Fixed-point-free motions

Fixed-Point-Free Motions. These include translations, screw rotations, and glide reflections. Because the primitive translation vector, Eq. 1.2, joins any two lattice points, an equivalent statement is that Eq. 1.2 represents the operation of translational symmetry bringing one lattice point into coincidence with another. However, we must choose the basis vectors (a, b, c) so as to include all lattice points, thus defining a... 

Defect structures of this type may be further subdivided according as to whether the mobile atoms are free to move without restraint or are restricted in their motion to rotation about a fixed point. 

The central region of the local mode representation of the phase space trajectories is called the resonance region. In this resonance region of phase space the trajectories ( 3 and 4) are not free to explore the full 0 < ip < n range and are threfore classified as normal mode trajectories. Points A and B are fixed points which he at the maximum and minimum E extremes, Ea(I) and Eb(I), of the resonance region for a particular value of I. Point A at Iz = 0 (vr = vi) and ip = 7t/2 (out-of-phase motion of the R and L oscillators) is stable and corresponds to a pure antisymmetric stretch. Point B at Iz = 0 (vr = vl) and ip = 0 and 7r (in-phase motion) is unstable (because it lies on the separatrix) and corresponds to a pure symmetric stretch. Quasiperiodic trajectories that circulate about a stable fixed point resemble the fixed point periodic trajectory. At E > Ea(I) no trajectories of any type can exist. At E < Er(I) the B-like trajectories vanish and are replaced by trajectories that circulate about the Ca, Cb fixed points and are therefore C-like. The Ca and Cb lines (Iz = / = 2, 0 < ip < 7r) are actually the north and south poles on the local mode polyad phase sphere (Fig. 9.13(c)). The stable fixed points he near the poles and trajectories la and 2a circulate about the fixed point near Ca and trajectories lb... 

Y, and Z are connected by bonds of fixed length joined at fixed valence angles, that atoms W, X, and Y are confined to fixed positions in the plane of the paper, and that torsional rotation 0 occurs about the X-Y bond which allows Z to move on the circular path depicted. If the rotation 0 is "free such that the potential energy is constant for all values of 0, then all points on the circular locus are equally probable, and the mean position of Z, i.e., the terminus of , lies at point z. The mean vector would terminate at z for any potential function symmetric in 0 for any potential function at all, except one that allows absolutely no rotational motion, the vector will terminate at a point that is not on the circle. Thus, the mean position of Z as seen from W is not any one of the positions that Z can actually adopt, and, while the magnitude ll may correspond to some separation that W and Z can in fact achieve, it is incorrect to attribute the separation to any real conformation of the entity W-X-Y-Z. Mean conformations tiiat would place Z at a position z relative to the fixed positions of W, X, and Y have been called "virtual" conformations.i9,20it is clear that such conformations can never be identified with any conformation that the molecule can actually adopt... 

Altmann remarked that, in free space, the Euclidean operations are not of physical interest. Therefore the Euclidean operations will be 2issimilated with the identity. Besides, he stated that the discrete symmetry operations are purely changes of labelling, especifically they are not motions of atoms. The Schrodinger subgroup may be then assimilated with the symmetry point group of the molecule in a fixed configuration. 

In a body-powered device, the jjerson uses his or her own muscular power to operate the prosthesis, usually via a cable link called a Bowden cable (Fig. 32.4). A Bowden cable consists of two parts, an outer housing and an inner tension cable. The housing is fixed at both ends and serves as a flexible bridge between two points, maintaining a constant length regardless of any motion. The cable is free to slide within the housing. 

The SPH particles in the previous simulations had been spatially fixed, so that the method was used in a similar way as a mesh-based approach. However, the main advantages of a mesh-free method rely in its Lagrangian way of view with discretisation points moving according to the fluid motion. We will shortly explain the underlying algorithm and numerical schemes. For a deeper insight, we refer to... 

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