We present an overview of four approaches of the finite automata use in stringology: deterministic finite automaton, deterministic simulation of nondeterministic finite automaton, finite automaton as a model of computation, and compositions of finite automata solutions. We also show how the finite automata can process strings build over more complex alphabet than just single symbols (degenerate symbols, strings, variables).

Methods for robust controller design, are an invaluable tool in the hands of the control engineer. Several methodologies been developed over the years and have been successfully applied for the solution of specific robust design problems. One of these methods, is based on the Finite Inclusions Theorem (FIT) and exploits properties of polynomials. This has led to the development of FIT-based algorithms for robust stabilization, robust asymptotic tracking and robust noise attenuation design. In this...

This paper considers the problem of determining linear relations from data affected by additive noise in the context of the Frisch scheme. The loci of solutions of the Frisch scheme and their properties are first described in the algebraic case. In this context two main problems are analyzed: the evaluation of the maximal number of linear relations compatible with data affected by errors and the determination of the linear relation actually linking the noiseless data. Subsequently the extension...

Up to present for modelling and analyzing of random phenomenons, some statistical distributions are proposed. This paper considers a new general class of distributions, generated from the logit of the gamma random variable. A special case of this family is the Gamma-Uniform distribution. We derive expressions for the four moments, variance, skewness, kurtosis, Shannon and Rényi entropy of this distribution. We also discuss the asymptotic distribution of the extreme order statistics, simulation issues,...

The energy in a square membrane Ω subject to constant viscous damping on a subset $\omega \subset \Omega$ decays exponentially in time as soon as ω satisfies a geometrical condition known as the “Bardos-Lebeau-Rauch” condition. The rate $\tau \left(\omega \right)$ of this decay satisfies $\tau \left(\omega \right)=2min(-\mu (\omega ),g(\omega \left)\right)$ (see Lebeau [Math. Phys. Stud.19 (1996) 73–109]). Here $\mu \left(\omega \right)$ denotes the spectral abscissa of the damped wave equation operator and $g\left(\omega \right)$ is a number called the geometrical quantity of ω and defined as follows. A ray in Ω is the trajectory generated by the free motion...

We revisit the classical problem of 'Darlington synthesis', or Darlington embedding. Although traditionally it is solved using analytic means, a more natural way to approach it is to use the geometric properties of a well-chosen Hankel map. The method yields surprising results. In the first place, it allows us to formulate necessary and sufficient conditions for the existence of the embedding in terms of systems properties of the transfer operation to be embedded. In addition, the approach allows...

This paper describes a new approach, the HeKatE methodology, to the design and development of complex rule-based systems for control and decision support. The main paradigm for rule representation, namely, eXtended Tabular Trees (XTT), ensures high density and transparency of visual knowledge representation. Contrary to traditional, flat rule-based systems, the XTT approach is focused on groups of similar rules rather than on single rules. Such groups form decision tables which are connected into...

Tumor growth and progression is a complex phenomenon dependent on the interaction of multiple intrinsic and extrinsic factors. Necessary for tumor development is a small subpopulation of potent cells, so-called cancer stem cells, that can undergo an unlimited number of cell divisions and which are proposed to divide symmetrically with a small probability to produce more cancer stem cells. We show that the majority of cells in a tumor must indeed be non-stem cancer cells with limited life span and...

One shows that the linearized Navier-Stokes equation in $𝒪\subset R^d,d\ge 2$, around an unstable equilibrium solution is exponentially stabilizable in probability by an internal noise controller ${\displaystyle V(t,\xi )=\sum _{i=1}^N{V}_i\left(t\right){\psi }_i\left(\xi \right){\dot{\beta }}_i\left(t\right)}$, $\xi \in 𝒪$, where ${\left\{{\beta }_i\right\}}_{i=1}^N$ are independent Brownian motions in a probability space and ${\left\{{\psi }_i\right\}}_{i=1}^N$ is a system of functions on $𝒪$ with support in an arbitrary open subset ${𝒪}_0\subset 𝒪$. The stochastic control input ${\left\{V_i\right\}}_{i=1}^N$ is found in feedback form. One constructs also a tangential boundary noise controller which exponentially stabilizes in probability the equilibrium...

One shows that the linearized Navier-Stokes equation in $𝒪\subset R^d,d\ge 2$, around an unstable equilibrium solution is exponentially stabilizable in probability by an internal noise controller ${\displaystyle V(t,\xi )=\sum _{i=1}^N{V}_i\left(t\right){\psi }_i\left(\xi \right){\dot{\beta }}_i\left(t\right)}$, $\xi \in 𝒪$, where ${\left\{{\beta }_i\right\}}_{i=1}^N$ are independent Brownian motions in a probability space and ${\left\{{\psi }_i\right\}}_{i=1}^N$ is a system of functions on $𝒪$ with support in an arbitrary open subset ${𝒪}_0\subset 𝒪$. The stochastic control input ${\left\{V_i\right\}}_{i=1}^N$ is found in feedback form. One constructs also a tangential boundary noise controller which exponentially stabilizes in probability the equilibrium solution. ...