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

Abstract:In this paper we study the 3- rings ,Idempotent of 3-ring and some other theorems .In the second section we introduce Ideals on 3-rings,center of 3-rings and theorems, Topological 3-rings and their properties: the set of open neighbourhoods of 0, its properties in topological 3-rings, Every topological 3- ring is a homogeneous algebra and other theorem.

Key words: Hausdorff space , Ring, p-ring, Topological space.

[1]. A.L.Foster,The theory of Boolean like rings,Trans.Amer.Math.Soc.,59 (1946),pp:166-187.
[2]. Jacobson. N, Basic algebra-2, Hindustan Publishing Corporation, 1994.
[3]. Jacobson. N; Totally disconnected locally compact rings, Amer. J. Math. 58 (1936), 433-449.
[4]. Koteswara Rao, P: A*-Algebra an If-Then-Else structures (Doctoral Thesis) 1994, Nagarjuna University, A.P., India.
[5]. M.H. McCoy, and Montgomery, A representation of generalized Boolean
[6]. rings. Duke Math. J. 3 (1937), 455–459. 6.M,H.Stone,The theory of representation of Boolean algebras,Trans.Amer.Math.Soc.40 (1936),pp:37-111.

Abstract: In this paper we prove a uniqueness theorem for a meromorphic function which is sharing one value and a small meromorphic functionwith its derivatives.

[1]. L. RUBEL and C. C. YANG, (1976) : Values shared by an entire function and its derivative in "Complex Analysis, Kentucky"
(Proc. Conf.), Lecture notes in Mathematics, Vol. 599, pp. 101-103, Springer Verlag, Berlin 1977.
[2]. E. MUES and N. STEINMETZ (1979) : Mermorphic functionen, die mit ihrer Ableitung wert teilen. Manuscripta Math 29 (1979), 195-206.
[3]. G. JANK, E. MUES and L. VOLKMANN (1986) : Meromorphic functionen, die mit ihrer ersten und zweiten ableitung einen
endlichen wert teilen, Complex variables 6, 51-71.
[4]. INDRAJIT LAHIRI (1998) : "Uniqueness of meromorphic functions sharing the same 1-points‟. Bull. Korean Math. Soc. 35, No. 2, pp. 375-385.

Paper Type	:	Research Paper
Title	:	Weak Convergence Theorem of Khan Iterative Scheme for Nonself I-Nonexpansive Mapping
Country	:	India
Authors	:	Poonam Lata Sagar || S. K. Malhotra
	:	10.9790/5728-10121114

Abstract: In this paper, we prove the weak convergence of a modified Khan iteration for nonself I - nonexpansive mapping in a Banach space which satisfies Opial's condition. Our result extends and improves these announced by S. Chornphrom and S.Phonin [Weak Converges Theorem of Noor iterative Scheme for Nonself I-Nonexpansive mapping, Thai Journal of Mathematics Volume 7(2009) no.2:311-317].
Keywords: Khan iterative scheme; weak convergence; nonself nonexpansive mapping; fixed point; Banach space.

[1] J. B. Baillon, Un theorem de type ergodique pour les contractions non lineaires dans un espace de Hilbert, Comptes Rendus de
l'Academie des Sciences de Paris, Serie A 280(1975), no. 22, 1511-1514.
[2] J. B. Diaz and F. T. Metcalf, On the set of sub sequential limit points of successive approximations, Transaction of the American
Mathematical Society 135(1969), 459-485.
[3] W. G. Dotson Jr., On the Mann iterative process, Transactions of the American Mathematical Society 149(1970), no. 1, 65-73.
[4] M.K.Ghosh and L.Debnath, Convergence of Ishikawa iterates of quasi -nonexpansive mappings, Journal of Mathematical
Analysis and Applications 207(1997), no. 1, 96-103.
[5] S. Ishikawa, Fixed points by a new iteration method, Proc. Am. Math. Soc.44(1974) 147-150.

