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• Open Access   Article

  Minimum Variance Unbiased Estimator of Software Reliability for Weibull Class Models

  B. Roopashri Tantri

  Research Paper | Journal-Paper (IJSRMSS)

  Vol.7 , Issue.6 , pp.1-5, Dec-2020

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  The quality of the software can be measured quantitatively using software reliability. Estimation of software reliability plays a major role in deciding about the performance of the software. Depending on the type of distribution of failure times, several classes of software reliability models exist. One such class of model is the Weibull class model. The estimation of reliability for the Weibull class model has been considered herein. Several researchers have estimated the reliability of various other classes of models. The method of least squares is the most sought after method. However, the method has certain drawbacks. Herein, two other methods of estimation of reliability, viz, the method of Maximum Likelihood Estimation (MLE) and the method of Minimum Variance Unbiased Estimation (MVUE) are considered obtained for Weibull class models. Further, a comparison of these two estimates of reliability has been carried out by using their statistical properties. The mean square errors of the two estimates of reliability have also been obtained. A few case studies have been considered. It is observed that the method of MVUE provides more accurate value than the method of MLE. Accordingly, both the developer and the user of the software may prefer MVUE of reliability over MLE of reliability while deciding the performance of the software.

  Key-Words / Index Term
  Maximum Likelihood Estimator, Mean square error, Minimum Variance Unbiased Estimator, Software reliability models, Weibull class models

  References
  [1] John D. Musa, Anthony Iannino, Kazuhira Okumoto, “Software Reliability Measurement, Prediction, Application", Mc-Graw Hill, International Edition, 1991.
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  [8] Bev Littlewood, “Forecasting software reliability”, Lecture Notes in Computer Science, Springer Verlag, Berlin no. 341, pp. 30-41, 1988.
  [9] Jason Allen Denton, “Accurate Software Reliability Estimation”, Ph.D. thesis, Computer Science, Colorado State University, 1999.
  [10] Taehyoun Kim, Kwangkyu Lee, Jongmoon Baik, “An effective approach to estimating the parameters of software reliability growth models using a real-valued genetic algorithm” Journal of Systems and Software, Vol 102, pp. 134-144, 2015.
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  Citation
  B. Roopashri Tantri, "Minimum Variance Unbiased Estimator of Software Reliability for Weibull Class Models," International Journal of Scientific Research in Mathematical and Statistical Sciences, Vol.7, Issue.6, pp.1-5, 2020

• Open Access   Article

  Generalized Maximum Entropy Estimation; Application to Seed Rice Data in Sierra Leone

  R.M. Bangura, A.R Fofie, M.Y. Turay, E.G. Kamara

  Research Paper | Journal-Paper (IJSRMSS)

  Vol.7 , Issue.6 , pp.6-11, Dec-2020

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  The maximum entropy method supports the subjective point of view with regards to probability, when drawing conclusions is plausible for incomplete information. The study of maximum entropy is of vital, when probabilities are given and we may want to measure the level of uncertainty of its outcome. Since matrix X is posed with problem of multicillinearity, we present an application of generalized maximum entropy procedure to seed rice data in Sierra Leone. We determined the estimation of three parameters based on generalized maximum entropy method and also provided a diagnostic fit for collinearity to obtain eigenvalues with their proportional variations. Analysis of variance is also presented to test the level of significance based on the GME procedure. With insufficient data, we also identified outliers with their corresponding observation points in the studentized residuals. Again, from the regression residual, a model was obtained, which shows the relationship between the regressors and the explained variable.

