The Riemann Mapping Theorem
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Geometric Algorithms
Geometric Algorithms primitive operations convex hull closest pair voronoi diagram References: Algorithms in C (2nd edition), Chapters 24-25 http://www.cs.princeton.edu/introalgsds/71primitives http://www.cs.princeton.edu/introalgsds/72hull 1 Geometric Algorithms Applications. • Data mining. • VLSI design. • Computer vision. • Mathematical models. • Astronomical simulation. • Geographic information systems. airflow around an aircraft wing • Computer graphics (movies, games, virtual reality). • Models of physical world (maps, architecture, medical imaging). Reference: http://www.ics.uci.edu/~eppstein/geom.html History. • Ancient mathematical foundations. • Most geometric algorithms less than 25 years old. 2 primitive operations convex hull closest pair voronoi diagram 3 Geometric Primitives Point: two numbers (x, y). any line not through origin Line: two numbers a and b [ax + by = 1] Line segment: two points. Polygon: sequence of points. Primitive operations. • Is a point inside a polygon? • Compare slopes of two lines. • Distance between two points. • Do two line segments intersect? Given three points p , p , p , is p -p -p a counterclockwise turn? • 1 2 3 1 2 3 Other geometric shapes. • Triangle, rectangle, circle, sphere, cone, … • 3D and higher dimensions sometimes more complicated. 4 Intuition Warning: intuition may be misleading. • Humans have spatial intuition in 2D and 3D. • Computers do not. • Neither has good intuition in higher dimensions! Is a given polygon simple? no crossings 1 6 5 8 7 2 7 8 6 4 2 1 1 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 2 18 4 18 4 19 4 19 4 20 3 20 3 20 1 10 3 7 2 8 8 3 4 6 5 15 1 11 3 14 2 16 we think of this algorithm sees this 5 Polygon Inside, Outside Jordan curve theorem. -
Arxiv:1911.03589V1 [Math.CV] 9 Nov 2019 1 Hoe 1 Result
A NOTE ON THE SCHWARZ LEMMA FOR HARMONIC FUNCTIONS MAREK SVETLIK Abstract. In this note we consider some generalizations of the Schwarz lemma for harmonic functions on the unit disk, whereby values of such functions and the norms of their differentials at the point z = 0 are given. 1. Introduction 1.1. A summary of some results. In this paper we consider some generalizations of the Schwarz lemma for harmonic functions from the unit disk U = {z ∈ C : |z| < 1} to the interval (−1, 1) (or to itself). First, we cite a theorem which is known as the Schwarz lemma for harmonic functions and is considered as a classical result. Theorem 1 ([10],[9, p.77]). Let f : U → U be a harmonic function such that f(0) = 0. Then 4 |f(z)| 6 arctan |z|, for all z ∈ U, π and this inequality is sharp for each point z ∈ U. In 1977, H. W. Hethcote [11] improved this result by removing the assumption f(0) = 0 and proved the following theorem. Theorem 2 ([11, Theorem 1] and [26, Theorem 3.6.1]). Let f : U → U be a harmonic function. Then 1 − |z|2 4 f(z) − f(0) 6 arctan |z|, for all z ∈ U. 1+ |z|2 π As it was written in [23], it seems that researchers have had some difficulties to arXiv:1911.03589v1 [math.CV] 9 Nov 2019 handle the case f(0) 6= 0, where f is harmonic mapping from U to itself. Before explaining the essence of these difficulties, it is necessary to recall one mapping and some of its properties. -
The Riemann Mapping Theorem Christopher J. Bishop
The Riemann Mapping Theorem Christopher J. Bishop C.J. Bishop, Mathematics Department, SUNY at Stony Brook, Stony Brook, NY 11794-3651 E-mail address: [email protected] 1991 Mathematics Subject Classification. Primary: 30C35, Secondary: 30C85, 30C62 Key words and phrases. numerical conformal mappings, Schwarz-Christoffel formula, hyperbolic 3-manifolds, Sullivan’s theorem, convex hulls, quasiconformal mappings, quasisymmetric mappings, medial axis, CRDT algorithm The author is partially supported by NSF Grant DMS 04-05578. Abstract. These are informal notes based on lectures I am giving in MAT 626 (Topics in Complex Analysis: the Riemann