WPA(Win Probability Added) 1 WPA WPA ( ) WPA WPA WPA WPA WPA
|
|
|
- えいじろう いちぞの
- 7 years ago
- Views:
Transcription
1
2 WPA(Win Probability Added) 1 WPA WPA ( ) WPA WPA WPA WPA WPA
3 1 1 2 WPA WPA(Win Probability Added) () WPA A m m A 29 A A
4 1 [7, 8, 9, 12, 14] 1959 Lindsey [13] 1 Bukiet, Harold and Palacios [4] 9 9 Bukiet [4] ( ) 2 Bukiet WPA(Win Probability Added) , , = WPA WPA WPA () Win Expectancy Finder web [18] [17] 1 1 ( ) p(1 ) x λ x λ x
5 () WPA 6 9 WPA 4 1 () ( ) WPA Bukiet, Harold and Palacios [4] [15] Bukiet WPA () WPA WPA WPA 2012 vs 6 ( ) 1 WPA WPA 1 WPA WPA WPA 0 [10] [16] WPA WPA 2 ( [15])
6 5 2 WPA 2.1 WPA(Win Probability Added) 1 WPA(Win Probability Added) 1 WPA 1 1 WPA WPA WPA WPA WPA WPA WPA 3 WPA WPA () 2.2 WPA 1 WPA [15] WPA t 1, t 2, t 3, i j i j 3 1 Leverage Index WPA Leverage Index 3
7 1: 1 i j (i, j) P P = P n i(i = 1, 2, 3) u n i ( u n 1 u n 2 u n 3 ) ( = u n 1 1 u2 n 1 u n 1 3 ) [4] (0 1 2 ) 8 ( ) : i j (i, j) P P A 0 B 0 C 0 D 0 P = O A 1 B 1 E 1 O O A 2 F 2 O O O 1 A i (i = 0, 1, 2) i i A i (i = 0, 1, 2) ( 2 ) 8 8 B i (i = 0, 1) 4
![2 1 [4] 25 1 25 3 (0 1 2 ) 8 ( ) 24 3 25 1 1: 25 0 1 2 3 4 5 6 7 8 1 9 10 11 12 13 14 15 16 2 17 18 19 20](/docs-images/41/22535782/images/page_7.jpg "21 22 23 24 3 25 1 i j (i, j) P P A 0 B 0 C 0 D 0 P = O A 1 B 1 E 1 O O A 2 F 2 O O O 1 A i (i = 0, 1, 2)")
8 3 C 0 3 D 0, E 1, F 2 3 D E F P 0 P P (16,13) A 1 (8,5) () 9 [4] P (16, 9) [4, 15]4 2 D Esopo and Lefkowitz [6] 5 2: D Esopo and Lefkowitz D Esopo and Lefkowitz [1] [4] P A B O O P = O A B O O O A F O O O D Esopo and Lefkowitz
9 C 0, D 0, E 0 0 A, B A, B, F 1 p s, p d, p t, p h, p w, p out A, B, F p h p s + p w p d p t p h 0 0 p t p s + p w 0 p d 0 p h p s p d p t p w A = p h p s p d p t 0 p w 0 0 p h 0 0 p t p s 0 p d p w p h 0 0 p t p s 0 p d p w p h p s p d p t p w p h 0 0 p t p s 0 p d p w B = p out I p out F =. B p out n s s 25 u n j (j ) P j 7 n j P j u n 1 u n u 0 = ( ) u n = u n 1 P j (1) n r (r = 0, 1, 2, ) s (r, s) U n 8 r R max U n (R max + 1) 25 u n (1) U n U n 1 P P (0), P (1), P (2), P (3), P (4) P (1) (i, j) i 1 j () () 1() () () ( r) P (r) (i, j) ˆr r P (ˆr) (i, j) 0 P P (0) + P (1) + P (2) + P (3) + P (4) 7 8 U n
10 D Esopo and Lefkowitz P (r) (r = 0, 1, 2, 3, 4) A (0) B O O P (0) = O A (0) B O O O A (0) F O O O 1 A (r) O O O P (r) = O A (r) O O O O A (r) O (r = 1, 2, 3, 4) O O O 0 0 p s + p w p d p t p s + p w 0 p d p w A (0) = p w p w p w p w p h p t p s p d p t A (1) = 0 p s p d p t p s 0 p d p s 0 p d p w p h p h A (2) = p h p t p t p s p d p t p s 0 p d A (3) = p h p h p h p t
11 A (4) = p h U n r U n r U n r n j P j U n r = U n 1 r 0 P (0) j + U n 1 r 1 P (1) j + U n 1 r 2 P (2) j + U n 1 r 3 P (3) j + U n 1 r 4 P (4) j (2) 1 U 0 = R max (2) U n U n 25 n 3 U n (r, 25) n r 1 n U U n 1 9 U n 9 R max U n n (R max + 1) 25 U n r U n r u n 25 U n 25 ( n ) (R max + 1) u n 25 1 u n 25 (1-)( n ) q (ij) i j 8
12 R q(ij) i j (R max + 1) p (m,n) m n R p(m,n) m n m (R max + 1) R (m) m (R max + 1) m q (ij) R q(ij) u n 25 1 n n u n 25 1 u n i n(n = 1, 2, ) n j 1 ( j ) q (ij) i(i = 1, 2,, 9) q (ij) u n 25 u n 1 25 n i n(n = 1, 2, ) n j 1 ( j ) () R q(ij) i(i = 1, 2,, 9) R q(ij) n 1 n u n ε i j i, j R q(ij) R max q (ij) R q(ij) 2 Step1. i = 1 Step2. n = 0, j = i Step3. n = n + 1 Step4. r = 0, 1, 2,, R max U n r = U n 1 r 0 P (0) j + U n 1 r 1 P (1) j + U n 1 r 2 P (2) j + U n 1 r 3 P (3) j + U n 1 r 4 P (4) j Step5. j = (j mod 9) + 1 Step6. q (ij) = q (ij) + ( u n 25 1 u n ), R q(ij) = R q(ij) + (u n 25 u n 1 25 ) Step7. u n ε Step3. Step8. Step9. j = 1, 2,, 9 R q(ij) i = i + 1 i 9 Step2. i = 10 2: q (ij) R q(ij) q (ij) m n p (m,n) 1 1 p (1,1) = 1, p (1,n) = 0 (n = 2, 3,, 9) p (m,n) (m = 2, 3,, 10) q (ij) 9 p (m,n) = p (m 1,k) q (kn) (m = 2, 3,, 10, n = 1,, 9) (3) k=1 9
