We study families of superelliptic curves with fixed automorphism groups. Such families are parametrized with invariants expressed in terms of the coefficients of the curves. Algebraic relations among such invariants determine the lattice of inclusions among the loci of superelliptic curves and their field of moduli. We give a Maple package of how to compute the normal form of an superelliptic curve and its invariants. A complete list of all superelliptic curves of genus $g\le 10$ defined over any field of characteristic $\ne 2$ is given in a subsequent paper [3].

The paper devoted to implementation of the public key algorithm based on directed algebraic graphs over finite commutative ring $K$ and their symmetries. First we expand the key space $K^n$ of graph based encryption algorithm in such way that arbitrary chosen plaintext can be converted to arbitrary chosen ciphertext. Second, we conjugate chosen encryption map, which is a composition of several “elementary” cubical polynomial automorphisms of a free module $K^n$ with special invertible affine transformation of $K^n$. Finally we compute symbolically corresponding cubic public map $g$ of $K^n$ onto $K^n$. We evaluate time for the generation of $g$, time of execution of public map, number of monomial expression in the list of corresponding public rules.

In this paper, we obtain the general solution and the generalized Hyers-Ulam stability for the following functional equation

$$f\left(\frac{\sum _{i=1}^{m}xi}{m}\right)+\sum _{\underset{i\ne j}{i=1}}^{m}f\left(\frac{x_j-x_i}{m}\right)=f\left(x_j\right).$$

This is applied to investigate derivations and their stability in proper Lie $C{Q}^{*}$-algebras.

Let $R$ be an arbitrary ring with identity and $M$ a right $R$-module with $S={\mathrm{End}}_R\left(M\right)$. In this paper, we introduce a class of modules that is a generalization of principally quasi-Baer rings and Baer modules. The module ${}_{S}M$ is called principally quasi-Baer if for any $m\in M,l_S\left(Sm\right)=Se$ for some $e^2=e\in S$. It is proved that (1) if ${}_{S}M$ is regular and semicommutative module or (2) if $M_R$ is principally semisimple and ${}_{S}M$ is abelian, then ${}_{S}M$ is a principally quasi-Baer module. The connection between a principally quasi-Baer module ${}_{S}M$ and polynomial extension, power series extension, Laurent polynomial extension, Laurent power series extension of ${}_{S}M$ is investigated.