Calculating the edge Wiener and edge Szeged indices of graphs

The edge Szeged and edge Wiener indices of graphs are new topological indices presented very recently. It is not difficult to apply a modification of the well-known cut method to compute the edge Szeged and edge Wiener indices of hexagonal systems. The aim of this paper is to propose a method for computing these indices for general graphs under some additional assumptions.     

Passage to diffusion chaos in a model of an ecological system

We carry out analytical and numerical analysis of a model of an ecological system described by a system of nonlinear partial differential equations of reaction-diffusion type. We find conditions for the bifurcation of periodic spatially homogeneous and inhomogeneous solutions from the thermodynamic branch of the system. We show that the passage to diffusion chaos in the model occurs, in complete agreement with the universal Feigenbaum-Sharkovskii-Magnitskii bifurcation theory, via a subharmonic cascade of bifurcations of stable limit cycles.     

Substitution systems and the three versions of distributional chaos

There are three types of distributional chaos, namely DC1, DC2 and DC3. In this paper we present two constant-length substitution systems, one is DC2 but not DC1, and the other is DC3 but not DC2. (In this paper, chaos means existence of an uncountable scrambled set of the corresponding type while the existing examples deal with single pairs of points only.)     

Assessment of an edge type settlement of above ground liquid storage tanks using a simple beam model

Analysis of deformation and bending moment distributions along sections of the bottom plate of a large unanchored cylindrical liquid storage tank with appreciable out-of-plane localized differential edge settlement is considered. The analysis uses approximate simple slender beam bending theory to model localized edge settlements of the plate and takes into account the effects of foundation compliance, initial settlement shape, shell and hydrostatic loadings and the shell-bottom plate junction stiffness. The obtained model is solved, in the elastic range, using a combined analytical–numerical procedure for the deflection and bending moment distributions along the beam. The obtained approximate solutions were displayed graphically for selected values of system parameters: edge settlement amplitude, plate thickness, foundation stiffness, and hydrostatic load. The maximum allowable edge displacement amplitudes based on the plate yielding stress predicted by the present study are compared for the selected values of system parameters with those recommended in the API standard 653.     

Bifurcations and chaos of a discrete mathematical model for respiratory process in bacterial culture

The discrete mathematical model for the respiratory process in bacterial culture obtained by Euler method is investigated. The conditions of existence for flip bifurcation and Hopf bifurcation are derived by using center manifold theorem and bifurcation theory, condition of existence of chaos in the sense of Marotto＇s definition of chaos is proved. The bifurcation diagrams, Lyapunov exponents and phase portraits are given for different parameters of the model, and the fractal dimension of chaotic attractor was also calculated. The numerical simulation results confirm the theoretical analysis and also display the new and complex dynamical behaviors compared with the continuous model. In particular~ we found that the new chaotic attractor, and new types of two or four coexisting chaotic attractors, and two coexisting invariant torus.     

A note on the edge cover chromatic index of multigraphs

Let G be a multigraph with vertex set V(G). An edge coloring C of G is called an edge-cover-coloring if each color appears at least once at each vertex v∈V(G). The maximum positive integer k such that G has a k-edge-cover-coloring is called the edge cover chromatic index of G and is denoted by . It is well known that , where μ(v) is the multiplicity of v and δ(G) is the minimum degree of G. We improve this lower bound to δ(G)−1 when 2≤δ(G)≤5. Furthermore we show that this lower bound is best possible.     

一般三角帐篷映射混沌性与两种混沌互不蕴含性

将三角帐篷映射推广为一般的n-三角帐篷映射,并且借助于一般Bernoulli移位映射,Banks定理与Li-Yorke定理,首先证明：对于任意的正整数n,n-三角帐篷映射既是Devaney混沌的,也是Li-Yorke混沌的.然后,利用所得到的结果,通过实例展示：Devaney混沌与Li-Yorke混沌的互不蕴含性.     

Convergence analysis of an adaptive edge element method for Maxwell's equations

An efficient and reliable a posteriori error estimate is derived for solving three-dimensional static Maxwell's equations by using the edge elements of first family. Based on the a posteriori error estimates, an adaptive finite element method is constructed and its convergence is established. Compared with the existing results, an important advantage of the new theory lies in its feature that the usual marking of elements based on the oscillation is not needed in our adaptive algorithm, while the linear convergence of the algorithm can be still demonstrated in terms of the reduction of the energy-norm error and the oscillation. Numerical examples are provided which demonstrate the effectiveness and robustness of the adaptive methods.     

Closed-loop suppression of chaos in nonlinear driven oscillators

Summary This paper discusses the suppression of chaos in nonlinear driven oscillators via the addition of a periodic perturbation. Given a system originally undergoing chaotic motions, it is desired that such a system be driven to some periodic orbit. This can be achieved by the addition of a weak periodic signal to the oscillator input. This is usually accomplished in open loop, but this procedure presents some difficulties which are discussed in the paper. To ensure that this is attained despite uncertainties and possible disturbances on the system, a procedure is suggested to perform control in closed loop. In addition, it is illustrated how a model, estimated from input/output data, can be used in the design. Numerical examples which use the Duffing-Ueda and modified van der Pol oscillators are included to illustrate some of the properties of the new approach.This work has been supported by CNPq (Brazil) under Grant 200597/90-6 and SERC (UK) under Grant GR/H 35286.     