mapping theorem) during Fall 2008 at Stony Brook. We will start with brief introduction to conformal mapping focusing on the Schwarz-Christoffel formula and how to compute the unknown parameters. In later chapters we will fill in some of the details of results and proofs in geometric function theory and survey various numerical methods for computing conformal maps, including a method of my own using ideas from hyperbolic and computational geometry. Contents Chapter 1. Introduction to conformal mapping 1 1. Conformal and holomorphic maps 1 2. M¨obius transformations 16 3. The Schwarz-Christoffel Formula 20 4. Crowding 27 5. Power series of Schwarz-Christoffel maps 29 6. Harmonic measure and Brownian motion 39 7. The quasiconformal distance between polygons 48 8. Schwarz-Christoffel iterations and Davis’s method 56 Chapter 2. The Riemann mapping theorem 67 1. The hyperbolic metric 67 2. Schwarz’s lemma 69 3. The Poisson integral formula 71 4. A proof of Riemann’s theorem 73 5. Koebe’s method 74 6. -
Nine Chapters of Analytic Number Theory in Isabelle/HOL
Nine Chapters of Analytic Number Theory in Isabelle/HOL Manuel Eberl Technische Universität München 12 September 2019 + + 58 Manuel Eberl Rodrigo Raya 15 library unformalised 173 18 using analytic methods In this work: only multiplicative number theory (primes, divisors, etc.) Much of the formalised material is not particularly analytic. Some of these results have already been formalised by other people (Avigad, Harrison, Carneiro, . ) – but not in the context of a large library. What is Analytic Number Theory? Studying the multiplicative and additive structure of the integers In this work: only multiplicative number theory (primes, divisors, etc.) Much of the formalised material is not particularly analytic. Some of these results have already been formalised by other people (Avigad, Harrison, Carneiro, . ) – but not in the context of a large library. What is Analytic Number Theory? Studying the multiplicative and additive structure of the integers using analytic methods Much of the formalised material is not particularly analytic. Some of these results have already been formalised by other people (Avigad, Harrison, Carneiro, . ) – but not in the context of a large library. What is Analytic Number Theory? Studying the multiplicative and additive structure of the integers using analytic methods In this work: only multiplicative number theory (primes, divisors, etc.) Some of these results have already been formalised by other people (Avigad, Harrison, Carneiro, . ) – but not in the context of a large library. What is Analytic Number Theory? Studying the multiplicative and additive structure of the integers using analytic methods In this work: only multiplicative number theory (primes, divisors, etc.) Much of the formalised material is not particularly analytic. -
Winding Numbers and Solid Angles
Palestine Polytechnic University Deanship of Graduate Studies and Scientific Research Master Program of Mathematics Winding Numbers and Solid Angles Prepared by Warda Farajallah M.Sc. Thesis Hebron - Palestine 2017 Winding Numbers and Solid Angles Prepared by Warda Farajallah Supervisor Dr. Ahmed Khamayseh M.Sc. Thesis Hebron - Palestine Submitted to the Department of Mathematics at Palestine Polytechnic University as a partial fulfilment of the requirement for the degree of Master of Science. Winding Numbers and Solid Angles Prepared by Warda Farajallah Supervisor Dr. Ahmed Khamayseh Master thesis submitted an accepted, Date September, 2017. The name and signature of the examining committee members Dr. Ahmed Khamayseh Head of committee signature Dr. Yousef Zahaykah External Examiner signature Dr. Nureddin Rabie Internal Examiner signature Palestine Polytechnic University Declaration I declare that the master thesis entitled "Winding Numbers and Solid Angles " is my own work, and hereby certify that unless stated, all work contained within this thesis is my own independent research and has not been