A New Variant of Gaussian Elimination

We describe an algorithm based on Gaussian elimination for solvingan nxn system of linear equations. It uses a pivoting strategywhich is particularly appropriate whenever the residuals canbe made small by solving for fewer than n of the unknowns asmay happen, for example, in interpolation problems.     

Gaussian Groups and Garside Groups, Two Generalisations of Artin Groups

It is known that a number of algebraic properties of the braidgroups extend to arbitrary finite Coxeter-type Artin groups.Here we show how to extend the results to more general groupsthat we call Garside groups. Define a Gaussian monoid to be a finitely generated cancellativemonoid where the expressions of a given element have boundedlengths, and where left and right lowest common multiples exist.A Garside monoid is a Gaussian monoid in which the left andright lowest common multiples satisfy an additional symmetrycondition. A Gaussian group is the group of fractions of a Gaussianmonoid, and a Garside group is the group of fractions of a Garsidemonoid. Braid groups and, more generally, finite Coxeter-typeArtin groups are Garside groups. We determine algorithmic criteriain terms of presentations for recognizing Gaussian and Garsidemonoids and groups, and exhibit infinite families of such groups.We describe simple algorithms that solve the word problem ina Gaussian group, show that these algorithms have a quadraticcomplexity if the group is a Garside group, and prove that Garsidegroups have quadratic isoperimetric inequalities. We constructnormal forms for Gaussian groups, and prove that, in the caseof a Garside group, the language of normal forms is regular,symmetric, and geodesic, has the 5-fellow traveller property,and has the uniqueness property. This shows in particular thatGarside groups are geodesically fully biautomatic. Finally,we consider an automorphism of a finite Coxeter-type Artin groupderived from an automorphism of its defining Coxeter graph,and prove that the subgroup of elements fixed by this automorphismis also a finite Coxeter-type Artin group that can be explicitlydetermined. 1991 Mathematics Subject Classification: primary20F05, 20F36; secondary 20B40, 20M05.     

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.     

Elliptic and Hyperelliptic Curves Over Supersimple Fields

It is proved that if F is an infinite field with characteristicdifferent from 2, whose theory is supersimple, and C is an ellipticor hyperelliptic curve over F with generic ‘modulus’,then C has a generic F-rational point. The notion of generityhere is in the sense of the supersimple field F.     

Connes-Amenability And Normal, Virtual Diagonals For Measure Algebras I

It is proved that the measure algebra M(G) of a locally compactgroup G is Connes-amenable if and only if G is amenable.     

Structural Projections on JBW*-Triples

A linear projection R on a Jordan*-triple A is said to be structuralprovided that, for all elements a, b and c in A, the equality{Rab Rc} = R{a Rbc} holds. A subtriple B of A is said to becomplemented if A = B + Ker(B), where Ker(B) = {aA: {B a B}= 0}. It is shown that a subtriple of a JBW*-triple is complementedif and only if it is the range of a structural projection. A weak* closed subspace B of the dual E* of a Banach space Eis said to be an N*-ideal if every weak* continuous linear functionalon B has a norm preserving extension to a weak* continuous linearfunctional on E* and the set of elements in E which attain theirnorm on the unit ball in B is a subspace of E. It is shown thata subtriple of a JBW*-triple A is complemented if and only ifit is an N*-ideal, from which it follows that complemented subtriplesof A are weak* closed, and structural projections on A are weak*continuous and norm non-increasing. It is also shown that everyN*-ideal in A possesses a triple product with respect to whichit is a JBW*-triple which is isomorphic to a complemented subtripleof A.     

Periodicity in Group Cohomology and Complete Resolutions

A group G is said to have periodic cohomology with period qafter k steps, if the functors Hⁱ(G, –) and H^i+q(G, –)are naturally equivalent for all i > k. Mislin and the authorhave conjectured that periodicity in cohomology after some stepsis the algebraic characterization of those groups G that admita finite-dimensional, free G-CW-complex, homotopy equivalentto a sphere. This conjecture was proved by Adem and Smith underthe extra hypothesis that the periodicity isomorphisms are givenby the cup product with an element in H^q(G,Z). It is expectedthat the periodicity isomorphisms will always be given by thecup product with an element in H^q(G,Z); this paper shows thatthis is the case if and only if the group G admits a completeresolution and its complete cohomology is calculated via completeresolutions. It is also shown that having the periodicity isomorphismsgiven by the cup product with an element in H^q(G,Z) is equivalentto silp G being finite, where silp G is the supremum of theinjective lengths of the projective ZG-modules. 2000 MathematicsSubject Classification 20J05, 57S25.     

On the Multiplicities of Graph Eigenvalues

Star complements and associated quadratic functions are usedto obtain a sharp upper bound for the order of a graph withan eigenspace of prescribed codimension. It is shown that forregular graphs the bound can be reduced by 1, and that thisreduced bound is attained by a regular graph G if and only ifG is an extremal strongly regular graph. 2000 Mathematics SubjectClassification 05C50.     

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.     

