Related papers: Variance of vector fields -- Definition and proper…

We define the concepts of topological particles and topological radiation. These are nothing more than connected components of defects of a vector field. To each topological particle we assign an index which is an integer which is conserved…

In the broken-symmetry phase of the electroweak theory there is no unique definition of the electromagnetic field tensor in cases where the magnitude of the Higgs field differs from a constant value. The meaning of the electromagnetic field…

Attention is drawn to the mathematical equality of rights of symmetrical constituents derived affinor of a vector field in relation to its antisymmetric constituents. In this regard, raises the question not only of equitable accounting, but…

This paper introduces vector copulas associated with multivariate distributions with given multivariate marginals, based on the theory of measure transportation, and establishes a vector version of Sklar's theorem. The latter provides a…

We discuss several aspects of the geometry of vector fields in (Poincare'-Dulac) normal form. Our discussion relies substantially on Michel theory and aims at a constructive approach to simplify the analysis of normal forms via a splitting…

Phase transitions with spontaneous symmetry breaking and vector order parameter are considered in multidimensional theory of general relativity. Covariant equations, describing the gravitational properties of topological defects, are…

This short report establishes some basic properties of smooth vector fields on product manifolds. The main results are: (i) On a product manifold there always exists a direct sum decomposition into horizontal and vertical vector fields.…

We introduce vectorial and topological continuities for functions defined on vector metric spaces and illustrate spaces of such functions. Also, we describe some fundamental classes of vector valued functions and extension theorems.

We extend to the $n$-dimensional ellipsoid contained in $\R^{n+1},$ the Darboux theory of integrability for polynomial vector fields in the $n$-dimensional sphere (Llibre et al., 2018). New results on the maximum number of invariant…

Recently, a geometrical characterization of vector spaces served to generalize them into a new class of algebras. Instead of the algebraic properties of the underlying fields, we generalized the recently discovered property of such spaces…

This paper presents the theory of holomorphic vector valued modular forms from a geometric perspective. More precisely, we define certain holomorphic vector bundles on the modular orbifold of generalized elliptic curves whose sections are…

We investigate the existence of coordinate transformations which bring a given vector field on a manifold equipped with an involutive distribution into the form of a second-order differential equation field with parameters. We define…

Darboux's theorem and Jouanolou's theorem deal with the existence of first integrals and rational first integrals of a polynomial vector field. These results are given in terms of the degree of the polynomial vector field. Here we show that…

A new $(1,1)$-dimensional super vector bundle which exists on any super Riemann surface is described. Cross-sections of this bundle provide a new class of fields on a super Riemann surface which closely resemble holomorphic functions on a…

This article examines how the physical presence of field energy and particulate matter can be interpreted in terms of the topological properties of space-time. The theory is developed in terms of vector and matrix equations of exterior…

We state the implications on the properties of vector mesons due to gauge invariance. In particular, we find that polarized vector mesons exhibit a property in the radiation distribution of order $\omega^{-1}$ in the photon energy, namely…

This monograph, written for educational purposes, serves as an introduction to the concept of integrability as it applies to systems of differential equations (both ordinary and partial) as well as to vector-valued fields. The general cases…

Smooth vector fields on $\mathbb{R}^n$ can be decomposed into the sum of a gradient vector field and divergence-free (solenoidal) vector field under suitable hypotheses. This is called the Helmholtz-Hodge decomposition (HHD), which has been…

We prove that vector bundles give equations of cactus varieties. We derive from it that equations coming from vector bundles are not enough to define secant varieties of Veronese varieties in general.

We give a global geometric decomposition of continuously differentiable vector fields on $\mathbb{R}^n$. More precisely, given a vector field of class $\mathcal{C}^{1}$ on $\mathbb{R}^{n}$, and a geometric structure on $\mathbb{R}^n$, we…