Volume-6 , Issue-1 , Feb 2019, ISSN 2348-4519 (Online) Go Back

• Open Access   Article

  Vertex Antimagic Edge Labeling of the Mongolian Tent Graph

  R. Sreenivasan, M.S. Paulraj

  Research Paper | Isroset-Journal (IJSRMSS)

  Vol.6 , Issue.1 , pp.229-233, Feb-2019

  CrossRef-DOI:   https://doi.org/10.26438/ijsrmss/v6i1.229233

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  Abstract
  In graph theory, the word ‘label’ to the vertices and the edges refer to the weights associated with the labels and edges. The concept of magic and antimagic labeling in graphs has been one of the much sought after topics in Graph Theory. Researchers have been using this concept effectively to different types of graphs. The impetus to apply the concept of vertex antimagic edge labeling is the definition of the Mongolian Tent Graph Mn. In this paper, we have shown that the Mongolian tent graph admits vertex antimagic edge labeling.

  Key-Words / Index Term
  Antimagic Labeling, Mongolian Tent Graph, Vertex Antimagic Edge Labeling

  References
  [1]. R. Bodendiek and G. Walther – “On number theoretical methods in Graph Labelings”, – Res. Exp. Math. 21 (1995), 3 – 25
  [2]. R. Bodendiek and G. Walther – “On arithmetic antimagic edge labelling of graphs”, Mitt. Math. Ges. Hamburg 17 (1998), 85 – 89
  [3]. J. A. Gallian – “A Dynamic survey of Graph Labeling”, Electronic Journal of Combinatorics, 18(2015), DS#6
  [4]. N. Hartsfield and G. Ringel – “Pearls in Graph Theory”, Academic Press, Boston – San Diego– New York – London, 1990
  [5]. Martin Baca and Mirka Miller – “Super Edge Antimagic Graphs – A Wealth of Problems and Some Solutions” – Brown Walker Press, June 2008 ISBN – 10: 1-59942-466-5 (e-book)ISBN – 13: 978-1-59942-466-8 (e-book)
  [6]. J.A. Macdougall, M. Miller, Slamin and W.D. Wallis – “Vertex Magic Total Labelling of Graphs” – Utilitas Math. 61 (2002), 3 – 21
  [7]. J.A. Macdougall, M. Miller and W.D. Wallis –“Vertex Magic Total Labelling of Wheels and Related Graphs” - Utilitas Math. 62 (2002), 175 – 183
  [8]. J. Sedlacek – “Problem 27, In” – Theory and itsApplications, Proc. Symp. Smolenice, 1963, 163 – 169
  [9]. J. Sedlacek – “On Magic Graphs” – Math. Slovaca 26 (1976), 329 – 335
  [10]. Sharmila Mary Arul, K. Subashini – “Cordial labeling of Mongolian Tent Mn” – International Journal of Pure and Applied Mathematics - Volume 106 No. 8 2016, 1-6 ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version)
  

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  R. Sreenivasan, M.S. Paulraj, "Vertex Antimagic Edge Labeling of the Mongolian Tent Graph," International Journal of Scientific Research in Mathematical and Statistical Sciences, Vol.6, Issue.1, pp.229-233, 2019

• Open Access   Article

  Solutions of Pell’s Equation Involving Jarasandha Numbers

  C.Saranya, G.Janaki

  Research Paper | Isroset-Journal (IJSRMSS)

  Vol.6 , Issue.1 , pp.234-236, Feb-2019

  CrossRef-DOI:   https://doi.org/10.26438/ijsrmss/v6i1.234236

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  We look for discovering non-trivial integer solutions to the pell’s equation involving jarasandha numbers. Also, we acquire the recurrence relations among the solutions.

