a procedure for simultaneously stabilizing a collection of single input linear systems using non linear state feedback control abstract this paper considers the problem of simultaneously stabilizing a finite collection of single input linear systems this stabilization is achieved using a single non linear state feedback controller the stability of the resulting closed loop system is established using a collection of quadratic lyapunov functions the main result of this paper is a sufficient condition for simultaneous stabilizability furthermore when this sufficient condition is satisfied the paper gives a formula for constructing the stabilizing feedback control law the paper includes an example in which the stabilization procedure is applied to the stabilization of an f4e fighter aircraft

phase transitions in dynamical random graphs we study a large time limit of a markov process whose states are finite graphs the number of the vertices is described by a supercritical branching process and the dynamics of edges is determined by the rates of appending and deleting we find a phase transition in our model similar to the one in the random graph model g n p we derive a formula for the line of critical parameters which separates two different phases one is where the size of the largest component is proportional to the size of the entire graph and another one where the size of the largest component is at most logarithmic with respect to the size of the entire graph in the supercritical phase we find the asymptotics for the size of the largest component

wavevector dependent susceptibility in z invariant pentagrid ising model we study the q dependent susceptibility q of a z invariant ferromagnetic ising model on a penrose tiling as first introduced by korepin using de bruijn s pentagrid for the rapidity lines the pair correlation function for this model can be calculated exactly using the quadratic difference equations from our previous papers its fourier transform q is studied using a novel way to calculate the joint probability for the pentagrid neighborhoods of the two spins reducing this calculation to linear programming since the lattice is quasiperiodic we find that q is aperiodic and has everywhere dense peaks which are not all visible at very low or high temperatures more and more peaks become visible as the correlation length increases that is as the temperature approaches the critical temperature

simultaneous stabilization via static output feedback and state feedback in this paper the simultaneous stabilization problem is considered using the matrix inequality approach some necessary and sufficient conditions for simultaneous stabilizability of strictly proper multi input multi output mimo plants via static output feedback and state feedback are obtained in the form of coupled algebraic riccati inequalities it is shown that any such stabilizing feedback gain is the solution of some coupled linear quadratic control problems where every cost functional has a suitable cross term a heuristic iterative algorithm based on the linear matrix inequality technique is presented to solve the coupled matrix inequalities the effectiveness of the approach is illustrated by numerical examples

scaling of load in communications networks we show that the load at each node in a preferential attachment network scales as a power of the degree of the node for a network whose degree distribution is p k k we show that the load is l k k with 1 implying that the probability distribution for the load is p l 1 l 2 independent of the results are obtained through scaling arguments supported by finite size scaling studies they contradict earlier claims but are in agreement with the exact solution for the special case of tree graphs results are also presented for real communications networks at the ip layer using the latest available data our analysis of the data shows relatively poor power law degree distributions as compared to the scaling of the load versus degree this emphasizes the importance of the load in network analysis

scaling properties of scale free evolving networks continuous approach the scaling behavior of scale free evolving networks arising in areas such as communications scientific citations collaborations etc is studied we derive universal scaling relations describing properties of such networks and indicate the limits of their validity we show that the main properties of scale free evolving networks may be described in the framework of a simple continuous approach the simplest models of networks growing according to a mechanism of preferential attachment of links to nodes are used we consider different forms of this preference and demonstrate that the range of preferential attachments producing scale free networks is wide we also obtain scaling relations for networks with nonlinear accelerating growth and describe the temporal evolution of the arising distributions size effects char22 the cutoffs of these distributions char22 introduce restrictions for the observation of power law dependences mainly we discuss the so called degree distribution i e the distribution of the number of connections of nodes a scaling form of the distribution of links between pairs of individual nodes for a growing network of citations is also studied we describe the effects of differences between nodes the aging of nodes changes the exponents of the distributions the appearance of a single node with high fitness changes the degree distribution of a network dramatically if its fitness exceeds some threshold value this node captures a finite part of all links of the network we show that permanent random damage to a growing scale free network char22 a permanent deletion of some links char22 radically changes the values of the scaling exponents results of other kinds of permanent damage are described

finite lattice expansion for the ising model on the penrose tiling low temperature series are calculated for the free energy magnetization susceptibility and field derivatives of the susceptibility in the ising model on the quasiperiodic penrose lattice the series are computed to an order of 25 and estimates of the critical exponents and are obtained from pade approximants

