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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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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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Gröbner basis, Zhuang-Zi algorithm and attacks of multivariable cryptosystems
Doktorová, Alice ; Tomáš, Jiří (referee) ; Kureš, Miroslav (advisor)
This diploma thesis is devoted to the multivariate cryptosystems. It includes an overview of commutative algebra with emphasis on Gröbner bases. Of all algorithms, especially the ones using Gröbner bases are studied, i.e. Buchberger's algorithm, which is already implemented in Wolfram Mathematica, and F4 algorithm, for which a program package has been created in the Wolfram Mathematica environment. Also Zhuang-Zi algorithm is described. To simplify its steps a program to compute the Lagrange interpolation polynomial has been created in Python.
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Surfaces with constant Gauss curvature
Zemanová, Silvie ; Kureš, Miroslav (referee) ; Doupovec, Miroslav (advisor)
This bachelor thesis deals with description of surfaces with constant Gaussian curvature and its main goal is to classify these surfaces. The first part is devoted to the classification of surfaces of revolution with constant Gaussian curvature. The next part consists of description of selected surfaces with zero Gaussian curvature, on which is shown that the same shape of the first fundamental form can be achieved. The last part deals with the classification of all surfaces with zero Gaussian curvature. For easier understanding of the text, the thesis includes images of selected surfaces.
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Invariants of jet groups and applications in continuum mechanics
Buriánek, Martin ; Doupovec, Miroslav (referee) ; Kureš, Miroslav (advisor)
This thesis is focused on jet groups and their matrix representations. The opening section deals with group representations, group actions on sets and invariants of actions. Another section explains terms such as smooth manifolds, Lie group and Lie algebra. The following part clarifies terms jet and jet group as a special example of Lie group. First of all, groups $G_1^r$ and $G_n^1$ are described, then description of group $G_n^2$ and its subgroups ensues. Representations of these jet groups are proposed. Finally, applications of jet groups in continuum mechanics are mentioned. The thesis is complemented with algorithm of chosen problems in program Wolfram Mathematica.
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Lie groups from the point of view of kinematics and applications in robotics
Kalenský, Jan ; Kureš, Miroslav (referee) ; Tomáš, Jiří (advisor)
This diploma thesis deals with the kinematic and robotic implications of Lie theory. In the introductory section, a manifold is defined as a basic element of configuration space. The main body of the thesis deals with the definition of a structure in the configuration space - Lie group. Tangent space with vector field including a structure of Lie algebra is defined to represent velocity. These two structures are connected using exponential mapping. The conclusion of the thesis focuses on fibre space, especially considering principal bundle and principal connection. Throughout the thesis, numerous examples are presented to illustrate the terms used.
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Rings of order p^2 and p^3
Haluza, Vít ; Hrdina, Jaroslav (referee) ; Kureš, Miroslav (advisor)
This Bachelor thesis deals with classification and studying of properties of rings of order p^2 and p^3 (p is prime). Terms such as ring ideal or polynomial over finite field are introduced and used in this thesis. Apart from abstract unspecified rings, some special types of finite rings are also mentioned and classified. Program package, which is able to automatically classify given ring of order p^2 is also part of this thesis.
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