distributed consensus for multi agent systems with delays and noises in transmission channels this paper studies the distributed consensus problem for linear discrete time multi agent systems with delays and noises in transmission channels due to the presence of noises and delays existing techniques such as the lifting technique and the stochastic lyapunov theory are no longer applicable to the analysis of consensus in this paper a novel technique is introduced to overcome the difficulties induced by the delays and noises a consensus protocol with decaying gains satisfying persistence condition is adopted necessary and sufficient conditions for strong consensus and mean square consensus are respectively given for non leader follower and leader follower cases under a fixed topology under dynamically switching topologies and randomly switching topologies sufficient conditions for strong consensus and mean square consensus are also obtained numerical examples are given to demonstrate the effectiveness of the proposed protocols

asynchronous distributed algorithms for solving linear algebraic equations two asynchronous distributed algorithms are presented for solving a linear equation of the form ax b with at least one solution the equation is simultaneously and asynchronously solved by m agents assuming that each agent knows only a subset of the rows of the partitioned matrix a b the estimates of the equation s solution generated by its neighbors and nothing more neighbor relationships are characterized by a time dependent directed graph whose vertices correspond to agents and whose arcs depict neighbor relationships each agent recursively updates its estimate of a solution at its own event times by utilizing estimates generated by its neighbors which are transmitted with delays the event time sequences of different agents are not assumed to be synchronized it is shown that for any matrix vector pair a b for which the equation has a solution and any repeatedly jointly strongly connected sequence of neighbor graphs defined on the merged sequence of all agents event times the algorithms cause all agents estimates to converge exponentially fast to the same solution to ax b the first algorithm requires a specific initialization step at each agent and the second algorithm works for arbitrary initializations explicit expressions for convergence rates are provided and a relation between local initializations and limiting consensus solutions is established which is used to solve the least 2 norm solution

opinion dynamics in social networks with hostile camps consensus vs polarization most of the distributed protocols for multi agent consensus assume that the agents are mutually cooperative and trustful and so the couplings among the agents bring the values of their states closer opinion dynamics in social groups however require beyond these conventional models due to ubiquitous competition and distrust between some pairs of agents which are usually characterized by repulsive couplings and may lead to clustering of the opinions a simple yet insightful model of opinion dynamics with both attractive and repulsive couplings was proposed recently by c altafini who examined first order consensus algorithms over static signed graphs this protocol establishes modulus consensus where the opinions become the same in modulus but may differ in signs in this paper we extend the modulus consensus model to the case where the network topology is an arbitrary time varying signed graph and prove reaching modulus consensus under mild sufficient conditions of uniform connectivity of the graph for cut balanced graphs not only sufficient but also necessary conditions for modulus consensus are given

reformulation of the covering and quantizer problems as ground states of interacting particles it is known that the sphere packing problem and the number variance problem closely related to an optimization problem in number theory can be posed as energy minimizations associated with an infinite number of point particles in d dimensional euclidean space r d interacting via certain repulsive pair potentials we reformulate the covering and quantizer problems as the determination of the ground states of interacting particles in r d that generally involve single body two body three body and higher body interactions this is done by linking the covering and quantizer problems to certain optimization problems involving the void nearest neighbor functions that arise in the theory of random media and statistical mechanics these reformulations which again exemplify the deep interplay between geometry and physics allow one now to employ theoretical and numerical optimization techniques to analyze and solve these energy minimization problems the covering and quantizer problems have relevance in numerous applications including wireless communication network layouts the search of high dimensional data parameter spaces stereotactic radiation therapy data compression digital communications meshing of space for numerical analysis and coding and cryptography among other examples in the first three space dimensions the best known solutions of the sphere packing and number variance problems or their dual solutions are directly related to those of the covering and quantizer problems but such relationships may or may not exist for d 4 depending on the peculiarities of the dimensions involved our reformulation sheds light on the reasons for these similarities and differences we also show that disordered saturated sphere packings provide relatively thin economical coverings and may yield thinner coverings than the best known lattice coverings in sufficiently large dimensions in the case of the quantizer problem we derive improved upper bounds on the quantizer error using sphere packing solutions which are generally substantially sharper than an existing upper bound in low to moderately large dimensions we also demonstrate that disordered saturated sphere packings yield relatively good quantizers finally we remark on possible applications of our results for the detection of gravitational waves

robust algorithm to generate a diverse class of dense disordered and ordered sphere packings via linear programming we have formulated the problem of generating dense packings of nonoverlapping nontiling nonspherical particles within an adaptive fundamental cell subject to periodic boundary conditions as an optimization problem called the adaptive shrinking cell asc formulation s torquato and y jiao phys rev e 80 041104 2009 because the objective function and impenetrability constraints can be exactly linearized for sphere packings with a size distribution in d dimensional euclidean space r d it is most suitable and natural to solve the corresponding asc optimization problem using sequential linear programming slp techniques we implement an slp solution to produce robustly a wide spectrum of jammed sphere packings in r d for d 2 3 4 5 and 6 with a diversity of disorder and densities up to the respective maximal densities a novel feature of this deterministic algorithm is that it can produce a broad range of inherent structures locally maximally dense and mechanically stable packings besides the usual disordered ones such as the maximally random jammed state with very small computational cost compared to that of the best known packing algorithms by tuning the radius of the influence sphere for example in three dimensions we show that it can produce with high probability a variety of strictly jammed packings with a packing density anywhere in the wide range 0 6 0 7408 where 18 0 7408 corresponds to the density of the densest packing we also apply the algorithm to generate various disordered packings as well as the maximally dense packings for d 2 4 5 and 6 our jammed sphere packings are characterized and compared to the corresponding packings generated by the well known lubachevsky stillinger ls molecular dynamics packing algorithm compared to the ls procedure our slp protocol is able to ensure that the final packings are truly jammed produces disordered jammed packings with anomalously low densities and is appreciably more robust and computationally faster at generating maximally dense packings especially as the space dimension increases

reaching a consensus in a dynamically changing environment convergence rates measurement delays and asynchronous events this paper uses recently established properties of compositions of directed graphs together with results from the theory of nonhomogeneous markov chains to derive worst case convergence rates for the headings of a group of mobile autonomous agents which arise in connection with the widely studied vicsek consensus problem the paper also uses graph theoretic constructions to solve modified versions of the vicsek problem in which there are measurement delays asynchronous events or a group leader in all three cases the conditions under which consensus is achieved prove to be almost the same as the conditions under which consensus is achieved in the synchronous delay free leaderless case

