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We present an example of a monotone two-parameter family of vector fields on a torus whose bifurcation diagram we demonstrate to be in the class of 'simplest' diagrams proposed by Baesens and MacKay (2018 Nonlinearity 31 2928–81). This shows that the proposed class is realisable. [ABSTRACT FROM AUTHOR]

A two-way segmentation model is proposed in this article. The model is used to solve the problem that the objective contour can not be completely extracted from image due to occlusion between objects within similar image groups or image sequences. The proposed model first decomposes incomplete contours into sub-segments using local features identified by seed points set along each path. Then, locate the occluded part of the target object and reconstruct the target. Finally, a new vector field is generated based on the reconstructed object from the proposed model, followed by iterative evolution. The experimental results show that the proposed algorithm can better handle the problem of occlusion or misleading features of targets in composite images and medical images. Not only does it facilitate subsequent measurement and analysis, but it also preserves the original shape of the object during the segmentation process without prior information. It is worth noting that the accuracy of the proposed model is robust to our initialization strategy. [ABSTRACT FROM AUTHOR]

The objective of this paper is to introduce a derivative free projection method designed to find the singularities of pseudomonotone vector fields with convex constraints on Hadamard manifolds. This innovative approach combines the hyperplane projection method with a novel search direction. The global convergence of the proposed method is established under certain conditions. Our method improves some existing results in the literature on Hadamard manifolds. Additionally, illustrative numerical examples are provided to demonstrate the practical efficacy of our method. [ABSTRACT FROM AUTHOR]

Quantum chemical topology addresses the study of the chemical structure by applying the tools of differential topology to scalar and vector fields obtained by quantum mechanics. Here, the magnetically induced isotropically averaged Lorentz force density was computed and topologically analyzed for 11 small molecules. Critical points (attractors, repellers, and saddles) were determined and trajectories connecting the attractors computed. It is shown that kinds and numbers of the critical points are to some extent transferable in similar molecules. CC bonds of different orders are endowed with critical points of different kinds close to their center. The sum of topological indices of the isolated critical points is influenced by the presence of repellers on the outer part of the molecules. [ABSTRACT FROM AUTHOR]

We consider fixed boundary flows with canonical interpretability as principal components extended on nonlinear Riemannian manifolds. We aim to find a flow with fixed start and end points for noisy multivariate datasets lying near an embedded nonlinear Riemannian manifold. In geometric terms, the fixed boundary flow is defined as an optimal curve that moves in the data cloud with two fixed end points. At any point on the flow, we maximize the inner product of the vector field, which is calculated locally, and the tangent vector of the flow. The rigorous definition is derived from an optimization problem using the intrinsic metric on the manifolds. For random datasets, we name the fixed boundary flow the random fixed boundary flow and analyze its limiting behavior under noisy observed samples. We construct a high-level algorithm to compute the random fixed boundary flow, and provide the convergence of the algorithm. We show that the fixed boundary flow yields a concatenate of three segments, one of which coincides with the usual principal flow when the manifold is reduced to the Euclidean space. We further prove that the random fixed boundary flow converges largely to the population fixed boundary flow with high probability. Finally, we illustrate how the random fixed boundary flow can be used and interpreted, and demonstrate its application in real datasets. for this article are available online. [ABSTRACT FROM AUTHOR]

As a newly developing method, tool path generation based on vector fields often requires pre-partitioning of the surface, leading to non-smooth paths at the boundaries between surface segments. To tackle this challenge, we propose a novel global tool path planning method based on vector fields. Our approach eliminates the need for pre-partitioning of the surface and incorporates considerations to achieve uniform variance of the scallop height while ensuring path smoothness. As a result, our method enables the generation of smooth paths globally while simultaneously achieving length optimality. Experimental results demonstrate the effectiveness of our proposed method, as it achieves both smoothness and length optimality compared to state-of-the-art methods. [ABSTRACT FROM AUTHOR]