13 p (m,n) m n m R p(m,n) (R max + 1) c (m 1,k,n) i c (m 1,k,n)i = R p(m 1,k)r R q(kn)i r (i = 0, 1,, R max ) r=0 R p(m 1,k)r m 1 k m 2 r R q(kn)i r k n i r c (m 1,k,n)i m 1 k m n m 1 i m n m 1 k p (m 1,k), q (kn), p (m,n) p (m 1,k) q (kn) p (m,n) R p(m,n) (m = 2, 3,, 10) R p(m,n) = 9 k=1 p (m 1,k) q (kn) p (m,n) c (m 1,k,n) (m = 2, 3,, 10, n = 1,, 9) (4) 1 0 ( R p(1,n) = ) T (n = 1, 2,, 9) R p(m,n) m R (m) 10 ( 9 ) 10 n p (10,n) R p(10,n) R (10) = 9 p (10,n) R p(10,n) (5) n= A B A 9 R (10) B 9 R (10) A i B i 1 A 9 A P W (9) P W (9) = R max i=1 i 1 R (10)i j=0 R (10)j (6) A (B ) P L(9) P D(9) 10
14 A R (11) 10 B R (11) m 1 (m = 10, 11, ) m () n p (m,n) ( p (m,n) ) (5) R (10) = 9 p (10,n) R p(10,n) (7) n=1 10 A n p tie(10,n) (n = 1, 2,, 9) R (10) p tie(10,n) = R max r=0 R p(10,n)r R (10)r (8) 9 10 n p (10,n) p (10,n) = p (10,n)p tie(10,n) P D(9) (9) (7)-(9) 9 10 B n p (10,n) m = 10, 11, p (m,n), p (m,n) m m A B 3 P W (m), P L(m), P D(m) 2.2.3, P W (m), P L(m), P D(m) (m = 10, 11, ) 12 A P W (9) + P D(9) (P W (10) + P D(10) (P W (11) + P D(11) P W (12) )) (10) P D(9) P D(10) P D(11) P D(12) (11) 15 A P W (9) +P D(9) (P W (10) +P D(10) (P W (11) +P D(11) (P W (12) +P D(12) (P W (13) +P D(13) (P W (14) +P D(14) P W (15) ))))) (12) P D(9) P D(10) P D(11) P D(12) P D(13) P D(14) P D(15) (13) B 11
15 3 1 WPA () WPA m 0 d 0 s 0 (s 0 = 1, 2,, 24)i (6) 10 R (10) (5) 10 n p (10,n) 10 R p(10,n) R (10) m 0 s 0 (s 0 = 1, 2,, 24)i 0 m 0 +1 n p (m0+1,n) R p(m0 +1,n) q (ij) R q(ij) (3) (4) m p (m,n) R p(m,n) p (m0 +1,n) R p(m0 +1,n) p (m0 +1,n) R p(m0 +1,n) q (ij) R q(ij) q (ij) R q(ij) i 1 j 1 i s 0 (s 0 = 1, 2,, 24) j 1 q s0(ij) R qs0 (ij) q (ij) = q 1(ij), R q(ij) = R q1(ij) p (m0 +1,n) p (m0+1,n) = q s0(i 0n) m 0 R p(m0 +1,n) R p(m0 +1,n) = R q s0 (i 0 n) R p(m0+1,n) R p (m,n) R (m) P W (9), P L(9), P D(9) d 0 m 0 d 0 s 0 i 0 q s0(ij) i s 0 j R qs0 (ij) i s 0 j (R max + 1) p s0i 0(m,n)m 0 s 0 i 0 m n R ps0 i 0 (m,n) m n m 0 s 0 i 0 m (R max + 1) 12
16 R s0 i 0 (m)m 0 s 0 i 0 m (R max + 1) U n, R max, U n r, u n p s 0i 0(m,n), R ps0 i 0 (m,n), R s 0i 0(m) p (m,n), R p(m,n), R (m) i 1 j 1 q 1(ij) R q1(ij) q (ij) R q(ij) 2 q 1(ij) R q1(ij) i s 0 j q s0 (ij) R qs0(ij) q (ij) R q(ij) Step3. U 0 Step1. n = 0, j = i 0 Step2. n = n + 1 Step3. r = 0, 1, 2,, R max U n r = U n 1 r 0 P (0) j + U n 1 r 1 P (1) j + U n 1 r 2 P (2) j + U n 1 r 3 P (3) j + U n 1 r 4 P (4) j 1 s U 0 = Step4. j = (j mod 9) + 1 Step5. Step6. Step q s0 (i 0 j) = q s0 (i 0 j) + ( u n 25 1 u25 n 1 1),R qs0 (i 0 j) = R q s0 (i 0 j) + (un 25 u n 1 u n ε Step2. j = 1, 2,, 9 R qs0(i 0j) 3: q s0 (i 0 j) R qs0 (i 0 j) 25 ) p s0 i 0 (m,n) p s0 i 0 (m 0 +1,n) = q s0 (i 0 n) (n = 1, 2,, 9) 9 p s0 i 0 (m,n) = p s0 i 0 (m 1,k)q 1(kn) (m = m 0 + 2, m 0 + 3,, 10, n = 1,, 9) k=1 R ps0 i 0 (m,n) R ps0i0(m0+1,n) = R qs0(i0n) (n = 1, 2,, 9) R ps0 i 0 (m,n) = 9 k=1 p s0 i 0 (m 1,k)q 1(kn) c (m 1,k,n) (m = m 0 + 2, m 0 + 3,, 10, n = 1,, 9) p s0i 0(m,n) c (m 1,k,n)i := i R ps0i 0(m 1,k) R q1(kn)i r (i = 0, 1,, R max ) r r=0 R s0 i 0 (10) R s0 i 0 (10) = 9 n=1 p s0 i 0 (10,n)R ps0 i 0 (10,n) 13
