Introduction:  90 Sr/ 90 Y source  has  been  used  for  the  intravascular  brachytherapy  to  prevent  coronary  restenosis in the patients who have undergone angioplasty. The aim of this research is to determine the  dose distribution of  90 Sr/ 90 Y source in a water phantom.  Materials  and  Methods:  In  the  present  work,  MCNP  code  has  been  applied  to  calculate  the  dose  distri...

The ropelength of a knot is the quotient of its length by its thickness. We consider a family of energy functions R for knots, depending on a power p, which approach ropelength as p increases. We describe a numerically computed trefoil knot which seems to be a local minimum for ropelength; there are nearby critical points for R, which are evidently local minima for large enough p. 1. Thickness ...

We use Weierstrass Point Theory and Frobenius orders to prove the uniqueness (up to isomorphism) of some optimal curves. This paper continues the study, begun in [FT] and [FGT], of curves over finite fields with many rational points, based on Stöhr-Voloch’s approach [SV] to the Hasse-Weil bound by way of Weierstrass Point Theory and Frobenius orders. Some of the results were announced in [T]. A...

We give a simple recipe for Lissajous curves that for (certain) numerical purposes can serve as a proxy for the cube [−1, 1]d . For a ∈ Zd >0 we let `a(t) := (cos(a1 t), cos(a2 t), · · · , cos(ad t)), t ∈ R. (1) denote the associated Lissajous curve with frequencies a1, · · · , ad . We note that such curves are given by the fundamental parameter interval t ∈ [0,π]. The recent articles [2] and [...

Abstract. We prove several results regarding some invariants of elliptic curves on average over the family of all elliptic curves inside a box of sides A and B. As an example, let E be an elliptic curve defined over Q and p be a prime of good reduction for E. Let eE (p) be the exponent of the group of rational points of the reduction modulo p of E over the finite field Fp. Let C be the family o...

A new method is presented for identifying and comparing closed, bounded, free-form curves that are defined by even implicit polynomial (IP) equations in the Cartesian coordinates x and y. The method provides a new expression for an IP involving a product of conic factors with unique conic factor centers. The critical points for an IP curve also are defined. The conic factor centers and the crit...

e # where fo#]m range over continuous and increasing functions with foEqpDHrIs"t#!foE]u4HrIs"87q#nmnEgp HvIw" and mnE]u4HrIw$A7 . The functions fo#]m are also called parametrization functions. For polygonal curves x and y , consisting of z and { edges, Alt and Godau [2] developed an algorithm which computes their Fréchet distance in time |XEBz {>}~D€5EBz { H]H . We will review this algorithm i...

In [Ch1], Chern gives a generalization of projective geometry by considering foliations on the Grassman bundle of p-planes Gr(p, R) → R by p-dimensional submanifolds that are integrals of the canonical contact differential system. The equivalence method yields an sl(n + 1, R)valued Cartan connection whose curvature captures the geometry of such foliation. In the flat case, the space of leaves o...

We study an infinite family of Mordell curves (i.e. the elliptic curves in the form y = x + n, n ∈ Z) over Q with three explicit integral points. We show that the points are independent in certain cases. We describe how to compute bounds of the canonical heights of the points. Using the result we show that any pair in the three points can always be a part of a basis of the free part of the Mord...

Montgomery’s ladder algorithm for elliptic curve scalar multiplication uses only the xcoordinates of points. Avoiding calculation of the y-coordinates saves time for certain curves. Montgomery introduced his method to accelerate Lenstra’s elliptic curve method for integer factoring. Bernstein extended Montgomery’s ladder algorithm by computing integer combinations of two points, thus accelerati...