System dimensionality

Dimensional Analysis. In the design of rather simple devices or systems, dimensional analysis can be used in conjunction with physical model experimental investigations to gain insight into the performance of a particular design concept. It is usually possible to define the performance of a simple device or system with a certain number of well chosen geometric and performance related variables that describe the device or system. Once these variables have been selected, dimensional analysis can be used to ... 

Packaging, Handling, Storage, and Transportation System Dimensional Constraints, Definition of , Ml L-STD-1306 (1972) 24) Anon,... 

Elution with salt pulses A multiple step elution is performed by the introduction of, for example, 5%, 10%, 25%, 50%, and 100% of 1.5 M sodium chloride in 19 mM phosphate buffer (pH 2.5) containing 5% methanol. Each step is for 10 min and run at 0.5 mL/min. This elution method compromises analytical system dimensionality, as the peak capacity of the ion-exchange chromatography (IEX) step is equal at most to the number of salt steps. However, in the second dimension only one or two columns are needed and there is no particular limitation in the second dimension separation time as peptides are eluted in portions in a controlled manner. However, the number of salt steps is limited by the total analysis time. In this case the multidimensional system is relatively simple. 

Fig. 1 Schematic drawing to show the concept of system dimensionality (a) bulk semiconductors, 3D (b) thin film, layer structure, quantum well, 2D (c) linear chain structure, quantum wire, ID (d) cluster, colloid, nanocrystal, quantum dot, OD. In the bottom, it is shown the corresponding density of states [A( )] versus energy (E) diagram (for ideal cases).

Statistical Mechanics of Ionic Systems, Dimensional Methods in (Blander). ... 

For a polyatomic reactant with many degrees of freedom the numerical calculations required to execute the program outlined above can easily achieve a scale that is impossible to handle even with a vectorized parallel processor supercomputer. The simplest approximation that reduces the scale of the numerical calculations is the neglect of some subset of the internal molecular motions, but this approximation usually leads to considerable error. A more sophisticated and intuitively reasonable approximation [72, 73] is to reduce the system dimensionality by placing constraints on the values of the internal molecular coordinates (instead of omitting them from the analysis). 

In all cases, the mass-transfer coefficient depends upon the diffusivity of the transferred material and the thickness of the effective film. The latter is largely determined by the Reynolds number of the moving fluid, that is, its average velocity, density, and viscosity, and some linear dimension of the system. Dimensional analysis gives the following relation ... 

The coefficients fco and kj, have been experimentally determined for many mass transfer systems and correlated with gas and liquid flow rates, liquid density and viscosity, the diffusivity of A in the gas and the liquid, and the physical dimensions of the systems. Dimensional analysis suggests that dimensionless quantities of the form... 

The steady states which are unstable using the static analysis discussed above are always unstable. However, steady states that are stable from a static point of view may prove to be unstable when the full dynamic analysis is performed. That is to say simply that branch 2 in Figure 4.8 is always unstable, while branches 1,3 in Figure 4.8 and branch 4 in Figure 4.8 can be stable or unstable depending upon the dynamic stability analysis of the system. As mentioned earlier, the analysis for the CSTR presented here is mathematically equivalent to that of a catalyst pellet using lumped parameter models or a distributed parameter model made discrete by a technique such as the orthogonal collocation technique. However, in the latter case, the system dimensionality will increase considerably, with n dimensions for each state variable, where n is the number of internal collocation points. 

For concentrated systems, dimensional analysis of the retraction caused by the interfacial tension makes it possible to express the second terms in Eq 7.98 as ... 

Dimensionality is an important term, which can be used to describe the nature of the sample, or of the multidimensional system. System dimensionality simply refers to the number of coupled separation stages or dimensions used in the multidimensional system. For example, a single column liquid chromatograph would have a dimensionality of 1, whilst a LC-GC system described above has a dimensionality of 2. On the other hand, sample dimensionality, introduced by Giddings in 1995, is used to describe the intrinsic nature of the sample, and can be defined... 

Theoretically, optimum separation is achieved when the sample dimensionality and system dimensionality are equivalent, resulting in an ordered separation (Figure 3). In the above example, only one separation dimension would be required to analyze the alkane sample. If, however, the dimensionality of the sample exceeds that of the system, sample components will not be resolved in an orderly fashion, but rather a disordered or chaotic separation will result. In Figure 3, three descriptors are required to define the sample - shape, pattern, and size. In a chemical sense, these might be molar mass, polarity, and molecular shape. As more dimensions of separation are applied, greater definition of the mixture components is achieved. Unfortunately, for very complex samples, the sample dimensionality will be... 

Note that for small errors, Eq. (19.31) converges to the derivative of activation function at the point of the output value. With an increase of system dimensionality, the chances for local minima decrease. It is believed that the described phenomenon, rather than a trapping in local minima, is responsible for convergency problems in the error backpropagation algorithm. 

Topological invariants moreover offer an intriguing explicit dependence from system dimensionality D that fixes the leading exponent of the polynomial forms that express the indices as a function of the lattice size in case of D= stmetures like GNR ", the following general laws hold W(L) D and being s=2D+l (Ori-Cataldo-Putz, 2011 ... 

Essentially, each of the above systems has two widely different time scales. If the initial transient is not of interest, the systems can be projected onto a one-dimensional subspace. The subspace is invariant in that no matter where one starts, after a fast transient, all trajectories get attracted to the subspace in which A and B are algebraically related to each other. In essence, what one achieves is dimension reduction of the reactant space through time scale separation. For large, complex systems sueh as oil refining, it is difficult to use the foregoing ad hoc approaches to reduce system dimensionality manually. Computer codes are available for mechanism reduction by means of the QSA/QEA and sensitivity analysis. ... 

Suppose that the process time scale (or the time window of interest) is bounded between and tmax (tmm reaction time scale spectrum. Then the species reacting on time scales longer than /niax remain dormant their concentrations are hardly different from their initial values. The state of these species may be treated as system parameters. On the other hand, species reacting with time scales shorter than tnim niay be considered relaxed. The relaxed state of these fast-reacting species may be treated as system initial conditions. These considerations naturally help reduce the system dimensionality. 

It can be seen that Eqn. (22.12) is a static model representation. PCA is useful for systems with many (correlated) variables. If a process has 40 measured variables, it is often possible to define a few principal components that capture most of the variance in the originally measured variables, thereby achieving a considerable system dimensionality reduction. 

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