Quantization scale

If the configuration in question is a viable physical approximation (i.e. i), then the support of the wavelet transform must be consistent with the quantization scale. That is a, 6 (og, oo), or uq < a -... 

On the other hand, for a spurious solution, high frequency noise (i.e. approximation errors in the MRF representation) conspire to make it satisfy the TPQ conditions. These solutions (i.e. ) must have the support of their wavelet transform lie outside of the quantization scale range. That is, Ow (flQ) oo), or Ow < aq. 

As noted previously, a similcir relation was used in the recent work of HMBB, which developed a TPQ analysis within the MRF representation, the physical solutions were identified by the extent to which the condition physQj- (aQ,cxi) was satisfied, where o[Pg.226]

Within a pure scalet representation, the quantization scale is defined differently to that indicated above. This is discussed in Sec. I.4.4.2. 

As clarified shortly, the quantization scale, aQ(r), within the scalet representation, corresponds to the scale where the scalet representation for the kinetic energy operator begins to converge to zero, at the respective turning point location, r, as required. 

The expression 9aS (o, 6) will also be zero at a = oo. FVom the asymptotic expansion in Eq.(1.111), we see that it will also be zero at the turning points, in the zero scale limit d S (a = 0,t/( )) = 0. Accordingly, 9aS a,Ti E)) will also have extremal values, in the scale variable. In particular, at either ends of the interval 0,ae-ff Ti E)), this expression will be zero, with no other zeroes lying within the interval. Its one extremum point, in this interval, is defined to be the quantization scale, oq, satisfying... 

The definition of the quantization scale may be conservative (i.e. smaller than the scale at which d S (a,r) starts its convergence to zero), however, it is the more accessible and natural definition from the scaling transform perspective. 

In addition, we develop a slightly more comprehensive analysis of the scaling transform and its derivatives, including the significance of the adopted definition for the quantization scale, as presented within our analysis of the quartic anharmonic oscillator. 

Defining the Quantization Scale for Arbitrary Scaling Functions... 

The quantization scale, aQ(r), within the scalet formalism, is the scale oi, for the scalet solutions that converge to solutions of the Schrodinger equation although we are particularly interested for the case corresponding to physical solutions. 

If the scalet solutions only converged to the Schrodinger equation, for the physical energies, then one could implement a TPQ analysis by simply imposing (approximate) zero kinetic energy conditions starting at the respective quantization scales, a (r(), for each of the turning points. 

It is for this case, as argued earlier (for the quartic anharmonic oscillator), that we can take the quantization scale, ag(T), to be defined as the smallest, nontrivial scale-root satisfying... 

As argued previously, we then have ag(T<) < Ou,(rf). This condition was used within the TPQ-MRF representation to discriminate between the physical and spurious solutions generated (HMBB (2000)). The effective quantization scale in that case corresponded to the smallest scale up to which the MRF generated solutions satisfied the scalet equation. 

The quantization scale, aQ Ti E)), is the smallest (nontrivial) root (in the scale variable) for the third order derivative of the scaling transform 5fS (aQ,r) = 0. It marks the point where the kinetic energy related expression, dlS a,r), begins to converge to zero. 

Unless one can estimate either the physical extremal scales, ae Tt E)), or the quantization scales, accurately, it becomes very difficult... 

For example, the measured pressure exerted by an enclosed gas can be thought of as a time-averaged manifestation of the individual molecules random motions. When one considers an individual molecule, however, statistical thermodynamics would propose its random motion or pressure could be quite different from that measured by even the most sensitive gauge which acts to average a distribution of individual molecule pressures. The particulate nature of matter is fundamental to statistical thermodynamics as opposed to classical thermodynamics, which assumes matter is continuous. Further, these elementary particles and their complex substmctures exhibit wave properties even though intra- and interparticle energy transfers are quantized, ie, not continuous. Statistical thermodynamics holds that the impression of continuity of properties, and even the soHdity of matter is an effect of scale. 

The above measurements all rely on force and displacement data to evaluate adhesion and mechanical properties. As mentioned in the introduction, a very useful piece of information to have about a nanoscale contact would be its area (or radius). Since the scale of the contacts is below the optical limit, the techniques available are somewhat limited. Electrical resistance has been used in early contact studies on clean metal surfaces [62], but is limited to conducting interfaces. Recently, Enachescu et al. [63] used conductance measurements to examine adhesion in an ideally hard contact (diamond vs. tungsten carbide). In the limit of contact size below the electronic mean free path, but above that of quantized conductance, the contact area scales linearly with contact conductance. They used these measurements to demonstrate that friction was proportional to contact area, and the area vs. load data were best-fit to a DMT model. 

Hardness indentations are a result of plastic, rather than elastic, deformation, so some discussion of the mechanisms by which this occurs is in order, especially since the traditional literature of the subject is confused about its fundamental nature. This confusion seems to have arisen because it was considered to be a continuous process for a great many years, and because some metals behave plastically on the macroscopic scale in a nearly time-independent fashion. During the twentieth century, it became well established that plastic deformation is fundamentally discontinuous (quantized), and a time-dependent flow process. 

A chemical reaction can be viewed as occurring via the formation of an excited state that can be any one of the degrees of freedom of the collection of N atoms. That is, translational, rotational, vibrational, and electronic excitation can lead to a chemical reaction. We often do not need to consider explicitly the quantized nature of rotational and vibrational energies in practical applications because of time scale considerations. For example, when a chemical reaction proceeds via a vibrationally excited state, in which the average lifetime typically is about 3 x 10" where T is in Kelvins... 

Figure 3. Watermark embedding using Costa s seheme with a scalar component codebook (SCS). The watermark letter e is embedded after dithered uniform scalar quantization of x and the addition of the scaled quantization error as watermark Wr,.

Classical mechanics does not apply to the atomic scale and does not take the quantized nature of molecular vibration energies into account. Thus, in contrast to ordinary mechanics where vibrators can assume any potential energy, quantum mechanical vibrators can only take on certain discrete energies. Transitions in vibrational energy levels can be brought about by radiation absorption, provided the energy of the radiation exactly matches the difference in energy levels between the vibrational quantum states and provided also that the vibration causes a fluctuation in dipole. 

---