Poincaré-Mirandov izrek

delo diplomskega seminarja

Barbara Lipnik (Author), Aleš Vavpetič (Mentor)

Abstract

Diplomsko delo obravnava posplošitev dobro poznanega izreka v eni dimenziji, in sicer izreka o vmesni vrednosti. Natančneje, njegove posledice, ki zagotovi obstoj ničle funkcije. Ta izrek je dobro poznan tudi kot izrek o ničli. Dijaki se z njim srečajo že v srednji šoli, študentom matematike je že nekaj samoumevnega. Izrek o ničli nam pove, da ima vsaka zvezna funkcija na zaprtem intervalu, če v robnih točkah zavzame nasprotno predznačeni vrednosti, vsaj eno ničlo. Izrek lahko, z nekaterimi modifikacijami, posplošimo na poljubno dimenzijo. V delu dokažemo, da ima vsaka zvezna preslikava na enotski kocki v $n$-dimenzionalnem evklidskem prostoru, pod določenim pogojem, vsaj eno ničlo. Pogoj, ki ga potrebujemo, je, da so komponentne funkcije te preslikave različno predznačene na ustreznih stranicah enotske kocke. V delu opišemo tudi ekvivalenco te posplošitve izreka o ničli oziroma Poincaré-Mirandovega izreka, in Brouwerjevega izreka o negibni točki. Predstavimo diskreten dokaz Poincaré-Mirandovega izreka, kjer si pomagamo s Steinhausovim izrekom o šahovnici. Predstavimo tudi možne posplošitve Poincaré-Mirandovega izreka na nekatere neskončno dimenzionalne prostore.

Keywords

matematika;ničle funkcije;negibne točke;preslikave;zveznost;Poincaré-Mirandov izrek;

Data

Language:	Slovenian
Year of publishing:	2021
Typology:	2.11 - Undergraduate Thesis
Organization:	UL FMF - Faculty of Mathematics and Physics
Publisher:	[B. Lipnik]
UDC:	515.1
COBISS:	78598147
Views:	824
Downloads:	70
Average score:	0 (0 votes)
Metadata:

Other data

Secondary language:	English
Secondary title:	The Poincaré-Miranda theorem
Secondary abstract:	This thesis deals with the generalization of a well-known theorem in one dimension, namely the Intermediate value theorem. Specifically, its corollary, which proves that a function has a root. This theorem is well known. Students learn about it already in high school. For math students, this is almost a self-evident result. The theorem says that every continuous function on a closed interval that changes sign at the boundary points has at least one root in this interval. With some modifications, this theorem can be generalized to an arbitrary dimension. We prove that every continuous map on a unit cube in a $n$-dimensional Euclidean space has, under a certain condition, at least one root. The assumption we need is that each component function of the map changes sign on the corresponding sides of the unit cube. Here, we also show the equivalence of this generalization of a one-dimensional theorem, namely the Poincaré-Miranda theorem, and The Brouwer fixed point theorem. We give a discrete proof of the Poincaré-Miranda theorem using the Steinhaus chessboard theorem. We present possible generalizations of the Poincaré-Miranda theorem to certain infinite-dimensional spaces.
Secondary keywords:	mathematics;roots of a function;fixed points;maps;continuity;Poincaré-Miranda theorem;
Type (COBISS):	Final seminar paper
Study programme:	0
Thesis comment:	Univ. v Ljubljani, Fak. za matematiko in fiziko, Oddelek za matematiko, Matematika - 1. stopnja
Pages:	28 str.
ID:	13525390