17 R s0a i 0A (10) A m 0A () s 0A i 0A 9 R s0b i 0B (10) B m 0B () s 0B i 0B 9 A d 0 A m 0A s 0A i 0A B m 0B s 0B i 0B 9 A P W sit(9) P W sit(9) = R max i=0 R s0a i 0A (10) i d 0 +i 1 j=0 R s0b i 0B (10) j (m 0A, s 0A, i 0A ) (m 0B, s 0B, i 0B ) (m 0A, s 0A, i 0A ) = (2, 1, 4), (m 0B, s 0B, i 0B ) = (1, 24, 6) 9 A (B ) P Lsit(9) P Dsit(9) P Lsit(9) = R max i=0 P Dsit(9) = R s0b i 0B (10) i R max i=0 d 0+i 1 j=0 R s0a i 0A (10) j ( ) R s0a i 0A (10) i R s0b i 0B (10) d0 +i d 0 A (7)-(9) m 1 (m = 10, 11,, 15) m () n p (m,n) ( p (m,n) ) m m A B 3 P W sit(m), P Lsit(m), P Dsit(m) 2.2.3, P W sit(m), P Lsit(m), P Dsit(m) (m = 10, 11, ) (10)-(13) P W sit(9) = 0, P Lsit(9) = 0, P Dsit(9) = 1 4 () WPA CPU Intel Core i5, 2.27GHz 4.00GBOS Windows7 Home Premium Matlab ε = 10 3, R max = 20 [19] [5] [2, 3, 11] 14
18 4.1 WPA WPA 1 WPA 2012 vs 6 1 WPA WPA WPA WPA (144 ) () 19 () () 19 () () () p w p w = () + () () 9 + () + () + () DH 9 () = () () () () () () () 15
19 ( ) ( ) ( ) WPA WPA : () () () : WPA : WPA : WPA = WPA = WPA WPA 3 WPA WPA 16
20 4: p s p d p t p h p w p out : 2012 WPA () WPA WPA 2 1 () () () () () () () () () () () () () () WPA WPA 6 WPA WPA( WPA) 1. WPA( WPA) 2. WPA WPA 10 WPA WPA WPA WPA 1 () WPA WPA 17
21 3. 6,7 6 7 WPA 1 2 6: WPA () WPA WPA WPA WPA 1 (0.352) (0.407) (0.262) (0.472) 2 (0.300) (0.305) (0.200) (0.268) 3 (0.224) (0.304) (0.014) (0.004) 7: WPA () WPA WPA WPA WPA 1 (0.274) (0.357) (0.211) (0.184) 2 ()(0.193) ()(0.265) (0.205) (0.183) 3 (0.170) (0.154) (0.102) (0.106) 8: p s p d p t p h p w p out WPA WPA 2 3 WPA WPA 2012 MVP ( 8 ) WPA () WPA WPA ()
22 WPA WPA WPA WPA WPA WPA 0 WPA [16] 9 [10] 2005 [16] WPA (0 ) P 0 1 P s 1 11 P f p 0 WPA (P s P 0 )p + (P f P 0 )(1 p) (14) (14) 0 p p out = p out =
23 9: 5 p s p d p t p h p w p out () ( 167/ 264) (0.848) : (1 ),(-2+2 ) : (24 ),(-2+2 ) +1(4 ),+2-2(2 ), (4 ) : (5 ),(-2+2 ) : (6,7 ),(-2+2 ) 0+2-1(6 ), (6 ), (7 )
24 14: (8 ),(-2+2 ) : (912 ),(-20 ) 0(9,12 ) ( ) () [2, 3, 11] 13 (p s, p d, p t, p h, p w, p out ) = (0.1919, , , , , ) (2012 ) ( ())
25 : (1,2 ),( : (3,4 ),(-2+2 ) +1, (2 ) ) 0+2 0(4 ),+1,+2 0(4 ),+1,+2 +1, : (58 ),( : (915 ),(-2 ) 0(8 ) 0+2 0(8 ) 0+2 0(8 ) 0+2 0(8 ) )
26 (2 ) (2 ) : 2 (68 ),(-2 20: 2 (15 ),(-2 +2 ) +2 ) -2+1, +2(1,2 ) ,-1, 0(3 ) -2+1, +2(13 ) -2(5 ), -1(5 ) -20(6,8 ), 0(7 ) -20,+1(6 ) -2+1,+2(6 ) , -20, +2(6,7 ) +1(6,7 ) , -2+1, -20, +2(6,7 ) +2(6,7 ) +1(6,7 ) -20(7 ), 0(8 ) 22: 2 (911 ),(-2 23: 2 (12 ),(-2,-1 0 ) ,+1(11 ) 0 0,+1 0,+1(11 ) 0 0,+1 0,+1 0,+1 0 ) , ( 8 ) ()
27 25: 2 (58 ),(-2 24: 2 (14 ),(-2 +2 ) +2 ) (8 ) -2+1,+2(7,8 ) : 2 (911 ),(0 27: 2 (12 +2 ) +1(9,10 ) 0,+1 0,+1 0, ,+1 0, ,+1,+2(11 ) 0,+1 ),(+1,+2 ) +2 +1, ,+2 +1,+2 +1,+2 +1,+2 +1,+2 +1,+2 +1,+2 +1,+2 +1, ( ) 5 () ( ) () WPA 0 () 4,5,6 4:
28 5: 6: 1 ( 4) 2 ( 5) 0,1 2 ( 6) ( 48/ 94) 0.673( 981/ 1457) ()
29 (1 ) (2 1 ) 28,29 28: 0 (18 ) , : 0 (912 ) , (1 ) (2 ) 26
30 28, () WPA WPA WPA 0 WPA 0 [1] Albert, J. and Bennett, J., :,, (2004). [2] , 27
31 [3] , [4] Bukiet, B., Harold, E.R. and Palacious, J.L.: A Markov Chain Approach to Baseball, Operations Research, 45, 1, (1997). [5], [6] D Esopo, D.A. and Lefkowitz, B.: The Distribution of Runs in the Game of Baseball, Optimal Strategies in Sports (edited by Ladany, S.P. and Machol, R.E.), 55-62, North-Holland, New York (1977). [7] : OR, --, 24, 4, (1979). [8], : - -, --, 49, 6, (2004). [9] James, B: The New Bill James Historical Baseball Abstract, Free Press, New York (2001). [10], :,, (2008). [11], [12] Lewis, M, : --,, (2004). [13] Lindsey, G.R.: Statistical Data Useful for the Operation of a Baseball Team, Operations Research, 7, 2, (1959). [14],, Student,,,,, : 1,, (2012). [15], :,, Vol.18, No.3, (2008). [16] : 9,, (2011). [17] : Win Probability Added in Sabermetrics,, 1703, 1-9 (2010). [18] Win Expectancy Finder, gregstoll/baseball/stats.php [19] Yahoo!JAPAN, 28
