Left Cotorsion Rings

It is proved that if R is an associative ring that is cotorsionas a left module over itself, and J is the Jacobson radicalof R, then the quotient ring R/J is a left self-injective vonNeumann regular ring and idempotents lift modulo J. In particular,if R is indecomposable, then it is a local ring. 2000 MathematicsSubject Classification 16D50, 16D90, 16L30.     

Zeros of Composite Meromorphic Functions and Ahlfors's Theory of Covering Surfaces

This paper improves a theorem of Bergweiler on the zeros offunctions f g — R with f meromorphic, g entire and Rrational. The proof yields a simple inequality for the numberof zeros and T(r, g). 2000 Mathematics Subject Classification30D35.     

On the Connectedness of Self-Affine Tiles

Let T be a self-affine tile in Rⁿ defined by an integral expandingmatrix A and a digit set D. The paper gives a necessary andsufficient condition for the connectedness of T. The conditioncan be checked algebraically via the characteristic polynomialof A. Through the use of this, it is shown that in R², for anyintegral expanding matrix A, there exists a digit set D suchthat the corresponding tile T is connected. This answers a questionof Bandt and Gelbrich. Some partial results for the higher-dimensionalcases are also given.     

Large Lattices Over Orders

Let K be the quotient field of a Dedekind domain R. We characterizethe R-orders in a separable K-algebra for which every R-projective-module decomposes into -lattices. Butler, Campbell and Kovácshave recently shown that the latter holds for the integral groupring of a cyclic group of prime order, as well as for lattice-finiteorders over a complete discrete valuation domain. 2000 MathematicsSubject Classification 16H05, 16G30 (primary), 05C65 (secondary).     

Stable Jacobson Radicals and Semiprime Smash Products

We prove that if H is a finite-dimensional semisimple Hopf algebraacting on a PI-algebra R of characteristic 0, and R is eitheraffine or algebraic over k, then the Jacobson radical of R isH-stable. Under the same hypotheses, we show that the smashproduct algebra R#H is semiprimitive provided that R is H-semiprime.More generally we show that the ‘finite’ Jacobsonradical is H-stable, and that R#H is semiprimitive providedthat R is H-semiprimitive and all irreducible representationsof R are finite-dimensional. We also consider R#H when R isan FCR-algebra. Finally, we prove a general relationship betweenstability of the radical and semiprimeness of R#H; in particularif for a given H, any action of H stabilizes the Jacobson radical,then also any action of H stabilizes the prime radical. 2000Mathematics Subject Classification 16W30, 16N20, 16R99, 16S40.     

Poincare Series of Multi-Filtered Algebras and Partitivity

It is proved that if an algebra R over a field can be endowedwith a pointed and finite-dimensional Nⁿ-filtration such thatthe associated Nⁿ-graded algebra T is semi-commutative, thenR is left and right finitely partitive. In order to do this,a multi-variable Poincaré series for every finitely generatedgraded T-module is considered and it is shown that this Poincaréseries is a rational function. The methods apply to some iteratedOre extensions such as quantum matrices and quantum Weyl algebrasas well as to the quantized enveloping algebra of sl(+1).     

POWERS OF OPERATORS DOMINATED BY STRICTLY SINGULAR OPERATORS

It is proved that every positive operator R on a Banach latticeE dominated by a strictly singular operator T:E E satisfiesthat the R⁴ is strictly singular. Moreover, if E is order continuousthen the R² is already strictly singular.     

On Groups and Rings Definable In O-Minimal Expansions of Real Closed Fields

Let R, >,+,• be a real closed field, and let M be ano-minimal expansion of R. We prove here several results regardingrings and groups which are definable in M. We show that everyM–definable ring without zero divisors is definably isomorphicto R, R((–l)) or the ring of quaternions over R. One corollaryis that no model of T_exp is interpretable in a model of T_an.     

On the Volume Ratio of Two Convex Bodies

Let K and L be two convex bodies in Rⁿ. The volume ratio vr(K,L) of K and L is defined by vr(K, L = inf(|K|/|T(L)|)^1/n, wherethe infimum is over all affine transformations T of Rⁿ for whichT(L) K. It is shown in this paper that vr(K, L) , where c > 0 is an absolute constant. This isoptimal up to the logarithmic term. 2000 Mathematics SubjectClassification 52A40, 46B07 (primary); 52A21, 52A20 (secondary).     

