mixed h2 h control of markovian jump time delay systems with uncertain transition probabilities this study addresses the robust stability problem of markovian jump time delay systems with mixed h2 and h control and uncertain transition probabilities in the discrete time domain the transition probabilities are considered to be uncertain but bounded by proposing a new lyapunov functional and applying the free weighting matrix technique we design the mode dependent mixed h2 h controller such that the resultant closed loop systems possess stochastic stability and are prescribed with h performance indices these results generalize several results reported in previous literature which consider the markovian transition probabilities to be known a priori or partially unknown transition probabilities numerical examples and a practical motion control system are presented to illustrate the effectiveness of this technique

dti segmentation using an information theoretic tensor dissimilarity measure in recent years diffusion tensor imaging dti has become a popular in vivo diagnostic imaging technique in radiological sciences in order for this imaging technique to be more effective proper image analysis techniques suited for analyzing these high dimensional data need to be developed in this paper we present a novel definition of tensor distance grounded in concepts from information theory and incorporate it in the segmentation of dti in a dti the symmetric positive definite spd diffusion tensor at each voxel can be interpreted as the covariance matrix of a local gaussian distribution thus a natural measure of dissimilarity between spd tensors would be the kullback leibler kl divergence or its relative we propose the square root of the j divergence symmetrized kl between two gaussian distributions corresponding to the diffusion tensors being compared and this leads to a novel closed form expression for the distance as well as the mean value of a dti unlike the traditional frobenius norm based tensor distance our distance is affine invariant a desirable property in segmentation and many other applications we then incorporate this new tensor distance in a region based active contour model for dti segmentation synthetic and real data experiments are shown to depict the performance of the proposed model

uniform convexity and smoothness and their applications in finsler geometry we generalize the alexandrov toponogov comparison theorems to finsler manifolds under suitable upper lower resp bounds on the flag and tangent curvatures together with the 2 uniform convexity smoothness resp of tangent spaces we show the 2 uniform convexity smoothness resp of finsler manifolds as applications we prove the almost everywhere existence of the second order differentials of semi convex functions and of c concave functions with the quadratic cost function

generalized space forms spaces with radially symmetric curvature at base point p are shown to be diffeomorphic to space forms furthermore they are either isometric to r n or s n under a radially symmetric metric to rp n with riemannian universal covering of s n equipped with a radially symmetric metric or else have constant curvature outside a metric ball of radius equal to the injectivity radius at p

simulating alternating tree automata by nondeterministic automata new results and new proofs of the theorems of rabin mcnaughton and safra abstract we give a proof that alternating tree automata can be simulated by nondeterministic tree automata which yields new complexity results and a unified proof of the theorems of rabin mcnaughton and safra we also give a simple axiomatic framework for uniformizing strategies

total curvatures of model surfaces control topology of complete open manifolds with radial curvature bounded below ii we prove as our main theorem the finiteness of topological type of a complete open riemannian manifold m with a base point p m whose radial curvature at p is bounded from below by that of a non compact model surface of revolution m which admits a finite total curvature and has no pair of cut points in a sector here a sector is by definition a domain cut off by two meridians emanating from the base point p m notice that our model m does not always satisfy the diameter growth condition introduced by abresch and gromoll in order to prove the main theorem we need a new type of the toponogov comparison theorem as an application of the main theorem we present a partial answer to milnor s open conjecture on the fundamental group of complete open manifolds

the algorithmic analysis of hybrid systems we present a general framework for the formal specification and algorithmic analysis of hybrid systems a hybrid system consists of a discrete program with an analog environment we model hybrid systems as finite automata equipped with variables that evolve continuously with time according to dynamical laws for verification purposes we restrict ourselves to linear hybrid systems where all variables follow piecewise linear trajectories we provide decidability and undecidability results for classes of linear hybrid systems and we show that standard program analysis techniques can be adapted to linear hybrid systems in particular we consider symbolic model checking and minimization procedures that are based on the reachability analysis of an infinite state space the procedures iteratively compute state sets that are definable as unions of convex polyhedra in multidimensional real space we also present approximation techniques for dealing with systems for which the iterative procedures do not converge

