Rayleigh waves having generalised lateral dependence

It is shown that the surface-guided elastic waves found by Kiselevfor isotropic materials and having displacements depending linearlyupon the Cartesian coordinate orthogonal to the sagittal planemay be generalised in many ways. For surface waves on any anisotropichalf-space, a simple procedure applied to the displacementswithin the standard surface wave having dependence eⁱ, where k · x – t and k is the (surface) wave vector,yields displacements depending linearly upon the surface cartesiancoordinate orthogonal to the group velocity vector. Moreover,repeated application of this (differentiation) procedure yieldsa hierarchy of waves having algebraic dependence of successivelyincreasing degree. For isotropic materials, substantial simplificationand generalization are possible. Solutions of all algebraicdegrees have identical depth dependence. This allows the solutionsto be constructed iteratively and motivates a search for generalsolutions having depth dependence of the normal displacementthe same as in the standard surface wave. The procedure givesa new derivation of the solutions found by Achenbach havingamplitude of the normal displacement of the surface given byany solution to the two-dimensional Helmholtz equation. Furthermore,exploiting the scale invariance (a property of surface waveson any homogeneous half-space) shows that in every surface-guideddisturbance of an elastic half-space, the elevation of the freesurface is a solution of the wave equation in two dimensions(the membrane equation). Using the paraxial approximation tothe membrane equation, high-frequency Rayleigh waves propagatingas narrow beams are described in terms of a scalar Gaussianbeam.     

The Asymptotic Behaviour of the Coefficients in Stokes's Series for Surface Gravity Waves

It is shown that the asymptotic behaviour of the coefficientsa_n at high order n and at large wave steepness ak is determinedmainly by the limiting form of the wave crest. In a lower rangeof n, a_n, decreases like n^–, corresponding to the Stokes120° corner flow. In an upper range, a_n, decreases exponentiallywith n. The transition occurs when n³ is O(1) where is relatedto the steepness ak of the waves by ² = 2.0[(ak)_max –ak].     

Asymptotic Interpolating Sequences in Uniform Algebras

T. Hosokawa, K. Izuchi and D. Zheng recently introduced theconcept of asymptotic interpolating sequences (of type 1) inthe unit disk for H(D). It is shown that these sequences coincidewith sequences that are interpolating for the algebra QA. Alsoa characterization is given of the interpolating sequences oftype 1 for H(D), and asymptotic interpolating sequences in thespectrum of H(D) are studied. The existence of asymptotic interpolatingsequences of type 1 for H() on arbitrary domains is verified.It is shown that any asymptotic interpolating sequence in auniform algebra eventually is interpolating.     

Asymptotic Expansions of Double and Multiple Integrals Occurring in Diffraction Theory

