Том 240, № 5 (2019)
- Год: 2019
- Статей: 20
- URL: https://journals.rcsi.science/1072-3374/issue/view/15016
Article
Quantum Markov States and Quantum Hidden Markov States
Аннотация
In a previous paper (Funct. Anal. Appl., 3 (2015), 205–209), we defined quantum Markov states. Here we recall this definition and present a proof of the results from that paper (which are given there without proofs). We give a definition of a quantum hidden Markov state generated by a function of a quantum Markov process and show how it is related to other definitions of such states. Our definitions work for quantum Markov fields on ℤN and on graphs. We consider an example with the Cayley tree.
On a Universal Borel Adic Space
Аннотация
We prove that the so-called uniadic graph and its adic automorphism are Borel universal, i.e., every aperiodic Borel automorphism is isomorphic to the restriction of this automorphism to a subset invariant under the adic transformation, the isomorphism being defined on a universal (with respect to the measure) set. We develop the concept of basic filtrations and combinatorial definiteness of automorphisms suggested in our previous paper.
Asymptotics of the Number of Geodesics in the Discrete Heisenberg Group
Аннотация
The study of the degenerate part of the absolute of the discrete Heisenberg group required solving a problem on the number of geodesics in this group and in its semigroup. Analytically, this problem reduces to the study of the asymptotic behavior of Gaussian q-binomial coefficients, and the required property is the almost multiplicativity of group characters. The problem has a natural formulation in terms of an (apparently, new) asymptotic property of Young diagrams.
The Boundary of the Refined Kingman Graph
Аннотация
We study the refined Kingman graph ????, first introduced by Gnedin, whose vertices are indexed by the set of compositions of positive integers and multiplicity function reflects the Pieri rule for quasisymmetric monomial functions. Gnedin identified the Martin boundary of ???? with the space Ω of sets of disjoint open subintervals of [0, 1]. We show that the minimal and Martin boundaries of ???? coincide.
The Limit Shape of a Probability Measure on a Tensor Product of Modules of the Bn Algebra
Аннотация
We study a probability measure on the integral dominant weights in the decomposition of the Nth tensor power of the spinor representation of the Lie algebra so(2n + 1). The probability of a dominant weight λ is defined as the dimension of the irreducible component of λ divided by the total dimension 2nN of the tensor power. We prove that as N →∞, the measure weakly converges to the radial part of the SO(2n+1)-invariant measure on so(2n+1) induced by the Killing form. Thus, we generalize Kerov’s theorem for su(n) to so(2n + 1).
A Remark on the Isomorphism Between the Bernoulli Scheme and the Plancherel Measure
Аннотация
We formulate a theorem of Romik and Śniady that establishes an isomorphism between the Bernoulli scheme and the Plancherel measure. Then we derive several combinatorial results as corollaries. The first one is related to measurable partitions; the other two are related to the Knuth equivalence. We also give several examples and one conjecture belonging to A. Vershik.
On the Group of Infinite p-Adic Matrices with Integer Elements
Аннотация
Let G be an infinite-dimensional real classical group containing the complete unitary group (or the complete orthogonal group) as a subgroup. Then G generates a category of double cosets (train), and any unitary representation of G can be canonically extended to the train. We prove a technical lemma on the complete group GL of infinite p-adic matrices with integer coefficients; this lemma implies that the phenomenon of an automatic extension of unitary representations to a train is valid for infinite-dimensional p-adic groups.
Asymptotics of Traces of Paths in the Young and Schur Graphs
Аннотация
Let G be a graded graph with levels V0, V1, . . .. Fix m and choose a vertex v in Vn where n ≥ m. Consider the uniform measure on the paths from V0 to v. Each such path has a unique vertex at the level Vm, so a measure \( {\nu}_v^m \) on Vm is induced. It is natural to expect that these measures have a limit as the vertex v goes to infinity in some “regular” way. We prove this (and compute the limit) for the Young and Schur graphs, for which regularity is understood as follows: the fraction of boxes contained in the first row and the first column goes to 0. For the Young graph, this was essentially proved by Vershik and Kerov in 1981; our proof is more straightforward and elementary.
