Tsinghua Lectures on Hypergeometric Functions (Unfinished and Comments Are Welcome)
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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. -
Analytic Vortex Solutions on Compact Hyperbolic Surfaces
Analytic vortex solutions on compact hyperbolic surfaces Rafael Maldonado∗ and Nicholas S. Mantony Department of Applied Mathematics and Theoretical Physics, Wilberforce Road, Cambridge CB3 0WA, U.K. August 27, 2018 Abstract We construct, for the first time, Abelian-Higgs vortices on certain compact surfaces of constant negative curvature. Such surfaces are represented by a tessellation of the hyperbolic plane by regular polygons. The Higgs field is given implicitly in terms of Schwarz triangle functions and analytic solutions are available for certain highly symmetric configurations. 1 Introduction Consider the Abelian-Higgs field theory on a background surface M with metric ds2 = Ω(x; y)(dx2+dy2). At critical coupling the static energy functional satisfies a Bogomolny bound 2 1 Z B Ω 2 E = + jD Φj2 + 1 − jΦj2 dxdy ≥ πN; (1) 2 2Ω i 2 where the topological invariant N (the `vortex number') is the number of zeros of Φ counted with multiplicity [1]. In the notation of [2] we have taken e2 = τ = 1. Equality in (1) is attained when the fields satisfy the Bogomolny vortex equations, which are obtained by completing the square in (1). In complex coordinates z = x + iy these are 2 Dz¯Φ = 0;B = Ω (1 − jΦj ): (2) This set of equations has smooth vortex solutions. As we explain in section 2, analytical results are most readily obtained when M is hyperbolic, having constant negative cur- vature K = −1, a case which is of interest in its own right due to the relation between arXiv:1502.01990v1 [hep-th] 6 Feb 2015 hyperbolic vortices and SO(3)-invariant instantons, [3]. -
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). -
Transformation Formulas for the Generalized Hypergeometric Function with Integral Parameter Differences
CORE Metadata, citation and similar papers at core.ac.uk Provided by Abertay Research Portal Transformation formulas for the generalized hypergeometric function with integral parameter differences A. R. Miller Formerly Professor of Mathematics at George Washington University, 1616 18th Street NW, No. 210, Washington, DC 20009-2525, USA [email protected] and R. B. Paris Division of Complex Systems, University of Abertay Dundee, Dundee DD1 1HG, UK [email protected] Abstract Transformation formulas of Euler and Kummer-type are derived respectively for the generalized hypergeometric functions r+2Fr+1(x) and r+1Fr+1(x), where r pairs of numeratorial and denominatorial parameters differ by positive integers. Certain quadratic transformations for the former function, as well as a summation theorem when x = 1, are also considered. Mathematics Subject Classification: 33C15, 33C20 Keywords: Generalized hypergeometric function, Euler transformation, Kummer trans- formation, Quadratic transformations, Summation theorem, Zeros of entire functions 1. Introduction The generalized hypergeometric function pFq(x) may be defined for complex parameters and argument by the series 1 k a1; a2; : : : ; ap X (a1)k(a2)k ::: (ap)k x pFq x = : (1.1) b1; b2; : : : ; bq (b1)k(b2)k ::: (bq)k k! k=0 When q = p this series converges for jxj < 1, but when q = p − 1 convergence occurs when jxj < 1. However, when only one of the numeratorial parameters aj is a negative integer or zero, then the series always converges since it is simply a polynomial in x of degree −aj. In (1.1) the Pochhammer symbol or ascending factorial (a)k is defined by (a)0 = 1 and for k ≥ 1 by (a)k = a(a + 1) ::: (a + k − 1). -
Computation of Hypergeometric Functions
Computation of Hypergeometric Functions by John Pearson Worcester College Dissertation submitted in partial fulfilment of the requirements for the degree of Master of Science in Mathematical Modelling and Scientific Computing University of Oxford 4 September 2009 Abstract We seek accurate, fast and reliable computations of the confluent and Gauss hyper- geometric functions 1F1(a; b; z) and 2F1(a; b; c; z) for different parameter regimes within the complex plane for the parameters a and b for 1F1 and a, b and c for 2F1, as well as different regimes for the complex variable z in both cases. In order to achieve this, we implement a number of methods and algorithms using ideas such as Taylor and asymptotic series com- putations, quadrature, numerical solution of differential equations, recurrence relations, and others. These methods are used to evaluate 1F1 for all z 2 C and 2F1 for jzj < 1. For 2F1, we