We quantize the interaction of gravity with Yang-Mills and spinor fields, hence offering a quantum theory incorporating all four fundamental forces of nature. Let as abbreviate the spatial Hamilton functions of the Standard Model by $H_{SM}$ and the Hamilton function of gravity by $H_G$. Working in a fiber bundle $E$ with base space $\socc=\R[n]$, where the fiber elements are Riemannian metrics, we can express the Hamilton functions in the form $H_G+H_{SM}=H_G+t^{-\frac23}\tilde H_{SM}$ if $n=3$, where $\tilde H_{SM}$ depends on metrics $\s_{ij}$ satisfying $\det{\s_{ij}}=1$. In the quantization process, we quantize $H_G$ for general $\s_{ij}$ but $\tilde H_{SM}$ only for $\s_{ij}=\de_{ij}$ by the usual methods of QFT. Let $v$ \resp $\psi$ be the spatial eigendistributions of the respective Hamilton operators, then, the solutions $u$ of the Wheeler-DeWitt equation are given by $u=wv\psi$, where $w$ satisfies an ODE and $u$ is evaluated at $(t,\de_{ij})$ in the fibers.

Discontinuous Galerkin (DG) methods combine features in finite element methods (weak formulation, finite dimensional solution and test function spaces) and in finite volume methods (numerical fluxes, nonlinear limiters) and are particularly suitable for simulating convection dominated problems, such as linear and nonlinear waves including shock waves. In this article we will give a brief survey of DG methods, emphasizing their applications in computational fluid dynamics (CFD). We will discuss essential ingredients and properties of DG methods, and will also give a few examples of recent developments of DG methods which have facilitated their applications in CFD.

The Orlicz–Legendre ellipsoids, which are in the framework of emerging dual Orlicz Brunn–Minkowski theory, are introduced for the first time. They are in some sense dual to the recently found Orlicz–John ellipsoids, and have largely generalized the classical Legendre ellipsoid of inertia. Several new affine isoperimetric inequalities are established. The connection between the characterization of Orlicz– Legendre ellipsoids and isotropy of measures is demonstrated.

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