2025-01-02 14:16:50 Pacfic
Geometric and Topological Concepts - 8d
Recent studies have delved into the fascinating realms of geometry and topology, exploring several intriguing concepts. One area of focus involves the behavior of trajectories within planar polynomial ordinary differential equations (ODEs). Researchers are investigating the relationship between the trajectory of these systems and the level sets defined by the polynomial function, specifically when the trajectory avoids equilibrium points.
Further research has also explored a probability paradox related to acute triangles. It has been demonstrated that the probability of forming an acute triangle using randomly selected points differs between circles and disks, as well as between spheres and balls. Specifically, the probability is lower on the boundary circle than within the disk and higher on the boundary sphere compared to inside the ball. In addition, it was highlighted how quaternions can be used to derive the equations of spherical trigonometry, illustrating their power in relating algebraic and geometrical constructs.
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References:
- Recent Questions - Mathematics Stack Exchange - Trajectory extending in planar polynomial ODE - 8d
- Recent Questions - MathOverflow - Paradox involving probability that a triangle is acute in circles and spheres - 8d
- What's new - Quaternions and spherical trigonometry - 8d