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1. Operational prediction of solar flares using a transformer-based framework

   https://doi.org/10.1038/s41598-023-40884-1  

   Abduallah, Yasser; Wang, Jason T.; Wang, Haimin; Xu, Yan (December 2023, Scientific Reports)

   Abstract Solar flares are explosions on the Sun. They happen when energy stored in magnetic fields around solar active regions (ARs) is suddenly released. Solar flares and accompanied coronal mass ejections are sources of space weather, which negatively affects a variety of technologies at or near Earth, ranging from blocking high-frequency radio waves used for radio communication to degrading power grid operations. Monitoring and providing early and accurate prediction of solar flares is therefore crucial for preparedness and disaster risk management. In this article, we present a transformer-based framework, named SolarFlareNet, for predicting whether an AR would produce a$$\gamma$$ $\gamma$-class flare within the next 24 to 72 h. We consider three$$\gamma$$ $\gamma$classes, namely the$$\ge$$ $\ge$M5.0 class, the$$\ge$$ $\ge$M class and the$$\ge$$ $\ge$C class, and build three transformers separately, each corresponding to a$$\gamma$$ $\gamma$class. Each transformer is used to make predictions of its corresponding$$\gamma$$ $\gamma$-class flares. The crux of our approach is to model data samples in an AR as time series and to use transformers to capture the temporal dynamics of the data samples. Each data sample consists of magnetic parameters taken from Space-weather HMI Active Region Patches (SHARP) and related data products. We survey flare events that occurred from May 2010 to December 2022 using the Geostationary Operational Environmental Satellite X-ray flare catalogs provided by the National Centers for Environmental Information (NCEI), and build a database of flares with identified ARs in the NCEI flare catalogs. This flare database is used to construct labels of the data samples suitable for machine learning. We further extend the deterministic approach to a calibration-based probabilistic forecasting method. The SolarFlareNet system is fully operational and is capable of making near real-time predictions of solar flares on the Web. 

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2. Rates of convergence for regression with the graph poly-Laplacian

   https://doi.org/10.1007/s43670-023-00075-5  

   Trillos, Nicolás García; Murray, Ryan; Thorpe, Matthew (December 2023, Sampling Theory, Signal Processing, and Data Analysis)

   Abstract In the (special) smoothing spline problem one considers a variational problem with a quadratic data fidelity penalty and Laplacian regularization. Higher order regularity can be obtained via replacing the Laplacian regulariser with a poly-Laplacian regulariser. The methodology is readily adapted to graphs and here we consider graph poly-Laplacian regularization in a fully supervised, non-parametric, noise corrupted, regression problem. In particular, given a dataset$$\{x_i\}_{i=1}^n$$ ${\left\{x_i\right\}}_{i=1}^n$and a set of noisy labels$$\{y_i\}_{i=1}^n\subset \mathbb {R}$$ ${\left\{y_i\right\}}_{i=1}^n\subset R$we let$$u_n{:}\{x_i\}_{i=1}^n\rightarrow \mathbb {R}$$ $u_n:{\left\{x_i\right\}}_{i=1}^n\to R$be the minimizer of an energy which consists of a data fidelity term and an appropriately scaled graph poly-Laplacian term. When$$y_i = g(x_i)+\xi _i$$ $y_i=g\left(x_i\right)+{\xi }_i$, for iid noise$$\xi _i$$ ${\xi }_i$, and using the geometric random graph, we identify (with high probability) the rate of convergence of$$u_n$$ $u_n$togin the large data limit$$n\rightarrow \infty $$ $n\to \infty$. Furthermore, our rate is close to the known rate of convergence in the usual smoothing spline model. 

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3. L-Systems and the Lovász Number

   https://doi.org/10.1007/s00493-025-00136-4  

   Linz, William (April 2025, Combinatorica)

   Abstract Given integers$$n> k > 0$$ $n>k>0$, and a set of integers$$L \subset [0, k-1]$$ $L\subset [0,k-1]$, anL-systemis a family of sets$$\mathcal {F}\subset \left( {\begin{array}{c}[n]\\ k\end{array}}\right) $$ $F\subset \left(\begin{array}{c}\left[n\right]\\ k\end{array}\right)$such that$$|F \cap F'| \in L$$ $|F\cap F^{\prime }|\in L$for distinct$$F, F'\in \mathcal {F}$$ $F,F^{\prime }\in F$.L-systems correspond to independent sets in a certain generalized Johnson graphG(n, k, L), so that the maximum size of anL-system is equivalent to finding the independence number of the graphG(n, k, L). TheLovász number$$\vartheta (G)$$ $\vartheta \left(G\right)$is a semidefinite programming approximation of the independence number$$\alpha $$ $\alpha$of a graphG. In this paper, we determine the leading order term of$$\vartheta (G(n, k, L))$$ $\vartheta \left(G\right(n,k,L\left)\right)$of any generalized Johnson graph withkandLfixed and$$n\rightarrow \infty $$ $n\to \infty$. As an application of this theorem, we give an explicit construction of a graphGonnvertices with a large gap between the Lovász number and the Shannon capacityc(G). Specifically, we prove that for any$$\epsilon > 0$$ $ϵ>0$, for infinitely manynthere is a generalized Johnson graphGonnvertices which has ratio$$\vartheta (G)/c(G) = \Omega (n^{1-\epsilon })$$ $\vartheta \left(G\right)/c\left(G\right)=\Omega \left(n^{1-ϵ}\right)$, which improves on all known constructions. The graphGa fortiorialso has ratio$$\vartheta (G)/\alpha (G) = \Omega (n^{1-\epsilon })$$ $\vartheta \left(G\right)/\alpha \left(G\right)=\Omega \left(n^{1-ϵ}\right)$, which greatly improves on the best known explicit construction. 

