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Differentiability of the inverse mapping
Konopecký, František ; Hencl, Stanislav (advisor) ; Honzík, Petr (referee)
Primary objective of the thesis is proof of the statement that if for ∈ ℕ a ≥ 1 a bilipschitz mapping belongs to +1, loc ∩ ,∞ loc then also its inverse −1 belongs to +1, loc . We prove a similar statement also for spaces loc . For this purpose we construct a new ordering of -th partial derivatives to generalized Jacobian matrix. Thanks to this matrix we are able to differentiate matrices in an applicable way. Generalized Jacobian matrix is projected so that there still holds the Chain rule and, in some way, also rules for matrices product differentiation. 1
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Lipschitz mappings of discrete sets
Kaluža, Vojtěch ; Matoušek, Jiří (advisor) ; Šámal, Robert (referee)
In this thesis we consider Feige's question of whether there always exists a constantly Lipschitz bijection of an n2 -element set S ⊂ Z2 onto a regular lattice of n × n points in Z2 . We propose a solution of this problem in case the points of the set S form a long rectangle or they are arranged in the shape of a square without a part of its interior points. The main part is a summary of Burago's and Kleiner's article [2] and the article by McMullen [12] dealing with the problem of existence of separated nets in R2 that are not bi-Lipschitz equivalent to the integer lattice. This problem looks similar to Feige's problem. According to these articles we construct a separated net that is not bi-Lipschitz equivalent to the integer lattice, using a positive bounded measurable function that is not the Jacobian of a bi-Lipschitz homeomorphism almost everywhere. We present McMullen's construction of such a function and we complete his proof of its correctness. 1
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Differentiability of the inverse mapping
Konopecký, František ; Hencl, Stanislav (advisor)
Primary objective of the thesis is proof of the statement that if for ∈ ℕ a ≥ 1 a bilipschitz mapping belongs to +1, loc ∩ ,∞ loc then also its inverse −1 belongs to +1, loc . We prove a similar statement also for spaces loc . For this purpose we construct a new ordering of -th partial derivatives to generalized Jacobian matrix. Thanks to this matrix we are able to differentiate matrices in an applicable way. Generalized Jacobian matrix is projected so that there still holds the Chain rule and, in some way, also rules for matrices product differentiation. 1
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Lipschitz mappings in the plane
Kaluža, Vojtěch ; Matoušek, Jiří (advisor) ; Šámal, Robert (referee)
In this thesis we consider an open question of Feige that asks whether there always exists a constantly Lipschitz bijection of an n2 -point subset of Z2 onto a regular grid [n] × [n] for every n ∈ N. We relate this question to an already resolved problem of the existence of a bounded positive measurable density in R2 that is not the Jacobian of any bilipschitz map. This problem was resolved by Burago and Kleiner [1], and independently, by McMullen [12]. We present the work of Burago and Kleiner, analyze its relation to Feige's problem and sug- gest a continuous formulation of Feige's question in a special case. Then we present the Burago-Kleiner density, make several observation about the properties of this density, and after that we construct a density that is everywhere nonrealizable as the Jacobian of a bilipschitz map. Subsequently, we discuss our continuous variant of Feige's question, provide several observation concerning it, and finally, we try to use the everywhere nonrealizable density constructed before to answer our continuous variant of Feige's question. However, this last task still remains incomplete. 1
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Differentiability of the inverse mapping
Konopecký, František ; Hencl, Stanislav (advisor)
Primary objective of the thesis is proof of the statement that if for ∈ ℕ a ≥ 1 a bilipschitz mapping belongs to +1, loc ∩ ,∞ loc then also its inverse −1 belongs to +1, loc . We prove a similar statement also for spaces loc . For this purpose we construct a new ordering of -th partial derivatives to generalized Jacobian matrix. Thanks to this matrix we are able to differentiate matrices in an applicable way. Generalized Jacobian matrix is projected so that there still holds the Chain rule and, in some way, also rules for matrices product differentiation. 1
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Lipschitz mappings of discrete sets
Kaluža, Vojtěch ; Matoušek, Jiří (advisor) ; Šámal, Robert (referee)
In this thesis we consider Feige's question of whether there always exists a constantly Lipschitz bijection of an n2 -element set S ⊂ Z2 onto a regular lattice of n × n points in Z2 . We propose a solution of this problem in case the points of the set S form a long rectangle or they are arranged in the shape of a square without a part of its interior points. The main part is a summary of Burago's and Kleiner's article [2] and the article by McMullen [12] dealing with the problem of existence of separated nets in R2 that are not bi-Lipschitz equivalent to the integer lattice. This problem looks similar to Feige's problem. According to these articles we construct a separated net that is not bi-Lipschitz equivalent to the integer lattice, using a positive bounded measurable function that is not the Jacobian of a bi-Lipschitz homeomorphism almost everywhere. We present McMullen's construction of such a function and we complete his proof of its correctness. 1
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Differentiability of the inverse mapping
Konopecký, František ; Hencl, Stanislav (advisor) ; Honzík, Petr (referee)
Primary objective of the thesis is proof of the statement that if for ∈ ℕ a ≥ 1 a bilipschitz mapping belongs to +1, loc ∩ ,∞ loc then also its inverse −1 belongs to +1, loc . We prove a similar statement also for spaces loc . For this purpose we construct a new ordering of -th partial derivatives to generalized Jacobian matrix. Thanks to this matrix we are able to differentiate matrices in an applicable way. Generalized Jacobian matrix is projected so that there still holds the Chain rule and, in some way, also rules for matrices product differentiation. 1
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