#### Volume-8 ~ Issue-5

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Paper Type |
: | Research Paper |

Title |
: | The Complex Quantum-State of Black-Hole and Thermostatistics. |

Country |
: | India |

Authors |
: | Dr. Narayan Kumar Bhadra |

: | 10.9790/5728-0850119 |

**Abstract:** In this paper the quantum aspects are described in-detail with the help of a new type of energy source called latent energy group SU(6) of the super unified theory of SU(11). The thermodynamics of general self gravitating systems created by the energy group SU(6) and some related topics such as complex space-time (i.e. Pseudo-Space-Time) are also briefly discussed. The thermodynamic connection is based on Hawking celebrated application of quantum theory to Black-Hole.

**Keywards:** finite difference, explicit, diffusion equation, soil moisture.

[1]. Hawking S.W(1984): The quantum state of the universe, Nucl. Phys. B239.257.

[2]. Hoyle F. and Naralikar J.V(1964): A new theory of gravitation. Proc. R. Soc., A282.191.

[3]. Bhadra N.K(2012):The complex Model of the Universe, IOSR-JM, ISSN: 2278-5728, vol.2, 4, pp-20; and The complex model of the quantum universe,vol.4, 1, pp-20.

[4]. Einstein, A. de-Sitter, W. (1932): On the relation between the expansion and mean density of the universe. Proc. Natl. Acad. Sci.,(USA), 18,213.

[5]. Davies, P.C.W. (1974): The Physics of Time Asymmetry(Surrey University Press/ University of California Press) – 1976a Proc. R. Soc. A. 351 139; -1976b Nature 263 377; -1977a Proc. R. Soc. A. 353 499; -1977b Space and Time in the Modern University (Cambridge : Cambridge University Press).

[6]. Mendeez, V. and Pavon, D.(1996): Gen. Rel. Grav., 28. 697; Birrell N. D and Davies, P.C.W 1978 Nature 272 35

[7]. Bekenstein, J.D.1973 Phys. Rev. D. 72333; Bertin, G., and Radicati, L.A 1976 Astrophys.206 825;.

[8]. Candelas P., and Seiama, D.W. 1977 Phys. Rev. Lett. 38 1372; Carr, B. 1977 Mon, Not. R. Astrom. Soc. 181 293; Carter B 1973 Black-Holes ed DeWitt and DeWitt (London: Gordon and Breach) Casimir HBG 1948 Proc. Kon. Ned. Akad. Wetens chap. 51 793.

[9]. Christensen, S.M. and Fulling, S.A. 1977 Phys. Rev. D15 2088;Einstein Albert (1987): Ideas and Opinions, Crown Publishers, New York, pp-348.

[10]. Halliwel, J.J. Hawking, S.W., (1985): The Origin of Structure in the Universe, Phys. Rev. D 31 1777. Hawking, S.W.(1985): The Arrow of Time in Cosmology Phys. Rev., D 32 2489. 1.

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Paper Type |
: | Research Paper |

Title |
: | Fixed points of self maps in dp – complete topological spaces |

Country |
: | India |

Authors |
: | V. Naga Raju, V. Srinivas |

: | 10.9790/5728-0852023 |

**Abstract:** The purpose of this paper is to prove some fixed point theorems in dp - complete topological spaces
which generalize the results of Troy L Hicks and B.E.Rhoades[6 ].

**Keywords :** dp - complete topological spaces, d-complete topological spaces, orbitally lower semi continuous
and orbitally continuous maps.

[1] J. Achari, Results on fixed point theorems, Maths. Vesnik 2 (15) (30) (1978), 219-221.

[2] Ciric, B. Ljubomir, A certain class of maps and fixed point theorems, Publ. L'Inst. Math. (Beograd) 20 (1976), 73-77.

[3] B. Fisher, Fixed point and constant mappings on metric spaces, Rend. Accad. Lincei 61(1976), 329 – 332.

