diff --git a/SL(3,Z) - SDP for Property (T).ipynb b/SL(3,Z) - SDP for Property (T).ipynb new file mode 100644 index 0000000..5f795a7 --- /dev/null +++ b/SL(3,Z) - SDP for Property (T).ipynb @@ -0,0 +1,32994 @@ +{ + "cells": [ + { + "cell_type": "code", + "execution_count": 1, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "data": { + "text/plain": [ + "3-element Array{String,1}:\n", + " \"/opt/julia-3c9d75391c/local/share/julia/site/v0.5\"\n", + " \"/opt/julia-3c9d75391c/share/julia/site/v0.5\" \n", + " \"./\" " + ] + }, + "execution_count": 1, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "push!(LOAD_PATH, \"./\");\n", + "LOAD_PATH" + ] + }, + { + "cell_type": "code", + "execution_count": 2, + "metadata": { + "ExecuteTime": { + "end_time": "2016-12-10T02:58:13.544392", + "start_time": "2016-12-10T01:58:13.443Z" + }, + "collapsed": false + }, + "outputs": [ + { + "name": "stderr", + "output_type": "stream", + "text": [ + "WARNING: Method definition redirect_stdout(Function, Any) in module Compat at /home/kalmar/.julia/v0.5/Compat/src/Compat.jl:1600 overwritten in module Compat at /home/kalmar/.julia/v0.5/Compat/src/Compat.jl:1600.\n", + "WARNING: Method definition isnull(Any) in module Compat at /home/kalmar/.julia/v0.5/Compat/src/Compat.jl:1678 overwritten in module Compat at /home/kalmar/.julia/v0.5/Compat/src/Compat.jl:1678.\n", + "WARNING: Method definition take!(Main.Base.AbstractIOBuffer) in module Compat at /home/kalmar/.julia/v0.5/Compat/src/Compat.jl:1698 overwritten in module Compat at /home/kalmar/.julia/v0.5/Compat/src/Compat.jl:1698.\n", + "WARNING: Method definition redirect_stderr(Function, Any) in module Compat at /home/kalmar/.julia/v0.5/Compat/src/Compat.jl:1600 overwritten in module Compat at /home/kalmar/.julia/v0.5/Compat/src/Compat.jl:1600.\n", + "WARNING: Method definition redirect_stdin(Function, Any) in module Compat at /home/kalmar/.julia/v0.5/Compat/src/Compat.jl:1600 overwritten in module Compat at /home/kalmar/.julia/v0.5/Compat/src/Compat.jl:1600.\n" + ] + } + ], + "source": [ + "workspace()\n", + "using GroupAlgebras\n", + "include(\"property(T).jl\");" + ] + }, + { + "cell_type": "code", + "execution_count": 3, + "metadata": { + "ExecuteTime": { + "end_time": "2016-12-10T02:58:13.546531", + "start_time": "2016-12-10T01:58:13.450Z" + }, + "collapsed": false + }, + "outputs": [], + "source": [ + "using JuMP\n", + "import SCS: SCSSolver\n", + "using LatexPrint" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Basis of $\\mathbb{R}[SL(3,\\mathbb{Z})]$\n", + "All elements of the group algebra $\\mathbb{R}[SL(3,\\mathbb{Z})]$ will be written as coefficients vector in the following (partial) basis of the group ring (in total 121 elements)" + ] + }, + { + "cell_type": "code", + "execution_count": 4, + "metadata": { + "ExecuteTime": { + "end_time": "2016-12-10T02:58:13.549348", + "start_time": "2016-12-10T01:58:13.457Z" + }, + "collapsed": false + }, + "outputs": [], + "source": [ + "GROUP = \"SL(3,Z)\";" + ] + }, + { + "cell_type": "code", + "execution_count": 5, + "metadata": { + "ExecuteTime": { + "end_time": "2016-12-10T02:58:13.620087", + "start_time": "2016-12-10T01:58:13.464Z" + }, + "collapsed": false + }, + "outputs": [], + "source": [ + "basis = read_GAP_raw_list(\"./basis.\"*GROUP);\n", + "convert_to_matrix(x) = hcat(x...);\n", + "basis = map(convert_to_matrix, basis);\n", + "\n", + "# lap(basis...);" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$\\left[\n", + "\\begin{array}{ccc}\n", + "1 & 0 & 0 \\\\\n", + "0 & 1 & 0 \\\\\n", + "0 & 0 & 1 \\\\\n", + "\\end{array}\n", + "\\right]\n", + ",\\left[\n", + "\\begin{array}{ccc}\n", + "1 & 0 & 0 \\\\\n", + "1 & 1 & 0 \\\\\n", + "0 & 0 & 1 \\\\\n", + "\\end{array}\n", + "\\right]\n", + ",\\left[\n", + "\\begin{array}{ccc}\n", + "1 & 0 & 0 \\\\\n", + "0 & 1 & 0 \\\\\n", + "1 & 0 & 1 \\\\\n", + "\\end{array}\n", + "\\right]\n", + ",\\left[\n", + 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\\\\\n", + "\\end{array}\n", + "\\right]\n", + ",\\left[\n", + "\\begin{array}{ccc}\n", + "1 & -1 & 0 \\\\\n", + "0 & 1 & 0 \\\\\n", + "1 & -1 & 1 \\\\\n", + "\\end{array}\n", + "\\right]\n", + ",\\left[\n", + "\\begin{array}{ccc}\n", + "1 & -1 & 0 \\\\\n", + "0 & 1 & 1 \\\\\n", + "0 & 0 & 1 \\\\\n", + "\\end{array}\n", + "\\right]\n", + ",\\left[\n", + "\\begin{array}{ccc}\n", + "1 & -1 & 0 \\\\\n", + "-1 & 2 & 0 \\\\\n", + "0 & 0 & 1 \\\\\n", + "\\end{array}\n", + "\\right]\n", + ",\\left[\n", + "\\begin{array}{ccc}\n", + "1 & -1 & 0 \\\\\n", + "0 & 1 & 0 \\\\\n", + "-1 & 1 & 1 \\\\\n", + "\\end{array}\n", + "\\right]\n", + ",\\left[\n", + "\\begin{array}{ccc}\n", + "1 & -2 & 0 \\\\\n", + "0 & 1 & 0 \\\\\n", + "0 & 0 & 1 \\\\\n", + "\\end{array}\n", + "\\right]\n", + ",\\left[\n", + "\\begin{array}{ccc}\n", + "1 & -1 & -1 \\\\\n", + "0 & 1 & 0 \\\\\n", + "0 & 0 & 1 \\\\\n", + "\\end{array}\n", + "\\right]\n", + ",\\left[\n", + "\\begin{array}{ccc}\n", + "1 & -1 & 0 \\\\\n", + "0 & 1 & -1 \\\\\n", + "0 & 0 & 1 \\\\\n", + "\\end{array}\n", + "\\right]\n", + ",\\left[\n", + "\\begin{array}{ccc}\n", + "1 & 0 & -1 \\\\\n", + "1 & 1 & -1 \\\\\n", + "0 & 0 & 1 \\\\\n", + "\\end{array}\n", + "\\right]\n", + ",\\left[\n", + "\\begin{array}{ccc}\n", + "1 & 0 & -1 \\\\\n", + "0 & 1 & 0 \\\\\n", + "1 & 0 & 0 \\\\\n", + "\\end{array}\n", + "\\right]\n", + ",\\left[\n", + "\\begin{array}{ccc}\n", + "1 & 0 & -1 \\\\\n", + "0 & 1 & 0 \\\\\n", + "0 & 1 & 1 \\\\\n", + "\\end{array}\n", + "\\right]\n", + ",\\left[\n", + "\\begin{array}{ccc}\n", + "1 & 0 & -1 \\\\\n", + "-1 & 1 & 1 \\\\\n", + "0 & 0 & 1 \\\\\n", + "\\end{array}\n", + "\\right]\n", + ",\\left[\n", + "\\begin{array}{ccc}\n", + "1 & 0 & -1 \\\\\n", + "0 & 1 & 0 \\\\\n", + "-1 & 0 & 2 \\\\\n", + "\\end{array}\n", + "\\right]\n", + ",\\left[\n", + "\\begin{array}{ccc}\n", + "1 & 0 & -1 \\\\\n", + "0 & 1 & 0 \\\\\n", + "0 & -1 & 1 \\\\\n", + "\\end{array}\n", + "\\right]\n", + ",\\left[\n", + "\\begin{array}{ccc}\n", + "1 & 0 & -2 \\\\\n", + "0 & 1 & 0 \\\\\n", + "0 & 0 & 1 \\\\\n", + "\\end{array}\n", + "\\right]\n", + ",\\left[\n", + "\\begin{array}{ccc}\n", + "1 & 0 & -1 \\\\\n", + "0 & 1 & -1 \\\\\n", + "0 & 0 & 1 \\\\\n", + "\\end{array}\n", + "\\right]\n", + ",\\left[\n", + "\\begin{array}{ccc}\n", + "1 & 0 & 0 \\\\\n", + "0 & 1 & -1 \\\\\n", + "1 & 0 & 1 \\\\\n", + "\\end{array}\n", + "\\right]\n", + ",\\left[\n", + "\\begin{array}{ccc}\n", + "1 & 0 & 0 \\\\\n", + "0 & 1 & -1 \\\\\n", + "0 & 1 & 0 \\\\\n", + "\\end{array}\n", + "\\right]\n", + ",\\left[\n", + "\\begin{array}{ccc}\n", + "1 & 1 & -1 \\\\\n", + "0 & 1 & -1 \\\\\n", + "0 & 0 & 1 \\\\\n", + "\\end{array}\n", + "\\right]\n", + ",\\left[\n", + "\\begin{array}{ccc}\n", + "1 & 0 & 0 \\\\\n", + "0 & 1 & -1 \\\\\n", + "-1 & 0 & 1 \\\\\n", + "\\end{array}\n", + "\\right]\n", + ",\\left[\n", + "\\begin{array}{ccc}\n", + "1 & 0 & 0 \\\\\n", + "0 & 1 & -1 \\\\\n", + "0 & -1 & 2 \\\\\n", + "\\end{array}\n", + "\\right]\n", + ",\\left[\n", + "\\begin{array}{ccc}\n", + "1 & -1 & 1 \\\\\n", + "0 & 1 & -1 \\\\\n", + "0 & 0 & 1 \\\\\n", + "\\end{array}\n", + "\\right]\n", + ",\\left[\n", + "\\begin{array}{ccc}\n", + "1 & 0 & 0 \\\\\n", + "0 & 1 & -2 \\\\\n", + "0 & 0 & 1 \\\\\n", + "\\end{array}\n", + "\\right]$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# The generating set\n", + "We set the generating set $S$ of $SL(3,\\mathbb{Z})$ to consist of the following $12$ elements: \n", + "* 3 elementary matrices\n", + "* their transpositions\n", + "* inverse elements to the above 6\n", + "\n", + "The basis above consists of all words of length less than or equal to $2$ in $S$ (i.e. $S$ and all double products of elements of $S$, excluding repetitions)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$\\left[\\left(\\begin{array}{rrr}\n", + "1 & 1 & 0 \\\\\n", + "0 & 1 & 0 \\\\\n", + "0 & 0 & 