  Key-Words / Index Term
  Generalized maximum entropy, Multicollinearity, Outliers, Residuals

  References
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  Citation
  R.M. Bangura, A.R Fofie, M.Y. Turay, E.G. Kamara, "Generalized Maximum Entropy Estimation; Application to Seed Rice Data in Sierra Leone," International Journal of Scientific Research in Mathematical and Statistical Sciences, Vol.7, Issue.6, pp.6-11, 2020

• Open Access   Article

  Prediction of Stock Price Movement Using Knowledge Graph Embeddings

  Eric Kwamena Asibu Yartey

  Research Paper | Journal-Paper (IJSRMSS)

  Vol.7 , Issue.6 , pp.12-19, Dec-2020

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  Given the advancement of artificial intelligence and increased computational capabilities to handle a massive amount of web Information, Knowledge graphs (KG) and Knowledge graphs (KG) embeddings have proven useful in modelling and representing factual information in various domains such as medical diagnosis, web search, semantic parsing, speech recognition, question answering, named entity disambiguation, link prediction, recommendation system among others. While TransE model has been adopted by some researchers in the field of finance to generate knowledge graph embeddings for their proposed stock prediction models, it has also been under criticism. Onuki et al. had claimed in their research work that their proposed KGML model outperforms TransE model in terms of predictive accuracy, learning speed, and convergence speed of learning. This study evaluates the effect that learned embedding vectors from TransE and KGML models have on the performance of a TCN stock prediction model. Results have shown that KG embeddings are powerful predictors, and the overall best KG embeddings are that of the TransE model.

  Key-Words / Index Term
  Knowledge Graph, Knowledge Graph Embeddings, TransE Model, KGML Model, Temporal Convolution Network (TCN), Stock Price Movement Prediction

  References
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  [9] P. Wang, H. Jiang, J. Xu and Q. Zhang, "Knowledge Graph Construction and Applications for Web Search and Beyond," Data Intelligence, Vol.1, No.4, pp.333-349, 2019.
  [10] L. Heck and H. Huang, "Deep learning of knowledge graph embeddings for semantic parsing of Twitter dialogs," in 2014 IEEE Global Conference on Signal and Information Processing (GlobalSIP), Atlanta, GA, USA, pp.597-601, 2014.
  [11] Moschitti, a. Tymoshenko, P. Alexopoulos, A. Walker, M. Nicosia, G. Vetere, A. Faraotti, M. Monti, J. Z. Pan, H. Wu and Y. Zhao, "Question Answering and Knowledge Graphs," in Exploiting Linked Data and Knowledge Graphs in Large Organisations, Springer, Cham, pp.181-212, 2017
  [12] Lin, S. Ong, C. Williams and Y. B. Zhang, "Named Entity Disambiguation with Knowledge Graphs," Institute for Applied Computational Science, Harvard university, USA, 2019.
  [13] H. Xiao, M. Huang and X. Zhu, "From One Point to a Manifold: Knowledge Graph Embedding for Precise Link Prediction," in Proceedings of the Twenty-Fifth International Joint Conference on Artificial Intelligence, New York, USA, pp.1315-1321, 2016.
  [14] Z. Sun, J. Yang, J. Zhang, A. Bozzon, L.-K. Huang and C. Xu, "Recurrent knowledge graph embedding for effective recommendation," in Proceedings of the 12th ACM Conference on Recommender Systems (RecSys `18), New York, USA, pp.297-305, 2018.
  [15] Y. Onuki, T. Murata, S. Nukui, S. Inagi, X. Qiu, M. Watanabe and H. Okamoto, "Relation prediction in knowledge graph by Multi-Label Deep Neural Network," Applied Network Science, Vol.4, No.20, 2019.
  [16] Z. Liu and X. Han, "Deep Learning in Knowledge Graph," in Deep Learning in Natural Language Processing, Springer, Singapore, pp.117-146, 2018.
  [17] Q. Wang, Z. Mao, B. Wang and L. Guo, "Knowledge Graph Embedding: A Survey of Approaches and Applications," IEEE Transactions on Knowledge and Data Engineering, Vol.29, No.12, pp.2724-2473, 2017.
  [18] Ringler and H. Paulheim, "One Knowledge Graph to Rule Them All? Analyzing the Differences Between DBpedia, YAGO, Wikidata & co.," in Kern-Isberner G., Fürnkranz J., Thimm M. (eds) KI 2017: Advances in Artificial Intelligence. KI 2017. Cham, pp.366-372, 2017.
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  [26] R. D. Almeida, G. Reynoso-Meza and M. T. A. Steiner, "Multi-objective optimization approach to stock market technical indicators," in 2016 IEEE Congress on Evolutionary Computation (CEC), Vancouver, BC, Canada, pp.3670-3677, 2016.
  [27] S. Bai, J. Z. Kolter and V. Koltun, "An Empirical Evaluation of Generic Convolutional and Recurrent Networks for Sequence Modeling", ArXiv, Vol.abs/1803.01271, 2018.
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  [30] L. Costabello, S. Pai, C. L. Van, R. McGrath, N. McCarthy and P. Tabacof, "AmpliGraph: a Library for Representation Learning on Knowledge Graphs," zenodo, 2019.
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  Citation
  Eric Kwamena Asibu Yartey, "Prediction of Stock Price Movement Using Knowledge Graph Embeddings," International Journal of Scientific Research in Mathematical and Statistical Sciences, Vol.7, Issue.6, pp.12-19, 2020

