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  1. Do algebraic numbers follow Khinchin's law?
    Author: Sibbertsen, Philipp; Lampert, Timm; Müller, Karsten; Taktikos, Michael
    Published: [2021]
    Publisher:  Wirtschaftswissenschaftliche Fakultät der Leibniz Universität Hannover, [Hannover]

    This paper argues that the distribution of the coefficients of the regular continued fraction should be considered for each algebraic number of degree >2 separately. For random numbers the coefficients are distributed by the Gauss-Kuzmin distribution... more

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    This paper argues that the distribution of the coefficients of the regular continued fraction should be considered for each algebraic number of degree >2 separately. For random numbers the coefficients are distributed by the Gauss-Kuzmin distribution (also called Khinchin's law). We apply the Kullback Leibler Divergence (KLD) to show that the Gauss-Kuzmin distribution does not fit well for algebraic numbers of degree > 2. Our suggestion to truncate the Gauss-Kuzmin distribution for finite parts fits slightly better, but its KLD is still much larger than the KLD of a random number. We consider differences regarding Khinchin's constant and Khinchin's approximation speed between random and algebraic numbers and conclude that laws concerning the random numbers do not automatically carry over to the algebraic numbers.

     
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    Source: Union catalogues
    Language: English
    Media type: Book
    Format: Online
    Other identifier:
    hdl: 10419/235234
    Series: Hannover economic papers (HEP) / Wirtschaftswissenschaftliche Fakultät der Leibniz Universität Hannover ; Nummer: 686 (May 2021)
    Subjects: continued fraction; truncated Gauss-Kuzmin distribution; Khinchin's constant; Kullback Leibler Divergence; algebraic number
    Scope: 1 Online-Ressource (circa 14 Seiten)
  2. Ludwig Wittgensteins Logisch-philosophische Abhandlung
    Entstehungsgeschichte und Herausgabe der Typoskripte und Korrekturexemplare
    Author: Graßhoff, Gerd; Lampert, Timm; Wittgenstein, Ludwig
    Published: 2004
    Publisher:  Springer, Wien [u.a.]

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  3. Ludwig Wittgensteins Logisch-philosophische Abhandlung
    Entstehungsgeschichte und Herausgabe der Typoskripte und Korrekturexemplare
    Author: Graßhoff, Gerd; Lampert, Timm
    Published: 2004
    Publisher:  Springer, Wien

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    Source: Union catalogues
    Language: German
    Media type: Book
    Format: Print
    ISBN: 9783211837825; 3211837825
    Other identifier:
    9783211837825
    Subjects: Analytische Philosophie; Textgeschichte
    Other subjects: Wittgenstein, Ludwig (1889-1951): Tractatus logico-philosophicus; (VLB-PF)BC: Paperback
    Scope: 465 S., 34 cm
  4. Ludwig Wittgensteins Logisch-philosophische Abhandlung
    Entstehungsgeschichte und Herausgabe der Typoskripte und Korrekturexemplare
    Author: Wittgenstein, Ludwig
    Published: 2004
    Publisher:  Springer, Wien [u.a.]

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    Content information
    Source: Staatsbibliothek zu Berlin
    Contributor: Graßhoff, Gerd (Hrsg.); Lampert, Timm (Hrsg.)
    Language: German; English
    Media type: Book
    Format: Print
    ISBN: 3211837825
    Other identifier:
    9783211837825
    RVK Categories: CI 5011 ; CI 5015
    Subjects: Wittgenstein, Ludwig <1889-1951>
    Other subjects: Wittgenstein, Ludwig <1889-1951>
    Scope: 465 S, 34 cm
    Notes:

    Faksimile
  5. Ludwig Wittgensteins Logisch-philosophische Abhandlung
    Entstehungsgeschichte und Herausgabe der Typoskripte und Korrekturexemplare
    Author: Graßhoff, Gerd; Lampert, Timm; Wittgenstein, Ludwig
    Published: 2004
    Publisher:  Springer, Wien [u.a.]

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    Content information
    Source: Philologische Bibliothek, FU Berlin; Staatsbibliothek zu Berlin
    Language: German
    Media type: Book
    ISBN: 3211837825
    RVK Categories: CI 5011
    DDC Categories: 100
    Subjects: Handschriften; Tractatus logico-philosophicus (Wittgenstein); Handschrift; Textgeschichte; Analytische Philosophie
    Other subjects: Wittgenstein, Ludwig <1889-1951>: Tractatus logico-philosophicus; Wittgenstein, Ludwig <1889-1951>; Wittgenstein, Ludwig (1889-1951): Tractatus logico-philosophicus
    Scope: 465 S.
  6. Ludwig Wittgensteins logisch-philosophische Abhandlung
    Entstehungsgeschichte und Herausgabe der Typoskripte und Korrekturexemplare
    Author: Graßhoff, Gerd
    Published: 2004
    Publisher:  Springer, Wien [u.a.]

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    Content information
    Source: Union catalogues
    Contributor: Lampert, Timm; Wittgenstein, Ludwig
    Language: German
    Media type: Book
    Format: Print
    ISBN: 3211837825
    Other identifier:
    9783211837825
    RVK Categories: CI 5011 ; CI 5015
    DDC Categories: 100
    Subjects: Analytische Philosophie; Textgeschichte; Sprachphilosophie
    Other subjects: Wittgenstein, Ludwig (1889-1951): Tractatus logico-philosophicus
    Scope: 465 S., 34 cm
  7. Roth's theorem implies a weakened version of the ABC conjecture for special cases
    Author: Sibbertsen, Philipp; Müller, Karsten; Lampert, Timm; Taktikos, Michael
    Published: [2022]
    Publisher:  [Wirtschaftswissenschaftliche Fakultät der Leibniz Universität Hannover], [Hannover]

    Enrico Bombieri proved that the ABC Conjecture implies Roth's theorem in 1994. This paper concerns the other direction. In making use of Bombieri's and Van der Poorten's explicit formula for the coefficients of the regular continued fractions of... more

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    Enrico Bombieri proved that the ABC Conjecture implies Roth's theorem in 1994. This paper concerns the other direction. In making use of Bombieri's and Van der Poorten's explicit formula for the coefficients of the regular continued fractions of algebraic numbers, we prove that Roth's theorem implies a weakened non-effective version of the ABC Conjecture in certain cases relating to roots.

     
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    Source: Union catalogues
    Language: English
    Media type: Book
    Format: Online
    Other identifier:
    hdl: 10419/268223
    Series: [Hannover economic papers (HEP)] / [Wirtschaftswissenschaftliche Fakultät der Leibniz Universität Hannover] ; [Nummer: 700 (Jul 2022)]
    Scope: 1 Online-Ressource (circa 8 Seiten)