Volume-5 , Issue-6 , Dec 2018, ISSN 2348-4519 (Online) Go Back

• Open Access   Article

  Oscillation Criteria of Second Order Nonlinear Impulsive Neutral Differential Equations

  R. Sakthivel

  Research Paper | Isroset-Journal (IJSRMSS)

  Vol.5 , Issue.6 , pp.222-227, Dec-2018

  CrossRef-DOI:   https://doi.org/10.26438/ijsrmss/v5i6.222227

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  Abstract
  In this paper, we study the oscillatory behavior of second order non-linear impulsive neutral differential equations. By using the generalized Ricatti transformation and the integral averaging technique, we obtained some new oscillation criteria. Examples are given to show how impulse perturbations greatly affect the oscillation behavior of the solutions.

  Key-Words / Index Term
  Oscillation, Second Order, Nonlinear, Impulsive, Neutral Differential Equations.

  References
  [1]. L. Berenzansky and E. Braverman, On oscillation of a second order impulsive delay differential equation, J. Math. Anal. Appl., 39 (2000), 217-225.
  [2]. E. M. Bonotto, L. P. Gimenes and M. Federson, Oscillation for a second order neutral differential equation with impulses, Cad. Mat., 9 (2008),169-190.
  [3]. L. P. Gimenes and M. Federson, Oscillation by impulses for a second order delay differential equation, Cad. Mat., 06 (2005), 181-191.
  [4]. K. Gopalsamy and B. G. Zhang, On delay differential equations with impulses, J. Math. Anal. Appl., 139 (1989), 110-122.
  [5]. I. Gyori and G. Ladas, Oscillation Theory of Delay Differential Equations with Applications, Clarendon Press, Oxford, (1991).
  [6]. C. Guo and Z. Xu, On the oscillation of second order linear impulsive differential equations, Diff. Eqns. Apps., 3 (2010), 319-330.
  [7]. Z. He and W. Ge, Oscillations of second order nonlinear impulsive ordinary differential equations, J. Copmut. Appl. Math., 158(2) (2003),397-406.
  [8]. V. Lakshmikantham, D. D. Bainov and P. S. Simeonov, Theory of Impulsive Differential Equations, World Scientific, Singapore 32 (1989).
  [9]. W. Li, C. Yang and H. Ji, Oscillation of a class of second order nonlinear ODE with impulses, Appl. Math. Comput., 138(2-3) (2003), 181-188.
  [10]. X. Li, Oscillation properties of second order delay differential – difference equations with impulses, Adv. Appl. Math. Anal., 3 (2008), 55-66.
  [11]. M. Peng, Oscillation caused by impulses, J. Math. Anal. Appl., 255 (2001), 163-176.
  [12]. M. Peng and W. Ge, Oscillation criteria for second order nonlinear differential equations with impulses, Compu. Math. Appl., 39 (2000), 217-255.
  [13]. M. Peng and R. P. Agarwal, Oscillation theorems of second order non linear neutral delay differential equations under impulsive perturbations, Indian. J. Pure. Appl. Math., 33(7) (2002), 1017-1029.
  [14]. CH. G. Philos, Oscillation theorems for linear differential equation of second order, Arch. Math., 53 (1989), 482-492.
  [15]. Y V. Rogovchenko, Oscillation criteria for certain nonlinear differential equations, J. Math. Anal. Appl., 229 (1999), 399-416.
  [16]. A. A. Soliman, R. A. Sallam, A. M. Hassan, Oscillation criteria of second order nonlinear neutral differential equations, Int. J. of App. Math. Research., 1 (3) (2012), 314 - 322.
  [17]. J. Yan, Oscillation properties of a second order impulsive delay differential equation, Comp. Math. Appl., 47 (2004), 253-258.
  [18]. Y. Zhang, A. Zhao and J. Yan, Oscillation criteria for impulsive delay differential equations, J. Math. Anal. Appl., 205 (1997), 461-470.
  [19]. A. Zhao and J. Yan, Existence of positive solutions for delay differential equations with impulses, J. Math. Anal. Appl., 210 (1997), 667-678.
  

