1 1.1 / Fik Γ= D n x / Newton Γ= µ vx y / Fouie Q = κ T x 1. fx, tdx t x x + dx f t = D f x 1 fx, t = 1 exp x 4πDt 4Dt lim fx, t =δx 3 t + dxfx, t = 1

Size: px
Start display at page:

Download "1 1.1 / Fik Γ= D n x / Newton Γ= µ vx y / Fouie Q = κ T x 1. fx, tdx t x x + dx f t = D f x 1 fx, t = 1 exp x 4πDt 4Dt lim fx, t =δx 3 t + dxfx, t = 1"

Transcription

1 Bownian Motion Einstein Einstein Langevin Eq Navie-Stokes Benoulli Reynolds Boltzmann Kaman-Howath Reynolds Kolmogoov Taylo Kolmogoov -5/3 K K6 K K ɛ Reynolds Reynolds Reynolds Reynolds Pandtl Hagen Poiseuille Kolmogoov MHD Kaman-Howath Navie-Stokes

2 1 1.1 / Fik Γ= D n x / Newton Γ= µ vx y / Fouie Q = κ T x 1. fx, tdx t x x + dx f t = D f x 1 fx, t = 1 exp x 4πDt 4Dt lim fx, t =δx 3 t + dxfx, t = 1 4 x =Dt W l, n n l n 1 W l, n = n C n+l 6 n! = 7 n + l/! n l/! πn exp l n l 1 8 n Stiling log n! n + 1 log n n + 1 log π 1.4 x = la, t = nτ 9 W x, n x = W l, n x a 1 = exp x x 1 4πDt 4Dt D = a τ 1.5 = [L] [T ] 11 n +1 x n x a x + a W x, n +1= 1 W x a, n+ 1 W x + a, n W t τ = a W x 13 n x n x n =x n x n 1 +x n 1 x n +...+x 1 x +x 14 ±a x n x n = na =Dt 15 /4/1 Bownian Motion.1 F x v = βfx β n t = Γ x = D n x x βfn 16

3 . Einstein Γ=D n βfn =, Ux = dxf x 17 x n = n exp Maxwell n = n exp Einstein βux D Ux kt D = βkt Bown Renolds β = 1 1 6πη Einstein D = kt 6πη, x =Dt = kt 3πη t x.3 Einstein Bow P = nkt f = 1 n kt 3 n x f = f f v = f/6πη Γ= D n x + nf 6πη = 4 D = kt/6πη.4 pq W l, n = n C l p l q n l 1 l np exp πnpq pqn 1 x vt exp πdt πdt 5 D = pqa /τ v = pa/τ Stiling.5 Langevin Eq. ζ mftf t ft =, ftft =Aδt t 6 Langevin Foke-Plank 1: m =,Ft = W x =, ζ dx dt W t m =,Ft W ζ dx dt x = F ζ t, W t = ft 7 x = A t 8 ζ = D W x 9 /4/17 = F t+ft 3 x = A t 31 ζ = D W x x βfxw 3 m,ft = Langevin d x dx dt = ζ + ft 33 dt <x > < v>=< dx/dt > d <x > dt + ζ d <x > = v = A dt ζ = kt 34 3

4 <v>=, v w < x >= A ζ t = kt ζ t 35 w t = w A x + ζvw 36 v w exp ζv /A t ζ 1 w W w exp v /kt, m,ft dv dt W exp x x /4Dt 37 = ζv + F x+ft 38 < x >= v t, < v >= ζv + F t, < x > < v x >=, < v >=A t 39 v w Foke-Plank w t.6 = v w x + v ζv F w + A w v 4 Langevin <vtvt >= A ζ e ζ t t,<vt >= kt 41 <vtv >= kt ζ β = 1 ζ = 1 kt ζ = 1 kt, < ftf >= ζktδt 4 dt < vtv > 43 dt < ftf > 44 m d dt <v>= mζ < v > +ee, J = ne < v > 45 < v > = ee/mζσ = J/E = ne /mζ jt = n i ev i d jt = ζjt+e dt m n ɛ i t 46 i=1 e4 m <ɛ i tɛ l t >= Aδ il δt t σ = 1 dt < jtj > 47 kt 1 σ = 1 dt < ɛtɛ > 48 kt.7 P = P t + t P t P t + t =bt P t+ft 49 ft bt Langevin d P t = ν P t+ft dt ν =1 bt/ t F t =ft/ t bt > 1 P 49 P t + t = bt P t + ft P t ft = 1 bt bt P 1/4/18 m d x dt + mω x = F t 5 Fouie mω x + mω 1 + iφ x = F 51 φ β = F ω x ω ω φ m ω ω + ω 4 5 φ x = v /ω kt ω φ mω ω ω + ω 4 53 φ φ ω ω ω ω 4

5 3 3.1 ρ + ρ v = 54 t ρ ρ = onst. 3. substantial mateial deivative 3.3 ρ v t v = 55 D Dt = + v 56 t + ρ v v = + 57 ρ v vdv = ρ v v nds 58 ρ gdv ρ v v v ρ t 59 σ xy :y x σ xy = µ dv x dy 6 σ nds = σ Navie-Stokes σ = p I τ σ ij = pδ ij τ ij 63 τ i j vi τ ij = µ + v j 64 x j x i µ Navie-Stokes τ = µ v 65 D v Dt = 1 ρ p + ν v + g 66 ν = µ/ρ 3.5 1/4/5 v N-S Eq. t + v v = 1 ρ p + ν v φ g = φ ω v v + ω v = t + P + φ ν ω 67 p/ρ = P ω t + v ω =ω v + ν ω Benoulli t =, g = φ, ν = v v ω = +P+φ 69 v +P+φ = onst. 7 ρ D v Dt = σ + ρ g ν = ωds = onst. 5

6 3.8 Reynolds N-S Eq. U, L N-S Eq. v = U v,x= Lx,y= Ly, z = Lz, t =L/Ut, P = ρu P Dv Dt = P + ν UL v 71 R = UL/ν:Reynolds numbe 3.9 1/5/ v / ρ D v Dt = ρ v D v Dt = v p τ + ρ g 7 54 ρv t ρv v = p v+p v τ v+ τ v + ρ v g µ vi τ v = + v j x j x i < U dv ρu + ρv / v q q p v τ v ρ g v 73 ρ D Dt U = q p v τ v Boltzmann δf δt f q, p, t Liouville df dt = f F δf + v f + t m v f = 76 δt Boltzmann n n fd v 77 g <g> fgd v <g> 78 fd v f, n, g g f t d v = t n<g> n t g f gv i d v = n<v i g> n v i g x i x i x i g F i f d v = n gf i 79 m v i m v i Boltzmann g t n<g> n + n <g v> t n< g v > n F m δf v g = g d v8 δt g =1 n t + n< v>= δf δt d v 81 6

7 8 g = m v mn < v >+ mn < v j v> n < F>= t x δf m v d v j δt 8 v =< v>+ v mn D Dt < v>= n< F> P τ + R 83 P = nm < v >/3 τ ij = nm < v i v j <v >δ ij /3 > R = m v δf δt d v 8 g = m v/ nm > t <v + nm <v v> n F v v mv δf = d v 84 δt v =< v>+ v nm t <v> + 3 p nm + <v> + 5 p < v>+τ < v>+ q = nf < v>+ R < v>+ Q + m δf <v> d v 85 δt q = nm <v v > andom motion Q = m δf v δt d v 81, 83 nm t <v> = m δf n <v> < v>+ d v δt mn < v > < v> < v>+n < F> < v> < v> p < v> τ + R < v> p + 3 t p< v>+p < v>+ τ < v>+ q = Q 87 3 ndt Dt + p < v>= q τ < v>+ Q 88 1/5/16 4 Reynolds 4.1 Reynolds Reynolds Navie-Stokes v t + v v = 1 ρ p + 1 ρ τ v = V + v V = v p = P + p τ = T + τ Vi T ij = µ + V j x j x i v,τ ij = µ i + v j x j x i Reynolds Navie-Stokes V V + v v = 1 ρ P 1 ρ T 89 v = v v 89 V V = 1 ρ P T + ρ v v 9 ρ v v Reynolds Boltzmann P ij = nm < v i v j > 4. Reynolds Reynolds τ 1 µ V 1 x µ U L 91 ρv 1 v ρv 9 ρv 1 v τ 1 v R 93 U Reynolds Reynolds 4.3 Pandtl ξ x = ξ V 1 x 1 v th x = 7

