characterization and computation of disturbance invariant sets for constrained switched linear systems with dwell time restriction this paper introduces the concept and characterization of disturbance dwell time invariance ddt invariance and constraint admissible ddt invariance caddt invariance for constrained systems with additive disturbance under dwell time switching using the characterization necessary and sufficient conditions for ddt invariance and algorithms for the computation of the minimal and maximal constraint admissible convex ddt invariant sets are provided numerical examples of such sets for several systems are also provided

controlled invariance and feedback laws we consider controlled invariance for cones translated cones polyhedra and various special polyhedral structures for any polyhedron if controlled invariance occurs then all and nothing but the admissible controls can be obtained by an inequative feedback controller for each special polyhedral structure we compare this feedback controller against piecewise affine linear and affine feedback controllers

multiple spectral kernel learning and a gaussian complexity computation multiple kernel learning mkl partially solves the kernel selection problem in support vector machines and similar classifiers by minimizing the empirical risk over a subset of the linear combination of given kernel matrices for large sample sets the size of the kernel matrices becomes a numerical issue in many cases the kernel matrix is of low efficient rank however the low rank property is not efficiently utilized in mkl algorithms here we suggest multiple spectral kernel learning that efficiently uses the low rank property by finding a kernel matrix from a set of gram matrices of a few eigenvectors from all given kernel matrices called a spectral kernel set we provide a new bound for the gaussian complexity of the proposed kernel set which depends on both the geometry of the kernel set and the number of gram matrices this characterization of the complexity implies that in an mkl setting adding more kernels may not monotonically increase the complexity while previous bounds show otherwise

crack paths in plane situations ii detailed form of the expansion of the stress intensity factors abstract in a previous paper we established the general form of the expansion of the stress intensity factors in powers of the crack extension length for a crack propagating in a two dimensional body along an arbitrary kinked and curved path the aim of the present paper is to calculate precisely the various functions of the geometric and mechanical parameters which appear in this expansion the functions involved in the case of a straight extension are identified by considering the special case of a crack composed of two straight branches placed in an infinite body loaded by uniform forces at infinity the problem is solved with the aid of muskhelishvili s formalism and conformal mapping the functions describing the effect of the curvature of the crack extension are determined by studying another special case identical to the first one except that the crack extension is curved the method of solution consists of using a perturbative procedure with respect to the curvature parameters to reduce the original problem to a simpler one involving a crack with two straight branches and solving again the latter problem by conformal mapping a numerical strategy using these results for the prediction of crack paths over arbitrary long distances is discussed in conclusion

stretched exponential fixation in stochastic ising models at zero temperature we study a class of continuous time markov processes which describes 1 spin flip dynamics on the hypercubic lattice d d 2 with initial spin configurations chosen according to the bernoulli product measure with density p of spins 1 during the evolution the spin at each site flips at rate c 0 or 0 0 q t exp t 1 d for large t in d 2 we obtain the complementary bound for arbitrary 0 q t exp t 1 2 for large t

a sharper threshold for bootstrap percolation in two dimensions two dimensional bootstrap percolation is a cellular automaton in which sites become infected by contact with two or more already infected nearest neighbours we consider these dynamics which can be interpreted as a monotone version of the ising model on an n n square with sites initially infected independently with probability p the critical probability pc is the smallest p for which the probability that the entire square is eventually infected exceeds 1 2 holroyd determined the sharp first order approximation pc 2 18 log n as n here we sharpen this result proving that the second term in the expansion is log n 3 2 o 1 and moreover determining it up to a poly log log n factor the exponent 3 2 corrects numerical predictions from the physics literature

