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1. Hyperuniformity and anti-hyperuniformity in one-dimensional substitution tilings

   https://doi.org/10.1107/S2053273318015528  

   Oğuz, Erdal C.; Socolar, Joshua E.; Steinhardt, Paul J.; Torquato, Salvatore (January 2019, Acta Crystallographica Section A Foundations and Advances)

   This work considers the scaling properties characterizing the hyperuniformity (or anti-hyperuniformity) of long-wavelength fluctuations in a broad class of one-dimensional substitution tilings. A simple argument is presented which predicts the exponent α governing the scaling of Fourier intensities at small wavenumbers, tilings with α > 0 being hyperuniform, and numerical computations confirm that the predictions are accurate for quasiperiodic tilings, tilings with singular continuous spectra and limit-periodic tilings. Quasiperiodic or singular continuous cases can be constructed with α arbitrarily close to any given value between −1 and 3. Limit-periodic tilings can be constructed with α between −1 and 1 or with Fourier intensities that approach zero faster than any power law. 

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2. Measurable Tilings by Abelian Group Actions

   https://doi.org/10.1093/imrn/rnad048  

   Grebík, Jan; Greenfeld, Rachel; Rozhoň, Václav; Tao, Terence (March 2023, International Mathematics Research Notices)

   Abstract Let $$X$$ be a measure space with a measure-preserving action $$(g,x) \mapsto g \cdot x$$ of an abelian group $$G$$. We consider the problem of understanding the structure of measurable tilings $$F \odot A = X$$ of $$X$$ by a measurable tile $$A \subset X$$ translated by a finite set $$F \subset G$$ of shifts, thus the translates $$f \cdot A$$, $$f \in F$$ partition $$X$$ up to null sets. Adapting arguments from previous literature, we establish a “dilation lemma” that asserts, roughly speaking, that $$F \odot A = X$$ implies $$F^{r} \odot A = X$$ for a large family of integer dilations $$r$$, and use this to establish a structure theorem for such tilings analogous to that established recently by the second and fourth authors. As applications of this theorem, we completely classify those random tilings of finitely generated abelian groups that are “factors of iid”, and show that measurable tilings of a torus $${\mathbb{T}}^{d}$$ can always be continuously (in fact linearly) deformed into a tiling with rational shifts, with particularly strong results in the low-dimensional cases $d=1,2$ (in particular resolving a conjecture of Conley, the first author, and Pikhurko in the $d=1$ case). 

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3. Structural characterization of a new subclass of panicum mosaic virus-like 3′ cap-independent translation enhancer

   https://doi.org/10.1093/nar/gkac007  

   Johnson, Philip Z.; Kasprzak, Wojciech K.; Shapiro, Bruce A.; Simon, Anne E. (February 2022, Nucleic Acids Research)

   Abstract Canonical eukaryotic mRNA translation requires 5′cap recognition by initiation factor 4E (eIF4E). In contrast, many positive-strand RNA virus genomes lack a 5′cap and promote translation by non-canonical mechanisms. Among plant viruses, PTEs are a major class of cap-independent translation enhancers located in/near the 3′UTR that recruit eIF4E to greatly enhance viral translation. Previous work proposed a single form of PTE characterized by a Y-shaped secondary structure with two terminal stem-loops (SL1 and SL2) atop a supporting stem containing a large, G-rich asymmetric loop that forms an essential pseudoknot (PK) involving C/U residues located between SL1 and SL2. We found that PTEs with less than three consecutive cytidylates available for PK formation have an upstream stem-loop that forms a kissing loop interaction with the apical loop of SL2, important for formation/stabilization of PK. PKs found in both subclasses of PTE assume a specific conformation with a hyperreactive guanylate (G*) in SHAPE structure probing, previously found critical for binding eIF4E. While PTE PKs were proposed to be formed by Watson–Crick base-pairing, alternative chemical probing and 3D modeling indicate that the Watson–Crick faces of G* and an adjacent guanylate have high solvent accessibilities. Thus, PTE PKs are likely composed primarily of non-canonical interactions. 

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4. A Geometric Approach to Inelastic Collapse

   Chazelle, B; Karntikoon, K; Zheng, Y. (January 2022, Computational geometry)

   We show how to interpret logarithmic spiral tilings as one-dimensional particle systems undergoing inelastic collapse. By deforming the spirals appropriately, we can simulate collisions among particles with distinct or varying coefficients of restitution. Our geometric constructions provide a strikingly simple illustration of a widely studied phenomenon in the physics of dissipative gases: the collapse of inelastic particles. 

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5. ABAQUS Code for the Simulation of Interlocked Irregular Square Tilings

   https://doi.org/10.4231/PDKT-Q131  

   Kim, Dong; Siegmund, Thomas (January 2021, Purdue University Research Repository)

   A set of three input files for ABAQUS models to simulate the response of interlocked irregular square tilings subjected to displacement loading. 

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