1\n", + "\\end{array}\\right), \\left(\\begin{array}{rrr}\n", + "1 & 0 & 1 \\\\\n", + "0 & 1 & 0 \\\\\n", + "0 & 0 & 1\n", + "\\end{array}\\right), \\left(\\begin{array}{rrr}\n", + "1 & 0 & 0 \\\\\n", + "0 & 1 & 1 \\\\\n", + "0 & 0 & 1\n", + "\\end{array}\\right), \\left(\\begin{array}{rrr}\n", + "1 & 0 & 0 \\\\\n", + "1 & 1 & 0 \\\\\n", + "0 & 0 & 1\n", + "\\end{array}\\right), \\left(\\begin{array}{rrr}\n", + "1 & 0 & 0 \\\\\n", + "0 & 1 & 0 \\\\\n", + "1 & 0 & 1\n", + "\\end{array}\\right), \\left(\\begin{array}{rrr}\n", + "1 & 0 & 0 \\\\\n", + "0 & 1 & 0 \\\\\n", + "0 & 1 & 1\n", + "\\end{array}\\right), \\\\\\left(\\begin{array}{rrr}\n", + "1 & -1 & 0 \\\\\n", + "0 & 1 & 0 \\\\\n", + "0 & 0 & 1\n", + "\\end{array}\\right), \\left(\\begin{array}{rrr}\n", + "1 & 0 & -1 \\\\\n", + "0 & 1 & 0 \\\\\n", + "0 & 0 & 1\n", + "\\end{array}\\right), \\left(\\begin{array}{rrr}\n", + "1 & 0 & 0 \\\\\n", + "0 & 1 & -1 \\\\\n", + "0 & 0 & 1\n", + "\\end{array}\\right), \\left(\\begin{array}{rrr}\n", + "1 & 0 & 0 \\\\\n", + "-1 & 1 & 0 \\\\\n", + "0 & 0 & 1\n", + "\\end{array}\\right), \\left(\\begin{array}{rrr}\n", + "1 & 0 & 0 \\\\\n", + "0 & 1 & 0 \\\\\n", + "-1 & 0 & 1\n", + "\\end{array}\\right), \\left(\\begin{array}{rrr}\n", + "1 & 0 & 0 \\\\\n", + "0 & 1 & 0 \\\\\n", + "0 & -1 & 1\n", + "\\end{array}\\right)\\right]$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Elements of $\\mathbb{R}[SL(3,\\mathbb{Z}]$\n", + "Every element \n", + "$$X = \\sum_g a_g g\\in \\mathbb{R}[SL(3,\\mathbb{Z}]$$ of the group ring will be represented as type (container) with two elements inside:\n", + "1. a vector of coefficients $(a_g)_g$\n", + "2. a matrix $P=p_{ij}$ which tells us how to compute product of the element with every other element in basis" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "We define the Laplacian element as\n", + "$$\\Delta_S = \\Delta = \\frac{1}{2}\\sum_{g\\in S} (1-g)^*\\cdot(1-g),$$\n", + "where $\\cdot$ denotes the group algebra multiplication and $*$ is a linear operator on $\\mathbb{R}[SL(3,\\mathbb{Z})]$ given by\n", + "\n", + "$$\n", + "\\begin{align}\n", + "g^* &\\mapsto g^{-1} \\quad \\text{for group elements}\\\\\n", + "a_g^* &\\mapsto a_g \\quad \\text{for coefficients.}\n", + "\\end{align}\n", + "$$\n", + "(Note: we work over $\\mathbb{R}$; over $\\mathbb{C}$ the $*$-involution is given by conjugation.)\n", + "\n", + "Example: \n", + "$$(1-x+y)^*\\cdot(1-x+y) = (1-x^{-1}+y^{-1})(1-x+y) = 1 - x + y - x^{-1} + 1 - x^{-1}y + y^{-1} - y^{-1}x + 1 = 3 - x - x^{-1} +y + y^{-1} - x^{-1}y - yx^{-1}$$" + ] + }, + { + "cell_type": "code", + "execution_count": 6, + "metadata": { + "ExecuteTime": { + "end_time": "2016-12-10T02:58:14.012260", + "start_time": "2016-12-10T01:58:13.472Z" + }, + "collapsed": false, + "scrolled": true + }, + "outputs": [], + "source": [ + "matrix_constraints = read_GAP_raw_list(\"./constraints.\"*GROUP);\n", + "product_matrix = create_product_matrix(matrix_constraints);\n", + "\n", + "Delta² = read_GAP_raw_list(\"./delta_sq.\"*GROUP);\n", + "Delta = read_GAP_raw_list(\"./delta.\"*GROUP);\n", + "\n", + "Δ = GroupAlgebraElement(Delta, product_matrix);\n", + "Δ² = GroupAlgebraElement(Delta², product_matrix);\n", + "@assert Δ*Δ == Δ²" + ] + }, + { + "cell_type": "code", + "execution_count": 7, + "metadata": { + "ExecuteTime": { + "end_time": "2016-12-10T02:58:14.605892", + "start_time": "2016-12-10T01:58:13.481Z" + }, + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Δ = Element of Group Algebra over Int64\n", + "[12,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]\n", + "Δ² = Element of Group Algebra over Int64\n", + "[156,-24,-24,-24,-24,-24,-24,-24,-24,-24,-24,-24,-24,1,2,1,1,1,2,2,1,1,1,2,1,2,1,1,1,2,2,1,1,1,1,1,2,1,1,1,2,2,1,1,1,1,1,2,1,1,1,2,2,1,1,1,1,1,2,1,1,1,2,2,1,1,1,1,2,1,1,1,2,1,1,1,1,2,1,1,1,2,1,1,1,1,2,1,1,1,1,1,1,1,1,2,1,1,1,1,1,1,1,1,2,1,1,1,1,1,1,1,1,2,1,1,1,1,1,1,1]\n" + ] + } + ], + "source": [ + "@show(Δ);\n", + "@show(Δ²);" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Eg. the the $1st$ coefficients of those vectors (respecively $12$ i $156$) corresponds to basis vector $\\operatorname{Id}$." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Semi-definite programming (optimalisation of a linear functional over the manifold of symmetric matrices)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Our main problem is to find (possibly large) constant $\\kappa$ such that $\\Delta^2 - \\kappa \\Delta$ can be decomposed in the group ring as a hermitian sum of squares. That is, there exist a finite set of elements $\\xi_1, \\xi_2,\\ldots, \\xi_n \\in \\mathbb{R}[SL(3,\\mathbb{Z})]$ such that\n", + "$$ \\Delta^2 - \\kappa\\Delta = \\sum_{i=1}^n \\xi_i^*\\xi.$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Such sum-of-squares decompositions correspond to symmetric, positive semi-definite matrices $A$ such that\n", + "$$\\left(1, x, x^{-1}, \\ldots \\right)^*A\\left(1, x, x^{-1}, \\ldots\\right)^T = \\Delta^2 - \\kappa\\Delta $$\n", + "for some basis $(1, x, x^{-1}, \\ldots )$ of $\\mathbb{R}[SL(3,\\mathbb{Z})]$. Indeed, since $A$ is symmetric and positive semi-definite, there exist a root $\\sqrt{A}$, satisfying $\\sqrt{A}\\sqrt{A}^T = A$. Then we can set\n", + "$$(\\xi_1,\\xi_2,\\ldots,\\xi_{n}) = \\left(1,x, x^{-1},\\ldots\\right)\\sqrt{A},$$\n", + "and it follows that\n", + "$$\\Delta^2-\\kappa\\Delta = \\left(1,x, x^{-1},\\ldots\\right)^* A \\left(1,x, x^{-1},\\ldots \\right)^T = \\Big(\\left(1,x, x^{-1},\\ldots\\right)\\sqrt{A}\\Big)^* \\Big(\\sqrt{A}\\left(1,x, x^{-1},\\ldots\\right)\\Big)^T =\\\\=\n", + "(\\xi_1,\\xi_2,\\ldots\\xi_{n})^*(\\xi_1,\\xi_2,\\ldots\\xi_{n})^T = \\sum_{i=1}^{n}\\xi_i^*\\xi_i\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "In our case we will try to find such $121\\times 121$ matrix $A$ (as our basis consists of $121$ elements) and therefore to express $\\Delta^2 - \\kappa\\Delta$ as sum of $121$ hermitian squares." + ] + }, + { + "cell_type": "markdown", + "metadata": { + "ExecuteTime": { + "end_time": "2016-12-10T00:58:21.801317", + "start_time": "2016-12-09T23:58:21.724Z" + } + }, + "source": [ + "First we define `SDP` problem:" + ] + }, + { + "cell_type": "code", + "execution_count": 8, + "metadata": { + "ExecuteTime": { + "end_time": "2016-12-10T02:58:16.814693", + "start_time": "2016-12-10T01:58:13.494Z" + }, + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "SL_3ZZ = Maximization problem with:\n", + " * 121 linear constraints\n", + " * 1 semidefinite constraint\n", + " * 7382 variables\n", + "Solver is SCS\n" + ] + } + ], + "source": [ + "SL_3ZZ = create_SDP_problem(matrix_constraints, Δ², Δ);\n", + "\n", + "const TOL=6\n", + "\n", + "setsolver(SL_3ZZ, SCSSolver(eps=10.0^-TOL));\n", + "@show(SL_3ZZ);" + ] + }, + { + "cell_type": "code", + "execution_count": 9, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "data": { + "text/latex": [ + "$$ \\begin{alignat*}{1}\\max\\quad & κ\\\\\n", + "\\text{Subject to} \\quad & A_{1,1} + A_{2,2} + A_{3,3} + A_{4,4} + A_{5,5} + A_{6,6} + A_{7,7} + A_{8,8} + A_{9,9} + A_{10,10} + A_{11,11} + A_{12,12} + A_{13,13} + A_{14,14} + A_{15,15} + A_{16,16} + A_{17,17} + A_{18,18} + A_{19,19} + A_{20,20} + A_{21,21} + A_{22,22} + A_{23,23} + A_{24,24} + A_{25,25} + A_{26,26} + A_{27,27} + A_{28,28} + A_{29,29} + A_{30,30} + A_{31,31} + A_{32,32} + A_{33,33} + A_{34,34} + A_{35,35} + A_{36,36} + A_{37,37} + A_{38,38} + A_{39,39} + A_{40,40} + A_{41,41} + A_{42,42} + A_{43,43} + A_{44,44} + A_{45,45} + A_{46,46} + A_{47,47} + A_{48,48} + A_{49,49} + A_{50,50} + A_{51,51} + A_{52,52} + A_{53,53} + A_{54,54} + A_{55,55} + A_{56,56} + A_{57,57} + A_{58,58} + A_{59,59} + A_{60,60} + A_{61,61} + A_{62,62} + A_{63,63} + A_{64,64} + A_{65,65} + A_{66,66} + A_{67,67} + A_{68,68} + A_{69,69} + A_{70,70} + A_{71,71} + A_{72,72} + A_{73,73} + A_{74,74} + A_{75,75} + A_{76,76} + A_{77,77} + A_{78,78} + A_{79,79} + A_{80,80} + A_{81,81} + A_{82,82} + A_{83,83} + A_{84,84} + A_{85,85} + A_{86,86} + A_{87,87} + A_{88,88} + A_{89,89} + A_{90,90} + A_{91,91} + A_{92,92} + A_{93,93} + A_{94,94} + A_{95,95} + A_{96,96} + A_{97,97} + A_{98,98} + A_{99,99} + A_{100,100} + A_{101,101} + A_{102,102} + A_{103,103} + A_{104,104} + A_{105,105} + A_{106,106} + A_{107,107} + A_{108,108} + A_{109,109} + A_{110,110} + A_{111,111} + A_{112,112} + A_{113,113} + A_{114,114} + A_{115,115} + A_{116,116} + A_{117,117} + A_{118,118} + A_{119,119} + A_{120,120} + A_{121,121} + 