*Ted Janssen, Gervais Chapuis, and Marc de Boissieu*

- Published in print:
- 2007
- Published Online:
- September 2007
- ISBN:
- 9780198567776
- eISBN:
- 9780191718335
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198567776.003.0003
- Subject:
- Physics, Crystallography

This chapter discusses mathematical models for studying the basic properties of crystals. Topics covered include model sets, introduction of tilings, aperiodic tilings, approximants, coverings, and ...
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This chapter discusses mathematical models for studying the basic properties of crystals. Topics covered include model sets, introduction of tilings, aperiodic tilings, approximants, coverings, and random tilings.Less

This chapter discusses mathematical models for studying the basic properties of crystals. Topics covered include model sets, introduction of tilings, aperiodic tilings, approximants, coverings, and random tilings.

*J. L. Ramírez Alfonsín*

- Published in print:
- 2005
- Published Online:
- September 2007
- ISBN:
- 9780198568209
- eISBN:
- 9780191718229
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198568209.003.0008
- Subject:
- Mathematics, Algebra, Combinatorics / Graph Theory / Discrete Mathematics

This chapter presents a number of applications of FP to a variety of problems. The complexity analysis of the Shellsort method was not well understood until J. Incerpi and R. Sedgewick nicely ...
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This chapter presents a number of applications of FP to a variety of problems. The complexity analysis of the Shellsort method was not well understood until J. Incerpi and R. Sedgewick nicely observed that FP can be used to obtain upper bounds for the running time of this fundamental sorting algorithm. This chapter starts by explaining this application. It then explains how FP may be applied to analyse Petri nets, to study partitions of vector spaces, to compute exact resolutions via Rødseth's method for finding the Frobenius number when n = 3, to investigate Algebraic Geometric codes via the properties of special semigroups and their corresponding conductors, and to study tiling problems. The chapter also discusses three applications of the denumerant. An application of the modular change problem to study nonhypohamiltonian graphs, and of the vector generalization to give a new method for generating random vectors are presented.Less

This chapter presents a number of applications of **FP** to a variety of problems. The complexity analysis of the Shellsort method was not well understood until J. Incerpi and R. Sedgewick nicely observed that **FP** can be used to obtain upper bounds for the running time of this fundamental sorting algorithm. This chapter starts by explaining this application. It then explains how **FP** may be applied to analyse Petri nets, to study partitions of vector spaces, to compute exact resolutions via Rødseth's method for finding the Frobenius number when n = 3, to investigate Algebraic Geometric codes via the properties of special semigroups and their corresponding conductors, and to study tiling problems. The chapter also discusses three applications of the denumerant. An application of the modular change problem to study nonhypohamiltonian graphs, and of the vector generalization to give a new method for generating random vectors are presented.

*Sergry V. Krivovichev*

- Published in print:
- 2009
- Published Online:
- May 2009
- ISBN:
- 9780199213207
- eISBN:
- 9780191707117
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199213207.003.0004
- Subject:
- Physics, Crystallography

This chapter deals with description of dense 2-D topologies using the concept of anion-topology introduced by P. C. Burns and co-authors for the analysis of uranyl layered structures. Anion ...
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This chapter deals with description of dense 2-D topologies using the concept of anion-topology introduced by P. C. Burns and co-authors for the analysis of uranyl layered structures. Anion topologies are considered as plane tilings and their classification is developed. Several cases of geometrical isomerism are discussed using the concept of orientation matrix introduced in Chapter 2.Less

This chapter deals with description of dense 2-D topologies using the concept of anion-topology introduced by P. C. Burns and co-authors for the analysis of uranyl layered structures. Anion topologies are considered as plane tilings and their classification is developed. Several cases of geometrical isomerism are discussed using the concept of orientation matrix introduced in Chapter 2.

*Walter Steurer and Julia Dshemuchadse*

- Published in print:
- 2016
- Published Online:
- September 2016
- ISBN:
- 9780198714552
- eISBN:
- 9780191782848
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198714552.003.0003
- Subject:
- Physics, Crystallography

Crystal structures, be they periodic or aperiodic, can be described in many different ways: as lattices, tilings, or coverings decorated by atoms or clusters on one hand, or as packings of structural ...
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Crystal structures, be they periodic or aperiodic, can be described in many different ways: as lattices, tilings, or coverings decorated by atoms or clusters on one hand, or as packings of structural subunits (AETs, clusters, modules, etc.) on the other hand. In this chapter the decomposition of crystal structures into AETs and clusters is discussed, as well as tilings and coverings. This ranges from the 1D quasiperiodic Fibonacci sequence to examples of 2D and 3D periodic as well as quasiperiodic tilings. The standard unit cell description of periodic crystal structures corresponds to a specific tiling approach with just a single prototile with the shape of a parallelepiped, i.e., the crystallographic unit cell. The detailed discussion of periodic and quasiperiodic tilings and packings can be quite helpful in understanding structural building principles of complex intermetallics, periodic as well as quasiperiodic ones.Less

Crystal structures, be they periodic or aperiodic, can be described in many different ways: as lattices, tilings, or coverings decorated by atoms or clusters on one hand, or as packings of structural subunits (AETs, clusters, modules, etc.) on the other hand. In this chapter the decomposition of crystal structures into AETs and clusters is discussed, as well as tilings and coverings. This ranges from the 1D quasiperiodic Fibonacci sequence to examples of 2D and 3D periodic as well as quasiperiodic tilings. The standard unit cell description of periodic crystal structures corresponds to a specific tiling approach with just a single prototile with the shape of a parallelepiped, i.e., the crystallographic unit cell. The detailed discussion of periodic and quasiperiodic tilings and packings can be quite helpful in understanding structural building principles of complex intermetallics, periodic as well as quasiperiodic ones.