also apply transformation formulae to generate approximations for all z 2 C. We discuss the results of numerical experiments carried out on the most effective methods and use these results to determine the best methods for each parameter and variable regime investigated. We find that, for both the confluent and Gauss hypergeometric functions, there is no simple answer to the problem of their computation, and different methods are optimal for different parameter regimes. Our conclusions regarding the best methods for computation of these functions involve techniques from a wide range of areas in scientific computing, which are discussed in this dissertation. We have also prepared MATLAB code that takes account of these conclusions. -
A Note on the 2F1 Hypergeometric Function
A Note on the 2F1 Hypergeometric Function Armen Bagdasaryan Institution of the Russian Academy of Sciences, V.A. Trapeznikov Institute for Control Sciences 65 Profsoyuznaya, 117997, Moscow, Russia E-mail: [email protected] Abstract ∞ α The special case of the hypergeometric function F represents the binomial series (1 + x)α = xn 2 1 n=0 n that always converges when |x| < 1. Convergence of the series at the endpoints, x = ±1, dependsP on the values of α and needs to be checked in every concrete case. In this note, using new approach, we reprove the convergence of the hypergeometric series for |x| < 1 and obtain new result on its convergence at point x = −1 for every integer α = 0, that is we prove it for the function 2F1(α, β; β; x). The proof is within a new theoretical setting based on a new method for reorganizing the integers and on the original regular method for summation of divergent series. Keywords: Hypergeometric function, Binomial series, Convergence radius, Convergence at the endpoints 1. Introduction Almost all of the elementary functions of mathematics are either hypergeometric or ratios of hypergeometric functions. Moreover, many of the non-elementary functions that arise in mathematics and physics also have representations as hypergeometric series, that is as special cases of a series, generalized hypergeometric function with properly chosen values of parameters ∞ (a ) ...(a ) F (a , ..., a ; b , ..., b ; x)= 1 n p n xn p q 1 p 1 q (b ) ...(b ) n! n=0 1 n q n X Γ(w+n) where (w)n ≡ Γ(w) = w(w + 1)(w + 2)...(w + n − 1), (w)0 = 1 is a Pochhammer symbol and n!=1 · 2 · .. -
Lauricella's Hypergeometric Function
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 7, 452-470 (1963) Lauricella’s Hypergeometric Function FD* B. C. CARLSON Institute for Atomic Research and D@artment of Physics, Iowa State University, Ames, Iowa Submitted by Richard Bellman I. INTRODUCTION In 1880 Appell defined four hypergeometric series in two variables, which were generalized to 71variables in a straightforward way by Lauricella in 1893 [I, 21. One of Lauricella’s series, which includes Appell’s function F, as the case 71= 2 (and, of course, Gauss’s hypergeometric function as the case n = l), is where (a, m) = r(u + m)/r(a). The FD f unction has special importance for applied mathematics and mathematical physics because elliptic integrals are hypergeometric functions of type FD [3]. In Section II of this paper we shall define a hypergeometric function R(u; b,, . ..) b,; zi, “., z,) which is the same as FD except for small but important modifications. Since R is homogeneous in the variables zi, ..., z,, it depends in a nontrivial way on only 71- 1 ratios of these variables; indeed, every R function with n variables is expressible in terms of an FD function with 71- 1 variables, and conversely. Although R and FD are therefore equivalent, R turns out to be more convenient for both theory and applica- tion. The choice of R was initially suggested by the observation that the Euler transformations of FD would be greatly simplified by introducing homo- geneous variables. -
Numerical Evaluation of the Gauss Hypergeometric Function with the Hypergeo Package
