#### Volume-6 ~ Issue-6

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Paper Type |
: | Research Paper |

Title |
: | Matrix Transformations on Some Difference Sequence Spaces |

Country |
: | Nigeria |

Authors |
: | Z. U. Siddiqui, A. Kiltho |

: | 10.9790/5728-0660104 |

**Abstract:** The sequence spaces πβ(π’,π£,Ξ), π0(π’,π£,Ξ) and π(π’,π£,Ξ) were recently introduced. The matrix classes (π π’,π£,Ξ :π) and (π π’,π£,Ξ :πβ) were characterized. The object of this paper is to further determine the necessary and sufficient conditions on an infinite matrix to characterize the matrix classes (π π’,π£,Ξ βΆππ ) and (π π’,π£,Ξ βΆ ππ). It is observed that the later characterizations are additions to the existing ones.

**Keywords-** Difference operators, Duals, Generalized weighted mean, Matrix transformations

[1] Ahmad, Z. U., and Mursaleen, KΕthe-Teoplitz Duals of Some New Sequence Spaces and Their Matrix Transformations, Pub. DΓ© L'institut MathΓ©matique Nouvelle sΓ©rie tome 42 (56), 1987, p. 57-61

[2] Altay, B., and F. BaΕar, Some Paranormed Sequence Spaces of Non-absolute Type Derived by Weighted Mean, J. math. Anal. Appl. 319, (2006), p. 494-508 [3] Altay B., and F. BaΕar, The Fine Spectrum and the Matrix Domain of the Difference Operator Ξ on the Sequence Space lp,0<π<1), Comm. Math. Anal. Vol. 2 (2), 2007, p. 1-11

[4] BaΕarir, M., and E. E. Kara, On Some Difference Sequence Spaces of Weighted Means and Compact Operators, Ann. Funct. Anal. 2, 2011, p. 114-129

[5] Boos, J., Classical and Modern Methods in Summability, Oxford Sci. Pub., Oxford, 2000

[6] Kizmaz, H., On Certain Sequence Spaces, Canad. Math. Bull. Vol. 24 (2), 1981, p. 169-176

[7] Maddox, I. J., Spaces of Strongly Summable Sequences, Quart. J. Math. Oxford,18 (2), 1967, p. 345-355

[8] Polat, H., and F. BaΕar, Some Euler Spaces of Difference Sequences of Order mβ, Acta Mathematica Scienta, 2007, 27B (2), p. 254-266

[9] Polat, H., Vatan K. and Necip S., Difference Sequence Spaces Derived by Generalized Weighted Mean, App. Math. Lett. 24 (5), 2011, p. 608-614

[10] Simons, S., The Sequences Spaces,l(pv) and m(pv), Proc. London Math. Soc. 15 (3), 1965, p. 422-436

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Paper Type |
: | Research Paper |

Title |
: | Jordan Higher (π,π)-Centralizer on Prime Ring |

Country |
: | Iraq |

Authors |
: | Salah M. Salih, Marwa M. Shaapan |

: | 10.9790/5728-0660511 |

**Abstract:** Let π
be a ring and π,π be an endomorphisms of π
, in this paper we will present and study the concepts of higher (π,π)-centralizer, Jordan higher(π,π)-centralizer and Jordan triple higher (π,π)-centralizer and their generalization on the ring. The main results are prove that every Jordan higher (π,π)-centralizer of prime ring π
is higher (π,π)-centralizer of π
and we prove let π
be a 2-torsion free ring,π πππ π are commutative endomorphism then every Jordan higher (π,π)-centralizer is Jordan triple higher (π,π)-centralizer. Mathematics Subject Classification: 16A12,16N60,16W25,16Y99.

**Keywords:** higher (π,π)-centralizer, Jordan higher (π,π)-cenralizer, Jordan triple higher (π,π)-centralizr

[1] E.Albas, "On π-Centralizers of Semiprime Rings", Siberian Mathematical Journal, Vol.48, No.2, pp.191-196, 2007.

[2] W.Cortes and C.Haetinger," On Lie Ideals Left Jordan π-centralizers of 2-torsion Free Rings", Math.J.Okayama Univ., Vol.51, No.1, pp.111-119, 2009.

[3] J.Vukman,"Centralizers in Prime and Semiprime Rings", Comment. Math.Univ. Carolinae, pp.231-240, 38(1997).

[4] J.Vakman, "An Identity Related to Centralizers in Semiprime Rings", Comment. Math.Univ.Carolinae, 40, pp.447-456, 3(1999).

