Volume in General Metric Spaces

A central question in the geometry of finite metric spaces is how well can an arbitrary metric space be “faithfully preserved” by a mapping into Euclidean space. In this paper we present an algorithmic embedding which obtains a new strong measure of faithful preservation: not only does it (approximately) preserve distances between pairs of points, but also the volume of any set of \(k\) points. Such embeddings are known as volume preserving embeddings. We provide the first volume preserving embedding that obtains constant average volume distortion for sets of any fixed size. Moreover, our embedding provides constant bounds on all bounded moments of the volume distortion while maintaining the best possible worst-case volume distortion. Feige, in his seminal work on volume preserving embeddings defined the volume of a set \(S = \{v_1, \ldots , v_k \}\) of points in a general metric space: the product of the distances from \(v_i\) to \(\{ v_1, \dots , v_{i-1} \}\) , normalized by \(\tfrac{1}{(k-1)!}\) , where the ordering of the points is that given by Prim’s minimum spanning tree algorithm. Feige also related this notion to the maximal Euclidean volume that a Lipschitz embedding of \(S\) into Euclidean space can achieve. Syntactically this definition is similar to the computation of volume in Euclidean spaces, which however is invariant to the order in which the points are taken. We show that a similar robustness property holds for Feige’s definition: the use of any other order in the product affects volume \(^{1/(k-1)}\) by only a constant factor. Our robustness result is of independent interest as it presents a new competitive analysis for the greedy algorithm on a variant of the online Steiner tree problem where the cost of buying an edge is logarithmic in its length. This robustness property allows us to obtain our results on volume preserving embedding.     

Adequacy of finite difference schemes for convection‐diffusion equations

Finite difference schemes for the numerical solution of singularly perturbed convection problems on uniform grids are studied in the limit case where the viscosity and the meshsize approach zero at the same time. The present error estimates are given in terms of order of magnitude in the above limit process and are useful in a priori choosing adequate schemes and meshsizes for boundary‐layer problems and problems with closed characteristics. Published 2002 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 18: 280–295, 2002; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/num.10007     

Upper bounds for convergence rates of acceleration methods with initial iterations

GMRES(n,k), a version of GMRES for the solution of large sparse linear systems, is introduced. A cycle of GMRES(n,k) consists of n Richardson iterations followed by k iterations of GMRES. Such cycles can be repeated until convergence is achieved. The advantage in this approach is in the opportunity to use moderate k, which results in time and memory saving. Because the number of inner products among the vectors of iteration is about k^2/2, using a moderate k is particularly attractive on message-passing parallel architectures, where inner products require expensive global communication. The present analysis provides tight upper bounds for the convergence rates of GMRES(n,k) for problems with diagonalizable coefficient matrices whose spectra lie in an ellipse in 0. The advantage of GMRES(n,k) over GMRES(k) is illustrated numerically.     

A 19F NMR STUDY OF THE HYDROLYSIS OF ORGANOPHOSPHORUS FLUORIDATES IN AQUEOUS SOLUTION Doubling of the 19F nmr in Diastereoisomeric Mixtures

Abstract

¹H and ¹⁹F nmr spectra of two series of organophosphorus esters containing a P–F bond were studied. 2-Fluoro-4-methyl 1,3,2-dioxaphosphorinane 2-oxide (2) was found to be a mixture of two diastereoisomers in the ratio of trans/cis = 4.

Stereospecific catalysis of phosphate and maleate anions in the hydrolysis of 2 was observed, leading to enrichment of the non-hydrolysed fluoridates with the trans isomer.

The ¹⁹F nmr spectra of O-2-butyl methylphosphonofluoridate (7) and O-pinacolyl methylphosphonofluoridate (8), showed doubling of the ¹⁹F nmr spectra, giving rise to two sets of double quartets of equal intensity. 1:1 mixtures of diastereoisomers account for the doubling of the resonance rather than sterically hindered rotation. The applicability of nmr spectroscopy to the study of stereospecific displacement reactions at tetrahedral phosphorus is discussed.     

Some Complexation Reactions of Cellulosic Ethers

Abstract

Cellulose ethers were prepared by the reaction of sodium cellulosate with 1-chloromethylnaphthalene, 2-bromomethyl-pyridine, 2-chloropyridine, crotyl bromide, propargyl chloride, and 1-naphthyl glycidyl ether and their complexation ability was investigated. No π-complex formation with picric acid or 2,4,6-trinitrobenzene took place. Results were attributed to steric hindrance from the cellulosic backbone. Complexation of some metal salts by the crotyl and propargyl ethers was investigated. Good results were obtained with mercuric chloride, aluminum chloride, and silver nitrate when propargyl cellulose was used.     

DFT and Empirical Considerations on Electrocatalytic Water/Carbon Dioxide Reduction by CoTMPyP in Neutral Aqueous Solutions**

A combined experimental and density functional theory (DFT) investigation was employed in order to examine the mechanism of electrochemical CO₂ reduction and H₂ formation from water reduction in neutral aqueous solutions. A water soluble cobalt porphyrin, cobalt [5,10,15,20-(tetra-N-methyl-4-pyridyl)porphyrin], (CoTMPyP), was used as catalyst. The possible attachment of different axial ligands as well as their effect on the electrocatalytic cycles were examined. A cobalt porphyrin hydride is a key intermediate which is generated after the initial reduction of the catalyst. The hydride is involved in the formation of H₂ and formate and acts as an indirect proton source for the formation of CO in these H⁺-starving conditions. The experimental results are in agreement with the computations and give new insights into electrocatalytic mechanisms involving water soluble metalloporphyrins. We conclude that in addition to the porphyrin's structure and metal ion center, the electrolyte surroundings play a key role in dictating the products of CO₂/H₂O reduction.     

Countable Primitive Groups

We give a complete characterization of countable primitive groups in several settings including linear groups, subgroups of mapping class groups, groups acting minimally on trees and convergence groups. The latter category includes as a special case Kleinian groups as well as subgroups of word hyperbolic groups. As an application we calculate the Frattini subgroup in many of these settings, often generalizing results that were only known for finitely generated groups. In particular, we answer a question of G. Higman and B.H. Neumann on the Frattini group of an amalgamated product. Received: January 2006, Revision: May 2006, Accepted: May 2006     

Computational acceleration of projection algorithms for the linear best approximation problem

This is an experimental computational account of projection algorithms for the linear best approximation problem. We focus on the sequential and simultaneous versions of Dykstra’s algorithm and the Halpern-Lions-Wittmann-Bauschke algorithm for the best approximation problem from a point to the intersection of closed convex sets in the Euclidean space. These algorithms employ different iterative approaches to reach the same goal but no mathematical connection has yet been found between their algorithmic schemes. We compare these algorithms on linear best approximation test problems that we generate so that the solution will be known a priori and enable us to assess the relative computational merits of these algorithms. For the simultaneous versions we present a new component-averaging variant that substantially accelerates their initial behavior for sparse systems.