explicit serre duality on complex spaces abstract in this paper we use recently developed calculus of residue currents together with integral formulas to give a new explicit analytic realization as well as a new analytic proof of serre duality on any reduced pure n dimensional paracompact complex space x at the core of the paper is the introduction of certain fine sheaves b x n q of currents on x of bidegree n q such that the dolbeault complex b x n becomes in a certain sense a dualizing complex in particular if x is cohen macaulay then b x n is an explicit fine resolution of the grothendieck dualizing sheaf

invariance principle for the random conductance model with unbounded conductances we study a continuous time random walk x in an environment of i i d random conductances e 1 we obtain heat kernel bounds and prove a quenched invariance principle for x this holds even when e e

abelian differentials on singular varieties and variations on a theorem of lie griffiths we compare various notions of meromorphic and holomorphic differential forms on singular analytic varieties in particular we find that every meromorphic form gives rise to a canonical principal value current with support in the variety demanding that this current be closed we obtain a useful analytic description of the dualizing sheaf we then go on to generalize the lie griffiths theorem on one hand it is shown that a rational trace is enough to ensure algebraicity of the data on the other hand we prove that zero trace implies that the form is holomorphic in the sense of currents in fact we prove a more general local version of the theorem

harnack inequalities on weighted graphs and some applications to the random conductance model we establish elliptic and parabolic harnack inequalities on graphs with unbounded weights as an application we prove a local limit theorem for a continuous time random walk x in an environment of ergodic random conductances taking values in 0 infty satisfying some moment conditions

homogeneous approximation recursive observer design and output feedback we introduce two new tools that can be useful in nonlinear observer and output feedback design the first one is a simple extension of the notion of homogeneous approximation to make it valid both at the origin and at infinity homogeneity in the bi limit exploiting this extension we give several results concerning stability and robustness for a homogeneous in the bi limit vector field the second tool is a new recursive observer design procedure for a chain of integrator combining these two tools we propose a new global asymptotic stabilization result by output feedback for feedback and feedforward systems

random walks on supercritical percolation clusters we obtain gaussian upper and lower bounds on the transition density qt x y of the continuous time simple random walk on a supercritical percolation cluster c1 in the euclidean lattice the bounds analogous to aronsen s bounds for uniformly elliptic divergence form diusions hold with constants ci depending only on p the percolation probability and d the irregular nature of the medium means that the bound for qt x only holds for t sx where the constant sx depends on the percolation congura tion

stability and performance for saturated systems via quadratic and nonquadratic lyapunov functions in this paper we develop a systematic lyapunov approach to the regional stability and performance analysis of saturated systems in a general feedback configuration the only assumptions we make about the system are well posedness of the algebraic loop and local stability problems to be considered include the estimation of the domain of attraction the reachable set under a class of bounded energy disturbances and the nonlinear l2 gain the regional analysis is established through an effective treatment of the algebraic loop and the saturation deadzone function this treatment yields two forms of differential inclusions a polytopic differential inclusion pdi and a norm bounded differential inclusion ndi that contain the original system adjustable parameters are incorporated into the differential inclusions to reflect the regional property the main idea behind the regional analysis is to ensure that the state remain inside the level set of a certain lyapunov function where the pdi or the ndi is valid with quadratic lyapunov functions conditions for stability and performances are derived as linear matrix inequalities lmis to obtain less conservative conditions we use a pair of conjugate non quadratic lyapunov functions the convex hull quadratic function and the max quadratic function these functions yield bilinear matrix inequalities bmis as conditions for stability and guaranteed performance level the bmi conditions cover the corresponding lmi conditions as special cases hence the bmi results are guaranteed to be as good as the lmi results in most examples the bmi results are significantly better than the lmi results

the bar partial equation on a non reduced analytic space let x be a possibly non reduced analytic space of pure dimension we introduce a notion of overline partial equation on x and prove a dolbeault grothendieck lemma we obtain fine sheaves mathcal a x q of 0 q currents so that the associated dolbeault complex yields a resolution of the structure sheaf mathscr o x our construction is based on intrinsic semi global koppelman formulas on x

residue currents with prescribed annihilator ideals on singular varieties given an ideal mathcal j on a complex manifold andersson and wulcan constructed explicitly a current r mathcal j such that the annihilator of r mathcal j is mathcal j generalizing the duality theorem for coleff herrera products we describe a way to generalize this construction to ideals on singular varieties

invariance principle for the random conductance model we study a continuous time random walk x in an environment of i i d random conductances mu e in 0 infty in mathbb z d we assume that mathbb p mu e 0 p c so that the bonds with strictly positive conductances percolate but make no other assumptions on the law of the e we prove a quenched invariance principle for x and obtain green s functions bounds and an elliptic harnack inequality

