positive curvature local and global symmetry and fundamental groups a 1 invariant torus action on a manifold m is a t k action on the universal covering which extends to the action of a semi direct product 1 m t k in particular the t k action is the lift of a t k action on m if is the identity map the main result asserts that if a compact manifold m n of positive sectional curvature admits a 1 invariant isometric t k action then the fundamental group has a cyclic subgroup of index w n this refines the main result in ro1

minimal entropy and collapsing with curvature bounded from below we show that if a closed manifold m admits an structure not necessarily polarized possibly of rank zero then its minimal entropy vanishes in particular this is the case if m admits a non trivial s1 action as a corollary we obtain that the simplicial volume of a manifold admitting an structure is zero we also show that if m admits an structure then it collapses with curvature bounded from below this in turn implies that m collapses with bounded scalar curvature or equivalently its yamabe invariant is non negative we show that structures of rank zero appear rather frequently every compact complex elliptic surface admits one as well as any simply connected closed 5 manifold we use these results to study the minimal entropy problem we show the following two theorems suppose that m is a closed manifold obtained by taking connected sums of copies of s4 p2 overline mathbb c p 2 s2 s2and the k3 surface then m has zero minimal entropy moreover m admits a smooth riemannian metric with zero topological entropy if and only if m is diffeomorphic to s4 p2 s2 s2 p2 overline mathbb c p 2 or p2 p2 finally suppose that m is a closed simply connected 5 manifold then m has zero minimal entropy moreover m admits a smooth riemannian metric with zero topological entropy if and only if m is diffeomorphic to s5 s3 s2 then on trivial s3 bundle over s2 or the wu manifold su 3 so 3

design of networked control systems with packet dropouts this note is concerned with stability and controller design of networked control systems ncss with packet dropouts new ncs models are provided considering both single and multiple packet transmissions both sensor to controller s c and controller to actuator c a packet dropouts are modeled and their history behavior is described by different independent markov chains in term of the given models sufficient conditions for stochastic stability are derived in the form of linear matrix inequalities lmis and corresponding control laws are given numerical examples illustrate the effectiveness of the results

robust h finite horizon filtering with randomly occurred nonlinearities and quantization effects in this paper the robust h finite horizon filtering problem is investigated for discrete time varying stochastic systems with polytopic uncertainties randomly occurred nonlinearities as well as quantization effects the randomly occurred nonlinearity which describes the phenomena of a nonlinear disturbance appearing in a random way is modeled by a bernoulli distributed white sequence with a known conditional probability a new robust h filtering technique is developed for the addressed ito type discrete time varying stochastic systems such a technique relies on the forward solution to a set of recursive linear matrix inequalities and is therefore suitable for on line computation it is worth mentioning that in the filtering process the information of both the current measurement and the previous state estimate is employed to estimate the current state finally a simulation example is exploited to show the effectiveness of the method proposed in this paper

a switched state feedback law for the stabilization of lti systems inspired by prior work in the design of switched feedback controllers for second order systems we develop a switched state feedback control law for the stabilization of linear time invariant lti systems of arbitrary dimension the control law operates by switching between two static gain vectors in such a way that the state trajectory is driven onto a stable n 1 dimensional hyperplane where n represents the system dimension we begin by briefly examining relevant geometric properties of the phase portraits in the case of two dimensional systems to develop intuition and we then show how these geometric properties can be expressed as algebraic constraints on the switched vector fields that are applicable to lti systems of arbitrary dimension we then derive necessary and sufficient conditions to ensure stabilizability of the resulting switched system using the proposed approach characterized primarily by simple conditions on eigenvalues and describe an explicit procedure for designing stabilizing controllers we then show how the newly developed control law can be applied to the problem of minimizing the maximal lyapunov exponent of the corresponding closed loop state trajectories and we illustrate the closed loop transient performance of these switched state feedback controllers via multiple examples

quantized output feedback control for networked control systems this paper addresses the problem of output feedback control for networked control systems ncss with limited communication capacity firstly we propose a new model to describe the non ideal network conditions and the input output state quantization of the ncss in a unified framework secondly based on our newly proposed model and an improved separation lemma the observer based controller is developed for the asymptotical stabilization of the ncss which are shown in terms of nonlinear matrices inequalities the nonlinear problems can be computed through solving a convex optimization problems and the observed and controller gains could be derived by solving a set of linear matrix inequalities thirdly two simulation examples are given to demonstrate the effectiveness of the proposed method

