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1. Restricted spaces of holomorphic sections vanishing along subvarieties

   https://doi.org/10.1007/s00209-025-03706-w  

   Coman, Dan; Marinescu, George; Nguyên, Viêt-Anh (May 2025, Mathematische Zeitschrift)

   Abstract LetXbe a compact normal complex space of dimensionnandLbe a holomorphic line bundle onX. Suppose that$$\Sigma =(\Sigma _1,\ldots ,\Sigma _\ell )$$ $\Sigma =({\Sigma }_1,\dots ,{\Sigma }_{\ell })$is an$$\ell $$ $\ell$-tuple of distinct irreducible proper analytic subsets ofX,$$\tau =(\tau _1,\ldots ,\tau _\ell )$$ $\tau =({\tau }_1,\dots ,{\tau }_{\ell })$is an$$\ell $$ $\ell$-tuple of positive real numbers, and let$$H^0_0(X,L^p)$$ $H_0^0(X,L^p)$be the space of holomorphic sections of$$L^p:=L^{\otimes p}$$ $L^p:=L^{\otimes p}$that vanish to order at least$$\tau _jp$$ ${\tau }_{j}p$along$$\Sigma _j$$ ${\Sigma }_j$,$$1\le j\le \ell $$ $1\le j\le \ell$. If$$Y\subset X$$ $Y\subset X$is an irreducible analytic subset of dimensionm, we consider the space$$H^0_0 (X|Y, L^p)$$ $H_0^0\left(X\right|Y,L^p)$of holomorphic sections of$$L^p|_Y$$ $L^p{|}_Y$that extend to global holomorphic sections in$$H^0_0(X,L^p)$$ $H_0^0(X,L^p)$. Assuming that the triplet$$(L,\Sigma ,\tau )$$ $(L,\Sigma ,\tau )$is big in the sense that$$\dim H^0_0(X,L^p)\sim p^n$$ $dim{H}_0^0(X,L^p)\sim p^n$, we give a general condition onYto ensure that$$\dim H^0_0(X|Y,L^p)\sim p^m$$ $dim{H}_0^0\left(X\right|Y,L^p)\sim p^m$. WhenLis endowed with a continuous Hermitian metric, we show that the Fubini-Study currents of the spaces$$H^0_0(X|Y,L^p)$$ $H_0^0\left(X\right|Y,L^p)$converge to a certain equilibrium current onY. We apply this to the study of the equidistribution of zeros inYof random holomorphic sections in$$H^0_0(X|Y,L^p)$$ $H_0^0\left(X\right|Y,L^p)$as$$p\rightarrow \infty $$ $p\to \infty$. 

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2. Measurement of non-prompt $${{\textrm{D}}^{0}}$$-meson elliptic flow in Pb–Pb collisions at $$\sqrt{s_{\textrm{NN}}} = 5.02$$ TeV

   https://doi.org/10.1140/epjc/s10052-023-12259-3  

   Acharya, S; Adamová, D; Aglieri_Rinella, G; Agnello, M; Agrawal, N; Ahammed, Z; Ahmad, S; Ahn, S U; Ahuja, I; Akindinov, A; et al (December 2023, The European Physical Journal C)

   Abstract The elliptic flow$$(v_2)$$ $\left(v_2\right)$of$${\textrm{D}}^{0}$$ ${\text{D}}^0$mesons from beauty-hadron decays (non-prompt$${\textrm{D}}^{0})$$ ${\text{D}}^0)$was measured in midcentral (30–50%) Pb–Pb collisions at a centre-of-mass energy per nucleon pair$$\sqrt{s_{\textrm{NN}}} = 5.02$$ $\sqrt{s_{\text{NN}}}=5.02$ TeV with the ALICE detector at the LHC. The$${\textrm{D}}^{0}$$ ${\text{D}}^0$mesons were reconstructed at midrapidity$$(|y|<0.8)$$ $\left(\right|y|<0.8)$from their hadronic decay$$\mathrm {D^0 \rightarrow K^-\uppi ^+}$$ $D^0\to K^{-}{\pi }^{+}$, in the transverse momentum interval$$2< p_{\textrm{T}} < 12$$ $2<p_{\text{T}}<12$ GeV/c. The result indicates a positive$$v_2$$ $v_2$for non-prompt$${{\textrm{D}}^{0}}$$ ${\text{D}}^0$mesons with a significance of 2.7$$\sigma $$ $\sigma$. The non-prompt$${{\textrm{D}}^{0}}$$ ${\text{D}}^0$-meson$$v_2$$ $v_2$is lower than that of prompt non-strange D mesons with 3.2$$\sigma $$ $\sigma$significance in$$2< p_\textrm{T} < 8~\textrm{GeV}/c$$ $2<p_{\text{T}}<8\text{GeV}/c$, and compatible with the$$v_2$$ $v_2$of beauty-decay electrons. Theoretical calculations of beauty-quark transport in a hydrodynamically expanding medium describe the measurement within uncertainties. 

