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1. The p-widths of a surface

   https://doi.org/10.1007/s10240-023-00141-7  

   Chodosh, Otis; Mantoulidis, Christos (June 2023, Publications mathématiques de l'IHÉS)

   Abstract The $$p$$ p -widths of a closed Riemannian manifold are a nonlinear analogue of the spectrum of its Laplace–Beltrami operator, which corresponds to areas of a certain min-max sequence of possibly singular minimal submanifolds. We show that the $$p$$ p -widths of any closed Riemannian two-manifold correspond to a union of closed immersed geodesics, rather than simply geodesic nets. We then prove optimality of the sweepouts of the round two-sphere constructed from the zero set of homogeneous polynomials, showing that the $$p$$ p -widths of the round sphere are attained by $$\lfloor \sqrt{p}\rfloor $$ ⌊ p ⌋ great circles. As a result, we find the universal constant in the Liokumovich–Marques–Neves–Weyl law for surfaces to be $$\sqrt{\pi }$$ π . En route to calculating the $$p$$ p -widths of the round two-sphere, we prove two additional new results: a bumpy metrics theorem for stationary geodesic nets with fixed edge lengths, and that, generically, stationary geodesic nets with bounded mass and bounded singular set have Lusternik–Schnirelmann category zero. 

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2. A Two-Stage Constrained Submodular Maximization

   https://doi.org/10.1007/978-3-030-27195-4_30  

   Ruiqi Yang, Shuyang Gu (August 2019, AAIM 2019)

   We consider a two-stage submodular maximization under p-matroid (or p-extendible) constraints. In the model, we are given a collection of submodular functions and some p-matroid (or extendible) system constraints for each of these functions, one need to choose a representative set with a cardinality constraint and simultaneously select a series of subsets that are restricted to the representative set for all functions, the aim is to maximize the average of the summarization of these function values. We extend the two-stage submodular maximization under single matroid to handle p-matroid (or p-extendible) constraints, and derive constant approximation ratio algorithms for the two problems, respectively. In the end, we empirically demonstrate the efficiency of our method on some datasets. 

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3. A three-layer Hele-Shaw problem driven by a sink

   https://doi.org/10.1017/jfm.2024.688  

   Zhao, Meng; Barua, Amlan K; Lowengrub, John S; Ying, Wenjun; Li, Shuwang (November 2024, Journal of Fluid Mechanics)

   In this paper, we investigate a sink-driven three-layer flow in a radial Hele-Shaw cell. The three fluids are of different viscosities, with one fluid occupying an annulus-like domain, forming two interfaces with the other two fluids. Using a boundary integral method and a semi-implicit time stepping scheme, we alleviate the numerical stiffness in updating the interfaces and achieve spectral accuracy in space. The interaction between the two interfaces introduces novel dynamics leading to rich pattern formation phenomena, manifested by two typical events: either one of the two interfaces reaches the sink faster than the other (forming cusp-like morphology), or they come very close to each other (suggesting a possibility of interface merging). In particular, the inner interface can be wrapped by the other to have both scenarios. We find that multiple parameters contribute to the dynamics, including the width of the annular region, the location of the sink, and the mobilities of the fluids. 

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4. KMT-2024-BLG-0404L: A triple microlensing system consisting of a star, a brown dwarf, and a planet

   https://doi.org/10.1051/0004-6361/202452760  

   Han, Cheongho; Udalski, Andrzej; Lee, Chung-Uk; Ryu, Yoon-Hyun; Albrow, Michael D; Chung, Sun-Ju; Gould, Andrew; Hwang, Kyu-Ha; Kil_Jung, Youn; Shin, In-Gu; et al (August 2025, Astronomy & Astrophysics)

   Aims. We have investigated the lensing event KMT-2024-BLG-0404. The light curve of the event exhibited a complex structure with multiple distinct features, including two prominent caustic spikes, two cusp bumps, and a brief discontinuous feature between the caustic spikes. While a binary-lens model captured the general anomaly pattern, it could not account for a discontinuous anomaly feature between the two caustic spikes. Methods. To explore the origin of the unexplained feature, we conducted more advanced modeling beyond the standard binary-lens framework. This investigation demonstrated that the previously unexplained anomaly was resolved by introducing an additional lens component with planetary mass. Results. The estimated masses of the lens components areM_p= 17.3_−8.8^+25.5M_Efor the planet, andM_h,A= 0.090_−0.046^+0.133M_⊙andM_h,B= 0.026_−0.013^+0.038M_⊙for the binary host stars. Based on these mass estimates, the lens system is identified as a planetary system where a Uranus-mass planet orbits a binary consisting of a late M dwarf and a brown dwarf. The distance to the planetary system is estimated to beD_L= 7.21_−0.97^+0.93kpc, with an 82% probability that it resides in the Galactic bulge. This discovery represents the ninth planetary system found through microlensing with a planet orbiting a binary host. Notably, it is the first case in which the host consists of both a star and a brown dwarf. 

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5. Folding points to a point and lines to a line

   Akitaya, Hugo A.; Ballinger, Brad; Demaine, Erik D.; Hull, Thomas C.; Schmidt, Christiane (August 2021, Proceedings of the 33rd Canadian Conference on Computational Geometry (CCCG 2021))

   He, Meng; Sheehy, Don (Ed.)

   We introduce basic, but heretofore generally unexplored, problems in computational origami that are similar in style to classic problems from discrete and computational geometry. We consider the problems of folding each corner of a polygon P to a point p and folding each edge of a polygon P onto a line segment L that connects two boundary points of P and compute the number of edges of the polygon containing p or L limited by crease lines and boundary edges. 

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