optimal simultaneous stabilization of linear single input systems via linear state feedback control simultaneous stabilization of a collection of linear time invariant systems is considered the aim is to develop a practical method for the design of a single feedback controller such that all the systems involved are stabilized with good transient responses hurwitz s necessary and sufficient conditions are used as the required set of constraints for simultaneous stability for transient behaviour we introduce the usual quadratic objective function as performance index for each system their sum is the objective function for the collection of systems and the problem is to minimize it by choosing a feedback gain vector subject to boundedness and the hurwitz stability constraints a computational technique is proposed for solving this problem the results obtained give good transient behaviour for illustration two numerical examples are solved

self avoiding walks and trails on the 3 122 lattice we find the generating function of self avoiding walks saws and trails on a semi regular lattice called the 3 12 2 lattice in terms of the generating functions of simple graphs such as saws self avoiding polygons and tadpole graphs on the hexagonal lattice since the growth constant for these graphs is known on the hexagonal lattice we can find the growth constant for both walks and trails on the 3 12 2 lattice a result of watson 1970 j phys c solid state phys 3 l28 30 then allows us to find the generating function and growth constant of neighbour avoiding walks on the covering lattice of the 3 12 2 lattice which is tetra valent a mapping into walks on the covering lattice allows us to obtain improved bounds on the growth constant for a range of lattices

alternative algorithms for solving nonlinear function and functional inequalities several sophisticated and efficient algorithms are now available in the literature for solving nonlinear function inequalities in a finite number of iterations this paper addresses an alternative approach to this class of problems the approach is relatively straightforward and easy to use essentially the nonlinear inequality constrained problem is reformulated as a standard unconstrained optimization problem via a differentiable transcription thus any existing efficient unconstrained optimization software packages can be used to solve the corresponding unconstrained optimization problem due to the special structure of the differentiable transcription a solution of the nonlinear inequality constraints can be obtained after a finite number of iterations in the process of solving the unconstrained optimization problem the main aim of this paper is however to extend the approach to solving nonlinear functional rather than function inequalities in a finite number of iterations for illustration several examples are computed to demonstrate the feasibility and versatility of the approach

algebraic design techniques for reliable stabilization in this paper we study two problems in feedback stabilization the first is the simultaneous stabilization problem which can be stated as follows given plants g 0 g 1 g l does there exist a single compensator c that stabilizes all of them the second is that of stabilization by a stable compensator or more generally a least unstable compensator given a plant g we would like to know whether or not there exists a stable compensator c that stabilizes g if not what is the smallest number of right half place poles counted according to their mcmillan degree that any stabilizing compensator must have we show that the two problems are equivalent in the following sense the problem of simultaneously stabilizing l 1 plants can be reduced to the problem of simultaneously stabilizing l plants using a stable compensator which in turn can be stated as the following purely algebraic problem given 2l matrices a 1 a l b 1 b l where a i b i are right coprime for all i does there exist a matrix m such that a i mb i is unimodular for all i conversely the problem of simultaneously stabilizing l plants using a stable compensator can be formulated as one of simultaneously stabilizing l 1 plants the problem of determining whether or not there exists an m such that a bm is unimodular given a right coprime pair a b turns out to be a special case of a question concerning a matrix division algorithm in a proper euclidean domain we give an answer to this question and we believe this result might be of some independent interest we show that given two n times m plants g 0 and g 1 we can generically stabilize them simultaneously provided either n or m is greater than one in contrast simultaneous stabilizability of two single input single output plants g 0 and g 1 is not generic

algorithm for simultaneous stabilization of single input systems via dynamic feedback using a sufficient algebraic stability criterion for polynomials an algorithm is deduced for the design of dynamic feedback which simultaneously stabilizes a set of single input systems the application of this algorithm to the stabilization problem of an aircraft is also presented

