Essentially Two-Dimensional Systems

Three examples will suffice to demonstrate this information Figure 3 shows the polyhedral units in the synthetic zeolite Linde Type A, which link to provide a three-dimensional interconnecting array of channels, Figure 4 illustrates the essentially two-dimensional system of channels in the mordenite framework, and Figure 5 shows the major channels in synthetic zeolite Linde Type L arranged as parallel one-dimensional channels and shown as a stereo pair. Table 6 lists the Atlas notations for these structures with explanations, including the symbols used in Tables 2-5. 

The most common type of machine vision system is one which is responsible for examining situations two-dimensionally. These two-dimensional systems view a scene in much the same way that a person views a photograph. Cues such as shapes, shadows, textures, glares, and colors within the scene allow this type of vision system to be very good at making decisions based on what essentially amounts to a flat picture. 

If we consider a gas in equilibrium with adsorbed molecules on a solid surface, at sufficiently low gas pressures the adsorbed molecules will form a dilute two-dimensional system with negligible interactions. This is analogous to the perfect gas state in three dimensions. Under these conditions we are interested in the behavior of essentially independent adsorbed molecules, the only forces of interest being those between adsorbed molecules and the adsorbent. The interaction of an adsorbed molecule with the adsorbent is (in the usual approximation of additive forces) the sum of the separate interactions of the adsorbed molecule with the atoms or molecules of the adsorbent in the immediate neighborhood of the adsorbed molecule. 

The present chapter is an attempt at conceptual synthesis concerning smectite-water systems, designed to help sharpen the experimental issues that should be addressed in the next generation of laboratory investigations. The focus of our discussion is on Li-, Na-, and K-montmorillonite, based on the results of our previous simulation studies carried out separately for these homoionic smectites (16-18). The emphasis in our present analysis is on comparison with respect to the type of interlayer cation, in order to respond more definitively to the question of whether the aqueous phases in low-order montmorillonite hydrates are essentially two-dimensional ionic solutions (7, 2). 

The problem with all these types of theorems is that they only relate to two-dimensional systems because they essentially utilise the fact that the plane is divided into two parts by a closed, sufficiently smooth curve (Jordan theorem). Therefore it would be desirable to have theorems relating to three-... 

Diffusivities of binary, ternary and multi-component liquid crystalline mixtures, e.g. of soap (potassium laurate (PL), water [25, 58], and lipid (dipalmitoylphosphatidylcho-line (DPPC) [25, 59] systems in lamellar, hexagonal, cubic, nematic and micellar mesophases [25,60,61] have been studied extensively by pulsed-field-gradient NMR [25] and optical techniques [62], partly because of their intimate relation to the structure and dynamical performance of biological membranes [18]. The main distinction from thermotropic phases is that for layered structures a noticeable diffusion occurs only within the layers (i.e. lateral, frequently written as Dl, but in our notation DjJ, whereas it is negligibly small and difficult to detect across the layers [60-62] (transverse migration, for bilayers denoted by flip-flop ) so the mobility is essentially two dimensional, and the anisotropy ratio is so great that it is seldom specified explicit-... 

The notion of roughness/structural stability can be extended to the highdimensional case without any problem. However, some other problems do arise here when we need to find out explicitly the necessary and sufficient conditions for roughness. We have remarked that Andronov and Pontryagin, as well as Peixoto, had used the classification of proper two-dimensional systems in an essential way. So, we must stop here to get acquainted with some basic notions and facts from the general theory of dynamical systems. 

As we have seen above, the dynamics near the homoclinic loop to a saddle with real leading eigenvalues is essentially two-dimensional. New phenomena appear when we consider the case of a saddle-focus. Namely, we take a C -smooth (r > 2) system with an equilibrium state O of the saddle-focus saddle-focus (2,1) type (in the notation we introduced in Sec. 2.7). In other words, we assume that the equilibrium state has only one positive characteristic exponent 7 > 0, whereas the other characteristic exponents Ai, A2,..., are with negative real parts. Besides, we also assume that the leading (nearest to the imaginary axis) stable exponents consist of a complex conjugate pair Ai and A2 ... 

The model consists of a two dimensional harmonic oscillator with mass 1 and force constants of 1 and 25. In Fig. 1 we show trajectories of the two oscillators computed with two time steps. When the time step is sufficiently small compared to the period of the fast oscillator an essentially exact result is obtained. If the time step is large then only the slow vibration persists, and is quite accurate. The filtering effect is consistent (of course) with our analytical analysis. Similar effects were demonstrated for more complex systems [7]. 

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