Differences Between Consecutive Primes

Let p_n be the nth prime. Then this paper is concerned with provingthe following result on the distribution of consecutive primes. The exponent of x in this theorem improves on the work of Heath-Brownwho proved (1) with exponent . Under the Riemann hypothesisone can prove (1) with exponent .The proof of the theorem startswith the Heath-Brown–Linnik identity which leads to aformula giving the number of primes in an interval in termsof coefficients of certain Dirichlet series. I then estimatethe coefficients by using, among other things, the informationwhich can be gained from Montgomery's mean value theorem andHuxley's version of the Hal' asz lemma. Furthermore, by usingfamiliar sieve arguments I am able to discard some of the coefficientsallowing us to gain an improvement over the previous resultof Heath-Brown. 1991 Mathematics Subject Classification: 11N05.     

Local Cohomology of BP*BP-Comodules

Given a spectrum X, we construct a spectral sequence of BP_*BP-comodulesthat converges to BP_*(L_nX), where L_nX is the Bousfield localizationof X with respect to the Johnson–Wilson theory E(n)_*.The E₂-term of this spectral sequence consists of the derivedfunctors of an algebraic version of L_n. We show how to calculatethese derived functors, which are closely related to local cohomologyof BP_*-modules with respect to the ideal I_n+1. 2000 MathematicsSubject Classification 55N22, 55P60, 16W30.     

A Generalization of Vinogradov's Mean Value Theorem

We obtain new upper bounds for the number of integral solutionsof a complete system of symmetric equations, which may be viewedas a multi-dimensional version of the system considered in Vinogradov'smean value theorem. We then use these bounds to obtain Weyl-typeestimates for an associated exponential sum in several variables.Finally, we apply the Hardy–Littlewood method to obtainasymptotic formulas for the number of solutions of the Vinogradov-typesystem and also of a related system connected to the problemof finding linear spaces on hypersurfaces. 2000 MathematicsSubject Classification 11D45, 11D72, 11L07, 11P55.     

Improving the accuracy of the mini-element approximation to Navier-Stokes equations

^** Email: blanca{at}imati.cnr.it^*** Email: frutos{at}mac.cie.uva.es^**** Corresponding author. Email: julia.novo{at}uam.es A technique to improve the accuracy of the mini-element approximationto incompressible the Navier–Stokes equations is introduced.Once the mini-element approximation has been computed at a fixedtime, the linear part of this approximation is postprocessedby solving a discrete Stokes problem. The bubble functions neededto stabilize the approximation to the Navier–Stokes equationsare not used at the postprocessing step. This postprocessingprocedure allows us to increase by one unit (up to a logarithmicterm) the H¹ norm rate of convergence of the velocity and correspondinglythe L² norm of the pressure. An error analysis of the algorithmis performed.     

Stability of uniformly Morse-Smale gradient-like numerical methods for flows

In Garay (1996, Numer. Math., 72, 449–479) and Li (1997b,SIAM J. Math. Anal., 28, 381–388), it was shown that thequalitative properties of a Morse–Smale gradient-likeflow are preserved by its numerical approximations. In thispaper, we show that the qualitative properties of a family ofuniformly Morse–Smale gradient-like numerical methodsare preserved by the approximated flow. The techniques usedin the study of the structural stability theorem for diffeomorphismsare the main tools for this work.     

Finite-difference approximation of a one-dimensional Hamilton-Jacobi/elliptic system arising in superconductivity

Finite-difference approximations to an elliptic–hyperbolicsystem arising in vortex density models for type II superconductorsare studied. The problem can be formulated as a non-local Hamilton–Jacobiequation on a bounded domain with zero Neumann boundary conditions.Monotone schemes are defined and shown to be stable. An L errorbound is proved for the approximations of the unique viscositysolution.     

A Petrov-Galerkin method with quadrature for elliptic boundary value problems

We propose and analyse a fully discrete Petrov–Galerkinmethod with quadrature, for solving second-order, variable coefficient,elliptic boundary value problems on rectangular domains. Inour scheme, the trial space consists of C² splines of degreer 3, the test space consists of C⁰ splines of degree r –2, and we use composite (r – 1)-point Gauss quadrature.We show existence and uniqueness of the approximate solutionand establish optimal order error bounds in H², H¹ and L² norms.     

Conservative upwind finite-element method for a simplified Keller-Segel system modelling chemotaxis

^** Email: saito{at}infsup.jp Finite-element approximation for a non-linear parabolic–ellipticsystem is considered. The system describes the aggregation ofslime moulds resulting from their chemotactic features and iscalled a simplified Keller–Segel system. Applying an upwindtechnique, first we present a finite-element scheme that satisfiesboth positivity and mass conservation properties. Consequently,if the triangulation is of acute type, our finite-element approximationpreserves the L¹ norm, which is an important property of theoriginal system. Then, under some assumptions on the regularityof a solution and on the triangulation, we establish error estimatesin L^p x W^1, with a suitable p > d, where d is the dimensionof a spatial domain. Our scheme is well suited for practicalcomputations. Some numerical examples that validate our theoreticalresults are also presented.     

