Terms Of Asymptotic Expansion Research Articles

Segregation of a liquid mixture by a radially oscillating bubble

A theoretical formulation is proposed for forced mass transport by pressure gradients in a liquid binary mixture around a spherical bubble undergoing volume oscillations in a sound field. Assuming the impermeability of the bubble wall to both species, diffusion driven by pressure gradients and classical Fick-diffusion must cancel at the bubble wall, so that an oscillatory concentration gradient arises in the vicinity of the bubble. The Péclet number pe is generally high in typical situations and Fick diffusion cannot restore equilibrium immediately, so that an asymptotic average concentration profile may progressively build up in the liquid over large times. Such a behaviour is reminiscent of the so-called rectified diffusion problem, leading to slow growth of a gas bubble oscillating in a sound field. A rigorous method formerly proposed by Fyrillas & Szeri (J. Fluid Mech. vol. 277, 1994, p. 381) to solve the latter problem is used to solve the present one. It is based on splitting the problem into a smooth part and an oscillatory part. The smooth part is solved by a multiple scales method and yields the slowly varying average concentration field everywhere in the liquid. The oscillatory part is obtained by matched asymptotic expansions in terms of the small parameter pe−1/2: the inner solution is required to satisfy the oscillatory balance between pressure diffusion and Fick diffusion at the bubble wall, while the outer solution is required to be zero. Matching both solutions yields a unique splitting of the problem. The final analytical solution, truncated to leading order, compares successfully to direct numerical simulation of the full convection–diffusion equation. The analytical expressions for both smooth and oscillatory parts are calculated for various sets of bubble parameters: driving pressure, frequency and ambient radius. The smooth problem always yields an average depletion of the heaviest species at the bubble wall, only noticeable for large molecules or nano-particles. For driving pressures sufficiently high to yield inertial oscillations of the bubble, the oscillatory problem predicts a periodic peak excess concentration of the heaviest species at the bubble wall at each collapse, lingering on several tens of the time of the characteristic duration of the bubble rebound. The two effects may compete for large molecules and practical implications of this segregation phenomenon are proposed for various processes involving acoustic cavitation.

Trajectory-based modelling of broad-band electromagnetic wavefields

SUMMARY A trajectory-based solution of Maxwell's equations is derived using the method of multiple scales. This time-domain technique utilizes an asymptotic expansion in terms of the ratio of the wave front length-scale to the length-scale of the heterogeneity. The approach provides an alternative to an expansion in terms of frequency and allows one to model electromagnetic wave propagation over a wide frequency band. At the lower end of the frequency band, the trajectory-based solution reduces to a previously derived diffusive solution. Similarly, at higher frequencies one obtains the ‘delta-like’ solution associated with hyperbolic wave propagation. However, the solution is also valid at intermediate frequencies which cannot be characterized as either diffusive or hyperbolic. A numerical illustration demonstrates the importance of both conduction and displacement currents at frequencies between 10 and 100 MHz. The amplitudes computed using the trajectory-based approach compare well with analytic results for a homogeneous whole-space. Using the technique I am able to model observations from a broad-band (3–300 kHz) experiment at the Richmond Field Station in California. In addition, ground penetrating radar waveforms in the 5–200 MHz range, gathered at the Boise Hydrogeophysical Research Site, are matched using the results of a radar velocity tomogram.

On the 2D models of plates and shells including the transversal shear

AbstractThe 2D Kirchhoff–Love (KL) theory and the Timoshenko–Reissner (TR) theory for thin shells made of the transversally isotropic, homogeneous material are discussed. For the cyclic‐symmetric deformations of shells of revolution the asymptotic analysis of stress–strain states is performed. Two simple linear problems for double‐periodic deformations of plates are studied applying the exact 3D theory and the 2D approximate theories. From these problems it follows that the KL theory is asymptotically correct because it gives the first term of asymptotic expansion of the 3D solution. The TR theory is asymptotically incorrect. It gives correctly the first term only and incorrectly the second term. But if the transversal shear modulus is comparatively small then this theory gives the main part of the second term. The case of the extremely small shear modulus is discussed. As an example the multi‐layered plate with alternating hard and soft isotropic layers is studied.

