We extend the applicability of Newton’s method, so we can approximate a locally unique solution of a nonlinear equation in a Banach space setting in cases not covered before. To achieve this, we find a more precise set containing the Newton iterates than

We study convergence in probability of weighted sums of independent random variables which are not necessarily identically distributed. The results obtained are applied to ratios of independent random variables and ratios of smallest order statistics.

In this paper, we consider a stochastic approximation procedure with semi-Markov switchings in an averaging scheme with a small parameter.

In the present paper, the variances in the space and momentum domains of the spherical Abel-Poisson wavelet, as well as the limit of the uncertainty product for $ρ\to0$, where $ρ$ is the scale parameter, are computed. The values of these quantities coincide with a certain accuracy with those of the spherical Poisson wavelet with $\tfrac{1}{2}$ substituted for the order parameter.

Some optimality results for hybrid control problems are presented. The hybrid model under study consists of two subdynamics, one of a standard type governed by an ordinary differential equation, and the other of a special type having a discrete evolution.

Interval estimation of the probability of success in a binomial model is considered. The classical confidence interval is compared with a confidence interval which uses information about non-homogeneity of the sample.

In the present paper, we consider large-scale continuous-time differential matrix Riccati equations having low rank right-hand sides. These equations are generally solved by Backward Differentiation Formula (BDF) or Rosenbrock methods leading to a large scale algebraic Riccati equation which has to be solved for each timestep. We propose a new approach, based on the reduction of the problem dimension prior to integration. We project the initial problem onto an extended block Krylov subspace and obtain a low-dimentional differential algebraic Riccati equation. The latter matrix differential problem is then solved by Backward Differentiation Formula (BDF) method and the obtained solution is used to reconstruct an approximate solution of the original problem. We give some theoretical results and a simple expression of the residual allowing the implementation of a stop test in order to limit the dimension of the projection space. Some numerical experiments will be given.

Let $ S_i,\,i=0,1$, be homeomorphisms of $I=[0,1]$ such that $S_i^{-1}(x)=(1-\epsilon _i)x+\epsilon _ig(x)$, $i=0,1$, for some reals $\epsilon _0 \lt 0$ and $\epsilon _1 \gt 0.$ Here $g$ is a $C^1(0,1)$ homeomorphism and $g(x) \lt x$ for $x\in (0,1).$ Let

A random sequence having two segments being the homogeneous Markov processes is registered. Each segment has his own transition probability law and the length of the segment is unknown and random. The transition probabilities of each process are known and a priori distribution of the disorder moment is given. The decision maker aim is to detect the moment of the transition probabilities change. The detection of the disorder rarely is precise. The decision maker accepts some deviation in estimation of the disorder moment. In the considered model the aim is to indicate the change point with fixed, bounded error with maximal probability. The case with various precision for over and under estimation of this point is analysed. The case when the disorder does not appears with positive probability is also included. The results insignificantly extends range of application, explain the structure of optimal detector in various circumstances and shows new details of the solution construction. The motivation for this investigation is the modelling of the attacks in the node of networks. The objectives is to detect one of the attack immediately or in very short time before or after it appearance with highest probability. The problem is reformulated to optimal stopping of the observed sequences. The detailed analysis of the problem is presented to show the form of optimal decision function.

We propose some estimators of the ratio parameter of a geometric process, i.e. of a stochastic process $\{X_{i},i=1,2,\ldots \}$ for which there exists a positive real number $a,$ called the ratio parameter, such that $\{Y_{i}=a^{i-1}X_{i},\,i=1,2,\l