DOCSLIB.ORG

By Lindsay Mooradian

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
By Lindsay Mooradian Knots by Lindsay Mooradian 1 1 Can you untie this Knot? Knot theory is the study of mathematical knots. A knot is a closed non-intersecting curve that is em- bedded in three dimensions. A break in a knot diagram represents a shadow, as knots are three dimen- sional objects being drawn on a two dimensional plane. The break helps us distinguish what strand is on top and what strand is below. Knot theorists are driven by the question: How do we tell if two knots are the same? 1 1 In a talk given by Candice Price at Harvey Mudd College in 2019, she discusses her research in knotted DNA structures. She uses well defined definitions and reasonings as to why knots are studied and the basics of knot theory. Candice Price (University of San Diego) on “Unravelling Biochemistry Mysteries: Knot theory applied to biochemistry” So, why do we study knots? Knot theory has real world applica- tions as simple as unknoting your headphones and as complex as understanding antibiotics and potentially curing cancer! 1 1 Price’s research involves knot theory because DNA can become knotted just like a mathematical knot. A topoisomerase I and II are proteins that are important for the replication transcription, unknotting, unlinking. When there is a knotted piece of DNA it needs to be untangled in order to replicate and make a new cell. This can cause cell death if the DNA can not become untangled. A type 11 to- poisomerase can come in when DNA is knotted and unlink these two pieces of DNA so it can replicate. Antibiotics can target the type 11s to not work so that those particular bad cells don’t replicate themselves and spread throughout the body. We can also target cancer cells, but when this happens we are telling topoisomerase 11s to not work at all, for good cells or bad cells. So, if we can advance our medicine to cure cancer it will be through knots and knot theory. Candice Price (University of San Diego) on “Unravelling Biochemisty Mysteries: Knot theory applied to biochemistry” We also study knots because of their strong history. One of the first records of knots being used in mathematics is in the Inca empire. 1 1 Quipu were recording knots used by many cultures from Andean South America pre-colonization. The knots were fashioned from cotton or camelid fiber strings. Quipu had numeric values and were used to count in the base ten system. Quipus were first noted around 1100 AD, but were replaced by European writing and numeral systems after the invasion of the Spanish Empire. D’altroy, Terence N. The Incas (2001) By the 1900s, knots in mathematics were gaining recognition again in mainstream academia. Peter Tait and John Newton Little published the first knot tabulation where they categorized knots up to ten crossings.1 1 Colberg writes about the beginnings of knot theory in academia. She emphasizes the very much diagram heavy upbringing of knot theory, as most mathematicians were first concerned with drawing and categorizing types of knots. The algebraic invariants and categorization came later on after many tables like the ones Tait and Little produced. Colberg, Erin. “A brief history of knot theory.” Página consultada a 8 (2017). The first book about knots, Knotentheorie was published by Kurt Reidemeister in 1932. 1His book was so revolutionary because he de- veloped the Reidemeister moves, which could be used to untangle a knot. 1 Kurt Reidemeister was born in Brunswick, Germany. After attending the University of Hamburg where he received his doctorate in Algebraic number theory he became a professor at the University of Konigsberg until he was dismissed by the Nazis for being politi- cally unsound. In 1927 he proved his three Reidemeister moves, which at the time was the most effective invariant. It is now the basis of knot theory and the first thing students of knot theory typically learn. Reidemeister, K. Knotentheorie. Julius Springer, 1932. We are able to look at the field of knot theory through knot di- agrams. In a knot diagram the string is a closed loop, as if you glued two ends of a shoelace together. If a knot can be untangled into a simple circle we categorize the knot as an “unknot”. If two knots are equivalent, their diagrams are related by a sequence of Reidemeister