(Basic of Proability Theory). (Probability Spacees ad Radom Variables , (Expectatios, Meas) (Weak Law

Size: px
Start display at page:

Download "(Basic of Proability Theory). (Probability Spacees ad Radom Variables , (Expectatios, Meas) (Weak Law"

Transcription

1 I (Radom Walks ad Percolatios) ( -2 ) (Preface),.,,,...,,.,,,,.,.,,.,,. (,.)

2 (Basic of Proability Theory). (Probability Spacees ad Radom Variables , (Expectatios, Meas) (Weak Law of Large Numbers) (Radom Walks) 4 2. (Markov Chais) d (d-dimesioal Radom Walks) (Oe-dimesioal ati-symmetric Radom Walks) (Percolatios) /3 p H 2/ ( 2.2(iii))

3 (Basic of Proability Theory). (Probability Spacees ad Radom Variables,, (Ω, F,P),, X,. (Ω, F,P) (probability space) Ω ( ω Ω ) F( 2 Ω ) Ω σ (σ-field); (2 Ω Ω ) (i) Ω F (ii) A F A c F (iii) A F ( =, 2,...) A F P = P (dω) (Ω, F) (probability measure), i.e., ; P : F [0, ]. (i) P (Ω) = (ii) A F ( =, 2,...) P ( A )= P (A )(σ ). (Ω, F,P),. (i) σ-., F σ- A, B, A F F, A B, A \ B, A B := (A \ B) (B \ A), A. lima = lim sup A := A, lima = lim if A := A F. N N N N (lim = if sup, lim = sup if.) (ii) P ( ) =0, A, B F; A B P (A) P (B) ( ). (iii) A k F (k =, 2,...,) P ( A k)= ( ) P (A k)( ). (iv) A F, A P A = lim P (A ). ( ) (v) A F, A P A = lim P (A ). ( ) (vi) A F ( ) P A P (A ). ( ) (vii) (Borel-Catelli ) A F ( ), P (A ) < P lim sup A = 0, i.e., ( ) P lim if Ac =. (Ω, F,P) X = X(ω) :Ω R {X a} := {ω Ω; X(ω) a} F ( a R). (radom variable). X k S = {a j } j, {X = a j } F ( j ). X k (Ω, F,P) (k =, 2,...,). {X k } (idepedet) P (X a,,x a )=P (X a ) P (X a ) ( a k R,k =,...,).

4 {X k } k N {X k } N. X k S = {a j } j, : P (X = b,,x = b )=P (X = b ) P (X = b ) (b k S, k =,...,). µ(a) =P (X A) X (distributio), F (x) =P (X x) X (distributio fuctio)..2, (Expectatios, Meas) X Z := Z {± }. X (expectatio, mea) EX = E[X] = XdP = X(ω)P (dω). () X 0 EX := P (X = )+ P (X = ). =0 (P (X = ) =0 P (X = ) =0. P (X = ) > 0 EX =.) (2) X X + := X 0, X := ( X) 0 ( X ± 0, X = X + X.) EX := EX + EX.,. EX = Z P (X = ), f : Z R, Ef(X) = f()p (X = ). (.) Z ;f()>0 ;f()<0 X, V (X) :=E[(X EX) 2 ]=E[X 2 ] E[X] 2.. (Chebichev ) p. a>0, P ( X a) E[ X p ] a p. [ ] P ( X a) =P ( X p a p ) p =. E X = P ( X = ) a Ω P ( X = ) a a P ( X = ) =ap ( X a)..2 X,...,X Z, E[Xk 2 ] < (k =,...,). X,...,X, E[X j X k ]=E[X j ]E[X k ](j k). 0(E[X k ]=0) ( ) 2 E X k = E[Xk 2 ]. [ ] () j k P (X j = m, X k = ) =P (X j = m)p (X k = ) E[X j X k ]= m, mp (X j = m, X k = ) = m, mp (X j = m)p (X k = ) =E[X j ]E[X k ]. ( ) 2 (2) X k = X j X k () j k E[X j X k ]=E[X j ]E[X k ]=0. X 2 k + j k 2

5 .3 (Weak Law of Large Numbers), /2.,..3 ( (Weak Law of Large Numbers)) X,X 2,... EX = m v := sup V (X ) < ɛ>0, ( ) ( ) lim P X k m ɛ =0, i.e., lim P X k m <ɛ =. [ ] {X } { X = X m} ( ). X k m = (X k m), X X m = 0, i.e., E[X ]=0 V (X )=E[X] 2, ( ) 2 E X k = E[Xk 2 ]= V (X k ) sup V (X )=v. ɛ>0, ( P ) X k ɛ = P ( ) X k ɛ E[( X k) 2 ] ɛ 2 2 v ɛ 2 2 v = ɛ 2 0 ( ).., P. ( lim ) X k = m =. X,X, ɛ>0, P ( X X ɛ) 0( ), X X i pr., X X. P (X X) =, X X, P -a.s., X X. (a.s. almost surely ).2, i.e., X X, P -a.s. X X i pr.. ( P (X X) = ( { P X X < } ) ( { = P X X } ) =0 k k k N N = k, lim P N ( N { X X k } ) = P k N N ( N N { X X k } ) =0 k, ε =/k, lim P ( X X /k) =0 m, N ; N,P( X X /k) < /m.) 3

6 2 (Radom Walks),,.,,. d, d. Z d ( j =(j,...,j d )) d (lattice). (X,P) d (simple radom walk),, 2d,. Y = X X ( ) {X 0,Y,Y 2,...}, {Y }, P (Y = k) =/(2d) ( k =), =0( k ). k =(k,...,k d ), k = k k2 d. {X 0,Y,Y 2,...}, k 0,k,...,k Z, P (X 0 = k 0,Y = k,...,y = k )=P(X 0 = k 0 )P (Y = k ) P(Y = k ). Z d {p k } k Z d (p k 0, p k =), (X,P), d. P (Y = k) =p k (,k Z d ). P j (X = k,...,x = k ):=P(X = k,...,x = k X 0 = j) P j (X,P j ) j d. 2. P (A B) :=P (A B)/P (B) P (B) > 0. A, B F P (A B) =P (A). 2. (Markov Chais),,.. 2. S,,.,,,.,.,,,,.. S, S (X,P)=(X (ω),p(dω)) ( =0,, 2,...) (Markov Chai) : 4

7 (M) [ ],j 0,j,...,j,k S, P (X + = k X 0 = j 0,X = j,...,x = j )=P (X + = k X = j ).. (M2) [ ],j,k S, P (X + = k X = j) =P (X = k X 0 = j).,. X 0 µ = {µ j }; µ j = P (X 0 = j) (iitial distributio),, j S, P (X 0 = j) = P P j, (X,P j ) j. ( P (X 0 = j) > 0, P j ( ) :=P ( X 0 = j),.) 2.2 P j ( ): P j (X = k,...,x = k ) := P (X = k,...,x = k X 0 = j) = P (X m+ = k,...,x m+ = k X m = j) (m 0). 0,j,k S, j,k = P (X = k X 0 = j), Q () =( j,k ) ( ) (-step trasitio probability (trasitio matrix)),, Q () Q =(q j,k ),, ( ). 2.. (i) j,k 0, k q() j,k =(j S), (ii), j 0,j,...,j S P (X 0 = j 0,X = j,...,x = j )=µ j0 q j0,j q j,j, (iii) m,, j,...,j m,k 0,k,...,k S P (X + = j,...,x +m = j m X 0 = k 0,X = k,...,x = k )=q k,j q j,j 2 q jm,j m. (iv) Q (0) = I := (δ jk )( ), Q () = Q ( ),, δ jk =(j = k), = 0 (j k). 2.2 µ = {µ j } X. P (X = k) = j S µ j j,k., j S (recurrece time): T j : T j = if{ ; X = j} (= if { } = ). j (recurret) j (trasiet) def P j (T j < ) =, def P j (T j < ) < 5

8 . j, T j, j (positive-recurret) def E j [T j ] <, j (ull-recurret) def E j [T j ]=, P j (T j < ) =. E j [T j ] T j P j, : E j [T j ]= mp j (T j = m)+ P j (T j = ). m= j (or,, ) (X ) (or,, ) {X } Q =(q j,k ) π = {π j } π (statioary distributio) def π k = j π jq j,k (k S), π (reversible distributio) def π k q k,j = π j q j,k (j, k S) (i) π,, X π. (ii) π, {X } :,j 0,...,j S, P (X 0 = j 0,...,X = j )=P(X 0 = j,...,x = j 0 ). {X } Q =(q j,k ) (irreducible) j, k,, j,k > 0.,,. (,,.), : 2.2 j, k S. (i) j : a) j,j =. =0 b) P j ({X } j )=. (ii) j : a) j,j <. =0 b) P j ({X } j )=0. (iii) {X },,,,,. (π j )[ k, j π jq j,k = π k ], π j =/E j [T j ] ( ). 6

9 (i), (ii) b), a), (iii). (iii). 2. O- {B k }, A, C, P (A B k)=p (A C) ( k ). P (A B k )=P(A C). O-2 m,, j,...,j m,k 0,k,...,k S P (X + = j,...,x +m = j m X 0 = k 0,X = k,...,x = k ) = P (X + = j,...,x +m = j m X = k ). 2. (i) j S P j ({X } j )=. (ii) j S P j ({X } j )=0..,,,., 0. m j T (m) j. P j (T (m) j T () j = T j, T (m) j = mi{ >T (m ) j ; X = j} (= if { } = ). < ) =P j (T j < ) m. s, t,, P j (T (m) j = s + t T (m ) j = s) =P j (T j = t) (, [ ]= P (X s+t = j, X s+u j ( u t )), {X u j} = k {X u S;k u j u = k u } {T (m ) j = s} {X,...,X s (= j)}, 2 O-, O-2.)., P j (T (m) j P j (T (m ) j = s, T (m) j P j (T (m) j < ) = P j (T (m ) j = = s + t) =P j (T (m ) j = s)p j (T j = t) s=m t= <T (m) j < ) P j (T (m ) j = P j (T (m ) j < )P j (T j < ) < ) =P j (T j < ) m. P j ({X } j ) = P j ( m P j (T j < ) =, 0. = s, T (m) j = s + t) {T (m) j < }) = lim m P j(t (m) j < ) = lim m P j(t j < ) m. [, ] 7