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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. On a Traveling Salesman Problem for Points in the Unit Cube

   https://doi.org/10.1007/s00453-024-01257-w  

   Balogh, József; Clemen, Felix Christian; Dumitrescu, Adrian (September 2024, Algorithmica)

   Abstract LetXbe ann-element point set in thek-dimensional unit cube$$[0,1]^k$$ ${[0,1]}^k$where$$k \ge 2$$ $k\ge 2$. According to an old result of Bollobás and Meir (Oper Res Lett 11:19–21, 1992) , there exists a cycle (tour)$$x_1, x_2, \ldots , x_n$$ $x_1,x_2,\dots ,x_n$through thenpoints, such that$$\left( \sum _{i=1}^n |x_i - x_{i+1}|^k \right) ^{1/k} \le c_k$$ ${\left({\sum }_{i=1}^n,{|x_i-x_{i+1}|}^k\right)}^{1/k}\le c_k$, where$$|x-y|$$ $|x-y|$is the Euclidean distance betweenxandy, and$$c_k$$ $c_k$is an absolute constant that depends only onk, where$$x_{n+1} \equiv x_1$$ $x_{n+1}\equiv x_1$. From the other direction, for every$$k \ge 2$$ $k\ge 2$and$$n \ge 2$$ $n\ge 2$, there existnpoints in$$[0,1]^k$$ ${[0,1]}^k$, such that their shortest tour satisfies$$\left( \sum _{i=1}^n |x_i - x_{i+1}|^k \right) ^{1/k} = 2^{1/k} \cdot \sqrt{k}$$ ${\left({\sum }_{i=1}^n,{|x_i-x_{i+1}|}^k\right)}^{1/k}=2^{1/k}·\sqrt{k}$. For the plane, the best constant is$$c_2=2$$ $c_2=2$and this is the only exact value known. Bollobás and Meir showed that one can take$$c_k = 9 \left( \frac{2}{3} \right) ^{1/k} \cdot \sqrt{k}$$ $c_k=9{\left(\frac{2}{3}\right)}^{1/k}·\sqrt{k}$for every$$k \ge 3$$ $k\ge 3$and conjectured that the best constant is$$c_k = 2^{1/k} \cdot \sqrt{k}$$ $c_k=2^{1/k}·\sqrt{k}$, for every$$k \ge 2$$ $k\ge 2$. Here we significantly improve the upper bound and show that one can take$$c_k = 3 \sqrt{5} \left( \frac{2}{3} \right) ^{1/k} \cdot \sqrt{k}$$ $c_k=3\sqrt{5}{\left(\frac{2}{3}\right)}^{1/k}·\sqrt{k}$or$$c_k = 2.91 \sqrt{k} \ (1+o_k(1))$$ $c_k=2.91\sqrt{k}(1+o_k\left(1\right))$. Our bounds are constructive. We also show that$$c_3 \ge 2^{7/6}$$ $c_3\ge 2^{7/6}$, which disproves the conjecture for$$k=3$$ $k=3$. Connections to matching problems, power assignment problems, related problems, including algorithms, are discussed in this context. A slightly revised version of the Bollobás–Meir conjecture is proposed. 

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5. Measurement of multidifferential cross sections for dijet production in proton–proton collisions at $$\sqrt{s} = 13\,\text {Te}\hspace{-.08em}\text {V} $$

   https://doi.org/10.1140/epjc/s10052-024-13606-8  

   Hayrapetyan, A; Tumasyan, A; Adam, W; Andrejkovic, J W; Bergauer, T; Chatterjee, S; Damanakis, K; Dragicevic, M; Valle, A_Escalante Del; Hussain, P S; et al (January 2025, The European Physical Journal C)

   Abstract A measurement of the dijet production cross section is reported based on proton–proton collision data collected in 2016 at$$\sqrt{s}=13\,\text {Te}\hspace{-.08em}\text {V} $$ $\sqrt{s}=13\mathrm{Te}V$by the CMS experiment at the CERN LHC, corresponding to an integrated luminosity of up to 36.3$$\,\text {fb}^{-1}$$ ${\mathrm{fb}}^{-1}$. Jets are reconstructed with the anti-$$k_{\textrm{T}} $$ $k_{\text{T}}$algorithm for distance parameters of$$R=0.4$$ $R=0.4$and 0.8. Cross sections are measured double-differentially (2D) as a function of the largest absolute rapidity$$|y |_{\text {max}} $$ ${\left|y\right|}_{\max }$of the two jets with the highest transverse momenta$$p_{\textrm{T}}$$ $p_{\text{T}}$and their invariant mass$$m_{1,2} $$ $m_{1,2}$, and triple-differentially (3D) as a function of the rapidity separation$$y^{*} $$ $y^{\ast }$, the total boost$$y_{\text {b}} $$ $y_b$, and either$$m_{1,2} $$ $m_{1,2}$or the average$$p_{\textrm{T}}$$ $p_{\text{T}}$of the two jets. The cross sections are unfolded to correct for detector effects and are compared with fixed-order calculations derived at next-to-next-to-leading order in perturbative quantum chromodynamics. The impact of the measurements on the parton distribution functions and the strong coupling constant at the mass of the$${\text {Z}} $$ $Z$boson is investigated, yielding a value of$$\alpha _\textrm{S} (m_{{\text {Z}}}) =0.1179\pm 0.0019$$ ${\alpha }_{\text{S}}\left(m_Z\right)=0.1179\pm 0.0019$. 

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