More Like this

1. Almost Everywhere Behavior of Functions According to Partition Measures

   https://doi.org/10.1017/fms.2023.130  

   Chan, William ; Jackson, Stephen ; Trang, Nam ( January 2024 , Forum of Mathematics, Sigma)

   This paper will study almost everywhere behaviors of functions on partition spaces of cardinals possessing suitable partition properties. Almost everywhere continuity and monotonicity properties for functions on partition spaces will be established. These results will be applied to distinguish the cardinality of certain subsets of the power set of partition cardinals.

   The following summarizes the main results proved under suitable partition hypotheses.•

   If$\kappa $is a cardinal,$\epsilon < \kappa $,${\mathrm {cof}}(\epsilon ) = \omega $,$\kappa \rightarrow _* (\kappa )^{\epsilon \cdot \epsilon }_2$and$\Phi : [\kappa ]^\epsilon _* \rightarrow \mathrm {ON}$, then$\Phi $satisfies the almost everywhere short length continuity property: There is a club$C \subseteq \kappa $and a$\delta < \epsilon $so that for all$f,g \in [C]^\epsilon _*$, if$f \upharpoonright \delta = g \upharpoonright \delta $and$\sup (f) = \sup (g)$, then$\Phi (f) = \Phi (g)$.

   •

   If$\kappa $is a cardinal,$\epsilon $is countable,$\kappa \rightarrow _* (\kappa )^{\epsilon \cdot \epsilon }_2$holds and$\Phi : [\kappa ]^\epsilon _* \rightarrow \mathrm {ON}$, then$\Phi $satisfies the strong almost everywhere short length continuity property: There is a club$C \subseteq \kappa $and finitely many ordinals$\delta _0, ..., \delta _k \leq \epsilon $so that for all$f,g \in [C]^\epsilon _*$, if for all$0 \leq i \leq k$,$\sup (f \upharpoonright \delta _i) = \sup (g \upharpoonright \delta _i)$, then$\Phi (f) = \Phi (g)$.

   •

   If$\kappa $satisfies$\kappa \rightarrow _* (\kappa )^\kappa _2$,$\epsilon \leq \kappa $and$\Phi : [\kappa ]^\epsilon _* \rightarrow \mathrm {ON}$, then$\Phi $satisfies the almost everywhere monotonicity property: There is a club$C \subseteq \kappa $so that for all$f,g \in [C]^\epsilon _*$, if for all$\alpha < \epsilon $,$f(\alpha ) \leq g(\alpha )$, then$\Phi (f) \leq \Phi (g)$.

   •

   Suppose dependent choice ($\mathsf {DC}$),${\omega _1} \rightarrow _* ({\omega _1})^{\omega _1}_2$and the almost everywhere short length club uniformization principle for${\omega _1}$hold. Then every function$\Phi : [{\omega _1}]^{\omega _1}_* \rightarrow {\omega _1}$satisfies a finite continuity property with respect to closure points: Let$\mathfrak {C}_f$be the club of$\alpha < {\omega _1}$so that$\sup (f \upharpoonright \alpha ) = \alpha $. There is a club$C \subseteq {\omega _1}$and finitely many functions$\Upsilon _0, ..., \Upsilon _{n - 1} : [C]^{\omega _1}_* \rightarrow {\omega _1}$so that for all$f \in [C]^{\omega _1}_*$, for all$g \in [C]^{\omega _1}_*$, if$\mathfrak {C}_g = \mathfrak {C}_f$and for all$i < n$,$\sup (g \upharpoonright \Upsilon _i(f)) = \sup (f \upharpoonright \Upsilon _i(f))$, then$\Phi (g) = \Phi (f)$.

   •

   Suppose$\kappa $satisfies$\kappa \rightarrow _* (\kappa )^\epsilon _2$for all$\epsilon < \kappa $. For all$\chi < \kappa $,$[\kappa ]^{<\kappa }$does not inject into${}^\chi \mathrm {ON}$, the class of$\chi $-length sequences of ordinals, and therefore,$|[\kappa ]^\chi | < |[\kappa ]^{<\kappa }|$. As a consequence, under the axiom of determinacy$(\mathsf {AD})$, these two cardinality results hold when$\kappa $is one of the following weak or strong partition cardinals of determinacy:${\omega _1}$,$\omega _2$,$\boldsymbol {\delta }_n^1$(for all$1 \leq n < \omega $) and$\boldsymbol {\delta }^2_1$(assuming in addition$\mathsf {DC}_{\mathbb {R}}$).