  Citation
  R. Sakthivel, "Oscillation Criteria of Second Order Nonlinear Impulsive Neutral Differential Equations," International Journal of Scientific Research in Mathematical and Statistical Sciences, Vol.5, Issue.6, pp.222-227, 2018

• Open Access   Article

  A Two Stage Group Sampling Plans based on truncated life tests for Type-II Generalized Log-Logistic distribution

  S.V.S.V.S.V. Prasad, K. Rosaiah, G. Srinivasa Rao

  Research Paper | Isroset-Journal (IJSRMSS)

  Vol.5 , Issue.6 , pp.228-243, Dec-2018

  CrossRef-DOI:   https://doi.org/10.26438/ijsrmss/v5i6.228243

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  Abstract
  This paper will illustrate the development of a two-stage group sampling plan based on the truncated life tests for the life time of the products follows type-II generalized log-logistic distribution (TGLLD). The decision to be made regarding the acceptance of lot is depends on the observed number of failures from each group either in first or second stage of sampling. At an accepted level of consumer’s risk and at a predefined test termination time, the design parameters like number of groups required and the acceptance number for each group at two stages are derived for the underlying lifetime distribution. Single stage group sampling plans are also considered as special cases of the proposed plan and both plans are compared with each other with respect to the values of average sample number and the operating characteristics

  Key-Words / Index Term
  Type-II generalized log-logistic distribution, average sample number, consumer’s risk, producer’s risk, operating characteristics

  References
  [1]. Aslam, M., and Shahbaz, M. Q., “Economic reliability test plans using the generalized exponential distribution”, Journal of Statistics, Vol.14, pp. 53-60,2007.
  [2]. Aslam, M., and Jun, C.-H., “Group acceptance sampling plans for truncated life Tests based on the inverse Rayleigh distribution and log-logistic distribution” Pakistan Journal of Statistics, 25, 107-119, 2009.
  [3]. Aslam, M. and Jun, C.-H., “A group acceptance sampling plan for truncated life test having Weibull distribution”, Journal of Applied Statistics, 39, 1021-1027, 2009.
  [4]. Aslam, M., and Jun, C.-H., “A group acceptance sampling plans for truncated life tests based on the inverse Rayleigh and log-logistic distributions”, Pakistan Journal of statistics, 25(2),1-13, 2009.
  [5]. Aslam, M., Debasis Kundu., Jun, C.H., and Ahmad, M.,”Time truncated group acceptance sampling plans for generalized exponential distribution”, Journal of Testing and Evaluation, 39(4):968-976, 2011.
  [6]. Aslam, M., Jun, C.H., Ahmad, M., “A two-stage group sampling plan based on truncated life tests for a general distribution”, J Stat. Comput. Simul., 81(12), 1927-1938, 2011.
  [7]. Folks, J. L. and Chhikara, R.,”The inverse Gaussian distribution and its statistical application”, Journal of The Royal Statistical Society Series B-Statistical Methodology, 20, 263-289, 1978.
  [8]. Gupta, S.S. and Groll, P.A., “Gamma distribution in acceptance sampling based on life tests”, J. Amer. Statist. Assoc, 56, p. 942 – 970, 1961.
  [9]. Kantam, R.R.L. and Rosaiah, K.,“Half logistic distribution in acceptance sampling based on life tests”, IAPQR Transactions, 23 (2), p. 117 –125, 1998.
  [10]. Kantam, R. R. L., Rosaiah, K. and Rao, G. S., “Acceptance sampling based on life tests: log-logistic models”, J. Appl. Statist, 28, p. 121–128, 2001.
  [11]. Ramaswamy, A.R.S. and Anburajan. P., “A two stage group acceptance sampling Plans Based on truncated life tests for Inverse and Generalized Rayleigh Distributions”, Global Journal of Advance Research, Vol.2, Issue-1, 172-183, 2015.
  [12]. Ramaswamy, A.R.S. and Anburajan. P., “A two stage group acceptance sampling Plans Based on truncated life tests using log-logistic and Gamma Distributions”, Journal of Progressive Research in Mathematics, Vol.2, Issue-2, 107-117, 2015.
  [13]. Razzaque, M. A., Hanif, M., Imran, A. A., Rafi, M., Ahmad, M., “Economic reliability two-stage group sampling plan for truncated life test having Weibul distribution”, European Journal of Scientific Research, Vol. 54 No.4, pp.593-599, 2011.
  [14] Rosaiah, K., and Kantam, R.R.L. and Santosh Kumar, Ch. “Reliability test plans for exponentiated log-logistic distribution”, Economic Quality Control, 21(2), 165-175, 2006.
  [15]. Rosaiah, K., Kantam. R.R.L., and Santosh Kumar, Ch.,“Exponentiated log-logistic distribution-an economic reliability test plan”, Pakistan Journal of Statistics, 23 (2), 147-156, 2007.
  [16]. Rosaiah, K., Kantam. R.R.L., Prasad, S.V.S.V.S.V. and Pratap Reddy. J.,” Reliability estimation in type-II generalized log-logistic distribution”, Int. J. Agricult. Stat. Sci., Vol. 4No.2, pp. 283-292, 2008.
  [17]. Rosaiah. K., Rao, G.S., and Prasad, S.V.S.V.S.V. “Group Acceptance Sampling Plans for Resubmitted Lots for Life Tests Based on Type-II Generalized Log-Logistic Distribution”, European Journal of Scientific Research, Vol. 136, No 1, pp.98-106, 2015.
  [18]. Rosaiah. K., Rao, G.S., and Prasad, S.V.S.V.S.V., ”A Group Acceptance Sampling Plans Based on Truncated Life Tests for Type-II Generalized Log-Logistic Distribution”, ProbStat Forum, Volume 09, Pages 88–94, 2016.
  [19]. Rao, G.S. “A hybrid group acceptance sampling plans for lifetimes based on log-logistic distribution”, Journal of Reliability and Statistical studies, 4(1), 31-40, 2011.
  [20]. Rao, G. S., R.R.L. Kantam., Rosaiah. K and Prasad, S.V.S.V.S.V. “Reliability test plans for type-II generalized log-logistic distribution”, JRSS Vol.5: Issue 1(2012): 55-64, 2012.
  [21]. Rao, G. S., R.R.L. Kantam, Rosaiah. K and Prasad, S.V.S.V.S.V., “An economic reliability test plan for generalized log-logistic distribution”, International Journal of Engineering and Applied Sciences, Vol. 3: 61-68, 2012.
  [22]. Rao, G. S., Rosaiah. K., Sridhar Babu M and Siva Kumar D C U. “A two –Stage group sampling plan based on truncated life tests for a exponentiated Frechet Distribution”, European Scientific Journal, Vol. 10, No. 33, 145-160, 2014.
  

