Search

This paper is devoted to an application of the methods of geometric function theory of a single complex variable and some results from the theory of conformal and quasiconformal mappings to a solution of the direct problem of the constructive theory of functions. Direct theorems are obtained for the theory of approximation of functions regular in regions bounded by quasiconformal curves admitting a geometric description. In particular, direct theorems are obtained for arbitrary bounded convex regions.Bibliography: 11 items.

A finite bordered Riemann surface R \mathcal {R} with s boundary components and interior genus g has first Betti number r = 2 g + s − 1 r = 2g + s - 1 . Let a be any interior point of R \mathcal {R} and e a {e_a} denote evaluation at a on the usual hypo-Dirichlet algebra associated with R \mathcal {R} . We establish some connections between the topological and, more strongly, the conformal type of R \mathcal {R} and the geometry of M a {\mathfrak {M}_a} the set of representing measures for e a {e_a} . For example, we show that if M a {\mathfrak {M}_a} has an isolated extreme point, then R \mathcal {R} must be a planar surface. Several questions posed by Sarason are answered through exhausting the possibilities for the case r = 2 r = 2 .

A shear plane theory of hot rolling is proposed, based on the well-known observation of localized deformation during the rolling process. Equations of simple form in terms of yield stress and geometric functions enable roll load and torque to be readily evaluated with or without tensions. Comparison with the experimental results of Wallquist suggests that the theory gives a more satisfactory indication of the influence of the R/h ratio on load and torque than does the Sims analysis. Equations are presented in four forms: the general case with tensions; the more common case without tensions; simplified equations which are valid for most practical conditions; and graphical functions for use with the Cook and McCrum yield stress data.

An arbitrary Riemann surface with signature can be represented as the quotient of a simply connected Riemann surface by a properly discontinuous group of conformal mappings with the natural projection satisfying some conditions. The representation is unique up to conjugation.

We are interested in the question how the spaces of solutions of elliptic equations vary according to the variations of underlying regions and coefficients of the equation. We will discuss this question for the case of equations Δu = Pu considered on noncompact Riemann surfaces R. Typically we ask the properties of mappings τx: (R,P) → dim PX(R) from the space Φ of pairs (R, P) of noncompact Riemann surfaces R and densities P on R, i.e. P(z)dxdy are 2-forms on R such that P(z)dxdy ≢ 0 and P(z) ≥ 0 are Hölder continuous with respect to local parameters z = x + iy, into cardinals, where PX(R) are the linear spaces of solutions of Δu = Pu on R with certain boundedness properties X.

On reviewing recently the proof which I gave for the Riemann mapping theorem for simply-connected Riemann surfaces several years ago [2], I observed that the argument which I used could be so modified that the assumption of a countable base could be completely eliminated. The problem of treating the Riemann mapping theorem without this assumption has been current for some time. The object of the present note is to give an account of a solution of this question. Of course, the classical theorem of Radó permits us to dispense with an attack on the Riemann mapping theorem which does not appeal to the countable base assumption. In this connection, we recall that Nevanlinna [4] has given a straightforward potential-theoretic treatment of the Radó theorem in which neither the Riemann mapping theorem (nor the notion of a universal covering) enters as they do in Radó’s proof. Nevertheless, a certain technical interest attaches to a direct treatment of the Riemann mapping theorem without the countable base assumption. An immediate byproduct of such a treatment is a simple proof of the Radó theorem which invokes the notion of a universal covering but in a manner different from that of Radó’s proof. Indeed, it suffices to note that a manifold has a countable base if the domain of a universal covering does.

In this paper we generalize to Riemann surfaces a theorem of Helson and Lowdenslager in (2) describing the closed subspaces of L2(﹛|z| = 1﹜) that are invariant under multiplication by eiθ.Let R be a region on a Riemann surface with boundary Γ consisting of a finite number of disjoint simple closed analytic curves such that R ⋃ Γ is compact and R lies on one side of Γ. Let dμ be the harmonic measure on Γ with respect to a fixed point t0 on R. We shall consider the closed subspaces of L2(Γ, dμ) that are invariant under multiplication by functions in A (R) = ﹛F|F continuous on , analytic on R}.

Let X X be a Riemann surface (compact or noncompact) with the property that the length of every closed geodesic is bounded away from zero. Then we show that sufficiently small complex structures on X X can be described without making use of Schwarzian derivatives or the theory of quasiconformal mappings. Instead, we use methods developed in Kuranishi’s work on the existence of locally complete families of deformations of compact complex manifolds. We introduce norms | | k |\quad {|_k} ( k k a positive integer) on the space of C ∞ ( 0 , p ) {C^\infty }(0,p) -forms with values in the tangent bundle on X X , which are similar to the usual Sobolev | | | | k ||\quad |{|_k} -norms. (In the compact case | | k |\quad {|_k} is equivalent to | | | | k ||\quad |{|_k} .) Then we prove that certain properties of | | | | k ||\quad |{|_k} , crucial for Kuranishi’s approach, are also satisfied by | | k |\quad {|_k} .

