note

[Mathematica覚書]

結合則を満たす２項演算子を定義する方法．例えば，2項演算子f:(g,h)->f(g,h)をg・h (CenterDot[g,h])と表記して，g1・g2・g3・...・gn を定義したければ， CenterDot[z__] := Fold[f, {z}]; とする．ここで，__はBlankSequence．

Stringの間に下付き添字を入れる方法．（LaTeX形式に変換なされないようだ．） (link)

ショートカットキー (link)

Pfaffianの実装例 (link)

(Mathematica 11.2)(12.1で修正済み) 対角成分のみ非ゼロの行列において，対角成分の一部にゼロを含む場合に，
SmithDecomposition[m][[1]].m.SmithDecomposition[m][[3]]==SmithDecomposition[m][[2]]
が不成立．例えば，
m = {{0, 0, 0}, {0, -2, 0}, {0, 0, 2}}

(Mathematica 11.2) 次の行列のLatticeReduceを取ると，フリーズする．
{{0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2, 0, 0, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {-1, 0, 0, 1, -1, 1, -1, 0, 0, 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, -2, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 2, 0, 0, -1, 1, -1, 1, 0, 0, 0, 0, 2, 0, 0, 1, 1, 1, 1, 1, 0, 0, 1}, {1, 0, 2, 0, -1, 1, 1, -1, 0, 0, 3, -1, 1, -1, 0, -1, 1, -1, 0, 0, -1, 3, 0, 0, 1, 1, 1, 1, 0, 1, 0, 1}, {1, 0, 2, 0, -1, 1, 1, -1, 0, 0, 3, -1, 0, -1, 1, -2, 1, 0, 0, 0, -1, 3, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2, -2, 0, 0, 2, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 3, 1, -2, 2, 1, -1, 0, 0, 4, -2, 1, -1, 0, -2, 1, -1, 0, 0, -2, 4, 0, 0, 1, 1, 1, 1, 0, 0, 0, 2}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2, 0, 2, 2, 0, 2, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, -1, 0, 1, -1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, -1, 0, 1, -1, 0, 1, 0, 0, -2, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, -2, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, -2, -2, 0, -2, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0}, {0, 0, 1, 1, 0, 0, -1, -1, 0, 0, 2, 0, -1, -1, 0, -2, 1, -1, 0, 0, 0, 0, 0, 0, 1, -1, -1, -1, 0, 0, 0, 0}, {0, 0, 1, 1, 0, 0, -1, -1, 0, 0, 0, 0, -1, -1, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, -1, 1, 1, -1, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -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, -2, -2, 2, -2, 2, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 2, 2, 0, 0, -2, -2, 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, -2, 0, 0, 2, 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, 2, 0, 0, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, -2, 2, 2, -2, 2, -2, -2, 0, 0, -2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, -2, 0, 0, 0, 0, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, -2, 2, 2, -2, 2, -2, -2, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0, -2, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, -2, 2, 0, -2, 2, 0, -2, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0, -2, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 1, -1, 1, -1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, -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, -1, -1, 0, 0, 1, 1, 0, 0, -2, 0, 1, 1, 0, 2, -1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0}, {-2, 0, -1, 1, 0, 0, -1, 1, 0, 0, -2, 0, -1, -1, 0, 0, 1, 1, 0, 0, 0, -2, 0, 0, -1, -1, -1, -1, 0, 0, 0, 0}, {0, -2, 1, 1, -2, 0, -1, -1, 0, 0, 0, -2, -1, -1, 0, -2, 1, 1, 0, 0, -2, 0, 0, 0, -1, -1, -1, -1, 0, 0, 0, 0}, {0, -2, 0, 0, 0, 0, -2, 0, 0, 0, -4, 0, 0, 0, 0, 2, 0, 2, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, -2, -1, 1, 0, 0, -1, -1, 0, 0, -2, 0, -1, -1, 0, 0, 1, 1, 0, 0, 0, -2, 0, 0, -1, -1, -1, -1, 0, 0, 0, 0}, {0, -4, 2, 2, -2, 2, -2, -2, 0, 0, -2, -2, 0, 0, 0, 0, 0, 2, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}}