  Citation
  S.V.S.V.S.V. Prasad, K. Rosaiah, G. Srinivasa Rao, "A Two Stage Group Sampling Plans based on truncated life tests for Type-II Generalized Log-Logistic distribution," International Journal of Scientific Research in Mathematical and Statistical Sciences, Vol.5, Issue.6, pp.228-243, 2018

• Open Access   Article

  Various Distance Measures on Pythagorean Vague Sets

  S.Vinnarasi, F. Nirmala Irudayam

  Research Paper | Isroset-Journal (IJSRMSS)

  Vol.5 , Issue.6 , pp.244-247, Dec-2018

  CrossRef-DOI:   https://doi.org/10.26438/ijsrmss/v5i6.244247

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  Abstract
  In this paper, we introduced and study a new concept of Pythagorean vague sets and deduce some of its properties. Based on these definition varies form of Pythagorean vague distances are proposed and studied with suitable illustration.

  Key-Words / Index Term
  In Pythagorean vague Sets, Hamming Pythagorean Vague Measures, Normalized Hamming Pythagorean Vague Measures, Euclidean Pythagorean Vague Measures, Normalized Euclidean Pythagorean Vague Measures

  References
  [1] K. Atanassov, “ Intuitionistic fuzzy sets”, VII ITKR`s Session, Sofia (September, 1983) (in Bugaria).
  [2] Chang.C.L, “Fuzzy topological spaces”, Journal of Mathematical Analysis and Applications, 24(1968), 182- 190.
  [3] Eulalia Szmidt and Jamuz Kacprzyk , “On Measuring Distance Between Intuitionistic Fuzzy Sets”, Fuzzy Sets and Systems, 111,505-518.
  [4] Khaista Rahman , Asad Ali., Muhammad Sajjiad Ali Khan , “Some Interval Valued Pythagorean Fuzzy Weighted Averaging Aggregation Operators and their Application to Multiple Attribute Decision Making” , Journal of Mathematics, 50(2), pp-113-129.
  [5] W.L. Gau and D.J. Buehrer, “Vague sets”, IEEE Transactions on Systems, Man and Cybernetics, 23, N0.20(1993), 610-614.
  [6] Yager RR. “Pythagorean fuzzy subsets”, In: Proc Joint IFSA World Congress and NAFIPS Annual Meeting, Edmonton, Canada; 2013. pp 57–61.
  [7] Yager RR, Abbasov AM. “Pythagorean membership grades, complex numbers, and decision making” , Int J Intell Syst 2013;28:436–452.
  [8] Yager RR. “Pythagorean membership grades in multi-criteria decision making”, IEEE Trans Fuzzy Syst 2014;22:958–965.
  [9] Xunjie Gou, Zeshui Xu, Peijia Ren , “The properties of continuous Pythagorean fuzzy information”, Int J Intell Syst (2016) vol 31, pp-401-424.
  [10] Xindong peng and Yong Yang , “Some Results for Pythagorean Fuzzy Sets”, Int J Intell Syst (2015), vol-30, pp-1133-1160.
  [11] L.A. Zadeh, “Fuzzy sets”, Information and Control, 8, (1965), 338-353.
  [12] Z.S .Xu and R.R, Yagar , “Some geomentric aggregation operators based on intuitionistic fuzzy sets”, Int J Gen Syst (2006) 35, 417-433.
  

