Using elliptic and parabolic regularity results in $L^p$-spaces and generalized Dirichlet form theory, we construct for every starting point weak solutions to SDEs in $\mathbb{R}^d$ up to their explosion times including the following conditions. For arbitrary but fixed $p>d$ the diffusion coefficient $A=(a_{ij})_{1\le i,j\le d}$ is locally uniformly strictly elliptic with functions $a_{ij}\in H^{1,p}_{loc}(\mathbb{R}^d)$ and the drift coefficient $\mathbf{G}=(g_1,\dots, g_d)$ consists of functions $g_i\in L^p_{loc}(\mathbb{R}^d)$. The solution originates by construction from a Hunt process with continuous sample paths on the one-point compactification of $\mathbb{R}^d$ and the corresponding SDE is by a known local well-posedness result pathwise unique up to an explosion time. Just under the given assumptions we show irreducibility and the strong Feller property on $L^{1}(\mathbb{R}^d,m)+L^{\infty}(\mathbb{R}^d,m)$ of its transition function, and the strong Feller property on $L^{q}(\mathbb{R}^d,m)+L^{\infty}(\mathbb{R}^d,m)$, $q=\frac{dp}{d+p}\in (d/2,p/2)$, of its resolvent, which both include the classical strong Feller property. We present moment inequalities and classical-like non-explosion criteria for the solution which lead to pathwise uniqueness results up to infinity under presumably optimal general non-explosion conditions. We further present explicit conditions for recurrence and ergodicity, including existence as well as uniqueness of invariant probability measures.

In this article, we establish a truncated non-integrated defect relation for meromorphic mappings from a complete Kähler manifold into a projective variety intersecting a family of hypersurfaces located in subgeneral position, where the truncation level of the defect is explicitly estimated. Our result generalizes and improves previous results. In particular, when the family of hypersurfaces located in general position, our result will implies the previous result of Min Ru-Sogome. In the last part of this paper we will apply our result to study the distribution of the Gauss maps of minimal surfaces.

We introduce entire multiple zeta functions, which are natural interpolations of symmetric multiple zeta values. Moreover we give further generalizations which interpolate these zeta functions, $\widehat{\mathcal{A}}$-finite multiple zeta values and $\widehat{\mathcal{S}}$-symmetric multiple zeta values. We also show that the correspondence which Kaneko--Zagier conjecture suggests holds on nonpositive integers for finite multiple zeta values and special values of these multiple zeta functions.

Self dual symmetric $R$-spaces have special curves, called circles, introduced by Burstall, Donaldson, Pedit and Pinkall in 2011, whose definition does not involve the choice of any Riemannian metric. We characterize the elements of the big transformation group $G$ of a self dual symmetric $R$-space $M$ as those diffeomorphisms of $M$ sending circles in circles. Besides, although these curves belong to the realm of the invariants by $G$, we manage to describe them in Riemannian geometric terms: Given a circle $c$ in $M$, there is a maximal compact subgroup $K$ of $G$ such that $c$, except for a projective reparametrization, is a diametrical geodesic in $M$ (or equivalently, a diagonal geodesic in a maximal totally geodesic flat torus of $M$), provided that $M$ carries the canonical symmetric $K$-invariant metric. We include examples for the complex quadric and the split standard or isotropic Grassmannians.

We exhibit a particular free subarrangement of a certain restriction of the Weyl arrangement of type $E_7$ and use it to give an affirmative answer to a recent conjecture by T. Abe on the nature of additionally free and stair-free arrangements.

Biconformal deformations in the presence of a conformal foliation by curves are exploited to study equivalence between 3-dimensional Ricci solitons. We show that a wide class of solitons are biconformally equivalent to the flat metric.