On nonlinear -widths

For characterization of best nonlinear approximation, DeVore,
Howard, and Micchelli have recently suggested the nonlinear -width of a subset in a normed linear space . We proved by a topological method that for and the well-known Aleksandrov -width in a Banach space the following inequalities hold: . Let be the unit ball of Besov space , of multivariate periodic functions. Then for approximation in , with some restriction on and , we established the asymptotic degree of these -widths: .

     

Universally signable graphs

In a graph, a chordless cycle of length greater than three is called a hole. Let be a {0, 1} vector whose entries are in one-to-one correspondence with the holes of a graphG. We characterize graphs for which, for all choices of the vector , we can pick a subsetF of the edge set ofG such that |F H| H (mod 2), for all holesH ofG and |F T| 1 for all trianglesT ofG. We call these graphsuniversally signable. The subsetF of edges is said to be labelledodd. All other edges are said to be labelledeven. Clearly graphs with no holes (triangulated graphs) are universally signable with a labelling of odd on all edges, for all choices of the vector . We give a decomposition theorem which leads to a good characterization of graphs that are universally signable. This is a generalization of a theorem due to Hajnal and Surányi [3] for triangulated graphs.This work was supported in part by NSF grants DMI-9424348 DMS-9509581 and ONR grant N00014-89-J-1063. Ajai Kapoor was also supported by a grant from Gruppo Nazionale Delle Ricerche-CNR. We also acknowledge the support of Laboratoire ARTEMIS, Université Joseph Fourier, Grenoble.     

Photocatalytic decarboxylative amidosulfonation enables direct transformation of carboxylic acids to sulfonamides

Sulfonamides feature prominently in organic synthesis, materials science and medicinal chemistry, where they play important roles as bioisosteric replacements of carboxylic acids and other carbonyls. Yet, a general synthetic platform for the direct conversion of carboxylic acids to a range of functionalized sulfonamides has remained elusive. Herein, we present a visible light-induced, dual catalytic platform that for the first time allows for a one-step access to sulfonamides and sulfonyl azides directly from carboxylic acids. The broad scope of the direct decarboxylative amidosulfonation (DDAS) platform is enabled by the efficient direct conversion of carboxylic acids to sulfinic acids that is catalyzed by acridine photocatalysts and interfaced with copper-catalyzed sulfur–nitrogen bond-forming cross-couplings with both electrophilic and nucleophilic reagents.

Sulfonamides are now accessible directly from carboxylic acids by a one-step, tricomponent decarboxylative amidosulfonation that provides the missing link between the two key functionalities.     

Extraction and spectrophotometric determination of uranium in phosphate fertilizers

An extraction and spectrophotometric method for determination of trace amounts of uranium in phosphate fertilizers is described. It is based on the extraction of uranium with trioctylphosphine oxide in benzene and the spectrophotometric determination of uranium with Arsenazo III in buffer-alcoholic medium. The maximum absorbance occurs at 655 nm with a molar absorptivity of 1.2·10⁴ l·mol^–1·cm^–1. Beer's law is obeyed over the range 0.6–15.0 g·ml^–1 of uranium(VI). The proposed method has been applied successfully to the analysis of phosphate fertilizers with phosphate concentrations of 45% P₂O₅.     

Born-Floquet theory of electron-atom collisions in the presence of a laser field

We present a theory of fast electron-atom collisions in the presence of a strong laser field, which treats the interaction of the laser field with both the projectile electron and the target atom in a fully non-perturbative way. The theory is illustrated by considering the laser-assisted “elastic” scattering of fast electrons by atomic hydrogen, for non-resonant as well as resonant cases.     

The influence of electrolyte composition on pH of nickel hydrate formation

The influence of the composition of nickel plating electrolytes containing various ligands (Ac⁻, Gly⁻, and Asp²⁻) on pH of nickel hydrate formation (pH_hydr) was studied. The data obtained were interpreted from the point of view of ionic equilibria in the systems studied. The nature of sparingly soluble precipitates formed in these electrolytes when pH_hydr was reached was determined. Sparingly soluble precipitates were found to be pure Ni(OH)₂ hydroxides or basic salts with the composition Ni(OH) _n Cl_{2 − n} , in which the content of chloride ions increased as the concentration of NiCl₂ grew. Organic substance anions were absent in the precipitates formed and only influenced pH_hydr through changes in the concentration of free nickel ions as a result of complex formation. A principle of selecting the composition of nickel plating electrolytes was suggested on the basis of the results obtained.     

GHP: A Maple Package for Performing Calculations in the Geroch-Held-Penrose Formalism

We present a new symbolic algebra package, written for Maple, for performing computations in the Geroch-Held-Penrose formalism. We demonstrate the essential features and capabilities of our package by investigating Petrov-D vacuum solutions of Einstein's field equations.     

On the colored Jones polynomial and the Kashaev invariant

We express the colored Jones polynomial as the inverse of the quantum determinant of a matrix with entries in the q-Weyl algebra of q-operators, evaluated at the trivial function (plus simple substitutions). The Kashaev invariant is proved to be equal to another special evaluation of the determinant. We also discuss the similarity between our determinant formula of the Kashaev invariant and the determinant formula of the hyperbolic volume of knot complements, hoping it will lead to a proof of the volume conjecture. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 11, No. 5, pp. 57–78, 2005.