a discrete event model of asynchronous quantised systems the paper deals with the discrete event modelling of continuous variable dynamical systems whose input and state signals can only be measured through quantisers a change of the quantised input or state value is called an input or a state event the paper concerns the general case in which these events may occur asynchronously to one another a nondeterministic automaton is used as the discrete event model of such quantised systems the paper presents an abstraction algorithm for determining the state transition relation of the automaton from the state space description of the continuous time continuous variable system and the signal space partitions introduced by the quantisers the abstraction is based on the reachability analysis of quantised systems a hierarchy of models with respect to the set of spurious behaviours is obtained by varying an abstraction parameter the discrete event models are proved to be complete models of the quantised system with the best possible transition relation

backstepping controller synthesis and characterizations of incremental stability incremental stability is a property of dynamical and control systems requiring the uniform asymptotic stability of every trajectory rather than that of an equilibrium point or a particular time varying trajectory similarly to stability lyapunov functions and contraction metrics play important roles in the study of incremental stability in this paper we provide characterizations and descriptions of incremental stability in terms of existence of coordinate invariant notions of incremental lyapunov functions and contraction metrics respectively most design techniques providing controllers rendering control systems incrementally stable have two main drawbacks they can only be applied to control systems in either parametric strict feedback or strict feedback form and they require these control systems to be smooth in this paper we propose a design technique that is applicable to larger classes of control systems including a class of non smooth control systems moreover we propose a recursive way of constructing contraction metrics for smooth control systems and incremental lyapunov functions which have been identified as a key tool enabling the construction of finite abstractions of nonlinear control systems the approximation of stochastic hybrid systems source code model checking for nonlinear dynamical systems and so on the effectiveness of the proposed results in this paper is illustrated by synthesizing controllers rendering two non smooth control systems incrementally stable the first example aims to show how to recursively construct the incremental lyapunov functions the second example aims to show the key role of the computed incremental lyapunov function in constructing a finite abstraction that is equivalent to the system under study

estimating the k function of a point process with an application to cosmology motivated by the study of an important data set for understanding the large scale structure of the universe this work considers the estimation of the reduced second moment function or k function of a stationary point process on r observed over a large number of segments of possibly varying lengths theory and simulation are used to compare the behavior of isotropic and rigid motion correction estimators and some modifications of these estimators these results generally support the use of modified versions of the rigid motion correction when applied to a catalog of astronomical objects known as absorbers the proposed methods confirm results from earlier analyses of the absorber catalog showing clear evidence of clustering up to 50 h 1 mpc and marginal evidence for clustering of matter on spatial scales beyond 100 h 1 mpc which is beyond the distance at which clustering of matter is now generally accepted to exist

square function non tangential maximal function estimates and the dirichlet problem for non symmetric elliptic operators a we consider divergence form elliptic operators l div a x defined in the half space rn 1 n 2 where the coefficient matrix a x is bounded measurable uniformly elliptic t independent and not necessarily symmetric we establish square function non tangential maximal function estimates for solutions of the homogeneous equation lu 0 and we then combine these estimates with the method of approximability to show that l harmonic measure is absolutely continuous with respect to surface measure i e n dimensional lebesgue measure on the boundary in a scale invariant sense more precisely that it belongs to the class a with respect to surface measure equivalently that the dirichlet problem is solvable with data in lp for some p previously these results had been known only in the case n 1

approximation metrics for discrete and continuous systems established system relationships for discrete systems such as language inclusion simulation and bisimulation require system observations to be identical when interacting with the physical world modeled by continuous or hybrid systems exact relationships are restrictive and not robust in this paper we develop the first framework of system approximation that applies to both discrete and continuous systems by developing notions of approximate language inclusion approximate simulation and approximate bisimulation relations we define a hierarchy of approximation pseudo metrics between two systems that quantify the quality of the approximation and capture the established exact relationships as zero sections our approximation framework is compositional for a synchronous composition operator algorithms are developed for computing the proposed pseudo metrics both exactly and approximately the exact algorithms require the generalization of the fixed point algorithms for computing simulation and bisimulation relations or dually the solution of a static game whose cost is the so called branching distance between the systems approximations for the pseudo metrics can be obtained by considering lyapunov like functions called simulation and bisimulation functions we illustrate our approximation framework in reducing the complexity of safety verification problems for both deterministic and nondeterministic continuous systems