digital idle speed control of automotive engines a safety problem for hybrid systems abstract we address the idle speed control problem in automotive electronics using hybrid methods to derive a digital control law with guaranteed properties associating a switching system with the hybrid system that describes the engine operation is crucial to developing a computationally feasible approach for switching systems with minimum and maximum dwell times and controlled resets we are able to derive digital control strategies with guaranteed properties that ensure safety the proposed methodology while motivated by the idle control problem is of general interest for hybrid systems for which minimum and maximum dwell times can be established in our modeling approach we do not assume synchronization between sampling time and switching time this is an important technical aspect in general and in particular for our application where there is no reason why sampling and switching should be synchronized some simulation results are included to demonstrate the effectiveness of the approach

metastability effects in bootstrap percolation bootstrap percolation models or equivalently certain types of cellular automata exhibit interesting finite volume effects these are studied at a rigorous level the authors find that for an initial configuration obtained by placing particles independently with probability p or 2 the density of the bootstrapped final configurations in the sequence of cubes l 2 l 2 d typically undergoes an abrupt transition as l is increased from being close to 0 to the value 1 with l fixed at a large value the mean final density as a function of p changes from 0 to 1 around a value which varies only slowly with l the pertinent parameter being lambda p1 d 1 ln l the driving mechanism is the capture of a critical droplet this behaviour is analogous to the decay of a metastable state near a first order phase transition for which the analysis offers some suggestive ideas

observability and controllability of piecewise affine and hybrid systems we prove in a constructive way the equivalence between piecewise affine systems and a broad class of hybrid systems described by interacting linear dynamics automata and propositional logic by focusing our investigation on the former class we show through counterexamples that observability and controllability properties cannot be easily deduced from those of the component linear subsystems instead we propose practical numerical tests based on mixed integer linear programming

computation of maximal safe sets for switching systems the problem of determining maximal safe sets and hybrid controllers is computationally intractable because of the mathematical generality of hybrid system models given the practical and theoretical relevance of the problem finding implementable procedures that could at least approximate the maximal safe set is important to this end we begin by restricting our attention to a special class of hybrid systems switching systems we exploit the structural properties of the graph describing the discrete part of a switching system to develop an efficient procedure for the computation of the safe set this procedure requires the computation of a maximal controlled invariant set we then restrict our attention to linear discrete time systems for which there is a wealth of results available in the literature for the determination of maximal controlled invariant sets however even for this class of systems the computation may not converge in a finite number of steps we then propose to compute inner approximations that are controlled invariant and for which a procedure that terminates in a finite number of steps can be obtained a tight bound on the error can be given by comparing the inner approximation with the classical outer approximation of the maximal controlled invariant set our procedure is applied to the idle speed regulation problem in engine control to demonstrate its efficiency

equivalence of switching linear systems by bisimulation a general notion of hybrid bisimulation is proposed for the class of switching linear systems connections between the notions of bisimulation based equivalence state space equivalence algebraic and input output equivalence are investigated an algebraic characterization of hybrid bisimulation and an algorithmic procedure converging in a finite number of steps to the maximal hybrid bisimulation are derived hybrid state space reduction is performed by hybrid bisimulation between the hybrid system and itself by specializing the results obtained on bisimulation also characterizations of simulation and abstraction are derived connections between observability bisimulation based reduction and simulation based abstraction are studied

a sharp threshold for a modified bootstrap percolation with recovery bootstrap percolation is a type of cellular automaton on graphs introduced as a simple model of the dynamics of ferromagnetism vertices in a graph can be in one of two states healthy or infected and from an initial configuration of states healthy vertices become infected by local rules while the usual bootstrap processes are monotone in the sets of infected vertices in this paper a modification is examined in which infected vertices can return to a healthy state vertices are initially infected independently at random and the central question is whether all vertices eventually become infected the model examined here is such a process on a square grid for which healthy vertices with at least two infected neighbours become infected and infected vertices with no infected neighbours become healthy sharp thresholds are given for the critical probability of initial infections for all vertices eventually to become infected

a d dimensional nucleation and growth model we analyze the relaxation time of a ferromagnetic d dimensional growth model on the lattice the model is characterized by d parameters which represent the activation energies of a site depending on the number of occupied nearest neighbours this model is a natural generalisation of the model studied by dehghanpour and schonmann probab theory relat fields 107 1 123 135 1997 where the activation energy of a site with more than two occupied neighbours is zero

