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Thesis (Ph.D.)--Auburn University, 2006. / Abstract. Vita. Includes bibliographic references (ℓ.28-29)

The aim of this research is to develop real-time object recognition systems which are robust and have good discrimination. An important aspect of this work is the development of a rotationally invariant optical correlator. Optical correlation systems are investigated for the purpose of high speed, high discriminant and distortion invariant pattern recognition. A photorefractive joint transform correlator (JTC) using Bismuth Silicon Oxide (BSO) as a non-linear recording medium and a liquid crystal television as a spatial light modulator is implemented. The underlying physics is considered, some specific techniques to improve the operation are proposed. The properties of photorefractive BSO are investigated for use as the dynamic holographic recording medium in information processing systems. The moving grating technique is used for edge-enhanced image reconstruction and for making the correlation peak sharper. The object and frequency space filtering methods are presented to improve the correlation performance, the discrimination, and to realise distortion invariant pattern recognition. Circular harmonic matched filters and phase-only filters with different expansion orders are involved in the photorefractive JTC for real-time rotationally invariant pattern recognition. These filters can also be used to track an object with different orientations. The coherent triple joint transform correlator employs a third beam to modify the Fourier spectrum and hence improves the correlation performance. In the incoherent triple JTC, the wavelet transform is used in the Fourier domain to achieve a high signalto-noise ratio, noise robustness as well as discrimination. Several wavelet functions are also used, after processing, in the conventional JTC for high-speed image feature extraction. The wavelet transform functions can also be used in the JTC with circular harmonic filters to improve the output quality of rotation invariant pattern recognition.

This thesis will examine Relative Local Cohomology. First we extend many well known theorems about Local Cohomology of finitely generated modules with respect to an ideal of a commutative noetherian rings so that they hold for non-finitely generated modules with respect to certain ideals of non-commutative non-noetherian rings. Then we show how similar results hold for Relative Local Cohomology. In particular we provide a relative version of the Local Duality Theorem. We then examine the links between Relative Homological Algebra and the concept of Structure Theorems and give a bound on the Castelnuovo-Mumford Regularity of rings of invariants based on the Cech Complex.

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Title: Algebraic Structures for Knot Coloring Author: Martina Vaváčková Department: Department of Algebra Supervisor: doc. RNDr. David Stanovský, Ph.D., Department of Algebra Abstract: This thesis is devoted to the study of the algebraic structures providing coloring invariants for knots and links. The main focus is on the relationship between these invariants. First of all, we characterize the binary algebras for arc and semiarc coloring. We give an example that the quandle coloring invariant is strictly stronger than the involutory quandle coloring invariant, and we show the connection between the two definitions of a biquandle, arising from different approaches to semiarc coloring. We use the relationship between links and braids to conclude that quandles and biquandles yield the same coloring invariants. Keywords: knot, coloring invariant, quandle, biquandle iii

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This dissertation consists of three more or less independent projects. In the first project, we find the microstates free entropy dimension of a large class of L1[0; 1]{ circular operators, in the presence of a generator of the diagonal subalgebra. In the second one, for each sequence {cn}n in l1(N), we de fine an operator A in the hyper finite II1-factor R. We prove that these operators are quasinilpotent and they generate the whole hyper finite II1-factor. We show that they have non-trivial, closed, invariant subspaces affiliated to the von Neumann algebra, and we provide enough evidence to suggest that these operators are interesting for the hyperinvariant subspace problem. We also present some of their properties. In particular, we show that the real and imaginary part of A are equally distributed, and we find a combinatorial formula as well as an analytical way to compute their moments. We present a combinatorial way of computing the moments of A*A. Finally, let fTkg1k =1 be a family of *-free identically distributed operators in a finite von Neumann algebra. In this paper, we prove a multiplicative version of the Free Central Limit Theorem. More precisely, let Bn = T*1T*2...T*nTn...T2T1 then Bn is a positive operator and B1=2n n converges in distribution to an operator A. We completely determine the probability distribution v of A from the distribution u of jTj2. This gives us a natural map G : M M with u G(u) = v. We study how this map behaves with respect to additive and multiplicative free convolution. As an interesting consequence of our results, we illustrate the relation between the probability distribution v and the distribution of the Lyapunov exponents for the sequence fTkg1k=1 introduced by Vladismir Kargin.