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Absolute Convergence

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Francois Denis – One of the best experts on this subject based on the ideXlab platform.

  • Absolute Convergence of rational series is semi decidable
    Information & Computation, 2011
    Co-Authors: Raphael Bailly, Francois Denis

    Abstract:

    This paper deals with Absolute Convergence of real-valued rational series, i.e. mappings r:Σ∗→R computed by weighted automata. An algorithm is provided, that takes a weighted automaton A as input and halts if and only if the corresponding series rA is Absolutely convergent: hence, Absolute Convergence of rational series is semi-decidable. A spectral radius-like parameter ρ|r| is introduced, which satisfies the following property: a rational series r is Absolutely convergent iff ρ|r|<1. We show that if r is rational, then ρ|r| can be approximated by convergent upper estimates. Then, it is shown that the sum Σw∈Σ∗|r(w)| can be estimated to any accuracy rate. This result can be extended to any sum of the form Σw∈Σ∗|r(w)|p, for any integer p.

  • Absolute Convergence of rational series is semi decidable
    Language and Automata Theory and Applications, 2009
    Co-Authors: Raphael Bailly, Francois Denis

    Abstract:

    We study real-valued Absolutely convergent rational series , i.e. functions $r: {\it\Sigma}^* \rightarrow {\mathbb R}$, defined over a free monoid ${\it\Sigma}^*$, that can be computed by a multiplicity automaton A and such that $\sum_{w\in {\it\Sigma}^*}|r(w)| . We prove that any Absolutely convergent rational series r can be computed by a multiplicity automaton A which has the property that r |A | is simply convergent, where r |A | is the series computed by the automaton |A | derived from A by taking the Absolute values of all its parameters. Then, we prove that the set ${\cal A}^{rat}({\it\Sigma})$ composed of all Absolutely convergent rational series is semi-decidable and we show that the sum $\sum_{w\in \Sigma^*}|r(w)|$ can be estimated to any accuracy rate for any $r\in {\cal A}^{rat}({\it\Sigma})$. We also introduce a spectral radius-like parameter ρ |r | which satisfies the following property: r is Absolutely convergent iff ρ |r | < 1.

Werner E G Muller – One of the best experts on this subject based on the ideXlab platform.

  • on the spectral side of arthur s trace formula Absolute Convergence
    Annals of Mathematics, 2011
    Co-Authors: Tobias Finis, Erez Lapid, Werner E G Muller

    Abstract:

    We derive a renement of the spectral expansion of Arthur’s trace formula. The expression is Absolutely convergent with respect to the trace norm.

Raphael Bailly – One of the best experts on this subject based on the ideXlab platform.

  • Absolute Convergence of rational series is semi decidable
    Information & Computation, 2011
    Co-Authors: Raphael Bailly, Francois Denis

    Abstract:

    This paper deals with Absolute Convergence of real-valued rational series, i.e. mappings r:Σ∗→R computed by weighted automata. An algorithm is provided, that takes a weighted automaton A as input and halts if and only if the corresponding series rA is Absolutely convergent: hence, Absolute Convergence of rational series is semi-decidable. A spectral radius-like parameter ρ|r| is introduced, which satisfies the following property: a rational series r is Absolutely convergent iff ρ|r|<1. We show that if r is rational, then ρ|r| can be approximated by convergent upper estimates. Then, it is shown that the sum Σw∈Σ∗|r(w)| can be estimated to any accuracy rate. This result can be extended to any sum of the form Σw∈Σ∗|r(w)|p, for any integer p.

  • Absolute Convergence of rational series is semi decidable
    Language and Automata Theory and Applications, 2009
    Co-Authors: Raphael Bailly, Francois Denis

    Abstract:

    We study real-valued Absolutely convergent rational series , i.e. functions $r: {\it\Sigma}^* \rightarrow {\mathbb R}$, defined over a free monoid ${\it\Sigma}^*$, that can be computed by a multiplicity automaton A and such that $\sum_{w\in {\it\Sigma}^*}|r(w)| . We prove that any Absolutely convergent rational series r can be computed by a multiplicity automaton A which has the property that r |A | is simply convergent, where r |A | is the series computed by the automaton |A | derived from A by taking the Absolute values of all its parameters. Then, we prove that the set ${\cal A}^{rat}({\it\Sigma})$ composed of all Absolutely convergent rational series is semi-decidable and we show that the sum $\sum_{w\in \Sigma^*}|r(w)|$ can be estimated to any accuracy rate for any $r\in {\cal A}^{rat}({\it\Sigma})$. We also introduce a spectral radius-like parameter ρ |r | which satisfies the following property: r is Absolutely convergent iff ρ |r | < 1.