Maximum curves and isolated points of entire functions

Given M(r; f) =maxjzj=r (jf(z)j) , curves belonging to the set of points M = fz : jf(z)j = M(jzj; f)g were de�ned by Hardy to be maximum curves. Clunie asked the question as to whether the set M could also contain isolated points. This paper shows that maximum curves consist of analytic arcs and determines a necessary condition for such curves to intersect. Given two entire functions f1(z) and f2(z), if the maximum curve of f1(z) is the real axis, conditions are found so that the real axis is also a maximum curve for the product function f1(z)f2(z). By means of these results an entire function of in�nite order is constructed for which the set M has an in�nite number of isolated points. A polynomial is also constructed with an isolated point.

Enhanced transmission in microwave arrays of periodic sub-wavelength apertures

In this paper we have conclusively proven that the "enhanced" optical transmission through a periodic array of sub-wavelength holes in metal films (Ebbessen's experiment) is the result of the array periodicity. This work has overturned the commonly accepted theory that the surface plasmons were responsible for the transmission enhancement. It was demonstrated that the reflectance, transmittance and frequency selectivity of the multilayered arrays can be efficiently modified by the aperture shapes.

Computing Zeros of Analytic Functions in the Complex Plane without using Derivatives

A new approach to evaluating all multiple complex roots of analytical function f(z) confined to the specified rectangular domain of complex plane has been developed and implemented in Fortran code. Generally f (z), despite being holomorphic function, does not have a closed analytical form thereby inhibiting explicit evaluation of its derivatives. The latter constraint poses a major challenge to implementation of the robust numerical algorithm. This work is at the instrumental level and provides an enabling tool for solving a broad class of eigenvalue problems and polynomial approximations.

Near-field distribution of optical transmission of periodic subwavelength holes in a metal film

Optical transmission of a two-dimensional array of subwavelength holes in a metal film has been numerically studied using a differential method. Transmission spectra have been calculated showing a significant increase of the transmission in certain spectral ranges corresponding to the excitation of the surface polariton Bloch waves on a metal surface with a periodic hole structure. Under the enhanced transmission conditions, the near-field distribution of the transmitted light reveals an intensity enhancement greater than 2 orders of magnitude in localized (similar to 40 nm) spots resulting from the interference of the surface polaritons Bragg scattered by the holes in an array.

Decompositions of spaces of measures

Let M be the Banach space of sigma-additive complex-valued measures on an abstract measurable space. We prove that any closed, with respect to absolute continuity norm-closed, linear subspace L of M is complemented and describe the unique complement, projection onto L along which has norm 1. Using this fact we prove a decomposition theorem, which includes the Jordan decomposition theorem, the generalized Radon-Nikodym theorem and the decomposition of measures into decaying and non-decaying components as particular cases. We also prove an analog of the Jessen-Wintner purity theorem for our decompositions.

On linear extension operators

We construct a countable-dimensional Hausdorff locally convex topological vector space $E$ and a stratifiable closed linear subspace $F$ subset of $E$ such that any linear extension operator from $C_b(F)$ to $C_b(E)$ is unbounded (here $C_b(X)$ stands for the Banach space of continuous bounded real-valued functions on $X$).

On effective linearly ordered sets of comparison functions

Both the existence and the non-existence of a linearly ordered (by certain natural order relations) effective set of comparison functions (=dense comparison classes) are compatible with the ZFC axioms of set theory.

Tailoring the AMC and EBG characteristics of periodic metallic arrays printed on grounded dielectric substrate

The artificial magnetic conductor (AMC) and electromagnetic band gap (EBG) characteristics of planar periodic metallic arrays printed on grounded dielectric substrate are investigated. The currents induced on the arrays are presented for the first time and their study reveals two distinct resonance phenomena associated with these surfaces. A new technique is presented to tailor the spectral position of the AMC operation and the EBG. Square patch arrays with fixed element size and variable periodicities are employed as working examples to demonstrate the dependence of the spectral AMC and EBG characteristics on array parameters. It is revealed that as the array periodicity is increased, the AMC frequency is increased, while the EBG frequency is reduced. This is shown to occur due to the different nature of the resonance phenomena and the associated underlying physical mechanisms that produce the two effects. The effect of substrate thickness is also investigated. Full wave method of moments (MoM) has been employed for the derivation of the reflection characteristics, the currents and the dispersion relations. A uniplanar array with simultaneous AMC and EBG operation is demonstrated theoretically and experimentally.