  Citation
  S.Vinnarasi, F. Nirmala Irudayam, "Various Distance Measures on Pythagorean Vague Sets," International Journal of Scientific Research in Mathematical and Statistical Sciences, Vol.5, Issue.6, pp.244-247, 2018

• Open Access   Article

  On Prime Labeling Of Some Union Graphs and Circulant Graphs

  S.K. Patel, J.B. Vasava

  Research Paper | Isroset-Journal (IJSRMSS)

  Vol.5 , Issue.6 , pp.248-254, Dec-2018

  CrossRef-DOI:   https://doi.org/10.26438/ijsrmss/v5i6.248254

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  Abstract
  A graph G of order n is said to admit prime labeling if its vertices can be labeled with distinct positive integers not exceeding n such that each pair of adjacent vertices have relatively prime labels. A graph G that admits a prime labeling is called a prime graph. In this paper we investigate for prime labeling of some union graphs and later study some necessary and sufficient conditions for prime labeling of certain circulant graphs.

  Key-Words / Index Term
  Prime labeling, Prime graphs, Independence number, Union of graphs, Circulant graphs

  References
  [1] J. Gross and J. Yellen, “Graph theory and its applications”, CRC press, 1999
  [2] D. M. Burton, “Elementary Number Theory”, Brown Publishers, Second Edition, 1990.
  [3] A. Tout, A. N. Dabboucy and K. Howalla, "Prime labeling of graphs", Nat. Acad. Sci. Lett., Vol. 11 , pp. 365–368, 1982.
  [4] M. Sundaram, R. Ponraj and S. Somasundaram, “Prime cordial labeling of graphs”, J. Indian Acad.Math., Vol. 27, pp. 373–390, 2005.
  [5] S. K. Patel and N. P. Shrimali, “Neighborhood-prime labeling”, International Journal of Mathematics and Soft Computing, Vol. 5, pp. 17–25, 2015.
  [6] J. A. Gallian, “A dynamic survey of graph labeling”, Electron. J. Combin., 2017.
  [7] Hung-Lin Fu and Kuo-Ching Huang, “On prime labelling”, Discrete Math.,Vol. 127, pp. 181–186, 1994.
  [8] M. A. Seoud and M. Z. Youssef, “On prime labelings of graphs”, Congr. Numer., Vol. 141, pp. 203-215, 1999.
  [9] S. K. Patel and Jayesh Vasava, “On prime labeling of some union graphs”, Kragujevac journal of Mathematics, Vol. 42, pp. 441-452, 2018.
  [10] M. A. Seoud, A. T. Diab, and E. A. Elsahawi, “On strongly-C harmonious, relatively prime, odd graceful and cordial graphs”, Proc. Math. Phys. Soc. Egypt, Vol. 73, pp. 33-55, 1998.
  

  Citation
  S.K. Patel, J.B. Vasava, "On Prime Labeling Of Some Union Graphs and Circulant Graphs," International Journal of Scientific Research in Mathematical and Statistical Sciences, Vol.5, Issue.6, pp.248-254, 2018

• Open Access   Article

  On Quasi-Reduced Modules

  K.H. Singh, N. Ansari, M. Dutta

  Research Paper | Journal-Paper (IJSRMSS)

  Vol.5 , Issue.6 , pp.255-258, Dec-2018

  CrossRef-DOI:   https://doi.org/10.26438/ijsrmss/v5i6.255258

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  Abstract
  This paper generalizes the concept of a reduced modules. We introduced the concept of quasi-reduced-I i.e, QR-I and quasi-reduced-II i.e, QR-II in modules and studied some of their properties.

