Shape convexity

In this respect, Kuipers made an important point (as illustrated in Fig. 3.10c), namely that layers of thickness x which cover the support to a fraction 6, have the same dispersion as hemispheres of radius 2 x, or spheres with a diameter 3x. Even more interesting is the fact that these three particle shapes with the same surface-to-volume ratio give virtually the same fp/fs intensity ratio in XPS when they are randomly oriented in a supported catalyst The authors tentatively generalized the mathematically proven result to the following statement that we quote literally For truly random samples the XPS signal of a supported phase which is present as equally sized but arbitrarily shaped convex particles is determined by the surface/volume ratio. Thus, in Kuipers model the XPS intensity ratio fp/fs is a direct measure of the dispersion, independent of the particle shape. As the mathematics of the model is beyond the scope of this book, the interested reader... 

As columns are overloaded for preparative work, peak shape often deviates from the Gaussian shape typical of analytical work. In preparative work, the peaks can assume a triangular shape because the adsorption isotherm is nonlinear. A typical isotherm is shown in Figure 6-37, where CM is the concentration of sample in the mobile phase and Cs is the concentration of sample in the stationary phase. At low concentration of sample (CM) there is a linear adsorption isotherm which results in Gaussian peak shapes. At a point when either the sample adsorption in the stationary phase or the sample solubility in the mobile phase becomes limited, the isotherm becomes nonlinear, assuming either a convex or a concave shape. Convex isotherms are the most common and result in peak tailing. Conversely, concave isotherms cause fronting of the peaks. 

Anisotropic dissolution of crystal surfaces results in the formation of surface contour whose geometric features depend on the crystal orientation [81]. During steady-state etching the etched surface profile exhibits a characteristic shape convex or concave [160]. A convex surface will be bounded by fast etching planes while a concave surface will be bounded by slow etching planes. 

Plots of Z vs. t for various soil reactions and other solid-fluid processt generally behave in a more complex fashion. They are S-shaped convex i small t, concave at large t, and linear at some intermediate range of t. Fc example, the plot of Z against t for the sorption of phosphate by a Typ Dystrochrept soil depicted in Fig. 1-6 illustrates this S-shaped behavior. Th property of the Z (Z) plot suggests that the kinetics of soil reactions ofte obey some complex function that can be approximated by Eq. [12] at sma t, by Eq. [13] at an intermediate t, and by Eq. [14] at large t. 

Briefly, these models assume that the orientation angles of differential sample surface elements in space are of equal probability. This assumption allows the use of geometric probability for deriving the signal ratios for supported particles of different shape. It was found that, for truly random samples, the XPS signal of a supported phase which is present as equally sized but arbitrarily shaped convex particles is determined by their surface/volume ratio (and hence dispersion). In other words, the surface/volume ratios found for the supported compounds can be interpreted in terms of dimensions of particles of arbitrary shape.. Hence, the way in which XPS sees1 at supported catalysts is similar to that in which the reactants do. 

Description Herbaceous perennial with tuber-like roots. Stems 70-130 cm tall, simple or branched. Leaf blades circular-cordate in outline, 5-9 cm long, 8-12 cm wide, palmatisect with 5 segments divided to the base segments pinnatifid with 2 or 3 linear lobes, lobes 3-5 mm wide. Inflorescence an apical raceme. Flowers irregular. Sepals 5, petaloid, violet upper sepal hood-shaped, convex, with a long beak. Petals 2, each with a spur. Fruit a follicetum with 3 foUicles. Seeds 4-5 mm long. 

FIG. 6-2 Bony shapes. At this articulation, an ovoid shape sits in a sellar shape. Convex and concave surlaces are shown. 

The sacroiliac articulations are kidney-shaped and convex ventrally. The sacral and iliac articulations seem to match in a crescent-shaped, convex-concave arrangement, but this is not true for the joints entire bony relationship. Horizontal sections from various levels of the sacroiliac articulation show that the convex-concave relationship exists only at the upper and middle portions. In the lower portion, the relationship is variously described as a flattened, planar joint or a reverse, concave-convex relationship (Fig. 57-1) anatomists differ in their descriptions of the sacroiliac articulations. 

The method has been extended to mixtures of hard spheres, to hard convex molecules and to hard spherocylinders that model a nematic liquid crystal. For mixtures m. subscript) of hard convex molecules of the same shape but different sizes. Gibbons [38] has shown that the pressure is given by... 

