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Tetris 3D - made of array LED diode
Paták, Pavel ; Pust, Radim (referee) ; Šebela, Radek (advisor)
This thesis deals with suggestion of device for playing tetris game in 3D. There is used 3D LED display – the ashlar where the LED are situated in its volume. Thesis describes possibilities of display connection with a view to minimal number of control pins. It also describes display mechanical construction. Motion of blocks in the game is controled by tilt of device, thesis describes way of tilt measuring. It also decribes in details possibilities of connection of the microcontroller and the display, comparison of connection ways, sellection of one of these solution and sellection of the microcontroller. There is also noted the structure of control software.
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DRM signal transmitter
Paták, Pavel ; Šebesta, Jan (referee) ; Lukeš, Zbyněk (advisor)
Master’s thesis deals with design and practical realisation of electronic circuits, which are needed for assembling of DRM signal transmitter for ham short waves bands. There is presented DRM standard as well as there are described the differencies between DRM for radio broadcast and for ham using. There is described design of input audio circuits, modulator, mixer, local generator, amplifier and filters. Principle of used SSB modulator is based on phase method, often called Tayloe modulator. This principle is analysed in detail including mathematical description, which was derived. It is possible to control the transmitter by program running at computer, communication takes place via USB. There is described establishment of communication in this work too.
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Kombinatorika matematických struktur
Paták, Pavel ; Krajíček, Jan (advisor) ; Thapen, Neil (referee)
The combinatorics of a first order mathematical structure is the class of all formulas valid in all in it definable structures. This notion was first introduced by Kraj'ček in [6]. In the present work we try to characterize and compare the combinatorics of several different prominent structures (reals, complex number, dense linear order, . . . ). We also study the question of algorithmical complexity, i.e. the question how hard it is to check whether a given formula lies in the combinatorics of a given structure. We prove that this question is corecursively enumeratively complete and therefore algorithmicaly undecidable in the case of models of complete theories without strict order property (SOP) and in the case of pseudofinite structures.
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Definability in mathematical structures
Paták, Pavel ; Krajíček, Jan (advisor) ; Jeřábek, Emil (referee)
Nazev pram: Defiuovatelnost v matrnnatickych struktnrneh Auiur: Pave! Patak Kat.odra: Katedra algebry Vedouci bakalafske prace: Prof. R.XDr. Jan Krajicek, DrSc. e-mail vedouciho: krajicek'Q'math.cas.cz AbytrakL: V pfedlo/.ene praci so zabyvame popisem definovatelnych nmozin v ruznych matematickych st.rukturaeh. Ukazujerne, zo defiuovatelne mnoziny v pfirozenych, celych a racionalnich cfslcch inohon byt volico kompliko- vann. naproti toinn dnfiiiovatolnr mnoziny ve .striiktnrach s {'liininari kvanti- fikatoru (reaina, komplexni cfsla,. - . ) JHOU jcdnoduclic. Vciinjoino se i pojinu modolovo I'iplnosti. S poinoci zfskanych poznatku a vo.ty o nplno.sti pak snadno dokazeme nektere obtizne vety jinych disciplin - alji,ebraickou Xnll- stollcnsatz a Artinovn charaktorizaci pozitivnr dofinitnicb racionalnfcli fimkoi, geometrickou Tarski-Seidenbergovn vetn a ninohc dalyi. Klicova slova: nmtematicke sl.rnktnry, dcliiiuvatelnost, eliminace kvantilika- toru Title: DcfinnViility in inatlioinatica.l structures Author: Pavel Patak Department: Department of Algebra Supervisor: Prof. RNDr. Jan Krajirek, DrSc. Supervisor's e-mail address: kra.jicokv'iJina.lb.cas.cz Abstract: In t,be present work we study the description of definable sets in various mathematical structures. We show that, the definable sets in natural, integer...
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Kombinatorika matematických struktur
Paták, Pavel
The combinatorics of a first order mathematical structure is the class of all formulas valid in all in it definable structures. This notion was first introduced by Krajíček in [6]. In the present work we try to characterize and compare the combinatorics of several different prominent structures (reals, complex number, dense linear order, . . . ). We also study the question of algorithmical complexity, i.e. the question how hard it is to check whether a given formula lies in the combinatorics of a given structure. We prove that this question is corecursively enumeratively complete and therefore algorithmicaly undecidable in the case of models of complete theories without strict order property (SOP) and in the case of pseudofinite structures.
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Balanced and almost balanced group presentations from algorithmic viewpoint
Skotnica, Michael ; Tancer, Martin (advisor) ; Paták, Pavel (referee)
In this thesis we study algorithmic aspects of balanced group presentations which are finite presentations with the same number of generators and relations. The main motivation is that the decidability of some problems, such as the triviality problem, is open for balanced presentations. First, we summarize known results on decision problems for general finite presen- tations and we show two group properties which are undecidable even for balanced presentations - the property of "being a free group"' and the property of "having a finite presentation with 12 generators". We also show reductions of some graph problems to the triviality problem for group presentations, such as determining whether a graph is connected, k-connected or connected including an odd cycle. Then we show a reduction of the determining whether a graph with the same number of vertices and edges is a cycle to the triviality problem for balanced presentations. On the other hand, there is also a limitation of reduction to balanced presentations. We prove that there is no balanced presentation with two generators a, b|ap(m) bq(m) , ar(m) bs(m) for p(m), q(m), r(m), s(m) ∈ Z[m] which describes the trivial group if and only if m is odd. In the last part of this thesis, we describe a relation between group presentations and topology. In addition,...
