卷 224, 编号 2 (2017)
- 年: 2017
- 文章: 20
- URL: https://journals.rcsi.science/1072-3374/issue/view/14837
Article
Announce of an Entropy Formula for a Class of Actions Coming From Gibbs Measures
摘要
An explicit formula for the sofic and Rokhlin entropy of a class of actions coming from Gibbs measures is announced. Bibliography: 22 titles.
On the Distribution of Points with Algebraically Conjugate Coordinates in a Neighborhood of Smooth Curves
摘要
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 ci = ci(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.
Multi-Dimensional Random Walks and Integrable Phase Models
摘要
We consider random multi-dimensional lattice walks bounded by a hyperplane, calling them walks over multi-dimensional simplicial lattices. We demonstrate that generating functions of these walks are dynamical correlation functions of a certain type of exactly solvable quantum phase models describing strongly correlated bosons on a chain. Walks over oriented lattices are related to the phase model with a non-Hermitian Hamiltonian, while walks over disoriented ones are related to the model with a Hermitian Hamiltonian. The calculation of the generating functions is based on the algebraic Bethe Ansatz approach to the solution of integrable models. The answers are expressed through symmetric functions. Continuous-time quantum walks bounded by a onedimensional lattice of finite length are also studied. Bibliography: 40 titles.
Numerical Study of the Asymptotics of Path Probabilities in a Markov Process Close to a Central One on the 3D Young Graph
摘要
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.
The Computational Complexity of the Initial Value Problem for the Three Body Problem
摘要
The paper is concerned with the computational complexity of the initial value problem (IVP) for a system of ordinary dynamical equations. A formal problem statement is given, containing a Turing machine with an oracle for getting the initial values as real numbers. It is proven that the computational complexity of the IVP for the three-body problem is not bounded by a polynomial. The proof is based on the analysis of oscillatory solutions of the Sitnikov problem, which have a complex dynamical behavior. These solutions contradict the existence of an algorithm that solves the IVP in polynomial time. Bibliography: 12 titles.
Special Representations of the Iwasawa Subgroups of Simple Lie Groups
摘要
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.
On the Ring of Local Unitary Invariants for Mixed X-States of Two Qubits
摘要
Entangling properties of a mixed two-qubit system can be described by local homogeneous unitary invariant polynomials in the elements of the density matrix. The structure of the corresponding ring of invariant polynomials for a special subclass of states, the so-called mixed X-states, is established. It is shown that for the X-states there is an injective ring homomorphism of the quotient ring of SU(2)×SU(2)-invariant polynomials modulo its syzygy ideal to the SO(2) × SO(2)-invariant ring freely generated by five homogeneous polynomials of degrees 1, 1, 1, 2, 2.
On the Generating Function of Discrete Chebyshev Polynomials
摘要
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.
A Set of 12 Numbers is Not Determined by the Set of its 4-Sums
摘要
We present two sets of 12 integers that have the same sets of 4-sums. The proof of the fact that a set of 12 numbers is uniquely determined by the set of its 4-sums published 50 years ago is wrong, and we demonstrate an incorrect calculation in it.
Dual Multiparameter Schur Q-Functions
摘要
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.
The Mallows Measures on the Hyperoctahedral Group
摘要
The Mallows measures on the symmetric group Sn form a deformation of the uniform distribution. These measures are commonly used in mathematical statistics, and in recent years they found applications in other areas of mathematics as well.
As shown by Gnedin and Olshanski, there exists an analog of the Mallows measures on the infinite symmetric group. These new measures are diffuse, and they are quasi-invariant with respect to the two-sided action of a countable dense subgroup.
The purpose of the present note is to extend the Gnedin–Olshanski construction to the infinite hyperoctahedral group. Along the way, we obtain some results for the Mallows measures on finite hyperoctahedral groups, which may be of independent interest.
Applying the Kirchhoff Relations in Proofs of Theorems on Graph Operations that Do Not Affect the Structure of the Sandpile Groups of Graphs
摘要
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.
Limiting Curves for Polynomial Adic Systems
摘要
We prove the existence of and describe limiting curves resulting from deviations in the partial sums in the ergodic theorem for cylinder functions and polynomial adic systems. For a general ergodic measure-preserving transformation and a summable function, we give a necessary condition for a limiting curve to exist. Our work generalizes results by É. Janvresse, T. de la Rue, and Y. Velenik and answers several questions from their work.
On Local Combinatorial Formulas for Chern Classes of a Triangulated Circle Bundle
摘要
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.
The Whishart–Pickrell Distributions and Closures of Group Actions
摘要
Consider probability distributions on the space of infinite Hermitian matrices Herm(∞) invariant with respect to the unitary group U(∞). We describe the closure of U(∞) in the semigroup of spreading maps (polymorphisms) of Herm(∞); this closure is a semigroup isomorphic to the semigroup of all contractive operators.
Diagonal Complexes for Punctured Polygons
摘要
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 Asn 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 Asn,k. We prove that it is a topological ball. We also show a natural cellular fibration Asn,k → Asn,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.
Asymptotics of the Jordan Normal Form of a Random Nilpotent Matrix
摘要
We study the Jordan normal form of an upper triangular matrix constructed from a random acyclic graph or a random poset. Some limit theorems and concentration results for the number and sizes of Jordan blocks are obtained. In particular, we study a linear algebraic analog of Ulam’s longest increasing subsequence problem.
Symbolic Generation of Painlevé Equations
摘要
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.
The Spectrum and Separability of Mixed Two-Qubit X-States
摘要
The separable mixed two-qubit X-states are classified in accordance with the degeneracies in the spectrum of density matrices. It is shown that there are four classes of separable X-states, among them: one four-dimensional family, a pair of two-dimensional families, and a single zero-dimensional maximally mixed state.
Efficient Absolute Factorization of Polynomials with Parametric Coefficients
摘要
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′dO(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.