8 <mv 1 ξ mv 1 > ρ m <mv 1 ξ mv 1 >v th ρ V 1 x ξv th = µ V 1 x 94 µ = ρξv th ρv l ρv = ρv 1 l ρv 1 + ρv 1 l ρv 1 ρ V 1 x l 95 l Pandtl ρ V1 x lv Reynolds ρv v V1 x v 1 v 1 v ρl V 1 V 1 96 x x ρl V 1 x ρlv 1, µ = ρv th ξ 97 ρlv µ = lv ν Reynolds n v =, v =n, v n t + n v = n t + v n = 99 γ n e γt 1 γn + v n = v = n L n n L nγ 11 L n l= n = n l L n 1 Γ=<n v > D n = D n L n 13 11,1 D = l γ = γ k 14 k Hagen Poiseuille N-S Eq. z ν v p d ρ = ν d + 1 d v z 1 dp d ρ dz = 15 v z = 1 dp 4ρν dz a 16 Hagen Poiseuille V a V = πv zd πa = a dp 17 8ρν dz λ dp dz = λ 1 ρv a 18 λ = R R = V a/ν Reynolds Reynoolds R< 3 R > 3 λ =.3164R 1/4 11 8

9 1/5/3 Kolmogoov Kolmogoov 5 kolmogoov 5/3 5.1 Kolmogoov ɛ ɛ d v dt v v v3 111 L L k L 1 µ v τ v ρ ρ v ν l 11 l l ɛ l 1 ɛ L K ɛ V 3 K L K ν V K l 113 ɛ, ν L K = T K = ν 3 1/4 ɛ ν 1/ ɛ V K = νɛ 1/ Kolmogoov Ek Ekdk = v 115 [Ek] = L3 T 116 [ɛ] = L T ɛ, k Ek k α ɛ β L 3 =[Ek] = L α T 3=β α, =3β α = 5 3, β = 3 L T 3 β 118 Ek ɛ /3 k 5/3 Kolomogoov 5/3 ν v l νv k νek k νk 5/3 k k νk 1/3 k 119 k Kolomogoov 5/3 5.3 MHD MagnetohydodynamiMHD v b Ek, F k MHD v A = B µρ T [ɛ] = [v3 ] [L] = [v3 ] B T = L3 T 4 1 B 1 9

10 Ek k α ɛ β L 3 L 3 β 1 =[Ek] = L α T T 4 B 3=3β α, =4β α = 3, β = 1 11 Ek ɛb 1/ k 3/ Kaihnann 6. Kaman-Howath A, B < u A p B >: < u A u B >: < u A u A u B >: 17 6 Kaman-Howath : : : B i =A i B ij =A 1 i j + B 1 δ ij B ijk =A i j k + B k δ ij +C j δ ik + D i δ jk Reynolds Kamann-Howath N-S Eq. t u i ν u i = u j 1 p 1 x j ρ x j OpeatoL, M, L 1 L u = M u u + L 1 p 13 L < u>= M < u u >+L 1 <p> 14 u Reynolds Pandtl < u u > < u> 14 L < u u >= M < u u u >+L 1 <p u> 15 L < u u u >= M < u u u u >+L 1 <p u u > 16 Kaman-Howath 15 Q ij Q ij = u ia u jb AB Q ij =Q ji A 1,B 1 1/5/3 =,, Q ij Q ll = u la u lb Q nn = u na u nb Q ij = i j Q ll Q nn +Q nn δ ij 19 S ij,k S ij,k = u ia u ja u kb =,, S ij,k = S ji,k C = D S S 11,1 = Q lll S,1 = S 33,1 = Q nnl S 1, = S 13,3 = Q lnn S ij,k = i j k 3 Q lll Q nnl Q lnn + kδ ij Q nnl + iδ jk + j δ ik Q lnn 13 1

11 S ij,k = S k.ij Kaman-Howath K-H Eq Navie-Stokes N-S Eq. A B N-S Eq. u i t + u i u k = 1 p + ν u i x k ρ x i u j t + x u j u k = 1 p + ν u j 13 k ρ x i u j, u i t ν Q ij = = S ik,j S i,jk k S ik,j + S jk,i T ij 133 k pu j = p u i = 6.3. i Q ij = Q nn = Q ll + Q ll = 1 Q ll Q ii = i S ik,j = Q nnl = 1 Q lll Q lnn = 1 4 Q lll S ik.i = k Kaman-Howath +3 Q ll Q lll i = j t Q ll ν + +3 Q ll Q lll [ t ν + 4 ] Q ll = + 4 Q lll 136 Kaman-Howath N-S Eq. t u i ν u i = u i u k 1 x k ρ p 137 x i u Al u Bl = Q ll u Al u Al u Bl = Q lll k Q ij =u ia u jb = + Φ ij ke i k d 3 k 138 S ij,k Φ ij,k 133 t Φ ij = ik α Φ iα,j +Φ jα,i νk Φ ij 139 t Ek, t =4πk6 Γk, t νk Ek, t 14 Ek, tdk = u i / Γk, t Φ iα,j /6/6 l e v e mve/ ρlev 3 e/ µ v y l e µl ev e = µl e vet t t ρl3 e v e µl e v e ρl e µ l e ν 6.4. Reynolds 141 Reynolds ρu i u j l E v E Reynolds ρve l E ρve 3 l E 11

12 ve 3 /l E ɛ ɛ K-H Eq.136 Q ll t + 4 Q lll 14 t E v E t E v3 E l E t E l E v E Kolmogoov K-H Eq.136 Q ll ν t + 4 Q ll 143 t K v K t K ν v K l K ɛ t K l K /v K v K ν l K lk R K v Kl K ν v K ν l K ɛ ν3 l 4 K 1 Kolomogoov 114 GOY i k u i k N-S Eq. d dt u i k+νk u i k= i k j δ il k ik l k u j pu l q p+ q= k 144 p + q = k GOYGledze-Ohkitani-Yamada k i = i k i =,..N 1 u i t i ii iii d dt + νk u i =+i 1 i u i+1 u i+ + i u i 1u i i u i 1u i i = k i, i = 1 k i 1, 3 i = 1 k i 1 N =1 N 1 =, = N 1 =3 = 3 1 = GOY Kolmogoov Taylo Q ll = MioSale : λ l Q ll Q ll Q ll MaoSale : Λ l Q ll d l K <λ l < Λ l 6.5 N-S Eq. 6.6 Kolmogoov -5/3 K41 ɛ K41 ɛ ɛ v p /ɛ/k p/3 k 6.7 K6 K41 1/6/13 ɛ ɛ ɛ x, t = 3 4π 3 ɛ x + y, tdy 146 y < 1

