If the i/o equation is reducible, then the realization of the reduced equation
is also considered a realization for the original system.
We substitute (6), (7), or (8) into the reduced equation
(5), balance the terms of the sech, tanh, and csch functions, and solve the resulting system of algebraic equations by using computerized symbolic packages.
Considering [u.sub.s](t) = [P.sub.s]u(t) and substituting the pullback limit [h.sup.(1).sub.[lambda]]([xi](t)) of the first lever system (10)-(11) into the above equation, we obtain reduced equation
We can now easily show that the solution of (5) converges to the solution of the associated reduced equation
as [epsilon] [right arrow] [0.sup.+] if the continuity of the g-Drazin inverse is assumed.
System (3) and (4) are, respectively, called the reduced equation
and the layer equation of system (1).
the change of variable v = dY/dX, gives the reduced equation
A number of constants will remain in the reduced equation
and these dimensionless parameters have the most pronounced effect on the yarn dynamics.
the reduced equation
of state obtained from the resulting partition function is given by:
As other explanatory variables are, in theory, uncorrelated with the error of the reduced equation
(obtained in the first stage of the test), the supposedly endogenous variable is not correlated with the error of the reduced equation
if, and only if, the error of the reduced equation
is uncorrelated with the error of the structural equation.
A PhD thesis of Baker  in 1911, devoted to the interior and exterior problems, referred only to Barbarin [3, 4], discussed the interior and exterior bisectors problems in a more direct fashion (without solving first, like Barbarin, the problem with two bisectors and an angle given), and studied the irreducibility and the group of the reduced equation
. He also considered various special cases.
We substitute (16)-(17) into the reduced equation
(15); we get
Tam, "An integrable hierarchy and Darboux transformations, bilinear Backlund transformations of a reduced equation
," Applied Mathematics and Computation, vol.