Projections of patterns and mode interactions

Castro, Sofia B. S. D.; Labouriau, Isabel S.; Oliveira, Juliane F.

doi:10.1080/14689367.2017.1401590

Cited by 2 publications

(3 citation statements)

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“…The projections of triclinic lattices above are good illustrations of the fact that the symmetries of the lattice of periods are not necessarily symmetries of the pattern. Projected patterns with period lattice H and with D 6 -symmetry are exhibited in [2] together with a general method for finding them.…”

Section: Discussion Of the Examplementioning

confidence: 99%

“…We list all the possible lattices L that lead to this result for some y 0 and describe what happens for other values of y 0 . The lattices are described up to symmetries of the form α + where α belongs to the holohedry of H. This particular example is interesting because of its connection to special patterns discussed in [2,6].…”

Section: Example -Projection Into a Hexagonal Latticementioning

confidence: 99%

“…In particular we obtain all the periods of the projection and compare them to the periods of the restriction of the pattern and to the projection of the lattice. This particular group of examples is interesting because it has been proposed as an explanation for certain special patterns, called black-eye patterns, as discussed in [2,6].…”

Section: Introductionmentioning

confidence: 99%

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Periodic Functions, Lattices and Their Projections

Labouriau¹,

Pinho²

2018

Preprint

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Functions whose symmetries form a crystallographic group in particular have a lattice of periods, and the set of their level curves forms a periodic pattern. We show how after projecting these functions, one obtains new functions with a lattice of periods that is not the projection of the initial lattice. We also characterise all the crystallographic groups in three dimensions that are symmetry groups of patterns whose projections have periods in a given two-dimensional lattice. The particular example of patterns that after projection have a hexagonal lattice of periods is discussed in detail.

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Section: Discussion Of the Examplementioning

confidence: 99%

Section: Example -Projection Into a Hexagonal Latticementioning

confidence: 99%

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Periodic Functions, Lattices and Their Projections

Labouriau¹,

Pinho²

2018

Preprint

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Bifurcations, symmetries and the notion of fixed subspace

Cicogna

2021

Rev. Math. Phys.

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In the context of stationary bifurcation problems admitting a symmetry, this paper is focused on the key notion of Fixed Subspace (FS), and provides a review of some applications aimed at detecting bifurcating solutions in various situations. We start recalling, in its commonly used simplified version, the old Equivariant Bifurcation Lemma (EBL), where the FS is one-dimensional; then we provide a first generalization in a typical case of non-semisimple critical eigenvalues, where the presence of the symmetry produces a non-trivial situation. Next, we consider the case of FSs of dimension [Formula: see text] in very different contexts. First, relying on the topological index theory and in particular on the Krasnosel’skii theorem, we provide a largely applicable statement of an extension of the EBL. Second, we propose a completely different and new application which combines symmetry properties with the notion of stability of bifurcating solutions. We also provide some simple examples, constructed ad hoc to illustrate the various situations.

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Projections of patterns and mode interactions

Cited by 2 publications

References 24 publications

Periodic Functions, Lattices and Their Projections

Periodic Functions, Lattices and Their Projections

Bifurcations, symmetries and the notion of fixed subspace

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