The inner loop was designed using linear algebraic method via solving a set of Diophantine equation
, while the outer loop was designed using LQG controller.
The specialization of the polynomial f (a, x) with a 2 Q is irreducible if and only if a is not one of the following forms with a rational solution (A, B) of the Diophantine equation
[A.sup.2] - 2[B.sup.2] = 1:
Spock: But again Feinstein's argument would not apply to this Diophantine equation
, precisely because this Diophantine equation
can be reduced via the Euclidean algorithm to the equation,
Both methods, the standard PSO algorithm and the proposed modification PSO + GI algorithm, are tested on the Diophantine equation
solver task (see Table 1 in Section 5.1 for test equations).
A note on the Diophantine equation
[x.sup.2] + [q.sup.m] = [c.sup.2n] Mou-Jie DENG Communicated by Shigefumi MORI, M.J.A.
By solving the Diophantine equation
(11), the polynomials R and S can be obtained.
By Diophantine equation
, we mean that Pythagorean triples a,b,c have been substituted with natural numbers such that for any n2, the equation (5) can be expressed as:
Since Peter had stumbled across 33 + 43 + 53 = 63, I looked for other solutions to the Diophantine equation
[a.sup.3] + [b.sup.3] + [c.sup.3] = [d.sup.3], and found that it was Euler who discovered the complete solution, with Ramanujan later finding a simpler form (see Berndt & Bhargava, 1993).
Some groups considered equation (1) as a diophantine equation
, determined all non-negative solutions and proved that these solutions are not realisable.
The general solution of the Diophantine equation
[x.sup.2] - [2y.sup.2] = -1 is
The only solutions to this Diophantine equation
are x - x' = sa + bt, y - y' = sb - at for t [member of] Z.
In , Stroeker investigated the Diophantine equation