computation of minimum time feedback control laws for discrete time systems with state control constraints the problem of finding a feedback law that drives the state of a linear discrete time system to the origin in minimum time subject to state control constraints is considered algorithms are given to obtain facial descriptions of the m step admissible sets these descriptions are then used to characterize the complete class of minimum time feedback laws moreover the characterization leads to a conceptually simple on line implementation the main ideas are illustrated with two simple examples

a time dependent hamilton jacobi formulation of reachable sets for continuous dynamic games we describe and implement an algorithm for computing the set of reachable states of a continuous dynamic game the algorithm is based on a proof that the reachable set is the zero sublevel set of the viscosity solution of a particular time dependent hamilton jacobi isaacs partial differential equation while alternative techniques for computing the reachable set have been proposed the differential game formulation allows treatment of nonlinear systems with inputs and uncertain parameters because the time dependent equation s solution is continuous and defined throughout the state space methods from the level set literature can be used to generate more accurate approximations than are possible for formulations with potentially discontinuous solutions a numerical implementation of our formulation is described and has been released on the web its correctness is verified through a two vehicle three dimensional collision avoidance example for which an analytic solution is available

computational tools for the safety control of a class of piecewise continuous systems with imperfect information on a partial order this paper addresses the two agent safety control problem for piecewise continuous systems with disturbances and imperfect state information in particular we focus on a class of systems that evolve on a partial order and whose dynamics preserve the ordering while the safety control problem with imperfect state information is prohibitive for general classes of nonlinear and hybrid systems the class of systems considered in this paper admits an explicit solution we compute this solution with linear complexity discrete time algorithms that are guaranteed to terminate the proposed algorithms are illustrated on a two vehicle collision avoidance problem and implemented on a hardware roundabout test bed

on the eshelby kostrov property for the wave equation in the plane this work deals with the linear wave equation considered in the whole plane r2 except for a rectilinear moving slit represented by a curve t x1 0 x1 t with t 0 along t either homogeneous dirichlet or neumann boundary conditions are imposed we discuss existence and uniqueness for these problems and derive explicit representation formulae for solutions these last have a simple geometrical interpretation and in particular allow to derive precise asymptotic expansions for solutions near the tip of the curve in the neumann case we thus recover a classical result in fracture dynamics namely the form of the stress intensity factor in crack propagation under antiplane shear conditions

metastable behavior for bootstrap percolation on regular trees we examine bootstrap percolation on a regular b 1 ary tree with initial law given by bernoulli p the sites are updated according to the usual rule a vacant site becomes occupied if it has at least occupied neighbors occupied sites remain occupied forever it is known that when b 2 the limiting density q q p of occupied sites exhibits a jump at some p t p t b 0 1 from q t q p t p t we investigate the metastable behavior associated with this transition explicitly we pick p p t h with h 0 and show that as h 0 the system lingers around the critical state for time order h 1 2 and then passes to fully occupied state in time o 1 the law of the entire configuration observed when the occupation density is q q t 1 converges as h 0 to a well defined measure

the evolution of stress intensity factors and the propagation of cracks in elastic media when a crack s propagates in an elastic medium the stress intensity factors evolve with the tip x s of s in this paper we derive formulae which describe the evolution of these stress intensity factors for a homogeneous isotropic elastic medium under plane strain conditions denoting by x s the stress potential is biharmonic and has zero traction along the crack s and by s the curvature of the crack at the tip x s we prove that the stress intensity factors a 1 s a 2 s as functions of s satisfy where are stress intensity factors of the tangential derivative of in the polar coordinate system at x s with 0 in the direction of the crack at x s the case of antiplane shearing is also briefly considered in this case is harmonic

