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1. Product of Simplices and sets of positive upper density in ℝ ^d

   https://doi.org/10.1017/S0305004117000184  

   LYALL, NEIL; MAGYAR, ÁKOS (July 2018, Mathematical Proceedings of the Cambridge Philosophical Society)

   Abstract We establish that any subset of ℝ d of positive upper Banach density necessarily contains an isometric copy of all sufficiently large dilates of any fixed two-dimensional rectangle provided d ⩾ 4. We further present an extension of this result to configurations that are the product of two non-degenerate simplices; specifically we show that if Δ k 1 and Δ k 2 are two fixed non-degenerate simplices of k 1 + 1 and k 2 + 1 points respectively, then any subset of ℝ d of positive upper Banach density with d ⩾ k 1 + k 2 + 6 will necessarily contain an isometric copy of all sufficiently large dilates of Δ k 1 × Δ k 2 . A new direct proof of the fact that any subset of ℝ d of positive upper Banach density necessarily contains an isometric copy of all sufficiently large dilates of any fixed non-degenerate simplex of k + 1 points provided d ⩾ k + 1, a result originally due to Bourgain, is also presented. 

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2. Topological k-Metrics

   https://doi.org/10.4230/LIPIcs.SoCG.2024.13  

   Barkan, Willow; Bennett, Huck; Nayyeri, Amir (January 2024, Schloss Dagstuhl – Leibniz-Zentrum für Informatik)

   Mulzer, Wolfgang; Phillips, Jeff M (Ed.)

   Metric spaces (X, d) are ubiquitous objects in mathematics and computer science that allow for capturing pairwise distance relationships d(x, y) between points x, y ∈ X. Because of this, it is natural to ask what useful generalizations there are of metric spaces for capturing "k-wise distance relationships" d(x_1, …, x_k) among points x_1, …, x_k ∈ X for k > 2. To that end, Gähler (Math. Nachr., 1963) (and perhaps others even earlier) defined k-metric spaces, which generalize metric spaces, and most notably generalize the triangle inequality d(x₁, x₂) ≤ d(x₁, y) + d(y, x₂) to the "simplex inequality" d(x_1, …, x_k) ≤ ∑_{i=1}^k d(x_1, …, x_{i-1}, y, x_{i+1}, …, x_k). (The definition holds for any fixed k ≥ 2, and a 2-metric space is just a (standard) metric space.) In this work, we introduce strong k-metric spaces, k-metric spaces that satisfy a topological condition stronger than the simplex inequality, which makes them "behave nicely." We also introduce coboundary k-metrics, which generalize 𝓁_p metrics (and in fact all finite metric spaces induced by norms) and minimum bounding chain k-metrics, which generalize shortest path metrics (and capture all strong k-metrics). Using these definitions, we prove analogs of a number of fundamental results about embedding finite metric spaces including Fréchet embedding (isometric embedding into 𝓁_∞) and isometric embedding of all tree metrics into 𝓁₁. We also study relationships between families of (strong) k-metrics, and show that natural quantities, like simplex volume, are strong k-metrics. 

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3. VLT/SPHERE imaging survey of the largest main-belt asteroids: Final results and synthesis

   https://doi.org/10.1051/0004-6361/202141781  

   Vernazza, P.; Ferrais, M.; Jorda, L.; Hanuš, J.; Carry, B.; Marsset, M.; Brož, M.; Fetick, R.; Viikinkoski, M.; Marchis, F.; et al (October 2021, Astronomy & Astrophysics)

   Context. Until recently, the 3D shape, and therefore density (when combining the volume estimate with available mass estimates), and surface topography of the vast majority of the largest ( D   ≥ 100 km) main-belt asteroids have remained poorly constrained. The improved capabilities of the SPHERE/ZIMPOL instrument have opened new doors into ground-based asteroid exploration. Aims. To constrain the formation and evolution of a representative sample of large asteroids, we conducted a high-angular-resolution imaging survey of 42 large main-belt asteroids with VLT/SPHERE/ZIMPOL. Our asteroid sample comprises 39 bodies with D   ≥ 100 km and in particular most D   ≥ 200 km main-belt asteroids (20/23). Furthermore, it nicely reflects the compositional diversity present in the main belt as the sampled bodies belong to the following taxonomic classes: A, B, C, Ch/Cgh, E/M/X, K, P/T, S, and V. Methods. The SPHERE/ZIMPOL images were first used to reconstruct the 3D shape of all targets with both the ADAM and MPCD reconstruction methods. We subsequently performed a detailed shape analysis and constrained the density of each target using available mass estimates including our own mass estimates in the case of multiple systems. Results. The analysis of the reconstructed shapes allowed us to identify two families of objects as a function of their diameters, namely “spherical” and “elongated” bodies. A difference in rotation period appears to be the main origin of this bimodality. In addition, all but one object (216 Kleopatra) are located along the Maclaurin sequence with large volatile-rich bodies being the closest to the latter. Our results further reveal that the primaries of most multiple systems possess a rotation period of shorter than 6 h and an elongated shape ( c ∕ a ≤ 0.65). Densities in our sample range from ~1.3 g cm −3 (87 Sylvia) to ~4.3 g cm −3 (22 Kalliope). Furthermore, the density distribution appears to be strongly bimodal with volatile-poor ( ρ ≥ 2.7 g cm −3 ) and volatile-rich ( ρ ≤ 2.2 g cm −3 ) bodies. Finally, our survey along with previous observations provides evidence in support of the possibility that some C-complex bodies could be intrinsically related to IDP-like P- and D-type asteroids, representing different layers of a same body (C: core; P/D: outer shell). We therefore propose that P/ D-types and some C-types may have the same origin in the primordial trans-Neptunian disk. 

