A coupled electron diffraction and rigid unit mode (RUM) study of the crystal chemistry of some zeotypic AlPO4 compounds

Electron diffraction and lattice dynamical calculations are used to investigate the unit cells, space group symmetries and inherent displacive flexibility of the room-temperature average structures of AlPO₄-8, AlPO₄-16 and AlPO₄-tridymite. The zero-frequency rigid unit modes (RUMs) of the idealized high-symmetry polymorphs thereof are also investigated along with their relationship to the lower-temperature polymorphism of these zeotypic aluminophosphates. The clear presence of satellite reflections in addition to the Bragg reflections (G) of the underlying Cmc2₁ parent structure in the case of AlPO₄-8 shows that the true unit cell of the room-temperature polymorph has a doubled c-axis due to a condensed RUM mode. Structured diffuse scattering is also observed which can be related to the thermal excitation of RUM modes. In the case of AlPO₄-tridymite, a complex F1 triclinic polymorph is observed and related to soft RUM modes while, in the case of AlPO₄-16, a soft q=0 RUM mode is shown to be responsible for an observed phase transition in the case of the all SiO₂ analogue of AlPO₄-16. A large number of additional zero-frequency RUM modes also exist in the case of AlPO₄-16.     

Construction of universal rotations from point-to-point transformations

For a desired range of offsets, universal rotations of arbitrary flip angle can be constructed based on point-to-point rotations of I(y) with half the flip angle. This approach allows, for example, creation of broadband or bandselective refocusing pulses from broadband or bandselective excitation pulses. Furthermore, universal rotations about any axis can be obtained from point-to-point transformations that can easily be optimized using optimal control algorithms. The construction procedure is demonstrated on the examples of a broadband refocusing pulse, a broadband 120(x) degrees rotation and a z-rotation with offset pattern.     

The first return time properties of an irrational rotation

If an ergodic system has positive entropy, then the Shannon-McMillan-Breiman theorem provides a relationship between the entropy and the size of an atom of the iterated partition. The system also has Ornstein-Weiss' first return time property, which offers a method of computing the entropy via an orbit. We consider irrational rotations which are the simplest model of zero entropy. We prove that almost every irrational rotation has the analogous properties if properly normalized. However there are some irrational rotations that exhibit different behavior.

     

On geometric interpretations of split quaternions

Quaternions are an important tool that provides a convenient and effective mathematical method for representing reflections and rotations in three-dimensional space. A unit timelike split quaternion represents a rotation in the Lorentzian space. In this paper, we give some geometric interpretations of split quaternions for lines and planes in the Minkowski 3-space with the help of mutual pseudo orthogonal planes. We classified mutual planes with respect to the casual character of the normals of the plane as follows; if the normal is timelike, then the mutual plane is isomorphic to the complex plane; if the normal is spacelike, then the plane is isomorphic to the hyperbolic number plane (Lorentzian plane); if the normal is lightlike, then the plane is isomorphic to the dual number plane (Galilean plane).     

The Vectorial Parameterization of Rotation

The parameterization of rotation is the subject of continuous research and development in many theoretical and applied fields of mechanics, such as rigid body, structural, and multibody dynamics, robotics, spacecraft attitude dynamics, navigation, image processing, and so on. This paper introduces the vectorial parameterization of rotation, a class of parameterization techniques encompassing many formulations independently developed to date for the analysis of rotational motion. The exponential map of rotation, the Rodrigues, Cayley, Gibbs, Wiener, and Milenkovic parameterization all are special cases of the vectorial parameterization. This generalization parameterization sheds additional light on the fundamental properties of these techniques, pointing out the similarities in their formal structure and showing their inter-relationships. Although presented in a compact manner, all of the formulae needed for a complete implementation of the vectorial parameterization of rotation are included in this paper.     

On a formulation for anisotropic elastoplasticity at finite strains invariant with respect to the intermediate configuration

The paper is concerned with a formulation of anisotropic finite strain inelasticity based on the multiplicative decomposition of the deformation gradient F=F_eF_p. A major feature of the theory is its invariance with respect to rotations superimposed on the inelastic part of the deformation gradient. The paper motivates and shows how such an invariance can be achieved. At the heart of the formulation is the mixed-variant transformation of the structural tensor, defined as the tensor product of the privileged directions of the material as given in a reference configuration, under the action of F_p. Issues related to the plastic material spin are discussed in detail. It is shown that, in contrast to the isotropic case, any flow function formulated purely in terms of stress quantities, necessarily exhibits a non-vanishing plastic material spin. The possible construction of spin-free rates is discussed as well, where it is shown that the flow rule must then depend not only on the stress but on the strain as well.     

Flow due to non-coaxial rotation of a porous disk and a second grade fluid rotating at infinity

An exact solution for the three-dimensional flow due to non-coaxial rotation of a porous disk and a second grade fluid at infinity is obtained. It is shown that for uniform suction or uniform blowing at the disk, an asymptotic profile exists for the velocity distribution. The velocity depends on two parameters: one of them is the suction parameter or blowing parameter and the other is the visco-elastic parameter. Furthermore, it is found that when the value of the visco-elastic parameter is fixed, the velocity decreases with an increase in the value of the suction parameter and when the value of the suction parameter is fixed, the velocity increases with an increase in the value of the visco-elastic parameter.     

Structure and methyl groups internal rotation of difluorodimethylsilane

The rotational spectra of the normal and ¹³C isotopomers of difluorodimethylsilane have been measured by molecular beam Fourier-transform microwave (FTMW) and free-jet absorption millimeter-wave (FJ-AMMW) spectroscopy. Each rotational transition is split into several component lines due to the two methyl groups internal rotations. The corresponding V₃ barrier has been determined to be 5.256(3) kJ mol⁻¹. A r₀ structure of the heavy atoms skeleton has been obtained. The silicon atom lies—accidentally—on the center of mass of the molecule.