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1. CscK metrics near the canonical class

   https://doi.org/10.1515/crelle-2024-0096  

   Guo, Bin; Jian, Wangjian; Shi, Yalong; Song, Jian (January 2025, Journal für die reine und angewandte Mathematik (Crelles Journal))

   Abstract Let 𝑋 be a Kähler manifold with semiample canonical bundle $K_X$K_{X}.It is proved in [W. Jian, Y. Shi and J. Song, A remark on constant scalar curvature Kähler metrics on minimal models,Proc. Amer. Math. Soc.147(2019), 8, 3507–3513] that, for any Kähler class 𝛾, there exists $\delta >0$\delta>0such that, for all $t\in (0,\delta )$t\in(0,\delta), there exists a unique cscK metric $g_t$g_{t}in $K_X+t\gamma$K_{X}+t\gamma.In this paper, we prove that ${\left\{(X,g_t)\right\}}_{t\in (0,\delta )}$\{(X,g_{t})\}_{t\in(0,\delta)}have uniformly bounded Kähler potentials, volume forms and diameters.As a consequence, these metric spaces are pre-compact in the Gromov–Hausdorff sense. 

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2. Global Well-Posedness for $$H^{-1}(\mathbb {R})$$ Perturbations of KdV with Exotic Spatial Asymptotics

   https://doi.org/10.1007/s00220-022-04522-7  

   Laurens, Thierry (November 2022, Communications in Mathematical Physics)

   Abstract Given a suitable solutionV(t, x) to the Korteweg–de Vries equation on the real line, we prove global well-posedness for initial data$$u(0,x) \in V(0,x) + H^{-1}(\mathbb {R})$$ $u(0,x)\in V(0,x)+H^{-1}\left(R\right)$. Our conditions onVdo include regularity but do not impose any assumptions on spatial asymptotics. We show that periodic profiles$$V(0,x)\in H^5(\mathbb {R}/\mathbb {Z})$$ $V(0,x)\in H^5(R/Z)$satisfy our hypotheses. In particular, we can treat localized perturbations of the much-studied periodic traveling wave solutions (cnoidal waves) of KdV. In the companion paper Laurens (Nonlinearity. 35(1):343–387, 2022.https://doi.org/10.1088/1361-6544/ac37f5) we show that smooth step-like initial data also satisfy our hypotheses. We employ the method of commuting flows introduced in Killip and Vişan (Ann. Math. (2) 190(1):249–305, 2019.https://doi.org/10.4007/annals.2019.190.1.4) where$$V\equiv 0$$ $V\equiv 0$. In that setting, it is known that$$H^{-1}(\mathbb {R})$$ $H^{-1}\left(R\right)$is sharp in the class of$$H^s(\mathbb {R})$$ $H^s\left(R\right)$spaces. 

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3. Violent Nonlinear Collapse in the Interior of Charged Hairy Black Holes

   https://doi.org/10.1007/s00205-024-02038-z  

   Van_de_Moortel, Maxime (October 2024, Archive for Rational Mechanics and Analysis)

   Abstract We construct a new one-parameter family, indexed by$$\epsilon $$ $ϵ$, of two-ended, spatially-homogeneous black hole interiors solving the Einstein–Maxwell–Klein–Gordon equations with a (possibly zero) cosmological constant$$\Lambda $$ $\Lambda$and bifurcating off a Reissner–Nordström-(dS/AdS) interior ($$\epsilon =0$$ $ϵ=0$). For all small$$\epsilon \ne 0$$ $ϵ\ne 0$, we prove that, although the black hole is charged, its terminal boundary is an everywhere-spacelikeKasner singularity foliated by spheres of zero radiusr. Moreover, smaller perturbations (i.e. smaller$$|\epsilon |$$ $\left|ϵ\right|$) aremore singular than larger ones, in the sense that the Hawking mass and the curvature blow up following a power law of the form$$r^{-O(\epsilon ^{-2})}$$ $r^{-O\left({ϵ}^{-2}\right)}$at the singularity$$\{r=0\}$$ $\{r=0\}$. This unusual property originates from a dynamical phenomenon—violent nonlinear collapse—caused by the almost formation of a Cauchy horizon to the past of the spacelike singularity$$\{r=0\}$$ $\{r=0\}$. This phenomenon was previously described numerically in the physics literature and referred to as “the collapse of the Einstein–Rosen bridge”. While we cover all values of$$\Lambda \in \mathbb {R}$$ $\Lambda \in R$, the case$$\Lambda <0$$ $\Lambda <0$is of particular significance to the AdS/CFT correspondence. Our result can also be viewed in general as a first step towards the understanding of the interior of hairy black holes. 

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4. Finding solutions with distinct variables to systems of linear equations over $$\mathbb {F}_p$$

   https://doi.org/10.1007/s00208-022-02391-y  

   Sauermann, Lisa (April 2022, Mathematische Annalen)

