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LetC be an analytic Jordan curve, letG be the interior ofC, and letU (w) be (at least) continuous onC. Here the solution of the Dirichlet problemu(w) which coincides withU(w) onC is approximated by harmonic polynomials. These harmonic polynomialsF n F(w) are determined by interpolatingU(w) in a given point system. For sufficiently greatn we prove |u(w)−F n (w)|≤K·logn·E n in\(\bar G\), where\(E_n = \mathop {\max }\limits_{w \in C} \left| {U(w) - h_n (w)} \right|\) andh n (w) is the harmonic polynomial of degreen of best approximation toU(w) onC andK is a constant independent ofn.

LetC be a closed Jordan curve in the complex plane and letf(z)=dz+a0+a1z−1+… be the analytic function mapping |z|>1 schlicht onto the exterior ofC (d>0 is the transfinite diameter ofC). Similar to the Fekete points a point system will be defined calledextremal points. One can use the Fekete points or the extremal points to approximated. The author has proved [3] that in the case of an analytic closed Jordan curve the approximation ofd by means of extremal points is much better than the approximation ofd by the use of Fekete points. Here we show how to approximated by means of extremal points in the case of a piecewise analytic, closed Jordan curve possessing corners of openingαπ (0

Gaier [3] has given an account of the different constructive methods of approximating the conformal mapping. Our method relates to the methods of extremal points by Fekete [1] and Leja [6]. These point systems can be used to approximate the conformal mapping. In the case of a domain bounded by a closed analytic Jordan curve, however, the point system we deal with has much better properties of approximating the conformal mapping than the other point systems. (Compare [9], Pommerenke [12].) We define an algorithm to compute our point system and we finally take three curves for which we compute the point system with the help of which we get an approximation of the conformal mapping. The numerical results are rather good.

We complete the results of [5] by the following theorem: Let G be a domain bounded by a closed piecewise smooth Jordan curve possessing corners with angles απ (o

In the theory of pseudoanalytic functions one can define (pseudoanalytic) rational functions, especially polynomials called “pseudopolynomials”. (See Bers [3], [4], Vekua [12]) Therefore it can be developed a theory of approximation and interpolation by rational functions. First results have been published by Bers [3] (Runge's theorem), Ismailov and Taglieva [8]. Let G be a domain of the complex plane bounded by a closed Jordan curve, let w(z) be pseudoanalytic in G. In this paper we deal with a relation between the behaviour of w(z) on C (Holder-continuity) and the degree of approximation of w(z) by pseudopolynomials. The results correspond to certain theorems of Curtiss, Sewell and Walsh in the theory of analytic functions.

A homeomorphism f f of the two-sphere S 2 {S^2} onto itself is defined to be pointwise almost periodic (p.a.p.) if the collection of orbit closures forms a decomposition of S 2 {S^2} . It is shown that if f : S 2 → S 2 f:{S^2} \to {S^2} is p.a.p. and orientation-reversing then the set of fixed points of f f is either empty or a simple closed curve; if f : S 2 → S 2 f:{S^2} \to {S^2} is p.a.p. orientation-preserving and has a finite number of fixed points, then f f is shown to have exactly two fixed points.

Given a function ƒ(z), continuous on C: |z| = 1 in the complex plane, there is a close analogy between approximation in the sense of least squares by polynomials on the unit circle and interpolation by polynomials in the nth roots of unity to the same function. For detailed discussion of the problem and its generalization for a suitable Jordan curve one can refer to Walsh (3) or to a recent paper by Curtiss (2). More recently, Curtiss (1) has considered the problem of interpolation by polynomials in non-equally spaced points on the unit circle and has pointed out the limitations inherent in the problem.

