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.     

Rational Control of Conformational Distributions and Mixed‐Valence Characteristics in Diruthenium Complexes

The electronic characteristics of mixed‐valence complexes are often inferred from the shape of the inter‐valence charge transfer (IVCT) band, which usually falls in the near infrared (NIR) region, and relationships derived from Marcus‐Hush theory. These analyses typically assume one single, dominant molecular conformation. The NIR spectra of the prototypical delocalised (Class III Robin–Day mixed‐valence) complexes [{Ru(pp)Cp’}₂(μ‐C≡C?C≡C)]⁺ ([ 1 ]⁺: Cp’=Cp, pp=(PPh₃)₂; [ 2 ]⁺: Cp’=Cp, pp=dppe; [ 3 ]⁺: Cp’=Cp*, pp=dppe) feature a ‘two‐band’ pattern, which complicates band‐shape analysis using these traditional methods. In the past, the appearance of sub‐bands within or near the IVCT transition has been attributed to vibronic effects or localised d‐d transitions. Quantum‐chemical modelling of a series of rotational conformers of [ 1 ]⁺–[ 3 ]⁺ reveals the two components that contribute to the NIR absorption band envelope to be a π‐π* transition and an MLCT transition. The MLCT components only gain appreciable intensity when the orientation of the half‐sandwich ruthenium ligand spheres deviates from idealised cis (Ω P?Ru?Ru?P=0°) or trans (Ω P?Ru?Ru?P=180°) conformations. The increased steric demand of the supporting ligands, together with some underlying inter‐phosphine ligand T‐shaped CH???π stacking interactions across the series [ 1 ]⁺ to [ 2 ]⁺ to [ 3 ]⁺ results in local minima biased towards such non‐idealised conformations of the metal‐ligand fragments (Ω P?Ru?Ru?P=33–153°). Experimentally, this is indicated by appearance of multiple bands within the IR (C≡C) band envelopes and increasing intensity of the higher‐energy MLCT transition(s) relative to the π‐π* transition across the series, and the appearance of a pronounced ‘two‐band’ pattern in the experimental NIR absorption envelopes. These conformational effects and the methods of analysis presented here, which combine analysis of IR and NIR spectra with quantum‐chemical calculations on a range of energetically similar conformational minima, are expected to be quite general for mixed‐valence systems.