Showing papers on "Asymptotic analysis published in 1983"
TL;DR: In this article, the authors established bounds on the number of modes which determine the solutions of the Navier-Stokes equations in 2-dimensional Rayleigh-Benard convection.
180 citations
TL;DR: In this paper, the authors studied convection in a rapidly rotating spherical layer with constant-temperature boundary conditions and found that the power-law dependences of the critical buoyancy number and the critical wavenumber on the rotation rate are borne out, although discrepancies in the actual values of these quantities do exist.
Abstract: Convection in a rapidly rotating spherical layer with constant-temperature boundary conditions is studied in a laboratory experiment. The asymptotic theory of Busse (1970) is extended to permit a comparison with the observations of the onset of convection and its properties. It is found that the prediction of the power-law dependences of the critical buoyancy number and the critical wavenumber on the rotation rate are borne out, although discrepancies in the actual values of these quantities do exist. Calculations on the basis of equations proposed by Roberts (1968) show that a thermal wind that is present in the basic state of the model has a stabilizing influence on the onset of convection. Stewartson layers not taken into account in the asymptotic analysis for vanishing Ekman number E appear to be responsible for the remaining disagreement between theoretical predictions and observations at finite values of E.
131 citations
TL;DR: In this article, the equations of radiative transfer are systematically analyzed by asymptotic methods, and the classical equilibrium diffusion approximation is recovered to lowest order, and next order analysis leads to the equilibrium diffusion differential equations and initial condition, but with a boundary condition containing a linear extrapolation distance α. This quantity is related to the solution of a canonical halfspace problem and is computed by deriving an appropriate variational principle.
Abstract: The equations of radiative transfer are systematically analyzed by asymptotic methods. To lowest order, the classical equilibrium diffusion approximation is recovered. The next order analysis leads to the equilibrium diffusion differential equations and initial condition, but with a boundary condition containing a linear extrapolation distance α. This quantity is related tothe solution of a canonical halfspace problem and is computed by deriving an appropriate variational principle. For the case of no scattering, an exact Wiener-Hopf solution is available. The FN solution technique is also applied to the problem of obtaining α with good results. Higher order asymptotic radiative transfer descriptions are discussed and, while not immediately constituting practical calculational techniques, do have implications for computing the parameters in the multiband treatment of the frequency variable.
124 citations
TL;DR: In this article, a model for flame propagation though a condensed combustible mixture in which the limiting component of the mixture melts during the reaction process is presented, and an asymptotic analysis, valid for large activation energies, is employed to derive a two-term expansion for the steady, planar adiabatic flame speed.
Abstract: A model is presented for flame propagation though a condensed combustible mixture in which the limiting component of the mixture melts during the reaction process. An asymptotic analysis, valid for large activation energies, is employed to derive a two-term expansion for the steady, planar adiabatic flame speed. A linear stability analysis is then used to show that for sufficiently large values of the activation energy and/or a special group of melting parameters, the steady, planar solution loses stability to various types of planar and nonplanar pulsating modes. The effect of melting is found to be destabilizing in the sense that these pulsating modes occur for lower values of the activation energy than would be the case for strictly solid fuel combustion.
103 citations
86 citations
TL;DR: In this article, a half space of combustible material is subjected to an arbitrary energy flux at the boundary where convection heat loss is also allowed, and an asymptotic analysis of the temperature growth reveals two conditions necessary for ignition to occur.
Abstract: A half space of combustible material is subjected to an arbitrary energy flux at the boundary where convection heat loss is also allowed. An asymptotic analysis of the temperature growth reveals two conditions necessary for ignition to occur. Cases of both large and order unity Lewis number are shown to lead to a nonlinear integral equation governing the thermal runaway. Some global and asymptotic properties of the integral equation are obtained.
34 citations
33 citations
TL;DR: In this article, a rescaling transformation bringing friction terms in the new equation is used to obtain the asymptotic solution of a one-dimensional, one-species beam.
Abstract: Rescaling transformations bringing friction terms in the new equation are used to obtain the asymptotic solution of a one-dimensional, one-species beam. It is shown that for all possible initial conditions this asymptotic solution coincides with the self-similar solution.
32 citations
TL;DR: In this article, the asymptotic estimate of the solution y ( t ) of linear difference equations with almost constant coefficients and the condition of equivalence between y( t ) and y 0 ( t ), where y 0 is the corresponding solution of linear equations with constant coefficients are given.
