#### Series-1 (Jul. – Aug. 2020)**Jul. – Aug. 2020 Issue Statistics**

**Jul. – Aug. 2020 Issue Statistics**

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**Index Page****Cover Paper**

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**Abstract: **The book on 'Vedic Mathematics' written by Jagadguru Sankaracarya Sri Bharati Krisna Tirthaji Maharaja contains 16 Sutras or formulas and 13 Sub-Sutras or corollaries to carry out various arithmetical operations. The sutra 'Urdhva Tiryagbhyam' is used for multiplication of two decimal numbers containing equal number of digits. The multiplication of two binary numbers having same number of bits can be also done using this sutra. Using this sutra, two polynomials can also be multiplied. The sutra 'Nikhilam Navatascaramm Dasatah' is applied to multiplication of decimal numbers, where the multiplicand and multiplier contain the same number of digits and the numbers are such that one can choose a base (in powers of ten) that is nearest to the numbers to be multiplied.The sub-sutra 'Anurupyena' is used to find the cube of a decimal number and also a binary number......

**Keywords:** Vedic mathematics, Urdhva tiryagbhyam, Anurupyena

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**Abstract: **Well test analysis of a horizontal well is complex and difficult to interpret. Most horizontal well mathematical models assume that horizontal wells are perfectly horizontal and are parallel to the top and bottom boundaries of the reservoir. As part of effort towards correct horizontal well test analysis, the purpose of this study is to develop a mathematical model using source and Green's functions for a horizontal well completed in an oil reservoir at late time flow period, where the reservoir is bounded by an edge and bottom constant pressure boundaries...

**Keywords:** Oil Reservoir, Horizontal Well, Two Constant Pressure Boundaries, Late Flow Period....

[1]. Gringarten, A. C., Ramey, H. J., "The Use of Source and Green‟s Function in Solving Unsteady-Flow Problem in Reservoir", SPEJ. Vol.13(05).pp.285-295.SPE3818. doi: 10.2118/3818-PA., October, 1973.

[2]. Carvalho, R.S., and Rosa, A.J., "A Mathematical Model for Pressure Evaluation in an infinite-Conductivity Horizontal well", SPE Formation Evaluation Vol.4 (4): pp.559-566. SPE 15967-PA. doi:10.2118/15967-PA., December,1989.

[3]. Ozkan Erdal and Raghavan Rajagopal, "Performance of Horizontal Wells Subject to Bottom Water Drive", SPE-18559-PA Vol.5(03), doi:10.2118/18559-PA., August,1990.

[4]. Lu, J. and Lin, T., "A Mathematical Model of Horizontal Wells Pressure Drawdown and Buildup", PETSOC-02-10-02. Vol.41(10). doi:10.2118/02-10-02., October,2002.

[5]. Al-Rbeawi, S. J. H. and Tiab, D., "Transient Pressure Analysis of Horizontal Wells in a Multi-Boundary System", Presented at the SPE Production and Operations Symposium, Oklahoma City, Oklahoma, 27–29 March. SPE-142316-MS. https://doi.org/10.2118/142316-MS., March,2011..

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**Abstract: **Studies have been done on the aspect ratio effect on natural convection turbulence using standard k-ε model but further studies showed thatk-ω SST model performed better than both k-ε and k-ω model in the whole enclosure. Thus, there was need to do a numerical study on the natural convection fluid flow in a rectangular enclosure full of air using SST k-ω model. The left vertical wall of the enclosure was maintained at a steady high temperature Th of 323K while the right wall at a steady cool temperature Tc of 303K with the remaining walls adiabatic. Time-averaged energy, momentum and continuity equations with the two equation SST k-omega turbulence model were used to generate isotherms, streamlines and velocity magnitudes for different aspect ratios of the enclosure so as to be able to investigate effect of aspect ratio on turbulence. It was shown that as the aspect ratio of

**Key terms** Convection: is heat transfer through movement of the heated sections of a fluid....

[1]. Aksel, M.H. 2003.Fluid mechanics. Ankara: METU Press.

[2]. ANSYS (2012), Inc., ANSYS Fluent 14.0 Theory Guide.

[3]. Awuor, K. O. (2013). Simulating Natural Turbulent Convection Fluid Flow in an Enclosure the Two-Equation Turbulent Models (Doctoral dissertation).

[4]. Awuor, K. O., & Gicheru, M. G. M. (2017). Numerical simulation of natural convection in rectangular enclosures. International Journal for Innovative Research in Multidisciplinary Field 3(7) pp. 306-313

[5]. Aydin, O.Ünal, A.&Ayhan, T. (1999). Natural convection in rectangular enclosures heated from one side and cooled from the ceiling. International journal of heat and mass transfer, 42(13),.

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**Abstract: **This paper establishes Non - Reducible Farey N - Subsequence of order (4, 6,...). This research formulates iterated function systems by using HB Operator and additionally shows the generalized Non - Reducible Farey N - Subsequence of order (4, 6,...)..

**Keywords:** Arithmetic Mediant, Farey 'N' subsequence, Hausdorff dimension, Invariant Measure, Iterated Function System, Markov Operator, Non - Reducible Farey Sequence

[1]. Don Redmond, Number Theory An Introduction, Marcel Dekker, Inc. 1996.

[2]. Gnanam A and Dinesh C., "Extraction of Cantor Middle 𝜔=25 ,37 from Non – Reducible Farey Subsequence", International Journal of Scientific Engineering and Research (IJSER), ISSN: 2347 – 3878.

[3]. Hansraj Gupta, Selected Topics in Number Theory, Abacus Press 1980.

[4]. Jahurul Islam Md and Shahidul Islam Md., "Invariant Measures for Iterated Function Systems of Generalized Cantor Sets", German Journal of Advanced Mathematical Sciences (GJAMS), Vol.1, Issue.2, pp. 31-47, 2016.

[5]. Thomas Koshy, "Elementary Number Theory with Applications", Academic Press, an Imprint of Elsevier, USA..

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**Abstract: **The model equations which exhibits the disease-free equilibrium 𝑬𝟎 state for COVID-19 coronavirus does not exist and hence does not satisfy the criteria for a locally or globally asymptotic stability when the basic reproductive number 𝑅0=1 for and endemic situation. This implies that the COVID-19 coronavirus does not have a curative vaccine yet and precautionary measures are advised through quarantine and observatory procedures. The basic reproductive number was found to be 𝑅0<1and hence shows that there is a chance of decline of secondary infections when the ratio between the incidence rate in the population and the total number of infected population quarantined with observatory..

