Cheung, Chun Ngai. / "August 2011." / Thesis (M.Phil.)--Chinese University of Hong Kong, 2011. / Includes bibliographical references (leaves 55-57). / Abstracts in English and Chinese. / Chapter 1 --- Introduction --- p.6 / Chapter 2 --- Coding the Lorenz knots --- p.10 / Chapter 2.1 --- Lorenz words --- p.10 / Chapter 3 --- Lorenz links and positive braids --- p.15 / Chapter 3.1 --- Lorenz braids --- p.15 / Chapter 3.2 --- Properties of Lorenz links as the closure of a positive braid --- p.17 / Chapter 4 --- T-Links and the braid index --- p.28 / Chapter 4.1 --- T-links --- p.30 / Chapter 4.2 --- Symmetries --- p.35 / Chapter 4.3 --- Trip number and the braid index --- p.39 / Chapter 5 --- Modular knots --- p.47 / Chapter 5.1 --- The Modular flow --- p.47 / Chapter 5.2 --- Modular Knots --- p.49 / Bibliography --- p.55

Thesis (B.A.)--Haverford College, Dept. of Mathematics, 2004. / Includes bibliographical references.

Thesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1981. / MICROFICHE COPY AVAILABLE IN ARCHIVES AND SCIENCE. / Bibliography: leaf 80. / by Mark David Baker. / Ph.D.

Diss. (sammanfattning) Uppsala : Univ., 2004. / Härtill 2 uppsatser.

Thesis (Ph. D.)--University of Oregon, 2004. / Typescript. Includes vita and abstract. Includes bibliographical references (leaves 123-125). Also available for download via the World Wide Web; free to University of Oregon users.

Diss. (sammanfattning) Uppsala : Univ., 2003. / Härtill 3 uppsatser.

This thesis covers improvements on the upperbounds for ropelength of a specific class of algebraic knots.

Fatigue, or failure of material due to repeated cyclic loading, is one of the most common causes of mechanical failures. The risk of fatigue in a load carrying component is often lowered by adding material, thereby reducing stresses. This increases the component weight, reducing the performance of the component and increasing its manufacturing cost. There is thus a need to design components to be as light as possible, while keeping the risk of fatigue at a low enough level, i.e. there is a need for optimization of the component subject to fatigue constraints.  This thesis deals with design against fatigue using topology optimization, which is a form of structural optimization where an optimal design is sought by using mathematical programming to decide which parts of a design domain should be filled with material, and which should not.  To predict fatigue, accurate representation of the geometry and accurate stress computation are of utmost importance. In this thesis, methods for imposing constraints such as minimum inner radii and minimum member sizes in the form of four new density filters are proposed. The filters are able to generate a very sharp representation of the structural boundary. A method for improving the accuracy of stress results at the structural boundary is also proposed, based on extrapolation of results from the interior of the structure. The method gives more accurate stresses, which affects the resulting structures when solving optimization problems.  A formulation for fatigue constraints in topology optimization is proposed, based on the weakest link integral. The formulation avoids the problem of choosing between accurate but costly local constraints, and efficient but approximate aggregated constraints, and gives a theoretical motivation for using expressions similar to the p-norm of stresses.  For verifying calculations of the fatigue probability of an optimized structure, critical plane criteria are commonly used. A new method for evaluating such criteria using optimization methods is proposed, and is proved to give results within a user given error tolerance. It is shown that compared to existing brute force methods, the proposed method evaluates significantly fewer planes in the search of the critical one. / <p>QC 20150504</p>

Mechanisation of Mathematics refers to use of computers to generate or check proofs in Mathematics. It involves translation of relevant mathematical theories from one system of logic to another, to render these theories implementable in a computer. This process is termed formalisation of mathematics. Two among the many ways of mechanising are: 1 Generating results using automated theorem provers. 2 Interactive theorem proving in a proof assistant which involves a combination of user intervention and automation. In the first part of this thesis, we reformulate the question of equivalence of two Links in first order logic using braid groups. This is achieved by developing a set of axioms whose canonical model is the braid group on infinite strands B∞. This renders the problem of distinguishing knots and links, amenable to implementation in first order logic based automated theorem provers. We further state and prove results pertaining to models of braid axioms. The second part of the thesis deals with formalising knot Theory in Higher Order Logic using the interactive proof assistant -Isabelle. We formulate equivalence of links in higher order logic. We obtain a construction of Kauffman bracket in the interactive proof assistant called Isabelle proof assistant. We further obtain a machine checked proof of invariance of Kauffman bracket.

This thesis compares Weibull weakest link theory with inclusion of one and all three principal stresses. Principal stresses needed for this theory were calculated with finite element method. The informational research is in the introduction of this thesis. It includes ceramic materials, Weibull weakest link theory, Gaussian quadrature over spherical surface and ceramic head of hip joint endoprothesis. Theoretical part is used for next calculations of probability of failure. The probability of failure of ceramic rod loaded by four-point bending is calculated in first calculations. This task is solved as contact in the next step. Next part of this thesis is about selection of suitable method of numerical integration. This method will be used in the calculation with all three principal stresses. The results of calculation with all three principal stresses are compared with the results of the calculation which includes only first principal stress. Firstly, this is done for cylindrical body and then used on head of hip joint endoprothesis. In the last part of this thesis, probability of fracture of head hip joint endoprothesis with shape deviation of nominal conicity is calculated.