Sur L'equation Diophantienne (xn-1)/(x-1)=yq, III

In this paper we study the diophantine equation of the title,which was first introduced by Nagell and Ljunggren during thefirst half of the twentieth century. We describe a method whichallows us, on the one hand when n is fixed, to obtain an upperbound for q, and on the other hand when n and q are fixed, toobtain upper bounds for x and y which are far sharper than thosederived from the theory of linear forms in logarithms. We alsoshow how these bounds can be used even when they seem too largefor a straightforward enumeration of the remaining possiblevalues of x. By combining all these techniques, we are ableto solve the equation in many cases, including the case whenn has a prime divisor less than 13, or the case when n has aprime divisor which is less than or equal to 23 and distinctfrom q. 2000 Mathematical Subject Classification: primary 11D41;secondary 11J86, 11Y50.     

Long Arithmetic Progressions in Sum-Sets and the Number x-Sum-Free Sets

In this paper we obtain optimal bounds for the length of thelongest arithmetic progression in various kinds of sum-sets.As an application, we derive a sharp estimate for the numberof sets A of residues modulo a prime n such that no subsum ofA equals x modulo n, where x is a fixed residue modulo n. 2000Mathematics Subject Classification 05A16, 11B25, 11P32.     

Explicit Error Bounds for Periodic Splines of Odd Order on a Uniform Mesh

The dependence relationships connecting equal interval splinesand their derivatives are analysed to obtain the form of theerror term when the spline is replaced by a general function.The defining equations for periodic splines of odd order ona uniform mesh are then expressed in terms of a positive definitecirculant matrix A and attainable bounds determined for thecondition number of A and for the norm of A^-1. In conjunctionwith the error term associated with the dependence relationships,this enables explicit error bounds to be established for thederivatives at the knots of the spline function. Some subsidiary results in the paper also relate to B-splineson a uniform mesh.     

Non-Constant Positive Steady States of a Predator-Prey System with Non-Monotonic Functional Response and Diffusion

This paper deals with non-constant positive steady-state solutionsof a predator-prey system with non-monotonic functional response,also called Holling type-IV interaction terms, and diffusionunder the homogeneous Neumann boundary condition. We first establishpositive upper and lower bounds for such solutions, and thenstudy their non-existence, global existence and bifurcation.2000 Mathematics Subject Classification 35J55, 92D25.     

New perturbation analyses for the Cholesky factorization

We present new perturbation analyses for the Cholesky factorizationA = R^T R of a symmetric positive definite matrix A. The analysesmore accurately reflect the sensitivity of the problem thanprevious normwise results. The condition numbers here are alteredby any symmetric pivoting used in PAP^T = R^TR, and both numericalresults and an analysis show that the standard method of pivotingis optimal in that it usually leads to a condition number veryclose to its lower limit for any given A. It follows that thecomputed R will probably have greatest accuracy when we usethe standard symmetric pivoting strategy. Initially we give a thorogh analysis to obtain both first-orderand strict normwise perturbation bounds which are as tight aspossible, leading to a definition of an optimal condition numberfor the problem. Then we use this approach to obtain reasonablyclear first-order and strict componentwise perturbation bounds. We complete the work by giving a much simpler normwise analysiswhich provides a somewhat weaker bound, but which allows usto estimate the condition of the problem quite well with anefficient computation. This simpler analysis also shows whythe factorization is often less sensitive than we previouslythought, and adds further insight into why pivoting usuallygives such good results. We derive a useful upper bound on thecondition of the problem when we use pivoting. This research was supported by the Natural Sciences and EngineeringResearch Ciuncil of Canada Grant OGP0009236. This research was supported in part by the US National ScienceFoundation under grant CCR 95503126.     

Mixed Boundary Value Problems for some Linear and Non-linear Elliptic Difference Equations

We obtain a priori bounds on difference quotients for some mixedboundary value problems associated with uniformly elliptic differenceequations over rectangular regions. These results are appliedto mildly quasi-linear and mildly non-linear problems to establishexistence and uniqueness criteria. Our estimates are usefulin the numerical computation of these solutions.     