[4] K.M. Ghosh, An extension of contractive mappings, JASSY 23(1977), 39-42.

[5] Troy. L. Hicks, Fixed point theorems for d-complete topological spaces I, Internet. J. Math & Math. Sci. 15 (1992), 435-440.

[6] Troy. L. Hicks and B.E. Rhoades, Fixed point theorems for d-complete topological spaces II, Math. Japonica 37, No. 5(1992), 847-

853.

[7] K. Iseki, An approach to fixed point theorems, Math. Seminar Notes, 3(1975), 193-202.

[8] S. Kasahara, On some generalizations of the Banach contraction theorem, Math. Seminar Notes, 3(1975), 161-169.

[9] S. Kasahara, Some fixed point and coincidence theorems in L-Spaces, Math. Seminar Notes, 3(1975), 181-187.

[10] M.S. Khan, Some fixed point theorems IV, Bull. Math. Roumanie 24 (1980), 43-47.

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**Abstract:** The main aim of this research work is to construct a statistical model for the determination of gateway prices of yam in Abia State, which could also be used to predict reliable and dependable future price values of yam in Abia State and Nigeria in general. It intends to estimate and isolate trend and seasonal components of time series present in the data, using Least Squares Dummy Variable Analysis Approach. The results obtained showed that monthly dummy variables are all positively related to gateway price of yam, which implies that as the months go by, the gateway prices of yam in Abia State keep increasing in arithmetic progression. Again, the monthly dummy variables are highly statistically significant at 5% level of significance, meaning that the gateway prices are highly influenced by the months especially, the festive and farming seasons. The deseasonalized values obtained were used to forecast the gateway price value of yam in Abia State, Nigeria for 2011.

**Key words: **deseasonalization, gateway price, yam and Dummy Variable.

[1]. Ahmed, H. Y. and Sobhi, M. R. "A Comparative Study for Estimation Parameters in Panel Data Model". Journal of Econometrics, 2009, pp 1 – 15.

[2]. Alper, C. E. and S. B. Aruoba, "Deseasonalizing Macroeconomic Data: A Caveat to Applied Researchers in Turkey " Journal of Econometrics, 2001, pp 1- 17

[3]. Box, G.E.P. and G.M. Jenkins "Time series Analysis Forecasting and Control". San Francisco: Holden Day 1976.

[4]. Coursey, J. "Production of Improved Planting Materials- Propagated by Seed, Agriculture in the Tropical" Longman, 2001, p264.

[5]. Francis, X. D. "Elements of Forecasting" 2nd ed., South- Western Publishing 2001.

[6]. Food and Agriculture Organization "Production Year Book Rome", 2008, p 39.

[7]. Gujarati, D. N.(2006), "Essentials of Econometrics" 4th ed. McGraw – Hill, New York.

[8]. Gujarati, D. N. and Porter, D. C. (2009), "Basic Econometrics", 5th ed. McGraw – Hill, New York.

[9]. Hylleberg, S., R. Engle, C. Granger and B. Yoo, "Seasonal Integration and cointegration" Journal of Econometrics, 1990, vol.44, pp 215 – 238

[10]. Hylleberg, S. "Modeling Seasonality Advanced Texts in Econometrics", Oxford University Press, New York, 1992, pp 476.

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Paper Type |
: | Research Paper |

Title |
: | Projecting the population size of Dhaka city with migration using growth rate method |

Country |
: | Bangladesh |

Authors |
: | Masuma Parvin |

: | 10.9790/5728-0852937 |

**Abstract:** This paper examines the effect of net migration on prospective population growth in Dhaka City for
the next several years. The paper deals with the urban challenges in Bangladesh focusing on rapid urban
growth in the megacity of Dhaka. Here Population of Dhaka city has been predicted with the help of an
ordinary differential equation model known as Malthusian Exponential population model which is
parameterized by growth rate. In order to include the immigrant population, we make necessary modification of
the model, which is again an exponential model where the growth rate 𝑅 is the sum of the actual growth rate
𝑎 and immigrant rate 𝑟 .We use fourth order Runge-Kutta scheme for the numerical solution of the
autonomous and non-autonomous case where we incorporate the growth rate as a function of time. We perform
error estimation of the numerical solution which justifies the correctness of the implementation by using
computer programming. The procedure used in this study is by comparing two projected population scenarios
one with constant growth rate and the other is time dependent growth rate based on the latest data collected
through surveys of population censuses and relevant studies.

