Bulletin of the London Mathematical Society - Current Issue

Volume 38 Issue 06

Fri, 01 Dec 2006 00:00:00 GMT

Bulletin of the London Mathematical Society, Volume 38 Issue 06

A well-established journal with over thirty five years' coverage extending across the whole range of pure mathematics, together with some more applied areas of analysis, theoretical computing and mathematical physics. The Bulletin publishes important short research articles, authoritative survey articles and advanced expositions, often of an extensive nature, reviewing all major developments in an important area over many years. Occasional biographical articles are also published on the lives and mathematical achievements of distinguished mathematicians. The Bulletin has a substantial book review section, including books in applied mathematics and statistics as well as pure mathematics.

ANALYTIC RANK OF GRAPH ALGEBRAS AND QUANTUM SPACES

Research Articles
JEONG HEE HONG, WOJCIECH SZYMANSKI,
Bulletin of the London Mathematical Society, Volume 38 Issue 06 , pp 1000-1004

Abstract
We show that the analytic rank, as defined by Murphy, of a unital graph C -algebra is either 1 or 0, depending on whether or not the underlying graph possesses an initial loop. As a corollary, we show that the analytic rank of certain quantum spaces (including some quantum spheres, projective spaces and lens spaces) is 1.

CANCELLATION DOES NOT IMPLY STABLE RANK ONE

Research Articles
ANDREW S. TOMS,
Bulletin of the London Mathematical Society, Volume 38 Issue 06 , pp 1005-1008

Abstract
A unital C -algebra $A$ is said to have cancellation of projections if the semigroup $D(A)$ of Murray von Neumann equivalence classes of projections in matrices over $A$ is cancellative. It has long been known that stable rank one implies cancellation for any $A$, and some partial converses have been established. In this paper it is proved that cancellation does not imply stable rank one for simple, stably finite C -algebras.

COMMON FIXED POINTS OF ONE-PARAMETER NONEXPANSIVE SEMIGROUPS

Research Articles
TOMONARI SUZUKI,
Bulletin of the London Mathematical Society, Volume 38 Issue 06 , pp 1009-1018

Abstract
In this paper, we show that the set of all common fixed points of a one-parameter nonexpansive semigroup is that of some single nonexpansive mapping. We next compare our result with Bruck s famous fixed-point theorem. We finally prove very simple convergence theorems to a common fixed point. In our discussion, we assume neither the strict convexity of the underlying space, nor the weak compactness of the domain of a nonexpansive semigroup.

ON SINGULARITY OF ENERGY MEASURES ON SELF-SIMILAR SETS II

Research Articles
MASANORI HINO, KENJI NAKAHARA,
Bulletin of the London Mathematical Society, Volume 38 Issue 06 , pp 1019-1032

Abstract
We provide a criterion for the energy measures of regular Dirichlet forms on post-critically finite self-similar sets to be singular with respect to every self-similar measure. A probabilistic interpretation of the singularity is also discussed.

UNIQUENESS AND NONEXISTENCE OF POSITIVE SOLUTIONS TO SEMIPOSITONE PROBLEMS

Research Articles
E. NORMAN DANCER, JUNPING SHI,
Bulletin of the London Mathematical Society, Volume 38 Issue 06 , pp 1033-1044

Abstract
We consider the uniqueness of the positive solution to a semilinear elliptic equation with Dirichlet boundary condition and the nonlinearity satisfying $f(0) and having asymptotic sublinear growth rate. A similar idea is also applied to the nonexistence of a positive solution to a superlinear problem.

SHARP GRADIENT ESTIMATE AND YAU'S LIOUVILLE THEOREM FOR THE HEAT EQUATION ON NONCOMPACT MANIFOLDS

Research Articles
PHILIPPE SOUPLET, QI S. ZHANG,
Bulletin of the London Mathematical Society, Volume 38 Issue 06 , pp 1045-1053

Abstract
We derive a sharp, localized version of elliptic type gradient estimates for positive solutions (bounded or not) to the heat equation. These estimates are related to the Cheng Yau estimate for the Laplace equation and Hamilton s estimate for bounded solutions to the heat equation on compact manifolds. As applications, we generalize Yau s celebrated Liouville theorem for positive harmonic functions to positive ancient (including eternal) solutions of the heat equation, under certain growth conditions. Surprisingly this Liouville theorem for the heat equation does not hold even in ${\mathbb R}^n$ without such a condition. We also prove a sharpened long-time gradient estimate for the log of the heat kernel on noncompact manifolds.

