iosr-Jm   Volume-9 ~ Issue-6

Abstract: An infinite slab subject to temperature variation is analyzed, the problem is formulated using conformal mapping and solved by mellin transform and method of residue. A closed form solution for the temperature distribution is obtained .A detailed verification of the solution is carried out and find to satisfying the Laplace equation.

[1] I.N Sneddon : The use of integral transforms, McGraw-Hill,new York,1972.
[2] G.N.Emenogu and J.N.Nnadi: Analysis of elastic wedge under out-of-plane stress volume 66,number 1, pp117-125(2012)
[3] Z. Szmydt and B.Ziemian, The Mellin transform and fuchsia type partial differential equations, mathematics and its applications (East Europe Series)56,kluwer Academic publishers Group, Dordrech,1992
[4] J.N.Nnadi: Anti plane shear analysis for a Non-homogeneous semi-infinite layer. Journal of the Nigerian Association of mathematical physis,volume7,p215(2003)

Abstract: An elastic homogeneous isotropic material with a right line inhomogeneity embedded in the material under Anti-shear is analyzed; the mathematical model of the problem is a boundary value problem formulated using the mellin transform and solved by the Wiener-Hoph Techniques. A closed form solution for displacement is obtained from which the stress intensity factor is calculated. The stress field were found to have square-root singularity at the inner tip. As a result of this, micro-cracking can initiate at the inner tip of the line inhomogeneity in the matrix depending on the applied loads. The outer tip showed no singularly.

[1] Z. M Xiao and H. Fan, Micro Crack initiation at tip of a rigid inhomogeneity. Journal of fracture Vol. 83: Pp 1-9, 1996
[2] J. Dundurs and X. Markenscoff: A Green's function formulation of anti-cracks and their interaction with load induced singularities. ASME journal of Applied Mechanics Vol. 56, Pp. 550-555, 1989
[3] H. Tada, P.C. Paris and G. R. Trwin. Stress intensity hand book. Del. Research corporation. Heller town pennsyvania 1993.
[4] P. C Paris and G. C. Sil, Stress analysis of cracks. Symposium on fracture Toughness, Testing and its application. ASTM Special Technical publication 381; Pp. 30-83; 1965
[5] M.G. Arfken: Mathematical methods for physicists Academic press Inc. New York. 1966.
[6] R. V. Churchill: Complex variables and applications 2nd edition, McGraw-Hill, New York 1960.

Abstract:Finite difference solution of partial differential equations must satisfy the requirement of convergence and stability if they are to be reasonably accurate. In this paper, we examined convergence and stability criterion of finite difference scheme for solving a partial differential equation. A case which is possible when the cumulative effect of all rounding errors is negligible. Consequently, we investigated the propagation of these errors as increases by applying the Fourier series method.

[1]. Smith, G.D; Numerical solution of Partial Differential Equations. Oxford University Press (1965), page 70
[2]. Greenspan. D and Casulli, V.; Numerical Analysis for applied mathematics, Science and Engineering, (1988) Oxford University Press.
[3]. James W. Brown, Ruel V. Churchill: Fourier series and Boundary Value problems. International Students Edition, McGraw-Hill Book company Inc. (1941).
[4] Murray R. Spiegel; Theory and problems of complex variables: Schaum's outline series, McGraw Hill Book company Singapore (1981).

Abstract: In the present paper, soft -open and soft -closed sets in soft topological spaces are defined over an initial universe with a set of parameters. A necessary and sufficient condition for a soft set to be soft -open set in soft topological space is stated and proved. A detailed study is carried out on properties of soft -interier and soft -closure of soft sets.

Keywords: Soft -open sets, soft -closed sets, soft -interior, soft -closure.

[1] N. Levine, Semi-open sets and semi-continuity in topological spaces, Amer. Math. Monthly, Vol. 70 (1963), 36-41.
[2] J. Mahanta and P. K. Das, On soft topological space via semi-open and semi-closed soft sets, arXiv [math.GN.], Vol. 1(2012), 1-9.
[3] D. Molodtsov, Soft set theory-First results, Comput. Math.Appli., Vol. 37 (1999), 19-31.
[4] O. Njastad, On some classes of nearly open sets, Pacific Journal of Mathematics, Vol. 15, No. 3, (1965), 961-970.
[5] M. Shabir and M. Naz, On soft topological spaces, Comput. Math.Appli., Vol. 61 (2011), 1786-1799.
[6] I. Zorlutuna, M. Akdag, W. K. Min and S. Atmaca, Remarks on soft topological spaces, Annals of Fuzzy Mathematics and Informatics, Vol. 3, No. 2 (2012), 171-185.

