Independent Vector Fields Research Articles

Tensors with Constant Components in the Constitutive Equations of a Hemitropic Micropolar Solids

The present paper is devoted to elastic potentials and the constitutive equations of mechanics of anisotropic micropolar solids, the kinematics of which can be specified by two independent vector fields: a contravariant field of translational displacements and a contravariant pseudovector field of microrotations of weight +1. The quadratic stress potential is represented by three constitutive tensors of the fourth rank, two of which are pseudotensor in nature and can be assigned weights –2 and –1. Such a solid is completely specified by the 171st micropolar elastic modulus. The main attention is focused on the model of a hemitropic (half-isotropic, demitropic) micropolar elastic solid characterized by nine constitutive constants. The components of the constitutive pseudo-tensor of weight ‒1 turn out to be sensitive to mirror reflection transformations in three-dimensional space. A peculiar algebraic structure of the constitutive tensors of a hemitropic solid, more precisely, their absolute analogues obtained by multiplying by integer powers of a pseudoscalar unity, is studied. It is shown that these tensors can always be constructed from isomers (isomer) of a tensor with constant components (generally insensitive to any transformations of the coordinate system) and one additional fourth-rank tensor constructed, in turn, from the components of the metric tensor.

Geometrical Aspects of Motion of Charged Particles in Magnetic and Killing Magnetic Fields and Their Corresponding Trajectories in Anti-De Sitter 3-Space

We inquire the motion of charged particles varying under the effect of Lorentz force produced by magnetic and Killing magnetic fields in anti-de Sitter 3-space . Primarily, we characterize magnetic trajectories in in terms of their Frenet apparatus. Thereafter, utilizing a geometrical model of split-quaternions for (where corresponds to a subspace of the Lie group H of split-quaternions) we find 6 independent unit Killing vector fields on which constitutes a basis for the corresponding 6D Lie algebra i(). We conclude with characterizations of Killing magnetic trajectories in with respect to quasi-slope, curvature and torsion or pseudo-torsion (depending on the situation).

Origin independent current density vector fields induced by time-dependent magnetic field. I. The LiH molecule.

The electronic current density, induced in a molecule by the optical magnetic field associated with a frequency-dependent monochromatic plane wave, assumed to be spatially uniform within the electric quadrupole approximation, has been studied by using a theoretical method based on a continuous translation of its origin. The induced electronic current density vector field designated by this procedure, invariant of the origin for any point of the molecular domain, is obtained via a computational scheme, formally annihilating the diamagnetic contribution of the conventional common-origin approach based on perturbation theory. In a preliminary application of the theoretical methods outlined in the present work, the simple molecule of lithium hydride has been investigated. Particular attention has been paid to the structure of induced electronic current density for several values of the magnetic field frequency by investigating equilibrium points of four different types, organized in stagnation lines, which constitute its stagnation graph, i.e., a topological fingerprint of the vector field conveying complete information in the condensed form, to verify the fulfillment of fundamental requirements, e.g., the continuity equation and the Poincaré-Hopf theorem on spherical and toroidal surfaces.

An application of a global lifting method for homogeneous Hörmander vector fields to the Gibbons conjecture

In this paper we exploit a global lifting method for homogeneous Hormander vector fields in order to extend the Gibbons conjecture to any second-order differential operator $$\mathcal {L}_X = \sum _{j = 1}^mX_j^2$$, where the $$X_j$$’s are linearly independent smooth vector fields on $$\mathbb {R}^n$$ satisfying Hormander’s rank condition and fulfilling a suitable homogeneity property with respect to a family of non-isotropic dilations. The class of these operators comprehends the sub-Laplacians on Carnot groups, the smooth Grushin-type operators and the smooth $$\Delta _\lambda $$-Laplacians studied by Franchi, Lanconelli and Kogoj. We also establish a comparison result for the solutions of the semi-linear equation $$\mathcal {L}_Xu+f(u) = 0$$ under suitable assumptions on the function f.

A note on classification of spatially homogeneous rotating space-times in f(R) theory of gravity according to their proper conformal vector fields

In this paper, we present a study of proper conformal vector fields of spatially homogeneous rotating space-times in the [Formula: see text] theory of gravity. A total of six cases have been discussed in detail. It is found that the conformally non-flat spatially homogeneous rotating space-times do not admit proper conformal vector fields, while in a conformally flat case the space-time admits 15 independent conformal vector fields.

From Hessian to Weitzenböck: manifolds with torsion-carrying connections

We investigate affine connections that have zero curvature but not necessarily zero torsion. Slightly generalizing from what is known as Weitzenbock connections, such non-flat connections (we call “pseudo-Weitzenbock connections”) can be constructed from any frame (a set of linearly independent vector fields), not just the orthonormal frames, with their torsions vanishing if and only if the frame is a coordinate frame. In such situations, the notion of biorthogonal frames generalizes the notion of biorthogonal coordinates of a Hessian manifold, with respect to any given Riemannian metric g (not necessarily Hessian). Our main theorem shows that the pair of pseudo-Weitzenbock connections, each adapted to one of the pair of g-biorthogonal frames, are g-conjugate to each other. As a result, the pseudo-Weitzenbock connection pair generalize dually flat connections characteristic of Hessian manifolds, by being both curvature-free yet admitting (generally unequal) torsions. These results allow us to construct a pseudo-Weitzenbock connection for the manifold of parametric statistical models and treat it as a “statistical manifold admitting torsion”.