  Key-Words / Index Term
  Reducedring, Armendarizring, Semi-primerin and Rigidrings

  References
  [1] M.B. Rege, S.C. Chhawchharia, “Armendariz rings ”, Proc. Japan Acad. Ser. A Math. Sci, Vol.31, pp.123-141, 2012.
  [2] M. Baser, “On Armendariz and quasi-Armendariz modules”, Note di Mathematica n., Vol.31, pp.123-141, 2012.
  [3] Kh.Hera Singh and A. M Bhuphang, “Semi-commutative and Duo Rings” Advances in Algebra Vol.8, pp. 1-12, 2015.
  [4] Y.Hirano, “Regular modules and V-modules”, Hiroshima Math.J., Vol.31, pp.123-141, 2012.
  [5] T.K. Lee, Y.Zhou, “Armendariz and reduced rings ”, Comm. Algebra, Vol.31, pp.123-141, 2012.
  [6] T.K. Lee, Y.Zhou, “Reduced modules”, Lecture Notes in Pure Appl. Math, Vol.31, pp.123-141, 2012.
  [7] Z. Liu, R. Zhao, “On weak Armendariz rings”,Comm. Algebra, Vol.31, pp.123-141, 2012.
  [8] R.R. Andruszikiwiecz, E.R. Puczlowski, “Right fully idempotent rings need not be left fully idempotent ”, Glasgow Math.J, Vol.37, pp.155-157, 1995.
  [9] L. Liang, L. Wang, Z. Liu, “On a generalization of semicommutative rings ”, Taiwanese J. of Math, Vol.31, pp.123-141, 2012.
  [10] G. Marks, “On 2-Primal ore Extensions”, Comm in Algebra, Vol.31, pp.123-141, 2012.
  [11] G. Marks, R. Mazurek, M. Ziembowski ,“A unified approach to the various generalization of Armendariz ring”, Bull.Aust.Math.Soc., Vol.31, pp.123-141, 2012.
  [12] M.B. Rege, A.M. Buhphang, “On reduced modules and rings ”, International Electronic Journal of Algebra, Vol.31, pp.123-141, 2012.
  [13] F.W. Anderson, K.R.Fuller, “Rings and Catagories of modules”, Springer-Verlag, 1974
  [14] A.M. Buhphang, M.B. Rege, “Semi-commutative modules and armendariz modules ”, Arab J.Math.Sc., Vol.8, pp.53-65, 2002.
  [15] B. Stenstrom, “Rings of Quotients-An Introduction to the Methods of Ring Theory”, Springer-Verlag, New York, 1975.
  

  Citation
  K.H. Singh, N. Ansari, M. Dutta, "On Quasi-Reduced Modules," International Journal of Scientific Research in Mathematical and Statistical Sciences, Vol.5, Issue.6, pp.255-258, 2018

• Open Access   Article

  Geo Chromatic Number of a Graph

  Beulah Samli. S, Robinson Chellathurai. S

  Research Paper | Isroset-Journal (IJSRMSS)

  Vol.5 , Issue.6 , pp.259-264, Dec-2018

  CrossRef-DOI:   https://doi.org/10.26438/ijsrmss/v5i6.259264

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  Abstract
  A set S⊆V(G) is called a geodetic set if every vertex of G lies on a shortest u-v path for some u,v∈S, the minimum cardinality among all geodetic sets is called geodetic number and is denoted by g(G). A set C⊆V(G) is called a chromatic set if C contains all p vertices of different colors in G,the minimum cardinality among all chromatic sets is called chromatic number and is denoted by χ(G). The combination of a geodetic set and a chromatic set gives a new concept, which is called a geo chromatic set (briefly χ_gc-set) of G. A geo chromatic set S_c⊆V(G) is both a geodetic set and a chromatic set. The geo chromatic number χ_gc (G) of G is the minimum cardinality among all geo chromatic sets of G. We determined the geo chromatic numbers of certain standard graphs and bounds of the geo chromatic number is proved. Also we illustrated that for positive integers x,y,z with 2≤x

