Convex bodies and multiplicities of ideals

We associate convex regions in ? ⁿ to m-primary graded sequences of subspaces, in particular m-primary graded sequences of ideals, in a large class of local algebras (including analytically irreducible local domains). These convex regions encode information about Samuel multiplicities. This is in the spirit of the theory of Gröbner bases and Newton polyhedra on the one hand, and the theory of Newton-Okounkov bodies for linear systems on the other hand. We use this to give a new proof as well as a generalization of a Brunn-Minkowski inequality for multiplicities due to Teissier and Rees-Sharp.     

Chemical Sensors and Microinstrumentation

The ¹H NMR spectra of racemic (erythro) methoxamine free base, 1, 1-(1-aminoethyl)-2,5-dimethoxybenzenemethanol, have been studied at or near ambient temperatures in CDCl₃ or CD₃CN with the added chiral lanthanide shift reagent (LSR), tris[3-(heptafluoropropylhydroxymethylene)-(+)-camphorato] europium (III), 2. Spectrometers operating at 60, 200 and 300 MHz were employed; some COSY spectra were acquired to support assignments. Enantiomeric shift differences (Δ Δ Δ) were observed for several nuclei of 1 with added 2 in either CDCl₃ or CD₃CN and an “anomalous” (upfield) shift was seen for the NCH signal. A high degree of signal broadening was seen for runs with either solvent, and surprising similarity was found for the slopes in plots of chemical shift versus [2]/[1] molar ratios for the different nuclei of 1 whether the hydrogen bond donor solvent (CDCl₃) or the hydrogen bond acceptor solvent (CD₃CN) was used. Together with the anomalous shift noted above, these results are interpreted as consistent with strong bidentate chelation of 2 by 1.     

On the Equivariant Cohomology of Subvarieties of a \mathfrak{B}-Regular Variety

By a $\mathfrak{B}$ -regular variety, we mean a smooth projective variety over $\mathbb{C}$ admitting an algebraic action of the upper triangular Borel subgroup $\mathfrak{B} \subset {\text{SL}}_{2} {\left( \mathbb{C} \right)}$ such that the unipotent radical in $\mathfrak{B}$ has a unique fixed point. A result of Brion and the first author [4] describes the equivariant cohomology algebra (over $\mathbb{C}$ ) of a $\mathfrak{B}$ -regular variety X as the coordinate ring of a remarkable affine curve in $X \times \mathbb{P}^{1}$ . The main result of this paper uses this fact to classify the $\mathfrak{B}$ -invariant subvarieties Y of a $\mathfrak{B}$ -regular variety X for which the restriction map i _Y : H ^*(X) → H ^*(Y) is surjective.     

Efficient and Convenient Deprotection of Thiocarbonyl to Carbonyl Compounds Using 3-Carboxypyridinium and 2,2′-Bipyridinium Chlorochromates in Solution,Dry Media,and under Microwave Irradiation

Summary. A synthetic utility of 3-carboxypyridinium (CPCC) and 2,2-bipyridinium (BPCC) chlorochromates in deprotection reactions is reported. Different types of thioamides, thioureas, thiono esters, and thioketones are deprotected to their corresponding carbonyl compounds with these reagents in good to excellent yields. The reactions were carried out in solution, under solvent-free conditions, and under microwave irradiation. The results show that with both reagents the rates of the reactions and the yields are usually highest under microwave irradiation.     

H2O2/SOCl2: a useful reagent system for the conversion of thiocarbonyls to carbonyl compounds

The H₂O₂/SOCl₂ reagent system has been used as a new and efficient reagent for deprotection of thiocarbonyls to carbonyl compounds. The salient features of this protocol are short reaction times, good chemoselectivity, clean reaction profiles, and simple work-up that preclude the use of toxic solvents.     

ZrCl4 as an Efficient Catalyst for Crossed‐Aldol Condensation of Cyclic Ketones with Aromatic Aldehydes in Refluxing Ethanol

The aldol condensation reaction of cyclic ketones with aromatic aldehydes in ethanol under reflux conditions using ZrCl₄ as a catalyst to afford the corresponding α,α′‐bis(substituted benzylidene and cinnamylidene)cycloalkanones in excellent yields has been described. No self‐condensation product was produced.     

Morse Theory and Euler Characteristic of Sections of Spherical Varieties

We generalize the formula for the Euler characteristic of a hypersurface in the torus (C^*)^d, due to D. Bernstein, Kusnirenko and Khovanskii, to the actions with spherical orbits. For this, we use a variant of classical Morse theory.