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This paper is concerned with the Riemann problem for bianalytic functions on a closed Jordan curve with a certain degree of fractality. We derive a solvability condition under which an explicit sol...

The standard cardioid is the set of points in the complex plane formed by reflecting the point 1 in every tangent to the unit circle. These points constitute a simple closed curve that is the bound...

We note an argument proving simultaneously Hopf's rotation angle theorem and the C1 Jordan curve theorem.

The existence of solutions of nonlinear Riemann–Hilbert problems for quasilinear -equations on the unit disc is considered. Let be a smooth function with bounded first derivatives and let be a family of Jordan curves in . Let be a smooth solution of the equation on such that belongs to the interior of the Jordan curve for every . Then there exists a smooth solution u of the equation on such that for every . Moreover, there is a sequence of solutions of this Riemann–Hilbert boundary value problem which uniformly on compact subsets of converges to .

We study linear mappings between C*-algebras A and B, which approximately satisfy Jordan multiplicativity condition and a *-preserving condition (that is, the so-called \epsilon-approximate Jordan *-homomorphisms). We first prove that every such a mapping is automatically continuous and we give the estimates of its norm, as well as the estimates of the norm of its inverse if it is bijective. If K(H_1) \subseteq A \subseteq B(H_1), K(H_2) \subseteq B \subseteq B(H_2), and \psi : A \to B is a bijective \epsilon-approximate Jordan *-homomorphism with sufficiently small \epsilon > 0, then either \psi^{; ; ; -1}; ; ; has a large norm, or \psi is close to a Jordan *-isomorphism, that is, to a mapping of the form X \mapsto UXU*, or X \mapsto UX^tU*, for some unitary U \in B(H_1, H_2). We also give the corresponding quantitative estimate.

We study Poletsky–Stessin Hardy spaces that are generated by continuous, subharmonic exhaustion functions on a domain , that is bounded by an analytic Jordan curve. Different from Poletsky and Stessin’s work these exhaustion functions are not necessarily harmonic outside of a compact set but have finite Monge–Ampere mass. We have showed that functions belonging to Poletsky–Stessin Hardy spaces have a factorization analogous to classical Hardy spaces and the algebra is dense in these spaces as in the classical case; however, contrary to the classical Hardy spaces, composition operators with analytic symbols on these Poletsky–Stessin Hardy spaces need not always be bounded

In two-dimensional Euclidean space, a simply connected unbounded domain whose boundary is a Jordan curve is called an unbounded Jordan domain. In this article, we discuss a piecewise linear finite element approximation of conformal mappings from the unit disk of the plane to unbounded Jordan domains. Some convergence results and error analysis are presented. Numerical examples are also given.

We study a pretopology on ℤ2 having the property that the Khalimsky topology is one of its quotient pretopologies. Using this fact, we prove an analogue of the Jordan curve theorem for this pretopology, thus showing that such a pretopology provides a large variety of digital Jordan curves. Some consequences of this result are discussed too.

This paper presents a decentralized controller to guide a group of aerial robots to converge to and to move along a simple closed curve specified in a three-dimensional environment. This curve may be considered as a perimeter to be surveilled by the robots. The solution presented in this paper is based on an artificial vector field modulated by a collision avoidance scheme and relies only on local sensing. Proofs of asymptotic stability of the proposed controller are devised for a team of kinematically controlled rotorcrafts. Experimental results with a group of autonomous quadrotors are presented to validate the applicability and performance of the approach.

Let G be a finite domain with z 0 ∈ G and bounded by a Jordan curve L ≔ ∂G. The Bieberbach polynomials π n , n = 1, 2, … , associated with the pair (G, z 0) can be used to approximate the conformal mapping φ0 from G to D(0, r 0) ≔ {w : |w

We consider the following singularly perturbed elliptic problem: where Ω is a bounded domain in ℝ2 with smooth boundary, ϵ > 0 is a small parameter, n denotes the outward normal of ∂Ω, and a, b are smooth functions that do not depend on ϵ. We assume that the zero set of a − b is a simple closed curve Γ, contained in Ω, and ∇(a − b) ≠ 0 on Γ. We will construct solutions u ϵ that converge in the Holder sense to max {a, b} in Ω, and their Morse index tends to infinity, as ϵ → 0, provided that ϵ stays away from certain critical numbers. Even in the case of stable solutions, whose existence is well established for all small ϵ > 0, our estimates improve previous results.

