Topic
Asymptotology
About: Asymptotology is a research topic. Over the lifetime, 1319 publications have been published within this topic receiving 35831 citations.
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TL;DR: This tutorial paper illustrates the principle of convergence to the asymptotic results is so fast that even for moderate n they yield results close to the true values by applying it to capacity calculations of multiple-antenna systems.
Abstract: Asymptotic theorems are very commonly used in probability. For systems whose performance depends on a set of n random parameters, asymptotic analyses for n → ∞ are often used to simplify calculations and obtain results yielding useful hints at the behavior of the system for finite n. These asymptotic analyses are especially useful whenever the convergence to the asymptotic results is so fast that even for moderate n they yield results close to the true values. This tutorial paper illustrates this principle by applying it to capacity calculations of multiple-antenna systems.
18 citations
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TL;DR: In this paper, the authors investigated the relation between the moments and the asymptotic behavior of solutions to the Burgers equation and showed that the convergence order increases by a similarity scale whenever the order of controlled moments is increased by one.
18 citations
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TL;DR: In this article, uniform asymptotic expansion of the Schwartz kernels of the Bremmer coupling series solution of the wave equation in inhomogeneous media is derived in the high frequency limit, taking into account critical scattering angle phenomena.
Abstract: The Bremmer coupling series solution of the wave equation, in generally inhomogeneous media, requires the introduction of pseudodifferential operators. In this paper, in two dimensions, uniform asymptotic expansions of the Schwartz kernels of these operators are derived. Also, we derive a uniform asymptotic expansion of the one-way propagator appearing in the series. We focus on designing closed-form representations, valid in the high-frequency limit, taking into account critical scattering-angle phenomena. Our expansion is not limited by propagation angle. In principle, the uniform asymptotic expansion of a kernel follows by matching its asymptotic behaviors away from and near its diagonal. The Bremmer series solver consists of three steps: directional decomposition into up- and downgoing waves, one-way propagation, and interaction of the counter-propagating constituents. Each of these steps is represented here by a kernel for which a uniform asymptotic expansion is found. The associated algorithm provid...
18 citations
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22 Jul 1991TL;DR: The asymptotic volume of Hamming spheres and Lee spheres for small alphabets is derived, and an asymPTotic relation between the covering radius and the dual distance of binary codes is derived.
Abstract: We use asymptotic estimates on coefficients of generating functions to derive anew the asymptotic behaviour of the volume of Hamming spheres and Lee spheres for small alphabets We then derive the asymptotic volume of Lee spheres for large alphabets, and an asymptotic relation between the covering radius and the dual distance of binary codes
17 citations
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TL;DR: In this article, it was shown that a divergent Taylor series for analytic functions at an isolated singularity does not represent the function in a full deleted neighborhood, but only in certain sectors.
Abstract: The principle of analytic continuation makes it possible to calculate effectively the corresponding convergent power series about all points where a holomorphic continuation exists. However, the nature of the function near its singularities cannot be so readily deduced from the series (1.1). Often series expansions about such singular points do exist, and sometimes it is possible to calculate them explicitly from the coefficients of the convergent expansions about a regular point. These expansions may even be power series. Nevertheless, they differ from the familiar convergent Taylor series in several decisive respects. The most important new feature is that they represent the function ƒ only in an asymptotic sense. To explain this concept, let us assume, for simplicity, that the singularity occurs at z = oo. To say that the function ƒ is asymptotically represented by the series 23r^o rZ~, in symbols ƒ(z)~ 2^r°l0 £r* ', as z—> oo, means that, for all N, the error committed in replacing ƒ (z) by the sum of the first N terms of the power series is 0(z~), as z—> oo. Such a series may well be divergent, in fact, it usually is. If so, another important feature enters the picture: A divergent asymptotic series for an analytic function at an isolated singularity never represents the function in a full deleted neighborhood, but only in certain sectors. There exists a substantial body of theory for the \"connection problem\" just described, namely the problem of finding asymptotic expansions about a singular point from a given convergent expansion for the same function about a regular point. I shall not say much
17 citations