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.     

Application of samarium diiodide (SmI2)-induced reduction of gamma-acetoxy-alpha,beta-enoates with alpha-specific kinetic electrophilic trapping for the synthesis of amino acid derivatives

Gamma-acetoxy-alpha,beta-enoates were easily reduced by samarium diiodide (SmI2) in THF to generate samarium dienolates which were kinetically trapped with ease at their alpha-positions by electrophiles (proton, aldehydes or ketones) to yield (E)-alkene dipeptide isosteres or gamma-amino acid derivatives in high chemical yields.     

SmI2-mediated reduction of gamma,gamma-difluoro-alpha,beta-enoates with application to the synthesis of functionalized (Z)-fluoroalkene-type dipeptide isosteres

A samarium diiodide (SmI(2))-mediated reduction of gamma,gamma-difluoro-alpha,beta-enoates (15, 29, and 34) was successfully applied to the synthesis of (Z)-fluoroalkene dipeptide isosteres (23, 30, and 35), which have served as potential dipeptide mimetics. Reduction of the gamma,gamma-difluoro-alpha,beta-enoates by SmI(2) proceeded via successive two-electron transfers to form dienolate species which upon kinetically controlled trapping with t-BuOH yielded Xaa-Gly-type fluoroalkene isosteres exemplified by 23, 30, and 35. Replacement of the t-BuOH kinetic trapping agent with aldehydes or ketones provided access to alpha-substituted fluoroalkene isosteres (43 and 45) through aldol reactions of Sm-dienolates with the carbonyl compounds. Of particular note, the use of the SmI(2)-HCHO reagent system with chiral enoate 34 provided D-Phe-psi[(Z)-CF[double bond]CH]-D/L-Ser isosteres (45), which could be converted to enantiomerically pure isosteres (49-52) that bore a variety of side chain functionalities at the alpha-position. This was achieved by a sequence of manipulations consisting of beta-lactone formation followed by chromatographic separation and ring-opening with soft nucleophiles. Included in the present work is the first utilization of a Rh-catalyzed Reformatsky reaction of chiral imines for the stereoselective preparation of alpha,alpha-difluoro-beta-amino acid derivatives (28 and 33). The appropriate choice of reagents (carbonyl compounds for kinetic trapping or ring-opening nucleophiles and imines for Reformatsky reactions) allows the presented methodology to yield various fluoroalkene isosteres possessing a wide range of side chain functionalities.     

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.     

On-line column preconcentration for the determination of cobalt in sea water by flow-injection chemiluminescence detection

A flow-injection method for the determination of dissolved cobalt(II) in sea water has been studied based on a combination of column preconcentration using 8-quinolinol immobilized on silica gel, fluoride containing metal alkoxide glass (8HQ-MAF) and chemiluminescence detection with a gallic acid-hydrogen peroxide system. Co(II) is selectively recovered from an acidified sample with 8-quinolinol immobilized on silica gel. After elution with dilute hydrochloric acid the resultant eluent is mixed with the reagent solutions, heated to 60^°C and then introduced into the CL cell. The analysis time including the 2-min sample load was 8 min per sea water sample with a corresponding detection limit of 0.62 ngl^-1 (3). The average standard deviation calculated for 10 replicate measurements of artificial sea water samples with a concentration of 10 ngl^-1 cobalt was ±2.1%. The method has been tested with the standard reference sea waters NASS and CASS.