The formal mathematical definition of a Jordan curve (a non-self-intersecting continuous loop in the plane) is so simple that one is often lead to the unimaginative view that a Jordan curve is noth...

We present a method for creating simple closed curves that divide the plane into two regions that, when coloured differently from one another, resemble knots and links. By cutting along these curves with a laser or water jet cutter, we obtain two-piece sculptures ideal for illustrating the Jordan curve theorem.

An n-Venn diagram is a Venn diagram on n sets, which is defined to be a collection of n simple closed curves (Jordan curves) C1,C2, . . . ,Cn in the plane such that any two intersect in finitely many points and each of the 2n sets of the form ∩C i i is nonempty and connected, where i is one of “interior” or “exterior.” Thus the Venn regions are all bounded except for the region exterior to all curves; each bounded region is the interior of a Jordan curve. See [6] for much more information on Venn diagrams. An n-Venn diagram is symmetric if each curve Ci is ρ i (C1), where ρ is a rotation of order n about some center (we use O for the fixed point of rotation ρ). We use Boolean notation for combinations of sets, with the 0-1 string e1e2 . . . en representing ∩C i i , where i is interior (respectively, exterior) if ei = 1 (respectively, 0). Thus 111 . . . 1 represents F , the full intersection of all the interiors, 000 . . . 0 is the intersection of all the exteriors (the unbounded region), and 100 . . . 0 represents the set of points interior to C1 and exterior to the others. In a symmetric Venn diagram, rotation of a region by ρ corresponds to a rightward cyclic shift of the Boolean string. The universally familiar three-circle Venn diagram is symmetric, as is the one on two sets using two circles. For about 40 years a major open question was whether symmetric n-Venn diagrams exist for all prime n. Henderson found one for n = 5 and also (unpublished) for n = 7. Much later, Hamburger [3] settled the case of 11, which was quite complicated, and then in 2004 Griggs, Killian, and Savage [1] found an approach that works for all primes. So we now have the strikingly beautiful theorem that a symmetric n-Venn diagram exists if and only if n is prime. But there is a small problem: Henderson’s proof, which appears to be very simple, has a gap. Here is the proof from [4]. Suppose 1 ≤ k ≤ n − 1. Since a symmetric n-Venn diagram is symmetric with respect to a rotation of 2π/n, the regions corresponding to the Boolean strings with k 1s must come in groups of size n, each group consisting of one such region and its images under repeated rotation by 2π/n. Therefore n divides (n k ) . This concludes the proof because the only n for which this is true for the specified k-values are the primes (an easy-to-prove fact of number theory; see [5]). This is a very seductive argument. The primeness arises in such a cute way that one wants it to be true. Thus the proof has been repeated in many papers in the decades since it was first published. Yet there are problems. The proof does not call upon the connectedness of the Venn regions. Without connectedness the result is false; see Figure 1 (due to Grunbaum [2]), which shows a diagram satisfying all of the conditions

(2007). The Jordan Curve Theorem, Formally and Informally. The American Mathematical Monthly: Vol. 114, No. 10, pp. 882-894.

1. INTRODUCTION. Let T be a tile in the plane. By calling T a tile, we mean that T is a topological disk whose boundary is a simple closed curve. But also implicit in the word “tile” is our intent to use congruent or reflected copies of T to cover the plane without gaps or overlapping; that is, we want to tessellate the plane with copies of T . In a minor abuse of language, one often speaks of T (as opposed to copies of it) as tiling or tessellating the plane, in the sense that T generates a tessellation. A tessellation by T may or may not be possible, so in order to learn something of T ’s abilities with regard to tessellating the plane, we perform the following procedure: around a centrally placed copy of T , we attempt to form a full layer, or corona ,o f congruent copies of T . We require as part of the definition that no point of T should be visible from the exterior of a corona to a Flatland creature in this plane. Also, we should form the corona without allowing gaps or overlapping, just as if we were building a tessellation. If a corona can be formed, then we attempt to surround this corona with yet another corona, and then another, and so on; if we get stuck, we go back and change a previously placed tile and try again. If T tessellates the plane, then this procedure will never end. On the other hand, if T does not tessellate the plane, and if we check all of the possible ways of forming a first corona, a second corona, and so forth, we will find that there is a maximum number of coronas that can be formed. This maximum number of layers that can be formed around a single centrally placed copy of T is called the Heesch number of T and is denoted by H (T ). We consider a few examples before proceeding. Consider first a regular hexagon. All bees know that a regular hexagon tessellates the plane, so H =∞ for a regular

