Vol 240, No 5 (2019)

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 ℤ^N and on graphs. We consider an example with the Cayley tree.

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.

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.

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 2^nN 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).

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.

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

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.

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.

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.

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.