The classical Baker-Campbell-Hausdorff formula gives a recursive way to compute the Hausdorff series H = ln(ee ) for noncommuting X, Y . Formally H lives in the graded completion of the free Lie algebra L generated by X, Y . We present a closed explicit formula for H = ln(ee ) in a linear basis of the graded completion of the free metabelian Lie algebra L̄ = L/[[L, L], [L, L]].

Crisp and L-fuzzy ambiguous representations of closed subsets of one space by closed subsets of another space are introduced. It is shown that, for each pair of compact Hausdorff spaces, the set of (crisp or L-fuzzy) ambiguous representations is a lattice and a compact Hausdorff Lawson upper semilattice. The categories of ambiguous and L-ambiguous representations are defined and investigated.

This paper investigates the fractals generated by the dynamical system of intuitionistic fuzzy contractions in the intuitionistic fuzzy metric spaces by generalizing the Hutchinson-Barnsley theory. We prove some existence and uniqueness theorems of fractals in the standard intuitionistic fuzzy metric spaces by using the intuitionistic fuzzy Banach contraction theorem. In addition to that, we an...

Abstract This paper aims to present the concept of multi-valued mappings in fuzzy cone metric spaces and prove some basic lemmas, a Hausdorff metric, fixed point results for set-valued cone-contraction mappings. We theorem rational type cone-contractions spaces. Our extend improve given literature.

We develop a matricial version of Rieffel’s Gromov-Hausdorff distance for compact quantum metric spaces within the setting of operator systems and unital C∗-algebras. Our approach yields a metric space of “isometric” unital complete order isomorphism classes of metrized operator systems which in many cases exhibits the same convergence properties as those in the quantum metric setting, as for e...

In this paper it is described a method to compute the distance between sets, that implies the formation of distance functions different from Hausdorff metric. Two functions with metric properties, which describe quantitatively distances between sets, are formed. First function can be used for sets arbitrary situated from each other. Second distance is more suited for sets clustered by rank link...

This paper presents an empirical evaluation of a number of recently developed Automatic Target Recognition algorithms for Forward-Looking Infrared (FLIR) imagery using a large database of real FLIR images. The algorithms evaluated are based on convolutional neural networks (CNN), principal component analysis (PCA), linear discriminant analysis (LDA), learning vector quantization (LVQ), modular ...

The main motivation of this paper arises from the study of Carnot–Carathéodory spaces, where the class of 1-rectifiable sets does not contain smooth non-horizontal curves; therefore a new definition of rectifiable sets including non-horizontal curves is needed. This is why we introduce in any metric space a new class of curves, called continuously metric differentiable of degree k, which are Hö...

*Correspondence: [email protected]; [email protected]; [email protected] 1Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), 126 Pracha Uthit Rd., Bang Mod, Thung Khru, Bangkok, 10140, Thailand 2Department of Mathematics and Statistics, Faculty of Science and Technology, Thammasat University Rangsit Center, Pathumthani, 1...

A Hausdorff measure version of W.M. Schmidt’s inhomogeneous, linear forms theorem in metric number theory is established. The key ingredient is a ‘slicing’ technique motivated by a standard result in geometric measure theory. In short, ‘slicing’ together with the Mass Transference Principle [3] allows us to transfer Lebesgue measure theoretic statements for lim sup sets associated with linear f...