It is known that the class of factorizing codes, i.e., codes satisfying the factorization conjecture formulated by Schützenberger, is closed under two operations: the classical composition of codes and substitution of codes. A natural question which arises is whether a finite set O of operations exists such that each factorizing code can be obtained by using the operations in O and starting with prefix or suffix codes. O is named here a complete set of operations (for factorizing codes). We show...

This is a part of a recent program by a number of authors to study information measures on open domain. In this series, this paper is devoted to the study of the functional equation (2) on an open domain. This functional equation is connected to the weighted entropy and weighted entropy of degree a.

The Shannon entropy has the sum form ∑f(pi) with f(x) = -x logx (x belonging to [0,1]). This together with the property of additivity leads to the 'sum' functional equation...

In this paper, we obtain all possible general solutions of the sum form functional equations $$\begin{array}{cc}\hfill \sum _{i=1}^k\sum _{j=1}^{\ell }f\left(p_i{q}_j\right)=& \sum _{i=1}^{k}g\left(p_i\right)\sum _{j=1}^{\ell }h\left(q_j\right)\hfill \\ \hfill \text{and}\sum _{i=1}^k\sum _{j=1}^{\ell }F\left(p_i{q}_j\right)=& \sum _{i=1}^{k}G\left(p_i\right)+\sum _{j=1}^{\ell }H\left(q_j\right)+\lambda \sum _{i=1}^{k}G\left(p_i\right)\sum _{j=1}^{\ell }H\left(q_j\right)\hfill \end{array}$$ valid for all complete probability distributions $(p_1,...,p_k)$, $(q_1,...,q_{\ell })$, $k\ge 3$, $\ell \ge 3$ fixed integers; $\lambda \in ℝ$, $\lambda \ne 0$ and $F$, $G$, $H$, $f$, $g$, $h$ are real valued mappings each having the domain $I=[0,1]$, the unit closed interval.