Constant Vector Field Research Articles

Explicit A Posteriori Error Representation for Variational Problems and Application to TV-Minimization

AbstractIn this paper, we propose a general approach for explicit a posteriori error representation for convex minimization problems using basic convex duality relations. Exploiting discrete orthogonality relations in the space of element-wise constant vector fields as well as a discrete integration-by-parts formula between the Crouzeix–Raviart and the Raviart–Thomas element, all convex duality relations are transferred to a discrete level, making the explicit a posteriori error representation –initially based on continuous arguments only– practicable from a numerical point of view. In addition, we provide a generalized Marini formula that determines a discrete primal solution in terms of a given discrete dual solution. We benchmark all these concepts via the Rudin–Osher–Fatemi model. This leads to an adaptive algorithm that yields a (quasi-optimal) linear convergence rate.

SOME CHARACTERIZATIONS OF TIMELIKE HELICES WITH THE $F$-CONSTANT VECTOR FIELD IN MINKOWSKI SPACE $E_{1}^{3}$}

In this paper we give characterizations of timelike normal, rectifying and osculating helices in Minkowski space $E_{1}^{3}.$ Moreover, we examine the characterization of timelike helices whose axis $U$\ perpendicular to the $F$-constant vector field $X$, which is a generalization of these helices$.$

Discrete Helmholtz Decompositions of Piecewise Constant and Piecewise Affine Vector and Tensor Fields

AbstractDiscrete Helmholtz decompositions dissect piecewise polynomial vector fields on simplicial meshes into piecewise gradients and rotations of finite element functions. This paper concisely reviews established results from the literature which all restrict to the lowest-order case of piecewise constants. Its main contribution consists of the generalization of these decompositions to 3D and of novel decompositions for piecewise affine vector fields in terms of Fortin–Soulie functions. While the classical lowest-order decompositions include one conforming and one nonconforming part, the decompositions of piecewise affine vector fields require a nonconforming enrichment in both parts. The presentation covers two and three spatial dimensions as well as generalizations to deviatoric tensor fields in the context of the Stokes equations and symmetric tensor fields for the linear elasticity and fourth-order problems. While the proofs focus on contractible domains, generalizations to multiply connected domains and domains with non-connected boundary are discussed as well.

Non-Abelian Carroll–Field–Jackiw term in a Rarita-Schwinger model

In this paper, we demonstrate the possibility of generating a non-Abelian Carroll–Field–Jackiw (CFJ) term in the theory of a non-Abelian gauge field coupled to a spin-3/2 field in the presence of the constant axial vector field. Applying two regularization schemes, we prove that this term is finite and ambiguous, particularly vanishing within the 't Hooft-Veltman scheme.

A generalization of the notion of helix

In this paper we generalize the notion of helix in the three-dimensional Euclidean space, which we define as that curve $\alpha$ for which there is an $F$-constant vector field $W$ along $\alpha$ that forms a constant angle with a fixed direction $V$ (called an axis of the helix). We find the natural equation and the geometric integration of helices $\alpha$ where the $F$-constant vector field $W$ is orthogonal to its axis.

On generalized derivations of polynomial vector fields Lie algebras

In this paper, we study the generalized derivation of a Lie sub-algebra of the Lie algebra of polynomial vector fields on $\mathbb{R}^n$ where $n\geq1$, containing all constant vector fields and the Euler vector field, under some conditions on this Lie sub-algebra.

Exact solution of generalized KG-oscillator under Lorentz-violating effects by a fixed vector field subject to interaction potential

In this work, the relativistic quantum oscillator under the Lorentz symmetry violation environment defined by an arbitrary constant vector field [Formula: see text] is analyzed. We solve the generalized Klein–Gordon oscillator with different vector field configurations and obtained the bound-states solutions. After that, we insert a static Coulomb-type scalar potential in the wave equation and solve the generalized Klein–Gordon oscillator analytically. We discuss the effects of Lorentz symmetry violation and the Coulomb-type potential on the energy spectrum of these oscillator fields.

Effects of Lorentz symmetry violation by a constant vector field subject to Cornell-type potential on relativistic quantum oscillator

In this paper, we analyze a relativistic quantum oscillator model under a Lorentz symmetry violation (LSV) environment defined by an arbitrary constant vector field [Formula: see text] subject to Cornell-type scalar potential. We first introduce the Cornell-type scalar potential by modifying the mass term via [Formula: see text], and then momentum vector [Formula: see text] in the wave equation. We solve the Klein–Gordon oscillator analytically for different vector field configurations and obtained the bound-states energy profiles and the wave function. We discuss the influences of a vector field on the energy eigenvalues and the wave function of these oscillators.

