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1. PLA-index: A k-mer Index Exploiting Rank Curve Linearity

   https://doi.org/10.4230/LIPIcs.WABI.2024.13  

   Abrar, Md Hasin; Medvedev, Paul (January 2024, Schloss Dagstuhl – Leibniz-Zentrum für Informatik)

   Pissis, Solon P; Sung, Wing-Kin (Ed.)

   Given a sorted list of k-mers S, the rank curve of S is the function mapping a k-mer from the k-mer universe to the location in S where it either first appears or would be inserted. An exciting recent development is the observation that, for certain datasets, the rank curve is predictable and can be exploited to create small search indices. In this paper, we develop a novel search index that first estimates a k-mer’s rank using a piece-wise linear approximation of the rank curve and then does a local search to determine the precise location of the k-mer in the list. We combine ideas from previous approaches and supplement them with an innovative data representation strategy that substantially reduces space usage. Our PLA-index uses an order of magnitude less space than Sapling and uses less than half the space of the PGM-index, for roughly the same query time. For example, using only 9 MiB of memory, it can narrow down the position of k-mer in the suffix array of the human genome to within 255 positions. Furthermore, we demonstrate the potential of our approach to impact a variety of downstream applications. First, the PLA-index halves the time of binary search on the suffix array of the human genome. Second, the PLA-index reduces the space of a direct-access lookup table by 76 percent, without increasing the run time. Third, we plug the PLA-index into a state-of-the-art read aligner Strobealign and replace a 2 GiB component with a PLA-index of size 1.5 MiB, without significantly effecting runtime. The software and reproducibility information is freely available at https://github.com/medvedevgroup/pla-index. 

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2. Multilayer dielectric reflector using low-index nanolattices

   https://doi.org/10.1364/OL.516147  

   Chen, I-te; Premnath, Vijay_Anirudh; Chang, Chih-Hao (February 2024, Optics Letters)

   Dielectric mirrors based on Bragg reflection and photonic crystals have broad application in controlling light reflection with low optical losses. One key parameter in the design of these optical multilayers is the refractive index contrast, which controls the reflector performance. This work reports the demonstration of a high-reflectivity multilayer photonic reflector that consists of alternating layers of TiO₂films and nanolattices with low refractive index. The use of nanolattices enables high-index contrast between the high- and low-index layers, allowing high reflectivity with fewer layers. The broadband reflectance of the nanolattice reflectors with one to three layers has been characterized with peak reflectance of 91.9% at 527 nm and agrees well with theoretical optical models. The high-index contrast induced by the nanolattice layer enables a normalize reflectance band of Δλ/λ_oof 43.6%, the broadest demonstrated to date. The proposed nanolattice reflectors can find applications in nanophotonics, radiative cooling, and thermal insulation. 

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3. Magnetic Mie resonances in negative-index medium

   https://doi.org/10.1063/5.0257357  

   Babicheva, Viktoriia_E (April 2025, APL Photonics)

   High-refractive-index nanoantennas have attracted significant attention lately because of the strong excitations of electric and magnetic resonances in these nanoantennas. Here, we theoretically investigate the excitation of multipolar Mie resonances in high-refractive-index nanoantennas that are immersed in a negative-index medium. Our analysis shows a significant enhancement of magnetic resonances in this case. Furthermore, the magnetic dipolar and quadrupolar resonances exhibit a π-shift compared to these magnetic resonances in a conventional medium, which stems from the “left-handedness” of the negative-index medium. As a result, the spectral regions where electric and magnetic resonances are in-phase or out-of-phase complement, or opposite, to those in a conventional medium. Most importantly, we demonstrate nanoantenna magnetic resonances in two practical cases of negative-index media realized with common materials, such as multilayer structures with surface waves with negative effective mode index and fishnet metamaterial. These findings represent significant progress toward the realization of hybrid emitting structures that exhibit transitions with both electric and magnetic dipolar characteristics and pave the way for greater flexibility in controlling radiation patterns from quantum emitters. 

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4. Ring graphs and Goldberg's bound on chromatic index

   https://doi.org/10.1002/jgt.22494  

   Cao, Yan; Chen, Guantao; He, Shushan; Jing, Guangming (August 2019, Journal of Graph Theory)

   Abstract Let be a multigraph with maximum degree and chromatic index . If is bipartite then . Otherwise, by a theorem of Goldberg, , where denotes the odd girth of . Stiebitz, Scheide, Toft, and Favrholdt in their book conjectured that if then contains as a subgraph a ring graph with the same chromatic index. Vizing's characterization of graphs with chromatic index attaining the Shannon's bound showed the above conjecture holds for . Stiebitz et al verified the conjecture for graphs with and . McDonald proved the conjecture when is divisible by . In this paper, we show that the chromatic index condition alone is not sufficient to give the conclusion in the conjecture. On the positive side, we show that the conjecture holds for every with , and the maximum degree condition is best possible. This positive result leans on the positive resolution of the Goldberg‐Seymour conjecture. 

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5. Burch Index, Summands of Syzygies and Linearity in Resolutions

   https://doi.org/10.1007/s41980-023-00755-x  

   Dao, Hailong; Eisenbud, David (February 2023, Bulletin of the Iranian Mathematical Society)

   Abstract The Burch index is a new invariant of a local ringRwhose positivity implies a kind of linearity in resolutions ofR-modules. We show that ifRhas depth zero and Burch index at least 2, then any non-free 7thR-syzygy contains the residue field as a direct summand. We compute the Burch index in various cases of interest. 

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