Curvature and helicity topological bounds for the magnetic energy of the streamlines magnetic structures of a kinematic dynamo flow are computed. The existence of the filament dynamos are determined by solving the magnetohydrodynamic equations for 3D flows and the solution is used to determine these bounds. It is shown that in the limit of zero resistivity filamentary dynamos always exists in t...

In 1920’s Marston Morse developed what is now known as Morse theory trying to study the topology of the space of closed curves on S2, see [7] and [5]. 80 years later a very similar problem about the topology of the space of closed and locally convex (i.e. without inflection points) curves on S2 is still widely open. The main difficulty is the absence of the covering homotopy principle for the m...

Abstract In this study, the position vector of a timelike curve $$\wp$$ ? is stated by linear combination its Serret Frenet frame with differentiable functions. The definition tangential dual using these Moreover, torque defined and investigated. New dynamically physical results are depending on direction tange...

We propose a general paradigm for generating optimal coordinate frame fields that may be exploited to annotate and display curves and surfaces. Parallel-transport framings, which work well for open curves, generally fail to have desirable properties for cyclic curves and for surfaces. We suggest that minimal quaternion measure provides an appropriate generalization of parallel transport. Our fu...

Some Definitions and Notations. In this section α : [0, L] → R will denote a curve in R that is parametrized by arclength and is of class C, k ≥ 2. We recall that the unit tangent to α at s is defined by −→t (s) := α′(s). (That it is a unit vector is just the definition of α being parametrized by arclength.) The curvature of α is the non-negative real-valued function k : [0, L] → R defined by k...

Motion planning under uncertainty is critical for safe self-driving. This paper proposes a unified obstacle avoidance framework that deals with 1) in ego-vehicle motion; and 2) prediction of dynamic obstacles from the environment. A two-stage traffic participant trajectory predictor comprising short-term long-term used layer to generate but not over-conservative trajectories ego-vehicle. The mo...

Motion-trajectory (MT) curves are used to introduce Frenet oscillations. Time-varying orientation of the motion plane that contains absolute velocity and acceleration vectors is defined in terms three Frenet–Euler angles; curvature, vertical-development, bank angles, referred as angles for brevity. The angle associated super-elevation plane, which measure deviation centrifugal inertia force fro...