random walks on infinite percolation clusters in models with long range correlations for a general class of percolation models with long range correlations on zdzd d 2d 2 introduced in j math phys 55 2014 083307 we establish regularity conditions of barlow ann probab 32 2004 3024 3084 that mesoscopic subballs of all large enough balls in the unique infinite percolation cluster have regular volume growth and satisfy a weak poincare inequality as immediate corollaries we deduce quenched heat kernel bounds parabolic harnack inequality and finiteness of the dimension of harmonic functions with at most polynomial growth heat kernel bounds and the quenched invariance principle of probab theory related fields 166 2016 619 657 allow to extend various other known results about bernoulli percolation by mimicking their proofs for instance the local central limit theorem of electron j probab 14 209 1 27 or the result of ann probab 43 2015 2332 2373 that the dimension of at most linear harmonic functions on the infinite cluster is d 1d 1 in terms of specific models all these results are new for random interlacements at every level in any dimension d 3d 3 as well as for the vacant set of random interlacements ann of math 2 171 2010 2039 2087 comm pure appl math 62 2009 831 858 and the level sets of the gaussian free field comm math phys 320 2013 571 601 in the regime of the so called local uniqueness which is believed to coincide with the whole supercritical regime for these models

surface order large deviations for high density percolation we derive surface order large deviation estimates for the volume of the largest cluster and for the volume of the largest region surrounded by a cluster of a bernoulli percolation process restricted to a big finite box with sufficiently large parameter we also establish a useful version of the isoperimetric inequality which is the main tool of our proofs

maximum rank of a legendrian web abstract we propose the legendrian web in a contact three manifold as a second order generalization of the planar web an abelian relation for a legendrian web is analogously defined as an additive equation among the first integrals of its foliations for a class of legendrian d webs defined by simple second order ode s we give an algebraic construction of d d 1 d 2 2 d 3 6 linearly independent abelian relations we then employ the method of local differential analysis and the theory of linear differential systems to show that d is the maximum rank of a legendrian d web in the complex analytic category we give a possible projective geometric interpretation of d as an analogue of castelnuovo bound for degree 2 d surfaces in the 3 quadric q 3 p 4 via the duality between p 3 and q 3 associated with the rank two complex simple lie group sp 2 c the legendrian 3 webs of maximum rank three are analytically characterized and their explicit local normal forms are found for an application we give an alternative characterization of a darboux super integrable metric as a two dimensional riemannian metric g which admits a mate metric g such that a legendrian 3 web naturally associated with the geodesic foliations of the pair g has maximum rank

stabilization of linear differential systems via hybrid feedback controls we study so called hybrid feedback stabilizers for an arbitrarily general system of linear differential equations we prove that under assumptions of controllability and observability there exists a hybrid feedback output control which makes the system asymptotically stable the control is designed by making use of a discrete automaton implanted into the system s dynamics in general the automaton has infinitely many locations but it gives rise to a uniform in some sense feedback control the approach we propose goes back to classical feedback control techniques combined with some ideas used in stability theory for equations with time delay

quasi optimal robust stabilization of control systems in this paper we investigate the problem of semiglobal minimal time robust stabilization of analytic control systems with controls entering linearly by means of a hybrid state feedback law it is shown that in the absence of minimal time singular trajectories the solutions of the closed loop system converge to the origin in quasi minimal time for a given bound on the controller with a robustness property with respect to small measurement noise external disturbances and actuator noise

weighted koppelman formulas and the equation on an analytic space let x be an analytic space of pure dimension we introduce a formalism to generate intrinsic weighted koppelman formulas on x that provide solutions to the partial derivative over bar equation we obtain new existence results for the partial derivative over bar equation as well as new proofs of various known results c 2011 elsevier inc all rights reserved

quenched invariance principle for simple random walk on percolation clusters we consider the simple random walk on the unique infinite cluster of super critical bond percolation in d with d 2 we prove that for almost every percolation configuration the path distribution of the walk converges weakly to that of non degenerate isotropic brownian motion our analysis is based on the consideration of a harmonic deformation of the infinite cluster on which the random walk becomes a square integrable martingale the size of the deformation expressed by the so called corrector is estimated by means of ergodicity arguments

genericity results for singular curves let m be a smooth manifold and cal d m m geq 2 be the set of rank m distributions on m endowed with the whitney c infty topology we show the existence of an open set o m dense in cal d m so that every nontrivial singular curve of a distribution d of o m is of minimal order and of corank one in particular for m geq 3 every distribution of o m does not admit nontrivial rigid curves as a consequence for generic sub riemannian structures of rank greater than or equal to three there does not exist nontrivial minimizing singular curves

