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1. ARRU Phase Picker: Attention Recurrent-Residual U-Net for Picking Seismic P - and S -Phase Arrivals

   https://doi.org/10.1785/0220200382  

   Liao, Wu-Yu; Lee, En-Jui; Mu, Dawei; Chen, Po; Rau, Ruey-Juin (March 2021, Seismological Research Letters)

   null (Ed.)

   Abstract Seismograms are convolution results between seismic sources and the media that seismic waves propagate through, and, therefore, the primary observations for studying seismic source parameters and the Earth interior. The routine earthquake location and travel-time tomography rely on accurate seismic phase picks (e.g., P and S arrivals). As data increase, reliable automated seismic phase-picking methods are needed to analyze data and provide timely earthquake information. However, most traditional autopickers suffer from low signal-to-noise ratio and usually require additional efforts to tune hyperparameters for each case. In this study, we proposed a deep-learning approach that adapted soft attention gates (AGs) and recurrent-residual convolution units (RRCUs) into the backbone U-Net for seismic phase picking. The attention mechanism was implemented to suppress responses from waveforms irrelevant to seismic phases, and the cooperating RRCUs further enhanced temporal connections of seismograms at multiple scales. We used numerous earthquake recordings in Taiwan with diverse focal mechanisms, wide depth, and magnitude distributions, to train and test our model. Setting the picking errors within 0.1 s and predicted probability over 0.5, the AG with recurrent-residual convolution unit (ARRU) phase picker achieved the F1 score of 98.62% for P arrivals and 95.16% for S arrivals, and picking rates were 96.72% for P waves and 90.07% for S waves. The ARRU phase picker also shown a great generalization capability, when handling unseen data. When applied the model trained with Taiwan data to the southern California data, the ARRU phase picker shown no cognitive downgrade. Comparing with manual picks, the arrival times determined by the ARRU phase picker shown a higher consistency, which had been evaluated by a set of repeating earthquakes. The arrival picks with less human error could benefit studies, such as earthquake location and seismic tomography. 

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2. Tree convolution for probability distributions with unbounded support

   Davis, Ethan; Jekel, David; Wang, Zhichao (January 2021, Alea)

   We develop the complex-analytic viewpoint on the tree convolutions studied by the second author and Weihua Liu in Jekel and Liu (2020), which generalize the free, boolean, monotone, and orthogonal convolutions. In particular, for each rooted subtree T of the N-regular tree (with vertices labeled by alternating strings), we define the convolution \boxplus_T (µ1, . . . , µN) for arbitrary probability measures µ1, . . . , µN on R using a certain fixed-point equation for the Cauchy transforms. The convolution operations respect the operad structure of the tree operad from Jekel and Liu (2020). We prove a general limit theorem for iterated T -free convolution similar to Bercovici and Pata’s results in the free case (Bercovici and Pata 1999), and we deduce limit theorems for measures in the domain of attraction of each of the classical stable laws. 

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3. Geometrization of the Satake transform for mod p Hecke algebras

   https://doi.org/10.1017/fms.2024.130  

   Cass, Robert; Xu, Yujie (January 2025, Forum of Mathematics, Sigma)

   Abstract We geometrize the modpSatake isomorphism of Herzig and Henniart–Vignéras using Witt vector affine flag varieties for reductive groups in mixed characteristic. We deduce this as a special case of a formula, stated in terms of the geometry of generalized Mirković–Vilonen cycles, for the Satake transform of an arbitrary parahoric modpHecke algebra with respect to an arbitrary Levi subgroup. Moreover, we prove an explicit formula for the convolution product in an arbitrary parahoric modpHecke algebra. Our methods involve the constant term functors inspired from the geometric Langlands program, and we also treat the case of reductive groups in equal characteristic. We expect this to be a first step toward a geometrization of a modpLocal Langlands Correspondence. 

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4. An anomalous fractional diffusion operator

   https://doi.org/10.1007/s13540-025-00394-5  

   Zheng, Xiangcheng; Ervin, V_J; Wang, Hong (April 2025, Fractional Calculus and Applied Analysis)

   Abstract In this article, using that the fractional Laplacian can be factored into a product of the divergence operator, a Riesz potential operator and the gradient operator, we introduce an anomalous fractional diffusion operator, involving a matrixK(x), suitable when anomalous diffusion is being studied in a non homogeneous medium. For the case ofK(x) a constant, symmetric positive definite matrix we show that the fractional Poisson equation is well posed, and determine the regularity of the solution in terms of the regularity of the right hand side function. 

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5. Fractional rheology-informed neural networks for data-driven identification of viscoelastic constitutive models

   https://doi.org/10.1007/s00397-023-01408-w  

   Dabiri, Donya; Saadat, Milad; Mangal, Deepak; Jamali, Safa (August 2023, Rheologica Acta)

   Abstract Developing constitutive models that can describe a complex fluid’s response to an applied stimulus has been one of the critical pursuits of rheologists. The complexity of the models typically goes hand-in-hand with that of the observed behaviors and can quickly become prohibitive depending on the choice of materials and/or flow protocols. Therefore, reducing the number of fitting parameters by seeking compact representations of those constitutive models can obviate extra experimentation to confine the parameter space. To this end, fractional derivatives in which the differential response of matter accepts non-integer orders have shown promise. Here, we develop neural networks that are informed by a series of different fractional constitutive models. These fractional rheology-informed neural networks (RhINNs) are then used to recover the relevant parameters (fractional derivative orders) of three fractional viscoelastic constitutive models, i.e., fractional Maxwell, Kelvin-Voigt, and Zener models. We find that for all three studied models, RhINNs recover the observed behavior accurately, although in some cases, the fractional derivative order is recovered with significant deviations from what is known as ground truth. This suggests that extra fractional elements are redundant when the material response is relatively simple. Therefore, choosing a fractional constitutive model for a given material response is contingent upon the response complexity, as fractional elements embody a wide range of transient material behaviors. 

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