Analytic Vortex Solutions on Compact Hyperbolic Surfaces
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UNIVERSIT´E PARIS 6 PIERRE ET MARIE CURIE Th`Ese De
UNIVERSITE´ PARIS 6 PIERRE ET MARIE CURIE Th`ese de Doctorat Sp´ecialit´e : Math´ematiques pr´esent´ee par Julien PAUPERT pour obtenir le grade de Docteur de l'Universit´e Paris 6 Configurations de lagrangiens, domaines fondamentaux et sous-groupes discrets de P U(2; 1) Configurations of Lagrangians, fundamental domains and discrete subgroups of P U(2; 1) Soutenue le mardi 29 novembre 2005 devant le jury compos´e de Yves BENOIST (ENS Paris) Elisha FALBEL (Paris 6) Directeur John PARKER (Durham) Fr´ed´eric PAULIN (ENS Paris) Rapporteur Jean-Jacques RISLER (Paris 6) Rapporteur non pr´esent a` la soutenance : William GOLDMAN (Maryland) Remerciements Tout d'abord, merci a` mon directeur de th`ese, Elisha Falbel, pour tout le temps et l'´energie qu'il m'a consacr´es. Merci ensuite a` William Goldman et a` Fr´ed´eric Paulin d'avoir accept´e de rapporter cette th`ese; a` Yves Benoist, John Parker et Jean-Jacques Risler d'avoir bien voulu participer au jury. Merci en particulier a` Fr´ed´eric Paulin et a` John Parker pour les nombreuses corrections et am´eliorations qu'ils ont sugg´er´ees. Pour les nombreuses et enrichissantes conversations math´ematiques durant ces ann´ees de th`ese, je voudrais remercier particuli`erement Martin Deraux et Pierre Will, ainsi que les mem- bres du groupuscule de g´eom´etrie hyperbolique complexe de Chevaleret, Marcos, Masseye et Florent. Merci ´egalement de nouveau a` John Parker et a` Richard Wentworth pour les conversations que nous avons eues et les explications qu'ils m'ont donn´ees lors de leurs visites a` Paris. -
Tsinghua Lectures on Hypergeometric Functions (Unfinished and Comments Are Welcome)
Tsinghua Lectures on Hypergeometric Functions (Unfinished and comments are welcome) Gert Heckman Radboud University Nijmegen [email protected] December 8, 2015 1 Contents Preface 2 1 Linear differential equations 6 1.1 Thelocalexistenceproblem . 6 1.2 Thefundamentalgroup . 11 1.3 The monodromy representation . 15 1.4 Regular singular points . 17 1.5 ThetheoremofFuchs .. .. .. .. .. .. 23 1.6 TheRiemann–Hilbertproblem . 26 1.7 Exercises ............................. 33 2 The Euler–Gauss hypergeometric function 39 2.1 The hypergeometric function of Euler–Gauss . 39 2.2 The monodromy according to Schwarz–Klein . 43 2.3 The Euler integral revisited . 49 2.4 Exercises ............................. 52 3 The Clausen–Thomae hypergeometric function 55 3.1 The hypergeometric function of Clausen–Thomae . 55 3.2 The monodromy according to Levelt . 62 3.3 The criterion of Beukers–Heckman . 66 3.4 Intermezzo on Coxeter groups . 71 3.5 Lorentzian Hypergeometric Groups . 75 3.6 Prime Number Theorem after Tchebycheff . 87 3.7 Exercises ............................. 93 2 Preface The Euler–Gauss hypergeometric function ∞ α(α + 1) (α + k 1)β(β + 1) (β + k 1) F (α, β, γ; z) = · · · − · · · − zk γ(γ + 1) (γ + k 1)k! Xk=0 · · · − was introduced by Euler in the 18th century, and was well studied in the 19th century among others by Gauss, Riemann, Schwarz and Klein. The numbers α, β, γ are called the parameters, and z is called the variable. On the one hand, for particular values of the parameters this function appears in various problems. For example α (1 z)− = F (α, 1, 1; z) − arcsin z = 2zF (1/2, 1, 3/2; z2 ) π K(z) = F (1/2, 1/2, 1; z2 ) 2 α(α + 1) (α + n) 1 z P (α,β)(z) = · · · F ( n, α + β + n + 1; α + 1 − ) n n! − | 2 with K(z) the Jacobi elliptic integral of the first kind given by 1 dx K(z)= , Z0 (1 x2)(1 z2x2) − − p (α,β) and Pn (z) the Jacobi polynomial of degree n, normalized by α + n P (α,β)(1) = . -
Arxiv:1711.01247V1 [Math.CO] 3 Nov 2017 Theorem 1.1
