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 SINGULARITIES OF QUADRATIC DIFFERENTIALS AND EXTREMAL TEICHMÜLLER MAPPINGS DEFINED BY DEHN TWISTS

2009 ◽  
Vol 87 (2) ◽  
pp. 275-288 ◽  
Author(s):  
C. ZHANG
Keyword(s):  
Riemann Surface ◽  
Finite Type ◽  
Quadratic Differential ◽  
Quadratic Differentials ◽  
Closed Geodesics ◽  
Dehn Twists ◽  
Holomorphic Quadratic Differential ◽  
Simple Closed Geodesics ◽  
Disk Component ◽  
Teichmüller Mapping

AbstractLet S be a Riemann surface of finite type. Let ω be a pseudo-Anosov map of S that is obtained from Dehn twists along two families {A,B} of simple closed geodesics that fill S. Then ω can be realized as an extremal Teichmüller mapping on a surface of the same type (also denoted by S). Let ϕ be the corresponding holomorphic quadratic differential on S. We show that under certain conditions all possible nonpuncture zeros of ϕ stay away from all closures of once punctured disk components of S∖{A,B}, and the closure of each disk component of S∖{A,B} contains at most one zero of ϕ. As a consequence, we show that the number of distinct zeros and poles of ϕ is less than or equal to the number of components of S∖{A,B}.

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.

scholarly journals The growth rate of trajectories of a quadratic differential

1990 ◽  
Vol 10 (1) ◽  
pp. 151-176 ◽  
Author(s):  
Howard Masur
Keyword(s):  
Growth Rate ◽  
Riemann Surface ◽  
Quadratic Differential ◽  
Compact Riemann Surface ◽  
Saddle Connection ◽  
Asymptotic Growth ◽  
Holomorphic Quadratic Differential ◽  
Asymptotic Growth Rate

AbstractSupposeqis a holomorphic quadratic differential on a compact Riemann surface of genusg≥ 2. Thenqdefines a metric, flat except at the zeroes. A saddle connection is a geodesic joining two zeroes with no zeroes in its interior. This paper shows the asymptotic growth rate of the number of saddles of length at mostTis at most quadratic inT. An application is given to billiards.

scholarly journals Bipartite graphs with close domination and k-domination numbers

2020 ◽  
Vol 18 (1) ◽  
pp. 873-885
Author(s):  
Gülnaz Boruzanlı Ekinci ◽  
Csilla Bujtás
Keyword(s):  
Positive Integer ◽  
Dominating Set ◽  
Domination Number ◽  
Bipartite Graphs ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Minimum Cardinality ◽  
Vertex Set ◽  
Necessary And Sufficient ◽  
The Given

Abstract Let k be a positive integer and let G be a graph with vertex set V(G) . A subset D\subseteq V(G) is a k -dominating set if every vertex outside D is adjacent to at least k vertices in D . The k -domination number {\gamma }_{k}(G) is the minimum cardinality of a k -dominating set in G . For any graph G , we know that {\gamma }_{k}(G)\ge \gamma (G)+k-2 where \text{Δ}(G)\ge k\ge 2 and this bound is sharp for every k\ge 2 . In this paper, we characterize bipartite graphs satisfying the equality for k\ge 3 and present a necessary and sufficient condition for a bipartite graph to satisfy the equality hereditarily when k=3 . We also prove that the problem of deciding whether a graph satisfies the given equality is NP-hard in general.

A General Analysis of Sequential Social Learning

2021 ◽  
Author(s):  
Itai Arieli ◽  
Manuel Mueller-Frank
Keyword(s):  
Social Learning ◽  
Utility Function ◽  
Posterior Probability ◽  
General Analysis ◽  
Sufficient Condition ◽  
Important Class ◽  
General Utility ◽  
Learning Game ◽  
Necessary And Sufficient ◽  
The Given

This paper analyzes a sequential social learning game with a general utility function, state, and action space. We show that asymptotic learning holds for every utility function if and only if signals are totally unbounded, that is, the support of the private posterior probability of every event contains both zero and one. For the case of finitely many actions, we provide a sufficient condition for asymptotic learning depending on the given utility function. Finally, we establish that for the important class of simple utility functions with finitely many actions and states, pairwise unbounded signals, which generally are a strictly weaker notion than unbounded signals, are necessary and sufficient for asymptotic learning.

scholarly journals Schiffer Comparison Operators and Approximations on Riemann Surfaces Bordered by Quasicircles

2020 ◽  
Author(s):  
Eric Schippers ◽  
Mohammad Shirazi ◽  
Wolfgang Staubach
Keyword(s):  
Riemann Surface ◽  
Bergman Space ◽  
Riemann Surfaces ◽  
Compact Riemann Surface ◽  
Dirichlet Space ◽  
Holomorphic Functions ◽  
Finite Collection ◽  
Approximation Theorems ◽  
Arbitrary Genus ◽  
Simply Connected

Abstract We consider a compact Riemann surface R of arbitrary genus, with a finite number of non-overlapping quasicircles, which separate R into two subsets: a connected Riemann surface $$\Sigma $$ Σ , and the union $$\mathcal {O}$$ O of a finite collection of simply connected regions. We prove that the Schiffer integral operator mapping the Bergman space of anti-holomorphic one-forms on $$\mathcal {O}$$ O to the Bergman space of holomorphic forms on $$\Sigma $$ Σ is an isomorphism onto the exact one-forms, when restricted to the orthogonal complement of the set of forms on all of R. We then apply this to prove versions of the Plemelj–Sokhotski isomorphism and jump decomposition for such a configuration. Finally we obtain some approximation theorems for the Bergman space of one-forms and Dirichlet space of holomorphic functions on $$\Sigma $$ Σ by elements of Bergman space and Dirichlet space on fixed regions in R containing $$\Sigma $$ Σ .

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 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.

Sign in / Sign up

Export Citation Format

Share Document