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Lelkova hypotéza
Bartošová, Dana ; Simon, Petr (advisor) ; Balcar, Bohuslav (referee)
Lelek's conjecture which states that metric continua with span zero are chainable has been one of the most widely investigated problems in continuum theory over the past 40 years. We broaden our field of interest to non-metric continua and prove that if there is a non-metric counterexample to Lelek's conjecture we can convert it to a metric one. For a continuum X we take the lattice of all of its closed subsets 2X and consider a countable elementary sublattice L of 2X that we represent by a metric continuum wL via the Wallman representation for distributive lattices. By means of set theory, we obtain an L such that X is not chainable if and only if wL is not chainable and X has span zero if and only if wL has span zero. In the proof of the latter we use Shelah's theorem stating that every two elementarily equivalent models have isomorphic ultrapowers.
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MDS codes conjecture
Kesely, Michal ; Drápal, Aleš (advisor) ; Lisoněk, Petr (referee)
In this thesis, we study some properties of MDS codes and we mainly focus on the MDS codes conjecture. In the first chapter we define MDS codes, show some examples and basic properties of MDS codes, for example a link between MDS codes and Latin squares or rectangles. Afterwards we state the MDS codes conjecture and prove it in several cases. In the third chapter we can observe the relationship between MDS codes and narcs in projective geometries. Finally we present those known cases, for which the MDS conjecture holds.
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Algebraický přístup k CSP
Bulín, Jakub ; Barto, Libor (advisor) ; Růžička, Pavel (referee)
For a finite relational structure A, the Constraint Satisfaction Problem with template A, or CSP(A), is the problem of deciding whether an input relational structure X admits a homomorphism to A. The CSP dichotomy conjecture of Feder and Vardi states that for any A, CSP(A) is either in P or NP-complete. In the first part we present the algebraic approach to CSP and summarize known results about CSP for digraphs, also known as the H-coloring problem. In the second part we study a class of oriented trees called special polyads. Using the algebraic approach we confirm the dichotomy conjecture for special polyads. We provide a finer description of the tractable cases and give a construction of a special polyad T such that CSP(T) is tractable, but T does not have width 1 and admits no near-unanimity polymorphisms.
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Narušuje trh bydlení pracovní mobilitu a zaměstnanost v České republice?
Konečný, Martin ; Stroukal, Dominik (advisor) ; Brožová, Dagmar (referee)
The main goal of the paper is to find out whether homeownership can have detrimental effect on employment in the Czech Republic. The Oswalds conjecture is tested on the set of panel data across Czech regions between the years of 2005--2012. By testing model similar to the Oswald's I receive the similar result that the rate of homeownership leads to higher rate of unemployment in following years. The second model tested in the paper does not support previous findings that regional rate of homeownership has negative effect on individual's probability of being unemployed. Possible refinements to the model are presented as inspiration for further research.
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Probabilistic Methods in Discrete Applied Mathematics
Fink, Jiří ; Loebl, Martin (advisor) ; Koubek, Václav (referee) ; Sereni, Jean-Sébastein (referee)
One of the basic streams of modern statistical physics is an effort to understand the frustration and chaos. The basic model to study these phenomena is the finite dimensional Edwards-Anderson Ising model. We present a generalization of this model. We study set systems which are closed under symmetric differences. We show that the important question whether a groundstate in Ising model is unique can be studied in these set systems. Kreweras' conjecture asserts that any perfect matching of the $n$-dimensional hypercube $Q_n$ can be extended to a Hamiltonian cycle. We prove this conjecture. The {\it matching graph} $\mg{G}$ of a graph $G$ has a vertex set of all perfect matchings of $G$, with two vertices being adjacent whenever the union of the corresponding perfect matchings forms a Hamiltonian cycle. We prove that the matching graph $\mg{Q_n}$ is bipartite and connected for $n \ge 4$. This proves Kreweras' conjecture that the graph $M_n$ is connected, where $M_n$ is obtained from $\mg{Q_n}$ by contracting all vertices of $\mg{Q_n}$ which correspond to isomorphic perfect matchings. A fault-free path in $Q_n$ with $f$ faulty vertices is said to be \emph{long} if it has length at least $2^n-2f-2$. Similarly, a fault-free cycle in $Q_n$ is long if it has length at least $2^n-2f$. If all faulty vertices are...
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Progress of motivation techniques
Kozlovský, Ondřej ; Chocholatý, Drahomír (advisor) ; Šebesta, Michal (referee)
The thesis consists of two parts, the theoretical and practical one. The former includes definitions and three basic types of approach to motivation that are then discussed. It also describes the historical development of motivation. This part is heading towards current state and its approaches, problems related to motivation and development trends. Upon these grounds the latter one builds to inspect how the theory is used in practice. It is discussed what aspects of motivation are used by three significant software companies, market leaders in their field. In each case the specific motivational programme is described and things in common are observed. Finally, the conjecture is verified, if and to what extent these players register and reflect trends in the development of motivational methods according to the theory.
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Gradientní modely
Bernát, Marek ; Kotecký, Roman (advisor) ; Zahradník, Miloš (referee)
We have investigated gradient models, one of them was a model with double-well potential and the other one a so called extended model. In dimension two we have calculated exact free energies of the disseminated edge configurations for the extended model and for arbitrary dimension we have derived bounds on these free energies. Combining these bounds with an argument on exstince of bad contours together with the estimate of the number of these contours and using the method of reflection positivity we have been able to show that at low temperatures there is a phase transition in the extended model. We have further shown that the phase transition exists also in the double-well model as long as a conjecture on estimates of mean energy holds. Besides these results the thesis also contains basic tools of statistical physics and facts from related fields, as well as basic results on gradient models, so that our work can serve as an introduction into these areas.
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