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This article concerns the life of Leonhard Euler and his achievements in theoretical mechanics. A number of topics are discussed related to the development of Euler's ideas and methods: divergent series and asymptotics of solutions of non-linear differential equations; the hydrodynamics of a perfect fluid and Hamiltonian systems; vortex theory for systems on Lie groups with left-invariant kinetic energy; energy criteria of stability; Euler's problem of two gravitating centres in curved spaces.

The main purpose of this paper is to prove the central limit theorem for the position at large times of a particle performing a discrete-time random walk on the lattice  when the particle interacts with a random `environment' (and starts out at a fixed point of the lattice). Two cases are considered for the distribution of the particle position: first, the distribution when the configuration of the `environment' (that is, of the random field) is fixed at all points of the `space-time' (the so-called quenched model), and, second, the distribution induced by the joint evolution of the environment and the particle position under the assumption that the pair forms a Markov chain (the annealed model). Two cases are considered also for the quenched model: the values of the field at all points of `space-time' are independent and identically distributed, or the values of the field at different times are linked by a homogeneous Markov chain. In the case of quenched models the central limit theorem with one and the same limit law is true for almost all configurations of the `environment', and in the case of annealed models it is true for any initial distribution of the field. Besides the central limit theorem, the paper briefly treats some other topics related to these models (decay of correlations, large deviations, `the field from the viewpoint of a particle', and so on).

This is an investigation of the problem of the asymptotic distribution of the Frobenius numbers of  relatively prime integers. For virtually definitive results are obtained. For it is shown that the distributions appearing form a compact set. An essential role is played by the limit theorem for logarithms of denominators of continued fractions of random numbers.

Methods are considered for applying an algebra with bilinear commutation relations to the theory of quantum integrable systems. This survey describes most of the results obtained in this area over the last twenty years, mainly in connection with the computation of correlation functions of quantum integrable systems. Methods for constructing eigenfunctions of the quantum transfer matrix and computing inner products and correlation functions are presented in detail. An example of application of the general scheme to the model of the Heisenberg chain is considered.