DEGREE-REGULAR TRIANGULATIONS OF SURFACES BASUDEB DATTA AND SUBHOJOY GUPTA Abstract. A degree-regular triangulation is one in which each vertex has iden- tical degree. Our main result is that any such triangulation of a (possibly non- compact) surface S is geometric, that is, it is combinatorially equivalent to a geodesic triangulation with respect to a constant curvature metric on S, and we list the possibilities. A key ingredient of the proof is to show that any two d-regular triangulations of the plane for d > 6 are combinatorially equivalent. The proof of this uniqueness result, which is of independent interest, is based on an inductive argument involving some combinatorial topology. 1. Introduction A triangulation of a surface S is an embedded graph whose complementary regions are faces bordered by exactly three edges (see x2 for a more precise definition). Such a triangulation is d-regular if the valency of each vertex is exactly d, and geometric if the triangulation are geodesic segments with respect to a Riemannian metric of constant curvature on S, such that each face is an equilateral triangle. Moreover, two triangulations G1 and G2 of S are combinatorially equivalent if there is a homeomorphism h : S ! S that induces a graph isomorphism from G1 to G2. The existence of such degree-regular geometric triangulations on the simply- connected model spaces (S2; R2 and H2) is well-known. Indeed, a d-regular geomet- ric triangulation of the hyperbolic plane H2 is generated by the Schwarz triangle group ∆(3; d) of reflections across the three sides of a hyperbolic triangle (see x3.2 for a discussion). -
Animal Testing
Animal Testing Adrian Dumitrescu∗ Evan Hilscher † Abstract an animal A. Let A′ be the animal such that there is a cube at every integer coordinate within the box, i.e., it A configuration of unit cubes in three dimensions with is a solid rectangular box containing the given animal. integer coordinates is called an animal if the boundary The algorithm is as follows: of their union is homeomorphic to a sphere. Shermer ′ discovered several animals from which no single cube 1. Transform A1 to A1 by addition only. ′ ′ may be removed such that the resulting configurations 2. Transform A1 to A2 . are also animals [6]. Here we obtain a dual result: we ′ give an example of an animal to which no cube may 3. Transform A2 to A2 by removal only. be added within its minimal bounding box such that ′ ′ It is easy to see that A1 can be transformed to A2 . the resulting configuration is also an animal. We also ′ We simply add or remove one layer of A1 , one cube O n present a ( )-time algorithm for determining whether at a time. The only question is, can any animal A be n a configuration of unit cubes is an animal. transformed to A′ by addition only? If the answer is yes, Keywords: Animal, polyomino, homeomorphic to a then the third step above is also feasible. As it turns sphere. out, the answer is no, thus our alternative algorithm is also infeasible. 1 Introduction Our results. In Section 2 we present a construction of an animal to which no cube may be added within its An animal is defined as a configuration of axis-aligned minimal bounding box such that the resulting collection unit cubes with integer coordinates in 3-space such of unit cubes is an animal. -
Space Complexity of Perfect Matching in Bounded Genus Bipartite Graphs
Space Complexity of Perfect Matching in Bounded Genus Bipartite Graphs Samir Datta1, Raghav Kulkarni2, Raghunath Tewari3, and N. Variyam Vinodchandran4 1 Chennai Mathematical Institute Chennai, India [email protected] 2 University of Chicago Chicago, USA [email protected] 3 University of Nebraska-Lincoln Lincoln, USA [email protected] 4 University of Nebraska-Lincoln Lincoln, USA [email protected] Abstract We investigate the space complexity of certain perfect matching problems over bipartite graphs embedded on surfaces of constant genus (orientable or non-orientable). We show that the prob- lems of deciding whether such graphs have (1) a perfect matching or not and (2) a unique perfect matching or not, are in the logspace complexity class SPL. Since SPL is contained in the logspace counting classes ⊕L (in fact in ModkL for all k ≥ 2), C=L, and PL, our upper bound places the above-mentioned matching problems in these counting classes as well. We also show that the search version, computing a perfect matching, for this class of graphs is in FLSPL. Our results extend the same upper bounds for these problems over bipartite planar graphs known earlier. As our main technical result, we design a logspace computable and polynomially bounded weight function which isolates a minimum weight perfect matching in bipartite graphs embedded on surfaces of constant genus. We use results from algebraic topology for proving the correctness of the weight function. 