fekete points and convergence towards equilibrium measures on complex manifolds building on bb1 we prove a general criterion for convergence of possibly singular bergman measures towards pluripotential theoretic equilibrium measures on complex manifolds the criterion may be formulated in terms of the growth properties of the unit balls of certain norms on holomorphic sections or equivalently as an asymptotic minimization property for generalized donaldson l functionals our result settles in particular a well known conjecture in pluripotential theory concerning the equidistribution of fekete points and it gives the convergence of bergman measures towards the equilibrium measure for bernstein markov measures applications to interpolation of holomorphic sections are also discussed

de rham complex for quantized irreducible flag manifolds it is shown that quantized irreducible flag manifolds possess a canonical q analogue of the de rham complex generalizing the well known situation for the standard podle quantum sphere this analogue is obtained as the universal differential calculus of a distinguished first order differential calculus the corresponding differential d can be written as a sum of differentials and the universal differential calculus corresponding to the first order differential calculi d and are given in terms of generators and relations relations to well known quantized exterior algebras are established the dimensions of the homogeneous components are shown to be the same as in the classical case the existence of a volume form is proven

the locally finite part of the dual coalgebra of quantized irreducible flag manifolds for quantized irreducible flag manifolds the locally finite part of the dual coalgebra is shown to coincide with a natural quotient coalgebra overline u of u q mathfrak g on the way the coradical filtration of overline u is determined a graded version of the duality between overline u and the quantized coordinate ring is established this leads to a natural construction of several examples of quantized vector spaces as an application covariant first order differential calculi on quantized irreducible flag manifolds are classified

an algorithm for computing fekete points in the triangle on the line and its tensor products fekete points are known to be the gauss lobatto quadrature points but unlike high order quadrature fekete points generalize to non tensor product domains such as the triangle thus fekete points might serve as an alternative to the gauss lobatto points for certain applications in this work we present a new algorithm to compute fekete points and give results up to degree 19 for the triangle for degree d 10 these points have the smallest lebesgue constant currently known the computations validate a conjecture of bos j approx theory 64 1991 pp 271 280 that fekete points along the boundary of the triangle are the one dimensional gauss lobatto points

le langage des espaces et des groupes quantiques we study the foundations of the differential calculus in quantum geometry the notions of differential quantum space and cone are introduced generalizing a construction of manin to a quantum cone we associate the quantum group of its linear automorphisms preserving the differentials and deduce a de rham complex on this group we give examples of differential calculi on quantum hyperplanes and quantum linear groups

a cardinal function algorithm for computing multivariate quadrature points we present a new algorithm for numerically computing quadrature formulas for arbitrary domains which exactly integrate a given polynomial space an effective method for constructing quadrature formulas has been to numerically solve a nonlinear set of equations for the quadrature points and their associated weights symmetry conditions are often used to reduce the number of equations and unknowns our algorithm instead relies on the construction of cardinal functions and thus requires that the number of quadrature points n be equal to the dimension of a prescribed lower dimensional polynomial space the cardinal functions allow us to treat the quadrature weights as dependent variables and remove them as well as an equivalent number of equations from the numerical optimization procedure we give results for the triangle where for all degrees d le 25 we find quadrature formulas of this form which have positive weights and contain no points outside the triangle seven of these quadrature formulas improve on previously known results

a numerical algorithm for the construction of efficient quadrature rules in two and higher dimensions we present a numerical algorithm for the construction of efficient high order quadratures in two and higher dimensions quadrature rules constructed via this algorithm possess positive weights and interior nodes resembling the gaussian quadratures in one dimension in addition rules can be generated with varying degrees of symmetry adaptable to individual domains we illustrate the performance of our method with numerical examples and report quadrature rules for polynomials on triangles squares and cubes up to degree 50 these formulae are near optimal in the number of nodes used and many of them appear to be new

on braiding and dyslexia abstract braided monoidal categories have important applications in knot theory algebraic quantum field theory and the theory of quantum groups and hopf algebras we will construct a new class of braided monoidal categories typical examples of braided monoidal categories are the category of modules over a quasitriangular hopf algebra and the category of comodules over a coquasitriangular hopf algebra we consider the notion of a commutative algebra a in such a category the category of left and or right a modules with the tensor product over a is again a monoidal category which is not necessarily braided however if we restrict this category to a special class of modules which we call dyslectic then this new category of dyslectic a modules turns out to be a braided monoidal category too and it is a coreflexive subcategory of all a modules

