iosr-Jm   Volume-10 ~ Issue-1 ~ Jan-Feb 2014

Abstract: This article investigates the strength and limitation of t-test and Wilcoxon Sign Rank test procedures on paired samples from related population. These tests are conducted under different scenario whether or not the basic parametric assumptions are met for different sample sizes. Hypothesis testing on equality of means require assumptions to be made about the format of the data to be employed. The test may depend on the assumption that a sample comes from a distribution in a particular family. Since there is doubt about the nature of data, non-parametric tests like Wilcoxon signed rank test and Sign test are employed. Random samples were simulated from normal, gamma, uniform and exponential distributions. The three tests procedures were applied on the simulated data sets at various sample sizes (small, moderate and large) and their Type I error and power of the test were studied in both situations under study.

Keywords: Parametric, t-test, Sign-test, Wilcoxon Sign Rank, Type I error, Power of a test.

[1] L. J. Kazmier, Schaum's Outline of Business Statistics 3rd Edition Mc Graw-Hill New York, 1996.
[2] G. M. Clarke and D. Cooke, The Nature of Parametric and Non-Parametric tests. A Basic Course in Statistics, 4th edn. Arnold, 1998.
[3] P. Sprent, Applied Non-parametric Statistical Methods. 2nd Edition. Chapman and Hall, 1993.
[4] M. Usman and A. I. Maksha, An Efficient Alternative To t-Test For One Sample Nonnormal Data, Journal of Applied Science & technology, Auchi Polytechnic, Nigeria, 2010.
[5] I. Akeyede and S. G. Akinyemi, Power Comparison of Sign and Wilcoxon Sign Rank Test Under Non Normal, African Journal of Physical Sciences, Devon Science Publication, 2010.

Abstract: The object of present paper is to establish an integral pertaining to a product of Fox's H-function, general polynomials given by Srivastava and H-function of several complex variables given by Srivastava and Panda with general arguments of quadratic nature.

[1]. Braaksma, B.L.J., Asymptotic expansions and analytic continuations for a class of Barnes integrals, Compositio Math. 15 (1963),
239-341.
[2]. Chaurasia, V.B.L. and Shekhawat, Ashok Singh, An integral involving general polynomials and the H-function of several complex
variables, Tamkang J. Math. 36(3) (2005), 255-260.
[3]. Chaurasia, V.B.L. and Godika, Anju, An integral involving certain product of special functions, Bull. Cal. Math. Soc. 91 (1999),
337-342.
[4]. Fox, C., The G and H-functions as symmetrical Fourier kernels, Trans. Amer. Math. Soc., 98, 395-429 (1961).
[5]. Goyal, S.P. and Mathur, S.L., On integrals involving H-function of two variables, Indian J. Pure & Appl. Math. 7 (1976), 347-358.
[6]. Gupta, K.C. and Jain, R., An integral involving a general polynomial and product of Fox's H-function having general arguments,
Ganita Sandesh, 3 (1989), 64-67.

Abstract:Squaring of circle is an unsolved problem with the official  value 3.1415926… with the new  value 1/4 (14- 2 ) it is done in this paper.

Keywords: Exact Pi value = 1/4 (14- 2 ), Squaring of circle, Hippocrates squaring of lunes.

.....................,

Abstract: In this work we study how to apply PALEY WIENER theorem to the Fourier transforms of functions with compact support.

[1]. Michael Reed and Barry Simon, Functional analysis, volume I of the series Methods of Modern Mathematical Physics, Academic Press, 1972.
[2]. Michael Reed and Barry Simon, Fourier analysis, selfadjointness,volume II of the series Methods of Modern Mathematical Physics, Academic Press, 1975.
[3]. Laurent Schwartz, Th´eorie des distributions, Hermann,. This is the first edition of the original two volumes in one, 1966.
[4]. Laurent Schwartz, M´ethodesmath´ematiques pour les sciences physiques, Hermann, 1966. Translated into English as Mathematics for the physical sciences.
[5]. Rudin, Walter, Real and complex analysis (3rd ed.), New York: McGraw-Hill, ISBN 978-0-07-054234-1, MR 924157, 1987.

