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1. Circulation and Energy Theorem Preserving Stochastic Fluids

   https://doi.org/10.1017/prm.2019.43  

   Drivas, Theodore D.; Holm, Darryl D. (December 2020, Proceedings of the Royal Society of Edinburgh: Section A Mathematics)

   null (Ed.)

   Abstract Smooth solutions of the incompressible Euler equations are characterized by the property that circulation around material loops is conserved. This is the Kelvin theorem. Likewise, smooth solutions of Navier–Stokes are characterized by a generalized Kelvin's theorem, introduced by Constantin–Iyer (2008). In this note, we introduce a class of stochastic fluid equations, whose smooth solutions are characterized by natural extensions of the Kelvin theorems of their deterministic counterparts, which hold along certain noisy flows. These equations are called the stochastic Euler–Poincaré and stochastic Navier–Stokes–Poincaré equations respectively. The stochastic Euler–Poincaré equations were previously derived from a stochastic variational principle by Holm (2015), which we briefly review. Solutions of these equations do not obey pathwise energy conservation/dissipation in general. In contrast, we also discuss a class of stochastic fluid models, solutions of which possess energy theorems but do not, in general, preserve circulation theorems. 

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2. On solenoidal-injective and injective ray transforms of tensor fields on surfaces

   https://doi.org/10.1515/jiip-2018-0067  

   Krishnan, Venkateswaran P.; Mishra, Rohit K.; Monard, François (January 2019, Journal of Inverse and Ill-posed Problems)

   Abstract We first give a constructive answer to the attenuated tensor tomography problem on simple surfaces. We then use this result to propose two approaches to produce vector-valued integral transforms, which are fully injective over tensor fields. The first approach is by construction of appropriate weights, which vary along the geodesic flow, generalizing the moment transforms. The second one is by changing the pairing with the tensor field to generate a collection of transverse ray transforms. 

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3. Extended super BMS algebra of celestial CFT

   https://doi.org/10.1007/JHEP09(2020)198  

   Fotopoulos, Angelos; Stieberger, Stephan; Taylor, Tomasz R.; Zhu, Bin (September 2020, Journal of High Energy Physics)

   null (Ed.)

   A bstract We study two-dimensional celestial conformal field theory describing four- dimensional $$ \mathcal{N} $$ N =1 supergravity/Yang-Mills systems and show that the underlying symmetry is a supersymmetric generalization of BMS symmetry. We construct fermionic conformal primary wave functions and show how they are related via supersymmetry to their bosonic partners. We use soft and collinear theorems of supersymmetric Einstein-Yang- Mills theory to derive the OPEs of the operators associated to massless particles. The bosonic and fermionic soft theorems are shown to form a sequence under supersymmetric Ward identities. In analogy with the energy momentum tensor, the supercurrents are shadow transforms of soft gravitino operators and generate an infinite-dimensional super- symmetry algebra. The algebra of $$ {\mathfrak{sbms}}_4 $$ sbms 4 generators agrees with the expectations based on earlier work on the asymptotic symmetry group of supergravity. We also show that the supertranslation operator can be written as a product of holomorphic and anti-holomorphic supercurrents. 

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4. Inverse Elastic Scattering for a Random Source

   https://doi.org/10.1137/18M1235119  

   Li, J.; Li, P. (January 2020, SIAM journal on mathematical analysis)

   Consider the inverse random source scattering problem for the two-dimensional time-harmonic elastic wave equation with a linear load. The source is modeled as a microlocally isotropic generalized Gaussian random function whose covariance operator is a classical pseudodifferential operator. The goal is to recover the principal symbol of the covariance operator from the displacement measured in a domain away from the source. For such a distributional source, we show that the direct problem has a unique solution by introducing an equivalent Lippmann--Schwinger integral equation. For the inverse problem, we demonstrate that, with probability one, the principal symbol of the covariance operator can be uniquely determined by the amplitude of the displacement averaged over the frequency band, generated by a single realization of the random source. The analysis employs the Born approximation, asymptotic expansions of the Green tensor, and microlocal analysis of the Fourier integral operators. 

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5. Generalized Wilson loop method for nonlinear light-matter interaction

   https://doi.org/10.1038/s41535-022-00472-4  

   Wang, Hua; Tang, Xiuyu; Xu, Haowei; Li, Ju; Qian, Xiaofeng (June 2022, npj Quantum Materials)

   Abstract Nonlinear light–matter interaction, as the core of ultrafast optics, bulk photovoltaics, nonlinear optical sensing and imaging, and efficient generation of entangled photons, has been traditionally studied by first-principles theoretical methods with the sum-over-states approach. However, this indirect method often suffers from the divergence at band degeneracy and optical zeros as well as convergence issues and high computation costs when summing over the states. Here, using shift vector and shift current conductivity tensor as an example, we present a gauge-invariant generalized approach for efficient and direct calculations of nonlinear optical responses by representing interband Berry curvature, quantum metric, and shift vector in a generalized Wilson loop. This generalized Wilson loop method avoids the above cumbersome challenges and allows for easy implementation and efficient calculations. More importantly, the Wilson loop representation provides a succinct geometric interpretation of nonlinear optical processes and responses based on quantum geometric tensors and quantum geometric potentials and can be readily applied to studying other excited-state responses. 

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