**Keywords: **Exponential model, growth rate method, migration, population prediction, urban structure.

[1] M. Braun, Differential Equation and their Application (4th ed. Narosha Publishing House, 1993).

[2] N. Hasan, Population Projection in Dhaka City Based on Malthusian Model, M.Sc. Thesis, Dept. of Mathematics, Jahangirnagar University, 2008.

[3] S.Hossain, Rapid Urban Growth and Poverty in Dhaka city, Bangladesh e-journal of sociology, 5(1), 2008, 57-80.

[4] B.B.S. (2007), Population Census – 2001(Community Series, Zilla: Dhaka), Bangladesh Bureau of Statistics, Planning Division, Ministry of Planning, Government of Bangladesh.

[5] C.M. Arun, Projections of Population, Enrolment and Teachers: Module on enrolment and population projections, Fellow, ORSM Unit, National Institute of Educational Planning and Administration, 17-B, January 18 2004.

[6] Walter, W., Ordinary Differential Equation (Springer., 305-309, 1998).

[7] Dhaka: Improving Living Conditions for the Urban Poor, Bangladesh Development Series, Paper No. 17, The World Bank Office, Dhaka, June 2007.

[8] N. Islam, Dhaka city: Some general Concerns, Centre for Urban Studies, Dhaka. 3(6), 1999, 71-82.

[9] M. A. Mabud, Bangladesh‟s Population Projection, 2001-2101, Organization for Population and Poverty Alleviation (OPPA), Dhaka, 2008

[10] S.A. Sinthia, Sustainable Urban Development of Slum Prone Area of Dhaka City, World Academy of Science, Engineering and Technology , 2013,75.

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Paper Type |
: | Research Paper |

Title |
: | Almost Closed and Continuous Maps |

Country |
: | India |

Authors |
: | Asha Gupta, Kamal Kishore |

: | 10.9790/5728-0853839 |

**Abstract:** The primary purpose of this paper is to establish the relation between continuous maps and almost
closed maps. In this endeavor some known results are improved.

**Key Words: **Almost closed, , almost compact, continuous, fibers, Frechet space.
AMS SUBJECT CLASSIFICATION CODES.54C05 ,54C10.

[1]. A. Wilansky, Topology for analysis Xerox College Publishing Lexington, Massachusetts, Toronto,1970.

[2] . W.J. Thron, Topological Structures , Holt, Rinehart and Winston.1966.

[3]. R.V.Fuller, Relations among continuous and various non continuous functions, Pacific J. Math. 25, (1968)495-509.

[4]. Asha Goel & G.L. Garg, On Maps:Continuous,Closed, Perfect, and with Closed Graph, International Journal of Mathematics &

Maths. Sciences, Florida, USA, Vol. 20, No. 2(1997) 405-408.

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Paper Type |
: | Research Paper |

Title |
: | Interval valued fuzzy generalized semi-preclosed mappings |

Country |
: | India |

Authors |
: | R. Jeyabalan, K. Arjunan |

: | 10.9790/5728-0854047 |

**Abstract:** In this paper, we introduce interval valued fuzzy generalized semi-preclosed mappings and interval valued
fuzzy generalized semi-preopen mappings. Also we investigate some of their properties.
AMS Subject Classification(2010): 54A40.

