Spectral analysis and computation for the Kuramoto-Sakaguchi integroparabolic equation

A spectral method is developed to numerically solve the so-calledKuramoto–Sakaguchi equation, which is a nonlinear integro-differentialequation of the parabolic type, governing the dynamical statisticalbehaviour of certain populations of nonlinearly coupled randomoscillators. The approach rests on explicit bounds for the spacederivatives of solutions, obtained via energy-like estimates.Bounds for the numerical approximations of solutions are given,and improved (sometimes appreciably) by means of an ‘aposteriori error analysis’. Plots are shown to illustratethe performance of the method, and comparison with a finitedifference approach is also made.     

On improving the Hashin-Shtrikman bounds for the effective properties of three-phase composite media

For a composite comprising an isotropic mixture of two isotropicdielectric materials, the Hashin-Shtrikman bounds for the overalldielectric constant tensor are attainable and hence are thebest possible. Considering instead a three-phase composite,the Hashin-Shtrikman bounds are the best that are known in termsof volume fractions alone, and yet, in the limit of vanishingvolume fraction of the material of greatest dielectric constant,the three-phase upper bound remains strictly greater than thetwo-phase bound. A similar comment applies to the lower bound,in relation to a small volume fraction of the material withthe smallest dielectric constant. Although this phenomenon mayreflect a limitation of the Hashin-Shtrikman methodology, itremains conceivable that some microgeometries exist for whichall the ‘third’ phase is positioned in regions ofhigh field concentration, so that it always has a large effect.This paper resolves this problem to some extent, by generatinga new upper bound that ranges continuously from the Hashin-Shtrikmantwo-phase bound to the Hashin-Shtrikman three-phase bound asthe volume fraction c₃ of the ‘third’ material increasesfrom zero. The Hashin-Shtrikman three-phase bound thus cannotbe optimal, at least when c₃ is small. The method of derivationof the new bound relies on an application of the theory of functionsof bounded mean oscillation, recently developed in the contextof bounding the behaviour of nonlinear composites.     

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.     

Uniform Convergence of Variational Methods for Elliptic Problems in N Dimensions

Earlier work in variational methods in the plane is extendedto N dimensions. Smoothness theorems are constructed and usedot convert means convergence to uniform convergence, establishingpointwise error bounds. A constrained variational method isused so that a priori bounds for the derivatives of the solutionmay be imposed on the approximating functions. In the non-linearproblem of compressible flow such a bound is provided by anassumption that the solution is subsonic.     

Solving two-point boundary value problems with spline functions

We consider a collocation method for the approximation of thesolution of the nonlinear two-point boundary value problem y'(x)=f(x,y(x)), y(a)=A, y(b)=B, using splines of degree m3. The methodwhich we shall use leads to a system of recurrence relationswhich can be solved by Newton's method. By obtaining asymptotic error bounds we verify a conjectureof Khalifa & Eilbeck, i.e. splines of even degree can giveeven better solutions than splines of odd degree in certaincases.     

Spectral decompositions in nonlinear coupled diffusion

The solutions of a coupled, linear and nonlinear diffusion equationin a semi-infinite medium are derived using series methods.In addition, perturbation techniques allied to the spectraldecomposition of matrices are used to simplify the analysisand to find semianalytic solutions. The discussion is motivatedby the transmission of heat, moisture, and solute through thestrongly nonlinear medium of soil. Under boundary conditionsrepresenting the daily or seasonal fluctuations, it is shownusing spectral decomposition, despite the nonlinearities, howthe period of oscillation is preserved on passage through themedium. It is also shown how n³ partial differential equationsmay be solved for each of the n coupled variables to determineclosed forms for the first- and second-order perturbation effects.Examples of the solutions are given for the case of the coupledtransport of heat and moisture in soil.     

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.     

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.     

Linear Bounds for the Degree of Subgroup Growth in Terms of the Hirsch Length

Let G be a residually-finite virtually soluble minimax group.We give upper and lower linear bounds for the degree of subgroupgrowth of G in terms of the Hirsch length of G. 2000 MathematicsSubject Classification 20F99.     

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.     

Bounds for the Singular Values of a Matrix

A lower-bound theorem is developed for the singular values ofa matrix A (and therefore for the eigenvalues of B = A^HA). Itis found that there is always a unique column scaling of A whichproduces the optimum bound. However, sharper bounds still maysometimes be obtained by taking advantage of matrix partitioning.It is shown that the resulting bounds may often (but not always)be better than those obtained by applying Gerschgorin's theoremto B. The equivalent upper-bound theorem is found to be weak.     

Observability of Nonlinear Systems

We consider the observability of systems of the form = Ax +Nx, y = Fx, where A is a linear operator and N and F are nonlinear.We show that if the system is linearized about an equilibriumpoint x_e and the linearized system is continuously initiallyobservable, then the nonlinear system is continuously initiallyobservable in some neighbourhood of x_e. We then look at conditionsunder which solutions of the nonlinear system can be extendedfor all time and consider the problem of stabilizing the systemby feedback controls such that the solutions are eventuallyin the observability neighbourhood of x_e. Finally, we applythese ideas to two systems: a wave equation and a diffusionequation with nonlinear perturbations and nonlinear observations.     