32 A m m A A.1 [15] m = 10, 11, p (m,n), p (m,n) m m A B 3 P W (m), P L(m), P D(m) m = 10, 11, 1. (7)-(9) p (m,n) p (m,n) 2. p (m+1,n) ( (3) ) 9 p (m+1,n) = p (m,k) q (kn) k=1 3. R p(m+1,n) ( (4) ) ( R p(m,n) = 9 p (m,k) q (kn) R p(m+1,n) = c (m,k,n)i := k=1 p (m+1,n) c (m,k,n) i R p(m,k)r R q(kn)i r r=0 ) T (n = 1, 2,, 9) 4. R (m+1) ( (5) ) 9 R (m+1) = p (m+1,n) R p(m+1,n) n= B q (ij), R q(ij) R (m+1) 6. (6) P W (m), P L(m), P D(m) ( ) T 4. R p(m,n) = (4) R p(m,n) 1 0 A.2 A d 0 A m 0A s 0A i 0A m 1 (m = 10, 11, ) m n p s0a i 0A (m,n) A d 0 B m 0B s 0B i 0B m 1 (m = 10, 11, ) m n p s0b i 0B (m,n) 29
33 m 1 (m = 10, 11, ) m () n p s0a i 0A (m,n)( p ) m s0b i 0B (m,n) m A B 3 P W sit(m), P Lsit(m), P Dsit(m) (i)9 10 n p s0a i 0A (10,n) p (m,n) ( p (m,n) ) (7)-(9) A d 0 B d 0 10 A n R max p ties0a i 0A (10,n) = r=0 R ps0a i 0A (10,n) r R s0b i 0B (10) d0 +r (15) (9) 9 10 n p s0a i 0A (10,n) = p s 0A i 0A (10,n)p ties0a i 0A (10,n) P Dsit(9) (16) A d B n p s0b i 0B (10,n) A.1 P W sit(10), P Lsit(10), P Dsit(10) R ps0a i 0A (m,n) R p s0b i 0B (m,n) 1 m = 11 p s0a i 0A (m,n), p s0b i 0B (m,n) P W sit(m), P Lsit(m), P Dsit(m) (ii)9 m 0A = m 0B + 1 m 0B 9 p s0a i 0A (m 0A,n) 9 (m 0B = 9) m 0A = 10, s 0A = 1 10 i 0A 9 10 n p s0a i 0A (10,n) p s0a i 0A (10,n) = p 1i0A (10,n) = { 1 (n = i 0A ) 0 () p s0b i 0B (10,n) (15),(16) A d A.1 P W sit(10), P Lsit(10), P Dsit(10) R ps0a i 0A (m,n) R p s0b i 0B (m,n) 1 m = 11 p s0a i 0A (m,n), p s0b i 0B (m,n) P W sit(m), P Lsit(m), P Dsit(m) 9 P W sit(m), P Lsit(m), P Dsit(m) 30
34 (iii)10 m 0A = m 0B m 0B 10 9 m A n p ties0a i 0A (m 0+1,n) A d 0 R max p ties0a i 0A (m 0 +1,n) = r=0 R ps0a i 0A (m 0 +1,n) r R s0b i 0B (m 0+1) d0+r (9) m 0 m n p s0a i 0A (m 0 +1,n) = p s 0A i 0A (m 0 +1,n)p ties0a i 0A (m 0 +1,n) P Dsit(m0 ) A d 0 p s0b i 0B (m 0 +1,n) A.1 P W sit(10), P Lsit(10), P Dsit(10) R ps0a i 0A (m,n) R p s0b i 0B (m,n) 1 m = m p s0a i 0A (m,n), p s0b i 0B (m,n) P W sit(m), P Lsit(m), P Dsit(m) 31
untitled
c 645 2 1. GM 1959 Lindsey [1] 1960 Howard [2] Howard 1 25 (Markov Decision Process) 3 3 2 3 +1=25 9 Bellman [3] 1 Bellman 1 k 980 8576 27 1 015 0055 84 4 1977 D Esopo and Lefkowitz [4] 1 (SI) Cover and
…}…‰…R…tŸA“½‡Ì−î‚b
2012 OR 2 ( ) 2012 OR 2 1 / 29 1 2 3 Google PageRank ( ) 2012 OR 2 2 / 29 41 1, 2004,, 1, 2 3, 2, 5 5, 6-4,? 42 2, 2, 3, 2,? ( ) 2012 OR 2 3 / 29 Exercise 41, 43, or, 70% 30%, 60% 40% ( ) 2012 OR 2 4 /
( ( )) : 1970 * ID * * *1 sabermetrics *2 Important Data identification *3 1
修士学位論文 題名 統計的手法によるメジャーリーグ野球の 打順最適化モデルの構築 指導教員福永力教授 平成 30 年 1 月 5 日 提出 首都大学東京大学院 理工学研究科 数理情報科学専攻 学修番号 16878321 氏名 下木健太 ( ( )) : 1970 *1 1990 ID *2 1 1 10 *3 1 2 3 4 2 1 2 1 *1 sabermetrics *2 Important
2
CONTENT S 01 02 04 06 08 10 20 25 25 30 38 49 53 58 60 64 74 8 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1 2 6 1 2 2002,000 200 7 Web 20 20 13 13 4 13 13F 13F 20 20 13 13 4 13 13F 13F 13 6 A B C A B 13 20 13 20
Grund.dvi
24 24 23 411M133 i 1 1 1.1........................................ 1 2 4 2.1...................................... 4 2.2.................................. 6 2.2.1........................... 6 2.2.2 viterbi...........................
