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V. V. Zudilin 《Mathematical Notes》2007,81(3-4):297-301
We prove two new series of Ramanujan type for 1/π2. 相似文献
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For each sN define the constant θs with the following properties: if an entire function g(z) of type t(g)<θs satisfies then g is a polynomial; conversely, for any δ>0 there exists an entire transcendental function g(z) satisfying the display conditin and t(g)<θs+δ. The result θ1=log2 is known due to Hardy and Pólya. We provide the upper bound θsπs/3 and improve earlier lower bounds due to Gelfond (1929) and Selberg (1941). 相似文献
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Derda R Tang SK Li SC Ng S Matochko W Jafari MR 《Molecules (Basel, Switzerland)》2011,16(2):1776-1803
The amplification of phage-displayed libraries is an essential step in the selection of ligands from these libraries. The amplification of libraries, however, decreases their diversity and limits the number of binding clones that a screen can identify. While this decrease might not be a problem for screens against targets with a single binding site (e.g., proteins), it can severely hinder the identification of useful ligands for targets with multiple binding sites (e.g., cells). This review aims to characterize the loss in the diversity of libraries during amplification. Analysis of the peptide sequences obtained in several hundred screens of peptide libraries shows explicitly that there is a significant decrease in library diversity that occurs during the amplification of phage in bacteria. This loss during amplification is not unique to specific libraries: it is observed in many of the phage display systems we have surveyed. The loss in library diversity originates from competition among phage clones in a common pool of bacteria. Based on growth data from the literature and models of phage growth, we show that this competition originates from growth rate differences of only a few percent for different phage clones. We summarize the findings using a simple two-dimensional "phage phase diagram", which describes how the collapse of libraries, due to panning and amplification, leads to the identification of only a subset of the available ligands. This review also highlights techniques that allow elimination of amplification-induced losses of diversity, and how these techniques can be used to improve phage-display selection and enable the identification of novel ligands. 相似文献
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Wadim Zudilin 《Journal of Number Theory》2009,129(8):1848-1857
We present several supercongruences that may be viewed as p-adic analogues of Ramanujan-type series for 1/π and 1/π2, and prove three of these examples. 相似文献
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The aim of the paper is to relate computational and arithmetic questions about Euler’s constant γ with properties of the values
of the q-logarithm function, with natural choice of~q. By these means, we generalize a classical formula for γ due to Ramanujan, together with Vacca’s and Gosper’s series for
γ, as well as deduce irrationality criteria and tests and new asymptotic formulas for computing Euler’s constant. The main
tools are Euler-type integrals and hypergeometric series.
2000 Mathematics Subject Classification Primary—11Y60; Secondary—11J72, 33C20, 33D15
The work of the second author is supported by an Alexander von Humboldt research fellowship
Dedication: To Leonhard Euler on his 300th birthday. 相似文献
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We propose hypergeometric constructions of simultaneous approximations to polylogarithms. These approximations suit for computing the values of polylogarithms and satisfy 4-term Apéry-like (polynomial) recursions. 相似文献
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Mathematical Notes - 相似文献
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