Search Results

In order to identify the essential items of the mode-coupling-theory scenarios, this chapter describes asymptotic expansions of the correlation functions for small frequencies and small separations of the coupling constants from their critical values. For generic glass transitions the leading-order asymptotic contributions are specified by two scaling laws. The slowing down of the dynamics is governed by two time scales, which exhibit power-law dependencies on the separation coordinates. Stretching is caused by the interplay of a critical power-law relaxation and a von-Schweidler-power-law decay. Logarithmic relaxation is the leading-order behaviour near higher-order singularities. The leading-order asymptotic corrections determine the range of validity of the leading-order formulas. Strong-coupling effects are shown to cause Cole–Cole relaxation processes for the critical decay near generic transitions and sublinear power-law variations of the mean-squared displacements near higher-order ones.

In this chapter the constructions of chapters 1 and 2 for are illustrated with explicit computations. The first part of the chapter presents various methods of computation commonly used in the literature and which have acted as guide posts to the development of trace and determinant methods in geometric analysis over the past few decades. There are basic equivalences, indispensable in both theoretical developments and exact computations, between coefficients in the asymptotic expansions of zeta traces, heat traces and resolvent traces. The middle part of the chapter presents and proves these identifications, and provides an application to the computation of relative zeta determinants. The final part of the chapter turns to residue trace and residue determinant computations. Based on a residue determinant formula for the spectral zeta function at zero, an elementary proof of the local Atiyah Singer index theorem is given.

Chapter 1 discusses the mathematical aspects of the asymptotic theory. It starts with basic definitions, using for this purpose the so-called coordinate asymptotic expansions. Then the chapter turns to asymptotic analysis of integrals and describe, among others, the method of steepest descent. However, the main attention is with parametric asymptotic expansions. These are used in a wide variety of fluid–dynamic problems, for which purpose a number of asymptotic techniques has been developed. We discuss in Chapter 1 the method of matched asymptotic expansions, the method of multiple scales, the method of strained coordinates, and the WKB method.

Functions of a complex variable are shown to be more complete and rigid than functions of a real variable. Analytic functions with well defined derivatives satisfy two Cauch–Riemann conditions. Multivalued functions can be made single-valued on a multi-sheet Riemann surface. The values of an analytic function in a region of the complex plane are completely defined by the knowledge of their values on a closed boundary of the region. Important properties and techniques of complex analysis are described. These include Taylor and Laurent expansions, contour integration and residue calculus, Green's functions, Laplace transforms and Bromwich integrals, dispersion relations and asymptotic expansions. Analytic functions are defined by their properties at the locations called singularities (poles and branch cuts) where they cease to be analytic. This feature makes analytic functions of particular interest in the construction of physical theories.

Chapter 24 examines the topic of hyper–asymptotic expansions and instantons. A number of quantum mechanics and quantum field theory (QFT) examples exhibit degenerate classical minima connected by quantum barrier penetration effects. The analysis of the large order behaviour, based on instanton calculus, shows that the perturbative expansion is not Borel summable, and does not define unique functions. An important issue is then what kind of additional information is required to determine the exact expanded functions. While the QFT examples are complicated, and their study is still at the preliminary stage, in quantum mechanics, in the case of some analytic potentials that have degenerate minima (like the quartic double–well potential), the problem has been completely solved. Some examples are described in Chapter 24. There, the perturbative, complete, hyper–asymptotic expansion exhibits the resurgence property. The perturbative expansion can be related to the calculation of the spectral equation via the complex WKB method.

This chapter deals with small-volume expansions of the displacement fields. It first introduces the notion of elastic moment tensor, a geometric quantity associated with a small-volume inclusion, before discussing some of its important properties such as symmetry and positive-definiteness. The asymptotic expansion of the displacement in the presence of a small-volume inclusion is expressed in terms of the elastic moment tensor. The chapter proceeds by deriving formulas for the elastic moment tensors under linear transformations and computes those associated with ellipses and balls. It also considers both the static and time-harmonic regimes and extends the small-volume asymptotic framework to anisotropic elasticity. Finally, it provides the leading-order terms in the asymptotic expansions of the solutions to the static and time-harmonic elasticity equations with respect to the size of a small inclusion.

Defects and impurities affect the vibrations of a solid, and hence properties such as infra-red spectra and thermal conductivity. There can be new, high-frequency, local modes; there can be new, low-frequency, resonances. This chapter analyses key results systematically, including some convenient simple limits. It also goes into detail concerning classical Green's functions, thermodynamic Green's functions, response functions, isotopic impurity, and asymptotic expansions. Local and resonance modes, Rayleigh scattering, and the peak theorem are also considered.

