Given a positive integer n and a point P on an elliptic curve E, the computation of nP , that is, the result of adding n times the point P to itself, called the scalar multiplication, is the central operation of elliptic curve cryptosystems. We present an algorithm that, using p processors, can compute nP in time O(log n+H(n)=p+ log p), where H(n) is the Hamming weight of n. Furthermore, if thi...

In this paper we recall two basic conjectures on the developables of convex projective curves, prove one of them and disprove the other in the first nontrivial case of curves in RP 3. Namely, we show i) that the tangent developable of any convex curve in RP 3 has degree 4 and ii) construct an example of 4 tangent lines to a convex curve in RP 3 such that no real line intersects all four of them...

Dynamic term structure models explain the yield curve variation well but perform poorly in pricing and hedging interest rate options. Most existing option pricing practices take the yield curve as given, thus having little to say about the fair valuation of the underlying interest rates. This paper proposes an m + n model structure that bridges the gap in the literature by successfully pricing ...

Until now the representation (i.e. plotting) of curve in Parallel Coordinates is constructed from the point ↔ line duality. The result is a " line-curve " which is seen as the envelope of it's tangents. Usually this gives an unclear image and is at the heart of the " over-plotting " problem; a barrier in the effective use of Parallel Coordinates. This problem is overcome by a transformation whi...

A Q-curve is an elliptic curve defined over Q that is isogenous to all its Galois conjugates. The term Q-curve was first used by Gross to denote a special class of elliptic curves with complex multiplication having that property, and later generalized by Ribet to denote any elliptic curve isogenous to its conjugates. In this paper we deal only with Q-curves with no complex multiplication, the c...

BACKGROUND Growth curve analysis is a statistical issue in life course epidemiology. Height in puberty involves a growth spurt, the timing and intensity of which varies between individuals. Such data can be summarized with individual Preece-Baines (PB) curves, and their five parameters then related to earlier exposures or later outcomes. But it involves fitting many curves. METHODS We present...

We study the Spohn conditional independence (CI) variety CX of an n-player game X for undirected graphical models on n binary random variables consisting one edge. For a generic game, we show that is smooth irreducible complete intersection curve (Nash curve) in Segre (P1)n−2×P3 and give explicit formula its degree genus. prove two universality theorems CX: The product any affine real algebraic...