Jordan Curve Research Articles

Free energy of the two-matrix model/dToda tau-function

We provide an integral formula for the free energy of the two-matrix model with polynomial potentials of arbitrary degree (or formal power series). This is known to coincide with the τ-function of the dispersionless two-dimensional Toda hierarchy. The formula generalizes the case of conformal maps of Jordan curves studied by Kostov, Krichever, Mineev-Weinstein, Wiegmann, Zabrodin and separately Takhtajan. Finally we generalize the formula found in genus zero to the case of spectral curves of arbitrary genus with certain fixed data.

A novel classification method based on hypersurface

The main idea of the support vector machine (SVM) classification approach is mapping the data into higher-dimensional linear space where the data can be separated by hyperplane. Based on the Jordan curve theory, a general nonlinear classification method by the use of hypersurface is proposed in this paper. The separating hypersurface is directly used to classify the data according to whether the number of intersections with the radial is odd or even. In contrast to the SVM approach, the proposed approach has no need for mapping from lower-dimensional space to higher-dimensional space. Furthermore, the approach does not use kernel functions and it can directly solve the nonlinear classification problem via the hypersurface. Numerical experiments showed that the proposed approach can efficiently and accurately solve the classification problems with a large amount of data.

Unavoidable Configurations in Complete Topological Graphs

A topological graph is a graph drawn in the plane so that its vertices are represented by points, and its edges are represented by Jordan curves connecting the corresponding points, with the property that any two curves have at most one point in common. We define two canonical classes of topological complete graphs, and prove that every topological complete graph with n vertices has a canonical subgraph of size at least clog1/8n, which belongs to one of these classes. We also show that every complete topological graph with n vertices has a non-crossing subgraph isomorphic to any fixed tree with at most clog1/6n vertices.

Bifurcation to an entire function

In this paper we will consider a bifurcation that occurs in the topology of the Julia set of meromorphic maps as they become entire. For some meromorphic maps the Julia set will be a Cantor set. We will investigate how the Julia set changes as these meromorphic maps approach the entire function whose Julia set is a Cantor bouquet. Other meromorphic maps have a Julia set that is a Jordan curve. Again we will study how this curve changes as these meromorphic maps approach the entire function whose Julia set is a Cantor bouquet.

On Macbeath-Singerman symmetries of Belyi surfaces with PSL(2,p) as a group of automorphisms

The famous theorem of Belyi states that the compact Riemann surface X can be defined over the number field if and only if X can be uniformized by a finite index subgroup Γ of a Fuchsian triangle group Λ. As a result such surfaces are now called Belyi surfaces. The groups PSL(2,q),q=pn are known to act as the groups of automorphisms on such surfaces. Certain aspects of such actions have been extensively studied in the literature. In this paper, we deal with symmetries. Singerman showed, using acertain result of Macbeath, that such surfaces admit a symmetry which we shall call in this paper the Macbeath-Singerman symmetry. A classical theorem by Harnack states that the set of fixed points of a symmetry of a Riemann surface X of genus g consists of k disjoint Jordan curves called ovals for some k ranging between 0 and g+1. In this paper we show that given an odd prime p, a Macbetah-Singerman symmetry of Belyi surface with PSL(2,p) as a group of automorphisms has at most

One-dimensional Singular Integral Equations

A class of singular integral equations having solution in closed form is considered. This class contains the characteristic equation and other equations associated with it. Equations are solved by their reduction to some Riemann boundary value problems in generalized Hölder spaces Hφ (Γ), where Γ is a closed rectifiable Jordan curve for which the Plemelj–Privalov theorem holds.

Analysis of an epidemic model with bistable equilibria using the Poincaré index

In most disease transmission models, disease eradication depends on a certain threshold quantity ( R 0 ), known as the basic reproductive number, such that if R 0<1 , then the disease-free equilibrium is stable and the disease can be eradicated from the population. However, several recent studies have shown that reducing R 0 to values less than unity is not sufficient to control the spread of a disease. These studies have considered epidemic models that have bistable equilibria, where both the disease-free equilibrium and an endemic equilibrium are locally asymptotically stable. This paper proposes a technique for analysing the models that exhibit such dynamics. It is shown that the stability of the equilibria for such models can be established using the Poincaré index of a piecewise Jordan curve defined as the boundary of a positively invariant region for the model. An application of this result to a vaccination model for the transmission dynamics of an infectious disease is given.

On One Theorem of S. Warschawski

Abstract A theorem of S. Warschawski on the derivative of a holomorphic function mapping conformally the circle onto a simply-connected domain bounded by the piecewise-Lyapunov Jordan curve is extended to domains with a non-Jordan boundary having interior cusps of a certain type.

Realization of Virasoro unitarizing measures on the set of Jordan curves

Two univalent functions are equivalent, f∼ g, if they have the same Schwarzian derivative. The equivalence relation ∼ being defined up to an homographic transformation, it gives an isomorphism between the manifold J of Jordan curves and the quotient manifold S . It permits to obtain vector fields on S and on J . The action of these vector fields on the Neretin polynomials is explicited. The existence of a unitarizing measure on the quotient manifold S is discussed and for such a measure, orthogonality relations for the Neretin polynomials are obtained. This work is a concrete realization on the complex space C ∞ of the abstract quotient Diff(S 1)/ SL(2, R ) considered in Airault et al., C. R. Acad. Sci. Paris, Ser. I 335 (2002) 621–626. To cite this article: H. Airault, V. Bogachev, C. R. Acad. Sci. Paris, Ser. I 336 (2003).