This theorem provides a complement to Theorem VI of [1] which states that if G is a continuous collection of nondegenerate compact continuous curves in the plane which is a compact continuum with respect to its elements as points, then G is a hereditary continuous curve such that the closure of its set of emanation points is totally disconnected. Some notation similar to that of [1 ] will be used. A simple chain is a finite collection x1, x2, * * *, xn of open discs (interiors of simple closed curves) such that xi x; exists if and only if I i-il < 1 and is a closed disc (a simple closed curve plus its interior) if it does exist. A (simple) chain C is said to simply cover a set M if C* contains M but for no proper subchain C' of C does the closure of C'* contain M.' Two chains, C and C', are said to be mutually exclusive if C* and C'* are mutually exclusive. The chain C is said to be a closed refinement of the chain C' provided that each link of C' contains the closure of some link of C and the closure of each link of C is contained in some link of C'. PROOF OF THEOREM. We note first that only a countable number of the elements of G contain triods. (See [5].) Denote the elements of some subset of G containing these elements by gl, g2, * C C . Each of the remaining elements of G is an arc or a simple closed curve. If G is not an arc or a simple closed curve with respect to its elements as points, it has an emanation element. Let H denote the closure of the set of emanation elements of G. It has been noted that, by Theorem VI of [1], H is totally disconnected. Let T be an arc of elements of G and suppose that the non-endelement t of T is an isolated element of T H. There is a subarc T1 of

The above two theorems suggest a procedure for showing that a given condition restricting the embedding of S implies S is tame from U. Namely, it may be possible to use the condition to slightly adjust a map f from a disk D into UUF so that the new image of D lies entirely in U while fI Bd D is unaltered. The facts that f(D)C\F is compact 0-dimensional and F lies in S-UXi may also be helpful while adjusting f. The above technique is employed in this paper to answer in the affirmative the following question asked in [I] and [5]. Is a 2-sphere in E3 tame if it can be pierced at each arc with a tame disk? Other illustrations of this procedure may be found in [7] and [8]. DEFINITION. A disk D is said to pierce sphere S at arc A CS if Bd A CBd D, Int A CInt D and the two components of D-A lie in different complementary domains of S. DEFINITION. If J is a simple closed curve in 2-sphere S and U is a complementary domain of S then J can be collared from U by A if A is an annulus such that JCBd A and A -JC U. The reader is referred to [21 and [3 1 for definitions of other terms used in this paper.

(1. 1) (1 + 2)oxx 2oxy4xy + (1 + ox)lo = 0. In this paper we deal with another kind of boundary value problem for the same equation. We are looking for a solution defined in a domain exterior to a simple closed curve P, possessing at the point at infinity a singularity, having on TP continuous partial derivatives, and satisfying either of the homogeneous boundary conditions (1.2) W: a4/9n = 0, 3: 4 = 0 on 'P. We shall not require, in general, that 45(x, y) be single-valued, but shall assume that (1.3) w = oxic/ is. We shall also prescribe the value of (1.4) qmax = max I w|. It will be convenient to associate with every solution of (1.1) the function (2)

A transformation, which maps the exterior or the interior of a simple closed curve with corners into the exterior or interior respectively of a simple smooth (corner-free) closed curve, is introduced. Symmetry properties are shown to be preserved by the transformation and a numerical procedure for applying the proposed transformation to an arbitrary curve is presented.

This paper considers an optimal-control problem for the dynamics of the Vidale-Wolfe advertising model, the optimal control being the rate of advertising expenditure to achieve a terminal market share within specified limits in a way that maximizes the present value of net profit streams over a finite horizon. First, the special polar cases of fixed and free end points are solved with and without an upper limit on advertising rate. The complete solution to the general problem is then constructed from these polar cases. The fixed-end-point case with no upper limit on the advertising rate is solved by using Green's theorem, while the other cases require additional use of switching-point analysis based on the maximum principle. The optimal control is characterized by a combination of bang-bang, impulse, and singular control, with the singular arc forming a turnpike.

It is the object of the present note to indicate that under suitable conditions (Theorem 1) various degrees of approximation of pn(z) to f(z) on C imply certain continuity properties of f(z), and (Theorem 2) under suitable conditions equations (4) and (5) in more precise form are valid even on C itself. We consider also (Theorems 3, 4, 5) approximation on C by analytic functions more general than polynomials pn(z), and prove the analogues of (4) and (5) in more precise form. We mention briefly (Theorems 6 and 7) approximation in a multiply-connected region.

Let P be a closed region in the plane bounded by a simple closed curve, and let S be a simplicial dissection of P. We may say that S is a dissection of P into a finite number α of triangles so that no vertex of any one triangle is an interior point of an edge of another. The triangles are ‘'topological” triangles and their edges are closed arcs which need not be straight segments. No two distinct edges of the dissection join the same two vertices, and no two triangles have more than two vertices in common.