  Key-Words / Index Term
  Geodetic number, chromatic number, Geo chromatic number

  References
  [1] J.A. Bondy and U.S.R. Murthy, “Graph Theory with
  Applications”, Macmillon London and Elsevier New York, 1976.
  [2] F. Buckley and F. Harary, “Distance in Graphs”, Addison – Wesly
  Publishing company, Redwood city, CA, 1990.
  [3] H. Escuardo, R. Gera, A. Hansberg, N. Jafari Rad and L. Volkmann, “Geodetic Domination in graphs”, J. Combin. Math Combin. Comput. 77 (2011), 89-101.
  [4] Gary Chatrand and P.Zhang, “Introduction to Graph Theory”,
  MacGraw Hill, 2005.
  [5] Gary Chatrand, F. Harary and P. Zhang, “On the geodetic number
  of a graph”, Networks, 39 (2002), 1-6.
  [6] Gary Chatrand, F. Harary and P. Zhang, “Geodetic sets in graphs”, Discussiones Mathematicae Graph Theory
  [7] A. Haneberg and L. Volkmann, “On the geodetic and geodetic domination numbers of a graph”, Discrete Mathematics 310 (2010), 2140 – 2146.
  [8] M. Mohammed Abdul Khayoom and P. Arul Paul Sudhahar, “Monophonic Chromatic Parameter in a Connected Graph”, International Journal of Mathematical Analysis 11 (2017), no.19, 911 – 920.
  

  Citation
  Beulah Samli. S, Robinson Chellathurai. S, "Geo Chromatic Number of a Graph," International Journal of Scientific Research in Mathematical and Statistical Sciences, Vol.5, Issue.6, pp.259-264, 2018

• Open Access   Article

  Limits in Sequence and Function Spaces

  A.N. Kumar, L.K. Sharan

  Research Paper | Isroset-Journal (IJSRMSS)

  Vol.5 , Issue.6 , pp.265-269, Dec-2018

  CrossRef-DOI:   https://doi.org/10.26438/ijsrmss/v5i6.265269

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  Abstract
  In this work we have developed the study of sequence spaces, by establishing some of the results. We have also extended the study by establishing a few results to the case of function space analogous to that for sequence spaces. We also construct some suitable sequence and function spaces. In Result and Discussion section , 1st part we have established some of the result on the sequence spaces and in 2nd part we have established a few results using the definitions of different limits in function spaces. These have done on account of previous work here we refer [Cooke, (1),chapter 10].

  Key-Words / Index Term
  Co-ordinate Convergent , Projective Convergent, Co-ordinate Limit ,Projective Limit , ,in case of sequence space and Dual Space, Perfect Space ,Parametric Convergent, Parametric Limit, Projective Convergent and Projective Limit in case of function space

  References
  [1] H.S. Allen (i) “Projective convergence and limit in sequence spaces” Proc. London Math. Soc,(2), 48, pp.310-338., 1944 (ii)“Transformation of sequences spaces”, Journal London Math soc, 31, pp.374-376. (1956),
  [2] R.G. Cooke (i)Infinite Matrices and Sequence Spaces , Macmillan , London , (1950) (ii)Linear Operators, Macmillan , London , (1950)
  [3] P. Dienes : The Taylor Series , Oxford (1931) .
  [4] Paul R. Halmos : (i)Naïve Set theory. D.Van Nostrand company, Inc London, Toronto, New Yark.(1960) (ii)Measure Theory, Narosa Publishing House, New Delhi (1981) (iii) Finite dimensional vector space.(Van Nostrand, Princeton, N.J.,(1958)
  [5] G.H.Hardy :Divergent Series , Oxford , (1956) .
  [6] I.N. Herstein : Topics in Algebra John wiley and Sons, New York Chichester Brisbane, Toronto, Singapore.
  [7] G. Kothe :Topological Vector Sp[ace I. 2nd Edition, springer Verlag Berlin Heidelberg. New York (1983)
  [8] G. Kothe, and O. Toeplitz : Lineare Raumemit unendli chvielen Koordinaten and Ringe unendlicher Matrizen , Crelie , 171 , pp.193-226 (1934)
  [9] I.J. Maddox : Elements of Functional Analysis. (Cambridge University press, Cambridge,(1970)).
  [10] I.P. Natanson:Theory of Functions of a real variable , Frederic Umgar , New York , (1955)
  [11] S.N.Prasad :Analogues of some results on sequence spaces for function spaces , Quart . J. of Math. , (Oxford), 2,II, (1960),310-320.
  [12] W. Rudin : (i)Principles of Mathematical Analysis. (Mc Graw-Hill Book Company, New York, (1964) (ii)Functional Analysis.(Tata Mc.Graw-Hill Publishing Company Ltd, New Delhi, (1974))
  [13] L.K.Sharan : “Some Contributions to the Theory of function spaces” Ph.D. Thesis , M.U. Bodh Gaya (1986).
  [14] A.N.Kumar : “Some contributions to the theory of limit and different convergences in function spaces” , Ph.D. Thesis,V.K.S.U. Ara . (2014).
  [15] A.N.Kumar: “On parametric and strong projective convergence“, International conference on Recent advances in mathematics and scienctific computing”India,pp.37, 2018.
  [16] A.N.Kumar:”Convergence in Dual Space” IOSR Journal of Mathematics , International Organization of Scienctific Research, Vol.13, Issue.2, pp.35-37, 2017.
  

