where u and v denote the parameters determining the curves and t is the parameter which determines the points on the curve. By this method of definition the equations of a congruence of curves are given a form similar to that usually adopted in the discussions of rectilinear congruences, so that one proceeds naturally to a generalization of some of the properties of the latter. At the same time...

Pulmonary pressure-volume curves (P-V curves) of patients with acute lung injury are commonly analyzed using a parametric algorithm with symmetrical properties. Some of the aspects observed after performing nonlinear regression for two models capable of fitting symmetric, respectively asymmetric data are discussed. One analyzed aspect was the algebraic complexity of the asymmetric model that do...

In this paper, we show explicitly the classes of elliptic and hyperelliptic curves of low genera dened over extension elds, which have weak coverings , i.e. their Weil restrictions can be attacked by either index calculus attacks to hyperelliptic curves or Diem's recent attack to non-hyperelliptic curves. In particular, we show how to construct such c o v erings from these curves and analyze de...

In this paper, a de Casteljau algorithm to compute (p, q)-Bernstein Bézier curves based on (p, q)integers is introduced. We study the nature of degree elevation and degree reduction for (p, q)-Bézier Bernstein functions. The new curves have some properties similar to q-Bézier curves. Moreover, we construct the corresponding tensor product surfaces over the rectangular domain (u, v) ∈ [0, 1]× [0...

We propose a direct method for solving the evolution of plane curves satisfying the geometric equation v= β(x, k, ν) where v is the normal velocity, k and ν are the curvature and tangential angle of a plane curve Γ ⊂ R2 at a point x ∈ Γ . We derive and analyze the governing system of partial differential equations for the curvature, tangential angle, local length and position vector of an evolv...

In this paper a class of quartic trigonometric Bézier curve, called QT Bézier curve, with two shape parameters is presented. These curves not only inherit most properties of the usual quartic Bézier curves in the polynomial space, but also enjoy some other advantageous properties for shape modelling. The shape of the curve can be adjusted by altering the values of shape parameters while the con...

We show that one can find two nonisomorphic curves over a field K that become isomorphic to one another over two finite extensions of K whose degrees over K are coprime to one another. More specifically, let K0 be an arbitrary prime field and let r > 1 and s > 1 be integers that are coprime to one another. We show that one can find a finite extension K of K0, a degree-r extension L of K, a degr...

This talk gives an introduction into the subject of Seiberg-Witten curves and their relation to integrable systems. We discuss some motivations and origins of this relation and consider explicit construction of various families of Seiberg-Witten curves in terms of corresponding integrable models.

Manjul Bhargava and I have recently proved a result on the average order of the 2-Selmer groups of the Jacobians of hyperelliptic curves of a fixed genus n ≥ 1 over Q, with a rational Weierstrass point [2, Thm 1]. A surprising fact which emerges is that the average order of this finite group is equal to 3, independent of the genus n. This gives us a uniform upper bound of 3 2 on the average ran...

The arithmetic of quaternions is recalled from a constructive point of view. A Hecke module is introduced, defined as a free abelian group on right ideal classes of a quaternion order, together with a natural action of Hecke operators. An equivalent construction in terms of Shimura curves is then introduced, and the quaternion construction is applied to the analysis of specific modular and Shim...