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1. Affine Stresses, Inverse Systems, and Reconstruction Problems

   https://doi.org/10.1093/imrn/rnad294  

   Murai, Satoshi; Novik, Isabella; Zheng, Hailun (December 2023, International Mathematics Research Notices)

   Abstract A conjecture of Kalai asserts that for $$d\geq 4$$, the affine type of a prime simplicial $$d$$-polytope $$P$$ can be reconstructed from the space of affine $$2$$-stresses of $$P$$. We prove this conjecture for all $$d\geq 5$$. We also prove the following generalization: for all pairs $(i,d)$ with $$2\leq i\leq \lceil \frac d 2\rceil -1$$, the affine type of a simplicial $$d$$-polytope $$P$$ that has no missing faces of dimension $$\geq d-i+1$$ can be reconstructed from the space of affine $$i$$-stresses of $$P$$. A consequence of our proofs is a strengthening of the Generalized Lower Bound Theorem: it was proved by Nagel that for any simplicial $(d-1)$-sphere $$\Delta $$ and $$1\leq k\leq \lceil \frac {d}{2}\rceil -1$$, $$g_{k}(\Delta )$$ is at least as large as the number of missing $(d-k)$-faces of $$\Delta $$; here we show that, for $$1\leq k\leq \lfloor \frac {d}{2}\rfloor -1$$, equality holds if and only if $$\Delta $$ is $$k$$-stacked. Finally, we show that for $$d\geq 4$$, any simplicial $$d$$-polytope $$P$$ that has no missing faces of dimension $$\geq d-1$$ is redundantly rigid, that is, for each edge $$e$$ of $$P$$, there exists an affine $$2$$-stress on $$P$$ with a non-zero value on $$e$$. 

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2. Trends in the Magnitude and Frequency of Extreme Rainfall Regimes in Florida

   https://doi.org/10.3390/w12092582  

   Mahjabin, Tasnuva; Abdul-Aziz, Omar I. (September 2020, Water)

   Trends in the extreme rainfall regimes were analyzed at 24 stations of Florida for four analysis periods: 1950–2010, 1960–2010, 1970–2010, and 1980–2010. A trend-free pre-whitening approach was utilized to correct data for autocorrelations. Non-parametric Mann-Kendall test and Theil-Sen approach were employed to detect and estimate trends in the magnitude of annual maximum rainfalls and in the number of annual above-threshold events (i.e., frequency). A bootstrap resampling approach was used to account for cross-correlations across sites and evaluate the global significance of trends at the 10% level (p-value ≤ 0.10). Dominant locally significant (p-value ≤ 0.10) increasing trends were found in the magnitudes of 1–12 h extreme rainfalls for the longest period, and in 6 h to 7 day rainfalls for the shortest period. The trends in 2–12 h rainfalls were also globally significant (i.e., exceeded the trends that could occur by chance). In contrast, globally significant decreasing trends were noted in the annual number of 1–3 h, 1–6 h, and 3–6 h extreme rainfalls during 1950–2010, 1960–2010, and 1980–2010, respectively. Trends in the number of 1–7 day extreme rainfalls were mixed, lacking global significance. Our findings would guide stormwater management in tropical/subtropical environments of Florida and around the world. 

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3. Nonvarying, affine and extremal geometry of strata of differentials

   https://doi.org/10.1017/S0305004123000567  

   CHEN, DAWEI (October 2023, Mathematical Proceedings of the Cambridge Philosophical Society)

   Abstract We prove that the nonvarying strata of abelian and quadratic differentials in low genus have trivial tautological rings and are affine varieties. We also prove that strata ofk-differentials of infinite area are affine varieties for allk. Vanishing of homology in degree higher than the complex dimension follows as a consequence for these affine strata. Moreover we prove that the stratification of the Hodge bundle for abelian and quadratic differentials of finite area is extremal in the sense that merging two zeros in each stratum leads to an extremal effective divisor in the boundary. A common feature throughout these results is a relation of divisor classes in strata of differentials as well as its incarnation in Teichmüller dynamics. 

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4. Searching for Regularity in Bounded Functions

   Iyer, Siddharth; Whitmeyer, Michael (July 2023, 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023))

   Given a function f on (F_2_^n, we study the following problem. What is the largest affine subspace U such that when restricted to U, all the non-trivial Fourier coefficients of f are very small? For the natural class of bounded Fourier degree d functions f : (F_2)^n → [−1, 1], we show that there exists an affine subspace of dimension at least ˜Ω (n^(1/d!) k^(−2)), wherein all of f ’s nontrivial Fourier coefficients become smaller than 2^(−k) . To complement this result, we show the existence of degree d functions with coefficients larger than 2^(−d log n) when restricted to any affine subspace of dimension larger than Ω(dn^(1/(d−1))). In addition, we give explicit examples of functions with analogous but weaker properties. Along the way, we provide multiple characterizations of the Fourier coefficients of functions restricted to subspaces of (F_2)^n that may be useful in other contexts. Finally, we highlight applications and connections of our results to parity kill number and affine dispersers. 

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5. Optimal boundary regularity for some singular Monge-Ampère equations on bounded convex domains

   https://doi.org/10.3934/dcds.2021188  

   Le, Nam Q. (January 2022, Discrete & Continuous Dynamical Systems)

   By constructing explicit supersolutions, we obtain the optimal global Hölder regularity for several singular Monge-Ampère equations on general bounded open convex domains including those related to complete affine hyperbolic spheres, and proper affine hyperspheres. Our analysis reveals that certain singular-looking equations, such as \begin{document}$$ \det D^2 u = |u|^{-n-2-k} (x\cdot Du -u)^{-k} $$\end{document} with zero boundary data, have unexpected degenerate nature. 

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