arXiv subjects
Publications and source records attributed to Piotr Pieranski.
The shape of the most tight trefoil knot with N=200640 vertices found with an appropriately modified finite element method is analyzed. The high number of vertices makes plots of its curvature and torsion very precise what allows the authors to formulate new, firmly justified conjectures concerning the shape of the ideal trefoil knot.
We reinterpret the distribution of the Hall potential in the Hall bar-with-a-hole that has been found and interpreted by Mani and Klitzing [Appl. Phys. Lett., 64 1262 (1994)]. Our reinterpretation explains all the "paradoxes" without resorting to new theoretical conceptions.
A physical interpretation of the rope simulated by the SONO algorithm is presented. Properties of the tight polygonal knots delivered by the algorithm are analyzed. An algorithm for bounding the ropelength of a smooth inscribed knot is shown. Two ways of calculating the ropelength of tight polygonal knots are compared. An analytical calculation performed for a model knot shows that an appropriately weighted average should provide a good estimation of the minimum ropelength for relatively small numbers of edges.
The question: "How many different trajectories are there on a single invariant torus within the phase space of an integrable Hamiltonian system?" is posed. A rigorous answer to the question is found both for the rational and the irrational tori. The relevant notion of non-measurable sets is discussed.