• Open Access   Article

  Deterministic inventory model for deteriorating items with price dependent demand rate and inflation

  Virali Vora, U.B. Gothi

  Research Paper | Conference-Paper (IJSRMSS)

  Vol.7 , Issue.6 , pp.20-28, Dec-2020

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  Abstract
  In this study, a deterministic inventory control model has been developed for physical perishable items that deteriorate with selling price dependent linear demand rate. Here, shortages are allowed to occur and are partially backlogged with inflation that incurs lost sale cost which is considered keeping in mind the criteria of modern era. A two parameter Weibull distribution and Pareto Type-I distribution are considered for the time to deterioration. The aim of the paper is to understand retailer’s replenishment decisions under practical circumstances such as having a holding cost as a linear function of time in an economic order quantity model. The model is solved analytically with an objective to find an optimal solution that minimizes the overall cost and maximizes the total profit. Numerical solution is derived by means of an application and a comprehensive sensitivity analysis of the optimal solution is performed to study the effects of the variation in various significant parameters.

  Key-Words / Index Term
  Selling price dependent demand rate, Weibull distribution, Pareto Type-I Distribution, Deterioration

  References
  [1] G.P. Samanta1, Jhuma Bhowmick, “A deterministic inventory system with Weibull distribution deterioration and ramp type demand rate”; Electronic Journal of Applied Statistical Analysis, Vol. 3, Issue 2, pp. 92 – 114, 2010.
  [2] Sandeep Kumar Chaudhary and R.P.Tripathi, “An EOQ Model for Weibull Distribution Deterioration with Exponential Demand under Linearly Time Dependent Shortages”; International Journal of Computational and Applied Mathematics, Vol 12, Issue 1, pp. 81-98, 2010.
  [3] Anchal Agarwal, S. R. Singh, “An EOQ Inventory Model For Two Parameter Weibull Deterioration With Time Dependent Demand And Shortages”; International Journal of Engineering Research & Technology (IJERT), Vol. 2 Issue 7, 1193-1200, 2013.
  [4] Virali Vora, U. B. Gothi, “Deterministic Inventory Model for Deteriorating Items under Ramp Type Demand Rate with Time Dependent Holding Cost”, Sankhya Vignan, NSV 14, 19-31, 2017.
  [5] A.K. Jalan, K. S. Chaudhari, “Structural properties of an inventory system with deterioration and trended demand”; International Journal of Systems Science, Volume 30, Issue 6, pp. 627-633, 2010.
  [6] Devyani Chatterji, U. B. Gothi, “An EPQ Model for Two-Parameter Weibully Deteriorated Items with Exponential Demand Rate and Completely Backlogged Shortages”, International Journal of Computer Science Trends and Technology (IJCST), Volume 3 Issue 6, pp. 38-47, 2015.
  [7] Rekha Rani Chaudhary and Vikas Sharma, “A Model for Weibull Deteriorate Items with Price Dependent Demand Rate and Inflation”; Indian Journal of Science and Technology, Vol 8(10), pp. 975–981, 2015.
  [8] Vinod Kumar Mishra, “Inventory model for time dependent holding cost and deterioration with salvage value and shortages”; The Journal of Mathematics and Computer Science, Vol. 4 No.1, pp. 37 – 47, 2012.
  [9] Pooja Khatri, U. B. Gothi, “An Inventory Model of Deteriorating Items under Inflation and Discount Rate with Power Demand and Time Dependent Holding Cost”; International Journal of Advanced Research in Statistics, Vol 4, Issue 3, pp. 3523-3532, 2017.
  [10] Shital S. Patel, “Different Deterioration Rates Two Warehouse Production Inventory Model with Time and Price Dependent Demand under Inflation and Permissible Delay in Payments”; International Journal of Engineering Science Invention (IJESI), Vol 7, Issue 4, pp. 53-62, 2018.
  [11] Kirtan Parmar, U. B. Gothi, “Epq model for deteriorating items under three parameter weibull distribution and time dependent ihc with shortages”; American Journal of Engineering Research, Volume-4, Issue-7, pp. 246-255, 2015.
  [12] N. K. Kaliraman, R. Raj, S. Chandra, H. Chaudhry, “An EPQ Inventory Model For Deteriorating Items With Weibull Deterioration Under Stock Dependent Demand”; International Journal of Scientific & Technology Research, Vol 4, Issue 01, pp. 232-236, 2015.
  [13] Saha,S. and Sen, N., “A study on inventory model with negatives exponential demand and probabilistic deterioration under backlogging”; Uncertain Supply Chain Management, Vol 5 Issue 2, pp. 77-88, 2017.
  [14] Ankit Bhojak, U. B. Gothi, “Production Inventory System under Weibull Amelioration, Pareto Deterioration and Exponentially Time Based Demand under Fully Backlogged Shortages”; International Journal of Scientific and Innovative Mathematical Research (IJSIMR) Volume 4, Issue 5, pp. 26-36, 2016.
  [15] Asoke Kumar Bhuniaa, Ali Akbar, “A deterministic inventory model for deteriorating items with selling price dependent demand and three-parameter Weibull distributed deterioration’; International Journal of Industrial Engineering Computations, Volume 5 Issue 3, pp. 497–510, 2014.