At present at I.N.S.T.N., Saclay and Faculté des Sciences, Paris, France Asymptotic expansions of double integrals of the type have been derived in terms of thereal parameter k by the method of stationary phase. The resultscan easily be extended to multi-dimensional integrals. In the first part of this paper a rigorous proof of the applicationof the method of stationary phase to double and multiple integralsis established with the aid of neutralizer or unitary functions.It is shown that the principal contributions to U(k) come fromsmall but otherwise arbitrary neighbourhoods of critical pointsof the integral, which may be located in the interior or onthe boundary of the domain of integration. These points areassociated with the phase or amplitude function. An explicitasymptotic series in the parameter k of the principal contributionis exhibited when the amplitude and the phase functions havein the neighbourhood of a critical point (x₁,y₁) a developmentof the form g(x,y) = (x–x₁)^0–1 (y–y₁)^µ0–1g₁(x,y), (x,y) = (x₁,y₁) + a _,0 (x–x₁ [1 + P(x,y) + b_0,(y–y₁[1+Q(x,y)]. The function g₁ is a regular function and P,Q can be developedin power series in the vicinity of the critical point and vanishat this point. The above expansion we shall call normal or canonicaland the critical point a normal or canonical critical pointof the integral. Although the assumption of the normal form expansion of theamplitude and phase functions is too restrictive for the generalcase, nevertheless it is found to be sufficiently broad to includemost of the important and interesting cases which occur in diffraction,scattering and other problems of mathematical physics. In Part II the principal contribution arising from a criticalpoint of normal type has been calculated in the form of a descendingpower series in the parameter k. It is shown, with the use ofmajorant functions, that the contribution due to the remainderpart of the series is of higher order in the parameter thanthat of the last term of the finite part, which proves the asymptoticcharacter of the series in the sense of Poincaré. Theresults derived here are in agreement with that of Part I. However,the new series has a decided advantage over that given in PartI if calculations are desired for even a few terms of the series,since the coefficients entering in the asymptotic expansionof the principal contribution are expressed directly in termsof the original functions g(x,y) and (x,y) and their derivatives,which is not the case in the formulas derived in Part I. In Part III explicit asymptotic expansions of the double integralare derived for several typical critical points associated withthe phase function. These are important in connection with thetheory of diffraction of optical instruments with large aberrationsand scattering problems. On account of their importance, eachcase has been treated in detail. In the appendices we have given an alternative proof of thetheorem announced in Part I and the derivation of the leadingterm due to a boundary stationary point. There will be foundalso a discussion of the more general integral where the parameterk appears implicitly in the phase function and not explicitlyas considered in the text. Integrals of this kind occur in manybranches of physics, especially when dealing with wave propagationin dispersive and absorbing media. Finally, we have concludedon the basis of our results that the Rubinowicz approach todiffraction and the stationary phase application to diffractionintegrals lead to similar mathematical results, although differentphysical interpretations, in diffraction phenomena, the formerleading to Young diffraction phenomena and the latter to Fresneldiffraction phenomena.     

On the extension of real regular embedding

Let X be a real nonsingular affine algebraic variety of dimensionk. It is proved that any two regular (algebraic) embeddingsX ⁿ are regularly equivalent, provided that n 4k + 2.     

On Second-Order Almost-Periodic Elliptic Operators

The paper considers second-order, strongly elliptic, operatorsH with complex almost-periodic coefficients in divergence formon R^d. First, it is proved that the corresponding heat kernelis Hölder continuous and Gaussian bounds are derived withthe correct small and large time asymptotic behaviour on thekernel and its Hölder derivatives. Secondly, it is establishedthat the kernel has a variety of properties of almost-periodicity.Thirdly, it is demonstrated that the kernel of the homogenization of H is the leading term inthe asymptotic expansion of t K_t.     

Asymptotic Solution of Hyperbolic Systems with Constant Coefficients

An integral representation of the exact solution of the initialvalue problem for the hyperbolic equation of the form is derived. Here A^o, A^v, B, and Care constant m x m matrices, u(t, X; ) is an m-component columnvector, and is a positive parameter. Various conditions areimposed on the coefficient matrices that permit the applicationof the method of stationary phase in several variables to theintegral representation of the exact solution. The leading termof the asymptotic expansion as of the exact solution is obtainedfor several types of initial data and source functions whichdepend on the parameter .     

On optimal convergence rates for higher-order Navier-Stokes approximations. I. Error estimates for the spatial discretization

^** Email: bause{at}am.uni-erlangen.de Due to the increasing use of higher-order methods in computationalfluid dynamics, the question of optimal approximability of theNavier–Stokes equations under realistic assumptions onthe data has become important. It is well known that the regularitycustomarily hypothesized in the error analysis for parabolicproblems cannot be assumed for the Navier–Stokes equations,as it depends on non-local compatibility conditions for thedata at time t = 0, which cannot be verified in practice. Takinginto account this loss of regularity at t = 0, improved convergenceof the order (min{h^(5/2)–,h³/t^(1/4)+}), for any >0, is shown locally in time for the spatial discretization ofthe velocity field by (non-)conforming finite elements of third-orderapproximability properties. The error estimate itself is provedby energy methods, but it is based on sharp a priori estimatesfor the Navier–Stokes solution in fractional-order spacesthat are derived by semigroup methods and complex interpolationtheory and reflect the optimal regularity of the solution ast 0.     