submitted for the award of any other degree at any institution, except where due acknowledgment is made in the text. Warda Farajallah Signature: Date: iv Statement of Permission to Use In presenting this thesis in partial fulfillment of the requirements for the master degree in mathematics at Palestine Polytechnic University, I agree that the library shall make it available to borrowers under rules of library. Brief quotations from this thesis are allowable without special permission, provided that accurate acknowledg- ment of the source is made. Permission for the extensive quotation from, reproduction, or publication of this thesis may be granted by my main supervisor, or in his absence, by the Dean of Graduate Studies and Scientific Research when, in the opinion either, the proposed use of the material is scholarly purpose. -
COMPLEX ANALYSIS–Spring 2014 1 Winding Numbers
COMPLEX ANALYSIS{Spring 2014 Cauchy and Runge Under the Same Roof. These notes can be used as an alternative to Section 5.5 of Chapter 2 in the textbook. They assume the theorem on winding numbers of the notes on Winding Numbers and Cauchy's formula, so I begin by repeating this theorem (and consequences) here. but first, some remarks on notation. A notation I'll be using on occasion is to write Z f(z) dz jz−z0j=r for the integral of f(z) over the positively oriented circle of radius r, center z0; that is, Z Z 2π it it f(z) dz = f z0 + re rie dt: jz−z0j=r 0 Another notation I'll use, to avoid confusion with complex conjugation is Cl(A) for the closure of a set A ⊆ C. If a; b 2 C, I will denote by λa;b the line segment from a to b; that is, λa;b(t) = a + t(b − a); 0 ≤ t ≤ 1: Triangles being so ubiquitous, if we have a triangle of vertices a; b; c 2 C,I will write T (a; b; c) for the closed triangle; that is, for the set T (a; b; c) = fra + sb + tc : r; s; t ≥ 0; r + s + t = 1g: Here the order of the vertices does not matter. But I will denote the boundary oriented in the order of the vertices by @T (a; b; c); that is, @T (a; b; c) = λa;b + λb;c + λc;a. If a; b; c are understood (or unimportant), I might just write T for the triangle, and @T for the boundary. -
INTRODUCTION to ALGEBRAIC TOPOLOGY 1 Category And
INTRODUCTION TO ALGEBRAIC TOPOLOGY (UPDATED June 2, 2020) SI LI AND YU QIU CONTENTS 1 Category and Functor 2 Fundamental Groupoid 3 Covering and fibration 4 Classification of covering 5 Limit and colimit 6 Seifert-van Kampen Theorem 7 A Convenient category of spaces 8 Group object and Loop space 9 Fiber homotopy and homotopy fiber 10 Exact Puppe sequence 11 Cofibration 12 CW complex 13 Whitehead Theorem and CW Approximation 14 Eilenberg-MacLane Space 15 Singular Homology 16 Exact homology sequence 17 Barycentric Subdivision and Excision 18 Cellular homology 19 Cohomology and Universal Coefficient Theorem 20 Hurewicz Theorem 21 Spectral sequence 22 Eilenberg-Zilber Theorem and Kunneth¨ formula 23 Cup and Cap product 24 Poincare´ duality 25 Lefschetz Fixed Point Theorem 1 1 CATEGORY AND FUNCTOR 1 CATEGORY AND FUNCTOR Category In category theory, we will encounter many presentations in terms of diagrams. Roughly speaking, a diagram is a collection of ‘objects’ denoted by A, B, C, X, Y, ··· , and ‘arrows‘ between them denoted by f , g, ··· , as in the examples f f1 A / B X / Y g g1 f2 h g2 C Z / W We will always have an operation ◦ to compose arrows. The diagram is called commutative if all the composite paths between two objects ultimately compose to give the same arrow. For the above examples, they are commutative if h = g ◦ f f2 ◦ f1 = g2 ◦ g1. Definition 1.1. A category C consists of 1◦. A class of objects: Obj(C) (a category is called small if its objects form a set). We will write both A 2 Obj(C) and A 2 C for an object A in C. -
The Chazy XII Equation and Schwarz Triangle Functions
