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2 results

  • 3 - Global Attraction to Stationary Orbits

  • Alexander Komech, Universität Wien, Austria, Elena Kopylova, Universität Wien, Austria
  • Book:
    Attractors of Hamiltonian Nonlinear Partial Differential Equations
    Published online:
    17 September 2021
    Print publication:
    30 September 2021, pp 91-113

  • Summary

    In this chapter we present in detail the first resultsonglobal attractionto stationary orbits obtained for a 1D Klein--Gordon equationcoupled to a nonlinear oscillator. The proofs rely on the concept of the omega-limit trajectory and a nonlinear analog of the Kato theorem on the absence of embedded eigenvalues, and on new theory of multipliers in the space of quasimeasures and a novel application of the Titchmarsh convolution theorem. Besides the formal proof, we give an informal explanation of the nonlinear radiative mechanism, which causes theglobal attraction: nonlinear energy transfer from lower to higher harmonics and subsequent dispersive radiation of energy to infinity. In conclusion, we specifythe general conjecture on global attractors, which summarizes all results obtained thus far.

  • Characterizations in a random record model with a nonidentically distributed initial record

  • Part of
  • Gadi Barlevy, H. N. Nagaraja
  • Journal:
    Journal of Applied Probability / Volume 43 / Issue 4 / December 2006
    Published online by Cambridge University Press:
    14 July 2016, pp. 1119-1136
    Print publication:
    December 2006
    • Article
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  • We consider a sequence, of random length M, of independent, continuous observations Xi, 1 ≤ i ≤ M, where M is geometric, X1 has cumulative distribution function (CDF) G, and Xi, i ≥ 2, have CDF F. Let N be the number of upper records and let Rn, n ≥ 1, be the nth record value. We show that N is independent of F if and only if G(x) = G0(F(x)) for some CDF G0, and that if E(|X2|) is finite then so is E(|Rn|), n ≥ 2, whenever N ≥ n or N = n. We prove that the distribution of N, along with appropriately chosen subsequences of E(Rn), characterize F and G and, along with subsequences of E(Rn - Rn-1), characterize F and G up to a common location shift. We discuss some applications to the identification of the wage offer distribution in job search models.