central limit theorem for linear eigenvalue statistics of random matrices with independent entries we consider n x n real symmetric and hermitian wigner random matrices n 1 2 w with independent modulo symmetry condition entries and the null sample covariance matrices n 1 x x with independent entries of m x n matrix x assuming first that the 4th cumulant excess 4 of entries of w and x is zero and that their 4th moments satisfy a lindeberg type condition we prove that linear statistics of eigenvalues of the above matrices satisfy the central limit theorem clt as n m m n c 0 oo with the same variance as for gaussian matrices if the test functions of statistics are smooth enough essentially of the class c 5 this is done by using a simple interpolation trick from the known results for the gaussian matrices and the integration by parts presented in the form of certain differentiation formulas then by using a more elaborated version of the techniques we prove the clt in the case of nonzero excess of entries again for essentially 5 test function here the variance of statistics contains an additional term proportional to 4 the proofs of all limit theorems follow essentially the same scheme

domination by product measures we consider families of 0 1 valued random variables indexed by the vertices of countable graphs with bounded degree first we show that if these random variables satisfy the property that conditioned on what happens outside of the neighborhood of each given site the probability of seeing a 1 at this site is at least a value p which is large enough then this random field dominates a product measure with positive density moreover the density of this dominated product measure can be made arbitrarily close to 1 provided that p is close enough to 1 next we address the issue of obtaining the critical value of p defined as the threshold above which the domination by positive density product measures is assured for the graphs which have as vertices the integers and edges connecting vertices which are separated by no more than k units this critical value is shown to be 1 k k k 1 k 1 and a discontinuous transition is shown to occur similar critical values of p are found for other classes of probability measures on 0 1 z for the class of k dependent measures the critical value is again 1 k k k 1 k 1 with a discontinuous transition for the class of two block factors the critical value is shown to be 1 2 and a continuous transition is shown to take place in this case thus both the critical value and the nature of the transition are different in the two block factor and 1 dependent cases

patterns in eigenvalues the 70th josiah willard gibbs lecture typical large unitary matrices show remarkable patterns in their eigenvalue distribution these same patterns appear in telephone encryption the zeros of riemann s zeta function a variety of physics problems and in the study of toeplitz operators this paper surveys these applications and what is currently known about the patterns

multirate sampling and input to state stable receding horizon control for nonlinear systems abstract in this paper a robust receding horizon control for multirate sampled data nonlinear systems with bounded disturbances is presented the proposed receding horizon control is based on the solution of bolza type optimal control problems for the approximate discrete time model of the nominal system low measurement rate is assumed it is shown that the multistep receding horizon controller that stabilizes the nominal approximate discrete time model also practically input to state stabilizes the exact discrete time system with disturbances

level spacings distribution for large random matrices gaussian fluctuations we study the level spacings distribution for eigenvalues of large n x n matrices from the classical compact groups in the scaling limit when the mean distance between nearest eigenvalues equals 1 defining by 77n s the number of nearest neighbors spacings greater than s 0 smaller than s 0 we prove functional limit theorem for the process 77n s e n s n1 2 giving weak convergence of this distribution to some gaussian random process on 0 x the limiting gaussian random process is universal for all classical compact groups it is holder continuous with any exponent less than 1 2 similar results can be obtained for the n level spacings distribution

clt for linear spectral statistics of large dimensional sample covariance matrices let bn 1 n t1 2nxnx nt1 2n where xn xij is n n with i i d complex standardized entries having finite fourth moment and t1 2n is a hermitian square root of the nonnegative definite hermitian matrix tn the limiting behavior as n with n n approaching a positive constant of functionals of the eigenvalues of bn where each is given equal weight is studied due to the limiting behavior of the empirical spectral distribution of bn it is known that these linear spectral statistics converges a s to a nonrandom quantity this paper shows their rate of convergence to be 1 n by proving after proper scaling that they form a tight sequence moreover if exppx211 0 and expp x11 4 2 or if x11 and tn are real and exppx411 3 they are shown to have gaussian limits

the fourier spectrum of critical percolation consider the indicator function f of a 2 dimensional percolation crossing event in this paper the fourier transform of f is studied and sharp bounds are obtained for its lower tail in several situations various applications of these bounds are derived in particular we show that the set of exceptional times of dynamical critical site percolation on the triangular grid in which the origin percolates has dimension frac 31 36 almost surely and the corresponding dimension in the half plane is frac 5 9 it is also proved that critical bond percolation on the square grid has exceptional times almost surely also the asymptotics of the number of sites that need to be resampled in order to significantly perturb the global percolation configuration in a large square is determined

