localization and diagonalization a review of functional integral techniques for low dimensional gauge theories and topological field theories localization techniques for functional integrals which have recently been used to perform calculations in and gain insight into the structure of certain topological field theories and low dimensional gauge theories are reviewed these are the functional integral counterparts of the mathai quillen formalism the duistermaat heckman theorem and the weyl integral formula respectively in each case the necessary mathematical background euler classes of vector bundles equivariant cohomology topology of lie groups is introduced and the finite dimensional integration formulae described then some applications to path integrals are discussed and an overview of the relevant literature is given the applications include supersymmetric quantum mechanics cohomological field theories phase space path integrals and two dimensional yang mills theory

detection of signals by information theoretic criteria a new approach is presented to the problem of detecting the number of signals in a multichannel time series based on the application of the information theoretic criteria for model selection introduced by akaike aic and by schwartz and rissanen mdl unlike the conventional hypothesis testing based approach the new approach does not requite any subjective threshold settings the number of signals is obtained merely by minimizing the aic or the mdl criteria simulation results that illustrate the performance of the new method for the detection of the number of signals received by a sensor array are presented

model order selection of damped sinusoids in noise by predictive densities we develop a procedure for the order selection of damped sinusoidal models based on the maximum a posteriori map criterion the proposed method merges the concept of predictive densities with bayesian inference to arrive at a complex multidimensional integral whose solution is achieved by way of the efficient monte carlo importance sampling technique the importance function a multivariate cauchy probability density is employed to produce stratified samples over the hypersurfaces support region centrality location parameters for the cauchy are resolved by exploiting the special structure of the compressed likelihood function clf and applying the fast maximum likelihood fml procedure of umesh and tufts simulation results allow for a comparison between our method and the singular value decomposition svd based information theoretic criteria of reddy and biradar see ieee trans signal processing vol 41 no 9 p 2872 81 1993

spatio temporal spectral analysis by eigenstructure methods this paper presents new algorithms for estimating the spatio temporal spectrum of the signals received by a passive array the algorithms are based on the eigenstructure of the covariance and spectral density matrices of the received signals they allow partial correlation between the sources and thus are applicable to certain kinds of multipath problems simulation results that illustrate the performance of the new algorithms are presented

subspace based channel estimation for code division multiple access communication systems we consider the estimation of channel parameters for code division multiple access cdma communication systems operating over channels with either single or multiple propagation paths the multiuser channel estimation problem is decomposed into a series of single user problems through a subspace based approach by exploiting the eigenstructure of the received signal s sample correlation matrix the observation space can be partitioned into a signal subspace and a noise subspace without prior knowledge of the unknown parameters the channel estimate is formed by projecting a given user s spreading waveform into the estimated noise subspace and then either minimizing the likelihood or minimizing the euclidean norm of this projection both of these approaches yield algorithms which are near far resistant and do not require a preamble

multiple emitter location and signal parameter estimation processing the signals received on an array of sensors for the location of the emitter is of great enough interest to have been treated under many special case assumptions the general problem considers sensors with arbitrary locations and arbitrary directional characteristics gain phase polarization in a noise interference environment of arbitrary covariance matrix this report is concerned first with the multiple emitter aspect of this problem and second with the generality of solution a description is given of the multiple signal classification music algorithm which provides asymptotically unbiased estimates of 1 number of incident wavefronts present 2 directions of arrival doa or emitter locations 3 strengths and cross correlations among the incident waveforms 4 noise interference strength examples and comparisons with methods based on maximum likelihood ml and maximum entropy me as well as conventional beamforming are included an example of its use as a multiple frequency estimator operating on time series is included

a universal prior for integers and estimation by minimum description length of the number of bits required to write down the observed data has been reformulated to extend the classical maximum likelihood principle the principle permits estimation of the number of the parameters in statistical models in addition to their values and even of the way the parameters appear in the models i e of the model structures the principle rests on a new way to interpret and construct a universal prior distribution for the integers which makes sense even when the parameter is an individual object truncated realvalued parameters are converted to integers by dividing them by their precision and their prior is determined from the universal prior for the integers by optimizing the precision 1 introduction in this paper we study estimation based upon the principle of minimizing the total number of binary digits required to rewrite the observed data when each observation is given with some precision instead of attempting at an absolutely shortest description which would be futile we look for the optimum relative to a class of parametrically given distributions this minimum description length mdl principle which we introduced in a less comprehensive form in 25 turns out to degenerate to the more familiar maximum likelihood ml principle in case the number of parameters in the models is fixed so that the description length of the parameters themselves can be ignored in another extreme case where the parameters determine the data it similarly degenerates to jaynes s principle of maximum entropy 14 but the main power of the new criterion is that it permits estimates of the entire model its parameters their number and even the way the parameters appear in the model i e the model structure hence there will be no need to supplement the estimated parameters with a separate hypothesis test to decide whether a model is adequately parameterized or perhaps over parameterized

