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Topic: Asymptotic expansion

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Exponential function

	Asymptotic analysis - Wikipedia, the free encyclopedia (Site not responding. Last check: 2007-10-09)
		The asymptotic point of view is basic in computer science, where the question is typically how to describe the resource implication of scaling-up the size of a computational problem, beyond the 'toy' level.
		An asymptotic expansion of a function f(x) is in practice an expression of that function in terms of an infinite series, the partial sums of which do not (necessarily have to) converge; but such that taking any initial partial sum provides an asymptotic formula for f.
		Asymptotic expansions typically arise in the approximation of certain integrals (saddle-point method, method of steepest descent) or in the approximation of probability distributions (Edgeworth series).
	en.wikipedia.org /wiki/Asymptotic_analysis (353 words)

	Landau notation - Wikipedia, the free encyclopedia
		More exactly, it is used to describe asymptotic upper bounds for the magnitude of a function in terms of another, usually simpler, function.
		Asymptotic expansion: application to approximation of functions, generalizing the idea of Taylor's formula.
		Main applications of Landau notations are found in complexity theory and asymptotic analysis.
	www.wikipedia.com /wiki/Landau%2Bsymbol (708 words)

	Asymptotic expansion - Wikipedia, the free encyclopedia
		In mathematics an asymptotic expansion, asymptotic series or Poincaré expansion is a formal series of functions which has the property that truncating the series after a finite number of terms provides an approximation to a given function as the argument of the function tends towards a particular, often infinite, point.
		The most common type of asymptotic expansion is a power series in either positive or negative terms.
		Methods of generating such expansions include the Euler-Maclaurin summation formula and integral transforms such as the Laplace and Mellin transforms.
	en.wikipedia.org /wiki/Asymptotic_expansion (282 words)

	Untitled (Site not responding. Last check: 2007-10-09)
		Uniform asymptotic expansion for the distribution applicable to both stationary and nonstationary cases is obtained.
		Unfortunately, most asymptotic expansions are nonconvergent, with the magnitude of successive terms tracking a J curve of initial decline followed by a steep rise.
		An uniform asymptotic expansion for the distribution of the least squares estimator for the autoregressive coefficient is derived.
	econwpa.wustl.edu:8089 /eps/em/papers/9604/9604001.html (4552 words)

	[No title] (Site not responding. Last check: 2007-10-09)
		(2) The behavior of the integrand at infinity determines whether the expansion of the Green function is genuinely asymptotic in the literal, pointwise sense, or is merely valid in a distributional (Ces\`aro-averaged) sense; this is the difference between the heat kernel and the Schr\"odinger kernel.
		In Section~\ref{sec5} we show that the moment asymptotic expansion, which is the basic building block in the asymptotic expansion of series and integrals,\cite{EK94} can be generalized to distributions, giving expansions that hold in an ``averaged'' or distributional sense explaining, for instance, the small-$t$ behavior of the Schr\"odinger propagator.
		The expansion of $G(t)$ as $t \rightarrow 0^{+}$ will be a distributional or ``averaged'' expansion, in general, but when $g$ has the behavior of the elements of ${\cal K}$ at $\infty$ it becomes a pointwise expansion.
	www.ma.utexas.edu /mp_arc/papers/97-551 (8980 words)

	CMUC
		In this paper we apply the asymptotic expansion method to obtain a nonlinear adaptive elastic rod model.
		We first consider the model derived in [2, 3] with the modifications proposed in [5], with a remodeling rate equation depending nonlinearly on the strain field and for a thin rod whose cross section is a function of a small parameter.
		Based on the asymptotic expansion method for the elastic case [6], we prove that, when the small parameter tends to zero the solution of the nonlinear adaptive elastic rod model converges to the leading term of its asymptotic expansion.
	www.mat.uc.pt /~cmuc/pubdetails.php?pub=779&lid=2 (153 words)

	Asymptotic expansion of certain n-dimensional Laplace integrals whose phase function has a degenerate critical point (Site not responding. Last check: 2007-10-09)
		In this paper, an asymptotic solution is derived for the case where the phase function is a polynomial in x, and the Hessian matrix is singular at the critical points which are therefore degenerate.
		This expansion is arrived at by exploiting certain geometrical relationships between the phase function and the coefficient function $/varphi(x)$.
		Expansions will be obtained under certain geometrical constraints for the 2-dimensional case, and under slightly more restricted conditions for the n-dimensional case.
	digitalcommons.stevens.edu /dissertations/AAI9903543 (163 words)

