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1. Gauss curvature-based unique signatures of individual large earthquakes and its implications for customized data-driven prediction

   https://doi.org/10.1038/s41598-022-12575-w  

   Cho, In Ho (May 2022, Scientific Reports)

   Abstract Statistical descriptions of earthquakes offer important probabilistic information, and newly emerging technologies of high-precision observations and machine learning collectively advance our knowledge regarding complex earthquake behaviors. Still, there remains a formidable knowledge gap for predicting individual large earthquakes’ locations and magnitudes. Here, this study shows that the individual large earthquakes may have unique signatures that can be represented by new high-dimensional features—Gauss curvature-based coordinates. Particularly, the observed earthquake catalog data are transformed into a number of pseudo physics quantities (i.e., energy, power, vorticity, and Laplacian) which turn into smooth surface-like information via spatio-temporal convolution, giving rise to the new high-dimensional coordinates. Validations with 40-year earthquakes in the West U.S. region show that the new coordinates appear to hold uniqueness for individual large earthquakes ($$M_w \ge 7.0$$ $M_w\ge 7.0$), and the pseudo physics quantities help identify a customized data-driven prediction model. A Bayesian evolutionary algorithm in conjunction with flexible bases can identify a data-driven model, demonstrating its promising reproduction of individual large earthquake’s location and magnitude. Results imply that an individual large earthquake can be distinguished and remembered while its best-so-far model can be customized by machine learning. This study paves a new way to data-driven automated evolution of individual earthquake prediction. 

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2. The $$L^p$$-Fisher–Rao metric and Amari–C̆encov $$\alpha $$-Connections

   https://doi.org/10.1007/s00526-024-02660-5  

   Bauer, Martin; Le Brigant, Alice; Lu, Yuxiu; Maor, Cy (February 2024, Calculus of Variations and Partial Differential Equations)

   Abstract We introduce a family of Finsler metrics, called the$$L^p$$ $L^p$-Fisher–Rao metrics$$F_p$$ $F_p$, for$$p\in (1,\infty )$$ $p\in (1,\infty )$, which generalizes the classical Fisher–Rao metric$$F_2$$ $F_2$, both on the space of densities$${\text {Dens}}_+(M)$$ ${\text{Dens}}_{+}\left(M\right)$and probability densities$${\text {Prob}}(M)$$ $\text{Prob}\left(M\right)$. We then study their relations to the Amari–C̆encov$$\alpha $$ $\alpha$-connections$$\nabla ^{(\alpha )}$$ ${\nabla }^{\left(\alpha \right)}$from information geometry: on$${\text {Dens}}_+(M)$$ ${\text{Dens}}_{+}\left(M\right)$, the geodesic equations of$$F_p$$ $F_p$and$$\nabla ^{(\alpha )}$$ ${\nabla }^{\left(\alpha \right)}$coincide, for$$p = 2/(1-\alpha )$$ $p=2/(1-\alpha )$. Both are pullbacks of canonical constructions on$$L^p(M)$$ $L^p\left(M\right)$, in which geodesics are simply straight lines. In particular, this gives a new variational interpretation of$$\alpha $$ $\alpha$-geodesics as being energy minimizing curves. On$${\text {Prob}}(M)$$ $\text{Prob}\left(M\right)$, the$$F_p$$ $F_p$and$$\nabla ^{(\alpha )}$$ ${\nabla }^{\left(\alpha \right)}$geodesics can still be thought as pullbacks of natural operations on the unit sphere in$$L^p(M)$$ $L^p\left(M\right)$, but in this case they no longer coincide unless$$p=2$$ $p=2$. Using this transformation, we solve the geodesic equation of the$$\alpha $$ $\alpha$-connection by showing that the geodesic are pullbacks of projections of straight lines onto the unit sphere, and they always cease to exists after finite time when they leave the positive part of the sphere. This unveils the geometric structure of solutions to the generalized Proudman–Johnson equations, and generalizes them to higher dimensions. In addition, we calculate the associate tensors of$$F_p$$ $F_p$, and study their relation to$$\nabla ^{(\alpha )}$$ ${\nabla }^{\left(\alpha \right)}$. 

