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i-manager's Journal on Mathematics (JMAT)

Current Issue Vol. 13 Issue 2

Mathematical Modeling of Higher Overtone Vibrational Frequencies in Dichlorine Monoxide
Some Types of Generalized Closed and Generalized Star Closed Sets in Topological Ordered Spaces
Optimizing Capsule Endoscopy Detection: A Deep Learning Approach with L-Softmax and Laplacian-SGD
Kernel Ideals in Semigroups
Modeling the Dynamics of Covid-19 with the Inclusion of Treatment, Vaccination and Natural Cure
Calculation of Combined Vibrational Frequencies in Cl₂O using Lie Algebraic Method

Most Cited

Volume 1 Issue 1 January - March 2012

Article

New Representation of Biharmonic Curves in Special 3-Dimensional Kenmotsu Manifold

Talat Körpinar* , Essin TURHAN**

*-** Firat University, Department of Mathematics, Elazi.

Körpinar, T., and Turhan, E. (2012). New Representation Of Biharmonic Curves In Special 3-Dimensional Kenmotsu Manifold. i-manager’s Journal on Mathematics, 1(1), 1-7. https://doi.org/10.26634/jmat.1.1.1662

Abstract

In this article, the authors study matrix representation of biharmonic curves in 3-dimensional Kenmotsu manifold. We characterize Frenet frame of the biharmonic curves in terms of their curvature and torsion in special 3-dimensional Kenmotsu manifold.

References

[1]. K. Arslan, R. Ezentas, C. Murathan, T. Sasahara, Biharmonic submanifolds 3-dimensional (k,m)-manifolds, Internat. J. Math. Math. Sci. 22 (2005), 3575-3586.

[2]. D.E. Blair: Contact Manifolds in Riemannian Geometry, Lecture Notes in Mathematics, Springer-Verlag 509, Berlin-New York, 1976.

[3]. R. Caddeo, S. Montaldo and C. Oniciuc: Biharmonic submanifolds of Sn , Israel J. Math., to appear.

[4]. R. Caddeo, S. Montaldo and P. Piu: Biharmonic curves on a surface, Rend. Mat., to appear.

[5]. B. Y. Chen: Some open problems and conjectures on submanifolds of finite type, Soochow J. Math. 17 (1991), 169--188.

[6]. S. Degla, L. Todjihounde: Biharmonic Reeb curves in Sasakian manifolds, arXiv:1008.1903.

[7]. J. Eells and L. Lemaire: A report on harmonic maps, Bull. London Math. Soc. 10 (1978), 1-68.

[8]. J. Eells and J.H. Sampson: Harmonic mappings of Riemannian manifolds, Amer. J. Math. 86 (1964), 109-160.

[9]. T. Hasanis and T. Vlachos: Hypersurfaces in E4 with harmonic mean curvature vector field, Math. Nachr. 172 (1995), 145- 169.

[10]. G. Y.Jiang: 2-harmonic isometric immersions between Riemannian manifolds, Chinese Ann. Math. Ser. A 7(2) (1986), 130--144.

[11]. G. Y. Jiang: 2-harmonic maps and their first and second variational formulas, Chinese Ann. Math. Ser. A 7(4) (1986), 389- -402.

[12]. K. Kenmotsu: A class of almost contact Riemannian manifolds, Tohoku Math. J., 24(1972), 93-103.

[13]. T. Körpinar and E. Turhan: Biharmonic curves in the special three-dimensional Kentmotsu manifold K with h-parallel Ricci tensor, Revista Notas de Matemática, 7(1) (2011), 14-24.

[14]. B. O'Neill: Semi-Riemannian Geometry, Academic Press, New York (1983).

[15]. P. Strzelecki: On biharmonic maps and their generalizations, Calc. Var. 18 (2003), 401-432.

[16]. E. Turhan and T. Körpinar: Characterize on the Heisenberg Group with left invariant Lorentzian metric, Demonstratio Mathematica 42 (2) 2009, 423-428.

[17]. C. Wang: Biharmonic maps from R4 into a Riemannian manifold, Math. Z. 247 (2004), 65-87.

[18]. C. Wang: Remarks on biharmonic maps into spheres, Calc. Var. 21 (2004), 221-242.

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Article

On the Uniform Approximation of Analytic Functions By Fejer Type Sums of Faber Series

T. TUNC* , M. KUCUKASLAN**

*-** Mersin University Faculty of Science and Literature, Department of Mathematics, Mersin, TURKEY.

