quantitative uniqueness of elliptic equations based on a variant of frequency function we improve the vanishing order of solutions for schr o dinger equations which describes quantitative behavior of strong uniqueness continuation property for the first time we investigate the quantitative uniqueness of higher order elliptic equations and show the vanishing order of solutions furthermore strong unique continuation is established for higher order elliptic equations using this variant of frequency function

finite approximations of switched homogeneous systems for controller synthesis in this note we demonstrate the use of a control oriented notion of finite state input output approximation to synthesize correct by design controllers for hybrid plants under sensor limitations specifically we consider the problem of designing stabilizing switching controllers for a pair of unstable homogeneous second order systems with binary output feedback in addition to yielding a deterministic finite state approximate model of the hybrid plant our approach allows one to efficiently establish a useable upper bound on the quality of approximation and leads to a discrete optimization problem whose solution immediately provides a certified finite state controller for the plant the resulting controller consists of a deterministic finite state observer and a corresponding full state feedback control law

large deviations of empirical neighborhood distribution in sparse random graphs consider the erd s renyi random graph on n vertices where each edge is present independently with probability lambda n with lambda 0 fixed for large n a typical random graph locally behaves like a galton watson tree with poisson offspring distribution with mean lambda here we study large deviations from this typical behavior within the framework of the local weak convergence of finite graph sequences the associated rate function is expressed in terms of an entropy functional on unimodular measures and takes finite values only at measures supported on trees we also establish large deviations for other commonly studied random graph ensembles such as the uniform random graph with given number of edges growing linearly with the number of vertices or the uniform random graph with given degree sequence to prove our results we introduce a new configuration model which allows one to sample uniform random graphs with a given neighborhood distribution provided the latter is supported on trees we also introduce a new class of unimodular random trees which generalizes the usual galton watson tree with given degree distribution to the case of neighborhoods of arbitrary finite depth these generalized galton watson trees turn out to be useful in the analysis of unimodular random trees and may be considered to be of interest in their own right

the cut metric random graphs and branching processes in this paper we study the component structure of random graphs with independence between the edges under mild assumptions we determine whether there is a giant component and find its asymptotic size when it exists we assume that the sequence of matrices of edge probabilities converges to an appropriate limit object a kernel but only in a very weak sense namely in the cut metric our results thus generalize previous results on the phase transition in the already very general inhomogeneous random graph model introduced by the present authors in random struct algorithms 31 3 122 2007 as well as related results of bollobas borgs chayes and riordan ann probab 38 150 183 2010 all of which involve considerably stronger assumptions we also prove corresponding results for random hypergraphs these generalize our results on the phase transition in inhomogeneous random graphs with clustering random struct algorithms 2010 to appear

some large deviation results for sparse random graphs we obtain a large deviation principle ldp for the relative size of the largest connected component in a random graph with small edge probability the rate function which is not convex in general is determined explicitly using a new technique the proof yields an asymptotic formula for the probability that the random graph is connected

approximately bisimilar symbolic models for nonlinear control systems control systems are usually modeled by differential equations describing how physical phenomena can be influenced by certain control parameters or inputs although these models are very powerful when dealing with physical phenomena they are less suited to describe software and hardware interfacing with the physical world for this reason there is a growing interest in describing control systems through symbolic models that are abstract descriptions of the continuous dynamics where each symbol corresponds to an aggregate of states in the continuous model since these symbolic models are of the same nature of the models used in computer science to describe software and hardware they provide a unified language to study problems of control in which software and hardware interact with the physical world furthermore the use of symbolic models enables one to leverage techniques from supervisory control and algorithms from game theory for controller synthesis purposes in this paper we show that every incrementally globally asymptotically stable nonlinear control system is approximately equivalent bisimilar to a symbolic model the approximation error is a design parameter in the construction of the symbolic model and can be rendered as small as desired furthermore if the state space of the control system is bounded the obtained symbolic model is finite for digital control systems and under the stronger assumption of incremental input to state stability symbolic models can be constructed through a suitable quantization of the inputs

