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1. From low to high-and lower-optimal regularity of the SMGTJ equation with Dirichlet and Neumann boundary control, and with point control, via explicit representation formulae

   https://doi.org/10.3934/eect.2022007  

   Triggiani, Roberto; Wan, Xiang (January 2022, Evolution Equations and Control Theory)

   We consider the linear third order (in time) PDE known as the SMGTJ-equation, defined on a bounded domain, under the action of either Dirichlet or Neumann boundary control \begin{document}$ g $$\end{document}. Optimal interior and boundary regularity results were given in [1], after [41], when \begin{document}$$ g \in L^2(0, T;L^2(\Gamma)) \equiv L^2(\Sigma) $$\end{document}, which, moreover, in the canonical case \begin{document}$$ \gamma = 0 $$\end{document}, were expressed by the well-known explicit representation formulae of the wave equation in terms of cosine/sine operators [19], [17], [24,Vol Ⅱ]. The interior or boundary regularity theory is however the same, whether \begin{document}$$ \gamma = 0 $$\end{document} or \begin{document}$$ 0 \neq \gamma \in L^{\infty}(\Omega) $$\end{document}, since \begin{document}$$ \gamma \neq 0 $$\end{document} is responsible only for lower order terms. Here we exploit such cosine operator based-explicit representation formulae to provide optimal interior and boundary regularity results with \begin{document}$$ g $$\end{document} "smoother" than \begin{document}$$ L^2(\Sigma) $$\end{document}, qualitatively by one unit, two units, etc. in the Dirichlet boundary case. To this end, we invoke the corresponding results for wave equations, as in [17]. Similarly for the Neumann boundary case, by invoking the corresponding results for the wave equation as in [22], [23], [37] for control smoother than \begin{document}$$ L^2(0, T;L^2(\Gamma)) $$\end{document}, and [44] for control less regular in space than \begin{document}$$ L^2(\Gamma) $$\end{document}$. In addition, we provide optimal interior and boundary regularity results when the SMGTJ equation is subject to interior point control, by invoking the corresponding wave equations results [42], [24,Section 9.8.2]. 

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2. The mean-field limit of the Lieb-Liniger model

   https://doi.org/10.3934/dcds.2022006  

   Rosenzweig, Matthew (January 2022, Discrete and Continuous Dynamical Systems)

   We consider the well-known Lieb-Liniger (LL) model for \begin{document}$ N $$\end{document} bosons interacting pairwise on the line via the \begin{document}$$ \delta $$\end{document} potential in the mean-field scaling regime. Assuming suitable asymptotic factorization of the initial wave functions and convergence of the microscopic energy per particle, we show that the time-dependent reduced density matrices of the system converge in trace norm to the pure states given by the solution to the one-dimensional cubic nonlinear Schrödinger equation (NLS) with an explict rate of convergence. In contrast to previous work [3] relying on the formalism of second quantization and coherent states and without an explicit rate, our proof is based on the counting method of Pickl [65,66,67] and Knowles and Pickl [44]. To overcome difficulties stemming from the singularity of the \begin{document}$$ \delta $$\end{document} potential, we introduce a new short-range approximation argument that exploits the Hölder continuity of the \begin{document}$$ N $$\end{document}-body wave function in a single particle variable. By further exploiting the \begin{document}$$ L^2 $$\end{document}-subcritical well-posedness theory for the 1D cubic NLS, we can prove mean-field convergence when the limiting solution to the NLS has finite mass, but only for a very special class of \begin{document}$$ N $$\end{document}$-body initial states. 

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3. Lateral Beam Shifts and Depolarization Upon Oblique Reflection from Dielectric Mirrors

   https://doi.org/10.1002/andp.202400028  

   Xiao, Yuzhe; Phuttitarn, Linipun; Graham, Trent Michael; Wan, Chenghao; Saffman, Mark; Kats, Mikhail A. (March 2024, Annalen der Physik)

   Abstract Dielectric mirrors comprising thin‐film multilayers are widely used in optical experiments because they can achieve substantially higher reflectance compared to metal mirrors. Here, potential problems are investigated that can arise when dielectric mirrors are used at oblique incidence, in particular for focused beams. It is found that light beams reflected from dielectric mirrors can experience lateral beam shifts, beam‐shape distortion, and depolarization, and these effects have a strong dependence on wavelength, incident angle, and incident polarization. Because vendors of dielectric mirrors typically do not share the particular layer structure of their products, several dielectric‐mirror stacks are designed and simulated, and then the lateral beam shift from two commercial dielectric mirrors and one coated metal mirror is also measured. This paper brings awareness of the tradeoffs between dielectric mirrors and front‐surface metal mirrors in certain optics experiments, and it is suggested that vendors of dielectric mirrors provide information about beam shifts, distortion, and depolarization when their products are used at oblique incidence. 

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4. Optical scattering measurements and implications on thermal noise in Gravitational Wave detectors test-mass coatings

   https://doi.org/https://doi.org/10.1016/j.physleta.2017.05.050  

   Glover, Lamar; Goff, Michael; Patel, Jignesh; Pinto, Innocenzo; Principe, Maria; Sadecki, Travis; Savage, Richard; Villarama, Ethan; Arriaga, Eddy; Barragan, Erik; et al (August 2018, Physics letters. A)

   Photographs of the LIGO Gravitational Wave detector mirrors illuminated by the standing beam were analyzed with an astronomical software tool designed to identify stars within images, which extracted hundreds of thousands of point-like scatterers uniformly distributed across the mirror surface, likely distributed through the depth of the coating layers. The sheer number of the observed scatterers implies a fundamental, thermodynamic origin during deposition or processing. These scatterers are a possible source of the mirror dissipation and thermal noise foreseen by V. Braginsky and Y. Levin, which limits the sensitivity of observatories to Gravitational Waves. This study may point the way towards the production of mirrors with reduced thermal noise and an increased detection range. 

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5. Capturing dynamics of time-varying data via topology

   https://doi.org/10.3934/fods.2021033  

   Xian, Lu; Adams, Henry; Topaz, Chad M.; Ziegelmeier, Lori (January 2022, Foundations of Data Science)

   One approach to understanding complex data is to study its shape through the lens of algebraic topology. While the early development of topological data analysis focused primarily on static data, in recent years, theoretical and applied studies have turned to data that varies in time. A time-varying collection of metric spaces as formed, for example, by a moving school of fish or flock of birds, can contain a vast amount of information. There is often a need to simplify or summarize the dynamic behavior. We provide an introduction to topological summaries of time-varying metric spaces including vineyards [19], crocker plots [55], and multiparameter rank functions [37]. We then introduce a new tool to summarize time-varying metric spaces: a crocker stack. Crocker stacks are convenient for visualization, amenable to machine learning, and satisfy a desirable continuity property which we prove. We demonstrate the utility of crocker stacks for a parameter identification task involving an influential model of biological aggregations [57]. Altogether, we aim to bring the broader applied mathematics community up-to-date on topological summaries of time-varying metric spaces. 

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