15 research outputs found

    LIPIcs, Volume 251, ITCS 2023, Complete Volume

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    LIPIcs, Volume 251, ITCS 2023, Complete Volum

    LIPIcs, Volume 261, ICALP 2023, Complete Volume

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    LIPIcs, Volume 261, ICALP 2023, Complete Volum

    LIPIcs, Volume 274, ESA 2023, Complete Volume

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    LIPIcs, Volume 274, ESA 2023, Complete Volum

    LIPIcs, Volume 248, ISAAC 2022, Complete Volume

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    LIPIcs, Volume 248, ISAAC 2022, Complete Volum

    Polyhedral and Tropical Geometry of Flag Positroids

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    A flag positroid of ranks r:=(r1<<rk)\boldsymbol{r}:=(r_1<\dots <r_k) on [n][n] is a flag matroid that can be realized by a real rk×nr_k \times n matrix AA such that the ri×rir_i \times r_i minors of AA involving rows 1,2,,ri1,2,\dots,r_i are nonnegative for all 1ik1\leq i \leq k. In this paper we explore the polyhedral and tropical geometry of flag positroids, particularly when r:=(a,a+1,,b)\boldsymbol{r}:=(a, a+1,\dots,b) is a sequence of consecutive numbers. In this case we show that the nonnegative tropical flag variety TrFlr,n0_{\boldsymbol{r},n}^{\geq 0} equals the nonnegative flag Dressian FlDrr,n0_{\boldsymbol{r},n}^{\geq 0}, and that the points μ=(μa,,μb)\boldsymbol{\mu} = (\mu_a,\ldots, \mu_b) of TrFlr,n0=_{\boldsymbol{r},n}^{\geq 0} = FlDrr,n0_{\boldsymbol{r},n}^{\geq 0} give rise to coherent subdivisions of the flag positroid polytope P(μ)P(\underline{\boldsymbol{\mu}}) into flag positroid polytopes. Our results have applications to Bruhat interval polytopes: for example, we show that a complete flag matroid polytope is a Bruhat interval polytope if and only if its (2)(\leq 2)-dimensional faces are Bruhat interval polytopes. Our results also have applications to realizability questions. We define a positively oriented flag matroid to be a sequence of positively oriented matroids (χ1,,χk)(\chi_1,\dots,\chi_k) which is also an oriented flag matroid. We then prove that every positively oriented flag matroid of ranks r=(a,a+1,,b)\boldsymbol{r}=(a,a+1,\dots,b) is realizable.Comment: 40 page

    Combinatorial Algorithms for Multidimensional Necklaces

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    A necklace is an equivalence class of words of length nn over an alphabet under the cyclic shift (rotation) operation. As a classical object, there have been many algorithmic results for key operations on necklaces, including counting, generating, ranking, and unranking. This paper generalises the concept of necklaces to the multidimensional setting. We define multidimensional necklaces as an equivalence classes over multidimensional words under the multidimensional cyclic shift operation. Alongside this definition, we generalise several problems from the one dimensional setting to the multidimensional setting for multidimensional necklaces with size (n1,n2,...,nd)(n_1,n_2,...,n_d) over an alphabet of size qq including: providing closed form equations for counting the number of necklaces; an O(n1n2...nd)O(n_1 \cdot n_2 \cdot ... \cdot n_d) time algorithm for transforming some necklace ww to the next necklace in the ordering; an O((n1n2...nd)5)O((n_1 \cdot n_2 \cdot ... \cdot n_d)^5) time algorithm to rank necklaces (determine the number of necklaces smaller than ww in the set of necklaces); an O((n1n2...nd)6(d+1)logd(q))O((n_1\cdot n_2 \cdot ... \cdot n_d)^{6(d + 1)} \cdot \log^d(q)) time algorithm to unrank multidimensional necklace (determine the ithi^{th} necklace in the set of necklaces). Our results on counting, ranking, and unranking are further extended to the fixed content setting, where every necklace has the same Parikh vector, in other words every necklace shares the same number of occurrences of each symbol. Finally, we study the kk-centre problem for necklaces both in the single and multidimensional settings. We provide strong approximation algorithms for solving this problem in both the one dimensional and multidimensional settings

    Ranking Bracelets in Polynomial Time

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    The main result of the paper is the first polynomial-time algorithm for ranking bracelets. The time-complexity of the algorithm is O(k^2 n^4), where k is the size of the alphabet and n is the length of the considered bracelets. The key part of the algorithm is to compute the rank of any word with respect to the set of bracelets by finding three other ranks: the rank over all necklaces, the rank over palindromic necklaces, and the rank over enclosing apalindromic necklaces. The last two concepts are introduced in this paper. These ranks are key components to our algorithm in order to decompose the problem into parts. Additionally, this ranking procedure is used to build a polynomial-time unranking algorithm

    Electronic Journal of Qualitative Theory of Differential Equations 2021

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    Plasmonic Nanoplatforms for Biochemical Sensing and Medical Applications

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    Plasmonics, the science of the excitation of surface plasmon polaritons (SPP) at the metal-dielectric interface under intense beam radiation, has been studied for its immense potential for developing numerous nanophotonic devices, optical circuits and lab-on-a-chip devices. The key feature, which makes the plasmonic structures promising is the ability to support strong resonances with different behaviors and tunable localized hotspots, excitable in a wide spectral range. Therefore, the fundamental understanding of light-matter interactions at subwavelength nanostructures and use of this understanding to tailor plasmonic nanostructures with the ability to sustain high-quality tunable resonant modes are essential toward the realization of highly functional devices with a wide range of applications from sensing to switching. We investigated the excitation of various plasmonic resonance modes (i.e. Fano resonances, and toroidal moments) using both optical and terahertz (THz) plasmonic metamolecules. By designing and fabricating various nanostructures, we successfully predicted, demonstrated and analyzed the excitation of plasmonic resonances, numerically and experimentally. A simple comparison between the sensitivity and lineshape quality of various optically driven resonances reveals that nonradiative toroidal moments are exotic plasmonic modes with strong sensitivity to environmental perturbations. Employing toroidal plasmonic metasurfaces, we demonstrated ultrafast plasmonic switches and highly sensitive sensors. Focusing on the biomedical applications of toroidal moments, we developed plasmonic metamaterials for fast and cost-effective infection diagnosis using the THz range of the spectrum. We used the exotic behavior of toroidal moments for the identification of Zika-virus (ZIKV) envelope proteins as the infectious nano-agents through two protocols: 1) direct biding of targeted biomarkers to the plasmonic metasurfaces, and 2) attaching gold nanoparticles to the plasmonic metasurfaces and binding the proteins to the particles to enhance the sensitivity. This led to developing ultrasensitive THz plasmonic metasensors for detection of nanoscale and low-molecular-weight biomarkers at the picomolar range of concentration. In summary, by using high-quality and pronounced toroidal moments as sensitive resonances, we have successfully designed, fabricated and characterized novel plasmonic toroidal metamaterials for the detection of infectious biomarkers using different methods. The proposed approach allowed us to compare and analyze the binding properties, sensitivity, repeatability, and limit of detection of the metasensing device
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