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Showing papers on "Asymptotology published in 1973"


Book ChapterDOI
01 Jan 1973
TL;DR: In this paper, the authors discuss the asymptotic behavior of the solution when the parameter tends to a limiting value of its range, where the solution is obtained first in a more or less explicit form, and is then simplified by the appropriate asymPT expansions when the parameters tend to one or both limiting values of its interval.
Abstract: Asymptotic methods play an important role in all branches of applied mathematics in the evaluation of the solutions of problems depending on a parameter with a certain range. Usually the solution is obtained first in a more or less explicit form, and is then simplified by the appropriate asymptotic expansions when the parameter tends to one or both limiting values of its interval. On the other hand, in elasticity one often encounters problems for which no explicit solution is available for arbitrary values of the parameter. Numerical techniques may enable us to obtain accurate numerical solutions for specific values of the parameter. Such numerical methods, however, are often less convenient to discuss the asymptotic behaviour of the solution when the parameter tends to a limiting value of its range.

296 citations


Journal ArticleDOI
TL;DR: Following Kasminski, this paper investigated the asymptotic behavior of solutions of linear time-independent Ito equations and gave a sufficient condition for stability of the zero solution in dimension 2.
Abstract: Following Kasminski, we investigate asymptotic behavior of solutions of linear time-independent Ito equations We first give a sufficient condition for asymptotic stability of the zero solution Then in dimension 2 we determine conditions for spiraling at a linear rate Finally we give applications to the Cauchy problem for the associated parabolic equation by the use of a tauberian theorem Introduction Consider a system of linear, constant coefficients differential equations dx I (0 1) -d=Ebxj x (I 01, he gave a necessary and sufficient condition that r(t) 0 as when t --> He did not examine, however, the behavior of {X(t), t > 01 The first study of the angular behavior in the case 1 2 Received by the editors March 13, 1972, AMS (MOS) subject classifications (1970) Primary 60H10, 60J60; Secondary 34D05, 34A30, 34C05

21 citations


Journal ArticleDOI
TL;DR: In this article, Kuzmak et al. extended the asymptotic method of G. B. Whitham to a more general class of nonlinear second order partial differential equations and obtained asymptic expansions of the solutions.
Abstract: By extending the asymptotic method of G. E. Kuzmak, J. C. Luke has shown how G. B. Whitham’s theory of nonlinear wave propagation can be derived directly from the partial differential equation without using the variational principle, in special cases. We apply the same method to a more general class of nonlinear second order partial differential equations or systems of first order equations containing a small parameter e and obtain asymptotic expansions of the solutions.

20 citations


Journal ArticleDOI
TL;DR: In this article, the authors discuss the importance of the asymptotic distribution of estimators in the context of small sample properties and apply it to their findings on the magnitudes and directions of the "biases" observed in Ordinary Least Squares estimates for their system.
Abstract: The theme of the volume is doggedly operational. In effect, one is encouraged to formulate each Econometric problem precisely as dictated by the requisite Economic Theory, and solve the resultant estimation problem directly, by numerically maximizing the relevant likelihood function, thus eschewing approximations of unknown magnitude and importance. The finite sample behaviour of the adopted estimator can then be simulated to provide evidence on the reliability of the empirical results. In this rather novel methodology, the large body of established theory on the asymptotic distributions of estimators is assigned a limited role and merits discussion in but one chapter. "The small sample problem stems from the fact that asymptotic results provide no clear guide to finite sample distributions" (p. 219), although ". . . one would like to know how large is 'large"' (p. 76). Agreed, and specifically investigating this should surely be an important objective of experimental work on small-sample properties. However, comparisons with asymptotic results materialize only for situations in which the Maximum Likelihood Estimator is believed consistent and asymptotically efficient, even though Goldfeld and Quandt (G-Q below) consider a most commendable number of examples of mis-specified models or incorrect estimator choices. One of the main purposes of this review article is to indicate how valuable asymptotic theory can be in clarifying simulation work by applying it to their findings on the magnitudes and directions of the "biases" observed in Ordinary Least Squares estimates for their system

16 citations



Journal ArticleDOI
TL;DR: The main thrust of this work will be to obtain conditions which ensure that to each bounded solution v of (1) there corresponds a (not necessarily unique) bounded solution u of (2) such that limt~®]u(t)-v(t) I = 0.
Abstract: are asymptotically equivalent, where each F and G is a continuous function from Ro × Y to Y. The major advantage of the present study is that not only is equation (I) not assumed to linear, but no differentiability hypotheses are placed on F. Our hypotheses on F will be such that solutions of (I) are unique and can be continued indefinitely rightward. The main thrust of our work will be to obtain conditions which ensure that to each bounded solution v of (1) there corresponds a (not necessarily unique) bounded solution u of (2) such that limt~®]u(t)-v(t) I = 0. Our results will be such that, if F is linear, they are implied by recent work of P. Talpalaru [15, Th6or6mes 2.1 and 3.1]. The primary tools used here will be the circle of ideas involving the H61der inequality developed by R. Conti [1], V. A. Staikos [14], and Talpalaru [15], and the notion of logarithmic derivative developed by T. Wa~,ewski [16] and S. M. Lozinskii [10], and most recently employed by R. H. Martin, Jr. and the present author [11], [6], [9], [7], and [8]. The reader familiar with [1 I], [6], [9], [7], and [8] will note that in none of those articles was a restriction placed on the dimension of Y. The restriction is necessary here so that we may apply the fixed-point theorem of J. Schauder [13] to Banach spaces of Y-valued functions. The type of Schauder argument we use is similar to that developed by C. Corduneanu [3].

