iosr-Jm   Volume-16 ~ Issue-3 ~ may – june 2020

Abstract: This paper presents some basic concepts of shift map and cantor set. We have proved some properties of shift map such as continuous, chaotic and homeomorphism. We have described the construction and the formula of the cantor ternary set, which is the most common modern construction. We selected a problem about cantor set of Logistic Function.

Keywords: Shift Map, chaotic dynamical system, Homeomorphism, Cantor Set

[1]. Robert L. Devaney, Boston University, "A First Course in Chaotic Dynamical Systems" ,Theory and Experiment.
[2]. Richard A. Holmgren, " A First Course in Discrete Dynamical Systems", Second Edition, ISBN 0-387-94780-9.
[3]. Shaver Christopher, Rockhurst University, "An Exploration of the Cantor Set", Rose-Hulman Undergraduate Mathematics Journal: Vol. 11:Iss. 1, Article 1.
[4]. W. Obeng-Denteh, Peter Amoako-Yirenkyi, James Owusu Asare, "Cantor's Ternary Set Formula-Basic Approach" , British Journal of Mathematics & Computer Science 13(1): 1-6, 2016, Article no. BJMCS. 21435, ISSN: 2231-0851.
[5]. Indranil Bhaumik, Binayak S. Choudhury, "A Note On The Generalized Shift Map", Gen.Math. Notes, Vol. 1, No. 2, December 2010, PP, 159-165. ISSN 2219-7184..

Abstract: If after 374 years the famous theorem of Fermat-Wiles was demonstrated in 150 pages by A. Wiles [4], The purpose of this article is to give a proofs both for the Fermat last theorem and the Beal conjecture by
using the Fermat class concept. Résumé : Si après 374 ans le célèbre théorème de Fermat-Wiles a été démontré en 150 pages par A. Wiles [4] , le but de cet article est de donner des démonstrations à la fois du dernier théorème de Fermat et de la
conjecture de Beal en utilisant la notion des classes de Fermat.

Keywords : Fermat, Fermat-Wiles theorem, Fermat's great theorem, Beal conjecture, Diophantine equation

[1]. M. Sghiar, La preuve de la conjecture abc, IOSR Journal of Mathematics, 14.4:22-26, 2018.
[2]. https://en.wikipedia.org/wiki/Beal conjecture
[3]. https://en.wikipedia.org/wiki/Fermat last theorem
[4]. Andrew Wiles, Modular elliptic curves and fermat's last theorem, Annal of mathematics, 10:443--551, september-december 1995.

Abstract: In this work, the Legendre Polynomial is used as a basis function in the reconstruction step of the Weighted Essentially Non Oscillatory (WENO) method for the numerical solution of two dimensional scalar conservation laws. The WENO method is a high order high accurate finite volume method that has been designed for problems that have piecewise smooth solutions but still contain some discontinuities. The most common basis used in the reconstruction step of the finite volume methods are polynomial basis. The minimum error property of the Legendre polynomial makes it a good choice for the basis function to be used. In this work, we used the two-dimensional Legendre polynomial as a basis function and the resulting method is called the Legendre-WENO (L-WENO) method. The reconstruction procedure for the L-WENO method is clearly highlighted. Ten cells were used for a cubic reconstruction on triangular meshes. Two numerical tests confirm the efficiency and accuracy of the L-WENO method..

Keywords: Finite Volume Method; Weighted Essentially Non Oscillatory (WENO) Scheme; Legendre Polynomials

[1]. Harten, A., Engquist, B., Osher, S. and Chakravarthy, S. R. (1987). Uniformly High Order Accurate Essentially Non-oscillatory Schemes, III. Journal of Computational Physics, 71(2): 231-303.
[2]. Liu, X. D., Osher, S. and Chan, T. (1994). Weighted Essentially Non-oscillatory Schemes. Journal of Computational Physics, 115(1): 200-212.
[3]. Jiang, G. S. and Shu, C. W. (1996). Efficient Implementation of Weighted ENO Schemes. Journal of Computational Physics, 126(1), 202-228.
[4]. Friedrich, O. (1998). Weighted Essentially Non-oscillatory Schemes for the Interpolation of Mean Values on Unstructured Grids. Journal of Computational Physics, 144(1): 194-212.
[5]. Xing, Y. and Shu, C. W. (2014). A Survey of High Order Schemes for the Shallow Water Equations. Journal of Mathematical Study, 47(3): 221-249..