THE DIRECT METHOD IN SOLITON THEORY (Cambridge Tracts in Mathematics 155) By R YOGO H IROTA (translated and edited by A TSUSHI N AGAI , J ON N IMMO and C LAIRE G ILSON ): 200 pp., £40.00 (US$65.00), ISBN 0-521-83660-3 (Cambridge University Press, 2004)

Book Reviews
PETER CLARKSON,
Bulletin of the London Mathematical Society, Volume 38 Issue 06 , pp 1054-1056

Abstract

PROGRESS AND PROBLEMS IN THE THEORY OF REGENERATIVE PHENOMENA

Research Articles
J. F. C. KINGMAN,
Bulletin of the London Mathematical Society, Volume 38 Issue 06 , pp 881-896

Abstract
Regenerative phenomena were introduced some forty years ago to address problems in the theory of continuous-time Markov processes. The early work in the theory left a number of difficult unsolved problems, in the classification of $p$-functions, oscillation and inequalities, the multiplicative theory, and the theory of unbounded semi-$p$-functions. Recent years have shown progress on all of these fronts, and this paper surveys these results, while drawing attention to significant problems that remain open.

OVERCONVERGENT REAL CLOSED QUANTIFIER ELIMINATION

Research Articles
L. LIPSHITZ, Z. ROBINSON,
Bulletin of the London Mathematical Society, Volume 38 Issue 06 , pp 897-906

Abstract
Let $K$ be the (real closed) field of Puiseux series in $t$ over ${\mathbf R}$ endowed with the natural linear order. Then the elements of the formal power series rings ${\mathbf R}[\![\xi_1,\dots,\xi_n]\!]$ converge $t$-adically on $[-t,t]^n$, and hence define functions $[-t,t]^n\to K$. Let ${\mathcal L}$ be the language of ordered fields, enriched with symbols for these functions. By Corollary 3.15, $K$ is o-minimal in ${\mathcal L}$. This result is obtained from a quantifier elimination theorem. The proofs use methods from non-Archimedean analysis.

VON NEUMANN'S PROBLEM AND LARGE CARDINALS

Research Articles
ILIJAS FARAH, BOBAN VELICKOVIC,
Bulletin of the London Mathematical Society, Volume 38 Issue 06 , pp 907-912

Abstract
It is a well-known problem of Von Neumann to discover whether the countable chain condition and weak distributivity of a complete Boolean algebra imply that it carries a strictly positive probability measure. It was shown recently by Balcar, Jech and Paz k, and by Veli kovi , that it is consistent with ZFC, modulo the consistency of a supercompact cardinal, that every ccc weakly distributive complete Boolean algebra carries a contiuous strictly positive submeasure that is, it is a Maharam algebra. We use some ideas of Gitik and Shelah and implications from the inner model theory to show that some large cardinal assumptions are necessary for this result.

THE BERGÉ–MARTINET CONSTANT AND SLOPES OF SIEGEL CUSP FORMS

Research Articles
CRIS POOR, DAVID S. YUEN,
Bulletin of the London Mathematical Society, Volume 38 Issue 06 , pp 913-924

Abstract
We give a theoretical lower bound for the slope of a Siegel modular cusp form that is as least as good as Eichler s lower bound. In degrees $n=5,6$ and 7 we show that our new bound is strictly better. In the process we find the forms of smallest dyadic trace on the perfect core for ranks $n \le 8$. In degrees $n=5,6$ and 7 we settle the value of the generalized Hermite constant $\gamma_n'$ introduced by Berg and Martinet and find all dual-critical pairs.

THE WEIERSTRASS SUBGROUP OF A CURVE HAS MAXIMAL RANK

Research Articles
MARTINE GIRARD, DAVID R. KOHEL, CHRISTOPHE RITZENTHALER,
Bulletin of the London Mathematical Society, Volume 38 Issue 06 , pp 925-931

Abstract
We show that the Weierstrass points of the generic curve of genus $g$ over an algebraically closed field of characteristic 0 generate a group of maximal rank in the Jacobian.

