Vector Field Research Articles

Electric Penrose, circular orbits and collisions of charged particles near charged black holes in Kalb–Ramond gravity

General relativity (GR) is a well-tested theory of gravity in strong and weak field regimes. Many modifications to this theory were obtained, including different scalar, vector, and tensor fields to the GR with non-minimal coupling to gravity. Kalb–Ramond (KR) gravity is also a modified theory formulated in the presence of a bosonic field. One astrophysical way to test gravity is by studying the motion of test particles in the spacetime of black holes (BH). In this work, we study the circular motion of charged particles and explore energetic processes around charged BHs in KR theory. First, we investigated the event horizon radius and analyzed horizon-no horizon regions in the BH charge and KR parameter space. Considering the Coulomb interaction, we derive and analyze the effective potential for charged particles around a charged KR BH. We investigate charged particles’ angular momentum and energy corresponding to circular orbits. We also investigate how the KR non-minimal coupling parameter affects the radius of the innermost stable circular orbits, the corresponding energy, and the angular momentum. We also investigated the electric Penrose process and charged-particle collisions near the KR BH. The presence of the nonzero KR parameter results in a decrease in the energy efficiency of the Penrose process. Also obtained is that the KR parameter’s positive (negative) values cause a decrease (increase) in the center of mass energy of colliding particles near the BH horizon.

Mathematical Assessment of Pollution Factors from Eddy Currents in the Durrës Seaport

Eddy currents, a significant phenomenon in electromagnetic fields, were first distinguished from general electric currents by Foucault. These currents, influenced by the magnetic field vector (H), magnetic flux density (B), and electric field (E), are present in various environments, including homogeneous and non-homogeneous areas, linear and non-linear environments, and open boundary problems. In marine environments, these currents, referred to as planetary currents, contribute to electrical pollution, impacting maritime infrastructure, machinery, and ecosystems. This study aims to assess the impact of eddy currents in the Durrës Seaport, Albania, focusing on how meteorological factors such as temperature, atmospheric pressure, and humidity influence pollution levels. Using the TES-92 Electro-smog meter and the PCE-THB 40 Thermo-hygrometer and barometer, we measured electromagnetic field characteristics and meteorological parameters. Our results highlight the significant impact of stray currents on marine environments, emphasizing the need for effective measures to minimize their effects and improve the quality and selection of materials and equipment in port facilities. This study provides valuable insights into the environmental and infrastructural challenges posed by electromagnetic pollution and proposes recommendations for mitigating these impacts.

Analysis of Stokes singularities using a lateral shear interferometer

Polarization and Poincaré singularities in the optical fields can be studied by analyzing the phase singularities of mathematically constructed Stokes vector fields. The wider applicability of the Stokes construction is found by exploring the generation and detection methods for various types of Stokes singularities and their analysis. Here, we detect and analyze all forms of the Stokes singularities through lateral shear interferometry. Specifically, the projections of a Stokes singularity on three pairs of orthogonal polarization basis states, defined by the eigen polarization states of Pauli’s matrices, are analyzed through unique fork patterns in the shearogram pairs. These interference patterns also provide the topological indices of the singularities. Such a self-referencing interferometric method also helps to remove the degeneracy in the Stokes index and polarization. Through both, simulations and experiments, we have analyzed specific beams represented by higher order Poincaré sphere and hybrid order Poincaré sphere topological constructs.

Stability analysis of homogeneous cooperative positive differential systems with time-varying delays and its generalization

This article is devoted to studying the asymptotic behavior of differential systems with bounded delays. We first focus on the analysis of homogeneous cooperative positive systems. Under the assumption that the vector field is homogeneous of a degree greater than 1, we show that the non-trivial solutions of the system converge to the origin at a polynomial rate. In the case when the degree of homogeneity equals 1, we prove that the solutions will decay at an exponential rate. As a generalization of these results, we consider nonlinear non-autonomous differential systems with time-varying delays that are bounded above by stable homogeneous positive systems. By some additional imposed conditions, in light of the comparison principle, we obtain the locally exponential stability and polynomial stability of the equilibrium point to these systems. Finally, specific examples and discussions are provided to illustrate the validity of the proposed theoretical results.

