A Hurwitz correspondence is a multi-valued self-map of the moduli space of genus zero curves with n marked points, M0,n. Considering two n-marked curves (C, a1, ..., an) and (D, b1, ..., bn), given a map ρ from {1, ..., n} to itself and some ramification data, one obtains a Hurwitz correspondence on M0,n which sends a marked curve (D, b1, ..., bn) to any of the marked curves (C, a1, ..., an) fo...

A C curve interpolation scheme for monotonic data has been developed. This scheme uses piecewise rational cubic functions. The two families of parameters, in the description of the rational interpolant, have been constrained to preserve the shape of the data. The monotone rational cubic spline scheme has a unique representation.

I Local geometry of smooth curves on singular surfaces 1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2 The type of a surface-curve pair . . . . . . . . . . . . . . . . . . . . . . . . . 8 3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 4 Classification of rational double point pairs . . . . . . . . . . . . . . ....

In Lecture 1 we defined an elliptic curve as a smooth projective curve of genus 1 with a distinguished rational point. An equivalent definition is that an elliptic curve is an abelian variety of dimension one. An abelian variety is a smooth projective variety that is also a group, where the group operation is defined by rational functions (ratios of polynomials). Remarkably, these constraints f...

 The Mordell-Weil theorem states that the group of rational points‎ ‎on an elliptic curve over the rational numbers is a finitely‎ ‎generated abelian group‎. ‎In our previous paper, H‎. ‎Daghigh‎, ‎and S‎. ‎Didari‎, On the elliptic curves of the form $ y^2=x^3-3px$‎, ‎‎Bull‎. ‎Iranian Math‎. ‎Soc‎.‎‎ 40 (2014)‎, no‎. ‎5‎, ‎1119--1133‎.‎, ‎using Selmer groups‎, ‎we have shown that for a prime $p...

In many applications it is desirable to analyze parametric curves for undesirable features like cusps and innection points. Previously known algorithms to analyze such features are limited to cubics and in many cases are for planar curves only. We present a general purpose method to detect cusps in polynomial or rational space curves of arbitrary degree. If a curve has no cusp in its deening in...

We present efficient and robust algorithms for intersecting a rational parametric freeform surface with a general swept surface. A swept surface is given as a one-parameter family of cross-sectional curves. By computing the intersection between a freeform surface and each cross-sectional curve in the family, we can solve the intersection problem. We propose two approaches, which are closely rel...

The rational solutions with as denominators powers of to the elliptic diophantine equation y x x are determined An idea of Yuri Bilu is applied which avoids Thue and Thue Mahler equations and deduces four term S unit equations with special properties that are solved by linear forms in real and p adic logarithms Introduction In a recent paper SW my colleague R J Stroeker and I determined the com...

by the mordell-weil theorem‎, ‎the group of rational points on an elliptic curve over a number field is a finitely generated abelian group‎. ‎there is no known algorithm for finding the rank of this group‎. ‎this paper computes the rank of the family $ e_p:y^2=x^3-3px $ of elliptic curves‎, ‎where p is a prime‎.

Regarding to the high between-band correlation and large volumes of hyperspectral data, feature reduction (either feature selection or extraction) is an important part of classification process for this data type. A variety of feature reduction methods have been developed using spectral and spatial domains. In this paper, a feature extracting technique is proposed based on rational function cur...