a complete greedy algorithm for infinite horizon sensor scheduling abstract in this paper we study the problem of scheduling sensors to estimate the state of a linear dynamical system the estimator is a kalman filter and our objective is to optimize the a posteriori error covariance over an infinite time horizon we focus on the case where a fixed number of sensors are selected at each time step and we characterize the exact conditions for the existence of a schedule with uniformly bounded estimation error covariance using this result we construct a scheduling algorithm that guarantees that the error covariance will be bounded if the existence conditions are satisfied we call such an algorithm complete finally we provide simulations to compare the performance of the algorithm against other known techniques

optimal multiple sensor scheduling for general scalar gauss markov systems with the terminal error in this work we study finite horizon multiple sensor scheduling for general scalar gauss markov systems extending previous results where only a class of systems are considered the scheduling objective is to minimize the terminal estimation error covariance only one sensor can transmit its measurement per time instant and each sensor has limited energy through building a comparison function and solving its monotone intervals an efficient algorithm is designed to construct the optimal schedule

kalman filtering with intermittent observations tail distribution and critical value in this paper we analyze the performance of kalman filtering for discrete time linear gaussian systems where packets containing observations are dropped according to a markov process modeling a gilbert elliot channel to address the challenges incurred by the loss of packets we give a new definition of non degeneracy which is essentially stronger than the classical definition of observability but much weaker than one step observability which is usually used in the study of kalman filtering with intermittent observations we show that the trace of the kalman estimation error covariance under intermittent observations follows a power decay law moreover we are able to compute the exact decay rate for non degenerate systems finally we derive the critical value for non degenerate systems based on the decay rate improving upon the state of the art

euclidean spaces as weak tangents of infinitesimally hilbertian metric measure spaces with ricci curvature bounded below we show that in any infinitesimally hilbertian cd k n space at almost every point there exists a euclidean weak tangent i e there exists a sequence of dilations of the space that converges to a euclidean space in the pointed measured gromov hausdorff topology the proof follows by considering iterated tangents and the splitting theorem for infinitesimally hilbertian cd 0 n spaces

on the equivalence of the entropic curvature dimension condition and bochner s inequality on metric measure spaces we prove the equivalence of the curvature dimension bounds of lott sturm villani via entropy and optimal transport and of bakry emery via energy and gamma 2 calculus in complete generality for infinitesimally hilbertian metric measure spaces in particular we establish the full bochner inequality on such metric measure spaces moreover we deduce new contraction bounds for the heat flow on riemannian manifolds and on mms in terms of the l 2 wasserstein distance

on stability in the presence of analog erasure channel between the controller and the actuator consider a discrete time networked control system in which the controller has direct access to noisy measurements of the output of the plant however information flows from the controller to the actuator via a channel that features bernoulli erasure events if an erasure occurs the channel outputs an erasure symbol otherwise it transmits a real finite dimensional vector we determine necessary and sufficient conditions for the stabilizability of an unstable linear time invariant finite dimensional plant given a minimal state space representation for the plant the necessary and sufficient conditions for stabilizability are expressed in terms of the probability of erasure at the channel and of the spectral radius of the one step state transition matrix there are two main results in the technical note the first result shows that if the actuator has processing capabilities then the necessary and sufficient conditions for stabilizability remain unchanged with or without acknowledgements from the actuator to the controller the second result shows that the stabilizability conditions are identical for two types of actuators type i processing at the actuator has access to the plant s model type ii processing at the actuator uses a universal algorithm that does not depend on the model of the plant thus neither the knowledge of the model of the plant at the actuator nor the presence of acknowledgements from the actuator to the controller can be used to alter or relax the necessary and sufficient conditions for stabilizability

locational optimization problems solved through voronoi diagrams abstract this paper reviews a class of continuous locational optimization problems where an optimal location or an optimal configuration of facilities is found in a continuum on a plane or a network that can be solved through the voronoi diagram eight types of continuous locational optimization problems are formulated and these problems are solved through the ordinary voronoi diagram the farthest point voronoi diagram the weighted voronoi diagram the network voronoi diagram the voronoi diagram with a convex distance function the line voronoi diagram and the area voronoi diagram

nodal length fluctuations for arithmetic random waves using the spectral multiplicities of the standard torus we endow the laplace eigenspaces with gaussian probability measures this induces a notion of random gaussian laplace eigenfunctions on the torus arithmetic random waves we study the distribution of the nodal length of random eigenfunctions for large eigenvalues and our primary result is that the asymptotics for the variance is nonuniversal our result is intimately related to the arithmetic of lattice points lying on a circle with radius corresponding to the energy