differential inequalities in multi agent coordination and opinion dynamics modeling abstract many distributed algorithms for multi agent coordination employ the simple averaging dynamics referred to as the laplacian flow besides the standard consensus protocols examples include but are not limited to algorithms for aggregation and containment control target surrounding distributed optimization and models of opinion formation in social groups in spite of their similarities each of these algorithms has been studied using separate mathematical techniques in this paper we show that stability and convergence of many coordination algorithms involving the laplacian flow dynamics follow from the general consensus dichotomy property of a special differential inequality the consensus dichotomy implies that any solution to the differential inequality is either unbounded or converges to a consensus equilibrium in this paper we establish the dichotomy criteria for differential inequalities and illustrate their applications to multi agent coordination and opinion dynamics modeling

dense packings of polyhedra platonic and archimedean solids understanding the nature of dense particle packings is a subject of intense research in the physical mathematical and biological sciences the preponderance of previous work has focused on spherical particles and very little is known about dense polyhedral packings we formulate the problem of generating dense packings of nonoverlapping nontiling polyhedra within an adaptive fundamental cell subject to periodic boundary conditions as an optimization problem which we call the adaptive shrinking cell asc scheme this optimization problem is solved here using a variety of multiparticle initial configurations to find the dense packings of each of the platonic solids in three dimensional euclidean space mathbb r 3 except for the cube which is the only platonic solid that tiles space we find the densest known packings of tetrahedra icosahedra dodecahedra and octahedra with densities 0 823 dots 0 836 dots 0 904 dots and 0 947 dots respectively it is noteworthy that the densest tetrahedral packing possesses no long range order unlike the densest tetrahedral packing which must not be a bravais lattice packing the densest packings of the other nontiling platonic solids that we obtain are their previously known optimal bravais lattice packings we also derive a simple upper bound on the maximal density of packings of congruent nonspherical particles and apply it to platonic solids archimedean solids superballs and ellipsoids provided that what we term the asphericity ratio of the circumradius to inradius is sufficiently small the upper bounds are relatively tight and thus close to the corresponding densities of the optimal lattice packings of the centrally symmetric platonic and archimedean solids our simulation results rigorous upper bounds and other theoretical arguments lead us to the conjecture that the densest packings of platonic and archimedean solids with central symmetry are given by their corresponding densest lattice packings this can be regarded to be the analog of kepler s sphere conjecture for these solids the truncated tetrahedron is the only non centrally symmetric archimedean solid the densest known packing of which is a non lattice packing with density at least as high as 23 24 0 958 text 333 dots we discuss the validity of our conjecture to packings of superballs prisms and antiprisms as well as to high dimensional analogs of the platonic solids in addition we conjecture that the optimal packing of any convex congruent polyhedron without central symmetry generally is not a lattice packing finally we discuss the possible applications and generalizations of the asc scheme in predicting the crystal structures of polyhedral nanoparticles and the study of random packings of hard polyhedra

comment on packing hyperspheres in high dimensional euclidean spaces it is shown that the numerical data in cond mat 0608362 are in very good agreement with the predictions of cond mat 0601573

nash wings and real prime divisors let v be an affine variety defined over an arbitrary real closed field r with affine coordinate ring a r v and function field k r v then open semialgebraic subsets of the regular points of v r define what we call oeometric preorders on k the study of affine semialgebraic geometry corresponds to the study of commutative algebra in k in the presence of such a preorder the theorem which motivated this paper is

a distributed algorithm for solving a linear algebraic equation a distributed algorithm is described for solving a linear algebraic equation of the form ax b assuming the equation has at least one solution the equation is simultaneously solved by m agents assuming each agent knows only a subset of the rows of the partitioned matrix matrix a b the current estimates of the equation s solution generated by its neighbors and nothing more each agent recursively updates its estimate by utilizing the current estimates generated by each of its neighbors neighbor relations are characterized by a time dependent directed graph bbn t whose vertices correspond to agents and whose arcs depict neighbor relations it is shown that for any matrix a for which the equation has a solution and any sequence of repeatedly jointly strongly connected graphs bbn t t 1 2 ldots the algorithm causes all agents estimates to converge exponentially fast to the same solution to ax b it is also shown that under mild assumptions the neighbor graph sequence must actually be repeatedly jointly strongly connected if exponential convergence is to be assured a worst case convergence rate bound is derived for the case when ax b has a unique solution it is demonstrated that with minor modification the algorithm can track the solution to ax b even if a and b are changing with time provided the rates of change of a and b are sufficiently small it is also shown that in the absence of communication delays exponential convergence to a solution occurs even if the times at which each agent updates its estimates are not synchronized with the update times of its neighbors a modification of the algorithm is outlined which enables it to obtain a least squares solution to ax b in a distributed manner even if ax b does not have a solution

second order global consensus in multiagent networks with random directional link failure in this paper we consider the second order globally nonlinear consensus in a multiagent network with general directed topology and random interconnection failure by characterizing the behavior of stochastic dynamical system with the corresponding time averaged system a criterion for the second order consensus is derived by constructing a lyapunov function for the time averaged network by associating the solution of random switching nonlinear system with the constructed lyapunov function a sufficient condition for second order globally nonlinear consensus in a multiagent network with random directed interconnections is also established it is required that the second order consensus can be achieved in the time averaged network and the lyapunov function decreases along the solution of the random switching nonlinear system at an infinite subsequence of the switching moments a numerical example is presented to justify the correctness of the theoretical results

multiagent systems with compasses this paper investigates agreement protocols over cooperative and cooperativeantagonistic multiagent networks with coupled continuous time nonlinear dynamics to guarantee convergence for such syste

robust consensus for continuous time multiagent dynamics this paper investigates consensus problems for continuous time multiagent systems with time varying communication graphs subject to input noise based on input to state stability and integral input to state stability robust consensus and integral robust consensus are defined with respect to l and l1 norms of the noise function respectively sufficient and or necessary connectivity conditions are obtained for the system to reach robust consensus or integral robust consensus under mild assumptions the results answer the question on how much interaction is required for a multiagent network to converge despite a certain amount of input disturbance the convergence time is obtained for the consensus computation on directed and k bidirectional graphs