10 ,. j, k S, f (m) j,k := P j(t k = m) (m ) Q jk (s) := =0 j,k s ( s < ), F jk (s) := m= f (m) j,k s ( s ). { j,k } 0, {f (m) j,k } m (geeratig fuctios). F jk () = P j (T k < ). 2. j, k S, : j,k = m= f (m) j,k q m k,k ( ), Q jk (s) =δ jk + F jk (s)q kk (s) ( s < ). {T k = m} = {X m = k, X s k ( s m )} m= f (m) j,k q( m) k,k = = = P j (T k = m)p j (X = k X m = k) m= P j (T k = m)p j (X = k T k = m) m= P j (X = k, T k = m) m= = P j (X = k) = j,k. Q jk (s) = δ jk + = δ jk + j,k s f (m) j,k q( m) k,k s m= = δ jk + F jk (s)q kk (s). 2.2 j S =0 j,j =. Q jj (s)( F jj (s)) = ( s < ) F jj () = P j (T j < ) lim Q jj (s) = s =0 j,j s. : =0 j,j ( P j(t j < )) =. 8

11 2.6 j k j S =0 k,j < ( k S), k S; =0 k,j = j :. ( q() k,j = F kj() q() j,j.) 2.2 j j k [i.e., ; j,k > 0] P k(t j < ) =., i, j S P i (T j < ) =q i,j +. (, k S;k j q i,k P k (T j < ) P i (X = k, T j = ) =q i,k P k (T j = ) P i (T j < ) = P i (X = k, T j = ) k S.) i = j j, k ; q j,k > 0, P k (T j < ) =., k 2 ; q k,k 2 > 0, i.e, q (2) j,k 2 > 0, P k2 (T j < ) =., j,k > 0 (k,...,k ); q j,k q k,k 2 q k2,k 3 q k,k > 0, : j, j k =0 k,j =. j, k S j k k j j k. 2.3 j, k S; j k, j,, k.,,,. l, m 0; q (l) j,k > 0,q(m) k,j > 0. j, =0 q (l+m+) j,j q (l) j,k q() k,k q(m) k,j ( 0) Q jj (s) s l+m q (l) j,k q(m) k,j Q kk(s). lim Q jj (s) = s =0 j,j < k,k <, k. j, k. 9

12 . 2. Q jj (s)( F jj (s)) = F jj (s) =Q jj (s)/q jj(s) 2. j Q jj lim (s) s Q jj (s) 2 = F jj ( ) =E j[t j ] <.. Q kk (s) s l+m q (m) k,j q(l) j,k Q jj(s), Q jj (s) (l + m + )s l+m+ q (l+m+) =0 s l+m q (l) j,k q(m) k,j Q kk(s) Q kk (s) Q kk (s) 2 Q jj (s) s 3(l+m) (q (l) j,k )3 (q (m) k,j )3 Q jj (s) 2. j,j E k [T k ]=F kk( ) Q kk = lim (s) s Q kk (s) 2 < k. j, k. 2.8 k S E k [T k ]=F kk ( ) , 2.7 j, k S, q() j,k =. j, k S, q() j,k <., j, k S, q() j,k,. 2.2 d (d-dimesioal Radom Walks) (X,P) d., {p k } k Z d Z d, {X 0,X X 0,X 2 X,...}, P (X X = k) =p k (,k Z d ). ( p k =/(2d).), d. Q =(q j,k ) q j,k = p k j [,,, ] 2.9 (X,P) d. (i) X + X (X 0,X,...,X ), i.e., P (X + X = j, X 0 = k 0,X = k,...,x = k ) = P (X + X = j)p (X 0 = k 0,X = k,...,x = k ). k 0,k,...,k Z d, X + X X. (ii) P (X + = j X 0 = k 0,X = k,...,x = k )=P(X + = j X = k )=p j k. {X }, q j,k = p k j. 0

13 (iii). ( j k := j k + + j d k d j k, j = k.), Q =(q j,k )=(p k j ),,,., : 2.3 d (i) d =, 2 (i.e., E j [T j ]= P j (T j < )), (ii) d ( ), 0,0. q (2+) 0,0 =0, q (2) 0,0.. ( 2.2.) 2.4 d Q =(q j,k ) (i) d =, 2 q (2) 0,0 { / π (d =) /(π) (d =2) (ii) d =3 C q (2) 0,0 C 3/2. 2.4,, : (d =3 (3/π) 3 /4) q (2) 0,0 2 d d d/2 (π) d/2 ( ). a b ( ) def a /b ( ). 2.0 {a }, {b }, a b ( ) c,c 2 > 0; c b a c 2 b ( ).. [ (Stirlig s formula)]! 2π +/2 e ( ). 2.4 d =, : ( ) q (2) 2 0,0 = 2 2. d =2 q (2) 0,0 = j,k 0;j+k= (2)! (j!k!) = ( 2 ) j=0 ( ) k

14 , j=0 d =3 ( ) 2 = k ( ) 2. q (2) 0,0 = j,k,m 0;j+k+m= (2)! (j!k!m!) 2 6 2, 3 q (2) (2)! 0,0 c 3 6 2!. c = max j,k,m 0;j+k+m= (j!k!m!). c,,. c c3 +3/2 3/2 e (c>0 ). (), 3 (m!) 3 ( =3m) c (m!) 2 ((m + )!) ( =3m +) (m!) ((m + )!) 2 ( =3m +2) (2),, c,c 2 > 0 c +/2 e! c 2 +/2 e (2), (), d =3 ( ). [ ( 2.2).] d =, 2,. 2.5 d =, 2 Z d (i.e., E 0 [T 0 ]= ). 2.3 (i) α> α s Γ(α +) ( s) α+ (s ). (ii) α = s = log s. α> log(/s) s (s ) : α = log. 0 x α s x dx = ( log s ) α Γ(α +). 2

15 F 00 (s) =Q 00 (s)/q 00(s) 2, d =, q (2) 0,0 / π ( ), s Q 00 (s) =+ Q 00 (s) = s 2 q (2) 0,0 + 2s 2 q (2) 0,0 s 2 Γ(/2) π π ( s 2 ), /2 2s 2 2 Γ(3/2) π π ( s 2 ). 3/2 F 00(s) = Q 00(s) Q 00 (s) 2 2 πγ(3/2) Γ(/2) 2 s (s ). s 2 E 0 [T 0 ] = lim F s 00(s) =. d =2 q (2) 0,0 /(π) ( ), s Q 00(s) E 0 [T 0 ] = lim s 2 Q 00 (s) π log s 2, 2 πs( s 2 ) 2 π( s 2 ) [ π( s 2 ) ( log ) ] 2 s 2 =. 2.3 (Oe-dimesioal ati-symmetric Radom Walks) Z {X } p (0 <p<), p. p /2, {X = X (p) }. d, d, ( 0 ). q j,j+ = q 0, = p, q j,j = q 0, = p, j,k = ( ) +j k p ( j+k)/2 ( p) (+j k)/2 ( + j k 2Z) 2 0 ( + j k 2Z +) [ + l, m, = l + m. k j =?.] 3

16 q (2) 0,0 = ( ) 2 (p( p)) (4p( p)) π ( ) 2.4. p /2 4p( p) < : 2.4 {X = X (p) } (0 <p<,p /2)., :. ( P lim ) X =2p =. 2.5 p>/2 j, u j (s) :=F j0 (s) = m sm P j (T 0 = m) (0<s<) lim j u j(s) =0,,. u (s) = psu 2 (s)+( p)s u j (s) = psu j+ (s)+( p)su j (s) (j 2) ( ) j 4p( p)s F j0 (s) = 2 (0 <s<) 2ps ( ) j p P j (T 0 < ) = (j ) p. [ {X = j +}, {X = j }., P j (T 0 = m) =P j (T 0 = m X = j +)P j (X = j +)+P j (T 0 = m X = j )P j (X = j ), P j (T 0 = m X = j ) j 2 P j (T 0 = m ), j = P (T 0 = m X =0)=. ] 2.6 u j := P j (T 0 < ) j Z. p>/2 j, u j = P j (T 0 < ) =, j = 0 u 0 = pu +( p), P 0 (T 0 < ) =u 0 = 2( p) <. 4

17 3 (Percolatios) 3. Z 2, B 2 = {{x, y}; x, y Z 2, x y =} 2, bod.,, p (0 p ) ope, p closed. X b = X (p) b = X (p) b (ω) b B 2, X = {X b ; b B 2 }. P p. S = {b B 2 ; X b =}, O C O, (ope cluster). C O C O. p H : Hammersley (critical probability) θ(p) =P p ( C O = ), : p H = if{p [0, ]; θ(p) > 0}. p T : Temperley χ(p) =E p [ C O ]=, : P p ( C O = )+ P p ( C O = ) p T = if{p [0, ]; χ(p) = }. p H p T, p H = p T.. 3. Z 2, p H, p T, p H = p T = /2, p c. p>p c,, p p c. 3.. Z d (d 3), p c =/2., /3 p H 2/3. θ(p) =P p ( C O = ) p., p,. 3. p H p T.,,,,.,. Ω=Ξ:={0, } B2 ω = ω(b); B 2 {0, }, (0 ) Ω ; (cylider set) A i,...,i b,...,b = {ω; ω(b )=i,...,ω(b )=i } (b k B 2,i k = 0 or,k =,...,) 5

18 C. F = B(Ξ) := σ(c) (C σ-field, C σ-field). B(Ξ) = {G 2 Ω ; G C σ-field}. P = P p cylider set, P p (A i,,i b,...,b )=p i+ +i ( p) ( i)+ +( i). (.), b B 2 (Ω b, F b,p b,p ) Ω b = {ω(b) =,ω(b) =0}, F b =2 Ω b, P b,p (ω(b) = ) = p,. X b = X (p) b X b (ω) =ω(b). X b P p (X b =)= P p (ω(b) = ) = p, X = {X b ; b B 2 } P p. X(ω) =ω. (.) θ(p) =P p ( C O = ) p. θ (p) :=P p ( C O ) θ(p) = lim θ (p) θ (p). b B 2, Z b Z b (ω) [0, ], {Z b }. Q Q(Z b p) =p = P p (X b =). S(p) ={b B 2 ; Z b p}, O C O (p), p θ (p) =P p ( C O ) =Q( C O (p) ) θ (p). 3.2 /3 p H 2/3 p H,. Peierls. 3.2 Hammersley p H /3 p H 2/3. = if{p [0, ]; θ(p) = P p ( C O = ) > 0},. γ = {x 0,b,x,b 2,...,b,x } (, path) (i) b = {x i,x i }, (ii) i j b i b j., γ = {b,...,b }. 3.2 [p H /3 ] p</3,θ(p) =0., γ ( ), P p (γ C O )=P p (X b =, b γ )=p. γ 4 3, P p ( γ C O ) 4 3 p (< if p</3). 6

