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1. Finite-dimensional boundary uniform stabilization of the Boussinesq system in Besov spaces by critical use of Carleman estimate-based inverse theory

   https://doi.org/10.1515/jiip-2020-0132  

   Lasiecka, Irena; Priyasad, Buddhika; Triggiani, Roberto (April 2021, Journal of Inverse and Ill-posed Problems)

   null (Ed.)

   Abstract We consider the 𝑑-dimensional Boussinesq system defined on a sufficiently smooth bounded domain and subject to a pair { v , u } \{v,\boldsymbol{u}\} of controls localized on { Γ ~ , ω } \{\widetilde{\Gamma},\omega\} .Here, 𝑣 is a scalar Dirichlet boundary control for the thermal equation, acting on an arbitrarily small connected portion Γ ~ \widetilde{\Gamma} of the boundary Γ = ∂ ⁡ Ω \Gamma=\partial\Omega .Instead, 𝒖 is a 𝑑-dimensional internal control for the fluid equation acting on an arbitrarily small collar 𝜔 supported by Γ ~ \widetilde{\Gamma} .The initial conditions for both fluid and heat equations are taken of low regularity.We then seek to uniformly stabilize such Boussinesq system in the vicinity of an unstable equilibrium pair, in the critical setting of correspondingly low regularity spaces, by means of an explicitly constructed, finite-dimensional feedback control pair { v , u } \{v,\boldsymbol{u}\} localized on { Γ ~ , ω } \{\widetilde{\Gamma},\omega\} .In addition, they will be minimal in number and of reduced dimension; more precisely, 𝒖 will be of dimension ( d - 1 ) (d-1) , to include necessarily its 𝑑-th component, and 𝑣 will be of dimension 1.The resulting space of well-posedness and stabilization is a suitable, tight Besov space for the fluid velocity component (close to L 3 ⁢ ( Ω ) \boldsymbol{L}^{3}(\Omega) for d = 3 d=3 ) and a corresponding Besov space for the thermal component, q > d q>d .Unique continuation inverse theorems for suitably over-determined adjoint static problems play a critical role in the constructive solution.Their proof rests on Carleman-type estimates, a topic pioneered by M. V. Klibanov since the early 80s. 

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2. The two-phase problem for harmonic measure in VMO and the chord-arc condition

   https://doi.org/10.1090/btran/197  

   Tolsa, Xavier; Toro, Tatiana (November 2024, Transactions of the American Mathematical Society, Series B)

   Let ${\mathrm{\Omega <\#comment/>}}^{+}\subset <\#comment/>{\mathbb{R}}^{n+1}$be a bounded $\mathrm{\delta <\#comment/>}$-Reifenberg flat domain, with $\mathrm{\delta <\#comment/>}>0$small enough, possibly with locally infinite surface measure. Assume also that ${\mathrm{\Omega <\#comment/>}}^{-<\#comment/>}={\mathbb{R}}^{n+1}\setminus <\#comment/>\stackrel{¯<\#comment/>}{{\mathrm{\Omega <\#comment/>}}^{+}}$is an NTA (non-tangentially accessible) domain as well and denote by ${\mathrm{\omega <\#comment/>}}^{+}$and ${\mathrm{\omega <\#comment/>}}^{-<\#comment/>}$the respective harmonic measures of ${\mathrm{\Omega <\#comment/>}}^{+}$and ${\mathrm{\Omega <\#comment/>}}^{-<\#comment/>}$with poles $p^{\pm <\#comment/>}\in <\#comment/>{\mathrm{\Omega <\#comment/>}}^{\pm <\#comment/>}$. In this paper we show that the condition that $\log <\#comment/>\frac{d{\mathrm{\omega <\#comment/>}}^{-<\#comment/>}}{d{\mathrm{\omega <\#comment/>}}^{+}}\in <\#comment/>\mathrm{VMO}<\#comment/>\left({\mathrm{\omega <\#comment/>}}^{+}\right)$is equivalent to ${\mathrm{\Omega <\#comment/>}}^{+}$being a chord-arc domain with inner unit normal belonging to $\mathrm{VMO}<\#comment/>\left({\mathcal{H}}^n{|}_{\mathrm{\partial <\#comment/>}{\mathrm{\Omega <\#comment/>}}^{+}}\right)$. 

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3. Evidence for the singly Cabibbo-suppressed decay $$ {\Omega}_c^0\to {\Xi}^{-}{\pi}^{+} $$ and search for $$ {\Omega}_c^0\to {\Xi}^{-}{K}^{+} $$ and Ω−K+ decays at Belle

   https://doi.org/10.1007/JHEP01(2023)055  

   Han, X.; Jia, S.; Yuan, L.; Shen, C. P.; Adachi, I.; Ahn, J. K.; Aihara, H.; Asner, D. M.; Aushev, T.; Ayad, R.; et al (January 2023, Journal of High Energy Physics)

