the reflexive and anti reflexive solutions of the matrix equation ax b abstract an n n complex matrix p is said to be a generalized reflection matrix if p h p and p 2 i an n n complex matrix a is said to be a reflexive or anti reflexive matrix with respect to the generalized reflection matrix p if a pap or a pap this paper establishes the necessary and sufficient conditions for the existence of and the expressions for the reflexive and anti reflexive with respect to a generalized reflection matrix p solutions of the matrix equation ax b in addition in corresponding solution set of the equation the explicit expression of the nearest matrix to a given matrix in the frobenius norm have been provided

symplectic surgery and the spinc dirac operator abstract letgbe a compact connected lie group and m a compact hamiltoniang space with moment mapj m g under the assumption that these data are pre quantizable one can construct an associated spinc dirac operator formula whose equivariant index yields a virtual representation ofg we prove a conjecture of guillemin and sternberg that if 0 is a regular value ofj the multiplicityn 0 of the trivial representation in the index space formula is equal to the index of the spinc dirac operator for the symplectic quotientm0 j 1 0 g this generalizes previous results for the case thatg tis abelian i e a torus

examples of non kahler hamiltonian torus actions an important question with a rich history is the extent to which the symplectic category is larger than the kahler category many interesting examples of non kahler symplectic manifolds have been constructed t m g however sufficiently large symmetries can force a symplectic manifolds to be kahler d kn in this paper we solve several outstanding problems by constructing the first symplectic manifold with large non trivial symmetries which does not admit an invariant kahler structure the proof that it is not kahler is based on the atiyah guillemin sternberg convexity theorem at gs using the ideas of this paper c woodward shows that even the symplectic analogue of spherical varieties need not be kahler w

towards a generalized singular value decomposition we suggest a form for and give a constructive derivation of the generalized singular value decomposition of any two matrices having the same number of columns we outline its desirable characteristics and compare it to an earlier suggestion by van loan siam j numer anal 13 1976 pp 76 83 the present form largely follows from the work of van loan but is slightly more general and computationally more amenable than that in the paper cited we also prove a useful extension of a theorem of stewart siam rev 19 1977 pp 634 662 on unitary decompositions of submatrices of a unitary matrix

multiplicity free hamiltonian actions need not be kahler multiplicity free actions are symplectic manifolds with a very high degree of symmetry delzant 2 showed that all compact multiplicity free torus actions admit compatible kahler structures and are therefore toric varieties in this note we show that delzant s result does not generalize to the non abelian case our examples are constructed by applying u 2 equivariant symplectic surgery to the flag variety u 3 t3 we then show that these actions fail a criterion which tolman 9 shows is necessary for the existence of a compatible kahler structure

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

singular value and generalized singular value decompositions and the solution of linear matrix equations abstract the solution of the linear matrix equations i axb cyd e and ii axb fxg e h are considered new necessary and sufficient conditions for the consistency of the equations are derived some using the generalized singular value decomposition special cases iii ax yd e and iv axb e are treated using the singular value decomposition numerical algorithms for the solutions are also suggested

bisymmetric and centrosymmetric solutions to systems of real quaternion matrix equations in this paper we consider bisymmetric and centrosymmetric solutions to certain matrix equations over the real quaternion algebra necessary and sufficient conditions are obtained for the matrix equation ax c and the following systemsa 1x c 1 a 1x c 1 xb 3 c 3 a 2x c 2 to have bisymmetric solutions and the systema 1x c 1 a 3xb 3 c 3 to have centrosymmetric solutions the expressions of such solutions of the matrix and the systems mentioned above are also given moreover a criterion for a quaternion matrix to be bisymmetric is established and some auxiliary results on other sets over are also mentioned

computing thecs decomposition of a partitioned orthonormal matrix this paper describes an algorithm for simultaneously diagonalizing by orthogonal transformations the blocks of a partitioned matrix having orthonormal columns

the gelfand cetlin system and quantization of the complex flag manifolds the construction of angle action variables for collective completely integrable systems is described and the associated bohr sommerfeld sets are determined the quantization method of sniatycki applied to such systems gives formulas for multiplicities for the gelfand cetlin system on complex flag manifolds we show that these formulas give the correct answers for the multiplicities of the associated representations

