iosr-Jm   Volume-15 ~ Issue-5 ~ Sep-Oct 2019

Abstract: Quasi Newton's methods are promising schemes for solving systems of nonlinear equations. In this paper, we continue in the spirit of quasi-Newton update and present an accelerated Broyden's-like method with improved Jacobian approximation for solving large-scale systems of nonlinear equations. The anticipation has been to further improve the performance of Broyden's update as well as reducing function values. The effectiveness of our proposed scheme is appraised through numerical comparison with some well known
Newton's like methods..

Key Word: Approximation, Broyden's, Equations, iterative, Single-point.

[1]. J. E. Dennis, Numerical methods for unconstrained optimization and nonlinear equations (Englewood Cliffs, New Jersey: Prince-Hall Inc., 1983)
[2]. I. D. L.Bogle, and J. D. Perkins, A New Sparsity Preserving Quasi-Newton Update for Solving Nonlinear Equations, SIAM Journal on Scientific and Statistical Computing, 1990 Vol. 11, No. 4 : 621-630 https://epubs.siam.org/doi/abs/10.1137/0911036
[3]. W. J.Leong, M. A. Hassan, and , M. Y. Waziri., A matrix-free quasi- Newton method for solving large-scale nonlinear systems, Computers and Mathematics with Applications 62(2011) 2354- 2363.
[4]. D.H. Li and M. Fukushima, A modified BFGS method and its global convergence in nonconvex minimization, Journal of Computational and Applied Mathematics, Vol. 129 (2001) 15-35.
[5]. M.Y. Waziri, W.J. Leong, M. Mamat, A Two-Step Matrix-Free Secant Method for Solving Large-scale Systems of Nonlinear Equations, Journal of Applied Mathematics: Vol. (2012), Article ID 348654, 9 pages doi:10.1155/2012/348654.

Abstract: In this paper we present a circuit network in the concept of application of graph theory and circuit models of graph are represented in logical connection by using truth table. We formulate the matrix method of adjacency and incidence of matrix followed by application of truth table.

Key Word: Adjacent Matrix, Network Circuit, Electrical Circuit, Representations of Graph Models

[1]. B. Bollobas, Modern Graph Theory, Springer 1998.
[2]. Introductory graph theory for Electrical and Electronics Engineers, IEEE Multidisciplinary Engineering education magazine.
[3]. Narasih Deo, Graph Theory and its Applications to Computer Science..

Abstract: In this paper the authors carried some self-standing studies to examinethe game of chess and its counting techniques associatedwith non-attacking bishop positions. The authors studied the chess boardfor bishop placement with forbidden positions and varied the diagonal movement to the direction𝜃𝑟=450(to the right) and 𝜃𝑙=1350(to the left). We discussed the general movement of a bishopas generating function of two diagonal sums. Furthermore, weconstructed the movement techniques of a bishop placements in the game of chess that generates a finite sum of terms 𝒳𝜃𝑟𝜇𝛽𝑢 𝑎𝑛𝑑 𝒳𝜃𝑙 𝜑𝛿𝑣 respectively. Finally, we applied it to combinatorial problems that generates' the diagonal movement of a bishop to give the sum of two algebraic expansion.

Key Word: Chess movements; Laurent series;Permutation;Puiseux expansion; Rings

[1]. Abigail, M. (2004). A block decomposition algorithm for computing rook polynomials,.
[2]. Artin, M. (1991). Algebra. Massachusetts Institute of Technology: Prentice Hall Upper Saddle River New Jersey 07458.
[3]. Barbeau, E. J. (2003). Polynomials. Springer, New York.
[4]. Berge, C. (1971). Principles of Combinatorics; vol. 72 in Mathematics in . New York: Science and Engineering a series of monographs and textbooks, Academic press, vol. 72.
[5]. Butler, F. (1985). Rook theory and cycle-counting permutation statistics. Advances in Applied Mathematics, 124-135.

Abstract: In this paper, we find the necessary and sufficient conditions for a Finsler space with the metric 𝐿=𝛼−𝛽2𝛼 to be a Berwald space and also to be a Berwald space, where 𝛼 is a Riemannian metric and 𝛽 is a differential one-form. Further, we study the conformal change of Berwald space with the above mentioned special 𝛼,𝛽 −𝑚𝑒𝑡𝑟𝑖𝑐.

