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Volume 51 Issue 02

Sat, 31 May 2008 23:00:00 GMT

Proceedings of the Edinburgh Mathematical Society, Volume 51 Issue 02

The Edinburgh Mathematical Society was founded in 1883 and over the years, has evolved into the principal society for the promotion of mathematics research in Scotland. The Society has published its Proceedings since 1884. This journal contains research papers on topics in a broad range of pure and applied mathematics, together with a number of topical book reviews.

ON THE REGULARITY CONJECTURE FOR THE COHOMOLOGY OF FINITE GROUPS

Research Articles
David J. Benson,
Proceedings of the Edinburgh Mathematical Society, Volume 51 Issue 02 , pp 273-284

Abstract
Let $K$ be a field of characteristic $p$ and let $G$ be a finite group of order divisible by $p$. The regularity conjecture states that the Castelnuovo Mumford regularity of the cohomology ring $H^*(G,K)$ is always equal to 0. We prove that if the regularity conjecture holds for a finite group $H$, then it holds for the wreath product $H\wr\mathbb{Z}/p$. As a corollary, we prove the regularity conjecture for the symmetric groups $\varSigma_n$. The significance of this is that it is the first set of examples for which the regularity conjecture has been checked, where the difference between the Krull dimension and the depth of the cohomology ring is large. If this difference is at most 2, the regularity conjecture is already known to hold by previous work.For more general wreath products, we have not managed to prove the regularity conjecture. Instead we prove a weaker statement: namely, that the dimensions of the cohomology groups are polynomial on residue classes (PORC) in the sense of Higman.

UNBOUNDED B-FREDHOLM OPERATORS ON HILBERT SPACES

Research Articles
M. Berkani, N. Castro-González,
Proceedings of the Edinburgh Mathematical Society, Volume 51 Issue 02 , pp 285-296

Abstract
This paper is concerned with the study of a class of closed linear operators densely defined on a Hilbert space $H$ and called B-Fredholm operators. We characterize a B-Fredholm operator as the direct sum of a Fredholm closed operator and a bounded nilpotent operator. The notion of an index of a B-Fredholm operator is introduced and a characterization of B-Fredholm operators with index $0$ is given in terms of the sum of a Drazin closed operator and a finite-rank operator. We analyse the properties of the powers $T^m$ of a closed B-Fredholm operator and we establish a spectral mapping theorem.

A NOTE ON RANDOM HOLOMORPHIC ITERATION IN CONVEX DOMAINS

Research Articles
Filippo Bracci,
Proceedings of the Edinburgh Mathematical Society, Volume 51 Issue 02 , pp 297-304

Abstract
We introduce a geometric condition of Bloch type which guarantees that a subset of a bounded convex domain in several complex variables is degenerate with respect to every iterated function system. Furthermore, we discuss the relations of such a Bloch-type condition with the analogous hyperbolic Lipschitz condition.

A CHARACTERIZATION OF COMPOSITION OPERATORS ON ALGEBRAS OF ANALYTIC FUNCTIONS

Research Articles
Daniel Carando,
Proceedings of the Edinburgh Mathematical Society, Volume 51 Issue 02 , pp 305-313

Abstract
We give a characterization of composition operators between algebras of analytic functions on a Banach space. We show (under fairly general conditions) that they are precisely the multiplicative operators that are transposes of operators of between the preduals of the algebras. The special cases of $H^\infty(U)$ and $H_{\mathrm{b}}(U)$ are considered. In these cases, the composition operators are those which are pointwise-to-pointwise continuous and or $\tau_0$-to-$\tau_0$ continuous (where $\tau_0$ is the compact-open topology). We obtain Banach Stone-type theorems for these algebras.

GREEN'S FUNCTIONS AND REGULARIZED TRACES OF STURM–LIOUVILLE OPERATORS ON GRAPHS

Research Articles
Sonja Currie, Bruce A. Watson,
Proceedings of the Edinburgh Mathematical Society, Volume 51 Issue 02 , pp 315-335

Abstract
Asymptotic approximations to the Green s functions of Sturm Liouville boundary-value problems on graphs are obtained. These approximations are used to study the regularized traces of the differential operators associated with these boundary-value problems. Various inverse spectral problems for Sturm Liouville boundary-value problems on graphs resembling those considered in Halberg and Kramer s A generalization of the trace concept (Duke Mathematics Journal 27 (1960), 607 617), for Sturm Liouville problems, and Pielichowski s An inverse spectral problem for linear elliptic differential operators (Universitatis Iagellonicae Acta Mathematica 27 (1988), 239 246), for elliptic boundary-value problems, are solved.