changing supply rates for input output to state stable discrete time nonlinear systems with applications we present results on changing supply rates for input output to state stable discrete time nonlinear systems our results can be used to combine two lyapunov functions none of which can be used to verify that the system has a certain property into a new composite lyapunov function from which the property of interest can be concluded the results are stated for parameterized families of discrete time systems that naturally arise when an approximate discrete time model is used to design a controller for a sampled data system we present several applications of our results i a lasalle criterion for input to state stability iss of discrete time systems ii constructing iss lyapunov functions for time varying discrete time cascaded systems iii testing iss of discrete time systems using positive semidefinite lyapunov functions iv observer based input to state stabilization of discrete time systems our results are exploited in a case study of a two link manipulator and some simulation results that illustrate advantages of our approach are presented

on uniform asymptotic stability of time varying parameterized discrete time cascades recently a framework for controller design of sampled data nonlinear systems via their approximate discrete time models has been proposed in the literature in this paper we develop novel tools that can be used within this framework and that are useful for tracking problems in particular results for stability analysis of parameterized time varying discrete time cascaded systems are given this class of models arises naturally when one uses an approximate discrete time model to design a stabilizing or tracking controller for a sampled data plant while some of our results parallel their continuous time counterparts the stability properties that are considered the conditions that are imposed and the the proof techniques that are used are tailored for approximate discrete time systems and are technically different from those in the continuous time context a result on constructing strict lyapunov functions from nonstrict ones that is of independent interest is also presented we illustrate the utility of our results in the case study of the tracking control of a mobile robot this application is fairly illustrative of the technical differences and obstacles encountered in the analysis of discrete time parameterized systems

exceptional planes of percolation consider the standard continuous percolation in 4 and choose the parameters so that the induced percolation on a fixed two dimensional linear subspace is critical although two dimensional critical percolation dies we show that there are exceptional two dimensional linear subspaces in which percolation occurs

optimization based stabilization of sampled data nonlinear systems via their approximate discrete time models we present results on numerical regulator design for sampled data nonlinear plants via their approximate discrete time plant models the regulator design is based on an approximate discrete time plant model and is carried out either via an infinite horizon optimization problem or via a finite horizon with terminal cost optimization problem in both cases we discuss situations when the sampling period t and the integration period h used in obtaining the approximate discrete time plant model are the same or they are independent of each other we show that using this approach practical and or semiglobal stability of the exact discrete time model is achieved under appropriate conditions

regularity conditions in the clt for linear eigenvalue statistics of wigner matrices abstract we show that the variance of centred linear statistics of eigenvalues of gue matrices remains bounded for large n for some classes of test functions less regular than lipschitz functions this observation is suggested by the limiting form of the variance which has previously been computed explicitly but it does not seem to appear in the literature we combine this fact with comparison techniques following tao vu and erdos yau et al and a littlewood paley type decomposition to extend the central limit theorem for linear eigenvalue statistics to functions in the holder class c 1 2 in the case of matrices of gaussian convolution type we also give a variance bound which implies the clt for test functions in the sobolev space h 1 and c 1 for general wigner matrices satisfying moment conditions if the additional assumption of the test function being supported away from the edge of the spectrum is made we prove the clt for test functions of regularity h 1 2 l and h 1 2 for gue and johansson matrices respectively previous results on the clt impose the existence and continuity of at least one classical derivative

survey constrained model predictive control stability and optimality model predictive control is a form of control in which the current control action is obtained by solving at each sampling instant a finite horizon open loop optimal control problem using the current state of the plant as the initial state the optimization yields an optimal control sequence and the first control in this sequence is applied to the plant an important advantage of this type of control is its ability to cope with hard constraints on controls and states it has therefore been widely applied in petro chemical and related industries where satisfaction of constraints is particularly important because efficiency demands operating points on or close to the boundary of the set of admissible states and controls in this review we focus on model predictive control of constrained systems both linear and nonlinear and discuss only briefly model predictive control of unconstrained nonlinear and or time varying systems we concentrate our attention on research dealing with stability and optimality in these areas the subject has developed in our opinion to a stage where it has achieved sufficient maturity to warrant the active interest of researchers in nonlinear control we distill from an extensive literature essential principles that ensure stability and use these to present a concise characterization of most of the model predictive controllers that have been proposed in the literature in some cases the finite horizon optimal control problem solved on line is exactly equivalent to the same problem with an infinite horizon in other cases it is equivalent to a modified infinite horizon optimal control problem in both situations known advantages of infinite horizon optimal control accrue

