Narrow Results

We study some geometric properties of helices of proper order 4 on a complex projective space which are of type 2 in the sense of submanifolds in a Euclidean space through the first standard embedding and which are generated by some Killing vector fields.

A well-order was introduced on the set of links by A. Kawauchi [3]. This well-order also naturally induces a well-order on the set of prime link exteriors and eventually induces a well-order on the set of closed connected orientable 3-manifolds. With respect to this order, we enumerated the prime links with lengths up to 10 and the prime link exteriors with lengths up to 9. Our present plan is to enumerate the 3–manifolds by using the enumeration of the prime link exteriors. In this paper, we show our latest result in this plan and as an application, give a new proof of the identification of Perko's pair.

Let be a field and let A be a finite-dimensional -algebra. We define the length of a finite generating set of this algebra as the smallest number k such that words of the length not greater than k generate A as a vector space, and the length of the algebra is the maximum of lengths of its generating sets. In this paper we study the connection between the length of an algebra and the lengths of its subalgebras. It turns out that the length of an algebra can be smaller than the length of its subalgebra. To investigate, how different the length of an algebra and the length of its subalgebra can be, we evaluate the difference and the ratio of the lengths of an algebra and its subalgebra for several representative families of algebras. Also we give examples of length computation of two and three block upper triangular matrix algebras.