A global Riesz decomposition theorem on trees without positive potentials

We study the potential theory of trees with nearest-neighbortransition probability that yields a recurrent random walk andshow that, although such trees have no positive potentials,many of the standard results of potential theory can be transferredto this setting. We accomplish this by defining a non-negativefunction H, harmonic outside the root e and vanishing only ate, and a substitute notion of potential which we call H-potential.We define the flux of a superharmonic function outside a finiteset of vertices, give some simple formulas for calculating theflux and derive a global Riesz decomposition theorem for superharmonicfunctions with a harmonic minorant outside a finite set. Wediscuss the connection of the H-potentials with other notionsof potentials for recurrent Markov chains in the literature.     

Order Stars and a Saturation Theorem for First-order Hyperbolics

We prove that stable numerical finite difference methods forfirst-order hyperbolics, which use s forward and r backwardsteps in the discretization of the space derivatives, are oforder at most 2 min{r+1, s}. This generalizes results of Strang(1964) and of Engquist & Osher (1980b). We also derive linearstability results for interpolatory finite differences. Thegiven analysis is based on a generalization of the theory oforder stars.     

Plain Representations of Lie Algebras

In this paper we study representations of finite dimensionalLie algebras. In this case representations are not necessarilycompletely reducible. As the general problem is known to beof enormous complexity, we restrict ourselves to representationsthat behave particularly well on Levi subalgebras. We call suchrepresentations plain (Definition 1.1). Informally, we showthat the theory of plain representations of a given Lie algebraL is equivalent to representation theory of finitely many finitedimensional associative algebras, also non-semisimple. The senseof this is to distinguish representations of Lie algebras thatare of complexity comparable with that of representations ofassociative algebras. Non-plain representations are intrinsicallymuch more complex than plain ones. We view our work as a steptoward understanding this complexity phenomenon. We restrict ourselves also to perfect Lie algebras L, that is,such that L = [L, L]. In our main results we assume that L isperfect and sl₂-free (which means that L has no quotient isomorphicto sl₂). The ground field F is always assumed to be algebraicallyclosed and of characteristic 0.     

Components and Periodic Points in Non-Archimedean Dynamics

In this paper we expand the theory of connected components innon-archimedean discrete dynamical systems. We define two typesof components and discuss their uses and applications in thestudy of dynamics of a rational function K(z) defined overa non-archimedean field K. We prove that some fundamental conjectures,including the No Wandering Domains conjecture, are equivalent,regardless of which definition of 'component' is used. We deriveseveral results on the geometry of our components and the existenceof periodic points within them. We also give a number of examplesof p-adic maps with interesting or pathological dynamics. 2000Mathematical Subject Classification: primary 37B99; secondary11S99, 30D05.     

Gleason Property and Extensions of States on Projection Logics

We prove that every state on the projection logic P(M) of avon Neumann algebra M not containing a direct summand of typeI₂ extends to a state of an arbitrary larger unital logic L.We also show that if a C^*-algebra enjoys the Gleason property,and if it possesses sufficiently many projections, then an analogousresult can be derived. Moreover, we prove that the extensionscan be taken linear in a complete order unit norm space associatedwith L. (Results of this paper generalize results of [22] andmay contribute to the noncommutative measure theory, convextheory of state spaces and foundations of quantum physics.)     

Asymptotic methods for spherically symmetric MHD α2-dynamos

We consider two models of spherically-symmetric MHD α²–dynamos; one with idealized boundary conditions (BCs); and one with physically realistic BCs. As it has been shown in our previous work, the eigenvalues λ of a model with idealized BCs and constant α–profile α₀ are linear functions of α₀ and form a mesh in the (α₀, λ)–plane. The nodes of the spectral mesh correspond to double-degenerate eigenvalues of algebraic and geometric multiplicity 2 (diabolical points). It was found that perturbations of the constant α –profile lead to a resonant unfolding of the diabolical points with selection rules of the resonant unfolding defined by the Fourier coefficients of the perturbations. In the present contribution we present new exact results on the spectrum of the model with physically realistic BCs and constant α. For non-degenerate (simple) eigenvalues perturbation gradients are found at any particular α₀. We briefly discuss the spectral behavior of the α²–dynamo operator over a family of homotopic deformations of the BCs between idealized ones and physically realistic ones. Furthermore, we demonstrate that although the spectral singularities are lifted, a memory about their locations remains deeply imprinted in the homotopic family of spectral deformations due to a hidden underlying invariance. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On Towers Approximating Homological Localizations

Our object of study is the natural tower which, for any givenmap f:AB and each space X, starts with the localization of Xwith respect to f and converges to X itself. These towers canbe used to produce approximations to localization with respectto any generalized homology theory E_*, yielding, for example,an analogue of Quillen's plus-construction for E_*. We discussin detail the case of ordinary homology with coefficients inZ/p or Z[1/p]. Our main tool is a comparison theorem for nullificationfunctors (that is, localizations with respect to maps of theform f:Apt), which allows us, among other things, to generalizeNeisendorfer's observation that p-completion of simply-connectedspaces coincides with nullification with respect to a Moorespace M(Z[1/p], 1).     

