wpmath0000002_18

Schwinger_model.html

  1. r r
  2. 1 / r 1/r

Scientific_law.html

  1. Δ E = 0 \Delta E=0
  2. d U = δ Q - δ W \mathrm{d}U=\delta Q-\delta W\,

Scramjet.html

  1. q = 1 2 ρ v 2 q=\frac{1}{2}\rho v^{2}
  2. R R
  3. R R
  4. Γ \Gamma
  5. Π e + Π f + 1 Γ = 1 \Pi_{e}+\Pi_{f}+\frac{1}{\Gamma}=1
  6. Π e = m empty m initial \Pi_{e}=\frac{m\text{empty}}{m\text{initial}}
  7. Π f = m fuel m initial \Pi_{f}=\frac{m\text{fuel}}{m\text{initial}}
  8. Γ = m initial m payload \Gamma=\frac{m\text{initial}}{m\text{payload}}
  9. Π e \Pi_{e}
  10. Π f \Pi_{f}
  11. Γ \Gamma
  12. D D
  13. D e D_{e}
  14. D e = ϕ e F D_{e}=\phi_{e}F
  15. ϕ e \phi_{e}
  16. F F
  17. D e D_{e}
  18. η 0 \eta_{0}
  19. η 0 = g 0 V 0 h P R I s p = Thrust Power Chemical energy rate \eta_{0}=\frac{g_{0}V_{0}}{h_{PR}}\cdot I_{sp}=\frac{\mbox{Thrust Power}~{}}{% \mbox{Chemical energy rate}~{}}
  20. g 0 g_{0}
  21. V 0 V_{0}
  22. I s p I_{sp}
  23. h P R h_{PR}
  24. η 0 \eta_{0}
  25. I s p I_{sp}
  26. Π f = 1 - exp [ - ( V i n i t i a l 2 2 - V i 2 2 ) + g d r η 0 h P R ( 1 - D + D e F ) ] \Pi_{f}=1-\exp\left[-\frac{\left(\frac{V_{initial}^{2}}{2}-\frac{V_{i}^{2}}{2}% \right)+\int{g}\,dr}{\eta_{0}h_{PR}\left(1-\frac{D+D_{e}}{F}\right)}\right]
  27. Π f = 1 - exp [ - g 0 r 0 ( 1 - 1 2 r 0 r ) η 0 h P R ( 1 - D + D e F ) ] \Pi_{f}=1-\exp\left[-\frac{g_{0}r_{0}\left(1-\frac{1}{2}\frac{r_{0}}{r}\right)% }{\eta_{0}h_{PR}\left(1-\frac{D+D_{e}}{F}\right)}\right]
  28. Π f = 1 - exp [ - g 0 R η 0 h P R ( 1 - ϕ e ) C L C D ] \Pi_{f}=1-\exp\left[-\frac{g_{0}R}{\eta_{0}h_{PR}\left(1-\phi_{e}\right)\frac{% C_{L}}{C_{D}}}\right]
  29. R R
  30. Π f = 1 - e - B R \Pi_{f}=1-e^{-BR}
  31. B = g 0 η 0 h P R ( 1 - ϕ e ) C L C D B=\frac{g_{0}}{\eta_{0}h_{PR}\left(1-\phi_{e}\right)\frac{C_{L}}{C_{D}}}
  32. C L {C_{L}}
  33. C D {C_{D}}

