Reduction in dimensionality

The number of special coordinates, or dimensionality of a problem, can be reduced using three basic strategies symmetry, aspect ratio and series resistances. 

Symmetry is the easiest to apply. It is based on the correct selection of the coordinate system for a given problem. For example, a temperature field with circular symmetry can be described using just the coordinates (r, z), instead of (x, y, z). In addition, symmetry can help to get rid of special variables that are not required by the conservation equations and interfacial conditions. For example, the velocity field in a tube, according to the Navier-Stokes and continuity equations, can have the functional form uz(r). 

The ratio of two linear dimensions of an object is called the aspect ratio. There are a number of possible simplifications when the aspect ratio of an object or region is large (or small). For example, for the classical fin approximation, the thickness of the fin is small compared with the length, therefore the temperature will be assumed to change in the direction of the length only. 

Finally, it is possible to reduce the dimensionality of a problem by determining which rate processes in series is the controlling step. As shown for l i -C 1, the convection controls the cooling process and conduction is so fast that the solid is considered isothermal, reducing the dimensionality from (x, y, z) to a zero dimensional problem or lumped mass method. 

Characteristic times are a key factor in formulating conduction or diffusion models, because they determine how fast a system can respond to changes imposed at a boundary. In other words, if the temperature or concentration is perturbed at some location, it is important to estimate the finite time required for the temperature or concentration changes to be noticed at a given distance from the original perturbation. The time involved in a stagnant medium is the characteristic time for conduction or diffusion, therefore this is the most widely used characteristic time in transport models [3, 6], 

---

Context-dependent situations often lead to a large-scale input dimension. Because the required number of training examples increases with the number of measured variables or features, reducing the input dimensionality may improve system performance. In addition, decision discriminants will be less complex (because of fewer dimensions in the data) and more easily determined. The reduction in dimensionality can be most readily achieved by eliminating redundancy in the data so that only the most relevant features are used for mapping to a given set of labels. 

By substituting the new variables into the original equations we will acquire information that allows the simplification of a specific model. Length and time scales, for example, can lead to geometrical simplifications such as a reduction in dimensionality. 

Temperature development in an extruder channel during melting. In this example, we illustrate reduction in dimensionality of the energy equation to find an equation that would reveal the change of the melt temperature through the gap between the solid bed and extruder barrel during melting, as schematically depicted in Fig. 5.10. To simplify the problem, we assume to have constant properties and a Newtonian viscosity. 

The thickness between the solid and the barrel is small compared to the screw channel, which indicates that a reduction in dimensionality can be performed. Initially, it can be assumed that the velocity field is unidirectional, i.e. ux(y). The energy equation is then reduced to,... 

There are several factor analysis procedures that result in a reduction in dimensionality. One such procedure simply eliminates some of the variables, which in Py-MS case would be the peaks for certain m/Zi values. To prevent the loss of valuable information in this procedure, an appropriate rule for this procedure must be established. Dedicated computer programs are available to perform such procedures, and commonly they have the following steps ... 

Section 5-2) and in the suppression of specific unwanted peaks (Section 5-1). Frequency selection within two-dimensional spectra (Chapter 6) results in a reduction in dimensionality, so that effects at a single frequency can be examined in detail. (A one-dimensional cut of a two-dimensional spectrum offers the twin advantages of reduced experimental time and decreased storage needs.)... 

The first three factors on the right-hand side of this relation are smaller than unity. The reduction in dimensionality that accompanies the transfer of reactant B from solution to an interface slows down the reaction rate. Using realistic values of the masses m and the radii rh one may predict a maximum estimate of a 50-fold reduction in the rate constant ks. This geometric disadvantage is sometimes compensated by a lower activation energy Ea at the surface. Astumian and Schelly (1984) estimate that Ea of the heterogeneous reaction must be 2.5 to 7.5 kJ mol-1 lower to compensate for the geometric effects. 

Consider a molecule diffusing in free space or a solute molecule diffusing in solution. Upon colliding with a surface, assume that the molecule is sufficiently entrained by surface forces that there results a reduction in dimensionality of its diffusion space from d = 3 to d — 2, and that in its subsequent motion the molecule is sterically constrained to follow the pathways defined by the lattice structure of the surface (or, perhaps, the boundary lines separating adjacent domains). If at some point in its trajectory the molecule becomes permanently immobilized, either because of physical binding at a site or because an irreversible reaction has occurred at that site, then, qualitatively, this sequence of events is descriptive of many diffusion-reaction processes in biology, chemistry and physics. 

