Domain shape invariant

Figure 7.10 Time-domain representation of an original speech signal (top), and of its time-stretched version (bottom), showing the loss of shape-invariance. Phase-vocoder...

Time-domain representation of a speech signal showing shape invariance 305... 

In Table 10.2 below we summarize the calculated values of the shape invariant r 2//m and aspect ratio rm/lm of string-like domains in different fe crystals formed under the fdb conditions. The experimental parameters used for these calculations are U = 3kV (applied voltage), R = 50 nm (typical radius of curvature of the tip apex), d = 0.5 nm (distance between the tip apex and the sample surface). The calculated aspect ratios rm/lm are quite different for the chosen fe. 

Table 10.2 Shape invariant and aspect ratio of domains grown under FDB effect in different FE materials...

Figure 5.16 A direction-independent SGIM characterization of a space curve C, regarded as a molecular backbone. On the left-hand side the shape globe S of radius R is shown enclosing the space curve C. The centre of the sphere is chosen as the centre of mass of chain molecule C. On the right-hand side the shape invariance domains of the sphere are shown, as defined by the knot types derived from the projections. There are only two knot types in this example unknots and trefoil knots.

Local Shape Invariance of Density Domains and the Transfer... 

Ranges DD Shape Invariance Domains of the Configuration Space... 

These domains represent a partitioning of M, hence the union of these DD shape invariance domains is the entire nuclear configuration space M,... 

The numerical value of the reference curvature b can be specified in absolute units or in units scaled relative to the size of the object G(a). If absolute units are used, then a relative convexity characterization of G(a) involves size information if an object G(a) is scaled twofold, then its shape remains the same, but with respect to a fixed, nonzero b value a different relative convexity characterization is obtained. That is, the pattern of relative shape domains Do(b)> D (b), and D2(b) defined with respect to some fixed, nonzero reference curvature value b (b K)) is size-dependent. On the other hand, if the reference curvature b is specified with respect to units proportional to the size of G(a), then a simple. scaling of the object does not alter the pattern of relative shape domains with respect to the scaled reference curvature b. In this case, the shape characterization is size-invariant, that is, a "pure" shape characterization is obtained. 

The resulting curvature domains Do(bK)> D (bK). and D2(bK) are not invariant with respect to the size of the G(a) objects (this size is dependent on the contour parameter a), nevertheless, the scaling is specific for the size of the nuclear arrangement K, hence these shape domains provide a valid shape comparison of MIDCO s or other molecular surfaces of molecules of different sizes. This approach is simpler than the fully size-invariant approach using the reference curvature be, where a new scaling factor r(G(a)) is required for each new MIDCO G(a). 

If a different reference curvature value b is chosen, then the shape matrix may be different, although the matrix is invariant within small enough intervals of the b values. As examples, the three different shape matrices s(0.0l,0), s(0.01,0.005), and s(0.01,-0.008) of the three shape domain partitionings of the ally I alcohol MIDCO G(O.OI) shown in Figure 5.6 are given below. The index... 

If the entire range of curvature parameter b is considered, then a list of the finite number of distinct shape matrices and those curvature values bj where a change of the shape matrix occurs, gives a detailed, numerical shape characterization of the MIDCO surface G(a). In the most general case of variations in the two parameters a and b, as well as in the nuclear configuration K, one can study the dynamic shape space invariance domains, the (a,b)-maps, and various projections of the invariance domains of shape matrices, following the principles [158] applied for the shape group invariance domains of the dynamic shape space D. 

Figure 5.9 The construction of Shape Globe Invariance Maps (SGIM s), of MIDCO relative convexity shape domain patterns for two reference curvature values, b = 0 and b < 0.

A practical implementation of the above approach is the following a global shape property of the molecule is assigned to each point of the sphere S, followed by the determination of those domains of S where this shape property is invariant. A pair of examples is shown in Figure 5.9, where the shape globe invariance domains of a MIDCO surface for two relative convexity shape domain partitionings (P) with respect to two reference curvatures, b = 0, and b < 0, are given. As... 

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