iosr-Jm   Volume-7 ~ Issue-3

Abstract: Every count ably normed nuclear space is isomorphic to a subspace of a nuclear Frechet space with basis and a continuous norm. The proof as given in section 2 is a modification of the Komura-Komura inbedding theorem .In this paper, we shall show that a nuclear Frechet space with a continuous norm is isomorphic to a subspace of a nuclear Frechet space with basis and a continuous norm if and only if it is countably normed.The concept of countably normedness is very important in constructing the examples of a nuclear Frechet space. Moreover, the space with basis can be chosen to be a quotient of (s)

Key words and phrases: Nuclear frechet space, Countably normed and Nuclear Kothe space.

[1] Adasch,N.: Topological vector spaces,lecture notes in maths, springer-verlag,1978.
[2] Grothendieck, A: Topological Vector spaces,Goedan and Breach,NewYork,1973.
[3] Holmstrom, L: A note on countably normed nuclear spaces, Pro. Amer. Math. Soc. 89(1983), p. 453.
[4] Litvinov, G.I.: Nuclear Space,Encyclopaedia of Mat Hematics, Kluwer Academic Publishers Spring-Verlag, 2001.
[5] Komura,T. and Y.Komura : Uber die Einbettung der nuclear Raume in (S)
N , Mathe,Ann.162,1966, pp. 284-288.

Abstract: Some popular recreational games have a group- theoretic foundation, and group are useful in modelling games that involve a series of discrete moves, with each move leading to a change in the board state. In this work we present to you our local game called "tsorry" ( meaning they line up straight). Which we carefully examine and found that it has a board and checkers, each possible moves has an element of group, where the effect of performing a sequence of moves corresponds to the product of those elements. The group under consideration here is Klein four- group.

Keyword: Checkerboard, Group theory, Games, Klein four – group, Possible moves and Tsorry Checkerboard,

[1] H. Wussing, The Genesis of the abstract group concept: A contribution to the History of the origin of Abstract Group theory, New
York: Dover Publications (2007) http://en.Wikipedia.org/wiki/Group(Mathematics)
[2] Y.K. Algon , "Peg solitaire and Group theory" Retrieved from google search August 2010 ( unpublished) p1 (2006)
[3] Kriloff, " A Group theory explanation for a Rubik's cube". (2005) Retrieved from http://groupprops.subwiki.org/wiki/ Group theory
in Games, August 2010
[4] C. Cecka, "Group Theory and a look at the Slide Puzzle" Retrieved from google search unpublished ( unpublished manuscript )
p1 (2003)
[5] Checker Board Games Retrieved on net August 2010 titled "checker board game review". (unpublished manuscript)
[6] J. Loy's, "The basic rules of Checkers", Jim Loy's Home page www.jimloy.com/checkers/rules2htm unpublished (unpublished
Manuscript)
[7] G.G. Hall , Applied group theory, American Elsevier publishing Co., Inc., New York, (1967) MR0219593, an elementary
introduction.
[8] E. Pavel, "Groups Around Us", Retrieved from Google search unpublished ( Unpublished manuscript) p4 (2009)
[9] Klein four- group, Retrieved from www.jmline.org/math/CourseNotes/GT.pdf unpublished (unpublished manuscript)

Abstract: This paper discusses the concepts and methodology used in the system theory and operations research in respect of planning and control of the sustainable development. The sustainable development of any industrial park is not achieved merely by the introduction of foreign capital and embedding into global value chains system. Any sustainable development problem can be represented mathematically using the concepts of transition of system from the given initial state to the final state. It is established in this paper that sustainable development represents a specific control problem. The aim is to keep the system in the prescribed feasible region of the state space. Further the analysis of planning and control problems of sustainable development has also shown that methods developed in the operations research such as multicriterion optimization, dynamic processes, simulation and goal programming etc. are more than adequate.

Keywords: Sustainability, sustainable development, systems approach, multiple criteria optimization, state transition.