kaplan meier estimators of distance distributions for spatial point processes when a spatial point process is observed through a bounded window edge effects hamper the estimation of characteristics such as the empty space function f the nearest neighbor distance distribution g and the reduced second order moment function k here we propose and study product limit type estimators of f g and k based on the analogy with censored survival data the distance from a fixed point to the nearest point of the process is right censored by its distance to the boundary of the window the resulting estimators have a ratio unbiasedness property that is standard in spatial statistics we show that the empty space function f of any stationary point process is absolutely continuous and so is the product limit estimator of f the estimators are strongly consistent when there are independent replications or when the sampling window becomes large we sketch a clt for independent replications within a fixed observation window and asymptotic theory for independent replications of sparse poisson processes in simulations the new estimators are generally more efficient than the border method estimator but for estimators of k somewhat less efficient than sophisticated edge corrections

the dirichlet problem in non smooth domains dirichlet problem for an elliptic operator 2s laj jaj with smooth coefficients aij in domains d in rm m 3 with non smooth boundaries the domains we treat are given locally in some cm coordinate system by the graph of a continuous function qs with vq5 e li for some p 1 p oo such domains are called li domains see section 1 for the precise definitions note that l domains are usually called lipschitz domains if d is an li domain and x e d we study the elliptic measure a7 associated with s and d at x thus for f a continuous function on ad the formula u x hf x f doh gives the perron wiener brelot generalized soluid

estimating the large scale structure of the universe using quasi stellar object carbon iv absorbers galaxies have long been known to form large clusters and cosmologists are interested in characterizing this clustering as a way of studying the large scale structure of the universe this work is motivated by a data catalog consisting of information on lines of sight from earth to distant quasi stellar objects qso s and the carbon iv absorbers that lie on them the absorbers are believed to be gas clouds near galaxies too far away to be easily observed thus the absorber catalog provides a unique and interesting way to examine the large scale structure of the universe on large scales previous studies have mainly used pairs of absorbers on the same lines of sight to obtain estimates describing the clustering of absorbers it is clear that absorbers on different lines of sight contain information about the degree of clustering we develop an adaptation of the rigid motion corrected estimator of the reduced second moment function for the absorber catalog taking into account the across line of sight info

homogenization of elliptic boundary value problems in lipschitz domains in this paper we study the lp boundary value problems for mathcal l u 0 in mathbb r d 1 where mathcal l rm div a nabla is a second order elliptic operator with real and symmetric coefficients assume that a is periodic in xd 1 and satisfies some minimal smoothness condition in the xd 1 variable we show that the lp neumann and regularity problems are uniquely solvable for 1 p 2 we also present a new proof of dahlberg s theorem on the lp dirichlet problem for 2 p dahlberg s original unpublished proof is given in the appendix as the periodic and smoothness conditions are imposed only on the xd 1 variable these results extend directly from mathbb r d 1 to regions above lipschitz graphs consequently by localization techniques we obtain uniform lp estimates for the dirichlet neumann and regularity problems on bounded lipschitz domains for a family of second order elliptic operators arising in the theory of homogenization the results on the neumann and regularity problems are new even for smooth domains

symbolic control of stochastic systems via approximately bisimilar finite abstractions symbolic approaches for control design construct finite state abstract models that are related to the original systems then use techniques from finite state synthesis to compute controllers satisfying specifications given in a temporal logic and finally translate the synthesized schemes back as controllers for the original systems such approaches have been successfully developed and implemented for the synthesis of controllers over non probabilistic control systems in this paper we extend the technique to probabilistic control systems modelled by controlled stochastic differential equations we show that for every stochastic control system satisfying a probabilistic variant of incremental input to state stability and for every given precision e 0 a finite state transition system can be constructed which is e approximately bisimilar to the original stochastic control system moreover we provide results relating stochastic control systems to their corresponding finite state transition systems in terms of probabilistic bisimulation relations known in the literature we demonstrate the effectiveness of the construction by synthesizing controllers for stochastic control systems over rich specifications expressed in linear temporal logic our technique enables automated correct by construction controller synthesis for stochastic control systems which are common mathematical models employed in many safety critical systems subject to structured uncertainty

