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• Open Access   Article

  An Exponential Strategy for Estimation of Population Mean in Systematic Sampling

  Rajesh Tailor and Asha Mishra

  Research Paper | Isroset-Journal (IJSRMSS)

  Vol.5 , Issue.3 , pp.73-78, Jun-2018

  CrossRef-DOI:   https://doi.org/10.26438/ijsrmss/v5i3.7378

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  Abstract
  Using exponential function, an estimator for mean has been developed and it’s Properties are studied. The mean squared error of the developed estimator has been derived. To compare developed estimator with existing estimators. An empirical comparative study is also conducted.

  Key-Words / Index Term
  population mean, Exponential estimator ,Mean squared error

  References
  [1]. Bahl, S. and Tuteja, R.K. (1991). Ratio and product type exponential estimator. Info. & Opti. Sci. XIII, 159- 163.
  [2]. Shukla, N. D. (1971). Systematic sampling and product method of estimation. Proceedings of all India Seminar on Demography and Statistics. B.H.U., Varanasi, India.
  [3]. Singh, M. P. (1967). Ratio-cum-product method of estimation. Metrika, 12, 34- 42.
  [4]. Singh, H. P. and Singh, R. (1998). Almost unbiased ratio and product type estimators in systematic sampling. Questiio, 22,3, 403-416.
  [5]. Singh, H. P., Tailor, R. and Jatwa, N. K. (2011). Modified ratio and product estimators for population mean in systematic sampling. J. Mod. Appl. Statist. Meth. 10 , 2,424-435.
  [6]. Swain, A. K. P. C. (1964). The use of systematic sampling ratio estimate. J. Ind. Statist. Assoc., 2,160- 164.
  

  Citation
  Rajesh Tailor and Asha Mishra, "An Exponential Strategy for Estimation of Population Mean in Systematic Sampling," International Journal of Scientific Research in Mathematical and Statistical Sciences, Vol.5, Issue.3, pp.73-78, 2018

• Open Access   Article

  Determination of Economic Ordering Policies under Trade credit with Application of Zero Acceptance number Single Sampling by Attributes involving Destructive Testing, Inspection Errors

  K.Pradeepa Veerakumari, H.M. Aruna

  Research Paper | Isroset-Journal (IJSRMSS)

  Vol.5 , Issue.3 , pp.79-83, Jun-2018

  CrossRef-DOI:   https://doi.org/10.26438/ijsrmss/v5i3.7983

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  Abstract
  Conventional Economic Order Quantity model tacitly assumes that the all items received are perfect. But, however, in practice, the items received may be imperfect. Inspection process helps to identify the defective items. 100% inspection is not feasible when the testing involved is destructive and costly. Acceptance sampling plan provides fool-proof solution to the problem. It is a trade-off between 100% inspection and zero inspection. Based on the above conditions, this paper incorporates the Economic Ordering Quantity model with permissible delay in payments and single sampling plan by attributes when the inspection requires destructive testing. Mathematical models are developed to characterize the scenario. Inspection errors are also included in the model. A hypothetical example is illustrated to validate the results in the proposed paper.

  Key-Words / Index Term
  Economic-Order Quantity; Single Sampling plan; Inspection errors; Trade-credit; Destructive testing

  References
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  Citation
  K.Pradeepa Veerakumari, H.M. Aruna, "Determination of Economic Ordering Policies under Trade credit with Application of Zero Acceptance number Single Sampling by Attributes involving Destructive Testing, Inspection Errors," International Journal of Scientific Research in Mathematical and Statistical Sciences, Vol.5, Issue.3, pp.79-83, 2018

• Open Access   Article

  Procedural Approach for Designing Skip-lot Sampling Plan of type SkSP-T through Minimum Sum of Risks

  K. Pradeepa Veerakumari, S. Suganya

  Research Paper | Isroset-Journal (IJSRMSS)