[5] J.Vakman, "Centralizers on Semiprime Rings", Comment. Math. Univ. Carolinae, 42, pp.237-245, 2(2001).

[6] B.Zalar, "On Centralizers of Semiprime Ring", Comment. Math. Univ. Carol.32, pp.609-614, 1991.

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Paper Type |
: | Research Paper |

Title |
: | Proposal Of New Conjecture For Solution Of Goldback's Puzzle |

Country |
: | India |

Authors |
: | Umasankar Dolai |

: | 10.9790/5728-0661213 |

**Abstract:**New conjecture about prime numbers is proposed for solving the criterion of Goldback's puzzle about natural numbers. It is found that Goldback's puzzle is a corollary of that new conjecture.

**Keywords β** Goldback's Puzzle, Prime Numbers Distribution, Remarks.π)-centralizr

[1] Paolo Giordano, The Solitude of Prime Numbers, Pamela Dorman Books (2010).

[2] Richard E. Crandall, Prime Numbers : A Computational Perspective, Springer (2005).

[3] Ribenboim, Paulo, The book of Prime Number Records, Sringer (1996).

[4] Matthew Watkins, Matt Tweea, Math Book : The Mystery of the Prime Numbers, Murray (2011).

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Paper Type |
: | Research Paper |

Title |
: | (ππ’, ππ£)β RGB Closed Sets in Bitopological Spaces |

Country |
: | Iraq |

Authors |
: | Bushra Jaralla Tawfeeq, Dunya Mohamed Hammed |

: | 10.9790/5728-0661422 |

**Abstract:** In this paper we introduce and study the concept of a new class of closed sets called (ππ, ππ)β regular generalized b- closed sets (briefly(ππ, ππ)β rgb-closed) in bitopological spaces.Further we define and study new neighborhood namely (ππ, ππ)β rgb- neighbourhood (briefly(ππ, ππ)β rgb-nhd) and discuss some of their properties in bitopological spaces. Also, we give some characterizations and applications of it.)-centralizr

[1] Ahmad Al-Omari and Mohd. Salmi Md. Noorani, On Generalized b-closed sets. Bull. Malays. Math. Sci. Soc(2) 32(1) (2009), 19-30 [2] Benchalli.S.S and Wali.R.S., On Rw-closed sets in topological spaces ,Bull. Malays. math. Sci. Soc(2) 30(2),(2007), 99-110

[3] K.chandrasekhara rao and K.kannan,regular generalized star closed sets in bitopological spaces ,Thai journal of Math.,vol.4,(2),(2006),341-349

[4] R. Devi, H. Maki and K. Balachandran, Semi-generalized closed maps and generalized semi- closed maps, Mem. Fac. Sci. Kochi Univ. Ser. A. Math. 14 (1993), 41β54.

[5]. J. Donchev, On generalizing Semi-pre open sets, Mem. Fac. Sci. Kochi Univ. Ser. A. Mat16 (1995), 53-48.

[6]. O.A. El-Tantawy and H.M. Abu-Donia, Generalized Separation Axioms in Bitopological Spaces, The Arabian Jl for Science and Engg.Vol.30,No.1A,117-129 (2005).

[7] T.Fukutake, On generalized closed sets in bitopological spaces, Bull. Fukuoka Univ. Ed. Part III, 35, 19-28 (1985).

[8] Futake, T., Sundaram, P., and SheikJohn.M., 2002, w -closed sets, w-open sets and w -continuity in bitopological spaces Bull.Fukuoka Univ.Ed.Vol.51.Part III 1-9.

[9]. Y. Gnanambal, On generalized pre-regular closed sets in topological spaces, Indian J. PureAppl. Math. 28 (1997), 351-360.

[10] D. Iyappan & N.Nagaveni, On semi generalized b-closed set, Nat. Sem. On Mat & Comp.Sci, Jan (2010), Proc.6.

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**Abstract:** Tan-cot method is applied to get exact soliton solutions of non-linear partial differential equations notably generalized Benjamin-Bona-Mahony, Zakharov-Kuznetsov Benjamin-Bona-Mahony, Kadomtsov-Petviashvilli Benjamin-Bona-Mahony and Korteweg-de Vries equations, which are important evolution equations with wide variety of physical applications. Elastic behavior and soliton fusion/fission is shown graphically and discussed physically as far as possible.

**Keywords:** Benjamin-Bona-Mahony equation, Tan-Cot method, soliton solutions.)-centralizr

[1] T. B. Benjamin, J. L. Bona, and J. J. Mahony, Model equations for long waves in non-linear dispersive systems, Philosophical Transactions of the Royal Society of London Series A, 272, 1972, 47-78.

