Vector Field Research Articles

Local models for smooth vector fields of the line

We present the local classification of singularities of smooth vector fields on the line, with respect to the equivalence relation of C1–conjugacy. Along the way, we recall the analogous classification, up to C0 and C∞ conjugacy. We also give the transversal unfoldings of the corresponding normal forms and treat the case where the changes of coordinates are tangent to the identity. Thus, a fairly complete description of the 1–d case is achieved.

Structure-preserving algorithms for guiding center dynamics based on the slow manifold of classical Pauli particle

The classical Pauli particle (CPP) serves as a slow manifold, substituting the conventional guiding center dynamics. Based on the CPP, we utilize the averaged vector field (AVF) method in the computations of drift orbits. Demonstrating significantly higher efficiency, this advanced method is capable of accomplishing the simulation in less than one-third of the time of directly computing the guiding center motion. In contrast to the CPP-based Boris algorithm, this approach inherits the advantages of the AVF method, yielding stable trajectories even achieved with a tenfold time step and reducing the energy error by two orders of magnitude. By comparing these two CPP algorithms with the traditional RK4 method, the numerical results indicate a remarkable performance in terms of both the computational efficiency and error elimination. Moreover, we verify the properties of slow manifold integrators and successfully observe the bounce on both sides of the limiting slow manifold with deliberately chosen perturbed initial conditions. To evaluate the practical value of the methods, we conduct simulations in non-axisymmetric perturbation magnetic fields as part of the experiments, demonstrating that our CPP-based AVF method can handle simulations under complex magnetic field configurations with high accuracy, which the CPP-based Boris algorithm lacks. Through numerical experiments, we demonstrate that the CPP can replace guiding center dynamics in using energy-preserving algorithms for computations, providing a new, efficient, as well as stable approach for applying structure-preserving algorithms in plasma simulations.

Finsler–Randers–Sasaki gravity and cosmology

We present for the first time a Friedmann-like construction in the framework of an osculating Finsler–Randers–Sasaki (F–R–S) geometry. In particular, we consider a vector field in the metric on a Lorentz tangent bundle, and thus the curvatures of horizontal and vertical spaces, as well as the extra contributions of torsion and non-linear connection, provide an intrinsic richer geometrical structure, with additional degrees of freedom, that lead to extra terms in the field equations. Applying these modified field equations at a cosmological setup we extract the generalized Friedmann equations for the horizontal and vertical space, showing that we obtain an effective dark energy sector arising from the richer underlying structure of the tangent bundle. Additionally, as it is common in Finsler-like constructions, we obtain an effective interaction between matter and geometry. Finally, we consider a specific model and we show that it can describe the sequence of matter and dark-energy epochs, and that the dark-energy equation of state can lie in the quintessence or phantom regimes, or cross the phantom divide.

A method to efficiently and rapidly approximate the vectorial fields generated by large area metasurfaces

In order to calculate the electromagnetic fields that are produced after light passes through a metasurface, simulation methods such as the Finite-Difference Time-Domain method are often employed. While these provide a good approximation to the fields, the level of detail at which the volume of space that the light is propagating in needs to be modelled and the time for which simulations need to run, mean that as the area of the metasurface is increased these simulations rapidly become unwieldy. In this paper we show how the result of a FDTD simulation of a unit cell can be used to generate a good approximation of the vectorial field that large area metasurfaces will generate, but using a fraction of the computational resources. This approach can provide an intermediate design step, allowing potentially interesting designs to be rapidly identified or discarded.

Quantifying local cardiac substrate heterogeneity from high density recordings: an experimental study

