Open Problems and Conjectures Related to the Theory of Mathematical Quasicrystals

Adiceam, Faustin; Damanik, David; Gähler, Franz; Grimm, Uwe; Haynes, Alan; Julien, Antoine; Navas, Andrés; Sadun, Lorenzo; Weiss, Barak

doi:10.1007/s40598-016-0046-6

Cited by 4 publications

(4 citation statements)

References 40 publications

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“…In that case, Laczkovich [29,28] (see also [9,37,40]) showed that Theorem 1.2. If n = 2, then X 1 is BD to a lattice if and only if there exist constants c 1 and c 2 such that, for all topological disks U , U 1 − ρ|U | ≤ c 1 + c 2 |∂U |, where |U | is the area of U and |∂U | is the perimeter of U .…”

Section: Introduction and Resultsmentioning

confidence: 99%

“…There has been a burst of activity in recent years studying the bounded displacement (BD) equivalence relation for tilings and Delone sets [16,19,17,18,24,1,39,38]. As we reported in our previous paper [24], there are many relevant papers contributing to the subject that predate the terminology BD such as [7,8,26].…”

Section: Discussionmentioning

confidence: 99%

“…Let Ω k (T) be the set of strongly PE k-cochains. The coboundary of a strongly PE cochain is strongly PE (albeit possibly with a slightly larger radius), yielding a complex (1) 0 → Ω 0 (T)…”

Section: Introductionmentioning

confidence: 99%

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Pattern Equivariant Mass Transport in Aperiodic Tilings and Cohomology

Kelly

Sadun

2020

International Mathematics Research Notices

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Suppose that we have a repetitive and aperiodic tiling T of R n , and two mass distributions f 1 and f 2 on R n , each pattern equivariant with respect to T. Under what circumstances is it possible to do a bounded transport from f 1 to f 2 ? When is it possible to do this transport in a strongly or weakly pattern-equivariant way? We reduce these questions to properties of theČech cohomology of the hull of T, properties that in most common examples are already well-understood.

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Section: Introduction and Resultsmentioning

confidence: 99%

Section: Discussionmentioning

confidence: 99%

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Pattern Equivariant Mass Transport in Aperiodic Tilings and Cohomology

Kelly

Sadun

2020

International Mathematics Research Notices

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“…These discoveries and production techniques generated an interest in the study of quasiperiodic environments, such that in 2011 the Nobel Prize in Chemistry was given to Schechtman for the discovery of quasicrystals [41]. Such interest continues to grow today, width many open problems from a mathematical point of view [42], and computational challenges, such as the statistical description of the quantum energy levels of a quasicrystal recently addressed with the use of random matrices [43], which also could be related with the length distribution of the free flights in a Lorentz gas [44].…”

Section: Introductionmentioning

confidence: 99%

Efficient algorithm for simulating particles in true quasiperiodic environments

Sosa¹,

Kraemer²

2022

J. Phys. A: Math. Theor.

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We introduce an algorithm based on generalized dual method (GDM) to efficiently study the dynamics of a particle in quasiperiodic environments without the need to use periodic approximations or to save the information of the vertices that make up the quasiperiodic lattice. We show that the computation time and the memory required to find the tile in which a particle is located as a function of the distance R to the center of symmetry remains constant in our algorithm, while using the GDM directly both quantities go like R 2. This allows us to perform realistic simulations with low consumption of computational resources. The algorithm can be used to study any quasiperiodic lattice that can be produced by the cut-and-project method. Using this algorithm, we have calculated the free path length distribution in quasiperiodic Lorentz gases reproducing previous results and for systems with high symmetries at the Boltzmann–Grad limit. We have found for the Boltzmann–Grad limit, that the distribution of free paths depends on the rank r of the quasiperiodic system and not on its symmetry. The distribution as a function of the free path length l appears to be a combination of exponential decay and a power-law behavior. The latter seems to become important only for probabilities less than ( 2 r − 2 r ( r + 1 ) ) − 1 , showing an exponential decaying free-path length distribution for r → ∞, similar to what is observed in disordered systems.

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