Covering radius

Fig. 2. Shape space covering by a small spherical region of sequence space. In order to find (at least) one sequence for every common structure, it is only necessary to search a relatively small sphere around an arbitrarily chosen reference point in sequence space. For example, the covering radius for RNA molecules of chain length n= 100 was determined to be rc = 15 the covering sphere thus contains about 4 x 1024 sequences compared to 1.6 10 sequences in the entire sequence space.

For a porous medium represented by the phase function /g(r), we define the covering radius rc(r) of the solid phase as the radius of the largest sphere (or disk in 2D porous media) placed entirely into the solid phase and covering the point with coordinates r. The value of rc is zero inside the pores. The covering radius rc can be found by the morphological operation called opening, i.e., the erosion followed by the dilatation. The dilatation of the solid-phase domain A by the spherical element Br with radius r is the set A B, covered by all translations of Br centered in A,... 

The important property of the opening (A, r) of the solid-phase domain A by the sphere/disk Br of radius r is that it contains only points that can be covered by fir, because the components that were too small to contain Br were removed by an erosion. The covering radius can thus be defined as... 

Fig. 5. Distribution of covering radius rc. The solid phase of the porous medium is covered by circles of radius rc and the color of circles depends on circle radius. The integral distribution of covering radius G(rc) is then displayed.

Fig. V-5. The repulsive force between crossed cylinders of radius R (1 cm) covered with mica and immersed in propylene carbonate solutions of tetraethylammonium bromide at the indicated concentrations. The dotted lines are from double-layer theory (From Ref. 51).

Because values of 8 cover a rather small range, the three-dimensional scheme is often reduced to two dimensions, with polymers and solvents represented on — 6 coordinates with a solubihty circle of radius R. 

Large fragments were scattered in a circle of approximately 400 m (13(X) ft) radius. A shell of 28(X) kg (6000 lb) landed at a distance of 150 m (500 ft), and a fragment weighing 1000 kg (2200 lb) landed 250 m (820 ft) away. A large amount of carbon dioxide was released, causing the immediate vicinity of the yard to be covered with solid carbon dioxide (dry ice). 

The interfacial activity is determined by the sterical properties of the molecule. At the interface the spatial demand A0 of the hydrophobic part of the molecule is higher because of the second chain of the internal sulfonate compared with the terminal sulfonate. Thus, the surface concentration of the surfactant molecules is lower. That means that the hydrocarbon chains are laterally oriented and therefore cover the interface between the solution surface and air more completely. Because the ratio of the spatial demand of the head group to the volume of the alkyl chain governs the radius of the micellar surface, it... 

A radius of 128 pm corresponds to 1.28 X 10-8 cm, and the molar mass of copper (from the periodic table on the inside front cover) is 63.55 g-mol. The predicted density is therefore... 

Comparisons of the accuracy and efficiency for three numerical procedures, the direct summation, DC-FFT-based method and MLMI, are made in this section. The three methods were applied to calculating normal surface deformations at different levels of grids, under the load of a uniform pressure on a rectangle area 2a X 2fo, or a Hertzian pressure on a circle area in radius a. The calculations were performed on the same personal computer, the computational domain was set as -1.5a=Sx 1.5a and -1.5a=Sy 1.5a, and covered... 

It can be seen that the prediction 6a underestimates any results. This is because axial expansion is unrealistic, as indicated in Figure 4.2.8. On the other hand, prediction 5a covers almost all the results, except when the value of Vg is smaller than lOm/s. This is probably because of the usage of the mean pressure averaged over twice the radius of the vortex core in the model by Asato et al. [16], which is in quantitative agreement with the present vortex ring whose core diameter is about 25% the ring diameter. 

Bartle et al. [286] described a simple model for diffusion-limited extractions from spherical particles (the so-called hot-ball model). The model was extended to cover polymer films and a nonuniform distribution of the extractant [287]. Also the effect of solubility on extraction was incorporated [288] and the effects of pressure and flow-rate on extraction have been rationalised [289]. In this idealised scheme the matrix is supposed to contain small quantities of extractable materials, such that the extraction is not solubility limited. The model is that of diffusion out of a homogeneous spherical particle into a medium in which the extracted species is infinitely dilute. The ratio of mass remaining (m ) in the particle of radius r at time t to the initial amount (mo) is given by ... 

Flat plates are used as covers for manways, and as the channel covers of heat exchangers. Formed flat ends, known as flange-only ends, are manufactured by turning over a flange with a small radius on a flat plate, Figure 13.9a. The comer radius reduces the abrupt... 

Indeed, if a radius of 300 miles is drawn at the point of convergence of the Iranian, Turkish and Armenian borders, just south of Yerevan, the territory thusly covered would include the following list of cross-border and internal conflicts that have occurred during the 1990 s ... 

It is possible to derive a simple particle dissolution model where diffu-sional film thickness is not explicitly required. However, the boundary layer concentration profile derived from this model will extend for distances which cover an order of magnitude of the initial particle radius. 

Rutile, Ti02, which has the structure shown in Figure 7.8, is an important chemical that is used in enormous quantities as the opaque white material to provide covering ability in paints. Because the Ti4+ ion is quite small (56 pm), the structure of Ti02 has only six O2- ions surrounding each Ti4+, as predicted by the radius ratio of 0.39. Therefore, each Ti-O bond has an electrostatic bond character of 2/3 because the six bonds to (ions total the valence of 4 for Ti. There can be only three bonds from Ti4+ to each ()2 ion because three such bonds would give the total valence of 2 for oxygen (3 X 2/3 = 2). 

This potential was developed to ensure that the molecules inside the sphere never escape and maintain a fully solvated system during molecular dynamics. Here, es, Rs, ew and Rw are the van der Waals constants for the solvent and the wall and rj is the distance between the molecule i and the center of the water sphere, Ro is the radius of the sphere. The quantities A, B and Rb are determined by imposing the condition that W and dW/dr, vanish at r, = Ro. The restraining potential W is set to zero for r, < R0. The van der Waals parameters Es, ew, Rs and Rw can also be specifically defined for different solvents. The constants Awaii and Cwan are computed using a well depth of es = ew = 0.1 kcal and the radius of Rs = Rw = 1.25 A. For the other set of simulations, especially for the hydride ion transfer, we applied periodic boundary conditions by using a spherical boundary shell of 10.0 A of TIP3P40 water to cover the edges of the protein. 

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