The Existence of Quadratic Differentials in Simply Connected Regions of the Complex Plane

1973 ◽  
Vol 25 (1) ◽  
pp. 83-91 ◽  
Author(s):  
E. Grassmann
Keyword(s):  
Variational Method ◽  
Complex Plane ◽  
Quadratic Differentials ◽  
Sufficient Condition ◽  
Main Difficulty ◽  
Extremum Problems ◽  
Simply Connected ◽  
Admissible Functions

The general coefficient theorem [2] and the extended general coefficient theorem [3] state that the existence of certain quadratic differentials is a sufficient condition for a function to be a solution of certain extremum problems. The purpose of this paper is to show that in the case of simply connected regions this condition is also necessary.We shall do this by a variational method of the Schiffer-Golusin-type. The main difficulty is, that the class of admissible functions for the general coefficient theorem is restricted and we must therefore have a method of variation with restrictions.

PRESCRIBED HORIZONTAL AND VERTICAL TREES PROBLEM OF QUADRATIC DIFFERENTIALS

2006 ◽  
Vol 08 (03) ◽  
pp. 381-399
Author(s):  
THOMAS KWOK-KEUNG AU ◽  
TOM YAU-HENG WAN
Keyword(s):  
Riemann Surface ◽  
Quadratic Differential ◽  
Quadratic Differentials ◽  
Sufficient Condition ◽  
Holomorphic Quadratic Differential ◽  
Simply Connected ◽  
The Given

A sufficient condition for the existence of holomorphic quadratic differential on a non-compact simply-connected Riemann surface with prescribed horizontal and vertical trees is obtained. In particular, for any pair of complete ℝ-trees of finite vertices with (n + 2) infinite edges, there exists a polynomial quadratic differential on ℂ of degree n such that the associated vertical and horizontal trees are isometric to the given pair.

scholarly journals Escaping Fatou components of transcendental self-maps of the punctured plane

2019 ◽  
pp. 1-25 ◽  
Author(s):  
DAVID MARTÍ-PETE
Keyword(s):  
Approximation Theory ◽  
Entire Functions ◽  
Holomorphic Functions ◽  
Sufficient Condition ◽  
Wandering Domain ◽  
Simply Connected ◽  
Fatou Components ◽  
Wandering Domains

Abstract We study the iteration of transcendental self-maps of $\,\mathbb{C}^*\!:=\mathbb{C}\setminus \{0\}$ , that is, holomorphic functions $f:\mathbb{C}^*\to\mathbb{C}^*$ for which both zero and infinity are essential singularities. We use approximation theory to construct functions in this class with escaping Fatou components, both wandering domains and Baker domains, that accumulate to $\{0,\infty\}$ in any possible way under iteration. We also give the first explicit examples of transcendental self-maps of $\,\mathbb{C}^*$ with Baker domains and with wandering domains. In doing so, we developed a sufficient condition for a function to have a simply connected escaping wandering domain. Finally, we remark that our results also provide new examples of entire functions with escaping Fatou components.

scholarly journals Positive Solutions of a Kind of Equations Related to the Laplacian andp-Laplacian

2014 ◽  
Vol 2014 ◽  
pp. 1-5 ◽  
Author(s):  
Fangfang Zhang ◽  
Zhanping Liang
Keyword(s):  
Variational Method ◽  
Bounded Domain ◽  
Positive Solutions ◽  
Eigenvalue Problems ◽  
Sufficient Condition ◽  
Nonlinear Eigenvalue Problems ◽  
Existence Of Positive Solutions ◽  
Nonlinear Eigenvalue

Positive solutions of a kind of equations related to the Laplacian andp-Laplacian on a bounded domain inRNwithN⩾1are studied by using variational method. A sufficient condition of the existence of positive solutions is characterized by the eigenvalues of linear and another nonlinear eigenvalue problems.

scholarly journals On the Boundary Behavior of Conformal Maps

1967 ◽  
Vol 30 ◽  
pp. 83-101 ◽  
Author(s):  
S.E. Warschawski
Keyword(s):  
Boundary Behavior ◽  
Mapping Function ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Conformal Maps ◽  
Simply Connected Domain ◽  
Definitive Solution ◽  
Necessary And Sufficient ◽  
Simply Connected ◽  
Fundamental Paper

Suppose Ω is a simply connected domain which is mapped conformally onto a disk. A much studied problem is the behavior of the mapping function at an accessible boundary point P of Ω, in particular the question, under what conditions the map is ‘ “conformai” at such a point (a) in the sense that angles are preserved as P is approached from Ω (“semi-conformality” at P) and (b) the dilatation at P is finite and positive. In his fundamental paper [8] in 1936, A. Ostrowski established a necessary and sufficient condition (depending on the geometry of the domain only) for the validity of the first property which subsumes all previous results and establishes a definitive solution of this problem.

