Change search
CiteExportLink to record
Permanent link

Direct link
Cite
Citation style
  • apa
  • harvard1
  • ieee
  • modern-language-association-8th-edition
  • vancouver
  • Other style
More styles
Language
  • de-DE
  • en-GB
  • en-US
  • fi-FI
  • nn-NO
  • nn-NB
  • sv-SE
  • Other locale
More languages
Output format
  • html
  • text
  • asciidoc
  • rtf
The n-dimensional Stern-Brocot tree
Blekinge Institute of Technology, School of Engineering, Department of Mathematics and Natural Sciences.
Responsible organisation
2012 (English)Report (Other academic)Alternative title
Stern-Brocots träd i n dimensioner (Swedish)
Abstract [en]

The n-dimensional Stern-Brocot tree consists of all sequences (p₁, ...,p_{n}) of positive integers with no common multiple. The relatively prime sequences can be generated branchwise from each other by simple vector summation, starting with an ON-base, and controlled by a generalized Euclidean algorithm.The tree induces a multiresolution partition of the first quadrant of the (n-1)-dimensional unit sphere, providing a direction approximation property of a sequence by its ancestors. Two matrix representations are introduced, where in both a matrix contains the parents of a sequence. From one of them the isomorphism of a subtree to the entire tree of dimension equal to the number of parents of the top sequence follows. A form of Fibonacci sequences turn out to be the sequences of fastest growing sums. The construction can be regarded an n-dimensional continued fraction, and it may invite further n-dimesional number theory.

Abstract [sv]

Stern-Brocots träd består av alla sekvenser (p₁, ...,p_{n}) av positiva heltal som saknar gemensam delare. De genereras gren för gren med vektoraddition, med en ON-bas som start, och kontrolleras av en generaliserad Euklidisk algoritm. Trädet inducerar en multiresolutionpartition av den första kvadranten av den (n-1)-dimensionella enhetssfären, vilket ger en riktningsapproximationsegenskap för en sekvens med dess föregångare i trädet. Två matrisrepresentationer presenteras, i båda innehåller matrisen alla föräldrar till en sekvens. Från en av dem följer isomorfismen av ett delträd med ett helt träd vars dimension är antalet föräldrar av toppsekvensen i delträdet. En typ av Fibonaccisekvenser visar sig vara sekvenserna som har snabbast växande summa. Konstruktionen kan ses som ett n-dimensionellt kedjebråk, och den kan vara en inbjudan till fortsatt n-dimensionell talteori. Nyckelord: Stern-Brocots träd, relativt prima tal, Fibonaccital, multiresolution partition, ON-bas, kedjebråk, Minkowskis frågeteckenfunktion.

Place, publisher, year, edition, pages
2012.
Series
Blekinge Tekniska Högskola Forskningsrapport, ISSN 1103-1581 ; 4
Keyword [en]
Stern-Brocot tree, relatively prime integers, relative primality, Fibonacci numbers, multiresolution partition, ON-base, continued fraction, Minkowski question mark function.
National Category
Mathematical Analysis Geometry
Identifiers
URN: urn:nbn:se:bth-00534Local ID: oai:bth.se:forskinfoA02A252219018A4EC1257A16003CC993OAI: oai:DiVA.org:bth-00534DiVA: diva2:834871
Available from: 2012-09-18 Created: 2012-06-07 Last updated: 2016-09-06Bibliographically approved

Open Access in DiVA

fulltext(425 kB)119 downloads
File information
File name FULLTEXT01.pdfFile size 425 kBChecksum SHA-512
cb32fa630546fb5870ce0e41a03c3ad82d6dac6ebf16c2e156b6c64c23093b947e8435ade6b503a4b24209d6617dfe208d54bf65cf8bbacf4c14b234a4a54b47
Type fulltextMimetype application/pdf

Search in DiVA

By author/editor
Lennerstad, Håkan
By organisation
Department of Mathematics and Natural Sciences
Mathematical AnalysisGeometry

Search outside of DiVA

GoogleGoogle Scholar
Total: 119 downloads
The number of downloads is the sum of all downloads of full texts. It may include eg previous versions that are now no longer available

Total: 264 hits
CiteExportLink to record
Permanent link

Direct link
Cite
Citation style
  • apa
  • harvard1
  • ieee
  • modern-language-association-8th-edition
  • vancouver
  • Other style
More styles
Language
  • de-DE
  • en-GB
  • en-US
  • fi-FI
  • nn-NO
  • nn-NB
  • sv-SE
  • Other locale
More languages
Output format
  • html
  • text
  • asciidoc
  • rtf