Synthesis of cyclotetrapeptides,am-toxin analogs,containing α-hydroxyalanine

Condensation of amide with α-keto acid to yield a α-hydroxyalanine (α-Hyala) or dehydroalanine residue was applied to syntheses of analogs of AM-toxins, cyclotetradepsipeptides. Cyclo(-α-Hyala-L-Ala-L-Val-L-Phe-), cyclo(-α-Hyala-L-Ala-L-Hmb-L-Phe-) and cyclo(-α-Hyala-L-Ala-L-Hmb-L-Tyr-) (Hmb, 2-hydroxy-3-methylbutanoic acid) were obtained from the corresponding pyruvyl-tripeptide amides in good yields by the treatment of anhydrous hydrogen fluoride.     

The horospherical geometry of surfaces in hyperbolic 4-space

We study some geometrical properties associated to the contacts of surfaces with hyperhorospheres inH ₊ ⁴ (−1). We introduce the concepts of osculating hyperhorospheres, horobinormals, horoasymptotic directions and horospherical points and provide conditions ensuring their existence. We show that totally semiumbilical surfaces have orthogonal horoasymptotic directions. Work partially supported by Grant-in-Aid for Scientific Research, JSPS, No. 12874007. Work partially supported by Grant-in-Aid for Scientific Research, JSPS, No. 12000266. Work partially supported by DGCYT grant no. BFM2003-02037.     

Homotopy classification of equivariant regular sections

In his paper [2], Bierstone proves the equivariant Gromov theorem which is an integrability theorem for “open regularity condition” of equivariant sections of a smooth G-fibre bundle under the assumption that all orbit bundles of base manifold are non-closed. Here, we prove the result without his assumption under a nice “open regularity condition” which we call “G-extensible”. One of the examples of “G-extensible condition” is given by notions of Thom-Boardman singularities.     

Great circular surfaces in the three-sphere

In this paper, we consider a special class of the surfaces in the 3-sphere defined by one-parameter families of great circles. We give a generic classification of singularities of such surfaces and investigate the geometric meanings from the view point of spherical geometry.     

Anti de Sitter horospherical flat timelike surfaces

We investigate a special timelike surfaces in Anti de Sitter 3-space.We call such a timelike surface an Anti de Sitter horospherical flat surface which belongs to a class of surfaces given by one parameter families of Anti de Sitter horocycle.We give a generic classification of singularities and study the geometric properties of such surfaces from the viewpoint of Legendrian singularity theory.     

Generic bifurcations of varieties

We study the zero point set of a parametrized smooth map germ. It is not only a natural generalization of Mather's theory of smooth map germs (cf. [11],[12]), but also it contains the bifurcation theory of stationary solutions of parametrized ordinary differential equations. One of our main results is a classification of parametrized smooth map germs under a certain equivalence relation.Dedicated to Professor Minoru Nakaoka on his 60th birthday     

Flat lightlike hypersurfaces in Lorentz–Minkowski 4-space

The lightlike hypersurfaces in Lorentz–Minkowski space are of special interest in Relativity Theory. In particular, the singularities of these hypersurfaces provide good models for the study of different horizon types. We introduce the notion of flatness for these hypersurfaces and study their singularities. The classification result asserts that a generic classification of flat lightlike hypersurfaces is quite different from that of generic lightlike hypersurfaces.