Vector Field Research Articles

A study of self-similar vector fields in bianchi type III spacetime via Rif tree approach

Abstract In this study, we use the Rif tree approach to explore self-similar vector fields in Bianchi type III spacetime. This work adopt a computer-based method to transform symmetry equations into an involutive form that is simplified and divides the integration problem into multiple cases, each represented as a tree structure. In some cases, the metric functions are subject to particular constraints. These conditions allow one to solve the system of equations governing self-similar symmetries and provide explicit formulations for the metrics and their corresponding self-similar vector fields. This approach is novel in that it covers not only the analysis of the direct integration strategy but also some metrics that are practically relevant. For a detailed investigation of the created models, stability and physical importance are also considered.

Modified TRIAD Method for Solving the Problem of a Moving Object Orientation

Currently, it is relevant to create moving objects orientation systems based on the integration of various types of primary information sensors. One way to solve the orientation problem is to use the TRIAD (Tri-Axial Orientation Determination) method which allows determining the matrix of direction cosines between two coordinates. The traditional TRIAD force method is based on the use of two reference vectors – of gravitational and geomagnetic fields, measured by accelerometers and magnetometers, respectively. The disadvantage of this method is – appearance of additional deviations during the accelerated movement of the object and influence of primary information sensors’ random errors. Modified TRIAD method which is based on measurements of three triads of sensors: magnetometers, accelerometers and gyroscopes was proposed in the article. Estimates of acceleration vectors of gravitational and geomagnetic fields were calculated taking into account gyroscope measurements. Then these estimates were combined with the accelerometers’ and magnetometers’ data. The complex gravitational and geomagnetic fields’ accelerations were used to form the direction cosine matrix by the TRIAD method. The suggested modified method can be used to implement free-form moving objects’ orientation systems, since it is 6–8 times more accurate compared to the classic TRIAD method. Attenuation of random sensor errors and disturbances due to object acceleration can be adjusted by use of weigh factors.

On a variant of Hilbert’s 16th problem

Abstract We study the number of limit cycles that a planar polynomial vector field can have as a function of its number m of monomials. We prove that the number of limit cycles increases at least quadratically with m and we provide good lower bounds for m ⩽ 10 .

Conjugate points of dynamic pairs and control systems

We study the geometry of dynamic pairs $(X,\VV)$ on a manifold $M$, where $X$ is a vector field and $\VV$ is a distribution on $M$, both satisfying a regularity condition. Special cases are pairs defined by systems of second order ODEs, geodesic sprays in Riemannian, Finslerian and Lagranian geometries, semi-Hamiltonian systems and control-affine systems. Analogs of conjugate points from the calculus of variations are defined for the pair $(X,\VV)$. The main results give estimates for the position of conjugate points in terms of a curvature operator, analogously to the Cartan--Hadamard and Bonet--Myers theorems. Contrary to classical cases, no metric is given a priori, the distribution $\VV$ may be nonintegrable and the curvature operator is defined in terms of $(X,\VV)$.

Basic curvature & the Atiyah cocycle in gauge theory

Abstract Motivated from target space covariant formulations of topological sigma models and from a graded-geometric approach to higher gauge theory, we study connections on Lie and Courant algebroids and on their description as differential graded (dg) manifolds. We revisit the notion of basic curvature for a connection on a Lie algebroid, which measures its compatibility with the Lie bracket and it appears in the BV-BRST differential of 2D gauge theories such as (twisted) Poisson and Dirac sigma models. We define a basic curvature tensor for connections on Courant algebroids and we show that in the description of a Courant algebroid as a QP2 manifold it appears naturally as part of the homological vector field together with the Gualtieri torsion of an induced generalised connection. Furthermore, we consider connections on dg manifolds and revisit the structure of gauge transformations in the graded-geometric approach to higher gauge theories from a manifestly covariant perspective. We argue that it is governed by the brackets of a Kapranov L ∞ [ 1 ] algebra, whose binary bracket is given by the Atiyah cocycle that measures the compatibility of the connection with the homological vector field. We also revisit some aspects of the derived bracket construction and show how to extend it to connections and tensors.

