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1. Étale structures and the Joyal–Tierney representation theorem in countable model theory

   https://doi.org/10.1017/bsl.2025.10086  

   Chen, Ruiyuan (July 2025, The Bulletin of Symbolic Logic)

   An étale structure over a topological space X is a continuous family of structures (in some first-order language) indexed over X. We give an exposition of this fundamental concept from sheaf theory and its relevance to countable model theory and invariant descriptive set theory. We show that many classical aspects of spaces of countable models can be naturally framed and generalized in the context of étale structures, including the Lopez-Escobar theorem on invariant Borel sets, an omitting types theorem, and various characterizations of Scott rank. We also present and prove the countable version of the Joyal–Tierney representation theorem, which states that the isomorphism groupoid of an étale structure determines its theory up to bi-interpretability; and we explain how special cases of this theorem recover several recent results in the literature on groupoids of models and functors between them. 

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2. Oscillatory Breuer–Major theorem with application to the random corrector problem

   https://doi.org/10.3233/ASY-191575  

   Nualart, David; Zheng, Guangqu (September 2020, Asymptotic Analysis)

   In this paper, we present an oscillatory version of the celebrated Breuer–Major theorem that is motivated by the random corrector problem. As an application, we are able to prove new results concerning the Gaussian fluctuation of the random corrector. We also provide a variant of this theorem involving homogeneous measures. 

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3. Stable anisotropic minimal hypersurfaces in

   https://doi.org/10.1017/fmp.2023.1  

   Chodosh, Otis; Li, Chao (January 2023, Forum of Mathematics, Pi)

   Abstract We show that a complete, two-sided, stable immersed anisotropic minimal hypersurface in $$\mathbf {R}^4$$ has intrinsic cubic volume growth, provided the parametric elliptic integral is $C^2$ -close to the area functional. We also obtain an interior volume upper bound for stable anisotropic minimal hypersurfaces in the unit ball. We can estimate the constants explicitly in all of our results. In particular, this paper gives an alternative proof of our recent stable Bernstein theorem for minimal hypersurfaces in $$\mathbf {R}^4$$ . The new proof is more closely related to techniques from the study of strictly positive scalar curvature. 

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4. Exploring AC Power Flow Resultants through the Lens of Multi-homogeneous Algebraic Geometry

   https://doi.org/10.1109/KPEC58008.2023.10215389  

   Barati, Masoud (April 2023, IEEE)

   The computational aspects of power systems have been exploring the steady states of AC power flow for several decades. In this paper, we propose a novel approach to AC power flow calculation using resultants and discriminants for polynomials, which are primarily compiled for quadratic power flow equations. In the case of AC power flow nonlinear systems, it is not possible to determine the number of isolated solutions. However, for polynomial systems, the theorem of Bézout is the primary theorem of algebraic geometry. This study considers a certain multi-homogeneous structure in an algebraic geometry system to demonstrate that the theorem of Bézout is indeed a generalization of the fundamental theorem, among other results. 

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5. Functional limit theorems for the euler characteristic process in the critical regime

   https://doi.org/10.1017/apr.2020.46  

   Thomas, Andrew M.; Owada, Takashi (March 2021, Advances in Applied Probability)

   null (Ed.)

   Abstract This study presents functional limit theorems for the Euler characteristic of Vietoris–Rips complexes. The points are drawn from a nonhomogeneous Poisson process on $$\mathbb{R}^d$$ , and the connectivity radius governing the formation of simplices is taken as a function of the time parameter t , which allows us to treat the Euler characteristic as a stochastic process. The setting in which this takes place is that of the critical regime, in which the simplicial complexes are highly connected and have nontrivial topology. We establish two ‘functional-level’ limit theorems, a strong law of large numbers and a central limit theorem, for the appropriately normalized Euler characteristic process. 

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