  Key-Words / Index Term
  Jarasandha numbers, Pell’s equation, brahmagupta lemma

  References
  [1] R.D.Carmichael, “History of theory of numbers and diophantine analysis”, Dover Publication, New york, 1959.
  [2] L.J.Mordell, “Diophantine Equations”, Academic press, London, 1969.
  [3] T.Nagell, “Introduction to Number theory”, Chelsea publishing company, Newyork, 1981.
  [4] H.W.Lenstra Jr, “Solving the pell equation”, Notice of the American Mathematical society, Vol 49, No.2, Pp: 182-192. Feb 2002.
  [5] G.Janaki and C.Saranya, “Special Pairs of Pythagorean Triangles and Jarasandha Numbers”, American International Journal of Research in Science, Technology, Engineering & Mathematics, issue-13, 118-120, Dec 2015-Feb 2016.
  [6] G.Janaki and C.Saranya, “Special Rectangles and Jarasandha Numbers”, Bulletin of Mathematics and Statistics Research, vol-4, issue-2, 63-67, Apr-June 2016.
  [7] G.Janaki and C.Saranya, “Special Pairs of Rectangles and Jarasandha Numbers”, Asian Journal of Science & Technology, vol-7, issue-5, 3015-3017, May 2016.
  [8] G.Janaki and C.Saranya, “Pythagorean Triangle with area/perimeter as a Jarasandha Number of orders 2 & 4”, International Research Journal of Engineering and Technology, vol 3, issue 7, 1259-1264, July 2016.
  [9] S.Vidhya and G.Janaki, “An Integral solution of negative pell’s equation involving two digit sphenic numbers”, International Journal of computer sciences and engineering, vol 6, Issue 7, Pp: 444-445, July 2018.
  

  Citation
  C.Saranya, G.Janaki, "Solutions of Pell’s Equation Involving Jarasandha Numbers," International Journal of Scientific Research in Mathematical and Statistical Sciences, Vol.6, Issue.1, pp.234-236, 2019

• Open Access   Article

  Orthogonal Modular Stability of Radical Cubic Functional Equation

  R. Murali, P. Divyakumari

  Research Paper | Isroset-Journal (IJSRMSS)

  Vol.6 , Issue.1 , pp.237-240, Feb-2019

  CrossRef-DOI:   https://doi.org/10.26438/ijsrmss/v6i1.237240

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  Abstract
  In this paper, authors newly introduce radical cubic type functional equation and obtain its general solution. Also, investigate the Hyers-Ulam-Rassias stability of introduced radical cubic type functional equation in modular space.

  Key-Words / Index Term
  Modular space, orthogonality, cubic functional equation, generalized Hyers-Ulam-Rassias stability

  References
  [1] T. Aoki, On the stability of the linear transformation in Banach
  spaces, J. Math. Soc.Japan, Vol 2, pp 64-66, 1950.
  [2] P.Gavruta, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl., Vol 184, pp 431-436, 1994.
  [3] D. H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci., Vol 27, pp 222-224, 1941.
  [4] Iz-iddine El-Fassi and Samir Kabbaj, On the generalized orthogonal stability of mixed type additive-cubic functional equations in modular spaces, Tbilisi Mathematical Journal, Vol 9, Issue 1, pp 231-243, 2016.
  [5] Hark-Mahn Kim and Young Soon Hong, Approximate Quadratic Mappings in Modular Spaces, International Journal of Pure and Applied Mathematics, Vol 116, Issue 1, pp 31-43, 2017.
  [6] J.M. Rassias, On approximately of approximately linear mappings by linear mappings, J. Funct. Anal. USA, Vol 46, pp 126-130, 1982.
  [7] K.Ravi, M. Arunkumar and J.M. Rassias, On the Ulam stability for the orthogonally general Euler-Lagrange type functional equation, International Journal of Mathematical Sciences, Vol 3, Issue 08, pp 36 – 47, Autumn 2008.
  [8] Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math.Soc.,Vol 72, pp 297-300, 1978.
  [9] Wanchitra Towanlong and Paisan Nakmahachalasint, A quadratic functional equation and its generalized Hyers-Ulam Rassias stability, Thai Journal of Mathematics, special Issue (Annual Meeting in Mathematics, 2008), pp 85-91.
  [10] S.M. Ulam, A collection of the mathematical problems, Interscience Publ., New York, 1960.
  [11] Kittipong Wongkum, Poom Kumam, Yeol Je Cho, Phatiphat Thounthong, Parin Chaipunya, On the generalized Ulam-Hyers-Rassias stability for quartic functional equation in modular spaces, J. Nonlinear Sci. Appl., 10(2017), 1399-1406.
  