nodes selection strategy in cooperative tracking problem in this paper a new optimization problem is addressed for node selection that has application potentials in input output switches for sensors in control system design and leader determination in social networks the purpose of the addressed problem is to develop a strategy for selecting a subset of nodes as controlled nodes in order to minimize certain objective function consisting of the convergence speed and the energy of control action over a finite time horizon for networks with fixed controlled nodes an upper bound of the objective function is obtained which is shown to be convex and independent of the time horizon for networks with switched controlled nodes a greedy algorithm is proposed to reduce the computation complexity resulting from the length of the time horizon where the nodes selection is carried out over divided small time intervals the cost gap is also analyzed between the strategy of optimizing over the whole time horizon and the strategy of optimizing over the small intervals finally the proposed nodes selection strategy is validated through simulations and two regions are found in which the number of optimal controlled nodes is determined

distributed leader follower flocking control for multi agent dynamical systems with time varying velocities abstract using tools from algebraic graph theory and nonsmooth analysis in combination with ideas of collective potential functions velocity consensus and navigation feedback a distributed leader follower flocking algorithm for multi agent dynamical systems with time varying velocities is developed where each agent is governed by second order dynamics the distributed leader follower algorithm considers the case in which the group has one virtual leader with time varying velocity for each agent i this algorithm consists of four terms the first term is the self nonlinear dynamics which determines the final time varying velocity the second term is determined by the gradient of the collective potential between agent i and all of its neighbors the third term is the velocity consensus term and the fourth term is the navigation feedback from the leader to avoid an impractical assumption that the informed agents sense all the states of the leader the new designed distributed algorithm is developed by making use of observer based pinning navigation feedback in this case each informed agent only has partial information about the leader yet the velocity of the whole group can still converge to that of the leader and the centroid of those informed agents having the leader s position information follows the trajectory of the leader asymptotically finally simulation results are presented to demonstrate the validity and effectiveness of the theoretical analysis surprisingly it is found that the local minimum of the potential function may not form a commonly believed lattice

randomized optimal consensus of multi agent systems in this paper we formulate and solve a randomized optimal consensus problem for multi agent systems with stochastically time varying interconnection topology the considered multi agent system with a simple randomized iterating rule achieves an almost sure consensus meanwhile solving the optimization problem min z r email protected i 1 nf i z in which the optimal solution set of objective function f i can only be observed by agent i itself at each time step simply determined by a bernoulli trial each agent independently and randomly chooses either taking an average among its neighbor set or projecting onto the optimal solution set of its own optimization component both directed and bidirectional communication graphs are studied connectivity conditions are proposed to guarantee an optimal consensus almost surely with proper convexity and intersection assumptions the convergence analysis is carried out using convex analysis we compare the randomized algorithm with the deterministic one via a numerical example the results illustrate that a group of autonomous agents can reach an optimal opinion by each node simply making a randomized trade off between following its neighbors or sticking to its own opinion at each time step

brief paper convergence speed in distributed consensus over dynamically switching random networks characterizing convergence speed is one of the most important research challenges in the design of distributed consensus algorithms for networked multi agent systems in this paper we consider a group of agents that communicate via a dynamically switching random information network each link in the network which represents the directed undirected information flow between any ordered unordered pair of agents could be subject to failure with a certain probability hence we model the information flow using dynamically switching random graphs we characterize the convergence speed for the distributed discrete time consensus algorithm over a variety of random networks with arbitrary weights in particular we propose the asymptotic and per step mean square convergence factors as measures of the convergence speed and derive the exact value for the per step mean square convergence factor numerical examples are also given to illustrate our theoretical results

on the realizability of pair correlation functions abstract the pair correlation function g 2 r provides a basic geometric descriptor for many particle systems it must obey two necessary conditions i non negativity for all distances r and ii non negativity of its associated structure factor s k for all k here we utilize an improved stochastic construction algorithm for particle configurations to establish conditions in which i and ii are also sufficient i e g 2 r is in fact realizable two types of target pair correlation functions have been investigated in one two and three dimensions for hard core particles specifically a unit step function and a contact plus step pair correlation function results indicate that the former target function is realizable up to a terminal density set by necessary condition ii at which the particle core packing fraction equals 2 d in d dimensions furthermore results are consistent with the proposition that for d 1 the contact plus step function is realizable up to a terminal density due to condition ii at which the packing fraction of cores is d 2 2 d 1 torquato and stillinger j phys chem b 106 2000 8354 11406

local density fluctuations hyperuniformity and order metrics questions concerning the properties and quantification of density fluctuations in point patterns continue to provide many theoretical challenges the purpose of this paper is to characterize certain fundamental aspects of local density fluctuations associated with general point patterns in any space dimension d our specific objectives are to study the variance in the number of points contained within a regularly shaped window ensuremath omega of arbitrary size and to further illuminate our understanding of hyperuniform systems i e point patterns that do not possess infinite wavelength fluctuations for large windows hyperuniform systems are characterized by a local variance that grows only as the surface area rather than the volume of the window we derive two formulations for the number variance i an ensemble average formulation which is valid for statistically homogeneous systems and ii a volume average formulation applicable to a single realization of a general point pattern in the large system limit the ensemble average formulation which includes both real space and fourier representations enables us to show that a homogeneous point pattern in a hyperuniform state is at a critical point of a type with appropriate scaling laws and critical exponents but one in which the direct correlation function rather than the pair correlation function is long ranged we also prove that the non negativity of the local number variance does not add a new realizability condition on the pair correlation the volume average formulation is superior for certain computational purposes including optimization studies in which it is desired to find the particular point pattern with an extremal or targeted value of the variance we prove that the simple periodic linear array yields the global minimum value of the average variance among all infinite one dimensional hyperuniform patterns we also evaluate the variance for common infinite periodic lattices as well as certain nonperiodic point patterns in one two and three dimensions for spherical windows enabling us to rank order the spatial patterns our results suggest that the local variance may serve as a useful order metric for general point patterns contrary to the conjecture that the lattices associated with the densest packing of congruent spheres have the smallest variance regardless of the space dimension we show that for d 3 the body centered cubic lattice has a smaller variance than the face centered cubic lattice finally for certain hyperuniform disordered point patterns we evaluate the direct correlation function structure factor and associated critical exponents exactly

