iosr-Jm   Volume-10 ~ Issue-3 ~ May-June 2014

Abstract: The object of this paper is to study a measure 𝜇 on (ℝn, Σ, ) and to introduce the concepts of measurable manifold (M, Σ1, 𝓣1) and measure manifold (M, Σ1, 𝓣1, 𝜇1). Here we introduce the concept of measurable charts and measureable atlases and define measure 𝜇1 restricted to them respectively, we extend the study of Heine Borel property on (M, Σ1, 𝓣1, 𝜇1), that is measure invariant.
Keywords: 𝜎-algebra, Heine Borel property, Measureable charts, Measureable Atlases, Measureable Manifold, Measure 𝜇, Measure chart, Measure Atlas, Measure Manifold.

[1]. Jonathan Weitsman, "Measure on Banach Manifold, Random Surfaces, And Non Perturbative String Field Theory with Cut-Offs".arXiv:0807.2069v1[math- ph] (2008) (pp-1 - 27)
[2]. Joaquim Ortega-Cerda and Bharthi Pridhnani, "Carleson Measures and Logivinenko-Sereda Sets On Compact Manifolds".arXiv:1004.2456v1[math.CA ] (2010) (pp-1 - 19)
[3]. T . Samuel , "A simple proof of vitali theorem for signed measure" The Mathematical Association of America, V-120-07, (2013) (PP-654 – 659).

[4]. Antoni A Kosinski ,Differential Manifolds (Academic Press, Har court Brace, Jovanorich Publisher Bostonsan Diego, New York, London)(1993).
[5]. C.Wayne Patty, Foundations of Topology ( Second Edition, Jones and Bartlett India. PVT.LTD New Delhi, Canada, London) (2012).

Abstract: A set of m Gaussian integers is called a complex Diophantine m-tuple with the property D (z) if the product of its any two distinct elements increased by z is a square of a Gaussian integer. In this paper, we present Gaussian-Diophantine quadruples with property D(1).Few examples of complex Diophantine quadruples with the property D (1) are presented.
Keywords: Diophantine quadruples, Integral solutions, Gaussian integers, Pell equation

[1]. Balker.A, Duvemport.H; "The equations
2 2 2 2 3x  2  y and 8x 7  z ",. Quart.J.Math.Oxford Ser, 1969,20
(2),129-137.
[2]. Hoggatt.V.E,Bergum.G.E,"A Problem of Fermat and the Fibonacci Sequence",Fibonacci Quart,1977;15:323-330.
[3]. Horadam.A.F,"Generalization of a result of Morgudo",Portugaliae Math,1987;44:131-136
[4]. Jones.B.E,"Asecond variation on a problem of DiophPantus and Devenport",Fibonacci Quart,1978; 16:155-165
[5]. Long.C,Bergum.G.E,"On a problem of Diophantus,in:Application of Fibonacci Numbers.(Philippou.A.N, Horadam.A.F,
Bergum.G.E,Eds.),Kluwer,Dordrecht,vol.2,1998,183-191

Abstract:The Fuzzy set theory has been applied in many fields such as management, engineering and almost in every business enterprise as well as day to day activities. Ordering fuzzy numbers plays an important role in approximate reasoning, optimization, forecasting, decision making, risk analysis, controlling, scheduling andvarious other usages in our day to day activities. This paper describes a ranking method for ordering fuzzy numbers based on Area, Mode, divergence, Spreads and Weights of generalized (non-normal) hexagonal fuzzynumbers.
Keywords: Ranking function, Hexagonal fuzzy numbers, Centroid points, Area

[1] Jain .R Decision making in the presence of fuzzy variables, IEEE Transactions on systems, Man and Cybernetics 6 (1976) 698-703
[2] Bortolan, G and Degani R. A review of some methods for ranking fuzzy subsets, fuzzy sets and systems 15(1985) 1-19.
[3] Chen S.J and Hwang C.L Fuzzy multiple attribute decision making, Springer, Berlin (1992)
[4] Lee E.S and Li .R.J, Comparison of fuzzy numbers based on the probability number of fuzzy events computers and mathematics with applications 15 (1988) 887-896
[5] Liou, T-S., Wang, M.J., (1992), Ranking fuzzy numbers with integral value. Fuzzy Sets and Systems, 50, pp.247-255.