  Citation
  R. Murali, P. Divyakumari, "Orthogonal Modular Stability of Radical Cubic Functional Equation," International Journal of Scientific Research in Mathematical and Statistical Sciences, Vol.6, Issue.1, pp.237-240, 2019

• Open Access   Article

  Cordial Labeling of an Arbitrary Supersubdivision of a Graph

  S. J. Gajjar

  Review Paper | Journal-Paper (IJSRMSS)

  Vol.6 , Issue.1 , pp.241-244, Feb-2019

  CrossRef-DOI:   https://doi.org/10.26438/ijsrmss/v6i1.241244

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  Abstract
  In this paper, we have investigated a condition that if for a graph G there exist a binary vertex labeling f:V(G)?{0,1} such that e_f (0)=0 then an arbitrary supersubdivision of G, G×P_n, G×C_2n, splitting graph of G, disjoint union of graphs, path union of some graphs and a graph obtained by attaching m pendent edges to every vertex of the graph G are cordial

  Key-Words / Index Term
  Graph Labeling, Cordial Labeling, An Arbitrary Suppersubdivision

  References
  [1] J. Gross and J. Yellen (2004). Handbook of Graph Theory, CRC Press.
  [2] D. M. Burton (2007). Elementary Number Theory. 6th ed.
  [3] J. A. Gallian, “A Dynamic Survey of Graph Labeling," The Electronic Journal of Combinatorics, 17 (2014), # DS6
  [4] I. Cahit, ”Cordial Graphs: A weaker version of graceful and harmonious Graphs", Ars Combinatoria, 23(1987), 201-207.
  [5] Y S Ho, S M Lee and S C Shee, ”Cordial labeling of unicyclic graphs and generalized Petersen graphs", Congress. Numer.,68(1989), 109-122.
  [6] M Andar, S Boxwala and N B Limaye, ”A Note on cordial labeling of multiple shells", Trends Math. (2002), 77-80.
  [7] S K Vaidya, G V Ghodasara, Sweta Srivastav, V J Kaneria, ”Some new cordial graphs", Int. J. of Sci. Comp.,2(1)(2008), 81-92.
  [8] S K Vaidya, Sweta Srivastav, G V Ghodasara, V J Kaneria, ”Cordial labeling for cycle with one chord and its related graphs", Indian J. of Math. and Math. Sciences, 4(2)(2008), 145-156.
  [9] G Sethuraman and P Selvaraju, ”Gracefullness of arbitreary supersubdivision of graphs", Indian J. pure. appl. math., 32(7)(2001), 1059-1064.
  [10] S K Vaidya, N. A. Dani, ”Cordial Labeling and Arbitrary SuperSubdivision of Some Graphs", International Journal of Information Science and Computer Mathematics, Vol. 2, 2010, no. 1, 51 - 60.
  [11] S K Vaidya, K. K. Kanani, ”Some New Results on Cordial Labeling in the Context of Arbitrary Supersubdivision of Graph", Applied Mathematical Sciences, Vol. 4, 2010, no. 47, 2323 - 2329.
  

  Citation
  S. J. Gajjar, "Cordial Labeling of an Arbitrary Supersubdivision of a Graph," International Journal of Scientific Research in Mathematical and Statistical Sciences, Vol.6, Issue.1, pp.241-244, 2019

• Open Access   Article

  Analysis of Non-Markovian Process by the Inclusion of Supplementary Variables

  Shakuntla, A.K.Lal, S.S.Bhatia, Rachna

  Research Paper | Isroset-Journal (IJSRMSS)

  Vol.6 , Issue.1 , pp.245-250, Feb-2019

  CrossRef-DOI:   https://doi.org/10.26438/ijsrmss/v6i1.245250

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  Abstract
  Certain stochastic processes with discrete states in continuous time can be converted into Markov process by the well-known method of including supplementary variables technique. In this paper we developed a mathematical model of the steel industry which manufactures the stainless steel plates and also made an attempt to improve its availability. The failure and repair rates of different subsystem are arbitrary distributed. Lagrange’s method for partial differential equations is used to solve system governing equations. Availability analysis of the system helped in identifying the contribution factors and assessing their impact on the system availability.

