Factor representation

Values of n have also been calculated for the sugar liquors reported by Peters and Phelps,3 who used an intensive filtration through a pad of purified asbestos. These values ranged from 5.4 to 8.5. About two thirds lay between 5.5 and 6.5. Therefore, it appears to be possible to express the wave-length dependency (of all types of commercial sugar) in the visible spectrum by a two-factor representation. The two factors may be the... 

The following SVD analysis is based on factorized representation with additional orthogonality conditions, see details e.g. [1,2,6,7],... 

The constants a( ), b E), A E Uj( ), U2( ) have been determined by fitting the results of calculations to these analytic expressions. Appendix E provides some values for the Berger formula constants. The best equations for the gamma buildup factor representation are based on the so-called geometric progression (G-P) form. The G-P function has the form... 

In Yang s analysis the effect of diameter comes into play mainly through the terminal velocity, which is present in the particle velocity and friction factor representation. By analyzing the system for the various diameters of particles given above one finds ... 

An approximate contour plot of the robustness function with respect to a varied total damper quantity has been constructed. By comparing the contour plots of the robustness function for various damper distributions, a total damper quantity can be derived which satisfies the performance criterion under the uncertainty of the structural parameters. It has also been shown that it is possible to clarify the relation of total quantities of passive dampers for the robustness function representation and the safety factor representation. It is expected that this leads to a more robust damper design. 

As stated in Section 6.1, multilinear PLS, as its two-way analogue, aims at finding components which, sequentially, maximize the product of the explained variance of the dependent block and the variance of the component itself. Therefore, if one looks at a factor representation which at the same time accounts for as much as possible of the information in the two blocks, it can happen that this condition is not completely fulfilled by the PLS criterion, which involves the maximization of the variance of Xw, instead of the maximization of the amount of X variance explained by the component model. Accordingly, with the aim of building a factor model whose components had the property of explaining both the variance of the independent and... 

Orientation factor representations have been developed for the second moments for each of these cases. In fact, two sets have been developed using Euler s angles by Stein (60) and by Nomura, Kawai, and co-workers (68). 

Figure B3.3.11. The classical ring polymer isomorphism, forA = 2 atoms, using/ = 5 beads. The wavy lines represent quantum spring bonds between different imaginary-time representations of the same atom. The dashed lines represent real pair-potential interactions, each diminished by a factor P, between the atoms, linking corresponding imaginary times.

The sum over all m is justified by the fact that the diagonal elements (t P ji ) vanish in a real representation. It is also evident from the factorization of n) and... 

The information content of a structure descriptor depends on two major factors a) the molecular representation of the compound b) the algorithm which is used for the calculation of the descriptor. 

The number of discrete points of /cicc( ) determines the resolution of the chirality code is a smoothing factor which in practice controls the width of the peaks obtained by a graphical representation versus u. An example of a chir-... 

Figure 2.1 served as the basis for our initial analysis of viscosity, and we return to this representation now with the stipulation that the volume of fluid sandwiched between the two plates is a unit of volume. This unit is defined by a unit of contact area with the walls and a unit of separation between the two walls. Next we consider a shearing force acting on this cube of fluid to induce a unit velocity gradient. According to Eq. (2.6), the rate of energy dissipation per unit volume from viscous forces dW/dt is proportional to the square of the velocity gradient, with t]q (pure liquid, subscript 0) the factor of proportionality ... 

The volumetric properties of fluids are conveniently represented by PVT equations of state. The most popular are virial, cubic, and extended virial equations. Virial equations are infinite series representations of the compressibiHty factor Z, defined as Z = PV/RT having either molar density, p[ = V ), or pressure, P, as the independent variable of expansion ... 

Rules may represent either guidelines based on experience, or compact descriptions of events, processes, and behaviors with the details and assumptions omitted. In either case, there is a degree of uncertainty associated with the appHcation of the rule to a given situation. Rule-based systems allow for expHcit ways of representing and dealing with uncertainty. This includes the representation of the uncertainty of individual rules, as weU as the computation of the uncertainty of a final conclusion based on the uncertainty of individual rules, and uncertainty in the data. There are numerous approaches to uncertainty within the rule-based paradigm (2,35,36). One of these approaches is based on what are called certainty factors. In this approach, a certainty factor (CF) can be associated with variable—value pairs, and with individual rules. The certainty of conclusions is then computed based on the CF of the preconditions and the CF for the rule. For example, consider the foUowing example. 

The corresponding states approach suggested by Pitzer et al. requires only the critical temperature and acentric factor of the compound. For a close approximation, an analytical representation of this method proposed by Reid et al. " for 0.6 [Pg.394]

Virial Equations of State The virial equation in density is an infinite-series representation of the compressiDility factor Z in powers of molar density p (or reciprocal molar volume V" ) about the real-gas state at zero density (zero pressure) ... 

FIG. 5-8 Graphical representation of the Colhiirn / factor for the heating and cooling of fluids inside tnhes. The curves for helow 2100 are based on Eq. (5-40). L is the length of each pass in feet. The curve for Nr above 10,000 is represented by Eq. (5-50c). 

With a reactive solvent, the mass-transfer coefficient may be enhanced by a factor E so that, for instance. Kg is replaced by EKg. Like specific rates of ordinary chemical reactions, such enhancements must be found experimentally. There are no generalized correlations. Some calculations have been made for idealized situations, such as complete reaction in the liquid film. Tables 23-6 and 23-7 show a few spot data. On that basis, a tower for absorption of SO9 with NaOH is smaller than that with pure water by a factor of roughly 0.317/7.0 = 0.045. Table 23-8 lists the main factors that are needed for mathematical representation of KgO in a typical case of the absorption of CO9 by aqueous mouethauolamiue. Figure 23-27 shows some of the complex behaviors of equilibria and mass-transfer coefficients for the absorption of CO9 in solutions of potassium carbonate. Other than Henry s law, p = HC, which holds for some fairly dilute solutions, there is no general form of equilibrium relation. A typically complex equation is that for CO9 in contact with sodium carbonate solutions (Harte, Baker, and Purcell, Ind. Eng. Chem., 25, 528 [1933]), which is... 

This allows for the equivalence between crossed cylinders and the particle on a plane problem. Likewise, the mechanics of two spheres can be described by an equivalently radiused particle-on-a-plane problem. The combination of moduli and the use of an effective radius greatly simplifies the computational representation and allows all the cases to be represented by the same formula. On the other hand, it opens the possibility of factors of two errors if the formula are used without realizing that such combinations have been made. Readers are cautioned to be aware of these issues in the formulae that follow. 

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