Volume-7 , Issue-1 , Feb 2020, ISSN 2348-4519 (Online) Go Back

• Open Access   Article

  Computational Thinking for Mathematics and Sciences: an Overview of Learning Approach

  Namrata Tripathi

  Research Paper | Journal-Paper (IJSRMSS)

  Vol.7 , Issue.1 , pp.69-74, Feb-2020

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  This study emphasis new approach in learning physical and mathematical phenomenon by using various computational Techniques (i.e MATLAB AND NEWTON software). These software provides a completely innovative way of learning and exploring multi objective programming, Elementary Mathematics, Mechanism, Thermodynamics, Electricity and Optics by creating a virtual 3D lab on a computer. These Techniques facilitate learner to establish a clear connection between the real world and its mathematical model as well as develops their model creating skill which is time saving. Sensitivity analysis is also represented for the given model. The virtual 3d-lab facilitates the students to understand the process of performing physical act frequently and also provides programming approach for scientific inquiry

  Key-Words / Index Term
  MATLAB, Newton Model, Thermodynamics, Isothermal, Adiabatic Process, Isobaric Process, Optimization, Linear Programming, Non Linear Programming and Quadratic Programming

  References
  [1] A. Bork, ” Computer –based instruction in physics,” Phys Today, September, Vol.24, 1981.
  [2] A.D.Belegundu,,Chandrupatla,T.R.,”Optimization concepts and applications in engineering”,Cambridge University Press, 2011.
  [3] Bower, M. James "Looking for Newton: realistic modeling in modern biology," Brains, Minds and Media Vol.1, No. 2 (2005).
  [4] GB.Dantzig,”Linear programming and extensions,”Princeton university press,1998.
  [5] Hestenes, D.,“Modeling games in the Newtonian world,”. American Journal of Physics, 60(8), pp.732-748, 1992.
  [6] http://newtonlab.com/English/newton/
  [7] LaViola Jr, Joseph J.,Robert C. Zeleznik. "Mathpad2: a system for the creation and exploration of mathematical sketches," In ACM SIGGRAPH 2006 Courses, pp. 33-es. 2006.

  Citation
  Namrata Tripathi, "Computational Thinking for Mathematics and Sciences: an Overview of Learning Approach," International Journal of Scientific Research in Mathematical and Statistical Sciences, Vol.7, Issue.1, pp.69-74, 2020

• Open Access   Article

  An Empirical Study on Effect of Smart Phones on Students Along With Social Life

  Sirisha Ch

  Research Paper | Journal-Paper (IJSRMSS)

  Vol.7 , Issue.1 , pp.75-79, Feb-2020

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  Now-a-days smart phones are being used by each and every one. Increasing technology, skills and capability every one using smart phones for their developing the relations and improve their knowledge. People always want to connect to the internet at any-time and any-where. For that, consumption of smart phones has become too high. Along with their books and school articles, most high school and college students and many younger students make their daily journey to institute with their trusty smart phone. Using of smart phones can crease some opportunities for students to increase their social skills and academic knowledge. At the same time it can be a harmful weapon for students to weakening in their communication skills and social life. The study deals, how effect of smart phones on students and their social life. The data has been collected through structured questionnaire and the sample size of the present study is 200. The current study was conducted in Vijayawada.

  Key-Words / Index Term
  Smart Phones, Social life, Health, Students

  References
  [1] Bhatt Mayank, “A study of mobile Phone Usage Among the Post Graduate Students”, Indian Journal of Marketing, April 2008, pp. 13-21.
  [2] Bedall-Hill, N. (2010). Postgraduates, field trips and mobile devices. In J. Traxler, & J. Wishart (Eds.), Making mobile learning work: Case studies of practice, ESCalate HEA Subject Centre for Education, pp. 18-22.
  [3] Faulkner Xristine and Fintan Culwin, “When fingers do the talking: a study of text messaging Original Research Article Interacting with Computers”, Vol. 17, Issue 2, March 2005, pp 167-185.
  [4] Farley Tom; ‘Mobile Telephone History” Telektronikk, April 3, 2005, pp. 22-34.
  [5] Gottman, J., Gonso, J., & Rasmussen, B.(1975). Social interaction, social competence, and friendship in children. Child Deveopment, 46(3), 709-718. Doi:10.2307/1128569.
  [6] K. Ketheeswaran and T. Mukunthan, study on “Usage of te Smart Phones for Learning Purposes by Students Who Follows ‘Disploma in Commonwealth Youth Development Programmes’ in the Colombo and Batticaloa Centres of the Open University of Sri Lanka, IOSR Journal Of Humanities and Social Science (IOSR-JHSS), Volume 21, Issue 5, Ver.2 pp-75-78, e-ISSN:2279-0837, p-ISSN:2279-0845, May 2016.
  [7] Selwyn Neil, “Schooling the Mobile Generation: the Future for schools in the Mobile-networked society,” British Journal of Sociology of Education, vol. 24 No.2, 2003, pp.131-143.

