flag algebras and some applications
play

Flag Algebras and Some Applications Bernard Lidick y Iowa State - PowerPoint PPT Presentation

Jul 24, 2023 •301 likes •1.72k views

Flag Algebras First try for Mantel More automatic approach Applications Flag Algebras and Some Applications Bernard Lidick y Iowa State University 50 th CzechSlovak Graph Theory Conference Bo z Dar Jun 5, 2015 (Joint results

  1. Flag Algebras First try for Mantel More automatic approach Applications Example - Mantel’s theorem, 1st try Theorem (Mantel 1907) 4 n 2 edges. A triangle-free graph contains at most 1 Assume edges are red and non-edges are blue. ≤ 1 Assume = 0. (We want to conclude 2 .) 13

  2. Flag Algebras First try for Mantel More automatic approach Applications Example - Mantel’s theorem, 1st try Theorem (Mantel 1907) 4 n 2 edges. A triangle-free graph contains at most 1 Assume edges are red and non-edges are blue. ≤ 1 Assume = 0. (We want to conclude 2 .) � 2 � 0 ≤ 1 − 2 v 13

  3. Flag Algebras First try for Mantel More automatic approach Applications Example - Mantel’s theorem, 1st try Theorem (Mantel 1907) 4 n 2 edges. A triangle-free graph contains at most 1 Assume edges are red and non-edges are blue. ≤ 1 Assume = 0. (We want to conclude 2 .) � 2 � � � 0 ≤ 1 − 2 v = 1 − 4 v + 4 v + 4 v v × v = v + v 13

  4. Flag Algebras First try for Mantel More automatic approach Applications Example - Mantel’s theorem, 1st try Theorem (Mantel 1907) 4 n 2 edges. A triangle-free graph contains at most 1 Assume edges are red and non-edges are blue. ≤ 1 Assume = 0. (We want to conclude 2 .) � 2 � � � 0 ≤ 1 = 1 � � 1 − 2 v 1 − 4 v + 4 v + 4 n n v v v 13

  5. Flag Algebras First try for Mantel More automatic approach Applications Example - Mantel’s theorem, 1st try Theorem (Mantel 1907) 4 n 2 edges. A triangle-free graph contains at most 1 Assume edges are red and non-edges are blue. ≤ 1 Assume = 0. (We want to conclude 2 .) � 2 � � � 0 ≤ 1 = 1 � � 1 − 2 v 1 − 4 v + 4 v + 4 n n v v v + 4 = 1 − 4 + 4 3 1 1 1 = � = � v ∈ V ( G ) v ∈ V ( G ) 3 | V ( G ) | | V ( G ) | v v 13

  6. Flag Algebras First try for Mantel More automatic approach Applications Example - Mantel’s theorem, 1st try Theorem (Mantel 1907) 4 n 2 edges. A triangle-free graph contains at most 1 Assume edges are red and non-edges are blue. ≤ 1 Assume = 0. (We want to conclude 2 .) � 2 � � � 0 ≤ 1 = 1 � � 1 − 2 v 1 − 4 v + 4 v + 4 n n v v v + 4 = 1 − 4 + 4 3 = 2 + 1 + 3 3 13

  7. Flag Algebras First try for Mantel More automatic approach Applications Example - Mantel’s theorem, 1st try Theorem (Mantel 1907) 4 n 2 edges. A triangle-free graph contains at most 1 Assume edges are red and non-edges are blue. ≤ 1 Assume = 0. (We want to conclude 2 .) � 2 � � � 0 ≤ 1 = 1 � � 1 − 2 v 1 − 4 v + 4 v + 4 n n v v v + 4 = 1 − 4 3 = 2 + 1 3 3 13

  8. Flag Algebras First try for Mantel More automatic approach Applications Example - Mantel’s theorem, 1st try Theorem (Mantel 1907) 4 n 2 edges. A triangle-free graph contains at most 1 Assume edges are red and non-edges are blue. ≤ 1 Assume = 0. (We want to conclude 2 .) � 2 � � � 0 ≤ 1 = 1 � � 1 − 2 v 1 − 4 v + 4 v + 4 n n v v v + 4 = 1 − 4 3 − 2 = 1 − 2 3 = 4 + 2 2 3 3 13

  9. Flag Algebras First try for Mantel More automatic approach Applications Example - Mantel’s theorem, 1st try Theorem (Mantel 1907) 4 n 2 edges. A triangle-free graph contains at most 1 Assume edges are red and non-edges are blue. ≤ 1 Assume = 0. (We want to conclude 2 .) � 2 � � � 0 ≤ 1 = 1 � � 1 − 2 v 1 − 4 v + 4 v + 4 n n v v v + 4 = 1 − 4 3 − 2 = 1 − 2 3 = 4 + 2 2 3 3 ≤ 1 − 2 13

