In this paper, we introduce λκ−closed sets and study its properties in generalized topological spaces.

‎‎The purpose of this paper is to introduce new methods for constructing generalized groups, generalized topological groups and top spaces. We study some properties of these structures and present some relative concrete examples. Moreover, we obtain generalized groups by using of Hilbert spaces and tangent spaces of Lie groups, separately.

It is argued that the geometric dual of a noncommutative operator algebra represents a notion of quantum state space which differs from existing notions by representing observables as maps from states to outcomes rather than from states to distributions on outcomes. A program of solving for an explicitly geometric manifestation of quantum state space by adapting the spectral presheaf, a constru...

Topological approximation of self-formation of artificial objects in micro-scale is well established. However expansion of the topological approximation into nanoand molecular scales requires new and more generalized concepts of self-formation. This paper represents a preliminary effort to define self-formation as an evolution of 4-D parametric space in 8-D topological space, as universal conce...

The Toda lattice equation is a nonlinear evolutionary differential-difference equation introduced by Toda [1] describing an infinite system of masses on a line that interact through an exponential force which is used to explain the well-known FermiPasta-Ulam phenomenon. It was soon realized that this equation is completely integrable, i.e. admits infinite conserved quantities. It has important ...

In [5], a representation of bounded lattices within so-called standard topological contexts has been developed. Based on the theory of formal concept analysis [14] it includes Stone’s representation of Boolean algebras by totally disconnected compact spaces [10], Priestley’s representation of bounded distributive lattices by totally order disconnected compact spaces [7] as well as Urquhardt’s r...

A new methodology that combines the generalized topological indices (GTI) and the down hill simplex optimisation procedure has been developed to search for optimised quantitative structure–property relationship models. The structural interpretation of the optimal molecular descriptors is attained analytically by means of the GTIs decomposition in terms of geodesics (shortest paths) in the molec...

We study a wide class of free-fermion systems on a hypercubic lattice in spatial dimensions d ≥ 1. When the system is insulating, the topological properties, e.g., the integral quantization of the topological invariant, are protected by certain symmetries of the Hamiltonian against disorder. This generic feature is characterized by a generalized index theorem which is a noncommutative analogue ...

The lattice definition of the two-dimensional topological quantum field theory [Fukuma, et al, Commun. Math. Phys. 161, 157 (1994)] is generalized to arbitrary (not necessarily orientable) compact surfaces. It is shown that there is a one-to-one correspondence between real associative ∗-algebras and the topological state sum invariants defined on such surfaces. The partition and n-point functio...

Locally solid Riesz spaces have been widely investigated in the past several decades; but locally solid topological lattice-ordered groups seem to be largely unexplored. The paper is an attempt to initiate a relatively systematic study of locally solid topological lattice-ordered groups. We give both Roberts-Namioka-type characterization and Fremlin-type characterization of locally solid topolo...