13 ɛ ln ɛ L l n L l 1 = L/Γ l = l 1 /Γ l n = L/Γ n ɛ = ɛ ɛ 1 ɛ ɛ n ɛ = ɛ n = ɛ ɛ 1 ɛ ɛ ɛ 1 ɛ n 1 ln ɛ = lnɛ +lne 1 +lne + +lne n 147 e i ɛ i /ɛ i 1 ln ɛ 1 P ɛ = exp ln ɛ m 148 πσ ɛ σ m =lnɛ σ, σ = A + µ ln L/ µ ln L/ ɛ /3 = ɛ /3 ɛ /3 P ɛ dɛ ɛ /3 ɛ n µ/9 L 149 v ɛ/k 1/3 Ek v k ɛ/3 k 5/3 k µ/9 15 K41-5/3 µ/9 v p /ɛ/k p/3 k µpp 1/ /k N N = k D D k D /k 3 ɛ ɛ ɛ = k3 k D ɛ = ɛ k 3 D ɛ /3 = ɛ /3 k D /k 3 = ɛ /3 k D 3/3 Ek = ɛ /3 k 5 /3 = ɛ /3 k D 3/3 k 5/ /6/ 7 Reynolds Reynolds x y U s x lx x Lx Ux, y L l L U y/lx U s x x, y U, V, u, v U s, L, l U x + V y = V = OU s l/lreynolds N-S Eq. y U V V +V x y + x uv+ y v = 1 P ρ y +ν U x + V y 15 y v = 1 P ρ y 153 L l, Reynolds Rynolds U s l L u l 156 x v = 1 P 154 ρ x N-S Eq. x U U x + V U y + x u v + y uv = ν U x + V y U U x + V U y + uv = 156 y O U s u = O L l ξ = y/l, lx, U s x U = U s fξ 157 uv = Us gξ

14 f = O1, g = Ol/L V y V = U dy = l x ξ 156 dus dx f U s dl l dx ξf dξ 159 ξ ξ g = l du s U s dx f dl dx ξff l du s U s dx f fdξ+ dl dx f ξf dξ 16 x l = onst., du s U s dx dl dx = onst. l x x y x Us l = onst. l x U s 1/ x ν T Reynolds R T U s l/ν T Reynolds U uv = ν T y Us g = ν T l f g = f 16 f = 1 ξ R T f + f fdξ f = seh ξ/ = 8 R T 161 e ξ/ +e ξ/ Navie-Stokes Diet Numeial Simulation DNS DNS Reynolds UL/ν Kolmogoov l K ν 3 /ɛ 1/4, t K ν/ɛ 1/ ɛ v 3 /l l k ν 3/4 L = R 3/4 UL T L/U t K ν 1/ T = R 1/ UL R 9/4 R 1/ = R.75 U = 1m/s 36km/h ν = 1 5 m /s L =m R /6/7 Kaman-Howath Reynolds Reynolds Rynolds 14

15 8.3 Reynolds x U y U y l u v U y l Reynolds uv = l U y U y 164 Businesque Reynolds uv = ν T U y ν T 165 Reynolds Reynolds 8.4 N-S Eq. U i t + U j U i = P x j x i ρ + ν x j x j U i u i t + U u i U i j + u j = p x j x j x i ρ + ν x i Reynolds x j u j u i 166 ui + u j x j x i x i u i u j u i u j 167 K 1 u i 168 u j ɛ τ ij = ν uj u i uj u i x i x i x j x i x j K t + U K U j i = R ij x i x i x j [ pu j ρ + U i u j uj νu i + u i x i x j 169 ] 17 ɛ ɛ ɛ t + U i = W H j + ν ɛ 171 x i x j ui u k W ν + u j u j Ui +ν u i u i u j x j x j x k x i x k x k x j x k ν U i x j x k H j ν ρ 8.5 +νu j U j x k U i x j x k 17 u j p u i u i + νu j 173 x k x k x k x k K = u i u i / Reynolds K ν T R ij = u i u j = 3 Kδ Ui ij ν T + U j 174 x j x i Pandtl u i u j l U U x x ν T lu l K ν T U x ν T = Kl 175 l K 17 K t + U U i U j j + R ij x j x i = νt K K3/ x j σ k x j l 176 σ K, σ k ɛ = K3/ l K N-S Eq.166 U i x i, σ K l 8.6 K ɛ ɛ K ɛ ɛ µ ν T = C µ K ɛ 171 ɛ t + U ɛ ɛ i = C ɛ1 x i K R U i ɛ ij C ɛ x j K + x j ɛ νt σ ɛ 177 ɛ x j

16 N-S Eq. Reynolds 174 K K t + U U i U j j + R ij x j x i = νt K ɛ 179 x j σ k x j C µ, C ɛ1 C ɛ, σ ɛ 8.7 Lage Eddy Simulation, 1, G G U = Gx x ux dx = G u 18 u i = U i + u Rynolds G u i u j = G U i U j +G u i U j + U i u j + u i u j = U i U j + L ij + M ij 181 N-S Eq. U i t + U i U j = 1 P x j ρ x i x j L ij x j M ij + U i L ij = G U i U j U i U j M ij = Ui 3 K δ ij + ν SG + U j x j x i L ij Lenad ν SG = C S 4/3 ɛ 1/3 u K = G i u j =C K /3 ɛ 1/3 184 C S, C K ɛ ɛ = M ij U i x j 185 Lage Eddy Simulation 9 1/7/4 Kolmogoov 9.1 Se.4.4 λ P. dx = λρv 186 d λ = R λ =.316 R 1/4 188 λ N-S Eq. v t + v v = 1 ρ P + ν v x λ 1 x, v λ v, t λ 3 t, P/ρ λ 4 P/ρ, ν λ 5 ν, λ 1,λ, N-S Eq. λ λ 1 3 = λ λ 1 1 = λ 1 1 λ 4 = λ 5 λ 1 λ λ 3 = λ 1 λ 1, λ 4 = λ, λ 5 = λ 1 λ x λ 1 x, v λ v, t λ 1 λ 1 t, P/ρ λ P/ρ, ν λ 1λ ν, 189 dp/ρ dx = Cd α V β ν γ C λ λ 1 1 = λ α 1 λβ λ 1λ γ 19 α = 1 γ, β = α 191 dp dx = Cd 1 γ V γ ν γ = C V d ν γ dv γ dp dx = C V d F R 16

17 9. τ τ MHD τ v mn + v v = p + j B t n + n v = t v = E + v e B = η j p e en B t = E p 3/ η T 3/ 19 n x λ 1 x, v λ v, t λ 3 t, n λ 4 n, B λ 5 B, E λ 6 E, P λ 7 P, j λ 8 j, η λ 9 η 193 λ 4 λ λ 1 3 = λ 4 λ λ 1 1 λ 7λ 1 1 = λ 8 λ 5 λ 4 λ 1 3 = λ 1 1 λ 4λ λ 6 = λ λ 5 = λ 9 λ 8 = λ 7 λ 1 1 λ 1 4 λ 8 = λ 1 1 λ 5 λ 5 λ 1 3 = λ 1 1 λ 6 λ 9 = λ 3/ 7 λ 3/ λ = λ P x λ 4 x, v λv, t λ 5 t, n λ 8 n, B λ 5 B, E λ 6 E, P λ 1, j λ 9 j, η λ 3 η 195 τ n T B a τ = Cn p T q B a s 196 C λ 5 = λ 8p λ q λ 5 λ 4s p + q s = τ n p T q B a p+ q na p T a q +Ba 5/4 +1 B τ = 1 B F na,t a, Ba 5/4 17

gr09.dvi

gr09.dvi .1, θ, ϕ d = A, t dt + B, t dtd + C, t d + D, t dθ +in θdϕ.1.1 t { = f1,t t = f,t { D, t = B, t =.1. t A, tdt e φ,t dt, C, td e λ,t d.1.3,t, t d = e φ,t dt + e λ,t d + dθ +in θdϕ.1.4 { = f1,t t = f,t {

More information

Gauss Gauss ɛ 0 E ds = Q (1) xy σ (x, y, z) (2) a ρ(x, y, z) = x 2 + y 2 (r, θ, φ) (1) xy A Gauss ɛ 0 E ds = ɛ 0 EA Q = ρa ɛ 0 EA = ρea E = (ρ/ɛ 0 )e

Gauss Gauss ɛ 0 E ds = Q (1) xy σ (x, y, z) (2) a ρ(x, y, z) = x 2 + y 2 (r, θ, φ) (1) xy A Gauss ɛ 0 E ds = ɛ 0 EA Q = ρa ɛ 0 EA = ρea E = (ρ/ɛ 0 )e 7 -a 7 -a February 4, 2007 1. 2. 3. 4. 1. 2. 3. 1 Gauss Gauss ɛ 0 E ds = Q (1) xy σ (x, y, z) (2) a ρ(x, y, z) = x 2 + y 2 (r, θ, φ) (1) xy A Gauss ɛ 0 E ds = ɛ 0 EA Q = ρa ɛ 0 EA = ρea E = (ρ/ɛ 0 )e z

More information

Z: Q: R: C: 3. Green Cauchy

Z: Q: R: C: 3. Green Cauchy 7 Z: Q: R: C: 3. Green.............................. 3.............................. 5.3................................. 6.4 Cauchy..................... 6.5 Taylor..........................6...............................

