Orthonormal rational function vectors

Summary In this paper, we develop a matrix framework to solve the problem of finding orthonormal rational function vectors with prescribed poles with respect to a certain discrete inner product that is defined by a set of data points and corresponding weight vectors w_i,j. Our algorithm for solving the problem is recursive, and it is of complexity If all data points are real or lie on the unit circle, then the complexity is reduced by an order of magnitude.     

Multi-level spectral galerkin method for the navier-stokes problem I : spatial discretization

A multi-level spectral Galerkin method for the two-dimensional non-stationary Navier-Stokes equations is presented. The method proposed here is a multiscale method in which the fully nonlinear Navier-Stokes equations are solved only on a low-dimensional space subsequent approximations are generated on a succession of higher-dimensional spaces j=2, . . . ,J, by solving a linearized Navier-Stokes problem around the solution on the previous level. Error estimates depending on the kinematic viscosity 0<ν<1 are also presented for the J-level spectral Galerkin method. The optimal accuracy is achieved when We demonstrate theoretically that the J-level spectral Galerkin method is much more efficient than the standard one-level spectral Galerkin method on the highest-dimensional space . The work of this author was supported in part by the NSF of China 10371095, City University of Hong Kong Research Project 7001093 Hong Kong and the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. CityU 1084/02P)     

Besov regularity and new error estimates for finite volume approximations of the p-laplacian

Summary. In [1], we have constructed a family of finite volume schemes on rectangular meshes for the p-laplacian and we proved error estimates in case the exact solution lies in W^2,p. Actually, W^2,p is not a natural space for solutions of the p-laplacian in the case p>2. Indeed, for general L^p data it can be shown that the solution only belongs to the Besov space In this paper, we prove Besov kind a priori estimates on the approximate solution for any data in L^p. We then obtain new error estimates for such solutions in the case of uniform meshes     

The Euler characteristic of local systems on the moduli of genus 3 hyperelliptic curves

For a partition ={₁₂₃0} of non-negative integers, we calculate the Euler characteristic of the local system on the moduli space of genus 3 hyperelliptic curves using a suitable stratification. For some of low degree, we make a guess for the motivic Euler characteristic of using counting curves over finite fields.Mathematics Subject Classification (1991): 14J15, 20B25     

Interior penalty method for the indefinite time-harmonic Maxwell equations

Summary In this paper, we introduce and analyze the interior penalty discontinuous Galerkin method for the numerical discretization of the indefinite time-harmonic Maxwell equations in the high-frequency regime. Based on suitable duality arguments, we derive a-priori error bounds in the energy norm and the L²-norm. In particular, the error in the energy norm is shown to converge with the optimal order (h^min{s,}) with respect to the mesh size h, the polynomial degree , and the regularity exponent s of the analytical solution. Under additional regularity assumptions, the L²-error is shown to converge with the optimal order (h⁺¹). The theoretical results are confirmed in a series of numerical experiments.Supported by the EPSRC (Grant GR/R76615).Supported by the Swiss National Science Foundation under project 21-068126.02.Supported in part by the Natural Sciences and Engineering Council of Canada.     

Order and algebra isomorphisms of spaces of regular operators

If E and F are real Banach lattices and there is an algebra and order isomorphism Φ:(E) →(F) between their respective ordered Banach algebras of regular operators then there is a linear order isomorphism U:E→F such that Φ(T) =UTU⁻¹ for all T ∈ (E).     

Preconditioning Landweber iteration in Hilbert scales

In this paper we investigate convergence of Landweber iteration in Hilbert scales for linear and nonlinear inverse problems. As opposed to the usual application of Hilbert scales in the framework of regularization methods, we focus here on the case s≤0, which (for Tikhonov regularization) corresponds to regularization in a weaker norm. In this case, the Hilbert scale operator L^−2s appearing in the iteration acts as a preconditioner, which significantly reduces the number of iterations needed to match an appropriate stopping criterion. Additionally, we carry out our analysis under significantly relaxed conditions, i.e., we only require instead of which is the usual condition for regularization in Hilbert scales. The assumptions needed for our analysis are verified for several examples and numerical results are presented illustrating the theoretical ones. supported by the Austrian Science Foundation (FWF) under grant SFB/F013     

Hausdorff measures, capacities and compact composition operators

It is shown that there exist analytic self-maps ϕ of the unit disc inducing compact composition operators on the Hardy space , 1 ≤ p < ∞ such that the Hausdorff dimension of the set is one; sharpening a classical result due to Schwartz. Moreover, the same holds in the weighted Dirichlet spaces with 0 < α < 1. As a consequence, we deduce that there exist symbols ϕ inducing compact composition operators on such that the α-capacity of E_ϕ is positive, which is no longer true for those just inducing Hilbert-Schmidt composition operators on . First author is partially supported by Plan Nacional I+D grant no. BFM2003-00034, and Gobierno de Aragón research group Análisis Matemático y Aplicaciones, ref. DGA E-64 . Second author is partially supported by Plan Nacional I+D grant no. BFM2002-00571 and Junta de Andalucía RNM-314.     

2-universal -lattices over real quadratic fields

Let be a real quadratic field with m a square-free positive rational integer, and be the ring of integers in F. An -lattice L on a totally positive definite quadratic space V over F is called r-universal if L represents all totally positive definite -lattices l with rank r over . We prove that there exists no 2-universal -lattice over F with rank less than 6, and there exists a 2-universal -lattice over F with rank 6 if and only if m=2, 5. Moreover there exists only one 2-universal -lattice with rank 6, up to isometry, over .     