exclusion sensitivity of boolean functions recently the study of noise sensitivity and noise stability of boolean functions has received considerable attention the purpose of this paper is to extend these notions in a natural way to a different class of perturbations namely those arising from running the symmetric exclusion process for a short amount of time in this study the case of monotone boolean functions will turn out to be of particular interest we show that for this class of functions ordinary noise sensitivity and noise sensitivity with respect to the complete graph exclusion process are equivalent we also show this equivalence with respect to stability after obtaining these fairly general results we study exclusion sensitivity of critical percolation in more detail with respect to medium range dynamics the exclusion dynamics due to its conservative nature is in some sense more physical than the classical i i d dynamics interestingly we will see that in order to obtain a precise understanding of the exclusion sensitivity of percolation we will need to describe how typical spectral sets of percolation diffuse under the underlying exclusion process

unconstrained receding horizon control of nonlinear systems it is well known that unconstrained infinite horizon optimal control may be used to construct a stabilizing controller for a nonlinear system we show that similar stabilization results may be achieved using unconstrained finite horizon optimal control the key idea is to approximate the tail of the infinite horizon cost to go using as terminal cost an appropriate control lyapunov function roughly speaking the terminal control lyapunov function clf should provide an incremental upper bound on the cost in this fashion important stability characteristics may be retained without the use of terminal constraints such as those employed by a number of other researchers the absence of constraints allows a significant speedup in computation furthermore it is shown that in order to guarantee stability it suffices to satisfy an improvement property thereby relaxing the requirement that truly optimal trajectories be found we provide a complete analysis of the stability and region of attraction operation properties of receding horizon control strategies that utilize finite horizon approximations in the proposed class it is shown that the guaranteed region of operation contains that of the clf controller and may be made as large as desired by increasing the optimization horizon restricted of course to the infinite horizon domain moreover it is easily seen that both clf and infinite horizon optimal control approaches are limiting cases of our receding horizon strategy the key results are illustrated using a familiar example the inverted pendulum where significant improvements in guaranteed region of operation and cost are noted

receding horizon control via bolza type optimization this paper deals with the stabilization problem of nonlinear control systems a variant of the receding horizon control method is proposed which is based on the solution of a certain bolza problem the stabilizing property of the method is proved in the case of lipschitz continuous value function this version of receding horizon scheme can be considered as a straightforward generalization of the so called fake riccati equation technique for nonlinear constrained systems

oded schramm s contributions to noise sensitivity we survey in this paper the main contributions of oded schramm related to noise sensitivity we will describe in particular his various works which focused on the spectral analysis of critical percolation and more generally of boolean functions his work on the shape fluctuations of first passage percolation and finally his contributions to the model of dynamical percolation

stabilization of sampled data nonlinear systems by receding horizon control via discrete time approximations abstract results on stabilizing receding horizon control of sampled data nonlinear systems via their approximate discrete time models are presented the proposed receding horizon control is based on the solution of bolza type optimal control problems for the parametrized family of approximate discrete time models this paper investigates both situations when the sampling period t is fixed and the integration parameter h used in obtaining approximate model can be chosen arbitrarily small and when these two parameters coincide but they can be adjusted arbitrary sufficient conditions are established which guarantee that the controller that renders the origin to be asymptotically stable for the approximate model also stabilizes the exact discrete time model for sufficiently small integration and or sampling parameters

receding horizon control of nonlinear systems the receding horizon control strategy provides a relatively simple method for determining feedback control for linear or nonlinear systems the method is especially useful for the control of slow nonlinear systems such as chemical batch processes where it is possible to solve sequentially open loop fixed horizon optimal control problems online the method has been shown to yield a stable closed loop system when applied to time invariant or time varying linear systems it is shown that the method also yields a stable closed loop system when applied to nonlinear systems

