Volume-9 , Issue-2 , Apr 2022, ISSN 2348-4519 (Online) Go Back

• Open Access   Article

  An Efficient Ratio-Cum-Product Estimator for Finite Population Mean

  K.B. Panda, P.P. Mohanty

  Research Paper | Journal-Paper (IJSRMSS)

  Vol.9 , Issue.2 , pp.1-6, Apr-2022

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  Abstract
  Use of auxiliary information in estimating population parameters with greater precision is a common practice in survey sampling literature. For example, ratio-cum-product estimator is the one which makes use of auxiliary information. Here, in this paper, ratio-cum-product method of estimation is carried forward with the introduction of a new estimator for finite population mean. The estimator, to first order of approximation, is found to be biased, but its mean square error is found to be less than that of the customary ratio-cum-product estimator under feasible condition. The performance of the estimator has been examined in double sampling. Empirical investigations have been carried out to demonstrate the superiority of the estimator.

  Key-Words / Index Term
  Auxiliary variable, Ratio-cum-product estimator, Double sampling, Bias, Mean square error

  References
  [1] D. N. Gujarati, “Basic Econometrics(Third Edition),” Mc-Graw-Hill, International Editions, Economic Series, 1995.
  [2] H. P. Singh, M. R. Espejo, “Double Sampling Ratio-product Estimator of a Finite Population Mean in Sample Surveys,” Journal of Applied Statistics, Vol. 34, No. 1, 71-85, January, 2007.
  [3] H. P. Henderson, P. F. Velleman, “Building Multiple Regression Models Interactivity,” Biometrics 37, 391-411, June 1981.
  [4] J. Johnston, “Econometric Methods(Second Edition),” Mc-Graw Hill, New York, International Student Edition, 1960.
  [5] K. B. Panda, M. Sen, “ Weighted Ratio-cum-Product Estimator for Finite Population Mean,” International Journal of Scientific Research in Mathematical and Statistical Sciences, Vol-5, Issue-4, pp.354-358, August, 2018.
  [6] M. N. Murthy, “Product Method of Estimation,” Sankhya, Series A, Vol. 26, 69-74. 1964.
  [7] M. P. Singh, “Ratio-Cum-Product Method of Estimation,” Metrika, 12, 34-42, 1967.
  [8] M. P. Singh, “ Comparison of Some Ratio-Cum-Product Estimators,” Sankhya, series B, 31, 375-378, 1969.
  [9] P. V. Sukhatme, B. V. Sukhatme, S. Sukhatmai, C. Asok, “ Sampling Theory of Surveys with Applications (Third revised edition),” Ames: Iowa State University Press, 1984.
  [10] R. A. Fisher, “The Use of Multiple Measurements in Taxonomic Problems,” Annals of Eugenics, 7, Part II, 179, 1936.
  [11] R. Tailor, B. Sharma, “A Modified Ratio-Cum-Product Estimator of Finite Population Mean Using Known Coefficient of Variation and Coefficient of Kurtosis,”. Statistics in Transition-new series, Vol. 10, No. 1, pp. 15—24, 2009.
  [12] S. Tailor, S.Chouhan, R. Tailor, N. Garg, “A Ratio-Cum-Product Estimator Of Population Mean In Stratified Random Sampling Using Two Auxiliary Variables,” STATISTICA, anno LXXII, n. 3, 2012.
  W. G. Cochran, “Sampling Techniques(Third Edition),” John Wiley &Sons, New York, February, 1977.

  Citation
  K.B. Panda, P.P. Mohanty, "An Efficient Ratio-Cum-Product Estimator for Finite Population Mean," International Journal of Scientific Research in Mathematical and Statistical Sciences, Vol.9, Issue.2, pp.1-6, 2022

• Open Access   Article

  A Study on Double Geo Chromatic Number of Line Graph of Some Graph Families

  R. Joseph Paul, S. Beulah Samli, U. Mary

  Research Paper | Journal-Paper (IJSRMSS)

  Vol.9 , Issue.2 , pp.7-11, Apr-2022

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  Abstract
  The double geo chromatic number of G is the minimum cardinality of a double geo chromatic set of G. If any u ? v geodesic, where u, v ? S includes at least two entire color classes of G, it is said to be a double geo chromatic set Sdg. This concept has been used to estimate the double geo chromatic numbers of line graphs and to derive certain general features and also some of the existing research has also been found to be inaccurate.

  Key-Words / Index Term
  Geodetic Number, Chromatic Number, Geo Chromatic Number, Double Geo Chromatic Number

