Geometrical structures of some non-distance models for asymmetric MDS are examined in error-free data measured at a ratio level and a unified geometrical interpretation of these models is provided. These include CASK (Canonical Analysis of SKew symmetry), DEDICOM, GIPSCAL, and HCM (Hermitian Canonical Model). It is shown that these models except for CASK as well as other possible models for square asymmetric proximity data matrix are expressible in terms of finite-dimensional complex Hilbert space under some general condition, and that differences in form of these models depend only on the bases chosen. It is also shown that the Hilbert space structure has an interesting property which traditional distance model does not. Finally it is shown that the general condition relates to an extension of the famous Young-Householder theorem to complex Hilbert space.

In this paper, we shall discuss boundedness of solutions of the equationunder suitable conditions. And we shall discuss asymptotic stability of a periodic solution and convergence of solutions for the equationfor a positive constant cand a periodic function e(t)under some restricted conditions.

In [3], M. W. Hirsch obtained some necessary conditions for the existence of an Anosov diffeomorphism on a differentiable manifold. As an application, he constructed many manifolds which do not admit Anosov diffeomorphisms.

In his study of non-linear differential equations of the second order, N. Levinson [3] defined the dissipative systems (D-systems) which arise in many important cases in practice. To a dissipative system a transformation T: R2 → R2 called the Poincaré transformation is associated. Levinson used the Poincaré transformation in the qualitative study of dissipative systems, and he [3] and Massera [5] obtained certain equalities between the number of subharmonic solutions of a dissipative systems under suitable conditions. We call these the Levinson-Massera’s equalities.

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