  Citation
  Sirisha Ch, "An Empirical Study on Effect of Smart Phones on Students Along With Social Life," International Journal of Scientific Research in Mathematical and Statistical Sciences, Vol.7, Issue.1, pp.75-79, 2020

• Open Access   Article

  Beta Expansion of Pisot Roots with pseudo co-factor

  G. Nagendra Kumar, E. Keshava Reddy

  Research Paper | Journal-Paper (IJSRMSS)

  Vol.7 , Issue.1 , pp.80-85, Feb-2020

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  A beta expansion is correspondent to the base 10 representation where base may be real number or non-integer. If is a pisot number, and P(x) is a monic polynomial then where R(x) is the companion polynomial of the expansion P(x) and Q(x) is the polynomial factor which is nonreciprocal converges to 2, it is the limit point of the pisot number. By using greedy expansion with base then the expansion will be either finite or periodic. If and are the limit points of the regular pisot number then the corresponding polynomials can be determined. Some Beta expansions for algebraic integers can be determined if pisot numbers hold the finite reversibly greedy (FRG) beta expansion. This paper gives the expansion for the pisot roots of which satisfies the Finite Reversibly Greedy (FRG) property.

  Key-Words / Index Term
  Pisot numbers, Beta expansion, FRG

  References
  [1] Victor J Katz, “A history of mathematics”, Harper Collins college publishers, New York 1993 an introduction.
  [2] A Renyi, “Representations of real numbers and their ergodic properties”,.Acta math.Acad.Sci.Hungar,8:477-493,1957.
  [3] K.G.Hare “Beta-Expansion of pisot and salem numbers.”In computer algebra 2006, pages 67-84 world sci. publ.,Hackensanck,NJ,2007.
  [4] C.L.Siegel,”Algrebraic integers whose conjugates lie in the unit circle”,Duke math.J.11(1944)597-602.
  [5] M.Amara,”Ensembles fermes de nombres algebriques”, Ann.Sci Ecole Norm.Sup(3) 83(1966) 215-270.
  [6] Maysum Panju,”Beta Expansion for regular pisot numbers”,arXiv:1103.2147v, math..NT,10 Mar 2011.

  Citation
  G. Nagendra Kumar, E. Keshava Reddy, "Beta Expansion of Pisot Roots with pseudo co-factor," International Journal of Scientific Research in Mathematical and Statistical Sciences, Vol.7, Issue.1, pp.80-85, 2020

• Open Access   Article

  A Stochastic Study on Different Oil Seeds Using DEA Approach

  B.R. Sreedhar, K. Muthyalappa, M. Amarnath

  Research Paper | Journal-Paper (IJSRMSS)

  Vol.7 , Issue.1 , pp.86-91, Feb-2020

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  This paper contains a number of to measure technical efficiency of Decision Making Units (DMU’s). This approach engages the linear programming technique (L.P.P) with parametric and non-parametric production frontiers in easy way. The parametric estimates cannot be subjected to significance tests due to the non-obtainability of standard errors (S.E’s). In this Paper we proposed a method of stochastic production frontiers- technical efficiency of Cobb-Douglas frontier production function as a linear programming problem (L.P.P).This method can be stretched in easy way to any parametric frontier production or cost function which is linear in parameters.