  Key-Words / Index Term
  Supplementary variable technique, Chapman- Kolmogorov, variable failure and repair rates, Lagrange’s method

  References
  [1] Krivtosv,V.V. “Recent advances in theory and applications of stochastic point process model in reliability engineering”, Reliability Engineering andSystem Safety, Vol.92,pp.549-551,2007.
  [2] Dayer,D. “Unification of reliability/availability models for markov systems”, IEEE Transaction on Reliability,Vol.38,pp.10-15,1985.
  [3] Kumar, D. “Availability of the feeding system in the sugar industry”, Micro electron reliability,Vol.28, Issue.6, pp.12-15,1988.
  [4] Islamov,R.T. “Using markov reliability modeling for multiple repairablesystems”, Reliability Engineering and System Safety,,pp 113-118,1994.
  [5] Tsai,Y.,Wang,K..and Tsai,L., “A study of availability centered preventive for multi component System”,Reliability Engineering and System Safety, Vol.84, pp. 261-270,2004.
  [6] Sharma,R.K. and Kumar,S. “Performance modeling in critical engineering system using RAM Analysis”, Reliability Engineering and System Safety, Vol.93, pp.891-897, 2008.
  [7] Guo,H., Watson,S.,Taver,P. and Xiang,J. “Reliability analysis of wind turbines with incomplete Failure data collected from the date of initial installation”,Reliability Engineering and System Safety, Vol.94,pp.1057-1063, 2009.
  [8] Reza Ahmadi, Mitra Fouladirad,“Maintenance planning of deteriorating production process”.eliability engineering and system safety, Vol.157, pp 108-118, 2017.
  [9] Gregory Levitin, Liudong Xing, “Reliability and Performance of multi state system”, Reliability Engineering and System Safety.Vol.166, pp. 1-2, 2017.
  [10] Parveen Gupta and Pooja Vinodiya,“Analysis of Reliability of Two Non Identical Units Cold Standby Repairable systems has two types of failure” International Journal of computer Sciences and Engineering ,Vol.6,Issue.11,pp907-913,2018.
  

  Citation
  Shakuntla, A.K.Lal, S.S.Bhatia, Rachna, "Analysis of Non-Markovian Process by the Inclusion of Supplementary Variables," International Journal of Scientific Research in Mathematical and Statistical Sciences, Vol.6, Issue.1, pp.245-250, 2019

• Open Access   Article

  The Generalized Zagreb Index of Capra Operation on Cycle

  Keerthi G. Mirajkar, Pooja B, M. R. Farahani

  Research Paper | Isroset-Journal (IJSRMSS)

  Vol.6 , Issue.1 , pp.251-254, Feb-2019

  CrossRef-DOI:   https://doi.org/10.26438/ijsrmss/v6i1.251254

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  Abstract
  The Generalized Zagreb index of a graph G is defined for arbitrary non-negative integer r and s as M_((r,s)) (G)=∑_(uv∈E(G) )▒‍[d_G^r (u)d_G^s (v)+d_G^s (u)d_G^r (v)]. In this paper, we obtain expressions for generalized Zagreb index of Capra operation of cycle C_n on n vertices, which generalizing the existing results of Sardar et al. (Open J. Math. Sci. 1 (2017) 44-51). As an application, this result enables us to find first Zagreb index, second Zagreb index and F-index of Capra operation of cycle C_n on n vertices.

