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Integral representation theorems in noncompact cases
Kraus, Michal ; Lukeš, Jaroslav (advisor) ; Malý, Jan (referee)
Classical Choquet's theory deals with compact convex subsets of locally convex spaces. This thesis discuss some aspects of generalization of Choquet's theory for a broader class of sets, for example those which are assumed to be only closed and bounded instead of compact. Because Radon measures are usually defined for locally compact topological spaces, and this is not the case of the closed unit ball in a Banach space of infinite dimension, there are used the so called Baire measures in this setting. This thesis particularly deals with the question of existence of resultants of these measures, with the properties of the resultant map, with the analogy of Bauer's characterization of extreme points and with some other concepts known from compact theory. By using some examples we show that many of these theorems doesn't hold in noncompact setting. We also mention forms of these theorems which can be proved.
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The equation div u = f
Mielec, Jaromír ; Malý, Jan (advisor) ; Kaplický, Petr (referee)
V této práci dáváme odpověď na otázku, zda rovnice div u = f má řešení u s gradi- entem v Lp (Rn ) pro každou pravou stranu f ∈ Lp (Rn ). Dokážeme, že je to pravda pro 1 < p < ∞ a zkonstruujeme protipříklady pro p = 1 a p = ∞. In this thesis, we answer the question whether the equation div u = f has a solution u with gradient in Lp (Rn ) for each f ∈ Lp (Rn ). We prove that this is true for 1 < p < ∞ and construct counterexamples for p = 1 and p = ∞. 1
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Use of a programmable logic controller in teaching process
Malý, Jan ; Ctibor, Jiří (referee) ; Červinka, Dalibor (advisor)
Bachelor thesis is divided into four parts. First part concentrates on programming languages used for programming a PLC. These are divided into graphical and text languages. Graphical languages consist of ladder diagram and function block diagram. Text languages are Structured text and instruction list. Each of these languages are described in subchapters. Second part deals with description of three laboratory tasks. The last part contains pictures a information about laboratory task’s made equipment.
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Use of a programmable logic controller in teaching process
Malý, Jan ; Knobloch, Jan (referee) ; Červinka, Dalibor (advisor)
Bachelor thesis is divided into four parts. First part concentrates on programming languages used for programming a PLC. These are divided into graphical and text languages. Graphical languages consist of ladder diagram and function block diagram. Text languages are Structured text and instruction list. Each of these languages are described in subchapters. Second part deals with description of three laboratory tasks. The last part contains pictures a information about laboratory task’s made equipment.
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Generalized ordinary differential equations in metric spaces
Skovajsa, Břetislav ; Malý, Jan (advisor)
The aim of this thesis is to build the foundations of generalized ordinary differ- ential equation theory in metric spaces. While differential equations in metric spaces have been studied before, the chosen approach cannot be extended to in- clude more general types of integral equations. We introduce a definition which combines the added generality of metric spaces with the strength of Kurzweil's generalized ordinary differential equations. Additionally, we present existence and uniqueness theorems which offer new results even in the context of Euclidean spaces.
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Nonabsolutely convergent integrals
Kuncová, Kristýna ; Malý, Jan (advisor)
Title: Nonabsolutely convergent integrals Author: Krist'yna Kuncov'a Department: Department of Mathematical Analysis Supervisor: prof. RNDr. Jan Mal'y, DrSc., Department of Mathematical Analysis Abstract: In this thesis we develop the theory of nonabsolutely convergent Hen- stock-Kurzweil type packing integrals in different spaces. In the framework of metric spaces we define the packing integral and the uniformly controlled inte- gral of a function with respect to metric distributions. Applying the theory to the notion of currents we then prove a generalization of the Stokes theorem. In Rn we introduce the packing R and R∗ integrals, which are defined as charges - additive functionals on sets of bounded variation. We provide comparison with miscellaneous types of integrals such as R and R∗ integral in Rn or MCα integral in R. On the real line we then study a scale of integrals based on the so called p-oscillation. We show that our indefinite integrals are a.e. approximately differ- entiable and we give comparison with other nonabsolutely convergent integrals. Keywords: Nonabsolutely convergent integrals, BV sets, Henstock-Kurzweil in- tegral, Divergence theorem, Analysis in metric measure spaces 1
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Nonabsolutely convergent integrals
Kuncová, Kristýna ; Malý, Jan (advisor)
