Gradient Vector Field Research Articles

Cosymplectic Geometry, Reductions, and Energy-Momentum Methods with Applications

Classical energy-momentum methods study the existence and stability properties of solutions of t-dependent Hamilton equations on symplectic manifolds whose evolution is given by their Hamiltonian Lie symmetries. The points of such solutions are called relative equilibrium points. This work devises a new cosymplectic energy-momentum method providing a new and more general framework to study t-dependent Hamilton equations. In fact, cosymplectic geometry allows for using more types of distinguished Lie symmetries (given by Hamiltonian, gradient, or evolution vector fields), relative equilibrium points, and reduction methods, than symplectic techniques. To make our work more self-contained and to fill some gaps in the literature, a review of the cosymplectic formalism and the cosymplectic Marsden–Weinstein reduction is included. Known and new types of relative equilibrium points are characterised and studied. Our methods remove technical conditions used in previous energy-momentum methods, like the Ad∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{Ad}^*$$\\end{document}-equivariance of momentum maps. Eigenfunctions of t-dependent Schrödinger equations are interpreted in terms of relative equilibrium points in cosymplectic manifolds. A new cosymplectic-to-symplectic reduction is developed and a new associated type of relative equilibrium points, the so-called gradient relative equilibrium points, are introduced and applied to study the Lagrange points and Hill spheres of a restricted circular three-body system by means of a not Hamiltonian Lie symmetry of the system.

Quantum Topological Atomic Properties of 44K molecules

We present a data set of quantum topological properties of atoms of 44K randomly selected molecules from the GDB-9 data set. These atomic properties were obtained as defined by the quantum theory of atoms in molecules (QTAIM) within an atomic basin, a region of real space bounded by zero-flux surfaces in the electron density gradient vector field. The wave function files were generated through DFT static calculations (B3LYP/6-31G), and the atomic properties were calculated using QTAIM. The calculated atomic properties include the energy of the atomic basin, the electronic population, the magnitude of the total dipole moment, and the magnitude of the total quadrupole moment. The atomic properties allow one to understand the chemical structure, reactivity, and molecular recognition. They can be incorporated into force fields for molecular dynamics or for predicting reactive sites. We believe that this data set could facilitate new studies in chemical informatics, machine learning applied to chemistry, and computational molecular design.

Ricci bi-conformal vector fields on Lorentzian five-dimensional two-step nilpotent Lie groups

In this paper, we completely classify Ricci bi-conformal vector fields on simply-connected five-dimensional two-step nilpotent Lie groups which are also connected and we show which of them are the Killing vector fields and gradient vector fields.

Distributed Control for a Multiagent System to Pass Through a Connected Quadrangle Virtual Tube

In order to guide the multi-agent system in a cluttered environment, a connected quadrangle virtual tube is designed for all agents to keep moving within it, whose basis is called the single trapezoid virtual tube. There is no obstacle inside the tube, namely the area inside the tube can be seen as a safety zone. Then, a distributed swarm controller is proposed for the single trapezoid virtual tube passing problem. This issue is resolved by a gradient vector field method with no local minima. Formal analyses and proofs are made to show that all agents are able to pass the single trapezoid virtual tube. Finally, a modified controller is put forward for convenience in practical use. For the connected quadrangle virtual tube, a modified switching logic is proposed to avoid the deadlock and prevent agents from moving outside the virtual tube. Finally, the effectiveness of the proposed method is validated by numerical simulations and real experiments.

Differential geometry for the optimal design of the contingent valuation method

AbstractThe contingent valuation method (CVM) is a widely used experimental method to measure the monetary value of goods. However, CVM estimates are sensitive to experiment design. In this study, we formulated the optimal design problem as a minimization problem of the Fisher information metric of a gradient vector field generated by using the statistical model of the CVM. Furthermore, a necessary and sufficient condition of the optimal design was proven.

The Maximum Locus of the Bloch Norm

Abstract For a Bloch function f in the unit ball in ℂ n , we study the maximal locus of the Bloch norm of f; namely, the set Lf where the Bergman length of the gradient vector field of f attains its maximum. We prove that for n ≥, the set Lf consists of a finite union of real analytic sets with dimensions at most 2n − 2. This is not the case for n = 1 as was proved earlier by Cima and Wogen. We also give some rigidity properties of the set Lf . In particular, we give some sufficient criteria for constructing extreme functions in the Little Bloch ball.

