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Integrálně-geometrická míra
Penkov, Deyvid ; Rataj, Jan (vedoucí práce) ; Campbell, Daniel Cameron (oponent)
Definujeme třídu měr nižších dimenzí pomocí průměrování Lebesgueovy míry z pro- jekce množiny přes všechny ortogonální projekce. Ukážeme některé vztahy s Hausdor- ffovou měrou a konstruujeme množiny, na kterých se tato míra od Hausdorffovy liší. Definujeme třídu měr nižších dimenzí pomocí průměrování Lebesgueovy míry z projekce množiny přes všechny ortogonální projekce. Ukážeme některé vztahy s Hausdorffovou měrou a konstruujeme množiny, na kterých se tato míra od Hausdorffovy liší. 1
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Optimalizace přistání na Měsíci
Campbell, Daniel ; Milota, Jaroslav (vedoucí práce) ; Bárta, Tomáš (oponent)
Nazev prace: Optimali/ace pristani na Mesici Autor: Daniel Campbell Katedra. (listav): Katedra inatematicke analy/y Vedouci bakalafske prace: Doc.RNDr. .laroslav Miluta. CSc. e-mail vedoueiho: Jaroslav.Milota'Q'mlT.cuni.c/ Abstrakt: V toto prat:i vytvoriine a /komnamo rnodol, kt.ory ])opisujo raketu pri pristani na inesiui. Urchin- za jakych okolnosti l/,e pristat a zda f'xiistnji1 kontrol. klery by niininiili/oval inno/stvi jjotrcbneho paliva pon- ziteho pri pristani. Pokud existuje. pak tento prvek najdome a doka/eme vlastnowl . Title:0ptimilisation of the moon-landing problem Author:Daniel Campbell Department:Katedra matematiuke analy/y Snpnrvisor:Doc. RNDr. Jaroslav Milota, CSc. Supervisor's e-mail address:Jarosla\.Milota'(PniJr.(.:nni.c/ Abstract: In this paper \vo are to rreate and examine a model, which describes (he motion of a rocket landing on the surface of the moon. We will determine under which circumstances it is possible to make (.he landing and determine whether there exists some way of landing that minimises fuel consumption. IT so we are to find this method and prove the desired property.
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Vlastnosti zobrazení s konečnou distorzí
Campbell, Daniel ; Hencl, Stanislav (vedoucí práce) ; Malý, Jan (oponent)
Zkoumáme spojitost zobrazení s konečnou distorzí, funkce které mají sloužit jako model elastických deformací při nelineární elas- ticitě. Zaměřujeme se na podmínky pro spojitost na vnitřní distorzi a navíc ukážeme, že jistý odhad modulu spojitosti je ostrý, t.j. nemůže být vylepšen. Uvedeme důkaz spojitosti pro zobrazení s konečnou distorzí za zjednodušených předpoklad˙u na distorzi. 1
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Properties of mappings of finite distortion
Campbell, Daniel ; Hencl, Stanislav (vedoucí práce)
In the following thesis we will be mostly concerned with questions related to the regularity of solutions to non-linear elasticity models in the calculus of variations. An important step in this is question is the approximation of Sobolev homeomorphisms by diffeomorphisms. We refine an approximation result of Hencl and Pratelli's which, for a given planar Sobolev (or Sobolev-Orlicz) homeomorphism, constructs a diffeomorphism arbitrarily close to the original map in uniform convergence and in terms of the Sobolev-Orlicz norm. Further we show, in dimension 4 or higher, that such an approximation result cannot hold in Sobolev spaces W1,p where p is too small by constructing a sense-preserving homeomorphism with Jacobian negative on a set of positive measure. The class of mappings referred to as mappings of finite distortion have been proposed as possible models for deformations of bodies in non-linear elasticity. In this context a key property is their continuity. We show, by counter-example, the surprising sharpness of the modulus of continuity with respect to the integrability of the distortion function. Also we prove an optimal regularity result for the inverse of a bi-Lipschitz Sobolev map in Wk,p and composition of Lipschitz maps in Wk,p comparable with the classical inverse mapping theorem. As a...