The methods of waypoint setting and point curve fitting which are not complex was usually adopted in the path planning of UAV, and the expected heading angle is given in segments. The expected heading angle does not automatically modified with changes in the route. The ability to automatically and quickly generate the expected heading and position is lacking. A vector field algorithm suitable for complex flight mission path planning is studied, and a simulation verification system to verify the effectiveness of its algorithm is developed in this paper. The concept of vector field is introduced in order to design a heading angle guidance law. An aircraft at any position can smoothly fly to the desired target path along the desired heading angle, and then complete the flight task along the desired target path. The implicit function equation is used to establish the desired path vector field, and the principle and method of path planning based on vector field are geometrically analyzed. According to the Lyapunov stability theory, the boundedness of the planned path error is proved. The distance error and the heading angle error can be stably converged to 0 with the help of the heading guidance law. Taking the path planning of the altitude hold mission for a quad tiltrotor UAV as an example, the desired path of smooth flight is planned using the vector field method and the piecewise transition strategy. The verification is conducted and some simulation results are given, which show that the vector field path planning algorithm is feasible and effective for the mission flight planning of an aircraft.

The main objective of this paper is to present a repository of image databases whose features are augmented with embedded vector field (VF) features. The repository is designed to provide the user with image databases that enhance machine learning (ML) classification. Also, six VFs are provided, and the user can embed them into her/his own image database with the help of software named ELPAC. Three of the VFs generate real-shaped singular points (SPs): springing, sinking, and saddle. The other three VFs generate seven kinds of SPs, which include the real-shaped SPs and four complex-shaped SPs: repelling and attracting (out and in) spirals and clockwise and counterclockwise orbits (centers). Using the repository, this work defines the locations of the SPs according to the image objects and the mappings between the SPs' shapes if separate VFs are embedded into the same image. Next, this paper produces recommendations for the user on how to select the most appropriate VF to be embedded in an image database so that the augmented SP shapes enhance ML classification. Examples of images with embedded VFs are shown in the text to illustrate, support, and validate the theoretical conclusions. Thus, the contributions of this paper are the derivation of the SP locations in an image; mappings between the SPs of different VFs; and the definition of an imprint of an image and an image database in a VF. The advantage of classifying an image database with an embedded VF is that the new database enhances and improves the ML classification statistics, which motivates the design of the repository so that it contains image features augmented with VF features. [ABSTRACT FROM AUTHOR]

From a fixed acute triangular base Δ A B C , all possible tetrahedra in three-dimensional real space are considered. The possible angles at the additional vertex P are shown to be bounded by certain inequalities, mostly linear inequalities. Together, these inequalities provide fairly tight bounds on the possible angle combinations at P. Four sets of inequalities are used for this purpose, though the inequalities in the first set are rather trivial. The inequalities in the second set can be established quickly, but do not seem to be known. The third and fourth set of inequalities are proved by studying scalar and vector fields on toroids. The first three sets of inequalities are linear in the angles at P, but the last set involves cosines of these angles. A generalization of the last two sets of inequalities is also proved, using the Poincaré–Hopf Theorem. Extensive testing of these results has been done using Mathematica and C++. The C++ code for this is listed in an appendix. While it has been demonstrated that the inequalities bound the possible combinations of angles at P, the results also reveal that additional inequalities, in particular linear inequalities, exist that would provided tighter bounds. [ABSTRACT FROM AUTHOR]

The problem of constructing n-linear (n ≥ 2) plane vector fields with an isolated critical point and given separatrices of prescribed types is considered. Such constructions are based on the use of vector algebra, the qualitative theory of second-order dynamic systems and classical methods for investigating their critical points. This problem is essentially an inverse problem of the qualitative theory of ordinary differential equations, and its solution can be used to synthesize mathematical models of controlled dynamical systems of various physical nature. [ABSTRACT FROM AUTHOR]