Tunc, T., and Kucukaslan, M. (2012). On The Uniform Approximation of Analytic Functions by Fejer Type Sums of Faber Series. i-manager’s Journal on Mathematics, 1(1), 8-11. https://doi.org/10.26634/jmat.1.1.1663

Abstract

In this paper, a submethod of De la Vallee-Poussin method is defined. The deviation of any function analytic in a region with bounded variation boundary from the polynomials obtained by applying the submethod to the Faber series of the function is estimated.

References

[1]. Abdullayev, F.G., Zaderey, N., & Zaderey, P. (2010). On the Approximation of Analytic Functions by Fejer Sums of Faber Series. Math. Analysis Differential Equations and Application, (pp. 13-18). Sunny Beach, Bulgaria.

[2]. Armitage, H.D., & Maddox, I.J. (1989). A new type of Cesaro Mean. Analysis 9, 195-204.

[3]. Faber, G. (1903). Über Polynomische Entwickelungen. Math. Ann., 57, 389-408.

[4]. Maddox, I.J. (1970). Elements of Functional Analysis. Cambridge at the University Press.

[5]. Sidon, S. (1939). Hinreichende Bedingungen fur den Fourier-carakter einer trigonometrisehen Reine H.J. London Math. Soc., 14(2), 158-160.

[6]. Stechkin, S.B. (1961). The Approximation of Priodic Functions by Fejer Sums. Collection of Articles on Linear Methods of Summation of Fourier Series, Trudy Math. Inst. Steklov., Acad.Sci. USSR, Moscow. 62, 48-60.

[7]. Suetin, P.K. (1998). Series of Faber Polynomials. Gordon and Breach Science Publishers.

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Research Paper

A New Hilbert-type Integral Inequality with the Homogeneous Kernel of 1/2 Degree Form and the Integral in Whole Plane

Xie Zitian* , Zheng Zeng**

* Department of Math. Zhaoqing University, Zhaoqing, Guangdong, P. R. China.

** Shaoguan University, Shaoguan, Guangdong, China.

Zitian, X., and Zheng, Z. (2012). A New Hilbert-Type Integral Inequality with The Homogeneous Kernel of ½ Degree Form and The Integral in Whole Plane. i-manager’s Journal on Mathematics, 1(1), 12-17. https://doi.org/10.26634/jmat.1.1.1664

Abstract

In this paper, the authors build a new Hilbert's inequality with the homogeneous kernel of 1/2 order and the integral in whole plane. The equivalent inequality is considered. The best constant factor is calculated using Complex Analysis.

References

[1]. Hardy G.H., Littlewood J.E., and P$\acute{o}$lya G., Inequalities, London, Cambridge University Press, Cambridge, 1952.

[2]. Hardy G.H., Note on a theorem of Hilbert concerning series of positive terms, Proceedings London Math. Soc., 1925,23(2):45-46.

[3]. Zitian Xie and Zheng Zeng, A Hilbert-type integral inequality whose kernel is a homogeneous form of degree -3, J.Math.Appl, 2008,(339):324-331.

[4]. Zitian Xie, Zheng Zeng, A Hilbert-type integral inequality with a non-homogeneous form and a best constant factor, Advances and Applications in Mathematical Sciens 2010,3(1), 61-71.

[5]. Zitian Xie , Zheng Zeng, The Hilbert-type integral inequality with the system kernel of degree homogeneous form, Kyungpook Mathematical Journal, 2010(50), 297-306.

[6]. Xie Zitian, A New Hilbert-type integral inequality with the homogeneous kernel of real number-degre, Journal of Jishou University(Natturnal Science Edition), 2011, 32(4), 26-30

[7]. Bicheng Yang, A new Hilbert-type integral inequality with some parameters, Journal of Jilin University (Science edition) 2008,46,(6): 1085-1090.

[8]. Zitian Xie and Bicheng Yang, A new Hilbert-type integral inequality with some parameters and its reverse, Kyungpook Mathematical Journal,2008,(48):93-100.

[9]. Zitian Xie and Juming Murong, A rewerse Hilbert-type inequality with some parameters, Journal of Jilin University (Science edition), 2008,46 (4):665-669.

[10]. Zitian Xie, A new reverse Hilbert-type inequality with a best constant facter, J.Math.Appl, 2008,(343):1154-1160.