More information

= hυ = h c λ υ λ (ev) = 1240 λ W=NE = Nhc λ W= N 2 10-16 λ / / Φe = dqe dt J/s Φ = km Φe(λ)v(λ)dλ THBV3_0101JA Qe = Φedt (W s) Q = Φdt lm s Ee = dφe ds E = dφ ds Φ Φ THBV3_0102JA Me = dφe ds M = dφ ds

More information

* 1 2014 7 8 *1 iii 1. Newton 1 1.1 Newton........................... 1 1.2............................. 4 1.3................................. 5 2. 9 2.1......................... 9 2.2........................

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

A = A x x + A y y + A, B = B x x + B y y + B, C = C x x + C y y + C..6 x y A B C = A x x + A y y + A B x B y B C x C y C { B = A x x + A y y + A y B B

A = A x x + A y y + A, B = B x x + B y y + B, C = C x x + C y y + C..6 x y A B C = A x x + A y y + A B x B y B C x C y C { B = A x x + A y y + A y B B 9 7 A = A x x + A y y + A, B = B x x + B y y + B, C = C x x + C y y + C..6 x y A B C = A x x + A y y + A B x B y B C x C y C { B = A x x + A y y + A y B B x x B } B C y C y + x B y C x C C x C y B = A

More information

『共形場理論』

『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 information

量子力学A

量子力学A c 1 1 1.1....................................... 1 1............................................ 4 1.3.............................. 6 10.1.................................. 10......................................

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

0.,,., m Euclid m m. 2.., M., M R 2 ψ. ψ,, R 2 M.,, (x 1 (),, x m ()) R m. 2 M, R f. M (x 1,, x m ), f (x 1,, x m ) f(x 1,, x m ). f ( ). x i : M R.,,

0.,,., m Euclid m m. 2.., M., M R 2 ψ. ψ,, R 2 M.,, (x 1 (),, x m ()) R m. 2 M, R f. M (x 1,, x m ), f (x 1,, x m ) f(x 1,, x m ). f ( ). x i : M R.,, 2012 10 13 1,,,.,,.,.,,. 2?.,,. 1,, 1. (θ, φ), θ, φ (0, π),, (0, 2π). 1 0.,,., m Euclid m m. 2.., M., M R 2 ψ. ψ,, R 2 M.,, (x 1 (),, x m ()) R m. 2 M, R f. M (x 1,, x m ), f (x 1,, x m ) f(x 1,, x m ).

More information

B

B B YES NO 5 7 6 1 4 3 2 BB BB BB AA AA BB 510J B B A 510J B A A A A A A 510J B A 510J B A A A A A 510J M = σ Z Z = M σ AAA π T T = a ZP ZP = a AAA π B M + M 2 +T 2 M T Me = = 1 + 1 + 2 2 M σ Te = M 2 +T

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

( ) 2002 1 1 1 1.1....................................... 1 1.1.1................................. 1 1.1.2................................. 1 1.1.3................... 3 1.1.4......................................

More information

2 p T, Q

2 p T, Q 270 C, 6000 C, 2 p T, Q p: : p = N/ m 2 N/ m 2 Pa : pdv p S F Q 1 g 1 1 g 1 14.5 C 15.5 1 1 cal = 4.1855 J du = Q pdv U ( ) Q pdv 2 : z = f(x, y). z = f(x, y) (x 0, y 0 ) y y = y 0 z = f(x, y 0 ) x x =

More information

CVMに基づくNi-Al合金の

CVMに基づくNi-Al合金の CV N-A (-' by T.Koyama ennard-jones fcc α, β, γ, δ β α γ δ = or α, β. γ, δ α β γ ( αβγ w = = k k k ( αβγ w = ( αβγ ( αβγ w = w = ( αβγ w = ( αβγ w = ( αβγ w = ( αβγ w = ( αβγ w = ( βγδ w = = k k k ( αγδ

More information

13Ad m in is t r a t ie e n h u lp v e r le n in g Ad m in is t r a t ie v e p r o b le m e n,p r o b le m e n in d e h u lp v e r le n in g I n d ic

13Ad m in is t r a t ie e n h u lp v e r le n in g Ad m in is t r a t ie v e p r o b le m e n,p r o b le m e n in d e h u lp v e r le n in g I n d ic 13D a t a b a n k m r in g R a p p o r t M ィC Aa n g e m a a k t o p 19 /09 /2007 o m 09 :3 1 u u r I d e n t if ic a t ie v a n d e m S e c t o r BJB V o lg n r. 06 013-00185 V o o r z ie n in g N ie

More information

untitled

untitled (a) (b) (c) (d) Wunderlich 2.5.1 = = =90 2 1 (hkl) {hkl} [hkl] L tan 2θ = r L nλ = 2dsinθ dhkl ( ) = 1 2 2 2 h k l + + a b c c l=2 l=1 l=0 Polanyi nλ = I sinφ I: B A a 110 B c 110 b b 110 µ a 110

More information

10:30 12:00 P.G. vs vs vs 2

10:30 12:00 P.G. vs vs vs 2 1 10:30 12:00 P.G. vs vs vs 2 LOGIT PROBIT TOBIT mean median mode CV 3 4 5 0.5 1000 6 45 7 P(A B) = P(A) + P(B) - P(A B) P(B A)=P(A B)/P(A) P(A B)=P(B A) P(A) P(A B) P(A) P(B A) P(B) P(A B) P(A) P(B) P(B

More information

1990 IMO 1990/1/15 1:00-4:00 1 N N N 1, N 1 N 2, N 2 N 3 N 3 2 x x + 52 = 3 x x , A, B, C 3,, A B, C 2,,,, 7, A, B, C

1990 IMO 1990/1/15 1:00-4:00 1 N N N 1, N 1 N 2, N 2 N 3 N 3 2 x x + 52 = 3 x x , A, B, C 3,, A B, C 2,,,, 7, A, B, C 0 9 (1990 1999 ) 10 (2000 ) 1900 1994 1995 1999 2 SAT ACT 1 1990 IMO 1990/1/15 1:00-4:00 1 N 1990 9 N N 1, N 1 N 2, N 2 N 3 N 3 2 x 2 + 25x + 52 = 3 x 2 + 25x + 80 3 2, 3 0 4 A, B, C 3,, A B, C 2,,,, 7,

More information

13 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 information

p.2/76

p.2/76 kino@info.kanagawa-u.ac.jp p.1/76 p.2/76 ( ) (2001). (2006). (2002). p.3/76 N n, n {1, 2,...N} 0 K k, k {1, 2,...,K} M M, m {1, 2,...,M} p.4/76 R =(r ij ), r ij = i j ( ): k s r(k, s) r(k, 1),r(k, 2),...,r(k,

