u V u V u u +( 1)u =(1+( 1))u =0 u = o u =( 1)u x = x 1 x 2. x n,y = y 1 y 2. y n K n = x 1 x 2. x n x + y x α αx x i K Kn α K x, y αx 1
|
|
- まいえ はしかわ
- 5 years ago
- Views:
Transcription
1 5 K K Q R C V V K K- 1) u, v V u + v V (a) u, v V u + v = v + u (b) u, v, w V (u + v)+w = u +(v + w) (c) u V u + o = u o V (d) u V u + u = o u V 2) α K u V u α αv V (a) α, β K u V (αβ)u = α(βv) (b) u V 1u = u (c) α, β K u V (α + β)u = αu + βu (d) α K U, v V α(u + v) =αu + αv V K-V 1) (c) o V o V 1) (c) o = o + o = o + o = o 1) (c) o V K- V 1) (d) u V u V u V 1) (d) u = u + o = u +(u + u )=(u + u)+u =(u + u )+u = o + u = u 46
2 u V u V u u +( 1)u =(1+( 1))u =0 u = o u =( 1)u x = x 1 x 2. x n,y = y 1 y 2. y n K n = x 1 x 2. x n x + y x α αx x i K Kn α K x, y αx 1 x 1 + y 1 x + y = x 2 + y 2, αx = αx 2..x n + y n αx n K K (m, n) M m,n (K) K- K n = M n,1 (K) [a, b] (a<b) C([a, b]) ϕ, ψ C([a, b]) ϕ + ψ C([a, b]) ϕ C([a, b]) λ R λϕ C([a, b]) (ϕ + ψ)(t) =ϕ(t)+ψ(t), (λϕ)(t) =λ ϕ(t) (aleqt b) C([a, b]) R K- V W V W V K-
3 ) W 2) w, w W w + w W 3) α K w W αw W K- V 1) V {o} V K- 2) W V K-V W [ ] 1) 2) W w W 0 K o =0 w W W K- V K-V W W K ) W V w W u =( 1)u V w K- V {v 1,v 2,,v r } { r } v 1,v 2,,v r K = α i v i α i K V K- [ ] W = v 1,,v r K V v, w v 1,,v r K v = r r α i v i, w = β i v i (α i,β i K) r v + w = (α i + β i )v i W α K r αv = (αα i )v i W
4 K- V K- W f : V W f V W K- 1) v, v V f(v + v = f(v)+f(v ) 2) α K v V 1) f(o) =ok- f V W 2) v V f( v) = f(v)k v V 0 v = o f(o) =f(0 v) =0 f(v) =o v =( 1)v f( v) =f(( 1)v) =( 1)f(v) = f(v) (m, n) A M m,n (K) f A : K n K m f A (x) =Ax f A K- K n K- K m K-f A A K K- V K- W K- f : V W Im(f) ={f(v) W v V }, Ker(f) ={v V f(v) =o} Im(f) W K-Ker(f) V K- Im(f), Ker(f) K- f [ ] Im(f) W w, w Im(f) w = f(v), w = f(v ) v, v V
5 50 5. f(v + v )==w + w w + w Im(f) α K f(αv) =αw αw Im(f) Im(f) W K- f(o) =o V Ker(f) Ker(f) V v, v Ker(f) f(v) = f(v ) = o f(v + v ) = o v + v Ker(f) α K f(αv) =αf(v) =o αv Ker(f) Ker(f) V K K- V K- W K- f : V W 1) f Im(f) =W 2) f Ker(f) ={o} [ ] 1) 2) f v Ker(f) f(v) =o f(o) =o f(v) =f(o) f v = o Ker(f) Ker(f) ={o} v, v V f(v) =f(v ) f(v v) =f(v ) f(v) =o v v Ker(f) Ker(f) v v = o v = v f K- V K- W K- f : V W f V W K- f : V W K- V,W K K- f : V W f f 1 : W V f 1 K- V,W K K- f : V W, g : W V f g =id W, g f =id V
6 K- V {v 1,v 2,v r } λ 1 v 1 + λ 2 v λ r v r = o λ i K λ 1 = λ 2 = = λ r =0 {v 1,v 2,,v r } K {v 1,v 2,,v r } K {v 1,v 2,,v r } K K- V {v 1,v 2,,v r } x 1 x 2 r x =.. Kr f(x) = x i v i V (5.1) x r K- f : K r V x 1 x 2 Ker(f) =.. Kr x 1v 1 + x 2 v x r v r = o x r ) {v 1,v 2,,v r } K (5.1) K- f K- V 1) {v 1,v 2,,v r } V K v i o (i =1, 2,,r) 2) v 1 V {v 1 } K v 1 o [ ] 1) v 1 = o λ 1 =1 λ 2 = = λ r =0 λ 1 v 1 + λ 2 v λ r v r = o {v 1,v 2,,v r } K 2) {v 1 } K v 1 o v 1 o 0 λ 1 K λ 1 v 1 = o λ 1 K v 1 = o {v 1 } K R 3
7 R- R 3 1) {v 1,v 2 } R 3 R {v 1,v 2 } 2) {v 1,v 2,v 3 } R 3 R {v 1,v 2,v 3 } K K n r v 1,v 2,,v r v j K n a 1j a 2j v j =. (a ij K). a nj K λ 1,λ 2,,λ r λ 1 v 1 + λ 2 v λ r v r = o λ j a 11 λ 1 + a 12 λ a 1r λ r =0 a 21 λ 1 + a 22 λ a 2r λ r =0.... a n1 λ 1 + a n2 λ a nr λ r =0 {v 1,v 2,,v r } K λ 1 = λ 2 = = λ r = rank(v 1,v 2,,v r )=r {v 1,v 2,,v r } K n K rank(v 1,v 2,,v r )=r (v 1,v 2,,v r ) v 1,v 2,,v r (n, r) K- V r K r V K dim K V r dim K V = dim K V < K- V K-
8 K- V dim K V = ) K- V dim K V K- K n K n dim K K n = n [ ] K n n e 1 = 0,e 2 = 0,,e n = (e 1,e 2,,e n )=I n rank(e 1,e 2 m,e n )= n {e 1.e 2..e n } K dim K K n n r >n K n r v 1,v 2,,v r rank(v 1,v 2,,v r ) n<r {v 1,v 2,,v r } K dim K K n = n K- V,W K- f : V W 1) f dim K V dim K W 2) f dim K V dim K W f K- dim K V = dim K W [ ] 1){w 1,w 2,,w r } W K f f(v i )=w i v i V {v 1,v 2,,v r } V K α 1 v 1 + α 2 v α r v r = o (α i K) K- f f(o) =o α 1 w 1 +α 2 w 2 + +α r w r = o α 1 = α 2 = cdots = α r =0 dim K V dim K W