A Farey tail for attractor black holes

The microstates of 4d BPS black holes in IIA string theory compactified on a Calabi-Yau manifold are counted by a (generalized) elliptic genus of a (0,4) conformal field theory. By exploiting a spectral flow that relates states with different charges, and using the Rademacher formula, we find that the elliptic genus has an exact asymptotic expansion in terms of semi-classical saddle-points of the dual supergravity theory. This generalizes the known "Black Hole Farey Tail" of [1] to the case of attractor black holes.

Another approach to asymptotic expansions for Euler’s approximations of semigroups

In [2], optimal bounds for the remainder terms in asymptotic expansions for Euler’s approximations of semigroups were derived. The approach was based on applications of the Fourier-Laplace transforms, which allowed one to reduce the problem to estimation of error terms in the Law of Large Numbers. In this paper, we propose an alternative (direct) approach based on application of certain integro-differential identities (the so-called multiplicative representations of differences). Such identities were introduced by Bentkus [3] and applied (see Bentkus and Paulauskas [4]) to derive the optimal convergence rates in Chernoff-type lemmas and Euler’s approximations of semigroups.

POLYHOMOGENEOUS SOLUTIONS OF NONLINEAR WAVE EQUATIONS WITHOUT CORNER CONDITIONS

The study of Einstein equations leads naturally to Cauchy problems with initial data on hypersurfaces which closely resemble hyperboloids in Minkowski space-time, and with initial data with polyhomogeneous asymptotics, that is, with asymptotic expansions in terms of powers of ln r and inverse powers of r. Such expansions also arise in the conformal method for analysing wave equations in odd space-time dimension. In recent work it has been shown that for non-linear wave equations, or for wave maps, polyhomogeneous initial data lead to solutions which are also polyhomogeneous provided that an infinite hierarchy of corner conditions holds. In this paper we show that the result is true regardless of corner conditions.

Penalized maximum likelihood and semiparametric second-order efficiency

We consider the problem of estimation of a shift parameter of an unknown symmetric function in Gaussian white noise. We introduce a notion of semiparametric second-order efficiency and propose estimators that are semiparametrically efficient and second-order efficient in our model. These estimators are of a penalized maximum likelihood type with an appropriately chosen penalty. We argue that second-order efficiency is crucial in semiparametric problems since only the second-order terms in asymptotic expansion for the risk account for the behavior of the “nonparametric component” of a semiparametric procedure, and they are not dramatically smaller than the first-order terms.

Zeta-function approach to Casimir energy with singular potentials

In the framework of zeta-function approach the Casimir energy for three simple model system: single delta potential, step function potential and three delta potentials are analyzed. It is shown that the energy contains contributions which are peculiar to the potentials. It is suggested to renormalize the energy using the condition that the energy of infinitely separated potentials is zero which corresponds to subtraction all terms of asymptotic expansion of zeta-function. The energy obtained in this way obeys all physically reasonable conditions. It is finite in the Dirichlet limit, and it may be attractive or repulsive depending on the strength of potential. The effective action is calculated, and it is shown that the surface contribution appears. The renormalization of the effective action is discussed.

An extension of the J-integral to piezoelectric materials

This paper provides an analysis of the crack propagation criterion for a thin piezoelectric plate with a symmetry of order six. On the basis of Gol’denveizer’s asymptotic integration method or Destuynder’s unidirectional zoom technique, we obtain an extension of the purely mechanical J-integral to piezoelectric materials, with a dependence of the gradient of energy of the plate only on zeroth order terms of asymptotic expansions.

Drag on a large spherical aggregate with self-similar structure: An asymptotic analysis

A porous particle with radially varying permeability is used to model a large self-similar (fractal-like) aggregate in the continuum fluid flow regime. The Stokes flow in the viscous fluid around the aggregate is coupled with the solution of the flow inside the aggregate. This internal flow is shown to be composed of a central core governed by Darcy equation and a thin shell at the aggregate edge where the flow is described by Brinkman's equation. Solutions to the governing equations in each of these regions are sought as asymptotic expansions in terms of a suitably small permeability parameter k. The hydrodynamic jumps across the Brinkman layer are systematically established and used to compute the viscous drag experienced by the aggregate. These conditions, which generalize the widely used Saffman (tangential flow) boundary conditions, are more broadly applicable to flows past/through fine-grained deposits of arbitrary shape, and many other configurations of technological interest.

