brief paper distributed containment control for lagrangian networks with parametric uncertainties under a directed graph in this paper we study the distributed containment control problem for networked lagrangian systems with multiple dynamic leaders in the presence of parametric uncertainties under a directed graph that characterizes the interaction among the leaders and the followers we propose a distributed adaptive control algorithm combined with distributed sliding mode estimators a necessary and sufficient condition on the directed graph is presented such that all followers converge to the dynamic convex hull spanned by the dynamic leaders asymptotically as a byproduct we show a necessary and sufficient condition on leaderless consensus for networked lagrangian systems under a directed graph numerical simulation results are given to show the effectiveness of the proposed control algorithms

the global dimension of schur algebras for gl2 and gl3 abstract we first define the notion of good filtration dimension and weyl filtration dimension in a quasi hereditary algebra we calculate these dimensions explicitly for all irreducible modules in sl 2 and sl 3 we use these to show that the global dimension of a schur algebra for gl 2 and gl 3 is twice the good filtration dimension to do this for sl 3 we give an explicit filtration of the modules by modules of the form f l where is a dominant weight and is p restricted

scaling and crossover in the one dimensional true self avoiding walk the authors apply scaling and crossover arguments to the one dimensional true self avoiding walk which avoids itself with strength g the problem is formulated in terms of a grand canonical ensemble a real space renormalisation group analysis shows that for large repulsion x e g is a relevant variable causing crossover from the self repelling chain src limit with crossover exponent phi 1 a physical interpretation of this result is given in terms of competition between the correlation length and the average distance before the walk turns back on itself the resulting flow suggests the existence of an intermediate attractive fixed point which makes the exponent v different from src and random walk values for all x between 0 and 1 in agreement with recent monte carlo results

edge consensus on complex networks a structural analysis abstractthis paper investigates the consensus taking place on the edges of networks by defining dynamics to edges discrete time and continuous time edge consensus protocols are proposed in which each edge regulates itself according to the information of itself and its neighbouring edges by mapping the original graph to its corresponding line graph conditions guarantee the reaching of edge consensus are presented furthermore the influence of network structure on the speed of reaching edge consensus is analysed by considering a scale free network with a tunable exponent between 2 and it is found that the convergence speed increases as the network becomes more heterogeneous in addition as original network becomes more homogeneous i e as the exponent increases from 2 the corresponding line graph becomes more heterogeneous till reaches 2 5 after that the line graph becomes more homogeneous

standardly stratified algebras and tilting abstract the concept of the characteristic tilting module and of the ringel dual for quasi hereditary algebras is generalized for the setting of standardly stratified algebras

maximum allowable loss probability for consensus of multi agent systems over random weighted lossy networks this note studies the consensus seeking problem for a team of general linear dynamical agents that communicate via a weighted random lossy network linear state feedback consensus protocols are applied and both the weights and feedback gain are treated as control parameters in the protocol it is shown that the weights and the link loss probabilities of a network have non negligible effects on the consensus seeking ability of multi agent systems firstly a weight condition characterized by the eigenvalues of the weighted laplacian matrix is given for systems over ideal communication networks without packet losses secondly based on stochastic stability analysis a maximum allowable loss probability bound is proposed for systems over random lossy networks as long as the link loss probabilities of the network are less than this bound and the mean topology has spanning trees there exist linear protocols solving the mean square consensus problem of the system

distributed containment control with multiple stationary or dynamic leaders in fixed and switching directed networks in this paper we study the problem of distributed containment control of a group of mobile autonomous agents with multiple stationary or dynamic leaders under both fixed and switching directed network topologies first when the leaders are stationary and all followers share an inertial coordinate frame we present necessary and sufficient conditions on the fixed or switching directed network topology such that all followers will ultimately converge to the stationary convex hull formed by the stationary leaders for arbitrary initial states in a space of any finite dimension when the directed network topology is fixed we partition the nonsymmetric laplacian matrix and explore its properties to derive the convergence results when the directed network topology is switching the commonly adopted decoupling technique based on the kronecker product in a high dimensional space can no longer be applied and we hence present an important coordinate transformation technique to derive the convergence results the proposed coordinate transformation technique also has potential applications in other high dimensional distributed control scenarios and might be used to simplify the analysis of a high dimensional system to that of a one dimensional system when the decoupling technique based on the kronecker product cannot be applied second when the leaders are dynamic and all followers share an inertial coordinate frame we propose a distributed tracking control algorithm without velocity measurements when the directed network topology is fixed we derive conditions on the network topology and the control gain to guarantee that all followers will ultimately converge to the dynamic convex hull formed by the dynamic leaders for arbitrary initial states in a space of any finite dimension when the directed network topology is switching we derive conditions on the network topology and the control gain such that all followers will ultimately converge to the minimal hyperrectangle that contains the dynamic leaders and each of its hyperplanes is normal to one axis of the inertial coordinate frame in any high dimensional space we also show via some counterexamples that it is in general impossible to find distribute containment control algorithms without velocity measurements to guarantee that all followers will ultimately converge to the convex hull formed by the dynamic leaders under a switching network topology in a high dimensional space simulation results are presented as a proof of concept

