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Banach Algebras
Machovičová, Tatiana ; Nechvátal, Luděk (referee) ; Franců, Jan (advisor)
By Banach algebra we mean Banach space enriched with a multiplication operation. It is a mathematical structure that is used in the non-periodic homogenization of composite materials. The theory of classical homogenization studies materials assuming the periodicity of the structure. For real materials, the assumption of a periodicity is not enough and is replaced by the so-called an abstract hypothesis based on a concept composed mainly of the spectrum of Banach algebra and Sigma convergence. This theory was introduced in 2004.
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Homogenization in Perforated Domains
Rozehnalová, Petra ; Bock, Igor (referee) ; Rohan, Eduard (referee) ; Franců, Jan (advisor)
Numerické řešení matematických modelů popisujících chování materiálů s jemnou strukturou (kompozitní materiály, jemně perforované materiály, atp.) obvykle vyžaduje velký výpočetní výkon. Proto se při numerickém modelování původní materiál nahrazuje ekvivalentním materiálem homogenním. V této práci je k nalezení homogenizovaného materiálu použita dvojškálová konvergence založena na tzv. rozvinovacím operátoru (anglicky unfolding operator). Tento operátor poprvé použil J. Casado-Díaz. V disertační práci je operátor definován jiným způsobem, než jak uvádí původní autor. To dovoluje pro něj dokázat některé nové vlastnosti. Analogicky je definován operátor pro funkce definované na perforovaných oblastech a jsou dokázány jeho vlastnosti. Na závěr je rozvinovací operátor použit k nalezení homogenizovaného řešení speciální skupiny diferenciálních problémů s integrální okrajovou podmínkou. Odvozené homogenizované řešení je ilustrováno na numerických experimentech.
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Continuous mathematical models of population dynamics
Pecka, Luboš ; Opluštil, Zdeněk (referee) ; Franců, Jan (advisor)
The aim of this thesis is to describe the most frequent models describing population dynamics and then to perform some numerical experiments in the MATLAB environment. These simulations should validate our theoretical results. The models are sorted from the basic models to the most complicated and are divided into the models which describe dynamics of one population and models of coexistence of two biological species. The master's thesis icludes also a program for drawing graphs and trajectories of solutions of models described in this thesis including a description of this MATLAB program.
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Modeling of surface waves
Zgabaj, Martin ; Haluza, Miloslav (referee) ; Franců, Jan (advisor)
This bachelor's thesis presents the derivation of basic characteristics of surface waves which includes the dispersion relation, the shape of the wave, the phase velocity, the group velocity and the movement of water particle. Thesis deals with both deep-water waves and shallow-water waves. Derived characteristics are examined in both cases. In the last part of thesis there are specific examples of those two types of waves.
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Mathematical pendulum
Kučerová, Barbora ; Dub, Petr (referee) ; Franců, Jan (advisor)
This bachelor’s thesis deals with the mathematical modeling of the behaviour of mathematical pendulum. The aim of this thesis is to find the equation of mathematical pendulum, compute the trajectories of solution and interpret their meaning, clasify the singular solution, plot the phase portrait in MATLAB software for the basic model and for the generalized model of the damped and driven pendulum.
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Nonlinear diffusion equations
Polášek, Radek ; Hájek, Jiří (referee) ; Franců, Jan (advisor)
This work is focused on various types of diusion equations. The diffusion equations are derived for various assumptions of media. We describe in detail Barenblatt solution for slow diffusion equation, whose features are demonstrated.
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New town hall for the district Brno-North
Franců, Jan ; Pechman, Tomáš (referee) ; Sochor, Jan (advisor)
The project design a city town hall for Brno north on the site of today's military barracks. Classical symmetry of the composition of the original building dominates the space above the terrain configuration. The buildings are oriented towards the slope and determine the nature of space. In this constellation any classical building would disturb this order. The volume of the building must be very subtle to preserve the harmony of the original composition. The main volume of the building is located below the surface of the square, however, the building is open towards most exposed corner. Sloping slab fall line is aiming to the city center and Špilberk Castle. The inclination is only 8 degrees to allow safe movement and to preserve the original space. The building preserves the street lines of surrounding blocks, but sloped slab exceeds the corner across the street line to draw attention. Parts of the building, requiring daylight had to be placed above the ground, but to preserve the original space the new volume must be subtle.
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Integral transforms and their applications
Béreš, Lukáš ; Štarha, Pavel (referee) ; Franců, Jan (advisor)
This bachelor's thesis deals with integral transforms and their applications. Its aim is to get together basic properties of Laplace and Fourier transforms and then illustrate their application in solving partial dierential equations, by calculating specic tasks with numerical experiments in MATLAB software.
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Stochastic ordinary differential equations
Bahník, Michal ; Kolářová, Edita (referee) ; Franců, Jan (advisor)
Diplomová práce se zabývá problematikou obyčejných stochastických diferenciálních rovnic. Po souhrnu teorie stochastických procesů, zejména tzv. Brownova pohybu je zaveden stochastický Itôův integrál, diferenciál a tzv. Itôova formule. Poté je definováno řešení počáteční úlohy stochastické diferenciální rovnice a uvedena věta o existenci a jednoznačnosti řešení. Pro případ lineární rovnice je odvozen tvar řešení a rovnice pro jeho střední hodnotu a rozptzyl. Závěr tvoří rozbor vybraných rovnic.
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