	Uniform Asymptotic Expansion of the Green's Function for the Two-dimensional Acoustic Equation , by Mathew J. Yedlin (Site not responding. Last check: 2007-10-09)
		Uniform Asymptotic Expansion of the Green's Function for the Two-dimensional Acoustic Equation, by Mathew J. Yedlin
		A uniform asymptotic expansion in the frequency domain is derived for the Green's function of the two-dimensional acousit equation.
		The expansion is uniform in that it is valid near the source region.
	sepwww.stanford.edu /oldreports/sep25/25_15_abs.html (124 words)

	The topological asymptotic (Site not responding. Last check: 2007-10-09)
		The topological expansion of $j$ is straightforwardly deduced from the asymptotic analysis of $T_{\varepsilon}$.
		Asymptotic expansions can be derived for other shapes than circular holes.
		At the first iteration, the two waveguides are not connected, and the topological asymptotic is computed in the free-space rectangular region shown in figure 1-a.
	mip.ups-tlse.fr /equipes/control/topological.html (776 words)

	Asymptotic Expansion For the Poisson Distribution (Site not responding. Last check: 2007-10-09)
		The purpose of this document is to develop an asymptotic expansion for the cumulative Poisson distribution function.
		The asymptotic expansion shows the connection between the cumulative Poisson and Normal distribution functions.
		This particular derivation of an asymptotic expansion for the Poisson distribution is based on Rob Johnson's asymptotic expansion for Gamma(x+1,x)/Gamma(x+1).
	members.aol.com /iandjmsmith/PoissonApprox.htm (220 words)

	asympt -- compute an asymptotic series expansion (Site not responding. Last check: 2007-10-09)
		The number of terms is counted from the lowest degree term on for finite expansion points, and from the highest degree term on for expansions around infinity, i.e., ``
		The actual number of terms in the resulting series expansion may differ from the requested number of terms.
		A different expansion is valid ``to the left'' of the expansion point:
	www.sciface.com /STATIC/DOC25/eng/stdlib/asympt.shtml (474 words)

	Asymptotic expansion of Fourier integrals via the method of stationary phase (Site not responding. Last check: 2007-10-09)
		One particularly useful method is the so-called method of stationary phase, a technique that may be used to develop an asymptotic expansion.
		In this paper, two asymptotic expansions are derived for $I w$ for the case where the region of integration is (a, b) = ( $-∞,∞$ ).
		Each expansion is derived using the properties and theories of distributions along with those of Fourier transform analysis.
	digitalcommons.stevens.edu /dissertations/AAI3088828 (325 words)

	Asymptotic expansion -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-09)
		In (A science (or group of related sciences) dealing with the logic of quantity and shape and arrangement) mathematics an asymptotic expansion, (Click link for more info and facts about asymptotic series) asymptotic series or (Click link for more info and facts about Poincaré expansion) Poincaré expansion is a formal series of functions
		See (Click link for more info and facts about asymptotic analysis) asymptotic analysis and (Click link for more info and facts about big O notation) big O notation for the notation.
		This expansion is valid for all complex s and is often used to compute the zeta function by using
	www.absoluteastronomy.com /encyclopedia/a/as/asymptotic_expansion.htm (461 words)

	A Symbolic Algorithm for Studying Adjustments to the Profile Likelihood -- from Mathematica Information Center
		A symbolic algorithm for the derivation of the asymptotic expansion of the maximum likelihood estimate was first introduced by Andrews & Stafford (1993).
		This algorithm is quite general and can be modified to give asymptotic expansions for the maximum likelihood estimate based on either the conditional, modified or adjusted profile likelihoods.
		Bartlett corrections of each adjusted likelihood, and asymptotic expansions for the first four moments of the maximum likelihood estimate based on the conditional and adjusted profile likelihoods, are derived.
	library.wolfram.com /infocenter/Articles/3749 (145 words)

