
Neuronal coding and metabolic cost of information
Bárta, Tomáš ; Košťál, Lubomír (advisor) ; Martinez, Dominique (referee) ; Nowotny, Thomas (referee)
For most neurons, the information the neuron passes on is contained within the times of sending out electrical pulses  socalled action potentials. It is still not fully understood how to read this "neural code". The efficient coding hypothesis proposes that due to evolutionary pressures sensory systems evolved to transmit and process information in the most efficient way possible. However, the notion of efficiency seems to be different in different sensory systems. Cortical neurons keep their firing rates low to minimize metabolic expenses. So do insect olfactory receptor neurons (ORNs, the first layer of the olfactory system). Neurons in the insect antennal lobe (the second layer of the olfactory system), on the other hand fully use the space of possible firing rates to encode the maximum information about the odor. In my thesis, I studied how can single cortical neurons and their populations transmit and process information, while keeping metabolic expenses low, and also how the insect olfactory system encodes information about odors encountered in the air. In the part of my thesis about metabolically efficient information transmission I focused mainly on the role of inhibitory neurons in efficient information transmission. Through mathematical analysis and Monte Carlo simulations of spiking...


Volterra's principle and its generalizations
Havelková, Alexandra ; Pražák, Dalibor (advisor) ; Bárta, Tomáš (referee)
The thesis concerns the Volterra principle and its validity in a more general sense. Volterra principle states that an improvement of the environment benefits predators rela tively more than prey. To begin with, there is attention paid to the LotkaVolterra model and in connection to it, the validity of the Volterra principle is discussed. Next, we de fine the term of Volterra principle formally and furthermore we formulate and prove a sufficient condition of its validity. Then the focus is shifted to more complicated models, specifically HollingTanner and Gause models. The Volterra principle is here discussed in the context of individual equilibria in the original sense and in a generalized one. With most of the equilibria, the results show the principle holds. With those equilibria that negate the Volterra principle, the stability of said equilibria is discussed along with validity of the Volterra principle in generalized sense. 1


Heat equation with dynamic boundary conditions
Gregor, Michal ; Bárta, Tomáš (advisor) ; Průša, Vít (referee)
In this thesis we deal with the solution of the heat equation with a dynamic boundary condition, which relates the time derivative at the boundary and the normal flow at the boundary. We first deal with the derivation of this dynamic boundary condition and its physical interpretation. Furthermore, we solve problems in one spacial dimension, where we learn the necessary techniques, which we will then use in solving more complex twodimensional problems on a square, an annulus and a circle. We proceed using the Fourier method. We convert the problem to finding eigenvalues and eigenfunctions of the Laplace operator, which satisfy a special boundary condition. We use the results from articles that say that such constructed eigenfunctions form a complete system in a suitable space of functions and so we are able to construct a general solution as their superposition.

 

Optimalizace přistání na Měsíci
Campbell, Daniel ; Milota, Jaroslav (advisor) ; Bárta, Tomáš (referee)
Nazev prace: Optimali/ace pristani na Mesici Autor: Daniel Campbell Katedra. (listav): Katedra inatematicke analy/y Vedouci bakalafske prace: Doc.RNDr. .laroslav Miluta. CSc. email vedoueiho: Jaroslav.Milota'Q'mlT.cuni.c/ Abstrakt: V toto prat:i vytvoriine a /komnamo rnodol, kt.ory ])opisujo raketu pri pristani na inesiui. Urchin za jakych okolnosti l/,e pristat a zda f'xiistnji1 kontrol. klery by niininiili/oval inno/stvi jjotrcbneho paliva pon ziteho pri pristani. Pokud existuje. pak tento prvek najdome a doka/eme vlastnowl . Title:0ptimilisation of the moonlanding problem Author:Daniel Campbell Department:Katedra matematiuke analy/y Snpnrvisor:Doc. RNDr. Jaroslav Milota, CSc. Supervisor's email address:Jarosla\.Milota'(PniJr.(.:nni.c/ Abstract: In this paper \vo are to rreate and examine a model, which describes (he motion of a rocket landing on the surface of the moon. We will determine under which circumstances it is possible to make (.he landing and determine whether there exists some way of landing that minimises fuel consumption. IT so we are to find this method and prove the desired property.