least squares solutions to the matrix equations ax b and xc d abstract the least squares solution and the least squares symmetric solution with the minimum norm of the matrix equations ax b and xc d are considered in this paper by the matrix differentiation and the spectral decomposition of matrices an explicit representation of such solution is given

the matrix equations ax b xc d with px sxp constraint abstract the matrix equations ax b xc d with the constraint px sxp are considered where p is a given hermitian matrix satisfying p 2 i and s 1 using an eigenvalue decomposition of p the constrained problem can be equivalently transformed to two independent unconstrained problems of matrix equations in similar forms and hence the constrained problem can be solved in terms of the eigenvectors of p in this paper a simple and eigenvector free formula of the general solutions to the constrained problem is presented using moore penrose generalized inverses of the coefficient matrices a and c a similar problem of the matrix equations with generalized constraint is discussed too we also consider the least norm solution of the constrained problems

positive solutions to the equations ax c and xb d for hilbert space operators abstract the paper studies the equation a x c for bounded linear operators between hilbert spaces gives conditions for the existence of hermitian solutions and positive solutions and obtains the formula for the general form of these solutions then the common hermitian and positive solutions to the equations a x c and x b d are studied and new representations of the general solutions are given many results for matrices are recovered as special cases and the results of phadke and thakare s v phadke n k thakare generalized inverses and operator equations linear algebra appl 23 1979 191 199 are corrected

symmetric and skew antisymmetric solutions to systems of real quaternion matrix equations in this paper we consider symmetric and skew antisymmetric solutions to certain matrix equations over the real quaternion algebra h first a criterion for a quaternion matrix to be symmetric and skew antisymmetric is given then necessary and sufficient conditions are obtained for the matrix equation ax c and the following system a 1x c 1xb 3 c 3 to have symmetric and skew antisymmetric solutions the expressions of such solutions of the matrix equation and the system mentioned above are also given

addendum to concentration and lack of observability of waves in highly heterogeneous media in 1 we introduced a class of 1 d wave equations with rapidly oscillating holder continuous coefficients for which the classical boundary observability property fails we also established that these examples could be used to contradict strichartz type inequalities for the wave equation with low regularity coefficients the object of this addendum is to further analyze this issue as we will see the argument in 1 only provides sharp counter examples to the strichartz estimates when the coefficient belongs to l we carefully analyze these counter examples for holder continuous coefficients we also give a new application of our construction which shows that some eigenfunction estimates for elliptic operators due to sogge can fail when coefficients are not smooth enough

stability of spectra of hodge de rham laplacians in this paper we investigate the stability of eigenspaces of the laplace operator acting on differential forms satisfying relative or absolute boundary conditions on a compact oriented riemannian manifold with boundary this includes in particular both neumann and dirichlet conditions for the laplace beltrami operator on functions more precisely our main result is that the gap between corresponding eigenspaces precise definition will be recalled below measured using the l norm converges to zero when smooth metrics g converge to g0 in the c 1 topology it is quite well known cf 3 or 14 that the eigenvalues of the laplacian vary continuously under c 0 continuous perturbations of the metric it is perhaps less well known but implicit in the work of cheeger 3 that eigenspaces vary continuously as subspaces of l2 when the metric is perturbed c 0 continuously we reprove this c 0 l2 stability in sect 4 for completeness and in order to be able to use certain notation conventions and partial results in the proof of c 1 l stability in sect 5 the second section of the paper contains a review of the hodge theory for the laplace operator with absolute and relative boundary conditions we also state here the sobolev embedding theorems and the basic a priori estimates for the square root d of the laplacian we need to work with d rather than d 2 since the coefficients of depend on the second derivatives of the metric tensor and we allow only c 1 continuous perturbations of the metric and do not assume any bounds on the second derivatives in the third section we review following kato 12 and osborn 16 general results from functional analysis concerning perturbation theory for compact operators on banach spaces that reduce proving convergence of eigenvalues and eigenspaces

the matrix equations ax c xb d for the pair of matrix equations ax c xb d this paper gives common solutions of minimum possible rank and also other feasible specified ranks

