continuous representation and control of hybrid systems this article introduces a continuous representation for controlled model and state jump hybrid systems such a representation is valuable in that it manages to cope with the difficulties related to discontinuities by replacing the study of these systems by that of continuous ones which is simple the studied hybrid systems are first introduced together with their associated continuous systems some localized and global convergences of the states of continuous systems to those of hybrid systems are proved then the continuous representation is used within the optimal control framework free end time and trajectory pursuit hybrid problems are studied as well as their respective continuous versions convergences of solutions between continuous and hybrid problems are established which shows that the resolution of a problem ensuing from the representation enables one to obtain an approximation of the solution of the initial hybrid problem finally the formalism of calculus of variations is used to obta

input output stabilization of linear systems on spl zopf a formal framework is set up for the discussion of generalized autoregressive with external input models of the form ay bu where a and b are linear operators with the main emphasis being on signal spaces consisting of bounded sequences parametrized by the integers different notions of stability are explored and topological notions such as the idea of a closed system are linked with questions of stabilizability in this very general context various problems inherent in using spl zopf as the time axis are analyzed in this operatorial framework

intrinsic difficulties in using the doubly infinite time axis for input output control theory points out that the natural definitions of stability and causality in input output control theory lead to certain inconsistencies when inputs and outputs are allowed to have support on the doubly infinite time axis in particular linear time invariant systems with right half plane poles cannot be considered to be both causal and stabilizable in contrast there is no such conflict when the semi infinite time axis is used

synthesis of optimal controllers for piecewise affine systems with sampled data switching this paper discusses the optimal control problem of the continuous time piecewise affine pwa systems with sampled data switching where the switching action is executed based upon a condition on the state at each sampling time first an algebraic characterization for the problem to be feasible is derived next an optimal continuous time controller is derived for a general class of pwa systems with sampled data switching for which the optimal control problem is feasible but whose subsystems in some modes may be uncontrollable in the usual sense finally as an application of the proposed approach the high speed and energy saving control problem of the cpu processing is formulated and the validity of the proposed methods is shown by numerical simulations

a numerical algorithm for optimal control of a class of hybrid systems differential transformation based approach a novel numerical algorithm based on differential transformation is proposed for optimal control of a class of hybrid systems with a predefined mode sequence from the necessary conditions for optimality of hybrid systems the hybrid optimal control problem is first converted into a two point boundary value problem tpbvp with additional transverse conditions at the switching times then we propose a differential transformation algorithm for solving the tpbvp which may have discontinuities in the state and or control input at the switching times using differential transformation the hybrid optimal control problem reduces to a problem of solving a system of algebraic equations the numerical solution is obtained in the form of a truncated taylor series by taking advantage of the special properties of the linear subsystems and a quadratic cost functional the differential transformation algorithm can be further simplified for the switched linear quadratic optimal control problem we analyse the error of

cooperation in two dimensional mixed games evolutionary game theory is a common framework to study the evolution of cooperation where it is usually assumed that the same game is played in all interactions here we investigate a model where the game that is played by two individuals is uniformly drawn from a sample of two different games using the master equation approach we show that the random mixture of two games is equivalent to play the average game when i the strategies are statistically independent of the game distribution and ii the transition rates are linear functions of the payoffs we also use monte carlo simulations in a two dimensional lattice and mean field techniques to investigate the scenario when the two above conditions do not hold we find that even outside of such conditions several quantities characterizing the mixed games are still the same as the ones obtained in the average game when the two games are not very different

fragmentation of random trees we study fragmentation of a random recursive tree into a forest by repeated removal of nodes the initial tree consists of n nodes and it is generated by sequential addition of nodes with each new node attaching to a randomly selected existing node as nodes are removed from the tree one at a time the tree dissolves into an ensemble of separate trees namely a forest we study statistical properties of trees and nodes in this heterogeneous forest and find that the fraction of remaining nodes m characterizes the system in the limit we obtain analytically the size density of trees of size s the size density has power law tail with exponent therefore the tail becomes steeper as further nodes are removed and the fragmentation process is unusual in that exponent increases continuously with time we also extend our analysis to the case where nodes are added as well as removed and obtain the asymptotic size density for growing trees

