What is a Quadrature Domain?

2005 ◽  
pp. 1-25 ◽  
Author(s):  
Björn Gustafsson ◽  
Harold S. Shapiro
Keyword(s):  
Quadrature Domain

scholarly journals Asymptotic properties of unbounded quadrature domains the plane

2015 ◽  
Vol 26 (2) ◽  
pp. 175-191 ◽  
Author(s):  
LAVI KARP
Keyword(s):  
Compact Support ◽  
Asymptotic Properties ◽  
Quadrature Domain ◽  
Quadrature Domains ◽  
Straight Line ◽  
Simply Connected

We prove that if Ω is a simply connected quadrature domain (QD) of a distribution with compact support and the point of infinity belongs to the boundary, then the boundary has an asymptotic curve that is a straight line, parabola or infinite ray. In other words, such QDs in the plane are perturbations of null QDs.

scholarly journals Stress fields around two pores in an elastic body: exact quadrature domain solutions

2015 ◽  
Vol 471 (2180) ◽  
pp. 20150240
Author(s):  
Darren Crowdy
Keyword(s):  
Elastic Body ◽  
Analytical Solutions ◽  
Plane Elasticity ◽  
Stress Fields ◽  
Quadrature Domain ◽  
Mathematical Approach ◽  
Quadrature Domains ◽  
Planar Domains ◽  
Elastic Bodies ◽  
Prime Function

Analytical solutions are given for the stress fields, in both compression and far-field shear, in a two-dimensional elastic body containing two interacting non-circular pores. The two complex potentials governing the solutions are found by using a conformal mapping from a pre-image annulus with those potentials expressed in terms of the Schottky–Klein prime function for the annulus. Solutions for a three-parameter family of elastic bodies with two equal symmetric pores are presented and the compressibility of a special family of pore pairs is studied in detail. The methodology extends to two unequal pores. The importance for boundary value problems of plane elasticity of a special class of planar domains known as quadrature domains is also elucidated. This observation provides the route to generalization of the mathematical approach here to finding analytical solutions for the stress fields in bodies containing any finite number of pores.

Meshless Solution of 2D Fluid Flow Problems by Subdomain Variational Method Using MLPG Method With Radial Basis Functions (RBFs)

2006 ◽  
Author(s):  
Mohammad Haji Mohammadi ◽  
A. Shamsai
Keyword(s):  
Fluid Flow ◽  
Variational Method ◽  
Radial Basis Functions ◽  
Basis Functions ◽  
Quadrature Domain ◽  
Two Dimensional ◽  
Matrix Assembly ◽  
Flow Problems ◽  
Mlpg Method ◽  
Radial Basis

This paper deals with the solution of two-dimensional fluid flow problems using the truly meshless Local Petrov-Galerkin (MLPG) method. The present method is a truly meshless method based only on a number of randomly located nodes. Radial basis functions (RBF) are employed for constructing trial functions in the local weighted meshless local Petrov-Galerkin method for two-dimensional transient viscous fluid flow problems. No boundary integration is needed, no element matrix assembly is required and no special treatment is needed to impose the essential boundary conditions due to satisfaction of kronecker delta property in RBFs. Three different radial basis functions (RBFs), i.e. Multiquadrics (MQ), Gaussian (EXP) and Thin Plate Splines (TPS) are examined and the selection of their shape parameters is studied based on closed-form solutions. The effect of quadrature domain size is also studied. The variational method is used for the development of discrete equations. The results are obtained for a two-dimensional model problem using three RBFs and compared with the results of finite element and exact methods. Results show that the proposed method is highly accurate and possesses no numerical difficulties.

scholarly journals Linear combinations of harmonic measures and quadrature domains of signed measures with small supports

1999 ◽  
Vol 42 (3) ◽  
pp. 433-444 ◽  
Author(s):  
Makoto Sakai
Keyword(s):  
Harmonic Functions ◽  
Positive Measure ◽  
Signed Measure ◽  
Quadrature Domain ◽  
Arbitrary Domain ◽  
Linear Combinations ◽  
Total Measure ◽  
Quadrature Domains ◽  
Harmonic Measures ◽  
Small Support

In this paper we discuss the shape of the quadrature domain of a signed measure for harmonic functions. It is known that the quadrature domain of a positive measure with small support is like a ball if the total measure is large enough. We show that, on the contrary, if the measure is not positive then the quadrature domain can be close to an arbitrary domain. This follows from a lemma concerning linear combinations of harmonic measures.

A Meshfree Weak-Strong-Form Method for Magnetohydrodynamic Flow in a Pipe

2012 ◽  
Vol 490-495 ◽  
pp. 1883-1887
Author(s):  
Xing Hui Cai ◽  
Cheng Ying Shi ◽  
Peng Xu ◽  
Man Lin Zhu ◽  
Guo Liang Wang
Keyword(s):  
Interpolation Method ◽  
Weak Form ◽  
Strong Form ◽  
Quadrature Domain ◽  
Point Collocation ◽  
Function Point ◽  
Point Interpolation Method ◽  
Point Interpolation ◽  
Local Weak Form ◽  
Natural Boundaries

In this paper, a meshfree weak-strong form method is presented to compute the fully developed magnetohydrodynmic flow in a pipe. The radial basis function point interpolation approximation is adopted to construct the shape functions. For the nodes whose local quadrature domain is intersect with the natural boundaries, a local weak form of radial point interpolation method is applied. Otherwise, a strong form of meshfree point collocation method is employed. Numerical simulations are carried out for fully developed magnetohydrodynmic flow in a rectangular pipe with arbitrary electrical conductivity.

scholarly journals Quadrature domains and the real quadratic family

2021 ◽  
Vol 25 (6) ◽  
pp. 104-125
Author(s):  
Kirill Lazebnik
Keyword(s):  
Congruence Subgroup ◽  
Mandelbrot Set ◽  
Quadrature Domain ◽  
Symmetric Polynomials ◽  
Schwarz Function ◽  
Content Type ◽  
Quadrature Domains ◽  
Hyperbolic Component ◽  
Simply Connected ◽  
Real Quadratic

We study several classes of holomorphic dynamical systems associated with quadrature domains. Our main result is that real-symmetric polynomials in the principal hyperbolic component of the Mandelbrot set can be conformally mated with a congruence subgroup of P S L ( 2 , Z ) \mathrm {PSL}(2,\mathbb {Z}) , and that this conformal mating is the Schwarz function of a simply connected quadrature domain.

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