Affine pseudo-planes with torus actions

An affine pseudo-plane X is a smooth affine surface defined over which is endowed with an -fibration such that every fiber is irreducible and only one fiber is a multiple fiber. If there is a hyperbolic -action on X and X is an -surface, we shall show that the universal covering is isomorphic to an affine hypersurface in the affine 3-space and X is the quotient of by the cyclic group via the action where and It is also shown that a -homology plane X with and a nontrivial -action is an affine pseudo-plane. The automorphism group is determined in the last section.     

Standard monomial bases, Moduli spaces of vector bundles, and Invariant theory

Consider the diagonal action of on the affine space where an algebraically closed field of characteristic We construct a "standard monomial" basis for the ring of invariants As a consequence, we deduce that is Cohen-Macaulay. As the first application, we present the first and second fundamental theorems for -actions. As the second application, assuming that the characteristic of K is we give a characteristic-free proof of the Cohen-Macaulayness of the moduli space of equivalence classes of semi-stable, rank 2, degree 0 vector bundles on a smooth projective curve of genus > 2. As the third application, we describe a K-basis for the ring of invariants for the adjoint action of on m copies of in terms of traces.     

Yangians and Mickelsson Algebras I

We study the composition of the functor from the category of modules over the Lie algebra to the category of modules over the degenerate affine Hecke algebra of GL_N introduced by I. Cherednik, with the functor from the latter category to the category of modules over the Yangian due to V. Drinfeld. We propose a representation theoretic explanation of a link between the intertwining operators on the tensor products of -modules, and the "extremal cocycle" on the Weyl group of defined by D. Zhelobenko. We also establish a connection between the composition of the functors, and the "centralizer construction" of the Yangian discovered by G. Olshanski.     

Intersection pairings on singular moduli spaces of bundles over a Riemann surface and their partial desingularisations

This paper studies intersection theory on the compactified moduli space of holomorphic bundles of rank n and degree d over a fixed compact Riemann surface of genus where n and d may have common factors. Because of the presence of singularities we work with the intersection cohomology groups defined by Goresky and MacPherson and the ordinary cohomology groups of a certain partial resolution of singularities of Based on our earlier work [25], we give a precise formula for the intersection cohomology pairings and provide a method to calculate pairings on The case when n = 2 is discussed in detail. Finally Witten's integral is considered for this singular case.     

Generating Self-Map Monoids of Infinite Sets

Let be a countably infinite set, the group of permutations of , and the monoid of self-maps of . Given two subgroups , let us write if there exists a finite subset such that the groups generated by and are equal. Bergman and Shelah showed that the subgroups which are closed in the function topology on S fall into exactly four equivalence classes with respect to . Letting denote the obvious analog of for submonoids of E, we prove an analogous result for a certain class of submonoids of E, from which the theorem for groups can be recovered. Along the way, we show that given two subgroups which are closed in the function topology on S, we have if and only if (as submonoids of E), and that for every subgroup (where denotes the closure of G in the function topology in S and its closure in the function topology in E).     

Approximation and Spanning in the Hardy Space,by Affine Systems

We show that every function in the Hardy space can be approximated by linear combinations of translates and dilates of a synthesizer , provided only that and satisfies a mild regularity condition. Explicitly, we prove scale averaged approximation for each ,

Fourier Inequalities and Moment Subspaces in Weighted Lebesgue Spaces

For define where Pointwise estimates and weighted inequalities describing the local Lipschitz continuity of are established. Sufficient conditions are found for the boundedness of from into and a spherical restriction property is proved. A study of the moment subspaces of is next developed in the one-variable case, for locally integrable, a.e. It includes a decomposition theorem and a complete classification of all possible sequences of moment subspaces in Characterizations are also given for each class. Applications related to the approximation and decomposition of are discussed.     

Distributional Point Values and Convergence of Fourier Series and Integrals

In this article we show that the distributional point values of a tempered distribution are characterized by their Fourier transforms in the following way: If and , and is locally integrable, then distributionally if and only if there exists k such that , for each a > 0, and similarly in the case when is a general distribution. Here means in the Cesaro sense. This result generalizes the characterization of Fourier series of distributions with a distributional point value given in [5] by . We also show that under some extra conditions, as if the sequence belongs to the space for some and the tails satisfy the estimate ,\ as , the asymmetric partial sums\ converge to . We give convergence results in other cases and we also consider the convergence of the asymmetric partial integrals. We apply these results to lacunary Fourier series of distributions.     

Kernel and Trace Operators for Extensions of Brandt Semigroups

Let S be an (ideal) extension of a Brandt semigroup S₀ by a Brandt semigroup S₁ and let denote the congruence lattice of S. For denote by and the least and the greatest congruences on S with the same kernel as respectively, and let and have the analogous meaning relative to trace. We establish necessary and sufficient conditions on S in order that one or more of the operators

The Uniqueness of Shift-Generated Duals for Frames in Shift-Invariant Subspaces

Given an invertible matrix B and a finite or countable subset of , we consider the collection generating the closed subspace of . If that collection forms a frame for , one can introduce two different types of shift-generated (SG) dual frames for X, called type I and type II SG-duals, respectively. The main distinction between them is that a SG-dual of type I is required to be contained in the space generated by the original frame while, for a type II SG-dual, one imposes that the range of the frame transform associated with the dual be contained in the range of the frame transform associated with the original frame. We characterize the uniqueness of both types of duals using the Gramian and dual Gramian operators which were introduced in an article by Ron and Shen and are known to play an important role in the theory of shift-invariant spaces.     