  Key-Words / Index Term
  DEA, DMUs, Peer, Technical Efficiency, Super Efficiency

  References
  [1] Farrell, M.J. “The Measurement of Productive Efficiency”, Journal of Royal Statistical society, Series-A, 120, pp.253-281, 1957.
  [2] Afriat, S. “Efficiency Estimation of Production Functions”, International Economic Review, 13, pp. 568-98, 1972.
  [3] Richmond, J. “Estimating the Efficiency of Production”, International Economic Review, 15, pp. 515-521, 1974.
  [4] Aigner, D.J., Lovell, C.A.K. and Schmidt, P. “Formulation and Estimation of Stochastic Frontier Production Function Models”, Journal of Econometrics, 6, pp. 21-37, 1977
  [5] Schmidt,P., Lovell,C.A.K “Estimating Stochastic Production and Cost Frontiers when Technical and Allocative inefficiencies are Correlated”, Journal of Econometrics, 13, pp.83-100, 1980.
  [6] Charnes,A., Cooper,W.W and Rhodes, E. “Evaluating program and Managerial Efficiency: An Application of Data Envelopment Analysis to Program Follow Through”, Management Science, 27, pp.668-697, 1981.
  [7] Kopp,R.J. “The Measurement of Productive Efficiency: A Reconsideration”, The Quarterly Journal of Economics, pp.477-500, 1981.
  [8] Schmidt, P. “Frontier Production Functions”, Econometric Reviews, 4, pp. 289-328, 1986.
  [9] Bauer,P.W. “Recent Developments in the Econometric Estimation of Frontiers”, Journal of Econometrics, 46, pp.39-56, 1990.
  [10] Kalirajan,K.P “On measuring Economic Efficiency”, Journal of Applied Econometrics, 5, pp.75-85, 1990.
  [11] Kumbhakar, S.C “Production Frontiers, Panel Data and Time Varying Technical inefficiency”, Journal of Econometrics, 46, pp.201-11, 1990.
  [12] Battese, G.E and Coelli, T.J. “Frontier Production Functions, Technical Efficiency and panel data: with application to Paddy Farmers in India”, Journal of Productivity Analysis, 3, pp. 153-69, 1992.
  [13] Coelli, T.J “A Computer Program for Frontier Production Function Estimation- FRONTIER, Version 2.0”, Economics letters 39, 29-32, 1992.
  [14] Kalirajan,K.P and Shand “Causality between Technical and Allocative Efficiencies-An Empirical Testing”, Journal of Economic studies, 19, pp.3-17, 1992.
  [15] Mark Ivaldi, SylvetteMonier –Dilhan, Michel Simioni “Stochastic production frontiers and panel data- A latent variable frame work”, European Journal of Operations Research, 80, pp. 534-547, 1995.
  [16] Bhatt, R “Data Envelopment Analysis, Journal of Health Management”, Vol.3, No.2, pp 309-328, 2000.
  [17] Banker, Cooper, Seiford, Thrall and Chu, “Returns to scale in different DEA models”, European Journal of Operations Research, 154, pp.345-362, 2004.

  Citation
  B.R. Sreedhar, K. Muthyalappa, M. Amarnath, "A Stochastic Study on Different Oil Seeds Using DEA Approach," International Journal of Scientific Research in Mathematical and Statistical Sciences, Vol.7, Issue.1, pp.86-91, 2020

• Open Access   Article

  Inventory Model with Genralized Pareto Rate of Replinishment Having Time Dependent Demand

  K. Srinivasaro, B. Punyavathi

  Research Paper | Journal-Paper (IJSRMSS)

  Vol.7 , Issue.1 , pp.92-106, Feb-2020

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  Abstract
  Generalized Pareto distribution gained lot of importance in income analysis and life testing experiments due to its long upper tail. The characteristics of the life time of a commodity in production processes dealing with deteriorated items match with the statistical characteristics of the Generalized Pareto distribution. Hence, in this paper we develop and analyze an economic production quantity model with generalized Pareto rate of production and deterioration. Here it is assumed that the production quantity is random and follows a Generalized Pareto distribution. It is further assumed that the life time of the commodity is random and follows a Generalized Pareto distribution. Considering that the demand rate is time dependent and follows a power pattern the instantaneous state of inventory at any given time in the cycle length under the assumptions of shortages are allowed and fully backlogged is derived. With plausible cost considerations the total production cost of the system is derived and minimized with respect to production up time, production down time and production quantity. The sensitivity analysis of the model reveals that the random production and random life time have significant influence on production. It is further observed that the deteriorating distribution and life time distribution parameters have tremendous influence on optimal operate policies of the system. This model also includes the model without shortages as a limiting case. This model is useful for analyzing the production systems dealing with deteriorating items.

