Let $K$ be a field of characteristic zero, and let $R=K[X_1,\dots ,X_n]$. Let $A_n\left(K\right)=K\langle X_1,\dots ,X_n,{\partial }_1,\dots ,{\partial }_n\rangle$ be the $n$th Weyl algebra over $K$. We consider the case when $R$ and $A_n\left(K\right)$ are graded by giving $deg{X}_i={\omega }_i$ and $deg{\partial }_i=-{\omega }_i$ for $i=1,\dots ,n$ (here ${\omega }_i$ are positive integers). Set $\omega ={\sum }_{k=1}^n{\omega }_k$. Let $I$ be a graded ideal in $R$. By a result due to Lyubeznik the local cohomology modules $H_I^i\left(R\right)$ are holonomic $\left(A_n\right(K\left)\right)$-modules for each $i\ge 0$. In this article we prove that the de Rham cohomology modules $H^{\ast }(\partial ;H_I^{\ast }(R\left)\right)$ are concentrated in degree $-\omega$; that is, $H^{\ast }(\partial ;H_I^{\ast }(R){)}_j=0$ for $j\ne -\omega$. As an application when $A=R/\left(f\right)$ is an isolated singularity, we relate $H^{n-1}(\partial ;H_{\left(f\right)}^1(R\left)\right)$ to $H^{n-1}\left(\partial \right(f);A)$, the $(n-1)$th Koszul cohomology of $A$ with respect to ${\partial }_1\left(f\right),\dots ,{\partial }_n\left(f\right)$.

The main aim of this article is to give sufficient conditions for a family of meromorphic mappings of a domain $D$ in ${\mathbb{C}}^n$ into ${\mathbb{P}}^N\left(\mathbb{C}\right)$ to be meromorphically normal if they satisfy only some very weak conditions with respect to moving hypersurfaces in ${\mathbb{P}}^N\left(\mathbb{C}\right)$, namely, that their intersections with these moving hypersurfaces, which moreover may depend on the meromorphic maps, are in some sense uniform. Our results generalize and complete previous results in this area, especially the works of Fujimoto, Tu, Tu-Li, Mai-Thai-Trang, and the recent work of Quang-Tan.

We consider the system ${\mathcal{F}}_4(a,b,c)$ of differential equations annihilating Appell’s hypergeometric series $F_4(a,b,c;x)$. We find the integral representations for four linearly independent solutions expressed by the hypergeometric series $F_4$. By using the intersection forms of twisted (co)homology groups associated with them, we provide the monodromy representation of ${\mathcal{F}}_4(a,b,c)$ and the twisted period relations for the fundamental systems of solutions of ${\mathcal{F}}_4$.

We show that the cyclic and epicyclic categories which play a key role in the encoding of cyclic homology and the lambda operations, are obtained from projective geometry in characteristic one over the infinite semifield of max-plus integers ${\mathbb{Z}}_{max}$. Finite-dimensional vector spaces are replaced by modules defined by restriction of scalars from the one-dimensional free module, using the Frobenius endomorphisms of ${\mathbb{Z}}_{max}$. The associated projective spaces are finite and provide a mathematically consistent interpretation of Tits’s original idea of a geometry over the absolute point. The self-duality of the cyclic category and the cyclic descent number of permutations both acquire a geometric meaning.

It is known that the Fatou set of the map $exp\left(z\right)/z$ defined on the punctured plane ${\mathbb{C}}^{\ast }$ is empty. We consider the $M$-set of $\lambda exp\left(z\right)/z$ consisting of all parameters $\lambda$ for which the Fatou set of $\lambda exp\left(z\right)/z$ is empty. We prove that the $M$-set of $\lambda exp\left(z\right)/z$ has infinite area. In particular, the Hausdorff dimension of the $M$-set is 2. We also discuss the area of complement of the $M$-set.

In this article we discuss three types of mean values of the Euler double zeta-function. To get the results, we introduce three approximate formulas for this function.