9 ) {v 1,v 2,,v r } V K w i = f(v i ) {w 1,w 2,,w r } W K α 1 w 1 + α 2 w α r w r = o (α i K) f K- f(α 1 v 1 + α 2 v α r v r )=o = f(o) f α 1 v 1 + α 2 v α r v r = o α 1 = α 2 = cdots = α r =0 dim K V dim K W K- V {v 1,v 2,,v n } {v 1,v 2,,v n } V K 1) {v 1,v 2,,v n } K 2) V = v 1,v 2,,v n K K- V o V K [ ] dim K V = n n 1 K {v 1,v 2,,v n } V v V {v 1,v 2,,v n,v} K α 1 v 1 + α 2 v α n v n + α n+1 v = o α i K i α i 0 α n+1 =0 {v 1,v 2,,v n } K α n+1 0 v =( α 1 /α n+1 )v 1 + +( α n /α n+1 )v n v v 1,v 2,,v n K V = v 1,v 2,,v n K {v 1,v 2,,v n } V K K- V {v 1,v 2,,v n } x 1 x 2 n f : K n. x i v i V x n
10 f K- K n V K- Im(f) = v 1,v 2,,v n K, x 1 x 2 Ker(f) =. x 1v 1 + x 2 v x n v n = o x n ) f V = v 1,v 2,,v n K 2) f {v 1,v 2,,v n } K 3) f {v 1,v 2,,v n } V K {v 1,v 2,,v n } V K K= K n K- V K dim K V = n K- V V K K- V {v 1,v 2,,v n,v n+1 } (n 1) 1) v 1,v 2,,v n,v n+1 K = v 1,v 2,,v n K, 2) v n+1 = α 1 v 1 + α 2 v α n v n α i K [ ] 1) 2) v n+1 v 1,v 2,,v n+1 K = v 1,v 2,,v n K 2) 1) v 1,v 2,,v n K v 1,v 2,,v n,v n+1 K v v 1,,v n,v n+1 K n+1 v = λ i v i (λ i K) = n (λ ; + λ n+1 α i )v i v v 1,,v n K
11 K- V {v 1,v 2,,v n } 1) V K {v 1,v 2,,v r } V K V K {v 1,,v r,v r+1,,v n } 2) V = v 1,v 2,,v m K (v i V ) V K {v 1,v 2,,v m } V K [ ] 1){v 1,v 2,,v r } V K {v 1,,v r,,v n } V K {v 1,,v r,,v n } V K V = v 1,,v r,,v n K v V {v 1,,v r,,v n,v} K α 1 v α r v r + + α n v n + α n+1 v = o α i K i α i 0 α n+1 =0 {v 1,,v r,,v n } K α n+1 0 v =( α 1 /α n+1 )v 1 + +( α r /α n+1 )v r + +( α n /α n+1 )v n v v 1,,v r,,v n K 2) {v 1,v 2,,v m } K V S V K S = {v 1,v 2,,v n } (n m) K K α 1 v 1 + α 2 v α n v = o α i K i α i 0 α n 0 v n =( α 1 /α n )v 1 +( α 2 /α n )v n + +( α n 1 /α n )v n V = v 1,v 2,,v n 1 K S
12 n K- V n {v 1,v 2,,v n } 1) {v 1,v 2,,v n } V K 2) {v 1,v 2,,v n } K 3) V = v 1,v 2,,v n K [ ] 1) 2), 3) 2) 1) {v 1,v 2,,v n } K ) {v 1,v 2,,v n } V V K n = dim K V {v 1,v 2,,v n } V K 3) 1) ) {v 1,v 2,,v n } V K n = dim K V {v 1,v 2,,v n } V K K- K- V,W W V dim K V = dim K W< V = W [ ] dim K W = dim K V = n {w 1,w 2,,w n } W K {w 1,w 2,,w n } V K V = w 1,w 2,,w n K = V K- K n K- K n n {v 1,v 2,,v n } K n K n (v 1,v 2,,v n ) det(v 1,v 2,,v n ) 0 [ ] {v 1,v 2,,v n } K- K n K {v 1,v 2,,v n } K {v 1,v 2,,v n } K rank(v 1,v 2,,v n )=n (v 1,v 2,,v n ) n rank(v 1,v 2,,v n )=n (v 1,v 2,,v n )
13 K- V K- W K- f : V W V K Im(f) K dim K Im(f) = dim K V dim K Ker(f) [ ] Im(f) ={o} dim K Im(f) =0 Ker(f) =V Ker(f) ={o} f V Im(f) K dim K Im(f) = dim K V Im(f) {o} Ker(f) {o} V K {v 1,v 2,,v n } V = v 1,v 2,,v n K Im(f) = f(v 1 ),f(v 2 ),,f(v n ) K ) Im(f) K Im(f) K {w 1,,w r } r = dim K Im(f) w i Im(f) w i = f(v i ) v i V Ker(f) K {u 1,,u s } s = dim K Ker(f) {v 1,,v r,u 1, cdots, u s } V K α i,β j K α 1 v α r v r + β 1 u β s u s = o (5.2) f s β ju j Ker(f) f(v i )=w i r α iw i = o {w 1,,w r } α 1 = = α r =0 (5.2) s β ju j = o {u 1,,u s } β 1 = = β s =0 {v 1,,v r,u 1,,u s } K v V f(v) Im(f) f(v) = r α iw i α i K u = v r α iv i V u Ker(f) u = s β ju j β j K v = r α i v i + s β j u j v 1,,v r,u 1,,u s K {v 1,,v r,u 1,,u s } V K dim K V = r + s = dim K Im(f) + Ker(f)
14 (m, n) A M m,n (K) K- f A : K n K m f A (x) =Ax Ker(f A ) = Ker(A) (m, n) A M m,n (K) K- f A dim K Ker(f A )=n rank(a), dim K Im(f A ) = rank(a) [ ] rank(a) =r dim K Ker(f A )=n r K- f A : K n K m dim K Im(f A )=r Ker(f A ) = Ker(A) = q r+1,q r+2,,q n K Q =(q 1,q,,q n ) n {q 1,q 2,,q n } K n K {q r+1,q r+2,,q n } K {q r+1,q r+2,,q n } Ker(f A ) K dim K Ker(f A )=n r (m, n)- A M m,n (K) Ker(F A ) Im(f A ) K rank(a) =r [ ] I r 0 PAQ = 0 0 m P n Q Q =(q 1,q 2,,q n ), (q i K n ) {q r+1,,q n } Ker(f A ) K Im(f A ) [ ] f A (x) =Ax = P 1 I r 0 Q 1 x (x K n ) 0 0 y = Q 1 x Q 1 x K n y K n P 1 =(v 1,v 2,,v m ) (v j K m )
