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.
An annulus-like set of the form K = B & times; T infinity is under consideration, where B is a closed ball in a Banach space V and T infinity is the where E is an infinite-dimensional Banach space and Z infinity is an abstract integer lattice in E. The main result is as follows: for a certain class of smooth maps Pi: K -* K we establish sufficient conditions for the existence and stability of an invariant toroidal manifold of the form A = {(v, p) E K: v = h(p) E V, p E T infinity}, where h(p) is a continuous function of p E T infinity. We also answer a number of related questions. First, we consider the problem of the Cm-smoothness of the manifold A for each positive integer m; second, we show that all trajectories of the map Pi with initial conditions in K tend to A and admit an asymptotic phase; third, we extend our results to semiflows and then apply the theory developed to integral networks of nonlinear oscillators.
Kleene iteration (star operator) is one of the most interesting algebraic operations arising in computer science. The studies of structures with this operation, Kleene algebras and their extensions, begin with the classical notion of regular expression describing formal languages. Subsequently, so-called action algebras (Pratt 1991, Kozen 1994), or Kleene algebras with division, were introduced. In these structures the Kleene star operator is combined with divisions compatible with a partial order (such operations had previously been introduced by Krull in 1924). This article gives a survey of results on algorithmic complexity for the logical theories of structures with Kleene iteration. Although the simplest of these theories, the theory of equality of regular expressions, is algorithmically decidable, some of its generalizations, such as Horn theories and fragments of these, as well as theories with division, almost immediately become undecidable. Particularly interesting is the case of *-continuous Kleene algebras, where iteration is defined as the limit of powers of an element (in the general case iteration is defined as a fixed point). In the language of logic, *-continuity corresponds to the omega rule, and the complexity of such theories can attain the level of H11-completeness. Bibliography: 83 titles.