  Key-Words / Index Term
  Capra Operation, Generalized Zagreb Index

  References
  M. Azari, A. Iranmanesh, “Generalized Zagreb index of graphs”, Studia Universitatis Babes-Bolyai, Chemia, Vol. 56, Issue. 3, pp. 59-70, 2011.
  L. B. Bhajantri, "Fuzzy Logic Based Fault Detection in Distributed Sensor Networks", International Journal of Scientific Research in Computer Science and Engineering, Vol. 6, Issue. 2, pp.27-32, 2018.
  B. Basavanagoud, Praveen Jakkannavar, “On the Zagreb polynomials of transformation graphs”, International Journal of Scientific Research in Mathematical and Statistical Sciences, Vol. 5, Issue. 6, pp. 328-335, 2018.
  A. Dress, G. Brinkmann, “Phantasmagorical fulleroids”, MATCH Commun. Math. Comput. Chem, Vol. 33, pp. 87-100, 1996.
  M. V. Diudea, “Nanoporous carbon allotropes by septupling map operations”, Journal of chemical information and modeling, Vol. 45, Issue. 4, pp. 1002-1009, 2005.
  M. V. Diudea, M. Stefu, P. E. John, A. Graovac, “Generalized operations on maps”, Croatica chemica acta, Vol. 79, Issue. 3, pp. 355-362, 2006.
  M. R. Farahani, M. P. Vlad, “On the Schultz, Modifed Schultz and Hosoya polynomials and Derived Indices of Capra-designed planar Benzenoids”, Studia UBB Chemia, Vol. 57, Issue. 4, pp. 55-63, 2012.
  M. R. Farahani, “Computing ABC4 index of Capra-designed planar benzenoid series Cak (C6)”, Advances in Materials and Corrosion, Vol. 1, Issue. 1, pp. 61-64, 2012.
  M. R. Farahani, “On the Schultz polynomial and Hosaya polynomial of Circumcoronene series of benzenoid”, Journal of Applied Mathematics & Informatics, Vol. 31, Issue. 5-6, pp. 595-608, 2013.
  M. R. Farahani, “A new version of Zagreb index of circumcoronene series of benzenoid”, Chemical Physics Research Journal, Vol. 6, Issue. 1, pp. 27-33, 2013.
  B. Furtula, I. Gutman, “A forgotten topological index”, J. Math. Chem. Vol. 53, pp. 1184-1190, 2015.
  M. Goldberg, “A class of multi-symmetric polyhedra”, Tohoku Mathematical Journal, First Series, Vol. 43, pp. 104-108, 1937.
  I. Gutman, N. Trinajstic`, “Graph theory and molecular orbitals:Total π-electron energy of alternant hydrocarbons”, Chem. Phys. Lett. Vol. 17, pp. 535-538, 1972.
  F. Harary, “Graph Theory”, Addison-Wesley, Mass 1969.
  H. H. Patel, Purvi Prajapati, "Study and Analysis of Decision Tree Based Classification Algorithms", International Journal of Computer Sciences and Engineering, Vol. 6, Issue. 10, pp.74-78, 2018.
  M. S. Sardar, S. Zafar, M. R. Farahani, “The generalized Zagreb index of capra-designed planar benzenoid series Ca_k (C_6)”, Open J. Math. Sci. Vol. 1, pp. 44-51, 2017.
  

  Citation
  Keerthi G. Mirajkar, Pooja B, M. R. Farahani, "The Generalized Zagreb Index of Capra Operation on Cycle," International Journal of Scientific Research in Mathematical and Statistical Sciences, Vol.6, Issue.1, pp.251-254, 2019

• Open Access   Article

  A Review on Univariate Bayesian Frailty Models

  S.G. Parekh, S.R. Patel

  Review Paper | Isroset-Journal (IJSRMSS)

  Vol.6 , Issue.1 , pp.255-258, Feb-2019

  CrossRef-DOI:   https://doi.org/10.26438/ijsrmss/v6i1.255258

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  Abstract
  The main idea of this review article is to review different univariate Bayesian frailty model, discussed by some authors, such as estimation of negative binomial distribution with Bayesian prior distribution as beta, Bayesian frailty estimation of Poisson distribution with gamma as frailty prior distribution, Bayesian frailty estimation of gamma distribution with gamma as frailty prior distribution, Bayesian frailty estimation of univariate normal distribution with frailty prior distribution as gamma and Bayesian estimation of inverse Gaussian distribution with uniform prior frailty model. In all these Bayesian estimation squared error loss function is used.

  Key-Words / Index Term
  Bayesian frailty model, Bayesian prior distribution, Beta, Negative Binomial, Gamma, Inverse Gaussian, Normal, Poisson, Uniform distribution