12 κ = 156\\\\\n", + " & A_{1,2} + A_{2,14} + A_{3,15} + A_{4,35} + A_{5,45} + A_{6,55} + A_{7,19} + A_{1,8} + A_{9,20} + A_{10,91} + A_{11,99} + A_{12,107} + A_{13,24} + A_{16,26} + A_{3,30} + A_{34,115} + A_{4,40} + A_{5,50} + A_{6,60} + A_{18,64} + A_{29,65} + A_{7,69} + A_{23,73} + A_{41,74} + A_{59,76} + A_{8,77} + A_{9,78} + A_{31,79} + A_{81,114} + A_{13,82} + A_{70,85} + A_{21,87} + A_{10,94} + A_{11,102} + A_{12,110} + A_{90,118} - κ = -24\\\\\n", + " & A_{1,3} + A_{2,15} + A_{3,25} + A_{4,26} + A_{5,46} + A_{6,56} + A_{7,65} + A_{8,30} + A_{1,9} + A_{10,31} + A_{11,100} + A_{12,108} + A_{13,115} + A_{19,29} + A_{2,20} + A_{21,91} + A_{16,35} + A_{4,41} + A_{32,42} + A_{5,51} + A_{27,52} + A_{6,61} + A_{7,70} + A_{40,74} + A_{8,78} + A_{34,82} + A_{37,83} + A_{69,85} + A_{9,86} + A_{10,87} + A_{88,96} + A_{24,90} + A_{79,94} + A_{11,103} + A_{12,111} + A_{13,118} - κ = -24\\\\\n", + " & A_{1,4} + A_{2,16} + A_{3,26} + A_{4,36} + A_{5,37} + A_{6,57} + A_{7,66} + A_{8,74} + A_{9,41} + A_{1,10} + A_{11,42} + A_{12,109} + A_{13,116} + A_{20,35} + A_{2,21} + A_{27,46} + A_{30,40} + A_{3,31} + A_{38,48} + A_{51,83} + A_{5,52} + A_{6,62} + A_{7,71} + A_{8,79} + A_{9,87} + A_{88,103} + A_{15,91} + A_{63,92} + A_{78,94} + A_{10,95} + A_{11,96} + A_{53,97} + A_{32,100} + A_{43,105} + A_{12,112} + A_{13,119} - κ = -24\\\\\n", + " & A_{1,5} + A_{2,17} + A_{3,27} + A_{4,37} + A_{5,47} + A_{6,48} + A_{7,67} + A_{8,75} + A_{9,83} + A_{10,52} + A_{1,11} + A_{12,53} + A_{13,117} + A_{2,22} + A_{26,46} + A_{3,32} + A_{4,42} + A_{43,109} + A_{38,57} + A_{6,63} + A_{54,64} + A_{7,72} + A_{49,73} + A_{8,80} + A_{51,87} + A_{9,88} + A_{62,92} + A_{10,96} + A_{31,100} + A_{59,101} + A_{41,103} + A_{11,104} + A_{12,105} + A_{106,114} + A_{97,112} + A_{13,120} - κ = -24\\\\\n", + " & A_{1,6} + A_{2,18} + A_{3,28} + A_{4,38} + A_{5,48} + A_{6,58} + A_{7,59} + A_{8,76} + A_{9,84} + A_{10,92} + A_{11,63} + A_{1,12} + A_{13,64} + A_{19,55} + A_{2,23} + A_{3,33} + A_{42,57} + A_{4,43} + A_{49,67} + A_{52,62} + A_{5,53} + A_{72,101} + A_{7,73} + A_{8,81} + A_{60,82} + A_{9,89} + A_{10,97} + A_{11,105} + A_{106,120} + A_{24,107} + A_{37,109} + A_{69,110} + A_{96,112} + A_{12,113} + A_{13,114} + A_{54,117} - κ = -24\\\\\n", + " & A_{1,7} + A_{2,19} + A_{3,29} + A_{4,39} + A_{5,49} + A_{6,59} + A_{7,68} + A_{8,69} + A_{9,85} + A_{10,93} + A_{11,101} + A_{12,73} + A_{1,13} + A_{18,55} + A_{20,70} + A_{2,24} + A_{30,65} + A_{3,34} + A_{4,44} + A_{48,67} + A_{5,54} + A_{60,76} + A_{6,64} + A_{81,110} + A_{8,82} + A_{9,90} + A_{10,98} + A_{72,105} + A_{11,106} + A_{23,107} + A_{12,114} + A_{15,115} + A_{53,117} + A_{78,118} + A_{63,120} + A_{13,121} - κ = -24\\\\\n", + " & A_{1,8} + A_{1,2} + A_{3,30} + A_{4,40} + A_{5,50} + A_{6,60} + A_{7,69} + A_{8,77} + A_{9,78} + A_{10,94} + A_{11,102} + A_{12,110} + A_{13,82} + A_{2,14} + A_{3,15} + A_{16,26} + A_{18,64} + A_{7,19} + A_{9,20} + A_{21,87} + A_{23,73} + A_{13,24} + A_{29,65} + A_{31,79} + A_{4,35} + A_{41,74} + A_{5,45} + A_{6,55} + A_{59,76} + A_{70,85} + A_{90,118} + A_{10,91} + A_{11,99} + A_{12,107} + A_{81,114} + A_{34,115} - κ = -24\\\\\n", + " & A_{1,9} + A_{2,20} + A_{1,3} + A_{4,41} + A_{5,51} + A_{6,61} + A_{7,70} + A_{8,78} + A_{9,86} + A_{10,87} + A_{11,103} + A_{12,111} + A_{13,118} + A_{2,15} + A_{16,35} + A_{24,90} + A_{3,25} + A_{4,26} + A_{27,52} + A_{19,29} + A_{8,30} + A_{10,31} + A_{32,42} + A_{34,82} + A_{37,83} + A_{40,74} + A_{5,46} + A_{6,56} + A_{7,65} + A_{69,85} + A_{79,94} + A_{21,91} + A_{88,96} + A_{11,100} + A_{12,108} + A_{13,115} - κ = -24\\\\\n", + " & A_{1,10} + A_{2,21} + A_{3,31} + A_{1,4} + A_{5,52} + A_{6,62} + A_{7,71} + A_{8,79} + A_{9,87} + A_{10,95} + A_{11,96} + A_{12,112} + A_{13,119} + A_{15,91} + A_{2,16} + A_{3,26} + A_{32,100} + A_{20,35} + A_{4,36} + A_{5,37} + A_{38,48} + A_{30,40} + A_{9,41} + A_{11,42} + A_{43,105} + A_{27,46} + A_{53,97} + A_{6,57} + A_{63,92} + A_{7,66} + A_{8,74} + A_{78,94} + A_{51,83} + A_{88,103} + A_{12,109} + A_{13,116} - κ = -24\\\\\n", + " & A_{1,11} + A_{2,22} + A_{3,32} + A_{4,42} + A_{1,5} + A_{6,63} + A_{7,72} + A_{8,80} + A_{9,88} + A_{10,96} + A_{11,104} + A_{12,105} + A_{13,120} + A_{2,17} + A_{3,27} + A_{31,100} + A_{4,37} + A_{38,57} + A_{41,103} + A_{26,46} + A_{5,47} + A_{6,48} + A_{49,73} + A_{51,87} + A_{10,52} + A_{12,53} + A_{54,64} + A_{59,101} + A_{62,92} + A_{7,67} + A_{8,75} + A_{9,83} + A_{97,112} + A_{43,109} + A_{106,114} + A_{13,117} - κ = -24\\\\\n", + " & A_{1,12} + A_{2,23} + A_{3,33} + A_{4,43} + A_{5,53} + A_{1,6} + A_{7,73} + A_{8,81} + A_{9,89} + A_{10,97} + A_{11,105} + A_{12,113} + A_{13,114} + A_{2,18} + A_{24,107} + A_{3,28} + A_{37,109} + A_{4,38} + A_{5,48} + A_{54,117} + A_{19,55} + A_{42,57} + A_{6,58} + A_{7,59} + A_{60,82} + A_{52,62} + A_{11,63} + A_{13,64} + A_{49,67} + A_{69,110} + A_{8,76} + A_{9,84} + A_{10,92} + A_{96,112} + A_{72,101} + A_{106,120} - κ = -24\\\\\n", + " & A_{1,13} + A_{2,24} + A_{3,34} + A_{4,44} + A_{5,54} + A_{6,64} + A_{1,7} + A_{8,82} + A_{9,90} + A_{10,98} + A_{11,106} + A_{12,114} + A_{13,121} + A_{15,115} + A_{2,19} + A_{23,107} + A_{3,29} + A_{4,39} + A_{5,49} + A_{53,117} + A_{18,55} + A_{6,59} + A_{63,120} + A_{30,65} + A_{48,67} + A_{7,68} + A_{8,69} + A_{20,70} + A_{72,105} + A_{12,73} + A_{60,76} + A_{78,118} + A_{9,85} + A_{10,93} + A_{11,101} + A_{81,110} - κ = -24\\\\\n", + " & A_{1,14} + A_{2,8} + A_{15,30} + A_{35,40} + A_{45,50} + A_{55,60} + A_{19,69} + A_{1,77} + A_{20,78} + A_{24,82} + A_{91,94} + A_{99,102} + A_{107,110} = 1\\\\\n", + " & A_{1,15} + A_{4,16} + A_{7,29} + A_{3,8} + A_{2,9} + A_{14,20} + A_{25,30} + A_{26,40} + A_{35,41} + A_{46,50} + A_{45,51} + A_{56,60} + A_{55,61} + A_{65,69} + A_{19,70} + A_{4,74} + A_{30,77} + A_{1,78} + A_{82,115} + A_{7,85} + A_{20,86} + A_{87,91} + A_{31,94} + A_{100,102} + A_{99,103} + A_{108,110} + A_{107,111} + A_{24,118} = 2\\\\\n", + " & A_{1,16} + A_{4,8} + A_{9,35} + A_{10,15} + A_{26,30} + A_{36,40} + A_{37,50} + A_{57,60} + A_{66,69} + A_{29,71} + A_{74,77} + A_{41,78} + A_{3,79} + A_{82,116} + A_{2,87} + A_{88,99} + A_{1,94} + A_{42,102} + A_{109,110} = 1\\\\\n", + " & A_{1,17} + A_{5,8} + A_{22,99} + A_{27,30} + A_{37,40} + A_{47,50} + A_{48,60} + A_{67,69} + A_{75,77} + A_{78,83} + A_{82,117} + A_{52,94} + A_{1,102} + A_{53,110} = 1\\\\\n", + " & A_{1,18} + A_{7,55} + A_{6,8} + A_{12,24} + A_{28,30} + A_{33,115} + A_{38,40} + A_{48,50} + A_{58,60} + A_{59,69} + A_{72,99} + A_{2,73} + A_{76,77} + A_{78,84} + A_{13,81} + A_{64,82} + A_{92,94} + A_{63,102} + A_{1,110} = 1\\\\\n", + " & A_{1,19} + A_{7,8} + A_{9,70} + A_{12,23} + A_{2,13} + A_{14,24} + A_{29,30} + A_{15,34} + A_{39,40} + A_{35,44} + A_{49,50} + A_{45,54} + A_{59,60} + A_{55,64} + A_{68,69} + A_{69,77} + A_{78,85} + A_{12,81} + A_{1,82} + A_{20,90} + A_{93,94} + A_{91,98} + A_{101,102} + A_{99,106} + A_{73,110} + A_{107,114} + A_{9,118} + A_{24,121} = 2\\\\\n", + " & A_{1,20} + A_{2,3} + A_{8,9} + A_{10,21} + A_{13,90} + A_{14,15} + A_{15,25} + A_{26,35} + A_{1,30} + A_{31,91} + A_{13,34} + A_{40,41} + A_{45,46} + A_{50,51} + A_{55,56} + A_{60,61} + A_{19,65} + A_{69,70} + A_{77,78} + A_{78,86} + A_{10,79} + A_{82,118} + A_{87,94} + A_{99,100} + A_{102,103} + A_{107,108} + A_{110,111} + A_{24,115} = 