On a subalgebra of the centre of a group ring II

Let p be a prime, G a finite group with p | |G| and F a field of characteristic p. By we denote the F-subspace of the centre of the group ring FG spanned by the p-regular conjugacy class sums. J. Murray proved that is an algebra, if G is a symmetric or alternating group. This can be used for the computation of the block idempotents of FG. We proved that is an algebra if the Sylow-p-subgroups of G are abelian. Recently, Y. Fan and B. Külshammer generalized this result to blocks with abelian defect groups. Here, we show that is an algebra if the Sylow-2-subgroups of G are dihedral. Therefore and are algebras for all primes p and all prime powers q. Furthermore we prove that is an algebra for the simple Suzuki-groups Sz(q), where q is a certain power of 2 and p is an arbitrary prime dividing |Sz(q)|. Received: 18 May 2007     

Selections of lattice-ordered Groups

It is well known that the quasitorsion class of archimedean -groups is the class of -groups G such that every closed convex -subgroup is a polar, and it is also well known that the class of -groups G such that every convex -subgroup is a polar is a torsion class. By defining a selection on -groups, these two results are generalized to show, whenever and are selections on -groups, the class of -groups G such that is a radical class. Three selections in particular — all convex -subgroups, all polars, and all closed convex -subgroups — and the radical classes determined by them are studied in some detail. Received March 7, 2006; accepted in final form August 29, 2006.     

Reconstruction and Subgaussian Operators in Asymptotic Geometric Analysis

We present a randomized method to approximate any vector from a set . The data one is given is the set T, vectors of and k scalar products , where are i.i.d. isotropic subgaussian random vectors in , and . We show that with high probability, any for which is close to the data vector will be a good approximation of , and that the degree of approximation is determined by a natural geometric parameter associated with the set T. We also investigate a random method to identify exactly any vector which has a relatively short support using linear subgaussian measurements as above. It turns out that our analysis, when applied to {−1, 1}-valued vectors with i.i.d. symmetric entries, yields new information on the geometry of faces of a random {−1, 1}-polytope; we show that a k- dimensional random {−1, 1}-polytope with n vertices is m-neighborly for The proofs are based on new estimates on the behavior of the empirical process when F is a subset of the L ₂ sphere. The estimates are given in terms of the γ ₂ functional with respect to the ψ ₂ metric on F, and hold both in exponential probability and in expectation. Received: November 2005, Revision: May 2006, Accepted: June 2006     

On an Essentially Algebraic Theory for Locally Presentable Categories

This paper presents a general construction, defining for each given strong generator in any locally finitely presentable category an essentially algebraic, finitary theory – maximal in a certain sense – such that is equivalent to the category of models of . For regular generators , generalization to the non-finitary case is easily done, and yields a new proof of the famous characterization of many-sorted quasivarieties.     

The Reduced Minimum Modulus in <Emphasis>C</Emphasis><Superscript>*</Superscript>-Algebras

We define the reduced minimum modulus of a nonzero element a in a unital C ^*-algebra by . We prove that . Applying this result to and its closed two side ideal , we get that dist , and for any if RR = 0, where and is the quotient homomorphism and . These results generalize corresponding results in Hilbert spaces.     

Some results on the qualitative properties of positive solutions of quasilinear elliptic equations

We consider the Dirichlet problem in Ω with zero Dirichlet boundary conditions. We prove local summability properties of and we exploit these results to give geometric characterizations of the critical set . We extend to the case of changing sign nonlinearities some results known in the case f(s) > 0 for s > 0. Berardino Sciunzi: Supported by MURST, Project “Metodi Variazionali ed Equazioni Differenziali Non Lineari”     

Aluthge Transforms of Complex Symmetric Operators

If denotes the polar decomposition of a bounded linear operator T, then the Aluthge transform of T is defined to be the operator . In this note we study the relationship between the Aluthge transform and the class of complex symmetric operators (T iscomplex symmetric if there exists a conjugate-linear, isometric involution so that T = CT*C). In this note we prove that: (1) the Aluthge transform of a complex symmetric operator is complex symmetric, (2) if T is complex symmetric, then and are unitarily equivalent, (3) if T is complex symmetric, then if and only if T is normal, (4) if and only if T ² = 0, and (5) every operator which satisfies T ² = 0 is necessarily complex symmetric. This work partially supported by National Science Foundation Grant DMS 0638789.     

