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Geometry of interval linear systems
Kotecký, Cyril ; Hladík, Milan (advisor) ; Rada, Miroslav (referee)
Interval linear systems of equalities and inequalities are linear systems, where the real numbered entries of the vectors and matrices are replaced with closed intervals of real numbers. The sets of solutions to these systems have interesting properties, mainly that they are unions of exponentially many convex polyhedra. This makes solving many problems hard, while on the other hand, the solution sets have a form that is convenient to analyze. This thesis deals with studying the geometry of such sets. We will begin by reviewing known properties of these sets, such as boundedness and connectedness. But mostly, we will focus on the conditions for convexity and the characterization of the convex hull, which are both known for the special case of systems with invertible interval matrices. Using polyhedral theory, we will broaden these results, mostly to general systems of interval linear inequalities. We will present illustrative examples, some serving as counter-examples in cases where generalizations are not possible. 1
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Interval linear systems with linear dependencies
Král, Ondřej ; Hladík, Milan (advisor) ; Rada, Miroslav (referee)
The main problem discussed in this thesis is about finding an enclo- sure of the solution set of an interval linear system with linear dependencies. We get familiar with definitions from interval arithmetic and analysis. Then we extend them to matrices and linear systems, where we introduce several modern approaches to finding an enclosure and divide them thematically. Most of them are implemented in MATLAB using INTLAB library. We compare their precision and computational time on Toeplitz, symmetric and random matrices. For depen- dencies we design our memory saving representation. The results are interpreted and the final function, which can compute either fast, sharp or memory efficient, is build on individual methods. Powered by TCPDF (www.tcpdf.org)
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Application of Branch and Bound Approach to Parametric Interval Linear Systems
Szabó, Adam ; Horáček, Jaroslav (advisor) ; Rada, Miroslav (referee)
This work is focused on parametric interval linear systems. By using branch and bound method and various pruning conditions, we first obtained their solution and then described it more precisely with n-dimensional boxes. We were acquainted with the basic concepts of intervals and linear systems. Subsequently, we processed the boxes obtained by multiple methods to opti- mize their number. Part of the work is also a comparison of various pruning conditions on parametric systems with the different number of parameters. Finally, our algorithms were implemented into the Lime interval package with the possibility of simple visualization of the obtained solutions. 1
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Visualisation of interval data
Mečiar, Martin ; Horáček, Jaroslav (advisor) ; Rada, Miroslav (referee)
The thesis is focused on visualisation, comparison and modification of outputs of interval solvers for solving a continuous constraint satisfaction problem. The author's designed solution for the approximation of outputs of solvers is presented in the thesis. The approximation of outputs of solvers is transformed into the problem of visual reallocation of the sets of outputs of solvers on a finer level than interval box level. Main part of the thesis is the program on added CD that allows visualisation, comparison and modification of outputs of interval solvers. The program is written in C++, but can be compiled as a MEX file for MATLAB. A user documentation and a technical documentation for the program are included in the thesis. The thesis shows several examples of program output in the devoted chapter. Powered by TCPDF (www.tcpdf.org)
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Interval linear systems with linear dependencies
Král, Ondřej ; Hladík, Milan (advisor) ; Rada, Miroslav (referee)
The main problem discussed in this thesis is about finding an enclo- sure of the solution set of an interval linear system with linear dependencies. We get familiar with definitions from interval arithmetic and analysis. Then we extend them to matrices and linear systems, where we introduce several modern approaches to finding an enclosure and divide them thematically. Most of them are implemented in MATLAB using INTLAB library. We compare their precision and computational time on Toeplitz, symmetric and random matrices. For depen- dencies we design our memory saving representation. The results are interpreted and the final function, which can compute either fast, sharp or memory efficient, is build on individual methods. Powered by TCPDF (www.tcpdf.org)
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Lying and manipulation in voting procedures
Kapusta, Matyáš ; Dlouhý, Martin (advisor) ; Rada, Miroslav (referee)
This thesis focuses on the phenomenon of strategic manipulation in collective decision making, i.e. situations where it is beneficial for certain members of the group to derive from their sincere preferences in the voting process. This manipulation complicates the process of collective decision making, leading to situations where lying becomes a successful strategy that some mambers of the group can use to gain an advantage over others. In the thesis, examples of strategic manipulation and a summary of up to date research in the field of social choice theory are included. The main objective is to create a simulation model that can be used to quantify the probability of a successful manipulation. A Monte Carlo simulation using this model is performed on ten voting procedures. Analysed procedures include the most common and intuitive ones, more sophisticated systems created by experts of social choice theory and the most modern attempts to improve democracy by Karel Janeček.
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Algorithms for various geometric problems over zonotopes and their applications in optimization and data analysis
Rada, Miroslav ; Černý, Michal (advisor) ; Vlach, Milan (referee) ; Kopa, Miloš (referee)
The thesis unifies the most important author's results in the field of algorithms concerning zonotopes and their applications in optimization and statistics. The computational-geometric results consist of a new compact output-sensitive algorithm for enumerating vertices of a zonotope, which outperforms the rival algorithm with the same complexity-theoretic properties both theoretically and empirically, and a polynomial algorithm for arbitrarily precise approximation of a zonotope with the Löwner-John ellipsoid. In the application area, the thesis presents a result, which connects linear regression model with interval outputs with the zonotope matters. The usage of presented geometric algorithms for solving a nonconvex optimisation problem is also discussed.
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The aplication of Benford's law in accounting and auditing
Kabourková, Lucie ; Procházka, David (advisor) ; Rada, Miroslav (referee)
This bachelor thesis focuses on the Benford's law, also called the First-Digit Law. The major objective of this study is to describe the application of the law customarily in accounting and auditing. It outlines possible complications in the practical application, but attempts to eliminate the aforementioned shortcomings. Selected financial statements served as subjects for the empirical part of the study. Financial statements are matched against the expected Benford's common distribution utilizing the Pearson's chi-squared test.
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Interval data and sample variance: computational aspects
Sokol, Ondřej ; Černý, Michal (advisor) ; Rada, Miroslav (referee)
This thesis deals with the calculation of the upper limit of the sample variance when the exact data are not known but intervals which certainly contain them are available. Generally, finding the upper limit of the sample variance knowing only interval data is an NP-hard problem, but under certain conditions imposed on the input data an appropriate efficient algorithm can be used. In this work algorithms were modified so that, even at the cost of exponential complexity, one can always find the optimal solution. The goal of this thesis is to compare selected algorithms for calculating the upper limit of sample variance over interval data from the perspective of the average computational complexity on the generated data. Using simulations it is shown that if the data meets certain conditions, the complexity of the average case is polynomial.
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Work activities as a motivation for choosing apprenticeship
RADA, Miroslav
My thesis deal with topic: "Work activities as a subject of motivation for an apprenticeship selection". The basic idea is an opportunity of cooperation between primary and secondary vocational schools in field of work activities. They do not take place in primary school facilities but in professional workshops of vocational school. I handle in detail making test tasks and didactic test, which will objectively judge (in a set group of pupils) whether doing work activities in vocational school´s workshop have some influence on choosing their next field of study.
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