Nonlinear Distance Scaling

Quality of distance measurements can be improved by nonlinear scaling (Gauss-ian-type or exponential functions) of the distances S(a,b). 

not only the similarity between two molecules is of interest, but also the similarity between one compound and a set of compounds (e. g., all pre-selected structures from a library). This problem can be addressed by summing all scaled S(a,b) values for all interesting molecule pairs [23]. 

---

In fluid dynamics the behavior in this system is described by the full set of hydrodynamic equations. This behavior can be characterized by the Reynolds number. Re, which is the ratio of characteristic flow scales to viscosity scales. We recall that the Reynolds number is a measure of the dominating terms in the Navier-Stokes equation and, if the Reynolds number is small, linear terms will dominate if it is large, nonlinear terms will dominate. In this system, the nonlinear term, (u V)u, serves to convert linear momentum into angular momentum. This phenomena is evidenced by the appearance of two counter-rotating vortices or eddies immediately behind the obstacle. Experiments and numerical integration of the Navier-Stokes equations predict the formation of these vortices at the length scale of the obstacle. Further, they predict that the distance between the vortex center and the obstacle is proportional to the Reynolds number. All these have been observed in our 2-dimensional flow system obstructed by a thermal plate at microscopic scales. ... 

The first theoretical attempts in the field of time-resolved X-ray diffraction were entirely empirical. More precise theoretical work appeared only in the late 1990s and is due to Wilson et al. [13-16]. However, this theoretical work still remained preliminary. A really satisfactory approach must be statistical. In fact, macroscopic transport coefficients like diffusion constant or chemical rate constant break down at ultrashort time scales. Even the notion of a molecule becomes ambiguous at which interatomic distance can the atoms A and B of a molecule A-B be considered to be free Another element of consideration is that the electric field of the laser pump is strong, and that its interaction with matter is nonlinear. What is needed is thus a statistical theory reminiscent of those from time-resolved optical spectroscopy. A theory of this sort was elaborated by Bratos and co-workers and was published over the last few years [17-19]. 

Fig. 4. Adjacent genes with conserved relative transcription orientation versus evolutionary distance. The plot shows the fraction of conserved gene pairs with the same relative direction of transcription between two species. Shown are pairs with a single direction of transcription, pairs with a divergent direction of transcription, and pairs with a convergent direction of transcription. The X axis represents the small subunit rRNA distance between the species (Fig. 3). The number of conserved adjacent pairs declines nonlinearly with phylogenetic distance and was therefore depicted on a logarithmic scale. Note that the vast majority of conserved adjacent pairs with a conserved relative transcription orientation are transcribed in the same direction. Divergently transcribed pairs are better conserved than convergently transcribed ones, hinting at the conservation of divergent promotors.

Table 1. Necessary nonlinearities for a nonlinear phase shift of 27rover a propagation distance of one centimeter for different laser sources. A refractive index of n = 1.5 is assumed for the nonlinear material. Note that the tabulated values are only estimates and correct only on the scale of one order of magnitude. But the consequences we conclude from these estimations are nevertheless correct and important...

Another piece of information that we wanted to extract from our experiments was connected with the dynamic behavior of spatial variables. If we consider three successive particles in the chain and we denote by the distance of the middle one from the center of mass of the other two and by the distance between these two, we can compute the normalized autocorrelation function of these two variables. They are shown in Fig. 9 as can be immediately observed, they decay to zero on a time scale which is much greater than that of the velocity variable. Also, the center of mass decays faster than R . In the next section we shall argue that this suggests that the virtual potential characterizing the itinerant oscillator model has to be assumed to be fluctuating around a mean shape, which, moreover, will be shown to be nonlinear and softer than its harmonic approximation. 

Diffusion is a nonlinear process in which the time t required for a species to diffuse scales quadratically with the distance x covered. A simple case of diffusion can be modeled in one dimension by the equation ... 

Like PCA, non-linear mapping (NLM), or multidimensional scaling, is a method for visualizing relationships between objects, which in the medicinal chemistry context often are compounds, but could equally be a number of measured activities. It is an iterative minimization procedure which attempts to preserve interpoint distances in multidimensional space in a 2D or 3D representation. Unlike PCA however, the axes are not orthogonal and are not clearly interpretable with respect to the original variables. Nonlinear mapping has been used to cluster aromatic and aliphatic substituents. ... 

Multidimensional scaling [98], nonlinear mapping [99], and other nonlinear dimensionality reduction approaches [100] intend to maintain pairwise distance or similarity relationships from the original high-dimensional space. This sometimes can better preserve close local relationships in a dataset, while detailed relationships between more dissimilar compounds are not maintained. 

---