The second variation operator of minimal submanifolds of Riemannian manifolds (the Jacobi operator) carries information about stability properties of the submanifold when it is thought of as a critical point for the area functional. When the ambient Riemannian manifold is a sphere S, Simons [S] characterized the totally geodesic submanifolds as the minimal submanifolds of S either with the lowe...

Using an unusual, yet natural invariant measure we show that there exists a sensitive cellular automaton whose perturbations propagate at asymptotically null speed for almost all configurations. More specifically, we prove that Lyapunov Exponents measuring pointwise or average linear speeds of the faster perturbations are equal to zero. We show that this implies the nullity of the measurable en...

We prove that a minimal automaton has a minimal adjacency matrix rank and a minimal adjacency matrix nullity using equitable partition (from graph spectra theory) and Nerode partition (from automata theory). This result naturally introduces the notion of matrix rank into a regular language L, the minimal adjacency matrix rank of a deterministic automaton that recognises L. We then define and fo...

Knot colorings are one of the simplest ways to distinguish knots, dating back Reidemeister and popularized by Fox. In this mostly expository article, we discuss knot invariants like colorability, determinant, number colorings, how these can be computed from either coloring matrix or Goeritz matrix. We give an elementary approach equivalence, without using any algebraic topology. also compute de...