︠b1c69134-d8e3-4f12-b765-f2a5cb776a94︠ () ︠0e0973fe-c3a8-46f3-ae6c-a17d5f601b81︠ print(range(10)) ︡69d4b7de-fae3-427c-85e8-8fca1b2142b1︡{"stdout":"[0, 1, 2, 3, 4, 5, 6, 7, 8, 9]\n"}︡{"done":true}︡ ︠c09f927c-108e-45bd-a930-33cf6b2c3ba6︠ binomial(34,10) ︡caff7419-dd92-466d-864f-ea9076c045de︡{"stdout":"131128140\n"}︡{"done":true} ︠8b930f07-eefe-4da4-a3e7-ffe45a1d2801︠ binomial(40,var('a')) ︡21979818-c46b-451c-af36-39d4a1c19837︡{"stdout":"binomial(40, a)\n"}︡{"done":true}︡ ︠2952af00-bb9d-4c0b-b695-06422e0480ec︠ G = graphs.PetersenGraph() show(G) ︡e5d1d4dc-9b35-4ebc-a096-8ef0fe7885a4︡{"d3":{"data":{"charge":0,"directed":false,"edge_labels":false,"edge_thickness":2,"gravity":0,"height":null,"link_distance":50,"link_strength":0,"links":[{"color":"#aaa","curve":0,"name":"","source":0,"strength":0,"target":1},{"color":"#aaa","curve":0,"name":"","source":0,"strength":0,"target":4},{"color":"#aaa","curve":0,"name":"","source":0,"strength":0,"target":5},{"color":"#aaa","curve":0,"name":"","source":1,"strength":0,"target":2},{"color":"#aaa","curve":0,"name":"","source":1,"strength":0,"target":6},{"color":"#aaa","curve":0,"name":"","source":2,"strength":0,"target":3},{"color":"#aaa","curve":0,"name":"","source":2,"strength":0,"target":7},{"color":"#aaa","curve":0,"name":"","source":3,"strength":0,"target":4},{"color":"#aaa","curve":0,"name":"","source":3,"strength":0,"target":8},{"color":"#aaa","curve":0,"name":"","source":4,"strength":0,"target":9},{"color":"#aaa","curve":0,"name":"","source":5,"strength":0,"target":7},{"color":"#aaa","curve":0,"name":"","source":5,"strength":0,"target":8},{"color":"#aaa","curve":0,"name":"","source":6,"strength":0,"target":8},{"color":"#aaa","curve":0,"name":"","source":6,"strength":0,"target":9},{"color":"#aaa","curve":0,"name":"","source":7,"strength":0,"target":9}],"loops":[],"nodes":[{"group":"0","name":"0"},{"group":"0","name":"1"},{"group":"0","name":"2"},{"group":"0","name":"3"},{"group":"0","name":"4"},{"group":"0","name":"5"},{"group":"0","name":"6"},{"group":"0","name":"7"},{"group":"0","name":"8"},{"group":"0","name":"9"}],"pos":[[6.123233995736766e-17,-1],[-0.9510565162951535,-0.3090169943749475],[-0.5877852522924732,0.8090169943749473],[0.5877852522924729,0.8090169943749476],[0.9510565162951536,-0.3090169943749472],[1.5308084989341916e-16,-0.5],[-0.4755282581475767,-0.1545084971874739],[-0.29389262614623674,0.4045084971874736],[0.2938926261462364,0.40450849718747384],[0.4755282581475769,-0.15450849718747348]],"vertex_labels":true,"vertex_size":7,"width":null},"viewer":"graph"}}︡{"html":"