Paper Type	:	Research Paper
Title	:	Cluster Information of Non-Sampled Area In Small Area Estimation
Country	:	Indonesia
Authors	:	Rahma Anisa, Anang Kurnia, Indahwati
	:	10.9790/5728-10121519

Abstract: Empirical Best Linear Unbiased Predictor (EBLUP) has been widely used to predict parameters in area with small or even zero sample size. The problem is when this model should be used to predict the parameters of non-sampled area. Ordinary EBLUP predicted the parameters using synthetic model which ignore the area random effects because lack of non-sampled area information. Thus, those prediction will be distorted based on a single line of the synthetic model. One of idea that developed in this paper is to modify the prediction model by adding cluster information by assuming that there are similiarities among particular areas.

1] A. Kurnia. "Prediksi Terbaik Empirik untuk Model Transformasi Logaritma di Dalam Pendugaan Area Kecil dengan Penerapan pada Data Susenas". Unpublished doctoral dissertation, Department of Statistics FMIPA-IPB, Indonesia, 2009.
[2] J.N.K. Rao. Small Area Estimation. New York: John Wiley & Sons, 2003.
[3] A. Saei and R. Chambers. "Empirical Best Linear Unbiased Prediction for Out of Sample Area". S3RI Methodology Working Papers, Southampton Statistical Sciences Research Institute, March 2005.
[4] K. Das, J. Jiang, and J.N.K. Rao. "Mean Square Error of Empirical Predictor". The Annals of Statistics, vol. 32, no. 2, pp. 818-840, June 2004.
[5] A.A. Mattjik and I.M. Sumertajaya. Sidik Peubah Ganda. Bogor: Department of Statistics FMIPA-IPB, 2011.
[6] R.A. Johnson and D.W. Wichern, D.W. Applied Multivariate Statistical Analysis 6th Edition. London: Prentice-Hall, 2007.

Paper Type	:	Research Paper
Title	:	An application of nested Newton-type algorithm for finite difference method solving Richards' equation
Country	:	Bangladesh
Authors	:	M Sayful Islam, M Khayrul Hasan
	:	10.9790/5728-10122032

Abstract:Richards' equation is frequently used to model flow in unsaturated porous media. This model captures physical effects, such as sharp fronts in fluid pressures and saturations, which is considered for practical application to an extensive variety of hydrologic evaluations, at a range of scales of simulation. The numerical solution of Richards' equation is difficult not only because of these physical effects but also because of the mathematical problems that occur in dealing with the nonlinearities. The mixed form of Richards' equation with a finite difference discretization leads to a nonlinear numerical model. In this work a nested Newton-type algorithm is briefly derived and it is suggested that nested Newton-type for finite difference methods can be effectively implemented as well as comparable with the results of nested Newton-type algorithm for finite volume and used in numerical models of Richards' equation. Two test problems are solved with a judicious choice of the initial guess and the quadratic convergence rate is obtained for any time step size for all flow systems.

[1] R. H. Brooks and A. T. Corey, Hydraulic properties of porous media (Hydrology Paper No.3, Civil Engineering, Colorado State
University, Fort Collins, CO, 1964)
[2] M. T. van Genuchten, A Closed-form Equation for Predicting the Hydraulic Conductivity of Unsaturated Soils, Soil Sci. Soc. Am.
J., 44, 1980, 892–898.
[3] L. Guarracino and F. Quintana, A third-order accurate time scheme for variably saturated groundwater flow modeling,
Communications in Numerical Methods in engineering, 20, 2004, 379-389.
[4] P. C. D. Milly, A mass-conservative procedure for time-stepping in models of unsaturated flow, Adv. Water Resources, 8, 1985, 32-
36.
[5] R. G. Baca, J. N. Chung, and D. J. Mulla, Mixed transform finite element method for solving the non-linear equation for flow in
variably saturated porous media, Int. J. Numer. Meth. Fluids, 24, 1997, 441-455.

Paper Type	:	Research Paper
Title	:	An Appliaction of Generalized Fuzzy Soft Matrices in Decision Making Problem
Country	:	India
Authors	:	B.K. Saikia , H. Boruah , P.K. Das
	:	10.9790/5728-10123341

Abstract:After development of fuzzy soft matrices, it has been applying in many fields of real life scenarios. The problems which are unable to solve by ordinary matrices can be solved by fuzzy soft matrices. In this paper our main aim is to define generalized fuzzy soft matrices and to study a few of its properties. Finally, we presented a decision making problem based on one of the operation of generalized fuzzy soft matrices.