  Citation
  A.N. Kumar, L.K. Sharan, "Limits in Sequence and Function Spaces," International Journal of Scientific Research in Mathematical and Statistical Sciences, Vol.5, Issue.6, pp.265-269, 2018

• Open Access   Article

  Semi Pre Zc-open sets in Generalized Topology

  RM. Sivagama Sundari, A.P. Dhana Balan

  Research Paper | Isroset-Journal (IJSRMSS)

  Vol.5 , Issue.6 , pp.270-272, Dec-2018

  CrossRef-DOI:   https://doi.org/10.26438/ijsrmss/v5i6.270272

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  Abstract
  In this paper, we introduce the concept of -spZc-open sets in generalized topology by using the concept of Zc-open sets in general topology. In this paper we have discussed some properties and theorems using -spZc-continuous, -spZc-irresolute in GTS through -spZc –open sets

  Key-Words / Index Term
  -spZc -open, -spZc –closed, -spZc-mapping, -spZc-continuous, -spZc-irressolute

  References
  [1] Csaszar,A.,“Generalized topology, generalized continuity”, Acta Mathematica Hungarica, Vol.96, Issue.4, pp.351-357, 2002.
  [2] RM.Sivagama Sundari, A.P.Dhanabalan, “Zc-open sets and Zs-open sets in topological spaces”, ijsr., Vol.3, Issue.5, 2016.
  [3] RM.Sivagama Sundari, A.P.Dhanabalan, “On Zc Separation Axioms and On g.Zc closed , Zc.g–closed sets”, ISRJ,Vol.6, Issue.8, 2016.
  [4] RM.Sivagama Sundari, A.P.Dhanabalan,“Zc-Continuous function in Topological Spaces”, GJTAMS, Vol.6, Issue.1, 2016.
  [5] RM.Sivagama Sundari, A.P.Dhanabalan, “semi pre Zc-connected spaces in general topology”, IJMTT, Vol.52, Issue.6, 2017.
  

  Citation
  RM. Sivagama Sundari, A.P. Dhana Balan, "Semi Pre Zc-open sets in Generalized Topology," International Journal of Scientific Research in Mathematical and Statistical Sciences, Vol.5, Issue.6, pp.270-272, 2018

• Open Access   Article

  Measure of Location using Data Depth Procedures

  R.Muthukrishnan, D.Gowri, N.Ramkumar

  Research Paper | Isroset-Journal (IJSRMSS)

  Vol.5 , Issue.6 , pp.273-277, Dec-2018

  CrossRef-DOI:   https://doi.org/10.26438/ijsrmss/v5i6.273277

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  Abstract
  Data depth is used to measure the depth or outlyingness of a given multivariate sample with respect to its underlying distribution. It can be lead to a natural center-outward ordering of sample points. The essence of depth function in multivariate analyses is to measure the degree of centrality of point relative to a data set or probability distribution. This work explores data depth procedures in order to find the measure of location, namely deepest or center point. Further, the various depth procedures are examined under real and simulation environment with the help of R software. The efficiency of various data depth procedures have been studied by computing average misclassification error in the context of discriminant analysis with numerical illustration

