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The Frölicher-Nijenhuis bracket and its applications in geometry and calculus of variations
Šramková, Kristína ; Tomáš, Jiří (referee) ; Kureš, Miroslav (advisor)
This Master's thesis clarifies the significance of Frölicher-Nijenhuis bracket and its applications in problems of physics. The basic apparatus for these applications is differential geometry on manifolds, tensor calculus and differential forms, which are contained in the first part of the thesis. The second part summarizes the basic theory of calculus of variations on manifolds and its selected applications in the field of physics. The last part of the thesis is devoted to the applications of Frölicher-Nijenhuis bracket in the derivation of Maxwell's equations and to the description of the geometry of ordinary differential equations.
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Minimal surfaces and their applications
Beran, Filip ; Šír, Zbyněk (advisor) ; Slavík, Antonín (referee)
The aim of this bachelor thesis is to explain basic qualities of minimal surfaces and to demonstrate some significant examples. The first chapter summarizes clas- sic concepts of differential geometry of curves and surfaces, which are essential for formulation of the surface minimization problem. Solving of this variational problem brings us back to local property of surface, zero mean curvature. In the rest of the second chapter we reveal which other properties this condition implies; one of the most important is the conformity of the Gauss map. Emphasizing the geometric view, in the third chapter we derive minimal surfaces of revolution and ruled minimal surfaces. Finally we construct isometric deformation of these one parameter surface families, catenoids and helicoids, to show nontrivial case of local isometry which is also typical for minimal surfaces. 1
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Mountain Pass Theorem and its applications
Příhoda, Vojtěch ; Černý, Robert (advisor) ; Vlasák, Václav (referee)
Cílem této práce je formulovat a dokázat Větu o horském sedle a ukázat příklady její aplikace tak, aby student třetího ročníku matematiky nebo fy- ziky, jenž absolvoval alespoň úvodní kurs funkcionální analýzy, byl schopen práci porozumět. D·kaz Věty o horském sedle provedeme s pomocí Ekelan- dova variačního principu, který si rovněž formulujeme a dokážeme. Následně si ukážeme dva příklady její aplikace na d·kaz existence netriviálního sla- bého řešení eliptické parciální diferenciální rovnice obsahující nelinearitu se subkritickým r·stem. 1
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Isoperimetric problem in economics
Volek, Mikoláš ; John, Oldřich (advisor) ; Bárta, Tomáš (referee)
The isoperimetric problem is one of the broad class of optimal control problems, which draw on the generalization of classical calculus developed in the mid-20th century. In the bachelor's thesis I lay down the mathematical framework that permits to rigorously prove both the necessary and sufficient conditions for the existence of a maximizer of the objective function. I analyze a simple problem from the field of project planning, which is a branch of applied economics. On the basis of a 1973 article by Cullingford and Prideaux I present an augmented cost function that involves the concept of the time value of money, which is key to proper economic reasoning. I give an explicit solution along with graphical depictions of the impact of a non-zero discount factor on project planing under the model in question. Finally, I introduce additional constraints and discuss the subproblem of multiple equality and non-equality constraints.
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