construction of manifolds of positive scalar curvature we prove that a regular neighborhood of a codimension 3 subcomplex of a manifold can be chosen so that the induced metric on its boundary has positive scalar curvature a number of useful facts concerning manifolds of positive scalar curvature follow from this construction for example we see that any finitely presented group can appear as the fundamental group of a compact 4 manifold with such a metric 0 outline of results we give a new method for constructing complete riemannian manifolds of positive scalar curvature and use it to continue the investigation of properties of positive scalar curvature our construction uses the idea that manifolds having spheres of dimension 2 as factors will admit metrics of positive scalar curvature if the spheres can be made to carry sufficient positive curvature to dominate any negative curvature most of the known methods for constructing manifolds of positive scalar curvature employ this same idea for example any manifold of the form m x s2 can be given a warped product metric of positive scalar curvature by suitably adjusting the radius of the s2 factor similarly by deforming the standard metric on s3 point in a small neighborhood of the point and using the s2 factor to carry positive curvature around the corner we can construct a complete metric of positive scalar curvature on r3 this same idea was used by gromov and lawson gl and schoen and yau sy in proving that codimension 3 surgeries on a manifold of positive scalar curvature yields a manifold which also carries positive scalar curvature in this paper we generalize the above techniques to cover any manifold formed as the boundary of a regular neighborhood of a subcomplex k of a manifold m if the codimension of k 3 this boundary looks locally like k x s2 and so should carry positive scalar curvature theorem 1 let m be an n dimensional riemannian manifold with a fixed smooth cell decomposition and k a codimension q 3 subcomplex of m then there is a regular neighborhood u of k in m so that the induced metric on the boundary du has positive scalar curvature an easy consequence of this theorem is the following corollary 2 let 7r be a finitely presented group then there exists a compact 4 manifold m of positive scalar curvature with 7rl m 7r this fact is interesting since it is generally believed that manifolds that are large in some sense should not adrrlit metrics of positive curvature corollary 2 receiv d by the editors septem er 25 1985 and ill revised form july 17 1986 1980 m lthf rb lti s s l jjf t l l n n ijl ltion 1985 rfvision primary 53c20 r 1988 americatl mathematic al society 0002 9947 88 1 00 25 per page

nodal intersections for random waves against a segment on the 3 dimensional torus abstract we consider random gaussian eigenfunctions of the laplacian on the three dimensional flat torus and investigate the number of nodal intersections against a straight line segment the expected intersection number against any smooth curve is universally proportional to the length of the reference curve times the wavenumber independent of the geometry we found an upper bound for the nodal intersections variance depending on the arithmetic properties of the straight line the considerations made establish a close relation between this problem and the theory of lattice points on spheres

optimal periodic sensor schedule for steady state estimation under average transmission energy constraint we consider periodic sensor scheduling for remote state estimation under average transmission energy constraint the sensor decides whether or not to send its data to a remote estimator in order to meet the transmission energy constraint the transmitted data are likely to be dropped due to the imperfect communication an optimal periodic schedule is found via the tools from the markov chain furthermore a sufficient condition of the system dynamics energy budget and packet drop rate under which the remote estimator is guaranteed to be stable is derived examples are provided to show the effectiveness of results

stochastic link activation for distributed filtering under sensor power constraint we consider the problem of link activation for distributed estimation with power constraint to satisfy the requirement of power consumption we propose a stochastic link activation scheme where each sensor equipped with a distributed estimator sends data to its neighboring sensors according to different probabilities first we design the optimal estimator gain of each sensor to minimize the state estimation error covariance then we find an upper bound of the expected state estimation error covariance and provide a sufficient condition to guarantee the stability of the proposed estimator finally we formulate the link activation problem as an optimization problem and convert it to a convex optimization

active classification for pomdps a kalman like state estimator author s zois ds levorato m mitra u abstract 1991 2012 ieee the problem of state tracking with active observation control is considered for a system modeled by a discrete time finite state markov chain observed through conditionally gaussian measurement vectors the measurement model statistics are shaped by the underlying state and an exogenous control input which influence the observations quality exploiting an innovations approach an approximate minimum mean squared error mmse filter is derived to estimate the markov chain system state to optimize the control strategy the associated mean squared error is used as an optimization criterion in a partially observable markov decision process formulation a stochastic dynamic programming algorithm is proposed to solve for the optimal solution to enhance the quality of system state estimates approximate mmse smoothing estimators are also derived finally the performance of the proposed framework is illustrated on the problem of physical activity detection in wireless body sensing networks the power of the proposed framework lies within its ability to accommodate a broad spectrum of active classification applications including sensor management for object classification and tracking estimation of sparse signals and radar scheduling