2\\\\\n", + " & A_{1,21} + A_{3,91} + A_{4,20} + A_{8,10} + A_{2,26} + A_{30,31} + A_{32,99} + A_{1,40} + A_{50,52} + A_{60,62} + A_{69,71} + A_{9,74} + A_{77,79} + A_{78,87} + A_{82,119} + A_{94,95} + A_{96,102} + A_{110,112} + A_{90,116} = 1\\\\\n", + " & A_{1,22} + A_{8,11} + A_{30,32} + A_{40,42} + A_{1,50} + A_{60,63} + A_{69,72} + A_{74,103} + A_{75,102} + A_{76,101} + A_{77,80} + A_{78,88} + A_{79,100} + A_{80,99} + A_{81,106} + A_{82,120} + A_{94,96} + A_{102,104} + A_{105,110} = 1\\\\\n", + " & A_{1,23} + A_{6,19} + A_{8,12} + A_{13,107} + A_{30,33} + A_{40,43} + A_{50,53} + A_{1,60} + A_{2,64} + A_{69,73} + A_{7,76} + A_{77,81} + A_{78,89} + A_{82,114} + A_{70,84} + A_{94,97} + A_{102,105} + A_{110,113} + A_{99,120} = 1\\\\\n", + " & A_{1,24} + A_{3,115} + A_{6,18} + A_{2,7} + A_{8,13} + A_{14,19} + A_{15,29} + A_{30,34} + A_{35,39} + A_{40,44} + A_{45,49} + A_{50,54} + A_{55,59} + A_{60,64} + A_{3,65} + A_{19,68} + A_{1,69} + A_{73,107} + A_{6,76} + A_{77,82} + A_{78,90} + A_{82,121} + A_{20,85} + A_{91,93} + A_{94,98} + A_{99,101} + A_{102,106} + A_{110,114} = 2\\\\\n", + " & A_{1,25} + A_{3,9} + A_{15,20} + A_{26,41} + A_{46,51} + A_{56,61} + A_{65,70} + A_{30,78} + A_{1,86} + A_{31,87} + A_{100,103} + A_{108,111} + A_{115,118} = 1\\\\\n", + " & A_{1,26} + A_{8,40} + A_{4,9} + A_{3,10} + A_{11,32} + A_{16,20} + A_{15,21} + A_{25,31} + A_{36,41} + A_{37,51} + A_{46,52} + A_{57,61} + A_{56,62} + A_{66,70} + A_{65,71} + A_{74,78} + A_{30,79} + A_{41,86} + A_{1,87} + A_{11,88} + A_{8,94} + A_{31,95} + A_{96,100} + A_{42,103} + A_{109,111} + A_{108,112} + A_{116,118} + A_{115,119} = 2\\\\\n", + " & A_{1,27} + A_{4,46} + A_{5,9} + A_{11,31} + A_{17,20} + A_{22,91} + A_{37,41} + A_{3,42} + A_{43,108} + A_{47,51} + A_{48,61} + A_{67,70} + A_{75,78} + A_{83,86} + A_{52,87} + A_{10,88} + A_{1,103} + A_{53,111} + A_{117,118} = 1\\\\\n", + " & A_{1,28} + A_{6,9} + A_{18,20} + A_{33,108} + A_{38,41} + A_{48,51} + A_{58,61} + A_{59,70} + A_{76,78} + A_{84,86} + A_{87,92} + A_{63,103} + A_{1,111} + A_{64,118} = 1\\\\\n", + " & A_{1,29} + A_{8,65} + A_{7,9} + A_{13,15} + A_{19,20} + A_{39,41} + A_{16,44} + A_{49,51} + A_{59,61} + A_{68,70} + A_{69,78} + A_{81,108} + A_{3,82} + A_{85,86} + A_{87,93} + A_{2,90} + A_{101,103} + A_{73,111} + A_{1,118} = 1\\\\\n", + " & A_{1,30} + A_{2,3} + A_{8,9} + A_{10,79} + A_{13,34} + A_{14,15} + A_{15,25} + A_{19,65} + A_{1,20} + A_{10,21} + A_{24,115} + A_{26,35} + A_{40,41} + A_{45,46} + A_{50,51} + A_{55,56} + A_{60,61} + A_{69,70} + A_{77,78} + A_{78,86} + A_{87,94} + A_{13,90} + A_{31,91} + A_{99,100} + A_{102,103} + A_{107,108} + A_{110,111} + A_{82,118} = 2\\\\\n", + " & A_{1,31} + A_{2,91} + A_{3,4} + A_{5,27} + A_{9,10} + A_{15,16} + A_{20,21} + A_{25,26} + A_{2,35} + A_{26,36} + A_{37,46} + A_{1,41} + A_{42,100} + A_{51,52} + A_{56,57} + A_{61,62} + A_{65,66} + A_{70,71} + A_{30,74} + A_{78,79} + A_{5,83} + A_{86,87} + A_{87,95} + A_{96,103} + A_{108,109} + A_{111,112} + A_{115,116} + A_{118,119} = 2\\\\\n", + " & A_{1,32} + A_{5,26} + A_{9,11} + A_{10,100} + A_{20,22} + A_{41,42} + A_{1,51} + A_{3,52} + A_{61,63} + A_{70,72} + A_{40,75} + A_{78,80} + A_{4,83} + A_{86,88} + A_{87,96} + A_{97,108} + A_{103,104} + A_{105,111} + A_{118,120} = 1\\\\\n", + " & A_{1,33} + A_{9,12} + A_{20,23} + A_{41,43} + A_{51,53} + A_{1,61} + A_{70,73} + A_{78,81} + A_{83,109} + A_{84,111} + A_{85,110} + A_{86,89} + A_{87,97} + A_{88,112} + A_{89,108} + A_{90,107} + A_{103,105} + A_{111,113} + A_{114,118} = 1\\\\\n", + " & A_{1,34} + A_{2,115} + A_{7,30} + A_{9,13} + A_{3,19} + A_{20,24} + A_{23,108} + A_{41,44} + A_{51,54} + A_{61,64} + A_{1,70} + A_{78,82} + A_{8,85} + A_{86,90} + A_{87,98} + A_{79,93} + A_{103,106} + A_{111,114} + A_{118,121} = 1\\\\\n", + " & A_{1,35} + A_{3,16} + A_{8,41} + A_{2,10} + A_{14,21} + A_{4,30} + A_{15,31} + A_{34,116} + A_{50,83} + A_{45,52} + A_{55,62} + A_{19,71} + A_{1,79} + A_{20,87} + A_{9,94} + A_{91,95} + A_{96,99} + A_{107,112} + A_{24,119} = 1\\\\\n", + " & A_{1,36} + A_{4,10} + A_{16,21} + A_{26,31} + A_{37,52} + A_{57,62} + A_{66,71} + A_{74,79} + A_{41,87} + A_{1,95} + A_{42,96} + A_{109,112} + A_{116,119} = 1\\\\\n", + " & A_{1,37} + A_{3,46} + A_{6,38} + A_{5,10} + A_{4,11} + A_{17,21} + A_{16,22} + A_{27,31} + A_{26,32} + A_{36,42} + A_{47,52} + A_{48,62} + A_{57,63} + A_{67,71} + A_{66,72} + A_{75,79} + A_{74,80} + A_{83,87} + A_{41,88} + A_{6,92} + A_{52,95} + A_{1,96} + A_{3,100} + A_{42,104} + A_{105,109} + A_{53,112} + A_{117,119} + A_{116,120} = 2\\\\\n", + " & A_{1,38} + A_{6,10} + A_{11,57} + A_{12,37} + A_{18,21} + A_{28,31} + A_{33,46} + A_{48,52} + A_{58,62} + A_{59,71} + A_{76,79} + A_{84,87} + A_{92,95} + A_{63,96} + A_{5,97} + A_{4,105} + A_{106,116} + A_{1,112} + A_{64,119} = 1\\\\\n", + " & A_{1,39} + A_{7,10} + A_{19,21} + A_{29,31} + A_{44,116} + A_{49,52} + A_{59,62} + A_{68,71} + A_{69,79} + A_{85,87} + A_{93,95} + A_{96,101} + A_{73,112} + A_{1,119} = 1\\\\\n", + " & A_{1,40} + A_{2,26} + A_{9,74} + A_{8,10} + A_{4,20} + A_{1,21} + A_{30,31} + A_{50,52} + A_{60,62} + A_{69,71} + A_{77,79} + A_{78,87} + A_{90,116} + A_{3,91} + A_{94,95} + A_{96,102} + A_{32,99} + A_{110,112} + A_{82,119} = 1\\\\\n", + " & A_{1,41} + A_{2,35} + A_{3,4} + A_{5,83} + A_{9,10} + A_{15,16} + A_{20,21} + A_{25,26} + A_{26,36} + A_{5,27} + A_{30,74} + A_{1,31} + A_{37,46} + A_{51,52} + A_{56,57} + A_{61,62} + A_{65,66} + A_{70,71} + A_{78,79} + A_{86,87} + A_{2,91} + A_{87,95} + A_{96,103} + A_{42,100} + A_{108,109} + A_{111,112} + A_{115,116} + A_{118,119} = 2\\\\\n", + " & A_{1,42} + A_{4,5} + A_{9,103} + A_{10,11} + A_{12,43} + A_{16,17} + A_{21,22} + A_{26,27} + A_{31,32} + A_{36,37} + A_{37,47} + A_{48,57} + A_{9,51} + A_{1,52} + A_{53,109} + A_{62,63} + A_{66,67} + A_{71,72} + A_{74,75} + A_{79,80} + A_{41,83} + A_{87,88} + A_{95,96} + A_{96,104} + A_{12,97} + A_{105,112} + A_{116,117} + A_{119,120} = 2\\\\\n", + " & A_{1,43} + A_{5,109} + A_{6,42} + A_{10,12} + A_{21,23} + A_{31,33} + A_{4,48} + A_{52,53} + A_{54,116} + A_{1,62} + A_{71,73} + A_{79,81} + A_{84,103} + A_{87,89} + A_{11,92} + A_{95,97} + A_{96,105} + A_{112,113} + A_{114,119} = 1\\\\\n", + " & A_{1,44} + A_{10,13} + A_{21,24} + A_{31,34} + A_{52,54} + A_{62,64} + A_{1,71} + A_{79,82} + A_{87,90} + A_{91,115} + A_{92,120} + A_{93,119} + A_{94,118} + A_{95,98} + A_{96,106} + A_{97,117} + A_{98,116} + A_{112,114} + A_{119,121} = 1\\\\\n", + " & A_{1,45} + A_{2,11} + A_{14,22} + A_{15,32} + A_{35,42} + A_{50,75} + A_{55,63} + A_{19,72} + A_{1,80} + A_{20,88} + A_{91,96} + A_{99,104} + A_{105,107} + A_{24,120} = 1\\\\\n", + " & A_{1,46} + A_{9,37} + A_{10,27} + A_{3,11} + A_{15,22} + A_{25,32} + A_{26,42} + A_{38,61} + A_{56,63} + A_{65,72} + A_{30,80} + A_{5,87} + A_{1,88} + A_{75,94} + A_{31,96} + A_{4,103} + A_{100,104} + A_{105,108} + A_{115,120} = 1\\\\\n", + " & A_{1,47} + A_{5,11} + A_{17,22} + A_{27,32} + A_{37,42} + A_{48,63} + A_{67,72} + A_{75,80} + A_{83,88} + A_{52,96} + A_{1,104} + A_{53,105} + A_{117,120} = 1\\\\\n", + " & A_{1,48} + A_{10,62} + A_{6,11} + A_{5,12} + A_{13,54} + A_{18,22} + A_{17,23} + A_{28,32} + A_{27,33} + A_{38,42} + A_{37,43} + A_{47,53} + A_{58,63} + A_{59,72} + A_{67,73} + A_{76,80} + A_{75,81} + A_{84,88} + A_{83,89} + A_{92,96} + A_{52,97} + A_{63,104} + A_{1,105} + A_{13,106} + A_{10,112} + A_{53,113} + A_{114,117} + A_{64,120} = 2\\\\\n", + " & A_{1,49} + A_{6,67} + A_{7,11} + A_{13,53} + A_{19,22} + A_{29,32} + A_{39,42} + A_{44,109} + A_{60,75} + A_{59,63} + A_{5,64} + A_{68,72} + A_{69,80} + A_{85,88} + A_{93,96} + A_{101,104} + A_{73,105} + A_{12,106} + A_{1,120} = 1\\\\\n", + " & A_{1,50} + A_{8,11} + A_{1,22} + A_{30,32} + A_{40,42} + A_{60,63} + A_{69,72} + A_{77,80} + A_{78,88} + A_{94,96} + A_{80,99} + A_{79,100} + A_{76,101} + A_{75,102} + A_{74,103} + A_{102,104} + A_{105,110} + A_{81,106} + A_{82,120} = 