Green's theorem, for line integrals in the plane, is well known, but proofs of it are often complicated. Verblunsky [1] and Potts [2] have given elegant proofs, which depend on a lemma on the decomposition of the interior of a closed rectifiable Jordan curve into a finite collection of subregions of arbitrarily small diameter. The following proof, for the case of Riemann integration, avoids this requirement by making a construction closely analogous to Goursat's proof of Cauchy's theorem. The integrability of Qx—Py is assumed, where P(x, y) and Q(x, y) are the functions involved, but not the integrability of the individual partial derivatives Qx, and py this latter assumption being made by other authors. However, P and Q are assumed differentiable, at points interior to the curve.

The approximation to complex-valued functions, continuous on a closed Jordan curve by polynomials of degree n, whose uniform norm on that curve is greater than or equal to some prescribed constant, is investigated. The limits for the resultant sequence of the best such deviations are found for a large class of functions.

It is proved that every centrally symmetric simple closed curve on the boundary of a centrally symmetric convex body in a three-dimensional linear space possesses an inscribed concentric affinely regular hexagon. This result is used to settle affirmatively a conjecture in [2] about the metric structure of the unit spheres of three-dimensional normed space.

A complete proof is given for Schnirelmann’s theorem on the existence of a square inC 2 Jordan curves. The following theorems are then proved, using the same method: 1. On every hypersurface inR n,C 3-diffeomorphic toS n−1, there exist 2n points which are the vertices of a regular 2 n -cellC n. 2. Every planeC′ Jordan curve can beC′ approximated by a curve on which there are 2N distinct points which are the vertices of a centrally symmetric 2N-gon (anglesπ not excluded). 3. On every planeC 2 curve there exist 5 distinct points which are the vertices of an axially symmetric pentagon with given base anglesa, π/2≦a

While a Jordan curve in Euclidean 3-space always spans a minimal surface of the type of the circular disc, the existence of a minimal surface of the type of the circular annulus, bounded by a system of two disjoint Jordan curves, very much depends on the configuration of these curves. A sufficient criterion for the existence has been given by J. Douglas in a famous paper [3] and, in different form, by R. Courant (see [2], esp. chapter IV). Roughly speaking, this condition guarantees the existence of a minimal surface of the type of the circular annulus, bounded by two Jordan curves Pi and P~ without common points, provided that these curves span a doubly-connected surface whose area is smaller than the sum of the areas ot the two simply-connected surfaces of least area bounded by P i and P~, respectively. Douglas' condition can be verified in many concrete cases. Nevertheless, even if Douglas' condition is violated, the problem of finding a minimal surface of the type of the annulus, bounded by two Jordan curves, may still be solvable. This fact is illustrated with the classical experiment where the curves Pi and P2 are coaxial unit circles in parallel planes, say the planes z = h and z = h (h ~ 0). (See for instance the presentation in (i. A. Bliss [1]). For h < ht : 0 . 5 2 7 7 . Douglas' condition is fulfilled. For ht ~< h ~ h~ = 0.6627... Douglas' condition is violated, but the circles Pi and r~ still span a doubly-connected minimal

In this paper we deal with trivalent planar maps in which the boundary of each country (or “face“) is a simple closed curve. One vertex is distinguished as the root and its three incident edges are distinguished as the first, second, and third major edges. We determine the average number of Hamiltonian polygons, passing through the first and second major edges, in such a “rooted map” of 2n vertices. Next we consider the corresponding problem for 3-connected rooted maps. In this case we obtain a functional equation from which the average can be computed for small values of n.

A simple closed curve is the simplest example of a compact, nondegenerate, homogeneous continuum. If a bounded, nondegenerate, homogeneous plane continuum has any local connectedness property, even of the weakest sort, it is known to be a simple closed curve [1, 2, 3].1 The recent discovery of a bounded, nondegenerate, homogenous plane continuum which does not separate the plane [4, 5] has given substance to the old question as to whether or not such a continuum must be indecomposable. Under certain conditions such a continuum must contain an indecomposable continuum [6]. It is the main purpose of this paper to show that every bounded homogeneous plane continuum which does not separate the plane is indecomposable. NOTATION. If M is a continuum and x is a point of M, U. will be used to denote the set of all points z of M such that M is aposyndetic at z with respect to x.2 It is evident that U. is an open subset of M.