Keywords:** coronavirus pandemic globally, COVID-19 coronavirus, mathematical modeling of infection disease, SEIRUS-model, parameter identification, statistical methods**..

[1]. Nesteruk, I. (2020). Statistics based predictions of coronavirus 2019-nCoV spreading in mainland China. MedRxiv.

[2]. Ming, W. K., Huang, J., & Zhang, C. J. (2020). Breaking down of healthcare system: Mathematical modelling for controlling the novel coronavirus (2019-nCoV) outbreak in Wuhan, China. bioRxiv.

[3]. Oduwole, H. K. and Kimbir, A. R. (2018). "Modelling vertical transmission and the effect of Antiretroviral Therapy (ART) on the dynamics of HIV/AIDS in an age-structured population in Nigeria. Journal of Natural and Applied Science-NasaraScientifique, Vol. 7 No. 1 pp. 51 – 78.

[4]. Victor, A. O. and Oduwole, H. K. (2020).Evaluating the deterministic SEIRUS mode for disease control in an age-structured population. Global Scientific Journal. Vol. 8. Issue 3.

[5]. Gerald T. (2012). Ordinary Differential Equations and Dynamical Systems. American Mathematical Society. URL: http://www.mat.univie.ac.at/~gerald/

#### Series-2

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**Index Page****Cover Paper**

**Abstract: **In this study, we use a Hilbert-space valued, time-inhomogeneous exponential to model the forward curve. The energy forward is represented as an element of Hilbert-space of function. Representing energy forward and futures contracts as a time-changing stochastic process in a Hilbert-space of functions shows clearly, that an arbitrage-free forward price can be derived from the buy-and hold strategy in the energy market thereby enabling investors in the market willing to be salvage from the market uncertainties as well as Arrow-Debreu situations to execute a spot or forward contracts depending on the time and place the market becomes favorable. With a clock measuring speed of evolution or data frequency for the energy stock market, the distribution of the increments of the L´evy process with the subordinator is subordinated to the distribution of increments of the L´evy process and the results was utilized to price forward contracts of a sample electricity commodity.

**Keywords:** Energy; subordination; Hilbert-space; jump-diffusion; prices; stochastic; futures; forwards; contract; volatility; l´evy process; energy; jumps.

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Paper Type |
: | Research Paper |

Title |
: | Scalar Field Influenced Robertson –Walker Model |

Country |
: | Bangladesh |

Authors |
: | Md. Ismail Hossain || Sarmin Islam |

: | 10.9790/5728-1604021926 |

**Abstract: **Earlier work was done by [1] considering when a dust(i.e., p 0 ) universe has always
2
1
q for
all values of spatial curvature where they have shown that if a scalar field similar to the one mediating inflation

is effective even now, then the Robertson – Walker model may have some interesting properties under certain
conditions. In this case even a spatially closed universe may expand forever. The critical density is also less

than in the standard Robertson – Walker model. In this work we have considered not only for.....

**Key Word:** Relativity, Scalar Field, Cosmology

[1] L.P.Sarkar and Banerji, Relativity and Gravitation, Vol. 25, No. 11, 1993

[2] C. Brans and R.H. Dicke. 1961. Mach's Principle and a Relativistic theory of Gravitations. Phys. Rev. 124, 125.

[3] R.H. Dicke 1962. Mach's Principle and invariance under Transformation of units, Phys. Rev. 125. 2163

[4] STEVEN WEINBERG, Gravitation & Cosmology, Principles Applications of the general theory of Relativity.

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**Abstract: **In this paper, the Homotopy Analysis Method (HAM) is applied to obtain the solution of fractional differential equations. The fractional derivatives are described in the Caputo sense. The solution obtained by this method has been compared with those obtained by Homotopy Perturbation Method (HPM) and the Variational Iteration Method (VIM). Results show that (HPM) and (VIM) are all special cases of the homotopy analysis method (HAM) when the nonzero convergence-control parameter ℏ=−1.

**Keywords:** Ordinary Fractional differential equation, Homotopy Analysis Method, Homotopy Perturbation Method, Variational Iteration Method

[1]. He JH. Variational iteration method for delay differential equations. Commun Nonlinear Sci Numer Simulat 1997; 2(4):235–6.

[2]. He JH. Approximate analytical solution for seepage flow with fractional derivatives in porous media. Comput Meth Appl Mech Eng 1998; 167:57–68.

[3]. He JH. Approximate solution of nonlinear differential equations with convolution product nonlinearities. Comput Meth Appl Mech Eng 1998; 167:69–73.

[4]. He JH. Variational iteration method—a kind of non-linear analytical technique: some examples. Int J Nonlinear Mech 1999; 34:699–708.

[5]. He JH. Variational iteration method for autonomous ordinary differential systems. Appl Math Comput 2000; 114:115–23..

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**Abstract: **In this paper, the homotopy analysis method (HAM) is applied to solve a time-fractional nonlinear partial differential equation. The fractional derivatives are described by Caputo's sense, and the (HAM) gives a series of solutions which converge rapidly within a few terms with the help of the nonzero convergence control parameter ℏ. After applying this method we reach the conclusion that the HAM is very efficient and accurate. Graphical representations of the solution obtained

**Keywords:** Fractional calculus, Homotopy Analysis Method, Time-fractional nonlinear partial differential equation..

[1]. Mehta MR, Dasgupta C, Ullal GR. A neural network model for kindling of focal epilepsy: Basic mechanism. Biological Cybernetics, 1993; 68(4): 335-340.

[2]. Wulsin DF, Fox EB, Litt B. Modeling the complex dynamics and changing correlations of epileptic events. Artificial Intelligence, 2014; 216: 55-75.

[3]. Jedynak M, Pons AJ, Garcia-Ojalvo J. Collective excitability in a mesoscopic neuronal model of epileptic activity. Physical Review E, 2018; 97: 012204.

[4]. Hodgkin AL, Huxley AA. A quantitative description of membrane current and its application to conduction and excitation in nerve. Journal of Physiology, 1952; 117: 500-544.