Abstract: In this paper, we introduce the notions of Irwg-continuous maps and Irwg-irresolute maps in ideal topological spaces. We investigate some of their properties.

Key words: Irwg- closed set, Irwg-continuous maps, Irwg-irresoluteness.

[1] T. R. Hamlett and D. Jankovic, Compactness with respect to an ideal, Boll. Un. Mat. Ita.,(7), 4-B, 849-861. 1990.
[2] E. Hayashi. Topologies defined by local properties. Math.Ann., 1964, 156: 205-215
[3] T. R. Hamlett and D. Jankovic, Ideals in topological spaces and the set operator, Boll.Un. Mat. Ita, 7, 863-874. 1990.
[4] T. R. Hamlett and D. Jankovic, Ideals in General Topology and Applications (Midletown,CT, 1988), 115-125, Lecture Notes in Pure and Appl. Math. Dekker, New York, 1990.

Abstract: We discuss the effect of radiation on unsteady MHD free connective flow through a porous medium in a vertical channel .The unsteadiness in the flow is due to the traveling thermal wave imposed on the wall y = L . A uniform magnetic field of strength Ho is applied normal to the boundaries. The coupled equations governing the flow and heat transfer have been solved by using a perturbation technique with the aspect ratio as perturbation parameter. The expression for the velocity, the temperature, the shear stress and the rate of heat transfer are derived and are analysed for different variations of the governing parameters G,R,M, and 

Key Words:-Dissipative fluid, Porous medium, Radiation

[1]. Brinkman,H.L :A calculation of the viscous force exerted by a flowing on a dense swam of particles.,Appl.Sci.Res,p.82,(1948)
[2]. Chawla,S.S and Singh,S : Oscillatory flow past a porous bed,Acta Mech.,V.34,pp.205-213,(1979)
[3]. Fand,R.M and Brucker,J : Int.J.Heat and Mass Transfer,pp.723-732,(1996)
[4]. Raptis,A,A,Paredikis,C and Zivanidis,G : J.Phys.D :Appl.Phys,V.14,pp.99-102,(1981)
[5]. Ravindra,M: Mhd convection flow through porous medium with non-uniform wall temperature.,Ph.D thesis, S.K.University, Anantapur, India,(1994)

Abstract: The purpose of this article is to obtain common fixed point theorems of multivalued operators in generalized metric spaces. Keywords: multivalued mapping, fixed point, common fixed point, generalized metric space. AMS Subject Classification (2000): 47H04, 47H10, 54H25.

[1] Alina Sintamarian , Common Fixed Point Theorems For Multivalued Mappings, Seminar on Fixed Point Theory Cluj-Napoca,
Volume 1, 2000, 93-102.
[2] A. Latif, I. Beg, Geometric fixed points for single and multivalued mappings, Demonstratio Math. 30 (4), 1997, 791-800.
[3] B E Rhoades, A Fixed Point Theorem For Generalized Metric Spaces, Internat. J. Math. & Math. Sci. Vol 19 No.3 (1996), 457-460.
[4] Dhage B C, Genaralized Metric Space and Mapping with Fixed Point Bull. Cal. Math. Soc. 84, (1992), 329-336.
[5] Dhage B C, Generalized Metric Spaces and Topological Structure I, An. Stiint. Univ. Al.I. Cuza Iasi. Mat(N.S), 46, (2000), 3-24.
[6] Dhage B C, On Generalized Metric Spaces and Topological Structure II, Pure. Appl. Math. Sci. 40, 1994, 37-41.
[7] Mustafa. Z and Sims. B, Some Remarks Concerning D- Metric Spaces, Proceedings of the International Conference on Fixed Point
Theory and Applications, Valencia (Spain), July (2003), 189-198.

