Framed curves in Euclidean space are used to investigate singular and important for singularity theory. In this study, framed four-dimensional introduced new results obtained. The relation of with Frenet is given Bishop-type frame the help Euler angles. addition, by using space, rectifying curves, osculating normal introduced. Also, some characterizations depending on curvatures

The aim of this study is to investigate the variations Bishop frame curvatures for null Cartan curves in semi-Riemannian manifolds. Killing equations terms along curve especially derived. are used interpret movement charged particles a magnetic field. particle motion through examined as an application study. It found that traces trajectory form cubic during its vector According results obtained...

In this paper, we give the parametric equation of Bishop frame for a timelike sweeping surface with unit speed curve in Minkowski 3-space. We introduce new geometric invariant to explain properties and local singularities surface. derive sufficient necessary conditions be developable ruled Afterwards, take advantage singularity theory classification Furthermore, some representative examples sho...

Chess is a deep and well-studied game with a wealth of content and data. Because of the relative simplistic nature of the game states, games themselves are relatively easy to record, and thus there are large databases of games, dating back to Francesco di Castellvi vs Narciso Vinyoles, 1475. The vast majority of published games are between masters and grandmasters, usually recorded by tournamen...