moves. There are three possible Reidemeister moves that we use to change a knot dia- gram. These moves change the diagram locally, leaving the rest of the diagram alone. A Reidemeister I move is defined by un- twisting or twisting a string. A Reidemeister II move is defined by moving one loop over another. A Reidemeister III move is defined by a slide, where a third string with can be moved over or under a crossing. Using these three moves we are now prepared to solve a problem in knot theory. First, we use a Reidemeister II move to untangle the crossing located in the dotted circle. We can see from the close up view of this section that it is a loop that we can pull under. Next, we use a Reidemeister II move again to undo the crossing in the middle of our knot. Looking at the close up version, we can see the loop. this time we pull it un- der in the opposite direction. Last, we are left with a circle with a twist in it. To un- twist the loop we use a Reidemeister I move. Now we are left with a complete circle--The unknot! Congratulations! You have just solved your first of many problems in knot theory! References Candice Price (University of San Diego) on “Unravelling Biochemistry Mysteries: Knot theory ap- plied to biochemistry” Colberg, Erin. “A brief history of knot theory.” Página consultada a 8, 2017. Epple, Moritz. “Knot Invariants in Vienna and Princeton during the 1920s: Epistemic Configura- tions of Mathematical Research.” Science in Context, 2004, www.maths.ed.ac.uk/~v1ranick/papers/ epple1.pdf. Goulding, Dominic. “Knot Theory: The Yang-Baxter Equation, Quantum Groups and Computation of the Homfly Polynomial.” Durham University, 2010 http://www.maths.dur.ac.uk/Ug/projects/ highlights/PR4/Goulding_Knot_Theory_report.pdf O’Conor, JJ. “Kurt Werner Friedrich Reidemeister.” Cardan Biography, 2014, www-history.mcs.st- and.ac.uk/history/Biographies/Reidemeister.html. Przytycki, Jozef H. “Classical Roots of Knot Theory.” Chnos, Solitons & Fracrals, 1998, https:// www.maths.ed.ac.uk/~v1ranick/papers/przytycki2.pdf Reidemeister, K. Knotentheorie. Julius Springer, 1932. “Reidemeister, Kurt Werner Friedrich.”. “Reidemeister, Kurt Werner Friedrich.” The Columbia Ency- clopedia, 6th Ed, Encyclopedia.com, 2019, www.encyclopedia.com/science/dictionaries-thesaurus- es-pictures-and-press-releases/reidemeister-kurt-werner-friedrich..
Load more

Recommended publications
  • Hyperbolic Structures from Link Diagrams
    University of Tennessee, Knoxville TRACE: Tennessee Research and Creative Exchange Doctoral Dissertations Graduate School 5-2012 Hyperbolic Structures from Link Diagrams Anastasiia Tsvietkova [email protected] Follow this and additional works at: https://trace.tennessee.edu/utk_graddiss Part of the Geometry and Topology Commons Recommended Citation Tsvietkova, Anastasiia, "Hyperbolic Structures from Link Diagrams. " PhD diss., University of Tennessee, 2012. https://trace.tennessee.edu/utk_graddiss/1361 This Dissertation is brought to you for free and open access by the Graduate School at TRACE: Tennessee Research and Creative Exchange. It has been accepted for inclusion in Doctoral Dissertations by an authorized administrator of TRACE: Tennessee Research and Creative Exchange. For more information, please contact [email protected]. To the Graduate Council: I am submitting herewith a dissertation written by Anastasiia Tsvietkova entitled "Hyperbolic Structures from Link Diagrams." I have examined the final electronic copy of this dissertation for form and content and recommend that it be accepted in partial fulfillment of the equirr ements for the degree of Doctor of Philosophy, with a major in Mathematics. Morwen B. Thistlethwaite, Major Professor We have read this dissertation and recommend its acceptance: Conrad P. Plaut, James Conant, Michael Berry Accepted for the Council: Carolyn R. Hodges Vice Provost and Dean of the Graduate School (Original signatures are on file with official studentecor r ds.) Hyperbolic Structures from Link Diagrams A Dissertation Presented for the Doctor of Philosophy Degree The University of Tennessee, Knoxville Anastasiia Tsvietkova May 2012 Copyright ©2012 by Anastasiia Tsvietkova. All rights reserved. ii Acknowledgements I am deeply thankful to Morwen Thistlethwaite, whose thoughtful guidance and generous advice made this research possible.