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Properties of mappings of finite distortion
Campbell, Daniel ; Hencl, Stanislav (vedoucí práce) ; Koskela, Pekka (oponent) ; Mora Corral, Carlos (oponent)
In the following thesis we will be mostly concerned with questions related to the regularity of solutions to non-linear elasticity models in the calculus of variations. An important step in this is question is the approximation of Sobolev homeomorphisms by diffeomorphisms. We refine an approximation result of Hencl and Pratelli's which, for a given planar Sobolev (or Sobolev-Orlicz) homeomorphism, constructs a diffeomorphism arbitrarily close to the original map in uniform convergence and in terms of the Sobolev-Orlicz norm. Further we show, in dimension 4 or higher, that such an approximation result cannot hold in Sobolev spaces W1,p where p is too small by constructing a sense-preserving homeomorphism with Jacobian negative on a set of positive measure. The class of mappings referred to as mappings of finite distortion have been proposed as possible models for deformations of bodies in non-linear elasticity. In this context a key property is their continuity. We show, by counter-example, the surprising sharpness of the modulus of continuity with respect to the integrability of the distortion function. Also we prove an optimal regularity result for the inverse of a bi-Lipschitz Sobolev map in Wk,p and composition of Lipschitz maps in Wk,p comparable with the classical inverse mapping theorem. As a...
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Properties of mappings of finite distortion
Campbell, Daniel ; Hencl, Stanislav (vedoucí práce)
In the following thesis we will be mostly concerned with questions related to the regularity of solutions to non-linear elasticity models in the calculus of variations. An important step in this is question is the approximation of Sobolev homeomorphisms by diffeomorphisms. We refine an approximation result of Hencl and Pratelli's which, for a given planar Sobolev (or Sobolev-Orlicz) homeomorphism, constructs a diffeomorphism arbitrarily close to the original map in uniform convergence and in terms of the Sobolev-Orlicz norm. Further we show, in dimension 4 or higher, that such an approximation result cannot hold in Sobolev spaces W1,p where p is too small by constructing a sense-preserving homeomorphism with Jacobian negative on a set of positive measure. The class of mappings referred to as mappings of finite distortion have been proposed as possible models for deformations of bodies in non-linear elasticity. In this context a key property is their continuity. We show, by counter-example, the surprising sharpness of the modulus of continuity with respect to the integrability of the distortion function. Also we prove an optimal regularity result for the inverse of a bi-Lipschitz Sobolev map in Wk,p and composition of Lipschitz maps in Wk,p comparable with the classical inverse mapping theorem. As a...
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Vlastnosti zobrazení s konečnou distorzí
Campbell, Daniel ; Hencl, Stanislav (vedoucí práce) ; Malý, Jan (oponent)
Zkoumáme spojitost zobrazení s konečnou distorzí, funkce které mají sloužit jako model elastických deformací při nelineární elas- ticitě. Zaměřujeme se na podmínky pro spojitost na vnitřní distorzi a navíc ukážeme, že jistý odhad modulu spojitosti je ostrý, t.j. nemůže být vylepšen. Uvedeme důkaz spojitosti pro zobrazení s konečnou distorzí za zjednodušených předpoklad˙u na distorzi. 1
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Optimalizace přistání na Měsíci
Campbell, Daniel ; Bárta, Tomáš (oponent) ; Milota, Jaroslav (vedoucí práce)
Nazev prace: Optimali/ace pristani na Mesici Autor: Daniel Campbell Katedra. (listav): Katedra inatematicke analy/y Vedouci bakalafske prace: Doc.RNDr. .laroslav Miluta. CSc. e-mail vedoueiho: Jaroslav.Milota'Q'mlT.cuni.c/ Abstrakt: V toto prat:i vytvoriine a /komnamo rnodol, kt.ory ])opisujo raketu pri pristani na inesiui. Urchin- za jakych okolnosti l/,e pristat a zda f'xiistnji1 kontrol. klery by niininiili/oval inno/stvi jjotrcbneho paliva pon- ziteho pri pristani. Pokud existuje. pak tento prvek najdome a doka/eme vlastnowl . Title:0ptimilisation of the moon-landing problem Author:Daniel Campbell Department:Katedra matematiuke analy/y Snpnrvisor:Doc. RNDr. Jaroslav Milota, CSc. Supervisor's e-mail address:Jarosla\.Milota'(PniJr.(.:nni.c/ Abstract: In this paper \vo are to rreate and examine a model, which describes (he motion of a rocket landing on the surface of the moon. We will determine under which circumstances it is possible to make (.he landing and determine whether there exists some way of landing that minimises fuel consumption. IT so we are to find this method and prove the desired property.
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