	Asymptotics of Implicit Functions and Computer Algebra
		If the function admits a power series expansion near the point of interest, then Lagrange's inversion theorem can be used, or Brent and Kung's fast algorithm for power series inversion (see section 2).
		In this section, we present an algorithm to compute a ``nested expansion'' for the asymptotic functional inverse of f.
		Such a nested expansion often (although not always) yields an asymptotic expansion of the function.
	pauillac.inria.fr /algo/seminars/sem96-97/salvy.html (1113 words)

	A Heavy-Traffic Expansion For Asymptotic Decay Rates Of Tail Probabilities In Multi-Channel Queues - Abate, Whitt ... (Site not responding. Last check: 2007-10-09)
		0.9: Heavy-Traffic Asymptotic Expansions For The Asymptotic Decay..
		9 the asymptotic behavior of the stationary distribution of Ma..
		3 Heavy-traffic asymptotic expansions for the asymptotic decay..
	citeseer.ist.psu.edu /490209.html (617 words)

	[No title] (Site not responding. Last check: 2007-10-09)
		Then the first term of the asymptotic expansion of $R^\s(\xi)$ is $R^\infty(\xi)=e\tilde{R}^\infty(\xi)r$, where $e:L^2(\U)\ds L^2(\OO)$ is the extension by zero in $\O$ and $r:L^2(\OO)\ds L^2(\U)$ is the restriction operator.
		The asymptotic expansions of the resolvent are given in Section 3.
		The second difficulty is that $Q^\s$ is not self-adjoint and the existence of an asymptotic expansion of the operator at any order does not imply the existence of the corresponding asymptotic expansion for the eigenvalues.
	www.ma.utexas.edu /mp_arc/papers/02-318 (4744 words)

	Abstract for Cowles Foundation Discussion Paper 1374 (Site not responding. Last check: 2007-10-09)
		For the p = 1 case, an explicit second order asymptotic expansion is found for a quadratic functional of the autocovariance sequences of stationary long memory time series.
		The order of magnitude of the second term in this expansion is shown to depend on the long memory parameters.
		It is demonstrated that the pole in the first order approximation is removed by the second order term, which provides a substantially improved approximation to the original functional.
	www.econ.yale.edu /cowles/P/ab/a13/a1374.htm (208 words)

	The topological asymptotic expansion for the Dirichlet problem (Site not responding. Last check: 2007-10-09)
		The topological sensitivity analysis provides an asymptotic expansion of a shape function with respect to the insertion of a small hole inside a domain.
		This expansion can then be used for shape optimization.
		In this paper, such an expansion is obtained for the Poisson equation with general shape functions and arbitrary shaped holes.
	mip.ups-tlse.fr /publi/rapp01/01.02.html (69 words)

	Asymptotic Expansion For the Binomial Distribution (Site not responding. Last check: 2007-10-09)
		The purpose of this document is to develop an asymptotic expansion for the cumulative Binomial distribution function.
		The asymptotic expansion shows the connection between the cumulative Binomial and Student's t-distribution functions.
		If we use the first 14 terms of this asymptotic expansion, the relative error is 5e-14 or less when i and n-i are > 50 and the mean is between 0.9i and 1.1i.
	members.aol.com /iandjmsmith/BinomialApprox.htm (184 words)

	Citebase - The strategy of regions for asymptotic expansion of two-loop vertex Feynman diagrams
		Authors: Smirnov, V. Rakhmetov, E. General prescriptions for evaluation of coefficients at arbitrary powers and logarithms in the asymptotic expansion of Feynman diagrams in the Sudakov limit are discussed and illustrated by two-loop examples.
		As in the case of previously known prescriptions for various limits of momenta and masses, the terms of the threshold expansion are associated with subgraphs of a given graph and a...
		Problems that arise in the application of general prescriptions of the so-called strategy of regions for asymptotic expansions of Feynman integrals in various limits of momenta and masses are discussed with the help of characteristic examples of two-loop diagrams.
	citebase.eprints.org /cgi-bin/citations?archiveID=oai:arXiv.org:hep-ph/9812529 (973 words)

	Asymptotics of Linear Recurrences with Rational Coefficients
		As already mentioned, the asymptotic expansion is governed by the successive ``layers'' of singularities of the rational function, sorted by increasing moduli.
		Note that, because of the trigonometric functions involved in the coefficients, the expansion (5) is not of Poincaré type.
		Tasks 1 and 2 have been solved, and we are now able to give the asymptotic expansion of the coefficients of a rational function by Algorithm 2.
	pauillac.inria.fr /algo/papers/html/GoSa96/GoSa96.html (4271 words)