1\\\\\n", + " & A_{1,51} + A_{3,52} + A_{4,83} + A_{9,11} + A_{20,22} + A_{5,26} + A_{1,32} + A_{40,75} + A_{41,42} + A_{61,63} + A_{70,72} + A_{78,80} + A_{86,88} + A_{87,96} + A_{10,100} + A_{103,104} + A_{105,111} + A_{97,108} + A_{118,120} = 1\\\\\n", + " & A_{1,52} + A_{4,5} + A_{9,51} + A_{10,11} + A_{12,97} + A_{16,17} + A_{21,22} + A_{26,27} + A_{31,32} + A_{36,37} + A_{37,47} + A_{41,83} + A_{1,42} + A_{12,43} + A_{48,57} + A_{62,63} + A_{66,67} + A_{71,72} + A_{74,75} + A_{79,80} + A_{87,88} + A_{95,96} + A_{9,103} + A_{96,104} + A_{105,112} + A_{53,109} + A_{116,117} + A_{119,120} = 2\\\\\n", + " & A_{1,53} + A_{4,109} + A_{5,6} + A_{7,49} + A_{11,12} + A_{17,18} + A_{22,23} + A_{27,28} + A_{32,33} + A_{37,38} + A_{42,43} + A_{47,48} + A_{4,57} + A_{48,58} + A_{59,67} + A_{1,63} + A_{64,117} + A_{72,73} + A_{75,76} + A_{80,81} + A_{83,84} + A_{88,89} + A_{52,92} + A_{96,97} + A_{7,101} + A_{104,105} + A_{105,113} + A_{114,120} = 2\\\\\n", + " & A_{1,54} + A_{7,48} + A_{11,13} + A_{12,117} + A_{22,24} + A_{32,34} + A_{42,44} + A_{63,64} + A_{1,72} + A_{5,73} + A_{80,82} + A_{88,90} + A_{62,93} + A_{96,98} + A_{6,101} + A_{104,106} + A_{105,114} + A_{75,110} + A_{120,121} = 1\\\\\n", + " & A_{1,55} + A_{8,59} + A_{2,12} + A_{13,18} + A_{14,23} + A_{15,33} + A_{35,43} + A_{50,67} + A_{45,53} + A_{19,73} + A_{1,81} + A_{6,82} + A_{20,89} + A_{91,97} + A_{99,105} + A_{7,110} + A_{107,113} + A_{24,114} + A_{84,118} = 1\\\\\n", + " & A_{1,56} + A_{3,12} + A_{15,23} + A_{25,33} + A_{26,43} + A_{46,53} + A_{61,84} + A_{65,73} + A_{30,81} + A_{1,89} + A_{31,97} + A_{100,105} + A_{108,113} + A_{114,115} = 1\\\\\n", + " & A_{1,57} + A_{5,38} + A_{10,63} + A_{4,12} + A_{16,23} + A_{26,33} + A_{36,43} + A_{51,84} + A_{6,52} + A_{37,53} + A_{71,101} + A_{66,73} + A_{74,81} + A_{41,89} + A_{1,97} + A_{42,105} + A_{11,112} + A_{109,113} + A_{114,116} = 1\\\\\n", + " & A_{1,58} + A_{6,12} + A_{18,23} + A_{28,33} + A_{38,43} + A_{48,53} + A_{59,73} + A_{76,81} + A_{84,89} + A_{92,97} + A_{63,105} + A_{1,113} + A_{64,114} = 1\\\\\n", + " & A_{1,59} + A_{2,55} + A_{5,67} + A_{7,12} + A_{6,13} + A_{19,23} + A_{18,24} + A_{29,33} + A_{28,34} + A_{39,43} + A_{38,44} + A_{49,53} + A_{48,54} + A_{58,64} + A_{68,73} + A_{69,81} + A_{76,82} + A_{85,89} + A_{84,90} + A_{93,97} + A_{92,98} + A_{101,105} + A_{63,106} + A_{2,107} + A_{73,113} + A_{1,114} + A_{5,117} + A_{64,121} = 2\\\\\n", + " & A_{1,60} + A_{2,64} + A_{7,76} + A_{8,12} + A_{6,19} + A_{1,23} + A_{30,33} + A_{40,43} + A_{50,53} + A_{70,84} + A_{69,73} + A_{77,81} + A_{78,89} + A_{94,97} + A_{99,120} + A_{102,105} + A_{13,107} + A_{110,113} + A_{82,114} = 1\\\\\n", + " & A_{1,61} + A_{9,12} + A_{20,23} + A_{1,33} + A_{41,43} + A_{51,53} + A_{70,73} + A_{78,81} + A_{86,89} + A_{87,97} + A_{103,105} + A_{90,107} + A_{89,108} + A_{83,109} + A_{85,110} + A_{84,111} + A_{88,112} + A_{111,113} + A_{114,118} = 1\\\\\n", + " & A_{1,62} + A_{4,48} + A_{11,92} + A_{10,12} + A_{21,23} + A_{31,33} + A_{6,42} + A_{1,43} + A_{52,53} + A_{71,73} + A_{79,81} + A_{87,89} + A_{95,97} + A_{84,103} + A_{96,105} + A_{5,109} + A_{112,113} + A_{114,119} + A_{54,116} = 1\\\\\n", + " & A_{1,63} + A_{4,57} + A_{5,6} + A_{7,101} + A_{11,12} + A_{17,18} + A_{22,23} + A_{27,28} + A_{32,33} + A_{37,38} + A_{42,43} + A_{47,48} + A_{48,58} + A_{7,49} + A_{52,92} + A_{1,53} + A_{59,67} + A_{72,73} + A_{75,76} + A_{80,81} + A_{83,84} + A_{88,89} + A_{96,97} + A_{104,105} + A_{4,109} + A_{105,113} + A_{114,120} + A_{64,117} = 2\\\\\n", + " & A_{1,64} + A_{6,7} + A_{8,60} + A_{11,120} + A_{12,13} + A_{18,19} + A_{23,24} + A_{28,29} + A_{33,34} + A_{38,39} + A_{43,44} + A_{48,49} + A_{53,54} + A_{58,59} + A_{59,68} + A_{69,76} + A_{11,72} + A_{1,73} + A_{81,82} + A_{84,85} + A_{89,90} + A_{92,93} + A_{97,98} + A_{63,101} + A_{105,106} + A_{8,110} + A_{113,114} + A_{114,121} = 2\\\\\n", + " & A_{1,65} + A_{2,29} + A_{9,69} + A_{3,13} + A_{7,20} + A_{21,93} + A_{15,24} + A_{25,34} + A_{26,44} + A_{46,54} + A_{61,76} + A_{56,64} + A_{30,82} + A_{1,90} + A_{31,98} + A_{100,106} + A_{108,114} + A_{8,118} + A_{115,121} = 1\\\\\n", + " & A_{1,66} + A_{4,13} + A_{16,24} + A_{26,34} + A_{36,44} + A_{37,54} + A_{57,64} + A_{71,93} + A_{74,82} + A_{41,90} + A_{1,98} + A_{42,106} + A_{109,114} + A_{116,121} = 1\\\\\n", + " & A_{1,67} + A_{11,59} + A_{12,49} + A_{5,13} + A_{22,55} + A_{17,24} + A_{27,34} + A_{37,44} + A_{47,54} + A_{48,64} + A_{75,82} + A_{83,90} + A_{52,98} + A_{7,105} + A_{1,106} + A_{93,112} + A_{53,114} + A_{6,120} + A_{117,121} = 1\\\\\n", + " & A_{1,68} + A_{7,13} + A_{19,24} + A_{29,34} + A_{39,44} + A_{49,54} + A_{59,64} + A_{69,82} + A_{85,90} + A_{93,98} + A_{101,106} + A_{73,114} + A_{1,121} = 1\\\\\n", + " & A_{1,69} + A_{2,7} + A_{3,65} + A_{6,76} + A_{8,13} + A_{14,19} + A_{15,29} + A_{6,18} + A_{19,68} + A_{20,85} + A_{1,24} + A_{30,34} + A_{35,39} + A_{40,44} + A_{45,49} + A_{50,54} + A_{55,59} + A_{60,64} + A_{77,82} + A_{78,90} + A_{91,93} + A_{94,98} + A_{99,101} + A_{102,106} + A_{73,107} + A_{110,114} + A_{3,115} + A_{82,121} = 2\\\\\n", + " & A_{1,70} + A_{3,19} + A_{8,85} + A_{9,13} + A_{20,24} + A_{7,30} + A_{1,34} + A_{41,44} + A_{51,54} + A_{61,64} + A_{79,93} + A_{78,82} + A_{86,90} + A_{87,98} + A_{103,106} + A_{23,108} + A_{111,114} + A_{2,115} + A_{118,121} = 1\\\\\n", + " & A_{1,71} + A_{10,13} + A_{21,24} + A_{31,34} + A_{1,44} + A_{52,54} + A_{62,64} + A_{79,82} + A_{87,90} + A_{95,98} + A_{96,106} + A_{112,114} + A_{91,115} + A_{98,116} + A_{97,117} + A_{94,118} + A_{93,119} + A_{92,120} + A_{119,121} = 1\\\\\n", + " & A_{1,72} + A_{5,73} + A_{6,101} + A_{11,13} + A_{22,24} + A_{32,34} + A_{42,44} + A_{7,48} + A_{1,54} + A_{62,93} + A_{63,64} + A_{75,110} + A_{80,82} + A_{88,90} + A_{96,98} + A_{104,106} + A_{105,114} + A_{12,117} + A_{120,121} = 1\\\\\n", + " & A_{1,73} + A_{6,7} + A_{8,110} + A_{11,72} + A_{12,13} + A_{18,19} + A_{23,24} + A_{28,29} + A_{33,34} + A_{38,39} + A_{43,44} + A_{48,49} + A_{53,54} + A_{58,59} + A_{59,68} + A_{8,60} + A_{63,101} + A_{1,64} + A_{69,76} + A_{81,82} + A_{84,85} + A_{89,90} + A_{92,93} + A_{97,98} + A_{105,106} + A_{113,114} + A_{11,120} + A_{114,121} = 2\\\\\n", + " & A_{1,74} + A_{2,4} + A_{3,40} + A_{10,78} + A_{14,16} + A_{15,26} + A_{19,66} + A_{20,41} + A_{9,21} + A_{24,116} + A_{27,50} + A_{8,31} + A_{35,36} + A_{37,45} + A_{55,57} + A_{71,85} + A_{1,91} + A_{42,99} + A_{107,109} = 1\\\\\n", + " & A_{1,75} + A_{2,5} + A_{14,17} + A_{15,27} + A_{16,46} + A_{17,50} + A_{18,54} + A_{19,67} + A_{20,83} + A_{21,51} + A_{22,45} + A_{23,49} + A_{24,117} + A_{35,37} + A_{45,47} + A_{48,55} + A_{52,91} + A_{1,99} + A_{53,107} = 1\\\\\n", + " & A_{1,76} + A_{2,6} + A_{12,69} + A_{13,60} + A_{14,18} + A_{15,28} + A_{19,59} + A_{20,84} + A_{7,23} + A_{24,64} + A_{33,65} + A_{35,38} + A_{45,48} + A_{55,58} + A_{91,92} + A_{63,99} + A_{1,107} + A_{8,114} + A_{50,117} = 1\\\\\n", + " & A_{1,77} + A_{2,8} + A_{1,14} + A_{15,30} + A_{19,69} + A_{20,78} + A_{24,82} + A_{35,40} + A_{45,50} + A_{55,60} + A_{91,94} + A_{99,102} + A_{107,110} = 1\\\\\n", + " & A_{1,78} + A_{2,9} + A_{3,8} + A_{4,74} + A_{7,85} + A_{14,20} + A_{1,15} + A_{4,16} + A_{19,70} + A_{20,86} + A_{24,118} + A_{25,30} + A_{26,40} + A_{7,29} + A_{30,77} + A_{31,94} + A_{35,41} + A_{45,51} + A_{46,50} + A_{55,61} + A_{56,60} + A_{65,69} + A_{87,91} + A_{99,103} + A_{100,102} + A_{107,111} + A_{108,110} + A_{82,115} = 2\\\\\n", + " & A_{1,79} + A_{2,10} + A_{4,30} + A_{9,94} + A_{14,21} + A_{15,31} + A_{3,16} + A_{19,71} + A_{20,87} + A_{24,119} + A_{1,35} + A_{8,41} + A_{45,52} + A_{55,62} + A_{50,83} + A_{91,95} + A_{96,99} + A_{107,112} + A_{34,116} = 1\\\\\n", + " & A_{1,80} + A_{2,11} + A_{14,22} + A_{15,32} + A_{19,72} + A_{20,88} + A_{24,120} + A_{35,42} + A_{1,45} + A_{55,63} + A_{50,75} + A_{91,96} + A_{99,104} + A_{105,107} = 