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

2012 ◽  
Vol 370 (1965) ◽  
pp. 1730-1739 ◽  
Author(s):  
Mark A. Peletier ◽  
Marco Veneroni
Keyword(s):  
Vector Field ◽  
Block Copolymers ◽  
Recent Work ◽  
Eikonal Equation ◽  
Unit Vector ◽  
Unit Vector Field ◽  
New Formulation ◽  
Projection Matrices

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.

scholarly journals Knots connected by wide ribbons

2019 ◽  
Vol 28 (12) ◽  
pp. 1950071
Author(s):  
Susan C. Brooks ◽  
Oguz Durumeric ◽  
Jonathan Simon
Keyword(s):  
Vector Field ◽  
Constant Width ◽  
Unit Vector ◽  
Outer Edge ◽  
Smooth Mapping ◽  
Unit Vector Field ◽  
Finite Set

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.

scholarly journals A new quaternion valued frame of curves with an application

Filomat ◽  
2021 ◽  
Vol 35 (1) ◽  
pp. 315-330
Author(s):  
Gizem Cansu ◽  
Yusuf Yaylı ◽  
İsmail Gök
Keyword(s):  
Vector Field ◽  
Unit Vector ◽  
Space Curve ◽  
Unit Vector Field ◽  
The Right

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.

Gauge freedom, anholonomy, and Hopf index of a three-dimensional unit vector field

1993 ◽  
Vol 47 (9) ◽  
pp. 5438-5441 ◽  
Author(s):  
Radha Balakrishnan ◽  
A. R. Bishop ◽  
R. Dandoloff
Keyword(s):  
Vector Field ◽  
Three Dimensional ◽  
Unit Vector ◽  
Gauge Freedom ◽  
Unit Vector Field ◽  
Dimensional Unit

scholarly journals Slant Curves in the Unit Tangent Bundles of Surfaces

ISRN Geometry ◽  
2013 ◽  
Vol 2013 ◽  
pp. 1-5 ◽  
Author(s):  
Zhong Hua Hou ◽  
Lei Sun
Keyword(s):  
Vector Field ◽  
Tangent Bundle ◽  
Unit Vector ◽  
Sasaki Metric ◽  
Geometric Properties ◽  
Unit Vector Field ◽  
Unit Tangent Bundle ◽  
Tangent Bundles ◽  
Slant Curves

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.

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

Mathematics ◽  
2020 ◽  
Vol 8 (6) ◽  
pp. 997
Author(s):  
Rejeb Hadiji ◽  
Carmen Perugia
Keyword(s):  
Asymptotic Behavior ◽  
Vector Field ◽  
Lower Bound ◽  
Unit Vector ◽  
Ginzburg Landau ◽  
Positive Weight ◽  
Unit Vector Field

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.

On the volume of a unit vector field on the three-sphere

1986 ◽  
Vol 61 (1) ◽  
pp. 177-192 ◽  
Author(s):  
Herman Gluck ◽  
Wolfgang Ziller
Keyword(s):  
Vector Field ◽  
Unit Vector ◽  
Three Sphere ◽  
Unit Vector Field

scholarly journals A topological lower bound for the energy of a unit vector field on a closed Euclidean hypersurface

2020 ◽  
Vol 125 (3) ◽  
pp. 203-213
Author(s):  
Fabiano G. B. Brito ◽  
Icaro Gonçalves ◽  
Adriana V. Nicoli
Keyword(s):  
Vector Field ◽  
Lower Bound ◽  
Unit Vector ◽  
Unit Vector Field ◽  
Euclidean Hypersurface

An improved gravitational formalism

1978 ◽  
Vol 56 (9) ◽  
pp. 1202-1203 ◽  
Author(s):  
Peter Rastall
Keyword(s):  
Vector Field ◽  
Scalar Potential ◽  
Unit Vector ◽  
Space Time ◽  
Unit Vector Field ◽  
Theory Of Gravity ◽  
Special Case

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.

scholarly journals Ideal, non-extended formulations for disjunctive constraints admitting a network representation

2021 ◽  
Author(s):  
Tamás Kis ◽  
Markó Horváth
Keyword(s):  
Piecewise Linear ◽  
Unit Vector ◽  
Linear Functions ◽  
Full Characterization ◽  
Network Representation ◽  
Disjunctive Constraints ◽  
New Formulation ◽  
Logarithmic Number

AbstractIn this paper we reconsider a known technique for constructing strong MIP formulations for disjunctive constraints of the form $$x \in \bigcup _{i=1}^m P_i$$ x ∈ ⋃ i = 1 m P i , where the $$P_i$$ P i are polytopes. The formulation is based on the Cayley Embedding of the union of polytopes, namely, $$Q := \mathrm {conv}(\bigcup _{i=1}^m P_i\times \{\epsilon ^i\})$$ Q : = conv ( ⋃ i = 1 m P i × { ϵ i } ) , where $$\epsilon ^i$$ ϵ i is the ith unit vector in $${\mathbb {R}}^m$$ R m . Our main contribution is a full characterization of the facets of Q, provided it has a certain network representation. In the second half of the paper, we work-out a number of applications from the literature, e.g., special ordered sets of type 2, logical constraints, the cardinality indicating polytope, union of simplicies, etc., along with a more complex recent example. Furthermore, we describe a new formulation for piecewise linear functions defined on a grid triangulation of a rectangular region $$D \subset {\mathbb {R}}^d$$ D ⊂ R d using a logarithmic number of auxilirary variables in the number of gridpoints in D for any fixed d. The series of applications demonstrates the richness of the class of disjunctive constraints for which our method can be applied.

Sign in / Sign up

Export Citation Format

Share Document