An a posteriori error estimator based on shifts for positive Hermitian eigenvalue problems

This work deals with an a posteriori error estimator for Hermitian positive eigenvalue problems. The proposed estimator is based on the residual and the definition of suitable shifts in the matrix spectrum. The mathematical properties (certification and sharpness) are investigated and some numerical experiments are proposed.     

Polynomials whose coefficients coincide with their zeros

In this paper we consider monic polynomials such that their coefficients coincide with their zeros. These polynomials were first introduced by S. Ulam. We combine methods of algebraic geometry and dynamical systems to prove several results. We obtain estimates on the number of Ulam polynomials of degree N. We provide additional methods to obtain algebraic identities satisfied by the zeros of Ulam polynomials, beyond the straightforward comparison of their zeros and coefficients. To address the question about the existence of orthogonal Ulam polynomial sequences, we show that the only Ulam polynomial eigenfunctions of hypergeometric type differential operators are the trivial Ulam polynomials \(\{x^N\}_{N=0}^\infty \). We propose a family of solvable N-body problems such that their stable equilibria are the zeros of certain Ulam polynomials.     

Ditocopheryl Sulfides and Disulfides: Synthesis and Antioxidant Profile

Symmetrical ditocopheryl disulfides ( Toc ) ₂ S₂ and symmetrical and unsymmetrical ditocopheryl sulfides ( Toc )₂ S were simply prepared under remarkably mild conditions with complete control of the regiochemistry by using δ-, γ-, and β-tocopheryl-N-thiophthalimides ( Toc-NSPht ) as common starting materials. The roles of sulfur atom(s), H-bond and aryl ring substitution pattern on the antioxidant profile of these new compounds, which were assembled by linking together two tocopheryl units, are also discussed.     

A formal total synthesis of eleutherobin using the ring-closing metathesis (RCM) reaction of a densely functionalized diene as the key step: investigation of the unusual kinetically controlled RCM stereochemistry

Asymmetric oxyallylation reactions and ring-closing metathesis have been used to synthesize compound 3, a key advanced intermediate used in the total synthesis of eleutherobin reported by Danishefsky and co-workers. The aldehyde 6, which is readily prepared from commercially available R-(-)-carvone in six steps in 30 % overall yield on multigram quantities, was converted into the diene 5 utilizing two stereoselective titanium-mediated Hafner-Duthaler oxyallylation reactions. The reactions gave the desired products (8 and 12) in high yields (73 and 83 %, respectively) as single diastereoisomers, with the allylic alcohol already protected as the p-methoxyphenyl (PMP) ether, which previous work has demonstrated actually aids ring-closing metathesis compared to other protective groups and the corresponding free alcohol. Cyclization under forcing conditions, using Grubbs' second-generation catalyst 13, gave the ten-membered carbocycle (E)-14 in 64 % yield. This result is in sharp contrast to similar, but less functionalized, dienes, which have all undergone cyclization to give the Z stereoisomers exclusively. A detailed investigation of this unusual cyclization stereochemistry by computational methods has shown that the E isomer of the ten-membered carbocycle is indeed less thermodynamically stable than the corresponding Z isomer. In fact, the selectivity is believed to be due to the dense functionality around the ruthenacyclobutane intermediate that favors the trans-ruthenacycle, which ultimately leads to the less stable E isomer of the ten-membered carbocycle under kinetic control. During the final synthetic manipulations the double bond of enedione (E)-16 isomerized to the more thermodynamically stable enedione (Z)-4, giving access to the advanced key-intermediate 3, which was spectroscopically and analytically identical to the data reported by Danishefsky and co-workers, and thereby completing the formal synthesis of eleutherobin.     

Synthetic strategies to alpha-trifluoromethyl and alpha-difluoromethyl substituted alpha-amino acids

The combination of the unique physical and chemical properties of fluorine with proteinogenic amino acids represents a new approach to the design of biologically active compounds including peptides with improved pharmacological parameters. Therefore, the development of routine synthetic methods which enable the effective and selective introduction of fluorine into the desired amino acids from readily available starting materials is of significant synthetic importance. The scope of this critical review is to summarize the most frequently employed strategies for the synthesis of alpha-difluoromethyl and alpha-trifluoromethyl substituted alpha-amino acids (114 references).