scholarly journals An Affine Model of a Riemann Surface Associated to a Schwarz–Christoffel Mapping

Axioms ◽  
2021 ◽  
Vol 10 (2) ◽  
pp. 49
Author(s):  
Richard Cushman
Keyword(s):  
Riemann Surface ◽  
Half Plane ◽  
Complex Plane ◽  
Riemannian Metric ◽  
Affine Model ◽  
Upper Half Plane

In this paper, we construct an affine model of a Riemann surface with a flat Riemannian metric associated to a Schwarz–Christoffel mapping of the upper half plane onto a rational triangle. We explain the relation between the geodesics on this Riemann surface and billiard motions in a regular stellated n-gon in the complex plane.

scholarly journals On Complex Explicit Formulae Connected with the Möbius Function of an Elliptic Curve

2014 ◽  
Vol 57 (2) ◽  
pp. 381-389
Author(s):  
Adrian Łydka
Keyword(s):  
Functional Equation ◽  
Meromorphic Function ◽  
Elliptic Curve ◽  
Explicit Formula ◽  
Half Plane ◽  
Complex Plane ◽  
Möbius Function ◽  
Mobius Function ◽  
Upper Half Plane ◽  
Explicit Formulae

AbstractWe study analytic properties function m(z, E), which is defined on the upper half-plane as an integral from the shifted L-function of an elliptic curve. We show that m(z, E) analytically continues to a meromorphic function on the whole complex plane and satisfies certain functional equation. Moreover, we give explicit formula for m(z, E) in the strip |ℑz| < 2π.

A Short Proof of an Interpolation Theorem

1974 ◽  
Vol 17 (1) ◽  
pp. 127-128 ◽  
Author(s):  
Edward Hughes
Keyword(s):  
Half Plane ◽  
Complex Plane ◽  
Simple Proof ◽  
Real Line ◽  
Short Proof ◽  
Interpolation Theorem ◽  
Operator Interpolation ◽  
Upper Half Plane ◽  
Positive Functions ◽  
Class Of Functions

In this note we give a simple proof of an operator-interpolation theorem (Theorem 2) due originally to Donoghue [6], and Lions-Foias [7].Let be the complex plane, the open upper half-plane, the real line, ℛ+ and ℛ- the non-negative and non-positive axes. Denote by the class of positive functions on which extend analytically to —ℛ-, and map into itself. Denote by ’ the class of functions φ such that φ(x1/2)2 is in .

ELLIPTIC AND AUTOMORPHIC DYNAMICAL SYSTEMS ON SURFACES

2006 ◽  
Vol 16 (04) ◽  
pp. 911-923 ◽  
Author(s):  
S. P. BANKS ◽  
SONG YI
Keyword(s):  
Dynamical Systems ◽  
Differential Equations ◽  
Half Plane ◽  
Complex Plane ◽  
Fundamental Domain ◽  
Function Theory ◽  
Theta Series ◽  
The Hyperbolic Metric ◽  
Upper Half Plane ◽  
Definition Of

We derive explicit differential equations for dynamical systems defined on generic surfaces applying elliptic and automorphic function theory to make uniform the surfaces in the upper half of the complex plane with the hyperbolic metric. By modifying the definition of the standard theta series we will determine general meromorphic systems on a fundamental domain in the upper half plane the solution trajectories of which "roll up" onto an appropriate surface of any given genus.

scholarly journals Local model for moduli space of affine vortices

2018 ◽  
Vol 29 (03) ◽  
pp. 1850020
Author(s):  
Sushmita Venugopalan ◽  
Guangbo Xu
Keyword(s):  
Half Plane ◽  
Complex Plane ◽  
Moduli Space ◽  
Smooth Manifold ◽  
American Mathematical Society ◽  
Fredholm Theory ◽  
Mathematical Society ◽  
Vortex Equation ◽  
Upper Half Plane ◽  
Funct Anal

We show that the moduli space of regular affine vortices, which are solutions of the symplectic vortex equation over the complex plane, has the structure of a smooth manifold. The construction uses Ziltener’s Fredholm theory results [A Quantum Kirwan Map: Bubbling and Fredholm Theory, Memiors of the American Mathematical Society, Vol. 230 (American Mathematical Society, Providence, RI, 2014), pp. 1–129]. We also extend the result to the case of affine vortices over the upper half plane. These results are necessary ingredients in defining the “open quantum Kirwan map” proposed by Woodward [Gauged Floer theory for toric moment fibers, Geom. Funct. Anal. 21 (2011) 680–749].

