the domain of attraction and the stability region for stochastic partial differential equations with delays abstract in this paper a stochastic partial differential equation with delays is considered on the basis of the properties of nonnegative matrices stochastic convolution and the inequality technique sufficient conditions for determining the domain of p th moment attraction and the p th moment asymptotic stability region are obtained an example is also discussed to illustrate the efficiency of the results obtained

two dimensional random interlacements and late points for random walks we define the model of two dimensional random interlacements us ing the simple random walk trajectories conditioned on never hitting the origin and then obtain some properties of this model also for the random walk on a large torus conditioned on not hitting the origin up to some time proportional to the mean cover time we show that the law of the vacant set around the origin is close to that of random interlacements at the corresponding level thus this new model pro vides a microscopic description of late points of the covering process

uniform asymptotic stability of abstract functional differential equations abstract this paper considers a class of abstract functional differential equations by using razumikhin technique we establish several stability criteria in terms of two measures which improve some earlier results found in the literature several examples are also worked out to demonstrate the advantages of our results

large deviations for occupation time profiles of random interlacements we derive a large deviation principle for the density profile of occupation times of random interlacements at a fixed level in a large box of mathbb z d d ge 3 as an application we analyze the asymptotic behavior of the probability that atypically high values of the density profile insulate a macroscopic body in a large box as a step in this program we obtain a similar large deviation principle for the occupation time measure of brownian interlacements at a fixed level in a large box of mathbb r d and we derive a new identity for the laplace transform of the occupation time measure which is based on the analysis of certain schrodinger semi groups

on pinned fields interlacements and random walk on mathbb z n mathbb z 2 we define two families of poissonian soups of bidirectional trajectories on mathbb z 2 which can be seen to adequately describe the local picture of the trace left by a random walk on the two dimensional torus mathbb z n mathbb z 2 started from the uniform distribution run up to a time of order n log n 2 and forced to avoid a fixed point the local limit of the latter was recently established in comets et al commun math phys 343 129 164 2016 our construction proceeds by considering somewhat in the spirit of statistical mechanics a sequence of finite volume approximations consisting of random walks avoiding the origin and killed at spatial scale n either using dirichlet boundary conditions or by means of a suitably adjusted mass by tuning the intensity u of such walks with n the occupation field can be seen to have a nontrivial limit corresponding to that of the actual random walk our construction thus yields a two dimensional analogue of the random interlacements model introduced in sznitman ann math 171 3 2039 2087 2010 in the transient case it also links it to the pinned free field in mathbb z 2 by means of a pinned ray knight type isomorphism theorem

quenched large deviations for simple random walks on percolation clusters including long range correlations we prove a quenched large deviation principle ldp for a simple random walk on a supercritical percolation cluster srwpc on mathbb z d d geq 2 the models under interest include classical bernoulli bond and site percolation as well as models that exhibit long range correlations like the random cluster model the random interlacement and the vacant set of random interlacements for d geq 3 and the level sets of the gaussian free field d geq 3 inspired by the methods developed by kosygina et al commun pure appl math 59 1489 1521 2006 for proving quenched ldp for elliptic diffusions with a random drift and by yilmaz commun pure appl math 62 8 1033 1075 2009 and rosenbluth quenched large deviations for multidimensional random walks in a random environment a variational formula ph d thesis nyu arxiv 0804 1444v1 for similar results regarding elliptic random walks in random environment we take the point of view of the moving particle and prove a large deviation principle for the quenched distribution of the pair empirical measures of the environment markov chain in the non elliptic case of srwpc via a contraction principle this reduces easily to a quenched ldp for the distribution of the mean velocity of the random walk and both rate functions admit explicit variational formulas the main difficulty in our set up lies in the inherent non ellipticity as well as the lack of translation invariance stemming from conditioning on the fact that the origin belongs to the infinite cluster we develop a unifying approach for proving quenched large deviations for srwpc based on exploiting coercivity properties of the relative entropies in the context of convex variational analysis combined with input from ergodic theory and invoking geometric properties of the supercritical percolation cluster

robust control for linear systems with structured state space uncertainty abstract a recent sufficient condition for the stability robustness of linear systems with time varying norm bounded state space uncertainty is extended to include the structure of the uncertainty our new result requires that the nominal eigenvalues lie to the left of a vertical line in the complex plane which is determined by a norm involving the structure of the uncertainty and the nominal closed loop eigenvector matrix therefore this robustness result is especially well suited to the design of control systems using eigenstructure assignment when the uncertainty is time invariant our norm is also an upper bound on the incremental eigenvalue perturbations we also consider the use of perron weightings to reduce conservatism and the extension of the results to discrete time systems