Homogenization of the Stefan Problem and Application to Magnetic Composite Media

The theory of homogenization (Bensoussan, Lions & Papanicolaou,1978) shows that u, the solution of the diffusion equation [with k(y) periodic in the space-variable y and q = cu a linearfunction of u] has a weak limit u for = 0. This theory allowsone to compute, for a given k, the conductivity tensor of ananisotropic but homogeneous medium in which, for unchanged initialand boundary conditions, u is the solution of the diffusionequation. We examine here the case where the relation between q and uis given by a maximal monotone graph (i.e. the Stefan problem),depending on the space variable in the same manner as k. Applicationsto eddy-current problems in magnetic composite media (steelcables, laminations) are suggested. A numerical example is given.     

Sur L'equation Diophantienne (xn-1)/(x-1)=yq, III

In this paper we study the diophantine equation of the title,which was first introduced by Nagell and Ljunggren during thefirst half of the twentieth century. We describe a method whichallows us, on the one hand when n is fixed, to obtain an upperbound for q, and on the other hand when n and q are fixed, toobtain upper bounds for x and y which are far sharper than thosederived from the theory of linear forms in logarithms. We alsoshow how these bounds can be used even when they seem too largefor a straightforward enumeration of the remaining possiblevalues of x. By combining all these techniques, we are ableto solve the equation in many cases, including the case whenn has a prime divisor less than 13, or the case when n has aprime divisor which is less than or equal to 23 and distinctfrom q. 2000 Mathematical Subject Classification: primary 11D41;secondary 11J86, 11Y50.     

Cerf Theory for Graphs

We develop a deformation theory for k-parameter families ofpointed marked graphs with fixed fundamental group F_n. Applicationsinclude a simple geometric proof of stability of the rationalhomology of Aut(F_n), computations of the rational homology insmall dimensions, proofs that various natural complexes of freefactorizations of F_n are highly connected, and an improvementon the stability range for the integral homology of Aut(F_n).     

Mixed block elimination for linear systems with wider borders

The paper is about the stable solution of possibly ill-conditionedbordered linear systems. Given stable solvers for matrix A andfor A^T, we prove that the Govaerts Mixed Block Elimination (BEM)method constitutes a stable solver for the matrix consistingof A or A^T with a border of width 1, and hence by recursionfor a border of any width. We express the algorithm in an efficient,iterative, form. We analyse its operation count, and verifythe theory by extensive numerical experiments. ^*Senior Research Associate of the Belgian National Fund of ScientificResearch NFWO.     

L1 Properties of Second order Elliptic Operators

We present a connected account of various spectral and regularityproperties of the semigroup associated with a fairly generalsymmetric second order elliptic operator on a Riemannian manifold.Our main goal is to relate the L² theory to the less well understoodL¹ theory, and hence to the approach via the theory of stochasticdifferential equations.     

The Ubiquity of Thompson's Group F in Groups of Piecewise Linear Homeomorphisms of the Unit Interval

In the 1960s, Richard J. Thompson introduced a triple of groupsF T G which, among them, supplied the first examples of infinite,finitely presented, simple groups [14] (see [6] for publisheddetails), a technique for constructing an elementary exampleof a finitely presented group with an unsolvable word problem[12], the universal obstruction to a problem in homotopy theory[8], and the first examples of torsion free groups of type FPand not of type FP [5]. In abstract measure theory, it has beensuggested by Geoghegan (see [3] or [9, Question 13]) that Fmight be a counterexample to the conjecture that any finitelypresented group with no non-cyclic free subgroup is amenable(admits a bounded, non-trivial, finitely additive measure onall subsets that is invariant under left multiplication). Recently,F has arisen in the theory of groups of diagrams over semigrouppresentations [10], and as the object of questions in the algebraof string rewriting systems [7]. For more extensive bibliographiesand more results on Thompson's groups and their generalizationssee [1, 4, 6]. A persistent peculiarity of Thompson's groups is their abilityto pop up in diverse areas of mathematics. This suggests thatthere might be something very natural about Thompson's groups.We support this idea by showing (Theorem 1.1 below) that PL_o(I),the group of piecewise linear (finitely many changes of slope),orientation-preserving, self-homeomorphisms of the unit interval,is riddled with copies of F: a very weak criterion implies thata subgroup of PL_o(I) must contain an isomorphic copy of F.     