If d d is a strongly convex metric on E 3 {E^3} and C C is a simple closed curve in E 3 {E^3} such that C C is the union of three line segments then C C is unknotted.

Consider the Cauchy transform \[T\varphi (z) = \frac{1}{{\pi i}}\int_\Gamma {\frac{{\varphi (\zeta )d\zeta }}{{\zeta - z}},} \quad z \in \Gamma ,\] in which $\Gamma $ is a simple closed curve with a continuously differentiable parameterization. In order to have direct methods for the numerical solution of equations involving $T\varphi (z)$, it is desirable to have numerical integration methods for evaluating $T\varphi (z)$. In this paper, numerical methods are investigated which are based on replacing $\varphi (z)$ by a uniformly convergent sequence $\varphi _n (z)$; if these approximations are “sufficiently smooth”, then the speed of convergence of $T_n \varphi $ to $T\varphi $ is “essentially the same” as the speed of convergence of $T\varphi _n $ to $\varphi $. Specific methods are analyzed, with $\varphi _n (z)$ defined as a piecewise linear or quadratic interpolating function to $\varphi (z)$ at a given set of nodes on $\Gamma $. Numerical examples conclude the paper.

We discuss some open questions especially concerning the coincidence set and the curve of separation in two-dimensional variational inequalities.

The purpose of this paper is to consider the class of continuous collections of mutually exclusive compact continuous curves in the plane. Throughout this paper we shall denote by G or G with a subscript or superscript a collection of this class with the property that G with respect to its elements as points is a nondegenerate compact closed point set. G has then the significance of being both a collection of continua and a point set itself. By a continuous curve will be meant a nondegenerate locally connected compact continuum. By a continuous collection will be meant a collection which is both upper and lower semi-continuous. By a (simple) chain will be meant a finite collection xi, x2, * *, x. of open discs (i.e. interiors of simple closed curves) such that i,.j exists if and only if j i-ijI ?1 and is the closure of an open disc (i.e. a 2-cell) if it does exist. The xi are called links of the chain. A subchain of a chain c is a chain whose links are links of c. A chain c will be said to simply cover a set M if c* contains M and if for no proper subchain c' of c does the closure of c'* contain M. Two chains will be said to be mutually exclusive if no link of either intersects any link of the other. A collection C' of sets is said to be a (closed) refinement of a collection C of sets if (the closure of) each element of C' is a subset of some element of C. An emanation point of a continuum M is a point which is the common part of each pair of some three nondegenerate subcontinua of M. A hereditary continuous curve is a continuous curve each of whose nondegenerate subcontinua is a continuous curve. It is immediately clear that if G is connected, G contains uncountably many elements and that only countably many can contain triods [1]. Except for a countable number of elements, each element of G must be either an arc or a simple closed curve. We denote the elements of G which are neither arcs nor simple closed curves by g1, g2, g3, * . . From the hypothesis of continuity of G it follows immediately that no element of a connected G contains a 2-cell.

The purpose of this note is to establish a bound on the number of edges on a face of an arrangement of curves in the plane, and to correct thereby an error in an earlier formulation by Grunbaum.

Let F ( z ) = { u ( z ) , v ( z ) , w ( z ) } , | z | > 1 F(z) = \{ u(z),v(z),w(z)\} ,|z| > 1 , represent a minimal surface spanning the curve Γ : { U ( s ) , V ( s ) , W ( s ) } , s \Gamma :\{ U(s),V(s),W(s)\} ,s being the arc length. Suppose Γ \Gamma has a tangent at a point P P . Then F ( z ) F(z) is differentiable at this point if U ′ ( s ) , V ′ ( s ) , W ′ ( s ) U’(s),V’(s),W’(s) satisfy a Dini condition at P P .