bootstrap percolation on complex networks we consider bootstrap percolation on uncorrelated complex networks we obtain the phase diagram for this process with respect to two parameters f the fraction of vertices initially activated and p the fraction of undamaged vertices in the graph we observe two transitions the giant active component appears continuously at a first threshold there may also be a second discontinuous hybrid transition at a higher threshold avalanches of activations increase in size as this second critical point is approached finally diverging at this threshold we describe the existence of a special critical point at which this second transition first appears in networks with degree distributions whose second moment diverges but whose first moment does not we find a qualitatively different behavior in this case the giant active component appears for any fg0 and pg0 and the discontinuous transition is absent this means that the giant active component is robust to damage and also is very easily activated we also formulate a generalized bootstrap process in which each vertex can have an arbitrary threshold

optimized robust control invariance for linear discrete time systems theoretical foundations this paper introduces the concept of optimized robust control invariance for discrete time linear time invariant systems subject to additive and bounded state disturbances a novel characterization of two families of robust control invariant sets is given the existence of a constraint admissible member of these families can be checked by solving a single and tractable convex programming problem in the generic linear convex case and a standard linear quadratic program when the constraints are polyhedral or polytopic the solution of the same optimization problem yields the corresponding feedback control law that is in general set valued a procedure for selection of a point valued nonlinear control law is provided

optimization of the l sup spl infin induced norm under full state feedback this paper considers the l sup spl infin optimal control problem i e minimization of the effects of disturbances as measured by the l sup spl infin induced norm earlier work showed that even in the case of full state feedback optimal and near optimal linear controllers may be dynamic and of arbitrarily high order however previous work by the author derived the existence of near optimal nonlinear controllers which are static this paper presents a constructive algorithm for such nonlinear controllers the main idea is to construct a certain subset of the state space such that achieving disturbance rejection is equivalent to restricting the state dynamics to this set a construction of this subset requires the solution of several finite linear programs with the number of variables being at least the number of states but less than the number of states plus the number of controls the concept of controlled invariance plays a central role throughout

bootstrap percolation in three dimensions by bootstrap percolation we mean the following deterministic process on a graph g given a set a of vertices infected at time 0 new vertices are subsequently infected at each time step if they have at least n previously infected neighbors when the set a is chosen at random the main aim is to determine the critical probability p c g r at which percolation infection of the entire graph becomes likely to occur this bootstrap process has been extensively studied on the d dimensional grid n d with 2 r d fixed it was proved by cerf and cirillo for d r 3 and by cerf and manzo in general that p c n d r 1 log r 1 n d r 1 where log r is an r times iterated logarithm however the exact threshold function is only known in the case d r 2 where it was shown by holroyd to be 1 o 1 2 18 log n in this paper we shall determine the exact threshold in the crucial case d r 3 and lay the groundwork for solving the problem for all fixed d and r

integrals partitions and cellular automata we prove that 1 0 log f x x dx 2 3ab where f x is the decreasing function that satisfies f a f b x a x b for 0 a b when a is an integer and b a 1 we deduce several combinatorial results these include an asymptotic formula for the number of integer partitions not having a consecutive parts and a formula for the metastability thresholds of a class of threshold growth cellular automaton models related to bootstrap percolation

ultimate boundedness control for uncertain discrete time systems via set induced lyapunov functions in this note linear discrete time systems affected by both parameter and input uncertainties are considered the problem of the synthesis of a feedback control assuring that the system state is ultimately bounded within a given compact set containing the origin with an assigned rate of convergence is investigated it is shown that the problem has a solution if and only if there exists a certain lyapunov function which does not belong to a preassigned class of functions e g the quadratic ones but it is determined by the target set in which ultimate boundedness is desired one of the advantages of this approach is that we may handle systems with control constraints no matching assumptions are made for systems with linearly constrained uncertainties it is shown that such a function may be derived by numerically efficient algorithms involving polyhedral sets the resulting compensator may be implemented as a linear variable structure control

