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Sobolev-type Spaces on Metric Measure Spaces
Malý, Lukáš ; Pick, Luboš (advisor) ; Malý, Jan (referee) ; Shanmugalingam, Nages (referee)
Title: Sobolev-Type Spaces on Metric Measure Spaces Author: RNDr. Lukáš Malý Department: Department of Mathematical Analysis Supervisor: Prof. RNDr. Luboš Pick, CSc., DSc., Department of Mathematical Analysis Abstract: is thesis focuses on function spaces related to rst-order analysis in abstract metric measure spaces. In metric spaces, we can replace distributional gra- dients, whose de nition depends on the linear structure of Rn , by upper gradients that control the functions' behavior along all recti able curves. is gives rise to the so-called Newtonian spaces. e summability condition, considered in the thesis, is expressed using a general Banach function lattice quasi-norm and so an extensive framework is built. Sobolev-type spaces (mainly based on the Lp norm) on metric spaces, and Newtonian spaces in particular, have been under intensive study since the mid- s. Standard toolbox for the theory is set up in this general setting and Newto- nian spaces are proven complete. Summability of an upper gradient of a function is shown to guarantee the function's absolute continuity on almost all curves. Ex- istence of a unique minimal weak upper gradient is established. Regularization of Newtonian functions via Lipschitz truncations is discussed in doubling Poincaré spaces using weak boundedness of maximal...
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Zobecněné obyčejné diferenciální rovnice v metrických prostorech
Skovajsa, Břetislav ; Malý, Jan (advisor) ; Pražák, Dalibor (referee)
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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Arthur I, Duke of Brittany and His Claim on the English Throne
Malý, Jan ; Drška, Václav (advisor) ; Suchánek, Drahomír (referee)
The death of king Richard The Lion Heart in 1199 caused considerable troubles to the Angevin empire, when there again raised for english medieval history very pressing question - who is legitimate successor to the throne? There were two possible pretendents, both had comparable claim to the crown. First of them was Richard's brother John, the second his nephew, at this time twelve years old duke of Brittany Arthur. Legal customs of this period theoretically admitted the succession of both men, because there were no unified successorial usage and every single part of the Angevin empire looked on this problem differently. While John was generally accepted without problems in Normandy and then he was crowned king of England, the toughest fight blazed out in Anjou, Maine and Touraine, where support was given to Arthur. He had also support of king of France Philip Augustus, who understood well, that Arthur is an ideal tool for his schemes to elimination and mastery over the Angevin empire. Whole long struggle between the nephew and his uncle was finsihed by Arthur's capture in the summer of 1202 and his subsequent death in 1203. However king John was not able to stop the dissolution of Plantagenet empire, which was reduced to the duchy of Aquitaine at the beginning of 13th century.
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Sobolev mappings and Luzin condition N
Matějka, Milan ; Hencl, Stanislav (advisor) ; Malý, Jan (referee)
A mapping f from R^{n} to R^{n} is said to satisfy the Luzin condition N if f maps sets of measure zero to sets of measure zero. It is known to be valid for mappings in the Sobolev space W^{1,p} for p > n and for p <= n there are counterexamples. The aim of this thesis is to summarize known results and study the validity of Luzin condition N for mappings in the Sobolev space W^{2,p}.
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First Baron War and Louis VIII as the Kinf of England (1215-1217). A view of sources.
Malý, Jan ; Drška, Václav (advisor) ; Suchánek, Drahomír (referee)
In 1215, king of England John the Lackland was forced to seal a document known as Magna Carta. It was a simple legal procedure. Nevertheless it was the beginning of conflict between royal power and english nobility, commonly known as first baron's war. It lasted from 1215 to 1217 and it culminated by the invasion of french crown prince Louis to England when english nobility offered him the crown. After the death of king John in autumn 1216, the original revolt, inspired mostly by personal hate against the ruler changed into the effort of prince Louis to achieve the title of English king. Most of original memebers of the oposition quickly switched sides and joined John's little son Henry (king Henry III) and Louis then had only the support of few leaders of rebelion. After a series of defeats in the first half of 1217, capetian prince was forced to abandon his goals and to retreat from the Isles. This work also follows the view of selected contemporary narrative sources, mainly to the activity of french prince in England and his attitude to the uprising.
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Nonabsolutely convergent integrals
Kuncová, Kristýna ; Malý, Jan (advisor) ; Rataj, Jan (referee)
Title: Nonabsolutely convergent integrals Author: Kristýna Kuncová Department: Department of Mathematical Analysis Supervisor: Prof. RNDr. Jan Malý, DrSc., Department of Mathematical Analysis Abstract: Our aim is to introduce an integral on a measure metric space, which will be nonabsolutely convergent but including the Lebesgue integral. We start with spaces of continuous and Lipschitz functions, spaces of Radon measures and their dual and predual spaces. We build up the so-called uniformly controlled integral (UC-integral) of a function with respect to a distribution. Then we investigate the relationship between the UC-integral with respect to a measure and the Lebesgue integral. Then we introduce another kind of integral, called UCN-integral, based on neglecting of small sets with respect to a Hausdorff measure. Hereafter, we focus on the concept of n-dimensional metric currents. We build the UC-integral with respect to a current and then we proceed to a very general version of Gauss-Green Theorem, which includes the Stokes Theorem on manifolds as a special case. Keywords: Nonabsolutely convergent integrals, Multidimensional integrals, Gauss-Green Theorem 1
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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
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