29 citations
TL;DR: In this paper, a stationary phase asymptotic analysis of the elastic Green's function for a semi-infinite anisotropic elastic solid was performed to calculate the focusing of surface phonons.
Abstract: Through stationary-phase asymptotic analysis of the elastic Green's function for a semi-infinite anisotropic elastic solid, we calculate the focusing of surface phonons. We present results for surface-phonon focusing on the (100), (110), and (111) surfaces of NaCl, Ge, and Cu. In addition we present results showing that pseudosurface waves will also be focused.
25 citations
TL;DR: In this article, the motion of a particle in a potential decreasing with time is considered, and different time and space rescaling are considered in order to obtain the asymptotic solutions.
Abstract: The motion of a particle in a potential decreasing with time as ‖X‖n is considered. Different time and space rescaling are considered in order to obtain the asymptotic solutions. The validity of adiabatic invariants is discussed. The classical critical case corresponds to the obtainment of self‐similar solutions for the quantum problem.
01 Jan 1983
TL;DR: In this article, a simple asymptotic elastic theory and a verification by the method of reflected caustics were used to determine the boundary of the core region of brittle fracture.
TL;DR: In this article, a simple model which can, in principle, be solved exactly was discussed, and a comparison of the two solutions showed that the matched asymptotic expansion technique determines correctly the final damping rate and frequency (at least to fourth order in a small parameter), but the phase information is spurious.
Abstract: Increasing interest has been shown in the method of matched asymptotic expansions for the construction and solution of the equations of motion in general relativity. This paper discusses a simple model which can, in principle, be solved exactly. A comparison of the two solutions shows that the matched asymptotic expansion technique determines correctly the final damping rate and frequency (at least to fourth order in a small parameter), but the phase information is spurious.
01 Dec 1983
TL;DR: This paper uses companding approximations to derive asymptotic criteria for the evaluation of quantizer performance and the design of optimum quantizers based on statistical measures of divergence at the output of the quantizer.
Abstract: Asymptotic analysis using companding approximations has proven to be a very useful technique for the performance analysis and design of minimum-distortion data quantizers. However, when quantized data is to be used for inferential (e.g., detection or estimation) purposes, the use of distortion-based performance criteria is inappropriate since the measures of quality in such problems are usually based on quantities such as error probability or mean-square estimation error. In this paper we use companding approximations to derive asymptotic criteria for the evaluation of quantizer performance and the design of optimum quantizers based on statistical measures of divergence at the output of the quantizer. Several applications of these results to signal detection and parameter estimation are considered.
TL;DR: Formal asymptotic solutions which are uniform with respect to the parameter are constructed for ordinary differential equations with symbols depending smoothly on the parameter and having a non-degenerate stationary point as discussed by the authors.
Abstract: Formal asymptotic solutions which are uniform with respect to the parameter are constructed for ordinary differential equations with symbols depending smoothly on the parameter and having a nondegenerate stationary point.Bibliography: 10 titles.
TL;DR: This paper deals with the limiting distribution of the level number of the jth leaf of a planted plane tree (where leaves are enumerated from left to right) for the so-called "simply generated families" introduced by Meir and Moon.
Abstract: This paper deals with the limiting distribution of the level number of the jth leaf of a planted plane tree (where leaves are enumerated from left to right) for the so-called “simply generated families” introduced by Meir and Moon. The mathematical apparatus is determined by the idea of an asymptotic analysis of a given sequence of numbers by studying the location and nature of the singularities of appropriate generating functions.
TL;DR: In this article, the summation procedure of the Pade type is applied to the perturbation expansion of the solution of the potential Kortewegde-Vries equation (K.V.), introduced by Rosales.
Abstract: The summation procedure of the Pade type is applied to the perturbation expansion of the solution of the potential Korteweg‐de‐Vries equation (K.d.V.), introduced by Rosales. For the N‐soliton solution without background the [(n−1)/n] Pade approximants are shown to exist for n≤N. Their asymptotic behavior is investigated and it is found that it corresponds to a system of n solitons with the leading velocity parameters. The analogous results for the K.d.V. then follow in agreement with some previous numerical observations.
TL;DR: In this article, the authors derived natural analogies of results by several authors concerning the asymptotic properties of nonlinear contraction semigroups in Hilbert space, where the authors considered the nonlinear Volterra equation.