technical results for the study of robustness of lagrange stability abstract we study the robustness of boundedness of solutions of nonlinear dynamical systems a sufficient coordinate free technical condition on the characterization of unmodelled effects is given this characterization with some analogy with the input to state stability of 16 has the potential to encompass most classical uncertainties in this context we establish lagrange stability results which look very much like a small gain theorem we illustrate the use of these technical results in the robust stabilization problem by a class of interconnected systems

characterizations of input to state stability for hybrid systems abstract we study input to state stability iss for a broad class of hybrid systems which are combinations of a differential equation on a constraint set and a difference equation on another constraint set for this class of hybrid systems we establish the equivalence of iss nonuniform iss and the existence of a smooth iss lyapunov function by additionally assuming that the right hand side of the differential equation has a convex property with respect to inputs moreover we demonstrate by examples that the equivalence may fail when such a convexity assumption is removed

gaussian bounds and parabolic harnack inequality on locally irregular graphs a well known theorem of delmotte is that gaussian bounds parabolic harnack inequality and the combination of volume doubling and poincare inequality are equivalent for graphs in this paper we consider graphs for which these conditions hold but only for sufficiently large balls and prove a similar equivalence

invariance principle for the random conductance model in a degenerate ergodic environment we study a continuous time random walk x on zd in an environment of random conductances taking values in 0 we assume that the law of the conductances is ergodic with respect to space shifts we prove a quenched invariance principle for x under some moment conditions of the environment the key result on the sublinearity of the corrector is obtained by moser s iteration scheme

flow stability of patchy vector fields and robust feedback stabilization the paper is concerned with patchy vector fields a class of discontinuous piecewise smooth vector fields that were introduced by the authors to study feedback stabilization problems we prove the stability of the corresponding solution set w r t a wide class of impulsive perturbations these results yield the robustness of patchy feedback controls in the presence of measurement errors and external disturbances

disorder entropy and harmonic functions we study harmonic functions on random environments with particular emphasis on the case of the infinite cluster of supercritical percolation on z d we prove that the vector space of harmonic functions growing at most linearly is d 1 dimensional almost surely in particular there are no non constant sublinear harmonic functions thus implying the uniqueness of the corrector the main ingredient of the proof is a quantitative annealed version of the kaimanovich vershik entropy argument this also provides bounds on the derivative of the heat kernel simplifying and generalizing existing results the argument applies to many different environments even reversibility is not necessary we also mention several open problems and conjectures on the behavior of harmonic functions on stationary random graphs

global effective versions of the briancon skoda huneke theorem we prove global effective versions of the briancon skoda huneke theorem our results extend to singular varieties a result of hickel on the membership problem in polynomial ideals in cn and a related theorem of ein and lazarsfeld for smooth projective varieties the proofs rely on known geometric estimates and new results on multivariable residue calculus

tissus algebriques exceptionnels in pirio and trepreau int math res not imrn doi 10 1093 imrn rnu066 2014 it has been proved that for r 1 n ge 2 and d ge r 1 n 1 2 a d web of t ype r n with maximal rank is algebraizable in the classical sense except maybe when n ge 3 and d r 2 n 1 1 in the present paper one considers this particular case under these hypotheses on n and d one constructs some examples of exceptional algebraic webs these are generalized algebraic webs of maximal rank that are not algebraizable in the classical sense

on the intersection of algebraic curves and hypersurfaces many classical theorems about the intersection of plane algebraic curves can be derived from and generalized by the algebraic theory of residues in higher dimensional spaces one obtains results about the intersection of curves and hypersurfaces in various degrees of generality this was pointed out f i in s gh k2 and hu in the present note we wish to prove higher dimensional analogues of theorems of waring g humbert and reiss for arbitrary curves in affine n space and their intersection with hypersurfaces see theorems 4 1 5 1 and 6 2 below by a curve hypersurface we mean a closed subscheme in projective or affine n space of pure dimension 1 pure codimension 1 having no embedded components our considerations are based on the residue theorem for complete integral algebraic curves

semi global stabilization by an output feedback law from a hybrid state controller this article suggests a design method of a hybrid output feedback for siso continuous systems we focus on continuous systems for which there exists a hybrid state feedback law a local hybrid stabilizability and a global complete uniform observability are assumed to achieve the stabilization of an equilibrium with a hybrid output feedback law this is an existence result moreover assuming the existence of a robust lyapunov function instead of a stabilizability assumption allows to design explicitly this hybrid output feedback law this last result is illustrated for linear systems with reset saturated controls

