replica treatment of non hermitian disordered hamiltonians we employ the fermionic and bosonic replicated nonlinear models to treat ginibre unitary symplectic and orthogonal ensembles of non hermitian random matrix hamiltonians using the saddle point approach combined with the borel resummation procedure we derive the exact large n results for microscopic density of states in all three ensembles we also obtain tails of the density of states as well as the two point function for the unitary ensemble

equivariance brst symmetry and superspace the structure of equivariant cohomology in non abelian localization formulas and topological field theories is discussed equivariance is formulated in terms of a nilpotent becchi rouet stora tyutin brst symmetry and another nilpotent operator which restricts the brst cohomology onto the equivariant or basic sector a superfield formulation is presented and connections to reducible batalin fradkin vilkovisky bfv quantization of topological yang mills theory are discussed

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

oversampled filter banks as error correcting codes theory and impulse noise correction oversampled filter banks ofbs provide an overcomplete representation of their input signal this paper describes how ofbs can be considered as error correcting codes acting on real or complex sequences very much like classical binary convolutional codes act on binary sequences the structured redundancy introduced by ofbs in subband signals can be used to increase robustness to noise in this paper we define the notions of code subspace syndrome and parity check polynomial matrix for ofbs furthermore we derive generic expressions for projection based decoding suitable for the case when a simple second order model completely characterizes the noise incurred by subband signals we also develop a nonlinear hypotheses test based decoding algorithm for the case when the noise in subbands is constituted by a gaussian background noise and impulsive errors a model that adequately describes the action of both quantization noise and transmission errors simulation results show that the algorithm effectively removes the effect of impulsive errors occurring with a probability of 10 sup 3

complex wishart matrices and conductance in mesoscopic systems exact results it is noted that the hypothesis of independent random complex elements for the off diagonal blocks say m2 and m3 of the transfer matrix describing conductance in a mesoscopic wire allows the eigenvalue distribution of the matrix product m3 m3 or m2 m2 to be computed exactly exact expressions in terms of double wronskian and toeplitz determinants are derived for the distribution of the smallest and second smallest eigenvalue of this and similar random matrix products

analysis of the effect of impulse noise on multicarrier and single carrier qam systems the paper first shows the equivalence of the bernoulli gaussian impulse noise model in the discrete time domain to the continuous time model of poisson arriving delta functions with random area distributed according to the power rayleigh probability density function this equivalence is then used to develop a closed form expression for the probability of error for single carrier qam that is easily evaluated furthermore the performance of multicarrier modulation mcm is also analyzed using the same impulse noise model and it is shown that in most cases mcm performs better than single carrier systems specifically when the probability of an impulse is not too high and the impulse power is moderate

a one dimensional analysis for the probability of error of linear classifiers for normally distributed classes computing the probability of error is an important problem in evaluating classifiers when dealing with normally distributed classes this problem becomes intricate due to the fact that there is no closed form expression for integrating the probability density function in this paper we derive lower and upper bounds for the probability of error for a linear classifier where the random vectors representing the underlying classes obey the multivariate normal distribution the expression of the error is derived in the one dimensional space independently of the dimensionality of the original problem based on the two bounds we propose an approximating expression for the error of a generic linear classifier in particular we derive the corresponding bounds and the expression for approximating the error of fisher s classifier our empirical results on synthetic data including up to two hundred dimensional featured samples show that the computations for the error are extremely fast and quite accurate it differs from the actual error in at most e 0 0184340683 the scheme has also been successfully tested on real life data sets drawn from the uci machine learning repository

riemannian symmetric superspaces and their origin in random matrix theory gaussian random matrix ensembles defined over the tangent spaces of the large families of cartan s symmetric spaces are considered such ensembles play a central role in mesoscopic physics as they describe the universal ergodic limit of disordered and chaotic single particle systems the generating function for the spectral correlations of each ensemble is reduced to an integral over a riemannian symmetric superspace in the limit of large matrix dimension such a space is defined as a pair g h m r where g h is a complex analytic graded manifold homogeneous with respect to the action of a complex lie supergroup g and m r is a maximal riemannian submanifold of the support of g h