Abstract: u(0) =x, (1.1) where A is a nonlinear maximal monotone (possibly multi-valued) operator on a real Hilbert space H, f~Z&,([0, ~4); H), x E D(A) and G is a given mapping G:C([O, ~;D(A))-+~‘(O, T;H), VO
01 Dec 1983
TL;DR: In this paper, the theory of large deviations is applied to the study of the asymptotic properties of the stochastic approximation algorithms (1.1 and 1.2).
Abstract: The theory of large deviations is applied to the study of the asymptotic properties of the stochastic approximation algorithms (1.1) and (1.2). The method provides a useful alternative to the currently used technique of obtaining rate of convergence results by studying the sequence {(Xn-?)/?an} (for (1.1)), where ? is a 'stable' point of the algorithm. Let G be a bounded neighborhood of ?, which is in the domain of attraction of ? for the 'limit ODE'. The process xn(?) is defined as a 'natural interpolation' of {Xj,j?n} with xn(0) = Xn, and interpolation intervals {aj,j?n}. Define ?G n = min{t:xn(t)?G}. Then it is shown (among other things) that Px{?G n ? T} ~ exp-nqV, where q depends on {an,cn}, and V depends on the b(?) cov ?n, and G. Such estimates imply that the asymptotic behavior is much better than suggested by the 'local linearization methods', and they yield much new insight into the asymptotic behavior. The technique is applicable to related problems in the asymptotic analysis of recursive algorithms, and requires weaker conditions on the dynamics than do the 'linearization methods'. The necessary basic background is provided and the optimal control problems associated with getting the V above are derived.
TL;DR: In this article, a closed-form solution for the concentrati pore radius and conversion profiles in a long, narrow pore for both cylindrical and slab geometries is presented.
TL;DR: In this article, the generalized anharmonic oscillator in three dimensions described by the potentials of the form ∑2m+1k=1bkr 2k was studied and the energy eigenvalues appeared as the real roots of the finite factor of the Hill determinant; the corresponding wave functions are Gaussian weighted polynomials.
Abstract: We study the generalized anharmonic oscillator in three dimensions described by the potentials of the form ∑2m+1k=1bkr 2k. An asymptotic analysis of the Schrodinger equation yields the leading asymptotic behavior of the energy eigenfunctions in terms of the dominant (m+1) coupling constants bk, m+1≤k≤2m+1. Using an ansatz which incorporates this asymptotic behavior, we reduce the eigenvalue equation to an (m+2)‐term difference equation. The corresponding Hill determinant may be made to factorize with a finite determinant as a factor if a set of constraints on the couplings is satisfied; an infinite sequence of such sets exists. The exact energy eigenvalues appear as the real roots of the finite factor of the Hill determinant; the corresponding wavefunctions are Gaussian weighted polynomials. We consider the potentials ∑31bkr 2k and ∑51bkr 2k explicitly; potentials of the form ∑2m1bjr j and ∑2m1bjr j+δ/r containing both even and odd terms are also considered. Finally, we show that this method of constructi...
01 Oct 1983
TL;DR: In this article, the authors explore the properties of simple search and choice behaviors, by exploiting the asymptotic properties of maxima of sequences of random variables, and show that convergence to a Logit model holds under considerably weaker assumptions than those commonly found in the literature to justify it.
Abstract: The paper explores the properties of some simple search and choice behaviors, by exploiting the asymptotic properties of maxima of sequences of random variables. Heterogeneity in the preference is introduced by means of additive random utilities, and the actor is assumed to choose points in a plane region, by sampling them according to a stochastic process. It is shown that asymptotic convergence to a Logit model holds under considerably weaker assumptions than those commonly found in the literature to justify it. This asymptotic property is treated in details for utility-maximizing behavior, and outlined for satisficing behavior. The asymptotic equivalence of the two behaviors suggests that progress in widening the family of asymptotically Logit-equivalence behaviors can be made with further research.
TL;DR: In this article, a numerical solution of the Navier-Stokes equation and an asymptotic analysis for Laminar steady compressible flow between close rotating thermally conducting axisymmetric disks with inflow was investigated by means of the numerical solution, obtained for small ϵ, E and H (Rossby and Ekman numbers, and height/radius, respectively) is valid for merged, close and separate boundary layers on the disks.