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1. Variability of relationship between the volume scattering function at 180° and the backscattering coefficient for aquatic particles

   https://doi.org/10.1364/AO.383229  

   Hu, Lianbo ; Zhang, Xiaodong ; Xiong, Yuanheng ; Gray, Deric_J ; He, Ming-Xia ( February 2020 , Applied Optics)

   Properly interpreting lidar (light detection and ranging) signal for characterizing particle distribution relies on a key parameter,${\mathrm{\chi <\#comment/>}}_p(\mathrm{\pi <\#comment/>})$, which relates the particulate volume scattering function (VSF) at 180° (${\mathrm{\beta <\#comment/>}}_p(\mathrm{\pi <\#comment/>})$) that a lidar measures to the particulate backscattering coefficient ($b_{\mathit{\text{bp}}}$). However,${\mathrm{\chi <\#comment/>}}_p(\mathrm{\pi <\#comment/>})$has been seldom studied due to challenges in accurately measuring${\mathrm{\beta <\#comment/>}}_p(\mathrm{\pi <\#comment/>})$and$b_{\mathit{\text{bp}}}$concurrently in the field. In this study,${\mathrm{\chi <\#comment/>}}_p(\mathrm{\pi <\#comment/>})$, as well as its spectral dependence, was re-examined using the VSFs measuredin situat high angular resolution in a wide range of waters.${\mathrm{\beta <\#comment/>}}_p(\mathrm{\pi <\#comment/>})$, while not measured directly, was inferred using a physically sound, well-validated VSF-inversion method. The effects of particle shape and internal structure on the inversion were tested using three inversion kernels consisting of phase functions computed for particles that are assumed as homogenous sphere, homogenous asymmetric hexahedra, or coated sphere. The reconstructed VSFs using any of the three kernels agreed well with the measured VSFs with a mean percentage difference$<<\#comment/>5\mathrm{\%<\#comment/>}$at scattering angles$<<\#comment/>{170}^{\circ <\#comment/>}$. At angles immediately near or equal to 180°, the reconstructed${\mathrm{\beta <\#comment/>}}_p(\mathrm{\pi <\#comment/>})$depends strongly on the inversion kernel.${\mathrm{\chi <\#comment/>}}_p(\mathrm{\pi <\#comment/>})$derived with the sphere kernels was smaller than those derived with the hexahedra kernel but consistent with${\mathrm{\chi <\#comment/>}}_p(\mathrm{\pi <\#comment/>})$estimated directly from high-spectral-resolution lidar andin situbackscattering sensor. The possible explanation was that the sphere kernels are able to capture the backscattering enhancement feature near 180° that has been observed for marine particles.${\mathrm{\chi <\#comment/>}}_p(\mathrm{\pi <\#comment/>})$derived using the coated sphere kernel was generally lower than those derived with the homogenous sphere kernel. Our result suggests that${\mathrm{\chi <\#comment/>}}_p(\mathrm{\pi <\#comment/>})$is sensitive to the shape and internal structure of particles and significant error could be induced if a fixed value of${\mathrm{\chi <\#comment/>}}_p(\mathrm{\pi <\#comment/>})$is to be used to interpret lidar signal collected in different waters. On the other hand,${\mathrm{\chi <\#comment/>}}_p(\mathrm{\pi <\#comment/>})$showed little spectral dependence.

    

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2. Phase-factor spectra of turbulent phase screens

   https://doi.org/10.1364/JOSAA.429928  

   Muschinski, Andreas ( August 2021 , Journal of the Optical Society of America A)

