Dissertation (Ph.D.)--Auburn University, 2006. / Abstract. Vita. Includes bibliographic references (p.79).

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Paul Erdo ̋s’s Empty Hexagon Problem asks if there exists a number H(6) such that for all sets of n ≥ H points in general position on the plane six of the points form the vertices of an empty convex hexagon. This problem is open.

Wave Number Selection and Defect Dynamics in Patterns with Hexagonal Symmetry Denis B. Semwogerere 108 Pages Directed by Dr. Michael F. Schatz We report quantitative measurements of wave number selection, secondary instability and defect dynamics in hexagonal patterns. A novel optical technique ("thermal laser writing") is used to imprint initial patterns with selected characteristics in a B뮡rd-Marangoni convection experiment. Initial patterns of ideal hexagons are imposed to determine the band of stable-pattern wave numbers. For small values of control parameter epsilon the measured stable band is found to agree quantitatively with theoretical predictions at the low-wave-number side of the band, and qualitatively at the high-wave-number side. Long-wavelength perturbations of ideal hexagonal patterns suggested by theory are imposed for epsilon=0.46 and their growth rates are measured to investigate the mechanisms of secondary instability. Our results suggest a transverse-phase instability limits stable hexagons at low wave number while a longitudinal-phase instability limits high-wave-number hexagons. Initial patterns containing an isolated penta-hepta defect are imprinted to study defect propagation directions and velocities. The experimental results agree well with theoretical predictions. The experimental investigations are discussed in the context of patterns with hexagonal symmetry formed under nonequilibrium external driving conditions.

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In this paper, the coarse mesh transport method is extended to hexagonal geometry. This stochastic-deterministic hybrid transport method calculates the eigenvalue and explicit pin fission density profile of hexagonal reactor cores. It models the exact detail within complex heterogeneous cores without homogenizing regions or materials, and neither block-level nor core-level asymmetry poses any limitations to the method. It solves eigenvalue problems by first splitting the core into a set of coarse meshes, and then using Monte Carlo methods to create a library of response expansion coefficients, found by expanding the angular current in phase-space distribution using a set of polynomials orthogonal on the angular half-space defined by mesh boundaries. The coarse meshes are coupled by the angular current at their interfaces. A deterministic sweeping procedure is then used to iteratively construct the solution. The method is evaluated using benchmark problems based on a gas-cooled, graphite-moderated high temperature reactor. The method quickly solves problems to any level of detail desired by the user. In this paper, it is used to explicitly calculate the fission density of individual fuel pins and determine the reactivity worth of individual control rods. In every case, results for the core multiplication factor and pin fission density distribution are found within several minutes. Results are highly accurate when compared to direct Monte Carlo reference solutions; errors in the eigenvalue calculations are on the order of 0.02%, and errors in the pin fission density average less than 0.1%.

Two ways of drawing cubes on dot paper connect students to life outside their classroom.

在這篇論文裡，我們研究Diffy六邊形。本文一開始將Diffy六邊形視為Ducci序列，然後我們討論關於Ducci序列的一些性質。然而，Diffy六邊形事實上是可以旋轉與翻轉的，但是，我們所考慮的Ducci序列並不具備這樣的性質。所以，在本文的最後，我們討論在考慮旋轉與翻轉情況下的Ducci序列。 / In this thesis, we study the Diffy Hexagons: Initially, we regard a Ducci sequence as a Diffy Hexagon game and discuss some properties about Ducci sequences. However, a Ducci sequence isn't actually a Diffy Hexagon game due to the fact that regular hexagons has some symmetries under rotations and reflections, but the Ducci sequences don't. So, we apply an identification in the end.

<p>Microparticles have been investigated not only as feedstock spherical or amorphous bulk materials used for shape molding, but also as agents that can perform work in the micron scale. The fabrication of microparticles with active properties of self-propulsion, self-assembly, and mobility with enhanced mechanical, thermal, and chemical properties is of particular interest for emerging technologies such as drug delivery, micro-robotics, micro energy generation/harvesting, and MEMS. Conventional fabrication methods can produce several complex particle shapes in one fabrication session or hundreds of spheroid shaped particles per second. Innovative techniques, as flow lithography, have demonstrated control over particle form and composition for continuous fabrication cycles. In recent years predefined shape polymer microparticles have been fabricated as well as ceramic microparticles through suspension processing with these set of techniques. Even though ceramic materials have been fabricated, there is still a strong need to increment the palette of available materials to be processed via flow lithography. We have pioneered the production of shaped ceramic microparticles by Stop-Flow Lithography (SFL) using preceramic polymers, providing control of particle size and shape in the range of 1 – 1000 μm. The principal arranged technique (SFL) combines aspects of PDMS-based microfluidics and photolithography for the continuous cyclable fabrication of microparticles with predefined shapes. The PDMS microchannel devices used were fabricated with vinyl film molds in a laminar hood avoiding the need for a cleanroom, procedure that reduced fabrication costs. After a fabrication session, the preceramic polymer microparticles were collected, washed, and dried before entering an inert atmosphere furnace for pyrolysis. Additionally, by treating the material initially as liquid polymer, special properties can be added by converting it into an emulsion or a suspension. Microparticles were functionalized by introducing porosity and magnetic nanoparticles in the preceramic polymer matrix. The porous characteristic of a particle leads to an increase in surface area, allowing the particle to be infiltrated with a catalyzer or act as a chemical/physical carrier, and the magnetic behavior of the particles allows a controllable trajectory with defined external magnetic fields. These two properties can be used to fabricate bifunctional microparticles to serve as drug carriers through human arteries and veins for drug delivery purposes. We successfully fabricated solid and functional ceramic microparticles in the 10 – 50 μm range with predefined shapes as hexagons, gears, triangles, and ovals. This system is an economical route to fabricate functional defined shape particles that can serve as microrobots to perform tasks in liquid media.</p>