Mathematica--A System for Doing Mathematics by Computer.

Lösung: Man kann die Ebene mit Sechsecken überdecken. Nimmt man die Mittelpunkte der Sechsecke als Punkte für ein Voronoidiagramm, dann haben alle genau 6 Nachbarn, und die Voronoi Polygone sind gerade die Sechsecke. Gäbe es eine Punkteverteilung, bei der jeder Punkt mehr als sechs Nachbarn hätte, dann könnte man die Punkte so hinschieben, dass die Voronoi Polygone regelmässige n-Ecke mit n > 6 wären. Eine solche Überdeckung der Ebene gibt es aber nicht (wie man sich leicht aus einer Betrachtung der Winkel überlegen kann).

Computational geometry

This paper attempts to describe some of the results obtained in the iteration theory of transcendental meromorphic functions, not excluding the case of entire functions. The reader is not expected to be familiar with the iteration theory of rational functions. On the other hand, some aspects where the transcendental case is analogous to the rational case are treated rather briefly here. For example, we introduce the different types of components of the Fatou set that occur in the iteration of rational functions but omit a detailed description of these types. Instead, we concentrate on the types of components that are special to transcendental functions (Baker domains and wandering domains).

/pdf/iteration-of-meromorphic-functions-5an8yxam4g.pdf

Iteration of meromorphic functions

/pdf/iteration-of-meromorphic-functions-4jswwprgfa.pdf

This article is an expository description of quadratic rational maps from the Riemann sphere to itself

/pdf/geometry-and-dynamics-of-quadratic-rational-maps-37auxehtzp.pdf

Geometry and Dynamics of Quadratic Rational Maps

We investigate Newton’s method to find roots of polynomials of fixed degree d, appropriately normalized: we construct a finite set of points such that, for every root of every such polynomial, at least one of these points will converge to this root under Newton’s map. The cardinality of such a set can be as small as 1.11 d log2d; if all the roots of the polynomial are real, it can be 1.30 d.

How to find all roots of complex polynomials by Newton's method

A recurring theme of this book has been computer-generated mysteries. Examples are sequences defined by simple rational recursions whose terms turn out to be integers with interesting but unexplained divisibility properties or geometric configurations that exist although there are no proofs of existence. In most of the examples, the reported mysteries have remained unsolved and in some cases may in fact be, in a suitable sense, unsolvable. It is therefore gratifying to be able to present an elegant solution of a previously described mystery. An especially pleasing feature of this solution is that the breakthrough became possible by drawing the right picture. Once the picture is drawn, it becomes clear what must be proved, after which further study of the picture gives the clue for constructing the proof. It turns out that at one point one needs to use the Jordan curve theorem for a special class of closed curves.

Further Travels with My Ant

A family of root-finding algorithms is constructed that combines knowledge of the branched covering structure of a polynomial with a path-lifting algorithm for finding individual roots. In particular, the family includes an algorithm that computes an $\epsilon$-factorization of a polynomial of degree $d$ that has an arithmetic complexity of $\Order{d(\log d)^2|\log\epsilon| +d^2(\log d)^2}$. At the present time, this complexity is the best known in terms of the degree.

pdf/polynomial-root-finding-algorithmsand-branched-covers-1nnk6dic4d.pdf

Polynomial Root-Finding Algorithmsand Branched Covers

We construct a family of root-finding algorithms which exploit the branched covering structure of a polynomial of degree $d$ with a path-lifting algorithm for finding individual roots. In particular, the family includes an algorithm that computes an $\epsilon$-factorization of the polynomial which has an arithmetic complexity of $\Order{d^2(\log d)^2 + d(\log d)^2|\log\epsilon|}$. At the present time (1993), this complexity is the best known in terms of the degree.

Polynomial root-finding algorithms and branched covers

John Milnor, best known for his work in differential topology, K-theory, and dynamical systems, is one of only three mathematicians to have won the Fields medal, the Abel prize, and the Wolf prize, and is the only one to have received all three of the Leroy P. Steele prizes. In honor of his eightieth birthday, this book gathers together surveys and papers inspired by Milnor's work, from distinguished experts examining not only holomorphic dynamics in one and several variables, but also differential geometry, entropy theory, and combinatorial group theory. The book contains the last paper written by William Thurston, as well as a short paper by John Milnor himself. Introductory sections put the papers in mathematical and historical perspective, color figures are included, and an index facilitates browsing. This collection will be useful to students and researchers for decades to come. The contributors are Marco Abate, Marco Arizzi, Alexander Blokh, Thierry Bousch, Xavier Buff, Serge Cantat, Tao Chen, Robert Devaney, Alexandre Dezotti, Tien-Cuong Dinh, Romain Dujardin, Hugo Garcia-Compean, William Goldman, Rotislav Grigorchuk, John Hubbard, Yunping Jiang, Linda Keen, Jan Kiwi, Genadi Levin, Daniel Meyer, John Milnor, Carlos Moreira, Vincente Munoz, Viet-Anh Nguyen, Lex Oversteegen, Ricardo Perez-Marco, Ross Ptacek, Jasmin Raissy, Pascale Roesch, Roberto Santos-Silva, Dierk Schleicher, Nessim Sibony, Daniel Smania, Tan Lei, William Thurston, Vladlen Timorin, Sebastian van Strien, and Alberto Verjovsky.

/pdf/frontiers-in-complex-dynamics-in-celebration-of-john-milnor-4qlqei14v5.pdf

Frontiers in Complex Dynamics: In Celebration of John Milnor's 80th Birthday

Handle everyday research tasks with reliable, citation-backed results

Your personal Research Agent to handle research tasks with citation-backed results

Popular Tasks used by Researchers

How can I help with your research?

Meet SciSpace

Get more enhanced response by uploading the PDFs you want me to reference.

No relevant PDFs in your library

SciSpace is the AI research assistant for academics. Run systematic literature reviews on 280M+ papers, and write papers with cited sources. Trusted by 1M+ students, PhDs & researchers.

SciSpace | AI for Research

Analyze PDFs

Code & Manuscripts

Funding & Grants

Literature & Patents

Medical & Clinical Data

Systematic Review

Visualize & Present

Web & Data

Build a Google Scholar-like website for your research.

Build a website

Create charts and images for your research

Create a Chart

Write a paper for submission to a journal

Draft a manuscript

Patent Search

Design eye-catching scientific posters in minutes.

Scientific Poster Generation

Systematic Literature Review

One task is running at the moment. Your messages will be shown right after.

Drag and drop or click here to browse

Loved by <highlight>1 million+</highlight> researchers

Extract a list of specific topics and their sources from unstructured text

Topics

Compare and analyze relevant papers that matches with your search

Papers

Get insights from PDFs and bookmarked papers from your library

My library

Recent searches

Try searching for:

Catch AI-generated content in scholarly and non-scholarly content

{ai} Detector

Ai Writer

Get PDF Summaries, highlighted text explanations 

Chat with PDF

Effortlessly create in-text citations and bibliographies in APA and 2,500 other formats

Citation generator

Get explanations, summaries, and answers on academic papers

Ease up your research workflow with {scispace}'s cohort of exciting AI tools

Elevate your academic writing skills and convey your ideas the way you want

Paraphraser

Explore our range of reading and writing tools

Your file is being prepared and should be ready in a few minutes. If it's a large file, it might take a bit longer. You can close this window, and we'll email you the file when it's done.

You have reached a maximum limit of <strong>{limit}</strong> columns in the table. Remove at least <strong>1</strong> column to add or create another one.

Scott Sutherland

Author Tools

Chat about Author