a new application of random matrices ext c _ mathrm red f_2 is not a group in the process of developing the theory of free probability and free entropy voiculescu introduced in 1991 a random matrix model for a free semicircular system since then random matrices have played a key role in von neumann algebra theory cf v8 v9 the main result of this paper is the following extension of voiculescui s random matrix result let x n 1 x n r be a system of r stochastically independent n i n gaussian self adjoint random matrices as in voiculescui s random matrix paper v4 and let x1 xr be a semi circular system in a c probability space then for every polynomial p in r noncommuting variables lim niaeia p x n 1 o x n r o p x1 xr for almost all o in the underlying probability space we use the result to show that the ext invariant for the reduced c algebra of the free group on 2 generators is not a group but only a semi group this problem has been open since anderson in 1978 found the first example of a c algebra a for which ext a is not a group

on representations of the elliptic quantum group e τ η sl 2 we describe representation theory of the elliptic quantum groupe sl 2 it turns out that the representation theory is parallel to the representation theory of the yangiany sl 2 and the quantum loop group u q widetilde sl 2 we introduce basic notions of representation theory of the elliptic quantum groupe sl 2 and construct three families of modules evaluation modules cyclic modules one dimensional modules we show that under certain conditions any irreducible highest weight module of finite type is isomorphic to a tensor product of evaluation modules and a one dimensional module we describe fusion of finite dimensional evaluation modules in particular we show that under certain conditions the tensor product of two evaluation modules becomes reducible and contains an evaluation module in this case the imbedding of the evaluation module into the tensor product is given in terms of elliptic binomial coefficients we describe the determinant element of the elliptic quantum group representation theory becomes special ifn m l wheren m l are integers we indicate some new features in this case

the finite criss cross method for hyperbolic programming in this paper the finite criss cross method is generalized to solve hyperbolic fractional linear programming problems just as in the case of linear or quadratic programming the criss cross method can be initialized with any not necessarily feasible basic solution it is known that if the feasible region of the problem is unbounded then some of the known algorithms fail to solve the problem our criss cross algorithm does not have such drawback finiteness of the procedure is proved under the usual mild assumptions some small numerical examples illustrate the main features of the algorithm and show that our method generates different iterates than other earlier published methods

an interior point method for fractional programs with convex constraints we present an interior point method for a class of fractional programs with convex constraints the proposed algorithm converges at a polynomial rate similarly as in the case of a convex problem even though fractional programs are only pseudo convex here the rate of convergence is measured in terms of the area of two dimensional convex setsck containing the origin and certain projections of the optimal points and the area ofck is reduced by a constant factorc 1 at each iteration the factorc depends only on the self concordance parameter of a barrier function associated with the feasible set we present an outline of a practical implementation of the proposed method and we report results of some preliminary numerical experiments

conformal blocks on elliptic curves and the knizhnik zamolodchikov bernard equations we give an explicit description of the vector bundle of wzw conformal blocks on elliptic curves with marked points as a subbundle of a vector bundle of weyl group invariant vector valued theta functions on a cartan subalgebra we give a partly conjectural characterization of this subbundle in terms of certain vanishing conditions on affine hyperplanes in some cases explicit calculations are possible and confirm the conjecture the friedan shenker flat connection is calculated and it is shown that horizontal sections are solutions of bernard s generalization of the knizhnik zamolodchikov equation

on polynomiality of the method of analytic centers for fractional problems we establish polynomial time convergence of the method of analytic centers for the fractional programming problemt min x g tb x a x k whereg n is a closed and bounded convex domain k m is a closed convex cone anda x g n b x g k are regular enough say affine mappings

multipliers of the fourier algebras of some simple lie groups and their discrete subgroups for any amenable locally compact group g the space of multipliers ma g of the fourier algebra a g coincides with the space b g of functions on g that are linear combinations of continuous positive definite functions we prove that ma g b g 0 for many non amenable connected groups more specifically we prove that moa g b g 0 for the classical complex lie groups and the general lorentz groups soo n 1 n 2 moa g is a certain subspace of ma g which we call the space of completely bounded multipliers of a g unlike ma g the space moa g has nice stability properties with respect to direct products of groups it is known that the fourier algebra of the free group on n generators n 2 2 admits an unbounded approximate unit pn which is bounded in the multiplier norm we extend this result to any closed subgroup of the general lorentz group soo n 1 moreover we show that for these groups pn can be chosen to be bounded with respect to the moa g norm by a duality argument we obtain that the reduced c algebra of every discrete subgroup of soo n 1 has the completely bounded approximation property in particular this property holds for c f2 the reduced c algebra of the free group on two generators we also prove

