Symbolic analogy

Pasteur s analogy was the key to the development of his vaccines, which are generally based on the concept of creating viral attacks on the human body, but attacks that are safe because the viruses used were weakened earlier. The body is able to defend itself and in the process becomes immune to attacks by truly vicious viruses. This example is also particularly appropriate for engineers, because it shows how an aggressive and potentially dangerous action may result in a definitely positive outcome. 

There is another and easier interpretation of the symbolic analogy, which is that it is simply a statement that synthetizes the entire problem using several abstract terms or describing all the essential elements of the system under consideration. Such use of the symbolic analogy can be best explained by an example involving the author. 

When the author was in his early twenties, he worked at the Department of Metal Structures at the Warsaw University of Technology, Poland. Once, he was invited by a senior faculty member, a world-class expert in the area of steel structures, to determine with him the mechanism by which a large steel structure had collapsed in a small town in northern Poland. This determination had various legal implications and was quite important for all parties involved in the incident. 

The structure to be investigated was a large steel gas tank with a floating roof. The roof structure was in the form of several steel arches, which 

Tensile stress reduction Resultant stress distribution 

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To illustrate the relation between the different flows and the two reaction velocities, we remark that the flows 23, 34, and 45 are obviously the velocity of the main reaction, r, while the flows 12 and 50 equals the velocity s of the side reaction. This is shown in Fig. 5 by means of letters and arrows. The diagram also shows the symbolic analogy between our flows and real physical flows. Thus we may speak of sources and sinks, 1 being a source and 0 a sink, and of translational and rotational flows for example, we may say that the flow s62 is a superposition of a translational flow ( — s) and a rotational flow (r). s may be assumed to be always positive. The case s = 0 is in principle the same as the one treated above (p. 322), where we may speak of catalysis with X2 as a catalyst. As the chain (23452) is broken in this case only by the reaction 21, the chain length then has its maximum, but its numerical value cannot be defined unless we know the kinetics of the reactions (12) and (21), which may be unknown compare the discussion in the literature of the hydrogen-bromine reaction (see also p. 334). 

Since the left-hand sides are identical, it is seen that, due to the phase difference in the definition of the F (/) function and the 3-j symbol [analogously to Eq. (18)], the two reduced matrix elements in Eqs. (28a) and (28b) are identical. The operator set as well as the sets of functions are supposed to transform according to irreducible representations, and the labels i and as introduce extra parameters when necessary. 

We shall designate the nucleus using nuclear symbols, analogous to the atomic symbols that were introduced in Section 2.1. The nuclear symbols consist of the elemental symbol, the atomic number (the number of protons in the nucleus), and the mass number, which is defined as the sum of neutrons and protons in the nucleus. 

The first two sign factors are associated with permutations of rows in the 97-symbols. Analogous simplifications are possible for the other reduced matrix elements the final formulae are collected in Table 11.9. 

A description of these symmetry elements and their symbols may be found in Figs. 2.6-2.S and in Tables 2.4 and 2.5. This correspondence allows us to characterize space groups with symbols analogous to those used for the plane groups (Section 2.7.1). The international symbolfor a space group is composed of ... 

Here the end of the sphere is characterized by the symbol /. Analogously, the third and fourth spheres are obtained and the HOSE code for all ofthe four spheres ofthe molecule in Figure 7.3 is... 

Figure 8.8 Symbolic analogy safe attack. (a) A beam under bending (transverse load), (b) A beam under bending and compression.

Structural members—all this systematic and purely quantitative effort was transformed into two words of our opinion a plastic collapse. We synthesized the mountain of data using our background knowledge, and finally the symbolic analogy simply emerged. This symbolic analogy became obviously the equivalent of all the documentation and of all this twisted steel and in very few words perfectly described the situation. 

These terms are analogous to those on p. 265 of [7], It will be noted that the symbol c has been reinstated as in Section VI.F, so as to facilitate the order of magnitude estimation in the nearly nonrelativistic limit. We now proceed based on Eq. (168) as it stands, since the transformation of Eq. (168) to modulus and phase variables and functional derivation gives rather involved expressions and will not be set out here. 

