Let X be a divergence-free vector field defined on a closed, connected Riemannian manifold. In this paper, we show the equivalence between the following conditions: • X is a divergence-free vector field satisfying the shadowing property. • X is a divergence-free vector field satisfying the Lipschitz shadowing property. • X is an expansive divergence-free vector field. • X has no singularities a...

The purpose of this paper is to demonstrate that the index of a vector field on a smooth manifold is an intrinsic invariant of the manifold. The paper is divided into three sections. The first section introduces the Inverse Function Theorem and provides the proof for the Implicit Function Theorem, laying the foundation for our discussion. In the second section we will introduce the concept of a...

Phase-contrast magnetic-resonance angiography (PC MRA) generates images of vascular structures as three-dimensional maps of the blood-flow velocity in a volume of interest. To improve visualization methods for PC MRA, radiologists can benefit from image-processing algorithms that can classify flow and stationary tissue. In this paper, I describe a vector-difference distribution (VDD): a statist...

Experimental confirmation discussed the effect of the immediate surroundings of a pulse-powered toroidal coil on biological material which was placed in an environment without the influence of electromagnetic force.

We present a technique called ow surface probes for interactive visualization of 3D steady vector elds. The user places in the ow eld an arbitrarily-shaped seed surface, which is then deformed over time by the local or global 3D vector eld, producing an animation sequence. This sequence and individual frames are used to probe the behavior of the ow eld.

Trajectory visualization is an important tool to capture intrinsic characteristics of vector fields. However, often this class of geometric visualization suffers from visual clutter due to the number of curves that occlude each other and may cover relevant features. Attempts trying to avoid this effect often require complex pre/post-processing for seeding, clustering, or filtering of relevant l...

The parameter changes resulting from a combination of Lorentz transformation are shown to form vector field flows. The exact, finite Thomas rotation angle is determined and interpreted intuitively. Using phase portraits, the parameters evolution can be clearly visualized. In addition to identifying the fixed points, we obtain an analytic invariant, which correlates the evolution of parameters. ...

We present in this paper a covariant quantization of the “massive” vector field on de Sitter (dS) space based on analyticity in the complexified pseudo-Riemanian manifold. The correspondence between unitary irreducible representations of the de Sitter group and the field theory on de Sitter space-time is essential in our approach. We introduce the Wightman-Gärding axiomatic for vector field on ...

We report work in progress on measuring “forces” underlying the semantic drift by comparing it with plate tectonics in geology. Based on a brief survey of energy as a key concept in machine learning, and the Aristotelian concept of potentiality vs. actuality allowing for the study of energy and dynamics in language, we propose a field approach to lexical analysis. Until evidence to the contrary...