[2] H. Zhang, G. M. Wei, and Y. T. Gao, On the general form of the Benjamin-Bona-Mahony equation in fluid mechanics, Czechoslovak Journal of Physics, 52, 2002, 373-377.

[3] B. Wang, and W. Yang, Finite-dimensional behaviour for the Benjamin-Bona-Mahony equation, Journal of Physics A: Mathematical and General, 30, 1997, 4877-4885.

[4] B. Wang, Regularity of attractors for the Benjamin-Bona-Mahony equation, Journal of Physics A: Mathematical and General, 31, 1998, 7635-7645.

[5] A. M. Wazwaz, Exact solutions of compact and noncompact structures for the KP-BBM equation, Applied Mathematics and Computation, 169, 2005, 700-712.

[6] A. M. Wazwaz, The tanh and the sine-cosine methods for a reliable treatment of the modied equal width equation and its variants, Communications in non-linear Science and Numerical Simulation, 11, 2006, 148-160.

[7] A. M. Wazwaz, Compact and noncompact physical structures for the ZKBBM equation, Applied Mathematics and Computation, 169, 2005, 713-725.

[8] M. A. Abdou, Exact periodic wave solutions to some non-linear evolution equations, Applied Mathematics and Computation, 6, 2008, 145-153.

[9] M. A. Karaca, and E. Hizel, Similarity reductions of Benjamin-Bona-Mahony equation, Applied Mathematical Sciences, 2, 2008, 463-469.

[10] D. J. Korteweg, and G. de Vries, on the change form of long waves advancing in a rectangular canal, and on a new type of long stationary waves, Philosophical Magazine, 39, 1895, 422-443

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Paper Type |
: | Research Paper |

Title |
: | On Spaces of Entire Functions Having Slow Growth Represented By Dirichlet Series |

Country |
: | India |

Authors |
: | Arvind Kumar |

: | 10.9790/5728-0662934 |

**Abstract:** In this paper spaces of entire function represented by Dirichlet Series have been considered. A norm has been introduced and a metric has been defined. Properties of this space and a characterization of continuous linear functionals have been established.

[1]. K.N. Awasthi : A study in the mean values and the growth of entire functions, Ph.D. Thesis, Kanpur University, 1969.

[2]. T. Hussain and P.K. Kamthan: Spaces of Entire functions represented by Dirichlet Series. Collect. Math., 19(3), 1968,203-216.

[3]. P.K. Kamthan: Proximate order (R) of Entire functions represented by Dirichlet Series. Collect. Math.,14(3), (1962),275-278.

[4]. P.K. Kamthan: FK βspaces for entire Dirichlet functions. Collect.Math., 20(1969), 271-280.

[5]. Q.I. Rahman : On the maximum modulus and the coefficients of a Dirichlet Series. Quart. J. Math. (2), 7(1956),96-99.

[6]. J.F.Ritt: On certain points in the theory of Dirichlet Series. Amer. J. Maths., 50(1928), 73-86.

[7]. G.S. Srivastava: A note on proximate order of entire functions represented by Dirichlet Series. Bull De L' Academie polonaise Des Sci 19(3)(1971),199-202.

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Paper Type |
: | Research Paper |

Title |
: | Cash Flow Valuation Mode Lin Discrete Time |

Country |
: | Ibadan |

Authors |
: | Olayiwola. M. A., Oni, N. O. |

: | 10.9790/5728-0663541 |

**Abstract:** This research consider the modelling of each cash flow valuation in discrete time. It is shown that
the value of cash flow can be modeled in three equivalent ways under same general assumptions. Also,
consideration is given to value process at a stopping time and/ or the cash flow process stopped at some
stopping times.

[1]. Akepe A. O (2008): Option Pricing on Multiple Assets M.Sc. Project University of Ibadan

[2]. Armerin F. (2002): Valuation of Cash flow in Discrete Time working paper Ayoola E. O: Lecture note on Advance Analysis Lebesgue Measure University of Ibadan, Unpublished

[3]. Ugbebor O. O (1991): Probability Distribution and Elementary Limit Theorems.