hierarchical trajectory optimization for a class of hybrid dynamical systems we discuss trajectory optimization for a class of hybrid systems with a natural hierarchical separation of discrete and continuous dynamics where the continuous dynamics do not change with the discrete state the trajectory optimization problem considered requires that a discrete state sequence and a continuous state trajectory must be both determined to minimize a single cost function such that the discrete state sequence also solves a symbolic planning problem we model this symbolic planning problem as a search on a planning graph and we introduce a family of graphs called lifted planning graphs parametrized by an integer h we define a family of continuous state trajectory optimization problems and associate them with edge costs in the lifted planning graphs next we present an algorithm for finding an optimal solution to the hybrid trajectory optimization problem which includes mapping paths in the lifted planning graphs to discrete state sequences and continuous state trajectories we show that the cost of optimal hybrid trajectories is a nonincreasing function of h and that there exists a finite h for which this cost attains a minimum we illustrate the proposed algorithm with numerical simulation results for two application examples an autonomous mobile vehicle and an autonomous robotic manipulator

role separating ordering in social dilemmas controlled by topological frustration three is a crowd is an old proverb that applies as much to social interactions as it does to frustrated configurations in statistical physics models accordingly social relations within a triangle deserve special attention with this motivation we explore the impact of topological frustration on the evolutionary dynamics of the snowdrift game on a triangular lattice this topology provides an irreconcilable frustration which prevents anticoordination of competing strategies that would be needed for an optimal outcome of the game by using different strategy updating protocols we observe complex spatial patterns in dependence on payoff values that are reminiscent to a honeycomb like organization which helps to minimize the negative consequence of the topological frustration we relate the emergence of these patterns to the microscopic dynamics of the evolutionary process both by means of mean field approximations and monte carlo simulations for comparison we also consider the same evolutionary dynamics on the square lattice where of course the topological frustration is absent however with the deletion of diagonal links of the triangular lattice we can gradually bridge the gap to the square lattice interestingly in this case the level of cooperation in the system is a direct indicator of the level of topological frustration thus providing a method to determine frustration levels in an arbitrary interaction network

when is a linear convolution system stabilizable lack of operator closedness and closability arguments are used to discuss the impossibility of finite lp lp gain stabilization of unstable finite dimensional linear convolution operator systems in the doubly infinite time axis case for any 1 p the presented analysis generalizes and refines a result due to georgiou and smith ieee trans automat control 40 1995 516 for the l2 setting

optimal stabilization using lyapunov measures numerical solutions for the optimal feedback stabilization of discrete time dynamical systems is the focus of this technical note set theoretic notion of almost everywhere stability introduced by the lyapunov measure weaker than conventional lyapunov function based stabilization methods is used for optimal stabilization the linear perron frobenius transfer operator is used to pose the optimal stabilization problem as an infinite dimensional linear program set oriented numerical methods are used to obtain the finite dimensional approximation of the linear program we provide conditions for the existence of stabilizing feedback controls and show the optimal stabilizing feedback control can be obtained as a solution of a finite dimensional linear program the approach is demonstrated on stabilization of period two orbit in a controlled standard map

optimal control of continuous time switched affine systems this paper deals with optimal control of switched piecewise affine autonomous systems where the objective is to minimize a performance index over an infinite time horizon we assume that the switching sequence has a finite length and that the decision variables are the switching instants and the sequence of operating modes we present two different approaches for solving such an optimal control problem the first approach iterates between a procedure that finds an optimal switching sequence of modes and a procedure that finds the optimal switching instants the second approach is inspired by dynamic programming and identifies the regions of the state space where an optimal mode switch should occur therefore providing a state feedback control law

convex dynamic programming for hybrid systems a classical linear programming approach to optimization of flow or transportation in a discrete graph is extended to hybrid systems the problem is finite dimensional if the state space is discrete and finite but becomes infinite dimensional for a continuous or hybrid state space it is shown how strict lower bounds on the optimal loss function can be computed by gridding the continuous state space and restricting the linear program to a finite dimensional subspace upper bounds can be obtained by evaluation of the corresponding control laws