Abstract Background and Purpose High-density (HD) grid catheters offer enhanced local electrical conduction characterization, owing to their unique grid arrangement. This arrangement enables the generation of orientation-independent EGMs, termed omnipolar EGMs (oEGMs), overcoming limitations associated with traditional unipolar and bipolar EGMs. This innovation holds potential for differentiating healthy tissue from scarred or arrhythmogenic areas. In this study, we introduce a novel metric, the Vector Field Heterogeneity (VFH), designed to quantify the heterogeneity of propagation wavefronts in local vector maps derived from oEGMs. Methods We developed and validated the VFH metric through a combination of simulations and experiment conducted on Langendorff-perfused rabbit hearts. Simulations involved creating a model that generated 100,000 simulated maps exhibiting various levels of disorganization. For experimental data, 68 propagation maps were utilized, including 29 from basal recordings, 39 from recordings with stimulation (4 or 6Hz), from five Langendorff-perfused rabbit hearts at 37ºC, using a multielectrode mapping catheter with 128 electrodes spaced 1mm apart. The VFH metric, based on vector field theory (see figure 2), was computed and compared to the widely adopted the Spatial Inhomogeneity Index (SI) from Statistical analysis was then performed to assess the distribution of heterogeneity values. Results The analysis of the VFH metric through simulations revealed a linear increase in VFH values with disorganization up to a 45º angle, followed by a plateau. Different catheter grid sizes demonstrated that larger grids reduced the standard deviation, with the VFH metric converging to a lower value of approximately 0.62 for larger grids. Experimental data indicated that for a 4x4 grid, VFH values averaged 0.35 without stimulation and decreased to 0.11 with stimulation. The VFH metric consistently outperformed the SI index in binary classification of basal vs stimulated conditions, as evidenced by higher Area Under Curve (AUC) values in both Receiver Operating Characteristic (ROC) and Precision-Recall (PR) analyses across various catheter sizes. For example, in the 4x4 grid, the AUROC for the VFH metric was 0.9752 compared to 0.8311 for the SI index, showcasing the VFH metric’s superior accuracy in assessing cardiac wavefront heterogeneity (see figure 2). Conclusion Vector Field Heterogeneity (VFH) emerges as an effective tool for characterizing cardiac tissue. It successfully differentiates between stimulated and non-stimulated tissue, accurately representing progressive disorganization in simulated maps. The VFH metric demonstrates superiority over the established Spatial Inhomogeneity (SI) index, making it a reliable parameter for assessing cardiac heterogeneity, especially on small high-density grids. It holds promise for clinical use in studying and understanding the cardiac substrate during arrhythmic conditions.Figure 1.Abstract Figure.Figure 2.Methods and Results.

Enhancing OpenEP: atrial conduction velocity and conduction velocity heterogeneity quantification through EP Workbench

Abstract Background Alterations in atrial conduction velocity have been implicated in the pathogenesis of atrial arrhythmias, with conduction velocity thought to be slower and more heterogeneous in areas promoting atrial fibrillation. However, clinical studies of conduction velocity are challenging since there are no available platforms that provide researchers and clinicians with the ability to quantify conduction velocity and analyse conduction heterogeneity from electroanatomic mapping data. Purpose In this paper we sought to address these limitations by enhancing the OpenEP Python library by (1) automating calculations using previously-published conduction velocity estimation algorithms; (2) providing an interactive graphical representing of conduction velocity and conduction heterogeneity and (3) implementing a histogram analysis tool to quantify conduction velocity and enable the identification of slow conduction regions. Method Two well-known and previously published methods for calculating cardiac conduction velocity, Triangulation and polynomial surface fitting, were implemented. Conduction velocity estimation was improved further by excluding regions with wave collisions and focal discharges. These regions were identified using the divergence of conduction velocity vector fields, with positive divergence representing focal sources and negative divergence representing areas of collision. Finally, calculated conduction velocities using electrogram data were interpolated over the surface mesh via Radial Basis Function (RBF) interpolation method. All algorithms were implemented in the OpenEP Python library (openep-py), with visualisation tools provided in the complementary graphical interface, EP Workbench. Results An extensible "Analysis" feature was created in openep-py, to provide conduction velocity and divergence calculations which are fully customisable through the graphical interface based on user preferences. The EP Workbench desktop application displays resulting cardiac surface maps to depict calculated conduction velocity and divergence fields, together with conventional activation and voltage maps (Figure 1). The histogram analysis tool can be used to identify regions of slow conduction velocity from electroanatomic mapping data. Using a threshold of <0.3m/s, two slow conducting channels are visualised on the posterior wall of a test case (Figure 2). The tool is not specific to conduction velocity and may also be used to quantify other data types in EP Workbench. Finally, computed conduction velocity and divergence values and surface maps are stored in the standard OpenEP data structure, allowing further analysis following data export. Conclusion We have extended OpenEP to provide a comprehensive set of tools for conduction velocity calculation and analysis which will facilitate investigation of the structural and functional basis of conduction velocity alterations in disease states.