the nevai condition and a local law of large numbers for orthogonal polynomial ensembles we consider asymptotics of orthogonal polynomial ensembles in the macroscopic and mesoscopic scales we prove both global and local laws of large numbers analogous to the recently proven local semicircle law for wigner matrices under fairly weak conditions on the underlying measure our main tools are a general concentration inequality for determinantal point processes with a kernel that is a selfadjoint projection and a strengthening of the nevai condition from the theory of orthogonal polynomials

stabilizing receding horizon control of nonlinear time varying systems a receding horizon control scheme for nonlinear time varying systems is proposed which is based on a finite horizon optimization problem with a terminal state penalty the penalty is equal to the cost that would be incurred over an infinite horizon by applying a locally stabilizing linear control law to the nonlinear system assuming only stabilizability of the linearized system around the desired equilibrium the new scheme ensures exponential stability of the equilibrium as the length of the optimization horizon goes from zero to infinity the domain of attraction moves from the basin of attraction of the linear controller toward the basin of attraction of the infinite horizon nonlinear controller stability robustness in the face of system perturbations is also established

solution of ordinary differential equations via nonlocal transformations the classical lie analysis provides a useful technique for the solution of ordinary differential equations via point symmetries transformations unfortunately the requirement that an nth order equation possesses at least an n dimensional solvable lie algebra of symmetries is only satisfied by a select number of equations we provide a class of second order ordinary differential equations with less than the required two lie point symmetries that can be solved via nonlocal transformations

limit theorems for linear spectrum statistics of orthogonal polynomial ensembles and their applications in random matrix theory in this paper we consider the asymptotic behavior of xfn n i 1nfn xi where xi i 1 n form orthogonal polynomial ensembles and fn is a real valued bounded measurable function under the condition that varxfn n the berry esseen be bound and cramer type moderate deviation principle mdp for xfn n are obtained by using the method of cumulants as two applications we establish the be bound and cramer type mdp for linear spectrum statistics of wigner matrix and sample covariance matrix in the complex cases these results show that in the edge case which means fn has a particular form f x i x n where n is close to the right edge of equilibrium measure and f is a smooth function xfn n behaves like the eigenvalues counting function of the corresponding wigner matrix and sample covariance matrix respectively

dynamical models for circle covering brownian motion and poisson updating we consider two dynamical variants of dvoretzky s classical problem of random interval coverings of the unit circle the latter having been completely solved by l shepp in the first model the centers of the intervals perform independent brownian motions and in the second model the positions of the intervals are updated according to independent poisson processes where an interval of length l is updated at rate l where 0 is a parameter for the model with brownian motions a special case of our results is that if the length of the nth interval is c n then there are times at which a fixed point is not covered if and only if c 2 and there are times at which the circle is not fully covered if and only if c 3 for the poisson updating model we obtain analogous results with c and c 1 instead we also compute the hausdorff dimension of the set of exceptional times for some of these questions

on stability of sets for sampled data nonlinear inclusions via their approximate discrete time models and summability criteria this paper consists of two main parts in the first part we provide a framework for stabilization of arbitrary not necessarily compact closed sets for sampled data nonlinear differential inclusions via their approximate discrete time models we generalize d nesic a r teel and p v kokotovic systems control lett 38 1999 pp 259 270 theorem 1 in several different directions we consider stabilization of arbitrary closed sets plants described as sampled data differential inclusions and arbitrary dynamic controllers in the form of difference inclusions our result does not require the knowledge of a lyapunov function for the approximate model which is a standing assumption in d nesic and a r teel ieee trans automat control 49 2004 pp 1103 1122 and d nesic a r teel and p v kokotovic systems control lett 38 1999 pp 259 270 theorem 2 we present checkable conditions that one can use to conclude semiglobal asymptotic stability or global exponential stability ges of the sampled data system via appropriate properties of its approximate discrete time model in the second part we present sufficient conditions for stability of parameterized difference inclusions that involve various summability criteria on trajectories of the system to conclude global asymptotic stability gas or ges and they represent discrete time counterparts of results given in a r teel e panteley and a loria math control signals systems 15 2002 pp 177 201 these summability criteria are not lyapunov based and they are tailored to be used within our above mentioned framework for stabilization of sampled data differential inclusions via their approximate discrete time models we believe that these tools will be a useful addition to the toolbox for controller design for sampled data nonlinear systems via their approximate discrete time models