Numerical evaluation of the Gauss hypergeometric function with the hypergeo package Robin K. S. Hankin Auckland University of Technology Abstract This paper introduces the hypergeo package of R routines, for numerical calculation of hypergeometric functions. The package is focussed on efficient and accurate evaluation of the hypergeometric function over the whole of the complex plane within the constraints of fixed-precision arithmetic. The hypergeometric series is convergent only within the unit circle, so analytic continuation must be used to define the function outside the unit circle. This short document outlines the numerical and conceptual methods used in the package; and justifies the package philosophy, which is to maintain transparent and verifiable links between the software and AMS-55. The package is demonstrated in the context of game theory. Keywords: Hypergeometric functions, numerical evaluation, complex plane, R, residue theo- rem. 1. Introduction k The geometric series k∞=0 tk with tk = z may be characterized by its first term and the con- k 1 stant ratio of successiveP terms tk+1/tk = z, giving the familiar identity k∞=0 z = (1 − z)− . Observe that while the series has unit radius of convergence, the rightP hand side is defined over the whole complex plane except for z = 1 where it has a pole. Series of this type may be generalized to a hypergeometric series in which the ratio of successive terms is a rational function of k: t +1 P (k) k = tk Q(k) where P (k) and Q(k) are polynomials. If both numerator and denominator have been com- pletely factored we would write t +1 (k + a1)(k + a2) ··· (k + ap) k = z tk (k + b1)(k + b2) ··· (k + bq)(k + 1) (the final term in the denominator is due to historical reasons), and if we require t0 = 1 then we write ∞ k a1,a2,...,ap tkz = aFb ; z (1) b1,b2,...,bq Xk=0 2 Numerical evaluation of the Gauss hypergeometric function with the hypergeo package − z when defined. -
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 -
Representations of Hypergeometric Functions for Arbitrary Parameter
Representations of hypergeometric functions for arbitrary parameter values and their use D.B. Karpa∗ and J.L. L´opezb aFar Eastern Federal University, Vladivostok, Russia bDpto. de Ingenier´ıa Matem´atica e Inform´atica, Universidad P´ublica de Navarra and INAMAT, Navarra, Spain Abstract. Integral representations of hypergeometric functions proved to be a very useful tool for studying their properties. The purpose of this paper is twofold. First, we extend the known representations to arbitrary values of the parameters and show that the extended representations can be interpreted as examples of regularizations of integrals containing Meijer’s G function. Second, we give new applications of both, known and extended representations. These include: inverse factorial series expansion for the Gauss type function, new information about zeros of the Bessel and Kummer type functions, connection with radial positive definite functions and generalizations of Luke’s inequalities for the Kummer and Gauss type functions. Keywords: generalized hypergeometric function, Meijer’s G function, integral representation, radial positive definite function, inverse factorial series, Hadamard finite part MSC2010: 33C20, 33C60, 33F05, 42A82, 65D20 1 Introduction Throughout the paper we will use the standard definition of the generalized hypergeometric function pFq as the sum of the series ∞ a (a ) (a ) (a ) arXiv:1609.02340v2 [math.CA] 5 Oct 2016 F z = F (a; b; z)= 1 n 2 n · · · p n zn (1) p q b p q (b ) (b ) (b ) n! n=0 1 n 2 n q n X · · · if p q, z C. If p = q + 1 the above series only converges in the open unit disk and F (z) is ≤ ∈ p q defined as its analytic continuation for z C [1, ). -
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. -
Hypergeometric Integral Transforms
Hypergeometric integral transforms François Rouvière April 15, 2018 The following integral formula plays an essential role in Cormack’s 1981 study [1] of the line Radon transform in the plane: s sT (p=s) rT (p=r) dp 2 n n = , n ; 0 < r < s; (1) 2 2 2 2 Z r s p p r p 2 Z where Tn denotes thep Chebyshevp polynomial de…ned by Tn(cos ) = cos n.A few years later, Cormack [2] extended his method to the Radon transform on certain hypersurfaces in Rn, using a similar formula with Chebyshev polynomials replaced by Gegenbauer’s. All these polynomials may be viewed as hypergeometric functions. Here we shall prove in Section 2 a more general integral formula (Proposition 3) and an inversion formula for a general hypergeometric integral transform (Theorem 4). These results are applied to Chebyshev polynomials in Section 3. For the reader’s convenience we give self-contained proofs, only sketched in the appendices to Cormack’s papers [1] and [2]. No previous knowledge of hypergeometric functions is assumed; the basics are given in Section 1. 1 Hypergeometric functions Let a; b; c; z denote complex numbers. For k N, let 2 (a)k := a(a + 1) (a + k 1) if k 1 , (a)0 := 1: For instance the classical binomial series may be written as k a z (1 z) = (a)k , z < 1; (2) k! j j k N X2 a where (1 z) denotes the principal value. It is useful to note that (a)k = 1 x s 1 (a + k)=(a) if a = N, where (s) := 0 e x dx is Euler’s Gamma function.