  References
  [1] .A. Ahangar, “Graph with large geodetic number,” Filomat, Vol.31, Issue 13, pp.4297-4304, 2017
  [2] S.B. Samli, S.R. Chellathurai, “Geo chromatic number of a graph,” International Journal of Scientific Research in Mathematical and Statistical Sciences, Vol.5, Issue 6, pp.259-264, 2018.
  [3] F. Buckley, F. Harary, “Distance in Graphs,” Addison - Wesly Publishing company, Redwood City, CA, 1990.
  [4] G. Chatrand, P. Zhang, “Introduction to Graph Theory,” MacGraw Hill, 2005.
  [5] G. Chatrand, F. Harary, P. Zhang, “On the geodetic number of a graph,” Networks, Vol.39, Issue 1, pp.1-6, 2002.
  [6] G. Chatrand, F. Harary, P. Zhang, “Geodetic sets in graphs,” Discuss. Math., Graph Theory, Vol.20, Issue 1, pp.129-138, 2000.
  [7] H. Escuardo, R. Gera, A. Hansberg, N. Jafari Rad, L. Volkmann, “Geodetic domination in graphs,” J. Comb. Math. Comb. Comput, Vol 77, pp.89-101, 2011.
  [8] A. Hansberg, L. Volkmann, “On the geodetic and geodetic domination numbers of a graph,” Discrete Math., Vol 310, Issue 15, pp.2140-2146, 2010.
  [9] F. Harary. “Graph Theory,” Addison - Wesley, 1969.
  [10] F. Harary, E. Loukakis, C. Tsouros, “The geodetic number of a graph,” Math. Comput. Modelling , Vol 17, Issue 11, pp.89-95, 1993.
  [11] M. Mohammed Abdul Khayoom, P. Arul Paul Sudhahar, “Monophonic chromatic parameter in a connected graph,” Int. J. Math. Anal., Ruse, Vol.11, Issue 19, pp.911-920, 2017.
  [12] A.P. Santhakumaran, J. John, “Edge geodetic number of a graph”, J. Discrete Math. Sci. and Cryptograpy, Vol.10, Issue 3, pp.415-432, 2007.
  [13] A.P. Santhakumaran, T. Jebaraj, “Double geodetic mumber of a graph,” Discuss. Math. Graph Theory, Vol. 32, Issue 1, pp. 109-119, 2012.
  [14] S. Beulah Samli, J. John, S. Robinson Chellathurai, “The Double Geo Chromatic Number of a graph,” Bulletin of the international Mathematical Virtual Institute, Vol.11, Issue 1, pp.25-38, 2021.
  [15] V.M. Goudar, K. S. Ashalatha, Venkatesh, M.H. Muddebihal , “On the Geodetic number of Line Graph,” Int. J. Contemp. Math. Sci, Vol. 7, Issue 46, pp.2289-2295, 2012.

  Citation
  R. Joseph Paul, S. Beulah Samli, U. Mary, "A Study on Double Geo Chromatic Number of Line Graph of Some Graph Families," International Journal of Scientific Research in Mathematical and Statistical Sciences, Vol.9, Issue.2, pp.7-11, 2022

• Open Access   Article

  A New Algorithm for Reaching an Optimal Solution to a Nonlinear Transportation Problem

  J. Opara, P.A. Esemokumo, R. Bekesuoyeibo

  Research Paper | Journal-Paper (IJSRMSS)

  Vol.9 , Issue.2 , pp.12-16, Apr-2022

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  The study was able to develop a technique to solve optimization transportation problems with nonlinear separable quadratic objective function with linear constraints. Transforming a nonlinear to linear approximation is not a new concept, but when it is to be employed, it starts with a large number of breakpoints for reason of solution accuracy. The algorithm employed only two breakpoints and also maintained accuracy of the solution. A numerical example was solved to demonstrate the effectiveness of the developed algorithm together with the time of execution. A Wolfram Mathematica Code for the Original Problem and that of the self developed algorithm was written with the time of execution and it was discovered that both of them gave the same optimal solution, but the time of execution (0.0312) for the self developed algorithm was far less than the time of execution (145.487) for the solution of the original problem using a system with 32 bit operating system, 2.00GB of RAM and processor Intel (R) (1.66GHz). Further investigation observed that the self developed algorithm can handle a higher number of variables and constraints with lesser time of execution, compared to the nonlinear.

  Key-Words / Index Term
  Nonlinear Transportation Problem, Optimal Solution, Separable Quadratic Objective Function, Maclaurin’s series expansion, Dichotomous Breakpoints

  References
  [1] F. L. Hitchcock, “The distribution of a product from several sources to numerous localities,”. Journal of Mathematical and Physics, Vol.20, Issue.3, pp. 224-230, 1941.
  [2] S.C. Inyama, “Operation Research: Introduction,” Supreme Publisher, Nigeria, pp. 226-232, 2007.
  [3] J. Opara, C. J. Ogbonna, S. O. Ihekuna, C. C. Ogbonna, “Proposed Piecewise Linear Approximation For Solving Nonlinear Transportation Problems,”. International Journal of Mathematics Trends and Technology, Vol.4, Issue.3, pp. 135-141, 2019.
  [4] S. S. Dey, A. Gupte, “Analysis of MILP techniques for the pooling problem,”. Operations Research, Vol.63, Issue.2, 412-427, 2015.
  [5] L. A. Wolsey, G. L. Nemhauser, “Integer and combinatorial optimization,”. John Wiley & Sons, USA, pp. 528–536, 1999.
  [6] J. Opara, P.A. Esemokumo, R. Bekesuoyeibo, “Reaching an Optimal Solution to a Nonlinear Programming Problem Involving a Separable Quadratic Objective Function,” In the Proceedings of the 2022 FNAS Conference on Scientific Innovation, Nigeria, pp. 121-132, 2022.
  [7] S. M. Abdul-Salam, “Transportation with volume discount - a case study of a logistic operator in Ghana,” Journal of Transport Literature, Vol.8, Issue.2, 7-37, 2014.
  [8] D. D. Ekezie, J. Opara, “The Application of Transportation Algorithm with Volume Discount on Distribution Cost,” Journal of Emerging Trends in Engineering and Applied Sciences, Vol.4, Issue.2, 258 –272, 2013.

  Citation
  J. Opara, P.A. Esemokumo, R. Bekesuoyeibo, "A New Algorithm for Reaching an Optimal Solution to a Nonlinear Transportation Problem," International Journal of Scientific Research in Mathematical and Statistical Sciences, Vol.9, Issue.2, pp.12-16, 2022

• Open Access   Article

  Analysis of Finite Capacity Queueing System with Multiple Vacations and Encouraged Customers

  S. Malik, R. Gupta

  Research Paper | Journal-Paper (IJSRMSS)

  Vol.9 , Issue.2 , pp.17-22, Apr-2022

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  In this paper, a single server queueing model of finite size with multiple vacations and the encouraged arrival of customers is analyzed. The steady-state solution is obtained by using the recursive technique. Whenever the system is empty i.e, the server is idle, then the server goes on vacation. Upon vacation completion, if he finds any customer waiting for the service, he will return to normal working conditions otherwise he will go on another vacation and so on. Some of the operating characteristics of the system like expected queue length, sojourn time, and probabilities of different states of the server are derived via recursive technique. The term encouraged customers arises due to a sudden increase in the number of customers because of the attractive off-season sales or festive season discounts given by the firms.

