Interaction of intelligent and autonomous systems – part II: realization of cognitive technical systems

In this study, the application of a meta-modelling technique to structure complex environments is demonstrated with two examples. A special situation-operator model developed to model the human-machine-interaction is applied to automated supervision within the HMI context and to realize flexible and situated interaction of autonomous systems. A concept to automated monitoring of human operators is introduced to illustrate the structuring of complex environments as the first example. In the second example, this modelling technique is applied to a mobile robot to autonomously build and update a mental model of the interaction with the environment as an example of a cognitive technical system.     

Small embeddings of partial directed triple systems and partial triple systems with even λ

In 1976, Lindner and Rosa (Ars Combin. 1 (1976), 159–166) showed that a partial triple system with λ > 1 can be embedded in a finite triple system with the same λ. This result is improved in the case when λ is even by embedding a partial triple system on υ symbols in a triple system on t symbols, t ≡ 0,1 (mod 3), for all t >/ 3(λυ² + υ(2 ? λ) + 1). In the process, it is shown that for any λ >/ 1, a partial directed triple system on υ symbols can be embedded in a directed triple system on t symbols, t ≡ 0, 1 (mod 3), for all t ? 6λv² + 6v(1 ? λ) + 3, thus generalizing a result of Hamm (Proceedings, 14th Southeastern Conf. on Combinatorics, Graph Theory, and Computing, Boca Raton, Florida, 1983).     

Stability and absence of binding for multi-polaron systems

We resolve several longstanding problems concerning the stability and the absence of multi-particle binding for N≥2 polarons. Fröhlich’s 1937 polaron model describes non-relativistic particles interacting with a scalar quantized field with coupling \(\sqrt{\alpha}\), and with each other by Coulomb repulsion of strength U. We prove the following: (i) While there is a known thermodynamic instability for U<2α, stability of matter does hold for U>2α, that is, the ground state energy per particle has a finite limit as N→∞. (ii) There is no binding of any kind if U exceeds a critical value that depends on α but not on N. The same results are shown to hold for the Pekar-Tomasevich model.     

Generalizations of Darbo’s fixed point theorem and solvability of integral and differential systems

In this paper, we establish some new generalizations of Darbo’s fixed point theorem for multivalued mappings. Moreover, we prove the existence of solutions for a class of integral equations by Darbo’s fixed point theorem and the existence of solutions for a class of differential inclusions using a generalization of Darbo’s fixed point theorem.     

Integrability and solvability of polynomial Liénard differential systems

We provide the necessary and sufficient conditions of Liouvillian integrability for Liénard differential systems describing nonlinear oscillators with a polynomial damping and a polynomial restoring force. We prove that Liénard differential systems are not Darboux integrable excluding subfamilies with certain restrictions on the degrees of the polynomials arising in the systems. We demonstrate that if the degree of a polynomial responsible for the restoring force is greater than the degree of a polynomial producing the damping, then a generic Liénard differential system is not Liouvillian integrable with the exception of linear Liénard systems. However, for any fixed degrees of the polynomials describing the damping and the restoring force we present subfamilies possessing Liouvillian first integrals. As a by-product of our results, we find a number of novel Liouvillian integrable subfamilies. In addition, we study the existence of nonautonomous Darboux first integrals and nonautonomous Jacobi last multipliers with a time-dependent exponential factor.     

Theβ- andγ-band systems of14NS and15NS

The spectra of¹⁴NS and¹⁵NS molecules are obtained in a 2450 mc./s. microwave oscillator discharge. Vibrational assignments of theβ- andγ-systems are studied in more detail. Several hitherto unrecorded bands are satisfactorily explained as belonging to higherv′,v″ levels in the Deslandres schemes of bothβ- andγ-systems. The observed isotope shifts (¹⁴NS-¹⁵NS) provide confirmatory evidence for the proposed vibrational analysis. In the case of theγ-system, thev′=2 level of the upper state appears to be strongly perturbed showing a shift of about 38 cm.^?1 from the expected position. In theβ-system, the isotopic shifts in the band-heads involving thev′=0 and 1 levels of the²Δ_5/2 of the upper state show small deviations from expected values. The reality of these small deviations is established beyond doubt by the occurrence of the effect on the 0–0 sub-band which exhibits the isotopic head in a wavelength direction opposite to the expected one.     

