The main purpose of this paper is to study feebly open and feebly closed mappings and we proved several results about that by using some concepts of topological feebly open and feebly closed sets , semi open (- closed ) set , gs-(sg-) closed set and composition of mappings.

The goal of the research is to introduce new types of maps called semi totally Bc-continuous map and totally Bc-continuous map furthermore, study its properties. Additionally, we study the relationship of these functions and other known mappings are discussed.

This work employs the conceptions of neutrosophic crisp a-open and semi-a-open sets to distinguish some novel forms of weakly neutrosophic crisp open mappings; for instance, neutrosophic crisp a-open mappings, neutrosophic crisp a*-open mappings, neutrosophic crisp a**-open mappings, neutrosophic crisp semi-a-open mappings, neutrosophic crisp semi-a*-open mappings, and neutrosophic crisp semi-a**-open mappings. Moreover, the close connections between these forms of weakly neutrosophic crisp open mappings and the viewpoints of neutrosophic crisp open mappings are explained. Additionally, various theorems and related features and notes are submitted.

      Some cases of common fixed point theory for classes of generalized nonexpansive maps are studied. Also, we show that the Picard-Mann scheme can be employed to approximate the unique solution of a mixed-type Volterra-Fredholm functional nonlinear integral equation.

        In our research, we introduced new concepts, namely ^*and **-light mappings, after we knew ^*and **-totally disconnected mappings through the use of -open sets.

Many examples, facts, relationships and results have been given to support our work.

Speech recognition is a very important field that can be used in many applications such as controlling to protect area, banking, transaction over telephone network database access service, voice email, investigations, House controlling and management ... etc. Speech recognition systems can be used in two modes: to identify a particular person or to verify a person’s claimed identity. The family speaker recognition is a modern field in the speaker recognition. Many family speakers have similarity in the characteristics and hard to identify between them. Today, the scope of speech recognition is limited to speech collected from cooperative users in real world office environments and without adverse microphone or channel impairments.

In this article, the additivity of higher multiplicative mappings, i.e., Jordan mappings, on generalized matrix algebras are studied. Also, the definition of Jordan higher triple product homomorphism is introduced and its additivity on generalized matrix algebras is studied.