Keywords: Fuzzy Soft Matrices, Generalized Fuzzy Soft Set, Generalized Fuzzy Soft Matrices.

[1] Chetia, B. and Das, P. K. : "Some Results of Intuitionistic Fuzzy Soft matrix Theory," Pelagia Research Library, Advances in
Applied Science Research, 2012,Vol. 3(1)412- 423.
[2] Cagman, N. and Enginoglu, S. :"Fuzzy soft Matrix Theory and Its Application in Decision Making," Iranian Journal of Fuzzy
systems, Vol.9(1), 2012.
[3] Cagman, N., Enginoglu, S. : "Soft Matrix Theory and It‟s Decision Making," Computers and Mathematics with applications,
59(2010), 3308-3314.
[4] Dinda, B., Bera, T. and Samanta, T. K. : "Generalized Intuitionistic Fuzzy Soft Sets and An Adjustable Approach to Decision
Making," Annals of Fuzzy Mathematics and Informatics, Vol. 4, No. 2, 2012, pp. 207-215.
[5] Dinda, B., Bera, T. and Samanta, T. K. : "Generalized Intuitionistic Fuzzy Soft Sets and Its Application in Decision Making,"
[math.GM], 2010.
[6] D. Pei and D. Miao: "From Soft Sets to Information Systems," Proceedings of the IEEE International Conference on Granular
Computing 2 (2005), 617-621.

Paper Type	:	Research Paper
Title	:	Optimal Choice of Splines and Knots in TPSPLINE and TRANSREG Procedures
Country	:	India
Authors	:	Rashmi Aggarwal, Suresh Kumar Sharma, Kanchan Jain
	:	10.9790/5728-10124252

Abstract: Multivariate functions observed with noise can be approximated by thin-plate smoothing splines. It does not depend on the assumptions of the parametric model. The amount of smoothing can be judged by generalized cross validation. If there is no prior knowledge about the model and the data is unable to represent a model with fixed number of parameters then TPSPLINE procedure is appropriate. With the increase in sample size, the model space also increases but this situation can be handled by thin-plate smoothing spline i.e. it is suitable for complicated situations. We worked on optimal choice of knots in TPSPLINE procedure. The procedure has been demonstrated by taking real life data set.

[1] Craven, P. and G. Wahba (1979). Smoothing noisy data with spline functions: Estimating the correct degree of smoothing by the method of generalized cross-validation. Numer. Math. 31, 377 - 403.
[2] De Leeuw, J., Young, F.W., and Takane, Y. (1976), "Additive Structure in Qualitative Data: An lternating Least Squares Approach with Optimal Scaling Features," Psychometrika, 41, 471–503.
[3] Eilers, P. H. C. and Marx, B. D. (1996), "Flexible Smoothing with B-splines and Penalities," Statistical Science, 11, 89 – 121, with discussion.
[4] Green, P.E., and Wind, Y. (1975), "New Way to Measure Consumers‟ Judgements," Harvard Business Review, July–August, 107–117.
[5] Kimeldorf, G. and G. Wahba (1970a), A correspondence between Bayesian estimation of stochastic processes and smoothing by splines. Ann. Math. Statist. 41, 495 - 502.
[6] Kimeldorf, G. and G. Wahba (1970b).Spline functions and stochastic processes.Sankhya Ser. A 32, 173 - 180.

Paper Type	:	Research Paper
Title	:	The Method of Steepest Descent for Feedforward Artificial Neural Networks
Country	:	Bangladesh
Authors	:	Muhammad Hanif, Md. Jashim Uddin, Md Abdul Alim
	:	10.9790/5728-10125361

Abstract: In this paper, we implement the method of Steepest Descent in single and multilayer feedforward artificial neural networks. In all previous works, all the update weight equations for single or multilayer feedforward artificial neural networks has been calculated by choosing a single activation function for various processing unit in the network.

