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1. 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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2. Polynomial Bounds for Chromatic Number. IV: A Near-polynomial Bound for Excluding the Five-vertex Path

   https://doi.org/10.1007/s00493-023-00015-w  

   Scott, Alex; Seymour, Paul; Spirkl, Sophie (October 2023, Combinatorica)

   Abstract A graphGisH-freeif it has no induced subgraph isomorphic toH. We prove that a$$P_5$$ $P_5$-free graph with clique number$$\omega \ge 3$$ $\omega \ge 3$has chromatic number at most$$\omega ^{\log _2(\omega )}$$ ${\omega }^{{log}_2\left(\omega \right)}$. The best previous result was an exponential upper bound$$(5/27)3^{\omega }$$ $(5/27)3^{\omega }$, due to Esperet, Lemoine, Maffray, and Morel. A polynomial bound would imply that the celebrated Erdős-Hajnal conjecture holds for$$P_5$$ $P_5$, which is the smallest open case. Thus, there is great interest in whether there is a polynomial bound for$$P_5$$ $P_5$-free graphs, and our result is an attempt to approach that. 

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3. Study of baryon number transport dynamics and strangeness conservation effects using Ω-hadron correlations

   https://doi.org/10.1007/s41365-024-01464-8  

   Dong, Wei-Jie; Yu, Xiao-Zhou; Ping, Si-Yuan; Wu, Xia-Tong; Wang, Gang; Huang, Huan-Zhong; Lin, Zi-Wei (July 2024, Nuclear Science and Techniques)

   Abstract In nuclear collisions at RHIC energies, an excess of$$\Omega$$ $\Omega$hyperons over$$\bar{\Omega }$$ $\overline{\Omega }$is observed, indicating that$$\Omega$$ $\Omega$has a net baryon number despitesand$$\bar{s}$$ $\overline{s}$quarks being produced in pairs. The baryon number in$$\Omega$$ $\Omega$may have been transported from the incident nuclei and/or produced in the baryon-pair production of$$\Omega$$ $\Omega$with other types of anti-hyperons such as$$\bar{\Xi }$$ $\overline{\Xi }$. To investigate these two scenarios, we propose to measure the correlations between$$\Omega$$ $\Omega$andKand between$$\Omega$$ $\Omega$and anti-hyperons. We use two versions, the default and string-melting, of a multiphase transport (AMPT) model to illustrate the method for measuring the correlation and to demonstrate the general shape of the correlation. We present the$$\Omega$$ $\Omega$-hadron correlations from simulated Au+Au collisions at$$\sqrt{s_\text{NN}} = 7.7$$ $\sqrt{s_{\text{NN}}}=7.7$and$$14.6 \ \textrm{GeV}$$ $14.6\text{GeV}$and discuss the dependence on the collision energy and on the hadronization scheme in these two AMPT versions. These correlations can be used to explore the mechanism of baryon number transport and the effects of baryon number and strangeness conservation on nuclear collisions. 

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4. Measurement of the branching fractions of $$ \overline{B} $$ → D(*)K−$$ {K}_{(S)}^{\left(\ast \right)0} $$ and $$ \overline{B} $$ → D(*)$$ {D}_s^{-} $$ decays at Belle II

   https://doi.org/10.1007/JHEP08(2024)206  

   Adachi, I; Aggarwal, L; Aihara, H; Akopov, N; Aloisio, A; Althubiti, N; Anh_Ky, N; Asner, D M; Atmacan, H; Aushev, T; et al (August 2024, Journal of High Energy Physics)

