lossless inverse scattering digital filters and estimation theory methods to construct rational solutions of the lossless inverse scattering lis problem for one port passive digital systems are described the first method is recursive and can be viewed as a generalization of the celebrated schur algorithm the second method is global and leads to a parametrization of what we call fundamental solutions from which all lis solutions may be constructed quite a few classical problems in estimation theory and network theory may be viewed as special cases of the lis problem with each fundamental solution there is a solution of a corresponding estimation problem leading to a prediction and a modeling filter for a given stochastic process

braid matrices and structure constants for minimal conformal models using the feigin fuchs representation of minimal conformal models in a form introduced recently by one of us the braid group representation matrices describing the analytic continuation properties of conformal blocks are computed in a suitable normalization their matrix elements are shown to essentially factorize into pairs of boltzmann weights of critical rsos models in a certain limit of the spectral parameter these boltzmann weights are related to quantum groupr matrices by the vertex sos transformation we show that the crossing symmetry of the four point function in left right symmetric models follows from a quantum group relation also called crossing symmetry this observation gives a simple way to evaluate the structure constants

semi infinite cohomology and string theory abstract we develop the theory of semi infinite cohomology of graded lie algebras first introduced by feigin we show that the relative semi infinite cohomology has a structure analogous to that of the de rham cohomology in kahler geometry we prove a vanishing theorem for a special class of modules and we apply our results to the case of the virasoro algebra and the fock module in this case the zero cohomology is identified as the physical subspace of the fock module and the no ghost theorem follows we reveal the profound relation of semi infinite cohomology theory to the gauge invariant free string theory constructed by banks and peskin we then indicate the connection between gauge invariant interacting string theories and the geometric realizations of the infinite dimensional lie algebras

qualitative properties of solutions to h optimization problems abstract let h denote the space of functions in l d 2 that have vanishing negative fourier coefficients suppose ei0 z is a nonnegative function of ei0 and z where 0 2 and z c this paper is concerned with solving ei0 f ei0 inf ei0 h ei0 h h with a function f h conditions on that will guarantee the existence uniqueness and continuity of solutions to the above equation are given

non euclidean geometrical aspects of the schur and levinson szego algorithms in this paper we address non euclidean geometrical aspects of the schur and levinson szego algorithms we first show that the lobachevski geometry is by construction one natural geometrical environment of these algorithms since they necessarily make use of automorphisms of the unit disk we next consider the algorithms in the particular context of their application to linear prediction then the schur resp levinson szego algorithm receives a direct resp polar spherical trigonometry st interpretation which is a new feature of the classical duality of both algorithms

quantum howe duality and invariant polynomials abstract we construct two examples of q deformed classical howe dual pairs sl 2 c so 3 c and sl 2 c sl n c moreover we obtain a noncommutative version of the first fundamental theorem of classical invariant theory our approach to these dualities differs from 9 and 11 furthermore we solve the tensor product decomposition problem for verma modules over u q sl 2 c provided q is not a root of unity

β γ systems and the deformations of the brst operator we describe the relation between simple logarithmic cfts associated with closed and open strings and their infinite metric limits corresponding to the systems this relation is studied on the level of the brst complex we show that the consideration of metric as a perturbation leads to a certain deformation of the algebraic operations of the lian zuckerman type on the vertex algebra associated with the systems the maurer cartan equations corresponding to this deformed structure in the quasi classical approximation lead to the nonlinear field equations as an explicit example we demonstrate that using this construction yang mills equations can be derived this gives rise to a nontrivial relation between the courant dorfman algebroid and homotopy algebras emerging from the gauge theory we also discuss a possible algebraic approach to the study of beta functions in sigma models

robust stabilization of distributed systems abstract this paper is devoted to the problem of robust stabilization for a class of distributed plants we consider the gain margin optimization problem and the robust stabilization problem for multiplicative perturbations using techniques from interpolation theory and complex variables we obtain explicit necessary and sufficient conditions for robust stabilizability a few examples are included to illustrate the results

