Metric marginal problems for set-valued or non-measurable variables

Summary In a separable metric space, if two Borel probability measures (laws) are nearby in a suitable metric, then there exist random variables with those laws which are nearby in probability. Specifically, by a well-known theorem of Strassen, the Prohorov distance between two laws is the infimum of Ky Fan distances of random variables with those laws. The present paper considers possible extensions of Strassen's theorem to two random elements one of which may be (compact) set-valued and/or non-measurable. There are positive results in finite-dimensional spaces, but with factors depending on the dimension. Examples show that such factors cannot entirely be avoided, so that the extension of Strassen's theorem to the present situation fails in infinite dimensions.This research was partially supported by a Guggenheim Fellowship, by National Science Foundation grant DMS 8505550 at MSRI-Berkeley, and other NSF grants     

Some variations on Kac's formula for Brownian motion

We show that under suitable conditions $$\begin{gathered} E_x f\left\{ {a + \int_0^t \beta \left[ {b + \int_0^s {\alpha \left( {X_r } \right)dr, c + s, X_s } } \right]ds, b + \int_0^t {\alpha \left( {X_s } \right)ds, c + t, X_t } } \right\} \hfill \\ = e^{tG} f\left[ {a, b, c, x} \right] \hfill \\ \end{gathered} $$ whereX _t is a Brownian motion andG is the generator of a (C ₀) contraction semigroupe ^tG.     

Three-photon near-threshold photoionization dynamics of isooctane

The electron survival probability following three-photon (9.3 eV total) near-threshold photoionization of neat isooctane is measured with sub-50 fs time resolution. The measured dynamics are nonexponential in time and are well described by a diffusion-controlled electron-cation recombination model. Excitation-power-dependent studies indicate that the unperturbed three-photon threshold ionization is only observed for pump irradiance below 0.5 TW cm2. At excitation fields above this level, the signal is no longer cubic in the excitation irradiance, and the observed electron survival probability dramatically changes, decaying as a single exponential in time.     

Synthesis of 2-methylcarbamazepine,a new internal standard for chromatographic assays of carbamazepine (tegretol®)

The synthesis of 2-methyl-5H-dibenz[b,f]azepine-5-carboxamide (2-methylcarbamazepine, 2-MCBZ, 8), a promising internal standard for chromatographic assays of the antiepileptic agent carbamazepine (CBZ, 1), is described. N-(p-Tolyl)anthranilic acid (2) was utilized as a starting material for the synthesis of a key compound, 2,9-dimethylacridine (4), which was converted in two steps to 2-methyl-9-hydroxymethylacridan (6). The acridan 6, in the presence of poly-phosphoric acid, was ring-expanded to form 2-methyl-5H-dibenz[b,f]azepine (7), this latter compound being converted by conventional reactions to its 5-carbamyl derivative, 2-MCBZ (8).     

How can enzymes be so efficient?

Dudley Williams and his colleagues discuss how ligands can gain binding energy to their receptors, and substrate transition states to their enzymes, by tightening the protein structures, with a decrease in their dynamic behaviour.     

Analysis of urinary nucleosides. I. Optimisation of high performance liquid chromatography/electrospray mass spectrometry

In order to optimise the analysis of urinary nucleosides by high performance liquid chromatography/mass spectrometry (HPLC/MS), the HPLC separation of these compounds was performed at different 'flow rates' and 0.2mL/min was found to give both a better separation and ionisation. The ionisation conditions were optimised to give the best intensity of the molecules quasi-molecular ions. The ion distribution profile and ionisation in both positive and negative mode were examined and the detection of the protonated molecule in positive mode chosen for further analysis. The limits of detection of the method developed are reported and representative LC/MS and LC/MS/MS spectra shown. Typical urinary nucleoside chromatograms are presented.     

Importance of structural tightening, as opposed to partially bound States, in the determination of chemical shift changes at noncovalently bonded interfaces

Two models (A and B) have been proposed to account for decreased downfield chemical shifts of a proton bound by noncovalent interactions at a ligand/antibiotic interface as the number of ligand/antibiotic interactions is decreased. In model A, the proton involved in the noncovalent bond suffers a smaller downfield shift because the bond is, with a relatively large probability, broken, and not because it is longer. In model B, the proton involved in the noncovalent bond suffers a smaller downfield shift because the bond is longer, and not because it is, with a relatively large probability, broken. We show that model A cannot account for the chemical shift changes. Model B accounts for the process of positively cooperative binding, in which noncovalent bonds are reduced in length and thereby increase the stability of the organized state.