On deformations of linear systems of differential equations and the Painlevé property

We consider systems of linear differential equations discussing some classical and modern results in the Riemann problem, isomonodromic deformations, and other related topics. Against this background, we illustrate the relations between such phenomena as the integrability, the isomonodromy, and the Painlevé property. The recent advances in the theory of isomonodromic deformations presented show perfect agreement with that approach.     

Methods for the investigation of dynamical systems with impulse action and “mortal” dynamical systems

The notions of dynamical systems with impulse action and mortal dynamical systems are introduced. Their connection with the idealizations of ordinary dynamical systems is considered. General methods for the investigation of these systems are worked out.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 11, pp. 1605–1613, November, 1991.     

Time symmetry preserving perturbations of systems,and Poincaré mappings

In the present paper, we obtain necessary and sufficient conditions under which two differential systems have the same symmetries described by a reflecting function. Under these conditions, the systems in question have a common shift operator along solutions of these systems on a symmetric time interval [?ω, ω]. Therefore, the mappings over the period [?ω, ω] coincide for such systems provided that these systems are 2ω-periodic.     

Infinite systems of noncolliding generalized meanders and Riemann–Liouville differintegrals

Yor’s generalized meander is a temporally inhomogeneous modification of the 2(ν + 1)-dimensional Bessel process with ν  >   ? 1, in which the inhomogeneity is indexed by $\kappa \in [0, 2(\nu+1))$ . We introduce the noncolliding particle systems of the generalized meanders and prove that they are Pfaffian processes, in the sense that any multitime correlation function is given by a Pfaffian. In the infinite particle limit, we show that the elements of matrix kernels of the obtained infinite Pfaffian processes are generally expressed by the Riemann–Liouville differintegrals of functions comprising the Bessel functions J _ν used in the fractional calculus, where orders of differintegration are determined by ν ? κ. As special cases of the two parameters (ν, κ), the present infinite systems include the quaternion determinantal processes studied by Forrester, Nagao and Honner and by Nagao, which exhibit the temporal transitions between the universality classes of random matrix theory.     

Interaction of intelligent and autonomous systems – part I: qualitative structuring of interaction

In this part I of the study, the development and the background of a meta-modelling technique to structure complex environments of systems is illustrated. A special situation-operator-model (SOM) developed to model the human-machine-interaction (HMI) is understood as a structuring approach modelling the interaction of intelligent systems in general. Part I of this study is structured as follows: Section 1 reflects on the term modelling with respect to the task: how to model interaction of unknown interaction partners, Section 2 briefly reviews the basics of the previously published foundations of the work and the SOM. The developed situation-operator-metamodelling approach is used to illustrate the connections between control loops and algorithms in Section 3, including a systematic view of systems' interaction mechanisms. In part II (part II available in this issue, pp. 319–339) of the contribution, the practical applications of the introduced approach are shown.     

Error estimates and stability of itô-type systems of nonlinear stochastic integro-differential equations

By developing an integral representation of a function of solution process of a system of Ito-type stochastic nonlinear integro-differential equations, the error estimates on absolute p-th deviation of a solution process with the solution of the mean of the corresponding deterministic system of integro-differential equations is analyzed. Furthermore, the p-th moment stability properties of the steady state of the system is studied. The obtained results would provide a partial solution to one of the fundamental problems in modelling of dynamic systems, namely, to what extent incorporating randomness in the system causes the change in behavior of the system relative to its deterministic version     

