Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · math/0301282

Hexagonal circle patterns with constant intersection angles and discrete Painleve and Riccati equations

S. I. Agafonov·A. I. Bobenko
View original source ↗Read source document ↗Export metadata ↓

Abstract

Hexagonal circle patterns with constant intersection angles mimicking holomorphic maps z^c and log(z) are studied. It is shown that the corresponding circle patterns are immersed and described by special separatrix solutions of discrete Painleve and Riccati equations. The general solution of the Riccati equation is expressed in terms of the hypergeometric function. Global properties of these solutions, as well as of the discrete z^c and log(z), are established.

Explore related subjects

math.CV

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

S. I. Agafonov, A. I. Bobenko. 2003-02-21. Hexagonal circle patterns with constant intersection angles and discrete Painleve and Riccati equations. https://doi.org/10.1063/1.1586966

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Log canonical singularities of plurisubharmonic functions

A plurisubharmonic weight is \emph{log canonical} if it is at the critical point of turning non-integrable. Given a log canonical plurisubharmonic weight, we show that locally there always exists a log canonical `holomorphic' weight having the same non-integrable locus. The proof uses the theory of quasimonomial valuations developed by Jonsson-Mustaţă and Xu, as well as the minimal model program over complex analytic spaces due to Fujino and Lyu-Murayama. As a consequence, we obtain that the non-integrable locus is seminormal.

math.CV↗
arXiv2026

Sobolev homeomorphisms and Jordan curves with positive area

Each homeomorphic parametrization of a Jordan curve via the unit circle $\mathbb{S}\subset \mathbb{S}^2$ extends to a homeomorphism of the entire $\mathbb{S}^2.$ We show that there exists a Jordan curve of positive area whose every parametrization via $\mathbb{S}$ admits a homeomorphic extension that has $p$-Sobolev regularity on $\mathbb{S}^2$ for all exponents $1\le p<2 .$ This is obtained via a new sharp extension result for the case of $p<2.$

math.CV↗
arXiv2026

Disproof of a conjecture of Ruscheweyh et al. on universal convexity of Ramanujan generating function

We disprove a conjecture of Ruscheweyh, Salinas and Sugawa concerning the universal convexity of the normalized generating function associated with the Ramanujan sequence $θ(n).$ More precisely, we prove that the function \[ σ(z)=\frac{1}{θ(1)-1/3}\sum_{n\geq1}(θ(n)-1/3)z^{n} \] is not universally convex on the slit domain $\mathbb{C}\backslash[1,\infty).$ The proof uses Watson\textquoteright s integral representation of $θ(n),$ asymptotic estimates for an auxiliary function $h,$ and the representation of derivatives of universally convex functions via logarithmic integrals. As a general extension of this result, we also show that the generalized Ramanujan generating function \[ σ_α(z)=\frac{1}{θ(1)-1/3}\sum_{n\geq1}(θ(n)-1/3)\frac{z^{n}}{n^α} \] is universally convex if and only if $α\geq1.$

math.CV↗