Unit circle, definition

Phase margin (PM) is defined as the angle between the negative real axis and a radial line drawn from the origin to the point where the GuB curve intersects the unit circle. See Fig. 13.7. The definition is more compact in equation form. 

The first effect is obvious from the definition of the logarithm of z as given in Eq. (19.44). Inside the unit circle, the magnitude of z is less than 1. Therefore the In I z I is negative. On the unit circle, In z = 0. So the unit circle in the z plane maps into the left half of the log-z plane. 

The admissible region for closed-loop eigenvalues is denoted as F, a system is called F-stable if all its eigenvalues are located in this region and an imcertain system is called robustly F-stable if all eigenvalues for all operating conditions are contained in F. The definition of F-stability permits arbitrary regions in the complex 5 -plane and does not imderlie any restrictions. It also includes the special cases of the left half-plane for Hurwitz stability and the unit circle for Schur stability. 

Equivalently, one may distribute the N oscillators on a unit circle (Fig. 5.2), associate a vector of unit length to each oscillator, take a vector sum and divide it by N. Clearly, a vanishes for the uniform distribution (5.4.10), and it is nonvanishing for the traveling density waves (5.4.11). From the definitions of n and the complex order parameter, (5.4.5) may be transformed into... 

Definition 5.2.2 A method is called stable at infinity, if oo is in its stability region, i.e. if the roots of the a polynomial lie in the unit circle and those on its boundary are simple. If there are no roots on the boundary of the unit circle the method is called strictly stable at infinity. 

Exercise 10.17 Suppose pr, p2 and pi are points on the unit sphere in that do not lie on one common great circle. (A great circle is the intersection of the unit sphere with a plane through the origin in ) Show that every point p on the sphere is uniquely defined by its distances from the three points pi, P2, p3. Interpret great circles in PCC ) physically i.e., give a definition in terms of experiments and probabilities. 

For r = 1, V = y so that y is the value of the solid harmonic in the surface of the unit sphere at points defined by the coordinates 6 and surface harmonics of degree /. The associated Legendre polynomials Pf(cosd) have l — m roots. Each of them defines a nodal cone that intersects a constant sphere in a circle. These nodes, as shown in Figure 20-5, are in the surface of the sphere and not at r = 0 as assumed in the definition of atomic orbitals. Surface harmonics are obviously undefined for r = 0. The linear combinations i/ i i/i i define one real and one imaginary function directed along the X and Y Cartesian axes respectively, but these functions (denoted and ipy) are no longer eigenfunctions of L, but of or Ly instead. 

Figure 1. Finite element model and the definitions of evaluated configurations of the interface cracks a) FE mesh with no cracks, b) the crack intercepting the interface between the brick and the mortar and c) the crack at the interface between the mortar and the brick unit. The region marked with the white circle corresponds to the location where the crack configurations (b) and (c) were analyzed.

Fig. 7 Definition of coarse-grained repeat units (symbolized by circles) for the 1,4-polybntadiene chain. The atomistic model is an united abun model of a random copolymer of 45% cis and 55% trans content, without vinyl units. Bmid lengths ate constrained ( = 1.53 A for the CH2-CH2 bond, i = 1.5A for the CH2-CH bond and 1.34A fiuthe CH-CH bond). From Strauch et al. [117]...

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