Jacobians with prescribed eigenvectors

Let Ω⊂Rn be open and let R be a partial frame on Ω; that is, a set of m linearly independent vector fields prescribed on Ω (m≤n). We consider the issue of describing the set of all maps F:Ω→Rn with the property that each of the given vector fields is an eigenvector of the Jacobian matrix of F. By introducing a coordinate independent definition of the Jacobian, we obtain an intrinsic formulation of the problem, which leads to an overdetermined PDE system, whose compatibility conditions can be expressed in an intrinsic, coordinate independent manner. To analyze this system we formulate and prove a generalization of the classical Frobenius integrability theorems. The size and structure of the solution set of this system depends on the properties of the partial frame; in particular, whether or not it is in involution. A particularly nice subclass of involutive partial frames, called rich frames, can be completely analyzed. The involutive, non-rich case is somewhat harder to handle. We provide a complete answer in the case of m=3 and arbitrary n, as well as some general results for arbitrary m. The non-involutive case is far more challenging, and we only obtain a comprehensive analysis in the case n=3, m=2. Finally, we provide explicit examples illustrating the various possibilities.

Sobolev spaces on Lie groups: Embedding theorems and algebra properties

Let G be a noncompact connected Lie group, denote with ρ a right Haar measure and choose a family of linearly independent left-invariant vector fields X on G satisfying Hörmander's condition. Let χ be a positive character of G and consider the measure μχ whose density with respect to ρ is χ. In this paper, we introduce Sobolev spaces Lαp(μχ) adapted to X and μχ (1<p<∞, α≥0) and study embedding theorems and algebra properties of these spaces. As an application, we prove local well-posedness and regularity results of solutions of some nonlinear heat and Schrödinger equations on the group.

Continuous solutions for divergence-type equations associated to elliptic systems of complex vector fields

In this paper, we characterize all the distributions F∈D′(U) such that there exists a continuous weak solution v∈C(U,Cn) (with U⊂Ω) to the divergence-type equationL1⁎v1+...+Ln⁎vn=F, where {L1,…,Ln} is an elliptic system of linearly independent vector fields with smooth complex coefficients defined on Ω⊂RN. In case where (L1,…,Ln) is the usual gradient field on RN, we recover the classical result for the divergence equation proved by T. De Pauw and W. Pfeffer. Its proof is based on the closed range theorem and inspired by [3] and [6] in the classical case. Our method slightly differs from theirs by relying on the Banach–Grothendieck theorem and introducing tools from pseudodifferential operators, useful in our local setting of a system of complex vector fields with variable coefficients.

Szekeres models: a covariant approach

We exploit the 1 + 1 + 2 formalism to covariantly describe the inhomogeneous and anisotropic Szekeres models. It is shown that an average scale length can be defined covariantly which satisfies a 2d equation of motion driven from the effective gravitational mass (EGM) contained in the dust cloud. The contributions to the EGM are encoded to the energy density of the dust fluid and the free gravitational field Eab. We show that the quasi-symmetric property of the Szekeres models is justified through the existence of 3 independent intrinsic Killing vector fields (IKVFs). In addition the notions of the apparent and absolute apparent horizons are briefly discussed and we give an alternative gauge-invariant form to define them in terms of the kinematical variables of the spacelike congruences. We argue that the proposed program can be used in order to express Sachs’ optical equations in a covariant form and analyze the confrontation of a spatially inhomogeneous irrotational overdense fluid model with the observational data.

Constructive Solution of Ellipticity Problem for the First Order Differential System

We built first order elliptic systems with any possible number of unknown functions and the maximum possible number of unknowns, i.e, in general. These systems provide the basis for studying the properties of any first order elliptic systems. The study of the Cauchy-Riemann system and its generalizations led to the identification of a class of elliptic systems of first-order of a special structure. An integral representation of solutions is of great importance in the study of these systems. Only by means of a constructive method of integral representations we can solve a number of problems in the theory of elliptic systems related mainly to the boundary properties of solutions. The obtained integral representation could be applied to solve a number of problems that are hard to solve, if you rely only on the non-constructive methods. Some analogues of the theorems of Liouville, Weierstrass, Cauchy, Gauss, Morera, an analogue of Green’s formula are established, as well as an analogue of the maximum principle. The used matrix operators allow the new structural arrangement of the maximum number of linearly independent vector fields on spheres of any possible dimension. Also the built operators allow to obtain a constructive solution of the extended problem ”of the sum of squares” known in algebra.

A Theoretical Design Approach for Passive Shimming of a Magic-Angle-Spinning NMR Magnet.