Difference equations and their applications in economical models
Ivanková, Kristýna ; John, Oldřich (advisor) ; Bárta, Tomáš (referee)
Ndzev prdc.e: Diferencni rovnice a jejich vyuziti v ekononrickych modelech Autor: Kristyna. Ivankova Katedra (uxtav): Katedra matematicke analyzy Vedouei bakaldrske prdce: Doc.RNDr. Oldfich John, C/Sc. email vedouciho: jorm@karlin.mff.cuni.cz Abstrakt: V praci studujeiue linearni diferencni rovnice prvniho fadu a je jich vyuziti pfi fonnulaci a liledani feseni inikroekononiickych a niakroekono mickycli inodclu. Modoly, ktere v praci uvadinio, jsou natolik zjednodusenc, aby byio inoznc ziskat jojich fesoni poinoci zkounianeho niateniatickeho aparatu. U dii'ereiicnich rovnic sc xaniefinie na feseni iechto rovnic pro spccialni prave strany. Ziskane vysledky nasledno pouzijcme v inikroeko nomickycli modelcch rovnuvahy trim. Zakladnim inodelein je zdc pavuci novy model, z nej jsou odvozeuy uiodoly s noririalni cenou a a adaptivninii ocekavanimi. V zavern prace so zabyvame zkoiiirianim stability a dynamiky multiplikatoru v inakroekononiickych modelcch uzavfenc ekonomiky (rnodely zdaneni, spofeiii a zvyky spotrebitelu) i otevfeue ekonomiky. Kltcovd slova: linearni dilerencni rovnice, ekonoinicke modely, pavucinovy model, mnltiplikatory, stabilHa feseni Title: DifTerence oquaLioiis and their upplic.ations in economical models Author: Kristyna Ivankova. Department: Department of mathematical analysis...

 
 

Defining the exponential function and logarithm
Franc, Tomáš ; Bárta, Tomáš (advisor) ; Veselý, Jiří (referee)
In this diploma thesis we will introduce six de nitions of the natural exponential function and ve de nitions of the natural logarithmic function. We will prove the de nitions' correctness, derive basic properties of both funcions and show the equivalence of all de nitions for each function. We will see how these funcions are de ned in some textbooks for universities and in textbooks for grammar schools. We will discuss the bene ts and drawbacks of all de nitions and will use the criteria such as required theory and difficulty or length of proofs. At the end of the thesis we will make some recommendations regarding de ning these functions at high schools and universities and we will give several suggestions for an additional research.


Limit behavior of the Nash equlibrium
Kovařík, Vojtěch ; Spurný, Jiří (advisor) ; Bárta, Tomáš (referee)
The subject of study of game theory  games  serves as mathematical models for reallife problems. In every game there are two or more players who aim to maximize their own profit by choosing their actions. A situation where no player can benefit from changing his own action alone has got particular importance in the study of games  it is called Nash equilibrium. Games with a finite number of players have certain advantages over those with an infinite number of players. For one, problems whose model is a game with a finite number of players are quite common. Moreover, one of the classical results of game theory is that (with certain additional assumptions) in every game with a finite number of players there exists a Nash equilibrium. On the other hand, when trying to describe the properties of a game with an infinite number of players we might be able to use calculus instead of going trough all possibilities (as is common for games with a finite number of players), which tends to be computationally demanding. However, if we want to use these advantages of games with an infinite number of players, it is important first to know whether there is any relationship between games with a finite and infinite number of players at all. The goal of this thesis is to define terms and to introduce tools which would allow...
