quadratic stabilizability of switched linear systems with polytopic uncertainties in this paper we consider quadratic stabilizability via state feedback for both continuous time and discrete time switched linear systems that are composed of polytopic uncertain subsystems by state feedback we mean that the switchings among subsystems are dependent on system states for continuous time switched linear systems we show that if there exists a common positive definite matrix for stability of all convex combinations of the extreme points which belong to different subsystem matrices then the switched system is quadratically stabilizable via state feedback for discrete time switched linear systems we derive a quadratic stabilizability condition expressed as matrix inequalities with respect to a family of non negative scalars and a common positive definite matrix for both continuous time and discrete time switched systems we propose the switching rules by using the obtained common positive definite matrix

the average distances in random graphs with given expected degrees abstract random graph theory is used to examine the small world phenomenon any two strangers are connected through a short chain of mutual acquaintances we will show that for certain families of random graphs with given expected degrees the average distance is almost surely of order log n log d where d is the weighted average of the sum of squares of the expected degrees of particular interest are power law random graphs in which the number of vertices of degree k is proportional to 1 k for some fixed exponent for the case of 3 we prove that the average distance of the power law graphs is almost surely of order log n log d however many internet social and citation networks are power law graphs with exponents in the range 2 3 for which the power law random graphs have average distance almost surely of order log log n but have diameter of order log n provided having some mild constraints for the average distance and maximum degree in particular these graphs contain a dense subgraph which we call the core having nc log log n vertices almost all vertices are within distance log log n of the core although there are vertices at distance log n from the core

complex networks modeled on the sierpinski gasket abstract in this paper we use the sierpinski gasket to construct evolving networks g t whose node set is the solid regular triangles in the construction of the sierpinski gasket up to the stage t and any two nodes are neighbors if and only if the corresponding solid triangles are in contact with each other on boundary using the encoding method we show that our evolving networks are scale free power law degree distribution and have the small world effect small average path length and high clustering coefficient

a deterministic small world network created by edge iterations abstract small world networks are ubiquitous in real life systems most previous models of small world networks are stochastic the randomness makes it more difficult to gain a visual understanding on how do different nodes of networks interact with each other and is not appropriate for communication networks that have fixed interconnections here we present a model that generates a small world network in a simple deterministic way our model has a discrete exponential degree distribution we solve the main characteristics of the model

stability and stabilization of switched linear systems with mode dependent average dwell time in this paper the stability and stabilization problems for a class of switched linear systems with mode dependent average dwell time mdadt are investigated in both continuous time and discrete time contexts the proposed switching law is more applicable in practice than the average dwell time adt switching in which each mode in the underlying system has its own adt the stability criteria for switched systems with mdadt in nonlinear setting are firstly derived by which the conditions for stability and stabilization for linear systems are also presented a numerical example is given to show the validity and potential of the developed techniques

random walks on deterministic scale free networks exact results we study the random walk problem on a class of deterministic scale free networks displaying a degree sequence for hubs scaling as a power law with an exponent gamma log 3 log 2 we find exact results concerning different first passage phenomena and in particular we calculate the probability of first return to the main hub these results allow to derive the exact analytic expression for the mean time to first reach the main hub whose leading behavior is given by tau approximately v1 1 gamma where v denotes the size of the structure and the mean is over a set of starting points distributed uniformly over all the other sites of the graph interestingly the process turns out to be particularly efficient we also discuss the thermodynamic limit of the structure and some local topological properties

brief paper stability of switched positive linear systems with average dwell time switching in this paper the stability analysis problem for a class of switched positive linear systems splss with average dwell time switching is investigated a multiple linear copositive lyapunov function mlclf is first introduced by which the sufficient stability criteria in terms of a set of linear matrix inequalities are given for the underlying systems in both continuous time and discrete time contexts the stability results for the splss under arbitrary switching which have been previously studied in the literature can be easily obtained by reducing mlclf to the common linear copositive lyapunov function used for the system under arbitrary switching those systems finally a numerical example is given to show the effectiveness and advantages of the proposed techniques

unconditionally stable finite difference scheme and iterative solution of 2d microscale heat transport equation abstract a two dimensional time dependent heat transport equation at the microscale is derived a second order finite difference scheme in both time and space is introduced and the unconditional stability of the finite difference scheme is proved a computational procedure is designed to solve the discretized linear system at each time step by using a preconditioned conjugate gradient method numerical results are presented to validate the accuracy of the finite difference scheme and the efficiency of the proposed computational procedure