    [Show full text]
  • Alternating Knots
    ALTERNATING KNOTS WILLIAM W. MENASCO Abstract. This is a short expository article on alternating knots and is to appear in the Concise Encyclopedia of Knot Theory. Introduction Figure 1. P.G. Tait's first knot table where he lists all knot types up to 7 crossings. (From reference [6], courtesy of J. Hoste, M. Thistlethwaite and J. Weeks.) 3 ∼ A knot K ⊂ S is alternating if it has a regular planar diagram DK ⊂ P(= S2) ⊂ S3 such that, when traveling around K , the crossings alternate, over-under- over-under, all the way along K in DK . Figure1 show the first 15 knot types in P. G. Tait's earliest table and each diagram exhibits this alternating pattern. This simple arXiv:1901.00582v1 [math.GT] 3 Jan 2019 definition is very unsatisfying. A knot is alternating if we can draw it as an alternating diagram? There is no mention of any geometric structure. Dissatisfied with this characterization of an alternating knot, Ralph Fox (1913-1973) asked: "What is an alternating knot?" black white white black Figure 2. Going from a black to white region near a crossing. 1 2 WILLIAM W. MENASCO Let's make an initial attempt to address this dissatisfaction by giving a different characterization of an alternating diagram that is immediate from the over-under- over-under characterization. As with all regular planar diagrams of knots in S3, the regions of an alternating diagram can be colored in a checkerboard fashion. Thus, at each crossing (see figure2) we will have \two" white regions and \two" black regions coming together with similarly colored regions being kitty-corner to each other.
    [Show full text]
  • Scottish Physics and Knot Theory's Odd Origins
    Scottish Physics and Knot Theory’s Odd Origins Daniel S. Silver My mother, a social worker and teacher, encouraged my interest in the mysteries of thought. My father, a physical chemist, fostered my appreciation of the history of science. The former chair of my department, prone to unguarded comment, once accused me of being too young to think about such things. As the years pass, I realize how wrong he is. Introduction. Knot theory is one of the most active research areas of mathematics today. Thousands of refereed articles about knots have been published during just the past ten years. One publication, Journal of Knot Theory and its Ramifications, published by World Scientific since 1992, is devoted to new results in the subject. It is perhaps an oversimplification, but not a grave one, to say that two mildly eccentric nineteenth century Scottish physicists, William Thomson (a.k.a. Lord Kelvin) and Peter Guthrie Tait were responsible for modern knot theory. Our purpose is not to debate the point. Rather it is to try to understand the beliefs and bold expectations that motivated Thomson and Tait. James Clerk Maxwell’s influence on his friend Tait is well known. However, Maxwell’s keen interest in the developing theory of knots and links has become clear only in recent years, thanks in great part to the efforts of M. Epple. Often Maxwell was a source of information and humorous encouragement. At times he attempted to restrain Tait’s flights from the path of science. Any discussion of the origin of knot theory requires consideration of the colorful trefoil of Scottish physicists: Thomsom, Tait and Maxwell.
    [Show full text]
  • Links, Quantum Groups, and TQFT's
    Abstract. The Jones polynomial and the Kauffman bracket are constructed, and their relation with knot and link theory is described. The quantum groups and tangle functor formalisms for understanding these invariants and their descendents are given. The quantum group Uq(sl2), which gives rise to the Jones polynomial, is constructed explicitly. The 3-manifold invariants and the axiomatic topological quantum field theories which arise from these link invariants at certain values of the parameter are constructed. LINKS, QUANTUM GROUPS AND TQFT’S STEPHEN SAWIN Introduction In studying a class of mathematical objects, such as knots, one usually begins by developing ideas and machinery to understand specific features of them. When invariants are discovered, they arise out of such understanding and generally give information about those features. Thus they are often immediately useful for proving theorems, even if they are difficult to compute. This is not the situation we find ourselves in with the invariants of knots and 3- manifolds which have appeared since the Jones polynomial. We have a wealth of invariants, all readily computable, but standing decidedly outside the traditions of knot and 3-manifold theory. They lack a geometric interpretation, and consequently have been of almost no use in answering questions one might have asked before their creation. In effect, we are left looking for the branch of mathematics from which these should have come organically. In fact, we have an answer of sorts. In [Wit89b], Witten gave a heuristic definition of the Jones polynomial in terms of a topological quantum field theory, following the arXiv:q-alg/9506002v2 31 Jul 1995 outline of a program proposed by Atiyah [Ati88].
