Two-dimensional Vector Field Research Articles

Harmonic flow-field representations of quantum bits and gates

We describe a general procedure for mapping arbitrary n-qubit states to two-dimensional (2D) vector fields. The mappings use complex rational function representations of individual qubits, producing classical vector field configurations that can be interpreted in terms of 2D inviscid fluid flows or electric fields. Elementary qubits are identified with localized defects in 2D harmonic vector fields, and multiqubit states find natural field representations via complex superpositions of vector field products. In particular, separable states appear as highly symmetric flow configurations, making them both dynamically and visually distinct from entangled states. The resulting real-space representations of entangled qubit states enable an intuitive visualization of their transformations under quantum logic operations. We demonstrate this for the quantum Fourier transform and the period finding process underlying Shor's algorithm, along with other quantum algorithms. Due to its generic construction, the mapping procedure suggests the possibility of extending concepts such as entanglement or entanglement entropy to classical continuum systems, and thus may help guide new experimental approaches to information storage and nonstandard computation. Published by the American Physical Society 2024

Extracting parity-violating gravitational waves from projected tidal force tensor in three dimensions

Gravitational waves (GWs) may be produced by various mechanisms in the early universe. In particular, if parity is violated, it may lead to the production of parity-violating GWs. In this paper, we focus on GWs on the scale of the large-scale structure. Since GWs induce tidal deformations of the shape of galaxies, one can extract such GW signals by observing images of galaxies in galaxy surveys. Conventionally the detection of such signals is discussed by considering the three-dimensional power spectra of the E/B-modes. Here, we develop a complementary new technique to estimate the contribution of GWs to the tidal force tensor field projected on the celestial sphere, which is a directly observable quantity. We introduce two two-dimensional vector fields constructed by taking the divergence and curl of the projected tidal field in three dimensions. Their auto-correlation functions naturally contain contributions of the scalar-type tidal field. However, we find that the divergence of the curl of the projected tidal field, which is a pseudo-scalar quantity, is free from the scalar contribution and thus enables us to extract GW signals. We also find that we can detect parity-violating signals in the GWs by observing the nonzero cross-correlation between the divergence of the projected tidal field and the curl of it. It roughly corresponds to measuring the cross-power spectrum of E and B-modes, but these are complementary to each other in the sense that our estimator can be naturally defined locally in position space. Finally we present expressions of the correlation functions in the form of Fourier integrals, and discuss the properties of the kernels specific to the GW case, which we call the overlap reduction function, borrowing the terminology used in the pulsar timing array experiments.

Two-dimensional vector field topology and scalar fields in viscous flows: Reconstruction methods

Two-dimensional vector field topology and scalar fields in viscous flows: Reconstruction methods

Transformations establishing equivalence across neural networks: When have two networks learned the same task?

Transformations are a key tool in the qualitative study of dynamical systems: transformations to a normal form, for example, underpin the study of instabilities and bifurcations. In this work, we test, and when possible establish, an equivalence between two different artificial neural networks by attempting to construct a data-driven transformation between them, using diffusion maps with a Mahalanobis-like metric. If the construction succeeds, the two networks can be thought of as belonging to the same equivalence class. We first discuss transformation functions between only the outputs of the two networks; we then also consider transformations that take into account outputs (activations) of a number of internal neurons from each network. Whitney's theorem dictates the number of (generic) measurements from one of the networks required to reconstruct each and every feature of the second network. The construction of the transformation function relies on a consistent, intrinsic representation of the network input space. We illustrate our algorithm by matching neural network pairs trained to learn (a) observations of scalar functions, (b) observations of two-dimensional vector fields, and (c) representations of images of a moving three-dimensional object (a rotating horse). We also demonstrate reconstruction of a network's input (and output) from minimal partial observations of intermediate neuron activations. The construction of equivalences across different network instantiations clearly relates to transfer learning and will also be valuable in establishing equivalence between different machine learning-based tools.