Abstract: This paper deals with the variations of annual rainfall in New- Bussa based on Markov Chain models. This principle was used to formulate a three state model for annual rainfall distribution in New- Bussa.The model is designed such that if given any of the three state in a given year, it is possible to determine quantitatively the probability of making transition to any other two states in the following year(s) and in the long-run. The result from the model shows that in the long-run 70% of the annual rainfall in New-Bussa will be State1, 27% will be state 2 and 3% of the annual rainfall will be State 3. The model could therefore be used to make a forecast of the annual rainfall pattern. This could provide some information for the Crop and Fish farmers, artisanal Fishermen , the Hydroelectric power generating station and the government that could be used to plan strategies to boost their production in New -Bussa and the environment.
Keywords: Markov Model, Transition probability, annual rainfall, crop and fish farmers

[1]. Allen, D.M. and Haan, C.T., (1975). Stochastic simulation of daily rainfall. Res. Rep. No. 82 Water Resour. Inst., Univ.,Lexington, Kentucky, USA
[2]. Akintunde A.A, Asiribo O.E. Adebanji A.O. Adelakun A.A Agwuegbo S.O.N. (2008) Stochastic modelling of daily precipitation in Abeokuta, proceedings of the third Conference on science and National Development pp108-118 www.unaab.edu.ng/journal/index.php/COLNAS/article/.../150/153. Accessed 15/10/2012.
[3]. Jimoh, O. D. and Webster, P. (1996). Optimum order of Markov chain for daily rainfall in Nigeria. Journal ofHydrology 185: 45-69.
[4]. RatnasinghamSrikanthan and Tom McMahon(2000):Stochastic Generation ofClimate Data, A Review:Technical Report 00/16, Cooperative Research Centre for Catchment Hydrology, Australia
[5]. Gabriel, K. R. and Neumann, J. (1962) A Markov chain model for daily rainfall occurrences at Tel Aviv. Quart. J. Roy. Met. Soc. 88:90-95

Abstract: In this note, anisotropic Bianchi type - I dark energy cosmological model with constant deceleration parameter and with time-dependent equation of state parameter  (t) in bimetric theory of gravitation have been investigated and studied the accelerating as well as decelerating phase of expansion of the universe. We have measured the matter energy density parameter M , the dark energy density parameter  and the equation of state parameter  in it. The parameters m , l and n affected the overall structure of the model.The time dependent equation of state parameter  (t) is goes over to the value which is independent of time and fully depends only on the values of m  0. Our measured values of M ,  ,  and Hubble parameter H< in our model are found to be consistent with the results of WMAP satellite, Bennett, C. L. et al. (2003) and SNe Ia data collaborated with CMBR anisotropy and galaxy clustering statistics, Tegmark, M., et al. (2004). The cosmological term  is a decreasing function of time and has a small positive value at present epoch which< matched with the results from recent supernovae Ia observations. The other geometrical and physical properties of the model have been traced out.
Keywords: Cosmology, Dark Matter, Dark Energy, Gravitational Theory

[1]. C.L. Bennett et al., Astrophysical Journal Supplement Series, 148, 2003, 1.
[2]. D.N. Spergel et al. (WMAP Collaboration), Astrophysical Journal Supplement Series, 148, 2003, 175.
[3]. M. Tegmark et al. (SDSS Collaboration), Physics Review D, 69, 2004, 103501.
[4]. K. Abazajian et al., Astronomical Journal, 29, 2005, 1755.
[5]. K. Abazajian et al. (SDSS Collaboration), Astronomical Journal, 128, 2004, 502.
[6]. K. Abazajian et al. (SDSS Collaboration), Astronomical Journal, 126, 2003, 2081.

