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1. Comparison between shadow imaging and in-line holography for measuring droplet size distributions

   https://doi.org/10.1007/s00348-023-03633-8  

   Erinin, Martin A; Néel, Baptiste; Mazzatenta, Megan T; Duncan, James H; Deike, Luc (May 2023, Experiments in Fluids)

   Kähler, C; Longmire, E; Westerweel, J (Ed.)

   Abstract A direct comparison of the droplet size and number measurements using in-line holography and shadow imaging is presented in three dynamically evolving laboratory scale experiments. The two experimental techniques and image processing algorithms used to measure droplet number and radii are described in detail. Droplet radii as low as$$r = 14$$ $r=14$ µm are measured using in-line holography and$$r = 50$$ $r=50$ µm using shadow imaging. The droplet radius measurement error is estimated using a calibration target (reticle) and it was found that the holographic technique is able to measure droplet radii more accurately than shadow imaging for droplets with$$r \le 625$$ $r\le 625$ µm. Using the measurements of droplet number and size we quantitatively cross-validate and assess the accuracy of the two measurement techniques. The droplet size distributions,N(r), are measured in all three experiments and are found to agree well between the two measurement techniques. In one of the laboratory experiments, simultaneous measurements of droplets ($$r \ge 14$$ $r\ge 14$ µm, using holography) and dry aerosols ($$0.07 \lessapprox r \lessapprox 2$$ $0.07⪅r⪅2$ µm, using an scanning mobility particle sizer and$$0.15 \lessapprox r \lessapprox 5$$ $0.15⪅r⪅5$ µm using an optical particle sizer) are reported, one of the first such comparison to the best of our knowledge. The total number and volume of droplets is found to agree well between both techniques in the three experiments. We demonstrate that a relatively simple shadow imaging technique can be just as reliable when compared to a more sophisticated holographic measurement technique over their common droplet radius measurement range. The agreement in results is shown to be valid over a large range of droplet concentrations, which include experiments with relatively sparse droplet concentrations as low as 0.02 droplets per image. Advantages and disadvantages for the two techniques are discussed in the context of our results. The main advantages to in-line holography are the greater accuracy in droplet radius measurement, greater spatial resolution, larger depth of field, and the high repetition rate and short pulse duration of the laser light source. In comparison, the main advantages to shadow imaging are the simpler experimental setup, image processing algorithm, and fewer computer resources necessary for image processing. Droplet statistics like number and size are found to be very reliable between the two methods for large range of droplet densities,$${\mathcal {P}}_{r>50}$$ $P_{r>50}$, ranging from$$10^{-4} \le {\mathcal {P}}_{r>50} \le 10^{-1}$$ ${10}^{-4}\le P_{r>50}\le {10}^{-1}$cm$$^{-3}$$ ${}^{-3}$, when the two techniques are implemented as shown in this paper. 

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2. Investigating $D^0$ meson production in p-Pb collisions at 5.02 TeV with a multi-phase transport model

   https://doi.org/10.1140/epjc/s10052-024-13265-9  

   Zhang, Chao; Zheng, Liang; Shi, Shusu; Lin, Zi-Wei (September 2024, The European Physical Journal C)

   Abstract We study the production of$$D^0$$ $D^0$meson inp+pandp-Pb collisions using the improved AMPT model considering both coalescence and independent fragmentation of charm quarks after the Cronin broadening is included. After a detailed discussion of the improvements implemented in the AMPT model for heavy quark production, we show that the modified AMPT model can provide a good description of$$D^0$$ $D^0$meson spectra inp-Pb collisions, the$$Q_{\textrm{pPb}}$$ $Q_{\text{pPb}}$data at different centralities and$$R_{\textrm{pPb}}$$ $R_{\text{pPb}}$data in both mid- and forward (backward) rapidities. We also studied the effects of nuclear shadowing and parton cascade on the rapidity dependence of$$D^{0}$$ $D^0$meson production and$$R_{\textrm{pPb}}$$ $R_{\text{pPb}}$. Our results indicate that using the same strength of the Cronin effect (i.e$$\delta $$ $\delta$value) as that obtained from the mid-rapidity data leads to a considerable overestimation of the$$D^0$$ $D^0$meson spectra and$$R_{\textrm{pPb}}$$ $R_{\text{pPb}}$data at high$$p_{\textrm{T}}$$ $p_{\text{T}}$in the backward rapidity. As a result, the$$\delta $$ $\delta$is determined via a$$\chi ^2$$ ${\chi }^2$fitting of the$$R_{\textrm{pPb}}$$ $R_{\text{pPb}}$data across various rapidities. This work lays the foundation for a better understanding of cold-nuclear-matter (CNM) effects in relativistic heavy-ion collisions. 

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3. Carleson’s $$\varepsilon ^{2}$$ conjecture in higher dimensions

   https://doi.org/10.1007/s00222-025-01337-w  

   Fleschler, Ian; Tolsa, Xavier; Villa, Michele (July 2025, Inventiones mathematicae)