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4. Search for CP violation in $$ {D}_{(s)}^{+}\to {h}^{+}{\pi}^0 $$ and $$ {D}_{(s)}^{+}\to {h}^{+}\eta $$ decays

   https://doi.org/10.1007/JHEP06(2021)019  

   Aaij, R.; Abellán Beteta, C.; Ackernley, T.; Adeva, B.; Adinolfi, M.; Afsharnia, H.; Aidala, C. A.; Aiola, S.; Ajaltouni, Z.; Akar, S.; et al (June 2021, Journal of High Energy Physics)

   A bstract Searches for CP violation in the two-body decays $$ {D}_{(s)}^{+}\to {h}^{+}{\pi}^0 $$ D s + → h + π 0 and $$ {D}_{(s)}^{+}\to {h}^{+}\eta $$ D s + → h + η (where h + denotes a π + or K + meson) are performed using pp collision data collected by the LHCb experiment corresponding to either 9 fb − 1 or 6 fb − 1 of integrated luminosity. The π 0 and η mesons are reconstructed using the e + e − γ final state, which can proceed as three-body decays π 0 → e + e − γ and η → e + e − γ , or via the two-body decays π 0 → γγ and η → γγ followed by a photon conversion. The measurements are made relative to the control modes $$ {D}_{(s)}^{+}\to {K}_{\mathrm{S}}^0{h}^{+} $$ D s + → K S 0 h + to cancel the production and detection asymmetries. The CP asymmetries are measured to be $$ {\displaystyle \begin{array}{c}{\mathcal{A}}_{CP}\left({D}^{+}\to {\pi}^{+}{\pi}^0\right)=\left(-1.3\pm 0.9\pm 0.6\right)\%,\\ {}{\mathcal{A}}_{CP}\left({D}^{+}\to {K}^{+}{\pi}^0\right)=\left(-3.2\pm 4.7\pm 2.1\right)\%,\\ {}\begin{array}{c}{\mathcal{A}}_{CP}\left({D}^{+}\to {\pi}^{+}\eta \right)=\left(-0.2\pm 0.8\pm 0.4\right)\%,\\ {}{\mathcal{A}}_{CP}\left({D}^{+}\to {K}^{+}\eta \right)=\left(-6\pm 10\pm 4\right)\%,\\ {}\begin{array}{c}{\mathcal{A}}_{CP}\left({D}_s^{+}\to {K}^{+}{\pi}^0\right)=\left(-0.8\pm 3.9\pm 1.2\right)\%,\\ {}\begin{array}{c}{\mathcal{A}}_{CP}\left({D}_s^{+}\to {\pi}^{+}\eta \right)=\left(0.8\pm 0.7\pm 0.5\right)\%,\\ {}{\mathcal{A}}_{CP}\left({D}_s^{+}\to {K}^{+}\eta \right)=\left(0.9\pm 3.7\pm 1.1\right)\%,\end{array}\end{array}\end{array}\end{array}} $$ A CP D + → π + π 0 = − 1.3 ± 0.9 ± 0.6 % , A CP D + → K + π 0 = − 3.2 ± 4.7 ± 2.1 % , A CP D + → π + η = − 0.2 ± 0.8 ± 0.4 % , A CP D + → K + η = − 6 ± 10 ± 4 % , A CP D s + → K + π 0 = − 0.8 ± 3.9 ± 1.2 % , A CP D s + → π + η = 0.8 ± 0.7 ± 0.5 % , A CP D s + → K + η = 0.9 ± 3.7 ± 1.1 % , where the first uncertainties are statistical and the second systematic. These results are consistent with no CP violation and mostly constitute the most precise measurements of $$ {\mathcal{A}}_{CP} $$ A CP in these decay modes to date. 

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5. Counting Simplices in Hypergraph Streams

   Chakrabarti, Amit; Haris, Themistoklis (October 2022, Leibniz international proceedings in informatics)

   We consider the problem of space-efficiently estimating the number of simplices in a hypergraph stream. This is the most natural hypergraph generalization of the highly-studied problem of estimating the number of triangles in a graph stream. Our input is a k-uniform hypergraph H with n vertices and m hyperedges, each hyperedge being a k-sized subset of vertices. A k-simplex in H is a subhypergraph on k+1 vertices X such that all k+1 possible hyperedges among X exist in H. The goal is to process the hyperedges of H, which arrive in an arbitrary order as a data stream, and compute a good estimate of T_k(H), the number of k-simplices in H. We design a suite of algorithms for this problem. As with triangle-counting in graphs (which is the special case k = 2), sublinear space is achievable but only under a promise of the form T_k(H) ≥ T. Under such a promise, our algorithms use at most four passes and together imply a space bound of O(ε^{-2} log δ^{-1} polylog n ⋅ min{(m^{1+1/k})/T, m/(T^{2/(k+1)})}) for each fixed k ≥ 3, in order to guarantee an estimate within (1±ε)T_k(H) with probability ≥ 1-δ. We also give a simpler 1-pass algorithm that achieves O(ε^{-2} log δ^{-1} log n⋅ (m/T) (Δ_E + Δ_V^{1-1/k})) space, where Δ_E (respectively, Δ_V) denotes the maximum number of k-simplices that share a hyperedge (respectively, a vertex), which generalizes a previous result for the k = 2 case. We complement these algorithmic results with space lower bounds of the form Ω(ε^{-2}), Ω(m^{1+1/k}/T), Ω(m/T^{1-1/k}) and Ω(mΔ_V^{1/k}/T) for multi-pass algorithms and Ω(mΔ_E/T) for 1-pass algorithms, which show that some of the dependencies on parameters in our upper bounds are nearly tight. Our techniques extend and generalize several different ideas previously developed for triangle counting in graphs, using appropriate innovations to handle the more complicated combinatorics of hypergraphs. 

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