[5]. Zetternerg LH, Kristiansson L, Mossberg K. Performance of a model for a local neural population. Biological Cybernetics, 1978; 31: 15-26.

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**Abstract: **Model: Small neuron clusters can demonstrate the hypersynchronized epileptiform regime which is very similar to the real human brain epilepsy. The model of Hodgkin-Huxley neuron driven by the stimulating current coming from the axon is used in our approach for a single cell. One of the neuronss plays a role of a control element acting autonomously: it tracks the outputs of other companions in the cluster and detects the epileptiform behavior in the collective dynamics. This control element possesses a feedback loop to some part of the cluster neurons and sends the signal to their inputs to suppress the epileptiform regime. Methods: The Kolesnikov's sub-optimal feedback algorithm is used together with our approach of the 'control back propagation' in the network. It consist....

**Key Word:** Hodgkin-Huxley neuron; Epileptiform Dynamics; Kolesnikov's Feedback

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Paper Type |
: | Research Paper |

Title |
: | Some special operators on Multi Interval Valued Fuzzy Soft
Matrix |

Country |
: | India |

Authors |
: | P.Rajarajeswari || J.Vanitha |

: | 10.9790/5728-1604024854 |

**Abstract:**In this paper, we have defined two Special Operators ⊕,⊗ on a new type of Matrix called Multi
Interval valued Fuzzy Soft Matrix. Also we have studied some of their properties. The concepts are illustrated

with suitable examples.

**Key Words:** Soft Set, Fuzzy Soft Set, Multi-Fuzzy Soft Set, Multi-Interval-Valued Fuzzy Soft Set, Multi
Interval Valued Fuzzy Soft Matrix.

[1]. Atanassov,K., Intuitionistc fuzzy sets, Fuzzy Sets and Systems,20(1)(1986),87-96.

[2]. Atanassov, K., Intuitionistic fuzzy sets, Physica-Verlag, Heidelberg, Newyork, 1999.

[3]. Baruah H.K.,Towards Forming a Field of Fuzzy Sets.International Journal of Energy, Information and
Communications,Vol.2,Issue 1(2011) 16-20.

[4]. Baruah H.K.,The Theory of Fuzzy Sets. Beliefs and Realities.International Journal of Energy, Information and
Communications,Vol.2,Issue 2(2011) 1-22.

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**Abstract: **This article, discusses about forming rational fraction through the sum of four Egyptian fractions
through various methods. Several these methods include the geometric series method, the pairing method, and

the splitting method. The determination of unit fraction to form the sum of four unit fractions, specifically based
on Simon Brown's article on the formation of the sum of three unit fractions and it is added with the splitting

method. The splitting method is a method for converting one fraction unit into a sum of two different unit
fractions. This splitting method is based on what is commonly referred to as splitting identity. By combining the

two methods, it is obtained the general form of the sum of four unit fractions.

**Key Word:**Egyptian Fractions, Geometry Series, Pairing Method, Splitting Method

[1]. D. M. Burton, The History of Mathematics, An Introduction, New York, 2007.

[2]. B. G. KartasasmitadanWahyudin, Mathematical History and Philosophy, Terbuka University, Jakarta, 2014, 1--47.

[3]. S. A. Rukmono, Building Egyptian fractions using approximation-based methods, Papers IF2251 Algorithmic Strategy, 2008, 1--4.

[4]. S. Brown, Bounds of the denominators of Egyptian fractions, World Applied Programming, 2 (2012), 425--430.

[5]. S. Brown, An alternative approach to estimating the bounds of the denominators of Egyptian fractions, Leonardo Journal of Sciences, 2013.

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Paper Type |
: | Research Paper |

Title |
: | Weak separation axioms in terms of R-I-open sets |

Country |
: | India |

Authors |
: | SANGEETHA M.V. || BABY CHACKO |

: | 10.9790/5728-1604026367 |

**Abstract: **Throughout this paper, we study some weak separation axioms in ideal topological spaces using R-I-open sets and certain properties of the same. 2010 Mathematics Subject Classification: 54A05

**Key Word:** R-I-R0 space, R-I-R1 space

[1]. A. S. Davis. Indexed systems of neighborhoods for general topological spaces, American Mathematical Society, 68(1961), 886–893.

[2]. D.Jankovic and T.R.Hamlett, New topologies from old via ideals, Amer.Math.Monthly, 97(1990), 295-310.

[3]. D. W. Hall, S. K. Murphy, and J. Rozycki. On spaces which are essentially T1, Journal of the Australian Mathematical Society, 12(1971), 451–455.

[4]. F. G. Arenas, J. Dontchev, and M. L. Puertas. Idealization of some weak separation axioms, Acta Mathematica Hungarica, 89(1)(2000), 47–53.

[5]. K. K. Dube. A note on R0 topological spaces, Matematickivesnik, 11(1974), 203–208.

#### Series-3

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**Index Page****Cover Paper**

**Abstract: **Background: This paper deals with the construction of a family of implicit one-block methods for the solution of
stiff problems using four different linear multistep methods.

Method: This is done by applying shift operator on the quadruple: Reversed Generalized Adams Moulton
(RGAM), Generalized Backward Differentiation Formula (GBDF), Top Order Method (TOM) and Backward

Differentiation Formula (BDF).
Results: The application of the shift operator on the quadruples is done in such a manner that the resultant oneblock

methods are self-starting and forms a family. Orders four and seven are L-stable.
Conclusion: Numerical experiments carried out using orders four, seven and ten of the family show that the

methods are good for solving stiff initial value problems.

**Keywords:** Stiff initial value problem; One-block methods; Self-starting; quadruple and shift operator.

[1]. Ajie, I. J., Ikhile, M. N. and Onumanyi, P.: Variable Order Implicit L(α)-Stable Block Methods for Stiff Ordinary Differential

Equations. Journal of Mathematical Sciences, 2014, vol. 3 No 1: National Mathematical Centre (NMC), Abuja, Nigeria.

[2]. Ajie, I.J., Utalor, K. and Onumanyi, P. A Family of High Order One-Block Methods for the Solution of Stiff Initial Value Problems.

Journal of Advances in Mathematics and Computer Science. 2019, 31(6), 1-14.

[3]. Amodio, P. and Mazzia, F. (1995). A boundary value approach to the numerical solution of ODE by multistep methods, J.