In this paper, a three-dimensional vector field model is proposed, whose dimensions are the Hansen Solubility Parameters: dispersion parameter (δD), polarity parameter (δP), and hydrogen bonding parameter (δH). The vector space that defines the field has the peculiarity of having a dispersion vector with a magnitude of 2 as its base vector, while the polarity and hydrogen bonding vectors have a magnitude of 1. A substance is characterised as a position vector, and the interaction between two substances is determined by calculating the vector difference of both, known as the interaction vector. The interaction among substances may involve solubility, swelling, cracking, surface tension, interface tension, and any physical phenomena where the intermolecular energies of dispersion, polarity or hydrogen bonding come into play. This paper studies free surface energy (surface and interfacial tension). It has been found that free surface energy is directly proportional to the square of the magnitude of the interaction vector. The proportionality constant, τ, is expressed in length units, has a value of 0.025 nm, and does not depend on the chemical nature of the substance or state of matter (solid, liquid or gas). The constant value τ appears universal and aligns with the thickness of interfaces, thereby supporting Guggenheim's hypothesis. This hypothesis asserts that interfaces possess actual thickness and are not merely mathematical surfaces, as originally postulated by Gibbs. Moreover, it also has been found that the interface thickness, τ, is approximately equal to half of the Bohr radius, a0, which is defined by universal constants. Because the solubility parameters of thousands of substances are known and can be easily determined from their molecular structure, a good approximation of the surface and interfacial tension of any given substance can now be calculated. It has also been found that the contact angles of sessile droplets in three-phased systems can be calculated from the interaction vectors of the implicated substances. [ABSTRACT FROM AUTHOR]

Quantum chemical topology addresses the study of the chemical structure by applying the tools of differential topology to scalar and vector fields obtained by quantum mechanics. Here, the magnetically induced isotropically averaged Lorentz force density was computed and topologically analyzed for 11 small molecules. Critical points (attractors, repellers, and saddles) were determined and trajectories connecting the attractors computed. It is shown that kinds and numbers of the critical points are to some extent transferable in similar molecules. CC bonds of different orders are endowed with critical points of different kinds close to their center. The sum of topological indices of the isolated critical points is influenced by the presence of repellers on the outer part of the molecules.

The limit cycles have a main role in understanding the dynamics of planar differential systems, but their study is generally challenging. In the last few years, there has been a growing interest in researching the limit cycles of certain classes of piecewise differential systems due to their wide uses in modeling many natural phenomena. In this paper, we provide the upper bounds for the maximum number of crossing limit cycles of certain classes of discontinuous piecewise differential systems (simply PDS) separated by a straight line and consequently formed by two differential systems. A linear plus cubic polynomial forms six families of Hamiltonian nilpotent centers. First, we study the crossing limit cycles of the PDS formed by a linear center and one arbitrary of the six Hamiltonian nilpotent centers. These six classes of PDS have at most one crossing limit cycle, and there are systems in each class with precisely one limit cycle. Second, we study the crossing limit cycles of the PDS formed by two of the six Hamiltonian nilpotent centers. There are systems in each of these 21 classes of PDS that have exactly four crossing limit cycles. [ABSTRACT FROM AUTHOR]

In this paper, we study generalized η-Ricci solitons with respect to the Schouten-van Kampen connection on trans-Sasakian manifolds. We give an example of generalized η-Ricci solitons on a trans-Sasakian manifold with respect to the Schoutenvan Kampen connection to prove our results. [ABSTRACT FROM AUTHOR]

Albeverio, Kondratiev, and Röckner have introduced a type of differential geometry, which we call lifted geometry, for the configuration space ΓX of any manifold X. The name comes from the fact that various elements of the geometry of ΓX are constructed via lifting of the corresponding elements of the geometry of X. In this note, we construct a general algebraic framework for lifted geometry which can be applied to various "infinite dimensional spaces" associated to X. In order to define a lifted geometry for a "space", one dose not need any topology or local coordinate system on the space. As example and application, lifted geometry for spaces of Radon measures on X, mappings into X, embedded submanifolds of X, and tilings on X, are considered. The gradient operator in the lifted geometry of Radon measures is considered. Also, the construction of a natural Dirichlet form associated to a random measure is discussed. It is shown that Stokes' Theorem appears as "differentiability" of "boundary operator" in the lifted geometry of spaces of submanifolds. It is shown that (generalized) action functionals associated with Lagrangian densities on X form the algebra of smooth functions in a specific lifted geometry for the path-space of X. [ABSTRACT FROM AUTHOR]