A dramatic reduction in dimensionality is often possible by converting a design equation from dimensioned to dimensionless form. Equation 1.62 contains the dependent variable a and the independent variable z. The process begins by selecting characteristic values for these variables. By characteristic value we mean some known parameter that has the same dimensions as the variable and that characterizes the system. Eor a PER, the variables are concentration and length. A characteristic value for concentration is flin and a characteristic value for length is L. These are used to define the dimensionless variables a = ala-m and zIL. The governing equation for a first-order reaction in an ideal PER becomes... 

Hund s cases are important because they tell the experimentalist what kind of patterns might be found in a spectrum, how to look for these patterns, and what inferences about quantum number assignments can be drawn from the patterns once they are detected. Hund s cases tell the dynamicist how to construct a reduced dimension picture of intramolecular processes. The reduction in dimensionality is based on the existence of approximate constants of motion, eigenvalues of operators that commute with most of the molecular Hamiltonian [see Sections 9.4.9 and 9.4.10]. Hund s cases, embodied in models of vectors precessing about other vectors, explain how information about molecule frame properties (e.g., a permanent magnetic dipole or an electric dipole transition moment) survives rotational averaging and becomes observable in the laboratory frame (and vice versa). 

One way to see the effect of the reduction in dimensionality that occurs as one goes from time series to phase space portrait to Poincare section is to consider the stability of a system as it is reflected in the stability of points in the Poincare section. The single point, which corresponds to the Poincare section of a simple limit cycle, can be treated in the same way an equilibrium point (or steady state point) is treated, even though, here, we are considering the stability of the periodic state, that is, the limit cycle. A small perturbation added to this point in the cross section will be found to decay back toward the point itself, if the limit cycle is stable, or to evolve away from the point, if the cycle is unstable. The stability properties of the point in the Poincare section are the same as the stability properties of the limit cycle to which it corresponds. Hence, a stable point in the section means that the limit cycle is stable, and an unstable one means that the limit cycle is unstable. And, furthermore, any bifurcations which occur for the point in the cross section also correspond to bifurcations which the limit cycle undergoes as a parameter is varied. 

SAMs illustrate a strategy for synthesis based on the idea of reduction in dimensionality. The generic idea underlying SAMs is to use a surface, or some other two-dimensional or pseudo two-dimensional system, as a template and to assemble molecules on it in reasonably predictable geometry using appropriate coordination chemistry to connect the surface with the adsorbed molecules. For this strategy to work, one needs ... 

As workers succeed in anchoring crown ether catalysts on surfaces, it will be interesting to see the extent to which catalyzed reaction rates are further enhanced by what is known as a "reduction in dimensionality"(28). The essence of this idea is that a substrate only needs to hunt for an "active site" in two dimensions once it sticks on the catalytic surface. This is in contrast to a longer three dimensional search for the catalyst in the case of homogeneous solutions. 

In general, the regime in which the solvation forces appear could be described as an entropically cooled boundary regime. The question arises if this boundary regime also exist without external pressure forces, i.e., solely because of surface interactions and a reduction in dimensionality. Winkler et al. predict with a MD simulation that hexadecane is well ordered, in crystalline like monolayers for strongly... 

The objective of a principal component analysis (PCA) is to transform a number of correlated variables into a smaller set of new, uncorrelated variables (factors or latent variables). The first few factors should then explain most of the relevant variation in the data set. To allow this reduction in dimensionality, the variables are characterized by a partial correlation. The new variables can then be generated through a linear combination of the original variables, i.e. the original matrix X is then the product of a score matrix P and the transpose ( ) of the loading matrix A ... 

Charge and proton relay through hydrogen bonds have been proposed to contribute to the catalytic efficiency of enzymes, and in this sense reversed micelles provide an appropriate model to delineate the importance of such factors at the enzyme active site. Micellar surfaces also provide a convenient means for the reduction in dimensionality, an important factor in enhancing reaction rates. They also serve as good models to demonstrate the feasibility of ultrafast proton transfer when the reactants are localized in a suitable environment such as membrane surfaces and other complex biomacromolecules. 

This aspect has been discussed in the weak and strong collision limits[l2,13], and for systems with many internal degrees of freedom it is certain that this reduction in dimensionality occurs at bath densities where the reaction rates... 

---