This paper presents a passive shimming design approach for a magic-angle-spinning (MAS) NMR magnet. In order to achieve a 1.5-T magic-angle field in NMR samples, we created two independent orthogonal magnetic vector fields by two separate coils: the dipole and solenoid. These two coils create a combined 1.5-T magnetic field vector directed at the magic angle (54.74° from the spinning axis). Additionally, the stringent magnetic field homogeneity requirement of the MAS magnet is the same as that of a solenoidal NMR magnet. The challenge for the magic-angle passive shimming design is to correct both the dipole and solenoid magnetic field spherical harmonics with one set of iron pieces, the so-called ferromagnetic shimming. Furthermore, the magnetization of the iron pieces is produced by both the dipole and solenoid coils. In our design approach, a matrix of 2 mm by 5 mm iron pieces with different thicknesses was attached to a thin-walled tube, 90-mm diameter and 40-mm high. Two sets of spherical harmonic coefficients were calculated for both the dipole and solenoid coil windings. By using the multiple-objective linear programming optimization technique and coordinate transformations, we have designed a passive shimming set that can theoretically reduce 22 lower-order spherical harmonics and improve the homogeneity of our MAS NMR magnet.

Uniqueness of black holes with bubbles in minimal supergravity

We generalize uniqueness theorems for non-extremal black holes with three mutually independent Killing vector fields in five-dimensional minimal supergravity in order to account for the existence of non-trivial two-cycles in the domain of outer communication. The black hole space-times we consider may contain multiple disconnected horizons and be asymptotically flat or asymptotically Kaluza–Klein. We show that in order to uniquely specify the black hole space-time, besides providing its domain structure and a set of asymptotic and local charges, it is necessary to measure the magnetic fluxes that support the two-cycles as well as fluxes in the two semi-infinite rotation planes of the domain diagram.

Differential geometry via infinitesimal displacements

We present a new formulation of some basic differential geometric notions on a smooth manifold M, in the setting of nonstandard analysis. In place of classical vector fields, for which one needs to construct the tangent bundle of M, we define a prevector field, which is an internal map from ∗ M to itself, implementing the intuitive notion of vectors as infinitesimal displacements. We introduce regularity conditions for prevector fields, defined by finite differences, thus purely combinatorial conditions involving no analysis. These conditions replace the more elaborate analytic regularity conditions appearing in previous similar approaches, e.g. by Stroyan and Luxemburg or Lutz and Goze. We define the flow of a prevector field by hyperfinite iteration of the given prevector field, in the spirit of Euler’s method. We define the Lie bracket of two prevector fields by appropriate iteration of their commutator. We study the properties of flows and Lie brackets, particularly in relation with our proposed regularity conditions. We note several simple applications to the classical setting, such as bounds related to the flow of vector fields, analysis of small oscillations of a pendulum, and an instance of Frobenius’ Theorem regarding the complete integrability of independent vector fields.

Method of characteristics and first integrals for systems of quasi-linear partial differential equations of first order

Given a set of independent vector fields on a smooth manifold, we discuss how to find a function whose zero-level set is invariant under the flows of the vector fields. As an application, we study the solvability of overdetermined partial differential equations: Given a system of quasi-linear PDEs of first order for one unknown function we find a necessary and sufficient condition for the existence of solutions in terms of the second jet of the coefficients. This generalizes to certain quasi-linear systems of first order for several unknown functions.

Vector inflation by kinetic coupled gravity

Vector inflation is a newly established model where inflation is driven by nonminimally coupled massive vector fields with a potential term. This model is similar to the model of chaotic inflation with scalar fields, except that for vector fields the isotropy of expansion is achieved either by considering a triplet of orthogonal vector fields or N randomly oriented independent vector fields. We introduce a new model of vector inflation where the vector field has no potential term and is nonminimally coupled to gravity through the kinetic term. This model shows that vector inflation without a potential term is possible and helps to avoid the fine tuning problem in inflationary models having a potential. The nonminimal coupling is established by introducing the Einstein tensor besides the metric tensor within the kinetic term of the vector field. A perturbation analysis is performed to confront the inflation under discussion with Planck and BICEP2 results.

On the linearly independent vector fields on Grassmann manifolds

In this paper are found θ(n) linearly independent vector fields on the Grassmann manifold Gk(V) of k-planes in n-dimensional Euclidean vector space if k is odd number, where θ(n) is the maximal number of linearly independent vector fields on Sn−1, i.e. skewsymmetric anticommuting complex structures on Rn.

A note on the construction of complex and quaternionic vector fields on spheres

A relationship between real, complex, and quaternionic vector fields on spheres is given by using a relationship between the corresponding standard inner products. The number of linearly independent complex vector fields on the standard (4n − 1)-sphere is shown to be twice the number of linearly independent quaternionic vector fields plus d ,w hered =1 or 3. DOI: 10.1134/S0001434613010148

A Note on Classification of Teleparallel Conformal Vector Fields in Cylindrically Symmetric Static Space-Times in the Teleparallel Theory of Gravitation

In this paper we explored teleparallel conformal vector fields in cylindrically symmetric static space-times in the teleparallel theory of gravitation by using the direct integration technique. It turns out that the dimension of teleparallel conformal vector fields are 8, 9, 10 or 11. The case VI in which the space-time becomes conformally flat admits eleven independent teleparallel conformal vector fields.

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.