Burt Rodin - Academia.edu (original) (raw)
Papers by Burt Rodin
Computational Methods and Function Theory, 1990
Without Abstract
Bulletin of the American Mathematical Society, 1966
Rocky Mountain Journal of Mathematics, 2017
Communications in Analysis and Geometry, 1993
In [R-S], the conjecture by W. Thurston [Th] that the hexagonal circle packings can be used to ap... more In [R-S], the conjecture by W. Thurston [Th] that the hexagonal circle packings can be used to approximate the Riemann mapping (in the topology of uniform convergence in compact subsets) is proved; and in [He], the derivatives of these approximations are shown to be convergent. We show in Section 1 that the methods used in [R-S] in the case of hexagonal packings can be easily extended to the case of nonhexagonal circle packing with bounded radii ratios. We note that Stephenson had taken the major steps toward such an extension in [Ste]. Although he follows the overall strategy of [R-S], he replaces certain key steps by parabolistic arguments which have an interesting nterpretation in terms of the flow of electricity in a network. In Section 2, we show that the method of [He] can be extended to a more general class of non hexagonal packings. Specifically, the restriction in [Ste] that the radii ratios be bounded can be replaced by the much weaker condition that the circle packings have uniformly bounded valence.
Journal of Differential Geometry, 1987
Journal of Differential Geometry, 1989
Indiana University Mathematics Journal, 1991
Journal d'Analyse Mathématique, 1991
The following subject areas are covered: topology and magnetic energy in incompressible perfectly... more The following subject areas are covered: topology and magnetic energy in incompressible perfectly conducting fluids; links of tori and the energy of incompressible flows; factoring the logarithmic spiral; inherent differentiability; a power law for the distortion of planar sets; strange actions of groups on spheres; solving Beltrami equations by circle packing; an estimate for hexagonal circle packings; the convergence of
Principal Functions, 1968
We begin by considering a harmonic function whose boundary behavior is a combination of L0- and L... more We begin by considering a harmonic function whose boundary behavior is a combination of L0- and L1-type behavior. This leads to a generalization of harmonic measure and capacity. In §2 we relate this function to some extremal length problems. The remaining sections treat applications to conformal mapping and stability problems.
Principal Functions, 1968
In this chapter the basic tools are created which will be used throughout the remainder of the bo... more In this chapter the basic tools are created which will be used throughout the remainder of the book. The central topic is the Main Existence Theorem for principal functions, given in §1. The hypotheses of this theorem require the existence of normal operators. That such operators always exist is a nontrivial fact; its proof is given in §2 by constructing the operators L0 and L1 on an arbitrary Riemann surface. The method used there, which is typical of such problems, consists of constructing operators on compact bordered subregions and passing to a limit.
Principal Functions, 1968
The existence of certain harmonic functions with prescribed singularities and prescribed boundary... more The existence of certain harmonic functions with prescribed singularities and prescribed boundary behavior is central to a variety of areas of complex function theory. It provides the unifying factor for all the topics of this book. Such functions will be called principal functions. For further orientation let us consider how they are constructed and some of their uses.
Principal Functions, 1968
In this chapter we shall discuss the problem of finding on a given harmonic space a harmonic func... more In this chapter we shall discuss the problem of finding on a given harmonic space a harmonic function which imitates the behavior of a given harmonic function on a neighborhood of the ideal boundary of the harmonic space.
Transactions of the American Mathematical Society, 1992
Figure 1.1 illustrates the fact that if a region is almost packed with circles of radius e in the... more Figure 1.1 illustrates the fact that if a region is almost packed with circles of radius e in the hexagonal pattern and if the unit disk is packed in an isomorphic pattern with circles of varying radii then, after suitable normalization, the correspondence of circles converges to the Riemann mapping function as e-► 0 (see [15]). In the present paper an inverse of this result is obtained as illustrated by Figure 1.2; namely, if the unit disk is almost packed with ecircles there is an isomorphic circle packing almost filling the region such that, after suitable normalization, the circle correspondence converges to the conformai map of the disk onto the region as e-> 0. Note that this set up yields an approximate triangulation of the region by joining the centers of triples of mutually tangent circles. Since this triangulation is intimately related to the Riemann mapping it may be useful for grid generation [18].
