Jordan Boundary Research Articles

Complete meromorphic curves with Jordan boundaries

We prove that given a finite set E in a bordered Riemann surface R, there is a continuous map h:R‾∖E→Cn (n≥2) such that h|R∖E:R∖E→Cn is a complete holomorphic immersion (embedding if n≥3) which is meromorphic on R and has effective poles at all points in E, and h|bR‾:bR‾→Cn is a topological embedding. In particular, h(bR‾) consists of the union of finitely many pairwise disjoint Jordan curves which we ensure to be of Hausdorff dimension one. We establish a more general result including uniform approximation and interpolation.

Israel's spatial and a-spatial strategy of dispossessing the Jordan Valley's Palestinian Inhabitants.

The Jordan Valley's territorial extent is defined for this study as an elongated stretch of border area, located west of Jordan's boundary with the West Bank, Palestine. This region along with the West Bank was conquered by the Israeli Army during the June 1967 War and has been held by Israel since. Almost 88% of its 200,000 Palestinian population then living in the region were expelled and turned overnight into refugees in Jordan and elsewhere. In the subsequent 56years since, the region, declared by Israel as a highly militarized zone, has often been discursively publicized as the front line of Israel's eastern defense. This paper seeks to refute such a notion that indirectly gave Israel justification to subject the remaining Palestinian inhabitants residing there (currently numbering ca. 60,000) to a harsh discriminatory regimen of surveillance and control under military occupation. At the same time, the Israeli government has over decades privileged the ca. 10,000 incoming Jewish settlers, now living in some 30 Israeli Jewish settlements in the Jordan Valley. Employing a grounded theory approach in our study provided an opportunity for the local Palestinians to have their voices heard, shedding needed analytical light on their difficult situation on the ground.

Analog of the Hadamard Theorem and Related Extremal Problems on the Class of Analytic Functions

We study several related extremal problems for analytic functions in a finitely connected domain \(G\) with rectifiable Jordan boundary \(\Gamma\). A sharp inequality is established between values of a function analytic in \(G\) and weighted means of its boundary values on two measurable subsets \(\gamma_{1}\) and \(\gamma_{0}=\Gamma\setminus\gamma_{1}\) of the boundary: \(|f(z_{0})|\leq\mathcal{C}\,\|f\|^{\alpha}_{L^{q}_{\varphi_{1}}(\gamma_{1})}\, \|f\|^{\beta}_{L^{p}_{\varphi_{0}}(\gamma_{0})},z_{0}\in G,0<q,p\leq\infty.\) The inequality is an analog of Hadamard’s three-circle theorem and the Nevanlinna brothers’ two-constant theorem. In the case of a doubly connected domain \(G\) and \(1\leq q,p\leq\infty\), we study the cases where the inequality provides the value of the modulus of continuity for a functional of analytic extension of a function from the part \(\gamma_{1}\) of the boundary to a given point of the domain. In these cases, the corresponding problem of optimal recovery of a function from its approximate boundary values on \(\gamma_{1}\) and the problem of the best approximation of a functional by bounded linear functionals are solved. The case of a simply connected domain \(G\) has been completely investigated previously.

On the Hilbert boundary-value problem for Beltrami equations with singularities

We investigate the Hilbert boundary-value problem for Beltrami equations \( \overline{\partial}f \) = μ∂f with singularities in generalized quasidisks D whose Jordan boundary ∂D consists of a countable collection of open quasiconformal arcs and, maybe, a countable collection of points. Such generalized quasicircles can be nowhere even locally rectifiable but include, for instance, all piecewise smooth curves, as well as all piecewise Lipschitz Jordan curves.Generally speaking, generalized quasidisks do not satisfy the standard (A)–condition in PDE by Ladyzhenskaya–Ural’tseva, in particular, the outer cone touching condition, as well as the quasihyperbolic boundary condition by Gehring–Martio that we assumed in our last paper for the uniformly elliptic Beltrami equations.In essence, here, we admit any countable collection of singularities of the Beltrami equations on the boundary and arbitrary singularities inside the domain D of a general nature. As usual, a point in \( \overline{D} \) is called a singularity of the Beltrami equation, if the dilatation quotient Kμ := (1 + |μ|)/(1 – |μ|) is not essentially bounded in all its neighborhoods.Presupposing that the coefficients of the problem are arbitrary functions of countable bounded variation and the boundary data are arbitrary measurable with respect to the logarithmic capacity, we prove the existence of regular solutions of the Hilbert boundary-value problem. As a consequence, we derive the existence of nonclassical solutions of the Dirichlet, Neumann, and Poincaré boundary-value problems for equations of mathematical physics with singularities in anisotropic and inhomogeneous media.

