Staircase boundary

An element along the "staircase" boundary between metals and nonmetals metalloids exhibit both metallic and nonmetallic properties. 

Furthermore, one should observe the termination of each row in the right-hand direction. This termination results in the staircase boundary. It is determined by the formulas situated at the extreme—right of each row, namely the formulas which pertain to the extremal polyhexes, as should be dear from the definition of these systems (Par. 3.3.4). When perfect extremal p-tuple coronoids are involved, then the corresponding staircase boundary reflects a part of the staircase boundary for benzenoids. It is specifically the part which corresponds to the benzenoids associated with the tuple coronoids in question. Thus, for instance, the staircase boundary for single coronoids has the same shape as the one for benzenoids when starting from C32H14 ovalene (cf. Table 5, where the start of this staircase boundary for benzenoids is indicated by thin lines). A staircase boundary of this kind, determined by perfect extremal coronoids or extremal benzenoids, shall be referred to as a perfect staircase (boundary). If imperfect extremal coronoids are involved we shall call it an imperfect staircase. 

In the above diagrams the staircase boundaries are indicated by heavy lines. For the portions where they are imperfect they are augmented by stippled lines to the shapes which conform with perfect boundaries. Hence it should be understood that the parenthesized formulas actually do not exist for the coronoids in question. The heavy formulas pertain to perfect extremal coronoids. 

The staircase boundary of the table of formulas can obviously consist of steps with exactly either one or two formulas of the horizontal part of the step. We shall presently refer to these steps as a short step and a long step respectively. The occurrence of a short (resp. long) step depends on the increment An for A(h) - A(h+1), whether it is 1 (resp. 2) cf. Par. 4.3.3. 

If X has a formula in a (long) step which is immediately followed by a short step of the staircase boundary, then X is also void of coves. This situation occurs for the constellations... 

In Sect. 5.7 the shape of the staircase boundary of the table of formulas was discussed. It was inferred that it consists exclusively of short (one-formula) steps and long (two-formula) steps. Here we shall give alternative demonstrations for the fact that there cannot be any "extra long steps" or "high steps". Suppose there was an extra long (three-formula) step as illustrated below. 

Whenever a cubic grid is mandatory—either due to coding limitations from the part of academic groups or due to the inherent properties of, e.g., LB techniques—and a staircase approach is to be avoided (e.g., for a revolving impeller axis) one can take refuge to some immersed boundary method (see, e.g., Mittal and Iaccarino, 2005). One may distinguish between... 

WMM represents a stable mode solving technique, where, similar to TMM, continuous geometrical profiles and gradient index media require a staircase-approximation. Accuracy is affected if boundaries are too close to the guiding structure, or if the number of ID modes is too low, which both is easy to check. So, WMM is a stable and accurate mode-solving technique. It... 

The large central block of the periodic table is occupied by the transition metals, which are mostly listed as Group B elements. Transition metals have properties that vary from extremely metallic, at the left side, to far less metallic, on the right side. The rightmost boundary of the metals is shaped like a staircase, shown in bold in Figure 4-1. 

In the periodic table for benzenoid hydrocarbons [7] the formulas C HS are arranged in an array with coordinates (ds, ,). The Dias parameters (ds) are found on a horizontal axis (increasing from left to right), while the numbers of internal vertices ( ) are on a vertical axis (increasing downwards). The table extends infinitely to the right and downwards. To the left the formulas form a line in the shape of an uneven staircase, which shall be referred to as the staircase-like boundary. 

An acute problem arises, however, concerning the limitation at the staircaselike boundary. In other words, where to stop writing up the formulas to the left This question is answered implicitly in Sect. 6.2 cf. Eqs, (12)—(15). In the subsequent sections a more detailed description of the staircase-like boundary is provided. 

The extreme-left benzenoids are defined by formulas on the staircase-like boundary so that, in each case, there is no formula in the same row to the left. Hence the extremal benzenoids without benzene form a subclass of the extreme-left benzenoids.But there exist extreme-left benzenoids, say x, which are not extremal. Their formulas are found one step up and one step to the right from every formula for the pericondensed protrusive benzenoids. Hence one obtains readily the following expression. 

It should be clear that strictly pericondensed benzenoids occur for formulas at unlimited distances from the staircase-like boundary. Consider, for instance, the homolog series of hydrocarbons as shown in Fig. 2. The systems have in general (for h 2, s > 8) the formulas ... 

The Dias parameter is constantly zero (ds = 0), which means that the formulas are found in the same column, where C10H8 (naphthalene) is at the top. There is no limitation as to how far down one can get in this way, moving steadily away from the staircase-like boundary. 

The C HS formulas on the staircase-like boundary are those of most interest in the present chapter. Therefore an extensive listing of these formulas is given here see Fig. 9. The values of the Dias parameter (ds), which are indicated in the figure, are supposed to facilitate the identification of the positions of the different formulas in the periodic table for benzenoid hydrocarbons the same is the case for the h values. 

Many important features of the staircase-like boundary can be deduced from the fundamental aufbau principle. 

Propositions 1 (a) The staircase-like boundary has no one-formula step. (b) The staircase-like boundary has no plateau of more than one formula for pericondensed benzenoid (s). 

Fig. 9. Formulas (C HS) on the staircase-like boundary of the periodic table for benzenoid hydrocarbons. The formula for benzene (C6H6) is added...

Suppose that C HS is the formula for an extreme-left benzenoid with a cove. By covering the cove (in the language of Section 2.2) a new benzenoid is created with one hexagon more and three more internal vertices. It has the formula C +1HS 1 which is situated in the periodic table relatively to C HS as indicated in the below scheme. At the same time the scheme shows the only possible shape of the staircase-like boundary in the vicinity of C HS and C +1Hs i, as can easily be deduced from Propositions 1 (a) and 1 (b). It is found that C HS is not compatible with an extremal benzenoid. That proves Proposition 2 (a). 

The known numbers of isomers for the constant-isomer series of benzenoids are listed in Tables 4 and 5 for the even- and odd-carbon atom formulas, respectively. In the former case (Table 4) the separate numbers for Kekulean (abbreviated Kek.) and non-Kekulean (abbr. Non-Kek.) systems are given. This listing goes far beyond the mapping with asterisks in Fig. 2, but moves only on the staircase-like boundary (cf. the formulas with black symbols in Fig. 9). 

Figure 2.2 Classical boundary conditions of common chromatographic problems, (a) Elution of a rectangular pulse, (b) Multiple gradient elution of a rectangular pulse, (c) Staircase frontal analysis, (d) Displacement. Dotted line component 1. Shaded line component 2.

Figure 3.37 Typical experimental chromatograms obtained in the determination of equilibrium isotherms by chromatographic methods, (a) Frontal analysis staircase. FACP on the diffuse rear boundary after the last frontal step, (b) ECP. Data recorded with an HP 1090 (Hewlett-Packard, Palo Alto, CA) liquid chromatograph. Reproduced with permission from S. Golshan-Shirazi and G. Guiochon, Anal Chem., 60 (1988) 2630 (Figs. 1 and 2), ( )1988 American Chemical Society.

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