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1. The eigensplitting of the fiber of the cyclotomic trace for the sphere spectrum

   https://doi.org/10.1090/tran/8822  

   Blumberg, Andrew; Mandell, Michael (January 2022, Transactions of the American Mathematical Society)

   Let p ∈ Z p\in {\mathbb {Z}} be an odd prime. We show that the fiber sequence for the cyclotomic trace of the sphere spectrum S {\mathbb {S}} admits an “eigensplitting” that generalizes known splittings on K K -theory and T C TC . We identify the summands in the fiber as the covers of Z p {\mathbb {Z}}_{p} -Anderson duals of summands in the K ( 1 ) K(1) -localized algebraic K K -theory of Z {\mathbb {Z}} . Analogous results hold for the ring Z {\mathbb {Z}} where we prove that the K ( 1 ) K(1) -localized fiber sequence is self-dual for Z p {\mathbb {Z}}_{p} -Anderson duality, with the duality permuting the summands by i ↦ p − i i\mapsto p-i (indexed mod p − 1 p-1 ). We explain an intrinsic characterization of the summand we call Z Z in the splitting T C ( Z ) p ∧ ≃ j ∨ Σ j ′ ∨ Z TC({\mathbb {Z}})^{\wedge }_{p}\simeq j \vee \Sigma j’\vee Z in terms of units in the p p -cyclotomic tower of Q p {\mathbb {Q}}_{p} . 

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2. Patches as an expressive medium for exploratory multi‐agent modelling

   https://doi.org/10.1111/bjet.13087  

   Brady, Corey (April 2021, British Journal of Educational Technology)

   Abstract This paper outlines the potential gains for Constructionist research and praxis in modelling that might be obtained by recognising the power of the Patch—a humble computational being in the NetLogo modelling environment that has been overshadowed by its more popular fellow agent, the Turtle. To contextualise this opportunity, I describe how Constructionist modelling has thrived by promoting forms of learning that rely on learners’ identifying with agents. I argue that patches are a neglected agent type in this multi‐agent modelling tradition, and that the possibilities for learners to adopt the patch perspective in support of exploratory forms of modelling and aesthetic expression have been under‐researched. Nevertheless, I show there are a variety of powerful ways for learners––both individually and in groups––to identify with patches. I describe ongoing research showing how taking an aesthetic approach to patches has the potential to support individuals and groups in powerful forms of learning with and about multi‐agent modelling. Practitioner notesWhat is already known about this topicTurtles (movable agents in Logo and Constructionist environments descended from Logo) can be ‘transitional objects’ that provide learners a way to make powerful ideas their own.These agents can be powerful ‘objects‐to‐think‐with’ in large part because they encourage learners to identify with them in a form of learning known as ‘syntonic learning’.Expressive activities that draw on learners’aestheticinterests can support their learning with and about computational representations.Multi‐agent modelling is a powerful extension of Logo‐based learning environments that provides access to powerful ideas about complex systems and their emergent properties.In the multi‐agent setting, individual learners and/or groups of learners can identify syntonically with agents to provide entry points for reasoning about complexity.What this paper addsPatches (non‐movable agents in the NetLogo modelling environment) are under‐represented in the research on multi‐agent modelling, and the potential for learners to adopt the patches’ perspective has been neglected.An aesthetically driven approach to patches can ground students’ understanding of their expressive value.Participatory activities in which learners play the role of patches (called ‘Stadium Card’ activities) can ground the patch perspective, so that learners can achieve a form of syntonicity and/or collectively adopt the perspective of patches in the aggregate.Participatory activities that blend intrinsic and extrinsic perspectives on the patch grid can further enhance learners’ facility with programming for patches and their understanding of patches’ collective expressive power.Implications for practice and/or policyBalancing the focus between turtles and patches can enrich the modelling toolbox of learners new to agent‐based modelling.Patchesdocapture important aspects of individual and collective experience, and so can be good objects‐to‐think‐with, especially when conceptualising phenomena at a larger scale.The expressive potential of the patch grid is an important topic for computer science as well (eg, through 2D cellular automata). This is a rich context for learning in itself, which can be made accessible to groups of learners through physical or virtual participatory role‐play.Moreover, physical or virtual grids of people‐patches may exhibit novel aggregate computational properties that could in turn become interesting areas for research in computer science. 

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3. Natural selection favours a larger eye in response to increased competition in natural populations of a vertebrate

   https://doi.org/10.1111/1365-2435.13334  

   Beston, Shannon M.; Walsh, Matthew R.; Higham, ed., Timothy (April 2019, Functional Ecology)

   Abstract Eye size varies notably across taxa. Much work suggests that this variation is driven by contrasting ecological selective pressures. However, evaluations of the relationship between ecological factors and shifts in eye size have largely occurred at the macroevolutionary scale. Experimental tests in nature are conspicuously absent.Trinidadian killifish,Rivulus hartii, are found across fish communities that differ in predation intensity. We recently showed that increased predation is associated with the evolution of a smaller eye. Here, we test how divergent predatory regimes alter the trajectory of eye size evolution using comparative mark–recapture experiments in multiple streams.We found that increases in eye size are associated with enhanced survival, irrespective of predation intensity. More importantly, eye size is associated with enhanced growth in communities that lack predators, while this trend is absent when predators are present.Such results argue that increased competition for food in sites that lack predators is the key driver of eye size evolution. Aplain language summaryis available for this article. 

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4. 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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5. A simple proof of reflexivity and separability of N^{1,p} Sobolev spaces

   https://doi.org/10.54330/afm.127419  

   Alvarado, Ryan; Hajłasz, Piotr; Malý, Lukáš (October 2022, Annales Fennici Mathematici)

   We present an elementary proof of a well-known theorem of Cheeger which states that if a metric-measure space \(X\) supports a \(p\)-Poincaré inequality, then the \(N^{1,p}(X)\) Sobolev space is reflexive and separable whenever \(p\in (1,\infty)\). We also prove separability of the space when \(p=1\). Our proof is based on a straightforward construction of an equivalent norm on \(N^{1,p}(X)\), \(p\in [1,\infty)\), that is uniformly convex when \(p\in (1,\infty)\). Finally, we explicitly construct a functional that is pointwise comparable to the minimal \(p\)-weak upper gradient, when \(p\in (1,\infty)\). 

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