An Access Theorem for Continuous Functions

Let f be a continuous function on an open subset of R² suchthat for every x there exists a continuous map : [–1,1] with (0) = x and f increasing on [–1, 1]. Thenfor every there exists a continuous map : [0, 1) suchthat (0) = y, f is increasing on [0; 1), and for every compactsubset K of , max{t : (t) K} < 1. This result gives an answerto a question posed by M. Ortel. Furthermore, an example showsthat this result is not valid in higher dimensions.     

TURAN'S EXTREMAL PROBLEM FOR POSITIVE DEFINITE FUNCTIONS ON GROUPS

We study the following question: given an open set , symmetricabout 0, and a continuous, integrable, positive definite functionf, supported in and with f(0) = 1, how large can f be? Thisproblem has been studied so far mostly for convex domains inEuclidean space. In this paper we study the question in arbitrarylocally compact abelian groups and for more general domains.Our emphasis is on finite groups as well as Euclidean spacesand ^d. We exhibit upper bounds for f assuming geometric propertiesof of two types: (a) packing properties of and (b) spectralproperties of . Several examples and applications of the maintheorems are shown. In particular, we recover and extend severalknown results concerning convex domains in Euclidean space.Also, we investigate the question of estimating f over possiblydispersed sets solely in dependence of the given measure m :=||of . In this respect we show that in and the integral is maximalfor intervals.     

Properties of Removable Singularities for Hardy Spaces of Analytic Functions

Removable singularities for Hardy spaces H^p() = {f Hol(): |f|^p u in for some harmonic u}, 0 < p < are studied. A setE = is a weakly removable singularity for H^p(\E) if H^p(\E) Hol(), and a strongly removable singularity for H^p(\E) if H^p(\E)= H^p(). The two types of singularities coincide for compactE, and weak removability is independent of the domain . The paper looks at differences between weak and strong removability,the domain dependence of strong removability, and when removabilityis preserved under unions. In particular, a domain and a setE that is weakly removable for all H^p, but not strongly removablefor any H^p(\E), 0 < p < , are found. It is easy to show that if E is weakly removable for H^p(\E)and q > p, then E is also weakly removable for H^q(\E). Itis shown that the corresponding implication for strong removabilityholds if and only if q/p is an integer. Finally, the theory of Hardy space capacities is extended, anda comparison is made with the similar situation for weightedBergman spaces.     

Exactness and Uniform Embeddability of Discrete Groups

A numerical quasi-isometry invariant R() of a finitely generatedgroup is defined whose values parametrize the difference between being uniformly embeddable in a Hilbert space and () being exact.     

Higher Generation Subgroup Sets and the Virtual Cohomological Dimension of Graph Products of Finite Groups

We introduce panels of stabilizer schemes (K, G_*) associatedwith finite intersection-closed subgroup sets of a given groupG, generalizing in some sense Davis' notion of a panel structureon a triangulated manifold for Coxeter groups. Given (K, G_*),we construct a G-complex X with K as a strong fundamental domainand simplex stabilizers conjugate to subgroups in . It turnsout that higher generation properties of in the sense of Abels-Holzare reflected in connectivity properties of X. Given a finite simplicial graph and a non-trivial group G()for every vertex of , the graph product G() is the quotientof the free product of all vertex groups modulo the normal closureof all commutators [G(), G(w)] for which the vertices , w areadjacent. Our main result allows the computation of the virtualcohomological dimension of a graph product with finite vertexgroups in terms of connectivity properties of the underlyinggraph .     

Positive Eigenfunctions of a Schrodinger Operator

The paper considers the eigenvalue problem where , and for some bounded open set R^N. Given >0, does there exist a value of >0 for which theproblem has a positive solution? It is shown that this occursif and only if lies in a certain interval (,₁) and that inthis case the value of is unique, =(). The properties of thefunction () are also discussed.     

Smooth Approximation of Sobolev Functions on Planar Domains

We examine two related problems concerning a planar domain .The first is whether Sobolev functions on can be approximatedby global C functions, and the second is whether approximationcan be done by functions in C() which, together with all derivatives,are bounded on . We find necessary and sufficient conditionsfor certain types of domains, such as starshaped domains, andwe construct several examples which show that the general problemis quite difficult, even in the simply connected case.     