   A bstract Using a data sample of 980 fb − 1 collected with the Belle detector at the KEKB asymmetric-energy e + e − collider, we study for the first time the singly Cabibbo-suppressed decays $$ {\Omega}_c^0\to {\Xi}^{-}{\pi}^{+} $$ Ω c 0 → Ξ − π + and Ω − K + and the doubly Cabibbo-suppressed decay $$ {\Omega}_c^0\to {\Xi}^{-}{K}^{+} $$ Ω c 0 → Ξ − K + . Evidence for an $$ {\Omega}_c^0 $$ Ω c 0 signal in the $$ {\Omega}_c^0 $$ Ω c 0 → Ξ − π + mode is reported with a significance of 4 . 5 σ including systematic uncertainties. The ratio of branching fractions to the normalization mode $$ {\Omega}_c^0 $$ Ω c 0 → Ω − π + is measured to be $$ \mathcal{B}\left({\Omega}_c^0\to {\Xi}^{-}{\pi}^{+}\right)/\mathcal{B}\left({\Omega}_c^0\to {\Omega}^{-}{\pi}^{+}\right)=0.253\pm 0.052\left(\textrm{stat}.\right)\pm 0.030\left(\textrm{syst}.\right). $$ B Ω c 0 → Ξ − π + / B Ω c 0 → Ω − π + = 0.253 ± 0.052 stat . ± 0.030 syst . . No significant signals of $$ {\Omega}_c^0\to {\Xi}^{-}{K}^{+} $$ Ω c 0 → Ξ − K + and Ω − K + modes are found. The upper limits at 90% confidence level on ratios of branching fractions are determined to be $$ \mathcal{B}\left({\Omega}_c^0\to {\Xi}^{-}{K}^{+}\right)/\mathcal{B}\left({\Omega}_c^0\to {\Omega}^{-}{\pi}^{+}\right)<0.070 $$ B Ω c 0 → Ξ − K + / B Ω c 0 → Ω − π + < 0.070 and $$ \mathcal{B}\left({\Omega}_c^0\to {\Omega}^{-}{K}^{+}\right)/\mathcal{B}\left({\Omega}_c^0\to {\Omega}^{-}{\pi}^{+}\right)<0.29. $$ B Ω c 0 → Ω − K + / B Ω c 0 → Ω − π + < 0.29 . 

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4. Galvin’s conjecture and weakly precipitous ideals

   https://doi.org/10.1090/proc/17299  

   Eisworth, Todd (June 2025, Proceedings of the American Mathematical Society)

   We investigate a combinatorial game on ${\mathrm{\omega <\#comment/>}}_1$and show that mild large cardinal assumptions imply that every normal ideal on ${\mathrm{\omega <\#comment/>}}_1$satisfies a weak version of precipitousness. As an application, we show that the Raghavan-Todorčević proof of a longstanding conjecture of Galvin (done assuming the existence of a Woodin cardinal) can be pushed through under much weaker large cardinal assumptions [Forum Math. Pi 8 (2020), p. e15]. 

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5. Hyperbolic families and coloring graphs on surfaces

   https://doi.org/https://doi.org/10.1090/btran/26  

   Postle, Luke; Thomas, Robin (December 2018, Transactions of the American Mathematical Society. Series B)

   We develop a theory of linear isoperimetric inequalities for graphs on surfaces and apply it to coloring problems, as follows. Let $ G$ be a graph embedded in a fixed surface $$ \Sigma $$ of genus $ g$ and let $$ L=(L(v):v\in V(G))$$ be a collection of lists such that either each list has size at least five, or each list has size at least four and $ G$ is triangle-free, or each list has size at least three and $ G$ has no cycle of length four or less. An $ L$-coloring of $ G$ is a mapping $$ \phi $$ with domain $ V(G)$ such that $$ \phi (v)\in L(v)$$ for every $$ v\in V(G)$$ and $$ \phi (v)\ne \phi (u)$$ for every pair of adjacent vertices $$ u,v\in V(G)$$. We prove if every non-null-homotopic cycle in $ G$ has length $$ \Omega (\log g)$$, then $ G$ has an $ L$-coloring, if $ G$ does not have an $ L$-coloring, but every proper subgraph does (``$ L$-critical graph''), then $$ \vert V(G)\vert=O(g)$$, if every non-null-homotopic cycle in $ G$ has length $$ \Omega (g)$$, and a set $$ X\subseteq V(G)$$ of vertices that are pairwise at distance $$ \Omega (1)$$ is precolored from the corresponding lists, then the precoloring extends to an $ L$-coloring of $ G$, if every non-null-homotopic cycle in $ G$ has length $$ \Omega (g)$$, and the graph $ G$ is allowed to have crossings, but every two crossings are at distance $$ \Omega (1)$$, then $ G$ has an $ L$-coloring, if $ G$ has at least one $ L$-coloring, then it has at least $$ 2^{\Omega (\vert V(G)\vert)}$$ distinct $ L$-colorings. We show that the above assertions are consequences of certain isoperimetric inequalities satisfied by $ L$-critical graphs, and we study the structure of families of embedded graphs that satisfy those inequalities. It follows that the above assertions hold for other coloring problems, as long as the corresponding critical graphs satisfy the same inequalities. 

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