A new quaternion valued frame of curves with an application

The aim of the paper is to obtain a new version of Serret-Frenet formulae for a quaternionic curve in R4 by using the method given by Bharathi and Nagaraj. Then, we define quaternionic helices in H named as quaternionic right and left X-helix with the help of given a unit vector field X. Since the quaternion product is not commutative, the authors ([4], [7]) have used by one-sided multiplication to find a space curve related to a given quaternionic curve in previous studies. Firstly, we obtain new expressions by using the right product and the left product for quaternions. Then, we generalized the construction of Serret-Frenet formulae of quaternionic curves. Finally, as an application, we obtain an example that supports the theory of this paper.

A Ginzburg–Landau Type Energy with Weight and with Convex Potential Near Zero

In this paper, we study the asymptotic behavior of minimizing solutions of a Ginzburg–Landau type functional with a positive weight and with convex potential near 0 and we estimate the energy in this case. We also generalize a lower bound for the energy of unit vector field given initially by Brezis–Merle–Rivière.

Knots connected by wide ribbons

A ribbon is a smooth mapping (possibly self-intersecting) of an annulus [Formula: see text] in 3-space having constant width [Formula: see text]. Given a regular parametrization [Formula: see text], and a smooth unit vector field [Formula: see text] based along [Formula: see text], for a knot [Formula: see text], we may define a ribbon of width [Formula: see text] associated to [Formula: see text] and [Formula: see text] as the set of all points [Formula: see text], [Formula: see text]. For large [Formula: see text], ribbons, and their outer edge curves, may have self-intersections. In this paper, we analyze how the knot type of the outer ribbon edge [Formula: see text] relates to that of the original knot [Formula: see text]. Generically, as [Formula: see text], there is an eventual constant knot type. This eventual knot type is one of only finitely many possibilities which depend just on the vector field [Formula: see text]. The particular knot type within the finite set depends on the parametrized curves [Formula: see text], [Formula: see text], and their interactions. We demonstrate a way to control the curves and their parametrizations so that given two knot types [Formula: see text] and [Formula: see text], we can find a smooth ribbon of constant width connecting curves of these two knot types.

Slant Curves in the Unit Tangent Bundles of Surfaces

Let (M,g) be a surface and let (U(TM),G) be the unit tangent bundle of M endowed with the Sasaki metric. We know that any curve Γ(s) in U(TM) consist of a curve γ(s) in M and as unit vector field X(s) along γ(s). In this paper we study the geometric properties γ(s) and X(s) satisfying when Γ(s) is a slant geodesic.

Stripe patterns and a projection-valued formulation of the eikonal equation

We describe recent work on striped patterns in a system of block copolymers. A by-product of the characterization of such patterns is a new formulation of the eikonal equation. In this formulation, the unknown is a field of projection matrices of the form P = e ⊗ e , where e is a unit vector field. We describe how this formulation is better adapted to the description of striped patterns than the classical eikonal equation, and illustrate this with examples.

An improved gravitational formalism

In earlier papers, a theory of gravity was developed in which the space-time metric depends on a unit vector field and a scalar potential. The formalism becomes much simpler if one uses just a vector field. The theory is equivalent to a special case of the earlier one.