Relativistic quantum oscillator model under the effects of the violation of Lorentz symmetry by an arbitrary fixed vector field

We study a spin-zero relativistic quantum oscillator model under the violation of Lorentz symmetry defined by an arbitrary constant vector field . We solve the Klein-Gordon oscillator using the Nikiforov-Uvarov method and obtain the bound-states solutions of the quantum system. We see that vector field configurations modify the energy profile of the oscillator fields.

Cyclic conformally flat hypersurfaces revisited

In this article we classify the conformally flat Euclidean hypersurfaces of dimension three with three distinct principal curvatures of $\mathbb{R}^4$, $\mathbb{S}^3\times \mathbb{R}$ and $\mathbb{H}^3\times \mathbb{R}$ with the property that the tangent component of the vector field $\partial/\partial t$ is a principal direction at any point. Here $\partial/\partial t$ stands for either a constant unit vector field in $\mathbb{R}^4$ or the unit vector field tangent to the factor $\mathbb{R}$ in the product spaces $\mathbb{S}^3\times \mathbb{R}$ and $\mathbb{H}^3\times \mathbb{R}$, respectively. Then we use this result to give a simple proof of an alternative classification of the cyclic conformally flat hypersurfaces of $\mathbb{R}^4$, that is, the conformally flat hypersurfaces of $\mathbb{R}^4$ with three distinct principal curvatures such that the curvature lines correspondent to one of its principal curvatures are extrinsic circles. We also characterize the cyclic conformally flat hypersurfaces of $\mathbb{R}^4$ as those conformally flat hypersurfaces of dimension three with three distinct principal curvatures for which there exists a conformal Killing vector field of $\mathbb{R}^4$ whose tangent component is an eigenvector field correspondent to one of its principal curvatures.

Unique reconstruction of simple magnetizations from their magnetic potential

Inverse problems arising in (geo)magnetism are typically ill-posed, in particular they exhibit non-uniqueness. Nevertheless, there exist nontrivial model spaces on which the problem is uniquely solvable. Our goal is here to describe such spaces that accommodate constraints suited for applications. In this paper we treat the inverse magnetization problem on a Lipschitz domain with fairly general topology. We characterize the subspace of L 2-vector fields that causes non-uniqueness, and identify a subspace of harmonic gradients on which the inversion becomes unique. This classification has consequences for applications and we present some of them in the context of geo-sciences. In the second part of the paper, we discuss the space of piecewise constant vector fields. This vector space is too large to make the inversion unique. But as we show, it contains a dense subspace in L 2 on which the problem becomes uniquely solvable, i.e. magnetizations from this subspace are uniquely determined by their magnetic potential.

Quasi-periodic incompressible Euler flows in 3D

We prove the existence of time-quasi-periodic solutions of the incompressible Euler equation on the three-dimensional torus T3, with a small time-quasi-periodic external force. The solutions are perturbations of constant (Diophantine) vector fields, and they are constructed by means of normal forms and KAM techniques for reversible quasilinear PDEs.

Kundt spacetimes in the Einstein-Gauss-Bonnet theory

We systematically investigate the complete class of vacuum solutions in the Einstein-Gauss-Bonnet gravity theory which belong to the Kundt family of non-expanding, shear-free and twist-free geometries (without gyratonic matter terms) in any dimension. The field equations are explicitly derived and simplified, and their solutions classified into three distinct subfamilies. Algebraic structures of the Weyl and Ricci curvature tensors are determined. The corresponding curvature scalars directly enter the invariant form of equation of geodesic deviation, enabling us to understand the specific local physical properties of the gravitational field constrained by the EGB theory. We also present and analyze several interesting explicit classes of such vacuum solutions, namely the Ricci type III spacetimes, all geometries with constant-curvature transverse space, and the whole pp-wave class admitting a covariantly constant null vector field. These exact Kundt EGB gravitational waves exhibit new features which are not possible in Einstein's general relativity.

Symmetries of the Kerr–Newman spacetime

It is shown that the only Killing vector fields admitted by the Kerr–Newman spacetime are those corresponding to the time independence and axial symmetries, and that the only conformal vector fields or projective collineations admitted by the Kerr–Newman spacetime are the Killing vector fields. In addition, it is shown that the Kerr–Newman spacetime admits no covariantly constant vector fields or recurrent vector fields, and is of holonomy type R15. It is also established that any Weyl conformal collineation, Weyl projective collineation or curvature collineation admitted by the Kerr–Newman spacetime are the Killing vector fields, and that the only Ricci or matter collineations admitted by the Kerr–Newman spacetime are again Killing vector fields. Finally, some analogous results are established for the Reissner–Nordström spacetime.

