eprintid: 1818
rev_number: 9
eprint_status: archive
userid: 1
dir: disk0/00/00/18/18
datestamp: 2001-11-29 00:00:00
lastmod: 2014-04-03 11:25:11
status_changed: 2012-08-14 15:02:51
type: doctoralThesis
metadata_visibility: show
creators_name: Gavrila, Caius
title: Functions with minimal number of critical points
title_de: Funktionen mit einer minimalen Anzahl kritischen Punkten
ispublished: pub
subjects: ddc-510
divisions: i-110400
adv_faculty: af-11
keywords: Critical Points , h-cobordism , Oriented Trees
cterms_swd: Morse-Theorie
cterms_swd: Ljusternik-Snirel'man-Kategorie
cterms_swd: Gerichteter Graph
abstract: In 1934 Lusternik and Schnirelmann introduced a new numerical topological invariant (the Lusternik-Schnirelmann category) during research into the global calculus of variation. They showed that this invariant carries important information about both the existence of critical points and the cardinality of the critical set. Further results on Lusternik-Schnirelmann category distract the attention from the original problem: finding crit(M) the minimal number of critical points of a smooth function on a given smooth manifold M. Only Cornea in 1998 analyzed the notion of crit, originally introduced by Takens in 1968. Inspired by Takens' paper and by Morse theory in the Smale setting we have analyzed the crit for products of manifolds. Using concepts from the graph theory we have found upper bounds of crit for products of special manifolds with generalized tori. The proofs of these results  are based on the fusing lemma, which establishes sufficient conditions to construct a triad function with at most one critical point from a triad function with two critical points. In order to compute the crit for products of lens spaces with spheres we prove that the ball category is a lower bound for the number of the critical points of a function. The thesis contains also a paragraph about crit of manifolds with boundary. 
abstract_translated_text: In 1934 Lusternik and Schnirelmann introduced a new numerical topological invariant (the Lusternik-Schnirelmann category) during research into the global calculus of variation. They showed that this invariant carries important information about both the existence of critical points and the cardinality of the critical set. Further results on Lusternik-Schnirelmann category distract the attention from the original problem: finding crit(M) the minimal number of critical points of a smooth function on a given smooth manifold M. Only Cornea in 1998 analyzed the notion of crit, originally introduced by Takens in 1968. Inspired by Takens' paper and by Morse theory in the Smale setting we have analyzed the crit for products of manifolds. Using concepts from the graph theory we have found upper bounds of crit for products of special manifolds with generalized tori. The proofs of these results  are based on the fusing lemma, which establishes sufficient conditions to construct a triad function with at most one critical point from a triad function with two critical points. In order to compute the crit for products of lens spaces with spheres we prove that the ball category is a lower bound for the number of the critical points of a function. The thesis contains also a paragraph about crit of manifolds with boundary. 
abstract_translated_lang: eng
class_scheme: msc
class_labels: 57R70, 55M30, 57R25, 57R50
date: 2000
date_type: published
id_scheme: DOI
id_number: 10.11588/heidok.00001818
ppn_swb: 1643256467
own_urn: urn:nbn:de:bsz:16-opus-18184
date_accepted: 2001-03-16
advisor: HASH(0x55fc36bbfa08)
language: eng
bibsort: GAVRILACAIFUNCTIONSW2000
full_text_status: public
citation:   Gavrila, Caius  (2000) Functions with minimal number of critical points.  [Dissertation]     
document_url: https://archiv.ub.uni-heidelberg.de/volltextserver/1818/1/KritischePunkte.pdf