情報教育と数学の関わり

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1 (Hideki Yamasaki) Hitotsubashi University $*$ 1 3 [1]. 1. How To [2]. [3]. $*E-$ -mail:yamasaki.hideki@r.hit-u.ac.jp

2 2 1, ( ) AND $(\wedge)$, $OR$ $()$, NOT $(\neg)$ ( ) [4] ( ) $=$ $<$ $=$ ( ) 69 3 $\sum_{n=1}^{\infty}\frac{1}{n}=\infty$ 2 ( ) ( 7 8 ) 15.4 $\sim$ $n$ $\log n$ $k$ $e^{k}$ 2 4 $)$. $1A$ B $B$ A ( )

3 $\cross$ $\subseteq$ $\cross$ $(i,j)$ $j$ ( $i$ ) ( ) SQL SELECT ( )FROM( )WHERE( ) ( ) 2 $\subseteq$ $ID\cross$ $\cross$ $\cross$ $ID\cross$ $ID\cross$

4 71 80 SELECT FROM WHERE $ID=$ $ID$ $\wedge$ $=$ $\wedge$ $\geqq$ 80 $\cross$ ( $ID=$ $ID\wedge$ $=$ $\wedge$ $)$ $\geqq$ 80 5 ( ) $(:=)$ ( ) 1. // 2. $\langle$ $\mapsto$ $\mapsto]$ 3. $\langle$ $\mapsto$] $arrow$ $\cdots$ ml $\langle$ 4.

5 72 $\mapsto]$ [ ] 5. $\langle$ ( $\langle$ ) $[$ $]$ 6. $\mapsto]$ $[$ $]$ $\cdots$ $\mapsto]$ 7. $\langle$ 8. ( ) ( ) [ ] 9. [ ] ( ) ( ) : : : - : : - :

6 73 (a,b), $(a,b], [a,b)$, m. n [m..n] : [a,b] (1 ): $-$ (2 ): $\cdots$ $\cdots$ : $\cdots$ : 1. $n$ $A$ $A^{-1}$ ( $n\cross m$ $B$ $i$ $j$ ) $[$ $B:=P(i,j)B]$ $k$ [1..m] $B_{i,k}$ $ab_{j,k}$ ($n\cross m$ $B$ a) $[$ $B:=Q(i, a)b]$ $i$ $k$ [1..m] $B_{i,k}$ $a$ ( $n\cross m$ $B$ $i$ $i$ $k$ [1..m] a) $[$ $B:=R(i,j, a)b]$ $B_{i,k}$ $ab_{j,k}$

7 74 $//n$ $A$ $A^{-1}$ $B:=n\cross 2n$ $A E$ $B$ // $E A^{-1}$ [1..n] $j$ $\langle B_{j..n,j}$ $//A$ $=ej$ $\langle ab_{i,j}=1\rangle$ $a$ $i\in b\cdot\cdot n$] ( $B$ a) $i$ ( $B$ $i$ $i$ ) $//B_{j,j}=1$ j$ $i$ $//B$ $j$ [1..n] $\langle ( $B$ $i$ $i$ $B_{i,j}$ ) $//B$ $i$ $=e_{j}$ $B$ // $E A^{-1}$ $A^{-1}:=B$ 2. $f(x)$ $\langle$ $f(x) $ ( ) $(a g(x)+bh(x)) $ $a$ $g(x) +bh(x) $ $(g(x)h(x)) $ $g(x) h(x)+g(x)h(x) $ $( \frac{g(x)}{h(x)}) $ $\frac{g(x) h(x)-g(x)h(x) }{h(x)^{2}}$ $(g(x)^{n}) $ $ng(x)^{n-1}g(x) $ $(h(g(x))) $ $h (g(x))g(x) $ $(h (g(x)))$ $\sin g(x)$ $\cos g(x)$ $\cos g(x)$ $\sin g(x)$ $a^{\prime g(x)}$ $\log$ $a$ $a^{g(x)}$ $\log_{a} g(x)$ $\frac{1}{\log ag(x)}$

8 75 $(h^{-1}(g(x))) $ $\frac{g(x) }{h(y)}$ $//h(y)=g(x)$ $($ $h(g(x)))$ $\sin g(x)$ $h (y)$ $\sqrt{1-g(x)^{2}}$ // $[-\pi/2, \pi/2],$ $h (y)=\cos(y)=\sqrt{1-h(y)^{2}}=\sqrt{1-g(x)^{2}}$ $\tan g(x)$ $h (y)$ $1+g(x)^{2}$ // $[-\pi/2,$ $\pi/2,h (y)=1/\cos^{2}(y)=1+h(y)^{2}=1+g(x)^{2}$ $x$ $a$ 1 $0$ $\Rightarrow$ ( ) ( ) ( $=$ ) ( ) 5.1 ( ) : ( ) ( )