stabilization of discrete time markovian jump linear systems via time delayed controllers this paper is concerned with the stabilization problem for a class of discrete time markovian jump linear systems with time delays both in the system state and in the mode signal the delay in the system state may be time varying the delay in the mode signal is manifested as a constant mismatch of the modes between the controller and the system we first show that the resulting closed loop system is a time varying delayed markovian jump linear system with extended state space then a sufficient condition is proposed for the design of a controller such that the closed loop system is stochastically stable finally numerical simulation is used to illustrate the developed theory

limits and degenerations of unitary conformal field theories in the present paper degeneration phenomena in conformal field theories are studied for this purpose a notion of convergent sequences of cfts is introduced properties of the resulting limit structure are used to associate geometric degenerations to degenerating sequences of cfts which as familiar from large volume limits of non linear sigma models can be regarded as commutative degenerations of the corresponding quantum geometries as an application the large level limit of the a series of unitary virasoro minimal models is investigated in detail in particular its geometric interpretation is determined

output feedback stabilization of networked control systems with random delays modeled by markov chains this note investigates the output feedback stabilization of networked control systems ncss the sensor to controller s c and controller to actuator c a random network induced delays are modeled as markov chains the focus is on the design of a two mode dependent controller that depends on not only the current s c delay but also the most recent available c a delay at the controller node the resulting closed loop system is transformed to a special discrete time jump linear system then the sufficient and necessary conditions for the stochastic stability are established further the output feedback controller is designed via the iterative linear matrix inequality lmi approach simulation examples illustrate the effectiveness of the proposed method

brief paper robust filtering with stochastic nonlinearities and multiple missing measurements this paper is concerned with the filtering problem for a class of discrete time uncertain stochastic nonlinear time delay systems with both the probabilistic missing measurements and external stochastic disturbances the measurement missing phenomenon is assumed to occur in a random way and the missing probability for each sensor is governed by an individual random variable satisfying a certain probabilistic distribution over the interval 01 such a probabilistic distribution could be any commonly used discrete distribution over the interval 01 the multiplicative stochastic disturbances are in the form of a scalar gaussian white noise with unit variance the purpose of the addressed filtering problem is to design a filter such that for the admissible random measurement missing stochastic disturbances norm bounded uncertainties as well as stochastic nonlinearities the error dynamics of the filtering process is exponentially mean square stable by using the linear matrix inequality lmi method sufficient conditions are established that ensure the exponential mean square stability of the filtering error and then the filter parameters are characterized by the solution to a set of lmis illustrative examples are exploited to show the effectiveness of the proposed design procedures

robust h control for a class of nonlinear discrete time delay stochastic systems with missing measurements this paper is concerned with the problem of robust h output feedback control for a class of uncertain discrete time delayed nonlinear stochastic systems with missing measurements the parameter uncertainties enter into all the system matrices the time varying delay is unknown with given low and upper bounds the nonlinearities satisfy the sector conditions and the missing measurements are described by a binary switching sequence that obeys a conditional probability distribution the problem addressed is the design of an output feedback controller such that for all admissible uncertainties the resulting closed loop system is exponentially stable in the mean square for the zero disturbance input and also achieves a prescribed h performance level by using the lyapunov method and stochastic analysis techniques sufficient conditions are first derived to guarantee the existence of the desired controllers and then the controller parameters are characterized in terms of linear matrix inequalities lmis a numerical example is exploited to show the usefulness of the results obtained

collapsing with a lower bound on the curvature operator cheeger and gromov showed that f structures are related to collapse with a double sided curvature bound we define fibered f structures and extend some of the cheeger gromov results to the setting of collapse with a lower bound on the curvature operator