  Key-Words / Index Term
  Generalized Pareto distribution, Random Production, Economic Production Quantity Model, Sensitivity Analysis, Production Scheduling, Deterioration

  References
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  [2]. Ruxian, LI., Lan, H. and Mawhinney, R. J, “A review on deteriorating inventory study”, Journal of Service Science Management, Vol. 3, No. 1, pp. 117-129, 2010.
  [3]. Pentico, D. W. and Drake, M. J, “A survey of deterministic models for the EOQ and EPQ with partial backordering”, European Journal of Operational Research, Vol. 214, Issue. 2, pp. 179-198, 2011.
  [4]. Ghare, P. M and Schrader, G. F, “A model for exponentially decaying inventories’, Journal of Industrial engineering”, Vol. 14, pp. 238-2430, 1963.
  [5]. Shah, Y. and Jaiswal, M. C, “An order level inventory model for a system with a constant rate of deterioration”, OPSEARCH, Vol. 14, pp. 174-184, 1977.
  [6]. Cohen, M. A,”Joint pricing and ordering for policy exponentially decaying inventories with known demand”, Naval Research Logistics. Q, Vol. 24, pp. 257-268, 1977.
  [7]. Aggarwal, S. P, “A note on an order level inventory model for system with constant rate of deterioration”, OPSEARCH, Vol. 15, No. 4, pp.184–187, 1978 .
  [8]. Dave, U and Shah, Y.K, “A probabilistic inventory model for deteriorating items with time proportional demand”, Journal of Operational Research Society, Vol. 32, pp. 137-142, 1982.
  [9]. Pal, M, ‘An inventory model for deteriorating items when demand is random”, Calcutta Statistical Association Bulletin, Vol. 39, pp. 201-207, 1990.
  [10]. Kalpakam, S and Sapna, K. P, “A lost sales (S-1, S) perishable inventory system with renewal demand”, Naval Research Logistics, Vol. 43, pp. 129-142, 1996.
  [11]. Giri, B. C and Chaudhuri, K. S,”An economic production lot-size model with shortages and time dependent demand”, IMA Journal of Management Mathematics, Vol. 10, No.3, pp. 203-211, 1999.
  [12]. Tadikamalla, P.R, “An EOQ inventory model for items with gamma distributed deteriorating”, AIIE TRANS, Vol. 10, pp. 100-103, 1978.
  [13]. Covert, R.P. and Philip, G.C, “An EOQ model for items with Weibull distribution deterioration”, AIIE. TRANS, Vol.5, pp. 323-326, 1973.
  [14]. Philip, G. C, “A generalized EOQ model for items with Weibull distribution”, AIIE TRANS, Vol. 16, pp. 159-162, 1974.
  [15]. Goel, V. P and Aggarwal, S. P, “Pricing and ordering policy with general Weibull rate deteriorating inventory”, Indian journal Pure and Applied Mathematics, Vol. 11(5), pp. 618 -627, 1980.
  [16]. Venkata Subbaiah, K., Srinivasa Rao, K. and Satyanarayana, B.V.S,”An inventory model for perishable items having demand rate dependent on stock level”, OPSEARCH, Vol. 41(4), pp. 222-235, 2004.
  [17]. Nirupama Devi, K., Srinivasa Rao, K. and Lakshminarayana, J, “Perishable inventory model with mixtures of Weibull distribution having demand as a power function of time”, Assam Statistical Review,Vol.15,No.2,pp.70-80,2001.
  [18]. Srinivasa Rao, K., Vivekananda Murthy, M and Eswara Rao, S, “An optimal ordering and pricing policies of inventory models for deteriorating items with generalized Pareto life time”, A Journal on Stochastic Process and Its Applications, Vol. 8(1), pp. 59-72, 2005.
  [19]. Xu, X. H. and Li, R. J, “A two-warehouse inventory model for deteriorating items with time-dependent demand”, Logistics Technology, No. 1, pp. 37-40, 2006.
