Steinerove trojke i planiranje eksperimenata

master's thesis

• Title (english): Steiner triplets and experiment planning
• Author: Mateo Hrelja
• Mentor: Anamari Nakić (mentor)
• Committee member: Anamari Nakić (predsjednik povjerenstva)
• Committee member: Andrea Aglić Aljinović (član povjerenstva)
• Committee member: Domagoj Kovačević (član povjerenstva)
• Granter: University of Zagreb; Faculty of Electrical Engineering and Computing; Zagreb
• Defense date and country: 2024-07-03, Croatia

Abstract

Steinerova trojka (V, B) je kombinatorni dizajn koji se sastoji od v-članog skupa točaka V i skupa tročlanih blokova B, uz uvjet da se svaki par iz V pojavljuje u točno jednom bloku iz B. Steinerove trojke STS(v) postoje ako i samo ako v ≡ 1, 3 (mod 6). U ovom radu, opisana su svojstva i dokazani uvjeti egzistencije Steinerovih trojki, prikazane metode njihove konstrukcije i dan osvrt na težak problem određivanja izomorfizma i klasifikacije Steinerovih trojki.
Opisani su Latinski kvadrati te simetrične idempotentne kvazigrupe i simetrične polu-idempotentne kvazigrupe. Ove strukture su neophodne za Boseovu konstrukciju i modificiranu Skolemovu konstrukciju Steinerovih trojki. Boseova konstrukcija koristi pritom idempotentnu kvazigrupu, dok Skolemova konstrukcija koristi polu-idempotentnu kvazigrupu. Istražen je pojam izomorfizma Steinerovih trojki. Opisan je otvoren problem enumeracije i klasifikacije Steinerovih trojki. Prikazana je primjena Steinerovih trojki u planiranju eksperimenata, te je opisana implementirana aplikacija koja koristi Steinerove trojke za efikasno planiranje izvođenja eksperimenata.

Abstract (english)

A Steiner triple system (V, B) is a combinatorial design consisting of a set V of v points and a set B of three-element blocks, with the condition that each pair from V appears in exactly one block from B. Steiner triple systems STS(v) exist if and only if v ≡ 1, 3 (mod 6). In this paper, the properties and existence conditions of Steiner triple systems are described, methods of their construction are presented, and an overview is given of the challenging problem of determining isomorphism and classification of Steiner triple systems.
Latin squares, as well as symmetric idempotent quasigroups and symmetric half-idempotent quasigroups, are described. These structures are necessary for Bose’s construction and the modified Skolem construction of Steiner triple systems. Bose’s construction uses an idempotent quasigroup, while Skolem’s construction uses a half-idempotent quasigroup. The concept of isomorphism of Steiner triple systems is explored. The open problem of enumeration and classification of Steiner triple systems is described. A practical application of Steiner triple systems in experiment design and planning is presented, and an implemented web-application that uses Steiner triple systems for efficient experimental planning is described.

• Keywords: teorija dizajna; kombinatorni dizajni; Steinerove trojke; planiranje eksperimenata
• Keywords (english): design theory; combinatorial designs; Steiner triple systems; experiment planning
• Language: croatian
• URN:NBN: urn:nbn:hr:168:571939
• Study programme: Title: Computing; Study programme type: university; Study level: graduate; Academic / professional title: magistar/magistra inženjer/inženjerka računarstva (magistar/magistra inženjer/inženjerka računarstva)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Closed access
• Created on: 2024-10-12 20:49:11