19 C O = N, N; γ C O, p</3, Borel- Catelli θ(p) =P p ( C O = ) P p { γ C O } =0. N N p H /3. [p H 2/3 ] p>2/3,θ(p) > 0. Z 2 (Z 2 ) := {(m +/2,+/2); m, Z}. (Z 2 ) b (B 2 ) b B 2. X b := X b {X b ; b (B 2 ) } = {X b ; b B 2 }. N, V N := {(m, ) Z 2 ; m := max( m, ) N}, V N. p>2/3 N = N(p) S = {b B 2 ; X b =}, P p (S V N ) 2 (3). C O < C O (B 2 ) (closed path) γ. O V N, P p (S V N ) = P p (V N (B 2 ) γ ) P p (X b =0, b γ ) γ ; V N, γ = k V N k 4(2N +) = 8N +4 ( 8N), γ [0,k] {0} (, {(j +/2, /2); j k} ) γ, k 4 3 k. P p (S V N ) 4k 3 k ( p) k. p>2/3 N, 0 ( ). N = N(p) (3). (3) {X b ; b V N }, {X b ; b V N } ( P p {Xb =, b V N } {S V N } ) = P p (X b =, b V N )P p (S V N ) 2 P p(x b =, b V N )= p4n 2 > 0. 2 P p ( C O = ) ( ), p>2/3 θ(p) > 0,, p H 2/ [0 p</3,θ(p) =0 p H /3] [2/3 < p,θ(p) > 0 p H 2/3]. 3.3 p>2/3 k3 k ( p) k <. k k 8N 3.4 θ(p). Z b S(p) ={b B 2 ; Z b p}, θ(p + h) =P ( ) S(p + h), h 0 S(p + h) S(p), i.e., S(p + h) =S(p) ( ), θ(p + h) θ(p) (h 0). h>0, θ h (p) (p ), θ (p) θ(p) ( ). 3.5 f (x) [0, ] f f ( ) f(x) [0, ]. 7

20 4.,. 4. ( 2.2 (iii)) 4.. π =(π j ) π j =/E j [T j ] > 0,. i, j S, 2. Q ij (s) =δ ij + F ij (s)q jj (s) i j s lim( s)q jj (s) = lim s s F jj (s) = F jj ( ) = E j [T j ]. lim( s)q ij (s) = F ij() s E j [T j ]., 2.2 F ij () = P i (T j < ) = i, j S, lim( s)q ij (s) = s E j [T j ] (=: π j ). E j [T j ] < 0 <π j. j S ( s)q ij (s) = Fatou j S π j, k S, j S π j q j,k lim if s ( s)q ij (s)q j,k j S lim( s) s =0 s q (+) i,k = lim s ( s)s (Q ik (s) δ ik ) = π k, k π j q j,k = π k (k S) j S. π j ( s)q jk (s) =( s) j S =0 s j S π j j,k = π k. (4) s Lebesgue j π j π k = π k j π j =. π =(π j ). π =(π j ), (4) s π k = F jk () π j E k [T k ] + π k E k [T k ] E k [T k ] j k. k; π k > 0 E k [T k ] <,. k S E k [T k ] <, 2.2 F jk () = P j (T k < ) =( j, k S) 8

21 ,., π k =/E k [T k ] > 0(k S). π =(π j ). 4. Fubii j S( s)q ij (s) =. 4.2 Fatou Lebsgue ( (Strog Law of Large Numbers)) X,X 2,... EX = m sup V (X ) < ( ) lim X k = m, a.s., i.e., P lim (X k m) =0 =. sup E[X] 4 < X,X 2,.... Borel f,...,f 4.3. E[f (X ) f (X )] = E[f (X )] E[f (X )]. (f k 0, f k = Ak (A k ),.) [ sup E[X 4 ] < ] X X = X m m = 0, i.e., E[X ]=0 ( ) 4 X k 0 Hölder E[Y 2 ] (E[Y 4 ]) /2 ( ) 4 E X k = E[Xk]+ 4 i j, i,j ( ) 4 ( E ) 4 X k = 4 E X k P ( lim ) X k =0 = E[Xi 2 ]E[Xj 2 ] 2 sup E[Xk] 4 k 2 sup E[Xk] 4 < k [], ( ),

(Basics of Proability Theory). (Probability Spacees ad Radom Variables,, (Ω, F, P ),, X,. (Ω, F, P ) (probability space) Ω ( ω Ω ) F ( 2 Ω ) Ω σ (σ-fi

(Basics of Proability Theory). (Probability Spacees ad Radom Variables,, (Ω, F, P ),, X,. (Ω, F, P ) (probability space) Ω ( ω Ω ) F ( 2 Ω ) Ω σ (σ-fi I (Basics of Probability Theory ad Radom Walks) 25 4 5 ( 4 ) (Preface),.,,,.,,,...,,.,.,,.,,. (,.) (Basics of Proability Theory). (Probability Spacees ad Radom Variables...............2, (Expectatios,

More information

(Basics of Proability Theory). (Probability Spacees ad Radom Variables,, (Ω, F, P ),, X,. (Ω, F, P ) (probability space) Ω ( ω Ω ) F ( 2 Ω ) Ω σ (σ-fi

(Basics of Proability Theory). (Probability Spacees ad Radom Variables,, (Ω, F, P ),, X,. (Ω, F, P ) (probability space) Ω ( ω Ω ) F ( 2 Ω ) Ω σ (σ-fi II (Basics of Probability Theory ad Radom Walks) (Preface),.,,,.,,,...,,.,.,,.,,. (Basics of Proability Theory). (Probability Spacees ad Radom Variables...............2, (Expectatios, Meas).............................

More information

II (Percolation) ( 3-4 ) 1. [ ],,,,,,,. 2. [ ],.. 3. [ ],. 4. [ ] [ ] G. Grimmett Percolation Springer-Verlag New-York [ ] 3

II (Percolation) ( 3-4 ) 1. [ ],,,,,,,. 2. [ ],.. 3. [ ],. 4. [ ] [ ] G. Grimmett Percolation Springer-Verlag New-York [ ] 3 II (Percolation) 12 9 27 ( 3-4 ) 1 [ ] 2 [ ] 3 [ ] 4 [ ] 1992 5 [ ] G Grimmett Percolation Springer-Verlag New-York 1989 6 [ ] 3 1 3 p H 2 3 2 FKG BK Russo 2 p H = p T (=: p c ) 3 2 Kesten p c =1/2 ( )

More information

(2 X Poisso P (λ ϕ X (t = E[e itx ] = k= itk λk e k! e λ = (e it λ k e λ = e eitλ e λ = e λ(eit 1. k! k= 6.7 X N(, 1 ϕ X (t = e 1 2 t2 : Cauchy ϕ X (t

(2 X Poisso P (λ ϕ X (t = E[e itx ] = k= itk λk e k! e λ = (e it λ k e λ = e eitλ e λ = e λ(eit 1. k! k= 6.7 X N(, 1 ϕ X (t = e 1 2 t2 : Cauchy ϕ X (t 6 6.1 6.1 (1 Z ( X = e Z, Y = Im Z ( Z = X + iy, i = 1 (2 Z E[ e Z ] < E[ Im Z ] < Z E[Z] = E[e Z] + ie[im Z] 6.2 Z E[Z] E[ Z ] : E[ Z ] < e Z Z, Im Z Z E[Z] α = E[Z], Z = Z Z 1 {Z } E[Z] = α = α [ α ]

More information

³ÎΨÏÀ

³ÎΨÏÀ 2017 12 12 Makoto Nakashima 2017 12 12 1 / 22 2.1. C, D π- C, D. A 1, A 2 C A 1 A 2 C A 3, A 4 D A 1 A 2 D Makoto Nakashima 2017 12 12 2 / 22 . (,, L p - ). Makoto Nakashima 2017 12 12 3 / 22 . (,, L p

More information

医系の統計入門第 2 版 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. このサンプルページの内容は, 第 2 版 1 刷発行時のものです.

医系の統計入門第 2 版 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.   このサンプルページの内容は, 第 2 版 1 刷発行時のものです. 医系の統計入門第 2 版 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/009192 このサンプルページの内容は, 第 2 版 1 刷発行時のものです. i 2 t 1. 2. 3 2 3. 6 4. 7 5. n 2 ν 6. 2 7. 2003 ii 2 2013 10 iii 1987

More information

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

More information

I (Analysis I) Lebesgue (Lebesgue Integral Theory) 1 (Seiji HIRABA) 1 ( ),,, ( )

I (Analysis I) Lebesgue (Lebesgue Integral Theory) 1 (Seiji HIRABA) 1 ( ),,, ( ) I (Analysis I) Lebesgue (Lebesgue Integral Theory) 1 (Seiji HIRABA) 1 ( ),,, ( ) 1 (Introduction) 1 1.1... 1 1.2 Riemann Lebesgue... 2 2 (Measurable sets and Measures) 4 2.1 σ-... 4 2.2 Borel... 5 2.3...

More information

微分積分 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. このサンプルページの内容は, 初版 1 刷発行時のものです.

