iosr-Jm   Volume-12 ~ Issue-1 ~ Jan-Feb 2016

Abstract: The notion of 2 - metric spaces was introduced by Gähler in the year 1963 and since then researchers are trying to study the concept of 2 - metric spaces in the fuzzy structures. Very recently , Dey and Saha made a very good contribution in the form of a book to study fixed point theory in 2 - metric spaces. In the present paper , we state and prove some fixed point theorems on fuzzy 2 - metric spaces due to Sharma by introducing the notion of 𝜀 - chain and (𝜀 ,𝜆) uniformly locally contractive mappings on fuzzy 2 - metric spaces. Our results extend the famous fixed point theorems due to R. Caccioppoli and M. Edelstein on classical metric spaces. We prove an important Lemma and deduce the Banach contraction theorem on fuzzy 2 - metric spaces as a corollary and also illustrate our results with examples. Mathematics Subject Classification: 54H25 and 47H10.

Keywords: Fuzzy 2 - metric spaces , contraction maps , 𝜀 - chain , uniformly locally contractive maps , fixed points.

[1]. S. H. Cho , On common fixed points in fuzzy metric spaces , Int. Math. Forum 1(10) (2006) 471 - 479.
[2]. S. H. Cho and J. H. Jung , On common fixed point theorems in fuzzy metric spaces , Int. Math. Forum 1(29) (2006) 1441 - 1451.
[3]. N. R. Das and M. L. Saha , Some fixed point theorems in fuzzy 2- metric spaces, J. Fuzzy Math. 21(4) (2013) 753 - 758.
[4]. N. R. Das and M. L. Saha , On fixed points in fuzzy metric spaces, Ann. Fuzzy Math. Inform. 7 (2) (2014) 313 - 318.
[5]. N. R. Das and M. L. Saha , Compatible mappings and common fixed points in fuzzy metric spaces, J. Fuzzy Math. 23(1) (2015) 199 - 208.
[6]. N. R. Das and M. L. Saha , On fixed points in complete fuzzy normed linear spaces, Ann. Fuzzy Math. Inform. 10 (4) (2015) 515 – 524.
[7]. N. R. Das and M. L. Saha , Non - commuting mappings and common fixed points in fuzzy metric spaces, Journal of Fuzzy Math. ,24 (1) (2016).
[8]. Zi - ke Deng , Fuzzy pseudo- metric spaces , J. Math. Anal. Appl. 86 (1982) 74 - 95.

Abstract: Let L (H) be the algebra of all bounded linear operators on a complex Hilbert space H. An operator T 𝜖 L (H) is called n power quasi normal operator if Tⁿ commutes with T*T that is,TⁿT*T =T*TTⁿ and it is denuded by [nQN]. In this paper we investigate some properties of n-power quasinormal operators. Also, the necessary and sufficient condition for a Binormal operator to be 2 power quasi normal operator is obtained. Mathematics Subject Classification: 47B20 Keywords: Self adjoint operator, n - power quasi normal operator, unitary and binormal operator..

[1] A. Brown , On a class of operators, Proc. Amer. Math. Soc, 4 (1953), 723 – 728.

[2] A.A.S. Jibril, on n – power normal operators, The Journal of Science and engineering, Volume 33, Number 2A, (2008) , 247 – 251.

[3] Ould Ahmed Mahmoud Sid Ahmed, On the class of n – power quasi normal operators on the Hilbert space, Bull. Of Math.Anal. Appl., Vol 3, 2 (2011), 213 – 228.

[4] Dipshikha Bhattacharya And Narendra Prasad, Quasi-P Normal operators linear operators on Hilbert space for which T+T* And T*T commute, Ultra Scientist Vol. 24(2)A, 269-272 (2012).

[5] Arun Bala, A note on Quasi normal operators, University of Delhi, 110007, 1976.

Abstract: In this work we show classical and known forms to carry out numerical interpolation and curve fitting. Each method is briefly explained and examples from Lagrange, Newton, Hermite, osculating polynomial, and Padé approximation are presented. In the least-squares line, data linearization method of exponential and power function exercises are solved. Our idea is to show the advantages of using MATLAB in the study of numerical analyses and to verify the minimal effort required in using this program to save time in making mathematical demonstrations and developments essential in the obtaining of each method.
Keywords: Polynomial approximation, interpolation, regression.ts.