stabilization of switched systems via composite quadratic functions this paper investigates stabilization of switched systems by using a lyapunov function composed of a family of continuously differentiable functions switching laws are constructed by using the directional derivatives along the trajectories of the subsystems conditions for stabilization are established with careful consideration of sliding modes and directional derivatives along sliding modes three types of composite quadratic lyapunov functions the max of quadratics the min of quadratics and the convex hull of quadratics are used for deriving matrix conditions of stabilization and for constructing switching laws dual stabilization results are established with respect to the pair of conjugate lyapunov functions the max of quadratics and the convex hull of quadratics conditions of stabilization are derived as bilinear matrix inequalities and solved with lmi based numerical tools relationship between the newly derived conditions and some existing conditions are investigated it is observed that the min of quadratics which is nondifferentiable and nonconvex may be a more convenient tool than the other two types of functions which are convex and or differentiable numerical examples are used to demonstrate the synthesis results and the advantage of the composite quadratic functions over existing multiple lyapunov functions in particular better results have been obtained when the number of quadratic functions is greater than the number of subsystems

high accuracy finite difference scheme for three dimensional microscale heat equation a compact finite difference scheme is developed to the three dimensional microscale heat transport equation this new scheme is fourth order in space and second order in time it is proved to be unconditionally stable with respect to initial values numerical results are provided for comparison testing purpose

switching rule design for affine switched systems with guaranteed cost and uncertain equilibrium condition this paper addresses the problem of determining switching rules for affine switched systems such that the system state is driven to a desired point and a guaranteed cost is minimized the switching rule is determined by solving an lmi problem and global asymptotic stability of the tracking error dynamics is guaranteed even if sliding motions occur on any switching surface of the system the potential of the results is illustrated on a true refrigeration system namely a domestic refrigerator where the purpose is to control the temperature in the fresh food and the freezer compartments

switching rule design for affine switched systems using a max type composition rule abstract this paper presents conditions for designing a switching rule that drives the state of the switched dynamic system to a desired equilibrium point the proposed method deals with the class of switched systems where each subsystem has an affine vector field and considers a switching rule using max composition the results guarantee global asymptotic stability of the tracking error dynamics even if sliding mode occurs at any switching surface of the system in addition the method does not require a hurwitz convex combination of the dynamic matrices of the subsystems two numerical examples are used to illustrate the results

a family of small world network models built by complete graph and iteration function abstract small world networks are popular in real life complex systems in the past few decades researchers presented amounts of small world models in which some are stochastic and the rest are deterministic in comparison with random models it is not only convenient but also interesting to study the topological properties of deterministic models in some fields such as graph theory theorem computer sciences and so on as another concerned darling in current researches community structure modular topology is referred to as an useful statistical parameter to uncover the operating functions of network so building and studying such models with community structure and small world character will be a demanded task hence in this article we build a family of sparse network space n t which is different from those previous deterministic models even though our models are established in the same way as them iterative generation by randomly connecting manner in each time step every resulting member in n t has no absolutely self similar feature widely shared in a large number of previous models this makes our insight not into discussing a class certain model but into investigating a group various ones spanning a network space somewhat surprisingly our results prove all members of n t to possess some similar characters a sparsity b exponential scale feature p k k and c small world property here we must stress a very screming but intriguing phenomenon that the difference of average path length a p l between any two members in n t is quite small which indicates this random connecting way among members has no great effect on a p l at the end of this article as a new topological parameter correlated to reliability synchronization capability and diffusion properties of networks the number of spanning trees on a representative member n b t of n t is studied in detail then an exact analytical solution for its spanning trees entropy is also obtained

scale free and small world properties of sierpinski networks abstract in this paper we construct the evolving networks from sierpinski carpet using the encoding approach in fractal geometry we consider the small similar copies of unit square as nodes of network where two nodes are neighbors if and only if their corresponding copies have common surface for our networks we check their scale free and small world effect by the self similar structures the exponent of power law on degree distribution is log 3 8 which is the hausdorff dimension of the carpet

the modeling of scale free networks abstract in order to explore further the mechanism responsible for scale free networks we introduce two extended models of the ba model the model a where the system incorporates the addition of new links between existing nodes a new node with new links and the rewiring of some links at every time step all sites are born with some initial attractiveness we calculate analytically the degree distribution the system self organizes into a scale free network the scaling exponent 2 the model b is a new model we consider that some old links are deleted with the anti preferential probability the result indicates that the system evolves itself into a scale free network the scaling exponent varies from 2 to 3