Two-dimensional Jacobians $$\textrm{det}\hspace{0.56905pt}$$ and $$\textrm{Det}\hspace{0.56905pt}$$ for bounded variation functions and applications

AbstractThe paper deals with the comparison in dimension two between the strong Jacobian determinant $$\textrm{det}\hspace{0.56905pt}$$ det and the weak (or distributional) Jacobian determinant $$\textrm{Det}\hspace{0.56905pt}$$ Det . Restricting ourselves to dimension two, we extend the classical results of Ball and Müller as well as more recent ones to bounded variation vector-valued functions, providing a sufficient condition on a vector-valued U in $$BV(\Omega )^2$$ B V ( Ω ) 2 such that the equality $$\textrm{det}\hspace{0.56905pt}(\nabla U)=\textrm{Det}\hspace{0.56905pt}(\nabla U)$$ det ( ∇ U ) = Det ( ∇ U ) holds either in the distributional sense on $$\Omega $$ Ω , or almost-everywhere in $$\Omega $$ Ω when U is in $$W^{1,1}(\Omega )^2$$ W 1 , 1 ( Ω ) 2 . The key-assumption of the result is the regularity of the Jacobian matrix-valued $$\nabla U$$ ∇ U along the direction of a given non vanishing vector field $$b\in C^1(\Omega )^2$$ b ∈ C 1 ( Ω ) 2 , i.e.$$\nabla U\, b$$ ∇ U b is assumed either to belong to $$C^0(\Omega )^2$$ C 0 ( Ω ) 2 with one of its coordinates in $$C^1(\Omega )$$ C 1 ( Ω ) , or to belong to $$C^1(\Omega )^2$$ C 1 ( Ω ) 2 . Two examples illustrate this new notion of two-dimensional distributional determinant. Finally, we prove the lower semicontinuity of a polyconvex energy defined for vector-valued functions U in $$BV(\Omega )^2$$ B V ( Ω ) 2 , assuming that the vector field b and one of the coordinates of $$\nabla U\, b$$ ∇ U b lie in a compact set of regular vector-valued functions.

Representations Of Lie Algebras of Vector Fields on Pairing Between Gauge Modules and Rudakov Modules

For an irreducible affine variety X over an algebraically closed field of characteristic zero, we define two new classes of modules over the Lie algebra of vector fields on X: gauge modules and Rudakov modules. These modules admit a compatible action of the algebra of functions on X. We prove general simplicity theorems for these two types of modules, demonstrating their irreducibility under specific conditions. Additionally, we establish a pairing between gauge modules and Rudakov modules, highlighting the connections and interactions between these two classes of modules. We have established that gauge modules and Rudakov modules, corresponding to simple glN -modules, remain irreducible as modules over the Lie algebra of vector fields unless they appear in the de Rham complex. Additionally, we have studied the irreducibility of tensor products of Rudakov modules, providing a comprehensive understanding of these module structures and their applications.

Generalized Hamiltonian systems on subvarieties: constant rank case

For the constraint variety in symplectic manifold, the solvable Hamiltonian vector fields on the constraint are investigated. According to P.A.M. Dirac [3], the space of solvable Hamiltonian systems is determined by the geometric restriction of the symplectic form to the constraint. Solvability condition of the generalized Hamiltonian systems is extended to singular varieties and applied under some assumption on singularities. The constraint being a smooth submanifold in a symplectic space was considered in [6]. In this paper, we investigate the solvability of generalized Hamiltonian systems and the constraint invariants on singular constraints in the constant rank case.

A robust second-order low-rank BUG integrator based on the midpoint rule

Dynamical low-rank approximation has become a valuable tool to perform an on-the-fly model order reduction for prohibitively large matrix differential equations. A core ingredient is the construction of integrators that are robust to the presence of small singular values and the resulting large time derivatives of the orthogonal factors in the low-rank matrix representation. Recently, the robust basis-update & Galerkin (BUG) class of integrators has been introduced. These methods require no steps that evolve the solution backward in time, often have favourable structure-preserving properties, and allow for parallel time-updates of the low-rank factors. The BUG framework is flexible enough to allow for adaptations to these and further requirements. However, the BUG methods presented so far have only first-order robust error bounds. This work proposes a second-order BUG integrator for dynamical low-rank approximation based on the midpoint quadrature rule. The integrator first performs a half-step with a first-order BUG integrator, followed by a Galerkin update with a suitably augmented basis. We prove a robust second-order error bound which in addition shows an improved dependence on the normal component of the vector field. These rigorous results are illustrated and complemented by a number of numerical experiments.

