the segal bargmann coherent state transform for compact lie groups abstract let k be an arbitrary compact connected lie group we describe on k an analog of the segal bargmann coherent state transform and we prove theorem 1 that this generalized coherent state transform maps l 2 k isometrically onto the space of holomorphic functions in l 2 g where g is the complexification of k and where is an appropriate heat kernel measure on g the generalized coherent state transform is defined in terms of the heat kernel on the compact group k and its analytic continuation to the complex group g we also define a k averaged version of the coherent state transform and we prove a similar result for it theorem 2 in the appendix we describe the coherent states which motivate the definitions of these transforms in section 9 we obtain inversion formulas for both the coherent state transform and the k averaged coherent state transform theorems 3 4 as a consequence we obtain formulas for certain bergman kernels on the complex group g theorems 5 6 in section 10 we derive a general paley wiener theorem for k and we exhibit a family of convolution transforms each of which is an isometric isomorphism of l 2 k onto a certain l 2 space of holomorphic functions on g these transforms include the k averaged coherent state transform as a special case we also discuss the analogous results for r 1 the results of section 10 are theorems 7 10 finally in section 11 we discuss the case of quotient spaces we prove theorem 11 that both the coherent state transform and the k averaged coherent state transform pass in a natural way from k to k h where h is a closed connected subgroup of k

dft calculation of the generalized and drazin inverse of a polynomial matrix a new algorithm is presented for the determination of the a generalized inverse and b drazin inverse of a polynomial matrix the proposed algorithms are based on the discrete fourier transform and thus are computationally fast in contrast to other known algorithms the above algorithms are implemented in the mathematica programming language and illustrated via examples

computation of the generalized inverse of a polynomial matrix and applications abstract the computation of the generalized inverse of a constant matrix is utilized in finding the generalized inverse and its laurent expansion of a nonregular polynomial matrix the proposed algorithm constitutes a generalization of the algorithm proposed by fragulis et al for regular polynomial matrices and gives rise to numerous applications in multivariable system analysis

the inverse segal bargmann transform for compact lie groups in this paper i give a new inversion formula theorem 1 for the generalized segal bargmann transform introduced in b c hall j funct anal 122 1994 103 151 the inversion formula may be viewed as a formula for theinverseheat operator for a compact lie group i then use this formula to give a new direct proof of the unitary of thek invariant form of the segal bargmann transform theorem 2 the proof of the inversion formula relies on an identity theorem 5 which relates the laplacian for a compact lie groupkto the laplacian for the non compact dual symmetric space kc k

finite generation of the cohomology of quotients of pbw algebras abstract in this article we prove finite generation of the cohomology of quotients of a pbw algebra a by relating it to the cohomology of quotients of a quantum symmetric algebra s which is isomorphic to the associated graded algebra of a the proof uses a spectral sequence argument and a finite generation lemma adapted from friedlander and suslin

on the geometry and quantization of symplectic howe pairs we study the orbit structure and the geometric quantization of a pair of mutually commuting hamiltonian actions on a symplectic manifold if the pair of actions fulfils a symplectic howe condition we show that there is a canonical correspondence between the orbit spaces of the respective moment images furthermore we show that reduced spaces with respect to the action of one group are symplectomorphic to coadjoint orbits of the other group in the kahler case we show that the linear representation of a pair of compact connected lie groups on the geometric quantization of the manifold is then equipped with a representation theoretic howe duality

quantum mechanics on a circle husimi phase space distributions and semiclassical coherent state propagators we discuss some basic tools for an analysis of one dimensional quantum systems defined on a cyclic coordinate space the basic features of the generalized coherent states the complexifier coherent states are reviewed these states are then used to define the corresponding quasi densities in phase space the properties of these generalized husimi distributions are discussed in particular their zeros furthermore the use of the complexifier coherent states for a semiclassical analysis is demonstrated by deriving a semiclassical coherent state propagator in phase space

cohomology of twisted tensor products abstract it is well known that the cohomology of a tensor product is essentially the tensor product of the cohomologies we look at twisted tensor products and investigate to which extent this is still true we give an explicit description of the ext algebra of the tensor product of two modules and under certain additional conditions describe an essential part of the hochschild cohomology ring of a twisted tensor product as an application we characterize precisely when the cohomology groups over a quantum complete intersection are finitely generated over the hochschild cohomology ring moreover both for quantum complete intersections and in related cases we obtain a lower bound for the representation dimension of the algebra