Time-reversibility and integrability of $ p:-q $ resonant vector fields

<abstract><p>We study the local analytical integrability in a neighborhood of $ p:-q $ resonant singular point of a two-dimensional vector field and its connection to time-reversibility with respect to the non-smooth involution $ \varphi(x, y) = (y^{p/q}, x^{q/p}). $ Some generalizations of the theory developed by Sibirsky for the $ 1:-1 $ resonant case to the $ p:-q $ resonant case are presented.</p></abstract>

Model-free autonomous control of four-wheel steering using artificial flow guidance

Four-wheel steering (4WS) is an effective technique to improve handling performance and lateral stability of road vehicles. Conventionally, controllers utilise the driver's actions and vehicle dynamics to coordinate front and rear-axle steering. This paper proposes a novel approach for 4WS controller design, based on the concept of Artificial Flow Guidance (AFG), which relies on a spatially distributed motion reference through a two-dimensional vector field. This field provides high-level guidance while lower-level steering controllers to control axle centres motions relative to the flow. These flow vectors, computed in real-time via simple geometric construction, can be pre-computed globally to evaluate the guidance algorithm's efficacy. When controlling only the front axle, this same approach can function as an autonomous driving system. Relying solely on a spatial reference field and control targets' velocities enables the controller to work in a simple and robust fashion, without using a reference vehicle dynamics model or lengthy parameter tuning. The proposed approach's effectiveness is validated through co-simulation with MATLAB/Simulink and the CarMaker simulation platform. AFG control performance is found to be at least comparable to that of more complex 4WS controllers using methods such as MPC; in the cases considered, AFG provides superior path-tracking performance.

Two-dimensional vector field of the phase energy spectrum application for motion identification

In the tasks of determining the location of objects and identifying movement, it is important to use the phase information contained in the phase-frequency spatial spectrum. In these problems, the phase-energy spectrum can be applied, which combines the advantages of the energy and phase spectra. A frame of a video sequence can be associated with a discrete vector field of the phase-energy spectrum; its components are located throughout the image. No less informative, but much simpler for analysis, is the discrete vector field of phase-energy characteristics. Its components are formed by the amplitudes of the harmonics of the phase-energy spectrum of a two-dimensional frame. To use the vector field of phase-energy characteristics in algorithms for identifying dynamic objects, information about interframe changes in its components is important. In this paper, the features of the change in the components of the vector field for the cases of horizontal and vertical movement of a rectangular object of constant brightness against a homogeneous background are investigated. Graphical representations of scalar fields of phase-energy functions are given. Characteristic regions are identified and their projections onto the coordinate plane are made. It is shown that two characteristic areas are the near and far hills, by analyzing which it is possible to determine the spatial coordinates of the object before the start of movement and the value of the object's inter-frame shift. The special areas of scalar fields of phase-energy functions are determined for the case of object motion in an arbitrary direction. It has been established that the presence of extended extrema in the graphs of phase-energy functions indicates a strictly vertical or horizontal movement of the object, and the presence of point extrema indicates the movement of the object “at an angle”, and the boundaries of the moving object and the magnitude of its displacement horizontally and vertically are completely determined by the coordinates of the point extrema. A number of options for finding the parameters of objects by extremums and/or by the boundaries of characteristic regions of phase-energy functions are proposed. The "boundary" and "extreme" approaches can be applied in vision systems as both an additional and a main tool for motion identification.

Microrobot Path Planning Based on the Multi-Module DWA Method in Crossing Dense Obstacle Scenario.

A hard issue in the field of microrobots is path planning in complicated situations with dense obstacle distribution. Although the Dynamic Window Approach (DWA) is a good obstacle avoidance planning algorithm, it struggles to adapt to complex situations and has a low success rate when planning in densely populated obstacle locations. This paper suggests a multi-module enhanced DWA (MEDWA) obstacle avoidance planning algorithm to address the aforementioned issues. An obstacle-dense area judgment approach is initially presented by combining Mahalanobis distance, Frobenius norm, and covariance matrix on the basis of a multi-obstacle coverage model. Second, MEDWA is a hybrid of enhanced DWA (EDWA) algorithms in non-dense areas with a class of two-dimensional analytic vector field methods developed in dense areas. The vector field methods are used instead of the DWA algorithms with poor planning performance in dense areas, which greatly improves the passing ability of microrobots over dense obstacles. The core of EDWA is to extend the new navigation function by modifying the original evaluation function and dynamically adjusting the weights of the trajectory evaluation function in different modules using the improved immune algorithm (IIA), thus improving the adaptability of the algorithm to different scenarios and achieving trajectory optimization. Finally, two scenarios with different obstacle-dense area locations were constructed to test the proposed method 1000 times, and the performance of the algorithm was verified in terms of step number, trajectory length, heading angle deviation, and path deviation. The findings indicate that the method has a smaller planning deviation and that the length of the trajectory and the number of steps can both be reduced by about 15%. This improves the ability of the microrobot to pass through obstacle-dense areas while successfully preventing the phenomenon of microrobots going around or even colliding with obstacles outside of dense areas.