This paper contains results of a somewhat varied nature, all obtained from a detailed examination of a special class of surfaces. These results include a method for the isometric embedding of Riemann surfaces in space, an application of this method for the construction of a hyperbolic surface in space, and several criteria in the problem of type. The first three sections are devoted to developing the necessary preliminary material. In ?1 we give an intuitive picture of the embedding process, and also the exact definitions to be used throughout the paper. ?2 contains a brief discussion of the surfaces to be considered, simply-connected covering surfaces of the plane having only algebraic singularities over finite points, which are denoted as surfaces of class A. A method is given for subdividing these surfaces into sheets, this subdivision being the main tool for the embedding. In ?3 we define and study the Euclidean metric on the surface. The embedding itself is carried out in ?4, where we show how a large class of Riemann covering surfaces can be represented isometrically as nonselfintersecting surfaces in 3-space. The interest in the embedding method is partly theoretical, since the existence of such an embedding is somewhat surprising in itself, and partly practical, in the sense tha,t by viewing a surface in the imbedded form one often obtains a much better intuitive picture of its internal structure. The author has indicated at the 1954 International Congress [8] how this embedding method could be used to settle a question on the existence of certain hyperbolic surfaces in 3-space. The details are carried out in ?5 with slight refinements so that the surface obtained is everywhere infinitely differentiable. The method used here for removing the singular points may be used in conjunction with the author's paper [9] to provide a different way of settling the original question. Finally, we turn to the problem of type for surfaces of class A. These surfaces were originally considered by Ahlfors [1 ] who introduced the function n(t) as a measure of the branching and found a sufficient condition for parabolic type. ?6 contains a number of new results on the use of the function n(t) in the determination of type.

Let F be an open Riemann surface spread over the z-plane. We say that F is of positive or null boundary, according as there exists a Green’s function on F or not, Let u(z) be a harmonic function on Fand be its Dirichlet integral As R. Nevanlinna proved, if F is of null boundary, there exists no one-valued non-constant harmonic function on F5 whose Dirichlet integral is finite, This Nevanlinna’s theorem was proved very simply by Kuroda.

A method of finding explicit expressions for the roots of a certain class of transcendental equations is discussed. In particular it is shown by determining a canonical solution of an associated Riemann boundary-value problem that expressions for the roots may be derived in closed form. The explicit solutions to two transcendental equations, tan β = ωβ and β tan β = ω, are discussed in detail, and additional specific results are given.

wherefi is of class HP on R,f6 is of class HP on R"\R, and s is an element of a certain specified finite dimensional space of functions meromorphic on R. The proof Heins gave for this result depends explicitly on the symmetry of the geometric configuration. In ?11 below we will obtain a generalization of this result, a generalization which requires no particular symmetry in the ambient compact surface. We will also establish a theorem which is an analogue of Laurent's theorem valid on compact Riemann surfaces. Our result contains as a special case a result of Alling [I]. ?1 is devoted to certain algebraic preliminaries needed in the subsequent analytic considerations.

Given a strong definition of "conformai map," there is a class of domains in C(X) conformally equivalent to the open unit ball which are in a specific way "interiors" of curves. 1. Introduction. In 1943 E. R. Lorch (5) extended the idea of "holomorphic function" to commutative Banach algebras with identity. Briefly, if A denotes the algebra and / is a mapping from A to A, then/ is called holomorphic at a point a EA if there is an algebra element/'(a) such that for hEA we have \\f(a + h)-f(a)-hf'(a)\\ = o(\\h\\) as ||fe|| tends to zero. A mapping is holomorphic on an open set if it is holomorphic at each point of the set. Lorch and others have shown that many classical theorems of complex function theory carry over to this more general setting. The Riemann mapping theorem cannot be extended without some qualification. Two open sets in a commutative Banach algebra will be called "conformally equivalent" if there is a one to one mapping between them which is holomorphic in both directions. A first try at a Riemann mapping theorem might be that the open unit ball of the algebra is conformally equivalent to every bounded homeomorph of itself. To this B. W. Glickfeld gives the following counterexample (2, p. 45). In the algebra C((0, l)) the set of functions/such that |/(x)

(1964). Riemann Integrals as Mappings in Elementary Applications. Mathematics Magazine: Vol. 37, No. 5, pp. 311-317.

Throughout this paper all functions are single-valued. Let R be a Riemann surface. We shall denote by φ∧ the least harmonic majorant of a function φ defined in R if it has the meaning.