Hyperfunctional Weights for Orthogonal Polynomials

The Chebychev polynomials associated to any given moments μ_{n ∞} ⁰ are formally orthogonal with respect to the formal δ-series $$w(x)= {\sum^\infty_0}(- 1)^{n}\mu_{{n}}\delta^{(n)}(x)/n!.$$ We show that this formal weight can be a true hyperfunctional weight if its Fourier transform is a slowly increasing holomorphic function in some tubular neighborhood of the real line. It provides a unifying treatment of real and complex orthogonality of Chebychev polynomials including all classical examples and characterizes Chebychev polynomials having Bessel type orthogonality.     

On the Reality of the Eigenvalues for a Class of -Symmetric Oscillators

 We study the eigenvalue problem with the boundary conditions that decays to zero as z tends to infinity along the rays , where is a real polynomial and . We prove that if for some we have for all , then the eigenvalues are all positive real. We then sharpen this to a larger class of polynomial potentials. In particular, this implies that the eigenvalues are all positive real for the potentials when with , and with the boundary conditions that decays to zero as z tends to infinity along the positive and negative real axes. This verifies a conjecture of Bessis and Zinn-Justin. Received: 16 January 2002 / Accepted: 1 May 2002 Published online: 6 August 2002     

Domain reversal behavior in perpendicular magnetic nanothin films

We report domain reversal behavior in perpendicular ferromagnetic nanothin films investigated by means of a novel magneto-optical microscope magnetometer, capable of grabbing domain reversal patterns in real time under an applied field as well as simultaneous measurements of 8000 local hysteresis loops with 400-nm special resolution. Three contrasting domain reversal behaviors are found to exist: wall-motion dominant, dendritic-growth dominant, and nucleation dominant reversal. Quantitative analysis reveals that the contrasting reversal behavior is mainly caused by a sensitive change in wall-motion speed and that the reversal ratio of wall-motion speed over nucleation rate is a governing parameter for the contrasting domain reversal dynamics. The activation volumes of the wall-motion and nucleation processes are found generally unequal, and the inequality is closely related with the domain dynamics. The domain reversal pattern is truly coincident with submicron-scale local coercivity variation and local switching time of domain evolution is exponentially dependent on local coercivity governed by a thermal activation relaxation process. The observed domain reversal behavior could be well predicted by a Monte Carlo simulation of a micromagnetic model based on the uniaxial magnetic anisotropy of nanothin films.     

The application of source methods in estimation of an interior sound field

In earlier studies, one has successfully developed three different source methods (SSM, similar source method; IPSM, internal parallel source method; ISM, internal source method) to estimate radiation and scattering sound fields. All these methods are estimations of exterior sound fields. In the present study, the similar source method is modified to estimate an interior sound field. The modification is to move the imaginary sources outside the boundary surface. In addition, general boundary conditions in terms of acoustic admittance or impedance are considered by introducing the suitable least square error functions. Finally, a two-dimensional interior sound field with alternative boundary conditions is evaluated to simulate sound fields inside a car. The results are in agreement with those of boundary integral method.