  10. Flag Algebras First try for Mantel More automatic approach Applications Example - stability for Mantel = 1 Assume = 0 and 2 . Goal is G = . 14

  11. Flag Algebras First try for Mantel More automatic approach Applications Example - stability for Mantel = 1 Assume = 0 and 2 . Goal is G = . − 2 0 ≤ 1 − 2 3 14

  12. Flag Algebras First try for Mantel More automatic approach Applications Example - stability for Mantel = 1 Assume = 0 and 2 . Goal is G = . − 2 0 ≤ 1 − 2 3 0 ≤ − 2 3 14

  13. Flag Algebras First try for Mantel More automatic approach Applications Example - stability for Mantel = 1 Assume = 0 and 2 . Goal is G = . − 2 0 ≤ 1 − 2 3 0 ≤ − 2 3 0 ≥ 14

  14. Flag Algebras First try for Mantel More automatic approach Applications Example - stability for Mantel = 1 Assume = 0 and 2 . Goal is G = . − 2 0 ≤ 1 − 2 3 0 ≤ − 2 3 0 ≥ Only and appear in G . 14

  15. Flag Algebras First try for Mantel More automatic approach Applications Example - stability for Mantel = 1 Assume = 0 and 2 . Goal is G = . − 2 0 ≤ 1 − 2 3 0 ≤ − 2 3 0 ≥ Only and appear in G . 14

  16. Flag Algebras First try for Mantel More automatic approach Applications Example - stability for Mantel = 1 Assume = 0 and 2 . Goal is G = . − 2 0 ≤ 1 − 2 3 0 ≤ − 2 3 0 ≥ Only and appear in G . 14

  17. Flag Algebras First try for Mantel More automatic approach Applications Example - stability for Mantel = 1 Assume = 0 and 2 . Goal is G = . − 2 0 ≤ 1 − 2 3 0 ≤ − 2 3 0 ≥ Only and appear in G . 14

  18. Flag Algebras First try for Mantel More automatic approach Applications Example - stability for Mantel = 1 Assume = 0 and 2 . Goal is G = . − 2 0 ≤ 1 − 2 3 0 ≤ − 2 3 0 ≥ Only and appear in G . 14

  19. Flag Algebras First try for Mantel More automatic approach Applications Example - stability for Mantel = 1 Assume = 0 and 2 . Goal is G = . − 2 0 ≤ 1 − 2 3 0 ≤ − 2 3 0 ≥ Only and appear in G . 14

  20. Flag Algebras First try for Mantel More automatic approach Applications Example - stability for Mantel = 1 Assume = 0 and 2 . Goal is G = . − 2 0 ≤ 1 − 2 3 0 ≤ − 2 3 0 ≥ Only and appear in G . 14

  21. Flag Algebras First try for Mantel More automatic approach Applications Example - stability for Mantel = 1 Assume = 0 and 2 . Goal is G = . − 2 0 ≤ 1 − 2 3 0 ≤ − 2 3 0 ≥ Only and appear in G . 14

  22. Flag Algebras First try for Mantel More automatic approach Applications Flag Algebras - formal approach • consider 2-edge-colored complete graphs G 1 , G 2 , . . . ( | G n | → ∞ ) 15

  23. Flag Algebras First try for Mantel More automatic approach Applications Flag Algebras - formal approach • consider 2-edge-colored complete graphs G 1 , G 2 , . . . ( | G n | → ∞ ) • p n ( F ) := probability that random | F | vertices of G n induces F 15

  24. Flag Algebras First try for Mantel More automatic approach Applications Flag Algebras - formal approach • consider 2-edge-colored complete graphs G 1 , G 2 , . . . ( | G n | → ∞ ) • p n ( F ) := probability that random | F | vertices of G n induces F • sequence ( G n ) is convergent if p n ( F ) converge for all F 15

  25. Flag Algebras First try for Mantel More automatic approach Applications Flag Algebras - formal approach • consider 2-edge-colored complete graphs G 1 , G 2 , . . . ( | G n | → ∞ ) • p n ( F ) := probability that random | F | vertices of G n induces F • sequence ( G n ) is convergent if p n ( F ) converge for all F • limit object – function q : all finite 2-edge-colored graphs → [0 , 1] 15

  26. Flag Algebras First try for Mantel More automatic approach Applications Flag Algebras - formal approach • consider 2-edge-colored complete graphs G 1 , G 2 , . . . ( | G n | → ∞ ) • p n ( F ) := probability that random | F | vertices of G n induces F • sequence ( G n ) is convergent if p n ( F ) converge for all F • limit object – function q : all finite 2-edge-colored graphs → [0 , 1] • q yields homomorphism from linear combinations of graphs to R 15