  References
  [1] Clayton, D., “A model for association in bivariate life tables and its applications to epidemiological studies of familial tendency in chronic disease incidence”, Biometrica, Vol.65, pp.141-151, 1979.
  [2] Vaupel JW, Manton KG, Stallard E., “The impact of heterogeneity in individual frailty on the dynamics of mortality”, Demography, Vol.16, pp.439-454, 1979
  [3] Cox D., “Regression models and life tables (with discussion)”, Journal of the Royal Statistical Society, Series B, Vol. 34, pp. 187-200, 1972.
  [4] Hougaard, P., “Survival models for heterogeneous populations derived from stable distrib¬utions”, Biometrika, Vol. 73, pp.387 – 396, 1986a.
  [5] Hougaard, P., “A class of multivariate failure timedistributions”, Biometrika, Vol. 73, pp. 671 – 678, 1986b.
  [6] Akaike H, “Information measures and model selection”,Bull. Int. statist. Inst. Vol.50, pp.277-290, 1983.
  [7] Le Cam L., “Maximum likelihood: an introduction”, Int.Statist. Rev. Vol.58, pp.153-172, 1990.
  [8] Joshi VM., “The censoring concept and the likelihood principle”, J. Statist. Plann. Inference, Vol.26, pp.109- 111, 1990.
  [9] Geyer CJ, Thompson EA. “Constrained Monte Carlo maximum likelihood for dependent data (with discussion)”, J. Roy. Statist. Soc., Ser B., Vol.54, pp.657-699, 1992.
  [10] Aalen, O.O. “Modelling heterogeneity in survival analysis by the compound Poisson distri¬bution”, Annals of Applied Probability Vol. 4, pp. 951 – 972, 1992.
  [11] McGilchrist, C.A., “REML estimation for survival models with frailty”, Biometrics, Vol.49, pp.221-225, 1993.
  [12] Yashin, A.I., Vaupel, J.W., Iachine, I.A. “Correlated individual frailty: An advantageous approach to survival analysis of bivariate data”, Working Paper Series: Population Studies of Aging 7, CHS, Odense University, 1993.
  [13] Yashin, A.I., Manton, K.G., Iachine, I.A.,“Genetic and environmental factors in duration studies: multivariate frailty models and estimation strategies”, Journal of Epidemiology and Bio-statistics, Vol.1, pp.115 – 120, 1996.
  [14] Xue, X., Brookmeyer, R., “Bivariate frailty model for the analysis of multivariate survival time”, Lifetime Data Analysis, Vol. 2, pp. 277 – 290, 1996.
  [15] Sahu, K.S., Dey, D.K., Aslanidou, H., Sinha, D., “A Weibull regression model with Gamma frailties for multivariate survival data”, Lifetime Data Analysis, Vol. 3, pp. 123 – 137, 1997.
  [16] Yashin, A.I., Begun, A., Iachine, I.A., “Genetic factors in susceptibility to death: a com¬parative analysis of bivariate survival models”, Journal of Epidemiology and Biostatistics, Vol. 4, pp.53-60, 1999a.
  [17] Ibrahim, J. G., Chen, M., and Sinha, D., “Bayesian Survival Analysis”, New York: Springer, 2001.
  [18] Ripatti, S., Larsen, K., Palmgren, J., “Maximum likelihoodinference for multivariate frailty models using an automated MCEM algorithm”, Lifetime Data Analysis, Vol. 8, pp.349 – 360, 2002.
  [19] Iachine, I.,“The Use of Twin and Family Survival Data in the Population Studies of Ag¬ing: Statistical Methods Based on Multivariate Survival Models”, Ph.D. Thesis. Monograph 8, Department of Statistics and Demography, University of Southern Denmark, 2002.
  [20] Wienke, A., Lichtenstein, P., Yashin, A.I.,“A bivariate frailty model with a cure fraction for modeling familial correlations in diseases”, Biometrics, Vol.59, pp.1178 – 1183, 2003a.
  [21] Santos dos, C. A. and Achcar, J. A., “A Bayesian analysis for multivariate survival data in the presence of covariates”, Journal of Statistical Theory and Applications, Vol. 9, pp.233-253, 2010.
  [22] Kheiri, S., Kimber, A., Meshkani M.R., “Bayesian analysis of an inverse Gaussian correlated frailty model”, Computat. Statist. Data Anal, Vol.51, pp.5317-5326, 2007.
  [23] Hangal, D., “Modeling survival data using frailty models”, CRC Press, 2011.
  [24] Parekh, S.G., Ghosh, D. K. and Patel, S.R., “On frailty models for kidney infection data with exponential baseline distribution”, International Journal of Applied Mathematics & Statistical Sciences (JAMSS), Vol.4, Issue5, pp.31-40, 2015.
  [25] Parekh, S.G. Ghosh, D. K. and Patel, S.R., “Some Bayesian Frailty models”, International Journal of Science and Research (IJSR), Vol.5, issue, 7, pp.1949-1952, 2016.
  [26] Parekh, S.G. and Patel, S.R., “On estimation of some Univariate Bayesian frailty models”, International Journal of Statistics and Applied Mathematics, Vol.3, issue 2,pp.199-205, 2018.
  