on chemical distances and shape theorems in percolation models with long range correlations in this paper we provide general conditions on a one parameter family of random infinite subsets of z d to contain a unique infinite connected component for which the chemical distances are comparable to the euclidean distance in addition we show that these conditions also imply a shape theorem for the corresponding infinite connected component by verifying these conditions for specific models we obtain novel results about the structure of the infinite connected component of the vacant set of random interlacements and the level sets of the gaussian free field as a byproduct we obtain alternative proofs to the corresponding results for random interlacements in the work of cern and popov on the internal distance in the interlacement set electron j probab 17 29 1 25 2012 and while our main interest is in percolation models with long range correlations we also recover results in the spirit of the work of antal and pisztora on the chemical distance for supercritical bernoulli percolation ann probab 24 2 1036 1048 1996 for bernoulli percolation finally as a corollary we derive new results about the chemical diameter of the largest connected component in the complement of the trace of the random walk on the torus

smooth lyapunov functions for hybrid systems part ii pre asymptotically stable compact sets it is shown that pre asymptotic stability which generalizes asymptotic stability of a compact set for a hybrid system satisfying mild regularity assumptions is equivalent to the existence of a smooth lyapunov function this result is achieved with the intermediate result that asymptotic stability of a compact set for a hybrid system is generically robust to small state dependent perturbations as a special case we state a converse lyapunov theorem for systems with logic variables and use this result to establish input to state stabilization using hybrid feedback control the converse lyapunov theorems are also used to establish semiglobal practical robustness to slowly varying weakly jumping parameters to temporal regularization to the insertion of jumps according to an ldquoaverage dwell timerdquo rule and to the insertion of flow according to a ldquoreverse average dwell timerdquo rule

a dolbeault grothendieck lemma on complex spaces via koppelman formulas let x be a complex space of pure dimension we introduce fine sheaves a x q of 0 q currents which coincides with the sheaves of smooth forms on the regular part of x so that the associated dolbeault complex yields a resolution of the structure sheaf hol x our construction is based on intrinsic and quite explicit semi global koppelman formulas

global stabilization by output feedback examples and counterexamples abstract we show by means of examples that global complete observability and global stabilizability by state feedback are not sufficient to guarantee global stabilizability by dynamic output feedback we show that a main obstruction is related to unboundedness unobservability this is that some unmeasured state components may escape in finite time whereas the measurements remain bounded

lyapunov conditions for input to state stability of impulsive systems this paper introduces appropriate concepts of input to state stability iss and integral iss for impulsive systems i e dynamical systems that evolve according to ordinary differential equations most of the time but occasionally exhibit discontinuities or impulses we provide a set of lyapunov based sufficient conditions for establishing these iss properties when the continuous dynamics are iss but the discrete dynamics that govern the impulses are not the impulses should not occur too frequently which is formalized in terms of an average dwell time adt condition conversely when the impulse dynamics are iss but the continuous dynamics are not there must not be overly long intervals between impulses which is formalized in terms of a novel reverse adt condition we also investigate the cases where i both the continuous and discrete dynamics are iss and ii one of these is iss and the other only marginally stable for the zero input while sharing a common lyapunov function in the former case we obtain a stronger notion of iss for which a necessary and sufficient lyapunov characterization is available the use of the tools developed herein is illustrated through examples from a micro electro mechanical system mems oscillator and a problem of remote estimation over a communication network

morse sard type results in sub riemannian geometry let m g be a sub riemannian manifold and x0 m assuming that chow s condition holds and that m endowed with the sub riemannian distance is complete we prove that there exists a dense subset n1 of m such that for every point x of n1 there is a unique minimizing path steering x0 to x this trajectory admitting a normal extremal lift if the distribution is everywhere of corank one we prove the existence of a subset n2 of m of full lebesgue measure such that for every point x of n2 there exists a minimizing path steering x0 to x which admits a normal extremal lift is nonsingular and the point x is not conjugate to x0 in particular the image of the sub riemannian exponential mapping is dense in m and in the case of corank one is of full lebesgue measure in m

random conductance models with stable like jumps heat kernel estimates and harnack inequalities abstract we establish two sided heat kernel estimates for random conductance models with non uniformly elliptic possibly degenerate stable like jumps on graphs these are long range counterparts of the well known two sided gaussian heat kernel estimates by m t barlow for nearest neighbor short range random walks on the supercritical percolation cluster unlike the cases of nearest neighbor conductance models we cannot use parabolic harnack inequalities since even elliptic harnack inequalities do not hold in the present setting as an application we establish the local limit theorem for the models