On a variance associated with the distribution of primes in arithmetic progressions

As is usual in prime number theory, write It is well known that when q is close to x the averagevalue of is about xlog q,and recently Friedlander and Goldston have shown that if then the first moment of V(x,q)-U(x,q)is small. In this memoir it is shown that the same is true forall moments. 2000 Mathematics Subject Classification: 11N13.     

X-ray photoelectron,X-ray emission and UV spectra of SiO2 calculated by the SCF Xα scattered wave method

Photoionization, X-ray emission and UV excitation energies are calculated for SiO₂ by the SCF Xα method and the transition-state procedure. In all cases agreement between calculation and experiment is good. The SiO^4?₄ cluster is found to be adequate for describing localized excitations in quartz. The valence orbitals of this cluster are found to be naturally separable into three sets (1) 3t₂ and 4a₁; O 2s, nonbonding: (2) 5a₁ and 4t₂; Si 3s, 3p and O 2p, sigma bonding: (3) 1e, 5t₂, 1t₁; O 2p nonbonding. The participation of Si 3d-type functions in the high-energy O 2p nonbonding orbitals is found to be very small.     

Enumeration of pairs of permutations

Let π = (a₁, a₂, …, a_n), ? = (b₁, b₂, …, b_n) be two permutations of $\text{Z}{}_{\mathrm{n}}\text{= {1, 2, …, n}}$. A rise of π is pair a_i, a_i+1 with a_i < a_i+1; a fall is a pair a_i, a_i+1 with a_i > a_i+1. Thus, for i = 1, 2, …, n ? 1, the two pairs a_i, a_i+1; b_i, b_i+1 are either both rises, both falls, the first a rise and the second a fall or the first a fall and the second a rise. These possibilities are denoted by RR, FF, RF, FR. The paper is concerned with the enumeration of pairs π, p with a given number of RR, FF, RF, FR. In particular if ω_n denotes the number of pairs with RR forbidden, it is proved that $\text{∑}{}^{\mathrm{\infty }}{}_0\text{ω}{}_{\mathrm{n}}\text{z}{}^{\mathrm{n}}\text{n!n!}\text{=}\text{1}\text{?(z)}$, $\text{?(z) = ∑}{}^{\mathrm{\infty }}{}_0\text{(-1)}{}^{\mathrm{n}}\text{z}{}^{\mathrm{n}}\text{n!n!}$. More precisely if ω(n, k) denotes the number of pairs π, p with exactly k occurences of RR(or FF, RF, FR) then $\text{1 + ∑}{}^{\mathrm{\infty }}{}_{\mathrm{n=1}}\text{z}{}^{\mathrm{n}}\text{n!n!}$$\text{∑}{}^{\mathrm{n?1}}{}_{\mathrm{k=0}}\text{ω(n, k)x}{}^{\mathrm{k}}\text{=}\text{(1 ? x)}\text{(?(z(1 ? x)) ? x)}$.     

Applications of enzyme-catalysed reactions in trace analysis-V Determination of zinc and calcium by their activation of the apo-enzyme of calf-intestinal alkaline phosphatase

The activation of the apo-enzyme of calf-intestinal alkaline phosphatase by zinc in the presence and absence of magnesium 4 x 10(-4)M) is applied to the determination of 6-65 ng and 65-650 ng of zinc, respectively. The activation of the apo-enzyme by calcium in the presence of 1.4 x 10(-7)M zinc is used to determine 1-4 mug of calcium. The mechanism of the re-activation effects is discussed.     

Enumeration of pairs of sequences by rises,falls and levels

The paper is concerned with the enumeration of pairs of sequences with given specification according to rises, falls and levels. Thus there are nine possibilities RR, ..., LL. Generating functions in the general case are very complicated. However in a number of special cases simple explicit results are obtained.Supported in part by NSF grant GP-37924X.     

Estimates of internal energies of vaporisation of some room temperature ionic liquids

Systematic variation in the bimolecular rate constant, k(q), for the reaction of singlet oxygen and 1,4-dimethylnaphthalene has led to estimates of the Hildebrand solubility parameter, cohesive pressures and internal energies of vaporisation of some room temperature ionic liquids.