symbolic computation of weighted moore penrose inverse using partitioning method abstract we propose a method and algorithm for computing the weighted moore penrose inverse of one variable rational matrices continuing this idea we develop an algorithm for computing the weighted moore penrose inverse of one variable polynomial matrix these methods and algorithms are generalizations of the method for computing the weighted moore penrose inverse for constant matrices originated in wang and chen g r wang y l chen a recursive algorithm for computing the weighted moore penrose inverse a mn j comput math 4 1986 74 85 and the partitioning method for computing the moore penrose inverse of rational and polynomial matrices introduced in stanimirovic and tasic p s stanimirovic m b tasic partitioning method for rational and polynomial matrices appl math comput 155 2004 137 163 algorithms are implemented in the symbolic computational package mathematica

phase space bounds for quantum mechanics on a compact lie group let k be a compact connected lie group and k bbb c its complexification i consider the hilbert space cal h l 2 left k bbb c nu t right of holomorphic functions introduced in h1 where the parameter t is to be interpreted as planck s constant in light of l s the complex group k bbb c may be identified canonically with the cotangent bundle of k using this identification i associate to each f in cal h l 2 left k bbb c nu t right a phase space probability density the main result of this paper is theorem 1 which provides an upper bound on this density which holds uniformly over all f and all points in phase space specifically the phase space probability density is at most a t left 2 pi t right n where n dim k and a t is a constant which tends to one exponentially fast as t tends to zero at least for small t this bound cannot be significantly improved

hermite functions on compact lie groups ii abstract the usual formula for hermite polynomials on rd is extended to a compact lie group g yielding an isometry of l2 g p1 where p1 is the heat kernel measure at time one with a natural completion of the universal enveloping algebra of g the existence of such an isometry was first established by l gross here we present a simplified proof as well as the explicit form of the isometry

coherent state transforms for spaces of connections abstract the segal bargmann transform plays an important role in quantum theories of linear fields recently hall obtained a non linear analog of this transform for quantum mechanics on lie groups given a compact connected lie groupgwith its normalized haar measure h the hall transform is an isometric isomorphism froml2 g h to h g c l2 g c whereg c the complexification ofg h g c the space of holomorphic functions ong c and an appropriate heat kernel measure ong c we extend the hall transform to the infinite dimensional context of non abelian gauge theories by replacing the lie groupgby a certain extension of the space a g of connections modulo gauge transformations the resulting coherent state transform provides a holomorphic representation of the holonomyc algebra of real gauge fields this representation is expected to play a key role in a non perturbative canonical approach to quantum gravity in 4 dimensions

the schrodinger equation and canonical perturbation theory lett0 ev be the schrodinger operator corresponding to the classical hamiltonianh0 ev whereh0 is thed dimensional harmonic oscillator with non resonant frequencies 1 d and the potentialv q1 qd is an entire function of order d 1 1 we prove that the algorithm of classical canonical perturbation theory can be applied to the schrodinger equation in the bargmann representation as a consequence each term of the rayleigh schrodinger series near any eigenvalue oft0 admits a convergent expansion in powers of of initial point the corresponding term of the classical birkhoff expansion moreover ifv is an even polynomial the above result and the kam theorem show that all eigenvalues n e oft0 ev such thatn coincides with a kam torus are given up to order e by a quantization formula which reduces to the bohr sommerfeld one up to first order terms in

some integral identities and inequalities for entire functions and their application to the coherent state transform let be an entire function on cn and for any h 0 and r 0 define fr z r e 2 z 2h let d h denote h n times lebesgue measure on cn frs2 2 d h n sh frsd h from this and a logarithmic sobolev inequality we easily deduce qnq fr q pnp fr p for all 0 p q t8 where the lp norms are taken with respect to the measure d h above we apply these results to the study of the spaces ap consisting of all entire functions satisfying z pe 2 z 2h d h obtaining sharp bounds for some associated operators and proving denseness of analytic polynomials in ap for 1 p we then apply our results to the coherent state transform extending and simplifying some previously known results