scholarly journals On boundaries of Schottky spaces

1976 ◽  
Vol 62 ◽  
pp. 97-124 ◽  
Author(s):  
Hiroki Sato
Keyword(s):  
Riemann Surface ◽  
Half Plane ◽  
Fuchsian Group ◽  
Compact Riemann Surface ◽  
Teichmüller Space ◽  
Schottky Group ◽  
Teichmuller Space ◽  
Upper Half Plane

Let S be a compact Riemann surface and let Sn be the surface obtained from S in the course of a pinching deformation. We denote by Γn the quasi-Fuchsian group representing Sn in the Teichmüller space T(Γ), where Γ is a Fuchsian group with U/Γ = S (U: the upper half plane). Then in the previous paper [7] we showed that the limit of the sequence of Γn is a cusp on the boundary ∂T(Γ). In this paper we will consider the case of Schottky space . Let Gn be a Schottky group with Ω(Gn)/Gn = Sn. Then the purpose of this paper is to show what the limit of Gn is.

scholarly journals The Calderon-Zygmund Operator and Its Relation to Asymptotic Estimates for Ordinary Differential Operators

2017 ◽  
Vol 63 (4) ◽  
pp. 689-702 ◽  
Author(s):  
A M Savchuk
Keyword(s):  
Half Plane ◽  
Complex Plane ◽  
Carleson Measure ◽  
Differential Operators ◽  
Fundamental System ◽  
Asymptotic Estimates ◽  
Zygmund Operator ◽  
Ordinary Differential Operators ◽  
Upper Half Plane

We consider the problem of estimating of expressions of the kind Υ(λ)=supx∈[0,1]∣∣∫x0f(t)eiλtdt∣∣. In particular, for the case f∈Lp[0,1], p∈(1,2], we prove the estimate ∥Υ(λ)∥Lq(R)≤C∥f∥Lp for any q>p′, where 1/p+1/p′=1. The same estimate is proved for the space Lq(dμ), where dμ is an arbitrary Carleson measure in the upper half-plane C+. Also, we estimate more complex expressions of the kind Υ(λ) arising in study of asymptotics of the fundamental system of solutions for systems of the kind y′=By+A(x)y+C(x,λ)y with dimension n as |λ|→∞ in suitable sectors of the complex plane.

scholarly journals Subgroups of infinite index in the modular group

1978 ◽  
Vol 19 (1) ◽  
pp. 33-43 ◽  
Author(s):  
W. W. Stothers
Keyword(s):  
Half Plane ◽  
Complex Plane ◽  
Modular Group ◽  
Infinite Index ◽  
Upper Half Plane ◽  
Extended Complex ◽  
Extended Complex Plane

The modular group Г is the group of integral bilinear transformations of the extended complex plane which preserve the upper half-plane. It has the presentation 〈x, y:x2 = y3 = 1〉, and the generators can be chosen so that u = xy maps z to z + 1.

scholarly journals Composition Operators from the Hardy Space to the Zygmund-Type Space on the Upper Half-Plane

2009 ◽  
Vol 2009 ◽  
pp. 1-8 ◽  
Author(s):  
Stevo Stević
Keyword(s):  
Hardy Space ◽  
Half Plane ◽  
Complex Plane ◽  
Composition Operator ◽  
Weighted Space ◽  
Composition Operators ◽  
Sufficient Conditions ◽  
Type Space ◽  
Upper Half Plane ◽  
Necessary And Sufficient

Here we introduce thenth weighted space on the upper half-planeΠ+={z∈ℂ:Im z>0}in the complex planeℂ. For the casen=2, we call it the Zygmund-type space, and denote it by&#x1D4B5;(Π+). The main result of the paper gives some necessary and sufficient conditions for the boundedness of the composition operatorCφf(z)=f(φ(z))from the Hardy spaceHp(Π+)on the upper half-plane, to the Zygmund-type space, whereφis an analytic self-map of the upper half-plane.

scholarly journals Basis of quadratic differentials for Riemann surfaces with automorphisms

1997 ◽  
Vol 39 (2) ◽  
pp. 193-210
Author(s):  
Gonzalo Riera
Keyword(s):  
Riemann Surface ◽  
Half Plane ◽  
Fuchsian Group ◽  
Riemann Surfaces ◽  
Compact Riemann Surface ◽  
Fundamental Region ◽  
Quadratic Differentials ◽  
Uniformization Theorem ◽  
Group A ◽  
Upper Half Plane

The uniformization theorem says that any compact Riemann surface S of genus g≥2 can be represented as the quotient of the upper half plane by the action of a Fuchsian group A with a compact fundamental region Δ.

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