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David Handelman

  1. Measures on Cantor sets: the good, the ugly, the bad.

    Wed, 01/11/2012 - 15:01 — SomNambul
    Authors: David Handelman, Sergey Bezuglyi
    Subjects: Dynamical Systems
    Abstract

    We translate Akin's notion of {\it good} (and related concepts) from measures
    on Cantor sets to traces on dimension groups, and particularly for invariant
    measures of minimal homeomorphisms (and their corresponding simple dimension
    groups), this yields characterizations and examples, which translate back to
    the original context. Good traces on a simple dimension group are characterized
    by their kernel having dense image in their annihilating set of affine
    functions on the trace space; this makes it possible to construct many examples
    with seemingly paradoxical properties.

  2. Log concavity of $(1+x)^m (1+ x^k)$.

    Wed, 02/16/2011 - 16:01 — SomNambul
    Authors: David Handelman
    Subjects: Combinatorics
    Abstract

    Let $P$ be the polynomial of the title, with $m$ and $k$ integers. Then $P$
    is strongly unimodal (that is, its sequence of coefficients is log concave) if
    and only if $m \geq k^2 -3$ if and only if $P$ has unimodal coefficients. We
    also show that in order to satisfy a condition concerning its behaviour on the
    unit circle, we must have $m$ of order $k^4$ or more.

  3. Real dimension groups.

    Wed, 02/16/2011 - 16:01 — SomNambul
    Authors: David Handelman
    Subjects: Rings and Algebras
    Abstract

    We show the characterization analogous to dimension groups of partially
    ordered real vector spaces with interpolation works, but sequential direct
    limits of simplicial vector spaces only under strong assumptions. We also
    provide and generalize a proof of a result of Fuchs asserting that the real
    polynomial algebra with pointwise ordering coming from an interval satisfies
    Riesz interpolation

  4. Simple archimedean dimension groups.

    Wed, 02/16/2011 - 16:01 — SomNambul
    Authors: David Handelman
    Subjects: Rings and Algebras
    Abstract

    We answer a question of Goodearl, by constructing for every metrizable
    Choquet simplex, a dimension group that is simple and archimedean and whose
    trace space is the desired Choquet simplex.

  5. In praise of order units.

    Thu, 09/30/2010 - 15:01 — SomNambul
    Authors: David Handelman
    Subjects: Algebraic Geometry
    Abstract

    We show that the ordered rings naturally associated to compact convex
    polyhedra with interior satisfy a positivity property known as order unit
    cancellation, and obtain other general positivity results as well.

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