#### Version-2 (Mar-Apr 2018)

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Paper Type |
: | Research Paper |

Title |
: | Finite Abelian Automata |

Country |
: | Kenya |

Authors |
: | Dr.K.Muthukumaran || S.Shanmugavadivoo |

: | 10.9790/5728-1402020104 |

**Abstract: **Finite Semigroup Automaton, Finite Monoid Automaton, Finite Group Automaton, Finite Abelian Automaton have been introduced. Cross Product of Finite Semigroup Automatons, Finite Monoid Automatons, Finite Group Automatons, Finite Abelian Automatons have been defined. If B1 = (Q1, Δ1, Ʃ1, δ1, p0, F1) and B2 = (Q2, Δ2, Ʃ2, δ2, q0, F2) are any two Finite Semigroup Automatons, then B1×B2 is also a finite Semigroup automaton. If B1 = (Q1, Δ1, Ʃ1, δ1, p0, F1) and B2 = (Q2, Δ2, Ʃ2, δ2, q0, F2) are any two Finite Moniod Automatons, then B1×B2 is also a finite Monoid automaton. If B1 = (Q1, Δ1, Ʃ1, δ1, p0, F1) and B2 = (Q2, Δ2, Ʃ2, δ2, q0, F2) are any two Finite Group Automatons, then B1×B2 is also a finite group automaton. If B1 = (Q1, Δ1, Ʃ1, δ1, p0, F1) and B2 = (Q2, Δ2, Ʃ2, δ2, q0, F2) are any two Finite Abelian Automatons, then B1×B2 is also a finite Abelian automaton. Some Propositions are found in a Finite Abelian Automaton.

**Keywords:** Finite Semigroup Automaton, Finite Monoid Automaton, Finite Group Automaton, Finite Abelian Automaton

[1] S.Shanmugavadivoo And Dr.K.Muthukumaran , "Ac Finite Binary Automata" Accepted In "Iosr Journal Of Mathematics", A Journal Of "International Organization Of Scientific Research"

[2] S.Shanmugavadivoo And Dr. M.Kamaraj, "Finite Binary Automata" "International Journal Of Mathematical Archive", 7(4),2016, Pages 217-223.

[3] John E. Hopcroft , Jeffery D.Ullman, Introduction To Automata Theory, Languages, And Computation, Narosa Publising House,.

[4] Zvi Kohavi, Switching And Finite Automata Theory, Tata Mcgraw-Hill Publising Co. Lid.

[5] John T.Moore, The University Of Florida /The University Of Western Ontario, Elements Of Abstract Algebra, Second Edition, The Macmillan Company, Collier-Macmillan Limited, London,1967.

[6] J.P.Tremblay And R.Manohar, Discrete Mathematical Structures With Applications To Computer Science, Tata Mcgraw-Hill Publishing Company Limited, New Delhi, 1997..

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Paper Type |
: | Research Paper |

Title |
: | The Convergence of the Approximated Derivative Function by Chebyshev Polynomials |

Country |
: | Libya |

Authors |
: | Souad Abumaryam |

: | 10.9790/5728-1402020511 |

**Abstract: **Let f(x) be a differentiable function on the interval [-1, 1]. Finding an approximation of the derivative of the function through values of the function at points 𝑥𝑗 𝑗=0𝑁 is a very interesting problem. It is also important for solving differential equation. In this paper, we study the error bound, in particular for first and second derivatives by Chebyshev polynomials. Moreover, a generalisation for error bound is found.

**Keywords:** Chebyshev polynomials, Chebyshev interpolation, Convergence rate, Error function

[1] Mason, J. And Handscomb, D, Chebyshev Polynomials. , CRC Press, 2003.

[2] Trefethen, N,Approximation Theory and Approximation Practice. , University of Oxford, 2012.

[3] Powell, M,Approximation Theory and Methods. , Cambridge University, 2004.

[4] Berrut, J, P. And Trefethen, L, N, Barycentric Lagrange Interpolation. , SIAM Review, 2004.

[5] Davis, P,Approximation and Approximation. , Blaisdell Publishing Company, 1965

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**Abstract: **This paper presents consistent and stable numerical solutionsof the three dimensional transient heat transfer problem with non-homogenous boundary conditions. Finite difference schemes; forward time, backward time, and Crank- Nicolson methods have been used to predict the temperature distribution acrossa photovoltaic solar cell with specified dimensions. Numerical predictionsusing the three schemes are compared with each other and tested against the exact solution of the same problem. Results of the numerical and analytical solutions are quite similar with levels of numerical errors kept very negligible. Furthermore, accuracy, consistency, convergence, and stability analysis of the finite difference solutions is investigated and presented in this study.Conditions for consistency, convergence and stability of each algorithm is derived in this paper. The methodology showed to be quite robust for such a time-dependent three-dimensional problem

**Keywords:** heat equation, finite difference, photovoltaic solar cell.

[1]. Fraunhofer Institute for Solar Energy Systems ISE, Photovoltaics Report, 2016.〈https://www.ise.fraunhofer.de/en/downloads-englisch/pdf-files-english/photovoltaics-report-slides.pdf〉.