    [Show full text]
  • KNOTS from Combinatorics of Knot Diagrams to Combinatorial Topology Based on Knots
    2034-2 Advanced School and Conference on Knot Theory and its Applications to Physics and Biology 11 - 29 May 2009 KNOTS From combinatorics of knot diagrams to combinatorial topology based on knots Jozef H. Przytycki The George Washington University Washington DC USA KNOTS From combinatorics of knot diagrams to combinatorial topology based on knots Warszawa, November 30, 1984 { Bethesda, March 3, 2007 J´ozef H. Przytycki LIST OF CHAPTERS: Chapter I: Preliminaries Chapter II: History of Knot Theory This e-print. Chapter II starts at page 3 Chapter III: Conway type invariants Chapter IV: Goeritz and Seifert matrices Chapter V: Graphs and links e-print: http://arxiv.org/pdf/math.GT/0601227 Chapter VI: Fox n-colorings, Rational moves, Lagrangian tangles and Burnside groups Chapter VII: Symmetries of links Chapter VIII: Different links with the same Jones type polynomials Chapter IX: Skein modules e-print: http://arxiv.org/pdf/math.GT/0602264 2 Chapter X: Khovanov Homology: categori- fication of the Kauffman bracket relation e-print: http://arxiv.org/pdf/math.GT/0512630 Appendix I. Appendix II. Appendix III. Introduction This book is about classical Knot Theory, that is, about the position of a circle (a knot) or of a number of disjoint circles (a link) in the space R3 or in the sphere S3. We also venture into Knot Theory in general 3-dimensional manifolds. The book has its predecessor in Lecture Notes on Knot Theory, which were published in Polish1 in 1995 [P-18]. A rough translation of the Notes (by J.Wi´sniewski) was ready by the summer of 1995.
    [Show full text]
  • Vortex Knots in Tangled Quantum Eigenfunctions
    Vortex knots in tangled quantum eigenfunctions Alexander J Taylor∗ and Mark R Dennisy H H Wills Physics Laboratory, University of Bristol, Tyndall Avenue, Bristol BS8 1TL, UK Tangles of string typically become knotted, from macroscopic twine down to long-chain macro- molecules such as DNA. Here we demonstrate that knotting also occurs in quantum wavefunctions, where the tangled filaments are vortices (nodal lines/phase singularities). The probability that a vortex loop is knotted is found to increase with its length, and a wide gamut of knots from standard tabulations occur. The results follow from computer simulations of random superpositions of degenerate eigen- states of three simple quantum systems: a cube with periodic boundaries, the isotropic 3-dimensional harmonic oscillator and the 3-sphere. In the latter two cases, vortex knots occur frequently, even in random eigenfunctions at relatively low energy, and are constrained by the spatial symmetries of the modes. The results suggest that knotted vortex structures are generic in complex 3-dimensional wave systems, establishing a topological commonality between wave chaos, polymers and turbulent Bose-Einstein condensates. Complexity in physical systems is often revealed in the sion of the Chladni problem to three dimensions. subtle structures within spatial disorder. In particular, the Despite numerous investigations of the physics of di- complex modes of typical 3-dimensional domains are not verse random filamentary tangles, including quantum usually regular, and at high energies behave according to turbulence[9], loop soups[10], cosmic strings in the early the principles of quantum ergodicity[1, 2]. The differ- universe[11], and optical vortices in 3-dimensional laser ences between chaotic and regular wave dynamics can be speckle patterns[14], the presence of knotted structures seen clearly in the Chladni patterns of a vibrating two- in generic random fields has not been previously empha- dimensional plate[2], in which the zeros (nodes) of the sized or systematically studied.
    [Show full text]
  • Writing@SVSU 2016–2017
    SAGINAW VALLEY STATE UNIVERSITY 2016-2017 7400 Bay Road University Center, MI 48710 svsu.edu ©Writing@SVSU 7400 Bay Road University Center, MI 48710 CREDITS Writing@SVSU is funded by the Office of the Dean of the College of Arts & Behavioral Sciences. Editorial Staff Christopher Giroux Associate Professor of English and Writing Center Assistant Director Joshua Atkins Literature and Creative Writing Major Alexa Foor English Major Samantha Geffert Secondary Education Major Sara Houser Elementary Education and English Language Arts Major Brianna Rivet Literature and Creative Writing Major Kylie Wojciechowski Professional and Technical Writing Major Production Layout and Photography Tim Inman Director of Marketing Support Jennifer Weiss Administrative Secretary University Communications Cover Design Justin Bell Graphic Design Major Printing SVSU Graphics Center Editor’s Note Welcome to the 2016-2017 issue of Writing@SVSU, our yearly attempt to capture a small slice of the good writing that occurs at and because of SVSU. Whether the pieces in Writing@SVSU are attached to a prize, the works you’ll find here emphasize that all writing matters, even when it’s not done for an English class (to paraphrase the title of an essay that follows). Given the political climate, where commentary is often delivered by our leaders through late- night or early-a.m. tweets, we hope you find it refreshing to read the pieces on the following pages. Beyond incorporating far more than 140 characters, these pieces encourage us to reflect at length, as they themselves were the product of much thought and reflection. Beyond being sources and the product of reflection, these texts remind us that the work of the university is, in part, to build bridges, often through words.