  Citation
  S.G. Parekh, S.R. Patel, "A Review on Univariate Bayesian Frailty Models," International Journal of Scientific Research in Mathematical and Statistical Sciences, Vol.6, Issue.1, pp.255-258, 2019

• Open Access   Article

  On Multivariate Bayesian frailty models: A Review

  S.G. Parekh, S.R. Patel

  Review Paper | Isroset-Journal (IJSRMSS)

  Vol.6 , Issue.1 , pp.259-261, Feb-2019

  CrossRef-DOI:   https://doi.org/10.26438/ijsrmss/v6i1.259261

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  Abstract
  In this paper we have reviewed three models one on: Bayesian frailty Bivariate Normal model, second on: Bayesian frailty estimation of Multivariate Normal model with prior Multivariate Normal model and third on: Bayesian frailty estimation of Compound Poisson model with prior frailty model as Poisson. In all these models Bayesian frailty estimators have been obtained.

  Key-Words / Index Term
  Bayesian frailty model, Bivariate Normal model, Compound Poisson model, Multivariate Normal model, Prior frailty model

  References
  [1] Clayton, D.G., “A model for association in bivariate life tables and its application in epi- demiological studies of familial tendency in chronic disease incidence”, Biometrika, Vol.65, pp.141 – 151, 1978.
  [2] Hougaard, P., “Survival models for heterogeneous populations derived from stable distrib¬utions”, Biometrika,Vol. 73, pp.387 – 396, 1986a.
  [3] Hougaard, P, “A class of multivariate failure time distributions”, Biometrika, Vol. 73, pp. 671 – 678, 1986b.
  [4] Sahu, K.S., Dey, D.K., Aslanidou, H., Sinha, D., “A Weibull regression model with Gamma frailties for multivariate survival data”, Lifetime Data Analysis, Vol. 3, pp.123 – 137, 1997.
  [5] Hanagal D.D., “Modeling survival data using frailty models”, Chapman and Hall, Boca Raton Heckman JJ, 2011.
  [6] Parekh, S.G., Ghosh, D. K. and Patel, S.R., “On frailty models for kidney infection data with exponential baseline distribution”, International Journal of Applied Mathematics & Statistical Sciences (IJAMSS), Vol.4, Issue.5, pp.31-40, 2015.
  [7] Yashin, A.I., Vaupel, J.W., Iachine, I.A., “Correlated individual frailty: An advantageous approach to survival analysis of bivariate data”, Working Paper Series: Population Studies of Aging 7, CHS, Odense University, 1993.
  [8] Yashin, A.I., Iachine, I., “Dependent hazards in multivariate survival problems”, Journal of Multivariate Analysis, Vol.71, pp.241 – 261, 1999.
  [9] Iachine, I. “The Use of Twin and Family Survival Data in the Population Studies of Ag¬ing: Statistical Methods Based on Multivariate Survival Models” Ph.D. Thesis.Monograph 8, Department of Statistics and Demography, University of Southern Denmark, 2002.
  [10] Wienke, A., Lichtenstein, P., Yashin, A.I., “A bivariate frailty model with a cure fraction for modeling familial correlations in diseases”, Biometrics, Vol.59, pp.1178 – 1183, 2003.
  [11] Ibrahim, J. G., Chen, M., and Sinha, D., “Bayesian Survival Analysis”. New York: Springer, 2001.
  [12] Kheiri, S., Kimber, A., Meshkani M.R., “Bayesian analysis of an inverse Gaussian correlated frailty model”, Computat. Statist. Data Anal. Vol.51, pp.5317-5326, 2007.
  [13] Santos dos, C. A. and Achcar, J. A., “A Bayesian analysis for multivariate survival data in the presence of covariates”, Journal of Statistical Theory and Applications, Vol. 9, pp.233-253, 2010.
  [14] Parekh, S.G., Ghosh, D. K. and Patel, S.R., “Some Bayesian frailty models”, International Journal of Science and Research (IJSR), Vol.5, issue.7, pp.1949-1952, 2016.
  [15] Parekh S.G. and Patel S.R., “On Multivariate Bayesian frailty models”, International Journal of Statistics and Applied Mathematics, Vol.3, issue2, pp.523-531, 2018.
  