orthonormal and biorthonormal filter banks as convolvers and convolutional coding gain convolution theorems for filter bank transformers are introduced both uniform and nonuniform decimation ratios are considered and orthonormal as well as biorthonormal cases are addressed all the theorems are such that the original convolution reduces to a sum of shorter decoupled convolutions in the subbands that is there is no need to have cross convolution between subbands for the orthonormal case expressions for optimal bit allocation and the optimized coding gain are derived the contribution to coding gain comes partly from the nonuniformity of the signal spectrum and partly from nonuniformity of the filter spectrum with one of the convolved sequences taken to be the unit pulse function e coding gain expressions reduce to those for traditional subband and transform coding the filter bank convolver has about the same computational complexity as a traditional convolver if the analysis bank has small complexity compared to the convolution itself

abcd of instantons we solve open image in new window 2 supersymmetric yang mills theories for an arbitrary classical gauge group i e su n so n sp n in particular we derive the prepotential of the low energy effective theory and the corresponding seiberg witten curves we manage to do this without resolving singularities of the compactified instanton moduli spaces

geometric quantization and multiplicities of group representations the heisenberg uncertainty principle says that it is impossible to determine simultaneously the position and momentum of a quantum mechanical particle this can be rephrased as follows the smallest subsets of classical phase space in which the presence of a quantum mechanical particle can be detected are its lagrangian submanifolds for this reason it makes sense to regard the lagrangian submanifolds of phase space as being its true points see weinstein 17 now let g be a compact lie group and g x x x a hamiltonian action of g on x see w for definitions it is well known that the fixed points of this action form a symplectic submanifold of x see for instance guillemin and sternberg 5 however what can one say about the fixed points of g we will show that they are also the points of a symplectic manifold xc this manifold is the marsden weinstein reduction of x with respect to the zero orbit in g and will be described in sect 2 it was introduced in a completely different context fl om ours by marsden and weinstein 12 problems in classical mechanics can often be reduced to the study of hamiltonian systems on symplectic manifolds and problems in quantum mechanics to the study of linear operators on hilbert space this fact has inspired a number of efforts to quantize symplectic geometry by devising schemes for associating hilbert space to symplectic manifolds the no go theorems of groenwald and van hove impose some embarrassing limitations on all such schemes however it seems to be a useful idea heuristically to think of every symplectic manifold x a ic 1 as being symbiotically associated with a hilbert space xq a t in such a way that the classical observables on the first space correspond to quantum observables on the second space the heuristics further suggests that if g is a group of symmetries of x la i j it should also be a group of symmetries of xq a tum in this heuristic spirit we will state the main conjecture of this paper

approximating the llr distribution for a class of soft output mimo detectors we present approximations of the llr distribution for a class of fixed complexity soft output mimo detectors such as the optimal soft detector and the soft output via partial marginalization detector more specifically in a mimo awgn setting we approximate the llr distribution conditioned on the transmitted signal and the channel matrix with a gaussian mixture model gmm our main results consist of an analytical expression of the gmm model including the number of modes and their corresponding parameters and a proof that in the limit of high snr this llr distribution converges in probability towards a unique gaussian distribution

frame theoretic analysis of oversampled filter banks we provide a frame theoretic analysis of oversampled finite impulse response fir and infinite impulse response fir uniform filter banks fbs our analysis is based on a new relationship between the fbs polyphase matrices and the frame operator corresponding to an fb for a given oversampled analysis fb we present a parameterization of all synthesis fbs providing perfect reconstruction we find necessary and sufficient conditions for an oversampled fb to provide a frame expansion a new frame theoretic procedure for the design of paraunitary fbs from given nonparaunitary fbs is formulated we show that the frame bounds of an fb can be obtained by an eigen analysis of the polyphase matrices the relevance of the frame bounds as a characterization of important numerical properties of an fb is assessed by means of a stochastic sensitivity analysis we consider special cases in which the calculation of the frame bounds and synthesis filters is simplified finally simulation results are presented

