Edwards curves are a particular form of elliptic curves that admit a fast, unified and complete addition law. Relations between Edwards curves and Montgomery curves have already been described. Our work takes the view of parameterizing elliptic curves given by their j-invariant, a problematic that arises from using curves with complex multiplication, for instance. We add to the catalogue the li...

A notion of dual curve for pseudoholomorphic curves in 4–manifolds turns out to be possible only if the notion of almost complex structure structure is slightly generalized. The resulting structure is as easy (perhaps easier) to work with, and yields many analogues of results in complex surface theory, using a description of the local geometry via Cartan’s method of equivalence. Duality then un...

So far we have discussed only ‘local’ properties of (plane) curves. These properties depend only on the behavior of a curve near a given point, and not on the ‘global’ shape of the curve. Now let us look at some global results about curves. The most famous, and perhaps the oldest, of these is the ‘isoperimetric inequality’, which relates the length of a certain ‘closed’ curve to the area it con...

‎‎In the category of Mordell curves (E_D:y^2=x^3+D) with nontrivial torsion groups we find curves of the generic rank two as quadratic twists of (E_1), ‎and of the generic rank at least two and at least three as cubic twists of (E_1). ‎Previous work‎, ‎in the category of Mordell curves with trivial torsion groups‎, ‎has found infinitely many elliptic curves with ...

The objective of the study in this paper is to define M₁,M₂-direction curves and M₁,M₂-donor curves of the spacelike curve γ via the Bishop frame of type-2 in E₁³. We obtained the necessary and sufficient conditions when the associated curves to be slant helices and general helices via the Bishop frame of type-2 in E₁³. After defining the spherical indicatrices of the associated curves, we obta...

1-type and biharmonic curves by using laplace operator in lorentzian 3-space arestudied and some theorems and characterizations are given for these curves.

Evidence of compressive growth of basilar membrane displacement can be observed in measurements of DPOAE. When the levels of the two stimulus tones are related by the formula L1=0.4 L2+39, then the DPOAE I/O function resembles the classic Fletcher-Munson (1933) loudness function. In order to understand how the DPOAE I/O function relates to the perception of loudness, we need to also understand ...