optimal control of markov decision processes with linear temporal logic constraints in this paper we develop a method to automatically generate a control policy for a dynamical system modeled as a markov decision process mdp the control specification is given as a linear temporal logic ltl formula over a set of propositions defined on the states of the mdp motivated by robotic applications requiring persistent tasks such as environmental monitoring and data gathering we synthesize a control policy that minimizes the expected cost between satisfying instances of a particular proposition over all policies that maximize the probability of satisfying the given ltl specification our approach is based on the definition of a novel optimization problem that extends the existing average cost per stage problem we propose a sufficient condition for a policy to be optimal and develop a dynamic programming algorithm that synthesizes a policy that is optimal for a set of ltl specifications

gradient flows on wasserstein spaces over compact alexandrov spaces we establish the existence of euclidean tangent cones on wasserstein spaces over compact alexandrov spaces of curvature bounded below by using this riemannian structure we formulate and construct gradient flows of functions on such spaces if the underlying space is a riemannian manifold of nonnegative sectional curvature then our gradient flow of the free energy produces a solution of the linear fokker planck equation

experiments with deterministic ω automata for formulas of linear temporal logic this paper addresses the problem of generating deterministic automata for formulas of linear temporal logic which can be solved by applying well known algorithms to construct a nondeterministic buchi automaton for the given formula on which we then apply a determinization algorithm we study here in detail safra s determinization algorithm present several heuristics that attempt to decrease the size of the resulting automata and report on experimental results

affine invariant segmentation by variational method affine invariance viewed as a simplified projective invariance is an essential requirement for analysis of natural scenes we propose an affine invariant analogue of mumford and shah energy functional to segment images and we discuss it we present a simple multiscale algorithm to minimize it based on region growing methods and we display our first numerical experiments

a fully automated framework for control of linear systems from temporal logic specifications we consider the following problem given a linear system and a linear temporal logic ltl formula over a set of linear predicates in its state variables find a feedback control law with polyhedral bounds and a set of initial states so that all trajectories of the closed loop system satisfy the formula our solution to this problem consists of three main steps first we partition the state space in accordance with the predicates in the formula and construct a transition system over the partition quotient which captures our capability of designing controllers second using a procedure resembling model checking we determine runs of the transition system satisfying the formula third we generate the control strategy illustrative examples are included

compactification and maximal diameter theorem for noncompact manifolds with radial curvature bounded below let m be a complete open n manifold with a base point p at which the radial sectional curvature along every minimizing geodesic emanating from p is bounded below by the radial curvature function of a model surface we discuss the maximal diameter theorem for the compactification of m by attaching the ideal boundary under certain conditions we prove that p becomes a pole and that m is isometric to the n model

scale sets image analysis this paper introduces a multi scale theory of piecewise image modelling called the scale sets theory and which can be regarded as a region oriented scale space theory the first part of the paper studies the general structure of a geometrically unbiased region oriented multi scale image description and introduces the scale sets representation a representation which allows to handle such a description exactly the second part of the paper deals with the way scale sets image analyses can be built according to an energy minimization principle we consider a rather general formulation of the partitioning problem which involves minimizing a two term based energy of the form c d where d is a goodness of fit term and c is a regularization term we describe the way such energies arise from basic principles of approximate modelling and we relate them to operational rate distorsion problems involved in lossy compression problems we then show that an important subset of these energies constitutes a class of multi scale energies in that the minimal cut of a hierarchy gets coarser and coarser as parameter increases this allows us to devise a fast dynamic programming procedure to find the complete scale sets representation of this family of minimal cuts considering then the construction of the hierarchy from which the minimal cuts are extracted we end up with an exact and parameter free algorithm to build scale sets image descriptions whose sections constitute a monotone sequence of upward global minima of a multi scale energy which is called the scale climbing algorithm this algorithm can be viewed as a continuation method along the scale dimension or as a minimum pursuit along the operational rate distorsion curve furthermore the solution verifies a linear scale invariance property which allows to completely postpone the tuning of the scale parameter to a subsequent stage for computational reasons the scale climbing algorithm is approximated by a pair wise region merging scheme however the principal properties of the solutions are kept some results obtained with mumford shah s piece wise constant model and a variant are provided and different applications of the proposed multi scale analyses are finally sketched

