Pythagorean Fuzzy Rough Connected Spaces

In this paper Pythagorean fuzzy rough set and Pythagorean fuzzy rough topological spaces are defined for the connected space. Then, the properties of connectedness are discussed with examples.

Repercussion of Conventional Complete Mandibular Denture Versus Single Implant Over-Denture on Retention and Biting Force

Aim: Aim of this study was the evaluate the retention and biting force of conventional complete denture and after placement of a single implant in the mandible for an implant retained over-denture Material and methods Eight completely edentulous patients were selected. A single implant was inserted at the symphysis of the mandible and left to osseointergrate. During the osseointegration period a conventional complete denture was fabricated and inserted. Retention and biting force was measured at insertion and after 3 months of service. After osseointegration attachment was connected, space was formed in the fitting surface of the existing mandibular denture and retention silicon was placed. Retention and biting force were measured at insertion and after 3 months follow-up. Data were collected and statistically analyzed. Results: The retention mean values for the complete denture was 2.420±0.360, however the retention mean values for the single implant over-denture was 6.395±0.289. F=3.80 with statically significance difference between the groups of P <0.01. The biting force mean values for the complete denture was 52.62±2.71, however the biting force mean values for single implant over-denture was 71.45±2.73. F= 1.790 with statically insignificance difference between the groups of P >0.01 Conclusion: Single implant overdenture improved the retention and the biting force when compared with the complete denture, this has improved the quality of life and happiness.

Infra -α- Compact and Infra -α- Connected Spaces

In 2016 Hakeem A. Othman and Md. Hanif Page introduced a new notion of set in general topology called an infra -α- open set and investigated its fundamental properties and studied the relationship between infra -α- open set and other topological sets. The objective of this paper is to introduce the new concepts called infra -α- compact space, countably infra -α- compact space, infra -α- Lindelof space, almost infra -α- compact space, mildly infra -α- compact space and infra -α- connected space in general topology and investigate several properties and characterizations of these new concepts in topological spaces.

New types of locally connected spaces via clopen set

In this paper, we define and study a new type of connected spaces called λco-connected space. It is remarkable that the class of λ-connected spaces is a subclass of the class of λco-connected spaces. We discuss some characterizations and properties of λco-connected spaces, λco components and λco-locally connected spaces

Some Results on mX-N-connected Space

In this essay, we utilize m - space to specify mX-N-connected, mX-N-hyper connected and mX-N-locally connected spaces and some functions by exploiting the intelligible mX-N-open set. Some instances and outcomes have been granted to boost our tasks.

Novel Idea of Gδ-α-Locally Continuous Functions

The new concepts of a neutrosophic Gδ set and neutrosophic Gδ-α-locally closed sets are introduced. Also, a neutrosophic εGδ-α-locally quasi neighborhood, neutrosophic Gδ-α-locally continuous function, neutrosophic Gδ-α-local T2 space, neutrosophic Gδ-α-local Urysohn space, neutrosophic Gδ-α-local connected space, and neutrosophic Gδ-α-local compact space are discussed and some interesting properties are established.

Recurrence Decompositions in Finsler Space

Finsler geometry is a kind of differential geometry originated by P. Finsler. Indeed, Finsler geometry has several uses in a wide variety and it is playing an important role in differential geometry and applied mathematics of problems in physics relative, manual footprint. It is usually considered as a generalization of Riemannian geometry. In the present paper, we introduced some types of generalized $W^{h}$ -birecurrent Finsler space, generalized $W^{h}$ -birecurrent affinely connected space and we defined a Finsler space $F_{n}$ for Weyl's projective curvature tensor $W_{jkh}^{i}$ satisfies the generalized-birecurrence condition with respect to Cartan's connection parameters $\Gamma ^{\ast i}_{kh}$, such that given by the condition (\ref{2.1}), where $\left\vert m\right. \left\vert n\right. $ is\ h-covariant derivative of second order (Cartan's second kind covariant differential operator) with respect to $x^{m}$ \ and $x^{n}$ ,\ successively, $\lambda _{mn}$ and $\mu _{mn~}$ are\ non-null covariant vectors field and such space is called as a generalized $W^{h}$ -birecurrent\ space and denoted briefly by $GW^{h}$ - $BRF_{n}$ . We have obtained some theorems of generalized $W^{h}$ -birecurrent affinely connected space for the h-covariant derivative of the second order for Wely's projective torsion tensor $~W_{kh}^{i}$ , Wely's projective deviation tensor $~W_{h}^{i}$ in our space. We have obtained the necessary and sufficient condition forsome tensors in our space.