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1. Higher Siegel–Weil formula for unitary groups: the non-singular terms

   https://doi.org/10.1007/s00222-023-01228-y  

   Feng, Tony; Yun, Zhiwei; Zhang, Wei (February 2024, Inventiones mathematicae)

   Abstract We construct special cycles on the moduli stack of hermitian shtukas. We prove an identity between (1) the$$r^{\mathrm{th}}$$ $r^{\mathrm{th}}$central derivative of non-singular Fourier coefficients of a normalized Siegel–Eisenstein series, and (2) the degree of special cycles of “virtual dimension 0” on the moduli stack of hermitian shtukas with$$r$$ $r$legs. This may be viewed as a function-field analogue of the Kudla-Rapoport Conjecture, that has the additional feature of encompassing all higher derivatives of the Eisenstein series. 

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2. The Mid-Pleistocene Transition: a delayed response to an increasing positive feedback?

   https://doi.org/10.1007/s00382-022-06544-2  

   Shackleton, J. D.; Follows, M. J.; Thomas, P. J.; Omta, A. W. (November 2022, Climate Dynamics)

   Abstract Glacial–interglacial cycles constitute large natural variations in Earth’s climate. The Mid-Pleistocene Transition (MPT) marks a shift of the dominant periodicity of these climate cycles from$$\sim 40$$ $\sim 40$to$$\sim 100$$ $\sim 100$ kyr. Recently, it has been suggested that this shift resulted from a gradual increase in the internal period (or equivalently, a decrease in the natural frequency) of the system. As a result, the system would then have locked to ever higher multiples of the external forcing period. We find that the internal period is sensitive to the strength of positive feedbacks in the climate system. Using a carbon cycle model in which feedbacks between calcifier populations and ocean alkalinity mediate atmospheric CO$$_2,$$ ${}_2,$we simulate stepwise periodicity changes similar to the MPT through such a mechanism. Due to the internal dynamics of the system, the periodicity shift occurs up to millions of years after the change in the feedback strength is imposed. This suggests that the cause for the MPT may have occurred a significant time before the observed periodicity shift. 

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3. Frobenius-like structure in Gaudin model

   https://doi.org/10.1142/S0129167X22500525  

   Mukhin, Evgeny; Varchenko, Alexander (June 2022, International Journal of Mathematics)

   We introduce a Frobenius-like structure for the [Formula: see text] Gaudin model. Namely, we introduce potential functions of the first and second kind. We describe the Shapovalov form in terms of derivatives of the potential of the first kind and the action of Gaudin Hamiltonians in terms of derivatives of the potential of the second kind. 

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4. Excitonic effects at the temperature-dependent direct bandgap of Ge

   https://doi.org/10.1063/5.0080158  

   Emminger, Carola; Samarasingha, Nuwanjula S.; Rivero Arias, Melissa; Abadizaman, Farzin; Menéndez, José; Zollner, Stefan (April 2022, Journal of Applied Physics)

   The temperature dependence of the complex dielectric function [Formula: see text] of bulk Ge near the direct bandgap was investigated with spectroscopic ellipsometry at temperatures between 10 and 710 K. Second derivatives of the dielectric function with respect to energy are obtained using a digital linear filter method. A model that incorporates excitonic effects using the Tanguy model for the Hulthén potential [C. Tanguy, Phys. Rev. B 60, 10660 (1999)] was used to fit the dielectric function and its second derivatives simultaneously. Using [Formula: see text] theory and literature values for effective masses, reasonable agreement with the experiment is obtained for [Formula: see text] up to room temperature using the direct bandgap and its broadening as the only adjustable parameters. 

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5. Smoothing and growth bound of periodic generalized Korteweg–De Vries equation

   https://doi.org/10.1142/S0219891621500260  

   Oh, Seungly; Stefanov, Atanas G. (December 2021, Journal of Hyperbolic Differential Equations)

   For generalized Korteweg–De Vries (KdV) models with polynomial nonlinearity, we establish a local smoothing property in [Formula: see text] for [Formula: see text]. Such smoothing effect persists globally, provided that the [Formula: see text] norm does not blow up in finite time. More specifically, we show that a translate of the nonlinear part of the solution gains [Formula: see text] derivatives for [Formula: see text]. Following a new simple method, which is of independent interest, we establish that, for [Formula: see text], [Formula: see text] norm of a solution grows at most by [Formula: see text] if [Formula: see text] norm is a priori controlled. 

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