Theory on Linear L-Fractional Differential Equations and a New Mittag–Leffler-Type Function

The L-fractional derivative is defined as a certain normalization of the well-known Caputo derivative, so alternative properties hold: smoothness and finite slope at the origin for the solution, velocity units for the vector field, and a differential form associated to the system. We develop a theory of this fractional derivative as follows. We prove a fundamental theorem of calculus. We deal with linear systems of autonomous homogeneous parts, which correspond to Caputo linear equations of non-autonomous homogeneous parts. The associated L-fractional integral operator, which is closely related to the beta function and the beta probability distribution, and the estimates for its norm in the Banach space of continuous functions play a key role in the development. The explicit solution is built by means of Picard’s iterations from a Mittag–Leffler-type function that mimics the standard exponential function. In the second part of the paper, we address autonomous linear equations of sequential type. We start with sequential order two and then move to arbitrary order by dealing with a power series. The classical theory of linear ordinary differential equations with constant coefficients is generalized, and we establish an analog of the method of undetermined coefficients. The last part of the paper is concerned with sequential linear equations of analytic coefficients and order two.

Conformal trajectories in three-dimensional space forms

We introduce the notion of conformal trajectories in three-dimensional Riemannian manifolds [Formula: see text]. Given a conformal vector field [Formula: see text], a conformal trajectory of [Formula: see text] is a regular curve [Formula: see text] in [Formula: see text] satisfying [Formula: see text], for some fixed nonzero constant [Formula: see text]. In this paper, we study the conformal trajectories in the space forms [Formula: see text], [Formula: see text] and [Formula: see text]. For (non-Killing) conformal vector fields in [Formula: see text] (respectively, in [Formula: see text]), we prove that conformal trajectories have constant curvature and its torsion is a linear combination of trigonometric (respectively, hyperbolic) functions on the arc-length parameter. In the case of Euclidean space [Formula: see text], we obtain the same result for the radial vector field and characterizing all conformal trajectories.

The dressing field method for diffeomorphisms: a relational framework

The dressing field method (DFM) is a tool to reduce gauge symmetries. Here we extend it to cover the case of diffeomorphisms. The resulting framework is a systematic scheme to produce Diff(M) -invariant objects, which has a natural relational interpretation. Its precise formulation relies on a clear understanding of the bundle geometry of field space. By detailing it, among other things we stress the geometric nature of field-independent and field-dependent diffeomorphisms, and highlight that the heuristic ‘extended bracket’ for field-dependent vector fields often featuring in the covariant phase space literature can be understood as arising from the Frölicher-Nijenhuis bracket. Furthermore, by articulating this bundle geometry with the covariant phase space approach, we give a streamlined account of the elementary objects of the (pre)symplectic structure of a Diff(M) -theory: Noether charges and their bracket, as induced by the standard prescription for the presymplectic potential and 2-form. We give conceptually transparent expressions allowing to read the integrability conditions and the circumstances under which the bracket of charge is Lie, and the resulting Poisson algebras of charges are central extensions of the Lie algebras of field-independent ( diff(M) ) and field-dependent vector fields.We show that, applying the DFM, one obtains a Diff(M) -invariant and manifestly relational formulation of a general relativistic field theory. Relying on results just mentioned, we easily derive the ‘dressed’ (relational) presymplectic structure of the theory. This reproduces or extends results from the gravitational edge modes and gravitational dressings literature. In addition to simplified technical derivations, the conceptual clarity of the framework supplies several insights and allows us to dispel misconceptions.