Diffraction, Interference, and Depolarization of Elastic Waves. Caustic and Penumbra

A theoretical study of polarization-spectral anomalies of wave fields, or their deviations from the predictions based on simple plane-wave models, is presented. A simple unified method of numerical simulation of anomalies of nonstationary wave fields near a caustic and in the penumbra is described. It uses both the leading and correcting terms in asymptotic expansions. Examples of calculation of displacements and average polarization ellipses are given. Qualitative properties of wave fields are discussed. A review of earlier research on polarization anomalies of elastic waves is given. Bibliography: 38 titles.

ON DELAY ESTIMATION FOR STOCHASTIC DIFFERENTIAL EQUATIONS

We present a review of some results concerning delay estimation by continuous time observations of solutions of stochastic differential equations in two asymptotics. The first one corresponds to small noise limit and the second to large samples limit. In both cases we describe the properties of the maximum likelihood estimator and Bayesian estimators with especial attention to asymptotic efficiency of the estimators. We show that the first asymptotic corresponds to regular problems of mathematical statistics and the second is close to non regular problems. In small noise asymptotics we give the next after the Gaussian term of asymptotic expansion of the maximum likelihood estimator.

Approximating the bias of the LSDV estimator for dynamic unbalanced panel data models

This paper extends the LSDV bias approximations in [Bun, B.J.G., Kiviet, J.F. 2003. On the diminishing returns of higher order terms in asymptotic expansions of bias. Economic letters, 79, 145-152.] to unbalanced panels. The approximations are obtained by modifying the within operator to accommodate the dynamic selection rule. They are accurate, with higher order terms bringing only decreasing improvements to the approximations. This removes an important cause for limited applicability of bias corrected LSDV estimators.

Transport properties of densely packed composites. Effect of shapes and spacings of inclusions

Summary We analyze transport properties of ∞uid/solid and solid/solid composites containing flnite arrays of closely spaced rigid inclusions when a host medium is either an elastic matrix or an incompressible ∞uid. The appropriate choice of the number of inclusions and the symmetry of a periodicity cell allows us to introduce simple, yet physically relevant models so that efiective characteristics of homogenized media can be investigated analytically. For various applied loads and shapes of (polydisperse) inclusions we demonstrate the spatial non-uniformity of geometric conflgurations corresponding to either lowest dissipation rate (for ∞uid/solid composites) or to minimal stifiness (for solid/solid composites). In order to flnd the optimal conflgurations, we use a unifled framework based on asymptotic expansions in terms of inter-inclusion distances. Furthermore, we compare efiective transport properties of composite materials containing inclusions with either ∞at or curved boundaries.

Sum rules and exact relations for quantal Coulomb systems

AbstractA complex response function describing a reaction of a multi‐particle system to a weak alternating external field is the boundary value of a Nevanlinna class function (i.e. a holomorphic function with non‐negative imaginary part in the upper half‐plane). Attempts of direct calculations of response functions based on standard approximations of the kinetic theory for real Coulomb condensed systems often result in considerable discrepancies with experiments and computer simulations. At the same time a relatively simple approach using only the exact values of leading asymptotic terms of the response function permits to restrict essentially a subset of Nevanlinna class functions containing this response function, and in this way to obtain sufficient data to explain and predict experimental results. Mathematical details of this approach are demonstrated on an example with the response function being the (external) dynamic electrical conductivity of cold dense hydrogen‐like plasmas. In particular, the exact values of the leading terms of asymptotic expansions of the conductivity are calculated. (© 2003 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Scaling of Structural Strength

Scaling of Structural Strength

Asymptotic relations in the Askey scheme for hypergeometric orthogonal polynomials