koji07-02.dvi
007 I II III 1,, 3, 4, 5, 6, 7 5 4 1 ε-n 1 ε-n ε-n ε-n. {a } =1 a ε N N a a N= a a
FA $*1$ $*$ 1, $*$2 : $*2$ : Takehiro Takano $*$ 1, Katsunori Ano*2 $*1$ : Graduate School of Engineering and Science, Shibaura Ins
Title マルコフ連鎖に基づく最適打順モデルによる FA 打者獲得戦略 ( 不確実 不確定性の下での数理意思決定モデルとその周辺 ) Author(s) 高野, 健大 ; 穴太, 克則 Citation 数理解析研究所講究録 (2016), 1990: 89-96 Issue Date 2016-04 URL http://hdl.handle.net/2433/224603 Right Type
a n a n ( ) (1) a m a n = a m+n (2) (a m ) n = a mn (3) (ab) n = a n b n (4) a m a n = a m n ( m > n ) m n 4 ( ) 552
3 3.0 a n a n ( ) () a m a n = a m+n () (a m ) n = a mn (3) (ab) n = a n b n (4) a m a n = a m n ( m > n ) m n 4 ( ) 55 3. (n ) a n n a n a n 3 4 = 8 8 3 ( 3) 4 = 8 3 8 ( ) ( ) 3 = 8 8 ( ) 3 n n 4 n n
717A 33 2 53 52B 429 430G 34 2005 5 5 2005 13 28 17 17 36 35 17 29 423G 424A 51A 7 41 7 54 5 59 51A4556 17 3 14 11 22 6 24 54 35 17 55 22 18 32 56F 57
Vol.13 CONTENTS P2.3s P4.5s P6.6s P7.7s P8.8s s 1 717A 33 2 53 52B 429 430G 34 2005 5 5 2005 13 28 17 17 36 35 17 29 423G 424A 51A 7 41 7 54 5 59 51A4556 17 3 14 11 22 6 24 54 35 17 55 22 18 32 56F 57G
スライド 1
19 9 9 26 39 38 100 300 30 37 1 2 3 4 5 6 7 2004 Vol.132(12) Web http://www.nsknet.or.jp/katoh/honbun.html http://www.hyo-med.ac.jp/department/ped/qa/rota.html http://kodomo-qq.jp/ http://www.guide.metro.tokyo.jp/
1 (1) (2)
1 2 (1) (2) (3) 3-78 - 1 (1) (2) - 79 - i) ii) iii) (3) (4) (5) (6) - 80 - (7) (8) (9) (10) 2 (1) (2) (3) (4) i) - 81 - ii) (a) (b) 3 (1) (2) - 82 - - 83 - - 84 - - 85 - - 86 - (1) (2) (3) (4) (5) (6)
- 2 -
- 2 - - 3 - (1) (2) (3) (1) - 4 - ~ - 5 - (2) - 6 - (1) (1) - 7 - - 8 - (i) (ii) (iii) (ii) (iii) (ii) 10 - 9 - (3) - 10 - (3) - 11 - - 12 - (1) - 13 - - 14 - (2) - 15 - - 16 - (3) - 17 - - 18 - (4) -
2 1980 8 4 4 4 4 4 3 4 2 4 4 2 4 6 0 0 6 4 2 4 1 2 2 1 4 4 4 2 3 3 3 4 3 4 4 4 4 2 5 5 2 4 4 4 0 3 3 0 9 10 10 9 1 1
1 1979 6 24 3 4 4 4 4 3 4 4 2 3 4 4 6 0 0 6 2 4 4 4 3 0 0 3 3 3 4 3 2 4 3? 4 3 4 3 4 4 4 4 3 3 4 4 4 4 2 1 1 2 15 4 4 15 0 1 2 1980 8 4 4 4 4 4 3 4 2 4 4 2 4 6 0 0 6 4 2 4 1 2 2 1 4 4 4 2 3 3 3 4 3 4 4
20 15 14.6 15.3 14.9 15.7 16.0 15.7 13.4 14.5 13.7 14.2 10 10 13 16 19 22 1 70,000 60,000 50,000 40,000 30,000 20,000 10,000 0 2,500 59,862 56,384 2,000 42,662 44,211 40,639 37,323 1,500 33,408 34,472
I? 3 1 3 1.1?................................. 3 1.2?............................... 3 1.3!................................... 3 2 4 2.1........................................ 4 2.2.......................................
$\text{ ^{ } }\dot{\text{ }}$ KATSUNORI ANO, NANZAN UNIVERSITY, DERA MDERA, MDERA 1, (, ERA(Earned Run Average) ),, ERA 1,,
併殺を考慮したマルコフ連鎖に基づく投手評価指標とそ Titleの 1997 年度日本プロ野球シーズンでの考察 ( 最適化のための連続と離散数理 ) Author(s) 穴太, 克則 Citation 数理解析研究所講究録 (1999), 1114: 114-125 Issue Date 1999-11 URL http://hdlhandlenet/2433/63391 Right Type Departmental
-1-1 1 1 1 1 12 31 2 2 3 4
2007 -1-1 1 1 1 1 12 31 2 2 3 4 -2-5 6 CPU 3 Windows98 1 -3-2. 3. -4-4 2 5 1 1 1 -5- 50000 50000 50000 50000 50000 50000 50000 50000 50000 50000-6- -7-1 Windows 2 -8-1 2 3 4 - - 100,000 200,000 500,000
2 2 MATHEMATICS.PDF 200-2-0 3 2 (p n ), ( ) 7 3 4 6 5 20 6 GL 2 (Z) SL 2 (Z) 27 7 29 8 SL 2 (Z) 35 9 2 40 0 2 46 48 2 2 5 3 2 2 58 4 2 6 5 2 65 6 2 67 7 2 69 2 , a 0 + a + a 2 +... b b 2 b 3 () + b n a
「産業上利用することができる発明」の審査の運用指針（案）
1 1.... 2 1.1... 2 2.... 4 2.1... 4 3.... 6 4.... 6 1 1 29 1 29 1 1 1. 2 1 1.1 (1) (2) (3) 1 (4) 2 4 1 2 2 3 4 31 12 5 7 2.2 (5) ( a ) ( b ) 1 3 2 ( c ) (6) 2. 2.1 2.1 (1) 4 ( i ) ( ii ) ( iii ) ( iv)
untitled
16 4 1 17 1 50 -1- -2- -3- -4- -5- -6- -7- 1 2-8- -9- -10- -11- Web -12- (1) (2)(1) (3) (4) (1)()(2) (3)(4) -13- -14- -15- -16- -17- -18- -19- -20- -21- -22- -23- (2)(1) (3) -24- -25- -26- -27- -28- -29-
