# root system

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## root system

the total below-ground vegetative system of vascular plants. see ROOT.
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In view of Theorem 3.1, to prove Theorem 1 (ii) it suffices to show that the Alexander polynomial of the twisted torus knot [K.sub.n,m] has a simple root on the unit circle.
These functions depend upon simple roots of a complete quadratic polynomial due to the inclusion of a second derivative spatial term in (24).
As a consequence, for an analytic IF -(z) we can say that (a) [PHI](z) is of order p if and only if [PHI](z) = [alpha] + O([(z - [alpha]).sup.p]), or, equivalently, if [PHI]([alpha]) = [alpha] and [[PHI].sup.(1)](z) = O([(z - [alpha]).sup.p-1]), and (b) if [alpha] is a simple root of f(z), then [PHI](z) is of order p if and only if [PHI](z) = [alpha] + O([f.sup.p](z)), or, equivalently, if [PHI]([alpha]) = [alpha] and [[PHI].sup.(1)](z) = O([f.sup.p-1](z)).
For each fixed simple root [alpha], let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
Assume [Florin],[Florin]' and [Florin]" are continuous on an interval [I.sub.[epsilon]]=([alpha]-[epsilon],[alpha]+[epsilon]) containing the simple root [alpha] ([Florin]'([alpha]) [not equal to] 0).
Jayakumar, Kalyanasundaram M., Generalized Power means Modification of Newton's Method for Simple Roots of Nonlinear Equation, Int.
Lemma 3 For every simple root [alpha] [member of] S, there exists a bijection [THETA] from the set of nonnesting partitions A [member of] NN([PHI](S)) with [alpha] [member of] A to NN([PHI](S\{[alpha]})), such that [absolute value of [THETA](A)] = [absolute value of A] - 1 and [absolute value of [THETA](A) [intersection] S\{[alpha]}] = [absolute value of A [intersection] S] - 1 for all A [member of] NN([PHI](S)).
We here investigate the comparison of the attained simple root finders in the complex plane using basins of attraction.
It is clear that [lambda] = 0 is a simple root of (28).
Let us consider the problem of approximating the simple root x* of the nonlinear equation involving a nonlinear univariate function f:
Orbital mechanics are simple in theory--it's all just gravity!--but from that simple root grow unexpected flowers of complexity and interest.
The operation began as a simple root canal, but after the veterinarian sedated Geil and took a closer look at his jaw, she realized that the tooth in question had been broken down the middle, past the gum line.

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