on the geometrical structure of symmetric matrices abstract the identity ray i for 0 can be seen as the center ray of the cone of symmetric and positive definite spd matrices in that sense the angle that any spd matrix forms with the identity plays a very important role to understand the geometrical structure of the cone in this work we extend this relationship and analyze the geometrical structure of symmetric matrices including the location of all orthogonal matrices not only the identity matrix this geometrical understanding leads to new results in the subspace of symmetric matrices we also extend some of the geometrical results for the case of general not necessarily symmetric nonsingular matrices

on the best constants in inequalities of the markov and wirtinger types for polynomials on the half line abstract the main topic of the paper is best constants in markov type inequalities between the norms of higher derivatives of polynomials and the norms of the polynomials themselves the norm is the l 2 norm with laguerre weight the leading term of the asymptotics of the constants is determined and tight bounds for the principal coefficient in this term which is the operator norm of a volterra operator are given for best constants in inequalities of the wirtinger type the limit is computed and an asymptotic formula for the error term is presented

vertexwise criteria for admissibility of alcoves we give a new description of the set rm adm mu of admissible alcoves as an intersection of certain obtuse cones of alcoves and we show this description may be given by imposing conditions vertexwise we use this to prove the vertexwise admissibility conjecture of pappas rapoport smithling the same idea gives simple proofs of two ingredients used in the proof of the kottwitz rapoport conjecture on existence of crystals with additional structure

deformed laguerre hahn orthogonal polynomials on the real line abstract one derives discrete dynamical systems related to laguerre hahn orthogonal polynomials one studies deformations of the recurrence relation coefficients of the orthogonal polynomials under a t dependence on the coefficients of the riccati differential equation for the related stieltjes function

appendix on parahoric subgroups we give the proofs of some simple facts on parahoric subgroups and on iwahori weyl groups used in t haines the base change fundamental lemma for central elements in parahoric hecke algebras preprint 2008 g pappas m rapoport twisted loop groups and their affine flag varieties adv math 219 2008 118 198 m rapoport a guide to the reduction modulo p of shimura varieties asterisque 298 2005 271 318

twisted loop groups and their affine flag varieties we develop a theory of affine flag varieties and of their schubert varieties for reductive groups over a laurent power series local field k t with k a perfect field this can be viewed as a generalization of the theory of affine flag varieties for loop groups to a twisted case a consequence of our results is that our construction also includes the flag varieties for kac moody lie algebras of affine type we also give a coherence conjecture on the dimensions of the spaces of global sections of the natural ample line bundles on the partial flag varieties attached to a fixed group over k t and some applications to local models of shimura varieties

geometrical properties of the frobenius condition number for positive definite matrices we study the geometrical properties of the frobenius condition number on the cone of symmetric and positive definite matrices this number related to the cosine of the angle between a given matrix and its inverse is equivalent to the classical 2 norm condition number but has a direct and natural geometrical interpretation in particular we establish bounds for the ratio between the angle that a matrix forms with the identity ray and the angle that the inverse of that matrix forms with the identity ray these bounds allow us to establish new lower bounds for the condition number that only require the trace and the frobenius norm of the matrix

new eigenvalue estimates for complex matrices abstract in a recent work 1 we have proved that all the eigenvalues j of a complex matrix a of order n lie in a disk with center at tr a n and radius n 1 n k 1 n k 2 tr a 2 n 1 2 in this paper we prove that the eigenvalues of a are contained in a rectangle with vertices on the boundary circle of the mentioned disk improving in this way the result in 1 we recall some inequalities which alloww us to bound the radius of the disk and the rectangle the bounds can be computed without knowing the eigenvalues of a

on the flatness of local models for the symplectic group abstract we investigate the bad reduction of certain shimura varieties associated to the symplectic group more precisely we look at a model of the shimura variety at a prime p with parahoric level structure at p we show that this model is flat as conjectured by rapoport and zink ann of math stud 141 1996 and that its special fibre is reduced a crucial ingredient is faltings theorem on the normality of schubert varieties in the affine flag variety

