A Discontinuous Colombeau Differential Calculus

Starting from the Colombeau Generalized Functions, the sharp topologies and the notion of generalized points, we introduce a new kind of differential calculus (for functions between totally disconnected spaces). We also define here the notions of holomorphic generalized functions (in this new framework) and generalized manifold. Finally we give an answer to a question raised in [6].Research partially supported by CNPq (Proc 300652/95-0).     

Numerical index of some polyhedral norms on the plane

We give explicit formulae for the numerical index of some (real) polyhedral spaces of dimension two. Concretely, we calculate the numerical index of a family of hexagonal norms, two families of octagonal norms and the family of norms whose unit balls are regular polygons with an even number of vertices.     

Strong Dunford–Pettis Sets and Spaces of Operators

Bibasic sequences of Singer are used to show that ℓ₁ embeds complementably in the Banach space X if and only if X^* contains a non-relatively compact strong Dunford–Pettis set. Spaces of operators and strongly additive vector measures are also discussed. An erratum to this article is available at .     

A Composition Theorem for Multiple Summing Operators

We prove that the composition S(u₁, …, u_n) of a multilinear multiple 2-summing operator S with 2-summing linear operators u_j is nuclear, generalizing a linear result of Grothendieck. Both authors were partially supported by DGICYT grant BMF2001-1284.     

Generalized Functions Valued in a Smooth Manifold

 Based on Colombeau’s theory of algebras of generalized functions we introduce the concepts of generalized functions taking values in differentiable manifolds as well as of generalized vector bundle homomorphisms. We study their basic properties, in particular with respect to some new point value concepts for generalized functions and indicate applications of the resulting theory in general relativity. Received February 13, 2002     

On a Definitizable Analog of the Trigonometric Moment Problem Generating an Indefinite Toeplitz Form

We prove the existence of an integro-polynomial representation for a sequence of numbers such that there exists a difference operator mapping this sequence to a sequence that generates the solvable trigonometric moment problem. A similar result related to the power moment problem was given in [12].     

Characterization of Finite Groups With Some <Emphasis Type="Italic">S</Emphasis>-quasinormal Subgroups

A subgroup of a finite group G is said to be S-quasinormal in G if it permutes with every Sylow subgroup of G. In this paper we give a characterization of a finite group G under the assumption that every subgroup of the generalized Fitting subgroup of prime order is S-quasinormal in G.     

Dirichlet and Bergman Spaces of Holomorphic Functions on the Unit Ball of ℂ<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

Let B denote the unit ball in ⁿ, n 1, and let and denote the volume measure and gradient with respect to the Bergman metric on B. In the paper we consider the weighted Dirichlet spaces , , and weighted Bergman spaces , , , of holomorphic functions f on B for which and respectively are finite, where and The main result of the paper is the following theorem.Theorem 1. Let f be holomorphic on B and .(a) If for some , then for all p, , with .(b) If for some p, , then for all with . Combining Theorem 1 with previous results of the author we also obtain the following.Theorem 2. Suppose is holomorphic in B. If for some p, , and , then . Conversely, if for some p, , then the series in * converges.     

Intersections and Unions of Orthogonal Polygons Starshaped Via Staircase n-Paths

For n ≥ 1, define p (n) to be the smallest natural number r for which the following is true: For any finite family of simply connected orthogonal polygons in the plane and points x and y in , if every r (not necessarily distinct) members of contain a common staircase n-path from x to y, then contains such a path. We show that p(1) = 1 and p(n) = 2 (n − 1) for n ≥ 2. The numbers p(n) yield an improved Helly theorem for intersections of sets starshaped via staircase n-paths. Moreover, we establish the following dual result for unions of these sets: Let be any finite family of orthogonal polygons in the plane, with simply connected. If every three (not necessarily distinct) members of have a union which is starshaped via staircase n-paths, then T is starshaped via staircase (n + 1)-paths. The number n + 1 in the theorem is best for every n ≥ 2.     

A self-sorting in-place fast Fourier transform algorithm suitable for vector and parallel processing

Summary. We propose a new algorithm for fast Fourier transforms. This algorithm features uniformly long vector lengths and stride one data access. Thus it is well adapted to modern vector computers like the Fujitsu VP2200 having several floating point pipelines per CPU and very fast stride one data access. It also has favorable properties for distributed memory computers as all communication is gathered together in one step. The algorithm has been implemented on the Fujitsu VP2200 using the basic subroutines for fast Fourier transforms discussed elsewhere. We develop the theory of index digit permutations to some extent. With this theory we can derive the splitting formulas for almost all mixed-radix FFT algorithms known so far. This framework enables us to prove these algorithms but also to derive our new algorithm. The development and systematic use of this framework is new and allows us to simplify the proofs which are now reduced to the application of matrix recursions. Received October 29, 1992 / Revised version received October 21, 1993     