Abstract: The study was done to probe the determinants of financial leverage of non-financial firms of service sector, listed on Karachi Stock Exchange, Pakistan for the period of 10 years form 2003-13. The sample of 57 companies was taken from listed non-financial firms on KSE. The sample was equally distributed and well organized to cover all service sectors of population. The regression model was applied on data while taking financial leverage as an independent variable. The study results and conclusions show that there exist a significant relationship.

Keywords: Financial leverage, KSE, Service Sector, Determinants

[1]. Agarwal, A., & Nagarajan,J (1990). Corporate Capital Structure: The case of equity firms: The journal of Finance, 2(4), 36-87.
[2]. Akhtar,S., & Oliver,B. (2009). Determinants of capital structure. International review of finance, 3(4), 234-296.
[3]. Allen, D.E. (1993). The pecking order hypothesis. Applied financial Economics, 3(2), 112-139.
[4]. Al-Najjar, B. (2011). The interrelationship between capital Structure and dividend policy. International Review of Applied Economics, 23(3),311-324.
[5]. Bufema, F.M. (2005). Determinants of capital structure. (Vol. 8) University of Liverpool.
[6]. Butters, J.K. (1949). Federal Income taxation and External vs Internal Financing. The Journal of finance, 3(34), 173-195.
[7]. Jensen, M. & Meckling, W. (1976). Managerial behavior, agency costs and ownership structure. Journal of financial economics, 2(4), 205-260.

Abstract: Writer identification problem has been largely studied in the field of image processing and it has found its application in forensic sciences and other related fields.Previous many works have been done in this area.Most of the problems encountered so far were script specefic inthe sense that they were dealing with the script of a perticulat language at a time e.g Engilsh, Spanish , Hindi etc. In this present work a more generalized problem has been addressed which could deal with the image samples written in diffrent scripts. Keywords : A.R Coefficient ,Classification, Image processing,Pattern Matching, Pattern recognition

[1] R. Plamondon and G. Lorette, "Automatic Signature Verification and Writer Identification— The State of the Art," Pattern Recognition, vol. 22 (2), pp. 107- 131, 1989.
[2] M. Bulacu and L. Schomaker, "Text-Independent Writer Identification and Verification Using Textural and Allographic Features," in IEEE Trans. Pattern- Anal. Mach. Intell. Vol. 29(4), pp. 701-717, 2007.
[3] K. Deguchi, "Two-dimensional Auto-regressive Model for Analysis and Sythesis of Gray-level textures," in Proc. Of the 1st Intll. Sym. for Science on Form, General Ed. S. Ishizaka, Eds. Y. Kato, R. Takaki, and J. Toriwaki, pp. 441-449, Tokyo, Japan, 1986.
[4] ITU-T recommendation T.82, "Information Technology-Coded representation of picture and audio information-Progressive bi-level image compression," March 1993.
[5] E. Grosicki, M. Carré, J. Brodin, and E. Geoffrois, "RIMES evaluation campaign for handwritten mail processing,", in Proc. of ICFHR, 2008.

Abstract: equal to 3.1415926… is an approximation. It has ruled the world for 2240 years. There is a necessity to find out the exact value in the place of this approximate value. The following method givesthe total area of the square, and also the total area of the inscribed circle. derived from this area is thus exact.

..................................,

Abstract: In this article, we study the existence of mild solutions for fractional semilinear differential equations with nonlocal conditions in separable Banach spaces. The result is obtained. The result is obtained using the Hausdorff measure of noncompactness and the Schauder fixed point theorem .

Keywords: Fractional differential equation , nonlocal conditions, Housdorff measure of noncompactness, mild solution.