15 60 5. y y 1,,y n [ ] f A (x) =P 1 I r 0 y = y 1 v y r vr 0 0 Im(f A )= v 1,,v r K P 1 {v 1,,v r } K {v 1,,v r } Im(f A ) K (m, n)- A M m,n (K) [ ] I r 0 PAQ = (r = rank(a)) 0 0 P, Q P 1 =(v 1,,v m ), Q =(q 1,,q n ) (v j K m,q i K n ) {v 1,,v r } Im(f A ) K {q r+1,,q n } Ker(f A ) K P, Q K- K n {v 1,v 2,,v m } dim K v 1,v 2,,v m K = rank(v 1,v 2,,v m ) (v 1,v 2,,v m ) {v 1,v 2,,v n } (n, m) [ ] A =(v 1,v 2,,v n ) M n,m (K) Im(f A )= v 1,v 2,,v m K K- V,W K- f : V W V K [v 1,v 2,,v n } K- x 1 x 2 n ϕ : K n V (. x j v j ) x n
16 W K {w 1,w 2,,w m } K- y 1 y 2 m ψ : K m W (. y i w i ) K- f : V W K- V,W K n, K m j =1,,n f(v j ) W {w 1,,w m } K f(v j )= y m m a ij w j (a ij K) v = n x jv j V f(v) = n x j f(v j )= n x j m a ij w i (5.3) m n = a ij x j w i (5.4) a 11 a 12 a 1n a 21 a 22 a 2n A = M mn(k) (5.5) a m1 a m2 a mn K- f A (x) =Ax (x K n ) f ϕ(x) =ψ f A (x) (x K n ) K n ϕ V f A K m ψ f W. K- f : V W f A : K n K m (5.5) {v 1,,v n }, {w 1,,w m } f f V,W K- f : V W K- U K-
17 62 5. g : U V f g U W K- U K- {u 1,,u l } f,g f g k =1,,l g(u k )= n m b jk v j (f g)(u k )= c ik w i (b jk,c ik K) (5.6) g, f g B = (b jk ) j,k M nl (K), C = (c ik ) i,k M ml (K) (f g)(u k )=f(g(u k )) = = n b jk m n b jk f(v j ) m n a ij w i = a ij b jk w i (5.6) c ik = n a ijb jk K- g : U V, f : V W A, B f g : U W AB
ver Web
ver201723 Web 1 4 11 4 12 5 13 7 2 9 21 9 22 10 23 10 24 11 3 13 31 n 13 32 15 33 21 34 25 35 (1) 27 4 30 41 30 42 32 43 36 44 (2) 38 45 45 46 45 5 46 51 46 52 48 53 49 54 51 55 54 56 58 57 (3) 61 2 3
More information2 (2016 3Q N) c = o (11) Ax = b A x = c A n I n n n 2n (A I n ) (I n X) A A X A n A A A (1) (2) c 0 c (3) c A A i j n 1 ( 1) i+j A (i, j) A (i, j) ã i
[ ] (2016 3Q N) a 11 a 1n m n A A = a m1 a mn A a 1 A A = a n (1) A (a i a j, i j ) (2) A (a i ca i, c 0, i ) (3) A (a i a i + ca j, j i, i ) A 1 A 11 0 A 12 0 0 A 1k 0 1 A 22 0 0 A 2k 0 1 0 A 3k 1 A rk
More information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 informationx V x x V x, x V x = x + = x +(x+x )=(x +x)+x = +x = x x = x x = x =x =(+)x =x +x = x +x x = x ( )x = x =x =(+( ))x =x +( )x = x +( )x ( )x = x x x R
V (I) () (4) (II) () (4) V K vector space V vector K scalor K C K R (I) x, y V x + y V () (x + y)+z = x +(y + z) (2) x + y = y + x (3) V x V x + = x (4) x V x + x = x V x x (II) x V, α K αx V () (α + β)x
More information> > <., vs. > x 2 x y = ax 2 + bx + c y = 0 2 ax 2 + bx + c = 0 y = 0 x ( x ) y = ax 2 + bx + c D = b 2 4ac (1) D > 0 x (2) D = 0 x (3
13 2 13.0 2 ( ) ( ) 2 13.1 ( ) ax 2 + bx + c > 0 ( a, b, c ) ( ) 275 > > 2 2 13.3 x 2 x y = ax 2 + bx + c y = 0 2 ax 2 + bx + c = 0 y = 0 x ( x ) y = ax 2 + bx + c D = b 2 4ac (1) D >
More information1. 1 A : l l : (1) l m (m 3) (2) m (3) n (n 3) (4) A α, β γ α β + γ = 2 m l lm n nα nα = lm. α = lm n. m lm 2β 2β = lm β = lm 2. γ l 2. 3
1. 1 A : l l : (1) l m (m 3) (2) m (3) n (n 3) (4) A 2 1 2 1 2 3 α, β γ α β + γ = 2 m l lm n nα nα = lm. α = lm n. m lm 2β 2β = lm β = lm 2. γ l 2. 3 4 P, Q R n = {(x 1, x 2,, x n ) ; x 1, x 2,, x n R}
More informationlinearal1.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
More information2 7 V 7 {fx fx 3 } 8 P 3 {fx fx 3 } 9 V 9 {fx fx f x 2fx } V {fx fx f x 2fx + } V {{a n } {a n } a n+2 a n+ + a n n } 2 V 2 {{a n } {a n } a n+2 a n+
R 3 R n C n V??,?? k, l K x, y, z K n, i x + y + z x + y + z iv x V, x + x o x V v kx + y kx + ky vi k + lx kx + lx vii klx klx viii x x ii x + y y + x, V iii o K n, x K n, x + o x iv x K n, x + x o x
More informationnewmain.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 informationkoji07-01.dvi
2007 I II III 1, 2, 3, 4, 5, 6, 7 5 10 19 (!) 1938 70 21? 1 1 2 1 2 2 1! 4, 5 1? 50 1 2 1 1 2 2 1?? 2 1 1, 2 1, 2 1, 2, 3,... 3 1, 2 1, 3? 2 1 3 1 2 1 1, 2 2, 3? 2 1 3 2 3 2 k,l m, n k,l m, n kn > ml...?
