S(N)1-like reactions of bicyclic alpha-amino ethers with sulfur,nitrogen, and carbon nucleophiles. Synthesis of 1-azabicyclo[3.2.2]nonanes functionalized at carbon C2 and C6 with complete stereocontrol

[reaction: see text] In the presence of nucleophiles, Lewis acid mediated cleavage of alpha-amino ethers derived from quincorine and quincoridine affords a variety of C2-substituted and C6-vinylated 1-azabicyclo[3.2.2]nonanes. These are enantiopure and are formed in S(N)1-like reactions with complete stereocontrol. There is no leakage into 2-Nu en route to product 1-Nu or vice versa. Me(3)SiCN provides new Strecker-type alpha-amino nitriles. In the presence of TTMPP-BF(3).OEt(2), the ketene acetal Me(2)C=C(OMe)OSiMe(3) delivers enantiopure bicyclic beta-amino acid esters.     

Kinetics and mechanisms of the reactions of OH with some oxygenated compounds of importance in tropospheric chemistry

The kinetics of the reactions have been studied in a discharge flow system under pseudo-first-order conditions. The OH concentration was monitored by laser induced fluorescence and helium was used as the carrier gas. Values of k₁ = (8.1 ± 1.7) × 10^?13, k₂ = (1.31 ± 0.26) × 10^?11, k₃ = (2.6 ± 0.5) × 10^?11, and k₄ = (2.5 ± 0.4) × 10^?11 cm³ molecule^?1 s^?1, at 298 K and 1 torr total pressure, were obtained. To validate the newly constructed system the rate constant for the reaction was determined in a similar manner. The value of k₅ = (6.7 ± 0.9) × 10^?12 cm³ molecule^?1 s^?1 at 298 K and 1 torr total pressure is in very good agreement with other literature values. The mechanisms for the atmospheric degradation of these compounds have been constructed to allow their incorporation in a photochemical trajectory computer model, to assess their impact on photochemical ozone creation in the troposphere. © 1994 John Wiley & Sons, Inc.     

Stochastic search for the conformations of bicyclic hydrocarbons

The stochastic search method was used to seek all of the conformations for 32 bicyclic hydrocarbons. Since the size of the random kick used is sufficient to invert the configurations of carbon atoms, the out, out, in, out, and in, in conformations were all found in a single stochastic run for each hydrocarbon. The lowest energy conformer obtained in each category is reported. A modification of the program to prevent inversion of configuration was developed. It was found, in some cases, that in, in and out, out isomers for some bicyclic hydrocarbons were interconverted by this modified program, presumably through the “homeomorphic isomerization” process described by Park and Simmons. A statistical formula for estimating the approximate chances of finding (or missing) any conformer as a function of the number of random kicks given is derived and presented.     

A symmetric numerical range for matrices

Summary For each normv on ⁿ, we define a numerical rangeZ _v, which is symmetric in the sense thatZ _v=Z_vD, wherev ^D is the dual norm.We prove that, fora ⁿⁿ,Z _v(a) contains the classical field of valuesV(a). In the special case thatv is thel ₁-norm,Z _v(a) is contained in a setG(a) of Gershgorin type defined by C. R. Johnson.Whena is in the complex linear span of both the Hermitians and thev-Hermitians, thenZ _v(a),V(a) and the convex hull of the usualv-numerical rangeV _v(a) all coincide. We prove some results concerning points ofV(a) which are extreme points ofZ _v(a).Part of this research was done while the authors were at the Mathematische Institut, Technische Universität, München, West Germany. The first author presented these results at the Seminar on Matrix Theory (Positivity and Norms) held in Munich in December, 1974. The second author also acknowledges support from the National Science Foundation under grant GP 37978X.     

The shadow of a collapsing dark star

The shadow of a black hole is usually calculated, either analytically or numerically, on the assumption that the black hole is eternal, i.e., that it has existed for all time. Here we ask the question of how this shadow comes about in the course of time when a black hole is formed by gravitational collapse. To that end we consider a star that is spherically symmetric, dark and non-transparent and we assume that it begins, at some instant of time, to collapse in free fall like a ball of dust. We analytically calculate the dependence on time of the angular radius of the shadow, first for a static observer who is watching the collapse from a certain distance and then for an observer who is falling towards the centre following the collapsing star.