strict monotonicity for critical points in percolation and ferromagnetic models when is the numerical value of the critical point changed by an enhancement of the process or of the interaction ferromagnetic spin models independent percolation and the contact process are known to be endowed with monotonicity properties in that certain enhancements are capable of shifting the corresponding phase transition in only an obvious direction e g the addition of ferromagnetic couplings can only increase the transition temperature the question explored here is whether enhancements do indeed change the value of the critical point we present a generally applicable approach to this issue for ferromagnetic ising spin systems with pair interactions of finite range ind 2 dimensions it is shown that the critical temperaturetc is strictly monotone increasing in each coupling with the first order derivatives bounded by positive functions which are continuous on the set of fullyd dimensional interactions for independent percolation with 0 pc 1 we prove that any essential enhancement of the process has an effect on the critical probability a result with applications to the question of the existence of entanglements and to invasion percolation with trapping

simultaneous stabilization via linear state feedback control the problem of simultaneously stabilizing a finite collection of plants via a single compensator utilizing full state information is considered the authors assume that the complete state can be measured and seek a static linear state feedback control law sufficient conditions for such full state gains to exist are given and they show via several examples how stabilizing compensators can be designed

the physical position neighbourhood evolving network model abstract many social technological biological and economical systems are properly described by evolved network models in this paper a new evolving network model with the concept of physical position neighbourhood connectivity is proposed and studied this concept exists in many real complex networks such as communication networks the simulation results for network parameters such as the first nonzero eigenvalue and maximal eigenvalue of the graph laplacian clustering coefficients average distances and degree distributions for different evolving parameters of this model are presented the dynamical behaviour of each node on the consensus problem is also studied it is found that the degree distribution of this new model represents a transition between power law and exponential scaling while the barabasi albert scale free model is only one of its special limiting cases it is also found that the time to reach a consensus becomes shorter sharply with increasing of neighbourhood scale of the nodes

dynamical random graphs with memory we study the large time dynamics of a markov process whose states are finite but unbounded graphs the number of vertices is described by a supercritical branching process and the edges follow a certain mean field dynamics determined by the rates of appending and deleting the older an edge is the lesser is the probability that it is still in the graph the lifetime of any edge is distributed exponentially we call its mean value common for all edges a parameter of memory since it shows for how long the system keeps a particular connection between the vertices in the graph we show that our model provides a bridge between two well known models when the parameter of memory goes to infinity this is a generalized model of random growth and when this parameter is zero i e no memory our model behaves as a random graph thus by introducing a general class of dynamical graphs we have a unified overview on rather different models and the relations between them we find all the critical values of the parameters at which our model exhibits phase transitions and describe the properties of the phase diagram finally we compare and discuss the efficiency of the corresponding networks

a markov chain based numerical method for calculating network degree distributions this paper establishes a relation between scale free networks and markov chains and proposes a computation framework for degree distributions of scale free networks we first find that under the ba model the degree evolution of individual nodes in a scale free network follows some non homogeneous markov chains exploring the special structure of these markov chains we are able to develop an efficient algorithm to compute the degree distribution numerically the complexity of our algorithm is o t 2 where t is the number of time steps for adding new nodes we use three examples to demonstrate the computation procedure and compare the results with those from the existing methods pacs 84 35 i 64 60 fr 87 23 ge

scale free property of complex ad hoc networks with accelerated growth a complex ad hoc network model with accelerated growth is proposed in this paper in the evolving process of the model at each time step with the addition probability c1c1 a new vertex with m t m t edges is added into the model which new adding edges are connected with the old vertices according to the preferential attachment rule and with the deletion probability c2 c1 c2 c1 a random chosen vertex is deleted and its associated edges are disappeared simultaneously the accelerated growth means that at each time step the number of new adding edges is the increasing function of the time tt we consider mainly m t t m t t where 0 10 1 to analyze the degree distribution of the model using the continuous approach we prove theoretically and numerically that the model follows the scale free degree distribution and generates both the barabasi and albert ba model and the model introduced by sarshar and roychowdhury sr in addition we present the appropriate numerical range of c1c1 c2c2 and according to theoretical analyzes and numerical simulation the model takes on a certain universality to complex evolving network and has the guiding significance to the application of the peer to peer networks and complex ad hoc networks

a strict inequality for the random triangle model the random triangle model on a graph g is a random graph model where the usual i i d measure is perturbed by a factor qt where q 1 is a constant and t is the number of triangles in the random subgraph here we consider the case where g is the usual two dimensional triangular lattice for which there exists a percolation threshold pc q such that the probability of getting an infinite connected component of retained edges is 0 for p pc q it has previously been shown that pc q is a decreasing function of q here we strengthen this by showing that pc q is strictly decreasing this confirms a conjecture by haggstrom and jonasson