[2]. W.Dai, R.Nassar, An unconditionally stable finite difference scheme for solving a 3D heat transport equation in a sub-microscale thin film, Computational and Applied Mathematics 145 (2002) 247–260.

[3]. G. Notton, C. Cristofari, M. Mattei, P. Poggi, Modelling of a double-glass photovoltaicmodule using finite differences, Appl. Therm. Eng. 25 (2005) 2854–2877,http://dx.doi.org/10.1016/j.applthermaleng.2005.02.008.

[4]. M. Mattei, G. Notton, C. Cristofari, M. Muselli, P. Poggi, Calculation of thePolycrystalline PV Module Temperature Using a Simple Method of Energy Balance,31, 2006, pp. 553–567. http://dx.doi.org/10.1016/j.renene.2005.03.010.

[5]. Two-dimensional finite difference-based model for coupled irradiation andheat transfer in photovoltaic modules, Shahzada Pamir Aly, Said Ahzi, Nicolas Barth, Benjamin W. Figgis..

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**Abstract: **In this paper, nonlinear difference equations for zooplankton–fish population model with noise is considered. The model is on predation of phytoplanktivore fishes on zooplankton, this is to understand the individual behaviour of the organisms as well as interaction with the environment.The model is a nonlinear logistic type of model incorporating nonlinear feeding functions. The conditions for the existence of the equilibrium points are obtained through some nonlinear equations and Diophantine equations. The conditions for local stability for the dual population investigated and results obtained .Simulation made for the dual populations when the ocean is polluted with chemical substances and oil spillage usingGaussian noise. The noise accounts for pollution of the ocean that may lead to species migration from the pollutants source. It is observed that the risk factor increases with time and makes the species to be endangered and some kind of chemo taxis effect is experienced whereby the survived species tend to migrate to region with lower concentrations of pollutants.

**Keywords:** Zooplankton-fish, model, stability, Simulation, pollution, environmental risk

[1] Arnfinn Langeland Interactions between zooplankton fish in a fertilizer lake. Holarctic Ecology 5: 273-310,Copenhagen 1982.

[2] Andrew M. Edwards, ''Adding detritus to a nutrient-phytoplankton-zooplankton model''. ADynamical systems approach. Journal

of Plankton Research (2001) 23 (4): 389-413. DOI:10.1093/PLANKT/23.4.389.

[3] Gonzalez E.J., Matsumura Tundisi J.G.Size, and dry weight of main zooplankton species in Bariri reservoir (SP,Brazil).Braz J.Biol

2008.Feb 68(1),69-75.

[4] Horst M, Birgit R.,Malaak Alexander B.M., Dmitry A.T. Sergiel V. P. Spatio- temporal pattern formation in coupled models of

plankton dynamics and fish school motion. Nonlinear Analysis Real World Application 1 (2000) 53-67.

[5] Katja Fennel, Martin Losch and Jens Schröter, Manfred Wenzel, Testing a marine ecosystem model: sensitivity analysis and

parameter optimization.Systems, Volume, February 2001, Pages 45–63..

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Paper Type |
: | Research Paper |

Title |
: | Extension of Some Theoremsin General Metric Spaces |

Country |
: | Iraq |

Authors |
: | Salwa Salman Abed || Ebtihal Nabil Jaluobe |

: | 10.9790/5728-1402023846 |

**Abstract: **We prove a version ofCaristi-Kirk - Browder Theorem and Park's Theorem (Park, 198) and (Park and Rhoades, 1983) in G-metric space. And then give some corollaries.

**Keywords:** G-metric spaces, fixed point.

[1] Mustafa, Z., Obiedat , H. Awawdeh, F. 2008. Some Fixed Point Theorem for Mapping on Complete G-metric Spaces. Fixed Point Theory and Applications, , article ID 189870, doi: 10.1155/2008/189870

[2] Mustafa, Z., Sims, B., 2006. A New Approach to Generalized Metric Spaces ,Journal of Nonlinear and Convex Analysis, 7 (2), 289–297.

[3] Park, S. 1983.On Extensions of The Caristi-kKirk Fixed Point Theorem, Korean math.sco.,19, 143-151.

[4] Park, S. Rhoades, B. 1983. Some Fixed Point Theorems for Expansion Mappings Jnanbha, 15, 151-156.

