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Discrete and Linear Structures in Enumeration
Teimoori Faal, Hossein ; Loebl, Martin (advisor) ; Klazar, Martin (referee) ; Kang, Mihyun (referee)
The central theme of this thesis is to nd the multiset version of the combinatorial identities arising from the cyclic decomposition of permutations of nite sets. The main contributions of author's work are as follows. In Chapter 1, we nd the appropriate multiset version of the Stirling cycle number. Then, using these new Stirling numbers, we give a new equivalent statement of the coin arrangements lemma which is an important trick in Sherman's proof of Feynman conjecture on two dimensional Ising model. We also present a new proof of the coin arrangements lemma. Finally, we show several relations of the coin arrangements lemma with various concepts in enumerative combinatorics. In Chapter 2, we rst give a new proof of the Witt identity which is an algebraic identity in the context of Lyndon words using the Bass' identity for zeta function of nite graphs. Then, we present a new proof of the Bass' identity by only slight modi cations to the approach that has been developed by Feynman and Sherman as the path method for combinatorial solution of two dimensional Ising problem. In Chapter 3, we give a multiset generalization of the well-known graph-theoretical interpretation of the determinant as a signed weighted sum over cycle covers. In Chapter 4, we nd a multiset generalization of the graph-theoretical...
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Regular and semi-regular solids in higher dimensions
Pekař, Vojtěch ; Surynková, Petra (advisor) ; Hromadová, Jana (referee)
This thesis deals with multidimensional objects, which are known as Platonic and Archimedean solids in common euclidean space. Although we describe especially four-dimensional figures and their relations with lesser grade, this text is formulated in such a way, that includes even different dimensions, if it is possible in particular instances. There exist a few works about this and similar topics in foreign, but usually they require a little basics of algebra teaching at university. Our approach uses methods similar to these, which are normally teaching in descriptive geometry and therefore includes a large number of pictures. The matter is therefore available to secondary school students, who want to increase their space imagination.
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Minimisation of Logical Functions
Horký, Miroslav ; Davidová, Olga (referee) ; Šeda, Miloš (advisor)
For minimisation of logical functions, laws of the Boolean algebra and the Karnaugh maps are mostly used. However, use of Karnaugh's maps is based on visual recognition of adjacent cells for functions with no more than 6 variables and, therefore, the method is not suitable for automated processing on computers. A direct application of the Boolean algebra laws is not restricted in this way, but there is no general algorithm defining the sequence of their application and thus this approach is not suitable for computer implementation either. The well-known method usable on computers is the algorithm proposed by E. J. McCluskey and W. Orman Quine.
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The issue of teaching sets of points with given properties at elementary school
HARAZIMOVÁ, Tereza
The aim of this diploma thesis is to describe various methods for examining loci of points of given properties. I have chosen both the classical approach and new technologies such as dynamic geometry software GeoGebra or a programme from CAS (Computer Algebra System). The topic loci of points of given properties is often very difficult for pupils to understand, therefore it is important to look at it from various points of view, to choose interesting application tasks and thus motivate learners in their studies. In this paper I present several suggestions for more attractive and amusing ways of teaching this issue at primary, secondary and university level of education.
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Morphisms of Dempster's Semigroup: A Revision and Interpretation
Daniel, Milan
Analyzing three new approaches to interpretation, definition, and measurement of conflicts of belief Functions (BFs), we previously observed a possibility of expression of a BF Bel as Dempster's sum of non-conflicting BF Bel0 with the same plausibility decisional support as the original BF Bel has and of indecisive BF BelS which does not prefer any of the elements of frame of discernment. Based on this observation, the theory of homomorphisms of Dempster's semigroup (the algebra of non-extremal BFs on 2-element frame of discernment with Dempster's rule as its binary operation) is updated in the present contribution. New homomorphisms are introduced; their interpretation and relation to the original ones are presented.
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Výnosová křivka neumožňující arbitráž
Dobiáš, Vladimír ; Kodera, Jan (advisor) ; Pelikán, Jan (referee) ; O Sullivan, Conall (referee)
We address the issue of market incompleteness in the time dimension. Specifically, we focus on interest rate markets and the yield curve extraction. The lack of information about interest rates manifest itself in a non-invertible linear system. The usual approach to circumvent this problem is by applying various curve fitting methods - both parametric and non-parametric. We argue in favor of a novel method relying on information theory, which reformulates the ill-posed linear algebra problem into a well-posed optimization problem, where the linear pricing equations are used as constraints. Local cross entropy is used to determine the optimal solution among the admissible solutions, while all the input prices reflected in constraints are perfectly matched. Large-scale optimization package called AMPL is used extensively throughout this work to obtain the optimal solution as well as to demonstrate the implementation details.