localization of eigenvalues in elliptic regions abstract we prove that the eigenvalues j of an n by n complex matrix a with its characteristic polynomial having real coefficients lie in the elliptic region defined by 2 tr a n 2 2 y 2 2 2 where n 1 n n 1 n re k 2 re te a 2 n 1 2 and n 1 n k 1 n im k 2 1 2 this region is intersected with the strip y 1 2 k 1 u im k 2 1 2 to obtain an improved eigenvalue localization region we also give bounds for the semiaxes which can be computed without knowing the eigenvalues of a when a has r n nonzero eigenvalues we obtain a smaller elliptic region containing such nonzero eigenvalues

finite perturbations of orthogonal polynomials in this paper we study orthogonal polynomials pn which arise from a given system of orthogonal polynomials pn by a finite perturbation of the recurrence coefficients i e the recurrence coefficients n n 1 of pn respectively n n 1 of pn are related to each other by n l n m and n 1 l n 1 m for n n where l and m are fixed nonnegative integers closed expressions for the polynomial pn in terms of pn and its associated polynomial pn 1 1 as well as for the orthogonality measure of pn in terms of the orthogonality measure of pn are given in fact the results are presented for the most general case when is an arbitrary not necessarily positive measure

on the flatness of models of certain shimura varieties of pel type consider a pel shimura variety associated to a unitary group that splits over an unramified extension of q p rapoport and zink have defined a model of the shimura variety over the ring of integers of the completion of the reflex field at a place lying over p with parahoric level structures at p we show that this model is flat as conjectured by rapoport and zink and that its special fibre is reduced

eigenvalue estimates for symmetric matrices abstract we use the fact that the set of symmetric positive semidefinite matrices of order n form a cone with a special structure in order to find bounds for the eigenvalues of a symmetric matrix

topological flatness of local models in the ramified case local models are schemes defined in terms of linear algebra which can be used to study the local structure of integral models of certain shimura varieties with parahoric level structure we investigate the local models for groups of the form open image in new window is a totally ramified extension as defined by pappas and rapoport and show that they are topologically flat in the linear case flatness can be deduced from this

new conditions for the additive inverse eigenvalue problem for matrices abstract this paper deals with the following inverse eigenvalue problem given an n by n real symmetric matrix a and a set of real numbers i 1 n find a diagonal matrix d such that a d has eigenvalues i for the solvability of this problem a number of necessary conditions and sufficient conditions are known in this work new necessary conditions are derived while some sufficient conditions are optimized in particular one sufficient condition due to morel is obtained by optimizing a sufficient condition discovered by laborde

on the support of the measure of orthogonality of a class of orthogonal polynomials for orthogonal polynomials of the formp nq n 1 x q nq n 1 x r nq n x xq n x q 0 x 0 q 1 x 1 where p n 0 q n 0 r email protected r and lim n p n lim n q n 1 2 a general sufficient condition is found such that the support of the measure of orthogonality is the entire interval 1 1 starting from this result more general cases of orthogonal polynomials are studied as a perturbation problem the results are applied to pollaczek polynomials random walk polynomials rwp neutron transport polynomials and generalized co recursive polynomials

on the cone of positive semidefinite matrices abstract a survey of some general properties of the cone of positive semidefinite matrices its faces two isometric isomorphisms and linear transformations on it is given

more estimates for eigenvalues and singular values abstract there is an interesting relation between the angle that a matrix forms with the identity and its eigenvalues then we show for symmetric matrices that some information about the matrix can be known if the angle is computed a lower bound for the maximum eigenvalue is obtained for symmetric positive semidefinite matrices this inequality and the upper bound for eigenvalues given in 5 provide us an interval in which must lie the maximum eigenvalue finally some results from 5 and the first part of this paper are generalized for the non symmetric case