  27. Flag Algebras First try for Mantel More automatic approach Applications Flag Algebras - formal approach • consider 2-edge-colored complete graphs G 1 , G 2 , . . . ( | G n | → ∞ ) • p n ( F ) := probability that random | F | vertices of G n induces F • sequence ( G n ) is convergent if p n ( F ) converge for all F • limit object – function q : all finite 2-edge-colored graphs → [0 , 1] • q yields homomorphism from linear combinations of graphs to R • the set of limit objects LIM = homomorphisms q : q ( F ) ≥ 0 15

  28. Flag Algebras First try for Mantel More automatic approach Applications Flag Algebras - formal approach • consider 2-edge-colored complete graphs G 1 , G 2 , . . . ( | G n | → ∞ ) • p n ( F ) := probability that random | F | vertices of G n induces F • sequence ( G n ) is convergent if p n ( F ) converge for all F • limit object – function q : all finite 2-edge-colored graphs → [0 , 1] • q yields homomorphism from linear combinations of graphs to R • the set of limit objects LIM = homomorphisms q : q ( F ) ≥ 0 � � � � • we optimize on LIM T = q ∈ LIM : q = 0 1 � � 2 ≥ q ∈ LIM T q max 15

  29. Flag Algebras First try for Mantel More automatic approach Applications More automatic approach • How to use computer to guess the right equation for you. � 2 � 0 ≤ 1 − 2 v 16

  30. Flag Algebras First try for Mantel More automatic approach Applications Example - Mantel’s theorem, 2nd try Theorem (Mantel 1907) 4 n 2 edges. A triangle-free graph contains at most 1 Assume edges are red and non-edges are blue. 17

  31. Flag Algebras First try for Mantel More automatic approach Applications Example - Mantel’s theorem, 2nd try Theorem (Mantel 1907) 4 n 2 edges. A triangle-free graph contains at most 1 Assume edges are red and non-edges are blue. ≤ 1 Assume = 0. (We want to conclude 2 .) 17

  32. Flag Algebras First try for Mantel More automatic approach Applications Example - Mantel’s theorem, 2nd try Theorem (Mantel 1907) 4 n 2 edges. A triangle-free graph contains at most 1 Assume edges are red and non-edges are blue. ≤ 1 Assume = 0. (We want to conclude 2 .) + 1 + 2 = 0 3 3 17

  33. Flag Algebras First try for Mantel More automatic approach Applications Example - Mantel’s theorem, 2nd try Theorem (Mantel 1907) 4 n 2 edges. A triangle-free graph contains at most 1 Assume edges are red and non-edges are blue. ≤ 1 Assume = 0. (We want to conclude 2 .) + 1 + 2 = 0 3 3 � � ≤ 2 + + 3 17

  34. Flag Algebras First try for Mantel More automatic approach Applications Example - Mantel’s theorem, 2nd try Theorem (Mantel 1907) 4 n 2 edges. A triangle-free graph contains at most 1 Assume edges are red and non-edges are blue. ≤ 1 Assume = 0. (We want to conclude 2 .) + 1 + 2 = 0 3 3 � � ≤ 2 + + 3 1 = + + + 17

  35. Flag Algebras First try for Mantel More automatic approach Applications Example - Mantel’s theorem, 2nd try Theorem (Mantel 1907) 4 n 2 edges. A triangle-free graph contains at most 1 Assume edges are red and non-edges are blue. ≤ 1 Assume = 0. (We want to conclude 2 .) + 1 + 2 = 0 3 3 � � ≤ 2 + + 3 1 = + + 17

  36. Flag Algebras First try for Mantel More automatic approach Applications Example - Mantel’s theorem, 2nd try Theorem (Mantel 1907) 4 n 2 edges. A triangle-free graph contains at most 1 Assume edges are red and non-edges are blue. ≤ 1 Assume = 0. (We want to conclude 2 .) + 1 + 2 = 0 3 3 � � ≤ 2 + + 3 1 = + + ≤ 2 3 17

  37. Flag Algebras First try for Mantel More automatic approach Applications Example - Mantel’s theorem, 2nd try ≤ 1 Assume = 0. (We want to conclude 2 .) + 1 + 2 = 0 3 3 18

  38. Flag Algebras First try for Mantel More automatic approach Applications Example - Mantel’s theorem, 2nd try ≤ 1 Assume = 0. (We want to conclude 2 .) + 1 + 2 = 0 3 3 Idea: find c 1 , c 2 , c 3 ∈ R such that for every graph G 0 ≤ c 1 + c 2 + c 3 . 18

  39. Flag Algebras First try for Mantel More automatic approach Applications Example - Mantel’s theorem, 2nd try ≤ 1 Assume = 0. (We want to conclude 2 .) + 1 + 2 = 0 3 3 Idea: find c 1 , c 2 , c 3 ∈ R such that for every graph G 0 ≤ c 1 + c 2 + c 3 . After summing together � 1 � � 2 � ≤ c 1 + 3 + c 2 + 3 + c 3 and � � (0 + c 1 ) , 1 3 + c 2 , 2 ≤ max 3 + c 3 . 18