[11]. Zheng Zeng and Zi-tian Xie, A Hilbert's inequality with a best constant factor, Journal of Inequalities and Applications, 2009, Article ID 820176, 8 pages doi:10.1155/2009/820176.

[12]. Xie Zitian ,Zeng Zheng, On generality of Hilbert's inequality with best constant factor, Natural Science Journal of Xiangtan University, 2010, 32(3): 1-4.

[13]. Zheng Zeng and Zitian Xie, On a new Hilbert-type integral inequality with the integral in whole plane, Journal of Inequalities and Applications ?vol. 2010, Article ID 256796, 8 pages, 2010. doi:10.1155/2010/256796.

[14]. Zitian Xie, Bicheng Yang, Zheng Zeng, A New Hilbert-type integral inequality with the homogeneous kernel of real number-degree, Journal of Jilin University(Science Edition), 2010, 48(6) : 941-945.

[15]. Xie Zitian, Zeng Zheng, A Hilbert-type integral inequality with the homogeneous kernel of real number-degree and its equivalent, Journal of Zhejiang University(Science Edition), 2011,38 (3),266-270.

[16]. Zitian Xie and Zheng Zeng, A Hilbert-type integral inequality with the integral in whole plane and its equivalent forms, Mathematica Aeterna, 2011,1 (8),577-586.

[17]. Xie Zitian, Zeng Zheng, On a Hilbert-type integral inequality with the homogeneous kernel of real number-degree and its operator form, Advances and Applications in Mathematical Sciences 2011,10(5),481-490

[18]. Xie Zitian, Zeng Zheng, A new Hilbert type integral inequality with the homogeneous kernel of degree 0, Journal of Southwest University(Natturnal Science Edition), 2011,23(8),137-141.

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Research Paper

The Effect of Slip Condition, Radiation and Chemical Reaction on Unsteady MHD Periodic Flow of a Viscous Fluid through Saturated Porous Medium in a Planer Channel

Tavva Sudhakar Reddy* , M. C. Raju, S. Vijaya Kumar Varma*

* Department of Mathematics, Shree Rama Educational Society Group of institutions, Tirupathi,Chittor District, Andhra Pradesh, India.

Department of Mathematics, Annamacharya Institute of Technology and Sciences Rajampet, Kadapa District, Andhra Pradesh, India.

* Department of Mathematics, Sri Venkateswara University, Tirupati, Andhra Pradesh, India.

Reddy, T.S., Raju, M.C., and Varma, S.V.K. (2012). The Effect of Slip Condition, Radiation and Chemical Reaction on Unsteady MHD Periodic Flow of a Viscous Fluid Through Saturated Porous Medium in a Planer Channel. i-manager’s Journal on Mathematics, 1(1), 18-28. https://doi.org/10.26634/jmat.1.1.1665

Abstract

In this paper the effect of slip condition, Chemical reaction, radiation and unsteady periodic flow of a viscous incompressible fluid through a porous medium in the presence of magnetic field is studied. The governing equations have been solved by general perturbation technique. The analytical solutions for velocity, temperature, concentration are presented and the effects of various physical parameters like Hartmann number M, Reynolds number Re,, Grashof number Gr, modified Grashof number Gm, permeability parameter k , the chemical reaction parameter kc, and Schmidt number on velocity, temperature and concentration are studied though graphs. The expression for skin friction is also derived and the effects of various physical parameters mentioned above are discussed. It is observed that the velocity of a fluid increases with an increase in slip parameter h, and it shows reverse effect in the case magnetic parameter M and concentration decreases with an increase in chemical reaction parameter kc.

References

[1]. Derek, C., Tretheway, D.C., Meinhart, C.D. (2002). Apparent fluid slip at hydrophobic micro channels walls, Physics of Fluid, 14,pp- L9-L12.

[2]. F. Soltani, and U. Yilmazer, Slip velocity and slip layer thickness in flow of concentrated suspensions, J. Appl. Polym. Sci. 70 (1998) 515–522.

[3]. W. Marques Jr., G.M. Kremer, and F.M. Shapiro, Coutte flow with slip and jump boundary conditions, Continumm Mech. Thermodynam. 12 (2000) 379–386.

[4]. Khaled, A.R.A., and Vafai, K. (2004) The effect of the slip condition on Stokes and Couette flows due to an oscillatory wall: exact solutions, International Journal of Nonlinear Mechanics, 39, pp. 795-809.

[5]. Makinde, O.D., and Osalusi, E. (2006) MHD steady flow in a channel with slip at the permeable boundaries , Romanian Journal of Physics 51, pp. 319-328.