on orthogonal polynomials with perturbed recurrence relations abstract orthogonal polynomials may be fully characterized by the following recurrence relation pn x x n 1 pn 1 x n 1pn 2 x with p0 x 1 p1 x x 0 and n 0 here we study how the structure and the spectrum of these polynomials get modified by a local perturbation in the and parameters of a co recursive k k co dilated k k and co modified k k k k nature for an arbitrary but fixed kth element 1 k specifically stieltjes functions differential equations and distributions of zeros as well as representations of the new perturbed polynomials in terms of the old unperturbed ones are given this type of problems is strongly related to the boundary value problems of finite difference equations and to the quantum mechanical study of physical many body systems atoms molecules nuclei and solid state systems

topological flatness of orthogonal local models in the split even case i local models are schemes defined in terms of linear algebra that were introduced by rapoport and zink to study the etale local structure of integral models of certain pel shimura varieties over p adic fields a basic requirement for the integral models or equivalently for the local models is that they be flat in the case of local models for even orthogonal groups genestier observed that the original definition of the local model does not yield a flat scheme in a recent article pappas and rapoport introduced a new condition to the moduli problem defining the local model the so called spin condition and conjectured that the resulting spin local model is flat we prove a preliminary form of their conjecture in the split iwahori case namely that the spin local model is topologically flat an essential combinatorial ingredient is the equivalence of admissibility and permissibility for two minuscule cocharacters in root systems of type d

schubert varieties in twisted affine flag varieties and local models abstract our concern in this paper is the dimension and inclusion relations of schubert varieties in twisted partial affine flag varieties in the end we apply our results to some local models of certain shimura varieties

orthogonality of linear combinations of two orthogonal polynomial sequences we find necessary and sufficient conditions for some linear combinations of two sequences of orthogonal polynomials to be again orthogonal

on perturbed szegő recurrences abstract the purpose of the present contribution is to investigate the effects of finite modifications of verblunsky coefficients on szeg recurrences more precisely we study the structural relations and the corresponding c functions of the orthogonal polynomials with respect to these modifications from the initial ones

on co polynomials on the real line abstract in this paper we study new algebraic and analytic aspects of orthogonal polynomials on the real line when finite modifications of the recurrence coefficients the so called co polynomials on the real line are considered we investigate the behavior of their zeros mainly interlacing and monotonicity properties furthermore using a transfer matrix approach we obtain new structural relations combining theoretical and computational advantages finally a connection with the theory of orthogonal polynomials on the unit circle is pointed out

perturbation of the coefficients in the recurrence relation of a class of polynomials let p n x n 0 be a system of polynomials satisfying the recurrence relation p 1 x 0 p 0 x 1 p n 1 x h n p n 1 x c n p n x xp n x where h n c n are real sequences and h n 0 n 0 1 2 the co recursive polynomials p n x n 0 satisfy the same recurrence relation except for n 1 where p 1 x x c 0 0 it is well known that the problem of d the zeros of p n x is equivalent to the problem of determining the eigenvalues of a generalized eigenvalue problem tf af where t and a are symmetric matrices in this paper the problem of determining the zeros of the co recursive polynomials is reduced to a perturbation problem of the operators t and a perturbed by perturbations of rank one a function 1 2 k is found k 1 2 n whose zeros are the zeros of p n x k are the zeros of the polynomial p n x of degree n for 0 this function unifies many results concerning interlacing between the zeros of p n x and p n x for 0 moreover we obtain from this function similar results in the unstudied case 0

fourth order differential equations satisfied by the generalized co recursive of all classical orthogonal polynomials a study of their distribution of zeros the unique fourth order differential equation satisfied by the generalized co recursive of all classical orthogonal polynomials is given for any but fixed level of recursivity up to now these differential equations were known only for each classical family separately and also for a specific recursivity level moreover we use this unique fourth order differential equation in order to study the distribution of zeros of these polynomials via their newton sum rules i e the sums of powers of their zeros which are closely related with the moments of such distribution both results are obtained with the help of two programs built in mathematica symbolic language