formal verification of stochastic max plus linear systems this work investigates the computation of finite abstractions of stochastic max plus linear smpl systems and their formal verification against general bounded time linear temporal specifications smpl systems are probabilistic extensions of discrete event mpl systems which are widely employed for modeling engineering systems dealing with practical timing and synchronization issues departing from the standard existing approaches for the analysis of smpl systems we newly propose to construct formal finite abstractions of a given smpl system the smpl system is first re formulated as a discrete time markov process then abstracted as a finite state markov chain mc the derivation of precise guarantees on the level of the introduced formal approximation allows us to probabilistically model check the obtained mc against bounded time linear temporal specifications which are of rather general applicability and to reliably export the obtained results over the original smpl system the approach is practically implemented on a dedicated software and is elucidated and run over numerical examples

three partition refinement algorithms we present improved partition refinement algorithms for three problems lexicographic sorting relational coarsest partition and double lexical ordering our double lexical ordering algorithm uses a new efficient method for unmerging two sorted sets

a unified translation of linear temporal logic to ω automata we present a unified translation of ltl formulas into nondeterministic buchi automata limit deterministic buchi automata ldba and deterministic rabin automata dra the translations yield automata of asymptotically optimal size double or single exponential respectively all three translations are derived from one single master theorem of purely logical nature the master theorem decomposes the language of a formula into a positive boolean combination of languages that can be translated into automata by elementary means all three translations are direct and no intermediate constructions are used

volume collapsed three manifolds with a lower curvature bound in this paper we determine the topology of three dimensional complete orientable riemannian manifolds with a uniform lower bound of sectional curvature whose volume is sufficiently small

total curvatures of model surfaces control topology of complete open manifolds with radial curvature bounded below i we investigate the finiteness structure of a complete non compact n dimensional riemannian manifold m whose radial curvature at a base point of m is bounded from below by that of a non compact von mangoldt surface of revolution with its total curvature greater than we show as our main theorem that all busemann functions on m are exhaustions and that there exists a compact subset of m such that the compact set contains all critical points for any busemann function on m as corollaries by the main theorem m has finite topological type and the isometry group of m is compact

sur une question d erdos et schinzel ii let f n denote a polynomial with integer coefficients and define h f x y z to be the number of integers not exceeding x for which f n has at least one divisor d such that y

the lipschitz continuity of the distance function to the cut locus let n be a closed submanifold of a complete smooth riemannian manifold m and uv the total space of the unit normal bundle of n for each v c uv let p v denote the distance from n to the cut point of n on the geodesic y with the velocity vector tyv 0 v the continuity of the function p on uv is well known in this paper we prove that p is locally lipschitz on which p is bounded in particular if m and n are compact then p is globally lipschitz on uy therefore the canonical interior metric 6 may be introduced on each connected component of the cut locus of n and this metric space becomes a locally compact and complete length space let n be an immersed submanifold of a complete cc riemannian manifold m and 7 uv n the unit normal bundle of n for each positive integer k and vector v c uv let a number k v denote the parameter value of fy where 7yv denotes the geodesic for which the velocity vector is v at t 0 such that yv ak v is the k th focal point conjugate point for the case in which n is a point of n along y counted with focal or conjugate multiplicities a unit speed geodesic segment 0 a m emanating from n is called an n segment if t d n y t on 0 a by the first variation formula an n segment is orthogonal to n a point 7y to on an n segment 7y v e uv is called a cut point of n if there is no n segment properly containing y o to for each v c uv let p v denote the distance from n to the cut point on y of n whitehead 27 investigated the structure of the conjugate locus and the cut locus of a point on a real analytic finsler manifold he determined the structure of the conjugate locus around a conjugate point for which the conjugate multiplicity is locally constant on its neighborhood cf also 25 and proved the continuity of the function p in compact symmetric spaces t sakai 19 and m takeuchi 23 determined the detailed structure of the cut locus of a point the detailed structure of the cut locus of a point in a 2 dimensional riemannian manifold has been investigated by poincare myers and others 7 11 13 hartman first tried to show the absolute continuity of the function p when m is 2 dimensional he proved in 8 that if p is of bounded variation then p is absolutely continuous recently hebda 11 and the first named author 13 independently proved ambrose s problem by showing that p is absolutely continuous on a closed arc on which p is bounded when n is a point in a 2 dimensional riemannian manifold therefore the cut locus of a point in a compact 2 dimensional received by the editors october 14 1998 and in revised form april 13 1999 2000 mathematics subject classification primary 53c22 secondary 28a78 supported in part by a grant in aid for scientific research from the ministry of education science sports and culture japan 2000 american mathematical society

