
Dynamical Properties of Ecological Models
Ráž, Adam ; Milota, Jaroslav (advisor) ; Pražák, Dalibor (referee)
Univerzita Karlova Abstract of the bachelor thesis Dynamical Properties of Ecological Models Adam Ráž, Praha 2011 In this thesis we study advantages and disadvantages of continuous mo delling of population coexistence. We analyze in detail the LotkaVolterra equations, which model the population dynamics of predatorprey interacti ons. We analyze the behavior in the absence of one species. We prove the existence of periodic solutions and analyze asymptotic properties of the mo del. We introduce an ecological stability of a model and in the terms of ecological stability we analyze the LotkaVolterra model. 1

 
 

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.

 

Control Problems in Epidemiology
Čížek, Pavel ; Milota, Jaroslav (advisor) ; Pražák, Dalibor (referee)
Na/.ev prace; Ki/cnf \) eh modeled] Aulor: Pavel Ci/ek Kaledra; Kaledra matemaliekc analy/.y Vcdouef bakala'rskc prace: Doc. KNDr. Jarosla\. CSc. cinait vedouci'ho: niilota(« karlin.mff.cuni.c/ Ahsirakl: V predlo/cne praci studujeme model maso\. inlekcm mikropara/iiicke epidcinie. Co oduv.eni" modeiu, klcre \ycha/( / hiologickyeh po/nulku o sludovaiicni lypu epideiuii. u\a\liine eho /a'kladm nialeniaiicke \htslmMi. /.axx'dcni'in ock(i\';ii:i' niotlei iransloriiiujcrnc. I);ile ^o /ah\\anie \laslnosUni feseni piivodm'ho i iransiorinox anelio niodelu a siahililou siaciona'nueli Inidii. PoroMuixanfni vysiedkii pro model puvodm a model iran>Jormovan\, /da je ocko\ain' /.a\edeno spra\nc. lA.idinie i jiiie mo/noMi jcho /a\'edem. N;is!cdnc se /.ahvvanie cenou Icehy a ockovant'. Illedamc linancne n e j \ i re^cnf ma/k\k \'clikou cast populace oekovat. Tiile; Isi/em \h modeled! Auihor: Pa\el C ' f / e k Departnicnl: Kaledra iimlematickr aiialv/y Sii)er\: Doe. RNDr. Jaio^lax Miloia. C'Se. Super\s email adress: nii]o{a('1''karliu.n)tT.cuni.c/ Ahsii'acl: In the present work \\ studx a model of a ma^action mieniparasiue epidoniic. After the deduction of llie motlel. \\lilcli issues from biologic in for uiafioii ahiui! ihe studied l\pe of epidemic. \\ mention basic maihematic.il chaiactcrislies. \\"e transform die...


Control of linear systems
Cesneková, Ivana ; Milota, Jaroslav (advisor) ; Honzík, Petr (referee)
The aim of this work is to look into the theory of linear systems via population model represented by partial differential equations with boundary and initial condition. Special attention is devoted to the strongly continuous semig roups on a complex Banach space. For this purpose, the notion of a homogeneous and inhomogeneous Cauchy problem is introduced and we solve our model in this abstract formulation. The system behaviour is based on properties of the resolvent set and spectrum. Controllability question limits to solve the uniformly exponen tially stability and the exponentially stabilizability. The point of this problem is in the case of the unstability to show exponencially stability of the system by using feedback. Keywords: control, differential equations, stability, controllability 1


Matrix calculus
Pekárková, Lenka ; Pražák, Dalibor (advisor) ; Milota, Jaroslav (referee)
Title: Matrix calculus Author: Lenka Pekárková Department: Department of Mathematical Analysis Supervisor: doc. RNDr. Dalibor Pražák, Ph.D., Department of Mathematical Analysis Abstract: This bachelor thesis deals with the exponential and logarithmic map, which are first defined and then their properties are proven. The exponential is considered on operators on Banach spaces in the text and its derivative, the differential equations which it solves and the formulae for the exponential of the sum of operators or the inverse of the exponential of an operator are presented. A logarithm is defined on the space of finite square matrices and its existence and uniqueness are analysed. Further the thesis focuses on the principle branch of the logarithm and properties of this map such as commutativity, properties of spectrum and monotony or the formula for the principle branch of the logarithm of the inverse. The root of a matrix is mentioned in the context of the logarithm. Keywords: Exponential of an operator, logarithm of a matrix, principle branch of the logarithm


Orthogonality in Banach spaces
Mašková, Alice ; Lukeš, Jaroslav (advisor) ; Milota, Jaroslav (referee)
In the present work we study properties of orthogonality in Hilbert spaces and possibilities of extending definition to more general type of spaces, Banach spaces. We concentrate mostly on BirkhoffJames orthogonality and investigate, which properties of Hilbert space orthogonality are still valid for Banach spaces, otherwise we provide counterexamples. As the orthogonality is generally not symmetric, we have to distinguish between right and left properties. We use BirkhoffJames orthogonality to characterize smooth and strictly convex Banach spaces. Then we study properties of Hilbert space orthogonal projection and its generalizations for Banach spaces.We study projections of norm equal one and minimal projections.


Matrix calculus
Pekárková, Lenka ; Pražák, Dalibor (advisor) ; Milota, Jaroslav (referee)
Title: Matrix calculus Author: Lenka Pekárková Department: Department of Mathematical Analysis Supervisor: doc. RNDr. Dalibor Pražák, Ph.D., Department of Mathematical Analysis Abstract: This bachelor thesis deals with the exponential and logarithmic map, which are first defined and then their properties are proven. The exponential is considered on operators on Banach spaces in the text and its derivative, the differential equations which it solves and the formulae for the exponential of the sum of operators or the inverse of the exponential of an operator are presented. A logarithm is defined on the space of finite square matrices and its existence and uniqueness are analysed. Further the thesis focuses on the principle branch of the logarithm and properties of this map such as commutativity, properties of spectrum and monotony or the formula for the principle branch of the logarithm of the inverse. The root of a matrix is mentioned in the context of the logarithm. Keywords: Exponential of an operator, logarithm of a matrix, principle branch of the logarithm