   Abstract We present measurements of the branching fractions of eight$$ {\overline{B}}^0 $$ ${\overline{B}}^0$→D^(*)+K⁻$$ {K}_{(S)}^{\left(\ast \right)0} $$ $K_{\left(S\right)}^{\left(\ast \right)0}$,B⁻→D^(*)0K⁻$$ {K}_{(S)}^{\left(\ast \right)0} $$ $K_{\left(S\right)}^{\left(\ast \right)0}$decay channels. The results are based on data from SuperKEKB electron-positron collisions at the Υ(4S) resonance collected with the Belle II detector, corresponding to an integrated luminosity of 362 fb⁻¹. The event yields are extracted from fits to the distributions of the difference between expected and observedBmeson energy, and are efficiency-corrected as a function ofm(K⁻$$ {K}_{(S)}^{\left(\ast \right)0} $$ $K_{\left(S\right)}^{\left(\ast \right)0}$) andm(D^(*)$$ {K}_{(S)}^{\left(\ast \right)0} $$ $K_{\left(S\right)}^{\left(\ast \right)0}$) in order to avoid dependence on the decay model. These results include the first observation of$$ {\overline{B}}^0 $$ ${\overline{B}}^0$→D⁺K⁻$$ {K}_S^0 $$ $K_S^0$,B⁻→D*⁰K⁻$$ {K}_S^0 $$ $K_S^0$, and$$ {\overline{B}}^0 $$ ${\overline{B}}^0$→D*⁺K⁻$$ {K}_S^0 $$ $K_S^0$decays and a significant improvement in the precision of the other channels compared to previous measurements. The helicity-angle distributions and the invariant mass distributions of theK⁻$$ {K}_{(S)}^{\left(\ast \right)0} $$ $K_{\left(S\right)}^{\left(\ast \right)0}$systems are compatible with quasi-two-body decays via a resonant transition with spin-parityJ^P= 1⁻for theK⁻$$ {K}_S^0 $$ $K_S^0$systems andJ^P= 1⁺for theK⁻K*⁰systems. We also present measurements of the branching fractions of four$$ {\overline{B}}^0 $$ ${\overline{B}}^0$→D^(*)+$$ {D}_s^{-} $$ $D_s^{-}$,B⁻→D^(*)0$$ {D}_s^{-} $$ $D_s^{-}$decay channels with a precision compatible to the current world averages. 

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5. Green function estimates on complements of low-dimensional uniformly rectifiable sets

   https://doi.org/10.1007/s00208-022-02379-8  

   David, Guy; Feneuil, Joseph; Mayboroda, Svitlana (March 2022, Mathematische Annalen)

   Abstract It has been recently established in David and Mayboroda (Approximation of green functions and domains with uniformly rectifiable boundaries of all dimensions.arXiv:2010.09793) that on uniformly rectifiable sets the Green function is almost affine in the weak sense, and moreover, in some scenarios such Green function estimates are equivalent to the uniform rectifiability of a set. The present paper tackles a strong analogue of these results, starting with the “flagship degenerate operators on sets with lower dimensional boundaries. We consider the elliptic operators$$L_{\beta ,\gamma } =- {\text {div}}D^{d+1+\gamma -n} \nabla $$ $L_{\beta ,\gamma }=-\text{div}{D}^{d+1+\gamma -n}\nabla$associated to a domain$$\Omega \subset {\mathbb {R}}^n$$ $\Omega \subset R^n$with a uniformly rectifiable boundary$$\Gamma $$ $\Gamma$of dimension$$d < n-1$$ $d<n-1$, the now usual distance to the boundary$$D = D_\beta $$ $D=D_{\beta }$given by$$D_\beta (X)^{-\beta } = \int _{\Gamma } |X-y|^{-d-\beta } d\sigma (y)$$ $D_{\beta }{\left(X\right)}^{-\beta }={\int }_{\Gamma }{|X-y|}^{-d-\beta }d\sigma \left(y\right)$for$$X \in \Omega $$ $X\in \Omega$, where$$\beta >0$$ $\beta >0$and$$\gamma \in (-1,1)$$ $\gamma \in (-1,1)$. In this paper we show that the Green functionGfor$$L_{\beta ,\gamma }$$ $L_{\beta ,\gamma }$, with pole at infinity, is well approximated by multiples of$$D^{1-\gamma }$$ $D^{1-\gamma }$, in the sense that the function$$\big | D\nabla \big (\ln \big ( \frac{G}{D^{1-\gamma }} \big )\big )\big |^2$$ $|D\nabla (ln(\frac{G}{D^{1-\gamma }})){|}^2$satisfies a Carleson measure estimate on$$\Omega $$ $\Omega$. We underline that the strong and the weak results are different in nature and, of course, at the level of the proofs: the latter extensively used compactness arguments, while the present paper relies on some intricate integration by parts and the properties of the “magical distance function from David et al. (Duke Math J, to appear). 

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