On the Expansion of Harmonic Functions in Series of Harmonic Polynomials Belonging to a Simply Connected Region

1935 ◽  
Vol 57 (4) ◽  
pp. 883
Author(s):  
O. J. Farrell
Keyword(s):  
Harmonic Functions ◽  
Connected Region ◽  
Harmonic Polynomials ◽  
In Series ◽  
Simply Connected Region ◽  
Simply Connected

A Conformal Proof of a Jordan Curve Problem

1969 ◽  
Vol 12 (5) ◽  
pp. 673-674 ◽  
Author(s):  
G. Spoar ◽  
N.D. Lane
Keyword(s):  
Jordan Curve ◽  
Connected Region ◽  
Inversive Plane ◽  
Simply Connected Region ◽  
Simply Connected

The following theorem appears in [1].Let R be a closed simply connected region of the inversive plane bounded by a Jordan curve J, and let J be divided into three closed arcs A1, A2, A3. Then there exists a circle contained in R and having points in common with all three arcs.

$\beta$-transformation, natural extension and invariant measure

2000 ◽  
Vol 20 (5) ◽  
pp. 1271-1285 ◽  
Author(s):  
GAVIN BROWN ◽  
QINGHE YIN
Keyword(s):  
Invariant Measure ◽  
Lebesgue Measure ◽  
Natural Extension ◽  
Connected Region ◽  
Two Dimensional ◽  
Constant Multiple ◽  
Unit Square ◽  
Simply Connected Region ◽  
Simply Connected ◽  
Dimensional Lebesgue Measure

For $\beta>1$, consider the $\beta$-transformation $T_\beta$. When $\beta$ is an integer, the natural extension of $T_\beta$ can be represented explicitly as a map on the unit square with an invariant measure: the corresponding two-dimensional Lebesgue measure. We show that, under certain conditions on $\beta$, the natural extension is defined on a simply connected region and an invariant measure is a constant multiple of the Lebesgue measure.We characterize those $\beta$ in terms of the $\beta$-expansion of one, and study the structure and size of the set of all such $\beta$.

Bounds for the Torsional Rigidity of Isotropic Beams

1962 ◽  
Vol 58 (2) ◽  
pp. 417-419 ◽  
Author(s):  
L. M. Milne-Thomson
Keyword(s):  
Cross Section ◽  
Isotropic Material ◽  
Torsional Rigidity ◽  
Connected Region ◽  
Closed Contour ◽  
Prismatic Beam ◽  
The Cross ◽  
Simply Connected Region ◽  
Simply Connected

Consider a cylindrical or prismatic beam of isotropic material. Let the cross-section of the beam be a simply-connected region S bounded by the closed contour C.

scholarly journals An Integral Equation Related to the Exterior Riemann- Hilbert Problem on Region with Corners

2014 ◽  
Vol 4 (2) ◽  
Author(s):  
Zamzana Zamzamir ◽  
Munira Ismail ◽  
Ali H. M. Murid
Keyword(s):  
Integral Equation ◽  
Hilbert Problem ◽  
Boundary Integral ◽  
Connected Region ◽  
Fredholm Alternative ◽  
Alternative Theorem ◽  
Smooth Region ◽  
Simply Connected Region ◽  
Simply Connected ◽  
Riemann Hilbert Problem

Nasser in 2005 gives the first full method for solving the Riemann-Hilbert problem (briefly the RH problem) for smooth arbitrary simply connected region for general indices via boundary integral equation. However, his treatment of RH problem does not include regions with corners. Later, Ismail in 2007 provides a numerical solution of the interior RH problem on region with corners via Nasser’s method together with Swarztrauber’s approach, but Ismail does not develop any integral equation related to exterior RH problem on region with corners. In this paper, we introduce a new integral equation related to the exterior RH problem in a simply connected region bounded by curves having a finite number of corners in the complex plane. We obtain a new integral equation that adopts Ismail’s method which does not involve conformal mapping. This result is a generalization of the integral equation developed by Nasser for the exterior RH problem on smooth region. The solvability of the integral equation in accordance with the Fredholm alternative theorem is presented. The proof of the equivalence of our integral equation to the RH problem is also provided.

Sign in / Sign up

Export Citation Format

Share Document