piecewise linear quadratic optimal control the use of piecewise quadratic cost functions is extended from stability analysis of piecewise linear systems to performance analysis and optimal control lower bounds on the optimal control cost are obtained by semidefinite programming based on the bellman inequality this also gives an approximation to the optimal control law an upper bound to the optimal cost is obtained by another convex optimization problem using the given control law a compact matrix notation is introduced to support the calculations and it is proved that the framework of piecewise linear systems can be used to analyze smooth nonlinear dynamics with arbitrary accuracy

shift invariant signal norms for fault detection and control two desirable properties of a signal norm to be used for the purpose of optimal control or fault detection are highlighted these properties are shift invariance and the permitting of persistent signals a class of norms called the window norms is analyzed among norms considered for control purposes they appear to be the only ones save the l norm that satisfy both properties the signal spaces defined by the window norms however include some interesting signals that are not in the l space the window norms have been found suitable for application in fault detection and are here also considered for optimal control it is shown that they can be taken as a support for the concept of l1 control but may also suggest a new class of optimal controllers

organization of growing random networks the organizational development of growing random networks is investigated these growing networks are built by adding nodes successively and linking each to an earlier node of degree k with an attachment probability a k when a k grows more slowly than linearly with k the number of nodes with k links n k t decays faster than a power law in k while for a k growing faster than linearly in k a single node emerges which connects to nearly all other nodes when a k is asymptotically linear n k t ensuremath sim mathrm tk ensuremath ensuremath nu with ensuremath nu dependent on details of the attachment probability but in the range 2l ensuremath nu l ensuremath infty the combined age and degree distribution of nodes shows that old nodes typically have a large degree there is also a significant correlation in the degrees of neighboring nodes so that nodes of similar degree are more likely to be connected the size distributions of the in and out components of the network with respect to a given node namely its descendants and ancestors are also determined the in component exhibits a robust s ensuremath 2 power law tail where s is the component size the out component has a typical size of order mathrm ln t and it provides basic insights into the genealogy of the network

stochastic win stay lose shift strategy with dynamic aspirations in evolutionary social dilemmas in times of plenty expectations rise just as in times of crisis they fall this can be mathematically described as a win stay lose shift strategy with dynamic aspiration levels where individuals aspire to be as wealthy as their average neighbor here we investigate this model in the realm of evolutionary social dilemmas on the square lattice and scale free networks by using the master equation and monte carlo simulations we find that cooperators coexist with defectors in the whole phase diagram even at high temptations to defect we study the microscopic mechanism that is responsible for the striking persistence of cooperative behavior and find that cooperation spreads through second order neighbors rather than by means of network reciprocity that dominates in imitation based models for the square lattice the master equation can be solved analytically in the large temperature limit of the fermi function while for other cases the resulting differential equations must be solved numerically either way we find good qualitative agreement with the monte carlo simulation results our analysis also reveals that the evolutionary outcomes are to a large degree independent of the network topology including the number of neighbors that are considered for payoff determination on lattices which further corroborates the local character of the microscopic dynamics unlike large scale spatial patterns that typically emerge due to network reciprocity here local checkerboard like patterns remain virtually unaffected by differences in the macroscopic properties of the interaction network

on three puzzles in robust control three surprising open topics puzzles in robust control are highlighted the first and the second puzzle deal with difficulties in generalizing h sub spl infin control theory to realistic persistent signal setups it seems difficult to generalize the theory to such setups and simultaneously keep the terminology robust h sub spl infin control intact these puzzles have implications also to control oriented model validation the third puzzle due to georgiou and smith 1995 deals with difficulties in the doubly infinite time axis formulations of input output stabilization theory