**Keywords:** interval valued fuzzy set, interval valued fuzzy topological space, interval valued fuzzy continuous
mapping, interval valued fuzzy generalized semi-preclosed set, interval valued fuzzy generalized semi-preopen set.

[1] Andrijevic. D, Semipreopen Sets, Mat.Vesnic, 38, (1986), 24-32.

[2] Bhattacarya. B., and Lahiri. B. K., Semi-generalized Closed Set in Topology, Indian Jour.Math.,29 (1987), 375-382.

[3] Chang. C. L., FTSs. JI. Math. Anal. Appl., 24(1968), 182-190.

[4] Dontchev. J., On Generalizing Semipreopen sets, Mem. Fac. sci. Kochi. Univ. Ser. A, Math.,16, (1995), 35-48.

[5] Ganguly. S and Saha. S, A Note on fuzzy Semipreopen Sets in Fuzzy Topological Spaces, Fuzzy Sets and System, 18, (1986), 83-96.

[6] Indira. R, Arjunan. K and Palaniappan. N, Notes on interval valued fuzzy rw-Closed, interval valued fuzzy rw-Open sets in interval valued

fuzzy topological space, International Journal of Computational and Applied Mathematics.,Vol .3,No.1(2013), 23-38.

[7] Jeyabalan. R, Arjunan. K, Notes on interval valued fuzzy generalaized semipreclosed sets, International Journal of Fuzzy Mathematics

and Systems.,Vol .3,No.3(2013), 215-224.

[8] Levine. N, Generalized Closed Sets in Topology, Rend. Circ. Math. Palermo,19,(1970),89-96.

[9] Mondal. T. K., Topology of Interval Valued Fuzzy Sets, Indian J. Pure Appl.Math.30 (1999), No.1, 23-38.

[10] Malghan. S. R and Benchalli. S. S, On FTSs, Glasnik Matematicki, Vol. 16(36) (1981), 313-325.

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**Abstract:** This paper concerns the numerical approximation of a class of second order linear ordinary
differential equations by Perturbed Integral Collocation Method.
The application of the method resulted to system of linear algebraic equations which are then solved by
Gaussian elimination method. The results obtained for some illustrative examples showed that the Perturbed
Integral Collocation Method is efficient, accurate and reliable.
Examples are given to illustrate the method.

**Key words:** Perturbed Integral Collocation, Polynomials and Chebyshev AMS Subject Classification: 65L10
+ Author to communicate concerning the paper

[1] Fox, L. (1962) . Numerical Solution of Ordinary Differential Equations and Partial Differential Equations. Pergamum, Oxford.

[2] Gerald, D. (2005). Numerical Methods in Engineering and Sciences. 7th ed., Kanna Publishers, Delhi.

[3] Hosseini Allahadi M. (2005). Solving Ordinary Differential Equation using the Perturbation term of the Tau Method over Semi -

Infinite Intervals. Far East J. Appli. Math. 4(), 295 - 303

[4] Taiwo, O . A and Olagunju, A. S. (2006). Perturbed Segmented Domain Collocation Tau Method for the Numerical Solution of

Second Order Boundary Value Problems. J. of NAMP, 10, 293 – 298

[5] Taiwo, O . A. and Olagunju, A . S. (2011). Chebyshev Methods for the Numerical Solution of 4th Order Differential Equations.

Pioneer J. of Mathematics and Mathematical Sciences. 3(1), 73 – 82.

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**Abstract:** In this paper, we employed Standard and Perturbed Integral Collocation methods to find numerical
solution of ordinary differential equations. Power series form of approximation is used as basis function and
Chebyshev Polynomial is used as perturbation term in the case of perturbed Integral Collocation Method.
Numerical Computations are carried out to illustrate the application of the methods and also the results
obtained by the methods are compared in terms of accuracy and computations involved in the two methods. Two
examples each of first and second orders linear integro differential equations are solved to demonstrate the
methods.