Schoenfliest was the first to formulate the converse of the fundamental theorem of C. Jordant that a simple closed curve? lying wholly within a plane decomposes the plane into an inside and an outside region. The statement of this converse theorem is as follows: Suppose K is a closed, bounded set of points lying in a plane S and that S K = Sl +-S2, where Si and S2 are point-sets such that (1) every two points of Si (i = 1, 2) can be joined by an arc lying entirely in Si (2) every arc joining a point of Si to a point of S2 contains at least one point of K (3) if 0 is a point of K and P is a point not belonging to K, then P can be joined to 0 by an arc that has no point except 0 in common with K. Every point-set that satisfies these conditions is a simple closed curve. Schoenflies used metrical hypotheses in his proof. Lennes gave a proof of this converse theorem using straight lines. R. L. Moore pointed out that a proof similar in large part to that of Lennes can be carried through with the use of arcs and closed curves on the basis of his system of postulates 23, thus furnishing a non-metrical proof of the converse theorem.? In all these proofs the condition numbered three, the condition of approachability (erreichbarkeit) plays a fundamental role. It is the purpose of the present paper to study the effect of substituting for the condition of approach

In this paper I propose to show that any plane satisfying Veblen's Axioms I-VIII, XI of his System of axioms for geometry^ is a number-plane, or in other words that any plane V satisfying these axioms contains a system of continuous curves such that, with reference to these curves regarded as straight lines, the plane V is an ordinary euclidean plane. In consequence, any discussion of analysis situs based on these axioms (as, for example, Veblen's proof I of the theorem that a Jordan curve divides its plane into just two parts) is no more general than one based on analytic hypotheses. This does not contradict the fact that the geometry based on axioms I-VIII, XI is much more general than euclidean geometry in the sense that the curves with respect to which the plane V is euclidean are not necessarily the straight lines referred to in these axioms. My argument also furnishes the answer to a problem proposed by Veblen in another paper. He states this problem as follows.§ "An interesting situation is obtained by introducing a postulate of uniformity among the hypotheses of plane analysis situs (cf. p. 84 of this volume). If the postulate is applied to the straight line, the line is necessarily a continuum but it is not obvious that other curves are. If it is applied to the plane, the segments

ion being made, in case the representation g is improper of the first kind, of the values of 0, at most denumerably infinite in number, where gi(0) is discontinuous. (19.9) rests on the fact (whose proof is trivial) that if fi(m)(t) tends uniformly to the continuous fi(t) when m-* oo, then if tm -t as m-> oo, we have lim fi(m) (t) =fi(t) for m-> oo . The assertion is now easily proved that if (19.10) Xi= =WFi(w) This content downloaded from 157.55.39.128 on Mon, 18 Jul 2016 05:43:48 UTC All use subject to http://about.jstor.org/terms 304 JESSE DOUGLAS [January are the harmonic functions determined by gi(O), then (19.1 1) Z1F 2 (w) = 0, t= 1 so that the surface (19.10) is minimal. For consider (18.2) without the factor w: (in)' 1 2eiG (19.12) F1 (w) = 2 i9 2gi(m) 6)dO. Since all the polygons P(m) are contained in a finite region of space, the functions gi(m)(0) are uniformly bounded; and if w is any fixed point interior to the unit circle, the denominator (eiO -W)2 remains superior in absolute value to a fixed positive quantity when ei0 describes C. Therefore the integrand in (19.12) remains uniformly bounded during the limit process (19.9); consequently the limit of the integral is equal to the integral of the limit: (19 .13) lim Fi (w) = Fi (w) . M-n+00 It is evident that in case g is improper of the first kind this result is not affected by the circumstance that the points of discontinuity of g,(G) are not considered in the limit relation (19.9), since these points, being at most denumerably infinite in number, form a set of zero measure. The result (19.11) now follows from (19.13) and the subsistence of (19.5) for every m. 20. The minimal surface is bounded by r. To show that the minimal surface whose existence is proved in the preceding section is bounded by r, we must prove that the representation (19.8) of r is proper. That it cannot be improper of the second kind is proved in ?18, which, being based on the relation (19.11), applies here with full validity. We cannot however apply the argument of ?17 to prove that (19.8) cannot be improper of the first kind. For although we would still have for a g of this kind A (g) = + so, it would not be true in the case of a general Jordan contour that A (g) sometimes takes finite values. We therefore use the following argument, based on the relation (19.11), to obtain the desired result. Suppose that under g the point P of C corresponds to the arc Q'Q" of r. Since r is a Jordan curve, Q' and Q" are distinct: and if ai denote the co6rdinates of Q', bi of Q", the distance Q'Q" or I with (20.1) 12 = (b.ai)2 i-a1 is not equal to zero. This content downloaded from 157.55.39.128 on Mon, 18 Jul 2016 05:43:48 UTC All use subject to http://about.jstor.org/terms 1931] THE PROBLEM OF PLATEAU 305 There is no loss of generality in supposing P to be at w = 1, for this may be achieved by a rotation of the unit circle, which changes nothing essential.