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.

function theory." (In later editions, he did include a proof of part of this theorem, to illustrate the power of the concept of winding number for closed curves. It is worth mentioning that the notion of winding number is never used in this article.) It turns out that the basic facts about topology in R2 can be explained efficiently by identifying the real plane with the complex plane (also known as the complex line). The key advantages in this approach, as we shall see, come not only from the direct presence of multiplication, but most of all from the availability of the exponential function. The big disadvantage, of course, is that some of the proofs do not transfer to the study of topology in R' for n > 2. It is the purpose of this article to provide an exposition of this method, which was introduced by Eilenberg [4], and in particular to provide a fairly short proof of the Alexander duality theorem in the plane, and thus an easy proof of the Jordan curve theorem. The Alexander duality theorem in the plane is proved in Eilenberg's paper, which however is rather long, including as it does his doctoral thesis. It is also to be found proved by this method in Dieudonne's book [3]. I believe the argument presented here to be simpler. Since first submitting this article to the MONTHLY, I have discovered that the proof given here of Theorem 23, which I had proudly believed to be my discovery, had in fact been found many years ago by Graham Allan, though never published. The reader of this article is assumed to have a basic knowledge of point set topology, to know what a group is, and to have a slight acquaintance with complex numbers, in particular to know that ez = ex(cosy + i sin y) when z = x + iy with x and y real numbers. The brief final section assumes an acquaintance with sheaf theory and Cech cohomology.

For meromorphic function F(z) new representation in the interior of a closed rectifiable Jordan curve passing through no poles of F(z), with the poles of F(z) σ I(C) deleted, is found. The representation is used to obtain new series expansions for meromorphic functions. Examples of such expansions are given. The expansions obtained can be of value both in theoretical usage and in numerical calculations of values of meromorphic functions, and in applications of such functions.

This paper considers boundaries of simply connected domains in C that admit an infinite number of intrinsic rotations about some point. The resuIts have applications to the unsolved problem of whether the boundary of a Siegel disk of a rational function must be a Jordan curve.

The second difference is important in the study of differentiability properties of functions of a real variable. We consider a similar quantity to describe the geometry of a Jordan curve C in the plane, with me aim of relating it to the existence ot tangents to C. Here we establish conditions that guarantee that C is a quasicircle or an asymptotically conformal curve.

We give an elementary proof of the Jordan Brouwer separation theorem for smooth hypersurfaces using the divergence theorem and the inverse function theorem.

In the present paper we study complex spline functions, which are defind as piecewise polynomial functions on a Jordan curve Г in the complex plane. Such splines we first introduced by Ahlberg and Nilson in 1965 [2] and further studied by Ahlberg., Nilson and Walsh [1], [3]-[5] and Schoenberg [9]. All these papers were mainly concerned with the interpolatory properties of complex splines. In contrast to the real case, there exist only very few results on complex B-splines; these are functions, which provide a numeically stable basis of the spline space S m.k . Although they are used in some papers (e.g. [6], [7], [9]), their main properties, for example the important recursion formula (3.8). were not proved until now. The reason for this might be that in the cited papers B-splines were defined as divided differences, applied to a certain truncated power function, a definition which is not at all easy to deal with. In this paper we present a more axiomatic approach to the B-splines B mv , which makes it po...

This paper contributes to the theory of quasiextremal distance domains. We present some new properties for these domains and point out results concerning the extension of quasiconformal homeomorphisms.For example, we establish a continuity property for mod(E, F; D) and use this to demonstrate that mod(E, F; D) = cap(E, F; D) whenever D is a QED domain and E, F are disjoint compacta in D. Our final result is that if each boundary component of a plane domain is either a point or a Jordan curve and if the domain satisfies a boundary quasiextremal distance property, then there exists a quasiconformal self-homeomorphism of the entire plane which maps the given domain conformally onto a circle domain.

Let be the unit disk, and let G be a doubly connected region bounded by the unit circle and a Jordan curve . We defined [9] a point system on C be an extremal property, which turned out to be very efficient to approximate the conformal mapping f of onto G. Here we show that is equally distributed on . If C is an analytic Jordan curve. Further, we show how to apply this point system to conformal mappings not only for regions of type G, but also to the case that the doubly connected region is bounded by two Jordan curves. As an example, we numerically approximate the conformai mapping of an annulus onto a doubly connected region bounded by “concentric squares”.