Key Word: FinslerSpace, Berwald Space, Conformal change

[1]. Benling Li, Yibing Shen and Zhongmin Shen, On a class of Douglas metrics, Comm. Korean Math. Soc., 14(3)(1999), 535-544. B.N Prasad. B.N Gupta and D.D. Singh, Conformal transformation in Finsler spaces with α,β −metric, Indian J. Pure and Appl. Math., 18(4)(1961), 290-301.
[2]. Hong-Suh Park and Eun-Seo Choi, Finsler spaces with an approximate Matsumoto metric of Douglas type, Comm. Korean Math. Soc., 14(3)(1999), 535-544.
[3]. M. Hashiguchi, S. Hojo and Matsumoto, On Landsberg spaces of dimension two with α,β −metric,Tensor, N.S., 57(1996), 145-153.
[4]. M. Matsumoto, Foundations of Finsler geometry and special Finsler spaces, Kaiseisha press, Otsu, Saikawa, Japan, (1986).
[5]. M. Matsumoto, The Berwald connection of a Finsler space with an α,β −metric, Tensor, N.S., 50(1991),18-21.

Abstract: In this paper, we find the necessary and sufficient conditions for a Finsler space with the metric 𝐿=𝛼−𝛽2𝛼 to be a Douglas space and also to be a Berwald space, where 𝛼 is a Riemannian metric and 𝛽 is a differential one-form. Further, we study the conformal change of Douglas space with the above mentioned special 𝛼,𝛽 −𝑚𝑒𝑡𝑟𝑖𝑐.

Key Word: Douglas Space, Berwald Space, Conformal change.

[1]. Benling Li, Yibing Shen and Zhongmin Shen, On a class of Douglas metrics, Comm. Korean Math. Soc., 14(3)(1999), 535-544. B.N Prasad. B.N Gupta and D.D. Singh, Conformal transformation in Finsler spaces with α,β −metric, Indian J. Pure and Appl. Math., 18(4)(1961), 290-301.
[2]. Hong-Suh Park and Eun-Seo Choi, Finsler spaces with an approximate Matsumoto metric of Douglas type, Comm. Korean Math. Soc., 14(3)(1999), 535-544.
[3]. M. Hashiguchi, S. Hojo and Matsumoto, On Landsberg spaces of dimension two with α,β −metric, Tensor, N.S., 57(1996), 145-153.
[4]. M. Matsumoto, Foundations of Finsler geometry and special Finsler spaces, Kaiseisha press, Otsu, Saikawa, Japan, (1986).
[5]. M. Matsumoto, The Berwald connection of a Finsler space with an α,β −metric, Tensor, N.S., 50(1991),18-21..

Abstract: Ciphertext policy attribute-based encryption(CP-ABE) scheme widely used in cloud storage for realizing the flexible and scalable fine grained data access control for secure data sharing with user's under certain credential or attribute's. However most of the CP-ABE scheme have the problems such as access policy complexity, low computational efficiency, efficient revocation cannot be performed. Where traditional attribute-based encryption fails to provide efficient keyword's search due to week encryption scheme. In this paper we proposed verifiable ciphertext policy..........𝑡𝑟𝑖𝑐.

Key Word: Attribute based encryption, Access control, Verifiability, Keyword search, Revocation,

[1]. Sahai A, Water B, Fuzzy identity-based encryption Proc. 24 Annual Int. Conf. Theory and Application of Cryptographic Techniques advance in Cryptography Eurocrypt, Lecturer notes in computer science Springer (2005) 3494:457-473.
[2]. Goyal V, Pandy O,Sahai A, and Water B, Attribute-based encryption for fine-grained access control of encrypted data. in proceeding of 13th ACM conference on computer and communications security (ACM 06) 2006:89-98.
[3]. Bethencourt J, Sahai A, and Water B, Ciphertext policy attribute-based encryption scheme. in proceeding IEEE Symposium on security and privacy.Berkely, 2007, 321-334.
[4]. Chase M, 2007 Multi Authority Attribute-based encryption Proc. 4th Theory of Cryptography Conf. TCC Lecture Notes in computer science Springer. 2007 4392:515-534.
[5]. Chase M, and Chow S.S.M., Improving Privacy and Security in Multi-Authority Attribute based Encryption. Proc. ACM Conf. Computer and Communication Security, ACM, CCS. 2009, 121-130..