In this paper, a three-dimensional vector field model is proposed, whose dimensions are the Hansen Solubility Parameters: dispersion parameter (δD), polarity parameter (δP), and hydrogen bonding parameter (δH). The vector space that defines the field has the peculiarity of having a dispersion vector with a magnitude of 2 as its base vector, while the polarity and hydrogen bonding vectors have a magnitude of 1. A substance is characterised as a position vector, and the interaction between two substances is determined by calculating the vector difference of both, known as the interaction vector. The interaction among substances may involve solubility, swelling, cracking, surface tension, interface tension, and any physical phenomena where the intermolecular energies of dispersion, polarity or hydrogen bonding come into play. This paper studies free surface energy (surface and interfacial tension). It has been found that free surface energy is directly proportional to the square of the magnitude of the interaction vector. The proportionality constant, τ, is expressed in length units, has a value of 0.025 nm, and does not depend on the chemical nature of the substance or state of matter (solid, liquid or gas). The constant value τ appears universal and aligns with the thickness of interfaces, thereby supporting Guggenheim’s hypothesis. This hypothesis asserts that interfaces possess actual thickness and are not merely mathematical surfaces, as originally postulated by Gibbs. Moreover, it also has been found that the interface thickness, τ, is approximately equal to half of the Bohr radius, a0, which is defined by universal constants. Because the solubility parameters of thousands of substances are known and can be easily determined from their molecular structure, a good approximation of the surface and interfacial tension of any given substance can now be calculated. It has also been found that the contact angles of sessile droplets in three-phased systems can be calculated from the interaction vectors of the implicated substances.

In many applications, such as atmospheric observation or disaster monitoring, cooperative control of a fleet of UAVs is crucial because it is effective in repeated tasks. In this work, we provide a workable and useful cooperative guiding algorithm for several fixed-wing UAVs to construct a path-following formation with communication delays. The two primary components of our concept are path-following (lateral guidance) and path formation (longitudinal guidance). The former is in charge of ensuring that, in the presence of wind disturbance, the lateral distance between the UAV and its targeted path converges using a well-known vector field technique. In the event of a communication delay, the latter ensures that several fixed-wing UAVs will create a predetermined formation shape. Furthermore, we provide a maximum delay bound that is dependent on the topology and a controller’s gain. Lastly, in order to confirm the viability and advantages of our suggested approach, we construct an effective platform for a hardware-in-the-loop (HIL) test.

Abstract Despite the continued efforts, a batch-insensitive tool that can both infer and predict the developmental dynamics using single-cell genomics is lacking. Here, I present scTour, a novel deep learning architecture to perform robust inference and accurate prediction of cellular dynamics with minimal influence from batch effects. For inference, scTour simultaneously estimates the developmental pseudotime, delineates the vector field, and maps the transcriptomic latent space under a single, integrated framework. For prediction, scTour precisely reconstructs the underlying dynamics of unseen cellular states or a new independent dataset. scTour’s functionalities are demonstrated in a variety of biological processes from 19 datasets.

In this paper, we discuss the concept of an analytic prolongation of a local Riemannian metric. We propose a generalization of the notion of completeness realized as an analytic prolongation of an arbitrary Riemannian metric. Various Riemannian metrics are studied, primarily those related to the structure of the Lie algebra 픤 of all Killing vector fields for a local metric. We introduce the notion of a quasi-complete manifold, which possesses the property of prolongability of all local isometries to isometries of the whole manifold. A classification of pseudo-complete manifolds of small dimensions is obtained. We present conditions for the Lie algebra of all Killing vector fields 픤 and its stationary subalgebra 픥 of a locally homogeneous pseudo-Riemannian manifold under which a locally homogeneous manifold can be analytically prolonged to a homogeneous manifold. [ABSTRACT FROM AUTHOR]

With the rapid growth of trajectory big data, there is a need for more efficient methods to extract, analyze, and visualize these data. However, existing research on trajectory big data visualization mainly focuses on displaying trajectories for a specific period or showing spatial distribution characteristics of trajectory points in a single time slice using clustering, filtering, and other techniques. Therefore, this paper proposes a vector field visualization model for trajectory big data, aiming to effectively represent the inherent movement trends in the data and provide a more intuitive visualization of urban traffic congestion trends. The model utilizes the motion information of vehicles to create a travel vector grid and employs WebGL technology for vector field visualization rendering. The vector field effects are effectively displayed by generating many particles and simulating their movements. Furthermore, this research also designs and implements congestion trend point identification and hotspot congestion analysis, thus validating the practicality and effectiveness of trajectory big data vector field visualization. The results indicate that compared to traditional visualization methods, the vector field visualization method can demonstrate the direction and density changes in traffic flow and predict future traffic congestion. This work provides valuable data references and decision support for urban traffic management and planning. [ABSTRACT FROM AUTHOR]