On the Radial Symmetry Property for Harmonic Functions

Let $\Gamma$ be a closed smooth Jordan curve in the complex plane $\mathbb{C}$ ,G be a bounded domain in $\mathbb{C}$ with the boundary $\Gamma$ , and let $\overline{G}=G\cup\Gamma$ . We study functions that are continuous in $\mathbb{C}\setminus G$ and harmonic in $\mathbb{C}\setminus\overline{G}$ that grow more slowly than the function $|z|^2$ at $z\to\infty$ . It is shown that, if in the class of such functions there exists a solution to the overdetermined Neumann boundary value problem in which the function equals zero on $\Gamma$ and $\mu$ -almost everywhere on $\Gamma$ there exists and equals one the normal derivative of this function, then the domain G is a disk (Theorem 1). In this case, the solution is unique and it coincides up to a constant with the fundamental solution for the Laplace operator in $\mathbb{C}$ with a singularity in the center of the disk G. The proof of Theorem 1 is based on the application of the conformal mapping of the exterior of the unit disk onto the domain $\mathbb{C}\setminus \overline{G}$ . This mapping allows us to reduce the original problem for the domain $\mathbb{C}\setminus \overline{G}$ to overdetermined boundary value problem for the exterior of the disk in which the main difficulty is the heterogeneity of the boundary condition for the normal derivative. To study this condition some subtle results were required on the boundary properties of a function that performs the indicated conformal mapping as well as some properties of the Hardy classes Hp in the unit disk. Theorem 2 of the paper shows that in the general case the conditions in Theorem 1 cannot be relaxed. It states the existence of a bounded domain $G\subset\mathbb{C}$ different from a disk with a smooth Jordan boundary $\Gamma$ and functions $f_1,f_2,f_3\in C(\mathbb{C}\setminus G)$ harmonic in $\mathbb{C}\setminus\overline{G}$ for each of which exactly one of the conditions of Theorem 1 is not satisfied.

The Calabi–Yau Property of Superminimal Surfaces in Self-Dual Einstein Four-Manifolds

In this paper, we show that if $(X,g)$ is an oriented four dimensional Einstein manifold which is self-dual or anti-self-dual then superminimal surfaces in $X$ of appropriate spin enjoy the Calabi-Yau property, meaning that every immersed surface of this type from a bordered Riemann surface can be uniformly approximated by complete superminimal surfaces with Jordan boundaries. The proof uses the theory of twistor spaces and the Calabi-Yau property of holomorphic Legendrian curves in complex contact manifolds.

Approximation by p(·)‐Faber polynomials in the variable Smirnov classes

Let be a bounded domain with regular Jordan boundary . In this work, ‐Faber polynomial series of functions in the variable exponent Smirnov class are defined and their approximation properties are investigated.

An analogue of the two-constants theorem and optimal recovery of analytic functions

Several related extremal problems for analytic functions in a simply connected domain with rectifiable Jordan boundary are treated. The sharp inequality is established between a value of an analytic function in the domain and the weighted integral norms of the restrictions of its boundary values to two measurable subsets and of the boundary of the domain. It is an analogue of the F. and R. Nevanlinna two-constants theorem. The corresponding problems of optimal recovery of a function from its approximate boundary values on and of the best approximation to the functional of analytic extension of a function from the part of the boundary into the domain are solved. Bibliography: 35 titles.