EVERY COUNTABLE GROUP HAS THE WEAK ROHLIN PROPERTY

Research Articles
E. GLASNER, J.-P. THOUVENOT, B. WEISS,
Bulletin of the London Mathematical Society, Volume 38 Issue 06 , pp 932-936

Abstract
We present a simple proof of the fact that every countable group ${\Gamma}$ is weak Rohlin, that is, there is in the Polish space $mathbb{A}_{\Gamma}$ of measure preserving ${\Gamma}$-actions an action $\mathbf{T}$ whose orbit in $\mathbb{A}_{\Gamma}$ under conjugations is dense. In conjunction with earlier results this in turn yields a new characterization of non-Kazhdan groups as those groups which admit such an action $\mathbf{T}$ which is also ergodic.

WANDERING DOMAINS IN NON-ARCHIMEDEAN POLYNOMIAL DYNAMICS

Research Articles
ROBERT L. BENEDETTO,
Bulletin of the London Mathematical Society, Volume 38 Issue 06 , pp 937-950

Abstract
We extend a recent result on the existence of wandering domains of polynomial functions defined over the $p$-adic field ${\mathbb{C}_p}$ to any algebraically closed complete non-archimedean field ${\mathbb{C}_K}$ with residue characteristic $p>0$. We also prove that polynomials with wandering domains form a dense subset of a certain one-dimensional family of degree $p+1$ polynomials in ${\mathbb{C}_K}[z]$.

BOUNDS FOR GRADIENT TRAJECTORIES AND GEODESIC DIAMETER OF REAL ALGEBRAIC SETS

Research Articles
D. D'ACUNTO, K. KURDYKA,
Bulletin of the London Mathematical Society, Volume 38 Issue 06 , pp 951-965

Abstract
Let $M\subset \mathbb{R}^n$ be a connected component of an algebraic set $\varphi^{-1}(0)$, where $\varphi$ is a polynomial of degree $d$. Assume that $M$ is contained in a ball of radius $r$. We prove that the geodesic diameter of $M$ is bounded by $2r\nu(n)d(4d-5)^{n-2}$, where $\nu(n)=2{\Gamma({1}/{2})\Gamma(({n+1})/{2})}{\Gamma({n}/{2})}^{-1}$. This estimate is based on the bound $r\nu(n)d(4d-5)^{n-2}$ for the length of the gradient trajectories of a linear projection restricted to $M$.

DEFORMATIONS OF FUNCTIONS AND $F$-MANIFOLDS

Research Articles
IGNACIO DE GREGORIO,
Bulletin of the London Mathematical Society, Volume 38 Issue 06 , pp 966-978

Abstract
We study deformations of functions on isolated singularities. A unified proof of the equality of Milnor and Tjurina numbers for functions on isolated complete intersections singularities and space curves is given. As a consequence, the base space of their miniversal deformations is endowed with the structure of an $F$-manifold, and we can prove a conjecture of V. Goryunov, stating that the critical values of the miniversal unfolding of a function on a space curve are generically local coordinates on the base space of the deformation.

EMBEDDING $\ell_{\infty}$ INTO THE SPACE OF BOUNDED OPERATORS ON CERTAIN BANACH SPACES

Research Articles
G. ANDROULAKIS, K. BEANLAND, S. J. DILWORTH, F. SANACORY,
Bulletin of the London Mathematical Society, Volume 38 Issue 06 , pp 979-990

Abstract
Sufficient conditions are given on a Banach space $X$ which ensure that $\ell_{\infty}$ embeds in ${\mathcal L}\, (X)$, the space of all bounded linear operators on $X$. A basic sequence $(e_n)$ is said to be quasisubsymmetric if for any two increasing sequences $(k_n)$ and $(\ell_n)$ of positive integers with $k_n \leq \ell_n$ for all $n$, $(e_{k_n})$ dominates $(e_{\ell_n})$. If a Banach space $X$ has a seminormalized quasisubsymmetric basis then $\ell_{\infty}$ embeds in ${\mathcal L}\, (X)$.

EXTENSIONS OF BOHR'S INEQUALITY

Research Articles
VERN I. PAULSEN, DINESH SINGH,
Bulletin of the London Mathematical Society, Volume 38 Issue 06 , pp 991-999

Abstract
We obtain operator-valued analogues of Bohr s inequality involving both the absolute values of operators and their norms, that when restricted to the scalar case imply the classical Bohr inequality. In the scalar case we extend Bohr s inequality to the case where one function is majorized by another function, to the Hardy space, $H^2(\mathbb{D})$, and to bounded analytic functions on the annulus.