We design an all-dielectric silicon metasurface adopting an elliptical-cylindrical unit structure. In the case of circularly polarized light incidence, we can obtain terahertz vector Bessel beams with radial or azimuthal polarization characteristics. We demonstrate that the spatial position polarization control of the beam can be achieved by rotating the unit structure with quarter-wave plate function. In addition, the transmission phase is used to offset the extra phase brought by the polarization control, and the function of converting the left-handed or right-handed circularly polarized incident light into a radially or azimuthally polarized Bessel beam is successfully realized at the frequency of 0.71 THz. This research provides feasibility for the generation of vector Bessel beams in the terahertz band. [ABSTRACT FROM AUTHOR]

In this paper, trajectory of time changing along a helical line is represented by parametric equations in Cartesian coordinates of Euclidean space. On the basis of a cycloidal sweep of a cylindrical surface onto a plane, analytical form of a helix is determined. On its basis, integral surface is determined, which is called the periodic characteristic of the diagonal differentiation operator and its connection with its linear characteristic is established. a) elements of new approach related to the periodic characteristic of diagonal differentiation operator are proposed, b) method for reducing integral along the periodic characteristic to an integral with linear characteristic, c) conditions establishing structure of the integral as sum of linear and multiperiodic functions. Some consequences of these results and recommendations of an algorithmic nature for further expansion of research in this direction are given. [ABSTRACT FROM AUTHOR]

Homoclinic tangencies and singular hyperbolicity are involved in the Palis conjecture for vector fields. Typical three dimensional vector fields are well understood by recent works. We study the dynamics of higher dimensional vector fields that are away from homoclinic tangencies. More precisely, we prove that for any dimensional vector field that is away from homoclinic tangencies, all singularities contained in its robustly transitive singular set are all hyperbolic and have the same index. Moreover, the robustly transitive set is C 1 -generically partially hyperbolic if the vector field cannot be accumulated by ones with a homoclinic tangency. [ABSTRACT FROM AUTHOR]

The Laplacian on the Heisenberg group as a sum of squares of the standard vector fields on the Heisenberg group is first made explicit to be a Laplace–Beltrami operator on the Heisenberg group, which is well known to be a Riemannian manifold. The dissipation of the heat generated by the Laplacian on the Heisenberg group is then given in terms of estimates reminiscent of the Hausdorff–Young inequalities for the Euclidean Fourier transform. The evolution of the quantum wave driven by the Laplacian on the Heisenberg group is given. Weyl transforms, which are pseudo-differential operators in disguise, are instrumental in deriving the explicit formula for the heat kernel of the Laplacian on the Heisenberg group. [ABSTRACT FROM AUTHOR]

Let $ (M, g) $ be an $ n $-dimensional (pseudo-)Riemannian manifold and $ TM $ be its tangent bundle $ TM $ equipped with the complete lift metric $ ^{C}g $. First, we define a Ricci quarter-symmetric metric connection $ \overline{\nabla } $ on the tangent bundle $ TM $ equipped with the complete lift metric $ ^{C}g $. Second, we compute all forms of the curvature tensors of $ \overline{\nabla } $ and study their properties. We also define the mean connection of $ \overline{\nabla } $. Ricci and gradient Ricci solitons are important topics studied extensively lately. Necessary and sufficient conditions for the tangent bundle $ TM $ to become a Ricci soliton and a gradient Ricci soliton concerning $ \overline{\nabla } $ are presented. Finally, we search conditions for the tangent bundle $ TM $ to be locally conformally flat with respect to $ \overline{\nabla } $.

