A note on total bounds on complex transport moduli of parametric two-phase media

We study a composite material consisting of two isotropic constituents with complex transport modulus depending on a real parameter. By starting from the parametric estimations reported by Baker [3], Bergman [5] and Milton [1] we propose a method of construction of the bound on an imaginary part of an effective modulus versus its real part and reversely, on a real part versus the imaginary one. The estimation obtained we call the total bound on a parametric modulus of two-phase medium. Illustrative examples of the numerical evaluations of low order total bounds are also provided.     

Convergence of eigenfunction expansions at points of discontinuity of the coefficients of a differential operator

The question of the convergence of expansions in the eigenfunctions of a differential operator with discontinuous coefficients at a point x₀ of discontinuity of the coefficients is studied. Given an arbitrary function f(x) in the class L₂, a corresponding function \(\tilde f_{x_o } (x)\) is constructed which is such that at the point x₀ the eigenfunction expansion of f(x) diverges with the expansion of \(\tilde f_{x_o } (x)\) into a Fourier trigonometric series.     

Pointwise bounds of orthogonal expansions on the real line via weighted Hilbert transforms

We study pointwise bounds of orthogonal expansions on the real line for a class of exponential weights of smooth polynomial decay at infinity. As a consequence of our main results, we establish pointwise bounds for weighted Hilbert transforms which are of independent interest. Mathematics subject classifications (2000)  41A10, 42C05     

The coefficients of the Laurent series expansions of real analytic functions

Letf be a real analytic function of a real variable such that 0 is an isolated (possibly essential) singularity off. In the existing literature the coefficients of the Laurent series expansion off around 0 are obtained by applying Cauchy's integral formula to the analytic continuation off on the complex plane. Here by means of a conformal mapping we derive a formula which determines the Laurent coefficients off solely in terms of the values off and the derivatives off at a real point of analyticity off. Using a more complicated mapping, we similarly determine the coefficients of the Laurent expansion off around 0 where now 0 is a singularity off which is not necessarily isolated.     

The coefficients of the Laurent series expansions of real analytic functions

Letf be a real analytic function of a real variable such that 0 is an isolated (possibly essential) singularity off. In the existing literature the coefficients of the Laurent series expansion off around 0 are obtained by applying Cauchy's integral formula to the analytic continuation off on the complex plane. Here by means of a conformal mapping we derive a formula which determines the Laurent coefficients off solely in terms of the values off and the derivatives off at a real point of analyticity off. Using a more complicated mapping, we similarly determine the coefficients of the Laurent expansion off around 0 where now 0 is a singularity off which is not necessarily isolated.     

Asymptotic expansions of matrix coefficients: The real rank one case

Let G be a real reductive Lie group of class $\text{H}$, and suppose that the split rank of G is one. We show that the asymptotic expansions of the Eisenstein integrals given in Harish-Chandra (1) give uniform approximation off of a certain naturally defined compact subset of $\text{A}\text{?}$, the unitary dual of A; G = KAN being an Iwasawa decomposition of G.     

Exact solutions for two-phase colloidal-suspension transport in porous media

Two-phase transport of colloids and suspensions occurs in numerous areas of chemical, environmental, geo-, and petroleum engineering. The main effects are particle capture by the rock and altering the flux by changing the suspended and retained concentrations. Multiple mechanisms of suspended particle capture are discussed. The mathematical model for m independent particle-capture mechanisms is considered, resulting in an (m + 2) × (m + 2) system of partial differential equations. Using the stream-function as an independent variable instead of time splits the system into an (m + 1) × (m + 1) auxiliary system, containing only concentrations and one lifting hydrodynamic equation for an unknown phase saturation. Introduction of the concentration potential linked with retention concentrations yields an exact solution of the auxiliary problem. The exact formulae allow for predicting the profiles and breakthrough histories for the suspended and retained concentrations, and phase saturations. The solution shows that for small retained concentrations, the suspended concentration is in a steady-state behind the concentration front, where all the retained concentrations are proportional to the mass of suspended particles that passed via a given reservoir cross-section. The maximum penetration depths for suspended and retained particles are the same and are equal to those for a single-phase flow.     

Inequalities for upper bounds of functionals

. . . , , .     

Explicit bounds of polynomial coefficients and counting points on Picard curves over finite fields

In this paper, we describe an algorithm for computing the order of the Jacobian varieties of Picard curves over finite fields. This is an extension of the algorithm of Matsuo, Chao and Tsujii (MCT) [K. Matsuo, J. Chao, S. Tsujii, An improved baby step algorithm for point counting of hyperelliptic curves over finite fields, in: LNCS vol. 2369, Springer-Verlag, 2005, pp. 461–474] for hyperelliptic curves. We study the characteristic polynomials and the Jacobian varieties of algebraic curves of genus three over finite fields. Based on this, we investigate the explicit computable bounds for coefficients of the characteristic polynomial and improve a part of the baby-step giant-step of the counting points algorithm. Usefulness of the proposed method is illustrated and verified by the simple examples.     

Accessory coefficients of the second-order Fuchs equation with real singular points

The author investigates the equations of Schwarz and Fuchs related with the conformal mapping of a semiplane onto a circular polygon. One investigates the question of the dependence of the accessory coefficients as functions of the singularities of the equations. As an intermediate result one constructs formulas for the Fourier coefficients of the Siegel-Selberg series and of the absolute Klein invariant. At the conclusion of the paper one considers the example of a rectangle for which the problem of the determination of the accessory coefficients is simplified.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 129, pp. 17–29, 1983.     

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.     

Lower bounds at infinity for solutions of differential equations with constant coefficients

Two extensions of a classical theorem of Rellich are proved: (1) LetP=P(−iϖ/ϖx) be a partial differential operator with constant coefficients in , let the manifolds contained in have codimension ≧k>0, and denote by Γ an open cone in intersecting each normal plane of every such manifold. If ,Pu=0 and it follows thatu=0. (2) Assume in addition that each irreducibe lfactor ofP van shes on a real hypersurface and that Γ contains both normal directions at some such point. If andP(D) u has compact support, the same condition withk=1 implies thatu has compact support. In both results the hypotheses on the cone Γ and on the operatorP are minimal.     

Error bounds for asymptotic expansions of Wilks’ lambda distribution

Let Λ=|S_e|/|S_e+S_h|, where S_h and S_e are independently distributed as Wishart distributions W_p(q,Σ) and W_p(n,Σ), respectively. Then Λ has Wilks’ lambda distribution Λ_p,q,n which appears as the distributions of various multivariate likelihood ratio tests. This paper is concerned with theoretical accuracy for asymptotic expansions of the distribution of T=-nlogΛ. We derive error bounds for the approximations. It is necessary to underline that our error bounds are given in explicit and computable forms.     

Uniform bounds for stably integral points on elliptic curves

We show that a conjecture by Lang and Vojta regarding integral points on varieties of logarithmic general type implies the existence of a uniform bound on the number of stably -integral points on an elliptic curve over a degree- number field.

     

Asymptotic expansions for the coefficients of analytic generating functions

PairsF(x), G(x) of analytic generating functions that satisfy relations such as 1+G(x)=exp(F(x)) are studied. It is shown that, ifF(x) satisfies fairly mild regularity conditions, such as those imposed by Hayman in his study of coefficients of some general classes of functions, thenG(x) satisfies the much stricter conditions imposed by Harris and Schoenfeld. This enables one to obtain complete asymptotic expansions for the coefficients ofG(x). Applications of this result are made to enumerations of trees.Dedicated to Professor Janos Aczél on his 60th birthday