simultaneous stabilization of linear single input single output plants the problem of designing a single compensator that simultaneously stabilizes several distinct plants is considered here using the fractional representation approach owing to saeks and murray 1982 1983 necessary and sufficient conditions for the existence of such a compensator are given and a design algorithm is proposed

spectra of real world graphs beyond the semicircle law results on the spectra of adjacency matrices corresponding to models of real world graphs we find that when the number of links grows as the number of nodes the spectral density of uncorrelated random matrices does not converge to the semicircle law furthermore the spectra of real world graphs have specific features depending on the details of the corresponding models in particular scale free graphs develop a trianglelike spectral density with a power law tail while small world graphs have a complex spectral density consisting of several sharp peaks these and further results indicate that the spectra of correlated graphs represent a practical tool for graph classification and can provide useful insight into the relevant structural properties of real networks

strict inequality for critical values of potts models and random cluster processes we prove that the critical value c of a ferromagnetic potts model is astrictly decreasing function of the strengths of interaction of the process this is achieved in the more general context of the random cluster representation of fortuin and kasteleyn by deriving and utilizing a formula which generalizes the technique known in percolation theory as russo s formula as a byproduct of the method we present a general argument for showing that at any given point on the critical surface of a multiparameter process the values of a certain critical exponent do not depend on the direction of approach of that point our results apply to all random cluster processes satisfying the fkg inequality

effect of node deleting on network structure abstract the ever increasing knowledge of the structure of various real world networks has uncovered their complex multi mechanism governed evolution processes therefore a better understanding of the structure and evolution of these networked complex systems requires us to describe such processes in a more detailed and realistic manner in this paper we introduce a new type of network growth rule which comprises addition and deletion of nodes and propose an evolving network model to investigate the effect of node deleting on network structure it is found that with the introduction of node deleting network structure is significantly transformed in particular degree distribution of the network undergoes a transition from scale free to exponential forms as the intensity of node deleting increases at the same time nontrivial disassortative degree correlation develops spontaneously as a natural result of network evolution in the model we also demonstrate that node deleting introduced in the model does not destroy the connectedness of a growing network so long as the increasing rate of edges is not excessively small in addition it is found that node deleting will weaken but not eliminate the small world effect of a growing network and generally it will decrease the clustering coefficient in a network

degree distribution and scaling in the connecting nearest neighbors model we present a detailed analysis of the connecting nearest neighbors model by vazquez phys rev e 67 056104 2003 we show that the degree distribution follows a power law but the scaling exponent can vary with the parameter setting moreover the correspondence of the growing version of the connecting nearest neighbors model to the particular random walk model and recursive search model is established

characterization of controllers in simultaneous stabilization this paper gives a convenient parametrization for the class of all stabilizing controllers for two or more plants the result represent a generalization of the youla parametrization of the class of all stabilizing controllers in terms of an arbitrary stable proper transfer function although as expected the additional constraints for simultaneous stabilization are not readily incorporated into h h2 optimization procedures as in the standard case there is immediate application of the theory for reduction of controllers which simultaneously stabilize two or more plants

general connectivity distribution functions for growing networks with preferential attachment of fractional power we study the general connectivity distribution functions for growing networks with preferential attachment pa of fractional power i k using simon s method we first show that the heart of the previously known methods of the rate equations for the connectivity distribution functions is nothing but simon s method for word problem secondly we show that for the case of fractional the z transformation of the rate equation provides a fractional differential equation of a new type which coincides with that for pa with linear power when 1 we show that to solve such a fractional differential equation we need to define a transcendental function a b c z that we call upsilon function most of all the previously known results are obtained consistently in the framework of a unified theory

self avoiding walks neighbour avoiding walks and trails on semiregular lattices we study self avoiding and neighbour avoiding walks and lattice trails on two semiregular lattices the lattice and the lattice for the lattice we find the exact connective constant for both self avoiding walks neighbour avoiding walks and trails for the lattice we generate long series which permit the accurate estimation of the connective constant for self avoiding walks and trails