sufficient conditions for stabilization of sampled data nonlinear systems via discrete time approximations given a parameterized by sampling period t family of approximate discrete time models of a sampled nonlinear plant and given a family of controllers stabilizing the family of plant models for all t sufficiently small we present conditions which guarantee that the same family of controllers semi globally practically stabilizes the exact discrete time model of the plant for sufficiently small sampling periods when the family of controllers is locally bounded uniformly in the sampling period the inter sample behavior can also be uniformly bounded so that the time varying sampled data model of the plant is uniformly semi globally practically stabilized the result justifies controller design for sampled data nonlinear systems based on the approximate discrete time model of the system when sampling is sufficiently fast and the conditions we propose are satisfied our analysis is applicable to a wide range of time discretization schemes and general plant models

generic behavior of the density of states in random matrix theory and equilibrium problems in the presence of real analytic external fields the equilibrium measure in the presence of an external field plays a role in a number of areas in analysis for example in random matrix theory the limiting mean density of eigenvalues is precisely the density of the equilibrium measure typical behavior for the equilibrium measure is 1 it is positive on the interior of a finite number of intervals 2 it vanishes like a square root at endpoints and 3 outside the support there is strict inequality in the euler lagrange variational conditions if these conditions hold then the limiting local eigenvalue statistics is loosely described by a bulk in which there is universal behavior involving the sine kernel and edge effects in which there is a universal behavior involving the airy kernel through techniques from potential theory and integrable systems we show that this regular behavior is generic for equilibrium measures associated with real analytic external fields in particular we show that for any one parameter family of external fields v c the equilibrium measure exhibits this regular behavior except for an at most countable number of values of c we discuss applications of our results to random matrices orthogonal polynomials and integrable systems 2000 john wiley sons inc

local time on the exceptional set of dynamical percolation and the incipient infinite cluster in dynamical critical site percolation on the triangular lattice or bond percolation on z2 we define and study a local time measure on the exceptional times at which the origin is in an infinite cluster we show that at a typical time with respect to this measure the percolation configuration has the law of kesten s incipient infinite cluster in the most technical result of this paper we show that on the other hand at the first exceptional time the law of the configuration is different we believe that the two laws are mutually singular but do not show this we also study the collapse of the infinite cluster near typical exceptional times and establish a relation between static and dynamic exponents analogous to kesten s near critical relation

which properties of a random sequence are dynamically sensitive consider a sequence of i i d random variables where each variable is refreshed i e replaced by an independent variable with the same law independently according to a poisson clock at any fixed time t the resulting sequence has the same law as at time 0 but there can be exceptional random times at which certain almost sure properties of the time 0 sequence are violated we prove that there are no such exceptional times for the law of large numbers and the law of the iterated logarithm so these laws are dynamically stable however there are times at which run lengths are exceptionally long that is run tests are dynamically sensitive we obtain a multifractal analysis of exceptional times for run lengths and for prediction in particular starting from an i i d sequence of unbiased random bits the random set of times t where log2 n bits among the first n bits can be predicted from their predecessors has hausdorff dimension 1 a s finally we consider simple random walk in the lattice zd and prove that transience is dynamically stable for d 5 and dynamically sensitive for d 3 4 moreover for d 3 4 the nonempty random set of exceptional times t where the walk is recurrent has hausdorff dimension 4 d 2 a s

a general framework to design stabilizing nonlinear model predictive controllers abstract we propose a new model predictive control mpc framework to generate feedback controls for time varying nonlinear systems with input constraints we provide a set of conditions on the design parameters that permits to verify a priori the stabilizing properties of the control strategies considered the supplied sufficient conditions for stability can also be used to analyse the stability of most previous mpc schemes the class of nonlinear systems addressed is significantly enlarged by removing the traditional assumptions on the continuity of the optimal controls and on the stabilizability of the linearized system some important classes of nonlinear systems including some nonholonomic systems can now be stabilized by mpc in addition we can exploit increased flexibility in the choice of design parameters to reduce the constraints of the optimal control problem and thereby reduce the computational effort in the optimization algorithms used to implement mpc

