About: Gradient ranges of bumps on the plane

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rdf:type	skos:Concept http://linked.opendata.cz/ontology/domain/vavai/Vysledek
Description	Obor hodnot gradientů $C^1$ hladké funkce $b$ v rovině je regulárně uzavřený (tj. je uzávěrem svého vnitřku) za předpokladu, že $b$ má neprázdný omezený nosič a gradient $\nabla b$ má modul spojitosti $\omega = \omega(t)$ který splňuje $\omega(t)/\sqrt{t} \to 0$ pro $t\searrow 0$. Za stejného předpokladu hladkosti má obor hodnot gradientu funkce $b \colon \Rn \to \R$ s neprázdným omezeným nosičem topologickou dimenzi alespoň 2 v hustě mnoha bodech. Je dokázána a použita nová věta typu Morse-Sard. (cs) We show that the gradient range of a $\C^1$-smooth function $b$ on the plane is regularly closed (i.e., it is the closure of its interior), provided $b$ has non-empty bounded support and the gradient $\grad b$ admits a modulus of continuity $\omega = \omega (t)$ that satisfies $\omega (t)/\sqrt{t} \to 0$ as $t \searrow 0$. Furthermore, under the same smoothness hypothesis, we show that the gradient range of a function $b \fcolon \Rn \to \R$ with non-empty bounded support has the topological dimension at least two at points of a dense subset. The proof relies on a new Morse-Sard type result. We show that the gradient range of a $\C^1$-smooth function $b$ on the plane is regularly closed (i.e., it is the closure of its interior), provided $b$ has non-empty bounded support and the gradient $\grad b$ admits a modulus of continuity $\omega = \omega (t)$ that satisfies $\omega (t)/\sqrt{t} \to 0$ as $t \searrow 0$. Furthermore, under the same smoothness hypothesis, we show that the gradient range of a function $b \fcolon \Rn \to \R$ with non-empty bounded support has the topological dimension at least two at points of a dense subset. The proof relies on a new Morse-Sard type result. (en)
Title	Gradient ranges of bumps on the plane Gradient ranges of bumps on the plane (en) Obory hodnot gradientů bumpů v rovině (cs)
skos:prefLabel	Gradient ranges of bumps on the plane Gradient ranges of bumps on the plane (en) Obory hodnot gradientů bumpů v rovině (cs)
skos:notation	RIV/00216208:11320/05:00000705!RIV06-MSM-11320___
http://linked.open.../vavai/riv/strany	1699;1706
http://linked.open...avai/riv/aktivita	P Z
http://linked.open...avai/riv/aktivity	P(GP201/02/D111), Z(MSM 113200007)
http://linked.open...iv/cisloPeriodika	6
http://linked.open...vai/riv/dodaniDat	2006
http://linked.open...aciTvurceVysledku	Kolář, Jan
http://linked.open.../riv/druhVysledku	J - Článek v odborném periodiku
http://linked.open...iv/duvernostUdaju	S - Úplné a pravdivé údaje nepodléhající ochraně podle zvláštních právních předpisů
http://linked.open...titaPredkladatele	Univerzita Karlova v Praze / Matematicko-fyzikální fakulta
http://linked.open...dnocenehoVysledku	522879
http://linked.open...ai/riv/idVysledku	RIV/00216208:11320/05:00000705
http://linked.open...riv/jazykVysledku	eng - angličtina
http://linked.open.../riv/klicovaSlova	Gradient; ranges; bumps; plane (en)
http://linked.open.../riv/klicoveSlovo	plane Gradient bumps ranges
http://linked.open...odStatuVydavatele	US - Spojené státy americké
http://linked.open...ontrolniKodProRIV	[BC473E04254B]
http://linked.open...i/riv/nazevZdroje	Proceedings of the American Mathematical Society
http://linked.open...in/vavai/riv/obor	BA
http://linked.open...ichTvurcuVysledku	1 (xsd:int)
http://linked.open...cetTvurcuVysledku	2 (xsd:int)
http://linked.open...vavai/riv/projekt	Real Analytic Methods in the Calculus of Variations
http://linked.open...UplatneniVysledku	2005
http://linked.open...v/svazekPeriodika	133
http://linked.open...iv/tvurceVysledku	Kolář, Jan
http://linked.open...n/vavai/riv/zamer	http://linked.opendata.cz/resource/domain/vavai/zamer/MSM%20113200007
issn	0002-9939
number of pages	8 (xsd:int)
http://localhost/t...ganizacniJednotka	11320
is http://linked.open...avai/riv/vysledek of	Gradient ranges of bumps on the plane

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