Abstract: The purpose of the study was to investigate factors that influence students, involvement in Participatory Problem Solving Approach (PPSA). To achieve this, one major question was answered: What factors influence students' involvement in participatory problem solving approach on derivatives/differentiation word problems of functions? The researcher used descriptive survey design for the study. A sample of 80 SHS3 students was drawn from Wa Senior High School in the Upper West Region. Data were collected using participatory problem solving approach questionnaires which was validated by experts and found to have a reliability index of 0.89. Data analyses were done using frequency counts and percentages. The results indicated that the student's intelligence quotient, home environment......

Keywords: Participatory Problem Solving Approach, influence, Factors, Differentiation word problems, problem solving

[1]. Adu, E., Assuah, C., & Asiedu-Addo, S. (2015). Students' errors in solving linear equation word problems: Case study of a Ghanaian senior high school. African Journal of Educational Studies in Mathematics and Sciences, 11, 17-29.
[2]. Aidoo, B., Boateng, K. S., Kissi, S. P., & Ofori, I. (2016). Effect of problem-based learning on students' achievement in chemistry. Journal of Education and Practice .
[3]. Albion, P., & Gibson, I. (2000). Problem Based as a Multimedia Design Framework in Teacher Education. Journal of Technology and Teacher Education, 8(4), 315-326.
[4]. Arı, A. A., & Katrancıb, Y. (2014). The Opinions of Primary Mathematics Student-Teachers on Problem-Based Learning Method. Procedia - Social and Behavioral Sciences, 116, 1826 – 1831.
[5]. Azer, S. A. (2009). Problem-based learning in the fifth, sixth, and seventh grades: Assessment of students' perceptions. Teaching and Teacher Education, 25(8), 1033–1042. doi:doi:10.1016/j.tate.2009.03.023.

Abstract: The notion of fuzzy normed vector space is an important notion in the fuzzy functional analysis with several properties and applications. The objective of this paper is to discuss some basic concept of fuzzy topological vector space together with𝛼-convergence and 𝛼-completeness of sequences and also to establish some properties of𝛼-convergence in fuzzy normed vector space..

Keywords:Fuzzy topology, fuzzy norm,𝛼-norm, 𝛼-convergence, 𝛼-Cauchyness, 𝛼-completeness.

[1] Bag, T. and S. K. Samanta (2003), "Finite dimensional fuzzy normed linear spaces."J. Fuzzy Math.,11, 687-705.
[2] Bag, T. and S. K. Samanta (2005), "Fuzzy Bounded Linear Operators."Fuzzy sets and systems,151, 512-547.
[3] Felbin, C. (1992), "Finite dimensional fuzzy normed linear spaces."Fuzzy sets and systems,48, 239-248.
[4] Kaleva, O. and S. Seikala (1984), "On fuzzy metric spaces." Fuzzy Sets and Systems,12, 215-229.
[5] Katsaras, A. K. (1991), "Fuzzy Topological Vector Spaces-I."Fuzzy sets and systems,6, 85-95..

Abstract: The general proof of Goldbach's conjecture in number theory is drawn in this paper by applying a specific bounding condition from Bertrand's postulate or Chebyshev's theorem and general concept of number theory..

Keywords: Bertrand's postulate & Chebyshev's theorem, Goldbach's conjecture, prime number, numbers series, number theory.

[1]. Fliegel, F. Henry, Robertson, S. Douglas (1989), Goldbach's Comet: the numbers related to Goldbach's Conjecture, Journal of Recreational Mathematics. 21(1): 1-7.
[2]. T. Estermann (1938), On Goldbach's problem: proof that almost all even positive integers are sum of two primes, Proc. London Math. Soc. 2. 44: 307-314. Doi: 10.1112/plms/s2-44.4.307.
[3]. M. Th. Rassias (2017), Goldbach's Problem: Selected Topics, Springer.
[4]. J. R. Chen (1973), On the representation of a larger even integer as the sum of a prime and the product of at most two primes, Sci. Sinica. 16: 157-176.
[5]. D. R. Heath-Brown, J. C. Puchta (2002), Integers represented as a sum of primes and powers of two, Assian Journal of Mathematics. 6(3): 535-565. doi: 10.4310/AJM.2002.

