Discrete Tiles Research Articles

Aperiodic spin chains at the boundary of hyperbolic tilings

In view of making progress towards establishing a holographic duality for theories defined on a discrete tiling of the hyperbolic plane, we consider a recently proposed boundary spin chain Hamiltonian with aperiodic couplings that are chosen such as to reflect the inflation rule, i.e. the construction principle, of the bulk tiling. As a remnant of conformal symmetry, the spin degrees of freedom are arranged in multiplets of the dihedral group under which the bulk lattice is invariant. For the boundary Hamiltonian, we use strong-disorder RG techniques and evaluate correlation functions, the entanglement entropy and mutual information for the case that the ground state is in an aperiodic singlet phase. We find that two-point functions decay as a power-law with exponent equal to one. Furthermore, we consider the case that the spin variables transform in the fundamental representation of SO(N), leading to a gapless system, and find that the effective central charge obtained from the entanglement entropy scales as lnN, reflecting the number of local degrees of freedom. We also determine the dependence of this central charge on the parameters specifying the bulk tiling. Moreover, we obtain an analytical expression for the mutual information, according to which there is no phase transition at any finite value of the distance between the two intervals involved.

A Continuous Single-Layer Discrete Tiling System for Online Detection of Corn Impurities and Breakage Rates

In order to improve the accuracy and efficiency of the methods that are used for the detection of impurities in and the breakage rate of harvested corn grains, we propose a classification and identification method using a feature threshold and a backpropagation (BP) neural network that is based on a genetic algorithm. We also constructed a continuous single-layer discrete tile detection system for application to harvested corn grains containing impurities and broken kernels. We conducted an evaluation of the proposed approach with a three-factor and three-level orthogonal experimental design. By setting the working parameters, we realized the continuous single-layer discrete tiling of the grains and 50 grain materials were collected on average in a single picture. In the static test, the error between the system monitoring value and the manual detection value was small, the maximum absolute errors of the breakage and impurity rates were 2.16% and 1.03%, and the average time that was required for each image recognition was 1.71 s. In the experimental environment, the maximum absolute error values of the breakage and impurity rates were 3.48% and 1.78%. The system’s identification accuracy and processing time meet the requirements of the online detection of corn characteristics in grain harvesting.

Perfect codes in Euclidean lattices

In the present paper, we investigate the existence of lattice perfect codes when considered as sublattices of other lattices under the Euclidean metric. We discuss the connection between a discrete tiling of a lattice and a continuous tilling of the n-dimensional space and equivalent characterizations of discrete tilings. We generalize bounds on the radius of perfect codes in a generic lattice, which were previously known for the cubic lattice, and we provide some new bounds. The new bounds are based on the packing and the covering densities and on the covering radius of the ambient lattice. An algorithm is presented for the search of perfect codes which is used to derive all perfect codes for a collection of ambient lattices in dimensions two and three. In contrast to the cubic lattice, these case studies show that by considering general ambient lattices one can find rich sets of perfect codes.

Tiling solutions for optimal biological sensing

Biological systems, from cells to organisms, must respond to the ever-changing environment in order to survive and function. This is not a simple task given the often random nature of the signals they receive, as well as the intrinsically stochastic, many-body and often self-organized nature of the processes that control their sensing and response and limited resources. Despite a wide range of scales and functions that can be observed in the living world, some common principles that govern the behavior of biological systems emerge. Here I review two examples of very different biological problems: information transmission in gene regulatory networks and diversity of adaptive immune receptor repertoires that protect us from pathogens. I discuss the trade-offs that physical laws impose on these systems and show that the optimal designs of both immune repertoires and gene regulatory networks display similar discrete tiling structures. These solutions rely on locally non-overlapping placements of the responding elements (genes and receptors) that, overall, cover space nearly uniformly. Les systèmes biologiques, depuis la cellule jusqu'à l'organisme, ne peuvent survivre et fonctionner que s'ils s'adaptent aux changements continuels de l'environnement. Ce n'est pas une tâche aisée, à cause de la nature aléatoire des signaux qu'ils reçoivent, de la nature collective et souvent autoorganisée des phénomènes qui contrôlent leur réponse, et aussi en raison de la limitation des ressources. Malgré la diversité des échelles et des fonctions qu'on observe dans le monde vivant, on peut faire apparaître quelques principes généraux qui gouvernent le comportement des systèmes biologiques. Je considère ici deux exemples très différents de problèmes biologiques : la transmission de l'information dans les réseaux de régulation des gènes et le système immunitaire adaptatif qui nous protège des agents pathogènes. Je discute les compromis que les lois physiques imposent à ces systèmes, et je montre que la structure optimale des systèmes immunitaires comme des réseaux de régulation des gènes est organisée de façon semblable, en forme de pavage discret. Ces solutions correspondent à des dispositions sans recouvrement des unités de réponse (gènes et récepteurs) qui remplissent l'espace de façon presque uniforme.

