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General Framework

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Sequences, Groups, and Number Theory

Part of the book series: Trends in Mathematics ((TM))

Abstract

This introductory chapter briefly presents some of the main notions that appear in the subsequent chapters of this book. We recap a few definitions and results from combinatorics on groups and words, formal language theory, morphic words, k-automatic and k-regular sequences, and dynamical systems. Our aim is not to be exhaustive. The reader can consult this chapter when studying other parts of this book.

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Notes

  1. 1.

    If the β-expansion of x/βd is d0 d1⋯, then using an extra decimal point, the expansion of x is conveniently written d0⋯dℓ−1•d ℓ dℓ+1⋯. Note that the presentation in Chapter 1 is not entirely consistent with our present treatment if x belongs to [0,  1/(β − 1)] ∖ [0,  1).

  2. 2.

    An R′-module M is Noetherian if every submodule of M is finitely generated. Let R′ be a Noetherian ring. An R′-module M is Noetherian if and only if it is finitely generated.

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Author information

Authors and Affiliations

  1. IRIF, UMR 8243, CNRS & Université Paris Diderot, Case 7014, F-75205, Paris Cedex 13, France

    Valérie Berthé

  2. Department of Mathematics, University of Liège, Allée de la découverte 12 (B37), B-4000, Liège, Belgium

    Michel Rigo

Authors
  1. Valérie Berthé
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  2. Michel Rigo
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Correspondence to Valérie Berthé .

Editor information

Editors and Affiliations

  1. IRIF, Université Paris Diderot, Paris, France

    Valérie Berthé

  2. Department of Mathematics, University of Liège, Liège, Belgium

    Michel Rigo

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Berthé, V., Rigo, M. (2018). General Framework. In: Berthé, V., Rigo, M. (eds) Sequences, Groups, and Number Theory. Trends in Mathematics. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-69152-7_1

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