Eigenvalues of Integral Operators Defined by Smooth Positive Definite Kernels

We consider integral operators defined by positive definite kernels , where X is a metric space endowed with a strictly-positive measure. We update upon connections between two concepts of positive definiteness and upgrade on results related to Mercer like kernels. Under smoothness assumptions on K, we present decay rates for the eigenvalues of the integral operator, employing adapted to our purposes multidimensional versions of known techniques used to analyze similar problems in the case where X is an interval. The results cover the case when X is a subset of endowed with the induced Lebesgue measure and the case when X is a subset of the sphere S^m endowed with the induced surface Lebesgue measure. The first author was partially supported by FAPESP, grants # 2005/56694–4, # 2007/58086–7.     

Some Nonlinear Elliptic Problems in Unbounded Domains

In this paper the results of some investigations concerning nonlinear elliptic problems in unbounded domains are summarized and the main difficulties and ideas related to these researches are described. The model problem

Domains of Existence of Polymonogenic Functions

The paper addresses the Levi problem for a system of n Fueter equations in a domain in quaternionic space . This problem relates to various conditions of convexity and pseudoconvexity of the boundary of the domain. Received: October, 2007, Accepted: February, 2008.     

Unbounded solutions of differential equations of second order

In this paper, the existence of unbounded solutions for the following nonlinear asymmetric oscillator

Some subordination results for classes of analytic functions defined by the Al-Oboudi–Al-Amoudi operator

In this paper, we investigate several interesting subordination results of some classes of analytic functions defined by means of the Al-Oboudi–Al-Amoudi operator. Received: 15 August 2008, Revised: 19 November 2008     

On the Positive Parts of Second Order Symmetric Pseudodifferential Operators

Using microlocalization, the positive and the negative parts for a class of second order formally self-adjoint pseudodifferential operators are constructed.      

A note on primes in short intervals

This paper is concerned with the number of primes in short intervals. We prove that , for θ > 1/2, with the assumption of an heuristic hypothesis weaker than the Lindel?f hypothesis. Received: 8 October 2007, Revised: 14 April 2008     

A note on a spanning 3-tree

A tree T is called a k-tree, if the maximum degree of T is at most k. In this paper, we prove that if G is an n-connected graph with independence number at most n + m + 1 (n≥1,n≥m≥0), then G has a spanning 3-tree T with at most m vertices of degree 3.     

Positive Elements and Automatic Continuity of Algebra Morphisms

We use positive elements of Hermitian algebras to give results on automatic continuity of algebra morphisms. Consequences and applications are also given.      

On orbits of automorphism groups II

Let G be a finite group, let A be a group of automorphisms of G and let C_G(A) denote the subgroup of fixed points of A in G. If the order of C_G(A) is coprime to the number of orbits of A in G, then C_G(A) is contained in the autocommutator subgroup [G, A]. The notion of class-avoiding automorphism is used to extend theorems of J. Thompson and P. Rowley. Received: 3 November 2008, Revised: 1 December 2008     

On the Wu metric in unbounded domains

We discuss the properties of the Wu pseudometric and present counterexamples for its upper semicontinuity that answers the question posed by Jarnicki and Pflug. We also give formulae for the Wu pseudometric in elementary Reinhardt domains. Received: 12 September 2007     

Certain invariants of extension algebras of C({\mathbb{T}}^{2})

In this paper, we compute certain invariants of extension algebras of the torus algebra by , where is the C*-algebra of compact operators on an infinite dimensional separable Hilbert space H. These extension algebras are also constructed up to isomorphism. Received: 5 July 2007, Revised: 14 February 2008     

A rigorous construction of maximum recoverable work

A simple and rigorous derivation of the maximum recoverable work is presented. In contrast to previous derivations it is based on simple and rigorous projectional methods. The principle holds if the stress has a non-trivial Newtonian component.      

Unbounded Toeplitz Operators

This partly expository article develops the basic theory of unbounded Toeplitz operators on the Hardy space H ², with emphasis on operators whose symbols are not square integrable. Unbounded truncated Toeplitz operators on coinvariant subspaces of H ² are also studied. In memory of Paul R. Halmos     

Smoluchowski-Kramers approximation for a general class of SPDEs

We prove that the so-called Smoluchowski-Kramers approximation holds for a class of partial differential equations perturbed by a non-Gaussian noisy term. Namely, we show that the solution of the one-dimensional semi-linear stochastic damped wave equations , u(0) = u₀, u_t (0) = v₀, endowed with Dirichlet boundary conditions, converges as the parameter μ goes to zero to the solution of the semi-linear stochastic heat equation , u(0) = u₀, endowed with Dirichlet boundary conditions. Dedicated to Giuseppe Da Prato on the occasion of his 70th birthday     

On end degrees and infinite cycles in locally finite graphs

We introduce a natural extension of the vertex degree to ends. For the cycle space C(G) as proposed by Diestel and Kühn [4, 5], which allows for infinite cycles, we prove that the edge set of a locally finite graph G lies in C(G) if and only if every vertex and every end has even degree. In the same way we generalise to locally finite graphs the characterisation of the cycles in a finite graph as its 2-regular connected subgraphs.     

Lempel–Ziv Factorization Using Less Time & Space

For 30 years the Lempel–Ziv factorization LZ _x of a string x = x[1..n] has been a fundamental data structure of string processing, especially valuable for string compression and for computing all the repetitions (runs) in x. Traditionally the standard method for computing LZ _x was based on Θ(n)-time (or, depending on the measure used, O(n log n)-time) processing of the suffix tree ST _x of x. Recently Abouelhoda et al. proposed an efficient Lempel–Ziv factorization algorithm based on an “enhanced” suffix array – that is, a suffix array SA _x together with supporting data structures, principally an “interval tree”. In this paper we introduce a collection of fast space-efficient algorithms for LZ factorization, also based on suffix arrays, that in theory as well as in many practical circumstances are superior to those previously proposed; one family out of this collection achieves true Θ(n)-time alphabet-independent processing in the worst case by avoiding tree structures altogether. The work of the first and third authors was supported in part by grants from the Natural Sciences & Engineering Research Council of Canada.     

On groups of type Φ

We define a group G to be of type Φ if it has the property that for every -module G, proj. G < ∞ iff proj. H G < ∞ for every finite subgroup H of G. We conjecture that the type Φ is an algebraic characterization of those groups G which admit a finite dimensional model for , the classifying space for the family of the finite subgroups of G. We also conjecture that the type Φ is equivalent to spli being finite, where spli is the supremum of the projective lengths of the injective -modules. Here we prove certain parts of these conjectures. The project is cofounded by the European Social Fund and National Resources–EPEAK II–Pythagoras. Received: 21 June 2006     

Note on limits of finite graphs

We prove that for any weakly convergent sequence of finite graphs with bounded vertex degrees, there exists a topological limit graphing.     

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.