The modified bordering method to evaluate eigenvalues and eigenvectors of normal matrices

A bordering procedure is here proposed to evaluate the eigensystem of hermitian matrices, and more in general of normal matrices, when the spectral decomposition is known of then–1×n–1 principal minor. The procedure is also applicable to special real and nonsymmetric matrices here named quasi-symmetric. The computational cost to write the characteristic polynomial isO(n ²), using a new set of recursive formulas. A modified Brent algorithm is used to find the roots of the polynomial. The eigenvectors are evaluated in a direct way with a computational cost ofO(n ²) for each one. Some numerical considerations indicate where numerical difficulties may occur. Numerical results are given comparing this method with the Givens-Householder one.     

A smooth method for the finite minimax problem

We consider unconstrained minimax problems where the objective function is the maximum of a finite number of smooth functions. We prove that, under usual assumptions, it is possible to construct a continuously differentiable function, whose minimizers yield the minimizers of the max function and the corresponding minimum values. On this basis, we can define implementable algorithms for the solution of the minimax problem, which are globally convergent at a superlinear convergence rate. Preliminary numerical results are reported.This research was partially supported by the National Research Program on Metodi di ottimizzazione per le decisioni, Ministero dell'Università e della Ricerca Scientifica e Tecnologica, Italy.     

Photodegradation of Polychlorinated Dibenzo-p-dioxins in Liquid Samples by Near UV Light Irradiation

Photodegradation of 1,2,3,4-tetrachlorodibenzo-p-dioxin (1,2,3,4-TCDD) in hexane solution was studied under controlled near-UV light exposure in the spectral region from 325 to 269 nm. GC-MS was used to detect the amount of unreacted dioxin and to characterize the relevant degradation products. Irradiation experiments carried out at a constant light energy (700 mJ) showed that the percentage of 1,2,3,4-TCDD left in the solution after irradiation changed from about 55 to 75%, with a minimum of 55% at 310 nm. Further irradiation experiments carried out at two wavelengths, namely 310 and 269 nm, and light energy ranging from 0 to 4000 mJ, showed that the photodegradation reaction of the TCDD always followed a pseudo-first-order kinetic, with a rate constant of 8 × 10⁻⁴and 5 × 10⁻⁴mJ⁻¹, respectively. These experiments also showed that trichloro- and dichloro-dibenzo-p-dioxins were produced with less than 15% of the initial quantity of TCDD, leading to the conclusion that the dechlorination process is a minor photolysis pathway.     

Zinc-catalyzed oxidation of 5-S-cysteinyldopa to 2,2'-bi(2H-1,4-benzothiazine): tracking the biosynthetic pathway of trichochromes, the characteristic pigments of red hair

Trichochromes, the peculiar pigments of red human hair, featuring the Delta(2,2)(')-bi(2H-1,4-benzothiazine) skeleton, are known to arise from cysteinyldopas, mainly the 5-S-isomer (5). However, the mode of formation and the direct precursors have remained largely undefined. To fill this gap, we investigated the oxidation of 5 in air or with chemical and enzymatic agents under biomimetic conditions. In the presence of zinc ions, which occur in epidermal tissues at significant concentrations, the reaction course is diverted toward the formation of a labile 3-carboxy-2H-1,4-benzothiazine intermediate (11), which was identified by direct NMR analysis. Structural formulation was supported by characterization of the analogous compound 13 isolated from oxidation of the model 5-methyl-3-S-cysteinylcatechol (12) after methylation. In the further stages of the oxidation, diastereomeric 2,2'-bi(2H-1,4-benzothiazine) 15 and 14 were obtained from 5 and 12, respectively, the reaction proceeding at a higher rate and to a greater extent in the presence of acids. The dimers were shown to readily convert to each other in the presence of acids. In the case of the methylated dimers 14, a 2,2'-bi(4H-1,4-benzothiazine) intermediate (16) was isolated and characterized. In acidic media, trichochrome C (1a), the most abundant in red human hair, was smoothly formed from aerial oxidation of 15, and under similar conditions, trichochrome-related products (17 and 18) were obtained from 14 prior to or after methylation. The presence of 1a and precursors 5 and 15 was investigated by HPLC analysis of red hair samples following mild proteolytic digestion. On the basis of these data, a likely biosynthetic route to trichochrome pigments of red human hair is depicted.