Small Solutions of Quadratic Diophantine Equations

We consider quadratic diophantine equations of the shape for a polynomial Q(X₁, ..., X_s) Z[X₁, ..., X_s] of degree 2.Let H be an upper bound for the absolute values of the coefficientsof Q, and assume that the homogeneous quadratic part of Q isnon-singular. We prove, for all s 3, the existence of a polynomialbound _s(H) with the following property: if equation (1) hasa solution x Z^s at all, then it has one satisfying For s = 3 and s = 4 no polynomial bounds _s(H) were previouslyknown, and for s 5 we have been able to improve existing boundsquite significantly. 2000 Mathematics Subject Classification11D09, 11E20, 11H06, 11P55.     

Bounds on the solution of the time-varying linear matrix differential equation P(t) = AH(t) P(t) + P(t) A(t) + Q(t)

^** Email: pagilla{at}ceat.okstate.edu We derive upper and lower bounds for the trace of the solutionof the time-varying linear matrix differential equation (t)= A^H(t) P(t) + P(t) A(t) + Q(t). A practical numerical exampleis given to verify the bounds. The bounds obtained are usefulsince the considered equation is encountered in a number ofapplications in systems and control theory.     

Bounds on Special Subsets in Graphs, Eigenvalues and Association Schemes

We give a bound on the sizes of two sets of vertices at a given minimum distance in a graph in terms of polynomials and the Laplace spectrum of the graph. We obtain explicit bounds on the number of vertices at maximal distance and distance two from a given vertex, and on the size of two equally large sets at maximal distance. For graphs with four eigenvalues we find bounds on the number of vertices that are not adjacent to a given vertex and that have µ common neighbours with that vertex. Furthermore we find that the regular graphs for which the bounds are tight come from association schemes.     

On The Greatest Prime Factor of axm+byn, II

We provide a new lower bound for the greatest prime factor ofthe norm of algebraic numbers of the form ax^m+byⁿ. The improvementconcerns the dependence on the number field containing a, b,x and y. 1991 Mathematics Subject Classification 11D61.     

Liftings and Mean Value Theorems for Automorphic L-Functions

A general mean value theorem for Dirichlet series, with a sharperror estimate near the boundary of the critical strip, is proved.Applications of this theorem to various automorphic L-functionsare discussed. Moreover, sharp upper bounds of mean square valuesof L-functions are obtained when they are attached to liftedforms, such as Doi–Naganuma and Ikeda lifts in the theoryof Siegel modular forms. 2000 Mathematics Subject Classification11F66, 11M41.     

Torsional Rigidity of Minimal Submanifolds

We prove explicit upper bounds for the torsional rigidity ofextrinsic domains of minimal submanifolds P^m in ambient Riemannianmanifolds Nⁿ with a pole p. The upper bounds are given in termsof the torsional rigidities of corresponding Schwarz symmetrizationsof the domains in warped product model spaces. Our main resultsare obtained via previously established isoperimetric inequalities,which are here extended to hold for this more general settingbased on warped product comparison spaces. We also characterizethe geometry of those situations in which the upper bounds forthe torsional rigidity are actually attained and give conditionsunder which the geometric average of the stochastic mean exittime for Brownian motion at infinity is finite. 2000 MathematicsSubject Classification 53C42, 58J65, 35J25, 60J65.     

Completely reducible hypersurfaces in a pencil

We study completely reducible fibers of pencils of hypersurfaces on Pn and associated codimension one foliations of Pn. Using methods from theory of foliations we obtain certain upper bounds for the number of these fibers as functions only of n. Equivalently this gives upper bounds for the dimensions of resonance varieties of hyperplane arrangements. We obtain similar bounds for the dimensions of the characteristic varieties of the arrangement complements.     

