| dc.contributor |
Zarzuela, Santiago |
|
| dc.creator |
Dediu, Catalin |
|
| dc.date |
2018-03-27T08:25:51Z |
|
| dc.date |
2018-03-27T08:25:51Z |
|
| dc.date |
2017-09-09 |
|
| dc.date.accessioned |
2024-12-16T10:26:20Z |
|
| dc.date.available |
2024-12-16T10:26:20Z |
|
| dc.identifier |
http://hdl.handle.net/2445/121133 |
|
| dc.identifier.uri |
http://fima-docencia.ub.edu:8080/xmlui/handle/123456789/20938 |
|
| dc.description |
Treballs finals del Màster en Matemàtica Avançada, Facultat de matemàtiques, Universitat de Barcelona, Any: 2017, Director: Santiago Zarzuela |
|
| dc.description |
[en] Let $\Delta$ be a triangulation of a $(d - 1)$-dimensional sphere with $n$ vertices. The Upper Bound Conjecture (UBC for short) gives an explicit bound of the number of $i$-dimensional faces of $\Delta$.
This question dates back to the beginning of the 1950’s, when the study of the efficiency of some linear programming techniques led to the following problem: Determine the maximal possible number of $i$-faces of d-polytope with $n$ vertices.
The first statement of the UBC was formulated in 1957 by Theodore Motzkin. The original result state that the number of $i$-dimensional faces of a $d$-dimensional polytope with n vertices are bound by a certain explicit number $f i (C(n, d))$ where $C(n, d)$ is a cyclic polytope and $f_{i}$ denotes the number of $i$-dimensional faces of the simplex. We say that $P$ is a polytope if it is the convex hull of a finite set of points in $\mathbb{R}^{d}$. Moreover, we say that $C(n, d)$ is a cyclic polytope if it is the convex hull of n distinct points on the moment curve $(t, t^{2},..., t{^d})$, $-\infty<t<\infty$. With this notation the Upper Bound
Conjecture (for convex polytopes) states that cyclic polytope maximizes the number of $i$-dimensional faces among all polytopes. |
|
| dc.format |
58 p. |
|
| dc.format |
application/pdf |
|
| dc.language |
eng |
|
| dc.rights |
cc-by-nc-nd (c) Catalin Dediu, 2017 |
|
| dc.rights |
http://creativecommons.org/licenses/by-nc-nd/3.0/es/ |
|
| dc.rights |
info:eu-repo/semantics/openAccess |
|
| dc.source |
Màster Oficial - Matemàtica Avançada |
|
| dc.subject |
Àlgebra commutativa |
|
| dc.subject |
Anells commutatius |
|
| dc.subject |
Treballs de fi de màster |
|
| dc.subject |
Geometria combinatòria |
|
| dc.subject |
Commutative algebra |
|
| dc.subject |
Commutative rings |
|
| dc.subject |
Master's theses |
|
| dc.subject |
Combinatorial geometry |
|
| dc.title |
On the proof of the upper bound theorem |
|
| dc.type |
info:eu-repo/semantics/masterThesis |
|