Asymptotic expansions for renewal measures in the plane

Summary Let P be a distribution in the plane and define the renewal measure R=P ^*nwhere * denotes convolution. The main results of this paper are three term asymptotic expansions for R far from the origin. As an application, expansions are obtained for distributions in linear boundary crossing problems.Research supported by NSF grants MCS-8102080 and DMS-8504708     

Freezing temperatures and densities of solutions of some hydrocarbons and sodium oleate inN-methylacetamide. Evidence for a solvophobic interaction

The freezing temperatures and densities (at 31°C) of solutions of octane, nonane, decane, 3,3-diethylpentane, and sodium oleate inN-methylacetamide (NMA) have been measured. The molality of the freezing solution was calculated from the density. The solubilities of octane, nonane, and decane inN-methylacetamide are also reported. Apparent molal volumes calculated from the densities are close to the values in the pure hydrocarbons and are not strong functions of the concentration. This indicates the absence of any unusual packing effect. The calculated free energies of transfer of the hydrocarbons from pure hydrocarbon to NMA solution are much less negative than the corresponding values for water, showing that the bulk solvophobic interaction inN-methylacetamide is smaller than in water. This is consistent with the freezing temperatures of sodium oleate which show that micelles do not form below 0.1 mole-kg^–1. The osmotic coefficients of the hydrocarbons calculated from the freezing temperatures showed negative deviations from ideality that were larger for the hydrocarbons with the higher molecular weights. Two estimates of the pairwise solvophobic interaction inN-methylacetamide indicate that it is also smaller than in water. The solvophobic effect in this solvent does not include the large entropy and enthalpy effects found in aqueous solutions.     

Sequence distribution of partially hydrolyzed poly(vinyl acetate)

Copolymers of vinyl acetate and vinyl alcohol were studied by differential thermal analysis. The melting points of the copolymers are not a simple function of the composition, but depend on the method of preparation of the copolymers. Partial saponification of poly(vinyl acetate) with sodium hydroxide leads to high melting, ordered copolymers, while reacetylation of poly(vinyl alcohol) leads to low melting, random copolymers. Catalytic alcoholysis of PVAc yields copolymers intermediate in melting point and order. The results are treated by assuming that equal melting points indicate similar sequence length distributions of crystallizable units.     

Kinetics and mechanism of the acid-catalysed decarboxylation of thecis-α- andcis-β-carbonato(3,6-dimethyl-1,8-diamino-3,6-diazaoctane)-cobalt(III) ions

Summary The acid-catalysed decarboxylation of thecis-- andcis--[CoL(CO₃)]⁺ complexes (L = 3,6-dimethyl-1,8-diamino-3,6-diazaoctane) have been studied over a range of HClO₄ concentrations and the temperatures 25, 35 and 45° at I = 1.0 mol dm^–3 (NaClO₄). The rate expression takes the form k_obs = k₀ + k₁[H⁺] where k_obs is the observed first order rate constant at constant hydrogen ion concentration. The k₀ term makes only a minor contribution to the overall reaction. Both complexes display solvent deuterium isotope effects ofca. 2.6 for the acid-catalysed decarboxylation, consistent with a rapid proton pre-equilibrium mechanism. Activation parameters have been determined and the mechanism of the reaction discussed. The magnitude of the solvent isotope effect is consistent with an A-1 type mechanism involving formation of a 5-coordinate intermediate.     

A criterion for the uniform distribution of sequences in compact metric spaces

LetX be a compact metric space, le μ be a non-negative normalized Borel measure onX and letf be a measurable bounded real-valued function defined onX such thatf is μ-almost everywhere continuous and different from zero. It is proved that a sequence (x _n ),n=1,2, … of points inX is μ-uniformly distributed if and only if for every Borel setE?X with μ(Bd(E))=0 we have \(\mathop {\lim }\limits_{N \to \infty } \frac{1}{N}\sum\limits_{n = 1}^N {f(x_n )} 1_E (x_n ) = \int\limits_E {f(x)d\mu (x)} ,\) where 1 _E denotes the characteristic function ofE andbdE the boundary ofE. Furthermore some quantitative aspects and generalizations of this theorem are discussed.     

A priori-Schranken für diskrete Lösungen monotoner Operatorgleichungen und Anwendungen

Summary Operator equationsTu=f are approximated by Galerkin's method, whereT is a monotone operator in the sense of Browder and Minty, so that existence results are available in a reflexive Banach spaceX. In a normed spaceY error estimates are established, which require a priori bounds for the discrete solutionsu _h in the norm of a suitable space . Sufficient conditions for the uniform boundedness u _{h Z} =O(1) ash0 are proved. Well-known error estimates in [3] for the special caseX=Y=Z are generalized by this. The theory is applied to quasilinear elliptic boundary value problems of order 2m in a bounded domain . The approximating subspaces are finite element spaces. Especially the caseX=W ₀ ^{m, p} (), 1<p<,Y=W ₀ ^{m. 2} (),Z=W ₀ ^{m. max (2,p)} ()W^m, () is treated. Some examples for 1<p<2 are considered. Forp2 a refined technique is introduced in the author's paper [7].

     

CHEMICALLY MODIFIED PHOTOSYNTHETIC BACTERIAL REACTION CENTERS: CIRCULAR DICHROISM, RAMAN RESONANCE, LOW TEMPERATURE ABSORPTION, FLUORESCENCE AND ODMR SPECTRA AND POLYPEPTIDE COMPOSITION OF BOROHYDRIDE TREATED REACTION CENTERS FROM Rhodobacter sphaeroides R26

Abstract— Reaction centers from Rhodobacter sphaeroides have been modified by treatment with sodium borohydride similar to the original procedure [Ditson et al., Biochim. Biophys. Acta 766 , 623 (1984)], and investigated spectroscopically and by gel electrophoresis.
(1) Low temperature (1.2 K) absorption, fluorescence, absorption- and fluorescence-detected ODMR, and microwave-induced singlet-triplet absorption difference spectra (MIA) suggest that the treatment produces a spectroscopically homogeneous preparation with one of the 'additional' bacteriochlorophylls being removed. The modification does not alter the zero field splitting parameters of the primary donor triplet (^TP870).
(2) From the circular dichroism and Raman resonance spectra in the1500–1800 cm^-1 region, the removed pigment is assigned to Bchl_M, e.g. the "extra" Bchl on the "inactive" M-branch.
(3) A strong coupling among all pigment molecules is deduced from the circular dichroism spectra, because pronounced band-shifts and/or intensity changes occur in the spectral components assigned to all pigments. This is supported by distinct differences among the MIA spectra of untreated and modified reaction centers, as well as by Raman resonance.
(4) The modification is accompanied by partial proteolytic cleavage of the M-subunit. The preparation is thus spectroscopically homogeneous, but biochemically heterogenous.