On the Existence of Solutions for Algebraically Stable Runge--Kutta Methods

An implicit Runge—Kutta method, applied to an initial-valueproblem, gives systems of algebraic equations. It is shown that,under natural assumptions concerning the differential system,these equations have unique solutions if the method satisfiesa condition related to algebraic stability. In particular, thiscondition is satisfied if the method is irreducible and (k,l)-algebraically stable for some l 0.     

Detecting K-Theory by Cyclic Homology

We discuss which part of the rationalized algebraic K-theoryof a group ring is detected via trace maps to Hochschild homology,cyclic homology, periodic cyclic or negative cyclic homology.2000 Mathematics Subject Classification 19D55.     

Real Abelian Varieties with Many Line Bundles

Let X and Y be affine nonsingular real algebraic varieties.A general problem in real algebraic geometry is to try to decidewhen a continuous map f: X Y can be approximated by regularmaps in the space of c⁰ mappings from X to Y, equipped withthe c⁰ topology. This paper solves this problem when X is theconnected component containing the origin of the real part ofa complex Abelian variety and Y is the standard 2-dimensionalsphere.     

On Cohen-Macaulay Posets, Koszul Algebras and Certain Modules Associated to Schubert Varieties

In the first part of this paper, some cohomological propertiesof the incidence algebra A_k(X) of a finite poset X over a fieldk are studied. In particular, it is shown that A_k(X) is Koszulif and only if every open interval of X is Cohen-Macaulay overk. In the second part, certain categories of representationsof a Borel subgroup B of a semisimple connected algebraic groupG are studied. It is shown that these categories have the samecohomological properties as certain Cohen-Macaulay posets relatedto the Weyl group of G, and the higher Ext-groups between certainB-modules are then computed, using results from the first part.     

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.     

Characteristic Zero Loop Space Homology of Two-Cones

Given a principal ideal domain R of characteristic zero, containing1/2, and a two-cone X of appropriate connectedness and dimension,we present sufficient algebraic conditions for the Hopf algebraFH(X; R) to be isomorphic with the universal enveloping algebraof an R-free graded Lie algebra; here, F stands for free part(that is, quotient by the R-torsion), H for homology, and forthe Moore loop space functor. 2000 Mathematics Subject Classification55P35, 55P62, 57T05, 17B70, 17B35.     

Cutting-plane theory: Algebraic methods

This paper is a survey, with new results, of the algebraic approach to cutting-planes. The new results are a subadditive dual program for integer programs, and a generalization of R. Gomory's linear inequality characterization of the cuts valid for the group problem. We also provide some results on the large family of alternate group problems, in addition to the one group problem that is conventionally thought of in connection with the algebraic approach. In terms of expository material, we shall provide very concise proofs of the most well-known results in the basic papers of the algebraic approach. (Note: This is a revision of Part I of MSRR no. 370).     

Towers of MU-Algebras and the Generalized Hopkins-Miller theorem

Our results are of three types. First, we describe a generalprocedure of adjoining polynomial variables to A-ring spectrawhose coefficient rings satisfy certain restrictions. A hostof examples of such spectra is provided by killing a regularideal in the coefficient ring of MU, the complex cobordism spectrum.Second, we show that the algebraic procedure of adjoining rootsof unity carries over in the topological context for such spectra.Third, we use the developed technology to compute the homotopytypes of spaces of strictly multiplicative maps between suitableK(n)-localizations of such spectra. This generalizes the famousHopkins–Miller theorem and gives strengthened versionsof various splitting theorems. 2000 Mathematics Subject Classification55N20, 55S35, 55T25 (primary), 16E40, 13D10 (secondary).     

Are Operads Algebraic Theories?

I exhibit a pair of non-symmetric operads that, while not themselvesisomorphic, induce isomorphic monads. The existence of sucha pair implies that if ‘algebraic theory’ is understoodas meaning ‘monad’ then operads cannot be regardedas algebraic theories of a special kind. 2000 Mathematics SubjectClassification 18D50 (primary), 18C10, 18C15 (secondary).     

From Klein to Painleve Via Fourier, Laplace and Jimbo

We describe a method for constructing explicit algebraic solutionsto the sixth Painlevé equation, generalising that ofDubrovin and Mazzocco. There are basically two steps. Firstwe explain how to construct finite braid group orbits of triplesof elements of SL₂(C) out of triples of generators of three-dimensionalcomplex reflection groups. (This involves the Fourier–Laplacetransform for certain irregular connections.) Then we adapta result of Jimbo to produce the Painlevé VI solutions.(In particular, this solves a Riemann–Hilbert problemexplicitly.) Each step is illustrated using the complex reflection groupassociated to Klein's simple group of order 168. This leadsto a new algebraic solution with seven branches. We also provethat, unlike the algebraic solutions of Dubrovin and Mazzoccoand Hitchin, this solution is not equivalent to any solutioncoming from a finite subgroup of SL₂(C). The results of this paper also yield a simple proof of a recenttheorem of Inaba, Iwasaki and Saito on the action of Okamoto'saffine D₄ symmetry group as well as the correct connection formulaefor generic Painlevé VI equations. 2000 Mathematics SubjectClassification 34M55, 34M40, 20F55.     

The Space of Loops on a Homogeneous Space and Rational Equivalence

We shall construct a map which is a rational equivalence with applications to phantom maps,spaces of the same n-type and genus sets. This answers in theaffirmative a question of C. McGibbon. 1991 Mathematics SubjectClassification 55Q05.     

