iosr-Jm   Volume-15 ~ Issue-6 ~ Nov – Dec 2019

Abstract: In this paper, we define Pell sequences * + over rings with identity and investigate the their properties. We obtain the generating functions and Binet's formulas for these sequences. Then we present some Hessenberg matrices with applications to these sequences and show that the determinants and permanents of these Hessenberg matrices are equal to th term of Pell sequences in rings with identity. Also, the terms of these sequences are derivated by the matrix.

[1]. Ocal A. A., Tuglu N. and Altinisik E. On the representation of 𝑘-generalized Fibonacci and Lucas numbers. Appl. Math. Comput..2005; 170(1): 584-596.
[2]. Horadam A. F. A Generalized Fibonacci Sequence. Amer. Math. Monthly. 1961; 68(5): 445-459.
[3]. Horadam, A. F. Pell Identities. The Fibonacci Quarterly, 1971; 9(3): 245-252, 263
[4]. Wall D. D. Fibonacci series modulo 𝑚. Amer. Math. Monthly. 1960; 67(6): 525-532.
[5]. DeCARLI D. J. A Generalized Fibonacci Sequence Over An Arbitrary Ring. Fibonacci Quart. 1970; 8(2):182-184,198..

Abstract: This paper is designed to develop high order compact finite differences schemes for solving Helmholtz equation using Pade approximation. The developed schemes are fourth order in one, two and three
dimensional cases. Test problems were conducted to validate the efficiency and accuracy of the schemes and results obtained from the proposed schemes are compared with the exact solution , the traditional second order
and any other fourth order schemes developed by Crank- Nicolson. The proposed schemes can be applied to any fractional problems. The results obtained have high degree of accuracy than usual second order difference
scheme and can be applied to any other type of partial differential equations.

Key Words: Helmholtz equation, Pade approximation, compact finite difference schemes, fourth order schemes and Numerical experiment

[1]. Alli. M. E. and Entesar Othman Lagha., (2013), Compact finite difference schemes for one- dimensional Helmholtz equation, Department of mathematics, Faculty of science , University Bulletin –ISSUE No. 15 – vol.2, Zawia.
[2]. Carey, G. F., and Spotz, W. F., (1997), High-order compact mixed methods, Communes Numerical Methods Eng 13 553–564.
[3]. Godehand, S., (2007), Compact Finite Difference Schemes of Sixth Order for the Helmholtz Equation, Journal of computational and Applied mathematics 203, 15-31.
[4]. Erlangga,Y and Turkel, E (2012)," iterate schemes for the high order discretization to the exterior Helmholtz equation", ESAIM: Mathematical modeling and Numerical Analysis vol.46 pp 647-660.
[5]. Eli. Turkel, Dan Gordon, Rachel Gordon and Semya Tsyskov (2017) Compact 2D and3D Sixth order schemes for the Helmholtz equation with variable number , School of Mathematics, Tel. Aviv University, Ramat 69978, Israel.

Abstract: This research work examinedMagnetohydrodynamics(MHD)flow over an infinite plate in a porous medium with constant suction, oscillatory viscoelastic and radiation effects. Effects of grashof numbers, viscoelastic parameter, radiation parameter,prandtl and Schmidt numbers are discussed.The results found for concentration, temperature and velocity are depicted graphically. The velocity flow field increaseswith increasing values of thermal grashof number, mass grashof number and magnetic field. Hence, the velocity flow field reduces with increasing values of Schmidt number, Prandtl number, reactive term, radiation parameter and porous parameter.

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Abstract: In this paper a new three-step consistent and zero stable implicit High Order Hybrid Methods has been developed for the solution of initial valued problems of ODEs. This work is based on the general High Order Hybrid algorithm developed by Shokri. The three-step method developed is shown to be of order 8, consistent, zero-stable and convergent. Simpson's block method was used to generate starting values for the implementation of the new methods. Results of numerical experiments to test the efficiency of the new methods compared to the exact solutions reveals that the new hybrid scheme computes favorably with most numerical methods and even better than some of the existing methods in literatures. Hence the new hybrid method proposed in this work is suitable for good approximation of the solution of initial valued problems of first order ordinary differential equations.

