An Analogy

There are no more stable states of the chair and there are only four energy levels. Fig. 2.6. The stable states of the chair are analogues to the stationary quantum states of Fig. 1.8.b,d, on p. 23, while unstable states of the chair on the floor are analogues of the non-stationary states of Fig. f.8.a,c. 

One way we might approach this problem is to take advantage of the fact that although the Volkswagen does not reflect light, it retains all of its other physical properties. We might, for example, take a basketball and throw it at the invisible object from some direction Jcq 

2 From the song The Purple People Eater by Sheb Wooley. 

4 University of California, Irvine Anteater (official mascot). 

FIGURE 1.2 A continuous object, here an imagined 1963 Volkswagen Beetle regally aligned in a defined parking location in three-dimensional space, with the supple lines amenable to description in terms of bouncing balls. 

Lets assume, however, that we can walk around the invisible Volkswagen and throw the basketballs from many, in fact all possible directions ko- Each time we note carefully how many balls bounce in which direction. Ultimately, we will know for every direction of our incoming beam of basketballs o how many are reflected in every direction k. 

---

Another design option that can be considered if a column will not fit is use of an intermediate reboiler or condenser. An intermediate condenser is illustrated in Fig. 14.5. The shape of the box is now altered because the intermediate condenser changes the heat flow through the column. The particular design shown in Fig. 14.5 would require that at least part of the heat rejected from the intermediate condenser be passed to the process. An analogous approach can be used to evaluate the possibilities for use of intermediate reboilers. Flower and Jackson," Kayihan, and Dhole and Linnhofl have presented procedures for the location of intermediate reboilers and condensers. 

The humidities for systems other than air and water are defined in an analogous manner. 

Used in virtually all organic chemistry analytical laboratories, gas chromatography has a powerful separation capacity. Using distillation as an analogy, the number of theoretical plates would vary from 100 for packed columns to 10 for 100-meter capillary columns as shown in Figure 2.1. 

Limited capabilities as the instrument carries out an analog measurement, on-line signal processing of data is almost impossible. All analysis tasks must be done off-line, and cost is increased (equipment, specialists). 

In the remainder of this paper, we exhibit the solution of the deconvolution problem in the frequency domain, but it is possible to establish an analogy with tlie temporal solution exposed by G. Demoment [5,6]. 

The shear viscosity is an important property of a Newtonian fluid, defined in terms of the force required to shear or produce relative motion between parallel planes [97]. An analogous two-dimensional surface shear viscosity ij is defined as follows. If two line elements in a surface (corresponding to two area elements in three dimensions) are to be moved relative to each other with a velocity gradient dvfdx, the required force is... 

An analogous procedure can be applied to a plane surface. The surface can be roughened by the successive application of one or another recipe, just as was done for the line in Fig. VII-6. One now has a fractal or self-similar surface, and in the limit Eq. VII-20 again applies, or... 

An analogous variation of frequency of the nonnal inodes occurs in poly atomics. 

The constant-temperature constant-pressure situation yields an analogous result. One can write for the reversible process... 

Kirkwood derived an analogous equation that also relates two- and tlnee-particle correlation fiinctions but an approximation is necessary to uncouple them. The superposition approximation mentioned earlier is one such approximation, but unfortunately it is not very accurate. It is equivalent to the assumption that the potential of average force of tlnee or more particles is pairwise additive, which is not the case even if the total potential is pair decomposable. The YBG equation for n = 1, however, is a convenient starting point for perturbation theories of inliomogeneous fluids in an external field. 

Kolm and Sham [25] decompose G[p] into the kinetic energy of an analogous set of non-interacting electrons with the same density p(r) as the interacting system. 

Trentler T J ef a/1995 Solution-liquid-solid growth of orystalline lll-V semioonduotors an analogy to vapor-liquid-solid growth Science 270 1791... 

In order to overcome these weaknesses, Pople and co-workers reverted to a more complete approach that they first proposed in 1965 [14], neglect of diatomic differential overlap (NDDO). In NDDO, all four-center integrals (pv are considered in which p and v are on one center, as are 2 and cr (but not necessarily on the same one as and v). Furthermore, integrals for which the two atomic centers are diEFer-ent are treated in an analogous way to the one-center integrals in INDO, resulting... 

A third approach is suggested by Hugo s formulation of material balances at the limit of bulk diffusion control, described in Section 11.3. Hugo found expressions for the fluxes by combining the stoichiometric conditions and the Stefan-Maxvell relations, and this led to no inconsistencies since there are only n - 1 independent Stefan-Maxwell relations for the n fluxes. An analogous procedure can be followed when the diffusion is of intermediate type, using the dusty gas model equations in the form (5.10) and (5.11). Equations (5.11), which have the following scalar form ... 

The charge transfer term arises from the transfer of charge (i.e. electrons) from occupied molecular orbitals on one molecule to unoccupied orbitals on the other molecule. This contribution is calculated as the difference between the energy of the supermolecule XY when this charge transfer is specifically allowed to occur, and an analogous calculation in which it is not. 

The number of terms of a complete polynomial of any given degree will hence correspond to the number of nodes in a triangular element belonging to this family. An analogous tetrahedral family of finite elements that corresponds to complete polynomials in terms of three spatial variables can also be constructed for three-dimensional analysis. 

Extension of the streamline Petrov -Galerkin method to transient heat transport problems by a space-time least-squares procedure is reported by Nguen and Reynen (1984). The close relationship between SUPG and the least-squares finite element discretizations is discussed in Chapter 4. An analogous transient upwinding scheme, based on the previously described 0 time-stepping technique, can also be developed (Zienkiewicz and Taylor, 1994). 

The formation of the (X-delocalized norbornyl cation via ionization of 2-norbornyl precusors in low-nucleophilicity, superacidic media, as mentioned, can be considered an analog of an intramolecnlar Friedel-Crafts alkylation in a saturated system. Indeed, deprotonation gives nortricyclane,... 

The following carbinols were prepared in an analogous way (note 2) in yields... 

Beyond pharmaceutical screening activity developed on aminothiazoles derivatives, some studies at the molecular level were performed. Thus 2-aminothiazole was shown to inhibit thiamine biosynthesis (941). Nrridazole (419) affects iron metabohsm (850). The dehydrase for 5-aminolevulinic acid of mouse liver is inhibited by 2-amino-4-(iS-hydroxy-ethyl)thiazole (420) (942) (Scheme 239). l-Phenyl-3-(2-thiazolyl)thiourea (421) is a dopamine fS-hydroxylase inhibitor (943). Compound 422 inhibits the enzyme activity of 3, 5 -nucleotide phosphodiesterase (944). The oxalate salt of 423, an analog of levamisole 424 (945) (Scheme 240),... 

The 2-anilinoselenazoles follow an analogous reaction path. The entry of bromine into the 5-position is in agreement as seen in Scheme 35. The 2-anilino-4-phenylselenazole gives a 5-bromo derivative by bromination. It is not identical with either of the bromo derivatives prepared by direct synthesis (Scheme 35) (99). 

The difference m odor between (R) and (S) carvone results from their different behavior toward receptor sites m the nose It is believed that volatile molecules occupy only those odor receptors that have the proper shape to accommodate them Because the receptor sites are themselves chiral one enantiomer may fit one kind of receptor while the other enantiomer fits a different kind An analogy that can be drawn is to hands and gloves Your left hand and your right hand are enantiomers You can place your left hand into a left glove but not into a right one The receptor (the glove) can accommodate one enantiomer of a chiral object (your hand) but not the other... 

---