    [Show full text]
  • Classical Roots of Knot Theory
    Chnos, Solrnons & Fracrals, Vol. 9. No. 415, pp. 531-545. 1YY8 Pergamon 0 1998 Published by Elsevier Science Ltd Printed in Great Britain. All rights reserved 0960-0779198 $19.02 --0.00 PII: SO960-0779(97)00107-O Classical Roots of Knot Theory J6ZEF H. PRZYTYCKIT Department of Mathematics, George Washington University, Washington, DC 20052, USA Abstract-Vandermonde wrote in 1771: “Whatever the twists and turns of a system of threads in space, one can always obtain an expression for the calculation of its dimensions, but this expression will be of little use in practice. The craftsman who fashions a braid, a net, or some knots will be concerned, not with questions of measurement, but with those of position: what he sees there is the manner in which the threads are interlaced.” We sketch in this essay the history of knot theory stressing the combinatorial aspect of the theory that is so visible in Jones type invariants. fQ 1998 Elsevier Science Ltd. All rights reserved Next he [Alexander] marched into Pisidia where he subdued any resistance which he encountered, and then made himself master of Phrygia. When he captured Gordium [in March 333 BC] which is reputed to have been the home of the ancient king Midas, he saw the celebrated chariot which was fastened to its yoke by the bark of the cornel-tree, and heard the legend which was believed by all barbarians that the fates had decreed that the man who untied the knot was destined to become the ruler of the whole world. According to most writers, the fastenings were so elaborately intertwined and coiled upon one another that their ends were hidden: in consequence Alexander did not know what to do, and in the end loosened the knot by cutting through it with his sword, whereupon the many ends sprang into view.
    [Show full text]
  • Chapter 1. a Century of Knot Theory 1 Chapter 1
    Chapter 1. A Century of Knot Theory 1 Chapter 1. A Century of Knot Theory Note. In this section we give some history of knot theory. Some of this information is from the Wikipedia page on ”Knot Theory” and the Wikipedia page on “The History of Knot Theory” (accessed 1/27/2021). Note. A mathematical theory of knots was first discussed by Alexandre-Th´eophile Vandermonde (February 28, 1735–January 1, 1796) in 1771 in his Remarques sur des Probl`emesde Situation. In this, he remarked that the topological properties are the important properties in discussing knots (interpret “situation” as “position”): “Whatever the twists and turns of a system of threads in space, one can always obtain an expression for the calculation of its dimensions, but this expression will be of little use in practice. The craftsman who fashions a braid, a net, or some knots will be concerned, not with ques- tions of measurement, but with those of position: what he sees there is the manner in which the threads are interlaced.” (from Wikipedia page on Vandermonde) Note. The linking number of two knots is related to the way the knots are inter- twined: From the Wikipedia “Linking Number” Page. Chapter 1. A Century of Knot Theory 2 In 1833, Johann Carl Friedrich Gauss (April 20, 1777–February 23, 1855) intro- duced the Gauss linking integral for computing linking numbers. His student, Johann B. Listing (July 25, 1808–December 24, 1882; Listing was the first to use the term “topology” in 1847), continued the study. Note. In 1877 Scottish physicist Peter Tait published a series of papers on the enumeration of knots (see P.G.