A Torsion-Free Milnor-Moore Theorem

Let X be the space of Moore loops on a finite, q-connected,n-dimensional CW complex X, and let R Q be a subring containing1/2. Let (R) be the least non-invertible prime in R. For a gradedR-module M of finite type, let FM = M/Torsion M. We show thatthe inclusion P FH_*(X;R) of the sub-Lie algebra of primitiveelements induces an isomorphism of Hopf algebras provided that (R) n/q. Furthermore, the Hurewiczhomomorphism induces an embedding of F(_*(X) R) in P, with P/F(_*(X)R)torsion. As a corollary, if X is elliptic, then FH_*(X;R) isa finitely generated R-algebra.     

Maximal Convex Subgroups of the Automorphism Group of a Doubly Transitive Chain

We characterise all maximal convex l-subgroups of A(), the automorphismgroup of a doubly transitive chain of countable coterminality.We then determine, up to isomorphism, all doubly transitiveactions of the l-group A(). Finally, assuming the ContinuumHypothesis, we show that there exists an unbounded chain ofproper prime subgroups of A().     

On the Asymptotics of the Growth of 2-Step Nilpotent Groups

Pansu has shown that the growth function of every virtuallynilpotent group with respect to any finite generating set hasasymptotics (n)n^d, where d is the degree of growth of . Thepaper refines his result in the special case of 2-step nilpotentgroups to obtain (n)=n^d+O(n^d–1).     

Algebraic invariants for Bestvina Brady groups

Bestvina–Brady groups arise as kernels of length homomorphismsG from right-angled Artin groups to the integers. Under someconnectivity assumptions on the flag complex , we compute severalalgebraic invariants of such a group N, directly from the underlyinggraph . As an application, we give examples of finitely presentedBestvina–Brady groups which are not isomorphic to anyArtin group or arrangement group.     

Compact Embeddings of Besov Spaces in Exponential Orlicz Spaces

Let 1 < p < , 0 < v < p', let be a bounded domainin Rⁿ, and denote by id the limiting compact embedding of theBesov space (Rⁿ) into the exponentialOrlicz space L_exp(t^v)(), mapping a function f onto its restrictionf|. In 1993 Triebel established, among others, two-sided estimatesfor the entropy numbers of id, which are even asymptoticallyoptimal for ‘small’ . The aim of the paper is toimprove the upper bounds in the case of ‘large’, where Triebel's estimates are not yet sharp, thus making afurther step towards the conjectured correct asymptotic behaviour.     

Fitted and Unfitted Finite-Element Methods for Elliptic Equations with Smooth Interfaces

This paper considers the finite-element approximation of theelliptic interface problem: -?(u) + cu = f in Rⁿ (n = 2 or3), with u = 0 on , where is discontinuous across a smoothsurface in the interior of . First we show that, if the meshis isoparametrically fitted to using simplicial elements ofdegree k - 1, with k 2, then the standard Galerkin method achievesthe optimal rate of convergence in the H¹ and L² norms overthe approximations _l⁴ of _l where _{l 2}. Second, since itmay be computationally inconvenient to fit the mesh to , weanalyse a fully practical piecewise linear approximation ofa related penalized problem, as introduced by Babuska (1970),based on a mesh that is independent of . We show that, by choosingthe penalty parameter appropriately, this approximation convergesto u at the optimal rate in the H¹ norm over _l⁴ and in the L²norm over any interior domain _l^* satisfying _l^* _l^** _l⁴ for somedomain _l^**. Present address: School of Mathematical and Physical Sciences,University of Sussex, Brighton BN1 9QH     

Minimal Geodesics on Manifolds with Discontinuous Metrics

The paper describes some qualitative properties of minimizerson a manifold M endowed with a discontinuous metric. The discontinuityoccurs on a hypersurface disconnecting M. Denote by ₁ and₂ the open subsets of M such that M\ =₁₂. Assume that and are endowed with metrics ·, · ₍₁₎ and ·,·₍₂₎, respectively, such that (i=1, 2) is convex or concave. The existence of a minimizerof the length functional on curves joining two given pointsof M is proved. The qualitative properties obtained allows therefraction law in a very general situation to be described.     

