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25998 results in Abstract analysis

Page 978 of 1300

  • The quantum zero space charge model for semiconductors

  • A. UNTERREITER
  • Journal:
    European Journal of Applied Mathematics / Volume 10 / Issue 4 / August 1999
    Published online by Cambridge University Press:
    01 August 1999, pp. 395-415

  • The thermal equilibrium state of a bipolar, isothermal quantum fluid confined to a bounded domain Ω⊂ℝd, d = 1, 2 or d = 3 is the minimizer of the total energy [Escr]ελ; [Escr]ελ involves the squares of the scaled Planck's constant ε and the scaled minimal Debye length λ. In applications one frequently has λ2[Lt]1. In these cases the zero-space-charge approximation is rigorously justified. As λ → 0, the particle densities converge to the minimizer of a limiting quantum zero-space-charge functional exactly in those cases where the doping profile satisfies some compatibility conditions. Under natural additional assumptions on the internal energies one gets an differential-algebraic system for the limiting (λ = 0) particle densities, namely the quantum zero-space-charge model. The analysis of the subsequent limit ε → 0 exhibits the importance of quantum gaps. The semiclassical zero-space-charge model is, for small ε, a reasonable approximation of the quantum model if and only if the quantum gap vanishes. The simultaneous limit ε = λ → 0 is analyzed.

  • Appendix D - An Extension of Lusin's Theorem

  • R. M. Dudley, Massachusetts Institute of Technology
  • Book:
    Uniform Central Limit Theorems
    Published online:
    20 May 2010
    Print publication:
    28 July 1999, pp 405-406

  • Summary

    Lusin's theorem says that for any measurable real-valued function ƒ, on [0, 1] with Lebesgue measure λ for example, and ε > 0, there is a set A with λ(A) < ε such that restricted to the complement of A, ƒ is continuous. Here [0, 1] can be replaced by any normal topological space and λ by any finite measure μ which is closed regular, meaning that for each Borel measurable set B, μ(B) = sup{μ(F): F closed, F ⊂ B) (RAP, Theorem 7.5.2). Recall that any finite Borel measure on a metric space is closed regular (RAP, Theorem 7.1.3).

    Proofs of Lusin's theorem are often based on Egorov's theorem (RAP, Theorem 7.5.1), which says that if measurable functions fn from a finite measure space to a metric space converge pointwise, then for any ε > 0 there is a set of measure less than ε outside of which the fn converge uniformly.

    Here, the aim will be to extend Lusin's theorem to functions having values in any separable metric space. The proof of Lusin's theorem in RAP, however, also relied on the Tietze-Urysohn extension theorem, which says that a continuous real-valued function on a closed subset of a normal space can be extended to be continuous on the whole space. Such an extension may not exist for some range spaces: for example, the identity from {0, 1} onto itself doesn't extend to a continuous function from [0, 1] onto {0, 1}; in fact there is no such function since [0, 1] is connected.

  • Appendix I - Modifications and Versions of Isonormal Processes

  • R. M. Dudley, Massachusetts Institute of Technology
  • Book:
    Uniform Central Limit Theorems
    Published online:
    20 May 2010
    Print publication:
    28 July 1999, pp 425-426

  • Summary

    Let T be any set and (Ω, A, P) a probability space. Recall that a real-valued stochastic process indexed by T is a function (t, ω) ↦ Xt(ω) from T × Ω into ℝ such that for each t ∈ T, Xt (·) is measurable from Ω into ℝ. A modification of the process is another stochastic process Yt defined for the same T and Ω such that for each t, we have P(Xt = Yt) = 1. A version of the process Xt, is a process Zt, t ∈ T, for the same T but defined on a possibly different probability space (Ω1, B, Q) such that Xt and Zt, have the same laws, that is, for each finite subset F of Clearly, any modification of a process is also a version of the process, but a version, even if on the same probability space, may not be a modification. For example, for an isonormal process L on a Hilbert space H, the process M(x) ≔ L(−x) is a version, but not a modification, of L.

    One may take a version or modification of a process in order to get better properties such as continuity. It turns out that for the isonormal process on subsets of Hilbert space, what can be done with a version can also be done by a modification, as follows.

    TheoremLet L be an isonormal process restricted to a subset C of Hilbert space. For each of the following two properties, if there exists a version M of L with the property, there also is a modification N with the property.

Page 978 of 1300