second order switching time optimization for nonlinear time varying dynamic systems this technical note gives a method for calculating the first and second derivatives of a cost function with respect to switching times for systems with piecewise second differentiable dynamics differential equations governing the linear and bilinear operators required for calculating the derivatives are presented example optimizations of linear and nonlinear systems are presented as evidence for the value of second order optimization methods one example converges in 33 iterations using second order methods whereas the first order algorithm requires over 30 000 iterations

feedback and optimal sensitivity model reference transformations multiplicative seminorms and approximate inverses in this paper the problem of sensitivity reduction by feedback is formulated as an optimization problem and separated from the problem of stabilization stable feedback schemes obtainable from a given plant are parameterized salient properties of sensitivity reducing schemes are derived and it is shown that plant uncertainty reduces the ability of feedback to reduce sensitivity the theory is developed for input output systems in a general setting of banach algebras and then specialized to a class of multivariable time invariant systems characterized by n times n matrices of h infty frequency response functions either with or without zeros in the right half plane the approach is based on the use of a weighted seminorm on the algebra of operators to measure sensitivity and on the concept of an approximate inverse approximate invertibility of the plant is shown to be a necessary and sufficient condition for sensitivity reduction an indicator of approximate invertibility called a measure of singularity is introduced the measure of singularity of a linear time invariant plant is shown to be determined by the location of its right half plane zeros in the absence of plant uncertainty the sensitivity to output disturbances can be reduced to an optimal value approaching the singularity measure in particular if there are no right half plane zeros sensitivity can be made arbitrarily small the feedback schemes used in the optimization of sensitivity resemble the lead lag networks of classical control design some of their properties and methods of constructing them in special cases are presented

intrinsic difficulties in stochastic control of unstable convolution operators on z stochastic control of causal linear unstable convolution operators on the full time axis is shown to be impossible the basic plant y gu and controller descriptions u cy via possibly unstable convolution operators or via their unstable transforms have been shown to possess severe limitations in stochastic control studies on the full time axis z and on z t in fact they have also similar limitations on n when g or c is noncausal and unstable such situations can appear in applications in which the independent variable is a spatial variable rather than time

a counterexample in power signals space the present author points out that in the paper by vidyasagar 1993 on nonlinear system analysis some of the results obtained are false the purpose of this paper is to give a counterexample it is shown that the set of power signals is not a vector space

new limit power function spaces to answer an open question we propose two function spaces one is banach and another is hilbert it is shown that the hilbert space is the largest one among those hilbert spaces in limit power function set whose members have associated fourier series in sense of a new basis and satisfy parseval s equality

optimal control of discrete time hybrid automata under safety and liveness constraints abstract in this contribution we address an optimal control problem for a class of discrete time hybrid automata under safety and liveness constraints the solution is based on a hierarchical decomposition of the problem where the low level controller enforces safety and liveness constraints while the high level controller exploits the remaining degrees of freedom for performance optimization lower level control is based on a discrete abstraction of the continuous dynamics the action of low level control can be interpreted as restricting invariants in the hybrid automaton representing the plant model a state feedback solution for the high level control is provided based on the off line construction of an appropriate partition of the state space

graphs closability and causality of linear time invariant discrete time systems in this paper a linear time invariant discrete time system is considered by regarding it as an unbounded operator on 2 n 2 z cn or cz the graphs of closed systems are characterized and properties such as causality and closability of systems are studied it is shown that the closure of a causal closable system on 2 n cn and cz is again causal whereas this is in general not the case for systems on 2 z necessary and sufficient conditions which guarantee that this property hold for systems on 2 z are given many of the results extend without change to the continuous time case

random networks with tunable degree distribution and clustering we present an algorithm for generating random networks with arbitrary degree distribution and clustering frequency of triadic closure we use this algorithm to generate networks with exponential power law and poisson degree distributions with variable levels of clustering such networks may be used as models of social networks and as a testable null hypothesis about network structure finally we explore the effects of clustering on the point of the phase transition where a giant component forms in a random network and on the size of the giant component some analysis of these effects is presented

