The duality of truth and falsity in a Boolean algebra of propositions is used to generate a duality of belief and disbelief. To each additive probability measure that represents belief there corresponds a dual additive measure that represents disbelief. The dual measure has its own peculiar calculus, in which, for example, measures are added when propositions are combined under conjunction. A V...

This paper develops slacks-based measure (SBM) and additive SBM (ASBM) to evaluate efficiency of decision making units (DMUs) in a two-stage structure with undesirable outputs and feedback variables from the internal perspective. The SBM model is linearized  for a specific weight and the ASBM model is reformulated as a second order cone program. The target values for all inputs, outputs (both d...

Measures A (set) function μ : K → [0,∞], with ∅ ∈ K ⊂ 2 and μ(∅) = 0 is called additive or finitely additive if A,B ∈ K, with A ∪ B ∈ K and A ∩ B = ∅ imply μ(A ∪ B) = μ(A) + μ(B). Similarly, μ is called σ-additive or countably additive if Ai ∈ K, with ⋃∞ i=1 Ai = A ∈ K and Ai ∩ Aj = ∅ for i 6= j imply μ(A) = ∑∞ i=1 μ(Ai). It is clear that if μ is σ-additive then μ is also additive, but the conv...

A problem which is appealing to the intuition in view of the relative frequency interpretation of probability is to define a measure on a countable space which assigns to each point the measure 0. Such a measure of course becomes trivial if it is countably additive. Finitely additive measures of this type have been discussed by R. C. Buck [l] and by E. F. Buck and R. C. Buck [2]. In a discussio...

According to a currently popular principle of classification, mathematics is the study of various "categories. " A category consists of certain "objects" (e.g., groups, topological spaces) and certain "mappings" (e.g., homomorphisms, continuous functions). One possible category has measure spaces for its objects and, correspondingly, measure-preserving transformations for its mappings. The usua...

Over the past few decades, numerous optimization-based methods have been proposed for solving the classification problem in data mining. Classic optimization-based methods do not consider attribute interactions toward classification. Thus, a novel learning machine is needed to provide a better understanding on the nature of classification when the interaction among contributions from various at...

We shall discuss further regularity properties of null-additive fuzzy measure on metric spaces following the previous results. Under the null-additivity condition, some properties of the inner/outer regularity and the regularity of fuzzy measure are shown. Also the strong regularity of fuzzy measure is discussed on complete separable metric spaces. As an application of strong regularity, we pre...