Vol 216, No 1 (2016)
- Year: 2016
- Articles: 10
- URL: https://journals.rcsi.science/1072-3374/issue/view/14747
Article
Group-Graded Systems and Algebras
Abstract
In the paper, we discuss some problems concerning the structural properties of crossed products. While expansions of C*-algebras under group actions have been studied rather extensively, known difficulties in the transition to irreversible dynamical systems require the development of new methods. The first step in this direction is to study actions of inverse semigroups, whose properties are closest to those of groups. The main tool to achieve the goal is the concept of grading. The detection of the grading structure in the corresponding constructions seems to be very promising.
Combinatorial Aspects of Correlation Functions of the XXZ Heisenberg Chain in Limiting Cases
Abstract
We discuss the connection between quantum integrable models and some aspects of enumerative combinatorics and the theory of partitions. As a basic example, we consider the spin XXZ Heisenberg chain in the limiting cases of zero and infinite anisotropy. The representation of the Bethe wave functions via Schur functions allows us to apply the theory of symmetric functions to calculating the thermal correlation functions as well as the form factors in the determinantal form. We provide a combinatorial interpretation of the correlation functions in terms of nests of self-avoiding lattice paths. The suggested interpretation is in turn related to the enumeration of boxed plane partitions. The asymptotic behavior of the thermal correlation functions is studied in the limit of small temperature provided that the characteristic parameters of the system are large enough. The leading asymptotics of the correlators are found to be proportional to the squared numbers of boxed plane partitions.
Orthogonal Pairs and Mutually Unbiased Bases
Abstract
The goal of our article is a study of related mathematical and physical objects: orthogonal pairs in sl(n) and mutually unbiased bases in ℂn. An orthogonal pair in a simple Lie algebra is a pair of Cartan subalgebras that are orthogonal with respect to the Killing form. The description of orthogonal pairs in a given Lie algebra is an important step in the classification of orthogonal decompositions, i.e., decompositions of the Lie algebra into a direct sum of Cartan subalgebras pairwise orthogonal with respect to the Killing form. One of the important notions of quantum mechanics, quantum information theory, and quantum teleportation is the notion of mutually unbiased bases in the Hilbert space ℂn. Two orthonormal bases {ei}i = 1n, {fj}j = 1nare mutually unbiased if and only if\( {\left|\left\langle {e}_i\left|{f}_j\right.\right\rangle \right|}^2=\frac{1}{n} \)for any i, j = 1,…, n. The notions of mutually unbiased bases in ℂnand orthogonal pairs in sl(n) are closely related. The problem of classification of orthogonal pairs in sl(n) and the closely related problem of classification of mutually unbiased bases in ℂnare still open even for the case n = 6. In this article, we give a sketch of our proof that there is a complex four-dimensional family of orthogonal pairs in sl(6). This proof requires a lot of algebraic geometry and representation theory. Further, we give an application of the result on the algebraic geometric family to the study of mutually unbiased bases. We show the existence of a real four-dimensional family of mutually unbiased bases in ℂ6, thus solving a long-standing problem. Bibliography: 24 titles.
Chip Removal for Computing the Number of Perfect Matchings
Abstract
We consider a transformation of a graph G that replaces an induced subgraph H of arbitrary size by a small new subgraph h. We choose h in such a way that the equality M(G) = xM(G′) holds (where G′ is the new graph and the factor x depends on the numbers of matchings of H and its subgraphs). We describe how one can construct h when G is a plane graph and H is a bipartite graph (with some restriction on the coloring of the vertices connecting it with the other part of the graph G). For a plane bipartite graph H with a small number of such vertices, we prove that the equality holds for an arbitrary graph G.
A Study of the Growth of the Maximum and Typical Normalized Dimensions of Strict Young Diagrams
Abstract
In this paper, we investigate the asymptotics of the normalized dimensions of strict Young diagrams. We describe the results of corresponding computer experiments. The strict Young diagrams parametrize the projective representations of the symmetric group Sn. So, the asymptotics of the normalized dimensions of diagrams gives us the asymptotics of the dimensions of projective representations. Sequences of strict diagrams of high dimension consisting of up to one million boxes were generated. It was proved by an exhaustive search that the first 250 diagrams of all these sequences have the maximum possible dimensions. Presumably, these sequences contain infinitely many diagrams of maximum dimension, and thus give the correct asymptotics of their growth. Also, we investigate the behavior of the normalized dimensions of typical diagrams with respect to the Plancherel measure on the Schur graph. The calculations strongly agree with A. M. Vershik’s conjecture on the convergence of the normalized dimensions of the maximum and Plancherel typical diagrams not only for the standard Young graph, but also for the Schur graph.
On Ergodic Decompositions Related to the Kantorovich Problem
Abstract
Let X be a Polish space, \( \mathcal{P} \)(X) be the set of Borel probability measures on X, and T : X → X be a homeomorphism. We prove that for the simplex Dom ⊆ \( \mathcal{P} \)(X) of all T -invariant measures, the Kantorovich metric on Dom can be reconstructed from its values on the set of extreme points. This fact is closely related to the following result: the invariant optimal transportation plan is a mixture of invariant optimal transportation plans between extreme points of the simplex. The latter result can be generalized to the case of the Kantorovich problem with additional linear constraints an the class of ergodic decomposable simplices.
On a Class of Operator Algebras Generated by a Family of Partial Isometries
Abstract
The paper provides a short overview of a series of articles devoted to C*-algebras generated by a self-mapping on a countable set. Such an algebra can be seen as a representation of the universal C*-algebra generated by a family of partial isometries satisfying a set of conditions. These conditions are determined by the initial mapping.
Limiting Curves for the Pascal Adic Transformation
Abstract
The paper generalizes results by E. Janvresse, T. de la Rue, and Y. Velenik and results by the second author on the fluctuations in the ergodic sums for the Pascal adic transformation in the case of an arbitrary ergodic invariant measure and arbitrary cylinder function.
Nonlinear Fokker–Planck–Kolmogorov Equations in Hilbert Spaces
Abstract
We study the Cauchy problem for the nonlinear Fokker–Planck–Kolmogorov equations for probability measures on a Hilbert space that corresponds to stochastic partial differential equations. Sufficient conditions for the uniqueness of probability solutions for a cylindrical diffusion operator and for a possibly degenerate diffusion operator are given. A new general existence result is established without explicit growth restrictions on the coefficients.
O(∞)- and Sp(∞)-Invariant Ergodic Measures on the Spaces of Infinite Antisymmetric and Quaternionic Antihermitian Matrices
Abstract
The ergodic measures for the actions of the infinite orthogonal group O(∞) and the infinite symplectic group Sp(∞) by conjugations on the spaces of infinite real antisymmetric and infinite quaternionic antihermitian matrices are classified, with the use of the Vershik–Olshanski “ergodic method.”