Vector Field Problem Research Articles

A FETI Preconditioner for Two Dimensional Edge Element Approximations of Maxwell's Equations on Nonmatching Grids

A class of FETI methods for the mortar approximation of a vector field problem in two dimensions is proposed. Edge element discretizations of lowest degree are considered. The method proposed can be employed with geometrically conforming and nonconforming partitions. Our numerical results show that its condition number increases only with the number of unknowns in each subdomain and is independent of the number of subdomains and the size of the problem.

The trace problem for vector fields satisfying Hörmander's condition

Trace theorems are proved for non-isotropic Sobolev and $L^p$ -Lipschitz spaces defined by vector fields satisfying Hormander's bracket condition of order 2. It is shown that the loss of regularity by traces is the same as in the classical case.

Singular Perturbations Arising in Hilbert's 16th Problem for Quadratic Vector Fields

A solution to the second part of Hilbert's 16th problem consists of finding out (or proving the existence of) an upper bound to the number of limit cycles in the family of polynomial planar vector fields. In this article, we indicate a way to tackle the singular perturbation problems that have to be studied in the quadratic case. In particular, for perturbations from the family (λx — y) (x + 1) ∂/∂x + (x + λy) (x + 1) ∂/∂y, we prove that the cyclicity of certain limit periodic sets is bounded by 1. The proposed method is applicable in any multi-parameter bifurcation problem and forms an extension to the known technique of “significant degeneration”, i.e. the rescaling of parameters by means of different weights.

Cyclicity of Graphics with Semi-Hyperbolic Points Inside Quadratic Systems

This paper is a part of the proof of the existential part of Hilbert's 16th problem for quadratic vector fields initiated in l2r. Its principal aim is the proof of the finite cyclicity of four elementary graphics among quadratic systems with two semi-hyperbolic points and surrounding a center, namely, the graphics (I^2_5), \ (I^2_{16b}), \ (H^1_9) and (H^1_{11}) (using the names introduced in l2r). The technique used is a refinement of the technique of Khovanskii as adapted to the finite cyclicity of graphics by Il'yashenko and Yakovenko, together with an equivalent of the Bautin trick to treat the center case. We show that the cyclicity of each of the first three graphics is equal to 2 and that the cyclicity of the fourth one is equal to three. We improve the known results about finite cyclicity of the graphics (H^1_8), \ (H^1_{10}), \ (I^2_4) by showing that their cyclicities are equal to 2.

On the vector field problem for product manifolds

On the vector field problem for product manifolds

Hilbert's 16-th problem for quadratic vector fields and cyclicity of graphics

We discuss here a method of proof of the existential part of Hilbert's 16th problem for quadratic vector fields initiated in 1991 [7]. The program consists in proving the finite cyclicity of 121 graphics appearing among quadratic systems. We briefly summarize some of the methods introduced for that purpose and describe some new developments obtained with Świrszcz and Żoldek [20] for graphics surrounding a center.

On A Minimization Problem for Vector Fields in L 1

This paper treats the problem of minimizing the norm of vector fields in L1 with prescribed divergence. The ridge of Ω. plays an important role in the analysis, and in the case where Ω ⊂ R2 is a polygonal domain, the ridge is thoroughly analysed and some examples are presented. In the case where Ω ⊂ Rn is a Lipschitz domain and the divergence is a finite positive Borel measure, the infimum is calculated, and it is shown that if an extremal exists, then it is of the form υ1 = −F▿d, where F is a nonnegative function and d(x) is the distance from x ∈ Ω to the boundary ∂Ω. Finally, if Ω ⊂ R2 is a polygonal domain and the measure is represented by a nonnegative continuous function, then an explicit expression for the extremal is given, and it is proven that this extremal is unique.

A finite element code for subsidence problems: Lagamine

A general concept for the numerical modelling of an aquifer subsidence has been presented. Its main advantage is the separation from the three-dimensional flow model (which is a scalar field with one variable per node) and the compaction model (which is a vector field problem coupled to a scalar field problem, with 4 variables per node). This separation is very interesting for a “plate-like aquifer” because the compaction problem is reduced to a series of unidimensional problems with a small number of degree of freedom.

