The Carrington storm (September 1/2, 1859) is one of the largest magnetic storms ever observed and it has caused global auroral displays in low-latitude areas, together with a series of multiple magnetic storms during August 28 and September 4, 1859. In this study, we revisit contemporary auroral observation records to extract information on their elevation angle, color, and direction to investigate this stormy interval in detail.We first examine their equatorward boundary of "auroral emission with multiple colors" based on descriptions of elevation angle and color. We find that their locations were 36.5° ILAT on August 28/29 and 32.7° ILAT on September 1/2, suggesting that trapped electrons moved to, at least, L~1.55 and L~1.41, respectively. The equatorward boundary of "purely red emission" was likely located at 30.8° ILAT on September 1/2.If "purely red emission" was a stable auroral red arc, it would suggest that trapped protons moved to, at least, L ~ 1.36. This reconstruction with observed auroral emission regions provides conservative estimations of magnetic storm intensities. We compare the auroral records with magnetic observations. We confirm that multiple magnetic storms occurred during this stormy interval, and that the equatorward expansion of the auroral oval is consistent with the timing of magnetic disturbances. It is possible that the August 28/29 interplanetary coronal mass ejections (ICMEs) cleared out the interplanetary medium, making the ICMEs for the Carrington storm on September 1/2 more geoeffective.

The study of historical great geomagnetic storms is crucial for assessing the possible risks to the technological infrastructure of a modern society, caused by extreme space-weather events. The normal benchmark has been the great geomagnetic storm of September 1859, the so-called 'Carrington Event'. However, there are numerous records of another great geomagnetic storm in February 1872. This storm, about 12 years after the Carrington Event, resulted in comparable Hayakawa et al. 2018, Astrophysical Journal, doi 10.3847/1538 2 magnetic disturbances and auroral displays over large areas of the Earth. We have revisited this great geomagnetic storm in terms of the auroral and sunspot records in the historical documents from East Asia. In particular, we have surveyed the auroral records from East Asia and estimated the equatorward boundary of the auroral oval to be near 24.3° invariant latitude (ILAT), on the basis that the aurora was seen near the zenith at Shanghai (20° magnetic latitude, MLAT). These results confirm that this geomagnetic storm of February 1872 was as extreme as the Carrington Event, at least in terms of the equatorward motion of the auroral oval. Indeed, our results support the interpretation of the simultaneous auroral observations made at Bombay (10° MLAT). The East Asian auroral records have indicated extreme brightness, suggesting unusual precipitation of highintensity, low-energy electrons during this geomagnetic storm. We have compared the duration of the East Asian auroral displays with magnetic observations in Bombay and found that the auroral displays occurred in the initial phase, main phase, and early recovery phase of the magnetic storm.

Deep learning has been applied to various tasks in the field of machine learning and has shown superiority to other common procedures such as kernel methods. To provide a better theoretical understanding of the reasons for its success, we discuss the performance of deep learning and other methods on a nonparametric regression problem with a Gaussian noise. Whereas existing theoretical studies of deep learning have been based mainly on mathematical theories of well-known function classes such as Hölder and Besov classes, we focus on function classes with discontinuity and sparsity, which are those naturally assumed in practice. To highlight the effectiveness of deep learning, we compare deep learning with a class of linear estimators representative of a class of shallow estimators. It is shown that the minimax risk of a linear estimator on the convex hull of a target function class does not differ from that of the original target function class. This results in the suboptimality of linear methods over a simple but non-convex function class, on which deep learning can attain nearly the minimax-optimal rate. In addition to this extreme case, we consider function classes with sparse wavelet coefficients. On these function classes, deep learning also attains the minimax rate up to log factors of the sample size, and linear methods are still suboptimal if the assumed sparsity is strong. We also point out that the parameter sharing of deep neural networks can remarkably reduce the complexity of the model in our setting.

In numerical integration, cubature methods are effective, especially when the integrands can be well-approximated by known test functions, such as polynomials. However, the construction of cubature formulas has not generally been known, and existing examples only represent the particular domains of integrands, such as hypercubes and spheres. In this study, we show that cubature formulas can be constructed for probability measures provided that we have an i.i.d. sampler from the measure and the mean values of given test functions. Moreover, the proposed method also works as a means of data compression, even if sufficient prior information of the measure is not available.

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