Toric cubes

A toric cube is a subset of the standard cube defined by binomial inequalities. These basic semialgebraic sets are precisely the images of standard cubes under monomial maps. We study toric cubes from the perspective of topological combinatorics. Explicit decompositions as CW-complexes are constructed. Their open cells are interiors of toric cubes and their boundaries are subcomplexes. The motivating example of a toric cube is the edge-product space in phylogenetics, and our work generalizes results known for that space.     

Synthesis and structural characterization of trivalent amino acid derived chiral phosphorus compounds

Reaction of the N-toluenesulfonyl derivatives of (S)-alanine, phenylalanine, and valine (4-6) with PhPCl(2) gave in high yield the 4-methyl, benzyl, and isopropyl derivatives (7-9) of 2-phenyl-1-p-toluenesulfonyl-1,3,2-oxazaphospholidin-5-one. The ratios of the (2S,4S)/(2R,4S) diastereomers (cis/trans isomers) were 1:1, 2:1, and 10:1 for the methyl, benzyl, and isopropyl derivatives 7a,b, 8a,b, and 9a,b, respectively. For 7a,b, both isomers could be crystallized, but for the others only the major isomers were isolable. The X-ray crystal structure of 9a shows that the isopropyl and phenyl groups are mutually cis and that the tolyl moiety is oriented s-trans to both the isopropyl and phenyl groups. Reaction of 6 with Cl(2)PCH(2)CH(2)PCl(2) (10) gave a 56:38:7 mixture of the cis/cis, cis/trans, and trans/trans diphosphorus heterocycles 11a-c. The major isomer could be crystallized and isolated free of the other diastereomers. Reaction of 6 with EtPCl(2) gave a 6:1 mixture of cis/trans isomers of the ethyl-substituted heterocycles 12a,b as an inseparable oil but allowed confirmation of the structure of 11a. Slow epimerization at phosphorus may occur by inversion but more likely by ring opening/closure, since 7b, 9a, and 11a give rise upon standing in solution to mixtures containing starting material and 7a, 9b, and 11b, respectively, along with the free amino acid derivatives 4 and 6. The NMR spectra, and in particular the coupling constants between the alpha-hydrogen atom of the amino acid moiety and phosphorus, were used to establish the identities of the cis and trans isomers. Reaction of 9a with (THF)W(CO)(5) gave the phosphorus-ligated adduct (9a)W(CO)(5) (13), and the IR spectrum of this complex shows that 9a is a strongly electron-withdrawing ligand. The geometry of the sulfonamide moiety is discussed in detail, as are the (1)H NMR coupling constants. The data are consistent with the presence of little steric interaction between the cis isopropyl and phosphorus substituent in 9a, 11a, and 12a and orientation of the tolyl moiety s-cis to the isopropyl group in 9b, 12b, and 13.     

S-oxygenation of thiocarbamides I: Oxidation of phenylthiourea by chlorite in acidic media

The oxidation of 1-phenyl-2-thiourea (PTU) by chlorite was studied in aqueous acidic media. The reaction is extremely complex with reaction dynamics strongly influenced by the pH of reaction medium. In excess chlorite concentrations the reaction stoichiometry involves the complete desulfurization of PTU to yield a urea residue and sulfate: 2ClO2- + PhN(H)CSNH2 + H2O --> SO4(2-) + PhN(H)CONH2 + 2Cl- + 2H+. In excess PTU, mixtures of sulfinic and sulfonic acids are formed. The reaction was followed spectrophotometrically by observing the formation of chlorine dioxide which is formed from the reaction of the reactive intermediate HOCl and chlorite: 2ClO2- + HOCl + H+ --> 2ClO2(aq) + Cl- + H2O. The complexity of the ClO2- - PTU reaction arises from the fact that the reaction of ClO2 with PTU is slow enough to allow the accumulation of ClO2 in the presence of PTU. Hence the formation of ClO2 was observed to be oligooscillatory with transient formation of ClO2 even in conditions of excess oxidant. The reaction showed complex acid dependence with acid catalysis in pH conditions higher than pKa of HClO2 and acid retardation in pH conditions of less than 2.0. The rate of oxidation of PTU was given by -d[PTU]/dt = k1[ClO2-][PTU] + k2[HClO2][PTU] with the rate law: -d[PTU]/dt = [Cl(III)](T)[PTU]0/K(a1) + [H+] [k1K(a1) + k2[H+]]; where [Cl(III)]T is the sum of chlorite and chlorous acid and K(a1) is the acid dissociation constant for chlorous acid. The following bimolecular rate constants were evaluated; k1 = 31.5+/-2.3 M(-1) s(-1) and k2 = 114+/-7 M(-1) s(-1). The direct reaction of ClO2 with PTU was autocatalytic in low acid concentrations with a stoichiometric ratio of 8:5; 8ClO2 + 5PhN(H)CSNH2 + 9H2O --> 5SO4(2-) + 5PhN(H)CONH2 + 8Cl- + 18H+. The proposed mechanism implicates HOCl as a major intermediate whose autocatalytic production determined the observed global dynamics of the reaction. A comprehensive 29-reaction scheme is evoked to describe the complex reaction dynamics.     

The possibility of semiconductor–metal transition in a spherical quantum dot

We observe an abrupt change in diamagnetic susceptibility at critical donor concentration for an $\text{ Al }_\mathrm{x}\text{ Ga }_\mathrm{1-x}\text{ As/GaAs }$ Al x Ga 1 ? x As/GaAs quantum dot system in the effective mass approximation indicating a possible semiconductor metal transition. The effect of confining potential and the laser intensity on the abrupt change in diamagnetic susceptibility has also been studied. The effect of nonparabolicity of the conduction band has been included in our calculations. Results are presented and discussed.     

Carbocyclization of E,E-1,4-Diphenyl-1,3-butadiene with Dichloroalkanes Mediated by Rieke Metals

The Rieke metal complexes of barium and strontium readily react with E,E-1,4-diphenyl-1,3-butadiene to form metal-diene reagents. Upon treatment with l .n-dichloroalkanes, these metallocycles are transformed into ring derivatives in excellent chemical yield.     

Two Equivalent Reduction of Copper(I) Complexes; Evidence of an Anionic Copper Species

Reactive copper solutions have been prepared by a cold temperature, two equivalent lithium naphthalenide reduction of copper(I) complexes. One equivalent of organic halides react with one equivalent of the anionic copper to yield organocopper reagents.