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

a volume comparison theorem for manifolds with asymptotically nonnegative curvature and its applications where dpo is the distance to po this class was first studied by u abresch abi ab2 because of the decay condition on the curvature faster than quadratic we expect the class of manifolds of asymptotically nonnegative curvature behaves similarly at infinity to manifolds of nonnegative curvature for the case of sectional curvature in abl and ab2 abresch showed that the complexity of topology of such manifolds is similar to that of nonnegative curvature namely they are all of finite topological type with uniform bound on the betti numbers in ka a kasue investigated the theory of harmonic functions for such manifolds again he demonstrated the expected similarity for ricci curvature the examples of j p sha and d g yang showed that we can not hope to control the topology of such manifolds as we do for sectional curvature very little is known for the case of asymptotically nonnegative ricci curvature except the work of abresch and gromoll ab which studied such manifolds under further conditions on the sectional curvature and diameter growth a question of recent interest stimulated by the works of p li and l f tam lt1 lt2 is whether open manifolds of asymptotically nonnegative ricci curvature has finitely many ends for the special case when a has compact support m l cai cai and z d liu liu recently gave a bound on the number of ends in lt2 under the stronger condition of f0o tn a t dt o0 li and tam also gave such a bound

dti segmentation by statistical surface evolution we address the problem of the segmentation of cerebral white matter structures from diffusion tensor images dti a dti produces from a set of diffusion weighted mr images tensor valued images where each voxel is assigned with a 3 times 3 symmetric positive definite matrix this second order tensor is simply the covariance matrix of a local gaussian process with zero mean modeling the average motion of water molecules as we will show in this paper the definition of a dissimilarity measure and statistics between such quantities is a nontrivial task which must be tackled carefully we claim and demonstrate that by using the theoretically well founded differential geometrical properties of the manifold of multivariate normal distributions it is possible to improve the quality of the segmentation results obtained with other dissimilarity measures such as the euclidean distance or the kullback leibler divergence the main goal of this paper is to prove that the choice of the probability metric i e the dissimilarity measure has a deep impact on the tensor statistics and hence on the achieved results we introduce a variational formulation in the level set framework to estimate the optimal segmentation of a dti according to the following hypothesis diffusion tensors exhibit a gaussian distribution in the different partitions we must also respect the geometric constraints imposed by the interfaces existing among the cerebral structures and detected by the gradient of the dti we show how to express all the statistical quantities for the different probability metrics we validate and compare the results obtained on various synthetic data sets a biological rat spinal cord phantom and human brain dtis

minimal counterexamples for linear time probabilistic verification abstract counterexamples for property violations have a number of important applications like supporting the debugging of erroneous systems and verifying large systems via counterexample guided abstraction refinement in this paper we propose the usage of minimal critical subsystems of discrete time markov chains and markov decision processes as counterexamples for violated regular properties minimality can thereby be defined in terms of the number of states or transitions this problem is known to be np complete for markov decision processes we show how to compute such subsystems using mixed integer linear programming and evaluate the practical applicability in a number of experiments they show that our method yields substantially smaller counterexample than using existing techniques

fair simulation relations parity games and state space reduction for buchi automata we give efficient algorithms improving optimal known bounds for computing a variety of simulation relations on the state space of a buchi automaton our algorithms are derived via a unified and simple parity game framework this framework incorporates previously studied notions like fair and direct simulation but also a new natural notion of simulation called delayed simulation which we introduce for the purpose of state space reduction we show that delayed simulation unlike fair simulation preserves the automaton language upon quotienting and allows substantially better state space reduction than direct simulation using our parity game approach which relies on an algorithm by jurdzinski we give efficient algorithms for computing all of the above simulations in particular we obtain an o mn3 time and o mn space algorithm for computing both the delayed and the fair simulation relations the best prior algorithm for fair simulation requires time and space o n6 our framework also allows one to compute bisimulations we compute the fair bisimulation relation in o mn3 time and o mn space whereas the best prior algorithm for fair bisimulation requires time and space o n10