PREFACE

This volume is formed by papers prepared by speakers of the G2R2-conference on“Graphs and Groups,Relations and Representations”held in Novosibirsk,Russia in August 2018.The topics of the papers reflect the wide range of themes covered by the lectures given at the conference.The first paper by Akihiro Munemasa and Keiji Ito of the volume extends the classical notion of Krein parameters to the class of coherent configurations where the commutativity condition is weakened.The second paper by Stefan Gyiirki is devoted to directed strongly regular graphs,providing a survey on this topic and presenting a list of small examples.The third paper by Clara Franchi,Alexander A.     

PERTURBATION EQUATION OF DLW HIERARCHY AND ITS HAMILTONIAN STRUCTURE

The first-order perturbation equation of DLW hierarchy is derived by virtue of zero-curvature equation. Then, Hamiltonian structure of the obtained system is given by means of component-trace identity.     

Coefficient of Partial Correlation and Its Calculation

This thesis offers the general concept of coefficient of partial correlation. Starting with regression analysis, the paper, by using samples, infers the general formula of expressing coefficient of partial correlation by way of simple correlation coefficient.     

Inequalities on the inner radius of univalency and the norm of pre-Schwarzian derivative

In this paper, we get a lower bound of inner radius of univalency of Schwarzian derivative by means of the norm of pre-Schwarzian derivative. Furthermore, we apply the theory of Universal Teichmuller Space to explain its geometric meaning which shows the relationship between the inner radius in Universal Teichmuller Space embedded by Schwarzian derivative and the norm defined in Universal Teichmuller Space embedded by pre-Schwarzian derivative.     

Upper Semicontinuity and Kolmogorov ε-Entropy of Global Attractor for κ-Dimensional Lattice Dynamical System Corresponding to Klein-Gordon-SchrSdinger Equation

In this paper, we establish the existence of a global attractor for a coupled κ-dimensional lattice dynamical system governed by a discrete version of the Klein-Gordon-SchrSdinger Equation. An estimate of the upper bound of the Kohnogorov ε-entropy of the global attractor is made by a method of element decomposition and the covering property of a polyhedron by balls of radii ε in a finite dimensional space. Finally, a scheme to approximate the global attractor by the global attractors of finite-dimensional ordinary differential systems is presented .     

Functions approximated by any sequence of interpolation polynomials

In this note, we seek for functions f which are approximated by the sequence of interpolation polynomials of f obtained by any prescribed system of nodes.     

Linear isometries of Hardy spaces

According to results established by DeLeeuw-Rudin-Wermer and by Forelli,all linear isometries of any Hardy space H~p(p≥1,p≠2)on the open unit discΔof C are represented by weighted composition operators defined by inner functions onΔ.After reviewing(and completing when p=∞)some of those results,the present report deals with a characterization of periodic and almost periodic semigroups of linear isometries of H~p.     

关于半环半群

This paper characterizes the multiplication operator semigroup of semiring semigroup by using the tensor product of semigroups, and by using the endomorphism semigroup, and considers some properties of the multiplication operator semigroup of semiring semigroup.     

Geometric meanings of the parameters on rational conic segments

Using algebraic and geometric methods,functional relationships between a point on a conic segment and its corresponding parameter are derived when the conic segment is presented by a rational quadratic or cubic Bézier curve.That is,the inverse mappings of the mappings represented by the expressions of rational conic segments are given.These formulae relate some triangular areas or some angles,determined by the selected point on the curve and the control points of the curve,as well as by the weights of the rational Bézier curve.Also,the relationship can be expressed by the corresponding parametric angles of the selected point and two endpoints on the conic segment,as well as by the weights of the rational Bézier curve.These results are greatly useful for optimal parametrization,reparametrization,etc.,of rational Bézier curves and surfaces.     

Number of Induced Matchings of Graphs

A matching M of a graph G is an induced matching if no two edges in M arejoined by an edge of G.Let iz(G) denote the total number of induced matchings of G,named iz-index.It is well known that the Hosoya index of a graph is the total number of matchings and the Hosoya index of a path can be calculated by the Fibonacci sequence.In this paper,we investigate the iz-index of graphs by using the Fibonacci-Narayana sequence and characterize some types of graphs with minimum and maximum iz-index,respectively.     

