The Size of a Political Club

We investigate the size of a political club that admits new members according to a rule that if only a few members vote against admission the candidate joins, but the members who voted against admission leave. For the simplest ergodic case, we use a modified Bessel function to give the generating function for the invariant distribution of the club size. This also allows us to give asymptotics for the moments as the probability of voting for admission approaches 1 and to prove global and local central limit theorems. We also calculate the mean recurrent time and determine formulas for calculating first passage times. We then show that for an expanded set of cases the club size process approaches the Ornstein-Uhlenbeck process as the probability of voting for admission approaches 1. This means that the club size will approach a Gaussian distribution for a wide set of formal and informal admission rules. Finally, we show a comparison between theoretical results and simulation runs for two cases.     

Nuclear spin relaxation and magnetic shielding tensors of atoms constituting cyanogen bromide molecule

The temperature dependence of the correlation time describing reorientation kinetics of cyanogen bromide in CDCl₃ solution has been determined on the basis of the linewidths of the ¹⁴N NMR signal. It has been found that the longitudinal spin relaxation of the ¹⁵N nucleus occurs by shielding anisotropy and spin-rotation mechanisms, whereas for the ¹³C nucleus these mechanisms are of lesser importance. In the latter case the scalar relaxation of the second kind due to carbon-bromine coupling is the predominant relaxation mechanism. The parameter values: ¹ J(¹³C—⁷⁹Br) = 349 ± 10 Hz, T ₁ (⁷⁹Br, 303 K) = 2.31 ± 0.22 × 10^?7 s, Δσ(¹⁵N) = 565 ± 16 ppm and Δσ(¹³C) = 276 ± 120 ppm have been determined from the relaxation data analysis. The shielding anisotropy parameters Δσ(¹⁵N) = 580 ± 50 ppm and Δσ(¹³C) = 274 ± 9 ppm have been independently determined using ¹³C and ¹⁵N NMR in liquid crystalline solvent. The experimentally determined shielding tensors for sp-hybridized atoms in the investigated compound and in a series of bromoacetylenes have been compared with the results of quantum mechanical calculations [GIAO, DFT B3LYP/6-311 + +G(2d,p)]. The ‘heavy atom effect’ shielding bromine-bonded carbons is of the order of — 25 ppm and concerns mainly the σ⊥ component.     

Lekner summation of dipolar interaction in quasi-two-dimensional simulations

The Lekner method for calculation of electrostatic interactions in periodically replicated simulation cells is extended to quasi-two-dimensional systems of particles with dipolar interactions. The electric field, potential energy, forces and torques are expressed through rapidly converging series of modified Bessel functions. The method contains no arbitrary parameters, and has no limitations on the simulation box width.     

The energetics of naphthalene derivatives. II. The isomeric 1— and 2—naphthyl acetates

The isomeric 1- and 2-naphthyl acetates (acetoxynaphthalenes) are at the confluence of diverse concepts, techniques and classes of organic compounds. Summing the results of literature measurements of the enthalpy of formation of their solids and of our new sublimation enthalpies reported herein, we derive gas phase enthalpies of formation of ?209.9 ± 1.4 and ?213.3 ± 1.3 kJ mol^?1 respectively. This corresponds to 2-naphthyl acetate being more stable than its 1-isomer by 3.4 ± 1.9 kJ mol^?1. We also performed MP2(full)/6-31G(d) calculations on these species, resulting in enthalpies of formation of ?212.9 ± 3.9 and ?212.2 ± 3.9 kJ mol^?1 for 1- and 2-naphthyl acetate and a difference of ?0.7 kJ mol^?1 respectively in satisfactory agreement with the above experimental results.     

小角度范围内弱束缚核17F弹性散射微分截面的奇异行为

在兰州放射性束流线(RIBLL)上完成了¹⁷F/¹⁷O+²⁰⁸Pb的弹性散射微分截面角分布测量. 分析了¹⁷F/¹⁷O弹性散射产物微分截面的对数(ln(dσ/dθ))随散射角平方(θ²)的依赖关系. 结果表明, 在所测量的角度范围内(6°—20°), ¹⁷O的这一依赖关系可以用一条直线很好地拟合, 而¹⁷F的这一依赖关系需要两条不同斜率的直线才能拟合. ¹⁷F数据拟合中的这种斜率改变可能起因于¹⁷F的奇异结构. 对其他实验组数据的分析支持以上的结论, 即在一定的角度范围内, 弱束缚核与稳定核相比, 弹性散射产物微分截面的对数与散射角平方的依赖关系有明显的差异, 这可以作为深入研究“晕核”和“皮核”的一个新探针.     

Magnetic shielding tensors of carbon nuclei in 3,5-dichlorophenylacetylene,a theoretical ab initio and experimental study

Ab initio GIAO-CHF and DFT(B3LYP) molecular geometry optimization and magnetic shielding tensor calculations of carbon nuclei of 3,5-dichlorophenylacetylene have been performed using 6–311G?? and 6–311+G(2d,p) basis sets. The isotropic ¹³C chemical shifts, needed for comparison, have been measured in C₆D₁₂ solution. The principal elements of the shielding tensor of the carbon nuclei in the investigated molecule in the solid state have been determined from an intensity analysis of the spinning sidebands in ¹H-¹³C CP/MAS NMR spectra. Shielding anisotropy parameters of the acetylenic carbons have been independently determined using the method based on proton-coupled ¹³C nematic phase spectra as well as from the interpretation of the ¹³C longitudinal relaxation rates. The latter data have been analysed assuming the molecular reorientation to be the rotational diffusion of an asymmetrical top, which has provided, apart from the diffusion coefficients, an additional check on the reliability of the theoretical calculation of the shielding tensors. In general, satisfactory agreement between the theoretical and experimental results has been achieved.