Abstract:Bihermitian matrices are studied as a generalization of hermitian matrices. Some of the properties of hermitian matrices are extended to bihermitian matrices. Some important results of hermitian matrices are generalized to bihermitian matrices.
Keywords: Hermitian matrix, skew-hermitian matrix, bimatrix, bihermitian matrix, skew bihermitian matrix.

[1] Frank Ayres. JR, Theory and Problems of Matrices, Schaum's outline series, SI (Metric Ed.),P.No.10-19
[2] Richard Bronson, Matrix Methods: An Introduction(II Ed.),P.No.422-427
[3] W.B.Vasantha Kandasamy,Florentin Samarandache,K.Ilanthendral,Introduction to bimatrices,2005.
[4] W.B.Vasantha Kandasamy,Florentin Samarandache,K.Ilanthendral ,Applications of bimatrices to some Fuzzy and Neutrosophic
models,Hexis.Phoenix,Arizone,2005.

Abstract:This article considers a Markov -modulated risk process with stochastic premium income and a constant dividend barrier. We derive the integro-differential equations satisfied by the expected discounted penalty function regulated by an external environment. Applications of the integral equations are given to be the discounted expectation of the deficit at ruin. Explicit solution of the expected deficit at ruin for the model is obtained for exponential distribution. Finally in two state model, numerical example illustrates the effect of the different parameters.

Keywords: Markov-modulated risk model; Constant dividend barrier; Stochastic income; Gerber-Shiu function; Deficit at ruin; integro-differential equation.

[1] J. M. Reinhard, "On a class of semi-Markov risk models obtained as classical risk models in a Markovian environment," Astin Bulletin, vol. 14, no. 1, pp. 23–43, 1984.
[2] S. Asmussen, "Risk theory in a Markovian environment‟‟, Scandinavian Acturial Journal, vol. 1989, no. 2,pp.69- 100,1989.
[3] Cai, J., Dickson, D.C.M. On the expected discounted penalty function at ruin of a surplus process with interest. Insurances: Mathematics and Economics, 30: 389–404 (2002).
[4] X. S. Lin, G. E. Willmot, and S. Drekic, "The classical risk model with a constant dividend barrier: analysis of the Gerber-Shiu discounted penalty function," Insurance: Mathematics & Economics, vol. 33, no. 3, pp. 551–566, 2003.
[5] H. U. Gerber, X.S.Lin, H.Yang, " A note on the dividends penalty identity and the optimal dividend barrier," Astin Bulletin, (2006)
[6] Landriault, D. Constant dividend barrier in a risk model with interclaim-dependent claim sizes. Insurance: Mathematics and Economics, 16: 31–38 (2007)
[7] D. J. Yao, R. M.Wang, and L. Xu, "On the expected discounted penalty function associated with the time of ruin for a risk model with random income," Chinese Journal of Applied Probability and Statistics, vol. 24, no. 3, pp. 319– 326, 2008.
[8] Hu Yang, Yuan-yuan Hao," A Ruin Model with Random Income and Dependence between Claim Sizes and Claim Intervals ," Acta Mathematicae Applicatae Sinica, English Series, Vol. 26, No. 4 (2010) 625–632.
[9] Hua Dong, Xiang Hua Zao, Zai Ming Liu. "Risk Process With Barrier And Random Income". Applied Mathematics E-Notes, 10(2010), 191-198.
[10] Liu Juan, Xu Jianceng, u Yiun. On the Expected discounted penalty function in a markov-dependent risk model with constant dividend barrier. Acta Mathematicae Scientia 2010, 30B950: 1481 1491.

Abstract: In this paper, we constructed a control operator, G, which enables a Conjugate Gradient Method (CGM) to be employed in solving continuous time linear regulator problems with delay parameter in the state variable. The control operator takes care of any of Mayer's, Lagrange's and Bolza's cost form of linear regulator problems. It is the desire of the authors of this paper that the application of this control operator will further improve on the result of the Conjugate Gradient Method in solving this class of optimal control problem.We consider linear (and quadratic) possibilistic programs and show that the possibility distribution of their objective function remains stable under small changes in the membership functions of the fuzzy number coefficients. We generalize Kovacs's results to PLES with(Lipschitzian) fuzzy numbers and flexible linear programs.
Key words: Linear Equality System, Fuzzy Numbers, Fuzzy set of feasible solutions, Zadeh's extension principle, Kovacs's results.

