The Jacobian and the Ginzburg-Landau energy

We study the Ginzburg-Landau functional for , where U is a bounded, open subset of . We show that if a sequence of functions satisfies , then their Jacobians are precompact in the dual of for every . Moreover, any limiting measure is a sum of point masses. We also characterize the -limit of the functionals , in terms of the function space B2V introduced by the authors in [16,17]: we show that I(u) is finite if and only if , and for is equal to the total variation of the Jacobian measure Ju. When the domain U has dimension greater than two, we prove if then the Jacobians are again precompact in for all , and moreover we show that any limiting measure must be integer multiplicity rectifiable. We also show that the total variation of the Jacobian measure is a lower bound for the limit of the Ginzburg-Landau functional. Received: 15 December 2000 / Accepted: 23 January 2001 / Published online: 25 June 2001     

Asymptotic expansions for Markov processes with lévy generators

This paper considers a deterministic flow inn-dimensional space, perturbed by a Markov jump process with small variance. Asymptotic expansions are obtained for certain functionals of Feynman—Kac type, in powers of a small parameter representing a noise intensity. The methods are analytical rather than probabilistic.The research of the first author was partly supported by AFOSR under Contract No. 91-0116-0, by ONR under Contract No. N0014-83-K-0542, and by the Institute for Mathematics and Its Applications with funds provided by the NSF and ONR. The second author's research was partly supported by NSF under Contract No. DMS-8702537, and by the Institute for Mathematics and Its Applications with funds provided by the NSF and ONR.     

Random walks generated by affine mappings

This article is concerned with Markov chains on ^m constructed by randomly choosing an affine map at each stage, and then making the transition from the current point to its image under this map. The distribution of the random affine map can depend on the current point (i.e., state of the chain). Sufficient conditions are given under which this chain is ergodic.     

Functionalization of saturated hydrocarbons. High temperature bromination of octahydropentalene. Part 19

The synthesis and thermal bromination of octahydropentalene was studied. The reaction afforded 1a,3a,4b,6b-tetrabromo-1,2,3,4,5,6-hexahydropentalene (14) with remarkable regio- and stereospecificity. The structure of the product was determined by ¹H and ¹³C NMR data and single X-ray structural analysis. The treatment of octahydropentalene with tenfold bromine gave the octabromopentalene derivative. The formation mechanism of the products is discussed.     

Bicyclic allenes : Generation and trapping of 6,7-benzobicyclo [3.2.1] octa-2,3-diene

Treatment of 3-bromo-6,7-benzobicyclo [3.2.1] octa-2,3-diene (9) with potassium tert.-butoxide produces the strained bicyclic allene (10) which is trapped by 1,3-diphenylbenzo[c]furan (DBI) to give five isomeric cycloadducts. In the absence of DBI, allene (10) gave rise to enol ether (18) by addition of tert.-butanol.     

Synthesis and structure of cyclopropano-annelated homosesquinorbornene derivatives containing pyramidalized double bonds: evidence for the sterical effect of a cyclopropyl group on the degree of C=C double-bond pyramidalization

[reaction: see text] endo- and exo-2,3,4,7-tetrahydro-1H-1,4-methanobenzocycloheptene-7-carboxylic acid ethyl esters have been synthesized, and their Diels-Alder cycloaddition reactions with maleic anhydride, dimethyl acetylenedicarboxylate and singlet oxygen have been investigated. The X-ray analysis of four adducts indicated the pyramidalization of the central double bond. Density functional theory calculations on the isolated products and model compounds showed excellent agreement between the experimental and theoretical determined butterfly angles. Furthermore, it has been shown that a cyclopropyl group fused to [2.2.2] system decreases significantly the degree of the pyramidalization which is attributed to the steric interactions between the cyclopropyl group and ethano bridge of the norbornene systems. Due to the instability of the bicyclic endoperoxides, their X-ray analysis could not be carried out. DFT calculations on model compounds showed increased bending in the case of the product obtained by the addition of singlet oxygen to endo-2,3,4,7-tetrahydro-1H-1,4-methanobenzocycloheptene-7-carboxylic acid ethyl ester.