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1. The L _p chord Minkowski problem

   https://doi.org/10.1515/ans-2022-0041  

   Xi, Dongmeng; Yang, Deane; Zhang, Gaoyong; Zhao, Yiming (January 2023, Advanced Nonlinear Studies)

   Abstract Chord measures are newly discovered translation-invariant geometric measures of convex bodies in R n {{\mathbb{R}}}^{n} , in addition to Aleksandrov-Fenchel-Jessen’s area measures. They are constructed from chord integrals of convex bodies and random lines. Prescribing the L p {L}_{p} chord measures is called the L p {L}_{p} chord Minkowski problem in the L p {L}_{p} Brunn-Minkowski theory, which includes the L p {L}_{p} Minkowski problem as a special case. This article solves the L p {L}_{p} chord Minkowski problem when p > 1 p\gt 1 and the symmetric case of 0 < p < 1 0\lt p\lt 1 . 

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2. Ti ₂ CT _x MXene‐based all‐optical modulator

   https://doi.org/10.1002/inf2.12052  

   Yi, Jun; Li, Jie; Huang, Shuohan; Hu, Longyu; Miao, Lili; Zhao, Chujun; Wen, Shuangchun; Mochalin, Vadym N.; Rao, Apparao M. (December 2019, InfoMat)

   Abstract MXenes, a new class of 2D transition metal carbides, nitrides, and carbonitrides, have attracted much attention due to their outstanding properties. Here, we report the broadband spatial self‐phase modulation of Ti₂CT_xMXene nanosheets dispersed in deionized water in the visible to near‐infrared regime, highlighting the broadband nonlinear optical (NLO) response of Ti₂CT_xMXene. Using ultrafast pulsed laser excitation, the nonlinear refractive indexn₂and the third‐order nonlinear susceptibilityof Ti₂CT_xMXene were measured to be ∼10⁻¹³m²/W and ∼ 10⁻¹⁰esu, respectively. Leveraging the large optical nonlinearity of Ti₂CT_xMXene, an all‐optical modulator in the visible regime was fabricated based on the spatial cross‐phase modulation effect. This work suggests that 2D MXenes are ideal broadband NLO materials with excellent prospects in NLO applications. image 

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3. Breaking the r _max Barrier: Enhanced Approximation Algorithms for Partial Set Multicover Problem

   https://doi.org/10.1287/ijoc.2020.0975  

   Ran, Yingli; Zhang, Zhao; Tang, Shaojie; Du, Ding-Zhu (October 2020, INFORMS Journal on Computing)

   null (Ed.)

   Given an element set E of order n, a collection of subsets [Formula: see text], a cost c S on each set [Formula: see text], a covering requirement r e for each element [Formula: see text], and an integer k, the goal of a minimum partial set multicover problem (MinPSMC) is to find a subcollection [Formula: see text] to fully cover at least k elements such that the cost of [Formula: see text] is as small as possible and element e is fully covered by [Formula: see text] if it belongs to at least r e sets of [Formula: see text]. This problem generalizes the minimum k-union problem (MinkU) and is believed not to admit a subpolynomial approximation ratio. In this paper, we present a [Formula: see text]-approximation algorithm for MinPSMC, in which [Formula: see text] is the maximum size of a set in S. And when [Formula: see text], we present a bicriteria algorithm fully covering at least [Formula: see text] elements with approximation ratio [Formula: see text], where [Formula: see text] is a fixed number. These results are obtained by studying the minimum density subcollection problem with (or without) cardinality constraint, which might be of interest by itself. 

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4. Parameterized Inapproximability of the Minimum Distance Problem over All Fields and the Shortest Vector Problem in All \({\ell_{{p}}}\) Norms

   https://doi.org/10.1137/23M1573021  

   Bennett, Huck; Cheraghchi, Mahdi; Guruswami, Venkatesan; Ribeiro, João (October 2024, SIAM Journal on Computing)
5. A crystallographic approach to the short-range ordering problem in V _1−x Mo _x O ₂ (0.50 ≤ x ≤ 0.60)

   https://doi.org/10.1039/d0tc01173h  

   Davenport, Matthew A.; Allred, Jared M. (January 2020, Journal of Materials Chemistry C)

   The V 1−x Mo x O 2 phase diagram has high structural and electronic complexity that is driven by strong, short-range correlations that compete with the long-range rutile crystal structure. The substitution regime near 50% Mo occupancy is no exception, but there has so far been no significant progress in determining the actual structure. Reported here is a combined study using single crystal X-ray diffraction, powder X-ray diffraction, and representational analysis to examine both the local and crystallographically averaged atomic structures simultaneously near x = 0.50. Between about x = 0.50 and 0.60, the average structure of V 1−x Mo x O 2 is the parent rutile phase, but the local symmetry is broken by atomic displacements that are best described using the orthorhombic subgroup Fmmm . This model is locally similar to the two-dimensionally ordered 2D-M2 phase recently reported in the compositional range 0.19 ≤ x ≤ 0.30, except the correlation length is much shorter in the 2D plane, and longer in the frustrated one, making it more isotropic. This work also extends the 2D-M2 phase regime up to x = 0.43, and suggests that the local- Fmmm phase observed here can be seen as the end result of the continued suppression of the 2D-M2 phase through enhanced geometric frustration between the intrinsic order parameters. This suggests that other doped-rutile phases with elusive structures may also be dominated by similar short-range correlations that are hidden in the diffuse scattering. 

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