Automatic differentiation: Reduced gradient and reduced Hessian matrix

This paper is concerned with automatic differentiation methods for computing the reduced gradient M ^t G and the reduced Hessian matrix M ^t HM. Hereby G is the gradient and H is the Hessian matrix of a real function F of n variables, and M is a matrix with n rows and k columns where kn. The reduced quantities are of particular interest in constrained optimization with objective function F. Two automatic differentiation methods are described, a standard method that produces G and H as intermediate results, and an economical method that takes a shortcut directly to the reduced quantites. The two methods are compared on the basis of the reqired computing time and storage. It is shown that the costs for the economical method are less than (k ²+3k+2)/(n ²+3n+2) times the expenses for the standard method.     

Electronic and diffusion-induced spin-lattice relaxation and the lattice expansion about interstitial12B in Al

The spin-lattice relaxation of¹²B in single-crystal aluminum was measured by NMR in the temperature range 290 to 700 K. The total relaxation rate was separated into two well understood contributions caused by the interaction with conduction electrons and by¹²B diffusion, and into a third unexplained part at high temperatures. From the comparison of the diffusional relaxation rate with theoretical calculations it was concluded that the nearest neighbours of¹²B, trapped in the octahedral interstitial site, are displaced away by 8.8 (30)%.     

Influence of the side chain protecting group on the stereoselectivity of the 1,3-dipolar cycloaddition of d-talo-configured nitrones

Two new five-membered cyclic d-talo-configured nitrones were synthesized from d-mannose, and examined in 1,3-dipolar cycloadditions with vinyl acetate and vinylene carbonate. The stereoselectivity of the cycloadditions depends greatly on the protecting group of the vicinal diol attached to the nitrone C-5 carbon atom. Methyl protection resulted in limited syn/anti-selectivity, giving mixtures of the two isomeric exo-syn and exo-anti isoxazolidines in comparable amounts. On the other hand, the cyclohexylidene-substituted nitrone reacted more selectively in favour of the syn isomer. The difference in the diastereoselectivity was attributed to the specific spatial orientation of the nitrone C-5 substituent.