Input–output control systems and dichotomy of variational difference equations

We propose a new and unified approach for the study of dichotomy of variational difference equations, establishing a link between control methods and basic techniques from interpolation theory. We obtain necessary and sufficient conditions for the existence of uniform dichotomy and, respectively, for uniform exponential dichotomy of variational difference equations in terms of the admissibility of general pairs of sequence spaces. We provide a classification of the main classes of sequence spaces where the input spaces and the output spaces may belong to, for each dichotomy property and prove that the hypotheses on the underlying sequence spaces cannot be removed. The obtained results extend the framework to the study of dichotomy of variational difference equations, hold without any requirement on the coefficients and are applicable to all systems of variational difference equations.     

Mean and variance optimization of non–linear systems and worst–case analysis

In this paper, we consider expected value, variance and worst–case optimization of nonlinear models. We present algorithms for computing optimal expected value, and variance policies, based on iterative Taylor expansions. We establish convergence and consider the relative merits of policies based on expected value optimization and worst–case robustness. The latter is a minimax strategy and ensures optimal cover in view of the worst–case scenario(s) while the former is optimal expected performance in a stochastic setting. Both approaches are used with a small macroeconomic model to illustrate relative performance, robustness and trade-offs between the alternative policies.     

On the completeness of systems of eigenfunctions and associated functions of differential operators of orders 2 ? �� and 1 ? ��

In the space of vector-functions, we consider a boundary-value problem for differential operators of fractional orders (2 ? ??) and (1 ? ??) and prove the completeness of the system of eigenfunctions and associated functions of this problem in the space $L_1 \left( {\left[ {0,1} \right],\,\mathbb{C}^p } \right)$ .     

Asymptotic equivalence of linear stochastic Itô systems and oscillation of solutions of linear second-order equations

We obtain conditions for the asymptotic equivalence of linear stochastic and deterministic systems and analyze the oscillation of solutions of the Itô stochastic equation of the second order of the form \(\ddot x + (p(t) + q(t)\dot W(t))x = 0\) on the half-line.     

Stability analysis and stabilization of linear symmetric matrix-valued continuous,discrete, and impulsive dynamical systems — A unified approach for the stability analysis and the stabilization of linear systems

Matrix-valued dynamical systems are an important class of systems that can describe important processes such as covariance/second-order moment processes, or processes on manifolds and Lie Groups. We address here the case of processes that leave the cone of positive semidefinite matrices invariant, thereby including covariance and second-order moment processes. Both the continuous-time and the discrete-time cases are first considered. In the LTV case, the obtained stability and stabilization conditions are expressed as differential and difference Lyapunov conditions which are equivalent, in the LTI case, to some spectral conditions for the generators of the processes. Convex stabilization conditions are also obtained in both the continuous-time and the discrete-time setting. It is proven that systems with constant delays are stable provided that the systems with zero-delays are stable—which mirrors existing results for linear positive systems. The results are then extended and unified into an impulsive formulation for which similar results are obtained. The proposed framework is very general and can recover and/or extend many of the existing results in the literature on linear systems related to (mean-square) exponential (uniform) stability. Several examples are discussed to illustrate this claim by deriving stability conditions for stochastic systems driven by Brownian motion and Poissonian jumps, Markov jump systems, (stochastic) switched systems, (stochastic) impulsive systems, (stochastic) sampled-data systems, and all their possible combinations.     

Synchronization and anti-synchronization of Lu and Bhalekar–Gejji chaotic systems using nonlinear active control

This paper aims at synchronization and anti-synchronization between Lu chaotic system, a member of unified chaotic system, and recently developed Bhalekar–Gejji chaotic system, a system which cannot be derived from the member of unified chaotic system. These synchronization and anti-synchronization have been achieved by using nonlinear active control since the parameters of both the systems are known. Lyapunov stability theory is used and required condition is derived to ensure the stability of error dynamics. Controller is designed by using the sum of relevant variables in chaotic systems. Simulation results suggest that proposed scheme is working satisfactorily.