[1]. E.B.Ammar and M.A.El-HadyKassem, on stability anslysis of multicriteria L.P Problems with fuzzy parameters, Fezzy Sets and
Syatems, 1999 PP.331-334.
[2]. J.F.Brule,Fuzzy Systems – a tutorial, 1985.
[3]. J.J.Buckley, solving possibilistic linear programming problems, Fezzy Sets and Syatems, 1989, PP. 329-341.
[4]. E. Canestrelli, S. Giove and R.Fuller, sensitivity analysis in possibilistic quadratic programming, Fezzy Sets and Syatems, 1996,
PP.51-56.
[5]. C.Carlsson and R. Fuller, optimization under fuzzy if-then rule, Fezzy Sets and Syatems, 1999.
[6]. P.Czyzak and R. Slowinski, possibilistic construction of fuzzy outranking relation for multiplecriteria ranking, Fezzy Sets and
Syatems, 1996, PP.123-131.

Abstract: In this paper, some types of Irregular intuitionistic fuzzy graphs and properties of neighbourly irregular ,highly irregular intuitionistic fuzzy graphs are studied. Some results on totally Irregular intuitionistic fuzzy graphs are established.
Key words: Intuitionistic fuzzy graph, degree, total degree, Intuitionistic fuzzy sub graph.

[1]. Atanassov KT. Intuitionistic fuzzy sets: theory and applications. Physica, New York, 1999.
[2]. Harary,F., Graph Theory, Addition Wesley, Third Printing, October 1972.
[3]. NagoorGani. A and Latha .S.R., On Irregular Fuzzy Graphs, Applied Mathematical Sciences, Vol.6, 2012, no.11,517-523.
[4]. Nagoor Gani. A and Shajitha Begum.S, Degree, Order and Size in Intuitionistic Fuzzy Graphs, International Journal of Algorithms, Computing and Mathematics,(3)3 (2010).
[5]. Parvathi, R. and Karunambigai, M.G., Intuitionistic Fuzzy Graphs, Computational Intelligence, Theory and applications, International Conference in Germany, Sept 18 -20, 2006.
[6]. Zimmermann, H.J., Fuzzy Set Theory and its Applications, Kluwer-Nijhoff,Boston, 1985.

Abstract: This paper, which is based on the experiment conducted by Ministry of Agriculture and Food Security, State Government of Osun within 2012 planting season, investigates the best variety of cowpea in each of the three senatorial zones of Osun State with a view to recommending variety of cowpea that will produce optimum yields. Four common varieties of cowpea: IAR341, IT8D994, IAR1696 and LOCAL were considered in the experiment with one farmland each selected from individual zone as block. Randomized Complete Block Design (RCBD) with the assumption of random effect model was used in collecting the data sets on grain yields of cowpea varieties. Observations were taken from Iwo, Ila and Esa-Oke Farmlands representing Osun West, Osun Central and Osun East Senatorial Zones. Bartlett's statistic was used to test for the validity of homogeneity of error variances for the data sets obtained from each zone. The test revealed that the assumption of equality of error variances was not violated. Shapiro-Wilk test was employed to check for validity of normality assumption on the data sets in each zone and, it was confirmed that departure from normality was not established.
Key words: ANOVA, Bartlett's statistic, Box Plots, DMRT, RCBD, Shapiro-Wilk test, Varieties of Cowpea.