Symmetry, Integrability and Geometry: Methods and Applications SIGMA 13 (2017), 095, 24 pages The Chazy XII Equation and Schwarz Triangle Functions Oksana BIHUN and Sarbarish CHAKRAVARTY Department of Mathematics, University of Colorado, Colorado Springs, CO 80918, USA E-mail: [email protected], [email protected] Received June 21, 2017, in final form December 12, 2017; Published online December 25, 2017 https://doi.org/10.3842/SIGMA.2017.095 Abstract. Dubrovin [Lecture Notes in Math., Vol. 1620, Springer, Berlin, 1996, 120{348] showed that the Chazy XII equation y000 − 2yy00 + 3y02 = K(6y0 − y2)2, K 2 C, is equivalent to a projective-invariant equation for an affine connection on a one-dimensional complex manifold with projective structure. By exploiting this geometric connection it is shown that the Chazy XII solution, for certain values of K, can be expressed as y = a1w1 +a2w2 +a3w3 where wi solve the generalized Darboux{Halphen system. This relationship holds only for certain values of the coefficients (a1; a2; a3) and the Darboux{Halphen parameters (α; β; γ), which are enumerated in Table2. Consequently, the Chazy XII solution y(z) is parametrized by a particular class of Schwarz triangle functions S(α; β; γ; z) which are used to represent the solutions wi of the Darboux{Halphen system. The paper only considers the case where α + β +γ < 1. The associated triangle functions are related among themselves via rational maps that are derived from the classical algebraic transformations of hypergeometric functions. The Chazy XII equation is also shown to be equivalent to a Ramanujan-type differential system for a triple (P;^ Q;^ R^). -
THE IMPACT of RIEMANN's MAPPING THEOREM in the World
THE IMPACT OF RIEMANN'S MAPPING THEOREM GRANT OWEN In the world of mathematics, scholars and academics have long sought to understand the work of Bernhard Riemann. Born in a humble Ger- man home, Riemann became one of the great mathematical minds of the 19th century. Evidence of his genius is reflected in the greater mathematical community by their naming 72 different mathematical terms after him. His contributions range from mathematical topics such as trigonometric series, birational geometry of algebraic curves, and differential equations to fields in physics and philosophy [3]. One of his contributions to mathematics, the Riemann Mapping Theorem, is among his most famous and widely studied theorems. This theorem played a role in the advancement of several other topics, including Rie- mann surfaces, topology, and geometry. As a result of its widespread application, it is worth studying not only the theorem itself, but how Riemann derived it and its impact on the work of mathematicians since its publication in 1851 [3]. Before we begin to discover how he derived his famous mapping the- orem, it is important to understand how Riemann's upbringing and education prepared him to make such a contribution in the world of mathematics. Prior to enrolling in university, Riemann was educated at home by his father and a tutor before enrolling in high school. While in school, Riemann did well in all subjects, which strengthened his knowl- edge of philosophy later in life, but was exceptional in mathematics. He enrolled at the University of G¨ottingen,where he learned from some of the best mathematicians in the world at that time. -
Algebraicity Criteria and Their Applications
Algebraicity Criteria and Their Applications The Harvard community has made this article openly available. Please share how this access benefits you. Your story matters Citation Tang, Yunqing. 