    

   more » « less
2. Production of light-flavor hadrons in pp collisions at $$\sqrt{s}~=~7\text { and }\sqrt{s} = 13 \, \text { TeV} $$

   https://doi.org/10.1140/epjc/s10052-020-08690-5  

   Acharya, S. ; Adamová, D. ; Adler, A. ; Adolfsson, J. ; Aggarwal, M. M. ; Rinella, G. Aglieri ; Agnello, M. ; Agrawal, N. ; Ahammed, Z. ; Ahmad, S. ; et al ( March 2021 , The European Physical Journal C)

   null (Ed.)

   Abstract The production of $$\pi ^{\pm }$$ π ± , $$\mathrm{K}^{\pm }$$ K ± , $$\mathrm{K}^{0}_{S}$$ K S 0 , $$\mathrm{K}^{*}(892)^{0}$$ K ∗ ( 892 ) 0 , $$\mathrm{p}$$ p , $$\phi (1020)$$ ϕ ( 1020 ) , $$\Lambda $$ Λ , $$\Xi ^{-}$$ Ξ - , $$\Omega ^{-}$$ Ω - , and their antiparticles was measured in inelastic proton–proton (pp) collisions at a center-of-mass energy of $$\sqrt{s}$$ s = 13 TeV at midrapidity ( $$|y|<0.5$$ | y | < 0.5 ) as a function of transverse momentum ( $$p_{\mathrm{T}}$$ p T ) using the ALICE detector at the CERN LHC. Furthermore, the single-particle $$p_{\mathrm{T}}$$ p T distributions of $$\mathrm{K}^{0}_{S}$$ K S 0 , $$\Lambda $$ Λ , and $$\overline{\Lambda }$$ Λ ¯ in inelastic pp collisions at $$\sqrt{s} = 7$$ s = 7  TeV are reported here for the first time. The $$p_{\mathrm{T}}$$ p T distributions are studied at midrapidity within the transverse momentum range $$0\le p_{\mathrm{T}}\le 20$$ 0 ≤ p T ≤ 20 GeV/ c , depending on the particle species. The $$p_{\mathrm{T}}$$ p T spectra, integrated yields, and particle yield ratios are discussed as a function of collision energy and compared with measurements at lower $$\sqrt{s}$$ s and with results from various general-purpose QCD-inspired Monte Carlo models. A hardening of the spectra at high $$p_{\mathrm{T}}$$ p T with increasing collision energy is observed, which is similar for all particle species under study. The transverse mass and $$x_{\mathrm{T}}\equiv 2p_{\mathrm{T}}/\sqrt{s}$$ x T ≡ 2 p T / s scaling properties of hadron production are also studied. As the collision energy increases from $$\sqrt{s}$$ s = 7–13 TeV, the yields of non- and single-strange hadrons normalized to the pion yields remain approximately constant as a function of $$\sqrt{s}$$ s , while ratios for multi-strange hadrons indicate enhancements. The $$p_\mathrm{{T}}$$ p T -differential cross sections of $$\pi ^{\pm }$$ π ± , $$\mathrm {K}^{\pm }$$ K ± and $$\mathrm {p}$$ p ( $$\overline{\mathrm{p}}$$ p ¯ ) are compared with next-to-leading order perturbative QCD calculations, which are found to overestimate the cross sections for $$\pi ^{\pm }$$ π ± and $$\mathrm{p}$$ p ( $$\overline{\mathrm{p}}$$ p ¯ ) at high $$p_\mathrm{{T}}$$ p T . 

   more » « less
3. Phase mixing for solutions to 1D transport equation in a confining potential

   https://doi.org/10.3934/krm.2022002  

   Chaturvedi, Sanchit ; Luk, Jonathan ( January 2022 , Kinetic and Related Models)

   Consider the linear transport equation in 1D under an external confining potential \begin{document}$ \Phi $\end{document}:

   \begin{document}$ \begin{equation*} {\partial}_t f + v {\partial}_x f - {\partial}_x \Phi {\partial}_v f = 0. \end{equation*} $\end{document}

   For \begin{document}$ \Phi = \frac {x^2}2 + \frac { \varepsilon x^4}2 $\end{document} (with \begin{document}$ \varepsilon >0 $\end{document} small), we prove phase mixing and quantitative decay estimates for \begin{document}$ {\partial}_t \varphi : = - \Delta^{-1} \int_{ \mathbb{R}} {\partial}_t f \, \mathrm{d} v $\end{document}, with an inverse polynomial decay rate \begin{document}$ O({\langle} t{\rangle}^{-2}) $\end{document}. In the proof, we develop a commuting vector field approach, suitably adapted to this setting. We will explain why we hope this is relevant for the nonlinear stability of the zero solution for the Vlasov–Poisson system in \begin{document}$ 1 $\end{document}D under the external potential \begin{document}$ \Phi $\end{document}.