  40. Flag Algebras First try for Mantel More automatic approach Applications Example - Mantel’s theorem, 2nd try ≤ 1 Assume = 0. (We want to conclude 2 .) + 1 + 2 = 0 3 3 Idea: find c 1 , c 2 , c 3 ∈ R such that for every graph G 0 ≤ c 1 + c 2 + c 3 . After summing together � 1 � � 2 � ≤ c 1 + 3 + c 2 + 3 + c 3 and � � (0 + c 1 ) , 1 3 + c 2 , 2 ≤ max 3 + c 3 . c 3 < 0 18

  41. Flag Algebras First try for Mantel More automatic approach Applications Candidates for c 1 , c 2 , c 3 � a � c � 0 (matrix is positive semidefinite) c b 19

  42. Flag Algebras First try for Mantel More automatic approach Applications Candidates for c 1 , c 2 , c 3 � � a � T � � � c 0 ≤ , , c b v v v v � a � c � 0 (matrix is positive semidefinite) c b 19

  43. Flag Algebras First try for Mantel More automatic approach Applications Candidates for c 1 , c 2 , c 3 � � a � T � � � c 0 ≤ , , c b v v v v ? ? ? ? + 1 + 1 = a + b 2 c 2 c v v v v ? ? v = 1 v × v = v × 2 v v � a � c � 0 (matrix is positive semidefinite) c b 19

  44. Flag Algebras First try for Mantel More automatic approach Applications Candidates for c 1 , c 2 , c 3 � � a � T � � � c 0 ≤ , , c b v v v v ? ? ? = a + b + c v v v ? ? v = 1 v × v = v × 2 v v � a � c � 0 (matrix is positive semidefinite) c b 19

  45. Flag Algebras First try for Mantel More automatic approach Applications Candidates for c 1 , c 2 , c 3 � � a � T � � � c 0 ≤ , , c b v v v v ? ? ? = a + b + c v v v � a � c � 0 (matrix is positive semidefinite) c b 19

  46. Flag Algebras First try for Mantel More automatic approach Applications Candidates for c 1 , c 2 , c 3 � � a � T � � � 0 ≤ 1 c � , , n c b v v v v v ? ? ? = 1 � a + b + c n v v v v � a � c � 0 (matrix is positive semidefinite) c b 19

  47. Flag Algebras First try for Mantel More automatic approach Applications Candidates for c 1 , c 2 , c 3 � � a � T � � � 0 ≤ 1 c � , , n c b v v v v v ? ? ? = 1 � a + b + c n v v v v + a + 2 c + b + 2 c = a + b 3 3 1 1 1 = � = � 3 v ∈ V ( G ) v ∈ V ( G ) | V ( G ) | | V ( G ) | v v � a � c 2 1 = � � 0 (matrix is positive semidefinite) v ∈ V ( G ) 3 | V ( G ) | v c b 19

  48. Flag Algebras First try for Mantel More automatic approach Applications Candidates for c 1 , c 2 , c 3 � � a � T � � � 0 ≤ 1 c � , , n c b v v v v v ? ? ? = 1 � a + b + c n v v v v + a + 2 c + b + 2 c = a 3 3 1 1 1 = � = � 3 v ∈ V ( G ) v ∈ V ( G ) | V ( G ) | | V ( G ) | v v � a � c 2 1 = � � 0 (matrix is positive semidefinite) v ∈ V ( G ) 3 | V ( G ) | v c b 19

  49. Flag Algebras First try for Mantel More automatic approach Applications Candidates for c 1 , c 2 , c 3 � � a � T � � � 0 ≤ 1 c � , , n c b v v v v v ? ? ? = 1 � a + b + c n v v v v + a + 2 c + b + 2 c = a 3 3 c 1 = a , c 2 = a + 2 c , c 3 = b + 2 c 3 3 � a � c � 0 (matrix is positive semidefinite) c b 19

  50. Flag Algebras First try for Mantel More automatic approach Applications Using c 1 , c 2 , c 3 + 1 + 2 = 3 3 + a + 2 c + b + 2 c 0 ≤ a 3 3 � a � c � 0 (matrix is positive semidefinite) c b 20

  51. Flag Algebras First try for Mantel More automatic approach Applications Using c 1 , c 2 , c 3 + 1 + 2 = 3 3 + a + 2 c + b + 2 c 0 ≤ a 3 3 � a � c � 0 (matrix is positive semidefinite) c b 20

