Reverse mathematics and rank functions for directed graphs

A rank function for a directed graph G assigns elements of a well ordering to the vertices of G in a fashion that preserves the order induced by the edges. While topological sortings require a one-to-one matching of vertices and elements of the ordering, rank functions frequently must assign several vertices the same value. Theorems stating basic properties of rank functions vary significantly in logical strength. Using the techniques of reverse mathematics, we present results that require the subsystems , , , and . Received: 29 December 1998     

Molecular Compton profiles from SCF-MO wavefunctions

Compton profiles are derived from ab initio SCF-MO wavefunctions calculated for cyclopropane, propene, aziridine, ethanimine, ethenamine, oxirane, acetaldehyde and formamide using Dunning's gaussian orbital basis. Good agreement with the experimental profile is found for formamide. The differences between the profiles for the cyclic and acyclic isomers are only slightly larger than the experimental accuracy that can presently be achieved. The cyclic compounds have slightly broader profiles and there are no significant differences arising from strain. Comparison with profiles derived from localized orbital contributions indicate that the localized orbital approach is essentially valid for these systems.     

First principles predictions of thermophysical properties of refrigerant mixtures

We present pair potentials for fluorinated methanes and their dimers with CO(2) based on ab initio potential energy surfaces. These potentials reproduce the experimental second virial coefficients of the pure fluorinated methanes and their mixtures with CO(2) without adjustment. Ab initio calculations on trimers are used to model the effects of nonadditive dispersion and induction. Simulations using these potentials reproduce the experimental phase-coexistence properties of CH(3)F within 10% over a wide range of temperatures. The phase coexistence curve of the mixture of CH(2)F(2) and CO(2) is reproduced with an error in the mole fractions of both phases of less than 0.1. The potentials described here are based entirely on ab initio calculations, with no empirical fits to improve the agreement with experiment.     

Generics for computable Mathias forcing

We study the complexity of generic reals for computable Mathias forcing in the context of computability theory. The n-generics and weak n-generics form a strict hierarchy under Turing reducibility, as in the case of Cohen forcing. We analyze the complexity of the Mathias forcing relation, and show that if G is any n  -generic with n≥2$n\ge 2$ then it satisfies the jump property G⁽ⁿ⁻¹⁾_≡TG^′⊕∅⁽ⁿ⁾$G^{(n-1)}{\equiv }_T{G}^{\prime }\oplus {\varnothing }^{(n)}$. We prove that every such G has generalized high Turing degree, and so cannot have even Cohen 1-generic degree. On the other hand, we show that every Mathias n-generic real computes a Cohen n-generic real.     

Revertants,Low Temperature,and Correctors Reveal the Mechanism of F508del-CFTR Rescue by VX-809 and Suggest Multiple Agents for Full Correction

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Sur la courbure d’une série de surfaces et de lignes

Annali di Matematica Pura ed Applicata (1858-1865) -     

Do active site conformations of small ligands correspond to low free-energy solution structures?

We compare the low free energy structures of ten small, polar ligands in solution to their conformations in their respective receptor active sites. The solution conformations are generated by a systematic search and the free energies of representative structures are computed with a continuum solvation model. Based on the values of torsion angles, we find little similarity between low energy solution structures of small ligands and their active site conformations. However, in nine out of ten cases, the positions of 'anchor points' (key atoms responsible for tight binding) in the lowest energy solution structures are very similar to the positions of these atoms in the active site conformations. A metric that more closely captures the essentials of binding supports the basic premise underlying pharmacophore mapping, namely that active site conformations of small flexible ligands correspond to their low energy structures in solution. This work supports the efforts of building pharmacophore models based on the information present in solution structures of small isolated ligands.     

Reverse Mathematics and Recursive Graph Theory

We examine a number of results of infinite combinatorics using the techniques of reverse mathematics. Our results are inspired by similar results in recursive combinatorics. Theorems included concern colorings of graphs and bounded graphs, Euler paths, and Hamilton paths.     

Sull’inversione quadrica delle curve piane

Annali di Matematica Pura ed Applicata (1858-1865) -     

Highly Selective Allylborations of Aldehydes Using α,α‐Disubstituted Allylic Pinacol Boronic Esters

α,α‐Disubstituted allylic pinacol boronic esters undergo highly selective allylborations of aldehydes to give tetrasubstituted homoallylic alcohols with exceptional levels of anti‐Z‐selectivity (>20:1). The scope of the reaction includes both acyclic and cyclic allylic boronic esters which lead to acyclic and exocyclic tetrasubstituted homoallylic alcohols. The use of β‐borylated allylic boronic esters gave fully substituted alkenes bearing a boronic ester which underwent further cross‐coupling enabling a highly modular and stereoselective approach to the synthesis of diaryl tetrasubstituted alkenes. Computational analysis revealed the origin of the remarkable selectivity observed.