Detection of gross errors by the chauvenet test for observations connected in a homogeneous markov chain

The Chauvenet rule for detection of gross errors is extended for the case of a stationary Gaussian Markov chain with unknown correlation. Bibliography:4 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 207, pp. 137–142, 1993. Translated by L. Khalfin.     

On the calculation of some mixed moments of aggregative characteristics of a homogeneous Markov chain for the polynomial scheme

Formulas are given for the calculation of some mixed moments of aggregative characteristics of a homogeneous Markov chain for the polynomial scheme. Translated fromStatisticheskie Metody Otsenivaniya i Proverki Gipotez, pp. 156–164, Perm, 1991.     

Strengthening theC t -closing lemma for dynamical systems and foliations on the torus

We prove a strengthenedC ^r -closing lemma (r≥1) for wandering chain recurrent trajectories of flows without equilibrium states on the two-dimensional torus and for wandering chain recurrent orbits of a diffeomorphism of the circle. The strengthenedC ^r -closing lemma (r≥1) is proved for a special class of infinitely smooth actions of the integer lattice ℤ ^k on the circle. The result is applied to foliations of codimension one with trivial holonomy group on the three-dimensional torus. Translated fromMatematicheskie Zametki, Vol. 61, No. 3, pp. 323–331, March, 1997. Translated by S. K. Lando     

Regularity criterion for a mathematical model of the laser

There exists a deep relationship between the nonexplosion conditions for Markov evolution in classical and quantum probability theories. Both of these conditions are equivalent to the preservation of the unit operator (total probability) by a minimal Markov semigroup. In this work, we study the Heisenberg evolution describing an interaction between the chain ofN two-level atoms andn-mode external Bose field, which was considered recently by J. Alli and J. Sewell. The unbounded generator of the Markov evolution of observables acts in the von Neumann algebra. For the generator of a Markov semigroup, we prove a nonexplosion condition, which is the operator analog of a similar condition suggested by R. Z. Khas’minski and later by T. Taniguchi for classical stochastic processes. For the operator algebra situation, this condition ensures the uniqueness and conservativity of the quantum dynamical semigroup describing the Markov evolution of a quantum system. In the regular case, the nonexplosion condition establishes a one-to-one relation between the formal generator and the infinitesimal operator of the Markov semigroup. Translated fromMatematicheskie Zemetki, Vol. 67, No. 5, pp. 788–796, May, 2000.     

Asymptotic analysis of the behavior of a multichannel queueing system functioning in a Markov medium

The asymptotic behavior of some multidimensional characteristics of two Markov queueing systems, in which an incoming flow of units and their service time depend on a small parameter ɛ and the state of the Markov medium where these queueing systems function, is investigated. Bibliography: 6 titles. Translated fromObchyslyuval’na ta Prykladna Matematyka, No. 76, 1992, pp. 91–98.     

Limit theorems for Markov random walks with a fixed number of certain transitions

The limit behavior of Markov chains with discrete time and a finite number of states (MCDT) depending on the number n of its steps has been almost completely investigated [1–4]. In [5], MCDT with forbidden transitions were investigated, and in [6], the sum of a random number of functionals of random variables related by a homogeneous Markov chain (HMC) was considered. In the present paper, we continue the investigation of the limit behavior of the MCDT with random stopping time which is determined by a Markov walk plan II with a fixed number of certain transitions [7, 8]. Here we apply a method similar to that of [6], which allows us to obtain, together with some generalizations of the results of [6], a number of new assertions. Translated fromStatisticheskie Metody Otsenivaniya i Proverki Gipotez, pp. 119–130, Perm, 1990.     

Completely integrable nonlinear dynamical systems of the Langmuir chains type

The solution of the Cauchy problem for semi-infinite chains of ordinary differential equations, studied first by O. I. Bogoyavlenskii in 1987, is obtained in terms of the decomposition in a multidimensional continuous fraction of Markov vector functions (the resolvent functions) related to the chain of a nonsymmetric operator; the decomposition is performed by the Euler-Jacobi-Perron algorithm. The inverse spectral problem method, based on Lax pairs, on the theory of joint Hermite-Padé approximations, and on the Sturm-Liouville method for finite difference equations is used. Translated fromMatematicheskie Zametki, Vol. 62, No. 4, pp. 588–602, October, 1997. Translated by A. M. Chebotarev     

The Perron–Frobenius theorem – a proof with the use of Markov chains

The Perron–Frobenius theorem for an irreducible nonnegative matrix is proved using the matrix graph and the ergodic theorem of the theory of Markov chains. Bibliography: 7 titles. Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 359, 2008, pp. 5–16.     

A functional limit theorem for the position of a particle in the Lorentz model

We consider a particle moving through a medium under a constant external field. The medium consists of immobile spherical obstacles of equal radii randomly distributed in ℝ³. When the particle collides with an obstacle, it reflects inelastically, with restitution coefficient α ∈, (0, 1). We study the asymptotics of X(t), the position of the particle at time t, as t → ∞. The main result is a functional limit theorem for X(t). Its proof is based on the functional CLT for Markov chains. Bibliography: 10 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 328, 2005, pp. 42–68. An erratum to this article is available at .     

Interpolation of nonlinear functionals by integral continued fractions

We constructively prove the theorem of existence of an interpolation integral chain fraction for a nonlinear functionalF:Q[0,1]→R ¹. Kiev University, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 3, pp. 364–375, March, 1999.     