微分積分 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.   このサンプルページの内容は, 初版 1 刷発行時のものです. 微分積分 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. ttp://www.morikita.co.jp/books/mid/00571 このサンプルページの内容は, 初版 1 刷発行時のものです. i ii 014 10 iii [note] 1 3 iv 4 5 3 6 4 x 0 sin x x 1 5 6 z = f(x, y) 1 y = f(x)

More information

1 a b = max{a, b}, a b = mi{a, b} a 1 a 2 a a 1 a = max{a 1,... a }, a 1 a = mi{a 1,... a }. A sup A, if A A A A A sup A sup A = + A if A = ± y = arct

1 a b = max{a, b}, a b = mi{a, b} a 1 a 2 a a 1 a = max{a 1,... a }, a 1 a = mi{a 1,... a }. A sup A, if A A A A A sup A sup A = + A if A = ± y = arct 27 6 2 1 2 2 5 3 8 4 13 5 16 6 19 7 23 8 27 N Z = {, ±1, ±2,... }, R =, R + = [, + ), R = [, ], C =. a b = max{a, b}, a b = mi{a, b}, a a, a a. f : X R [a < f < b] = {x X; a < f(x) < b}. X [f] = [f ],

More information

III III 2010 PART I 1 Definition 1.1 (, σ-),,,, Borel( ),, (σ-) (M, F, µ), (R, B(R)), (C, B(C)) Borel Definition 1.2 (µ-a.e.), (in µ), (in L 1 (µ)). T

III III 2010 PART I 1 Definition 1.1 (, σ-),,,, Borel( ),, (σ-) (M, F, µ), (R, B(R)), (C, B(C)) Borel Definition 1.2 (µ-a.e.), (in µ), (in L 1 (µ)). T III III 2010 PART I 1 Definition 1.1 (, σ-),,,, Borel( ),, (σ-) (M, F, µ), (R, B(R)), (C, B(C)) Borel Definition 1.2 (µ-a.e.), (in µ), (in L 1 (µ)). Theorem 1.3 (Lebesgue ) lim n f n = f µ-a.e. g L 1 (µ)

More information

Lebesgue Fubini L p Banach, Hilbert Höld

Lebesgue Fubini L p Banach, Hilbert Höld II (Analysis II) Lebesgue (Applications of Lebesgue Integral Theory) 1 (Seiji HIABA) 1 ( ),,, ( ) 1 1 1.1 1 Lebesgue........................ 1 1.2 2 Fubini...................... 2 2 L p 5 2.1 Banach, Hilbert..............................

More information

,. Black-Scholes u t t, x c u 0 t, x x u t t, x c u t, x x u t t, x + σ x u t, x + rx ut, x rux, t 0 x x,,.,. Step 3, 7,,, Step 6., Step 4,. Step 5,,.

,. Black-Scholes u t t, x c u 0 t, x x u t t, x c u t, x x u t t, x + σ x u t, x + rx ut, x rux, t 0 x x,,.,. Step 3, 7,,, Step 6., Step 4,. Step 5,,. 9 α ν β Ξ ξ Γ γ o δ Π π ε ρ ζ Σ σ η τ Θ θ Υ υ ι Φ φ κ χ Λ λ Ψ ψ µ Ω ω Def, Prop, Th, Lem, Note, Remark, Ex,, Proof, R, N, Q, C [a, b {x R : a x b} : a, b {x R : a < x < b} : [a, b {x R : a x < b} : a,

More information

6.1 (P (P (P (P (P (P (, P (, P.

6.1 (P (P (P (P (P (P (, P (, P. (011 30 7 0 ( ( 3 ( 010 1 (P.3 1 1.1 (P.4.................. 1 1. (P.4............... 1 (P.15.1 (P.16................. (P.0............3 (P.18 3.4 (P.3............... 4 3 (P.9 4 3.1 (P.30........... 4 3.

More information

ii 3.,. 4. F. ( ), ,,. 8.,. 1. (75% ) (25% ) =7 24, =7 25, =7 26 (. ). 1.,, ( ). 3.,...,.,.,.,.,. ( ) (1 2 )., ( ), 0., 1., 0,.

ii 3.,. 4. F. ( ), ,,. 8.,. 1. (75% ) (25% ) =7 24, =7 25, =7 26 (. ). 1.,, ( ). 3.,...,.,.,.,.,. ( ) (1 2 )., ( ), 0., 1., 0,. (1 C205) 4 10 (2 C206) 4 11 (2 B202) 4 12 25(2013) http://www.math.is.tohoku.ac.jp/~obata,.,,,..,,. 1. 2. 3. 4. 5. 6. 7. 8. 1., 2007 ( ).,. 2. P. G., 1995. 3. J. C., 1988. 1... 2.,,. ii 3.,. 4. F. ( ),..

More information

TOP URL 1

TOP URL   1 TOP URL http://amonphys.web.fc.com/ 3.............................. 3.............................. 4.3 4................... 5.4........................ 6.5........................ 8.6...........................7

More information

st.dvi

st.dvi 9 3 5................................... 5............................. 5....................................... 5.................................. 7.........................................................................

More information

(JAIST) (JSPS) PD URL:

(JAIST) (JSPS) PD URL: (JAIST) (JSPS) PD URL: http://researchmap.jp/kihara Email: kihara.takayuki.logic@gmail.com 2012 9 5 ii 2012 9 4 7 2012 JAIST iii #X X X Y X

More information

B [ 0.1 ] x > 0 x 6= 1 f(x) µ 1 1 xn 1 + sin sin x 1 x 1 f(x) := lim. n x n (1) lim inf f(x) (2) lim sup f(x) x 1 0 x 1 0 (

B [ 0.1 ] x > 0 x 6= 1 f(x) µ 1 1 xn 1 + sin sin x 1 x 1 f(x) := lim. n x n (1) lim inf f(x) (2) lim sup f(x) x 1 0 x 1 0 ( . 28 4 14 [.1 ] x > x 6= 1 f(x) µ 1 1 xn 1 + sin + 2 + sin x 1 x 1 f(x) := lim. 1 + x n (1) lim inf f(x) (2) lim sup f(x) x 1 x 1 (3) lim inf x 1+ f(x) (4) lim sup f(x) x 1+ [.2 ] [, 1] Ω æ x (1) (2) nx(1

More information

S I. dy fx x fx y fx + C 3 C vt dy fx 4 x, y dy yt gt + Ct + C dt v e kt xt v e kt + C k x v k + C C xt v k 3 r r + dr e kt S Sr πr dt d v } dt k e kt

S I. dy fx x fx y fx + C 3 C vt dy fx 4 x, y dy yt gt + Ct + C dt v e kt xt v e kt + C k x v k + C C xt v k 3 r r + dr e kt S Sr πr dt d v } dt k e kt S I. x yx y y, y,. F x, y, y, y,, y n http://ayapin.film.s.dendai.ac.jp/~matuda n /TeX/lecture.html PDF PS yx.................................... 3.3.................... 9.4................5..............

More information

() 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 (

() 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

More information

ii p ϕ x, t = C ϕ xe i ħ E t +C ϕ xe i ħ E t ψ x,t ψ x,t p79 やは時間変化しないことに注意 振動 粒子はだいたい このあたりにいる 粒子はだいたい このあたりにいる p35 D.3 Aψ Cϕdx = aψ ψ C Aϕ dx

ii p ϕ x, t = C ϕ xe i ħ E t +C ϕ xe i ħ E t ψ x,t ψ x,t p79 やは時間変化しないことに注意 振動 粒子はだいたい このあたりにいる 粒子はだいたい このあたりにいる p35 D.3 Aψ Cϕdx = aψ ψ C Aϕ dx i B5 7.8. p89 4. ψ x, tψx, t = ψ R x, t iψ I x, t ψ R x, t + iψ I x, t = ψ R x, t + ψ I x, t p 5.8 π π π F e ix + F e ix + F 3 e 3ix F e ix + F e ix + F 3 e 3ix dx πψ x πψx p39 7. AX = X A [ a b c d x

More information

6.1 (P (P (P (P (P (P (, P (, P.101

6.1 (P (P (P (P (P (P (, P (, P.101 (008 0 3 7 ( ( ( 00 1 (P.3 1 1.1 (P.3.................. 1 1. (P.4............... 1 (P.15.1 (P.15................. (P.18............3 (P.17......... 3.4 (P................ 4 3 (P.7 4 3.1 ( P.7...........

More information

6 2 2 x y x y t P P = P t P = I P P P ( ) ( ) ,, ( ) ( ) cos θ sin θ cos θ sin θ, sin θ cos θ sin θ cos θ y x θ x θ P

6 2 2 x y x y t P P = P t P = I P P P ( ) ( ) ,, ( ) ( ) cos θ sin θ cos θ sin θ, sin θ cos θ sin θ cos θ y x θ x θ P 6 x x 6.1 t P P = P t P = I P P P 1 0 1 0,, 0 1 0 1 cos θ sin θ cos θ sin θ, sin θ cos θ sin θ cos θ x θ x θ P x P x, P ) = t P x)p ) = t x t P P ) = t x = x, ) 6.1) x = Figure 6.1 Px = x, P=, θ = θ P

More information

..3. Ω, Ω F, P Ω, F, P ). ) F a) A, A,..., A i,... F A i F. b) A F A c F c) Ω F. ) A F A P A),. a) 0 P A) b) P Ω) c) [ ] A, A,..., A i,... F i j A i A

..3. Ω, Ω F, P Ω, F, P ). ) F a) A, A,..., A i,... F A i F. b) A F A c F c) Ω F. ) A F A P A),. a) 0 P A) b) P Ω) c) [ ] A, A,..., A i,... F i j A i A .. Laplace ). A... i),. ω i i ). {ω,..., ω } Ω,. ii) Ω. Ω. A ) r, A P A) P A) r... ).. Ω {,, 3, 4, 5, 6}. i i 6). A {, 4, 6} P A) P A) 3 6. ).. i, j i, j) ) Ω {i, j) i 6, j 6}., 36. A. A {i, j) i j }.

More information

2016 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 1 16 2 1 () X O 3 (O1) X O, O (O2) O O (O3) O O O X (X, O) O X X (O1), (O2), (O3) (O2) (O3) n (O2) U 1,..., U n O U k O k=1 (O3) U λ O( λ Λ) λ Λ U λ O 0 X 0 (O2) n =

More information

newmain.dvi

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

More information

( )/2 hara/lectures/lectures-j.html 2, {H} {T } S = {H, T } {(H, H), (H, T )} {(H, T ), (T, T )} {(H, H), (T, T )} {1

( )/2   hara/lectures/lectures-j.html 2, {H} {T } S = {H, T } {(H, H), (H, T )} {(H, T ), (T, T )} {(H, H), (T, T )} {1 ( )/2 http://www2.math.kyushu-u.ac.jp/ hara/lectures/lectures-j.html 1 2011 ( )/2 2 2011 4 1 2 1.1 1 2 1 2 3 4 5 1.1.1 sample space S S = {H, T } H T T H S = {(H, H), (H, T ), (T, H), (T, T )} (T, H) S

More information

S I. dy fx x fx y fx + C 3 C dy fx 4 x, y dy v C xt y C v e kt k > xt yt gt [ v dt dt v e kt xt v e kt + C k x v + C C k xt v k 3 r r + dr e kt S dt d

S I. dy fx x fx y fx + C 3 C dy fx 4 x, y dy v C xt y C v e kt k > xt yt gt [ v dt dt v e kt xt v e kt + C k x v + C C k xt v k 3 r r + dr e kt S dt d S I.. http://ayapin.film.s.dendai.ac.jp/~matuda /TeX/lecture.html PDF PS.................................... 3.3.................... 9.4................5.............. 3 5. Laplace................. 5....