on the upper estimate of the heat kernel of a complete riemannian manifold let m be a complete non compact riemannian manifold whose sectional curvature is bounded between two constants k and k then one expects that the heat diffusion in such a manifold behaves like the heat diffusion in euclidean space the purpose of this paper is to give a justification of such a statement in 5 j cheeger and the third author found a lower estimate of the heat kernel by comparing it with the heat kernel of the space form whose curvature is the lower bound of the curvature of the manifold this lower estimate is sharp if we insist the dependence should be on the lower bound of the ricci curvature alone it remains to give an upper estimate of the heat kernel one does not however expect to have a comparison theorem for the upper bound because it is more sensitive to the geometry of the manifold in fact the heat kernel of the upper hialf space and the heat kernel of the complete hyperbolic manifold with finite volume have quite different behavior this is reflected by the fact that the laplacian has no discrete spectrum in the first case while infinite number of discrete eigenvalues may exist in the latter case what we will prove here is that in any case the heat kernel has to decay in a manner similar to the euclidean heat kernel thus we will prove that for any constant c 4 there exists c1 depending on c t the bound of the curvature of m and x so that for all t e 0 t

variance constrained cal h _ infty filtering for a class of nonlinear time varying systems with multiple missing measurements the finite horizon case this paper is concerned with the robust h finite horizon filtering problem for a class of uncertain nonlinear discrete time varying stochastic systems with multiple missing measurements and error variance constraints all the system parameters are time varying and the uncertainty enters into the state matrix the measurement missing phenomenon occurs in a random way and the missing probability for each sensor is governed by an individual random variable satisfying a certain probabilistic distribution in the interval the stochastic nonlinearities under consideration here are described by statistical means which can cover several classes of well studied nonlinearities sufficient conditions are derived for a finite horizon filter to satisfy both the estimation error variance constraints and the prescribed h performance requirement these conditions are expressed in terms of the feasibility of a series of recursive linear matrix inequalities rlmis simulation results demonstrate the effectiveness of the developed filter design scheme

robust stability and stabilization of uncertain discrete time markovian jump linear systems this note deals with robust stability and control of uncertain discrete time linear systems with markovian jumping parameters systems with polytopic type parameter uncertainty in either the state space model matrices or in the transition probability matrix of the markov process are considered this note develops methods of robust stability analysis and robust stabilization in the mean square sense which are dependent on the system uncertainty the design of both mode dependent and mode independent control laws is addressed the proposed methods are given in terms of linear matrix inequalities numerical examples are provided to demonstrate the effectiveness of the derived results

l2 l filter design for discrete time singular markovian jump systems with time varying delays the problem of l 2 l filter design for discrete time singular systems with markovian jump and time varying delays is investigated in this paper by using the delay partitioning technique a delay dependent condition is established to guarantee the filtering error systems to be stochastically admissible and achieve a prescribed l 2 l performance index based on the derived condition the full order and reduced order filters with mode independent characterization are designed in a unified framework the corresponding filter parameters can be obtained by solving a set of linear matrix inequalities the reported results not only depend upon the delay but also depend upon the partitioning which aims at reducing the conservatism two numerical examples are provided to illustrate the effectiveness of the proposed methods

three manifolds with many flat planes we discuss the rigidity or lack thereof imposed by dierent no tions of having an abundance of zero curvature planes on a complete riemann ian 3 manifold we prove a rank rigidity theorem for complete 3 manifolds showing that having higher rank is equivalent to having reducible universal covering we also study 3 manifolds such that every tangent vector is con tained in a at plane including examples with irreducible universal covering and discuss the eect of nite volume and real analiticity assumptions

exponential stability of stochastic delay interval systems with markovian switching in the past few years a lot of research has been dedicated to the stability of interval systems as well as the stability of systems with markovian switching however little research has been on the stability of interval systems with markovian switching which is the topic of this paper the system discussed is the stochastic delay interval system with markovian switching it is a very advanced system and takes all the features of interval systems ito equations and markovian switching as well as time lag into account the theory developed is applicable in many different and complicated situations so the importance of the paper is clear

a riemann singularity theorem for integral curves we prove results generalizing the classical riemann singularity theorem to the case of integral singular curves the main result is a computation of the multiplicity of the theta divisor of an integral nodal curve at an arbitrary point we also suggest a general formula for the multiplicity of the theta divisor of a singular integral curve at a point and present some evidence that this formula should hold our results give a partial answer to a question posed by lucia caporaso in a recent paper