Statistical conformal Killing vector fields for FLRW space-time

The classification of conformal Killing vector fields for FLRW space-time from Riemannian point of view was done by Maartens-Maharaj in []. In this paper, we introduce conformal Killing vector fields from a new point of view for the FLRW space-time. In particular, we consider three cases for the conformal factor. Then, it is shown that there exist nine conformal vector fields on FLRW in total, such that six of them are Killing and the rest being non-Killing conformal vector fields. Consequently, by recalling the concept of statistical conformal Killing vector fields introduced in [], we classify statistical structures whith repsect to which these vector fields are conformal Killing. We also obtain the form of affine connections that feature a vanishing Lie derivative with respect to these conformal Killing vector fields. Imposing the torsion-free and the Codazzi conditions on these connections, we study statistical structures on FLRW. Finally, for torsionful connections we study the vanishing of the Lie derivative of the torsion tensor with respect to these conformal Killing vector fields and derive the conditions under which this is valid.

Automated segmentation and recognition of C. elegans whole-body cells.

Motivation: Accurate segmentation and recognition of Caenorhabditis elegans cells are critical for various biological studies, including gene expression, cell lineages and cell fates analysis at single-cell level. However, the highly dense distribution, similar shapes, and inhomogeneous intensity profiles of whole-body cells in 3D fluorescence microscopy images make automatic cell segmentation and recognition a challenging task. Existing methods either rely on additional fiducial markers or only handle a subset of cells. Given the difficulty or expense associated with generating fiducial features in many experimental settings, a marker-free approach capable of reliably segmenting and recognizing C. elegans whole-body cells is highly desirable. We report a new pipeline, called Automated Segmentation and Recognition (ASR) of cells, and applied it to 3D fluorescent microscopy images of L1-stage C. elegans with 558 whole-body cells. A novel displacement vector field based deep learning model is proposed to address the problem of reliable segmentation of highly crowded cells with blurred boundary. We then realize the cell recognition by encoding and exploiting statistical priors on cell positions and structural similarities of neighboring cells. To the best of our knowledge, this is the first method successfully applied to the segmentation and recognition of C. elegans whole-body cells. The ASR segmentation module achieves an F1-score of 0.8956 on a dataset of 116 C. elegans image stacks with 64728 cells (Accuracy 0.9880, AJI 0.7813). Based on the segmentation results, the ASR recognition module achieved an average accuracy of 0.8879. We also show ASR's applicability to other cell types, e.g. platynereis and rat kidney cells. The code is available at https://github.com/reaneyli/ASR. Supplementary data are available at Bioinformatics online.

Concept of Processing and Interpretation of Spatiotemporal Vector Data

The purpose of this work is to develop an approach to complex processing and interpretation of spatio-temporal vector data. In this regard, it is necessary to analyze existing methods for visualizing vector and tensor fields and organizing a database for their storage and develop a concept which contains the main advantages of these methods and allows one to interpret spatial data represented as a tensor. The developed concept proposes an approach to field visualization based on drawing superellipse glyphs at each point in space. Axes of glyphs correspond to the rank of the visualized tensor, and attribute values are expressed by varying the color gradient in its monochrome representation. The architecture of a web-based application, which implements the storage of spatial data in a hybrid DBMS, and interpretation of the necessary data upon request is also proposed. For the physical storage of tensor information, a hybrid relational-hierarchical model is proposed, in which the representation of direct spatial data and the corresponding metadata, including those characterizing the source of information, are separated. It is advisable to present the final result and intermediate data in GeoJSON format. This format is actually currently the standard for presenting geospatial data and is supported by almost all software platforms and libraries aimed at solving problems of this kind.

On tiling spherical triangles into quadratic subpatches

Various interpolation and approximation methods arising in several practical applications in geometric modeling deal, at a particular step, with the problem of computing suitable rational patches (of low degree) on the unit sphere. Therefore, we are concerned with the construction of a system of spherical triangular patches with prescribed vertices that globally meet along common boundaries. In particular, we investigate various possibilities for tiling a given spherical triangular patch into quadratically parametrizable subpatches. We revisit the condition that the existence of a quadratic parameterization of a spherical triangle is equivalent to the sum of the interior angles of the triangle being π, and then circumvent this limitation by studying alternative scenarios and present constructions of spherical macro-elements of the lowest possible degree. Applications of our method include algorithms relying on the construction of (interpolation) surfaces from prescribed rational normal vector fields.

