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Foundations of Fractional Calculus on Time Scales
Dolník, Matej ; Nechvátal, Luděk (oponent) ; Kisela, Tomáš (vedoucí práce)
The bachelor thesis concerns fractional calculus on time scales, more precisely, it introduces fractional calculus on time scales and also investigates the property of uniqueness of the axiomatic definition of the power functions. After introducing basic concepts, the subject of discussion is mostly generalized Laplace transform as well as proof of uniqueness of generalized Laplace transform, which is used as a tool to proving the uniqueness of fractional power functions on time scales.
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Autonomous systems of differential equations - classical vs fractional ones
Glozigová, Anna ; Zatočilová, Jitka (oponent) ; Nechvátal, Luděk (vedoucí práce)
The main preoccupation of this thesis is an in-depth study and comparison of two fields of differential equations with a greater focus on a non-integer order which during the last decades has proven not only to become more popular because of its applications but also more complex, thus demanding more special approach. This thesis is also provided with multiple examples, experiments, and simulations in order to verify or invalidate the theoretical results.
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Discretely normed orders of quaternionic algebras
Horníček, Jan ; Skula, Ladislav (oponent) ; Kureš, Miroslav (vedoucí práce)
This thesis summarizes author's research on the field of theory of the quaternion algebras, their isomorphisms and maximal orders. The new point of view to this issue is received by using the concept of the discrete norm. The three following statements could be taken as the main results of the thesis: - Proof of the uniqueness of the discrete norm for integers, for the orders of the quadratic field extension and also for the orders of quaternion algebra - Theorem, which enables us to construct isomorphisms between quaternion algebras in explicit matrix form - Proof of the existence of infinitely many mutually distinct orders of the quaternion algebra Results given in this thesis will be also used in a scientific article.
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Log-optimal investment
Flimmel, Samuel ; Lachout, Petr (vedoucí práce) ; Kopa, Miloš (oponent)
Tvorba portfólia je častou a dôležitou úlohou vo finančnom sektore. V práci predstavujeme jeden z matematických modelov, ktorý túto úlohu pomáha riešiť. Predpokladmi modelu sú logaritmická úžitková funkcia a ergodická stacionarita sledovaného trhu. Ide o nízky počet predpokladov, preto je model celkom jednoduchý a prehľadný. V práci sa venujeme popisu modelu a dokázaniu vlastností zameraných najmä na jeho použitie. Rozoberáme prípad známej distribúcie trhu a ponúkame návod k získaniu portfólia. Venujeme sa aj prípadu neznámej distribúcii trhu, kde ponúkame jeden z možných postupov. Pomocou empirickej distribúcie sme schopní získať asymptoticky žiadaný výsledok. Práca je zakončená simulovanou aplikáciou na reálne hodnoty. Skúmame v nej správanie pri použití postupu s empirickou distribúciou na kratších časových úsekoch.
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Autonomous systems of differential equations - classical vs fractional ones
Glozigová, Anna ; Zatočilová, Jitka (oponent) ; Nechvátal, Luděk (vedoucí práce)
The main preoccupation of this thesis is an in-depth study and comparison of two fields of differential equations with a greater focus on a non-integer order which during the last decades has proven not only to become more popular because of its applications but also more complex, thus demanding more special approach. This thesis is also provided with multiple examples, experiments, and simulations in order to verify or invalidate the theoretical results.
|
|
| |
|
|
Log-optimal investment
Flimmel, Samuel ; Lachout, Petr (vedoucí práce) ; Kopa, Miloš (oponent)
Tvorba portfólia je častou a dôležitou úlohou vo finančnom sektore. V práci predstavujeme jeden z matematických modelov, ktorý túto úlohu pomáha riešiť. Predpokladmi modelu sú logaritmická úžitková funkcia a ergodická stacionarita sledovaného trhu. Ide o nízky počet predpokladov, preto je model celkom jednoduchý a prehľadný. V práci sa venujeme popisu modelu a dokázaniu vlastností zameraných najmä na jeho použitie. Rozoberáme prípad známej distribúcie trhu a ponúkame návod k získaniu portfólia. Venujeme sa aj prípadu neznámej distribúcii trhu, kde ponúkame jeden z možných postupov. Pomocou empirickej distribúcie sme schopní získať asymptoticky žiadaný výsledok. Práca je zakončená simulovanou aplikáciou na reálne hodnoty. Skúmame v nej správanie pri použití postupu s empirickou distribúciou na kratších časových úsekoch.
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Discretely normed orders of quaternionic algebras
Horníček, Jan ; Skula, Ladislav (oponent) ; Kureš, Miroslav (vedoucí práce)
This thesis summarizes author's research on the field of theory of the quaternion algebras, their isomorphisms and maximal orders. The new point of view to this issue is received by using the concept of the discrete norm. The three following statements could be taken as the main results of the thesis: - Proof of the uniqueness of the discrete norm for integers, for the orders of the quadratic field extension and also for the orders of quaternion algebra - Theorem, which enables us to construct isomorphisms between quaternion algebras in explicit matrix form - Proof of the existence of infinitely many mutually distinct orders of the quaternion algebra Results given in this thesis will be also used in a scientific article.
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Foundations of Fractional Calculus on Time Scales
Dolník, Matej ; Nechvátal, Luděk (oponent) ; Kisela, Tomáš (vedoucí práce)
The bachelor thesis concerns fractional calculus on time scales, more precisely, it introduces fractional calculus on time scales and also investigates the property of uniqueness of the axiomatic definition of the power functions. After introducing basic concepts, the subject of discussion is mostly generalized Laplace transform as well as proof of uniqueness of generalized Laplace transform, which is used as a tool to proving the uniqueness of fractional power functions on time scales.
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