8 research outputs found

    The number of minima in a discrete sample

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    The number of times is considered that the minimum occurs in a sample from a discrete distribution. The special case of the geometric distribution is considered in some detail, and applied to the computation of the expected maximum of a sample from the Cantor distribution

    Melnikov potential for exact symplectic maps

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    The splitting of separatrices of hyperbolic fixed points for exact symplectic maps of nn degrees of freedom is considered. The non-degenerate critical points of a real-valued function (called the Melnikov potential) are associated to transverse homoclinic orbits and an asymptotic expression for the symplectic area between homoclinic orbits is given. Moreover, if the unperturbed invariant manifolds are completely doubled, it is shown that there exist, in general, at least 44 primary homoclinic orbits (4n4n in antisymmetric maps). Both lower bounds are optimal. Two examples are presented: a 2n2n-dimensional central standard-like map and the Hamiltonian map associated to a magnetized spherical pendulum. Several topics are studied about these examples: existence of splitting, explicit computations of Melnikov potentials, transverse homoclinic orbits, exponentially small splitting, etc

    Learning Opportunities 1996/1997

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    MASTER LIST OF APPROVED COURSES - REVISED NOV 9
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