optimal scheduling of scalar gauss markov systems with a terminal cost function in this technical note we consider the problem of optimal measurement scheduling for a particular class of gauss markov systems these type of scheduling problems arise in applications such as multi target tracking and sensor management general solutions to such problems in the gauss markov framework are still the subject of ongoing research here for the first time we present a set of results for scalar systems where we consider optimality in the context of minimizing a terminal cost complete proofs are given in each case in some cases proof outlines have been previously available other cases are presented here for the first time for the class of problems considered we demonstrate that simple index policies are optimal we further examine practical problems in which suboptimal solutions may suffice numerical examples are presented for each case

local curvature dimension condition implies measure contraction property abstract we prove that for non branching metric measure spaces the local curvature condition cd loc k n implies the global version of mcp k n the curvature condition cd k n introduced by the second author and also studied by lott and villani is the generalization to metric measure space of lower bounds on ricci curvature together with upper bounds on the dimension this paper is the following step of bacher and sturm 2010 1 where it is shown that cd loc k n is equivalent to a global condition cd k n slightly weaker than the usual cd k n it is worth pointing out that our result implies sharp bishop gromov volume growth inequality and sharp poincare inequality

sharp and rigid isoperimetric inequalities in metric measure spaces with lower ricci curvature bounds we prove that if x mathsf d mathfrak m is a metric measure space with mathfrak m x 1 having in a synthetic sense ricci curvature bounded from below by k 0 and dimension bounded above by n in 1 infty then the classic levy gromov isoperimetric inequality together with the recent sharpening counterparts proved in the smooth setting by milman for any k in mathbb r n ge 1 and upper diameter bounds holds i e the isoperimetric profile function of x mathsf d mathfrak m is bounded from below by the isoperimetric profile of the model space moreover if equality is attained for some volume v in 0 1 and k is strictly positive then the space must be a spherical suspension and in this case we completely classify the isoperimetric regions finally we also establish the almost rigidity if the equality is almost attained for some volume v in 0 1 and k is strictly positive then the space must be mgh close to a spherical suspension to our knowledge this is the first result about isoperimetric comparison for non smooth metric measure spaces satisfying ricci curvature lower bounds examples of spaces fitting our assumptions include measured gromov hausdorff limits of riemannian manifolds satisfying ricci curvature lower bounds alexandrov spaces with curvature bounded from below finsler manifolds endowed with a strongly convex norm and satisfying ricci curvature lower bounds the result seems new even in these celebrated classes of spaces

control from an interior hypersurface we consider a compact riemannian manifold m possibly with boundary and m m an interior hypersurface possibly with boundary we study observation and control from for both the wave and heat equations for the wave equation we prove controllability from in time t under the assumption t gcc that all generalized bicharacteristics intersect transversally in the time interval 0 t for the heat equation we prove unconditional controllability from as a result we obtain uniform lower bounds for the cauchy data of laplace eigenfunctions on under t gcc and unconditional exponential lower bounds on such cauchy data

optimal sensor scheduling for multiple linear dynamical systems we consider the design of an optimal collision free sensor schedule for a number of sensors which monitor different linear dynamical systems correspondingly at each time only one of all the sensors can send its local estimate to the remote estimator a preliminary work for the two sensor scheduling case has been studied in the literature the generalization into multiple sensor scheduling case is shown to be nontrivial we first find a necessary condition of the optimal solution which can significantly reduce the feasible optimal solution space without loss of performance by modelling a finite state markov decision process mdp problem we can numerically search an asymptotic periodic schedule which is proven to be optimal some simple but effective suboptimal schedules for any systems are proposed we also find a lower bound of the optimal cost which enables us to quantify the performance gap between any suboptimal schedule and an optimal one

moderate deviations of a random riccati equation the paper characterizes the invariant filtering measures resulting from kalman filtering with intermittent observations in which the observation arrival is modeled as a bernoulli process with packet arrival probability our prior work showed that for 0 the sequence of random conditional error covariance matrices converges weakly to a unique invariant distribution this paper shows that as approaches one the family 0 satisfies a moderate deviations principle with good rate function i 1 as 1 the family converges weakly to the dirac measure p concentrated on the fixed point of the associated discrete time riccati operator 2 the probability of a rare event an event bounded away from p under decays to zero as a power law of 1 as 1 and 3 the best power law decay exponent is obtained by solving a deterministic variational problem involving the rate function i for specific scenarios the paper develops computationally tractable methods that lead to efficient estimates of rare event probabilities under

