Validity of running criterion

This paper describes an approach for interactive visualization of mixed scalar and vector fields, in which vector icons are generated from pre-voxelized icon templates and volume-rendered together with the volumetric scalar data. This approach displays simultaneously the global structure of the scalar field and the detailed features of the vector field. Interactive visualization is achieved with incremental image update, by re-rendering only a small portion of the image wherever and whenever a change occurs. This technique supports a set of interactive visualization tools, including change of vector field visualization parameters, real-time animation of vector icons advected within the scalar field, a zooming lens, and a local probe. Subject Terms: data visualisation; interactive systems; graphical user interfaces; rendering (computer graphics); interactive visualization; scalar fields; vector fields; vector icons; pre-voxelized icon templates; volume-rendered; volumetric scalar data; incremental image update; real-time animation; zooming lens; local probe

Feature-aware streamline generation of planar vector fields via topological methods

We discuss new models of an “affine” theory of gravity in multidimensional space-times with symmetric connections. We use and develop ideas of Weyl, Eddington, and Einstein, in particular, Einstein’s proposal to specify the space-time geometry by the use of the Hamilton principle. More specifically, the connection coefficients are determined using a “geometric” Lagrangian that is an arbitrary function of the generalized (nonsymmetric) Ricci curvature tensor (and, possibly, of other fundamental tensors) expressed in terms of the connection coefficients regarded as independent variables. Such a theory supplements the standard Einstein gravity with dark energy (the cosmological constant, in the first approximation), a neutral massive (or tachyonic) vector field (vecton), and massive (or tachyonic) scalar fields. These fields couple only to gravity and can generate dark matter and/or inflation. The new field masses (real or imaginary) have a geometric origin and must appear in any concrete model. The concrete choice of the geometric Lagrangian determines further details of the theory, for example, the nature of the vector and scalar fields that can describe massive particles, tachyons, or even “phantoms.” In “natural” geometric theories, which are discussed here, dark energy must also arise. We mainly focus on intricate relations between geometry and dynamics while only very briefly considering approximate cosmological models inspired by the geometric approach.

This paper discusses the reconstruction of the two-dimensional (2D) vector field topology (VFT) from a 2D scalar potential field and vice versa. The physical foundation of the proposed reconstruction method is the convection-type equation coupling a 2D vector field (e.g., skin friction) with a 2D potential field (e.g., surface pressure, temperature, or scalar concentration) in viscous flows. To reconstruct the VFT, a variational method is applied to this inverse problem, and then, an approximate method is proposed based on the linear superposition of some elemental potential field structures with simple analytical forms (source, vortex, saddle, etc.). As examples, the proposed method is applied to swept shock-wave/boundary-layer interaction and near-wall turbulence. Furthermore, in a reversed process to reconstruct a 2D potential field from a 2D vector field, a similar variational method is applied, and an approximate method with a constant source term in the convection-type relation is proposed, which is particularly applicable to reconstruction of a surface pressure field from global skin friction measurements in aerodynamics experiments. The significance of this work is that the complex VFT can be reconstructed based on a scalar potential field by using a semi-analytical approach. The proposed method can be used in fluid mechanics and other disciplines such as computer graphics and data visualization.

Recent advances in vector field topologymake it possible to compute its multi-scale graph representations for autonomous 2D vector fields in a robust and efficient manner. One of these representations is a Morse Connection Graph (MCG), a directed graph whose nodes correspond to Morse sets, generalizing stationary points and periodic trajectories, and arcs - to trajectories connecting them. While being useful for simple vector fields, the MCG can be hard to comprehend for topologically rich vector fields, containing a large number of features. This paper describes a visual representation of the MCG, inspired by previous work on graph visualization. Our approach aims to preserve the spatial relationships between the MCG arcs and nodes and highlight the coherent behavior of connecting trajectories. Using simulations of ocean flow, we show that it can provide useful information on the flow structure. This paper focuses specifically on MCGs computed for piecewise constant (PC) vector fields. In particular, we describe extensions of the PC framework that make it more flexible and better suited for analysis of data on complex shaped domains with a boundary. We also describe a topology simplification scheme that makes our MCG visualizations less ambiguous. Despite the focus on the PC framework, our approach could also be applied to graph representations or topological skeletons computed using different methods.

