Search results for "Unit vector"
showing 10 items of 12 documents
Minimal unit vector fields
2002
We compute the first variation of the functional that assigns each unit vector field the volume of its image in the unit tangent bundle. It is shown that critical points are exactly those vector fields that determine a minimal immersion. We also find a necessary and sufficient condition that a vector field, defined in an open manifold, must fulfill to be minimal, and obtain a simpler equivalent condition when the vector field is Killing. The condition is fulfilled, in particular, by the characteristic vector field of a Sasakian manifold and by Hopf vector fields on spheres.
Volume, energy and generalized energy of unit vector fields on Berger spheres: stability of Hopf vector fields
2005
We study to what extent the known results concerning the behaviour of Hopf vector fields, with respect to volume, energy and generalized energy functionals, on the round sphere are still valid for the metrics obtained by performing the canonical variation of the Hopf fibration.
Integration of finite displacement interface element in reference and current configurations
2017
In the present paper the non-linear behaviour of a solid body with embedded cohesive interfaces is examined in a finite displacements context. The principal target is the formulation of a two dimensional interface finite element which is referred to a local reference frame, defined by normal and tangential unit vectors to the interface middle surface. All the geometric operators, such as the interface elongation and the reference frame, are computed as function of the actual nodal displacements. The constitutive cohesive law is defined in terms of Helmholtz free energy for unit undeformed interface surface and, in order to obtain the same nodal force vector and stiffness matrix by the two i…
General duality in vector optimization
1993
Vector minimization of a relation F valued in an ordered vector space under a constraint A consists in finding x 0 ∊ A w,0 ∊ Fx$0 such that w,0 is minimal in FA. To a family of vector minimization problemsminimize , one associates a Lagrange relation where ξ belongs to an arbitrary class Ξ of mappings, the main purpose being to recover solutions of the original problem from the vector minimization of the Lagrange relation for an appropriate ξ. This ξ turns out to be a solution of a dual vector maximization problem. Characterizations of exact and approximate duality in terms of vector (generalized with respect to Ξ) convexity and subdifferentiability are given. They extend the theory existin…
Unit Vector Fields that are Critical Points of the Volume and of the Energy: Characterization and Examples
2007
In the last few years, many works have appeared containing examples and general results on harmonicity and minimality of vector fields in different geometrical situations. This survey will be devoted to describe many of the known examples, as well as the general results from where they are obtained.
Some Improvements on Relativistic Positioning Systems
2018
[EN] We make some considerations about Relativistic Positioning Systems (RPS). Four satellites are needed to position a user. First of all we define the main concepts. Errors should be taken into account. Errors depend on the Jacobian transformation matrix. Its Jacobian is proportional to the tetrahedron volume whose vertexes are the four tips of the receiver-satellite unit vectors. If the four satellites are seen by the user on a circumference in the sky, then, the Jacobian and the tetrahedron volume vanish. The users we consider are spacecraft. Spacecraft to be positioned cannot be close to a null Jacobian satellites-user configuration. These regions have to be avoided choosing an appropr…
Intraband and interband spin-orbit torques in noncentrosymmetric ferromagnets
2015
Intraband and interband contributions to the current-driven spin-orbit torque in magnetic materials lacking inversion symmetry are theoretically studied using Kubo formula. In addition to the current-driven field-like torque ${\bf T}_{\rm FL}= \tau_{\rm FL}{\bf m}\times{\bf u}_{\rm so}$ (${\bf u}_{\rm so}$ being a unit vector determined by the symmetry of the spin-orbit coupling), we explore the intrinsic contribution arising from impurity-independent interband transitions and producing an anti-damping-like torque of the form ${\bf T}_{\rm DL}= \tau_{\rm DL}{\bf m}\times({\bf u}_{\rm so}\times{\bf m})$. Analytical expressions are obtained in the model case of a magnetic Rashba two-dimension…
Description and evolution of anisotropy in superfluid vortex tangles with counterflow and rotation
2006
We examine several vectorial and tensorial descriptions of the geometry of turbulent vortex tangles. We study the anisotropy in rotating counterflow experiments, in which the geometry of the tangle is especially interesting because of the opposite effects of rotation, which orients the vortices, and counterflow, which randomizes them. We propose to describe the anisotropy and the polarization of the vortex tangle through a tensor, which contains the first and second moments of the distribution of the unit vector ${\mathbf{s}}^{\ensuremath{'}}$ locally tangent to the vortex lines. We use an analogy with paramagnetism to estimate the anisotropy, the average polarization, the polarization fluc…
Monte Carlo investigation of a model for a three-dimensional orientational glass with short-range gaussian interaction
1987
The analogue of the Edwards-Anderson model for isotropic vector spin glasses, but taking quadrupoles instead of unit vectors at each lattice site of the considered simple cubic lattice, is studied as a model for an orientational glass. We study both the case where the quadrupole moment can orient in a three-dimensional space (m=3) and the case where the orientation is restricted to a plane (m=2), but otherwise the Hamiltonian is fully isotropic. ℋ= $$ - \sum\limits_{\left\langle {i,j} \right\rangle } {J_{ij} } \left[ {\left( {\sum\limits_{\mu = 1}^m {S_i^\mu S_j^\mu } } \right)^2 - \frac{1}{m}} \right]$$ , whereJ ij is a random gaussian interaction between nearest neighbors, andS i μ the μ'…
An invariant analytic orthonormalization procedure with an application to coherent states
2007
We discuss a general strategy which produces an orthonormal set of vectors, stable under the action of a given set of unitary operators Aj, j=1,2,n, starting from a fixed normalized vector in H and from a set of unitary operators. We discuss several examples of this procedure and, in particular, we show how a set of coherentlike vectors can be produced and in which condition over the lattice spacing this can be done. © 2007 American Institute of Physics.