detectability and stabilizability of time varying discrete time linear systems the concepts of detectability and stabilizability are explored for time varying systems we study duality invariance under feedback an extended version of the lemma of lyapunov existence of stabilizing feedback laws linear quadratic filtering and control and the existence of approximate canonical forms

kalman filtering with intermittent observations motivated by navigation and tracking applications within sensor networks we consider the problem of performing kalman filtering with intermittent observations when data travel along unreliable communication channels in a large wireless multihop sensor network the effect of communication delays and loss of information in the control loop cannot be neglected we address this problem starting from the discrete kalman filtering formulation and modeling the arrival of the observation as a random process we study the statistical convergence properties of the estimation error covariance showing the existence of a critical value for the arrival rate of the observations beyond which a transition to an unbounded state error covariance occurs we also give upper and lower bounds on this expected state error covariance

riemannian ricci curvature lower bounds in metric measure spaces with finite measure in prior work 4 of the first two authors with savare a new riemannian notion of lower bound for ricci curvature in the class of metric measure spaces x d m was introduced and the corresponding class of spaces denoted by rcd k this notion relates the cd k n theory of sturm and lott villani in the case n to the bakry emery approach in 4 the rcd k property is defined in three equivalent ways and several properties of rcd k spaces including the regularization properties of the heat flow the connections with the theory of dirichlet forms and the stability under tensor products are provided in 4 only finite reference measures m have been considered the goal of this paper is twofold on one side we extend these results to general finite spaces on the other we remove a technical assumption appeared in 4 concerning a strengthening of the cd k condition this more general class of spaces includes euclidean spaces endowed with lebesgue measure complete noncompact riemannian manifolds with bounded geometry and the pointed metric measure limits of manifolds with lower ricci curvature bounds

scheduling parallel kalman filters for multiple processes in this paper we investigate the problem of scheduling parallel kalman filters for multiple processes where each process is observed by a kalman filter and at each time step only one kalman filter could obtain observation due to practical constraints to solve the problem two novel notions permissible consecutive observation loss pcol and least consecutive observation lco are introduced as criteria to describe feasible observation sequences for a process ensuring desired estimation qualities then two methods namely the threshold method and the periodic method are proposed to calculate pcol and lco for each process based on the derived pcol and lco requirements we develop two algorithms that are applicable to different situations sxy algorithm from the pinwheel problem for the case of lco 1 and tree search algorithm for general cases also to reduce the computational complexity of tree search algorithm several useful pruning conditions are obtained

nodal intersections for random eigenfunctions on the torus we investigate the number of nodal intersections of random gaussian laplace eigenfunctions on the standard two dimensional flat torus arithmetic random waves with a fixed smooth reference curve with nonvanishing curvature the expected intersection number is universally proportional to the length of the reference curve times the wavenumber independent of the geometry our main result prescribes the asymptotic behavior of the nodal intersections variance for smooth curves in the high energy limit remarkably it is dependent on both the angular distribution of lattice points lying on the circle with radius corresponding to the given wavenumber and the geometry of the given curve in particular this implies that the nodal intersection number admits a universal asymptotic law with arbitrarily high probability

the codimension one homology of a complete manifold with nonnegative ricci curvature in this paper we prove that a complete noncompact manifold with nonnegative ricci curvature has a trivial codimension one homology unless it is a flat normal bundle over a compact totally geodesic submanifold in particular we prove the conjecture that a complete noncompact manifold with positive ricci curvature has a trivial codimension one integer homology we also have a corollary stating when the codimension two integer homology of such a manifold is torsion free

submodularity and greedy algorithms in sensor scheduling for linear dynamical systems this paper focuses on sensor scheduling for state estimation which consists of a network of noisy sensors and a discrete time linear system with process noise as an energy constraint only a subset of sensors can take a measurement at each time step these measurements are fused into a common state estimate using a kalman filter and the goal is to schedule the sensors to minimize the estimation error at a terminal time a simple approach is to greedily choose sensors at each time step to minimize the estimation error at the next time step recent work has shown that this greedy algorithm outperforms other well known approaches results have been established to show that the estimation error is a submodular function of the sensor schedule submodular functions have a diminishing returns property that ensures the greedy algorithm yields near optimal performance as a negative result we show that most commonly used estimation error metrics are not in general submodular functions this disproves an established result we then provide sufficient conditions on the system for which the estimation error is a submodular function of the sensor schedule and thus the greedy algorithm yields performance guarantees