It has been recently pointed out that several orthogonal polynomials of the Askey table admit asymptotic expansions in terms of Hermite and Laguerre polynomials [López and Temme, Meth. Appl. Anal. 6 (1999) 131–146; J. Comp. Appl. Math. 133 (2001) 623–633]. From those expansions, several known and new limits between polynomials of the Askey table were obtained in [López and Temme, Meth. Appl. Anal. 6 (1999) 131–146; J. Comp. Appl. Math. 133 (2001) 623–633]. In this paper, we make an exhaustive analysis of the three lower levels of the Askey scheme which completes the asymptotic analysis performed in [López and Temme, Meth. Appl. Anal. 6 (1999) 131–146; J. Comp. Appl. Math. 133 (2001) 623–633]: (i) We obtain asymptotic expansions of Charlier, Meixner–Pollaczek, Jacobi, Meixner, and Krawtchouk polynomials in terms of Hermite polynomials. (ii) We obtain asymptotic expansions of Meixner–Pollaczek, Jacobi, Meixner, and Krawtchouk polynomials in terms of Charlier polynomials. (iii) We give new proofs for the known limits between polynomials of these three levels and derive new limits.

Two-dimensional wavelike equilibrium states in the wall injection-induced planar Taylor flow

Restricted accessMoreSectionsView PDF ToolsAdd to favoritesDownload CitationsTrack Citations ShareShare onFacebookTwitterLinked InRedditEmail Cite this article Griffond J. 2003Two-dimensional wavelike equilibrium states in the wall injection-induced planar Taylor flowProc. R. Soc. Lond. A.459749–771http://doi.org/10.1098/rspa.2002.1035SectionRestricted accessTwo-dimensional wavelike equilibrium states in the wall injection-induced planar Taylor flow J. Griffond J. Griffond ONERA–Toulouse, Aerodynamics and Energetics Modelling Department, 2, av. Ed. Belin, BP 4025, 31055 Toulouse Cedex 4, France [email protected] Google Scholar Find this author on PubMed Search for more papers by this author J. Griffond J. Griffond ONERA–Toulouse, Aerodynamics and Energetics Modelling Department, 2, av. Ed. Belin, BP 4025, 31055 Toulouse Cedex 4, France [email protected] Google Scholar Find this author on PubMed Search for more papers by this author Published:08 March 2003https://doi.org/10.1098/rspa.2002.1035AbstractIn this paper, the existence of particular nonlinear instabilities is shown in the sidewall injection–induced planar Taylor flow in rectangular channels with a solid headwall and an open aft end. These two–dimensional wave systems are asymptotic solutions of the nonlinear problem as the abscissa (distance to the headwall) tends to infinity. They are quasi–linearly spatially amplified and represent, in fact, a kind of equilibrium with the linearly accelerated base flow. The problem is solved by an asymptotic expansion in terms of the inverse–squared abscissa and also by a less rigorous but more convenient method. The results are successfully compared to direct numerical simulations. It is then suggested that these states play a role in the behaviour of a cold–flow set–up called VECLA, even if no accurate comparison with experimental work is possible. Previous ArticleNext Article VIEW FULL TEXT DOWNLOAD PDF FiguresRelatedReferencesDetailsCited by Volkov K (2006) Application of the parameter-expansion method to calculation of two-phase flows in channels with injection, Journal of Engineering Physics and Thermophysics, 10.1007/s10891-006-0220-0, 79:6, (1169-1178), Online publication date: 1-Nov-2006. This Issue08 March 2003Volume 459Issue 2031 Article InformationDOI:https://doi.org/10.1098/rspa.2002.1035Published by:Royal SocietyPrint ISSN:1364-5021Online ISSN:1471-2946History: Published online08/03/2003Published in print08/03/2003 License: Citations and impact KeywordsTaylor FlowEquilibrium StatesNonlinear Stability

On the diminishing returns of higher-order terms in asymptotic expansions of bias

The relative magnitudes are compared of successive terms in a higher-order asymptotic expansion of the bias of the LSDV estimator in dynamic panels. We find that the leading term accounts for the major part of the actual bias in small samples. This implies that bias correction procedures can be based on relatively simple bias approximation formulas.

Asymptotics of Quantum Mechanical Atom-Ion Systems

We have analyzed and reduced a general (quantum-mechanical) atom-ion diatomic exchange energy formulation into fundamental mathematical forms, namely a particular class of single and double definite integrals. These are of importance in the charge exchange of molecular processes in atmospheric physics and eventually of interest to matters related to climate. These integrals have been evaluated in terms of asymptotic expansions, with precise schemes for their numerical evaluation. Dividends for algebraic aspects concerning identities of hypergeometric functions as well as summation techniques for divergent series are also discussed in this context.