The structured sensitivity of Vandermonde-like systems

Summary We consider a general class of structured matrices that includes (possibly confluent) Vandermonde and Vandermonde-like matrices. Here the entries in the matrix depend nonlinearly upon a vector of parameters. We define, condition numbers that measure the componentwise sensitivity of the associated primal and dual solutions to small componentwise perturbations in the parameters and in the right-hand side. Convenient expressions are derived for the infinity norm based condition numbers, and order-of-magnitude estimates are given for condition numbers defined in terms of a general vector norm. We then discuss the computation of the corresponding backward errors. After linearising the constraints, we derive an exact expression for the infinity norm dual backward error and show that the corresponding primal backward error is given by the minimum infinity-norm solution of an underdetermined linear system. Exact componentwise condition numbers are also derived for matrix inversion and the least squares problem, and the linearised least squares backward error is characterised.     

MOP-phosphonites: a novel ligand class for asymmetric catalysis

Chiral phosphonite ligands (S,R(b))-5a, (S,S(b))-5b, (R,R(b))-6a and (R,S(b))-6b are introduced, comprising a MOP-type backbone with a binol-based binaphthyl group bound to the phosphorus. Their reaction with [Pd(η(3)-C(4)H(7))Cl](2) affords η(3)-methallylpalladium chloride complexes 7a/b and 8a/b which have been isolated and structurally characterised. Solid-state and solution studies indicate subtle differences in their coordination behaviour, which ultimately affects their efficacy in the asymmetric hydrosilylation of styrene.     

Stiffness of ODEs

It is argued that even for a linear system of ODEs with constant coefficients, stiffness cannot properly be characterized in terms of the eigenvalues of the Jacobian, because stiffness is a transient phenomenon whereas the significance of eigenvalues is asymptotic. Recent theory from the numerical solution of PDEs is adapted to show that a more appropriate characterization can be based upon pseudospectra instead of spectra. Numerical experiments with an adaptive ODE solver illustrate these findings.Supported by a Dundee University Research Initiatives Grant.Supported by NSF Grant DMS-9116110.     

Water-soluble hydroxyalkylated phosphines: examples of their differing behaviour toward ruthenium and rhodium

The reaction of P(CH2OH)3 (I) and P(C6H5)(CH2OH)2 (II) with RuCl3 in methanol eliminates two equivalents of formaldehyde to yield the mixed tertiary and secondary phosphine complexes all-trans-[RuCl2(P(CH2OH)3)2(P(CH2OH)2H)2] (1) and [RuCl2(P(C6H5)(CH2OH)2)2(P(C6H5)(CH2OH)H)2] (2), respectively. There is a high degree of hydrogen-bonding interactions between the hydroxymethyl groups in 1 and 2, although the phenyl groups of the latter reduce the extent of the network compared to 1. The generation of these mixed secondary and tertiary phosphine complexes is unprecedented. Under the same reaction conditions, the tris(hydroxypropyl)phosphine III formed no ruthenium complex. The reaction of P(CH2OH)3, P(C6H5)(CH2OH)2 and P{(CH2)3OH}3 with [RhCl(1,5-cod)]2 in an aqueous/dichloromethane biphasic medium yielded [RhH2(P(CH2OH)3)4]+ (3), [RhH2(P(C6H5)(CH2OH)2)4]+ (4) and [Rh(P(C6H5)(CH2OH)2)4]+ (5) and [Rh(P{(CH2)3OH}3)4]+ (6), respectively. Treating 5 with dihydrogen rapidly gave 4. The hydroxypropyl compound 6 formed the corresponding dihydride much more slowly in aqueous solution, although [RhH2(P{(CH2)3OH}3)4]+ (7) was readily formed by reaction with dihydrogen. Two separate reaction pathways are therefore involved; for P(CH2OH)3 and to a lesser extent P(C6H5)(CH2OH)2, the hydride source in the product is likely to be the aqueous solvent or the hydroxyl protons, whilst for P{(CH2)3OH}3 an oxidative addition of H2 is favoured. The protic nature of and was illustrated by the H-D exchange observed in d2-water. Dihydrides 3 and 4 reacted with carbon monoxide to yield the dicarbonyl cations [Rh(CO)2(P(CH2OH)3)3]+ (8) and [Rh(CO)2(P(C6H5)(CH2OH)2)3]+ (9). The analogous experiment with [RhH2(P{(CH2)3OH}3)4]+ resulted in phosphine exchange, although our experimental evidence points to the possibility of more than one fluxional process in solution.     

