Inhomogeneous quantum groups and their quantized universal enveloping algebras

The quantum group IGL _q (N), the inhomogenization of GL _q (N), is formulated with -matrices. Theq-deformed universal enveloping algebra is constructed as the algebra of regular functionals in this formulation and contains the partial derivatives of the covariant differential calculus on the quantum space.     

Electromagnetic duality transformations in supersymmetric field theories

We explore the properties of theories which are invariant under electromagnetic duality.N=2 supergravity Yang-Mills systems are discussed in more details. In particular, a symplectic and coordinate covariant framework is established which allows one to discuss classical and quantum duality symmetries (T andS dualities), as they emerge in superstrings in four dimensions withN=2 spacetime supersymmetry.This article is dedicated to the memory of my friend Julian Schwinger     

h-Deformation of GL(1∣1)

An h-deformation of a (graded) Hopf algebra of functions on supergroup GL(11) is introduced via a contraction of GL _q (11). The deformation parameter h is odd (Grassmann). A related differential calculus on h-superplane is presented.     

On the dimension of some modular irreducible representations of the symmetric group

We compute the dimension of some irreducible representations of the symmetric groups in characteristic p (Theorem 2). The representations considered here are associated with Young diagrams m: m ₁m₂...m_lsuch that m ₁–m_l(p–l). The formula is based on a variant of Verlinde's formula which computes some tensor product multiplicities of indecomposable modules for GL₁(F _p ).     

The Hermite-like description of two-magnon states of 1D quantumS = 1/2 chains with elliptic exchange is complete

In this Letter, we give an example of Navier-Stokes flow with a viscosityv, which converges to Euler flow inL ² norm asv 0. Our Navier-Stokes flow has a discontinuity on the boundary at the initial time, the so-called initial layer, and its initial data has singularity asymptotically asv 0 near the boundary. Even with these singularities, it can be said that our Navier-Stoke flow can converge to some Euler flow inL ² norm.     

Thomae Type Formulae For Singular Z N Curves

We give an elementary and rigorous proof of the Thomae type formula for the singular curves . To derive the Thomae formula we use the traditional variational method which goes back to Riemann, Thomae and Fuchs. An important step of the proof is the use of the Szegö kernel computed explicitly in algebraic form for non-singular 1/N-periods. The proof inherits principal points of Nakayashiki’s proof (Nakayashiki in Publ. Res. Inst. Math. Sci 33(6) 987–1015, 1997) obtained for non-singular Z _N curves.     

Krichever-Novikov algebras for more than two points: Explicit generators

For the Krichever-Novikov algebras and modules of meromorphic vector fields and forms with more than two poles on higher genus Riemann surfaces, explicit expressions of the generators are given. These are expressions in terms of theta functions and prime forms for g1, expressions in terms of the Weierstraß function for g=1 and rational expressions for g=0.     

An algebra related to the fusion rules of Wess-Zumino-Witten models

We introduce a new family of examples of quantum groups. Its irreducible representations are indexed by level l dominant integral weights of affine Lie algebra s(n), and their branching rules with respect to the truncated tensor product is given by the fusion rules of SU(n) _l -Wess-Zumino-Witten model in conformal field theory.     

Retrieval Properties of Bidirectional Associative Memories

We study a two-layer neural network made of N and M(N) neurons, producing a two-way association search for a family of p(N) patterns, where each pattern is a pair of two independent sub-categories of information having respectively N and M(N) components. In terms of the ratio =lim _N M(N)/N, we study the retrieval capability of this network and show that there exists, at least, three regimes of association for which we determine the evolution of the threshold _c () of the storage capacity =lim _N p(N)/N.     

Tau Functions for the Dirac Operator on the Cylinder

The goal of the present paper is to calculate the determinant of the Dirac operator with a mass in the cylindrical geometry. The domain of this operator consists of functions that realize a unitary one-dimensional representation of the fundamental group of the cylinder with n marked points. The determinant represents a version of the isomonodromic -function, introduced by M. Sato, T. Miwa and M. Jimbo. It is calculated by comparison of two sections of the det*-bundle over an infinite-dimensional grassmannian. The latter is composed of the spaces of boundary values of some local solutions to the Dirac equation. The principal ingredients of the computation are the formulae for the Green function of the singular Dirac operator and for the so-called canonical basis of global solutions on the 1-punctured cylinder. We also derive a set of deformation equations satisfied by the expansion coefficients of the canonical basis in the general case and find a more explicit expression for the -function in the simplest case n=2.Acknowledgement I would like to thank A. I. Bugrij and V. N. Roubtsov for constant support and numerous stimulating discussions. I am grateful to S. Pakuliak for his lectures on the infinite-dimensional grassmannians and boson-fermion correspondence. I would also like to express my gratitude to J. Palmer, whose clear ideas made this work possible.     

Remarks on a quantum state extension problem

The problem is considered of finding, for a given pair of states on C ^*-algebras A ₁ A ₂ and A ₂ A ₃, a joint extension to A ₁ A ₂ A ₃. The fact that, in contrast to classical probability, such an extension may fail to exist, is related to the fact that different convex decompositions of the same quantum state need not have a common refinement. Improved necessary criteria for extensibility in terms of Bell's inequalities are derived, and are compared to the necessary and sufficient criteria, as well as to entropic bounds in the simplest case.     