7 citations


01 Jan 1973
TL;DR: In this article, it is shown that for f bounded, locally integrable and positive, it is known that a unique positive solution u of (i.i.I) exists and this solution has been studied in various asymptotic limits.
Abstract: Here ¢ > 0 is a constant and f is a given function. Some background and references related to this physical problem are given in [4]° For f bounded, locally integrable and positive, it is known that a unique positive solution u of (i.I) exists° This solution has been studied in various asymptotic limits [3], [4], [6]° Of particular interest here are the limits t ~ for s fixed and ¢ ~ 0 for all t ~ O. We shall discuss both of these cases.

6 citations


Journal ArticleDOI
TL;DR: In this paper, the authors gave the complete asymptotic expansion of the measure of approximation of the Abel-Poisson integral for functions of the Lipschitz class.
Abstract: We give the complete asymptotic expansion of the measure of approximation of the Abel-Poisson integral for functions of Lipschitz class.

5 citations


Journal ArticleDOI
TL;DR: In this article, asymptotic expansions in powers of the coupling constant λ for self-coupled boson fields with space cut-off polynomial interaction in two space-time dimensions were constructed.
Abstract: We construct the asymptotic expansions in powers of the coupling constant λ for the asymptotic fields and the scattering operatorS for self-coupled boson fields with space cut-off polynomial interaction in two space-time dimensions. These asymptotic expansions are then used to prove thatS*S=SS*=1 in the sense of asymptotic power series inλ on a dense set of states. The results apply also, under the additional assumption of an ultraviolet cut-off, to large classes of boson-boson, fermion-boson and fermion-fermion interactions as well as to boson nonpolynomial interactions (in all space-time dimensions).

3 citations




11 Sep 1973
TL;DR: In this article, an asymptotic expansion for the distribution of M-estimators in the case of scale known, but with different basic assumptions, so as to include some of the M-stimators originally proposed, is presented.
Abstract: : An asymptotic expansion for the distribution of M-estimators in the case of scale known, but with different basic assumptions, so as to include some of the M-estimators originally proposed. Also, a formal expansion for the case of scale unknown. Finally, several numerical results comparing, in part, values obtained with the Monte Carlo results of work done in Princeton.





Journal ArticleDOI
01 Sep 1973
TL;DR: In this article, a method is proposed to derive rigorous asymptotic expansions of the field in both the illumination and shadow of a plane boundary in a general stratified medium with monotonically increasing refractive index.
Abstract: The asymptotic treatment of high-frequency scalar wave problems has in the past been rather unsatisfactory. Typically, the integral representations which arose were evaluated by stationary phase, or as a series of residues. The justification of these methods was usually heuristic and formal. In this paper, a method is advanced which, it is claimed, may be applied to any one-parameter separation of variables problem. The method assumes an integral representation whose contour of integration is the real axis. It is then only necessary to deform this contour in the neighbour-hood of the real axis to derive rigorous asymptotic expansions of the field in both the illumination and shadow. The method is applied to the particular example of scattering by a plane boundary in a general stratified medium with monotonically increasing refractive index.

Journal ArticleDOI
TL;DR: In this paper, a number of asymptotics related to the time parameter for t → ∞ relaxation times, heavy traffic theory, restricted accessibility with large bounds, approximation by diffusion processes, exponential and regular variation of the tail of the waiting time distribution, limit theorems and extreme value theorem.
Abstract: For the GI⧸G⧸1 queueing system a number of asymptotic results are reviewed. Discussed are asymptotics related to the time parameter for t → ∞ relaxation times, heavy traffic theory, restricted accessibility with large bounds, approximation by diffusion processes, exponential and regular variation of the tail of the waiting time distribution, limit theorems and extreme value theorems.


01 Jul 1973
TL;DR: In this paper, a general asymptotic boundary value solution for the N-bodies problem is proposed, which can be used as a basis for formulating a number of analytical two-point boundary value solutions.
Abstract: Previously published asymptotic solutions for lunar and interplanetary trajectories have been modified and combined to formulate a general analytical solution to the problem on N-bodies. The earlier first-order solutions, derived by the method of matched asymptotic expansions, have been extended to second order for the purpose of obtaining increased accuracy. The derivation of the second-order solution is summarized by showing the essential steps, some in functional form. The general asymptotic solution has been used as a basis for formulating a number of analytical two-point boundary value solutions. These include earth-to-moon, one- and two-impulse moon-to-earth, and interplanetary solutions. The results show that the accuracies of the asymptotic solutions range from an order of magnitude better than conic approximations to that of numerical integration itself. Also, since no iterations are required, the asymptotic boundary value solutions are obtained in a fraction of the time required for comparable numerically integrated solutions. The subject of minimizing the second-order error is discussed, and recommendations made for further work directed toward achieving a uniform accuracy in all applications.