  52. Flag Algebras First try for Mantel More automatic approach Applications Using c 1 , c 2 , c 3 + 1 + 2 = 3 3 + a + 2 c + b + 2 c 0 ≤ a 3 3 � a , 1 + a + 2 c , 2 + b + 2 c � ≤ max . 3 3 � a � c � 0 (matrix is positive semidefinite) c b 20

  53. Flag Algebras First try for Mantel More automatic approach Applications Using c 1 , c 2 , c 3 + 1 + 2 = 3 3 + a + 2 c + b + 2 c 0 ≤ a 3 3 � a , 1 + a + 2 c , 2 + b + 2 c � ≤ max . 3 3 Try � a � � � c 1 / 2 − 1 / 2 = . c b − 1 / 2 1 / 2 20

  54. Flag Algebras First try for Mantel More automatic approach Applications Using c 1 , c 2 , c 3 + 1 + 2 = 3 3 + a + 2 c + b + 2 c 0 ≤ a 3 3 � a , 1 + a + 2 c , 2 + b + 2 c � ≤ max . 3 3 Try � a � � � c 1 / 2 − 1 / 2 = . c b − 1 / 2 1 / 2 It gives � 1 2 , 1 6 , 1 � = 1 ≤ max 2 . 2 20

  55. Flag Algebras First try for Mantel More automatic approach Applications Optimizing a , b , c � a , 1 + a + 2 c , 2 + b + 2 c � ≤ max 3 3  Minimize d     subject to a ≤ d     1+ a +2 c  ≤ d  3 ( SDP ) 2+ b +2 c ≤ d  3   � �  a c   � 0    c b  ( SDP ) can be solved on computers using CSDP or SDPA. Rounding may be needed for exact results. 21

  56. Flag Algebras First try for Mantel More automatic approach Applications Applications Recall is the probability that 3 randomly chosen vertices form a red triangle. 22

  57. Flag Algebras First try for Mantel More automatic approach Applications J. Balogh P. Hu Hypercubes and posets H. Liu B. L. Application to sparse structure. 23

  58. Flag Algebras First try for Mantel More automatic approach Applications Hypercube Q n is n -dimensional hypercube ( n -cube) Q 1 Q 2 Q 3 24

  59. Flag Algebras First try for Mantel More automatic approach Applications Hypercube Q n is n -dimensional hypercube ( n -cube) Q 1 Q 2 Q 3 Problem (Erd˝ os 1984) What is the maximum number of edges in a subgraph of Q n with no Q 2 ? 24

  60. Flag Algebras First try for Mantel More automatic approach Applications Hypercube Q n is n -dimensional hypercube ( n -cube) Q 1 Q 2 Q 3 Problem (Erd˝ os 1984) What is the maximum number of edges in a subgraph of Q n with no Q 2 ? maximize subject to = 0 24

  61. Flag Algebras First try for Mantel More automatic approach Applications Lower bound Conjecture (Erd˝ os 1984) In Q n where n → ∞ : ≤ 1 = 0 ⇒ 2 . 25

  62. Flag Algebras First try for Mantel More automatic approach Applications Lower bound Conjecture (Erd˝ os 1984) In Q n where n → ∞ : ≤ 1 = 0 ⇒ 2 . As posets Q 7 Q 7 By removing every second layer, ≥ 1 / 2. 25

  63. Flag Algebras First try for Mantel More automatic approach Applications Results about hypercubes If = 0 then Theorem (Chung 1992) ≤ 0 . 62284 . 26

  64. Flag Algebras First try for Mantel More automatic approach Applications Results about hypercubes If = 0 then Theorem (Chung 1992) ≤ 0 . 62284 . Theorem (Thomason and Wagner 2009) ≤ 0 . 62256 . 26

  65. Flag Algebras First try for Mantel More automatic approach Applications Results about hypercubes If = 0 then Theorem (Chung 1992) ≤ 0 . 62284 . Theorem (Thomason and Wagner 2009) ≤ 0 . 62256 . ≤ 0 . 62083 . 26

  66. Flag Algebras First try for Mantel More automatic approach Applications Results about hypercubes If = 0 then Theorem (Chung 1992) ≤ 0 . 62284 . Theorem (Thomason and Wagner 2009) ≤ 0 . 62256 . ≤ 0 . 62083 . Theorem (Balogh, Hu, L. , Liu 2014; Baber 2014+) ≤ 0 . 6068 . (Uses Q 3 instead of Q 2 .) 26

  67. Flag Algebras First try for Mantel More automatic approach Applications Related results - boolean lattice Let B n denote n -dimensional boolean lattice. Let F be a subposet of B n not containing ♦ . B 7 27

