# International Journal of Pure Mathematics

**E-**ISSN: 2313-0571

Volume 7, 2020

Notice: As of 2014 and for the forthcoming years, the publication frequency/periodicity of NAUN Journals is adapted to the 'continuously updated' model. What this means is that instead of being separated into issues, new papers will be added on a continuous basis, allowing a more regular flow and shorter publication times. The papers will appear in reverse order, therefore the most recent one will be on top.

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**Volume 7, 2020**

Title of the Paper: **Beta- Star- Continuity and Beta- Star- Contra- Continuity**

Authors: **Raja Mohammad Latif**

Pages: **43-66**

**http://doi.org/10.46300/91019.2020.7.6**

Abstract: In 2014 Mubarki, Al-Rshudi, and Al- Juhani introduced and studied the notion of a set in general topology called β*-open set and investigated its fundamental properties and studied the relationships between β*-open set and other topological sets including β*-continuity in topological spaces. We introduce and investigate several properties and characterizations of a new class of functions between topological spaces called β*- open, β*- closed, β*- continuous and β*- irresolute functions in topological spaces. We also introduce slightly β*- continuous, totally β*- continuous and almost β*- continuous functions between topological spaces and establish several characterizations of these new forms of functions. Furthermore, we also introduce and investigate certain ramifications of contra continuous and allied functions, namely, contra β*- continuous, and almost contra β*-continuous functions along with their several properties, characterizations and natural relationships. Moreover, we introduce new types of closed graphs by using β*- open sets and investigate its properties and characterizations in topological spaces.

Title of the Paper: **Introduction to Hyperspaces**

Authors: **Mark Burgin**

Pages: **37-42**

**http://doi.org/10.46300/91019.2020.7.5**

Abstract: The development of mathematics brought mathematicians to infinite structures. This process started with transcendent real numbers and infinite sequences going through infinite series to transfinite numbers to nonstandard numbers to hypernumbers. From mathematics, infinity came to physics where physicists have been trying to get rid of infinity inventing a variety of techniques for doing this. In contrast to this, mathematicians as well as some physicists suggested ways to work with infinity introducing new mathematical structures such distributions and extrafunctions. The goal of this paper is to extend mathematical tools for treating infinity by considering hyperspaces and developing their theory.

Title of the Paper: ** M – STAR – Irresolute Topological Vector Spaces**

Authors: **Raja Mohammad Latif**

Pages: **20-36**

**http://doi.org/10.46300/91019.2020.7.4**

Abstract: In 2016 A. Devika and A. Thilagavathi introduced a new class of sets called M*-open sets and investigated some properties of these sets in topological spaces. In this paper, we introduce and study a new class of spaces, namely M*-irresolute topological vector spaces via M*-open sets. We explore and investigate several properties and characterizations of this new notion of M*-irresolute topological vector space. We give several characterizations of M*-Hausdorff space. Moreover, we show that the extreme point of the convex subset of M*-irresolute topological vector space X lies on the boundary.

Title of the Paper: ** The Dаrbоuх Theory and Geodesics in the Metric Space with Torsion**

Authors: **Yaremenko Mykola Ivanovich**

Pages: **8-19**

**http://doi.org/10.46300/91019.2020.7.3**

Abstract: The geometry of n Y^{n} space is generated congruently together by the metric tensor and the torsion tensor. In the presented article has been obtained an analog of the Dаrbоuх theory in the n Y^{n} space, also studied the deduction of the equation of the geodesic lines on the hypersurface that embedded in such spaces, showed that in the n Y^{n} space the structure of the curvature tensor has special features and for curvature tensor obtained Ricci - Jacobi identity. We establish that the equations of the geodesics have additional summands, which are caused by the presence of torsion in the space. In n Y^{n} space, the variation of the length of the geodesic lines is proportional to the product of metric and torsion tensors g_{ij}S^{j}_{pk}. We have introduced the second fundamental tensor π_{αβ} for the hypersurface n Y^{n-1} and established its structure, which is fundamentally different from the case of the Riemannian spaces with zero torsion. Furthermore, the results on the structure of the curvature tensor have been obtained.

Title of the Paper: ** The New Methodological Approach to Division of Angle, With Change of Dimensionality and Perspective for the Sake of Improved Efficiency, Faster Insight to the Essence of the Process and Problem-solving in Real Time and Under Outdoor Conditions**

Authors: **D. Knežević, M. Laban**

Pages: **4-7**

**http://doi.org/10.46300/91019.2020.7.2**

Abstract: The dividing of an angle at a certain proportion in an unambiguous way was shown by an analytical approach, taking into account the prospects and dimensionality [3]. Looking at the problem from the standpoint of antique principles and simultaneous contemporary approach to descriptive geometry proved to be necessary modernization of the tools used in solving of the practical construction problems by drawing. Importance of drawing on paper respective software tools used on PCs and tablets was emphasized. The fastest and most reliable solutions mostly were presented by simple models. The application of this methodological approach is seen in a system for monitoring remote objects [1, 2], since the method allows easier and faster transformations (addition, subtraction, multiplication and division) of the values of the measured angular widths with respect to: the reference direction and the relative value of angle covers between multiple detected objects

Title of the Paper: ** The Producing Functionals and its Applications**

Authors: **Muhammadyusuf Yunusi **

Pages: **1-3**

**http://doi.org/10.46300/91019.2020.7.1**

Abstract: The report is devoted to one class so-called producing functionals and corresponding with its modeling generating spaces, some transformations building models equations and corresponding spaces are considered and investigated. It is shown that these transformations any point of Euclidean space E^{m-1} (generating spaces M^{m-1}) transfer into some corresponding points E^{m} (or M^{m}) and conversely. It is also proposed one class general equations which are described many processes of chancing and generating spaces. Besides it is shown that processes of generating spaces are determined by some no depending prescribes points. Given others applications and from economics and physics.