Dyadic points

Because we could have carried out the analysis after applying a few levels of refinement, the discontinuity found in this way can, in principle, be found at limit points corresponding to control points at any level of refinement. These points are called dyadic points. They are dense, but not as dense as the rationals, let alone the reals. Between any two of them, however close, you can find a point which is not a dyadic. 

In the particular case shown as an example here, at the mark points constructed in this way other than at abscissae of original vertices, the fourth difference turns out to be zero, and so the discontinuities do not occur except at limit points corresponding to original control points. That ties up with our knowledge of B-splines, but it is a very special property. In general, subdivision schemes give limit curves with discontinuities of some derivative at all dyadic points. 

It then turns out that the discontinuities found at such points may be totally different from those at the dyadic points. We can always make schemes of higher arity by considering two or more refinements as a single step. We call this squaring or taking a higher power of the scheme. 

There is an apparent hole in this argument. We are using only dyadic points for looking at the convergence. How do we know that all differences in between also contract As in section 14.1.2 above, the answer is in the enclosure property. If the first differences contract, then a long enough sequence of consecutive first differences to define a span must also contract, and enclosure then says that the span itself must contract23. 

This shows that the limit curve is continuous, not just at a mark point, or at dyadic points, but at all points of the curve. 

Because of the fractal nature of the definition, it is not possible in general to evaluate the subdivision curve exactly, except at dyadic points with a small denominator. We have to settle for an approximation within some required tolerance. Because that tolerance can be chosen in the light of the precision needs of the application, this is indeed good enough. 

Each of the methods described under rendering above can be applied directly to evaluation. The third, Hermite, form is probably most relevant to applications requiring high accuracy. In fact where the second derivative can also be evaluated exactly at dyadic points, a quintic Hermite interpolant can be used to give an even higher rate of approximation. 

It should be pointed out that after coordinate normalization, functions of interest are evaluated in the closed intervals [0, 1], rather than in [—oo, oo] or other intervals. In this case, some modified interpolation functions can be constructed to interpolate the values in dyadic points outside [0, 1] to the desired interval (6,25). 

Other classically chaotic scattering systems have been shown to have repellers described by a symbolic dynamics similar to (4.10). One of them is the three-disk scatterer in which a point particle undergoes elastic collisions on three hard disks located at the vertices of an equilateral triangle. In this case, the symbolic dynamics is dyadic (M = 2) after reduction according to C)V symmetry. Another example is the four-disk scatterer in which the four disks form a square. The C4 symmetry can be used to reduce the symbolic dynamics to a triadic one based on the symbols 0,1,2), which correspond to the three fundamental periodic orbits described above [14]. 

This equation shows that the stress contribution tensor is essentially a dyadic product of the end-to-end vector r and the statistical force /, which is exerted by the chain on the considered end-point. The angular brackets indicate the averaging with the aid of the mentioned distribution function. Eq. (2.25) can be explained as follows Factor rt in the brackets gives the probability that the mentioned statistical force actually contributes to the stress. This factor gives the projection of the end-to-end vector of the chain on the normal of the considered sectional plane. If a unit area plane is considered, as is usual in stress-analysis, the said projection gives that part of the unit of volume, from which molecules possessing just this projection, actually contribute to the stress on the sectional plane. 

The hypothesis of small deformations means that c/.v. the change in die displacement vector when we go from P tu the neighboring point Q, is very small compared m dr. the position vector of Q relative to P. Consequently, the scalar components of the dyadic Vs arc al) very snlull compared lo unity. The geometrical meaning of ihe dyadic Vs is obtained by separating it into its symmetric part S = j(Vs + sV) and iis antisymmetric part R = - I x (V x si. where 1 is the unity dyadic. The antisymmetric part is interpreted as follows if at some point M the symmetric part vanishes, ilien we have for die neighborhood ul M the relation... 

The rotational motion of the rigid set of mass points about any axis through its center of mass in the absence of exterior forces is known as the free rotation of the rigid body. The planar moment tensor for this motion, with the position vectors ra referred to an arbitrary basis system, can be compactly written as a dyadic (T denotes transposition) [8,32],... 

Consider the centers of the identical spherical particles of radii a to be instantaneously located at the lattice points R . As such, the simplest geometric state exists, in which only one particle is contained within each unit cell. When the latter suspension is sheared, the three basic lattice vectors 1( (1 = 1,2, 3) (or, equivalently, the dyadic L) become functions of time t. Under a homogeneous deformation, the lattice composed of the sphere centers remains spatially periodic, although its instantaneous spatially periodic configuration necessarily changes with time. 

