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1. Non-volatile electrically programmable integrated photonics with a 5-bit operation

   https://doi.org/10.1038/s41467-023-39180-3  

   Chen, Rui ; Fang, Zhuoran ; Perez, Christopher ; Miller, Forrest ; Kumari, Khushboo ; Saxena, Abhi ; Zheng, Jiajiu ; Geiger, Sarah J. ; Goodson, Kenneth E. ; Majumdar, Arka ( December 2023 , Nature Communications)

   Scalable programmable photonic integrated circuits (PICs) can potentially transform the current state of classical and quantum optical information processing. However, traditional means of programming, including thermo-optic, free carrier dispersion, and Pockels effect result in either large device footprints or high static energy consumptions, significantly limiting their scalability. While chalcogenide-based non-volatile phase-change materials (PCMs) could mitigate these problems thanks to their strong index modulation and zero static power consumption, they often suffer from large absorptive loss, low cyclability, and lack of multilevel operation. Here, we report a wide-bandgap PCM antimony sulfide (Sb₂S₃)-clad silicon photonic platform simultaneously achieving low loss (<1.0 dB), high extinction ratio (>10 dB), high cyclability (>1600 switching events), and 5-bit operation. These Sb₂S₃-based devices are programmed via on-chip silicon PIN diode heaters within sub-ms timescale, with a programming energy density of$$\sim 10\,{fJ}/n{m}^{3}$$$~10fJ/n{m}^3$. Remarkably, Sb₂S₃is programmed into fine intermediate states by applying multiple identical pulses, providing controllable multilevel operations. Through dynamic pulse control, we achieve 5-bit (32 levels) operations, rendering 0.50 ± 0.16 dB per step. Using this multilevel behavior, we further trim random phase error in a balanced Mach-Zehnder interferometer.

    

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2. New approximations for monotone submodular maximization with knapsack constraint

   https://doi.org/10.1007/s10878-024-01214-x  

   Du, Hongmin_W ; Li, Xiang ; Wang, Guanghua ( October 2024 , Journal of Combinatorial Optimization)

   Given a monotone submodular set function with a knapsack constraint, its maximization problem has two types of approximation algorithms with running time$$O(n^2)$$$O\left(n^2\right)$and$$O(n^5)$$$O\left(n^5\right)$, respectively. With running time$$O(n^5)$$$O\left(n^5\right)$, the best performance ratio is$$1-1/e$$$1-1/e$. With running time$$O(n^2)$$$O\left(n^2\right)$, the well-known performance ratio is$$(1-1/e)/2$$$(1-1/e)/2$and an improved one is claimed to be$$(1-1/e^2)/2$$$(1-1/e^2)/2$recently. In this paper, we design an algorithm with running$$O(n^2)$$$O\left(n^2\right)$and performance ratio$$1-1/e^{2/3}$$$1-1/e^{2/3}$, and an algorithm with running time$$O(n^3)$$$O\left(n^3\right)$and performance ratio 1/2.

    

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3. Spherical convex hull of random points on a wedge

   https://doi.org/10.1007/s00208-023-02704-9  

   Besau, Florian ; Gusakova, Anna ; Reitzner, Matthias ; Schütt, Carsten ; Thäle, Christoph ; Werner, Elisabeth_M ( August 2023 , Mathematische Annalen)

   Consider two half-spaces$$H_1^+$$$H_1^{+}$and$$H_2^+$$$H_2^{+}$in$${\mathbb {R}}^{d+1}$$$R^{d+1}$whose bounding hyperplanes$$H_1$$$H_1$and$$H_2$$$H_2$are orthogonal and pass through the origin. The intersection$${\mathbb {S}}_{2,+}^d:={\mathbb {S}}^d\cap H_1^+\cap H_2^+$$$S_{2,+}^d:=S^d\cap H_1^{+}\cap H_2^{+}$is a spherical convex subset of thed-dimensional unit sphere$${\mathbb {S}}^d$$$S^d$, which contains a great subsphere of dimension$$d-2$$$d-2$and is called a spherical wedge. Choosenindependent random points uniformly at random on$${\mathbb {S}}_{2,+}^d$$$S_{2,+}^d$and consider the expected facet number of the spherical convex hull of these points. It is shown that, up to terms of lower order, this expectation grows like a constant multiple of$$\log n$$$logn$. A similar behaviour is obtained for the expected facet number of a homogeneous Poisson point process on$${\mathbb {S}}_{2,+}^d$$$S_{2,+}^d$. The result is compared to the corresponding behaviour of classical Euclidean random polytopes and of spherical random polytopes on a half-sphere.