On the Structure of Minimal Left Ideals in the Largest Compactification of a Locally Compact Group

This paper is centred around a single question: can a minimalleft ideal L in G^LUC, the largest semi-group compactificationof a locally compact group G, be itself algebraically a group?Our answer is no (unless G is compact). In deriving this conclusion,we obtain for nearly all groups the stronger result that nomaximal subgroup in L can be closed. A feature of our work isthat completely different techniques are required for the connectedand totally disconnected cases. For the former, we can relyon the extensive structure theory of connected, non-compact,locally compact groups to derive the solution from the commutativecase, using some reduction lemmas. The latter directly involvestopological dynamics; we construct a compact space and an actionof G on it which has pathological properties. We obtain otherresults as tools towards our main goal or as consequences ofour methods. Thus we find an extension to earlier work on therelationship between minimal left ideals in G^LUC and H^LUC whenH is a closed subgroup of G with G/H compact. We show that thedistal compactification of G is finite if and only if the almostperiodic compactification of G is finite. Finally, we use ourmethods to show that there is no finite subset of G^LUC invariantunder the right action of G when G is an almost connected groupor an IN-group.     

Buckling of an axisymmetric vesicle under compression: the effects of resistance to shear

We consider the axisymmetric deformation of an initially spherical,porous vesicle with incompressible membrane having finite resistanceto in-plane shearing, as the vesicle is compressed between parallelplates. We adopt a thin-shell balance-of-forces formulationin which the mechanical properties of the membrane are describedby a single dimensionless parameter, C, which is the ratio ofthe membrane's resistance to shearing to its resistance to bending.This results in a novel free-boundary problem which we solvenumerically to obtain vesicle shapes as a function of plateseparation, h. For small deformations, the vesicle contactseach plate over a small circular area. At a critical value ofplate separation, h_TC, there is a transcritical bifurcationfrom which a new branch of solutions emerges, representing buckledvesicles which contact each plate along a circular curve. Forthe values of C investigated, we find that the transcriticalbifurcation is subcritical and that there is a further saddle-nodebifurcation (fold) along the branch of buckled solutions ath = h_SN (where h_SN > h_TC). The resulting bifurcation structureis commensurate with a hysteresis loop in which a sudden transitionfrom an unbuckled solution to a buckled one occurs as h is decreasedthrough h_TC and a further sudden transition, this time froma buckled solution to an unbuckled one, occurs as h is increasedthrough h_SN. We find that h_SN and h_TC increase with C, thatis, vesicles that resist shear are more prone to buckling.     

Cluster-tilted algebras

We introduce a new class of algebras, which we call cluster-tilted. They are by definition the endomorphism algebras of tilting objects in a cluster category. We show that their representation theory is very close to the representation theory of hereditary algebras. As an application of this, we prove a generalised version of so-called APR-tilting.

     

Kazhdan-Lusztig Cells, q-Schur Algebras and James' Conjecture

We consider the Dipper–James q-Schur algebra S_q(n, r)_k,defined over a field k and with parameter q 0. An understandingof the representation theory of this algebra is of considerableinterest in the representation theory of finite groups of Lietype and quantum groups; see, for example, [6] and [11]. Itis known that S_q(n, r)_k is a semisimple algebra if q is nota root of unity. Much more interesting is the case when S_q(n,r)_k is not semisimple. Then we have a corresponding decompositionmatrix which records the multiplicities of the simple modulesin certain ‘standard modules’ (or ‘Weyl modules’).A major unsolved problem is the explicit determination of thesedecomposition matrices.     

Spectral Analysis of Higher-Order Differential Operators II: Fourth-Order Equations

We investigate the location and nature of the spectrum of thefourth-order self-adjoint equation (p₀ y')'+(p₁ y')'+qy=zwy subject to certain asymptotic assumptions on the coefficients.The main tools are the theory of asymptotic integration andthe Titchmarsh–Weyl M-matrix. Asymptotic integration yieldsasymptotic formulae for the solutions of the differential equationwhich are then used to derive properties of the M-matrix. Thecharacterisation of spectral properties in terms of the boundarybehaviour of M leads to the desired results.