Rates of Recurrence for Zq and Rq Extensions of Subshifts of Finite Type

In this note we give new asymptotic formulae for certain countingfunctions associated to the periodic behaviour of Z^q and R^qextensions of subshifts of finite type. In the case of the Z^qextensions, these strengthen previous estimates of Marcus andTuncel [9]. For both types of extension, our results complementthe central limit type results of Lalley [6]. Our proof requiresthe application of ideas from thermodynamic formalism. Whilstdeveloping this approach, in Section 2, we take the opportunityto present a counter-example to a related conjecture of Coelho-Filho[2].     

An Introduction to Lattice Rules and their Generator Matrices

For the one-dimensional quadrature of a naturally periodic functionover its period, the trapezoidal rule is an excellent choice,its efficiency being predicted theoretically and confirmed inpractice. However, for s-dimensional quadrature over a hypercube,the s-dimensional product trapezoidal rule is not generallycost effective even for naturally periodic functions. The searchfor more effective rules has led first to number theoretic rulesand then more recently to lattice rules. This survey outlinesthe motivation for and present results of this theory. It isparticularly designed to introduce the reader to lattice rules.     

Locally Nilpotent p-Groups whose Proper Subgroups are Hypercentral or Nilpotent-by-Chernikov

Let G be a group and P be a property of groups. If every propersubgroup of G satisfies P but G itself does not satisfy it,then G is called a minimal non-P group. In this work we studylocally nilpotent minimal non-P groups, where P stands for ‘hypercentral’or ‘nilpotent-by-Chernikov’. In the first case weshow that if G is a minimal non-hypercentral Fitting group inwhich every proper subgroup is solvable, then G is solvable(see Theorem 1.1 below). This result generalizes [3, Theorem1]. In the second case we show that if every proper subgroupof G is nilpotent-by-Chernikov, then G is nilpotent-by-Chernikov(see Theorem 1.3 below). This settles a question which was consideredin [1–3, 10]. Recently in [9], the non-periodic case ofthe above question has been settled but the same work containsan assertion without proof about the periodic case. The main results of this paper are given below (see also [13]).     

Periodic Points for Expansive Actions of Zd on Compact Abelian Groups

In this note we show that the periodic points of an expansiveZ^d action on a compact abelian group are uniformly distributedwith respect to Haar measure if the action has completely positiveentropy. In the general expansive case, we show that any measureobtained as the distribution of periodic points along some sequenceof periods necessarily has maximal entropy but need not be Haarmeasure.     

A transform method for linear evolution PDEs on a finite interval

We study initial boundary value problems for linear scalar evolutionpartial differential equations, with spatial derivatives ofarbitrary order, posed on the domain {t > 0, 0 < x <L}. We show that the solution can be expressed as an integralin the complex k-plane. This integral is defined in terms ofan x-transform of the initial condition and a t-transform ofthe boundary conditions. The derivation of this integral representationrelies on the analysis of the global relation, which is an algebraicrelation defined in the complex k-plane coupling all boundaryvalues of the solution. For particular cases, such as the case of periodic boundaryconditions, or the case of boundary value problems for even-orderPDEs, it is possible to obtain directly from the global relationan alternative representation for the solution, in the formof an infinite series. We stress, however, that there existinitial boundary value problems for which the only representationis an integral which cannot be written as an infinite series.An example of such a problem is provided by the linearized versionof the KdV equation. Similarly, in general the solution of odd-orderlinear initial boundary value problems on a finite intervalcannot be expressed in terms of an infinite series.     

Isotopy Stable Dynamics Relative to Compact Invariant Sets

Let f be an orientation-preserving homeomorphism of a compactorientable manifold. Sufficient conditions are given for thepersistence of a collection of periodic points under isotopyof f relative to a compact invariant set A. Two main applicationsare described. In the first, A is the closure of a single discreteorbit of f, and f has a Smale horseshoe, all of whose periodicorbits persist; in the second, A is a minimal invariant Cantorset obtained as the limit of a sequence of nested periodic orbits,all of which are shown to persist under isotopy relative toA. 1991 Mathematics Subject Classification: 58F20, 58F15.     

Perturbation Solutions of the Whittaker-Hill Equation

When the Helmholtz equation ²V+k²V = 0 is separated in the generalparaboloidal co-ordinate system, the three ordinary differentialequations obtained each take, after a suitable change of variable,the form of the Whittaker-Hill equation. For the case k²<0,a considerable amount is known about the periodic solutionsof this equation. For k²>0, however, very little is so farknown. In this paper solutions of the Whittaker-Hill equationfor small positive k² are derived. These are the first explicitsolutions to be obtained for the case k²>0, and they couldbe employed to solve the Dirichlet or Neumann problem for ageneral paraboloid when k² is small. Three limiting cases arenoted, involving the reduction of the solutions to Mathieu functionsand the reduction of the co-ordinate system to the rotation-paraboloidalsystem.     