Self-consistent solution to the back-reaction problem for vector fields in two dimensions

We describe vector fields propagating in a two-dimensional spatially flat Robertson-Walker background using a curved space-time generalization of the Stueckelberg formalism. We prove that the energy-momentum tensor expectation value in the vacuum defined through energy minimization is renormalizable and yields the usual anomalous trace in the massless limit of vector mesons. Further on we study the back-reaction problem using the semiclassical Einstein equations. In the massive case we found that the physical solution requires the cosmological constant to vanish but not necessarily with a vanishing curvature. However, for certain initial conditions of the scale factor, the de Sitter metric is a consistent solution with the cosmological constant depending on powers of the curvature scalar greater than one. In addition, matter is continuously being created at a steady state.

PDE techniques for solving the problem of radar scattering by a body of revolution

The authors discuss two partial differential equation (PDE) techniques, namely the finite difference (FD) method and the finite element method (FEM), for solving the problem of radar scattering by a body of revolution (BOR). The formulation of the BOR problem is based upon the use of the coupled azimuthal potentials (CAPs), introduced by M.A. Morgan et al. (1977), that allow one to reduce the vector field problem to the solution of two coupled scalar wave equations. An approach to FD/FEM mesh truncation is developed on the basis of a multipole representation for the CAPs in the asymptotic region. Numerical results are presented for some representative geometries to illustrate the application of the PDE methods and some observations regarding the comparative performance of the two methods are included. >

Massive vector field and initial singularity problem in a charged universe

We find that a massive U (1) vector field A μ in a charged Robertson-Walker universe for k=+1, 0, and −1 can solve the initial singularity problem even at the level of classical gravity because the energy density becomes negative as the scale factor a of the metric approaches zero. Namely, our universe might have started to expand from a finite size a 0. The condition that the universe expands from a 0 larger than the Planck scale gives a constraint on the charge density at t=0.

Dirac equation in external vector fields: Separation of variables

The method of separation of variables in the Dirac equation in the external vector fields is developed through the search for exact solutions. The essence of the method consists of the separation of the first-order matricial differential operators that define the dependence of the Dirac bispinor on the related variables, but commutation of such operators with the operator of the equations or between them is not assumed. This approach, which is perfectly justified in the presence of gravitational fields, permits one to prove rigorous theorems about necessary and sufficient conditions on the field functions that allow one to separate variables in the Dirac equation. In analogous investigations by other authors [Bagrov et al., Exact solutions of Relativistic Wave Equations (Nauka, Novosibirst, 1982)] for electromagnetic fields an essential demand related to the operators that define the dependence of the bispinor on the separated variables is the demand for the commutation of a complete set of operators between them or with the operators of the Dirac equation. For this reason a series of possibilities that do not satisfy this demand escape the attention of these other authors. The present work liquidates this gap, solving the problem for external vector fields in general.

Isotropic vector fields on spheres, spinor structures on spheres and projective spaces

In a previous paper “Fields of totally isotropic subspaces and almost complex structures” [1c], we constructed on the sphere S 4, in the complexified tangent bundle, a global field of totally isotropic planes to obtain a counterexample nullifying a statement about some manifolds and vector bundles; we proved that if the complexified tangent bundle T c( M) of a 2 r-dimensional C ∞ manifold M is a Withney sum T c(M) = η ⊕ η′ , where η, η′ are r-complex subbundles such that for any x belonging to M, η′ x = η macr; x , M does not have necessarily an almost complex structure. This article has two aims. First, we construct on any sphere global fields of isotropic vectors, nowhere zero, cross-sections of the complexified tangent bundle. We give for n ≥ 3 two independent methods, one with continuous hypothesis only, the second method with smooth conditions, valid for any dimension n ≥ 2. Our developments bring a new proof of the statement given in [1c] and a more precise and new result: the C ∞ complexified tangent bundle of any sphere is trivializable. Our second aim is to give an elementary approach to the existence conditions for spinor structures on spheres and projective spaces, using naturally our methods developed in numerous papers, particularly in [1d] and [1e]. However, the major part of the paper deals with the isotropic vector field problem on spheres and the reader unfamilar with spinors will be able to read the Secs. I, II and III, except some details at the end. We think that this paper will contribute to explaining some delicate questions and remove some errors or sophisms and also point out the role of the elementary geometry in spinor structures for particularly simple cases.