from finite covariance windows to modeling filters a convex optimization approach the trigonometric moment problem is a classical moment problem with numerous applications in mathematics physics and engineering the rational covariance extension problem is a constrained version of this problem with the constraints arising from the physical realizability of the corresponding solutions although the maximum entropy method gives one well known solution in several applications a wider class of solutions is desired in a seminal paper georgiou derived an existence result for a broad class of models in this paper we review the history of this problem going back to carath eodory as well as applications to stochastic systems and signal processing in particular we present a convex optimization problem for solving the rational covariance extension problem with degree constraint given a partial covariance sequence and the desired zeros of the shaping filter the poles are uniquely determined from the unique minimum of the corresponding optimization problem in this way we obtain an algorithm for solving the covariance extension problem as well as a constructive proof of georgiou s existence result and his conjecture a generalized version of which we have recently resolved using geometric methods we also survey recent related results on constrained nevanlinna pick interpolation in the context of a variational formulation of the general moment problem

topological representations of the quantum groupuq sl2 we define a topological action of the quantum groupuq sl2 on a space of homology cycles with twisted coefficients on the configuration space of the punctured disc this action commutes with the monodromy action of the braid groupoid which is given by ther matrix ofuq sl2

a new approach to spectral estimation a tunable high resolution spectral estimator traditional maximum entropy spectral estimation determines a power spectrum from covariance estimates here we present a new approach to spectral estimation which is based on the use of filter banks as a means of obtaining spectral interpolation data such data replaces standard covariance estimates a computational procedure for obtaining suitable pole zero arma models from such data is presented the choice of the zeros ma part of the model is completely arbitrary by suitable choices of filter bank poles and spectral zeros the estimator can be tuned to exhibit high resolution in targeted regions of the spectrum

new perspectives on the brst algebraic structure of string theory motivated by the descent equation in string theory we give a new interpretation for the action of the symmetry charges on the brst cohomology in terms of what we callthe gerstenhaber bracket this bracket is compatible with the graded commutative product in cohomology and hence gives rise to a new class of examples of what mathematicians call agerstenhaber algebra the latter structure was first discussed in the context of hochschild cohomology theory 11 off shell in the chiral brst complex all the identities of a gerstenhaber algebra hold up to homotopy applying our theory to thec 1 model we give a precise conceptual description of the brst gerstenhaber algebra of this model we are led to a direct connection between the bracket structure here and the anti bracket formalism in bv theory 29 we then discuss the bracket in string backgrounds with both the left and the right movers we suggest that the homotopy lie algebra arising from our gerstenhaber bracket is closely related to the hla recently constructed by witten zwiebach finally we show that our constructions generalize to any topological conformal field theory

constructions of vertex operator coalgebras via vertex operator algebras the notion of vertex operator coalgebra is presented which corre sponds to the family of correlation functions modeling one string propagating in space time splitting into n strings in conformal field theory this notion is in some sense dual to the notion of vertex operator algebra we prove that any vertex operator algebra equipped with a non degenerate virasoro preserving bilinear form gives rise to a corresponding vertex operator coalgebra we then explicitly calculate the vertex operator coalgebra structure and unique bilin ear form for the heisenberg algebra case which corresponds to considering free bosons in conformal field theory

quantum group as semi infinite cohomology we obtain the quantum group slq 2 as semi infinite cohomology of the virasoro algebra with values in a tensor product of two braided vertex operator algebras with complementary central charges c bar c 26 each braided voa is constructed from the free fock space realization of the virasoro algebra with an additional q deformed harmonic oscillator degree of freedom the braided voa structure arises from the theory of local systems over configuration spaces and it yields an associative algebra structure on the cohomology we explicitly provide the four cohomology classes that serve as the generators of slq 2 and verify their relations we also discuss the possible extensions of our construction and its connection to the liouville model and minimal string theory

semi infinite homology and 2d gravity i in 33 we studied the constraint problem for two dimensional quantum gravity in the conformal gauge in this gauge we proposed an ansatz for the gravitational sector using this ansatz we established a striking connection between the matrix models and continuum 2d gravity we also announced several results on semi infinite homology of the virasoro algebra with coefficients in a suitable class of positive energy modules in this article we will provide details of the proof of the announced results

dual fixed point property on spaces of continuous functions under equivalent norms abstract a banach space x is said to have the fixed point property if for every closed convex bounded subset c x and for every t c c nonexpansive there is a fixed point a banach space is said to have the dual fixed point property if its dual space x verifies the fixed point property it is a fact that c 0 k fails the dual fixed point property for every locally compact hausdorff set k in this article we study the fulfilment of the dual fixed point property under equivalent norms we prove that c 0 k can be renormed to have the dual fixed point property if and only if k is a countable set in this case we prove that this renorming can be achieved as close to the standard norm on c 0 k as we like for the banach mazur distance we further obtain the corresponding weak fixed point property stability coefficient regarding the duality c 0 k c 0 k it will be displayed that the value of this stability coefficient strongly depends on the distribution of the accumulation points of k finally we prove that the dual fixed point property is completely unstable for the banach mazur distance in most of nonreflexive banach spaces

