exact large deviation bounds up to t c for the ising model in two dimensions we prove an upper large deviation bound for the block spin magnetization in the 2d ising model in the phase coexistence region the precise rate given by the wulff construction is shown to hold true for all c combined with the lower bounds derived in i those results yield an exact second order large deviation theory up to the critical temperature

robust consensus tracking control for multiagent systems with initial state shifts disturbances and switching topologies this paper deals with the consensus tracking control issues of multiagent systems and aims to solve them as accurately as possible over a finite time interval through an iterative learning approach based on the iterative rule distributed algorithms are proposed for every agent using its nearest neighbor knowledge for which the robustness problem is addressed against initial state shifts disturbances and switching topologies these uncertainties are dynamically changing not only along the time axis but also the iteration axis it is shown that the matrix norm conditions can be developed to achieve the convergence of the considered consensus tracking objectives for which necessary and sufficient conditions are presented in terms of linear matrix inequalities to guarantee their feasibility in the sense of the spectral norm furthermore simulation examples are given to illustrate the effectiveness and robustness of the obtained consensus tracking results

distributed tracking control for linear multiagent systems with a leader of bounded unknown input this technical note considers the distributed tracking control problem of multiagent systems with general linear dynamics and a leader whose control input is nonzero and not available to any follower based on the relative states of neighboring agents two distributed discontinuous controllers with respectively static and adaptive coupling gains are designed for each follower to ensure that the states of the followers converge to the state of the leader if the interaction graph among the followers is undirected the leader has directed paths to all followers and the leader s control input is bounded a sufficient condition for the existence of the distributed controllers is that each agent is stabilizable simulation examples are given to illustrate the theoretical results

the survival of one dimensional contact processes in random environments consider the inhomogeneous contact process on z1 with recovery rate 8 k at site k and infection rates a k and p k at site k due to the presence of infected neighbors at k 1 and k 1 respectively a special case of the main result in this paper is the following suppose that the environment is chosen in such a way that the 8 k s a k s and p k s are all mutually independent with the 8 k s having a common distribution and the a k s and p k s having a common distribution then the process survives if

large deviations for the 2d ising model a lower bound without cluster expansions we show that a lower large deviation bound for the block spin magnetization in the 2d ising model can be pushed all the way forward toward its correct wulff value for all c

brief paper leader following consensus of second order agents with multiple time varying delays in this paper a leader following consensus problem of second order multi agent systems with fixed and switching topologies as well as non uniform time varying delays is considered for the case of fixed topology a necessary and sufficient condition is obtained for the case of switching topology a sufficient condition is obtained under the assumption that the total period over which the leader is globally reachable is sufficiently large we not only prove that a consensus is reachable asymptotically but also give an estimation of the convergence rate an example with simulation is presented to illustrate the theoretical results

approach to equilibrium of glauber dynamics in the one phase region i the attractive case various finite volume mixing conditions in classical statistical mechanics are reviewed and critically analyzed in particular somefinite size conditions are discussed together with their implications for the gibbs measures and for the approach to equilibrium of glauber dynamics inarbitrarily large volumes it is shown that dobrushin shlosman s theory ofcomplete analyticity and its dynamical counterpart due to stroock and zegarlinski cannot be applied in general to the whole one phase region since it requires mixing properties for regions ofarbitrary shape an alternative approach based on previous ideas of oliveri and picco is developed which allows to establish results on rapid approach to equilibrium deeply inside the one phase region in particular in the ferromagnetic case we considerably improve some previous results by holley and aizenman and holley our results are optimal in the sene that for example they show for the first time fast convergence of the dynamicsfor any temperature above the critical one for thed dimensional ising model with or without an external field in part ii we extensively consider the general case not necessarily attractive and we develop a new method based on renormalizations group ideas and on an assumption of strong mixing in a finite cube to prove hypercontractivity of the markov semigroup of the glauber dynamics

eigenfunction expansions of ultradifferentiable functions and ultradistributions in this paper we give a global characterisation of classes of ultradifferentiable functions and corresponding ultradistributions on a compact manifold x the characterisation is given in terms of the eigenfunction expansion of an elliptic operator on x this extends the result for analytic functions on compact manifolds by seeley in 1969 and the characterisation of gevrey functions and gevrey ultradistributions on compact lie groups and homogeneous spaces by the authors 2014

the complete convergence theorem holds for contact processes on open clusters of ℤd ℤ we study contact processes on open clusters of half space our result shows that the complete convergence theorem holds

