Endomorphism Rings and Isogenies Classes for Drinfeld Modules of Rank 2 Over Finite Fields

Let $\Phi$ be a Drinfeld $\mathbf{F}_{q}[T]$-module of rank 2, over a finite field $L$, a finite extension of $n$ degrees of a finite field with $q$ elements $\mathbf{F}_{q}$. Let $m$ be the extension degrees of $L$ over the field $\mathbf{F}_{q}[T]/P$, $P$ is the $\mathbf{F}$-characteristic of $L$, and $d$ the degree of the polynomial $P$. We will discuss about a many analogies points with elliptic curves. We start by the endomorphism ring of a Drinfeld $\mathbf{F}_{q}[T]$-module of rank 2, End$_{L}\Phi$, and we specify the maximality conditions and non maximality conditions as a $\mathbf{F}_{q}[T]$-order in the ring of division End$_{L}\Phi \otimes _{\mathbf{F}_{q}[T]}$, in the next point we will interested to the characteristic polynomial of a Drinfeld module of rank 2 and used it to calculate the number of isogeny classes for such module, at last we will interested to the Characteristic of Euler-Poincare $\chi_{\Phi}$ and we will calculated the cardinal of this ideals.Comment: 1

Facial expressions emotional recognition with NAO robot

Human-robot interaction research is diverse and covers a wide range of topics. All aspects of human factors and robotics are within the purview of HRI research so far as they provide insight into how to improve our understanding in developing effective tools, protocols, and systems to enhance HRI. For example, a significant research effort is being devoted to designing human-robot interface that makes it easier for the people to interact with robots. HRI is an extremely active research field where new and important work is being published at a fast pace. It is crucial for humanoid robots to understand the emotions of people for efficient human robot interaction. Initially, the robot detects human face by Viola- Jones technique. Later, facial distance measurements are accumulated by geometric based facial distance measurement method. Then facial action coding system is used to detect movements of measured facial points. Finally, measured facial movements are evaluated to get instant emotional properties of human face in this research; it has been specifically applied to NAO humanoid robot