Invariance domains

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. 

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... 

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.

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 nuclear configuration space M is provided with a metric that allows one to introduce a measure of volume V for various subsets of M. The relative importance of a given property can be characterized by the volume of the subset of M where this property is present. The relative importance of a given shape as specified by the associated matrix DfXj) can be expressed by volumes V Mo,k) of invariance domains Mojr... 

From this inequality, we can conclude that there exist nontrivial steady states inside the invariant domain. If Bi, i = 0, two nontrivial steady states merge, resulting in one steady state, a saddle node. If the initial conditions satisfy the inequality xother hand, the initial condition satisfies the inequality. v >y, all... 

We now show how the average number density can be calculated from the sample paths in prediscretized (time-invariant) domains of the particle state space. This calculation is considerably easier for the case where particle state does not vary with time during the quiescence period. Denote the... 

Broadly, the self-similar solution identifies invariant domains in the space of the independent variables along which the solution remains the same or contains a part that is the same. Consider, for example, the number density function / (x, t) that may satisfy a population balance equation such as (3.2.8) or (3.3.5). By a self-similar solution of either of these equations we mean one to be of the form... 

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