[1]. Luis Vázquez, Salvador Jiménez, Carlos Aguirre y Pedro J. Pascual.Métodos Numéricos para física e ingeniería.Edit. McGraw Hill pp 85-118. Spain (2009)
[2]. John H. Mathews and Kurtis D. Fink. Métodos numéricos con MATLAB. Edit. Prentice Hall pp 203-263.Spain (2000).
[3]. Bai Yu and Zhang Meng. Visual teaching of numerical analysis based on MATLAB. The 1st International conference of Information science and Engineering pp 3552-3555 (2009)
[4]. Eisa San et al. Numericalcurve length calculationusing polynomial interpolation.Journal of Mathematical Imaging and Vision 49 2 pp 377-383 (2014)
[5]. Roberto Cavoretto. A numerical algorithm for multidimensional modeling of scattering data points.Computational and Applied Mathematics.(2013)
[6]. Robert Plato Concise Numerical mathematics. Edit. American Mathematical Society. USA (2000)

Abstract: In the present paper a prey-predator model with disease that spreads among the predator species only is proposed and investigated. It is assumed that the disease is horizontally transmitted by contact between the infected predator and the susceptible predator. The local and global stability analyses are carried out. The persistence conditions of the model are established. Local bifurcation analyses are performed. Numerical simulation is used extensively to detect the occurrence of Hopf bifurcation and confirm our obtained analytical outcomes.
Keywords: Prey-Predator, Stability, Local bifurcation, Hopf bifurcation, Persistence.

1] H.I. Freedman, Deterministic mathematical method in population ecology, (Marcel Dekker, New York, 1980).
[2] J.D. Murray, Mathematical biology, (Springer-Verlag, New York, 1989).
[3] R. Xu, M.A.J. Chaplain, and F.A. Davidson. Persistence and global stability of a ratio-dependent predator–prey model with stage
structure. Applied Mathematics and Computation 158, 2004, 729–744.
[4] X. Tian and R. Xu, Global dynamics of a predator-prey system with Holling type II functional response. Nonlinear Analysis:
Modelling and Control, 16 (2), 2011, 242–253
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471–476.
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pathogen virulence. PLoS ONE 4 (8), 2009, 6761.

Abstract:This study has proposed a stochastic model for assessment of control measures on the intensity of bacterial spread over the plants of different types as well as of different age groups in the same species of the plant. Deriving the stochastic differential equations through bivariate point processes is the core objective of the study. The developed differential equations are used for deriving several statistical measures based on the proposed parameters of bacteria growth, spread and loss. Sensitivity analysis is carried out for observing the patterns of statistical measures at changing values of one parameter and at fixed values of the remaining
parameters.
Keywords: Bacterial Diseases, Bivariate Stochastic Processes, Stochastic Differential Equations.

[1]. Anderson, H. and Britton, T. (2000) Stochastic Epidemic Models and Their Statistical Analysis. Springer Verlag, New York.
[2]. Bailey, N.T.J. (1975) The Mathematical Theory of Infectious Diseases and Its Applications. 2nd Edn. Grin, London.
[3]. Kermack, W.O. and McKendrick, A.G. (1927) A contribution to the mathematical theory of epidemics. Proceedings of the Royal
Society, Series A 115, 700-721.
[4]. Segarra, J., Jeger, M. and Van den Bosch, F. (2001). Epidemic Patterns and Dynamics of Plant Disease. Phytopathology, Vol. 91,
pp. 1001-1010.
[5]. Tirupathi Rao, P. and Srinivasa Rao, K. (2006). Two Stage Stochastic Model for Cancer Cell Growth. Indian Journal of
Mathematical Sciences, Vol. 2(2), pp. 153-168.