   Abstract Let us fix a primepand a homogeneous system ofmlinear equations$$a_{j,1}x_1+\dots +a_{j,k}x_k=0$$ $a_{j,1}{x}_1+\cdots +a_{j,k}{x}_k=0$for$$j=1,\dots ,m$$ $j=1,\cdots ,m$with coefficients$$a_{j,i}\in \mathbb {F}_p$$ $a_{j,i}\in F_p$. Suppose that$$k\ge 3m$$ $k\ge 3m$, that$$a_{j,1}+\dots +a_{j,k}=0$$ $a_{j,1}+\cdots +a_{j,k}=0$for$$j=1,\dots ,m$$ $j=1,\cdots ,m$and that every$$m\times m$$ $m\times m$minor of the$$m\times k$$ $m\times k$matrix$$(a_{j,i})_{j,i}$$ ${\left(a_{j,i}\right)}_{j,i}$is non-singular. Then we prove that for any (large)n, any subset$$A\subseteq \mathbb {F}_p^n$$ $A\subseteq F_p^n$of size$$|A|> C\cdot \Gamma ^n$$ $\left|A\right|>C·{\Gamma }^n$contains a solution$$(x_1,\dots ,x_k)\in A^k$$ $(x_1,\cdots ,x_k)\in A^k$to the given system of equations such that the vectors$$x_1,\dots ,x_k\in A$$ $x_1,\cdots ,x_k\in A$are all distinct. Here,Cand$$\Gamma $$ $\Gamma$are constants only depending onp,mandksuch that$$\Gamma $\Gamma <p$. The crucial point here is the condition for the vectors$$x_1,\dots ,x_k$$ $x_1,\cdots ,x_k$in the solution$$(x_1,\dots ,x_k)\in A^k$$ $(x_1,\cdots ,x_k)\in A^k$to be distinct. If we relax this condition and only demand that$$x_1,\dots ,x_k$$ $x_1,\cdots ,x_k$are not all equal, then the statement would follow easily from Tao’s slice rank polynomial method. However, handling the distinctness condition is much harder, and requires a new approach. While all previous combinatorial applications of the slice rank polynomial method have relied on the slice rank of diagonal tensors, we use a slice rank argument for a non-diagonal tensor in combination with combinatorial and probabilistic arguments. 

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5. Search for CP violation in $${{{\textrm{D}}}^{{0}}} \rightarrow {{\textrm{K}} _{\text {S}}^{{0}}} {{\textrm{K}} _{\text {S}}^{{0}}} $$ decays in proton–proton collisions at $$\sqrt{s} = 13\,\text {Te}\hspace{-.08em}\text {V} $$

   https://doi.org/10.1140/epjc/s10052-024-13244-0  

   Hayrapetyan, A; Tumasyan, A; Adam, W; Andrejkovic, J W; Bergauer, T; Chatterjee, S; Damanakis, K; Dragicevic, M; Hussain, P S; Jeitler, M; et al (December 2024, The European Physical Journal C)

   Abstract A search is reported for charge-parity$$CP$$ $\mathrm{CP}$violation in$${{{\textrm{D}}}^{{0}}} \rightarrow {{\textrm{K}} _{\text {S}}^{{0}}} {{\textrm{K}} _{\text {S}}^{{0}}} $$ ${\text{D}}^0\to {\text{K}}_{\text{S}}^0{\text{K}}_{\text{S}}^0$decays, using data collected in proton–proton collisions at$$\sqrt{s} = 13\,\text {Te}\hspace{-.08em}\text {V} $$ $\sqrt{s}=13\text{Te}\text{V}$recorded by the CMS experiment in 2018. The analysis uses a dedicated data set that corresponds to an integrated luminosity of 41.6$$\,\text {fb}^{-1}$$ ${\text{fb}}^{-1}$, which consists of about 10 billion events containing a pair of b hadrons, nearly all of which decay to charm hadrons. The flavor of the neutral D meson is determined by the pion charge in the reconstructed decays$${{{\textrm{D}}}^{{*+}}} \rightarrow {{{\textrm{D}}}^{{0}}} {{{\mathrm{\uppi }}}^{{+}}} $$ $\text{D}_{}^{\ast +}\to {\text{D}}^0{\pi }^{+}$and$${{{\textrm{D}}}^{{*-}}} \rightarrow {\overline{{\textrm{D}}}^{{0}}} {{{\mathrm{\uppi }}}^{{-}}} $$ $\text{D}_{}^{\ast -}\to {\overline{\text{D}}}^0{\pi }^{-}$. The$$CP$$ $\mathrm{CP}$asymmetry in$${{{\textrm{D}}}^{{0}}} \rightarrow {{\textrm{K}} _{\text {S}}^{{0}}} {{\textrm{K}} _{\text {S}}^{{0}}} $$ ${\text{D}}^0\to {\text{K}}_{\text{S}}^0{\text{K}}_{\text{S}}^0$is measured to be$$A_{CP} ({{\textrm{K}} _{\text {S}}^{{0}}} {{\textrm{K}} _{\text {S}}^{{0}}} ) = (6.2 \pm 3.0 \pm 0.2 \pm 0.8)\%$$ $A_{\mathrm{CP}}\left({\text{K}}_{\text{S}}^0{\text{K}}_{\text{S}}^0\right)=(6.2\pm 3.0\pm 0.2\pm 0.8)\%$, where the three uncertainties represent the statistical uncertainty, the systematic uncertainty, and the uncertainty in the measurement of the$$CP$$ $\mathrm{CP}$asymmetry in the$${{{\textrm{D}}}^{{0}}} \rightarrow {{\textrm{K}} _{\text {S}}^{{0}}} {{{\mathrm{\uppi }}}^{{+}}} {{{\mathrm{\uppi }}}^{{-}}} $$ ${\text{D}}^0\to {\text{K}}_{\text{S}}^0{\pi }^{+}{\pi }^{-}$decay. This is the first$$CP$$ $\mathrm{CP}$asymmetry measurement by CMS in the charm sector as well as the first to utilize a fully hadronic final state. 

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