[1] Food and Agriculture Organization (FAO). 'Agriculture Food and Nutrition for Africa-A Reserve Book for Teachers of Agriculture.' Food and Nutrition Division Chapter 4. Rome: FAO, 2012.
[2] International Institute of Tropical Agriculture (IITA), Ibadan, Oyo State, Nigeria. Cowpea Crop – IITA, August, 2013 (www.iita.org.cowpea).
[3] Boys, Kathryn. Adoption and Economic Impact Implications of Storage Technology and Improved Cowpea Varieties in the North Central Peanut Basin of Senegal. Master of Science Thesis. West Lafayette, IN: Department of Agricultural Economics, Purdue University, 2005.
[4] Food and Agriculture Organization (FAO). Core Production Data for Statistical Analysis 'http://faostat.fao.org/site/340/default.aspx (accessed June 25, 2011).
[5] Douglas C. Montgomery (1997): Design and Analysis of Experiment, 5th Edition, John Willey and Sons Inc.
[6] International Institute of Tropical Agriculture (IITA), Ibadan, Oyo State, Nigeria. Cowpea Crop – IITA, August, 2013 (www.iita.org.cowpea).
[7] Fulton, Joan. Trip Report: June 27 – July 22, 2006, Bean/Cowpea CRSP. West Lafayette, IN: Purdue University, 2006.
[8] Singh, B. B., D.R. Mohan Raj, K.E. Dashiell, and L.E.N. Jackai, ens. Advances in Cowpea Research, Ibadan, Nigeria: International Institute of Tropical Agriculture and Japan International Research Centre for Agriculture and Sciences, 1997.
[9] Ogunleye Timothy A., Olawuwo Simeon and Olaleye M. Olugbenga. 'Economic Analysis and Modelling of Local Government Expenditure on Income in the South West Zone of Nigeria' (www.arpapress.com/Vol. 17, Issue 1) October, 2013.
[10] Development Core Team (2012): R Development Core Team – 2006, R: A Language and Environment for Statistical Computing, Vienna, Austria, R Foundation for Statistical Computing.

Abstract: This study compares results obtained from the application of Classical Least Squares with that obtained from the two major biased estimation methods: Ridge and Principal Components Regressions in multicollinear situations using gynecological data from University College Hospital, Ibadan, Oyo State, Nigeria. Numerical values of baby's weights (less than 2.5kg) at birth were considered as response variable while mother's age, weight and height, as well as preterm delivery, multiple pregnancies, parity, graphidity, maternal infections such as malaria, tuberculosis, sexually transmitted diseases, anaemia/shortage of blood, intra-uterine infections, congenital abnormalities, etc and fetal infections serve as explanatory variables. Regression method is used as the statistical tool. A number of assumptions of the regression analysis were inspected. Normality assumption was confirmed by plotting Normal Q-Q Plot and Histogram of the Standardized Residuals.
Key words: Classical Least Squares, Ridge and Principal Component Regressions, Normal Q-Q Plot, Fligner-Killeen and Farrar-Glauber Tests, Durbin-Watson statistic and Shrinkage estimator.

[1]. Berghella V. Prevention of Recurrent Fetal Growth Restriction. Obs and Gyn 2007; 110(4): 904-112
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applications to the outer continental shelf bidding data, Technometrics (23): 351-361
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362(9398): 1779-1784.
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Abstract: In the present paper, we investigate the existence of solutions of neutral integrodifferential equations with infinite delay. The results are obtained by using Schaefer's fixed point theorem and rely on a priori bounds of solution and the inequality established by Pachpatte.
Key words: Existence: Nonlinear Neutral integrodifferential equations: Schaefer's fixed point theorem: Pachpatte's inequality.

[1]. Abderrazak B., Hassane B. and Abdelhakim M., Neutral Volterra Integrodifferential Equations with Infinite Delays, Int. Journal of
Math. Analysis, Vol. 3, 2009, no. 4, pp 187 – 196.
[2]. Balachandran K. and llamaran S., Existence and uniqueness of mild and strong solutions of a semilinear evolution nonlocal
condition. India J. Pure Appl. Math., 1994, 25, pp 411 – 418.
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Pure Appl. Math., 1996, 27(s), pp 443 – 449.
[4]. Balachandran K. and Rajagounder R. K., Existence of Solutions of Integrodifferential Evolution Equations with Time Varying
Delays, Applied Mathematics E-Notes, 7 (2007), pp 1 – 8.
[5]. Eke A. N. and Jackreece P. C., Existence Result for Nonlinear Neutral Integro-diffential Equations with Distributed Delays,
American Journal of Science and Industrial Research, 2011, 2(3), pp 369 - 375.
[6]. Ferenc I., An existence theorem for Volterra Integrodifferential Equations with Infinite Delay, Electronic Journal of Differential
Equations, Vol. 2003, (2003), no. 4, pp 1 – 9.
[7]. Ntouyas, S. K. and Tsamatos Ch., Global existence for semilinear evolution equations with nonlocal conditions, J. Math. Anal.
Appl., 1997, 210, pp 447 – 457.
[8]. Pazy A., Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1933.
[9]. Pachpatte B. G., Inequalities for Differential and Integral Equations, Academic press, 1998.
[10]. Rupali S. J. and Dhakne M. B., On Global Existence of Solutions for Abstract Nonlinear Functional Integrodifferential Equations
with Nonlocal condition, Contemporary Mathematics and Statistics, (2013) 1, pp 44 – 53.

