spectra of complex networks we propose a general approach to the description of spectra of complex networks for the spectra of networks with uncorrelated vertices and a local treelike structure exact equations are derived these equations are generalized to the case of networks with correlations between neighboring vertices the tail of the density of eigenvalues ensuremath rho ensuremath lambda at large ensuremath lambda is related to the behavior of the vertex degree distribution p k at large k in particular as p k ensuremath sim k ensuremath ensuremath gamma ensuremath rho ensuremath lambda ensuremath sim ensuremath lambda 1 ensuremath 2 ensuremath gamma we propose a simple approximation which enables us to calculate spectra of various graphs analytically we analyze spectra of various complex networks and discuss the role of vertices of low degree we show that spectra of locally treelike random graphs may serve as a starting point in the analysis of spectral properties of real world networks e g of the internet

on the relationship between continuous and discrete time quantum walk quantum walk is one of the main tools for quantum algorithms defined by analogy to classical random walk a quantum walk is a time homogeneous quantum process on a graph both random and quantum walks can be defined either in continuous or discrete time but whereas a continuous time random walk can be obtained as the limit of a sequence of discrete time random walks the two types of quantum walk appear fundamentally different owing to the need for extra degrees of freedom in the discrete time case

just in time control of timed event graphs update of reference input presence of uncontrollable input a linear system theory has been developed for the class of discrete event systems subject to synchronization this paper presents the just in time control of such systems when reference input is updated and or in the presence of uncontrollable input s the proposed controls are the solutions to an optimization problem under equality constraint

model predictive control for uncertain max min plus scaling systems in this paper we extend the classical min max model predictive control framework to a class of uncertain discrete event systems that can be modelled using the operations maximization minimization addition and scalar multiplication and that we call max min plus scaling mmps systems provided that the stage cost is an mmps expression and considering only linear input constraints then the open loop min max model predictive control problem for mmps systems can be transformed into a sequence of linear programming problems hence the min max model predictive control problem for mmps systems can be solved efficiently despite the fact that the system is non linear a min max feedback model predictive control approach using disturbance feedback policies is also presented which leads to improved performance compared to the open loop approach

adaptive dynamic programming for finite horizon optimal control of discrete time nonlinear systems with varepsilon error bound in this paper we study the finite horizon optimal control problem for discrete time nonlinear systems using the adaptive dynamic programming adp approach the idea is to use an iterative adp algorithm to obtain the optimal control law which makes the performance index function close to the greatest lower bound of all performance indices within an error bound the optimal number of control steps can also be obtained by the proposed adp algorithms a convergence analysis of the proposed adp algorithms in terms of performance index function and control policy is made in order to facilitate the implementation of the iterative adp algorithms neural networks are used for approximating the performance index function computing the optimal control policy and modeling the nonlinear system finally two simulation examples are employed to illustrate the applicability of the proposed method

erratum apollonian networks simultaneously scale free small world euclidean space filling and with matching graphs phys rev lett 94 018702 2005

how the site degree influences quantum probability on inhomogeneous substrates we investigate the effect of the node degree and energy e on the electronic wave function for regular and irregular structures namely regular lattices disordered percolation clusters and complex networks we evaluate the dependency of the quantum probability for each site on its degree for a class of biregular structures formed by two disjoint subsets of sites sharing the same degree the probability p k e of finding the electron on any site with k neighbors is independent of e 0 a consequence of an exact analytical result that we prove for any bipartite lattice for more general nonbipartite structures p k e may depend on e as illustrated by an exact evaluation of a one dimensional semiregular lattice p k e is large for small values of e when k is also small and its maximum values shift towards large values of e with increasing k numerical evaluations of p k e for two different types of percolation clusters and the apollonian network suggest that this observed feature might be generally valid

discrete time quantum walk on the apollonian network this work discusses the discrete time quantum walk dtqw on the first five generations of the apollonian network this structure is constructed in a geometric recurrent way and is characterized by among other features a scale free distribution of node degrees the dtqw formalism requires a node dependent coin operator that for each node has as many different output states as that node degree so that each local coin operator is expressed by the fourier operator the dtqw time evolution matrix has a larger rank than that of the network adjacency matrix or the matrix representation of the continuous time quantum walk ctqw results for the time evolution return time to specific nodes and asymptotic probability of site occupancy are obtained such dtqw specific features are discussed and compared to those obtained for the classical random walk and ctqw on the same lattice