boundary value problems for second order elliptic operators satisfying a carleson condition let be a lipschitz domain in n n 2 and l div a be a second order elliptic operator in divergence form we establish the solvability of the dirichlet regularity problem with boundary data in h1 p and of the neumann problem with lp data for the operator l on lipschitz domains with small lipschitz constant we allow the coefficients of the operator l to be rough obeying a certain carleson condition with small norm these results complete the results of dindos petermichl and pipher 2007 where the lp dirichlet problem was considered under the same assumptions and dindos and rule 2010 where the regularity and neumann problems were considered on two dimensional domains 2016 wiley periodicals inc

symbolic models for nonlinear control systems alternating approximate bisimulations symbolic models are abstract descriptions of continuous systems in which symbols represent aggregates of continuous states in the last few years there has been a growing interest in the use of symbolic models as a tool for mitigating complexity in control design in fact symbolic models enable the use of well known algorithms in the context of supervisory control and algorithmic game theory for controller synthesis since the 1990s many researchers faced the problem of identifying classes of dynamical and control systems that admit symbolic models in this paper we make further progress along this research line by focusing on control systems affected by disturbances our main contribution is to show that incrementally globally asymptotically stable nonlinear control systems with disturbances admit symbolic models

symbolic models for networks of control systems in this note we propose symbolic models for networks of discrete time nonlinear control systems if each subsystem composing the network admits an incremental input to state stable lyapunov function and if some small gain theorem type conditions are satisfied a network of symbolic models each one associated with each subsystem composing the network is proposed and shown to be approximately bisimilar to the original network with any desired accuracy

convergence rates in l 2 for elliptic homogenization problems we study rates of convergence of solutions in l2 and h1 2 for a family of elliptic systems mathcal l varepsilon with rapidly oscillating coefficients in lipschitz domains with dirichlet or neumann boundary conditions as a consequence we obtain convergence rates for dirichlet neumann and steklov eigenvalues of mathcal l varepsilon most of our results which rely on the recently established uniform estimates for the l2 dirichlet and neumann problems in kenig and shen math ann 350 867 917 2011 commun pure appl math 64 1 44 2011 are new even for smooth domains

layer potentials and boundary value problems for elliptic systems in lipschitz domains various layer potential operators are constructed for general elliptic systems of partial differential equations with constant coefficients these operators are used to study the boundary value problems for these systems for a bounded lipschitz domain in rn n 3 it is shown that if the coefficients of the system satisfy the legendre hadamard condition and a symmetry condition then there exists a unique solution u of the system that solves the dirichlet problem for given data g in lp for p in a neighborhood of 2 also the oblique derivative problem for the system is studied furthermore some regularity results for the dirichlet problem are obtained

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

equivalence of dynamical systems by bisimulation a general notion of bisimulation is defined for linear input state output systems using analogies with the theory of concurrent processes a characterization of bisimulation and an algorithm for computing the maximal bisimulation relation is derived using geometric control theory bisimulation is shown to be a notion which unifies the concepts of state space equivalence and state space reduction and which allows to study equivalence of systems with nonminimal state space dimension the notion of bisimulation is especially powerful for nondeterministic dynamical systems and leads in this case to a notion of equivalence which is finer than equality of external behavior for abstractions of systems it is shown how the results specialize to previously obtained results by other authors extensions of the main results to the nonlinear case are provided

symbolic models for nonlinear control systems affected by disturbances in the last few years there has been a growing interest in the use of symbolic models for the formal verification and control design of purely continuous or hybrid systems symbolic models are abstract descriptions of continuous systems where one symbol corresponds to an aggregate of continuous states in this article we face the problem of deriving symbolic models for nonlinear control systems affected by disturbances the main contribution of this article is in proposing symbolic models that can be effectively constructed and that approximate nonlinear control systems affected by disturbances in the sense of alternating approximate bisimulation

layer potential methods for elliptic homogenization problems in this paper we use the method of layer potentials to study l 2 boundary value problems in a bounded lipschitz domain for a family of second order elliptic systems with rapidly oscillating periodic coefficients defining l e div a e 1 x under the assumption that a x is elliptic symmetric periodic and holder continuous we establish the solvability of the l 2 dirichlet regularity and neumann problems for l e u e 0 in with optimal estimates uniform in e 0