decay of two point functions for d 1 dimensional percolation ising and potts models with d dimensional disorder let left j left langle x y right rangle right left langle x y right rangle subset z d and left k x right x in z d be independent sets of nonnegative i i d r v s denoting a pair of nearest neighbors inzd let 0 we consider the random systems 1 a bond bernoulli percolation model onzd 1 with random occupation probabilities open image in new window 2 ferromagnetic random ising potts models onzd 1 in the ising case the hamiltonian is h beta sum limits t sum limits left langle x y right rangle j left langle x y right rangle sigma x t sigma y t gamma sum limits x sum limits t k x sigma x t sigma x t 1 for such d 1 dimensional systems withd dimensional disorder we prove i for anyd 1 if and are small then with probability one the two point functions decay exponentially in thed dimensional distance and faster than polynomially in the remaining dimension ii ifd 2 then with probability one we have long range order for either and with sufficiently large of sufficiently large and any

non perturbative criteria for gibbsian uniqueness for spin systems with an internal symmetry we provide sufficient conditions for unicity of the gibbs state and or complete analyticity by comparison to random cluster models

periodic distributions and periodic elements in modulation spaces abstract we characterize periodic elements in gevrey classes gelfand shilov distribution spaces and modulation spaces in terms of estimates of involved fourier coefficients and by estimates of their short time fourier transforms if q 1 is a suitable weight and e 0 e is the set of all e periodic elements then we prove that the dual of m q e 0 e equals m 1 q e 0 e by suitable extensions of bessel s identity

distributed robust control of linear multi agent systems with parameter uncertainties this article considers the distributed robust control problems of uncertain linear multi agent systems with undirected communication topologies it is assumed that the agents have identical nominal dynamics while subject to different norm bounded parameter uncertainties leading to weakly heterogeneous multi agent systems distributed controllers are designed for both continuous and discrete time multi agent systems based on the relative states of neighbouring agents and a subset of absolute states of the agents it is shown for both the continuous and discrete time cases that the distributed robust control problems under such controllers in the sense of quadratic stability are equivalent to the h control problems of a set of decoupled linear systems having the same dimensions as a single agent a two step algorithm is presented to construct the distributed controller for the continuous time case which does not involve any conservatism and meanwhile decouples the feedback gain design from the commun

a new proof of localization in the anderson tight binding model we give a new proof of exponential localization in the anderson tight binding model which uses many ideas of the frohlich martinelli scoppola and spencer proof but is technically simpler particularly the probabilistic estimates

chaotic time dependence in a disordered spin system stochastic ising and voter models on d are natural examples of markov processes with compact state spaces when the initial state is chosen uniformly at random can it happen that the distribution at time t has multiple subsequence limits as t yes for the d 1 voter model with random rates vmrr which is the same as a d 1 rate disordered stochastic ising model at zero temperature if the disorder distribution is heavy tailed no at least in a weak sense for the vmrr when the tail is light or d 2 these results are based on an analysis of the localization properties of random walks with random rates

relaxation to equilibrium for two dimensional disordered ising systems in the griffiths phase we consider glauber type dynamics for two dimensional disordered magnets of ising type we prove that if the disorder averaged influence of the boundary condition is sufficiently small in the equilibrium system then the corresponding glauber dynamics is ergodic with probability one and the disorder average c t of time autocorrelation function satisfies for large t for the standard two dimensional dilute ising ferromagnet with i i d random nearest neighbor couplings taking the values 0 or j 0 0 our results apply even if the active bonds percolate and j 0 is larger than the critical value j c of the corresponding pure ising model for the same model we also prove that in the whole griffiths phase the previous upper bound is optimal this implies the existence of a dynamical phase transition which occurs when j crosses j c

random walks in a random field of decaying traps we study random walks on d d 1 containing traps subject to decay the initial trap distribution is random in the course of time traps decay independently according to a given lifetime distribution we derive a necessary and sufficient condition under which the walk eventually gets trapped with probability 1 we prove bounds and asymptotic estimates for the survival probability as a function of time and for the average trapping time these are compared with some well known results for nondecaying traps

random walks on lattices with randomly distributed traps i the average number of steps until trapping for a random walk on a lattice with a random distribution of traps we derive an asymptotic expansion valid for smallq for the average number of steps until trapping whereq is the probability that a lattice point is a trap we study the case of perfect traps where the walk comes to an end and the extension obtained by letting the traps be imperfect i e by giving the walker a finite probability to remain free when stepping on a trap several classes of random walks of varying dimensionality are considered and special care is taken to show that the expansion derived is exact up to and including the last term calculated the numerical accuracy of the expansion is discussed