: Interactive visualization allows scientists and engineers to analyze massive data sets in new ways. They can prune useless data, identify important features, and see the science behind the data. We address computational tasks of image generation, dynamical visualization, and interactive manipulation and quantitative querying. The data sets comprise of scalar fields (volumetric histological scans of spinal cord injuries), vector fields (numerical simulation of turbulent flows) and tensor fields (Reynolds stresses produced by vortices in the flow). Image generation is the core of visualization. Dynamical visualization brings out time dependent features that are hard to recognize in static images. Interactive manipulation and querying allows users to focus and quantify key features in large data sets.

It is shown that ``${B}^{\ensuremath{\wedge}}F$'' theory in 3+1 dimensions with a Kalb-Ramond field is related by Buscher's duality transformation to two different versions of massive gauge-invariant spin-one theories of St\"uckelberg-type, one involving vector and scalar fields and the other antisymmetric tensor and vector fields. A similar construction for massive spin-zero theory is also shown. The implication of this equivalence to the five-dimensional theories from which these theories can be obtained is discussed.

This paper reports the simulated results of modified anhysteretic remanence (MAR) in rotating vectorial AC bias fields, by a two-axis vibrating sample magnetometer (VSM). The effects of perpendicular component and rotation of the head fields on the MAR are clarified. It is found that a rotating vectorial AC bias head field needs about 1.2 times smaller magnitude than a scalar AC bias head field to acquire the same level of MAR, and that MAR in vectorial fields has a large perpendicular component. The effects on MAR of the sample orientation and the direction of measurement with respect to the sample shape are also discussed.

Topological abstractions offer a method to summarize the behavior of vector fields, but computing them robustly can be challenging due to numerical precision issues. One alternative is to represent the vector field using a discrete approach, which constructs a collection of pairs of simplices in the input mesh that satisfies criteria introduced by Forman's discrete Morse theory. While numerous approaches exist to compute pairs in the restricted case of the gradient of a scalar field, state-of-the-art algorithms for the general case of vector fields require expensive optimization procedures. This paper introduces a fast, novel approach for pairing simplices of two-dimensional, triangulated vector fields that do not vary in time. The key insight of our approach is that we can employ a local evaluation, inspired by the approach used to construct a discrete gradient field, where every simplex in a mesh is considered by no more than one of its vertices. Specifically, we observe that for any edge in the input mesh, we can uniquely assign an outward direction of flow. We can further expand this consistent notion of outward flow at each vertex, which corresponds to the concept of a downhill flow in the case of scalar fields. Working with outward flow enables a linear-time algorithm that processes the (outward) neighborhoods of each vertex one-by-one, similar to the approach used for scalar fields. We couple our approach to constructing discrete vector fields with a method to extract, simplify, and visualize topological features. Empirical results on analytic and simulation data demonstrate drastic improvements in running time, produce features similar to the current state-of-the-art, and show the application of simplification to large, complex flows.

The theory of a neutral vector field is studied by introducing an auxiliary scalar field interacting with it. The vector field propagator of the theory is the same as that obtained by Yang and Lee using the $\ensuremath{\xi}$-limiting process. The parameter $\ensuremath{\xi}$ now enters as an interaction parameter. We show, further, that it is a gauge parameter and the physical content of the theory is independent of its value. Using a canonical quantization scheme, it is found that the contribution of the scalar field to the energy of the system is negative, and one is forced to introduce an indefinite metric. To have a positive norm of the states for arbitrary interactions involving production of these hypothetical scalar particles, the values of $\ensuremath{\xi}$ have to be restricted so that the energy of the system is less than ${m}^{2}{\ensuremath{\xi}}^{\ensuremath{-}1}$, $m$ being the mass of the vector meson. Most of the formalism can be easily extended to the theory of charged vector mesons. The main purpose of the theory is to investigate the possibility of renormalizing the interactions of the vector field with currents whose divergences do not vanish. It is found that an extra term in the interaction Lagrangian makes the auxiliary-field equation independent of the current and hence of the additional interaction. It is shown that the $S$ matrix cannot produce these particles and that there need be no restrictions on the parameter $\ensuremath{\xi}$ for this case so long as one deals with initial states having no scalar mesons.

Cosmological implication of a generalized model of two scalar and two vector fields, in which both scalar fields are non-minimally coupled to each vector field, is studied in this paper. In particular, we will seek a set of new anisotropic power-law inflationary solutions to this model. Additionally, the stability of the obtained solutions will be examined by using the dynamical system approach. As a result, we will show that this set of solutions turns out to be stable and attractive during the inflationary phase as expected due to the existence of the unusual couplings between two scalar and two vector fields. Notably, we will point out that the existence of phantom field will lead to an instability of the corresponding anisotropic power-law inflation.