Investigation of aluminum dust cloud dispersion characteristics in an explosion hazard testing device using laser-based particle and flow diagnostics

Monitoring the dust cloud dispersion process inside testing equipment like the MIKE3 minimum ignition energy device is useful for assessing the ignitability characteristics of different dust materials. However, it is difficult to comprehensively capture the relevant micro-scale particle properties and macro-scale flow behavior of a dust cloud dispersion using a single measurement method. Thus, the objective of this work is to combine two complementary laser diagnostic techniques to obtain quantitative particle and flow properties from dust clouds in the MIKE3 glass tube. For this experiment, thirty repeated runs for each method were conducted using small (15-mg) aluminum powder samples. No ignition electrodes are included in the test geometry to establish a baseline flow configuration for the glass tube, although measurements are still made at the characteristic time delay and location of ignition tests. The first technique, digital in-line holography (DIH) is used for three-dimensional micro-scale particle diagnostics, yielding particle number (n = 78 ± 20), size (D̅ = 18.9 ± 0.6 µm), and three-component velocity (v̅x = 0.05 ± 0.05 m/s, v̅y = 0.60 ± 0.09 m/s, v̅z = − 0.07 ± 0.16 m/s) measurements. The second laser diagnostic technique, particle image velocimetry (PIV), is used for macro-scale flow diagnostics, yielding two-dimensional flow velocity and vorticity vector field measurements. Particle sizes are observed to follow a lognormal distribution (μ = 2.89, σ = 0.31), and a discrepancy between in-situ and ex-situ sizing results is identified for particle sizes larger than ~55 µm. Two-dimensional particle velocities follow a Weibull distribution (θ = 0.86, β = 2.12), and reasonable agreement is found between the DIH and PIV velocity measurements.

Continuity Equation and Characteristic Flow for Scalar Hencky Plasticity

AbstractWe investigate uniqueness issues for a continuity equation arising out of the simplest model for plasticity, Hencky plasticity. The associated system is of the form where 𝜇 is a nonnegative measure and 𝜎 a two‐dimensional divergence‐free unit vector field. After establishing the Sobolev regularity of that field, we provide a precise description of all possible geometries of the characteristic flow, as well as of the associated solutions. © 2022 Courant Institute of Mathematics and Wiley Periodicals LLC.

Full-volume three-component intraventricular vector flow mapping by triplane color Doppler

Objective. Intraventricular vector flow mapping (iVFM) is a velocimetric technique for retrieving two-dimensional velocity vector fields of blood flow in the left ventricular cavity. This method is based on conventional color Doppler imaging, which makes iVFM compatible with the clinical setting. We have generalized the iVFM for a three-dimensional reconstruction (3D-iVFM). Approach. 3D-iVFM is able to recover three-component velocity vector fields in a full intraventricular volume by using a clinical echocardiographic triplane mode. The 3D-iVFM problem was written in the spherical (radial, polar, azimuthal) coordinate system associated to the six half-planes produced by the triplane mode. As with the 2D version, the method is based on the mass conservation, and free-slip boundary conditions on the endocardial wall. These mechanical constraints were imposed in a least-squares minimization problem that was solved through the method of Lagrange multipliers. We validated 3D-iVFM in silico in a patient-specific CFD (computational fluid dynamics) model of cardiac flow and tested its clinical feasibility in vivo in patients and in one volunteer. Main results. The radial and polar components of the velocity were recovered satisfactorily in the CFD setup (correlation coefficients, = 0.99 and 0.78). The azimuthal components were estimated with larger errors ( = 0.57) as only six samples were available in this direction. In both in silico and in vivo investigations, the dynamics of the intraventricular vortex that forms during diastole was deciphered by 3D-iVFM. In particular, the CFD results showed that the mean vorticity can be estimated accurately by 3D-iVFM. Significance. Our results tend to indicate that 3D-iVFM could provide full-volume echocardiographic information on left intraventricular hemodynamics from the clinical modality of triplane color Doppler.