an approximate simulation approach to symbolic control this paper introduces a methodology for the symbolic control of nonlinear systems based on an approximate notion of simulation relation this notion generalizes existing exact notions of simulation and is completely characterized in terms of known stabilizability concepts equipped with this notion we show how under certain stabilizability assumptions we can construct finite or symbolic models for nonlinear control systems synthesizing controllers for the original control system can then be done by using supervisory control techniques on the finite models and by refining the resulting finite controllers to hybrid controllers enforcing the specification on the original continuous control system the proposed design methodology can be seen as a correct by design way of obtaining both the feedback control laws as well as the control software responsible for deciding which law is executed and when

sharp vanishing order of solutions to stationary schrodinger equations on carnot groups of arbitrary step abstract based on a variant of the frequency function approach of almgren 1 under appropriate assumptions we establish an optimal upper bound on the vanishing order of solutions to stationary schrodinger equations associated to sub laplacian on carnot groups of arbitrary step such a bound provides a quantitative form of strong unique continuation and can be thought of as a subelliptic analogue of the recent results obtained by bakri 3 and zhu 27 for the standard laplacian

stabilization with relaxed controls cannot in general be stabilized using a continuous closed loop control u x even if each state separately can be driven asymptotically to the origin an example is analyzed in section 2 in this paper we examine the possibility of stabilizing such systems with a continuous closed loop relaxed control we find indeed that the family of systems stabilizable with relaxed controls is larger than the family of those stabilizable with ordinary controls an even larger class is obtained if the continuity of the closed loop at the origin is not required the latter class includes all one dimensional systems for which states can be driven asymptotically to the origin this result does not hold in two dimensional systems and we provide a counter example it should be pointed that relaxed control type stabilization is used both in theory and in practice the method is known as dither we shall comment on the similarities lyapunov functions for the system 1 help us in the construction of the continuous closed loop stabilizers in fact we find that the existence of a smooth lyapunov function is equivalent to the existence of a stabilizing closed loop which is continuous except possibly at the origin an additional condition on the lyapunov function implies the continuity at the origin as well we present these results in section 4 after a brief introduction of closed loop relaxed controls notations and terminology in section 3 prior to that in section 2 we discuss an example illustrating the power of relaxed controls in the particular case of systems linear in the controls relaxed controls can be replaced by ordinary controls this is discussed in section 5 the role of lyapunov functions in the stability and stabilization theories is of course well known examples of systems with lyapunov functions are available in the literature we display some in section 6 along with general comments on the construction applications and counterexamples including one which cannot be continuously stabilized yet possesses a nonsmooth lyapunov function closed loop stabilization with ordinary controls is analyzed extensively in the literature see sontag 8 sussmann ll and references therein lyapunov functions techniques in stabil

on large deviation properties of erdos renyi random graphs we show that large deviation properties of erdos renyi random graphs can be derived from the free energy of the q state potts model of statistical mechanics more precisely the legendre transform of the potts free energy with respect to ln q is related to the component generating function of the graph ensemble this generalizes the well known mapping between typical properties of random graphs and the q 1 limit of the potts free energy for exponentially rare graphs we explicitly calculate the number of components the size of the giant component the degree distributions inside and outside the giant component and the distribution of small component sizes we also perform numerical simulations which are in very good agreement with our analytical work finally we demonstrate how the same results can be derived by studying the evolution of random graphs under the insertion of new vertices and edges without recourse to the thermodynamics of the potts model

an input output construction of finite state rho mu approximations for control design we consider discrete time plants that interact with their controllers via fixed discrete alphabets for this class of systems and in the absence of exogenous inputs we propose a general methodical approach for constructing a sequence of finite state approximate models starting from finite length sequences of input and output signal pairs we explicitly derive conditions under which the proposed construct used in conjunction with a particular generalized structure satisfies desirable properties of approximations thereby leading to nominal deterministic finite state machine models that can be used in certified by design controller synthesis we also show that the cardinality of the min imal disturbance alphabet that can be used in this setting equals that of the sensor output alphabet finally we show that the pro posed construct satisfies a relevant semi completeness property

