Spectra of graph neighborhoods and scattering

Let (G)_>0 be a family of ‘-thin’ Riemannian manifoldsmodeled on a finite metric graph G, for example, the -neighborhoodof an embedding of G in some Euclidean space with straight edges.We study the asymptotic behavior of the spectrum of the Laplace–Beltramioperator on G, as 0, for various boundary conditions. We obtaincomplete asymptotic expansions for the kth eigenvalue and theeigenfunctions, uniformly for kC^–1, in terms of scatteringdata on a non-compact limit space. We then use this to determinethe quantum graph which is to be regarded as the limit object,in a spectral sense, of the family (G). Our method is a directconstruction of approximate eigenfunctions from the scatteringand graph data, and the use of a priori estimates to show thatall eigenfunctions are obtained in this way.     

Grothendieck's Inequalities for Real and Complex JBW*-Triples

We prove that, if and >0, if V and W are complex JBW*-triples (with preduals V_* andW_*, respectively), and if U is a separately weak*-continuousbilinear form on V x W, then there exist norm-one functionals₁, ₂ V_* and ₁, ₂ W_* satisfying for all (x, y) V x W. Here, for a norm-one functional on acomplex JB*-triple V, |·| stands for the prehilbertianseminorm on V associated to given by for all x W, where z V^** satisfies z = |z| =1. We arrive at this form of ‘Grothendieck's inequality’through results of C.-H. Chu, B. Iochum, and G. Loupias, andan amended version of the ‘little Grothendieck's inequality’for complex JB*-triples due to T. Barton and Y. Friedman. Wealso obtain extensions of these results to the setting of realJB*-triples. 2000 Mathematical Subject Classification: 17C65,46K70, 46L05, 46L10, 46L70.     

Exponential Sums and the Riemann Zeta Function V

A Van der Corput exponential sum is S = exp (2 i f(m)) wherem has size M, the function f(x) has size T and = (log M) / log T < 1. There are different bounds for S in differentranges for . In the middle range where is near 1/over 2, . This bounds the exponent of growthof the Riemann zeta function on its critical line Re s = 1/over2. Van der Corput used an iteration which changed at each step.The Bombieri–Iwaniec method, whilst still based on meansquares, introduces number-theoretic ideas and problems. TheSecond Spacing Problem is to count the number of resonancesbetween short intervals of the sum, when two arcs of the graphof y = f'(x) coincide approximately after an automorphism ofthe integer lattice. In the previous paper in this series [Proc.London Math. Soc. (3) 66 (1993) 1–40] and the monographArea, lattice points, and exponential sums we saw that coincidenceimplies that there is an integer point close to some ‘resonancecurve’, one of a family of curves in some dual space,now calculated accurately in the paper ‘Resonance curvesin the Bombieri–Iwaniec method’, which is to appearin Funct. Approx. Comment. Math. We turn the whole Bombieri–Iwaniec method into an axiomatisedstep: an upper bound for the number of integer points closeto a plane curve gives a bound in the Second Spacing Problem,and a small improvement in the bound for S. Ends and cusps ofresonance curves are treated separately. Bounds for sums oftype S lead to bounds for integer points close to curves, andanother branching iteration. Luckily Swinnerton-Dyer's methodis stronger. We improve from 0.156140... in the previous paperand monograph to 0.156098.... In fact (32/205 + , 269/410 +) is an exponent pair for every > 0. 2000 Mathematics SubjectClassification 11L07 (primary), 11M06, 11P21, 11J54 (secondary).     

Algebraic K-Theory In Low Degree and The Novikov Assembly Map

We prove that the Novikov assembly map for a group factorizes,in ‘low homological degree’, through the algebraicK-theory of its integral group ring. In homological degree 2,this answers a question posed by N. Higson and P. Julg. As adirect application, we prove that if is torsion-free and satisfiesthe Baum-Connes conjecture, then the homology group H₁(; Z)injects in and in , for any ring A such that . If moreover B is of dimension lessthan or equal to 4, then we show that H₂(; Z) injects in and in , where A is as before, and ₂ is generated by the Steinberg symbols{,}, for . 2000 Mathematical Subject Classification: primary 19D55, 19Kxx,58J22; secondary: 19Cxx, 19D45, 43A20, 46L85.     