consensus of high order multi agent systems with switching topologies abstract in this paper a consensus problem is investigated for high order multi agent systems with switching communication networks through which only output information instead of full state information can be transmitted to neighbors based on self state feedback and neighbor output feedback a new consensus protocol is proposed which can realize arbitrary convergence rate furthermore as an application a nonlinear heading consensus protocol is designed for a multi vehicle model finally numerical simulations are given to illustrate the theoretical results

exactly solvable disordered sphere packing model in arbitrary dimensional euclidean spaces we introduce a generalization of the well known random sequential addition rsa process for hard spheres in d dimensional euclidean space mathbb r d we show that all of the n particle correlation functions of this nonequilibrium model in a certain limit called the ghost rsa packing can be obtained analytically for all allowable densities and in any dimension this represents the first exactly solvable disordered sphere packing model in an arbitrary dimension the fact that the maximal density ensuremath phi ensuremath infty 1 2 d of the ghost rsa packing implies that there may be disordered sphere packings in sufficiently high d whose density exceeds minkowski s lower bound for bravais lattices the dominant asymptotic term of which is 1 2 d indeed we report on a conjectural lower bound on the density whose asymptotic behavior is controlled by 2 ensuremath 0 778 65 dots d thus providing the putative exponential improvement on minkowski s 100 year old bound our results suggest that the densest packings in sufficiently high dimensions may be disordered rather than periodic implying the existence of disordered classical ground states for some continuous potentials

precise algorithms to compute surface correlation functions of two phase heterogeneous media and their applications the quantitative characterization of the microstructure of random heterogeneous media in d dimensional euclidean space r d via a variety of n point correlation functions is of great importance since the respective infinite set determines the effective physical properties of the media in particular surface surface f ss and surface void f sv correlation functions obtainable from radiation scattering experiments contain crucial interfacial information that enables one to estimate transport properties of the media e g the mean survival time and fluid permeability and complements the information content of the conventional two point correlation function however the current technical difficulty involved in sampling surface correlation functions has been a stumbling block in their widespread use we first present a concise derivation of the small r behaviors of these functions which are linked to the mean curvature of the system then we demonstrate that one can reduce the computational complexity of the problem without sacrificing accuracy by extracting the necessary interfacial information from a cut of the d dimensional statistically homogeneous and isotropic system with an infinitely long line accordingly we devise algorithms based on this idea and test them for two phase media in continuous and discrete spaces specifically for the exact benchmark model of overlapping spheres we find excellent agreement between numerical and exact results we compute surface correlation functions and corresponding local surface area variances for a variety of other model microstructures including hard spheres in equilibrium decorated stealthy patterns as well as snapshots of evolving pattern formation processes e g spinodal decomposition it is demonstrated that the precise determination of surface correlation functions provides a powerful means to characterize a wide class of complex multiphase microstructures

convergence rates of distributed average consensus with stochastic link failures we consider a distributed average consensus algorithm over a network in which communication links fail with independent probability in such stochastic networks convergence is defined in terms of the variance of deviation from average we first show how the problem can be recast as a linear system with multiplicative random inputs which model link failures we then use our formulation to derive recursion equations for the second order statistics of the deviation from average in networks with and without additive noise we give expressions for the convergence behavior in the asymptotic limits of small failure probability and large networks we also present simulation free methods for computing the second order statistics in each network model and use these methods to study the behavior of various network examples as a function of link failure probability

kinetics and fractal properties of the random sequential adsorption of line segments the random sequential adsorption rsa of infinitely thin line segments is studied by computer simulation it is shown that the exponent for the time dependence of the surface coverage increases beyond the value 0 33 reported by sherwood 1990 to a value of 0 38 at later times it is also argued that the fractal dimension of the adsorbed lines is about 1 8 making use of a previous result on the rsa of rectangular objects

hermitian normalized laplacian matrix for directed networks abstract in this paper we extend and generalize the spectral theory of undirected networks towards directed networks by introducing the hermitian normalized laplacian matrix for directed networks in order to start we discuss the courant fischer theorem for the eigenvalues of hermitian normalized laplacian matrix based on the courant fischer theorem we obtain a similar result towards the normalized laplacian matrix of undirected networks for each i 1 2 n any eigenvalue of hermitian normalized laplacian matrix i 0 2 moreover we prove some special conditions if 0 or 2 is an eigenvalue of the hermitian normalized laplacian matrix l x on top of that we investigate the symmetry of the eigenvalues of l x and the edge version for the eigenvalue interlacing result finally we present two expressions for the coefficients of the characteristic polynomial of the hermitian normalized laplacian matrix as an outlook we sketch some novel and intriguing problems to which our apparatus could generally be applied

chebyshev polynomials in distributed consensus applications in this paper we analyze the use of chebyshev polynomials in distributed consensus applications it is well known that the use of polynomials speeds up the convergence to the consensus in a significant way however existing solutions only work for low degree polynomials and require the topology of the network to be fixed and known we propose a distributed algorithm based on the second order difference equation that describes the chebyshev polynomials of first kind the contributions of our algorithm are three i since the evaluation of chebyshev polynomials is stable there is no limitation in the degree of the polynomial ii instead of the knowledge of the whole network topology it only requires a partial knowledge or an approximation to it iii it can be applied to time varying topologies in the paper we characterize the main properties of the algorithm for both fixed and time varying communication topologies theoretical results as well as experiments with synthetic data show the benefits of using our algorithm compared to existing methods

convergence speed of unsteady distributed consensus decay estimate along the settling spanning trees results for estimating the convergence rate of nonstationary distributed consensus algorithms are provided on the basis of qualitative mainly topological as well as basic quantitative information lower bounds on the matrix entries the results appear to be tight in a number of instances and are illustrated through simple as well as more sophisticated examples the main idea is to follow propagation of information along certain spanning trees which arise in the communication graph