a generalized entropy criterion for nevanlinna pick interpolation with degree constraint we present a generalized entropy criterion for solving the rational nevanlinna pick problem for n 1 interpolating conditions and the degree of interpolants bounded by n the primal problem of maximizing this entropy gain has a very well behaved dual problem this dual is a convex optimization problem in a finite dimensional space and gives rise to an algorithm for finding all interpolants which are positive real and rational of degree at most n the criterion requires a selection of a monic schur polynomial of degree n it follows that this class of monic polynomials completely parameterizes all such rational interpolants and it therefore provides a set of design parameters for specifying such interpolants the algorithm is implemented in a state space form and applied to several illustrative problems in systems and control namely sensitivity minimization maximal power transfer and spectral estimation

mobius deconvolution on the hyperbolic plane with application to impedance density estimation in this paper we consider a novel statistical inverse problem on the poincare or lobachevsky upper complex half plane here the riemannian structure is hyperbolic and a transitive group action comes from the space of 2 x 2 real matrices of determinant one via mobius transformations our approach is based on a deconvolution technique which relies on the helgason fourier calculus adapted to this hyperbolic space this gives a minimax nonparametric density estimator of a hyperbolic density that is corrupted by a random mobius transform a motivation for this work comes from the reconstruction of impedances of capacitors where the above scenario on the poincare plane exactly describes the physical system that is of statistical interest

cascade equivalence of linear systems abstract this paper concerns the characterization of a linear system with respect to output feedback as well as to slightly more general transformations called cascade transformations with a transfer function t s we associate a finite tuplee of numbers that is invariant under cascade transformation

how to regularize singular vectors and kill the dynamical weyl group abstract let g be a simple lie algebra and let m be the verma module over g with highest weight for a finite dimensional g module u we introduce a notion of a regularizing operator acting in u which makes the meromorphic family of intertwining operators m m u holomorphic and conjugates the dynamical weyl group operators aw end u to constant operators we establish fundamental properties of regularizing operators including uniqueness and prove the existence of a regularizing operator in the case g sl 3

important moments in systems and control the moment problem matured from its various special forms in the late 19th and early 20th centuries to a general class of problems that continues to exert profound influence on the development of analysis and its applications to a wide variety of fields in particular the theory of systems and control is no exception where the applications have historically been to circuit theory optimal control robust control signal processing spectral estimation stochastic realization theory and the use of the moments of a probability density many of these applications are also still works in progress in this paper we consider the generalized moment problem expressed in terms of a basis of a finite dimensional subspace p of the banach space c a b and a positive sequence c but with a new wrinkle inspired by the applications to systems and control we seek to parameterize solutions which are positive rational measures in a suitably generalized sense our parameterization is given in terms of smooth objects in particular the desired solution space arises naturally as a manifold which can be shown to be diffeomorphic to a euclidean space and which is the domain of some canonically defined functions the analysis of these functions and related maps yields interesting corollaries for the moment problem and its applications which we compare to those in the recent literature and which play a crucial role in part of our proof

non euclidian metrics and the robust stabilization of systems with parameter uncertainty this paper considers from a complex function theoretic point of view certain kinds of robust synthesis problems in particular we use a certain kind of metric on the disk the hyperbolic metric which allows us to reduce the problem of robust stabilization of systems with many types of real and complex parameter variations to an easily solvable problem in non euclidian geometry it is shown that several apparently different problems can be treated in a unified general framework a new result on the gain margin problem for multivariable plants is also given finally we apply our methods to systems with real zero or pole variations

robust stabilization some extensions of the gain margin maximization problem considers robust stabilization for siso systems with linear dependence on a real uncertain parameter which may occur either in the numerator or denominator of the plant transfer function the authors show necessary and sufficient conditions for the existence of robust controllers their structure and parameterization can be obtained from the corresponding nevanlinna pick interpolation problems this work extends the previously known case of an uncertain gain parameter