optimal periodic scheduling of sensor networks a branch and bound approach abstract a periodic scheduling problem for sensor networks with communication constraints is considered for state estimation the solvability of the problem is first discussed and a necessary and sufficient condition is presented based on the notion of periodic detectability since the calculation of the average prediction error variance requires the computation of the symmetric periodic positive semidefinite stabilizing spps solutions to the periodic riccati equations a moving approximate cost function is proposed which gradually converges to the exact cost function also it is shown that the upper bound of the approximation error is independent of the spps solutions and converges to zero exponentially based on these results a branch and bound based algorithm is proposed to compute the optimal periodic schedule and the idea is to iteratively trim the set of schedules that are potentially robust optimal with respect to the approximation error if the optimal schedule is unique the algorithm solves the periodic scheduling problem by exploring a finite number of nodes moreover given an arbitrary nonzero suboptimality specification the algorithm results in a suboptimal schedule set containing all the optimal schedules at a manageable computation effort a numerical example is presented to illustrate the proposed results

metric measure spaces with riemannian ricci curvature bounded from below in this paper we introduce a synthetic notion of riemannian ricci bounds from below for metric measure spaces x d m which is stable under measured gromov hausdorff convergence and rules out finsler geometries it can be given in terms of an enforcement of the lott sturm and villani geodesic convexity condition for the entropy coupled with the linearity of the heat flow besides stability it enjoys the same tensorization global to local and local to global properties in these spaces which we call rcd k spaces we prove that the heat flow which can be equivalently characterized either as the flow associated to the dirichlet form or as the wasserstein gradient flow of the entropy satisfies wasserstein contraction estimates and several regularity properties in particular bakry emery estimates and the l lip feller regularization we also prove that the distance induced by the dirichlet form coincides with d that the local energy measure has density given by the square of cheeger s relaxed slope and as a consequence that the underlying brownian motion has continuous paths all these results are obtained independently of poincare and doubling assumptions on the metric measure structure and therefore apply also to spaces which are not locally compact as the infinite dimensional ones

distributed adaptive quantization for wireless sensor networks from delta modulation to maximum likelihood we consider distributed parameter estimation using quantized observations in wireless sensor networks wsns where due to bandwidth constraint each sensor quantizes its local observation into one bit of information a conventional fixed quantization fq approach which employs a fixed threshold for all sensors incurs an estimation error growing exponentially with the difference between the threshold and the unknown parameter to be estimated to address this difficulty we propose a distributed adaptive quantization aq approach which with sensors sequentially broadcasting their quantized data allows each sensor to adaptively adjust its quantization threshold three aq schemes are presented 1 aq fs that involves distributed delta modulation dm with a fixed stepsize 2 aq vs that employs dm with a variable stepsize and 3 aq ml that adjusts the threshold through a maximum likelihood ml estimation process the ml estimators associated with the three aq schemes are developed and their corresponding cramer rao bounds crbs are analyzed we show that our 1 bit aq approach is asymptotically optimum yielding an asymptotic crb that is only pi 2 times that of the clairvoyant sample mean estimator using unquantized observations

universal covers for hausdorff limits of noncompact spaces we prove that if y is the gromov hausdorff limit of a sequence of complete manifolds m n i with a uniform lower bound on ricci curvature then y has a universal cover

optimal sensor scheduling in batch processes using convex relaxations and tchebycheff systems theory in several industrial processes measurements are limited due to time or cost constraints hence scheduling of these measurements to obtain maximally informative and accurate estimates of the states becomes important the focus of this contribution is to determine a schedule of measurements that maximizes the quality of the estimates at the end of a finite time horizon this work finds applications in batch chemical processes the estimate covariance matrix as obtained by applying standard kalman filtering theory is approximated and the original problem is relaxed and reformulated as a cone program for a certain class of systems the approximated covariance matrix is parameterized in terms of a point belonging to a moment space induced by a tchebycheff system the theory of tchebycheff systems is used to determine an improved upper bound on the guaranteed minimum number of measurements a tractable solution methodology to obtain the optimal cost and a parsimonious discrete schedule using the theory of cone programming generalized barrier functions and tchebycheff systems is described