Sharp probability estimates for generalized Smirnov statistics

We give sharp, uniform estimates for the probability that the empirical distribution function for n uniform-[0,1] random variables stays to one side of a given line. Author’s address: Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 West Green Street, Urbana, IL 61801, USA     

The Schur Algorithm for Generalized Schur Functions II: Jordan Chains and Transformations of Characteristic Functions

 In the first paper of this series (Daniel Alpay, Tomas Azizov, Aad Dijksma, and Heinz Langer: The Schur algorithm for generalized Schur functions I: coisometric realizations, Operator Theory: Advances and Applications 129 (2001), pp. 1–36) it was shown that for a generalized Schur function s(z), which is the characteristic function of a coisometric colligation V with state space being a Pontryagin space, the Schur transformation corresponds to a finite-dimensional reduction of the state space, and a finite-dimensional perturbation and compression of its main operator. In the present paper we show that these formulas can be explained using simple relations between V and the colligation of the reciprocal s(z)⁻¹ of the characteristic function s(z) and general factorization results for characteristic functions. Received October 31, 2001; in revised form August 21, 2002 RID="a" ID="a" Dedicated to Professor Edmund Hlawka on the occasion of his 85th birthday     

Circularity of Finite Groups without Fixed Points

Let be a fixed point free group given by the presentation where and are relative prime numbers, t = /s and s = gcd( – 1,), and is the order of modulo . We prove that if (1) = 2, and (2) is embeddable into the multiplicative group of some skew field, then is circular. This means that there is some additive group N on which acts fixed point freely, and |((a)+b)((c)+d)| 2 whenever a,b,c,d N, a0c, are such that (a)+b(c)+d.     

Moments of Brownian Motions on Lie Groups

Let (B_t)_{t ≥ 0} be a Brownian motion on with the corresponding Gaussian convolution semigroup (μ_t)_{t ≥ 0} and generator L. We show that algebraic relations between L and the generators of the matrix semigroups lead to for t → s, k ≥ 1, and all coordinates i,j. These relations will form the basis for a martingale characterization of (B_t)_{t ≥ 0} in terms of generalized heat polynomials. This characterization generalizes a corresponding result for the Brownian motion on in terms of Hermite polynomials due to J. Wesolowski and may be regarded as a variant of the Lévy characterization without continuity assumptions.     

Extendible elements of the alternating groups

An element σ of A_n, the Alternating group of degree n, is extendible in S_n, the Symmetric group of degree n, if there exists a subgroup H of S_n but not A_n whose intersection with A_n is the cyclic group generated by σ. A simple number-theoretic criterion, in terms of the cycle-decomposition, for an element of A_n to be extendible in S_n is given here.     

Some Problems on Homomorphisms and Real Function Algebras

 In this paper we solve a problem about the representation of all homomorphisms on a real function algebra as point evaluations and another two about function algebras in which homomorphisms are point evaluations on sequences in the algebra. (Received 4 December 2000; in revised form 2 April 2001)     

The Segal Algebra {\bf S}_0({\Bbb R}^d) and Norm Summability of Fourier Series and Fourier Transforms

A general summability method, the so-called θ-summability is considered for multi-dimensional Fourier series. Equivalent conditions are derived for the uniform and L ₁-norm convergence of the θ-means σ _n ^θ f to the function f. If f is in a homogeneous Banach space, then the preceeding convergence holds in the norm of the space. In case θ is an element of Feichtinger’s Segal algebra , then these convergence results hold. Some new sufficient conditions are given for θ to be in . A long list of concrete special cases of the θ-summation is listed. The same results are also provided in the context of Fourier transforms, indicating how proofs have to be changed in this case. This research was supported by Lise Meitner fellowship No M733-N04 and the Hungarian Scientific Research Funds (OTKA) No T043769, T047128, T047132.     

A counterexample on non-archimedean regularity

A non-regular inductive sequence of non-archimedean reflexive Fréchet spaces is constructed. On the other hand, it is proved that every inductive sequence of reflexive Banach spaces over a spherically complete field is regular. Also, some applications are given. Research partially supported by Ministerio de Educación y Ciencia, MTM2006-14786. Authors’ addresses: N. De Grande-De Kimpe, Groene Laan 36 (302) B 2830, Willebroek, Belgium; C. Perez-Garcia, Department of Mathematics, Facultad de Ciencias, Universidad de Cantabria, Avda. de los Castros s/n, 39071 Santander, Spain     

The Asymptotic Behavior of the Joint Linear Complexity Profile of Multisequences

We prove a conjecture on the asymptotic behavior of the joint linear complexity profile of random multisequences over a finite field. This conjecture was previously shown only in the special cases of single sequences and pairs of sequences. We also establish an asymptotic formula for the expected value of the nth joint linear complexity of random multisequences over a finite field. Some more precise results are shown for triples of sequences.     

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.