|
| |
|
| |
|
|
Sobolev embedding theorem on domains without Lipschitz boundary
Roskovec, Tomáš ; Hencl, Stanislav (advisor) ; Honzík, Petr (referee)
We study the Sobolev embeddings theorem and formulate modified theorems on domains with nonlipschitz boundary. The Sobolev embeddings the- orem on a domain with Lipschitz boundary claims f ∈ W1,p ⇒ f ∈ Lp∗ (p) , kde p∗ (p) = np n − p . The function p∗ (p) is continuous and even smooth. We construct a domain with nonlipschitz boundary and function of the optimal embedding i.e. analogy of p∗ (p) is not continous. In the first part, according to [1], we construct the domain with the point of discontinuity for p = n = 2. Though we used known construction of domain, we prove this by using more simple and elegant methods. In the second part of thesis we suggest the way how to generalize this model domain and shift the point of discontinuity to other point than p = n = 2.
|
|
|
Typical continuous and integrable functions
Hruška, David ; Hencl, Stanislav (advisor) ; Pražák, Dalibor (referee)
In this thesis we use the Baire categories to define the concept of "typical functions". Then we prove several theorems generally asserting that a typical function from a space of functions having some nice property does not have a stronger property. In particular we prove that a typical continuous or Hölder continuous function is nowhere differentiable, a typical continuous monotone function does not satisfy the Luzin (N) condition and a typical integrable function is nowhere continuous. Powered by TCPDF (www.tcpdf.org)
|
|
|
Vlastnosti zobrazení s konečnou distorzí
Campbell, Daniel ; Hencl, Stanislav (advisor) ; Malý, Jan (referee)
We study the continuity of mappings of finite distortion, a set of mappings intended to model elastic deformations in non-linear elasticity. We focus on continuity criteria for the inner-distortion function and prove that certain modulus of continuity estimates are sharp, i.e. cannot be im- proved. We also give a proof of the continuity of mappings of finite distortion under simplified conditions on the integrability of the distortion function. 1
|
|
|
Spaces of functions with fractional derivatives on interval
Lopata, Jan ; Kaplický, Petr (advisor) ; Hencl, Stanislav (referee)
In literature we can find a variety of ways to introduce Sobolev space W1,1 on bounded and open interval. In this thesis we will put them in context. We will show that completion of set of function with continuous first derivative, the space of functions with weak derivative and space of absolutely continuous functions are isometrically isomorphic. Furthemore, we will demonstrate that the Sobolev space W1,∞ is isometrically isomorphic to space of Lipschitz functions. We will also show several trivial and nontrivial embeddings for Besov spaces. Finnaly, we will examine the question, whether functions from Besov space are, given some parameters, included in set of continuous functions. 1
|
|
|
Absolutely continuous function and functions of bounded variation
Hladký, Filip ; Hencl, Stanislav (advisor) ; Kurka, Ondřej (referee)
In this thesis we will study relationship between space of absolutely continuous func- tions and space of functions with bounded variation. In first three chapters we will study properties of absolutely continuous functions and functions with bounded variation and we will show nessesary and sufficient condition for functions with bounded variation to be absolutely continuous. Moreover we will show one part of fundamental theorem of calculus for Lebesgue's integral. In the last chapter we will study relationship between absolutely continuous mappings and mappings with bounded variation from Rn to Rm. 1
|
|
|
Properties of Cantor function
Fiala, Martin ; Hencl, Stanislav (advisor) ; Holický, Petr (referee)
Properties of Cantor function Author: Martin Fiala Supervisor: Stanislav Hencl Abstract: In the present thesis we study main properties of the Can- tor function (sometimes called Cantor Devil's staircase in popular lit- erature), named after significant german mathematician Georg Cantor ( 3 March 1845 in St Petersburg, 6 Jan 1918 in Halle). 1
|
|
|
Modelování spotových cen elektrické energie
Šmíd, Vítězslav ; Honzík, Petr (advisor) ; Hencl, Stanislav (referee)
We describe a single-period vector autoregressive model with parameter restrictions and find a consistent estimator of the parameters. We apply several restricted models to electricity prices in two markets. The datasets are comprised of the settlement prices of day-ahead auctions in which market participants bid on next day's electricity deliveries in 24 separate hourly blocks. We therefore model the data as a time series in R^24. To avoid overfitting we crossvalidate all models using sliding windows of training and testing data. We find that simple models perform better because they are more resilient in volatile periods than more comprehensive models. We suggest that model performance could be improved by incorporating exogenous data, such as weather conditions. Powered by TCPDF (www.tcpdf.org)
|
|
|
Three lakes problem
Šulc, Dominik ; Hencl, Stanislav (advisor) ; Vejnar, Benjamin (referee)
Cílem této práce je nalezení řešení problému tří jezer a podrobný d·kaz jeho správnosti. Problém tří jezer (Lakes of Wada) je úloha, která spočívá v sestrojení tří otevřených souvislých množin v rovině, které se neprotínají a mají společnou hranici. Ukážeme, že takové množiny existují a že kromě uvedených vlastností mohou být dokonce obloukově souvislé. 1
|
guest ::
