Log-lightning computation of capacity and Green's function

See Video Abstract (click the "Video Abstract" button next to the "PDF" button) A basic measure of the size of a set E in the complex plane is the logarithmic capacity cap(E). Capacities are known analytically for a few simple shapes like ellipses, but in most cases they must be computed numerically. We explore their computation by the new "log-lightning'' method based on reciprocal-log approximations in the complex plane. For a sequence of 16 examples involving both connected and disconnected sets E, we compute capacities to 8–15 digits of accuracy at great speed in MATLAB. The convergence is almost-exponential with respect to the number of reciprocal-log poles employed, so it should be possible to compute many more digits if desired in Maple or another extended-precision environment. This is the first systematic exploration of applications of the log-lightning method, which opens up the possibility of solving Laplace problems with an efficiency not achievable by previous methods. The method computes not just the capacity, but also the Green's function and its harmonic conjugate. It also extends to "domains of negative measure" and other Riemann surfaces.

Uniformization of Cantor sets with bounded geometry

In this note we provide a quasisymmetric taming of uniformly perfect and uniformly disconnected sets that generalizes a result of MacManus [Rev. Mat. Iberoamericana 15 (1999), pp. 267–277] from 2 to higher dimensions. In particular, we show that a compact subset of R n \mathbb {R}^n is uniformly perfect and uniformly disconnected if and only if it is ambiently quasiconformal to the standard Cantor set C \mathcal {C} in R n + 1 \mathbb {R}^{n+1} .

Some Properties Via GN −Preopen Sets in Grill Topological Spaces

This paper introduces and investigates the notions of GN -disconnected sets, GN -connected, GN -precompact sets, and separation axioms via GN -preopen sets. They will be introduced in grill topological spaces by using the class of GN -preopen sets.

A note on the removability of totally disconnected sets for analytic functions

UDC 517.537.38 We prove that each totally disconnected closed subset E of a domain G in the complex plane is removable for analytic functions f ( z ) defined in G ∖ E and such that for any point z 0 ∈ E the real or imaginary part of f ( z ) vanishes at z 0 .

ON THE CLASSIFICATION OF FRACTAL SQUARES

In the previous paper [K. S. Lau, J. J. Luo and H. Rao, Topological structure of fractal squares, Math. Proc. Camb. Phil. Soc. 155 (2013) 73–86], Lau, Luo and Rao completely classified the topological structure of so called fractal square [Formula: see text] defined by [Formula: see text], where [Formula: see text]. In this paper, we further provide simple criteria for the [Formula: see text] to be totally disconnected, then we discuss the Lipschitz classification of [Formula: see text] in the case [Formula: see text], which is an attempt to consider non-totally disconnected sets.

A RIGOROUS DETERMINATION OF THE OVERALL PERIOD IN THE STRUCTURE OF A CHAOTIC ATTRACTOR

There are many examples of nonconnected chaotic attractors consisting of several components. The determination of an overall period of such a system is typically done only by a numerical integration of the system. In this letter, we provide a rigorous proof that the exact value of the overall period of a particular 2-D chaotic attractor from an iterated map is two once the attractor has been partitioned and quantized into disconnected sets. As far as we know, there are no examples of a rigorous proof for such a property in the current literature.