  68. Flag Algebras First try for Mantel More automatic approach Applications Related results - boolean lattice Let B n denote n -dimensional boolean lattice. Let F be a subposet of B n not containing ♦ . Theorem n � � | F | ≤ ( c + o (1)) , where ⌊ n / 2 ⌋ c ≤ 2 . 3 [Griggs, Lu 2009] c ≤ 2 . 284 [Axenovich, Manske, Martin 2012] c ≤ 2 . 273 [Griggs, Li, Lu 2011] c ≤ 2 . 25 [Kramer, Martin, Young 2013] B 7 27

  69. Flag Algebras First try for Mantel More automatic approach Applications Related results - boolean lattice Let B n denote n -dimensional boolean lattice. Let F be a subposet of B n not containing ♦ . Theorem n � � | F | ≤ ( c + o (1)) , where ⌊ n / 2 ⌋ c ≤ 2 . 3 [Griggs, Lu 2009] c ≤ 2 . 284 [Axenovich, Manske, Martin 2012] c ≤ 2 . 273 [Griggs, Li, Lu 2011] c ≤ 2 . 25 [Kramer, Martin, Young 2013] B 7 If F is a subposet of only the middle three layers of B n , then c ≤ 2 . 1547 [Manske, Shen 2013] c ≤ 2 . 15121 [Balogh, Hu, L., Liu 2014] 27

  70. Flag Algebras First try for Mantel More automatic approach Applications Related results - boolean lattice Let B n denote n -dimensional boolean lattice. Let F be a subposet of B n not containing ♦ . Theorem n � � | F | ≤ ( c + o (1)) , where ⌊ n / 2 ⌋ c ≤ 2 . 3 [Griggs, Lu 2009] c ≤ 2 . 284 [Axenovich, Manske, Martin 2012] c ≤ 2 . 273 [Griggs, Li, Lu 2011] c ≤ 2 . 25 [Kramer, Martin, Young 2013] B 7 If F is a subposet of only the middle three layers of B n , then c ≤ 2 . 1547 [Manske, Shen 2013] c ≤ 2 . 15121 [Balogh, Hu, L., Liu 2014] c = 2 [Kramer, Martin 2015, announced] 27

  71. Flag Algebras First try for Mantel More automatic approach Applications J. Balogh P. Hu B. L. Permutations O. Pikhurko B. Udvari J. Volec Application with exact result. 28

  72. Flag Algebras First try for Mantel More automatic approach Applications Permutations and extremal problems Problem What is the minimum number of monotone subsequences of size k in a permutation of [ n ] ? 29

  73. Flag Algebras First try for Mantel More automatic approach Applications Permutations and extremal problems Problem What is the minimum number of monotone subsequences of size k in a permutation of [ n ] ? (5,4,1),(5,4,2),(5,4,3) k = 3 (1,2,3) n = 5 (5,4,1,2,3) 29

  74. Flag Algebras First try for Mantel More automatic approach Applications Permutations and extremal problems Problem What is the minimum number of monotone subsequences of size k in a permutation of [ n ] ? (5,4,1),(5,4,2),(5,4,3) k = 3 (1,2,3) n = 5 (5,4,1,2,3) (1,2,3) (4,5,1,2,3) 29

  75. Flag Algebras First try for Mantel More automatic approach Applications Conjecture Conjecture (Myers 2002) The number of monotone subsequences of length k is minimized by a permutation on [ n ] with k − 1 increasing runs of as equal lengths as possible. k = 4 , n = 15 30

  76. Flag Algebras First try for Mantel More automatic approach Applications Extremal case is not unique 31

Recommend

Flag Algebras (hopefully simple basics) Bernard Lidick y University of Illinois at

Flag Algebras (hopefully simple basics) Bernard Lidick y University of Illinois at

Flag algebras First try for Mantel More automatic approach Additional constraints Flag Algebras (hopefully simple basics) Bernard Lidick y University of Illinois at Urbana-Champaign Apr 29, 2014 Flag algebras First try for Mantel More

1.25k views • 104 slides
Pride flags Sadie, Yael, &amp; Yossi Original Pride Flag Traditional Gay Pride Flag Philly

Pride flags Sadie, Yael, &amp; Yossi Original Pride Flag Traditional Gay Pride Flag Philly

Pride flags Sadie, Yael, &amp; Yossi Original Pride Flag Traditional Gay Pride Flag Philly Pride Flag Transgender Flag Progress Pride Flag Lesbian Flag Non-binary Flag Gender Fluidity flag Bisexual flag Pansexual Flag Asexual flag THANK

563 views • 13 slides
Quaternion Algebras Properties and Applications Rob Eimerl 1 Department of Mathematics

Quaternion Algebras Properties and Applications Rob Eimerl 1 Department of Mathematics

Quaternion Algebras Applications of Quaternion Algebras Quaternion Algebras Properties and Applications Rob Eimerl 1 Department of Mathematics University of Puget Sound May 5th, 2015 Quaternion Algebras Applications of Quaternion Algebras

563 views • 20 slides
Flag Basics Agenda A1. The Pledge of Allegiance A2. The National Anthem A3. Flag

Flag Basics Agenda A1. The Pledge of Allegiance A2. The National Anthem A3. Flag

California Cadet Corps Curriculum on The Flag Flag Basics Agenda A1. The Pledge of Allegiance A2. The National Anthem A3. Flag Smarts A4. Folding the Flag A5. History of the Flag A6. Raising the Flag A7.

