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1 ( ) yoshidat@math.sci.hokudai.ac.jp 2008

2 II.

3 ( ) William Jones The Sanscrit Language (1786) two three ten father mother brother dvi tri dasa pitar matar bhratar duo treis deka pater meter phrater duo tres decem pater mater frater ( ) Schlegel 1808, Bopp 1816 (Schleicher 1861) (Schmidt 1872) 1

4 ( ) p t k d g ph th p t c(=k) d g f f/d p t š d j bh dh f θ/ h(x) t k b d 2

5 ( ) name boy ( ) ( ) ( ) (aiu) (a a ei i, ou u) 3

6 ( ) ( ) ( ) (12 ) ( 12 ) SOV SOV 4

7 1(hi)-2(hu), 3(mi)-6(mu) 4(yo)-8(ya), 5(itu)-10(to) ( 1913) ) ) ) ) ( ) ( ) ( ) ( ) 5

8 ( ) Polya Oswalt Swadesh 6

9 (1) Polya p 10 ( ) x 10 ( ) P (x) = 10C r p r (1 p) 10 r n n!, nc r = = r r! (n r)! r=x ( ) 1 one en en een ein un uno uno jeden egy 2 two tra to twee zwei deux dos due dwa ketto 3 three tre tre drie drei trois tres tre trzy harom 4 four frya fire vier vier quatre cuatro quattro cztery negy 5 five fem fem vijf funf cinq cinco cinque piec ot 6 six sex seks zes sechs six seis sei szesc hat 7 seven sju syv zeven sieben sept siete sette siedem het 8 eight atta otte acht acht huit ocho otto osiem nyolc 9 nine nio ni negen neun neuf nueve ove dziewiec kilenc 10 ten tio ti tien zehn dix diez dieci dziesisc tiz 7

10 ( P (x) < 0.05 ) x P (x) p = = , 0, 5,, 4 a, b, c,, z 8

11 (2) R.Oswalt ( ) (1970) {WA i }, {WB i } (i = 1,, n) f(i), g(i) WA i, WB i x 0 f(i) = g(i) i x 0, x 1,, x n 1 ( ) x 0 m = (x 0 + x x n 1 )/n s 2 = ((x 0 m) 2 + (x 1 m) (x n 1 m) 2 )/n z = (x 0 m)/s Q n (z) = 1 e t2 /2 dt 2π z Q n (z) < 0.05 (z > 2.33) 5 9

12 1 all mina 2 ash ËaËi 3 bark kaëa 4 belly Ëara woman me 100 yellow kï mot5n ts5i k@ptsir p5i. ky@tsip nurw 1 all mina (mot5n) f(i), g(i) i x 0 := {i f(i) = g(i)} x k := {i f(i) = g(i + k)} i + k modn LB LA x 1,, x n 1 x 0 ( ) 10

13 x m 53 x 0 p = z = s = m = x 0 = 53 ( 200 ) 11

14 ( ) (A) O(m 2 n 2 ) (2 m 100 n 200 ) (B) P (A) m s (B) P 12

15 N := {1, 2,, n} ( ) S n : n (n ) G S n 1 G( ) σ, τ G στ G G i, j N σ G σ(i) = j f, g : N Λ x[f, g] := {i N f(i) = g(i)} ( ) x(π) := x[f, gπ] (π G) m := 1 G π G x(π), s 2 := 1 G π G (x(π) m) 2 a λ := f 1 (λ), b λ := g 1 (λ) a λ, b λ λ LA, LB C n := π 0 = (1, 2,, n) Oswalt 13

16 G m = 1 n a λ b λ λ Λ {π G π(i) = j} = G /n ( i, j N). x[f, gπ] = {(π, i) f(i) = g(π(i))} π G = {(π, i, j) f(i) = g(j), π(i) = j} = {(i, j) f(i) = g(j)} G /n = G f 1 (λ) g 1 (λ) n λ Λ 14

17 s 2 = 1 n 1 m(m + n) 1 n(n 1) a λ b λ (a λ + b λ ) λ {(i, j) i j} S n LA, LB λ p λ = a λ /n, q λ = b λ /n LA, LB p = m/n = p λ q λ s 2 = n2 n 1 { p(1 + p) } p λ q λ (p λ + q λ ) Cn 15

18 1 n! π S n ( x(π) t ) = (n t)! n! Σt λ =t λ ( aλ tλ ) ( bλ tλ ) t λ! {x(π) π S n } {a λ }, {b λ } 16

19 J\K k m n p r t w y - k m n p r t w y x 0 = 53, m = , s = , z = , P = x 0 =

20 (A) (2 m 100 n 200 ) ( ) O(m 2 n 2 ) ( ) O(m 2 n) {a λ }, {b λ } x 0 (B) P x(π) = x[f, gπ], π S n ( ) P P (x 0 ) = {π S n x(π) x 0 }/n! 18

