ON HOLOMORPHIC FUNCTIONS ON RIEMANN SURFACES AND THE RIEMANN-HILBERT PROBLEM

Analysis ◽  
1989 ◽  
Vol 9 (3) ◽  
Author(s):  
Karlheinz Schüffler
Keyword(s):  
Riemann Surfaces ◽  
Hilbert Problem ◽  
Holomorphic Functions ◽  
Riemann Hilbert Problem

Debonded arc-shaped interface anticracks

2017 ◽  
Vol 23 (10) ◽  
pp. 1407-1419
Author(s):  
Xu Wang
Keyword(s):  
Hilbert Problem ◽  
Holomorphic Functions ◽  
Degenerate Case ◽  
Infinite Matrix ◽  
Vector Form ◽  
Cauchy Integrals ◽  
The Matrix ◽  
Elastic Inhomogeneity ◽  
Riemann Hilbert Problem

We analytically investigate a debonded arc-shaped anticrack lying on the interface between a circular elastic inhomogeneity and an infinite matrix when subjected to uniform remote in-plane stresses. One side of the anticrack is perfectly bonded to either the inhomogeneity or the matrix, whereas its other side has become fully debonded. Through the introduction of two sectionally holomorphic functions, the problem is reduced to a non-homogeneous Riemann–Hilbert problem of vector form that can be solved through a decoupling procedure and through evaluation of the Cauchy integrals. Solutions to both the non-degenerate case of distinct eigenvalues and the degenerate case of identical eigenvalues are derived.

Additive Riemann–Hilbert Problem in Line Bundles Over ℂℙ1

2006 ◽  
Vol 49 (1) ◽  
pp. 72-81 ◽  
Author(s):  
Roman J. Dwilewicz
Keyword(s):  
Line Bundle ◽  
Projective Space ◽  
Complex Projective Space ◽  
Real Hypersurface ◽  
Hilbert Problem ◽  
Holomorphic Functions ◽  
Riemann Function ◽  
Line Bundles ◽  
Complex Projective ◽  
Riemann Hilbert Problem

AbstractIn this note we consider -problem in line bundles over complex projective space ℂℙ1 and prove that the equation can be solved for (0, 1) forms with compact support. As a consequence, any Cauchy-Riemann function on a compact real hypersurface in such line bundles is a jump of two holomorphic functions defined on the sides of the hypersurface. In particular, the results can be applied to ℂℙ2 since by removing a point from it we get a line bundle over ℂℙ1.

scholarly journals On the Modular Operator of Mutli-component Regions in Chiral CFT

2021 ◽  
Author(s):  
Stefan Hollands
Keyword(s):  
Field Theory ◽  
Integral Equation ◽  
Hilbert Problem ◽  
Matrix Elements ◽  
New Approach ◽  
Conformal Field ◽  
The Matrix ◽  
Kms Condition ◽  
Modular Operator ◽  
Riemann Hilbert Problem

AbstractWe introduce a new approach to find the Tomita–Takesaki modular flow for multi-component regions in general chiral conformal field theory. Our method is based on locality and analyticity of primary fields as well as the so-called Kubo–Martin–Schwinger (KMS) condition. These features can be used to transform the problem to a Riemann–Hilbert problem on a covering of the complex plane cut along the regions, which is equivalent to an integral equation for the matrix elements of the modular Hamiltonian. Examples are considered.

scholarly journals Padé approximants on Riemann surfaces and KP tau functions

2021 ◽  
Vol 11 (4) ◽  
Author(s):  
Marco Bertola
Keyword(s):  
Riemann Surface ◽  
Riemann Surfaces ◽  
Hilbert Problem ◽  
Line Bundles ◽  
Fixed Degree ◽  
Tau Function ◽  
Tau Functions ◽  
Deformation Parameters ◽  
Higher Genus ◽  
Geometric Solutions

AbstractThe paper has two relatively distinct but connected goals; the first is to define the notion of Padé approximation of Weyl–Stiltjes transforms on an arbitrary compact Riemann surface of higher genus. The data consists of a contour in the Riemann surface and a measure on it, together with the additional datum of a local coordinate near a point and a divisor of degree g. The denominators of the resulting Padé-like approximation also satisfy an orthogonality relation and are sections of appropriate line bundles. A Riemann–Hilbert problem for a square matrix of rank two is shown to characterize these orthogonal sections, in a similar fashion to the ordinary orthogonal polynomial case. The second part extends this idea to explore its connection to integrable systems. The same data can be used to define a pairing between two sequences of line bundles. The locus in the deformation space where the pairing becomes degenerate for fixed degree coincides with the zeros of a “tau” function. We show how this tau function satisfies the Kadomtsev–Petviashvili hierarchy with respect to either deformation parameters, and a certain modification of the 2-Toda hierarchy when considering the whole sequence of tau functions. We also show how this construction is related to the Krichever construction of algebro-geometric solutions.

Sign in / Sign up

Export Citation Format

Share Document