Distance dimension

We underline these results and the implied concepts quoting from a comprehensive review on this subject (Simon 1983). We remember indeed that, ever since it was experimentally possible to determine atomic distances in molecules and crystals, efforts have been made to draw conclusions about the nature of the chemical bonding, and to compare interatomic distances (dimensions) in the compounds with those in the chemical elements. Distances between atoms in an element can be measured with high precision. As such, however, they cannot be simply used in predicting interatomic distances in the compounds. In a rational procedure, reference values (atomic radii) have to be extracted from the individual (interatomic distances) measured values. Various functions have been suggested for this purpose. In the specific case of the metals it has been pointed out that interatomic distances depend primarily on the number of ligands and on the number of valence electrons of the atoms (Pearson 1972). 

To facilitate calculations, eqns 1.31-1.33 were converted from time to distance dimensions. It was established that the total variance of the peak includes the variance due to diffusion and due to electromigration diffusion ... 

The operation of proximity sensors can be based on a wide range of principles, including capacitance, induction, Hall and magnetic effects variable reluctance, linear variable differential transformer (LVDT), variable resistor mechanical and electromechanical limit switches optical, photoelectric, or fiber-optic sensors laser-based distance, dimension, or thickness sensors air gap sensors ultrasonic and displacement transducers. Their detection ranges vary from micrometers to meters, and their applications include the measurement of position, displacement, proximity, or operational limits in controlling moving components of valves and dampers. Either linear or angular position can be measured ... 

Whereas the complete RDE vector describes a probability distribution, the individual vector components are related to the relative frequencies of atom distances in the molecule. Thus, the individual g(r) values are plotted in a frequency dimension whereas r lies in the distance dimension (Eigure 5.1). The smoothing parameter B can be interpreted as a temperatnre factor that is, is the root mean square... 

Consequently, the resolution of the distance dimension affects the accuracy of the probability dimension leading to the following effects ... 

The weight and normalization functions are available for one- and two-dimensional (2D) descriptors. However, multidimensional calculations are performed technically in one dimension that is, each descriptor contains multiple one-dimensional vectors, such as [Xq,Xi,..., x , yoJu- ] Consequently, distance-related functions like transforms are performed only in the distance dimension, whereas general functions like weighting and normalization are calculated for an entire descriptor for example, normalization takes place on the entire vector instead of on the individual vectors of the first dimension. 

This two-dimensional RDF descriptor is calculated depending on the distance r and an additional property p. In this case, p is a property difference calculated in the same fashion as the Cartesian distance r, in fact, p can be regarded as a property distance. Mnch in the same way as B influences the resolution of the distance dimension, the property-smoothing parameter D affects the resolution — and, thus, the half-peak width — in the property dimension. D is measured in inverse squared units [p l of the property p. ... 

The probability-weight properties p and pj are, and should be, primarily independent of the property p that dehnes the second dimension of the descriptor. In other words, the distance dimension g(r) is separated from the property dimension g(p ), and, additionally, the probability is weighted by the property p. Figure 5.17 shows an example of a 2D RDF descriptor calcnlated for a simple molecule. 

FIGURE 5.17 Two-dimensional RDF descriptor of ethene calculated with Cartesian distances in the first and the partial atomic charge as property for the second dimension. Instead of the one-dimensional descriptor with four peaks, the six distances occurring in ethene are clearly divided into the separate property and distance dimensions. 

Figure 5.18 shows a 2D RDF descriptor of a complex molecule. A direct interpretation wonld be more sophisticated however, the valne of separating distance and property becomes more obvious. 2D RDF descriptors of the distance dimension n and the property dimension m can be treated like a one-dimensional vector containing m descriptors of length n. This makes it easy to compare two-dimensional descriptors using the same algorithms as for one-dimensional vectors. 

The dimension of W(v) is (distance)T, e.g. ym2. For comparison with particle parameters, the square root of the Wiener spectrum values where used. This gives a function of linear distance dimension and is analogous to use of an RMS value such as granularity. Figure 7 shows examples of these experimental Wiener spectra, plotted as the square root of W(v) vs. log (v), for toners fractions 18 and parent toner. Note that the bimodal parent toner yields a spectrum with an inflection point, another supportive observation for the correlation of image parameters and toner PSD. 

The distributed parameter component can be introduced into the larger system being simulated in one of two ways either it can be introduced as an integral part through finite differencing in the distance dimension (or dimensions), or else it can be kept as a separate computational entity that communicates with the main simulation only at specified communication intervals. 

Although using a Lennard-Jones soft core has been proven to not be the optimal path, it has been shown to be relatively close to the global optimum in the space of k pathways [61]. Some additional optimization may still be possible Rodinger et al. proposed a pathway based on adding a fourth distance dimension [67] as the particle is taken to infinity in this dimension, it is alchemically decoupled from the environment. This is similar to previous alchemical approaches where the extra dimension is treated as a dynamical variable [68,69], but in this case it is treated solely as an alchemical parameter. By transformation of the interval e [0,oo) to k 6 [0,1], this can easily be shown to be a type of soft core, but with a different functional dependence on the alchemical parameter k than previously proposed methods. The Beutler et al. soft core functional form is not particularly efficient for simultaneous removal of Coulombic and Lennard-Jones potentials the 4D pathway appears to be more efficient, as do other proposed pathways [70], and comparisons between the methods might still lead to improvements in efficiency. 

Table 7.2 provides examples of the factors to consider in each distance dimension. It serves as a checklist for identifying distance factors for proposed supply chain initiatives. 

Size No. of HtKil No. of Root Minimum Pre- Distance Distance Dimension Radius... 

---