Showing papers in "Journal of The London Mathematical Society-second Series in 2002"

Iterative Algorithms for Nonlinear Operators

Abstract: Iterative algorithms for nonexpansive mappings and maximal monotone operators are investigated. Strong convergence theorems are proved for nonexpansive mappings, including an improvement of a result of Lions. A modification of Rockafellar’s proximal point algorithm is obtained and proved to be always strongly convergent. The ideas of these algorithms are applied to solve a quadratic minimization problem.

Sharp weighted estimates for multilinear commutators

Abstract: Multilinear commutators with vector symbol defined by \[ T_{\vec{b}}(f)(x)=\int_{{\bb R}^n}\Bigg[\prod\limits^m_{j=1}(b_j(x)-b_j(y))\Bigg]K(x,y)f(y)dy \] are considered, where is a Calderon–Zygmund kernel. The following a priori estimates are proved for . For , there exists a constant such that \[ \|\dot{T}_{{\vec{b}}}(f)\|_{L^P(w)}\le C\|\vec{b}\|\|M_{L(\log\,L)^{1/r}}(f)\|_{L^P(w)} \] and \[ \sup_{t>0}\frac{1}{\Phi(\frac{1}{t})}w(\{y\in{\bb R}^n:|T_{\vec{b}}f(y)|>t\})\le C\sup_{t>0}\frac{1}{\Phi(\frac{1}{t})}w(\{y\in{\bb R}^n:M_{L(\log\,L)^{1/r}}(\|\vec{b}\|f)(y)>t\}), \] where \begin{eqnarray*} &\|\vec{b}\|=\prod\limits^m_{j=1}\|b_j\|_{osc_{\exp L}^r j},\\ &\Phi(t)=t\log^{1/r}(e+t),\quad \frac{1}{r}=\frac{1}{r_1}+\cdots+\frac{1}{r_m}, \end{eqnarray*} and is an Orlicz type maximal operator. This extends, with a different approach, classical results by Coifman.As a corollary, it is deduced that the operators are bounded on when , and that they satisfy corresponding weighted -type estimates with .

Exponential sums equations and the schanuel conjecture

Abstract: A uniform version of the Schanuel conjecture is discussed that has some model-theoretical motivation. This conjecture is assumed, and it is proved that any ‘non-obviously-contradictory’ system of equations in the form of exponential sums with real exponents has a solution.

The Spectral Set Conjecture and Multiplicative Properties of Roots of Polynomials

Abstract: Fuglede's conjecture states that a set tiles by translations if and only if has an orthogonal basis of exponentials. New partial results are obtained supporting the conjecture in dimension 1.

Transitive Permutation Groups Without Semiregular Subgroups

Abstract: A transitive nite permutation group is called elusive if it contains no nontrivial semiregular subgroup. The purpose of the paper is to collect known information about elusive groups. The main results are recursive constructions of elusive permutation groups, using various product operations and ane group constructions. A brief historical introduction and a survey of known elusive groups are also included. In a sequel, Giudici has determined all the quasiprimitive elusive groups. Part of the motivation for studying this class of groups was a conjecture due to Maru si c, Jordan and Klin asserting that there is no elusive 2-closed permutation group. It is shown that the constructions given will not build counterexamples to this conjecture.

Finsler Metrics of Constant Positive Curvature on the Lie Group S3

Abstract: Guided by the Hopf fibration, a family (indexed by a positive constant ) of right invariant Riemannian metrics on the Lie group is singled out. Using the Yasuda–Shimada paper as an inspiration, a privileged right invariant Killing field of constant length is determined for each . Each such Riemannian metric couples with the corresponding Killing field to produce a -global and explicit Randers metric on . Employing the machinery of spray curvature and Berwald's formula, it is proved directly that the said Randers metric has constant positive flag curvature , as predicted by Yasuda–Shimada. It is explained why this family of Finslerian space forms is not projectively flat.

The Amenability of Measure Algebras

Abstract: In this paper we shall prove that the measure algebra M(G) of a locally compact group G is amenable as a Banach algebra if and only if G is discrete and amenable as a group. Our contribution is to resolve a conjecture by proving that M(G) is not amenable in the case where the group G is not discrete. Indeed, we shall prove a much stronger result: the measure algebra of a non-discrete, locally compact group has a non-zero, continuous point derivation at a certain character on the algebra.