Tuning the high reflectance Goos-Hänchen shift based on quasi-bound states in the continuum through vanadium dioxide material asymmetric metasurface

To dynamically tune the high-reflectance Goos and Hänchen (GH) shift, a vanadium dioxide metasurface is proposed in this paper, whose material asymmetry parameter can be tuned by the ambient temperature. The material symmetry and geometric symmetry of the metasurface are broken by controlling the material asymmetry parameter γ and incident angle θ, and three symmetry-protected quasi-bound states in the continuum (QBICs) resonances with high-quality factors are excited. Based on the electric/magnetic field vector and multipole decomposition, the three QBIC resonances are excited due to the magnetic quadrupole. The QBIC resonances are applied to dynamically tune the high-reflectance GH shift, and the influence laws of γ and θ on the high-reflectance GH shift are analyzed.

Construction of Planar Vector Fields with a Nonsimple Critical Point of Prescribed Topological Structure

Construction of Planar Vector Fields with a Nonsimple Critical Point of Prescribed Topological Structure

Left-Invariance for Smooth Vector Fields and Applications

Let X={X0,…,Xm}\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$X = \\{X_0,\\ldots ,X_m\\}$$\\end{document} be a family of smooth vector fields on an open set Ω⊆RN\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Omega \\subseteq \\mathbb {R}^N$$\\end{document}. Motivated by applications to the PDE theory of Hörmander operators, for a suitable class of open sets Ω\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Omega $$\\end{document}, we find necessary and sufficient conditions on X for the existence of a Lie group (Ω,∗)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(\\Omega ,*)$$\\end{document} such that the operator L=∑i=1mXi2+X0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L=\\sum _{i = 1}^mX_i^2+X_0$$\\end{document} is left-invariant with respect to the operation ∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$*$$\\end{document}. Our approach is constructive, as the group law is constructed by means of the solution of a suitable ODE naturally associated to vector fields in X. We provide an application to a partial differential operator appearing in the Finance.

DiffuseRT: predicting likely anatomical deformations of patients undergoing radiotherapy.

Predicting potential deformations of patients can improve radiotherapy treatment planning. Here, we introduce new deep-learning models that predict likely anatomical changes during radiotherapy for head and neck cancer patients.

Denoising diffusion probabilistic models (DDPMs) were developed to generate fraction-specific anatomical changes based on a reference cone-beam CT (CBCT), the fraction number and treatment dose. Three distinct DDPMs were developed: (1) the image model was trained to directly generate likely future CBCTs, (2) the deformable vector field (DVF) model was trained to generate DVFs that deform a reference CBCT and (3) the hybrid model was trained similarly to the DVF model, but without relying on an external deformable registration algorithm. The models were trained on 9 patients with longitudinal CBCT images (224 CBCTs) and evaluated on 5 patients (152 CBCTs). 

The generated images mainly exhibited random positioning shifts and small anatomical changes for early fractions. For later fractions, all models predicted weight losses in accordance with the training data. The distributions of volume and position changes of the body, esophagus, and parotids generated with the image and hybrid models were more similar to the ground truth distribution than the DVF model, evident from the lower Wasserstein distance achieved with the image (0.26) and hybrid model (0.25) compared to the DVF model (0.36). Generating several images for the same fraction did not yield the expected variability since the ground truth anatomical changes were only in 70% of the fractions within the 95% bounds predicted with the best model. Using the generated images for robust optimization of simplified proton therapy plans improved the worst-case clinical target volume V95 with 7% compared to optimizing with 3 mm set-up robustness while maintaining a similar integral dose.

In conclusion, the newly developed DDPMs generate distributions similar to the real anatomical changes and have the potential to be used for robust anatomical optimization.

A decomposition of light’s spin angular momentum density

Light carries intrinsic spin angular momentum (SAM) when the electric or magnetic field vector rotates over time. A familiar vector equation calculates the direction of light’s SAM density using the right-hand rule with reference to the electric and magnetic polarisation ellipses. Using Maxwell’s equations, this vector equation can be decomposed into a sum of two distinct terms, akin to the well-known Poynting vector decomposition into orbital and spin currents. We present the first general study of this spin decomposition, showing that the two terms, which we call canonical and Poynting spin, are chiral analogies to the canonical and spin momenta of light in its interaction with matter. Like canonical momentum, canonical spin is directly measurable. Both canonical and Poynting spin incorporate spatial variation of the electric and magnetic fields and are influenced by optical vortices. The decomposition allows us to show that a linearly polarised vortex beam, which has no total SAM, can nevertheless exert longitudinal chiral pressure due to equal and opposite canonical and Poynting spins.