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 the Neumann Problem for some Linear and Non-linear Uniformly Elliptic Difference Equations

In this paper we obtain a priori bounds on difference quotientsto the solution of the second boundary value problem for somelinear uniformly elliptic difference equations over rectangularregions. These results are applied to the constructive solutionof mildly non-linear and mildly quasi-linear uniformly ellipticdifference equations.     

Three-point bounds for the overall properties of a nonlinear composite dielectric

The Hashin-Shtrikman methodology for nonlinear composite problemsrelies on the use of a comparison medium and in many of theexamples studied so far the comparison medium has been takento be homogeneous. A related approach originated by P. PonteCastaeda employs a comparison medium which is itself a linearcomposite with the same microgeometry as the nonlinear composite.When the method is applicable, the bounds for the nonlinearproblem then involve bounds for the energy of the linear comparisoncomposite which could include, for example, three-point informationabout the microstructure of the composite. It is, however, onlypossible to obtain at most one bound using a linear comparisonmaterial. A recent approach involves the use of a nonlinearcomparison medium and trial fields with the property of boundedmean oscillation. In this paper the approach is extended byusing a nonlinear comparison composite so that both upper andlower bounds can be obtained which incorporate three-point information.The development is in the context of bounding the propertiesof nonlinear dielectric composites, although it has wider application.     

Optimal error estimates for the hp-version interior penalty discontinuous Galerkin finite element method

We consider the hp-version interior penalty discontinuous Galerkinfinite-element method (hp-DGFEM) for second-order linear reaction–diffusionequations. To the best of our knowledge, the sharpest knownerror bounds for the hp-DGFEM are due to Rivière et al.(1999,Comput. Geosci., 3, 337–360) and Houston et al.(2002,SIAM J. Numer. Anal., 99, 2133–2163). These are optimalwith respect to the meshsize h but suboptimal with respect tothe polynomial degree p by half an order of p. We present improvederror bounds in the energy norm, by introducing a new functionspace framework. More specifically, assuming that the solutionsbelong element-wise to an augmented Sobolev space, we deducefully hp-optimal error bounds.     

Symmetric Groups as Products of Abelian Subgroups

A proof is given that the full symmetric group over any infiniteset is the product of finitely many Abelian subgroups. In fact,289 subgroups suffice. Sharp bounds are also obtained on theminimal number k, such that the finite symmetric group S_n isthe product of k Abelian subgroups. Using this, S_n is provedto be the product of 72n^1/2(log n)^3/2 cyclic subgroups. 2000Mathematics Subject Classification 20B30, 20D40.     

Domain decomposition preconditioners of Neumann-Neumann type for hp-approximations on boundary layer meshes in three dimensions

We develop and analyse Neumann–Neumann methods for hpfinite-element approximations of scalar elliptic problems ongeometrically refined boundary layer meshes in three dimensions.These are meshes that are highly anisotropic where the aspectratio typically grows exponentially with the polynomial degree.The condition number of our preconditioners is shown to be independentof the aspect ratio of the mesh and of potentially large jumpsof the coefficients. In addition, it only grows polylogarithmicallywith the polynomial degree, as in the case of p approximationson shape-regular meshes. This work generalizes our previousone on two-dimensional problems in Toselli & Vasseur (2003a,submitted to Numerische Mathematik, 2003c to appear in Comput.Methods Appl. Mech. Engng.) and the estimates derived here canbe employed to prove condition number bounds for certain typesof FETI methods.     

Rigidity of Continuous Coboundaries

We consider the functional equation F_oT–F=f, where T isa measure-preserving transformation and f is a continuous function.We show that if there is an L function F which satisfies thisequation, then F is constrained to satisfy a number of regularityconditions, and, in particular, if T is a one-sided Bernoullishift, then we show that there is a continuous function F satisfyingthis equation. We show that this is not the case for the two-sidedshift. 1991 Mathematics Subject Classification 28D05, 58F11.