Uniform Eigenvalue Estimates for Time-Frequency Localization Operators

Abstract: Time-variantfilters based on Calderon and Gabor reproducingformulas are important tools in time­ frequencyanalysis.The paper studiesthe behaviorof the eigenvaluesof these filters.Optimal two-sided estimates of the number of eigenvaluescontainedin the interval (151,02), where 0 < 01 < 152 < 1, arc obtained.The estimatescoverlarge classesof localizationdomainsand generatingfunctions. 1. Introduction and statements of the results Calderon- Toeplitz and Gabor- Toeplitz operators arise naturally in two contexts: (i) Toephtz operators on Fock and Bergman spaces of holomorphic functions; (ii) time-variant filters based on Calderon and Gabor reproducing formulas. This paper is concerned with the eigenvalues of a subclass of Calderon- Toeplitz and Gabor- Toeplitz operators which have characteristic functions of bounded domains as symbols. Operators of this class are called time-frequency localiza­ tion operators. The basic idea of functional calculus is that the operators resemble the main algebraic features of their symbols. We consider symbols that are idem­ potent with respect to pointwise multiplication, so it is natural to expect that the corresponding operators are at least approximately idempotent. It is easy to verify that time-frequency localization operators are compact, self-adjoint and bounded by 1. In view of these facts and the above-mentioned correspondence principle, one is inclined to think that localization operators should resemble finite dimensional orthogonal projections. We show that this expectation is correct for Gabor- Toeplitz operators and that it is false for Calderon- Toeplitz operators. We identify the basic geometric features responsible for these two different behaviors. Our principal results are two-sided estimates of the number of eigenvalues inside the plunge region corresponding to 61, (52, where 0 < 61 < (52 < 1. The plunge region consists of the set of indices of the eigenvalues contained inside the open interval (61,62). The eigen­ val ues are ordered non-increasingly. Our work generalizes and improves previ ous results of Daubechies, Paul, Ramanathan and Topiwala [6, 8, 21].

Singular Integrals with Rough Kernels in L log L(Sn−1)

Abstract: A systematic treatment is given of several classes of singular integrals. Their boundedness is proved when their kernels are given by functions in .

Cohomology of Smooth Schubert Varieties in Partial Flag Manifolds

Abstract: The fact that smooth Schubert varieties in partial flag manifolds are iterated fiber bundles over Grassmannians is used to give a simple presentation for their integral cohomology ring, generalizing Borel's presentation for the cohomology of the partial flag manifold itself. More generally, such a presentation is shown to hold for a larger class of subvarieties of the partial flag manifolds (which are called subvarieties defined by inclusions). The Schubert varieties which lie within this larger class are characterized combinatorially by a pattern avoidance condition.

Votre Lévy Rampe-T-IL?

Abstract: The paper gives a necessary and sufficient criterion on the Levy measure that determines whether a Levy process creeps. (The author says that a Levy process creeps if it can continuously pass a fixed level.)

Approximation by algebraic integers and hausdorff dimension

Abstract: The paper computes the Hausdorff dimension of sets of real numbers which are close to infinitely many real algebraic integers of bounded degree. It also investigates the distribution of real algebraic integers of bounded degree, which are proved to be evenly spaced.

Systems of Inequalities and Numerical Semigroups

Abstract: A one-to-one correspondence is described between the setS(m) of numerical semigroups with multiplicity m and the set of non-negative integer solutions of a system of linear Diophantine inequalities. This correspondence infers in S(m) a semigroup structure and the resulting semigroup is isomorphic to a subsemigroup of Nm−1. Finally, this result is particularized to the symmetric case.

Multiple Solutions to p-Laplacian Problems with Asymptotic Nonlinearity as up−1 at Infinity

Abstract: The paper studies the existence of multiple solutions to the following p-Laplacian type elliptic problem (p > 1):

The Noncommutative Singer–Wermer Conjecture and ϕ-Derivations

Abstract: The question of when a $\phi$ -derivation on a Banach algebra has quasinilpotent values, and how this question is related to the noncommutative Singer–Wermer conjecture, is discussed.

