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1. An inviscid free boundary fluid-wave model

   https://doi.org/10.1007/s00028-023-00888-w  

   Kukavica, Igor ; Tuffaha, Amjad ( May 2023 , Journal of Evolution Equations)

   We consider the local existence and uniqueness of solutions for a system consisting of an inviscid fluid with a free boundary, modeled by the Euler equations, in a domain enclosed by an elastic boundary, which evolves according to the wave equation. We derive a priori estimates for the local existence of solutions and also conclude the uniqueness. Both, existence and uniqueness are obtained under the assumption that the Euler data belongs to$$H^{r}$$$H^r$, where$$r>2.5$$$r>2.5$, which is known to be the borderline exponent for the Euler equations. Unlike the setting of the Euler equations with vacuum, the membrane is shown to stabilize the system in the sense that the Rayleigh–Taylor condition does not need to be assumed.

    

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2. On the connection between uniqueness from samples and stability in Gabor phase retrieval

   https://doi.org/10.1007/s43670-023-00079-1  

   Alaifari, Rima ; Bartolucci, Francesca ; Steinerberger, Stefan ; Wellershoff, Matthias ( January 2024 , Sampling Theory, Signal Processing, and Data Analysis)

   Gabor phase retrievalis the problem of reconstructing a signal from only the magnitudes of its Gabor transform. Previous findings suggest a possible link between unique solvability of the discrete problem (recovery from measurements on a lattice) and stability of the continuous problem (recovery from measurements on an open subset of$$\mathbb {R}^2$$$R^2$). In this paper, we close this gap by proving that such a link cannot be made. More precisely, we establish the existence of functions which break uniqueness from samples without affecting stability of the continuous problem. Furthermore, we prove the novel result that counterexamples to unique recovery from samples aredensein$$L^2(\mathbb {R})$$$L^2\left(R\right)$. Finally, we develop an intuitive argument on the connection between directions of instability in phase retrieval and certain Laplacian eigenfunctions associated to small eigenvalues.

    

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3. A positive proportion of cubic fields are not monogenic yet have no local obstruction to being so

   https://doi.org/10.1007/s00208-024-03054-w  

   Alpöge, Levent ; Bhargava, Manjul ; Shnidman, Ari ( December 2024 , Mathematische Annalen)

   We show that a positive proportion of cubic fields are not monogenic, despite having no local obstruction to being monogenic. Our proof involves the comparison of 2-descent and 3-descent in a certain family of Mordell curves$$E_k :y^2 = x^3 + k$$$E_k:y^2=x^3+k$. As a by-product of our methods, we show that, for every$$r \ge 0$$$r\ge 0$, a positive proportion of curves$$E_k$$$E_k$have Tate–Shafarevich group with 3-rank at least r.

    

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4. Spin density matrix elements in exclusive $$\rho ^0$$ meson muoproduction

   https://doi.org/10.1140/epjc/s10052-023-11359-4  

   Alexeev, G D ; Alexeev, M G ; Alice, C ; Amoroso, A ; Andrieux, V ; Anosov, V ; Augsten, K ; Augustyniak, W ; Azevedo, C_D R ; Badelek, B ; et al ( October 2023 , The European Physical Journal C)

   We report on a measurement of Spin Density Matrix Elements (SDMEs) in hard exclusive$$\rho ^0$$${\rho }^0$meson muoproduction at COMPASS using 160 GeV/cpolarised$$ \mu ^{+}$$${\mu }^{+}$and$$ \mu ^{-}$$${\mu }^{-}$beams impinging on a liquid hydrogen target. The measurement covers the kinematic range 5.0 GeV/$$c^2$$$c^2$$$< W<$$$<W<$17.0 GeV/$$c^2$$$c^2$, 1.0 (GeV/c)$$^2$$${}^2$$$< Q^2<$$$<Q^2<$10.0 (GeV/c)$$^2$$${}^2$and 0.01 (GeV/c)$$^2$$${}^2$$$< p_{\textrm{T}}^2<$$$<p_{\text{T}}^2<$0.5 (GeV/c)$$^2$$${}^2$. Here,Wdenotes the mass of the final hadronic system,$$Q^2$$$Q^2$the virtuality of the exchanged photon, and$$p_{\textrm{T}}$$$p_{\text{T}}$the transverse momentum of the$$\rho ^0$$${\rho }^0$meson with respect to the virtual-photon direction. The measured non-zero SDMEs for the transitions of transversely polarised virtual photons to longitudinally polarised vector mesons ($$\gamma ^*_T \rightarrow V^{ }_L$$${\gamma }_T^{\ast }\to V_L^{}$) indicate a violation ofs-channel helicity conservation. Additionally, we observe a dominant contribution of natural-parity-exchange transitions and a very small contribution of unnatural-parity-exchange transitions, which is compatible with zero within experimental uncertainties. The results provide important input for modelling Generalised Parton Distributions (GPDs). In particular, they may allow one to evaluate in a model-dependent way the role of parton helicity-flip GPDs in exclusive$$\rho ^0$$${\rho }^0$production.

    

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5. The Dimension of Divisibility Orders and Multiset Posets

   https://doi.org/10.1007/s11083-023-09653-7  

   Haiman, Milan ( November 2023 , Order)

   The Dushnik–Miller dimension of a posetPis the leastdfor whichPcan be embedded into a product ofdchains. Lewis and Souza isibility order on the interval of integers$$[N/\kappa , N]$$$[N/\kappa ,N]$is bounded above by$$\kappa (\log \kappa )^{1+o(1)}$$$\kappa {(log\kappa )}^{1+o\left(1\right)}$and below by$$\Omega ((\log \kappa /\log \log \kappa )^2)$$$\Omega \left({(log\kappa /loglog\kappa )}^2\right)$. We improve the upper bound to$$O((\log \kappa )^3/(\log \log \kappa )^2).$$$O({(log\kappa )}^3/{(loglog\kappa )}^2).$We deduce this bound from a more general result on posets of multisets ordered by inclusion. We also consider other divisibility orders and give a bound for polynomials ordered by divisibility.

    

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