I have recently indicated [1, 2, 3, 4] some cases of best approximation of a meromorphic function f(z) by rational functions R.~(z) of a given type (n, v) having some free poles (that is, poles not prescribed in position), where it is proved that the free poles approach necessarily (n ~ ~) the poles of f (z) . The object of the present paper is to indicate (w that the methods already introduced for the case that the prescribed poles of the R.v(z) lie at infinity admit extensions that apply to the more general case that the prescribed poles of the R.v(z) do not lie at infinity nor in fact in a finite number of points. We study also (w the problem of approximation by meromorphic functions, bounded with the exception of v free poles.

In this paper, we propose to use the word acyclic in place of the phrase containing no simple closed curve. That is, an acyclic continuous curve is a continuous curve containing no simple closed curve. Acyclic continuous curves have been studied by Mazurkiewicz,T R. L. Wilder,? R. L. Moore,lI and the author.? As a result of Theorem 1, it follows that certain internal properties of an acyclic continuous curve which have been proved by the above authors for plane curves, are also possessed by curves in n-dimensional space. However, in the present paper, only plane point sets are considered, unless otherwise stated.

In 1920 very little was known about the class of homogeneous, bounded continua in the plane. At that time Knaster and Kuratowski [1] raised the question:' Is every such (nondegenerate) continuum a simple closed curve? Mazurkiewicz [2] showed such a continuum is a simple closed curve if it is locally connected, and I showed this is the case if the continuum is aposyndetic [3]. H. J. Cohen [4] proved that if a homogeneous, bounded, plane continuum contains a simple closed curve, it is a simple closed curve. And finally I proved that every homogeneous, compact continuum lying in but not separating a plane is indecomposable [5]. So the class of homogeneous, bounded, plane continua may be typed as follows: Type 1. Those which do not separate the plane. (These must all be indecomposable, and continua of Type 1 other than degenerate ones are known to exist [6 and 7].) Type 2. Those which are decomposable. (These must all separate the plane, and continua of Type 2 other than simple closed curves are known to exist [8].) Type 3. Those which separate the plane but are indecomposable. (Whether any of this type exists is not known. However, see [9, Example 2, pp. 48-49].) It is the purpose of this paper to show that each homogeneous, bounded, plane continuum of Type 2 is either a simple closed curve or becomes one under a natural aposyndetic decomposition,2 the elements of the decomposition being mutually homeomorphic continua of Type 1. In other words, thinking of a plane as an upper semicontinuous collection of continua (each lying in but not separating a given plane), every continuum of Type 2 is the sum of the elements of a simple closed curve lying in a plane of elements of Type 1.

In a recently published paper [l], the concept of complex cubic spline on a rectifiable Jordan curve was discussed together with the corresponding extension to the associated analytic spline. We introduce here complex polynomial splines which represent, simultaneously, extensions of complex cubic splines and of periodic (real) polynomial splines [2, 31. This generalization, together with the modifications in the methods of analysis which it necessitates, leads to new properties and serves to shed more light upon the structure of the spline itself. We discuss complex polynomial splines with some of their elementary properties, the associated analytic splines in various representations. We then examine in particular multiple interpolation at a point by analytic splines. In this regard, we both develop and extend some previously announced results [4].