The terms in the expansion derived from derivatives with respect to X are identical to those obtained by taking the corresponding derivatives of the charge density p(X) itself. The dyadic Wp is the Hessian matrix of p, whose eigenvectors and eigenvalues determine the properties of the critical points in the charge distribution. The trace of this term is the Laplacian of the charge density, V p. 

The derivatives with respect to x sample the off-diagonal behaviour of F > and generate terms related to the current density j and the quantum stress tensor er. The first-order term is proportional to the current density, and this vector field is the x complement of the gradient vector field Vp. The second-order term is proportional to the stress tensor. Considered as a real symmetric matrix, its eigenvalues and eigenvectors will characterize the critical points in the vector field J and its trace determines the kinetic energy densities jK(r) and G(r). The cross-term in the expansion is a dyadic whose trace is the divergence of the current density. 

A further striking example is given by the Pyrococcus abyssi Sm core (PA Sm). In the free state and in complex with RNA, the PA Sm has point symmetry 72 and consists of a sandwich of two heptameric rings in the same orientation and dyadically related [20]. In the two states, the folding of the monomers differs only slightly, so that the corresponding molecular forms of PA Sm heptamer (central hole and envelope) are the same. 

All the preceding equations are valid for any choice of origin. If the coupling and rotation dyadics are known at any point O they may be computed at any other point P by means of the origin displacement theorems (B22)... 

For centrally symmetric bodies such as spheres, ellipsoids, and the like, it is intuitively obvious that the rotation and coupling dyadics will adopt their simplest and most symmetrical forms when expressed in terms of an origin at the center of symmetry. It is natural, therefore, to inquire as to the existence of a corresponding point for a body of arbitrary shape. Such an inquiry, based on Eqs. (51) and (52), discloses the fact (B22) that every body possesses a unique point, termed its center of reaction (R), at which the coupling dyadic is symmetric, i.e.. 

It is at this point that the dyadics assume their most physically significant forms. In general, this point does not coincide with the centroid of the body, though it does for centrally symmetric bodies. If the coupling dyadic is known at any point O the location of the center of reaction can be determined from the expression ... 

If a body possesses three mutually perpendicular planes of reflection symmetry, its center of reaction lies at the point of intersection of these planes. The coupling dyadic is zero at this point, whereas the translation dyadic and rotation dyadic at R adopt the forms shown in Eqs. (44) and (45), in which the principal axes of translation and rotation (at R) coincide and lie normal to the three symmetry planes. An ellipsoid is an example of such a body [see Eqs. (58)-(60)]. 

Only six coefficients are required to characterize the coupling dyadic at the center of reaction. But then an additional three scalars are required to specify the location of this point, so that the total number of independent scalars required for a complete characterization is still nine. Similarly, three scalars suffice for the translation dyadic if we refer them to the principal axes of translation [see Eq. (44)], but then three additional scalars (e.g., an appropriate set of Eulerian angles) are required to specify the orientations of these axes. So it comes down to the same thing—namely, that six scalars are required. The same is true of the rotation dyadic at any point, and of the coupling dyadic at the center of reaction. 

If the particle is a solid of revolution its center of reaction lies along the axis (say the Rxy axis). At this point the coupling dyadic vanishes, while both the translation and rotation dyadics at any point along the axis each adopt the general form... 

If a body possesses two distinct axes of helicoidal symmetry they must intersect. The point of intersection is then the center of reaction of the body. The three resistance dyadics are then isotropic at R ... 

Ko are the translation and coupling dyadics for the particle in the unbounded fluid. Also, Wo is the wall-effect dyadic (B20, C20) evaluated at the point in the fluid presently occupied by O. In general, is a constant, symmetric dyadic which depends only upon the size and shape of the boundaries and upon the location of O relative to the bounding walls. In particular, Wp is independent of yt, Uq, co, of the size and shape of the particle, and of the orientation of the particle relative to the boundaries. At each point O in space it is, therefore, an intrinsic geometric property of the container boundaries... 

This rapid rate of attenuation of the velocity field is characteristic of all bodies for which the coupling dyadic vanishes at the center of reaction, providing that the body rotates about an axis through this point. However, it is only for axially symmetric bodies rotating about their symmetry axes that such motions may be stable. 

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