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.     

A short synthesis of 4-amino-3-hydroxybutyric acid (GABOB) via allyl cyanide

4-Amino-3-hydroxybutyric acid was synthesized from allyl cyanide in four steps in an overall yield of 38%. Ultrasonically promoted epoxidation of allyl cyanide with m-chloroperoxybenzoic acid giving oxiranylacetonitrile was used as a key step.     

Sodium p‐nitro­benzoxasulfamate monohydrate

The title compound, alternatively named sodium 6‐nitro‐3H‐1,2,3‐benzoxa­thia­zole 2,2‐dioxide monohydrate, Na⁺·C₆H₃­N₂O₅S^?·H₂O, consists of chains of NaO₇ units, with the seven donor‐O atoms coming from two water mol­ecules and five p‐­nitro­benzoxasulfamate anions. The seven‐coordinate geometry around the Na⁺ ion is described as monocapped trigonal prismatic, but with a large distortion from ideal geometry. Each triangular face is defined by one O atom each from a water mol­ecule, a nitro group and a sulfonyl group. An O atom from a sulfonyl group caps one of the square faces of the trigonal prism in an unsymmetrical fashion. The water mol­ecules and one sulfonyl O atom are involved in bridging adjacent units, as is the nitro group of the anion. The sulfamate ions adopt an antiparallel alignment between the NaO₇ units and are connected to each other by C—H?O and π–π interactions. The three‐dimensional crystal structure is stabilized by a network of strong O—H?N hydrogen bonds.     

Ultrafast dynamics of resonance energy transfer in cryptochrome

In this communication, we report the ultrafast dynamics of resonance energy transfer in a blue-light photoreceptor, Vibrio cholerae cryptochrome. The transfer was observed to occur in 60 ps. We also studied the local rigidity and solvation around the binding site of the photoantenna molecule. The results for the first time show energy transfer in cryptochrome suggesting some mechanistic similarities between photolyase that repairs damaged DNA and cryptochrome that mediates blue-light signaling.     

All-Solid-State PVC Membrane Ag+-Selective Electrodes Based on Diaza-18-Crown-6 Compounds

Two diaza-crown ether compounds were synthesized and evaluated as Ag⁺-selective carriers in polyvinylchloride (PVC) membrane electrodes of solid-state type. The all-solid-state PVC membrane electrode based on N,N-Dibenzyl-dibenzo-diaza-18-crown-6 exhibited a super-Nernstian response (75±10mV per decade) over the concentration range of 1×10^–1 to 7×10^–6M of Ag⁺ ion and a detection limit of 3×10^–6M, at a wide range of pH (pH 4–7). The response time of the electrode was fast (less than 10s), and it can be used for three months without any significant deviation in potential. The proposed all-solid-state PVC membrane electrodes revealed high selectivity toward Ag⁺ ion with respect to alkali, alkaline earth, heavy and transition metal ions. A flow-through cell of all-solid-state PVC membrane Ag⁺-selective electrode based on N,N-Dibenzyl-dibenzo-diaza-18-crown-6 has also been prepared and applied for flow-injection analysis of Ag⁺ ion in solution.     

A boundary-value problem for Hamilton-Jacobi equations in hilbert spaces

We study a Hamilton-Jacobi equation in infinite dimensions arising in optimal control theory for problems involving both exit times and state-space constraints. The corresponding boundary conditions for the Hamilton-Jacobi equation, of mixed nature, have been derived and investigated in [19], [2], [5], and [15] in the finite-dimensional case. We obtain a uniqueness result for viscosity solutions of such a problem and then prove the existence of a solution by showing that the value function is continuous.The work of P. Cannarsa was partially supported by the Italian National Project Equazioni Differenziali e Calcolo delle Variazioni. H. M. Soner's work was supported by National Science Foundation Grant DMS-90-02249.