(Mathematica 11.2) 以下を実行すると，警告なく強制終了する．
tmp = {{-(1/8), 1/2, 3/8}, {-(1/8), 1/2, 1/2}, {-(1/8), 1/2, 5/ 8}, {-(1/8), 1/2, 3/4}, {-(1/8), 5/8, 3/8}, {-(1/8), 5/8, 1/ 2}, {-(1/8), 5/8, 5/8}, {-(1/8), 5/8, 3/4}, {0, 1/2, 1/4}, {0, 1/ 2, 3/8}, {0, 1/2, 1/2}, {0, 1/2, 5/8}, {0, 1/2, 3/4}, {0, 5/8, 3/ 8}, {0, 5/8, 1/2}, {0, 5/8, 5/8}, {0, 5/8, 3/4}, {0, 3/4, 5/ 8}, {0, 3/4, 3/4}, {0, 7/8, 5/8}, {0, 7/8, 3/4}, {1/8, 1/2, 1/ 4}, {1/8, 1/2, 3/8}, {1/8, 1/2, 1/2}, {1/8, 1/2, 5/8}, {1/8, 1/2, 3/4}, {1/8, 5/8, 3/8}, {1/8, 5/8, 1/2}, {1/8, 5/8, 5/8}, {1/8, 5/ 8, 3/4}, {1/8, 5/8, 7/8}, {1/8, 3/4, 5/8}, {1/8, 3/4, 3/4}, {1/8, 3/4, 7/8}, {1/8, 7/8, 5/8}, {1/8, 7/8, 3/4}, {1/8, 7/8, 7/8}, {1/ 4, 1/2, 1/4}, {1/4, 1/2, 3/8}, {1/4, 1/2, 1/2}, {1/4, 1/2, 5/ 8}, {1/4, 1/2, 3/4}, {1/4, 5/8, 3/8}, {1/4, 5/8, 1/2}, {1/4, 5/8, 5/8}, {1/4, 5/8, 3/4}, {3/8, 1/2, 1/4}, {3/8, 1/2, 3/8}, {3/8, 1/ 2, 1/2}, {3/8, 1/2, 5/8}, {3/8, 1/2, 3/4}, {3/8, 5/8, 3/8}, {3/8, 5/8, 1/2}, {3/8, 5/8, 5/8}, {3/8, 5/8, 3/4}, {1/2, 1/2, 1/4}, {1/ 2, 1/2, 3/8}, {1/2, 1/2, 1/2}, {1/2, 1/2, 5/8}, {1/2, 1/2, 3/ 4}, {1/2, 5/8, 3/8}, {1/2, 5/8, 1/2}, {1/2, 5/8, 5/8}, {1/2, 5/8, 3/4}, {5/8, 1/2, 1/4}, {5/8, 1/2, 3/8}, {5/8, 5/8, 3/8}, {3/4, 1/ 2, 1/4}};
ConvexHullMesh[tmp]

Mathematica 11.2のConvexhullmeshは有理数成分の座標に対して正しい凸包を与えない場合があるので，COnvexhullmeshを用いる際は，ランダムに微小な摂動を加えて使うと良い．例えば，以下の点集合に対してConvexhullmeshは正しい凸包を与えない．
{{0, 1/2, 1/8}, {0, 1/2, 0}, {0, 1/2, -(1/8)}, {0, 0, 0}, {0, 0, 3/8}, {0, 0, -(3/8)}, {1/2, 0, (1/8)}, {1/2, 0, 0}, {1/2, 0, -(1/8)}, {1/4, 1/4, -(1/4)}, {1/4, 1/4, 1/4}, {0, 1/4, -(1/4)}, {0, 1/4, 1/4}, {1/4, 0, -(1/4)}, {1/4, 0, 1/4}}
しかし，微小な摂動を与えると数値的には正しい凸包が得られる．

[home]