[1] Aliyu, I . M. (2012) A computational error estimation of the Integral Formulation of the Tau Method for some class of ordinary differential equations. Ph. D Thesis (Unpublished), University of Ilorin, Nigeria.

[2] Asady, B. and Kajani, M.T. (2005). Direct Method for solving Integro Differential Equation using Hybrid Fourier and Block Pulse Function. Intern. J . Computer Math., 82(7), 989-995.

[3] Ayad, A. (1996). Spline Approximation for the First Order Fredhom Integral Differential Equations. University Babes Bolyai, Studia, Mathematics, 41(3), 1-8

[4] Behiri, S . H. and Hashish H. (2002). Wavelet Methods for the numerical solution of Fredholm Integro Differential Equations. Intern. J. Applied Mathematics. 11(1), 27-33

[5] Danfu, H . and Xafeng, R. C. (2001). Numerical solution of Integro Differential Equations by using Walet Operational Matrix of Integration second kind. Applied Math. Computer. 194(2), 460-466

[6] Golbadai, H.and Javidi, M. (2007). Application of He's Homotopy Perturbation Method for the nth order Integro Differential Equations. Applied Math. Computer, 190(2), 1409 – 1419

[7] Hussein, J. Omar A. and Al-Shara, S (2008). Numerical Solution of Linear Integro Differential Equations. J. Mathematics and Statistics. 4(4), 250 – 254

[8] Lekestani, M. Razzaghi, M. and Dhglhan. M. (2005). Semi Orthogonal Spline Wavelet's Approximation for Fredholm Integro Differential Equations. Hindawi Publishing Corporation, Mathematical Problem in Engineering. 2, 1-12

[9] Maleknejad, K. and Mirzace, F. (2006). Numerical solution of Integro Differential Equations using Rationalized Haar Functions Method. Keynote Intern. J. Syst. Math. 35, 1735-1744

[10] Raji, M. T. (2013). Collocation Approximation for solving Special Linear and Nonlinear Integro Differential Equations. Ph.D. Thesis, (Unpublished), University of Ilorin, Ilorin, Nigeria.

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Paper Type |
: | Research Paper |

Title |
: | Hypergeometric functions of three variables in terms of integral representations |

Country |
: | India |

Authors |
: | Showkat Ahmad |

: | 10.9790/5728-0856773 |

**Abstract:** Complete triple Hypergeometric functions of the second order which were denoted
by 𝐻𝐴1 , 𝐻𝐵1 , 𝐻𝐶1 . Each of these triple Hypergeometric functions has been investigated extensively in many
different ways including for example in the problem of finding their integral representations of one kind or
other. Here in this paper we aim at presenting further integral representations for each triple Hypergeometric
functions 𝐻𝐴1 , 𝐻𝐵1 , 𝐻𝐶1.

**Keywords: **Beta and Gamma functions, Picard's integral formula, Generalized Hypergeometric functions,
Gauss Hypergeometric functions, triple Hypergeometric functions, Exton's Hypergeometric functions, Appell
functions.

[1]. Carlson, B. C. (1963). Lauricella's hypergeometric function 𝐹𝐷. J. Math. Anal. Appl. 7, 452-470.

[2]. Hasanov, A. Srivastava, H. M and Turaev, M. (2006). Decomposition formulas for some triple hypergeometric function. J. Math. Anal. Apple. 324, 955- 969.

[3]. Saran, S. (1954). Hypergeometric functions of three variables. Ganita, 5, 71- 91; see also Corrigendum, Ganita, 7 (1956), 65.

[4]. Srivastava, H. M (1964).Hypergeometric function of three variables. Ganita 15, 97-108.

[5]. Srivastava, H. M (1967). Some integrals representing triple hypergeometric functions, Rend. Circ. Mat. Palermo (Ser. 2), 16, 99-115.

[6]. Srivastava, H.M. and Manocha, H. L. (1984). A Treatise on generating Functions. Halsted Press, John wiley and sons, New York, Chichester, Brisbane and Toronto.

