Some consequences of a well known theorem on conics

Some Consequences of a Well Known Theorem on Conics

On a Theorem of Intersecting Conics

Given two conics over an infinite field that intersect at the origin, a line through the origin will, in general intersect both conic sections once more each, at points C and D. As the line varies we find that the midpoint of C and D traces out a curve, which is typically a quartic. Intuitively, this locus is the “average” of the two conics from the perspective of an observer at the origin. We ...

On Computing Gaussian Curvature of Some Well Known Distributions

The Gaussian curvature of the surface at the point p is the product of the maximum and minimum curvatures in the family. The objective of this paper is to provide a deeper and broader understanding of the meaning of Gaussian curvature, using some more general alternative computational methods. We define the coefficients of the expected Fisher Information Matrix as the coefficients of the first ...

Some Consequences of the Karpenko-merkurjev Theorem

We use a recent theorem of N. A. Karpenko and A. S. Merkurjev to settle several questions in the theory of essential dimension. 2010 Mathematics Subject Classification: 20D15, 20C15, 20G15

Some Results on Baer's Theorem

Baer has shown that, for a group G, finiteness of G=Zi(G) implies finiteness of ɣi+1(G). In this paper we will show that the converse is true provided that G=Zi(G) is finitely generated. In particular, when G is a finite nilpotent group we show that |G=Zi(G)| divides |ɣi+1(G)|d′ i(G), where d′i(G) =(d( G /Zi(G)))i.