iosr-Jm   Volume-14 ~ Issue-3 ~ May-June 2018

Abstract: The paper considers Fermat's theorem as a special case of the problem of least (smallest) deviation. It is proved that the theorem is true, since error can not vanish on the set of rational numbers.

Keywords: Number theory, natural numbers, Fermat's last theorem

[1] Romanov V.N. Elementary way of proving Fermat's theorem. // Int. Journal of Engineering Research and Application, 2018, Vol. 5,
Issue 1, pp.57 – 68.
[2] Romanov V.N. Investigation of the fundamental problems of number theory. Publishing house "Asterion", 2015 (in Russian).
[3] Wiles A. Modular Elliptic Curves and Fermat's Last Theorem. // Annals of Mathematics, 1995, Vol. 142, pp. 443-551.

Abstract: Critical thinking is one of the important indicators for students to be competing in the world of work and personal life, students must have the ability to solve problems and must be able to think critically. One effort to improve critical thinking skills, especially in math lessons is to implement problem-solving learning. This article describes the implementation of problem-solving learning in improving students' critical thinking skills of grade VIII SMP Negeri 5 Bulukumba obtained through classroom action research that is conducted for 2 cycles with the result that with the application of learning problem solving can improve the ability of critical thinking as seen from the indicators the success of 35 people or 87.5% of students have completed learning mathematics and as many as 5 people or 12.5% of students have not finished learning math.........

Keywords: Critical Thinking, Problem-Solving

[1] Djamarah & Zain. Teaching and Learning Strategies. Jakarta: Rineka Cipta. 2006.
[2] Fachrurazi, Application of Problem Based Learning to Improve the Ability of Critical Thinking and Student Matter Communications, Jurnal UPI. Special edition 1(1), 2011.
[3] Fisher, Alec. Critical Thinking An Introduction. Jakarta : Erlangga. 2009.
[4] Fitria, Hanifah. T. "Improving the Ability of Critical Thinking of Junior High School Students through Elicitina Activities Model (MEAS)" "Approach in Mathematics Learning". Bandung : Repositori UPI.Edu.
[5] Kamdi, W dkk. (2007). Innovative Learning Models. Malang : Universitas Negeri Malang.

Abstract: The problem of Fitting conic sections to given data in the plane is one which is of great interest and arises in many applications, e.g. computer graphics, statistics, coordinate metrology, aircraft industry, metrology, astronomy, refractometry, and petroleum engineering [7, 2, 3]. In this paper, we present several methods which have been suggested for Fitting ellipses to data in the plane. We will look particularly at one method, by giving examples and using Matlab to solve these problem.

[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 Helmuth Spath pp. 259-264, Library of Congress, USA, 1997. ISBN 0-89871-393-5.
[5] Kolman, B., Elementary linear algebra, 1991. ISBN 0-02-366045-7.
[6] Schwarz, H. R., Numerical Analysis: A comprehensive introduction, Great Britain, 1989. ISBN 0471 92064 9 pp. 294-329.
[7] The MathWorks, Inc. MATLAB REFERENCE GUIDE, 1992. PP. 205.

Abstract: The 𝑓 and Bregman divergences are used to generate two universal portfolios in a model-free stock market. The logarithm of the estimated next-day wealth return is approximated by 𝑘 terms of its Taylor series. The condition for the two universal portfolios to be identical is derived. A sufficient condition for the two universal portfolios to be identical is demonstrated. The Helmbold family of universal portfolios are both 𝑓-divergence and Bregman-divergence universal portfolios. An empirical study of the Type 2 Helmbold universal portfolio is presented. There is empirical evidence that the investor's wealth can be increased by using this type of universal portfolio.

Keywords - Bregman divergence, 𝑓-divergence, Investment, Universal portfolio

[1] T. M. Cover, Universal Portfolios, Mathematical Finance, vol. 1, pp. 1-29, Jan 1991.
[2] T. M. Cover and E. Ordentlich, Universal portfolios with side information, IEEE Transactions on Information Theory, vol.42, no.2, pp.348-363, Mar. 1996.
[3] C. P. Tan, Performance bounds for the distribution-generated universal portfolios, Proc. 59th ISI World Statistics Congress, Hong Kong, 5327-5332, 2013.
[4] D. P. Helmbold, R. E. Shapire, Y. Singer and M. K. Warmuth, On-line portfolio selection using multiplicative updates, Mathematical Finance, vol.8, no.4, pp.325-347, Oct. 1998.