the grouped continuous model for multivariate ordered categorical variables and covariate adjustment the grouped continuous model for multivariate ordered categorical data is described this is based on partitioning an underlying multivariate normal distribution straightforward maximum likelihood estimation is really feasible only for one and two way tables we introduce an estimation system based on maximum likelihood estimation in the one and two way marginal tables of higher order tables this is computationally feasible and an example involving aspects of bird colouring is given the approach is extended to provide a regression model for multivariate ordered categorical data with an estimation scheme again based on the one and two way marginal tables the above example is developed to investigate the covariate effect of time the asymptotic efficiency of these sampling schemes is discussed it appears that they have high efficiency

correcting wavelet codes using interleaving and trapping a rate k n wavelet code a form of real convolutional code that encodes k numbers into n numerical code symbols using the present and past input numbers may be used to transport numerical data with redundancy factor rm n rm k such codes offer detection and correction within roundoff tolerances wavelet codes can be processed using two different algebraic representations vector matrix or polyphase polynomial when an interleaver intersperses the code symbols any burst of errors within the interelaver s frame length appears in only one component of the polyphase format a special extended polyphase parity check matrix exposes the error bursts directly for redundancies greater than 1 or a deconvolver operation achieves the same when the redundancy factor is 1 the trapped errors correct the symbols by subtraction simulations verify the effectiveness of this trapping technique

supersymmetry for systems with unitary disorder circular ensembles a generalized hubbard stratonovitch transformation relating an integral over random unitary matrices to an integral over efetov s unitary model manifold is introduced this transformation adapts the supersymmetry method to disordered and chaotic systems that are modelled not by a hamiltonian but by their scattering matrix or time evolution operator in contrast to the standard method no saddle point approximation is made and no massive modes have to be eliminated this first paper on the subject applies the generalized hubbard stratonovitch transformation to dyson s circular unitary ensemble it is shown how a supersymmetric variant of the harish chandra itzykson zuber formula can be used to compute in the large n limit the n level correlation function for any n non trivial applications to random network models quantum chaotic maps and lattice gauge theory are expected

the infinitesimal lefschetz formulas a heat equation proof abstract in this paper the lefschetz formulas of atiyah and bott ann of math 86 1967 374 407 88 1968 451 491 and atiyah and singer ann of math 87 1968 546 604 93 1971 119 138 are proved by heat equation methods in infinitesimal form i e without using the usual localization on the fixed point set the proof is obtained by a homotopic deformation of the dirac operator and uses probabilistic methods the main applications are potentially in noncompact situations where fixed point formulas do not always hold while their infinitesimal analogues are still valid

oversampled filter banks perfect reconstruction oversampled filter banks are equivalent to a particular class of frames in l sup 2 z these frames are the subject of this paper first the necessary and sufficient conditions of a filter bank for implementing a frame or a tight frame expansion are established as well as a necessary and sufficient condition for perfect reconstruction using fir filters after an fir analysis complete parameterizations of oversampled filter banks satisfying these conditions are given further we study the condition under which the frame dual to the frame associated with an fir filter bank is also fir and give a parameterization of a class of filter banks satisfying this property then we focus on non subsampled filter banks non subsampled filter banks implement transforms similar to continuous time transforms and allow for very flexible design we investigate the relations of these filter banks to continuous time filtering and illustrate the design flexibility by giving a procedure for designing maximally flat two channel filter banks that yield highly regular wavelets with a given number of vanishing moments

