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1. Revisiting closed asymptotic couples

   https://doi.org/10.1017/S0013091522000219  

   Aschenbrenner, Matthias; van den Dries, Lou; van der Hoeven, Joris (May 2022, Proceedings of the Edinburgh Mathematical Society)

   Abstract Every discrete definable subset of a closed asymptotic couple with ordered scalar field $${\boldsymbol {k}}$$ is shown to be contained in a finite-dimensional $${\boldsymbol {k}}$$ -linear subspace of that couple. It follows that the differential-valued field $$\mathbb {T}$$ of transseries induces more structure on its value group than what is definable in its asymptotic couple equipped with its scalar multiplication by real numbers, where this asymptotic couple is construed as a two-sorted structure with $$\mathbb {R}$$ as the underlying set for the second sort. 

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2. Dynamics of closed singularities

   https://doi.org/10.5802/aif.3343  

   Colding, Tobias Holck; Minicozzi II, William P. (January 2019, Annales de l'Institut Fourier)

   null (Ed.)
3. Closed-Form Diffusion Models

   Scarvelis, Christopher; Borde, Haitz; Solomon, Justin (April 2025, Transactions on machine learning research)

   Score-based generative models (SGMs) sample from a target distribution by iteratively transforming noise using the score function of the perturbed target. For any finite training set, this score function can be evaluated in closed form, but the resulting SGM memorizes its training data and does not generate novel samples. In practice, one approximates the score by training a neural network via score-matching. The error in this approximation promotes generalization, but neural SGMs are costly to train and sample, and the effective regularization this error provides is not well-understood theoretically. In this work, we instead explicitly smooth the closed-form score to obtain an SGM that generates novel samples without training. We analyze our model and propose an efficient nearest-neighbor-based estimator of its score function. Using this estimator, our method achieves competitive sampling times while running on consumer-grade CPUs. 

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4. Liouville closed HT-fields

   Kaplan, Elliot (April 2023, Journal of algebra)

   Let T be a complete, model complete o-minimal theory extending the theory of real closed ordered fields. An HT-field is a model K of T equipped with a T-derivation ∂ such that the underlying ordered differential field of (K,∂) is an H-field. We study HT-fields and their extensions. Our main result is that if T is power bounded, then every HT-field K has either exactly one or exactly two minimal Liouville closed HT-field extensions up to K-isomorphism. The assumption of power boundedness can be relaxed to allow for certain exponential cases, such as T = Th(Ran,exp). 

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5. Extremal Union-Closed Set Families

   https://doi.org/10.1007/s00373-019-02087-2  

   Chen, Guantao; van der Holst, Hein; Kostochka, Alexandr; Li, Nana (November 2019, Graphs and Combinatorics)