Astroid

Abstract: Axisymmetric toroidal wave fronts are pertinent to the near forward and backward scattering by objects that have rotational symmetry. For spheres, the wave field produced by such a wave front is known as glory scattering. As the wave front propagates, some porton of it becomes focused on an axis, forming a structurally unstable line caustic. A specific class of harmonic perturbations of the wave front shape is considered that leads to unfoldings of the axial caustic. When the wave front shape is perturbed to have two‐fold rotational and mirror symmetry, the unfolded caustic is a four‐cusped astroid curve. The three‐fold symmetric perturbed wave front propagates to produce a hypocycloid caustic with three cusps. In general, perturbed wave fronts with p‐fold rotational and mirror symmetry have caustics of cusped stars, with p cusps when p is odd, and 2p cusps when p is even. These wave front perturbations have applications to scattering from symmetric, slightly nonspherical, homogeneous objects such as spheroids. Wave fields are computed using a Fresnel approximation of the diffraction integral. The wave field patterns associated with astroid caustics are displayed. They have features similar to Pearcey patterns. Applications to backscattering from spheroids, distorted torii, and axicon reflectors are noted. Certain inverse problems are considered. An inequality is given for determining the magnitude of wave front perturbations needed to cause a significant change in the wave field from that of a spherical scatterer. The merging of rays as the observation direction moves across the caustic is discussed using concepts from catastrophe optics. A novel expression in polar coordinates is given for the Hessian associated with propagation.

Abstract: Under a perpendicular spin-transfer-torque (STT) polarizer and in-plane (IP) bias magnetic field, three-dimensional instability thresholds of the critical strength of STT and the field are derived from an effective one-dimensional free energy. The modified astroids derived from the STT and IP fields are quite different from the classical Stoner-Wohlfarth astroid. We find that the STT breaks the symmetry of the astroid seriously when the orientation of bias field is along the easy and hard axes. In particular, the modified astroid not only separates the region with two stable states from the region with only one stable state, but also delimits the region with a dynamical stable state (no stable equilibrium state) when the amplitude of the STT is larger than a critical value required to switch the magnetization at zero-bias field. Finally, the nucleation field, coercivity, and precessional critical field for uniaxial anisotropy are rigorously determined including the effect of the STT, and the hysteresis loops for various orientations of a bias field are computed and discussed.

Abstract: The Hessian topology has just begun to be developed (in connection with the study of parabolic curves on smooth surfaces in Euclidean or projective space), in contrast to the symplectic and contact topologies related to it. For instance, it is not known how many (compact) parabolic curves can belong to the graph of a polynomial of a given (even of the fourth) degree in two variables or to a smooth algebraic surface of a given degree. The astroid is a hypocycloid with four cusp points. A hyperbolic polynomial is a homogeneous polynomial whose second differential has the signature (+,–) at any non-zero point. Hyperbolic polynomials and functions are connected with Morse theory and Sturm theory and with hypocycloids via caustics (and wave fronts) of periodic functions. The astroid is the caustic of the cosine of a double angle. The caustic of any periodic function has at least four cusp points, and if there are four of them, as is the case for the astroid, then these points form a parallelogram. The theory developed in this paper, based on the study of envelopes and inequalities between derivatives of smooth functions, proves that hyperbolic polynomials of degree four form a connected set and those of degree six form a disconnected set. These topological generalizations of the Sturm and Hurwitz theorems about the zeros of Fourier series give algebraic-geometric results on caustics and wave fronts as well and also establish relationships between these results and the Morse theory of anti-Rolle functions (whose zeros alternate with those of their derivatives).

Abstract: The crossover of the hysteresis branches in the classical coherent rotation model is analyzed. The analysis is extended when the truncated Stoner-Wohlfarth astroid is used. The physical explanation of the crossover occurrence is given. More accurate expressions of the anisotropy free energy density are required to better describe the behavior of the system when the angle between the easy axis and the magnetic moment of the particle is near 90/spl deg/. A simple approximate formula for the hysteresis loop branches, which are in good agreement with those obtained when the discussed corrections are made, is given. The effect of the use of the Stoner-Wohlfarth astroid in the calculus of the equilibrium state in some vector Preisach-type models is analyzed.