  Key-Words / Index Term
  Data Depth, Location and Linear discriminant analysis

  References
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  [11] Y. Vardi, and C. Zhang, “The Multivariate L1 Median and Associated Data Depth”, Proceedings of the National Academy of Science USA, Vol.97, pp.1423-1426, 2000.
  [12] Y.J. Zuo, and R. Serfling, “General notions of statistical depth function”, The Annals of Statistics Vol.28, pp.461–482, 2000.
  [13] Y. Zuo, “Projection-based depth functions and associated medians”, The Annals of statistics, Vol.31, pp.1460-1490, 2003.
  [14] R. Serfling, “Depth functions in nonparametric multivariate inference”. In: Liu, R., Serfling, R., Souvaine, D. (eds.), Data Depth: Robust Multivariate Analysis, Computational Geometry and Applications, American Mathematical Society, pp.1–16, 2006.
  [15] X. Liu, and Y. Zuo, “Computing projection depth and its associated estimators”, Statistics and Computing Vol.24, pp.51–63, 2014.
  [16] R. Muthukrishnan, and G. Poonkuzhali, “Computing Median with Data Depth in Multivariate Data”, Journal of Modern Sciences, Vol.7, Issue.2, pp.11-19, 2015.
  [17] R. Muthukrishnan, M. Vadivel, and N. Ramkumar, “Projection based Data Depth Procedure with application in Discriminant Analysis”. International Journal of Research in Advent Technology, Vol.6, Issue.5, pp.824-832, 2018.
  [18] R. Muthukrishnan, and G. Poonkuzhali, “Robust Depth based weighted Estimator with Application in Discriminant Analysis”, International Journal of Scientific Research in Mathematical and Statistical Sciences, Vol.5, Issue.3, pp.96-101, 2018.
  [19] R Core Team, R: “A language and environment for statistical computing”, Vienna, Austria, 2018.
  

  Citation
  R.Muthukrishnan, D.Gowri, N.Ramkumar, "Measure of Location using Data Depth Procedures," International Journal of Scientific Research in Mathematical and Statistical Sciences, Vol.5, Issue.6, pp.273-277, 2018

• Open Access   Article

  Convergence of Weighted (0, 2; 0)-Interpolation on the Unit Circle

  S. Bahadur, S. Bano

  Research Paper | Isroset-Journal (IJSRMSS)

  Vol.5 , Issue.6 , pp.278-281, Dec-2018

  CrossRef-DOI:   https://doi.org/10.26438/ijsrmss/v5i6.278281

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  In this paper, we study the convergence of weighted Lacunary Pál-type interpolation on two pairwise disjoint sets of nodes obtained by projecting vertically the zeros of Legendre polynomial and its second derivative with boundary points of real line on the unit circle. Here the Lacunary data is prescribed on first set of nodes, whereas the Lagrange data on the other one.

  Key-Words / Index Term
  Legendre polynomial, Weighted interpolation, Pál-type interpolation, Explicit forms

  References
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  [3]. H.P. Dikshit, “ Pál - type Interpolation on Non - Uniformly Distributed Nodes on the Unit Circle”, Journal of computational and Applied Mathematics, Vol. 155, Issue 2, pp. 253-261, 2003.
  [4]. S. Xie, “(0,1,3) interpolation on unit circle, Acta. Math. Sinica”, Vol. 39, Issue 5, pp. 690-700, 1996.
  [5]. S. Bahadur and K. K. Mathur, “Weighted (0,2)*-interpolation on unit circle” , International Journal of Applied Mathematics and Statistics, Vol. 20, Issue M11, pp. 73-78, 2011.
  [6]. S. Bahadur and M. Shukla, “ (0,2) - Interpolation on Unit Circle ”, International Journal of Advancements in Research and Technology, Vol. 3, Issue 1, 2014.
  [7]. S. Bahadur and S. Bano, “On Weighted (0,2) - Interpolation”, Global Journal of Pure and Applied Mathematics, Vol. 13, Issue 2, pp. 319-329, 2017.
  [8]. S. Bahadur, “Pál – type (0; 0, 2) − Interpolation on the Unit Circle”, Int. J. Math. Sci., Vol. 11, no. 1-2, pp. 103-111, 2012.
  [9]. S. Bahadur, “(0, 2; 0)- Interpolation on the Unit Circle”, Italian Journal of Pure and App. Math., Vol. 32, pp. 57-66, 2014 .
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  [11]. S. Bahadur and S. Bano,“On Weighted Pa ́l Type (0,2)- Iterpolation on the Unit Circle”, (communicated.)
  

  Citation
  S. Bahadur, S. Bano, "Convergence of Weighted (0, 2; 0)-Interpolation on the Unit Circle," International Journal of Scientific Research in Mathematical and Statistical Sciences, Vol.5, Issue.6, pp.278-281, 2018