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1. Sharp critical thresholds in a hyperbolic system with relaxation

   https://doi.org/10.3934/dcds.2021098  

   Bhatnagar, Manas; Liu, Hailiang (January 2021, Discrete & Continuous Dynamical Systems)

   null (Ed.)

   We propose and study a one-dimensional \begin{document}$$ 2\times 2 $$\end{document} hyperbolic Eulerian system with local relaxation from critical threshold phenomena perspective. The system features dynamic transition between strictly and weakly hyperbolic. For different classes of relaxation we identify intrinsic critical thresholds for initial data that distinguish global regularity and finite time blowup. For relaxation independent of density, we estimate bounds on density in terms of velocity where the system is strictly hyperbolic. 

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2. The Joint Simon task is not joint for capuchin monkeys

   https://doi.org/10.1038/s41598-024-55885-x  

   Martínez, Mayte; Babb, Matthew H.; Range, Friederike; Brosnan, Sarah F. (March 2024, Scientific Reports)

   Abstract Human cooperation can be facilitated by the ability to create a mental representation of one’s own actions, as well as the actions of a partner, known as action co-representation. Even though other species also cooperate extensively, it is still unclear whether they have similar capacities. The Joint Simon task is a two-player task developed to investigate this action co-representation. We tested brown capuchin monkeys (Sapajus [Cebus] apella), a highly cooperative species, on a computerized Joint Simon task and found that, in line with previous research, the capuchins' performance was compatible with co-representation. However, a deeper exploration of the monkeys’ responses showed that they, and potentially monkeys in previous studies, did not understand the control conditions, which precludes the interpretation of the results as a social phenomenon. Indeed, further testing to investigate alternative explanations demonstrated that our results were due to low-level cues, rather than action co-representation. This suggests that the Joint Simon task, at least in its current form, cannot determine whether non-human species co-represent their partner’s role in joint tasks. 

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3. Uniformly Acute Triangulations of PSLGs

   https://doi.org/10.1007/s00454-023-00524-x  

   Bishop, Christopher J (October 2023, Discrete & Computational Geometry)

   We show that any PSLG has an acute conforming triangulation with an upper angle bound that is strictly less than 90 degrees and that depends only on the minimal angle occurring in the PSLG. In fact, all angles are inside the interval I_0= [theta_0, 90 -\theta_0/2] for some fixed theta_0>0 independent of the PSLG except for triangles T containing a vertex v where the PSLG has an interior angle theta_v < \theta_0; then T is an isosceles triangle with angles in I_v = [theta_v, 90 -\theta_v/2]. 

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4. Proof of the Michael–Simon–Sobolev inequality using optimal transport

   https://doi.org/10.1515/crelle-2023-0054  

   Brendle, Simon; Eichmair, Michael (August 2023, Journal für die reine und angewandte Mathematik (Crelles Journal))

   Abstract We give an alternative proof of the Michael–Simon–Sobolev inequality using techniques from optimal transport. The inequality is sharp for submanifolds of codimension 2. 

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5. Min–max theory for free boundary minimal hypersurfaces II: general Morse index bounds and applications

   https://doi.org/10.1007/s00208-020-02096-0  

   Guang, Qiang; Li, Martin Man-chun; Wang, Zhichao; Zhou, Xin (April 2021, Mathematische Annalen)

   null (Ed.)

   Abstract For any smooth Riemannian metric on an $$(n+1)$$ ( n + 1 ) -dimensional compact manifold with boundary $$(M,\partial M)$$ ( M , ∂ M ) where $$3\le (n+1)\le 7$$ 3 ≤ ( n + 1 ) ≤ 7 , we establish general upper bounds for the Morse index of free boundary minimal hypersurfaces produced by min–max theory in the Almgren–Pitts setting. We apply our Morse index estimates to prove that for almost every (in the $$C^\infty $$ C ∞ Baire sense) Riemannan metric, the union of all compact, properly embedded free boundary minimal hypersurfaces is dense in M . If $$\partial M$$ ∂ M is further assumed to have a strictly mean convex point, we show the existence of infinitely many compact, properly embedded free boundary minimal hypersurfaces whose boundaries are non-empty. Our results prove a conjecture of Yau for generic metrics in the free boundary setting. 

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