reachability analysis of discrete time systems with disturbances this paper presents new results that allow one to compute the set of states that can be robustly steered in a finite number of steps via state feedback control to a given target set the assumptions that are made in this paper are that the system is discrete time nonlinear and time invariant and subject to mixed constraints on the state and input a persistent disturbance dependent on the current state and input acts on the system existing results are not able to address state and input dependent disturbances and the results in this paper are therefore a generalization of previously published results one of the key aims of this paper is to present results such that one can perform the relevant set computations using polyhedral algebra and computational geometry software provided the system is piecewise affine and the constraints are polygonal existing methods are only applicable to piecewise affine systems that either have no control inputs or no disturbances whereas the results in this paper remove this limitation some simple examples are also given that show that even if all the relevant sets are convex and the system is linear convexity of the set of controllable states cannot be guaranteed

threshold growth dynamics we study the asymptotic shape of the occupied region for monotone deterministic dynamics in d dimensional euclidean space parametrized by a threshold 0 and a borel set n r d with positive and finite lebesgue measure if a n denotes the oocupied set of the dynamics at integer time n then a n 1 is obtained by adjoining any point x for which the volume of overlap between x n and a n exceeds except in some degenerate cases we prove that n 1 a n converges to a unique limiting shape l starting from any bounded initial region a 0 that is suitably large moreover l is computed as the polar transform for 1 w where w is an explicit width function that depends on n and

brief paper stability results for linear parameter varying and switching systems in this paper stabilizability problems for discrete and continuous time linear systems with switching or polytopic uncertainties are dealt with both state feedback and full information controllers are considered possibly with an integral action several cases will be analyzed depending on the control the uncertainty and the presence of uncertainties in the input matrix a complete overview of the stabilizability implications among these different cases will be given

bootstrap percolation on homogeneous trees has 2 phase transitions we study the threshold bootstrap percolation model on the homogeneous tree with degree b 1 2 b and initial density p it is known that there exists a nontrivial critical value for p which we call pf such that a for p pf the final bootstrapped configuration is fully occupied for almost every initial configuration and b if p pc then for almost every initial configuration there are infinite clusters of occupied vertices in the final bootstrapped configuration moreover we show that 3 for p pc the distribution of the occupied cluster size in the final bootstrapped configuration has an exponential tail 4 at p pc the expected occupied cluster size in the final bootstrapped configuration is infinite 5 the probability of percolation of occupied vertices in the final bootstrapped configuration is continuous on 0 pf and analytic on pc pf admitting an analytic continuation from the right at pc and only in the case b also from the left at pf

on the behavior of some cellular automata related to bootstrap percolation convergence to total occupancy and irc the threshold for this convergence to occur exponentially fast we locate these critical points for all the bootstrap percolation models showing that they are both 0 when 1 d for certain rules in which the orientation is important we show that 0 pc ir 1 by relating these systems to oriented site percolation finally these oriented models are used to obtain an estimate for a critical exponent of these models 1 introduction the fields of interacting particle systems mathematical statistical mechanics and percolation have benefited very much from their interrelations here we study a family of models which has arisen in the interface among these areas cellular automata such as those studied in this article may be considered as interacting particle systems see liggett 1985 for a survey of this field the relations between the models that we consider and percolation will become clear in many of the proofs given but can already be guessed from the fact that some of these systems are known as bootstrap percolation finally relations with statistical mechanics while not so explicit in this article were clearly present for instance in the article by chalupa leath and reich 1979 where bootstrap percolation was introduced in connection to disordered magnetic systems also in aizenman and lebowitz 1988 the motivation for studying these systems came from the nonequilibrium statistical mechanics problem of metastability our main concern in this article will be with the critical behavior of our models that is how their behavior changes qualitatively as some parameters cross certain values critical points as usual one of the main tools in the analysis of such phenomena will be a sort of renormalization procedure by