  [20]. Rong, M., Mahapatra, N.K. and Maiti, M,”A two-warehouse inventory model for a deteriorating item with partially/fully backlogged shortage and fuzzy lead time”, European Journal Of Operational Research, Vol. 189, Issue. 1, pp. 59-75, 2008.
  [21]. Srinivasa Rao, K., Srinivas. Y., Narayana, B. V. S. and Gopinath, Y, ‘Pricing and ordering policies of an inventory model for deteriorating items having additive exponential lifetime”, Indian Journal of Mathematics and Mathematical Sciences, Vol. 5(1), pp. 9-16, 2009.
  [22]. Chakrabarti, T and Chaudhuri, K. S. (1997). ‘An EOQ model for deteriorating items with a linear trend in demand and shortages in all cycles’, International Journal of production economics, Vol.49, No.3, pp. 205-213.
  [23]. Chang, H. J and Lin, W. F. (2010). ‘A partial backlogging inventory model for non-instantaneous deteriorating items with stock dependent consumption rate under inflation’, Yugoslav Journal of Operations Research, Vol.20 (1), pp. 35-54.
  [24]. Biswajit Sarkar, “An EOQ model with delay in payments and stock dependent demand in the presence of imperfect production”, Mathematics and Computation, Vol. 218(17), pp. 8295-8308, 2012.
  [25]. Mukherjee, S. P and Pal, M, “An order level production inventory policy for items subject to general rate of deterioration”, IAPQR Transactions, Vol. 11, pp. 75-85, 1986.
  [26]. Gowsami, A., Mahata, G. C. and Prakash, O. M., “Optimal retailer replenishment decisions in the EPQ model for deteriorating items with two level of trade credit financing”, International Journal of Mathematics in Operational Research, Vol.2, No.1, pp. 17-39, 2010.
  [27]. Goyal, S. K and Giri, B. C, “The production- inventory problem of a product with varying demand, production and deterioration rates”, European Journal of Operational Research, Vol. 147, No. 3, pp. 549-557, 2003.
  [28]. Panda, R. M. and Chaterjee, E,”On a deterministic single items model with static demand for deteriorating item subject to discrete time variabl”, IAPQR Trans, Vol. 12, pp. 41-49, 1987.
  [29]. Mandal, B. N. and Phaujdar, S,”An inventory model for deteriorating items and stock dependent consumption rate”, Journal of Operational Research Society, Vol. 40, No. 5, pp. 483- 488, 1989.
  [30]. Sana, S. S., Goyal, S. K. and Chaudhuri, K. S, “A production inventory model for a deteriorating item with trended demand and shortages”, European Journal of Operational Research, Vol. 157, No. 2, pp. 357-371,2004.
  [31]. Perumal, V. and Arivarignan, G,”A production model with two rates of productions and back orders”, International Journal of Management system, Vol. 18, pp. 109-119, 2002.
  [32]. Pal, M. and Mandal, B,”An EOQ model for deteriorating inventory with alternating demand rates”, Journal Of Applied Mathematics and Computing, Vol. 4, No.2, pp. 392-397, 1997.
  [33]. Sen, S. and Chakrabarthy, T, “An order- level inventory model with variable rate of deterioration and alternating replenishing rates considering shortages”, OPSEARCH, Vol. 44(1), pp. 17-26, 2007.
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  Citation
  K. Srinivasaro, B. Punyavathi, "Inventory Model with Genralized Pareto Rate of Replinishment Having Time Dependent Demand," International Journal of Scientific Research in Mathematical and Statistical Sciences, Vol.7, Issue.1, pp.92-106, 2020