Descent Calculations for the Elliptic Curves of Conductor 11

Let A be any one of the three elliptic curves over Q with conductor11. We show that A has Mordell–Weil rank zero over itsfield of 5-division points. In each case we also compute the5-primary part of the Tate–Shafarevich group. Our calculationsmake use of the Galois equivariance of the Cassels–Tatepairing. 2000 Mathematics Subject Classification 11G05, 11Y40,11R23.     

Pattern formation in a 2D simple chemical system with general orders of autocatalysis and decay

In this paper, we investigate pattern formation in a coupledsystem of reaction–diffusion equations in two spatialdimensions. These equations arise as a model of isothermal chemicalautocatalysis with termination in which the orders of autocatalysisand termination, m and n, respectively, are such that 1 <n < m. We build on the preliminary work by Leach & Wei(2003, Physica D, 180, 185–209) for this coupled systemin one spatial dimension, by presenting rigorous stability analysisand detailed numerical simulations for the coupled system intwo spatial dimensions. We demonstrate that spotty patternsare observed over a wide parameter range.     

Global existence and blow-up for the solutions to nonlinear parabolic-elliptic system modelling chemotaxis

In this paper we study the global in-time and blow-up solutionsfor the simplified Keller–Segel system modelling chemotaxis.We prove that there is a critical number which determines theoccurrence of blowup in the two-dimensional case for 1 <p < 2. In three- or higher-dimensional cases, we show thatthe radial symmetrical solution will blow up if 1 < p <N/N–2 (N 3) for non-negative initial value.     

Congruences of Modular Forms and Tensor Product L-Functions

Two normalised Hecke eigenforms f and g are considered in thispaper, of level one and weights k' > k, lying in a p-adicfamily, such that f g(mod p^r as q-expansions. Interpretingthe congruence in terms of the associated Galois representationsleads to the existence of non-trivial global p^r-torsion forthe motive associated with the tensor product L-function ats = k' – 1. (It must be assumed that the (mod p) Galoisrepresentation attached to g is irreducible.) This contributesa factor of p^r to the denominator of the Bloch–Kato conjecturalformula for the L-value. The p-part of the numerator is considered,using recent work of Diamond, Flach and Guo. Using Shimura'sRankin–Selberg integral formula and the Clausen–vonStaudt theorem, the ratio of L-values is examined at s = k'– 1, for the tensor product and a quadratic twist; confirmationis given (under certain conditions) that, at p, this is as predictedby Bloch and Kato. 2000 Mathematics Subject Classification 11F33,11F67, 11G40.     

Non-linear robust boundary control of the Kuramoto-Sivashinsky equation

^** Email: krsakthivel{at}rediffmail.com^*** Email: hiroshi{at}ces.kyutech.ac.jp This paper considers the problem of robust global stabilizationof the Kuramoto–Sivashinsky equation subject to Neumannand Dirichlet boundary conditions. The aim is to derive non-linearrobust boundary control laws which make the system robustlyglobally asymptotically stable in spite of uncertainty in boththe instability parameter and the anti-diffusion parameter.A unique approach this paper introduces for achieving the requiredrobustness is spatially dependent scaling of uncertain elementsin Lyapunov-based stabilization. An important advantage of thisapproach is flexibility to obtain robust control laws with smallcontrol effort. The control laws can be implemented as Dirichlet-likeboundary control as well as Neumann-like boundary control. Furthermore,it is shown that they guarantee the stability and boundednessin terms of both L₂ and L.     

Rees Algebras of Modules

We study Rees algebras of modules within a fairly general framework.We introduce an approach through the notion of Bourbaki idealsthat allows the use of deformation theory. One can talk aboutthe (essentially unique) generic Bourbaki ideal I(E) of a moduleE which, in many situations, allows one to reduce the natureof the Rees algebra of E to that of its Bourbaki ideal I(E).Properties such as Cohen–Macaulayness, normality and beingof linear type are viewed from this perspective. The known numericalinvariants, such as the analytic spread, the reduction numberand the analytic deviation, of an ideal and its associated algebrasare considered in the case of modules. Corresponding notionsof complete intersection, almost complete intersection and equimultiplemodules are examined in some detail. Special consideration isgiven to certain modules which are fairly ubiquitous becauseinteresting vector bundles appear in this way. For these modulesone is able to estimate the reduction number and other invariantsin terms of the Buchsbaum–Rim multiplicity. 2000 MathematicsSubject Classification 13A30 (primary), 13H10, 13B21 (secondary)     

Equilibrium attractivity of Runge-Kutta methods

For dissipative differential equations y' = f (y) it is knownthat contractivity of the exact solution is reproduced by algebraicallystable Runge–Kutta methods. In this paper we investigatewhether a different property of the exact solution also holdsfor Runge–Kutta solutions. This property, called equilibriumattractivity, means that the norm of the righthand side f neverincreases. It is a property dual to algebraic stability sinceneither is sufficient for the other, in general. We derive sufficientalgebraic conditions for Runge–Kutta methods and proveequilibrium attractivity of the high-order algebraically stableRadau-IIA and Lobatto-IIIC methods and the Lobatto-IIIA collocationmethods (which are not algebraically stable). No smoothnessassumptions on f and no stepsize restrictions are required but,except for some simple cases, f has to satisfy certain additionalproperties which are generalizations of the simple one-sidedLipschitz condition using more than two argument points. Thesemultipoint conditions are discussed in detail.     

