On categories and foundations of the theory of nonlinear control dynamical systems: VII

In the present paper, we show how the solution of an important problem of control theory (invariance under perturbations) can be obtained with the use of the categorical notion of a quotient system.     

Directed cyclic Hamiltonian cycle systems of the complete symmetric digraph

In this paper, we prove that directed cyclic Hamiltonian cycle systems of the complete symmetric digraph, , exist if and only if n is odd with n≠15 and n≠p^α for p an odd prime and α≥2 or with n≠2p^α for p an odd prime and α≥1. We also show that directed cyclic Hamiltonian cycle systems of the complete symmetric digraph minus a set of n/2 vertex-independent digons, (K_n−I)^∗, exist if and only if .     

Optimal difference systems of sets and partition-type cyclic difference packings

Difference systems of sets (DSSs) are combinatorial structures which were introduced by Levenshtein in connection with code synchronization. In this paper, we give some recursive constructions of DSSs by using partition-type cyclic difference packings, and obtain new infinite classes of optimal DSSs.     

On the stability of steady motions of systems with cyclic coordinates

A method of solving stabilization problems by isolating a controlled subsystem of possibly smaller dimension [1, 2] is developed further. The stabilizing action is determined by the solution of an optimal stabilization problem [3] for a linear controlled subsystem. The control that is found is implemented in the form of a feedback loop that uses an estimate [4] of the state vector (or part of it) constructed by measuring the perturbations of the positional coordinates. The stability of the unperturbed motion in a complete closed system is established by reducing the problem to a special case of the theory of critical cases [5, 6] or to the problem of stability under constantly acting perturbations [6].     

Controllability and observability in the problem of stabilizing mechanical systems with cyclic coordinates

An approach based on linear control theory is used to solve the problem of stabilizing the steady motions of holonomic mechanical systems in which only cyclic coordinates are controllable [1–3]. The most general structure of forces acting on the system is considered and it is assumed that the constraints imposed are time-independent. The set of new criteria of controllability and observability based on the reduction of the problem under consideration is obtained. The reduction enables one to reduce the investigation of these problems to an analysis of a problem of less dimensions.     

Language and Mathematics - Key Factors influencing the Comprehension Process in reality-based Tasks

Solving reality-based tasks is an important goal in mathematics instruction and is anchored in education standards determined by mathematical modeling skills. These tasks demand a serious examination of the real-world as well as text comprehension to successfully solve them. Therefore, this study empirically reconstructed the comprehension process during the solution of reality-based tasks and examined how it correlates with process-, person-, and task-related attributes. Fifty-five seventh graders using the Think Aloud Method solved reality-based tasks that were varied in their linguistic and situational complexity level. Their mathematical performance as well as their reading ability were measured. Based on detailed analyses of solution processes combined with the performance data, we point out the relevance of comprehension activities and empirically identify factors that influence the comprehension process.     

On a class of cyclic methods for the numerical integration of stiff systems of O.D.E.s

A class of cyclic linear multistep methods suitable for the approximate numerical integration of stiff systems of first order ordinary differential equations is developed. Particular attention is paid to the problem of deriving schemes which are almostA-stable, self starting, have relatively high orders of accuracy and contain a built in error estimate. These requirements demand that the linear multistep methods which are used are solved iteratively rather than directly in the usual way and an efficient method for doing this is suggested. Finally the algorithms are illustrated by application to a particular test problem.     

Attainability of the minimal exponential growth rate for free products of finite cyclic groups

We consider free products of two finite cyclic groups of orders 2 and n, where n is a prime power. For any such group ℤ₂ * ℤ _n = 〈a, b | a ² = b ⁿ = 1〉, we prove that the minimal growth rate α _n is attained on the set of generators {a, b} and explicitly write out an integer polynomial whose maximal root is α _n . In the cases of n = 3, 4, this result was obtained earlier by A. Mann. We also show that under sufficiently general conditions, the minimal growth rates of a group G and of its central extension [(G)\tilde]\tilde G coincide and that the attainability of one implies the attainability of the other. As a corollary, the attainability is proved for some cyclic extensions of the above-mentioned free products, in particular, for groups 〈a, b | a ² = b ⁿ 〉, which are groups of torus knots for odd n.     

