Pole

Where Ui denotes input number i and there is an implied summation over all the inputs in the expression above A, Bj, C, D, and F are polynomials in the shift operator (z or q). The general structure is defined by giving the time delays nk and the orders of the polynomials (i.e., the number of poles and zeros of the dynamic models trom u to y, as well as of the noise model from e to y). Note that A(q) corresponds to poles that are common between the dynamic model and the noise model (useful if noise enters system close to the input). Likewise Fj(q) determines the poles that are unique for the dynamics from input number i and D(q) the poles that are unique for the noise N(t). 

Calculations of mutual locations of poles and zeros for these TF models allow to trace dynamics of moving of the parameters (poles and zeros) under increasing loads. Their location regarding to the unit circle could be used for prediction of stability of the system (material behavior) or the process stationary state (absence of AE burst ) [7]. 

Figure 6. Location of poles and zeros for visco-elasto-plastic material (left) and brittle material (right) under loading close to fiacture.

To be easily attracted by the poles created perpendicularly to the defect, particles must satisfy precise conditions concerning dimensions, shape, density and magnetic property. 

The method seirsibility depeird essentially on the poles force attraction which exists at the position of the defect. This attraction force depend on the value of the leakage field, so of the magnetic exciting field which has created them. 

The preceding upper limit to particle size can be exceeded if more than one bubble is attached to the particle, t A matter relating to this and to the barrier that exists for a bubble to attach itself to a particle is discussed by Leja and Poling [63] see also Refs. 64 and 65. The attachment of a bubble to a surface may be divided into steps, as illustrated in Figs. XIII-8a-c, in which the bubble is first distorted, then allowed to adhere to the surface. Step 1, the distortion step, is not actually unrealistic, as a bubble impacting a surface does distort, and only after the liquid film between it and the surface has sufficiently thinned does... 

If the contact angle is zero, as in Fig. XIII-8e, there should be no tendency to adhere to a flat surface. Leja and Poling [63] point out, however, that, as shown in Fig. XIII-8/, if the surface is formed in a hemispherical cup of the same radius as the bubble, then for step la, the free energy change of attachment is... 

J. Leja and G. W. Poling, Preprint, International Mineral Processing Congress, London, April 1960. 

The multipole moment of rank n is sometimes called the 2"-pole moment. The first non-zero multipole moment of a molecule is origin independent but the higher-order ones depend on the choice of origin. Quadnipole moments are difficult to measure and experimental data are scarce [17, 18 and 19]. The octopole and hexadecapole moments have been measured only for a few highly syimnetric molecules whose lower multipole moments vanish. Ab initio calculations are probably the most reliable way to obtain quadnipole and higher multipole moments [20, 21 and 22]. 

Note the r dependence of these tenns the charge-indiiced-dipole interaction varies as r, the dipole-indiiced-dipole as and the quadnipole-mduced-dipole as In general, the interaction between a pennanent 2 -pole moment and an induced I -pole moment varies as + L + l) gQ enough r, only the leading tenn is important, with higher tenns increasing in importance as r decreases. The induction forces are clearly nonadditive because a third molecule will induce another set of miiltipole moments in tlie first two, and these will then interact. Induction forces are almost never dominant since dispersion is usually more important. 

Due to the nomialization integral, equation (A33.93). f(x) caimot be non-zero for arbitrarily large v f(x) must vanish for v greater than some cut-off value v which must be a pole of the integrand in equation (A3.3.98)... 

The evaluation of the integral in equation (A3.11.35) needs to be done carefiilly as there is a pole at /d = k. A standard trick to do it involves replacing k by /r+ie where e is a small positive constant that will be set to zero in the end. This reduces equation (A3.11.35) to... 

In this equation, the gradient term U(qx)Wtta (Rx)U(qx) Vr,z (Rx) = W > (R x) Vr x (Rx) still appears and, as mentioned before, introduces numerical inefficiencies in its solution. Even though a truncated Bom-Huang expansion was used to obtain Eq. (53), wJja (Rx), although no longer zero, has no poles at conical intersection geometries [as opposed to the full W (Rx) matrix]. 

The —(/i /2p)W (Rx) matrix does not have poles at conical intersection geometries [as opposed to W (R )] and furthermore it only appears as an additive term to the diabatic energy matrix (q ) and does not increase the computational effort for the solution of Eq. (55). Since the neglected gradient term is expected to be small, it can be reintroduced as a first-order perturbation afterward, if desired. 

Hie poling function typically adopts the following general functional form ... 

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