This paper is devoted to study of sufficient conditions under which a transcendental meromorphic function has no unbounded Fatou components and to extension of some results for entire functions to meromorphic functions. Actually, we shall mainly discuss non-existence of unbounded wandering domains of a meromorphic function. The case for a composition of finitely many meromorphic functions with at least one of them being transcendental can be also investigated in terms of the argument of this paper.

In the present paper the problems of existence and uniqueness of almost periodic solutions for impulsive Hopfield neural networks are considered. The impulses are in fixed moments of time and by using the technique of estimation of the Cauchy’s matrix new sufficient conditions for the exponential stability of the unique almost periodic solution of such systems are given.

We generalise the variant of the Babylonian tower theorem for vector bundles on projective spaces proved by I. Coandă and G. Trautmann (2006) to the case of principal $G$-bundles over projective spaces, where $G$ is a linear algebraic group defined over an algebraically closed field. In course of the proofs some new insight into the structure of such principal $G$-bundles is obtained.

We prove that the Klein cubic threefold $F$ is the only smooth cubic threefold which has an automorphism of order $11$. We compute the period lattice of the intermediate Jacobian of $F$ and study its Fano surface $S$. We compute also the set of fibrations of $S$ onto a curve of positive genus and the intersection between the fibres of these fibrations. These fibres generate an index $2$ sub-group of the Néron-Severi group and we obtain a set of generators of this group. The Néron-Severi group of $S$ has rank $25=h^{1,1}$ and discriminant $11^{10}$.

We extend a locally nilpotent derivation $\delta$ on a ring $B$ to $B$-modules $M$ and investigate the properties of such $B$-modules $M$. It turns out that the various geometric properties of the associated $G_a$-action on Spec $B$ can be reflected on the properties of $M$.

Let $G$ be a connected and simply connected, nilpotent Lie group. In this paper, we show that the cortex of $G$ is a semi-algebraic set by means of a geometric characterization. It is also shown that the cortex is the image under a linear projection of a countable union of a semi-algebraic sets lying in the tensor product $T$($\mathfrak{g}$)$\otimes$ $\mathfrak{g}$*.

We consider a class of linear higher order hyperbolic equations with a single degenerate point. We give sufficient conditions in order for the Cauchy problem to be well-posed in Gevrey classes and in the $C^\infty$ class.

Let $R$ be a commutative Noetherian ring, and let $I$ and $J$ be two ideals of $R$. Assume that $R$ is local with the maximal ideal ${\mathfrak{m}}$, we mainly prove that (i) there exists an equality \[{\text{inf}}\{i\, \mid H_{I,J}^i(M)\, {\text{ is not Artinian}} \}={\text{inf}}\{ {\text{depth}}M_{\mathfrak{p}} \mid \, {\mathfrak{p}}\in W(I, J)\backslash \{{\mathfrak{m}}\} \}\] for any finitely generated $R-$module $M$, where $W(I, J)=\{{\mathfrak{p}} \in {\text{Spec}}(R) \mid \, I^n \subseteq {\mathfrak{p}}+J\,\, {\text{for some positive integer}} \,n \}$; (ii) for any finitely generated $R-$module $M$ with ${\text{dim}}M=d$, $H_{I,J}^d(M)$ is Artinian. Also, we give a characterization to the supremum of all integers $r$ for which $H_{I,J}^r(M) \neq 0$.

We consider a Markov chain with values in [0,$\infty$)$^{\mathbb{z}d}$. The Markov chain includes some interesting examples such as the oriented site percolation, the directed polymers in random environment, and a time discretization of the binary contact process. We prove a central limit theorem for “the spatial distribution of population” when $d\geq 3$ and a certain square-integrability condition for the total population is satisfied. This extends a result known for the directed polymers in random environment to a large class of models.