Mixed products of multiplicative systems without discontinuities of the second kind

Mixed products of multiplicative evolution operator systems with values in a Banach ring R that satisfy the orthogonality condition are studied with the aid of their infinitesimal additive systems.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 3, pp. 332–340, March, 1990.     

The relative Brauer group and generalized cross products for a cyclic covering of affine space

We study algebras and divisors on a normal affine hypersurface defined by an equation of the form zⁿ=f(_x1,…,_xm)$z^n=f(x_1,\dots ,x_m)$. The coordinate ring is T=k[_x1,…,_xm,z]/(zⁿ−f)$T=k[x_1,\dots ,x_m,z]/(z^n-f)$, and if R=k[_x1,…,_xm][f⁻¹]$R=k[x_1,\dots ,x_m]\left[f^{-1}\right]$ and S=R[z]/(zⁿ−f)$S=R\left[z\right]/(z^n-f)$, then S$S$ is a cyclic Galois extension of R$R$. We show that if the Galois group is G$G$, the natural map H¹(G,Cl(T))→H¹(G,Pic(S))$H^1(G,Cl\left(T\right))\to H^1(G,Pic\left(S\right))$ factors through the relative Brauer group B(S/R)$B(S/R)$ and that all of the maps are onto. Sufficient conditions are given for H¹(G,Cl(T))$H^1(G,Cl\left(T\right))$ to be isomorphic to B(S/R)$B(S/R)$. As an example, all of the groups, maps, divisors and algebras are computed for an affine surface defined by an equation of the form zⁿ=(y−_a1x)?(y−_anx)(x−1)$z^n=(y-a_{1}x)?(y-a_{n}x)(x-1)$.     

The use of the Sherman–Morrison–Woodbury formula to solve cyclic block tri-diagonal and cyclic block penta-diagonal linear systems of equations

The article presents a new theoretical viewpoint of Batista’s algorithms for solving cyclic tri-diagonal (and penta-diagonal) linear systems. The theory is based on the Sherman–Morrison–Woodbury formula.     

Controllability and observability in the problem of stabilizing steady motions of non-holonomic mechanical systems with cyclic coordinatest

The approach to the solution of stabilization problems for steady motions of holonomic mechanical systems [1, 2] based on linear control theory, combined with the theory of critical cases of stability theory, is used to solve the analogous problems for non-holonomic systems. It is assumed that the control forces may affect both cyclic and positional coordinates, where the number r of independent control inputs may be considerably less than the number n of degrees of freedom of the system, unlike in many other studies (see, e.g., [3–5]), in which as a rule r = n. Several effective new criteria of controllability and observability are formulated, based on reducing the problem to a problem of less dimension. Stability analysis is carried out for the trivial solution of the complete non-linear system, closed by a selected control. This analysis is a necessary step in solving the stabilization problem for steady motion of a non-holonomic system (unlike holonomic systems), since in most cases such a system is not completely controllable.     

Actions of symbolic dynamical systems on C*-algebras. II. Simplicity of C*-symbolic crossed products and some examples

A C*-symbolic dynamical system ${(\mathcal{A}, \rho, \Sigma)}A C*-symbolic dynamical system (A, r, S){(\mathcal{A}, \rho, \Sigma)} consists of a unital C*-algebra A{\mathcal{A}} and a finite family { r_a }_{a ? S}{\{ \rho_\alpha \}_{\alpha \in \Sigma}} of endomorphisms ρ _α of A{\mathcal{A}} indexed by symbols α of Σ satisfying some conditions. The endomorphisms r_a, a ? S{\rho_\alpha, \alpha \in \Sigma } yield both a subshift Λ and a C*-algebra of a Hilbert C*-bimodule. The obtained C*-algebra is regarded as a crossed product of A{\mathcal{A}} by the subshift Λ. We will study simplicity condition of these C*-algebras. Some examples such as irrational rotation Cuntz–Krieger algebras will be studied.     

The theory of the numerical-analytic method: Achievements and new trends of development. VII

For the numerical-analytic method suggested by A. M. Samoilenko in 1965, we analyze the application to abstract differential equations, implicit equations, and control problems. Institute of Mathematics, Ukrainian Academy of Sciences, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 9, pp. 1244–1261, September, 1999.