Scaling axis

Manzano C, Soe WH, Wong HS, Ample F, Gourdon A, Chandrasekhar N, Joachim C (2009) Step-by-step rotation of a molecule-gear mounted on an atomic-scale axis. Nat Mater 8 576... 

The left plot in Figure 2 shows the experimentally derived reactivity course with untreated, respectively, HCl washed charcoal as a fimction of the conversion degree the right plot those of untreated, respectively, Na COs impregnated charcoal on a 14 fold larger vertical scale axis. 

Fig, 12.1. (Taken from [17]) Reproducing kernel of the Morlet wavelet for three different scales (a) s=8, (b) s=32 and (c) s=128. The width in time and in scale direction increases linearly with scale (i.e. in scale direction it appears constant on a logarithmic scale axis)... 

Fig. 12,3. (Taken from [17]) Smoothing according to the reproducing kernel to provide a constant variance for all scales, (a) In scale direction, the length of the smoothing window stays constant (for a logarithmic scale axis), (b) In time direction, the length of the smoothing window increases linearly with scale.

In Fig. 3 and Fig. 4, several peaks are concentrated in the initial values of the scale axis and over the same values in the time axis. The heights of these peaks present small differences. Although the time-frequency domain is presented instead of the time-domain, noise is a predominantly feature ofthese graphs. In addition, there is no visually detectable pattern to support the process of diagnosis. 

Some of the methods used in TED are short time Fourier transform (STFT), wavelet transform (WT) and wavelet packet transform (WPT). In general, the difference between these methods is the partitioning of the time-scale axis. In short-time Fourier transform (STFT), the EMG signal is mapped into frequency components that present within an interval of time (window). A suitable window size must be determined prior to this as small window will give good time resolution but poor frequency resolution and vice versa. The partitioning ratio of the STFT is fixed once specified, each cell has an identical aspect ratio. To overcome the resolution problem in STFT, WT was developed. 

Havel Ivan Milos (1938-) Czech kybemet., expert in cognitive sciences, inventor of spatial-scale axis... 

These curves are drawn for all properties having for the ordinate axis an appropriate scale of the property, and for the abscissa the yield in volume or weight. 

Figure 11 Single detector signal provided by the inspection of inner circonferential notch tube sample, and corresponding scalogram. Time axis (in s) and frequency axis (in Hz) have been scaled according to the speed of evolution of the detector in ihe tube (500 mm/s).

Although MAS is very widely applied to non-integer spin quadnipolar nuclei to probe atomic-scale structure in solids, such as distinguishing AlO and AlOg enviromnents [21], simple MAS about a single axis caimot produce a completely averaged isotropic spectrum. As the second-order quadnipole interaction contains both... 

Replotting the data on a lorgarithmic time scale as shown in Fig. 4.8b has an interesting effect. Figure 4.8b shows that this modification produces a far more uniform set of S curves. As a matter of fact, if the various curves are shifted along the horizontal axis, they may be superimposed over a wide portion of the transformation. The arrows in Fig. 4.8b show this displacement of the data at 126 and 130°C to correspond to the data at 128°C. This superpositioning is examined in the example below. 

Plot a family of curves, each of different n, with composition as the y axis and O2 absorbed as the x axis. Evaluate by Eq. (5.30) for n = 1, 2, 3, and 4 and 0.1 < p < 0.9 in increments of 0.1. Plot these results on y axis) on a separate graph drawn to the same scale as the experimental results. Compare your calculated curves with the experimental curves with respect to each of the following points (1) coordinates used, (2) general shape of curves, and (3) labeling of curves. 

Comparison of Alignment Charts and Cartesian Graphs. There are typically fewer lines on an alignment chart as compared to Cartesian plots. This reduces error introduced by interpolation and inconsistency between scales. For example, to find a point (x,j) on a Cartesian graph one draws two lines, one perpendicular to each axis, and these reference lines intersect at the point x,j). This point (x,j) may correspond to some finite value found by rea ding a contour map represented by a family of curves corresponding to different values of the function. 

Construction of Alignment Charts. Of the ways to constmct alignment charts, the bmte force method, which requires some idea of the geometry for the chart, is the easiest method to use. The mathematical method, which uses parametric equations of scale to determine the placement and scale of each axis, is the most accurate, but the most difficult to apply. 

Each of the parametric equations that can be formed from an expression represents an axis on the chart. Each set of parametric equations must simultaneously agree with the equation they represent. In other words, on a line drawn through any two variables, a third variable can be found which satisfies the parametric equations. So an evaluation of the chart requites that values produced by each parametric equation be on the chart as a line. A determinant can be used to determine whether or not points are collinear. The parametric equations must be evaluated so they always produce values which he on a line. By replacing the x andjy points with parametric equations of scale for the chart, it is possible to create any diagram. This method can be used to determine the placement of the axes, because the parametric equations can be transformed into equations of scale. 

If necessary, the fit can be improved by increasing the order of the polynomial part of Eq. (9-89), so that this approach provides a veiy flexible method of simulation of a cumulative-frequency distribution. The method can even be extended to J-shaped cui ves, which are characterized by a maximum frequency at x = 0 and decreasing frequency for increasing values of x, by considering the reflexion of the cui ve in the y axis to exist. The resulting single maximum cui ve can then be sampled correctly by Monte Carlo methods if the vertical scale is halved and only absolute values of x are considered. 

Ternary-phase equilibrium data can be tabulated as in Table 15-1 and then worked into an electronic spreadsheet as in Table 15-2 to be presented as a right-triangular diagram as shown in Fig. 15-7. The weight-fraction solute is on the horizontal axis and the weight-fraciion extraciion-solvent is on the veriical axis. The tie-lines connect the points that are in equilibrium. For low-solute concentrations the horizontal scale can be expanded. The water-acetic acid-methylisobutylketone ternary is a Type I system where only one of the binary pairs, water-MIBK, is immiscible. In a Type II system two of the binary pairs are immiscible, i.e. the solute is not totally miscible in one of the liquids. 

---