If it be assumed that the ionising characteristics of nitric acid are similar to those of the organic indicators used to define the scales of acidity, then a correspondence between the acidity-dependence of nitration and would suggest the involvement of the nitronium ion, whereas a correspondence with Hq would support the h)rpothesis that the nitric acidium ion were active. The analogies with and Hg are expressed in the first and last pairs of the followii equations respectively. The symbol AQ represents anthraquinone, the indicator originally used in this way for comparison with the acidity dependence of the rate of nitration of nitrobenzene ... 

By analogy with Eq. (3.1), we seek a description for the relationship between stress and strain. The former is the shearing force per unit area, which we symbolize as as in Chap. 2. For shear strain we use the symbol y it is the rate of change of 7 that is involved in the definition of viscosity in Eq. (2.2). As in the analysis of tensile deformation, we write the strain AL/L, but this time AL is in the direction of the force, while L is at right angles to it. These quantities are shown in Fig. 3.6. It is convenient to describe the sample deformation in terms of the angle 6, also shown in Fig. 3.6. For distortion which is independent of time we continue to consider only the equilibrium behavior-stress and strain are proportional with proportionality constant G ... 

The concentration dependence of s is eliminated by making measurements at several different concentrations and then extrapolating to zero concentration. The limiting value is given by the symbol s°. This is the sedimentation analog of D°. 

Although it is less often done, I have used an analogous symbolism for pure vibrational transitions for the sake of consistency. Here N refers to a vibrational (infrared or Raman) transition from a lower state with vibrational quantum number v" to an upper state v in the vibration numbered N. 

A mass (weight) density function, given the symbol m and having dimensions mass/(volume)(length), can be defined analogously to population density letting AM be the mass of crystals per unit system volume in the size range E to L + AL,... 

Tritium [15086-10-9] the name given to the hydrogen isotope of mass 3, has symbol or more commonly T. Its isotopic mass is 3.0160497 (1). Moletecular tritium [10028-17-8], is analogous to the other hydrogen isotopes. The tritium nucleus is energetically unstable and decays radioactively by the emission of a low-energy P particle. The half-life is relatively short (- 12 yr), and therefore tritium occurs in nature only in equiUbrium with amounts produced by cosmic rays or man-made nuclear devices. 

Mutual Diffusivity, Mass Diffusivity, Interdiffusion Coefficient Diffusivity is denoted by D g and is defined by Tick s first law as the ratio of the flux to the concentration gradient, as in Eq. (5-181). It is analogous to the thermal diffusivity in Fourier s law and to the kinematic viscosity in Newton s law. These analogies are flawed because both heat and momentum are conveniently defined with respec t to fixed coordinates, irrespective of the direction of transfer or its magnitude, while mass diffusivity most commonly requires information about bulk motion of the medium in which diffusion occurs. For hquids, it is common to refer to the hmit of infinite dilution of A in B using the symbol, D°g. 

Figure 8 A joint principal coordinate projection of the occupied regions in the conformational spaces of linear (Ala) (triangles) and its conformational constraint counterpart, cyclic-CAla) (squares), onto the optimal 3D principal axes. The symbols indicate the projected conformations, and the ellipsoids engulf the volume occupied by the projected points. This projection shows that the conformational volume accessible to the cyclic analog is only a small subset of the conformational volume accessible to the linear peptide, (Adapted from Ref. 41.)...

If the letter symbols for sets are replaced by numbers, tlie commutative and associative laws become familiar laws of aritlimetic. In Boolean algebra tlie first of tlie two distributive laws, Eq. (19.3.5), lias an analogous counterpart in arithmetic. Tlie second, Eq. (19.3.6), does not. In risk analysis. Boolean algebra is used to simplify e. pressions for complicated events. For example, consider tlie event... 

To treat acid-base equilibria involving zwitterions, it is convenient to consider the cation stable at low pH to be a diprotic add (analogous to H2C03), which ionizes in two steps. Using the symbols C+, Z, and A- to stand for the cation, zwitterion, and anion, respectively, we have... 

It is clear that the analogy between these statements and the corresponding ones of Section 8.6 permits us at once to take over the results of Sections 8.7 and 8.8 by a mere reinterpretation of the symbols. Thus, the analogs of Eq. (8-50) can be derived by steps identical to those used in Section 8.7 ... 

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