on the nodal sets of toral eigenfunctions we study the nodal sets of eigenfunctions of the laplacian on the standard d dimensional flat torus the question we address is can a fixed hypersurface lie on the nodal sets of eigenfunctions with ar bitrarily large eigenvalue in dimension two we show that this happens only for segments of closed geodesics in higher dimensions certain cylindrical sets do lie on nodal sets corresponding to arbitrarily large eigenvalues our main result is that this cannot happen for hypersur faces with nonzero gauss kronecker curvature in dimension two the result follows from a uniform lower bound for the l 2 norm of the restriction of eigenfunctions to the curve proved in an earlier paper 1 in high dimensions we currently do not have this bound instead we make use of the real analytic nature of the flat torus to study variations on this bound for restrictions of eigenfunctions to suitable submanifolds in the complex domain in all of our results we need an arithmetic ingredient concerning the cluster structure of lattice points on the sphere we also present an independent proof for the two dimensional case relying on the abc theorem in function fields

sensor selection based on generalized information gain for target tracking in large sensor networks in this paper sensor selection problems for target tracking in large sensor networks with linear equality or inequality constraints are considered first we derive an equivalent kalman filter for sensor selection i e generalized information filter then under a regularity condition we prove that the multistage look ahead policy that minimizes either the final or the average estimation error covariances of next multiple time steps is equivalent to a myopic sensor selection policy that maximizes the trace of the generalized information gain at each time step moreover when the measurement noises are uncorrelated between sensors the optimal solution can be obtained analytically for sensor selection when constraints are temporally separable when constraints are temporally inseparable sensor selections can be obtained by approximately solving a linear programming problem so that the sensor selection problem for a large sensor network can be dealt with quickly although there is no guarantee that the gap between the performance of the chosen subset and the performance bound is always small numerical examples suggest that the algorithm is near optimal in many cases finally when the measurement noises are correlated between sensors the sensor selection problem with temporally inseparable constraints can be relaxed to a boolean quadratic programming problem which can be efficiently solved by a gaussian randomization procedure along with solving a semi definite programming problem numerical examples show that the proposed method is much better than the method that ignores dependence of noises

on a stochastic sensor selection algorithm with applications in sensor scheduling and sensor coverage in this note we consider the following problem suppose a set of sensors is jointly trying to estimate a process one sensor takes a measurement at every time step and the measurements are then exchanged among all the sensors what is the sensor schedule that results in the minimum error covariance we describe a stochastic sensor selection strategy that is easy to implement and is computationally tractable the problem described above comes up in many domains out of which we discuss two in the sensor selection problem there are multiple sensors that cannot operate simultaneously e g sonars in the same frequency band thus measurements need to be scheduled in the sensor coverage problem a geographical area needs to be covered by mobile sensors each with limited range thus from every position the sensors obtain a different view point of the area and the sensors need to optimize their trajectories the algorithm is applied to these problems and illustrated through simple examples

ricci curvature for metric measure spaces via optimal transport we dene a notion of a measured length space x having nonnegative n ricci curvature for n 2 1 1 or having1 ricci curvature bounded below byk fork2 r the denitions are in terms of the displacement convexity of certain functions on the associated wasserstein metric space p2 x of probability measures we show that these properties are preserved under measured gromov hausdor limits we give geometric and analytic consequences this paper has dual goals one goal is to extend results about optimal transport from the setting of smooth riemannian manifolds to the setting of length spaces a second goal is to use optimal transport to give a notion for a measured length space to have ricci curvature bounded below we refer to 11 and 44 for background material on length spaces and optimal transport respectively further bibliographic notes on optimal transport are in appendix f in the present introduction we motivate the questions that we address and we state the main results to start on the geometric side there are various reasons to try to extend notions of curvature from smooth riemannian manifolds to more general spaces a fairly general setting is that of length spaces meaning metric spaces x d in which the distance between two points equals the inmum of the lengths of curves joining the points in the rest of this introduction we assume that x is a compact length space alexandrov gave a good notion of a length space having curvature bounded below by k with k a real number in terms of the geodesic triangles in x in the case of a riemannian manifold m with the induced length structure one recovers the riemannian notion of having sectional curvature bounded below by k length spaces with alexandrov curvature bounded below by k behave nicely with respect to the gromovhausdor topology on compact metric spaces modulo isometries they form