   The optical phase$\mathrm{\varphi <\#comment/>}$is a key quantity in the physics of light propagating through a turbulent medium. In certain respects, however, the statistics of the phasefactor,$\mathrm{\psi <\#comment/>}=\exp <\#comment/>(i\mathrm{\varphi <\#comment/>})$, are more relevant than the statistics of the phase itself. Here, we present a theoretical analysis of the 2D phase-factor spectrum$F_{\mathrm{\psi <\#comment/>}}(\mathit{\kappa <\#comment/>})$of a random phase screen. We apply the theory to four types of phase screens, each characterized by a power-law phase structure function,$D_{\mathrm{\varphi <\#comment/>}}(r)=(r/r_c{)}^{\mathrm{\gamma <\#comment/>}}$(where$r_c$is the phase coherence length defined by$D_{\mathrm{\varphi <\#comment/>}}(r_c)=1{\mathrm{r}\mathrm{a}\mathrm{d}}^2$), and a probability density function$p_{\mathrm{\alpha <\#comment/>}}(\mathrm{\alpha <\#comment/>})$of the phase increments for a given spatial lag. We analyze phase screens with turbulent ($\mathrm{\gamma <\#comment/>}=5/3$) and quadratic ($\mathrm{\gamma <\#comment/>}=2$) phase structure functions and with normally distributed (i.e., Gaussian) versus Laplacian phase increments. We find that there is a pronounced bump in each of the four phase-factor spectra$F_{\mathrm{\psi <\#comment/>}}(\mathrm{\kappa <\#comment/>})$. The precise location and shape of the bump are different for the four phase-screen types, but in each case it occurs at$\mathrm{\kappa <\#comment/>}\sim <\#comment/>1/r_c$. The bump is unrelated to the well-known “Hill bump” and is not caused by diffraction effects. It is solely a characteristic of the refractive-index statistics represented by the respective phase screen. We show that the second-order$\mathrm{\psi <\#comment/>}$statistics (covariance function, structure function, and spectrum) characterize a random phase screen more completely than the second-order$\mathrm{\varphi <\#comment/>}$counterparts.

    

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3. The Vlasov–Poisson–Landau system in the weakly collisional regime

   https://doi.org/10.1090/jams/1014  

   Chaturvedi, Sanchit ; Luk, Jonathan ; Nguyen, Toan ( January 2023 , Journal of the American Mathematical Society)

   Consider the Vlasov–Poisson–Landau system with Coulomb potential in the weakly collisional regime on a$3$-torus, i.e.$\begin{align*} \partial _t F(t,x,v) + v_i \partial _{x_i} F(t,x,v) + E_i(t,x) \partial _{v_i} F(t,x,v) = \nu Q(F,F)(t,x,v),\\ E(t,x) = \nabla \Delta ^{-1} (\int _{\mathbb R^3} F(t,x,v)\, \mathrm {d} v - {{\int }\llap {-}}_{\mathbb T^3} \int _{\mathbb R^3} F(t,x,v)\, \mathrm {d} v \, \mathrm {d} x), \end{align*}$with$\nu \ll 1$. We prove that for$\epsilon >0$sufficiently small (but independent of$\nu$), initial data which are$O(\epsilon \nu ^{1/3})$-Sobolev space perturbations from the global Maxwellians lead to global-in-time solutions which converge to the global Maxwellians as$t\to \infty$. The solutions exhibit uniform-in-$\nu$Landau damping and enhanced dissipation.

   Our main result is analogous to an earlier result of Bedrossian for the Vlasov–Poisson–Fokker–Planck equation with the same threshold. However, unlike in the Fokker–Planck case, the linear operator cannot be inverted explicitly due to the complexity of the Landau collision operator. For this reason, we develop an energy-based framework, which combines Guo’s weighted energy method with the hypocoercive energy method and the commuting vector field method. The proof also relies on pointwise resolvent estimates for the linearized density equation.

    

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4. Stable Big Bang formation for Einstein’s equations: The complete sub-critical regime

   https://doi.org/10.1090/jams/1015  

   Fournodavlos, Grigorios ; Rodnianski, Igor ; Speck, Jared ( July 2023 , Journal of the American Mathematical Society)