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Paper Type |
: | Research Paper |

Title |
: | Extension of Some Theorems In General Metric Spaces |

Country |
: | Iraq |

Authors |
: | Salwa Salman AbedEbtihal Nabil Jaluobe |

: | 10.9790/5728-1402024750 |

**Abstract: **We prove a version of Caristi-Kirk - BrowderTheorem and Park's Theorem [3,4] in G-metric space. And then give some corollaries..

**Keywords:** G-metric spaces,fixed point

[1]. Z. Mustafa and B.Sims,"A New Approach to Generalized Metric Spaces" ,Journal of Nonlinear and Convex Analysis, 7 (2), (2006 ). 289–297.

[2]. Z.Mustafa, H.Obiedat, F.Awawdeh, "Some Fixed Point Theorem for Mapping on Complete G-metric Spaces", Fixed Point Theory and Applications, 2008, article ID 189870, doi: 10.1155/2008/189870

[3]. S. Park,"On Extensions of The Caristi-kKirk Fixed Point Theorem", Korean math.sco.,19, 1983, 143-151.

[4]. S.Park, B, Rhoades, "Some Fixed Point Theorems for Expansion Mappings" Jnanbha, 15(1983),151-156..

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**Abstract: **in this paper,we introduce a new concept of common fixed point theorem in fuzzy metric spaces and occasionally weakly compatible mappings .

**Keywords:** fixed point , common fixed point theorem , fuzzy metric space , occasionally weakly compatible mappings AMS subject classification : Primary :47H10, Secondary :45G10.

[1] Aage, C. T. and Salunke, J. N., "On Fixed Point Theorems in Fuzzy Metric Spaces." Int. J. Open Problems Compt. Math., 3 (2), (2010), 123-131.

[2] Alaca, C., Altun, I. and Turkoglu, D., "On Compatible Mapping of Type (I) and (II) in Intuitionistic Fuzzy Metric Spaces." Korean Mathematical Society, 23(3), (1986), 771-779.

[3] Atanssove, K., "Intuitionstic Fuzzy Sets." Fuzzy Sets and Systems, 2(1), (1986), 87-96.

[4] Banach, S. "Surles Operation Dansles Ensembles Abstraites Etleur Application Integrals." Fund. Math., 3, (1922), 133-181.

[5] Balasubramaniam, P., Muralisankar S., Pant R.P., "Common Fixed Points of Four Mappings in a Fuzzy Metric Space." J. Fuzzy Math., 10 (2), (2002), 379-384...

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**Abstract: **In this paper we introduce a generalized MATHEMATICA code for solving one dimensional second order ordinary boundary value problems by Modified Galerkin's technique of finite element methodusing the standard basis functions of different degrees and any number of elements

**Keywords:** Finite element method,Modified Galerkin's method, Standard shape function,MATHEMATICA

[1]. Lewis, P. E., Ward J. P., 1991, The Finite Element Method, Addison-Wesley Publishers Ltd., Great Britain.

[2]. Fish J., Belytschko T., 2007, A First Course in Finite Elements, John Wiley & Sons Ltd, England.

[3]. Reddy J. N., 1993, An Introduction to The Finite Element Method, MacGraw-Hill Inc.

[4]. Smith I. M., Griffiths D. V., 1982, Programming The Finite Element Method, John Wiley & Sons Ltd, England.

[5]. Zienkiewicz O.C., Taylor R.L., 2000, The Finite Element Method, Butterworth-Heinemann, England....

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Paper Type |
: | Research Paper |

Title |
: | Properties of Square Roots of the Jacobsthal Matrix of Order 3x3 |

Country |
: | Turkey |

Authors |
: | Fikri Köken || Saadet Arslan |

: | 10.9790/5728-1402026167 |

**Abstract: **In this study, a certain Jacobsthal matrix J of order............

**Keywords:** Jacobsthal number, Jacobsthal Lucas number, Jacobsthal Matrices, Square Root Matrices

[1] N. J. A. Sloane, A Handbook of Integer Sequences, (New York: Academic Press, 1973).

[2] A.F. Horadam, Jacobsthal representation numbers, The Fibonacci Quarterly, 34, No. 1, 1996, 40 - 54.

[3] G. Berzsenyi. Solution to Problem H-274. The Fibonacci Quarterly 17. No. 1, 1979, 95.

[4] F. Koken and D. Bozkurt, On The Jacobsthal numbers by matrix methods, Int. J. Cont. Math. Sci., V.3, N.13, 2008, 605-614.

[5] F. Koken, D. Bozkurt, On The Jacobsthal-Lucas Numbers by Matrix Methods, Int. J. Cont.. Math. Sci., V.3, N.33, 2008, 1629-1633.

