The main goal of this paper is to study the maximum number of limit cycles that bifurcate from the period annulus of the cubic centers that have a rational first integral of degree 2 when they are perturbed inside the class of all cubic polynomial differential systems using the averaging theory. The computations of this work have been made with Mathematica and Maple.

We show that a generally convergent root-finding algorithm for cubic polynomials defined by C. McMullen is of order 3, and we give generally convergent algorithms of order 5 and higher for cubic polynomials. We study the Julia sets for these algorithms and give a universal rational map and Julia set to explain the dynamics.

We classify the possible torsion structures of rational elliptic curves over cubic fields. Along the way we find a previously unknown torsion structure over a cubic field, Z/21Z, which corresponds to a sporadic point on X1(21) of degree 3, which is the lowest possible degree of a sporadic point on a modular curve X1(n).

A cubic polynomial lter is presented for edge preserving noise smoothing. The lter coeecients are adaptable and can be optimized to improve the l-ter result. Experiments show that the image quality achieved with the optimized cubic lter is comparable to that of a more complex rational lter.

This paper presents an algorithm to solve the approximate implicitization of planar parametric curves using cubic algebraic splines. It applies piecewise cubic algebraic curves to give a global G2 continuity approximation to planar parametric curves. Approximation error on approximate implicitization of rational curves is given. Several examples are provided to prove that the proposed method is...

A cubic polynomial lter is presented for edge preserving noise smoothing. The lter coeecients are adaptable and can be optimized to improve the l-ter result. Experiments show that the image quality achieved with the optimized cubic lter is comparable to that of a more complex rational lter.

In this paper we study the linearizability problem of the two-dimensional complex quasi-cubic system ż = z + (zw)(a30z + a21zw + a12zw + a03w), ẇ = −w − (zw)(b30w + b21wz+b12wz +b03z), where z,w, ai j, bi j ∈ C and d is a real number. We find a transformation to change the quasi-cubic system into an equivalent quintic system and then obtain the necessary and sufficient linearizability condition...

The techniques for parametrizing nonsingular cubic surfaces have shown to be of great interest in recent years. This paper is devoted to the rational parametrization of nonsingular cubic blending surfaces. We claim that these nonsingular cubic blending surfaces can be parametrized using the symbolic computation due to their excellent geometric properties. Especially for the specific forms of th...

The Pythagorean hodograph (PH) curves are polynomial parametric curves whose hodograph (derivative) components satisfy the Pythagorean condition. Lots of research works had been done with PH curves relevant topics due that PH curves own many remarkable properties, e.g., its offset curves have exact rational representations. In this paper, ball curves with rational polynomial offset are studied....