Topic
Remainder
About: Remainder is a research topic. Over the lifetime, 3443 publications have been published within this topic receiving 35379 citations. The topic is also known as: dividend mod divisor.
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TL;DR: In this article, Narens et al. studied the scale type of concatenation structures and showed that concatenations are all isomorphic to numerical ones for which the operation can be written x∘y = yf(x y ), where f is strictly increasing and f(x) x is strictly decreasing (unit structures).
178 citations
TL;DR: In this article, the classical Hardy inequalities with optimal constants in the Sobolev spaces W 1,p 0 and in higher-order Soboleve spaces on a bounded domain Ω ⊂ R can be refined by adding remainder terms which involve L p norms.
Abstract: We show that the classical Hardy inequalities with optimal constants in the Sobolev spaces W 1,p 0 and in higher-order Sobolev spaces on a bounded domain Ω ⊂ R can be refined by adding remainder terms which involve L p norms. In the higher-order case further L p norms with lower-order singular weights arise. The case 1 < p < 2 being more involved requires a different technique and is developed only in the space W 1,p 0.
173 citations
170 citations
TL;DR: In this article, a modified static remainder approximation is proposed to reduce the computation time of the summation over empty states by using a modified version of the static remainder algorithm, which yields accurate self-energies for both bulk and molecular systems requiring a small fraction of the typical number of empty orbitals.
Abstract: Ab initio GW calculations are a standard method for computing the spectroscopic properties of many materials. The most computationally expensive part in conventional implementations of the method is the generation and summation over the large number of empty orbitals required to converge the electron self-energy. We propose a scheme to reduce the summation over empty states by the use of a modified static remainder approximation, which is simple to implement and yields accurate self-energies for both bulk and molecular systems requiring a small fraction of the typical number of empty orbitals.
157 citations
TL;DR: The way the author has proceeded in order to prove these results is a rather complicated and involved one, and this by necessity since a proof by induction after v starting from the case v = 1, presents overwhelming difficulties in keeping track of the estimates of the remainder terms in their dependence of the two parameters v and x as mentioned in this paper.
Abstract: While the results of Sathe's paper [J. Indian Math. Soc. 17 (1953), 63-141; 18 (1954), 27-81] are very beautiful and highly interesting, the way the author has proceeded in order to prove these results is a rather complicated and involved one, and this by necessity since a proof by induction after v starting from the case v = 1, presents overwhelming difficulties in keeping track of the estimates of the remainder terms in their dependence of the two parameters v and x.
149 citations