Title: Nonabsolutely convergent integrals Author: Krist'yna Kuncov'a Department: Department of Mathematical Analysis Supervisor: prof. RNDr. Jan Mal'y, DrSc., Department of Mathematical Analysis Abstract: In this thesis we develop the theory of nonabsolutely convergent Hen- stock-Kurzweil type packing integrals in different spaces. In the framework of metric spaces we define the packing integral and the uniformly controlled inte- gral of a function with respect to metric distributions. Applying the theory to the notion of currents we then prove a generalization of the Stokes theorem. In Rn we introduce the packing R and R∗ integrals, which are defined as charges - additive functionals on sets of bounded variation. We provide comparison with miscellaneous types of integrals such as R and R∗ integral in Rn or MCα integral in R. On the real line we then study a scale of integrals based on the so called p-oscillation. We show that our indefinite integrals are a.e. approximately differ- entiable and we give comparison with other nonabsolutely convergent integrals. Keywords: Nonabsolutely convergent integrals, BV sets, Henstock-Kurzweil in- tegral, Divergence theorem, Analysis in metric measure spaces 1
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Nonabsolutely convergent integrals
Kuncová, Kristýna ; Malý, Jan (advisor) ; Slavík, Antonín (referee) ; Tvrdý, Milan (referee)
Title: Nonabsolutely convergent integrals Author: Krist'yna Kuncov'a Department: Department of Mathematical Analysis Supervisor: prof. RNDr. Jan Mal'y, DrSc., Department of Mathematical Analysis Abstract: In this thesis we develop the theory of nonabsolutely convergent Hen- stock-Kurzweil type packing integrals in different spaces. In the framework of metric spaces we define the packing integral and the uniformly controlled inte- gral of a function with respect to metric distributions. Applying the theory to the notion of currents we then prove a generalization of the Stokes theorem. In Rn we introduce the packing R and R∗ integrals, which are defined as charges - additive functionals on sets of bounded variation. We provide comparison with miscellaneous types of integrals such as R and R∗ integral in Rn or MCα integral in R. On the real line we then study a scale of integrals based on the so called p-oscillation. We show that our indefinite integrals are a.e. approximately differ- entiable and we give comparison with other nonabsolutely convergent integrals. Keywords: Nonabsolutely convergent integrals, BV sets, Henstock-Kurzweil in- tegral, Divergence theorem, Analysis in metric measure spaces 1
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Weighted inequalities and properties of operators and embeddings on function spaces
Slavíková, Lenka ; Pick, Luboš (advisor) ; Pérez, Carlos (referee) ; Malý, Jan (referee)
The present thesis is devoted to the study of various properties of Banach func- tion spaces, with a particular emphasis on applications in the theory of Sobolev spaces and in harmonic analysis. The thesis consists of four papers. In the first one we investigate higher-order embeddings of Sobolev-type spaces built upon rearrangement-invariant Banach function spaces. In particular, we show that optimal higher-order Sobolev embeddings follow from isoperimetric inequal- ities. In the second paper we focus on the question when the above-mentioned Sobolev-type space is a Banach algebra with respect to a pointwise multiplica- tion of functions. An embedding of the Sobolev space into the space of essentially bounded functions is proved to be the answer to this question in several standard as well as nonstandard situations. The third paper is devoted to the problem of validity of the Lebesgue differentiation theorem in the context of rearrangement- invariant Banach function spaces. We provide a necessary and sufficient condition for the validity of this theorem given in terms of concavity of certain functional depending on the norm in question and we find also alternative characterizations expressed in terms of properties of a maximal operator related to the norm. The object of the final paper is the boundedness of the...
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Eliptické rovnice v nereflexivních prostorech funkcí
Maringová, Erika ; Bulíček, Miroslav (advisor) ; Malý, Jan (referee)
In the work we modify the well-known minimal surface problem to a very special form, where the exponent two is replaced by a general positive parameter. To the modified problem we define four notions of solution in nonreflexive Sobolev space and in the space of functions of bounded variation. We examine the relationships between these notions to show that some of them are equivalent and some are weaker. After that we look for assumptions needed to prove the existence of solution to the problem in the sense of definitions provided. We outline that in the setting of spaces of functions of bounded variation the solution exists for any positive finite parameter and that if we accept some restrictions on the parameter then the solution exists in the Sobolev space, too. We also provide counterexample indicating that if the domain is non-convex, the solution in Sobolev space need not exist. Powered by TCPDF (www.tcpdf.org)
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