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Paper Type |
: | Research Paper |

Title |
: | A Perfect Implantation of Euler's Phi-Function in Hill Cipher Cryptosystem |

Country |
: | Bangladesh |

Authors |
: | Mohiuddin Ahmed |

: | 10.9790/5728-1402026873 |

**Abstract: **In This Paper, we have discussed the RSA public key crypto system proposed by R, River, A, Shamir & L. Adeleman and Hill Chiper cryptosystem proposed by Lester S. Hill for encrypte the text. To Discuss these
we use Euler-phi function, congruence and simple Matrix Application in cryptography to decrypt and encrypt
the message. In Our analyses we combined two cryptosystem which is more secure than conventional
cryptographic system such as Ceaser cipher. We use both secret key cryptography and public key cryptography which differ from conventional cryptography. In our thesis we use two keys for encryption & two for decryption. Decryption two keys effect inverse operations encryption keys. Therefore these keys are related each other.

**Keywords:** Cryptography, Congruence, Euler' phi function, RSA cryptosystem, Matrix

[1] M.R Adhikari & Avishek Adhikari, Introduction to linear algebra with application to basic cryptography (New Delhi 2007):)

[2] Niven IHerbert S.Z Hugh L.M An introduction to the theory of numbers (5th edition Willy and Sons 1980)

[3] Burton M.D Elementary Number theory (2nd edition New Delhi W.M.C Brown publishers 1989 )

Chapters in Books:

[4] L.M Adleman, R.l Rivest, A shamir A method for obtaining digital signature and public key cryptosystems (Comm. Of ACM21(1978)120- 126)

Thesis:

[5] W, Diffine M.E Hellman-New direction in cryotography(IEEE Trans. Information Thesis 22(1976)) volume 644-654.

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Paper Type |
: | Research Paper |

Title |
: | The Algebraic Least Squares Fitting Of Ellipses |

Country |
: | Saudi Arabia |

Authors |
: | Abdellatif Bettayeb |

: | 10.9790/5728-1402027483 |

**Abstract: **Fitting ellipses to a set of given points in the plane is a problem that arises in many application areas, e.g. computer graphics [9], [3], coordinate metrology [2], petroleum engineering [8]. In this paper, we
present several algorithms which the ellipse for which the sum of the squares of the distances to the given points is minimal. These algorithms are compared with classical and iterative methods. Ellipses is represented algebraically i.e. by an equation of the form F(x) = 0. The algorithm computes a continuous function closely approximating the ellipses, for which the sum of the squares to the given set of points is minimized. We will look particularly at one method, by giving examples and using Matlab to solve these problems and then compares the efficiency of them.

[1] Dixon, L. C. W.,Spedicato, E. and Szego, G. P., Nonlinear Optimization: theory and algorithms, Birkhauser Boston, 1980. ISBN 3-7643-3020-1. pp. 91-102.

[2] Fletcher, R. G., Practical Methods of Optimization, John Wiley and Sons, 1995. ISBN 0 471 91547 5 pp. 111-136.

[3] Gander, W., Golub, G. H. and Strebel, R., Fitting of circles and ellipses: least square solution,BIT, 34(1994), pp. 556-577.

[4] Huffel Sabine Van, Recent Advances in total least squares techniques and errors in variables modeling, " Orthogonal Least Squares Fitting by Conic Sections " by HelmuthSpath pp. 259-264, Library of Congress, USA, 1997. ISBN 0-89871-393-5.

[5] Lay, D. C., Linear Algebra and its applications, Addison-Wesley, 1994. ISBN 0-201-52031-1.

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**Abstract: **Explaining what a tensor is is not an easy task; worse still is trying to visualize it. However, in a very
simple way, we can say that a tensor can be defined as a set of entities that satisfy some basic rules, something
similar to the vectors. The vectors satisfy the rules of the vector space, while the tensors obey the rules of a tensorial space. The vector space is contained in the tensorial space. The mathematical entity called a tensor is a generalization of the concept of a vector. Aliases, numbers, vectors and matrices, are examples of tensors, and the difference between them is that each has a certain order. Number is a "order tensor 0", vector is a "order tensor 1", arrays are "order 2 tensors" and so on. The whole modern formulation of physics is based on tensor calculus, for these mathematical entities better describe the physical quantities in question, just as a vector better describes a displacement than a scalar number would describe. In fact a tensor serves, mathematically, to simplify a physical information.

**Keywords**: tensor, physics, applications

[1] Zhu Han, Mingyi Hong, Dan Wang, Signal Processing and Networking for Big Data Applications Cambridge University Press, 2017 – 474p

[2] NEARING, J. Mathematical Tools for Physics. Dover Publications (2009)

[3] VON RÜCKERT, E. What are tensors?, text available in https://ask.fm/wolfedler/answers, access in 15/062017

[4] SOARES, D. , Brazilian Journal of Physics Teaching , v. 35, n. 3, 3302, 2013

[5] GURTIN, M.E. An Introduction to Continuum Mechanics, Academic Press, New York, 1981