On singular foliations tangent to a given hypersurface

We consider a class of singular foliations in the sense of Androulidakis and Skandalis that we call transverse order k foliations. These have a finite number of leaves: one hypersurface (the singular leaf) together with the components of its complement (open leaves). The positive integer parameter k encodes the “order of tangency” of the leafwise vector fields to L . We show that a loop in the singular leaf induces a well-defined holonomy transformation at the level of (k-1) -jets. The resulting holonomy invariant can be used to give a complete classification of these foliations and obtain concrete descriptions of their associated groupoids and algebras.

A physical classification of Killing magnetic fields in Thurston geometries

In recent years, numerous studies have appeared that considered Killing vectors of three‐dimensional Riemannian manifolds as magnetic fields, since these vector fields are divergenceless by definition. The existence of adivergenceless vector field modeled as a magnetic field does not imply that it is physically realizable. In this study, we propose a physical classification scheme based on the divergences of the integral that defines the energy of a Killing magnetic field. We consider all the Killing magnetic fields of Thurston geometries studied in the literature and classify them as either physical or nonphysical.

Quadratic Killing tensors on symmetric spaces which are not generated by Killing vector fields

Quadratic Killing tensors on symmetric spaces which are not generated by Killing vector fields

Lie–Hamilton systems on Riemannian and Lorentzian spaces from conformal transformations and some of their applications

Abstract We propose a generalization of two classes of Lie–Hamilton systems on the Euclidean plane to two-dimensional curved spaces, leading to novel Lie–Hamilton systems on Riemannian spaces (flat 2-torus, product of hyperbolic lines, sphere and hyperbolic plane), pseudo-Riemannian spaces (anti-de Sitter, de Sitter, and Minkowski spacetimes), as well as to semi-Riemannian spaces (Newtonian or non-relativistic spacetimes). The vector fields, Hamiltonian functions, symplectic form and constants of the motion of the Euclidean classes are recovered by a contraction process. The construction is based on the structure of certain subalgebras of the so-called conformal algebras of the two-dimensional Cayley–Klein spaces. These curved Lie–Hamilton classes allow us to generalize naturally the Riccati, Kummer–Schwarz and Ermakov equations on the Euclidean plane to curved spaces, covering both the Riemannian and Lorentzian possibilities, and where the curvature can be considered as an integrable deformation parameter of the initial Euclidean system.

Chuprov invariant for vector-scalar fields of multipole sources in a shallow sea

A computational and theoretical study of the properties of the well-known waveguide invariant S.D. Chuprova (IC) was carried out in a plane-parallel Pekeris waveguide. Unlike earlier works, in which predominantly non-directional (monopole) sources were used as a source, and sound pressure fields (scalar fields) were studied, in this work not only scalar, but also vector fields formed in the waveguide by directional - combined multipole sources with directivity in both horizontal and vertical planes. A differential equation has been obtained that makes it possible to fairly accurately calculate the IC values under different conditions of signal propagation and different depths of sources and receivers. This makes it possible, in a simpler way than “full computer modeling,” to predict the invariance (stability) of the IC when varying both the hydrophysical conditions in the waveguide and the geometry of the experiment. It is shown that the directionality of sources in the horizontal plane has virtually no effect on the properties of the IC, and the directionality in the vertical plane leads to a shift in the fan structure of the signal amplitude fields, but has little effect on the IC values. The properties of the fan structure change in a similar way when using vertical projections of the oscillatory velocity vector - despite the fact that another analytical relation, different from scalar fields, is used to calculate the IC, the IC value is close to (+1) at all frequencies and distances, except those at which new modes or dislocations arise. At these frequencies and in these zones, alternating emissions with different signs and magnitudes occur. It is concluded that the stability of IC allows the application of signal processing algorithms developed for scalar fields and non-directional sources to vector-scalar fields generated, including using directional sources.