1\\\\\n", + " & A_{1,81} + A_{2,12} + A_{6,82} + A_{7,110} + A_{14,23} + A_{15,33} + A_{13,18} + A_{19,73} + A_{20,89} + A_{24,114} + A_{35,43} + A_{45,53} + A_{1,55} + A_{8,59} + A_{50,67} + A_{84,118} + A_{91,97} + A_{99,105} + A_{107,113} = 1\\\\\n", + " & A_{1,82} + A_{2,13} + A_{7,8} + A_{9,118} + A_{12,81} + A_{14,24} + A_{15,34} + A_{1,19} + A_{20,90} + A_{12,23} + A_{24,121} + A_{29,30} + A_{35,44} + A_{39,40} + A_{45,54} + A_{49,50} + A_{55,64} + A_{59,60} + A_{68,69} + A_{69,77} + A_{9,70} + A_{73,110} + A_{78,85} + A_{91,98} + A_{93,94} + A_{99,106} + A_{101,102} + A_{107,114} = 2\\\\\n", + " & A_{1,83} + A_{3,5} + A_{10,51} + A_{11,41} + A_{15,17} + A_{22,35} + A_{25,27} + A_{26,37} + A_{30,75} + A_{31,52} + A_{4,32} + A_{46,47} + A_{48,56} + A_{65,67} + A_{61,92} + A_{9,96} + A_{1,100} + A_{53,108} + A_{115,117} = 1\\\\\n", + " & A_{1,84} + A_{3,6} + A_{15,18} + A_{25,28} + A_{26,38} + A_{27,62} + A_{28,61} + A_{29,55} + A_{30,76} + A_{31,92} + A_{32,57} + A_{33,56} + A_{34,60} + A_{46,48} + A_{56,58} + A_{59,65} + A_{63,100} + A_{1,108} + A_{64,115} = 1\\\\\n", + " & A_{1,85} + A_{2,70} + A_{3,7} + A_{13,78} + A_{15,19} + A_{18,61} + A_{9,24} + A_{25,29} + A_{26,39} + A_{30,69} + A_{31,93} + A_{8,34} + A_{44,74} + A_{46,49} + A_{56,59} + A_{65,68} + A_{100,101} + A_{73,108} + A_{1,115} = 1\\\\\n", + " & A_{1,86} + A_{3,9} + A_{15,20} + A_{1,25} + A_{26,41} + A_{30,78} + A_{31,87} + A_{46,51} + A_{56,61} + A_{65,70} + A_{100,103} + A_{108,111} + A_{115,118} = 1\\\\\n", + " & A_{1,87} + A_{3,10} + A_{4,9} + A_{8,94} + A_{11,88} + A_{15,21} + A_{16,20} + A_{25,31} + A_{1,26} + A_{30,79} + A_{31,95} + A_{11,32} + A_{36,41} + A_{37,51} + A_{8,40} + A_{41,86} + A_{42,103} + A_{46,52} + A_{56,62} + A_{57,61} + A_{65,71} + A_{66,70} + A_{74,78} + A_{96,100} + A_{108,112} + A_{109,111} + A_{115,119} + A_{116,118} = 2\\\\\n", + " & A_{1,88} + A_{3,11} + A_{4,103} + A_{5,87} + A_{15,22} + A_{25,32} + A_{26,42} + A_{10,27} + A_{30,80} + A_{31,96} + A_{9,37} + A_{38,61} + A_{1,46} + A_{56,63} + A_{65,72} + A_{75,94} + A_{100,104} + A_{105,108} + A_{115,120} = 1\\\\\n", + " & A_{1,89} + A_{3,12} + A_{15,23} + A_{25,33} + A_{26,43} + A_{30,81} + A_{31,97} + A_{46,53} + A_{1,56} + A_{65,73} + A_{61,84} + A_{100,105} + A_{108,113} + A_{114,115} = 1\\\\\n", + " & A_{1,90} + A_{3,13} + A_{7,20} + A_{8,118} + A_{15,24} + A_{25,34} + A_{26,44} + A_{2,29} + A_{30,82} + A_{31,98} + A_{46,54} + A_{56,64} + A_{1,65} + A_{9,69} + A_{61,76} + A_{21,93} + A_{100,106} + A_{108,114} + A_{115,121} = 1\\\\\n", + " & A_{1,91} + A_{2,4} + A_{8,31} + A_{9,21} + A_{14,16} + A_{15,26} + A_{35,36} + A_{37,45} + A_{3,40} + A_{20,41} + A_{42,99} + A_{27,50} + A_{55,57} + A_{19,66} + A_{1,74} + A_{10,78} + A_{71,85} + A_{107,109} + A_{24,116} = 1\\\\\n", + " & A_{1,92} + A_{4,6} + A_{5,62} + A_{12,96} + A_{16,18} + A_{26,28} + A_{33,100} + A_{36,38} + A_{37,48} + A_{41,84} + A_{42,63} + A_{11,43} + A_{49,71} + A_{10,53} + A_{57,58} + A_{59,66} + A_{74,76} + A_{1,109} + A_{64,116} = 1\\\\\n", + " & A_{1,93} + A_{4,7} + A_{16,19} + A_{26,29} + A_{35,70} + A_{36,39} + A_{37,49} + A_{38,67} + A_{39,71} + A_{40,65} + A_{41,85} + A_{42,101} + A_{43,72} + A_{44,66} + A_{57,59} + A_{66,68} + A_{69,74} + A_{73,109} + A_{1,116} = 1\\\\\n", + " & A_{1,94} + A_{2,87} + A_{3,79} + A_{4,8} + A_{10,15} + A_{1,16} + A_{26,30} + A_{29,71} + A_{9,35} + A_{36,40} + A_{37,50} + A_{41,78} + A_{42,102} + A_{57,60} + A_{66,69} + A_{74,77} + A_{88,99} + A_{109,110} + A_{82,116} = 1\\\\\n", + " & A_{1,95} + A_{4,10} + A_{16,21} + A_{26,31} + A_{1,36} + A_{37,52} + A_{41,87} + A_{42,96} + A_{57,62} + A_{66,71} + A_{74,79} + A_{109,112} + A_{116,119} = 1\\\\\n", + " & A_{1,96} + A_{3,100} + A_{4,11} + A_{5,10} + A_{6,92} + A_{16,22} + A_{17,21} + A_{26,32} + A_{27,31} + A_{36,42} + A_{1,37} + A_{6,38} + A_{41,88} + A_{42,104} + A_{3,46} + A_{47,52} + A_{48,62} + A_{52,95} + A_{53,112} + A_{57,63} + A_{66,72} + A_{67,71} + A_{74,80} + A_{75,79} + A_{83,87} + A_{105,109} + A_{116,120} + A_{117,119} = 2\\\\\n", + " & A_{1,97} + A_{4,12} + A_{6,52} + A_{11,112} + A_{16,23} + A_{26,33} + A_{36,43} + A_{37,53} + A_{5,38} + A_{41,89} + A_{42,105} + A_{1,57} + A_{10,63} + A_{66,73} + A_{74,81} + A_{51,84} + A_{71,101} + A_{109,113} + A_{114,116} = 1\\\\\n", + " & A_{1,98} + A_{4,13} + A_{16,24} + A_{26,34} + A_{36,44} + A_{37,54} + A_{41,90} + A_{42,106} + A_{57,64} + A_{1,66} + A_{74,82} + A_{71,93} + A_{109,114} + A_{116,121} = 1\\\\\n", + " & A_{1,99} + A_{2,5} + A_{14,17} + A_{15,27} + A_{35,37} + A_{22,45} + A_{16,46} + A_{45,47} + A_{48,55} + A_{23,49} + A_{17,50} + A_{21,51} + A_{52,91} + A_{53,107} + A_{18,54} + A_{19,67} + A_{1,75} + A_{20,83} + A_{24,117} = 1\\\\\n", + " & A_{1,100} + A_{4,32} + A_{3,5} + A_{9,96} + A_{15,17} + A_{25,27} + A_{22,35} + A_{26,37} + A_{11,41} + A_{46,47} + A_{48,56} + A_{10,51} + A_{31,52} + A_{53,108} + A_{61,92} + A_{65,67} + A_{30,75} + A_{1,83} + A_{115,117} = 1\\\\\n", + " & A_{1,101} + A_{5,7} + A_{12,72} + A_{13,63} + A_{17,19} + A_{27,29} + A_{37,39} + A_{44,57} + A_{47,49} + A_{48,59} + A_{52,93} + A_{53,73} + A_{6,54} + A_{67,68} + A_{69,75} + A_{83,85} + A_{22,107} + A_{11,114} + A_{1,117} = 1\\\\\n", + " & A_{1,102} + A_{5,8} + A_{1,17} + A_{27,30} + A_{37,40} + A_{47,50} + A_{48,60} + A_{52,94} + A_{53,110} + A_{67,69} + A_{75,77} + A_{78,83} + A_{22,99} + A_{82,117} = 1\\\\\n", + " & A_{1,103} + A_{3,42} + A_{5,9} + A_{10,88} + A_{17,20} + A_{1,27} + A_{11,31} + A_{37,41} + A_{4,46} + A_{47,51} + A_{48,61} + A_{52,87} + A_{53,111} + A_{67,70} + A_{75,78} + A_{83,86} + A_{22,91} + A_{43,108} + A_{117,118} = 1\\\\\n", + " & A_{1,104} + A_{5,11} + A_{17,22} + A_{27,32} + A_{37,42} + A_{1,47} + A_{48,63} + A_{52,96} + A_{53,105} + A_{67,72} + A_{75,80} + A_{83,88} + A_{117,120} = 1\\\\\n", + " & A_{1,105} + A_{5,12} + A_{6,11} + A_{10,112} + A_{13,106} + A_{17,23} + A_{18,22} + A_{27,33} + A_{28,32} + A_{37,43} + A_{38,42} + A_{47,53} + A_{1,48} + A_{52,97} + A_{53,113} + A_{13,54} + A_{58,63} + A_{59,72} + A_{10,62} + A_{63,104} + A_{64,120} + A_{67,73} + A_{75,81} + A_{76,80} + A_{83,89} + A_{84,88} + A_{92,96} + A_{114,117} = 2\\\\\n", + " & A_{1,106} + A_{5,13} + A_{6,120} + A_{7,105} + A_{17,24} + A_{27,34} + A_{37,44} + A_{47,54} + A_{48,64} + A_{12,49} + A_{52,98} + A_{53,114} + A_{22,55} + A_{11,59} + A_{1,67} + A_{75,82} + A_{83,90} + A_{93,112} + A_{117,121} = 1\\\\\n", + " & A_{1,107} + A_{2,6} + A_{7,23} + A_{8,114} + A_{14,18} + A_{15,28} + A_{35,38} + A_{45,48} + A_{50,117} + A_{55,58} + A_{19,59} + A_{13,60} + A_{63,99} + A_{24,64} + A_{33,65} + A_{12,69} + A_{1,76} + A_{20,84} + A_{91,92} = 1\\\\\n", + " & A_{1,108} + A_{3,6} + A_{15,18} + A_{25,28} + A_{26,38} + A_{46,48} + A_{29,55} + A_{33,56} + A_{32,57} + A_{56,58} + A_{59,65} + A_{34,60} + A_{28,61} + A_{27,62} + A_{63,100} + A_{64,115} + A_{30,76} + A_{1,84} + A_{31,92} = 1\\\\\n", + " & A_{1,109} + A_{4,6} + A_{10,53} + A_{11,43} + A_{16,18} + A_{26,28} + A_{36,38} + A_{37,48} + A_{57,58} + A_{59,66} + A_{5,62} + A_{42,63} + A_{64,116} + A_{49,71} + A_{74,76} + A_{41,84} + A_{1,92} + A_{12,96} + A_{33,100} = 1\\\\\n", + " & A_{1,110} + A_{2,73} + A_{6,8} + A_{13,81} + A_{1,18} + A_{12,24} + A_{28,30} + A_{38,40} + A_{48,50} + A_{7,55} + A_{58,60} + A_{59,69} + A_{63,102} + A_{64,82} + A_{76,77} + A_{78,84} + A_{92,94} + A_{72,99} + A_{33,115} = 1\\\\\n", + " & A_{1,111} + A_{6,9} + A_{18,20} + A_{1,28} + A_{38,41} + A_{48,51} + A_{58,61} + A_{59,70} + A_{63,103} + A_{64,118} + A_{76,78} + A_{84,86} + A_{87,92} + A_{33,108} = 1\\\\\n", + " & A_{1,112} + A_{4,105} + A_{5,97} + A_{6,10} + A_{18,21} + A_{28,31} + A_{12,37} + A_{1,38} + A_{33,46} + A_{48,52} + A_{11,57} + A_{58,62} + A_{59,71} + A_{63,96} + A_{64,119} + A_{76,79} + A_{84,87} + A_{92,95} + A_{106,116} = 