the normal law under linear restrictions simulation and estimation via minimax tilting summary simulation from the truncated multivariate normal distribution in high dimensions is a recurrent problem in statistical computing and is typically only feasible by using approximate markov chain monte carlo sampling we propose a minimax tilting method for exact independently and identically distributed data simulation from the truncated multivariate normal distribution the new methodology provides both a method for simulation and an efficient estimator to hitherto intractable gaussian integrals we prove that the estimator has a rare vanishing relative error asymptotic property numerical experiments suggest that the scheme proposed is accurate in a wide range of set ups for which competing estimation schemes fail we give an application to exact independently and identically distributed data simulation from the bayesian posterior of the probit regression model

heckman kostant and steinberg formulas for symplectic manifolds abstract let m be a compact symplectic manifold on which a compact connected lie group k acts in a hamiltonian fashion in part i we derive a formula for j m which generalizes the heckman formula for co adjoint orbits here j is the moment map associated with the action of k on m and m the liouville measure on m in part ii we derive a quantum analogue of this formula which extends to the symplectic setting classical multiplicity formulas of kostant and steinberg

12 24 and beyond abstract we generalize the well known 12 and 24 theorems for reflexive polytopes of dimension 2 and 3 to any smooth reflexive polytope our methods apply to a wider category of objects here called reflexive gkm graphs that are associated with certain monotone symplectic manifolds which do not necessarily admit a toric action as an application we provide bounds on the betti numbers for certain monotone hamiltonian spaces which depend on the minimal chern number of the manifold

correlation functions of the bc calogero sutherland model the bc type calogero sutherland model csm is an integrable extension of the ordinary a type csm that possesses a reflection symmetry point the bc cs mi sr elated to the chiral classes of random matrix ensembles rmes in exactly the same way as the a csm is related to the dyson classes we first develo pt he fermionic replica model formalism suitable to treat all chiral rmes by exploiting generalized colour flavour transformation we then extend the method to find the exact asymptotics of the bc csm density profile consistency of our result with the c 1g aussian conformal field theory description is verified the emerging friedel oscillations structure and sum rules are discussed in details we also compute the distribution of the particle nearest to the reflection point

the calogero sutherland model and generalized classical polynomials multivariable generalizations of the classical hermite laguerre and jacobi polynomials occur as the polynomial part of the eigenfunctions of certain schrodinger operators for calogero sutherland type quantum systems for the generalized hermite and laguerre polynomials the multidimensional analogues of many classical results regarding generating functions differentiation and integration formulas recurrence relations and summation theorems are obtained we use this and related theory to evaluate the global limit of the ground state density obtaining in the hermite case the wigner semi circle law and to give an explicit solution for an initial value problem in the hermite and laguerre case

some properties of the calogero sutherland model with reflections we prove that the calogero sutherland model with reflections the model possesses a property of duality relating the eigenfunctions of two hamiltonians with different coupling constants we obtain a generating function for their polynomial eigenfunctions the generalized jacobi polynomials the symmetry of the wavefunctions for certain particular cases associated with the root systems of the classical lie groups and is also discussed

bohr sommerfeld conditions for integrable systems with critical manifolds of focus focus type we present a detailed study in the semi classical regime h of microlocal properties of systems of two commuting h pseudo dierential operators ph ph such that the joint principal symbol p p p has a special kind of singularity called a focusfocus singularity typical examples include the quantum spherical pendulum or the quantum champagne bottle in the spirit of colin de verdiere and parisse we show that such systems have a universal behavior described by singular quantization conditions of bohr sommerfeld type these conditions are used to give a precise description of the joint spectrum of such systems including the phenomenon of quantum monodromy and dierent formulations of the counting function for the joint eigenvalues close to the singularity in which a logarithm of the semi classical constant h appears thanks to numerical computations done by ms child for the case of the champagne bottle we are able to accurately illustrate our statements

non hamiltonian actions with isolated fixed points we construct a non hamiltonian symplectic circle action on a closed connected six dimensional symplectic manifold with exactly 32 fixed points