distributed nonlinear control of mobile autonomous multi agents abstract this paper studies the distributed nonlinear control of mobile autonomous agents with variable and directed topology a new distributed nonlinear design scheme is presented for multi agent systems modeled by double integrators with the new design the outputs of the controlled agents asymptotically converge to each other as long as a mild connectivity condition is satisfied moreover the velocity derivative of the output of each agent can be restricted to be within any specified neighborhood of the origin which is of practical interest for systems under such physical constraint the new design is still valid if one of the agents is a leader and the control objective is to achieve leader following as an illustration of the generality and effectiveness of the presented methodology the formation control of a group of unicycle mobile robots with nonholonomic constraints is revisited instead of assuming the point robot model the unicycle model is transformed into two double integrators by dynamic feedback linearization and the proposed distributed nonlinear design method is used to overcome the singularity problem caused by the nonholonomic constraint by properly restricting the velocities simulation results are included to illustrate the theoretical results

coordination of multiagents interacting under independent position and velocity topologies we consider the coordination control for multiagent systems in a very general framework where the position and velocity interactions among agents are modeled by independent graphs different algorithms are proposed and analyzed for different settings including the case without leaders and the case with a virtual leader under fixed position and velocity interaction topologies as well as the case with a group velocity reference signal under switching velocity interaction it is finally shown that the proposed algorithms are feasible in achieving the desired coordination behavior provided the interaction topologies satisfy the weakest possible connectivity conditions such conditions relate only to the structure of the interactions among agents while irrelevant to their magnitudes and thus are easy to verify rigorous convergence analysis is preformed based on a combined use of tools from algebraic graph theory matrix analysis as well as the lyapunov stability theory

cavity approach to sphere packing in hamming space in this paper we study the hard sphere packing problem in the hamming space by the cavity method we show that both the replica symmetric and the replica symmetry breaking approximations give maximum rates of packing that are asymptotically the same as the lower bound of gilbert and varshamov consistently with known numerical results the replica symmetric equations also suggest a crystalline solution where for even diameters the spheres are more likely to be found in one of the subspaces even or odd of the hamming space these crystalline packings can be generated by a recursive algorithm which finds maximum packings in an ultra metric space finally we design a message passing algorithm based on the cavity equations to find dense packings of hard spheres known maximum packings are reproduced efficiently i n non trivial ranges of dimensions and number of spheres

generalized sarymsakov matrices within the set of stochastic indecomposable aperiodic sia matrices the class of sarymsakov matrices is the largest known subset that is closed under matrix multiplication and more critically whose compact subsets are all consensus sets this paper shows that a larger subset with these two properties can be obtained by generalizing the standard definition for sarymsakov matrices the generalization is achieved by introducing the notion of the sia index of a stochastic matrix whose value is 1 for sarymsakov matrices and then exploring those stochastic matrices with larger sia indices in addition to constructing the larger set this paper introduces another class of generalized sarymsakov matrices which contains matrices that are not sia and studies their products sufficient conditions are provided for an infinite product of matrices from this class converging to a rank one matrix finally as an application of the results just described and to confirm their usefulness a necessary and sufficient combinatorial condition the avoiding set condition for deciding whether or not a compact set of stochastic matrices is a consensus set is revisited in addition a necessary and sufficient combinatorial condition is established for deciding whether or not a compact set of doubly stochastic matrices is a consensus set

exponential consensus of general linear multi agent systems under directed dynamic topology this paper aims to investigate the consensus control of generic linear multi agent systems mass under directed dynamic topology nonnegative matrix theory in particular the product properties of infinite row stochastic matrices which are widely used for multiple integrator agents is explored to deal with the convergence analysis of generic linear mass it is finally shown that the exponential consensus can be reached under very relaxed conditions i e the directed interaction topology is only required to be repeatedly jointly rooted and the exponentially unstable mode of each individual system is weak enough moreover a least convergence rate and a bound for the unstable mode of the individual agent system both of which are independent of the switching mode can be explicitly specified

consensus in directed networks of agents with nonlinear dynamics this technical note studies the consensus problem for cooperative agents with nonlinear dynamics in a directed network both local and global consensus are defined and investigated techniques for studying the synchronization in such complex networks are exploited to establish various sufficient conditions for reaching consensus the local consensus problem is first studied via a combination of the tools of complex analysis local consensus manifold approach and lyapunov methods a generalized algebraic connectivity is then proposed to study the global consensus problem in strongly connected networks and also in a broad class of networks containing spanning trees for which ideas from algebraic graph theory matrix theory and lyapunov methods are utilized

the multi agent rendezvous problem part 2 the asynchronous case this paper is concerned with the collective behavior of a group of n 1 mobile autonomous agents labelled 1 through n which can all move in the plane each agent is able to continuously track the positions of all other agents currently within its sensing region where by an agent s sensing region we mean a closed disk of positive radius r centered at the agent s current position the multi agent rendezvous problem is to devise local control strategies one for each agent which without any active communication between agents cause all members of the group to eventually rendezvous at a single unspecified location this paper describes a family of unsynchronized strategies for solving the problem correctness is established appealing to the concept of analytic synchronization

exponential synchronization of complex networks of linear systems and nonlinear oscillators a unified analysis a unified approach to the analysis of synchronization for complex dynamical networks i e networks of partial state coupled linear systems and networks of full state coupled nonlinear oscillators is introduced it is shown that the developed analysis can be used to describe the difference between the state of each node and the weighted sum of the states of those nodes playing the role of leaders in the networks thus making it feasible to consider the error dynamics for the whole network system different from the other various methods given in the existing literature the analysis employed in this paper is demonstrated successfully in not only providing the consistent convergence analysis with much simpler form but also explicitly specifying the convergence rate

behaviors of networks with antagonistic interactions and switching topologies in this paper we study the discrete time consensus problem over networks with antagonistic and cooperative interactions a cooperative interaction between two nodes takes place when one node receives the true state of the other while an antagonistic interaction happens when the former receives the opposite of the true state of the latter we adopt a quite general model where the node communications can be either unidirectional or bidirectional the network topology graph may vary over time and the cooperative or antagonistic relations can be time varying it is proven that the limits of all the node states exist and the absolute values of the node states reach consensus if the switching interaction graph is uniformly jointly strongly connected for unidirectional topologies or infinitely jointly connected for bidirectional topologies these results are independent of the switching of the interaction relations we construct a counterexample to indicate a rather surprising fact that quasi strong connectivity of the interaction graph i e the graph contains a directed spanning tree is not sufficient to guarantee the consensus in absolute values even under fixed topologies based on these results we also propose sufficient conditions for bipartite consensus to be achieved over the network with joint connectivity finally simulation results using a discrete time kuramoto model are given to illustrate the convergence results showing that the proposed framework is applicable to a class of networks with general nonlinear dynamics