orthogonal polynomials with a constant recursion formula and an application to harmonic analysis on calcule la mesure pour laquelle une suite de polynomes avec une formule de recurrence constante est orthogonale

bounds for singular values using traces abstract bounds for the singular values ratios of singular values and rank of a square matrix a involving tr a tr a 2 and tr a h a are presented some of the results are analogous to the well known bounds for the eigenvalues ratios of eigenvalues and rank of a involving tr a and tr a 2

a decreasing sequence of eigenvalue localization regions abstract we construct a decreasing sequence of rectangles rp in such a way that all the eigenvalues of a complex matrix a are contained in each rectangle when a is a matrix with real spectrum or a normal matrix each rp can be obtained without knowing the eigenvalues of a

bounds for the spectral radius and the largest singular value abstract we derive an increasing sequence of lower bounds for the spectral radius of a matrix with real spectrum and progressively improved bounds for the largest singular value of a complex matrix we also find estimates for the rank of normal matrices with real spectrum and for the rank of normal nonnegative matrices including some sufficient condition for such matrices to be invertible

alcoves associated to special fibers of local models the special fiber of the local model of a pel shimura variety with iwahori type level structure admits a cellular decomposition the set of strata is in a natural way a finite subset of the affine weyl group determined by the shimura data in this paper we generalize and give a new proof of a theorem of kottwitz and rapoport concerning the description of this set for the case of linear and symplectic groups for higher rank groups not of type an we produce counterexamples showing the situation is not as nice in general we also recover a theorem of deodhar characterizing the bruhat order on sn the local geometry can be approached using the rapoport zink local model m j 13 this protective op scheme is a local model for the singularities in the special fiber of sh g x9 k 9 but it is defined in terms of the pair g p using linear algebra and is somewhat easier to deal with than the shimura variety itself in this paper we address certain combinatorial questions which arise in the study of the special fiber m f of m to fix ideas consider the case where g is split it turns out that m f can be regarded as a finite dimensional union of iwahori orbits in the affine flag variety for g fp t see 5 and 8 it follows that m f has a stratification indexed by a finite set perm p of the extended affine weyl group w g for g equivalently perm p can be regarded as a finite set of alcoves in the affine coxeter complex determined by a splitting for g let wo denote the finite weyl group of g for each translation in the wo orbit

topological flatness of local models for ramified unitary groups i the odd dimensional case abstract local models are certain schemes defined in terms of linear algebraic moduli problems which give etale local neighborhoods of integral models of certain p adic pel shimura varieties defined by rapoport and zink when the group defining the shimura variety ramifies at p the local models and hence the shimura models as originally defined can fail to be flat and it becomes desirable to modify their definition so as to obtain a flat scheme in the case of unitary similitude groups whose localizations at q p are ramified quasi split gu n pappas and rapoport have added new conditions the so called wedge and spin conditions to the moduli problem defining the original local models and conjectured that their new local models are flat we prove a preliminary form of their conjecture namely that their new models are topologically flat in the case n is odd

computation of newton sum rules for associated and co recursive classical orthogonal polynomials abstract by using some preceding results of buendia et al in alfaro et al eds orthogonal polynomials and their applications lecture notes in mathematics vol 1329 springer berlin 1986 pp 222 235 ricci j math phys 34 1993 4884 4891 natalini calcolo 31 1994 127 144 and differential equations of associated see belmedhi ronveaux rend mat 11 1991 313 326 zarzo et al j comput appl math 49 1993 349 359 and co recursive see ronveaux marcellan j comput appl math 25 1989 105 109 ronveaux et al j comput appl 59 1995 295 328 orthogonal polynomials we obtain numerical results for newton sum rules of associated and co recursive laguerre hermite and jacobi polynomials

