Definition Let K be a field. An elliptic curve over K is a pair (X, O) where X is a smooth absolutely integral proper curve over K with genus one and O is a K-rational point on X. Terminology (see Liu [Liu] and Stichtenoth [Sti]) A curve over K is an equidimensional variety over K of dimension one. A variety over K is a separated scheme of finite type over Spec(K). Examples A triangle with equa...

in this paper, we give some characterizations for legendre spherical, legendre normal and legendre rectifying curves in the 3-dimensional sasakian space. furthermore, we show that legendre spherical curves are also legendre normal curves. in particular, we prove that the inverse of curvature of a legendre rectifying curve is a non-constant linear function of the arclength parameter.

Let C : Y 2 = anX + · · · + a0 be a hyperelliptic curve with the ai rational integers, n ≥ 5, and the polynomial on the right irreducible. Let J be its Jacobian. We give a completely explicit upper bound for the integral points on the model C, provided we know at least one rational point on C and a Mordell–Weil basis for J(Q). We also explain a powerful refinement of the Mordell–Weil sieve whic...

An integral-generic factoring algorithm is, loosely speaking, a constant sequence of ring operations that computes an integer whose greatest common divisor with a given integral random variable n, such as an RSA public key, is non-trivial. Formal definitions for generic factoring will be stated. Integral-generic factoring algorithms seem to include versions of trial division and Lenstra’s ellip...

We study the integral expression of a knot invariant obtained as the second coefficient in the perturbative expansion of Witten’s Chern-Simons path integral associated with a knot. One of the integrals involved turns out to be a generalization of the classical Crofton integral on convex plane curves and it is related with invariants of generic plane curves defined by Arnold recently with deep m...

A new heuristic method of geometric Hermite interpolation is presented to construct a planar cubic rational Bézier curve with two points and two unit tangent directions. The integral, which shows the change rate of the curvature, is taken as the energy function to measure the fairness of the parameter curve. Hence the curvature of the new curve is more stable. Since that the energy function of ...

The purpose of this paper is two-fold: first, Grassmann numbers are defined outside of path integral, and secondly spinor degrees of freedom are defined as a variation of vierbines along space-filling curve.

We prove that the product of an Enriques surface and a very general curve of genus at least 1 does not satisfy the integral Hodge conjecture for 1-cycles. This provides the first examples of smooth projective complex threefolds of Kodaira dimension zero for which the integral Hodge conjecture fails, and the first examples of non-algebraic torsion cohomology classes of degree 4 on smooth project...

It is well known that life expectancy can be expressed as an integral of the survival curve. The reverse that the survival function can be expressed as an integral of life expectancy is also true. 1. Relationship Let e(a) denote remaining life expectancy at age a and let `(a) denote the proportion surviving to age a (the survival function). Then (1) l(a) = e(0) e(a) exp ( − ∫ a