Quantum spaces

In this paper, a more general concept of quantum space is given by modifying the original concept defined by Borceux and Bossche. We show that a quantum space is a topological analogue of a quantale defined by Mulvey, and also a non-commutative generalization of the Zariski spectrum of a commutative ring. But quantum spaces are not good enough to have much of the properties of topological spaces, such as product spaces and quotient spaces.     

Test for autocorrelation change in discretely observed Ornstein-Uhlenbeck processes driven by Lévy processes

In this paper,we consider the problem of testing for an autocorrelation change in discretely observed Ornstein-Uhlenbeck processes driven by Lévy processes.For a test,we propose a class of test statistics constructed by an iterated cumulative sums of squares of the difference between two adjacent observations.It is shown that each of the test statistics weakly converges to the supremum of the square of a Brownian bridge.The test statistics are evaluated by some empirical results.     

收缩投影法与涉及Hemi-相对相对非扩张映象的广义平衡问题的强收敛性

Motivated by the recent result obtained by Takahashi and Zembayashi in 2008,we prove a strong convergence theorem for finding a common element of the set of solutions of a generalized equilibrium problem and the set of fixed points of a hemi-relatively nonexpansive mapping in a Banach space by using the shrinking projection method.The main results obtained in this paper extend some recent results.     

Exact Synchronization for a Coupled System of Wave Equations with Dirichlet Boundary Controls

In this paper, the exact synchronization for a coupled system of wave equations with Dirichlet boundary controls and some related concepts are introduced. By means of the exact null controllability of a reduced coupled system, under certain conditions of compatibility, the exact synchronization, the exact synchronization by groups, and the exact null controllability and synchronization by groups are all realized by suitable boundary controls.     

Lagrange插值逼近导数的平均收敛

<正>We consider the rate of mean convergence of derivatives by Lagrange interpolation operators L_n(f,x) based on the zeros of Chebyshev polynomials of the first kind.A sharp estimate of the derivative of L_n(f,x)—f(x) in terms of the error of best approximation by polynomials of degree n is derived.     

A branching particle system approximation for nonlinear stochastic filtering

The optimal filter π = {π t,t ∈ [0,T ]} of a stochastic signal is approximated by a sequence {π n t } of measure-valued processes defined by branching particle systems in a random environment(given by the observation process).The location and weight of each particle are governed by stochastic differential equations driven by the observation process,which is common for all particles,as well as by an individual Brownian motion,which applies to this specific particle only.The branching mechanism of each particle depends on the observation process and the path of this particle itself during its short lifetime δ = n 2α,where n is the number of initial particles and α is a fixed parameter to be optimized.As n →∞,we prove the convergence of π n t to π t uniformly for t ∈ [0,T ].Compared with the available results in the literature,the main contribution of this article is that the approximation is free of any stochastic integral which makes the numerical implementation readily available.     

Stabilization of nonuniform Euler-Bernoulli beam with locally distributed feedbacks

In this article, we study the stabilization problem of a nonuniform Euler-Bernoulli beam with locally distributed feedbacks. Firstly, using the semi-group theory, we establish the well-posedness of the associated closed loop system. Then by proving the uniqueness of the solution of a related ordinary differential equations, we derive the asymptotic stability of the closed loop system. Finally, by means of the piecewise frequency domain multiplier method, we prove that the corresponding closed loop system can be exponentially stabilized by only one of the two distributed feedback controls proposed in this paper.     

Covariant Projective Extensions

The theory of crossed products of C~*-algebras by groups of automorphisms is a well-developed area of the theory of operator algebras. Given the importance and the success ofthat theory, it is natural to attempt to extend it to a more general situation by, for example,developing a theory of crossed products of C~*-algebras by semigroups of automorphisms, or evenof endomorphisms. Indeed, in recent years a number of papers have appeared that are concernedwith such non-classical theories of covariance algebras, see, for instance [1-3].     

Uniform Compact Attractors for a Nonlinear Non-Autonomous Equation of Viscoelasticity

In this paper we prove the regularity, exponential stability of global solutions and existence of uniform compact attractors of semiprocesses, generated by the global solutions, of a two-parameter family of operators for a nonlinear onedimensional non-autonomous equation of viscoelasticity. We employ the properties of the analytic semigroup to show the compactness for the semiprocess generated by the global solutions.