	Asymptotic Expansion Techniques and its Application to the Nonlinear Circuits Analysis (Site not responding. Last check: 2007-10-09)
		The criterion of the applicability of asymptotics expansion (8) and also the criterion of the conversion efficiency is
		The asymptotic expansion of the (10) consists of the sum of contributions of maxima of
		The theorem about extra parameters which can be found in [3] states that the asymptotic expansions we use are uniform over the domains of these parameters, where the topology of the maxima of
	atm563.phys.msu.su /ilyushin/ieee/IEE.htm (1037 words)

	Hyperasymptotics -- from Mathematica Information Center
		We develop a technique for systematically reducing the exponentially small (superasymptotic) remainder of an asymptotic expansion truncated near its least term, for solutions of ordinary differential equations of Schrödinger type where one transition point dominates.
		This is achieved by repeatedly applying Borel summation to a resurgence formula discovered by Dingle, relating the late to the early terms of the original expansion.
		Each such 'hyperseries' involves the terms of the original asymptotic series for the particular function being approximated, together with terminating integrals that are universal in form, and is half the length of its predecessor.
	library.wolfram.com /infocenter/Articles/3153 (169 words)

	An asymptotic expansion for the expectation of an age-dependent branching process with a submultiplicative estimate of ...
		An asymptotic expansion for the expectation of an age-dependent branching process with a submultiplicative estimate of the remainder, M. Sgibnev
		An asymptotic expansion for the expectation of an age-dependent branching process with a submultiplicative estimate of the remainder
		An asymptotic expansion for the expected number, μ(t), of particles of an age-dependent branching process is obtained with a general submultiplicative estimate for the remainder term.
	projecteuclid.org /Dienst/UI/1.0/Display/euclid.jap/1005091046 (187 words)

	The Topological Asymptotic Expansion for the Dirichlet Problem
		In this paper, such an expansion is obtained for the Poisson equation for a large class of cost functions and arbitrarily shaped holes.
		In contrast, the asymptotic expansion is independent of the shape in the two-dimensional case.
		A numerical example illustrates the use of the asymptotic expansion, which yields a minimizing sequence of classical domains in a case where no classical solution exists.
	epubs.siam.org /sam-bin/dbq/article/38419 (233 words)

	Asymptotic Expansion of Central Limit Theorem
		Essentially, the method is to build an asymptotic expansion about a gaussian kernel and relies upon the relations of cumulative moments to the CLT.
		In the following we lay the ground work and then derive the expansion; and then follow up (section 7) with some observations in the frequency domain relating to the issue ``how large does n have to be for the CLT to apply?'' But first an overview of the terrain.
		We have repeated the discoveries of Tchebychev-Hermite and constructed an asymptotic expansion that allows approximate computation of multiple self-convolution integrals merely from a knowledge of the moments of a density function.
	www.triumf.ca /people/baartman/dl/shane/synkro-rad3.html (4507 words)

	[No title] (Site not responding. Last check: 2007-10-09)
		asympt is used to compute an asymptotic expansion of f when x tends to x0.
		If asympt cannot compute an asymptotic expansion, then a symbolic expression of type "asympt" is returned.
		The number of terms is counted from the lowest degree term on for finite expansion points, and from the highest degree term on for expansions around infinity, i.e., “order” has to be regarded as a “relative truncation order”.
	www.sciface.com /STATIC/DOC30/eng/stdlib_asympt.html (585 words)

	Applied Asymptotic Expansions in Momenta and Masses (Springer Tracts in Modern Physics) by Vladimir A. Smirnov [ISBN: ... (Site not responding. Last check: 2007-10-09)
		The book presents asymptotic expansions of Feynman integrals in various limits of momenta and masses, and their applications to problems of physical interest.
		The problem of expansion is systematically solved by formulating universal prescriptions that express terms of the expansion using the original Feynman integral with its integrand expanded into a Taylor series in appropriate momenta and masses.
		Knowledge of the structure of the asymptotic expansion at the diagrammatic level is key in understanding how to perform expansions at the operator level.
	www.gettextbooks.com /isbn_3540423346.html (142 words)

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