Special simplices and Gorenstein toric rings

Athanasiadis [Ehrhart polynomials, simplicial polytopes, magic squares and a conjecture of Stanley, J. Reine Angew. Math., to appear.] studies an effective technique to show that Gorenstein sequences coming from compressed polytopes are unimodal. In the present paper we will use such the technique to find a rich class of Gorenstein toric rings with unimodal h-vectors arising from finite graphs.     

The upper and lower second order directional derivatives of a sup-type function

The purpose of this paper is to give a formula for expressing the second order directional derivatives of the sup-type functionS(x) = sup{f(x, t); t T} in terms of the first and second derivatives off(x, t), whereT is a compact set in a metric space and we assume thatf, f/x and ² f/x ² are continuous on ⁿ × T. We will give a geometrical meaning of the formula. We will moreover give a sufficient condition forS(x) to be directionally twice differentiable.     

Electron spin resonance study of polymerization of butadiene with tris(acetylacetonato)titanium and triethylaluminum

ESR spectra of homogeneous catalyst derived from tris(acetylacetonato)titanium(III) and triethylaluminum were observed at several temperatures from ?78°C, to +25°C, at molar ratios of aluminum to titanium of 1–108. At ?78°C, this catalyst yields a violet complex which shows an ESR signal with a g value of 1.959 and is associated with the first intermediate. At ?40°C to ?30°C, this signal decreases, and two signals with g values of 1.947 and 1.960 are observed. The latter two signals diminish at ?5°C to +10°C, while two kinds of new signals with g values of 1.965 and 1.969 appear overlapping each other. The structures of the species corresponding to these five signals are discussed on the basis of the ESR spectra, the intensity change, and the unpaired spin distribution. A new signal with a g value of 1.978 is found in the presence of butadiene at 25°C at Al/Ti > 8 and is assigned to a growing end of polybutadiene with this catalyst. The polymer yield increases remarkedly at Al/Ti molar ratios greater than 10. The microstructure of the resulting polymer consists almost completely of 1,2 units. The structure of the growing end is proposed to be a titanium (III) species containing two 1-substituted allyl groups, by comparison with the structure ascribed to the growing end of polybutadiene with n-butyl titanate-triethylaluminum catalyst.     

Electron spin resonance study on polymerization of alkyl acrylates with n-butyl titanate(IV)–triethylaluminum catalyst

n-Butyl titanate(IV)–triethylaluminum catalyst at Al/Ti molar ratios greater than 6 polymerizes methyl and n-butyl acrylates at ?78°C. The polymerization system which includes methyl acrylate at ?78°C, gives two ESR signals with g factors of 1.958 and 1.961 that overlap each other. The absorption intensity of the latter signal is approximately proportional to the polymer chain concentration calculated from polymer yield and the molecular weight. The polymerization system at Al/Ti ratios smaller than 3 has no catalytic activity on the polymerization and shows only the ESR signal with the g factor of 1.958. On the basis of these facts the ESR signal with the g factor of 1.961 is attributed to the active growing end of poly(methyl acrylate) with this catalyst. The character of this active growing end is discussed.     

Trifluoromethyl‐Radical‐Mediated Carbonylation of Alkanes Leading to Ethynyl Ketones

The carbonylation of alkanes 1 under radical‐reaction conditions was examined by using ethynyl triflone A as the unimolecular chain‐transfer (UMCT) reagent. Good to moderate yields of ethynyl ketones 2 were prepared by means of this three‐component coupling reaction. Higher CO pressures as well as lower concentrations of triflone A improved the efficiency of the reaction over the direct addition, the latter leading to alkylated ethynes 3 . In contrast to the reaction with A , the reaction of cyclohexane ( 1a ) with allyl triflone B (= ethyl 2‐methylene‐3‐[(trifluoromethyl)sulfonyl]propanoate) in the presence of CO gave a mixture of carbonylation products, including 8a formed from two molecules each of cyclohexane, CO, and allyl triflone B .     

Total synthesis of cis-solamin and its inhibitory action with bovine heart mitochondrial complex I

A convergent total synthesis of cis-solamin (1a) and its diastereomer (1b) was accomplished. A key reaction of this approach was the use of VO(acac)₂-catalyzed diastereoselective epoxidation of (Z)-bis-homoallylic alcohol 3 followed by spontaneous cyclization for the cis-THF ring formation. By comparison of the optical rotation of the two possible diastereomers, it is suggested that the absolute configuration of natural cis-solamin is 1a. Inhibitory action of synthetic 1a and 1b with bovine heart mitochondrial complex I are reported.     

Total synthesis of cis-solamin

[structure: see text] A convergent total synthesis of cis-solamin and its diastereomer was accomplished using VO(acac)2-catalyzed diastereoselective epoxidation followed by cyclization of bis-homoallylic alcohol as the key step. By comparison of the optical rotation of two possible diastereomers, it is suggested that the absolute configuration of natural cis-solamin is 1a.     

Polymerization of coordinated monomers VII. A kinetic study on the alternating copolymerization of methyl methacrylate with styrene using stannic chloride

The alternating copolymerization of methyl methacrylate with styrene in the presence of stannic chloride at ?50°C in toluene was kinetically investigated both under photoirradiation and with the tri-n-butylboron-benzoyl peroxide initiator. The concentrations of the binary and ternary molecular complexes in the copolymerization solution were estimated by use of the equilibrium constants. The rates are found to be proportional to the 1.5th and 1.0th orders of the concentration of the ternary molecular complex composed of stannic chloride, methyl methacrylate, and styrene, under photoirradiation and with initiator, respectively. The conversion increases proportionally with the polymerization time, while the degree of polymerization is constant irrespective of the time. The rates depend linearly upon the square root of the intensity of the incident light and upon the concentration of tri-n-butylboron, respectively. The alternating copolymerization is confirmed experimentally to precede the homopolymerization of the monomer charged in large excess both under photoirradiation and with initiator. The kinetic results indicate consistently that the alternating copolymerization proceeds through the homopolymerization of the ternary molecular complex in the steady state with a bimolecular termination. Both the conventional radical mechanism and the double complex mechanism are unsuitable for the present alternating copolymerization.     

Polymerization of coordinated monomers. III. Copolymerization of acrylonitrile–zinc chloride,methacrylonitrile–zinc chloride or methyl methacrylate–zinc chloride complex with styrene

The 1:1 or 2:1 complex of acrylonitrile, methacrylonitrile, or methyl methacrylate with ZnCl₂ was copolymerized with styrene at the temperature of 0–30°C without any initiator. The structure of the copolymer from methyl methacrylate complex and styrene was examined by NMR spectroscopy. The complexes of acrylonitrile or methacrylonitrile with ZnCl₂ gave a copolymer containing about 50 mole-% styrene units. The complexes of methyl methacrylate yielded an alternating copolymer when the feed molar ratio of methyl methacrylate to styrene was small, but with increasing feed molar ratio the resulting copolymer consisted of about 2 moles of methyl methacrylate per mole of styrene. The formation of a charge-transfer complex of styrene with a monomer coordinated to zinc atom was inferred from the ultraviolet spectra. The regulation of the copolymerization was considered to be effected by the charge-transfer complex. The copolymer resulting from the 2:1 methyl methacrylate–zinc chloride complex had no specific tacticity, whereas the copolymer from the 1:1 complex was richer in coisotacticity than in cosyndiotacticity. The change of the composition of the copolymer and its specific tacticity in the polymerization of the methyl methacrylate complex is related to the structure of the complex.