  Key-Words / Index Term
  Queueing model, Vacations, Encouraged customers, Performance measure, Poisson distribution

  References
  [1] V. Levy, U. Yechiali, “Utilization of idle time in an M/G/1 queueing system”, Management Science, Vol.22, Issue 2, pp.202–211, 1995.
  [2] B. Doshi, “Single server queues with vacation: A survey”, Queueing Systems, Vol. 1, pp. 29 – 66, 1986.
  [3] U. Chatterjee, S.P. Mukherjee, “GI/M/1 queue with server vacations”, Journal of the Operational Research Society, Vol. 41, Issue 1, pp. 83–87, 1990.
  [4] H. Takagi, “Queueing Analysis: A Foundation of Performance Evaluation, Volume 1: Vacation and Priority Systems, Part 1”, Elsevier Science Publishers, North-Holland, Amsterdam, 1991.
  [5] J.G. William, P.P. Wang, M.A. Scott, “Vacation queueing model with service breakdowns”, Applied Mathematical Modelling. Vol. 24, pp. 391-400, 2000.
  [6] G. Choudhury, ‘Analysis of the M^X/G/1 Queueing System with Vacation Times”, Sankhya Indian J Stat., Vol. 64, Issue 1, pp. 37-49, 2002.
  [7] Y. Baba, “Analysis of a GI/M/1 queue with multiple working vacations”, Operations Research Letters, Vol. 33, Issue. 2, pp. 201–209, 2005.
  [8] A.D. Banik, U.C. Gupta, S.S. Pathak, “On the GI/M/1/N queue with multiple working vacations-analytic analysis and computation”, Applied Mathematical Modelling, Vol 31, Issue 9, pp. 1701–1710, 2007.
  [9] L.D.Servi, S.G. Finn, “M/M/1 queues with working vacations (M/M/1/WV)”, Perform. Evaluation, Vol.50, pp. 41-52, 2002.
  [10] N. Tian., Z.G. Zhang, “Vacation queueing models: Theory and applications”, Springer, New York, 2006.
  [11] E. Altman, U. Yechiali, “Analysis of customers’ impatience in queues with server vacations”, Queueing System, Vol. 52, pp. 261–279, 2006.
  [12] J.C. Ke, C.H. Wu, Z.G. Zhang, “Recent developments in vacation queueing models: a short survey”, International Journal of Operational Research, Vol.7, pp.3–8, 2010.
  [13] N.K. Jain, R. Kumar, B.K. Som, “An M/M/1/N Queuing system with reverse balking”, American Journal of Operational Research, Vol.4, Issue 2, pp.17-20, 2014.
  [14] R. Gupta, “Cost Optimization of Queueing System with Working Vacation, Setup, Feedback, Reneging, and Retention of Reneged Customers”, J. Sci. Res., Vol.14, Issue 1, pp.257-268, 2022.
  [15] P. Gupta, N. Kumar, “Cost optimization of single server retrial queueing model with Bernoulli schedule working vacation, vacation interruption and balking”, J. Math. Computer. Sci., Vol.11, Issue 3, pp.2508-2523, 2021.
  [16] P. Gupta, “Study of feedback retrial queueing system with working vacation, setup time, and perfect repair”, Ratio Mathematica, Vol. 41, pp. 291-307, 2021.
  [17] B.K. Som, S. Seth, “An M/M/1/N Queuing system with Encouraged Arrivals”, Global Journal of Pure and Applied Mathematics, Vol.13, Issue 7, pp.3443-3453, 2017.
  [18] O.C. Ibe, O.A. Isijola, “M/M/1 multiple vacation queueing systems with differentiated vacations”, Modeling and Simulation in Engineering, Vol.2014, 6-pages, 2014.
  [19] R. Gupta, S. Malik, “Study of feedback queueing system with unreliable waiting server under multiple differentiated vacation policy”, Ratio Mathematica, Vol.40, pp.146-161, 2021.
  [20] P. Gupta, N. Kumar, “Analysis of classical retrial queue with differentiated vacation and state-dependent arrival rate”, Ratio Mathematica, Vol. 40, pp.47-66, 2021.

  Citation
  S. Malik, R. Gupta, "Analysis of Finite Capacity Queueing System with Multiple Vacations and Encouraged Customers," International Journal of Scientific Research in Mathematical and Statistical Sciences, Vol.9, Issue.2, pp.17-22, 2022

• Open Access   Article

  Application of Bivariate Regression Model Data Comparison Using Information Measures

  P.A. Esemokumo, J. Opara, R. Bekesuoyeibo

  Research Paper | Journal-Paper (IJSRMSS)

  Vol.9 , Issue.2 , pp.23-27, Apr-2022

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  Abstract
  This study is on the application of bavariate regression model data comparison using information measures. Data set for the study was extracted from Esemokumo and Opara (2018), which was on weight of soap (Y) and the number of days it had been used (Z). Ten Regression models: Linear, Quadratic, Polynomial, Logarithmic, Hyperbolic, Power, S-curve, Growth, Exponential and Compound were studied. The E-views package was employed in the analysis of data. Three information measures for model selection known as; AIC, SIC, and HQIC were employed for best model selection. The results revealed that the Growth and Exponential regression models outperformed the other eight regression models utilized, while the least performed regression model is the hyperbolic. Hence, future studies should look at a similar work with different bivariate data sets with the assumptions fully examined.