Let be an -dimensional (pseudo-)Riemannian manifold and be its tangent bundle equipped with the complete lift metric . First, we define a Ricci quarter-symmetric metric connection on the tangent bundle equipped with the complete lift metric . Second, we compute all forms of the curvature tensors of and study their properties. We also define the mean connection of . Ricci and gradient Ricci solitons are important topics studied extensively lately. Necessary and sufficient conditions for the tangent bundle to become a Ricci soliton and a gradient Ricci soliton concerning are presented. Finally, we search conditions for the tangent bundle to be locally conformally flat with respect to . [ABSTRACT FROM AUTHOR]

We use the theory of calibrations to write an equation of a minimal volume vector field on a given Riemann surface. [ABSTRACT FROM AUTHOR]

In this paper we propose and justify a new method calculation of the mapping degree of n‑dimensional vector field on the unit sphere of the space , . The essence of the proposed method is that the calculation of the mapping degree of vector field is reduced to the calculation of the mapping degree of its tangent component on the components of the set, where the vector field has an obtuse angle with the unit vector field. In the special case, for the gradient of a smooth positively homogeneous function, we derive a formula for calculation of the mapping degree through the Eulerian characteristic of the set of points where the function is negative. [ABSTRACT FROM AUTHOR]

Unless another thing is stated one works in the C∞ category and manifolds have empty boundary. Let X and Y be vector fields on a manifold M. We say that Y tracks X if [Y, X] = fX for some continuous function f: M→ R. A subset K of the zero set Z(X) is an essential block for X if it is non-empty, compact, open in Z(X) and its Poincaré-Hopf index does not vanishes. One says that X is non-flat at p if its ∞-jet at p is non-trivial. A point p of Z(X) is called a primary singularity of X if any vector field defined about p and tracking X vanishes at p. This is our main result: consider an essential block K of a vector field X defined on a surface M. Assume that X is non-flat at every point of K. Then K contains a primary singularity of X. As a consequence, if M is a compact surface with non-zero characteristic and X is nowhere flat, then there exists a primary singularity of X.

Objective Aiming at the cluster control problem of small underactuated autonomous underwater vehicles (AUVs), a formation control strategy based on an improved RRT* algorithm is designed.MethodPaths planned by the RRT* algorithm are steep and difficult to track, with slow convergence speed, so an improved method is proposed to solve the above problems. First, a deviation function is added to bring the random sampling points closer to the target point, then the sampling points are connected smoothly using a Dubins curve. By rerouting within the variable radius range and designing the cost function in relation to the curve length and obstacle avoidance, the best path is chosen. According to the cost and minimum value, multiple AUVs are assigned a rendezvous point, and the speed of multiple AUVs is coordinated to complete the minimum rendezvous time constraint. A segmented vector field construction method based on the Dubins path is then designed, enabling multiple AUVs to track the planned path and reach the target rendezvous point with the direction remaining the same.ResultsThe simulation results show that the average path length of multiple AUV formations is shortened by 26.6% and the average assembly time is shortened by 21.7%. ConclusionThe improved algorithm proposed herein has high path planning quality and can successfully complete formation assembly tasks.

Rock fracturing is often accompanied by electromagnetic phenomenon. As a vector field, in addition to the intensity that is widely concerned, the generated electromagnetic field also has obvious directionality. To this end, a set of electromagnetic antennas capable of simultaneous three-axis measurement is used to monitor the electromagnetic vector field generated from rock fracturing based on Brazilian tests. The signal amplitude on each axis can represent the magnitude of actual magnetic flux density component on the three axes. The intensity and directional characteristics of electromagnetic signals received at different positions are studied using vector synthesis. The directionality of electromagnetic radiation measured using a three-axis electromagnetic antenna shows that the direction of the magnetic flux intensity generated by rock fracturing tends to be parallel to the crack surface, and the measured signal intensity is greater in a direction closer to the crack surface.

Our purpose is to study the variational inequality problem for a vector field on Hadamard spaces. The existence and uniqueness of the solutions to the variational inequality problem associated with a vector field in Hadamard spaces are studied.

