Browsing by Author "Dr. Mehmet Ozturk, Committee Member"

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    Automating the Enumeration of Sequences Defined by Digraphs
    (2005-09-20) D'Souza, Erwin Francis; Dr. Carla Savage, Committee Chair; Dr. Jon Doyle, Committee Member; Dr. Erich Kaltofen, Committee Member; Dr. Mehmet Ozturk, Committee Member
    We consider sequences of nonnegative integers $S=(s_1,s_2,ldots, s_n)$ defined by systems of constraints represented as weighted directed graphs in which edge $(s_a,s_b)$ of weight $w$ indicates constraint $s_a~geq~s_b~+~w$. We propose a set of seven rules and a decomposition technique for obtaining multivariate and single-variable generating functions for families of such graphs. Our method compares to existing techniques by offering an elegant and intuitive approach to obtaining generating functions and recurrences, albeit only for a subset of the partition and composition enumeration problems addressed by other techniques. The decomposition technique we propose remains relevant, nevertheless, to a wide range of applications, including several well-known ones. Moreover, our objective is to obtain recurrences for generating functions so as to assist the formulation and proof of their closed-form solutions. For integer sequences defined by directed graphs with $w in 0,1]$, we prove that our technique holds sufficient. We describe the formulation of finite-variable generating function recurrences from multi-variable ones and provide a set of rules to determine the variables chosen. The construction tree is introduced as the tree representation of the construction of a generating function from the decomposition of a weighted directed graph. Given such a construction tree, automation of the process of building the multi-variable and finite-variable recurrences is possible, and we implement it as a computer program. Finally, we apply our techniques and tools to a wide range of famous problems, including 2-rowed plane partitions, up-down compositions and hexagonal plane partitions, as well as some new problems, and obtain recurrences to each. We find that our methods are not only effective but also easy and simple to use.
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    Implantable Devices for a Retinal Prosthesis: Design and Electromagnetic and Thermal Effects
    (2009-12-16) Singh, Vinit; Dr. Gianluca Lazzi, Committee Chair; Dr. Zhilin Li, Committee Member; Dr. Griff Bilbro, Committee Member; Dr. Mehmet Ozturk, Committee Member
    A retinal prosthesis, wherein electrical stimulation is provided to the retina of a person inflicted with outer retinal degenerative diseases such as Retinitis Pigmentosa and Age-related Macular degeneration, has been clinically tested and has succeeded in providing limited vision; such as shape recognition. It is hoped that an increased electrode count will improve visual acuity. The retinal prosthesis considered in this work is a dual-unit system with an external (outside the human body) unit and an internal (inside the body) unit with a wireless link for power and data transfer between them. Such a system poses possible health risks due to the incident electromagnetic energy of the wireless link and the power dissipated by the internal components, particularly the processing chip which drives the electrodes responsible for eliciting a neural response from the retina. Tissue damage via heating is one the primary concerns for such a system making it necessary to obtain via simulation and in-vivo and in-vitro experiments, accurate estimates of thermal elevation due to the operation of the such devices. In this work, numerical methods have been developed to compute temperature increases and electromagnetic effects due to the prosthesis components in anatomically correct human head models. The explicit and the Alternating-Direction Implicit (ADI) Finite-Difference Time-Domain (FDTD) have been used. Further, a hybrid explicit-ADI method was developed for the heat equation which provided simulation speedup of more than 10x over conventional methods for the models considered. FDTD methods were employed to compute the induced current densities and Specific Absorption Rate (SAR) in the human head due the inductive link comprising the primary coil (external) and a secondary coil (internal). Different orientations of the primary coil were considered in a frequency range of 1 MHz-20 MHz to provide guidelines for choosing eventual frequency and power parameters to conform to international safety standards. A novel displacement field excitation method was used for the spiral primary coil and verified with analytical results. In an effort to reduce the size of the internal unit and to allow integration of a patch antenna (for a separate data link), and the active devices on a single substrate, a 3-D trench inductor geometry was investigated. To enable patterning of structured surface, a custom experimental setup was designed and maintained to process a positive tone PEPR2400 electro-depositable photoresist.