[1] J. Banas, K.Goebel , Measure of Noncompactness in Banach spaces, Lect. Notes Pure Appl. Math., vol.60, Marcel Dekker, New York, 1980.
[2] E. Bazhlekova, Fractional Evolution Equations in Banach spaces, Ph.D. Thesis, Eindohoven University of Technology, 2001.
[3] D.Bothe, Multivalued perturbations of m-accretive differential inclusions, Israel J. Math. 108, 1998, 109-138.
[4] L.Byszewski, Theorems about the existence and uniqueness of solutions of a semilinear evolution nonlocal Cauchy problem, J.Math. Anal. Appl. 162, 1991, 494-505.
[5] Z. Fan, G. Li, Existence results for semiliinear differential equations with nonlocal and impulsive conditions, J.Funct. Anal. 258, 2010, 1709-1727.
[6] X.Fu, K. Ezzinbi, Existence of solutions of a semilinear functional differential evolution equations with nonlocal conditions, Nonlinear Anal. 54, 2003, 215-227.
[7] M.Hahn, K.Kobayashi, S.Umarov, Fokker–Planck-Kolmogorov equations associated with time changed fractional Brownian motion, Proc.Amer.Math.Soc.13, 2011, 691-705.
[8] D.Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Math., vol.840, Springer-verlag, New York, Berlin, 1981.

Abstract: In this paper we have established a new theorem on | , , | n k A p  -summability which gives some new and interesting results and previous known results as a corollary.

Keywords: | , | n N p -summability, | |k A -summability, | , |k A  -summability, | , , | n k A p  -summability and infinite series. Mathematical classification :40D25, 40E05, 40F05 & 40C10.

[1] BOR, H; On local property of | , , | n k N p  -summability of factored Fourier series, J. Math. Anal. Appl. 179 (1993).
[2] OZARSLAN, H.S.; On absolute matrix summability methods, Mathematical Commun. 12, (2007).
[3] RHOADES, B,E. and SAVAS, E.; On | |k A -summability factors, Alco, Math. Hung, (2006).
[4] SAVAS, E.; A summability factor theorem for absolute summability involving  -quasi -monotone and almost increasing
sequence, Math. Comput. Model, 48, (2008).

Abstract: The uncovering of company and financial difficulties is a theme which has particularly susceptible to financial ratio analysis. Rating and predictive the show of the top ranking companies on the basis of certain financial ratios based on Statistical Data Mining (SDM) techniques. In this paper we used three different Statistical Data Mining methods and they are Factor Analysis, K-means Clustering techniques and Multivariate Disriminant Analaysis.

[1]. Altman E. I (1968), Financial Ratios, Discriminant Analysis and the Prediction of Corporate Bankruptcy, The Journal of Finance, vol.23, pp. 589-609.
[2]. Altman E. I (1983), Multidimensional Graphics and Bankruptcy Prediction: A comment, Journal of Accounting Research, Volume 21, Number 1, U.S.A
[3]. Chandrasekaran R et.al,. (2013), Indian Industrial Position on the basis of Financial Ratios: A Data Mining Approach, International Journal of Statistika nad Mathematika, ISSN: 2277-2793, Volume 7, Issue 3, pp. 54-59, 2013., India.
[4]. Chandrasekaran R et.al,. (2013),Financial Ratios Performance of Major Indian Industries to evaluate their Performances using Multivariate Analyses and Perceptual Mapping, International Journal of Scientific and Engineering Research, Volume 4, Issue 2, ISS No. 2229 – 5518, 2013, Houston, TX77043, USA.
[5]. Mahmoud Ezzamel, Judith and Cecilio (1987), Financial Patterns of UK Manufacturing Companies, Journal of Business Finance and Accounting, 14(4), Winter 1987, pp. 519- 536.
[6]. Richard A Johnson and Dean W. Wichern (1992), Applied Multivariate Statistical Analysis, 3/ed, Prentice-Hall of India Private Limited, New Delhi.

Abstract: On January 1st 2012, Nigerians were greeted with removal of petroleum fuel subsidy which resulted in increase on pump prices of petroleum product. Some arguments have it that it will be good for the economy while other argued that it will not favour the average Nigerian, especially those on grassroots. In January 2014 similar speculation on total removal of subsidy on all petroleum products was still on leading to brief scarcity in some areas. a multivariate study of the implications of these prices change on some economic variables reveals that PMS has significant impact on all economic variable studied.