a framework for stabilization of nonlinear sampled data systems based on their approximate discrete time models a unified framework for design of stabilizing controllers for sampled data differential inclusions via their approximate discrete time models is presented both fixed and fast sampling are considered in each case sufficient conditions are presented which guarantee that the controller that stabilizes a family of approximate discrete time plant models also stabilizes the exact discrete time plant model for sufficiently small integration and or sampling periods previous results in the literature are extended to cover 1 continuous time plants modeled as differential inclusions 2 general approximate discrete time plant models 3 dynamical discontinuous controllers modeled as difference inclusions and 4 stability with respect to closed arbitrary not necessarily compact sets

fluctuations of linear eigenvalues statistics for wigner matrices edge case in this note we consider the fluctuation theorem for x f n n sum i 1 n f lambda i i lambda i ge theta n where lambda i i 1 ldots n are eigenvalues from a wigner matrix and theta n rightarrow 2 we prove that in the edge case x f n n behaves like the counting function of wigner matrix our results can be viewed as a complement of bao et al j stat phys 150 1 88 129 2013

limiting laws of linear eigenvalue statistics for hermitian matrix models we study the variance and the laplace transform of the probability law of linear eigenvalue statistics of unitary invariant matrix models of n n hermitian matrices as n assuming that the test function of statistics is smooth enough and using the asymptotic formulas by deift et al commun pure appl math 52 1325 1425 1999 for orthogonal polynomials with varying weights we show first that if the support of the density of states of the model consists of q 2 intervals then in the global regime the variance of statistics is a quasiperiodic function of n as n generically in the potential determining the model we show next that the exponent of the laplace transform of the probability law is not in general 1 2 variance as it should be if the central limit theorem would be valid and we find the asymptotic form of the laplace transform of the probability law in certain cases

upper bounds on fourier entropy given a function f 0 1 n rightarrow mathbb r its fourier entropy is defined to be sum s widehat f 2 s log widehat f 2 s where hat f denotes the fourier transform of f this quantity arises in a number of applications especially in the study of boolean functions an outstanding open question is a conjecture of friedgut and kalai 1996 called the fourier entropy influence fei conjecture asserting that the fourier entropy of any boolean function f is bounded above up to a constant factor by the total influence average sensitivity of f

exceptional times and invariance for dynamical random walks consider a sequence open image in new window of i i d random variables associate to each xi 0 an independent mean one poisson clock every time a clock rings replace that x variable by an independent copy and restart the clock in this way we obtain i i d stationary processes xi t t 0 i 1 2 whose invariant distribution is the law of x1 0

hiv dynamics analysis and robust multirate mpc based treatment schedules abstract analysis and control of human immunodeficiency virus hiv infection have attracted the interests of mathematicians and control engineers during the recent years several mathematical models exist and adequately explain the interaction of the hiv infection and the immune system up to the stage of clinical latency as well as viral suppression and immune system recovery after treatment therapy however none of these models can completely exhibit all that is observed clinically and account the full course of infection besides model inaccuracies that hiv models suffer from some disturbances uncertainties from different sources may arise in the modelling in this paper we study the basic properties of a 6 dimensional hiv model that describes the interaction of hiv with two target cells cd4 t cells and macrophages the disturbances are modelled in the hiv model as additive bounded disturbances highly active antiretroviral therapy haart is used the control input is defined to be dependent on the drug dose and drug efficiency we developed treatment schedules for hiv infected patients by using robust multirate model predictive control mpc based method the mpc is constructed on the basis of the approximate discrete time model of the nominal model we established a set of conditions which guarantee that the multirate mpc practically stabilizes the exact discrete time model with disturbances the proposed method is applied to the stabilization of the uninfected steady state of the hiv model the results of simulations show that after initiation of haart with a strong dosage the viral load drops quickly and it can be kept under a suitable level with mild dosage of haart moreover the immune system is recovered with some fluctuations due to the presence of disturbances

dynamical stability of percolation for some interacting particle systems and epsilon stability in this paper we will investigate dynamic stability of percolation for the stochastic ising model and the contact process we also introduce the notion of downwards and upwards epsilon stability which will be a key tool for our analysis

on the 1 n expansion for some unitary invariant ensembles of random matrices we present a version of the 1 n expansion for random matrix ensembles known as matrix models the case where the support of the density of states of an ensemble consists of one interval and the case where the density of states is even and its support consists of two symmetric intervals is treated in these cases we construct the expansion scheme for the jacobi matrix determining a large class of expectations of symmetric functions of eigenvalues of random matrices prove the asymptotic character of the scheme and give an explicit form of the first two terms this allows us in particular to clarify certain theoretical physics results on the variance of the normalized traces of the resolvent of random matrices we also find the asymptotic form of several related objects such as smoothed squares of certain orthogonal polynomials the normalized trace and the matrix elements of the resolvent of the jacobi matrices etc

