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1. Curvature Sets Over Persistence Diagrams

   https://doi.org/10.1007/s00454-024-00634-0  

   Gómez, Mario; Mémoli, Facundo (April 2024, Discrete & Computational Geometry)

   Abstract We study a family of invariants of compact metric spaces that combines the Curvature Sets defined by Gromov in the 1980 s with Vietoris–Rips Persistent Homology. For given integers$$k\ge 0$$ $k\ge 0$and$$n\ge 1$$ $n\ge 1$we consider the dimensionkVietoris–Rips persistence diagrams ofallsubsets of a given metric space with cardinality at mostn. We call these invariantspersistence setsand denote them as$${\textbf{D}}_{n,k}^{\textrm{VR}}$$ $D_{n,k}^{\text{VR}}$. We first point out that this family encompasses the usual Vietoris–Rips diagrams. We then establish that (1) for certain range of values of the parametersnandk, computing these invariants is significantly more efficient than computing the usual Vietoris–Rips persistence diagrams, (2) these invariants have very good discriminating power and, in many cases, capture information that is imperceptible through standard Vietoris–Rips persistence diagrams, and (3) they enjoy stability properties analogous to those of the usual Vietoris–Rips persistence diagrams. We precisely characterize some of them in the case of spheres and surfaces with constant curvature using a generalization of Ptolemy’s inequality. We also identify a rich family of metric graphs for which$${\textbf{D}}_{4,1}^{\textrm{VR}}$$ $D_{4,1}^{\text{VR}}$fully recovers their homotopy type by studying split-metric decompositions. Along the way we prove some useful properties of Vietoris–Rips persistence diagrams using Mayer–Vietoris sequences. These yield a geometric algorithm for computing the Vietoris–Rips persistence diagram of a spaceXwith cardinality$$2k+2$$ $2k+2$with quadratic time complexity as opposed to the much higher cost incurred by the usual algebraic algorithms relying on matrix reduction. 

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2. Spin density matrix elements in exclusive $$\rho ^0$$ meson muoproduction

   https://doi.org/10.1140/epjc/s10052-023-11359-4  

   Alexeev, G D; Alexeev, M G; Alice, C; Amoroso, A; Andrieux, V; Anosov, V; Augsten, K; Augustyniak, W; Azevedo, C_D R; Badelek, B; et al (October 2023, The European Physical Journal C)

   Abstract We report on a measurement of Spin Density Matrix Elements (SDMEs) in hard exclusive$$\rho ^0$$ ${\rho }^0$meson muoproduction at COMPASS using 160 GeV/cpolarised$$ \mu ^{+}$$ ${\mu }^{+}$and$$ \mu ^{-}$$ ${\mu }^{-}$beams impinging on a liquid hydrogen target. The measurement covers the kinematic range 5.0 GeV/$$c^2$$ $c^2$$$< W<$$ $<W<$17.0 GeV/$$c^2$$ $c^2$, 1.0 (GeV/c)$$^2$$ ${}^2$$$< Q^2<$$ $<Q^2<$10.0 (GeV/c)$$^2$$ ${}^2$and 0.01 (GeV/c)$$^2$$ ${}^2$$$< p_{\textrm{T}}^2<$$ $<p_{\text{T}}^2<$0.5 (GeV/c)$$^2$$ ${}^2$. Here,Wdenotes the mass of the final hadronic system,$$Q^2$$ $Q^2$the virtuality of the exchanged photon, and$$p_{\textrm{T}}$$ $p_{\text{T}}$the transverse momentum of the$$\rho ^0$$ ${\rho }^0$meson with respect to the virtual-photon direction. The measured non-zero SDMEs for the transitions of transversely polarised virtual photons to longitudinally polarised vector mesons ($$\gamma ^*_T \rightarrow V^{ }_L$$ ${\gamma }_T^{\ast }\to V_L^{}$) indicate a violation ofs-channel helicity conservation. Additionally, we observe a dominant contribution of natural-parity-exchange transitions and a very small contribution of unnatural-parity-exchange transitions, which is compatible with zero within experimental uncertainties. The results provide important input for modelling Generalised Parton Distributions (GPDs). In particular, they may allow one to evaluate in a model-dependent way the role of parton helicity-flip GPDs in exclusive$$\rho ^0$$ ${\rho }^0$production. 

