Principal coordinates

Note that 0" < A< 60". The invariants A , and form a cylindrical coordinate system relative to the principal coordinates, with axial coordinate / A equally inclined to the principal coordinate axes, with radial coordinate /3t, and with angular coordinate The plane A" = 0 is called the II plane. Because the principal values can be ordered arbitrarily, the representation of A through its invariants in n plane coordinates has six-fold symmetry. 

In general, two related techniques may be used principal component analysis (PCA) and principal coordinate analysis (PCoorA). Both methods start from the n X m data matrix M, which holds the m coordinates defining n conformations in an m-dimensional space. That is, each matrix element Mg is equal to q, the jth coordinate of the /th conformation. From this starting point PCA and PCoorA follow different routes. 

Principal coordinate analysis (PCoorA) [37], on the other hand, operates on the square n X n matrix, reflecting the relationships between the conformations. The... 

Figure 8 A joint principal coordinate projection of the occupied regions in the conformational spaces of linear (Ala) (triangles) and its conformational constraint counterpart, cyclic-CAla) (squares), onto the optimal 3D principal axes. The symbols indicate the projected conformations, and the ellipsoids engulf the volume occupied by the projected points. This projection shows that the conformational volume accessible to the cyclic analog is only a small subset of the conformational volume accessible to the linear peptide, (Adapted from Ref. 41.)...

CM Becker. Principal coordinate maps of molecular potential energy surfaces. J Comput Chem 19 1255-1267, 1998. 

Figure 4 The energy landscape of the pnon protein (Pi P) (residues 124-226) in vacuum, obtained by principal coordinate analysis followed by the minimal energy envelope procedure. Two large basins are seen. One basin is associated with the native Pi P conformation the other is associated with partially unfolded conformations.

Ethical assessment made by the principal/coordinating investigator... 

Principal coordinates analysis 31.6.1 Distances defined from data... 

Principal coordinates analysis (PCoA) is applied to distance tables rather than to original data tables, as is the case with principal components analysis (PCA). 

Using D as input we apply principal coordinates analysis (PCoA) which we discussed in the previous section. This produces the nxn factor score matrix S. The next step is to define a variable point along they th coordinate axis, by means of the coefficient kj and to compute its distance d kj) from all n row-points ... 

Starting from the transition state it was expected the reaction would evolve either forward to the products or backward to the reactants. During the unconstrained CPMD simulations, however, the system was always found to evolve towards the reactants. Because of this it was necessary to apply constrained dynamics to the principal coordinate reaction (the distance between WAT oxygen and GTP y-phosphorus) this enabled investigation of the system evolution towards the products (Fig. 2.7). 

The lower path is somewhat more complicated. The first step in the path involves either PCA (83) or principal-coordinate analysis (PCO) (83). This step can be followed by optimization of a function that minimizes the error between the proximity measure computed in the reduced-dimension and full coordinate systems if desired. Xie et al. (84) recently published an interesting paper along these lines. Kruscal stress (79) is a widely used function in this regard... 

Stokes Postulates Stokes s postulates provide the theory to relate the strain-rate to the stress. As a result the forces may be related to the velocity field, leading to viscous-force terms in the Navier-Stokes equations that are functions of the velocity field. Working in the principal coordinates facilitates the development of the Stokes postulates. 

There is always a particular set of coordinates, called the principal coordinates, for which the shear components vanish the strain-rate tensor can be written as... 

Developing the stress-strain-rate relationships is greatly facilitated in the principal coordinate directions. Since isotropy requires that the constitutive relationships be independent of coordinate orientation, the principal-direction relationships can be transformed to any other coordinate directions. At every point in a flow field the strain-rate and stress state... 

The principal coordinates provide an extraordinarily useful conceptual framework within which to develop the fundamental relationships between stress and strain rate. For practical application, however, it is essential that a common coordinate system be used for all points in the flow. The coordinate system is usually chosen to align as closely as possible with the natural boundaries of a particular problem. Thus it is essential that the stress-strain-rate relationships can be translated from the principal-coordinate setting (which, in general, is oriented differently at all points in the flow) to the particular coordinate system or control-volume orientation of interest. Accomplishing this objective requires developing a general transformation for the rotation between the principal axes and any other set of axes. 

Perhaps surprisingly, it turns out that the complex series of operations represented by Eq. 2.151 leads to a relatively simple result that is independent of the particular principal-coordinate directions. The stress tensor in a given coordinate system is related to the strain-rate tensor in the same coordinate system as... 

In this equation the velocity components align with the (z, r, 6) coordinates, and not the principal coordinates. 

In the principal coordinates, of course, there are only three nonzero components of the stress and strain-rate tensors. Upon rotation, all nine (six independent) tensor components must be determined. The nine tensor components are comprised of three vector components on each of three orthogonal planes that pass through a common point. Consider that the element represented by Fig. 2.16 has been shrunk to infinitesimal dimensions and that the stress state is to be represented in some arbitrary orientation (z, r, 6), rather than one aligned with the principal-coordinate direction (Z, R, 0). We seek to find the tensor components, resolved into the (z, r, 6) coordinate directions. 

As an illustration, we determine the stress components on the z face illustrated in Fig. 2.18. The cosines of the three angles between each of the principal coordinates... 

Fig. 2.18 Relationship at a point between an arbitrary coordinate system (z, r, 6) that is rotated relative to the principal coordinates (Z, R, ). The z-face plane is shown for the purposes of illustrating the three components of the stress tensor on that face.

When rotated into the principal coordinates, all the shear stresses vanish, and it is reasonable to think of the average normal compressive stress as a pressure. It is apparent from Eq. 2.183, however, that the average compressive stress is not equal to the thermodynamic pressure p as evaluated from an equation of state. Stokes made this interesting observation and recognized its concomitant dilemma in his famous 1845 paper. He hypothesized that the dilemma could be resolved by assuming that... 

The final objective of this chapter was to develop quantitative relationships between a fluid s strain-rate and stress fields. Expressions for the strain rates were developed in terms of velocities and velocity gradients. Then, using Stokes s postulates, the stress field was found to be proportional to the strain rates and a physical property of the fluid called viscosity. The fact that the stress tensor and strain-rate tensor share the same principal coordinates is an important factor in applying Stokes s postulates. The stress-strain-rate relationships are fundamental to the Navier-Stokes equations, which describe conservation of momentum in fluids. 

Using scalar and vector products among the eigenvectors (direction cosines), demonstrate that the principal coordinates remain orthogonal. 

Referring to Fig. A.2, assume that the principal coordinates align with z, r, and O. The unit vectors (direction cosines) just determined correspond with the row of the transformation matrix N. Thus, if the principal stress tensor is... 

To solve the diffusion equation in the principal coordinate system (i.e., Eq. 4.61), the Cartesian space can now be stretched or contracted along the principal axes by scaling ... 

A consequence of Neumann s symmetry principle is that direct tensor Onsager coefficients (such as in the diffusivity tensor) must be symmetric. This is equivalent to the addition of a center of symmetry (an inversion center) to a material s point group. Thus, the direct tensor properties of crystalline materials must have one of the point symmetries of the 11 Laue groups. Neumann s principle can impose additional relationships between the diffusivity tensor coefficients Dij in Eq. 4.57. For a hexagonal crystal, the diffusivity tensor in the principal coordinate system has the form... 

The rotation, R, required to rotate the original (xi,X2) coordinate system into the principal coordinate system must satisfy... 

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