method of centers for minimizing generalized eigenvalues abstract we consider the problem of minimizing the largest generalized eigenvalue of a pair of symmetric matrices each of which depends affinely on the decision variables although this problem may appear specialized it is in fact quite general and includes for example all linear quadratic and linear fractional programs many problems arising in control theory can be cast in this form the problem is nondifferentiable but quasiconvex so methods such as kelley s cutting plane algorithm or the ellipsoid algorithm of shor nemirovsky and yudin are guaranteed to minimize it in this paper we describe relevant background material and a simple interior point method that solves such problems more efficiently the algorithm is a variation on huard s method of centers using a self concordant barrier for matrix inequalities developed by nesterov and nemirovsky nesterov and nemirovsky have also extended their potential reduction methods to handle the same problem since the problem is quasiconvex but not convex devising a nonheuristic stopping criterion i e one that guarantees a given accuracy is more difficult than in the convex case we describe several nonheuristic stopping criteria that are based on the dual of a related convex problem and a new ellipsoidal approximation that is slightly sharper in some cases than a more general result to nesterov and nemirovsky the algorithm is demonstrated on an example determining the quadratic lyapunov function that optimizes a decay rate estimate for a differential inclusion

an interior point method for generalized linear fractional programming we develop an interior point polynomial time algorithm for a generalized linear fractional problem the latter problem can be regarded as a nonpolyhedral extension of the usual linear fractional programming typical example which is of interest for control theory is the minimization of the generalized eigenvalue of a pair of symmetric matrices linearly depending on the decision variables

q deformed kzb heat equation completeness modular properties and sl 3 z abstract we study the properties of one dimensional hypergeometric integral solutions of the q difference quantum analogue of the knizhnik zamolodchikov bernard equations on tori we show that they also obey a difference kzb heat equation in the modular parameter give formulae for modular transformations and prove a completeness result by showing that the associated fourier transform is invertible these results are based on sl 3 z transformation properties parallel to those of elliptic gamma functions

quadratic forms in unitary operators abstract let u 1 u n be unitary matrices on l 2 denote by the matrix a defined by a i i j j k 1 n u k i j u k i j acting as a bounded operator on l 2 n n in other words a is the sum of the kronecker products of u k with their complex conjugates we show the following sharp inequality as an application we show that the natural representation of u n n 1 acting on l 2 of the unit sphere in c n and restricted to mean zero functions satisfies for any choice 1 n in u n the lower bound this extends a result due to lubotzky phillips and sarnak who proved this with so 3 in the place of u n

a system of difference equations with elliptic coefficients and bethe vectors an elliptic analogue of theq deformed knizhnik zamolodchikov equations is introduced a solution is given in the form of a jackson type integral of bethe vectors of the xyz type spin chains

the q deformed knizhnik zamolodchikov bernard heat equation we introduce a q deformation of the genus one sl 2 knizhnik zamolodchikov bernard heat equation we show that this equation for the dependence on the moduli of elliptic curves is compatible with the qkzb equations which give the dependence on the marked points

generalized fractional programming algorithms and numerical experimentation abstract several algorithms to solve the generalized fractional program are summarized and compared numerically in the linear case these algorithms are iterative procedures requiring the solution of a linear programming problem at each iteration in the linear case the most efficient algorithm is obtained by marrying the newton approach within the dinkelbach approach for fractional programming

extended trigonometric cherednik algebras and nonstationary schrodinger equations with delta potentials we realize an extended version of the trigonometric cherednik algebra as affine dunkl operators involving heaviside functions we use the quadratic casimir element of the extended trigonometric cherednik algebra to define an explicit nonstationary schrodinger equation with delta potential we use coordinate bethe ansatz methods to construct solutions of the nonstationary schrodinger equation in terms of generalized bethe wave functions it is shown that the generalized bethe wave functions satisfy affine difference knizhnik zamolodchikov equations as functions of the momenta the relation to the vector valued root system analogs of the quantum bose gas on the circle with delta function interactions is indicated

on the von neumann economic growth problem we study the complexity of the von neumann economic growth problem gamma ast max gamma mid exists y ne 0 y ge 0 b gamma ay ge 0 where a and b are given two nonnegative and rational m n matrices and a has no all zero column let the binary data length of a and b be l we develop an interior point algorithm to generate a i such that 2 1 i in om nl minm nt iterations where each iteration solves a system of m n linear equations