Fast convolution with radial kernels at nonequispaced knots

Summary. We develop a new algorithm for the fast evaluation of linear combinations of radial functions based on the recently developed fast Fourier transform at nonequispaced knots. For smooth kernels, e.g. the Gaussian, our algorithm requires arithmetic operations. In case of singular kernels an additional regularization procedure must be incorporated and the algorithm has the arithmetic complexity if either the points y_j or the points x_k are reasonably uniformly distributed. We prove error estimates to obtain clues about the choice of the involved parameters and present numerical examples for various singular and smooth kernels in two dimensions.Mathematics Subject Classification (2000): 65T40, 65T50, 65F30Revised version received December 3, 2003     

On an eigenvalue problem related to the critical exponent

In this paper we study the eigenvalue problemwhere is a smooth bounded domain, and u is a positive solution of the problemsuch thatwhere S is the best Sobolev constant for the embedding of H¹₀() into L²*(), We prove several estimates for the eigenvalues _i, of (I), i=2,..,N+2 and some qualitative properties of the corresponding eigenfunctions.Supported by M.I.U.R., project Variational methods and nonlinear differential equations.     

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      

Infinitely many solutions of a semilinear problem for the Heisenberg Laplacian on the Heisenberg group

We study the semilinear equationwhere is the Heisenberg Laplacian and is the Heisenberg group. The function f C²(×, ) is supposed to satisfy some (subcritical) growth conditions and to be left invariant under the action of the subgroup of consisting of points with integer coordinates.. We show the existence of infinitely many solutions in the space S₁²(), which is the Heisenberg analogue of the Sobolev space W^1,2(^N).Mathematics Subject Classification (2000): 22E30, 22E27     

Diophantine inequalities involving several power sums

Let denote the ring of power sums, i.e. complex functions of the form for some and _iA, where is a multiplicative semigroup. Moreover, let We consider Diophantine inequalities of the form where >1 is a quantity depending on the dominant roots of the power sums appearing as coefficients in F(n,y), and show that all its solutions have y parametrized by some power sums from a finite set. This is a continuation of the work of Corvaja and Zannier [4–6] and of the authors [10, 18] on such problems.Mathematics Subject Classification (2000):11D45,11D61Revised version: 6 May 2004     

Estimates for the -Neumann problem and nonexistence of C 2 Levi-flat hypersurfaces in

Let be a pseudoconvex domain with C² boundary in , n 2. We prove that the -Neumann operator N exists for square-integrable forms on . Furthermore, there exists a number ₀>0 such that the operators and the Bergman projection are regular in the Sobolev space W ( ) for <₀. The -Neumann operator is used to construct -closed extension on for forms on the boundary b. This gives solvability for the tangential Cauchy-Riemann operators on the boundary. Using these results, we show that there exist no non-zero L²-holomorphic (p, 0)-forms on any domain with C² pseudoconcave boundary in with p > 0 and n 2. As a consequence, we prove the nonexistence of C² Levi-flat hypersurfaces in .This paper is a revision of our preprint (May 2003) formerly titled Estimates for the -Neumann problem and nonexistence of Levi-flat hypersurfaces in where the nonexistence of C², Levi-flat hypersurfaces is proved for >0.All three authors are partially supported by NSF grants.An erratum to this article can be found at     

On the logarithmic plurigenera of complements of plane curves

Let B be a (not necessarily irreducible) plane curve in ². In the present article, we prove that if and only if Moreover, we determine the curve B when and Mathematics Subject Classification (2000): 14R05, 14H50, 14J26     

A new condensation principle

We generalize (A), which was introduced in [Sch], to larger cardinals. For a regular cardinal >₀ we denote by (A) the statement that and for all regular >,is stationary in It was shown in [Sch] that can hold in a set-generic extension of L. We here prove that can hold in a set-generic extension of L as well. In both cases we in fact get equiconsistency theorems. This strengthens results of [Rä00] and [Rä01]. is equivalent with the existence of 0^#.Mathematics Subject Classification (1991): Primary 03E55, 03E15, Secondary 03E35, 03E60     

Integral varieties of the canonical cone structure on <Emphasis Type="Italic">G</Emphasis>/<Emphasis Type="Italic">P</Emphasis>

The canonical cone structure on a compact Hermitian symmetric space G/P is the fiber bundle where is the cone of the highest weight vectors under the action of the reductive part of P. It is known that the cone coincides with the cone of the vectors tangent to the lines in G/P passing through x, when we consider G/P as a projective variety under its homogeneous embedding into the projective space of the irreducible representation space V of G with highest weight associated to P. A subvariety X of G/P is said to be an integral variety of at all smooth points xG/P. Equivalently, an integral variety of is a subvariety of G/P whose embedded projective tangent space at each smooth point is a linear space We prove a kind of rigidity of the integral varieties under some dimension condition. After making a uniform setting to study the problem, we apply the theory of Lie algebra cohomology as a main tool. Finally we show that the dimension condition is necessary by constructing counterexamples.     

Approximation of Integral Operators by Variable-Order Interpolation

Summary. We employ a data-sparse, recursive matrix representation, so-called -matrices, for the efficient treatment of discretized integral operators. We obtain this format using local tensor product interpolants of the kernel function and replacing high-order approximations with piecewise lower-order ones. The scheme has optimal, i.e., linear, complexity in the memory requirement and time for the matrix-vector multiplication. We present an error analysis for integral operators of order zero. In particular, we show that the optimal convergence (h) is retained for the classical double layer potential discretized with piecewise constant functions.Corrigendum This revised version was published online in February 2005 due to typesetting mistakes in the author correction process.     

