We prove an asymptotically tight bound (asymptotic with respect to the number of polynomials for fixed degrees and number of variables) on the number of semi-algebraically connected components of the realizations of all realizable sign conditions of a family of real polynomials. More precisely, we prove that the number of semi-algebraically connected components of the realizations of all realiz...

For a sequence \begin{document}$ (a_n) $\end{document} of complex numbers we consider the cubic parabolic polynomials f_n(z) = z^3+a_n z^2+z and (F_n) iterates F_n f_n\circ\dots\circ f_1 . The Fatou set \mathcal{F}_0 is all z\in\hat{\mathbb{C}} such that normal. complement called Julia denoted by \mathcal{J}_0 aim this paper to study some properties As particular case, when constant, a_n , then...

This paper presents a method of parameterization for online freehand drawing objects based on a piecewise cubic Bezier curve approximation. The target is to represent sketches in a compact format within a certain error tolerance with lower computation to be practically adaptable for the online graphics input. A set of user’s intended breakpoints in digital ink is firstly produced in terms of pe...

It is an important problem to compute the geodesics on a surface in many fields. To find the geodesics in practice, however, the traditional discrete algorithms or numerical approaches can only find a list of discrete points. The first author proposed in 2010 a new, elegant and accurate method, the geodesic-like method, for approximating geodesics on a regular surface. This paper will present b...

Bezier curve are very strong for variety of application. Specifically in image processing it applies to object recognition, face recognition, and human gait recognition. It also works on fingerprint and other biometric system recognition. This paper presents a work to recognize digital images of human frontal faces using the approximated Bezier curve and an intelligent process of learning using...

Our previous work studied a size-aware type system for functional programs with non-monotonic polynomial size dependencies. In that approach output sizes depended only on input sizes. That is rather restrictive since in many cases the size of the output can differ for different input data of the same size. In this paper we remove that limitation by presenting value-sensitive size dependencies v...