On Simultaneously Badly Approximable Numbers

Abstract: For any pair $i,j\ge 0$ with $i+j=1$ let ${\mathbf Bad}(i,j)$ denote the set of pairs $(\alpha,\beta)\in {\bb R}^2$ for which $\max\{\|q\alpha\|^{1/i}\|q\beta\|^{1/j}\}>c/q$ for all $q\in {\bb N}$ . Here $c=c(\alpha,\beta)$ is a positive constant. If $i=0$ the set ${\mathbf Bad}(0, 1)$ is identified with ${\bb R}\times {\mathbf Bad}$ where ${\mathbf Bad}$ is the set of badly approximable numbers. That is, ${\mathbf Bad}(0, 1)$ consists of pairs $(\alpha, \beta)$ with $\alpha\in {\bb R}$ and $\beta\in {\mathbf Bad}$ If $j=0$ the roles of $\alpha$ and $\beta$ are reversed. It is proved that the set ${\mathbf Bad}(1,0)\cap {\mathbf Bad} (0,1)\cap {\mathbf Bad}(i,j)$ has Hausdorff dimension 2, that is, full dimension. The method easily generalizes to give analogous statements in higher dimensions.

Derivations on real and complex jb*-triples

Abstract: At the regional conference held at the University of California, Irvine, in 1985 [24], Harald Upmeier posed three basic questions regarding derivations on JB*-triples:(1) Are derivations automatically bounded?(2) When are all bounded derivations inner?(3) Can bounded derivations be approximated by inner derivations?These three questions had all been answered in the binary cases. Question 1 was answered affirmatively by Sakai [17] for C*-algebras and by Upmeier [23] for JB-algebras. Question 2 was answered by Sakai [18] and Kadison [12] for von Neumann algebras and by Upmeier [23] for JW-algebras. Question 3 was answered by Upmeier [23] for JB-algebras, and it follows trivially from the Kadison–Sakai answer to question 2 in the case of C*-algebras.In the ternary case, both question 1 and question 3 were answered by Barton and Friedman in [3] for complex JB*-triples. In this paper, we consider question 2 for real and complex JBW*-triples and question 1 and question 3 for real JB*-triples. A real or complex JB*-triple is said to have the inner derivation property if every derivation on it is inner. By pure algebra, every finite-dimensional JB*-triple has the inner derivation property. Our main results, Theorems 2, 3 and 4 and Corollaries 2 and 3 determine which of the infinite-dimensional real or complex Cartan factors have the inner derivation property.

Torsional rigidity and expected lifetime of Brownian motion

Abstract: Let $D$ be an open set in euclidean space ${\bb R}^m$ with non-empty boundary $\partial D$ , and let $p_D : D \times D \times; [0,\infty)) \longrightarrow {\bb R}$ be the Dirichlet heat kernel for the parabolic operator ${-}\Delta + \partial/\partial t$ , where ${-}\Delta$ is the Dirichlet laplacian on $L^2(D)$ . Since the Dirichlet heat kernel is non-negative, we may define the (open) set function \renewcommand{\theequation}{1.1} \begin{equation} P_D = \int olimits^{\infty}_0 \int olimits_D \int olimits_D p_D (x,y;t)\,dx\,dy\,dt. \end{equation} We say that $D$ has finite torsional rigidity if $P_D < \infty$ . It is well known that if $D$ has finite volume, then $D$ has finite torsional rigidity [ 11 ]. As we shall see, the converse is not true. The main purpose of this paper is to obtain necessary and sufficient conditions on the geometry of $D$ to guarantee finite torsional rigidity and to gain some understanding of the behaviour of the expected lifetime of brownian motion in a certain natural class of domains that do not have finite torsional rigidity.

The wp-bailey tree and its implications

Abstract: The object of the paper is a thorough analysis of the WP-Bailey tree, a recent extension of classical Bailey chains. The paper begins by observing how the WP-Bailey tree naturally requires a finite number of classical -hypergeometric transformation formulas. It then shows how to move beyond this closed set of results, and in the process, heretofore mysterious identities of Bressoud are explicated. Next, WP-Bailey pairs are used to provide a new proof of a recent formula of Kirillov. Finally, the relation between the approach in the paper and that of Burge is discussed.

Strongly Flat Covers

Abstract: It is proved that all modules over an integral domain have strongly flat cover if and only if every flat -module is strongly flat. The domains satisfying this property are characterized by the property that all their proper quotients are perfect rings, and are called almost perfect. They are exactly the -local domains which are locally almost perfect. Various relevant classes of modules admitting or not admitting strongly flat cover are exhibited.

Isometries of Elliptic 3-Manifolds

Abstract: The closed 3-manifolds of constant positive curvature were classified long ago by Seifert and Threlfall. Using well-known information about the orthogonal group O(4), we calculate their full isometry groups Isom(M), determine which elliptic 3-manifolds admit Seifert fiberings that are invariant under all isometries, and verify that the inclusion of Isom(M) to Diff(M) is a bijection on path components.