Key Words: Offstep point, High Order, Consistent, Ordinary Differential Equation, Hybrid Methods

[1]. Adee, S.O., Onumanyi, P., Sirisena, U.W. &Yahaya, Y.A. (2005). Note on Starting the Numerov Method More Accurately by a Hybrid Formula of Order Four for Initial Value Problems. Journal of Computational and Applied Mathematics, 175, 369-373.
[2]. Butcher J.C. (1965). A Modified Multistep Method for the Numerical Integration of Ordinary Differential Equations, J. Assoc. Comput. Math, 12(124-135).
[3]. ChangY.F. andCorlissG. (1994).ATOMFT: Solving ODEs and DAES using Taylor series,Computers Math.Applic., .28 209-233.
[4]. Chollom, J. P., Ndam, J. N. &Kumleng, G.M. (2007). On Some Properties of the Block Linear Multistep Methods. Science World Journal, 2(3), 11-17.
[5]. Chollom, J.P (2010). Numerical Solution of Ordinary Differential Equations (ODE) from Reformulated Blocks 3-step Adams- Bashforth Method. African Journal online vol.18,No.2

Abstract: A sub-model for the transmission dynamics of HIV/TB co-infection with controls in a HIV endemic area was formulated using differential equations. The impact of post-exposure prophylaxis was considered
given that both HIV and TB patients are receiving other treatment. The parameters responsible for the diseases spread were analyzedin other to find the most sensitive of them all. The effective basic reproduction number, 0 R
of the systems was obtained and shown that the disease will spread only if R0 1 and would die off with time, if 1. 0 R  Numerical Simulations carried out using MATLAB on the model showedthat with the judicious and
increased use of post-exposure prophylaxis, the disease will phase out rapidly in the population.

Key Words: Post-exposure prophylaxis, basic reproduction number, endemic equilibrium point, sensitivity index.

[1]. AIDS gov. (2015). Post-exposure prophylaxis. https://www.aids.gov/hiv-aids-basics/prevention/reduce-your-risk/post-exposure.
[2]. Asogwa C.C., Mbah G.C.E., Aniaku S.E., Mbah E.C.(2019). YuMathematical Model of the Transmission Dynamics of HIV/TB Co-Infection with controls in a HIV Endemic Area. IOSR Journal of Mathematics, e-ISSN: 2278-5728, p-ISSN: 2319-765X.
Volume 15, Issue2 Ser.II, pp01-11.
[3]. Avert (2017). Information on AIDS: HIV and AIDS in Nigeria. https://www.avert.org>sub-sahara-africa.
[4]. Bosena Tebeje, Chevnet Hailu (2010): Assesment of HIV Post Prophylaxis use Among Health Workers of Government Health Institutions in Jimma zone, Oromiya Region Southwest Ethiopia.
[5]. Crampin A.C., Kasimba S., Mwaungulu N.J, Dacombe R., Floyd S., Glynn J.R., Fine P.E.M. (2011). Married to M. Tuberculosis: Risk of infection and disease in spouses of smear positive TB Patients. Trop. Med. Int. Health.16 (7): 811-818.

Abstract: This paper shows that a completely continuous operator is continuous but every continuous operator is not completely continuous whereas continuous operator of finite rank is completely continuous operator.