c sup algebra of strong limit power functions motivated by problems in robust control of power signal set h sub 2 e g h sub 2 is not a vector space fourier analysis cannot be carried out on h sub 2 in general we study those functions in h sub 2 which we call strong limit power we show that the set of all such functions is a sufficiently large c sup algebra fourier analysis is carried out on the functions in particular the uniqueness of the fourier expansion of a strong limit power function is established finally we point out how to analyze and reconstruct such functions

partition of networks into basins of attraction we study partition of networks into basins of attraction based on a steepest ascent search for the node of highest degree each node is associated with or attracted to its neighbor of maximal degree as long as the degree is increasing a node that has no neighbors of higher degree is a peak attracting all the nodes in its basin maximally random scale free networks exhibit different behavior based on their degree distribution exponent gamma for small gamma broad distribution networks are dominated by a giant basin whereas for large gamma narrow distribution there are numerous basins with peaks attracting mainly their nearest neighbors we derive expressions for the first two moments of the number of basins we also obtain the complete distribution of basin sizes for a class of hierarchical deterministic scale free networks that resemble random nets finally we generalize the problem to regular networks and lattices where all degrees are equal and thus the attractiveness of a node must be determined by an assigned weight rather than the degree we derive the complete distribution of basins of attraction resulting from randomly assigned weights in one dimensional chains

controllability and observability of hybrid systems abstract a special class of hybrid systems is considered whose state space is a cross product space of a euclidean space and a finite state space such models may be used to represent systems subject to known abrupt parameter variations such as commu tated networks they may also be used to approximate some types of time varying systems controllability observability and stability of hybrid systems are inves tigated in particular a necessary and sufficient algebraic condition a simple algebraic criterion and a computationally simple algebraic sufficient test for controllability and observability are derived moreover a simple sufficient stability condition is provided

rate equation approach for correlations in growing network models we propose a rate equation approach to compute two vertex correlations in scale free growing network models based on the preferential attachment mechanism the formalism based on previous work of szabo et al fphys rev e 67 056102 s2002dg for the clustering spectrum measuring three vertex correlations is based on a rate equation in the continuous degree and time approximation for the average degree of the nearest neighbors of vertices of degree k with an appropriate boundary condition we study the properties of both two and three vertex correlations for linear preferential attachment models and also for a model yielding a large clustering coefficient the expressions obtained are checked by means of extensive numerical simulations the rate equation proposed can be generalized to more sophisticated growing network models and also extended to deal with related correlation measures as an example we consider the case of a recently proposed model of weighted networks for which we are able to compute a weighted two vertex correlation function taking into account the strength of the interactions between connected vertices

a time domain approach to model validation in this paper we offer a novel approach to control oriented model validation problems the problem is to decide whether a postulated nominal model with bounded uncertainty is consistent with measured input output data our approach directly uses time domain input output data to validate uncertainty models the algorithms we develop are computationally tractable and reduce to generally nondifferentiable convex feasibility problems or to linear programming problems in special cases we give analytical solutions to these problems

l 1 optimal feedback controllers for mimo discrete time systems the problem considered in this paper is the design of a closed loop system consisting of a mimo discrete time plant and compensator in such a way that the system is internally stable and optimally tracks all persistent bounded inputs the solution consists of two parts first the calculation of the minimum value of the performance index a weighted transfer function which is done by solving a linear programming problem and second construction of the optimal transfer function by solving a set of linear equations it is shown that in general the discrete time problem will have a rational solution and thus appears to be of considerable practical significance

stabilization and optimization of switched linear systems in this paper we establish the equivalence among switched convergency asymptotic stabilizability and exponential stabilizability for force free switched linear systems and discuss the implication to the infinite time horizon optimal switching problem we show that for a general cost function under mild assumptions the finiteness of the optimal cost is equivalent to the asymptotic stabilizability of the switched linear system finally we prove the equality between the optimal costs for the switched system and for the relaxed differential inclusion

