Finiteness Properties of Certain Metabelian Arithmetic Groups in the Function Field Case

Consider the group scheme where R is an arbitrary commutative ring with 1 0 and a unitx R* acts on R by multiplication. We will study the finiteness properties of subgroups of G(O_S)where O_S is an S-arithmetic subring of a global function field.The subgroups we are interested in are of the form where Q is a subgroup of O_S*. The finiteness propertiesof these metabelian groups can be expressed in terms of the-invariant due to R. Bieri and R. Strebel. Theorem A. Let S be a finite set of places of a global functionfield (regarded as normalized discrete valuations) and O_S thecorresponding S-arithmetic ring. Let Q be a subgroup of O_S*.Then Q is finitely generated and for all integers n 1 the followingare equivalent:

(1) O_S Q is of type FP_n;

(2) O_S is n-tameas a ZQ-module;

(3) each p S restricts to a non-trivial homomorphism and the set is n-tame.

If these conditions hold for at least one n 1 then the identity holds.} Theorem B. Let r denote the rank of Q. Then the followinghold:

(1) the group O_S Q is not of type FP_r+1};

(2) if Qhas maximum rank r = |S| –1 then the group O_S Q is oftype FP_r.

In particular, is of type FP_{|S| –1} but not of type FP_|S|. 1991 Mathematics SubjectClassification: 20E08, 20F16, 20G30, 52A20.     

On O0-boundedness and -stability properties with perturbing Liapunov functional

In this paper we will extend the perturbing Liapunov functionmethod to systems of functional differential equations and discussØ₀-equiboundedness and Ø₀-equistability propertiesvia the concept of perturbing Liapunov functional.     

Exponential stability and maximal attractors for a one-dimensional nonlinear thermoviscoelasticity

This paper is concerned with the global existence, exponentialstability of solutions and associated nonlinear C₀-semigroupas well as the existence of maximal attractors in Hⁱ (i = 1,2, 4) for a nonlinear one-dimensional thermoviscoelasticitydescribing a kind of solid-like material. Some new ideas andmore delicated estimates are employed to prove the global existenceand exponential stability of solutions. The important featurefor the existence of maximal attractors in Hⁱ₊ (i = 1, 2, 4)is that the metric spaces H¹₊, H²₊ and H⁴₊ we work with arethree incomplete metric spaces, as can be seen from the physicalconstraints, i.e. > 0 and u > 0, with and u being absolutetemperature and deformation gradient (strain). For any positiveparameters ₁, ₂, ..., ₅ verifying some conditions, a sequenceof closed subspaces Hⁱ Hⁱ+ (i = 1, 2, 4) is found, and theexistence of maximal attractors in Hⁱ (i = 1, 2, 4) is established.     

The Volume of Hyperbolic Alternating Link Complements

If a hyperbolic link has a prime alternating diagram D, thenwe show that the link complement's volume can be estimated directlyfrom D. We define a very elementary invariant of the diagramD, its twist number t(D), and show that the volume lies betweenv₃(t(D) – 2)/2 and v₃(10t(D) – 10), where v₃ isthe volume of a regular hyperbolic ideal 3-simplex. As a consequence,the set of all hyperbolic alternating and augmented alternatinglink complements is a closed subset of the space of all completefinite-volume hyperbolic 3-manifolds, in the geometric topology.2000 Mathematics Subject Classification 57M25, 57N10.     

Arbitrary-norm hyperplane separation by variable neighbourhood search

^** Email: alejandro.karam{at}hec.ca^*** Email: gilles.caporossi{at}gerad.ca^**** Email: pierre.hansen{at}gerad.ca We consider the problem of separating two sets of points ina Euclidean space with a hyperplane that minimizes the sum ofp-norm distances to the plane of points lying on the ‘wrong’side of the plane. A variable neighbourhood search heuristicis used to determine the plane coefficients. For a set of exampleswith L₁-norm, L₂-norm and L-norm, for which the exact solutioncan be computed, we show that our algorithm finds it in mostcases and gets good approximations in the others. The use ofour heuristic solutions for problems in these norms can dramaticallyaccelerate exact algorithms. Our method can be applied on verylarge instances that are intractable by exact algorithms. Sincethe proposed approach works for truly arbitrary norms (otherthan the traditional 1, 2 and ), we can explore for the firsttime the effects of the choice of p on the generalization propertiesof p-norm hyperplane separation.     