Optical Singularities in Photonic Microstructures with Rosette Symmetries: A Unified Theoretical Scheme

Optical singularities are topological defects of electromagnetic fields; they include phase singularity in scalar fields, polarization singularity in vector fields, and three-dimensional (3D) singularities such as optical skyrmions. The exploitation of photonic microstructures to generate and manipulate optical singularities has attracted wide research interest in recent years, with many photonic microstructures having been devised to this end. Accompanying these designs, scattered phenomenological theories have been proposed to expound the working mechanisms behind individual designs. In this work, instead of focusing on a specific type of microstructure, we concentrate on the most common geometric features of these microstructures—namely, symmetries—and revisit the process of generating optical singularities in microstructures from a symmetry viewpoint. By systematically employing the projection operator technique in group theory, we develop a widely applicable theoretical scheme to explore optical singularities in microstructures with rosette (i.e., rotational and reflection) symmetries. Our scheme agrees well with previously reported works and further reveals that the eigenmodes of a symmetric microstructure can support multiplexed phase singularities in different components, such as out-of-plane, radial, azimuthal, and left- and right-handed circular components. Based on these phase singularities, more complicated optical singularities may be synthesized, including C points, V points, L lines, Néel- and bubble-type optical skyrmions, and optical lattices, to name a few. We demonstrate that the topological invariants associated with optical singularities are protected by the symmetries of the microstructure. Lastly, based on symmetry arguments, we formulate a so-called symmetry matching condition to clarify the excitation of a specific type of optical singularity. Our work establishes a unified theoretical framework to explore optical singularities in photonic microstructures with symmetries, shedding light on the symmetry origin of multidimensional and multiplexed optical singularities and providing a symmetry perspective for exploring many singularity-related effects in optics and photonics.

Collocation methods for nonlinear differential equations on low-rank manifolds

We introduce new methods for integrating nonlinear differential equations on low-rank manifolds. These methods rely on interpolatory projections onto the tangent space, enabling low-rank time integration of vector fields that can be evaluated entry-wise. A key advantage of our approach is that it does not require the vector field to exhibit low-rank structure, thereby overcoming significant limitations of traditional dynamical low-rank methods based on orthogonal projection. To construct the interpolatory projectors, we develop a sparse tensor sampling algorithm based on the discrete empirical interpolation method (DEIM) that parameterizes tensor train manifolds and their tangent spaces with cross interpolation. Using these projectors, we propose two time integration schemes on low-rank tensor train manifolds. The first scheme integrates the solution at selected interpolation indices and constructs the solution with cross interpolation. The second scheme generalizes the well-known orthogonal projector-splitting integrator to interpolatory projectors. We demonstrate the proposed methods with applications to several tensor differential equations arising from the discretization of partial differential equations.

A Novel Recognition Method for Complex Power Quality Disturbances Based on Rotation Vector and Fuzzy Transfer Field

A Novel Recognition Method for Complex Power Quality Disturbances Based on Rotation Vector and Fuzzy Transfer Field

POLYCORE: Polygon-based contour refinement for improved Intravascular Ultrasound Segmentation

Segmentation of the coronary vessel wall in intravascular ultrasound is a fundamental step in guiding coronary intervention. However, it is an challenging task, even for highly skilled cardiologists, due to image artefacts and shadowed regions caused by calcified plaque, guide wires and vessel side branches. Recently, dense-based neural networks have been applied to this task, however, they often fail to predict anatomically plausible contours in these low-signal areas. We propose a novel methodology called Polygon-based Contour Refiner (POLYCORE) that addresses topological error in dense-based segmentation networks using a relational inductive bias through higher-order connections between vertices to learn anatomically rational contours. Our approach remedies the over-smoothing phenomena common in polygon networks by introducing a new vector field refinement module which enables pixel-level detail to be added in an iterative process. POLYCORE is enhanced with augmented polygon aggregation which we show is more effective than typical dense-based test-time augmentation strategies. We achieve state-of-the-art results on two diverse datasets, observing particular improvements when segmenting the lumen structure and in topologically-challenging regions containing shadow artefacts. Our source code is available here: http://orcid.org/https://github.com/kitbransby/POLYCORE.