[7]. Srivastava, H. M. and Karlsson, P. W. (1985). Multiple Gaussian hypergeometric series. Halsted Press (Ellis Horwood limited, Chichester, Brisbane and Toronto.

[8]. Turaev, M. (2009). Decomposition formulas for Srivastava's hypergeometric- ric function 𝐻𝐴 on Saran function. J. Compute. Appl. Math, 233, 842- 846.

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**Abstract:** The effect of heat transfer on unsteady MHD oscillatory flow of Jeffrey fluid in a horizontal channel with chemical reaction has been studied. The temperature prescribed at plates is uniform and asymmetric. A perturbation method is employed to solve the momentum and energy equations. The skin frictions, the Nusselt numbers and Sherwood are evaluated using perturbation technique. The effects of various dimensionless parameters on velocity and temperature profiles are considered and discussed in details through graphs and tables. It is found that, the velocity u increases with increase in Gr, Gc, N and Re. The velocity also increases with decrease in λ1,Ha, Sc and ω. The velocity only decreases with increase in Pe. It is also observed that the temperature θ decreases with increase in N, Re and ω. Decrease in Schmidt number Sc, chemical parameter Kc and frequency of oscillation ω increase the species concentration or the concentration boundary layer thickness of the flow field.

**Keywords:** Heat Transfer, Unsteady, MHD, Jeffrey Fluid, Oscillatory Flow

[1]. Aruna Kumari B., Ramakrishna Prasad K., Kavita K. (2012): Slip Effects on MHD Oscillatory Flow of Jeffrey Fluid in a Channel with Heat Transfer. Int. J. Math. Arch. 3 (8):2903 – 2911.

[2]. Asadulla M., Umar K., Nareed A., Raheela M. and Mohyud – Din S.T. (2013): MHD Flow of a Jeffrey Fluid in Converging and Diverging Channels. Int. J. Mod. Math. Sci. 6 (2): 92 – 106.

[3]. Aziz A. and Na T.Y. (1984): Perturbation Methods in Heat Transfer. Hemisphere, New York.

[4]. Badari Narayana C.H, Sreenadh S. and Devaki P. (2012): Oscillatory Flow of Jeffrey Fluid in Elastic tube of Variable Cross – Section. Adv. Appl. Sci. Res. 3 (2): 671 – 677.

[5]. Bharali A. and Borkakati A.K. (1980): The Flow and Heat Transfer between Two Horizontal Parallel Plates. Journal of the Physical Society of Japan. 49 (5): 2091.

[6]. Bharali A. and Borkakati A.K. (1980): The Heat Transfer in an Axisymmetric Flow between Two Parallel Porous Disk under the Effect of a Transverse Magnetic Field. Journal of the Physical Society of Japan. 52 (1): 6.

[7]. Bodosa G. and Borkakati A.K. (2003): MHD Couette Flow with Heat Transfer between Two Horizontal Plates in the Presence of Uniform Transverse Magnetic Field. Theor. Appl. Mech. 30 (1): 1 – 9.

[8]. Devika B., Satga Narayana P.V. and Venkatarmana S. (2013): MHD Oscillatory Flow of a Viscous Elastic Fluid in a Porous Channel with Chemical Reaction. Int. J. Eng. Sci. Inv. 2(2): 26 – 35.

[9]. Israel – Cookey C., Nwaigwe C. (2010): Unsteady MHD Flow of a Radiating Fluid over a Moving Heated Porous Plate with Time – Dependent Suction. Am. J. Sci. Ind. Res. 1 (1): 88 – 95.

[10]. Israel – Cookey C., Omubo – Pepple V.B. and Tamunoberetonari I. (2010): On Steady Hydromagnetic Flow of a Radiating Viscous Fluid through a Horizontal Channel in a Porous medium. Am. J. Sci. Ind. Res. 1 (2): 303 – 308.