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Using algebra in geometry
Paták, Pavel ; Růžička, Pavel (advisor) ; Šmíd, Dalibor (referee) ; Blagojevic, Pavle (referee)
Using algebra in geometry Pavel Paták Department: Department of Algebra Supervisor: Mgr. Pavel Růžička, Ph.D., Department of Algebra 1 Abstract In this thesis, we develop a technique that combines algebra, algebraic topology and combinatorial arguments and provides non-embeddability results. The novelty of our approach is to examine non- embeddability arguments from a homological point of view. We illustrate its strength by proving two interesting theorems. The first one states that k-dimensional skeleton of b 2k+2 k + k + 3 -dimensional simplex does not embed into any 2k-dimensional manifold M with Betti number βk(M; Z2) ≤ b. It is the first finite upper bound for Kühnel's conjecture of non-embeddability of simplices into manifolds. The second one is a very general topological Helly type theorem for sets in Rd : There exists a function h(b, d) such that the following holds. If F is a finite family of sets in Rd such that ˜βi ( G; Z2) ≤ b for any G F and every 0 ≤ i ≤ d/2 − 1, then F has Helly number at most h(b, d). If we are only interested whether the Helly numbers are bounded or not, the theorem subsumes a broad class of Helly types theorems for sets in Rd . Keywords: Homological Non-embeddability, Helly Type Theorem, Kühnel's conjecture of non-embeddability of ske- leta of simplices into manifolds
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Kombinatorika matematických struktur
Paták, Pavel
The combinatorics of a first order mathematical structure is the class of all formulas valid in all in it definable structures. This notion was first introduced by Krajíček in [6]. In the present work we try to characterize and compare the combinatorics of several different prominent structures (reals, complex number, dense linear order, . . . ). We also study the question of algorithmical complexity, i.e. the question how hard it is to check whether a given formula lies in the combinatorics of a given structure. We prove that this question is corecursively enumeratively complete and therefore algorithmicaly undecidable in the case of models of complete theories without strict order property (SOP) and in the case of pseudofinite structures.
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Kombinatorika matematických struktur
Paták, Pavel ; Krajíček, Jan (advisor) ; Thapen, Neil (referee)
The combinatorics of a first order mathematical structure is the class of all formulas valid in all in it definable structures. This notion was first introduced by Kraj'ček in [6]. In the present work we try to characterize and compare the combinatorics of several different prominent structures (reals, complex number, dense linear order, . . . ). We also study the question of algorithmical complexity, i.e. the question how hard it is to check whether a given formula lies in the combinatorics of a given structure. We prove that this question is corecursively enumeratively complete and therefore algorithmicaly undecidable in the case of models of complete theories without strict order property (SOP) and in the case of pseudofinite structures.
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Definability in mathematical structures
Paták, Pavel ; Jeřábek, Emil (referee) ; Krajíček, Jan (advisor)
Nazev pram: Defiuovatelnost v matrnnatickych struktnrneh Auiur: Pave! Patak Kat.odra: Katedra algebry Vedouci bakalafske prace: Prof. R.XDr. Jan Krajicek, DrSc. e-mail vedouciho: krajicek'Q'math.cas.cz AbytrakL: V pfedlo/.ene praci so zabyvame popisem definovatelnych nmozin v ruznych matematickych st.rukturaeh. Ukazujerne, zo defiuovatelne mnoziny v pfirozenych, celych a racionalnich cfslcch inohon byt volico kompliko- vann. naproti toinn dnfiiiovatolnr mnoziny ve .striiktnrach s {'liininari kvanti- fikatoru (reaina, komplexni cfsla,. - . ) JHOU jcdnoduclic. Vciinjoino se i pojinu modolovo I'iplnosti. S poinoci zfskanych poznatku a vo.ty o nplno.sti pak snadno dokazeme nektere obtizne vety jinych disciplin - alji,ebraickou Xnll- stollcnsatz a Artinovn charaktorizaci pozitivnr dofinitnicb racionalnfcli fimkoi, geometrickou Tarski-Seidenbergovn vetn a ninohc dalyi. Klicova slova: nmtematicke sl.rnktnry, dcliiiuvatelnost, eliminace kvantilika- toru Title: DcfinnViility in inatlioinatica.l structures Author: Pavel Patak Department: Department of Algebra Supervisor: Prof. RNDr. Jan Krajirek, DrSc. Supervisor's e-mail address: kra.jicokv'iJina.lb.cas.cz Abstract: In t,be present work we study the description of definable sets in various mathematical structures. We show that, the definable sets in natural, integer...
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