1.78k views • 158 slides
Explicit realization of affine vertex algebras and their applications Dra zen Adamovi c

Explicit realization of affine vertex algebras and their applications Dra zen Adamovi c

Notre Dame 2015 Explicit realization of affine vertex algebras and their applications Dra zen Adamovi c University of Zagreb, Croatia Supported by CSF, grant. no. 2634 Conference on Lie algebras, vertex operator algebras and related

586 views • 39 slides
Tense MV-algebras and related functions Jan Paseka Michal Botur Department of Mathematics and

Tense MV-algebras and related functions Jan Paseka Michal Botur Department of Mathematics and

Introduction Basic notions and definitions Dyadic numbers and MV-terms Filters, ultrafilters and the term tr Semistates on MV-algebras Functions between MV-algebras and their construction The main theorem and its applications Tense MV-algebras

1.59k views • 121 slides
Flag Algebra Methods (more formal approach) Bernard Lidick y 6th Lake Michigan Workshop on

Flag Algebra Methods (more formal approach) Bernard Lidick y 6th Lake Michigan Workshop on

Flag Algebra Methods (more formal approach) Bernard Lidick y 6th Lake Michigan Workshop on Combinatorics and Graph Theory Apr 7, 2019 Outline Flag Algebras definitions First try for Mantels theorem More automatic

733 views • 48 slides
Cluster algebras and applications Bernhard Keller Universit Paris Diderot Paris 7 DMV

Cluster algebras and applications Bernhard Keller Universit Paris Diderot Paris 7 DMV

Preliminaries: The Dynkin diagrams Definitions: quiver mutation, cluster algebras Application in Lie theory, after B. Leclerc et al. Application to discrete dynamical systems: periodicity Cluster algebras and applications Bernhard Keller

797 views • 22 slides
Algebras birational to generic Sklyanin algebras Sue Sierra University of Edinburgh*

Algebras birational to generic Sklyanin algebras Sue Sierra University of Edinburgh*

Algebras birational to generic Sklyanin algebras Sue Sierra University of Edinburgh* Noncommutative and non-associative structures, braces and applications Malta 2018 Throughout, k is an algebraically closed field. Definition Let a , b , c

574 views • 20 slides
Flemish Aerospace Group (FLAG) Together for a stronger Flemish aviation industry FLAG Members

Flemish Aerospace Group (FLAG) Together for a stronger Flemish aviation industry FLAG Members

Flemish Aerospace Group (FLAG) Together for a stronger Flemish aviation industry FLAG Members FLAG Member Locations FLAG growth path FLAG ambition 2,5 2030: 2 Turnover + 100% 1,5 Employment + 50% 1 0,5 0 Turnover Employment 2016

498 views • 7 slides
A Factorization Algorithm for G -Algebras and Applications ACA 2016 Kassel Germany

A Factorization Algorithm for G -Algebras and Applications ACA 2016 Kassel Germany

A Factorization Algorithm for G -Algebras and Applications ACA 2016 Kassel Germany Albert Heinle Symbolic Computation Group David R. Cheriton School of Computer Science University of Waterloo Canada 20160804 1 / 30

802 views • 42 slides
Multiplicities of irreducible characters of table algebras and applications to association

Multiplicities of irreducible characters of table algebras and applications to association

Multiplicities of irreducible characters of table algebras and applications to association schemes Bangteng Xu Department of Mathematics and Statistics Eastern Kentucky University June 2 5, 2014 Modern Trends in Algebraic Graph Theory

738 views • 27 slides
Results on potential algebras: contraction algebras and Sklyanin algebras N.K.Iyudu Malta, March

Results on potential algebras: contraction algebras and Sklyanin algebras N.K.Iyudu Malta, March

Results on potential algebras: contraction algebras and Sklyanin algebras N.K.Iyudu Malta, March 2018 1 We consider finiteness conditions and ques- tions of growth of noncommutative algebras, known as A con s. They appear in M.Wemyss work on

214 views • 17 slides
Applications of non-associative Hopf algebras to Loop theory Jos e M. P erez-Izquierdo

Applications of non-associative Hopf algebras to Loop theory Jos e M. P erez-Izquierdo

Applications of non-associative Hopf algebras to Loop theory Jos e M. P erez-Izquierdo Dpto. Matem aticas y Computaci on Universidad de La Rioja, Spain Loops 2019 Conference Budapest University of Technology and Economics, Hungary

856 views • 64 slides
The Flag Known as Big Red All of the evidence presented in this report is sourced, proven fact.

