Inequalities for irregular Gabor frames

In [C.K. Chui and X.L. Shi, Inequalities of Littlewood-Paley type for frames and wavelets, SIAM J. Math. Anal., 24 (1993), 263–277], the authors proved that if is a Gabor frame for with frame bounds A and B, then the following two inequalities hold: and . In this paper, we show that similar inequalities hold for multi-generated irregular Gabor frames of the form , where Δ _k and Λ _k are arbitrary sequences of points in and , 1 ≤ k ≤ r. Corresponding author for second author Authors’ address: Lili Zang and Wenchang Sun, Department of Mathematics and LPMC, Nankai University, Tianjin 300071, China     

Geometric progressions in sumsets over finite fields

Given two sets , the set of d dimensional vectors over the finite field with q elements, we show that the sumset contains a geometric progression of length k of the form vΛ ^j , where j = 0,…, k − 1, with a nonzero vector and a nonsingular d × d matrix Λ whenever . We also consider some modifications of this problem including the question of the existence of elements of sumsets on algebraic varieties.     

Smooth Universal Taylor Series

Let be a simply connected domain in , such that is connected. If g is holomorphic in Ω and every derivative of g extends continuously on , then we write g ∈ A^∞ (Ω). For g ∈ A^∞ (Ω) and we denote . We prove the existence of a function f ∈ A^∞(Ω), such that the following hold:

Covering and packing in ${\Bbb Z}^n$ and ${\Bbb R}^n$, (I)

Given a finite subset of an additive group such as or , we are interested in efficient covering of by translates of , and efficient packing of translates of in . A set provides a covering if the translates with cover (i.e., their union is ), and the covering will be efficient if has small density in . On the other hand, a set will provide a packing if the translated sets with are mutually disjoint, and the packing is efficient if has large density. In the present part (I) we will derive some facts on these concepts when , and give estimates for the minimal covering densities and maximal packing densities of finite sets . In part (II) we will again deal with , and study the behaviour of such densities under linear transformations. In part (III) we will turn to . Authors’ address: Department of Mathematics, University of Colorado at Boulder, Campus Box 395, Boulder, Colorado 80309-0395, USA The first author was partially supported by NSF DMS 0074531.     

A Characteristic Property of Injective Holomorphic Germs

Let X ₀ be the germ at 0 of a complex variety and let be a holomorphic germ. We say that f is pseudoimmersive if for any such that , we have . We prove that f is pseudoimmersive if and only if it is injective. Some results about the real case are also considered.     

Regularity in a Singular Biharmonic Dirichlet Problem

We consider the singular biharmonic equation with Dirichlet boundary conditions u = f₀ and ∂_nu = f₁ on . In our setup the boundary values f_j (j = 0,1) are elements in two homogeneous Banach spaces B_j (j = 0,1) on . We give a sufficient condition on the spaces B_j (j = 0,1) to ensure that the solution u of this Dirichlet problem has the appropriate boundary values f_j (j = 0,1) in the sense of convergence in spaces B_j (j = 0,1). Our results also apply in the unweighted case.     

Classification of Möbius Isoparametric Hypersurfaces in ${\Bbb S}^5$

Let M ⁿ be an immersed umbilic-free hypersurface in the (n + 1)-dimensional unit sphere , then M ⁿ is associated with a so-called M?bius metric g, a M?bius second fundamental form B and a M?bius form Φ which are invariants of M ⁿ under the M?bius transformation group of . A classical theorem of M?bius geometry states that M ⁿ (n ≥ 3) is in fact characterized by g and B up to M?bius equivalence. A M?bius isoparametric hypersurface is defined by satisfying two conditions: (1) Φ ≡ 0; (2) All the eigenvalues of B with respect to g are constants. Note that Euclidean isoparametric hypersurfaces are automatically M?bius isoparametrics, whereas the latter are Dupin hypersurfaces. In this paper, we determine all M?bius isoparametric hypersurfaces in by proving the following classification theorem: If is a M?bius isoparametric hypersurface, then x is M?bius equivalent to either (i) a hypersurface having parallel M?bius second fundamental form in ; or (ii) the pre-image of the stereographic projection of the cone in over the Cartan isoparametric hypersurface in with three distinct principal curvatures; or (iii) the Euclidean isoparametric hypersurface with four principal curvatures in . The classification of hypersurfaces in with parallel M?bius second fundamental form has been accomplished in our previous paper [7]. The present result is a counterpart of the classification for Dupin hypersurfaces in up to Lie equivalence obtained by R. Niebergall, T. Cecil and G. R. Jensen. Partially supported by DAAD; TU Berlin; Jiechu grant of Henan, China and SRF for ROCS, SEM. Partially supported by the Zhongdian grant No. 10531090 of NSFC. Partially supported by RFDP, 973 Project and Jiechu grant of NSFC.     