Difference Equations Appl.vol.1 pp. 353-367.

[4]. Brugnano, L. and Trigiante, D., Solving differential problems by multistep initial and boundary value methods. Published by Gordon and Beach Science Publishers, 1998.

[5]. Butcher, J. C. and Jackiewicz, Z.. Construction of GLM with Runge-Kutta stability properties Numer Alg., 2004, 36, pp. 53-72..

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Paper Type |
: | Research Paper |

Title |
: | On (f,g)-Derivations in BG-algebras |

Country |
: | Indonesia |

Authors |
: | Wellya Aziz || Sri Gemawati || Leli Deswita |

: | 10.9790/5728-1604031420 |

**Abstract: **In this paper, we discuss (l,r)-f-derivation, (r,l)-f-derivation, and (f,g)-derivation in BG-algebra, and investigate some of related properties. Also, the notions of left f-derivation and left (f,g)-derivation in BG-algebra are introduced and some of related properties are investigated.

**Keyword: **BG-algebra, (l,r)-f-derivation, (r,l)-f-derivation5, (f,g)-derivation

[1] J. Neggers and H. S. Kim, On B-algebras, Matematicki Vesnik, 54, 2002, 21-29.

[2] C. B. Kim and H. S. Kim, On BG-algebras, Demonstratio Mathematica, 41, 2008, 497-505.

[3] Y. B. Jun and X. L. Xin, On derivations of BCI-algebras, Information Sciences, 159, 2004, 167-176.

[4] H. A. S. Abujabal and N. O. Al-Shehri , On Left Derivations of BCI-algebras, Soochow Journal Of Mathematics, 33, 2007, 435-444.

[5] J. Zhan and Y. L. Liu, f-derivations of BCI-algebras, International Journal of Mathematics and Mathematical Sciences, 11, 2005, 1675-1684..

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Paper Type |
: | Research Paper |

Title |
: | A Review on the Theory of Continued Fractions |

Country |
: | India |

Authors |
: | M.Narayan Murty || Padala Navina |

: | 10.9790/5728-1604032132 |

**Abstract: **In this article, the theory of continued fractions is presented. There aretwo types of continued fraction, one is the finite continued fraction and the other is the infinite continued fraction. A rational number can be expressed as a finite continued fraction. The value of an infinite continued fraction is an irrational number. The ratio of two successive Fibonacci numbers, which is a rational number, can be written as a simple finite continued fraction. The golden ratio can be expressed as an infinite continued fraction. The concept of golden ratio finds application in architecture. Using the convergents of finite continued fraction, the relation between Fibonacci numbers can be calculated and Linear Diophantine equations will be solved.

**Keywords:** Continued fraction, Convergent, Fibonacci numbers, Golden ratio.

[1]. Burton, David M., Elementary Number Theory, Tata Me-Graw Hill Publishing Company Ltd. (2007)

[2]. Narayan Murty M., Some facts about Fibonacci numbers, Lucas numbers and goldenratio, TheMathematics Student, Indian Mathematical Society, 87 No.1-2, (2018), 133 -144.

[3]. Ramaswamy A.M.S., Contributions of Srinivasa Ramanujan to number theory, Research Gate, (2016), 1-6.

[4]. Sury B., On the front cover, Resonance, Indian Academy of Sciences, 23 No.7,(2018),729..

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Paper Type |
: | Research Paper |

Title |
: | Restriction of Soft Setsand Soft Mappings and Some of its Real LifeApplications |

Country |
: | Iraq |

Authors |
: | Majd Hamid Mahmood |

: | 10.9790/5728-1604033338 |

**Abstract: **In this search we define the restriction of soft set in two methods,first method ,the restriction of soft set (F,E) on A E (E is the set of parameters ),second method,the restriction of soft set (F,E) with respect to A X (X is the universal set) , introduce some practical examples .Then ,we introduce and study restriction of soft mappings ..

**Key words **: restricted soft set,soft topology ,image processing, restriction of soft mappings

[1] Hussain S., Ahmad B., "Some properties of soft topological spaces" , Computers and Mathematics with Applications , Vol. 62 ,(2011) , pp. 4058–4067 .

file:///C:/Users/Zian/Downloads/1-s2.0-S0898122111008297-main.pdf

[2] Irfan Ali M., Feng F. , Liu X. , Min W.K., Shabir M." On some new operations in soft set theory , Computers and Mathematics with Applications " , Vol. 57 ,(2009), pp.1547–1553.

https://dl.acm.org/doi/10.1016/j.camwa.2008.11.009 [3] Mahmood M.H. , "On Soft Topological Spaces", LAP Lambert Academic Publishing ISBN-13 978-613-9-94966-3, (2018) . https://www.lap-publishing.com/catalog/details/store/gb/book/978-613-9-94966-3/on-soft-topological-spaces

[4] Maji P.K.,Biswas R., Roy A. R., "Soft set theory", Computers and Mathematics with Applications, Vol.45, (2003) , pp.555-562 .

https://reader.elsevier.com/reader/sd/pii/S0898122103000166?token=C26B6C38AC667EB1B38F83FEF424E00526F8481E97205AE55191E3B4BE6286AC5F9B512D943FC3375C08ADC8AC9048E3

[5] 7Molodtsov D.,"Soft set theory – first results", Computers and Mathematics with Applications, Vol. 37, (1999), pp.19-31 . https://core.ac.uk/download/pdf/82496757.pdf

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Paper Type |
: | Research Paper |

Title |
: | Riemannian Curvature Tensor on Trans -Sasakian Manifold |

Country |
: | India |

Authors |
: | Savita Verme |

: | 10.9790/5728-1604033941 |

**Abstract: **Background: Oubina, J.A.[1] defined and initiated the study of Trans-Sasakian manifolds. Blair [2] ,Prasad and Ojha [3], Hasan Shahid [4] and some other authors have studied different properties of C-R-Sub –manifolds of Trans-Sasakian manifolds. Golab, S. [5] studied the properties of semi-symmetric and Quarter symmetric connections in Riemannian manifold. Yano,K.[6] has defined contact conformal connection and studied some of its properties in a sasakian manifold. Mishra and Pandey [7] have studied the properties in Quarter symmetric metric F-connections in an almost Grayan manifold. Result :In this paper we have studied Riemannian curvature tensor on Trans-Sasakian manifold. Following the patterns of Yano [6], we have proved that a Trans –Sasakian manifold......

