Suppose, by contradiction, that equation (3) is nonoscillatory and let w be a solution of the associated Riccati equation (6) and v = h p w - G the associated solution of the modified Riccati equation (9).
Suppose (q^{ast}) and (q_{L}^{ast}) are, respectively, the solutions of the continuous optimization problem (2.12) and its discrete counterpart (3.3), (u(q^{ast})) and (u_{L}(q^{ast}_{L})) are the state solutions of (2.9) and (3.2) associated to (q^{ast}) and (q_{L}^{ast}), respectively, (z(q^{ast})) and (z_{L}(q^{ast}_{L})) are the associated solutions of (2.22) and (3.6), respectively.
If is the distinguished solution of (2.1), then the associated solution of (1.1) given by the formula.
This means that any solution of (3.13) which starts with the initial condition blows down to at a finite time, which is a zero point of the associated solution of (3.7).
Denote by the associated solution of (2.1) and let be a solution of (2.1) generated by another solution (linearly independent of ) of (1.1).
A hex-to-hex surface (upper right) and associated solution of Poisson's equation (bottom left) are shown in Fig. 13.
Hence, one can guess the "correct" value of ρ, solve n + 1 nominal problems and take the best of the associated solutions as the optimal robust solution.
Its effective use lies in interpreting the data collected from a nanoindentation experiment with an associated analytical/numerical solution of the corresponding problem configuration.
To explain the connection between short-range order of liquid Ca Sb alloys and the strong deviation from ideality in the thermodynamic properties, two thermodynamic models were invoked and reconciled: the regular associated solution model, assuming the presence of a CaSb2 associate, and the molecular interaction volume model (MIVM).
For alcohol-aliphatic hydrocarbon solutions, the athermal associated solution theory of Wiehe and Bagley is modified by introducing a physical interaction term of the Scatchard Hildebrand type expression.
