The Gaussian polytope \mathcal P_{n,d}$\mathcal P_{n,d}$ is the convex hull of n independent standard normally distributed points in \mathbb{R}^d$\mathbb{R}^d$ . We derive explicit expressions for the probability that \mathcal P_{n,d}$\mathcal P_{n,d}$ contains a fixed point x\in\mathbb{R}^d$x\in\mathbb{R}^d$ as a function of the Euclidean norm of x, and the probability that \mathcal P_{n,d}$\mathcal P_{n,d}$ contains the point \sigma X$\sigma X$ , where \sigma\geq 0$\sigma\geq 0$ is constant and X is a standard normal vector independent of \mathcal P_{n,d}$\mathcal P_{n,d}$ . As a by-product, we also compute the expected number of k-faces and the expected volume of \mathcal P_{n,d}$\mathcal P_{n,d}$ , thus recovering the results of Affentranger and Schneider (Discr. and Comput. Geometry, 1992) and Efron (Biometrika, 1965), respectively. All formulas are in terms of the volumes of regular spherical simplices, which, in turn, can be expressed through the standard normal distribution function \Phi(z)$\Phi(z)$ and its complex version \Phi(iz)$\Phi(iz)$ . The main tool used in the proofs is the conic version of the Crofton formula.

This work is concerned with Bielawski’s hyperkähler slices in the cotangent bundles of homogeneous affine varieties. One can associate such a slice with the data of a complex semisimple Lie group G$G$, a reductive subgroup H\subseteq G$H\subseteq G$, and a Slodowy slice S\subseteq \mathfrak{g}:=\text{Lie}(G)$S\subseteq \mathfrak{g}:=\text{Lie}(G)$, defining it to be the hyperkähler quotient of T^{\ast }(G/H)\times (G\times S)$T^{\ast }(G/H)\times (G\times S)$ by a maximal compact subgroup of G$G$. This hyperkähler slice is empty in some of the most elementary cases (e.g., when S$S$ is regular and (G,H)=(\text{SL}_{n+1},\text{GL}_{n})$(G,H)=(\text{SL}_{n+1},\text{GL}_{n})$, n\geqslant 3$n\geqslant 3$), prompting us to seek necessary and sufficient conditions for non-emptiness.

We give a spherical-geometric characterization of the non-empty hyperkähler slices that arise when S=S_{\text{reg}}$S=S_{\text{reg}}$ is a regular Slodowy slice, proving that non-emptiness is equivalent to the so-called \mathfrak{a}$\mathfrak{a}$-regularity of (G,H)$(G,H)$. This \mathfrak{a}$\mathfrak{a}$-regularity condition is formulated in several equivalent ways, one being a concrete condition on the rank and complexity of G/H$G/H$. We also provide a classification of the \mathfrak{a}$\mathfrak{a}$-regular pairs (G,H)$(G,H)$ in which H$H$ is a reductive spherical subgroup. Our arguments make essential use of Knop’s results on moment map images and Losev’s algorithm for computing Cartan spaces.

The solution of the pressure Poisson equation in spherical polar coordinates using a higher order compact (HOC) scheme effectively captures low pressure values in the wake region for viscous flow past a sphere. In the absence of an exact solution, the fourth-order of accuracy of the results is illustrated. Low pressure circular contours occur in the wake region when the Reynolds number Re = 161, which is a lower value than previously identified in the literature, and closed pressure contours appear in two regions when Re = 250.