Abstract: In this paper, the two parameter Weibull probability distributions are embedded in a larger family obtained by introducing an additional parameter. We generalize the two parameter Weibull distributions using the quadratic rank transmutation map by Shaw et al [15] to develop a transmuted Weibull distribution. We provide a comprehensive description of the mathematical properties of the subject distribution along with its reliability behavior. The usefulness of the transmuted Weibull distribution for modeling reliability is illustrated using real data. Psychocutaneous diseases constitute a large proportion of psychosomatic dis-orders, with psoriasis being one of the most typical cases.

[1]. Arnetz BB, Fjellner B, Eneroth P, Kallner A, (1991) Endocrine and dermatological concomitants of mental stress. Acta Derm Venereol Suppl (Stockh) 156: 9-12.

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[4]. Buske-Kirschbaum A, EbrechtM, KernS, Hellhammer DH, (2006) Endocrine stress responses in TH1-mediated chronic inflammatory skin disease ( psoriasis vulgaris ) do they parallel stress – induced endocrine changes in TH2 mediatedinflammatorydermatoses(atopicdermatitis)? Psychoneuroendocrinology 31: 439-446

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Abstract:In this paper various mathematical models developed to date accounts for only that portion of BOD which is in dissolved form and not the least in settleable form. These models also do not account for the storage zone in a scattered way in rivers and hence do not represents the actual situation caused by the discharge of partially treated/ untreated waste water, which contains a significant portion of BOD in settleable form into the water body with a large width where the water becomes stagnant anywhere, due to rag/garbage in rivers. The present paper represents a model to predict the concentration of total BOD when partially treated/untreated
waste water is discharged into the Damodar river having staggered storage zones and thus address the above stated situation.
Keywords: Damodar river, Mathematical modeling, Biochemical oxygen demand, Settleable BOD,Main zone,
Scattered storage zone.

[1] V. V. Vlatschii, Modeling of river flow using GIS_Tehnology, Bulletin of the Orenburg State University, Russia, No. 9,104 - 109,
(2010).
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470-473,(2003).
[3] B. Tyagi, S. Gakkhar and D.S. Bhargava, Mathematical modeling of stream DO-BOD accounting for settleable BOD and
periodically varying BOD source, Environmental Software, Elsevier, U.K.14, , 461- 471,(1999).
[4] S.C. Chapra and R.L. Runkel, Modeling impact of storage zones on stream Dissolved oxygen, Journal of Environmental
Engineering, ASCE 125, 415-419,(1999).
[5] R.L. Runkel, One dimensional transport with inflow and storage (OTIS): A solute transport model for streams and rivers, U.S.
Geological Survey, Water Resources Investigation Report, 98-4018, (1998),

Abstract: In this paper we introduce the notation of symmetric skew 4-reverse derivation of semiprime ring and we consider 𝑅 be a non-commutative 2,3-torsion free semiprime ring, 𝐼 be a non zero two sided ideal of 𝑅, 𝛼 be an anti-automorphism of 𝑅, and 𝐷:𝑅4→𝑅 be a symmetric skew 4-reverse derivation associated with theanti-automorphism𝛼. Suppose that the trace function 𝑓 is commuting on 𝐼 and 𝑓 𝑦 ,𝛼 𝑦 ∈𝑍, for all 𝑦∈𝐼, then 𝑓(𝑦),𝛼(𝑦) =0, for all 𝑦∈𝐼. Keywords: Semi prime ring, Derivation, Bi derivation,Reverse derivation,Symmetric Skew 3- derivation,Symmetric Skew 4-derivation,Symmetric skew 4-Reverse derivation and Anti-automorphism.

[1] AjdaFosner:Prime and Semiprime rings with symmetric skew 3-derivations, Aequat. Math.87(2014), 191-200.

[2] Bresar.M. andVukman.J: On some additive mappings in rings with involution, Aequationes Math. 38(1989),178-185.

[3] FaizaShujat,Abuzaid Ansari:Symmetric skew 4-derivationson Prime rings, J.Math.Comput.Sci.4(2014),No.4,649-656.

[4] Jaya Subba Reddy .C: Prime rings with symmetric skew 3-reverse derivations, International Journal of Mathematics and Computer Applications Research,Vol.4, Issue 6, (2014),69-74.

[5] Posner. E.C: Derivations in prime rings, Proc. Amer. Math.Soc.8(1957),1093-1100).