"}︡{"done":true}︡ ︠89212c9b-a2bc-46e7-b094-7e3563113aa8︠ G.chromatic_polynomial() ︡b8fa41ee-32a6-4a58-ac22-68ece9f30d8c︡{"stdout":"x^10 - 15*x^9 + 105*x^8 - 455*x^7 + 1353*x^6 - 2861*x^5 + 4275*x^4 - 4305*x^3 + 2606*x^2 - 704*x\n"}︡{"done":true}︡ ︠fe6e9bde-46ac-4a71-9d91-6ff50031c62f︠ version() ︡551d8e5b-43df-4a37-b09b-e3f731c07958︡{"stdout":"'SageMath version 7.3, Release Date: 2016-08-04'\n"}︡{"done":true}︡ ︠8b515572-e9f6-4726-a89b-b294aad4edd3︠ %time for t in range(10000): M1 = Matroid(MatrixSpace(GF(2),8,30).random_element()) M2 = Matroid(MatrixSpace(GF(2),8,30).random_element()) X = M1.is_isomorphic(M2) ︡00d2b9fd-2209-47ce-96b1-e06f6bbc7f1e︡{"stdout":"CPU time: 3.48 s, Wall time: 5.16 s"}︡{"stdout":"\n"}︡{"done":true}︡ ︠95a5444f-d12b-4e43-91b3-53289af35732︠ 21100*5 ︡77d5161f-cc9c-477a-a7dd-8bb54e37d827︡{"stdout":"105500\n"}︡{"done":true}︡ ︠777fe290-0e09-46a3-a2f5-9b545a36295a︠ _/3600 ︡dfdd2372-9eb2-4d96-ac8d-93b2fdeb9602︡{"stdout":"1055/36\n"}︡{"done":true} ︠0947cc1e-80de-4b66-9193-1946a6512245︠ N(_) ︡853494c0-49d4-4990-8d54-6a9b2190e53a︡{"stdout":"29.3055555555556\n"}︡{"done":true} ︠5c1634c3-2bbb-46bf-be3b-06601d5b68d8︠ M1 = Matroid(field=GF(2), reduced_matrix=[[1,1,0,0],[1,0,1,0],[1,0,0,1],[0,1,1,0],[0,1,0,1],[0,0,1,1],[1,1,1,1]]).dual() M1.representation() ︡1681e7f1-c6ec-4f69-8f09-227663381624︡{"stdout":"[1 0 0 0 1 1 1 0 0 0 1]\n[0 1 0 0 1 0 0 1 1 0 1]\n[0 0 1 0 0 1 0 1 0 1 1]\n[0 0 0 1 0 0 1 0 1 1 1]\n"}︡{"done":true} ︠770752b4-1151-4582-bcbd-27310bfa4fae︠ list(Subsets([1,2,3],2)) ︡f2eef21f-a746-45a3-aeed-688554e5a92e︡{"stdout":"[{1, 2}, {1, 3}, {2, 3}]\n"}︡{"done":true} ︠a73f16a3-f602-4930-96b1-b747dd32fc28︠ M2 = Matroid(field=GF(2), reduced_matrix=[[]]) def universal_template_matroid(k, n): """ Generate the universal matroid conforming to Phi_k, with a n-vertex complete graph, for a total rank of n-1+k. """ num_elts = n * (n-1) / 2 + k + (k+1)*(n-1) A = Matrix(GF(2), n-1+k, num_elts) i = 0 # identity in front for j in range(n-1): A[j+k,j] = 1 i = n-1 # all pairs for S in Subsets(range(k,n+k-1),2): for j in S: A[j,i] = 1 i = i + 1 # Top-identity from template for j in range(k): A[j,i] = 1 i = i + 1 # rest of template thing for p in range(k): for j in range(n-1): A[p, i] = 1 A[k+j,i] = 1 i = i + 1 for j in range(n-1): for p in range(k): A[p,i] = 1 A[k+j,i] = 1 i = i + 1 return Matroid(field=GF(2), matrix=A) ︡7d0b28bc-f179-47b5-94b0-88d361e576ed︡ ︠8e1f6693-a445-48e9-9581-e8f7d76f8fcf︠ universal_template_matroid(2,5) ︡00eadc02-d138-4495-8c4b-d10283fce15a︡{"stdout":"Binary matroid of rank 6 on 24 elements, type (3, 1)\n"}︡ ︠7f15949c-7775-48c2-9d2f-51810eb18555︠ # Question: is the (2,5) matroid contained in the 4-template? M1 = universal_template_matroid(2,5) M2 = universal_template_matroid(4,4) for e in M2.groundset(): if (M2 \ e).is_isomorphic(M1): print e break print "Nope" ︡7e7320ec-35d9-4880-b8f9-b7c2c0f36da1︡{"stdout":"Nope\nNope\nNope\nNope\nNope\nNope\nNope\nNope\nNope\nNope\nNope\nNope\nNope\nNope\nNope\nNope\nNope\nNope\nNope\nNope\nNope\nNope\nNope\nNope\nNope\n"}︡ ︠118c5014-16e9-4bde-b2ff-f2b4aabb2bc4︠ M2 = universal_template_matroid(4,5) M2.size() ︡174ce876-670a-43d0-b580-e7ef050cd90d︡{"stdout":"34\n"}︡ ︠55fd5320-807d-464a-8faa-79682e75650a︠ 131000000/200 ︡734d2441-3e6c-44ff-9088-a5e8ca0bc21a︡{"stdout":"655000\n"}︡ ︠8efd2e8f-7ec7-43ec-9985-1f599c7f83b0︠ i = 0 for S in Subsets(range(34),10): if i % 700000 == 0: print i i = i + 1 if (M2 \ S).is_isomorphic(M1): print S break ︡7210f197-9912-41d3-be8e-90dbc6d77c24︡{"stdout":"0\n700000"}︡{"stdout":"\n1400000"}︡{"stdout":"\n2100000"}︡{"stdout":"\n2800000"}︡{"stdout":"\n3500000"}︡{"stdout":"\n4200000"}︡{"stdout":"\n4900000"}︡{"stdout":"\n5600000"}︡{"stdout":"\n6300000"}︡{"stdout":"\n7000000"}︡{"stdout":"\n7700000"}︡{"stdout":"\n8400000"}︡{"stdout":"\n9100000"}︡{"stdout":"\n9800000"}︡{"stdout":"\n10500000"}︡{"stdout":"\n11200000"}︡{"stdout":"\n11900000"}︡{"stdout":"\n12600000"}︡{"stdout":"\n13300000"}︡{"stdout":"\n14000000"}︡{"stdout":"\n14700000"}︡{"stdout":"\n15400000"}︡{"stdout":"\n16100000"}︡{"stdout":"\n16800000"}︡{"stdout":"\n17500000"}︡{"stdout":"\n18200000"}︡{"stdout":"\n18900000"}︡{"stdout":"\n19600000"}︡{"stdout":"\n20300000"}︡{"stdout":"\n21000000"}︡{"stdout":"\n21700000"}︡{"stdout":"\n22400000"}︡{"stdout":"\n23100000"}︡{"stdout":"\n23800000"}︡{"stdout":"\n24500000"}︡{"stdout":"\n25200000"}︡{"stdout":"\n25900000"}︡{"stdout":"\n26600000"}︡{"stdout":"\n27300000"}︡{"stdout":"\n28000000"}︡{"stdout":"\n28700000"}︡{"stdout":"\n29400000"}︡{"stdout":"\n30100000"}︡{"stdout":"\n30800000"}︡{"stdout":"\n31500000"}︡{"stdout":"\n32200000"}︡{"stdout":"\n32900000"}︡{"stdout":"\n33600000"}︡{"stdout":"\n34300000"}︡{"stdout":"\n35000000"}︡{"stdout":"\n35700000"}︡{"stdout":"\n36400000"}︡{"stdout":"\n37100000"}︡{"stdout":"\n37800000"}︡{"stdout":"\n38500000"}︡{"stdout":"\n39200000"}︡{"stdout":"\n39900000"}︡{"stdout":"\n40600000"}︡{"stdout":"\n41300000"}︡{"stdout":"\n42000000"}︡{"stdout":"\n42700000"}︡{"stdout":"\n43400000"}︡{"stdout":"\n44100000"}︡{"stdout":"\n44800000"}︡{"stdout":"\n45500000"}︡{"stdout":"\n46200000"}︡{"stdout":"\n46900000"}︡{"stdout":"\n47600000"}︡{"stdout":"\n