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1. An algorithmic hypergraph regularity lemma

   https://doi.org/10.1002/rsa.20739  

   Nagle, Brendan; Rödl, Vojtěch; Schacht, Mathias (December 2017, Random Structures & Algorithms)

   Abstract Szemerédi 's Regularity Lemma is a powerful tool in graph theory. It asserts that all large graphs admit bounded partitions of their edge sets, most classes of which consist of uniformly distributed edges. The original proof of this result was nonconstructive, and a constructive proof was later given by Alon, Duke, Lefmann, Rödl, and Yuster. Szemerédi's Regularity Lemma was extended to hypergraphs by various authors. Frankl and Rödl gave one such extension in the case of 3‐uniform hypergraphs, which was later extended tok‐uniform hypergraphs by Rödl and Skokan. W.T. Gowers gave another such extension, using a different concept of regularity than that of Frankl, Rödl, and Skokan. Here, we give a constructive proof of a regularity lemma for hypergraphs. 

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2. Hölder regularity of the Boltzmann equation past an obstacle

   https://doi.org/10.1002/cpa.22167  

   Kim, Chanwoo; Lee, Donghyun (April 2024, Communications on Pure and Applied Mathematics)

   Abstract Regularity and singularity of the solutions according to the shape of domains is a challenging research theme in the Boltzmann theory. In this paper, we prove an Hölder regularity in for the Boltzmann equation of the hard‐sphere molecule, which undergoes the elastic reflection in the intermolecular collision and the contact with the boundary of a convex obstacle. In particular, this Hölder regularity result is a stark contrast to the case of other physical boundary conditions (such as the diffuse reflection boundary condition and in‐flow boundary condition), for which the solutions of the Boltzmann equation develop discontinuity in a codimension 1 subset (Kim [Comm. Math. Phys. 308 (2011)]), and therefore the best possible regularity is BV, which has been proved by Guo et al. [Arch. Rational Mech. Anal. 220 (2016)]. 

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3. Global Regularity of Skew Mean Curvature Flow for Small Data in d ≥ 4 Dimensions

   https://doi.org/10.1093/imrn/rnad104  

   Huang, Jiaxi; Li, Ze; Tataru, Daniel (May 2023, International Mathematics Research Notices)

   Abstract The skew mean curvature flow is an evolution equation for a $$d$$ dimensional manifold immersed into $$\mathbb {R}^{d+2}$$, and which moves along the binormal direction with a speed proportional to its mean curvature. In this article, we prove small data global regularity in low-regularity Sobolev spaces for the skew mean curvature flow in dimensions $$d\geq 4$$. This extends the local well-posedness result in [7]. 

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4. Equivalent regular partitions of three‐uniform hypergraphs

   https://doi.org/10.1002/rsa.21229  

   Nagle, Brendan; Rödl, Vojtěch; Schacht, Mathias (May 2024, Random Structures & Algorithms)

   Abstract Theregularity methodwas pioneered by Szemerédi for graphs and is an important tool in extremal combinatorics. Over the last two decades, several extensions to hypergraphs were developed which were based on seemingly different notions ofquasirandomhypergraphs. We consider the regularity lemmata for three‐uniform hypergraphs of Frankl and Rödl and of Gowers, and present a new proof that the concepts behind these approaches are equivalent. 

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5. Beyond Dyadic Coordination: Multimodal Behavioral Irregularity in Triads Predicts Facets of Collaborative Problem Solving

   https://doi.org/10.1111/cogs.12787  

   Amon, Mary_Jean; Vrzakova, Hana; D'Mello, Sidney_K (October 2019, Cognitive Science)

   Abstract We hypothesize that effective collaboration is facilitated when individuals and environmental components form a synergy where they work together and regulate one another to produce stable patterns of behavior, or regularity, as well as adaptively reorganize to form new behaviors, or irregularity. We tested this hypothesis in a study with 32 triads who collaboratively solved a challenging visual computer programming task for 20 min following an introductory warm‐up phase. Multidimensional recurrence quantification analysis was used to examine fine‐grained (i.e., every 10 s) collective patterns of regularity across team members' speech rate, body movement, and team interaction with the shared user interface. We found that teams exhibited significant patterns of regularity as compared to shuffled baselines, but there were no systematic trends in regularity across time. We also found that periods of regularity were associated with a reduction in overall behavior. Notably, the production of irregular behavior predicted expert‐coded metrics of collaborative activity, such as teams' ability to construct shared knowledge and effectively negotiate and coordinate execution of solutions, net of overall behavioral production and behavioral self‐similarity. Our findings support the theory that groups can interact to form interpersonal synergies and indicate that information about system‐level dynamics is a viable way to understand and predict effective collaborative processes. 

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