Artinian Rings and the H-Condition

It is proved that a ring R is right Artinian if and only if,for each countably generated right R-module M, there existsa finite subset F of M such that the annihilator of M in R equalsthe annihilator of F in R. 2000 Mathematics Subject Classification16P20.     

New perturbation analyses for the Cholesky factorization

We present new perturbation analyses for the Cholesky factorizationA = R^T R of a symmetric positive definite matrix A. The analysesmore accurately reflect the sensitivity of the problem thanprevious normwise results. The condition numbers here are alteredby any symmetric pivoting used in PAP^T = R^TR, and both numericalresults and an analysis show that the standard method of pivotingis optimal in that it usually leads to a condition number veryclose to its lower limit for any given A. It follows that thecomputed R will probably have greatest accuracy when we usethe standard symmetric pivoting strategy. Initially we give a thorogh analysis to obtain both first-orderand strict normwise perturbation bounds which are as tight aspossible, leading to a definition of an optimal condition numberfor the problem. Then we use this approach to obtain reasonablyclear first-order and strict componentwise perturbation bounds. We complete the work by giving a much simpler normwise analysiswhich provides a somewhat weaker bound, but which allows usto estimate the condition of the problem quite well with anefficient computation. This simpler analysis also shows whythe factorization is often less sensitive than we previouslythought, and adds further insight into why pivoting usuallygives such good results. We derive a useful upper bound on thecondition of the problem when we use pivoting. This research was supported by the Natural Sciences and EngineeringResearch Ciuncil of Canada Grant OGP0009236. This research was supported in part by the US National ScienceFoundation under grant CCR 95503126.     

The Ziegler Spectrum of a Locally Coherent Grothendieck Category

The general theory of locally coherent Grothendieck categoriesis presented. To each locally coherent Grothendieck categoryC a topological space, the Ziegler spectrum of C, is associated.It is proved that the open subsets of the Ziegler spectrum ofC are in bijective correspondence with the Serre subcategoriesof coh C the subcategory of coherent objects of C. This is aNullstellensatz for locally coherent Grothendieck categories.If R is a ring, there is a canonical locally coherent Grothendieckcategory RC (respectively, CR) used for the study of left (respectively,right) R-modules. This category contains the category of R-modulesand its Ziegler spectrum is quasi-compact, a property used toconstruct large (not finitely generated) indecomposable modulesover an artin algebra. Two kinds of examples of locally coherentGrothendieck categories are given: the abstract category theoreticexamples arising from torsion and localization and the examplesthat arise from particular modules over the ring R. The dualitybetween coh-(RC) and coh-CR is shown to give an isomorphismbetween the topologies of the left and right Ziegler spectraof a ring R. The Nullstellensatz is used to give a proof ofthe result of Crawley-Boevey that every character : K₀(coh-C) Z is uniquely expressible as a Z-linear combination of irreduciblecharacters. 1991 Mathematics Subject Classification: 16D90,18E15.     

Quantum Markov Processes with a Christensen-Evans Generator in a Von Neumann Algebra

Let A be a unital von Neumann algebra of operators on a complexseparable Hilbert space H₀, and let {T_t, t 0} be a uniformlycontinuous quantum dynamical semigroup of completely positiveunital maps on A. The infinitesimal generator L of {T_t} is abounded linear operator on the Banach space A. For any Hilbertspace K, denote by B(K) the von Neumann algebra of all boundedoperators on K. Christensen and Evans [3] have shown that Lhas the form [formula] where is a representation of A in B(K) for some Hilbert spaceK, R: H₀ K is a bounded operator satisfying the ‘minimality’condition that the set {(RX–(X)R)u, uH₀, XA} is totalin K, and K₀ is a fixed element of A. The unitality of {T_t}implies that L(1) = 0, and consequently K₀=iH–R^*R, whereH is a hermitian element of A. Thus (1.1) can be expressed as [formula] We say that the quadruple (K, , R, H) constitutes the set ofChristensen–Evans (CE) parameters which determine theCE generator L of the semigroup {T_t}. It is quite possible thatanother set (K', ', R', H') of CE parameters may determine thesame generator L. In such a case, we say that these two setsof CE parameters are equivalent. In Section 2 we study thisequivalence relation in some detail. 1991 Mathematics SubjectClassification 81S25, 60J25.     