In a 4-dimensional real space, we discuss linearly homogeneous hypersurfaces that are the orbits of 3-dimensional solvable Lie algebras. A complete list of inequivalent representations of a decomposable 3-dimensional algebra in space is given and a list of all (up to linear transformations) 3-dimensional orbits of such representations. Most of the resulting hypersurfaces have trivial affine stabilizers and are new in the problem of describing affinely homogeneous varieties. [ABSTRACT FROM AUTHOR]

With the rapid development of digital earth, there are higher demands for the virtual of true feeling marine environments and the reveal of the rules hidden in multi-source data. First, an efficient 3D Delaunay algorithm which improves the speed by establishing relative relationship of nodes and optimizing the new tetrahedral construction is proposed to visualize the large scale dataset of scalar field, then, an adaptive visualization method to resolve the problem of overlapping effects caused by excessive noise or omitting important details caused by less noise is put forward to visualize the vector field dataset, finally, the integration of scalar field dataset and vector field dataset are visualized by the presented fusion platform. The main contributions are: (1) the improved 3D Delaunay algorithm can visualize marine temperature dataset efficiently. (2) the adaptive LIC visualization method can visualize flowing cloud data and storm surge data legibly. (3) the proposed fusion platform can visualize temperature field data and wind field data perfectly. [ABSTRACT FROM AUTHOR]

We propose a new symmetry reduction method for (1+1)-dimensional differential-difference equations (DDEs), namely, the λ-symmetry reduction method of solving ordinary differential equations is generalized to DDEs. Order-reduction processes are a consequence of the invariance of the given DDE under vector fields of the new class. These vector fields satisfy a new prolongation formula. A simple example of order-reduction is provided to illustrate the application.

Solar flare forecasting is one of major components of operational space weather forecasting. Complex active regions (ARs) are the main source producing major flares, but only a few studies are carried out to establish flare forecasting models for these ARs. In this study, four deep learning models, called Complex Active Region Flare Forecasting Model (CARFFM)-1, −2, −3, and −4, are established. They take AR longitudinal magnetic fields, AR vector magnetic fields, AR longitudinal magnetic fields and the total unsigned magnetic flux in the neutral line region, AR vector magnetic fields and the total unsigned magnetic flux in the neutral region as input, respectively. These four models can predict the production of M-class or above flares in the complex ARs for the next 48 h. Through comparing the performance of the models, CARFFM-4 has the best forecasting ability, which has the most abundant input information. It is suggested that more valuable and rich input can improve the model performance.

The limit cycles have a main role in understanding the dynamics of planar differential systems, but their study is generally challenging. In the last few years, there has been a growing interest in researching the limit cycles of certain classes of piecewise differential systems due to their wide uses in modeling many natural phenomena. In this paper, we provide the upper bounds for the maximum number of crossing limit cycles of certain classes of discontinuous piecewise differential systems (simply PDS) separated by a straight line and consequently formed by two differential systems. A linear plus cubic polynomial forms six families of Hamiltonian nilpotent centers. First, we study the crossing limit cycles of the PDS formed by a linear center and one arbitrary of the six Hamiltonian nilpotent centers. These six classes of PDS have at most one crossing limit cycle, and there are systems in each class with precisely one limit cycle. Second, we study the crossing limit cycles of the PDS formed by two of the six Hamiltonian nilpotent centers. There are systems in each of these 21 classes of PDS that have exactly four crossing limit cycles.

In this paper, a model of a pulsating universe is investigated with periodic function of the scale factor together with a homogeneous, isotropic and spatially flat Friedman-Robertson-Walker metric. The parameters of the pulsating universe model coincide with observation parameters of the modern universe, taking into account the different contributions of various types of matter to its evolution. In the resulting solution, during periods of accelerated expansion of the universe, the scalar field with negative pressure makes a greater contribution to the expansion growth. The fermionic field and the vector field, which have positive pressure values during this period, prevent the growth of this expansion. During periods of compression, the fermionic field and the vector field have a greater influence on the dynamics of the evolution of the universe. In the resulting periodic solution, the universe goes through endless cycles of expansion and contraction. The observed modern accelerated expansion of the universe corresponds to one of these cycles. Cosmographic parameters are calculated - parameters of deceleration q, jerk j and snap s and energy conditions for the model under research. These parameters make it possible to link model-independent results obtained from cosmography with theoretically sound assumptions of gravity. [ABSTRACT FROM AUTHOR]