[6]. Anil Kumar, C.L. Varshney and Sajjanlal. (2010) perturbation technique to unsteady periodic Flow of viscous fluid through a planer channel, Journal of Enng. and Technology, Vol.2(4),pp.73-81,April 2010 .

[7]. Soundalgekar, V.M., and Takhar, H.S. “Radioactive convective flow past a semi-infinite vertical plate'', Modeling Measure and Cont., 51, 31-40, 1992.

[8]. Tahkar, H.S., Gorla, S.R., Soundalgekar, V.M. “Radiation effects on MHD free convection flow of a radiating gas past a semi-infinite vertical plate”, Int. J. Numerical Methods heat fluid flow, 6, 77-83, 1996.

[9]. Hossain, A.M., Alim, M.A., and Rees, D.A.S. “Effect of radiation on free convection from a porous a vertical plate”, Int.J.Heat Mass Transfer,42, 181-191, 1999.

[10]. Muthucumarswamy, R., and Kumar, G.S. “Heat and Mass Transfer effects on moving vertical plate in the presence of thermal radiation, Theoret. Appl. Mach., 31(1), 35-46,2004.

[11]. Magyari E, Pop I and Keller B. “ Analytical solutions for unsteady free convection flow through a porous media”., J.Eng. math., 48, 93-104, 2004.

[12]. Chamaka A.J. Hydro magnetic combined heat and mass transfer by natural convection from a permeable surface embedded in a fluid saturated porous medium”., Int. J. Num. Methods for heat and fluid flow, 10(5), 455-476,2000.

[13]. Mzumdar, M.K., and Deka, R.K. “MHD flow past an impulsively started infinite vertical plate in presence of thermal radiation”., Rom. Journ. Phys., Vol. 52, Nos. 5-7. p. 565-573, 2007.

[14]. Cussler, E.L. “Diffusion Mass transfer in Fluid systems, Cambridge University press, London, 1988.

[15]. Takhar, H.S., Chamkha, A.J., and Nath, G. “ Flow and Mass Transfer on a stretching sheet with a magnetic field and chemically reactive species, Int.J.Eng.Sci., 38, No.12, 1303-1314, 2000.

[16]. Muthucumarswamy, R.,and Ganesan, P. “Effect of chemical reaction and injection on flow characteristics in an unsteady upward motion of an isothermal plate”, J.Appl.mech.Tech. Phys., 42, No.4, 665-671, 2001.

[17]. Chamkha A.J., “MHD flow of a uniformly stretched vertical permeable surface in the presence of heat generation/absorption and a chemical reaction, Int.Commun.Heat mass transfer”,30,N0.3, 413-422,2003.

[18]. Ram, P.C. Effects of Hall and ion-slips currents on free convective heat generating flow ina rotating fluid, Int. J . of energy research 19,371-376,1995.

[19]. Mohamed A. Seddeek and Emad M. AboelDahab., “Radiation effects on unsteady MHD free convection with hall current near an infinite vertical porous plate”, IJMMS 26:4, 249-255 ,2001.

[20]. R.C. Chaudhary and Preeti Jain, “Hall effect on MHD mixed convection flow of a viscoelastic fluid past an infinite vertical porous plate with mass transfer and radiation” ,Theoret. Appl. Mech., Vol.33, No.4, pp. 281-309, 2006.

[21]. Ahmed, N., Kalita, H., and. Barua, D.P. “Unsteady MHD free convective flow past a vertical porous plate immersed in a porous medium with Hall current, thermal diffusion and heat source”, International Journal of Engineering, Science and Technology Vol. 2, No. 6, 2010, pp. 59-74.

[22]. Chaudhary, R.C., Jain, P. “Hall effect on mhd mixed convection flow of a viscoelastic fluid past an infinite vertical porous plate with mass transfer and radiation”, Ukr. J. Phys. 2007. V. 52, N 10.pp. 1001-1010.

[23]. Abdulla, I.A. “Analytic solution of heat and mass transfer over a permeable stretching plate affected by chemical reaction, internal heating, Dufour-Soret effect and Hall effect”, Thermal Science, Vol.13, No.2, pp. 183-197, 2009.

[24]. Raju, M.C., Reddy, N.A., and Varma, S.V.K. “Hall current effects on unsteady MHD flow between stretching sheet and an oscillating porous upper parallel plate with constant suction”, Thermal science, Vol. 15, No.2 , pp. 527-536, 2011.

