Distance coefficients

Far distance - coefficients much smaller with stable conditions. 

When molecules are represented by high-dimensional descriptors such as 2D fingerprints or several hundred topological indices, then the diversity of a library of compounds is usually calculated using a function based on the pairwise (dis)similarities of the molecules. Pairwise similarity can be quantified using a similarity or distance coefficient. The Tanimoto coefficient is most often used with binary fingerprints and is given by the formula below ... 

Distance coefficient (e.g., linked to fish and benthos field surveys) that reaches a maximum value of 1 for two sites that are entirely different and a minimum value of 0 for two sites that possess identical descriptors. It measures the amount of association between sites. Volume 2(4). 

A variety of different compound selection methods have been developed. These techniques are dependent on the use of molecular descriptors which are numerical values that characterize the properties of molecules. In addition, many compound selection methods are based on quantifying the degree of similarity or dissimilarity of compounds based on molecular descriptors. This requires the use of similarity or distance coefficients. 

The similarity, or dissimilarity, of a pair of compounds is quantified using a similarity or distance coefficient that is applied to the descriptor representation of the molecules. As mentioned, the Tanimoto coefficient is commonly used when the molecules are represented by binary bitstrings. The Tanimoto similarity between compounds A and B, SAB is ... 

Distance-based metrics quantify the diversity of a set of compounds as a function of their pairwise (dis)similarities in a descriptor space. It is important to mention that distance coefficients are analogous to distances in multidimensional geometric space, although they are usually not equivalent to such distances. For a distance coefficient to be described as a metric, it must possess the following four properties (1) Distance values must be nonzero and the distance from an object to itself must be zero. (2) Distance values must be symmetric. (3) Distance values must obey the triangular inequality. (4) Distances between nonidentical objects must be greater than zero. A coefficient containing only the first three properties is dubbed a pseudometric, and one without the third property is a nonmetric. 

There are many different types of similarity indexes, including the association coefficients (e.g., Tanimoto coefficient [27], Jaccard coefficient [38], Hodgkin-Richards coefficient [39,40]), the correlation coefficients or cosinelike indexes, and the distance coefficients or dissimilarity indexes (e.g., Hamming distance) [26],... 

In Fig. 8a the spin-coupled and final spin-coupled VB potentials are compared for the ground state. Fig. 8b shows the associated spin-coupling coefficients. It can be seen that the spin-coupled calculation by itself yields a potential energy curve with all the essential features of the final result. The spin-coupling coefficients display much of the essential chemistry of bond formation in BH. At large distances, coefficients are C2 I and 0, which shows that the two 2s-like orbitals stemming from B are coupled to a... 

Table 1.2. Distance Coefficients for Electron and Energy Transfer in Some Simple Systems... 

Distance is complementary to similarity. A few lines have been discussed on distance coefficients in the previous sections. The complemerrtary relationship between the similarity and distance coefficients allows the ealcirlation of one from the value provided for the other by subtracting it form one, that is. 

Distance coefficients are also called as distance matrices when they obey the criteria discussed previously. Hamming distance and Soergel Distance are examples of metric distance eoeffieients. 

In order to determine the matrix thresholds, we present an expression of the coefficients dispersion that is related to the flattening of the cloud of the points around the central axis of inertia. The aim is to measure the distance to the G barycentre in block 3. So, we define this measure Square of Mean Distance to the center of Gravity as follow ... 

This definition is in terms of a pool of liquid of depth h, where z is distance normal to the surface and ti and k are the liquid viscosity and thermal diffusivity, respectively [58]. (Thermal diffusivity is defined as the coefficient of thermal conductivity divided by density and by heat capacity per unit mass.) The critical Ma value for a system to show Marangoni instability is around 50-100. 

It is known that even condensed films must have surface diffusional mobility Rideal and Tadayon [64] found that stearic acid films transferred from one surface to another by a process that seemed to involve surface diffusion to the occasional points of contact between the solids. Such transfer, of course, is observed in actual friction experiments in that an uncoated rider quickly acquires a layer of boundary lubricant from the surface over which it is passed [46]. However, there is little quantitative information available about actual surface diffusion coefficients. One value that may be relevant is that of Ross and Good [65] for butane on Spheron 6, which, for a monolayer, was about 5 x 10 cm /sec. If the average junction is about 10 cm in size, this would also be about the average distance that a film molecule would have to migrate, and the time required would be about 10 sec. This rate of Junctions passing each other corresponds to a sliding speed of 100 cm/sec so that the usual speeds of 0.01 cm/sec should not be too fast for pressurized film formation. See Ref. 62 for a study of another mechanism for surface mobility, that of evaporative hopping. 

The electrostatic potential generated by a molecule A at a distant point B can be expanded m inverse powers of the distance r between B and the centre of mass (CM) of A. This series is called the multipole expansion because the coefficients can be expressed in temis of the multipole moments of the molecule. With this expansion in hand, it is... 

In principle, simulation teclmiques can be used, and Monte Carlo simulations of the primitive model of electrolyte solutions have appeared since the 1960s. Results for the osmotic coefficients are given for comparison in table A2.4.4 together with results from the MSA, PY and HNC approaches. The primitive model is clearly deficient for values of r. close to the closest distance of approach of the ions. Many years ago, Gurney [H] noted that when two ions are close enough together for their solvation sheaths to overlap, some solvent molecules become freed from ionic attraction and are effectively returned to the bulk [12]. 

R), i.e. there is no effect due to caging of the encounter complex in the common solvation shell. There exist numerous modifications and extensions of this basic theory that not only involve different initial and boundary conditions, but also the inclusion of microscopic structural aspects [31]. Among these are hydrodynamic repulsion at short distances that may be modelled, for example, by a distance-dependent diffiision coefficient... 

Inter-atomic two-centre matrix elements (cp the hopping of electrons from one site to another. They can be described [7] as linear combmations of so-called Slater-Koster elements [9], The coefficients depend only on the orientation of the atoms / and m. in the crystal. For elementary metals described with s, p, and d basis fiinctions there are ten independent Slater-Koster elements. In the traditional fonnulation, the orientation is neglected and the two-centre elements depend only on the distance between the atoms [6]. (In several models [6,... 

---