an exponential upper bound for the survival probability in a dynamic random trap model we consider a symmetric translation invariant random walk on thed dimensional lattice d the walker moves in an environment of moving traps when the walker hits a trap he is killed the configuration of traps in the course of time is a reversible markov process satisfying a level 2 large deviation principle under some restrictions on the entropy function we prove an exponential upper bound for the survival probability i e mathop lim sup limits t to infty frac 1 t log mathbb p t geqslant t 0 wheret is the survival time of the walker as an example our results apply to a random walk in an environment of traps that perform a simple symmetric exclusion process

optimal design for synchronization of cooperative systems state feedback observer and output feedback this technical note studies synchronization of identical general linear systems on a digraph containing a spanning tree a leader node or command generator is considered which generates the desired tracking trajectory a framework for cooperative tracking control is proposed including full state feedback control observer design and dynamic output feedback control the classical system theory notion of duality is extended to networked systems it is shown that unbounded synchronization regions that achieve synchronization on arbitrary digraphs containing a spanning tree can be guaranteed by using linear quadratic regulator based optimal control and observer design methods at each node

localization in the ground state of the ising model with a random transverse field we study the zero temperature behavior of the ising model in the presence of a random transverse field the hamiltonian is given by h j sum limits left langle x y right rangle sigma 3 x sigma 3 y sum limits x h x sigma 1 x wherej 0 x y zd 1 3 are the usual pauli spin 1 2 matrices andh h x x zd are independent identically distributed random variables we consider the ground state correlation function 3 x 3 y and prove 1 letd be arbitrary for anym 0 andj sufficiently small we have for almost every choice of the random transverse fieldh and everyx zd that left langle sigma 3 x sigma 3 y right rangle leqq c x h e m left x y right for ally zd withcxh 2 letd 2 ifj is sufficiently large then for almost every choice of the random transverse fieldh the model exhibits long range order i e mathop overline lim limits left y right to infty left langle sigma 3 x sigma 3 y right rangle 0 for anyx zd

percolation techniques in disordered spin flip dynamics relaxation to the unique invariant measure we consider lattice spin systems with short range but random and unbounded interactions we give criteria for ergodicity of spin flip dynamics and estimate the speed of convergence to the unique invariant measure we find for this convergence a stretched exponential in time for a class of directed dynamics such as in the disordered toom or stavskaya model for the general case we show that the relaxation is faster than any power in time no assumptions of reversibility are made the methods are based on relating the problem to an oriented percolation problem contact process and for the general case using a slightly modified version of the multiscale analysis of e g klein 1993

dobrushin kotecký shlosman theorem up to the critical temperature we develop a non perturbative version of the dobrushin koteck shlosman theory of phase separation in the canonical 2d ising ensemble the results are valid for all temperatures below critical

second order large deviation estimates for ferromagnetic systems in the phase coexistence region we consider thed dimensional ising model with ferromagnetic nearest neighbor interaction at inverse temperature let m lambda lambda 1 sum limits i in lambda sigma i be the magnetization inside ad dimensional hyper cube be the gibbs state andm be the spontaneous magnetization for such thatm 0 we find a sufficient condition easily verified to hold for large for m a b to decay exponentially with d 1 d when m b m 1 a b ford 2 this sufficient condition is the exponential decay of a connectivity function we also prove a partial converse to this result obtain a sharper result for the magnetization ond 1 dimensional cross sections of the model and prove a similar result ford 2 m a b m and large when free boundary conditions are chosen outside

wave equation for sums of squares on compact lie groups in this paper we investigate the well posedness of the cauchy problem for the wave equation for sums of squares of vector fields on compact lie groups we obtain the loss of regularity for solutions to the cauchy problem in local sobolev spaces depending on the order to which the hormander condition is satisfied but no loss in globally defined spaces we also establish gevrey well posedness for equations with irregular coefficients and or multiple characteristics as in the sobolev spaces if formulated in local coordinates we observe well posedness with the loss of local gevrey order depending on the order to which the hormander condition is satisfied

