We introduce the consistent topological structure and neighborhood structure into the logical framework for providing the logical foundation and logical normalization for the approximate reasoning. We present the concept of the formulae mass, theknowledge mass and the approximating knowledge closure of the knowledge library by means of topological closure. We obtain the fundamental framework of...

The biochemical networks can present the relationships between genes and gene products, proteins, metabolites and etc. The exploration of these networks helps to understand cellular processes, functions or properties of biological system. The growing size of interaction models of biological system building elements request determination of the most important topological measurements of given ta...

We present a theory of topological edge states in onedimensional resonant photonic crystals with compound unit cell. We demonstrate how the structure, despite being onedimensional, can be characterized by topological indices. Contrary to conventional electronic topological states the modes under consideration are radiative, i.e., they decay in time due to the light escape through the structure ...

We study algebraic properties of the Brandt λ-extensions of monoids with zero and non-trivial homomorphisms between the Brandt λ-extensions of monoids with zero. We introduce finite, compact topological Brandt λextensions of topological semigroups and countably compact topological Brandt λ-extensions of topological inverse semigroups in the class of topological inverse semigroups and establish ...

in this paper we find sufficient conditions on primitive inverse topological semigroup s under which: the inversion inv : (h(s)) (h(s)) is continuous; we show that every topologically periodic countable compact primitive inverse topological semigroups with closed h-classes is topologically isomorphic to an orthogonal sum p i2= bi (gi) of topological brandt extensions bi (gi) of countably compac...

Finite simple graphs are a playground for classical areas of mathematics. We illustrate this by looking at some results.

Irregular pyramids organise several topological partitions which can be deduced from a partition (called the base of the pyramid) by successive unions of regions. In this paper, we introduce the redundancy pyramid structure. This structure accounts for redundant topological structures present in several topological partitions. We apply it to the problem of segmentation fusion, where we show tha...

Monte Carlo studies of pure glue SU(3) gauge theory using the overlap-based topological charge operator have revealed a laminar structure in the QCD vacuum consisting of extended, thin, coherent, locally 3-dimensional sheets of topological charge embedded in 4D space, with opposite sign sheets interleaved. In this talk I discuss the interpretation of these Monte Carlo results in terms of our cu...

We give the operadic formulation of (weak, strong) topological vertex algebras, which are variants of topological vertex operator algebras studied recently by Lian and Zuckerman. As an application, we obtain a conceptual and geometric construction of the Batalin-Vilkovisky algebraic structure (or the Gerstenhaber algebra structure) on the cohomology of a topological vertex algebra (or of a weak...