Existence Of Vector Fields Research Articles

On the existence of vector fields with nonnegative divergene in rearrangement-invariant spaces

We investigate the existence of solutions of \begin{equation} divF=\mu,\text{ on } \mathbf{R}^{d}. \tag{*} \end{equation} Here, $\mu\geq0$ is a Radon measure, and we look for a solution $F\in X(\mathbf{R}^{d}\rightarrow\mathbf{R}^{d})$, where $X$ is a rearrangement-invariant space. We first prove the equivalence of the following assertions: (i) (*) has a solution for some nontrivial $\mu$; (ii) the function $x\mapsto |x|^{1-d}1_{B^{c}}(x)$ belongs to $X$. Here, $B$ is the unit ball in $\mathbf{R}^{d}$. We next investigate the solvability of (*) when $\mu$ is fixed. A sufficient condition is that $I_{1}\mu\in X$, where $ I_{1}\mu$ is the 1-Riesz potential of $\mu$. This condition turns out to be also necessary when the Boyd indexes of $X$ belong to $(0,1)$. Our analysis generalizes the one of Phuc and Torres (2008) when $X=L^{p}$.

Signatures of anisotropic sources in the squeezed-limit bispectrum of the cosmic microwave background

The bispectrum of primordial curvature perturbations in the squeezed configuration, in which one wavenumber, k3, is much smaller than the other two, k3 << k1 ≈ k2, plays a special role in constraining the physics of inflation. In this paper we study a new phenomenological signature in the squeezed-limit bispectrum: namely, the amplitude of the squeezed-limit bispectrum depends on an angle between k1 and k3 such that Bζ(k1,k2,k3) → 2∑LcLPL(k̂1·k̂3)Pζ(k1)Pζ(k3), where PL are the Legendre polynomials. While c0 is related to the usual local-form fNL parameter as c0 = 6fNL/5, the higher-multipole coefficients, c1, c2, etc., have not been constrained by the data. Primordial curvature perturbations sourced by large-scale magnetic fields generate non-vanishing c0, c1, and c2. Inflation models whose action contains a term like I(ϕ)2F2 generate c2 = c0/2. A recently proposed ``solid inflation'' model generates c2 >> c0. A cosmic-variance-limited experiment measuring temperature anisotropy of the cosmic microwave background up to ℓmax = 2000 is able to measure these coefficients down to δc0 = 4.4, δc1 = 61, and δc2 = 13 (68% CL). We also find that c0 and c1, and c0 and c2, are nearly uncorrelated. Measurements of these coefficients will open up a new window into the physics of inflation such as the existence of vector fields during inflation or non-trivial symmetry structure of inflaton fields. Finally, we show that the original form of the Suyama-Yamaguchi inequality does not apply to the case involving higher-spin fields, but a generalized form does.

Spheres with more than 7 vector fields: All the fault of Spin(9)

We give an interpretation of the maximal number of linearly independent vector fields on spheres in terms of the Spin(9) representation on R16. This casts an insight on the role of Spin(9) as a subgroup of SO(16) on the existence of vector fields on spheres, parallel to the one played by complex, quaternionic and octonionic structures on R2, R4 and R8, respectively.

Vector fields with a given set of singular points

Theorems on the existence of vector fields with given sets of indexes of isolated singular points are proved for the cases of closed manifolds, pairs of manifolds, manifolds with boundary, and gradient fields. It is proved that, on a two-dimensional manifold, an index of an isolated singular point of the gradient field is not greater than one.

On the existence of vector fields with a given set of singular points on a two-dimensional closed oriented manifold

We study the possibility of constructing locally gradient and arbitrary vector fields with a given set of singular points on a two-dimensional closed oriented manifold. The sum of the indices of the vector field at these points is equal to the Euler characteristic of the manifold.

On the symmetry groups of dynamic systems

The existence of vector fields which commute with the vector field of the initial system and are defined in the entire phase space is discussed. The phase fluxes of these fields are well-known to be symmetry groups of a dynamic system, since they map the set of all its solutions into itself. Obstacles to the existence of non-trivial symmetry groups are the generation of a large number of non-degenerate periodic solutions, and the transversal intersection of asymptotic surfaces. The symmetry groups of systems of a “normal” type, which play an important part in perturbation theory, are examined in detail. The general results are applied, in particular, to Hamiltonian systems. It is shown that the equations of rotation of a heavy asymmetric rigid body with a fixed point do not have a non-trivial symmetry group if the centre of mass of the body is not the same as the point of suspension. In particular, there is no supplementary many-valued analytic integral which is independent of the classical energy and area integrals.