Deformable vector fields warping for modelling of irregular breathing.

Purpose
4D computed tomography (4DCT) is the clinical standard to image organ motion in radiotherapy, although it is limited in imaging breathing variability. We propose a method to transfer breathing motion across longitudinal imaging datasets to include intra-patient variability and verify its performance in lung cancer patients. 
Methods 
Five repeated control 4DCTs for 6 non-small cell lung cancer patients were combined into multi-breath datasets (m4DCT) by merging stages of deformable image registration to isolate respiratory motion. The displacement of the centre of mass of the primary tumour and its volume changes were evaluated to quantify intra-patient differences. Internal target volumes defined on the m4DCT were compared with those conventionally drawn on the 4DCT.
Results 
Motion analysis suggests no discontinuity at the junction between successive breaths, confirming the method's ability to merge repeated imaging into a continuum. Motion (variability) is primarily in superior-inferior direction and goes from 14.4 mm (8.7 mm) down to 0.1 mm (0.6 mm), respectively for tumours located in the lower lobes or most apical ones. On average, up to 65% and 74% of the tumour volume was subject to expansion or contraction in the inhalation and exhalation phases. These variations lead to an enlargement of the ITV up to 8% of its volume in our dataset.
Conclusion 
4DCT can be extended to model variable breathing motion by adding synthetic phases from multiple time-resolved images. The inclusion of this improved knowledge of patients' breathing allows better definition of treatment volumes and their margins for radiation therapy.&#xD.

3D imaging of magnetic domains in Nd2Fe14B using scanning hard X-ray nanotomography.

Nanoscale structural and electronic heterogeneities are prevalent in condensed matter physics. Investigating these heterogeneities in 3D has become an important task for understanding material properties. To provide a tool to unravel the connection between nanoscale heterogeneity and macroscopic emergent properties in magnetic materials, scanning transmission X-ray microscopy (STXM) is combined with X-ray magnetic circular dichroism. A vector tomography algorithm has been developed to reconstruct the full 3D magnetic vector field without any prior noise assumptions or knowledge about the sample. Two tomographic scans around the vertical axis are acquired on single-crystalline Nd2Fe14B pillars tilted at two different angles, with 2D STXM projections recorded using a focused 120 nm X-ray beam with left and right circular polarization. Image alignment and iterative registration have been implemented based on the 2D STXM projections for the two tilts. Dichroic projections obtained from difference images are used for the tomographic reconstruction to obtain the 3D magnetization distribution at the nanoscale.

Equilibria and bifurcations in contact dynamics

We provide a systematic study of equilibria of contact vector fields and the bifurcations that occur generically in 1-parameter families, and express the conclusions in terms of the Hamiltonian functions that generate the vector fields. Equilibria occur at points where the zero-level set of the Hamiltonian function is either singular or is tangent to the contact structure. The eigenvalues at an equilibrium have an interesting structure: there is always one particular real eigenvalue of any equilibrium, related to the contact structure, that we call the principal coefficient, while the other eigenvalues arise in quadruplets, similar to the symplectic case except they are translated by a real number equal to half the principal coefficient. There are two types of codimension 1 equilibria, named Type I, arising where the zero-set of the Hamiltonian is singular, and Type II where it is not, but there is a degeneracy related again to the principal coefficient and the contact of the zero-level set of the Hamiltonian with the contact structure. Both give rise generically to saddle-node bifurcations. Some special features include: (i) for Type II singularities, Hopf bifurcations cannot occur in dimension 3, but they may in dimension 5 or more; (ii) for Type I singularities, a fold-Hopf bifurcation can occur with codimension 1 in any dimension, and (iii) again for Type I, and in dimension at least 5, a fold-multi-Hopf bifurcation (where several pairs of eigenvalues pass through the imaginary axis simultaneously together with one through the origin) may also occur with codimension 1.