high temperature expansion for ising models on quasiperiodic tilings we consider high temperature expansions for the free energy of zero field ising models on planar quasiperiodic graphs for the penrose and the octagonal ammann beenker tiling we compute the expansion coefficients up to 18th order as a by product we obtain exact vertex averaged numbers of self avoiding polygons on these quasiperiodic graphs in addition we analyse periodic approximants by computing the partition function via the kac ward determinant it turns out that the series expansions alone do not yield reliable estimates of the critical exponents this is due to the limitation on the order of the series caused by the number of graphs that have to be taken into account and more seriously to rather strong fluctuations in the behaviour of the coefficients nevertheless our results are compatible with the commonly accepted conjecture that the models under consideration belong to the same universality class as those on periodic two dimensional lattices

long paths and cycles in dynamical graphs we study the large time dynamics of a markov process whose states are finite directed graphs the number of the vertices is described by a supercritical branching process and the edges follow a certain mean field dynamics determined by the rates of appending and deleting we find sufficient conditions under which asymptotically a s the order of the largest component is proportional to the order of the graph a lower bound for the length of the longest directed path in the graph is provided as well we derive an explicit formula for the limit as time goes to infinity of the expected number of cycles of a given finite length finally we study the phase diagram

some results on simultaneous stabilization this paper considers the simultaneous stabilization problem and gives an answer to it in terms of the poles and zeros on the right half plane which is considered as an extension of the conditions previously reported

human sexual contact network as a bipartite graph abstract a simple model to encapsulate the essential growth properties of the web of human sexual contacts is presented in the model only heterosexual connection is considered and represented by a random growing bipartite graph where both male female contact networks grow simultaneously the time evolution of the model is analysed by a rate equation approach leading to confirm that male and female sexual contact distributions decay as power laws with exponents depending on influx and charisma of the sexes

dynamic random networks in dynamic populations we consider a random network evolving in continuous time in which new nodes are born and old may die and where undirected edges between nodes are created randomly and may also disappear the node population is markovian and so is the creation and deletion of edges given the node population each node is equipped with a random social index and the intensity at which a node creates new edges is proportional to the social index and the neighbour is either chosen uniformly or proportional to its social index in a modification of the model we derive properties of the network as time and the node population tends to infinity in particular the degree distribution is shown to be a mixed poisson distribution which may exhibit a heavy tail e g power law if the social index distribution has a heavy tail the limiting results are verified by means of simulations and the model is fitted to a network of sexual contacts

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

scaling function and universal amplitude combinations for self avoiding polygons we analyse new data for self avoiding polygons saps on the square and triangular lattices enumerated by both perimeter and area providing evidence that the scaling function is the logarithm of an airy function the results imply universal amplitude combinations for all area moments and suggest that rooted saps may satisfy a q algebraic functional equation

walks on the penrose lattice the authors performed extensive monte carlo simulations for different types of walks random walks ideal chains and self avoiding walks on the penrose quasilattice the critical exponent nu for each process is found to be the same as for periodic two dimensional lattices thus universality seems to hold also for the penrose tiling

network model of human language abstract the phenomenon of human language is widely studied from various points of view it is interesting not only for social scientists antropologists or philosophers but also for those interested in the network dynamics in several recent papers word web or language as a graph has been investigated r f cancho r sole the small world of human language proc r soc london b 268 2001 2261 2265 a e motter p s de moura lai ying cheng p dasgupta topology of the conceptual network of language phys rev e 65 2002 r 065102 m steyvers j b tenenbaum the large scale structure of semantic networks statistical analysis and a model of semantic growth cogn sci 29 2005 41 78 in this paper i revise recent studies of syntactical word web r f cancho r sole the small world of human language proc r soc london b 268 2001 2261 2265 s n dorogovtsev j f f mendes language as an evolving word web proc r soc london b 268 2001 2603 2606 i present a model of growing network in which such processes as node addition edge rewiring and new link creation are taken into account i argue that this model is a satisfactory minimal model explaining measured data r f cancho r sole the small world of human language proc r soc london b 268 2001 2261 2265 m markosova p nather language as a graph in v kvasnicka p trebatick j pospichal eds mind intelligence and life kelemen stu bratislava 2007 pp 298 307 in slovak

root lattices and quasicrystals it is shown that root lattices and their reciprocals might serve as the right pool for the construction of quasicrystalline structure models all noncrystallographic symmetries observed so far are covered in minimal embedding with maximal symmetry

