Distance function

These workers used binary solvent systems over a range of mole fractions to determine, for each solute, the constants a and b of equation (8.2). For methyl and phenacyl esters, TLC was used, while overpressured layer chromatography (OPLC) was used for dansyl amino acids. Nurok and co-workers (11) also evaluated how the quality of a simulated separation varies with changing solvent strength by using the inverse distance function (IDF) or planar response function (PRF), as follows ... 

Defining a distance function d i,j) = miiipat/is of links (i,j path) and a d -size neighborhood of f as Afd i) = /c such that d i, k) < d, we wish to study the discrete time-evolution of S(f) in which the site variables undergo transitions of the general form... 

The Use of the Interatomic Distance Function in Discussing the Electronic Structure of Molecules.—The observed interatomic distances listed in Table II are interpreted with the aid of the curve of Fig. 1 to lead to the assignment of fractional double bond character as given in the last column. It is seen that a single bond between conjugated double bonds (cyclopentadiene) or... 

Interatomic Distances in Polynuclear Aromatic Hydrocarbons.—As a further example of the use of the interatomic distance function we shall discuss the polynuclear aromatic hydrocarbons. 

Considering the success of fragment methods, which apply additive models for log P predichon, one can assume that addihve approaches may also sahsfactory work for MLP. Indeed, similar to the Generalized Born model, one can consider fragments of molecules as centers of some potenhal functions and use an empirically defined distance function ) to calculate the MLP value by ... 

The CLIP approach was developed by the group of Testa [55]. It is based on the atomic lipophilic system of Broto and Moreau [56] and uses a modified exponential distance function of which differs from the e function of Fauchere et al. [45]. In addition the authors restricted the distance function at 4A to avoid influence of too distant elements. The most recent version of CLIP uses a Fermi-like distance function, which does not need any cutoff values [57]. Another implementation of the approach is available in the VEGA software provided by Pedretti et al. [58]. 

We consider an nxn table D of distances between the n row-items of an nxp data table X. Distances can be derived from the data by means of various functions, depending upon the nature of the data and the objective of the analysis. Each of these functions defines a particular metric (or yardstick), and the graphical result of a multivariate analysis may largely depend on the particular choice of distance function. 

It can be regarded as a special case of the squared weighted Euclidean distance (Section 30.2.2.1), A property of (weighted) Euclidean distance functions is that the distances between row-items D are invariant under column-centering of the table X ... 

The Minkowski distance defines a class of distance functions which are characterized by the parameter r (Section 30.2.3.2) ... 

Given the original nxp measurement table X, one derives the table of nxn distances D between the n row-items, using a particular distance function tp ... 

Since the specification of topologies implies that all open sets are defined, the concept of continuity can also be generalized to topological spaces, even if distance functions are not given. 

Fig. 1. The level sets of distance function for a smooth interface over a Cartesian grid.

Even if the initial value of the level-set function (x,0) is set to be the distance function, the level set function r/j may not remain as a distance function at t >() when the advection equation, Eq. (3), is solved for . Thus, a redistance scheme is needed to enforce the condition of V0 = 1. An iterative procedure was designed (Sussman et al., 1998) to reinitialize the level-set function at each time step so that the level-set function remains as a distance function while maintaining the zero level set of the level-set function. This is achieved by solving for the steady-state solution of the equation (Sussman et al., 1994, 1998 Sussman and Fatemi, 1999) ... 

More recently, the concept of the molecular lipophilicity potential (MLP), initially introduced by Audry and coworkers (2), has attracted increasing attention. The method involves mapping the local lipophilicity at points in 3D space around a chemical compound through the use of a parametrized fragmental system coupled to an empirical distance function. Despite the absence of any physical basis for the distance-dependent functions introduced, and the difficulties inher-... 

The function fct d) has no physical basis. According to Heiden (6), it should fulfill only two conditions it should be smooth and continuous and have finite values for d < dcut oS. The value of dclH 0[f should be larger than the van der Waals radius of any atom in the molecule under consideration. In the program LipoDyn, the Ghose and Crippen parameters (18) as well as the Broto and Moreau parameter set (16) are implemented, with the the exponential function used by Fauchere (cxp(-d)) (3), the hyperbolic function defined by Audry 1/(1 + d) (2) and the parameterized Fermi distance function used by Testa and coworkers (1). 

In order to back-calculate log Poct for each molecule, the guidelines proposed by Testa et al. (1) have been followed. Two parameters, 2 MLP and 2 MLP, the total of positive and negative MLP values, respectively, were calculated using various combinations of atomic parameters and distance functions, for the set of 91 rigid structures. The best log Pocl re-predictions (Fig. 5) are obtained using the Fermi distance function and the Broto and Moreau parameters ... 

Fig. 5. Estimation of log Poct using MLP calculation with the atomic fragmental system of Ghose and Crippen and the Fauchere distance function on the 91 selected WDI rigid molecules.

The distance z has been derived for any trajectory with a preselected entrance angle 0, i.e., a principal trajectory as defined in connection with Fig. 4.4 is not yet specified. In order to fix such a principal trajectory, which goes from the source to the image, one has to find the focusing properties of the analyser. In other words, one has to investigate how the distance function z( ) behaves for small changes in. For this purpose one starts with a Taylor expansion of z in A9 around a still arbitrarily selected 0 value ... 

Once the structure is partitioned into three-dimensional hyperplanes of simultaneity further features can be added. One such feature is a distance function d(p,q) defined on any two points within the same hyper-plane. This distance function provides the hyper-plane with a metric. And with such a metric, the space will exhibit what contempo-... 

A second form of motion that extension so characterized can underlie is increase and decrease. Suppose that there is a cubical composite, A, of form Fx and region rt at time tr And suppose that there are three lines—x, y, and z—of A by which one measures the volume of A. Hie volume of A would thus be equal to the product of the lengths of x, y and z. The lengths of x, y, and z, though, would be determined by the distance function applied to the endpoints of those lines—call them VW Za,zb. In other words, the volume of A would be d(xj,xb)... 

Other measures of properties in 3D, such as Molecular Lipophilicity Potential (MLPot) and Molecular Hydrogen Bond Potential (MHBP), have been used to characterize 3D properties. They are defined for points on a molecular surface created around the molecule and calculated from the summation of contributions from the substructural fragments making up the molecule weighted by the distance function. The hydrogen bond potentials include an angle-dependent function. 

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