The Effect of Sidewalls on Overstable Convection in a Rotating Fluid

A simple two-dimensional model is used to demonstrate some interestingeffects which arise when Chandrasekhar's (1962) theory of overstableconvection in an infinite rotating fluid layer is modified totake account of lateral walls. The aim of the investigationis to determine how sidewalls aligned with the convective rollsaffect the critical Rayleigh number and frequency of oscillationand also how the overstable eigensolutions are related to thepreviously determined stationary solutions of the equations(Daniels, 1977). For containers of large aspect ratio, L, thecritical Rayleigh number for overstability is R_o+O(L^–1)(where R_o is the value for the infinite layer) and in the neighbourhoodof this single perturbed value it is found that there is aninfinite spectrum of overstable eigenvalues with frequencieswhich differ by O(L^–1). The O(L^–1) correction toR_o is determined analytically for the case of small Prandtlnumber and rapid rotation.     

Long-Time Behavior of Solutions of the Fast Diffusion Equations with Critical Absorption Terms

This paper is devoted to the long-time behavior of solutionsto the Cauchy problem of the porous medium equation u_t = (u^m)– u^p in Rⁿ x (0,) with (1 – 2/n)₊ < m < 1and the critical exponent p = m + 2/n. For the strictly positiveinitial data u(x,0) = O(1 + |x|)^–k with n + mn(2 –n + nm)/(2[2 – m + mn(1 – m)]) k < 2/(1 –m), we prove that the solution of the above Cauchy problem convergesto a fundamental solution of u_t = (u^m) with an additional logarithmicanomalous decay exponent in time as t .     

On the Solution of Some Elliptic Difference Equations

Iterative methods for the solution of some nonlinear ellipticdifference systems, approximating the first boundary value problemare considered. If h > 0 is the network step in the spaceof variables x = (x₁, x₂,..., x_p) and 2m is the order of theoriginal boundary value problem, then the iterative methodsproposed give solution of accuracy with the expenditure ofO(|In | h^–(p+m–)) and O(|In | |In h| h^–p)arithmetic operations in the case of a general region and arectangular parallelepiped respectively. In the case p = 2 theestimate O(|In | h^{–[2+ (m/2)]}) is obtained if the regionis made up of rectangles with sides parallel to the co-ordinateaxes.     

A Fourth-order Tridiagonal Finite Difference Method for General Two-point Boundary Value Problems with Non-linear Boundary Conditions

We present a fourth-order finite difference method for the generalsecond-order nonlinear differential equation y^" = f(x, y, y‘)subject to non-linear two-point boundary conditions g₁(y(a), — y^’()) = 0, g₂(y(b), y'(b)) = 0. When both the differential equation and the boundary conditionsare linear, the method leads to a tridiagonal linear system.We show that the finite difference method is O(h⁴)-convergent.Numerical examples are given to illustrate the method and itsfourth-order convergence. The present paper extends the methodgiven in Chawla (1978) to the case of non-linear boundary conditions.     

On Extensions of Myers' Theorem

Let M be a compact Riemannian manifold, and let h be a smoothfunction on M. Let p^h(x) = inf||–₁(Ric_x(,)–2Hess(h_x(,)).Here Ric_x denotes the Ricci curvature at x and Hess(h) is theHessian of h. Then M has finite fundamental group if ^h–p^h<0. Here ^h =:+2L_h is the Bismut-Witten Laplacian. This leadsto a quick proof of recent results on extension of Myers' theoremto manifolds with mostly positive curvature. There is also asimilar result for noncompact manifolds.     

Compact High-Order Finite Differences for Interface Problems in One Dimension

This paper is concerned with the construction and analysis ofcompact finite difference approximations to the model linearsource problem –(pu')' + qu = f where the functions p,q, and f can have jump discontinuities at a finite number ofpoints. Explicit formulae that give O(h²) O(h³) and O(h⁴) accuracyare derived, and a procedure for computing three-point schemesof any prescribed order of accuracy is presented. A rigoroustruncation and discretization error analysis is offered. Numericalresults are also given.     

