We fix two mistakes in the paper “The average analytic rank of elliptic curves with prescribed torsion” and remove the moment conditions of the main theorem therein.
We fix two mistakes in the paper “The average analytic rank of elliptic curves with prescribed torsion” and remove the moment conditions of the main theorem therein.
Missing data in multiple variables is a common issue. We investigate the applicability of the framework of graphical models for handling missing data to a complex longitudinal pharmacological study of children with HIV treated with an efavirenz-based regimen as part of the CHAPAS-3 trial. Specifically, we examine whether the causal effects of interest, defined through static interventions on multiple continuous variables, can be recovered (estimated consistently) from the available data only. So far, no general algorithms are available to decide on recoverability, and decisions have to be made on a case-by-case basis. We emphasize the sensitivity of recoverability to even the smallest changes in the graph structure, and present recoverability results for three plausible missingness-directed acyclic graphs (m-DAGs) in the CHAPAS-3 study, informed by clinical knowledge. Furthermore, we propose the concept of a "closed missingness mechanism": if missing data are generated based on this mechanism, an available case analysis is admissible for consistent estimation of any statistical or causal estimand, even if data are missing not at random. Both simulations and theoretical considerations demonstrate how, in the assumed MNAR setting of our study, a complete or available case analysis can be superior to multiple imputation, and estimation results vary depending on the assumed missingness DAG. Our analyses demonstrate an innovative application of missingness DAGs to complex longitudinal real-world data, while highlighting the sensitivity of the results with respect to the assumed causal model.
We investigate a semilinear problem for a fractional diffusion equation with variable order Caputo fractional derivative (left( partial _t^{beta (t)} uright) (t)) subject to homogeneous Dirichlet boundary conditions. The right-hand side of the governing PDE is nonlinear (Lipschitz continuous) and it contains a weakly singular Volterra operator. The whole process takes place in a bounded Lipschitz domain in ({{mathbb {R}}}^d). We establish the existence of a unique solution in (Cleft( [0,T],L^{2} (varOmega )right) ) if (u_0in L^{2} (varOmega )). Moreover, if (mathcal {L}^{gamma }u_0in L^{2} (varOmega )) for some (0<gamma <1-frac{delta }{beta (0)}) ((delta ) depends on the right-hand-side of the PDE) then (mathcal {L}^{gamma }uin Cleft( {[}0,T{]},L^{2} (varOmega )right) ).
For h-FEM discretisations of the Helmholtz equation with wavenumber k, we obtain k-explicit analogues of the classic local FEM error bounds of Nitsche and Schatz (Math. Comput. 28(128), 937–958 1974), Wahlbin (1991, §9), Demlow et al.(Math. Comput. 80(273), 1–9 2011), showing that these bounds hold with constants independent of k, provided one works in Sobolev norms weighted with k in the natural way. We prove two main results: (i) a bound on the local (H^1) error by the best approximation error plus the (L^2) error, both on a slightly larger set, and (ii) the bound in (i) but now with the (L^2) error replaced by the error in a negative Sobolev norm. The result (i) is valid for shape-regular triangulations, and is the k-explicit analogue of the main result of Demlow et al. (Math. Comput. 80(273), 1–9 2011). The result (ii) is valid when the mesh is locally quasi-uniform on the scale of the wavelength (i.e., on the scale of (k^{-1})) and is the k-explicit analogue of the results of Nitsche and Schatz (Math. Comput. 28(128), 937–958 1974), Wahlbin (1991, §9). Since our Sobolev spaces are weighted with k in the natural way, the result (ii) indicates that the Helmholtz FEM solution is locally quasi-optimal modulo low frequencies (i.e., frequencies (lesssim k)). Numerical experiments confirm this property, and also highlight interesting propagation phenomena in the Helmholtz FEM error.
In Bénard-Rayleigh convection we consider the pattern defect in orthogonal domain walls connecting a set of convective rolls with another set of rolls orthogonal to the first set. This is understood as an heteroclinic orbit of a reversible system where the x - coordinate plays the role of time. This appears as a perturbation of the heteroclinic orbit proved to exist in a reduced 6-dimensional system studied by a variational method in Buffoni et al. (J Diff Equ, 2023, https://doi.org/10.1016/j.jde.2023.01.026), and analytically in Iooss (Heteroclinic for a 6-dimensional reversible system occuring in orthogonal domain walls in convection. Preprint, 2023). We then prove for a given amplitude (varepsilon ^2), and an imposed symmetry in coordinate y, the existence of a one-parameter family of heteroclinic connections between orthogonal sets of rolls, with wave numbers (different in general) which are linked with an adapted shift of rolls parallel to the wall.
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