RESEARCH IN MATHEMATICS EDUCATION: SOME ISSUES AND SOME EMERGING INFLUENCES

In this article, I introduce a typology of forms of algebraic thinking. In the first part, I argue that the form and generality of algebraic thinking are characterised by the mathematical problem at hand and the embodied and other semiotic resources that are mobilised to tackle the problem in analytic ways. My claim is based not only on semiotic considerations but also on new theories of cognition that stress the fundamental role of the context, the body and the senses in the way in which we come to know. In the second part, I present some concrete examples from a longitudinal classroom research study through which the typology of forms of algebraic thinking is illustrated.     

Linearformen mit algebraischen Koeffizienten

In a recent paper I generalized the Thue-Siegel-Roth-Schmidt theorem on simultaneous rational approximation of n real algebraic numbers to include also the p-adic case. In the present paper I shall carry the argument further and prove analogous results for systems of real and p-adic linear forms with algebraic coefficients.     

Existence of complete vector topologies with prescribed conditions

In every infinite-dimensional vector space we build a complete, non-locally convex linear Hausdorff topology under the prescribed condition that it shall be compatible with a bounded algebraic structure for the space, that is, it admits a bounded Hamel basis whose coefficient linear functionals are continuous. As a byproduct, we obtain the existence of non-locally convex topologies having the approximate and fixed point property as prescribed conditions. The obstruction of bounded Hamel bases by complete vector topologies on linear space of countable dimension is also discussed.     

The Beginnings of the Theory of Hopf Algebras

We consider issues related to the origins, sources and initial motivations of the theory of Hopf algebras. We consider the two main sources of primeval development: algebraic topology and algebraic group theory. Hopf algebras are named from the work of Heinz Hopf in the 1940’s. In this note we trace the infancy of the subject back to papers from the 40’s, 50’s and 60’s in the two areas mentioned above. Many times we just describe—and/or transcribe parts of—some of the relevant original papers on the subject.     

Fibrations spinorielles et twisteurs généralisés

In the first part my purpose is to find existence conditions for a spin structure on a manifold in a purely differential geometric scope, without the machinery of the algebraic topology and avoiding systematically any matricial formalism. A spin structure can be considered as aG-structure (using a classical terminology). Groups, called by myself “groups of spinoriality” play an essential part in this problem. We study them with a few details in the second part of chapter I. The second part is devoted to introduce the notion of twistors, generalizing studies already made in particular cases. Twistorial fibrations naturally go with spin fibrations. Broadly, twistors of order 2 are for the conformal group of signature (p, q), that are spinors for the group of isometry of signature (p+1,q+1).     

Burns' Equivariant Tamagawa Invariant T{Omega}loc(N/Q,1) for some Quaternion Fields

Inspired by the work of Bloch and Kato in [2], David Burns constructedseveral ‘equivariant Tamagawa invariants’ associatedto motives of number fields. These invariants lie in relativeK-groups of group-rings of Galois groups, and in [3] Burns gaveseveral conjectures (see Conjecture 3.1) about their values.In this paper I shall verify Burns' conjecture concerning theinvariant T^loc(N/Q,1) for some families of quaternion extensionsN/Q. Using the results of [9] I intend in a subsequent paperto verify Burns' conjecture for those families of quaternionfields which are not covered here.     

Stability of Runge-Kutta Methods for Trajectory Problems

A solution of a system of m autonomous differential equationsdefines a trajectory in m-dimensional space and, in particular,may give a closed orbital path. Typical trajectories are describedby a model nonlinear problem introduced in this article. Forthis problem, a trajectory lies on a surface characterized bya real symmetric matrix. It is shown that some Runge-Kutta methodspossess a property which ensures that, for this model problem,the numerical solution lies on the same surface as the trajectory.When m = 2, the numerical solution lies on the trajectory. Thisproperty is related to algebraic stability. A weaker propertysuffices for normalized differential systems.     

Witt groups and unipotent elements in algebraic groups

Let G be a semisimple algebraic group defined over an algebraicallyclosed field K of good characteristic p>0. Let u be a unipotentelement of G of order p^t, for some t N. In this paper it isshown that u lies in a closed subgroup of G isomorphic to theit Witt group W^t(K), which is a t-dimensional connected abelianunipotent algebraic group. 2000 Mathematics Subject Classification:20G15.     

Euler Characteristics in Relative K-Groups

Suppose that M is a finite module under the Galois group ofa local or global field. Ever since Tate's papers [17, 18],we have had a simple and explicit formula for the Euler–Poincarécharacteristic of the cohomology of M. In this note we are interestedin a refinement of this formula when M also carries an actionof some algebra A, commuting with the Galois action (see Proposition5.2 and Theorem 5.1 below). This refinement naturally takesthe shape of an identity in a relative K-group attached to A(see Section 2). We shall deduce such an identity whenever wehave a formula for the ordinary Euler characteristic, the keystep in the proof being the representability of certain functorsby perfect complexes (see Section 3). This representabilitymay be of independent interest in other contexts. Our formula for the equivariant Euler characteristic over Aimplies the ‘isogeny invariance’ of the equivariantconjectures on special values of the L-function put forwardin [3], and this was our motivation to write this note. Incidentally,isogeny invariance (of the conjectures of Birch and Swinnerton-Dyer)was also a motivation for Tate's original paper [18]. I am verygrateful to J-P. Serre for illuminating discussions on the subjectof this note, in particular for suggesting that I consider representability.I should also like to thank D. Burns for insisting on a mostgeneral version of the results in this paper. 2000 MathematicsSubject Classification 19A99, 18G99, 11R34.