Zeta Functions for Curves and Log Canonical Models

The topological zeta function and Igusa's local zeta functionare respectively a geometrical invariant associated to a complexpolynomial f and an arithmetical invariant associated to a polynomialf over a p-adic field. When f is a polynomial in two variables we prove a formula forboth zeta functions in terms of the so-called log canonicalmodel of f^-1{0} in A². This result yields moreover a conceptualexplanation for a known cancellation property of candidate polesfor these zeta functions. Also in the formula for Igusa's localzeta function appears a remarkable non-symmetric ‘q-deformation’of the intersection matrix of the minimal resolution of a Hirzebruch-Jungsingularity. 1991 Mathematics Subject Classification: 32S5011S80 14E30 (14G20)     

Cyclic homology of Hopf crossed products

We obtain a mixed complex, simpler than the canonical one, given the Hochschild, cyclic, negative and periodic homology of a crossed product E=A#fH, where H is an arbitrary Hopf algebra and f is a convolution invertible cocycle with values in A. We actually work in the more general context of relative cyclic homology. Specifically, we consider a subalgebra K of A which is stable under the action of H, and we find a mixed complex computing the Hochschild, cyclic, negative and periodic homology of E relative to K. As an application we obtain two spectral sequences converging to the cyclic homology of E relative to K. The first one works in the general setting and the second one (which generalizes those previously found by several authors) works when f takes its values in K.     

Finitary Power Semigroups of Infinite Groups are not Finitely Generated

For a semigroup S, the finitary power semigroup of S, denotedP_f(S), consists of all finite subsets of S under the usual multiplication.The main result of this paper asserts that P_f(G) is not finitelygenerated for any infinite group G. 2000 Mathematics SubjectClassification 20M05 (primary), 20M30, 20F99 (secondary).     

On Sets Where Iterates of a Meromorphic Function Zip Towards Infinity

For a transcendental meromorphic function f, various propertiesof the set [formula] were obtained in [8] and [9]. Here we establish analogous propertiesfor the smaller sets [formula] introduced in [5], and [formula] We deduce a symmetry result for Julia sets J(f), and also indicatesome techniques for showing that certain invariant curves liein I'(f), Z(f) and J(f). 2000 Mathematics Subject Classification30D05, 37F10, 37F50.     

The periodic points and the invariant set of an ϵ-contractive map

It is shown that the invariant set of an ϵ-contractive map f on a compact metric space X is the same as the set of periodic points of f. Furthermore, the set of periodic points of f is finite and, only assuming that X is locally compact, there is at most one periodic point in each component X. The theorems are applied to prove a known fixed-point theorem, a result concerning inverse limits, a result about periodic points of compositions, and a result showing that ϵ-contractive maps on continua are really contraction maps with a change in metric. It is shown that all our results hold for locally contractive maps on compact metric spaces.     

High-accuracy Tridiagonal Finite Difference Approximations for Non-linear Two-point Boundary Value Problems

We discuss the construction of finite difference approximationsfor the non-linear two-point boundary value problem: y" = f(x,y), y(a)=A, y(b)=B. In the case of linear differential equations,the resulting finite difference schemes lead to tridiagonallinear systems. Approximations of orders higher than four involvederivatives of f. While several approximations of a particularorder are possible, we obtain the "simplest" of these approximationsleading to two high-accuracy methods of orders six and eight.These two methods are described and their convergence is established;numerical results are given to illustrate the order of accuracyachieved.     

Type 2 Semi-Algebras of Continuous Functions

A semi-algebra of continuous functions is a cone A of continuousreal functions on a compact Hausdorff space X such that A containsthe products of its elements. A cone A is said to be of typen if fA implies fⁿ(1 + f)^–1 A. Uniformly closed semi-algebrasof types 0 and 1 have long been characterized in a manner analogousto the Stone–Weierstrass theorem, but, except for thecase when A is generated by a single function, little has beenknown about type 2. Here, progress is reported on two problems.The first is the characterization of those continuous linearfunctionals on C(X) that determine semi-algebras of type 2.The second is the determination of the type of the tensor productof two type 1 semi-algebras. 1991 Mathematics Subject Classification:46J10.     