**Key words:** Riemannian curvature tensor, Trans-Sasakian manifold,C-R-Sub –manifolds of Trans-Sasakian manifolds,semi-symmetric and Quarter symmetric connections in Riemannian manifold, almost Grayan manifold

[1] Obina,J.A. : New classes of almost contact metric structure publ.Math.32(1985),pp 187-193.

[2] Blair, D.E.: Contact manifold in Riemannian geometric lecture note in Math. Vol.509, Springer Verlag, N.4(1978).

[3] Prasad, S. and Ojha, R.H.: C-Rsubmanifolds of Trans –Sasakian manifold, Indian Journal of pure and Applied Math.24(1993)(7 and 8),pp.427-434.

[4] Hasan Shahid,M.: C-R sub manifolds of Trans –Sasakian manifold, Indian Journal of pure and Applied Math. Vol.22 (1991),pp.1007-1012.

[5] Golab,S.: On semi-symmetric and quarter symmetric linear connections;Tensor,N.S.;29(1975).

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**Abstract: **The purpose of this study was to determine the effect of GeoGebra on conceptual and procedural knowledge of
Geometry. The study involved about 147 Students of Abetifi Presbyterian College of Education. A total of eighty

– four (84) students were involved in both the Pre-intervention and the Post intervention methods for both
Science 𝐵 and Maths 𝐹 classes and 63 students were in the control group. The data was collected using the Pre

and the Post interventions of GeoGebra method and Lecture Methods respectively with the assistance of paired
sampled 𝑡 - test. The results of the Pre – test between Science 𝐵 and Maths 𝐹 classes recorded

t 0.628, 0.534 0.05 which indicates that we retain the 0 H and reject the a H . The Pre- test and Posttest
between Science B class recorded t 14.351.......

**Keywords:** GeoGebra, conceptual knowledge, procedural knowledge, phase base.

[1]. Suan, S.J. (2014). Factors Affecting Underachievement in Mathematics. Proceeding of the Global Summit on Education GSE. Kuala Lumpur, MALAYSIA

[2]. Atiyah, M. (2001) Mathematics in the 20th Century: geometry versus algebra, Mathematics Today, 37(2), 46- 53.

[3]. Burns and Grove (1993). Business research methods. New York, NY: McGraw-Hill guide. Edinburgh: Scottish Council for Research in Education.

[4]. Ministry of Education and Sports (2006). The Ghana information and Communications Technology (ICT) in Education Policy.

[5]. Senechal, M. (1990). Shape. In L.A. Steen (ed.), On the Shoulders of Giants; New Approaches to Numeracy (pp. 139-182). Washington DC: National Research Council.

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Paper Type |
: | Research Paper |

Title |
: | Mathematical Modeling of COVID-19 Transmission with Mass Testing and Contact Tracing |

Country |
: | Kenya |

Authors |
: | Mutuguta John Wanjau |

: | 10.9790/5728-1604035564 |

**Abstract: **Corona virus disease or COVID-19 is an infectious disease whose etiological agent has been identified as a novel corona virus known as Severe Acute Respiratory Syndrome Corona virus 2 (SARS CoV 2). Symptoms of the disease include fever, fatigue, loss of smell and taste, dry cough and breathing difficulties in severe cases. The disease is mainly transmitted through discharge from the nose or mouth when an infectious person coughs or sneezes. In this paper, we used a 5-compartmental model incorporating isolation of positive cases to investigate the effect of mass testing and contact tracing on the transmission of COVID-19. The stability analysis of the model showed that the disease-free equilibrium was asymptotically stable when the basic reproduction number is less than unity. Further we performed the sensitivity analysis of the model. The purpose of the sensitivity analysis was to compare the impact of the mass testing to that of contact tracing with the aim of advising the disease control practitioners on the best......

**Keywords;**COVID-19, SARS Coronavirus 2, contact tracing, mass testing, sensitivity analysis, equilibrium, stability 2010 Mathematics Subject Classification: 97M60, 00A71, 46N60, 92D30, 93A30

[1] Saroj Kumar Chandra, Avaneesh Singh and Manish Kumar Bajpai. Mathematical model for social distancing parameter for early
estimation of COVID-19 spread. medRxiv preprint, 2020.

[2] O. Diekmann, J. A. P. Heesterbeek and M. G. Roberts. The construction of the next generation matrices for compartmental epidemic

models. Journal of Royal Society Interface, 7:873–885, 2010.

[3] P. Van den Driesche. Reproduction numbers of infectious disease models. Infectious Disease Modeling, pages 1–16, 2017.

[4] Government of Kenya. Kenya population situation analysis. United Nations Population Fund Kenya County Office, 2013.
www.unfpa.org.

[5] S.M. Kassa, H.J.B. Njagarah and Y.A. Terefe.Analysis of the mitigation startegies for covi-19: From mathematical modeling
perspective. medRxiv preprint, 2020..

#### Series-4

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**Index Page****Cover Paper**

**Abstract: **We define the primal and dual linear programming problems involving interval numbers as the way of traditional linear programming problems. We discuss the solution concepts of primal and dual linear programming problems involving interval numbers without converting them to classical linear programming problems. By introducing arithmetic operations between interval numbers, we prove the weak and strong duality theorems. Complementary slackness theorem is also proved. A numerical example is provided to illustrate the theory developed in this paper.

**Keywords;** Closed intervals, Linear Programming, Weak Duality, Strong Duality, Complementary Slackness

[1]. G. Ramesh and K. Ganesan, Duality theory for interval Linear Programming Problems, IOSR Journal of Mathematics, 2012, Vol. 4, No. 4, pp. 39-47.

[2]. E.B. Bajalinove, Linear Fractional Programming: Theory, Methods, Applications and Software, Kluwer Academic Publishers, 2003.

[3]. S. Effati, M. Pakdaman, "Solving the Interval-Valued Linear Fractional Programming Problem", American Journal of Computational Mathematics, 2012, Vol. 2, pp. 51-55.

[4]. G. Alefeld and J. Herzberger, Introduction to Interval Computations, Academic Press, New York 1983.

[5]. Atanu Sengupta, Tapan Kumar Pal, Theory and Methodology: On comparing interval numbers, European Journal of Operational Research, 27 (2000), 28 - 43.