Mixed finite-element methods for Hamilton-Jacobi-Bellman-type equations

The numerical solution of Dirichlet's problem for a second-orderelliptic operator in divergence form with arbitrary nonlinearitiesin the first- and zero-order terms is considered. The mixedfinite-element method is used. Existence and uniqueness of theapproximation are proved and optimal error estimates in L² aredemonstrated for the relevant functions. Error estimates arealso derived in L^q, 2q+     

Witt groups and unipotent elements in algebraic groups

Let G be a semisimple algebraic group defined over an algebraicallyclosed field K of good characteristic p>0. Let u be a unipotentelement of G of order p^t, for some t N. In this paper it isshown that u lies in a closed subgroup of G isomorphic to theit Witt group W^t(K), which is a t-dimensional connected abelianunipotent algebraic group. 2000 Mathematics Subject Classification:20G15.     

Rate of convergence of numerical approximations to homoclinic bifurcation points

For x=f (x, ), x Rⁿ, R, having a hyperbolic or semihyperbolicequilibrium p(), we study the numerical approximation of parametervalues ^* at which there is an orbit homoclinic to p(). We approximate^* by solving a finite-interval boundary value problem on J=[T_–,T₊], T_–<0<T₊, with boundary conditions that sayx(T_–) and x(T₊) are in approximations to appropriate invariantmanifolds of p(). A phase condition is also necessary to makethe solution unique. Using a lemma of Xiao-Biao Lin, we improve,for certain phase conditions, existing estimates on the rateof convergence of the computed homoclinic bifurcation parametervalue , to the true value ^*. The estimates we obtain agree withthe rates of convergence observed in numerical experiments.Unfortunately, the phase condition most commonly used in numericalwork is not covered by our results.     

A qualocation method for a unidimensional single phase semilinear Stefan problem

Based on straightening the free boundary, a qualocation methodis proposed and analysed for a single phase unidimensional Stefanproblem. This method may be considered as a discrete versionof the H¹-Galerkin method in which the discretization is achievedby approximating the integrals by a composite Gauss quadraturerule. Optimal error estimates are derived in L(W^j,), j = 0,1,and L (H^j), j = 0,1,2, norms for a semidiscrete scheme withoutany quasi-uniformity assumption on the finite element mesh.     

Algebraic cobordisms of a Pfister quadric

In this article we compute the ring of algebraic cobordismsof a Pfister quadric. This is a rare example of a non-cellularvariety where such a computation is known. We consider the algebraiccobordisms * of Levine and Morel, as well as the MGL^{2*, *} ofVoevodsky. The methods of computation in these two cases arequite different. However, the results do agree (which supportsthe expectation that the two theories actually coincide). Weshow that the restriction homomorphism in our case is injectivefor any field extension E/F.     

Numerical approximation of the boundary control for the wave equation with mixed finite elements in a square

Sorin Micu ^{This paper studies the numerical approximation of the boundarycontrol for the wave equation in a square domain. It is knownthat the discrete and semi-discrete models obtained by discretizingthe wave equation with the usual finite-difference or finite-elementmethods do not provide convergent sequences of approximationsto the boundary control of the continuous wave equation as themesh size goes to zero. Here, we introduce and analyse a newsemi-discrete model based on the space discretization of thewave equation using a mixed finite-element method with two differentbasis functions for the position and velocity. The main theoreticalresult is a uniform observability inequality which allows usto construct a sequence of approximations converging to theminimal L2-norm control of the continuous wave equation. Wealso introduce a fully discrete system, obtained from our semi-discretescheme, for which we conjecture that it provides a convergentsequence of discrete approximations as both h and t, the timediscretization parameter, go to zero. We illustrate this factwith several numerical experiments.}     

The Second Dual Algebra of the Measure Algebra of a Compact Group

It is shown that for every compact group G, L¹(G)^ is uniqueand minimal among all the closed subsets I of M(G)** such thatI is a proper (0, M(G)**) algebraic ideal, and such that I issolid with respect to absolute continuity; that is, n L¹(G)^whenever n M(G)** and n << µ L¹(G)^. 1991 MathematicsSubject Classification 43A20, 43A22.     

