Browsing by Author "Lech, Jacek"

Now showing 1 - 10 of 10

Access status: Open Access ,
Komutatory dyfeomorfizmów klasy $C^{r}$
(Data obrony: 2008) Lech, Jacek
Wydział Matematyki Stosowanej
The aim of that dissertation is to prove that some compact diffeo-morphism groups on a foliated manifold and on a manifold with corners are perfect. The main idea is to use modifications of the Mather-Epstein method. The paper has four chapters. The first includes basie notions and properties which are essential in the proof. It contains facts about moduli of continuity and seminorms in funetion spaces on a standard manifold. In the rest of the chapter one can find theorems about topological properties of diffeomorphism groups on $\mathbb{R}$. In the second chapter the notion of a $C^{r,s}$, -mapping is introduced and some properties of such mappings are shown. Next, estimations of seminorms are reformulated into the case of manifold $\mathbb{R}^{n}, \mathbb{F}_{k}$, where $\mathbb{F}_{k}$ means the product foliation. Let $\mathbb{R, F}$ be a foliated manifold and let $Diff^{r, s} (\mathbb{M, F)}$ be the group of $C^{r,s}$ -diffeomorphisms on $\mathbb{M, F}$ acting along leaves and diffeotopic to the Id through $C^{r,s}$ -diffeotopies with compact supports. It is shown in the paper that it is a topological group. Moreover, the fragmenta-tion property in the case of foliated mauifold is proved. The proof of the main theorem is presented in chapter three. The main construetion has a few steps. The most important are rolling-up operator and Mather's operator. Ali maps in considerations agree with the foliation structure. Using this construetion and Schauder-Tichonoff fixed point theorem it is shown that the group $Diff^{r, s} (\mathbb{M, F)}$ is perfect whenever r is finite, $r \neq n + 1$, where dim $M = n$. At the end of the chapter there are remarks about the case $r = n + 1$ which is still open. It also contains the description how the perfectness of a diffeomorphism group on a foliated manifold is connected with the perfectness of an analogical group on a standard manifold. The last chapter includes considerations about the perfectness of the identity component of the compact $C^{\infty}$-diffeomorphism group $Diff^{\infty}(\mathbb{M})$ on a manifold with corners. It is assumed that the manifold has no vertices. First, there are presented basic notions about a manifold with corners and preliminary properties. Next, one can find the construction of Mather's operator with using Epstein method. Mather's operator may be used along edges of the manifold only. Hence it is introduced another method called bump decomposition. The main proof use both of them. It is ałso used Schauder-Tichonoff fixed point theorem. In the last section there are remarks about the case of a manifold with corners which has vertices. It is shown that the group $Diff^{\infty}(\mathbb{M})$ on such manifold is not perfect.
Access status: Restricted ,
Macierz Seiferta i wyznacznik węzłów
(Data obrony: 2013-07-09) Sadowska, Katarzyna
Wydział Matematyki Stosowanej
Access status: Restricted ,
Model FitzHugh-Nagumo
(Data obrony: 2013-07-09) Śleziak, Agnieszka
Wydział Matematyki Stosowanej
Access status: Restricted ,
Nawias Kauffmana grafu stowarzyszonego z diagramami Gaussa: podstawowe własności i zastosowania
(Data obrony: 2013-10-25) Jurzak, Angelika
Wydział Matematyki Stosowanej
Access status: Open Access ,
On the boundedness of equivariant homeomorphism groups
(Wydawnictwa AGH, 2018) Lech, Jacek; Michalik, Ilona; Rybicki, Tomasz
Given a principal $G$-bundle $\pi:M\to B$, let $\mathcal{H}_G(M)$ be the identity component of the group of $G$-equivariant homeomorphisms on $M$. The problem of the uniform perfectness and boundedness of $\mathcal{H}_G(M)$ is studied. It occurs that these properties depend on the structure of $\mathcal{H}(B)$, the identity component of the group of homeomorphisms of $B$, and of $B$ itself. Most of the obtained results still hold in the $C^r$ category.
Access status: Open Access ,
On the perfectness of C∞,s-diffeomorphism groups on a foliated manifold
(2008) Lech, Jacek
The notion of $C^{r,s}$ and $C^{\infty,s}$-diffeomorphisms is introduced. It is shown that the identity component of the group of leaf preserving $C^{\infty,s}$-diffeomorphisms with compact supports is perfect. This result is a modification of the Mather and Epstein perfectness theorem.
Access status: Open Access ,
On the structure of certain nontransitive diffeomorphism groups on open manifolds
(2012) Kowalik, Agnieszka; Lech, Jacek; Michalik, Ilona
It is shown that in some generic cases the identity component of the group of leaf preserving diffeomorphisms (with not necessarily compact support) on a foliated open manifold is perfect. Next, it is proved that it is also bounded, i.e. bounded with respect to any bi-invariant metric. It follows that the group is uniformly perfect as well.
Access status: Restricted ,
Rzut ortogonalny w przestrzeni Hilberta
(Data obrony: 2013-07-09) Hanejko, Małgorzata
Wydział Matematyki Stosowanej
Access status: Restricted ,
Symetrie i całkowalność równania Riccatiego
(Data obrony: 2013-07-09) Kurpiel, Łukasz
Wydział Matematyki Stosowanej
Access status: Restricted ,
Węzłowe modele i struktury w biologii
(Data obrony: 2013-07-09) Gacek, Natalia
Wydział Matematyki Stosowanej