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Paper Type |
: | Research Paper |

Title |
: | A new application of Adomian Decomposition Method |

Country |
: | |

Authors |
: | Somaia Ali Alaqel || Yahya Qaid Hasan |

: | 10.9790/5728-1604041115 |

**Abstract: **In this work, we applied a powerful technique to approximate the solutions of singular initial value problems of second order this technique is based on a new application of Adomian decomposition method. We presented Few illustrative examples which indicate that the approach is accurate, reliable and e_cient.

**Key words:** Ordinary di_erential equations, Adomian Decomposition Method, second-order singular initial value problems.

[1]. Adomian G. A review of the decomposition in applied and some recent results for nonlinear equation, Math Comput Model. 1999;13:17-43.

[2]. Adomian G. and Rach, R. Inversion of nonlinear stochastic operators, J Math Anal Appl 1983;91:39-46.

[3]. Adomian G. Solving frontier problem of physics: the decomposition method. Kluwer academic publishers London; 1994.

[4]. Hasan Y. Q., Zhu L. M. Modi ed Adomian decomposition method for singular initial value problems in the second order ordinary differential equations. Surveys in Mathematics and its Applications 2008;3: 183- 193, .

[5]. Othman S.G.,Hassan Y.Q. New development of Adomian decomposition method for second order ordinary differential equations. IJMS 2020;6:28-49

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Paper Type |
: | Research Paper |

Title |
: | Picard Sequence and Fixed Point Results on G-Metric Space |

Country |
: | India |

Authors |
: | Vijay Dadhore || Savita Tiwari || Devkrishna Magarde |

: | 10.9790/5728-1604041623 |

**Abstract: **In this paper we introduced picard sequence and fixed point result in G-metric spaces. We have utilized these
concepts to deduce certain fixed point theorems in G-metric space . our theorem extend and improve the results
of Sumitra and Ranjeth kumar [3], B. Singh and S. Jain [4,5,6,7] and Urmila Mishra et al.[10] in the settings
of G-metric space.

Mathematics Subject Classification : 47H10, 54H25.

**Keywords:** Continuous mapping ,Cauchy sequence, G-metric space , Metric space

[1]. Hussain, N., Khaleghizadeh, S., Salimi, P., Abdou, A. A. N., A New Approach to Fixed Point Results in Triangular Intuitionistic

Fuzzy Metric Spaces, Abstract and Applied Analysis, Volume 2014, Article ID 690139, 16 pages,

http://dx.doi.org/10.1155/2014/690139.

[2]. Hussain, N., Salimi, P., Parvaneh, V., Fixed point results for various contractions in parametric and fuzzy b-metric spaces, J.

Nonlinear Sci. Appl. 8 (2015), 719-739.

[3]. O. Kada, T. Suzuki and W. Takahashi, Nonconvex minimization theorems and fixed point theorems in complete metric spaces,

Math. Japon. 44 (1996) 381-391

[4]. M. Aamri and D.El. Moutawakil, Some new common fixed point theorems under strict ontractive conditions , J. Math. Anal. Appl.,

270 (2002), 181-188.

[5]. M. Abbas and B.E. Rhoades, Common fixed point results for noncommuting mappings without continuity in generalized metric

spaces, Appl. Math. Comput., (2009), doi:10.1016/j.amc.2009.04.085..

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**Abstract: **In this paper, we evaluate a finite integral involving the product of a general class of polynomials and the multivariable H-function. Also we reduce the H-function of several variables to the product of whittaker function and multivariable H-function to a generalized hyper-geometric function of several variable.

**Keywords:** H-function, Multivariable H-function, Contour Integral, Hyper-geometric function.

[1]. GuptaK.(2007), A study of modified H-transform and fractional integral operator, Kyungpook Math. J., 47, 519-527.

[2]. GuptaK. C., Jain R. and AgarwalP.(1995), On composition of multidimentional integral operators involving general polynomials and a multivariable H-function, Kyungpook Math. J., 35,151-162.

[3]. KilbasA.A. and SaigoM.(2004), H-transform: Theory and Applications, Chapman and Hall/CRC, Boca Raton, London, New York and Washington, D. C.,

[4]. KilbasA.A. and SrivastavaH.M.andTrujillo J.J(2006), Theory and Applications of Fractional Differential Equations, North-Holland Mathematical Studies, Vol. 204, Elsevier Science Publishers, Amasterdam,

[5]. MukherjeeR.(2004), A Study of General Sequence of Functions, Multidimentional Fractional Calculus Operator and the General H-function of One Variables with applications, ph.D Thesis, University of Rajasthan, India..

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**Abstract: **In this paper, a new fifth-stage fourth-order Runge–Kutta formula was derived for solving initial value problems (IVPs) in Ordinary Differential Equation, which was implemented and compared with classical Runge-Kutta formula through the computation of some tested initial value problemsin other to determine the level of performance, consistency and accuracy.We discovered that errors are minimal with our new method.Errors in the new method and that of classical Runge-Kutta method were plotted with MATLAB to determine their curves.Our new method was tested further on the Kermack-McKendrick SIR model for the course of an epidemic in a population which computes the number of Susceptible, Infected, and Recovered people in a population at any time. The codes and plots were built in Python. We called this method BO4 method

**Keywords**: Initial value problems, Taylor series, consistency, convergence, stability, and error analysis

[1]. Agbeboh And Ehielmua (2012): The Modified Fourth Order Kutta's Algorithm for Solving Initial Value Problems in Ordinary

Differential Equations. A synopsis of master's theses of School of Post Graduate Studies, Ambrose Alli University Ekpoma, Edo

State Nigeria.

[2]. Agbebohand Ehielmua (2019): On the derivation of a new fifth order implicit Runge-Kuttascheme for stiff problems in Ordinary

Differential Equation. Journal of the Nigeria mathematics Society. Vol.38, 1ssue.247-258,2019.

[3]. AgbebohG.U,Ukpebor, L.A. And Esehkaigbe(2009): "A modified sixth stage fourth-orderRunge-Kutta method for solving initial

value problems in ordinary differential equations" Journal of mathematical science Vil.20,pp,97-100.2009

[4]. Agbeboh G.U AndEhielmua.M. (2014): Modified Kutta's algorithm" JNAMP, Vol.25,pp.105-114, 2014

[5]. Agbeboh G.U. (2013): "On the stability analysis of a geometric 4th order Runge-Kutta formula" mathematical theory and modeling

Vol. 3, pp. 90-105, 2013.