Abstract: Tetrahedron is the only 3-simplex convex polyhedron having four faces, and its shape has a wide application in science and technology. In this article, using graph theory and combinatorics, a study on a special type of tetrahedron called coxeter Andreev's tetrahedron has been facilitated and it has been found that there are exactly one, four and thirty coxeter Andreev's tetrahedrons having respectively two edges of order n  6 , one edge of order n  6 and no edge of order n  6, nN upto symmetry.
Key words: Planar graph, Dihedral angles, Coxeter tetrahedron.

[1] en.wikipedia.org/wiki/simplex, access in October, 2013
[2] en.wikipedia.org/wiki/tetrahedron#Applications access in October, 2013.
[3] John G. Ratcliffe, Foundations of Hyperbolic Manifolds, ©1994 by Springer-Verlag, New York, Inc.
[4] Chris Godsil, Gordon Royle, Algebraic Graph Theory, Springer International Edition.
[5] Gil Kalai, Polytope Skeletons and Paths, ©1997 by CRC Press LLC.
[6] Dipankar Mondal, Introduction to Reflection Groups, April 26, 2013, Triangle Group (Course Project).
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[9] J. Mcleod, Hyperbolic Coxeter Pyramids, Advances in Pure Mathematics, Scientific Research, 2013, 3, 78-82
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Abstract: In this paper, we explore new applications of the exp(( )) -expansion method for finding exact traveling wave solutions of generalized Klein-Gordon Equation and right-handed nc-Burgers equation. By means of this method three new solutions of each equations is obtained including the hyperbolic functions, exponential functions and rational function solutions. The proposed method is very effective, efficient and applicable mathematical tools for nonlinear evolution equations (NLEEs). So this method can be used for many other nonlinear evolution equations.
Keywords: The exp (( )) - expansion method; generalized Klein-Gordon Equation and right-handed nc- Burgers equation; nonlinear partial differential equation; traveling wave solutions.

[1]. M.J. Ablowitz and P. A. Clarkson, Soliton, nonlinear evolution equations and inverse scattering (Cambridge University Press, New
York, 1991).
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HirotaCSatsuma coupled KdV system, Appl. Anal. 85 (2006) 751-768.
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Abstract: In this paper a Mathematical model is proposed and analysed to study a dynamical behaviour of exploited system consisting of two preys and a predator which is being harvested. It has been assumed susceptible and infected prey populations are predated by predated species. A local stability of the model has been carried out. It has been observed that the harvesting activity of the predator taken into the consideration. The population size of the prey decreased and naturally a stable equilibrium model becomes unstable.
Keywords: Prey-predator system, diseased prey, local stability, harvesting effort.

[1]. A Martin and S. Ruan, Prey Predator Models with time delay and prey harvesting, J.Math. Biol 43 (2001), 247-267.
[2]. B. Dube and R.K.Upadhyay, Persistence and extinction of one prey and two predator system, J. Non Linear Anal. Appl : Model
Control 9(4) (2004), 307-329.
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Appl. Sci. 23 (2000), 347-356.
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Abstract: In the literature of information theory several types of coding theorems involving fuzzy entropy functions exists. In this paper, some new fuzzy coding theorems have been obtained involving utilities. The fuzzy coding theorems obtained here are not only new but also generalizes some well known results available in the literature.
Key Words: Fuzzy Set, Fuzzy Entropy Function, Fuzzy Useful Entropy Function, Fuzzy code word length and Fuzzy average useful codeword length.

[1]. Belis, M. and Guiasu, S. [1968]: 'A quantitative and qualitative measure of information in cybernetic system‟, IEEE Transaction on information theory, Vol.IT-14, pp. 593-594.
[2]. Boekee, E. and Van Der Lubbe, J.C.A. [1980]: "The R- Norm Information measure‟‟, Information and Control, Vol. 45, pp. 136-155.
[3]. De Luca and S. Termini [1972]: A Definition of Non-probabilistic Entropy in the Setting of fuzzy sets theory, Information and Control, Vol.20, pp. 301-312.
[4]. Guiasu, S. and Picard,C.F. [1971]: Borne inferieure de la longer de certain codes, C.R. Academic Sciences, Paris,Vol. 273, pp. 248-251
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Abstract: The object of the present paper is to study some properties of W2- curvature tensor in an Lorentzian para-Sasakian manifolds.
Key Words: Lorentzian para-Sasakian manifold, W2-curvature tensor.

