Combinatorial properties of finite geometric lattices

Let L be a finite geometric lattice of dimension n, and let w(k) denote the number of elements in L of rank k. Two theorems about the numbers w(k) are proved: first, w(k) ≥ w(1) for k = 2, 3, ..., n-1. Second, w(k) = w(1) if and only if k = n-1 and L is modular. Several corollaries concerning the "matching" of points and dual points are derived from these theorems.

Both theorems can be regarded as a generalization of a theorem of de Bruijn and Erdös concerning ʎ= 1 designs. The second can also be considered as the converse to a special case of Dilworth's theorem on finite modular lattices.

These results are related to two conjectures due to G. -C. Rota. The "unimodality" conjecture states that the w(k)'s form a unimodal sequence. The "Sperner" conjecture states that a set of non-comparable elements in L has cardinality at most max/k {w(k)}. In this thesis, a counterexample to the Sperner conjecture is exhibited.

Dynamic readout of subdiffraction-limited pit arrays with a silver superlens

Recent theoretical and experimental results suggested that the silver superlens could be constructed through controlling silver thin film thickness and preparation conditions, and applied in subdiffraction-limited optical imaging and optical lithography. In this work, we report another significant application of silver superlens-ultrahigh density optical data storage. With the silver superlens the subdiffraction-limited pit arrays on an optical disk are dynamically read out and the carrier-to-noise ratio can reach 25 dB for the thin film thickness of 46 nm. The readout laser power and readout velocity have little effect on the carrier-to-noise ratio. Additionally, in our experiment the silver thin film thickness needs to be controlled in the range from 20 to 80 nm.

On the dynamic readout of sub-diffraction-limit pit arrays with a silver thin film

Fast moving arrays of periodic sub-diffraction-limit pits were dynamically read out via a silver thin film. The mechanism of the dynamic readout is analysed and discussed in detail, both experimentally and theoretically. The analysis and experiment show that, in the course of readout, surface plasmons can be excited at the silver/air interface by the focused laser beam and amplified by the silver thin film. The surface plasmons are transmitted into the substrate/silver interface with a large enhancement. The surface waves at the substrate/silver interface are scattered by the sinusoidal pits of sub-diffraction-limit size. The scattered waves are collected by a converging lens and guided into the detector for the readout.

Nano-polypyrrole supercapacitor arrays prepared by layer-by-layer assembling method in anodic aluminum oxide templates

A novel method for preparing nano-supercapacitor arrays, in which each nano-supercapacitor consisted of electropolymerized Polypyrrole (PPy) electrode / porous TiO2 separator / chemical polymerized PPy electrode, was developed in this paper. The nano-supercapacitors were fabricated in the nano array pores of anodic aluminum oxide template using the bottom-up, layer-by-layer synthetic method. The nano-supercapacitor diameter was 80 nm, and length 500 nm. Based on the charge/discharge behavior of nano-supercapacitor arrays, it was found that the PPy/TiO2/PPy array supercapacitor devices performed typical electrochemical supercapacitor behavior. The method introduced here may find application in manufacturing nano-sized electrochemical power storage devices in the future for their use in the area of microelectronic devices and microelectromechanical systems.