on the statistical mechanics approach in the random matrix theory integrated density of states we consider the ensemble of random symmetricn n matrices specified by an orthogonal invariant probability distribution we treat this distribution as a gibbs measure of a mean field type model this allows us to show that the normalized eigenvalue counting function of this ensemble converges in probability to a nonrandom limit asn and that this limiting distribution is the solution of a certain self consistent equation

asymptotic properties of receding horizon optimal control problems the asymptotic behavior of receding horizon optimal control problems with terminal cost chosen as a control liapunov function is analyzed for regulator as well as disturbance attenuation problems both the continuous as well as the discrete time cases are treated further the approximation of the continuous time optimal control problem by the discrete time receding horizon problems is studied

the central limit theorem for local linear statistics in classical compact groups and related combinatorial identities author s soshnikov alexander abstract we discuss clt for the global and local linear statistics of random matrices from classical compact groups the main part of our proofs are certain combinatorial identities much in the spirit of works by kac and spohn

a quasi infinite horizon nonlinear model predictive control scheme with guaranteed stability we present in this paper a novel nonlinear model predictive control scheme that guarantees asymptotic closed loop stability the scheme can be applied to both stable and unstable systems with input constraints the objective functional to be minimized consists of an integral square error ise part over a finite time horizon plus a quadratic terminal cost the terminal state penalty matrix of the terminal cost term has to be chosen as the solution of an appropriate lyapunov equation furthermore the setup includes a terminal inequality constraint that forces the states at the end of the finite prediction horizon to lie within a prescribed terminal region if the jacobian linearization of the nonlinear system to be controlled is stabilizable we prove that feasibility of the open loop optimal control problem at time t 0 implies asymptotic stability of the closed loop system the size of the region of attraction is only restricted by the requirement for feasibility of the optimization problem due to the input and terminal inequality constraints and is thus maximal in some sense

constant depth circuits fourier transform and learnability in this paper boolean functions in 4c0 are studied using harmonic analysis on the cube the main result is that an aco boolean function has almost all of its power spectrum on the low order coefficients an important ingredient of the proof is hastad s switching lemma 8 this result implies several new properties of functions in 4c functions in ac have low average sensitivity they may be approximated well by a real polynomial of low degree and they cannot be pseudorandom function generators perhaps the most interesting application is an o n poiyiog n time algorithm for learning func tions in aco the algorithm observes the behavior of an ac function on o npo y og n randomly chosen inputs and derives a good approximation for the fourier transform of the function this approximation allows the algorithm to predict with high probability the value of the function on other randomly chosen inputs

formulas relating kl stability estimates of discrete time and sampled data nonlinear systems abstract we provide an explicit kl stability or input to state stability iss estimate for a sampled data nonlinear system in terms of the kl estimate for the corresponding discrete time system and a k function describing inter sample growth it is quite obvious that a uniform inter sample growth condition plus an iss property for the exact discrete time model of a closed loop system implies uniform iss of the sampled data nonlinear system our results serve to quantify these facts by means of comparison functions our results can be used as an alternative to prove and extend results in 1 or extend some results in 4 to a class of nonlinear systems finally the formulas we establish can be used as a tool for some other problems which we indicate

universality of the local eigenvalue statistics for a class of unitary invariant random matrix ensembles this paper is devoted to the rigorous proof of the universality conjecture of random matrix theory according to which the limiting eigenvalue statistics ofn n random matrices within spectral intervals ofo n 1 is determined by the type of matrix real symmetric hermitian or quaternion real and by the density of states we prove this conjecture for a certain class of the hermitian matrix ensembles that arise in the quantum field theory and have the unitary invariant distribution defined by a certain function the potential in the quantum field theory satisfying some regularity conditions

quenched voronoi percolation we prove that the probability of crossing a large square in quenched voronoi percolation converges to 1 2 at criticality confirming a conjecture of benjamini kalai and schramm from 1999 the main new tools are a quenched version of the box crossing property for voronoi percolation at criticality and an efron stein type bound on the variance of the probability of the crossing event in terms of the sum of the squares of the influences as a corollary of the proof we moreover obtain that the quenched crossing event at criticality is almost surely noise sensitive