Mixed hp-DGFEM for incompressible flows II: Geometric edge meshes

We consider the Stokes problem of incompressible fluid flowin three-dimensional polyhedral domains discretized on hexahedralmeshes with hp-discontinuous Galerkin finite elements of typeQ_k for the velocity and Q_k–1 for the pressure. We provethat these elements are inf-sup stable on geometric edge meshesthat are refined anisotropically and non-quasiuniformly towardsedges and corners. The discrete inf-sup constant is shown tobe independent of the aspect ratio of the anisotropic elementsand is of O(k^–3/2) in the polynomial degree k, as in thecase of conforming Q_k–Q_k–2 approximations on thesame meshes.     

Hypersurfaces in a Unit Sphere Sn+1(1) with Constant Scalar Curvature

The paper considers n-dimensional hypersurfaces with constantscalar curvature of a unit sphere S^n–1(1). The hypersurfaceS^k(c₁)xS^n–k(c₂) in a unit sphere Sⁿ⁺¹(1) is characterized,and it is shown that there exist many compact hypersurfaceswith constant scalar curvature in a unit sphere Sⁿ⁺¹(1) whichare not congruent to each other in it. In particular, it isproved that if M is an n-dimensional (n > 3) complete locallyconformally flat hypersurface with constant scalar curvaturen(n–1)r in a unit sphere Sⁿ⁺¹(1), then r > 1–2/n,and (1) when r (n–2)/(n–1), if then M is isometric to S¹xS^n–1(c),where S is the squared norm of the second fundamental form ofM; (2) there are no complete hypersurfaces in Sⁿ⁺¹(1) with constantscalar curvature n(n–1)r and with two distinct principalcurvatures, one of which is simple, such that r = (n–2)/(n–1)and      

(k, l)-Algebraic Stability of Runge-Kutta Methods

For ordinary differential equations satisfying a one-sided Lipschitzcondition with Lipschitz constant v, the solutions satisfy with l=hv, so that, in the case of Runge-Kutta methods, estimatesof the form ||y_n||²k(l)||y_n–1||² are desirable. Burrage(1986) has investigated the behaviour of the error-boundingfunction k for positive l for the family of s-stage Gauss methodsof order 2s, and has shown that k(l)=exp 2l+O(l³) (l0) for s3.In this paper, we extend the analysis of k to any irreduciblealgebraically stable Runge-Kutta method, and obtain resultsabout the maximum order of k as an approximation to exp 2l.As a particular example, we investigate the function k for allalgebraically stable methods of order 2s–1.     

Shapiro's Cyclic Sum

Shapiro's cyclic sum is defined by , If K is the cone in Rⁿ of points withnon-negative coordinates, it is shown that the minimum of Ein K is a fixed point of T², where T is the non-linear operatordefined by (Tx)_i = x_n–i+1/(x_n–i+2 + x_n–i+3)²for i = 1,2,...,n. It is conjectured that Tx = S^kx, where Sis the shift operator in Rⁿ, and a proof is given under someadditional hypotheses. One of the consequences is a simple proofthat at the minimum point, a_i(x) = a_n–i+1–k(x) fori = 1,2,...,n.     

On the Local and Superlinear Convergence of Quasi-Newton Methods

This paper presents a local convergence analysis for severalwell-known quasi-Newton methods when used, without line searches,in an iteration of the form to solve for x^* such that Fx^* = 0. The basic idea behind theproofs is that under certain reasonable conditions on x_o, Fand x_o, the errors in the sequence of approximations {H_k} toF'(x^*)^–1 can be shown to be of bounded deterioration inthat these errors, while not ensured to decrease, can increaseonly in a controlled way. Despite the fact that H_k is not shownto approach F'(x^*)^–1, the methods considered, includingthose based on the single-rank Broyden and double-rank Davidon-Fletcher-Powellformulae, generate locally Q-superlinearly convergent sequences{x_k}.     

