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1. Bounds for exit times of Brownian motion and the first Dirichlet eigenvalue for the Laplacian

   https://doi.org/10.1090/tran/8903  

   Bañuelos, Rodrigo; Mariano, Phanuel; Wang, Jing (August 2023, Transactions of the American Mathematical Society)

   For domains in R d \mathbb {R}^d , d ≥ 2 d\geq 2 , we prove universal upper and lower bounds on the product of the bottom of the spectrum for the Laplacian to the power p > 0 p>0 and the supremum over all starting points of the p p -moments of the exit time of Brownian motion. It is shown that the lower bound is sharp for integer values of p p and that for p ≥ 1 p \geq 1 , the upper bound is asymptotically sharp as d → ∞ d\to \infty . For all p > 0 p>0 , we prove the existence of an extremal domain among the class of domains that are convex and symmetric with respect to all coordinate axes. For this class of domains we conjecture that the cube is extremal. 

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2. Limit models in strictly stable abstract elementary classes

   https://doi.org/10.1002/malq.202200075  

   Boney, Will; VanDieren, Monica_M (December 2024, Mathematical Logic Quarterly)

   Abstract In this paper, we examine the locality condition for non‐splitting and determine the level of uniqueness of limit models that can be recovered in some stable, but not superstable, abstract elementary classes. In particular we prove the following. Suppose that is an abstract elementary class satisfyingthe joint embedding and amalgamation properties with no maximal model of cardinality ,stability in ,,continuity for (i.e., if and is a limit model witnessed by for some limit ordinal and there exists so that does not ‐split over for all , then does not ‐split over ). Then for and limit ordinals both with cofinality , if satisfies symmetry for (or just ‐symmetry), then, for any and that are and ‐limit models over , respectively, we have that and are isomorphic over . Note that no tameness is assumed. 

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3. Attention and platypuses

   https://doi.org/10.1002/wcs.1600  

   Shomstein, Sarah; Zhang, Xiaoli; Dubbelde, Dick (April 2022, WIREs Cognitive Science)

   Abstract This perspective piece discusses a set of attentional phenomena that are not easily accommodated within current theories of attentional selection. We call these phenomena attentional platypuses, as they allude to an observation that within biological taxonomies the platypus does not fit into either mammal or bird categories. Similarly, attentional phenomena that do not fit neatly within current attentional models suggest that current models are in need of a revision. We list a few instances of the “attentional platypuses” and then offer a new approach, that we term dynamically weighted prioritization, stipulating that multiple factors impinge onto the attentional priority map, each with a corresponding weight. The interaction between factors and their corresponding weights determines the current state of the priority map which subsequently constrains/guides attentional allocation. We propose that this new approach should be considered as a supplement to existing models of attention, especially those that emphasize categorical organizations. This article is categorized under:Psychology > AttentionPsychology > Perception and PsychophysicsNeuroscience > Cognition 

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4. Global meta-analysis shows pervasive phosphorus limitation of aboveground plant production in natural terrestrial ecosystems

   https://doi.org/10.1038/s41467-020-14492-w  

   Hou, Enqing; Luo, Yiqi; Kuang, Yuanwen; Chen, Chengrong; Lu, Xiankai; Jiang, Lifen; Luo, Xianzhen; Wen, Dazhi (January 2020, Nature Communications)

   Abstract Phosphorus (P) limitation of aboveground plant production is usually assumed to occur in tropical regions but rarely elsewhere. Here we report that such P limitation is more widespread and much stronger than previously estimated. In our global meta-analysis, almost half (46.2%) of 652 P-addition field experiments reveal a significant P limitation on aboveground plant production. Globally, P additions increase aboveground plant production by 34.9% in natural terrestrial ecosystems, which is 7.0–15.9% higher than previously suggested. In croplands, by contrast, P additions increase aboveground plant production by only 13.9%, probably because of historical fertilizations. The magnitude of P limitation also differs among climate zones and regions, and is driven by climate, ecosystem properties, and fertilization regimes. In addition to confirming that P limitation is widespread in tropical regions, our study demonstrates that P limitation often occurs in other regions. This suggests that previous studies have underestimated the importance of altered P supply on aboveground plant production in natural terrestrial ecosystems. 

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5. Computing the Caratheodory Number of a Point

   Bereg, Sergey; Haghpanah, Mohammadreza (August 2020, Canadian Conference on Computational Geometry)

   Caratheodory’s theorem says that any point in the convex hull of a set $$P$$ in $R^d$ is in the convex hull of a subset $P'$ of $$P$$ such that $$|P'| \le d + 1$$. For some sets P, the upper bound d + 1 can be improved. The best upper bound for P is known as the Caratheodory number [2, 15, 17]. In this paper, we study a computational problem of finding the smallest set $P'$ for a given set $$P$$ and a point $$p$$. We call the size of this set $P'$, the Caratheodory number of a point p or CNP. We show that the problem of deciding the Caratheodory number of a point is NP-hard. Furthermore, we show that the problem is k-LDT-hard. We present two algorithms for computing a smallest set $P'$, if CNP= 2,3. Bárány [1] generalized Caratheodory’s theorem by using d+1 sets (colored sets) such that their convex hulls intersect. We introduce a Colorful Caratheodory number of a point or CCNP which can be smaller than d+1. Then we extend our results for CNP to CCNP. 

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