A Hilbert Basis Theorem for Quantum Groups

A noncommutative version of the Hilbert basis theorem is usedto show that certain R-symmetric algebras S_R(V) are Noetherian.This result applies in particular to the coordinate ring ofquantum matrices A_R(V) associated with an R-matrix R operatingon the tensor square of a vector space V, to show that, undera natural set of hypotheses on R, the algebra A_R(V) is Noetherianand its augmentation ideal has a polynormal set of generators.As a corollary we deduce that these properties hold for thegeneric quantized function algebras R_q[G] over any field ofcharacteristic zero, for G an arbitrary connected, simply connected,semisimple group over C. That R_q[G] is Noetherian recovers aresult due to Joseph [10], with a different proof.1991 MathematicsSubject Classification 17B37, 16P40.     

On the Auslander-Reiten Periodicity of Self-Injective Algebras

This paper investigates how to relate the syzygy periodicityof a self-injective algebra A to its Auslander–Reitenperiodicity. Moreover, a characterization is provided of theAuslander–Reiten bounded A–A-bimodules that areperiodic. 2000 Mathematics Subject Classification 16G70, 16E40(primary), 16G20 (secondary).     

Error Bounds for Hermite Interpolation by Quadratic Splines on an {alpha}-Triangulation

For l, an -triangulation F of a planar domain is such that,for every T F, there holds 1 R_T/2r_T , where R_T (resp. r_T)denotes the radius of the circumscribed (resp. inscribed) circleof the triangle T. When T is varying in F the centre of itsinscribed circle is varying in a compact interior to T and itsorthogonal projections on the sides are varying in compact intervalsinterior to these sides. Precise results are given about thesizes of these compacts and are used for the computation oferror constants in the problem of Hermite interpolation by Powell-Sabinquadratic finite elements, bringing to the fore their dependenceon the parameter .     

The (Q, q)-Schur Algebra

In this paper we use the Hecke algebra of type B to define anew algebra S which is an analogue of the q-Schur algebra. Weshow that S has ‘generic’ basis which is independentof the choice of ring and the parameters q and Q. We then constructWeyl modules for S and obtain, as factor modules, a family ofirreducible S-modules defined over any field. 1991 MathematicsSubject Classification: 16G99, 20C20, 20G05.     

A Note on Two-Weight Inequalities for Maximal Functions and Singular Integrals

We deal with weighted inequalities of the type [formula] where: T is either the Hardy–Littlewood maximal operator or asingular integral operator; G is any measurable subset of Rⁿ; f is any measurable function, vanishing outside G, such thatTf is well-defined; v and w are weights, that is, nonnegativelocally integrable functions on G; p, q(1, ). 1991 Mathematics Subject Classification 42B20, 42B25.     

Plane with A{infty}-Weighted Metric not Bilipschitz Embeddable to Rn

A planar set G R² is constructed that is bilipschitz equivalentto (G,d^z), where (G, d) is not bilipschitz embeddable to anyuniformly convex Banach space. Here, Z (0, 1) and d^z denotesthe zth power of the metric d. This proves the existence ofa strong A weight in R², such that the corresponding deformedgeometry admits no bilipschitz mappings to any uniformly convexBanach space. Such a weight cannot be comparable to the Jacobianof a quasiconformal self-mapping of R². 2000 Mathematics SubjectClassification 54E40 (primary); 30C62, 30C65, 28A80 (secondary).     

Marked Schottky Groups and Representations of Surface Groups

We give a complete condition for any n elements of PSL(2, R)to generate a Fuchsian group which is a Schottky group on thatset of generators, and apply it to a question of Bowditch onrepresentations of surface groups into PSL(2, R). 2000 MathematicalSubject Classification: 20H10, 32G15.