[1] Adams, W.M., "The future of sustainability, rethinking environment and development in the twenty first century", Report IUCN, 2006. [2] Allen, W., "Learning for sustainability", Sustainable Development, 2007.

[3] Donald Brown, 'The role of economics in sustainabledevelopment and environmental protection', in Sustainable Development: Science, Ethics and Public Policy, edited by John Lemons and Donald Brown, Kluwer Academic Publishers, Dordrecht, 1995, pp. 52-63.

[4] ESCAP. 1994. UN Economic and Social Commission for Asia and the Pacific Committee on Environment and Sustainable Development. Note by the Secretariat. E/ESCAP/ESD (2)/4.

[5] Galloppin, G., "A systems approach to sustainability and sustainable development", UN Publication, Santiago, Chile, 2003.

[6] Hanley, N., Shorgen, J., and White. B., Environmental Economics in Theory and Practice,Palgrave, London, 2007.

[7] Harris, J.,"Basic principles of sustainable development", Tufts University, Medford MA,USA, 2000.

[8] Munasinghe, M. (1993) Environmental Economics and Sustainable Development, World Bank Environment Paper Number 3, The World Bank, Washington, D.C. [9] D. Meadows, D. Medows, J. Randers, and W. Berens III, The Limits to Growth, Universe Books, New York, 1972.

[10] Pearce, D., "An intellectual history of environmental economics", Annual Review of Energy and Environment, 27 (2002) 57-81.

Abstract: This paper present valuation models which is define as the expected discounted value of a stream of each flows at a time. Three equivalent forms of this value process is established with each of which has its own merits. Local dynamics of the values process is also considered.

Keywords: Cash Flow, Valuation, Continuous, Deflator, Filtration.

Armerin, F. (2002), 'Valuation of Cash Flows in Continuous Time', Working paper Bias, B., et al (1997), 'Financial Mathematics', Lecture Notes in Mathematics, 1656, Springer – Verlag Delbean, F., et al (1994),

'A General Version of the Fundamental Theorem of Asset Pricing', Math. Annalen, Vol. 300, 463-520 Duffie, D., et al (1995), 'Black's consol rate conjecture', The Annals of Applied Probability, Vol. 5, No. 2, 356-382 Harrison, J.M., et al (1979),

'Martingales and Arbitrage in Multi-period Securities Markets', Journal of Economic

Theory Hubalek F., (2001), 'The Limitations of No-Arbitrage Arguments for Real Options', International Journal of Theoretical and Applied Finance Vol. 4, No. 2

Abstract: The new Bianchi type-IX cosmological models with binary mixture of perfect fluid and anisotropic dark energy have been studied. The unique solution of field equations is obtained by assuming that the energy conservation equation of the perfect fluid and dark energy vanishes separately together with a special law for the mean Hubble parameter (Berman, 1983) which yields a constant value of the deceleration parameter and generates two types of solutions, one is of power law type and other is of the exponential type. To have a general description of an anisotropic dark energy component in terms of its equation of state (de)  , a skewness parameters  have been introduced. The statefinder diagnostic pair (i.e. r, s parameter) is used to characterize different phases of the universe. The various geometric and kinematic properties of the model and the behavior of the anisotropy of the dark energy have been discussed. Keywords - Anisotropic Dark Energy, Perfect Fluid, Bianchi type-IX Universe, Statefinder parameters.