Screened_Poisson_equation.html

  1. [ Δ - λ 2 ] u ( 𝐫 ) = - f ( 𝐫 ) , \left[\Delta-\lambda^{2}\right]u(\mathbf{r})=-f(\mathbf{r}),
  2. Δ \Delta
  3. n = 3 n=3
  4. u ( 𝐫 ) ( Poisson ) = d 3 r f ( 𝐫 ) 4 π | 𝐫 - 𝐫 | . u(\mathbf{r})_{(\,\text{Poisson})}=\iiint\mathrm{d}^{3}r^{\prime}\frac{f(% \mathbf{r}^{\prime})}{4\pi|\mathbf{r}-\mathbf{r}^{\prime}|}.
  5. [ Δ - λ 2 ] G ( 𝐫 ) = - δ 3 ( 𝐫 ) . \left[\Delta-\lambda^{2}\right]G(\mathbf{r})=-\delta^{3}(\mathbf{r}).
  6. G ( 𝐤 ) = d 3 r G ( 𝐫 ) e - i 𝐤 𝐫 G(\mathbf{k})=\iiint\mathrm{d}^{3}r\;G(\mathbf{r})e^{-i\mathbf{k}\cdot\mathbf{% r}}
  7. [ k 2 + λ 2 ] G ( 𝐤 ) = 1. \left[k^{2}+\lambda^{2}\right]G(\mathbf{k})=1.
  8. G ( 𝐫 ) = 1 ( 2 π ) 3 d 3 k e i 𝐤 𝐫 k 2 + λ 2 . G(\mathbf{r})=\frac{1}{(2\pi)^{3}}\;\iiint\mathrm{d}^{3}\!k\;\frac{e^{i\mathbf% {k}\cdot\mathbf{r}}}{k^{2}+\lambda^{2}}.
  9. k r k_{r}
  10. G ( 𝐫 ) = 1 2 π 2 r 0 + d k r k r sin k r r k r 2 + λ 2 . G(\mathbf{r})=\frac{1}{2\pi^{2}r}\;\int_{0}^{+\infty}\mathrm{d}k_{r}\;\frac{k_% {r}\,\sin k_{r}r}{k_{r}^{2}+\lambda^{2}}.
  11. G ( 𝐫 ) = e - λ r 4 π r . G(\mathbf{r})=\frac{e^{-\lambda r}}{4\pi r}.
  12. u ( 𝐫 ) = d 3 r G ( 𝐫 - 𝐫 ) f ( 𝐫 ) = d 3 r e - λ | 𝐫 - 𝐫 | 4 π | 𝐫 - 𝐫 | f ( 𝐫 ) . u(\mathbf{r})=\int\mathrm{d}^{3}r^{\prime}G(\mathbf{r}-\mathbf{r}^{\prime})f(% \mathbf{r}^{\prime})=\int\mathrm{d}^{3}r^{\prime}\frac{e^{-\lambda|\mathbf{r}-% \mathbf{r}^{\prime}|}}{4\pi|\mathbf{r}-\mathbf{r}^{\prime}|}f(\mathbf{r}^{% \prime}).
  13. ( Δ - 1 ρ 2 ) u ( 𝐫 ) = - f ( 𝐫 ) \left(\Delta_{\perp}-\frac{1}{\rho^{2}}\right)u(\mathbf{r}_{\perp})=-f(\mathbf% {r}_{\perp})
  14. Δ = \Delta_{\perp}=\nabla\cdot\nabla_{\perp}
  15. = - 𝐁 B \nabla_{\perp}=\nabla-\frac{\mathbf{B}}{B}\cdot\nabla
  16. 𝐁 \mathbf{B}
  17. ρ \rho
  18. G ( 𝐤 ) = d 2 r G ( 𝐫 ) e - i 𝐤 𝐫 . G(\mathbf{k_{\perp}})=\iint d^{2}r~{}G(\mathbf{r}_{\perp})e^{-i\mathbf{k}_{% \perp}\cdot\mathbf{r}_{\perp}}.
  19. ( k 2 + 1 ρ 2 ) G ( 𝐤 ) = 1 \left(k_{\perp}^{2}+\frac{1}{\rho^{2}}\right)G(\mathbf{k}_{\perp})=1
  20. G ( 𝐫 ) = 1 4 π 2 d 2 k e i 𝐤 𝐫 k 2 + 1 / ρ 2 . G(\mathbf{r}_{\perp})=\frac{1}{4\pi^{2}}\;\iint\mathrm{d}^{2}\!k\;\frac{e^{i% \mathbf{k}_{\perp}\cdot\mathbf{r}_{\perp}}}{k_{\perp}^{2}+1/\rho^{2}}.
  21. 𝐤 = ( k r cos ( θ ) , k r sin ( θ ) ) \mathbf{k}_{\perp}=(k_{r}\cos(\theta),k_{r}\sin(\theta))
  22. k r k_{r}
  23. G ( 𝐫 ) = 1 2 π 0 + d k r k r J 0 ( k r r ) k r 2 + 1 / ρ 2 = 1 2 π K 0 ( r / ρ ) . G(\mathbf{r}_{\perp})=\frac{1}{2\pi}\;\int_{0}^{+\infty}\mathrm{d}k_{r}\;\frac% {k_{r}\,J_{0}(k_{r}r_{\perp})}{k_{r}^{2}+1/\rho^{2}}=\frac{1}{2\pi}K_{0}(r_{% \perp}\,/\,\rho).

