
Bellman's Lost in a Forest Problem and Its Analysis
Haviger, Vojtěch ; Hoderová, Jana (referee) ; Čermák, Jan (advisor)
This thesis is focused on the Bellman’s problem of searching for the shortest escape path from the planar, closed and convex set with a nonempty interior. After introduction of some notions needed to understand and solve the problem, the thesis deals with discusions of the shortest escape paths for considered shapes of the given set (circular disc, circular sector, infinite strip, rectangle, regular polygon, triangle, halfplane, circle). Finally, the obtained results are summarized, and extended by some open problems.


Characterization of convex sets
Lžičař, Jiří ; Lachout, Petr (advisor) ; Kozmík, Václav (referee)
The idea of convexity is very important especially for probability theory, optimization and stochastic optimization. Convexity is a unique set pro perty in many ways, which is worth to be studied. Various properties of convex sets are generally known, such as the ones related to separability. It however becomes apparent that the definition of convexity is very interesting, since it is possible to replace the definition by various collections of properties which are equivalent to it. There also exist set operations preserving convexity and another ones which preserve it when supported by another requirements. 1


Modifications of Whitney's $C^1$ extension theorem.
Dovhoruk, Olesya ; Zajíček, Luděk (advisor) ; Holický, Petr (referee)
Title: Modifications of Whitney's C1 extension theorem. Author: Olesya Dovhoruk Department: Department of Mathematical Analysis Supervisor: prof. RNDr. Luděk Zajíček, DrSc., Department of Mathematical Ana lysis Abstract: This work deals with modifications of the Whitney's C1 extension theorem on a special closed set M in Rn . The work investigates of whether it is possible to skip some of the assumptions of the Whitney's theorem. It turns out that if we do not assume the continuity of a function f : M → R, which is being extended from a general closed set M ⊂ Rn , then f is continuous from the remaining assumptions in the Whitney's theorem, but if we skip the continuity of a function d, which features in the Whitney's theorem (and plays a role of the generalised differential of the function f), then d is continuous from the remaining assumptions, but just for n = 1. Further, some proposals based on modifications of the assumptions of the Whit ney's theorem are proved. For instance, the theorem of an existence of a C1 extension of a function f : (a, b)×[0, c), f ∈ C1 ((a, b)×(0, c)), for which is valid that the function gradf has finite limits in (a, b) × {0}. Another similar result of the work is the existence of a C1 extension of a function f : M ⊂ Rn → R, where M = M◦ = ∅ is a compact and convex set and...
