Stereochemistry of the thermal isomerizations of (2S,3R-methoxymethyl-2,3-dideuterio-1-(dideuteriomethylene)cyclopropane

The (2S,3R) isomer of 2-methoxymethyl-2,3-dideuterio-1-(dideuteriomethylene)cyclopropane has been synthesized and heated at 198.8°: from the experimentally observed mol fractions of the eight isomers of 2,3,α,α,- and 2,3,3,α - tetradeuterio - 2 - methoxymethyl - 1 - methylenecyclopropane in the pyrolysis product mixture have been derived rate constants for seven distinct modes of isomerization. One-center thermal epimerizations at C(2) and C(3) and the C(2)C(3) two-center epimerization are of kinetic importance. Only two of four observable stereochemical modes for carbon [1,3] shifts are seen: there is inversion of stereochemistry at the migrating C atom, while the C(3)H trans to C(2)-CH₂OCH₃ in starting material becomes syn 4 : 1 C(α)-H in the [1,3] shift product. Stereomutation at C(2) does not occur along the reaction coordinate for [1,3] carbon shifts.     

Kinetic theory of reacting systems

In this paper transport processes of reacting systems are investigated, based on the Boltzmann equations. The Boltzmann equations are solved by means of Grad's moment method to thirteen moments and some formal results are obtained for transport properties. It is shown that the rate coefficients are quadratic functions of hydrodynamic fluxes and are in the form

where

are the scalar moments associated with the reaction and q, J, Π are heat flux, material flux and traceless symmetric stress tensor. k⁽⁰⁾_i is the usual local equilibrium formula for reaction rate constant. Iterative solutions for the equations of change for

, q, J and Π are obtained from which transport coefficients are calculated for the reacting system. It is shown that the solutions, when specialized to nonreacting mixtures, lead to results for the transport coefficients which are exactly in agreement with the Chapman-Enskog theory results. The modifications of the transport coefficients due to reactions are obtained from the iterative solutions and the bracket integrals necessary for their calculations are explicitly given in an appendix.     

Synthesis of [18F]F-γ-T-3, a Redox-Silent γ-Tocotrienol (γ-T-3) Vitamin E Analogue for Image-Based In Vivo Studies of Vitamin E Biodistribution and Dynamics

Vitamin E, a natural antioxidant, is of interest to scientists, health care pundits and faddists; its nutritional and biomedical attributes may be validated, anecdotal or fantasy. Vitamin E is a mixture of tocopherols (TPs) and tocotrienols (T-3s), each class having four substitutional isomers (α-, β-, γ-, δ-). Vitamin E analogues attain only low concentrations in most tissues, necessitating exacting invasive techniques for analytical research. Quantitative positron emission tomography (PET) with an F-18-labeled molecular probe would expedite access to Vitamin E’s biodistributions and pharmacokinetics via non-invasive temporal imaging. (R)-6-(3-[¹⁸F]Fluoropropoxy)-2,7,8-trimethyl-2-(4,8,12-trimethyltrideca-3,7,11-trien-1-yl)-chromane ([¹⁸F]F-γ-T-3) was prepared for this purpose. [¹⁸F]F-γ-T-3 was synthesized from γ-T-3 in two steps: (i) 1,3-di-O-tosylpropane was introduced at C6-O to form TsO-γ-T-3, and (ii) reaction of this tosylate with [¹⁸F]fluoride in DMF/K₂₂₂. Non-radioactive F-γ-T-3 was synthesized by reaction of γ-T-3 with 3-fluoropropyl methanesulfonate. [¹⁸F]F-γ-T-3 biodistribution in a murine tumor model was imaged using a small-animal PET scanner. F-γ-T-3 was prepared in 61% chemical yield. [¹⁸F]F-γ-T-3 was synthesized in acceptable radiochemical yield (RCY 12%) with high radiochemical purity (>99% RCP) in 45 min. Preliminary F-18 PET images in mice showed upper abdominal accumulation with evidence of renal clearance, only low concentrations in the thorax (lung/heart) and head, and rapid clearance from blood. [¹⁸F]F-γ-T-3 shows promise as an F-18 PET tracer for detailed in vivo studies of Vitamin E. The labeling procedure provides acceptable RCY, high RCP and pertinence to all eight Vitamin E analogues.     

凯雷—徐氏数、卡里兹数及高次0差渐近展开的精化

In this paper, we generalize Cayley-L. C. Hsu Number and give it the recurrence formula. We show that Carlitz Number may be represented by the linear Combination of Cayley-L. C. Hsu Numbers. In the last part, we give a refinement of an asymptotic expansion for ΔⁿO^n+k.     

Asymptotic solutions of Fokker-Planck equations for nonlinear irreversible processes and generalized Onsager-Machlup theory

Fokker-Planck equations for nonlinear processes are solved asymptotically in the limit k→0 where k is the Boltzmann constant. It is shown that the leading asymptotic solutions for conditional (two-gate) distribution functions simply correspond to generalizations of the Onsager-Machlup theory to nonlinear processes. The asumptotic solution method used in the paper is similar to the well-known W.K.B. method in quantum mechanics. A stability criterion of nonlinear irreversible processes is also considered and compared with the Glansdorff-Prigogine stability criterion.     

Catalan and Motzkin numbers modulo 4 and 8

In this paper, we compute the congruences of Catalan and Motzkin numbers modulo 4 and 8. In particular, we prove the conjecture proposed by Deutsch and Sagan that no Motzkin number is a multiple of 8.     

On the Congruences of Some Combinatorial Numbers

In this paper, we apply Lucas' theorem to evaluate the congruences of several combinatorial numbers, including the central Delannoy numbers and a class of Apéry-like numbers, the numbers of noncrossing connected graphs, the numbers of total edges of all noncrossing connected graphs on n vertices, etc. One of these results verifies a conjecture given by Deutsch and Sagan recently. In the end, we use an automaton to explain the idea of our approach.     

Spatial filters for differentiation

A grating is copied in monochromatic coherent light. The copy plate is not quite in contact with the grating. Therefore, the fringe contrast on the copy plate will vary across the field. This copy plate can serve as spatial filter for performing first order or higher order differentiations, if the geometry of the copy process was chosen properly.     

On surface integrals for triple collision integrals in kinetic theory of gases

The triple collision integrals in kinetic theory of gases are re-examined in terms of mass-normalized coordinate systems and new surface integrals are obtained from them. The five-dimensional surface integrals obtained are more symmetric than those known up to know and appear suggestive of further generalization to more complex problems of many-body systems.