Reactions. -A relatively inexpensive catalyst easily prepared from L-valine is applied to asymmetric aldol and cross-aldol reactions [reaction of (VI)]. The reaction proceeds with excellent stereoselectivity under mild, environmentally friendly conditions. Noteworthy, the aldol reaction occurs at the less-hindered face of the enamine in contrast to the cross-aldol reaction. -(TANIMURA, Y.; YASUNAGA, K.; ISHIMARU*, K.; Eur.

From the biharmonic equation of the plane problem in the polar coordinate system and taking into account the variable-separable form of the partial solutions, a homogeneous ordinary differential equation (ODE) of the fourth order is deduced. Our study is based on the investigation of the behavior of the coefficients of the above fourth order ODE, which are functions of the radial coordinate r. According to the proposed investigation additional terms, φ¯−m(r,θ)(1≤m≤n) other than the usually tabulated in the Michell solution (1899, “On the Direct Determination of Stress in an Elastic Solid, With Application to the Theory of Plates,” Proc. Lond. Math. Soc., 31, pp. 100–124) are found. Finally the stress and the displacement fields due to each one additional term of φ¯−m(r,θ) are determined.

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In the present paper the elastostatic problem of a generally anisotropic and angularly inhomogeneous plane wedge loaded at its apex by a concentrated force, is studied in linear elasticity. At first the self-similarity property is formulated and the stress field of the inhomogeneous anisotropic self-similar wedge problem, is deduced. The wedge is radially separated and the plane wedge problem is reformulated by the introduction of a characteristic vector. Furthermore, the angular distribution of the load is determined. The multi-material wedge problem in terms of a formulation based on the isotropic angularly inhomogeneous wedge, is confronted, and necessary conditions that ensure the self-similarity property, are found. Finally, the "similar" elastostatic wedge problems and the involution between stresses, are studied. (2000): 74B05, 74K30, 34B05, 51N15. Mathematics Subject Classifications