Abstract: Hippocrates has squared lunes, circle and a semicircle. He is the first man and a last man. S. Ramanujan is the second mathematician who has squared a circle upto a few decimals of  equal to 3.1415926… The squaring of curvature entities implies that lune, circle are finite entities having a finite magnitude to be represented by a finite number.
Keywords: Squaring, lune, circle, Hippocrates, S. Ramanujan, , algebraic number

[1]. T. Dantzig (1955), The Bequest of the Greeks, George Allen & Unwin Ltd., London.
[2]. P. Dedron and J. Itard (1973) Mathematics and Mathematicians, Vol.2, translated from French, by J.V. Field, The Open University
Press, England.
[3]. W.W. Rouse Ball (1960), A short Account of the History of Mathematics, Dover Publications, New York.
[4]. W.G.H. Kustner and M.H.H. Kastner (1975). The VNR Concise Encyclopedia of Mathematics, Van Nostrand Rusinhold Company.
[5]. R.D. Sarva Jagannadha Reddy (2014). Pi of the Circle at www.rsjreddy.webnode.com

Abstract: Unsteady MHD Poiseuille flow between two infinite parallel porous plates in an inclined magnetic field with heat transfer has been studied. The pressure gradient is considered negligible. The lower plate alone is porous. The momentum and energy equations were solved by variable separable technique. The fluid velocity is expressed in terms of Hartmann number while temperature is in terms of Prandlt number. The effects of various magnetic inclinations to velocity and skin friction with various Prandlt number to temperature and Nulsselt number are discussed by graphs and tables.
Key Words: Magnetohydrodynamics(MHD) flow, Porous Plate, Heat Transfer

[1]. A.K. Al-Hadhrami, L. Elliot, M.D.Ingham, X.Wen,(2003): Flow through horizontal channels of porous materials, Inter. J. of Energy Research 27, 875-889.Raptis, C. Perkidis, (1999): Radiation and Free Convention flow past a moving plate, Int, J. of Applied Mechanics and Engineering, 4(4):817-821.
[2]. Gbadeyan, J.A. Idowu, A.S., Areo, A.O. and Olaleye. O.P.(2010):The radiative effect on velocity, magnetic and temperature fields of a magneto hydrodynamic oscillatory flow past a limiting surface with variable suction. Journal of Mathematical Sciences.21 (4), 395-411
[3]. C.B. Singh,(1993): MHD Steady flow of liquid between two parallel plates, In: Proc. of First Conference of Kenya Mathematical Society, 24-26.
[4]. C.B.Singh, P.c. Ram,(1978): Unsteady MHD Fluid Flow through a channel: Journal of Scientific Research, 28(2).
[5]. D.G.Drake, (1965): Flow in a channel due to a periodic pressure gradient, Quart. Journal of Mech. And Appl. Maths., 18(1).

Abstract: This paper concerns an important application of the (123)-avoiding class of Aunu Permutation studied by earlier by Usman, Magami and Garba under various topics such as enumeration, thin cyclic desighn and association schemes. .In this report, some method of construction are used using accents and descents as enumerated by positive and negative units in a manner that resembles, in some sense, the Motzkin numbers. This method produced some graph models which when superimposed produced the required design of a slanted flag. This paper therefore uncovers yet another rich property of this class of permutation pattern: applications in design theory which interested researchers can further explore.
Key Words: Design theory, Permutation, avoiding pattern, Flag Algorithm, Special class

[1]. Ibrahim A.A and Audu M.S. (2005) Some Group theoretic Properties of Certain Class of (123) and (132)-Avoiding Patterns
Numbers: an enumeration scheme African journal Natural Sciences Vol. 8: 79-84
[2]. Ibrahim A.A. (2005) On the Combinatorics ofn Succession Using a 5-Element Sample. Abacus, Journal of Mathematics
Association of NIgeria., .32(2B):410-415
[3]. Ibrahim A.A. (2008) Some Transformation Schemes Involving the Special (132) – avoiding Permutation Patterns and a Binary
Coding: An Algorithmic Approach Asian Journal of Algebra 1 (1):10-14, Asian Network for Scientific Information (ANSI)
[4]. Sloane N.J.S (2006) Integer Sequence A119626 Online Encyclopeadia of Integer sequence,
http://www.research.att.com/~njas/sequence/?q=119626&sort=0&language=english&go=search
[5]. Ibrahim A.A and Isah G.A (2010) On the Use of Automat Models in the Study of a Class of (123)-avoiding Permutation Patterns:
Applications in Circuit Design and in Number Theory (In process)

