Harmonization defined

In the MSE approach, the size and field homogeneity of the FOV are proportional to number of zeroed inner spherical harmonics, and number of vanished outer harmonics defines the size of the system footprint. We recall that the number of inner and outer spherical harmonics made to vanish refers to the order and degree of the design. Once the order and degree are specified based on the requirements of the FOV and the stray field, and the magnet domain has been established, then the MSE current density map is calculated. 

In this equation the symbols X (a,b) denote the so-called nonnormalized spherical harmonics defined as... 

Equations (4.329) for a solid assembly and (4.332) for a magnetic suspension are solved by expanding W with respect to the appropriate sets of functions. Convenient as such are the spherical harmonics defined by Eq. (4.318). In this context, the internal spherical harmonics used for solving Eq. (4.329) are written Xf (e, n). In the case of a magnetic fluid on this basis, a set of external harmonics is added, which are built on the angles of e with h as the polar axis. Application of a field couples [see the kinetic equation (4.332)] the internal and external degrees of freedom of the particle so that the dynamic variables become inseparable. With regard to this fact, the solution of equation (4.332) is constructed in the functional space that is a direct product of the internal and external harmonics ... 

Here, the following assumptions are made the radial function Ri r) is the same for all basis functions of the same ul quantum number , and its dependence on a shell quantum number n is of no consequence. The coefficients a/ describe the contribution of s, p, d,. .. character to the hybrid, and the bim govern the shape and orientation of that contribution. Sim are the real surface harmonics, defined in terms of the spherical harmonics (Y m). 

N = number of ions/unit volume, also demagnetization factor 07 (-f) = operator equivalent of spherical harmonic, defined by Elliott and Stevens (1953)... 

Phase angle of the nth harmonic defined by Equation 1.6 Sphericity... 

Let us consider the evaluation of the scaled real solid harmonics defined in (9.13.43) and (9.13.44). We first note that the real regular solid harmonics are related to the standard functions of Section 6.4 as... 

The inverse problem would be well defined if we knew the temperature or the harmonic function... 

A different approach comes from the idea, first suggested by Flelgaker et al. [77], of approximating the PES at each point by a harmonic model. Integration within an area where this model is appropriate, termed the trust radius, is then trivial. Normal coordinates, Q, are defined by diagonalization of the mass-weighted Flessian (second-derivative) matrix, so if... 

Another option is a q,p) = p and b q,p) = VU q). This guarantees that we are discretizing a pure index-2 DAE for which A is well-defined. But for this choice we observed severe difficulties with Newton s method, where a step-size smaller even than what is required by explicit methods is needed to obtain convergence. In fact, it can be shown that when the linear harmonic oscillator is cast into such a projected DAE, the linearized problem can easily become unstable for k > . Another way is to check the conditions of the Newton-Kantorovich Theorem, which guarantees convergence of the Newton method. These conditions are also found to be satisfied only for a very small step size k, if is small. 

A mapping is said to be symplectic or canonical if it preserves the differential form dp A dq which defines the symplectic structure in the phase space. Differential forms provide a geometric interpretation of symplectic-ness in terms of conservation of areas which follows from Liouville s theorem [14]. In one-degree-of-freedom example symplecticness is the preservation of oriented area. An example is the harmonic oscillator where the t-flow is just a rigid rotation and the area is preserved. The area-preserving character of the solution operator holds only for Hamiltonian systems. In more then one-degree-of-freedom examples the preservation of area is symplecticness rather than preservation of volume [5]. 

Various other ways to incorporate the out-of-plane bending contribution are possible. For e3plane bend involves a cakulation of the angle between a bond from the central atom and the plane defined by I he central atom and the other two atoms (Figure 4.10). A value of 0° corresponds to all four atoms being coplanar. A third approach is to calculate the height of the central atom above a plane defined by the other three atoms (Figure 4.10). With these two definitions the deviation of the out-of-plane coordinate (be it an angle or a distance) can be modelled Lt ing a harmonic potential of the form... 

