Abstract: We present our solutions to two long standing open problems, one from probability theory formulated by Malyshev in 1970 and another one from a crossroad of geometry and dynamics, going back to Darboux in 1879. The Malyshev problem is of finding effective, explicit necessary and sufficient conditions in the closed form to characterize all random walks in the quarter plane with a finite group of the random walk of order 2n, for all n ≥ 2. We also describe all n-periodic Darboux transformations for 4-bar link problems for all n ≥ 2, thus completely solving the Darboux problem, that he solved for n = 2. This is based on a joint work with Milena Radnovic.

---

Last modified: Mon Mar 16 14:20:52 EDT 2026