Reduced dimensionality

Often a degree of freedom moves very slowly for example, a heavy-atom coordinate. In that case, a plausible approach is to use a sudden approximation, i.e. fix that coordinate and do reduced dimensionality quantum-dynamics simulations on the remaining coordinates. A connnon application of this teclmique, in a three-dimensional case, is to fix the angle of approach to the target [120. 121] (see figure B3.4.14). 

Wang D and Bowman J M 1992 Reduced dimensionality quantum calculations of mode specificity in OH + H2 H2O + HJ. Chem. Phys. 96 8906... 

Transport of France reduces dimensional change by using a thin inner liner of Invar. Another patented French system accommodates this change by means of the flexibility of thin metal which is creased. The creases run in two direc tions, and the form of the crossings of the creases is a feature of the system. 

Weathering. This generally occurs as a result of the combined effect of water absorption and exposure to ultra-violet radiation (u-v). Absorption of water can have a plasticizing action on plastics which increases flexibility but ultimately (on elimination of the water) results in embrittlement, while u-v causes breakdown of the bonds in the polymer chain. The result is general deterioration of physical properties. A loss of colour or clarity (or both) may also occur. Absorption of water reduces dimensional stability of moulded articles. Most thermoplastics, in particular cellulose derivatives, are affected, and also polyethylene, PVC, and nylons. 

N concentrations. Years ago, this fact was useful since numerical solutions to ODEs required substantial computer time. They can now be solved in literally the blink of an eye, and there is little incentive to reduce dimensionality in sets of ODEs. However, the theory used to reduce dimensionality also gives global stoichiometric equations that can be useful. We will therefore present it briefly. 

In the adiabatic bend approximation (ABA) for the same reaction,18 the three radial coordinates are explicitly treated while an adiabatic approximation was used for the three angles. These reduced dimensional studies are dynamically approximate in nature, but nevertheless can provide important information characterizing polyatomic reactions, and they have been reviewed extensively by Clary,19 and Bowman and Schatz.20 However, quantitative determination of reaction probabilities, cross-sections and thermal reaction rates, and their relation to the internal states of the reactants would require explicit treatment of five or the full six degrees-of-freedom in these four-atom reactions, which TI methods could not handle. Other approximate quantum approaches such as the negative imaginary potential method16,21 and mixed classical and quantum time-dependent method have also been used.22... 

The analytic potential energy surfaces, used for the Cl + CH3Clb and Cl + CHjBr trajectory studies described here, should be viewed as initial models. Future classical and quantum dynamical calculations of SN2 nucleophilic substitution should be performed on quantitative potential energy functions, derived from high-level ab initio calculations. By necessity, the quantum dynamical calculations will require reduced dimensionality models. However, by comparing the results of these reduced dimensionality classical and quantum dynamical calculations, the accuracy of the classical dynamics can be appraised. It will also be important to compare the classical and quantum reduced dimensionality and classical complete dimensionality dynamical calculations with experiment. 

To tackle these problems successfully, new concepts will be required for developing systematic modeling techniques that can describe parts of the chemical supply chain at different levels of abstraction. A specific example is the integration of molecular thermodynamics in process simulation computations. This would fulfill the objective of predicting the properties of new chemical products when designing a new manufacturing plant. However, such computations remain unachievable at the present time and probably will remain so for the next decade. The challenge is how to abstract the details and description of a complex system into a reduced dimensional space. 

Bowman, J. M. Reduced dimensionality theory of quantum reactive scattering,. /.Phys. Cliem., 95 (1991), 4960-4968... 

The peculiarity of PCA is in a representation of the data set onto a subspace of reduced dimensionality where the statistical properties of the original data set are preserved. 

In another interesting development, Gumhalter and Brenig (1995a,b) proposed that, in H-Ni(llO) and H-Cu(llO) systems, adatom-induced reconstruction of the surface gives rise to quasi-one-dimensional surface states, which mediate the indirect interaction. The greatly reduced dimensionality... 

Mechanical Calculations of the Rate Constant for the H2 + OH — H + H20 Reaction Full-Dimensional Results and Comparison to Reduced Dimensionality Models. 

Reduced dimensional parameters (strain parameters and near-neighbours diagrams) By comparing the space-filling theoretical curves and the actual values of intermetallic phases it has been observed that an incompressible sphere model of the atom gives only a rough description when discussing metallic structures. 

Recently, however, NMR techniques have been developed that can be used to considerably accelerate the measurement of large three-dimensional data sets [31,32], These reduced-dimensionality techniques have already been successfully used in in-cell NMR applications [16]. In the future these techniques might enable the complete backbone assignment of a protein in its natural environment and might enable detailed investigation of different folding states. 

If applicable, EIM may help to reduce dimensionality, which decreases the numerical burden by more than an order of magnitude, definitely. Evidently, the use of FDTD in integrated optics design is restricted to small device structures, only. But, FDTD-calculations may offer a deeper insight... 

In this case the exchange coefficient k has units of s" (rather than the usual cm/s) because of the reduced dimensionality of the system, i.e., fluxes are written in terms of surface concentrations rather than volumetric concentration. 

HALF-REACTIONS NERNST EQUATION REDUCED CONCENTRATION Reduced dimensionality of diffusion,... 

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