[25]. Muthucumarswamy, R. “Effects of chemical reaction on a moving isothermal vertical surface with suction”, Acta Mech.155, 65-70,2002.

[26]. Manivannan, K., Muthucumarswamy, R., Thangaraj, V. “Radiation and chemical reaction effects on isothermal vertical oscillating plate with variable mass diffusion”, Thermal science, Vol.13, No.2, pp.155-162, 2009.

[27]. Sharma P.R, Kumar N, Sharma P., “Influence of chemical reaction and radiation on unsteady MHD free convection flow and mass transfer through viscous incompressible fluid past a heated vertical plate immersed in porous medium in the presence of heat source”, Appl. Math. Sciences, Vol.5, No.46, 2249-2260, 2011.

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Research Paper

On some dynamical properties of the discontinuous dynamical system represents the Logistic equation with different delays

A.M.A. El-Sayed* , M.E. Nasr**

* Faculty of Science, Alexandria University, Alexandria, Egypt.

** Faculty of Science, Benha University, Benha, Egypt.

El-Sayed, A.M.A., and Nasr, M.E. (2012). On Some Dynamical Properties of The Discontinuous Dynamical System Represents the Logistic Equation with Different Delays. i-manager’s Journal on Mathematics, 1(1), 29-33. https://doi.org/10.26634/jmat.1.1.1667

Abstract

In this work the authors are concerned with the discontinuous dynamical system representing the problem of the logistic retarded functional equation. The existence of a unique solution which is continuously dependence on the initial data will be proved. The local stability at the equilibrium points will be studied. The bifurcation analysis and chaos will be discussed.

References

[1]. L. Berezansky and E. Braverman. (2006). On stability of some linear and nonlinear delay differential equations, J. Math. Anal. Appl.,314, 9-15.

[2]. A.El-Sayed, A.El-Mesiry and H. EL-Saka. (2007). On the fractional-order logistic equation, Applied Mathematics Letters, 20, 817-823.

[3]. A. M. A. El-Sayed and M. E. Nasr. (2011). Existence of uniformly stable solutions of nonautonomous discontinuous dynamical systems, J. Egypt Math. Soc. Vol. 19(1).

[4]. H.EL-Saka, E. Ahmed, M. I. Shehata, A. M. A. El-Sayed. (2009). On stability, persistence, and Hopf bifurcation in fractional order dynamical systems, Nonlinear Dyn, 56, 121-126.

[5]. J.Hale, S.M. Verduyn Lunel. (1993). Introduction to functional differential equations, Springer-Verlag, New York.

[6]. Y.Hamaya and A. Redkina. (2006). On global asymptotic stability of nonlinear stochastic difference equations with delays, Internat. J.Diff., 1, 101-118.

[7]. Y. Kuang. (1993). Dely differential equations with applications in population dynamics, Aca-demic Press, New York.

[8]. J.D. Murray. (2002). Mathematical Biology I: An Introduction, Springer-Verlag, Berlin, 3rd edition.

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Research Paper

Surfaces in R³ with density

Lakehal BELARBI* , Mohamed Belkhelfa**

*-** Laboratoire de physique quantique de la matière et modélisation mathématiques de matière, (LPQ3M), Université de Mascara, Algérie.

Belarbi, L., and Belkhelfa, M. (2012). Surfaces In R3 With Density. i-manager’s Journal on Mathematics, 1(1), 34-48. https://doi.org/10.26634/jmat.1.1.1845

Abstract

In this paper, the authors write the equation of minimal surfaces in R with linear density (in the case j(x,y,z) = x, j(x,y,z) = y and j(x,y,z) = z), and they characterize some solutions of the equation of minimal graphs in R3 with linear j 3 density = e , and they write the j -Gauss curvature and the j - mean curvature formulae of the revolution surfaces in R with radial density .

References

[1]. V. Bayle, Propriétés de concavité du profil isopérimétrique et applications. Thèse de doctorat, Université de Grenoble, 2003.tel-00004617.

[2]. I. Corwin, N. Hoffman, S. Hurder, V. Sesum, and Y. Xu, Differential geometry of manifolds with density,Rose-Hulman Und.Math.J., 7(1)(2006).

[3]. C. Costa, Surfaces minimales dans R , Séminaire de théorie spectrale et géométrie,tome 7(1988-1989), p.53-91.

[4]. M.P. do Carmo, Differential geometry of curves and surfaces, Prentice-Hall, Englewood Cliffs, NJ,1976.