lethargy results in lti system modelling modelling of linear time invariant lti systems is studied in an approximate modelling framework motivated by modern robust control theory as convolution type lti systems are represented by their unit impulse responses i e through a sequence of real complex numbers this paper provides also system theoretic results on modelling of sequences and their associated transfer functions the modelling results derived in this paper are mostly of lethargy type i e they establish limitations in approximate modelling of lti systems non existence of coprime factorizations is established for a number of different types of lti systems this leads to the somewhat surprising conclusion that many lti systems are not even stabilizable and that an even larger class of lti systems cannot be well approximated by rational transfer function models in addition it is discussed that often the rate of rational approximation of lti systems is not essentially better than the rate of approximation in some basis such as finite impulse response fir models furthermore different definitions of the important finite power signal setup are studied and new results given

focusing on unbounded input output models on the basic one operator input output plant model y gu has fundamental limitations in feedback and closed loop modelling problems on the full time axis when g is an unbounded linear operator it has been a deeply rooted misunderstanding that there is no essential difference between singly infinite and doubly infinite time axis formulations of feedback problems using this basic plant model we present here several new types of results on the limitations of the one operator plant model y gu in discrete time feedback and modelling problems on the full time axis a detailed analysis of existence of solutions in two concrete feedback problems is also provided we discuss also the errors in the classical transform domain analysis of the unbounded plant mapping case on the full time axis an important conclusion of the present and related earlier work is that the two operator plant model ay bu should be seen as the fundamental input output plant model this model allows the modeling of open loop unstable

convex duality and nonlinear optimal control problems in nonlinear optimal control can be reformulated as convex optimization problems over a vector space of linear functionals in this way methods of convex analysis can be brought to bear on the task of characterizing solutions to such problems the result is a necessary and sufficient condition of optimality that generalizes well known sufficient conditions referred to as verification theorems in dynamic programming as a byproduct we obtain a representation of the minimum cost in terms of the upper envelope of subsolutions to the hamilton jacobi equation it is a striking illustration of the wide range of problems to which convex analysis and in particular convex duality is applicable the approach applied to parametric problems in the calculus of variations was pioneered by l c young lectures on the calculus of variations and optimal control theory w b saunders philadelphia pa 1969 as recent work has shown however it is equally fruitful when applied in optimal control thi

the evolutionary game of pressure or interference resistance and collaboration in this paper we extend the framework of the evolutionary inspection game put forward recently by the author and coworkers to a large class of conflict interactions to address the pressure executed by the major player or principal on the large group of small players who can resist this pressure or collaborate with the major player we prove rigorous results on the convergence of various markov decision models of interacting small agents including evolutionary growth i e pairwise in groups and by coalition formation to a deterministic evolution on the distributions of the state spaces of small players paying main attention to situations with an infinite state space of small players we supply precise rates of convergence the theoretical results of the paper are applied to the analysis of the processes of inspection corruption cyber security counter terrorism banks and firms merging strategically enhanced preferential attachment and many other

evolutionary mixed games in structured populations cooperation and the benefits of heterogeneity evolutionary games on networks traditionally involve the same game at each interaction here we depart from this assumption by considering mixed games where the game played at each interaction is drawn uniformly at random from a set of two different games while in well mixed populations the random mixture of the two games is always equivalent to the average single game in structured populations this is not always the case we show that the outcome is in fact strongly dependent on the distance of separation of the two games in the parameter space effectively this distance introduces payoff heterogeneity and the average game is returned only if the heterogeneity is small for higher levels of heterogeneity the distance to the average game grows which often involves the promotion of cooperation the presented results support preceding research that highlights the favorable role of heterogeneity regardless of its origin and they also emphasize the importance of the population structure in amplifying facilitators of cooperation

greedy connectivity of geographically embedded graphs we introduce a measure of greedy connectivity for geographical networks graphs embedded in space and where the search for connecting paths relies only on local information such as a node s location and that of its neighbors constraints of this type are common in everyday life applications greedy connectivity accounts also for imperfect transmission across established links and is larger the higher the proportion of nodes that can be reached from other nodes with a high probability greedy connectivity can be used as a criterion for optimal network design

