The nonlinear stochastic control problem related with flow control is considered. The state of the link is described by controlled hidden Markov process while the loss flow is described by the counting process with the intensity depending on the current transmission rate and unobserved link state. The control is the transmission rate and have to be chosen as non-anticipating process depending on the observation of the loss process. The aim of the control is to achieve the maximum of some utility function taking into account the losses of transmitted information. Originally the problem belongs to a class of stochastic control with incomplete information, however, the optimal filtering equations giving the estimation of the current link state based on the observation of the loss process give the opportunity to reduce the problem to the standard stochastic control problem with full observations. Then, a necessary optimality condition is derived in the form of stochastic maximum principle which allows us to obtain explicit analytic expressions for the optimal control in some particular cases. The optimal and suboptimal controls are investigated and compared with the flow control schemes which is used in TCP/IP networks. In particular, the optimal control demonstrates a much smoother behavior than the currently used TCP/IP congestion control.

We consider a recursive algorithm to construct an aggregated estimator from a finite number of base decision rules in the classification problem. The estimator approximately minimizes a convex risk functional under the l1-constraint. It is defined by a stochastic version of the mirror descent algorithm which performs descent of the gradient type in the dual space with an additional averaging. The main result of the paper is an upper bound for the expected accuracy of the proposed estimator. This bound is of the order with an explicit and small constant factor C, where M is the dimension of the problem and t stands for the sample size. A similar bound is proved for a more general setting, which covers, in particular, the regression model with squared loss.

A multiserver on-demand system is considered in which each call has three interdependent random characteristics: the required number of servers, capacity, and service time. The total capacity of calls and the total number of servers in the system are limited. The type of a call is defined by the number of servers required for its service. We find a stationary distribution of the number of calls in the system, as well as the loss probability for a call of each type.

Eigenvalue densities of complex noncentral Wishart matrices are investigated to study an open problem in information theory. Specifically, the largest, smallest, and joint eigenvalue densities of complex noncentral Wishart matrices are derived. These densities are expressed in terms of complex zonal polynomials and invariant polynomials. A connection between the complex Wishart matrix theory and information theory is given. This facilitates evaluation of the most important information-theoretic measure, the so-called ergodic channel capacity. In particular, the capacity of multiple-input multiple-output (MIMO) Rician distributed channels is investigated. We consider both spatially correlated and uncorrelated MIMO Rician channels and derive exact and easily computable tight upper bound formulas for ergodic capacities. Numerical results are also given, which show how the channel correlation degrades the capacity of the communication system.

A nonlinear stochastic control problem related to ow control is considered. It is assumed that the state of a link is described by a controlled hidden Markov process with a finite state set, while the loss ow is described by a counting process with intensity depending on a current transmission rate and an unobserved link state. The control is the transmission rate, and it has to be chosen as a nonanticipating process depending on the observation of the loss process. The aim of the control is to achieve the maximum of some utility function that takes into account losses of the transmitted information. Originally, the problem belongs to the class of stochastic control problems with incomplete information; however, optimal filtering equations that provide estimation of the current link state based on observations of the loss process allow one to reduce the problem to a standard stochastic control problem with full observations. Then a necessary optimality condition is derived in the form of a stochastic maximum principle, which allows us to obtain explicit analytic expressions for the optimal control in some particular cases. Optimal and suboptimal controls are investigated and compared with the ow control schemes used in TCP/IP (Transmission Control Protocols/Internet Protocols) networks. In particular, the optimal control demonstrates a much smoother behavior than the TCP/IP congestion control currently used.

We develop and study the concept of similarity functions for q-ary sequences. For the case q = 4, these functions can be used for a mathematical model of the DNA duplex energy [1,2], which has a number of applications in molecular biology. Based on these similarity functions, we define a concept of DNA codes [1]. We give brief proofs for some of our unpublished results [3] connected with the well-known deletion similarity function [4–6]. This function is the length of the longest common subsequence; it is used in the theory of codes that correct insertions and deletions [5]. Principal results of the present paper concern another function, called the similarity of blocks. The difference between this function and the deletion similarity is that the common subsequences under consideration should satisfy an additional biologically motivated [2] block condition, so that not all common subsequences are admissible. We prove some lower bounds on the size of an optimal DNA code for the block similarity function. We also consider a construction of close-to-optimal DNA codes which are subcodes of the parity-check one-error-detecting code in the Hamming metric [7].