1998 ACM Subject Classification Computational Complexity Keywords and phrases perfect matching, bounded genus graphs, isolation problem Digital Object Identifier 10.4230/LIPIcs.STACS.2011.579 1 Introduction The perfect matching problem and its variations are one of the most well-studied prob- lems in theoretical computer science. -
Applications of Elliptic Functions in Classical and Algebraic Geometry
APPLICATIONS OF ELLIPTIC FUNCTIONS IN CLASSICAL AND ALGEBRAIC GEOMETRY Jamie Snape Collingwood College, University of Durham Dissertation submitted for the degree of Master of Mathematics at the University of Durham It strikes me that mathematical writing is similar to using a language. To be understood you have to follow some grammatical rules. However, in our case, nobody has taken the trouble of writing down the grammar; we get it as a baby does from parents, by imitation of others. Some mathe- maticians have a good ear; some not... That’s life. JEAN-PIERRE SERRE Jean-Pierre Serre (1926–). Quote taken from Serre (1991). i Contents I Background 1 1 Elliptic Functions 2 1.1 Motivation ............................... 2 1.2 Definition of an elliptic function ................... 2 1.3 Properties of an elliptic function ................... 3 2 Jacobi Elliptic Functions 6 2.1 Motivation ............................... 6 2.2 Definitions of the Jacobi elliptic functions .............. 7 2.3 Properties of the Jacobi elliptic functions ............... 8 2.4 The addition formulæ for the Jacobi elliptic functions . 10 2.5 The constants K and K 0 ........................ 11 2.6 Periodicity of the Jacobi elliptic functions . 11 2.7 Poles and zeroes of the Jacobi elliptic functions . 13 2.8 The theta functions .......................... 13 3 Weierstrass Elliptic Functions 16 3.1 Motivation ............................... 16 3.2 Definition of the Weierstrass elliptic function . 17 3.3 Periodicity and other properties of the Weierstrass elliptic function . 18 3.4 A differential equation satisfied by the Weierstrass elliptic function . 20 3.5 The addition formula for the Weierstrass elliptic function . 21 3.6 The constants e1, e2 and e3 ..................... -
V. Analytic Continuation and Riemann Sur- Faces
MTH6111 Complex Analysis 2009-10 Lecture Notes c Shaun Bullett 2009 V. Analytic Continuation and Riemann Sur- faces This section will introduce various ideas that are important for more advanced work in complex analysis. 1. Continuation along paths A holomorphic function f defined on some open disc containing z0, has a Taylor series ∞ (n) X f (z0) (z − z )n n! 0 n=0 We can use this series to define f at any points inside the disc D(z0, r0) of convergence of the series where f has not already been defined. Now choosing a new point z1 near the edge of D(z0, r0) we can repeat the same procedure for the Taylor series ∞ (n) X f (z1) (z − z )n n! 1 n=0 and this will enable us to define f in a new disc D(z1, r1), overlapping the first disc. This process is known as analytic continuation of f. An analytic function is one that can be written as the sum of a power series: note that we can only carry out the continuation process because of Taylor’s Theorem, which says that a holomorphic function is an analytic function. The most interesting examples of this process occur when f is one branch of a multi-valued function, and we follow the analytic continuation of this branch along some path, or even round a loop back to where we started. This is best illustrated with an example. √ Let f(z) = z. Around the point z = 1, one branch of this function can be defined by the power series: (1/2) (1/2)(−1/2) (1/2)(−1/2)(−3/2) (1 + (z − 1))1/2 = 1 + (z − 1) + (z − 1)2 + (z − 1)3 + .. -