[5]. M. Gromov, Isoperimetry of waists and concentration of maps, Geom.Func.Anal13(2003), 285-215.

[6]. N. Minh, D.T. Hieu, Ruled minimal surfaces in R3 with density ez, Pacific J. Math. 243 (2009), no. 2, 277—285.

[7]. F. Morgan, Geometric measure theory, ABeginer'sGuide,fourthedition, Academic Press.2009.

[8]. F. Morgan, Manifolds with density, Notices Amer.Math.Soc., 52 (2005), 853-858.

[9]. C. Rosales, A. Cañete, V. Bayle, F. Morgan. On the isoperimetric problem in Euclidean space with density.Cal.Var.Partial Differential Equations.31(1) (2008) 27-46.

i-manager's Journal on Mathematics (JMAT)

Current Issue Vol. 13 Issue 2

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Most Cited

Volume 1 Issue 1 January - March 2012

New Representation of Biharmonic Curves in Special 3-Dimensional Kenmotsu Manifold

Talat Körpinar* , Essin TURHAN**

*-** Firat University, Department of Mathematics, Elazi.

Körpinar, T., and Turhan, E. (2012). New Representation Of Biharmonic Curves In Special 3-Dimensional Kenmotsu Manifold. i-manager’s Journal on Mathematics, 1(1), 1-7. https://doi.org/10.26634/jmat.1.1.1662

Abstract

References

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On the Uniform Approximation of Analytic Functions By Fejer Type Sums of Faber Series

T. TUNC* , M. KUCUKASLAN**

*-** Mersin University Faculty of Science and Literature, Department of Mathematics, Mersin, TURKEY.

Tunc, T., and Kucukaslan, M. (2012). On The Uniform Approximation of Analytic Functions by Fejer Type Sums of Faber Series. i-manager’s Journal on Mathematics, 1(1), 8-11. https://doi.org/10.26634/jmat.1.1.1663

Abstract

References

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A New Hilbert-type Integral Inequality with the Homogeneous Kernel of 1/2 Degree Form and the Integral in Whole Plane

Xie Zitian* , Zheng Zeng**

* Department of Math. Zhaoqing University, Zhaoqing, Guangdong, P. R. China. ** Shaoguan University, Shaoguan, Guangdong, China.

Zitian, X., and Zheng, Z. (2012). A New Hilbert-Type Integral Inequality with The Homogeneous Kernel of ½ Degree Form and The Integral in Whole Plane. i-manager’s Journal on Mathematics, 1(1), 12-17. https://doi.org/10.26634/jmat.1.1.1664

Abstract

References

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The Effect of Slip Condition, Radiation and Chemical Reaction on Unsteady MHD Periodic Flow of a Viscous Fluid through Saturated Porous Medium in a Planer Channel

Tavva Sudhakar Reddy* , M. C. Raju**, S. Vijaya Kumar Varma***

Abstract

References

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On some dynamical properties of the discontinuous dynamical system represents the Logistic equation with different delays

A.M.A. El-Sayed* , M.E. Nasr**

* Faculty of Science, Alexandria University, Alexandria, Egypt. ** Faculty of Science, Benha University, Benha, Egypt.

El-Sayed, A.M.A., and Nasr, M.E. (2012). On Some Dynamical Properties of The Discontinuous Dynamical System Represents the Logistic Equation with Different Delays. i-manager’s Journal on Mathematics, 1(1), 29-33. https://doi.org/10.26634/jmat.1.1.1667

Abstract

References

Full Article (HTML)

Pdf

Surfaces in R3 with density

Lakehal BELARBI* , Mohamed Belkhelfa**

*-** Laboratoire de physique quantique de la matière et modélisation mathématiques de matière, (LPQ3M), Université de Mascara, Algérie.

Belarbi, L., and Belkhelfa, M. (2012). Surfaces In R3 With Density. i-manager’s Journal on Mathematics, 1(1), 34-48. https://doi.org/10.26634/jmat.1.1.1845

Abstract

References

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* Department of Math. Zhaoqing University, Zhaoqing, Guangdong, P. R. China.

** Shaoguan University, Shaoguan, Guangdong, China.

Tavva Sudhakar Reddy* , M. C. Raju, S. Vijaya Kumar Varma*

* Faculty of Science, Alexandria University, Alexandria, Egypt.

** Faculty of Science, Benha University, Benha, Egypt.

Surfaces in R³ with density