Euler characteristics of the real points of certain varieties of algebraic tori

Let G be a complex connected reductive group which is definedover , let be its Lie algebra, and let be the variety of maximaltori of G. For (), let be the variety of tori in whose Liealgebra is orthogonal to with respect to the Killing form.We show, using the Fourier–Sato transform of conical sheaveson real vector bundles, that the ‘weighted Euler characteristic’of () is zero unless is nilpotent, in which case it equals(–1)^{(dim )/2}. Here ‘weighted Euler characteristic’means the sum of the Euler characteristics of the connectedcomponents, each weighted by a sign ± 1 which dependson the real structure of the tori in the relevant component.This is a real analogue of a result over finite fields whichis connected with the Steinberg representation of a reductivegroup.     

Unknotting Tunnels and Seifert Surfaces

Let K be a knot with an unknotting tunnel and suppose thatK is not a 2-bridge knot. There is an invariant = p/q Q/2Z,with p odd, defined for the pair (K, ). The invariant has interesting geometric properties. It is oftenstraightforward to calculate; for example, for K a torus knotand an annulus-spanning arc, (K, ) = 1. Although is definedabstractly, it is naturally revealed when K is put in thinposition. If 1 then there is a minimal-genus Seifert surfaceF for K such that the tunnel can be slid and isotoped to lieon F. One consequence is that if (K, ) 1 then K > 1. Thisconfirms a conjecture of Goda and Teragaito for pairs (K, )with (K, ) 1. 2000 Mathematics Subject Classification 57M25,57M27.     

General Uniqueness Results and Variation Speed for Blow-Up Solutions of Elliptic Equations

Let be a smooth bounded domain in R^N. We prove general uniquenessresults for equations of the form – u = au – b(x)f(u) in , subject to u = on . Our uniqueness theorem is establishedin a setting involving Karamata's theory on regularly varyingfunctions, which is used to relate the blow-up behavior of u(x)with f(u) and b(x), where b 0 on and a certain ratio involvingb is bounded near . A key step in our proof of uniqueness usesa modification of an iteration technique due to Safonov. 2000Mathematics Subject Classification 35J25 (primary), 35B40, 35J60(secondary).     

Cohen-Macaulay Properties of Thom-Boardman Strata I: Morin's Ideal

Thom–Boardman strata ^I are fundamental tools in studyingsingularities of maps. The Zariski closures of the strata ^Iare components of the set of zeros of the ideals ^I defined by B. Morin using iterated jacobian extensions in his paper‘Calcul jacobien’ (Ann. Sci. École Norm.Sup.} 8 (1975) 1–98). In this paper, we consider the questionof when the Morin ideals ^I define Cohen–Macaulay spaces.We determine all I=(i₁...,i_k) such that ^I defines a Cohen–Macaulayspace alongthe stratum. 1991 Mathematics Subject Classification: 13D25, 14B05, 14M12, 58C25.     

Multivariate R-O Varying Measures Part I: Uniform Bounds

A finite Borel measure µ on R^d is called R-O varying withindex F if there exist a GL(R^d)-valued function f varying regularlywith index (–F), an increasing function k: (0, ) (0,) with k(t) and k(t + 1)/k(t) c 1 as t , and a -finitemeasure on R^d\0 such that R-O varying measures generalize regularly varying measures introducedby Meerschaert (see M. M. Meerschaert, ‘Regular variationin R^k’, Proc. Amer. Math. Soc. 102 (1988) 341–348)and have numerous applications in limit theorems for probabilitymeasures. For an R-O varying measure µ and – < let denote the tail- andtruncated moment functions of µ in the direction || =1. The purpose of this paper is to show that R-O variation ofa measure implies sharp bounds on the growth rate of the tail-and truncated moment functions depending on the real parts ofthe eigenvalues of the index F along a compact set of directions.Furthermore, bounds on the ratio of these functions for certainvalues of a and b are obtained. 1991 Mathematics Subject Classification:60B10, 28C15.     

Chaotic Expansion of Elements of the Universal Enveloping Algebra of a Lie Algebra Associated with a Quantum Stochastic Calculus

The functional Ito formula, in the form df() = f( + d ) –f(),is formulated and proved in the context of a Lie algebra L associatedwith a quantum (non-commutative) stochastic calculus. Here fis an element of the universal enveloping algebra U of L, andf() + d() – f() is given a meaning using the coproductstructure of U even though the individual terms of this expressionhave no meaning. The Ito formula is equivalent to a chaoticexpansion formula for f() which is found explicitly. 1991 MathematicsSubject Classification: primary 81S25; secondary 60H05; tertiary18B25.     