  Vol.5 , Issue.3 , pp.85-88, Jun-2018

  CrossRef-DOI:   https://doi.org/10.26438/ijsrmss/v5i3.8588

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  Abstract
  The principal objective of this paper is to evaluate optimal plan parameters for Skip-lot Sampling Plan of type SkSP-T with single sampling plan as reference plan involving minimum sum risks. The designing technique includes the assessment of Acceptable Quality Level (AQL) and Limiting Quality Level (LQL) taking in to account the proper balancing of Producer and Consumer risks. Tables are simulated for various parametric values of SkSP-T with Single Sampling Plan.

  Key-Words / Index Term
  SkSP-T, Minimum sum of risks, AQL, LQL, Single Sampling plan

  References
  [1] Dodge H. F, “Skip-Lot sampling plan”, Industrial Quality Control, 11(5), pp. 3-5, 1955.
  [2] Dodg.H. F, Perry. R L, “A System of Skip-lot Plans for Lot-by-lot inspection”, American society for Quality Control Technical Conference Transactions, Chicago, Illinois, pp.469-477, 1971.
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  [5] Balamurali S, Chi-Hyuck Jun, “ A New System of Skip-lot Sampling Plans having a Provision for Reducing Normal Inspection”, Applied Stochastic Models in Business Industry, Vol.27, pp. 348-363, 2011.
  [6] Saminathan Balamurali, Muhammad Aslam, and Chi-Hyuck Jun, “A New System of Skip-Lot Sampling Plans including Resampling”, The Scientific World Journal, pp.1-6, 2014.
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  [9] Balamurali S, “Modified Tightened Three level Continuous sampling plan”, Economic Quality Control, Vol. 17, p. 221-234, 2002.
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  [11] Pradeepa Veerakumari. K and Suganya, “A New System of SkSP-T with Single Sampling Plan as Reference Plan”, Research Journal of Mathematics and Statistics, Vol.4 (4), 1-6, 2016.
  [12] Pradeepa Veerakumari. K and Suganya. S, “SkSP-T with Double Sampling Plan (DSP) as Reference Plan using Fuzzy Logic Optimization Techniques”, International Journal of Pure and Applied mathematics, Vol.117, No.12, 409-418, 2017.
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  [14] Soundararajan, V, “Sampling Inspection plans when the sample size is fixed”, Journal of Madras University, Section B, 44, 9-19, 1981.
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  [16] Soundararajan, V and Vijayaraghavan, R., “On Designing Multi Deferred State Sampling plans Involving Minimum Risks”, Journal of Applied Statistics, Vol. 16, No.1, pp.87-94, 1989.
  [17] Subramani, K, “Studies on Designing Attribute Acceptance Sampling Plans with Emphasis on Chain Sampling Plans”, PhD Thesis, Bharathiar University, Coimbatore, Tamil Nadu, 1991.
  

  Citation
  K. Pradeepa Veerakumari, S. Suganya, "Procedural Approach for Designing Skip-lot Sampling Plan of type SkSP-T through Minimum Sum of Risks," International Journal of Scientific Research in Mathematical and Statistical Sciences, Vol.5, Issue.3, pp.85-88, 2018

• Open Access   Article

  Economic scheme for designing Skip lot sampling plans involving Cost functions using Graphical Evaluation Review Technique

  K. Pradeepa Veerakumari, M. Kokila

  Research Paper | Isroset-Journal (IJSRMSS)

  Vol.5 , Issue.3 , pp.89-95, Jun-2018

  CrossRef-DOI:   https://doi.org/10.26438/ijsrmss/v5i3.8995

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  Abstract
  The main thrust of this paper is to design optimum cost procedure for skip lot sampling plan of type SkSP-2 and SkSP-3 with single sampling as reference plan using GERT techniques. This favors producer as well as consumer. Simulation has been done and tables are constructed for various iterations of parameters of SkSP-2 and SkSP-3 with single sampling plan.