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

newman ziff algorithm for the bootstrap percolation application to the archimedean lattices abstract we propose very efficient algorithms for the bootstrap percolation and the diffusion percolation models by extending the newman ziff algorithm of the classical percolation m e j newman and r m ziff 2000 27 using these algorithms and the finite size scaling we calculated with high precision the percolation threshold and critical exponents in the eleven two dimensional archimedean lattices we present the condition for the continuous percolation transition in the bootstrap percolation and the diffusion percolation and show that they have the same critical exponents as the classical percolation within error bars in two dimensions we conclude that the bootstrap percolation and the diffusion percolation almost certainly belong to the same universality class as the classical percolation

the sharp threshold for bootstrap percolation in all dimensions in r neighbour bootstrap percolation on a graph g a typically random set a of initially infected vertices spreads by infecting at each time step vertices with at least r already infected neighbours this process may be viewed as a monotone version of the glauber dynamics of the ising model and has been extensively studied on the d dimensional grid n d the elements of the set a are usually chosen independently with some density p and the main question is to determine p c n d r the density at which percolation infection of the entire vertex set becomes likely in this paper we prove for every pair d ge r ge 2 that there is a constant l d r such that p c n d r l d r o 1 log r 1 n d r 1 as n o infty where log r denotes an r times iterated logarithm we thus prove the existence of a sharp threshold for percolation in any fixed number of dimensions moreover we determine l d r for every pair d r

higher order corrections for anisotropic bootstrap percolation we study the critical probability for the metastable phase transition of the two dimensional anisotropic bootstrap percolation model with 1 2 neighbourhood and threshold r 3 the first order asymptotics for the critical probability were recently determined by the first and second authors here we determine the following sharp second and third order asymptotics begin aligned p c big l 2 mathcal n scriptscriptstyle 1 2 3 big frac log log l 2 12 log l frac log log l log log log l 3 log l frac left log frac 9 2 1 pm o 1 right log log l 6 log l end aligned we note that the second and third order terms are so large that the first order asymptotics fail to approximate p c even for lattices of size well beyond 10 10 1000

on positively invariant sets for discrete time linear systems with disturbance an application of maximal disturbance sets this note concerns the investigation of maximal sets with respect to which a set is positively invariant and given a set a some properties of spl delta p i sets are analyzed the approach uses one step of the backward recursion introduced in d alessandro and de santis 1991 where feasibility and optimality conditions for linear dynamical input and or state constrained systems were analyzed in section ii the author gives the main definitions and some basic results in the subsequent section iii some properties of spl delta p i sets are investigated the particular case of polyhedral sets is analyzed in section iv in section v the author shows an application to linear systems with parametric uncertainties in the dynamic matrix a the author develops a technique to determine a set in the parameters space such that for any value of the parameter in the set a given set in the state space is positively invariant in the last section a numerical example is developed

admissible sets and feedback control for discrete time linear dynamical systems with bounded controls and states for linear discrete time dynamical systems necessary and sufficient conditions are given for a control sequence belonging to a polyhedral constraint set to stabilize the system under the condition that the state is restricted to a polyhedral state constraint set a variable structure linear state feedback controller given in terms of the controls at the vertices of the polyhedral state constraint set is presented

polluted bootstrap percolation with threshold two in all dimensions in the polluted bootstrap percolation model the vertices of a graph are independently declared initially occupied with probability p or closed with probability q at subsequent steps a vertex becomes occupied if it is not closed and it has at least r occupied neighbors on the cubic lattice mathbb z d of dimension d ge 3 with threshold r 2 we prove that the final density of occupied sites converges to 1 as p and q both approach 0 regardless of their relative scaling our result partially resolves a conjecture of morris and contrasts with the d 2 case where gravner and mcdonald proved that the critical parameter is q p 2

