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.     

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.     

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.     

Movement and Separation of Subsets of Points Under Group Actions

Let G be a permutation group on a set , and let m and k be integerswhere 0<m<k. For a subset of , if the cardinalities ofthe sets ^g\, for gG, are finite and bounded, then is said tohave bounded movement, and the movement of is defined as move()=max_gG|^g\|. If there is a k-element subset such that move()m, it is shown that some G-orbit has length at most (k²–m)/(k–m).When combined with a result of P. M. Neumann, this result hasthe following consequence: if some infinite subset has boundedmovement at most m, then either is a G-invariant subset withat most m points added or removed, or nontrivially meets aG-orbit of length at most m²+m+1. Also, if move ()m for allk-element subsets and if G has no fixed points in , then either||k+m (and in this case all permutation groups on have thisproperty), or ||5m–2. These results generalise earlierresults about the separation of finite sets under group actionsby B. J. Birch, R. G. Burns, S. O. Macdonald and P. M. Neumann,and groups in which all subsets have bounded movement (by theauthor).     

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 .     

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).     

The model theory of the field of reals with a subgroup of the unit circle

We describe definable sets in the field of reals augmented bya predicate for a finite rank multiplicative group of complexnumbers contained in the unit circle . This structure interpretsthe quotient-space / which, for infinite cyclic, is relatedto the quantum torus. Every definable set is proved to be aBoolean combination of existentially definable sets. We givea complete set of axioms for the theory of such a structure.     

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.     

BOUNDARY CONCENTRATION IN RADIAL SOLUTIONS TO A SYSTEM OF SEMILINEAR ELLIPTIC EQUATIONS

We study concentration phenomena for the system in the unit ball B₁ of ³ with Dirichlet boundaryconditions. Here , , > 0 and p > 1. We prove the existenceof positive radial solutions (, ) such that concentrates ata distance (/2)|log | away from the boundary B₁ as the parameter tends to 0. The approach is based on a combination of Lyapunov–Schmidtreduction procedure together with a variational method.     

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.     

Character Quotients for Coprime Acting Groups

Let the finite group A be acting on a finite group G with (|A|,|G|)=1. Let be the semidirect product of A and G. Let be acharacter of irreducible after restriction to G. In a previouspaper by Brian Hartley and the author, we proved that the restrictionof to S belongs to the set C(S) obtained by running over all that arise in this manner, by assuming, in addition, that Gis a product of extraspecial groups. This was proved in general,assuming only some condition on the Green functions of groupsof Lie type that is not as yet fully verified. In the presentpaper, we define the map Q(): SC by Q()(s)=|C_G(s)|/(s). We provethat Q()C(S) under the same hypotheses. In particular, the characterquotient Q() is an ordinary character.     

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.     

On the Total Coloring of Graphs Embeddable in Surfaces

The paper shows that any graph G with the maximum degree (G) 8, which is embeddable in a surface of Euler characteristic() 0, is totally ((G)+2)-colorable. In general, it is shownthat any graph G which is embeddable in a surface and satisfiesthe maximum degree (G) (20/9) (3–())+1 is totally ((G)+2)-colorable.     

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.     

Numerical Forms

Let I be the graded ring of homogeneous rational polynomialsin n-variables which are numerical over Z. Then I is a subringof , the divided polynomial algebra over Z in the n-variables.A consequence of the main theorem is that for any positive integerk, the image of the induced homomorphism I(Z/kZ) (Z/kZ) isa finite graded ring.     

Geometry of Critical Loci

Let :(Z,z)(U,0) be the germ of a finite (that is, proper with finite fibres)complex analytic morphism from a complex analytic normal surfaceonto an open neighbourhood U of the origin 0 in the complexplane C². Let u and v be coordinates of C² defined on U. Weshall call the triple (, u, v) the initial data. Let stand for the discriminant locus of the germ , that is,the image by of the critical locus of . Let ()_A be the branches of the discriminant locus at O whichare not the coordinate axes. For each A, we define a rational number d by where I(–, –) denotes the intersection number at0 of complex analytic curves in C². The set of rational numbersd, for A, is a finite subset D of the set of rational numbersQ. We shall call D the set of discriminantal ratios of the initialdata (, u, v). The interesting situation is when one of thetwo coordinates (u, v) is tangent to some branch of , otherwiseD = {1}. The definition of D depends not only on the choiceof the two coordinates, but also on their ordering. In this paper we prove that the set D is a topological invariantof the initial data (, u, v) (in a sense that we shall definebelow) and we give several ways to compute it. These resultsare first steps in the understanding of the geometry of thediscriminant locus. We shall also see the relation with thegeometry of the critical locus.     

Stability and Asymptotic Stability of Functional-Differential Equations

We investigate asymptotic behaviour of solutions of the functional-differentialequation where f and g arelocally Lipschitz functions, C is a continuous matrix and thesmooth lag function obeys 0 (t) t for t 0. We transformthe equation into a delay equation with an infinity of delaysand use a theorem of Söderlind to derive sufficient conditionsfor stability and for asymptotic stability in the case lim_t (t) = . The situation is qualitatively different when lim_t (t) = * < and we outline stability conditions for thatcase by employing direct techniques.     

Calderon-Zygmund Operators on Mixed Lebesgue Spaces and Applications to Null Forms

The boundedness of Calderón–Zygmund operators isproved in the scale of the mixed Lebesgue spaces. As a consequence,the boundedness of the bilinear null forms Q_{i j} (u,) =_i u_j - _j u_i , Q₀(u,)=u_{t t} -_x u· _x on various space–timemixed Sobolev–Lebesgue spaces is shown.     

Non-Normal Pairs of Non-Euclidean Crystallographic Groups

Let be a non-Euclidean crystallographic group. is said tobe non-maximal if there exists a non-Euclidean crystallographicgroup ' such that ' and the dimension of the Teichmüllerspace of equals the dimension of the Teichmüller spaceof '. The full list of such pairs of groups is computed in thecase when is non-normal in '. The corresponding problem forFuchsian groups was solved by Singerman. 2000 Mathematics SubjectClassification 20H10 (primary), 30F10 (secondary).     

Asymptotics for Fractional Nonlinear Heat Equations

The Cauchy problem is studied for the nonlinear equations withfractional power of the negative Laplacian where (0,2), with critical = /n and sub-critical (0,/n)powers of the nonlinearity. Let u₀ L^1,a L C, u₀(x) 0 in Rⁿ, = . The case of not small initial data is of interest. It is proved that the Cauchy problemhas a unique global solution u C([0,); L L^1,a C) and the largetime asymptotics are obtained.