The theory of electrical networks, in its current state, covers a number of areas of contemporary mathematics and mathematical physics including the combinatorics of paths, forests and groves on graphs, discrete harmonic analysis, problems of random walks, exactly solved models in statistical mechanics, cluster varieties related to spaces of totally positive matrices, discrete integrable systems, algebraic structures similar to Zamolodchikov's tetrahedron equation, and many others. The main aim of this survey is to present some of these topics, classical and recently discovered ones alike. Bibliography: 114 titles.

the outstanding researcher, member of the Russian Academy of Sciences Dmitry Treschev observed his 60th birthday.He made a lasting contribution to the dynamics of Hamiltonian systems, perturbation theory, Arnold diffusion, stability, integrability, chaos, and KAM theory.Treschev published more than 110 scientific articles and 3 monographs.Dmitry Valerievich Treschev was born on October 26th 1964, in the town of Olenegorsk in Murmansk Oblast, in a family of a military officer.

This survey sums up a cycle of papers devoted to the construction of finite-dimensional moduli spaces points in which are certain special Lagrangian submanifolds of compact complex simply connected algebraic varieties. The starting point for this construction was the idea, due to A. Tyurin, to treat Largrangian submanifolds (or equivalence classes of such submanifolds) as mirror counterparts of stable vector bundles. Our constructions are based on the programme of abelian Lagrangian algebraic geometry developed by A. Tyurin and Gorodentsev 25 years ago. Since this programme was in its turn based on the Bohr-Sommerfeld Lagrangian geometry known in geometric quantization, we call our construction special Bohr-Sommerfeld geometry. The definitions arising in the course of work turn out to be closely connected with the theory of Weinstein domains, Eliashberg's conjectures, and many other concepts in symplectic geometry. The core conjecture that arose in our work and is confirmed by the available examples states that each moduli space of this type is in its turn an algebraic variety. Bibliography: 13 titles.

Optimal flows in transport networks, affine-nonlinear deformations of networks, and replicator dynamical systems generated by Schur functions A

The stability of equilibria is considered for systems whose kinetic energy is a pseudo-Riemannian metric on the configuration space. Equilibria are critical points of the potential energy. For a linear system with two degrees of freedom the stability diagram is plotted and the bifurcations of eigenvalues are indicated. Points of maximum and minimum of the potential energy are unstable equilibria in the pseudo-Euclidean case. The same conclusion holds for nonlinear analytic systems with two degrees of freedom. Conditions for stability are indicated for multidimensional linear systems in a pseudo-Euclidean space. In particular, an equilibrium is stable if and only if the linear equations of motion can be reduced to a ‘natural’ system with positive definite kinetic energy and, in addition, the potential energy takes a strict minimum at this equilibrium. The influence of dissipative and gyroscopic forces on the stability of equilibria in a pseudo-Riemannian space is investigated. The instability of an isolated equilibrium is proved in the case when dissipative forces with full energy dissipation are added. The instability degree is calculated for linear dissipative systems. Conditions for the stability of linear systems in the case when large gyroscopic forces are applied to them are indicated. Bibliography: 40 titles.