We consider stochastic partial differential equations, on $\mathbb{R}^{d}$ $(d\geq 1)$, driven by a Gaussian noise white in time and colored in space. Assuming pathwise uniqueness holds, we establish various strong stability results. As consequence, we give an application to the convergence of the Picard successive approximation. Finally, we show that in the sense of Baire category, almost all stochastic partial differential equations with continuous and bounded coefficients have the properties of existence and pathwise uniqueness of solutions as well as the continuous dependence on the coefficients.

Let $P$ be the maximal parabolic subgroup of $\text{PGL}(d, {\mathbb C})$ defined by invertible matrices $(a_{ij})_{i,j=1}^d$ with $a_{dj}\,=\, 0$ for all $j\, \in\, [1\, ,d-1]$. Take a holomorphic parabolic geometry $(M\, ,E_P\, ,\omega)$ of type $\text{PGL}(d,{\mathbb C})/P$. Assume that $M$ is a complex projective manifold. We prove the following: If there is a nonconstant holomorphic map $f\, :\, {\mathbb C} {\mathbb P}^1\,\longrightarrow \, M$, then $M$ is biholomorphic to the projective space ${\mathbb C}{\mathbb P}^{d-1}$.

We consider normal affine surfaces $X$ with étale endomorphisms. It is proved that if $X$ has at least one singular point which is not a quotient singular point then such an endomorphism is an isomorphism.

We study the moduli spaces and compute the fundamental groups of plane sextics of torus type with at least two type-${\bf E}_6$ singular points. As a simple application, we compute the fundamental groups of $125$ other sextics, most of which are new.

We define the notion of universal lift of a projective complex based on non-commutative parameter algebras, and prove its existence and uniqueness. We investigate the properties of parameter algebras for universal lifts.

Let $G$ be a locally compact group and let $K$ be a compact subgroup of $Aut(G)$, the group of automorphisms of $G$. $(G,K)$ is a Gelfand pair if the algebra $L_{K}^{1}(G)$ of K-invariant integrable functions on $G$ is commutative under convolution. In this paper, we give some charactezations of this algebra in the nilpotent case, which generalize some results obtained by C. Benson, J. Jenkins, G. Ratcliff in [1] and obtain a new criterion for Gelfand pairs.

The aim of the present paper is to provide a \emph{global} presentation of the theory of special Finsler manifolds. We introduce and investigate\emph{ globally} (or intrinsically, free from local coordinates) many of the most important and most commonly used special Finsler manifolds\,: locally Minkowskian, Berwald, Landesberg, general Landesberg, $P$-reducible, $C$-reducible, semi-$C$-reducible, quasi-$C$-reducible, $P^{*}$-Finsler, $C^{h}$-recur-rent, $C^{v}$-recurrent, $C^{0}$-recurrent, $S^{v}$-recurrent, $S^{v}$-recurrent of the second order, $C_{2}$-like, $S_{3}$-like, $S_{4}$-like, $P_{2}$-like, $R_{3}$-like, $P$-symmetric, $h$-isotropic, of scalar curvature, of constant curvature, of $p$-scalar curvature, of $s$-$ps$-curvature.

The global definitions of these special Finsler manifolds are introduced. Various relationships between the different types of the considered special Finsler manifolds are found. Many local results, known in the literature, are proved globally and several new results are obtained. As a by-product, interesting identities and properties concerning the torsion tensor fields and the curvature tensor fields are deduced.

Although our investigation is entirely global, we provide; for comparison reasons, an appendix presenting a local counterpart of our global approach and the {\it{local}} definitions of the special Finsler spaces considered.

The theta map sends code polynomials into the ring of Siegel modular forms of even weights. Explicit description of the image is known for $g\leq 3$ and the surjectivity of the theta map follows. Instead it is known that this map is not surjective for $g\geq 5$. In this paper we discuss the possibility of an embedding between the associated projective varieties. We prove that this is not possible for $g\geq 4$ and consequently we get the non surjectivity of the graded rings for the remaining case $g=4$.

Let $X$ be a projective variety over an algebraically closed field $k$ of characteristic 0.We consider categories of rational maps from $X$ to commutative algebraic groups, and ask for objects satisfying the universal mapping property.A necessary and sufficient condition for the existence of such universal objects is given, as well as their explicit construction, using duality theory of generalized 1-motives. An important application is the Albanese of a singular projective variety, which was constructed by Esnault, Srinivas and Viehweg as a universal regular quotient of a relative Chow group of 0-cycles of degree 0 modulo rational equivalence.We obtain functorial descriptions of the universal regular quotient and its dual 1-motive.

If a Griffiths domain $D$ is a symmetric Hermitian domain, the toroidal compactification of the quotient space $\Gamma\backslash D$, associated to a projective fan and a discrete subgroup $\Gamma$ of ${\rm Aut}(D)$, was constructed by Mumford et al. Kazuya Kato and Sampei Usui studied extensions of $\Gamma\backslash D$ for a Griffiths domain $D$ in general, and introduced a notion of \lq\lq complete fan\rq\rq$\;$ as a generalization of a notion of projective fan. The existence of complete fans is expected. In this paper, we give an example of $D$ which has no complete fan.