A continuous path of singular masas in the hyperfinite II1 factor

Using methods of Tauer, we exhibit an uncountable family ofsingular masas in the hyperfinite II₁ factor R all with Pukánszkyinvariant {1}, no pair of which is conjugate by an automorphismof R. This is done by introducing an invariant (A) for a masaA in a II₁ factor N as the maximal size of a projection eA forwhich A e contains non-trivial centralizing sequences for eNe. The masas produced give rise to a continuous map from theinterval [0, 1] into the singular masas in R equipped with thed_{, 2}-metric. A result is also given showing that the Pukánszky invariantis d_{, 2}-upper semi-continuous. As a consequence, the sets ofmasas with Pukánszky invariant {n} are all closed.     

Fixed Point Index in Hyperspaces: A Conley-Type Index for Discrete Semidynamical Systems

Let X be a locally compact metric absolute neighbourhood retractfor metric spaces, U X be an open subset and f: U X be a continuousmap. The aim of the paper is to study the fixed point indexof the map that f induces in the hyperspace of X. For any compactisolated invariant set, K U, this fixed point index produces,in a very natural way, a Conley-type (integer valued) indexfor K. This index is computed and it is shown that it only dependson what is called the attracting part of K. The index is usedto obtain a characterization of isolating neighbourhoods ofcompact invariant sets with non-empty attracting part. Thisindex also provides a characterization of compact isolated minimalsets that are attractors.     

A quasi-periodic Poincaré's theorem

We study the persistence of invariant tori on resonant surfaces of a nearly integrable Hamiltonian system under the usual Kolmogorov non-degenerate condition. By introducing a quasi-linear iterative scheme to deal with small divisors, we generalize the Poincaré theorem on the maximal resonance case (i.e., the periodic case) to the general resonance case (i.e., the quasi-periodic case) by showing the persistence of majority of invariant tori associated to non-degenerate relative equilibria on any resonant surface.The first author was partially supported by NSFC grant 19971042, the National 973 Project of China: Nonlinearity, and the outstanding young's project of the Ministry of Education of China.The second author was partially supported by NSF grant DMS9803581.Mathematics Subject Classification (2000): Primary 58F05, 58F27, 58F30     

Algebras Generated by Holomorphic and Harmonic Functions on the Disc

Let E be a subset of the boundary of the open unit disc D, andlet A be the algebra of bounded holomorphic functions on D thatextend continuously to D E. It is shown that if f is a boundedharmonic function on D that extends continuously to D E andis not holomorphic, then the uniformly closed algebra A[f] generatedby A and f contains . This result contains as special cases a result on the disc algebradue to irka and a result on H(D due to Axler and Shields. Astronger form of the result, in which f is allowed to have discontinuitieson a small subset of E, is also established. 2000 MathematicsSubject Classification 46J10, 46J15, 30H05.     

Dimensions of Julia Sets of Hyperbolic Meromorphic Functions

It is known that, if f is a hyperbolic rational function, thenthe Hausdorff, packing and box dimensions of the Julia set,J(f), are equal. In this paper it is shown that, for a hyperbolictranscendental meromorphic function f, the packing and upperbox dimensions of J(f) are equal, but can be strictly greaterthan the Hausdorff dimension of J(f). 2000 Mathematics SubjectClassification 30D05, 37F10, 37F15, 37F35, 37F50.     

The Modulus of the Image Annuli Under Univalent Harmonic Mappings and a Conjecture of Nitsche

The object of the paper is to show that if f is a univalent,harmonic mapping of the annulus A(r, 1) = {z : r < |z| <1} onto the annulus A(R, 1), and if s is the length of the segmentof the Grötzsch ring domain associated with A(r, 1), thenR < s. This gives the first, quantitative upper bound ofR, which relates to a question of J. C. C. Nitsche that he raisedin 1962. The question of whether this bound is sharp remainsopen.     