A Characterization of the Topological Group of p-Adic Integers

It is proved that a compact Hausdorff group is topologicallyisomorphic to the topological group of p-adic integers, forsome prime number p, if and only if all of its non-trivial properclosed subgroups are topologically isomorphic.     

A Characterization of Convexity-Preserving Maps from a Subset of a Vector Space into Another Vector Space

Let V and X be Hausdorff, locally convex, real, topologicalvector spaces with dim V > 1. It is shown that a map froman open, connected subset of V onto an open subset of X is homeomorphicand convexity-preserving if and only if is projective.     

Ovoids of PG(3,q), q Even, with a Conic Section

It is shown that if a plane of PG(3,q), q even, meets an ovoidin a conic, then the ovoid must be an elliptic quadric. Thisis proved by using the generalized quadrangles T₂(C) (C a conic),W(q) and the isomorphism between them to show that every secantplane section of the ovoid must be a conic. The result thenfollows from a well-known theorem of Barlotti.     

Linking, Legendrian Linking and Causality

The set N of all null geodesics of a globally hyperbolic (d+ 1)-dimensional spacetime (M, g) is naturally a smooth (2d– 1)-dimensional contact manifold. The sky of an eventx in M is the subset X of N consisting of all null geodesicsthrough x, and is an embedded Legendrian submanifold of N diffeomorphicto S^{(d – 1)}. It was conjectured by Low that for d = 2two events x and y are causally related if and only if X andY are linked (in an appropriate sense). We use the contact structureand knot polynomial calculations to prove this conjecture incertain particular cases, and suggest that for d = 3 smoothlinking should be replaced with Legendrian linking.     

ELEMENTS WITH THE SAME NORMAL CLOSURE IN A METABELIAN GROUP

It is known that in a free group, two elements a, b have thesame normal closure if and only if the images of these elementshave the same order in every finite quotient of the free group(Klyachko, 1999) or if and only if a is conjugate to b or b^–1(Magnus, 1931). Here, we prove that the result of Klyachko remainstrue in a finitely generated soluble group of derived lengthd 2. An example shows that the property fails when d = 3. Also,we give a counterpart of Magnus's result in the context of metabeliannilpotent group.     

Minimal Immersions of Kahler Manifolds into Euclidean Spaces

It is proved here that a minimal isometric immersion of a Kähler-Einsteinor homogeneous Kähler-manifold into an Euclidean spacemust be totally geodesic. As an application, it is shown thatan open subset of the real hyperbolic plane RH² cannot be minimallyimmersed into the Euclidean space. As another application, aproof is given that if an irreducible Kähler manifold isminimally immersed in a Euclidean space, then its restrictedholonomy group must be U(n), where n = dim_CM. 2000 MathematicsSubject Classification 53B25 (primary); 53C42 (secondary).     

Power-Bounded Operators and Related Norm Estimates

It is considered whether implies that the operator T is power-bounded. It is shown that thisis so if L<1/e, but it does not necessarily hold if L=1/e.As part of the methods, a result of Esterle is improved, showingthat if (T)={1} and T{}I, then The constant 1/e is sharp. Finally, a way to create many generalizationsof Esterle's result is described, and also many conditions aregiven on an operator which imply that its norm is equal to itsspectral radius.     

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.     

Invariants of Finite Group Schemes

Let G be a finite group scheme operating on an algebraic varietyX, both defined over an algebraically closed field k. The paperfirst investigates the properties of the quotient morphism X- X/G over the open subset of X consisting of points whose stabilizershave maximal index in G. Given a G-linearized coherent sheafon X, it describes similarly an open subset of X over whichthe invariants in the sheaf behave nicely in some way. The pointsin X with linearly reductive stabilizers are characterized inrepresentation theoretic terms. It is shown that the set ofsuch points is nonempty if and only if the field of rationalfunctions k(X) is an injective G-module. Applications of theseresults to the invariants of a restricted Lie algebra g operatingon the function ring k[X] by derivations are considered in thefinal section. Furthermore, conditions are found ensuring thatthe ring k[X]^g is generated over the subring of pth powers ink[X], where p=char,k>0, by a given system of invariant functionsand is a locally complete intersection.     

On Maximal Regularity and Semivariation of Cosine Operator Functions

It is proved that a cosine operator function C(·), withgenerator A, is locally of bounded semivariation if and onlyif u'(t) = Au(t)+f(t), t>0, u(0), u'(0)D(A), has a strongsolution for every continuous function f, if and only if thefunction , is twice continuously differentiable for every continuous function f, that is, C(·)has the maximal regularity property if and only if A is a boundedoperator. Some other characterisations of bounded generatorsof cosine operator functions are also established in terms oftheir local semivariations.     

Schur Superalgebras in Characteristic p, II

It is proved that if a Schur superalgebra is not semisimple,then it is neither cellular nor quasi-hereditary (Theorem 2),and it has infinite global dimension (Corollary 18). The algebraS(m|n, r) with m, n 1 is semisimple if and only if p, the characteristicof the ground field, is zero or greater than r, or when m =n = 1 and p does not divide r. 2000 Mathematics Subject Classification17A70 (primary), 20C30 (secondary).