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Abstract In 1970, Schneider introduced the$$m$$ $m$th order difference body of a convex body, and also established the$$m$$ $m$th-order Rogers–Shephard inequality. In this paper, we extend this idea to the projection body, centroid body, and radial mean bodies, as well as prove the associated inequalities (analogues of Zhang’s projection inequality, Petty’s projection inequality, the Busemann–Petty centroid inequality and Busemann’s random simplex inequality). We also establish a new proof of Schneider’s$$m$$ $m$th-order Rogers–Shephard inequality. As an application, a$$m$$ $m$th-order affine Sobolev inequality for functions of bounded variation is provided. 

Abstract The paper introduces a finite element method for an Eulerian formulation of partial differential equations governing the transport and diffusion of a scalar quantity in a time-dependent domain. The method follows the idea from[C. Lehrenfeld and M. Olshanskii,An Eulerian finite element method for PDEs in time-dependent domains,ESAIM Math. Model. Numer. Anal. 53 2019, 2, 585–614]of a solution extension to realise the Eulerian time-stepping scheme. However, a reformulation of the partial differential equation is suggested to derive a scheme which conserves the quantity under consideration exactly on the discrete level. For the spatial discretisation, the paper considers an unfitted finite element method. Ghost-penalty stabilisation is used to realise the discrete solution extension and gives a scheme robust against arbitrary intersections between the mesh and geometry interface. The stability is analysed for both first- and second-order backward differentiation formula versions of the scheme. Several numerical examples in two and three spatial dimensions are included to illustrate the potential of this method. 

Abstract We study equivariant geometry and rationality of moduli spaces of points on the projective line, for twists associated with permutations of the points. 

Abstract We consider a parabolic–parabolic interface problem and construct a loosely coupled prediction-correction scheme based on the Robin–Robin splitting method analyzed in [J. Numer. Math., 31(1):59–77, 2023]. We show that the errors of the correction step converge at $$\mathcal O((\varDelta t)^{2})$$, under suitable convergence rate assumptions on the discrete time derivative of the prediction step, where $$\varDelta t$$ stands for the time-step length. Numerical results are shown to support our analysis and the assumptions. 

Abstract We consider discontinuous Galerkin methods for an elliptic distributed optimal control problem, and we propose multigrid methods to solve the discretized system.We prove that the 𝑊-cycle algorithm is uniformly convergent in the energy norm and is robust with respect to a regularization parameter on convex domains.Numerical results are shown for both 𝑊-cycle and 𝑉-cycle algorithms. 

I introduce a novel associative memory model named Correlated Dense Associative Memory (CDAM), which integrates both auto- and hetero-association in a unified framework for continuous-valued memory patterns. Employing an arbitrary graph structure to semantically link memory patterns, CDAM is theoretically and numerically analysed, revealing four distinct dynamical modes: auto-association, narrow hetero-association, wide hetero-association, and neutral quiescence. Drawing inspiration from inhibitory modulation studies, I employ anti-Hebbian learning rules to control the range of hetero-association, extract multi-scale representations of community structures in graphs, and stabilise the recall of temporal sequences. Experimental demonstrations showcase CDAM’s efficacy in handling real-world data, replicating a classical neuroscience experiment, performing image retrieval, and simulating arbitrary finite automata.