renormalisation of a diffusion on an anisotropic chain a direct renormalisation method is used to calculate the index v which characterises the mean square displacement of a walk on a one dimensional anisotropic homogeneous lattice rn2 1 3 approximately nv for large number of steps n the two parameter exact analysis is carried out for arbitrary choice of the cell size the results are shown to be independent of the cell size

brief paper target containment control of multi agent systems with random switching interconnection topologies in this paper the distributed containment control is considered for a second order multi agent system guided by multiple leaders with random switching topologies the multi leader control problem is investigated via a combination of convex analysis and stochastic process the interaction topology between agents is described by a continuous time irreducible markov chain a necessary and sufficient condition is obtained to make all the mobile agents almost surely asymptotically converge to the static convex leader set moreover conditions on the tracking estimation are provided for the convex target set determined by moving multiple leaders

some homological properties of category o iii we prove that thick category o associated to a semi simple complex finite dimensional lie algebra is extension full in the category of all modules we also prove the weak alexandru conjecture both for regular blocks of thick category o and the associated categories of harish chandra bimodules but disprove it for singular blocks

finitistic dimension of properly stratified algebras abstract we prove that the finitistic dimension of a properly stratified algebra having a simple preserving duality and for which every tilting module is cotilting equals twice the projective dimension of the characteristic tilting module as a corollary we get that the global dimension of a quasi hereditary algebra with duality equals twice the projective dimension of the characteristic tilting module as another corollary we obtain an affirmative answer to the conjecture of erdmann and parker finally we calculate the finitistic dimension of the blocks of certain parabolic generalizations of the category o

random walk renormalisation and the spectral dimension of harmonic models on a class of hierarchical lattice random walk recursion relations are developed on a simple example of hierarchical lattice and are found to be the generating functions for first passage walks with exact renormalisation of a general harmonic model the spectral dimension f for the bond hierarchy class is shown to be f 2ln g ln g ln lambda r where g and lambda r are respectively the aggregation number and resistance eigenvalue the behaviour of f across families of hierarchies is discussed it is noted that the renormalisation group has limitations in its abilities to give results for the above models on hierarchies and that f so found may not maintain on hierarchies its usual significance for random walk statistics

on the rendezvous problem for multiple nonholonomic agents in this note a decentralized feedback control strategy that drives a system of multiple nonholonomic unicycles to a rendezvous point in terms of both position and orientation is introduced the proposed nonholonomic control law is discontinuous and time invariant and using tools from nonsmooth lyapunov theory and graph theory the stability of the overall system is examined similarly to the linear case the convergence of the multi agent system relies on the connectivity of the communication graph that represents the inter agent communication topology the control law is first defined in order to guarantee connectivity maintenance for an initially connected communication graph moreover the cases of static and dynamic communication topologies are treated as corollaries of the proposed framework

on the good filtration dimension of weyl modules for a linear algebraic group let g be a linear algebraic group over an algebraically closed field of characteristic p whose corresponding root system is irreducible in this paper we calculate the weyl filtration di mension of the induced g modules and the simple g modules l fora regular weight we use this to calculate some ext groups of the form ext extl l and ext where are regular and is the weyl module of highest weight we then deduce the projective dimensions and injective dimensions for l and for a regular weight in associated generalised schur algebras we also deduce the global dimension of the schur algebras for gln s n r when p n and for s mp p with m an integer

a necessary and sufficient condition for consensus over random networks we consider the consensus problem for stochastic discrete time linear dynamical systems the underlying graph of such systems at a given time instance is derived from a random graph process independent of other time instances for such a framework we present a necessary and sufficient condition for almost sure asymptotic consensus using simple ergodicity and probabilistic arguments this easily verifiable condition uses the spectrum of the average weight matrix finally we investigate a special case for which the linear dynamical system converges to a fixed vector with probability 1

finitistic dimension and tilting modules for stratified algebras abstract we calculate the finitistic dimension of certain stratified algebras in terms of the projective dimension of the characteristic tilting module this includes in particular quasi hereditary algebras whose koszul dual is again quasi hereditary stratified algebras which are quotients of quasi hereditary algebras over complete local commutative rings and stratified algebras associated with harish chandra bimodules for complex semi simple finite dimensional lie algebras