Vector Field Heterogeneity as a Novel Omnipolar Mapping Metric for Functional Substrate Characterisation in Scar-related Ventricular Tachycardias

BackgroundVector Field Heterogeneity (VFH) is a novel omnipolar metric to quantify local propagation heterogeneities that may identify functionally critical sites for ablation in scar-related VT. ObjectiveAssess the diagnostic value of VFH to identify abnormal propagation patterns during ventricular substrate mapping and compare VFH in VT isthmus sites (IS), low-voltage bystander (LVA) and normal voltage areas (NVA). MethodsSubstrate maps acquired with a 16-pole grid catheter in patients with scar-related VT were segmented into sites corresponding to IS, LVA and NVA (defined as omnipolar voltages >&<1.5mV) based on corresponding VT activation maps. For each 4×4 electrode-array acquisition, omnipolar-derived vector maps of the directions of electrical propagation were computed offline and a VFH value per clique ranging from 0 (perfect planar wave) to 1 (maximal disorganisation) derived. Results16 patients were studied (56.3% ischemic) evaluating 9 endocardial/7 epicardial substrate maps. VFH metric at IS was 0.57±0.26, LVA 0.52±0.34, NVA 0.16±0.24. VFH at IS was higher (more disorganised) than at LVA (p<0.001). VFH in NVA was significantly lower (i.e. more homogenous) compared to IS and LVA (p<0.001, respectively). Within the isthmus, highest heterogeneity was recorded at entry (0.61±0.24), lowest at exit sites (0.44±0.27). ConclusionVFH mapping identified a significant increase in electrical heterogeneity at IS compared to bystanders with highest heterogeneity at the entrance. Absolute differences were small and substantial overlap between sites were recorded precluding its use as a stand-alone mapping approach. VFH may be integrated in existing mapping strategies as complementary information to support identification of ablation targets.

Solar active region evolution and imminent flaring activity through color-coded visualization of photospheric vector magnetograms

Context. The emergence of magnetic flux, its transition to complex configurations, and the pre-eruptive state of active regions are probed using photospheric magnetograms. Aims. Our aim is to pinpoint different evolutionary stages in emerging active regions, explore their differences, and produce parameters that could advance flare prediction using color-coded maps of the photospheric magnetic field. Methods. The three components of the photospheric magnetic field vector are combined to create color-combined magnetograms (COCOMAGs). From these, the areas occupied by different color hues are extracted, creating appropriate time series (color curves). These COCOMAGs and color curves are used as proxies of the active region evolution and its complexity. Results. The morphology of COCOMAGs showcases typical features of active regions, such as sunspots, plages, and sheared polarity inversion lines. The color curves represent the area occupied by photospheric magnetic field of different orientation and contain information pertaining to the evolutionary stages of active regions. During emergence, most of the region area is dominated by horizontal or highly inclined magnetic field, which is gradually replaced by more vertical magnetic field. In complex regions, large parts are covered by highly inclined magnetic fields, appearing as abrupt color changes in COCOMAGs. The decay of a region is signified by a domination of vertical magnetic field, indicating a gradual relaxation of the magnetic field configuration. The color curves exhibit a varying degree of correlation with active region complexity. Particularly the red and magenta color curves, which represent strong, purely horizontal magnetic field, are good indicators of future flaring activity. Conclusions. Color-combined magnetograms facilitate a comprehensive view of the evolution of active regions and their complexity. They offer a framework for the treatment of complex observations and can be used in pattern recognition, feature extraction, and flare-prediction schemes.

Statistical inference with a manifold-constrained RNA velocity model uncovers cell cycle speed modulations.

Across biological systems, cells undergo coordinated changes in gene expression, resulting in transcriptome dynamics that unfold within a low-dimensional manifold. While low-dimensional dynamics can be extracted using RNA velocity, these algorithms can be fragile and rely on heuristics lacking statistical control. Moreover, the estimated vector field is not dynamically consistent with the traversed gene expression manifold. To address these challenges, we introduce a Bayesian model of RNA velocity that couples velocity field and manifold estimation in a reformulated, unified framework, identifying the parameters of an explicit dynamical system. Focusing on the cell cycle, we implement VeloCycle to study gene regulation dynamics on one-dimensional periodic manifolds and validate its ability to infer cell cycle periods using live imaging. We also apply VeloCycle to reveal speed differences in regionally defined progenitors and Perturb-seq gene knockdowns. Overall, VeloCycle expands the single-cell RNA sequencing analysis toolkit with a modular and statistically consistent RNA velocity inference framework.