[1]. Arinze P. E (2011), "The Impact Of Oil Price On The Nigerian Economy", JORIND (9)1 June, ISSN 1596-8303. www.Transcampus.Org/Journals.
[2]. Ghalayini, L (2011), "The Interaction between Oil Price and Economic Growth" Middle Eastern Finance and Economics, ISSN: 1450-2889 Issue 13 Http://Www.Eurojournals.Com/MEFE.Htm
[3]. Onyemaechi J. O, (2012), "Economic Implications Of Petroleum Policies In Nigeria: An Overview", American International Journal Of Contemporary Research Vol. 2 No. 5; May 2012
[4]. Aliyu,S.U. (2009), "Impact Of Oil Price Shock And Exchange Rate Volatility On Economic Growth In Nigeria: An Empirical Investigation", Associate Professor Of Economics At Bayero University Kano Research Department Of The Central Bank Of Nigeria
[5]. Sharma A, Singh, G and Gupta P, et al (2012), "Impact Of Crude Oil Price On Indian Economy", International Journal Of Social Sciences And Inter Disciplinary Studies. Vol. 1(4), ISSN: 277-3630

Abstract: Life table is a key summary tool for assessing and comparing mortality conditions prevailing in populations. It is also a veritable tool for the study of mortality experience in a population. Hence, this paper has extended its application to infant survival pattern, using data on infant mortality collected from University of Nigeria Teaching hospital Enugu for a period of ten years;

[1] Kaliaperumal V.G. , Venkataswamy Reddy. M, Channabasavanna S.M, Shivaji Rao and Joseph D.M. ( 1987)An Application of Life Table methods in the study of length of stay. Indian Journal of Psychiatry, 29(4), October 1987, 325-331
[2] Anousschkar van der meulen (2012) Life tables and Survival Analysis. Statistics Method, 2012. Statistics NetherlandsThe hagues/Heerlern Google Assessed on 8th January,2014.
[3] Wei Zhang (1993) Survival Data Analysis-Life table methods. Synergic Resources Corporation NESUG '93 Proceedings, Google Assessed on 8th January, 2014
[4]. Li, N. and S. Tuljapurkar(2014). The Probabilistic Life Table and Its Applications. Google Assessed on 8th January,2014.
[5] Leonard K. Atuhaire (2011) Application of Current-Status Survival Analysis Methodology to Estimation of Age at First Sex: Uganda. The African Statistical Journal, 12, May 2011.

Abstract: Path Analysis is the statistical technique used to examine causal relationships between two or more variables; the technique was used to develop an exploratory Path model of global warming, using Carbon dioxide (CO2) emission as a surrogate. Some factors which are assumed to have some causal relationship with CO2 emission were considered. The correlation analysis result shows significant relationships between CO2 emission and the selected factors which include; Energy consumption,

[1] WRIGHT, S. (1921).Correlation and causation. Journal of Agricultural Research 20, 557-85.
[2] Duncan, O.D.(1966).Path analysis: sociological examples. American Journal of Sociology 72, 1-16.
[3] Shipley, B. (1997). Exploratory path analysis with applications in ecology and evolution. The American Naturalist. 149 (6),1113-1138.
[4] Ullman. J.B. (1996). Structural equation modeling: Using Multivariate Statistics.3, (B.G. Tabachnick and L.S. Fidell, Edition, HarperCollins College Publishers, New York) 709-819.
[5] United States Environmental Protection Agency, Carbon Dioxide Emission.http://www.epa.gov/climatechange/ghgemissions/gases/co2.html Assessed 14th January,2014.

