In every one of the above integrals, the phase function F(t, u) has a unique
stationary point at t = 0 with [F.sup.(n)](0,u) = 0, n = 0,1, 2,...,m - 1 and [F.sup.(m)](0,u) = [f.sup.(m)](a)[u.sup.(m)] [not equal to] 0.
However, this indicates that a segment around the
stationary point is also excluded from the calculation.
The optimized geometries and frequencies of the
stationary point and the minimum-energy paths are calculated by using the DFT (B3LYP) methods with 6-311 basis sets.
Letting (2) equal zero, we can find the
stationary point for the objective function.
Define [partial derivative]f(i)/[partial derivative][P.sub.j] = f(i,j), [[alpha].sub.k] = [Y.sub.exp]([t.sub.k]) - f(k) and for an independent variation of the variables (P, k) at the
stationary point, we have
where S([f.sub.[tau]], [f.sub.a]; [r.sub.0], [x.sub.0]) is the 2-D spectrum, [f.sub.[tau]] and [f.sub.a] are range frequency and Doppler frequency respectively, [W.sub.r](*) and [W.sub.a](*) are the envelopes of range spectrum and azimuth spectrum respectively, [f.sub.c] is the carrier frequency, [t.sup.*] is the azimuth
stationary point and can be written as
Maybe not as well as Ford had hoped as they were fighting from a virtual
stationary point against rivals with a huge head start, but the first generation model shifted 45,000 units in the UK until the second generation model appeared in 2013.
It has now resumed direct motion, having reached its second
stationary point at the end of January.
Analogous to the inhibited variational models in which one has stationary solutions, our objective is to present results that establish the necessary and sufficient conditions for the existence of a
stationary point.
The canonical analysis of the response surface was performed with SAS to determine the shape of the fitted response and the estimated
stationary point. According to the model, the predicted response at the
stationary point ([X.sub.1] = 76.25, [X.sub.2] = 1.38, and [X.sub.3] = 3.08, shown in Table 6) was 109.93 g/L.