Existence Theorems for the Linear, Space-inhomogeneous Transport Equation

This paper studies existence problems in L¹ for the linear,space-inhomogeneous Boltzmann equation with periodic or (perfectly)absorbing boundary conditions under realistic assumptions onthe cross-sections. By an iteration technique, solutions arefirst constructed to an integral equation variant of the transportequation in the case of bounded impact parameters and an L¹type of cross-sections. They are then used to study the existenceof solutions of a measure form of the transport equation inthe case of unbounded impact parameters. These solutions conservemass. Estimates of their higher moments are also given. In particularthe results hold for inverse kth-power forces with 3 < k 5.     

On Hill's Equation with a Singular Complex-Valued Potential

In this paper Hill's equation y' + qy = Ey, where q is a complex-valuedfunction with inverse square singularities, is studied. Resultson the dependence of solutions to initial value problems onthe parameter E and the initial point x₀, on the structure ofthe conditional stability set, and on the asymptotic distributionof (semi-)periodic and Sturm-Liouville eigenvalues are obtained.It is proved that a certain subset of the set of Floquet solutionsis a line bundle on a certain analytic curve in C². We establishnecessary and sufficient conditions for q to be algebro-geometric,that is, to be a stationary solution of some equation in theKorteweg-de Vries (KdV) hierarchy. To do this a distinctionbetween movable and immovable Dirichlet eigenvalues is employed.Finally, an example showing that the finite-band property doesnot imply that q is algebro-geometric is given. This is in contrastto the case where q is real and non-singular. 1991 MathematicsSubject Classification: 34L40, 14H60.     

High-accuracy P-stable Methods for y' = f(t, y)

We obtain a one-parameter family of sixth-order P-stable methodsfor the numerical integration of periodic or near-periodic differentialequations that are defined by initial-value problems of theform: y^" = f(t, y), y(t₀)= y₀, y^'(t₀)= y^’₀. Our P-stablemethods are symmetric and involve three function evaluationsper step (periteration, in case f(t, y) is non-linear in y).For non-linear problems, starting values for the solution ofthe implicit equations by modified Newton's method are suggestedand illustrated by an example.     

Modular Subgroup Arithmetic and a Theorem of Philip Hall

A surprising relationship is established in this paper, betweenthe behaviour modulo a prime p of the number S_n G of index nsubgroups in a group G, and that of the corresponding subgroupnumbers for a normal subgroup in G normal subgroup in p-powerorder. The proof relies, among other things, on a twisted versiondue to Philip Hall of Frobenius' theorem concerning the equationx^m=1 in finite groups. One of the applications of this result,presented here, concerns the explicit determination modulo pof S_n G in the case when G is the fundamental group of a treeof groups all of whose vertex groups are cyclic of p-power order.Furthermore, a criterion is established (by a different technique)for the function S_n G to be periodic modulo p. 2000 MathematicsSubject Classification 20E06, 20F99 (primary); 05A15, 05E99(secondary).     

On the Analytic Order-Preserving Discrete-Time Dynamical Systems in Rn with every Fixed Point Stable

This paper studies the asymptotic behaviour of an analytic order-preservingdiscrete-time dynamical system in Rⁿ, which is usually generatedby a periodic cooperative system. The author proves that forsuch a dynamical system, if every fixed point is Liapunov stableand every positive semi-orbit has compact closure, then everypositive semi-orbit converges. This result does not requirethe assumption ‘strongly’ and gives an affirmativeanswer to the conjecture proposed by the author in [17] forthe analytic case.     

Weakly Almost Periodic Functions on N with a Negative Base

Weakly almost periodic compactifications have been seriouslystudied for over 30 years. In the pioneering papers of de Leeuwand Glicksberg [4] and [5], the approach adopted was operator-theoretic.The current definition is more likely to be created from theperspective of universal algebra (see [1, Chapter 3]). For adiscrete group or semigroup S, the weakly almost periodic compactificationwS is the largest compact semigroup which (i) contains S asa dense subsemigroup, and (ii) has multiplication continuousin each variable separately (where largest means that any othercompact semigroup with the properties (i) and (ii) is a quotientof wS). A third viewpoint is to envisage wS as the Gelfand spaceof the C*-algebra of bounded weakly almost periodic functionson S (for the definition of such functions, see below). In this paper, we are concerned only with the simplest semigroup(N, +). The three approaches described above give three methodsof obtaining information about wN. An early striking resultabout wN, that it contains more than one idempotent, was obtainedby T. T. West using operator theory [13]. He considered theweak operator closure of the semigroup {T, T², T³, ...} of iteratesof a single operator T on the Hilbert space L²(µ) fora particular measure µ on [0, 1]. Brown and Moran, ina series of papers culminating in [2], used sophisticated techniquesfrom harmonic analysis to produce measures µ that permittedthe detection of further structure in wN; in particular, theyfound 2^cdistinct idempotents. However, for many years, no otherway of showing the existence of more than one idempotent inwN was found. The breakthrough came in 1991, and it was made by Ruppert [11].In his paper, he created a direct construction of a family ofweakly almost periodic functions which could detect 2^c differentidempotents in wN. His method was very ingenious (he used aunique variant of the p-adic expansion of integers) and rathercomplicated. Our main aim in this paper is to construct weaklyalmost periodic functions which are easy to describe and soappear more ‘natural’ than Ruppert's. We also showthat there are enough functions of our type to distinguish 2^cidempotentsin wN.     