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**Abstract: **This paper is concerned with analytical determination of (a) the normal displacement and Airy stress function of an imperfect viscously damped toroidal shell segment subjected to a step load and (b) the dynamic buckling load of the structure using perturbation technique in asymptotic procedures. The continuously differentiable imperfection is assumed in the form of a two – term Fourier series expansion while homogeneous initial and boundary conditions are assumed. The formulation contains a small parameter depicting the amplitude of imperfection and upon which a multi – timing regular perturbation procedure is initiated, while the light viscous damping is of some order of imperfection. Simply – supported boundary conditions are assumed and in the final analysis, a uniformly valid asymptotic formula for determining the dynamic buckling load is determined nontrivially. The dynamic buckling.....

[1]. Stein, M. and J. A. McElman, J. A. (1965), Buckling of segments of toroidal shells, AIAA Jnl. 3, 1704.

[2]. Hutchinson, J. W. (1967), Initial post buckling behavior of toroidal shell segment, Int. J. Solids Struct. 3, 97 -115.

[3]. Oyesanya, M. O. (2002), Asymptotic analysis of imperfection sensitivity of toroidal shell segment modal imperfections, J. Nigerian Ass. Maths. Physics 6, 197 – 206.

[4]. Oyesanya, M. O. (2005), Influence of extra terms on asymptotic analysis of imperfection sensitivity of toroidal shell segment with random imperfection, Mechanics research Communications 32, 444 – 453.

[5]. Ette, A. M., Chukwuchekwa, J. U., Udo – Akpan, I. U. and Ozoigbo, G. E. (2019), On the normal response and buckling of a toroidal shell segment pressurized by a static compressive load, IJMTT, 65(10), 15 – 26.

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**Index Page****Cover Paper**

**Abstract: **The purpose of this study focused on modeling the population of Ethiopia using different models and estimating the models parameters via least square method. The models, that were applied for the population growth, were Malthus growth model, Logistic growth model and General growth model. To identify the models which performed effectively in prediction of the actual population, the measure accuracy has been used, such that the models satisfying the criteria of the measure of accuracy is the best statistical model. The results of the analysis were presented using tables and graphical form which are very good to perform comparison for the effectiveness of the models. In this study.......

**Keywords:** Logistic model, parameter estimation, Malthus model, General model, selection methods

[1]. Alemayehu B and Yihunie L, 2014, Projecting Ethiopian demographics from 2012-2050 Using the spectrum suite of models, Ethiopian public health association.

[2]. Ali LE, Khan BR and Sams IS, 2015, Brief study on census and predicted population of Bangladesh using logistic population model. Ann. Pure Appl. Math.10(1): 41-47.

[3]. Al-Eideh BM and Al-Omar HO, 2019, Population projection model using exponential growth function with a birth and death diffusion growth rate processes. Eur. J. Sci. Res. 151(3): 271-276

[4]. Amare Wubishet Ayele and Mulugeta Aklilu Zewdie, 2017,Modeling and Forecasting Ethiopian Human Population Size and it`s Pattern, International Journal of Social Sciences Arts and Humanities Vol. 4 No. 3. 2017. Pp. 71-82

[5]. Andongwisye J Mwakisisile1 and Allen R Mushi, 2019, Mathematical Model for Tanzania Population Growth, Tanzania Journal of Science 45(3): 346-354, ISSN 0856-1761, e-ISSN 2507-7961

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**Abstract: **In this article, a new distribution is introduced, which is generated from the truncated Cauchy power-G family
of distribution named as truncated Cauchy power- inverse exponential distribution (TCP-IE). We have explored
various statistical and mathematical properties, shapes and behavior of the proposed distribution through
probability density function (PDF) plot, cumulative distribution function (CDF) plot, and hazard rate function.

We illustrated the estimation of the parameters and their corresponding confidence interval using the maximum
likelihood estimation (MLE) method for the (TCP-IE) distribution. Two real data sets are taken to assess the

suitability and applicability of purposed......

**Keywords:**Truncated Cauchy power-G family, Inverse Exponential distribution, Hazard function,Maximum
likelihood estimation.

[1]. Aldahlan, M. A., Jamal, F., Chesneau, C., Elgarhy, M., &Elbatal, I. (2020). The truncated Cauchy power family of distributions with inference and applications. Entropy, 22(3), 346.

[2]. Alzaatreh, A., Famoye, F., & Lee, C. (2013).Weibull-Pareto distribution and its applications. Commun. Stat. Theory Methods. 42, 1673–1691.

[3]. Alzaatreh, A., Mansoor, M., Tahir, M.H., Zubair, M., Ghazali, S.A. (2016). The gamma half-Cauchy distribution:Properties and applications. Hacet. J. Math. Stat. 45, 1143–1159.

[4]. Alizadeh, M., Altun, E., Cordeiro, G. M., &Rasekhi, M. (2018). The odd power Cauchy family of distributions: properties, regression models and applications. Journal of statistical computation and simulation, 88(4), 785-807.

[5]. Ashani, Z. N. &Bakar, M. R. A. (2016). A Skewed Truncated Cauchy Logistic Distribution and its Moments. In International Mathematical Forum (Vol. 11, No. 20, pp. 975-988).

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**Abstract: **The purpose of the present paper is to investigate new subclass of harmonic univalent function in the unit disc 𝑈= 𝑧∈ℂ: 𝑧 <1 by using derivative operator. Also, we obtain coefficient inequalities and distortion theorems for this subclass. 2000 Mathematics Subject Classification: 30C45, 30C50.

**Keywords: **Harmonic, Univalent, Coefficient inequalities, Derivative operator..

[1]. Avci Y. and Zlotkiewicz E., (1990), On harmonic univalent mappings, Ann. Univ. Mariae Curie- Sklodowska Sect.A 44, 1-7.

[2]. Clunie J. and Sheil-Small, T. (1984), Harmonic univalent functions, Ann. Acad. Sci. Fenn. Ser.AI Math., 9, 3-25.