optimal rejection of persistent bounded disturbances in this paper we formulate the problem of optimal disturbance rejection in the case where the disturbance is generated as the output of a stable system in response to an input which is assumed to be of unit amplitude but is otherwise arbitrary the objective is to choose a controller that minimizes the maximum amplitude of the plant output in response to such a disturbance mathematically this corresponds to requiring uniformly good disturbance rejection over all time since the problem of optimal tracking is equivalent to that of optimal disturbance rejection if a feedback controller is used see 7 sect 5 6 the theory presented here can also be used to design optimal controllers that achieve uniformly good tracking over all time rather than a tracking error whose l 2 norm is small as is the case with the currently popular h infty theory the present theory is a natural counterpart to the existing theory of optimal disturbance rejection the so called h infty theory which is based on the assumption that the disturbance to be rejected is generated by a stable system whose input is square integrable and has unit energy it is shown that the problem studied here has quite different features from its predecessor complete solutions to the problem are given in several important cases including those where the plant is minimum phase or when it has only a single unstable zero in other cases procedures are given for obtaining bounds on the solution and for obtaining suboptimal controllers

a mathematical model for generating bipartite graphs and its application to protein networks complex systems arise in many different contexts from large communication systems and transportation infrastructures to molecular biology most of these systems can be organized into networks composed of nodes and interacting edges here we present a theoretical model that constructs bipartite networks with the particular feature that the degree distribution can be tuned depending on the probability rate of fundamental processes we then use this model to investigate protein domain networks a protein can be composed of up to hundreds of domains each domain represents a conserved sequence segment with specific functional tasks we analyze the distribution of domains in homo sapiens and arabidopsis thaliana organisms and the statistical analysis shows that while a the number of domain types shared by k proteins exhibits a power law distribution b the number of proteins composed of k types of domains decays as an exponential distribution the proposed mathematical model generates bipartite graphs and predicts the emergence of this mixing of a power law and b exponential distributions our theoretical and computational results show that this model requires 1 growth process and 2 copy mechanism

zero pearson coefficient for strongly correlated growing trees we obtained pearson s coefficient of strongly correlated recursive networks growing by preferential attachment of every new vertex by m edges we found that the pearson coefficient is exactly zero in the infinite network limit for the recursive trees m 1 if the number of connections of new vertices exceeds one mg1 then the pearson coefficient in the infinite networks equals zero only when the degree distribution exponent ensuremath gamma does not exceed 4 we calculated the pearson coefficient for finite networks and observed a slow power law like approach to an infinite network limit our findings indicate that pearson s coefficient strongly depends on size and details of networks which makes this characteristic virtually useless for quantitative comparison of different networks

towards effective payoffs in the prisoner s dilemma game on scale free networks we study the transition towards effective payoffs in the prisoner s dilemma game on scale free networks by introducing a normalization parameter guiding the system from accumulated payoffs to payoffs normalized with the connectivity of each agent we show that during this transition the heterogeneity based ability of scale free networks to facilitate cooperative behavior deteriorates continuously eventually collapsing with the results obtained on regular graphs the strategy donations and adaptation probabilities of agents with different connectivities are studied results reveal that strategies generally spread from agents with larger towards agents with smaller degree however this strategy adoption flow reverses sharply in the fully normalized payoff limit surprisingly cooperators occupy the hubs even if the averaged cooperation level due to partly normalized payoffs is moderate

bounded power signal spaces for robust control and modeling the nonlinear space of signals allowing wiener s generalized harmonic analysis gha the linear bounded power signal spaces of beurling marcinkiewicz and wiener and a new linear bounded power space are studied from a control and systems theory perspective specifically it is shown that the system power gain is given by the h infty norm of the system transfer function in each of these spaces for a large class of power stable finite and infinite dimensional systems the gha setup is shown to possess several limitations for the purpose of robustness analysis which motivates the use of the other more general nonstationary signal spaces the natural double sided time axis versions of bounded power signal spaces are shown to break the symmetry between hardy space h infty methods and bounded power operators e g the system transfer function being in h infty does not imply that a causal linear time invariant lti system is bounded as an operator on any of the double sided versions of the studied bounded power signal spaces