exponential decay for subcritical contact and percolation processes we study the contact process together with a version of the percolation process with one continuously varying coordinate it is proved here that the radius of the infected cluster has an exponentially decaying tail throughout the subcritical phase the same is true of the lebesgue measure in space time of this cluster certain critical exponent inequalities are derived and the critical point of the percolation process in two dimensions is determined exactly

leader following consensus of multi agent systems under fixed and switching topologies the leader following consensus problem of higher order multi agent systems is considered in the system the dynamics of each agent and the leader is a linear system the control of each agent using local information is designed and detailed analysis of the leader following consensus is presented for both fixed and switching interaction topologies which describe the information exchange between the multi agent systems the design technique is based on algebraic graph theory riccati inequality and lyapunov inequality simulations indicate the capabilities of the algorithms

the supercritical phase of percolation is well behaved we prove a general result concerning the critical probabilities of subsets of a lattice l it is a consequence of this result that the critical probability of a percolation process on l equals the limit of the critical probability of a slice of l as the thickness of the slice tends to infinity this verification of one of the standard hypotheses of the subject settles many questions concerning supercritical percolation

relaxation of disordered magnets in the griffiths regime we study the relaxation to equilibrium of discrete spin systems with random finite range not necessarily ferromagnetic interactions in the griffiths regime we prove that the speed of convergence to the unique reversible gibbs measure is almost surely faster than any stretched exponential at least if the probability distribution of the interaction decays faster than exponential e g gaussian furthermore if the interac tion is uniformly bounded the average over the disorder of the time autocorrelation function goes to equilibrium as exp k log t d d 1 in d 1 in agreement with previous results obtained for the dilute ising model

optimal flow through the disordered lattice author s aldous d abstract consider routing traffic on the n n torus simultaneously between all source destination pairs to minimize the cost ec e f 2 e where f e is the volume of flow across edge e and the c e form an i i d random environment we prove existence of a rescaled n limit constant for minimum cost by comparison with an appropriate analogous problem about minimum cost flows across a m m subsquare of the lattice institute of mathematical statistics 2007

phase transition in dependent percolation in this paper we discuss two different models of dependent percolation on the graph 2 these models can be thought of as percolation in a random environment they were inspired by the work of mccoy and wu 7 8 on the ising model in a random environment as well as other models of particle systems in a random environment 9 5 6 3 we show that both models of dependent percolation exhibit phase transitions this proves a version of stability for percolation on 2 and proves a conjecture of jonasson mossel and peres 4 who proved a similar result on 3

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

surface order large deviations for ising potts and percolation models we derive uniform surface order large deviation estimates for the block magnetization in finite volume ising or potts models with plus or free or a combination of both boundary conditions in the phase coexistence regime ford 3 the results are valid up to a limit of slab thresholds conjectured to agree with the critical temperature our arguments are based on the renormalization of the random cluster model withq 1 andd 3 and on corresponding large deviation estimates for the occurrence in a box of a largest cluster with density close to the percolation probability the results are new even for the case of independent percolation q 1 as a byproduct of our methods we obtain further results in the fk model concerning semicontinuity inp andq of the percolation probability the second largest cluster in a box and the tail of the finite cluster size distribution

brief paper distributed discrete time coordinated tracking with a time varying reference state and limited communication this paper studies a distributed discrete time coordinated tracking problem where a team of vehicles communicating with their local neighbors at discrete time instants tracks a time varying reference state available to only a subset of the team members we propose a pd like discrete time consensus algorithm to address the problem under a fixed communication graph we then study the condition on the communication graph the sampling period and the control gain to ensure stability and give the quantitative bound of the tracking errors it is shown that the ultimate bound of the tracking errors is proportional to the sampling period the benefit of the proposed pd like discrete time consensus algorithm is also demonstrated through comparison with an existing p like discrete time consensus algorithm simulation results are presented as a proof of concept

stretched exponential relaxation in the biased random voter model we study the relaxation properties of the voter model with i i d random bias we prove under mild conditions that the disorder averaged relaxation of this biased random voter model is faster than a stretched exponential with exponent d d where 0 2 depends on the transition rates of the non biased voter model under an additional assumption we show that the above upper bound is optimal the main ingredient of our proof is a result of donsker and varadhan

upper and lower bounds on dynamic correlations in the griffiths phase dilute ferromagnetic spin systems are considered in the griffiths phase below the percolation threshold it is shown that for relaxational dynamics the spin autocorrelation functions are bounded both above and below by functions of the form exp a 1n t d d 1 ising systems and exp bt12 heisenberg systems establishing these as the correct asymptotic forms

