A Note on Ideal Spaces of Banach Algebras

In a previous paper, the second author introduced a compacttopology _r on the space of closed ideals of a unital Banachalgebra A. If A is separable, then _r is either metrizable orelse neither Hausdorff nor first countable. Here it is shownthat _r is Hausdorff if A is C¹[0, 1], but that if A is a uniformalgebra, then _r is Hausdorff if and only if A has spectral synthesis.An example is given of a strongly regular, uniform algebra forwhich every maximal ideal has a bounded approximate identity,but which does not have spectral synthesis. 1991 MathematicsSubject Classification 46H10.     

Examples of asymptotic completeness in translation invariant systems with an unbounded number of particles

A tagged particle interacting with a finite but unbounded number of particles is considered. Some examples of asymptotic completeness are given which are uniform in the number of particles.     

Approximation and Extension in Normed Spaces of Infinitely Differentiable Functions

We consider questions of rational and polynomial approximation,and related extension questions, for various normed spaces ofinfinitely differentiable functions on perfect compact subsetsof the complex plane C and the real line R. We obtain an approximationtheorem for compact planar sets which are, in a precise sense,locally radially self-absorbing. All smoothly-bounded compactsets are of this type. We give a variety of results and counterexampleson extension, mainly in the one-dimensional case. We also provepolynomial approximation theorems for totally-disconnected sets,linear sets, and some others.     

Inhomogeneous cosmologies: The cosmic peeling-off property of gravity

Asymptotic expansions are used to study the late-time behavior of the Riemann tensor for a large class of inhomogeneous cosmological models. These models include the unidirectional inhomogeneous generalizations of Bianchi type I–VII universes. The cosmic analog of the peeling-off theorem, familiar from general relativity, is formulated. It is also shown that chaotic behavior near the initial singularity transforms itself into pure gravitational radiation at late times.     

THE ORIGIN OF ENTROPY IN THE UNIVERSE

We discuss the entropy generation in quantum tunneling of a relativistic particle under the influence of a time-varying force with the help of squeezing formalism. It is shown that if one associates classical coarse grained entropy to the phase space volume, there is an inevitable entropy growth due to the changes in position and momentum variances. The entropy change can be quantified by a simple expression S=ln cosh 2r, where r, is the squeeze parameter measuring the height and width of the potential barrier. We suggest that the universe could have acquired its initial entropy in a quantum squeeze from nothing and briefly discuss the implications of our proposal.     

eFindSite: Improved prediction of ligand binding sites in protein models using meta-threading, machine learning and auxiliary ligands

Molecular structures and functions of the majority of proteins across different species are yet to be identified. Much needed functional annotation of these gene products often benefits from the knowledge of protein–ligand interactions. Towards this goal, we developed eFindSite, an improved version of FINDSITE, designed to more efficiently identify ligand binding sites and residues using only weakly homologous templates. It employs a collection of effective algorithms, including highly sensitive meta-threading approaches, improved clustering techniques, advanced machine learning methods and reliable confidence estimation systems. Depending on the quality of target protein structures, eFindSite outperforms geometric pocket detection algorithms by 15–40 % in binding site detection and by 5–35 % in binding residue prediction. Moreover, compared to FINDSITE, it identifies 14 % more binding residues in the most difficult cases. When multiple putative binding pockets are identified, the ranking accuracy is 75–78 %, which can be further improved by 3–4 % by including auxiliary information on binding ligands extracted from biomedical literature. As a first across-genome application, we describe structure modeling and binding site prediction for the entire proteome of Escherichia coli. Carefully calibrated confidence estimates strongly indicate that highly reliable ligand binding predictions are made for the majority of gene products, thus eFindSite holds a significant promise for large-scale genome annotation and drug development projects. eFindSite is freely available to the academic community at http://www.brylinski.org/efindsite.     

Reaction of Dimethyl Acetylene-Dicarobxylate With Triazinone

Abstract

The addition of dimethyl acetylenedicarboxylate (DMAD) to 6-menthy-1, 2,4-triazlne-3(2H)thione-5(4H)-one afforded 2-methoxylcarboxy-7-methyl-1,3-thiazino[3,2-b][1,2,4]triazine-4,8-dione.