Real Interpolation and Two Variants of Gehring's Lemma

Let be a fixed open cube in Rⁿ. For r[1, ) and [0, ) we define where Q is a cube in Rⁿ (with sides parallel to the coordinateaxes) and _Q stands for the characteristic function of the cubeQ. A well-known result of Gehring [5] states that if (1.1) for some p(1, ) and c(0, ), then there exist q(p, ) and C=C(p,q, n, c)(0, ) such that for all cubes Q, where |Q| denotes the n-dimensional Lebesguemeasure of Q. In particular, a function fL¹() satisfying (1.1)belongs to L^q(). In [9] it was shown that Gehring's result is a particular caseof a more general principle from the real method of interpolation.Roughly speaking, this principle states that if a certain reversedinequality between K-functionals holds at one point of an interpolationscale, then it holds at other nearby points of this scale. Usingan extension of Holmstedt's reiteration formulae of [4] andresults of [8] on weighted inequalities for monotone functions,we prove here two variants of this principle involving extrapolationspaces of an ordered pair of (quasi-) Banach spaces. As an applicationwe prove the following Gehring-type lemmas.     

POSITIVE SOLUTIONS FOR A NONLINEAR ELLIPTIC PROBLEM WITH STRONG LACK OF COMPACTNESS

This paper deals with the lack of compactness in the nonlinearelliptic problem – u + u = |u|^p–2u in , u > 0in , u = 0 on , when is un unbounded domain in ⁿ and 2 <p < 2n/(n–2). In particular, a domain is provided where non-converging Palais–Smale sequences existat every energy level. Nevertheless, it is proved that the problemhas infinitely many solutions on .     

Extremal Subgroups in Chevalley Groups

Throughout this paper G(k) denotes a Chevalley group of rankn defined over the field k, where n3. Let be the root systemassociated with G(k) and let ={₁, ₂, ..., _n} be a set of fundamentalroots of , with ⁺ being the set of positive roots of with respectto . For and ⁺, let n() be the coefficient of in the expressionof as a sum of fundamental roots; so =n(). Also we recall thatht(), the height of , is given by ht()=n(). The highest rootin ⁺ will be denoted by . We additionally assume that the Dynkindiagram of G(k) is connected.     

On Artin's Braid Group And Polyconvexity In The Calculus Of Variations

Let R² be a bounded Lipschitz domain and let be a Carathèodory integrand such that F(x,·) is polyconvex for L²-a.e. x . Moreover assume thatF is bounded from below and satisfies the condition as det for L²-a.e. x . The paper describes the effect of domain topologyon the existence and multiplicity of strong local minimizersof the functional wherethe map u lies in the Sobolev space W_id^1,p (, R²) with p 2and satisfies the pointwise condition u(x) >0 for L²-a.e.x . The question is settled by establishing that F[·]admits a set of strong local minimizers on that can be indexed by the group P_n Zⁿ, the directsum of Artin's pure braid group on n strings and n copies ofthe infinite cyclic group. The dependence on the domain topologyis through the number of holes n in and the different mechanismsthat give rise to such local minimizers are fully exploitedby this particular representation.     

Some Compactness Results in Real Interpolation for Families of Banach Spaces

The paper shows that, if the operator T:A()B() is compact foralmost every , then is compact when or is the interpolation functor constructed for infinitefamilies of Banach spaces and S satisfies certain conditions.     

Von Neumann Eta-Invariants and C*-Algebra K-Theory

Let M be a smooth, compact, oriented, odd-dimensional Riemannianmanifold and let M be anormal covering of M. It is proved that the relative von Neumanneta-invariant ₍₂₎() of Cheeger and Gromov is a homotopy invariant when is torsion-free, discreteand the Baum–Connes assembly map µ_max:K₀(B) K₀(C*)is an isomorphism.