symbolic and recursive computation of different types of generalized inverses abstract we propose a method and algorithm for recursive computation of different classes of generalized inverses of a given one variable rational matrix and corresponding algorithm for polynomial matrix these methods and algorithms are generalizations of the method for computing the generalized inverses for constant matrices originated in f e udwadia r e kalaba a unified approach for the recursive determination of generalized inverses comp math appl 37 1999 125 130 and the partitioning method for computing the generalized inverses of rational and polynomial matrices introduced in p s stanimirovic m b tasic partitioning method for rational and polynomial matrices appl math comput 155 2004 137 163 m b tasic p s stanimirovic m d petkovic symbolic computation of weighted moore penrose inverse using partitioning method appl math comput 189 2007 615 640 algorithms are implemented in the symbolic computational package mathematica

complex time evolution in geometric quantization and generalized coherent state transforms abstract for the cotangent bundle t k of a compact lie group k we study the complex time evolution of the vertical tangent bundle and the associated geometric quantization hilbert space l 2 k under an infinite dimensional family of hamiltonian flows for each such flow we construct a generalized coherent state transform cst which is a unitary isomorphism between l 2 k and a certain weighted l 2 space of holomorphic functions for a particular set of choices we show that this isomorphism is naturally decomposed as a product of a heisenberg type evolution for complex time within l 2 k followed by a polarization changing geometric quantization evolution for complex time in this case our construction yields the usual generalized segal bargmann transform of hall we show that the infinite dimensional family of hamiltonian flows can also be understood in terms of thiemann s complexifier method which generalizes the construction of adapted complex structures

yang mills theory and the segal bargmann transform we use a variant of the segal bargmann transform to study canonically quantized yang mills theory on a space time cylinder with a compact structure group k the non existent lebesgue measure on the space of connections is approximated by a gaussian measure with large variance the segal bargmann transform is then a unitary map from the l 2 space over the space of connections to a holomorphicl 2 space over the space of complexified connections with a certain gaussian measure this transform is given roughly bye t1a 2 followed by analytic continuation here 1a is the laplacian on the space of connections and is the hamiltonian for the quantized theory on the gauge trivial subspace consisting of functions of the holonomy around the spatial circle the segal bargmann transform becomes e t1 k 2 followed by analytic continuation where 1k is the laplacian for the structure group k this result gives a rigorous meaning to the idea that 1a reduces to 1k on functions of the holonomy by letting the variance of the gaussian measure tend to infinity we recover the standard realization of the quantized yang mills theory on a space time cylinder namely 1 1k is the hamiltonian and l 2 k is the hilbert space as a byproduct of these considera tions we find a new one parameter family of unitary transforms from l 2 k to certain holomorphicl 2 spaces over the complexification of k this family of transformations interpolates between the two previously known unitary transformations our work is motivated by results of landsman and wren and uses probabilistic techniques similar to those of gross and malliavin

holomorphic sobolev spaces associated to compact symmetric spaces using gutzmer s formula due to lassalle we characterise the images of soblolev spaces under the segal bargmann transform on compact riemannian symmetric spaces we also obtain necessary and sufficient conditions on a holomorphic function to be in the image of smooth functions and distributions under the segal bargmann transform

the segal bargmann transform for path groups letkbe a connected lie group of compact type and letw k denote the set of continuous paths ink starting at the identity and with time interval 0 1 thenw k forms an infinite dimensional group under the operation of pointwise multiplication let denote the wiener measure onw k we construct an analog of the segal bargmann transform forw k letkcbe the complexification ofk w kc the set of continuous paths inkcstarting at the identity and the wiener measure onw kc our transform is a unitary map ofl2 w k onto the holomorphic subspace ofl2 w kc by analogy with the classical transform our transform is given by convolution with the wiener measure followed by analytic continuation we prove that the transform forw k is nicely related by means of the ito map to the classical segal bargmann transform for the path space in the lie algebra ofk

interpolation algorithm for computing drazin inverse of polynomial matrices in this paper we introduce an interpolation method for computing the drazin inverse of a given polynomial matrix this method is an extension of the known method from a schuster p hippe inversion of polynomial matrices by interpolation ieee trans automat control 37 3 1992 363 365 applicable to usual matrix inverse also we improve our interpolation method using a more effective estimation of degrees of polynomial matrices generated in leverrier faddeev method algorithms are implemented and tested in the symbolic package mathematica