seeking consensus in networks of linear agents communication noises and markovian switching topologies the stochastic consensus problem of linear multi input multi output mimo multi agent systems mass with communication noises and markovian switching topologies is studied in this technical note the agent s full state is first estimated by the state observer and then the estimated state is exchanged with neighbor agents through a noisy communication environment the communication topology is randomly switching and the switching law is described by a continuous time markovian chain then a consensus protocol is proposed for this mas and some sufficient conditions are obtained for ensuring the mean square and almost sure consensus in addition if the communication topology is fixed some necessary and sufficient conditions for the mean square consensus can be obtained according to whether or not each agent in the system has parents

consensus seeking over random weighted directed graphs we examine the consensus problem for a group of agents that communicate via a stochastic information network communication among agents is modeled as a weighted directed random graph that switches periodically the existence of any edge is probabilistic and independent from the existence of any other edge we further allow each edge to be weighted differently sufficient conditions for asymptotic almost sure consensus are presented for the case of positive weights and for the case of arbitrary weights

bipartite containment tracking of signed networks abstract this paper considers a class of bipartite containment tracking problems for leader following networks associated with signed digraphs it admits multiple leaders which can be not only stationary but also dynamically changing via interactions between them and their neighbored leaders it is shown that the followers can converge to the convex hull containing each leader s trajectory as well as its symmetric trajectory which is the same as it in modulus but different in sign in particular if the adjacency weight matrix of the digraph is not signed but nonnegative then the leader following network achieves the traditional containment tracking simulation tests are performed to illustrate the observed bipartite containment tracking performances of signed networks

leader follower containment control over directed random graphs the leader follower consensus problem for multi agent systems over directed random graphs is investigated motivated by the fact that inter agent communication can be subject to random failure when agents perform tasks in a complex environment a directed random graph is used to model the random loss of communication between agents where the connection of the directed edge in the graph is assumed to be probabilistic and evolves according to a two state markov model in the leader follower network the leaders maintain a constant desired state and the followers update their states by communicating with local neighbors over the random communication network based on convex properties and a stochastic version of lasalle s invariance principle almost sure convergence of the followers states to the convex hull spanned by the leaders states is established for the leader follower random network a numerical simulation is provided to demonstrate the developed result

containment control of multi agent systems in a noisy communication environment the containment problem of first order and second order integral multi agent systems with communication noises is investigated in this paper motivated by the previous work a time varying gain a t is introduced to attenuate the noise effect it is proved that the proposed protocol can solve the mean square containment problem if the following conditions hold 1 the communication topology graph has a spanning forest whose roots are leaders of the multi agent system 2 0 a t dt 3 lim t a t 0 moreover if a t is uniformly continuous these conditions are necessary for ensuring the mean square containment in addition the proposed protocol is also extended to allow the case where each agent has its own gain a i t it is proved that the mean square containment problem can still be solved if all agent dependent gains are infinitesimals of the same order as time goes to infinity finally two simulation examples are provided to demonstrate the effectiveness of the proposed protocols

containment control in mobile networks in this paper the problem of driving a collection of mobile robots to a given target destination is studied in particular we are interested in achieving this transfer in an orderly manner so as to ensure that the agents remain in the convex polytope spanned by the leader agents while the remaining agents only employ local interaction rules to this aim we exploit the theory of partial difference equations and propose hybrid control schemes based on stop go rules for the leader agents non zenoness liveness and convergence of the resulting system are also analyzed

controllable subspace of edge dynamics in complex networks abstract for the edge dynamics in some real networks it may be neither feasible nor necessary to be fully controlled an accompanying issue is that when the external signal is applied to a few nodes or even a single node how many edges can be controlled in this paper for the edge dynamics system we propose a theoretical framework to determine the controllable subspace and calculate its generic dimension based on the integer linear programming this framework allows us not only to analyze the control centrality i e the ability of a node to control but also to uncover the controllable centrality i e the propensity of an edge to be controllable the simulation results and analytic calculation show that dense and homogeneous networks tend to have larger control centrality of nodes and controllable centrality of edges but the negatively correlated in and out degrees of nodes or edges can reduce the two centrality the positive correlation between the control centrality of node and its out degree leads to that the distribution of control centrality instead of that of controllable centrality is encoded by the out degree distribution of networks meanwhile the positive correlation indicates that the nodes with high out degree tend to play more important roles in control

synchronization and convergence of linear dynamics in random directed networks recently methods in stochastic control are used to study the synchronization properties of a nonautonomous discrete time linear system x k 1 g k x k where the matrices g k are derived from a random graph process the purpose of this note is to extend this analysis to directed graphs and more general random graph processes rather than using lyapunov type methods we use results from the theory of inhomogeneous markov chains in our analysis these results have been used successfully in deterministic consensus problems and we show that they are useful for these problems as well sufficient conditions are derived that depend on the types of graphs that have nonvanishing probabilities for instance if a scrambling graph occurs with nonzero probability then the system synchronizes