Systems with Parameters, or Efficiently Solving Systems of Polynomial Equations 33 Years Later. II
Аннотация
Consider a system of polynomial equations with parametric coefficients over an arbitrary ground field. We show that the variety of parameters can be represented as a union of strata. For values of the parameters from each stratum, the solutions of the system are given by algebraic formulas depending only on this stratum. Each stratum is a quasiprojective algebraic variety with degree bounded from above by a subexponential function in the size of the input data. The number of strata is also subexponential in the size of the input data. Thus, here we avoid double exponential upper bounds on the degrees and solve a long-standing problem
On the Moduli Space of Wigner Quasiprobability Distributions for N-Dimensional Quantum Systems
Аннотация
A mapping between operators on the Hilbert space of an N-dimensional quantum system and Wigner quasiprobability distributions defined on the symplectic flag manifold is discussed. The Wigner quasiprobability distribution is constructed as a dual pairing between the density matrix and the Stratonovich–Weyl kernel. It is shown that the moduli space of Stratonovich–Weyl kernels is given by the intersection of the coadjoint orbit space of the group SU(N) and a unit (N − 2)-dimensional sphere. The general considerations are exemplified by a detailed description of the moduli space of 2, 3, and 4-dimensional systems.
On Difference Schemes Approximating First-Order Differential Equations and Defining a Projective Correspondence Between Layers
Аннотация
It is well known that there are remarkable differential equations that can be integrated in CAS, but admit several inequivalent approaches to their description. In the present paper, we discuss remarkable differential equations in another sense, namely, equations for which there exist finite difference schemes that preserve the algebraic properties of solutions exactly. It should be noted that this class of differential equations coincides with the class introduced by Painlevé. In terms of the Cauchy problem, a differential equation of this class defines an algebraic correspondence between the initial and final values. For example, the Riccati equation y′ = p(x)y2 + q(x)y + r(x) defines a one-to-one correspondence between the initial and final values of y on the projective line. However, the standard finite difference schemes do not preserve this algebraic property of the exact solution. Moreover, the scheme that defines a one-to-one correspondence between the layers actually describes the solution not only before moving singularities but also after them and preserves algebraic properties of equations, such as the anharmonic ratio. After a necessary introduction (Secs. 1 and 2), we describe such a one-to-one scheme for the Riccati equation and prove its properties mentioned above.
Relations Between Second-Order Fuchsian Equations and First-Order Fuchsian Systems
Аннотация
Each component of any solution of a Fuchsian differential system satisfies a Fuchsian differential equation. The set of Fuchsian systems is fibered into equivalence classes. Each class consists of systems with similar sets of matrix residues, the conjugation matrix being the same for all elements of the set. We investigate the corresponding classes of scalar equations.
An Algorithm for Decomposing Representations of Finite Groups Using Invariant Projections
Аннотация
We describe an algorithm for decomposing permutation representations of finite groups over fields of characteristic zero into irreducible components. The algorithm is based on the fact that the components of the invariant inner product in invariant subspaces are operators of projecting to these subspaces. This allows us to reduce the problem to solving systems of quadratic equations. The current implementation of the suggested algorithm allows us to split representations with dimensions up to hundreds of thousands. Computational examples are given.
On the Consistency Analysis of Finite Difference Approximations
Аннотация
Finite difference schemes are widely used in applied mathematics to numerically solve partial differential equations. However, for a given solution scheme, it is usually difficult to evaluate the quality of the underlying finite difference approximation with respect to the inheritance of algebraic properties of the differential problem under consideration. In this paper, we present an appropriate quality criterion of strong consistency for finite difference approximations to systems of nonlinear partial differential equations. This property strengthens the standard requirement of consistency of difference equations with differential ones. We use a verification algorithm for strong consistency, which is based on the computation of difference Gröbner bases. This allows for the evaluation and construction of solution schemes that preserve some fundamental algebraic properties of the system at the discrete level. We demonstrate the suggested approach by simulating a Kármán vortex street for the two-dimensional incompressible viscous flow described by the Navier–Stokes equations.
Foliation of the Space \( \mathfrak{s}{\mathfrak{l}}^{\ast}\left(n,\mathbb{R}\right) \) into Coadjoint Orbits
Аннотация
We suggest a method for constructing parameters on the coadjoint orbits in \( \mathfrak{s}{\mathfrak{l}}^{\ast}\left(n,\mathbb{R}\right) \). The method is based on the fact that parameters are invariant under the action of vector fields normal to the tangent space of the orbit with respect to the Killing form. The construction reduces to solving a homogeneous system of linear equations.
Automorphisms of Types and Their Applications
Аннотация
We outline recent results in the theory of type isomorphisms and automorphisms and present several practical applications of these results that can be useful in the contexts of programming and data security. Bibliography: 27 titles.