Global convergence and the Powell singular function

The Powell singular function was introduced 1962 by M.J.D. Powell as an unconstrained optimization problem. The function is also used as nonlinear least squares problem and system of nonlinear equations. The function is a classic test function included in collections of test problems in optimization as well as an example problem in text books. In the global optimization literature the function is stated as a difficult test case. The function is convex and the Hessian has a double singularity at the solution. In this paper we consider Newton’s method and methods in Halley class and we discuss the relationship between these methods on the Powell Singular Function. We show that these methods have global but linear rate of convergence. The function is in a subclass of unary functions and results for Newton’s method and methods in the Halley class can be extended to this class. Newton’s method is often made globally convergent by introducing a line search. We show that a full Newton step will satisfy many of standard step length rules and that exact line searches will yield slightly faster linear rate of convergence than Newton’s method. We illustrate some of these properties with numerical experiments.     

The frequency-dependent dipole polarizability of the mercury dimer from four-component relativistic density-functional theory

The frequency-dependent dipole polarizability of Hg(2) is calculated using response theory within four-component relativistic density-functional theory [using the local-density approximation (LDA) and the hybrid functional B3LYP] including corrections for the basis-set superposition error. The anisotropic component of the polarizability tensor agrees well with the values obtained from collision-induced Raman spectroscopy carried out at a wavelength of 488 nm. The values obtained from the two density functionals agree closely with the experimentally derived anisotropy component of the dipole polarizability, despite their rather large differences in the dimer potential-energy curves (LDA is strongly overbinding while B3LYP is purely repulsive). The first two refractivity virial coefficients for the generalized Clausius-Mossotti function are derived.     

A Review of Finite Approximations,Archimedean and Non-Archimedean

We give a review of finite approximations of quantum systems, both in an Archimedean and a non-Archimedean setting. Proofs will generally be omitted. In the Appendix we present some numerical results.     

One-pot copper(I)-catalyzed synthesis of 3,5-disubstituted isoxazoles

[reaction: see text] 3,5-Disubstituted isoxazoles are obtained in good yields by a convenient one-pot, three-step procedure utilizing a regioselective copper(I)-catalyzed cycloaddition reaction between in situ generated nitrile oxides and terminal acetylenes. Most functional groups do not interfere with the reaction, which can be performed in aqueous solvents without protection from oxygen. Since all reagents are used in stoichiometric amounts, formation of byproducts is minimized.     

Prediction of physical properties of hydrocarbon mixtures by partial-least-squares calibration of carbon-13 nuclear magnetic resonance data

By using the partial-least-squares (PLS) method, bulk properties of 12-component synthetic mixtures containing n-alkanes, iso-alkanes, cyclo-alkanes and aromatics are calibrated against intensities and chemical shifts of ¹³C-NMR spectra. The standard error of prediction (SEP) for the determinations of density, refractive index, mean molecular weight and carbon-type distribution was found to be less than 3.2% of the observed range. The SEP for excess densities is significantly larger, especially for values based on chemical shift data. The chemical shift variation supplies unique chemical information on solute/solvent interactions.     

Metabolism of carotenoids in salmonids. Part 2. Distribution and absolute configuration of idoxanthin in various organs and tissues of one atlantic salmon (Salmo salar,L.) fed with astaxanthin

The content of total carotenoids and the ratio astaxanthin/idoxanthin ( = 3,3′-dihydroxy-β,β-carotene-4,4′-dione/3,3′,4′-trihydroxy-β,β-caroten-4-one) in varoius organs and tissues of one Atlantic salmon (Salmo salar, L.) reared indoors in a tank were analyzed after feeding ‘racemic’ ((3R,3′R)/(3R,3′S; meso)/(3S,3′S) 1:2:) astaxanthin (90 mg/kg feed) during one yera. Configurational analysis of astaxanthin was carried out via the (?)-dicamphanate derivative and that of idoxanthin after reaction with (+)-(S)-l-(l-naphthyl)ethyl isocyanate. Separation of all eight optical isomers of idoxanthin-tricarbamate derivatives by HPLC is described. In salmon, enzymatic reduction of astaxanthin was found to be sterospecific leading to th (4′R)-hydroxy group irrespective of the configuration at C(3′), thus resulting in four different stereoisomers of idoxanthin formed from (3R,3′R), (3R,3′S; meso)-, and (3S3′S)-astaxanthin present in the diet.     

A high yielding one-pot method for the preparation of salen ligands

Phenols are converted to salicylaldehydes with paraformaldehyde, MgCl₂-Et₃N in THF, and subsequently treated with (+)-(R,R)-1,2-diammoniumcyclohexane mono-(+)-tartrate salt affording the corresponding salen ligands in high yields. The reactions are conveniently carried out as a one-pot procedure.     

ortho-Formylation of oxygenated phenols

Oxygenated phenols are mono-formylated using a mixture of paraformaldehyde, MgCl₂, and Et₃N in THF. In all cases but one, only one regioisomer of the salicylaldehyde is obtained in good to high yield.     

Graph coloring in the estimation of sparse derivative matrices: Instances and applications

We describe a graph coloring problem associated with the determination of mathematical derivatives. The coloring instances are obtained as intersection graphs of row partitioned sparse derivative matrices. The size of the graph is dependent on the partition and can be varied between the number of columns and the number of nonzero entries. If solved exactly our proposal will yield a significant reduction in computational cost of the derivative matrices. The effectiveness of our approach is demonstrated via a practical problem from computational molecular biology. We also remark on the hardness of the generated coloring instances.