scholarly journals Some remarks on the spaceR2(E)

1983 ◽  
Vol 6 (3) ◽  
pp. 459-466
Author(s):  
Claes Fernström
Keyword(s):  
Compact Subset ◽  
Complex Plane ◽  
Rational Functions ◽  
Compact Set ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Point Evaluation ◽  
Bounded Point Evaluation ◽  
Point Evaluations ◽  
Necessary And Sufficient

LetEbe a compact subset of the complex plane. We denote byR(E)the algebra consisting of the rational functions with poles offE. The closure ofR(E)inLp(E),1≤p<∞, is denoted byRp(E). In this paper we consider the casep=2. In section 2 we introduce the notion of weak bounded point evaluation of orderβand identify the existence of a weak bounded point evaluation of orderβ,β>1, as a necessary and sufficient condition forR2(E)≠L2(E). We also construct a compact setEsuch thatR2(E)has an isolated bounded point evaluation. In section 3 we examine the smoothness properties of functions inR2(E)at those points which admit bounded point evaluations.

Some remarks on universal functions and Taylor series

2000 ◽  
Vol 128 (1) ◽  
pp. 157-175 ◽  
Author(s):  
G. COSTAKIS
Keyword(s):  
Complex Plane ◽  
Taylor Series ◽  
Connected Domain ◽  
Constructive Proof ◽  
Universal Functions ◽  
Baire’S Theorem ◽  
Simply Connected Domain ◽  
Universal Taylor Series ◽  
Simply Connected

We derive properties of universal functions and Taylor series in domains of the complex plane. For some of our results we use Baire's theorem. We also give a constructive proof, avoiding Baire's theorem, of the existence of universal Taylor series in any arbitrary simply connected domain.

scholarly journals A Characterization of Invariant Affine Connections

1960 ◽  
Vol 16 ◽  
pp. 35-50 ◽  
Author(s):  
Bertram Kostant
Keyword(s):  
Riemannian Manifold ◽  
Simple Proof ◽  
General Theorem ◽  
Transitive Group ◽  
Complete Riemannian Manifold ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Necessary And Sufficient ◽  
Simply Connected

1. Introduction and statement of theorem. 1. In [1] Ambrose and Singer gave a necessary and sufficient condition (Theorem 3 here) for a simply connected complete Riemannian manifold to admit a transitive group of motions. Here we shall give a simple proof of a more general theorem — Theorem 1 (the proof of Theorem 1 became suggestive to us after we noted that the Tx of [1] is just the ax of [6] when X is restricted to p0, see [6], p. 539).

scholarly journals Derivatives, Products, and Pullbacks in Forman's Combinatorial Differential Forms

2020 ◽  
Vol 3 (1) ◽  
pp. 7-16
Author(s):  
Vu Quang Huynh ◽  
Thach Phu Nguyen ◽  
Phuong Van Phan
Keyword(s):  
Connected Domain ◽  
Differential Forms ◽  
Sufficient Condition ◽  
Simply Connected Domain ◽  
Definition Of ◽  
Simply Connected

We study derivatives, closedness, and exactness of 0-forms and 1-forms in the theory of combinatorial differential forms constructed by Robin Forman. We give an example of a closed but not exact 1-form on a non-simply connected domain. We give a sufficient condition on the domain for a closed 1-form to be exact. We show that the product of forms proposed by Forman is not anti-commutative. We propose a definition of pullbacks of forms and show that this operation has several properties analogous to pullbacks on smooth forms.

A Frequency Domain Criterion for Robustly Stabilizing a Family of Interval Plants

1995 ◽  
Vol 117 (1) ◽  
pp. 98-104 ◽  
Author(s):  
Yongdong Zhao ◽  
Suhada Jayasuriya
Keyword(s):  
Frequency Domain ◽  
Complex Plane ◽  
Closed Loop ◽  
Robust Stabilization ◽  
Frequency Function ◽  
Sufficient Condition ◽  
Interval Plants ◽  
Loop Stability

Considered in this paper is the problem of robust stabilization of a family of interval plants by a single fixed compensator. A frequency domain sufficient condition determines the closed-loop stability. In particular if a specially constructed frequency function does not, at any frequency ω ∈ [0, ∞], intersect the box [−1, 1] × [−1, 1] in the complex plane then the entire closed loop family is stable. Also identified is a class of interval plants for which the sufficient condition in fact does become necessary.

Sign in / Sign up

Export Citation Format

Share Document