optimal periodic sensor scheduling in networks of dynamical systems we consider the problem of finding optimal time periodic sensor schedules for estimating the state of discrete time dynamical systems we assume that multiple sensors have been deployed and that the sensors are subject to resource constraints which limits the number of times each can be activated over one period of the periodic schedule we seek an algorithm that strikes a balance between estimation accuracy and total sensor activations over one period we make a correspondence between active sensors and the nonzero columns of the estimator gain we formulate an optimization problem in which we minimize the trace of the error covariance with respect to the estimator gain while simultaneously penalizing the number of nonzero columns of the estimator gain this optimization problem is combinatorial in nature and we employ the alternating direction method of multipliers admm to find its locally optimal solutions numerical results and comparisons with other sensor scheduling algorithms in the literature are provided to illustrate the effectiveness of our proposed method

optimization of observations a stochastic control approach we study a stochastic control problem for the optimization of observations in a partially observable stochastic system using a method of discontinuous time transformation we associate with the original problem with unbounded controls a problem that has bounded controls this latter problem allows us to construct nearly optimal nonanticipative lipschitz markov controls with finite observation power for the original problem since the controlled observation equation may degenerate we also derive a corresponding filtering result and show a separation property of the optimal controls

sensor selection for kalman filtering of linear dynamical systems complexity limitations and greedy algorithms abstract we consider the problem of selecting an optimal set of sensors to estimate the states of linear dynamical systems specifically the goal is to choose at design time a subset of sensors satisfying certain budget constraints from a given set in order to minimize the trace of the steady state a priori or a posteriori error covariance produced by a kalman filter we show that the a priori and a posteriori error covariance based sensor selection problems are both np hard even under the additional assumption that the system is stable we then provide bounds on the worst case performance of sensor selection algorithms based on the system dynamics and show that greedy algorithms are optimal for a certain class of systems however as a negative result we show that certain typical objective functions are not submodular or supermodular in general while this makes it difficult to evaluate the performance of greedy algorithms for sensor selection outside of certain special cases we show via simulations that these greedy algorithms perform well in practice

on complete manifolds with nonnegative ricci curvature complete open riemannian manifolds mn g with nonnegative sectional curvature are well understood the basic results are toponogov s splitting theorem and the soul theorem cg1 the splitting theorem has been extended to manifolds of nonnegative ricci curvature cg2 on the other hand the soul theorem does not extend even topologically according to recent examples in gm2 a different method to construct manifolds which carry a metric with ric 0 but no metric with nonnegative sectional curvature has been given by l berard bergery bb this leads to the question cf also y1 is there any finiteness result for complete riemannian manifolds with ric 0 the answer is certainly affirmative in the low dimensional special cases n 2 where all notions of curvature coincide and n 3 where nonnegative ricci curvature has been studied by means of stable minimal surfaces msy sy on the other hand j p sha and d g yang shy have constructed complete manifolds with strictly positive ricci curvature in higher dimensions for example they can choose the underlying space to be r4 x s3 with infinitely many copies of s3 x cp 2 attached to it by surgery cf also shy 1 it is therefore clear that any finiteness result for arbitrary dimensions requires additional assumptions the purpose of this paper is to establish the following main result

on the geometry of metric measure spaces ii we introduce a curvature dimension condition cd k n for metric measure spaces it is more restrictive than the curvature bound underline text curv left m text d m right k introduced in sturm k t 2006 on the geometry of metric measure spaces i acta math 196 65 131 which is recovered as the borderline case cd k the additional real parameter n plays the role of a generalized upper bound for the dimension for riemannian manifolds cd k n is equivalent to text ric m left xi xi right k left xi right 2 and dim m n the curvature dimension condition cd k n is stable under convergence for any triple of real numbers k n l the family of normalized metric measure spaces m d m with cd k n and diameter l is compact condition cd k n implies sharp version of the brunn minkowski inequality of the bishop gromov volume comparison theorem and of the bonnet myers theorem moreover it implies the doubling property and local scale invariant poincare inequalities on balls in particular it allows to construct canonical dirichlet forms with gaussian upper and lower bounds for the corresponding heat kernels

approximate optimal periodic scheduling of multiple sensors with constraints a constrained periodic multiple sensor scheduling problem is considered in this paper for each sensor constraints on dwell time and activation times are imposed at each time instant only one sensor can update its measurement with the estimator and the objective is to minimize the average state estimation error an approximation framework is proposed to calculate the objective function which transforms the original scheduling problem into an approximate optimal scheduling problem aosp an upper bound on the approximation error is presented to evaluate the performance of the framework to solve the aosp a necessary condition is first proposed on the optimal schedules when no constraints on activation times exist a dynamic programming based algorithm is devised to identify the optimal schedule with polynomial computational complexity when activation time constraints exist we show that the aosps can be solved by solving traveling salesman problems examples are provided to illustrate the proposed results