This result enables us to investigate the extreme crack shape problem. The formulation of the last one is as follows. Let C Hq 0, 1) be a convex, closed and bounded set. Assume that for every -0 G the graph y = %j) x) describes the crack shape. Consequently, for a given -0 G there exists a unique solution of the problem... 

Consider the problem of finding the extreme crack shapes. The setting of this problem is as follows. Let C be a convex, closed and... 

In transforming bis-ketone 45 to keto-epoxide 46, the elevated stereoselectivity was believed to be a consequence of tbe molecular shape — tbe sulfur ylide attacked preferentially from tbe convex face of the strongly puckered molecule of 45. Moreover, the pronounced chemoselectivity was attributed to tbe increased electropbilicity of the furanone versus the pyranone carbonyl, as a result of an inductive effect generated by tbe pair of spiroacetal oxygen substituents at tbe furanone a-position. ... 

Most thermoplastics can be printed some thermosets. Handles flat, concave, or convex surfaces, including round or tubular shapes. 

It is important to note here that if an element does not radiate directly to any part of its own surface, the shape factor with respect to itself, F]t, Fu and so on, is zero. This applies to any convex surface for which, therefore, Fu =0. 

A second approach considers that the regions of equivalent parameter values must enclose parameters for which the loss function is nearly the same or at any rate less different than some threshold. In other words, the equivalence regions should take the form 015(0) < c 5(6) for some appropriate constant of. Note that in this case the shape of the regions would not necessarily be ellipsoidal, or even convex In fact, we might postulate in general the existence of multiple minima surrounded by disjoint equivalence neigh-... 

In water, Scatchard plots showed clear concave-shaped curves whatever the pectin origin (figure 4A). Nevertheless, differences between sugar-beet and citrus pectins appeared in presence of ionic strength. While citrus pectins exhibited convex-shaped curves whatever the metal ion, sugar-beet pectins display convexe curvature for Cu2+ and Pb2+ but concave-shaped curves for the other three cations (figure 4B, in the case of Ni2+). 

An anticooperative mode of interactions was assumed in case of concave-shaped Scatchard plots, as alrea% proposed by other authors (Mattai Kwak, 1986 Gamier et al, 1994). A convexe curvature of the plots indicated a cooperative binding process (figure 4). 

Flexural strength is determined using beam-shaped specimens that are supported longways between two rollers. The load is then applied by either one or two rollers. These variants are called the three-point bend test and the four-point bend test, respectively. The stresses set up in the beam are complex and include compressive, shear and tensile forces. However, at the convex surface of the beam, where maximum tension exists, the material is in a state of pure tension (Berenbaum Brodie, 1959). The disadvantage of the method appears to be one of sensitivity to the condition of the surface, which is not surprising since the maximum tensile forces occur in the convex surface layer. 

Figure 7.4 Influence of nanorod shape on its optical extinction properties, as simulated using the discrete dipole approximation, (a) different aspect ratios, fixed volume, (b) fixed aspect ratio, variable volume, (c) aspect ratio and volume fixed, variable end cap geometry, (d) convexity of...

Unlike solid electrodes, the shape of the ITIES can be varied by application of an external pressure to the pipette. The shape of the meniscus formed at the pipette tip was studied in situ by video microscopy under controlled pressure [19]. When a negative pressure was applied, the ITIES shape was concave. As expected from the theory [25a], the diffusion current to a recessed ITIES was lower than in absence of negative external pressure. When a positive pressure was applied to the pipette, the solution meniscus became convex, and the diffusion current increased. The diffusion-limiting current increased with increasing height of the spherical segment (up to the complete sphere), as the theory predicts [25b]. Importantly, with no external pressure applied to the pipette, the micro-ITIES was found to be essentially flat. This observation was corroborated by numerous experiments performed with different concentrations of dissolved species and different pipette radii [19]. The measured diffusion current to such an interface agrees quantitatively with Eq. (6) if the outer pipette wall is silanized (see next section). The effective radius of a pipette can be calculated from Eq. (6) and compared to the value found microscopically [19]. 

Formation of products and intermediate species, as well as disappearance of reactants during the photocatalytic reactions can be discerned by the evolution of positive (i.e., concave shape) bands and negative (i.e., convex shape) bands, respectively. 

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