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

A knot K is a polygonal simple closed curve in S3 which does not bound a disk in S\ A solid torus T is a 3manifold homeomorphic to S1 x D\ The boundary of T is a torus, a 2-manifold homeomorphic to S x S 1. A meridian of T is a simple closed curve on dT which bounds a disk in T but is not homologous to zero on dT. A meridianal disk of T is a disk D in T such that D Π 3Γ = dD and 3D is a meridian of T. A longitude of Γ is a simple closed curve on 3Γ which is transverse to a meridian of T and is null-homologo us in S3-T. A meridianlongitude pair for T is an ordered pair (Af, L) of curves such that ikf is a meridian of T and L is a longitude of T transverse to ikf. π^dT) ~ Z x Z with generators M and L. gilf + pL is the homotopy class of a simple closed

The most general continuous plalne curve without multiple points may be defined as a set of points which can be referred in a one-to-one manner and continuously to the points of a segment of a right line, inclusive of the extremities of the segment, if the curve is niot closed; and to the points of the circumference of a circle, if the curve is closed.f Such a curve is called a Jordan curve. It may be represented analytically by the equations

It has been shown by Mengert that a necessary and sufficient condition that a plane continuum M contains no domain is that for each point P of M and each positive number e there exists a simple closed curve J of diameter less than e which encloses P, and such that M AJ is totally disconnected. In the present paper it is shown that if M is a continuum which contains no domain then there exists a set G of simple closed curves filling the whole plane and indeed topologically equivalent to the set of all polygons, such that the common part of M and any curve of the set G is vacuous or totally disconnected. Additional results are obtained for the special case where M is a continuous curve. I wish to acknowledge my indebtedness to Professor R. L. Moore, and to thank him. Credit is due him for the suggestion of most of the theorems of this paper, and for many helpful criticisms of the proofs.

An extended version of the Jordan curve theorem [i] states that if X and Y are respectively an n-manifold and an (n+ 1)-sphere and X is contained in Y, then Y-X is the union of two disjoint, open, connected sets each having X as point set boundary. It is shown here that it is possible to relax the requirement that X be an n-manifold in such a way that the conclusions continue to hold. Actually, topological conditions on X will be given that are necessary and sufficient for X to separate the (n+1)-sphere Sn+1 in the stated manner. A cohomology theory is assumed to be defined on the category of compact pairs and to satisfy the continuity property as well as the axioms of Eilenberg and Steenrod [2]. Spanier [3 ] has given one of several ways of showing the existence of such a theory having an arbitrary abelian coefficient group G (discrete). For such a cohomology theory an n-manifold X contained in Sn+ has the following well known properties which are assumed here without proof: (1) Hn(X) is isomorphic to G; (2) Hn(A) = 0 for every closed proper subset A of X. It will be shown that for any closed subset X of Sn+l, (1) and (2) are necessary and sufficient conditions for Sn+l -X to be the union of two disjoint, open, connected sets each having X as point set boundary. For n =0, Hn(X) is taken to be the reduced zero-dimensional group. The material in ?1 is contained in the Tulane University lecture notes of A. D. Wallace.

Introduction. In 1954 C. E. Capel proved [1] the following theorems: Let S be the inverse limit of a sequence of arcs (simple closed curves) where the bounding maps are onto and monotone. Then S is an arc (simple closed curve). It may be noted that if f is a monotone map of an arc (simple closed curve) onto itself, then f is the uniform limit of a sequence of onto homeomorphisms.2 We call such a map a nearhomeomorphism. In this paper we prove the following two theorems: (1) If S is the inverse limit of a sequence of copies of a given compact metric space X and the bonding maps are near-homeomorphisms, then S is homeomorphic to X. (2) Let f: XY, g: Y-*X, where f, g are maps and X, Y are compact metric spaces. Suppose fg and gf are nearhomeomorphisms. Then X is homeomorphic to Y. The second theorem follows directly from the first. In order to establish the first theorem we develop an approximation theorem which has interest in its own right. DEFINITIONS AND NOTATION. Let Xi be a sequence of compact metric spaces, and for i ? 2 letf* map Xi into Xi-,. Then the subspace3 S= {zEz -II Xilfij(zj) =zi} of I1 Xi is the limit space of the inverse system (Xi, fi); in notation S = Lim (Xi, ft). Let f map X into Y where X, Y are compact metric spaces. Then for e>O:L(e, f)=Sup13 cn.