Spectral reordering of a range-dependent weighted random graph

^** Email: aas96106{at}maths.strath.ac.uk Grindrod (2002. Phys. Rev. E, 66, 0667021–0667027) posedthe problem of reordering a range-dependent random graph andshowed that it is relevant to the analysis of data sets frombioinformatics. Reordering under a random graph hypothesis canbe regarded as an extension of clustering and fits into thegeneral area of data mining. Here, we consider a generalizationof Grindrod's model and show how an existing spectral reorderingalgorithm that has arisen in a number of areas may be interpretedfrom a maximum likelihood range-dependent random graph viewpoint.Looked at this way, the spectral algorithm, which uses eigenvectorinformation from the graph Laplacian, is found to be automaticallytuned to an exponential edge density. The connection is precisefor optimal reorderings, but is weaker when approximate reorderingsare computed via relaxation. We illustrate the performance ofthe spectral algorithm in the weighted random graph contextand give experimental evidence that it can be successful forother edge densities. We conclude by applying the algorithmto a data set from the biological literature that describescortical connectivity in the cat brain.     

A new family of macrocycles produced by sequential Claisen-Schmidt condensations

[reaction: see text] Members of a new family of macrocycles have been synthesized in one step, from simple building blocks, by sequential Claisen-Schmidt condensations.     

Coupled perturbed modes and internal solitary waves

Coupled perturbed mode theory combines conventional coupled modes and perturbation theory. The theory is used to directly calculate mode coupling in a range-dependent shallow water problem involving propagation through continental shelf internal solitary waves. The solitary waves considered are thermocline depressions, separating well-mixed upper and lower layers. The method is fast and accurate. Results highlight mode coupling associated with internal solitary waves, and mode capture or loss to and from the discrete mode spectrum.     

Asymmetry through time dependency

Given a single network of interactions, asymmetry arises when the links aredirected. For example, if protein A upregulates protein B and protein Bupregulates protein C, then (in the absence of any further relationships between them) Amay affect C but not vice versa. This type of imbalance is reflected in the associatedadjacency matrix, which will lack symmetry. A different type of imbalance can arise wheninteractions appear and disappear over time. If A meets B today and B meets C tomorrow,then (in the absence of any further relationships between them) A may pass a message ordisease to C, but not vice versa. Hence, even when each interaction is a two-way exchange,the effect of time ordering can introduce asymmetry. This observation is very closelyrelated to the fact that matrix multiplication is not commutative. In this work, wedescribe a method that has been designed to reveal asymmetry in static networks and showhow it may be combined with a measure that summarizes the potential information flowbetween nodes in the temporal case. This results in a new method that quantifies theasymmetry arising through time ordering. We show by example that the new tool can be usedto visualize and quantify the amount of asymmetry caused by the arrow of time.     

{(E)‐ and {(Z)‐2‐[α,β‐bis­(methoxy­carbonyl)­vinyl]­cyclo­pentadien‐1‐yl­idene}tri­phenyl­phospho­rane

The Ramirez yl­ide undergoes electrophilic substitution with di­alkyl acetyl­ene­di­carboxyl­ates, yielding a mixture of the Z and E adducts. The crystal structure analyses of the two adducts formed using di­methyl­acetyl­ene, viz. di­methyl (E)‐ and (Z)‐1‐[2‐(tri­phenyl­phospho­ranyl­idene)­cyclo­pentadien‐1‐yl]­ethyl­ene­di­carboxyl­ate, both C₂₉H₂₅O₄P, explain an unusual chemical shift observed for the vinyl H atom of the Z adduct, which had previously precluded a definitive assignment of the isomers. In addition, the structures explain why only one of the isomers reacts further with acetyl­ene esters to produce azulenes with a rare substitution pattern.