distributed finite time containment control for second order nonlinear multi agent systems this paper considers the finite time containment control problem for a class of second order nonlinear multi agent systems with external disturbances under the directed topology graph the considered nonlinear systems are composed of multiple dynamic leaders and they have only a subset of the followers with their accessing to the leaders firstly by using the one hop neighbors information a second order finite time distributed observer estimator protocol is constructed to achieve accurate estimation of the weighted average of the velocities and positions of the leaders furthermore it is proved that the proposed estimator protocol can guarantee accurate estimation of the weighted average of the velocities and positions of the leaders in the finite time then we design a new continuous nonlinear control protocol by means of the nonsingular terminal sliding mode it is shown that by using proposed control protocol the states of the followers can converge to the convex hull spanned by those of the leaders in the finite time in addition the finite convergence time can be also explicitly estimated finally the effectiveness of the obtained theoretical results is illustrated by the numerical simulation

the phase boundary in dilute and random ising and potts ferromagnets the authors present an approach to dilute ising and potts models based on the fortuin kasteleyn random cluster representation which is simultaneously rigorous intuitive and surprisingly simple their analysis yields with no dimensional restrictions or other caveats the following asymptotic form of the phase boundary for the regular dilute model in which bonds have constant ferromagnetic coupling j with probability p and are vacant with probability 1 p the critical temperature scales as exp j ktc p approximately mod p pc mod implying that the crossover exponent is phi 1 if the constant couplings are replaced by a distribution f j with mass near j 0 quite different crossover behaviour is observed for example if f j approximately jalpha then for p near pc tc p approximately mod p pc mod 1 alpha

coexistence of infinite clusters ii ising percolation in two dimensions we show a strong type of conditionally mixing property for the gibbs states ofd dimensional ising model when the temperature is above the critical one by using this property we show that there is always coexistence of infinite and clusters when is smaller than c andh 0 in two dimensions it is also possible to show that this coexistence region extends to some non zero external field case i e for every c there exists someh c 0 such that h h c implies the coexistence of infinite clusters with respect to the gibbs state for h

dynamic structure factor in a random diffusion model let xt 0 denote random walk in the random waiting time model i e simple random walk with jump ratew 1 xt where w x x d is an i i d random field we show that under some mild conditions theintermediate scattering functionf q t e0 e iqx l q d is completely monotonic int e0 denotes double expectation w r t walk and field we also show that thedynamic structure factors q w 2 0 cos t f q t exists for 0 and is strictly positive ind 1 2 it diverges as 1 1 2 resp ln in the limit 0 ind 3 its limit value is strictly larger than expected from hydrodynamics this and further results support the conclusion that the hydrodynamic region is limited to smallq and small such that d q 2 whered is the diffusion constant

strong decay to equilibrium in one dimensional random spin systems we consider a spin system on a lattice with finite range possibly unbounded random interactions we show that for such systems the glauber dynamics cannot decay to equilibrium exponentially fast inl2 even at high temperatures additionally for one dimensional systems with unbounded random couplings we prove that with probability one the corresponding glauber dynamics has a fast subexponential decay to equilibrium in the uniform norm provided that the distribution of random couplings satisfies some exponential bound

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

approach to equilibrium of glauber dynamics in the one phase region ii the general case we develop a new method based on renormalization group ideas block decimation procedure to prove under an assumption of strong mixing in a finite cube 0 a logarithmic sobolev inequality for the gibbs state of a discrete spin system as a consequence we derive the hypercontractivity of the markov semigroup of the associated glauber dynamics and the exponential convergence to equilibrium in the uniform norm in all volumes multiples of the cube 0

extinction of contact and percolation processes in a random environment we consider the inhomogeneous percolation process on z d r defined as follows along each vertical line x r we put cuts at times given by a poisson point process with intensity x and between each pair of adjacent vertical lines x r and y r we place bridges at times given by a poisson point process with intensity x y we say that x t and y s are connected or in the same cluster if there is a path from x t to y s made out of uncut segments of vertical lines and bridges if we consider only oriented percolation we have the graphical representation of the inhomogeneous d dimensional contact process we consider these percolation and contact processes in a random environment by taking x x z d and x y x y z d x y 2 1 to be independent families of independent identically distributed strictly positive random variables we use and for representative random variables we prove extinction i e no percolation of these percolation and contact processes for almost every and if and satisfy e log 1 2d 2 1 1 1 d 1 2d and if e log 1 is sufficiently small