On the automorphism groups of cayley graphs of finite simple groups

Abstract: Let G be a nite nonabelian simple group and let be a connected undirected Cayley graph for G. The possible structures for the full automorphism group Aut are specied. Then, for certain nite simple groups G, a sucient condition is given under which G is a normal subgroup of Aut. Finally, as an application of these results, several new half-transitive graphs are constructed. Some of these involve the sporadic simple groups G =J 1 ,J 4, Ly and BM, while others fall into two innite families and involve the Ree simple groups and alternating groups. The two innite families contain examples of half-transitive graphs of arbitrarily large valency.

Autoduality of the Compactified Jacobian

Abstract: The following autoduality theorem is proved for an integral projective curve C in any characteristic. Given an invertible sheaf L of degree 1, form the corresponding Abel map AL:C→, which maps C into its compactified Jacobian, and form its pullback map , which carries the connected component of 0 in the Picard scheme back to the Jacobian. If C has, at worst, points of multiplicity 2, then is an isomorphism, and forming it commutes with specializing C. Much of the work in the paper is valid, more generally, for a family of curves with, at worst, points of embedding dimension 2. In this case, the determinant of cohomology is used to construct a right inverse to . Then a scheme-theoretic version of the theorem of the cube is proved, generalizing Mumford's, and it is used to prove that is independent of the choice of L. Finally, the autoduality theorem is proved. The presentation scheme is used to achieve an induction on the difference between the arithmetic and geometric genera; here, special properties of points of multiplicity 2 are used.

Donsker's Delta Function and the Covariance between Generalized Functionals

Abstract: A classical space and a nonstandard space are used as models for white noise. The expectation of the product of certain generalized functionals is first proved to be finite. Then the covariance and the correlation between Donsker's delta functions are calculated and given meaning.

Harmonic diffeomorphisms of noncompact surfaces and teichmüller spaces

Abstract: Let g : M → N be a quasiconformal harmonic diffeomorphism between noncompact Riemann surfaces M and N. In this paper we study the relation between the map g and the complex structures given on M and N. In the case when M and N are of finite analytic type we derive a precise estimate which relates the map g and the Teichmuller distance between complex structures given on M and N. As a corollary we derive a result that every two quasiconformally related finitely generated Kleinian groups are also related by a harmonic diffeomorphism. In addition, we study the question of whether every quasisymmetric selfmap of the unit circle has a quasiconformal harmonic extension to the unit disk. We give a partial answer to this problem. We show the existence of the harmonic quasiconformal extensions for a large class of quasisymmetric maps. In particular it is proved that all symmetric selfmaps of the unit circle have a unique quasiconformal harmonic extension to the unit disk.

Character Theory of Symmetric Groups and Subgroup Growth of Surface Groups

Abstract: Results from the character theory of symmetric groups are used to obtain an asymptotic estimate for the subgroup growth of fundamental groups of closed 2-manifolds. The main result implies an affirmative answer, for the class of groups investigated, to a question of Lubotzky's concerning the relationship between the subgroup growth of a one-relator group and that of a free group of appropriately chosen rank. As byproducts, an interesting statistical property of commutators in symmetric groups and the fact that in a 'large' surface group almost all finite index subgroups are maximal are obtained, among other things. The approach requires an asymptotic estimate for the sum Sigma 1/(chi(lambda)(1))(s) taken over all partitions lambda of n with fixed s greater than or equal to 1, which is also established.

Spectral synthesis and operator synthesis for compact groups

Abstract: LetG be a compact group andC(G )b e theC-algebra of continuous complex-valued functions onG. The paper constructs an imbedding of the Fourier algebra A(G )o fG into the algebra V(G )= C(G) h C(G) (Haagerup tensor product) and deduces results about parallel spectral synthesis, generalizing a result of Varopoulos. It then characterizes which diagonal sets in GG are sets of operator synthesis with respect to the Haar measure, using the denition of operator synthesis due to Arveson. This result is applied to obtain an analogue of a result of Froelich: a tensor formula for the algebras associated with the pre-orders dened by closed unital subsemigroups of G.

Oscillation for First Order Superlinear Delay Differential Equations

Abstract: Some almost sharp sufficient conditions of oscillation and nonoscillation are obtained for the superlinear delay differential equation