On mappings related to the gradient of the conformal radius

We establish a criterion for the gradient ?R(D, z) of the conformal radius of a convex domain D to be conformal: the boundary ?D must be a circle. We obtain estimates for the coefficients K(r) for the K(r)-quasiconformal mappings ?R(D, z), D(r) ? D, 0 < r < 1, and supplement the results of Avkhadiev and Wirths concerning the structure of the boundary under diffeomorphic mappings of the domain D.     

Variation of the conformal radius

We study the change of the conformal radiusr(U) of a simply connected planar domainU versus the subdomainU _ε consisting of the points of distance at least ε to ∂U. We show that the smallest exponent λ such thatr(U)-r(U _t)=0(e ^λ) satisfies 0.59<λ<0.91. We also show that a well-known conjecture implies . Partially supported by NSF Grant DMS-9970398.     

The multicovering radius problem for some types of discrete structures

The covering radius problem is a question in coding theory concerned with finding the minimum radius r such that, given a code that is a subset of an underlying metric space, balls of radius r over its code words cover the entire metric space. Klapper (IEEE Trans. Inform. Theory 43:1372–1377, 1997) introduced a code parameter, called the multicovering radius, which is a generalization of the covering radius. In this paper, we introduce an analogue of the multicovering radius for permutation codes (Des. Codes Cryptogr. 41:79–86, cf. 2006) and for codes of perfect matchings (cf. 2012). We apply probabilistic tools to give some lower bounds on the multicovering radii of these codes. In the process of obtaining these results, we also correct an error in the proof of the lower bound of the covering radius that appeared in (Des. Codes Cryptogr. 41:79–86, cf. 2006). We conclude with a discussion of the multicovering radius problem in an even more general context, which offers room for further research.     

Automatic continuity for linear surjective mappings decreasing the local spectral radius at some fixed vector

Let X be a complex Banach space and denote by L( X){mathcal{L}left( Xright)} the space of bounded linear operators on X. Let e be a nonzero element of X. We prove that if φ is a linear and surjective mapping from L( X){ mathcal{L}left( Xright) } into itself which decreases the local spectral radius at e, then φ is automatically continuous.     

New extremal properties for constructing conformal mappings

Summary It is well known that for a given simply connected regionR containing zero the uniform norm attains its minimum in the class of all holomorphic functions normalized byf(0)=0 andf(0)=1 only for the conformal mappingfRD(r)={z|z|}. It is shown that this theorem is still valid if one replaces the ordinary modulus | | on by any other norm on . For instance it is possible to obtain direct mappings ofR onto parallelograms, rectangles and ellipses. For the special norms |₁ and | this leads to a simple and fast computational technique involving linear programming methods. Several numerical examples are given.     

Conjugate function method for numerical conformal mappings

We present a method for numerical computation of conformal mappings from simply or doubly connected domains onto so-called canonical domains, which in our case are rectangles or annuli. The method is based on conjugate harmonic functions and properties of quadrilaterals. Several numerical examples are given.     

A numerical algorithm for the Stokes problem based on an integral equation for the pressure via conformal mappings

Summary In this paper we present an algorithm for solving numerically the Stokes problem in the plane. The known algorithms are all based on certain discretization schemes for the analytic equations. In contrast to this recent work our algorithm uses an explicit analytic solution of a certain approximating problem, which can easily be solved numerically up to machine accuracy. On the one hand this analytic formula is based on a complex representation of all solutions of the Stokes differential equations, and on the other hand it is based on the conformal mapping of the given domain on the unit disc. Therefore, a central prerequisite of our corresponding program is a program for computing this conformal mapping.     

Partial linearization for noninvertible mappings

A Hartman-Grobman result for noninvertible mappings is proved. It is assumed that the spectrum of the linearized mapping contains zero but is disjoint from the complex unit circle. In the infinite-dimensional case, additional spectral conditions are assumed. These additional conditions are satisfied if the linearized mapping is completely continuous. The proof combines Hartman'sC ¹ linearization method for contractive mappings [10] with our previous result [1] on linearization of the expanding part.The paper was written while the second author was an Alexander-von-Humboldt Research Fellow in the Mathematical Institute of the University of Augsburg, Germany.     

On sums of linear and bounded mappings

We give a necessary and sufficient condition in order that a mapf from a real linear space into a real Banach space should have the formf =L + r withL being a linear operator andr being a bounded perturbation.     

Bohr radius for locally univalent harmonic mappings

We consider the class of all sense‐preserving harmonic mappings of the unit disk , where h and g are analytic with , and determine the Bohr radius if any one of the following conditions holds:

• 1. h is bounded in .
• 2. h satisfies the condition in with .
• 3. both h and g are bounded in .
• 4. h is bounded and .

We also consider the problem of determining the Bohr radius when the supremum of the modulus of the dilatation of f in is strictly less than 1. In addition, we determine the Bohr radius for the space of analytic Bloch functions and the space of harmonic Bloch functions. The paper concludes with two conjectures.