[1] S. Perlmutter et al., "Discovery of Supernovae explosion at half the age of the Universe", Nature, Vol. 391, No. 2, pp. 51 -54, 1998.
doi:10.1038/34124
[2] A. G. Reiss, et al., "Observational Evidence from super-novae for an Accelerating Universe and a Cosmological Constant," The
Astrophysical Journal, Vol. 116, No. 3, 1998, pp. 1009-1038. doi:10.1086/300499.
[3] C. L. Benett et al., "First-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Preliminary Maps and Basic
Results", Astrophys. J. Suppl. Ser., Vol. 148, No. 1, pp. 1, 2003. doi:10.1086/377253
[4] Hawkins S. et al., "The 2dF galaxy redshift survey: correlation functions, peculiar velocities and the matter density of the universe",
Monthly Notices of the Royal Astronomical Society , Vol. 346 No. 1, pp. 78–96, 2003. doi.10.1046/j.1365-2966.2003.07063.x .
[5] R. R. CALDWELL, M. DORAN, "COSMIC MICROWAVE BACKGROUND AND SUPERNOVA CONSTRAINTS ON QUINTESSENCE:
CONCORDANCE REGIONS AND TARGET MODELS", PHYSICS REVIEW D, VOL. 69, NO. 10, 103517, 2004. DOI.
10.1103/PHYSREVD.69.103517 .
[6] Z. Y. HUANG, B. WANG, E. ABDALLA, "HOLOGRAPHIC EXPLANATION OF WIDE-ANGLE POWER CORRELATION SUPPRESSION IN THE
COSMIC MICROWAVE BACKGROUND RADIATION" , JOURNAL OF COSMOLOGY AND ASTROPARTICLE PHYSICS, VOL. 2006, NO. 05,
DOI:10.1088/1475-7516/2006/05/013
[7] S. D. TADE, M. M. SAMBHE, "BIANCHI TYPE-I COSMOLOGICAL MODELS FOR BINARY MIXTURE OF PERFECT FLUID AND DARK
ENERGY", ASTROPHYSICS AND SPACE SCIENCE, VOL. 338, PP. 179-185, 2012. DOI: 10.1007/S10509-011-0910-8
[8] S. KUMAR, O. AKARSU, "BIANCHI TYPE-II MODELS IN THE PRESENCE OF PERFECT FLUID AND DARK ENERGY", THE EUROPEAN
PHYSICAL JOURNAL PLUS, VOL. 127, PP. 64, 2012. DOI: 10.1110/EPJP/I2012-12064-4
[9] S. D. KATORE ET AL., "BIANCHI TYPE-III COSMOLOGICAL MODEL WITH BINARY MIXTURE OF PERFECT FLUID AND DARK ENERGY",
BULGARIAN JOURNAL OF PHYSICS, VOL. 38, PP. 390-399, 2011.
[10] T. SINGH, R. CHAUBEY, "BIANCHI TYPE-V COSMOLOGICAL MODELS WITH PERFECT FLUID AND DARK ENERGY", ASTROPHYSICS AND
SPACE SCIENCE, VOL. 319, PP. 149-154, 2009. DOI: 10.1007/S10509-008-9959-4

Abstract: This paper examines the problem of MHD steady mass transfer flow of a polar fluid through a porous medium bounded by an infinite vertical porous plate in slip flow regime. In the mass transfer equation effect of thermal diffusivity which plays an important role in the flow is also considered. The exact solutions for velocity field, angular velocity field, temperature distributions and concentration field are obtained. The expression for the skin-friction, the rate of heat transfer are also derived. Effects of rotational parameter ( α ), couple stress parameter ( β) and other parameter entered into the problem are examined with the help of graphs. It is found that the velocity of the fluid is increased when the rotational parameter is increased but decreased in case of couple stress parameter.

Key words: Magnetic field, Mass transfer, Porous medium, Rotational velocity, Thermal diffusion.