time evolution of the degree distribution of model a of random attachment growing networks abstract for random growing networks barabas and albert proposed a kind of model in barabas et al physica a 272 1999 173 i e model a in this paper for model a we give the differential format of master equation of degree distribution and obtain its analytical solution the obtained result p k t is the time evolution of degree distribution p k t is composed of two terms at given finite time one term decays exponentially the other reflects size effect at infinite time the degree distribution is the same as that of barabas and albert in this paper we also discuss the normalization of degree distribution p k t in detail

simultaneous stabilization of linear single input systems by linear state feedback control a necessary and sufficient condition for the existence of a linear state feedback controller which will simultaneously stabilize a finite collection of single input linear systems is presented this condition can be checked numerically through the solution of a non linear programming problem presented here and can be used to determine whether or not a particular simultaneous stabilization problem has a solution if the stabilization problem has a solution the method provides a means of constructing a suitable controller the proposed method is very versatile it is a simple matter to limit the size of elements of the feedback gain vector and thus ensure that the control action is not excessive in addition for simultaneous stabilization problems which have a range of possible solutions the proposed method makes it possible to select one which yields good performance

size and area of square lattice polygons we use the finite lattice method to calculate the radius of gyration the first and second area weighted moments of self avoiding polygons on the square lattice the series have been calculated for polygons up to perimeter 82 analysis of the series yields high accuracy estimates confirming theoretical predictions for the value of the size exponent and certain universal amplitude combinations furthermore a detailed analysis of the asymptotic form of the series coefficients provide the firmest evidence to date for the existence of a correction to scaling exponent 3 2

growing networks with mixed attachment mechanisms networks grow and evolve when new nodes and links are added in there are two methods to add the links uniform attachment and preferential attachment we take account of the addition of links with mixed attachment between uniform attachment and preferential attachment in proportion by using numerical simulations and analysis based on a continuum theory we obtain that the degree distribution p k has an extended power law form p k k k0 when the number of edges k of a node is much larger than a certain value k0 the degree distribution reduces to the power law form p k k and when k is much smaller than k0 the degree distribution degenerates into the exponential form it has been found that degree distribution possesses this extended power law form for many real networks such as the movie actor network the citation network of scientific papers and diverse protein interaction networks

algebraic techniques for enumerating self avoiding walks on the square lattice the authors describe a new algebraic technique for enumerating self avoiding walks on the rectangular lattice the computational complexity of enumerating walks of n steps is of order 3n 4 times a polynomial in n and so the approach is greatly superior to direct counting techniques they have enumerated walks of up to 39 steps as a consequence they are able to accurately estimate the critical point critical exponent and critical amplitude

are randomly grown graphs really random we analyze a minimal model of a growing network at each time step a new vertex is added then with probability ensuremath delta two vertices are chosen uniformly at random and joined by an undirected edge this process is repeated for t time steps in the limit of large t the resulting graph displays surprisingly rich characteristics in particular a giant component emerges in an infinite order phase transition at ensuremath delta 1 8 at the transition the average component size jumps discontinuously but remains finite in contrast a static random graph with the same degree distribution exhibits a second order phase transition at ensuremath delta 1 4 and the average component size diverges there these dramatic differences between grown and static random graphs stem from a positive correlation between the degrees of connected vertices in the grown graph older vertices tend to have higher degree and to link with other high degree vertices merely by virtue of their age we conclude that grown graphs however randomly they are constructed are fundamentally different from their static random graph counterparts

optimal simultaneous stabilization of descriptor systems via output feedback in this paper we consider the optimal simultaneous stabilization problem for a collection of linear time invariant descriptor systems the objective function for the collection of systems is defined to be the sum of all quadratic objective functions associated with individual systems the problem is to choose an output feedback gain such that the objective function is minimized firstly the set of all the output feedback gains for which the objective function is finite is characterized and proven to be a compact set secondly an iterative algorithm based on the gradient flow idea is derived for solving this problem and some properties on the gradient flow are obtained to guarantee the success of this method thirdly an initial value for the gradient flow equation is found by solving a standard unconstrained optimization problem finally two numerical examples are solved for illustration