More information

行列代数2010A

行列代数2010A a ij i j 1) i +j i, j) ij ij 1 j a i1 a ij a i a 1 a j a ij 1) i +j 1,j 1,j +1 a i1,1 a i1,j 1 a i1,j +1 a i1, a i +1,1 a i +1.j 1 a i +1,j +1 a i +1, a 1 a,j 1 a,j +1 a, ij i j 1,j 1,j +1 ij 1) i +j a

More information

熊本県数学問題正解

熊本県数学問題正解 00 y O x Typed by L A TEX ε ( ) (00 ) 5 4 4 ( ) http://www.ocn.ne.jp/ oboetene/plan/. ( ) (009 ) ( ).. http://www.ocn.ne.jp/ oboetene/plan/eng.html 8 i i..................................... ( )0... (

More information

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

More information

18 ( ) I II III A B C(100 ) 1, 2, 3, 5 I II A B (100 ) 1, 2, 3 I II A B (80 ) 6 8 I II III A B C(80 ) 1 n (1 + x) n (1) n C 1 + n C

18 ( ) I II III A B C(100 ) 1, 2, 3, 5 I II A B (100 ) 1, 2, 3 I II A B (80 ) 6 8 I II III A B C(80 ) 1 n (1 + x) n (1) n C 1 + n C 8 ( ) 8 5 4 I II III A B C( ),,, 5 I II A B ( ),, I II A B (8 ) 6 8 I II III A B C(8 ) n ( + x) n () n C + n C + + n C n = 7 n () 7 9 C : y = x x A(, 6) () A C () C P AP Q () () () 4 A(,, ) B(,, ) C(,,

More information

No δs δs = r + δr r = δr (3) δs δs = r r = δr + u(r + δr, t) u(r, t) (4) δr = (δx, δy, δz) u i (r + δr, t) u i (r, t) = u i x j δx j (5) δs 2

No δs δs = r + δr r = δr (3) δs δs = r r = δr + u(r + δr, t) u(r, t) (4) δr = (δx, δy, δz) u i (r + δr, t) u i (r, t) = u i x j δx j (5) δs 2 No.2 1 2 2 δs δs = r + δr r = δr (3) δs δs = r r = δr + u(r + δr, t) u(r, t) (4) δr = (δx, δy, δz) u i (r + δr, t) u i (r, t) = u i δx j (5) δs 2 = δx i δx i + 2 u i δx i δx j = δs 2 + 2s ij δx i δx j

More information

II A A441 : October 02, 2014 Version : Kawahira, Tomoki TA (Kondo, Hirotaka )

II A A441 : October 02, 2014 Version : Kawahira, Tomoki TA (Kondo, Hirotaka ) II 214-1 : October 2, 214 Version : 1.1 Kawahira, Tomoki TA (Kondo, Hirotaka ) http://www.math.nagoya-u.ac.jp/~kawahira/courses/14w-biseki.html pdf 1 2 1 9 1 16 1 23 1 3 11 6 11 13 11 2 11 27 12 4 12 11

More information

24 I ( ) 1. R 3 (i) C : x 2 + y 2 1 = 0 (ii) C : y = ± 1 x 2 ( 1 x 1) (iii) C : x = cos t, y = sin t (0 t 2π) 1.1. γ : [a, b] R n ; t γ(t) = (x

24 I ( ) 1. R 3 (i) C : x 2 + y 2 1 = 0 (ii) C : y = ± 1 x 2 ( 1 x 1) (iii) C : x = cos t, y = sin t (0 t 2π) 1.1. γ : [a, b] R n ; t γ(t) = (x 24 I 1.1.. ( ) 1. R 3 (i) C : x 2 + y 2 1 = 0 (ii) C : y = ± 1 x 2 ( 1 x 1) (iii) C : x = cos t, y = sin t (0 t 2π) 1.1. γ : [a, b] R n ; t γ(t) = (x 1 (t), x 2 (t),, x n (t)) ( ) ( ), γ : (i) x 1 (t),

More information

‚åŁÎ“·„´Šš‡ðŠp‡¢‡½‹âfi`fiI…A…‰…S…−…Y…•‡ÌMarkovŸA“½fiI›ð’Í

‚åŁÎ“·„´Šš‡ðŠp‡¢‡½‹âfi`fiI…A…‰…S…−…Y…•‡ÌMarkovŸA“½fiI›ð’Í Markov 2009 10 2 Markov 2009 10 2 1 / 25 1 (GA) 2 GA 3 4 Markov 2009 10 2 2 / 25 (GA) (GA) L ( 1) I := {0, 1} L f : I (0, ) M( 2) S := I M GA (GA) f (i) i I Markov 2009 10 2 3 / 25 (GA) ρ(i, j), i, j I

More information

I

I I io@hiroshima-u.ac.jp 27 6 A A. /a δx = lim a + a exp π x2 a 2 = lim a + a = lim a + a exp a 2 π 2 x 2 + a 2 2 x a x = lim a + a Sic a x = lim a + a Rect a Gaussia Loretzia Bilateral expoetial Normalized

More information

n ξ n,i, i = 1,, n S n ξ n,i n 0 R 1,.. σ 1 σ i .10.14.15 0 1 0 1 1 3.14 3.18 3.19 3.14 3.14,. ii 1 1 1.1..................................... 1 1............................... 3 1.3.........................

More information

入試の軌跡

入試の軌跡 4 y O x 4 Typed by L A TEX ε ) ) ) 6 4 ) 4 75 ) http://kumamoto.s.xrea.com/plan/.. PDF) Ctrl +L) Ctrl +) Ctrl + Ctrl + ) ) Alt + ) Alt + ) ESC. http://kumamoto.s.xrea.com/nyusi/kumadai kiseki ri i.pdf

More information

(I) (II) 2 (I) 2 (II) 2 (III) (I) (II) (II) : 2 Typeset by Akio Namba usig Powerdot. 2 / 47

(I) (II) 2 (I) 2 (II) 2 (III) (I) (II) (II) : 2 Typeset by Akio Namba usig Powerdot. 2 / 47 4 Typeset by Akio Namba usig Powerdot. / 47 (I) (II) 2 (I) 2 (II) 2 (III) (I) (II) (II) : 2 Typeset by Akio Namba usig Powerdot. 2 / 47 (I) (II) 2 (I) 2 (II) 2 (III) (I) (II) (II) : 2 (radom variable):

More information

.3. (x, x = (, u = = 4 (, x x = 4 x, x 0 x = 0 x = 4 x.4. ( z + z = 8 z, z 0 (z, z = (0, 8, (,, (8, 0 3 (0, 8, (,, (8, 0 z = z 4 z (g f(x = g(

.3. (x, x = (, u = = 4 (, x x = 4 x, x 0 x = 0 x = 4 x.4. ( z + z = 8 z, z 0 (z, z = (0, 8, (,, (8, 0 3 (0, 8, (,, (8, 0 z = z 4 z (g f(x = g( 06 5.. ( y = x x y 5 y 5 = (x y = x + ( y = x + y = x y.. ( Y = C + I = 50 + 0.5Y + 50 r r = 00 0.5Y ( L = M Y r = 00 r = 0.5Y 50 (3 00 0.5Y = 0.5Y 50 Y = 50, r = 5 .3. (x, x = (, u = = 4 (, x x = 4 x,

More information

201711grade1ouyou.pdf

201711grade1ouyou.pdf 2017 11 26 1 2 52 3 12 13 22 23 32 33 42 3 5 3 4 90 5 6 A 1 2 Web Web 3 4 1 2... 5 6 7 7 44 8 9 1 2 3 1 p p >2 2 A 1 2 0.6 0.4 0.52... (a) 0.6 0.4...... B 1 2 0.8-0.2 0.52..... (b) 0.6 0.52.... 1 A B 2

More information

x () g(x) = f(t) dt f(x), F (x) 3x () g(x) g (x) f(x), F (x) (3) h(x) = x 3x tf(t) dt.9 = {(x, y) ; x, y, x + y } f(x, y) = xy( x y). h (x) f(x), F (x

x () g(x) = f(t) dt f(x), F (x) 3x () g(x) g (x) f(x), F (x) (3) h(x) = x 3x tf(t) dt.9 = {(x, y) ; x, y, x + y } f(x, y) = xy( x y). h (x) f(x), F (x [ ] IC. f(x) = e x () f(x) f (x) () lim f(x) lim f(x) x + x (3) lim f(x) lim f(x) x + x (4) y = f(x) ( ) ( s46). < a < () a () lim a log xdx a log xdx ( ) n (3) lim log k log n n n k=.3 z = log(x + y ),

More information

1 1.1 ( ). z = a + bi, a, b R 0 a, b 0 a 2 + b 2 0 z = a + bi = ( ) a 2 + b 2 a a 2 + b + b 2 a 2 + b i 2 r = a 2 + b 2 θ cos θ = a a 2 + b 2, sin θ =

1 1.1 ( ). z = a + bi, a, b R 0 a, b 0 a 2 + b 2 0 z = a + bi = ( ) a 2 + b 2 a a 2 + b + b 2 a 2 + b i 2 r = a 2 + b 2 θ cos θ = a a 2 + b 2, sin θ = 1 1.1 ( ). z = + bi,, b R 0, b 0 2 + b 2 0 z = + bi = ( ) 2 + b 2 2 + b + b 2 2 + b i 2 r = 2 + b 2 θ cos θ = 2 + b 2, sin θ = b 2 + b 2 2π z = r(cos θ + i sin θ) 1.2 (, ). 1. < 2. > 3. ±,, 1.3 ( ). A

More information

V 0 = + r pv (H) + qv (T ) = + r ps (H) + qs (T ) = S 0 X n+ (T ) = n S n+ (T ) + ( + r)(x n n S n ) = ( + r)x n + n (d r)s n = ( + r)v n + V n+(h) V

V 0 = + r pv (H) + qv (T ) = + r ps (H) + qs (T ) = S 0 X n+ (T ) = n S n+ (T ) + ( + r)(x n n S n ) = ( + r)x n + n (d r)s n = ( + r)v n + V n+(h) V I (..2) (0 < d < + r < u) X 0, X X = 0 S + ( + r)(x 0 0 S 0 ) () X 0 = 0, P (X 0) =, P (X > 0) > 0 0 H, T () X 0 = 0, X (H) = 0 us 0 ( + r) 0 S 0 = 0 S 0 (u r) X (T ) = 0 ds 0 ( + r) 0 S 0 = 0 S 0 (d r)

More information

S K(S) = T K(T ) T S K n (1.1) n {}}{ n K n (1.1) 0 K 0 0 K Q p K Z/pZ L K (1) L K L K (2) K L L K [L : K] 1.1.