Properties of Higher-Dimensional Shintani Generating Functions and Cocycles on Pgl3(Q)

In an earlier paper the second author used the formal, algebraicproperties of 2-dimensional Shintani generating functions toconstruct a 1-cocycle on PGL₂{Q}. We aim to generalise theseresults by using such functions in dimension n to obtain an(n–1)-cocycle on PGL_n{Q}, presumably related to the Bernoulliand Eisenstein cocycles of R. Sczech. By improving our methodswe achieve this goal for n=3. For n>3 we encounter obstaclesrelated to degenerate configurations of hyperplanes in n-space.Nevertheless, we obtain partial results closely connected toreciprocity laws for certain n-dimensional Dedekind sums. 1991Mathematics Subject Classification: 11F20, 11F75.     

On Perfect Isometries for Tame Blocks

Any 2-block of a finite group G with a quaternion defect groupQ₈ is Morita equivalent to the corresponding block of the centraliserH of the unique involution of Q₈ in G; this answers positivelyan earlier question raised by M. Broué. 2000 MathematicsSubject Classification 20C20.     

Finite-dimensional H{infty} control of flexible beam equation systems

In this paper, we consider H control problem with measurementfeedback for flexible beam equation systems. The aim is to constructa finite-dimensional H controller with a given level for theflexible beam equation system. For that purpose, we first formulatethe system as an infinite-dimensional system in l² and derivea finite-dimensional reduced-order system for the infinite-dimensionalsystem. Then, an H controller with level d less than is constructedfor the reduced-order model. The finite-dimensional controllertogether with a residual mode filter plays a role of a finite-dimensionalH controller with level for the original flexible beam equationsystem, if the order of the residual mode filter is chosen sufficientlylarge.     

A posteriori L2 error estimation on anisotropic tetrahedral finite element meshes

A new a posteriori L₂ norm error estimator is proposed for thePoisson equation. The error estimator can be applied to anisotropictetrahedral or triangular finite element meshes. The estimatoris rigorously analysed for Dirichlet and Neumann boundary conditions. The lower error bound relies on specifically designed anisotropicbubble functions and the corresponding inverse inequalities.The upper error bound utilizes non-standard anisotropic interpolationestimates. Its proof requires H² regularity of the Poisson problem,and its quality depends on how good the anisotropic mesh resolvesthe anisotropy of the problem. This is measured by a so-called‘matching function’. A numerical example supports the anisotropic error analysis.     

On approximation theorems for controllability of non-linear parabolic problems

^** Email: anil{at}math.iitb.ac.in^*** Email: mcj{at}math.iitb.ac.in^**** Email: akp{at}math.iitb.ac.in In this paper, we consider the following control system governedby the non-linear parabolic differential equation of the form: [graphic: see PDF] where A is a linear operator with dense domain and f(t, y)is a non-linear function. We have proved that under Lipschitzcontinuity assumption on the non-linear function f(t, y), theset of admissible controls is non-empty. The optimal pair (u^*,y^*) is then obtained as the limit of the optimal pair sequence{(u_n^*, y_n^*)}, where u_n^* is a minimizer of the unconstrainedproblem involving a penalty function arising from the controllabilityconstraint and y_n^* is the solution of the parabolic non-linearsystem defined above. Subsequently, we give approximation theoremswhich guarantee the convergence of the numerical schemes tooptimal pair sequence. We also present numerical experimentwhich shows the applicability of our result.     

The A-Polynomial has Ones in the Corners

1. Definition of the A-polynomial The A-polynomial was introduced in [3] (see also [5]), and wepresent an alternative definition here. Let M be a compact 3-manifoldwith boundary a torus T. Pick a basis , µ of ₁T, whichwe shall refer to as the longitude and meridian. Consider thesubset R_U of the affine algebraic variety R = Hom (₁M, SL₂C)having the property that () and (µ) are upper triangular.This is an algebraic subset of R, since one just adds equationsstating that the bottom-left entries in certain matrices arezero. There is a well-defined eigenvalue map given by taking the top-left entries of () and (µ).1991 Mathematics Subject Classification 57M25, 57M50.     

On The Profinite Topology on a Free Group

If F is a free abstract group, its profinite topology is thecoarsest topology making F into a topological group, such thatevery group homomorphism from F into a finite group is continuous.It was shown by M. Hall Jr that every finitely generated subgroupof F is closed in that topology. Let H₁, H₂, ..., H_n be finitelygenerated subgroups of F. J.-E. Pin and C. Reutenauer have conjecturedthat the product H₁ H₂ ... H_n is a closed set in the profinitetopology of F; also, they have shown that this conjecture impliesa conjecture of J. Rhodes on finite semigroups. In this paperwe give a positive answer to the conjecture of Pin and Reutenauer.Our method is based on the theory of profinite groups actingon graphs.     