approximations to multivariate normal rectangle probabilities based on conditional expectations abstract two new approximations for multivariate normal probabilities for rectangular regions based on conditional expectations and regression with binary variables are proposed one is a second order approximation that is much more accurate but also more numerically time consuming than the first order approximation a third approximation based on the moment generating function of a truncated multivariate normal distribution is proposed for orthant probabilities only its accuracy is between the first and second order approximations when the dimension is less than seven and the correlations are not large all of the approximations get worse as correlations get larger these new approximations offer substantial improvements on previous approximations they also compare favorably with the methods of genz for numerical evaluation of the multivariate normal integral the approximation methods should be especially useful within a quasi newton routine for parameter estimation in discrete models that involv

stochastic equivariant cohomologies and cyclic cohomology we give two stochastic diffeologies on the free loop space which allow us to define stochastic equivariant cohomology theories in the chen souriau sense and to establish a link with cyclic cohomology with the second one we can establish a stochastic fixed point theorem

wavelet codes detection and correction using kalman estimation wavelet codes encoded by uniform oversampled filter banks where the wavelet coefficients represent code symbols are decoded employing kalman fixed lag estimation procedures the code symbol error estimates are based on the mean squared error mse criterion and are first used to detect the presence of large numerical errors then the correction of symbols uses these kalman estimators directly the wavelet code symbols are corrupted by low levels of processing noise continuously and occasionally by large disruptive errors e g impulsive noise the modeling variables have time varying statistics allowing both types of errors to be handled naturally kalman time varying tracking filters which use wavelet syndromes as their measurement inputs develop the fixed lag smoothed estimates for the larger errors the syndromes are also modeled as corrupted by processing noise modeling computational failure errors large and small hypothesis detection methods are used to locate large errors and their statistics must include kalman mismatch characteristics to be valid estimators from several adjacent lag positions assist in the detection procedures simulation results show correction accuracy in the time domain as well as on a mse basis large classes of wavelet codes are designed using binary convolutional codes as starting points from which the proper filter weights for encoding and syndrome evaluating are determined

the fixed point theorem in equivariant cohomology in this paper we study the s equivariant de rham cohomology of infinite dimensional s manifolds our main example is the free loop space lx where x is a finite dimensional manifold with the circle acting by rotating loops we construct a new form of equivariant cohomology h which agrees with the usual periodic equivariant cohomology in finite dimensions and we prove a suitable analogue of the classical fixed point theorem which is valid for loop spaces lx this gives a cohomological framework for studying differential forms on loop spaces and we apply these methods to various questions which arise from the work of witten 16 atiyah 2 and bismut 5 in particular we show following atiyah in 2 that the a polynomial of x arises as an equivariant characteristic class in the theory h of the normal bundle to x considered as the space of constant loops in lx

correlation structure and variable selection in generalized estimating equations via composite likelihood information criteria the method of generalized estimating equations gee is popular in the biostatistics literature for analyzing longitudinal binary and count data it assumes a generalized linear model for the outcome variable and a working correlation among repeated measurements in this paper we introduce a viable competitor the weighted scores method for generalized linear model margins we weight the univariate score equations using a working discretized multivariate normal model that is a proper multivariate model because the weighted scores method is a parametric method based on likelihood we propose composite likelihood information criteria as an intermediate step for model selection the same criteria can be used for both correlation structure and variable selection simulations studies and the application example show that our method outperforms other existing model selection methods in gee from the example it can be seen that our methods not only improve on gee in terms of interpretability and efficiency but also can change the inferential conclusions with respect to gee copyright 2016 john wiley sons ltd