Investigation of heat treatment process using Particle Image Velocimetry

Martensitic hardening is a technology widely used in mechanical engineering practice to increase the hardness and abrasion resistance of ferrous alloys. To achieve martensitic hardening, the workpiece must be cooled the workpiece very quickly, which can be achieved through using various refrigerants. The quality and the properties of these coolants have a great influence on the success of the process, so their investigation is essential. Throughout the investigation method – according to ISO 9950 standard –various aqueous suspensions used as refrigerants are studied by placing a heated cylindrical specimen in a refrigerated container (which will be described later) and by recording the temperature change over time. The flows generated in the tank during the heat treatment influence the nature of the heat transfer, thus the hardening process itself. The task is to map these flow conditions to give a comprehensive description of the qualification process in question. With the advancement of measurement technologies, there are more and more possibilities to determine the velocity of flowing media, even at several points simultaneously. Using the available PIV (Particle Image Velocity) system, the two-dimensional vector field of the flow can be determined on a complete plane surface. In the present research, the applicability of seeding particles required for the PIV measurement technique is also investigated as part of the mentioned project, establishing further research possibilities. The classification of eight different seeding particles is presented. As a final result, seeding particles suitable in the current arrangement for proper flow display are listed.

Analytic properties of the complete formal normal form for the Bogdanov–Takens singularity * *Supported by the Polish NCN OPUS Grant 2017/25/B/ST1/00931

In Stróżyna E and Żołądek H (2015 Moscow Math. J. 15 141) a complete formal normal forms for germs of two-dimensional holomorphic vector fields with nilpotent singularity was obtained. That classification is quite nontrivial (7 cases), but it can be divided into general types like in the case of the elementary singularities. One could expect that also the analytic properties of the normal forms for the nilpotent singularities are analogous to the case of the elementary singularities. This is really true. In the cases analogous to the focus and the node the normal form is analytic. In the case analogous to the nonresonant saddle the normal form is often nonanalytic due to the small divisors phenomenon. In the cases analogous to the resonant saddles (including saddle-nodes) the normal form is nonanalytic due to bad properties of some homological operators associated with the first nontrivial term in the orbital normal form.

Connecting boundary layer dynamics with extreme bulk dissipation events in Rayleigh-Bénard flow (a) (a)Contribution to the Focus Issue Turbulent Thermal Convection edited by Mahendra Verma and Jörg Schumacher.

We study the connection between extreme events of thermal and kinetic energy dissipation rates in the bulk of three-dimensional Rayleigh-Bénard convection and the wall shear stress patterns at the top and the bottom plates that enclose the layer. Zero points of this two-dimensional vector field stand for detachments of strong thermal plumes. If their position at the opposite plates and a given time is close then they can be considered as precursors for high-amplitude bulk dissipation events triggered by plume collisions or close passings. This scenario requires a breaking of the synchronicity of the boundary layer dynamics at both plates which is found to be in line with a transition of the bulk derivative statistics from Gaussian to intermittent. Our studies are based on three-dimensional high-resolution direct numerical simulations for moderate Rayleigh numbers between and .

DeepFlux for Skeleton Detection in the Wild

The medial axis, or skeleton, is a fundamental object representation that has been extensively used in shape recognition. Yet, its extension to natural images has been challenging due to the large appearance and scale variations of objects and complex background clutter that appear in this setting. In contrast to recent methods that address skeleton extraction as a binary pixel classification problem, in this article we present an alternative formulation for skeleton detection. We follow the spirit of flux-based algorithms for medial axis recovery by training a convolutional neural network to predict a two-dimensional vector field encoding the flux representation. The skeleton is then recovered from the flux representation, which captures the position of skeletal pixels relative to semantically meaningful entities (e.g., image points in spatial context, and hence the implied object boundaries), resulting in precise skeleton detection. Moreover, since the flux representation is a region-based vector field, it is better able to cope with object parts of large width. We evaluate the proposed method, termed DeepFlux, on six benchmark datasets, consistently achieving superior performance over state-of-the-art methods. Finally, we demonstrate an application of DeepFlux, augmented with a skeleton scale estimation module, to detect objects in aerial images. This combination yields results that are competitive with models trained specifically for object detection, showcasing the versatility and effectiveness of mid-level representations in high-level tasks. An implementation of our method is available at https://github.com/YukangWang/DeepFlux .

Vector fields created by Fourier transform of video signals

The power spectrum and autocorrelation are widely used in video signals processing. The phase-frequency spectrum contains more information than the amplitude spectrum. However, the amplitude spectrum is used more often in the video signal process. It is possible to combine the advantages of these two spectra to obtain a vector function such as phase-energy spectrum. This spectrum is sensitive even to a small difference in pixel brightness but at the same time it is as stable as the power spectrum. In this paper, we explored the properties of the phase-power spectrum and related amplitude of partials vector field. We showed that phase-power spectrum has conservative as well as solenoidal components and so it creates curl’s vector field. We also found the components of phase-power spectra that contain pixels with a different degree of contrast on the uniform background. We showed the possibility to use solenoidal component of the phase-power vector field to find the circulation (i.e. field work). We concluded that the circulation along any of brightness contour constants, which is symmetrical to the origin, is equal zero. We explored two-dimensional vector field of the phase-power spectrum. Finally, we determined an effect of pixels with different degree of brightness and their relative position to the image’s edges on vector field’s components formation.

