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 system:

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.