resolutions and hilbert series of determinantal varieties and unitary highest weight modules abstract let g k be a hermitian symmetric pair and let g and k denote the corresponding complexified lie algebras let g k p p be the usual decomposition of g as a k module there is a natural correspondence between k c orbits in p and a distinguished family of unitarizable highest weight modules for g called the wallach representations we denote by yk the closure of the k c orbit in p that is associated to the kth wallach representation in this article we give explicit formulas for the numerator polynomials of the hilbert series of the varieties yk by using bgg resolutions of unitarizable highest weight modules a preliminary result gives a new branching formula for a certain two parameter family of finite dimensional representations of the even orthogonal groups our work is an extension of previous work by enright and willenbring

differential calculi on some quantum prehomogeneous vector spaces this paper is devoted to study of differential calculi over quadratic algebras which arise in the theory of quantum bounded symmetric domains we prove that in the quantum case dimensions of the homogeneous components of the graded vector spaces of k forms are the same as in the classical case this result is well known for quantum matrices the quadratic algebras which we consider in the present paper are q analogs of the polynomial algebras on prehomogeneous vector spaces of commutative parabolic type this enables us to prove that the de rham complex is isomorphic to the dual of a quantum analog of the generalized bernstein gelfand gelfand resolution lie goups and their representations summer school of the bolyai janos mathematical society edited by i m gelfand halsted new york 1975 pp 21 64

generalized gaussian quadrature rules on arbitrary polygons in this paper we present a numerical algorithm based on group theory and numerical optimization to compute efficient quadrature rules for integration of bivariate polynomials over arbitrary polygons these quadratures have desirable properties such as positivity of weights and interiority of nodes and can readily be used as software libraries where numerical integration within planar polygons is required we have used this algorithm for the construction of symmetric and non symmetric quadrature rules over convex and concave polygons while in the case of symmetric quadratures our results are comparable to available rules the proposed algorithm has the advantage of being flexible enough so that it can be applied to arbitrary planar regions for the integration of generalized classes of functions to demonstrate the efficiency of the new quadrature rules we have tested them for the integration of rational polygonal shape functions over a regular hexagon for a relative error of 10 8 in the computation of stiffness matrix entries one needs at least 198 evaluation points when the region is partitioned whereas 85 points suffice with our quadrature rule copyright 2009 john wiley sons ltd

computing approximate fekete points by qr factorizations of vandermonde matrices we propose a numerical method implemented in matlab for computing approximate fekete points on compact multivariate domains it relies on the search of maximum volume submatrices of vandermonde matrices computed on suitable discretization meshes and uses a simple greedy algorithm based on qr factorization with column pivoting the method gives also automatically an algebraic cubature formula provided that the moments of the underlying polynomial basis are known numerical tests are presented for the interval and the square which show that approximate fekete points are well suited for polynomial interpolation and cubature

rounding error analysis of the classical gram schmidt orthogonalization process this paper provides two results on the numerical behavior of the classical gram schmidt algorithm the first result states that provided the normal equations associated with the initial vectors are numerically nonsingular the loss of orthogonality of the vectors computed by the classical gram schmidt algorithm depends quadratically on the condition number of the initial vectors the second result states that provided the initial set of vectors has numerical full rank two iterations of the classical gram schmidt algorithm are enough for ensuring the orthogonality of the computed vectors to be close to the unit roundoff level

polynomial interpolation and cubature over polygons we have implemented a matlab code to compute discrete extremal sets of fekete and leja type on convex or concave polygons together with the corresponding interpolatory cubature formulas the method works by qr and lu factorizations of rectangular vandermonde matrices on weakly admissible meshes wams of polygons constructed by polygon quadrangulation

geometric weakly admissible meshes discrete least squares approximations and approximate fekete points using the concept of geometric weakly admissible meshes see 2 below together with an algorithm based on the classical qr factorization of matrices we compute efficient points for discrete multivariate least squares approximation and lagrange interpolation

