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.     

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.     

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.     

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.     

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.