1\\\\\n", + " & A_{1,113} + A_{6,12} + A_{18,23} + A_{28,33} + A_{38,43} + A_{48,53} + A_{1,58} + A_{59,73} + A_{63,105} + A_{64,114} + A_{76,81} + A_{84,89} + A_{92,97} = 1\\\\\n", + " & A_{1,114} + A_{2,107} + A_{5,117} + A_{6,13} + A_{7,12} + A_{18,24} + A_{19,23} + A_{28,34} + A_{29,33} + A_{38,44} + A_{39,43} + A_{48,54} + A_{49,53} + A_{2,55} + A_{58,64} + A_{1,59} + A_{63,106} + A_{64,121} + A_{5,67} + A_{68,73} + A_{69,81} + A_{73,113} + A_{76,82} + A_{84,90} + A_{85,89} + A_{92,98} + A_{93,97} + A_{101,105} = 2\\\\\n", + " & A_{1,115} + A_{3,7} + A_{8,34} + A_{9,24} + A_{15,19} + A_{25,29} + A_{26,39} + A_{46,49} + A_{56,59} + A_{18,61} + A_{65,68} + A_{30,69} + A_{2,70} + A_{73,108} + A_{44,74} + A_{13,78} + A_{1,85} + A_{31,93} + A_{100,101} = 1\\\\\n", + " & A_{1,116} + A_{4,7} + A_{16,19} + A_{26,29} + A_{36,39} + A_{37,49} + A_{57,59} + A_{40,65} + A_{44,66} + A_{38,67} + A_{66,68} + A_{69,74} + A_{35,70} + A_{39,71} + A_{43,72} + A_{73,109} + A_{41,85} + A_{1,93} + A_{42,101} = 1\\\\\n", + " & A_{1,117} + A_{6,54} + A_{5,7} + A_{11,114} + A_{17,19} + A_{22,107} + A_{27,29} + A_{37,39} + A_{47,49} + A_{44,57} + A_{48,59} + A_{13,63} + A_{67,68} + A_{69,75} + A_{12,72} + A_{53,73} + A_{83,85} + A_{52,93} + A_{1,101} = 1\\\\\n", + " & A_{1,118} + A_{2,90} + A_{3,82} + A_{7,9} + A_{13,15} + A_{16,44} + A_{19,20} + A_{1,29} + A_{39,41} + A_{49,51} + A_{59,61} + A_{8,65} + A_{68,70} + A_{69,78} + A_{73,111} + A_{85,86} + A_{87,93} + A_{101,103} + A_{81,108} = 1\\\\\n", + " & A_{1,119} + A_{7,10} + A_{19,21} + A_{29,31} + A_{1,39} + A_{49,52} + A_{59,62} + A_{68,71} + A_{69,79} + A_{73,112} + A_{85,87} + A_{93,95} + A_{96,101} + A_{44,116} = 1\\\\\n", + " & A_{1,120} + A_{5,64} + A_{7,11} + A_{12,106} + A_{19,22} + A_{29,32} + A_{39,42} + A_{1,49} + A_{13,53} + A_{59,63} + A_{6,67} + A_{68,72} + A_{69,80} + A_{73,105} + A_{60,75} + A_{85,88} + A_{93,96} + A_{101,104} + A_{44,109} = 1\\\\\n", + " & A_{1,121} + A_{7,13} + A_{19,24} + A_{29,34} + A_{39,44} + A_{49,54} + A_{59,64} + A_{1,68} + A_{69,82} + A_{73,114} + A_{85,90} + A_{93,98} + A_{101,106} = 1\\\\\n", + " & A_{i,j} free \\quad\\forall i \\in \\{1,2,\\dots,120,121\\}, j \\in \\{1,2,\\dots,120,121\\}\\\\\n", + " & κ \\geq 0\\\\\n", + "\\end{alignat*}\n", + " $$" + ], + "text/plain": [ + "Maximization problem with:\n", + " * 121 linear constraints\n", + " * 1 semidefinite constraint\n", + " * 7382 variables\n", + "Solver is SCS" + ] + }, + "execution_count": 9, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "SL_3ZZ" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "This includes:\n", + "1. defining $\\kappa$ as a variable\n", + "2. setting maximizing $\\kappa$ as the objective\n", + "3. defining matrix $A$ of $121\\times 121$ variables $a_{i,j}$ and adding constraints:\n", + " * $A\\succeq 0$, i.e. $A$ is positive semi-definite\n", + " * $(1, x, \\ldots)^* A (1, x, \\ldots)^T = \\Delta^2 - \\kappa\\Delta$ (this is achieved on the matrix: entries which contribute to a given poistion in the resulting vector are constrained by the appropriate coefficient in $\\Delta^2 - \\kappa\\Delta$." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Numeric solution\n", + "Which takes $< 30s$ (depending on the precision and processor&memory)" + ] + }, + { + "cell_type": "code", + "execution_count": 10, + "metadata": { + "ExecuteTime": { + "end_time": "2016-12-10T02:59:01.038061", + "start_time": "2016-12-10T01:58:13.520Z" + }, + "collapsed": false, + "scrolled": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "solution_status = :Optimal\n", + "κ = 7.100471806882364\n", + "----------------------------------------------------------------------------\n", + "\tSCS v1.1.8 - Splitting Conic Solver\n", + "\t(c) Brendan O'Donoghue, Stanford University, 2012-2015\n", + "----------------------------------------------------------------------------\n", + "Lin-sys: sparse-direct, nnz in A = 17465\n", + "eps = 1.00e-06, alpha = 1.80, max_iters = 20000, normalize = 1, scale = 5.00\n", + "Variables n = 7382, constraints m = 14884\n", + "Cones:\tprimal zero / dual free vars: 121\n", + "\tlinear vars: 1\n", + "\tsd vars: 14762, sd blks: 2\n", + "Setup time: 1.37e-02s\n", + "----------------------------------------------------------------------------\n", + " Iter | pri res | dua res | rel gap | pri obj | dua obj | kap/tau | time (s)\n", + "----------------------------------------------------------------------------\n", + " 0| inf inf -nan -inf inf inf 1.90e-02 \n", + " 100| 2.56e-03 1.30e-02 5.25e-03 -7.04e+00 -7.12e+00 2.31e-16 1.01e+00 \n", + " 200| 3.08e-04 1.59e-03 9.57e-05 -7.07e+00 -7.07e+00 2.34e-16 2.11e+00 \n", + " 300| 4.23e-05 1.34e-03 3.46e-05 -7.09e+00 -7.09e+00 2.34e-16 3.16e+00 \n", + " 400| 4.19e-05 1.25e-03 2.11e-05 -7.09e+00 -7.09e+00 2.36e-16 4.27e+00 \n", + " 500| 4.21e-05 1.17e-03 2.61e-05 -7.10e+00 -7.10e+00 2.37e-16 5.33e+00 \n", + " 600| 3.17e-05 1.77e-04 2.36e-05 -7.10e+00 -7.10e+00 2.39e-16 6.48e+00 \n", + " 700| 1.73e-06 2.86e-05 1.06e-06 -7.10e+00 -7.10e+00 2.39e-16 7.65e+00 \n", + " 800| 1.74e-07 5.93e-06 8.33e-08 -7.10e+00 -7.10e+00 2.39e-16 8.76e+00 \n", + " 900| 4.04e-08 1.36e-06 3.18e-09 -7.10e+00 -7.10e+00 2.39e-16 9.80e+00 \n", + " 940| 2.24e-08 7.63e-07 4.00e-09 -7.10e+00 -7.10e+00 2.39e-16 1.02e+01 \n", + "----------------------------------------------------------------------------\n", + "Status: Solved\n", + "Timing: Solve time: 1.02e+01s\n", + "\tLin-sys: nnz in L factor: 39791, avg solve time: 2.20e-04s\n", + "\tCones: avg projection time: 1.05e-02s\n", + "----------------------------------------------------------------------------\n", + "Error metrics:\n", + "dist(s, K) = 1.7899e-09, dist(y, K*) = 1.0969e-09, s'y/m = 3.7798e-14\n", + "|Ax + s - b|_2 / (1 + |b|_2) = 2.2377e-08\n", + "|A'y + c|_2 / (1 + |c|_2) = 7.6300e-07\n", + "|c'x + b'y| / (1 + |c'x| + |b'y|) = 3.9957e-09\n", + "----------------------------------------------------------------------------\n", + "c'x = -7.1005, -b'y = -7.1005\n", + "============================================================================\n" + ] + } + ], + "source": [ + "solution_status = solve(SL_3ZZ);\n", + "\n", + "κ = SL_3ZZ.objVal;\n", + "A = getvalue(getvariable(SL_3ZZ, :A));; # Pobiera wartość numeryczną\n", + "\n", + "@show solution_status;\n", + "@show κ;" + ] + }, + { + "cell_type": "code", + "execution_count": 15, + "metadata": { + "ExecuteTime": { + "end_time": "2016-12-10T02:59:02.715122", + "start_time": "2016-12-10T01:58:13.536Z" + }, + "collapsed": false + }, + "outputs": [ + { + "data": { + "image/svg+xml": [ + "\n", + "\n", + "\n", + " \n", + " j\n", + " \n", + " \n", + " 50\n", + " 100\n", + " \n", + " \n", + " \n", + " 0\n", + " 3\n", + " 2\n", + " -1\n", + " 1\n", + " 4\n", + " 5\n", + " 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230\n", + " 240\n", + " 250\n", + " \n", + " \n", + " i\n", + " \n", + "\n", + "\n", + " \n", + " \n", + "\n", + "\n", + "\n", + "\n", + "\n" + ], + "text/plain": [ + "Plot(...)" + ] + }, + "execution_count": 15, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "using Gadfly\n", + "Gadfly.spy(A)\n", + "# imshow(A)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Computed coordinates of $\\Delta^2 - \\kappa\\Delta$ in the basis (at the beginning)" + ] + }, + { + "cell_type": "code", + "execution_count": 16, + "metadata": { + "ExecuteTime": { + "end_time": "2016-12-10T02:59:03.357062", + "start_time": "2016-12-10T01:58:13.563Z" + }, + "collapsed": false + }, + "outputs": [ + { + "name": "stderr", + "output_type": "stream", + "text": [ + "WARNING: Scalars and coefficients ring are not the same! Trying to promote...\n", + "WARNING: Adding elements with different base rings!