The inner feedback loop will be designed based on linear algebraic method, by solving a set of Diophantine equations
, while the outer loop will be designed using linear quadratic Gaussian (LQG) controller, which is one of the robust controllers.
His topics are integers, modular arithmetic, quadratic reciprocity and primitive roots, secrets, arithmetic functions, algebraic numbers, rational and irrational numbers, diophantine equations
, elliptic curves, dynamical systems, and polynomials.
The numbers of the elements of different ranks satisfy the diophantine equations
Lemma 2 lists all possible trigonometric Diophantine equations
with up to six.
The PSO + GI algorithm was tested on several Diophantine equations
and the results were compared to the standard PSO and SHC.
Something like 900 years passed by between the time of Pythagoras and the time of Diophantus who worked with what is now called Diophantine equations
. If we look at the symbols they used as numerals, we can readily understand how difficult it must have been to perform simple computations, a necessity if one is to recognize the various properties of the number system.
Substituting (24) into the system of Diophantine equations
(1), we get all integral solutions, namely, (x, y, z) = (16561, [+ or -]6761, [+ or -]91), (71, [+ or -]29, [+ or -]6), (17, [+ or -]7, [+ or -]3), (7, [+ or -]3, [+ or -]2), (1, [+ or -]1, [+ or -]1), and (-1, [+ or -]1, 0).
In order to obtain optimal predictive value of [phi](k + j), the following Diophantine equations
are considered firstly:
Sandor, Geometric theorems, diophantine equations
and arithmetic functions, New Mexico, 2002.
However, other equations (e.g., Diophantine equations
) could be included in the curriculum in specific regions.
Such equations are called Diophantine equations
in the honour of Diophantus who studied them many centuries ago.
Key words and phrases : Diophantine equations
; Elliptic Curves.