An experimental investigation of a square supersonic jet and impinging jet on an inclined plate

Supersonic free jets and impinging jets are found in many engineering applications, such as short and vertical take-off and landing vehicles, cold gas dynamic spray processes, hot surface cooling mechanisms, and turbomachinery systems. The flow characteristics of a supersonic square jet discharging into the ambient and a supersonic jet impinging on a 45° inclined surface were experimentally investigated for nozzle-pressure-ratios (NPRs) of 4.8 and 5.9. Experimental measurements of impinging jets were acquired for nozzle-to-plate distances of 0.82Dj and 1.8Dj, where Dj is the jet hydraulic diameter. The velocity fields in the central plane of the jet were obtained using planar particle image velocimetry. The flow characteristics of the supersonic jets, including mean velocity and turbulent kinetic energy, were computed from the acquired two-dimensional two-component velocity vector fields, and statistical profiles were compared for different NPRs and nozzle-to-plate distances. For supersonic free jets, the acquired statistical results revealed the presence of multiple shock cells along the streamwise direction. Impinging jet measurements revealed the presence of shock cells in the vicinity of the nozzle outlet, oblique plate shocks near the impingement location, and several tail shocks along the streamwise direction. Spatial turbulent velocity cross correlations were calculated for various points located along the shear layers to investigate the characteristics of turbulent features, such as the shape, orientation, and integral length scales of the studied configurations. In addition, a proper orthogonal decomposition analysis was applied to the instantaneous velocity fields to identify the statistically dominant flow structures that play an important role in the flow field characteristics of supersonic free jets and supersonic impinging jets.

Measuring the criticality of a Hopf bifurcation

This work is based on the observation that the first Poincaré–Lyapunov constant is a quadratic function of the coefficients of the two-dimensional vector field at a Hopf bifurcation. From a given parameter point, we find the distance to the “Hopf quadric.” This distance provides a measure of the criticality of the Hopf bifurcation. The viability of the approach is demonstrated through numerical examples.

A simple model of Keratocyte membrane dynamics: The case of motionless living cell

We perform an analytical investigation of the cell interface dynamics in the framework of a minimal phase field model of cell motility suggested in Ziebert et al. (2012), which consists of two coupled evolution equations for the order parameter and a two-dimensional vector field describing the actin network polarization (orientation). We derive a closed evolutionary integro-differential equation governing the cell interface dynamics. The equation includes the normal velocity of the membrane, curvature, volume relaxation, and a parameter that is determined by the non-equilibrium effects in the cytoskeleton. This equation can be simplified to obtain a Burgers-like equation. A condition on the system parameters for the existence of a stationary cell shape is obtained.

Topological defects at the boundary of neutron P23 superfluids in neutron stars

We study surface effects of neutron $^{3}P_{2}$ superfluids in neutron stars. $^{3}P_{2}$ superfluids are in uniaxial nematic (UN), D$_{2}$ biaxial nematic (BN), or D$_{4}$ BN phase, depending on the strength of magnetic fields from small to large. We suppose a neutron $^{3}P_{2}$ superfluid in a ball with a spherical boundary. Adopting a suitable boundary condition for $^{3}P_{2}$ condensates, we solve the Ginzburg-Landau equation to find several surface properties for the neutron $^{3}P_{2}$ superfluid. First, the phase on the surface can be different from that of the bulk, and symmetry restoration or breaking occurs in general on the surface. Second, the distribution of the surface energy density has an anisotropy depending on the polar angle in the sphere, which may lead to the deformation of the geometrical shape of the surface. Third, the order parameter manifold induced on the surface, which is described by two-dimensional vector fields induced on the surface from the condensates, allows topological defects (vortices) on the surface, and there must exist such defects even in the ground state thanks to the Poincar\'{e}-Hopf theorem: although the numbers of the vortices and antivortices depend on the bulk phases, the difference between them is topologically invariant (the Euler number $\chi=2$) irrespective of the bulk phases. These vortices, which are not extended to the bulk, are called boojums in the context of liquid crystals and helium-3 superfluids. The surface properties of the neutron $^{3}P_{2}$ superfluid found in this paper may provide us useful information to study neutron stars.