[4]. University of Ibadan External Studies Programme University Of Ibadan

[5]. Elliott R. J (1982): Stochastic Calculus and Applications, Springerverlag

[6]. Pratt, S. et al (2000): Valuing a Business. McGraw-Hill Professional. McGraw Hill

[7]. Williams, D. (1999): Probability with Martingales, Cambridge University Press Ξ¦Ksendal, B. (1998): Stochastic differential Equation, Fifth Edition, Springer - verlag

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Paper Type |
: | Research Paper |

Title |
: | Simultaneous Triple Series Equations Involving Konhauser Biorthogonal Polynomials |

Country |
: | India |

Authors |
: | P. K. Mathur |

: | 10.9790/5728-0664245 |

**Abstract:** Biorthogonal polynomials are of great interest for Physicists.Spencer and Fano [9] used the biorthogonal polynomials (for the case k = 2) in carrying out calculations involving penetration of gamma rays through matter.In the present paper an exact solution of simultaneous triple series equations involving Konhauser-biorthogonal polynomials of first kind of different indices is obtained by multiplying factor technique due to Noble.[4] This technique has been modified by Thakare [10, 11] to solve dual series equations involving orthogonal polynomials which led to disprove a possible conjecture of Askey [1] that a dual series equation involving Jacobi polynomials of different indices can not be solved. In this paper the solution of simultaneous triple series equations involving generalized Laguerre polynomials also have been discussed as a charmfull particular case.

**Key Words:** 45F 10 Simultaneous Triple Series Equations, 33C45 Konhasure bi-orthogonal polynomials, Laguerre polynomials,42C O5 Orthogonal Functions and polynomials, General Theory, 33D45 Basic Orthogonal polynomials, 26A33 Fractional derivatives and integrals.

[1]. Askey, Richard: Dual equations and classical orthogonal polynomials, J. Math. Anal. Applic., 24, pp. 677- 685, (1968).

[2]. Konhasure, J.D.E. : Some properties of biorthogonal polynomials, Ibid, 11, pp.242- 260, (1965).

[3]. Konhasure, J.D.E. : Biorthogonal polynomials suggested by the Laguerre polynomials, Pacific J. Math. , 21, pp. 303- 314, (1967).

[4]. Noble, B.: Some dual series equations involving Jacobi polynomials,Proc. Camb.

[5]. phil. Soc. , 59, pp. 363- 372, (1963).

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**Abstract:** Study of buyer-vendor integrated system, in general, has two major features; determining delivery schedule with supply quantity in each shipment and minimization of total incremental cost. Researchers in this area concentrate on latter part, which probably may not justify both the features. Many models developed so far, without considering both the features, begin with some pre-determined shipment pattern and establish cost minimization but may not establish stability in supply or shipment size that may vary in reality. In fact along with cost minimization stability in shipment size should also be a dominant feature of the doubly effective inventory model. Stable supply within a normal limit of small fluctuation will allow the carrying charges to be considered constant. The importance of the model lies in making all shipment size dependent on EOQ. The shipment size gets stable after two or three shipments and achieves optimization of total incremental cost.

[1]. Benerjee, A, (1986) ,' A joint economic lot size model for purchaser and vendor β, Decision Science 17,292. 311

[2]. Goyal. S. K (1977),'Determination of optimum production quantity for a two- stage production systemβ , Operational Research Quartely 28, 865-870

[3]. Goyal. S. K. (1988) ,β A joint economic lot size model for purchaser and vendor; A comment , Decision Sciences 19,236-241

[4]. Goyal S. K.( 1995), β a one β vendor multi-buyer integrated inventory Modelβ . A Commentβ European Journal if Operations Research 82, 209- 210

[5]. Lu. L (1995) β A one vendor multi buyer integrated inventory Modelsβ European Journal of Operations Research.

[6]. Jha.Pradeep.j ( 2009) , β Buyer Vendor Integrated System( time Scaling and Normalization), Ph.D. Thesis to North _Gujarat University, 5, 56-75

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Paper Type |
: | Research Paper |

Title |
: | Numerical solution of heat equation through double interpolation |

Country |
: | India |

Authors |
: | P. Kalyani, P. S. Ramachandra Rao |

: | 10.9790/5728-0665862 |

**Abstract:**In this article an attempt is made to find the solution of one-dimensional Heat equation with initial and boundary conditions using the techniques of numerical methods, and the finite differences. Applying Bender-Schmidt recurrence relation formula we found u(x ,t) values at lattice points. Further using the double interpolation we found the solution of Heat equation as double interpolating polynomial.

**Keywords -** Boundary Value Problem , Finite Difference method, Double Interpolation.

[1] Bender.C.M and Orszag.S.A., Advanced mathematics methods for scientist and engineers ( McGraw-Hill. New York,1978).

[2] Collatz, L.,The numerical treatment of differential equations ( Springer Verlag,Berlin,1996).

[3] Na,T.Y., Computational methods in engineering boundary value problems (Academic Press, New York,1979).

