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外積と行列式

数学

最近、電磁気の勉強をしていて、ベクトルの外積が行列式で表されることを知りました。


ベクトルの外積とは、ベクトルの掛け算のようなもので、\vec{A} \times \vec{B}と表されます。
\vec{A}と\vec{B}のなす角を\thetaとすると、

\vec{A}\times\vec{B}=\left|\vec{A}\right|\left|\vec{B}\right|\sin{\theta} と定義されます。
内積のcosがsinになった感じですね。


特に二次元の時は、\vec{A}=\left(a_x, a_y\right)、\vec{B}=\left(b_x,b_y\right)とすると
\vec{A}\times\vec{B}=a_x b_y - a_y b_x
となり、これは二つのベクトルがなす平行四辺形の面積に等しいです。


…と、ここまでは知っていたのですが
三次元のベクトルの場合、ベクトルの外積はスカラでなくベクトルになります。
しかもそれが、
\vec{A}\times\vec{B}=\left| \begin{array}{ccc}\vec{i} & \vec{j} & \vec{k} \\a_x & a_y & a_z \\b_x & b_y & b_z \end{array} \right|
という、非常に綺麗な行列式に纏まることにびっくり。
(\vec{i},\vec{j},\vec{k}はそれぞれx方向, y方向, z方向の基本ベクトルです)


証明とかは出来ませんので省きますが、この事実に感銘を受けました。

無限の猿定理

数学

「無限の猿定理」と呼ばれる，確率論の定理があります。

「ランダムに文字列を打ち続けていれば，そのうちシェイクスピアの作品を打ち出す」という定理です。
確かに時間が無限大にあったら，そのうちあらゆる文字列が出てきそうではあります。

ところで，実際にプリマス大学という大学で，6匹のクロザルの檻の中にキーボードを入れてタイプさせたそうです。
5ページ程に渡る文字列が入力されたとか。

その時の文字列はこちら。

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異常にsが多いですね…。

11の倍数判定問題

数学

ICPCアジア地区予選の模擬選があったので、参加資格は無いが問題だけ貰い解いてみた。
その中に11の倍数を判定する問題があったので、
11の倍数というものは10進数表示した整数にどのような特徴で現れるのかを調べてみた。

　
まず、1桁の場合。
数をa_0と置くと、これが11の倍数であるのは
a_0=0
の場合だけである。
　
次に、2桁の場合。
数を10a_1+a_0と置くと、これが11の倍数であるのは
a_1=a_0
の場合だけである。
　
更に、3桁の場合。
数を100a_2+10a_1+a_0と置く。
直感では分からないので、100a_2+10a_1+a_0=11(10b_1+b_0)となるような数b_1, b_0を考える。
右辺を展開して整理すると、
\,11(10b_1+b_0)\\=110b_1+11b_0\\=100b_1+10b_1+10b_0+b_0\\=100b_1+10(b_1+b_0)+b_0
左辺と比較すると、
b_1=a_2,\,b_1+b_0=a_1,\,b_0=a_0
が得られる。
真ん中の式に左右の式を代入してやると、
a_0+a_2=a_1
となる。
つまり、100の位の数と1の位の数の合計が10の位の数となるわけだ。
実際には、10(b_1+b_0)が繰り上がりを起こすことがあるため、
a_0+a_2\eq a_1\,(mod\,11)とすれば全ての場合に対して正しくなる。
　
4桁の場合は、説明は省略するけれど
1000a_3+100a_2+10a_1+a_0は11の倍数ならa_0+a_2\eq a_1+a_3\,(mod\,11)が成り立つ。
ここで、
1桁の場合の11の倍数である条件を、a_0\eq 0\,(mod\,11)　と変形し、
2桁の場合の11の倍数である条件を、a_0\eq a_1\,(mod\,11)　と変形する。

そしてこれを一般化してわかることは、
「任意の整数nが11の倍数である時、nの偶数桁の数字の合計と奇数桁の数字の合計は、11を法として合同である」ということ(証明省略)。
そしてこれは実は必要十分条件であるので(また証明省略)、この命題の逆を取って
「任意の整数nの偶数桁の数字の合計と奇数桁の数字の合計が11を法として合同であるとき、nは11の倍数である」が成り立つ。
こういうこと。
　
　
ちなみに、Wikipediaの11のページにちょっと違う形で普通に書いてあった。。

相加相乗平均の定理

数学

数学ガールを読んで発見。
高校では、相加相乗平均の関係式を
\frac{a+b}{2} \geq \sqrt{ab}
こう習うのだが、これを任意の数列aに対して

\Large\(\Large\sum_{k=1}^{n}a_k \Large\)\cdot \Large\frac{\Large 1}{\Large n} \geq \Large\(\Large\prod_{k=1}^{n}a_k\Large\)^{\frac{1}{n}} \hspace{50} (n\geq\1)

と拡張できる。
n=2の時、高校で習う式と同じになる。
この左辺と右辺の式の形が類似している所に感動した。
　
// tex難しいのぉ