We show that, to some extent, the behavior of an entire function of several complex variables is reflected by its behavior on algebraic subvarieties.

We give algebraic characterizations for a reduction system to be respectively Noetherian and confluent, and for a Noetherian reduction system to be confluent. The characterization of a confluent reduction system $(A,\to )$ is based on a relationship between the confluence of the system and the exactness of the colimit functor colim : $A{b}^{\mathcal{A}}\to Ab$ where $\mathcal{A}$ is the small category with objects the elements of $A$ and arrows $a\to b$ whenever $b$ is a consequence of $a$ in the system.

We study the distribution of the values of certain additive functions restricted to those integers with a fixed number of prime divisors.

We establish some new properties and identities of Generalized Gaussian Numbers (GGN) which are defined recently in [10, 11] parallel to those of Gaussian coefficients. We present generating functions and some properties which are very useful for GGN. We obtain some family of sequences which are unimodal and present the log-concavity property of GGN. Finally, we give a connection of GGN to the Rogers-Szego polynomials.

We classify non-symplectic automorphisms of order $6$ on elliptic $\mathrm{K}3$ surfaces which commute with a given elliptic fibration. We show how their study can be reduced to the study of non-symplectic automorphisms of order $3$ and to a local analysis of the fibers. In particular, we determine the possible fixed loci and give their location on the singular fibers. When the Picard lattice is fixed, we show that $\mathrm{K}3$ surfaces come in mirror pairs.