Degrees of freedom defined

Let us stress a very important difference between the representations of Stokes operators (137) and (157). If the former is valid only for the electric dipole photons, the latter describes an arbitrary multipole radiation with any X and j. The similarity in the operator structure and quantum phase properties is caused by the same number of degrees of freedom defining the representation of the SU(2) subalgebra in the Weyl-Heisenberg algebra. 

The behavior for 50 Hz to 100 Hz was qualitatively different from the behavior for 1 kHz to 10 MHz. Inclusion of the degrees of freedom defined in Chapter 4 improved the quality of the fit for tan 8 measured at 50 Hz to 100 Hz. The more numerous the degrees of freedom (especially of the side groups) were, the more a polymer was able to dissipate energy in... 

Let us consider a dynamical system with two degrees of freedom, defined by the set of two second order differential equations... 

In Eq. (d), u represents the degrees of freedom defined in Problem 9.14, while u represents the degrees of freedom defined in Problem 9.16. 

If the Hamiltonian were truly separable in the degrees of freedom defined by r and R, the solution of the Schrodinger equation would, of course, be a product of a function of r and a function of R. The separability one has in mind when discussing electronic systems is of a different origin, however. It relies on the different masses of the underlying particles and hence on their different velocities. The fast electrons can instantly adjust to the slow nuclei and, consequently, their wavefunction depends parametrically on the coordinates of the nuclei. The total wavefunction thus reads ... 

The extended system method was developed by Nose l and subsequently by Hoover who considered the thermal reservoir to be an integral part of the system. The inclusion of the reservoir requires an additional degree of freedom, defined as s, be added to the system. The potential energy for this additional degree of freedom is calculated as... 

While in the bulk of materials local electroneutrality has to be fulfilled, at interfaces space charge zones occur which provide additional degrees of freedom defined by the contact chemistry. So it is the individual preference of a given charge carrier for a certain phase or the boundary core itself that is to the fore. The tradeoff of this additional flexibility is the occurrence of an electrical field which confines the effects to the immediate vicinity of the interface. 

For conducting numerous penetrant diffusion simulations, it isn t reasonable to inspect Fig. 2 by eye to choose when to stop adding degrees of freedom an automated procedure is required instead. One approach [88] is to continue until all degrees of freedom defined by atoms within the penetrant neighbor list have been added. The remaining atoms are affected only indirectly the penetrant interacts with atom i, which then... 

Table 5.5 The application of the reduction formula to the nine basis vector representation of H2O atomic degrees of freedom defined in Figure 5.2.

Having explored the major degrees of freedom, the material and energy balance is now fixed, and hence the hot and cold streams which contribute to the heat exchanger network are firmly defined. The remaining task is to complete the design of the heat exchanger network. 

Once the initial network structure has been defined, then loops, utility paths, and stream splits offer the degrees of freedom for manipulating network cost in multivariable optimization. During the optimization, there is no constraint that temperature differences should be larger than or that there should not be heat transfer... 

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]. 

Spanned by tbc atoms 4, 2, and 1, and 2, 1, and 3 (tlic ry-planc), Except of the first three atoms, each atom is described by a set of three internal coordinates a distance from a previously defined atom, the bond angle formed by the atom with two previous atoms, and the torsion angle of the atom with three previous atoms. A total of 3/V - 6 internal coordinates, where N is the number of atoms in the molecule, is required to represent a chemical structure properly in 3D space. The number (,3N - 6) of internal coordinates also corresponds to the number of degrees of freedom of the molecule. 

Defining the sample s variance with a denominator of n, as in the case of the population s variance leads to a biased estimation of O. The denominators of the variance equations 4.8 and 4.12 are commonly called the degrees of freedom for the population and the sample, respectively. In the case of a population, the degrees of freedom is always equal to the total number of members, n, in the population. For the sample s variance, however, substituting X for p, removes a degree of freedom from the calculation. That is, if there are n members in the sample, the value of the member can always be deduced from the remaining - 1 members andX For example, if we have a sample with five members, and we know that four of the members are 1, 2, 3, and 4, and that the mean is 3, then the fifth member of the sample must be... 

---