Finite cyclicity of loops and cusps

The 16th Hilbert problem for polynomial planar vector fields is a consequence of the following conjecture: any analytic deformation of the planar vector field germ along a limit set has a finite cyclicity (a finite bound for the number of limit cycles near the limit periodic set). The property of finite cyclicity is established for the simplest singular limit sets loops made by homoclinic connection at a hyperbolic saddle point and cuspidal singular points.

A contribution to the topological classification of homogeneous vector fields in R3

In this paper we study the topological classification problem for homogeneous polynomial vector fields of degree k > 2 in three variables. In 1960 Markus [7] (see also [6]) classified topologically the quadratic homogeneous vector fields in two variables. Subsequent works relative to the differentiable classification of these vector fields can be found in Date [4], Newton [lo], Samardzija [13], Vulpe and Sibirskii [14]. Let us now consider a homogeneous vector field Q = (Q,, Q2, Q,), of degree k > 2 in R3 with an isolated singularity at 0 E R3, i.e., all the Qi, i = 1, 2, 3, are homogeneous polynomials of degree k, with an isolated common zero at 0 E R3. The vector field defined as Qr(.?) = Q(i) (X, Qi(.%) + X2Q2(X) +X3 Q&f)). X, X = (2, ,X2, X3) E S2, is tangent to the 2-sphere S2, and for any regular orbit y of Qr the set

Approximation of potential and solenoidal vector fields

In order to formulate the approximation problem for potential vector fields, we introduce some notations. By Gp(~) we shall denote the closure in Lp(~) of the subspace consisto ing of the restrictions to ~ of the vector fields from G~(R~)----{v(x):v(x)=V~(x), ~(x)~C~(R,~)}, where v(x) =(v~ .... , v~), x= (x~ ..... x~), ~R '~ is an arbitrary domain, n /> 2, ] < p < ~. Let G~(~)={v(x): v(x)=V~(x)~Lp(~)}. Obviously, Gp(~)~Gp(~) . The inverse inclusion need not hold.

Clifford bundles, immersions of manifolds and the vector field problem

Clifford bundles, immersions of manifolds and the vector field problem

Immersions of Complex Projective Spaces and the Generalized Vector Field Problem

Immersions of Complex Projective Spaces and the Generalized Vector Field Problem

GENERALIZED HOMOLOGY AND THE GENERALIZED VECTOR FIELD PROBLEM

Journal Article GENERALIZED HOMOLOGY AND THE GENERALIZED VECTOR FIELD PROBLEM Get access DONALD DAVIS DONALD DAVIS Stanford University and University of CaliforniaSan Diego Search for other works by this author on: Oxford Academic Google Scholar The Quarterly Journal of Mathematics, Volume 25, Issue 1, 1974, Pages 169–193, https://doi.org/10.1093/qmath/25.1.169 Published: 01 January 1974 Article history Received: 23 March 1973 Published: 01 January 1974

Alternative Representations of Source‐Excited Vector and Scalar Fields

Radiation from electromagnetic sources in layered regions is reviewed and illustrated for the case of planar stratification. Mter a reduction of the vector field problem to equivalent scalar Green's function problems, the latter are expressed in alternative forms by the systematic application of one‐dimensional characteristic Green's function techniques. Solutions are derived in this manner without the intervention of the classical Watson procedure and are interpretable as guided‐wave representations along various coordinate directions. Considerations pertaining to scalar Green's functions apply directly, upon suitable reinterpretation of the electromagnetic parameters, to other fields such as the acoustic which are derivable from scalar potentials.