Let $(V,0)$ be a germ of a complete intersection variety in ${ℂ}^{n+k}$, $n\>0$, having an isolated singularity at $0$ and $X$ be the germ of a holomorphic vector field having an isolated zero at $0$ and tangent to $V$. We show that in this case the homological index and the GSV-index coincide. In the case when the zero of $X$ is also isolated in the ambient space ${ℂ}^{n+k}$ we give a formula for the homological index in terms of local linear algebra.

Let $p^{\text{'}}$ and $q^{\text{'}}$ be points in ${ℝ}^n$. Write $p^{\text{'}}\sim q^{\text{'}}$ if $p^{\text{'}}-q^{\text{'}}$ is a multiple of $(1,...,1)$. Two different points $p$ and $q$ in ${ℝ}^n/\sim$ uniquely determine a tropical line $L(p,q)$ passing through them and stable under small perturbations. This line is a balanced unrooted semi-labeled tree on $n$ leaves. It is also a metric graph. If some representatives $p^{\text{'}}$ and $q^{\text{'}}$ of $p$ and $q$ are the first and second columns of some real normal idempotent order $n$ matrix $A$, we prove that the tree $L(p,q)$ is described by a matrix $F$, easily obtained from $A$. We also...

The function $f\left(z\right)=z^p+{\sum }_{k=1}^{\infty }{a}_{p+k}{z}^{p+k}$ (p ∈ ℕ = 1,2,3,...) analytic in the unit disk E is said to be in the class $K_{n,p}\left(h\right)$ if ($D^{n+p}f)/\left(D^{n+p-1}f\right)\prec h$, where $D^{n+p-1}f=\left(z^p\right)/\left({(1-z)}^{p+n}\right)*f$ and h is convex univalent in E with h(0) = 1. We study the class $K_{n,p}\left(h\right)$ and investigate whether the inclusion relation $K_{n+1,p}\left(h\right)\subseteq K_{n,p}\left(h\right)$ holds for p > 1. Some coefficient estimates for the class are also obtained. The class $A_{n,p}(a,h)$ of functions satisfying the condition $a*\left(D^{n+p}f\right)/\left(D^{n+p-1}f\right)+(1-a)*\left(D^{n+p+1}f\right)/\left(D^{n+p}f\right)\prec h$ is also studied.

We study the simultaneous linearizability of $d$–actions (and the corresponding $d$-dimensional Lie algebras) defined by commuting singular vector fields in ${ℂ}^n$ fixing the origin with nontrivial Jordan blocks in the linear parts. We prove the analytic convergence of the formal linearizing transformations under a certain invariant geometric condition for the spectrum of $d$ vector fields generating a Lie algebra. If the condition fails and if we consider the situation where small denominators...

Let $T:L^p\left(\Omega \right)\to L^p\left(\Omega \right)$ be a positive contraction, with $1\<p\<\infty$. Assume that $T$ is analytic, that is, there exists a constant $K\ge 0$ such that $\parallel T^n-T^{n-1}\parallel \le K/n$ for any integer $n\ge 1$. Let $2\<q\<\infty$ and let $v^q$ be the space of all complex sequences with a finite strong $q$-variation. We show that for any $x\in L^p\left(\Omega \right)$, the sequence ${\left(\left[T^n\left(x\right)\right]\left(\lambda \right)\right)}_{n\ge 0}$ belongs to $v^q$ for almost every $\lambda \in \Omega$, with an estimate $\parallel {\left(T^n\left(x\right)\right)}_{n\ge 0}{\parallel }_{L^p\left(v^q\right)}\le C{\parallel x\parallel }_p$. If we remove the analyticity assumption, we obtain an estimate $\parallel {\left(M_n\left(T\right)x\right)}_{n\ge 0}{\parallel }_{L^p\left(v^q\right)}\le C{\parallel x\parallel }_p$, where $M_n\left(T\right)={(n+1)}^{-1}{\sum }_{k=0}^n{T}^k$ denotes the ergodic average of $T$. We also obtain similar results for strongly continuous semigroups...

Let $X$ be a complex manifold of dimension at least $2$ which has an exhaustion function whose Levi form has at each point at least $2$ strictly positive eigenvalues. We construct proper holomorphic discs in $X$ through any given point and in any given direction.