Key Words: Compact operator, Reiszoerator, Completely Continuous operator, Bounded operator

[1]. G. Kothe, Topological vector spaces, I, II New York Heidelberg Berlin. (1963, 1979)
[2]. A.P. Robertson and W. Robertson, Topological vector spaces Cambridge Uni. Peris (1964)
[3]. G.K. Palei. N.P. Sah, BIBECHANA, 8 (2012)
[4]. A.E. Taylor, Introduction to Functional Analysis, New York John Wiley and Sons Inc London (1956)
[5]. A. Gruthendicek,Topological vector spaces, New York London (1973)

Abstract: In this study, a dynamical system of ordinary differential equations has been formulated to describe to describe the dynamics of human population subjected to HIV disease. This study classifies human population into six compartments as susceptible class, primary class, asymptomatic class, symptomatic class, treatment class and AIDS (SPAJTV). A well-possedness of formulated dynamical system has been verified. Additionally, parametric expression of basic reproduction number has been constructed using next generation matrix method. The equilibrium points of formulated dynamical system are identified. Both Global stability and local stability of disease free equilibrium point has been analyzed. The local stability of endemic equilibrium point also analyzed using a reproduction number and Routh Hurwitz principle. A disease free equilibrium point is locally and globally stable for a reproduction number less than unity and unstable for greater than unity. Sensitivity analysis computation shows that death and recruitment rates are more sensitive in reproduction number. Finally, numerical solutions of the model equations are simulated using MATLAB. The results and observations have been included in the text of this paper lucidly.

Key Words: Global Stability, Local Stability, Basic Reproduction Number, Routh Hurwitz criterion, well-possedness, Simulation.

[1] Luboobi et al. 2011. The Role of HIV Positive Immigrants and Dual Protection in a Co-Infection of Malaria and HIV/AIDS. Applied Mathematical Sciences, Vol. 5, 2011, no. 59, 2919 – 2942.
[2] Vyambwer M.S. 2014. Mathematical modeling of the HIV/AIDS epidemic and the effect of public health education. Department of Mathematics and Applied Mathematics, University of the Western Cape.
[3] W.S. Ronald and H. James.1996. Mathematical biology: An Introduction with Maple andMatlab. Springer Dordrecht Heidelberg, Boston.
[4] Kumama Regassa and Purnachandra Rao Koya. Modeling and Analysis of Population Dynamics of Human Cells Pertaining to HIV/AIDS with Treatment, American Journal of Applied Mathematics. Vol. 7, No. 4, 2019, Pp. 127 – 136. Doi: 10.11648/j.ajam.20190704.14.
[5] Kumama Regassa Cheneke, Geremew Kenassa Edessa, Purnachandra Rao Koya. 2019. Global Stability and Sensitivity Analysis of the Dynamics of Human Population Subjected to HIV/AIDS with Treatment. IOSR Journal of Mathematics (IOSR-JM). Volume 15, Issue 6 Ser. III (Nov – Dec 2019), PP 34-51.DOI: 10.9790/5728-1506033451.

Abstract: Cauchy-Schwartz's inequality is a kind of important inequality widely used in mathematics, and often serves as an important basis to bridge the assumption between conditions and conclusions.The n-dimensional
Euclidean space provides a strong theoretical basis for the establishment of Cauchy-Schwartz's inequality, that is, the Cauchy-Schwartz inequality can be establishedin the n-dimensional Euclidean space to establish the
n-dimensional Euclidean space as the Cauchy-Schwartz inequality. The establishment provides a strong theoretical basis and can provide rigorous proof of Schwartz's inequality in probability space

Key Words: Cauchy-Schwartz inequality, Euclidean space, Probability space

[1]. XiaoliSun. The promotion and application of Cauchy-Schwartz inequality [D].Hefei University of Technology, 2013.
[2]. ShisongPei, YimingCheng, XiaolongXiao. Probability Theory and Mathematical Statistics [M].Beijing Higher Education Press.
[3]. QiongWang. Application of Probability Method in Proof of Inequality[J]. Journal of Tibet University (Chinese Edition), 2002 (02): 75-78.

Abstract: In this paper, we define permutation graphs on......

Key Words: inversion numbers, co-inversion, permutation graph, 1  -non deranged permutations.

[1] Aremu K.O., Ibrahim A.H., Buoro S. and Akinola F.A.Pattern Popularity in 1.....