Edge detection is the process that attempts to characterize the intensity changes in the image in terms of the physical processes that have originated them. A critical, intermediate goal of edge detection is the detection and characterization of significant intensity changes. This paper discusses this part of the edge d6tection problem. To characterize the types of intensity changes derivatives of different types, and possibly different scales, are needed. Thus, we consider this part of edge detection as a problem in numerical differentiation.

In this paper we present the solution to a longstanding problem of differential geometry: Lie’s third theorem for Lie algebroids. We show that the integrability problem is controlled by two computable obstructions. As applications we derive, explain and improve the known integrability results, we establish integrability by local Lie groupoids, we clarify the smoothness of the Poisson sigma-model for Poisson manifolds, and we describe other geometrical applications.

CONTENTS 1 Introduction 1 2 The Basics of Predictor-Corrector Path Following 3 3 Aspects of Implementations 7 4 Applications 15 5 Piecewise-Linear Methods 34 6 Complexity 41 7 Available Software 44 References 48 1. Introduction Continuation, embedding or homotopy methods have long served as useful theoretical tools in modern mathematics. Their use can be traced back at least to such venerated works as those of Poincar'e (1881--1886), Klein (1882-- 1883) and Bernstein (1910). Leray and Schauder (1934) refined the tool and presented it as a global result in topology, viz., the homotopy invariance of degree. The use of deformations to solve nonlinear systems of equations Partially supported by the National Science Foundation via grant # DMS-9104058 y Preprint, Colorado State University, August 2 E. Allgower and K. Georg may be traced back at least to Lahaye (1934). The classical embedding methods were the

Abstract. For a large class of semiclassical pseudodifferential operators, including Schrödinger operators, P(h) = −h 2 ∆g + V (x), on compact Riemannian manifolds, we give logarithmic lower bounds on the mass of eigenfunctions outside neighbourhoods of generic closed hyperbolic orbits. More precisely we show that if A is a pseudodifferential operator which is microlocally equal to the identity near the hyperbolic orbit and microlocally zero away from the orbit, then ‖u ‖ ≤ C ( √ log(1/h)/h)‖P(h)u ‖ + C √ log(1/h)‖(I − A)u ‖. This generalizes earlier estimates of Colin de Verdière-Parisse [CVP] obtained for a special case, and of Burq-Zworski [BuZw] for real hyperbolic orbits. 1.

We define the concept of pseudo symplectic capacities that is a mild generalization of that of the symplectic capacities. In particular a typical pseudo symplectic capacity, whose special case is a variant of the Hofer-Zehnder symplectic capacity, is constructed and estimated in terms of the Gromov-Witten invariants. An example is also given to illustrate that using the pseudo symplectic capacity may get better estimation result than doing the Hofer-Zehnder capacity. Among various potential applications of these estimations three typical applications are given. The first one is to derive some general nonsqueezing theorems that generalize and unite many past versions. The second is to give an alternate generalization of Lalonde-McDuff theorem on length minimizing Hamiltonian paths to a general closed symplectic manifold. In the third application we give the new symplectic packing obstructions for a wider class of symplectic manifolds.

this paper, has been directed toward extending their results beyond Axiom A.

We show that a nearly integrable hamiltonian system has invariant tori of all dimensions smaller than the number of degrees of freedom provided that certain nondegeneracy conditions are met. The tori we construct are generated by the resonances of the system and are topologically different from the orbits that are present in the integrable system. We also show that the tori we construct have stable and unstable manifolds and point out how to construct other types of interesting orbits. The method of proof is a combination of different perturbation methods.

The main goal of this paper is a global continuation theorem for homoclinic solutions of autonomous ordinary differential equations with two real parameters. In one-parameter flows, Hopf bifurcation serves as a starting point for global paths of periodic orbits. B-points, alias Arnol'd-Bogdanov-Takens points, play an analogous role for paths of homoclinic orbits in two-parameter flows. In fact, a path of homoclinic orbits emanating from a B-point can be continued in phase space until it terminates at another B-point, or becomes unbounded, or approaches a region with chaotic dynamics. This result is obtained via a new topological invariant for homoclinic orbits, based on approximation of the homoclinic orbit by nearby periodic orbits. Several local bifurcation results for homoclinic and heteroclinic orbits are reviewed, along the way, to illustrate scope, significance, and limitations of the global approach. The paper concludes with an extensive discussion, including "non-generic" aspec...

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Abstract — We consider the time-dependent nonlinear system ˙q(t) = u(t)X(q(t)) + (1 − u(t))Y (q(t)), where q ∈ R 2, X and Y are two smooth vector fields, globally asymptotically stable at the origin and u: [0, ∞) → {0, 1} is an arbitrary measurable function. Analysing the topology of the set where X and Y are parallel, we give some sufficient and some necessary conditions for global asymptotic stability, uniform with respect to u(.). Such conditions can be verified without any integration or construction of a Lyapunov function, and they are robust under small perturbations of the vector fields.