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We describe the outcome of a large international interlaboratory study of the measurement of particle number concentration of colloidal nanoparticles, project 10 of the technical working area 34, "Nanoparticle Populations" of the Versailles Project on Advanced Materials and Standards (VAMAS). A total of 50 laboratories delivered results for the number concentration of 30 nm gold colloidal nanoparticles measured using particle tracking analysis (PTA), single particle inductively coupled plasma mass spectrometry (spICP-MS), ultraviolet-visible (UV-Vis) light spectroscopy, centrifugal liquid sedimentation (CLS) and small angle X-ray scattering (SAXS). The study provides quantitative data to evaluate the repeatability of these methods and their reproducibility in the measurement of number concentration of model nanoparticle systems following a common measurement protocol. We find that the population-averaging methods of SAXS, CLS and UV-Vis have high measurement repeatability and reproducibility, with between-labs variability of 2.6%, 11% and 1.4% respectively. However, results may be significantly biased for reasons including inaccurate material properties whose values are used to compute the number concentration. Particle-counting method results are less reproducibile than population-averaging methods, with measured between-labs variability of 68% and 46% for PTA and spICP-MS respectively. This study provides the stakeholder community with important comparative data to underpin measurement reproducibility and method validation for number concentration of nanoparticles.

Let $��_1:\mathcal{X} \to ��$ be a flat family of smooth, projective curves of genus $g \ge 2$, degenerating to an irreducible nodal curve $X_0$ with exactly one node. Fix an invertible sheaf $\mathcal{L}$ on $\mathcal{X}$ of relative odd degree. Let $��_2:\mathcal{G}(2,\mathcal{L}) \to ��$ be the relative Gieseker moduli space of rank $2$ semi-stable vector bundles with determinant $\mathcal{L}$ over $\mathcal{X}$. Since $��_2$ is smooth over $��^*$, there exists a canonical family $\widetilde��_i:\mathbf{J}^i_{\mathcal{G}(2, \mathcal{L})_{��^*}} \to ��^{*}$ of $i$-th intermediate Jacobians i.e., for all $t \in ��^*$, $(\widetilde��_i)^{-1}(t)$ is the $i$-th intermediate Jacobian of $��_2^{-1}(t)$. There exist different N��ron models $\overline��_i:\overline{\mathbf{J}}_{\mathcal{G}(2, \mathcal{L})}^i \to ��$ extending $\widetilde��_i$ to the entire disc $��$, constructed by Clemens, Saito, Schnell, Zucker and Green-Griffiths-Kerr. In this article, we prove that in our setup, the N��ron model $\overline��_i$ is canonical in the sense that the different N��ron models coincide and is an analytic fiber space which graphs admissible normal functions. We also show that for $1 \le i \le \max\{2,g-1\}$, the central fiber of $\overline��_i$ is a fibration over product of copies of $J^k(\mathrm{Jac}(\widetilde{X}_0))$ for certain values of $k$, where $\widetilde{X}_0$ is the normalization of $X_0$. In particular, for $g \ge 5$ and $i=2, 3, 4$, the central fiber of $\overline��_i$ is a semi-abelian variety. Furthermore, we prove that the $i$-th generalized intermediate Jacobian of the (singular) central fibre of $��_2$ is a fibration over the central fibre of the N��ron model $\overline{\mathbf{J}}^i_{\mathcal{G}(2, \mathcal{L})}$. In fact, for $i=2$ the fibration is an isomorphism., to appear in Revista Matem\'atica Complutense

Let $R$ be a complete discrete valuation ring with fraction field of characteristic $0$ and algebraically closed residue field of characteristic $p>0$. Let $X_R \to \mathrm{Spec}(R)$ be a smooth projective morphism of relative dimension $1$. We prove that, given a line bundle $\mathcal{L}_R$ the moduli space of Gieseker stable torsion-free sheaves of rank $r\geq 2$ over $X_R$, with determinant $\mathcal{L}_R$, is smooth over $R$., 14 pages, To appear in Proceedings of the conference on "Analytic and Algebraic Geometry of Bundles"

In this note we prove the classical decomposition theorems for certain subgroups of the automorphism group of the Bruhat-Tits tree associated to $\mathrm{PGL}_{2}(F)$, where $F$ is a local field. The results can be used to understand the representation theory of these subgroups. Moreover, we can also use the results to obtain similar decompositions for \textit{any} closed transitive subgroup of the automorphism group of \textit{any} Bruhat-Tits tree., Comment: 14 pages