Let [Formula: see text] be the maximal order of a number field. Belcher showed in the 1970s that every algebraic integer in [Formula: see text] is the sum of pairwise distinct units, if the unit equation [Formula: see text] has a non-trivial solution [Formula: see text]. We generalize this result and give applications to signed double-base digit expansions.

More than 50 methods have been developed to draw unequal probability samples with fixed sample size. All these methods require the sum of the inclusion probabilities to be an integer number. There are cases, however, where the sum of desired inclusion probabilities is not an integer. Then, classical algorithms for drawing samples cannot be directly applied. We present two methods to overcome th...

The variable sum exdeg index of a graph G is defined as $SEI_a(G)=sum_{uin V(G)}d_G(u)a^{d_G(u)}$, where $aneq 1$ is a positive real number,  du(u) is the degree of a vertex u ∈ V (G). In this paper, we characterize the graphs with the extremum variable sum exdeg index among all the graphs having a fixed number of vertices and cut edges, for every a>1.

A ring $R$ is a strongly clean ring if every element in $R$ is the sum of an idempotent and a unit that commutate. We construct some classes of strongly clean rings which have stable range one. It is shown that such cleanness of $2 imes 2$ matrices over commutative local rings is completely determined in terms of solvability of quadratic equations.

-A neural network approach for solving fuzzy linear programming problems is proposed in which fuzzy concepts are not used. This approach is totally based on the level-sum method introduced Pandian [11]. The nearest optimal solution of the fuzzy linear programming original problem can be obtained by the proposed approach. The large scale fuzzy linear programming problems can be solved efficientl...