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3. Multipliers on bi-parameter Haar system Hardy spaces

   https://doi.org/10.1007/s00208-024-02887-9  

   Lechner, R.; Motakis, P.; Müller, P_F_X; Schlumprecht, Th (May 2024, Mathematische Annalen)

   Abstract Let$$(h_I)$$ $\left(h_I\right)$denote the standard Haar system on [0, 1], indexed by$$I\in \mathcal {D}$$ $I\in D$, the set of dyadic intervals and$$h_I\otimes h_J$$ $h_I\otimes h_J$denote the tensor product$$(s,t)\mapsto h_I(s) h_J(t)$$ $(s,t)\mapsto h_I\left(s\right)h_J\left(t\right)$,$$I,J\in \mathcal {D}$$ $I,J\in D$. We consider a class of two-parameter function spaces which are completions of the linear span$$\mathcal {V}(\delta ^2)$$ $V\left({\delta }^2\right)$of$$h_I\otimes h_J$$ $h_I\otimes h_J$,$$I,J\in \mathcal {D}$$ $I,J\in D$. This class contains all the spaces of the formX(Y), whereXandYare either the Lebesgue spaces$$L^p[0,1]$$ $L^p[0,1]$or the Hardy spaces$$H^p[0,1]$$ $H^p[0,1]$,$$1\le p < \infty $$ $1\le p<\infty$. We say that$$D:X(Y)\rightarrow X(Y)$$ $D:X\left(Y\right)\to X\left(Y\right)$is a Haar multiplier if$$D(h_I\otimes h_J) = d_{I,J} h_I\otimes h_J$$ $D(h_I\otimes h_J)=d_{I,J}{h}_I\otimes h_J$, where$$d_{I,J}\in \mathbb {R}$$ $d_{I,J}\in R$, and ask which more elementary operators factor throughD. A decisive role is played by theCapon projection$$\mathcal {C}:\mathcal {V}(\delta ^2)\rightarrow \mathcal {V}(\delta ^2)$$ $C:V\left({\delta }^2\right)\to V\left({\delta }^2\right)$given by$$\mathcal {C} h_I\otimes h_J = h_I\otimes h_J$$ $C{h}_I\otimes h_J=h_I\otimes h_J$if$$|I|\le |J|$$ $\left|I\right|\le \left|J\right|$, and$$\mathcal {C} h_I\otimes h_J = 0$$ $C{h}_I\otimes h_J=0$if$$|I| > |J|$$ $\left|I\right|>\left|J\right|$, as our main result highlights: Given any bounded Haar multiplier$$D:X(Y)\rightarrow X(Y)$$ $D:X\left(Y\right)\to X\left(Y\right)$, there exist$$\lambda ,\mu \in \mathbb {R}$$ $\lambda ,\mu \in R$such that$$\begin{aligned} \lambda \mathcal {C} + \mu ({{\,\textrm{Id}\,}}-\mathcal {C})\text { approximately 1-projectionally factors through }D, \end{aligned}$$ $\begin{array}{c}\lambda C+\mu (\text{Id}-C)\text{approximately 1-projectionally factors through}D,\end{array}$i.e., for all$$\eta > 0$$ $\eta >0$, there exist bounded operatorsA, Bso thatABis the identity operator$${{\,\textrm{Id}\,}}$$ $\text{Id}$,$$\Vert A\Vert \cdot \Vert B\Vert = 1$$ $\Vert A\Vert ·\Vert B\Vert =1$and$$\Vert \lambda \mathcal {C} + \mu ({{\,\textrm{Id}\,}}-\mathcal {C}) - ADB\Vert < \eta $$ $\Vert \lambda C+\mu (\text{Id}-C)-ADB\Vert <\eta$. Additionally, if$$\mathcal {C}$$ $C$is unbounded onX(Y), then$$\lambda = \mu $$ $\lambda =\mu$and then$${{\,\textrm{Id}\,}}$$ $\text{Id}$either factors throughDor$${{\,\textrm{Id}\,}}-D$$ $\text{Id}-D$. 

