By direct computation, we have and are eigenvalues of with corresponding eigenvectors and respectively.
By direct computation, we get (f"(x)=nAx^{n-2}geq0) and (lim_{xrightarrowinfty}f(x)=+infty).
In view of Lemma 1 and Remark 1, by direct computation, we obtain explicit formulas for (S_{n,p}(t^{i};q;x)), (i=1,2) as follows.
By direct computation, (f1) implies that (H t, z)) is even and satisfies begin{aligned}& nabla H t, z)=alpha(t z+obigl |z|bigr) quadmbox{as } |z|rightarrow0, end{aligned} (3.1) begin{aligned}& nabla H t, z)=beta(t z+obigl |z|bigr) quadmbox{as } |z|rightarrow infty, end{aligned} (3.2) uniformly for (tin[0, 2N]).
By direct computation, we have (Delta x(t_{k})=varphi_{k} (x(t_{k}) )) and also (x 0)=alpha).
By direct computation, we get (Deltaphi(t)=t 3t-3-6eta)), (Deltaphi (t geq0) for (t>1+2eta) and (Deltaphi(t)leq0) for (0< tleq 1+2eta).
Note that for the case (a_{1} < a_{2}), OFF-BANG switching cannot be ruled out using this argument, and the synthesis has to be constructed by direct computation.
Because the model signals depend linearly on the amplitude b, it is sufficient to maximize the magnitude of the inner product (cross correlation) between z and χ, the amplitude estimate is then determined by direct computation.
By direct computation, L α is a bounded self-adjoint linear operator on E and if z(t) ∈ SE L α z ( t ) = ∑ m = 1 ∞ 1 1 + β m 2 ( β m 2 A N - 1 - α I ) ( a m cos β m t + b m sin β m t ).
By direct computation, we have the Hopf bifurcation critical value (tau _{0}approx2.5181) when (n=0) and (omega_{0}approx0.635415).
By direct computation one checks that (varphi ^+=chi _{ 0,infty )}varphi in H^s({mathbb {R}})) if and only if (s