on the variation in the cohomology of the symplectic form of the reduced phase space is called the momentum mapping of the hamiltonian t action given 1 1 the condition 1 2 just means that t acts along the fibers of j for the basic definitions and properties of non commutative hamiltonian group actions see am the results of this paper can easily be extended to hamiltonian actions of arbitrary compact connected lie groups by applying our results to the action of its maximal toms and using the equivariance of the momentum mapping for some more details see the remarks at the end of sect 2 we will assume throughout this paper that the momentum map is proper that is j i u is compact for each compact subset u of t now let r be a regular value of j that is t j tmm t is surjective for all meye j 1 4 then ye is a smooth submanifold of m compact because

fundamentals of statistical signal processing estimation theory minimum variance unbiased estimation cramer rao lower bound linear models general minimum variance unbiased estimation best linear unbiased estimators maximum likelihood estimation least squares method of moments the bayesian philosophy general bayesian estimators linear bayesian estimators kalman filters summary of estimators extension for complex data and parameters

the supersymmetric technique for random matrix ensembles with zero eigenvalues the supersymmetric technique is applied to computing the average spectral density near zero energy in the large n limit of the random matrix ensembles with zero eigenvalues b diii odd and the chiral ensembles classes aiii bdi and cii the supersymmetric calculations reproduce the existing results obtained by other methods the effect of zero eigenvalues may be interpreted as reducing the symmetry of the zero energy supersymmetric action by breaking a certain abelian symmetry

convexity properties of the moment mapping ii be its associated momen t mapping see w for definitions the set x is the union of co adjoint orbits the main result of this paper is a description of the orbit structure of this set to describe this result we first note that the coadjoint orbits in q can be parametr ized as follows let t be a caf tan subalgebra of g and let b be a positive definite g invariant bilinear form on by means of b we get a g equivariant identification q let t be the subspace of q corresponding to t then every co adjoint orbit intersects i in a weyl group orbit so there is a one one correspondence between co adjoint orbits in q and weyl group orbits in t to get an even sharper parametrization let t be a weyl chamber in then every weyl group orbit intersects t in a single point so t is a modul i space for the co adjoint orbits two orbits in x are said to be of the same type if they are isomorphic as homogeneous g spaces since x is compact there are at most a finite number of different types of orbits see 1113 let h be a closed subgroup of t of codimension r and let cit be the set of all orbits in x of type g h

on the construction of certain 6 dimensional symplectic manifolds with hamiltonian circle actions let m w be a connected compact 6 dimensional symplectic manifold equipped with a semi free hamiltonian s 1 action such that the fixed point set consists of isolated points or surfaces assume dim h 2 m 3 in an earlier paper we defined a certain invariant of such spaces which consists of fixed point data and twist type and we divided the possible values of these invariants into six types in this paper we construct such manifolds with these types as a consequence we have a precise list of the values of these invariants

the calogero moser equation system and the ensemble average in the gaussian ensembles from random matrix theory it is known that for special values of the coupling constant the calogero moser cm equation system is nothing but the radial part of a generalized harmonic oscillator schrodinger equation this allows an immediate construction of the solutions by means of a rodriguez relation the results are easily generalized to arbitrary values of the coupling constant by this the cm equations become nearly trivial as an application an expansion for eitr xy in terms of eigenfunctions of the cm equation system is obtained where x and y are matrices taken from one of the gaussian ensembles and the brackets denote an average over the angular variables

likelihood inference in a correlated probit regression model correlated binary observations arise in a variety of applications for example in animal studies the term litter effect is used to describe the greater alikeness of responses within a litter as compared to that between litters at a given set of experimental conditions in a setting that motivated this work cells on individual atomic bomb survivors were scored as to the presence or absence of chromosomal aberrations the number of aberrant cells varied among subjects at a specified radiation exposure estimate age at exposure city of exposure and sex in a manner that substantially exceeds that consistent with a binomial error structure otake prentice 1984 this extra binomial variation or overdispersion reflects the fact that the binary response of cells from a given survivor tend to be more alike than are the responses of cells from distinct survivors having the same estimated radiation exposure level and other covariate values such overdispersion may result from individual differences in susceptibility to radiation damage from omitted covariates or most plausibly in this setting from substantial random errors in the estimated radiation exposure levels failure to acknowledge overdispersion may lead to serious underestimation of the standard errors associated with regression parameter estimates and to unduly precise inferences