order selection for ar models by predictive least squares a criterion is presented for selecting the order of autoregressive models that unlike the existing criteria is amenable to online or adaptive operation it is based on the predictive least squares pls principle and is implemented in a computationally efficient way by predictive lattice filters the consistency of the criterion is proved and its performance is demonstrated by computer simulations assuming the data to be generated by an ar model of order p the order selection criterion should select the correct order p with probability that converges to 1 as the sample size grows to infinity it is proved that the pls criterion is indeed consistent thereby giving a solid justification for the criterion simulation results that demonstrate the performance of the pls criterion in comparison to h akaike s aic 1973 and the mdl criteria of j rissanen 1978 and g schwarz 1978 are given

the determination of optimum structures for the state space representation of multivariate stochastic processes when identifying a model for a multivariate stationary stochastic process an important problem is that of determining the structure of the state variable model several overlapping parameterizations can usually be fitted to a given process and the question arises as to which structure leads to the most accurate parameter estimates the accuracy of parameter estimates is often measured by the determinant of the fisher information matrix we show that all admissible structures will give asymptotically the same value to this criterion for finite data some structures may still be better than others and two heuristic structure estimation methods are analyzed some simulation results are also presented

symmetry properties of higher numerical ranges abstract let a be a linear operator on a finite dimensional unitary space v of dimension n the k th higher numerical range of a denoted by w k a is the totality of complex numbers tr pap where p runs over all k dimensional orthogonal projections on v we show that w k a is a polygon with the real axis as a line of symmetry k 1 n if and only if a is normal with a real characteristic polynomial we also construct several nonnormal examples to investigate the extent to which the symmetry of all of the w k a is required

order selection of autoregressive models the problem of determining the order of autoregressive models by bayesian predictive densities is addressed a criterion employing noninformative prior densities of the model parameters is derived simulation results which demonstrate the good performance of the criterion are presented comparisons with four popular approaches verify its superiority in many cases

estimating the parameters of exponentially damped sinusoids and pole zero modeling in noise we have presented techniques 1 6 based on linear prediction lp and singular value decomposition svd for accurate estimation of closely spaced frequencies of sinusoidal signals in noise in this note we extend these techniques to estimate the parameters of exponentially damped sinusoidal signals in noise the estimation procedure presented here makes use of backward prediction in addition to svd first the method is applied to data consisting of one and two exponentially damped sinusoids the choice of one and two signal components facilitates the comparison of estimation error in pole damping factors and pole frequencies to the appropriate cramer rao cr bounds and to traditional methods of linear prediction second our method is applied to an example due to steiglitz 8 in which the data consists of noisy values of the impulse response samples composed of many exponentially damped sinusoids of a linear system having both poles and zeros the poles of the system are accurately determined by our method and the zeros are obtained subsequently using shanks method

noncommutative geometry and gauge theory on fuzzy sphere the differential algebra on the fuzzy sphere is constructed by applying connes scheme the u 1 gauge theory on the fuzzy sphere based on this differential algebra is defined the local u 1 gauge transformation on the fuzzy sphere is identified with the left u n 1 transformation of the field where a field is a bimodule over the quantized algebra the interaction with a complex scalar field is also given

bayesian detection and estimation of cisoids in colored noise the problem of estimating the number of cisoids in colored noise is addressed it is assumed that the noise can be modeled by an autoregression whose order has also to be estimated a new criterion is proposed for estimating the number of cisoids and the autoregressive model order as well as a new algorithm for estimating the cisoidal frequencies in the derivation a bayesian methodology and subspace decomposition are employed the proposed criterion significantly outperforms the popular mdl and aic as applied in a paper by nagesha and kay in addition an algorithm that reduces the computational complexity of the solution is developed computer simulations that demonstrate the performance of the criterion are included

