Analysis and construction of rational curve parametrizations with non-ordinary singularities

In this paper, we provide a method that allows to construct parametric curves having (or not) non-ordinary singularities and having (or not) neighboring points. This method is based on a characterization of the non-ordinary singularities and neighboring points by means of linear equations involving the given parametrization. As a consequence, we obtain an algorithm that constructs a parametrization which contains a given point, P, as a singularity as well as some additional information as for instance, the order of P, parameters corresponding to P, multiplicity of each parameter and the singularities in the first neighborhood of the singularity P.Ministerio de Economía y Competitivida

The T-function of a parametric curve

In this paper, we introduce the T–function, T(s), which is a polynomial defined by means of a univariate resultant constructed from a given parametrization P(t) ∈ K(t) n , n ≥ 2 of an algebraic space curve C. It is shown that T(s) = Qn i=1 HPi (s) mi−1 , where HPi (s), i = 1, . . . , n are polynomials (the fibre functions) whose roots are the fibre of the ordinary singularities Pi ∈ C of multiplicity mi , i = 1, . . . , n of C. Therefore, a complete classification of the singularities of C, via the factorization of a resultant, is obtained.Agencia Estatal de Investigació

On the problem of proper reparametrization for rational curves and surfaces

A rational parametrization of an algebraic curve (resp. surface) establishes a rational correspondence of this curve (resp. surface) with the affine or projective line (resp. affine or projective plane). This correspondence is a birational equivalence if the parametrization is proper. So, intuitively speaking, a rational proper parametrization trace the curve or surface once. We consider the problem of computing a proper rational parametrization from a given improper one. For the case of curves we generalize, improve and reinterpret some previous results. For surfaces, we solve the problem for some special surface's parametrizations

Puerto Rican and African-American Males: Current Challenges, Promising Strategies

Before the beginning of the next century, the Hispanic, African-American, and other “minority” populations in the United States are expected to increase at a faster rate than the white population. In fact, the Census Bureau projects that Latinos will become the largest minority and, together with African Americans, will constitute one-fourth (25.5 percent) of the U.S. labor force by the year 2010. However, despite some gains, increases in population have not been proportionate to increases in voting and buying power—or to comparable increases in economic success or socioeconomic stability—for a significant proportion of either Latinos or African Americans. Moreover, inaccurate perceptions have influenced public understanding of both groups. For example. despite the existence of factual information to the contrary, major opinion polls indicate that Americans perceive Latinos to be lazy, unpatriotic, and mostly recent immigrants. Similarly, while recent research has documented factors that have contributed to the poor socioeconomic status of many African Americans, newspapers and other media continue to depict African Americans as unemployed, prone to violence, and/or on welfare. The public image of Latino and African-American young males, in particular, is especially poor and has been adversely affected by the increases in both teenage pregnancy and female-headed households, high unemployment rates, and other social and economic problems affecting both groups