on the kakutani ito segal gross and segal bargmann hall isomorphisms recently gross has shown that the kakutani ito segal isomorphism theorem has an extension from the setting of gaussian measure on a vector space to heat kernel measure pt on a simply connected lie group g of compact type the isomorphism relates l2 pt to a certain completion of the universal enveloping algebra of g lie g gross proves this result using the kakutani ito segal theorem and an infinite dimensional calculus associated to g valued brownian motion hijab has greatly simplified and clarified gross proof hiiab s proof avoids most but not all of the infinite dimensional analysis in the original proof in this paper we will build on hijab s proof to give a completely finite dimensional non probabilistic proof of gross isomorphism theorem the proof given here relies heavily on hall s beautiful extension of the segal bargmann transformation to the setting of compact lie groups this theorem relating l2 pt to a certain l2 space of holomorphic functions l2 gc h on the complexified lie group gc will also be generalized to lie groups of compact type in the process it is shown how to characterize in terms of summability conditions on all the derivatives at the identity in gc those holomorphic functions which are in l2 gc

extensions of faddeev s algorithms to polynomial matrices starting from algorithms introduced in ky m vu an extension of the faddeev s algorithms in proceedings of the ieee multi conference on systems and control on september 3 5th 2008 san antonio tx which are applicable to one variable regular polynomial matrices we introduce two dual extensions of the faddeev s algorithm to one variable rectangular or singular matrices corresponding algorithms for symbolic computing the drazin and the moore penrose inverse are introduced these algorithms are alternative with respect to previous representations of the moore penrose and the drazin inverse of one variable polynomial matrices based on the leverrier faddeev s algorithm complexity analysis is performed algorithms are implemented in the symbolic computational package mathematica and illustrative test examples are presented

geometric quantization and the generalized segal bargmann transform for lie groups of compact type let k be a connected lie group of compact type and let t k be its cotangent bundle this paper considers geometric quantization of t k first using the vertical polarization and then using a natural kahler polarization obtained by identifying t k with the complexified group k the first main result is that the hilbert space obtained by using the kahler polarization is naturally identifiable with the generalized segal bargmann space introduced by the author from a different point of view namely that of heat kernels the second main result is that the pairing map of geometric quantization coincides with the generalized segal bargmann transform introduced by the author this means that the pairing map in this case is a constant multiple of a unitary map for both results it is essential that the half form correction be included when using the kahler polarization

semiclassical approximation for schrodinger operators on a two sphere at high energy let h s v be a schrodinger operator acting in l2 s with s the two dimensional unit sphere s the spherical laplacian and v a smooth potential approximate eigenfunctions and eigenvalues for h are obtained involving expansions in inverse powers of the classical angular momentum variables provided that these variables are in a region of phase space where the corresponding classical hamiltonian is nearly integrable the analysis is carried out in a bargmann representation where s becomes a quadratic expression in the sum of two quantum harmonic oscillator hamiltonians and v becomes a pseudodifferential operator

holomorphic sobolev spaces and the generalized segal bargmann transform we consider the generalized segal bargmann transform ct for a compact group k introduced in b c hall j funct anal 122 1994 103 151 let kc denote the complexification of k we give a necessary and sufficient pointwise growth condition for a holomorphic function on kc to be in the image under ct of c 1 k we also characterize the image under ct of sobolev spaces on k the proofs make use of a holomorphic version of the sobolev embedding theorem

effective partitioning method for computing weighted moore penrose inverse we introduce a method and an algorithm for computing the weighted moore penrose inverse of multiple variable polynomial matrix and the related algorithm which is appropriated for sparse polynomial matrices these methods and algorithms are generalizations of algorithms developed in m b tasic p s stanimirovic m d petkovic symbolic computation of weighted moore penrose inverse using partitioning method appl math comput 189 2007 615 640 to multiple variable rational and polynomial matrices and improvements of these algorithms on sparse matrices also these methods are generalizations of the partitioning method for computing the moore penrose inverse of rational and polynomial matrices introduced in p s stanimirovic m b tasic partitioning method for rational and polynomial matrices appl math comput 155 2004 137 163 m d petkovic p s stanimirovic symbolic computation of the moore penrose inverse using partitioning method internat j comput math 82 2005 355 367 to the case of weighted moore penrose inverse algorithms are implemented in the symbolic computational package mathematica