S K(S) = T K(T ) T S K n (1.1) n {}}{ n K n (1.1) 0 K 0 0 K Q p K Z/pZ L K (1) L K L K (2) K L L K [L : K] 1.1. () 1.1.. 1. 1.1. (1) L K (i) 0 K 1 K (ii) x, y K x + y K, x y K (iii) x, y K xy K (iv) x K \ {0} x 1 K K L L K ( 0 L 1 L ) L K L/K (2) K M L M K L 1.1. C C 1.2. R K = {a + b 3 i a, b Q} Q( 2, 3) = Q( 2

More information

ルベーグ積分 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. このサンプルページの内容は, 初版 1 刷発行時のものです.

ルベーグ積分 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.   このサンプルページの内容は, 初版 1 刷発行時のものです. ルベーグ積分 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/005431 このサンプルページの内容は, 初版 1 刷発行時のものです. Lebesgue 1 2 4 4 1 2 5 6 λ a

More information

ii 3.,. 4. F. (), ,,. 8.,. 1. (75%) (25%) =7 20, =7 21 (. ). 1.,, (). 3.,. 1. ().,.,.,.,.,. () (12 )., (), 0. 2., 1., 0,.

ii 3.,. 4. F. (), ,,. 8.,. 1. (75%) (25%) =7 20, =7 21 (. ). 1.,, (). 3.,. 1. ().,.,.,.,.,. () (12 )., (), 0. 2., 1., 0,. 24(2012) (1 C106) 4 11 (2 C206) 4 12 http://www.math.is.tohoku.ac.jp/~obata,.,,,.. 1. 2. 3. 4. 5. 6. 7.,,. 1., 2007 (). 2. P. G. Hoel, 1995. 3... 1... 2.,,. ii 3.,. 4. F. (),.. 5... 6.. 7.,,. 8.,. 1. (75%)

More information

f(x) = f(x ) + α(x)(x x ) α(x) x = x. x = f (y), x = f (y ) y = f f (y) = f f (y ) + α(f (y))(f (y) f (y )) f (y) = f (y ) + α(f (y)) (y y ) ( (2) ) f

f(x) = f(x ) + α(x)(x x ) α(x) x = x. x = f (y), x = f (y ) y = f f (y) = f f (y ) + α(f (y))(f (y) f (y )) f (y) = f (y ) + α(f (y)) (y y ) ( (2) ) f 22 A 3,4 No.3 () (2) (3) (4), (5) (6) (7) (8) () n x = (x,, x n ), = (,, n ), x = ( (x i i ) 2 ) /2 f(x) R n f(x) = f() + i α i (x ) i + o( x ) α,, α n g(x) = o( x )) lim x g(x) x = y = f() + i α i(x )

More information

Microsoft Word - 表紙.docx

Microsoft Word - 表紙.docx 黒住英司 [ 著 ] サピエンティア 計量経済学 訂正および練習問題解答 (206/2/2 版 ) 訂正 練習問題解答 3 .69, 3.8 4 (X i X)U i i i (X i μ x )U i ( X μx ) U i. i E [ ] (X i μ x )U i i E[(X i μ x )]E[U i ]0. i V [ ] (X i μ x )U i i 2 i j E [(X i

More information

X G P G (X) G BG [X, BG] S 2 2 2 S 2 2 S 2 = { (x 1, x 2, x 3 ) R 3 x 2 1 + x 2 2 + x 2 3 = 1 } R 3 S 2 S 2 v x S 2 x x v(x) T x S 2 T x S 2 S 2 x T x S 2 = { ξ R 3 x ξ } R 3 T x S 2 S 2 x x T x S 2

More information

量子力学 問題

量子力学 問題 3 : 203 : 0. H = 0 0 2 6 0 () = 6, 2 = 2, 3 = 3 3 H 6 2 3 ϵ,2,3 (2) ψ = (, 2, 3 ) ψ Hψ H (3) P i = i i P P 2 = P 2 P 3 = P 3 P = O, P 2 i = P i (4) P + P 2 + P 3 = E 3 (5) i ϵ ip i H 0 0 (6) R = 0 0 [H,

More information

waseda2010a-jukaiki1-main.dvi

waseda2010a-jukaiki1-main.dvi November, 2 Contents 6 2 8 3 3 3 32 32 33 5 34 34 6 35 35 7 4 R 2 7 4 4 9 42 42 2 43 44 2 5 : 2 5 5 23 52 52 23 53 53 23 54 24 6 24 6 6 26 62 62 26 63 t 27 7 27 7 7 28 72 72 28 73 36) 29 8 29 8 29 82 3

More information

離散最適化基礎論 第 11回 組合せ最適化と半正定値計画法

離散最適化基礎論 第 11回  組合せ最適化と半正定値計画法 11 okamotoy@uec.ac.jp 2019 1 25 2019 1 25 10:59 ( ) (11) 2019 1 25 1 / 38 1 (10/5) 2 (1) (10/12) 3 (2) (10/19) 4 (3) (10/26) (11/2) 5 (1) (11/9) 6 (11/16) 7 (11/23) (11/30) (12/7) ( ) (11) 2019 1 25 2

More information

I A A441 : April 15, 2013 Version : 1.1 I Kawahira, Tomoki TA (Shigehiro, Yoshida )

I A A441 : April 15, 2013 Version : 1.1 I   Kawahira, Tomoki TA (Shigehiro, Yoshida ) I013 00-1 : April 15, 013 Version : 1.1 I Kawahira, Tomoki TA (Shigehiro, Yoshida) http://www.math.nagoya-u.ac.jp/~kawahira/courses/13s-tenbou.html pdf * 4 15 4 5 13 e πi = 1 5 0 5 7 3 4 6 3 6 10 6 17

More information

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 = 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

More information

A

A A 2563 15 4 21 1 3 1.1................................................ 3 1.2............................................. 3 2 3 2.1......................................... 3 2.2............................................

More information

基礎から学ぶトラヒック理論 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. このサンプルページの内容は, 初版 1 刷発行時のものです.

基礎から学ぶトラヒック理論 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.   このサンプルページの内容は, 初版 1 刷発行時のものです. 基礎から学ぶトラヒック理論 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/085221 このサンプルページの内容は, 初版 1 刷発行時のものです. i +α 3 1 2 4 5 1 2 ii 3 4 5 6 7 8 9 9.3 2014 6 iii 1 1 2 5 2.1 5 2.2 7

More information

d ϕ i) t d )t0 d ϕi) ϕ i) t x j t d ) ϕ t0 t α dx j d ) ϕ i) t dx t0 j x j d ϕ i) ) t x j dx t0 j f i x j ξ j dx i + ξ i x j dx j f i ξ i x j dx j d )

d ϕ i) t d )t0 d ϕi) ϕ i) t x j t d ) ϕ t0 t α dx j d ) ϕ i) t dx t0 j x j d ϕ i) ) t x j dx t0 j f i x j ξ j dx i + ξ i x j dx j f i ξ i x j dx j d ) 23 M R M ϕ : R M M ϕt, x) ϕ t x) ϕ s ϕ t ϕ s+t, ϕ 0 id M M ϕ t M ξ ξ ϕ t d ϕ tx) ξϕ t x)) U, x 1,...,x n )) ϕ t x) ϕ 1) t x),...,ϕ n) t x)), ξx) ξ i x) d ϕi) t x) ξ i ϕ t x)) M f ϕ t f)x) f ϕ t )x) fϕ

More information

all.dvi

all.dvi 5,, Euclid.,..,... Euclid,.,.,, e i (i =,, ). 6 x a x e e e x.:,,. a,,. a a = a e + a e + a e = {e, e, e } a (.) = a i e i = a i e i (.) i= {a,a,a } T ( T ),.,,,,. (.),.,...,,. a 0 0 a = a 0 + a + a 0

More information

1 1.1 H = µc i c i + c i t ijc j + 1 c i c j V ijklc k c l (1) V ijkl = V jikl = V ijlk = V jilk () t ij = t ji, V ijkl = V lkji (3) (1) V 0 H mf = µc

1 1.1 H = µc i c i + c i t ijc j + 1 c i c j V ijklc k c l (1) V ijkl = V jikl = V ijlk = V jilk () t ij = t ji, V ijkl = V lkji (3) (1) V 0 H mf = µc 013 6 30 BCS 1 1.1........................ 1................................ 3 1.3............................ 3 1.4............................... 5 1.5.................................... 5 6 3 7 4 8

More information

untitled

untitled 2 : n =1, 2,, 10000 0.5125 0.51 0.5075 0.505 0.5025 0.5 0.4975 0.495 0 2000 4000 6000 8000 10000 2 weak law of large numbers 1. X 1,X 2,,X n 2. µ = E(X i ),i=1, 2,,n 3. σi 2 = V (X i ) σ 2,i=1, 2,,n ɛ>0

More information

populatio sample II, B II? [1] I. [2] 1 [3] David J. Had [4] 2 [5] 3 2

populatio sample II, B II?  [1] I. [2] 1 [3] David J. Had [4] 2 [5] 3 2 (2015 ) 1 NHK 2012 5 28 2013 7 3 2014 9 17 2015 4 8!? New York Times 2009 8 5 For Today s Graduate, Just Oe Word: Statistics Google Hal Varia I keep sayig that the sexy job i the ext 10 years will be statisticias.