Spectral triples and characterization of aperiodic order

We construct spectral triples for compact metric spaces (X, d). This provides us with a new metric d̄s on X. We study its relation with the original metric d. When X is a subshift space, or a discrete tiling space, and d satisfies certain bounds, we advocate that the property of d̄s and d to be Lipschitz equivalent is a characterization of high order. For episturmian subshifts, we prove that d̄s and d are Lipschitz equivalent if and only if the subshift is repulsive (or power-free). For Sturmian subshifts this is equivalent to linear recurrence. For repetitive tilings we show that if their patches have equi-distributed frequencies, then the two metrics are Lipschitz equivalent. Moreover, we study the zeta-function of the spectral triple and relate its abscissa of convergence to the complexity exponent of the subshift or the tiling. Finally, we derive Laplace operators from the spectral triples and compare our construction with that of Pearson and Bellissard.

Speech perception from a crudely quantized spectrogram: A figure‐ground analogy

Recent research into auditory chimeras has shown that comprehensible speech can be generated by superimposing the frequency‐dependent modulation envelope of a target speech signal onto the fine structure of music, speech, or almost any other broadband waveform. In this experiment, we show that, in certain cases, intelligibility can be achieved with an even simpler stimulus that preserves only the rough outline of the energy distribution of the target speech waveform. This stimulus was generated by 1) dividing the target speech spectrogram into discrete tiles that were 1/3rd octave band wide and 7.8 ms long; 2) identifying the minimum number of tiles necessary to capture 90% of the energy in the target waveform; and 3) filtering a uniform, broadband noise to activate only those tiles that met the threshold energy criterion in the original stimulus. The resulting stimulus captures the overall outline of the distribution of energy in the target speech, but contains no information about the relative distribution of energy within the spectrogram. In this paper, we discuss the use of this stimulus in a series of auditory figure‐ground experiments where the target speech signal differs from the background by only a single parameter (relative level, ITD, ILD, etc.).

Modeling Very Complex Geometric Assemblies: The Use of Discrete and Hierarchical Models

AbstractVirtual Reality (VR) Systems allow immersive inspection of very complex environments, with direct interaction with the objects in the virtual scene. Low cost and affordable VR systems are becoming essential in application areas like Industrial Design, Medical Applications and Cultural Heritage and require sophisticated techniques for model inspection and interaction. Discrete geometric models can be very useful for some involved geometry processing algorithms like model repair, occlusion culling and multi-resolution and level of detail. The interest of these techniques and their potential in present and future VR applications will be discussed. This paper addresses the interest of discrete geometric models in volume-based geometric simplification, the use of discrete tiles for impostorbased extreme simplification, topology preserving model repair techniques based on discrete membranes and specific algorithms for recovering polygonal shapes from discrete volume models. The use of out-of-core octree...

A “regular” pentagonal tiling of the plane

The paper introduces conformal tilings, wherein tiles have specified conformal shapes. The principal example involves conformally regular pentagons which tile the plane in a pattern generated by a subdivision rule. Combinatorial symmetries imply rigid conformal symmetries, which in turn illustrate a new type of tiling self-similarity. In parallel with the conformal tilings, the paper develops discrete tilings based on circle packings. These faithfully reflect the key features of the theory and provide the tiling illustrations of the paper. Moreover, it is shown that under refinement the discrete tiles converge to their true conformal shapes, shapes for which no other approximation techniques are known. The paper concludes with some further examples which may contribute to the study of tilings and shinglings being carried forward by Cannon, Floyd, and Parry.

Transfer-matrix analysis of a two-dimensional quasicrystal.

We investigate the quasicrystalline state of a two-dimensional binary alloy in a discrete tiling approximation. Through transfer-matrix calculations we determine the configurational entropy over a range of concentrations. We find that the entropy density is maximized by a state with tenfold symmetry at the quasicrystal concentration. Derivatives of the entropy density at its maximum yield values for the phason elastic constants. Our results confirm the existence of quasi-long-range translational order in equilibrium quasicrystalline alloys and lend support to the random-tiling model of quasicrystals.