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4. 3D Mirror Symmetry for Instanton Moduli Spaces

   https://doi.org/10.1007/s00220-023-04831-5  

   Koroteev, Peter; Zeitlin, Anton M. (September 2023, Communications in Mathematical Physics)

   Abstract We prove that the Hilbert scheme ofkpoints on$${\mathbb {C}}^2$$ $C^2$($$\hbox {Hilb}^k[{\mathbb {C}}^2]$$ ${\text{Hilb}}^k\left[C^2\right]$) is self-dual under three-dimensional mirror symmetry using methods of geometry and integrability. Namely, we demonstrate that the corresponding quantum equivariant K-theory is invariant upon interchanging its Kähler and equivariant parameters as well as inverting the weight of the$${\mathbb {C}}^\times _\hbar $$ $C_{ħ}^{\times }$-action. First, we find a two-parameter family$$X_{k,l}$$ $X_{k,l}$of self-mirror quiver varieties of type A and study their quantum K-theory algebras. The desired quantum K-theory of$$\hbox {Hilb}^k[{\mathbb {C}}^2]$$ ${\text{Hilb}}^k\left[C^2\right]$is obtained via direct limit$$l\longrightarrow \infty $$ $l⟶\infty$and by imposing certain periodic boundary conditions on the quiver data. Throughout the proof, we employ the quantum/classical (q-Langlands) correspondence between XXZ Bethe Ansatz equations and spaces of twisted$$\hbar $$ $ħ$-opers. In the end, we propose the 3d mirror dual for the moduli spaces of torsion-free rank-Nsheaves on$${\mathbb {P}}^2$$ $P^2$with the help of a different (three-parametric) family of type A quiver varieties with known mirror dual. 

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5. VC-Dimension of Hyperplanes Over Finite Fields

   https://doi.org/10.1007/s00373-025-02909-6  

   Ascoli, Ruben; Betti, Livia; Cheigh, Justin; Iosevich, Alex; Jeong, Ryan; Liu, Xuyan; McDonald, Brian; Milgrim, Wyatt; Miller, Steven_J; Romero_Acosta, Francisco; et al (March 2025, Graphs and Combinatorics)

   Abstract Let$$\mathbb {F}_q^d$$ $F_q^d$be thed-dimensional vector space over the finite field withqelements. For a subset$$E\subseteq \mathbb {F}_q^d$$ $E\subseteq F_q^d$and a fixed nonzero$$t\in \mathbb {F}_q$$ $t\in F_q$, let$$\mathcal {H}_t(E)=\{h_y: y\in E\}$$ $H_t\left(E\right)=\{h_y:y\in E\}$, where$$h_y:E\rightarrow \{0,1\}$$ $h_y:E\to \{0,1\}$is the indicator function of the set$$\{x\in E: x\cdot y=t\}$$ $\{x\in E:x·y=t\}$. Two of the authors, with Maxwell Sun, showed in the case$$d=3$$ $d=3$that if$$|E|\ge Cq^{\frac{11}{4}}$$ $\left|E\right|\ge C{q}^{\frac{11}{4}}$andqis sufficiently large, then the VC-dimension of$$\mathcal {H}_t(E)$$ $H_t\left(E\right)$is 3. In this paper, we generalize the result to arbitrary dimension by showing that the VC-dimension of$$\mathcal {H}_t(E)$$ $H_t\left(E\right)$isdwhenever$$E\subseteq \mathbb {F}_q^d$$ $E\subseteq F_q^d$with$$|E|\ge C_d q^{d-\frac{1}{d-1}}$$ $\left|E\right|\ge C_d{q}^{d-\frac{1}{d-1}}$. 

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