x, y x 3 y xy 3 x 2 y + xy 2 x 3 + y 3 = x 3 y xy 3 x 2 y + xy 2 x 3 + y 3 = 15 xy (x y) (x + y) xy (x y) (x y) ( x 2 + xy + y 2) = 15 (x y)
x, y x 3 y xy 3 x 2 y + xy 2 x 3 + y 3 = 15 1 1977 x 3 y xy 3 x 2 y + xy 2 x 3 + y 3 = 15 xy (x y) (x + y) xy (x y) (x y) ( x 2 + xy + y 2) = 15 (x y) ( x 2 y + xy 2 x 2 2xy y 2) = 15 (x y) (x + y) (xy
(1) (2) (1) (2) 2 3 {a n } a 2 + a 4 + a a n S n S n = n = S n
. 99 () 0 0 0 () 0 00 0 350 300 () 5 0 () 3 {a n } a + a 4 + a 6 + + a 40 30 53 47 77 95 30 83 4 n S n S n = n = S n 303 9 k d 9 45 k =, d = 99 a d n a n d n a n = a + (n )d a n a n S n S n = n(a + a n
n 2 + π2 6 x [10 n x] x = lim n 10 n n 10 k x 1.1. a 1, a 2,, a n, (a n ) n=1 {a n } n=1 1.2 ( ). {a n } n=1 Q ε > 0 N N m, n N a m
![n 2 + π2 6 x [10 n x] x = lim n 10 n n 10 k x 1.1. a 1, a 2,, a n, (a n ) n=1 {a n } n=1 1.2 ( ). {a n } n=1 Q ε > 0 N N m, n N a m](/thumbs/94/118912662.jpg "n 2 + π2 6 x [10 n x] x = lim n 10 n n 10 k x 1.1. a 1, a 2,, a n, (a n ) n=1 {a n } n=1 1.2 ( ). {a n } n=1 Q ε > 0 N N m, n N a m")
1 1 1 + 1 4 + + 1 n 2 + π2 6 x [10 n x] x = lim n 10 n n 10 k x 1.1. a 1, a 2,, a n, (a n ) n=1 {a n } n=1 1.2 ( ). {a n } n=1 Q ε > 0 N N m, n N a m a n < ε 1 1. ε = 10 1 N m, n N a m a n < ε = 10 1 N
1 n 1 1 2 2 3 3 3.1............................ 3 3.2............................. 6 3.2.1.............. 6 3.2.2................. 7 3.2.3........................... 10 4 11 4.1..........................
目次
00D80020G 2004 3 ID POS 30 40 0 RFM i ... 2...2 2. ID POS...2 2.2...3 3...5 3....5 3.2...6 4...9 4....9 4.2...9 4.3...0 4.4...4 4.3....4 4.3.2...6 4.3.3...7 4.3.4...9 4.3.5...2 5...23 5....23 5.....23
: : : : ) ) 1. d ij f i e i x i v j m a ij m f ij n x i =
1 1980 1) 1 2 3 19721960 1965 2) 1999 1 69 1980 1972: 55 1999: 179 2041999: 210 211 1999: 211 3 2003 1987 92 97 3) 1960 1965 1970 1985 1990 1995 4) 1. d ij f i e i x i v j m a ij m f ij n x i = n d ij
() n C + n C + n C + + n C n n (3) n C + n C + n C 4 + n C + n C 3 + n C 5 + (5) (6 ) n C + nc + 3 nc n nc n (7 ) n C + nc + 3 nc n nc n (
3 n nc k+ k + 3 () n C r n C n r nc r C r + C r ( r n ) () n C + n C + n C + + n C n n (3) n C + n C + n C 4 + n C + n C 3 + n C 5 + (4) n C n n C + n C + n C + + n C n (5) k k n C k n C k (6) n C + nc
土木工事の
.1 ... 2 -...2 -...3 -...4 -...5 -...7... 8 -...8 -...9... 10 -... 10 -... 13 -... 14... 17 -... 17... 19... 19... 24... 24... 26 -... 26... 27... 27... 29... 30... 31 -... 31 -... 33 -... 35 1 1 2 3 4
Q2 2.6% 37.6% 42.4% 16.7%.6% Q2 0% 20% 40% 60% 80% 100% (n=1379) A (n=33) 6.1% 39.7% 30.3% 43.2% 48.5% 15% 15.2% 1.6% 0.5% 0% B (n=33) (n=8) (n=30) 18.2% 12.5% 6.7% 27.3% 33.3% 50% 36.4% 12.5% 40% 15.2%
I II
I II I I 8 I I 5 I 5 9 I 6 6 I 7 7 I 8 87 I 9 96 I 7 I 8 I 9 I 7 I 95 I 5 I 6 II 7 6 II 8 II 9 59 II 67 II 76 II II 9 II 8 II 5 8 II 6 58 II 7 6 II 8 8 I.., < b, b, c, k, m. k + m + c + c b + k + m log
cm H.11.3 P.13 2 3-106-
H11.3 H.11.3 P.4-105- cm H.11.3 P.13 2 3-106- 2 H.11.3 P.47 H.11.3 P.27 i vl1 vl2-107- 3 h vl l1 l2 1 2 0 ii H.11.3 P.49 2 iii i 2 vl1 vl2-108- H.11.3 P.50 ii 2 H.11.3 P.52 cm -109- H.11.3 P.44 S S H.11.3
2016 Institute of Statistical Research
2016 Institute of Statistical Research 2016 Institute of Statistical Research 2016 Institute of Statistical Research 2016 Institute of Statistical Research 2016 Institute of Statistical Research 2016 Institute
16 16 16 1 16 2 16 3 24 4 24 5 25 6 33 7 33 33 1 33 2 34 3 34 34 34 34 34 34 4 34-1 - 5 34 34 34 1 34 34 35 36 36 2 38 38 41 46 47 48 1 48 48 48-2 - 49 50 51 2 52 52 53 53 1 54 2 54 54 54 56 57 57 58 59
ito.dvi
1 2 1006 214 542 160 120 160 1 1916 49 1710 55 1716 1 2 1995 1 2 3 4 2 3 1950 1973 1969 1989 1 4 3 3.1 3.1.1 1989 2 3.1.2 214 542 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27