dynamical percolation on general trees haggstrom et al ann inst h poincare probab stat 33 4 497 528 1997 have introduced a dynamical version of percolation on a graph g when g is a tree they derived a necessary and sufficient condition for percolation to exist at some time t in the case that g is a spherically symmetric tree peres and steif in probab theory relat fields 111 1 141 165 1998 derived a necessary and sufficient condition for percolation to exist at some time t in a given target set d the main result of the present paper is a necessary and sufficient condition for the existence of percolation at some time t in d in the case that the underlying tree is not necessary spherically symmetric this answers a question of yuval peres personal communication we present also a formula for the hausdorff dimension of the set of exceptional times of percolation

semiclassical asymptotics of orthogonal polynomials riemann hilbert problem and universality in the matrix model we derive semiclassical asymptotics for the orthogonal polynomials pn z on the line with respect to the exponential weight exp inv z where v z is a double well quartic polynomial in the limit when n n 1 we assume that n n cri for some 0 where cr is the critical value which separates orthogonal polynomials with two cuts from the ones with one cut simultaneously we derive semiclassical asymptotics for the recursive coe cients of the orthogonal polynomials and we show that these coe cients form a cycle of period two which drifts slowly with the change of the ratio n n the proof of the semiclassical asymptotics is based on the methods of the theory of integrable systems and on the analysis of the appropriate matrix riemannhilbert problem as an application of the semiclassical asymptotics of the orthogonal polynomials we prove the universality of the local distribution of eigenvalues in the matrix model with the double well quartic interaction in the presence of two cuts contents

toward infinite horizon optimality in nonlinear model predictive control a new model predictive control scheme with guaranteed stability is presented for constrained discrete time nonlinear systems the open loop optimization problem does not involve explicit terminal constraints and employs a nonstandard terminal cost function the closed loop system is shown to be infinite horizon optimal provided that the terminal cost exactly captures the infinite horizon optimal value in a neighborhood of the origin stability and optimality are proven for a set of initial states which is invariant and approaches the set of all controllable states as the prediction horizon increases

on the scaling limits of planar percolation we prove tsirelson s conjecture that any scaling limit of the critical planar percolation is a black noise our theorems apply to a number of percolation models including site percolation on the triangular grid and any subsequential scaling limit of bond percolation on the square grid we also suggest a natural construction for the scaling limit of planar percolation and more generally of any discrete planar model describing connectivity properties

matrosov theorem for parameterized families of discrete time systems a version of matrosov s theorem for parameterized discrete time time varying systems is presented the theorem is a discrete time version of the continuous time result in loria et al 2002 d persistency of excitation a necessary and sufficient condition for uniform attractivity 2002 submitted for publication our result facilitates controller design for sampled data nonlinear systems via their approximate discrete time models an application of the theorem to establishing uniform asymptotic stability of systems controlled by model reference adaptive controllers designed via approximate discrete time plant models is presented

the incipient infinite cluster in two dimensional percolation letpp be the probability measure on the configurations of occupied and vacant vertices of a two dimensional graphg under which all vertices are independently occupied respectively vacant with probabilityp respectively 1 p letph be the critical probability for this system andw the occupied cluster of some fixed vertexw0 we show that for many graphsg such as mathbb z 2 or its covering graph which corresponds to bond percolation on mathbb z 2 the following two conditional probability measures converge and have the same limit v say

the ising magnetization exponent on z2 is 1 15 we prove that for the ising model defined on the plane mathbb z 2 at beta beta c the average magnetization under an external magnetic field h 0 behaves exactly like begin aligned langle sigma 0 rangle beta c h asymp h frac 1 15 end aligned the proof which is surprisingly simple compared to an analogous result for percolation i e that theta p p p c 5 36 o 1 on the triangular lattice kesten in commun math phys 109 1 109 156 1987 smirnov and werner in math res lett 8 5 6 729 744 2001 relies on the ghs inequality as well as the rsw theorem for fk percolation from duminil copin et al commun pure appl math 64 1165 1198 2011 the use of ghs to obtain inequalities involving critical exponents is not new in this paper we show how it can be combined with rsw to obtain matching upper and lower bounds for the average magnetization