The parametric cubic curve is a well-developed subject in the field of computer-aided design. Yet there are still some aspects of its nature that need to be more fully investigated. In this paper, formulations of composite PC curves are presented in an order of increasing rigorousness and order of accuracy. A unique PC curve formulation is developed for the practical use in circle approximation...

This paper develops a framework for shape analysis of tree-like structures with the following common features: (1) a main branch viewed as a parameterized curve in R3, and (2) a random number of secondary branches, each one of them a parameterized curve in R3, emanating from the main branch at arbitrary points. In this framework, comparisons of objects is based on shapes-scales-orientations of ...

The tumour-node metastasis (TNM) classification system is anatomically based. We investigated whether the addition of simple physiological variables, age and body mass index (BMI), would affect survival curves, i.e. a composite anatomical and physiological staging system. We retrospectively analysed a prospectively validated thoracic surgery database (n = 1981). Cox multivariate analysis was pe...

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...

objectives: this in-vitro study aimed to compare the push-out bond strength of composite to prepared and unprepared intracanal dentin using a 5th generation bonding agent in primary anterior teeth after etching for seven and 15 seconds. materials and methods: sixty primary anterior teeth were randomly divided into four groups. in groups one and two, intracanal dentin remained intact while in gr...