  Key-Words / Index Term
  AIC, SCI, HQIC, Regression models, Model Comparison

  References
  [1] D.C. Montgomery, E.A. Peck, G.G. Vining, “Introduction to Linear Regression Analysis,”. Wiley & Sons, Hoboken, pp. 134-146, 2006.
  [2] N.J.D. Nagelkerke, “A note on a general definition of the coefficient of determination,”. Biometrika, pp. 691–692, 1991. doi: 10.1093/biomet/78.3.691.
  [3] L. Magee, “R2 measures based on Wald and likelihood ratio joint significance tests,”. Amer Stat., Vol.44, Issue.2, pp. 250–253, 1990.
  [4] S. A. Juliano, F.M. Williams, “A comparison of methods for estimating the functional response parameters of the random predator equation,”. J. Anim Ecol, Vol. 56, Issue. 4, pp.641–653, 1987.
  [5] A. Spiess, N. Neumeyer, “An evaluation of R2 as an inadequate measure for nonlinear models in pharmacological and biochemical research: a Monte Carlo approach”. BMC Pharmacol, Vol.10, Issue 2, pp. 13-21, 2010.
  [6] E. C. Enyoh, A.V. Wirnkor, C.E. Duru, E.C. Enyoh, B. Setiawan, “Curve Estimation Modeling For Predictions of the Novel Corona-virus (COVID-19) Epidemic in Nigeria,”. Healthy-Mu Journal, Vol. 4, Issue. 1, 2020.
  [7] P. A. Esemokumo, R. Bekesuoyeibo, “Simple Regression Models’ Comparison using Goodness-of-fit Measures”. 17th multidisciplinary academic conference at FUTA for Transformation Agenda for Third World Communities in Evolving as Global Developed Nation: Multidisciplinary Approach, Nigeria, 2022.
  [8] H. Akaike, "A new look at the statistical model identification,". IEEE Transactions on Automatic Control, Vol.19, Issue.6, 716–723, 1974.
  [9] G. Schwarz, “Estimating the dimension of a model,”. The Annals of Statistics, Vol. 6, 461-464, 1978.
  [10] E.J. Hannan, B.G. Quinn, "The Determination of the order of an autoregression,". Journal of the Royal Statistical Society, Series B, Vol. 41, 190–195, 1979.

  Citation
  P.A. Esemokumo, J. Opara, R. Bekesuoyeibo, "Application of Bivariate Regression Model Data Comparison Using Information Measures," International Journal of Scientific Research in Mathematical and Statistical Sciences, Vol.9, Issue.2, pp.23-27, 2022

• Open Access   Article

  Observations on the Ternary Quadratic Diophantine Equation 9(x2 + y2)-17xy+4x+4y+16=84z2

  C. Saranya, T. Vennila

  Research Paper | Journal-Paper (IJSRMSS)

  Vol.9 , Issue.2 , pp.28-30, Apr-2022

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  Abstract
  The Ternary Quadratic Diophantine Equation representing non-homogeneous cone is analyzed for its non-zero distinct integer points on it. Six different patterns of integral solutions satisfying the cone under consideration are obtained. A few interesting relations between the solutions and some special number patterns are presented.

  Key-Words / Index Term
  Ternary non-homogeneous Quadratic, Diophantine equation, integral solutions

  References
  [1] Batta.B and Singh.A.N, “History of Hindu Mathematics”, Asia Publishing House, 1938.
  [2] Carmichael, R.D., “The Theory of Numbers and Diophantine Analysis”, Dover Publications, New York , 1959.
  [3] Dickson,.L.E., “History of the theory of numbers”, Chelsia Publishing Co., Vol.II, New York, 1952.
  [4] Mollin.R.A., “All solutions of the Diophantine equation ”, For East J.Math. Sci., Social Volume, Part III, pages 257-293, 1998.
  [5] Mordell.L.J., “Diophantine Equations”, Academic Press, London 1969.
  [6] Telang.S.G., “Number Theory”, Tata McGraw-Hill Publishing Company, New Delhi 1996.
  [7] Nigel.P.Smart, “The Algorithmic Resolutions of Diophantine Equations”, Cambridge University Press, London 1999.
  [8] Gopalan.M.A., Manju Somnath, and Vanitha.N., “Integral Solutions of ”, Acta Ciencia Indica, Volume 33;number 4, pages 1287-1290, 2007.
  [9] Gopalan.M.A., Vidhyalakshmi.S and Umarani.J., “On ternary Quadratic Diophantine equation ”, Sch.J.Eng.Tech. 2(2A); 108- 112,2014
  [10] Gopalan.M.A., Vidhyalakshmi.S and Shanthi.J “On ternary quadratic Diophantine equation ”, International Journal of Innovative Science, Engineering &Technology, Vol 1 , Issue 6, 2014.
  [11] Janaki.G and Saranya.C., “Observations on the Ternary Quadratic Diophantine Equation ”, International Journal of Innovative Research in Science, Engineering and Technology, Vol-5, Issue-2, 2060-2065, Feb 2016.
  [12] Janaki.G, and Saranya.C., “Integral Solutions of the non-homogeneous heptic equation with five unknowns ”, International Journal of Engineering Science and Computing, Vol. 6, Issue 5, 5347-5349, May, 2016.
  [13] Janaki. G, and Saranya.C, Integral solutions of the Ternary cubic equation , International Research Journal of Engineering and Technology, vol 4, issue 3, 665-669, March 2017.
  [14] Janaki. G, and Saranya.C, “Integral solutions of the homogeneous Biquadratic Diophantine equation, ”, International Journal for Research in Applied Science & Engineering Technology, vol.5, issue VIII, 1123-1127, Aug 2017.
  [15] Saranya.C, and kayathri.P Integral solutions of the Ternary Cubic Equation International Journal for Research in Applied and Engineering Technology, vol.10, Issue 1 Jan 2022.