All known proofs of the easy part of (2) (that K5 and K3 3 cannot be embedded in the plane) are based on (1). [1] contains a short proof of the easy part of (2) based only on the restriction of (1) to polygonal curves. Conversely, it is shown in [2] how (1) is an easy consequence of the easy part of (2). Thus there is a close relation between the two theorems, and H. Wilf raised the question if this relationship holds in other topological spaces as well. Here we shall answer the question in the affirmative for those topological spaces that cannot be separated by a simple arc. For spaces of low connectivity the relationship disappears. Every graph (which is defined in the next section) may be thought of as a topological space where the edges are simple arcs. Then a nonseparating, simple closed curve is the same as what in [3] is called a nonseparating induced cycle. The investigations in [3] show that the existence of such cycles depends more on edge density than planarity (or nonplanarity). One basic result in [3] says that every finite connected graph of minimum valency > 3 has a nonseparating induced cycle. For infinite graphs not much is known about nonseparating induced cycles.

The holomorphic solution φ of the Hilbert boundary value problem Re (a + ib) φ = g on γ in a disk D bounded by the simple closed curve γ has been solved in the space of Holder-continuously differentiable functions C1, C1,α (D) by many authors. Under the assumptionthat g belongs to the Slobodecky space Ws,p (γ), s = 1 − 1/p,1 < p < ∞ it is shown here that the problem has a uniquesolution in the Sobolev space W1,p: (D). An a-priori estimate for the norm of in W1 p(D)is given.

Let be a given bijective analytic mapping of the circumference C of the unit circle onto itself. By the uniformization theorem, there exists an analytic Jordan curve Г with the following property: there are two bijective conformal mappings F (interior of C→ interior of Г) and G (exterior of C → exterior of Г) such that for all . We consider this conformal welding under the assumption that G(∞) = ∞. Our main result is a relation (Theorem 2) which will allow—at least in certain cases—an estimate of the rotation of Г in terms of the given function Ф.

For F analytic in let for and an infinite set. Δ(F) consists of a Jordan curve γ⊂D and a finite number of points in the bounded component Г of . It is shown that .

The Jordan curve theorem is one of those frustrating results in topology: it is intuitively clear but quite hard to prove. In this note we will look at two discrete analogs of the Jordan curve theorem that are easy to prove by an induction argument coupled with some geometric intuition. One of the surprises is that when we discretize the plane we get two Jordan curve theorems rather than one, a consequence of the interplay between two natural products in the category of graphs. Topology in this context has been studied by Farmer in [2]. To state the discrete versions, we need to know what the discrete analog of the plane is and what plays the role of a simple closed curve. Since the plane is the topological product of two lines, we take as our discrete analog the product of two discrete lines. We will use undirected graphs for our analogs of spaces, with vertices for points and edges connecting points which are to be thought of as touching.

(1969). The Jordan Curve Theorem for Piecewise Smooth Curves. The American Mathematical Monthly: Vol. 76, No. 6, pp. 605-610.

1. We shall consider here the isoperimetric problemfor convex regions in the plane and its extension to more general sets. Part I is completely elementary as regards the methods used and presupposes next to nothing on the part of the reader. In it we prove the existence and uniqueness of the solution to the isoperimetric problem in the plane, restricted to convex regions. The uniqueness of a solution, if one exists, can quite easily be extended to regions bounded by a Jordan curve, but the existence becomes a little more difficult to show. In the second part we consider the existence proof for these more general regions, and al-so formulate and prove a very general isoperimetric theorem valid for arbitrary measurable sets in the plane. This part presupposes a very little knowledge of point set topology and measure theory, and a few statements are left unproved. Some of the places whiere tile words "clearly" or "obviously" are used are admittedly actual gaps in the exposition. Tle author feels, however, that these gaps can be easily filled in by the intelligent reader by straightforward, if perhaps long, reasoning.

Let Pn(z) be a polynomial* of degree n in z = x+iy and let j P (z) j < M on a set E, where M is a constant independent of n and z. The author has shownt that if the set E is bounded by an analytic Jordan curve C then j P ' (z)1 < K(C) MAln, where K(C) is a constant depending only on C. If C is the unit circle we know by a theorem of M. Rieszt that K(C) = 1. Here we will show that if C is an ellipse with semi-axes a and b. a _ b, then K(C) ? 1/b. Let us suppose that the ellipse C has its vertices in the points a and -a on the real axis. Then for a point z = x +iy on C we have x = a cos 0, y = b sin 0, and