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Author Spotlight: Introducing the Tile/SED/Array Interface for Rapid Field of View Positioning in Tissue Imaging
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Tiling and weak tiling in .

Gergely Kiss1,2, Dávid Matolcsi2, Máté Matolcsi1,3

  • 1Alfréd Rényi Institute of Mathematics, HUN-REN, Reáltanoda u. 13-15, 1053 Budapest, Hungary.

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|December 7, 2023
PubMed
Summary
This summary is machine-generated.

This study explores tiling and spectral sets in finite abelian groups, introducing a novel 4-tuple generalization for elementary p-groups. Researchers characterize these structures in specific group types, advancing understanding of spectral properties.

Keywords:
Abelian p-groupsFuglede’s conjectureSpectral setsTranslational tilesWeak tiling

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Area of Science:

  • Harmonic Analysis
  • Group Theory
  • Abstract Algebra

Background:

  • Tiling and spectral set problems are fundamental in harmonic analysis.
  • Finite abelian groups provide a structured framework for studying these concepts.
  • Existing research focuses on specific group structures and their tiling properties.

Purpose of the Study:

  • To investigate the relationship between tiling, weak tiling, and spectral sets in finite abelian groups.
  • To introduce and analyze a generalized object, a 4-tuple of functions, as a common generalization of tiles and spectral sets.
  • To characterize these generalized objects in specific types of finite abelian groups.

Main Methods:

  • Introduction of an averaging procedure for function tuples in elementary p-groups.
  • Development of a framework for analyzing 4-tuples of functions.
  • Characterization techniques applied to specific group structures (e.g., Z_p x Z_p).

Main Results:

  • A novel 4-tuple of functions is introduced, generalizing tiles and spectral sets.
  • Complete characterization of these 4-tuples is achieved for the group Z_p x Z_p.
  • Partial results are presented for the group Z_p x Z_p x Z_p.

Conclusions:

  • The study establishes a unified approach to studying tiling and spectral properties in finite abelian groups.
  • The introduced 4-tuple provides a powerful tool for further research in this area.
  • The findings contribute to a deeper understanding of spectral sets and tiling theory in abstract algebraic structures.