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In this article we modified the definition of hyperbolic measure given by R. Kumar and K. Sharma by introducing plus hyperbolic infinity and minus hyperbolic infinity. Also we introduced the concept of hyperbolic valued signed measure and proved some theorems on it. AMS Subject Classification (2010) : 28A12.

In this paper the notion of the bipolar pseudo-integral is introduced. This new type of integral is based on bi-capacities, belonging to the class of set functions which are a generalization of the classical signed measure. The discrete bipolar pseudo-integral is defined and its main properties are considered.

The aim of this paper is to investigate extremum problems with pay-off the total variational distance metric subject to linear functional constraints both defined on the space of probability measures, as well as related problems. Utilizing concepts from signed measures, the extremum probability measures of such problems are obtained in closed form, by identifying the partition of the support set and the mass of these extremum measures on the partition. The results are derived for abstract spaces, specifically, complete separable metric spaces, while the high level ideas are also discussed for denumerable spaces endowed with the discrete topology. © 2013 IEEE. 1204 1209

In this paper, first of all, for generalizing the notion of monotonicity, a most important key word, called signed monotonicity, is proposed, it is a generalization of not only the non-negative monotonicity but also the revised monotonicity, furthermore, the signed measure satisfying signed monotonicity, called a generalized signed fuzzy measure, is also proposed. It is proved that this new measure is a generalization of not only the non-negative fuzzy measure but also the signed fuzzy measure, and the well-known signed additive measure is not only a signed fuzzy measure but also a generalized signed fuzzy measure, moreover, the author points out that the signed measure satisfying Lambda-rule , called generalized Lambda-measure is not a signed fuzzy measure and generalized signed fuzzy measure, since it does not satisfy the revised monotonicity and signed monotonicity. At last, a new signed measure satisfying signed monotonicity based on Sugeno’s Lambda-measure, called extensional sign Lambda-fuzzy measure is also proposed, it is a generalized signed fuzzy measure and more useful than above mentioned signed measures based on Lambda-measure for using to analysis multi-criteria decision making problem, since the multi-criteria decision making problem needs to consider the signed monotoniccity. Some related properties are also discussed.

In this paper, a new concept of generalized lambda-fuzzy measure is introduced, and it is a non-negative, extended real-valued set function satisfying generalized sigma - lambda - rule, and is not necessarily monotone. The relationships between this new concept and lambda-fuzzy measure, a signed measure and so on, are discussed, and it is worth noting that any generalized lambda-fuzzy measure is a signed measure under some conditions. Moreover, an application of generalized lambda-fuzzy measure in image enhancement is given, and the corresponding experiment shows that different values of lambda have different influences on an image.

Recently, Wang and Ray introduced a signed measure for formal languages, called a language measure, to evaluate performance of strings generated by discrete event systems and a synthesis method of an optimal supervisor based on the language measure has been studied. In order to apply the method, exact information about a language measure of a controlled discrete event system is needed. From the practical point of view, it is not always possible to know the information a priori.In such a situation, a learning-based approach is useful to obtain an optimal supervisor with respect to the language measure. This paper considers a synthesis method of an optimal supervisor with respect to a language measure. First, we clarify the relationship between the Bellman equation in reinforcement learning and performance of the language generated by the controlled discrete event systems. Next, using the relationship, we propose a learning method of the optimal supervisor where costs of disabling events are taken into consideration. Finally, by computer simulation, we illustrate an efficiency of the proposed method.