CONSTRUCTION OF ELLIPTIC DIFFUSIONS WITH REFLECTING BOUNDARY CONDITION AND AN APPLICATION TO CONTINUOUS N -PARTICLE SYSTEMS WITH SINGULAR INTERACTIONS

Research Articles
Torben Fattler, Martin Grothaus,
Proceedings of the Edinburgh Mathematical Society, Volume 51 Issue 02 , pp 337-362

Abstract
We give a Dirichlet form approach for the construction and analysis of elliptic diffusions in $\bar{\varOmega}\subset\mathbb{R}^n$ with reflecting boundary condition. The problem is formulated in an $L^2$-setting with respect to a reference measure $\mu$ on $\bar{\varOmega}$ having an integrable, $\mathrm{d} x$-almost everywhere (a.e.) positive density $\varrho$ with respect to the Lebesgue measure. The symmetric Dirichlet forms $(\mathcal{E}^{\varrho,a},D(\mathcal{E}^{\varrho,a}))$ we consider are the closure of the symmetric bilinear forms\begin{gather*}\mathcal{E}^{\varrho,a}(f,g)=\sum_{i,j=1}^n\int_{\varOmega}\partial_ifa_{ij} \partial_jg\,\mathrm{d}\mu,\quad f,g\in\mathcal{D},\\\mathcal{D}=\{f\in C(\bar{\varOmega})\mid f\in W^{1,1}_{\mathrm{loc}}(\varOmega),\ \mathcal{E}^{\varrho,a}(f,f)in $L^2(\bar{\varOmega},\mu)$, where $a$ is a symmetric, elliptic, $n\times n$-matrix-valued measurable function on $\bar{\varOmega}$. Assuming that $\varOmega$ is an open, relatively compact set with boundary $\partial\varOmega$ of Lebesgue measure zero and that $\varrho$ satisfies the Hamza condition, we can show that $(\mathcal{E}^{\varrho,a},D(\mathcal{E}^{\varrho,a}))$ is a local, quasi-regular Dirichlet form. Hence, it has an associated self-adjoint generator $(L^{\varrho,a},D(L^{\varrho,a}))$ and diffusion process $\bm{M}^{\varrho,a}$ (i.e. an associated strong Markov process with continuous sample paths). Furthermore, since $1\in D(\mathcal{E}^{\varrho,a})$ (due to the Neumann boundary condition) and $\mathcal{E}^{\varrho,a}(1,1)=0$, we obtain a conservative process $\bm{M}^{\varrho,a}$ (i.e. $\bm{M}^{\varrho,a}$ has infinite lifetime). Additionally, assuming that $\sqrt{\varrho}\in W^{1,2}(\varOmega)\cap C(\bar{\varOmega})$ or that $\varrho$ is bounded, $\varOmega$ is convex and $\{\varrho=0\}$ has codimension at least 2, we can show that the set $\{\varrho=0\}$ has $\mathcal{E}^{\varrho,a}$-capacity zero. Therefore, in this case we can even construct an associated conservative diffusion process in $\{\varrho>0\}$. This is essential for our application to continuous $N$-particle systems with singular interactions. Note that for the construction of the self-adjoint generator $(L^{\varrho,a},D(L^{\varrho,a}))$ and the Markov process $\bm{M}^{\varrho,a}$ we do not need to assume any differentiability condition on $\varrho$ and $a$. We obtain the following explicit representation of the generator for $\sqrt{\varrho}\in W^{1,2}(\varOmega)$ and $a\in W^{1,\infty}(\varOmega)$:$$L^{\varrho,a}=\sum_{i,j=1}^n\partial_i(a_{ij}\partial_j)+\partial_i(\log\varrho)a_{ij}\partial_j.$$Note that the drift term can be singular, because we allow $\varrho$ to be zero on a set of Lebesgue measure zero. Our assumptions in this paper even allow a drift that is not integrable with respect to the Lebesgue measure.

TORSION UNITS IN INTEGRAL GROUP RINGS OF CERTAIN METABELIAN GROUPS

Research Articles
Martin Hertweck,
Proceedings of the Edinburgh Mathematical Society, Volume 51 Issue 02 , pp 363-385

Abstract
It is shown that any torsion unit of the integral group ring $\mathbb{Z}G$ of a finite group $G$ is rationally conjugate to an element of $\pm G$ if $G=XA$ with $A$ a cyclic normal subgroup of $G$ and $X$ an abelian group (thus confirming a conjecture of Zassenhaus for this particular class of groups, which comprises the class of metacyclic groups).