the pinwheel tilings of the plane prototiles in euclidean n space en for n 2 the prototiles are usually required to be rather nice topologically at least homeomorphs of the closed unit ball one then makes arbitrarily many congruent copies called tiles of these prototiles and considers all ways called tilings that such tiles may provide a simultaneous covering and packing of en a tiling is thus an unordered collection of tiles for which the union is all of en but such that the interiors of each pair of tiles do not intersect wang s original problem was to determine if it was possible to have a finite set of prototiles with associated tilings of space for which all the tilings were nonperiodic a tiling is nonperiodic if it is not invariant under any single simultaneous translation of its tiles this was settled in the affirmative in the thesis of wang s student robert berger 1

packing hyperspheres in high dimensional euclidean spaces we present a study of disordered jammed hard sphere packings in four five and six dimensional euclidean spaces using a collision driven packing generation algorithm we obtain the first estimates for the packing fractions of the maximally random jammed mrj states for space dimensions d 4 5 and 6t o be mrj0 46 0 31 and 0 20 respectively to a good approximation the mrj density obeys the scaling form mrj c1 2 d c2d 2 d where c1 2 72 and c2 2 56 which appears to be consistent with the highdimensional asymptotic limit albeit with different coefficients calculations of the pair correlation function g2r and structure factor sk for these states show that short range ordering appreciably decreases with increasing dimension consistent with a recently proposed decorrelation principle which among other things states that unconstrained correlations diminish as the dimension increases and vanish entirely in the limit d as in three dimensions where mrj0 64 the packings show no signs of crystallization are isostatic and have a power law divergence in g2r at contact with power law exponent 0 4 across dimensions the cumulative number of neighbors equals the kissing number of the conjectured densest packing close to where g2r has its first minimum additionally we obtain estimates for the freezing and melting packing fractions for the equilibrium hard sphere fluid solid transition f0 32 and m0 39 respectively for d 4 and f0 20 and m0 25 respectively for d 5 although our results indicate the stable phase at high density is a crystalline solid nucleation appears to be strongly suppressed with increasing dimension

estimates of the optimal density of sphere packings in high dimensions the problem of finding the asymptotic behavior of the maximal density max of sphere packings in high euclidean dimensions is one of the most fascinating and challenging problems in discrete geometry one century ago minkowski obtained a rigorous lower bound on max that is controlled asymptotically by 1 2d where d is the euclidean space dimension an indication of the difficulty of the problem can be garnered from the fact that exponential improvement of minkowski s bound has proved to be elusive even though existing upper bounds suggest that such improvement should be possible using a statistical mechanical procedure to optimize the density associated with a test pair correlation function and a conjecture concerning the existence of disordered sphere packings s torquato and f h stillinger exp math 15 307 2006 the putative exponential improvement on max was found with an asymptotic behavior controlled by 1 2 0 77865 d using the same methods we investigate whether this exponential im

robust distributed average consensus via exchange of running sums we consider a multi component system in which each component node can send receive information to from sets of neighboring nodes via communication links edges that form a fixed strongly connected possibly directed communication topology digraph we analyze a class of distributed iterative algorithms that allow the nodes to asymptotically compute the exact average of their initial values despite a variety of challenging scenarios including possible packet drops in the communication links and imprecise knowledge of the network the algorithms in this class run the two linear iterations of the so called ratio consensus algorithm modified so that messages sent by one node to another are encoded as running sums this convolutional encoding allows the receiving node l to infer information about past messages that node j meant to send to node l but may have been lost due to packet drops imprecise knowledge of the network unknown out neighborhoods can be handled at the cost of memory and communication overhead by also having each node track the progress of running sums of other nodes and forward to its out neighboring nodes the updated value of one such running sum that it randomly selects our analysis relies on augmenting the digraph that describes the communication topology by introducing additional virtual nodes and showing that the dynamics of each of the two iterations in the augmented digraph is mathematically equivalent to a finite inhomogeneous markov chain almost sure convergence to exact average consensus is then established via weak ergodicity analysis of the resulting inhomogeneous markov chain

random sequential addition of hard spheres in high euclidean dimensions sphere packings in high dimensions have been the subject of recent theoretical interest employing numerical and theoretical methods we investigate the structural characteristics of random sequential addition rsa of congruent spheres in d dimensional euclidean space mathbb r d in the infinite time or saturation limit for the first six space dimensions 1 ensuremath le d ensuremath le 6 specifically we determine the saturation density pair correlation function cumulative coordination number and the structure factor in each of these dimensions we find that for 2 ensuremath le d ensuremath le 6 the saturation density ensuremath phi s scales with dimension as ensuremath phi s c 1 2 d c 2 d 2 d where c 1 0 202 phantom rule 0 2em 0ex 048 and c 2 0 973 phantom rule 0 2em 0ex 872 we also show analytically that the same density scaling is expected to persist in the high dimensional limit albeit with different coefficients a byproduct of this high dimensional analysis is a relatively sharp lower bound on the saturation density for any d given by ensuremath phi s ensuremath ge d 2 1 ensuremath s 0 2 d 1 where s 0 0 1 is the structure factor at k 0 i e infinite wavelength number variance in the high dimensional limit we demonstrate that a pal asti type conjecture the saturation density in mathbb r d is equal to that of the one dimensional problem raised to the d th power cannot be true for rsa hyperspheres we show that the structure factor s k must be analytic at k 0 and that rsa packings for 1 ensuremath le d ensuremath le 6 are nearly hyperuniform consistent with the recent decorrelation principle we find that pair correlations markedly diminish as the space dimension increases up to six we also obtain kissing contact number statistics for saturated rsa configurations on the surface of a d dimensional sphere for dimensions 2 ensuremath le d ensuremath le 5 and compare to the maximal kissing numbers in these dimensions we determine the structure factor exactly for the related ghost rsa packing in mathbb r d and demonstrate that its distance from hyperuniformity increases as the space dimension increases approaching a constant asymptotic value of 1 2 our work has implications for the possible existence of disordered classical ground states for some continuous potentials in sufficiently high dimensions

