Ion flotation studies of the chlorocomplexes of some platinum group metals

The anionic chlorocomplexes of Au(III), Pt(IV), Pd(II), Ir(IV), Ir(III) and Rh(III) can be floated from aqueous solutions with cationic surfactants of the type RNR'₃Br. The flotation behavior of each metal is reported with respect to variations of hydrochloric acid and sodium chloride concentrations, the R and R' chain lengths, initial surfactant concentrations and initial metal ion concentrations. The flotation behavior of the metals is compared to the anion-exchange selectivity coefficients and a flotation selectivity sequence of Au(III) > Pt(IV), Ir(IV), Pd(II) > Ir(III) > Rh(III) is generally observed. Nearly 100% of Au(III), Pt(IV), Ir(IV) and Pd(II) can be recovered from dilute solutions using the ion flotation procedures.     

Weighted moments for a supercritical branching process in a varying or random environment

Let W be the limit of the normalized population size of a supercritical branching process in a varying or random environment. By an elementary method, we find sufficient conditions under which W has finite weighted moments of the form EWpl(W), where p > 1, l 0 is a concave or slowly varying function.     

On the parameterized complexity of short computation and factorization

A completeness theory for parameterized computational complexity has been studied in a series of recent papers, and has been shown to have many applications in diverse problem domains including familiar graph-theoretic problems, VLSI layout, games, computational biology, cryptography, and computational learning [ADF,BDHW,BFH, DEF,DF1-7,FHW,FK]. We here study the parameterized complexity of two kinds of problems: (1) problems concerning parameterized computations of Turing machines, such as determining whether a nondeterministic machine can reach an accept state in steps (the Short TM Computation Problem), and (2) problems concerning derivations and factorizations, such as determining whether a word can be derived in a grammar in steps, or whether a permutation has a factorization of length over a given set of generators. We show hardness and completeness for these problems for various levels of the hierarchy. In particular, we show that Short TM Computation is complete for . This gives a new and useful characterization of the most important of the apparently intractable parameterized complexity classes. Received August 1, 1994     

Diastereoselective magnesium halide-catalyzed anti-aldol reactions of chiral N-acyloxazolidinones.

A chiral auxilliary-based direct aldol reaction is reported. The reactions are catalytic in magnesium salts and are facilitated by silylation with chlorotrimethylsilane. The adducts isolated are in high diastereoselectivity (up to 32:1 dr) and favor the anti-aldol diastereomer B. Reactions are operationally simple and can be run under ambient atmosphere without rigorous exclusion of water. Many of the adducts are highly crystalline and a single diastereomer can be isolated without chromatography.     

Pseudo-jump inversion,upper cone avoidance,and strong jump-traceability

There are noncomputable c.e. sets, computable from every c.e. set relative to which ∅^′${\varnothing }^{\prime }$ is strongly jump-traceable. This yields a natural pseudo-jump operator, increasing on all sets, which cannot be inverted back to a minimal pair or even avoiding an upper cone.     

Magnesium halide-catalyzed anti-aldol reactions of chiral N-acylthiazolidinethiones

[reaction: see text] Diastereoselective direct aldol reactions of chiral N-acylthiazolidinethiones occur in high yield with preference for the illustrated anti diastereomer. This reaction is catalyzed by 10% MgBr2.OEt2 in the presence of triethylamine and chlorotrimethylsilane. Yields range from 56 to 93% with diastereoselectivity up to 19:1 for a variety of N-acylthiazolidinethiones and unsaturated aldehydes.     

Computability-theoretic and proof-theoretic aspects of partial and linear orderings

Szpilrajn’s Theorem states that any partial orderP=〈S,<p〉 has a linear extensionP=〈S,<L〉. This is a central result in the theory of partial orderings, allowing one to define, for instance, the dimension of a partial ordering. It is now natural to ask questions like “Does a well-partial ordering always have a well-ordered linear extension?” Variations of Szpilrajn’s Theorem state, for various (but not for all) linear order typesτ, that ifP does not contain a subchain of order typeτ, then we can chooseL so thatL also does not contain a subchain of order typeτ. In particular, a well-partial ordering always has a well-ordered extension.We show that several effective versions of variations of Szpilrajn’s Theorem fail, and use this to narrow down their proof-theoretic strength in the spirit of reverse mathematics.     

Degree Spectra of Relations on Boolean Algebras

We show that every computable relation on a computable Boolean algebra is either definable by a quantifier-free formula with constants from (in which case it is obviously intrinsically computable) or has infinite degree spectrum.