a new approach to absolute continuity of elliptic measure with applications to non symmetric equations in the late 1950s and early 1960s the work of de giorgi degi and nash n and then moser mo initiated the study of regularity of solutions to divergence form elliptic equations with merely bounded measurable coefficients weak solutions in a domain 0 a priori only in a sobolev space w 1 loc 0 were shown to be ho lder continuous of some order depending just on ellipticity and maximum principles and harnack inequalities were established the dirichlet problem for such operators with continuous data on the boundary was established in lsw this in turn paved the way for a more systematic and detailed study of the properties of the elliptic measures d l associated to l div a on a domain 0 the classical properties of existence of non tangential limits of solutions fatou type theorems and comparison principles appeared in cfms but owed a great deal to the earlier work of carleson ca and hunt and wheeden h w on harmonic functions in lipschitz domains all the results mentioned above were carried out for elliptic operators l div a where the matrix a aij has bounded measurable coefficients and is symmetric however it turns out that the symmetry of the matrix is not needed to get these results morrey mor first observed this in connection with the de giorgi nash moser theory for the results in cfms this fact has not been formally observed until now with appropriate reformulation in terms of adjoint operators and adjoint green s functions the results of cfms are valid without the symmetry assumption see section 1 doi 10 1006 aima 1999 1899 available online at http www idealibrary com on

boundary estimates in homogenization of systems of elasticity on lipschitz domains l p theory abstract the primary purpose of this paper is to investigate a family of elliptic systems of linear elasticity with rapidly oscillating periodic coefficients arising in the theory of homogenization in lipschitz domains as a consequence for d 4 we prove that the l p neumann and l p dirichlet boundary value problems for systems of second order linear elasticity are uniquely solvable for 2 d 1 d 1 p 2 and 2 p 2 d 1 d 3 respectively

the lp boundary value problems on lipschitz domains abstract let be a bounded lipschitz domain in r n we develop a new approach to the invertibility on l p of the layer potentials associated with elliptic equations and systems in as a consequence for n 4 and 2 n 1 n 1 e p 2 where e 0 depends on we obtain the solvability of the l p neumann type boundary value problems for second order elliptic systems the analogous results for the biharmonic equation are also established

asymptotically optimal estimation for the reduced second moment measure of point processes summary a modification of the rigid motion correction estimator for the reduced second moment measure of a stationary isotropic point process is shown to possess an asymptotic optimality property as the observation region grows furthermore a modification of the translation correction estimator for stationary anisotropic point processes has a similar defined as k t a e number of other points within t of x point at x where t 0 xe rd for a stationary poisson process in two dimensions k t xt so deviations of k t from irt2 are one way of describing deviations from the poisson model to estimate k t based on observing n in some bounded region a c rd we need to account for the fact that for points within t of the boundary of a we do not get to observe the number of other points within t of them ripley 1988 ch 3 and stein 1991 describe various estimators that correct for this edge effect while the estimator suggested by stein 1991 appears to work fairly well its motivation is somewhat ad hoc this paper describes how to construct an estimator motivated more clearly by the goal of attempting to minimize the mean squared error of the estimator when n is poisson subject to an unbiasedness constraint furthermore among a large class of estimators the estimator developed here is proven to come very close to minimizing the mean squared error when a is large and n is poisson the estimator is obtained by applying the haijek projection lemma hajek 1968 lemma 4 1 to the rigid motion correction due to miles 1974 and ohser stoyan 1981 the same approach also works for stationary but possibly anisotropic processes in which case the recommended estimator is obtained by applying the hajek projection lemma to the translation correction due to ohser stoyan 1981