Optimal Recovery of a Derivative of an Analytic Function from Values of the Function Given with an Error on a Part of the Boundary

We study several related extremal problems for functions analytic in a simply connected domain G with a rectifiable Jordan boundary Γ, in particular, the problem of optimal recovery of the derivative at a point z0 ∈ G from limit boundary values given with an error on a measurable part γ1 of the boundary Γ for the class Q of functions with limit boundary values bounded by 1 on γ0 = Γ\γ1 as well as the problem of the best approximation of the derivative at a point z0 ∈ G by bounded linear functionals in L∞ (γ1) on the class Q.

Area Minimizing Discs in Metric Spaces

We solve the classical problem of Plateau in the setting of proper metric spaces. Precisely, we prove that among all disc-type surfaces with prescribed Jordan boundary in a proper metric space there exists an area minimizing disc which moreover has a quasi-conformal parametrization. If the space supports a local quadratic isoperimetric inequality for curves we prove that such a solution is locally H\"older continuous in the interior and continuous up to the boundary. Our results generalize corresponding results of Douglas and Morrey from the setting of Euclidean space and Riemannian manifolds to that of proper metric spaces.

Every bordered Riemann surface is a complete conformal minimal surface bounded by Jordan curves: Figure. 5.1.

In this paper we find approximate solutions of certain Riemann-Hilbert boundary value problems for minimal surfaces in $\mathbb{R}^n$ and null holomorphic curves in $\mathbb{C}^n$ for any $n\ge 3$. With this tool in hand we construct complete conformally immersed minimal surfaces in $\mathbb{R}^n$ which are normalized by any given bordered Riemann surface and have Jordan boundaries. We also furnish complete conformal proper minimal immersions from any given bordered Riemann surface to any smoothly bounded, strictly convex domain of $\mathbb{R}^n$ which extend continuously up to the boundary; for $n\ge 5$ we find embeddings with these properties.

Computation of level lines of 4-/8-connectedness

We present a topdown algorithm to compute level line trees of 4-/8-connectedness. As a boundary of a level set component, a level line of an image is a Jordan boundary of intensity value on instant interior greater/less than on instant exterior. The interior of a Jordan boundary assumes 4-connectedness and the exterior 8-connectedness, or the inverse. All level lines form a tree structure. The running time of the algorithm is O(n+t), where n is the size of the input image and t is the total length of all level lines. The efficiency of the algorithm is illustrated by experiments.

On the polynomial approximation of a conformal mapping of a domain with nonzero corner

Let G be a bounded domain with a Jordan boundary that is smooth at all points except a single point at which it forms a nonzero corner. We prove Korevaar’s conjecture on the order of polynomial approximation of a conformal mapping of this domain into a disk. We also obtain a pointwise estimate for the error of approximation.

Bounds for the moduli of continuity for conformal mappings of domains near their accessible boundary arcs

The paper presents bounds for the moduli of continuity of conformal mappings of a bounded simply connected domain with an arbitrary Jordan boundary onto a bounded simply connected domain with an arbitrary Jordan boundary, the 'quality' of boundaries being taken into account. For a Jordan curve (simple arc or a closed contour), its quality is characterized in general by its modulus of oscillation, and if it has finite length, by a more sensitive modulus of rectifiability—these purely metric concepts were introduced by the author in 1996. Theorems on the behaviour of conformal mappings of simply connected domains of arbitrary nature near open accessible boundary arcs are established.Bibliography: 18 titles.

Comprehensive process-driven boundary making model:a case study of the Jordan-Israel boundary

Surveyors had a central role in boundary-making processes from at least the 19th century. The latest technological developments in this field have further widened their involvement in the process and changed the concepts and practices implemented. The Process Model presented here incorporates the new technical means available for modern surveyors in accomplishing a more stable and sustainable boundary through a structured procedure. The roots of the traditional theory of boundary making were put down a century ago by Curzon[6], Holdich[9] and others referring mainly to the three-stage process of allocation, delimitation and demarcation. This article renews the existing model of boundary making process adding to it two additional stages: boundary documentation and boundary maintenance to build up a comprehensive approach. The direct involvement of the authors in boundary making processes between Israel and its neighboring countries facilitated assessment of the model elements and their full implementation in the Israel–Jordan Boundary making from negotiations, through demarcation in the field up to continuing bi-lateral maintenance operations.