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Homomorphic encryption and coding theory
Půlpánová, Veronika ; Hojsík, Michal (advisor) ; Šťovíček, Jan (referee)
Title: Homomorphic encryption and coding theory Author: Veronika Půlpánová Department: Department of algebra Supervisor: RNDr. Michal Hojsík, Ph.D., Department of algebra Abstract: The current mainstream in fully homomorphic encryption is the appro- ach that uses the theory of lattices. The thesis explores alternative approaches to homomorphic encryption. First we present a code-based homomorphic encrypti- on scheme by Armknecht et. al. and study its properties. Then we describe the family of cryptosystems commonly known as Polly Cracker and identify its pro- blematic aspects. The main contribution of this thesis is the design of a new fully homomorphic symmetric encryption scheme based on Polly Cracker. It proposes a new approach to overcoming the complexity of the simple Polly Cracker - based cryptosystems. It uses Gröbner bases to generate zero-dimensional ideals of po- lynomial rings over finite fields whose factor rings are then used as the rings of ciphertexts. Gröbner bases equip these rings with a multiplicative structure that is easily algorithmized, thus providing an environment for a fully homomorphic cryptosystem. Keywords: Fully homomorphic encryption, Polly Cracker, coding theory, zero- dimensional ideals
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Errors-in-variables models
Fürjesová, Ida ; Pešta, Michal (advisor) ; Anděl, Jiří (referee)
This thesis analyzes an errors-in-variables model. It compares parameter estimation methods least squares and total least squares. The main difference between these methods lies in approach to the measurements errors. The first part of the bachelor thesis focuses on theoretical aspect of methods. It defines basic terms and shows differences in the methods graphically. Thesis also demonstrates algebraic solutions of the estimation methods. The theoretical part ends up with statistical properties of the estimating techniques. The thesis compares methods least squares and total least squares according to the size of mean square error by simulation study.
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Využití imsetů při učení bayesovských sítí
Vomlel, Jiří ; Studený, Milan
This paper describes a modification of the greedy equivalence search (GES) algorithm. The presented modification is based on the algebraic approach to learning. The states of the search space are standard imsets. Each standard imset represents an equivalence class of Bayesian networks. For a given quality criterion the database is represented by the respective data imset. This allows a very simple update of a given quality criterion since the moves between states are represented by differential imsets. We exploit a direct characterization of lower and upper inclusion neighborhood, which allows an efficient search for the best structure in the inclusion neighborhood. The algorithm was implemented in R and is freely available.
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Evolutionary Optimization of Control Algorithms
Weisser, Roman ; Šeda, Miloš (referee) ; Zelinka,, Ivan (referee) ; Ošmera, Pavel (advisor)
The dissertation thesis deals with Evolution optimization of control algorithms. The first part of the thesis describes the principles and partial methods of evolution optimization methods especially those used in two-level transplant evolution method. Later the grammatical evolution method is described, which modified algorithm became impulse for creation of transplant evolution method. The transplant evolution method and its two-level modification are new evolutionary algorithms proposed in this work, which were used for optimization of structure and parameters of general controllers control algorithms. The transplant evolution algorithm and its extended two-level modification are described in detail in next chapters. The proper settings of evolutionary algorithms are important for minimization the time of optimization and for finds results approaching the global optimum. For proper setting the parameters of differential evolution was created meta-evolution algorithm that is described in chapter named meta-evolution. The basic concepts of control, chosen methods of system identification and controller parameters settings are described in next part. This part describes algorithms of digital controllers and some specific methods uses in digital control. The demonstrations of control algorithm optimizations of various types of controllers are showed in experimental part. The optimized algorithms of general controllers are compared with various types of PSD controllers which were set by various algebraic methods or differential evolution for various models of systems. In the conclusion of this work is stated a recommendation for further development of evolutionary optimization of controllers are focusing on parallel and distributed computing.
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