Roots of unity

A unitary matrix may therefore be considered a kind of square root of unity, often complex-valued. Unitary matrices with all real elements are called orthogonal O, and satisfy a property analogous to (S9.1-12) ... 

Let oin— 1, where n is a positive integer. The roots of this equation are the nth roots of unity. The number 1 has absolute value 1 and its phase can... 

E. This double -1 point is yet another codimension-two bifurcation, which will be discussed in detail later. Another period 1 Hopf curve extends from point F through points G and H. F is another double -1 point and, as one moves away from F along the Hopf curve, the angle at which the complex multipliers leave the unit circle decreases from it. The points G and H correspond to angles jt and ixr respectively and are hard resonances of the Hopf bifurcation because the Floquet multipliers leave the unit circle at third and fourth roots of unity, respectively. Points G and H are both important codimension-two bifurcation points and will be discussed in detail in the next section. The Hopf curves described above are for period 1 fixed points. Subharmonic solutions (fixed points of period greater than one) can also bifurcate to tori via Hopf bifurcations. Such a curve exists for period 2 and extends from point E to K, where it terminates on a period 2 saddle-node curve. The angle at which the complex Floquet multipliers leave the unit circle approaches zero at either point of the curve. 

Among all of the points on the period 1 Hopf curve, some will have complex Floquet multipliers A with a phase angle 6 of mln)2u with n = 3 or 4 (i.e. third or fourth roots of unity) and are called hard reasonances. Because these points are fixed points for Fn that have multipliers equal to A" = 1, it is not surprising to find that subharmonic fixed points of period n are involved in addition to the bifurcating period 1 fixed point. 

FIGURE 8 (a) Detail of the tip of the 3/1 resonance horn illustrating typical way in which period 3 resonance horns close around a point with Floquet multipliers at the third root of unity (point F). (fa) and (c) The saddle-node pairings change from section AA to section BB and the unstable manifolds of the period 3 saddles no longer make up a phase locked torus. (d) A one-parameter vertical cut through the third root of unity point. The three saddles coalesce with the period one focus that is undergoing the Hopf bifurcation. 

FIGURE 9 Stroboscopic phase portraits for the points on figure 8 labelled (a)-(e). (a) Below the third root of unity point (labelled F in figure 8) the phase portrait is structurally a period three phase locked torus (b) above point F, the period I fixed point in the centre is now stable and the phase locked torus has disappeared (c)-(e) before, during, and after a period 3 homoclinic bifurcation to the right of point F oil cut, = 3.97 for each, and A/Ao = 5.90, 5.93 and 5.95 for (c)-(e) respectively. The period 3 phase locked torus is transformed to a free torus as the stable manifold of each saddle crosses the unstable manifold of an adjacent saddle. 

The PFs gi gy] are a set ofg2 complex numbers, which by convention are all chosen to be square roots of unity. (For vector representations the PFs are all unity.) PFs have the following properties (Altmann (1977)) ... 

In the last section of this chapter, in Section 8.7, we present identities about roots of unity in integral domains. The results will be useful in Section 12.4 where we shall investigate Coxeter sets of cardinality 2. 

A right triangle whose sides are relatively prime integers, primitive root of unity... 

The complex number z is a primitive nth root of unity if zn=l but zk is not equal to 1 for any positive integer k less than n. probability... 

We introduce the following finite group schemes ("roots of unity )... 

In the first case, we should in the first instance have to investigate each of the electrons 1 and 2 in the field of the two force-centres a and b. The proper functions are then symmetrical or anti-symmetrical with respect to the medial plane M, i.e. in passing from a to b they are multiplied by 1, whereas we saw above that in the case of several equal equidistant hollows the proper functions are multiplied by an arbitrary root of unity in passing from one atom to the next. If we assume that the hollows are very deep or very far apart, the proper functions may be represented approximately in terms of the proper... 

William Rowan Hamilton (1856). Memorandum respending a new system of Roots of Unity. Philosophical Magazine 12, 4, 446. 

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