Rate of convergence of regularization procedures and finite element approximations for the total variation flow

Summary We derive rates of convergence for regularization procedures (characterized by a parameter ) and finite element approximations of the total variation flow, which arises from image processing, geometric analysis and materials sciences. Practically useful error estimates, which depend on only in low polynomial orders, are established for the proposed fully discrete finite element approximations. As a result, scaling laws which relate mesh parameters to the regularization parameter are also obtained. Numerical experiments are provided to validate the theoretical results and show efficiency of the proposed numerical methods.     

Fast Runge-Kutta approximation of inhomogeneous parabolic equations

The result after N steps of an implicit Runge-Kutta time discretization of an inhomogeneous linear parabolic differential equation is computed, up to accuracy ɛ, by solving only linear systems of equations. We derive, analyse, and numerically illustrate this fast algorithm.     

New shape functions for triangular p-FEM using integrated Jacobi polynomials

In this paper, the second order boundary value problem −∇·((x,y)∇u)=f is discretized by the Finite Element Method using piecewise polynomial functions of degree p on a triangular mesh. On the reference element, we define integrated Jacobi polynomials as interior ansatz functions. If is a constant function on each triangle and each triangle has straight edges, we prove that the element stiffness matrix has not more than nonzero matrix entries. An application for preconditioning is given. Numerical examples show the advantages of the proposed basis.     

On a class of weak Tchebycheff systems

In this paper we study the approximation power, the existence of a normalized B-basis and the structure of a degree-raising process for spaces of the formrequiring suitable assumptions on the functions u and v. The results about degree raising are detailed for special spaces of this form which have been recently introduced in the area of CAGD.     

Commutativity of Almost <Emphasis Type="Italic">F</Emphasis>-algebras

Let A be an Archimedean vector lattice, let be its Dedekind completion and let B be a Dedekind complete vector lattice. If Ψ ₀:A × A→ B is a positive orthosymmetric bimorphism, then there exists a positive bimorphism extension Ψ of Ψ ₀ to × in B which is orthosymmetric. This leads to a new and short proof of the commutativity of the almost f-algebras multiplications.     

Ribbon tableaux and the Heisenberg algebra

In [LLT] Lascoux, Leclerc and Thibon introduced symmetric functions which are spin and weight generating functions for ribbon tableaux. This article is aimed at studying these functions in analogy with Schur functions. In particular we will describe: a Pieri and dual-Pieri formula for ribbon functions, a ribbon Murnaghan-Nakayama formula, ribbon Cauchy and dual Cauchy identities, and a -algebra isomorphism _n:(q)(q) which sends each to .Our study of the functions will be connected to the Fock space representation F of via a linear map :F(q) which sends the standard basis of F to the ribbon functions. Kashiwara, Miwa and Stern [KMS] have shown that a copy of the Heisenberg algebra H acts on F commuting with the action of . Identifying the Fock Space of H with the ring of symmetric functions (q) we will show that is in fact a map of H-modules with remarkable properties. The study of this map will lead to our identities concerning ribbon tableaux generating functions. We will also give a combinatorial proof that the ribbon Murnaghan-Nakayama and Pieri rules are formally equivalent.     

Holomorphic dynamics near germs of singular curves

Let M be a two dimensional complex manifold, p ∈ M and a germ of holomorphic foliation of M at p. Let be a germ of an irreducible, possibly singular, curve at p in M which is a separatrix for . We prove that if the Camacho-Sad-Suwa index Ind then there exists another separatrix for at p. A similar result is proved for the existence of parabolic curves for germs of holomorphic diffeomorphisms near a curve of fixed points.     

The cofinality of the saturated uncountable random graph

Assuming CH, let be the saturated random graph of cardinality ₁. In this paper we prove that it is consistent that and can be any two prescribed regular cardinals subject only to the requirement      

Hybrid cross approximation of integral operators

The efficient treatment of dense matrices arising, e.g., from the finite element discretisation of integral operators requires special compression techniques. In this article we use the -matrix representation that approximates the dense stiffness matrix in admissible blocks (corresponding to subdomains where the underlying kernel function is smooth) by low-rank matrices. The low-rank matrices are assembled by a new hybrid algorithm (HCA) that has the same proven convergence as standard interpolation but also the same efficiency as the (heuristic) adaptive cross approximation (ACA).     

On the approximation order of extremal point methods for hyperbolic minimal energy problems

Summary. Let be an analytic Jordan curve in the unit disk We regard the hyperbolic minimal energy problem where () denotes the set of all probability measures on . There exist several extremal point discretizations of *, among others introduced by M. Tsuji (Tsuji points) or by K. Menke (hyperbolic Menke points). In the present article, it is proven that hyperbolic Menke points approach the images of roots of unity under a conformal map from onto geometrically fast if the number of points tends to infinity. This establishes a conjecture of K. Menke. In particular, explicit bounds for the approximation error are given. Finally, an effective method for the numerical determination of * providing a geometrically shrinking error bound is presented.Mathematics Subject Classification (1991): 30C85, 30E10, 31C20The notation Menke points has been introduced by D. Gaier.     

The Rees algebra of a positive normal affine semigroup ring

Let R be a positive normal affine semigroup ring of dimension d and let be the maximal homogeneous ideal of R. We show that the integral closure of is equal to for all n ∈ℕ with n ≥ d − 2. From this we derive that the Rees algebra R[t] is normal in case that d ≤ 3. If emb dim(R) = d + 1, we can give a necessary and sufficient condition for R[t] to be normal.