arXiv subjects
Publications and source records attributed to Krzysztof Kowalski.
We consider a multi-type branching random walk with displacements that have either regularly varying or semi-exponential tails. We investigate the asymptotic behavior of the rightmost particle in irreducible and reducible regimes and identify the correct normalization for different settings.
We consider a modification of classical branching random walk, where we add i.i.d. perturbations to the positions of the particles in each generation. In this model, which was introduced and studied by Bandyopadhyay and Ghosh (2023), perturbations take form $\frac 1 θ\log\frac X E$, where $θ$ is a positive parameter, $X$ has arbitrary distribution $μ$ and $E$ is exponential with parameter 1, independent of $X$. Working under finite mean assumption for $μ$, they proved almost sure convergence of the rightmost position to a constant limit, and identified the weak centered asymptotics when $θ$ does not exceed certain critical parameter $θ_0$. This paper complements their work by providing weak centered asymptotics for the case when $θ> θ_0$ and extending the results to $μ$ with regularly varying tails. We prove almost sure convergence of the rightmost position and identify the appropriate centering for the weak convergence, which is of form $αn + c \log n$, with constants $α$, $c$ depending on the ratio of $θ$ and $θ_0$. We describe the limiting distribution and provide explicitly the constants appearing in the centering.
The nonlinear generalization of the von Neumann equation preserving convexity of the state space is studied in the nontrivial case of the qutrit. This equation can be cast into the integrable classical Riccati system of nonlinear ordinary differential equations. The solutions of such system are investigated in both the linear case corresponding to the standard von Neumann equation and the nonlinear one referring to the generalization of this equation. The analyzed dynamics of the qutrit is rich and includes quasiperiodic motion, multiple equilibria and limit cycles.
Based on the Hilbert space approach to the theory of nonlinear dynamical systems developed by the author a hypothesis is formulated concerning the "quantal" criterion for classical ordinary differential systems to exhibit chaotic behaviour.
It is demonstrated that very general nonlinear dynamical systems covering all cases arising in practice can be brought down to rate equations of chemical kinetics
It is demonstrated that nonlinear dynamical systems with analytic nonlinearities can be brought down to the abstract Schrödinger equation in Hilbert space with boson Hamiltonian. The Fourier coefficients of the expansion of solutions to the Schrödinger equation in the particular occupation number representation are expressed by means of the classical orthogonal polynomials. The introduced formalism amounts a generalization of the classical methods for linearization of nonlinear differential equations such as the Carleman embedding technique and Koopman approach.
A new method of determining Bäcklund transformations for nonlinear partial differential equations of the evolution type is introduced. Using the Hilbert space approach the problem of finding Bäcklund transformations is brought down to the solution of an abstract equation in Hilbert space.