Journal d'Analyse Mathématique, 1967
Journal d'Analyse Mathématique, 1986
Journal d'Analyse Mathématique, 1966
Inventiones Mathematicae, 1987
Duke Mathematical Journal, 1966
Computational Methods and Function Theory, 1990
Without Abstract
Bulletin of the American Mathematical Society, 1966
Rocky Mountain Journal of Mathematics, 2017
Communications in Analysis and Geometry, 1993
In [R-S], the conjecture by W. Thurston [Th] that the hexagonal circle packings can be used to ap... more In [R-S], the conjecture by W. Thurston [Th] that the hexagonal circle packings can be used to approximate the Riemann mapping (in the topology of uniform convergence in compact subsets) is proved; and in [He], the derivatives of these approximations are shown to be convergent. We show in Section 1 that the methods used in [R-S] in the case of hexagonal packings can be easily extended to the case of nonhexagonal circle packing with bounded radii ratios. We note that Stephenson had taken the major steps toward such an extension in [Ste]. Although he follows the overall strategy of [R-S], he replaces certain key steps by parabolistic arguments which have an interesting nterpretation in terms of the flow of electricity in a network. In Section 2, we show that the method of [He] can be extended to a more general class of non hexagonal packings. Specifically, the restriction in [Ste] that the radii ratios be bounded can be replaced by the much weaker condition that the circle packings have uniformly bounded valence.
Journal of Differential Geometry, 1987
Journal of Differential Geometry, 1989
Indiana University Mathematics Journal, 1991
Journal d'Analyse Mathématique, 1991
The following subject areas are covered: topology and magnetic energy in incompressible perfectly... more The following subject areas are covered: topology and magnetic energy in incompressible perfectly conducting fluids; links of tori and the energy of incompressible flows; factoring the logarithmic spiral; inherent differentiability; a power law for the distortion of planar sets; strange actions of groups on spheres; solving Beltrami equations by circle packing; an estimate for hexagonal circle packings; the convergence of
Principal Functions, 1968
We begin by considering a harmonic function whose boundary behavior is a combination of L0- and L... more We begin by considering a harmonic function whose boundary behavior is a combination of L0- and L1-type behavior. This leads to a generalization of harmonic measure and capacity. In §2 we relate this function to some extremal length problems. The remaining sections treat applications to conformal mapping and stability problems.
Principal Functions, 1968
In this chapter the basic tools are created which will be used throughout the remainder of the bo... more In this chapter the basic tools are created which will be used throughout the remainder of the book. The central topic is the Main Existence Theorem for principal functions, given in §1. The hypotheses of this theorem require the existence of normal operators. That such operators always exist is a nontrivial fact; its proof is given in §2 by constructing the operators L0 and L1 on an arbitrary Riemann surface. The method used there, which is typical of such problems, consists of constructing operators on compact bordered subregions and passing to a limit.
Principal Functions, 1968
The existence of certain harmonic functions with prescribed singularities and prescribed boundary... more The existence of certain harmonic functions with prescribed singularities and prescribed boundary behavior is central to a variety of areas of complex function theory. It provides the unifying factor for all the topics of this book. Such functions will be called principal functions. For further orientation let us consider how they are constructed and some of their uses.
Principal Functions, 1968
In this chapter we shall discuss the problem of finding on a given harmonic space a harmonic func... more In this chapter we shall discuss the problem of finding on a given harmonic space a harmonic function which imitates the behavior of a given harmonic function on a neighborhood of the ideal boundary of the harmonic space.
Transactions of the American Mathematical Society, 1992
Figure 1.1 illustrates the fact that if a region is almost packed with circles of radius e in the... more Figure 1.1 illustrates the fact that if a region is almost packed with circles of radius e in the hexagonal pattern and if the unit disk is packed in an isomorphic pattern with circles of varying radii then, after suitable normalization, the correspondence of circles converges to the Riemann mapping function as e-► 0 (see [15]). In the present paper an inverse of this result is obtained as illustrated by Figure 1.2; namely, if the unit disk is almost packed with ecircles there is an isomorphic circle packing almost filling the region such that, after suitable normalization, the circle correspondence converges to the conformai map of the disk onto the region as e-> 0. Note that this set up yields an approximate triangulation of the region by joining the centers of triples of mutually tangent circles. Since this triangulation is intimately related to the Riemann mapping it may be useful for grid generation [18].
Journal d'Analyse Mathématique, 1967
Journal d'Analyse Mathématique, 1986
Journal d'Analyse Mathématique, 1966
Inventiones Mathematicae, 1987
Duke Mathematical Journal, 1966