Minimum-Width Double-Strip and Parallelogram Annulus

In this paper, we study the problem of computing a minimum-width double-strip or parallelogram annulus that encloses a given set of n points in the plane. A double-strip is a closed region in the plane whose boundary consists of four parallel lines and a parallelogram annulus is a closed region between two edge-parallel parallelograms. We present several first algorithms for these problems. Among them are O(n^2) and O(n^3 log n)-time algorithms that compute a minimum-width double-strip and parallelogram annulus, respectively, when their orientations can be freely chosen

Conformal Modulus of the Exterior of Two Rectilinear Slits

We study moduli of planar ring domains whose complements are linear segments and establish formulas for their moduli in terms of the Weierstrass elliptic functions. Numerical tests are carried out to illustrate our results

Limits in PMF of Teichmuller geodesics

We consider the limit set in Thurston's compactification PMF of Teichmueller space of some Teichmueller geodesics defined by quadratic differentials with minimal but not uniquely ergodic vertical foliations. We show that a) there are quadratic differentials so that the limit set of the geodesic is a unique point, b) there are quadratic differentials so that the limit set is a line segment, c) there are quadratic differentials so that the vertical foliation is ergodic and there is a line segment as limit set, and d) there are quadratic differentials so that the vertical foliation is ergodic and there is a unique point as its limit set. These give examples of divergent Teichmueller geodesics whose limit sets overlap and Teichmueller geodesics that stay a bounded distance apart but whose limit sets are not equal. A byproduct of our methods is a construction of a Teichmueller geodesic and a simple closed curve $\gamma$ so that the hyperbolic length of the geodesic in the homotopy class of gamma varies between increasing and decreasing on an unbounded sequence of time intervals along the geodesic.Comment: 39 pages, 4 figure

Holomorphic Motions and Extremal Annuli

Holomorphic motions, soon after they were introduced, became an important subject in complex analysis. It is now an important tool in the study of complex dynamical systems and in the study of Teichmuller theory. This thesis serves on two purposes: an expository of the past developments and a discovery of new theories. First, I give an expository account of Slodkowski\u27s theorem based on the proof given by Chirka. Then I present a result about infinitesimal holomorphic motions. I prove the |ε log ε| modulus of continuity for any infinitesimal holomorphic motion. This proof is a very well application of Schwarz\u27s lemma and the estimate of Agard\u27s formula for the hyperbolic metric on the thrice punctured sphere. One application of this result is that, after the integration of an infinitesimal holomorphic motion, it leads to the Holder continuity property of a quasiconformal homeomorphism. This will be presented in Chapter 3. Second, I compare the proofs given by both Slodkowski and Chirka. Then I construct a different extension of a holomorphic motion in the frame work of Slodkowsk\u27s proof by using the method in Chirka\u27s proof. This gives some opportunity for me to discuss the uniqueness in the extension problem for a holomorphic motion. This will be presented in Chapter 4. Third, I discuss the universal holomorphic motion for a closed subset of the Riemann sphere and the lifting property in the Teichmuller theory. One application of this discussion is the proof of the coincidence of Teichmuller\u27s metric and Kobayashi\u27s metric, a result due to Royden and Gardiner, given by Earle, Kra, and Krushkal by using Slodkowski\u27s theorem. This will be presented in Chapters 5 and 6. Fourth, I study the complex structure of the universal asymptotically conformal Teichmuller space. I give a direct and new proof of the coincidence of Teichmuller\u27s metric and Kobayashi\u27s metric on the universal asymptotically conformal Teichmuller space, a result previously proved by Earle, Gardiner, and Lakic. The main technique that I have used in this proof is Strebel\u27s frame mapping theorem. This will be presented in Chapter 7. Finally, in Chapter 8, I study extremal annuli on a Riemann sphere with four points removed. By using the measurable foliation theory, the Weierstrass Pfunction, and the variation formula for the modulus of an annulus, I prove that the Mori annulus maximize the modulus for the two army problem in the chordal distance on the Riemann sphere. Gardiner and Masur\u27s minimum axis is also discussed in this chapter. Most of the results in this thesis have been published in several research papers jointly with Fred Gardiner, Jun Hu, Yunping Jiang, and Sudeb Mitra

High-order adaptive methods for computing invariant manifolds of maps

The author presents efficient and accurate numerical methods for computing invariant manifolds of maps which arise in the study of dynamical systems. In order to decrease the number of points needed to compute a given curve/surface, he proposes using higher-order interpolation/approximation techniques from geometric modeling. He uses B´ezier curves/triangles, fundamental objects in curve/surface design, to create adaptive methods. The methods are based on tolerance conditions derived from properties of B´ezier curves/triangles. The author develops and tests the methods for an ordinary parametric curve; then he adapts these methods to invariant manifolds of planar maps. Next, he develops and tests the method for parametric surfaces and then he adapts this method to invariant manifolds of three-dimensional maps

A TEM investigation of controlled magnetic behaviour in thin ferromagnetic films

Understanding the magnetic behaviour of thin film elements is of major importance for the magnetic sensor and storage industries, but also for fundamental micromagnetics. To store digital information, each memory element must support two distinct remanent magnetisation configurations that can be switched between using an applied field. In magnetoresistive random access memory (MRAM), a low switching field and reproducible reversal behaviour are desirable properties. The low field keeps the power consumption to a minimum and the reproducility enables efficient writing and read back of data. However, simple geometric structures are able to support a variety of metastable remanent configurations which can be problematic for device applications. For example, with rectangular elements, the switching fields are history dependent, and there is the possibility of flux-closure formation on repeated switching. This means different field strengths may be required to reverse the magnetisation of the same bit (binary digit) during different field cycles, and the information stored in a cell could be accidentally lost. In addition, the miniaturisation of these elements faces the problem that the coercivity is inversely proportional to element width for a given thickness; a factor which limits their use in high density arrays. The optimum geometry for supporting the stored information is therefore an important issue. In this thesis, different element shapes designed to tackle these problems have been investigated using transmission electron microscopy (TEM) backed by micromagnetic simulations. It has been found that variations in element geometry and symmetry can lead to a greater control of the states that can be formed. Alongside this work on patterned elements, continuous film multilayer samples in the form of magnetic tunnel junctions (MTJs) have also been studied. These multilayer structures serve as storage cells in MRAM devices so their successful operation is of the utmost importance to the development of this technology. At the most basic level, MTJs comprise two ferromagnetic layers separated by a layer of electrical insulator. Whilst one magnetic layer is fixed (pinned layer), the other is free to switch direction when an external field is applied (free layer). Ideally the free layer hysteresis loop would be centred at zero field, but because of magnetostatic interactions caused by layer roughness, the ferromagnets couple to one another and the hysteresis loop is offset. This shift means that the fields required to switch the cell in opposite directions are different. In collaboration with Philips Research in Eindhoven, the magnetic and physical structure of new MTJ stacks incorporating an artifical antiferromagnet (AAF) in the free layer were studied using TEM. An AAF consists of two ferromagnetic layers coupled anti-parallel through a thin layer of non-magnetic metal, typically Ru. These samples were found to reduce the offset field by up to 36% when compared to the basic MTJ stack. Whilst this research is valuable to the magnetic storage industry, the information it provides on these complicated magnetic systems is equally beneficial for solid state physics