9 ( ) $d_{i}\cross d_{i+1}$ $k,$ $i$ $A_{i}$ $A_{i}\cdot A_{i+1}\cdots A_{i+k}$ $A_{0}\cdot A_{1}\cdots A_{n}$ Google [5] Web Google Web $L$ $N(=81$ $)$ 0,1 $L_{i,j}:=\{\begin{array}{l}1 i i 0 ow.\end{array}$ $(e\succ$ $\Leftrightarrow$ ) ( 1) $\alpha\in[0,1)$ $N$ $G(\alpha)$ $G(\alpha)_{i,j}:=$ $\{\begin{array}{ll}\alpha L_{i,j}/c_{i}+(1-\alpha)/N :=\sum_{j}h_{i,j}>0 1/N c_{i}=0 \end{array}$ $\alpha$ 2. $p$ $pg(\alpha)=p$ $G$ ( $N^{3}$ $\sum_{j}p_{j}=1$ ) 2 $G(\alpha)$ [5]

10 $\alpha$ 77 $x$ $\lim_{n}xg(\alpha)^{n}=p$ $c_{i}:= \sum_{j}l_{i,j}$ $( xg(\alpha))_{j}=\sum_{i}x_{i}g(\alpha)_{i,j}=\alpha\sum_{l_{i,j}=1}x_{i}/c_{i}+(\alpha\sum_{c_{i}=0}x_{i}+1-\alpha)/n$ 2 1 $( \alpha\sum_{c_{i}=0}x_{i}+1-\alpha)/n$ $i$ $C_{i}$ $xg(\alpha)$ 10 $N$ $G(\alpha)$ 1 $xg(\alpha)^{n}arrow p$ $\alpha^{n}arrow 0$ $\alpha=0.85$ $\sim$ Google 6.2 RSA RSA RSA [3]. 1. $P$ $x\neq$ Omod $p$ $x^{p-1}=1mod p$ $k$ $p,$ $q$ $x^{k(p-1)(q-1)+1}=xmod pq$ 2. $+$ mod $p,$ $q$ $(e,pq)$ $ed=$ lmod $(p-1)(q-1)$ $(d,pq)$ $((e,pq)$ $(d,pq))$ $)$ $(y^{d}mod pq$ $)$ $(x^{e}mod pq$

11 78 3. $(e,pq)$ $p,$ $q$ $(e,pq)$ $(d,pq)$ 6.3 $\Sigma,$ $\Sigma$ $\Sigma$ $\Sigma^{*}:=\{a_{1}a_{2}\ldots a_{n} a_{1},$ $a_{2},$ $\ldots,$ $a_{n}\in$ $\Sigma$ $n\geq 0\}$ ( ) $\Sigma^{*}$ ( ) $\Sigma^{*}$ ( ) ( ) $\Sigma^{*}$ $(\cup, \cap, \cdot C)$ $\Sigma_{1},$ $\Sigma_{2},$ $\Sigma_{n}$ $\ldots,$ $n$ $\prod\sigma_{i}^{*}:=\sigma_{1}^{*}\cross\sigma_{2}^{*}\cross\cdots\cross\sigma_{n}^{*}$ $n$ ( ) $\prod\sigma_{i}^{*}$ ( ) $\prod\sigma_{i}^{*}$ ( ) ( ) $n$ $n$ $\prod\sigma_{i}^{*}$ $n$ 2

12 79 $n$ [6] free group fully ordered fully ordered group fully ordered 7 How To $\sim$ [1], (2004), http: $//www$.mext.go.$jp/a$-menu/shotou/zyouhou/ htm [2], ISSUE BRIEF NUMBER 604(2008), http: $//www$. ndl. gojp/jp/data/publication/issue/0604. pdf [3] (2008) [4] A. K. Dewdney, A Tinkertoy computer that plays tic-tac-toe, SCIENTIFIC AMER- ICAN October 1989 http: $//www$. rci.rutgers. edu/cfs$/472_{-}htm1/$ Intro$/TinkertoyComputer/$ TinkerToy.html [5] Langbille Meyer Google PageRank (2009) [6] Tero Harju and Juhani Karhumaki, The Equivalence Problem of Multitape Finite Automata, http: //citeseerx.ist.psu.edu/viewdoc/download?doi $= $. 1849&rep $=$repl&type$=pdf$