Abstract: A graph is said to be an edge sum graph if the edges of G can be labeled with distinct positive integers such that the sum of all the labels incident on a vertex is again an edge label of G and if the sum of any collection of edges is a label of an edge in G, then they are incident on a vertex. The edge sum number< E (G) of a graph G is the smallest number r of edges which added to G result in an edge sum graph. In this< paper, we prove that E 3,4  (J )  3.
Keywords: Sum graph, edge sum graph, sum number, edge sum number

[1] D. Berstrnd, F. Harary et. al. "The Sum number of a complete graph", Bull. Malaysian Math. Soc., (1989) 25-28.
[2] M. N. Elligham, "Sum graphs from trees", Ars Comb. 35 (1993), 335-349.
[3] T. Hao, "On sum graphs", J. Comb. Math. Combin. Comput. 6 (1989), 207-212.
[4] F. Harary, Graph Theory, Addison-Wesley, Reading, Mass. (1968).
[5] F. Harary, "Sum graphs and difference graphs", Cong. Number., 72, (1990), 101-108.
[6] M. Miller et. al., "The sum number of the covktail party graph", Bulletin of the Institute of Combinatorics and its applications, vol. 22 (1998), 79-90.

Abstract: In this paper we introduce the notion of Irregularity and total Irregularity of any Irregular Intuitionistic Fuzzy graphs and some results are present. Also Density, Balanced Irregular Intuitionistic Fuzzy graphs are discussed and some of their properties are established.
Keywords: Intuitionistic fuzzy graph, degree, Irregularity, Density of IIFG, Balanced IIFG

[1] Albertson,M.O. The Irregularity of a graph, Arscombin. 46(1997), 219-225.
[2] AL-Hawary. T, Complete fuzzy graph, International J. Math. Combinatory 4(2011),26-34
[3] Atanassov. KT. Intuitionistic fuzzy sets: theory and applications. Physica, New York, 1999.
[4] Parvathi, R. and Karunambigai, M.G., Intuitionistic Fuzzy Graphs, Computational Intelligence, Theory and applications, International Conference in Germany, Sept 18 -20, 2006.
[5] A.NagoorGani and S.R. Latha, On Irregular Fuzzy Graphs, Applied Mathematical Sciences,Vol.6, 2012, no.11,517-523.
[6] A.NagoorGani and S.ShajithaBegum, Degree,Order and Size in Intuitionistic Fuzzy Graphs,International Journal of Algorithms, Computing and Mathematics,(3)3 (2010)..

Abstract: In this paper, we prove some common coupled fixed point results using rational type contractive conditions in complex valued generalized metric spaces.
Keywords: Common coupled fixed point, Weakly increasing map, Complex valued generalized metric space,< Partially ordered set

[1] A. Azam, B. Fisher and M. Khan, Common fixed point theorems in complex valued metric spaces, Numerical Functional Analysis
and Optimization, 32 (3) (2011), 243-253.
[2] T. Ghana Bhaskar and V. Lakshmikantham, Fixed point theorems in partially ordered metric spaces and applications, Nonlinear
Analysis: Theory, Methods & Applications 65(1) (2006), 1379-1393.
[3] V. Lakshmikantham and L. Ciric, Coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces,
Nonlinear Analysis 70(12) (2009), 4341-4349.
[4] J.X. Fang, Common fixed point theorems of compatible and weakly compatible maps in Menger spaces, Nonlinear Analysis 71(5-6)
(2009), 1833-1843.
[5] M. Abbas, M.A. Khan and S. Radenovic, Common coupled fixed point theorems incone metric spaces for w-compatible mappings,
Applied Mathematics and Computation 217 (1) (2010), 195-202.

Abstract: Spectrum of pi is broadened in this article and made to be applicable to regular polygons. By this, several equations of area and volumes are generalized. Also, a new method is suggested to find the value of arc trigonometric functions like arcsin.
Keywords: apothem, arcsin, cylinder, pi, pyramid,volume.

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