There is a Group of Every Strong Symmetric Genus

Let G be a finite group. The strong symmetric genus ⁰(G) isthe minimum genus of any Riemann surface on which G acts, preservingorientation. For any non-negative integer g, there is at leastone group of strong symmetric genus g. For g2, one such grouphas the form Z_k x D_n for some k and n. 2000 Mathematics SubjectClassification 57M60 (primary), 20H10, 30F99 (secondary).     

Characterization of the Mod 3 Cohomology of the Compact, Connected, Simple, Exceptional Lie Groups of Rank 6

It is shown that the mod 3 cohomology of a 1-connected, homotopyassociative mod 3 H-space that is rationally equivalent to theLie group E₆ is isomorphic to that of E₆ as an algebra. Moreover,it is shown that the mod 3 cohomology of a nilpotent, homotopy-associativemod 3 H-space that is rationally equivalent to E₆, and whosefundamental group localized at 3 is non-trivial, is isomorphicto that of the Lie group Ad E₆ as a Hopf algebra over the mod3 Steenrod algebra. It is also shown that the mod 3 cohomologyof the universal cover of such an H-space is isomorphic to thatof E₆ as a Hopf algebra over the mod 3 Steenrod algebra. 2000Mathematics Subject Classification 57T05, 57T10, 57T25.     

Moments of Some Stopping Rules

Let X_n for n1 be independent random variables with . Set . Define T_k,c,m=inf{nm:|k!S_k,n|>cn^k/2}.We study critical values c_k,p for k2 and p>0, such that for c<c_k,p and all m, and for c>c_k,p and all sufficientlylarge m. In particular, c_1,1=c_2,1=1, c_3,1=2 and c_4,1=3 undercertain moment conditions on X₁, when X_n are identically distributed.We also investigate perturbed stopping rules of the form T_h,m=inf{nm:h(S_1,n/n^1/2)<_nor >_n} for continuous functions h and random variables _naand _nb with a<b. Related stopping rules of the Wiener processare also considered via the Uhlenbeck process.     

Projective Prime Ideals and Localisation in PI-Rings

The results here generalise [2, Proposition 4.3] and [9, Theorem5.11]. We shall prove the following. THEOREM A. Let R be a Noetherian PI-ring. Let P be a non-idempotentprime ideal of R such that P_R is projective. Then P is leftlocalisable and R_P is a prime principal left and right idealring. We also have the following theorem. THEOREM B. Let R be a Noetherian PI-ring. Let M be a non-idempotentmaximal ideal of R such that M_R is projective. Then M has theleft AR-property and M contains a right regular element of R.     

Deformation of Complete Intersections in the Plane

We study deformations of zero-dimensional complete intersectionsin the plane, and prove the following results. (1) Two complexnon-singular curves intersecting at r points with multiplicitiesd₁,...,d_r can be deformed into curves intersecting (at somepoints) with multiplicities d'₁,...,d'_s which are arbitraryprescribed partitions of the numbers d₁,...,d_r. (2) Two realcurves intersecting with multiplicity at most 2 at each of theirreal common points can be deformed so that all real multipleintersection points split into real simple intersection points.1991 Mathematics Subject Classification 14M10, 14P05.     

Z4-Kerdock Codes, Orthogonal Spreads, and Extremal Euclidean Line-Sets

When m is odd, spreads in an orthogonal vector space of type⁺(2m + 2,2) are related to binary Kerdock codes and extremalline-sets in 2^{m + 1} with prescribed angles. Spreads in a 2m-dimensionalbinary symplectic vector space are related to Kerdock codesover Z₄ and extremal line-sets in with prescribed angles. These connections involve binary, realand complex geometry associated with extraspecial 2-groups.A geometric map from symplectic to orthogonal spreads is shownto induce the Gray map from a corresponding Z₄-Kerdock codeto its binary image. These geometric considerations lead tothe construction, for any odd composite m, of large numbersof Z₄-Kerdock codes. They also produce new Z₄-linear Kerdockand Preparata codes. 1991 Mathematics Subject Classification:primary 94B60; secondary 51M15, 20C99.     

Morita Equivalent Principal 3-Blocks of the Chevalley Groups G2(q)

The principal 3-block of a Chevalley group G₂(q) with q a powerof 2 satisfying q 2 or 5 mod 9 and the principal 3-block ofG₂(2) are Morita equivalent. 2000 Mathematical Subject Classification:20C05, 20C20, 20C33.