  Citation
  C. Saranya, T. Vennila, "Observations on the Ternary Quadratic Diophantine Equation 9(x2 + y2)-17xy+4x+4y+16=84z2," International Journal of Scientific Research in Mathematical and Statistical Sciences, Vol.9, Issue.2, pp.28-30, 2022

• Open Access   Article

  Generalized Composite Regression with Ratio and Product Estimators in Double Sampling

  Megha, Sangeeta Malik

  Research Paper | Journal-Paper (IJSRMSS)

  Vol.9 , Issue.2 , pp.31-35, Apr-2022

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  Abstract
  In Sample survey Auxiliary information is widely used for estimation procedure to find out population parameters of interest in small area. Various sampling schemes are used by several authors. Double sampling scheme is effective in term of cost and application. In this paper a generalized composite regression with ratio estimator where weights assigned to both regression and ratio estimator and a generalized composite regression with product estimator where weights assigned to both regression and product estimators with double sampling scheme in sample survey using of auxiliary data are proposed. The variance of proposed estimator is presented. Hypothetical environments for which the proposed estimators are showing better results than simple mean, regression estimator and respectively with ratio and product estimator is obtained. Empirical study is performed on the known data set and the Percent Relative efficiency (PRE) of the proposed estimators has shown the degree of superiority with respect to existing estimators.

  Key-Words / Index Term
  Auxiliary information, Estimators, Variance, Double Random Sampling

  References
  [1] Sukhatme, P.V. and Sukhatme, B.V., Sampling Theory of Surveys with Applications, Asia Publishing House, New Delhi, 1971.
  [2] Murthy, M.N., Sampling Theory and Methods, Calcutta, India: Statistical Publishing Society, 1967.
  [3] Cochran, W.G., Sampling Techniques. 3rd edition, John Wiley: New York, 1977.
  [4] C. KADILAR, H. CINGI, “An Improvement in Estimating The Population Mean By Using The Correlation Coefficient”, Hacettepe Journal of Mathematics and Statistics, 35, 2006.
  [5] Singh, H.P., Kumar, S. and Kozak, M., “Improved estimation of finite population mean using sub- sampling to deal with non-response in two phase sampling scheme” Commun. in Statist.-Theo. And Meth., 39, 791-802, 2010.
  [6] Singh, H.P., Tailor, R. and Tailor R., “Estimation of Finite Population Mean in Two-Phase Sampling with Known Coefficient of Variation of An Auxiliary Character”, Statistica , LXXII no.1, 2012.
  [7] Tailor, R. and Malik, K. A., “Ratio Type Estimator of Population Mean in Double Sampling, Mathematical Reviews”; MathSciNet; Page No. 34-39, 2013.
  [8] Sharma, Prayas et al. “Some Exponential Ratio-Product Type Estimators Using Information on Auxiliary Attributes Under Second Order Approximation.” International Journal of Statistics and Economics, 12(3): Page No 58–89, 2013.
  [9] Hazra, R., Combination of Two Ratio Type Estimator for Estimating Population Mean Using Auxiliary Variable With Double Sampling in Presence of Nonresponse. Bull. Math. & Stat. R, 2015.
  [10] Bahl,Shashi and Malik Sangeeta, “Generalised Synthetic Estimator Using Double Sampling Scheme and Auxiliary Information”; Mathematics Journal of interdisciplinary sciences (2278-9561), Vol. 4; Page No 15-21, 2015.
  [11] Damodar N. Gujrati Basic Econometrics, Fourth edition, Mc Graw Hill.India, Page No 443, 2004.
  [12] Onyango R., Oduor B., Odundo F., “Enhanced Estimation of Population Mean in Presence of Errors on a Survey Variable in Stratified Two-Phase Sampling”, International Journal of Scientific Research in Mathematical and Statistical Sciences Volume-9, Issue-1, pp.32-39, February 2022.

  Citation
  Megha, Sangeeta Malik, "Generalized Composite Regression with Ratio and Product Estimators in Double Sampling," International Journal of Scientific Research in Mathematical and Statistical Sciences, Vol.9, Issue.2, pp.31-35, 2022

• Open Access   Article

  Numerical Approximation of Nonlinear, Non-Newtonian Arterial Blood Flow in human Circulatory System

  E.C. Akah, J. Chukwuchekwa, A.V. Onuche

  Research Paper | Journal-Paper (IJSRMSS)

  Vol.9 , Issue.2 , pp.36-42, Apr-2022

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  Abstract
  In this paper, we investigate the e?ects of uncertainties of parameters involved in the elastic and viscoelastic constitutive equation for blood ?ow model recently proposed by Bertaglia et al (2020). An IMEX ?nite volume stochastic collocation scheme which guarantees spectral convergence in the stochastic space is imminent. When the variability of flow rate and velocity waveforms is compared to the variability of pressure and area, the concluding ones are significantly more susceptible to the parametric errors underpinning the mechanical characterization of vessel walls. The scheme is consistent with the equilibrium limit, which corresponds to the asymptotic elastic behavior for small relaxation times. Simulations done considering both the simple elastic and the more realistic viscoelastic constitutive law show that the great uncertainty of the viscosity parameter plays a key role in the forecast of pressure waveforms, adding to the confidence interval of this variable. In-vivo recorded patient-specific pressure data lies within the confidence interval of the output gotten with the proposed methodology and anticipations of the computed pressures are comparable to the recorded waveforms.