intersection theory on moduli spaces of holomorphic bundles of arbitrary rank on a riemann surface let n and d be coprime positive integers and define m n d to be the moduli space of semi stable holomorphic vector bundles of rank n degree d and fixed determinant on a compact riemann surface e this moduli space is a compact kifhler manifold which has been studied from many different points of view for more than three decades see for instance narasimhan and seshadri 41 the subject of this article is the characterization of the intersection pairings in the cohomology ring1 h m n d a set of generators of this ring was described by atiyah and bott in their seminal 1982 paper 2 on the yangmills equations on riemann surfaces where in addition inductive formulas for the betti numbers of m n d obtained earlier using number theoretic methods 13 25 were rederived by poincare duality knowledge of the intersection pairings between products of these generators or equivalently knowledge of the evaluation on the fundamental class of products of the generators completely determines the structure of the cohomology ring in 1991 donaldson 15 and thaddeus 47 gave formulas for the intersection pairings between products of these generators in h m 2 1 in terms of bernoulli numbers then using physical methods witten 50 found formulas for generating functions from which could be extracted the intersection pairings between products of these generators in h m n d for general rank n these generalized his rigorously proved formulas 49 for the symplectic volume of m n d for instance the symplectic volume of m 2 1 is given by

conditions for the generalized numerical range to be real abstract necessary and sufficient conditions are given for the c numerical range of a matrix a to be a subset of the real axis in particular it is shown that both a and c must be translates of hermitian matrices

birkhoff s theorem and multidimensional numerical range abstract we show that under certain conditions birkhoff s theorem on doubly stochastic matrices remains valid for countable families of discrete probability spaces which have nonempty intersections using this result we study the relation between the spectrum of a self adjoint operator a and its multidimensional numerical range it turns out that the multidimensional numerical range is a convex set whose extreme points are sequences of eigenvalues of the operator a every collection of eigenvalues which can be obtained by the rayleigh ritz formula generates an extreme point of the multidimensional numerical range however it may also have other extreme points

on posterior distributions for signals in gaussian noise with unknown covariance matrix a bayesian approach to estimate parameters of signals embedded in complex gaussian noise with unknown color is presented the study specifically focuses on a bayesian treatment of the unknown noise covariance matrix making up a nuisance parameter in such problems by integrating out uncertainties regarding the noise color an enhanced ability to estimate both the signal parameters as well as properties of the error is exploited several noninformative priors for the covariance matrix such as the reference prior the jeffreys prior and modifications to this are considered some of the priors result in analytical solutions whereas others demand numerical approximations in the linear signal model connections are made between the standard adaptive maximum likelihood aml estimate and a bayesian solution using the jeffreys prior with adjustments to the jeffreys prior correspondence to the regularized solution is also established this in turn enables a formal treatment of the regularization parameter simulations indicate that significant improvements compared to the aml estimator can be obtained by considering both the derived regularized solutions as well as the one obtained using the reference prior the simulations also indicate the possibility of enhancing the predictions of properties of the error as uncertainties in the noise color are acknowledged

localization for yang mills theory on the fuzzy sphere we present a new model for yang mills theory on the fuzzy sphere in which the configuration space of gauge fields is given by a coadjoint orbit in the classical limit it reduces to ordinary yang mills theory on the sphere we find all classical solutions of the gauge theory and use nonabelian localization techniques to write the partition function entirely as a sum over local contributions from critical points of the action which are evaluated explicitly the partition function of ordinary yang mills theory on the sphere is recovered in the classical limit as a sum over instantons we also apply abelian localization techniques and the geometry of symmetric spaces to derive an explicit combinatorial expression for the partition function and compare the two approaches these extend the standard techniques for solving gauge theory on the sphere to the fuzzy case in a rigorous framework

instanton expansion of noncommutative gauge theory in two dimensions we show that noncommutative gauge theory in two dimensions is an exactly solvable model a cohomological formulation of gauge theory defined on the noncommutative torus is used to show that its quantum partition function can be written as a sum over contributions from classical solutions we derive an explicit formula for the partition function of yang mills theory defined on a projective module for an arbitrary noncommutativity parameter which is manifestly invariant under gauge morita equivalence the energy observables are shown to be smooth functions of the construction of noncommutative instanton contributions to the path integral is described in some detail in general there are infinitely many gauge inequivalent contributions of fixed topological charge along with a finite number of quantum fluctuations about each instanton the associated moduli spaces are combinations of symmetric products of an ordinary two torus whose orbifold singularities are not resolved by noncommutativity in particular the weak coupling limit of the gauge theory is independent of and computes the symplectic volume of the moduli space of constant curvature connections on the noncommutative torus