Notes on the Numerical Solution of the Biharmonic Equation

It is shown that the solution of the differenced form of theDirichlet problem for the biharmonic equation by iteration isachieved in O(h^–1)log h² steps of iteration where h isa step-size parameter from the mesh.     

On the Sloan iteration applied to integral equations of the first kind

When the piecewise constant collocation method is used to solvean integral equation of the first kind with logarithmic kernel,the convergence rate is O(h) in the L₂ norm. In this note weshow that O(h³) or O(h⁵) convergence in any Sobolev norm (andthus, for example, in L) may be obtained by a simple cheap postprocessingof the original collocation solution. The construction of thepostprocessor is based on writing the first kind equation asa second kind equation, and applying the Sloan iteration tothe latter equation. The theoretical convergence rates are verifiedin a numerical example.     

Necessary and Sufficient Conditions for Exponential Stability and Ultimate Boundedness of Systems Governed by Stochastic Partial Differential Equations

Consider the following infinite dimensional stochastic evolutionequation over some Hilbert space H with norm |·|: It is proved that under certain mild assumptions, the strongsolution X_t(x₀)VHV*, t 0, is mean square exponentially stableif and only if there exists a Lyapunov functional (·,·):HxR₊R¹ which satisfies the following conditions: (i)c₁|x|²–k₁e^–µ₁t(x,t)c₂|x|²+k₂+k₂e^–µ₂t; (ii) L(x,t)–c₃(x,t)+k₃e^–µ₃t, xV, t0; where L is the infinitesimal generator of the Markov processX_t and c_i, k_i, µ_i, i = 1, 2, 3, are positive constants.As a by-product, the characterization of exponential ultimateboundedness of the strong solution is established as the nulldecay rates (that is, µ_i = 0) are considered.     

Implicit finite difference schemes for a linear model of well-reservoir coupling

Standard reservoir models usually consider wells as Dirac measuresover an interval length. Moreover, the well-reservoir couplingis taken into account under quite simplified assumptions. Mostrecently, however, attention has been drawn to the fact thatin some situations, such as those related to non-vertical wells,these simplifications do not allow us to model some relevantmechanisms of the coupled flow. Therefore, more complex alternativemodels have been proposed recently in the oil reservoir simulationliterature. A linearized version for the well-reservoir couplingcan be written, in an appropriate functional setting, in theform U'(t) + AU(t) = F(t). In this work we discuss implicitin time discretizations of this equation, of the form { Uⁿ⁺¹_h - Uⁿ_h÷+A_hUⁿ⁺¹_h = Fⁿ⁼¹_h, U⁰_h=U_0.h We propose two different approximations, corresponding to first-and second-order spatial truncation errors, and we establishthe convergence of both approximations.     

Binary Quadratic Forms with Large Discriminants and Sums of Two Squareful Numbers II

Let F = (F₁, ..., F_m) be an m-tuple of primitive positive binaryquadratic forms and let U_F(x) be the number of integers notexceeding x that can be represented simultaneously by all theforms F_j, j = 1, ... , m. Sharp upper and lower bounds for U_F(x)are given uniformly in the discriminants of the quadratic forms. As an application, a problem of Erds is considered. Let V(x)be the number of integers not exceeding x that are representableas a sum of two squareful numbers. Then V(x) = x(log x)^–+o(1)with = 1 – 2^–1/3 = 0.206....     

Determination of a Convex Body from Minkowski Sums of its Projections

For a convex body K in R^d and 1 K d – 1, let P_K (K)be the Minkowski sum (average) of all orthogonal projectionsof K onto k-dimensional subspaces of R^d. It is Known that theoperator P_k is injective if kd/2, k=3 for all d, and if k =2, d 14. It is shown that P_2k (K) determines a convex body K among allcentrally symmetric convex bodies and P_2k+1(K) determines aconvex body K among all bodies of constant width. Correspondingstability results are also given. Furthermore, it is shown thatany convex body K is determined by the two sets P_k (K) and P_k'(K) if 1 < k < k'. Concerning the range of P_k , 1 k d–2, it is shown that its closure (in the Hausdorff-metric)does not contain any polytopes other than singletons.     