temporal logic motion planning for dynamic robots in this paper we address the temporal logic motion planning problem for mobile robots that are modeled by second order dynamics temporal logic specifications can capture the usual control specifications such as reachability and invariance as well as more complex specifications like sequencing and obstacle avoidance our approach consists of three basic steps first we design a control law that enables the dynamic model to track a simpler kinematic model with a globally bounded error second we built a robust temporal logic specification that takes into account the tracking errors of the first step finally we solve the new robust temporal logic path planning problem for the kinematic model using automata theory and simple local vector fields the resulting continuous time trajectory is provably guaranteed to satisfy the initial user specification

safety controller synthesis using human generated trajectories this paper focuses on the task of safety controller synthesis that is designing a controller that will take a system from any point within a compact set of initial states to a point inside a set of acceptable goal states while never entering any state that is deemed unsafe to do this we use a human generated trajectory based approach we introduce the control autobisimulation function which is the analog of the control lyapunov function for approximate bisimulation we consider a class of hybrid systems and use this function to determine a set of admissible feedback control laws that guarantee trajectory robustness for underlying dynamics that are linear affine feedback linearizable and differentially flat this property ensures that any trajectory of the closed loop system that is initialized within some neighborhood of a nominal trajectory will stay within some tube of the nominal trajectory when given the same input this feedback control and input can be used as the controller for some subset of the initial states we demonstrate how to combine multiple trajectories into a synthesized controller that satisfies the safety problem

symbolic models for nonlinear control systems without stability assumptions finite state models of control systems were proposed by several researchers as a convenient mechanism to synthesize controllers enforcing complex specifications most techniques for the construction of such symbolic models have two main drawbacks either they can only be applied to restrictive classes of systems or they require the exact computation of reachable sets in this paper we propose a new abstraction technique that is applicable to any nonlinear sampled data control system as long as we are only interested in its behavior in a compact set moreover the exact computation of reachable sets is not required the effectiveness of the proposed results is illustrated by synthesizing a controller to steer a vehicle

quantitative uniqueness for elliptic equations at the boundary of c1 dini domains abstract based on a variant of the frequency function approach of almgren we establish an optimal upper bound on the vanishing order of solutions to variable coefficient schrodinger equations at a portion of the boundary of a c 1 dini domain such bound provides a quantitative form of strong unique continuation at the boundary it can be thought of as a boundary analogue of an interior result recently obtained by bakri and zhu for the standard laplacian

on quantitative uniqueness for elliptic equations we address the question of quantitative uniqueness for the equation delta u v u with either periodic or dirichlet boundary conditions in a disk we construct solutions u corresponding to potential functions v such that u vanishes of order mathrm const vert v vert l infty 2 3 the example also shows sharpness of recently obtained bounds in the case of a parabolic equation u t delta u v u

asynchronous l complete approximations abstract this paper extends the l complete approximation method developed for time invariant systems to a larger system class ensuring that the resulting approximation can be realized by a finite state machine to derive the new abstraction method called asynchronous l complete approximation an asynchronous version of the well known concepts of state property memory span and l completeness is introduced extending the behavioral systems theory in a consistent way

counting nodal lines which touch the boundary of an analytic domain we consider the zeros on the boundary of a neumann eigenfunction of a real analytic plane domain we prove that the number of its boundary zeros is o where 2 we also prove that the number of boundary critical points of either a neumann or dirichlet eigenfunction is o it follows that the number of nodal lines of components of the nodal set which touch the boundary is of order this upper bound is of the same order of magnitude as the length of the total nodal line but is the square root of the courant bound on the number of nodal components in the interior more generally the results are proved for piecewise analytic domains

verification of hybrid automata diagnosability with measurement uncertainty the problem of system diagnosability verification is concerned with whether a fault in the system operation can be diagnosed by using the system model and observations of the system output in this paper we investigate the delta d delta m diagnosability of hybrid automata which characterizes the maximum delay for diagnosing faults since their first occurrence given the measurement uncertainty in observing the system output