More information

II Brown Brown

II Brown Brown II 16 12 5 1 Brown 3 1.1..................................... 3 1.2 Brown............................... 5 1.3................................... 8 1.4 Markov.................................... 1 1.5

More information

2 1 1 α = a + bi(a, b R) α (conjugate) α = a bi α (absolute value) α = a 2 + b 2 α (norm) N(α) = a 2 + b 2 = αα = α 2 α (spure) (trace) 1 1. a R aα =

2 1 1 α = a + bi(a, b R) α (conjugate) α = a bi α (absolute value) α = a 2 + b 2 α (norm) N(α) = a 2 + b 2 = αα = α 2 α (spure) (trace) 1 1. a R aα = 1 1 α = a + bi(a, b R) α (conjugate) α = a bi α (absolute value) α = a + b α (norm) N(α) = a + b = αα = α α (spure) (trace) 1 1. a R aα = aα. α = α 3. α + β = α + β 4. αβ = αβ 5. β 0 6. α = α ( ) α = α

More information

211 kotaro@math.titech.ac.jp 1 R *1 n n R n *2 R n = {(x 1,..., x n ) x 1,..., x n R}. R R 2 R 3 R n R n R n D D R n *3 ) (x 1,..., x n ) f(x 1,..., x n ) f D *4 n 2 n = 1 ( ) 1 f D R n f : D R 1.1. (x,

More information

1 Abstract 2 3 n a ax 2 + bx + c = 0 (a 0) (1) ( x + b ) 2 = b2 4ac 2a 4a 2 D = b 2 4ac > 0 (1) 2 D = 0 D < 0 x + b 2a = ± b2 4ac 2a b ± b 2

1 Abstract 2 3 n a ax 2 + bx + c = 0 (a 0) (1) ( x + b ) 2 = b2 4ac 2a 4a 2 D = b 2 4ac > 0 (1) 2 D = 0 D < 0 x + b 2a = ± b2 4ac 2a b ± b 2 1 Abstract n 1 1.1 a ax + bx + c = 0 (a 0) (1) ( x + b ) = b 4ac a 4a D = b 4ac > 0 (1) D = 0 D < 0 x + b a = ± b 4ac a b ± b 4ac a b a b ± 4ac b i a D (1) ax + bx + c D 0 () () (015 8 1 ) 1. D = b 4ac

More information

4 4 4 a b c d a b A c d A a da ad bce O E O n A n O ad bc a d n A n O 5 {a n } S n a k n a n + k S n a a n+ S n n S n n log x x {xy } x, y x + y 7 fx

4 4 4 a b c d a b A c d A a da ad bce O E O n A n O ad bc a d n A n O 5 {a n } S n a k n a n + k S n a a n+ S n n S n n log x x {xy } x, y x + y 7 fx 4 4 5 4 I II III A B C, 5 7 I II A B,, 8, 9 I II A B O A,, Bb, b, Cc, c, c b c b b c c c OA BC P BC OP BC P AP BC n f n x xn e x! e n! n f n x f n x f n x f k x k 4 e > f n x dx k k! fx sin x cos x tan

More information

( )

( ) 18 10 01 ( ) 1 2018 4 1.1 2018............................... 4 1.2 2018......................... 5 2 2017 7 2.1 2017............................... 7 2.2 2017......................... 8 3 2016 9 3.1 2016...............................

More information

TOP URL 1

TOP URL   1 TOP URL http://amonphys.web.fc.com/ 1 19 3 19.1................... 3 19.............................. 4 19.3............................... 6 19.4.............................. 8 19.5.............................

More information

untitled

untitled 1 kaiseki1.lec(tex) 19951228 19960131;0204 14;16 26;0329; 0410;0506;22;0603-05;08;20;0707;09;11-22;24-28;30;0807;12-24;27;28; 19970104(σ,F = µ);0212( ); 0429(σ- A n ); 1221( ); 20000529;30(L p ); 20050323(

More information

v v = v 1 v 2 v 3 (1) R = (R ij ) (2) R (R 1 ) ij = R ji (3) 3 R ij R ik = δ jk (4) i=1 δ ij Kronecker δ ij = { 1 (i = j) 0 (i

v v = v 1 v 2 v 3 (1) R = (R ij ) (2) R (R 1 ) ij = R ji (3) 3 R ij R ik = δ jk (4) i=1 δ ij Kronecker δ ij = { 1 (i = j) 0 (i 1. 1 1.1 1.1.1 1.1.1.1 v v = v 1 v 2 v 3 (1) R = (R ij ) (2) R (R 1 ) ij = R ji (3) R ij R ik = δ jk (4) δ ij Kronecker δ ij = { 1 (i = j) 0 (i j) (5) 1 1.1. v1.1 2011/04/10 1. 1 2 v i = R ij v j (6) [

More information

名称未設定

名称未設定 2007 12 19 i I 1 1 3 1.1.................... 3 1.2................................ 4 1.3.................................... 7 2 9 2.1...................................... 9 2.2....................................

More information

() Remrk I = [0, ] [x i, x i ]. (x : ) f(x) = 0 (x : ) ξ i, (f) = f(ξ i )(x i x i ) = (x i x i ) = ξ i, (f) = f(ξ i )(x i x i ) = 0 (f) 0.

() Remrk I = [0, ] [x i, x i ]. (x : ) f(x) = 0 (x : ) ξ i, (f) = f(ξ i )(x i x i ) = (x i x i ) = ξ i, (f) = f(ξ i )(x i x i ) = 0 (f) 0. () 6 f(x) [, b] 6. Riemnn [, b] f(x) S f(x) [, b] (Riemnn) = x 0 < x < x < < x n = b. I = [, b] = {x,, x n } mx(x i x i ) =. i [x i, x i ] ξ i n (f) = f(ξ i )(x i x i ) i=. (ξ i ) (f) 0( ), ξ i, S, ε >

More information

A11 (1993,1994) 29 A12 (1994) 29 A13 Trefethen and Bau Numerical Linear Algebra (1997) 29 A14 (1999) 30 A15 (2003) 30 A16 (2004) 30 A17 (2007) 30 A18

A11 (1993,1994) 29 A12 (1994) 29 A13 Trefethen and Bau Numerical Linear Algebra (1997) 29 A14 (1999) 30 A15 (2003) 30 A16 (2004) 30 A17 (2007) 30 A18 2013 8 29y, 2016 10 29 1 2 2 Jordan 3 21 3 3 Jordan (1) 3 31 Jordan 4 32 Jordan 4 33 Jordan 6 34 Jordan 8 35 9 4 Jordan (2) 10 41 x 11 42 x 12 43 16 44 19 441 19 442 20 443 25 45 25 5 Jordan 26 A 26 A1

More information

4. ϵ(ν, T ) = c 4 u(ν, T ) ϵ(ν, T ) T ν π4 Planck dx = 0 e x 1 15 U(T ) x 3 U(T ) = σt 4 Stefan-Boltzmann σ 2π5 k 4 15c 2 h 3 = W m 2 K 4 5.

4. ϵ(ν, T ) = c 4 u(ν, T ) ϵ(ν, T ) T ν π4 Planck dx = 0 e x 1 15 U(T ) x 3 U(T ) = σt 4 Stefan-Boltzmann σ 2π5 k 4 15c 2 h 3 = W m 2 K 4 5. A 1. Boltzmann Planck u(ν, T )dν = 8πh ν 3 c 3 kt 1 dν h 6.63 10 34 J s Planck k 1.38 10 23 J K 1 Boltzmann u(ν, T ) T ν e hν c = 3 10 8 m s 1 2. Planck λ = c/ν Rayleigh-Jeans u(ν, T )dν = 8πν2 kt dν c

More information

main.dvi

main.dvi SGC - 70 2, 3 23 ɛ-δ 2.12.8 3 2.92.13 4 2 3 1 2.1 2.102.12 [8][14] [1],[2] [4][7] 2 [4] 1 2009 8 1 1 1.1... 1 1.2... 4 1.3 1... 8 1.4 2... 9 1.5... 12 1.6 1... 16 1.7... 18 1.8... 21 1.9... 23 2 27 2.1

More information

16 B

16 B 16 B (1) 3 (2) (3) 5 ( ) 3 : 2 3 : 3 : () 3 19 ( ) 2 ax 2 + bx + c = 0 (a 0) x = b ± b 2 4ac 2a 3, 4 5 1824 5 Contents 1. 1 2. 7 3. 13 4. 18 5. 22 6. 25 7. 27 8. 31 9. 37 10. 46 11. 50 12. 56 i 1 1. 1.1..

More information

D = [a, b] [c, d] D ij P ij (ξ ij, η ij ) f S(f,, {P ij }) S(f,, {P ij }) = = k m i=1 j=1 m n f(ξ ij, η ij )(x i x i 1 )(y j y j 1 ) = i=1 j

D = [a, b] [c, d] D ij P ij (ξ ij, η ij ) f S(f,, {P ij }) S(f,, {P ij }) = = k m i=1 j=1 m n f(ξ ij, η ij )(x i x i 1 )(y j y j 1 ) = i=1 j 6 6.. [, b] [, d] ij P ij ξ ij, η ij f Sf,, {P ij } Sf,, {P ij } k m i j m fξ ij, η ij i i j j i j i m i j k i i j j m i i j j k i i j j kb d {P ij } lim Sf,, {P ij} kb d f, k [, b] [, d] f, d kb d 6..

More information

読めば必ずわかる 分散分析の基礎 第2版

読めば必ずわかる 分散分析の基礎 第2版 2 2003 12 5 ( ) ( ) 2 I 3 1 3 2 2? 6 3 11 4? 12 II 14 5 15 6 16 7 17 8 19 9 21 10 22 11 F 25 12 : 1 26 3 I 1 17 11 x 1, x 2,, x n x( ) x = 1 n n i=1 x i 12 (SD ) x 1, x 2,, x n s 2 s 2 = 1 n n (x i x)

More information

1 variation 1.1 imension unit L m M kg T s Q C QT 1 A = C s 1 MKSA F = ma N N = kg m s 1.1 J E = 1 mv W = F x J = kg m s 1 = N m 1.

1 variation 1.1 imension unit L m M kg T s Q C QT 1 A = C s 1 MKSA F = ma N N = kg m s 1.1 J E = 1 mv W = F x J = kg m s 1 = N m 1. 1.1 1. 1.3.1..3.4 3.1 3. 3.3 4.1 4. 4.3 5.1 5. 5.3 6.1 6. 6.3 7.1 7. 7.3 1 1 variation 1.1 imension unit L m M kg T s Q C QT 1 A = C s 1 MKSA F = ma N N = kg m s 1.1 J E = 1 mv W = F x J = kg m s 1 = N

More information

1 1 3 1.1 (Frequecy Tabulatios)................................ 3 1........................................ 8 1.3.....................................

1 1 3 1.1 (Frequecy Tabulatios)................................ 3 1........................................ 8 1.3..................................... 1 1 3 1.1 (Frequecy Tabulatios)................................ 3 1........................................ 8 1.3........................................... 1 17.1................................................