  Key-Words / Index Term
  SkSP-2, SkSP-3, Graphical Evaluation Review Technique, Optimum cost, Single sampling plan

  References
  [1]. S. Balamurali, C.H. Jun, “A new system of skip-lot sampling plans having a provision for reducing normal inspection”, Applied Stochastic Models in Business and Industry. Vol.27, Issue 3, pp. 348–363, 2011.
  [2]. S. Chakraborthy, U. K. Rathie, “Analysis of inspection error through GERT”, Sankhya series B, Vol.51, Issue 1, pp.125-133. 1989.
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  [7]. H.F. Dodge, R L. Perry. “A system of skip-lot plans for lot-by-lot inspection”, American Society for Quality Control Technical Conference Transactions, Chicago, Illinois, pp. 469-477, 1971.
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  [11]. C. Kandasamy, K. Govindaraju, “Selection of CSP-T plan”, Communications in Statistics: Simulation and Computation. Vol. 22, Issue 1, pp. 265-283, 1993.
  [12]. T.Y. Lin, “Optimal inspection policy for CSP-T with return cost using GERT”. Journal of the Chinese Institute of Industrial Engineers. Vol. 25, Issue 6, pp. 446-456, 2008.
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  [16]. H. Ohta, “GERT analysis of chain sampling inspection plans”. Bulletin of University of Osaka prefecture. Series. A Vol. 27, Issue 2, pp. 167-174, 1978.
  [17]. R.L. Perry,. “Skip-lot sampling plans”. Journal of Quality Technology. Vol. 5, Issue 3, pp.123-130, 1973.
  [18]. K. PradeepaVeerakumari, R. Resmi, “Designing optimum plan parameters for continuous sampling plan of type(CSP-2) through GERT analysis”. Journal of Statistics and management Systems, Vol. 19, Issue 2, pp. 303-311, 2016.
  [19]. K. Pradeepa Veerakumari, S. Azarudheen, “Evaluation of Single Sampling Plan under the Conditions of Intervened Poisson distribution”, Communications in Statistics- Simulation and Computation, pp.1-9, 2017.
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  Citation
  K. Pradeepa Veerakumari, M. Kokila, "Economic scheme for designing Skip lot sampling plans involving Cost functions using Graphical Evaluation Review Technique," International Journal of Scientific Research in Mathematical and Statistical Sciences, Vol.5, Issue.3, pp.89-95, 2018

• Open Access   Article

  Robust Depth based weighted Estimator with Application in Discriminant Analysis

  R. Muthukrishnan, G. Poonkuzhali

  Research Paper | Isroset-Journal (IJSRMSS)

  Vol.5 , Issue.3 , pp.98-101, Jun-2018

  CrossRef-DOI:   https://doi.org/10.26438/ijsrmss/v5i3.98101

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  Abstract
  Data depth concept used to measure the deepness of a given point in the entire multivariate data cloud. It leads to center-outward ordering of sample points used rather than usual smallest to largest rank. The ordering starts from middle and moves in all directions. Multivariate location and scatter can be computed by using the depth value of each data point. Various depth procedures have been established by many authors. In this paper, a new depth procedure is proposed, namely Modified Mahalanobis Depth (MMD), which calculates depth based on robust distance with Minimum Covariance Determinant (MCD) approach and a weight function is established to determine the location and scale. The superiority of the proposed depth based procedure over existing depth procedures has been studied in simulated environment using R software with respect to application in discriminant analysis. The proposed depth procedure performs well when compared with the existing procedures even with higher contamination levels and larger sample sizes.

