Axial symmetry

Abstract: An overview of the recent developments in the field of cylindrical vector beams is provided. As one class of spatially variant polarization, cylindrical vector beams are the axially symmetric beam solution to the full vector electromagnetic wave equation. These beams can be generated via different active and passive methods. Techniques for manipulating these beams while maintaining the polarization symmetry have also been developed. Their special polarization symmetry gives rise to unique high-numerical-aperture focusing properties that find important applications in nanoscale optical imaging and manipulation. The prospects for cylindrical vector beams and their applications in other fields are also briefly discussed.

Abstract: A large number of investigations has been devoted to the problem of the formation and the development of a crack during brittle fracture of solids. The first of these was the well-known work of Griffith [l] devoted to the determination of the critical length of a crack at a given load, i.e. the length of a crack at which it begins to widen catastrophically. Assuming an elliptical form of a crack forming in an infinite body subjected to an infinitely homogeneous tension, Griffith obtained an expression for the critical length of a crack as that corresponding to the total.of the full increase in energy (equal to the sum of the surface energy plus the elastic energy released due to the formation of the crack).

Abstract: The structure of certain axially symmetric circulations in a stably stratified, differentially heated, rotating Boussinesq fluid on a sphere is analyzed. A simple approximate theory [similar to that introduced by Schneider (1977)] is developed for the case in which the fluid is sufficiently inviscid that the poleward flow in the Hadley cell is nearly angular momentum conserving. The theory predicts the width of the Hadley cell, the total poleward heat flux, the latitude of the upper level jet in the zonal wind, and the distribution of surface easterlies and westerlies. Fundamental differences between such nearly inviscid circulations and the more commonly studied viscous axisymmetric flows are emphasized. The theory is checked against numerical solutions to the model equations.

Abstract: A theory of the energy states and the electromagnetic transitions between them is developed for nuclei which do not possess axial symmetry. It is shown that violation of axial symmetry does not significantly change the rotational states of axial nuclei and leads to the appearance of new energy states. The reduced probabilities for E2 and M1 transitions between various rotational states are computed.

Abstract: The propagation of free harmonic waves along a hollow circular cylinder of infinite extent is discussed within the framework of the linear theory of elasticity. A characteristic equation appropriate to the circular hollow cylinder is obtained by use of the Helmholtz potentials for arbitrary values of the physical parameters involved. Axially symmetric waves, the limiting modes of infinite wavelength, and a special family of equivoluminal modes are derived and discussed as degenerate cases of the general equations.