Let $R$ be a locality over a field $k$ , i. e., $R$ is a quotient ring of an affine domain $A$ over a field $k$ with respect to a prime ideal.We have then two types of differential theoretic characterization for $R$ to be a regular local ring.
Free edge stress singularities on and near the interface of dissimilar materials were investigated using a characteristic equation deduced in terms of Airy's stress function and the boundary element method. The values of the order of stress singularities determined by boundary element analyses under uniform tension agreed well with the values calculated by using the characteristic equation. The stress singularities disappear for certain combinations of wedge angles of the pair of materials. The stress distribution calculated by the boundary element method in the vicinity of the intersection of the surfaces and the interface agreed well with that obtained by using the stress function.
The scaffold has been used to form a three-dimensional tissue in tissue engineering. It has been known that not only the mechanical properties of scaffold, but also the mechanical stress affect the cell function and tissue structure. However, it is difficult to observe the effect of mechanical stress on each cell in the case of the three-dimensional scaffold. In this study, the effect of tensile strain on the cell adhesion and cell proliferation ratio by using the planar nylon mesh scaffold was investigated. It was found that the cell proliferation ratio increased in case of large strain. It was suggested that the high stress effectively worked to cell proliferation.
Introduction. Adopting the terminology of [1], a ring A is called winvariant (or strongly n-invariant) if A satisfies the condition: Given a ring B and indeterminates Xly • , ! „ ; Yly •••, Yny if A[Xly ~ yXn] is isomorphic to B[Yly •••, YJ, then A is isomorphic to B (or if any ring B and any isomorphism φ: A[Xly ...9XH]->B(Y1, — , y j are given, then φ(A)=B). If a ring A is ^-invariant (or strongly w-invariant) for every integer n ^ l , A is said to be invariant (or strongly invariant). Several types of rings are known to be invariant or strongly invariant respectively (cf. [1], [2], [8] etc.). However we have not any good criteria for a ring to be invariant or strongly invariant, and it is tempting to look for criteria of this kind. A purpose of the present paper is to give sufficient or necessary conditions for a ring to be strongly invariant in terms of locally finite (or locally finite iterative) higher derivations. The present paper consists of three parts. In the first section, definitions of locally finite (or locally finite iterative) higher derivations are recalled, and several results which follow easily from definitions are given. In the second section, sufficient or necessary conditions for strong invariance are given. In the final section we shall see how well these conditions work in giving examples and counter-examples. In the appendix we shall prove a Lemma on a ring which has a locally finite iterative higher derivation. Our terminology is essentially the same as that of [1].