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3. Characterizing Schwarz maps by tracial inequalities

   https://doi.org/10.1007/s11005-023-01636-4  

   Carlen, Eric; Müller-Hermes, Alexander (February 2023, Letters in Mathematical Physics)

   Abstract Let$$\phi $$ $\varphi$be a positive map from the$$n\times n$$ $n\times n$matrices$$\mathcal {M}_n$$ $M_n$to the$$m\times m$$ $m\times m$matrices$$\mathcal {M}_m$$ $M_m$. It is known that$$\phi $$ $\varphi$is 2-positive if and only if for all$$K\in \mathcal {M}_n$$ $K\in M_n$and all strictly positive$$X\in \mathcal {M}_n$$ $X\in M_n$,$$\phi (K^*X^{-1}K) \geqslant \phi (K)^*\phi (X)^{-1}\phi (K)$$ $\varphi \left(K^{\ast }{X}^{-1}K\right)⩾\varphi {\left(K\right)}^{\ast }\varphi {\left(X\right)}^{-1}\varphi \left(K\right)$. This inequality is not generally true if$$\phi $$ $\varphi$is merely a Schwarz map. We show that the corresponding tracial inequality$${{\,\textrm{Tr}\,}}[\phi (K^*X^{-1}K)] \geqslant {{\,\textrm{Tr}\,}}[\phi (K)^*\phi (X)^{-1}\phi (K)]$$ $\text{Tr}\left[\varphi \left(K^{\ast }{X}^{-1}K\right)\right]⩾\text{Tr}\left[\varphi {\left(K\right)}^{\ast }\varphi {\left(X\right)}^{-1}\varphi \left(K\right)\right]$holds for a wider class of positive maps that is specified here. We also comment on the connections of this inequality with various monotonicity statements that have found wide use in mathematical physics, and apply it, and a close relative, to obtain some new, definitive results. 

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4. Finding solutions with distinct variables to systems of linear equations over $$\mathbb {F}_p$$

   https://doi.org/10.1007/s00208-022-02391-y  

   Sauermann, Lisa (April 2022, Mathematische Annalen)

   Abstract Let us fix a primepand a homogeneous system ofmlinear equations$$a_{j,1}x_1+\dots +a_{j,k}x_k=0$$ $a_{j,1}{x}_1+\cdots +a_{j,k}{x}_k=0$for$$j=1,\dots ,m$$ $j=1,\cdots ,m$with coefficients$$a_{j,i}\in \mathbb {F}_p$$ $a_{j,i}\in F_p$. Suppose that$$k\ge 3m$$ $k\ge 3m$, that$$a_{j,1}+\dots +a_{j,k}=0$$ $a_{j,1}+\cdots +a_{j,k}=0$for$$j=1,\dots ,m$$ $j=1,\cdots ,m$and that every$$m\times m$$ $m\times m$minor of the$$m\times k$$ $m\times k$matrix$$(a_{j,i})_{j,i}$$ ${\left(a_{j,i}\right)}_{j,i}$is non-singular. Then we prove that for any (large)n, any subset$$A\subseteq \mathbb {F}_p^n$$ $A\subseteq F_p^n$of size$$|A|> C\cdot \Gamma ^n$$ $\left|A\right|>C·{\Gamma }^n$contains a solution$$(x_1,\dots ,x_k)\in A^k$$ $(x_1,\cdots ,x_k)\in A^k$to the given system of equations such that the vectors$$x_1,\dots ,x_k\in A$$ $x_1,\cdots ,x_k\in A$are all distinct. Here,Cand$$\Gamma $$ $\Gamma$are constants only depending onp,mandksuch that$$\Gamma $\Gamma <p$. The crucial point here is the condition for the vectors$$x_1,\dots ,x_k$$ $x_1,\cdots ,x_k$in the solution$$(x_1,\dots ,x_k)\in A^k$$ $(x_1,\cdots ,x_k)\in A^k$to be distinct. If we relax this condition and only demand that$$x_1,\dots ,x_k$$ $x_1,\cdots ,x_k$are not all equal, then the statement would follow easily from Tao’s slice rank polynomial method. However, handling the distinctness condition is much harder, and requires a new approach. While all previous combinatorial applications of the slice rank polynomial method have relied on the slice rank of diagonal tensors, we use a slice rank argument for a non-diagonal tensor in combination with combinatorial and probabilistic arguments. 

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5. Twisted-bilayer FeSe and the Fe-based superlattices

   https://doi.org/10.21468/SciPostPhys.15.3.081  

   Eugenio, Paul Myles; Vafek, Oskar (January 2023, SciPost Physics)

   We derive BM-like continuum models for the bands of superlattice heterostructures formed out of Fe-chalcogenide monolayers: (I) a single monolayer experiencing an external periodic potential, and (II) twisted bilayers with long-range moire tunneling. A symmetry derivation for the inter-layer moire tunnelling is provided for both the\Gamma $\Gamma$andM $M$high-symmetry points. In this paper, we focus on moire bands formed from hole-band maxima centered on\Gamma $\Gamma$, and show the possibility of moire bands withC=0 $C=0$or±1 $\pm 1$topological quantum numbers without breaking time-reversal symmetry. In theC=0 $C=0$region for\theta→0 $\theta \to 0$(and similarly in the limit of large superlattice period for I), the system becomes a square lattice of 2D harmonic oscillators. We fit our model to FeSe and argue that it is a viable platform for the simulation of the square Hubbard model with tunable interaction strength. 

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