On the nilpotent * - Fourier transform

LetG be a nilpotent Lie group. The adapted nilpotent Fourier transform was introduced by D. Arnal and J. C. Cortet,:L(G) C (V,L(^2d )), whereL(G) is the Schwartz space ofG andV × ^2k is aG-invariant Zariski open set ing ^* the dual of the Lie algebra ofG. We prove the surjectivity of this transformation, which allows us to extend it to distribution spaces.     

Laser-Induced Flourescence Excitation Spectroscopy and Photophysics of Naphthalene Bichromophoric Molecules in Supersonic Jets

We present studies of interchromophore interactions under supersonic jet conditions in a large number of dinaphthyl bichromophoric molecules by measuring their laser-induced fluorescence (LIF) excitation spectra. The molecules are composed of two naphthalene chromophores connected by an n-methylene bridge. The length of the bridge was varied as a function of the number of methylene units (n = 0, 1, 2, 4, 6), of the general type NnN(i,j), were N denotes naphthalene moiety, n the number of methylene units in the bridge, and (i,j) are the or positions of the bridge at each of the chromophores. We obtained high-quality LIF spectra of these bichromophoric olecules. In the molecules N1N(2,2), N1N(1,2), N2N(2,2), and N2N(1,2), the spectrum is characterized by an intense 0–0 region, with series of low-frequency progressions. These progressions are assigned as vibrational modes of the bridge. The appearance of several series of progressions is explained either by the excitation of different chromophores (in the mixed molecules) or by the excitation of different populated conformers. The spectrum of N4N(1,1) is different in several aspects from these spectra. The origin is shifted farther to the red, to 31,402 cm^–1. Low-frequency progressions or other transitions are not observed near the origin, but typical intrachromophore naphthalene vibrations are intense. The spectra of N6N(1,1) and N6N(2,2) are also characterized by intense intrachromophore vibrations, however, the spectrum of N6N(2,2) is very complicated due to many populated conformations, while that of N6N(1,1) is more simple.     

Generalized Apollonian packings

In this paper we generalize the classical two-dimensional Apollonian packing of circles to the case where the circles are no more tangent. We introduce two elements ofSL(2,) as generators:R andT that are hyperbolic rotations of 2/3 and 2/N (N=2,3,4....), around two distinct points. The limit set of the discrete group generated byR andT provides, forN=7,8,9,.... a generalization of the Apollonian packing (which is itself recovered forN=). The valuesN=2,3,4,5 produce a very different result, giving rise to the rotation groups of the cube forN=2 and 4, and the icosahedron forN=3 and 5. ForN=6 the group is no longer discrete. To further analyze this structure forN7, we move to the Minkowski space in which the group acts on a one sheeted hyperboloid. The circles are now represented by points on this variety and generate a crystal on it.Laboratoire de l'Institut de Recherche Fondamentale du Commissariat à l'Energie Atomique     

Complete integrability of relativistic Calogero-Moser systems and elliptic function identities

Poincaré-invariant generalizations of the Galilei-invariant Calogero-MoserN-particle systems are studied. A quantization of the classical integralsS ₁, ...,S _N is presented such that the operators ₁, ..., _N mutually commute. As a corollary it follows thatS ₁, ...,S _N Poisson commute. These results hinge on functional equations satisfied by the Weierstrass - and -functions. A generalized Cauchy identity involving the -function leads to anN×N matrixL whose symmetric functions are proportional toS ₁, ...,S _N .     

SU(2,2¦N) · 〈N′〉 superunified theory

We establish a superspaceU for supergauge actions, a conformal supergroupSU(2,2¦N), and a conformal extendedSU(2,2¦N) supergravity theory. Using the Lagrangian Higgs evolution mechanism under the supergroupSU(2,2¦N) SU(N) acting on the superspaceU, we advance anSU(2,2¦N)SUN superunified theory of a superunited system, discuss the Lagrangian evolution of the superunified theory, and give the fiber bundle geometry of all the above mechanisms.     

Ground-state structure in a highly disordered spin-glass model

We propose a new Ising spin-glass model on Z ^d of Edwards-Anderson type, but with highly disordered coupling magnitudes, in which a greedy algorithm for producing ground states is exact. We find that the procedure for determining (infinite-volume) ground states for this model can be related to invasion percolation with the number of ground states identified as 2 ^N , whereN=N(d) is the number of distinct global components in the invasion forest. We prove thatN(d)= if the invasion connectivity function is square summable. We argue that the critical dimension separatingN=1 andN= isd _c=8. WhenN(d)=, we consider free or periodic boundary conditions on cubes of side lengthL and show that frustration leads to chaoticL dependence withall pairs of ground states occurring as subsequence limits. We briefly discuss applications of our results to random walk problems on rugged landscapes.     

The equilibrium thermodynamics of a spin-boson model

We consider the equilibrium thermodynamics of a Dicke-type model forN identical spins of arbitrary magnitude interacting linearly and homogeneously with a boson field in a volumeV _N, in the limitN,V _N, withN/V _N=const. The system exhibits a second-order phase transition; complete information on the spin polarizations and their correlations is obtained. The proofs use a general result on the free energy of quantum spin systems based on the large deviation principle and the Berezin-Lieb inequalities.     

A characterization of Robertson-Walker spaces by null sectional curvature

ForN a null vector andA a vector perpendicular toN, define the null sectional curvature, with respect to TV, of the planeN A ask _N(N A) = R(N,A)A,N<A,A.Then Robertson-Walker metrics can be locally characterized as those for whichk _n at each point is a constant for all the null plans at that point (in each null direction,N must be appropriately chosen). A global characterization of Robertson-Walker spaces is achieved by adding completeness and causality hypotheses.