More information

タ 縺29135 タ 縺5 [ y 1 x i R 8 x j 1 7,5 2 x , チ7192, (2) チ41299 f 675

タ 縺29135 タ 縺5 [ y 1 x i R 8 x j 1 7,5 2 x , チ7192, (2) チ41299 f 675 139ィ 48 1995 3. 753 165, 2 6 86 タ7 9 998917619 4381 縺48 縺55 317832645 タ5 縺4273 971927, 95652539358195 45 チ5197 9 4527259495 2 7545953471 129175253471 9557991 3.9. タ52917652 縺1874ィ 989 95652539358195 45

More information

第86回日本感染症学会総会学術集会後抄録(I)

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 information

http://www2.math.kyushu-u.ac.jp/~hara/lectures/lectures-j.html 2 N(ε 1 ) N(ε 2 ) ε 1 ε 2 α ε ε 2 1 n N(ɛ) N ɛ ɛ- (1.1.3) n > N(ɛ) a n α < ɛ n N(ɛ) a n

http://www2.math.kyushu-u.ac.jp/~hara/lectures/lectures-j.html 2 N(ε 1 ) N(ε 2 ) ε 1 ε 2 α ε ε 2 1 n N(ɛ) N ɛ ɛ- (1.1.3) n > N(ɛ) a n α < ɛ n N(ɛ) a n http://www2.math.kyushu-u.ac.jp/~hara/lectures/lectures-j.html 1 1 1.1 ɛ-n 1 ɛ-n lim n a n = α n a n α 2 lim a n = 1 n a k n n k=1 1.1.7 ɛ-n 1.1.1 a n α a n n α lim n a n = α ɛ N(ɛ) n > N(ɛ) a n α < ɛ

More information

PII S (96)

PII S (96) C C R ( 1 Rvw C d m d M.F. Pllps *, P.S. Hp I q G U W C M H P C C f R 5 J 1 6 J 1 A C d w m d u w b b m C d m d T b s b s w b d m d s b s C g u T p d l v w b s d m b b v b b d s d A f b s s s T f p s s

More information

nsg04-28/ky208684356100043077

nsg04-28/ky208684356100043077 δ!!! μ μ μ γ UBE3A Ube3a Ube3a δ !!!! α α α α α α α α α α μ μ α β α β β !!!!!!!! μ! Suncus murinus μ Ω! π μ Ω in vivo! μ μ μ!!! ! in situ! in vivo δ δ !!!!!!!!!! ! in vivo Orexin-Arch Orexin-Arch !!

More information

現代物理化学 1-1(4)16.ppt

現代物理化学 1-1(4)16.ppt (pdf) pdf pdf http://www1.doshisha.ac.jp/~bukka/lecture/index.html http://www.doshisha.ac.jp/ Duet -1-1-1 2-a. 1-1-2 EU E = K E + P E + U ΔE K E = 0P E ΔE = ΔU U U = εn ΔU ΔU = Q + W, du = d 'Q + d 'W

More information

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

(Basic of Proability Theory). (Probability Spacees ad Radom Variables , (Expectatios, Meas) (Weak Law I (Radom Walks ad Percolatios) 3 4 7 ( -2 ) (Preface),.,,,...,,.,,,,.,.,,.,,. (,.) (Basic of Proability Theory). (Probability Spacees ad Radom Variables...............2, (Expectatios, Meas).............................

More information

縺 縺8 縺, [ 縺 チ : () () () 4 チ93799; () "64": ィャ 9997ィ

縺 縺8 縺, [ 縺 チ : () () () 4 チ93799; () 64: ィャ 9997ィ 34978 998 3. 73 68, 86 タ7 9 9989769 438 縺48 縺 378364 タ 縺473 399-4 8 637744739 683 6744939 3.9. 378,.. 68 ィ 349 889 3349947 89893 683447 4 334999897447 (9489) 67449, 6377447 683, 74984 7849799 34789 83747

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

213 2 katurada AT meiji.ac.jp http://nalab.mind.meiji.ac.jp/~mk/pde/ 213 9, 216 11 3 6.1....................................... 6.2............................. 8.3................................... 9.4.....................................

More information

研修コーナー

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 information

学習内容と日常生活との関連性の研究-第2部-第6章

学習内容と日常生活との関連性の研究-第2部-第6章 378 379 10% 10%10% 10% 100% 380 381 2000 BSE CJD 5700 18 1996 2001 100 CJD 1 310-7 10-12 10-6 CJD 100 1 10 100 100 1 1 100 1 10-6 1 1 10-6 382 2002 14 5 1014 10 10.4 1014 100 110-6 1 383 384 385 2002 4

More information

untitled

untitled V. 8 9 9 8.. SI 5 6 7 8 9. - - SI 6 6 6 6 6 6 6 SI -- l -- 6 -- -- 6 6 u 6cod5 6 h5 -oo ch 79 79 85 875 99 79 58 886 9 89 9 959 966 - - NM /6 Nucl Ml SI NM/6/685 85co /./ /h / /6/.6 / /.6 /h o NM o.85

More information

I 1 V ( x) = V (x), V ( x) = V ( x ) SO(3) x = R x: R SO(3) Lorentz R t JR = J: J = diag(1, 1, 1, 1) x = x + a Poincarré ( ) 2

I 1 V ( x) = V (x), V ( x) = V ( x ) SO(3) x = R x: R SO(3) Lorentz R t JR = J: J = diag(1, 1, 1, 1) x = x + a Poincarré ( ) 2 III 1 2005 Jan 30th, 2006 I : II : I : [ I ] 12 13 9 (Landau and Lifshitz, Quantum Mechanics chapter 12, 13, 9: Pergamon Pr.) [ ] ( ) (H. Georgi, Lie algebra in particle physics, Perseus Books) [ ] II

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

ISTC 3

ISTC 3 B- I n t e r n a t i o n a l S t a n d a r s f o r Tu b e r c u l o s i s C a r (ÏS r c ) E d is i k e - 3 ) a =1 / < ' 3 I n t e r n a t i o n a l s t a n d a r d s f o r T B C a r e e «l i s i k e 3

More information

nsg02-13/ky045059301600033210

nsg02-13/ky045059301600033210 φ φ φ φ κ κ α α μ μ α α μ χ et al Neurosci. Res. Trpv J Physiol μ μ α α α β in vivo β β β β β β β β in vitro β γ μ δ μδ δ δ α θ α θ α In Biomechanics at Micro- and Nanoscale Levels, Volume I W W v W

More information

II 2014 2 (1) log(1 + r/100) n = log 2 n log(1 + r/100) = log 2 n = log 2 log(1 + r/100) (2) y = f(x) = log(1 + x) x = 0 1 f (x) = 1/(1 + x) f (0) = 1

II 2014 2 (1) log(1 + r/100) n = log 2 n log(1 + r/100) = log 2 n = log 2 log(1 + r/100) (2) y = f(x) = log(1 + x) x = 0 1 f (x) = 1/(1 + x) f (0) = 1 II 2014 1 1 I 1.1 72 r 2 72 8 72/8 = 9 9 2 a 0 1 a 1 a 1 = a 0 (1+r/100) 2 a 2 a 2 = a 1 (1 + r/100) = a 0 (1 + r/100) 2 n a n = a 0 (1 + r/100) n a n a 0 2 n a 0 (1 + r/100) n = 2a 0 (1 + r/100) n = 2

More information

No2 4 y =sinx (5) y = p sin(2x +3) (6) y = 1 tan(3x 2) (7) y =cos 2 (4x +5) (8) y = cos x 1+sinx 5 (1) y =sinx cos x 6 f(x) = sin(sin x) f 0 (π) (2) y