simultaneous stabilization for a collection of single input nonlinear systems this paper presents a novel method for designing a controller that simultaneously stabilizes a collection of single input nonlinear systems the control lyapunov function approach is used to derive necessary and sufficient conditions for the existence of time invariant simultaneously stabilizing state feedback controllers additionally a universal formula for constructing a continuous simultaneously stabilizing controller when the provided sufficient condition is satisfied is presented for any collection of second order and third order feedback linearizable systems in canonical form global simultaneous stabilization via a single state feedback controller is shown to be always possible two examples are included for illustration

degree correlations in growing networks with deletion of nodes in this paper we study the degree distribution and the two node degree correlations in growing networks generated via a general linear preferential attachment of new nodes together with a uniformly random deletion of nodes by using a continuum approach we show that under some suitable combinations of parameters deletion rate and node attractiveness the degree distribution not only loses its scale free character but can even be supported on a small range of degrees moreover we obtain new results on two vertex degree correlations showing that for degree distributions with finite variance such correlations can change under a nonselective removal of nodes

self avoiding walks and polygons on quasiperiodic tilings we enumerate self avoiding walks and polygons counted by perimeter on the quasiperiodic rhombic penrose and ammann beenker tilings thereby considerably extending previous results in contrast to similar problems on regular lattices these numbers depend on the chosen initial vertex we compare different ways of counting and demonstrate that suitable averaging improves convergence to the asymptotic regime this leads to improved estimates for critical points and exponents which support the conjecture that self avoiding walks on quasiperiodic tilings belong to the same universality class as self avoiding walks on the square lattice for polygons the obtained enumeration data do not allow us to draw decisive conclusions about the exponent

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

fractional representation algebraic geometry and the simultaneous stabilization problem an explicit relationship between the fractional representation approach feedback system design and the algebro geometric approach to system theory is formulated and used to derive a global solution to the feedback system problem these techniques are then applied to the simultaneous stabilization problem yielding a natural geometric criterion for a set of plants to be simultaneously stabilized by a single compensator

a scale free network with limiting on vertices abstract we propose and analyze a random graph model which explains a phenomena in the economic company network in which company may not expand its business at some time due to the limiting of money and capacity the random graph process is defined as follows at any time step t i with probability k and independently of other time step each vertex v i i t 1 is inactive which means it cannot be connected by more edges where k is the degree of v i at the time step t ii a new vertex v t is added along with m edges incident with v t at one time and its neighbors are chosen in the manner of preferential attachment we prove that the degree distribution p k of this random graph process satisfies p k c 1 k 3 0 1 0 if is a constant 0 and p k c 2 k 3 if l 0 as l where c 1 c 2 are two positive constants the analytical result is found to be in good agreement with that obtained by numerical simulations furthermore we get the degree distributions in this model with m varying functions by simulation

modulated scale free network in euclidean space a random network is grown by introducing at unit rate randomly selected nodes on the euclidean space a node is randomly connected to its ith predecessor of degree k i with a directed link of length l using a probability proportional to k i l alpha our numerical study indicates that the network is scale free for all values of alpha alpha c and the degree distribution decays stretched exponentially for the other values of alpha the link length distribution follows a power law d l approximately l delta where delta is calculated exactly for the whole range of values of alpha

some results on simultaneous stabilization with multiple domains of stability abstract in this paper the problem of simultaneous stabilization in the case where each plant has a different domain of stability associated with it is studied specifically it is supposed that one is given plants p0 p1 pr together with domains of stability d0 dr and the objective is to find a common controller c such that the poles of the closed loop transfer matrix h pi c are all inside the corresponding domain of stability di for i 0 r this formulation of the simultaneous stabilization problem generalizes earlier problem formulations wherein it was assumed that all the domains of stability were the same and is believed to be a more realistic formulation of the problem of preserving system stability in the event of sensor or actuator failure in the paper necessary and sufficient conditions for the existence of a common controller are derived in the case where one of the domains say d0 is a subset of the rest the case of two plants p0 p1 with d0 a subset of d1 is studied in great detail and the following result is established the two plants can be simultaneously stabilized in the above generalized sense if and only if they can be simultaneously stabilized with respect to the larger of the two regions more precisely there exists a controller c such that the poles of h pi c lie inside di for i 0 1 if and only if there exists a controller such that the poles of bothh p0 c and of h p1 c lie inside the larger region d1