Representation of maxitive measures: an overview

Idempotent integration is an analogue of Lebesgue integration where $\sigma$-maxitive measures replace $\sigma$-additive measures. In addition to reviewing and unifying several Radon--Nikodym like theorems proven in the literature for the idempotent integral, we also prove new results of the same kind.Comment: 40 page

Properties of several metric spaces of fuzzy sets

This paper discusses the properties the spaces of fuzzy sets in a metric space equipped with the endograph metric and the sendograph metric, respectively. We first give some relations among the endograph metric, the sendograph metric and the $\Gamma$-convergence, and then investigate the level characterizations of the endograph metric and the $\Gamma$-convergence. By using the above results, we give some relations among the endograph metric, the sendograph metric, the supremum metric and the $d_p^*$ metric, $p\geq 1$. On the basis of the above results, we present the characterizations of total boundedness, relative compactness and compactness in the space of fuzzy sets whose $\alpha$-cuts are compact when $\alpha>0$ equipped with the endograph metric, and in the space of compact support fuzzy sets equipped with the sendograph metric, respectively. Furthermore, we give completions of these metric spaces, respectively

Review on computational methods for Lyapunov functions

Lyapunov functions are an essential tool in the stability analysis of dynamical systems, both in theory and applications. They provide sufficient conditions for the stability of equilibria or more general invariant sets, as well as for their basin of attraction. The necessity, i.e. the existence of Lyapunov functions, has been studied in converse theorems, however, they do not provide a general method to compute them. Because of their importance in stability analysis, numerous computational construction methods have been developed within the Engineering, Informatics, and Mathematics community. They cover different types of systems such as ordinary differential equations, switched systems, non-smooth systems, discrete-time systems etc., and employ di_erent methods such as series expansion, linear programming, linear matrix inequalities, collocation methods, algebraic methods, set-theoretic methods, and many others. This review brings these different methods together. First, the different types of systems, where Lyapunov functions are used, are briefly discussed. In the main part, the computational methods are presented, ordered by the type of method used to construct a Lyapunov function

One-Class Classification: Taxonomy of Study and Review of Techniques

One-class classification (OCC) algorithms aim to build classification models when the negative class is either absent, poorly sampled or not well defined. This unique situation constrains the learning of efficient classifiers by defining class boundary just with the knowledge of positive class. The OCC problem has been considered and applied under many research themes, such as outlier/novelty detection and concept learning. In this paper we present a unified view of the general problem of OCC by presenting a taxonomy of study for OCC problems, which is based on the availability of training data, algorithms used and the application domains applied. We further delve into each of the categories of the proposed taxonomy and present a comprehensive literature review of the OCC algorithms, techniques and methodologies with a focus on their significance, limitations and applications. We conclude our paper by discussing some open research problems in the field of OCC and present our vision for future research.Comment: 24 pages + 11 pages of references, 8 figure