Abstract: For high school mathematics teaching, modern education technology plays a great role, in the present stage of the epidemic, we seem to be more inseparable from modern education technology. However, it is not objective to blindly advocate modern educational technology, and its contribution to traditional educational technology should not be abandoned. According to the advantages and disadvantages of modern educational technology and traditional educational technology, and the effective integration in high school mathematics teaching, the following discussion is carried out in this paper.

Keywords: modern educational technology; Traditional educational techniques; Mathematics teaching in senior high school.

[1]. C. M. Lindvall, Instructionaldesign. Institute of Education Sciences, 1968-sep-3: ED036159
[2]. Yeaman, R. J. Andrew, Deconstructing modern educational technology. Educational Technology, 1994,34(2):15-24.
[3]. K. E. German, N. A. Salavatov, N. Gladkikh, et al. Modern educational technology. Modern Educational Technologies. Part 3, 2019.
[4]. J. E. Susskind. PowerPlint's power in the classroom: enhancing students' self-efficacy and attitudes. Computer &Education, 2005, 45(2): 203-215.
[5]. H. Y. Fan, Thethinking of using multimedia technology to improve the quality of mathematics teaching. Advanced Materials Research Vols: 189-193.

Abstract: The Bernoulli polynomial matrix is expressed by Bn 𝑥 , with each entry being a Bernoulli polynomial. The k-Fibonacci matrix is represented by 𝐹𝑛 𝑘 , with each entry being a k-Fibonacci number, whose first term is 0, the second term is 1, and the next term depends on a positive integer k. In this paper, we discuss about relation between the Bernoulli polynomial matrix and k-Fibonacci matrix. The results are define two new matrix, 𝐶𝑛 𝑥 and 𝐷𝑛 𝑥 such that Bn 𝑥 =𝐹𝑛 𝑘 𝐶𝑛 𝑥 =𝐷𝑛 𝑥 𝐹𝑛 𝑘 .

Keywords: Bernoulli number, Bernoulli polynomial, Bernoulli matrix, Bernoulli polynomial matrix, k-Fibonacci number, k-Fibonacci matrix.

[1] P. Sebah dan X. Gourdon, Introduction on Bernoulli's numbers, numbers.computation.free.fr/Constants/constants.html, 2002.
[2] T. Ernst, q-Pascal and q-Bernoulli matrices and umbral approach, Department of Mathematics Uppsala Universty, 23, 2008. 1-18.
[3] M. Can and M. C. Dagli, Extended Bernoulli and Stirling matrices and related combinatorial identities, Linear Algebra and its Applications, 444, 2014, 114–131.
[4] N. Tuglu dan S. Kus, q-Bernoulli matrices and their some properties, Gazi University Journal of Science, 28 (2), 2015, 269–273.
[5] Z. Zhang dan J. Wang, Bernoulli matrix and its algebraic properties, Discrete Applied Mathematics, 154, 2006, 1622–1632..

Abstract: In this Paper, some problems associated with numerical weather prediction are discussed. we have been able to simulate some finite difference schemes to predict weather trends of Abuja. By analyzing the results from these schemes, it has shown that the best scheme in the finite difference method that gives a close accurate weather forecast is the trapezoidal scheme when comparing with sunshine, Rainfall and windspeed. We use the trapezoidal scheme to stimulate the numerical weather data obtained from federal Airports Authority. Finally using Matlab (2012a) to acquire subsequent numerical tendency

[1]. Comblen, R., Lambrechts, J., Remacle, J. F. and Legat, V., (2010). Practical evaluation of five partly discontinous finite element pairs for the non-conservative shallow atmospheric weather equations. Intrenational Journal for Numerical Methods in Fluids 63 (6), 701 - 724
[2]. Cotter, C. and Ham, D., (2011). Numerical wave propagation for the triangular PIDG-P2 finite element pair. Journal of Computational Physics 230 (8), 2806-2820.
[3]. Cotter, C., Shipton, J., (2012). Mixed Finite Elements for Numerical Weather Prediction. Journal of Computational Physics 231 (21), 7076-7091.
[4]. Cotter, C. J., Ham, D. A. and Pain, C.C., (2009) A mixed discontinuous/continuous finite element pair for shallow-water ocean modelling. Ocean Modelling 26, 86-90.
[5]. Danilov, S., (2010). On Utility of Triangular C-grid Type Discretization for Numerical Modeling of Large-Scale Ocean Ows. Ocean Dynamics 60 (6), 1361-1369