The Flag Known as Big Red All of the evidence presented in this report is sourced, proven fact.

The Flag Known as Big Red All of the evidence presented in this report is sourced, proven fact. The issues: 1. Is the Big Red flag in question the same flag that posted on Morris Island on 9 Jan 1861? 2. What is the link between Cpl Bakers

515 views • 22 slides
Wreath Lie Algebras Cristina Di Pietro Cristina Di Pietro 1 Lie Algebras, their

Wreath Lie Algebras Cristina Di Pietro Cristina Di Pietro 1 Lie Algebras, their

Lie Algebras, their Classification and Applications, University of Trento- July 2005 Lie Algebras, their Classification and Applications University of Trento 25-27 July 2005 Wreath Lie Algebras Cristina Di Pietro Cristina Di Pietro 1

436 views • 22 slides
Units of Group Algebras, their Subgroups and Applications to Coding Theory Leo Creedon Institute

Units of Group Algebras, their Subgroups and Applications to Coding Theory Leo Creedon Institute

Units of Group Algebras, their Subgroups and Applications to Coding Theory Leo Creedon Institute of Technology Sligo Ireland Joint work with Fergal Gallagher and Ian McLoughlin Groups St. Andrews Birmingham August 6, 2017 1 / 1 Conference

606 views • 60 slides
Chiral asymmetry in the weak interaction Some applications of Clifford Algebras to the Standard

Chiral asymmetry in the weak interaction Some applications of Clifford Algebras to the Standard

Chiral asymmetry in the weak interaction Some applications of Clifford Algebras to the Standard Model Cristi Stoica September, 2019 Department of Theoretical Physics, National Institute of Physics and Nuclear Engineering Horia Hulubei,

982 views • 86 slides
Separable Automorphisms on Matrix Algebras over Finite Field Extensions. Applications to Ideal

Separable Automorphisms on Matrix Algebras over Finite Field Extensions. Applications to Ideal

Separable Automorphisms on Matrix Algebras over Finite Field Extensions. Applications to Ideal Codes. 1 J. Gmez-Torrecillas , F. J. Lobillo and G. Navarro Department of Algebra and CITIC, University of Granada Dep. of Computer

350 views • 22 slides
Memory Model COS 597C 10/5/2010 Example a = Flag = 0 Thread a = 26; Flag = 1; 2

Memory Model COS 597C 10/5/2010 Example a = Flag = 0 Thread a = 26; Flag = 1; 2

Memory Model COS 597C 10/5/2010 Example a = Flag = 0 Thread a = 26; Flag = 1; 2 Memory Model COS 597C, Fall 2010 Example a = Flag = 0 Thread a = 26; Flag = 1; Compiler Transformation Flag = 1; a =

1.09k views • 69 slides
FLAG-ERA Presentation FLAG-ERA JTC 2017 Project Kick-off Seminar March 21-22, 2018 Edouard

FLAG-ERA Presentation FLAG-ERA JTC 2017 Project Kick-off Seminar March 21-22, 2018 Edouard

Seminar Introduction FLAG-ERA Presentation FLAG-ERA JTC 2017 Project Kick-off Seminar March 21-22, 2018 Edouard Geoffrois, FLAG-ERA Coordinator French National Research Agency (ANR) edouard.geoffrois@agencerecherche.fr Supported by the

327 views • 22 slides
WEL ELS S Fl Flag Pre g Presen sentati tation Introduction to Flag Presentation The face of

WEL ELS S Fl Flag Pre g Presen sentati tation Introduction to Flag Presentation The face of

WEL ELS S Fl Flag Pre g Presen sentati tation Introduction to Flag Presentation The face of missions is changing, and the LWMS would like to reflect some of those changes in our presentation of flags . As women who have watched our sons and

245 views • 8 slides
Matrix Product Operators: Algebras and Applications Frank Verstraete Ghent University and

Matrix Product Operators: Algebras and Applications Frank Verstraete Ghent University and

Matrix Product Operators: Algebras and Applications Frank Verstraete Ghent University and University of Vienna Nick Bultinck, Jutho Haegeman, Michael Marien Burak Sahinoglu, Dominic Williamson Ignacio Cirac, David Perez Garcia, Norbert Schuch

733 views • 37 slides
Presenting the Flag Agenda B1. Color Guard B2. Taps B3. The California Flag

Presenting the Flag Agenda B1. Color Guard B2. Taps B3. The California Flag

California Cadet Corps Curriculum on The Flag Presenting the Flag Agenda B1. Color Guard B2. Taps B3. The California Flag COLOR GUARD B1a. Serve as a rifleman on a color guard B1b. Serve as a flag bearer on a color guard

391 views • 38 slides

More recommend