  Citation
  Virali Vora, U.B. Gothi, "Deterministic inventory model for deteriorating items with price dependent demand rate and inflation," International Journal of Scientific Research in Mathematical and Statistical Sciences, Vol.7, Issue.6, pp.20-28, 2020

• Open Access   Article

  Modeling of Big Chili Supply Response Using Bayesian Method

  M. Fajar, Y.G. Winarti

  Research Paper | Journal-Paper (IJSRMSS)

  Vol.7 , Issue.6 , pp.29-33, Dec-2020

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  Abstract
  This study aims to estimate the response model of Big Chili offerings with the Bayesian method so that information elasticity of price (production) derived from posterior hyperparameter can be obtained. The method used in this study is a supply response model that adopts the Nerlove model and it is estimated with the Bayesian method. The data used in this study are Big Chili production (kg), harvested area (hectares), and Big Chili prices of producer level (IDR/kg) with the period 2008 - 2018 monthly sourced from Statistics Indonesia. The Bayesian method can be applied in the estimation of the Nerlove Model of The Big Chili supply. However, the resulting coefficient of determination is low by 21.05%. The reason is thought to be the use of prior that have a bias effect on posterior distribution and/or there is a nonlinear relationship to the variables in the model. However, only two variables were not significant from the five predictor variables, namely the price of producer level of Big Chili at time t-1 and the production of Big Chili at time t-2. The estimation results of price elasticity in the short and long-term were 8.49% and 2.50%, respectively, which are the inelastic category. It shows that farmers are not responsive to prices. Because the costs of cultivation are high, so it causes the profits obtained by farmers not so much , even though the farm-level prices increase. It becomes insignificant for income farmers.

