Geometrical Spines of Lens Manifolds

We introduce the concept of ‘geometrical spine’for 3-manifolds with natural metrics, in particular, for lensmanifolds. We show that any spine of L_p,q that is close enoughto its geometrical spine contains at least E(p,q) – 3vertices, which is exactly the conjectured value for the complexityc(L_p,q). As a byproduct, we find the minimal rotation distance(in the Sleator–Tarjan–Thurston sense) between atriangulation of a regular p-gon and its image under rotation.     

DESINGULARIZATIONS OF CALABI-YAU 3-FOLDS WITH CONICAL SINGULARITIES. II. THE OBSTRUCTED CASE

This is the second of two papers studying Calabi–Yau 3-foldswith conical singularities and their desingularizations. Inour first paper [Y.-M. Chan, Quart. J. Math. 57 (2006), 151–181]we constructed the desingularization of the conically singularmanifold M₀ by gluing an asymptotically conical (AC) Calabi–Yau3-fold Y into M₀ at the singular point, thus obtaining a 1-parameterfamily of compact, non-singular Calabi–Yau 3-folds M_tfor small t > 0. During the gluing process one may encountera kind of cohomological obstruction to defining a 3-form _t onM_t which interpolates between the 3-form ₀ on M₀ and the scaled3-form t³ _Y on Y if the rate at which the AC Calabi–Yau3-fold Y converges to the Calabi–Yau cone is equal to– 3. The first paper [3] studied the simpler case <–3 where there is no obstruction. This paper extends theresult in the first one by considering a more complicated situtationwhen = –3. Assuming the existence of singular Calabi–Yaumetrics on compact complex 3-folds with ordinary double points,our result in this paper can be applied to repairing such kindsof singularities, which is an analytic version of Friedman'sresult giving necessary and sufficient conditions for smoothingordinary double points.     

Euler Characteristics of Coxeter Groups, PL-Triangulations of Closed Manifolds, and Cohomology of Subgroups of Artin Groups

The motivation for the theory of Euler characteristics of groups,which was introduced by C. T. C. Wall [21], was topology, butit has interesting connections to other branches of mathematicssuch as group theory and number theory. This paper investigatesEuler characteristics of Coxeter groups and their applications.In his paper [20], J.-P. Serre obtained several fundamentalresults concerning the Euler characteristics of Coxeter groups.In particular, he obtained a recursive formula for the Eulercharacteristic of a Coxeter group, as well as its relation tothe Poincaré series (see 3). Later, I. M. Chiswell obtainedin [10] a formula expressing the Euler characteristic of a Coxetergroup in terms of orders of finite parabolic subgroups (Theorem1). These formulae enable us to compute Euler characteristicsof arbitrary Coxeter groups. On the other hand, the Euler characteristics of Coxeter groupsW happen to be intimately related to their associated complexesF_W, which are defined by means of the posets of nontrivial parabolicsubgroups of finite order (see 2.1 for the precise definition).In particular, it follows from the recent result of M. W. Davis[13] that if F_W is a product of a simplex and a generalizedhomology 2n-sphere, then the Euler characteristic of W is zero(Corollary 3.1). The first objective of this paper is to generalizethe previously mentioned result to the case when F_W is a PL-triangulationof a closed 2n-manifold which is not necessarily a homology2n-sphere. In other words (as given below in Theorem 3), ifW is a Coxeter group such that F_W is a PL-triangulation of aclosed 2n-manifold, then the Euler characteristic of W is equalto 1–(F_W)/2.     

A fully discrete stabilized finite-element method for the time-dependent Navier-Stokes problem

A fully discrete stabilized finite-element method is presentedfor the two-dimensional time-dependent Navier–Stokes problem.The spatial discretization is based on a finite-element spacepair (X_h, M_h) for the approximation of the velocity and thepressure, constructed by using the Q₁ – P₀ quadrilateralelement or the P₁ – P₀ triangular element; the time discretizationis based on the Euler semi-implicit scheme. It is shown thatthe proposed fully discrete stabilized finite-element methodresults in the optimal order error bounds for the velocity andthe pressure.     

