Vol 57, No 6 (2016)

Considering the spectral Neumann problem for the Laplace operator on a doubly periodic square grid of thin circular cylinders (of diameter ε ≪ 1) with nodes, which are sets of unit size, we show that by changing or removing one or several semi-infinite chains of nodes we can form additional spectral segments, the wave passage bands, in the essential spectrum of the original grid. The corresponding waveguide processes are localized in a neighborhood of the said chains, forming I-shaped, V-shaped, and L-shaped open waveguides. To derive the result, we use the asymptotic analysis of the eigenvalues of model problems on various periodicity cells.

We study the problem of exponential dichotomy for the systems of linear difference equations with periodic coefficients. Some criterion is established for exponential dichotomy in terms of solvability of a special boundary value problem for a system of discrete Lyapunov equations. We also give estimates for dichotomy parameters.

Under study are the simple infinite-dimensional abelian Jordan superalgebras not isomorphic to the superalgebra of a bilinear form. We prove that the even part of such superalgebra is a differentially simple associative commutative algebra, and the odd part is a finitely generated projective module of rank 1. We describe unital simple Jordan superalgebras with associative nil-semisimple even part possessing two even elements which induce a nonzero derivation.

We study the limit behavior of sequences of cyclic systems of ordinary differential equations that were invented for the mathematical description of multistage synthesis. The main construction of the article is the distribution function of initial data. It enables us to indicate necessary and sufficient existence conditions as well as completely describe the structure and all typical properties of the limits of solutions to the integro-differential equations of “convolution” type to which the systems of cyclic synthesis are easily reduced. All notions, methods, and problems under discussion belong to such classical areas as real function theory, Euler integrals, and asymptotic analysis.

We prove that the problem of tabularity over Johansson’s minimal logic J is decidable. Describing all pretabular extensions of the minimal logic, we find that there are seven of them and show that they are all recognizable over J. We find axiomatizations and semantic characterizations of all seven pretabular logics.

Denote the set of all holomorphic mappings of a genus 3 Riemann surface S₃ onto a genus 2 Riemann surface S₂ by Hol(S₃, S₂). Call two mappings f and g in Hol(S₃, S₂) equivalent whenever there exist conformal automorphisms α and β of S₃ and S₂ respectively with f ◦ α = β ◦ g. It is known that Hol(S₃, S₂) always consists of at most two equivalence classes.

We obtain the following results: If Hol(S₃, S₂) consists of two equivalence classes then both S₃ and S₂ can be defined by real algebraic equations; furthermore, for every pair of inequivalent mappings f and g in Hol(S₃, S₂) there exist anticonformal automorphisms α− and β− with f ◦ α− = β− ◦ g. Up to conformal equivalence, there exist exactly three pairs of Riemann surfaces (S₃, S₂) such that Hol(S₃, S₂) consists of two equivalence classes.

Under study is the two-dimensional Schrödinger operator connected with the family of Hamiltonian-minimal Lagrangian surfaces in ℂP².

We classify the quasiendomorphism algebras of rank 4 torsion-free abelian groups quasidecomposable as the direct sum of strongly indecomposable rank 2 groups.