Ordinarily, the process of decision making by a committee through voting is modelled by a monotonic game the range of whose characteristic function is restricted to f0; 1g: The decision rule that governs the collective action of a voting body induces a hierarchy in the set of players in terms of the a-priori in‡uence that the players have over the decision making process. In order to determine ...

The most popular approach to decision-making in the setting of fuzzy sets is the maximin ranking of solutions. This method is natural when interpreting the fuzzy sets as flexible constraints that cannot compensate with one another. However the obtained ranking of solutions is very coarse. Two kinds of refinements to this ordering are introduced: a partial ordering according to the least satisfi...

Let s(t), 0 t T, be a smooth curve and let a sequence of points in two dimensions. An algorithm is given that calculates to the constraints 0 t 1 t 2 t n T. Further, the nal value of the objective function is best lexicographically, when the distances kx i ?s(t i)k 2 , i=1; 2; : : : ; n, are sorted into decreasing order. The algorithm nds the global solution to this calculation. Usually the mag...

In this paper, we present an innovative approach for ranking profiles of capability sets on the basis of equity. An interesting way of capturing the notion of equity is to take into account the extent to which each of the different functioning vectors is shared by the population under consideration (of size n). This is done by defining the ‘common capability sets’ enjoyed by k individuals as th...

A graph is closed when its vertices have a labeling by [n] such that the binomial edge ideal JG has a quadratic Gröbner basis with respect to the lexicographic order induced by x1 > · · · > xn > y1 > · · · > yn. In this paper, we generalize this notion and study the so called m-closed graphs. We find equivalent condition to 3-closed property of an arbitrary tree T . Using it, we classify a clas...

We generalize the main result of [K–K–S] concerning the convex embeddings of a chain Γ in a lexicographic power ∆Γ. For a fixed non-empty chain ∆, we derive necessary and sufficient conditions for the existence of non-empty solutions Γ to each of the lexicographic functorial equations (∆) ' Γ , (∆) ' Γ and (∆) ' Γ .

Helly’s theorem says that if every d + 1 elements of a given finite set of convex objects in R have a common point, then there is a point common to all of the objects in the set. We define three new types of Helly theorems: discrete Helly theorems—where the common point should belong to an a-priori given set, lexicographic Helly theorems—where the common point should not be lexicographically gr...

A primitive word w is a Lyndon word if w is minimal among all its conjugates with respect to some lexicographic order. A word w is bordered if there is a nonempty word u such that w = uvu for some word v. A right extension of a word w of length n is a word wu where all factors longer than n are bordered. A right extension wu of w is called trivial if there exists a positive integer k such that ...

We present anO(n) time algorithm for finding the lexicographically largest fixed-density necklace of length n. Then we determine whether or not a given string can be extended to a fixed-density necklace of length n in O(n) time. Finally, we give an O(n) algorithm that finds the largest fixed-density necklace of length n that is less than or equal to a given string. The efficiency of the latter ...