  Key-Words / Index Term
  Data Depth, Mahalanobis Distance, MCD, Robust Distance, Weight Function, Discriminat Analysis, R Software

  References
  [1]. T.W. Tukey, “Mathematics and picturing data,” In Proceedings of the International Congress on Mathematics (R. D. James, ed.), Canadian Math. Congress, 1975, Vol. 2, pp. 523–531
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  [3]. R.Y. Liu, “On a notion of data depth based on random simplices,” Annals of Statistics Vol. 18, pp. 405–414, 1990.
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  [7]. Y. Vardi, C.H. Zhang,. “The multivariate L1-median and associated data depth,” Proceedings of the National Academy of Sciences, U.S.A., 2000,Vol. 97, pp. 1423-1426.
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  [14]. R. Dyckerhoff, C. Ley, D. Paindaveine, “Depth-based runs tests for bivariate central symmetry,” Annals of the Institute of Statistical Mathematics, Vol. 67, Issue 5, pp. 917-941, 2015.
  [15]. M. Hubert, P.J. Rousseeuw, P. Segaert, “Multivariate and functional classification using depth and distance,” Advances in Data Analysis and Classification.Vol. 11, Issue 3, pp. 445-466, 2017.
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  Citation
  R. Muthukrishnan, 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.98-101, 2018

• Open Access   Article

  A Bingham plastic and peripheral layer model of blood flow in the presence of stenosed artery

  Rajesh Shrivastava, R.S. Chandel, Ajay Kumar, Keerty Shrivastava and Sanjeet Kumar

  Research Paper | Isroset-Journal (IJSRMSS)

  Vol.5 , Issue.3 , pp.102-110, Jun-2018

  CrossRef-DOI:   https://doi.org/10.26438/ijsrmss/v5i3.102110

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  Abstract
  The present work deals with the blood flow through the stenosis assuming that flowing blood is represented by a two layered model. We considered a two-layered blood flow model to study the axisymmetric flow of blood by assuming the core layer as Bingham plastic (non-Newtonian fluid) and the peripheral layer (Newtonian fluid) in through the stenosis in the arteries. The analytical expressions for flow rate, resistance to flow and wall shear stress have been developed in this model. We have depicts the effect of stenosis on resistance to flow and wall shear stress. This study gives an insight into the effects of slip velocity on the volumetric flow rate, resistance to flow and wall shear stress.

  Key-Words / Index Term
  Stenosis, blood flow, core layer, peripheral layer and Bingham plastic

  References
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  [15] M.S. Moayeri and Zendehbudi, “The effect of elastic property of the wall on flow characteristics through an arterial stenosis”, J. of Biomechanics, Vol. 36, pp. 525-535, 2003.
  [16] R.K. Naeem, S. Younus and Dania, “Inverse solutions for unsteady in-compressible couple fluid flow”, IJAMM, Vol. 6, pp. 1-17, 2010.
  [17] G. Radhakrishnamacharya and P. Shrinivasa Rao, “Flow of a magnetic fluid through a non-uniform wavy tube”, PNASI, Vol. 76, pp. 241-245, 2007.
  [18] B.R.P Rao, “Effect of stenosis on non-Newtonian flow of the blood in an artery”, Bull. Math., Vol. 42, pp. 283-294, 1980.
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  [21] P Sinha and C. Singh, “Effect of couple stresses on the blood flow through an artery with mild stenosis”, Biorheology, vol. 15, pp. 3-4, 1984.
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  Citation
  Rajesh Shrivastava, R.S. Chandel, Ajay Kumar, Keerty Shrivastava and Sanjeet Kumar, "A Bingham plastic and peripheral layer model of blood flow in the presence of stenosed artery," International Journal of Scientific Research in Mathematical and Statistical Sciences, Vol.5, Issue.3, pp.102-110, 2018

• Open Access   Article

  Bulk queue model with two phase’s service, unreliable server and m phase of repair

  Ashu Vij

  Research Paper | Isroset-Journal (IJSRMSS)