splittings and cr structures for manifolds with nonpositive sectional curvature let m n denote the universal covering space of a compact riemannian manifold m n with sectional curvature 1 k mn 0 we show that a collection of deck transformations of m n satisfying certain metric dependent conditions determines an open dense subset of m n at every point of which there exists a local isometric splitting with nontrivial flat factor such a collection which we call an abelian structure also gives rise to an essentially canonical cr structure in the sense of buyalo i e an atlas for an injective f structure for which additional conditions hold it follows in particular that the minimal volume of m n vanishes we show that an abelian structure exists if the injectivity radius at all points of m n is less than e n 0 this yields a conjecture of buyalo as well as a strengthened version of the conclusion of gromov s gap conjecture in our special situation in addition we observe that abelian structures on nonpositively curved manifolds have certain stability properties under suitably controlled changes of metric

robust h sub spl infin filtering for stochastic time delay systems with missing measurements in this paper the robust hinfin filtering problem is studied for stochastic uncertain discrete time delay systems with missing measurements the missing measurements are described by a binary switching sequence satisfying a conditional probability distribution we aim to design filters such that for all possible missing observations and all admissible parameter uncertainties the filtering error system is exponentially mean square stable and the prescribed hinfin performance constraint is met in terms of certain linear matrix inequalities lmis sufficient conditions for the solvability of the addressed problem are obtained when these lmis are feasible an explicit expression of a desired robust hinfin filter is also given an optimization problem is subsequently formulated by optimizing the hinfin filtering performances finally a numerical example is provided to demonstrate the effectiveness and applicability of the proposed design approach

a boundary of the set of the riemannian manifolds with bounded curvatures and diameters on etudie la structure de la fermeture de l ensemble constitue par des classes d isometrie de varietes de riemann a courbures et diametres bornes

the almost cyclicity of the fundamental groups of positively curved manifolds recall that a pure f structure is a kind of generalized torus action the main result asserts that if a compact positively curved manifold mn admits an invariant pure f structure such that each orbit has positive dimension then the fundamental group has a finite cyclic subgroup with index less than wn a constant depending only on n as an application we conclude that for all 0 1 the fundamental group of a pinched n manifold either has a cyclic subgroup with index less than wn or has order less than w n a constant depending only on n and in particular this substantially improves the main result in ro1

on the discriminant locus of a lagrangian fibration let x rightarrow mathbb p n be an irreducible holomorphic symplectic manifold of dimension 2n fibred over mathbb p n matsushita proved that the generic fibre is a holomorphic lagrangian abelian variety in this article we study the discriminant locus delta subset mathbb p n parametrizing singular fibres our main result is a formula for the degree of leading to bounds on the degree when x is a fourfold

brief paper stability and stabilization of markovian jump linear systems with partly unknown transition probabilities in this paper the stability and stabilization problems of a class of continuous time and discrete time markovian jump linear system mjls with partly unknown transition probabilities are investigated the system under consideration is more general which covers the systems with completely known and completely unknown transition probabilities as two special cases the latter is hereby the switched linear systems under arbitrary switching moreover in contrast with the uncertain transition probabilities studied recently the concept of partly unknown transition probabilities proposed in this paper does not require any knowledge of the unknown elements the sufficient conditions for stochastic stability and stabilization of the underlying systems are derived via lmis formulation and the relation between the stability criteria currently obtained for the usual mjls and switched linear systems under arbitrary switching are exposed by the proposed class of hybrid systems two numerical examples are given to show the validity and potential of the developed results

the second twisted betti number and the convergence of collapsing riemannian manifolds let mi overset d gh longrightarrow x denote a sequence of n manifolds converging to a compact metric space x in the gromov hausdorff topology such that the sectional curvature is bounded in absolute value and dim x n we prove the following stability result if the fundamental groups of mi are torsion groups of uniformly bounded exponents and the second twisted betti numbers of mi vanish then there is a manifold m and a sequence of diffeomorphisms from m to a subsequence of mi such that the distance functions of the pullback metrics converge to a pseudo metric in c0 norm furthermore m admits a foliation with leaves diffeomorphic to flat manifolds not necessarily compact such that a vector is tangent to a leaf if and only if its norm converges to zero with respect to the pullback metrics these results lead to a few interesting applications