Hypersurfaces and Codazzi tensors

In this paper we deal with the following problem. Let (M ⁿ ,〈,〉) be an n-dimensional Riemannian manifold and an isometric immersion. Find all Riemannian metrics on M ⁿ that can be realized isometrically as immersed hypersurfaces in the Euclidean space . More precisely, given another Riemannian metric on M ⁿ , find necessary and sufficient conditions such that the Riemannian manifold admits an isometric immersion into the Euclidean space . If such an isometric immersion exists, how can one describe in terms of f? Author’s address: Thomas Hasanis and Theodoros Vlachos, Department of Mathematics, University of Ioannina, 45110 Ioannina, Greece     

Jordan Triple Product Homomorphisms

Nondegenerate mappings that preserve Jordan triple product on are characterized. Here, n ≥ 3 and is an arbitrary field. This work was supported by grants from the Ministry of Science of Slovenia.     

On the Dimension of Coinvariants of Permutation Representations

Let be a univariate, separable polynomial of degree n with roots x ₁,…,x _n in some algebraic closure of the ground field . It is a classical problem of Galois theory to find all the relations between the roots. It is known that the ideal of all such relations is generated by polynomials arising from G-invariant polynomials, where G is the Galois group of f(Z). Namely: The action of G on the ordered set of roots induces an action on by permutation of the coordinates and each defines a relation P − P(x ₁,…,x _n ) called a G-invariant relation. These generate the ideal of all relations. In this note we show that the ideal of relations admits an H-basis of G-invariant relations if and only if the algebra of coinvariants has dimension ‖G‖ over . To complete the picture we then show that the coinvariant algebra of a transitive permutation representation of a finite group G has dimension ‖G‖ if and only if G = Σ _n acting via the tautological permutation representation.     

Constant angle surfaces in $ {\Bbb S}^2\times {\Bbb R} $

In this article we study surfaces in for which the unit normal makes a constant angle with the -direction. We give a complete classification for surfaces satisfying this simple geometric condition.     

Tameness of Pseudovariety Joins Involving ${\sf R}$

In this paper, we establish several decidability results for pseudovariety joins of the form , where is a subpseudovariety of or the pseudovariety . Here, (resp. ) denotes the pseudovariety of all -trivial (resp. -trivial) semigroups. In particular, we show that the pseudovariety is (completely) κ-tame when is a subpseudovariety of with decidable κ-word problem and is (completely) κ-tame. Moreover, if is a κ-tame pseudovariety which satisfies the pseudoidentity x₁ ⋯ x_ry^ω+1zt^ω = x₁ ⋯ x_ryzt^ω, then we prove that is also κ-tame. In particular the joins , , , and are decidable. Partial support by FCT, through the Centro de Matemática da Universidade do Porto, is also gratefully acknowledged. Partial support by FCT, through the Centro de Matemática da Universidade do Minho, is also gratefully acknowledged.     

Punishing Factors for Finitely Connected Domains

Let Ω and Π be two finitely connected hyperbolic domains in the complex plane and let R(z, Ω) denote the hyperbolic radius of Ω at z and R(w, Π) the hyperbolic radius of Π at w. We consider functions f that are analytic in Ω and such that all values f(z) lie in the domain Π. This set of analytic functions is denoted by A(Ω, Π). We prove among other things that the quantities are finite for all if and only if ∂Ω and ∂Π do not contain isolated points. This work was supported by a grant of the Deutsche Forschungsgemeinschaft for F. G. Avkhadiev.     