  Key-Words / Index Term
  Big Chili, Supply, Nerlove Model, Price Elasticity, Bayesian, Prior

  References
  [1] H. Askari, J.T. Cummings, “Estimating agricultural supply response with the Nerlove model: A survey,” International Economic Review 18 (2), pp. 257-292, 1977.
  [2] J. Berger, L. Pericchi, J. Varshavsky, “Bayes factors and marginal distributions in invariant situations,” Sankhya: The Indian Journal of Statistics, Series A 60, pp. 307-321, 1998.
  [3] M. Braulke, “A Note on The Nerlove Model of Agricultural Supply Response,” International Economic Review 23(1), pp. 241-246, 1982.
  [4] E. Edison, D. Renate, D. Denmar, “Dynamic Supply Response: Implications for Indonesia Soybean Crop,” Dinasti International Journal of Economics, Finance, and Accounting, 1(4), pp. 568-580, 2020.
  [5] M. Fajar, Pemodelan Respon Penawaran Komoditas Cabai Besar (edisi revisi), DOI: 10.13140/RG.2.2.20789.63209, 2019.
  
  [6] R. Leaver, “Measuring The Supply Response Function of Tobacco in Zimbabwe,” Contributed Paper Presented at the 41st Annual Conference of the Agricultural Economic Association of South Africa (AEASA), 2003.
  [7] R.E. Kass, A.E. Raftery, ”Bayes factors,” Journal of the American Statistical Association, 90(430), pp. 773-795, 1995.
  [8] J.M. Marin, C. Robert, “Essential Bayesian with R: Second Edition” Springer, New York, 2014.
  [9] M. Nerlove, “The dynamics of supply: Estimation of farmers’ response to price,” The Johns Hopkins Press, Baltimore, United States of America, 1958.
  [10] M. Nerlove, M., W. Addison, “Statistical estimation of long run elasticities of supply and demand,” Journal of Farm Economics, 40, pp. 861 – 880, 1958.
  [11] N. Rastogi, S. Rastogi, M. Darbari, "A Novel Software Reliability Prediction Algorithm Using Fuzzy Attribute Clustering and Naïve Bayesian Classification", International Journal of Computer Sciences and Engineering, Vol.7(2), pp.73-82, 2019.
  [12] A. Zellner, “An Introduction to Bayesian Econometrics,” John Wiley, New York, 1971.
  [13] A. Zellner,”Basic Issues in Econometrics,” University of Chicago Press, Chicago,1984.

  Citation
  M. Fajar, Y.G. Winarti, "Modeling of Big Chili Supply Response Using Bayesian Method," International Journal of Scientific Research in Mathematical and Statistical Sciences, Vol.7, Issue.6, pp.29-33, 2020

• Open Access   Article

  Stability Analysis of Endemic Equilibrium Points of Malaria and Dengue Fever Co-infection Model

  T.J. Oluwafemi, N.I. Akinwande, E. Azuaba

  Research Paper | Journal-Paper (IJSRMSS)

  Vol.7 , Issue.6 , pp.34-45, Dec-2020

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  Abstract
  Malaria-dengue co-infection is becoming a public health challenge with the growing report of the co-infection. Much has been done on analyzing malaria, dengue, and other infection but much has not been done to study the co-infection of malaria and dengue fever. A mathematical model to understand the transmission of malaria and dengue infection was formulated using a system of ordinary differential equations. We computed the basic reproduction number and establish the equilibria point. The endemic equilibrium point was further investigated. The result shows that with the reproduction number greater than one, the endemic equilibrium point was found to be locally asymptotically stable and globally asymptotically stable.

  Key-Words / Index Term
  Malaria, Dengue Fever, Co-infection, Endemic Equilibrium, Stability Analysis