[1] D. E. Blair, Contact manifolds in Riemannian geometry, Lecture Notes in Mathematics, 509 Springer- Verlag, Berlin, 1976.
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[5] K. Matsumoto, On Lorentzian Paracontact manifolds, Bulletin of the Yamagata University. Natural Science, 12(2) (1989), 151-156.

Abstract: In this paper, we define the concept of a bipolar fuzzy cosets of a bipolar fuzzy HX subgroup, bipolar anti fuzzy HX subgroup, bipolar fuzzy middle coset of a bipolar fuzzy HX subgroup and bipolar anti fuzzy HX subgroup, conjugate bipolar fuzzy HX subgroup, conjugate bipolar anti fuzzy HX subgroup and discussed some of its properties with the examples. Further we define the level subsets and lower level subsets of bipolar fuzzy cosets of a bipolar fuzzy and bipolar anti fuzzy HX subgroup and discussed some of its properties.
Keywords: bipolar fuzzy HX subgroup, bipolar anti fuzzy HX subgroup, bipolar fuzzy cosets of a bipolar fuzzy HX subgroup, bipolar fuzzy middle cosets, conjugate bipolar fuzzy HX subgroup .

[1]. R. Biswas, Fuzzy subgroups and anti-fuzzy subgroups, Fuzzy Sets and Systems , 35(1990), 121-124.
[2]. V.N. Dixit ,A. Rajesh Kumar, Naseem Ajmal, Level subgroups and union of Fuzzy Subgroups , Fuzzy Sets and Systems, 37, 359 - 371 (1990).
[3]. Li Hongxing, HX group, BUSEFAL, 33 October 1987, 31 – 37.
[4]. Luo Chengzhong , Mi Honghai , Li Hongxing, Fuzzy HX group, BUSEFAL , 41 – 14, Oct 1989. 97 – 106.
[5]. M. Marudai, V. Rajendran , New Constructions on Bipolar Anti Q-fuzzy Groups and Bipolar Anti Q-fuzzy d-ideals under (t,s) norms, Advances in Fuzzy Mathematics , Volume 6, Number 1 (2011), pp .145 – 153.

Abstract: In this paper, we define a new algebraic structure of an induced fuzzy k-ideal of a HX ring and some related properties are investigated. The purpose of this study is to implement the fuzzy set theory and ring theory in fuzzy HX k-ideals of a HX ring. We also establish the relation between fuzzy HX ideals and fuzzy HX k-ideals of a HX ring . Characterizations of level subsets of a fuzzy HX k-ideal of a HX ring are given. We also discussed the relation between a given fuzzy HX k-ideals of a HX ring and its level HX k-ideals and investigate the conditions under which a given HX ring has a properly inclusive chain of HX k-ideals. In particular, we formulate how to structure a fuzzy HX k-ideal of a HX ring by a given chain of HX k-ideals.
Keywords: fuzzy ideal, fuzzy k-ideal, fuzzy HX ring , fuzzy HX ideal, fuzzy HX k-ideal, level HX k-ideals.

[1] Bing-Xueyao And Yubin-Zhong, The Construction Of Power Ring, Fuzzy Information And Engineering (ICFIE),ASC 40,Pp.181-187,2007.
[2] Bing-Xueyao And Yubin-Zhong, Upgrade Of Algebraic Structure Of Ring, Fuzzy Information And Engineering (2009)2:219-228.
[3] Dheena.P And Mohanraaj.G, T- Fuzzy Ideals In Rings ,International Journal Of Computational Cognition, Volume 9, No.2, 98-101, June 2011 .
[4] Dheena.P And Mohanraaj.G, On Intuitionistic Fuzzy K Ideals Of Semirings , International Journal Of Computational Cognition, Volume 9, No.2, 45-50, June 2011
[5] Dutta T. K. And Biswas . B.K., Fuzzy K-Ideals Of Semirings,Bull, Culcutta Math.Soc.,87,91-96 .