No2 4 y =sinx (5) y = p sin(2x +3) (6) y = 1 tan(3x 2) (7) y =cos 2 (4x +5) (8) y = cos x 1+sinx 5 (1) y =sinx cos x 6 f(x) = sin(sin x) f 0 (π) (2) y No1 1 (1) 2 f(x) =1+x + x 2 + + x n, g(x) = 1 (n +1)xn + nx n+1 (1 x) 2 x 6= 1 f 0 (x) =g(x) y = f(x)g(x) y 0 = f 0 (x)g(x)+f(x)g 0 (x) 3 (1) y = x2 x +1 x (2) y = 1 g(x) y0 = g0 (x) {g(x)} 2 (2) y = µ

More information

2 σ γ l σ ο 4..5 cos 5 D c D u U b { } l + b σ l r l + r { r m+ m } b + l + + l l + 4..0 D b0 + r l r m + m + r 4..7 4..0 998 ble4.. ble4.. 8 0Z Fig.4.. 0Z 0Z Fig.4.. ble4.. 00Z 4 00 0Z Fig.4.. MO S 999

More information

I No. sin cos sine, cosine : trigonometric function π : π =.4 : n =, ±, ±, sin + nπ = sin cos + nπ = cos sin = sin : cos = cos :. sin. sin. sin + π si

I No. sin cos sine, cosine : trigonometric function π : π =.4 : n =, ±, ±, sin + nπ = sin cos + nπ = cos sin = sin : cos = cos :. sin. sin. sin + π si I 8 No. : No. : No. : No.4 : No.5 : No.6 : No.7 : No.8 : No.9 : No. : I No. sin cos sine, cosine : trigonometric function π : π =.4 : n =, ±, ±, sin + nπ = sin cos + nπ = cos sin = sin : cos = cos :. sin.

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 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

8.1 Fubini 8.2 Fubini 9 (0%) 10 (50%) 10.1 10.2 Carathéodory 10.3 Fubini 1 Introduction [1],, [2],, [3],, [4],, [5],, [6],, [7],, [8],, [1, 2, 3] 1980

8.1 Fubini 8.2 Fubini 9 (0%) 10 (50%) 10.1 10.2 Carathéodory 10.3 Fubini 1 Introduction [1],, [2],, [3],, [4],, [5],, [6],, [7],, [8],, [1, 2, 3] 1980 % 100% 1 Introduction 2 (100%) 2.1 2.2 2.3 3 (100%) 3.1 3.2 σ- 4 (100%) 4.1 4.2 5 (100%) 5.1 5.2 5.3 6 (100%) 7 (40%) 8 Fubini (90%) 2006.11.20 1 8.1 Fubini 8.2 Fubini 9 (0%) 10 (50%) 10.1 10.2 Carathéodory

More information

4.6: 3 sin 5 sin θ θ t θ 2t θ 4t : sin ωt ω sin θ θ ωt sin ωt 1 ω ω [rad/sec] 1 [sec] ω[rad] [rad/sec] 5.3 ω [rad/sec] 5.7: 2t 4t sin 2t sin 4t

4.6: 3 sin 5 sin θ θ t θ 2t θ 4t : sin ωt ω sin θ θ ωt sin ωt 1 ω ω [rad/sec] 1 [sec] ω[rad] [rad/sec] 5.3 ω [rad/sec] 5.7: 2t 4t sin 2t sin 4t 1 1.1 sin 2π [rad] 3 ft 3 sin 2t π 4 3.1 2 1.1: sin θ 2.2 sin θ ft t t [sec] t sin 2t π 4 [rad] sin 3.1 3 sin θ θ t θ 2t π 4 3.2 3.1 3.4 3.4: 2.2: sin θ θ θ [rad] 2.3 0 [rad] 4 sin θ sin 2t π 4 sin 1 1

More information

17 3 31 1 1 3 2 5 3 9 4 10 5 15 6 21 7 29 8 31 9 35 10 38 11 41 12 43 13 46 14 48 2 15 Radon CT 49 16 50 17 53 A 55 1 (oscillation phenomena) e iθ = cos θ + i sin θ, cos θ = eiθ + e iθ 2, sin θ = eiθ e

More information

Γ Ec Γ V BIAS THBV3_0401JA THBV3_0402JAa THBV3_0402JAb 1000 800 600 400 50 % 25 % 200 100 80 60 40 20 10 8 6 4 10 % 2.5 % 0.5 % 0.25 % 2 1.0 0.8 0.6 0.4 0.2 0.1 200 300 400 500 600 700 800 1000 1200 14001600

More information

5 36 5................................................... 36 5................................................... 36 5.3..............................

5 36 5................................................... 36 5................................................... 36 5.3.............................. 9 8 3............................................. 3.......................................... 4.3............................................ 4 5 3 6 3..................................................

More information

F8302D_1目次_160527.doc

F8302D_1目次_160527.doc N D F 830D.. 3. 4. 4. 4.. 4.. 4..3 4..4 4..5 4..6 3 4..7 3 4..8 3 4..9 3 4..0 3 4. 3 4.. 3 4.. 3 4.3 3 4.4 3 5. 3 5. 3 5. 3 5.3 3 5.4 3 5.5 4 6. 4 7. 4 7. 4 7. 4 8. 4 3. 3. 3. 3. 4.3 7.4 0 3. 3 3. 3 3.

More information

1 1 H Li Be Na M g B A l C S i N P O S F He N Cl A e K Ca S c T i V C Mn Fe Co Ni Cu Zn Ga Ge As Se B K Rb S Y Z Nb Mo Tc Ru Rh Pd Ag Cd In Sn Sb T e

1 1 H Li Be Na M g B A l C S i N P O S F He N Cl A e K Ca S c T i V C Mn Fe Co Ni Cu Zn Ga Ge As Se B K Rb S Y Z Nb Mo Tc Ru Rh Pd Ag Cd In Sn Sb T e No. 1 1 1 H Li Be Na M g B A l C S i N P O S F He N Cl A e K Ca S c T i V C Mn Fe Co Ni Cu Zn Ga Ge As Se B K Rb S Y Z Nb Mo Tc Ru Rh Pd Ag Cd In Sn Sb T e I X e Cs Ba F Ra Hf Ta W Re Os I Rf Db Sg Bh

More information

untitled

untitled .m 5m :.45.4m.m 3.m.6m (N/mm ).8.6 σ.4 h.m. h.68m h(m) b.35m θ4..5.5.5 -. σ ta.n/mm c 3kN/m 3 w 9.8kN/m 3 -.4 ck 6N/mm -.6 σ -.8 3 () :. 4 5 3.75m :. 7.m :. 874mm 4 865mm mm/ :. 7.m 4.m 4.m 6 7 4. 3.5

More information

3章 問題・略解

3章 問題・略解 S S W R S O( l) O( ) c Jg g J Jg S R J 7. K.9 JK S W S R S JK S S R J 7. K.9JK 4 (a) -Tice 7.K T ice T N 77 K S R.9 JK 4. JK T T ice N.6JK S W S R S JK S S.6JK R (b) S R JK S.6 JK T T ice N 6 O( c) O(

More information

[ ] = L [δ (D ) (x )] = L D [g ] = L D [E ] = L Table : ħh = m = D D, V (x ) = g δ (D ) (x ) E g D E (Table )D = Schrödinger (.3)D = (regularization)

[ ] = L [δ (D ) (x )] = L D [g ] = L D [E ] = L Table : ħh = m = D D, V (x ) = g δ (D ) (x ) E g D E (Table )D = Schrödinger (.3)D = (regularization) . D............................................... : E = κ ............................................ 3.................................................

