pairs of companion matrices and their simultaneous reduction to complementary triangular forms abstract a polynomial approach is described to deal with problems of the following type given two companion matrices a and z when does there exist a nonsingular matrix s such that s 1 as is upper triangular and s 1 zs is lower triangular and the eigenvalues of a and z appear in the main diagonals of s 1 as and s 1 zs in a prescribed ordering such problems are related to complete minimal factorizations in systems theory

semi classical properties of berezin toeplitz operators with c k symbol we obtain the semi classical expansion of the kernels and traces of toeplitz operators with ck symbol on a symplectic manifold we also give a semi classical estimate of the distance of a toeplitz operator to the space of self adjoint and multiplication operators we obtain the semi classical expansion of the kernels and traces of toeplitz operators with ck symbol on a symplectic manifold we also give a semi classical estimate of the distance of a toeplitz operator to the space of self adjoint and multiplication operators

simultaneous reduction to triangular and companion forms of pairs of matrices the case rank i az 1 abstract this paper is concerned with simultaneous reduction to triangular and companion forms of pairs of matrices a z with rank i az 1 special attention is paid to the case where a is a first and z is a third companion matrix two types of simultaneous triangularization problems are considered 1 the matrix a is to be transformed to upper triangular and z to lower triangular form 2 both a and z are to be transformed to the same upper triangular form the results on companions are made coordinate free by characterizing the pairs a z for which there exists an invertible matrix s such that s 1 as is of first and s 1 zs is of third companion type one of the main theorems reads as follows if rank i az 1 and 1 for every eigenvalue of a and every eigenvalue of z then a and z admit simultaneous reduction to complementary triangular forms

simultaneous reduction to triangular forms after extension with zeroes abstract this paper is concerned with pairs of m m matrices a z for which there exists an invertible m m matrix s such that s 1 as is upper triangular and s1zs is lower triangular such pairs are said to admit simultaneous reduction to complementary triangular forms in particular if such a similariy s does not exist for the pair a z tollowing question is taken into consideration does there exist a positive integer r such that the pair of m r m r matrices a o o o r z o o o r admits simultaneous reduction to complementary triangular forms so that the pair a z obtains the property after extension with zeroes it is shown that the answer to this question is mixed an example of a pair is given which obtains the property after extension with one zero this example is put into contrast with a large number of results that answers the question negatively i e that describes situations where the pair of extended matrices admits simultaneous reduction to complementary triangular forms if and only if the original pair does

simultaneous reduction of sets of matrices under similarity abstract let s be a set of n n matrices over a field f and a the algebra generated by s over f the problem of deciding whether the elements of s can be simultaneously reduced to block triangular form with the diagonal blocks of some specified size is considered and an account is given of various methods used to attack the problem most of the techniques use representation theory to obtain information on a the problems of simultaneous triangularization existence of common eigenvectors etc are also considered the aim of the paper is to survey the methods used to attack these problems and to give some typical results the paper does not contain many new results

semiclassical spectral analysis of toeplitz operators on symplectic manifolds the case of discrete wells we consider toeplitz operators associated with the renormalized bochner laplacian on high tensor powers of a positive line bundle on a compact symplectic manifold we study the asymptotic behavior in the semiclassical limit of low lying eigenvalues and the corresponding eigensections of a self adjoint toeplitz operator under assumption that its principal symbol has a non degenerate minimum with discrete wells as an application we prove upper bounds for low lying eigenvalues of the bochner laplacian in the semiclassical limit

common eigenvectors and quasicommutativity of sets of simultaneously triangularizable matrices abstract a set of simultaneously triangularizable square matrices over an arbitrary field is considered if the matrices are also quasicommutative then they have a common eigenvector for every distinct set of corresponding eigenvalues conversely if the set of matrices has this common eigenvector property hereditarily i e for every set of corresponding blocks in every simultaneous block triangularization then the matrices are quasicommutative

