Simple Zeros of the Riemann Zeta-Function

Assuming the Riemann Hypothesis, Montgomery showed by meansof his pair correlation method that at least two-thirds of thezeros of Riemann's zeta-function are simple. Later he and Taylorimproved this to 67.25 percent and, more recently, Cheer andGoldston increased the percentage to 67.2753. Here we proveby a new method that if the Riemann and Generalized LindelöofHypotheses hold, then at least 70.3704 percent of the zerosare simple and at least 84.5679 percent are distinct. Our methoduses mean value estimates for various functions defined by Dirichletseries sampled at the zeros of the Riemann zeta-function. 1991Mathematics Subject Classification: 11M26.     

Existence and Uniqueness of Fixed Points for Markov Operators and Markov Processes

This paper concerns a Markov operator T on a space L₁, and aMarkov process P which defines a Markov operator on a spaceM of finite signed measures. For T, the paper presents necessaryand sufficient conditions for:

a the existence of invariant probabilitydensities (IPDs)

b the existence of strictly positive IPDs,and

c the existence and uniqueness of IPDs.

Similar resultson invariant probability measures for P are presented. The basicapproach is to pose a fixed-point problem as the problem ofsolving a certain linear equation in a suitable Banach space,and then obtain necessary and sufficient conditions for thisequation to have a solution. 1991 Mathematics Subject Classification:60J05, 47B65, 47N30.     

库仑加线性位基态解析解

应用新发展的单一轨迹积分方法求解库仑加线性位的基态量子波函数,得到基态能量和波函数的一般解析表达式,并讨论了解的收敛性.应用此方法讨论了重夸克偶素系统.     

Study of small chlorine‐doped potassium clusters by thermal ionization mass spectrometry

The theoretical calculations have predicted that nonmetal‐doped potassium clusters can be used in the synthesis of a new class of charge‐transfer salts which can be considered as potential building blocks for the assembly of novel nanostructured material. In this work, K_nCl (n = 2–6) and K_nCl_n?1 (n = 3 and 4) clusters were produced by vaporization of a solid potassium chloride salt in a thermal ionization mass spectrometry. The ionization energies (IEs) were measured, and found to be 3.64 ± 0.20 eV for K₂Cl, 3.67 ± 0.20 eV for K₃Cl, 3.62 ± 0.20 eV for K₄Cl, 3.57 ± 0.20 eV for K₅Cl, 3.69 ± 0.20 eV for K₆Cl, 3.71 ± 0.20 eV for K₃Cl₂ and 3.72 ± 0.20 eV for K₄Cl₃. The K_nCl⁺ (n = 3–6) clusters were detected for the first time in a cluster beam generated by the thermal ionization source of modified design. Also, this work is the first to report experimentally obtained values of IEs for K_nCl⁺ (n = 3–6) and K_nCl_n?1⁺ (n = 3 and 4) clusters. The ionization energies for K_nCl⁺ and K_nCl_n?1⁺ clusters are much lower than the 4.34 eV of the potassium atom; hence, these clusters should be classified as ‘superalkali’ species. Copyright © 2012 John Wiley & Sons, Ltd.