Frobenius Problem and the Covering Radius of a Lattice

Let and let be relatively prime integers. The Frobenius number of this N-tuple is defined to be the largest positive integer that cannot be expressed as are non-negative integers. The condition that implies that such a number exists. The general problem of determining the Frobenius number given N and is NP-hard, but there have been a number of different bounds on the Frobenius number produced by various authors. We use techniques from the geometry of numbers to produce a new bound, relating the Frobenius number to the covering radius of the null-lattice of this N-tuple. Our bound is particularly interesting in the case when this lattice has equal successive minima, which, as we prove, happens infinitely often.     

Autocorrelation Functions as Translation Invariants in L1 and L2

The Adler-Konheim theorem [Proc. Amer. Math. Soc. 13 (1962), 425-428] states that the collection of nth-order autocorrelation functions is a complete set of translation invariants for real-valued L¹ functions on a locally compact abelian group. It is shown here that there are proper subsets of that also form a complete set of translation invariants, and these subsets are characterized. Specifically, a subset is complete if and only if it contains infinitely many even-order autocorrelation functions. In addition, any infinite subset of is complete up to a sign. While stated here for functions on the proofs presented hold for functions on any locally compact abelian group that is not compact, in particular, on and the integer lattice      

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 .     

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

Fourier transforms related to a root system of rank 1

We introduce an algebra consisting of difference-reflection operators and multiplication operators that can be considered as a q = 1 analogue of Sahi's double affine Hecke algebra related to the affine root system of type . We study eigenfunctions of a Dunkl-Cherednik-type operator in the algebra , and the corresponding Fourier transforms. These eigenfunctions are nonsymmetric versions of the Wilson polynomials and the Wilson functions.     

Bethe algebra and the algebra of functions on the space of differential operators of order two with polynomial kernel

We show that the algebra of functions on the scheme of monic linear second-order ordinary differential operators with prescribed n + 1 regular singular points, prescribed exponents at the singular points, and having the kernel consisting of polynomials only, is isomorphic to the Bethe algebra of the Gaudin model acting on the vector space Sing of singular vectors of weight Λ^(∞) in the tensor product of finite-dimensional polynomial -modules with highest weights .      

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.     

Smaller Coresets for k-Median and k-Means Clustering

In this paper we show that there exists a -coreset for k-median and k-means clustering of n points in which is of size independent of n. In particular, we construct a -coreset of size for k-median clustering, and of size for k-means clustering.     

Invariance Theorems in Approximation Theory and their Applications

Let B be a closed linear subspace of a Banach space F and let be a group of continuous linear operators , where G is a compact topological group. We prove that if is invariant under , then under some conditions on f, F, B, and G, there exists an element of best approximation to f that has the same property. As applications, we compute the bivariate Bernstein constant for polynomial approximation of and solve a Braess problem on the exponential order of decay of the error of polynomial approximation of . Other examples and applications are discussed as well.     

Many Valued Topologies and Lower Semicontinuity

On the basis of join-continuous semigroups , this paper demonstrates among other things that on every complete ortholattice there exists an L-valued topology such that order convergence is L-topological (this is not always the case with respect to ordinary topologies). Further, the concept of lower semicontinuous lattice-valued maps permits an extension of the omega-functor to the general setting of L-valued topological spaces where the importance of lies in the replacement of ordinary topologies by L-valued topologies. It is shown that has a right adjoint functor iff the underlying lattice is continuous.     

Typical separating invariants

It is shown that a trivial version of polarization is sufficient to produce separating systems of polynomial invariants: If two points in the direct sum of the G-modules W and m copies of V can be separated by polynomial invariants, then they can be separated by invariants depending only on variables of type V; when G is reductive, invariants depending only on variables suffice. A similar result is valid for rational invariants. Explicit bounds on the number of type V variables in a complete system of typical separating invariants are given for the binary polyhedral groups, and this is applied to the invariant theory of binary forms.     

Hardy Spaces for Laguerre Expansions

Let be the standard Laguerre functions of type a. We denote . Let and be the semigroups associated with the orthonormal systems and . We say that a function f belongs to the Hardy space associated with one of the semigroups if the corresponding maximal function belongs to . We prove special atomic decompositions of the elements of the Hardy spaces.     

Temperley-Lieb Immanants

We use the Temperley-Lieb algebra to define a family of totally nonnegative polynomials of the form . The cone generated by these polynomials contains all totally nonnegative polynomials of the form , where, are matrix minors. We also give new conditions on the sets I,...,K′ which characterize differences of products of minors which are totally nonnegative. Received September 30, 2004     

Quantitative Robust Uncertainty Principles and Optimally Sparse Decompositions

In this paper we develop a robust uncertainty principle for finite signals in which states that, for nearly all choices such that

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.     

The Chow ring of punctual Hilbert schemes on toric surfaces

Let X be a smooth projective toric surface, and the Hilbert scheme parametrizing the length d zero-dimensional subschemes of X. We compute the rational Chow ring . More precisely, if is the two-dimensional torus contained in X, we compute the rational equivariant Chow ring and the usual Chow ring is an explicit quotient of the equivariant Chow ring. The case of some quasi-projective toric surfaces such as the affine plane are described by our method too.     

On Polynomially Normal Lattice Configurations

In this paper we extend Champernowne’s construction of a normal sequence in base b to the case and obtain an explicit construction of the generic point of the shift transformation of the set . We prove that the intersection of the constructed configuration with an arbitrary polynomial curve in the plane is a normal sequence in base b.     

Localized Polynomial Frames on the Ball

Almost exponentially localized polynomial kernels are constructed on the unit ball in with weights , by smoothing out the coefficients of the corresponding orthogonal projectors. These kernels are utilized to the design of cubature formulas on with respect to and to the construction of polynomial tight frames in (called needlets) whose elements have nearly exponential localization.