model predictive control for perturbed max plus linear systems abstract model predictive control mpc is a popular controller design technique in the process industry conventional mpc uses linear or nonlinear discrete time models recently we have extended mpc to a class of discrete event systems that can be described by a model that is linear in the max algebra in our previous work we have only considered mpc for the deterministic noise free case without modeling errors in this paper we extend our previous results on mpc for max plus linear systems to cases with noise and or modeling errors we show that under quite general conditions the resulting optimization problems can be solved very efficiently

free electron gas in the apollonian network multifractal energy spectrum and its thermodynamic fingerprints we study the free electron gas in an apollonian network within the tight binding framework the scale free and small world character of the underlying lattice is known to result in a quite structured energy spectrum with deltalike singularities gaps and minibands after an exact numerical diagonalization of the corresponding adjacency matrix of the network with a finite number of generations we employ a scaling analysis of the moments of the density of states to characterize its multifractality and report the associated singularity spectrum the fractal nature of the energy spectrum is also shown to be reflected in the thermodynamic behavior by logarithmic modulations on the temperature dependence of the specific heat the absence of chiral symmetry of the apollonian network leads to distinct thermodynamic behaviors due to electrons and holes thermal excitations

robust dynamic programming for min max model predictive control of constrained uncertain systems we address min max model predictive control mpc for uncertain discrete time systems by a robust dynamic programming approach and develop an algorithm that is suitable for linearly constrained polytopic systems with piecewise affine cost functions the method uses polyhedral representations of the cost to go functions and feasible sets and performs multiparametric programming by a duality based approach in each recursion step we show how to apply the method to robust mpc and give conditions guaranteeing closed loop stability finally we apply the method to a tutorial example a parking car with uncertain mass

finite horizon min max control of max plus linear systems in this note we provide a solution to a class of finite horizon min max control problems for uncertain max plus linear systems where the uncertain parameters are assumed to lie in a given convex and compact set and it is required that the closed loop input and state sequence satisfy a given set of linear inequality constraints for all admissible uncertainty realizations we provide sufficient conditions such that the value function is guaranteed to be convex and continuous piecewise affine and such that the optimal control policy is guaranteed to be continuous and piecewise affine on a polyhedral domain

analytical approach to directed sandpile models on the apollonian network we investigate a set of directed sandpile models on the apollonian network which are inspired by the work of dhar and ramaswamy phys rev lett 63 1659 1989 on euclidian lattices they are characterized by a single parameter q which restricts the number of neighbors receiving grains from a toppling node due to the geometry of the network two and three point correlation functions are amenable to exact treatment leading to analytical results for avalanche distributions in the limit of an infinite system for q 1 2 the exact recurrence expressions for the correlation functions are numerically iterated to obtain results for finite size systems when larger values of q are considered finally a detailed description of the local flux properties is provided by a multifractal scaling analysis

q state potts model on the apollonian network the q state potts model is studied on the apollonian network with monte carlo simulations and the transfer matrix method the spontaneous magnetization correlation length entropy and specific heat are analyzed as a function of temperature for different number of states q different scaling functions in temperature and q are proposed a quantitative agreement is found between results from both methods no critical behavior is observed in the thermodynamic limit for any number of states

interval analysis and dioid application to robust controller design for timed event graphs this paper deals with feedback controller synthesis for timed event graphs where the number of initial tokens and time delays are only known to belong to intervals we discuss here the existence and the computation of a robust controller set for uncertain systems that can be described by parametric models the unknown parameters of which are assumed to vary between known bounds each controller is computed in order to guarantee that the closed loop system behavior is greater than the lower bound of a reference model set and is lower than the upper bound of this set the synthesis presented here is mainly based on dioid interval analysis and residuation theory

spectral properties of the apollonian network abstract a first characterization of the spectral properties of the apollonian network is presented results are based on the numerically evaluated eigenvalue spectra of adjacency matrices for the first nine generations of the network the main features refer to the presence of several gaps on the eigenvalue axis and a very large number of degenerated eigenvalues the increase in the degeneracy of a family of eigenvalues is found to obey a very simple empirical relation