a small world and scale free network generated by sierpinski pentagon abstract the sierpinski pentagon is used to construct evolving networks whose nodes are all solid regular pentagons in the construction of the sierpinski pentagon up to the stage t and any two nodes are neighbors if and only if the intersection of corresponding pentagons is non empty and non singleton we show that such networks have the small world and scale free effects but are not fractal scaling

farey graphs as models for complex networks farey sequences of irreducible fractions between 0 and 1 can be related to graph constructions known as farey graphs these graphs were first introduced by matula and kornerup in 1979 and further studied by colbourn in 1982 and they have many interesting properties they are minimally 3 colorable uniquely hamiltonian maximally outerplanar and perfect in this paper we introduce a simple generation method for a farey graph family and we study analytically relevant topological properties order size degree distribution and correlation clustering transitivity diameter and average distance we show that the graphs are a good model for networks associated with some complex systems

geometric fractal growth model for scale free networks we introduce a deterministic model for scale free networks whose degree distribution follows a power law with the exponent ensuremath gamma at each time step each vertex generates its offspring whose number is proportional to the degree of that vertex with proportionality constant m ensuremath 1 mg1 we consider the two cases first each offspring is connected to its parent vertex only forming a tree structure second it is connected to both its parent and grandparent vertices forming a loop structure we find that both models exhibit power law behaviors in their degree distributions with the exponent ensuremath gamma 1 mathrm ln 2m ensuremath 1 mathrm ln m thus by tuning m the degree exponent can be adjusted in the range 2l ensuremath gamma l3 we also solve analytically a mean shortest path distance d between two vertices for the tree structure showing the small world behavior that is d ensuremath sim mathrm ln n mathrm ln overline k where n is system size and overline k is the mean degree finally we consider the case that the number of offspring is the same for all vertices and find that the degree distribution exhibits an exponential decay behavior

a fractal growth model exploring the connection pattern of hubs in complex networks abstract fractal is ubiquitous in many real world networks previous researches showed that the strong disassortativity between the hub nodes on all length scales was the key principle that gave rise to the fractal architecture of networks although fractal property emerged in some models there were few researches about the fractal growth model and quantitative analyses about the strength of the disassortativity for fractal model in this paper we proposed a novel inverse renormalization method named box based preferential attachment bpa to build the fractal growth models in which the preferential attachment was performed at box level the proposed models provided a new framework that demonstrated small world fractal transition also we firstly demonstrated the statistical characteristic of connection patterns of the hubs in fractal networks the experimental results showed that given proper growing scale and added edges the proposed models could clearly show pure small world or pure fractal or both of them it also showed that the hub connection ratio showed normal distribution in many real world networks at last the comparisons of connection pattern between the proposed models and the biological and technical networks were performed the results gave useful reference for exploring the growth principle and for modeling the connection patterns for real world networks

stability and stabilizability of switched linear systems a survey of recent results during the past several years there have been increasing research activities in the field of stability analysis and switching stabilization for switched systems this paper aims to briefly survey recent results in this field first the stability analysis for switched systems is reviewed we focus on the stability analysis for switched linear systems under arbitrary switching and we highlight necessary and sufficient conditions for asymptotic stability after a brief review of the stability analysis under restricted switching and the multiple lyapunov function theory the switching stabilization problem is studied and a variety of switching stabilization methods found in the literature are outlined then the switching stabilizability problem is investigated that is under what condition it is possible to stabilize a switched system by properly designing switching control laws note that the switching stabilizability problem has been one of the most elusive problems in the switched systems literature a necessary and sufficient condition for asymptotic stabilizability of switched linear systems is described here

an unconditionally stable three level explicit difference scheme for the schro dinger equation with a variable coefficient a three level explicit difference scheme for the schrodinger equation with a variable coefficient is established it is proved by the discrete energy method that the scheme is unconditionally stable

mean field theory for scale free random networks random networks with complex topology are common in nature describing systems as diverse as the world wide web or social and business networks recently it has been demonstrated that most large networks for which topological information is available display scale free features here we study the scaling properties of the recently introduced scale free model that can account for the observed power law distribution of the connectivities we develop a mean eld method to predict the growth dynamics of the individual vertices and use this to calculate analytically the connectivity distribution and the scaling exponents the mean eld method can be used to address the properties of two variants of the scale free model that do not display power law scaling c 1999 elsevier science b v all rights reserved