[6] Samman.M. andAlyamani.N: Derivations and reverse derivations in semi prime rings, International Mathematical Forum, 2,No.39(2007),1895-1902.

Abstract: The effects of temperature dependent viscosity and thermal conductivity on a two dimensional unsteady laminar MHD free convective flow over a vertical plate immersed in a porous medium with viscous dissipation and heat generation have been studied in the current work. The governing boundary layer equations are converted into a system of coupled nonlinear ordinary partial differential equations by using non-dimensional parameters and then solved numerically using finite difference method. Effects of variable viscosity, variable thermal conductivity, and the other parameters engaged in the study on the velocity, temperature and concentration profiles are investigated graphically. Skin friction and Nusselt number profiles have been also illustrated graphically.

Keywords: MHD, Porous plate, Variable viscosity, Variable thermal conductivity, viscous dissipation, Heat and Mass transfer.

[1] Postelnicu, A., (2004), The influence of magnetic field on heat and mass transfer by natural convection from vertical surfaces in porous medium with Soret and Dufour effects, Int. J. of Heat and mass transfer, vol.47, p1467-1472.
[2] Alam, M. A., Alim, Md., and Chowdhury, M.K., (2007), The viscous dissipation effects on MHD natural convection boundary layer over a sphere of an electrically conducting fluid in the presence of heat generation, Non-linear Analysis: Modeling and control, vol.12(4), p447-459.
[3] Seddeek, M.A., and Salama, F.A., (2007), The effects of temperature dependent viscosity and thermal conductivity with variable suction on MHD convective heat transfer past a vertical moving porous plate with variable suction, Computational Material Science, vol.40(2), p186-192.
[4] Kairi, R. R., and Murthy P. V. S. N., (2013), The Soret effect with the influence of variable viscosity and natural convection from a melting vertical surface in a non-Darcy porous medium, J. of Porous Media, vol.16(2), p97-104.

Abstract:in this paper we present several kinds of methods that allow us to compute the exponential matrix tA e exactly. These methods include calculating eigenvalues and Laplace transforms are well known, and are mentioned here for completeness. Other method, not well known is mentioned in the literature, that don't including the calculation of eigenvectors, and which provide general formulas applicable to any matrix.

Keywords: Exponential matrix, functions of matrix, Lagrange-Sylvester interpolation, Putzer Spectral formula, Laplace transform, Commuting Matrix, Non-commuting Matrix.

[1] P. J. Antsaklis & A. N. Michel; A Linear Systems Primer. Birkhauser, Boston (2007).
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Abstract: In this paper, we have mentioned some properties of radical ideals ring with nilpotent ideals. Mainly,
we have focused on "the Jacobson radical of an Artinian ring R is nilpotent. In fact, J R is the largest
nilpotent (left or right or 2-sided) ideal of R and consequently, NR  JR". Finally, we have discussed
many of theorem on nilpotent ideals.
Keywords: Artinian ring, Jacobson radical, Nil radical, Primary ideal, Tertiary radical.

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Abstract:Queueing Theory is a mathematical approach to the studyof waiting lines. Long waiting time in a health care system indicates the lack in management of the system. As a result of this, a health care centre lost the patients satisfaction and also it decreases the enhancement of the system. This paper is an attempt to analyse the use of queueing theory in a local health care clinic and the calculations performed in this paper isbased upon the actual observed data collected from a local health care clinic located in Muzaffarpur city of Bihar, India. The paper summarizes a range of queueing theory results in the following areas. Traffic intensity, Average waiting time in queue, Average of time spent in system, Average queue length, Average number of individuals in the system. The central objective of this paper is to reduce the waiting time of patients and also to increase the efficiency of the clinic. The paper also considers the clinic as a single server queueing system following Poisson arrivalbased upon the disciplinefirst come first serve and exponential service rate.
Keywords: exponential service, Poisson arrival, queueing theory, waiting line, waiting time.

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Abstract: In this paper, we study a nonlinear boundary value problem ( bvp) which generalizes capillarity problem. An existence and uniqueness result is obtained using the knowledge of range for nonlinear operator. Ours extends the result in [12].

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