More informationuntitled
0. =. =. (999). 3(983). (980). (985). (966). 3. := :=. A A. A A. := := 4 5 A B A B A B. A = B A B A B B A. A B A B, A B, B. AP { A, P } = { : A, P } = { A P }. A = {0, }, A, {0, }, {0}, {}, A {0}, {}.
More information漸化式のすべてのパターンを解説しましたー高校数学の達人・河見賢司のサイト
https://www.hmg-gen.com/tuusin.html https://www.hmg-gen.com/tuusin1.html 1 2 OK 3 4 {a n } (1) a 1 = 1, a n+1 a n = 2 (2) a 1 = 3, a n+1 a n = 2n a n a n+1 a n = ( ) a n+1 a n = ( ) a n+1 a n {a n } 1,
More informationx, 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 informationX 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 information1 Ricci V, V i, W f : V W f f(v ) = Imf W ( ) f : V 1 V k W 1
1 Ricci V, V i, W f : V W f f(v = Imf W ( f : V 1 V k W 1 {f(v 1,, v k v i V i } W < Imf > < > f W V, V i, W f : U V L(U; V f : V 1 V r W L(V 1,, V r ; W L(V 1,, V r ; W (f + g(v 1,, v r = f(v 1,, v r
More information20 9 19 1 3 11 1 3 111 3 112 1 4 12 6 121 6 122 7 13 7 131 8 132 10 133 10 134 12 14 13 141 13 142 13 143 15 144 16 145 17 15 19 151 1 19 152 20 2 21 21 21 211 21 212 1 23 213 1 23 214 25 215 31 22 33
More information1 911 9001030 9:00 A B C D E F G H I J K L M 1A0900 1B0900 1C0900 1D0900 1E0900 1F0900 1G0900 1H0900 1I0900 1J0900 1K0900 1L0900 1M0900 9:15 1A0915 1B0915 1C0915 1D0915 1E0915 1F0915 1G0915 1H0915 1I0915
More informationII 2 3.,, A(B + C) = AB + AC, (A + B)C = AC + BC. 4. m m A, m m B,, m m B, AB = BA, A,, I. 5. m m A, m n B, AB = B, A I E, 4 4 I, J, K
II. () 7 F 7 = { 0,, 2, 3, 4, 5, 6 }., F 7 a, b F 7, a b, F 7,. (a) a, b,,. (b) 7., 4 5 = 20 = 2 7 + 6, 4 5 = 6 F 7., F 7,., 0 a F 7, ab = F 7 b F 7. (2) 7, 6 F 6 = { 0,, 2, 3, 4, 5 },,., F 6., 0 0 a F
More informationDecember 28, 2018
e-mail : kigami@i.kyoto-u.ac.jp December 28, 28 Contents 2............................. 3.2......................... 7.3..................... 9.4................ 4.5............. 2.6.... 22 2 36 2..........................
