Some remarks on the symmetric rank-one update

We consider the symmetric rank-one, quasi-Newton formula. The hereditary properties of this formula do not require quasi-Newton directions of search. Therefore, this formula is easy to use in constrained optimization algorithms; no explicit projections of either the Hessian approximations or the parameter changes are required. Moreover, the entire Hessian approximation is available at each iteration for determining the direction of search, which need not be a quasi-Newton direction. Theoretical difficulties, however, exist. Even for a positive-definite, quadratic function with no constraints, it is possible that the symmetric rank-one update may not be defined at some iteration. In this paper, we first demonstrate that such failures of definition correspond to either losses of independence in the directions of search being generated or to near-singularity of the Hessian approximation being generated. We then describe a procedure that guarantees that these updates are well-defined for any nonsingular quadratic function. This procedure has been incorporated into an algorithm for minimizing a function subject to box constraints. Box constraints arise naturally in the minimization of a function with many minima or a function that is defined only in some subregion of the space.     

Universal Abelian H-spaces

The question of the existence of universal homotopy commutative and homotopy associative H-spaces (called Abelian H-spaces) is studied. Such a space T(X) would prolong a map from X into an Abelian H-space to a unique H-map from T into X. Examples of such pairs (X,T) are given and conditions are discussed which limit the possible spaces X for which such a T can exist. Contrary to published assertions, the Anick spaces are shown not to be universal Abelian H-spaces for the corresponding Moore spaces; however conditions are discussed which could lead to a universal property with respect to a more limited range of targets, and a restricted universal property is proven.     

On decompositions in homotopy theory

We first describe Krull-Schmidt theorems decomposing spaces and simply-connected co- spaces into atomic factors in the category of pointed nilpotent -complete spaces of finite type. We use this to construct a 1-1 correspondence between homotopy types of atomic spaces and homotopy types of atomic co- spaces, and construct a split fibration which connects them and illuminates the decomposition. Various properties of these constructions are analyzed.

     

Oxygenation of zinc dialkyldithiocarbamate complexes: isolation, characterization, and reactivity of the stoichiometric oxygenates

S-oxygenation of dithiocarbamate (DTC) complexes has been implicated in their function as industrial anti-oxidants, as well as in their use as pesticides and most recently in their cumulative toxicity, but little is known of the species generated. Several S-oxygenated derivatives of N,N-disubstituted DTCs have been synthesized, characterized by a variety of methods, and their structure and reactivity examined. Low-temperature reaction of bis(N,N-diethyldithiocarbamato)zinc(II), Zn(deDTC)2 1, with oxygenating reagents (hydrogen peroxide, m-chloroperbenzoic acid, urea hydrogen peroxide) yields mono-oxygenated DTC complexes (N,N-peroxydiethyldithiocarbamato)(N,N-diethyldithiocarbamato)zin(II), Zn(O-deDTC)(deDTC), 2 and bis(N,N-peroxydiethyldithiocarbamato)zinc(II), Zn(O-deDTC)2, 3. The tetraoxygenated derivative bis(N,N-diethylthiocarbamoylsulfinato)zinc(II), Zn(O(2)-deDTC)2, 4, was cleanly obtained by initial reaction of the DTC salts with stoichiometric oxidant prior to complexation with Zn(II). X-ray crystallographic analysis of 2, 3, and 4 show that the peroxydithiocarbamate ligands are S,O-bound. Similar derivatives were obtained from the homoleptic dimethyl and pyrollidine DTC Zn complexes. These oxygenated species display unique 1H and 13C NMR variable-temperature spectra, as the symmetry of DTC ligand is broken upon oxygenation; total line shape analysis (TLSA) was used to compare the energetic parameters for rotation about the C-N bond in several derivatives. Compounds 2, 3, and 4 were deoxygenated by alkyl phosphine, regenerating the parent dithiocarbamate 1. The peroxydithiocarbamate complexes were susceptible to base-catalyzed hydrolytic decomposition, giving ligand-based products indicative of S-oxidation and S-extrusion.     

A novel heterocyclic atom exchange reaction with Lawesson's reagent: a one-pot synthesis of dithiomaltol

A one-pot reaction of maltol with Lawesson's reagent generates dithiomaltol, a thiopyran-4-thione, via an unusual heterocyclic atom exchange (HCAE) reaction; only pyrones with proton or aliphatic substituents undergo the HCAE substitution.     

A Convenient One-Pot Synthesis of Di-t-butylphosphinic Chloride

A convenient one-pot synthesis of (t-Bu)₂P(O)Cl (1) from m-chloroperbenzoic acid and (t-Bu)₂PCl is described. The byproduct m-chlorobenzoic acid is neutralized by addition of Et₃N. Complex 1 crystallizes in the monoclinic space group P2₁/c (No. 14), with a = 5.9637(2) Å, b = 11.4734(4) Å, c = 16.2400(5) Å and β = 107.7160(14).