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1. On finite energy monopoles on $$C x \Sigma$$.

   Wang, Donghao (July 2019, ArXiv.org)

   Let $$X=\mathbb{C}\times\Sigma$$ be the product of the complex plane and a compact Riemann surface. We establish a classification theorem of solutions to the Seiberg-Witten equation on $$X$$ with finite analytic energy. The spin bundle $$S^+\to X$$ splits as $$L^+\oplus L^-$$. When $$2-2g\leq c_1(S^+)[\Sigma]<0$$, the moduli space is in bijection with the moduli space of pairs $$((L^+,\bar{\partial}), f)$$ where $$(L^+,\bar{\partial})$$ is a holomorphic structure on $L^+$ and $$f: \mathbb{C}\to H^0(\Sigma, L^+,\bar{\partial})$$ is a polynomial map. Moreover, the solution has analytic energy $$-4\pi^2d\cdot c_1(S^+)[\Sigma]$$ if $$f$$ has degree $$d$$. When $$c_1(S^+)=0$$, all solutions are reducible and the moduli space is the space of flat connections on $$\bigwedge^2 S^+$$. We also estimate the decay rate at infinity for these solutions. 

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2. Navigating the space of seismic anisotropy for crystal and whole-Earth scales

   https://doi.org/10.1093/gji/ggaf134  

   Gupta, Aakash; Tape, Carl; Brown, J_Michael; Becker, Thorsten_W (April 2025, Geophysical Journal International)

   SUMMARY Evidence of seismic anisotropy is widespread within the Earth, including from individual crystals, rocks, borehole measurements, active-source seismic data, and global seismic data. The seismic anisotropy of a material determines how wave speeds vary as a function of propagation direction and polarization, and it is characterized by density and the elastic map, which relates strain and stress in the material. Associated with the elastic map is a symmetric $$6 \times 6$$ matrix, which therefore has 21 parameters. The 21-D space of elastic maps is vast and poses challenges for both theoretical analysis and typical inverse problems. Most estimation approaches using a given set of directional wave speed measurements assume a high-symmetry approximation, typically either in the form of isotropy (2 parameters), vertical transverse isotropy (radial anisotropy: 5 parameters), or horizontal transverse isotropy (azimuthal anisotropy: 6 parameters). We offer a general approach to explore the space of elastic maps by starting with a given elastic map $$\mathbf {T}$$. Using a combined minimization and projection procedure, we calculate the closest $$\Sigma$$-maps to $$\mathbf {T}$$, where $$\Sigma$$ is one of the eight elastic symmetry classes: isotropic, cubic, transverse isotropic, trigonal, tetragonal, orthorhombic, monoclinic and trivial. We apply this approach to 21-parameter elastic maps derived from laboratory measurements of minerals; the measurements include dependencies on pressure, temperature, and composition. We also examine global elasticity models derived from subduction flow modelling. Our approach offers a different perspective on seismic anisotropy and motivates new interpretations, such as for why elasticity varies as a function of pressure, temperature, and composition. The two primary advances of this study are (1) to provide visualization of elastic maps, including along specific pathways through the space of model parameters, and (2) to offer distinct options for reducing the complexity of a given elastic map by providing a higher-symmetry approximation or a lower-anisotropic version. This could contribute to improved imaging and interpretation of Earth structure and dynamics from seismic anisotropy. 

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3. Large-scale geometry of pure mapping class groups of infinite-type surfaces

   https://doi.org/10.1090/proc/17181  

   Hill, Thomas (April 2025, Proceedings of the American Mathematical Society)

   The work of Mann and Rafi [Geom. Topol. 27 (2023), pp. 2237–2296] gives a classification of surfaces $\mathrm{\Sigma <\#comment/>}$when $\mathrm{M}\mathrm{a}\mathrm{p}\left(\mathrm{\Sigma <\#comment/>}\right)$is globally CB, locally CB, and CB generated under the technical assumption of tameness. In this article, we restrict our study to the pure mapping class group and give a complete classification without additional assumptions. In stark contrast with the rich class of examples of Mann–Rafi, we prove that $\mathrm{P}\mathrm{M}\mathrm{a}\mathrm{p}\left(\mathrm{\Sigma <\#comment/>}\right)$is globally CB if and only if $\mathrm{\Sigma <\#comment/>}$is the Loch Ness monster surface, and locally CB or CB generated if and only if $\mathrm{\Sigma <\#comment/>}$has finitely many ends and is not a Loch Ness monster surface with (nonzero) punctures. 

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4. Breakdown of Regularity of Scattering for Mass-Subcritical NLS

   https://doi.org/10.1093/imrn/rnaa072  

   Lee, Gyu Eun (April 2020, International Mathematics Research Notices)

   null (Ed.)

   Abstract We study the scattering problem for the nonlinear Schrödinger equation $$i\partial _t u + \Delta u = |u|^p u$$ on $$\mathbb{R}^d$$, $$d\geq 1$$, with a mass-subcritical nonlinearity above the Strauss exponent. For this equation, it is known that asymptotic completeness in $L^2$ with initial data in $$\Sigma$$ holds and the wave operator is well defined on $$\Sigma$$. We show that there exists $$0<\beta <p$$ such that the wave operator and the data-to-scattering-state map do not admit extensions to maps $$L^2\to L^2$$ of class $$C^{1+\beta }$$ near the origin. This constitutes a mild form of ill-posedness for the scattering problem in the $L^2$ topology. 

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5. Using kinetics to avoid sigma phase formation on hyper duplex stainless weld cladding

   https://doi.org/10.1080/13621718.2023.2246719  

   Acuna, Andres; Ramirez, Antonio; Riffel, Kaue Correa (December 2023, Science and Technology of Welding and Joining)

   The Hyper Duplex Stainless Steel HDSS enhanced corrosion resistance and toughness relies upon high alloying to obtain a balanced ferrite and austenite volume and pitting resistance equivalent number PREn. However, during welding, sigma phase precipitates might form, hindering corrosion and mechanical performance. Therefore, a kinetics model is developed to avoid the sigma phase's formation during welding and validated using physical simulation, finite element analysis (FEA), welding, and SEM characterisation. The sigma phase kinetics model produced calculated and validated temperature-time-transformation (TTT) and continuous-cooling-transformation (CCT) curves from which a 4°C/s cooling rate was found as a cooling rate threshold for sigma phase formation in this new material. Three-layered gas tungsten arc welding GTAW cladded mockup with 53 beads produced 24°C/s minimum cooling rate. Moreover, microscopy, mechanical, and corrosion testing attested it as a sigma-free weld. 

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