More informationDynkin 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 informationx () 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 information6 19,,,
6 19,,, 15 6 19 4-2 à A si A s n + a n s n 1 + + a 2 s + a 1 à 0 1 0 0 1 0 0 0 1 a 1 a 2 a n 1 a n à ( 1, λ i, λ i 2,, λ i n 1 ) T ( λ i, λ 2 i,, λ n 1 i, a 1 a 2 λ i a n λ ) n 1 T i ( ) λ i 1, λ i,, λ
More informationJanuary 27, 2015
e-mail : kigami@i.kyoto-u.ac.jp January 27, 205 Contents 2........................ 2.2....................... 3.3....................... 6.4......................... 2 6 2........................... 6
More informationO x y z O ( O ) O (O ) 3 x y z O O x v t = t = 0 ( 1 ) O t = 0 c t r = ct P (x, y, z) r 2 = x 2 + y 2 + z 2 (t, x, y, z) (ct) 2 x 2 y 2 z 2 = 0
9 O y O ( O ) O (O ) 3 y O O v t = t = 0 ( ) O t = 0 t r = t P (, y, ) r = + y + (t,, y, ) (t) y = 0 () ( )O O t (t ) y = 0 () (t) y = (t ) y = 0 (3) O O v O O v O O O y y O O v P(, y,, t) t (, y,, t )
More informationI ( ) 1 de Broglie 1 (de Broglie) p λ k h Planck ( Js) p = h λ = k (1) h 2π : Dirac k B Boltzmann ( J/K) T U = 3 2 k BT
I (008 4 0 de Broglie (de Broglie p λ k h Planck ( 6.63 0 34 Js p = h λ = k ( h π : Dirac k B Boltzmann (.38 0 3 J/K T U = 3 k BT ( = λ m k B T h m = 0.067m 0 m 0 = 9. 0 3 kg GaAs( a T = 300 K 3 fg 07345
More information2016
2016 1 G x x G d G (x) 1 ( ) G d G (x) = 2 E(G). x V (G) 2 ( ) 1.1 1: n m on-off ( 1 ) off on 1: on-off ( on ) G v v N(v) on-off G S V (G) N(v) S { 3 G v S v S G G = 1 OK ( ) G 2 3.1 u S u u u 1 G u S
More information2000年度『数学展望 I』講義録
2000 I I IV I II 2000 I I IV I-IV. i ii 3.10 (http://www.math.nagoya-u.ac.jp/ kanai/) 2000 A....1 B....4 C....10 D....13 E....17 Brouwer A....21 B....26 C....33 D....39 E. Sperner...45 F....48 A....53
More information13 0 1 1 4 11 4 12 5 13 6 2 10 21 10 22 14 3 20 31 20 32 25 33 28 4 31 41 32 42 34 43 38 5 41 51 41 52 43 53 54 6 57 61 57 62 60 70 0 Gauss a, b, c x, y f(x, y) = ax 2 + bxy + cy 2 = x y a b/2 b/2 c x
More information2 2 1?? 2 1 1, 2 1, 2 1, 2, 3,... 1, 2 1, 3? , 2 2, 3? k, l m, n k, l m, n kn > ml...? 2 m, n n m
2009 IA I 22, 23, 24, 25, 26, 27 4 21 1 1 2 1! 4, 5 1? 50 1 2 1 1 2 1 4 2 2 2 1?? 2 1 1, 2 1, 2 1, 2, 3,... 1, 2 1, 3? 2 1 3 1 2 1 1, 2 2, 3? 2 1 3 2 3 2 k, l m, n k, l m, n kn > ml...? 2 m, n n m 3 2
More information/ 2 n n n n x 1,..., x n 1 n 2 n R n n ndimensional Euclidean space R n vector point R n set space R n R n x = x 1 x n y = y 1 y n distance dx,
1 1.1 R n 1.1.1 3 xyz xyz 3 x, y, z R 3 := x y : x, y, z R z 1 3. n n x 1,..., x n x 1. x n x 1 x n 1 / 2 n n n n x 1,..., x n 1 n 2 n R n n ndimensional Euclidean space R n vector point 1.1.2 R n set
More informationi I II I II II IC IIC I II ii 5 8 5 3 7 8 iii I 3........................... 5......................... 7........................... 4........................ 8.3......................... 33.4...................
More informationv 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 informationy π π O π x 9 s94.5 y dy dx. y = x + 3 y = x logx + 9 s9.6 z z x, z y. z = xy + y 3 z = sinx y 9 s x dx π x cos xdx 9 s93.8 a, fx = e x ax,. a =
[ ] 9 IC. dx = 3x 4y dt dy dt = x y u xt = expλt u yt λ u u t = u u u + u = xt yt 6 3. u = x, y, z = x + y + z u u 9 s9 grad u ux, y, z = c c : grad u = u x i + u y j + u k i, j, k z x, y, z grad u v =
More information16 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[ ] 0.1 lim x 0 e 3x 1 x IC ( 11) ( s114901) 0.2 (1) y = e 2x (x 2 + 1) (2) y = x/(x 2 + 1) 0.3 dx (1) 1 4x 2 (2) e x sin 2xdx (3) sin 2 xdx ( 11) ( s
[ ]. lim e 3 IC ) s49). y = e + ) ) y = / + ).3 d 4 ) e sin d 3) sin d ) s49) s493).4 z = y z z y s494).5 + y = 4 =.6 s495) dy = 3e ) d dy d = y s496).7 lim ) lim e s49).8 y = e sin ) y = sin e 3) y =
More informationn ( (
1 2 27 6 1 1 m-mat@mathscihiroshima-uacjp 2 http://wwwmathscihiroshima-uacjp/~m-mat/teach/teachhtml 2 1 3 11 3 111 3 112 4 113 n 4 114 5 115 5 12 7 121 7 122 9 123 11 124 11 125 12 126 2 2 13 127 15 128
More information2016 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 informationn 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
More informationall.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.5 z = a + b + c n.6 = a sin t y = b cos t dy d a e e b e + e c e e e + e 3 s36 3 a + y = a, b > b 3 s363.7 y = + 3 y = + 3 s364.8 cos a 3 s365.9 y =,
[ ] IC. r, θ r, θ π, y y = 3 3 = r cos θ r sin θ D D = {, y ; y }, y D r, θ ep y yddy D D 9 s96. d y dt + 3dy + y = cos t dt t = y = e π + e π +. t = π y =.9 s6.3 d y d + dy d + y = y =, dy d = 3 a, b
More informationA 2 3. m S m = {x R m+1 x = 1} U + k = {x S m x k > 0}, U k = {x S m x k < 0}, ϕ ± k (x) = (x 0,..., ˆx k,... x m ) 1. {(U ± k, ϕ± k ) 0 k m} S m 1.2.