Most of the molecules we shall be interested in are polyatomic. In polyatomic molecules, each atom is held in place by one or more chemical bonds. Each chemical bond may be modeled as a harmonic oscillator in a space defined by its potential energy as a function of the degree of stretching or compression of the bond along its axis (Fig. 4-3). The potential energy function V = kx j2 from Eq. (4-8), or W = ki/2) ri — riof in temis of internal coordinates, is a parabola open upward in the V vs. r plane, where r replaces x as the extension of the rth chemical bond. The force constant ki and the equilibrium bond distance riQ, unique to each chemical bond, are typical force field parameters. Because there are many bonds, the potential energy-bond axis space is a many-dimensional space. 

A molecule may show both electrical and mechanical anharmonicity, but the latter is generally much more important and it is usual to define a harmonic oscillator as one which is harmonic in the mechanical sense. It is possible, therefore, that a harmonic oscillator may show electrical anharmonicity. 

The vibrational temperature, defined for a diatomic harmonic oscillator by the temperature in Equation (5.22), is considerably higher because of the low efficiency of vibrational cooling. A vibrational temperature of about 100 K is typical although, in a polyatomic molecule, it depends very much on the nature of the vibration. 

Thus the average cost per share for John is the arithmetic mean of pi, po,. . . , pn, whereas that for Mary is the harmonic mean of these n numbers. Since the harmonic mean is less than or equal to the arithmetic mean for any set of positive numbers and the two means are equal only i pi=po = =pn, we conclude that the average cost per share for Mary is less than that for John if two of the prices Pi are distinct. One can also give a proof based on the Gaiichy-Schwarz inequality, To this end, define the vectors... 

This can be defined by the most severe external fault at which the schetne will remain inoperative. It should also remain inoperative in healthy conditions. That is it should be immune to the momentary voltage or current transients and normal harmonic contents in the circulating current. Series LC-filter circuits are generally provided with the relay coil to suppress the harmonics and to detect the fault current more precisely. 

The second component is caused by the different harmonic quantities present in the system when the supply voltage is non-linear or the load is nonlinear or both. This adds to the fundamental current, /,- and raises it to Since the active power component remains the same, it reduces the p.f of the system and raises the line losses. The factor /f/Zh is termed the distortion factor. In other words, it defines the purity of the sinusoidal wave shape. 

It should be ensured that under no condition of system disturbance w ould the filter circuit become capacitive when it approaches near resonance. To achieve this, the filter circuits may be tuned to a little less than the defined harmonic frequency. Doing so will make the L and hence Xl always higher than Xc, since... 

A term used in PFC is total harmonic distortion. This is defined as... 

We now want to study the consequences of such a model with respect to the optical properties of a composite medium. For such a purpose, we will consider the phenomenological Lorentz-Drude model, based on the classical dispersion theory, in order to describe qualitatively the various components [20]. Therefore, a Drude term defined by the plasma frequency and scattering rate, will describe the optical response of the bulk metal or will define the intrinsic metallic properties (i.e., Zm((a) in Eq.(6)) of the small particles, while a harmonic Lorentz oscillator, defined by the resonance frequency, the damping and the mode strength parameters, will describe the insulating host (i.e., /((0) in Eq.(6)). 

Fig. 7. Model calculations for the reflectivity (a) and the optical conductivity (b) for a simple (bulk) Drude metal and an effective medium of small metallic spherical particles in a dielectric host within the MG approach. The (bulk) Drude and the metallic particles are defined by the same parameters set the plasma frequency = 2 eV, the scattering rate hr = 0.2 eV. A filling factor/ = 0.5 and a dielectric host-medium represented by a Lorentz harmonic oscillator with mode strength fttOy, 1 = 10 eV, damping ftF] = I eV and resonance frequency h(H = 15 eV were considered for the calculations.

As mentioned in Section 2.2.3, the out-of-plane energy may also be described by an improper torsional angle. For the example shown in Figure 2.6, a torsional angle ABCD may be defined, even though there is no bond between C and D. The out-of-plane oop may then be described by an angle for example as a harmonic function... 

---