Vol 224, No 2 (2017)

Let φ : ℝ → ℝ be a continuously differentiable function on a finite interval J ⊂ ℝ, and let α = (α₁, α₂) be a point with algebraically conjugate coordinates such that the minimal polynomial P of α₁, α₂ is of degree ≤ n and height ≤ Q. Denote by \( {M}_{\varphi}^n\left(Q,\gamma, J\right) \) the set of points α such that |φ(α₁) − α₂| ≤  c₁Q^−γ. We show that for 0 < γ < 1 and any sufficiently large Q there exist positive values c₂ < c₃, where c_i = c_i(n), i = 1, 2, that are independent of Q and such that \( {c}_2\cdot {Q}^{n+1-\upgamma}<\#{M}_{\varphi}^n\left(Q,\upgamma, J\right)<{c}_3\cdot {Q}^{n+1-\upgamma}. \) Bibliography: 17 titles.

The article is devoted to the study of the asymptotics of the probabilities of paths in a certain Markov process on the 3D Young graph. We introduce a normalized dimension of paths and study the growth and oscillations of normalized dimensions along greedy trajectories of this process using computer experiments. Bibliography: 9 titles.

In the paper, a family of representations of maximal solvable subgroups of the simple Lie groups O(p, q), U(p, q), and Sp(p, q), where 1 ≤ p ≤ q, is introduced. These subgroups are called the Iwasawa subgroups of the corresponding simple groups. The main property of these representations is the existence of nontrivial 1-cohomology with values in the representations. For groups of rank 1, the representations from this family are unitary; for ranks greater than 1, they are nonunitary. The paper continues a series of our previous papers and serves as an introduction to the theory of nonunitary current groups.

We give a closed form for the generating function of the discrete Chebyshev polynomials. It is the MacWilliams transform of Jacobi polynomials together with a binomial multiplicative factor. It turns out that the desired closed form is a solution to a special case of the Heun differential equation, and that it implies combinatorial identities that appear quite challenging to prove directly.

For the Schur Q-functions there is a Cauchy identity, which shows a duality between the Schur P- and Q-functions. We will be interested in the multiparameter Schur Q-functions, which were introduced by V. N. Ivanov, and we will give dual analogs of the multiparameter Schur Q(P)-functions, with a corresponding multiparameter Cauchy identity.

New proofs of theorems on graph operations that do not affect the structure of the sandpile groups of graphs are suggested. The proofs are based on the isomorphism between the sandpile group and the Kirchhoff group of a graph.

A principal circle bundle over a PL polyhedron can be triangulated and thus obtains combinatorics. The triangulation is assembled from triangulated circle bundles over simplices. To every triangulated circle bundle over a simplex we associate a necklace (in the combinatorial sense). We express rational local formulas for all powers of the first Chern class in terms of expectations of the parities of the associated necklaces. This rational parity is a combinatorial isomorphism invariant of a triangulated circle bundle over a simplex, measuring the mixing by the triangulation of the circular graphs over vertices of the simplex. The goal of this note is to sketch the logic of deducing these formulas from Kontsevitch’s cyclic invariant connection form on metric polygons.

It is known that taken together, all collections of nonintersecting diagonals in a convex planar n-gon give rise to a (combinatorial type of a) convex (n − 3)-dimensional polytope As_n called the Stasheff polytope, or associahedron. In the paper, we act in a similar way by taking a convex planar n-gon with k labeled punctures. All collections of mutually nonintersecting and mutually nonhomotopic topological diagonals yield a complex As_n,k. We prove that it is a topological ball. We also show a natural cellular fibration Asn,k → As_n,k−1. A special example is delivered by the case k = 1. Here the vertices of the complex are labeled by all possible permutations together with all possible bracketings on n distinct entries. This hints to a relationship with M. Kapranov’s permutoassociahedron.

A symbolic generation of Painlevé equations is developed on the basis of the antiquantization of deformed Heun-class equations. The corresponding CAS Maple package is presented, along with examples of its use. The particular cases of reduced confluent Heun equations are discussed.

Consider a polynomial with parametric coefficients. We show that the variety of parameters can be represented as a union of strata. For values of the parameters from each stratum, the decomposition of this polynomial into absolutely irreducible factors is given by algebraic formulas depending only on the stratum. Each stratum is a quasiprojective algebraic variety. This variety and the corresponding output are given by polynomials of degrees at most D with D = d′d^O(1) where d′, d are bounds on the degrees of the input polynomials. The number of strata is polynomial in the size of the input data. Thus, here we avoid double exponential upper bounds for the degrees and solve a long-standing problem.