Divisors of Bernoulli Sums

Let where are independent Bernoulli random variables. In relation with the divisor problem, we evaluate the almost sure asymptotic order of the sums , where and is a sequence of positive integers. Received: May 23, 2007. Revised: June 8, 2007.     

On Separation of Points from Additive Subgroups of Banach Spaces by Continuous Characters and Positive Definite Functions

Let G be an additive subgroup of a normed space X. We say that a point is weakly separated (resp. -separated) from G if it can be separated from G by a continuous character (resp. by a continuous positive definite function). Let T : X → Y be a continuous linear operator. Consider the following conditions: (ws) if , then x is weakly separated from G; (ps) if , then x is -separated from G; (wp) if Tx is -separated from T(G), then x is weakly separated from G. By (resp. , ) we denote the class of operators T : X → Y which satisfy (ws) (resp. (ps), (wp)) for all and all subgroups G of X. The paper is an attempt to describe the above classes of operators for various Banach spaces X, Y. It is proved that if X, Y are Hilbert spaces, then is the class of Hilbert-Schmidt operators. It is also shown that if T is a Hilbert-to-Banach space operator with finite ℓ-norm, then .      

On meromorphic solutions of f 3 + g 3 = 1

We show that meromorphic solutions f, g of f ³ + g ³ = 1 in must be constant, if and have the same zeros (counting multiplicities). Received: 12 March 2007     

A remark on left invariant metrics on compact Lie groups

We show that a left invariant metric on a compact Lie group G with Lie algebra has some negative sectional curvature if it is obtained by enlarging a biinvariant metric on a subalgebra , unless the semi-simple part of is an ideal of This answers a question raised in [8]. Received: 7 May 2007     

Boolean Functions with small Spectral Norm

Let be a boolean function, and suppose that the spectral norm of f is at most M. Then where and each H _j is a subgroup of . This result may be regarded as a quantitative analogue of the Cohen-Helson-Rudin structure theorem for idempotent measures in locally compact abelian groups. Received: May 2006 Accepted: January 2007     

On values of n 2 + 1 free of large prime factors

Let P ⁺(m) denote the greatest prime factor of the positive integer m. Improving and simplifying work of Dartyge [3] we show that

On the stability of <Emphasis Type="Italic">t</Emphasis>-convex functions

Summary.  A real-valued function f defined on an open convex set is called (d, t)-convex if it satisfies

A Real Analyticity Result for Symmetric Functions of the Eigenvalues of a Domain-Dependent Neumann Problem for the Laplace Operator

Let Ω be an open connected subset of for which the imbedding of the Sobolev space W ^1,2(Ω) into the space L ²(Ω) is compact. We consider the Neumann eigenvalue problem for the Laplace operator in the open subset (Ω) of , where is a Lipschitz continuous homeomorphism of Ω onto (Ω). Then we prove a result of real analytic dependence for symmetric functions of the eigenvalues upon variation of . This paper represents an extension of a part of the work performed by P.D. Lamberti in his PhD Thesis at the University of Padova under the guidance of M. Lanza de Cristoforis.     

Asymptotic behaviour for a nonlocal diffusion equation on a lattice

In this paper we study the asymptotic behaviour as t → ∞ of solutions to a nonlocal diffusion problem on a lattice, namely, with t ≥ 0 and . We assume that J is nonnegative and verifies . We find that solutions decay to zero as t → ∞ and prove an optimal decay rate using, as our main tool, the discrete Fourier transform.      

Synchronizing and Collapsing Words

A word w over a finite alphabet Σ is called k-collapsing if for each finite deterministic automaton the inequality holds provided that for some word , depending on . A word over the alphabet Σ is called k-synchronizing if it is a reset word for all synchronizing automata with k + 1 states and input alphabet Σ. The aim of this work is to give the motivations of the interest in these words and to provide an overview of some results in this area. Received: May 2007     

Application of the Trace Inequality to the Poisson Equation

The purpose of this paper is to show that solutions of the Poisson equation

Multivalued evolution equations with nonlocal initial conditions in Banach spaces

We prove the existence of integral solutions to the nonlocal Cauchy problem in a Banach space X, where is m-accretive and such that –A generates a compact semigroup, has nonempty, closed and convex values, and is strongly-weakly upper semicontinuous with respect to its second variable, and . The case when A depends on time is also considered.