2016. Algebraicity Criteria and Their Applications. Doctoral dissertation, Harvard University, Graduate School of Arts & Sciences. Citable link http://nrs.harvard.edu/urn-3:HUL.InstRepos:33493480 Terms of Use This article was downloaded from Harvard University’s DASH repository, and is made available under the terms and conditions applicable to Other Posted Material, as set forth at http:// nrs.harvard.edu/urn-3:HUL.InstRepos:dash.current.terms-of- use#LAA Algebraicity criteria and their applications A dissertation presented by Yunqing Tang to The Department of Mathematics in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the subject of Mathematics Harvard University Cambridge, Massachusetts May 2016 c 2016 – Yunqing Tang All rights reserved. DissertationAdvisor:ProfessorMarkKisin YunqingTang Algebraicity criteria and their applications Abstract We use generalizations of the Borel–Dwork criterion to prove variants of the Grothedieck–Katz p-curvature conjecture and the conjecture of Ogus for some classes of abelian varieties over number fields. The Grothendieck–Katz p-curvature conjecture predicts that an arithmetic differential equation whose reduction modulo p has vanishing p-curvatures for all but finitely many primes p,hasfinite monodromy. It is known that it suffices to prove the conjecture for differential equations on P1 − 0, 1, . We prove a variant of this conjecture for P1 0, 1, , which asserts that if the equation { ∞} −{ ∞} satisfies a certain convergence condition for all p, then its monodromy is trivial. -
Chapter 2 Complex Analysis
Chapter 2 Complex Analysis In this part of the course we will study some basic complex analysis. This is an extremely useful and beautiful part of mathematics and forms the basis of many techniques employed in many branches of mathematics and physics. We will extend the notions of derivatives and integrals, familiar from calculus, to the case of complex functions of a complex variable. In so doing we will come across analytic functions, which form the centerpiece of this part of the course. In fact, to a large extent complex analysis is the study of analytic functions. After a brief review of complex numbers as points in the complex plane, we will ¯rst discuss analyticity and give plenty of examples of analytic functions. We will then discuss complex integration, culminating with the generalised Cauchy Integral Formula, and some of its applications. We then go on to discuss the power series representations of analytic functions and the residue calculus, which will allow us to compute many real integrals and in¯nite sums very easily via complex integration. 2.1 Analytic functions In this section we will study complex functions of a complex variable. We will see that di®erentiability of such a function is a non-trivial property, giving rise to the concept of an analytic function. We will then study many examples of analytic functions. In fact, the construction of analytic functions will form a basic leitmotif for this part of the course. 2.1.1 The complex plane We already discussed complex numbers briefly in Section 1.3.5. -
The Measurable Riemann Mapping Theorem
CHAPTER 1 The Measurable Riemann Mapping Theorem 1.1. Conformal structures on Riemann surfaces Throughout, “smooth” will always mean C 1. All surfaces are assumed to be smooth, oriented and without boundary. All diffeomorphisms are assumed to be smooth and orientation-preserving. It will be convenient for our purposes to do local computations involving metrics in the complex-variable notation. Let X be a Riemann surface and z x iy be a holomorphic local coordinate on X. The pair .x; y/ can be thoughtD C of as local coordinates for the underlying smooth surface. In these coordinates, a smooth Riemannian metric has the local form E dx2 2F dx dy G dy2; C C where E; F; G are smooth functions of .x; y/ satisfying E > 0, G > 0 and EG F 2 > 0. The associated inner product on each tangent space is given by @ @ @ @ a b ; c d Eac F .ad bc/ Gbd @x C @y @x C @y D C C C Äc a b ; D d where the symmetric positive definite matrix ÄEF (1.1) D FG represents in the basis @ ; @ . In particular, the length of a tangent vector is f @x @y g given by @ @ p a b Ea2 2F ab Gb2: @x C @y D C C Define two local sections of the complexified cotangent bundle T X C by ˝ dz dx i dy WD C dz dx i dy: N WD 2 1 The Measurable Riemann Mapping Theorem These form a basis for each complexified cotangent space. The local sections @ 1 Â @ @ Ã i @z WD 2 @x @y @ 1 Â @ @ Ã i @z WD 2 @x C @y N of the complexified tangent bundle TX C will form the dual basis at each point.