A continuum (i.e., a compact connected nondegenerate metric space) M M is said to be λ \lambda connected if any two of its points can be joined by a hereditarily decomposable continuum in M M . Here we prove that a plane continuum is λ \lambda connected if and only if it cannot be mapped continuously onto Knaster’s chainable indecomposable continuum with one endpoint. Recent results involving aposyndesis and decompositions to a simple closed curve are extended to λ \lambda connected continua.

L. M. Blumenthal has defined the concept of congruence indices for point sets in his book Theory and applications of distance geometry, Clarendon Press, Oxford, 1953. Blumenthal shows that the circle has congruence indices (3, 1) and asks if this characterizes the circle among the class of simple closed curves. In this paper it is established that various classes of simple closed curves do not have congruence indices ( 3 , n ) (3,n) for any n n . Included in these classes are the polygons, simple closed curves with convex interiors and a straight line segment contained in the curve, and simple closed curves with continuous nonconstant radius of curvature on some arc. Thus any noncircular simple closed curve with congruence indices (3, 1) must be very pathological. It is shown that if a simple closed curve has positive planar Lebesgue measure, then it fails to have congruence indices ( 3 , n ) (3,n) for any n n .

My paper On the foundations of plane analysis situst contains three sets of postulates, 24, 22, and 3, expressed in terms of the undefined notions point and region. In the present paper I will show that every space S that satisfies :1 or 22 is a number plane, that is to say there exists, between S and a twodimensional euclidean space S', a one-to-one correspondence that preserves limits.t This signifies that if P is a point and M is a point-set in S, and P' and M' are the corresponding point and point-set in S', then P is a limit point of M in the sense defined on page 132 of the above mentioned paper if, and only if, P' is a limit point of M' in the sense that every circle in S' that encloses P' encloses also a point of M' distinct from P'. It follows that 24 and :2 are both categorical with respect to ? point and limit point as defined on page 132. Moreover between every space S, satisfying 21, and a two-dimensional euclidean space S' there exists a one-to-one correspondence preserving point and region if in S' the term region is interpreted to mean Jordan region. That is to say if the set of points M is a region in S then the set 1M1' of corresponding points in S' is the interior of a simple closed curve and conversely. Thus 21 is absolutelyll categorical. The system 22 is satisfied if in ordinary euclidean space of two dimensions the term region is interpreted as signifying Jordan region. It is however also satisfied if in such a space region is so interpreted

Let r be a recifiable closed Jordan curve in the euclidean 3-space IR3, and denote by Ar the infimum of the areas of all surfaces bounded by r. Then for every real number H with\(\left| H \right| \leqslant 0.52 \cdot \sqrt \pi /\sqrt {A_\Gamma }\) we show the existence of a surface with boundary curve r having constant mean curvature H (except in possible branching points). This improves a theorem of WENTE. Given an isolated minimal surface bounded by r for sufficiently small |H| we further prove the existence of a surface of constant mean curvature with boundary curve r which is close to the minimal surface.

1. Poincare has proved that there exist 3-dimensional mnanifolds with identical Betti numbers and coefficients of torsion but which are nevertheless distinguishable in the sense of Analysis Situs by the fact that they have different groups.t It is proposed to set up an example of two 3-dimensional manifolds which are by no means equivalent but which cannot even be differentiated by their groups. Since the manifolds have the same group, they also have the same Betti numbers and coefficients of torsion, for in the 2-sided 3-dimensional case, these other invariants are functions of the group alone. 2. To clarify certain points that will arise later on in the discussion, we shall begin by recalling explicitly several well-known and quite obvious properties of an anchor ring, A, lying in ordinary 3-dimensional space. (a) The ring A may be made simply connected by means of two cuts, one along a generating circle, the other along a circle which intersects the first in one and only one point. If, therefore, we denote by a and b respectively the operations of describing these two curves in preassigned senses, the group of the surface will be the group generated by the operations a and b connected by the single relation aba-1 b-1-1. In the discussion that follows, the two circles determining a and b will also be referred to by the symbols for the corresponding group operations, when this can be done without confusion. (b) The circle a bounds a 2-cell that reduces the interior of the anchor ring to a simply connected piece. Any other simple closed curve with this property may be deformed continuously on the surface of the ring into coincidence with a.+ (c) A surface which in its initial position coincides with the anchor ring A