[4] Rama ChandraRao,P.S., Solution of a Class of Boundary Value Problems using Numerical Integration, Indian journal of mathematics and mathematical sciences.vol.2.No2,2006,pp.137-146.

[5] Rama Chandra Rao,P.S., Solution of fourth order of boundary value problems using spline functions, Indian Journal of Mathematics and Mathematical Sciences.vol.2.No1,2006, pp.47-56

[6] Ravikanth,A.S.V., Numerical treatment of singular boundary value problems, Ph.D. Thesis, National Institute of Technology, Warangal, India,2002.

[7] K.W. Mortan and D.F. Mayers. Numerical solution of partial differential equations: An introduction ( Cambridge University press, Cambridge,England.1994).

[8] Jeffery Cooper, Introduction to partial differential equations with matlab (Birkhauser, Bosten, 1998).

[9] Clive A.J,Fletcher, Computational techniques for fluid dynamics (Springer-Verlag,Berlin,1998).

[10] J.B. Scorborough, Numerical mathematical analysis ( Oxford and IBH Publishing Co. Pvt. Ltd.,1966)

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Paper Type |
: | Research Paper |

Title |
: | An EOQ Model for Weibull Deteriorating Items With Price Dependent Demand |

Country |
: | India |

Authors |
: | Sushil Kumar, U. S. Rajput |

: | 10.9790/5728-0666368 |

**Abstract:** In the present paper we developed an economic order quantity model for Weibull deteriorating items with price dependent demand rate together with a replenishment policy for profit maximization. The demand rate is a continuous and differentiable function of price. The variable items deteriorate with time shortages are allowed and completely back-ordered. Further it is illustrated with the help of numerical examples.

**Keywords: **Weibull distribution, Price-dependent demand rate and Varying rate of deterioration.

[1]. R.P.Covert and G.C. Philip, American institute of Industrial Engineering transaction 5,323(1973).

[2]. G.C. Philip, AIIE transaction 6, 159(1974).

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[4]. Y.K. Shah and M.C. Jaiswal, Opsearch 14,174(1977).

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[6]. M. Roy Choudhary and K. S. Chaudhary, Opsearch 20, 99 (1983).

[7]. P. Ghare and G. Schrader, journal of Industrial Engineering 14, 238 (1963).

[8]. P.R. Tadikamalla, AIIE Transaction 10, 108(1978).

[9]. S. Mukhopadhyay, R. N. Mukherjee, and K. S Chaudhuri, Computers and Industrial Engineering 47,339(2004).

[10]. S. Mukhopadhyay, R. N. Mukherjee, and K. S. Chaudhuri, International Journal of Mathematical Education in science and

Technology 36,25(2005).

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**Abstract:** An order level inventory model for decaying items with inventory level dependent demand rate. We have considered two cases: first is, model started with no shortages and second is model started from shortages. We have also taken the concept of inflation in this study. Finally, a numerical example for illustration is provided with sensitivity analysis.

[1] Benkherouf, L. (1997): A deterministic order level inventory model for deteriorating items with two storage facilities. I.J.P.E., 48 (2), 167β175.

[2] Bhunia, A.K. and Maiti, M. (1998): A two-warehouse inventory model for deteriorating items with a linear trend in demand and shortages. J.O.R.S., 49 (3), 287β292.

[3] Goswami, A. and Chaudhuri, K.S. (1992): An Economic order quantity model for items with two levels of storage for a linear trend in demand. J.O.R.S., 43, 157-167.

[4] Hsieh, T.P., Dye, C.Y. and Ouyang, L.Y. (2008): Determining optimal lot size for a two-warehouse system with deterioration and shortages using net present value. E.J.O.R., 191 (1), 180-190.

[5] Lee, C.C. (2006): Two-warehouse inventory model with deterioration under FIFO dispatching policy. E.J.O.R., 174 (2), 861-873.

[6] Murdeshwar, T.M. and Sathe, Y.S. (1985): Some aspects of lot size models with two levels of storage. Opsearch, 22, 255-262.

[7] Pakkala, T.P.M., Achary, K.K. (1992): A deterministic inventory model for deteriorating items with two warehouses and finite replenishment rate. E.J.O.R., 57, 71-76.

[8] Pakkala, T.P.M., Achary, K.K. (1994): Two level storage inventory model for deteriorating items with bulk release rule. Opsearch, 31, 215-227.

[9] Sarma, K.V.S. (1983): A deterministic inventory model with two level of storage and an optimum release rule. Opsearch, 20(3), 175-180.

[10] Sarma, K.V.S. (1987): A deterministic order level inventory model for deteriorating items with two storage facilities. E.J.O.R., 29, 70-73.