More information

(2) N elec = D p,q p,q χ q χ p dr = p,q D p,q S q,p Mulliken PA D Mull p = p = group A D p,p 1 + D p,q S q,p p q p [ r A D Mull p ] group χ p G Mull A

(2) N elec = D p,q p,q χ q χ p dr = p,q D p,q S q,p Mulliken PA D Mull p = p = group A D p,p 1 + D p,q S q,p p q p [ r A D Mull p ] group χ p G Mull A 7 - (Electron-Donor Acceptor) : Charge-Transfer ( CT) ( (Charge-Transfer) - (electron donor-electron acceptor) [1][2][3][4] Van der Waals CT [5] Population Analysis population analysis ( ), observable

More information

1 180m g 10m/s 2 2 6 1 3 v 0 (t=0) z max t max t z = z max 1 2 g(t t max) 2 (6) 1.3 2 3 3 r = (x, y, z) e x, e y, e z r = xe x + ye y + ze z. (7) v =

1 180m g 10m/s 2 2 6 1 3 v 0 (t=0) z max t max t z = z max 1 2 g(t t max) 2 (6) 1.3 2 3 3 r = (x, y, z) e x, e y, e z r = xe x + ye y + ze z. (7) v = 1. 2. 3 3. 4. 5. 6. 7. 8. 9. I http://risu.lowtem.hokudai.ac.jp/ hidekazu/class.html 1 1.1 1 a = g, (1) v = g t + v 0, (2) z = 1 2 g t2 + v 0 t + z 0. (3) 1.2 v-t. z-t. z 1 z 0 = dz = v, t1 dv v(t), v

More information

Part. 4. () 4.. () 4.. 3 5. 5 5.. 5 5.. 6 5.3. 7 Part 3. 8 6. 8 6.. 8 6.. 8 7. 8 7.. 8 7.. 3 8. 3 9., 34 9.. 34 9.. 37 9.3. 39. 4.. 4.. 43. 46.. 46..

Part. 4. () 4.. () 4.. 3 5. 5 5.. 5 5.. 6 5.3. 7 Part 3. 8 6. 8 6.. 8 6.. 8 7. 8 7.. 8 7.. 3 8. 3 9., 34 9.. 34 9.. 37 9.3. 39. 4.. 4.. 43. 46.. 46.. Cotets 6 6 : 6 6 6 6 6 6 7 7 7 Part. 8. 8.. 8.. 9..... 3. 3 3.. 3 3.. 7 3.3. 8 Part. 4. () 4.. () 4.. 3 5. 5 5.. 5 5.. 6 5.3. 7 Part 3. 8 6. 8 6.. 8 6.. 8 7. 8 7.. 8 7.. 3 8. 3 9., 34 9.. 34 9.. 37 9.3.

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

O1-1 O1-2 O1-3 O1-4 O1-5 O1-6

O1-1 O1-2 O1-3 O1-4 O1-5 O1-6 O1-1 O1-2 O1-3 O1-4 O1-5 O1-6 O1-7 O1-8 O1-9 O1-10 O1-11 O1-12 O1-13 O1-14 O1-15 O1-16 O1-17 O1-18 O1-19 O1-20 O1-21 O1-22 O1-23 O1-24 O1-25 O1-26 O1-27 O1-28 O1-29 O1-30 O1-31 O1-32 O1-33 O1-34 O1-35

More information

24.15章.微分方程式

24.15章.微分方程式 m d y dt = F m d y = mg dt V y = dy dt d y dt = d dy dt dt = dv y dt dv y dt = g dv y dt = g dt dt dv y = g dt V y ( t) = gt + C V y ( ) = V y ( ) = C = V y t ( ) = gt V y ( t) = dy dt = gt dy = g t dt

More information

2 Three-wave Painlevé VI 21 -Wilson three-wave Painlevé VI Gauss -Wilson [KK3] n 3 3 t = t 1 t 2 t 3 -Wilson W z; t := I + W 1 z + W 2 z 2 + z; t := 0

2 Three-wave Painlevé VI 21 -Wilson three-wave Painlevé VI Gauss -Wilson [KK3] n 3 3 t = t 1 t 2 t 3 -Wilson W z; t := I + W 1 z + W 2 z 2 + z; t := 0 1473 : de nouvelles perspectives 2006 2 pp 102 119 VI q 1 Tetsuya Kikuchi Sabro Kakei Drinfel d-sokolov Painlevé [KK1] [KK2] [KK3] [KIK] [ ] [ ] [KK3] three-wave equation Painlevé VI q q Drinfel d-sokolov

More information

1.1 ft t 2 ft = t 2 ft+ t = t+ t 2 1.1 d t 2 t + t 2 t 2 = lim t 0 t = lim t 0 = lim t 0 t 2 + 2t t + t 2 t 2 t + t 2 t 2t t + t 2 t 2t + t = lim t 0

1.1 ft t 2 ft = t 2 ft+ t = t+ t 2 1.1 d t 2 t + t 2 t 2 = lim t 0 t = lim t 0 = lim t 0 t 2 + 2t t + t 2 t 2 t + t 2 t 2t t + t 2 t 2t + t = lim t 0 A c 2008 by Kuniaki Nakamitsu 1 1.1 t 2 sin t, cos t t ft t t vt t xt t + t xt + t xt + t xt t vt = xt + t xt t t t vt xt + t xt vt = lim t 0 t lim t 0 t 0 vt = dxt ft dft dft ft + t ft = lim t 0 t 1.1

More information

d dt A B C = A B C d dt x = Ax, A 0 B 0 C 0 = mm 0 mm 0 mm AP = PΛ P AP = Λ P A = ΛP P d dt x = P Ax d dt (P x) = Λ(P x) d dt P x =

d dt A B C = A B C d dt x = Ax, A 0 B 0 C 0 = mm 0 mm 0 mm AP = PΛ P AP = Λ P A = ΛP P d dt x = P Ax d dt (P x) = Λ(P x) d dt P x = 3 MATLAB Runge-Kutta Butcher 3. Taylor Taylor y(x 0 + h) = y(x 0 ) + h y (x 0 ) + h! y (x 0 ) + Taylor 3. Euler, Runge-Kutta Adams Implicit Euler, Implicit Runge-Kutta Gear y n+ y n (n+ ) y n+ y n+ y n+

More information

M41 JP Manual.indd

M41 JP Manual.indd i ii iii iv v vi vii 1 No / A-B EQ 2 MIC REC REC00001.WAV Stereo CH:01 0:00:00 1:50:00 3 4 5 6 7 8 9 10 11 12 1 1 F F A A 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 Φ 35 36 37 38

More information

7) ẋt) =iaω expiωt) ibω exp iωt) 9) ẋ0) = iωa b) = 0 0) a = b a = b = A/ xt) = A expiωt) + exp iωt)) = A cosωt) ) ) vt) = Aω sinωt) ) ) 9) ) 9) E = mv

7) ẋt) =iaω expiωt) ibω exp iωt) 9) ẋ0) = iωa b) = 0 0) a = b a = b = A/ xt) = A expiωt) + exp iωt)) = A cosωt) ) ) vt) = Aω sinωt) ) ) 9) ) 9) E = mv - - m k F = kx ) kxt) =m d xt) dt ) ω = k/m ) ) d dt + ω xt) = 0 3) ) ) d d dt iω dt + iω xt) = 0 4) ω d/dt iω) d/dt + iω) 4) ) d dt iω xt) = 0 5) ) d dt + iω xt) = 0 6) 5) 6) a expiωt) b exp iωt) ) )

More information

橡博論表紙.PDF

橡博論表紙.PDF Study on Retaining Wall Design For Circular Deep Shaft Undergoing Lateral Pressure During Construction 2003 3 Study on Retaining Wall Design For Circular Deep Shaft Undergoing Lateral Pressure During Construction

More information

1 1 1 1 1 1 2 f z 2 C 1, C 2 f 2 C 1, C 2 f(c 2 ) C 2 f(c 1 ) z C 1 f f(z) xy uv ( u v ) = ( a b c d ) ( x y ) + ( p q ) (p + b, q + d) 1 (p + a, q + c) 1 (p, q) 1 1 (b, d) (a, c) 2 3 2 3 a = d, c = b

More information

LHC ALICE (QGP) QGP QGP QGP QGP ω ϕ J/ψ ALICE s = ev + J/ψ

LHC ALICE (QGP) QGP QGP QGP QGP ω ϕ J/ψ ALICE s = ev + J/ψ 8 + J/ψ ALICE B597 : : : 9 LHC ALICE (QGP) QGP QGP QGP QGP ω ϕ J/ψ ALICE s = ev + J/ψ 6..................................... 6. (QGP)..................... 6.................................... 6.4..............................