Approximation of 1/x by exponential sums in [1, {infty})

^** Email: braess{at}num.rub.de^*** Email: wh{at}mis.mpg.de Approximations of 1/x by sums of exponentials are well studiedfor finite intervals. Here the error decreases like (exp(–ck))with the order k of the exponential sum. In this paper we investigateapproximations of 1/x in the interval [1, ). We prove estimatesof the error by and confirm this asymptotic estimate by numerical results. Numericalresults lead to the conjecture that the constant in the exponentequals .     

On the Error Term for the Fourth Moment of the Riemann Zeta-Function

Let E₂(T) denote the error term in the asymptotic formula for^T₀|(+it)|⁴dt. It is proved that there exist constants A>0,B>1 such that for TT₀>0 every interval [T, BT] containspoints T₁, T₂ for which and that ^T₀|E₂(t)|^adt>>T^1+(a/2) for any fixed a1. Theseresults complement earlier results of Motohashi and Ivi that^T₀E₂(t)dt<<T^3/2 and that ^T₀E²₂(t)dt<<T²log^CT forsome C>0. Omega-results analogous to the above ones are obtainedalso for the error term in the asymptotic formula for the Laplacetransform of |(+it)|⁴.     

Extensions of Asymptotic Fields Via Meromorphic Functions

An asymptotic field is a special type of Hardy field in which,modulo an oracle for constants, one can determine asymptoticbehaviour of elements. In a previous paper, it was shown inparticular that limits of real Liouvillian functions can therebybe computed. Let denote an asymptotic field and let f . Weprove here that if G is meromorphic at the limit of f (whichmay be infinite) and satisfies an algebraic differential equationover R(x), then (G o f) is an asymptotic field. Hence it ispossible (modulo an oracle for constants) to compute asymptoticforms for elements of (G o f). An example is given to show thatthe result may fail if G has an essential singularity at limf.     

Waves in the Presence of an Infinite Dock with Gap

After extending some completeness results of an earlier paper,the two-dimensional problem of the infinite dock with gap isconsidered. With the frequency and 2 the non-dimensional lengthof the gap, eigenvalues of K = ²/g are first computed and thentheir apparent asymptotic form is established not only to orderK^–1 but also up to terms in K^–2.     

Ruled Special Lagrangian 3-Folds In C3

This is the fifth in a series of papers constructing explicitexamples of special Lagrangian submanifolds in C^m. A submanifoldof C^m is ruled if it is fibred by a family of real straightlines in C^m. This paper studies ruled special Lagrangian 3-foldsin C³, giving both general theory and families of examples.Our results are related to previous work of Harvey and Lawson,Borisenko, and Bryant. Special Lagrangian cones in C³ are automaticallyruled, and each ruled special Lagrangian 3-fold is asymptoticto a unique special Lagrangian cone. We study the family ofruled special Lagrangian 3-folds N asymptotic to a fixed specialLagrangian cone N0. We find that this depends on solving a linearequation, so that the family of such N has the structure ofa vector space. We also show that the intersection of N0 withthe unit sphere S⁵ in C³ is a Riemann surface, and constructa ruled special Lagrangian 3-fold N asymptotic to N0 for eachholomorphic vector field w on . As corollaries of this we writedown two large families of explicit special Lagrangian 3-foldsin C³ depending on a holomorphic function on C, which includemany new examples of singularities of special Lagrangian 3-folds.We also show that each special Lagrangian T2-cone N0 can beextended to a 2-parameter family of ruled special Lagrangian3-folds asymptotic to N0, and diffeomorphic to T²xR. 2000 Mathematical Subject Classification: 53C38, 53D12.