  Vol.5 , Issue.3 , pp.111-124, Jun-2018

  CrossRef-DOI:   https://doi.org/10.26438/ijsrmss/v5i3.111124

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  Abstract
  In the present investigation, we have considered state dependent unreliable bulk queueing model with two phase of service and m phase of repair. The server provides first phase of service to each arriving unit while second phase is optional, provided as per demand of the unit. The server is subject to breakdown at any instant of services; whenever the server gets break down during the service (any phase), then it is immediately send for repair. The repair of server done in m different phases. We obtain the steady state queue size distribution in explicit and closed form in terms of the probability generating functions for the number of customers in the queue. Various performance indices viz. average number of customers in queue and system, long run probabilities of server being in different states, availability, mean time to failure etc. Some special cases of interest are discussed by setting appropriate parameters. To validate the analytical results, numerical results are obtained by taking an illustration.

  Key-Words / Index Term
  Queue, Bulk, Essential service, Optional service, Supplementary variable, Server break down, Queue length

  References
  [1]. Aissani, A. and Artalejo, J. R., (1998). On the single server retrial queue subject to breakdowns, Queueing Systems, Vol. 30, pp.309-321.
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  [3]. Baba, Y.(1986), On the queue with vacation times. Operational research letters, vol.5,93-98.
  [4]. Borthakur, A. and Choudhury, G. (1997). On a batch arrival Poisson queue with generalized vacation, Sankhya, Series B, 59, 369-383.
  [5]. Chakravarthy, S.R. (2016). Queueing modelwith optional cooperative services, European Journal of Operational Research, 248(3), 997-1008.
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  [7]. Choudhury, G. (1998). On a batch arrival Poisson queue with a random set up time and a vacation period. Computers Oper. Res., Vol.25,pp. 1013-1026.
  [8]. Choudhury, G. and Tadj, L. (2009): An queue with two phases of service subject to the server breakdown and delayed repair, Applied Mathematical Modelling, 33, 2699-2709.
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  [11]. Huang, C.,Lee, W.C., Ke, J.C. and Liu, T.H. (2014). Optimization analysis of an unreliable multi-server queue with controllable repair policy, Computers & Operations Research, Vol.49(sept) ,P.P.83-96.
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  [14]. Ke,J.C. ,(2006) On system under NT policies with breakdowns, startup and closedown, Appl. Mathemat. Model. 30 pp. 49–66.
  [15]. Ke,J.C. ,(2006) An queue under hysteretic vacation policy with an early startup and unreliable server, Mathemat. Meth. Oper. Res.Vol. 63 (2)pp. 357–369.
  [16]. Ke, J. C.,(2007) Batch arrival queues under vacation policies with server breakdowns and startup/closedown times, Applied Mathematical Modelling, Vol. 31, No. 7, pp.1282-1292.
  [17]. Kuo, C.C. and Ke, J.C.(2016). Comparative analysis of standby systems with unreliable server and switching failure, Reliability Engineering & System Safety, Volume 145, January 2016, Pages 74-82.
  [18]. Madan, K. C.,(2000) An queue with second optional service, Queueing Systems, Vol.34, pp.37-46.
  [19]. Madan,K.C.,(2003) An queue with time homogeneous breakdown and deterministic repair times, Soochow J. Mathemat.Vol. 29 pp.103–110.
  [20]. Maraghi,F.A.,Madan,K.C and Dowman,K.D(2009),Batch arrival queueing system
  with random breakdowns and bernoulli Schedule server vacations having general vacation time distribution, International journal of information and management sciences,Vol.20,pp.55-70.
  [21]. Medhi, J. (2002): A single server poisson arrival queue with a second optional channel, Queueing Systems, 42, 239–242.
  [22]. Rajadurai,P.,Saravanarajan,M. and Chandrasekaran, V. (2014). Analysis of an retrial queueing system with balking, optional re-service under modified vacation policy and service interruption. Ain Shams Engineering Journal, 5(3),935-950.
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  Citation
  Ashu Vij, "Bulk queue model with two phase’s service, unreliable server and m phase of repair," International Journal of Scientific Research in Mathematical and Statistical Sciences, Vol.5, Issue.3, pp.111-124, 2018