Boundaries in digital spaces

Intuitively, a boundary in an N-dimensional digital space is a connected component of the (N − 1)-dimensional surface of a connected object. In this paper we make these concepts precise, and show that the boundaries so specified have properties that are intuitively desirable. We provide some efficient algorithms for tracking such boundaries. We illustrate that the algorithms can be used, in particular, for computer graphic display of internal structures (such as the skull and the spine) in the human body based on the output of medical imaging devices (such as CT scanners). In the process some interesting mathematical results are proven regarding “digital Jordan boundaries,” such as a specification of a local condition that guarantees the global condition of “Jordanness.”

Uniform estimates for polynomial approximation in domains with corners

Let G ⊂ C be a domain with a Jordan boundary ∂ G , consisting of l smooth curves Γ j , such that { z j } ≔ Γ j - 1 ∩ Γ j ≠ ∅ , j = 1 , … , l , where Γ 0 ≔ Γ l . Denote by α j π , 0 < α j ⩽ 2 , the angles at z j 's between the curves Γ j - 1 and Γ j , exterior with respect to G. Let Φ be a conformal mapping of the exterior C ⧹ G ¯ of G ¯ = G ∪ ∂ G onto the exterior of the unit disk, normed by Φ ′ ( ∞ ) > 0 . We assume that there is a neighborhood U of G ¯ , such that 0 < c ( G ) ⩽ ϕ ( z ) Φ ′ ( z ) ⩽ C ( G ) , z ∈ U ⧹ G ¯ , where ϕ ( z ) ≔ ∏ j = 1 l z - z j 1 - 1 α j , z ∈ C , z ≠ z j if α j ⩽ 1 . Set g G ≔ sup g ( z ) : z ∈ G . Then we prove Theorem. Let r ∈ N and 0 ⩽ β ⩽ r . If a function f is analytic in G and f ( r ) ϕ β G < + ∞ , then for each n ⩾ lr there is an algebraic polynomial P n of degree < n , such that ∥ ( f - P n ) ϕ β - r ∥ G ⩽ c ( r , G ) n r ∥ f ( r ) ϕ β ∥ G .

Approximation by p -Faber polynomials in the weighted Smirnov class Ep( G,ω ) and the Bieberbach polynomials

Let G\subset C be a finite domain with a regular Jordan boundary L . In this work, the approximation properties of a p -Faber polynomial series of functions in the weighted Smirnov class E p (G,ω) are studied and the rate of polynomial approximation, for f∈ E p ( G,ω) by the weighted integral modulus of continuity, is estimated. Some application of this result to the uniform convergence of the Bieberbach polynomials π n in a closed domain \overline G with a smooth boundary L is given.

On Integral Representations of an Axisymmetric Potential and the Stokes Flow Function in Domains of the Meridian Plane. II

We obtain new integral representations for an axisymmetric potential and the Stokes flow function in an arbitrary simply-connected domain of the meridian plane. The boundary properties of these integral representations are studied for domains with closed rectifiable Jordan boundary.

A justification of a fast surface tracking algorithm

In many three-dimensional (3D) imaging applications in science, medicine, and engineering, object information captured in a 3D digital image is visualized and analyzed based on object surfaces extracted from the image. Recently Gordon and Udupa ( Comput. Vision Graphics Image Process. 45, 1989, 196–214) reported a fast algorithm for tracking such surfaces, whose correct behavior has only been a conjecture but observed nonetheless in all uses of this algorithm in a variety of biomedical applications. In the present paper, we prove the correctness of their algorithm by showing that the surfaces generated by the algorithm correspond to connected components of boundary elements and that they possess Jordan boundary properties.