A A 1 A 5 A 6 1 2 3 4 5 6 7 1 1.1 1.1 (). Hausdorff M R m M M {U α } U α R m E α ϕ α : U α E α U α U β = ϕ α (ϕ β ϕβ (U α U β )) 1 : ϕ β (U α U β ) ϕ α (U α U β ) C M a m dim M a U α ϕ α {x i, 1 i m} {U,
More information) ] [ h m x + y + + V x) φ = Eφ 1) z E = i h t 13) x << 1) N n n= = N N + 1) 14) N n n= = N N + 1)N + 1) 6 15) N n 3 n= = 1 4 N N + 1) 16) N n 4
1. k λ ν ω T v p v g k = π λ ω = πν = π T v p = λν = ω k v g = dω dk 1) ) 3) 4). p = hk = h λ 5) E = hν = hω 6) h = h π 7) h =6.6618 1 34 J sec) hc=197.3 MeV fm = 197.3 kev pm= 197.3 ev nm = 1.97 1 3 ev
More informationAC Modeling and Control of AC Motors Seiji Kondo, Member 1. q q (1) PM (a) N d q Dept. of E&E, Nagaoka Unive
AC Moeling an Control of AC Motors Seiji Kono, Member 1. (1) PM 33 54 64. 1 11 1(a) N 94 188 163 1 Dept. of E&E, Nagaoka University of Technology 163 1, Kamitomioka-cho, Nagaoka, Niigata 94 188 (a) 巻数
More information2S III IV K A4 12:00-13:30 Cafe David 1 2 TA 1 appointment Cafe David K2-2S04-00 : C
2S III IV K200 : April 16, 2004 Version : 1.1 TA M2 TA 1 10 2 n 1 ɛ-δ 5 15 20 20 45 K2-2S04-00 : C 2S III IV K200 60 60 74 75 89 90 1 email 3 4 30 A4 12:00-13:30 Cafe David 1 2 TA 1 email appointment Cafe
More information『共形場理論』
T (z) SL(2, C) T (z) SU(2) S 1 /Z 2 SU(2) (ŜU(2) k ŜU(2) 1)/ŜU(2) k+1 ŜU(2)/Û(1) G H N =1 N =1 N =1 N =1 N =2 N =2 N =2 N =2 ĉ>1 N =2 N =2 N =4 N =4 1 2 2 z=x 1 +ix 2 z f(z) f(z) 1 1 4 4 N =4 1 = = 1.3
More information1. 2 P 2 (x, y) 2 x y (0, 0) R 2 = {(x, y) x, y R} x, y R P = (x, y) O = (0, 0) OP ( ) OP x x, y y ( ) x v = y ( ) x 2 1 v = P = (x, y) y ( x y ) 2 (x
. P (, (0, 0 R {(,, R}, R P (, O (0, 0 OP OP, v v P (, ( (, (, { R, R} v (, (, (,, z 3 w z R 3,, z R z n R n.,..., n R n n w, t w ( z z Ke Words:. A P 3 0 B P 0 a. A P b B P 3. A π/90 B a + b c π/ 3. +
More informationB ver B
B ver. 2017.02.24 B Contents 1 11 1.1....................... 11 1.1.1............. 11 1.1.2.......................... 12 1.2............................. 14 1.2.1................ 14 1.2.2.......................
More information.2 ρ dv dt = ρk grad p + 3 η grad (divv) + η 2 v.3 divh = 0, rote + c H t = 0 dive = ρ, H = 0, E = ρ, roth c E t = c ρv E + H c t = 0 H c E t = c ρv T
NHK 204 2 0 203 2 24 ( ) 7 00 7 50 203 2 25 ( ) 7 00 7 50 203 2 26 ( ) 7 00 7 50 203 2 27 ( ) 7 00 7 50 I. ( ν R n 2 ) m 2 n m, R = e 2 8πε 0 hca B =.09737 0 7 m ( ν = ) λ a B = 4πε 0ħ 2 m e e 2 = 5.2977
More informatione a b a b b a a a 1 a a 1 = a 1 a = e G G G : x ( x =, 8, 1 ) x 1,, 60 θ, ϕ ψ θ G G H H G x. n n 1 n 1 n σ = (σ 1, σ,..., σ N ) i σ i i n S n n = 1,,
01 10 18 ( ) 1 6 6 1 8 8 1 6 1 0 0 0 0 1 Table 1: 10 0 8 180 1 1 1. ( : 60 60 ) : 1. 1 e a b a b b a a a 1 a a 1 = a 1 a = e G G G : x ( x =, 8, 1 ) x 1,, 60 θ, ϕ ψ θ G G H H G x. n n 1 n 1 n σ = (σ 1,
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
.. 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 informationnumb.dvi
11 Poisson kanenko@mbkniftycom alexeikanenko@docomonejp http://wwwkanenkocom/ , u = f, ( u = u+f u t, u = f t ) 1 D R 2 L 2 (D) := {f(x,y) f(x,y) 2 dxdy < )} D D f,g L 2 (D) (f,g) := f(x,y)g(x,y)dxdy (L
More information2 A id A : A A A A id A def = {(a, a) A A a A} 1 { } 1 1 id 1 = α: A B β : B C α β αβ : A C αβ def = {(a, c) A C b B.((a, b) α (b, c) β)} 2.3 α
20 6 18 1 2 2.1 A B α A B α: A B A B Rel(A, B) A B (A B) A B 0 AB A B AB α, β : A B α β α β def (a, b) A B.((a, b) α (a, b) β) 0 AB AB Rel(A, B) 1 2 A id A : A A A A id A def = {(a, a) A A a A} 1 { } 1
More informationA
A 2563 15 4 21 1 3 1.1................................................ 3 1.2............................................. 3 2 3 2.1......................................... 3 2.2............................................
More informationN cos s s cos ψ e e e e 3 3 e e 3 e 3 e
3 3 5 5 5 3 3 7 5 33 5 33 9 5 8 > e > f U f U u u > u ue u e u ue u ue u e u e u u e u u e u N cos s s cos ψ e e e e 3 3 e e 3 e 3 e 3 > A A > A E A f A A f A [ ] f A A e > > A e[ ] > f A E A < < f ; >
More information20 4 20 i 1 1 1.1............................ 1 1.2............................ 4 2 11 2.1................... 11 2.2......................... 11 2.3....................... 19 3 25 3.1.............................
More information21 2 26 i 1 1 1.1............................ 1 1.2............................ 3 2 9 2.1................... 9 2.2.......... 9 2.3................... 11 2.4....................... 12 3 15 3.1..........