• Open Access   Article

  Indicatrix Given by a Square Change of (α, β)-metric

  Mohammad Rafee, Avdhesh Kumar, G.C. Chaubey

  Research Paper | Isroset-Journal (IJSRMSS)

  Vol.5 , Issue.3 , pp.125-128, Jun-2018

  CrossRef-DOI:   https://doi.org/10.26438/ijsrmss/v5i3.125128

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  Abstract
  Characteristic properties of indicatrix has been studied by so many authors in different situations. In 1984, C. Shibata introduced the concept of β-change in Finsler geometry. Using the theory of β-change, we study the behavior of indicatrix given by square change of famous Z. Shen’s square metric.

  Key-Words / Index Term
  Indicatrix, β-change, Square change, conformal change, Square metric

  References
  [1] S.H. Abed, “Conformal β-change in Finsler space”, arXiv:math602404v2[math.DG] 22Feb. 2006.
  [2] S.I. Amari, H. Nagaoka, “Methods of information geometry”, AMS Transaction of Math. Monographs, 191, USA, 2000.
  [3] P.L. Antonelli, R.S. Ingarden and M. Matsumoto, “ The Theory of Sprays and Finsler spaces with Applications to Physics and Biology”, Kluwer Academic Publishers, Netherland, 1993.
  [4] M. Hashiguchi, “On the hv-curvature tensors of Finsler spaces”, Rep. Fac. Sci. Kagoshima Vol. 4, 1-5, 1971.
  [5] S.W. Ikada, “On some properties of Finsler spaces based on the indicatrices”, Pub. Math. Debreen, Vol. 28, 129-136, 1981.
  [6] F. Ikada, “On S3- and S4-like Finsler spaces with the T-tensor of a special form”, Tensor N.S., Vol. 35, pp.345-351, 1981.
  [7] R. Ingarden, “Geometry of Thermodynamics”, Diff. Geo. Methods in Theor. Phy., World Scintific, Singapore, 1987.
  [8] M. Kityama, “Indicatrices of Randers change”, Tensor N.S., Vol. 69, pp.121-126, 2008.
  [9] V. Kropina, “On projective Finsler spaces with certain special form”, Nauch Doklody vyss, Vol. 2, pp.38-42, 1959.
  [10] M.Matsumoto, “On the Finsler spaces with curvature tensor of special forms”, Tensor N.S., Vol. 22, pp.201-204, 1971.
  [11] M. Matsumoto, “On the indicatrices of a Finsler space”, Period. Math. Hungarica, Vol. 8, pp.185-191, 1977.
  [12] M. Matsumoto,”Foundations of Finsler geometry and special Finsler spaces”, Otsu, Saikawa, Japan: Kaiseisha Press, 1986.
  [13] M. Matsumoto, “Theory of Finsler space with with (α,β)-metric”, Rep. on Math. Phy., Vol. 31, pp.43-88, 1992.
  [14] G. Shanker,”Characteristic properties of indicatrix under a Kropina change of Finsler metric” Preprint, 2017.
  [15] Z. Shen,”Lectures on Finsler geometry” World Scientific, USA, 2001.
  [16] C. Shibata,”On invariant tensors of β-change of Finsler metric” J. Math. Kyoto Univ., Vol. 24, pp.163-188, 1984.
  [17] M.Yoshida,”On anR3-like Finsler spaces and its special cases”, Tensor N.S., Vol. 34, pp.157-166, 1980.
  