  Key-Words / Index Term
  Stochastic partial differential equations, Viscoelastic fluids, collocation method, spectral convergence in stochastic space

  References
  [1] U.M. Ascher, S.J. Ruuth, R.J. Spiteri “Implicit-Explicit Runge-Kutta Methods for Time-Dependent Partial Differential Equations” Appl Numer Math, vol. 25, pp 2-3 , 1997.
  [2] M. Abramowitz and I.A. Stegun “Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables”. Dover, New York, 9th edition, (1972).
  [3] J. Alastruey, K.H. Parker, and S.J. Sherwin, “Arterial pulse wave haemodynamics”. In Proceedings BHR Group’s 11th International Conference on Pressure Surges, pages 401–443, 2012.
  [4] G. Bertaglia, V. Cale?, and A. Valiani “Modeling blood ?ow in viscoelastic vessels: the 1D augmented ?uid–structure interaction system”. Computer Methods in Applied Mechanics and Engineering, 360(C):112772, 2020.
  [5] G. Bertaglia, M. Ioriatti, A. Valiani, M. Dumbser, and V. Cale? “Numerical methods for hydraulic transients in visco-elastic pipes”. Journal of Fluids and Structures, Vol. 81, pp 230–254, 2018.
  [6] G. Bertaglia, A. Navas-Montilla, A. Valiani, M.L. Monge Garc?a, J. Murillo, and V. Cale?, “Computational hemodynamics in arteries with the one-dimensional augmented ?uid-structure interaction system”: viscoelastic parameters estimation and comparison with in-vivo data. Journal of Biomechanics, 100(C):109595 2020.
  [7] P. Chen, A. Quarteroni, and G. Rozza “Simulation-based uncertainty quanti?cation of human arterial network hemodynamics”. International Journal for Numerical Methods in Biomedical Engineering, Vol 29, pp 698–721, 2013.
  [8] J.D. Cole “On a quasi-linear parabolic equation occurring in aerodynamics”. Quarterly of Applied Mathematics, Vol 9, Issue 3, pp 225–236, 1951.
  [9] V.G. Eck, W.P. Donders, J. Sturdy, J. Feinberg, T. Delhaas, L.R. Hellevik, and W. Huberts “A guide to uncertainty quanti?cation and sensitivity analysis for cardiovascular applications”. International Journal for Numerical Methods in Biomedical Engineering, page e02755, 2016.
  [10] L. Formaggia, D. Lamponi, and A. Quarteroni “One-dimensional models for blood ?ow in arteries”. Journal of Engineering Mathematics, Vol 47, Issue 3-4, pp 251–276, 2003.
  [11] L. Formaggia, A. Quarteroni, and A. Veneziani “Cardiovascular Mathematics: Modeling and simulation of the circulatory system”. Springer, 2009.
  [12] E. Hopf “The partial di?erential equation ut +uux = µxx.” Communications on Pure and Applied Mathematics, Vol. 3, Issue 3, pp 201–230, 1950.
  [13] L.O. Müller, and E.F. Toro “Well-balanced high-order solver for blood ?ow in networks of vessels with variable properties”. International Journal for Numerical Methods in Biomedical Engineering, Vol. 29, Issue 12, pp 1388–1411, 2013.
  [14] W.W. Nichols, M.F. O’Rourke, and C. Vlachlopoulos C. “McDonald’s Blood Flow in Arteries”. Hodder Arnold, 6th edition, 2011.
  [15] L. Pareschi, G. Russo “Implicit-Explicit Runge-Kutta schemes for stiff systems of differential equations”, Recent Trends in Numerical Analysis, Vol. 3, pp 269-289, 2000.
  16] L. Pareschi and G. Russo “Implicit-explicit Runge-Kutta schemes and applications to hyperbolic systems with relaxation”, Journal of Scienti?c Computing, Vol. 25, Issue ½, pp 129–155, 2005.
  [17] G. Poette, B. Despres, and D. Lucor “Uncertainty quanti?cation for systems of conservation laws”. Journal of Computational Physics, Vol. 228, Issue 7, pp 2443–2467, 2009.
  [18] A. Quarteroni and L. Formaggia “Mathematical Modelling and Numerical Simulation of the Cardiovascular System”, vol. 12. Elsevier B.V, 2004.
  [19] P. Salvi “Pulse Waves: How vascular hemodynamics a?ects blood pressure”. Springer Verlag, 2012.
  [20] E.F. Toro “Riemann Solvers and Numerical Methods for Fluid Dynamics”. Springer Verlag, 3rd edition, 2009.
  [21] E.F. Toro and A. Siviglia “Flow in collapsible tubes with discontinuous mechanical properties”: Mathematical model and exact Solutions. Communications in Computational Physics, Vol. 13, Issue 2, pp 361– 385, 2013.
  [22] D. Valdez-Jasso “Analysis of Viscoelastic Wall Properties in Ovine Arteries”. IEEE Transactions on Biomedical Engineering, Vol. 56, Issue 2, pp 210–219, 2009.
  [23] N. Westerhof, N. Stergiopulos, M.I.M. Noble, and B.E. Westerhof “Snapshots of Hemodynamics”: An Aid for Clinical Research and Graduate Education. Springer, 3rd edition, 2019.
  [24] D. Xiu “Numerical Methods for Stochastic Computations - A Spectral Method Approach”. Princeton University Press, New Jersey, 2010.
  [25] D. Xiu and J.S. Hesthaven “High-Order Collocation Methods for Di?erential Equations with Random Inputs”. SIAM Journal on Scienti?c Computing, Vol. 27, issue 3, pp 1118–1139, 2005.
  [26] D. Xiu and G.E. Karniadakis “The Wiener-Askey polynomial chaos for stochastic di?erential equations”. SIAM Journal on Scienti?c Computing, Vol. 24, Issue 2, pp 619–644, 2002.