numerical integration over polygons using an eight node quadrilateral spline finite element in this paper a cubature formula over polygons is proposed and analysed it is based on an eight node quadrilateral spline finite element c j li r h wang a new 8 node quadrilateral spline finite element j comp appl math 195 2006 54 65 and is exact for quadratic polynomials on arbitrary convex quadrangulations and for cubic polynomials on rectangular partitions the convergence of sequences of the above cubatures is proved for continuous integrand functions and error bounds are derived some numerical examples are given by comparisons with other known cubatures

stratified nested and related quadrature rules the stratified nested quadrature procedure due to laurie is discussed together with an alternative computational procedure which leads to the concept of hybrid gkp rules in the context of the approximation of stratified nested sequences the work of krogh and van snyder on the representation of the gkp rules is considered and a generalisation of this employing hybrid rules of special form is discussed

on selecting a maximum volume sub matrix of a matrix and related problems given a matrix email protected r m x n n vectors in m dimensions we consider the problem of selecting a subset of its columns such that its elements are as linearly independent as possible this notion turned out to be important in low rank approximations to matrices and rank revealing qr factorizations which have been investigated in the linear algebra community and can be quantified in a few different ways in this paper from a complexity theoretic point of view we propose four related problems in which we try to find a sub matrix email protected r m x k of a given matrix email protected r m x n such that i s m a x c the largest singular value of c is minimum ii s m i n c the smallest singular value of c is maximum iii k c email protected m a x c s m i n c the condition number of c is minimum and iv the volume of the parallelepiped defined by the column vectors of c is maximum we establish the np hardness of these problems and further show that they do not admit ptas we then study a natural greedy heuristic for the maximum volume problem and show that it has approximation ratio 2 o k l o g k our analysis of the greedy heuristic is tight to within a logarithmic factor in the exponent which we show by explicitly constructing an instance for which the greedy heuristic is 2 w k from optimal when a has unit norm columns a related problem is to select the maximum number of vectors with a given volume we show that if the optimal solution selects k columns then greedy will select w k logk columns providing a logk approximation

numerical integration over arbitrary polygonal domains based on schwarz christoffel conformal mapping this paper presents a new numerical integration technique oil arbitrary polygonal domains the polygonal domain is mapped conformally to the unit disk using schwarz christoffel mapping and a midpoint quadrature rule defined oil this unit disk is used this method eliminates the need for a two level isoparametric mapping usually required moreover the positivity of the jacobian is guaranteed numerical results presented for a few benchmark problems in the context of polygonal finite elements show that the proposed method yields accurate results

tensor product gauss lobatto points are fekete points for the cube tensor products of gauss lobatto quadrature points are frequently used as collocation points in spectral element methods unfortunately it is not known if gauss lobatto points exist in non tensor product domains like the simplex in this work we show that the n dimensional tensor product of gauss lobatto quadrature points are also fekete points this suggests a way to generalize spectral methods based on gauss lobatto points to non tensor product domains since fekete points are known to exist and have been computed in the triangle and tetrahedron in one dimension this result was proved by fejer in 1932 but the extension to higher dimensions in non trivial

least squares polynomial approximation on weakly admissible meshes disk and triangle we construct symmetric polar wams weakly admissible meshes with low cardinality for least squares polynomial approximation on the disk these are then mapped to an arbitrary triangle numerical tests show that the growth of the least squares projection uniform norm is much slower than the theoretical bound and even slower than that of the lebesgue constant of the best known interpolation points for the triangle as opposed to good interpolation points such meshes are straightforward to compute for any degree the construction can be extended to polygons by triangulation

computing multivariate fekete and leja points by numerical linear algebra we discuss and compare two greedy algorithms that compute discrete versions of fekete like points for multivariate compact sets by basic tools of numerical linear algebra the first gives the so called approximate fekete points by qr factorization with column pivoting of vandermonde like matrices the second computes discrete leja points by lu factorization with row pivoting moreover we study the asymptotic distribution of such points when they are extracted from weakly admissible meshes

hidden symmetry of the differential calculus on the quantum matrix space a standard bicovariant differential calculus on the quantum matrix space is considered our main result is proving that the module differential algebra is in fact a module differential algebra