The tower of K-theory of truncated polynomial algebras

Let A be a regular noetherian F_p-algebra. The relative K-groupsK_q(A[x]/(x^m),(x)) and the Nil-groups Nil_q(A[x]/(x^m)) were evaluatedby the author and Ib Madsen in terms of the big de Rham–Wittgroups W_rA^q of the ring A. In this paper, we evaluate the mapsof relative K-groups and Nil-groups induced by the canonicalprojection f: A[x]/(x^m) A[x]/(xⁿ). The result depends stronglyon the prime p. It generalizes earlier work by Stienstra onthe groups in degrees 2 and 3. Received February 28, 2007.     

(1, 2)-Symplectic Metrics, Flag Manifolds and Tournaments

It has recently been shown by Mo and Negreiros that a necessarycondition for an invariant almost complex structure on the complexfull flag manifold F(n) to admit a (1, 2)-symplectic invariantmetric is that its associated tournament be cone-free. In thispaper, a canonical stair-shaped form is given for such tournaments,and this is applied to show that the condition is also sufficient;in the process, all the associated (1, 2)-symplectic metricsare described. 2000 Mathematics Subject Classification 53C55,05C20, 58E20.     

Exactness of Free and Amenable Groups by the Construction of Ozawa Kernels

Using properties of their Cayley graphs, specific examples ofOzawa kernels are constructed for both free and amenable groups,thus showing that these groups satisfy Property O. It is deducedboth that these groups are exact and satisfy Yu's Property A.2000 Mathematics Subject Classification 20F65 (primary), 46l05(secondary).     

Cyclic Spaces for Grassmann Derivatives and Additive Theory

Let A be a finite subset of Z_p (where p is a prime). Erdösand Heilbronn conjectured (1964) that the set of sums of the2-subsets of A has cardinality at least min(p, 2|A| —3). We show here that the set of sums of all m-subsets of Ahas cardinality at least min {p,m(|A| — m)+ 1}. In particular,we answer affirmatively the above conjecture. We apply thisresult to the problem of finding the smallest n such that forevery subset 5 of cardinality n and every xZ_p there is a subsetof S with sum equal to x. On this last problem we improve theknown results due to Erdös and Heilbronn and to Olson. The above result will be derived from the following generalproblem on Grassmann spaces. Let F be a field and let V be afinite dimensional vector space of dimension d over F. Let pbe the characteristic of F in nonzero characteristic and otherwise. Let Df be the derivative of a linear operatorfon V, restrictedto the mth Grassmann space ^mV. We show that there is a cyclicsubspace for the derivative with dimension at least min {p,m(n–m)+ 1}, where n is the maximum dimension of the cyclic subspacesof f. This bound is sharp and is reached when f has d distincteigenvalues forming an arithmetic progression.     

Almost Invariant Half-spaces of Algebras of Operators

Given a Banach space X and a bounded linear operator T on X, a subspace Y of X is almost invariant under T if TY í Y+F{TY\subseteq Y+F} for some finite-dimensional “error” F. In this paper, we study subspaces that are almost invariant under every operator in an algebra \mathfrak A{\mathfrak A} of operators acting on X. We show that if \mathfrak A{\mathfrak A} is norm closed then the dimensions of “errors” corresponding to operators in \mathfrak A{\mathfrak A} must be uniformly bounded. Also, if \mathfrak A{\mathfrak A} is generated by a finite number of commuting operators and has an almost invariant half-space (that is, a subspace with both infinite dimension and infinite codimension) then \mathfrak A{\mathfrak A} has an invariant half-space.     

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).     

Uniform Embeddings of Free Products into Uniformly Convex Banach Spaces

Let A and B be countable discrete groups, and let = A * B betheir free product. We show that if A and B are uniformly embeddableinto a uniformly convex Banach space, then so is . 2000 MathematicsSubject Classification 46L89, 20F65.