Convergence of the gradient flow of renormalized volume to convex cores with totally geodesic boundary

We consider the Weil–Petersson gradient vector field of renormalized volume on the deformation space of convex cocompact hyperbolic structures on (relatively) acylindrical manifolds. In this paper we prove the conjecture that the flow has a global attracting fixed point at the unique structure $M_{\rm geod}$ with minimum convex core volume.

Integration of vector fields on cell complexes and Morse theory

In this paper, we investigate vector fields on polyhedral complexes and their associated trajectories. We study vector fields which are analogues of the gradient vector field of a function in the smooth case. Our goal is to define a nice theory of trajectories of such vector fields, so that the set of them captures the topology of the polyhedral complex, as in classical Morse theory. Since we do not assume the polyhedral complex to be a manifold, the definition of vector fields on it is very different from the smooth case. Nevertheless, we will show that there exist nice classes of functions and metrics which give gradient vector fields with desired properties. Our construction relies on Forman's discrete Morse theory. In particular, the class of functions we use is an improvement of Forman's discrete Morse functions. A notable feature of our theory is that our gradient vector fields are defined purely from functions and metrics as in the smooth case, contrary to the case of discrete Morse theory where we need the data of dimension of cells. This allows us to implement several useful constructions which were not available in the discrete case.

Small Maritime Target Detection Using Gradient Vector Field Characterization of Infrared Image

Infrared small maritime target detection under strong ocean waves, a challenging task, plays a key role in maritime distress target search and rescue applications. Many methods based on directionality or gradient properties have proven to perform well for infrared images with heterogeneous scenarios. However, they tend to perform poorly when facing strong ocean wave background, mainly due to the following: 1) infrared images have low signal-to-clutter ratio with low intensity for small targets; 2) some waves have high local contrast that may be similar to or higher than targets. To solve these issues, a new method based on gradient vector field characterization (GVFC) of infrared images is proposed. First, we construct the gradient vector field and coarsely extract suspected targets. Then, gradient vector distribution measure (GVDM) is presented, which comprehensively integrates a synergistic homogeneity test based on Kolmogorov–Smirnov test with absolute difference standard deviation for gradient direction angle and regression analysis for gradient modulus. The proposed GVDM takes advantage of pixel-level gradient distribution property to further filtrate refined suspected targets. Moreover, gradient modulus horizontal local dissimilarity is proposed to measure the diversity of gradient modulus in horizontal direction between targets and waves, so as to enhance target saliency and suppress residual clutter simultaneously, which achieves preferable performance. Finally, a simple adaptive threshold is applied to confirm targets. Extensive experiments implemented on infrared maritime images with strong ocean waves demonstrate that the proposed method is superior to the state-of-the-art methods with respect to robustness and detection accuracy.

Infrared Small Target Detection Based on Gradient Correlation Filtering and Contrast Measurement

Infrared small target detection under complex backgrounds, especially in dense cloud and changeable clutter scenes, has always been a challenging research task. In order to improve the detection ability of small targets under complex backgrounds, an infrared small target detection method based on gradient correlation filtering and gradient contrast measurement (GCF-CM) is proposed in this paper. The infrared gradient vector field (IGVF) of the original image is first constructed through the facet model. Then, considering the unique gradient characteristics of small targets, a gradient correlation filtering (GCF) method is proposed to filter small targets and background clutters. Meanwhile, a gradient contrast measurement (GCM) method is designed to further enhance the intensity of the small target. Finally, after fusing the two response maps, an adaptive threshold is adopted to extract small targets. Experimental results demonstrate that the proposed method can improve the intensity of the small target and suppress clutter sufficiently. In comparison with other excellent methods, the proposed method exhibits a robust detection performance.

Geometry of weighted Lorentz–Finsler manifolds II: A splitting theorem

We show an analog of the Lorentzian splitting theorem for weighted Lorentz–Finsler manifolds: If a weighted Berwald spacetime of nonnegative weighted Ricci curvature satisfies certain completeness and metrizability conditions and includes a timelike straight line, then it necessarily admits a one-dimensional family of isometric translations generated by the gradient vector field of a Busemann function. Moreover, our formulation in terms of the [Formula: see text]-range introduced in our previous work enables us to unify the previously known splitting theorems for weighted Lorentzian manifolds by Case and Woolgar–Wylie into a single framework.