A bifurcation problem governed by the boundary condition II

In this work we consider the problem u = a(x)u^p in on , where is a smooth bounded domain, isthe outward unit normal to , is regarded as a parameter and0 < p < 1. We consider both cases where a(x) > 0 in or a(x) is allowed to vanish in a whole subdomain ₀ of . Ourmain results include existence of non-negative non-trivial solutionsin the range 0 < < ₁, where ₁ is characterized by meansof an eigenvalue problem, uniqueness and bifurcation from infinityof such solutions for small , and the appearance of dead coresfor large enough .     

On the laplace equation with non-linear dynamical boundary conditions

The main part of the paper deals with local existence and globalexistence versus blow-up for solutions of the Laplace equationin bounded domains with a non-linear dynamical boundary condition.More precisely, we study the problem consisting in: (1) theLaplace equation in (0, ) x ; (2) a homogeneous Dirichlet condition(0, ) x ₀; (3) the dynamical boundary condition ; (4) the initial condition u(0, x) = u₀ (x) on . Here is a regular and bounded domain in Rⁿ, with n 1, and₀ and ₁ endow a measurable partition of . Moreover, m>1,2 p < r, where r = 2 (n – 1) / (n – 2) whenn 3, r = when n = 1,2, and u₀ H^1/2 , u₀ = 0 on ₀. The final part of the paper deals with a refinement of a globalnon-existence result by Levine, Park and Serrin, which is appliedto the previous problem. 2000 Mathematics Subject Classification35K55 (primary), 35K90, 35K77 (secondary).     

Global Branch of Solutions for Non-Linear Schrodinger Equations with Deepening Potential Well

We consider the stationary non-linear Schrödinger equation where > 0 and the functionsf and g are such that and for some bounded open set R^N. We use topological methods to establish the existenceof two connected sets D^± of positive/negative solutionsin R x W², ^p R^N where that cover the interval (, ()) in the sense that and furthermore, The number () is characterized as the unique value of in theinterval (, ) for which the asymptotic linearization has a positiveeigenfunction. Our work uses a degree for Fredholm maps of indexzero. 2000 Mathematics Subject Classification 35J60, 35B32,58J55.     

Unicity of Types for Supercuspidal Representations of GLN

Let F be a non-Archimedean local field, with the ring of integerso_F. Let G = GL_N(F), K = GL_N (o_F), and be a supercuspidal representationof G. We show that there exists a unique irreducible smoothrepresentation of K, such that the restriction to K of a smoothirreducible representation ' of G contains if and only if 'is isomorphic to ° det, where is an unramified quasicharacterof F^x. Moreover, we show that contains with the multiplicity1. As a corollary we obtain a kind of inertial local Langlandscorrespondence. 2000 Mathematics Subject Classification 22E50.     

Decomposition of weighted Triebel-Lizorkin and Besov spaces on the ball

Weighted Triebel–Lizorkin and Besov spaces on the unitball B^d in ^d with weights w_µ(x)=(1–|x|²)^µ–1/2,µ0, are introduced and explored. A decomposition schemeis developed in terms of almost exponentially localized polynomialelements (needlets) {}, {} and it is shown that the membershipof a distribution to the weighted Triebel–Lizorkin orBesov spaces can be determined by the size of the needlet coefficients{f, } in appropriate sequence spaces.     

On the Asphericity of Length Five Relative Group Presentations

We investigate asphericity of the relative group presentation G,t |atbtctdtet=1 and prove it aspherical provided thesubgroupof G generated by ab^–1, bc^–1, cd^–1, de^–1is neither finite cyclic nor a finite triangle group. We alsoprove a similar result for the closely related relative grouppresentation G,s,t | sßst=1=tts^–1. 2000 MathematicsSubject Classification: 20F05, 57M05.     

Dynamics of projective morphisms having identical canonical heights

Let , :^{N N} be morphisms of degree at least 2 whose canonicalheights and are identical. We draw various conclusions aboutthe Green functions, Julia sets, and canonical local heightsof and . We use this information to completely characterize and in the following cases: (i) and are polynomial mapsin one variable; (ii) is the dth-power map; (iii) is a Lattèsmap.     

On Decomposition Numbers and Branching Coefficients for Symmetric and Special Linear Groups

In this paper we find the multiplicities dim L()_– where is an arbitrary root and L() is an irreducible SL_n-module withhighest weight . We provide different bases of the correspondingweight spaces and outline some applications to the symmetricgroups. In particular we describe certain composition multiplicitiesin the modular branching rule. 1991 Mathematics Subject Classification:20C05, 20G05.