a new method for computing the solutions of differential equation systems using generalized inverse via maple in j jones et al j symbolic comput 25 1998 99 124 an algorithm for computing the generalized inverse of a singular rational matrix has been implemented in maple in this paper we present a new symbolic algorithm to compute the initial value problem of higher order differential equations with constant coefficients using generalized inverse the algorithm is subsequently implemented in the symbolic computational package maple

coherent states on spheres we describe a family of coherent states and an associated resolution of the identity for a quantum particle whose classical configuration space is the d dimensional sphere sd the coherent states are labeled by points in the associated phase space t sd these coherent states are not of perelomov type but rather are constructed as the eigenvectors of suitably defined annihilation operators we describe as well the segal bargmann representation for the system the associated unitary segal bargmann transform and a natural inversion formula although many of these results are in principle special cases of the results of hall and stenzel we give here a substantially different description based on ideas of thiemann and of kowalski and rembielinski all of these results can be generalized to a system whose configuration space is an arbitrary compact symmetric space we focus on the sphere case in order to carry out the calculations in a self contained and explicit way

berezin toeplitz quantization on lie groups let k be a connected compact semisimple lie group and kc its complexification the generalized segal bargmann space for kc is a space of square integrable holomorphic functions on kc with respect to a k invariant heat kernel measure this space is connected to the schrodinger hilbert space l 2 k by a unitary map the generalized segal bargmann transform this paper considers certain natural operators on l 2 k namely mul tiplication operators and differential operators conjugated by the general ized segal bargmann transform the main results show that the resulting operators on the generalized segal bargmann space can be represented as toeplitz operators the symbols of these toeplitz operators are expressed in terms of a certain subelliptic heat kernel on kc i also examine some of the results from an infinite dimensional point of view based on the work of l gross and p malliavin

on the bks pairing for kahler quantizations of the cotangent bundle of a lie group a natural one parameter family of kahler quantizations of the cotangent bundle t k of a compact lie group k taking into account the half form correction was studied in fmmn in the present pa per it is shown that the associated blattner kostant sternberg bks pairing map is unitary and coincides with the parallel transport of the quantum connection introduced in our previous work from the point of view of adpw the bks pairing map is a composition of uni tary coherent state transforms of k introduced in ha1 continuity of the hermitian structure on the quantum bundle in the limit when one of the kahler polarizations degenerates to the vertical real polar ization leads to the unitarity of the corresponding bks pairing map this is in agreement with the unitarity up to scaling with respect to a rescaled inner product of this pairing map established by hall

computation of moore penrose generalized inverses of matrices with meromorphic function entries abstract in this paper given a field with an involutory automorphism we introduce the notion of moore penrose field by requiring that all matrices over the field have moore penrose inverse we prove that only characteristic zero fields can be moore penrose and that the field of rational functions over a moore penrose field is also moore penrose in addition for a matrix with rational functions entries with coefficients in a field k we find sufficient conditions for the elements in k to ensure that the specialization of the moore penrose inverse is the moore penrose inverse of the specialization of the matrix as a consequence we provide a symbolic algorithm that given a matrix whose entries are rational expression over c of finitely many meromorphic functions being invariant by the involutory automorphism computes its moore penrose inverve by replacing the functions by new variables and hence reducing the problem to the case of matrices with complex rational function entries

dft calculation for the 2 inverse of a polynomial matrix with prescribed image and kernel in this paper we first give the finite algorithm for generalized inverse a t s 2 of a matrix a over an integral domain and based on it and the discrete fourier transform present an algorithm for calculating 2 inverses of a polynomial matrix with prescribed image and kernel and the algorithm is implemented in the mathematica programming language and expands the algorithms in 13

geometric quantization complex structures and the coherent state transform abstract it is shown that the heat operator in the hall coherent state transform for a compact lie group k j funct anal 122 1994 103 151 is related with a hermitian connection associated to a natural one parameter family of complex structures on t k the unitary parallel transport of this connection establishes the equivalence of geometric quantizations of t k for different choices of complex structures within the given family in particular these results establish a link between coherent state transforms for lie groups and results of hitchin comm math phys 131 1990 347 380 and axelrod et al j differential geom 33 1991 787 902

harmonic analysis with respect to heat kernel measure this paper surveys developments over the last decade in harmonic analysis on lie groups relative to a heat kernel measure these include analogs of the hermite expansion the segal bargmann transform and the taylor ex pansion some of the results can be understood from the standpoint of geo metric quantization others are intimately related to stochastic analysis