The Fundamental Polygon 3 3. Method Two: Sewing Handles and Mobius Strips 13 Acknowledgments 18 References 18
THE CLASSIFICATION OF SURFACES CASEY BREEN Abstract. The sphere, the torus, and the projective plane are all examples of surfaces, or topological 2-manifolds. An important result in topology, known as the classification theorem, is that any surface is a connected sum of the above examples. This paper will introduce these basic surfaces and provide two different proofs of the classification theorem. While concepts like triangulation will be fundamental to both, the first method relies on representing surfaces as the quotient space obtained by pasting edges of a polygon together, while the second builds surfaces by attaching handles and Mobius strips to a sphere. Contents 1. Preliminaries 1 2. Method One: the Fundamental Polygon 3 3. Method Two: Sewing Handles and Mobius Strips 13 Acknowledgments 18 References 18 1. Preliminaries Definition 1.1. A topological space is Hausdorff if for all x1; x2 2 X, there exist disjoint neighborhoods U1 3 x1;U2 3 x2. Definition 1.2. A basis, B for a topology, τ on X is a collection of open sets in τ such that every open set in τ can be written as a union of elements in B. Definition 1.3. A surface is a Hausdorff space with a countable basis, for which each point has a neighborhood that is homeomorphic to an open subset of R2. This paper will focus on compact connected surfaces, which we refer to simply as surfaces. Below are some examples of surfaces.1 The first two are the sphere and torus, respectively. The subsequent sequences of images illustrate the construction of the Klein bottle and the projective plane. -
Local Symmetry Preserving Operations on Polyhedra
Local Symmetry Preserving Operations on Polyhedra Pieter Goetschalckx Submitted to the Faculty of Sciences of Ghent University in fulfilment of the requirements for the degree of Doctor of Science: Mathematics. Supervisors prof. dr. dr. Kris Coolsaet dr. Nico Van Cleemput Chair prof. dr. Marnix Van Daele Examination Board prof. dr. Tomaž Pisanski prof. dr. Jan De Beule prof. dr. Tom De Medts dr. Carol T. Zamfirescu dr. Jan Goedgebeur © 2020 Pieter Goetschalckx Department of Applied Mathematics, Computer Science and Statistics Faculty of Sciences, Ghent University This work is licensed under a “CC BY 4.0” licence. https://creativecommons.org/licenses/by/4.0/deed.en In memory of John Horton Conway (1937–2020) Contents Acknowledgements 9 Dutch summary 13 Summary 17 List of publications 21 1 A brief history of operations on polyhedra 23 1 Platonic, Archimedean and Catalan solids . 23 2 Conway polyhedron notation . 31 3 The Goldberg-Coxeter construction . 32 3.1 Goldberg ....................... 32 3.2 Buckminster Fuller . 37 3.3 Caspar and Klug ................... 40 3.4 Coxeter ........................ 44 4 Other approaches ....................... 45 References ............................... 46 2 Embedded graphs, tilings and polyhedra 49 1 Combinatorial graphs .................... 49 2 Embedded graphs ....................... 51 3 Symmetry and isomorphisms . 55 4 Tilings .............................. 57 5 Polyhedra ............................ 59 6 Chamber systems ....................... 60 7 Connectivity .......................... 62 References -
Wythoffian Skeletal Polyhedra
Wythoffian Skeletal Polyhedra by Abigail Williams B.S. in Mathematics, Bates College M.S. in Mathematics, Northeastern University A dissertation submitted to The Faculty of the College of Science of Northeastern University in partial fulfillment of the requirements for the degree of Doctor of Philosophy April 14, 2015 Dissertation directed by Egon Schulte Professor of Mathematics Dedication I would like to dedicate this dissertation to my Meme. She has always been my loudest cheerleader and has supported me in all that I have done. Thank you, Meme. ii Abstract of Dissertation Wythoff's construction can be used to generate new polyhedra from the symmetry groups of