Cohen-Macaulay Properties of Thom-Boardman Strata I: Morin's Ideal

Thom–Boardman strata ^I are fundamental tools in studyingsingularities of maps. The Zariski closures of the strata ^Iare components of the set of zeros of the ideals ^I defined by B. Morin using iterated jacobian extensions in his paper‘Calcul jacobien’ (Ann. Sci. École Norm.Sup.} 8 (1975) 1–98). In this paper, we consider the questionof when the Morin ideals ^I define Cohen–Macaulay spaces.We determine all I=(i₁...,i_k) such that ^I defines a Cohen–Macaulayspace alongthe stratum. 1991 Mathematics Subject Classification: 13D25, 14B05, 14M12, 58C25.     

On the global stability of a temporal discretization scheme for the Navier-Stokes equations

The time discretization by a linear backward Euler scheme forthe non-stationary viscous incompressible Navier–Stokesequations with a non-zero external force in a bounded 2D domainwith no-slip boundary condition or periodic boundary conditionis studied. Improved global stability results are obtained. The boundedness of the solution sequence in V and D(A) normsuniform with respect to &t for t [0, ) is proved. A similarresult in the V norm was previously obtained by (Geveci, 1989Math. Comp., 53, 43–53) for the non-forced system. A differentapproach is used here. As a corollary, the global attractorfor the approximation scheme is proved to exist, which is boundedin both V and D(A) spaces, thus compact in both H and V spaces.Applying the same techniques developed here, we are able toimprove the main result of (Hill and Süli 2000 IMA J. Numer.Anal., 20, 633–667) by showing that besides the existenceof a global attractor, the whole solution sequence is uniformlybounded in V as well, which is of significance from the pointof view of computing. As a corollary of local convergence results,upper semi-continuity of the attractor with respect to the numericalperturbation induced by the linear scheme is also establishedin both H and V spaces. Finally, some preliminary estimates,which are to our knowledge the first of their kind, on the dimensionsof the attractors in H and V spaces are also obtained.     

A fully discrete stabilized finite-element method for the time-dependent Navier-Stokes problem

A fully discrete stabilized finite-element method is presentedfor the two-dimensional time-dependent Navier–Stokes problem.The spatial discretization is based on a finite-element spacepair (X_h, M_h) for the approximation of the velocity and thepressure, constructed by using the Q₁ – P₀ quadrilateralelement or the P₁ – P₀ triangular element; the time discretizationis based on the Euler semi-implicit scheme. It is shown thatthe proposed fully discrete stabilized finite-element methodresults in the optimal order error bounds for the velocity andthe pressure.     

S-adic L-Functions Attached to the Symmetric Square of a Newform

In order to apply the ideas of Iwasawa theory to the symmetricsquare of a newform, we need to be able to define non-archimedeananalogues of its complex L-series. The interpolated p-adic L-functionis closely connected via a "Main Conjecture" with certain Selmergroups over the cyclotomic Z_p-extension of Q. In the p-ordinarycase these functions are well understood. In this article we extend the interpolation to an arbitraryset S of good primes (not necessarily satisfying ordinarityconditions). The corresponding S-adic functions can be characterisedin terms of certain admissibility criteria. We also allow interpolationat particular primes dividing the level of the newform. One interesting application is to the symmetric square of amodular elliptic curve E defined over Q. Our constructions yieldp-adic L-functions at all primes of stable or semi-stable reduction.If p is ordinary or multiplicative the corresponding analyticfunction is bounded; if p is supersingular our function behaveslike log²(1 + T). 1991 Mathematics Subject Classification: 11F67,11F66, 11F33, 11F30     

Representations of Ariki-Koike Algebras and Grobner-Shirshov Bases

In this paper, we investigate the structure of Ariki–Koikealgebras and their Specht modules using Gröbner–Shirshovbasis theory and combinatorics of Young tableaux. For a multipartition, we find a presentation of the Specht module S given by generatorsand relations, and determine its Gröbner–Shirshovpair. As a consequence, we obtain a linear basis of S consistingof standard monomials with respect to the Gröbner–Shirshovpair. We show that this monomial basis can be canonically identifiedwith the set of cozy tableaux of shape . 2000 Mathematics SubjectClassification 16Gxx, 05Exx.