on the most compact regular lattices in large dimensions a statistical mechanical approach in this paper i will consider the computation of the maximum density of regular lattices in large dimensions using an approach based on statistical mechanics the starting point will be some theorems of rogers which are virtually unknown in the community of physicists using his approach one can find many similarities and differences with the problem of computing the entropy of a system of hard spheres the relation between the two problems is investigated in detail some conjectures are presented further investigation is needed in order to check their consistency

consensus seeking in multiagent systems under dynamically changing interaction topologies this note considers the problem of information consensus among multiple agents in the presence of limited and unreliable information exchange with dynamically changing interaction topologies both discrete and continuous update schemes are proposed for information consensus this note shows that information consensus under dynamically changing interaction topologies can be achieved asymptotically if the union of the directed interaction graphs have a spanning tree frequently enough as the system evolves

reaching a consensus in a dynamically changing environment a graphical approach this paper presents new graph theoretic results appropriate for the analysis of a variety of consensus problems cast in dynamically changing environments the concepts of rooted strongly rooted and neighbor shared are defined and conditions are derived for compositions of sequences of directed graphs to be of these types the graph of a stochastic matrix is defined and it is shown that under certain conditions the graph of a sarymsakov matrix and a rooted graph are one and the same as an illustration of the use of the concepts developed in this paper graph theoretic conditions are obtained which address the convergence question for the leaderless version of the widely studied vicsek consensus problem

degree fluctuations and the convergence time of consensus algorithms we consider a consensus algorithm in which every node in a sequence of undirected b connected graphs assigns equal weight to each of its neighbors under the assumption that the degree of each node is fixed except for times when the node has no connections to other nodes we show that consensus is achieved within a given accuracy e on n nodes in time b 4n3 bln 2n e because there is a direct relation between consensus algorithms in time varying environments and in homogeneous random walks our result also translates into a general statement on such random walks moreover we give a simple proof of a result of cao spielman and morse that the worst case convergence time becomes exponentially large in the number of nodes n under slight relaxation of the degree constancy assumption

statistical properties of determinantal point processes in high dimensional euclidean spaces the goal of this paper is to quantitatively describe some statistical properties of higher dimensional determinantal point processes with a primary focus on the nearest neighbor distribution functions toward this end we express these functions as determinants of n n matrices and then extrapolate to n this formulation allows for a quick and accurate numerical evaluation of these quantities for point processes in euclidean spaces of dimension d we also implement an algorithm due to hough et al for generating configurations of determinantal point processes in arbitrary euclidean spaces and we utilize this algorithm in conjunction with the aforementioned numerical results to characterize the statistical properties of what we call the fermi sphere point process for d 1 4 this homogeneous isotropic determinantal point process discussed also in a companion paper s torquato a scardicchio and c e zachary j stat mech theory exp 2008 p11019 i s the high dimensional generalization of the distribution of eigenvalues on the unit circle of a random matrix from the circular unitary ensemble in addition to the nearest neighbor probability distribution we are able to calculate voronoi cells and nearest neighbor extrema statistics for the fermi sphere point process and we discuss these properties as the dimension d is varied the results in this paper accompany and complement analytical properties of higher dimensional determinantal point processes developed in a prior paper

new upper bounds on sphere packings i we continue the study of the linear programming bounds for sphere packing introduced by cohn and elkies we use theta series to give another proof of the principal theorem and present some related results and conjectures this article is in the arxiv as arxiv math mg 0110010

analysis and applications of spectral properties of grounded laplacian matrices for directed networks abstract in depth understanding of the spectral properties of grounded laplacian matrices is critical for the analysis of convergence speeds of dynamical processes over complex networks such as opinion dynamics in social networks with stubborn agents we focus on grounded laplacian matrices for directed graphs and show that their eigenvalues with the smallest real part must be real lower and upper bounds for such eigenvalues are provided utilizing tools from nonnegative matrix theory for those eigenvectors corresponding to such eigenvalues we discuss two cases when we can identify the vertex that corresponds to the smallest eigenvector component we then discuss an application in leader follower social networks where the grounded laplacian matrices arise naturally with the knowledge of the vertex corresponding to the smallest eigenvector component for the smallest eigenvalue we prove that by removing or weakening specific directed couplings pointing to the vertex having the smallest eigenvector component all the states of the other vertices converge faster to that of the leading vertex this result is in sharp contrast to the well known fact that when the vertices are connected together through undirected links removing or weakening links does not accelerate and in general decelerates the converging process

convergence rate for discrete time multiagent systems with time varying delays and general coupling coefficients multiagent systems mass are ubiquitous in our real world there is an increasing attention focusing on the consensus or synchronization problem of mass over the past decade although there are numerous results reported on the convergence of a discrete time mas based on the infinite products of matrices few results are on the convergence rate because of the switching topology the traditional eigenvalue analysis and the lyapunov function methods are both invalid for the convergence rate analysis of an mas with a switching topology therefore the estimation of the convergence rate for a discrete time mas with time varying delays remains a difficult problem to overcome the essential difficulty of switching topology this paper aims at developing a contractive set approach to analyze the convergence rate of a discrete time mas in the presence of time varying delays and generalized coupling coefficients using the proposed approach we obtain an upper bound of the convergence rate under the condition of joint connectivity in particular the proposed method neither requires the nonnegative property of the coupling coefficients nor the basic assumption of a uniform lower bound for all positive coupling coefficients which have been widely applied in the existing works on this topic as an application of the main results we will show that the classical vicsek model with time delays can realize synchronization if the initial topology is connected

coordinated path following in the presence of communication losses and time delays this paper addresses the problem of steering a group of vehicles along given spatial paths while holding a desired time varying geometrical formation pattern the solution to this problem henceforth referred to as the coordinated path following cpf problem unfolds in two basic steps first a path following pf control law is designed to drive each vehicle to its assigned path with a nominal speed profile that may be path dependent this is done by making each vehicle approach a virtual target that moves along the path according to a conveniently defined dynamic law in the second step the speeds of the virtual targets also called coordination states are adjusted about their nominal values so as to synchronize their positions and achieve indirectly vehicle coordination in the problem formulation it is explicitly considered that each vehicle transmits its coordination state to a subset of the other vehicles only as determined by the communications topology adopted it is shown that the system that is obtained by putting together the pf and coordination subsystems can be naturally viewed as either the feedback or the cascade connection of the latter two using this fact and recent results from nonlinear systems and graph theory conditions are derived under which the pf and the coordination errors are driven to a neighborhood of zero in the presence of communication losses and time delays two different situations are considered the first captures the case where the communication graph is alternately connected and disconnected brief connectivity losses the second reflects an operational scenario where the union of the communication graphs over uniform intervals of time remains connected uniformly connected in mean to better root the paper in a nontrivial design example a cpf algorithm is derived for multiple underactuated autonomous underwater vehicles auvs simulation results are presented and discussed

