Study Of Vector Fields Research Articles

The Product Formula and Evolution Families of Nonexpansive Mappings

In this work we obtain a product formula type for a two-parameters commuting family of nonexpansive mappings on $${\mathbb {D}}$$ . This is established by following the techniques used by Simeon Reich and David Shoikhet in the study of one-parameter semigroups of holomorphic and nonexpansive self-mappings in $${\mathbb {D}}$$ . Also, we stablish such a formula for the family of non-linear resolvent of a strongly $$\rho $$ -monotone functions on $${\mathbb {D}}$$ and its relation with evolution families of nonexpansive mappings on $${\mathbb {D}}$$ . It is worthy mentioning that the product formula is linked with semigroup of linear and nonlinear operators. Also it is associated with the study of vector fields and flows, but in the literature it is established a product formula for time independent flow.

Nonlinear Stability of Relative Equilibria and Isomorphic Vector Fields

We present applications of the notion of isomorphic vector fields to the study of nonlinear stability of relative equilibria. Isomorphic vector fields were introduced by Hepworth [Theory Appl. Categ. 22 (2009), 542-587] in his study of vector fields on differentiable stacks. Here we argue in favor of the usefulness of replacing an equivariant vector field by an isomorphic one to study nonlinear stability of relative equilibria. In particular, we use this idea to obtain a criterion for nonlinear stability. As an application, we offer an alternative proof of Montaldi and Rodr\'iguez-Olmos's criterion [arXiv:1509.04961] for stability of Hamiltonian relative equilibria.

Invariant Vector Fields and Groupoids

We use the notion of isomorphism between two invariant vector fields to shed new light on the issue of linearization of an invariant vector field near a relative equilibrium. We argue that the notion is useful in understanding the passage from the space of invariant vector fields in a tube around a group orbit to the space invariant vector fields on a slice to the orbit. The notion comes from Hepworth's study of vector fields on stacks.

Polynomial parametrization of nonsingular complete intersection curves

In this paper we give a new method to compute a polynomial parametrization, in case it exists, of an affine nonsingular complete intersection curve. Our method is based on the study of vector fields on nonsingular algebraic curves. In contrast to the existing methods, our algorithm does not use any projection, and no sample point on the curve is needed. It is also important to stress the fact that the algorithm produces a parametrization with coefficients in the ground field.

Two theorems by Helmholtz

Why isitthat div and curl crop up everywhere in the study of vector fields? These theorems by Helmholtz explain.

Vector fields tangent to a Reeb foliation on S3

The main subject of this work is the study of vector fields on the tridimensional sphere S3 tangent to a Reeb folitation o [l] and generic with respect to their ,nonwandering set. In some sense we are continuing research by Oliva [2] and Mello [3] who treated questions about the qualitative and generic study of singularities in Appell conservative fields (see the Introduction of [3]). We use results of Sotomayor [4] concerning bifurcations.

Potentials for tangent tensor fields on spheroids

In three-dimensional Euclidean space let S be a closed simply connected, smooth surface (spheroid). Let \(\hat n\) be the outward unit normal to S, ▽Sthe surface gradient on S, IS the metric tensor on S, gij the four covariant components of IS (i,j = 1, 2), hijthe four covariant components of -\(\hat n\)xIS, and Dicovariant differentiation on S. It is well known that for any tangent vector field u on S there exist scalars ϕ and ψ on S, unique to within additive constants, such that \(u = \nabla _s \varphi - \hat n \times \nabla _s \psi \); the covariant components of u are \(u_i = D_i \varphi + h_i^j D_j \psi \). This theorem is very useful in the study of vector fields in spherical coordinates. The present paper gives an analogous theorem for real second-order tangent tensor fields F on S: for any such F there exist scalar fields H, L, M, N such that the covariant components of F are $$F_{ij} = H h{}_{ij} + Lg_{ij} + E_{ij} (M,N),$$ where $$E_{ij} (M,N) = ( - \nabla _s ^2 M)g_{ij} + 2D_i D_j M + (h_i ^k D_j + h_j ^k D_i )D_k N$$ It is shown that H and L are uniquely determined by F but that M and N are not. The set of complex scalar fields ℳ′ = M′+iN′ such that Eij(M′, N′)=0 is shown to constitute a four-dimensional complex linear space \(\mathfrak{W}\). The scalars M and N which help to generate a given F are uniquely determined by F and the condition that, for every ℳ′ in \(\mathfrak{W}\), $$\mathop \smallint \limits_s (M - iN)\mathcal{M}\prime dS = 0$$ The real linear space of second-order tangent tensor fields on S which have simultaneously the form E(M, 0) and the form E(0, N) is shown to have dimension zero on a sphere, dimension four on a non-spherical, intrinsically axisymmetric spheroid (a spheroid whose isometries form a compact, one parameter group), and dimension six on a spheroid which is not intrinsically axisymmetric. Applications of the representation theorem to tensor problems in spherical coordinates are briefly discussed.

Vector Fields and Infinitesimal Transformations on Almost-Hermitian Manifolds with Boundary

Many authors have made interesting and important contributions to the study of vector fields or infinitesimal transformations on compact orientable Riemannian manifolds and Hermitian manifolds without boundary. Recently, Hsiung (6, 7, 8) has extended some of these results to compact orientable Riemannian manifolds with boundary. The purpose of this paper is to continue Hsiung's work by studying vector fields and infinitesimal transformations on almost-Hermitian manifolds with boundary.