sharp metastability threshold for two dimensional bootstrap percolation in the bootstrap percolation model sites in an l by l square are initially independently declared active with probability p at each time step an inactive site becomes active if at least two of its four neighbours are active we study the behaviour as p 0 and l simultaneously of the probability i l p that the entire square is eventually active we prove that i l p 1 if and i l p 0 if where 2 18 we prove the same behaviour with the same threshold for the probability j l p that a site is active by time l in the process on the infinite lattice the same results hold for the so called modified bootstrap percolation model but with threshold 2 6 the existence of the thresholds settles a conjecture of aizenman and lebowitz 3 while the determination of their values corrects numerical predictions of adler stauffer and aharony 2

exact solution of a jamming transition closed equations for a bootstrap percolation problem jamming or dynamical arrest is a transition at which many particles stop moving in a collective manner in nature it is brought about by for example increasing the packing density changing the interactions between particles or otherwise restricting the local motion of the elements of the system the onset of collectivity occurs because when one particle is blocked it may lead to the blocking of a neighbor that particle may then block one of its neighbors these effects propagating across some typical domain of size named the dynamical correlation length when this length diverges the system becomes immobile even where it is finite but large the dynamics is dramatically slowed such phenomena lead to glasses gels and other very long lived nonequilibrium solids the bootstrap percolation models are the simplest examples describing these spatio temporal correlations we have been able to solve one such model in two dimensions exactly exhibiting the precise evolution of the jamming correlations on approach to arrest we believe that the nature of these correlations and the method we devise to solve the problem are quite general both should be of considerable help in further developing this field

brief doubly invariant equilibria of linear discrete time games in this paper the notion of doubly invariant di equilibrium is introduced the concept extends controlled and robustly controlled invariance notions to the context of two person dynamic games each player tries to keep the state in a region of state space independently of the actions of the rival player the paper gives existence conditions criteria and algorithms for the determination of di equilibria of linear dynamic games in discrete time two examples illustrate the results the first one is in the area of fault tolerant controller synthesis the second is an application to macroeconomics

the second term for two neighbour bootstrap percolation in two dimensions in the r neighbour bootstrap process on a graph g vertices are infected in each time step if they have at least r already infected neighbours motivated by its close connections to models from statistical physics such as the ising model of ferromagnetism and kinetically constrained spin models of the liquid glass transition the most extensively studied case is the two neighbour bootstrap process on the two dimensional grid n 2 around 15 years ago in a major breakthrough holroyd determined the sharp threshold for percolation in this model and his bounds were subsequently sharpened further by gravner and holroyd and by gravner holroyd and morris in this paper we strengthen the lower bound of gravner holroyd and morris by proving that the critical probability p c big n 2 2 big for percolation in the two neighbour model on n 2 satisfies p c big n 2 2 big frac pi 2 18 log n frac theta 1 log n 3 2 the proof of this result requires a very precise understanding of the typical growth of a critical droplet and involves a number of technical innovations we expect these to have other applications for example to the study of more general two dimensional cellular automata and to the r neighbour process in higher dimensions

finite size scaling in three dimensional bootstrap percolation we consider the problem of bootstrap percolation on a three dimensional lattice and we study its finite size scaling behavior bootstrap percolation is an example of cellular automata defined on the d dimensional lattice 1 2 l d in which each site can be empty or occupied by a single particle in the starting configuration each site is occupied with probability p occupied sites remain occupied forever while empty sites are occupied by a particle if at least among their 2d nearest neighbor sites are occupied when d is fixed the most interesting case is the one d this is a sort of threshold in the sense that the critical probability p c for the dynamics on the infinite lattice z d switches from zero to one when this limit is crossed finite size effects in the three dimensional case are already known in the cases 2 in this paper we discuss the case 3 and we show that the finite size scaling function for this problem is of the form f l const in in l we prove a conjecture proposed by a c d van enter

the time of bootstrap percolation with dense initial sets let r n in r neighbour bootstrap percolation on the vertex set of a graph g vertices are initially infected independently with some probability p at each time step the infected set expands by infecting all uninfected vertices that have at least r infected neighbours when p is close to 1 we study the distribution of the time at which all vertices become infected given t t n o logn loglogn we prove a sharp threshold result for the probability that percolation occurs by time t in d neighbour bootstrap percolation on the d dimensional discrete torus t d n moreover we show that for certain ranges of p p n the time at which percolation occurs is concentrated either on a single value or on two consecutive values we also prove corresponding results for the modified d neighbour rule