symbolic models for stochastic switched systems stochastic switched systems are a relevant class of stochastic hybrid systems with probabilistic evolution over a continuous domain and control dependent discrete dynamics over a finite set of modes in the past few years several different techniques have been developed to assist in the stability analysis of stochastic switched systems however more complex and challenging objectives related to the verification of and the controller synthesis for logic specifications have not been formally investigated for this class of systems as of yet with logic specifications we mean properties expressed as formulae in linear temporal logic or as automata on infinite strings this paper addresses these complex objectives by constructively deriving approximately equivalent bisimilar symbolic models of stochastic switched systems more precisely this paper provides two different symbolic abstraction techniques one requires state space discretization but the other one does not require any space discretization which can be potentially more efficient than the first one when dealing with higher dimensional stochastic switched systems both techniques provide finite symbolic models that are approximately bisimilar to stochastic switched systems under some stability assumptions on the concrete model this allows formally synthesizing controllers switching signals that are valid for the concrete system over the finite symbolic model by means of mature automata theoretic techniques in the literature the effectiveness of the results are illustrated by synthesizing switching signals enforcing logic specifications for two case studies including temperature control of a six room building

necessary and sufficient conditions for the solvability of the l p dirichlet problem on lipschitz domains we study the homogeneous elliptic systems of order 2 ell with real constant coefficients on lipschitz domains in mathbb r n n ge 4 for any fixed p 2 we show that a reverse holder condition with exponent p is necessary and sufficient for the solvability of the dirichlet problem with boundary data in l p we also obtain a simple sufficient condition as a consequence we establish the solvability of the l p dirichlet problem for n ge 4 and 2 epsilon p frac 2 n 1 n 3 epsilon the range of p is known to be sharp if ell ge 2 and 4 le n le 2 ell 1 for the polyharmonic equation the sharp range of p is also found in the case n 6 7 if ell 2 and n 2 ell 2 if ell ge 3

hierarchical hybrid control systems a lattice theoretic formulation the notion of dynamical consistency is extended to hybrid systems so as to define the set of dynamically consistent hybrid partition machines associated with a continuous system s following the formulation of the notions of between block and in block controllable hybrid partition machines the lattice hibc s of hybrid in block controllable partition machines is defined and investigated based on these constructions the definition and properties of hierarchical hybrid control systems are presented together with an example

approximately bisimilar symbolic models for randomly switched stochastic systems in the past few years there has been a growing interest in the use of symbolic models for control systems the main reason is the possibility to leverage algorithmic techniques over symbolic models to synthesize controllers that are valid for the concrete control systems such controllers can enforce complex logical specifications that are otherwise hard if not impossible to establish on the concrete models with classical control techniques examples of such specifications include those expressible via linear temporal logic or as automata on infinite strings a relevant goal in this research line is in the identification of classes of systems that admit symbolic models in particular continuous time systems with stochastic or hybrid dynamics have been only recently considered due to their rather general and complex dynamics in this work we make progress in this direction by enlarging the class of stochastic hybrid systems admitting finite symbolic models specifically we show that randomly switched stochastic systems satisfying some incremental stability assumption admit such models

boundary korn inequality and neumann problems in homogenization of systems of elasticity this paper is concerned with a family of elliptic systems of linear elasticity with rapidly oscillating periodic coefficients arising in the theory of homogenization we establish uniform optimal regularity estimates for solutions of neumann problems in a bounded lipschitz domain with l2 boundary data the proof relies on a boundary korn inequality for solutions of systems of linear elasticity and uses a large scale rellich estimate obtained in shen anal pde arxiv 1505 00694v2

bayesian semiparametric intensity estimation for inhomogeneous spatial point processes summary in this work we propose a fully bayesian semiparametric method to estimate the intensity of an inhomogeneous spatial point process the basic idea is to first convert intensity estimation into a poisson regression setting via binning data points on a regular grid and then model the log intensity semiparametrically using an adaptive version of gaussian markov random fields to smooth the corresponding counts the inference is carried by an efficient markov chain monte carlo simulation algorithm compared to existing methods for intensity estimation for example parametric modeling and kernel smoothing the proposed estimator not only provides inference regarding the dependence of the intensity function on possible covariates but also uses information from the data to adaptively determine the amount of smoothing at the local level the effectiveness of using our method is demonstrated through simulation studies and an application to a rainforest dataset

forward completeness unboundedness observability and their lyapunov characterizations a finite dimensional continuous time system is forward complete if solutions exist globally for positive time this paper shows that forward completeness can be characterized in a necessary and sufficient manner by means of smooth scalar growth inequalities moreover a version of this fact is also proved for systems with inputs and a generalization is also provided for systems with outputs and a notion unboundedness observability of relative completeness we apply these results to obtain a bound on reachable states in terms of energy like estimates of inputs