least squares solutions to the equations ax b xc d with some constraints abstract the least squares solutions to the equations ax b xc d with some constraints such as orthogonality symmetric orthogonality symmetric idempotence are considered by one time singular value decomposition or eigenvalue decomposition of the matrix product generated by the coefficient matrices a c and right hand side matrices b d the constrained solutions are constructed simply similar strategy is applied to the equations with corresponding p commuting constraints with given symmetric matrix p numerical examples that show the efficiency of the proposed methods are presented

integrated density of states for random metrics on manifolds we study ergodic random schrodinger operators on a covering manifold where the randomness enters both via the potential and the metric we prove measurability of the random operators almost sure constancy of their spectral properties the existence of a self averaging integrated density of states and a pastur ubin type trace formula

the skew symmetric orthogonal solutions of the matrix equation ax b abstract an n n real matrix x is said to be a skew symmetric orthogonal matrix if xt x and xtx i using the special form of the c s decomposition of an orthogonal matrix with skew symmetric k k leading principal submatrix this paper establishes the necessary and sufficient conditions for the existence of and the expressions for the skew symmetric orthogonal solutions of the matrix equation ax b in addition in corresponding solution set of the equation the explicit expression of the nearest matrix to a given matrix in the frobenius norm have been provided furthermore the procrustes problem of skew symmetric orthogonal matrices is considered and the formula solutions are provided finally an algorithm is proposed for solving the first and third problems numerical experiments show that it is feasible

perturbation analysis of the canonical correlations of matrix pairs abstract we present a perturbation analysis of the canonical correlations of matrix pairs absolute perturbation bounds for normwise as well as componentwise perturbations are derived we demonstrate by way of examples that small relative perturbation of a matrix pair does not necessarily imply small relative perturbation in its canonical correlations and we identify a class of matrix pairs for which good relative perturbation bounds can be derived under certain conditions

gkm theory and hamiltonian non kahler actions in dimension 6 abstract using the classification of 6 dimensional manifolds by wall jupp and ubr we observe that the diffeomorphism type of simply connected compact 6 dimensional integer gkm t 2 manifolds is encoded in their gkm graph as an application we show that the 6 dimensional manifolds on which tolman and woodward constructed hamiltonian non kahler t 2 actions with finite fixed point set are diffeomorphic to eschenburg s twisted flag manifold su 3 t 2 in particular they admit a noninvariant kahler structure

harmonic analysis on constant curvature surfaces with point singularities let rn2 denote the n fold covering space of the plane with the origin deleted 0 n parametrized by polar coordinates r with periodic of period 2 n and equipped with the flat riemannian metric dr2 r2 d 2 there is a natural harmonic analysis with the fourier transform given by f c 12 0 f r j v r eiv r dr d fourier inversion formula f c 0 f r j v r eiv d dv and plancherel formula f 22 2 0 f v 2 d dv in the case n and analogous expressions with v restricted to the values kn when n is finite this harmonic analysis realizes the joint spectral decompositions of the self adjoint operators i and where is the dirichlet or friedrichs extension of the laplace beltrami operator defined on d this paper gives a detailed development of this harmonic analysis including the effect of basic differentiation and multiplication operators on the fourier transform the theory of l2 sobolev spaces hk and the distinctions between h2 the domain of and the space of functions f with both f and f in l2 in the process it is necessary to study certain generalizations of the hilbert transform that are related to the theory of hankel transforms two applications of this harmonic analysis are given to the study of a natural radon transform or x ray transform on rn2 and to the solution of the heat equation on rn2 the latter also gives information about the heat equation in a sector of the plane with dirichlet or neumann boundary conditions and about brownian motion in the plane taking into account the winding number of a brownian path with respect to the origin analogous problems for spaces of non zero constant curvature are discussed and in this context some higher dimensional spaces arise naturally

least squares solution with the minimum norm for the matrix equation axb gxh c d based on the projection theorem in hilbert space by making use of the generalized singular value decomposition and the canonical correlation decomposition an analytical expression of the least squares solution for the matrix equation axb gxh c d with the minimum norm is derived an algorithm for finding the minimum norm solution is described and some numerical results have been given