  Citation
  E.C. Akah, J. Chukwuchekwa, A.V. Onuche, "Numerical Approximation of Nonlinear, Non-Newtonian Arterial Blood Flow in human Circulatory System," International Journal of Scientific Research in Mathematical and Statistical Sciences, Vol.9, Issue.2, pp.36-42, 2022

• Open Access   Article

  Non-Linear Fractional Integrodifferential Systems with Delays: A Controllability Study

  E. Onyeocha, C.A. Nse, I.J. Njoku

  Research Paper | Journal-Paper (IJSRMSS)

  Vol.9 , Issue.2 , pp.43-48, Apr-2022

  Abstract

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  References

  Citation

  Abstract
  A dynamical system which can be acted on using suitable controls is said to be a control system. This work present results for the controllability of a class of non-linear implicit neutral fractional integro-differential equations using a fixed point method. This research aims to present novel results on the controllability of nonlinear fractional integro-differential systems. A neutral system in this paper refers to a dynamical system, in which there is a delay/relay while the system runs, i.e., the time and system functionality are not constant. The Schauder’s fixed point theory is used to establish the controllability results for the system. Furthermore, in order to establish non-compactness, the Arzela-Ascoli theorem was used and with that the controllability conditions for the dynamical system with delay was established. The study could be extended to consider systems without delay using the Schauder fixed point theory and also, other methods such as the algebraic and functional analytic method could be applied to this problem. The results of this study could find applications in Engineering.

  Key-Words / Index Term
  Controllability; Neutral fractional integro-differential systems; Schauder fixed point theorem; Fixed point; Arzela-Ascoli theorem

