We study the topological spectrum of a seminormed ring $R$ which we define as space prime ideals $\mathfrak{p}$ such that equals kernel some bounded power-multiplicative seminorm. For any show is quasi-compact sober space. When perfectoid Tate construct natural homeomorphism between and its tilt $R^{\flat}$. As an application, prove integral domain if only domain.

The convergence rate of a multigrid method for the numerical solution of the Poisson equation on a uniform grid is estimated. The results are independent of the shape of the domain as long as it is convex and polygonal. On the other hand, pollution effects become apparent when the domain contains reentrant corners. To estimate the smoothing of the Gauss-Seidel relaxation, the smoothness is meas...

We propose a method to quantify the complexity of conditional probability measures by a Hilbert space seminorm of the logarithm of its density. The concept of Reproducing Kernel Hilbert Spaces (RKHS) is a flexible tool to define such a seminorm by choosing an appropriate kernel. We present several examples with artificial datasets where our kernel-based complexity measure is consistent with our...

and Applied Analysis 3 Baeumer et al. 8, 13 have proved existence and uniqueness of a strong solution for 1.2 using the semigroup theory when f x, t, u is globally Lipschitz continuous. Furthermore, when f x, t, u is locally Lipschitz continuous, existence of a unique strong solution has also been shown by introducing the cut-off function. Finite difference methods have been studied in 14–16 fo...

We describe a time-dependent functional involving the relative entropy and $\dot{H}^1$ seminorm, which decreases along solutions to spatially homogeneous Landau equation with Coulomb potential. The study of this monotone sheds light on competition between dissipation nonlinearity for equation. It enables us obtain new results concerning regularity/blowup issues