Sylvester-Gallaijev izrek in njegova posplošitev za metrične prostore

diplomsko delo

Tanja Omerzel (Author), Matjaž Kovše (Mentor)

Abstract

Diplomsko delo v prvem poglavju obravnava Sylvester-Gallaijev izrek: predstavitev, formulacijo, možne posplošitve in dokaze ter zgodovino. Začetek slednje sega v konec 19. stoletja, ko je James Joseph Sylvester ustvaril temelje, ki so se kasneje razvili v oblikovanje Sylvester-Gallaijevega izreka. Sam je namreč pri raziskovanju raznih konfiguracij, sestavljenih iz mrež, ugotovil, da ni možno določiti končnega števila točk tako, da bo vsaka premica, ki poteka skozi dve točki, šla še skozi tretjo iz iste množice, razen, če vse točke ležijo na isti premici. Njegovo ugotovitev je kasneje v afini realni ravnini dokazal Tibor Gallai, katerega dokaz pa ni ostal osamljen. V diplomskem delu sta podana še dokaza L. M. Kellya in R. Ste inberga. Ob vsem naštetem so v prvem poglavju, predvsem zaradi lažjega razumevanja dokazov Sylvester-Gallaijevega izreka, predstavljene osnovne značilnosti evklidske, afine in projektivne geometrije in Motzkinov izrek kot ena izmed mnogih posplošitev Sylvester-Gallaijevega izreka. Drugo poglavje v uvodnem delu zajema definicije pojmov, kot so: metrični prostor, premica v poljubnem metričnem prostoru, trojna relacija, vmesnost. Sicer je v celoti namenjeno posplošitvi Sylvester-Gallaijevega izreka za metrične prostore, tako imenovanemu Sylvester-Chvátalovemu izreku. Vašek Chvátal je namreč razširil pojem premic v poljubnih metričnih prostorih in podal domnevo, ki posplošuje Sylvester-Gallaijev izrek. Chvátalova domneva je bila potrjena kot izrek, ki smo ga, zaradi lažjega dokazovanja, tudi s pomočjo primerov, razdelili na dva dela: Če za vsake 3 točke iz M velja, da ležijo na neki skupni premici, potem ta premica vsebuje vse točke iz M. Oziroma, če obstajajo 3 točke iz M, ki ne ležijo na skupni premici, potem obstaja premica, ki vsebuje natanko 2 točki.

Keywords

matematika;metrični prostori;izreki;evklidska geometrija;afina geometrija;projektivna geometrija;diplomska dela;

Data

Language:	Slovenian
Year of publishing:	2011
Source:	Maribor
Typology:	2.11 - Undergraduate Thesis
Organization:	UM FNM - Faculty of Natural Sciences and Mathematics
Publisher:	[T. Omerzel]
UDC:	51(043.2)
COBISS:	18641672
Views:	1907
Downloads:	118
Average score:	0 (0 votes)
Metadata:

Other data

Secondary language:	English
Secondary title:	SYLVESTER-GALLAI THEOREM AND ITS GENERALIZATION FOR METRIC SPACES
Secondary abstract:	This diploma thesis in first chapter beginns with Sylvester-Gallai theorem: its presentation, formulation, generalizations, proofs and history. James Joseph Sylvester has been investigated a figure which is a discrete set of points having the property that the line joining any two of them contains not only these two but infinitely many. It was probably the investigation of such configurations that led Sylvester to propose his problem of 1893: Prove that it is not possible to arrange any finite number of real points so that a right line through every two of them shall pass through a third, unless they all lie in the same right line. A Sylvester question was solved by Gallai, who gave an affine proof, proves has been followed by a Steinberg's proof in the projective plane and Kelly's Euclidean proof. First chapter also describes features of Euclidean, affine, projective plane and Motzkin generalization as one of many generalizations of Sylvester-Gallai theorem. The second chapter begins with definitions of concepts like metric space, a line in arbitrary metric space, a ternary relation, betwenness. The main purpose of the chapter is to present and prove Sylvester-Chvátal theorem. Vašek Chvátal extended the notion of lines to arbitrary metric spaces and made a conjecture that generalizes the Sylvester-Gallai theorem. Proof splits into two parts: If every three points of M are contained in some line, then some line consists of all points of M and if some three points of M are contained in no line, then some line consists of precisely two points.
Secondary keywords:	Sylvester-Gallai theorem;Euclidean geometry;Affine geometry;Projective geometry;metric space;Sylvester-Chvátal theorem;
URN:	URN:SI:UM:
Type (COBISS):	Undergraduate thesis
Thesis comment:	Univ. v Mariboru, Fak. za naravoslovje in matematiko, Oddelek za matematiko in računalništvo
Pages:	40 f.
Keywords (UDC):	mathematics;natural sciences;naravoslovne vede;matematika;mathematics;matematika;
ID:	19514