S‐((Phenylsulfonyl)difluoromethyl)thiophenium Salts: Carbon‐Selective Electrophilic Difluoromethylation of β‐Ketoesters, β‐Diketones,and Dicyanoalkylidenes

S‐((Phenylsulfonyl)difluoromethyl)thiophenium salts were designed and prepared by a triflic acid catalyzed intramolecular cyclization of ortho‐ethynyl aryldifluoromethyl sulfanes. The thiophenium salts were found to be efficient as electrophilic difluoromehtylating reagents for introduction of a CF₂H group to sp³‐hybridized carbon nucleophiles such as of β‐ketoesters and dicyanoalkylidenes. The (phenylsulfonyl)difluoromethyl group can be readily transformed into CF₂H under mild reaction conditions. Enantioselective electrophilic difluoromethylation was also achieved in the presence of bis(cinchona) alkaloids.     

Synthesis of Both Enantiomers of Nine‐Membered CF3‐Substituted Heterocycles Using a Single Chiral Ligand: Palladium‐Catalyzed Decarboxylative Ring Expansion with Kinetic Resolution

The two enantiomers of trifluoromethyl‐benzo[c][1,5]oxazonines, (R)‐ 4 and (S)‐ 4 , can be selectively accessed with high enantiopurity by the Pd‐catalyzed ring‐expansion reaction of trifluoromethyl‐benzo[d][1,3]oxazinones ( 1 ) with vinyl ethylene carbonates ( 3 ) using one antipode of a chiral ligand. Initially, the reaction proceeds by a double decarboxylative ring‐expansion with kinetic resolution of 1 in the presence of a Pd‐catalyst/chiral ligand to provide (R)‐ 4 with high enantiopurity. At the same time, the nonreactive antipode of 1 , (S)‐ 1 , which was recovered with an impeccable s factor of up to 713 and an ideal chemical yield, was transferred into the antipode of the products, (S)‐ 4 , with high enantiopurity by a second run of the Pd‐catalyzed double decarboxylation reaction, but this time without any chiral auxiliary. Thus, both antipodes of the chiral trifluoromethyl heterocycles 4 can be obtained in excellent enantiopurity using only a single antipode of the chiral catalyst.     

An addition formula for hyperbolic space

The addition formula for Gegenbauer polynomials on the Euclidean sphere is generalized to the unit hyperboloid. Applications are given in terms of new bounds on the cardinality of few-distance sets in Lobatchevski space.     

A note on the spherical embeddings of strongly regular graphs

A primitive symmetric association scheme of class 2 is naturally embedded as a two-distance set in the unit sphere of Euclidean space, with respect to the primitive idempotent E₁ of the Bose–Mesner algebra of the association scheme. Then it is shown that the ratio of the two distances of the two-distance set is instantly read from the character table (i.e., the first eigen matrix P) of the association scheme.     

On some spherical t-designs

Let G be a finite subgroup of the orthogonal group O(d). It is shown that many spherical t-designs are constructed from G, if some particular irreducible representations of O(d) remain irreducible when restricted to G.     

Uniqueness of certain association schemes

We prove the uniqueness of the two association schemes which appear in recent work of Henry Cohn and others in connection with their study of universally optimal spherical codes in Euclidean spaces: one is the class 4 association scheme with 40 vertices in and the other one is the class 3 association scheme with 64 vertices in . We prove the uniqueness mainly by geometric considerations with some computational help.     

Exploring the intermediates of photochemical CO2 reduction: reaction of Re(dmb)(CO)3 COOH with CO2

We have investigated the reaction of Re(dmb)(CO)(3)COOH with CO(2) using density functional theory, and propose a mechanism for the production of CO. This mechanism supports the role of Re(dmb)(CO)(3)COOH as a key intermediate in the formation of CO. Our new experimental work supports the proposed scheme.