\n" + ] + }, + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Δ² - κ * Δ = Element of Group Algebra over Float64\n", + "[70.7943,-16.8995,-16.8995,-16.8995,-16.8995,-16.8995,-16.8995,-16.8995,-16.8995,-16.8995,-16.8995,-16.8995,-16.8995,1.0,2.0,1.0,1.0,1.0,2.0,2.0,1.0,1.0,1.0,2.0,1.0,2.0,1.0,1.0,1.0,2.0,2.0,1.0,1.0,1.0,1.0,1.0,2.0,1.0,1.0,1.0,2.0,2.0,1.0,1.0,1.0,1.0,1.0,2.0,1.0,1.0,1.0,2.0,2.0,1.0,1.0,1.0,1.0,1.0,2.0,1.0,1.0,1.0,2.0,2.0,1.0,1.0,1.0,1.0,2.0,1.0,1.0,1.0,2.0,1.0,1.0,1.0,1.0,2.0,1.0,1.0,1.0,2.0,1.0,1.0,1.0,1.0,2.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,2.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,2.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,2.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0]\n" + ] + } + ], + "source": [ + "@show (Δ² - κ*Δ);" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Checking the correctness" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Let us check that the computed (approximated!) floating point matrix $A$ actually is positively semi-definite (up to some floating point error!):" + ] + }, + { + "cell_type": "code", + "execution_count": 28, + "metadata": { + "ExecuteTime": { + "end_time": "2016-12-10T02:59:04.584117", + "start_time": "2016-12-10T01:58:13.572Z" + }, + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "rounded_eigenvalues_A = [-2.0e-9 -1.0e-9 -1.0e-9 -1.0e-9 -1.0e-9 -1.0e-9 -1.0e-9 -1.0e-9 -1.0e-9 -1.0e-9 -1.0e-9 -1.0e-9 -1.0e-9 -1.0e-9 -1.0e-9 -1.0e-9 -1.0e-9 -1.0e-9 -1.0e-9 -1.0e-9 -1.0e-9 -1.0e-9 -1.0e-9 -1.0e-9 -1.0e-9 -1.0e-9 -1.0e-9 -1.0e-9 -1.0e-9 -1.0e-9 -0.0 -0.0 -0.0 -0.0 -0.0 -0.0 -0.0 -0.0 -0.0 -0.0 -0.0 -0.0 -0.0 -0.0 -0.0 -0.0 -0.0 -0.0 -0.0 -0.0 -0.0 -0.0 -0.0 -0.0 -0.0 -0.0 -0.0 -0.0 -0.0 -0.0 -0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.0e-9 1.0e-9 1.0e-9 1.0e-9 1.0e-9 1.0e-9 1.0e-9 1.0e-9 1.0e-9 1.0e-9 1.0e-9 1.0e-9 1.0e-9 1.0e-9 1.0e-9 1.0e-9 1.0e-9 1.0e-9 1.0e-9 2.0e-9 2.0e-9 2.0e-9 5.85665 5.85665 6.15125 6.15125 6.15125 6.48966 6.48966 6.48966 6.61277 6.61277 7.93277]\n" + ] + } + ], + "source": [ + "rounded_eigenvalues_A = round(sort(eigvals(A)),TOL)'\n", + "@show rounded_eigenvalues_A;" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Since $A$ is also symmetric we can find its root $\\sqrt{A}$, i.e. a matrix satisfying $\\sqrt{A}\\sqrt{A}^T = A$:" + ] + }, + { + "cell_type": "code", + "execution_count": 29, + "metadata": { + "ExecuteTime": { + "end_time": "2016-12-10T02:59:06.911316", + "start_time": "2016-12-10T01:58:13.579Z" + }, + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "A == Symmetric(A) = true\n" + ] + } + ], + "source": [ + "@show A == Symmetric(A)\n", + "A_sqrt = real(sqrtm(Symmetric(A)));" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Warning\n", + "Due to floating-point arithmetic, knowledge of $\\sqrt{A}$ only gives an approximate decomposition of $\\Delta^2 - \\kappa\\Delta$ into sum of hermitian squares. Approximate i.e. if we set $X = (1, x, x^{-1}, \\ldots)$, then \n", + "\n", + "$$\\left\\| (\\Delta^2 - \\kappa\\Delta) - X^*AX^{T}\\right\\|_\\infty < 1.1\\cdot 10^{-6}.$$" + ] + }, + { + "cell_type": "code", + "execution_count": 30, + "metadata": { + "ExecuteTime": { + "end_time": "2016-12-10T02:59:07.528636", + "start_time": "2016-12-10T01:58:13.587Z" + }, + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "floating_point_distance = 2.138915533578256e-8\n" + ] + }, + { + "name": "stderr", + "output_type": "stream", + "text": [ + "WARNING: Scalars and coefficients ring are not the same! Trying to promote...\n", + "WARNING: Adding elements with different base rings!\n" + ] + } + ], + "source": [ + "floating_SOS = resulting_SOS(A_sqrt, Δ)\n", + "\n", + "floating_point_distance = norm(floating_SOS - (Δ²-κ*Δ), Inf)\n", + "@show floating_point_distance;" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Checking Rationally" + ] + }, + { + "cell_type": "code", + "execution_count": 31, + "metadata": { + "ExecuteTime": { + "end_time": "2016-12-10T02:59:14.057069", + "start_time": "2016-12-10T01:59:12.690Z" + }, + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "rational_distance = 2.478455540080956980681875675141175225375506936376126899286240020874535316025564e-08\n" + ] + }, + { + "name": "stderr", + "output_type": "stream", + "text": [ + "WARNING: redefining constant EOI\n" + ] + } + ], + "source": [ + "A_sqrt_rational = Array{Rational{BigInt},2}(round((10^TOL)*A_sqrt,0))\n", + "\n", + "Δ_rational = convert(Rational{BigInt}, Δ )\n", + "Δ²_rational = convert(Rational{BigInt}, Δ²);\n", + "@assert Δ_rational*Δ_rational == Δ²_rational\n", + "\n", + "κ_rational = rationalize(BigInt, κ)\n", + "# @show(norm(Δ-Δ_rational, Inf))\n", + "# @show(norm(Δ²-Δ²_rational, Inf))\n", + "# @show(norm(κ-κ_rational))\n", + "# @show(maxabs((10^TOL)*A_sqrt-A_sqrt_rational)/10^TOL);\n", + "\n", + "const EOI = Δ²_rational - κ_rational*Δ_rational\n", + "\n", + "rational_SOS = resulting_SOS(A_sqrt_rational, Δ)//10^(2*TOL) \n", + "rational_distance = norm(rational_SOS - EOI, Inf)\n", + "@show rational_distance;" + ] + }, + { + "cell_type": "code", + "execution_count": 32, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "A_sqrt_corrected = correct_to_augmentation_ideal(A_sqrt_rational);" + ] + }, + { + "cell_type": "code", + "execution_count": 33, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "corrected_rational_distance = 6890048458160433089//278140993900000000000000000\n", + "float(corrected_rational_distance) = 2.477178340937981774070305427207290927854845779351333510856487954758832836686713e-08\n", + "corrected_rational_SOS = Element of Group Algebra over Rational{BigInt}\n", + 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+ ] + } + ], + "source": [ + "corrected_rational_SOS = resulting_SOS(A_sqrt_corrected, Δ_rational)//10^(2*TOL)\n", + "corrected_rational_distance = maxabs((corrected_rational_SOS - EOI).coordinates)\n", + "@show corrected_rational_distance;\n", + "@show float(corrected_rational_distance)\n", + "@show corrected_rational_SOS;" + ] + }, + { + "cell_type": "code", + "execution_count": 38, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "norm(floating_SOS - (Δ² - κ * Δ),Inf) = 1.2806626159544976e-7\n", + "norm(rational_SOS - (Δ² - κ * Δ),Inf) = 1.166245585359110984718427062034606933593749999999999999999999999999999995038896e-06\n", + "norm(corrected_rational_SOS - (Δ² - κ * Δ),Inf) = 1.166384273185079156291407049740364737022477545932654873301004029779386648940086e-06\n" + ] + }, + { + "name": "stderr", + "output_type": "stream", + "text": [ + "WARNING: Scalars and coefficients ring are not the same! Trying to promote...\n", + "WARNING: Adding elements with different base rings!\n", + "WARNING: Scalars and coefficients ring are not the same! Trying to promote...\n", + "WARNING: Adding elements with different base rings!\n", + "WARNING: Adding elements with different base rings!\n", + "WARNING: Scalars and coefficients ring are not the same! Trying to promote...\n", + "WARNING: Adding elements with different base rings!\n", + "WARNING: Adding elements with different base rings!\n" + ] + } + ], + "source": [ + "@show norm(floating_SOS - (Δ²-κ*Δ), Inf)\n", + "@show norm(rational_SOS - (Δ²-κ*Δ), Inf)\n", + "@show norm(corrected_rational_SOS - (Δ²-κ*Δ), Inf);" + ] + } + ], + "metadata": { + "kernelspec": { + "display_name": "Julia 0.5.0", + "language": "julia", + "name": "julia-0.5" + }, + "language_info": { + "file_extension": ".jl", + "mimetype": "application/julia", + "name": "julia", + "version": "0.5.0" + }, + "toc": { + "toc_cell": false, + "toc_number_sections": true, + "toc_threshold": 6, + "toc_window_display": false + } + }, + "nbformat": 4, + "nbformat_minor": 1 +} diff --git a/SL(3,ZZ)_GAP_Group_Constraints.gws b/SL(3,ZZ)_GAP_Group_Constraints.gws new file mode 100644 index 0000000..6ba826c Binary files /dev/null and b/SL(3,ZZ)_GAP_Group_Constraints.gws differ diff --git a/basis.SL(3,Z) b/basis.SL(3,Z) new file 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