We study arithmetic progression in the x-coordinate of rational points on genus two curves. As we know, there are two models for the curve C of genus two: C : y = f5(x) or C : y = f6(x), where f5, f6 ∈ Q[x], deg f5 = 5, deg f6 = 6 and the polynomials f5, f6 do not have multiple roots. First we prove that there exists an infinite family of curves of the form y = f(x), where f ∈ Q[x] and deg f = ...

background: monte carlo and experimental relative dose determination in a water phantom, due to a high dose rate (hdr) 192ir source is presented for real energy spectrum and monochromatic at 356 kev. materials and methods: the dose distribution has been calculated around the 192ir located in the center of 30 cm ×30 cm ×30 cm water phantom using mcnp4c code by monte carlo method. relative dose v...

Previous results on genera g of Fq2 -maximal curves are improved: (1) Either g ≤ ⌊(q − q + 4)/6⌋ , or g = ⌊(q − 1)/4⌋ , or g = q(q − 1)/2 . (2) The hypothesis on the existence of a particular Weierstrass point in [2] is proved. (3) For q ≡ 1 (mod 3), q ≥ 13, no Fq2 -maximal curve of genus (q−1)(q−2)/3 exists. (4) For q ≡ 2 (mod 3), q ≥ 11, the non-singular Fq2 -model of the plane curve of equat...

various methods for parametric interpolation of nurbs curves have been proposed in the past. however, the errors caused by the approximate nature of the nurbs interpolator were rarely taken into account. this paper proposes an integrated look-ahead algorithm for parametric interpolation along nurbs curves. the algorithm interpolates the sharp corners on the curve with the pythagorean-hodograph ...

In this paper, we give a ‘direct’ construction of the endomorphism ring of supersingular elliptic curves over a prime field Fp from ‘ideal classes’ of Q( √−p). We use the result to prove that the result of Kaneko on ‘minimal’ CM liftings of such supersingular elliptic curves is a best possible result. We also prove that the result of Elkies on ‘minimal’ CM liftings of all supersingular elliptic...

In this paper, we look at long geometric progressions on different model of elliptic curves, namely Weierstrass curves, Edwards and twisted Edwards curves, Huff curves and general quartics curves. By a geometric progression on an elliptic curve, we mean the existence of rational points on the curve whose x-coordinate (or y-coordinate) are in geometric progression. We find infinite families of t...

We construct examples of families of curves of genus 2 or 3 over Q whose Jacobians split completely and have various large rational torsion subgroups. For example, the rational points on a certain elliptic surface over P of positive rank parameterize a family of genus-2 curves over Q whose Jacobians each have 128 rational torsion points. Also, we find the genus-3 curve 15625(X + Y 4 + Z)− 96914...

We construct examples of families of curves of genus 2 or 3 over Q whose Jacobians split completely and have various large rational torsion subgroups. For example, the rational points on a certain elliptic surface over P of positive rank parameterize a family of genus-2 curves over Q whose Jacobians each have 128 rational torsion points. Also, we find the genus-3 curve 15625(X4 + Y 4 + Z4)− 969...

We give digital analogues of classical theorems of topology for continuous functions defined on spheres, for digital simple closed curves. In particular, we show the following: 1) A digital simple closed curve S of more than 4 points is not contractible, i.e., its identity map is not nullhomotopic in S ; 2) Let X and Y be digital simple closed curves, each symmetric with respect to the origin, ...

We show that if f : X −→ Y is a finite, separable morphism of smooth curves defined over a finite field Fq, where q is larger than an explicit constant depending only on the degree of f and the genus of X , then f maps X(Fq) surjectively onto Y (Fq) if and only if f maps X(Fq) injectively into Y (Fq). Surprisingly, the bounds on q for these two implications have different orders of magnitude. T...