The steepest descent algorithm with exact line searches (Cauchy algorithm) is inefficient, generating oscillating step lengths and a sequence of points converging to the span of the eigenvectors associated with the extreme eigenvalues. The performance becomes very good if a short step is taken at every (say) 10 iterations. We show a new method for estimating short steps, and propose a method al...

In order to investigate the behavior of rare earth elements (REEs) and uranium (U) in marine organism, the concentrations of REEs and U in some brown algae samples taken on the coast of Niigata Prefecture were determined. In addition, laboratory model experiment to uptake these elements using living and dried algae (Undaria pinnatifida and Sargassum hemiphyllum) was also carried out to survey t...

In this paper, a notion of principal $2$-bundle over Lie groupoid has been introduced. For such $2$-bundles, we produced short exact sequence VB-groupoids, namely, the Atiyah sequence. Two notions connection structures viz. strict connections and semi-strict on arising respectively, from retraction up to natural isomorphism have We constructed class $\mathbb{G}=[G_1\rightrightarrows G_0]$-bundl...

As our second example we consider the case where TT is a differentiable submersion (in the usual sense) of codimension 1. Thus we assume that X, B are C°°-manifolds of dimensions n + 1 and n, respectively, and that TT is a regular differentiate map. Such a submersion is orientable if the induced line element field on X is orientable, and w will be called simple if w~(b) is connected for all bÇEB.

Let Bn(RP ) (respectively Pn(RP )) denote the braid group (respectively pure braid group) on n strings of the real projective plane RP 2 . In this paper we study these braid groups, in particular the associated pure braid group short exact sequence of Fadell and Neuwirth, their torsion elements and the roots of the ‘full twist’ braid. Our main results may be summarised as follows: first, the pu...

It was shown by van Rees [18] that a latin square of order n cannot have more than n(n− 1)/18 latin subsquares of order 3. He conjectured that this bound is only achieved if n is a power of 3. We show that it can only be achieved if n ≡ 3 mod 6. We also state several conditions that are equivalent to achieving the van Rees bound. One of these is that the Cayley table of a loop achieves the van ...

In this paper, we consider the problem of computing a shortest path of bounded curvature amidst obstacles in the plane. More precisely, given prescribed initial and nal conngurations (i.e. positions and orientations) and a set of obstacles in the plane, we want to compute a shortest C 1 path joining those two conngurations, avoiding the obstacles, and with the further constraint that, on each C...

In this article, we show the existence of certain exact sequences with respect to two homology theories, called d-homology and extended d-homology. We present sufficient conditions for the existence of long exact extended d- homology sequence. Also we give some illustrative examples.