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Introduction E9F5FC

Understandable FFFFFF

Questions FFFFC0

Notes EEEEEE

Software

## Book.Polytopes istorija

2019 vasario 03 d., 22:38 atliko AndriusKulikauskas -
Pridėtos 2-3 eilutės:

[++++多胞形++++]
2018 rugsėjo 11 d., 14:52 atliko AndriusKulikauskas -
Pridėtos 1-2 eilutės:
[[Binomial theorem]]
2018 rugsėjo 05 d., 22:01 atliko AndriusKulikauskas -
Pridėtos 3-7 eilutės:
* Study the slicing of a hypercube.
** Read [[https://www.maa.org/sites/default/files/pdf/upload_library/22/Allendoerfer/1992/0025570x.di021171.02p0016b.pdf | Cube Slices, Pictorial Triangles, and Probability]] by Don Chakerian and Dave Logothetti.
** How does it relate to the slicing of a cross-polytope?
** Relate it to the other three families of polytopes.
** [[https://www.youtube.com/watch?v=KYaCtHPCARc | Infinite Series: Dissecting Hypercubes with Pascal's Triangle]]
2018 rugsėjo 05 d., 11:24 atliko AndriusKulikauskas -
Pridėta 5 eilutė:
* Relate polytopes and convexity.
2018 rugsėjo 01 d., 12:24 atliko AndriusKulikauskas -
Pakeistos 70-71 eilutės iš

Lecture 11:
į:
Lecture 17: Triangulation of d-cross-polytope.

Lecture 18: Counting lattice points in polytopes.

Lecture 19: Partition functions.

Lecture 20: Generating functions for cones.
2018 rugsėjo 01 d., 12:20 atliko AndriusKulikauskas -
Ištrintos 17-21 eilutės:
Concepts
* Building polytopes: pyramids (adding the center).
* Vertex figures (converting the vertices to faces).
Pakeistos 30-31 eilutės iš
* Caratheodory's theorem
į:
* [[https://en.wikipedia.org/wiki/Carath%C3%A9odory%27s_theorem_(convex_hull) | Caratheodory's theorem]]

Lecture 5 and 6: [[https://en.wikipedia.org/wiki/Farkas%27_lemma | Farkas' lemma]] versions 1 to 4.

Lecture 7: Faces of polytopes. Face of P in direction c is the set of all x in P where c-x is maximal.
* [[https://en.wikipedia.org/wiki/Affine_space | Affine spaces]].
* Dim(face) = Dim(Aff(face))
* f-vector gives the number of faces in each dimension
* f-poly the generated function where the lowest coefficient gives the number of vertices and the highest coefficient gives the volume

Lecture 8: Construction of faces
* Building polytopes: pyramids (adding the center).
* Vertex figures (converting the vertices to faces).
* Face lattice

Lecture 9: Face lattice
* P and Q are combinatorially isomorphic if their face lattices are isomorphic.
* Polar (Dual) polytopes. Dual = c in dual space where c x <=1 for all x in P.
* If 0 is in P, then P equals its dual's dual.

Lecture 10:
* Dual faces.
* The face lattices of P and its dual are opposites.
* Simple and simplicial polytopes.
* P is simple iff its dual is simplicial.

Lecture 11: [[https://en.wikipedia.org/wiki/Cyclic_polytope | The cyclic polytope.]]

Lecture 12: Graphs of polytopes

Lecture 13: How good is linear programming?
* Hirsch conjecture is false.

Lecture 14: Balinski's theorem: P is a d-polytope implies G(P) is d-connected.

Lecture 15: If P is simple, then G(P) determines P combinatorially.

Lecture 16: Complexes, subdivisions, triangulations.
* Every P has a triangulation.

Lecture 11:
2018 rugsėjo 01 d., 09:47 atliko AndriusKulikauskas -
Pakeistos 20-36 eilutės iš
* Vertex figures (converting the vertices to faces).
į:
* Vertex figures (converting the vertices to faces).

Lecture 1: Course is a combinatorial focus on convex polytopes and hyperplane arrangements.
* Euler's theorem: 1 - v + e - f + 1 = 0
* Steinitz's theorem: A 3-polytope exists iff 1 - v + e - f + 1 = 0, v <= 2f - 4, f <= 2v - 4.
* Regular polytopes.

Lecture 2: Definition of polytope P as convex hull of vertices. Sum of lambda x vertex where lambdas are nonnnegative and sum to 1.

Lecture 3: Intersections and products of polytopes are also polytopes.
* Main theorem of polytopes: Polytopes = convex hulls of finitely many points = bounded intersections of halfspaces.

Lecture 4: V-polyhedron intersected with affine plane is a V-polyhedron.
* Dual polytope consists of a where a<=1-x for all x in P.
* Caratheodory's theorem
2018 rugsėjo 01 d., 08:33 atliko AndriusKulikauskas -
Pakeista 12 eilutė iš:
** [[FedericoArdila-2010-Polytopes-Lectures.pdf | Lecture notes: Polytopes]]
į:
** [[Attach:FedericoArdila-2010-Polytopes-Lectures.pdf | Lecture notes: Polytopes]]
2018 rugsėjo 01 d., 08:33 atliko AndriusKulikauskas -
Pridėta 12 eilutė:
** [[FedericoArdila-2010-Polytopes-Lectures.pdf | Lecture notes: Polytopes]]
2018 rugpjūčio 30 d., 18:32 atliko AndriusKulikauskas -
Pakeistos 4-6 eilutės iš
į:
* Relate Schur functions and simplices by way of the Vandermonde determinant.

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2018 rugpjūčio 30 d., 18:28 atliko AndriusKulikauskas -
Pridėtos 1-5 eilutės:
>>bgcolor=#FFFFC0<<

* Relate the Vandermonde determinant to the q-analogue of simplices.
2018 rugpjūčio 28 d., 17:53 atliko AndriusKulikauskas -
Pridėta 6 eilutė:
*** [[https://www.amazon.com/Lectures-Polytopes-Graduate-Texts-Mathematics/dp/B000FJBNX4/ref=mt_other?_encoding=UTF8&me=&qid= | G. Ziegler. Lectures on polytopes]] 8 USD.
2018 rugpjūčio 28 d., 17:50 atliko AndriusKulikauskas -
Ištrinta 0 eilutė:
Pridėtos 4-5 eilutės:
** [[http://math.sfsu.edu/federico/Clase/Polytopes/lectures.html | Lectures: Polytopes]]
** [[http://math.sfsu.edu/federico/Clase/Polytopes/texts.html | Texts: Polytopes]]
2018 rugpjūčio 28 d., 17:46 atliko AndriusKulikauskas -
Pakeistos 1-2 eilutės iš
Videos
*
[[https://www.youtube.com/playlist?list=PL-XzhVrXIVeQ298S6uCyoDGWNActWwnzZ | Polytopes - Federico Ardila]]
į:

Federico Ardila
*
* [[http://math.sfsu.edu/federico/teaching.html | Teaching - online courses
]]
* [[http://math.sfsu.edu/federico/research.html | Research talks]]
2018 rugpjūčio 28 d., 17:41 atliko AndriusKulikauskas -
Pridėtos 1-6 eilutės:
Videos
* [[https://www.youtube.com/playlist?list=PL-XzhVrXIVeQ298S6uCyoDGWNActWwnzZ | Polytopes - Federico Ardila]]

Concepts
* Building polytopes: pyramids (adding the center).
* Vertex figures (converting the vertices to faces).

#### Polytopes

Naujausi pakeitimai

 Puslapis paskutinį kartą pakeistas 2019 vasario 03 d., 22:38