Some central limit theorems for doubly restricted partitions

Let P_{K, L}(N) be the number of unordered partitions of a positive integer N into K or fewer positive integer parts, each part not exceeding L. A distribution of the form

Ʃ/N≤x P_K,L(N)

is considered first. For any fixed K, this distribution approaches a piecewise polynomial function as L increases to infinity. As both K and L approach infinity, this distribution is asymptotically normal. These results are proved by studying the convergence of the characteristic function.

The main result is the asymptotic behavior of P_K,K(N) itself, for certain large K and N. This is obtained by studying a contour integral of the generating function taken along the unit circle. The bulk of the estimate comes from integrating along a small arc near the point 1. Diophantine approximation is used to show that the integral along the rest of the circle is much smaller.

A high-speed image sensing technique with adjustable frame rate based on an ordinary CCD

In the sinusoidal phase modulating interferometer technique, the high-speed CCD is necessary to detect the interference signals. The reason of ordinary CCD's low frame rate was analyzed, and a novel high-speed image sensing technique with adjustable frame rate based on ail ordinary CCD was proposed. And the principle of the image sensor was analyzed. When the maximum frequency and channel bandwidth were constant, a custom high-speed sensor was designed by using the ordinary CCD under the control of the special driving circuit. The frame rate of the ordinary CCD has been enhanced by controlling the number of pixels of every frame; therefore, the ordinary of CCD can be used as the high frame rate image sensor with small amount of pixels. The multi-output high-speed image sensor has the deficiencies of low accuracy, and high cost, while the high-speed image senor with small number of pixels by using this technique can overcome theses faults. The light intensity varying with time was measured by using the image sensor. The frame rate was LIP to 1600 frame per second (f/s), and the size of every frame and the frame rate were adjustable. The correlation coefficient between the measurement result and the standard values were higher than 0.98026, and the relative error was lower than 0.53%. The experimental results show that this sensor is fit to the measurements of sinusoidal phase modulating interferometer technique. (c) 2007 Elsevier GmbH. All rights reserved.

The oculomotor system: (1) Vertical-horizontal interaction and signal recognition, (2) Time delays and power spectra

In the first section of this thesis, two-dimensional properties of the human eye movement control system were studied. The vertical - horizontal interaction was investigated by using a two-dimensional target motion consisting of a sinusoid in one of the directions vertical or horizontal, and low-pass filtered Gaussian random motion of variable bandwidth (and hence information content) in the orthogonal direction. It was found that the random motion reduced the efficiency of the sinusoidal tracking. However, the sinusoidal tracking was only slightly dependent on the bandwidth of the random motion. Thus the system should be thought of as consisting of two independent channels with a small amount of mutual cross-talk.

These target motions were then rotated to discover whether or not the system is capable of recognizing the two-component nature of the target motion. That is, the sinusoid was presented along an oblique line (neither vertical nor horizontal) with the random motion orthogonal to it. The system did not simply track the vertical and horizontal components of motion, but rotated its frame of reference so that its two tracking channels coincided with the directions of the two target motion components. This recognition occurred even when the two orthogonal motions were both random, but with different bandwidths.

In the second section, time delays, prediction and power spectra were examined. Time delays were calculated in response to various periodic signals, various bandwidths of narrow-band Gaussian random motions and sinusoids. It was demonstrated that prediction occurred only when the target motion was periodic, and only if the harmonic content was such that the signal was sufficiently narrow-band. It appears as if general periodic motions are split into predictive and non-predictive components.

For unpredictable motions, the relationship between the time delay and the average speed of the retinal image was linear. Based on this I proposed a model explaining the time delays for both random and periodic motions. My experiments did not prove that the system is sampled data, or that it is continuous. However, the model can be interpreted as representative of a sample data system whose sample interval is a function of the target motion.

It was shown that increasing the bandwidth of the low-pass filtered Gaussian random motion resulted in an increase of the eye movement bandwidth. Some properties of the eyeball-muscle dynamics and the extraocular muscle "active state tension" were derived.

Computational Microscopy: Breaking the Limit of Conventional Optics

Computational imaging is flourishing thanks to the recent advancement in array photodetectors and image processing algorithms. This thesis presents Fourier ptychography, which is a computational imaging technique implemented in microscopy to break the limit of conventional optics. With the implementation of Fourier ptychography, the resolution of the imaging system can surpass the diffraction limit of the objective lens's numerical aperture; the quantitative phase information of a sample can be reconstructed from intensity-only measurements; and the aberration of a microscope system can be characterized and computationally corrected. This computational microscopy technique enhances the performance of conventional optical systems and expands the scope of their applications.

Verificação à fadiga de pontes rodoviárias mistas (aço-concreto)

As pontes rodoviárias mistas (aço-concreto) estão sujeitadas às ações dinâmicas variáveis, por exemplo, devido ao tráfego de veículos sobre a superfície irregular do pavimento. Estas ações dinâmicas podem gerar a nucleação de fraturas ou mesmo a sua propagação sobre a estrutura. A correta consideração desses aspectos objetivou o desenvolvimento de uma metodologia de análise, com a finalidade de avaliar os níveis dos esforços e tensões oriundos do tráfego dos veículos sobre a superfície irregular do pavimento e, bem como, proceder uma verificação à fadiga de obras de arte rodoviárias em aço e mistas (aço-concreto). Para tal, as técnicas para a contagem de ciclos de tensão e a aplicação das regras de dano acumulado foram analisadas através de curvas do tipo S-N, associadas a diversas normas de projeto. A ponte rodoviária mista (aço-concreto) investigada neste estudo é constituída por quatro vigas de aço longitudinais e por um tabuleiro de concreto armado. O modelo numérico-computacional, desenvolvido para a análise dinâmica da ponte, foi concebido com base em técnicas usuais de discretização através do método dos elementos finitos. Simulam-se as almas das vigas de aço e as lajes de concreto do tabuleiro através de elementos finitos de casca. As mesas dessas vigas, transversinas e os enrijecedores são modelados por elementos de viga tridimensionais. Os veículos são representados a partir de sistemas "massa-mola-amortecedor". O tráfego dessas viaturas é considerado mediante a simulação de comboios semi-infinitos, deslocando-se com velocidade constante sobre a ponte. As conclusões da presente investigação versam acerca da vida útil de serviço dos elementos estruturais de pontes mistas (aço-concreto).

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.