  References
  [1] G. Geleta, T. Ketema, “Severe Malaria Associated with Plasmodium falciparum and P. vivax among Children in Pawe Hospital, Northwest Ethiopia”, Malaria Research and Treatment, Hindawi Publishing Corporation, 2016.
  [2] World Health Organization, World Malaria Report, Geneva, WHO Pres, 2018.
  [3] J.P. Messina, et al. A global compendium of human dengue virus occurrence. Sci. Data 1, 140004 (2014).
  [4] B.R. Murphy, S.S. Whitehead, “Immune response to dengue virus and prospects for a vaccine”. Annual Review of Immunology, vol. 29, pp. 587-619, 2011.
  [5] S.E. Chong, Z.R.H. Mohamad, S. Suraiya, K.T. Lee, J.A. Lim, J. Malar, “The danger of accepting a single diagnosis: case report of concurrent Plasmodium Knowlesi malaria and dengue infection”, Malaria Journal, vol. 16, issue 2, 2017.
  [6] M.R.K. Rao, N.P. Rabindra, K.D. Manoj, “Prevalence of dengue viral and malaria parasitic co-infections in an epidemic district, Angul of Odisha, India: Aneco-epidemiological and cross-sectional study for the prospective aspects of public health”, Journal of Infection and Public Health, Vol. 9, pp 421—428, 2016.
  [7] E.A. Bakare, C.R. Nwozo, “Mathematical Analysis of Malaria-Schistosomiasis Co-infection model”, Epidemiology Research International.. Article ID 3854902, 19 pages. Hindawi Publishing Corporation. http://dx.doi.org/10.1155/2016/3854902 , 2016.
  [8] J. Amoah-Mensah, I.K. Dontwi, E. Bonyah, ”Stability Analysis of Zika-Malaria Co-infection Model for Model Endemic Region”, Journal of Advances in Mathematics and Computer Science, Vol. 26, Issue 1, 1-22, 2018.
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  Citation
  T.J. Oluwafemi, N.I. Akinwande, E. Azuaba, "Stability Analysis of Endemic Equilibrium Points of Malaria and Dengue Fever Co-infection Model," International Journal of Scientific Research in Mathematical and Statistical Sciences, Vol.7, Issue.6, pp.34-45, 2020

• Open Access   Article

  A Generalization to ? e x (ƒ (x) + ƒ1(x))dx

  Toyesh Prakash Sharma

  Research Paper | Journal-Paper (IJSRMSS)

  Vol.7 , Issue.6 , pp.46-50, Dec-2020

  Abstract

  Full-Text HTML

  References

  Citation

  Abstract
  With the help of this paper, the author is providing a generalized expression of well-known integral i.e. integral of e^x(f(x)+f`(x))dx, for concluding the main result author used concept of integral by parts, the concept of mathematical induction, differentiation, integration, etc. by using author’s generalization we can easily solve many time-consuming integrals which may a good thing for solvers to find the given integrals in less duration of time respectively. Generally, peoples are not focusing on finding generalizations to given integral as a problem but, yes, they can solve given integral, some of the persons try to find generalizations of given integral but due to very long procedure, they drop their idea.

  Key-Words / Index Term
  Integral by parts, Mathematical induction, differentian, Integration, exponentials etc

  References
  [1] Amit M Agrawal, “ Integral Calulus” Ch-1 Indefinate integral session 4 integral by parts publisher- Arhint p.22 ISBN-978-93-13191-91-9.
  [2] NCERT- Mathematics Textbook for class XII Ch-7 integrals Ex-7.6. P.326 . ISBN-81-7450-653-5
  [3] NCERT- Mathematics Textbook for class XII Ch-7 integrals Ex-7.6. P.323-328. ISBN-81-7450-653-5
  [4] Joseph Edwards: Integral Calculus for beginners, Ch-IV “ Integral by Parts.. Publisher-arihant P 32 ISBN-978-93-5094-145-4.
  [5] R.D Sharma “Mathematics class-XII” Vol. 1 Ch- 19 Indefinate integral Ex-19.26, publication Danpat Pai Publications rivised edition 2020 ISBN-978-81-941926-5-7. p.19.137-19.151
  [6] NCERT- Mathematics Textbook for class XI Ch-4 Principles of Mathematical induction. P.86
  [7] NCERT- Mathematics Textbook for class XII Ch-7 integrals Example 21 before Ex-7.6. P.326. ISBN-81-7450-653-5
  [8] Socratic Q and A: “How to find integral of excosxdx?.

  Citation
  Toyesh Prakash Sharma, "A Generalization to ? e x (ƒ (x) + ƒ1(x))dx," International Journal of Scientific Research in Mathematical and Statistical Sciences, Vol.7, Issue.6, pp.46-50, 2020