Abstract: Predator-prey models lie in a theoretical background of integrated plant protection. The crucial question is whether one is able to identify parameters of such a model on a base of field data. The contribution proposes a method of parameter identification for Gause-type predator-prey model. The method is tested on simulated data, in particular, the problem of distinguishing between distinct types of predator functional response (Ivlev and Holling type ones) and of prey growth rate (with limited and unlimited carrying capacity) is tested. Obtained results show that data do not allow one to find such subtilities of the model. Nevertheless, the proposed method is reliable enough for investigation of the system equilibrium and its stability.

[1] Britton, N., F.: Essential Mathematical Biology. Springer, London Berlin Heidelberg, 2003.
[2] Hluchý, M., Pospíšil, Z.: Use of the predatory mite Typhlodromus pyri Scheuten (Acari: Phytoseiidae) for biological protection of grape vines from phytophagous mites. In DUSBÁBEK, F., BUKVA, V. (eds.): Modern Acarology, Vol. 2. Prague: Academia and SPB Academic Publishing bv, The Hague, 1991, pp. 655–660.
[3] McCallum, H.: Population Parameters. Estimation for Ecological Models. Blackwell Science Ltd., Oxford London Edinburgh Malden Carlton Paris, 2000.
[4] Svirezhev, Yu., M., Logofet, D., O.: Stability of biological communities [Russian]. Nauka, Moscow, 1978.

Abstract: Despite the availability of the measles vaccine since 1963, the infectious disease is still endemic in many parts of the world including developing and developed nations. It was eliminated from the U.S. in 2000 but due to importation secondary transmissions has been occurring since 2008. With a greater than 90% attack rate measles is not only a health but also an economic problem as it spread very quick leading to a lot of persons been hospitalized or seeking medical help. The disease is endemic in Nigeria with an annual incidence rate of 18.2 per 100, 000 children and a case fatality rate of 1.2% as at 2008. The disease is one of the leading causes of child mortality worldwide. Its prevalence and severity in Nigeria might hamper the fourth millennium development goal by 2015 (To reduce by two thirds, between 1990 and 2015, the under five mortality rate).

[1] Wikipedia.MMR vaccine controversy.Available fromhttp://en.wikipedia.org/wiki/MMR_vaccine_controversy. Accessed 5th may 2013.
[2] U. Nnebe-Agumadu, Measles Control in Nigeria: The case for two dose vaccine policy. Nigerian Journal of Paediatrics, 32(3) 41 – 45, 2005.
[3] Public Health agency of Canada. Measles.Available from http://www.phac-aspc.gc.ca/im/vpd-mev/measles-rougeole-eng.php.Accessedon 5th may 2013.
[4] World Health organization. Measles fact sheet.Available from http://www.who.int/mediacentre/factsheets/fs286/en/.Accessedon 5th May 2013.
[5] R.F. Grais, M.J. Ferrari, C. Dubray, O.N. Bjørnstad, B.T. Grenfell, A. Djibo, F. Fermon, P.J. Guerin, Estimating transmission intensity for a measles epidemic in Niamey, Niger: lessons for intervention, Royal society of tropical medicine and hygiene, 100, 867 – 873, 2006.

Abstract: This study aims to find the risk factors of Diabetes Mellitus (DM) and to find the best model among Poisson, Negative Binomial (NB) and Zero-Inflated Negative Binomial (ZINB) regression models. In the count data, the existence of overdispersed data is a common situation for modeling approach. Overdispersion occurs when the data has greater value of variance compared to its mean. The Poisson regression model is a good starting step to model the data but does not account for overdispersion.

[1]. A. C. Cameron and P. K. Trivedi, Regression analysis of count data (Cambridge University Press, 1998).
[2]. J. Cohen, P. Cohen, S. G. West, and L. S. Aiken, Applied multiple regression/correlation analysis for the behavioral sciences 3rd ed (Mahwah, NJ: Lawrence Erlbaum Associates, 2003).
[3]. H. C. Chin, and M. A. Quddus, Applying the random effect negative binomial model to examine traffic accident occurrence at
signalized intersections, Accident Analysis and Prevention, 35(2), 2003, 253-259.
[4]. V. N. Shankar, F. Mannering, and W. Barfield, Effect of roadway geometric and environmental factors on rural freeway accident frequencies, Accident Analysis and Prevention, 27(3), 1995, 371-389.
[5]. D. Lambert, Zero-inflated Poisson regression, with an application to defects in manufacturing, Journal of Techonometrics, 34(1), 1992, 1-14.