Graph-theoretic algorithms for Kolmogorov operators: approximating solutions and their gradients in elliptic and parabolic problems on manifolds

We employ kernel-based approaches that use samples from a probability distribution to approximate a Kolmogorov operator on a manifold. The self-tuning variable-bandwidth kernel method (Berry and Harlim in Appl Comput Harmon Anal 40(1):68–96, 2016) computes a large, sparse matrix that approximates the differential operator. Here, we use the eigendecomposition of the discretization to (i) invert the operator, solving a differential equation, and (ii) represent gradient vector fields on the manifold. These methods only require samples from the underlying distribution and, therefore, can be applied in high dimensions or on geometrically complex manifolds when spatial discretizations are not available. We also employ an efficient k-d tree algorithm to compute the sparse kernel matrix, which is a computational bottleneck.

Hybrid gradient vector fields for path-following guidance

Guidance path-planning and following are two core technologies used for controlling un-manned aerial vehicles (UAVs) in both military and civilian applications. However, only a few approaches treat both the technologies simultaneously. In this study, an innovative hybrid gradient vector fields for path-following guidance (HGVFs-PFG) algorithm is proposed to control fixed-wing UAVs to follow a generated guidance path and oriented target curves in three-dimensional space, which can be any combination of straight lines, arcs, and helixes as motion primitives. The algorithm aids the creation of vector fields (VFs) for these motion primitives as well as the design of an effective switching strategy to ensure that only one VF is activated at any time to ensure that the complex paths are followed completely. The strategies designed in earlier studies have flaws that prevent the UAV from following arcs that make its turning angle too large. The proposed switching strategy solves this problem by introducing the concept of the virtual way-points. Finally, the performance of the HGVFs-PFG algorithm is verified using a reduced-order autopilot and four representative simulation scenarios. The simulation considers the constraints of the aircraft, and its results indicate that the algorithm performs well in following both lateral and longitudinal control, particularly for curved paths. In general, the proposed technical method is practical and competitive.

Nonparametric Bayesian inference for reversible multidimensional diffusions

We study nonparametric Bayesian models for reversible multidimensional diffusions with periodic drift. For continuous observation paths, reversibility is exploited to prove a general posterior contraction rate theorem for the drift gradient vector field under approximation-theoretic conditions on the induced prior for the invariant measure. The general theorem is applied to Gaussian priors and p-exponential priors, which are shown to converge to the truth at the optimal nonparametric rate over Sobolev smoothness classes in any dimension.

Numerical investigation on thermal–hydraulic characteristics of the helically corrugated tube in intermediate heat exchangers for very-high-temperature gas-cooled reactors

A three-dimensional numerical simulation is conducted to investigate the flow and heat transfer characteristics of the helically corrugated tube (HCT) intermediate heat exchanger (IHX) for very-high-temperature gas-cooled reactors (VHTRs). Helium is the working fluid with Reynolds number (Re) ranging from 10,000 to 20,000 and Prandtl numbers of 0.66. The heat transfer enhancement mechanism is analyzed in detail with the use of the ANSYS Fluent 18.0 computational fluid dynamics program. The results show that the helical corrugation disrupts the boundary layer development, strengthens the secondary flow intensity, and improves the synergy between the velocity vector field and temperature gradient vector field, thus enhancing the heat transfer. Moreover, the effect of geometric parameters characterized by dimensionless corrugation height e/d (0.02, 0.04, and 0.06) and dimensionless corrugation pitch p/d (0.5, 1.0, and 1.5) on the thermal–hydraulic performance are examined. The results indicate that increasing e/d and reducing p/d can enhance heat transfer. The Nusselt number of HCTs is increased by 17.25 %-81.46 % compared to a smooth tube, while the friction factor is also increased by 38.01 %-361.69 %. The comprehensive performance represented by the performance evaluation criterion is improved by 3.26 %–11.84 %. In the present study, e/d = 0.04 and p/d = 0.5 are the optimal HCT structures to obtain better comprehensive performance for the IHX design.