• Open Access   Article

  About New Sets and Topologies in Ideal Topological Spaces

  Anjali Srivastava, Rohit Gaur

  Research Paper | Journal-Paper (IJSRMSS)

  Vol.7 , Issue.1 , pp.107-110, Feb-2020

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  Abstract
  In this research article we discuss some properties of ideal topological spaces commonly denoted as (X,τ,I) and give a few characterizations of ideal topological spaces. With the help of the existing topology τ on X and the ideal I on X, a new topology called τ* is introduced and the properties of (X,τ) as well as (X, τ*) are discussed in details.

  Key-Words / Index Term
  LI-perfect sets, RI-perfect sets, CI-perfect sets, RI-open sets

  References
  [1] T. R. Hamlett and D. Jankovic, “Ideals in general topology”, General Topology and Applications, pp. 115–125, 1988.
  [2] E.Hayashi, “Topologies defined by local properties”, Mathematische Annal en, Vol.156, No.3, pp.205-215, 1964.
  [3] D.jankovic and T.R. Hamlett, “New topologies from old via ideals”, Amer. math. Monthly, Vol. 97, pp 295-310, 1990.
  [4] D.jankovic and T.R. Hamlett, “Compatible extensions of ideals”, Boll. Un.mat.ital, Vol. 7, No. 6-B, pp. 453-465, 1992.
  [5] Kuratowski, “Topology”, Vol.1, Academic Press, New York, 1966.
  [6] K.D. Joshi, “Introduction to General Topology”. New AGE INTERNATIONAL (P) LIMITED, New Delhi-110002 (1983).
  [7] Ümit Karabıyık1, Aynur Keskin Kaymakcı. “Some New Regular Generalized Closed Sets in Ideal Topological” Spaces”, EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 9, No. 4, pp. 434-442, 2016, ISSN 1307-5543.
  [8] J.Sebastian Lawrence and R.Manoharan. “NEW SETS AND TOPOLOGIES IN IDEAL TOPOLOGICAL SPACES”, International Journal of Pure and Applied Mathematics, Vol. 118, No. 20, pp. 93-98, 2018, ISSN: 1314-3395.
  [9] R.Manoharan and P.thangavelu. “Some New Sets and Topologies in Ideal Topological Spaces”. Chinese Journal of Mathematics, Vol. 10, 6 pages, 2013.
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  [11] Rodyna A. Hosny, “Notes on Soft Perfect Sets”, Int. Journal of Math. Analysis, Vol. 8, No. 17, pp. 803 – 812, 2014.

  Citation
  Anjali Srivastava, Rohit Gaur, "About New Sets and Topologies in Ideal Topological Spaces," International Journal of Scientific Research in Mathematical and Statistical Sciences, Vol.7, Issue.1, pp.107-110, 2020

• Open Access   Article

  I-Rough Continuous Functions

  B. P. Mathew, S. J. John

  Research Paper | Journal-Paper (IJSRMSS)

  Vol.7 , Issue.1 , pp.111-120, Feb-2020

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  Abstract
  This paper is an attempt to introduce the concept of functions in rough universe and to investigate its properties. I-rough functions between two arbitrary rough universes are introduced and the conditions for an I-rough function to be an I-rough continuous function with respect to I-rough topological spaces are studied. I rough functions in a rough universe are the generalization of functions in set theory. In addition the paper also introduce I-rough open mapping and I-rough closed mapping between two I-rough topological spaces and its properties are studied.

  Key-Words / Index Term
  Rough universe, I-rough sets, I-rough topological spaces, I-rough continuous function, I-rough open mapping, I-rough closed mapping

  References
  [1] Iwinski, T. B., “Algebraic approach to rough sets”, Bull. Pol. Acd. Sci., Math., 35, pp.673-683, 1987.
  [2] Kondo, M., & Dudek, W. A., “Topological Structures of Rough Sets Induced by Equivalence Relations”, JACIII, 10(5), pp. 621-624, 2006.
  [3] Lashin, E. F., Kozae, A. M., Khadra, A. A., & Medhat, T., “Rough set theory for topological spaces”, International journal of approximate reasoning, 40(1-2), pp. 35-43, 2005.
  [4] Mathew, B. P., & John, S. J., “I-Rough topological spaces”, International Journal of Rough Sets and Data Analysis (IJRSDA), 3(1), pp. 98-113, 2016.
  [5] Mathew, B. P., & John, S. J, “Some special properties of i-rough topological spaces”, Annals of Pure and Applied Mathematics, 12(2), pp. 111-122, 2016.
  [6] Munkres, J. R. Topology. 2nd. PHI learning private limited, 2000.
  [7] Pawlak, Z., “Rough sets”, International journal of computer & information sciences, 11(5), pp. 341-356, 1982.
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  [9] Pawlak, Z. Rough sets: Theoretical aspects of reasoning about data (Vol. 9). Springer Science & Business Media, 2012.
  [10] Thuan, N. D., Covering rough sets from a topological point of view. arXiv preprint arXiv:1207.6560, 2012
  [11] Willard, S., General topology. Courier Corporation, 2004.
  [12] Zhu, W, “Topological approaches to covering rough sets”. Information sciences, 177(6), pp. 1499-1508, 2007.
  [13] Yao, Y. Y., “Two views of the theory of rough sets in finite universes”. International journal of approximate reasoning, 15(4), pp. 291-317, 1996.
  [14] Yao, Y. Y., “Constructive and algebraic methods of the theory of rough sets”. Information sciences, 109(1-4), pp. 21-47, 1998.

  Citation
  B. P. Mathew, S. J. John, "I-Rough Continuous Functions," International Journal of Scientific Research in Mathematical and Statistical Sciences, Vol.7, Issue.1, pp.111-120, 2020