Hall-like behaviour of higher rank Chern-Simons theory of fractons

Fracton phases of matter constitute an interesting point of contact between condensed matter and high-energy physics. The limited mobility property of fracton quasi-particles finds applications in many different contexts, including quantum information, spin liquids, elasticity, hydrodynamics, gravity and holography. In this paper we adopt a field theoretical approach to investigate the three dimensional action of a rank-2 symmetric tensor field invariant under the covariant fracton symmetry. The theory appears as a non-topological higher rank generalization of the ordinary Chern-Simons model, depending only on the traceless part of the tensor gauge field. After defining a field strength, a rank-2 traceless “electric” field and a “magnetic” vector field are identified, in analogy with the standard Chern-Simons ones. Once matter is introduced, a Hall-like behaviour with fractonic features emerges. In particular, our model shows a Hall-like dipole current, together with a vectorial “flux-attachment” relation for dipoles. This gives a possible starting point for a fracton-vortex duality. A gauge-fixing term is then introduced, from which propagators are computed and the counting of the degrees of freedom is performed. Finally, the energy-momentum tensor is shown to be conserved and the integrated energy density is proved to be zero, which reminds the topological nature of the standard Chern-Simons model.

Path Planning for Mobile Robots Based on the Improved DAPF-QRRT* Strategy

The rapidly exploring random tree star (RRT*) algorithm is widely used to solve path planning problems. However, the RRT* algorithm and its variants fall short of achieving a balanced consideration of path quality and safety. To address this issue, an improved discretized artificial potential field-QRRT* (IDAPF-QRRT*) path planning strategy is introduced. Initially, the APF method is integrated into the Quick-RRT* (Q-RRT*) algorithm, utilizing the attraction of goal points and the repulsion of obstacles to optimize the tree expansion process, swiftly achieving superior initial solutions. Subsequently, a triangle inequality-based path reconnection mechanism is introduced to create and reconnect path points, optimize the path length, and accelerate the generation of sub-optimal paths. Finally, by refining the traditional APF method, a repulsive orthogonal vector field is obtained, achieving the orthogonalization between repulsive and attractive vector fields. This places key path points within the optimized vector field and adjusts their positions, thereby enhancing path safety. Compared to the Q-RRT* algorithm, the DPF-QRRT* algorithm achieves a 37.66% reduction in the time taken to achieve 1.05 times the optimal solution, and the IDAPF-QRRT* strategy nearly doubles generated path safety.

Full Stokes-vector Inversion of the Solar Mg ii h and k Lines

The polarization of the Mg ii h and k resonance lines is the result of the joint action of scattering processes and the magnetic field–induced Hanle, Zeeman, and magneto-optical effects, thus holding significant potential for the diagnostic of the magnetic field in the solar chromosphere. The Chromospheric LAyer Spectro-Polarimeter sounding-rocket experiment, carried out in 2019, successfully measured at each position along the 196″ spectrograph slit the wavelength variation of the four Stokes parameters in the spectral region of this doublet around 280 nm, both in an active-region plage and in a quiet region close to the limb. We consider some of these CLASP2 Stokes profiles and apply to them the recently developed HanleRT Tenerife Inversion Code, which assumes a one-dimensional model atmosphere for each spatial pixel under consideration (i.e., it neglects the effects of horizontal radiative transfer). We find that the nonmagnetic causes of symmetry breaking, due to the horizontal inhomogeneities and the gradients of the horizontal components of the macroscopic velocity in the solar atmosphere, have a significant impact on the linear polarization profiles. By introducing such nonmagnetic causes of symmetry breaking as parameters in our inversion code, we can successfully fit the Stokes profiles and provide an estimation of the magnetic field vector. For example, in the quiet region pixels, where no circular polarization signal is detected, we find that the magnetic field strength in the upper chromosphere varies between 1 and 20 G.