   Abstract In this paper we prove a higher dimensional analogue of Carleson’s$$\varepsilon ^{2}$$ ${\epsilon }^2$conjecture. Given two arbitrary disjoint Borel sets$$\Omega ^{+},\Omega ^{-}\subset \mathbb{R}^{n+1}$$ ${\Omega }^{+},{\Omega }^{-}\subset R^{n+1}$, and$$x\in \mathbb{R}^{n+1}$$ $x\in R^{n+1}$,$$r>0$$ $r>0$, we denote$$ \varepsilon _{n}(x,r) := \frac{1}{r^{n}}\, \inf _{H^{+}} \mathcal{H}^{n} \left ( ((\partial B(x,r)\cap H^{+}) \setminus \Omega ^{+}) \cup (( \partial B(x,r)\cap H^{-}) \setminus \Omega ^{-})\right ), $$ ${\epsilon }_n(x,r):=\frac{1}{r^n}\underset{H^{+}}{inf}{H}^n\left(\right(\left(\partial B\right(x,r)\cap H^{+})\setminus {\Omega }^{+})\cup (\left(\partial B\right(x,r)\cap H^{-})\setminus {\Omega }^{-}\left)\right),$where the infimum is taken over all open affine half-spaces$$H^{+}$$ $H^{+}$such that$$x \in \partial H^{+}$$ $x\in \partial H^{+}$and we define$$H^{-}= \mathbb{R}^{n+1} \setminus \overline{H^{+}}$$ $H^{-}=R^{n+1}\setminus \overline{H^{+}}$. Our first main result asserts that the set of points$$x\in \mathbb{R}^{n+1}$$ $x\in R^{n+1}$where$$ \int _{0}^{1} \varepsilon _{n}(x,r)^{2} \, \frac{dr}{r}< \infty $$ ${\int }_0^1{\epsilon }_n{(x,r)}^2\frac{dr}{r}<\infty$is$$n$$ $n$-rectifiable. For our second main result we assume that$$\Omega ^{+}$$ ${\Omega }^{+}$,$$\Omega ^{-}$$ ${\Omega }^{-}$are open and that$$\Omega ^{+}\cup \Omega ^{-}$$ ${\Omega }^{+}\cup {\Omega }^{-}$satisfies the capacity density condition. For each$$x \in \partial \Omega ^{+} \cup \partial \Omega ^{-}$$ $x\in \partial {\Omega }^{+}\cup \partial {\Omega }^{-}$and$$r>0$$ $r>0$, we denote by$$\alpha ^{\pm }(x,r)$$ ${\alpha }^{\pm }(x,r)$the characteristic constant of the (spherical) open sets$$\Omega ^{\pm }\cap \partial B(x,r)$$ ${\Omega }^{\pm }\cap \partial B(x,r)$. We show that, up to a set of$$\mathcal{H}^{n}$$ $H^n$measure zero,$$x$$ $x$is a tangent point for both$$\partial \Omega ^{+}$$ $\partial {\Omega }^{+}$and$$\partial \Omega ^{-}$$ $\partial {\Omega }^{-}$if and only if$$ \int _{0}^{1} \min (1,\alpha ^{+}(x,r) + \alpha ^{-}(x,r) -2) \frac{dr}{r} < \infty . $$ ${\int }_0^{1}min(1,{\alpha }^{+}(x,r)+{\alpha }^{-}(x,r)-2)\frac{dr}{r}<\infty .$The first result is new even in the plane and the second one improves and extends to higher dimensions the$$\varepsilon ^{2}$$ ${\epsilon }^2$conjecture of Carleson. 

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4. Violent Nonlinear Collapse in the Interior of Charged Hairy Black Holes

   https://doi.org/10.1007/s00205-024-02038-z  

   Van_de_Moortel, Maxime (October 2024, Archive for Rational Mechanics and Analysis)

   Abstract We construct a new one-parameter family, indexed by$$\epsilon $$ $ϵ$, of two-ended, spatially-homogeneous black hole interiors solving the Einstein–Maxwell–Klein–Gordon equations with a (possibly zero) cosmological constant$$\Lambda $$ $\Lambda$and bifurcating off a Reissner–Nordström-(dS/AdS) interior ($$\epsilon =0$$ $ϵ=0$). For all small$$\epsilon \ne 0$$ $ϵ\ne 0$, we prove that, although the black hole is charged, its terminal boundary is an everywhere-spacelikeKasner singularity foliated by spheres of zero radiusr. Moreover, smaller perturbations (i.e. smaller$$|\epsilon |$$ $\left|ϵ\right|$) aremore singular than larger ones, in the sense that the Hawking mass and the curvature blow up following a power law of the form$$r^{-O(\epsilon ^{-2})}$$ $r^{-O\left({ϵ}^{-2}\right)}$at the singularity$$\{r=0\}$$ $\{r=0\}$. This unusual property originates from a dynamical phenomenon—violent nonlinear collapse—caused by the almost formation of a Cauchy horizon to the past of the spacelike singularity$$\{r=0\}$$ $\{r=0\}$. This phenomenon was previously described numerically in the physics literature and referred to as “the collapse of the Einstein–Rosen bridge”. While we cover all values of$$\Lambda \in \mathbb {R}$$ $\Lambda \in R$, the case$$\Lambda <0$$ $\Lambda <0$is of particular significance to the AdS/CFT correspondence. Our result can also be viewed in general as a first step towards the understanding of the interior of hairy black holes. 

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5. Counterexamples to maximal regularity for operators in divergence form

   https://doi.org/10.1007/s00013-024-02014-9  

   Bechtel, Sebastian; Mooney, Connor; Veraar, Mark (August 2024, Archiv der Mathematik)

   Abstract In this paper, we present counterexamples to maximal$$L^p$$ $L^p$-regularity for a parabolic PDE. The example is a second-order operator in divergence form with space and time-dependent coefficients. It is well-known from Lions’ theory that such operators admit maximal$$L^2$$ $L^2$-regularity on$$H^{-1}$$ $H^{-1}$under a coercivity condition on the coefficients, and without any regularity conditions in time and space. We show that in general one cannot expect maximal$$L^p$$ $L^p$-regularity on$$H^{-1}(\mathbb {R}^d)$$ $H^{-1}\left(R^d\right)$or$$L^2$$ $L^2$-regularity on$$L^2(\mathbb {R}^d)$$ $L^2\left(R^d\right)$. 

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