exploiting symmetries within constraint satisfaction search abstract symmetry often appears in real world constraint satisfaction problems but strategies for exploiting it are only beginning to be developed here a framework for exploiting symmetry within depth first search is proposed leading to two heuristics for variable selection and a domain pruning procedure these strategies are then applied to two highly symmetric combinatorial problems namely the ramsey problem and the generation of balanced incomplete block designs experimental results show that these general purpose strategies can compete with and in some cases outperform previous more ad hoc procedures

common eigenvectors of two matrices abstract a computable criterion is given for two square matrices to possess a common eigenvector as well as a criterion for one matrix to have an eigenvector lying in a given subspace some applications are discussed

quantized nambu poisson manifolds and n lie algebras we investigate the geometric interpretation of quantized nambu poisson structures in terms of noncommutative geometries we describe an extension of the usual axioms of quantization in which classical nambu poisson structures are translated to n lie algebras at quantum level we demonstrate that this generalized procedure matches an extension of berezin toeplitz quantization yielding quantized spheres hyperboloids and superspheres the extended berezin quantization of spheres is closely related to a deformation quantization of n lie algebras as well as the approach based on harmonic analysis we find an interpretation of nambu heisenberg n lie algebras in terms of foliations of rn by fuzzy spheres fuzzy hyperboloids and noncommutative hyperplanes some applications to the quantum geometry of branes in m theory are also briefly discussed

gl and geometric quantization an axiomatic approach to the approximation of infinite dimensional algebras is presented examples illustrating the need for a rigorous treatment of this subject geometric quantization is employed to construct systematicallysu n approximations of diffeomorphism algebras which first appeared in the theory of relativistic membranes

complementary triangular forms of pairs of matrices realizations with prescribed main matrices and complete factorization of rational matrix functions abstract the problem considered is the following given two square matrices a and z when does there exist an invertible matrix s such that s 1 as is upper triangular and s 1 zs is lower triangular several sufficient conditions for such an s to exist are presented in some cases the conditions are also necessary attention is paid to the order in which the eigenvalues of a and z may appear on the diagonals of s 1 as and s 1 zs respectively there is an intimate connection with the notions of realization and minimal factorization from systems theory

factorization and job scheduling a connection via companion based matrix functions a connection is made between two sets of problems the first set involves factorization problems of specific rational matrix functions the companion based matrix functions the second set is concerned with variants of the two machine flow shop problem 2mfsp from job scheduling theory in particular it is shown that with each companion based matrix function one can associate an instance of 2mfsp and vice versa the latter can be done in such a way that the factorization properties of the companion based matrix function correspond to the combinatorial properties of the instance of 2mfsp

high power asymptotics of some weighted harmonic bergman kernels abstract for weights which are either radial on the unit ball or depend only on the vertical coordinate on the upper half space we describe the asymptotic behaviour of the corresponding weighted harmonic bergman kernels with respect to as this can be compared to the analogous situation for the holomorphic case which is of importance in the berezin quantization as well as in complex geometry

upper triangularization of matrices by permutations and lower triangular similarity transformations abstract let a be an arbitary square matrix as is well known there exists an invertible matrix s such that s 1as is upper triangular the present paper is concerned with the observation that s can always be chosen in the form s l where is a permutation matrix and l is lower triangular assuming that the eigenvalues of a are given the matrices l and u l 1 1a l are constructed in an explicit way the construction gives insight into the freedom one has in choosing the permutation matrix two cases where can be chosen to be the identity matrix are discussed a diagonable a a low order toeplitz matrix there is a connection with systems theory

hermite polynomials and quasi classical asymptotics we study an unorthodox variant of the berezin toeplitz type of quantization scheme on a reproducing kernel hilbert space generated by the real hermite polynomials and work out the associated quasi classical asymptotics