Lorentz symmetry breaking in supersymmetric quantum electrodynamics

Lorentz symmetry is one of the fundamental symmetries of nature; however, it can be broken by several proposals such as quantum gravity effects, low energy approximations in string theory and dark matter. In this paper, Lorentz symmetry is broken in supersymmetric quantum electrodynamics using aether superspace formalism without breaking any supersymmetry. To break the Lorentz symmetry in three-dimensional quantum electrodynamics, we must use the [Formula: see text] aether superspace. A new constant vector field is introduced and used to deform the deformed generator of supersymmetry. This formalism is required to fix the unphysical degrees of freedom that arise from the quantum gauge transformation required to quantize this theory. By using Yokoyama’s gaugeon formalism, it is possible to study these gaugeon transformations. As a result of the quantum gauge transformation, the supersymmetric algebra gets modified and the theory is invariant under BRST symmetry. These results could aid in the construction of the Gravity’s Rainbow theory and in the study of superconformal field theory. Furthermore, it is demonstrated that different gauges in this deformed supersymmetric quantum electrodynamics can be related to each other using the gaugeon formalism.

On the Dirac oscillator subject to a Coulomb-type central potential induced by the Lorentz symmetry violation

We investigated the relativistic oscillator model for spin-1/2 fermionic fields, known as the Dirac oscillator, in a background of breaking the Lorentz symmetry governed by a constant vector field inserted in the Dirac equation by non-minimal coupling, where we proposed two possible scenarios of Lorentz symmetry violation which induce a Coulomb-type potential. Thus, in the search for solutions of bound states, we determine, analytically, the relativistic energy profile for the Dirac oscillator, where an interesting quantum effect arises: The frequency of the Dirac oscillator is determined by the quantum numbers of the system and the parameters that characterize the scenarios of Lorentz symmetry violation.

On a massive scalar field subject to the relativistic Landau quantization in an environment of aether-like Lorentz symmetry violation

We analyze the effects of the aether-like Lorentz symmetry violation on a massive scalar field subject to a uniform magnetic field, where, analytically, we determine the relativistic Landau levels and note that this background governed by a constant vector field modifies the energy spectrum of the relativistic quantum system. We extend our analysis by including a hard-wall potential in the system, and, under some conditions, we determine the energy spectrum of the scalar field and show that the presence of this confinement potential drastically modifies the relativistic energy levels in this possible scenario of breaking of the Lorentz symmetry.

Modified f(R,T) cosmology with observational constraints in Lyra’s geometry

In this paper, we have investigated modified [Formula: see text] cosmological models with observational constraints in Lyra’s geometry. We have studied the models in two cases: in the first one it is taken as constant displacement vector field and in second case it is taken as time-dependent. We have established a relationship among energy parameters [Formula: see text], respectively, called as matter, anisotropy and dark energy (DE) density parameters in Bianchi type-I space-time in Lyra’s geometry. We have compared our models with observational data sets with union 2.1 compilation SNe Ia data and [Formula: see text] data sets and have found best fit values of various energy parameters for [Formula: see text]. We have found that the model with constant displacement vector field is the best fit to the study of the whole evolution of the model and the gauge function [Formula: see text] of Lyra geometry behaves like a DE candidate. Also, the various physical and kinematical parameters have been studied in this study.

On the linearization of vector fields on a torus with prescribed frequency

In this paper we are mainly concerned with the linearization of the flow with prescribed frequency for analytic perturbation of constant vector fields on a torus under weaker non-degeneracy condition and non-resonant condition. As is well known the perturbation of constant vector fields may induce a shift of frequency, when Kolmogorov's non-degeneracy condition is violated. By introducing external parameters and using the polynomial structure to truncate, we prove that if the frequency mapping has the nonzero Brouwer's topological degree at some non-resonant frequency, then the conjugated vector fields will have a linear flow with this frequency.

Geometry Processing Problems Using the Total Variation of the Normal Vector Field

AbstractThis paper presents a novel approach to the formulation and solution of discrete geometry processing problems including mesh denoising. The main quantity of interest is the piecewise constant unit normal vector field, which we consider on piecewise flat, triangulated surfaces. In a similar fashion as one does for images defined on ‘flat’ domains, our goal is to remove noise while preserving shape features such as sharp edges. To this end, we model a denoising problem via a quadratic vertex tracking term and a regularizer based on the total variation of the normal vector field. Since the latter has values on the unit sphere, its total variation reduces to a finite sum of the geodesic distances of neighboring normals. We solve the model numerically by applying split‐Bregmann iterations and provide results for the ‘fandisk’ benchmark.