Decompositions of finitely generated nilpotent groups of class 2

In this paper we prove that rational indecomposability is a genus property for finitely generated torsion-free nilpotent groups of class 2. We use this result to determine the genus of finitely generated torsion-free nilpotent groups of class 2 which decompose as a direct product of rationally indecomposable groups. Received: 3 November 2005     

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     

From Lucid to TransLucid: Iteration, Dataflow, Intensional and Cartesian Programming

We present the development of the Lucid language from the Original Lucid of the mid-1970s to the TransLucid of today. Each successive version of the language has been a generalisation of previous languages, but with a further understanding of the problems at hand. The Original Lucid (1976), originally designed for purposes of formal verification, was used to formalise the iteration in while-loop programs. The pLucid language (1982) was used to describe dataflow networks. Indexical Lucid (1987) was introduced for intensional programming, in which the semantics of a variable was understood as a function from a universe of possible worlds to ordinary values. With TransLucid, and the use of contexts as firstclass values, programming can be understood in a Cartesian framework.      

Special matchings and Coxeter groups

In the recent paper [Adv. Applied Math., 38 (2007), 210–226] it is proved that the special matchings of permutations generate a Coxeter group. In this paper we generalize this result to a class of Coxeter groups which includes many Weyl and affine Weyl groups. Our proofs are simpler, and shorter, than those in [loc. cit.] All authors are partially supported by EU grant HPRN-CT-2001-00272. Received: 30 October 2006     

Blow-up and critical exponents for parabolic equations with non-divergent operators: dual porous medium and thin film operators

Our first basic model is the fully nonlinear dual porous medium equation with source

The Nielsen number for free fundamental groups and maps without remnant

For Y any space that has the homotopy type of a wedge of finitely many circles, and for g : Y → Y a map, the Nielsen number of g, N(g), is a homotopy invariant lower bound for the size of the fixed point set of any map homotopic to g. Such a map g has k-remnant if, roughly, there is limited cancellation in any product g _♯(u)g _♯(v) where g _♯ is the induced homomorphism and u, v ∈ π₁(Y) and |u| = |v| = k. We prove that such maps are (k + 1)-characteristic, meaning that in order to determine the Nielsen classes of fixed points, we need only test whether a limited, specified, set of elements z ∈ π₁(Y) are solutions to the equation z = W ⁻¹ _x f _♯(z)W _y , with x and y fixed points that are represented in the fundamental group by W _x and W _y , respectively. The number of elements to be tested is profoundly decreased by using abelianization as well. This work is a significant extension of Wagner’s results involving maps with remnant and Wagner’s algorithm. Our proofs involve new concepts and techniques. We present an algorithm for N(g) for any map g with k-remnant, and we provide examples for which no algebraic techniques previously known would work. One example shows that for any k there is a map that does not have (k − 1)-remnant but does have k-remnant. Dedicated to Edward Fadell for inspirational teaching and guidance as the thesis advisor of the first author     

Pro-p groups of rank 3 and the question of Iwasawa

For a prime p > 3 we determine all pro-p groups that satisfy d(G) = d(H) = 3 for all open subgroups H of G. Received: 29 July 2008     

The Van den Bergh duality and the modular symmetry of a Poisson variety

We consider a smooth Poisson affine variety with the trivial canon-ical bundle over . For such a variety the deformation quantization algebra obeys the conditions of the Van den Bergh duality theorem and the corresponding dualizing module is determined by an outer automorphism of intrinsic to . We show how this automorphism can be expressed in terms of the modular class of the corresponding Poisson variety. We also prove that the Van den Bergh dualizing module of the deformation quantization algebra is free if and only if the corresponding Poisson structure is unimodular.      

Refinements of the generalised trapezoid and Ostrowski inequalities for functions of bounded variation

Refinements of the generalised trapezoid and Ostrowski inequalities for functions of bounded variation are given. Applications for the trapezoid and mid-point inequalities are also provided. Received: 19 May 2008     

Smoothness of Hessenberg and Bidiagonal Forms

Our purpose in this work is to explore smoothness properties of transformations of a matrix valued function A to Hessenberg and bidiagonal form. The interplay with the rank of associated Krylov functions is exploited to clarify what one should expect for smooth functions A satisfying generic properties. This work was supported in part under INDAM-GNCS and MIUR Rome-Italy.     

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.      

Field induced aggregation in electrorheological suspension: kernel form and self similar solutions

Within the framework of the study of the fibrillation mechanism in an electrorheological (ER) suspension, this work presents a comparison between the self similar solutions when the kernel is K_i,j ~ (i⁻¹ + j⁻¹) and the behaviour of the chains growth. Till now, the field induced chains formation has only been studied by numerical or experimental methods. The work of Fournier and Lauren?ot (Communications in Mathematical Physics 256 2005) on the Smoluchowski’s equation allows us to present an analytical solution for the field induced pearl chains in a colloidal ER suspension. René Limage: Chercheur indépendant, dipl?mé de l’Université de Liége.     

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     

Symmetric generation of Coxeter groups

We provide involutory symmetric generating sets of finitely generated Coxeter groups, fulfilling a suitable finiteness condition, which in particular is fulfilled in the finite, affine and compact hyperbolic cases.      

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     

Colored graphs without colorful cycles

A colored graph is a complete graph in which a color has been assigned to each edge, and a colorful cycle is a cycle in which each edge has a different color. We first show that a colored graph lacks colorful cycles iff it is Gallai, i.e., lacks colorful triangles. We then show that, under the operation mon ≡ m + n − 2, the omitted lengths of colorful cycles in a colored graph form a monoid isomorphic to a submonoid of the natural numbers which contains all integers past some point. We prove that several but not all such monoids are realized. We then characterize exact Gallai graphs, i.e., graphs in which every triangle has edges of exactly two colors. We show that these are precisely the graphs which can be iteratively built up from three simple colored graphs, having 2, 4, and 5 vertices, respectively. We then characterize in two different ways the monochromes, i.e., the connected components of maximal monochromatic subgraphs, of exact Gallai graphs. The first characterization is in terms of their reduced form, a notion which hinges on the important idea of a full homomorphism. The second characterization is by means of a homomorphism duality. The first author would like to express his thanks for support by project LN 00A056 of the Ministry of Education of the Czech Republic. The second author would like to express his thanks for support by project LN 00A056 of the Ministry of Education of the Czech Republic, by the NSERC of Canada and by the Gudder Trust of the University of Denver.     

The distribution of values of the enumerating function of finite, non-isomorphic abelian groups in short intervals

The result is: The distribution of values of the enumerating function of finite, non-isomorphic abelian groups in short intervals is similar to the distribution of square-free numbers in short intervals. Moreover, a new estimate of the error term in the corresponding asymptotic formula is given, which improves former estimates.      

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

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.