global synchronization for discrete time stochastic complex networks with randomly occurred nonlinearities and mixed time delays in this paper the problem of stochastic synchronization analysis is investigated for a new array of coupled discrete time stochastic complex networks with randomly occurred nonlinearities rons and time delays the discrete time complex networks under consideration are subject to 1 stochastic nonlinearities that occur according to the bernoulli distributed white noise sequences 2 stochastic disturbances that enter the coupling term the delayed coupling term as well as the overall network and 3 time delays that include both the discrete and distributed ones note that the newly introduced rons and the multiple stochastic disturbances can better reflect the dynamical behaviors of coupled complex networks whose information transmission process is affected by a noisy environment e g internet based control systems by constructing a novel lyapunov like matrix functional the idea of delay fractioning is applied to deal with the addressed synchronization analysis problem by employing a combination of the linear matrix inequality lmi techniques the free weighting matrix method and stochastic analysis theories several delay dependent sufficient conditions are obtained which ensure the asymptotic synchronization in the mean square sense for the discrete time stochastic complex networks with time delays the criteria derived are characterized in terms of lmis whose solution can be solved by utilizing the standard numerical software a simulation example is presented to show the effectiveness and applicability of the proposed results

on the topology of positively curved 4 manifolds with symmetry soit m une variete compacte a courbure positive orientable a 4 dimensions si m a un champ de vecteurs de killing non trivial alors m est homeomorphe a s 4 ou cp 2

reciprocally convex approach to stability of systems with time varying delays whereas the upper bound lemma for matrix cross product introduced by park 1999 and modified by moon park kwon and lee 2001 plays a key role in guiding various delay dependent criteria for delayed systems the jensen inequality has become an alternative as a way of reducing the number of decision variables it directly relaxes the integral term of quadratic quantities into the quadratic term of the integral quantities resulting in a linear combination of positive functions weighted by the inverses of convex parameters this paper suggests the lower bound lemma for such a combination which achieves performance behavior identical to approaches based on the integral inequality lemma but with much less decision variables comparable to those based on the jensen inequality lemma

h_ infty filtering of discrete time switched systems with state delays via switched lyapunov function approach in this note the problem of hinfin filtering for a class of discrete time switched systems with state delays is investigated attention is focused on the design of a stable filter guaranteeing a prescribed noise attenuation level in the hinfin sense by using switched lyapunov functionals sufficient conditions for the solvability of this problem are obtained in terms of linear matrix inequalities lmis by solving which a desired hinfin filter can be constructed a numerical example is provided to demonstrate the effectiveness of the proposed techniques

positively curved manifolds with maximal discrete symmetry rank let m be a closed simply connected n manifold of positive sectional curvature we determine its homeomorphism or homotopic type if m also admits an isometric elementary p group action of large rank our main results are there exists a constant p n 0 such that 1 if m 2 n admits an effective isometric inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i action for a prime p p n then k n and implies that m 2 n is homeomorphic to a sphere or a complex projective space 2 if m 2 n 1 admits an isometric s 1 inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 02i action for a prime p p n then k n and implies that m is homeomorphic to a sphere 3 for m in 1 or 2 if n 7 and k 3 n 4 2 then m is homeomorphic to a sphere or homotopic to a complex projective space

robust cal h _ infty filtering for markovian jump systems with randomly occurring nonlinearities and sensor saturation the finite horizon case this paper addresses the robust h filtering problem for a class of discrete time varying markovian jump systems with randomly occurring nonlinearities and sensor saturation two kinds of transition probability matrices for the markovian process are considered namely the one with polytopic uncertainties and the one with partially unknown entries the nonlinear disturbances are assumed to occur randomly according to stochastic variables satisfying the bernoulli distributions the main purpose of this paper is to design a robust filter over a given finite horizon such that the h disturbance attenuation level is guaranteed for the time varying markovian jump systems in the presence of both the randomly occurring nonlinearities and the sensor saturation sufficient conditions are established for the existence of the desired filter satisfying the h performance constraint in terms of a set of recursive linear matrix inequalities simulation results demonstrate the effectiveness of the developed filter design scheme