zero temperature glauber dynamics on mathbb z d we study zero temperature glauber dynamics on mathbb z d which is a dynamic version of the ising model of ferromagnetism spins are initially chosen according to a bernoulli distribution with density p and then the states are continuously and randomly updated according to the majority rule this corresponds to the sudden quenching of a ferromagnetic system at high temperature with an external field to one at zero temperature with no external field define p c mathbb z d to be the infimum over p such that the system fixates at with probability 1 it is a folklore conjecture that p c mathbb z d 1 2 for every 2 le d in mathbb n we prove that p c mathbb z d to 1 2 as d

on the robust controlled invariant abstract the robust controlled invariant is defined as a tool to characterize systems subject to parameter changes and to set up a geometric approach to robustness in the large in multi variable control problems as a typical application the robust disturbance decoupling problem an extended version of the well known disturbance decoupling is presented and a necessary and sufficient condition for it to have a solution is derived

constrained control of switching systems a positive invariant approach this paper presents conditions for the stabilization of switching discrete time linear systems with constrained control by using a positive invariance approach a state feedback control law is used to construct the stabilizing controller a numerical example is presented to illustrate the technique

bootstrap percolation on the hypercube in the bootstrap percolation on the n dimensional hypercube in the initial position each of the 2n sites is occupied with probability p and empty with probability 1 p independently of the state of the other sites every occupied site remains occupied for ever while an empty site becomes occupied if at least two of its neighbours are occupied if at the end of the process every site is occupied we say that the initial position spans the hypercube we shall show that there are constants c1 c2 0 such that for open image in new window the probability of spanning tends to 1 as n while for open image in new window the probability tends to 0 furthermore we shall show that for each n the transition has a sharp threshold function

sharp metastability threshold for an anisotropic bootstrap percolation model bootstrap percolation models have been extensively studied during the two past decades in this article we study the following anisotropic boot strap percolation model the neighborhood of a point m n is the set m 2 n m 1 n m n 1 m 1 n m 2 n m n 1 at time 0 sites are occupied with probability p at each time step sites that are occupied remain occupied while sites that are not occupied become occupied if and only if three of more sites in their neighborhood are occupied we prove that it exhibits a sharp metastability threshold this is the first mathematical proof of a sharp threshold for an anisotropic bootstrap percolation model

survey paper set invariance in control the properties of positively invariant sets are involved in many different problems in control theory such as constrained control robustness analysis synthesis and optimization in this paper we provide an overview of the literature concerning positively invariant sets and their application to the analysis and synthesis of control systems

on some growth models with a small parameter we consider the behavior of the asymptotic speed of growth and the asymptotic shape in some growth models when a certain parameter becomes small the basic example treated is the variant of richardson s growth model on d in which each site which is not yet occupied becomes occupied at rate 1 if it has at least two occupied neighbors at rate 1 if it has exactly 1 occupied neighbor and of course at rate 0 if it has no occupied neighbor occupied sites remain occupied forever starting from a single occupied site this model has asymptotic speeds of growth in each direction as time goes to infinity and these speeds determine an asymptotic shape in the usual sense it is proven that as tends to 0 the asymptotic speeds scale as 1 d and the asymptotic shape when renormalized by dividing it by 1 d converges to a cube other similar models which are partially oriented are also studied

a nucleation and growth model we consider the following simple nucleation and growth model on the lattice d starting with all sites unoccupied a site becomes occupied at rate e if it has no occupied neighbors at rate e if it has 1 occupied neighbor and at rate 1 if it has 2 or more occupied neighbors occupied sites remain occupied forever the parameters are fixed and we are interested in the behavior of the system as we show that the relaxation time of this system scales as e c where c max d 1