[1]. Brinkman, H.C., 1974. A Calculation of viscous force Extend by a flowing fluid on a Dense Swarm of Particles, Journal of Applied Science, A1, 27-34.
[2]. Chawla, S.S., and Singh, S., 1979.Oscillatory flow past a porous Bed. Acta Mechanica, 34, 205-213.
[3]. Yamamota, K., and Yoshida, Z., 1974. Flow through a Porous Wall With Convection Acceleration, Journal of Phys Soc. Japan, 37, 774-779.
[4]. Nield, D.A., and Bejan, A., 1998. Convection in Porous Media, 2nd edition, Springer-Verlag, Berlin
[5]. Aero, E.L. Bulganian, A.N., and Kuvshinski, E.N., 1965.Asymmetric Hydrodynamics, Journal of Applied Mathematics and Mechanics, 29, 333-346.
[6]. Lukaszewicz, G., 1999. Micropolar fluids,Theory and Applications, Birkhuaser, Berlin.
[7]. Raptis, A., Peredikis, C., and Tzivanidis, G., 1981. Free convection flow through a Porous Medium Bounded by a Vertical Surface, Journal of Phys. D. Appl. Phys., 14L, 99-102.
[8]. Raptis, A., and Takhar, H.S., 1999. Polar fluid through a Porous Bed, Acta Mechanica, 135, 91-93.
[9]. Saxena, S.S., and Dubey, G.K., 2011. Unsteady MHD Heat and Mass Transfer free Convection flow of Polar fluids Past a Vertical Moving Porous plate in a Porous Medium with Heat Generation and Thermal diffusion, Advances in Applied Science Research, 2(4), 259-278.
[10]. Jain, N.C., Chaudhary, D., and Jat, R.N., 2010. Effects of Radiation and Couple stress parameters on unsteady Mangetopolar free Convection Flow with Mass Transfer and Thermal Radiation in Slip flow regime, Journal of Energy, Heat and Mass Transfer, 32, 333-346.

Abstract: In this paper some particular harmonic numbers n, where n divides σ(n) completely have been taken. Three types of numbers prime factorised as say, 𝑝𝑚𝑞,2𝑘𝑝𝑚𝑞,2𝑘𝑝1𝑝2……𝑝𝑚 have been discussed and some propositions have been developed to understand the properties of these type of numbers. It has been observed that harmonic numbers n having 𝑔.𝑐.𝑑. 𝑛,𝜎 𝑛 =𝑛 of the form 𝑝𝑞,𝑝2𝑞,𝑝𝑚𝑞 does not exist if p,q both of them are odd primes. Further some useful properties of the harmonic numbers of the form 2𝑘𝑝𝑚𝑞, for some values of m have been discussed with the help of some propositions. After that an algorithm has been proposed to get the numbers of the form 2𝑘𝑝1𝑝2……𝑝𝑚, where n divides σ(n) completely.

Keywords: Harmonic number, Mersene prime, Ore's conjecture, Perfect number

[1] D.Callan, Solution to Problem 6616, Amer.Math.Monthly 99 (1992),783-789,MR1542194

[2] G.L.Cohen, Numbers whose positive divisors have small integral harmonic mean. Mathematics of computation.Vol. 66, No.218,1997, PP 883-891.

[3] G.L.Cohen and R.M.Soril. Odd harmonic number exceed 1024, Mathematics of Computation, S 0025- 5718(10)02337-9

[4] T. Goto and S.Shobata . All numbers whose positive divisors have integral harmonic mean up to 300, Mathematics of Computation, Vol. 73, No. 245, PP 473-491 [5] T. Goto and K.Okeya. All Harmonic numbers less than 1014, Japan J. Indust.Appl. Math.24(2007), 275-288 [6] M.Gracia. On numbers with integral harmonic mean, Amer. Math. Monthly, 61(1954), 89-96. MR 15:506d

[7] R.K.Guy. Unsolved Problems in Number Theory, 3rd edition. Springer-Verlag, New York, 2004.

[8] G.H.Hardy and E.M.Wright. An Introduction to the Theory of Numbers, fourth edition, Oxford (1962). MR0067125(16:673c)

[9] O.Ore. On the averages of the divisors of a number, Amer.Math.Monthly, 55 (1948), PP 615-619 [10] C. Pomerance. On a problem of Ore: harmonic numbers, unpublished manuscript (1973) see abstract *709-A5, Notice Amer.Math.soc. 20(1973),a-648

Abstract: In this paper, soliton solutions to four Nonlinear Evolution Equations (NLEEs) namely Boussinesq Equation (BE), Gardner Equation (GE), (coupled) Generalized Boussinesq- Burgers Equations (GBBE) and Mikhailov-Shabat system of Equations (MSE) are obtained by the Rational Sine-Cosine Method. It has been demonstrated that the method is a convenient and effective one for solving a wide class of NLEEs encountered in various areas of Nonlinear Physical Sciences. PACS NOS.:02.30Jr, 5.45Yv, 87.10Ed. AMS Mathematics Subject Classification(2010) Nos.:35C08, 35L05, 35Q51, 37K40.