stability analysis and control synthesis for switched systems a switched lyapunov function approach this paper addresses the problem of stability analysis and control synthesis of switched systems in the discrete time domain the approach followed in this paper looks at the existence of a switched quadratic lyapunov function to check asymptotic stability of the switched system under consideration two different linear matrix inequality based conditions allow to check the existence of such a lyapunov function the first one is classical while the second is new and uses a slack variable which makes it useful for design problems these two conditions are proved to be equivalent for stability analysis investigating the static output feedback control problem we show that the second condition is in this case less conservative the reduction of the conservatism is illustrated by a numerical evaluation

rationality of geometric signatures of complete 4 manifolds according to cg4 and cfg the complete manifolds with bounded sectional curvature and finite volume admit positive rank f structures near infinity in this paper we show that in dimension four if the manifolds also have bounded covering geometry near infinity then there exist f structures with special topological properties f structures with these properties cannot be constructed solely by means of the general methods in cg4 and cfg using these special properties we prove a conjecture of cheeger gromov on the rationality of the geometric signatures in the four dimensional case

a mass for alf manifolds we prove a positive mass theorem on alf manifolds i e complete noncompact manifolds that are asymptotic to a circle fibration over a euclidean base with fibers of asymptotically constant length

h sub spl infin control for networked systems with random communication delays this note is concerned with a new controller design problem for networked systems with random communication delays two kinds of random delays are simultaneously considered i from the controller to the plant and ii from the sensor to the controller via a limited bandwidth communication channel the random delays are modeled as a linear function of the stochastic variable satisfying bernoulli random binary distribution the observer based controller is designed to exponentially stabilize the networked system in the sense of mean square and also achieve the prescribed h sub spl infin disturbance attenuation level the addressed controller design problem is transformed to an auxiliary convex optimization problem which can be solved by a linear matrix inequality lmi approach an illustrative example is provided to show the applicability of the proposed method

obstructions to positive curvature and symmetry we discuss new obstructions to positive sectional curvature and symmetry the main result asserts that the index of the dirac operator twisted with the tangent bundle vanishes on a 2 connected manifold of dimension 8 if the manifold admits a metric of positive sectional curvature and isometric effective s1 action the proof relies on the rigidity theorem for elliptic genera and properties of totally geodesic submanifolds

ends of finite volume nonpositively curved manifolds we study complete finite volume n manifolds m of bounded nonpositive sectional curvature a classical theorem of m gromov says that if such m has negative curvature then it is homeomorphic to the interior of a compact manifold with boundary and we denote this boundary partial m if n ge 3 we prove that the universal cover of the boundary widetilde partial m and also the pi 1m cover of the boundary partial widetilde m have vanishing n 2 dimensional homology for n 4 the first of these recovers a result of nguy tilde hat mathrm e n phan saying that each component of the boundary partial m is aspherical for any n ge 3 the second of these implies the vanishing of the first group cohomology group with group ring coefficients h 1 b pi 1m mathbb z pi 1m 0 a consequence is that pi 1m is freely indecomposable these results extend to manifolds m of bounded nonpositive curvature if we assume that m is homeomorphic to the interior of a compact manifold with boundary our approach is a form of homological collapse for ends of finite volume manifolds of bounded nonpositive curvature this paper is very much influenced by earlier yet still unpublished work of nguy tilde hat mathrm e n phan

collapsing riemannian manifolds while keeping their curvature bounded i soit y n une variete de riemann connexe complete et p y n le rayon d injectivite i p de l application exponentielle en p est defini comme etant le plus petit r tel que exp p b r p n est pas un diffeomorphisme sur son image on etudie la situation dans laquelle la taille du rayon d injectivite est petite relativement a la courbure

stability and stabilization of delayed t s fuzzy systems a delay partitioning approach this paper proposes a new approach namely the delay partitioning approach to solving the problems of stability analysis and stabilization for continuous time delay takagi sugeno fuzzy systems based on the idea of delay fractioning a new method is proposed for the delay dependent stability analysis of fuzzy time delay systems due to the instrumental idea of delay partitioning the proposed stability condition is much less conservative than most of the existing results the conservatism reduction becomes more obvious with the partitioning getting thinner based on this the problem of stabilization via the so called parallel distributed compensation scheme is also solved both the stability and stabilization results are further extended to time delay fuzzy systems with time varying parameter uncertainties all the results are formulated in the form of linear matrix inequalities lmis which can be readily solved via standard numerical software the advantage of the results proposed in this paper lies in their reduced conservatism as shown via detailed illustrative examples the idea of delay partitioning is well demonstrated to be efficient for conservatism reduction and could be extended to solving other problems related to fuzzy delay systems