total energy shaping control of mechanical systems simplifying the matching equations via coordinate changes total energy shaping is a controller design methodology that achieves asymptotic stabilization of mechanical systems endowing the closed loop system with a lagrangian or hamiltonian structure with a desired energy function that qualifies as lyapunov function for the desired equilibrium the success of the method relies on the possibility of solving two pdes which identify the kinetic and potential energy functions that can be assigned to the closed loop particularly troublesome is the partial differential equation pde associated to the kinetic energy which is nonlinear and inhomogeneous and the solution that defines the desired inertia matrix must be positive definite in this note we prove that we can eliminate or simplify the forcing term in this pde by modifying the target dynamics and introducing a change of coordinates in the original system furthermore it is shown that in the particular case of transformation to the lagrangian coordinates the possibility of simplifying the pdes is determined by the interaction between the coriolis and centrifugal forces and the actuation structure the examples of pendulum on a cart and furuta s pendulum are used to illustrate the results

the distribution of free path lengths in the periodic lorentz gas and related lattice point problems the periodic lorentz gas describes the dynamics of a point particle in a periodic array of spherical scatterers and is one of the fundamental models for chaotic diffusion in the present paper we investigate the boltzmann grad limit where the radius of each scatterer tends to zero and prove the existence of a limiting distribution for the free path length we also discuss related problems such as the statistical distribution of directions of lattice points that are visible from a fixed position

on the correlations of directions in the euclidean plane let r v x y q denote the repartition of the v level correlation measure of the finite set of directions p x y p where p x y is the fixed point x y 0 1 2 and p is an integer lattice point in the square q q 2 we show that the average of the pair correlation repartition r 2 x y q over x y in a fixed disc d 0 converges as q oo more precisely we prove for every a r and 0 1 10 the estimate formula math we also prove that for each individual point x y 0 1 2 the 6 level correlation r 6 x y q diverges at any point a r 5 as q and we give an explicit lower bound for the rate of divergence

min max control of constrained uncertain discrete time linear systems for discrete time uncertain linear systems with constraints on inputs and states we develop an approach to determine state feedback controllers based on a min max control formulation robustness is achieved against additive norm bounded input disturbances and or polyhedral parametric uncertainties in the state space matrices we show that the finite horizon robust optimal control law is a continuous piecewise affine function of the state vector and can be calculated by solving a sequence of multiparametric linear programs when the optimal control law is implemented in a receding horizon scheme only a piecewise affine function needs to be evaluated on line at each time step the technique computes the robust optimal feedback controller for a rather general class of systems with modest computational effort without needing to resort to gridding of the state space

statistical properties of the periodic lorentz gas multidimensional case in 1981 bunimovich and sinai established the statistical properties of the planar periodic lorentz gas with finite horizon our aim is to extend their theory to the multidimensional lorentz gas in that case the markov partitions of the bunimovich sinai type the main tool of their theory are not available we use a crude approximation to such partitions which we call markov sieves their construction in many dimensions is essentially different from that in two dimensions it requires more routine calculations and intricate arguments we try to avoid technical details and outline the construction of the markov sieves in mostly qualitative heuristic terms hoping to carry out our plan in full detail elsewhere modulo that construction our proofs are conclusive in the end we obtain a stretched exponential bound for the decay of correlations the central limit theorem and donsker s invariance principle for multidimensional periodic lorentz gases with finite horizon

criticality on networks with topology dependent interactions weighted scale free networks with topology dependent interactions are studied it is shown that the possible universality classes of critical behavior which are known to depend on topology can also be explored by tuning the form of the interactions at fixed topology for a model of opinion formation simple mean field and scaling arguments show that a mapping gamma gamma mu 1 mu describes how a shift of the standard exponent gamma of the degree distribution can absorb the effect of degree dependent pair interactions j ij proportional to k i k j mu where k i stands for the degree of vertex i this prediction is verified by extensive numerical investigations using the cavity method and monte carlo simulations the critical temperature of the model is obtained through the bethe peierls approximation and with the replica technique the mapping can be extended to nonequilibrium models such as those describing the spreading of a disease on a network