[3]. Darus M., Sangle N.D. (2011), On certain class of harmonic univalent functions defined by generalized derivative operator, Int. J. Open Problems Compt. Math., Vol.4, No. 2,83-96.

[4]. Dixit K.K. and Porwal Saurabh (2010), A subclass of harmonic univalent functions with positive coefficients, Tamkang J. Math., 41(3), 261-269.

[5]. Duren P.(2004), Harmonic Mappings in the Plane, Cambridge Tracts in Mathematics, Vol.156, Cambridge University Press, Cambridg..

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Paper Type |
: | Research Paper |

Title |
: | Some Identities Satisfied By Ternary Semirings |

Country |
: | India |

Authors |
: | Koneti.Rajani || Cheerla .Meenakumari || Dr.G.Shobhalatha |

: | 10.9790/5728-1604052936 |

**Abstract: **In this paper we study about the ternary semirings satisfying some identities and introduce the concept of IMP in ternary semirings. By using these identities we characterize various properties of ternary semirings.

**Keywords: **Ternary Semiring, Singular, Band, Rectangular band, Zeroid, Mono-ternarysemiring, IMP, Periodic, Quasi-seperative, Weakly –seperative

[1]. Dutta.T.K,Kar.S,On Regular ternary Semirings,Advances in Algebra,Proceedings of the ICM Satellite Conference in algebra and Related Topics,World Scientific,New jersey (2003),343-355

[2]. Lehmer.D.H. A ternary analogue of abelian groups, Amer.J.Maths.,59(1932),329-338.

[3]. Dutta.T.K, kar.S, A note on regular ternary semirings, kyung-pook Math.J.,46(2006),357-365.

[4]. Jonathan S.Golan, Semirings and Affine equations over them, Theory and Applications, Kluwer Academic.

[5]. A.Rajeswari, G.Shobhalatha, On Some Structures Of Semirings, Shodhganga.inflibet.net thesis(2016)..

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Paper Type |
: | Research Paper |

Title |
: | Juvenile Delinquency in India: Incidences and Forecasting |

Country |
: | India |

Authors |
: | Prof. Jiten Hazarika || Daalima Goswami |

: | 10.9790/5728-1604053742 |

**Abstract: **Juvenile in conflict with law is a matter of grave concern as this age group people belong to the most productive period of life. This phase is undoubtedly very fragile, and has severe consequences, which may last for long. The involvement of juveniles in any kind of delinquencies is indeed disheartening. There are multiple reasons, and so the nature of crime is. In this paper, it has been tried to cover the incidences of juvenile delinquencies in India for a given period of time. A detail account of such cases has been provided, giving an insight into the type of offences juveniles have been committing over the years. Moreover, an attempt has been made to forecast cases in this line for understanding future trend.

**Keywords: **Juvenile Delinquencies, Crime Heads, ARIMA, Forecasting, Incidences.

[1]. Baligar M. P. 2014. Trends and patterns of juvenile delinquency in Karnataka a sociological study, PhD Thesis, Karnataka University, Karnataka

[2]. Haveripet Prakash (2013). Causes and consequences of juvenile delinquency in India. Recent Research in Science and Technology, 5(3), 29-31.

[3]. Sharma B. R., Dhillon Sangeet, Bano Sarmadi (2009). Review of Juvenile delinquency in India- a cause for concern. Journal of Indian Academy of Forensic Medicine, 31(1).

[4]. Tsay Ruey S. (2005). Analysis of Financial Time Series. Wiley India..

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**Abstract: **We conduct a study on Topological Manifold and some of its properties Theorems, structures on topological manifold and its structural characteristics. The main properties in Topological Manifold is connectedness, path connectedness, compactness, boundedness etc, which are the module in the real time manifold. These modules are use in day to day life to prevents spared of corona virus in the world Manifold. Actually Real World is space- time Manifold. Cut points are Used on this manifold as ,To maintain the social distance, to follow the rules of lock down , Also these topological properties are studied the path connected ,locally connected ,locally path connected with cut point, punctured point and punctured space on real World Manifold. Cut point and punctured points are additional properties of manifold helps for quarantine, curfew, social distance, isolation and lock down in society. The punctured space is quarantine space which prevents spread of corona virus in world Manifold.

**Keywords: **Connectedness, locally connected, path connected, Locally path connected, cut point, punctured points, corona virus, quarantine, lockdown curfew, isolation. Boundedness.

[1]. H.G. Haloli, "The structural relation between the topological manifold I- connectedness", IOSR Journal of Engineering (IOSRJEN) -3(7), 2013, 43-54.

[2]. H.G. Haloli, "The topological property of topological manifold-compactness with cut point", International journal of scientific and engineering research -4(8), 2013, 1374-1380.

[3]. H.G. Haloli,"The structural relation between tangent space and covering space", International journal of advance research IJOAR-1(7), 2013, 17-26.

[4]. H.G. Haloli,"Connectedness and punctured space in fiber bundle space", International journal of engineering research and technology,-2(6), 2013,43-54.

[5]. H.G. Haloli,(2020), "Cut points and puncture points on Topological Manifold", IOSR Journal of Engineering (IOSRJM).

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Paper Type |
: | Research Paper |

Title |
: | Application of Fuzzy Matrix Theory In COVID-19 Pandemic |

Country |
: | India |

Authors |
: | Anjan Mukherjee || Abhik Mukherjee |

: | 10.9790/5728-1604055256 |

**Abstract: **This article explores knowledge discovering in COVID-19 pandemic. It discusses about the people who affected severely in COVID-19 . It identifies of which age group of people's associated with the COVID-19 impactions. We use fuzzy matrix theory to determine the risk factor. In introduction we give the major signs and symptoms of corona virus disease 2019. This affects different people in different ways.

**Keywords: **Fuzzy Matrix, Raw Data, corona virus, Surge

[1]. Zadeh, L.A., Fuzzy sets, Information and control.,8,338-353(1965).

[2]. Thomason,M.G., Convergence of power of a fuzzy matrix, J. Math. Anal. Appl.,57,476-480(1977).

[3]. World Health Organization, nCoV Situation Report-22 on 12 February, 2020. source/corona virus /situation-reports/, 2019.

[4]. Imperial College London. Report 2: estimating the potential total number of novel corona virus cases in Wuhan City, China. Jan. disease-analysis/news--wuhan coronavirus,2020.