We present a lattice Boltzmann algorithm for simulating magnetohydrodynamics, and extend it to simulate the Jeffery equation that describes the rotating orientations of axisymmetric particles in a dilute suspension. Both systems involve material vector fields that evolve through the curl of another vector field. Both systems thus require an underlying kinetic formulation using vector fields, in contrast to the scalar fields used in the Boltzmann equation, and in lattice Boltzmann algorithms for hydrodynamics. Simulating Jeffery’s equation requires extra gradient terms that cannot be written in conservation form. These gradients are obtained locally at grid points using the non-equilibrium parts of the kinetic vector fields representing the particle orientations, and the kinetic scalar fields representing the suspending fluid. The kinetic formulation is discretised using a Strang splitting between advection to neighbouring grid points and local algebraic operations at grid points.

The calculus of differential forms is used to discuss the local integration theory of a general set of autonomous first order ordinary differential equations. Geometrically, such a set is a vector field ${\bf V}$ in the space of dependent variables. Integration consists of seeking associated geometric structures invariant along ${\bf V}$: scalar fields (first integrals), forms (such as symplectic tensors), vectors (CT generators, Lie’s invariance generators), integrals over subspaces (which are not limited to Poincaré’s absolute and relative invariants). Most classical results are easily understood as simple relations between these. It is shown first that to any field ${\bf V}$ can be associated a Hamiltonian structure of forms (if, when dealing with an odd number of dependent variables, an arbitrary equation of constraint is also added). Families of integral invariants are an immediate consequence. Poisson brackets are isomorphic to Lie products of associated CT-generating vector fields. Hamilton’s variational principle follows from the fact that the maximal regular integral manifolds of a closed set of forms (here a Hamiltonian structure) must include the characteristics of the set (here the trajectories of ${\bf V}$). Auxiliary vector fields are also carefully discussed: the hierarchy of invariance generators of (i) the vector field ${\bf V}$, of (ii) its trajectories, of (iii) any associated Hamiltonian structure of forms. An Appendix summarizes the manipulation of vectors and forms, as a tool for applied mathematics.

Multi-component seismic data contain a great deal of vector field information that reflects the situation of the underground medium. However, the processing methods used for multi-component seismic data are still being developed, and effectively retaining and using this information is the difficulty and the focus of the task. Currently, the main-stream processing techniques of multi-component seismic data treat the individual components independently as a scalar field; in this way, they do not excavate the vector features of the wavefield, thus restricting the potential utilities of the effective information. Research into processing methods that are suitable for use with the vector field, which can better retain and use the orientations and the relative amplitude relationship between multi-component seismic data, is urgently needed and represent an important direction for the current development of multi-component seismic data processing techniques. In this paper, we introduce and summarize several existing vector pre-processing techniques, including polarization filtering, de-noising using vector order statistics, group sparse representation, and vector separation of compressional waves and shear waves, to help scholars develop more effective vector field processing methods and to promote the development of vector processing techniques for multi-component seismic data.

In shallow water exist the stable and significant interference characteristics of low frequency sound propagation, which contain the information of the sound source state and waveguide peculiarity. A simplified mapping method for describing the scalar and vector sound field interference structure radiated by a moving target, and an indicatory mechanism of the target state implicated in the energy distribution of the mapping domain are investigated in this paper. The mapping characteristics of two-dimensional Fourier transform of the vector sound field time (space) frequency interference spectrum are analyzed theoretically. Relations among waveguide invariant, range-rate, heading angle, and energy ridge slope of the mapping domain for time-frequency interference spectrum produced by a uniformly moving target are derived. Indication of target attacking or moving away, and the degree of threatening through symbols or the absolute value change of mapping domain’s ridge slope are demonstrated. Then numerical simulation and sea trial research are carried out. Experimental results with theoretical analysis and simulation results are in good agreement with each other. Research results show that the scalar and vector field time (space) frequency interference structure can be simplified by the two-dimensional Fourier transform. The mapping domain ridges, range-rate, heading angle and waveguide invariant show an analytic relationship among them. Variation embodied in the form of scalar and vector field interference structure obtained after mapping are more consistent with each other. The ridge of mapping domain can indicate the moving state of target concisely.