Regularity of gradient vector fields giving rise to finite Caccioppoli partitions

For a finite set A subseteq mathbb {R}^n, consider a function u in mathrm {BV}_mathrm {loc}^2(mathbb {R}^n) such that nabla u in A almost everywhere. If A is convex independent, then it follows that u is piecewise affine away from a closed, countably {mathcal {H}}^{n - 1}-rectifiable set. If A is affinely independent, then u is piecewise affine away from a closed {mathcal {H}}^{n - 1}-null set.

Higher connectivity of the Morse complex

The Morse complex M ( Δ ) \mathcal {M}(\Delta ) of a finite simplicial complex Δ \Delta is the complex of all gradient vector fields on Δ \Delta . In this paper we study higher connectivity properties of M ( Δ ) \mathcal {M}(\Delta ) . For example, we prove that M ( Δ ) \mathcal {M}(\Delta ) gets arbitrarily highly connected as the maximum degree of a vertex of Δ \Delta goes to ∞ \infty , and for Δ \Delta a graph additionally as the number of edges goes to ∞ \infty . We also classify precisely when M ( Δ ) \mathcal {M}(\Delta ) is connected or simply connected. Our main tool is Bestvina–Brady Morse theory, applied to a “generalized Morse complex.”

Observer-based hybrid control for global attitude tracking on SO(3) with input quantisation

This paper studies the global attitude stabilization of a rigid body, a task that is subjected to topological obstacles. As a result of these obstructions, continuous feedbacks cannot globally stabilise the rigid body attitude. Therefore, this paper presents an observer-based hybrid controller to overcome these restrictions. Consequently, a new kind of synergistic potential function is designed which induces a gradient vector field to globally stabilise a given set. Moreover, the gradient of the proposed potential functions is utilised to derive a hybrid observer. Furthermore, this paper considers two types of constraints: angular velocity constraint and torque constraint. Afterward, these constraints are formulated in terms of the Linear Matrix Inequalities (LMI) optimisation problem to perform constraints satisfaction. Besides, this paper introduces a novel hybrid quantiser to deal with the problem of the low-price wireless network. Finally, a comparative study in simulations is provided to assess the performance of the proposed scheme.

Constraint optimization and quantum control landscapes

We develop the embedded gradient vector field method, introduced in [8, 9], for the case of the special unitary group regarded as a constraint submanifold of the unitary group . The optimization problem associated to the trace fidelity cost function defined on that appears in the context of quantum control landscapes is completely solved using the embedded gradient vector field method. We prove that for N ⩾ 5, the landscape is not -trap free, there are always kinematic local extrema that are not global extrema.

Unraveling the sequence of electron flow along the cyclocondensation reaction between ciprofloxacin and thiosemicarbazide through the bonding evolution theory

We have theoretically conducted a comprehensive investigation on the cyclocondensation reaction between ciprofloxacin and thiosemicarbazide at the MN15/6–311++G(d,p) level of approximation. In order to revisit and understand the sequence of electronic flow rearrangement, as described in terms of electron pair distribution, within the framework of Bonding Evolution Theory (BET) approach as provided by the application of Thom's elementary Catastrophe Theory (CT) to the changes, along the intrinsic reaction coordinate, of the gradient vector field of the electron localization function (ELF). This reaction has two channels (a and b) and each one takes place via three steps. The CDFT results show that ciprofloxacin and thiosemicarbazide have an electrophilic and nucleophilic characters respectively and therefore allowing this reaction to have a polar character. All the transition state (TS) of all reaction steps have been localized and characterized. In addition, the analysis of activation energy predicts the formation of ciprofloxacin thiosemicarbazone 3a (channels a) as a main product in good agreement with experimental outcomes. The BET analysis results along channel a reveal that the mechanism for each reaction step is divided into four structural stability domains. During the first step, a new N2–C2 bond occurs at the SSD-II, followed by a rupture of the H1–N2 single bond (SSD-III) illustrating the restoration of lone pairs of the N2 nitrogen atom, and finally, the formation of a new H1–O1 single bond. For the second step, the process involves the breaking of O1–C2 and N2–H3 bonds at the SSD-II and SSD-III, respectively, followed by the formation of O1–H3 bond at the SSD-IV. For the last step, it is noted the formation of C4–N8 bond at the SSD-II, followed by the breaking of N8–H9 and C4–O6 bonds simultaneously at the SSD-III with water elimination at the last domain (SSD-IV).