agreeing asynchronously this paper formulates and solves a version of the widely studied vicsek consensus problem in which each member of a group of n 1 agents independently updates its heading at times determined by its own clock it is not assumed that the agents clocks are synchronized or that the ldquoeventrdquo times between which any one agent updates its heading are evenly spaced nor is it assumed that heading updates must occur instantaneously using the concept of ldquoanalytic synchronizationrdquo together with several key results concerned with properties of ldquocompositionsrdquo of directed graphs it is shown that the conditions under which a consensus is achieved are essentially the same as those applicable in the synchronous case provided the notion of an agent s neighbor between its event times is appropriately defined however in sharp contrast with the synchronous case where for analysis an n dimensional state space model is adequate for the asynchronous version of the problem a 2n dimensional state space model is required it is explained how to analyze this model despite the fact that unlike the synchronous case the stochastic matrices involved do not have all positive diagonal entries

second order consensus for multi agent systems with switching topology and communication delay in this paper two kinds of consensus problems for second order agents under directed and arbitrarily switching topologies are investigated that is the cases without and with communication delay for the former by constructing a new kind of digraph and employing a new graphic method we can specify the least convergence rate for all the agents to reach consensus for the latter in virtue of a matrix inequality method a sufficient condition in the form of feasible matrix inequalities is presented for all the agents to reach consensus this on the other hand shows that consensus can be reached if the delay is small enough finally two numerical examples are given to demonstrate the effectiveness and advantages of the proposed results

on attack resilient distributed formation control in operator vehicle networks this paper tackles a distributed formation control problem where a group of vehicles is remotely controlled by a network of operators each operator vehicle pair is attacked by an adversary who corrupts the commands sent from the operator to the vehicle from the point of view of operators each adversary follows an attacking strategy linearly parameterized by some potentially time varying matrix which is unknown a priori in particular we consider two scenarios depending upon whether adversaries can adapt their attacking tactics online to assure mission completion in such a hostile environment we propose two novel attack resilient distributed control algorithms that allow operators to adjust their policies on the fly by exploiting the latest collected information about adversaries both algorithms enable vehicles to asymptotically achieve the desired formation from any initial configuration and initial estimate of the adversaries strategies it is further shown that the sequence of the distances t

geometrical ambiguity of pair statistics point configurations point configurations have been widely used as model systems in condensed matter physics materials science and biology statistical descriptors such as the n body distribution function g n are usually employed to characterize point configurations among which the most extensively used is the pair distribution function g 2 an intriguing inverse problem of practical importance that has been receiving considerable attention is the degree to which a point configuration can be reconstructed from the pair distribution function of a target configuration although it is known that the pair distance information contained in g 2 is in general insufficient to uniquely determine a point configuration this concept does not seem to be widely appreciated and general claims of uniqueness of the reconstructions using pair information have been made based on numerical studies in this paper we present the idea of the distance space called the mathbb d space the pair distances of a specific point configuration are then represented by a single point in the mathbb d space we derive the conditions on the pair distances that can be associated with a point configuration which are equivalent to the realizability conditions of the pair distribution function g 2 moreover we derive the conditions on the pair distances that can be assembled into distinct configurations i e with structural degeneracy these conditions define a bounded region in the mathbb d space by explicitly constructing a variety of degenerate point configurations using the mathbb d space we show that pair information is indeed insufficient to uniquely determine the configuration in general we also discuss several important problems in statistical physics based on the mathbb d space including the reconstruction of atomic structures from experimentally obtained g 2 and a recently proposed decorrelation principle the degenerate configurations have relevance to open questions involving the famous traveling salesman problem

on discrete time convergence for general linear multi agent systems under dynamic topology this note aims to develop the nonnegative matrix theory in particular the product properties of infinite row stochastic matrices which is widely used for multiple integrator agents to deal with the convergence analysis of general discrete time linear multi agent systems mass with the proposed approach it is finally shown both theoretically and by simulation that the consensus for all the agents can be reached exponentially fast under relaxed conditions i e the individual uncoupled system is allowed to be strictly unstable in the discrete time sense and it is only required that the joint of the communication topologies has a spanning tree frequently enough moreover a least convergence rate as well as an upper bound for the strictly unstable mode which are independent of the switching mode of the system are specified as well

toy model for the mean field theory of hard sphere liquids we investigate a toy model of liquid based on simplified hypernetted chain equations in very large spatial dimension d the model does not exhibit a phase transition but several regimes of behavior when d infinity can be observed in different intervals of the density

stability of multiagent systems with time dependent communication links we study a simple but compelling model of network of agents interacting via time dependent communication links the model finds application in a variety of fields including synchronization swarming and distributed decision making in the model each agent updates his current state based upon the current information received from neighboring agents necessary and or sufficient conditions for the convergence of the individual agents states to a common value are presented thereby extending recent results reported in the literature the stability analysis is based upon a blend of graph theoretic and system theoretic tools with the notion of convexity playing a central role the analysis is integrated within a formal framework of set valued lyapunov theory which may be of independent interest among others it is observed that more communication does not necessarily lead to faster convergence and may eventually even lead to a loss of convergence even for the simple models discussed in the present paper

convergence and state reconstruction of time varying multi agent systems from complete observability theory we study continuous time consensus dynamics for multi agent systems with undirected switching interaction graphs we establish a necessary and sufficient condition for exponential asymptotic consensus based on the classical theory of complete observability the proof is remarkably simple compared to similar results in the literature and the conditions for consensus are mild this observability based method can also be applied to the case where negatively weighted edges are present additionally as a by product of the observability based arguments we show that the nodes initial value can be recovered from the signals on the edges up to a shift of the network average