More information

meiji_resume_1.PDF

meiji_resume_1.PDF β β β (q 1,q,..., q n ; p 1, p,..., p n ) H(q 1,q,..., q n ; p 1, p,..., p n ) Hψ = εψ ε k = k +1/ ε k = k(k 1) (x, y, z; p x, p y, p z ) (r; p r ), (θ; p θ ), (ϕ; p ϕ ) ε k = 1/ k p i dq i E total = E

More information

「産業上利用することができる発明」の審査の運用指針(案)

「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)

More information

n Y 1 (x),..., Y n (x) 1 W (Y 1 (x),..., Y n (x)) 0 W (Y 1 (x),..., Y n (x)) = Y 1 (x)... Y n (x) Y 1(x)... Y n(x) (x)... Y n (n 1) (x) Y (n 1)

n Y 1 (x),..., Y n (x) 1 W (Y 1 (x),..., Y n (x)) 0 W (Y 1 (x),..., Y n (x)) = Y 1 (x)... Y n (x) Y 1(x)... Y n(x) (x)... Y n (n 1) (x) Y (n 1) D d dx 1 1.1 n d n y a 0 dx n + a d n 1 y 1 dx n 1 +... + a dy n 1 dx + a ny = f(x)...(1) dk y dx k = y (k) a 0 y (n) + a 1 y (n 1) +... + a n 1 y + a n y = f(x)...(2) (2) (2) f(x) 0 a 0 y (n) + a 1 y

More information

Dirac 38 5 Dirac 4 4 γ µ p µ p µ + m 2 = ( p µ γ µ + m)(p ν γ ν + m) (5.1) γ = p µ p ν γ µ γ ν p µ γ µ m + mp ν γ ν + m 2 = 1 2 p µp ν {γ µ, γ ν } + m

Dirac 38 5 Dirac 4 4 γ µ p µ p µ + m 2 = ( p µ γ µ + m)(p ν γ ν + m) (5.1) γ = p µ p ν γ µ γ ν p µ γ µ m + mp ν γ ν + m 2 = 1 2 p µp ν {γ µ, γ ν } + m Dirac 38 5 Dirac 4 4 γ µ p µ p µ + m 2 p µ γ µ + mp ν γ ν + m 5.1 γ p µ p ν γ µ γ ν p µ γ µ m + mp ν γ ν + m 2 1 2 p µp ν {γ µ, γ ν } + m 2 5.2 p m p p µ γ µ {, } 10 γ {γ µ, γ ν } 2η µν 5.3 p µ γ µ + mp

More information

Feynman Encounter with Mathematics 52, [1] N. Kumano-go, Feynman path integrals as analysis on path space by time slicing approximation. Bull

Feynman Encounter with Mathematics 52, [1] N. Kumano-go, Feynman path integrals as analysis on path space by time slicing approximation. Bull Feynman Encounter with Mathematics 52, 200 9 [] N. Kumano-go, Feynman path integrals as analysis on path space by time slicing approximation. Bull. Sci. Math. vol. 28 (2004) 97 25. [2] D. Fujiwara and

More information

ii 3.,. 4. F. (), ,,. 8.,. 1. (75% ) (25% ) =9 7, =9 8 (. ). 1.,, (). 3.,. 1. ( ).,.,.,.,.,. ( ) (1 2 )., ( ), 0. 2., 1., 0,.

ii 3.,. 4. F. (), ,,. 8.,. 1. (75% ) (25% ) =9 7, =9 8 (. ). 1.,, (). 3.,. 1. ( ).,.,.,.,.,. ( ) (1 2 )., ( ), 0. 2., 1., 0,. 23(2011) (1 C104) 5 11 (2 C206) 5 12 http://www.math.is.tohoku.ac.jp/~obata,.,,,.. 1. 2. 3. 4. 5. 6. 7.,,. 1., 2007 ( ). 2. P. G. Hoel, 1995. 3... 1... 2.,,. ii 3.,. 4. F. (),.. 5.. 6.. 7.,,. 8.,. 1. (75%

More information

OABC OA OC 4, OB, AOB BOC COA 60 OA a OB b OC c () AB AC () ABC D OD ABC OD OA + p AB + q AC p q () OABC 4 f(x) + x ( ), () y f(x) P l 4 () y f(x) l P

OABC OA OC 4, OB, AOB BOC COA 60 OA a OB b OC c () AB AC () ABC D OD ABC OD OA + p AB + q AC p q () OABC 4 f(x) + x ( ), () y f(x) P l 4 () y f(x) l P 4 ( ) ( ) ( ) ( ) 4 5 5 II III A B (0 ) 4, 6, 7 II III A B (0 ) ( ),, 6, 8, 9 II III A B (0 ) ( [ ] ) 5, 0, II A B (90 ) log x x () (a) y x + x (b) y sin (x + ) () (a) (b) (c) (d) 0 e π 0 x x x + dx e

More information

1 1 sin cos P (primary) S (secondly) 2 P S A sin(ω2πt + α) A ω 1 ω α V T m T m 1 100Hz m 2 36km 500Hz. 36km 1

1 1 sin cos P (primary) S (secondly) 2 P S A sin(ω2πt + α) A ω 1 ω α V T m T m 1 100Hz m 2 36km 500Hz. 36km 1 sin cos P (primary) S (secondly) 2 P S A sin(ω2πt + α) A ω ω α 3 3 2 2V 3 33+.6T m T 5 34m Hz. 34 3.4m 2 36km 5Hz. 36km m 34 m 5 34 + m 5 33 5 =.66m 34m 34 x =.66 55Hz, 35 5 =.7 485.7Hz 2 V 5Hz.5V.5V V

More information

1 12 ( )150 ( ( ) ) x M x 0 1 M 2 5x 2 + 4x + 3 x 2 1 M x M 2 1 M x (x + 1) 2 (1) x 2 + x + 1 M (2) 1 3 M (3) x 4 +

1 12 ( )150 ( ( ) ) x M x 0 1 M 2 5x 2 + 4x + 3 x 2 1 M x M 2 1 M x (x + 1) 2 (1) x 2 + x + 1 M (2) 1 3 M (3) x 4 + ( )5 ( ( ) ) 4 6 7 9 M M 5 + 4 + M + M M + ( + ) () + + M () M () 4 + + M a b y = a + b a > () a b () y V a () V a b V n f() = n k= k k () < f() = log( ) t dt log () n+ (i) dt t (n + ) (ii) < t dt n+ n

More information

Dynkin Serre Weyl

Dynkin Serre Weyl Dynkin Naoya Enomoto 2003.3. paper Dynkin Introduction Dynkin Lie Lie paper 1 0 Introduction 3 I ( ) Lie Dynkin 4 1 ( ) Lie 4 1.1 Lie ( )................................ 4 1.2 Killing form...........................................

More information

(2016 2Q H) [ ] R 2 2 P = (a, b), Q = (c, d) Q P QP = ( ) a c b d (a c, b d) P = (a, b) O P ( ) a p = b P = (a, b) p = ( ) a b R 2 {( ) } R 2 x = x, y

(2016 2Q H) [ ] R 2 2 P = (a, b), Q = (c, d) Q P QP = ( ) a c b d (a c, b d) P = (a, b) O P ( ) a p = b P = (a, b) p = ( ) a b R 2 {( ) } R 2 x = x, y (2016 2Q H) [ ] R 2 2 P = (a, b), Q = (c, d) Q P QP = a c b d (a c, b d) P = (a, b) O P a p = b P = (a, b) p = a b R 2 { } R 2 x = x, y R y 2 a p =, c q = b d p + a + c q = b + d q p P q a p = c R c b

More information

数学Ⅱ演習(足助・09夏)

数学Ⅱ演習(足助・09夏) II I 9/4/4 9/4/2 z C z z z z, z 2 z, w C zw z w 3 z, w C z + w z + w 4 t R t C t t t t t z z z 2 z C re z z + z z z, im z 2 2 3 z C e z + z + 2 z2 + 3! z3 + z!, I 4 x R e x cos x + sin x 2 z, w C e z+w

More information

9. 05 L x P(x) P(0) P(x) u(x) u(x) (0 < = x < = L) P(x) E(x) A(x) P(L) f ( d EA du ) = 0 (9.) dx dx u(0) = 0 (9.2) E(L)A(L) du (L) = f (9.3) dx (9.) P

9. 05 L x P(x) P(0) P(x) u(x) u(x) (0 < = x < = L) P(x) E(x) A(x) P(L) f ( d EA du ) = 0 (9.) dx dx u(0) = 0 (9.2) E(L)A(L) du (L) = f (9.3) dx (9.) P 9 (Finite Element Method; FEM) 9. 9. P(0) P(x) u(x) (a) P(L) f P(0) P(x) (b) 9. P(L) 9. 05 L x P(x) P(0) P(x) u(x) u(x) (0 < = x < = L) P(x) E(x) A(x) P(L) f ( d EA du ) = 0 (9.) dx dx u(0) = 0 (9.2) E(L)A(L)

More information

(iii) 0 V, x V, x + 0 = x. 0. (iv) x V, y V, x + y = 0., y x, y = x. (v) 1x = x. (vii) (α + β)x = αx + βx. (viii) (αβ)x = α(βx)., V, C.,,., (1)

(iii) 0 V, x V, x + 0 = x. 0. (iv) x V, y V, x + y = 0., y x, y = x. (v) 1x = x. (vii) (α + β)x = αx + βx. (viii) (αβ)x = α(βx)., V, C.,,., (1) 1. 1.1...,. 1.1.1 V, V x, y, x y x + y x + y V,, V x α, αx αx V,, (i) (viii) : x, y, z V, α, β C, (i) x + y = y + x. (ii) (x + y) + z = x + (y + z). 1 (iii) 0 V, x V, x + 0 = x. 0. (iv) x V, y V, x + y

More information

<4D6963726F736F667420506F776572506F696E74202D208376838C835B83938365815B835683878393312E707074205B8CDD8AB78382815B83685D>

<4D6963726F736F667420506F776572506F696E74202D208376838C835B83938365815B835683878393312E707074205B8CDD8AB78382815B83685D> i i vi ii iii iv v vi vii viii ix 2 3 4 5 6 7 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

More information

SC-85X2取説

SC-85X2取説 I II III IV V VI .................. VII VIII IX X 1-1 1-2 1-3 1-4 ( ) 1-5 1-6 2-1 2-2 3-1 3-2 3-3 8 3-4 3-5 3-6 3-7 ) ) - - 3-8 3-9 4-1 4-2 4-3 4-4 4-5 4-6 5-1 5-2 5-3 5-4 5-5 5-6 5-7 5-8 5-9 5-10 5-11

More information