Abstract: Aspects of an organism's defense to infections are the main problems of practical immunology. Understanding the regularities in immune response provide the researchers and clinicians new powerful tools for the simulation of immune system in order to increase its efficiency in the struggle against antigen invasion. Such general regularities are revealed, as a rule, on the basis of analysis of the main components of an organism's vital activities along with the system of immune defense. In this connection the construction of models of immune response to an antigen irritant seems to be a right tactic in the cognition of above regularities, that is why this monograph is dedicated to the analysis of the facts accumulated in immunology as a united system on the basis of logical concepts and mathematical models..............

[1]. Saxena, R., Voight, B. F., Lyssenko, V., Burtt, N. P., de Bakker, P. I., Chen, H., ... & Hughes, T. E. (2007). Genome-wide
association analysis identifies loci for type 2 diabetes and triglyceride levels. Science, 316(5829), 1331-1336.
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[3]. Kim, P. S., Levy, D., & Lee, P. P. (2009). Modeling and simulation of the immune system as a self-regulating network. Methods in
enzymology, 467, 79-109.
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[5]. Hartwell, L. H., Hopfield, J. J., Leibler, S., & Murray, A. W. (1999). From molecular to modular cell biology. Nature,
402(6761supp), C47..

Abstract: A collocation method for approximating second order boundary value problems in ordinary differential equations was developed using the probabilists' Hermite polynomial of degree eight as basis function. Five examples were carefully chosen to include homogeneous and nonhomogeneous boundary value problems with constant and variable coefficients. The boundary value problems involve Dirichlet, Neumann and Robin boundary conditions. The collocation method when applied to the boundary value problems provided a good approximation of the analytical solutions. However, the boundary value problems with Dirichlet boundary condition gave a better approximation of the analytical solution as compared to the boundary value problems with Neumann and Robin conditions. We also observed that the accuracy of the collocation method increased as more terms of the probabilists' Hermite polynomial were used as basis function.

Key Word: Collocation, Probabilists' Hermite Polynomial, Physicists' Hermite polynomial

[1]. Tesfy, G., Venketeswara, R. J. Araya, A. and Tesfary, D. (2012). Boundary Value Problems and Approximate Solutions. Mathematics and Computer Science, 4(1):102-114.
[2]. Taiwo, O. A., Raji, M. T. and Adeniran, P. O. (2018). Numerical Solution of Higher Order Boundary Value Problem using Collocation Methods. International Journal of Science and Research Publications, 8(3):361-367.

[3]. Ganaie, I. A. Arora, S. and Kukreja, V. K. (2014). Cubic Hermite Collocation Method for Solving Boundary Value Problems with Dirichlet, Neumann and Robin Conditions. International Journal of Engineering Mathematics: 1-8.
[4]. Li, Z. Qiao, Z. and Tang, T. (2018). Numerical Solution of Differential Equations: Introduction to Finite Difference and Finite Element Methods. United States of America: Cambridge University Press. P1.
[5]. Sauer, T. (2012). Numerical Analysis (2nd ed). United State of America: Pearson University Inc. Pp 348-373.

Abstract: This section presents a mathematical simulation that includes two different effectorresponses that fight a viral infection independently: CTL (responsible for cell mediated immune responses) and antibodies (responsible for humoral immune responses).Since it is assumed that both responses rely on antigenic stimulation, the simulationscapture the competition dynamics.This is because the virus population is a resource that both CTL and antibodies require for survival. Competition can result either in the exclusion of one branch of the immune system, or both branches may coexist. We have examined and simulated five different immune dynamics of immune system responses...........

[1]. Choo, Q. L., Kuo, G., Weiner, A. J., Overby, L. R., Bradley, D. W., & Houghton, M. (1989). Isolation of a cDNA clone derived
from a blood-borne non-A, non-B viral hepatitis genome. Science, 244(4902), 359-362.
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[3]. Wodarz, D. (2003). Hepatitis C virus dynamics and pathology: the role of CTL and antibody responses. Journal of General
Virology, 84(7), 1743-1750.
[4]. Perelson, A. S. (2009). Simulation and prediction of the adaptive immune response to influenza A virus infection. Journal of
virology, 83(14), 7151-7165.
[5]. Kim, P. S., Levy, D., & Lee, P. P. (2009). Modeling and simulation of the immune system as a self-regulating network. Methods in
enzymology, 467, 79-109..