for all t [member of] l, then there exists a unique continuously differentiable function
[v.sub.0] : I [right arrow] X such that [v'.sub.0](t)+p(t)v0(t)+q(t) = 0 for all t [member of] l and
Due to the strong coupling between the rigid body states and the flexible modes in the AHV model ((1), (2), (3), (4), (5), (6), and (7)), here the dynamic characteristics of strong nonlinearity and strong coupling in the original model are regarded as completely unknown continuous differentiable functions
((16) and (44)) by referring to the method of .
In general, the fractional derivative of order w of any differentiable function
f(l) is defined in the form :
Taking into account that ([theta] - [eta])/[psi] is an infinitely differentiable function
with compact support and that F(([eta](1 - [phi]))/[psi]) [member of] [L.sub.[??]]([R.sup.d]), we get by Theorem 2.2 from (3.2) and (3.6) for each g [member of] [B.sup.p.sub.[sigma]]([R.sup.d]) the estimate
Bi-quaternion of differentiable function
of x=([x.sub.1],[x.sub.2],[x.sub.3]) is defined as :
In the case of a differentiable function
F(y, z), we use subscript letters to denote derivatives [F.sub.y](y, z) = [partial derivative]F/[partial derivative]y, [F.sub.yz] (y,z) = [[partial derivative].sup.2]F/[partial derivative]y [partial derivative]z, [F.sub.yy](y,z) = [[partial derivative].sup.2]F/[partial derivative]y[partial derivative]z, each evaluated at (y, z).
Let f(x) be a real-valued differentiable function
defined for x [element of] [R.sup.n].
For mathematical convenience, f(x) must be a continuous and differentiable function
Coordinate geometry provides a useful way of representing functions on real numbers: calibrated horizontal and vertical axes have their zero-mark where they meet, the origin; then a line, possibly neither straight nor smooth nor uninterrupted, represents a function f according to familiar conventions.(5) Using these conventions, a continuous function can only be represented by an uninterrupted or gapless line, a line drawn without lifting the nib off the paper, so to speak; and a differentiable function
can only be represented by a line which is not only uninterrupted but also smooth, without sharp bends or sudden changes of direction.
The basis of the proof was Sard's theorem on the set of critical values of a differentiable function
, of which I learned in the late sixties in a first encounter with Steve Smale, whose path had remained separate from mine during the period of campus turbulence that began in September 1964.
Following the notation of Riley  and Spence , each applicant has an underlying ability level, n, which falls within the interval [n(0), n(1)] and the proportion of the applicant pool with ability less than n is assumed to be a differentiable function