  References
  [1] J-M , Coron, “Control and non-linearlity, Mathematical surveys and monographs”, ISSN 0076-5376, vol. 136, 2007.
  [2] A.E. Bashirov, K.R. Kerimov, “On controllability conception for stochastic systems”, SIAM Journal on Control and Optimization, Vol. 35, Issue 2, pp. 384–398, 1997.
  [3] A.E. Bashirov, N.I. Mahmudov, On concepts of controllability for deterministic and stochastic systems”, SIAM Journal on Control and Optimization, Vol. 37, Issue 6, pp. 1808–1821, 1999.
  [4] M. Benchohra, A. Ouahab, “Controllability results for functional semilinear differential inclusions in Frechet spaces”, Nonlinear Analysis: Theory, Methods & Applications, Vol. 61, Issue 3, pp. 405–423, 2005.
  [5] L. Gorniewicz, S. Ntouyas, D. O’Regan, “Controllability of semilinear differential equations and inclusions via semigroup theory in Banach spaces”, Reports on Mathematical Physics, Vol. 56, Issue 3, pp. 437–470, 2005.
  [6] A. Babiarz, J. Klamka, M. Niezabitowski, “Schauder’s fixed–point theorem in approximate controllability problems”, Int. J. Appl. Math. Comput. Sci., Vol. 26, Issue 2, pp. 263–275, 2016.
  [7] M. Nawaz, J. Wei, J. Sheng, A. Ullah, K. Niazi, L.C. Yang, “On the controllability of nonlinear fractional system with control delay”, Hacet. J. Math. Stat., pp. 1 – 9, 2018.
  [8] P.C. Jackreece, C.A. Godspower, H.C. Ugorji, “Controllability of Fractional-order Dynamical System with Time Varying-Delays”, General Letters in Mathematics, Vol. 5, Issue 3, pp. 148 -156, 2018.
  [9] K. Balachandran, S. Divya, “Controllability of Nonlinear Implicit Fractional Integrodifferential Systems”, Int. J. Appl. Math. Comput. Sci., Vol. 24, Issue 4, pp. 713–722, 2014.
  [10] K. Mourad, E. Fateh, D. Baleanu, “Stochastic fractional perturbed control systems with fractional Brownian motion and Sobolev stochastic non-local conditions”, Collect. Math. Vol. 69, pp. 283–296, 2018.
  [11] M. Caputo, “Linear model of dissipation whose Q is almost frequency independent, Part II”, Geophys. J. Royal Astronomical Soc., Vol. 13, Issue 5, pp. 529-539, 1967.
  [12] W.E. Olmstead, R.A. Handelsaman, “Diffusion in a semi-infinite region with nonlinear surface dissipation”, SIAM Review, Vol. 18, Issue 2, pp. 275-291, 1976.
  [13] J. Sabatier, O.P. Agarwal, J.A. Tenreiro Machado, (Eds.),. “Advances in fractional calculus: Theoretical Developments and Applications in Physics and Engineering”, Springer-Verlag, New York, NY, 2007.
  [14] R.C. Mittal, R. Nigan, “Solution of fractional integrodifferential equations by Adomian Decomposition method”, Int. J. Appl. Math. Mech., Vol. 4, Issue 2, pp. 87-94, 2008.
  [15] E.A. Rawashdeh, “Legendre wavelets method for fractional integrodifferential equations”, Appl. Math. Sci., Vol. 5, Issue 2, pp. 2467-2474, 2011.
  [16] D.J.N. Xiao-Li, “Controllability of Nonlinear Fractional Delay Dynamical Systems with prescribed controls”, Nonlinear Analysis: Modelling and Control, Vol. 23, Issue 1, pp. 1-18, 2018.
  [17] H. Zhang, J. Cao, W. Jiang, “Controllability criteria for Linear Fractional Differential Systems with State Delay and Impulse”, Journal of Applied Mathematics, (2013).
  [18] J. Klamka, “On the Global Controllability of Perturbed nonlinear Systems”, IEEE Transactions on Automatic Control, Vol. 20, Issue 1, pp. 170-172, 1975.
  [19] J. Klamka, “On the Local Controllability of Perturbed Nonlinear Control systems”, IEEE Transactions on Automatic control, Vol. 20, Issue 2, pp. 289-291, 1975.
  [20] J. Klamka, “Constrained Controllability of Semilinear Systems with Delayed Controls”, Bulletin of the Polish Academy of Science: Technical Science, Vol. 56, Issue 4, pp. 333-337, 2008.
  [21] J. Klamka, “Constrained Controllability of Dynamical systems”, International Journal of Applied Mathematics and Computer science, Vol. 9, Issue 2, pp. 231-244, 1999.
  [22] K. Balachandran, J.P. Dauer, “Controllability of Nonlinear Systems via Fixed point Theorem”, Journal of Optimization Theory and Applications, Vol. 5, Issue 3, pp. 345- 352, 1987.
  [23] K. Balachandran, P. Balasubramaniam, “A Note on Controllability of Nonlinear Volterra Integrodifferential Systems”, Kybernetika, Vol. 28, Issue 4, pp. 284-291, 1992.
  [24] K. Balachandran, P. Balasubramaniam, “Controllability of Nonlinear Neutral Volterra Integrodifferential Systems”, Journal of Australian Mathematical Society, Vol. 36, Issue 1, pp. 107-116, 1994.
  [25] K. Balachandran, “Controllability of Nonlinear Systems with Implicit Derivatives”, IMA Journal of Mathematical Control and Information, Vol. 5, Issue 2, pp. 77-83, 1988.
  [26] C. Dacka, “On the Controllability of a Class of Nonlinear Systems”, IEEE Transactions on Automatic Control, Vol. 25, Issue 2, pp. 263-266, 1980.
  [27] T.L. Guo, “Controllability and Observability of Impulsive Fractional Linear Time- Invariant Systems”, Computers and Mathematics with Applications, Vol. 64, Issue 10, pp. 3171-3182, 2012.
  [28] R.E. Kalman, Y.C. Ho, K.S. Narendra, “Controllability of Linear Dynamical Systems”, Contributions to Differential Equations, Vol. 1, Issue 2, pp. 189-213, 1963.
  [29] V. Govindaraj, R.K. George, “Controllability of fractional dynamical systems: A functional analytic approach”, Mathematical Control and Related Fields, Vol. 7, Issue 4, pp. 537–562.
  [30] A. Dabbouchi, D. Baleanu, “Exact Null Controllability for Fractional Nonlinear Integrodifferential equations via Implicit Evolution systems”, Journal of Applied Mathematics, Article ID 931975, 1-17, 2012.
  [31] N.I. Mahmudov, “Approximate Controllability of Fractional Sobolev-type Evolution Equation in Banach Spaces”, Abstract and Applied analysis, pp. 1-9, 2013.
  [32] J. Schauder, “Der Fixpunktsatz in Functionalraumen”, Studia Math., Vol. 2, pp. 171-180, 1930.
  [33] F.E. Browder, “A further generalization of the Schauder fixed point theorem”, Duke Math. J.. Vol. 32, pp. 575–578, 1965.
  [34] F.E. Browder, “Asymptotic fixed point theorems”, Math. Ann. Vol. 185, pp. 38–60, 1970.
  [35] D. Bugajewski, “Weak solutions of integral equations with weakly singular kernel in Banach spaces”, Comm. Math. Vol. 34, pp. 49–58, 1994.
  [36] K. Deimling, “Nonlinear Functional Analysis”, Springer-Verlag, Berlin-Heidelberg, 1985.
  [37] J. Dugundji, A. Granas, “Fixed Point Theory”, Polish Scientific Publishers, Warszawa, 1982.
  [38] L. Górniewicz, D. Rozp?och-Nowakowska, “On the Schauder Fixed Point Theorem”, Topology in Nonlinear Analysis, Banach Center Publications, vol. 35, Polish Academy of Sciences, Warszawa, 1996.
  [39] R.P. Agarwal, S. Arshad, D. O’Regan, V., Lupulescu, “A Schauder fixed point theorem in semilinear spaces and applications”, Fixed Point Theory Appl., Vol. 2013, Issue 306, 2013.
  [40] J. Dauer, N., Mahmudov, “Approximate controllability of semilinear functional equations in Hilbert spaces”, Journal of Mathematical Analysis and Applications, Vol. 273, Issue 2, pp. 310–327, 2002.
  [41] J. Dauer, N., Mahmudov, M., Matar, “Approximate controllability of backward stochastic evolution equations in Hilbert spaces”, Journal of Mathematical Analysis and Applications Vol. 323, Issue 1, pp. 42–56, 2006.
  [42] L. Kexue, P. Jigen, “Laplace transform and fractional differential equations”, Applied Mathematics Letters, Vol. 24, Issue 12, pp. 2013–2019, 2011.
  [43] A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, “Theory and Applications of Fractional Differential Equations”, Elsevier, Amsterdam, 2006.
  [44] K.S. Miller, B. Ross, “An Introduction to the Fractional Calculus and Fractional Differential Equations”, Wiley, New York, NY, 1993.
  [45] K.B. Oldham, J. Spanier, “The Fractional Calculus”, Academic Press, London, 1974.
  [46] I. Podlubny, “Fractional Differential Equations”, Academic Press, New York, NY, 1999.
  [47] S.G. Samko, A.A., Kilbas, O.I., Marichev, “Fractional Integrals and Derivatives; Theory and Applications”, Gordon and Breach, Amsterdam, 1993.
  [48] K. Kaczorek, “Selected Problems of Fractional Systems Theory”, Springer, Berlin, 2011.
  [49] K. Balachandran, J. Kokila, “Constrained controllability of fractional dynamical systems”, Numerical Functional Analysis and Optimization, Vol. 34, Issue 11, pp. 1187–1205, 2016.

  Citation
  E. Onyeocha, C.A. Nse, I.J. Njoku, "Non-Linear Fractional Integrodifferential Systems with Delays: A Controllability Study," International Journal of Scientific Research in Mathematical and Statistical Sciences, Vol.9, Issue.2, pp.43-48, 2022