similarity invariants for pairs of upper triangular toeplitz matrices abstract the problem considered is that of simultaneous reduction to simple forms of pairs of upper triangular toeplitz matrices in this context simple matrices are those that are equal to or differ only slightly from a power of the upper triangular nilpotent jordan block the problem is related to that of complementary triangularization of pairs of matrices and therefore has a background in systems theory the main results are concerned with existence and uniqueness of the reduced form also a similarity invariant for pairs of upper triangular toeplitz matrices is obtained the invariant consists of a pair involving a complex number and an integer the results can be generalized to a class of matrices strictly larger than that formed by the upper triangular toeplitz matrices

quantum riemann surfaces i the unit disc we construct a non commutative algebra c mu bar u which is a quantum deformation of the algebra of continuous functions on the closed unit disc bar u c mu bar u is generated by the toeplitz operators on a suitable hilbert space of holomorphic functions onu

generalized bergman kernels and geometric quantization in geometric quantization it is well known that if f is an observable and f a polarization on a symplectic manifold m then the condition xf leaves f invariant where xf denotes the hamiltonian vector field associated to f is sufficient to guarantee that one does not have to compute the bks kernel explicitly in order to know the corresponding quantum operator it is shown in this paper that this condition on f can be weakened to xf leaves f f invariant and the corresponding quantum operator is then given implicitly by formula 4 8 in particular when f is a positive kahler polarization all observables can be quantized directly and moreover an explicit formula for the corresponding quantum operator is derived theorem 5 8 applying this to the phase space r2n one obtains a quantization prescription which ressembles the normal ordering of operators in quantum field theory when we translate this prescription to the usual position representation of quantum mechanics the result i

on similarity of matrices over commutative rings the paper studies the problem on matrix similarity over a commutative rings the conditions are determined under which the matrix is similar to a companion or diagonal matrices

convergence of the wick star product we construct a frechet space as a subspace of c omega mathbb c n where the wick star product converges and is continuous the resulting frechet algebra mathcal a is studied in detail including a representation of mathcal a in the bargmann fock space and a discussion of star exponentials and coherent states

enhancement schemes for constraint processing backjumping learning and cutset decomposition abstract researchers in the areas of constraint satisfaction problems logic programming and truth maintenance systems have suggested various schemes for enhancing the performance of the backtracking algorithm this paper defines and compares the performance of three such schemes backjumping learning and cycle cutset the backjumping and the cycle cutset methods work best when the constraint graph is sparse while the learning scheme mostly benefits problem instances with dense constraint graphs an integrated strategy is described which utilizes the distinct advantages of each scheme experiments show that in hard problems the average improvement realized by the integrated scheme is 20 25 higher than any of the individual schemes

the gauss hypergeometric function f a b c z for large c we consider the gauss hypergeometric function f a b 1 c 2 z for a b c c c 2 3 4 and arg 1 z we derive a convergent expansion of f a b 1 c 2 z in terms of rational functions of a b c and z valid for b z c bz and c b z c bz this expansion has the additional property of being asymptotic for large c with fixed a uniformly in b and z with bounded b c moreover the asymptotic character of the expansion holds for a larger set of b c and z specified below

large parameter cases of the gauss hypergeometric function we consider the asymptotic behavior of the gauss hypergeometric function when several of the parameters a b c are large we indicate which cases are of interest for orthogonal polynomials jacobi but also meixner krawtchouk etc which results are already available and which cases need more attention we also consider a few examples of 3f2 functions of unit argument to explain which difficulties arise in these cases when standard integrals or differential equations are not available

quasicomplete factorization and the two machine flow shop problem a connection is made between the two machine flow shop problem 2mfsp from job scheduling theory and the issue of quasicomplete factorization of rational matrix functions a quasicomplete factorization is a factorization into elementary i e degree one factors such that the number of factors is minimal for a companion based matrix function w the number of factors in a quasicomplete factorization of w is related in a simple way to the minimum makespan of an instance j of 2mfsp which can be associated to w as a consequence of this result variants of the 2mfsp and other types of factorization can be related too