Homology and derived p-series of groups

We give homological conditions that ensure that a group homomorphisminduces an isomorphism modulo any term of the derived p –series, in analogy to Stallings's 1963 result for the p-lowercentral series. In fact, we prove a stronger theorem that isanalogous to Dwyer's extensions of Stallings’ results.It follows that spaces that are _p-homology equivalent have isomorphicfundamental groups modulo any term of their p-derived series.Various authors have related the ranks of the successive quotientsof the p-lower central series and of the derived p-series ofthe fundamental group of a 3-manifold M to the volume of M,to whether certain subgroups of ₁(M) are free, to whether finiteindex subgroups of ₁(M) map onto non-abelian free groups, andto whether finite covers of M are ‘large’ in variousother senses.     

Cohomology Actions and Centralisers in Unitary Reflection Groups

In an earlier work, the second author proved a general formulafor the equivariant Poincaré polynomial of a linear transformationg which normalises a unitary reflection group G, acting on thecohomology of the corresponding hyperplane complement. Thisformula involves a certain function (called a Z-function below)on the centraliser CG(g), which was proved to exist only incertain cases, for example, when g is a reflection, or is G-regular,or when the centraliser is cyclic. In this work we prove theexistence of Z-functions in full generality. Applications includereduction and product formulae for the equivariant Poincarépolynomials. The method is to study the poset L(C_G(g)) of subspaceswhich are fixed points of elements of C_G(g). We show that thisposet has Euler characteristic 1, which is the key propertyrequired for the definition of a Z-function. The fact aboutthe Euler characteristic in turn follows from the ‘join-atom’property of L(C_G(g)), which asserts that if [X₁,..., X_k} isany set of elements of L(C_G(g)) which are maximal (set theoretically)then their setwise intersection lies in L(C_G(g)). 2000 Mathematical Subject Classification:primary 14R20, 55R80; secondary 20C33, 20G40.     

Asymptotic behavior of positive solutions of some quasilinear elliptic problems

We discuss the asymptotic behavior of positive solutions ofthe quasilinear elliptic problem –_pu = a u^p–1–b(x)u^q, u| = 0, as q p – 1 + 0 and as q , via a scale argument.Here _p is the p-Laplacian with 1 < p and q > p –1. If p = 2, such problems arise in population dynamics. Ourmain results generalize the results for p = 2, but some technicaldifficulties arising from the nonlinear degenerate operator–_p are successfully overcome. As a by-product, we cansolve a free boundary problem for a nonlinear p-Laplacian equation.     

Essentially infinite colourings of hypergraphs

We consider edge colourings of the complete r-uniform hypergraphK_n^(r)on n vertices. How many colours may such a colouring haveif we restrict the number of colours locally? The local restrictionis formulated as follows: for a fixed hypergraph H and an integerk we call a colouring (H, k)-local if every copy of H in thecomplete hypergraph K_n^(r) receives at most k different colours. We investigate the threshold for k that guarantees that every(H, k)-local colouring of K_n^(r) must have a globally boundednumber of colours as n , and we establish this threshold exactly.The following phenomenon is also observed: for many H (at leastin the case of graphs), if k is a little over this threshold,the unbounded (H, k)-local colourings exhibit their colourfulnessin a ‘sparse way’; more precisely, a bounded numberof colours are dominant while all other colours are rare. Hencewe study the threshold k₀ for k that guarantees that every (H,k)-local colouring n of K_n^(r) with k k₀ must have a globallybounded number of colours after the deletion of up to n^r edgesfor any fixed > 0 (the bound on the number of colours isallowed to depend on H and only); we think of such colouringsn as ‘essentially finite’. As it turns out, everyessentially infinite colouring is closely related to a non-monochromaticcanonical Ramsey colouring of Erdös and Rado. This secondthreshold is determined up to an additive error of 1 for everyhypergraph H. Our results extend earlier work for graphs byClapsadle and Schelp [‘Local edge colorings that are global’,J. Graph Theory 18 (1994) 389–399] and by the first twoauthors and Schelp [‘Essentially infinite colourings ofgraphs’, J. London Math. Soc. (2) 61 (2000) 658–670].We also consider a related question for colourings of the integersand arithmetic progressions.