Abstract: We give some fixed point theorems for two weakly increasing self-mappings T and S satisfying contractive type conditions by using Delbosco's set in order G-metric spaces. 2000 AMS Mathematics subject classification: 47H10, 54H25.

Keywords and Phrases: Delbosco's Set, contraction, weakly increasing mappings, fixed point, ordered Gmetric spaces

[1]. Delbosco's, D. A Unified approach for all contractive mappings, Inst. Math. Univ. Torino, Report No. 19, 1981.
[2]. Cho. Y. J., Fixed points for compatible mappings of type (A), Math. Japonica 38 (3), (1993), 497-508.
[3]. Constantin, A., Common Fixed Points of weakly Commuting mappings in 2-metric spaces, Math. Japonica 36 (3), (1991), 507-514.
[4]. Constantin, A., On Fixed Points in non-composite metric space, publ. Math. Debrecen 40 (3-4) (1992), 297-302.
[5]. E. S. Wolk, "Continuous convergence in partially ordered Sets," General Topology and its Applications, Vol. 5, no. 3, pp. 221-234, 1975.
[6]. B. Monjardet, Metries on partially ordered Sets-a Survej," Dicrete Mathematics, Vol. 35, pp. 173-184, 1981.
[7]. A. C. M. Ran and M. C. B. Recurings, "A fixed point theorems in partially ordered sets and some applications to matrix Mathematical Society, Vol. 132, no. 5, pp. 1435-1443, 2004.

Abstract:This paper is mainly concerned with a generalization of the concept of a normal multi-fuzzy subgroup and a normal multi-anti fuzzy subgroup. We introduce the concept of a normal multi-fuzzy subgroup of a multi-fuzzy subgroup and examine its basic properties. Also we introduce the concept of a normal multi-anti fuzzy subgroup of a multi-anti fuzzy subgroup and examine its basic properties. We use them to develop results concerning multi-fuzzy subgroups of a multi-fuzzy subgroup and multi-anti fuzzy subgroups of a multi-anti fuzzy subgroup. Mathematics Subject Classification: MSC: 20N25; 03E72; 08A72

Keywords: Fuzzy set, multi-fuzzy set, fuzzy subgroup , multi-fuzzy subgroup, anti fuzzy subgroup, multi-anti fuzzy subgroup, normal multi-fuzzy subgroup, normal multi-anti fuzzy subgroup, multi-fuzzy coset.

[1] Das. P.S., Fuzzy groups and level subgroups, Journal of Mathematical Analysis and Applications, 84 (1981), 264-269.
[2] Liu W.J., Fuzzy invariant subgroup and fuzzy ideal, Fuzzy Sets and Systems, 8(1981),133-139.
[3] Mukharjee N.P. and Bhattacharya P., Fuzzy normal subgroups and fuzzy cosets, Information Sciences, 34 (1984), 225-239.
[4] Muthuraj.R and Balamurugan.S, Multi-anti fuzzy group and its Lower level subgroups, Int. Journal of Engineering Research and Applications, Vol. 3, Issue 6, Nov-Dec 2013, pp.1498-1501.
[5] Muthuraj.R, Sitharselvam.P.M and Muthuraman.M.S, Anti Q-fuzzy group and its Lower level subgroups, International Journal of Computer Applications(0975-8887), Volume 3-No. 3,June2010,16-20.
[6] Muthuraj.R and Balamurugan.S, Multi-fuzzy group and its level subgroups, Gen. Math. Notes, Vol. 17,No. 1, July, 2013, pp. 74-81.