newmain.dvi
数論 サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます. http://www.morikita.co.jp/books/mid/008142 このサンプルページの内容は, 第 2 版 1 刷発行当時のものです. Daniel DUVERNEY: THÉORIE DES NOMBRES c Dunod, Paris, 1998, This book is published
A_chapter5.dvi
5. f,g,h f d,g d,h d. f f = y f d = f(, ȳ) = ȳ = y f f d, f. g g = g d = = g = g d, g. h h = yz y ȳz h d =( y z)( y)( ȳ z) h = h d, h. 5.2 f d ((f d ) d =(f(, 2,..., n )) d = f(, 2,..., n )=f. 5.3 f f
linearal1.dvi
19 4 30 I 1 1 11 1 12 2 13 3 131 3 132 4 133 5 134 6 14 7 2 9 21 9 211 9 212 10 213 13 214 14 22 15 221 15 222 16 223 17 224 20 3 21 31 21 32 21 33 22 34 23 341 23 342 24 343 27 344 29 35 31 351 31 352
,..,,.,,.,.,..,,.,,..,,,. 2
A.A. (1906) (1907). 2008.7.4 1.,.,.,,.,,,.,..,,,.,,.,, R.J.,.,.,,,..,.,. 1 ,..,,.,,.,.,..,,.,,..,,,. 2 1, 2, 2., 1,,,.,, 2, n, n 2 (, n 2 0 ).,,.,, n ( 2, ), 2 n.,,,,.,,,,..,,. 3 x 1, x 2,..., x n,...,,
BEVEL(http://homepage3.nifty.com/bevel/) Cute Sister TRPG X PDF
Cute Sister TRPG 16 8 19 BEVEL(http://homepage3.nifty.com/bevel/) Cute Sister TRPG X PDF Cute Sister TRPG 1 I 8 1 About Cute Sister TRPG 8 1.1 intoroduction........................................... 8
1
1 100 50/60Hz 100 50/60Hz 2 3 4 Windows 4.0SP3 Windows98 4.0SP3 800600 256 1024768 SETUP.EXE 5 [ ] 6 [ OK] [ ] 7 [ OK] [ ] 8 [ OK] [ ] 9 [ ] [ ] [ ] Enter Enter Windows Alt MP-60 10 [ ] F4 [ ] [ ] [ ]
2012 A, N, Z, Q, R, C
2012 A, N, Z, Q, R, C 1 2009 9 2 2011 2 3 2012 9 1 2 2 5 3 11 4 16 5 22 6 25 7 29 8 32 1 1 1.1 3 1 1 1 1 1 1? 3 3 3 3 3 3 3 1 1, 1 1 + 1 1 1+1 2 2 1 2+1 3 2 N 1.2 N (i) 2 a b a 1 b a < b a b b a a b (ii)
II No.01 [n/2] [1]H n (x) H n (x) = ( 1) r n! r!(n 2r)! (2x)n 2r. r=0 [2]H n (x) n,, H n ( x) = ( 1) n H n (x). [3] H n (x) = ( 1) n dn x2 e dx n e x2
![II No.01 [n/2] [1]H n (x) H n (x) = ( 1) r n! r!(n 2r)! (2x)n 2r. r=0 [2]H n (x) n,, H n ( x) = ( 1) n H n (x). [3] H n (x) = ( 1) n dn x2 e dx n e x2](/thumbs/101/149159646.jpg "II No.01 [n/2] [1]H n (x) H n (x) = ( 1) r n! r!(n 2r)! (2x)n 2r. r=0 [2]H n (x) n,, H n ( x) = ( 1) n H n (x). [3] H n (x) = ( 1) n dn x2 e dx n e x2")
II No.1 [n/] [1]H n x) H n x) = 1) r n! r!n r)! x)n r r= []H n x) n,, H n x) = 1) n H n x) [3] H n x) = 1) n dn x e dx n e x [4] H n+1 x) = xh n x) nh n 1 x) ) d dx x H n x) = H n+1 x) d dx H nx) = nh
2001 Miller-Rabin Rabin-Solovay-Strassen self-contained RSA RSA RSA ( ) Shor RSA RSA 1 Solovay-Strassen Miller-Rabin [3, pp
200 Miller-Rabin 2002 3 Rabin-Solovay-Strassen self-contained RSA RSA RSA ( ) Shor 996 2 RSA RSA Solovay-Strassen Miller-Rabin [3, pp. 8 84] Rabin-Solovay-Strassen 2 Miller-Rabin 3 4 Miller-Rabin 5 Miller-Rabin
http://www2.math.kyushu-u.ac.jp/~hara/lectures/lectures-j.html 2 N(ε 1 ) N(ε 2 ) ε 1 ε 2 α ε ε 2 1 n N(ɛ) N ɛ ɛ- (1.1.3) n > N(ɛ) a n α < ɛ n N(ɛ) a n
http://www2.math.kyushu-u.ac.jp/~hara/lectures/lectures-j.html 1 1 1.1 ɛ-n 1 ɛ-n lim n a n = α n a n α 2 lim a n = 1 n a k n n k=1 1.1.7 ɛ-n 1.1.1 a n α a n n α lim n a n = α ɛ N(ɛ) n > N(ɛ) a n α < ɛ
P01_表紙
INDEX MEDIA DATA 2017 2 MEDIA DATA 2017 3 4.9 6.1 3.4 3.9 11.7 11.4 11.5 10.9 7.7 4.9 5.7 6.5 4.4 3.9 2.4 3.3 MEDIA DATA 2017 4 70.6 69.2 67.3 63.7 11.6 8.9 10.4 6.9 16.6 15.6 15.1 12.7 9.7 8.6 9.0 14.2
44 4 I (1) ( ) (10 15 ) ( 17 ) ( 3 1 ) (2)
(1) I 44 II 45 III 47 IV 52 44 4 I (1) ( ) 1945 8 9 (10 15 ) ( 17 ) ( 3 1 ) (2) 45 II 1 (3) 511 ( 451 1 ) ( ) 365 1 2 512 1 2 365 1 2 363 2 ( ) 3 ( ) ( 451 2 ( 314 1 ) ( 339 1 4 ) 337 2 3 ) 363 (4) 46
<4D F736F F D B B83578B6594BB2D834A836F815B82D082C88C60202E646F63>
確率的手法による構造安全性の解析 サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます. http://www.morikita.co.jp/books/mid/55271 このサンプルページの内容は, 初版 1 刷発行当時のものです. i 25 7 ii Benjamin &Cornell Ang & Tang Schuëller 1973 1974 Ang Mathematica
