On the order of a class of meromorphic functions

This paper proves the following result: Letf(z) be a meromorphic function in thez-plane with a deficient value, and δ(θ _k )(k=1,2, ...,q;0≤θ ₁<θ₂<...<θ _q<θ _q+1=θ ₁+2π) beq rays (1≤q<∞) starting at the origin, and letn≥3 be an integer such that for any given positive numberε,0<ε<π/2, $$\overline {\mathop {\lim }\limits_{r \to \infty } } \frac{{\log ^ + n\left\{ { \cup _{k = 1}^q \Omega \left( {\theta _k + \varepsilon ,\theta _{k + 1} - \varepsilon ,r} \right),f\prime f^n = 1} \right\}}}{{\log r}} \leqslant v< \infty ,$$ whereΝ is a constant independent ofε. IfΜ<∞, then we have $$\lambda \leqslant \frac{\pi }{\omega } + v,$$ whereΜ andλ denote the lower order and order off(z), respectively,Ω=minθ _k+1 ?θ _k ;1≤k≤q, andn(E, f=a) is the number of zeros off(z)?a inE with multiple zeros being counted with their multiplicities.     

Full wave electromagnetic simulation of electrooptic high-speed probes

This paper outlines the process of modelling external electrooptic probes mounted on coplanar waveguides (CPW). The techniques used to describe the electromagnetic properties of the external probes are the finite difference time domain (FDTD) method and the finite difference transmission line method (FDTLM). These full wave techniques are time domain in nature that must be Fourier transformed to describe important frequency domain characteristics such as scattering parameters. The optical retardation of the probe is related to the full wave analysis through a commonly used grid size that results in a generalized basis for a complete electrooptic system analysis including a unique definition of an electrooptic transfer function,H(). Following the field simulation, the properties of the probe (invasiveness, retardation, signal distortion and group delay) are presented. Procedures for optimizing models for probe tips are discussed and related to a first-order model that has been developed. The results indicate that these probes can be simulated successfully on moderately sized Unix work stations and that the optimization of probe models must include the full wave simulation in the definition of the necessary gradients for the optimization process.     

A Class of Bilinear forms on Dirichlet type Spaces

This paper characterizes the boundedness of certain bilinearforms on Dirichlet type spaces. The main tool is the Littlewood-Paleytheory. Applications to the preduals of the related functionspaces including BMO and Bloch spaces, and the boundedness ofcertain commutators are also indicated.