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1. Geometry of smooth extremal surfaces

   https://doi.org/10.1016/j.jalgebra.2024.02.003  

   Brosowsky, Anna; Page, Janet; Ryan, Tim; Smith, Karen E (May 2024, Journal of Algebra)

   We study the geometry of smooth projective surfaces defined by Frobenius forms, a class of homogenous polynomials in prime characteristic recently shown to have minimal possible F-pure threshold among forms of the same degree. We call these surfaces extremal surfaces, and show that their geometry is reminiscent of the geometry of smooth cubic surfaces, especially non-Frobenius split cubic surfaces. For instance, extremal surfaces have many lines but no triangles, hence many “star points” analogous to Eckardt points on a cubic surface. We generalize the classical notion of a double six for cubic surfaces to a double 2d on an extremal surface of degree d. We show that, asymptotically in d, smooth extremal surfaces have at least (1/16)d^{14} double 2d's. A key element of the proofs is the large automorphism group of an extremal surface, which we show to act transitively on many associated sets, such as the set of triples of skew lines on the extremal surface. 

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2. Selecting the independent coordinates of manifolds with large aspect ratios

   Chen, Yu-Chia; Meila, Marina (December 2019, Advances in neural information processing systems)

   Many manifold embedding algorithms fail apparently when the data manifold has a large aspect ratio (such as a long, thin strip). Here, we formulate success and failure in terms of finding a smooth embedding, showing also that the problem is pervasive and more complex than previously recognized. Mathematically, success is possible under very broad conditions, provided that embedding is done by carefully selected eigenfunctions of the Laplace-Beltrami operator Δ_M. Hence, we propose a bicriterial Independent Eigencoordinate Selection (IES) algorithm that selects smooth embeddings with few eigenvectors. The algorithm is grounded in theory, has low computational overhead, and is successful on synthetic and large real data. 

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3. Monodromy of Kodaira fibrations of genus 3

   https://doi.org/10.1002/mana.202000189  

   Flapan, Laure (November 2022, Mathematische Nachrichten)

   Abstract A Kodaira fibration is a non‐isotrivial fibration from a smooth algebraic surfaceSto a smooth algebraic curveBsuch that all fibers are smooth algebraic curves of genusg. Such fibrations arise as complete curves inside the moduli space of genusgalgebraic curves. We investigate here the possible connected monodromy groups of a Kodaira fibration in the case and classify which such groups can arise from a Kodaira fibration obtained as a general complete intersection curve inside a subvariety of parametrizing curves whose Jacobians have extra endomorphisms. 

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4. Nonparametric Density Estimation under Distribution Drift

   Mazzetto, A.; Upfal, E. (January 2023, Proceedings of Machine Learning Research)

   We study nonparametric density estimation in non-stationary drift settings. Given a sequence of independent samples taken from a distribution that gradually changes in time, the goal is to compute the best estimate for the current distribution. We prove tight minimax risk bounds for both discrete and continuous smooth densities, where the minimum is over all possible estimates and the maximum is over all possible distributions that satisfy the drift constraints. Our technique handles a broad class of drift models and generalizes previous results on agnostic learning under drift. 

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5. Autocovariance function estimation via difference schemes for a semiparametric change point model with m‐dependent errors

   https://doi.org/10.1111/anzs.70002  

   Levine, Michael; Tecuapetla‐Gómez, Inder (April 2025, Australian & New Zealand Journal of Statistics)

   We discuss a broad class of difference‐based estimators of the autocovariance function in a semiparametric regression model where the signal consists of the sum of a smooth function and another stepwise function whose number of jumps and locations are unknown (change points) while the errors are stationary and ‐dependent. We establish that the influence of the smooth part of the signal over the bias of our estimators is negligible; this is a general result as it does not depend on the distribution of the errors. We show that the influence of the unknown smooth function is negligible also in the mean squared error (MSE) of our estimators. Although we assumed Gaussian errors to derive the latter result, our finite sample studies suggest that the class of proposed estimators still show small MSE when the errors are not Gaussian. Our simulation study also demonstrates that, when the error process is mis‐specified as an AR instead of an ‐dependent process, our proposed method can estimate autocovariances about as well as some methods specifically designed for the AR(1) case, and sometimes even better than them. We also allow both the number of change points and the magnitude of the largest jump grow with the sample size . In this case, we provide conditions on the interplay between the growth rate of these two quantities as well as the vanishing rate of the modulus of continuity (of the signal's smooth part) that ensure consistency of our autocovariance estimators. As an application, we use our approach to provide a better understanding of the possible autocovariance structure of a time series of global averaged annual temperature anomalies. Finally, the R package dbacf complements this article. 

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