asymptotic constancy of solutions of linear parabolic volterra difference equations abstract in this paper we obtain some sufficient conditions under which every solution of a class of linear parabolic volterra difference equations of neutral type tends to a constant as n we also consider the asymptotic constancy of solutions for a class of linear ordinary volterra difference equations of neutral type the results obtained improve and generalize some known results

invariant and attracting sets of volterra difference equations with delays will be concerned with the invariant and asymptotic properties of volterra differ ence equations with delays sufficient conditions for determining the invariant and attracting sets of the equations are obtained examples are given to illustrate the obtained results 2003 elsevier science ltd all rights reserved

on the fragmentation of a torus by random walk we consider a simple random walk on a discrete torus input amssym bbb z n bbb z d with dimension d 3 and large side length n for a fixed constant u 0 we study the percolative properties of the vacant set consisting of the set of vertices not visited by the random walk in its first und steps we prove the existence of two distinct phases of the vacant set in the following sense if u 0 is chosen large enough all components of the vacant set contain no more than log n u vertices with high probability as n tends to infinity on the other hand for small u 0 there exists a macroscopic component of the vacant set occupying a nondegenerate fraction of the total volume nd in dimensions d 5 we additionally prove that this macroscopic component is unique by showing that all other components have volumes of order at most log n u our results thus solve open problems posed by benjamini and sznitman who studied the small u regime in high dimension the proofs are based on a coupling of the random walk with random interlacements on input amssym bbb z d among other techniques the construction of this coupling employs a refined use of discrete potential theory by itself this coupling strengthens a result by windisch 2011 wiley periodicals inc

global dynamics of hopfield neural networks involving variable delays abstract in this article a model describing dynamics of hopfield neural networks involving variable delays is considered existence and uniqueness of the equilibrium point under fairly general and easily verifiable conditions are also established further we derive sufficient criteria of global asymptotic stability gas of the equilibrium point

interlacing eigenvalues and graphs abstract we give several old and some new applications of eigenvalue interlacing to matrices associated to graphs bounds are obtained for characteristic numbers of graphs such as the size of a maximal co clique the chromatic number the diameter and the bandwidth in terms of the eigenvalues of the standard adjacency matrix or the laplacian matrix we also deal with inequalities and regularity results concerning the structure of graphs and block designs

asymptotic behavior of nonlinear difference equations with delays abstract this paper deals with asymptotic behavior of nonlinear difference equations with delays sufficient conditions for attracting set and basin of attraction of the equations are obtained examples are given to show the applicability of the proposed approach

domain of attraction of nonlinear discrete systems with delays abstract in this paper we obtain the methods to determine the domain of exponential attraction of a class of nonlinear discrete systems with delays examples are given to show the applicability of the proposed approach

on the relationship between the spectral radius and stability radius for discrete systems the relationship between the spectral radius and the stability radius is established via elementary calculation the result gives a discrete counterpart of the existing result for continuous systems and provides an upper bound for the stability radius the main result of the calculation is summarized in the form of a theorem

upper bound on the disconnection time of discrete cylinders and random interlacements we study the asymptotic behavior for large n of the disconnection time t n of a simple random walk on the discrete cylinder n d x z when d 2 we explore its connection with the model of random interlacements on d 1 recently introduced in ann math in press and specifically with the percolative properties of the vacant set left by random interlacements as an application we show that in the large n limit the tail of t n n 2d is dominated by the tail of the first time when the supremum over the space variable of the brownian local times reaches a certain critical value as a by product we prove the tightness of the laws of t n n 2d when d 2

robust eigenstructure assignment for the extended medium range air to air missile abstract a new sufficient condition for the robust stability of a linear time invariant system subject to linear time varying structured state space uncertainty is presented this sufficient condition is a sum of terms each of which involves the ith right eigenvector the ith left eigenvector and the real part of the ith eigenvalue this new robustness result is less conservative than an earlier result which involves the modal matrix and the real part of the eigenvalue which is closest to the imaginary axis in the complex plane a robust design is computed for the extended medium range air to air missile and the new design is compared with an earlier orthogonal projection eigenstructure assignment design the new design method uses the matlabtm optimization toolbox to minimize the integral of the deviations of the integrated roll rate which is approximately the bank angle from zero the integral is approximated by a sum of discrete samples of the integrated roll rate while normalizing the integration width to unity constraints are placed on the real part of the dutch roll and roll mode eigenvalues the damping ratios of the dutch roll and roll mode the aileron and rudder deflection rates and the sufficient condition for robust stability the new design satisfies the robustness condition while also yielding an improved transient response

