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.     

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.     

Binding of an inhibitor of the p53/MDM2 interaction to MDM2

The mode of action of the secondary metabolite chlorofusin, which antagonises the interaction between p53 and MDM2, involves direct binding to the N-terminal domain of MDM2.     

A predictive model for unimolecular reactions: Reactions of unsaturated carbonium ions

The major slow unimolecular reactions undergone by C₄H₇⁺, C₅H₉⁺ and C₆H⁺₁₁ are discussed in terms of a potential surface approach and the organic chemist's concept of mechanism. It is shown that the observed decompositions which do not involve σ-bond formation in the dissociation step are precisely those expected from the model. Further use of the model correctly predicts the slow reactions of C₇H⁺₁₃ which have not previously been reported. The approach also permits useful limits to be set on the transition state energies for reactions involving σ-bond formation in the dissociation step (H₂,CH₄ loss). It is concluded that stepwise addition of ethylene to the allyl cation is preferred to a concerted 4-electron process which is symmetry forbidden.     

Some new tests for multivariate normality

Summary A family of statistics is presented that can be used for testing goodness of fit to a parametric family. These statistics include Mardia's measure of multivariate kurtosis and Moore and Stubblebine's test for multivariate normality. The asymptotic distribution of the statistics is found under mild hypotheses on the parametric family and, in the case of multivariate normality, the distribution is shown to be independent of the true parameter. A class of tests for multivariate normality is presented and the performance of two such tests in the bivariate case is found in simulations.The research of this author was carried out in part while at M.I.T. and then at Bell Communications ResearchThe research of this author was partially supported by National Science Foundation Grants