wigner dyson statistics from the replica method we compute the correlation functions of the eigenvalues in the gaussian unitary ensemble using the fermionic replica method we show that non trivial saddle points which break replica symmetry must be included in the calculation in order to correctly reproduce the exact asymptotic behaviour of the correlation functions at a large distance

universal correlations of trapped one dimensional impenetrable bosons we calculate the asymptotic behaviour of the one body density matrix of one dimensional impenetrable bosons in finite size geometries our approach is based on a modification of the replica method from the theory of disordered systems we obtain explicit expressions for oscillating terms similar to fermionic friedel oscillations these terms are universal and originate from the strong short range correlations between bosons in one dimension

when is a pure state of three qubits determined by its single particle reduced density matrices using techniques from symplectic geometry we prove that a pure state of three qubits is up to local unitaries uniquely determined by its one particle reduced density matrices exactly when their ordered spectra belong to the boundary of the so called kirwan polytope otherwise the states with given reduced density matrices are parameterized up to local unitary equivalence by two real variables given inevitable experimental imprecision this means that already for three qubits a pure quantum state can never be reconstructed from single particle tomography we moreover show that the knowledge of the reduced density matrices is always sufficient if one is given the additional promise that the quantum state is not convertible to the greenberger horne zeilinger state by stochastic local operations and classical communication and discuss generalizations of our results to an arbitrary number of qubits

exact results and universal asymptotics in the laguerre random matrix ensemble for various values of the parameters in the laguerre random matrix ensemble the distribution of the smallest eigenvalue and the scaled n level distribution function are calculated exactly in terms of generalized hypergeometric functions in certain cases these functions are expressed as multidimensional integrals from which asymptotic formulas are calculated and predictions of universal behavior verified

localization formulas superconnections and the index theorem for families in this paper we give a new proof of the localization formulas of berline and vergne 9 and duistermaat and heckman 18 when interpreted in the framework of atiyah 2 the probabilistic heat equation proof of the index theorem given in our paper 12 appears as the rigorous infinite dimensional version of this new proof of the localization formulas in finite dimensions the results of quillen 25 on superconnections are briefly presented the heat equation proofs 15 of the index theorem for families are described it is shown that in this framework the superconnections formalism is the operator theoretic description of integration along the fiber in the loop space

multiplicities formula for geometric quantization part ii 1 introduction let p be a compact manifold let h be a compact lie group acting on the right on p we assume that the stabilizer of each element y e p is a finite subgroup of h the space m p h is an orbifold and every orbifold can be presented this way if h acts freely then m is a manifold if ae graded space of h invariant solutions of the h horizontal dirac operator on p twisted by the line bundle let m o 9 be the moment map for the g action assume 0 is a regular value of let mred be the reduced orbifold of m that is mred o g consider the reduced orbifold line bundle rd lu lto g on m oa in the ease where both g andh are torus we prove here the formula

ginzburg weinstein via gelfand zeitlin let u n be the unitary group and u n the dual of its lie al gebra equipped with the kirillov poisson structure in their 1983 paper guillemin sternberg introduced a densely defined hamil tonian action of a torus of dimension n 1 n 2 on u n with mo ment map given by the gelfand zeitlin coordinates a few years later flaschka ratiu described a similar multiplicative gelfand zeitlin system for the poisson lie group u n by the ginzburg weinstein theorem u n is isomorphic to u n as a poisson manifold flaschka ratiu conjectured that one can choose the ginzburg weinstein diffeomorphism in such a way that it intertwines the linear and nonlinear gelfand zeitlin systems our main result gives a proof of this conjecture and produces a canonical ginzburg weinstein diffeomorphism