on the two dimensional stochastic ising model in the phase coexistence region near the critical point we consider the two dimensional stochastic ising model in finite square with free boundary conditions at inverse temperature 0 and zero external field using duality and recent results of ioffe on the wulff construction close to the critical temperature we extend some of the results obtained by martinelli in the low temperature regime to any temperature below the critical one in particular we show that the gap in the spectrum of the generator of the dynamics goes to zero in the thermodynamic limit as an exponential of the side length of with a rate constant determined by the surface tension along one of the coordinate axes we also extend to the same range of temperatures the result due to shlosman on the equilibrium large deviations of the magnetization with free boundary conditions

convergence to equilibrium of random ising models in the griffiths phase we consider glauber type dynamics for disordered ising spin systems with nearest neighbor pair interactions in the griffiths phase we prove that in a nontrivial portion of the griffiths phase the system has exponentially decaying correlations of distant functions with probability exponentially close to 1 this condition has in turn been shown elsewhere to imply that the convergence to equilibrium is faster than any stretched exponential and that the average over the disorder of the time autocorrelation function goes to equilibrium faster than exp k log t d d 1 we then show that for the diluted ising model these upper bounds are optimal

a long time tail for random walk in random scenery consider a simple random walk on d whose sites are colored black or white independently with probabilityq resp 1 q walk and coloring are independent letn k be the number of steps by the walk between itskth and k 1 th visits to a black site i e the length of itskth white run and let k e n k q 1 our main result is a proof that lim k k d 2 k 1 q q d 2 2 d 2 d 2 since it is known thatq 1 k e n 1 n k 1 b e n 1 b e n k 1 b withb the event that the origin is black exhibits a long time tail in the run length autocorrelation function numerical calculations of k 1 k 100 ind 1 2 and 3 show that there is an oscillatory behavior of k for smallk this damps exponentially fast following which the power law sets in fairly rapidly we prove that if the coloring is not independent but is convex in the sense of fkg then the decay of k cannot be faster than

distributed consensus tracking of multi agent systems with nonlinear dynamics under a reference leader this paper investigates the distributed consensus tracking problem for multi agent systems with lipschitz type dynamics under a reference leader it is assumed that the leader state information is only available to a subset of followers while the bounded reference input of the leader s is unavailable to any follower to achieve consensus tracking a class of discontinuous protocols based on the relative information between the neighbouring agents are proposed furthermore as extensions of the former result the robust and adaptive consensus tracking problems are studied for the case where there exist parameter uncertainties and external disturbances in the network finally the effectiveness of the theoretical result is demonstrated through a network of single link manipulators

sharpness of the phase transition in percolation models the equality of two critical points the percolation threshold ph and the point p where the cluster size distribution ceases to decay exponentially is proven for all translation invariant independent percolation models on homogeneous d dimensional lattices d 1 the analysis is based on a pair of new nonlinear partial differential inequalities for an order parameter m h which for h q reduces to the percolation density p at the bond density p l e in the single parameter case these are 1 m hdm dh m2 mdm d and 2 dm d j mdm dh inequality 1 is intriguing in that its derivation provides yet another hint of a 3 structure in percolation models moreover through the elimination of one of its derivatives 1 yields a pair of ordinary differential inequalities which provide information on the critical exponents and one of these resembles an ising model inequality of fr hlich and sokal and yields the mean field bound 5 2 and the other implies the result of chayes and chayes that an inequality identical to 2 is known for ising models where it provides the basis for newman s universal relation 5 1 1 and for certain extrapolation principles which are now made applicable also to independent percolation these results apply to both finite and long range models with or without orientation and extend to periodic and weakly inhomogeneous systems

distributed coordinated tracking with a dynamic leader for multiple euler lagrange systems in this note we study a distributed coordinated tracking problem for multiple networked euler lagrange systems the objective is for a team of followers modeled by full actuated euler lagrange equations to track a dynamic leader whose vector of generalized coordinates is time varying under the constraints that the leader is a neighbor of only a subset of the followers and the followers have only local interaction we consider two cases i the leader has a constant vector of generalized coordinate derivatives and ii the leader has a varying vector of generalized coordinate derivatives in the first case we propose a distributed continuous estimator and an adaptive control law to account for parametric uncertainties in the second case we propose a model independent sliding mode control algorithm simulation results on multiple networked two link revolute joint arms are provided to show the effectiveness of the proposed control algorithms