ON AN ORDER-BASED CONSTRUCTION OF A TOPOLOGICAL GROUPOID FROM AN INVERSE SEMIGROUP

Research Articles
Daniel H. Lenz,
Proceedings of the Edinburgh Mathematical Society, Volume 51 Issue 02 , pp 387-406

Abstract
We show how to construct a topological groupoid directly from an inverse semigroup and prove that it is isomorphic to the universal groupoid introduced by Paterson. We then turn to a certain reduction of this groupoid. In the case of inverse semigroups arising from graphs (respectively, tilings), we prove that this reduction is the graph groupoid introduced by Kumjian \et (respectively, the tiling groupoid of Kellendonk). We also study the open invariant sets in the unit space of this reduction in terms of certain order ideals of the underlying inverse semigroup. This can be used to investigate the ideal structure of the associated reduced $C^\ast$-algebra.

ON EIGENVALUE PROBLEMS FOR ELLIPTIC HEMIVARIATIONAL INEQUALITIES

Research Articles
Zhenhai Liu, Guifang Liu,
Proceedings of the Edinburgh Mathematical Society, Volume 51 Issue 02 , pp 407-419

Abstract
This paper is devoted to the Dirichlet problem for quasilinear elliptic hemivariational inequalities at resonance as well as at non-resonance. Using Clarke s notion of the generalized gradient and the property of the first eigenfunction, we also build a Landesman Lazer theory in the non-smooth framework of quasilinear elliptic hemivariational inequalities.

BIMINIMAL IMMERSIONS

Research Articles
E. Loubeau, S. Montaldo,
Proceedings of the Edinburgh Mathematical Society, Volume 51 Issue 02 , pp 421-437

Abstract
We study biminimal immersions: that is, immersions which are critical points of the bienergy for normal variations with fixed energy. We give a geometrical description of the Euler Lagrange equation associated with biminimal immersions for both biminimal curves in a Riemannian manifold, with particular attention given to the case of curves in a space form, and isometric immersions of codimension 1 in a Riemannian manifold, in particular for surfaces of a three-dimensional manifold. We describe two methods of constructing families of biminimal surfaces using both Riemannian and horizontally homothetic submersions.

ON THE HOLLAND–WALSH CHARACTERIZATION OF BLOCH FUNCTIONS

Research Articles
Miroslav Pavlovic,
Proceedings of the Edinburgh Mathematical Society, Volume 51 Issue 02 , pp 439-441

Abstract
It is proved that the Bloch norm of an arbitrary $C^1$-function defined on the unit ball $\mathbb{B}_n\subset\mathbb{R}^n$ is equal to$$\sup_{x,y\in\mathbb{B}_n,\,x\neq y}{(1-|x|^2)^{1/2}(1-|y|^2)^{1/2}}\frac{|f(x)-f(y)|}{|x-y|}.$$

BOUNDEDNESS OF FOURIER INTEGRAL OPERATORS ON HARDY SPACES

Research Articles
Marco M. Peloso, Silvia Secco,
Proceedings of the Edinburgh Mathematical Society, Volume 51 Issue 02 , pp 443-463

Abstract
For $0, let $h^p(\mathbb{R}^n)$ denote the local Hardy space. Let $\mathcal{F}$ be a Fourier integral operator defined by the oscillatory integral$$\mathcal{F}f(x)=\iint_{\mathbb{R}^{2n}}\exp(2\pi\mathrm{i}(\phi(x,\xi)-y\cdot\xi))b(x,y,\xi)f(y)\,\mathrm{d} y\,\mathrm{d}\xi,$$where $\phi$ is a $\mathcal{C}^\infty$ non-degenerate real phase function, and $b$ is a symbol of order $\mu$ and type $(\rho,1-\rho)$, $\sfrac12, vanishing for $x$ outside a compact set of $\mathbb{R}^n$. We show that when $p\le1$ and $\mu\le-(n-1)(1/p-1/2)$ then $\mathcal{F}$ initially defined on Schwartz functions in $h^p(\mathbb{R}^n)$ extends to a bounded operator $\mathcal{F}:h^p(\mathbb{R}^n)\rightarrow h^p(\mathbb{R}^n)$. The range of $p$ and $\mu$ is sharp. This result extends to the local Hardy spaces the seminal result of Seeger \et for the $L^p$ spaces. As immediate applications we prove the boundedness of smooth Radon transforms on hypersurfaces with non-vanishing Gaussian curvature on the local Hardy spaces.Finally, we prove a local version for the boundedness of Fourier integral operators on local Hardy spaces on smooth Riemannian manifolds of bounded geometry.