The Rotation Number Approach to Eigenvalues of the One-Dimensional p-Laplacian with Periodic Potentials

The paper studies the periodic and anti-periodic eigenvaluesof the one-dimensional p-Laplacian with a periodic potential.After a rotation number function () has been introduced, itis proved that for any non-negative integer n, the endpointsof the interval ^–1(n/2) in R yield the corresponding periodicor anti-periodic eigenvalues. However, as in the Dirichlet problemof the higher dimensional p-Laplacian, it remains open if theseeigenvalues represent all periodic and anti-periodic eigenvalues.The result obtained is a partial generalization of the spectrumtheory of the one-dimensional Schrödinger operators withperiodic potentials.     

Long-wavelength thermal convection in a weak shear flow

The problem of thermal convection in an imposed shear flow isexamined for a horizontal layer of fluid between poorly conductingboundaries. The horizontal scale H of the convective motionnear its onset is much greater than the depth h of the fluidlayer, with h/H being proportional to the one-fourth power ofa Biot number appearing in the condition applied to the temperatureat the horizontal boundaries. It is known that an asymptoticexpansion in powers of h/H yields a nonlinear long-wavelengthevolution equation for the depth-averaged temperature fieldthat is spatially isotropic in the absence of an imposed shearflow, but is strongly anisotropic for ‘strong’ shear.We derive in this paper a nonlocal long-wavelength equationthat bridges these two cases, and that contains each case inthe zero-shear and large-shear limits. Using this evolutionequation, we show how the shear flow stabilizes the longitudinalrolls to the zigzag instability, and how a preference for asquare planfonn on a periodic square lattice gives way to apreference for longitudinal rolls near onset. The longitudinalrolls may then become unstable as the Rayleigh number is increased.The analytical work is illustrated by some numerical simulationsof the full three-dimensional Boussinesq Navier-Stokes equations.The problem of pattern selection on a hexagonal lattice is alsodiscussed, and some new results are presented.     

Homogenization of the Stefan Problem and Application to Magnetic Composite Media

The theory of homogenization (Bensoussan, Lions & Papanicolaou,1978) shows that u, the solution of the diffusion equation [with k(y) periodic in the space-variable y and q = cu a linearfunction of u] has a weak limit u for = 0. This theory allowsone to compute, for a given k, the conductivity tensor of ananisotropic but homogeneous medium in which, for unchanged initialand boundary conditions, u is the solution of the diffusionequation. We examine here the case where the relation between q and uis given by a maximal monotone graph (i.e. the Stefan problem),depending on the space variable in the same manner as k. Applicationsto eddy-current problems in magnetic composite media (steelcables, laminations) are suggested. A numerical example is given.     

The Existence of Periodic and Subharmonic Solutions of Subquadratic Second Order Difference Equations

By critical point theory, a new approach is provided to studythe existence of periodic and subharmonic solutions of the secondorder difference equation where f C(R x R^m, R^m), f(t+M,z)+f(t,z) for any (t, z)R x R^mand M is a positive integer. This is probably the first timecritical point theory has been applied to deal with the existenceof periodic solutions of difference systems.     

RIGIDITY OF PERIODIC DIFFEOMORPHISMS OF HOMOTOPY K3 SURFACES

In this paper, we show that homotopy K3 surfaces do not admita periodic diffeomorphism of odd prime order 3 acting triviallyon cohomology. This gives a negative answer for period 3 toProblem 4.124 in Kirby's problem list. In addition, we givean obstruction in terms of the rationality and the sign of thespin numbers to the non-existence of a periodic diffeomorphismof odd prime order acting trivially on cohomology of homotopyK3 surfaces. The main strategy is to calculate the Seiberg–Witteninvariant for the trivial spin^c structure in the presence ofsuch a Z_p-symmetry in two ways: (1) the new interpretation ofthe Seiberg–Witten invariants of Furuta and Fang, and(2) the theorem of Morgan and Szabó on the Seiberg–Witteninvariant of homotopy K3 surfaces for the trivial Spin^c structure.As a consequence, we derive a contradiction for any periodicdiffeomorphism of prime order 3 acting trivially on cohomologyof homotopy K3 surfaces.     

Increased Accuracy Cubic Spline Solutions to Two-Point Boundary Value Problems

The relation between finite difference approximation and cubicspline solutions of a two-point boundary value problem for thedifferential equation y' +f(x)y^'+g(x)y = r(x) has been consideredin a previous paper. The present paper extends the analysisto the integral equation formulation of the problem. It is shownthat an improvement in accuracy (local truncation error O(h⁶)rather than O(h⁴)) now results from a cubic spline approximationand that for the particular case f(x) 0 the resulting recurrencerelations have a form and accuracy similar to the well-knownNumerov formula. For this case also a formula with local truncationerror O(h⁸) is derived.     

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.