Application of some known methods of code construction (such as the Vasil'ev, Plotkin, and Mollard methods) to transitive codes satisfying certain auxiliary conditions yields infinite classes of large-length transitive codes, in particular, at least ⌊k/2⌋2 nonequivalent perfect transitive codes of length n = 2 k − 1, k > 4. A similar result is valid for extended perfect transitive codes.

The problem of constructing asymptotic bounds for multiple packings in the space of q-ary sequences of length n is considered. For the zero rate, tightness of the expurgation bound is proved.

We study coset weight distributions of binary primitive (narrow-sense) BCH codes of length n = 2 m (m odd) with minimum distance 8. In the previous paper [1], we described coset weight distributions of such codes for cosets of weight j = 1, 2, 3, 5, 6. Here we obtain exact expressions for the number of codewords of weight 4 in terms of exponential sums of three types, in particular, cubic sums and Kloosterman sums. This allows us to find the coset distribution of binary primitive (narrow-sense) BCH codes with minimum distance 8 and also to obtain some new results on Kloosterman sums over finite fields of characteristic 2.

The structure of symmetry groups of Vasilev codes is studied. It is proved that the symmetry group of an arbitrary perfect binary non-full-rank Vasilev code of length n is always nontrivial; for codes of rank n - log(n + 1) +1, an attainable upper bound on the order of the symmetry group is obtained.

We give new proofs of asymptotic upper bounds of coding theory obtained within the frame of Delsarte’s linear programming method. The proofs rely on the analysis of eigenvectors of some finite-dimensional operators related to orthogonal polynomials. Examples of the method considered in the paper include binary codes, binary constant-weight codes, spherical codes, and codes in projective spaces.

An ensemble of codes defined by parity-check matrices composed of M × M permutation matrices is considered. This ensemble is a subensemble of the ensemble of low-density parity-check (LDPC) codes considered by Gallager [1]. We prove that, as M , the minimum distance of almost all codes in the ensemble grows linearly with M. We also show that in several cases the asymptotic minimum-distance-to-block-length ratio for almost all codes in the ensemble satisfies Gallagers bound [1].

We suggest and experimentally investigate a method to construct forecasting algorithms based on data compression methods (or the so-called archivers). By the example of predicting currency exchange rates we show that the precision of thus obtained predictions is relatively high.

Jayness entropy concentration theorem states that, for most words 1 ...N of length N such that $$\mathop \Sigma \limits_{i = 1}{\rm N} \;f(\omega _i ) \approx vN$$ , empirical frequencies of values of a function f are close to the probabilities that maximize the Shannon entropy given a value v of the mathematical expectation of f. Using the notion of algorithmic entropy, we define the notions of entropy for the Bose and Fermi statistical models of unordered data. New variants of Jayness concentration theorem for these models are proved. We also present some concentration properties for free energy in the case of a nonisolated isothermal system. Exact relations for the algorithmic entropy and free energy at extreme points are obtained. These relations are used to obtain tight bounds on uctuations of energy levels at equilibrium points.

In this paper we give a sufficient condition for additivity of the minimum output entropy for a pair of given channels and an analytic verification of this condition for specific quantum channels breaking a closely related multiplicativity property [1, 2]. This yields validity of the additivity conjecture for these channels, a result obtained by a different method in [3]. Our proof relies heavily upon certain concavity properties of the output entropy, which are of independent interest.

A new attack (called “gradient statistical”) on block ciphers is suggested and experimentally investigated. We demonstrate the possibility of applying it to ciphers for which no attacks are known except for the exhaustive key search.

We propose an adaptive algorithm for tracking historical volatility. The algorithm borrows ideas from nonparametric statistics. In particular, we assume that the volatility is a several times differentiable function with a bounded highest derivative. We propose an adaptive algorithm with a Kalman filter structure, which guarantees the same asymptotics (well known from statistical inference) with respect to the sample size n, n → ∞. The tuning procedure for this filter is simpler than for a GARCH filter.

We consider a multiple communication channel system having channels with different transmission rates. We give a solution for the problem (of interest for such a system) of finding the probability densities and probability distribution functions of the N-busy period length. A solution in the case of identical channels (servers) was given in [1].