removing point singularities of riemannian manifolds we study the behavior of geodesics passing through a point singularity of a riemannian manifold in particular we show that if the curvature does not blow up too rapidly near the singularity then the singularity is at worst an orbifold singularity the idea is to construct the exponential map centered at a singularity since there is no tangent space at the singularity a surrogate is needed we show that the vector space of radially parallel vector fields is well defined and that there is a correspondence between unit radially parallel vector fields and geodesics emanating from the singular point

stability and stabilizability of switched linear systems a survey of recent results during the past several years there have been increasing research activities in the field of stability analysis and switching stabilization for switched systems this paper aims to briefly survey recent results in this field first the stability analysis for switched systems is reviewed we focus on the stability analysis for switched linear systems under arbitrary switching and we highlight necessary and sufficient conditions for asymptotic stability after a brief review of the stability analysis under restricted switching and the multiple lyapunov function theory the switching stabilization problem is studied and a variety of switching stabilization methods found in the literature are outlined then the switching stabilizability problem is investigated that is under what condition it is possible to stabilize a switched system by properly designing switching control laws note that the switching stabilizability problem has been one of the most elusive problems in the switched systems literature a necessary and sufficient condition for asymptotic stabilizability of switched linear systems is described here

analysis and synthesis of randomly switched systems with known sojourn probabilities in this paper a new approach is proposed and investigated for the stability analysis and stabilizing controller design of randomly switched linear discrete systems the approach is based on sojourn probabilities and it is assumed that these probabilities are known a prior a new lyapunov functional is constructed and two main theorems are proved in this paper theorem 1 gives a sufficient condition for a switched system with known sojourn probabilities to be mean square stable theorem 2 gives a sufficient condition for the design of a stabilizing controller the applications of these theorems and the corresponding corollary and lemma are demonstrated by three numerical examples finally some future research is proposed

variance constrained filtering for uncertain stochastic systems with missing measurements in this note we consider a new filtering problem for linear uncertain discrete time stochastic systems with missing measurements the parameter uncertainties are allowed to be norm bounded and enter into the state matrix the system measurements may be unavailable i e missing data at any sample time and the probability of the occurrence of missing data is assumed to be known the purpose of this problem is to design a linear filter such that for all admissible parameter uncertainties and all possible incomplete observations the error state of the filtering process is mean square bounded and the steady state variance of the estimation error of each state is not more than the individual prescribed upper bound it is shown that the addressed filtering problem can effectively be solved in terms of the solutions of a couple of algebraic riccati like inequalities or linear matrix inequalities the explicit expression of the desired robust filters is parameterized and an illustrative numerical example is provided to demonstrate the usefulness and flexibility of the proposed design approach

nilpotent structures and invariant metrics on collapsed manifolds let mn be a complete riemannian manifold of bounded curvature say iki 0 we put mn w n c u fn c where 1n e consists of those points at which the injectivity radius of the exponential map is e the complementary set n c is called the e collapsed part of mn if x e 4 n r e then the metric ball bx r is quasi isometric with small distortion to the flat ball b0 r in the euclidean space rn after slightly

collapsing of abelian fibered calabi yau manifolds we study the collapsing behavior of ricci flat kahler metrics on a projective calabi yau manifold which admits an abelian fibration when the volume of the fibers approaches zero we show that away from the critical locus of the fibration the metrics collapse with locally bounded curvature and along the fibers the rescaled metrics become flat in the limit the limit metric on the base minus the critical locus is locally isometric to an open dense subset of any gromov hausdorff limit space of the ricci flat metrics we then apply these results to study metric degenerations of families of polarized hyperkahler manifolds in the large complex structure limit in this setting we prove an analogue of a result of gross and wilson for k3 surfaces which is motivated by the strominger yau zaslow picture of mirror symmetry