  Citation
  S.G. Parekh, S.R. Patel, "On Multivariate Bayesian frailty models: A Review," International Journal of Scientific Research in Mathematical and Statistical Sciences, Vol.6, Issue.1, pp.259-261, 2019

• Open Access   Article

  The LMS Method and Its Extention for Constructing Smoothed Centile Curves of Weight for Age for 5-10 Years Children

  S. S. Shinde, S V Kakade

  Research Paper | Isroset-Journal (IJSRMSS)

  Vol.6 , Issue.1 , pp.262-275, Feb-2019

  CrossRef-DOI:   https://doi.org/10.26438/ijsrmss/v6i1.262275

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  Abstract
  Now–a-days, reference centile curves are widely used in medical professions for monitoring growth of an individual child. The requirement of centile curves in spite of simple reference range arises when the response variable strongly dependents on some covariate such as age. So the distribution of study variable changes according age. The LMS method & its extension LMSP method (BCPE distribution) & LMST method (BCT distribution) are popular methods for constructing smoothed centile curves. LMS method deals with skewed and normal peaked data, while LMSP & LMST are flexible methods for skewed & kurtotic data. The LMS method summarizes the changing distribution of study variable by smoothed curves of parameters. That is skewness parameter (L), median (M), and coefficient of variation (S). These smoothed parameters are obtained by method of maximization of penalized likelihood function. The smoothing parameters, their respective smoothed curves and final smoothed centiles can be obtained within a special software GAMLSS. The present study is carried out on weight of 5-10 years English medium school boys from Kolhapur district of Maharashtra. We assumed children going to English medium school are from well-to-do family & are healthy. So the growth curves being developed remains to be standard & may be used as reference growth curves. These centile curves generated by using Box Cox t distribution applying log link function for µ (BCTo) are assessed for goodness of fit.

  Key-Words / Index Term
  Hight for age, weight for age, BMI for age, Growth curves for 5-10 year age group

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  Citation
  S. S. Shinde, S V Kakade, "The LMS Method and Its Extention for Constructing Smoothed Centile Curves of Weight for Age for 5-10 Years Children," International Journal of Scientific Research in Mathematical and Statistical Sciences, Vol.6, Issue.1, pp.262-275, 2019

• Open Access   Article

  About Secured Isolate Inclusive Sets in Graphs

  D.K. Thakkar, N.J. Savaliya

  Research Paper | Isroset-Journal (IJSRMSS)

  Vol.6 , Issue.1 , pp.276-281, Feb-2019

  CrossRef-DOI:   https://doi.org/10.26438/ijsrmss/v6i1.276281

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  Abstract
  In this paper we introduce in new concept call secured isolate inclusive sets in graphs. We also define maximum secured isolate inclusive set and we define the cardinality of a maximum secured isolate inclusive set to be the secured isolate inclusive number of the graph G. We observe that a 1-maximal isolate inclusive set cannot be a secured isolate inclusive set. We also observe that if a secured isolate inclusive set contains a pendent vertex then it is an isolate in this set. It v & u are pendent vertices which are adjacent and M is a maximum secured isolate inclusive set of G then v∈M & u∉M or u∈M & v∉M.

  Key-Words / Index Term
  isolate inclusive set, maximum isolate inclusive set, 1-maximal isolate inclusive set, secured isolate inclusive set, maximum secured isolate inclusive set, maximal secured isolate inclusive set, external private neighbourhood, maximum independent set, independent number.

  References
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  [2] D.K.Thakkar and N.J.Savaliya, “On Isolate Inclusive sets in Graphs “, International Journal of Innovation in Science and Mathematical, Vol. 5, Issue 3, pp. 74-76, 2017.
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  Citation
  D.K. Thakkar, N.J. Savaliya, "About Secured Isolate Inclusive Sets in Graphs," International Journal of Scientific Research in Mathematical and Statistical Sciences, Vol.6, Issue.1, pp.276-281, 2019