| 21. | Combining the two claims above reveals that the Perron-Frobenius eigenvalue " r " is simple root of the characteristic polynomial.
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| 22. | Then the number of sign changes ( in either chain ) around decreases by one, since is a simple root of.
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| 23. | The graph is tangent to this axis at the multiple roots of " f " and not tangent at the simple roots.
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| 24. | Although a given root system has more than one possible set of simple roots, the Weyl group acts transitively on such choices.
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| 25. | If is a simple root of the polynomial, then Laguerre's method converges cubically whenever the initial guess is close enough to the root.
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| 26. | Using one additional Horner evaluation, the value of the second derivative is used to obtain methods of cubic convergence order for simple roots.
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| 27. | If ? is a set of simple roots for ?, then ? ( " is a set of simple roots for ? ( ".
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| 28. | Note that while the Weyl group is abstractly isomorphic to the Coxeter group, a specific isomorphism depends on an ordered choice of simple roots.
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| 29. | Even though he spent most of his life outside Cephalonia, he knew the local culture intimately and he never forgot his own simple roots.
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| 30. | If ? is a set of simple roots for ?, then ? ( " is a set of simple roots for ? ( ".
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