Key words: Soliton, NLEEs, BE, GE, GBBE, MSE, Rational Sine-Cosine Method, Nonlinear Physical Sciences, Plasma Physics, QFT.

[1] Marwan Alquran, Kamel Al-Khaled and HasanAnabeh, New Soliton Solutions for Systems of Nonlinear Evolution Equations by the Rational Sine-Cosine Method, Stud. Math.Sci. 3(1)(2011) 1 – 9.

[2] HasibunNaher and Farah Aini Abdullah, The basic G′/G Expansion Method for the Fourth Order Boussinesq Equation, Applied Mathematics, 3(10)(2012), 1144 – 1152.

[3] J. L. Bona, M. Chen and J. C. Saut,Boussinesq Equation and other Systems for Small Amplitude Long Waves in Nonlinear Dispersive Media I: Derivation and Linear Theory. J. Nonlinear Sci. 12 (2002) 283 – 318.

[4] M.A. Abdel – Razek , A. K. Seddeck and Nassar H. Abdel All: New Exact Jacobi Elliptic Function Solutions for Nonlinear Equations using F-expansion method, Stud.Math.Sci..2(1)(2011) 88-95.

[5] Peter A. Clarkson, Rational Solutions of the Boussinesq Equation, ANAL. APPL.6(4)(2008) 349 – 369.

[6] Sh. SadighBehzadi, Numerical Solution of Boussinesq Equation using Modified AdomianDecomposition and Homotopy Analysis Methods, IJM2C (01) (01)(2011)45 – 58.

[7] TianBao Dan, Qiu Yan-Hong and Chen Ning, Exact Solutions for a class of BoussinesqEquation, Appl. Math. Sci.3 (6) (2009) 257 – 265.

[8] Bin Zheng, Travelling Wave Solutions For Some Nonlinear Evolution Equations By The First Integral Method, WSEAS TRANSACTIONS on MATHEMATICS 10 (8) (1011) 249– 258.

[9] Mohammad Najafi, MaliheNajafi and SomayeArbabi, Application of the He's Semi-Inverse Method for Solving the Gardner Equation, Int.J. Modern Math. Sci. 5(2)(2013)59– 66.

[10] Mohammad Najafi, MaliheNajafi and SomayeArbabi, New Exact Traveling Wave Solutions of Gardner Equation, Int. J. Modern App. Physics 2(1) (2013) 34-47.

Abstract: Patterns of Pythagorean triangles, in each of which the ratio Area/ Perimeter is represented by some polygonal number. A few interesting relations among the sides are also given.

Keyword: Polygonal number, Pyramidal number, Centered polygonal number, Centered pyramidal number, Special number

[1]. Albert H.Beiler, Recreations in the Theory of Numbers, (Dover Publications, New York, 1963).
[2]. Bhatia B.L., and Supriya Mohanty, Nasty numbers and their characterizations, (Mathematical Education, P.34-37, July – Sep
1985).
[3]. Dickson L.E., History of the Theory of number, (Chelesa publishing Company, New York, Vol II, 1952).
[4]. Malik S.B., Basic Number theory, (Vikas Publishing house pvt Ltd., New Delhi, 1998).
[5]. Mordell L.J., Diophantine equations, (Academic press, New York, 1969).
[6]. Ivan Niven, Zuckermann, Herbert.S, and Montgomery, Hugh.L, An introduction to the Theory of Numbers, (John Wiley and Sons,
Inc, New York, 2004).
[7]. Gopalan M.A., and Devibala.S., Pythagorean Triangle: A Tressure house, proceeding of the KMA national seminar on Algebra,
Number theory and applications to coding and Cryptanalysis, Little Flower College, Guruvayur, 2004, Pp.16-18.
[8]. Gopalan M.A., and Anbuselvi.R., A Special Pythagorean Triangle, Acta Ciencia Indica, VolXXXIM, No.1, 2005, Pp.53.
[9]. Gopalan M.A., and Devibala.S., On a Pythagorean Triangle Problem, Acta Ciencia indica, Vol.XXXIIM, No.4, 2006, Pp.1451.
[10]. Gopalan M.A., and Gnanam.A, A Special Pythagorean Triangle, Acta Ciencia Indica, VolXXXIII M, No.4, 2007, Pp. 1435.