On the structural instability of non-hyperbolic limit cycles on planar polynomial vector fields

On the structural instability of non-hyperbolic limit cycles on planar polynomial vector fields

Vector field path following for a micro flapping-wing robot

Vector field path following for a micro flapping-wing robot

Shaking a container full of perfect liquid; a tractable case, a torus shell, exhibits a virtual wall

Manipulation (‘shaking’) of a rigid container filled with incompressible liquid starting from stationary generally results in some displacement, or mixing, of the liquid within it. If the liquid also has zero viscosity, a ‘perfect’ or ‘Euler’ liquid, Kelvin’s theorems dramatically simplify the flow analysis. Response is instantaneous; stop the container and all liquid motion stops. In fact an arbitrary manipulation can be considered as alternating infinitesimal translations and rotations of the container. Relative to the container, the liquid is stationary during every translation. Infinitesimal rotations (an infinitesimal vector along the rotation axis) resolve into three orthogonal components in the container frame. Each generates its own infinitesimal liquid displacement vector field. Their combined consequences are usually obscure. Rather than a volume flow, a surface flow in 3D is considerably easier, the liquid slipping freely in a shell, sandwiched between two nested closed surfaces with constant infinitesimal gap. The closedness avoids extra boundaries. The two dimensionality admits a scalar streamfunction determined by the container angular velocity vector. Manipulation of the container angular velocity at will, usually leads to an infinitely rich variety of area preserving re-configurations of the liquid. In particular, any chosen point of the liquid can be moved, in the shell container frame, to any other chosen point. However, for a torus (shell) with a small enough hole (diameter < 0.195 torus diameter), there exists a virtual wall, a hypothetical axial cylinder intersecting the torus. No matter how the torus is manipulated, liquid inside the cylinder stays inside; outside stays outside. The analysis, solving for the streamfunction, is based on the relative vorticity, and the conformal mapping of a torus to a (periodic) rectangle, which lead to a fairly simple convolution integral formula for the flow.

Resolvent of vector fields and Lefschetz numbers

Resolvent of vector fields and Lefschetz numbers

Conformally flat almost cosymplectic 3-manifolds

In this paper, we consider an almost cosymplectic 3-manifold M such that its scalar curvature is invariant along the Reeb vector field. We prove that if the Reeb vector field of M is harmonic, then M is conformally flat if and only if it is locally isometric to the product ℝ × N 2(c), where N 2(c) is a Kähler surface of constant sectional curvature c. Almost cosymplectic 3-manifolds on which the Reeb vector field satisfies the h-a condition together with conformal flatness are also classified.

A Variational Surface-Evolution Approach to Optimal Transport over Transitioning Compact Supports with Domain Constraints

We examine the optimal mass transport problem in Rn between densities with transitioning compact support by considering the geometry of a continuous interpolating support boundary Γ in space-time within which the mass density evolves according to the fluid dynamical framework of Benamou and Brenier. We treat the geometry of this space-time embedding in terms of points, vectors, and sets in Rn+1=R×Rn and blend the mass density and velocity as well into a space-time solenoidal vector field W|Ω→Rn+1 over a compact set Ω⊂Rn+1. We then formulate a joint optimization for W and its support using the shaped gradient of the space-time surface Γ outlining the support boundary ∂Ω. This easily accommodates spatiotemporal constraints, including obstacles or mandatory regions to visit.

Engineering the Band Topology in a Rhombohedral Trilayer Graphene Moiré Superlattice.

Moiré superlattices have become a fertile playground for topological Chern insulators, where the displacement field can tune the quantum geometry and Chern number of the topological band. However, in experiments, displacement field engineering of spontaneous symmetry-breaking Chern bands has not been demonstrated. Here in a rhombohedral trilayer graphene moiré superlattice, we use a thermodynamic probe and transport measurement to monitor the Chern number evolution as a function of the displacement field. At a quarter filling of the moiré band, a novel Chern number of three is unveiled to compete with the well-established number of two upon turning on the electric field and survives when the displacement field is sufficiently strong. The transition can be reconciled by a nematic instability on the Fermi surface due to the pseudomagnetic vector field potentials associated with moiré strain patterns. Our work opens more opportunities to active control of Chern numbers in van der Waals moiré systems.