Rational Points on Elliptic Curves y^2=x^3+a^3 in f_{p} where p{\equiv}1(mod6) is Prime

In this work, we consider the rational points on elliptic curves over finite fields F_{p}. We give results concerning the number of points on the elliptic curve y^2{\equiv}x^3+a^3(mod p)where p is a prime congruent to 1 modulo 6. Also some results are given on the sum of abscissae of these points. We give the number of solutions to y^2{\equiv}x^3+a^3(modp), also given in ([1], p.174), this time by means of the quadratic residue character, in a different way, by using the cubic residue character. Using the Weil conjecture, one can generalize the results concerning the number of points in F_{p} to F_{p^{r}}.Comment: 9 pages; Keywords: Elliptic curves over finite fields, rational point

Penerapan Metode Bermain Peran untuk Meningkatkan Pembelajaran Bahasa Indonesia Aspek Berbicara

Indonesian language (IN) is one of the obligatory lessons in Senior High School. The learning of Indonesian language is pointed to the increasing ability of students. This aim to increase students ability to communicate in Indonesian language well and properly. The communication ability is expected especially in oral and written. Besides, students are hoped to be able to appreciate literary works. The objective of this research is to identify and to describe the implementation of role play method in the efforts to increase the activity of Indonesian language learning in speaking aspects and to identify and to get the description whether the implementation of the method can increase process and result speaking aspect of Indonesian language learning of students class XI IPS 1, Semester 1, SMAN 3 Bengkalis. The method used was classroom action method and the subject was class XII IPS 1, semester 1, SMAN 3 Bengkalis. The collection of data and information were carried out by observation, journal and document study. The result of the research showed that (1) The implementation of of role play method can increase the activities of speaking aspects in Indonesian language learning. On the 1st cycle1, 1st meeting, the percentage of students activities was 73% with the good category (3,6), while on the second meeting, it was 82% with the very good category (3,7). On the first meeting of the second cycle, the percentage of students activities became 83% with the very good category (4,3), while on the second meeting was 96% with the very good category (5,0), (2) The implementation of role play method is able to increase the result of speaking aspect in learning Indonesian language. On the first cycle, the students reserved effort was 73% while on the second cycle was 77%. Meanwhile, based on attachment 3, the classical completeness on cycle 1 reached 86%, on cycle 2, it increased 96%. Based on the result of the research, the researcher proposed that it would be better for the Indonesian language teachers to implement role play method for the materials that are suitable with . speaking aspects. The reason is that this method is proved to be able to increase the process and result of students learning. When implementing this method, teachers are hoped to give guidance, motivation, and explanation that more focused so that students can learn optimally

A-stable 2-point block extended backward differentiation formula for solving stiff ordinary differential equations

This paper focuses on derivation of a 2-point block extended backward differentiation formula (BEBDF) for the integration of stiff ordinary differential equations. The formula derived computes two solution values simultaneously in a block and uses an extra future point, thereby having more advantage than the conventional block backward differentiation formula (BBDF). The stability region covered the entire negative half plane, proving that the method constructed is A-stable. To validate the method derived, some problems known to be stiff were solved and the results obtained are compared with the existing BBDF in terms of maximum error and computation time. The comparison shows that the error growth for the new method is smaller compared to 2-point BBDF

New explicit trigonometrically-fitted fourth-order and fifth-order Runge-Kutta-Nyström methods for periodic initial value problems

In this work, we derive two new Runge-Kutta-Nyström (RKN) methods for solving periodic second order ordinary differential equations. We constructed these methods based on the RKN method of three-stage fourth-order derived by Garcia and RKN method of four-stage fifth-order derived by Hairer. The numerical results show that the efficiency of the new methods is more than the other existing methods

An implicit 2-point block extended backward differentiation formula for integration of stiff initial value problems

A class of implicit 2-point block extended backward differentiation formula (BEBDF) of order 4 is presented. The stability region of the method is constructed and shown to be A – stable. Results obtained are compared with an existing block backward differentiation formula (BBDF). The comparison shows that using constant step size and the same number of integration steps, our method achieves greater accuracy than the 2-point BBDF and is suitable for solving stiff initial value problems

Comparison of forecasting performance between MODWT-GARCH(1,1) and MODWT-EGARCH(1,1) models: Evidence from African stock markets

Many researchers documented that if stock markets' returns series are significantly skewed, linear-GARCH(1,1) grossly underestimates the forecast values of the returns. However, this study showed that the linear Maximal Overlap Discreet Wavelet Transform MODWT-GARCH(1,1) actually gives an accurate forecast value of the returns. The study used the daily returns of four African countries' stock market indices for the period January 2, 2000, to December 31, 2014. The Maximal Overlap Discreet Wavelet Transform-GARCH(1,1) model and the Maximal Overlap Discreet Wavelet Transform-EGARCH(1,1) model are exhaustively compared. The results show that although both models fit the returns data well, the forecast produced by the Maximal Overlap Discreet Wavelet Transform-EGARCH(1,1) model actually underestimates the observed returns whereas the Maximal Overlap Discreet Wavelet Transform-GARCH(1,1) model generates an accurate forecast value of the observed returns