Counting components ofthe null-cone on tuples

For a finite-dimensional representation of a group G, the diagonal action of G on p-tuples of elements of M, is usually poorly understood. The algorithm presented here computes a geometric characteristic of this action in the case where G is connected and reductive, and is a morphism of algebraic groups: The algorithm takes as input the weight system of M, and it returns the number of irreducible components of the null-cone of G on for large p. The paper concludes with a theorem that if the characteristic is zero and G is semisimple, then only few M have the property that is small for all p.     

Coefficients of Orthogonal Polynomials on the Unit Circle and Higher-Order Szego Theorems

Let be a nontrivial probability measure on the unit circle the density of its absolutely continuous part, its Verblunsky coefficients, and its monic orthogonal polynomials. In this paper we compute the coefficients of in terms of the . If the function is in , we do the same for its Fourier coefficients. As an application we prove that if and if is a polynomial, then with and S the left-shift operator on sequences we have

Numerical Integration over Spheres of Arbitrary Dimension

In this paper we study the worst-case error (of numerical integration) on the unit sphere for all functions in the unit ball of the Sobolev space where More precisely, we consider infinite sequences of m(n)-point numerical integration rules where: (i) is exact for all spherical polynomials of degree and (ii) has positive weights or, alternatively to (ii), the sequence satisfies a certain local regularity property. Then we show that the worst-case error (of numerical integration) in has the upper bound where the constant c depends on s and d (and possibly the sequence This extends the recent results for the sphere by K. Hesse and I.H. Sloan to spheres of arbitrary dimension by using an alternative representation of the worst-case error. If the sequence of numerical integration rules satisfies an order-optimal rate of convergence is achieved.     

Interassociates of the Free Commutative Semigroup on n Generators

The interassociates of the free commutative semigroup on n generators, for n > 1, are identified. For fixed n, let (S, ·) denote this semigroup. We show that every interassociate can be written in the form , depending only on a n-tuple . Next, if and are isomorphic interassociates of (S, ·) such that , for x_ii and x_j in the generating set of S, then . Moreover, if and only if is a permutation of .     

Duality and Biorthogonality for Weyl-Heisenberg Frames

Let and let In this paper we investigate the relation between the frame operator and the matrix whose entries are given by for Here , for any We show that is bounded as a mapping of into if and only if is bounded as a mapping of into Also we show that if and only if where denotes the identity operator of and respectively, and Next, when generates a frame, we have that has an upper frame bound, and the minimal dual function can be computed as The results of this paper extend, generalize, and rigourize results of Wexler and Raz and of Qian, D. Chen, K. Chen, and Li on the computation of dual functions for finite, discrete-time Gabor expansions to the infinite, continuous-time case. Furthermore, we present a framework in which one can show that certain smoothness and decay properties of a generating a frame are inherited by In particular, we show that when generates a frame Schwartz space). The proofs of the main results of this paper rely heavily on a technique introduced by Tolimieri and Orr for relating frame bound questions on complementary lattices by means of the Poisson summation formula.     

Homogenization of Non-Linear Variational Problems with Thin Low-Conducting Layers

This paper deals with the homogenization of a sequence of non-linear conductivity energies in a bounded open set The energy density is of the same order as where is periodic, u is a vector-valued function in and The conductivity is equal to 1 in the "hard" phases composed by two by two disjoint-closure periodic sets while tends uniformly to 0 in the "soft" phases composed by periodic thin layers which separate the hard phases. We prove that the limit energy, according to γ-convergence, is a multi-phase functional equal to the sum of the homogenized energies (of order 1) induced by the hard phases plus an interaction energy (of order 0) due to the soft phases. The number of limit phases is less than or equal to N and is obtained by evaluating the γ-limit of the rescaled energy of density in the torus. Therefore, the homogenization result is achieved by a double γ-convergence procedure since the cell problem depends on ε.     

On the Bernstein Constants of Polynomial Approximation

Assume is not an integer. In papers published in 1913 and 1938, S.~N.~Bernstein established the limit

Asymptotics of Laurent Polynomials of Odd Degree Orthogonal with Respect to Varying Exponential Weights

Let denote the linear space over spanned by . Define the (real) inner product , where V satisfies: (i) V is real analytic on ; (ii) ; and (iii) . Orthogonalisation of the (ordered) base with respect to yields the even degree and odd degree orthonormal Laurent polynomials , and . Define the even degree and odd degree monic orthogonal Laurent polynomials: and . Asymptotics in the double-scaling limit such that of (in the entire complex plane), , and (in the entire complex plane) are obtained by formulating the odd degree monic orthogonal Laurent polynomial problem as a matrix Riemann-Hilbert problem on , and then extracting the large-n behaviour by applying the non-linear steepest-descent method introduced in [1] and further developed in [2],[3].