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Tensors and their applications
Korbel, Filip ; Tomáš, Jiří (referee) ; Doupovec, Miroslav (advisor)
The goal of this thesis is to provide an overview of basic terms and results of tensor calculus. We introduce tensors as multilinear mappings and introduce basic tensor operations. In the next section, we give some examples of tensors, especially from the field of differential geometry.
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Curves in D^3_1
Navrátil, Dušan ; Doupovec, Miroslav (referee) ; Kureš, Miroslav (advisor)
The Bachelor thesis deals with research on curves in three-dimensional space with Lorentzian inner product. The emphasis is on detailed analysis of dual numbers and dual Lorentzian space properties. Main part of this work is focused on dual functions, their differentiability, both arc length reparametrization and Frenet equations of dual curves and examples of such a curves. Within this work, many statements were derived generalizing from Minkowski space. Properties of dual rectifying curves were described in last section and finally we showed relation between these curves, dual unit spherical curves and ruled surfaces in Minkowski space.
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Dynamics of Snake Robots
Kubiena, Jaromír ; Doupovec, Miroslav (referee) ; Návrat, Aleš (advisor)
In this thesis, we deal with the mathematical description of the kinematics and the dynamics of mechanical systems. Then we focus on the particular mechanical system which is the Square robot with four legs with active joints and passive wheels, which moves on horizontal plane. The kinematics of the mechanical system is described by the control matrix, then we use it to express the equations of motion. We compute the dynamics the robot by using Lagrange equations. We verify that the mechanical system is nonholonomic constrained and we verify controllability by using Lie bracket and distribution. We find the singular postures of the robot.
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Spherical geometry
Kokh, Konstantin ; Vašík, Petr (referee) ; Doupovec, Miroslav (advisor)
This bachelor thesis deals with the description of sphere and spherical geometry. The second chapter defines the mathematical apparatus that we will need in the next part of the work. The third part begins with describing the sphere from the point of view of differential geometry of curves and planes. In the middle, we will show the conformal map of the sphere to the plane and the equiareal map of the sphere to the cylinder. Then we will describe the basic properties of spherical geometry. In the end, we will compare the properties of Euclidean geometry and spherical geometry.
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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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Hyperbolic geometries
Brdečková, Johanka ; Tomáš, Jiří (referee) ; Doupovec, Miroslav (advisor)
The present thesis deals with hyperbolic geometry. We derive parametric equations of the curve tractrix and the surface pseudosphere. Then we discuss two models of hyperbolic geometry, which are derived from the parametrization of pseudosphere.
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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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Tensors and their applications in mechanics
Adejumobi, Mudathir ; Doupovec, Miroslav (referee) ; Tomáš, Jiří (advisor)
The tensor theory is a branch of Multilinear Algebra that describes the relationship between sets of algebraic objects related to a vector space. Tensor theory together with tensor analysis is usually known to be tensor calculus. This thesis presents a formal category treatment on tensor notation, tensor calculus, and differential manifold. The focus lies mainly on acquiring and understanding the basic concepts of tensors and the operations over them. It looks at how tensor is adapted to differential geometry and continuum mechanics. In particular, it focuses more attention on the application parts of mechanics such as; configuration and deformation, tensor deformation, continuum kinematics, Gauss, and Stokes' theorem with their applications. Finally, it discusses the concept of surface forces and stress vector.
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Curves in D^3_1
Navrátil, Dušan ; Doupovec, Miroslav (referee) ; Kureš, Miroslav (advisor)
The Bachelor thesis deals with research on curves in three-dimensional space with Lorentzian inner product. The emphasis is on detailed analysis of dual numbers and dual Lorentzian space properties. Main part of this work is focused on dual functions, their differentiability, both arc length reparametrization and Frenet equations of dual curves and examples of such a curves. Within this work, many statements were derived generalizing from Minkowski space. Properties of dual rectifying curves were described in last section and finally we showed relation between these curves, dual unit spherical curves and ruled surfaces in Minkowski space.
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Special Surfaces
Ochodnický, Erik ; Vašík, Petr (referee) ; Doupovec, Miroslav (advisor)
The aim of this thesis is to create an overview of special surfaces and to define their characteristics. Categories of surfaces that I found the most important are surfaces of revolution, minimal, with constant Gaussian curvature, and finally Clairaut surfaces. For every category I'll introduce, in my opinion, the most important examples of surfaces along with their parametrizations and I'll describe them. Surfaces will be accompanied by images, created in MATLAB. In the last part I'm going to focus on Clairot patches, on finding geodesics on these surfaces and their description. I'll show numerous original images of geodesics on diverse surfaces.
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