dc.contributor.advisor | Kráľ, Daniel | |
dc.creator | Mach, Lukáš | |
dc.date.accessioned | 2017-05-08T13:54:21Z | |
dc.date.available | 2017-05-08T13:54:21Z | |
dc.date.issued | 2011 | |
dc.identifier.uri | http://hdl.handle.net/20.500.11956/49611 | |
dc.description.abstract | V t\'eto pr\'aci pod\'ame p\v rehled o n\v ekter\'ych ned\'avn\'ych v\'ysledc\' ich o skoc\'ich v hypergrafech v oblasti exterm\'aln\'i kombinatoriky. \v C\'islo $\alpha \in [0, 1)$ je skok pro $r$, pokud pro ka\v zd\'e $\epsilon > 0$ a ka\v zd\'e cel\'e \v c\'islo $m \ge r$ jak\'ykoliv $r$-graf na $N > N(\epsilon, m)$ vrcholech a s alespo\v n $(\alpha + \epsilon) {N \choose r}$ hranami obsahuje podgraf na $m$ vrcholech s alespo\v n $(\alpha + c) {m \choose r}$ hranami, kde $c := c(\alpha)$ z\'avis\' i pouze na $\alpha$. Baber a Talbot \cite{Baber} ned\'avno uk\'azali prvn\'i p\v r\'iklad existence skoku pro $r = 3$ v intervalu $[2/9, 1)$. Jejich v\'ysledek pou\v z\'iv\'a kalkul flag algeber \cite{Raz07}, kter\'y vede k re\v sen\'i probl\'emu semidefinitn\'i optimalizace. Sou\v c\'ast\'i pr\'ace je softwarov\'a implementace jejich metody. | cs_CZ |
dc.description.abstract | We give an overview of recent progress in the research of hypergraph jumps -- a problem from extremal combinatorics. The number $\alpha \in [0, 1)$ is a jump for $r$ if for any $\epsilon > 0$ and any integer $m \ge r$ any $r$-graph with $N > N(\epsilon, m)$ vertices and at least $(\alpha + \epsilon) {N \choose r}$ edges contains a subgraph with $m$ vertices and at least $(\alpha + c) {m \choose r}$ edges, where $c := c(\alpha)$ does depend only on $\alpha$. Baber and Talbot \cite{Baber} recently gave first examples of jumps for $r = 3$ in the interval $[2/9, 1)$. Their result uses the framework of flag algebras \cite{Raz07} and involves solving a semidefinite optimization problem. A software implementation of their method is a part of this work. | en_US |
dc.language | English | cs_CZ |
dc.language.iso | en_US | |
dc.publisher | Univerzita Karlova, Matematicko-fyzikální fakulta | cs_CZ |
dc.subject | extremální kombinatorika | cs_CZ |
dc.subject | flag algebra | cs_CZ |
dc.subject | skoky v hypergrafech | cs_CZ |
dc.subject | extremal combinatorics | en_US |
dc.subject | flag algebra | en_US |
dc.subject | hypergraph jumps | en_US |
dc.title | Extremální vlastnosti hypergrafů | en_US |
dc.type | diplomová práce | cs_CZ |
dcterms.created | 2011 | |
dcterms.dateAccepted | 2011-09-19 | |
dc.description.department | Department of Applied Mathematics | en_US |
dc.description.department | Katedra aplikované matematiky | cs_CZ |
dc.description.faculty | Faculty of Mathematics and Physics | en_US |
dc.description.faculty | Matematicko-fyzikální fakulta | cs_CZ |
dc.identifier.repId | 91634 | |
dc.title.translated | Extremální vlastnosti hypergrafů | cs_CZ |
dc.contributor.referee | Kaiser, Tomáš | |
dc.identifier.aleph | 001387643 | |
thesis.degree.name | Mgr. | |
thesis.degree.level | navazující magisterské | cs_CZ |
thesis.degree.discipline | Discrete Models and Algorithms | en_US |
thesis.degree.discipline | Diskrétní modely a algoritmy | cs_CZ |
thesis.degree.program | Computer Science | en_US |
thesis.degree.program | Informatika | cs_CZ |
uk.thesis.type | diplomová práce | cs_CZ |
uk.taxonomy.organization-cs | Matematicko-fyzikální fakulta::Katedra aplikované matematiky | cs_CZ |
uk.taxonomy.organization-en | Faculty of Mathematics and Physics::Department of Applied Mathematics | en_US |
uk.faculty-name.cs | Matematicko-fyzikální fakulta | cs_CZ |
uk.faculty-name.en | Faculty of Mathematics and Physics | en_US |
uk.faculty-abbr.cs | MFF | cs_CZ |
uk.degree-discipline.cs | Diskrétní modely a algoritmy | cs_CZ |
uk.degree-discipline.en | Discrete Models and Algorithms | en_US |
uk.degree-program.cs | Informatika | cs_CZ |
uk.degree-program.en | Computer Science | en_US |
thesis.grade.cs | Výborně | cs_CZ |
thesis.grade.en | Excellent | en_US |
uk.abstract.cs | V t\'eto pr\'aci pod\'ame p\v rehled o n\v ekter\'ych ned\'avn\'ych v\'ysledc\' ich o skoc\'ich v hypergrafech v oblasti exterm\'aln\'i kombinatoriky. \v C\'islo $\alpha \in [0, 1)$ je skok pro $r$, pokud pro ka\v zd\'e $\epsilon > 0$ a ka\v zd\'e cel\'e \v c\'islo $m \ge r$ jak\'ykoliv $r$-graf na $N > N(\epsilon, m)$ vrcholech a s alespo\v n $(\alpha + \epsilon) {N \choose r}$ hranami obsahuje podgraf na $m$ vrcholech s alespo\v n $(\alpha + c) {m \choose r}$ hranami, kde $c := c(\alpha)$ z\'avis\' i pouze na $\alpha$. Baber a Talbot \cite{Baber} ned\'avno uk\'azali prvn\'i p\v r\'iklad existence skoku pro $r = 3$ v intervalu $[2/9, 1)$. Jejich v\'ysledek pou\v z\'iv\'a kalkul flag algeber \cite{Raz07}, kter\'y vede k re\v sen\'i probl\'emu semidefinitn\'i optimalizace. Sou\v c\'ast\'i pr\'ace je softwarov\'a implementace jejich metody. | cs_CZ |
uk.abstract.en | We give an overview of recent progress in the research of hypergraph jumps -- a problem from extremal combinatorics. The number $\alpha \in [0, 1)$ is a jump for $r$ if for any $\epsilon > 0$ and any integer $m \ge r$ any $r$-graph with $N > N(\epsilon, m)$ vertices and at least $(\alpha + \epsilon) {N \choose r}$ edges contains a subgraph with $m$ vertices and at least $(\alpha + c) {m \choose r}$ edges, where $c := c(\alpha)$ does depend only on $\alpha$. Baber and Talbot \cite{Baber} recently gave first examples of jumps for $r = 3$ in the interval $[2/9, 1)$. Their result uses the framework of flag algebras \cite{Raz07} and involves solving a semidefinite optimization problem. A software implementation of their method is a part of this work. | en_US |
uk.publication.place | Praha | cs_CZ |
uk.grantor | Univerzita Karlova, Matematicko-fyzikální fakulta, Katedra aplikované matematiky | cs_CZ |
dc.identifier.lisID | 990013876430106986 | |