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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01 Jan 1993
7 citations
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7 citations
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TL;DR: In this paper, the problem of representing odd integers as the sum of a prime and a power of two is investigated using numerical computations, and the density of representable numbers is calculated up to 231 and the results are extrapolated in order to estimate the asymptotic density.
Abstract: The problem of representing odd integers as the sum of a prime and a power of two is investigated using numerical computations. The density of representable numbers is calculated up to 231 and the results are extrapolated in order to estimate the asymptotic density. A probabilistic model (suggested by Bombieri) is used to get an independent estimate for the asymptotic density. Either approach suggests 0.434... as a reasonable approximation for the asymptotic density.
7 citations
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TL;DR: The mathematical framework of singular analysis is introduced and a novel asymptotic parametrix construction for Hamiltonians of many-particle Coulomb systems is discussed, which corresponds to the construction of an approximate inverse of a Hamiltonian operator with remainder given by a so-called Green operator.
Abstract: The primary motivation for systematic bases in first principles electronic structure simulations is to derive physical and chemical properties of molecules and solids with predetermined accuracy. This requires a detailed understanding of the asymptotic behaviour of many-particle Coulomb systems near coalescence points of particles. Singular analysis provides a convenient framework to study the asymptotic behaviour of wavefunctions near these singularities. In the present work, we want to introduce the mathematical framework of singular analysis and discuss a novel asymptotic parametrix construction for Hamiltonians of many-particle Coulomb systems. This corresponds to the construction of an approximate inverse of a Hamiltonian operator with remainder given by a so-called Green operator. The Green operator encodes essential asymptotic information and we present as our main result an explicit asymptotic formula for this operator. First applications to many-particle models in quantum chemistry are presented in order to demonstrate the feasibility of our approach. The focus is on the asymptotic behaviour of ladder diagrams, which provide the dominant contribution to short-range correlation in coupled cluster theory. Furthermore, we discuss possible consequences of our asymptotic analysis with respect to adaptive wavelet approximation.
7 citations
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TL;DR: In this article, various asymptotic phenomena exhibited by solutions of singularly perturbed Robin boundary value problems are studied in the case when the right-hand side grows faster than the square of the derivative.
7 citations