In multi-label classification, where the evaluation of predictions is less straightforward than in single-label various meaningful, though different, loss functions have been proposed. Ideally, learning algorithm should be customizable towards a specific choice performance measure. Modern implementations boosting, most prominently gradient boosted decision trees, appear to appealing from this p...

This paper studies Lebesgue integral of a fuzzy closed set-valued stochastic process with respect to the time t. Firstly, a progressively measurable fuzzy closed set-valued stochastic process is discussed and an almost everywhere problem in the former Aumann type Lebesgue integral of the level-set process is pointed out. Secondly, a new definition of the Lebesgue integral by decomposable closur...

In this article, we shall discuss some recent developments and applications of the local spectral theory for linear operators on Banach spaces. Special emphasis will be given to those parts of operator theory, where spectral theory, harmonic analysis, and the theory of Banach algebras overlap and interact. Along this line, we shall present the recent progress of the theory of quotients and rest...

Decomposability of an algebraic structure into the union of its sub-structures goes back to G. Scorza's Theorem of 1926 for groups. An analogue of this theorem for rings has been recently studied by A. Lucchini in 2012. On the study of this problem for non-group semigroups, the first attempt is due to Clifford's work of 1961 for the regular semigroups. Since then, N.P. Mukherjee in 1972 studied...

For a tree T , the graph X is T -decomposable if there exists a partition of the edge set of X into isomorphic copies of T . In 1963, Ringel conjectured that K2m+1 can be decomposed by any tree with m edges. Graham and Häggkvist conjectured more generally that every 2m-regular graph can be decomposed by any tree with m edges. Fink showed in 1994 that for any directed tree T with m arcs, the dir...

In this paper, the well-known Egoroff’s theorem in classical measure theory is established on Sugeno fuzzy measure spaces. Taylor’s theorem, which concerns almost everywhere convergence of measurable function sequence in classical measure theory, is also generalized. The converse problem of the theorems are discussed, and a necessary and sufficient condition for the Egoroff’s theorem is obtaine...

This chapter approaches the image retrieval system on the base of the colors of image. It creates fuzzy signature to describe the color of image on color space HSV and builds fuzzy Hamming distance (FHD) to evaluate the similarity between the images. In order to reduce the storage space and speed up the search of similar images, it aims to create S-tree to store fuzzy signature relies on FHD an...