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1. Perturbations of elliptic operators in 1-sided chord-arc domains. Part I: Small and large perturbation for symmetric operators

   https://doi.org/http://dx.doi.org/10.1090/tran/7536  

   Cavero, J.; Hofmann, S.; Martell, J.M. (January 2019, Transactions of the American Mathematical Society)

   Let $$\Omega\subset\re^{n+1}$$, $$n\ge 2$$, be a 1-sided chord-arc domain, that is, a domain which satisfies interior Corkscrew and Harnack Chain conditions (these are respectively scale-invariant/quantitative versions of the openness and path-connectedness), and whose boundary $$\partial\Omega$$ is $$n$$-dimensional Ahlfors regular. Consider $$L_0$$ and $$L$$ two real symmetric divergence form elliptic operators and let $$\omega_{L_0}$$, $$\omega_L$$ be the associated elliptic measures. We show that if $$\omega_{L_0}\in A_\infty(\sigma)$$, where $$\sigma=H^n\lfloor_{\partial\Omega}$$, and $$L$$ is a perturbation of $$L_0$$ (in the sense that the discrepancy between $$L_0$$ and $$L$$ satisfies certain Carleson measure condition), then $$\omega_L\in A_\infty(\sigma)$$. Moreover, if $$L$$ is a sufficiently small perturbation of $$L_0$$, then one can preserve the reverse H√∂lder classes, that is, if for some $$1<\infty$$, one has $$\omega_{L_0}\in RH_p(\sigma)$$ then $$\omega_{L}\in RH_p(\sigma)$$. Equivalently, if the Dirichlet problem with data in $$L^{p'}(\sigma)$$ is solvable for $$L_0$$ then so it is for $$L$$. These results can be seen as extensions of the perturbation theorems obtained by Dahlberg, Fefferman-Kenig-Pipher, and Milakis-Pipher-Toro in more benign settings. As a consequence of our methods we can show that for any perturbation of the Laplacian (or, more in general, of any elliptic symmetric operator with Lipschitz coefficients satisfying certain Carleson condition) if its elliptic measure belongs to $$A_\infty(\sigma)$$ then necessarily $$\Omega$$ is in fact an NTA domain (and hence chord-arc) and therefore its boundary is uniformly rectifiable. 

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2. High resolution diagnostic tools for superconducting radio frequency cavities

   https://doi.org/10.1063/5.0117868  

   Parajuli, I.; Ciovati, G.; Delayen, J. R. (November 2022, Review of Scientific Instruments)

   Superconducting radio-frequency (SRF) cavities are one of the fundamental building blocks of modern particle accelerators. To achieve the highest quality factors (1010–1011), SRF cavities are operated at liquid helium temperatures. Magnetic flux trapped on the surface of SRF cavities during cool-down below the critical temperature is one of the leading sources of residual RF losses. Instruments capable of detecting the distribution of trapped flux on the cavity surface are in high demand in order to better understand its relation to the cavity material, surface treatments and environmental conditions. We have designed, developed, and commissioned two high-resolution diagnostic tools to measure the distribution of trapped flux at the surface of SRF cavities. One is a magnetic field scanning system, which uses cryogenic Hall probes and anisotropic magnetoresistance sensors that fit the contour of a 1.3 GHz cavity. This setup has a spatial resolution of ∼13μm in the azimuthal direction and ∼1 cm along the cavity contour. The second setup is a stationary, combined magnetic and temperature mapping system, which uses anisotropic magnetoresistance sensors and carbon resistor temperature sensors, covering the surface of a 3 GHz SRF cavity. This system has a spatial resolution of 5 mm close to the iris and 11 mm at the equator. Initial results show a non-uniform distribution of trapped flux on the cavities’ surfaces, dependent on the magnitude of the applied magnetic field during field-cooling below the critical temperature. 

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3. 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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4. Maps from Feigin and Odesskii's elliptic algebras to twisted homogeneous coordinate rings

   https://doi.org/10.1017/fms.2020.60  

   Chirvasitu, Alex; Kanda, Ryo; Smith, S. Paul (January 2021, Forum of Mathematics, Sigma)

   null (Ed.)

   Abstract The elliptic algebras in the title are connected graded $$\mathbb {C}$$ -algebras, denoted $$Q_{n,k}(E,\tau )$$ , depending on a pair of relatively prime integers $$n>k\ge 1$$ , an elliptic curve E and a point $$\tau \in E$$ . This paper examines a canonical homomorphism from $$Q_{n,k}(E,\tau )$$ to the twisted homogeneous coordinate ring $$B(X_{n/k},\sigma ',\mathcal {L}^{\prime }_{n/k})$$ on the characteristic variety $$X_{n/k}$$ for $$Q_{n,k}(E,\tau )$$ . When $$X_{n/k}$$ is isomorphic to $E^g$ or the symmetric power $S^gE$ , we show that the homomorphism $$Q_{n,k}(E,\tau ) \to B(X_{n/k},\sigma ',\mathcal {L}^{\prime }_{n/k})$$ is surjective, the relations for $$B(X_{n/k},\sigma ',\mathcal {L}^{\prime }_{n/k})$$ are generated in degrees $$\le 3$$ and the noncommutative scheme $$\mathrm {Proj}_{nc}(Q_{n,k}(E,\tau ))$$ has a closed subvariety that is isomorphic to $E^g$ or $S^gE$ , respectively. When $$X_{n/k}=E^g$$ and $$\tau =0$$ , the results about $$B(X_{n/k},\sigma ',\mathcal {L}^{\prime }_{n/k})$$ show that the morphism $$\Phi _{|\mathcal {L}_{n/k}|}:E^g \to \mathbb {P}^{n-1}$$ embeds $E^g$ as a projectively normal subvariety that is a scheme-theoretic intersection of quadric and cubic hypersurfaces. 

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5. Integrability and renormalizability for the fully anisotropic SU(2) principal chiral field and its deformations

   https://doi.org/10.1007/JHEP08(2024)239  

   Kotousov, Gleb A; Shabetnik, Daria A (August 2024, Journal of High Energy Physics)

   A<sc>bstract</sc> For the class of 1 + 1 dimensional field theories referred to as the non-linear sigma models, there is known to be a deep connection between classical integrability and one-loop renormalizability. In this work, the phenomenon is reviewed on the example of the so-called fully anisotropic SU(2) Principal Chiral Field (PCF). Along the way, we discover a new classically integrable four parameter family of sigma models, which is obtained from the fully anisotropic SU(2) PCF by means of the Poisson-Lie deformation. The theory turns out to be one-loop renormalizable and the system of ODEs describing the flow of the four couplings is derived. Also provided are explicit analytical expressions for the full set of functionally independent first integrals (renormalization group invariants). 

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