    

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4. Perfectly Packing a Square by Squares of Nearly Harmonic Sidelength

   https://doi.org/10.1007/s00454-023-00523-y  

   Tao, Terence ( July 2023 , Discrete & Computational Geometry)

   A well-known open problem of Meir and Moser asks if the squares of sidelength 1/nfor$$n\ge 2$$$n\ge 2$can be packed perfectly into a rectangle of area$$\sum _{n=2}^\infty n^{-2}=\pi ^2/6-1$$${\sum }_{n=2}^{\infty }{n}^{-2}={\pi }^2/6-1$. In this paper we show that for any$$1/2$1/2<t<1$, and any$$n_0$$$n_0$that is sufficiently large depending on t, the squares of sidelength$$n^{-t}$$$n^{-t}$for$$n\ge n_0$$$n\ge n_0$can be packed perfectly into a square of area$$\sum _{n=n_0}^\infty n^{-2t}$$${\sum }_{n=n_0}^{\infty }{n}^{-2t}$. This was previously known (if one packs a rectangle instead of a square) for$$1/2$1/2<t\le 2/3$(in which case one can take$$n_0=1$$$n_0=1$).

    

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5. On a Traveling Salesman Problem for Points in the Unit Cube

   https://doi.org/10.1007/s00453-024-01257-w  

   Balogh, József ; Clemen, Felix Christian ; Dumitrescu, Adrian ( September 2024 , Algorithmica)

   LetXbe ann-element point set in thek-dimensional unit cube$$[0,1]^k$$${[0,1]}^k$where$$k \ge 2$$$k\ge 2$. According to an old result of Bollobás and Meir (Oper Res Lett 11:19–21, 1992) , there exists a cycle (tour)$$x_1, x_2, \ldots , x_n$$$x_1,x_2,\dots ,x_n$through thenpoints, such that$$\left( \sum _{i=1}^n |x_i - x_{i+1}|^k \right) ^{1/k} \le c_k$$${\left({\sum }_{i=1}^n,{|x_i-x_{i+1}|}^k\right)}^{1/k}\le c_k$, where$$|x-y|$$$|x-y|$is the Euclidean distance betweenxandy, and$$c_k$$$c_k$is an absolute constant that depends only onk, where$$x_{n+1} \equiv x_1$$$x_{n+1}\equiv x_1$. From the other direction, for every$$k \ge 2$$$k\ge 2$and$$n \ge 2$$$n\ge 2$, there existnpoints in$$[0,1]^k$$${[0,1]}^k$, such that their shortest tour satisfies$$\left( \sum _{i=1}^n |x_i - x_{i+1}|^k \right) ^{1/k} = 2^{1/k} \cdot \sqrt{k}$$${\left({\sum }_{i=1}^n,{|x_i-x_{i+1}|}^k\right)}^{1/k}=2^{1/k}·\sqrt{k}$. For the plane, the best constant is$$c_2=2$$$c_2=2$and this is the only exact value known. Bollobás and Meir showed that one can take$$c_k = 9 \left( \frac{2}{3} \right) ^{1/k} \cdot \sqrt{k}$$$c_k=9{\left(\frac{2}{3}\right)}^{1/k}·\sqrt{k}$for every$$k \ge 3$$$k\ge 3$and conjectured that the best constant is$$c_k = 2^{1/k} \cdot \sqrt{k}$$$c_k=2^{1/k}·\sqrt{k}$, for every$$k \ge 2$$$k\ge 2$. Here we significantly improve the upper bound and show that one can take$$c_k = 3 \sqrt{5} \left( \frac{2}{3} \right) ^{1/k} \cdot \sqrt{k}$$$c_k=3\sqrt{5}{\left(\frac{2}{3}\right)}^{1/k}·\sqrt{k}$or$$c_k = 2.91 \sqrt{k} \ (1+o_k(1))$$$c_k=2.91\sqrt{k}(1+o_k\left(1\right))$. Our bounds are constructive. We also show that$$c_3 \ge 2^{7/6}$$$c_3\ge 2^{7/6}$, which disproves the conjecture for$$k=3$$$k=3$. Connections to matching problems, power assignment problems, related problems, including algorithms, are discussed in this context. A slightly revised version of the Bollobás–Meir conjecture is proposed.

    

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