This chapter considers the perturbations of the displacement (or traction) vector that are due to the presence of a small crack with homogeneous Neumann boundary conditions in an elastic medium. It derives an asymptotic formula for the boundary perturbations of the displacement as the length of the crack tends to zero. Using analytical results for the finite Hilbert transform, the chapter derives an asymptotic expansion of the effect of a small Neumann crack on the boundary values of the solution. It also derives the topological derivative of the elastic potential energy functional and proves a useful representation formula for the Kelvin matrix of the fundamental solutions of Lamé system. Finally, it gives an asymptotic formula for the effect of a small linear crack in the time-harmonic regime.

The idea of resonant nonlinear interactions of waves, and of resonant wave triads, is first explained using the example of Rossby waves, and then used to highlight a mechanism of excitation of wave-guide modes, by impinging free waves at the oceanic shelf, and at the equator. Physics and mathematics of the mechanism, which is related to the phenomena of parametric resonance and wave modulation, are explained in detail in both cases. The resulting modulation equations, of Ginzburg–Landau or nonlinear Schrodinger type, are obtained by multi-scale asymptotic expansions and elimination of resonances, after the explanation of this technique. The chapter thus makes a link between geophysical fluid dynamics and other branches of nonlinear physics. A variety of nonlinear phenomena including coherent structure formation is displayed. The resonant excitation of wave-guide modes provides an efficient mechanism of energy transfer to the wave guides from the large to the small.

This chapter presents some recent results on the elasticity equations with high contrast coefficients. It first sets up the problems for finite and extreme moduli before discussing the incompressible limit of elasticity equations. It then provides a complete asymptotic expansion with respect to the compressional modulus and considers the limiting cases of holes and hard inclusions. It proves that the energy functional is uniformly bounded and demonstrates that the potentials on the boundary of the inclusion are also uniformly bounded. It also shows that these potentials converge as the bulk and shear moduli tend to their extreme values and that similar boundedness and convergence result holds true for the boundary value problem.

This chapter describes the papers Dot presented to the University of London for her graduate degrees. While the papers were pure mathematics, sponges had shaped her work just as they shaped their spicules. Dot's thesis subject was “asymptotic expansions,” a subfield of infinite series. Her adviser, John Nicholson, put her to work on them because he needed them—he and Arthur Dendy, his biologist colleague at King's. Dendy was the world's leading spongeologist and the author of the authoritative Studies on the Comparative Anatomy of Sponges.

This chapter describes the use of time-reversal imaging techniques for optimal control of extended inclusions. It first considers the problem of reconstructing shape deformations of an extended elastic target before reconstructing the perturbations from boundary measurements of the displacement field. As for small-volume inclusions, direct imaging algorithms based on an asymptotic expansion for the perturbations in the data due to small shape deformations is introduced. The chapter also presents a concept equivalent to the polarization tensor for small-volume inclusions as well as the nonlinear optimization problem for reconstructing the shape of an extended inclusion from boundary displacement measurements. Finally, it develops iterative algorithms to address the nonlinearity of the problem.

After analysis of general properties of horizontal motion in primitive equations and introduction of principal parameters, the key notion of geostrophic equilibrium is introduced. Quasi-geostrophic reductions of one- and two-layer rotating shallow-water models are obtained by a direct filtering of fast inertia–gravity waves through a choice of the time scale of motions of interest, and by asymptotic expansions in Rossby number. Properties of quasi-geostrophic models are established. It is shown that in the beta-plane approximations the models describe Rossby waves. The first idea of the classical baroclinic instability is given, and its relation to Rossby waves is explained. Modifications of quasi-geostrophic dynamics in the presence of coastal, topographic, and equatorial wave-guides are analysed. Emission of mountain Rossby waves by a flow over topography is demonstrated. The phenomena of Kelvin wave breaking, and of soliton formation by long equatorial and topographic Rossby waves due to nonlinear effects are explained.

The fundamental process of geostrophic adjustment is treated by the method of multi-scale asymptotic expansions in Rossby number and fast-time averaging (which is explained), first in the barotropic one-layer case, and then in the baroclinic two-layer case. Together with the standard quasi-geostrophic regime of parameters, the frontal (or semi-) geostrophic regime is considered. Dynamical separation of slow and fast motions is demonstrated in both regimes. The former obey quasi-geostrophic or frontal-geostrophic equations, thus providing formal justification of the heuristic derivation of Chapter 5. Fast motions are inertia-gravity waves in quasi-geostrophic case, and inertial oscillations in the frontal-geostrophic case. Geostrophic adjustment is also considered in the presence of coastal, topographic, and equatorial wave-guides, and, again, separation of fast and slow motions is demonstrated, the latter now including long Kelvin waves in the first case, long topographic waves in the second case, and long Kelvin and Rossby waves in the third case.