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1. Clustering with Neighborhoods is Hard for Squares

   Klimenko, Georgiy; Raichel, Benjamin (July 2023, Proceedings of the 35th Canadian Conference on Computational Geometry)

   Pankratov, Denis (Ed.)

   In the clustering with neighborhoods problem one is given a set S of disjoint convex objects in the plane and an integer parameter k ≥ 0, and the goal is to select a set C of k center points in the plane so as to minimize the maximum distance of an object in S to its nearest center in C. Previously [HKR21] showed that this problem cannot be approximated within any factor when S is a set of disjoint line segments, however, when S is a set of disjoint disks there is a roughly 8.46 approximation algorithm and a roughly 6.99 approximation lower bound. In this paper we investigate this significant discrepancy in hardness between these shapes. Specifically, we show that when S is a set of axis aligned squares of the same size, the problem again is hard to approximate within any factor. This surprising fact shows that the discrepancy is not due to the fatness of the object class, as one might otherwise naturally suspect. 

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2. Interface Dependent Coexistence of Two‐Dimensional Electron and Hole Gases in Mn‐doped InAs/GaSb

   https://doi.org/10.1002/admi.202400630  

   Riney, Logan; Bac, Seul‐Ki; Zhukovskyi, Maksym; Orlova, Tatyana; Fields, Shelby_S; Wang, Jiashu; Kotte, Tommy; Graf, David; Bennett, Steven_P; Liu, Xinyu; et al (October 2024, Advanced Materials Interfaces)

   Abstract The interface of common III‐V semiconductors InAs and GaSb can be utilized to realize a two‐dimensional (2D) topological insulator state. The 2D electronic gas at this interface can yield Hall quantization from coexisting electrons and holes. This anomaly is a determining factor in the fundamental origin of the topological state in InAs/GaSb. Here, the coexistence of electrons and holes in InAs/GaSb is tied to the chemical sharpness of the interface. Magnetotransport, in samples of Mn‐doped InAs/GaSb cleaved from wafers grown at a spatially inhomogeneous substrate temperature, is studied. It is reported that the observation of quantum oscillations and a quantized Hall effect whose behavior, exhibiting coexisting electrons and holes, is tuned by this spatial nonuniformity. Through transmission electron microscopy measurements, it is additionally found that samples that host this co‐existence exhibit a chemical intermixing between group III and group V atoms that extends over a larger thickness about the interface. The issue of intermixing at the interface is systematically overlooked in electronic transport studies of topological InAs/GaSb. These findings address this gap in knowledge and shed important light on the origin of the anomalous behavior of quantum oscillations seen in this 2D topological insulator. 

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3. Bernstein–Sato functional equations, V-filtrations, and multiplier ideals of direct summands

   https://doi.org/10.1142/S0219199721500838  

   Àlvarez Montaner, Josep; Hernández, Daniel J.; Jeffries, Jack; Núñez-Betancourt, Luis; Teixeira, Pedro; Witt, Emily E. (December 2022, Communications in Contemporary Mathematics)

   This paper investigates the existence and properties of a Bernstein–Sato functional equation in nonregular settings. In particular, we construct [Formula: see text]-modules in which such formal equations can be studied. The existence of the Bernstein–Sato polynomial for a direct summand of a polynomial over a field is proved in this context. It is observed that this polynomial can have zero as a root, or even positive roots. Moreover, a theory of [Formula: see text]-filtrations is introduced for nonregular rings, and the existence of these objects is established for what we call differentially extensible summands. This family of rings includes toric, determinantal, and other invariant rings. This new theory is applied to the study of multiplier ideals and Hodge ideals of singular varieties. Finally, we extend known relations among the objects of interest in the smooth case to the setting of singular direct summands of polynomial rings. 

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4. Analysis and Evaluation of Piecewise Linear Systems With Coulomb Friction Using a Hybrid Symbolic-Numeric Computational Method

   https://doi.org/10.1115/DETC2021-69430  

   Shahhosseini, Amir; Tien, Meng-Hsuan; D’Souza, Kiran (August 2021, ASME 2021 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference)

   A general formulation of piecewise linear systems with discontinuous force elements is provided in this paper. It has been demonstrated that this class of nonlinear systems is of great importance due to their ability to accurately model numerous scientific and engineering phenomena. Additionally, it is shown that this class of nonlinear systems can demonstrate a wide spectrum of nonlinear motions and in fact, the phenomenon of weak chaos is observed in a mechanical assembly for the first time. Despite such importance, efficient methods for fast and accurate evaluation of piecewise linear systems’ responses are lacking and the methods of the literature are either incompatible, very slow, very inaccurate, or bear a combination of the aforementioned deficiencies. To overcome this shortcoming, a novel symbolic-numeric method is presented in this paper that is able to obtain the analytical response of piecewise linear systems with discontinuous elements in an efficient manner. Contrary to other efficient methods that are based on stationary steady state dynamics, this method will not experience failure upon the occurrence of complex motion and is able to capture the entirety of the dynamics. 

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5. Homologous Self‐Assembled Superlattices: What Causes their Periodic Polarity Switching?

   https://doi.org/10.1002/apxr.202300039  

   Thakur, Varun; Benafsha, Dor; Turkulets, Yury; Azulay, Almog_R; Liang, Xin; Goldman, R_S; Shalish, Ilan (November 2023, Advanced Physics Research)

   Abstract Quantum semiconductor structures are commonly achieved by bandgap engineering, which relies on the ability to switch from one semiconductor to another during their growth. Growth of a superlattice is typically demanding technologically. In contrast, accumulated evidence points to a tendency among a certain class of multiple‐cation binary oxides to self‐assemble spontaneously as superlattice structures. This class is dubbed the homologous superlattices. For a famous example, when a mixture of indium and zinc is oxidized, the phases of In‐O and ZnO separate in an orderly periodic manner, along the ZnO polar axis, with polarity inversion taking place between consecutive ZnO sections. The same structure is observed when the indium is replaced with other metals, and perhaps even in ZnO alone. This peculiar self‐assembled structure is attracting research over the past decade. The purpose of this study is to gain understanding of the physics underlying the formation of this unique structure. Here, an explanation is proposed for the long‐standing mystery of this intriguing self‐assembly in the form of an electrostatic growth phenomenon and a test of the proposed model is carried out on experimental data. 

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