unique continuation for elliptic operators a geometric variational approach dans un sous ensemble connexe ouvert de r n n 3 on considere l equation d ordre 2 lu div a x u b x u v x u 0 ou a x a id j x est une matrice symetrique reelle on etudie les proprietes d unicite de l operateur

concentration of measures via size biased couplings let y be a nonnegative random variable with mean and finite positive variance 2 and let ys defined on the same space as y have the y size biased distribution characterized by e yf y mu e f y s quad rm for all functions f rm for which these expectations exist under a variety of conditions on y and the coupling of y and ys including combinations of boundedness and monotonicity one sided concentration of measure inequalities such as p left frac y mu sigma ge t right le rm exp left frac t 2 2 a bt right quad rm for all t 0 hold for some explicit a and b the theorem is applied to the number of bulbs switched on at the terminal time in the so called lightbulb process of rao et al sankhy 69 137 161 2007

exceptional rotations of random graphs a vc theory in this paper we explore maximal deviations of large random structures from their typical behavior we introduce a model for a high dimensional random graph process and ask analogous questions to those of vapnik and chervonenkis for deviations of averages how rich does the process have to be so that one sees atypical behavior in particular we study a natural process of erdos renyi random graphs indexed by unit vectors in rd we investigate the deviations of the process with respect to three fundamental properties clique number chromatic number and connectivity in all cases we establish upper and lower bounds for the minimal dimension d that guarantees the existence of exceptional directions in which the random graph behaves atypically with respect to the property for each of the three properties four theorems are established to describe upper and lower bounds for the threshold dimension in the subcritical and supercritical regimes

bounds for eigenfunctions of the laplacian on compact riemannian manifolds suppose that is an eigenfunction of with eigenvalue 0 it is proved that c1 n 14 2 where n is the dimension of m and c1 depends only upon a bound for the absolute value of the sectional curvature of m and a lower bound for the injectivity radius of m it is then shown that if m admits an isometric circle action and the metric is generic one has exceptional sequences of eigenfunctions satisfying the complementary bounds k c2 n 18k 2

the swendsen wang process does not always mix rapidly the swendsen wang process provides one possible dynamics for the q state potts model computer simulations of this process are widely used to estimate the expectations of various observables random variables of a potts system in the equilibrium or gibbs distribution the legitimacy of such simulations depends on the rate of convergence of the process to equilibrium as measured by the mixing time empirical observations suggest that the mixing time of the swendsen wang process is short in many instances of practical interest although proofs of this desirable behavior are known only for some very special cases nevertheless we show that there are occasions when the mixing time of the swendsen wang process is exponential in the size of the system this undesirable behavior is related to the phenomenon of first order phase transitions in potts systems with q 2 states

stochastic processes in random graphs we study the asymptotics of large moderate and normal deviations for the connected components of the sparse random graph by the method of stochastic processes we obtain the logarithmic asymptotics of large deviations of the joint distribution of the number of connected components of the sizes of the giant components and of the numbers of the excess edges of the giant components for the supercritical case we obtain the asymptotics of normal deviations and the logarithmic asymptotics of large and moderate deviations of the joint distribution of the number of components of the size of the largest component and of the number of the excess edges of the largest component for the critical case we obtain the logarithmic asymptotics of moderate deviations of the joint distribution of the sizes of connected components and of the numbers of the excess edges some related asymptotics are also established the proofs of the large and moderate deviation asymptotics employ methods of idempotent probability theory as a byproduct of the results we provide some additional insight into the nature of phase transitions in sparse random graphs

nodal sets of eigenfunctions on riemannian manifolds soit m n une variete connexe compacte a metrique riemannienne c soit le laplacien de m soit f une fonction propre reelle de de valeur propre on demontre que la fonction propre f s annule au plus a l ordre c 1 2 pour tout point dans m