on ratios of harmonic functions abstract let u and v be harmonic functions in r n with the same zero set z we show that the ratio f of such functions is always well defined and is real analytic moreover it satisfies the maximum and minimum principles for n 3 we also prove the harnack inequality and the gradient estimate for the ratios of harmonic functions namely sup k f c inf k f sup k f c inf k f for any compact subset k of where the constant c depends on k z only in dimension two the first inequality follows from the boundary harnack principle and the second from the gradient estimate recently obtained by mangoubi it is an open question whether these inequalities remain true in higher dimensions n 4

hausdorff convergence and universal covers we prove that if y is the gromov hausdorff limit of a sequence of compact manifolds m n i with a uniform lower bound on ricci curvature and a uniform upper bound on diameter then y has a universal cover we then show that for i sufficiently large the fundamental group of mi has a surjective homeomorphism onto the group of deck transforms of y finally in the non collapsed case where the mi have an additional uniform lower bound on volume we prove that the kernels of these surjective maps are finite with a uniform bound on their cardinality a number of theorems are also proven concerning the limits of covering spaces and their deck transforms when the mi are only assumed to be compact length spaces with a uniform upper bound on diameter

convergence of pointed non compact metric measure spaces and stability of ricci curvature bounds and heat flows aim of this paper is to discuss convergence of pointed metric measure spaces in absence of any compactness condition we propose various definitions show that all of them are equivalent and that for doubling spaces these are also equivalent to the well known measured gromov hausdorff convergence then we show that the curvature conditions cd k and rcd k are stable under this notion of convergence and that the heat flow passes to the limit as well both in the wasserstein and in the l framework we also prove the variational convergence of cheeger energies in the naturally adapted mosco sense and the convergence of the spectra of the laplacian in the case of spaces either uniformly bounded or satisfying the rcd k condition with k 0 when applied to riemannian manifolds our results allow for sequences with diverging dimensions

on kalman filtering for detectable systems with intermittent observations we consider the problem of kalman filtering when observations are available according to a bernoulli process it is known that there exists a critical probability pc such that if measurements are available with probability greater than pc then the expected prediction covariance is bounded for all initial conditions otherwise it is unbounded for some initial conditions we show that when the system observation matrix restricted to the observable subspace is invertible the known lower bound on pc is in fact the exact critical probability this result is based on a novel decomposition of positive semidefinite matrices

lower bounds on ricci curvature and the almost rigidity of warped products the basic rigidity theorems for manifolds of nonnegative or positive ricci curvature are the volume cone implies metric cone theorem the maximal diameter theorem cg and the splitting theorem cg each asserts that if a certain geometric quantity volume or diameter is as large as possible relative to the pertinent lower bound on ricci curvature then the metric on the manifold in question is a warped product metric of a particular type in this paper we provide quantitative generalizations of the above mentioned results among the applications are the splitting theorem for gromovhausdorff limit spaces x where mn x ricmn i see fy other applications include the assertion that for complete manifolds m with ricmf 0 and euclidean volume growth all tangent cones at infinity are metric cones compare bkn ct p1 via resealing arguments there are also strong consequences for the local structure of manifolds whose ricci curvature satisfies a fixed lower bound and for their gromov hausdorff limits some of these are announced in ccol for a more detailed discussion see cco2 cco3 cco4 our work further develops and significantly extends techniques which were introduced in col co2 and significantly extended in co3 in order to prove certain stability conjectures of anderson cheeger gromov and perelman the results of col co3 were announced in co4 we briefly review some of those results let dgh denote the gromov hausdorff distance between metric spaces see glp let s denote the unit sphere and recall that s is the unique complete

localization and tensorization properties of the curvature dimension condition for metric measure spaces abstract this is an addendum to the paper k bacher k t sturm localization and tensorization properties of the curvature dimension condition for metric measure spaces j funct anal 259 2010 28 56 we prove the tensorization property for the curvature dimension condition add some detailed calculations including explicit dependence of constants and comment on assumptions and conjectures concerning the local to global statement in bacher and sturm 2010 1 and villani 2009 6 respectively

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

centroidal voronoi tessellations applications and algorithms a centroidal voronoi tessellation is a voronoi tessellation whose generating points are the centroids centers of mass of the corresponding voronoi regions we give some applications of such tessellations to problems in image compression quadrature finite difference methods distribution of resources cellular biology statistics and the territorial behavior of animals we discuss methods for computing these tessellations provide some analyses concerning both the tessellations and the methods for their determination and finally present the results of some numerical experiments