48300000"}︡{"stdout":"\n49000000"}︡{"stdout":"\n49700000"}︡{"stdout":"\n50400000"}︡{"stdout":"\n51100000"}︡{"stdout":"\n51800000"}︡{"stdout":"\n52500000"}︡{"stdout":"\n53200000"}︡{"stdout":"\n53900000"}︡{"stdout":"\n54600000"}︡{"stdout":"\n55300000"}︡{"stdout":"\n56000000"}︡{"stdout":"\n56700000"}︡{"stdout":"\n57400000"}︡{"stdout":"\n58100000"}︡{"stdout":"\n58800000"}︡{"stdout":"\n59500000"}︡{"stdout":"\n60200000"}︡{"stdout":"\n60900000"}︡{"stdout":"\n61600000"}︡{"stdout":"\n62300000"}︡{"stdout":"\n63000000"}︡{"stdout":"\n63700000"}︡{"stdout":"\n64400000"}︡{"stdout":"\n65100000"}︡{"stdout":"\n65800000"}︡{"stdout":"\n66500000"}︡{"stdout":"\n67200000"}︡{"stdout":"\n67900000"}︡{"stdout":"\n68600000"}︡{"stdout":"\n69300000"}︡{"stdout":"\n70000000"}︡{"stdout":"\n70700000"}︡{"stdout":"\n71400000"}︡{"stdout":"\n72100000"}︡{"stdout":"\n72800000"}︡{"stdout":"\n73500000"}︡{"stdout":"\n74200000"}︡{"stdout":"\n74900000"}︡{"stdout":"\n75600000"}︡{"stdout":"\n76300000"}︡{"stdout":"\n77000000"}︡{"stdout":"\n77700000"}︡{"stdout":"\n78400000"}︡{"stdout":"\n79100000"}︡{"stdout":"\n79800000"}︡{"stdout":"\n80500000"}︡{"stdout":"\n81200000"}︡{"stdout":"\n81900000"}︡{"stdout":"\n82600000"}︡ ︠81c007b6-7cab-4c41-80cb-1993e4c83fdbi︠ %md This is *very important* text, ok? $\frac{1}{\pi}$ ︡aa908959-bfc7-453d-ae30-1aa7ba2c2aaf︡{"done":true,"md":"This is *very important* text, ok? $\\frac{1}{\\pi}$"} ︠fdc3e4c9-69c3-49b9-b3dc-5b2742e06e4bi︠ %md # Heading 1 ## Subsection etc.... ︡92c40169-f5dd-43e0-a044-e425296fa4bb︡{"done":true,"md":"# Heading 1\n\n## Subsection\n\netc...."} ︠1260d7bb-c0ec-4266-9bcb-0efed91b5768︠ G = graphs.RandomGNP(6,0.6) G.show() ︡9bebd419-c63a-49f5-aad3-31bc029b4131︡{"file":{"filename":"/projects/45f4aab5-7698-4ac8-9f63-9fd307401ad7/smc/src/data/projects/4db1aa11-edb4-4f73-9a51-7bbb431f435d/.sage/temp/compute5-us/26332/tmp_c_wf7E.svg","show":true,"text":null,"uuid":"4a75818c-347f-466d-8087-0d1562f232fd"},"once":false}︡{"html":""}︡{"done":true}︡ ︠0ab4752c-8d59-4320-a725-0462d4e105d7︠ %md ** Lemma A.1 ** _Let $A$ be the following matrix: \[ \begin{bmatrix} 1 & 1 & 0\\ 1 & 0 & 1\\ 0 & 1 & 1 \end{bmatrix}. \] Then $A$ is pretty._ ︡5e7168f4-81ac-4494-8128-53cad3c60cd0︡{"done":true,"md":"** Lemma A.1 ** _Let $A$ be the following matrix:\n\\[\n\\begin{bmatrix}\n1 & 1 & 0\\\\\n1 & 0 & 1\\\\\n0 & 1 & 1\n\\end{bmatrix}.\n\\]\nThen $A$ is pretty._"} ︠b5faee74-fdd3-4c12-b2d0-f89522a9a230i︠ ︡486eaa92-ea18-4a91-b4c2-ddbe309056d0︡ ︠79242f47-c60d-4c1c-acc0-8f6b41d6e757︠ show(integrate(sin(x^pi))) ︡b80e5b69-ea9a-40ae-be12-4b19d362bd99︡{"html":"