extremal geometry of a brownian porous medium the path w 0 t of a brownian motion on a d dimensional torus mathbb t d run for time t is a random compact subset of mathbb t d we study the geometric properties of the complement mathbb t d setminus w 0 t as t rightarrow infty for d ge 3 in particular we show that the largest regions in mathbb t d setminus w 0 t have a linear scale varphi d t d log t d 2 kappa d t 1 d 2 where kappa d is the capacity of the unit ball more specifically we identify the sets e for which mathbb t d setminus w 0 t contains a translate of varphi d t e and we count the number of disjoint such translates furthermore we derive large deviation principles for the largest inradius of mathbb t d setminus w 0 t as t rightarrow infty and the varepsilon cover time of mathbb t d as varepsilon downarrow 0 our results which generalise laws of large numbers proved by dembo et al electron j probab 8 15 1 14 2003 are based on a large deviation estimate for the shape of the component with largest capacity in mathbb t d setminus w rho t 0 t where w rho t 0 t is the wiener sausage of radius rho t with rho t chosen much smaller than varphi d t but not too small the idea behind this choice is that mathbb t d setminus w 0 t consists of lakes whose linear size is of order varphi d t connected by narrow channels we also derive large deviation principles for the principal dirichlet eigenvalue and for the maximal volume of the components of mathbb t d setminus w rho t 0 t as t rightarrow infty our results give a complete picture of the extremal geometry of mathbb t d setminus w 0 t and of the optimal strategy for w 0 t to realise extreme events

p attracting and p invariant sets for a class of impulsive stochastic functional differential equations in this paper a nonlinear and nonautonomous impulsive stochastic functional differential equation is considered by establishing an l operator differential inequality and stochastic analysis technique we obtain the p attracting set and p invariant set of the impulsive stochastic functional differential equation an example is also discussed to illustrate the efficiency of the obtained results

gumbel fluctuations for cover times in the discrete torus this work proves that the fluctuations of the cover time of simple random walk in the discrete torus of dimension at least three with large side length are governed by the gumbel extreme value distribution this result was conjectured for example in aldous and fill reversible markov chains and random walks on graphs in preparation we also derive some corollaries which qualitatively describe how covering happens in addition we develop a new and stronger coupling of the model of random interlacements introduced by sznitman ann math 2 171 3 2039 2087 2010 and random walk in the torus this coupling is used to prove the cover time result and is also of independent interest

necessary and sufficient conditions for stability of time varying discrete interval matrices sufficient conditions for the stability of time varying discrete interval systems are presented the spectral radius of the maximum absolute matrix of all interval matrices is employed as opposed to utilizing the matrix norm or lyapunov theorem for special cases our result becomes a necessary and sufficient condition for stability furthermore the proposed conditions are inclusive of the previous results of stability of time invariant interval systems

vacant set of random interlacements and percolation we introduce a model of random interlacements made of a countable collection of doubly infinite paths on d d 3 a nonnegative parameter u measures how many trajectories enter the picture this model describes in the large n limit the microscopic structure in the bulk which arises when considering the disconnection time of a discrete cylinder n d 1 x by simple random walk or the set of points visited by simple random walk on the discrete torus n d at times of order u n d in particular we study the percolative properties of the vacant set left by the interlacement at level u which is an infinite connected translation invariant random subset of d we introduce a critical value u such that the vacant set percolates for u u our main results show that u is finite when d 3 and strictly positive when d 7

percolation for the vacant set of random interlacements we investigate random interlacements on d d 3 this model recently introduced in 8 corresponds to a poisson cloud on the space of doubly infinite trajectories modulo time shift tending to infinity at positive and negative infinite times a nonnegative parameter u measures how many trajectories enter the picture our main interest lies in the percolative properties of the vacant set left by random interlacements at level u we show that for all d 3 the vacant set at level u percolates when u is small this solves an open problem of 8 where this fact has only been established when d 7 it also completes the proof of the nondegeneracy in all dimensions d 3 of the critical parameter u of 8 2008 wiley periodicals inc

i i r volterra filtering with application to bilinear systems a new nonlinear filtering technique by means of infinite impulse response iir volterra functionals is developed it yields the projection onto the closed class of finite volterra series with separable kernels of arbitrary degree k such an optimal estimator is finitely realizable as a bilinear system with parameters that are computable off line moreover if the original system model is itself bilinear this computation is finitely recursive through higher moments of degree 2 k two simple illustrating examples are developed i estimation of the covariance of the internal white noise driving a linear system and ii filtering of a non gaussian linear system driven by a poisson process the robustness with respect to the observation noise distribution is finally examined