21 N = {1,, n} G f, g : N Λ x(σ) = {i N f(i) = g(σi)}, σ G x 0 = x(1) (1) m {a λ }, {b λ } Oswalt m, s 2 {a λ }, {b λ } (2) E n (N N ) x(σ, τ ) = {i N f(σi) = g(τ i)}, σ, τ E n f, g B(n, p) (p = (1/n) 2 a λ b λ ) Polya p = (1/2n) 2 (a λ + b λ ) 2 19

22 u ij = u i,λ(i) I = N, J = N N i I j J λ:i J i I F (θ) := x 0 j,k = λ,µ = 1 n 2n {(σ, τ ) x(σ, τ ) = x} n 2n θ x = 1 σ,τ i N θ δ(fσ(i), gτ (i)) = i N n 2n 1 n 2 σ,τ E n θ x(σ,τ ) j,k N a λ b µ θ δ(λ,µ) = n 2 n 2 p(1 θ), p := 1 n 2 a λ b λ F (θ) = (1 p + pθ) n ( ) λ θ δ(f(j), g(k)) 20

23 (3) ( ) m = m + (m x 0 )/(n 1) (e.g. ) G S n σ G, π S n πσπ 1 G x[fπσ, gπ] = x[fπσπ 1, g] S n (n 5) S n, A n (4) N (x y = z ) x s := {i N f(i) = g(s i)}(s N) m = (1/n) a λ b λ i j := i + j 1 (mod n) 21

24 S n E n x(π) := {i N f(i) = g(πi)}, π S n, S n = n! x(σ, τ ) := {i N f(σi) = g(τ i)}, σ, τ E n, E n = n n x(π) π S n Fisher π1,, πn 1,, n x(σ, τ ) σ, τ E n σ1, σ2, τ 1, τ 2, 1,, n HGD 1) (1a) (1b) Fisher (2a) (2b) 1) λ x λ,λ ( ) x λ,µ ( ) 22

25 (1a) P - (1b) (1c) P - (2a) P - (2b) (3a) (3b) 23

26 A,B 200 ( ) f, g : N Λ Λ A, B {a λ }, {b λ } A, B x 0 x 0 B(n, p) n n ( ) n P (x 0 ) = p x x (1 p) n x, p := 1 x=x 0 n 2 a λ b λ λ p 24

27 A,B x AB, m AB, s 2 AB f, g, h : N Λ A,B,C a λ := f 1 (λ), b λ := g 1 (λ), c λ := h 1 (λ) λ (multi-metric) 1 x 0 = x AB + x BC + x CA 2 x 0 = x ABC := x[f, g, h] = {i N f(i) = g(i) = h(i)} 25

28 m = 1 (a λ b λ + b λ c λ + c λ a λ ) n λ s 2 = s 2 AB + s2 BC + s2 CA m = 1 n 2 a λ b λ c λ λ s 2 = 2n 1 (n 1) 2m 2 + n2 2n 1 (n 1) 2m n 2 (n 1) 2 a λ b λ c λ (a λ +b λ +c λ ) λ 26

29 (J) (A) (K) J A J K A K JAK(1) JAK(2) x m s γ γ z P ( ) P ( ) P ( ) ( ) < ( ) < ( ) P - 27

30 (JAK(2)) ( ) ( 1 ) 3 28

31 p 2 F 0 P (x x 0 ) = {π S n x[f, gπ] x 0 }/n! f, g : N Λ, a λ := f 1 (λ), b λ := g 1 (λ) F (u) := ( ) ( ) aλ bλ k! u ( k n k k = k λ k 0 k 0 p(x) = ( ) ( 1) k x k q(k) x k x ) k! q(k) u k P (x x 0 ) = p(x 0 ) + p(x 0 + 1) + + p(n) : x 0 = 53 P (x 53) = P (x 0 53) = ( )

32 Swadesh LA, LB T ( ) x 0 t LA(t) LA(T ) = LA = LB r (r 0.8) Swadesh x 0 (t) = x 0 (0)r t x 0 = nr 2T T m(t) LA(0) LA(0) x 0 (t) LA(0) LA(t) Swadesh x 0 (T ) m(t ) x 0 (0) m(0) = x rt, 0 m x 0 (0) m(0) = r2t ( )

33 ( 2 ) ( ) ( ) ( ) ( ) ( ) ( ) 31

34 1984/ R.Gray-Q.Atkinson, Language-tree divergence times support the Anatolian theory of Indo-European origin, NATURE 426 (2003),

35 J\K k m n p r t w y - k m n p r t w y K\A k m n p r t w y - k m n p r t w y J\A k m n p r t w y - k m n p r t w y H=h=p=b=f=v=x t=sh=ts=d=s=z k=g=q=ng r=l 33

Part () () Γ Part ,

Part () () Γ Part , Contents a 6 6 6 6 6 6 6 7 7. 8.. 8.. 8.3. 8 Part. 9. 9.. 9.. 3. 3.. 3.. 3 4. 5 4.. 5 4.. 9 4.3. 3 Part. 6 5. () 6 5.. () 7 5.. 9 5.3. Γ 3 6. 3 6.. 3 6.. 3 6.3. 33 Part 3. 34 7. 34 7.. 34 7.. 34 8. 35