An alternative digital topology

Abstract We define a new topology on Z2 with respect to which, in difference to the commonly used Khalimsky topology, the curves fulfilling an analogy of the Jordan curve theorem can turn at the acute angle π/4. We also present a natural graph on Z2 all circuits in which fulfill the analogy of the Jordan curve theorem with respect to the topology defined.

Zero Distribution of Bergman Orthogonal Polynomials for Certain Planar Domains

Abstract. Let G be a simply connected domain in the complex plane bounded by a closed Jordan curve L and let P n , n≥ 0 , be polynomials of respective degrees n=0,1,··· that are orthonormal in G with respect to the area measure (the so-called Bergman polynomials). Let ϕ be a conformal map of G onto the unit disk. We characterize, in terms of the asymptotic behavior of the zeros of P n 's, the case when ϕ has a singularity on L . To investigate the opposite case we consider a special class of lens-shaped domains G that are bounded by two orthogonal circular arcs. Utilizing the theory of logarithmic potentials with external fields, we show that the limiting distribution of the zeros of the P n 's for such lens domains is supported on a Jordan arc joining the two vertices of G . We determine this arc along with the distribution function.

Differentiability properties of some nonlinear operators associated to the conformal welding of Jordan curves in Schauder spaces

Differentiability properties of some nonlinear operators associated to the conformal welding of Jordan curves in Schauder spaces

A bundle view of boundary-value problems: generalizing the Gardner–Jones bundle

Holomorphic families of linear ordinary differential equations on a finite interval with prescribed parameter-dependent boundary conditions are considered from a geometrical viewpoint. The Gardner–Jones bundle, which was introduced for linearized reaction–diffusion equations, is generalized and applied to this abstract class of λ-dependent boundary-value problems, where λ is a complex eigenvalue parameter. The fundamental analytical object of such boundary-value problems (BVPs) is the characteristic determinant, and it is proved that any characteristic determinant on a Jordan curve can be characterized geometrically as the determinant of a transition function associated with the Gardner–Jones bundle. The topology of the bundle, represented by the Chern number, then yields precise information about the number of eigenvalues in a prescribed subset of the complex λ-plane. This result shows that the Gardner–Jones bundle is an intrinsic geometric property of such λ-dependent BVPs. The bundle framework is applied to examples from hydrodynamic stability theory and the linearized complex Ginzburg–Landau equation.

Foliations of Hyperbolic Space by Constant Mean Curvature Surfaces Sharing Ideal Boundary

Let γ be a Jordan curve in S 2, considered as the ideal boundary of H 3. Under certain circumstances, it is known that for any c E (−1, 1), there is a disc of constant mean curvature c embedded in H 3 with γ as its ideal boundary. Using analysis and numerical experiments, we examine whether or not these surfaces in fact foliate H 3, and to what extent the known conditions on the curve can be relaxed.

Positive Solutions of Nonlinear Elliptic Equations in Unbounded Domains in R 2

We study the existence of positive solutions of the nonlinear equation Δu+f(⋅,u)=0, in D with u=0 on ∂D, where D is an unbounded domain in R2 with a compact nonempty boundary ∂D consisting of finitely many Jordan curves. The aim is to prove an existence result for the above equation in a general setting by using potential theory.

Existence and Uniqueness of F-Minimal Surfaces

We extend a classical result of Rado and Kneser concerning uniqueness ofminimal surfaces bounded by a given closed Jordan curve Γ inℝ3 to the case of extremals for certain geometric variationalintegrals. Using standard elliptic PDE theory, this gives the existenceand uniqueness of embedded F-minimal surfaces for suitable boundarycurves that project simply onto the boundary of a plane convex domain.

On a theorem of Gyula Pál

The proof of the following theorem of Gy. Pal is simplified: Each locally connected planar continuum with at least two points is the orthogonal projection of a closed Jordan curve of the Euclidean 3-space.

The Douglas problem for parametric double integrals

Let \(\) be a parametric variational double integral and γ ⊂ ℝn be a system of several distinct Jordan curves. We prove the existence of multiply connected, conformally parametrized minimizers of \(\) spanned in γ by solving the Douglas problem for parametric functionals on multiply connected schlicht domains. As a by-product we obtain a simple isoperimetric inequality for multiply connected \(\)-minimizers, and we discuss regularity results up to the boundary which follow from corresponding results for the Plateau problem.

Non-rectifiable limit sets of dimension one

We construct quasiconformal deformations of convergence type Fuchsian groups such that the resulting limit set is a Jordan curve of Hausdorff dimension 1, but having tangents almost nowhere. It is known that no divergence type group has such a deformation. The main tools in this construction are (1) a characterization of tangent points in terms of Peter Jones' \beta 's, (2) a result of Stephen Semmes that gives a Carleson type condition on a Beltrami coefficient which implies rectifiability and (3) a construction of quasiconformal deformations of a surface which shrink a given geodesic and whose dilatations satisfy an exponential decay estimate away from the geodesic.

Bounds on the chromatic number of intersection graphs of sets in the plane

In this paper we show that the chromatic number of intersection graphs of congruent geometric figures obtained by translations of a fixed figure in the plane is bounded by the clique number. Further, in the paper we prove that the triangle-free intersection graph of a finite number of compact connected sets with piecewise differentiable Jordan curve boundaries is planar and hence, is 3-colorable.