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4. Block mapping class groups and their finiteness properties

   https://doi.org/10.1007/s10711-025-00985-9  

   Aramayona, J; Aroca, J; Cumplido, M; Skipper, R; Wu, X (April 2025, Geometriae Dedicata)

   Abstract Given$$g \in \mathbb N \cup \{0, \infty \}$$ $g\in N\cup \{0,\infty \}$, let$$\Sigma _g$$ ${\Sigma }_g$denote the closed surface of genusgwith a Cantor set removed, if$$g<\infty $$ $g<\infty$; or the blooming Cantor tree, when$$g= \infty $$ $g=\infty$. We construct a family$$\mathfrak B(H)$$ $B\left(H\right)$of subgroups of$${{\,\textrm{Map}\,}}(\Sigma _g)$$ $\text{Map}\left({\Sigma }_g\right)$whose elements preserve ablock decompositionof$$\Sigma _g$$ ${\Sigma }_g$, andeventually like actlike an element ofH, whereHis a prescribed subgroup of the mapping class group of the block. The group$$\mathfrak B(H)$$ $B\left(H\right)$surjects onto an appropriate symmetric Thompson group of Farley–Hughes; in particular, it answers positively. Our main result asserts that$$\mathfrak B(H)$$ $B\left(H\right)$is of type$$F_n$$ $F_n$if and only ifHis. As a consequence, for every$$g\in \mathbb N \cup \{0, \infty \}$$ $g\in N\cup \{0,\infty \}$and every$$n\ge 1$$ $n\ge 1$, we construct a subgroup$$G <{{\,\textrm{Map}\,}}(\Sigma _g)$$ $G<\text{Map}\left({\Sigma }_g\right)$that is of type$$F_n$$ $F_n$but not of type$$F_{n+1}$$ $F_{n+1}$, and which contains the mapping class group of every compact surface of genus$$\le g$$ $\le g$and with non-empty boundary. 

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5. $1/f$ Noise in the Heliosphere: A Target for PUNCH Science

   https://doi.org/10.1007/s11207-024-02401-z  

   Wang, Jiaming; Matthaeus, William H; Chhiber, Rohit; Roy, Sohom; Pradata, Rayta A; Pecora, Francesco; Yang, Yan (December 2024, Solar Physics)

   Abstract We present a broad review of$$1/f$$ $1/f$noise observations in the heliosphere, and discuss and complement the theoretical background of generic$$1/f$$ $1/f$models as relevant to NASA’s Polarimeter to UNify the Corona and Heliosphere (PUNCH) mission. First observed in the voltage fluctuations of vacuum tubes, the scale-invariant$$1/f$$ $1/f$spectrum has since been identified across a wide array of natural and artificial systems, including heart rate fluctuations and loudness patterns in musical compositions. In the solar wind the interplanetary magnetic field trace spectrum exhibits$$1/f$$ $1/f$scaling within the frequency range from around$$\unit[2 \times 10^{-6}]{Hz}$$to around$$\unit[10^{-3}]{{Hz}}$$at 1 au. One compelling mechanism for the generation of$$1/f$$ $1/f$noise is the superposition principle, where a composite$$1/f$$ $1/f$spectrum arises from the superposition of a collection of individual power-law spectra characterized by a scale-invariant distribution of correlation times. In the context of the solar wind, such a superposition could originate from scale-invariant reconnection processes in the corona. Further observations have detected$$1/f$$ $1/f$signatures in the photosphere and corona at frequency ranges compatible with those observed at 1 au, suggesting an even lower altitude origin of$$1/f$$ $1/f$spectrum in the solar dynamo itself. This hypothesis is bolstered by dynamo experiments and simulations that indicate inverse cascade activities, which can be linked to successive flux tube reconnections beneath the corona, and are known to generate$$1/f$$ $1/f$noise possibly through nonlocal interactions at the largest scales. Conversely, models positing in situ generation of$$1/f$$ $1/f$signals face causality issues in explaining the low-frequency portion of the$$1/f$$ $1/f$spectrum. Understanding$$1/f$$ $1/f$noise in the solar wind may inform central problems in heliospheric physics, such as the solar dynamo, coronal heating, the origin of the solar wind, and the nature of interplanetary turbulence. 

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