the regular polyhedra. In this dissertation we examine all polyhedra that can be generated through this construction from the 48 regular polyhedra. We also examine when the construction produces uniform polyhedra and then discuss other methods for finding uniform polyhedra. iii Acknowledgements I would like to start by thanking Professor Schulte for all of the guidance he has provided me over the last few years. He has given me interesting articles to read, provided invaluable commentary on this thesis, had many helpful and insightful discussions with me about my work, and invited me to wonderful conferences. I truly cannot thank him enough for all of his help. I am also very thankful to my committee members for their time and attention. Additionally, I want to thank my family and friends who, for years, have supported me and pretended to care everytime I start talking about math. Finally, I want to thank my husband, Keith. -
13. Trigonometric Functions, Elliptic Functions 1. Reconsideration of Sinx
(February 24, 2021) 13. Trigonometric functions, elliptic functions Paul Garrett [email protected] http:=/www.math.umn.edu/egarrett/ [This document is http:=/www.math.umn.edu/egarrett/m/complex/notes 2020-21/13 elliptic.pdf] 1. Reconsideration of sin x 2. Construction of singly-periodic functions 3. Arc length of ellipses: elliptic integrals, elliptic functions 4. Construction of doubly-periodic functions 5. Fields of elliptic functions 1. Reconsideration of sin x Although we already know quite a bit about trigonometric functions and their role in calculus, their treatment can be redone to emphasize parallels for elliptic integrals and elliptic functions. [1.1] Integral defining arcsin x The length of arc of a piece of a circle is x2 + y2 = 1 is Z x Z x r p d p 2 arc length of circle fragment from 0 to x = 1 + y02 dt = 1 + 1 − t2 dt 0 0 dt x s 1 x r x Z · (−2t)2 Z t2 Z dt = 1 + 2p dt = 1 + dt = p = arcsin x 2 2 2 0 1 − t 0 1 − t 0 1 − t [1.2] Periodicity The periodicity of sin x comes from the multi-valuedness of arcsin x. The multi-valuedness is ascertainable from this integral, since the path from 0 to x can meander through the complex plane, going around the two points ±1 special for the integral. The basic unit of this multi-valuedness is Z dζ = ±2π (counter-clockwise circular path γ enclosing both ±1) p 2 γ 1 − ζ That is, a path integral from 0 to x could go from 0 to x along the real axis, but then add to the path a vertical line segment from x out to a circle of radius 2, traverse the circle an arbitrary number of times, come back along the same segment (thus cancelling the contribution from this segment). -
1. Basic Properties of Elliptic Function Let Ω1,Ω2 Be Two Complex Numbers
1. Basic Properties of Elliptic Function Let !1;!2 be two complex numbers such that the ratio !2=!1 is not purely real. Without loss of generality, we may assume Im τ > 0; where τ = !2=!1: A function f on C is called a doubly periodic function with periods !1;!2 if f(z + !1) = f(z); f(z + !2) = f(z): A doubly periodic function is called an elliptic function if it is meromorphic. Let P be the parallelogram P = fx!1 + y!2 : 0 ≤ x < 1; 0 ≤ y < 1g: We call P a fundamental period-parallelogram for f: The set Λ = fm!1 + n!2 : m; n 2 Zg is called the period lattice for f: Remark. The set Λ is a free abelian group generate by f!1;!2g: Two complex numbers z1; z2 are said to be congruent modulo Λ if z1 − z2 2 Λ: The congruence of z1; z2 is expressed by the notation z1 ≡ z2 mod Λ: Notice that any two points in P are not congruent. In general, a fundamental parallelogram for Λ is a parallelogram P in C such that distinct points of P are not congruent and [ C = (! + P ): !2Λ A cell P for an elliptic function f is a fundamental parallelogram such that f has no zeros and no poles on @P: Theorem 1.1. An elliptic function without poles is a constant. Proof. An elliptic function f with poles is a bounded entire function. By Liouville's theorem, f must be a constant. Proposition 1.1. The number of zeros / poles of an elliptic function in any cell is finite.