Abstract: This paper is concerned with the transient unsteady state thermoelastic problem of thin circular annular fin due to heat source .The transient heat conduction equation is solved by Marchi Zgrablich and Laplace transform with radiation boundary condition. The solution of the problem in the form of infinite series of Bessel function. To determine temperature distribution for heating process and their stresses .Numerical calculations are carried out by using mathematica software.

Keywords - Transient distribution, integral transform, heat source, circular annular fin, and transient heating

[1] Boley, B.A. and Weiner, J.H., Theory of thermal stresses, Johan Wiley and Sons, New York, (1960)
[2] Carrier,W.H. and Aderson ,S.W., The resistance of heat flow through finned tubing ,heating piping and Air conditioning ,Vol.10 ,1944,pp.304-320.
[3] Carslaw, S.H., and Jaeger J.C., Conduction of heat in solid, second edition, oxford University Press, New york (1959)
[4] Crandall S.H., DahlN.C. and Lardner T.J. ,An introduction to the mechanics of solids ,McGraw-Hill, New york(1978)
[5] Gardner K.A., Efficiency of extended surfaces, transaction of American society of mechanical engineers, vol.67, 1945, pp621-631.
[6] Harper, D.R., and Brown W.B. , Mathematica; equation for Heat conduction in Fins of air –cooled Engines .NACA Report No.158,(1922).
[7] Marchi E. and Zgrablich, J., Heat conduction in hollow cylinder with radiation processing Edinburg math .Soc. Vol.14 1964(Series 11) part 2pp.159-164.
[8] Murray W.M., Heat dissipation through an annular disc or fin of uniform thickness, Journal of applied mechanics Vol.5Transaction of the American society of mechanical engineers VOl.60, 1938, pA-78
[9] Wu, S.S., Analysis of transient thermal stresses in annular fin, Journal of thermal stresses, vol 20, 1997,pp591-615.
[10] Deshmukh K.C. and Kedar D.G., Estimation of temperature distribution and thermal stresses in a thick circular plate ,Afre.J.Math Comp.Sci.Res.4 (B) pp389-395, 2011

Abstract: The analysis of the effect of MHD oscillatory flow of elastico Viscous blood through porous medium in a stenosed artery has been presented. Here we assume that the blood behaves as a elastico-viscous fluid(Rivlin-Ericksen type). The expressions for velocity of blood, instantaneous flow rate, wall shear stress and resistive impedance have been obtained. Results obtained have been discussed graphically.

