This paper is devoted to the existence, uniqueness and comparison theorem on unbounded solutions of a scalar backward stochastic differential equation (BSDE) whose generator grows (with respect both unknown variables y z) in super-linear way like |y||ln|y||?+|z||ln|z||? for some ??[0,1] ??0. Let k be maximum ?, ?+1/2 2?. For following four different ranges growth power parameter k: k=1/2, k?(1/...

We consider the Lax representation of the new two-component coupled integrable system recently discovered by the author. Connection of the hierarchy of infinitely many Lax pairs with each other is presented.

This is an example of application Ziglin-Morales-Ramis algebraic studies in Hamiltonian integrability, more specifically the result by Morales, Ramis and Sim\'o on higher-order variational equations, to well-known Friedmann-Robertson-Walker cosmological model. A previous paper author formalises said systems such a way allowing simple expression notable elements differential Galois group needed ...

To demonstrate more visibly the close relation between thecontinuity and integrability, a new proof for the Banach-Zareckitheorem is presented on the basis of the Radon-Nikodym theoremwhich emphasizes on measure-type properties of the Lebesgueintegral. The Banach-Zarecki theorem says that a real-valuedfunction $F$ is absolutely continuous on a finite closed intervalif and only if it is continuo...

We review some essential aspects of classically integrable systems. The detailed outline of the lectures consists of