improving bounds for eigenvalues of complex matrices using traces abstract let a be a complex matrix of order n with eigenvalues j j 1 2 n and m be any integer satisfying rank a m n the bound for j 2 by kress de vries and wegmann is strengthened furthermore new bounds are presented to estimate the spectral radius of a using m and traces of a a 2 a a and a a aa we also improve some wolkowicz styan bounds and previous localization of eigenvalues in rectangular or elliptic regions using traces several simple lower bounds for the spectral radius are given involving tr a tr a 2 tr a 3 and m

decomposition of perturbed chebyshev polynomials we characterize polynomial decomposition f email protected q with r email protected c x of perturbed chebyshev polynomials defined by the recurrencef 0 x b f 1 x x c f n 1 x x d f n x af n 1 x n 1 where a b c email protected r and a 0 these polynomials generalize the chebyshev polynomials which are obtained by setting a 14 c d 0 and email protected 1 2 at the core of the method two algorithms for polynomial decomposition are provided which allow to restrict the investigation to the resolution of six systems of polynomial equations in three variables the final task is then carried out by the successful computation of reduced grobner bases with maple 10 some additional data for the calculations are available on the author s web page

sieved orthogonal polynomials vii generalized polynomial mappings systems of symmetric orthogonal polynomials whose recurrence relations are given by compatible blocks of second order difference equations are studied in detail applications are given to the theory of the recently discovered sieved orthogonal polynomials the connection with polynomial mappings is examined an example of a family of orthogonal polynomials having discrete masses in the interior of the spectrum is included

on the capacity of sets of divergence associated with the spherical partial integral operator in this article we study the pointwise convergence of the spherical partial integral operator s r f x b 0 r f y e 2 ix y dy when it is applied to functions with a certain amount of smoothness in particular for f l p r n n 1 2 n p 2 p 2n n 1 we prove that s r f x g g x c p quasieverywhere on r n where g l p r n is such that f g g almost everywhere a weaker version of this result in the range 0 n 1 2 as well as some related localisation principles are also obtained for 1 p 2 1 n and 0 2 p n 1 2p we construct a function f l p r n such that s r f x diverges everywhere

on co recursive orthogonal polynomials and their application to potential scattering abstract let pn x n 0 be a system of polynomials orthogonal with respect to a positive definite moment functional and satisfying the recurrence relation pn x x cn pn 1 x npn 2 x n 1 2 where p0 x 1 and p 1 x 0 the corresponding co recursive orthogonal polynomials p n x n 0 satisfy the same recurrence relations except for n 1 where now p 1 x x c 1 0 and p 0 x 1 the p n are orthogonal with respect to a moment functional which is positive definite for 0 and quasi definite for p n x separation theorems true interval of orthogonality etc can be determined from those of the pn x these polynomials occur in the l2 solution of the radial schrodinger equation for a separable potential where pn x is the tchebichef polynomial of the second kind in the case of s waves

a general method for deriving some semi classical properties of perturbed second degree forms we present a new general method and the corresponding symbolic algorithm psdf for deriving some semi classical properties of perturbed second degree forms namely the stieltjes function the stieltjes equation the functional equation the class a structure relation and the second order linear differential equation we give new explicit results for some perturbed of order 3 of the second kind chebyshev polynomials

computing the alternating trace of frobenius on the sheaves of nearby cycles on local models for gl4 and gl5 abstract consider a shimura variety defined over some number field and assume we have a model over the ring of integers at some prime of bad reduction it is then interesting to know the alternating trace of frobenius on the invariants under the inertia group of the sheaf of nearby cycles since these traces are related to the local factor of the hasse weil zeta function in this article we compute the semi simple alternating trace of frobenius for shimura varieties associated to the groups gu 2 2 and gu 3 2 with level structure of iwahori type by investigating the equations of the local model defined by rapoport and zink and by performing explicit blowing ups