Local geometrical measures

The word shape is loaded with strong personal intuitions of what each of us thinks defines an object in three-dimensional (3D) space. Sometimes, we may view an object as a distribution of points in space and conceive its shape in terms of distances and angles, that is, local geometrical measures. More often, we associate the notion of shape with a solid object defined by an envelope surface. In this case, a precise geometrical characterization may lose its relevance, and it may be sufficient to characterize shape in a more qualitative fashion. Global topological measures are better suited here. Therefore, we... 

Figure 3 Radon concentration in the indoor air of detached houses. The size of the symbol indicates the magnitude of the local geometric mean. Measurements have been made in about 4,450 houses in 183 localities.

As for the WCA system, the fraction of sixfold-ordered particles in DRPs (0.547 0.004) is significantly less than the fraction of six-coordinated particles (/g = 0.7861 0.0018), confirming that the sixfold bond orientational order parameter is a more sensitive indicator of local geometrical disorder than is the coordination number. The average size of ordered clusters in DRPs is (s) = 30.4 0.8, and the normalized average number of ordered clusters is Ac/N = 0.0180 0.0005. These values are comparable to those measured in the dense WCA liquid near freezing (see Figs. 52 and 53). 

The force measured between crossed cylinders (F ), as in the SEA, and between spheres (Fg), as in the MASIF, a distance D apart is normalized by the local geometric mean radius (R). This quantity is related to die fiee energy of interaction per unit area between flat surfaces (W) according to the Deijaguin approximation (30) ... 

Figure 18 Force normalized by local geometric mean radius as a function of surface separation between glass across a concentrated emulsion solution (20 wt % oil and 1.2 wt % phospholipid). The thinner lines correspond to the force measured on separation the dashed line represents the calculated force between two spherical surfaces connected by a capillary condensate in the full equilibrium case [Eq. (25)] and the dotted line represents the force between two spherical surfaces connected by a capillary condensate in the non-equilibrium case [Eq. (26)]. (From Ref. 81, with permission.)...

Barsky, B.A. The Beta-spline a local representation based on shape parameters and fundamental geometric measures. Ph.D. dissertation, university of Utah, Decem.ber. 1981. 

Measure Wall Thickness This window is used for the dialog to calibrate the algorithm aceording formula (3) and for point wise measurements after calibration. The row Ideal indicates the nominal wall thickness used, IQI indicates the wall thickness values used for calibration and the detected optical density. Local can be used for noise reduction and compensation of geometric effects. 

Since both the temperature dependence of the characteristic ratio and that of the density are known, the prediction of the scaling model for the temperature dependence of the tube diameter can be calculated using Eq. (53) the exponent a = 2.2 is known from the measurement of the -dependence. The solid line in Fig. 30 represents this prediction. The predicted temperature coefficient 0.67 + 0.1 x 10-3 K-1 differs from the measured value of 1.2 + 0.1 x 10-3 K-1. The discrepancy between the two values appears to be beyond the error bounds. Apparently, the scaling model, which covers only geometrical relations, is not in a position to simultaneously describe the dependences of the entanglement distance on the volume fraction or the flexibility. This may suggest that collective dynamic processes could also be responsible for the formation of the localization tube in addition to the purely geometric interactions. 

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