Numerical Methods for y' =f(x, y) via Rational Approximations for the Cosine

To investigate stability and phase lag, a numerical method isapplied to the test equation y' = –²y. Frequently, thecharacteristic equation of the resulting recurrence relationhas the form ²– 2R_nm(v²) + 1 = 0, where v = h, with hthe steplength, and R_nm(v²) is a rational approximation forcos v. In this paper, properties of such approximations areused to provide a general framework for the study of stabilityintervals and orders of dispersion of a variety of one- andtwo-step methods. Upper bounds on the intervals of periodicityof explicit methods with maximum order of dispersion are established.It is shown that the order of dispersion of a P-stable method,for given n and m, cannot exceed 2m; a consequence is that,of the Pad? approximants for cos v, only the [0/2m] approximantshave modulus less than unity for all v² >0. A complete characterizationof P-stable methods of fourth order corresponding to the rationalapproximation R₂₂(v²) is followed by several results for methodswhich have finite intervals of periodicity; in particular, weidentify methods which have order of dispersion 6 or 8 withlarge intervals of periodicity. There is also a detailed discussionof P-stable methods of sixth order corresponding to the rationalapproximation R₃₃(v²).     

A multigrid-type method for thin plate spline interpolation on a circle

We consider the problem of thin plate spline interpolation ton equally spaced points on a circle, where the number of datapoints is sufficiently large for work of O(n³ to be unacceptable.We develop an iterative multigrid-type method, each iterationcomprising ngrid stages, and n being an integer multiple of2^ngrid–1. We let the first grid, V₁ be the full set ofdata points, V say, and each subsequent (coarser) grid, V_k,k=2, 3,...,ngrid, contain exactly half of the data points ofthe preceding (finer) grid, these data points being equallyspaced. At each stage of the iteration, we correct our current approximationto the thin plate spline interpolant by an estimate of the interpolantto the current residuals on V_k, where the correction is constructedfrom Lagrange functions of interpolation on small local subsetsof p data points in V_k. When the coarsest grid is reached, however,then the interpolation problem is solved exactly on its q=n/2^ngrid–1points. The iterative process continues until the maximum residualdoes not exceed a specified tolerance. Each iteration has the effect of premultiplying the vector ofresiduals by an n x n matrix R, and thus convergence will dependupon the spectral radius, (R), of this matrix. We investigatethe dependence of the spectral radius on the values of n, p,and q. In all the cases we have considered, we find (R) <<1, and thus rapid convergence is assured.     

Serre's Theorem on the Cohomology Algebra of a p-Group

The purpose of this note is to give a proof of a theorem ofSerre, which states that if G is a p-group which is not elementaryabelian, then there exist an integer m and non-zero elementsx₁, ..., x_m H¹ (G, Z/p) such that with ß the Bockstein homomorphism. Denote by m_G thesmallest integer m satisfying the above property. The theoremwas originally proved by Serre [5], without any bound on m_G.Later, in [2], Kroll showed that m_G p^k – 1, with k =dim_Z/pH¹ (G, Z/p). Serre, in [6], also showed that m_G (p^k –1)/(p – 1). In [3], using the Evens norm map, Okuyamaand Sasaki gave a proof with a slight improvement on Serre'sbound; it follows from their proof (see, for example, [1, Theorem4.7.3]) that m_G (p + 1)p^k–2. However, m_G can be sharpenedfurther, as we see below. For convenience, write H^*(G, Z/p) = H^*(G). For every x_i H¹(G),set 1991 Mathematics SubjectClassification 20J06.