More information

(a) (b) 1: (a) ( ) (b) ( ) : ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( 2 ) 2 2

(a) (b) 1: (a) ( ) (b) ( ) : ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( 2 ) 2 2 (2) 1 1 4 ( beresit ) ( ) ( ) ( ) 1 Makio Uwaha. E-mail:uwaha@nagoya-u.jp; http://slab.phys.nagoya-u.ac.jp/uwaha/ 1 (a) (b) 1: (a) ( ) (b) ( ) : ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( 2 ) 2 2 20 [ ]

More information

2

2 16 1050026 1050042 1 2 1 1.1 3 1.2 3 1.3 3 2 2.1 4 2.2 4 2.2.1 5 2.2.2 5 2.3 7 2.3.1 1Basic 7 2.3.2 2 8 2.3.3 3 9 2.3.4 4window size 10 2.3.5 5 11 3 3.1 12 3.2 CCF 1 13 3.3 14 3.4 2 15 3.5 3 17 20 20 20

More information

( ) 24 1 ( 26 8 19 ) i 0.1 1 (2012 05 30 ) 1 (), 2 () 1,,, III, C III, C, 1, 2,,, ( III, C ),, 1,,, http://ryuiki.agbi.tsukuba.ac.jp/lec/12-physics/ E104),,,,,, 75 3,,,, 0.2, 1,,,,,,,,,,, 2,,, 1000 ii,

More information

<4D F736F F D B B83578B6594BB2D834A836F815B82D082C88C60202E646F63>

<4D F736F F D B B83578B6594BB2D834A836F815B82D082C88C60202E646F63> 単純適応制御 SAC 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/091961 このサンプルページの内容は, 初版 1 刷発行当時のものです. 1 2 3 4 5 9 10 12 14 15 A B F 6 8 11 13 E 7 C D URL http://www.morikita.co.jp/support

More information

3 3.3. I 3.3.2. [ ] N(µ, σ 2 ) σ 2 (X 1,..., X n ) X := 1 n (X 1 + + X n ): µ X N(µ, σ 2 /n) 1.8.4 Z = X µ σ/ n N(, 1) 1.8.2 < α < 1/2 Φ(z) =.5 α z α

3 3.3. I 3.3.2. [ ] N(µ, σ 2 ) σ 2 (X 1,..., X n ) X := 1 n (X 1 + + X n ): µ X N(µ, σ 2 /n) 1.8.4 Z = X µ σ/ n N(, 1) 1.8.2 < α < 1/2 Φ(z) =.5 α z α 2 2.1. : : 2 : ( ): : ( ): : : : ( ) ( ) ( ) : ( pp.53 6 2.3 2.4 ) : 2.2. ( ). i X i (i = 1, 2,..., n) X 1, X 2,..., X n X i (X 1, X 2,..., X n ) ( ) n (x 1, x 2,..., x n ) (X 1, X 2,..., X n ) : X 1,

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

Big Bang Planck Big Bang 1 43 Planck Planck quantum gravity Planck Grand Unified Theories: GUTs X X W X 1 15 ev 197 Glashow Georgi 1 14 GeV 1 2

Big Bang Planck Big Bang 1 43 Planck Planck quantum gravity Planck Grand Unified Theories: GUTs X X W X 1 15 ev 197 Glashow Georgi 1 14 GeV 1 2 12 Big Bang 12.1 Big Bang Big Bang 12.1 1-5 1 32 K 1 19 GeV 1-4 time after the Big Bang [ s ] 1-3 1-2 1-1 1 1 1 1 2 inflationary epoch gravity strong electromagnetic weak 1 27 K 1 14 GeV 1 15 K 1 2 GeV

More information

WECPNL = LA +10log10 N 27 N = N 2 + 3N3 + 10( N1 + N 4) L A N N N N N 1 2 3 4 Lden Lden Lden Lden LAE L pa pa 2 a /10 LpA = 20 log 10 ( pa = p 10 ) n na p0 p na n an n p0 2 Lp p L p

More information

0 1-4. 1-5. (1) + b = b +, (2) b = b, (3) + 0 =, (4) 1 =, (5) ( + b) + c = + (b + c), (6) ( b) c = (b c), (7) (b + c) = b + c, (8) ( + b)c = c + bc (9

0 1-4. 1-5. (1) + b = b +, (2) b = b, (3) + 0 =, (4) 1 =, (5) ( + b) + c = + (b + c), (6) ( b) c = (b c), (7) (b + c) = b + c, (8) ( + b)c = c + bc (9 1-1. 1, 2, 3, 4, 5, 6, 7,, 100,, 1000, n, m m m n n 0 n, m m n 1-2. 0 m n m n 0 2 = 1.41421356 π = 3.141516 1-3. 1 0 1-4. 1-5. (1) + b = b +, (2) b = b, (3) + 0 =, (4) 1 =, (5) ( + b) + c = + (b + c),

More information

<4D F736F F D F8DE98BCA8CA797A78FAC8E9988E397C3835A E815B82CC8A E646F63>

<4D F736F F D F8DE98BCA8CA797A78FAC8E9988E397C3835A E815B82CC8A E646F63> ˆ Ñ Ñ vìéê d Ê ÍÉÂÊÊÊ ÆÂ Æ Ç ÇÂÊ ~ÌÈÉ ÇÉÂÿ Â ss ÊÌ Ë sê~ Ê ÆÂ ~ÌÊÎÌÈÊÈÌÂ ÊÂ Ê ~ÊÉÆÉÊÂ ÇÉÉ ÇÈÂ Â Â Â xâîööð ÊÇÈÍÉÊÉÉÂÇÊÉÌÂÉÌÊÉÌÊÂ Ê Ê u Ç ÌÉÉÇÉÂ Ã ÃÊ ÈÂ ÊÆÇÍÃw ÃÎ v Êv ÊÑ Ñ vêî Í}ÌÂ Ã ÃÇÍÂ Ê vê u Ç ÇÆÉÊÎ

More information

AHPを用いた大相撲の新しい番付編成

AHPを用いた大相撲の新しい番付編成 5304050 2008/2/15 1 2008/2/15 2 42 2008/2/15 3 2008/2/15 4 195 2008/2/15 5 2008/2/15 6 i j ij >1 ij ij1/>1 i j i 1 ji 1/ j ij 2008/2/15 7 1 =2.01/=0.5 =1.51/=0.67 2008/2/15 8 1 2008/2/15 9 () u ) i i i

More information

176 B B.1: ( ) ( ) ( ) (2 2 ) ( ) ( ) ( ) (quantitative nondestructive evaluation:qnde) (1) X X X X CT(computed tomography)

176 B B.1: ( ) ( ) ( ) (2 2 ) ( ) ( ) ( ) (quantitative nondestructive evaluation:qnde) (1) X X X X CT(computed tomography) B 1) B.1 B.1.1 ( ) B.1 1 50 100 m B.1.2 (nondestructive testing:ndt) (nondestructive inspection:ndi) (nondestructive evaluation:nde) 175 176 B B.1: ( ) ( ) ( ) (2 2 ) ( ) ( ) ( ) (quantitative nondestructive

More information