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2000 Mathematics Subject Classification 05D10 (primary), 05C35(secondary). The first author was partially supported by NSF grants CCR 0225610and DMS 0505550. The second author was partially supported byFAPESP and CNPq through a Temático–ProNEx project(Proc. FAPESP 2003/09925–5) and by CNPq (Proc. 306334/2004–6and 479882/2004–5). The third author was partially supportedby NSF grant DMS 0300529. The fourth author was partly supportedby the DFG within the European graduate program ‘Combinatorics,Geometry, and Computation’ (No. GRK 588/2) and by DFGgrant SCHA 1263/1–1. This work was supported in part bya CAPES/DAAD collaboration grant.     

Exotic smooth structures on Formula

Motivated by Stipsicz and Szabó's exotic 4-manifoldswith b₂⁺ = 3 and b₂^– = 8, we construct a family of simplyconnected smooth 4-manifolds with b₂⁺ = 3 and b₂^– = 8.As a corollary, we conclude that the topological 4-manifold     

Recurrence and asymptotics for orthonormal rational functions on an interval

^{Let µ be a positive bounded Borel measure on a subsetI of the real line and = {₁, ..., _n} a sequence of arbitrary ‘complex’poles outside I. Suppose {₁, ..., _n} is the sequence of rationalfunctions with poles in orthonormal on I with respect to µ. First, we are concernedwith reducing the number of different coefficients in the three-termrecurrence relation satisfied by these orthonormal rationalfunctions. Next, we consider the case in which I = [–1, 1] and µ satisfies the Erdos–Turán conditionµ' > 0 a.e. on I (where µ' is the Radon–Nikodymderivative of the measure µ with respect to the Lebesguemeasure) to discuss the convergence of _n+1(x)/_n(x) as n tendsto infinity and to derive asymptotic formulas for the recurrencecoefficients in the three-term recurrence relation. Finally,we give a strong convergence result for _n(x) under the morerestrictive condition that µ satisfies the Szeg condition(1 – x2)–1/2 log µ'(x) L1([– 1, 1]).}     

A Class of Pencils of Matrices

A linear machine is one in which the time dependent input yis related to the output z by P(D). z = S(D). y where P andS are polynomials in D = d/dt with constant coefficients. Fornumerical computation it is necessary to replace this relationby a set of simultaneous first order differential equationsand this paper shows how to construct such equations by methodswhich extend the results of Gilder (1961). Attention is restrictedto those sets of equations that are of a special form (see (1))which is characterized by the matrix operating on the dependentvariables. This matrix forms a pencil, being linear in D, andthree theorems are given to show how such matrix pencils maybe constructed from the polynomials. The theorems also statethat any matrix pencil with the required properties can be transformedinto the canonical forms given in the theorems by pre- and post-multiplicationby suitable constant non-singular matrices. Thus the variablesof any set of equations having the required properties are linearcombinations of the variables of the equations given by thetheorems. In the paper it is assumed that the degree of P(D)is greater than that of S(D), as otherwise z would be replacedby z₁+Q(D) . y, where Q is the quotient of S(D)/P(D). Also,as the algebriac manipulations are independent of the natureof the polynomials, D is replaced by an indeterminate x andthe coefficients considered to be from an arbitrary field. Fortechnical reasons we rename y and z, y_o and y_n–_m respectively.     