N N 1,, N 2 N N N N N 1,, N 2 N N N N N 1,, N 2 N N N 8 1 6 3 5 7 4 9 2 1 12 13 8 15 6 3 10 4 9 16 5 14 7 2 11 7 11 23 5 19 3 20 9 12 21 14 22 1 18 10 16 8 15 24 2 25 4 17 6 13 8 1 6 3 5 7 4 9 2 1 12 13
untitled
( ) 200133 3 3 3 3, 7 347 57 10 i ii iii -1- -2- -3- -4- 90011001700mm -5- 4.2 1991 73.5 44.4 7.4 10.5 10.5 7.4 W 3 H 2.25 H 2.25 7.4 51.8 140.6 88.8 268.8m 5,037.9m 2 2mm 16cm916cm 10.5 W 3 H 2.25 62.8
P072-076.indd
3 STEP0 STEP1 STEP2 STEP3 STEP4 072 3STEP4 STEP3 STEP2 STEP1 STEP0 073 3 STEP0 STEP1 STEP2 STEP3 STEP4 074 3STEP4 STEP3 STEP2 STEP1 STEP0 075 3 STEP0 STEP1 STEP2 STEP3 STEP4 076 3STEP4 STEP3 STEP2 STEP1
STEP1 STEP3 STEP2 STEP4 STEP6 STEP5 STEP7 10,000,000 2,060 38 0 0 0 1978 4 1 2015 9 30 15,000,000 2,060 38 0 0 0 197941 2016930 10,000,000 2,060 38 0 0 0 197941 2016930 3 000 000 0 0 0 600 15
1
1 2 3 4 5 6 7 8 9 0 1 2 6 3 1 2 3 4 5 6 7 8 9 0 5 4 STEP 02 STEP 01 STEP 03 STEP 04 1F 1F 2F 2F 2F 1F 1 2 3 4 5 http://smarthouse-center.org/sdk/ http://smarthouse-center.org/inquiries/ http://sh-center.org/
平成27年度三菱重工グループ保険 フルガードくん（シニア）
TEL 0120-004-443 TEL 045-200-6560 TEL 042-761-2328 TEL 0120-539-022 TEL 042-762-0535 TEL 052-565-5211 TEL 077-552-9161 TEL 0120-430-372 TEL 0120-45-9898 TEL 0120-63-0051 TEL 0120-252-892 TEL 083-266-8041
i ii i iii iv 1 3 3 10 14 17 17 18 22 23 28 29 31 36 37 39 40 43 48 59 70 75 75 77 90 95 102 107 109 110 118 125 128 130 132 134 48 43 43 51 52 61 61 64 62 124 70 58 3 10 17 29 78 82 85 102 95 109 iii
Microsoft Word - ランチョンプレゼンテーション詳細.doc
PS1-1-1 PS1-1-2 PS1-1-3 PS1-1-4 PS1-1-5 PS1-1-6 PS1-1-7 PS1-1-8 PS1-1-9 1 25 12:4514:18 25 12:4513:15 B PS1-1-10 PS1-2-1 PS1-2-2 PS1-2-3 PS1-2-4 PS1-2-5 PS1-2-6 25 13:1513:36 B PS1-2-7 PS1-3-1 PS1-3-2
2 of 46 07.2.10 4:30 PM
1 of 46 07.2.10 4:30 PM 2 of 46 07.2.10 4:30 PM 3 of 46 07.2.10 4:30 PM 4 of 46 07.2.10 4:30 PM 5 of 46 07.2.10 4:30 PM 6 of 46 07.2.10 4:30 PM 7 of 46 07.2.10 4:30 PM 8 of 46 07.2.10 4:30 PM 9 of 46 07.2.10
*p145-174_Œâ‡í‡ê‡é
*p145-174_ 問 われる 09.1.16 10:34 PM ページ145 2007 200708 146 147 a s 148 a s d f g 153 a s d 158 a s d f g h j 166 a s d f 171 2009 145 *p145-174_ 問 われる 09.1.16 10:34 PM ページ146 45 2007 2008 146 *p145-174_
T T T T A 0 1 A 1 A P (A 1 ) = C 1 6 C 8C 3 = 15 8, P (A ) = C 6 C 1 8C 3 = 3 8 T 5 B P (A 1 B) = =
4 1.. 3. 4. 1. 1 3 4 5 6 1 3 4 5 6. 1 1 1 A B P (A B) = P (A) + P (B) P (C) = P (A) P (B) 3. 1 1 P (A) = 1 P (A) A A 4. A B P A (B) = n(a B) n(a) = P (A B) P (A) 50 015 016 018 1 4 5 8 8 3 T 1 3 1 T T
untitled
9118 154 B-1 B-3 B- 5cm 3cm 5cm 3m18m5.4m.5m.66m1.3m 1.13m 1.134m 1.35m.665m 5 , 4 13 7 56 M 1586.1.18 7.77.9 599.5.8 7 1596.9.5 7.57.75 684.11.9 8.5 165..3 7.9 87.8.11 6.57. 166.6.16 7.57.6 856 6.6.5
応用数学III-4.ppt
III f x ( ) = 1 f x ( ) = P( X = x) = f ( x) = P( X = x) =! x ( ) b! a, X! U a,b f ( x) =! " e #!x, X! Ex (!) n! ( n! x)!x! " x 1! " x! e"!, X! Po! ( ) n! x, X! B( n;" ) ( ) ! xf ( x) = = n n!! ( n
Bulletin of JSSAC(2014) Vol. 20, No. 2, pp (Received 2013/11/27 Revised 2014/3/27 Accepted 2014/5/26) It is known that some of number puzzles ca
Bulletin of JSSAC(2014) Vol. 20, No. 2, pp. 3-22 (Received 2013/11/27 Revised 2014/3/27 Accepted 2014/5/26) It is known that some of number puzzles can be solved by using Gröbner bases. In this paper,
取扱説明書 [d-01H]
![取扱説明書 [d-01H]](/thumbs/42/23082695.jpg "取扱説明書 [d-01H]")
d-01h 1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 1 2 19 3 1 2 4 3 20 4 21 1 2 3 4 22 1 2 1 2 1 23 1 1 2 24 25 26 1 1 1 2 27 1 2 3 28 29 1 2 1 2 3 30 1 2 3 4 5 1 2 3 31 1 2 3 4 32 33 34 1 35 1 36 37
25 7 18 1 1 1.1 v.s............................. 1 1.1.1.................................. 1 1.1.2................................. 1 1.1.3.................................. 3 1.2................... 3