More informationAI n Z f n : Z Z f n (k) = nk ( k Z) f n n 1.9 R R f : R R f 1 1 {a R f(a) = 0 R = {0 R 1.10 R R f : R R f 1 : R R 1.11 Z Z id Z 1.12 Q Q id
1 1.1 1.1 R R (1) R = 1 2 Z = 2 n Z (2) R 1.2 R C Z R 1.3 Z 2 = {(a, b) a Z, b Z Z 2 a, b, c, d Z (a, b) + (c, d) = (a + c, b + d), (a, b)(c, d) = (ac, bd) (1) Z 2 (2) Z 2? (3) Z 2 1.4 C Q[ 1] = {a + bi
More informationPart () () 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 informationτ τ
1 1 1.1 1.1.1 τ τ 2 1 1.1.2 1.1 1.1 µ ν M φ ν end ξ µ ν end ψ ψ = µ + ν end φ ν = 1 2 (µφ + ν end) ξ = ν (µ + ν end ) + 1 1.1 3 6.18 a b 1.2 a b 1.1.3 1.1.3.1 f R{A f } A f 1 B R{AB f 1 } COOH A OH B 1.3
More informationS 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() x + y + y + x dy dx = 0 () dy + xy = x dx y + x y ( 5) ( s55906) 0.7. (). 5 (). ( 6) ( s6590) 0.8 m n. 0.9 n n A. ( 6) ( s6590) f A (λ) = det(a λi)
0. A A = 4 IC () det A () A () x + y + z = x y z X Y Z = A x y z ( 5) ( s5590) 0. a + b + c b c () a a + b + c c a b a + b + c 0 a b c () a 0 c b b c 0 a c b a 0 0. A A = 7 5 4 5 0 ( 5) ( s5590) () A ()
More informationLINEAR ALGEBRA I Hiroshi SUZUKI Department of Mathematics International Christian University
LINEAR ALGEBRA I Hiroshi SUZUKI Department of Mathematics International Christian University 2002 2 2 2 2 22 2 3 3 3 3 3 4 4 5 5 6 6 7 7 8 8 9 Cramer 9 0 0 E-mail:hsuzuki@icuacjp 0 3x + y + 2z 4 x + y
More informationII R n k +1 v 0,, v k k v 1 v 0,, v k v v 0,, v k R n 1 a 0,, a k a 0 v 0 + a k v k v 0 v k k k v 0,, v k σ k σ dimσ = k 1.3. k
II 231017 1 1.1. R n k +1 v 0,, v k k v 1 v 0,, v k v 0 1.2. v 0,, v k R n 1 a 0,, a k a 0 v 0 + a k v k v 0 v k k k v 0,, v k σ kσ dimσ = k 1.3. k σ {v 0,...,v k } {v i0,...,v il } l σ τ < τ τ σ 1.4.
More information1.500 m X Y m m m m m m m m m m m m N/ N/ ( ) qa N/ N/ 2 2
1.500 m X Y 0.200 m 0.200 m 0.200 m 0.200 m 0.200 m 0.000 m 1.200 m m 0.150 m 0.150 m m m 2 24.5 N/ 3 18.0 N/ 3 30.0 0.60 ( ) qa 50.79 N/ 2 0.0 N/ 2 20.000 20.000 15.000 15.000 X(m) Y(m) (kn/m 2 ) 10.000
More information研修コーナー
l l l l l l l l l l l α α β l µ l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l
More informationD 24 D D D
5 Paper I.R. 2001 5 Paper HP Paper 5 3 5.1................................................... 3 5.2.................................................... 4 5.3.......................................... 6
More information直交座標系の回転
b T.Koama x l x, Lx i ij j j xi i i i, x L T L L, L ± x L T xax axx, ( a a ) i, j ij i j ij ji λ λ + λ + + λ i i i x L T T T x ( L) L T xax T ( T L T ) A( L) T ( LAL T ) T ( L AL) λ ii L AL Λ λi i axx
More information第86回日本感染症学会総会学術集会後抄録(I)
κ κ κ κ κ κ μ μ β β β γ α α β β γ α β α α α γ α β β γ μ β β μ μ α ββ β β β β β β β β β β β β β β β β β β γ β μ μ μ μμ μ μ μ μ β β μ μ μ μ μ μ μ μ μ μ μ μ μ μ β
More information20 6 4 1 4 1.1 1.................................... 4 1.1.1.................................... 4 1.1.2 1................................ 5 1.2................................... 7 1.2.1....................................
More information(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
More information行列代数2010A
(,) A (,) B C = AB a 11 a 1 a 1 b 11 b 1 b 1 c 11 c 1 c a A = 1 a a, B = b 1 b b, C = AB = c 1 c c a 1 a a b 1 b b c 1 c c i j ij a i1 a i a i b 1j b j b j c ij = a ik b kj b 1j b j AB = a i1 a i a ik
More informationŁ½’¬24flNfix+3mm-‡½‡¹724
571 0.0 31,583 2.0 139,335 8.9 310,727 19.7 1,576,352 100.0 820 0.1 160,247 10.2 38,5012.4 5,7830.4 9,5020.6 41,7592.7 77,8174.9 46,425 2.9 381,410 24.2 1,576,352 100.0 219,332 13.9 132,444 8.4 173,450
More information/ n (M1) M (M2) n Λ A = {ϕ λ : U λ R n } λ Λ M (atlas) A (a) {U λ } λ Λ M (open covering) U λ M λ Λ U λ = M (b) λ Λ ϕ λ : U λ ϕ λ (U λ ) R n ϕ
4 4.1 1 2 1 4 2 1 / 2 4.1.1 n (M1) M (M2) n Λ A = {ϕ λ : U λ R n } λ Λ M (atlas) A (a) {U λ } λ Λ M (open covering) U λ M λ Λ U λ = M (b) λ Λ ϕ λ : U λ ϕ λ (U λ ) R n ϕ λ U λ (local chart, local coordinate)
More information_0212_68<5A66><4EBA><79D1>_<6821><4E86><FF08><30C8><30F3><30DC><306A><3057><FF09>.pdf
More information
1 Euclid Euclid Euclid
II 2000 1 Euclid 1 1.1..................................... 1 1.2..................................... 8 1.3 Euclid............. 19 1.4 3 Euclid............................ 22 2 28 2.1 Lie Lie..................................
