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.     

Homoleptic, four-coordinate azadipyrromethene complexes of d10 zinc and mercury

Tetraarylazadipyrromethenes, and especially their boron chelates, are a growing class of chromophores that are photoactive toward red light. The coordination chemistry of these ligands remains to be explored. Reported here are four-coordinate zinc(II) and mercury(II) complexes of tetraarylazadipyrromethene ligands. The new complexes contain two azadipyrromethenes bound per d(10) metal center and are characterized by (1)H NMR, optical absorption spectroscopy, X-ray diffraction crystallography, and elemental analysis. Solid-state structures show that these bis-chelate complexes distort significantly from idealized D2d symmetry. AM1 geometry optimizations indicate relaxation energies in the range of 6.8-15.2 kcal mol(-1); interligand pi-stacking provides an added energetic impetus for distortion. The absorption spectra show a marked increase in the absorption intensity in the red region and, in the case of the zinc(II) complexes, the development of a second distinct absorption band in this region, which is red-shifted by ca. 40-50 nm relative to the free ligand. Semiempirical INDO/S computations indicate that these low-energy optical absorptions derive from allowed excitations among ligand-based orbitals that derive from the highest occupied molecular orbital and lowest unoccupied molecular orbital of the free azadipyrromethene.     

Stereoselective conjugate additions of Grignard reagents to cyclopentadienones

Conjugate addition reactions of Grignard reagents with cyclopentadienones having trimethylsilyl groups at the α and α′ positions, and 1-hydroxyethyl substitutents at the β and β′ positions, are reported; excellent stereoselectivity was observed and the relative stereochemistry of three of the products was determined by X-ray methods.