Noncolliding Jacobi processes as limits of Markov chains on the Gelfand–Tsetlin graph

We introduce a stochastic dynamics related to the measures that arise in harmonic analysis on the infinite–dimensional unitary group. Our dynamics is obtained as a limit of a sequence of natural Markov chains on the Gelfand–Tsetlin graph. We compute the finite-dimensional distributions of the limit Markov process, the generator and eigenfunctions of the semigroup related to this process. The limit process can be identified with the Doob h-transform of a family of independent diffusions. The space-time correlation functions of the limit process have a determinantal form. Bibliography: 21 titles. Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 360, 2008, pp. 91–123.     

On the lindblad equation with unbounded time-dependent coefficients

We prove newa priori estimates for the resolvent of a minimal quantum dynamical semigroup. These estimates simplify well-known conditions sufficient for conservativity and impose continuity conditions on the time-dependent operator coefficients ensuring the existence of conservative solutions of the Markov evolution equations. Translated fromMatematicheskie Zametki, Vol. 61, No. 1, pp. 125–140, January, 1997. Translated by A. M. Chebotarev     

Extensions of dynamical systems and the martingale approximation method

Let T be a measure-preserving transformation of a probability space (X, F, μ) and let A be the generator of a μ-symmetric Markov process with state space X. Under the assumption that A is an “eigenvector” for T an extension of T is constructed in terms of A. By means of this extension a version of the central limit theorem is proved via approximation by martingales. Bibliography: 5 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 216, 1994, pp. 10–19. Translated by V. Sudakov.     

Generation of the Exact Distribution and Simulation of Matched Nucleotide Sequences on a Phylogenetic Tree

Nucleotide sequences are often generated by Monte Carlo simulations to address complex evolutionary or analytic questions but the simulations are rarely described in sufficient detail to allow the research to be replicated. Here we briefly review the Markov processes of substitution in a pair of matching (homologous) nucleotide sequences and then extend it to k matching nucleotide sequences. We describe calculation of the joint distribution of nucleotides of two matching sequences. Based on this distribution, we give a method for simulation of the divergence matrix for n sites using the multinomial distribution. This is then extended to the joint distribution for k nucleotide sequences and the corresponding 4 ^k divergence array, generalizing Felsenstein (Journal of Molecular Evolution, 17, 368–376, 1981), who considered stationary, homogeneous and reversible processes on trees. We give a second method to generate matched sequences that begins with a random ancestral sequence and applies a continuous Markov process to each nucleotide site as in Rambaut and Grassly (Computer Applications in the Biosciences, 13, 235–238, 1997); further, we relate this to an equivalent approach based on an embedded Markov chain. Finally, we describe an approximate method that was recently implemented in a program developed by Jermiin et al. (Applied Bioinformatics, 2, 159–163, 2003). The three methods presented here cater for different computational and mathematical limitations and are shown in an example to produce results close to those expected on theoretical grounds. All methods are implemented using functions in the S-plus or R languages.     

Construction of the extremal function for a functional on the classH ω (n)

For a functional on the classH _ω ⁽ⁿ⁾ ,n≥3, we construct the extremal function on which the upper bound obtained by A. I. Stepanets is attained. Translated fromMatematicheskie Zametki, Vol. 61, No. 4, pp. 519–529, April, 1997. Translated by N. K. Kulman     

Erdős measures,sofic measures,and Markov chains

We consider the random variable ζ = ξ₁ρ+ξ₂ρ²+…, where ξ₁, ξ₂, … are independent identically distibuted random variables taking the values 0 and 1 with probabilities P(ξ_i = 0) = p₀, P(ξ_i = 1) = p₁, 0 < p₀ < 1. Let β = 1/ρ be the golden number. The Fibonacci expansion for a random point ρζ from [0, 1] is of the form η₁ρ + η₂ρ² + … where the random variables η_k are {0, 1}-valued and η_kη_k+1 = 0. The infinite random word η = η₁η₂ … η_n … takes values in the Fibonacci compactum and determines the so-called Erdős measure μ(A) = P(η ∈ A) on it. The invariant Erdős measure is the shift-invariant measure with respect to which the Erdős measure is absolutely continuous. We show that the Erdős measures are sofic. Recall that a sofic system is a symbolic system that is a continuous factor of a topological Markov chain. A sofic measure is a one-block (or symbol-to-symbol) factor of the measure corresponding to a homogeneous Markov chain. For the Erdős measures, the corresponding regular Markov chain has 5 states. This gives ergodic properties of the invariant Erdős measure. We give a new ergodic theory proof of the singularity of the distribution of the random variable ζ. Our method is also applicable when ξ₁, ξ₂, … is a stationary Markov chain with values 0, 1. In particular, we prove that the distribution of ζ is singular and that the Erdős measures appear as the result of gluing together states in a regular Markov chain with 7 states. Bibliography: 3 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 326, 2005, pp. 28–47.     

Two analogs of the Rudin-Carleson theorem

In this article, a new asymptotic estimate for embeddings of singular measures into H^1,∞ is proved and an application to multidimensional analogs of the theorem of Rudin-Carleson is given. Bibliography: 5 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 247, 1997, pp. 200–209. Translated by S. V. Kislyakov.     

An estimate of the optimalZ * value for threshold payments in the case when the insurance requirements assume two values

We give upper and lower bounds for the optimal value ofZ ^*. A special case is considered. Translated fromMetody Matematicheskogo Modelirovaniya, 1998, pp. 171–177.     

Generalization of the Mukhamadiev theorem on the invertibility of functional operators in the space of bounded functions

We establish necessary and sufficient conditions for the invertibility of the linear bounded operator d ^m / dt ^m + A in the space of functions bounded on ℝ. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 3, pp. 398–412, March, 2008.     

Rate function of large deviation for a class of nonhomogeneous markov chains on supercritical percolation network

We model an epidemic with a class of nonhomogeneous Markov chains on the supercritical percolation network on ℤ ^d . The large deviations law for the Markov chain is given. Explicit expression of the rate function for large deviation is obtained.