More information1 8, : 8.1 1, 2 z = ax + by + c ax by + z c = a b +1 x y z c = 0, (0, 0, c), n = ( a, b, 1). f = n i=1 a ii x 2 i + i<j 2a ij x i x j = ( x, A x), f =
1 8, : 8.1 1, z = ax + by + c ax by + z c = a b +1 x y z c = 0, (0, 0, c), n = ( a, b, 1). f = a ii x i + i
More informationVI VI.21 W 1,..., W r V W 1,..., W r W W r = {v v r v i W i (1 i r)} V = W W r V W 1,..., W r V W 1,..., W r V = W 1 W
3 30 5 VI VI. W,..., W r V W,..., W r W + + W r = {v + + v r v W ( r)} V = W + + W r V W,..., W r V W,..., W r V = W W r () V = W W r () W (W + + W + W + + W r ) = {0} () dm V = dm W + + dm W r VI. f n
More informationzz + 3i(z z) + 5 = 0 + i z + i = z 2i z z z y zz + 3i (z z) + 5 = 0 (z 3i) (z + 3i) = 9 5 = 4 z 3i = 2 (3i) zz i (z z) + 1 = a 2 {
04 zz + iz z) + 5 = 0 + i z + i = z i z z z 970 0 y zz + i z z) + 5 = 0 z i) z + i) = 9 5 = 4 z i = i) zz i z z) + = a {zz + i z z) + 4} a ) zz + a + ) z z) + 4a = 0 4a a = 5 a = x i) i) : c Darumafactory
More information15 mod 12 = 3, 3 mod 12 = 3, 9 mod 12 = N N 0 x, y x y N x y (mod N) x y N mod N mod N N, x, y N > 0 (1) x x (mod N) (2) x y (mod N) y x
A( ) 1 1.1 12 3 15 3 9 3 12 x (x ) x 12 0 12 1.1.1 x x = 12q + r, 0 r < 12 q r 1 N > 0 x = Nq + r, 0 r < N q r 1 q x/n r r x mod N 1 15 mod 12 = 3, 3 mod 12 = 3, 9 mod 12 = 3 1.1.2 N N 0 x, y x y N x y
More informationII 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
More informationSO(3) 49 u = Ru (6.9), i u iv i = i u iv i (C ) π π : G Hom(V, V ) : g D(g). π : R 3 V : i 1. : u u = u 1 u 2 u 3 (6.10) 6.2 i R α (1) = 0 cos α
SO(3) 48 6 SO(3) t 6.1 u, v u = u 1 1 + u 2 2 + u 3 3 = u 1 e 1 + u 2 e 2 + u 3 e 3, v = v 1 1 + v 2 2 + v 3 3 = v 1 e 1 + v 2 e 2 + v 3 e 3 (6.1) i (e i ) e i e j = i j = δ ij (6.2) ( u, v ) = u v = ij
More informationSO(3) 7 = = 1 ( r ) + 1 r r r r ( l ) (5.17) l = 1 ( sin θ ) + sin θ θ θ ϕ (5.18) χ(r)ψ(θ, ϕ) l ψ = αψ (5.19) l 1 = i(sin ϕ θ l = i( cos ϕ θ l 3 = i ϕ
SO(3) 71 5.7 5.7.1 1 ħ L k l k l k = iϵ kij x i j (5.117) l k SO(3) l z l ± = l 1 ± il = i(y z z y ) ± (z x x z ) = ( x iy) z ± z( x ± i y ) = X ± z ± z (5.118) l z = i(x y y x ) = 1 [(x + iy)( x i y )
More information+ 1 ( ) I IA i i i 1 n m a 11 a 1j a 1m A = a i1 a ij a im a n1 a nj a nm.....
+ http://krishnathphysaitama-uacjp/joe/matrix/matrixpdf 1 ( ) I IA i i i 1 n m a 11 a 1j a 1m A = a i1 a ij a im a n1 a nj a nm (1) n m () (n, m) ( ) n m B = ( ) 3 2 4 1 (2) 2 2 ( ) (2, 2) ( ) C = ( 46
More informationALGEBRA I Hiroshi SUZUKI Department of Mathematics International Christian University
ALGEBRA I Hiroshi SUZUKI Department of Mathematics International Christian University 2004 1 1 1 2 2 1 3 3 1 4 4 1 5 5 1 6 6 1 7 7 1 8 8 1 9 9 1 10 10 1 E-mail:hsuzuki@icu.ac.jp 0 0 1 1.1 G G1 G a, b,
More information2,., ,. 8.,,,..,.,, ,....,..,... 4.,..
Contents 1. 1 2. 2 3. 2 4. 2 5. 3 6. 3 7. 3 8. 4 9. 5 10. 6 11. 8 12. 9 13. - 10 14. 12 15. 13 16. 14 17. 14 18. 15 19. 15 20. 16 21. 16 References 16 1......, 1 2,.,. 4. 2. 2.,. 8.,,,..,.,,... 3....,....,..,...
More information137
136 137 16 12 16 17 3 18 138 8 125 144 269 9 125 144 269 10 125 144 269 11 125 144 269 12 125 143 268 13 124 146 270 14 124 147 271 15 124 149 273 16 124 149 273 17 124 152 276 4 1 3 3 3 3 3 3 8 2,641
More informationmeiji_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 information80 4 r ˆρ i (r, t) δ(r x i (t)) (4.1) x i (t) ρ i ˆρ i t = 0 i r 0 t(> 0) j r 0 + r < δ(r 0 x i (0))δ(r 0 + r x j (t)) > (4.2) r r 0 G i j (r, t) dr 0
79 4 4.1 4.1.1 x i (t) x j (t) O O r 0 + r r r 0 x i (0) r 0 x i (0) 4.1 L. van. Hove 1954 space-time correlation function V N 4.1 ρ 0 = N/V i t 80 4 r ˆρ i (r, t) δ(r x i (t)) (4.1) x i (t) ρ i ˆρ i t
More information(2018 2Q C) [ ] 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
(2018 2Q C) [ ] 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() 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