New results on systems of generalized vector quasi-equilibrium problems

In this paper, we firstly prove the existence of the equilibrium for the generalized abstract economy. We apply these results to show the existence of solutions for systems of vector quasi-equilibrium problems with multivalued trifunctions. Secondly, we consider the generalized strong vector quasi-equilibrium problems and study the existence of their solutions in the case when the correspondences are weakly naturally quasi-concave or weakly biconvex and also in the case of weak-continuity assumptions. In all situations, fixed-point theorems are used.Comment: 24 page

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

Spectral geometry with a cut-off: topological and metric aspects

Inspired by regularization in quantum field theory, we study topological and metric properties of spaces in which a cut-off is introduced. We work in the framework of noncommutative geometry, and focus on Connes distance associated to a spectral triple (A, H, D). A high momentum (short distance) cut-off is implemented by the action of a projection P on the Dirac operator D and/or on the algebra A. This action induces two new distances. We individuate conditions making them equivalent to the original distance. We also study the Gromov-Hausdorff limit of the set of truncated states, first for compact quantum metric spaces in the sense of Rieffel, then for arbitrary spectral triples. To this aim, we introduce a notion of "state with finite moment of order 1" for noncommutative algebras. We then focus on the commutative case, and show that the cut-off induces a minimal length between points, which is infinite if P has finite rank. When P is a spectral projection of $D$, we work out an approximation of points by non-pure states that are at finite distance from each other. On the circle, such approximations are given by Fejer probability distributions. Finally we apply the results to Moyal plane and the fuzzy sphere, obtained as Berezin quantization of the plane and the sphere respectively.Comment: Reference added. Minor corrections. Published version. 38 pages, 2 figures. Journal of Geometry and Physics 201

Fuzziness in Chang's fuzzy topological spaces

It is known that fuzziness within the concept of openness of a fuzzy set in a Chang's fuzzy topological space (fts) is absent. In this paper we introduce a gradation of openness for the open sets of a Chang jts (X, $\mathcal{T}$) by means of a map $\sigma\;:\; I^{x}\longrightarrow I\left(I=\left[0,1\right]\right)$, which is at the same time a fuzzy topology on X in Shostak 's sense. Then, we will be able to avoid the fuzzy point concept, and to introduce an adeguate theory for $\alpha$-neighbourhoods and $\alpha-T_{i}$ separation axioms which extend the usual ones in General Topology. In particular, our $\alpha$-Hausdorff fuzzy space agrees with $\alpha${*} -Rodabaugh Hausdorff fuzzy space when (X, $\mathcal{T}$) is interpreservative or $\alpha$-locally minimal