   For$(t,x) \in (0,\infty )\times \mathbb {T}^{\mathfrak {D}}$, the generalized Kasner solutions (which we refer to as Kasner solutions for short) are a family of explicit solutions to various Einstein-matter systems that, exceptional cases aside, start out smooth but then develop a Big Bang singularity as$t \downarrow 0$, i.e., a singularity along an entire spacelike hypersurface, where various curvature scalars blow up monotonically. The family is parameterized by the Kasner exponents$\widetilde {q}_1,\cdots ,\widetilde {q}_{\mathfrak {D}} \in \mathbb {R}$, which satisfy two algebraic constraints. There are heuristics in the mathematical physics literature, going back more than 50 years, suggesting that the Big Bang formation should be dynamically stable, that is, stable under perturbations of the Kasner initial data, given say at$\lbrace t = 1 \rbrace$, as long as the exponents are “sub-critical” in the following sense:$\underset {\substack {I,J,B=1,\cdots , \mathfrak {D}\\ I > J}}{\max } \{\widetilde {q}_I+\widetilde {q}_J-\widetilde {q}_B\}>1$. Previous works have rigorously shown the dynamic stability of the Kasner Big Bang singularity under stronger assumptions: (1) the Einstein-scalar field system with$\mathfrak {D}= 3$and$\widetilde {q}_1 \approx \widetilde {q}_2 \approx \widetilde {q}_3 \approx 1/3$, which corresponds to the stability of the Friedmann–Lemaître–Robertson–Walker solution’s Big Bang or (2) the Einstein-vacuum equations for$\mathfrak {D}\geq 38$with$\underset {I=1,\cdots ,\mathfrak {D}}{\max } |\widetilde {q}_I| > 1/6$. In this paper, we prove that the Kasner singularity is dynamically stable forallsub-critical Kasner exponents, thereby justifying the heuristics in the literature in the full regime where stable monotonic-type curvature-blowup is expected. We treat in detail the$1+\mathfrak {D}$-dimensional Einstein-scalar field system for all$\mathfrak {D}\geq 3$and the$1+\mathfrak {D}$-dimensional Einstein-vacuum equations for$\mathfrak {D}\geq 10$; both of these systems feature non-empty sets of sub-critical Kasner solutions. Moreover, for the Einstein-vacuum equations in$1+3$dimensions, where instabilities are in general expected, we prove that all singular Kasner solutions have dynamically stable Big Bangs under polarized$U(1)$-symmetric perturbations of their initial data. Our results hold for open sets of initial data in Sobolev spaces without symmetry, apart from our work on polarized$U(1)$-symmetric solutions.

   Our proof relies on a new formulation of Einstein’s equations: we use a constant-mean-curvature foliation, and the unknowns are the scalar field, the lapse, the components of the spatial connection and second fundamental form relative to a Fermi–Walker transported spatial orthonormal frame, and the components of the orthonormal frame vectors with respect to a transported spatial coordinate system. In this formulation, the PDE evolution system for the structure coefficients of the orthonormal frame approximately diagonalizes in a way that sharply reveals the significance of the Kasner exponent sub-criticality condition for the dynamic stability of the flow: the condition leads to the time-integrability of many terms in the equations, at least at the low derivative levels. At the high derivative levels, the solutions that we study can be much more singular with respect to$t$, and to handle this difficulty, we use$t$-weighted high order energies, and we control non-linear error terms by exploiting monotonicity induced by the$t$-weights and interpolating between the singularity-strength of the solution’s low order and high order derivatives. Finally, we note that our formulation of Einstein’s equations highlights the quantities that might generate instabilities outside of the sub-critical regime.

    

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5. Braid group action and quasi-split affine 𝚤quantum groups I

   https://doi.org/10.1090/ert/657  

   Lu, Ming ; Wang, Weiqiang ; Zhang, Weinan ( October 2023 , Representation Theory of the American Mathematical Society)

   This is the first of our papers on quasi-split affine quantum symmetric pairs$\big (\widetilde {\mathbf U}(\widehat {\mathfrak g}), \widetilde {{\mathbf U}}^\imath \big )$, focusing on the real rank one case, i.e.,$\mathfrak g = \mathfrak {sl}_3$equipped with a diagram involution. We construct explicitly a relative braid group action of type$A_2^{(2)}$on the affine$\imath$quantum group$\widetilde {{\mathbf U}}^\imath$. Real and imaginary root vectors for$\widetilde {{\mathbf U}}^\imath$are constructed, and a Drinfeld type presentation of$\widetilde {{\mathbf U}}^\imath$is then established. This provides a new basic ingredient for the Drinfeld type presentation of higher rank quasi-split affine$\imath$quantum groups in the sequels.

    

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