Abandoning all unfinished projects set in the near, middle, and deep past, I intend to complete as soon as I can — certainly in this autumn of 2003 — a detailed description, in the future definite, of the second Administration of George W. Bush.
A function that is (strictly) conditionally positive definite of order m on ({mathbb {R}}^d) is also (strictly) conditionally positive definite of any higher order; therefore, Lemma 3 and Theorem 1 are satisfied for any (mu ge m).
The design idea is to apply a suitable Lyapunov function and the property of positive definite of a matrix to satisfy the direct Lyapunov theorem.
V ( X ) is always positive definite, and therefore the negative definite of its derivative should be examined; it means W ( X ) in equation (17) should always be positive definite so that V ˙ ( X ) would be negative definite.
In this equation, the free-weighting matrix N and the scalar η indicate the relationship between the terms in our system and guarantee the negative definite of stability criteria.
end{cases} (57) According to [28], the eigenvalues of the matrix (B^{T}B) all lie in the interval ([0,8)). So, to guarantee the positive definite of Q, we should set (gammatau >0). In fact, the scheme (57) is the third case of Algorithm 2 in [17].
Error estimate of the proposed method was devoted in Sect. 4. Finally, numerical examples are carried out to verify the theoretical results in Sect. 5. Let (varphi) be a continuous function on ({mathbb {R}}^d) which is conditionally positive definite of order m.
Then, for every function (fin W_{2}^{m}), we have begin{aligned} |f-mathcal {P}f|_{m,L_2(Omega )}le |f-mathcal {P}f|_{varphi }le |f|_{varphi }, end{aligned} where as before stated, (varphi) is strictly conditionally positive definite of order m.