Positive Solutions of Nonlocal Boundary Value Problems: A Unified Approach

We give a unified approach for studying the existence of multiplepositive solutions of nonlinear differential equations of theform where g, f are non-negativefunctions, subject to various nonlocal boundary conditions.We study these problems via new results for a perturbed integralequation, in the space C[0, 1], of the form where [u], ß[u] are linear functionalsgiven by Stieltjes integrals but are not assumed to be positivefor all positive u. This means we actually cover many more differentialequations than the simple equation written above. Previous resultshave studied positive functionals only, but even for positivefunctionals our methods give improvements on previous work.The well-known m-point boundary value problems are special casesand we obtain sharp conditions on the coefficients, which allowssome of them to have opposite signs. We also use some optimalassumptions on the nonlinear term.     

Round-off Errors and Stability in Variational Calculations

We consider the problem of the stability of a variational solutionof a linear, inhomo-geneous operator equation, in the presenceof "round-off" errors in the various inner products involved.For systems which are asymptotically diagonal (see Delves &Mead, 1971; Freeman, Delves & Reid, 1974) we produce boundson the error induced by the round-off noise, which show thatat least for sufficiently small C the solution method is stablein these cases in the sense of Mikhlin (1971) provided thatthe system is normalized ("nice" in the sense of Delves &Mead, 1971). Un-normalized A.D. systems may not be stable, butare relatively stable in a sense defined here. In addition tothese error bounds we produce estimates of the distributionof the round-off errors amongst the expansion coefficients a_i^(N).A numerical example suggests that relative stability is sufficientto ensure good variational behaviour of the calculation.     

Automatic Hessians by reverse accumulation

Let n be the number of independent variables of a function f,and let W and S respectively be the time and space bounds forthe joint evaluation of {f, f} using automatic differentiationwith reverse accumulation. In this note, we examine an extensionof the technique of reverse accumulation which allows the automaticextraction of the Hessian of f. The method allows the parallelevaluation of all rows of the Hessian matrix in about 2W timeunits and 3S space units on each of n processors, or sequentialrow-by-row evaluation in about 2nW time units and 3S space unitson a single processor. The approach described here is intendedfor use with operator overloading (for example in Ada) and allowsthe conventional coding of the target function f.     

An orthogonal spline collocation alternating direction implicit method for second-order hyperbolic problems

On a rectangular region, we consider a linear second-order hyperbolicinitial-boundary value problem involving a mixed derivativeterm, continuous variable coefficients and non-homogeneous Dirichletboundary conditions. In comparison to the alternating directionimplicit Laplace-modified method of Fernandes (1997), we formulateand analyse a new parameter-free alternating direction implicitscheme in which the standard central difference formula is usedfor the time approximation and orthogonal spline collocationis used for the spatial discretization. We establish unconditionalstability of the scheme, and its optimal order in the discretemaximum norm in time and the H¹ norm in space. Numerical experimentsindicate that the new scheme, which has the same order as themethod of Fernandes (1997, Numer. Math., 77, 223–241),is more accurate. We also show that the new scheme is easilygeneralized to the second-order hyperbolic problems on rectangularpolygons. Extensions of the scheme to problems with discontinuouscoefficients, nonlinear problems, and problems with other boundaryconditions are also discussed.     

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.     

On the Mislin Genus of Symplectic Groups

In this paper we give lower bounds for the Mislin genus of thesymplectic groups Sp(m). This result appears to be the exactanalogue of Zabrodsky's theorem concerning the special unitarygroups SU(n). It is achieved by the determination of the stablegenus of the quasi-projective quaternionic spaces QH(m), followingthe approach of McGibbon. It leads to a symplectic version ofZabrodsky's conjecture, saying that these lower bounds are infact the exact cardinality of the genus sets. The genus of Sp(2)is well known to contain exactly two elements. We show thatthe genus of Sp(3) has exactly 32 elements and see that theconjecture is true in these two cases. Independently, we also show that any homotopy type in the genusof Sp(m) fibers over the sphere S⁴m–1 with fiber in thegenus of Sp(m–1), and that any homotopy type in the genusof SU(n) fibers over the sphere S²n–1 with fiber in thegenus of SU(n–1). Moreover, these fibrations are principalwith respect to some appropriate loop structures on the fibers.These constructions permit us to produce particular spaces realizingthe lower bounds obtained. 2000 Mathematics Subject Classification55P60 (primary), 55P15, 55R35 (secondary)     

The Minimax Solution of Linear Equations Subject to Linear Constraints

This paper presents an efficient active set method for computingthe l_m solution of linear equations subject to linear constraints.The search direction of the proposed method is obtained viathe solution of a linear least-squares problem. It is shownthat the steepest-descent direction can be obtained by solvingthe same least-squares problem with simple bounds on the variables.This direction is used to prevent cycling at degenerate deadpoints. Numerical experiments illustrate the feasibility ofthe new method.     

Lower Bounds for Morse Index of Constant Mean Curvature Tori

In the paper, three lower bounds are given for the Morse indexof a constant mean curvature torus in Euclidean3-space, in termsof its spectral genus g. The first two lower bounds grow linearlyin g and are stronger for smaller values of g, while the thirdgrows quadratically in g but is weaker for smaller values ofg. 2000 Mathematics Subject Classification 53A10, 53A35.