upper triangularization of matrices by lower triangular similarities abstract this paper is concerned with the following questions given a square matrix a when does there exist an invertible lower triangular matrix l such that l 1 al is upper triangular and if so what can be said about the order in which the eigenvalues of a may appear on the diagonal of l 1 al the motivation for considering these questions comes from systems theory in fact they arise in the study of complete factorizations of rational matrix functions there is also an intimate connection with the problem of complementary triangularization of pairs of matrices discussed elsewhere by the first author

automatically improving constraint models in savile row abstract when solving a combinatorial problem using constraint programming cp or satisfiability sat modelling and formulation are vital and difficult tasks even an expert human may explore many alternatives in modelling a single problem we make a number of contributions in the automated modelling and reformulation of constraint models we study a range of automated reformulation techniques finding combinations of techniques which perform particularly well together we introduce and describe in detail a new algorithm x cse to perform associative commutative common subexpression elimination ac cse in constraint problems significantly improving existing cse techniques for associative and commutative operators such as we demonstrate that these reformulation techniques can be integrated in a single automated constraint modelling tool called savile row whose architecture we describe we use savile row as an experimental testbed to evaluate each reformulation on a set of 50 problem classes with 596 instances in total our recommended reformulations are well worthwhile even including overheads especially on harder instances where solver time dominates with a sat solver we observed a geometric mean of 2 15 times speedup compared to a straightforward tailored model without recommended reformulations using a cp solver we obtained a geometric mean of 5 96 times speedup for instances taking over 10 seconds to solve

berezin toeplitz quantization on kaehler manifolds we study the berezin toeplitz quantization on kaehler manifolds we explain first how to compute various associated asymptotic expansions then we compute explicitly the first terms of the expansion of the kernel of the berezin toeplitz operators and of the composition of two berezin toeplitz operators as application we estimate the norm of donaldson s q operator

toeplitz quantization of kahler manifolds ang gl n n limits for general compact kahler manifolds it is shown that both toeplitz quantization and geometric quantization lead to a well defined by operator norm estimates classical limit this generalizes earlier results of teh authors and klimek and lesniewski obtained for the tours and higher genus riemann surfaces respectively we thereby arrive at an approximation of the poisson algebra by a sequence of finitedimensional matrix algebrasgl n n

berezin transform on the harmonic fock space abstract we show that the berezin transform associated to the harmonic fock segal bargmann space on c n has an asymptotic expansion analogously as in the holomorphic case the proof involves a computation of the reproducing kernel which turns out to be given by one of horn s hypergeometric functions of two variables and an ad hoc determination of the asymptotic behaviour of the resulting integrals to which the ordinary stationary phase method is not directly applicable

companion based matrix functions description and minimal factorization companion based matrix functions are rational matrix functions admitting a minimal realization involving state space matrices that are first companions necessary and sufficient conditions are given for a rational matrix function to be companion based minimal factorization of such functions is discussed in detail it is shown that the property of being companion based is hereditary with respect to minimal factorization also the issue of minimal factorization is reduced to a division problem for pairs of monic polynomials of the same degree in this context a connection with the euclidean algorithm is made the results apply to canonical wiener hopf factorization as well as to complete factorization the analysis of the latter leads to a combinatorial problem involving the eigenvalues of the state space matrices the algorithmic aspects of this problem are intimately related to the two machine flow shop problem and johnson s rule from job scheduling theory

categorical perspective on quantization of poisson algebra we propose a generalization of quantization using a categorical approach for a fixed poisson algebra quantization categories are defined as subcategories of the r module category equipped with the structure of classical limits we then construct the generalized quantization categories including matrix regularization strict deformation quantization prequantization and poisson enveloping algebra it is shown that the categories of strict deformation quantization prequantization and matrix regularization with certain conditions are equivalent categories on the other hand the categories of poisson enveloping algebra are not equivalent to the other categories