RETRACTIVE TRANSFERS AND p -LOCAL FINITE GROUPS

Research Articles
Kári Ragnarsson,
Proceedings of the Edinburgh Mathematical Society, Volume 51 Issue 02 , pp 465-487

Abstract
In this paper we explore the possibility of defining $p$-local finite groups in terms of transfer properties of their classifying spaces. More precisely, we consider the question, posed by Haynes Miller, of whether an equivalent theory can be obtained by studying triples $(f,t,X)$, where $X$ is a $p$-complete, nilpotent space with a finite fundamental group, $f:BS\to X$ is a map from the classifying space of a finite $p$-group, and $t$ is a stable retraction of $f$ satisfying Frobenius reciprocity at the level of stable homotopy. We refer to $t$ as a retractive transfer of $f$ and to $(f,t,X)$ as a retractive transfer triple over $S$.In the case where $S$ is elementary abelian, we answer this question in the affirmative by showing that a retractive transfer triple $(f,t,X)$ over $S$ does indeed induce a $p$-local finite group over $S$ with $X$ as its classifying space.Using previous results obtained by the author, we show that the converse is true for general finite $p$-groups. That is, for a $p$-local finite group $(S,\mathcal{F},\mathcal{L})$, the natural inclusion $\theta:BS\to X$ has a retractive transfer $t$, making $(\theta,t,|\mathcal{L}|^{\wedge}_p)$ a retractive transfer triple over $S$. This also requires a proof, obtained jointly with Ran Levi, that $|\mathcal{L}|^{\wedge}_p$ is a nilpotent space, which is of independent interest.

PROPERTIES OF THE MAXIMAL OPERATORS ASSOCIATED WITH BASES OF RECTANGLES IN $\mathbb{R}^3$

Research Articles
Alexander Stokolos,
Proceedings of the Edinburgh Mathematical Society, Volume 51 Issue 02 , pp 489-494

Abstract
This paper is an attempt to understand a phenomenon of maximal operators associated with bases of three-dimensional rectangles of dimensions $(t,1/t,s)$ within a framework of more general Soria bases. The Jessen Marcinkiewicz Zygmund Theorem implies that the maximal operator associated with a Soria basis continuously maps $L\log^2L$ into $L^{1,\infty}$. We give a simple geometric condition that guarantees that the $L\log^2L$ class cannot be enlarged. The proof develops the author s methods applied previously in the two-dimensional case and is related to theorems of C rdoba, Soria and Fefferman and Pipher.

GLOBAL ATTRACTIVITY IN A PREDATOR–PREY SYSTEM WITH PURE DELAYS

Research Articles
X. H. Tang, Xingfu Zou,
Proceedings of the Edinburgh Mathematical Society, Volume 51 Issue 02 , pp 495-508

Abstract
We consider a delay predator prey system without instantaneous negative feedback and establish some conditions for global attractivity of the positive equilibrium of the system which generalize and improve some of the existing ones. When the system is decoupled, one of the main results reduces to the well-known Wright 3/2 stability condition for the delayed logistic equation.

APPLICATIONS OF VARIATIONAL METHODS TO BOUNDARY-VALUE PROBLEM FOR IMPULSIVE DIFFERENTIAL EQUATIONS

Research Articles
Yu Tian, Weigao Ge,
Proceedings of the Edinburgh Mathematical Society, Volume 51 Issue 02 , pp 509-527

Abstract
In this paper, we investigate the existence of positive solutions to a second-order Sturm Liouville boundary-value problem with impulsive effects. The ideas involve differential inequalities and variational methods.

ORLICZ–POINCARÉ INEQUALITIES

Research Articles
Feng-Yu Wang,
Proceedings of the Edinburgh Mathematical Society, Volume 51 Issue 02 , pp 529-543

Abstract
Corresponding to known results on Orlicz Sobolev inequalities which are stronger than the Poincar inequality, this paper studies the weaker Orlicz Poincar inequality. More precisely, for any Young function $\varPhi$ whose growth is slower than quadric, the Orlicz Poincar inequality$$\|f\|_\varPhi^2\le C\E(f,f),\qquad\mu(f):=\int f\,\mathrm{d}\mu=0$$is studied by using the well-developed weak Poincar inequalities, where $\E$ is a conservative Dirichlet form on $L^2(\mu)$ for some probability measure $\mu$. In particular, criteria and concrete sharp examples of this inequality are presented for $\varPhi(r)=r^p$ $(p\in[1,2))$ and $\varPhi(r)= r^2\log^{-\delta}(\mathrm{e} +r^2)$ $(\delta>0)$. Concentration of measures and analogous results for non-conservative Dirichlet forms are also obtained. As an application, the convergence rate of porous media equations is described.