[1] Adby, P.R. and Dempster, M.A.H. 1974. Introduction to Optimization Methods. Haisted Press, New York.
[2] Battiti, R. 1992. First and second-order methods for learning: Between Steepest Descent and Newton's methods. Neural
Computation 4: 141-166.
[3] Johansson, E.M., Dowla, F.U. and Goodman, D.M. 1992. Backpropagation learning for multi-layer feed-forward neural networks
using the conjugate gradient method. Intl. J. Neural Systems, 2: 291-301 Judd, J.S. 1990. Neural Network Design and the
complexity of Learning. MIT Press, Cambridge, MA
[5] Muhammad Hanif. 2002. Optimality of Unconstrained Nonlinear Programming Problem. M.Phil Thesis, University of Chittagong,
Chittagong, Bangladesh.
[6] Mehra] P., and Wah, B. W. 1992. Artificial neural networks: concepts and theory IEEE Comput. Society Press,.
[7]. X. Yu, M. O. Efee, and O. Kaynak.2002. A general backpropagation algorithm for feed-forward neural networks learning. IEEE
Trans. Neural Networks. 13:251–254.

Paper Type	:	Research Paper
Title	:	Extension of Sub Compatible Maps in Fuzzy Metric Spaces
Country	:	India
Authors	:	S.K. Malhotra, Vineeta Singh
	:	10.9790/5728-10126265

Abstract:The present paper is the extension of results of sub compatibility and sub sequential continuity in fuzzy metric spaces which are weaker than occasionally weak compatibility and reciprocal continuity. Mathematics Subject Classification (2000). 47H10 , 54H25. Keywords: Compatible maps, Weak Compatible maps, Occasionally weak compatible maps, Sub compatible maps, , fixed points and fuzzy metric space space

[1] M. Abbas and B. E. Rhoades, Common fixed point theorems for hybrid pairs of occasionally weakly compatible mappings satisfying generalized condition of integral type, Fixed point theory appl., Art. ID 54101, 9pp.
[2] M. A. Al-Thagafi and N. Shahzad, A note on occasionally weakly compatible maps, Int. J. Math. Anal. 3(2)(2009) , 55-58.
[3] H. Bouhadjera, A. Djoudi and B. Fisher, A unique common fixed point theorem for occasionally weakly compatible maps, Surveys in Mathematics and its Appli. 3(2008), 177-182.
[4] H. Bouhadjera and C. G. Thobie, Common fixed point theorems for pairs of sub compatible maps, Hal-00356516. 1(2009),1-16.
[5] H. Bouhadjera and C. G. Thobie, Common fixed point theorems for occasionally weakly compatible maps, ArXiv. 0812.373 [math. FA]. 2(2009): 123:131.
[6] H. Chandra and A. Bhatt, Fixed point theorem for occasionally weakly compatible maps in probabilistic semi-metric space, Int. J. Math. Anal. 3(12)(2008), 563-570.

Paper Type	:	Research Paper
Title	:	Square Sum Graph Associated with a Sequence of Positive Integers
Country	:	India
Authors	:	Reena Sebastian, K.A Germina
	:	10.9790/5728-10126670

Abstract: A (p,q)-graph G is said to be square sum, if there exists a bijection f: V(G)→{0,1,2,..,p-1} such that the induced function f*:E(G)→ N defined by f*(uv)= (f(u))2+(f(v))2, for every uv E(G) is injective. In this paper, a recursive construction of infinite families of square sum graphs associate with a sequence of positive integers are studied. That is for any sequence of positive integers (a1,a2,..,an) with ai≥ 2, i=1,2,..,n we associate some square sum graphs. In particular we obtain the result of level joined planar grid are square sum as the special case.

Keywords: Square sum graphs, Level joined planar grid.

[1]. J.A.Gallian, A dynamic survey of graph labeling, The Electronic Journal of Combinatorics (DS6),2005
[2]. F. Harary, \textbf{Graph Theory}, Addison-Wesley Pub. Comp., Reading, Massachusetts, 1969.
[3]. B.D.Acharya, \newblock {\em Personal Communication}, \newblock September 2011.
[4]. Ajitha V, \emph{Studies in Graph Theory-Labeling of Graphs}, Ph D thesis (2007), Kannur Univeristy, Kannur.
[5]. B.D.Acharya and Hegde, \emph{Strongly Multiplicative graphs}, Discrete Mathematics 93(1991),123-129.
[6]. Germina K.A and Reena Sebastian,(2013). Further results on square sum graphs, International Mathematical Forum,(8) 47-57.
[7]. .Reena Sebastian and Germina K.A, Maximal square sum subgraph of a complete graph, accepted.