    

   more » « less
4. Multiplicity and rapidity dependence of $$\textrm{K}^{*}(\mathrm {{\textbf {892}}})^{0}$$ and $$\mathrm {\phi ({\textbf {1020}})}$$ production in p–Pb collisions at $$\sqrt{s_{\textrm{NN}}} =$$ 5.02 TeV

   https://doi.org/10.1140/epjc/s10052-023-11449-3  

   Acharya, S. ; Adamová, D. ; Adler, A. ; Aglieri Rinella, G. ; Agnello, M. ; Agrawal, N. ; Ahammed, Z. ; Ahmad, S. ; Ahn, S. U. ; Ahuja, I. ; et al ( June 2023 , The European Physical Journal C)

   Abstract The transverse-momentum $$(p_{\textrm{T}})$$ ( p T ) spectra of K $$^{*}(892)^{0}~$$ ∗ ( 892 ) 0 and $$\mathrm {\phi (1020)}~$$ ϕ ( 1020 ) measured with the ALICE detector up to $$p_{\textrm{T}} $$ p T  = 16 GeV/ c in the rapidity range $$-1.2< y < 0.3,$$ - 1.2 < y < 0.3 , in p–Pb collisions at the center-of-mass energy per nucleon–nucleon collision $$\sqrt{s_{\textrm{NN}}} = 5.02$$ s NN = 5.02  TeV are presented as a function of charged particle multiplicity and rapidity. The measured $$p_{\textrm{T}} $$ p T distributions show a dependence on both multiplicity and rapidity at low $$p_{\textrm{T}} $$ p T whereas no significant dependence is observed at high $$p_{\textrm{T}} $$ p T . A rapidity dependence is observed in the $$p_{\textrm{T}} $$ p T -integrated yield (d N /d y ), whereas the mean transverse momentum $$\left( \langle p_{\textrm{T}} \rangle \right) $$ ⟨ p T ⟩ shows a flat behavior as a function of rapidity. The rapidity asymmetry ( $$Y_{\textrm{asym}}$$ Y asym ) at low $$p_{\textrm{T}} $$ p T (< 5 GeV/ c ) is more significant for higher multiplicity classes. At high $$p_{\textrm{T}} $$ p T , no significant rapidity asymmetry is observed in any of the multiplicity classes. Both K $$^{*}(892)^{0}~$$ ∗ ( 892 ) 0 and $$\mathrm {\phi (1020)}~$$ ϕ ( 1020 ) show similar $$Y_{\textrm{asym}}$$ Y asym . The nuclear modification factor $$(Q_{\textrm{CP}})$$ ( Q CP ) as a function of $$p_{\textrm{T}} $$ p T shows a Cronin-like enhancement at intermediate $$p_{\textrm{T}} $$ p T , which is more prominent at higher rapidities (Pb-going direction) and in higher multiplicity classes. At high $$p_{\textrm{T}}$$ p T (> 5 GeV/ $$c$$ c ), the $$Q_{\textrm{CP}}$$ Q CP values are greater than unity and no significant rapidity dependence is observed. 

   more » « less
5. Global Gradient Estimate for a Divergence Problem and Its Application to the Homogenization of a Magnetic Suspension

   Dang, T ( April 2022 , Association for Women in Mathematics series)

   Español, M (Ed.)

   "This paper generalizes the results obtained by the authors in Dang et al. (SIAM J. Appl. Math. 81(6):2547--2568, 2021) concerning the homogenization of a non-dilute suspension of magnetic particles in a viscous flow. More specifically, in this paper, a restrictive assumption on the coefficients of the coupled equation, made in Dang et al. (SIAM J. Appl. Math. 81(6):2547--2568, 2021), that significantly narrowed the applicability of the homogenization results obtained is relaxed and a new regularity of the solution of the fine-scale problem is proven. In particular, we obtain a global L∞-bound for the gradient of the solution of the scalar equation −divax∕$\epsilon$∇$\phi$$\epsilon$(x)=f(x){\$}{\$}- {\backslash}operatorname {\{}{\{}{\backslash}mathrm {\{}div{\}}{\}}{\}} {\backslash}left [ {\backslash}mathbf {\{}a{\}} {\backslash}left ( x/{\backslash}varepsilon {\backslash}right ){\backslash}nabla {\backslash}varphi ^{\{}{\backslash}varepsilon {\}}(x) {\backslash}right ] = f(x){\$}{\$}, uniform with respect to microstructure scale parameter $\epsilon${\thinspace}≪{\thinspace}1 in a small interval (0, $\epsilon$0), where the coefficient a is only piecewise H{\"o}lder continuous. Thenceforth, this regularity is used in the derivation of the effective response of the given suspension discussed in Dang et al. (SIAM J. Appl. Math. 81(6):2547--2568, 2021)." 

   more » « less