Real Hypersurfaces of Type B in Complex Two-Plane Grassmannians

In this paper we give a characterization of real hypersurfaces of type B, that is, a tube over a totally real totally geodesic in complex two-plane Grassmannians with the shape operator A satisfying Aφ + φA = kφ, k is non-zero constant, for the structure tensor φ.     

Density of the Set of Generators of Wavelet Systems

Given a function ψ in the affine (wavelet) system generated by ψ, associated to an invertible matrix a and a lattice Γ, is the collection of functions In this paper we prove that the set of functions generating affine systems that are a Riesz basis of ${\cal L}^2({\Bbb R}^d)$ is dense in We also prove that a stronger result is true for affine systems that are a frame of In this case we show that the generators associated to a fixed but arbitrary dilation are a dense set. Furthermore, we analyze the orthogonal case in which we prove that the set of generators of orthogonal (not necessarily complete) affine systems, that are compactly supported in frequency, are dense in the unit sphere of with the induced metric. As a byproduct we introduce the p-Grammian of a function and prove a convergence result of this Grammian as a function of the lattice. This result gives insight in the problem of oversampling of affine systems.     

Conjugates of Rational Equivariant Holomorphic Maps of Symmetric Domains

Let be an equivariant holomorphic map of symmetric domains associated to a homomorphism of semisimple algebraic groups defined over . If and are torsion-free arithmetic subgroups with , the map induces a morphism : of arithmetic varieties and the rationality of is defined by using symmetries on and as well as the commensurability groups of and . An element determines a conjugate equivariant holomorphic map of which induces the conjugate morphism of . We prove that is rational if is rational.     

On products of <Emphasis Type="Italic">k</Emphasis> atoms

Let H be an atomic monoid. For let denote the set of all with the following property: There exist atoms (irreducible elements) u ₁, …, u _k , v ₁, …, v _m ∈ H with u ₁· … · u _k = v ₁ · … · v _m . We show that for a large class of noetherian domains satisfying some natural finiteness conditions, the sets are almost arithmetical progressions. Suppose that H is a Krull monoid with finite cyclic class group G such that every class contains a prime (this includes the multiplicative monoids of rings of integers of algebraic number fields). We show that, for every , max which settles Problem 38 in [4]. Authors’ addresses: W. Gao, Center for Combinatorics, Nankai University, Tianjin 300071, P.R. China; A. Geroldinger, Institut für Mathematik und Wissenschaftliches Rechnen, Karl-Franzens-Universit?t Graz, Heinrichstra?e 36, 8010 Graz, Austria     

Embedding metric spaces into normed spaces and estimates of metric capacity

Let be an arbitrary real normed space of finite dimension d ≥ 2. We define the metric capacity of as the maximal such that every m-point metric space is isometric to some subset of (with metric induced by ). We obtain that the metric capacity of lies in the range from 3 to , where the lower bound is sharp for all d, and the upper bound is shown to be sharp for d ∈ {2, 3}. Thus, the unknown sharp upper bound is asymptotically linear, since it lies in the range from d + 2 to . Research supported by the German Research Foundation, Project AV 85/1-1.     

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.     

Probability measures on [SIN] groups and some related ideals in group algebras

In 1921, Blichfeldt gave an upper bound on the number of integral points contained in a convex body in terms of the volume of the body. More precisely, he showed that , whenever is a convex body containing n + 1 affinely independent integral points. Here we prove an analogous inequality with respect to the surface area F(K), namely . The proof is based on a slight improvement of Blichfeldt’s bound in the case when K is a non-lattice translate of a lattice polytope, i.e., K = t + P, where and P is an n-dimensional polytope with integral vertices. Then we have . Moreover, in the 3-dimensional case we prove a stronger inequality, namely . Authors’ addresses: Martin Henk, Institut für Algebra und Geometrie, Universit?t Magdeburg, Universit?tsplatz 2, D-39106 Magdeburg, Germany; J?rg M. Wills, Mathematisches Institut, Universit?t Siegen, ENC, D-57068 Siegen, Germany