On certain categories of modules for affine Lie algebras

In this paper, we exploit basic formal variable techniques to study certain categories of modules for an (untwisted) affine Lie algebra , motivated by Chari-Pressleys work on certain integrable modules. We define and study two categories and of -modules using generating functions, where is proved to contain the well known evaluation modules and to unify highest weight modules, evaluation modules and their tensor product modules. We classify integrable irreducible -modules in categories and and we determine the isomorphism classes of those irreducible modules. Finally we prove a result that relates fusion rules in the context of vertex operator algebras with integrable irreducible modules of Chari-Pressley.in final form: 12 November 2003Partially supported by a NSA grant and a grant from Rutgers Research Council.     

Associated primes and cofiniteness of local cohomology modules

Let be an ideal of Noetherian ring R and let s be a non-negative integer. Let M be an R-module such that is finite R-module. If s is the first integer such that the local cohomology module is non -cofinite, then we show that is finite. In particular, the set of associated primes of is finite. Let be a local Noetherian ring and let M be a finite R-module. We study the last integer n such that the local cohomology module is not -cofinite and show that n just depends on the support of M.The research of the first author was supported in part by a grant from IPM (No. 83130114).The second author was supported by a grant from University of Tehran (No. 6103023/1/01).     

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.     

Topologically transitive extensions of bounded operators

Let X be any Banach space and T a bounded operator on X. An extension of the pair (X,T) consists of a Banach space in which X embeds isometrically through an isometry i and a bounded operator on such that When X is separable, it is additionally required that be separable. We say that is a topologically transitive extension of (X, T) when is topologically transitive on , i.e. for every pair of non-empty open subsets of there exists an integer n such that is non-empty. We show that any such pair (X,T) admits a topologically transitive extension , and that when H is a Hilbert space, (H,T) admits a topologically transitive extension where is also a Hilbert space. We show that these extensions are indeed chaotic.Mathematics Subject Classification (2000): 47 A 16     

<Emphasis Type="Italic">BMO</Emphasis> spaces related to Schrödinger operators with potentials satisfying a reverse Hölder inequality

We identify the dual space of the Hardy-type space related to the time independent Schrödinger operator =–+V, with V a potential satisfying a reverse Hölder inequality, as a BMO-type space . We prove the boundedness in this space of the versions of some classical operators associated to (Hardy-Littlewood, semigroup and Poisson maximal functions, square function, fractional integral operator). We also get a characterization of in terms of Carlesson measures.Mathematics Subject Classification (2000): 35J10, 42B35, 42B30This research was partially supported by the European Commission, within the IHP Network HARP 2002-2006, contract number HPRN-CT-2001-00273-HARP. Second author also supported by Programa Ramón y Cajal and grant BMF2001-0189 MCyT (Spain). The first and the fifth author supported by grants 5P03A05020 and 5P03A02821 from KBN (Poland) and by Foundation of Polish Sciences Subsidy 3/99. The third and the fourth author supported by grant MCyT BMF2002-04013-C02-02.     

Analysis of the -Neumann problem along a straight edge

We show there exists an L^p solution, for p (2,), to the -Neumann problem on an edge domain in ² for (0,1)-forms, and we explicitly compute the singularities, which are of complex logarithmic and arctangent type, along the edge, of the solution.Mathematics Subject Classification (2000): 32W05, 35B65     

A class of <Emphasis Type="Italic">p</Emphasis>-adic Galois representations arising from abelian varieties over

Let V be a p-adic representation of the absolute Galois group G of that becomes crystalline over a finite tame extension, and assume p2. We provide necessary and sufficient conditions for V to be isomorphic to the p-adic Tate module V_p() of an abelian variety defined over . These conditions are stated on the filtered (,G)-module attached to V.Mathematics Subject Classification (2000): 14F30, 11G10, 11F80, 14G20, 14F20     

L p -Estimates for the Navier–Stokes Equations for Steady Compressible Flow

We consider the Navier–Stokes equations for compressible isentropic flow in the steady three-dimensional case. The pressure and the kinetic energy are estimated uniformly in L^q with being the density. This is an improvement of known estimates in the case Mathematics Subject Classification (2000): 35Q30, 76N10     

The optimal rate of convergence of the <Emphasis Type="Italic">p</Emphasis>-version of the boundary element method in two dimensions

Summary. We introduce the Jacobi-weighted Besov and Sobolev spaces in the one-dimensional setting. In the framework of these spaces, we analyze lower and upper bounds for approximation errors in the p-version of the boundary element method for hypersingular and weakly singular integral operators on polygons. We prove the optimal rate of convergence for the p-version in the energy norms of and respectively.Mathematics Subject Classification (2000): 65N38This author is supported by NSERC of Canada under Grant OGP0046726 and partially supported by the FONDAP Program (Chile) on Numerical Analysis during his visit of the Universidad de Concepción in 2001.This author is supported by Fondecyt project no. 1010220 and by the FONDAP Program (Chile) on Numerical Analysis.Revised version received January 28, 2004     

Modified Ariki-Koike algebras and cyclotomic <Emphasis Type="Italic">q</Emphasis>-Schur algebras

The modified Ariki-Koike algebra is a variation of the original Ariki-Koike algebra over an integral domain R. When R is a rational function field over the independent parameters, But for general R, is not isomorphic to , and has a simpler structure than . In this paper, we construct a cellular basis of which has a similar property as the cellular basis of introduced by Dipper-James-Mathas. By comparing these two cellular bases, we obtain some estimate on the decomposition numbers of in terms of the decomposition numbers of . We also prove the integral form of the Schur-Weyl reciprocity between a certain quantum algebra U_q and on the tensor space