Two-dimensional sketching

Fig. 9. Two-dimensional sketch of the 3N-dimensional configuration space of a protein. Shown are two Cartesian coordinates, xi and X2, as well as two conformational coordinates (ci and C2), which have been derived by principle component analysis of an ensemble ( cloud of dots) generated by a conventional MD simulation, which approximates the configurational space density p in this region of configurational space. The width of the two Gaussians describe the size of the fluctuations along the configurational coordinates and are given by the eigenvalues Ai.

Figure 8. Two-dimensional sketch of water management in a PEM fuel cell whereby the membrane—electrode assembly separates the anode feed channel from the cathode, and a diagram of water uptake profiles in anode and cathode channels.

Two-dimensional sketch of a grain boundary. The crowding at atom positions indicated by S can be relieved if they are occupied by small atoms. Open positions, indicated by L, attract large atoms. 

Fig. 51. Hypothetical two dimensional sketches of three different polymer network structures...

Figure 2. (a) Two-dimensional sketch of the most general wedge-shaped confinement that... 

Consider a molecular dynamics simulation of a system consisting of K particles in a cubic box of volume V=L. Periodic boundary conditions are typically used to minimize surface effects. Periodic boundary condi-tions correspond to densely filling space with identical copies of the simulation box (see Fig. 9.28 for a two-dimensional sketch of periodic boundary conditions). As particles leave the simulation box from one side, they automatically reenter it from the opposite side. 

Figure 8 Two-dimensional sketch illustrating the MIF filtering step in ALMOND. MIFs for the N1 probe (hydrogen bond donor, green) and the O probe (hydrogen bond acceptor, red), top, are processed at the same energy level, e.g., —2 kcal/mol, to yield a few hundred representative nodes, bottom. The grey lines illustrate distances used in the auto- (Nl-Nl) and cross- (Nl-O) correlograms by the MACC-2 encoding algorithm. (See color plate at end of chapter.)...

The next step is to come up with a bookkeeping scheme to keep all the (x, y, z) coordinate information connected to the identity of each atom. The easiest approach may be to write the (x, y, z) value as a coordinate triple on the same pieces of paper used for the two-dimensional sketch of the biopolymer, right next to each atom. This associates the (x, y, z) value with the atom it is attached to. 

Fig. 8.2. A two-dimensional sketch of a curved surface advancing through a material that provides a simple basis for interpreting the second and third terms on the right side of (8.2).

When carbon forms four single bonds, they are arranged tetrahedrally around the carbon atom the molecular geometry is tetrahedral (Fig. 21.1). Recall from Chapter 13 that it is not possible to represent this three-dimensional shape accurately in a two-dimensional sketch. Thus the four bonds radiating from each carbon atom in... 

Let s examine this relationship between two-dimensional sketches and three-dimensional molecules a bit more carefully. Compare the following two Lewis diagrams for C3HgBrCl and the ball-and-stick diagrams beneath them ... 

To begin the first meeting, the physicists made two sketches of the disc diyer machine on the whiteboard (see Figure 6) that visualized mechanical features physical processes and production data parameters. The upper sketch illustrates a crude three-dimensional representation of the machine. The physicists understood the internal physical construction of the dryer to be a big cavity containing rotating discs product inlet outlet and the internal distribution of product. Beneath the 3D sketch, we see a two dimensional sketch where the round discs are lepiesented with lines separating the dryer s interior into discrete volumes. The physicists attached... 

Figure 2.13. Symmetrised two-dimensional INADEQUATE experiment with isopinocampheol (2) [ CDshCO, 250 mg in 0.3 ml, 25 °C, 50 MHz, 256 scans and exp.], (a) Stacked plot of the section between 8c = 20.9 and 48.2 (b) complete contour plot with cross signal pairs labelled a-k for the 11 CC bonds of the molecule to facilitate the assignments sketched in formula 2...

When using air curtains the edge effects are neglected and the flow is treated as two-dimensional. The different parts of a two-dimensional jet are sketched in Fig. 10.63. 

Here the constant C takes care of the relative importance of the second derivative influence. Instead of solving a front problem in the coordinates (x,t) (physical space) we perform the calculations in the computational space (C t). For one dimensional problems this adaptive grid transformation proved to be very successful. We can perform a transformation in a similar spirit for a two dimensional domain (x,y,t) -> A general sketch of this transformation... 

What is the feasible region for x given the following constraints Sketch the feasible region for the two-dimensional problems. 

Suppose you are given the task of preparing a ternary (three-component) solvent system such that the total volume be 1.00 liter. Write the equality constraint in terms of x X2, and Xj, the volumes of each of the three solvents. Sketch the three-dimensional factor space and clearly draw within it the planar, two-dimensional constrained feasible region. (Hint try a cube and a triangle after examining Figure 2.16.)... 

Fig. 16. Bifurcation diagram of temporal dissipative structures, c (maximal amplitude of the oscillation minus the homogeneous steady-state value) is sketched versus B for a two-dimensional system with zero flux boundary conditions. The first bifurcation occurs at B = Bn and corresponds to a stable homogeneous oscillation. At B, two space-dependent unstable solutions bifurcate simultaneously. They become stable at B a and Bfb. Notice that as it is generally the case Bfa Bfb.

FIG. 1.2 A simplified sketch of a hypothetical protein molecule embedded in a bilayer (a biological membrane). The bilayer shown is a two-dimensional cross section of a membrane. The bundle of cylinders shown represents the helices of a protein. The cylinders are part of the same protein and are joined together by other segments (not shown) of the protein protruding out of the bilayer on either side. 

Identification of area as the two-dimensional equivalent of volume is a straightforward geometrical concept. That tt should be interpreted as the two-dimensional equivalent of pressure is not so evident, however, even though the notion was introduced without discussion in Chapter 6, Section 6.6. Figure 7.3 helps to clarify this equivalency as well as suggest how to compare quantitatively two- and three-dimensional pressures. The figure sketches a possible profile of the air-water surface with an adsorbed layer of amphipathic molecules present. In... 

FIG. 7.6 Composite two-dimensional pressure ir versus area a isotherm, which includes a wide assortment of monolayer phenomena. Note that the scale of the figure is not uniform so that all features may be included on one set of coordinates. The sketches of the surfactants show the orientations of the molecules in each phase at various stages of compression. 

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