    [Show full text]
  • On the Crossing Number of Positive Knots and Braids and Braid Index Criteria of Jones and Morton-Williams-Franks
    TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 354, Number 10, Pages 3927{3954 S 0002-9947(02)03022-2 Article electronically published on June 10, 2002 ON THE CROSSING NUMBER OF POSITIVE KNOTS AND BRAIDS AND BRAID INDEX CRITERIA OF JONES AND MORTON-WILLIAMS-FRANKS A. STOIMENOW Abstract. We give examples of knots with some unusual properties of the crossing number of positive diagrams or strand number of positive braid rep- resentations. In particular, we show that positive braid knots may not have positive minimal (strand number) braid representations, giving a counterpart to results of Franks-Williams and Murasugi. Other examples answer questions of Cromwell on homogeneous and (partially) of Adams on almost alternating knots. We give a counterexample to, and a corrected version of, a theorem of Jones on the Alexander polynomial of 4-braid knots. We also give an example of a knot on which all previously applied braid index criteria fail to estimate sharply (from below) the braid index. A relation between (generalizations of) such examples and a conjecture of Jones that a minimal braid representation has unique writhe is discussed. Finally, we give a counterexample to Morton's conjecture relating the genus and degree of the skein polynomial. 1. Introduction The braid index b(L) of a knot or link L is defined to be the minimal number of strands of a braid whose closure is the link. (That such a braid always exists was first shown by Alexander [3].) To determine the braid index of L, one seeks general lower and upper estimates on b(L).
    [Show full text]
  • Fun with Knots: Invariants and Applications
    Fun with knots: Invariants and applications Mark Ioppolo School of Mathematics and Statistics University of Western Australia August 18, 2011 Mark Ioppolo Fun with knots: Invariants and applications 1867: Mathematician PG Tait demonstrated that vortex lines in a fluid (lines along r × ~v) can form loops. Physicist Lord Kelvin proposed atoms were vortices in the ether, ie: knotted electric field lines. Figure: Vortices in the air around aeroplane wings Origins of knot theory Early 19th century: Maxwell's equations suggested light was an electromagnetic wave travelling in the 0luminiferous ether0. Physicist Lord Kelvin proposed atoms were vortices in the ether, ie: knotted electric field lines. Figure: Vortices in the air around aeroplane wings Origins of knot theory Early 19th century: Maxwell's equations suggested light was an electromagnetic wave travelling in the 0luminiferous ether0. 1867: Mathematician PG Tait demonstrated that vortex lines in a fluid (lines along r × ~v) can form loops. Figure: Vortices in the air around aeroplane wings Origins of knot theory Early 19th century: Maxwell's equations suggested light was an electromagnetic wave travelling in the 0luminiferous ether0. 1867: Mathematician PG Tait demonstrated that vortex lines in a fluid (lines along r × ~v) can form loops. Physicist Lord Kelvin proposed atoms were vortices in the ether, ie: knotted electric field lines. Origins of knot theory Early 19th century: Maxwell's equations suggested light was an electromagnetic wave travelling in the 0luminiferous ether0. 1867: Mathematician PG Tait demonstrated that vortex lines in a fluid (lines along r × ~v) can form loops. Physicist Lord Kelvin proposed atoms were vortices in the ether, ie: knotted electric field lines.
    [Show full text]
  • Using Knot Theory to Model and Analyze Dna Replication and Recombination
    USING KNOT THEORY TO MODEL AND ANALYZE DNA REPLICATION AND RECOMBINATION by Megan K. Farrell, B.S. A thesis submitted to the Graduate Council of Texas State University in partial fulfillment of the requirements for the degree of Master of Science with a Major in Mathematics December 2018 Committee Members: David Snyder, Chair Eugene Curtin Anton Dochtermann COPYRIGHT by Megan K. Farrell 2018 FAIR USE AND AUTHOR'S PERMISSION STATEMENT Fair Use This work is protected by the Copyright Laws of the United States (Public Law 94{553, section 107). Consistent with fair use as defined in the Copyright Laws, brief quotations from this material are allowed with proper acknowledgment. Use of this material for financial gain without the author's express written permission is not allowed. Duplication Permission As the copyright holder of this work I, Megan K. Farrell, refuse permission to copy in excess of the \Fair Use" exemption without my written permission. DEDICATION In memory of my mother, who always supported my pursuit of mathematics. ACKNOWLEDGMENTS Thanks to my advisor, Dr. David Snyder, for all the assistance and encouragement he gave me during my mathematical studies. Thanks to my mentor, Dr. Ray Treinen for his support and guidance through my academic career, and for motivating me to continue my mathematical studies. Thanks to Dr. Eugene Curtin for his excellent teaching skills that piqued the interest of an adolescent mathematician. Lastly, thanks to Dr. Anton Dochtermann for his assistance on this committee and for his wonderful instruction in my graduate academics. v TABLE OF CONTENTS Page ACKNOWLEDGEMENTS..............................................v LIST OF FIGURES...................................................
    [Show full text]