Title Cantorov skup i primjene Title (english) Cantor set and applications Author Tina Brumnić Mentor Maja Resman (mentor)Committee member Maja Resman (predsjednik povjerenstva)Committee member Eduard Marušić-Paloka (član povjerenstva)Committee member Marko Erceg (član povjerenstva)Committee member Juraj Šiftar (član povjerenstva)Granter University of Zagreb Defense date and country 2022-09-29, Croatia Scientific / art field, NATURAL SCIENCES Abstract Ovaj diplomski rad bavi se raznim analitičkim konstrukcijama Cantorovog skupa. U prvom poglavlju upoznajemo se s Cantorovim skupom i konstruiramo ga na tzv. geometrijski način te objašnjavamo neka njegova svojstva (duljinu, kardinalitet i gustoću u skupu \(\mathbb{R}\)). U drugom poglavlju bavimo se prvom konstrukcijom Cantorovog skupa preko tzv. verižnih razlomaka. Definiramo verižne razlomke, dokazujemo postojanje konačnog prikaza svakog racionalnog broja u obliku verižnog razlomka i postojanje razvoja u verižni razlomak za svaki realni broj, odnosno, dokazujemo konvergenciju beskonačnih verižnih razlomaka. Naposlijetku dokazujemo da je, za cijeli broj \(k\), \(k\geq2\), skup svih realnih brojeva \(\alpha\), takvih da je \(0\leq \alpha\leq k^{-1}\) i takvih da razvoj broja \(\alpha\), u verižni razlomak ne sadrži parcijalni kvocijent manji od k, Cantorov skup. U trećem poglavlju konstruiramo Cantorov skup kao homeomorfnu sliku prstena cijelih \(p\)-adskih brojeva \(\mathbb{Z}_p\), \(p\) prost broj. Prvo definiramo \(p\)-adske brojeve i cijele \(p\)-adske brojeve. Definiramo \(p\)-adsku normu i \(p\)-adsku udaljenost na polju \(\mathbb{Q}\) te dokazujemo da polje \(\mathbb{Q}\) s \(p\)-adskom normom nije potpuno. Upotpunjujemo polje \(\mathbb{Q}\) poljem klasa ekvivalencije Cauchyjevih nizova, \(\mathbb{Q}_p \). Nakon toga, pokazujemo ekvivalenciju polja \(\mathbb{Q}_p \) kao skupa klasa ekvivalencija nizova i beskonačnih \(p\)-adskih ekspanzija koje konvergiraju u \(p\)-adskoj normi na \(\mathbb{Q}_p \). Konačno, pokazujemo postojanje homeomorfizma izmedu prstena cijelih \(p\)-adskih brojeva \(\mathbb{Z}_p\) i Cantorovog skupa u naslijeđenoj euklidskoj topologiji.
Abstract (english) This thesis deals with various analytic constructions of the Cantor set. In the first chapter, we get acquainted with the Cantor set and construct it on the so-called geometric way and we explain some of its properties (length, cardinality and density in the set \(\mathbb{R}\)). In the second chapter, we deal with the first construction of the Cantor set through the so-called continued fractions. We define continued fractions, we prove the existence of a finite representation of every rational number in the form of a continued fraction and the existence of a development into a continued fraction for every real number, that is, we prove the convergence of infinite continued fractions. Finally, we prove that, for an integer \(k\), \(k\geq2\), the set of all real numbers \(\alpha\), such that \(0\leq \alpha\leq k^{-1 }\) and such that the development of the number \(\alpha\) into a continued fraction does not contain a partial quotient smaller than k, the Cantor set. In the third chapter, we construct the Cantor set as a homeomorphic image of the ring of \(p\)-adic integers \(\mathbb{Z}_p\), \(p\) a prime number. First we define \(p\)-adic numbers and whole \(p\)-adic numbers. We define the \(p\)-adic norm and the \(p\)-adic distance on the field \(\mathbb{Q}\) and prove that the field \(\mathbb{Q}\) with the \(p\)-adic the norm is not complete. We complete the field \(\mathbb{Q}\) with the field of equivalence classes of Cauchy sequences, \(\mathbb{Q}_p \). After that, we show the equivalence of the field \(\mathbb{Q}_p \) as a set of equivalence classes of sequences and infinite \(p\)-adic expansions that converge in the \(p\)-adic norm on \(\mathbb{Q} _p\). Finally, we show the existence of a homeomorphism between the ring of \(p\)-adic integers \(\mathbb{Z}_p\) and the Cantor set in the inherited Euclidean topology.
Keywords
Cantorov skup
verižni razlomci
Cauchyjevi nizovi
euklidska topologija
Keywords (english)
Cantor set
continued fractions
Cauchy series
Euclidean topology
Language croatian URN:NBN urn:nbn:hr:217:645741 Study programme Title: Mathematics Education; specializations in: Mathematics Education Type of resource Text File origin Born digital Access conditions Open access Terms of use Created on 2022-10-25 10:19:36
