K-Riemannov integral

master's thesis

• Author: Ante Nikšić
• Mentor: Sanja Varošanec (mentor)
• Committee member: Sanja Varošanec (predsjednik povjerenstva)
• Committee member: Slaven Kožić (član povjerenstva)
• Committee member: Igor Ciganović (član povjerenstva)
• Committee member: Ljiljana Arambašić (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2019-09-25, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

Ovaj se diplomski rad bavi definiranjem \(\mathbb{K}\)-Riemannovog integrala. Kako bismo lakše definirali pojam \(\mathbb{K}\)-Riemannovog integrala u radu smo ponovili važna svojstva i definicije vezane za Riemannov integral kao što su: gornja i donja Darbouxova suma, infimum, supremum, gornje i donje međe, integrabilnost funkcije, monotonost funkcije i ograničenost funkcije. Promatrajući integral na polju \(\mathbb{K}\) kao potpolju od \(\mathbb{R}\) konstrukcija \(\mathbb{K}\)-Riemannovog integrala slična je konstrukciji Riemannovog integrala. U radu je također defininan pojam radijalne \(\mathbb{K}\)-derivacije te smo saznali postoji li poveznica između radijalne \(\mathbb{K}\)-derivacije i \(\mathbb{K}\)-Riemannovog integrala. Na kraju rada smo proširili klasične Hermite-Hadamardove nejednakosti na slučaj kada se uobičajeni Riemannov integral zamijeni s \(\mathbb{K}\)-Riemannovim integralom i pojam konveksnosti zamijeni s \(\mathbb{K}\)-konveksnosti.

Abstract (english)

This master thesis introduces definition of \(\mathbb{K}\)-Riemann integral. In the beginning, we repeat some important properties and definitions regarding Riemann integral: upper and lower Darboux sum, infimum, supremum, upper and lower bound, function integrability, monotone function. Looking at integral on \(\mathbb{K}\) field as subfield of \(\mathbb{R}\), construction of \(\mathbb{K}\)-Riemann integral is similar to Riemann integral construction. In this thesis we also define radial \(\mathbb{K}\)-derivation and we find out a connection between radial \(\mathbb{K}\)-derivation and \(\mathbb{K}\)-Riemann integral. At the end of this thesis we extended classical Hermite-Hadamard inequalities to the case when the classical Riemann integral is replaced by \(\mathbb{K}\)-Riemann integral and the notion of convexity is replaced by \(\mathbb{K}\)-convexity.

• Keywords: Riemannov integral; radijalna derivacija; Hermite-Hadamardova nejednakost
• Keywords (english): Riemann integral; radial derivation; Hermite-Hadamard inequality
• Language: croatian
• URN:NBN: urn:nbn:hr:217:368877
• Study programme: Title: Mathematics Education; specializations in: Mathematics Education; Course: Mathematics Education; Study programme type: university; Study level: graduate; Academic / professional title: magistar/magistra edukacije matematike (magistar/magistra edukacije matematike)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Open access
• Created on: 2020-02-14 12:16:06