Edge agreement of second-order multi-agent system with dynamic quantization via the directed edge Laplacian

This work explores the edge agreement problem of second-order multi-agent systems with dynamic quantization under directed communication. To begin with, by virtue of the directed edge Laplacian, we propose a model reduction representation of the closed-loop multi-agent system depending on the spanning tree subgraph. Considering the limitations of the finite bandwidth channels, the quantization effects of second-order multi-agent systems under directed graph are considered. The static quantizers generally contain a fixed quantization interval and infinite quantization level, which are, to some extent, inefficient and impractical. To further reduce the bit depth (number of bits available) and to obtain better precision, the dynamic quantized communication strategy referring to zooming in-zooming out scheme is required. Based on the reduced model associated with the essential edge Laplacian, the asymptotic stability of second-order multi-agent systems under dynamic quantized effects with only finite quantization level can be guaranteed. Finally, the simulation of altitude alignment of micro air vehicles is provided to verify the theoretical results.     

On the existence of edge cuts leaving several large components

We characterize graphs of large enough order or large enough minimum degree which contain edge cuts whose deletion results in a graph with a specified number of large components. This generalizes and extends recent results due to Ou [Jianping Ou, Edge cuts leaving components of order at least m, Discrete Math. 305 (2005), 365-371] and Zhang and Yuan [Z. Zhang, J. Yuan, A proof of an inequality concerning k-restricted edge connectivity, Discrete Math. 304 (2005), 128-134].     

Verticum-type systems applied to ecological monitoring

In the paper ecological interaction chains of the type resource - producer - primary user - secondary consumer are considered. The dynamic behaviour of these four-level chains is modelled by a system of differential equations, the linearization of which is a verticum-type system introduced for the study of industrial verticums. Applying the technique of such systems, for the monitoring of the considered ecological system, an observer system is constructed, which makes it possible to recover the whole state process from the partial observation of the ecological interaction chain.     

Collective behavior models with vision geometrical constraints: Truncated noises and propagation of chaos

We consider large systems of stochastic interacting particles through discontinuous kernels which has vision geometrical constrains. We rigorously derive a Vlasov–Fokker–Planck type of kinetic mean-field equation from the corresponding stochastic integral inclusion system. More specifically, we construct a global-in-time weak solution to the stochastic integral inclusion system and derive the kinetic equation with the discontinuous kernels and the inhomogeneous noise strength by employing the 1-Wasserstein distance.     

Some bounds on the number of colors in interval and cyclic interval edge colorings of graphs

An interval$t$-coloring of a multigraph $G$ is a proper edge coloring with colors $1,\dots ,t$ such that the colors of the edges incident with every vertex of $G$ are colored by consecutive colors. A cyclic interval$t$-coloring of a multigraph $G$ is a proper edge coloring with colors $1,\dots ,t$ such that the colors of the edges incident with every vertex of $G$ are colored by consecutive colors, under the condition that color $1$ is considered as consecutive to color $t$. Denote by $w\left(G\right)$ ($w_c\left(G\right)$) and $W\left(G\right)$ ($W_c\left(G\right)$) the minimum and maximum number of colors in a (cyclic) interval coloring of a multigraph $G$, respectively. We present some new sharp bounds on $w\left(G\right)$ and $W\left(G\right)$ for multigraphs $G$ satisfying various conditions. In particular, we show that if $G$ is a $2$-connected multigraph with an interval coloring, then $W\left(G\right)\le 1+?\frac{\left|V\left(G\right)\right|}{2}?(\Delta \left(G\right)?1)$. We also give several results towards the general conjecture that $W_c\left(G\right)\le \left|V\left(G\right)\right|$ for any triangle-free graph $G$ with a cyclic interval coloring; we establish that approximate versions of this conjecture hold for several families of graphs, and we prove that the conjecture is true for graphs with maximum degree at most $4$.     

Global attractivity of an almost periodic N-species nonlinear ecological competitive model

By using comparison theorem and constructing suitable Lyapunov functional, we study the following almost periodic nonlinear N-species competitive Lotka-Volterra model:

     

Dependence of polynomial chaos on random types of forces of KdV equations

In this study, one-dimensional stochastic Korteweg–de Vries equation with uncertainty in its forcing term is considered. Extending the Wiener chaos expansion, a numerical algorithm based on orthonormal polynomials from the Askey scheme is derived. Then dependence of polynomial chaos on the distribution type of the random forcing term is inspected. It is numerically shown that when Hermite (Laguerre or Jacobi) polynomial chaos is chosen as a basis in the Gaussian (Gamma or Beta, respectively) random space for uncertainty, the solution to the KdV equation converges exponentially. If a proper polynomial chaos is not used, however, the solution converges with slower rate.     

An extension of the edge covering problem

In this paper we define a weightedr-covering problem, and show that there exists an optimum solution of ther-covering problem which can be decomposed into the sum of a rounded down solution of its linear relaxation and an optimal solution of a weighted edge covering problem on a reduced graph. Vertexr-packing problem can also be reduced to ther-covering problem.