The Product Separability of the Generalized Free Product of Cyclic Groups

Let G be a group endowed with its profinite topology, then Gis called product separable if the profinite topology of G isHausdorff and, whenever H₁, H₂, ..., H_n are finitely generatedsubgroups of G, then the product subset H₁ H₂ ... H_n is closedin G. In this paper, we prove that if G=FxZ is the direct productof a free group and an infinite cyclic group, then G is productseparable. As a consequence, we obtain the result that if Gis a generalized free product of two cyclic groups amalgamatinga common subgroup, then G is also product separable. These resultsgeneralize the theorems of M. Hall Jr. (who proved the conclusionin the case of n=1, [3]), and L. Ribes and P. Zalesskii (whoproved the conclusion in the case of that G is a finite extensionof a free group, [6]).     

Some New Subgroups of the Baby Monster

In this paper we show how to construct maximal subgroups L₂(31)and L₂(49)·2 of the Baby Monster. We also show that eachis unique up to conjugacy.     

Constructions of stable equivalences of Morita type for finite-dimensional algebras III

In this paper, we provide a new method to produce stable equivalencesof Morita type. Our main results can be stated as follows. LetA and B be two finite-dimensional k-algebras over a field k.Suppose that two bimodules _AM_B and _BN_A define a stable equivalenceof Morita type between A and B and that R is a generator forA-modules. Then there is a stable equivalence of Morita typedefined by X and Y between the endomorphism algebra End_A(R)of the module R and the endomorphism algebra End_B(N_AR) of themodule N_AR. If M and N satisfy the property that both (N_A–,M_B–) and (M_B–, N_A–) are adjoint pairs of functors,then so do the modules X and Y. Moreover, we show that the self-injectivedimension and the Gorenstein property are invariant under stableequivalences of Morita type with the above-mentioned adjointproperty.     

Estimates for Commutators of Orthogonal Fourier Series

In this paper we study weighted norm inequalities for the commutators[b, S_n] where b is a BMO function and S_n denotes the nth partialsum of the Fourier series relative to a system of orthogonalpolynomials on [–1, 1] with respect to general weights.Results about generalized Jacobi and Bessel Fourier series areobtained.     

A remark on the representation of Zolotarev polynomials

Zolotarev polynomials are the polynomials that have minimaldeviation from zero on [–1, 1] with respect to the norm||xⁿ – x^n–1 + a_n–2 x^n–2 + ... + a₁x+ a_n|| for given and for all a_k . This note complements the paper of F. Pehersforfer [J. LondonMath. Soc. (1) 74 (2006) 143–153] with exact (not asymptotic)construction of the Zolotarev polynomials with respect to thenorm L₁ for || < 1 and with respect to the norm L₂ for || 1 in the form of Bernstein–Szegö orthogonal polynomials.For all in L₁ and L₂ norms, the Zolotarev polynomials satisfyexactly (not asymptotically) the triple recurrence relationof the Chebyshev polynomials.     

Artin's map in stable homology

Using a recent theorem of Galatius, we identify the map on stablehomology induced by Artin's injection of the braid group β_ninto the automorphism group of the free group Aut F_n.     

On the non-existence of exceptional automorphisms on Shimura curves

We study the group of automorphisms of Shimura curves X₀(D,N) attached to an Eichler order of square-free level N in anindefinite rational quaternion algebra of discriminant D>1.We prove that, when the genus g of the curve is greater thanor equal to 2, Aut (X₀(D, N)) is a 2-elementary abelian groupwhich contains the group of Atkin–Lehner involutions W₀(D,N) as a subgroup of index 1 or 2. It is conjectured that Aut(X₀(D, N))=W₀(D, N) except for finitely many values of (D, N)and we provide criteria that allow us to show that this is indeedoften the case. Our methods are based on the theory of complexmultiplication of Shimura curves and the Cerednik–Drinfeldtheory on their rigid analytic uniformization at primes p| D.