quantization towards a comparison between methods in this paper it is shown that the procedure of geometric quantiztion applied to kahler manifolds gives the following result the hilbert space h consists roughly speaking of holomorphic functions on the phase space m and to each classical observable f i e a real function on m is associated an operator f on h as follows first multiply by f 1 4 drf dr being the laplace de rham operator on the kahler manifold m and then take the holomorphic part see g m tuynman j math phys 27 573 1987 this result is correct on compact kahler manifolds and correct modulo a boundary term md on noncompact kahler manifolds in this way these results can be compared with the quantization procedure of berezin math ussr izv 8 1109 1974 9 341 1975 commun math phys 40 153 1975 which is strongly related to quantization by products e g see c moreno and p ortega navarro amn inst h poincare sec a 38 215 1983 lett math phys 7 181 1983 c moreno lett math phys 11 3

berezin transform of two arguments abstract we describe the asymptotic behavior of the berezin transform of two arguments which generalizes the standard notion of the berezin transform the asymptotic analysis we give is in the setting of the holomorphic and the harmonic fock spaces respectively

weighted bergman kernels and quantization let be a bounded pseudoconvex domain in c n two positive functions on such that log log are plurisubharmonic and z a point at which log is smooth and strictly plurisubharmonic we show that as k the bergman kernels with respect to the weights k have an asymptotic expansion for x y near z where x y is an almost analytic extension of phi x x x and similarly for further if in addition is of finite type behave reasonably at the boundary and log log are strictly plurisubharmonic on we obtain also an analogous asymptotic expansion for the berezin transform and give applications to the berezin quantization finally for smoothly bounded and strictly pseudoconvex and a smooth strictly plurisubharmonic defining function for we also obtain results on the berezin toeplitz quantization

upper triangular similarity of upper triangular matrices abstract we consider the following equivalence relation in the set of all complex upper triangular n n matrices a and b are called u similar if there exists an invertible upper triangular matrix s such that a s 1 bs if a b are u similar then they must have the same diagonal and the same jordan form it is known that for n 6 there are infinitely many mutually non u similar nilpotent upper triangular matrices with the same jordan form we introduce an appropriate generalization of the jordan block called an irreducible matrix and we prove that each upper triangular matrix is u similar to a generalized direct sum of irreducible blocks where the location and the order of the blocks is fixed and each block is determined uniquely up to u similarity

identification of berezin toeplitz deformation quantization we give a complete identification of the deformation quantization which was obtained from the berezin toeplitz quantization on an arbitrary compact kahler manifold the deformation quantization with the opposite star product proves to be a differential deformation quantization with separation of variables whose dassifying form is explicitly calculated its characteristic dass which dassifies star products up to equivalence is obtained the proof is based on the microlocal description of the szego kernel of a strictly pseudoconvex domain given by boutet de monvel and sjostrand

deformation estimates for the berezin toeplitz quantization deformation estimates for the berezin toeplitz quantization ofcn are obtained these estimates justify the description of ccr k as a first order quantum deformation of ap c0 where ccr is the usualc algebra of boson canonical commutation relations k is the full algebra of compact operators ap is the algebra of almost periodic functions andc0 is the algebra of continuous functions which vanish at infinity

asymptotic analysis via calculus of hypergeometric functions abstract the generalized hypergeometric function satisfies many identities or transforms which can be used to establish their asymptotic behavior for large argument and even in some cases for large parameters we will show that using just three transforms alone valid for a large class of multivariate hypergeometric functions we can use a similar calculus to compute asymptotic expansions even in higher dimensions

transfer functions and operator theory abstract the notion of a transfer function from systems theory serves as an important tool in dealing with certain problems in operator and matrix theory this is illustrated with a variety of material taken from existing literature a striking feature of the results is their high degree of explicitness

berezin transform on harmonic bergman spaces on the real ball abstract we provide the full asymptotic expansion of the harmonic berezin transform on the unit ball in r n purely by means of transformations of hypergeometric functions and function s hypergeometrization

circular symmetry and the trace formula let m be a compact n dimensional manifold without boundary and let p be a positive first order elliptic self adjoint classical pseudodifferential operator on lz m the hilbert space of half densities on m the spectrum of p is discrete and accumulates only at we denote the eigenvalues with multiplicities by 2o 21 2 recall that the singularities of the inverse fourier transform of the spectral measure