Most real-life situations need some sort of approximation to fit mathematical models. The beauty of using topology in approximation is achieved via obtaining approximation for qualitative subgraphs without coding or using assumption. The aim of this paper is to apply near concepts in the -closure approximation spaces. The basic notions of near approximations are introduced and sufficiently illustrated. Near approximations are considered as mathematical tools to modify the approximations of graphs. Moreover, proved results, examples, and counterexamples are provided.

The primary objective of this paper is to present a new concept of fibrewise topological spaces over B is said to be fibrewise slightly topological spaces over B. Also, we introduce the concepts of fibrewise slightly perfect topological spaces, filter base, contact point, slightly convergent, slightly directed toward a set, slightly adherent point, slightly rigid, fibrewise slightly weakly closed, H.set, fibrewise almost slightly perfect, slightly∗ .continuous fibrewise slightly∗ topological spaces respectively, slightly Te, locally QHC, In addition, we state and prove several propositions related to these concepts.

Form the series of generalization of the topic of supra topology is the generalization of separation axioms . In this paper we have been introduced (S * - SS *) regular spaces . Most of the properties of both spaces have been investigated and reinforced with examples . In the last part we presented the notations of supra *- -space ( =0,1) and we studied their relationship with (S * - SS *) regular spaces.

Abstract. In this study, we shall research the fibrewise micro ideal topological spaces over Ḃ, as well as the relationship between fibrewise micro ideal topological spaces over Ḃ and fibrewise micro topological spaces over Ḃ. At first present introduces a novel notion from fibrewise micro ideal topological spaces over Ḃ, and differentiates it from fibrewise micro topological spaces over Ḃ. Some fundamental characteristics from these spaces are studied. Then show discussed the fibrewise micro ideal closed and micro ideal open topologies. Many propositions relating to these ideas are offered. In the next part will study defines and investigates novel conceptions from fibrewise micro ideal topological spaces over Ḃ, particularly f

The theory of Topological Space Fiber is a new and essential branch of mathematics, less than three decades old, which is created in forced topologies. It was a very useful tool and played a central role in the theory of symmetry. Furthermore, interdependence is one of the main things considered in topology fiber theory. In this regard, we present the concept of topological spaces α associated with them and study the most important results.

The main purpose of this paper is to introduce a some concepts in fibrewise bitopological spaces which are called fibrewise , fibrewise -closed, fibrewise −compact, fibrewise -perfect, fibrewise weakly -closed, fibrewise almost -perfect, fibrewise ∗-bitopological space respectively. In addition the concepts as - contact point, ij-adherent point, filter, filter base, ij-converges to a subset, ij-directed toward a set, -continuous, -closed functions, -rigid set, -continuous functions, weakly ijclosed, ij-H-set, almost ij-perfect, ∗-continuous, pairwise Urysohn space, locally ij-QHC bitopological space are introduced and the main concept in this paper is fibrewise -perfect bitopological spaces. Several theorems and characterizations c

In this paper, we introduce the concept of fuzzy n-fold KUideal in KU-algebras, which is a generalization of fuzzy KU-ideal of KUalgebras and we obtain a few properties that is similar to the properties of fuzzy KU-ideal in KU-algebras, see [8]. Furthermore, we construct some algorithms for folding theory applied to KU-ideals in KU-algebras.

          The use of silicon carbide is increasing significantly in the fields of research and technology. Topological indices enable data gathering on algebraic graphs and provide a mathematical framework for analyzing the chemical structural characteristics. In this paper, well-known degree-based topological indices are used to analyze the chemical structures of silicon carbides. To evaluate the features of various chemical or non-chemical networks, a variety of topological indices are defined. In this paper, a new concept related to the degree of the graph called "bi-distance" is introduced, which is used to calculate all the additive as well as multiplicative degree-based indices for the isomer of silicon carbide, Si₂

Nano crystalline copper sulphide (Cu2S) thin films pure and 3% Bi doped were deposited on glass substrate by thermal evaporation technique of thickness 400±20 nm under a vacuum of ~ 2 × 10− 5 mbar to study the influence of annealing temperatures ( as-deposited, and 573) K on structural, surface morphology and optical properties of (Cu2S and Cu2S:3%Bi). (XRD) X-ray diffraction analysis showed (Cu2S and Cu2S:3%Bi) films before and after annealing are polycrystalline and hexagonal structure. AFM measurement approves that (Cu2S and Cu2S:3%Bi) films were Nano crystalline with grain size of (105.05-158.12) nm. The optical properties exhibits good optical absorption for Cu2S:3%Bi films. Decreased of optical band gap from 2.25 to 2 eV after dop

Cadmium Oxide and Bi doped Cadmium Oxide thin films are prepared by using the chemical spray pyrolysis technique a glass substrate at a temperature of (400?C) with volumetric concentration (2,4)%. The thickness of all prepared films is about (400±20) nm. Transmittance and Absorbance spectra are recorded in the wave length ranged (400-800) nm. The nature of electronic transitions is determined, it is found out that these films have directly allowed transition with an optical energy gap of (2.37( eV for CdO and ) 2.59, 2.62) eV for (2% ,4%) Bi doped CdO respectively. The optical constants have been evaluated before and after doping.