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Systems of Polynomial Equations in Economics
Šramková, Kristína ; Tomáš, Jiří (referee) ; Kureš, Miroslav (advisor)
Bachelor thesis is based on application of mathematical apparatus for the analysis of economic models, in particular models that lead to a system of polynomial equations. One of the parts is a summary of basic knowledge of algebra focused on Gröbner basis. Hereinafter are discussed economic models in which solution Gröbner basis are applied using the program Wolfram Mathematica. Own software package is implemented into this program as a concept of solution to simplify the calculation and work with models.
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Mathematical principles of Robotics
Pivovarník, Marek ; Kureš, Miroslav (referee) ; Hrdina, Jaroslav (advisor)
Táto diplomová práca sa zaoberá matematickými aparátmi popisujúcimi doprednú a inverznú kinematiku robotického ramena. Pre popis polohy koncového efektoru, teda doprednej kinematiky, je potrebné zaviesť špeciálnu Euklidovskú grupu zobrazení. Táto grupa môže byť reprezentovaná pomocou matíc alebo pomocou duálnych kvaterniónov. Problém inverznej kinematiky, kedy je potrebné z určenej polohy koncového efektoru dopočítať kĺbové parametre robotického ramena, je v tejto práci riešený pomocou exponenciálnych zobrazení a Grobnerovej bázy. Všetky spomenuté popisy doprednej a inverznej kinematiky sú aplikované na robotické rameno s troma rotačnými kĺbami. Odvodené postupy sú následne implementované a vizualizované v prostredí programu Mathematica.
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Non-commutative Gröbner bases
Požárková, Zuzana ; Šťovíček, Jan (advisor) ; Stanovský, David (referee)
In the presented work we define non-commutative Gröbner bases including the necessary basis of non- commutative algebra theory and notion admissible ordering. We present non-commutative variant of the Buchberger algorithm and study how the algorithm can be improved. Analogous to the Gebauer-Möller criteria lead us to detect almost all unnecessary obstructions in the non-commutative case. The obstructions are graphically ilustrated. The Buchberger algorithm can be improved within redundant polynomials. This work is a summary and its specification of the results of some known authors engaged in this field. Presented definitions are ilustrated on examples. We perform proves of some of the statements which have been proven differently by other authors. Powered by TCPDF (www.tcpdf.org)
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Determining primeness of an ideal?
Stejskal, Adam ; Šťovíček, Jan (advisor) ; Šaroch, Jan (referee)
We present an algorithm for determining whether an ideal in a polynomial ring is prime or not. We use the Gröbner bases as a main tool for operations with ideals. We show an analogue of Buchberger's algorithm for computing a Gröbner basis for an ideal in polynomials over a ring, which not need to be a field. We also show a relation between prime ideals in polyno- mials over a ring R and prime ideals in polynomials over a quotient ring of R modulo a prime ideal. We are primarilly discussing the issues of theoretical corectness, but we also present the conditions of actual computability. 1
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Non-commutative Gröbner bases
Požárková, Zuzana ; Šťovíček, Jan (advisor) ; Stanovský, David (referee)
In the presented work we define non-commutative Gröbner bases including the necessary basis of non- commutative algebra theory and notion admissible ordering. We present non-commutative variant of the Buchberger algorithm and study how the algorithm can be improved. Analogous to the Gebauer-Möller criteria lead us to detect almost all unnecessary obstructions in the non-commutative case. The obstructions are graphically ilustrated. The Buchberger algorithm can be improved within redundant polynomials. This work is a summary and its specification of the results of some known authors engaged in this field. Presented definitions are ilustrated on examples. We perform proves of some of the statements which have been proven differently by other authors. Powered by TCPDF (www.tcpdf.org)
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Determining primeness of an ideal?
Stejskal, Adam ; Šťovíček, Jan (advisor) ; Šaroch, Jan (referee)
We present an algorithm for determining whether an ideal in a polynomial ring is prime or not. We use the Gröbner bases as a main tool for operations with ideals. We show an analogue of Buchberger's algorithm for computing a Gröbner basis for an ideal in polynomials over a ring, which not need to be a field. We also show a relation between prime ideals in polyno- mials over a ring R and prime ideals in polynomials over a quotient ring of R modulo a prime ideal. We are primarilly discussing the issues of theoretical corectness, but we also present the conditions of actual computability. 1
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Systems of Polynomial Equations in Economics
Šramková, Kristína ; Tomáš, Jiří (referee) ; Kureš, Miroslav (advisor)
Bachelor thesis is based on application of mathematical apparatus for the analysis of economic models, in particular models that lead to a system of polynomial equations. One of the parts is a summary of basic knowledge of algebra focused on Gröbner basis. Hereinafter are discussed economic models in which solution Gröbner basis are applied using the program Wolfram Mathematica. Own software package is implemented into this program as a concept of solution to simplify the calculation and work with models.
|
|
|
Mathematical principles of Robotics
Pivovarník, Marek ; Kureš, Miroslav (referee) ; Hrdina, Jaroslav (advisor)
Táto diplomová práca sa zaoberá matematickými aparátmi popisujúcimi doprednú a inverznú kinematiku robotického ramena. Pre popis polohy koncového efektoru, teda doprednej kinematiky, je potrebné zaviesť špeciálnu Euklidovskú grupu zobrazení. Táto grupa môže byť reprezentovaná pomocou matíc alebo pomocou duálnych kvaterniónov. Problém inverznej kinematiky, kedy je potrebné z určenej polohy koncového efektoru dopočítať kĺbové parametre robotického ramena, je v tejto práci riešený pomocou exponenciálnych zobrazení a Grobnerovej bázy. Všetky spomenuté popisy doprednej a inverznej kinematiky sú aplikované na robotické rameno s troma rotačnými kĺbami. Odvodené postupy sú následne implementované a vizualizované v prostredí programu Mathematica.
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