A decomposition of light’s spin angular momentum density

Light carries intrinsic spin angular momentum (SAM) when the electric or magnetic field vector rotates over time. A familiar vector equation calculates the direction of light’s SAM density using the right-hand rule with reference to the electric and magnetic polarisation ellipses. Using Maxwell’s equations, this vector equation can be decomposed into a sum of two distinct terms, akin to the well-known Poynting vector decomposition into orbital and spin currents. We present the first general study of this spin decomposition, showing that the two terms, which we call canonical and Poynting spin, are chiral analogies to the canonical and spin momenta of light in its interaction with matter. Like canonical momentum, canonical spin is directly measurable. Both canonical and Poynting spin incorporate spatial variation of the electric and magnetic fields and are influenced by optical vortices. The decomposition allows us to show that a linearly polarised vortex beam, which has no total SAM, can nevertheless exert longitudinal chiral pressure due to equal and opposite canonical and Poynting spins.

Caratheodory periodic perturbations of degenerate systems

We study the structure of the set of harmonic solutions to T-periodically perturbed coupled differential equations on differentiable manifolds, where the perturbation is allowed to be of Caratheodory-type regularity. Employing degree-theoretic methods, we prove the existence of a noncompact connected set of nontrivial T-periodic solutions that, in a sense, emanates from the set of zeros of the unperturbed vector field. The latter is assumed to be ''degenerate'': Meaning that, contrary to the usual assumptions on the leading vector field, it is not required to be either trivial nor to have a compact set of zeros. In fact, known results in the ``nondegenerate case can be recovered from our ones. We also provide some illustrating examples of Lienard- and \(\phi\)-Laplacian-type perturbed equations. For more information see https://ejde.math.txstate.edu/Volumes/2024/39/abstr.html

A unified approach to inversion formulae for vector and tensor ray and radon transforms and the Natterer inequality

Most derivations of inversion formulae for x-ray or Radon transform are based on the projection theorem, where for fixed direction the Fourier transform of x-ray or Radon transform is calculated and compared with the Fourier transform of the searched-for function. In contrast to this we start here off from the searched-for field, calculate its Fourier transform for fixed direction, which is now a vector or tensor field, that we then expand in a suitable direction dependent basis. The expansion coefficients are recognized as the Fourier transform of longitudinal, transversal or mixed ray transforms or vectorial Radon transform respectively. The inverse Fourier transform of the searched-for field then directly leads to inversion formulae for those transforms applying problem adapted backprojections. When considering the Helmholtz decomposition of the field we immediately find inversion formulae for those transversal or longitudinal transforms. First inversion formulae for the longitudinal ray transform, similar to those given by Natterer (1986 The Mathematics of Computerized Tomography (Teubner and Wiley)) for x-ray tomography, were given by Natterer-Wübbeling in 2001, Natterer and Wübbeling (2001 Mathematical Methods in Image Reconstruction (SIAM)), but then not pursued by other authors. In this paper, we present the above described method and derive in a unified way inversion formulae for the ray transforms treated in Louis (2022 Inverse Problems 38 065008) containing the results from Louis (2022 Inverse Problems 38 065008) as special cases. Additionally we present new inversion formulae for the vectorial Radon transform. As a consequence the inversion formulae directly give Plancherel’s formulae for the vectorial or tensorial transforms. Together with the Natterer inequality, which is independent of the ray or Radon transforms, we present the Natterer stability of those vectorial and tensorial transforms.

Homogeneous Projective Coordinates for the Bondi–Metzner–Sachs Group

This paper studies the Bondi–Metzner–Sachs group in homogeneous projective coordinates because it is then possible to write all transformations of such a group in a manifestly linear way. The 2-sphere metric, the Bondi–Metzner–Sachs metric, asymptotic Killing vectors, generators of supertranslations as well as boosts and rotations of Minkowski spacetime are all re-expressed in homogeneous projective coordinates. Lastly, the integral curves of vector fields which generate supertranslations are evaluated in detail. This work paves the way for more advanced applications of the geometry of asymptotically flat spacetime in projective coordinates by virtue of the tools provided from complex analysis in several variables and projective geometry.