  Citation
  Mohammad Rafee, Avdhesh Kumar, G.C. Chaubey, "Indicatrix Given by a Square Change of (α, β)-metric," International Journal of Scientific Research in Mathematical and Statistical Sciences, Vol.5, Issue.3, pp.125-128, 2018

• Open Access   Article

  Efficient Ratio-type Exponential Estimator for Population Variance

  K.B. Panda, M. Sen and P. Das

  Research Paper | Isroset-Journal (IJSRMSS)

  Vol.5 , Issue.3 , pp.129-132, Jun-2018

  CrossRef-DOI:   https://doi.org/10.26438/ijsrmss/v5i3.129132

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  Abstract
  In this paper, a new exponential ratio type estimator has been proposed for estimating the population variance using auxiliary information. To the first order of approximation, i.e., to o(n-1), the expressions for the bias and the mean square error of the proposed exponential ratio-type estimator have been derived. The optimum value of the characterizing scalar, which minimizes the MSE of proposed estimator, has been obtained. With this optimum value, the expression for minimum MSE of the proposed estimator has been arrived at. The proposed estimator has been compared theoretically with sample variance, traditional ratio estimator due to Isaki [1], and exponential ratio- type estimator due to Singh et.al.[3] and it is found that, under practical conditions, the proposed estimator fares better than its competing estimators. An empirical investigation has been carried out to demonstrate the efficiency of the proposed estimator.

  Key-Words / Index Term
  Auxiliary variable, single-phase sampling, mean square error, bias

  References
  [1] Isaki, C.T., ‘Variance estimation using auxiliary information’, Journal of the American Statistical Association 78, pp-117-123, 1983.
  [2] Panda, K.B and Sen., M (2017).Efficient Ratio-Type and Product-Type exponential estimators, Int.J.Agricult.Stat.Sci. Vol.13, No.2, pp-639-644, 2017.
  [3] Singh, R., Chauhan, P., Sawan,N. & Smarandache,F.,’Improved exponential estimator for population variance using two auxiliary variables’, Italian Journal of pure ans Applied Mathematics 28,pp-101-108, 2011.
  [4] Sukhatme, P.V. and Sukhatme, B.V. Sampling Theory of Surveys with Applications. Iowa State University Press, Ames., 1970.
  

  Citation
  K.B. Panda, M. Sen and P. Das, "Efficient Ratio-type Exponential Estimator for Population Variance," International Journal of Scientific Research in Mathematical and Statistical Sciences, Vol.5, Issue.3, pp.129-132, 2018

• Open Access   Article

  Finite to Infinite - An Expository on Nonstandard Analysis

  Alagu. S, R. Kala

  Review Paper | Isroset-Journal (IJSRMSS)

  Vol.5 , Issue.3 , pp.133-135, Jun-2018

  CrossRef-DOI:   https://doi.org/10.26438/ijsrmss/v5i3.133135

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  Abstract
  Nonstandard analysis is a branch of Mathematics introduced by Abraham Robinson[1] in 1966. In 1977, Edward Nelson[2] gave an axiomatic approach to Non-standard analysis. In many instances, analysis on infinite sets can be reduced to a finiteness argument using Nonstandard methods. In this expository article, we present an introduction to the theory and indicate an application to infinite graphs. The application is to a result by De Bruijn and Paul R. Erdos that it is enough to study colouring problem only in finite graphs.

  Key-Words / Index Term
  Nonstandard analysis, Graphs, Colouring

  References
  [1] Abraham Robinson, Nonstandard Analysis, North Holland Publishing Company, 1966.
  [2] Edward Nelson, Internal Set theory, a new approach to NSA, Bull. Amer. Math. Soc.,83(1977) pp.1165-1198.
  [3] Alain Robert, Nonstandard Analysis, John Wiley and sons, 1985.
  [4] A.E.Hurd and P.A.Loeb, An Introduction to Non Standard Real Analysis, Academic Press (1985).
  

  Citation
  Alagu. S, R. Kala, "Finite to Infinite - An Expository on Nonstandard Analysis," International Journal of Scientific Research in Mathematical and Statistical Sciences, Vol.5, Issue.3, pp.133-135, 2018