The theorem for semisimple Hopf algebras

We give an algebraic version of a result of G. I. Kac, showing that a semisimple Hopf algebra of dimension , where is a prime and , over an algebraically closed field of characteristic 0 contains a non-trivial central group-like. As an application we prove that, if , is isomorphic to a group algebra.

     

Fan-type theorem for path-connectivity

A graphG is said to bek-path-connected if every pair of distinct vertices inG are joined by a path of length at leastk. We prove that if max{deg _G ^x , deg _G ^y }k for every pair of verticesx,y withd _G (x,y)=2 in a 2-connected graphG, whered _G (x,y) is the distance betweenx andy inG, thenG isk-path-connected.     

Chains of strongly non-reflexive dual groups of integer-valued continuous functions

Answering a question of Eklof-Mekler (Almost free modules, set-theoretic methods, North-Holland, Amsterdam, 1990), we prove: (1) If there exists a non-reflecting stationary set of consisting of ordinals of cofinality for each , then there exist abelian groups such that and for each . (2) There exist abelian groups such that for each and for each . The groups are the groups of -valued continuous functions on a topological space and their dual groups.

     

Synthesis of Six-Membered Compounds by Environmentally Friendly Cyclization Using Indirect Electrolysis

[Ni(cyclam)](ClO(4))(2)-catalyzed indirect electroreduction of olefinic bromides produced six-membered compounds in low to high yields. The synthetic intermediate 49 of Ipecac and Corynanthe alkaloids was obtained in 88% yield in a highly stereoselective manner. Lactam 66, the synthetic precursor of tacamonine, was prepared in 49% yield as a mixture of two diastereoisomers. The electrolysis of the bromoacetates gave the debrominated compounds in good yields.     

Identification of Shiga toxins in Shiga toxin-producing Escherichia coli using immunoprecipitation and high-performance liquid chromatography-electrospray ionization mass spectrometry

We developed a rapid and reliable identification method for Shiga toxins in Shiga toxin-producing Escherichia coli (STEC) using immunoprecipitation and high-performance liquid chromatography-electrospray ionization mass spectrometry (HPLC-ESI-MS). Polyclonal antisera specific for Shiga toxin 1 (Stx1) and Shiga toxin 2 (Stx2) were raised in rabbits so as to be used for the immunoprecipitation. The immunoprecipitaion was carried out by mixing sample solutions with 50 microl each of the antisera to Stx1 and Stx2 followed by allowing the mixed solutions to stand for 30 min. The quantity required to obtain the immunoprecipitate was more than 0.5 microg of Shiga toxins. HPLC-ESI-MS analysis of the resulting immunoprecipitates provided accurate molecular weight information on Shiga toxins, leading to direct evidence for the presence of these toxins. It requires at most two days to perform our procedure from toxin extraction to measurement of HPLC-ESI-MS whereas the previous method using isolation procedures required about two weeks to complete. The usefulness of the present method has been demonstrated by identifying Stx1, Stx2 and a variant of Stx2 (Stx2e) in the immunoprecipitates prepared from STEC strains.     

Development of versatile cis- and trans-dicarbon-substituted chiral cyclopropane units: synthesis of (1S,2R)- and (1R,2R)-2-aminomethyl-1-(1H-imidazol-4-yl)cyclopropanes and their enantiomers as conformationally restricted analogues of histamine

The cyclopropane ring can be used effectively in restricting the conformation of biologically active compounds to improve activity and also to investigate bioactive conformations. We designed (1S,2R)- and (1R,2R)-2-aminomethyl-1-(1H-imidazol-4-yl)cyclopropanes (1 and 2, respectively) and their enantiomers (ent-1 and ent-2) as conformationally restricted analogues of histamine. The four types of chiral cyclopropanes bearing two differentially functionalized carbon substituents in a cis or trans relationship on a cyclopropane ring, (1S,2R)-2-(tert-butyldiphenylsilyloxy)methyl-1-formylcyclopropane (7) and (1R,2R)-2-(tert-butyldiphenylsilyloxy)methyl-1-formylcyclopropane (8) and their enantiomers (ent-7 and ent-8), were developed as the key intermediates for synthesizing 1, 2, ent-1, and ent-2. The reaction between (R)-epichlorohydrin [(R)-12] and phenylsulfonylacetonitrile (13a) in the presence of NaOEt in EtOH followed by treatment with acid gave the chiral cyclopropane lactone 11a with 98% ee in 82% yield. Compound 11a was converted into both the cis- and trans-chiral cyclopropane units 7 and 8, respectively, via reductive desulfonylation with Mg/MeOH as the key step. The corresponding enantiomers, the cis-substituted ent-7 and the trans-substituted ent-8, were also prepared starting from (S)-epichlorohydrin [(S)-12]. The four conformationally restricted target histamine analogues 1, 2, ent-1, and ent-2 were successfully synthesized from 7, 8, ent-7, and ent-8, respectively. The chiral cyclopropane units 7, 8, ent-7, and ent-8 should be useful as versatile intermediates for synthesizing various compounds having an asymmetric cyclopropane structure.