl group. In the excited state, -naphthol interacts much more weakly. An analysis of the dependences of positions of electronic-vibrational transitions on the solvent basicity and acidity demonstrates their nonlinearity. For the parameters describing the solvation ability of the solvent, linear dependences are observed for the following series of solvents: isopropanol, ethanol, acetonitrile, and water. The possible values of the acid number (pA*) for the excited -naphthol molecule fall within the interval from 0.4 to 1.4 depending on the chosen maximum of the absorption band. In fluorescence spectra, the band intensities differ for neutral and ionic modifications of the molecule. The fluorescence efficiency of the neutral molecule is by an order of magnitude higher than that of the ionic modification. Possible channels of quenching of the ionic modification are discussed.     

The α-efficiency of corals and its importance for ESR-dating

The uncertainty in an ESR age results almost completely from the error in the evaluation of the annual dose. In the case of corals, α-efficiency has a high influence on D_E (Paleodose). The published k_α-values vary from 0.05 to 0.2, so the correct determination of k_α is crucial. In this experiment, based on direct irradiation with an ²⁴¹Am-source, we used seven samples (20 aliquots for each sample) with different ages. The first aim was to improve the measurement conditions to reach a reliable data base for the α-efficiency calculation and the second was the question of age dependency of k_α. With respect to the results of our experiments we propose an α-efficiency value for corals of 0.06–0.07 which leads to younger ages of about 7–10% less than with the k_α factor of 0.05 we used in previous studies.     

Free-radical copolymerisation of acrylamides,acrylates, and α-olefins

We report the results of a joint theoretical and experimental investigation into the copolymerisation of acrylamides and acrylates with α-olefins in free-radical processes. The transition-state structures of models for free-radical homo- and copolymerisation involving acrylamide, methylacrylamide, methacrylate, methyl methacrylate, and ethylene have been determined using density functional theory. The reaction energies and barrier heights comport with the experimentally observed properties, including the prevalence of monomer alternation, the realised stereospecificity, and the reaction yield. Continuum solvation models have been applied to determine the sensitivity of the relative energies to the bulk solvent properties. Experimentally, a Lewis acid catalyst is demonstrated to increase the incorporation of nonpolar 1-alkenes in copolymerisations with polar acrylamides and acrylates. In the presence of the Lewis acid, scandium (III) trifluoromethanesulfonate, the copolymerisation of 1-hexene and acrylamide results in an 8.5 mol % incorporation, up from 3.9 mol % in the absence of the Lewis acid. Computations incorporating Mg²⁺ as a model Lewis acid elucidate the mechanism of this catalysis. In the addition of methacrylate to a methyl methacrylate radical terminated polymer, the Lewis acid binds to the carbonyls on both promoting isotactic addition, while for the addition of an alkene to the same polymer, the Lewis acid binds to the polymer, reducing the barrier for alkenyl addition inductively by withdrawing electron density. We have demonstrated the ability of computational studies to aid experimentalists in the synthesis of new copolymers with desired properties.     

Poles of Zeta and Eta Functions for Perturbations¶of the Atiyah–Patodi–Singer Problem

The zeta and eta functions of a differential operator of Dirac-type on a compact n-dimensional manifold, provided with a well-posed pseudodifferential boundary condition, have been shown in [G99] to be meromorphic on ℂ with simple or double poles on the real axis. Extending results from [G99] we show how perturbations of the boundary condition of order −J affect the poles; in particular they preserve a possible regularity of zeta at 0 and a possible simple pole of eta at 0 when J≥n. This applies to perturbations of spectral boundary conditions, also when the structure is non-product and the problem is non-selfadjoint. Received: 4 October 1999 / Accepted: 7 July 2000     

Conformational Studies of α-Substituted Carbonyl Compounds by I. R. Spectroscopy. II. α-Heterosubstituted N,N-Diethylacetamides

Abstract

The α-heterosubstituted N, N?diethylacetamides(1–8) cis/gauche conformations population ratio have been estimated through the analysis of the carbonyl group stretching vibration in the carbon tetrachloride and in chloroform. The substituent steric properties determine the cis/gauche ratio which decreases as the substituent size increases. For most of the studied amides, the gauche rotamer carbonyl band exhibits larger shifts to lower frequencies than in the corresponding α-heterosubstituted propanones¹, but for the α-chloro-(4) and for the α-bromo-N,N?diethylacetamide(5) a reverse trend is observed. This behavior can be related to changes in the πC=O C—X Dihedral angle, an hyperconjugative effect being sugges ted to explain the frequency shifts of the gauche rotamers.