[1]. Bhattacharya, T.K. : Simple blood flow problem in the presence of magnetic field, Jour. of M.A.C.T., Vol. 23, p-17 (1990).
[2]. Daly, B.J.: Pulsatile flow through canine femoral arteries with lumen constrictions, Jour. of Biomechnics, Vol. 9,p-465 (1976).
[3]. Halder, K.: Oscillatory flow of blood in a stenosed artery in the presence of magnetic field, Bulletin of Mathematic Biology, Vol. 49(3), p-279 (1987).
[4]. Kumar, A. and Singh, K.K.: Oscillatory flow of blood in a stenosed artery in the presence of magnetic field, Acta Ciencia Indica, Vol. XXXIVM, No. 2, p-805 (2008).
[5]. Kumar, S.: Oscillatory MHD flow of blood in a stenosed artery, Proc. 65th Annual Conference of IMS, APS Univ., Rewa (1999).
[6]. Liang, S. B. and Ma, H. B.: Oscillating motions of slug flow in capillary tubes, Int. Comm. Heat Mass Transfer, Vol. 31(3), p. 365. (2004).
[7]. Mazumdar, H.P. et. al.: Some effects of magnetic field on a Newtonian fluid through a circular tube, Jour. Pure Appl. Math. Vol. 27(5), p. 519 (1996).
[8]. Newman, D.L., Westerhof, N. and Sippema: The oscillatory flow in a rigid tube with stenosis, Jour. Biomechanics, Vol.12, p. 229 (1979).
[9]. Quadrio, M. and Sibilla, S.: Numerical simulation of turbulent flow in a pipe oscillating around its axis, Jour. of Fluid Mechanics, Vol. 424, p-217 (2000).
[10]. Rathod, V.P., Tanveer. S., Rani, I.S. and Rajput, G.G.: Steady blood flow with periodic body acceleration and magnetic field through an exponentially diversing vessel, Acta Ciencia Indica,Vol. XXX M, No.1, p. 5 (2004).

Abstract: The quintic Diophantine equation with five unknowns given by 5 5 3 3 2 2 2 x  y  xy(x  y )  34((x  y)(z w )P is analyzed for its infinitely many non-zero distinct integral solutions. A few interesting relations between the solutions and special numbers namely, centered polygonal numbers, centered pyramidal numbers, jacobsthal numbers, lucas numbers and kynea numbers are presented.

Keywords: Quintic equation with five unknowns, Integral solutions, centered polygonal numbers, centered pyramidal numbers.

[1]. L.E.Dickson, History of Theory of Numbers, Vol.11, Chelsea Publishing company, New York (1952).
[2]. L.J.Mordell, Diophantine equations, Academic Press, London(1969).
[3]. Carmichael ,R.D.,The theory of numbers and Diophantine Analysis,Dover Publications, New York (1959)
[4]. M.A.Gopalan & A.Vijayashankar, An Interesting Diophantine problem
3 3 5 x  y  2z Advances in Mathematics, Scientific
Developments and Engineering Application, Narosa Publishing House, Pp 1-6, 2010.
[5]. M.A.Gopalan & A.Vijayashankar, Integral solutions of ternary quintic Diophantine equation
2 2 5 x  (2k 1)y  z ,International Journal of Mathematical Sciences 19(1-2), 165-169,(jan-june 2010)
[6]. M.A.Gopalan,G.Sumathi & S.Vidhyalakshmi, Integral solutions of non-homogeneous Ternary quintic equation in terms of pells
sequence
3 3 5 x  y  xy(x  y)  2z Journal of Applied Mathematical Sciences,vol.6,No.1,59-62,April.2013.
[7]. M.A.Gopalan,G.Sumathi and S.Vidhyalakshmi ''Integral solutions of non- homogeneous quintic equation with three unknowns
2 2 2 5 x y xy x y 1 (k 3) z n        ''International J. of innovative Research in science engineering and
tech.Vol.2,issue.4,920-925,April.2013.
[8]. S.Vidhyalakshmi, K.Lakshmi and M.A.Gopalan, Observations on the homogeneous quintic equation with four unknowns
5 5 5 2 2 2 x  y  2z 5(x  y)(x  y )w ,accepted for Publication in International Journal of Multidisciplinary
Research Academy (IJMRA).
[9]. M.A.Gopalan,S.Vidhyalakshmi and A.Kavitha On the quintic equation with four unknowns x y k t z w 4 4 2 2 4   2(  )
accepted for publication in Bessel J. of Mathematics10.M.A.Gopalan,G.Sumathi and S.Vidhyalakshmi,''Integral solutions of the
non--homogeneous quintic equation with four unknowns
( ) 52 4 (1 7 ) 5 5 4 4 4 2 x  y  x  y z  w z  z  w ''Bessel J. of Mathematics,3(1),175-180,2003.