Seafloor_spreading.html

  1. T 1 Θ ( - z ) T_{1}\cdot\Theta(-z)
  2. T t = κ 2 T = κ 2 T 2 z + κ 2 T 2 x \frac{\partial T}{\partial t}=\kappa\nabla^{2}T=\kappa\frac{\partial^{2}T}{% \partial^{2}z}+\kappa\frac{\partial^{2}T}{\partial^{2}x^{\prime}}
  3. κ \kappa
  4. x = x + v t x=x^{\prime}+vt
  5. T x = 1 v T t \frac{\partial T}{\partial x^{\prime}}=\frac{1}{v}\cdot\frac{\partial T}{% \partial t}
  6. T t = κ 2 T = κ 2 T 2 z + κ v 2 2 T 2 t \frac{\partial T}{\partial t}=\kappa\nabla^{2}T=\kappa\frac{\partial^{2}T}{% \partial^{2}z}+\frac{\kappa}{v^{2}}\frac{\partial^{2}T}{\partial^{2}t}
  7. v v
  8. T t = κ 2 T 2 z \frac{\partial T}{\partial t}=\kappa\frac{\partial^{2}T}{\partial^{2}z}
  9. T ( t = 0 ) = T 1 Θ ( - z ) T(t=0)=T_{1}\cdot\Theta(-z)
  10. z 0 z\leq 0
  11. erf \operatorname{erf}
  12. T ( x , z , t ) = T 1 erf ( z 2 κ t ) T(x^{\prime},z,t)=T_{1}\cdot\operatorname{erf}(\frac{z}{2\sqrt{\kappa t}})
  13. h ( t ) = h 0 + α e f f 0 [ T ( z ) - T 1 ] d z = h 0 - 2 π α e f f T 1 κ t h(t)=h_{0}+\alpha_{eff}\int_{0}^{\infty}[T(z)-T_{1}]dz=h_{0}-\frac{2}{\sqrt{% \pi}}\alpha_{eff}T_{1}\sqrt{\kappa t}
  14. α e f f \alpha_{eff}
  15. κ \kappa
  16. L 2 / T L^{2}/T
  17. α e f f \alpha_{eff}
  18. α \alpha
  19. α e f f = α ρ ρ - ρ w \alpha_{eff}=\alpha\cdot\frac{\rho}{\rho-\rho_{w}}
  20. ρ 3.3 g / c m 3 \rho\sim 3.3g/cm^{3}
  21. ρ 0 = 1 g / c m 3 \rho_{0}=1g/cm^{3}
  22. κ 8 10 - 7 \kappa\sim 8\cdot 10^{-7}
  23. α 4 10 - 5 \alpha\sim 4\cdot 10^{-5}
  24. h ( t ) h 0 - 350 t h(t)\sim h_{0}-350\sqrt{t}
  25. h ( t ) h 0 - 390 t h(t)\sim h_{0}-390\sqrt{t}

Secchi_disk.html

  1. I z I 0 = e - k z {I_{z}\over I_{0}}=e^{-kz}

Second_law_of_thermodynamics.html

  1. d S dS
  2. δ Q δQ
  3. T T
  4. d S = δ Q T \mathrm{d}S=\frac{\delta Q}{T}\!
  5. Δ Q = Q ( 1 η - 1 ) \Delta Q=Q\left(\frac{1}{\eta}-1\right)
  6. δ Q = T d S \delta Q=TdS
  7. δ Q T 0. \oint\frac{\delta Q}{T}\leq 0.
  8. η = A q H = q H - q C q H = 1 - q C q H ( 1 ) \eta=\frac{A}{q_{H}}=\frac{q_{H}-q_{C}}{q_{H}}=1-\frac{q_{C}}{q_{H}}\qquad(1)
  9. q C q H = f ( T H , T C ) ( 2 ) . \frac{q_{C}}{q_{H}}=f(T_{H},T_{C})\qquad(2).
  10. f ( T 1 , T 3 ) = q 3 q 1 = q 2 q 3 q 1 q 2 = f ( T 1 , T 2 ) f ( T 2 , T 3 ) . f(T_{1},T_{3})=\frac{q_{3}}{q_{1}}=\frac{q_{2}q_{3}}{q_{1}q_{2}}=f(T_{1},T_{2}% )f(T_{2},T_{3}).
  11. T 1 T_{1}
  12. f ( T 2 , T 3 ) = f ( T 1 , T 3 ) f ( T 1 , T 2 ) = 273.16 f ( T 1 , T 3 ) 273.16 f ( T 1 , T 2 ) . f(T_{2},T_{3})=\frac{f(T_{1},T_{3})}{f(T_{1},T_{2})}=\frac{273.16\cdot f(T_{1}% ,T_{3})}{273.16\cdot f(T_{1},T_{2})}.
  13. T = 273.16 f ( T 1 , T ) T=273.16\cdot f(T_{1},T)\,
  14. f ( T 2 , T 3 ) = T 3 T 2 , f(T_{2},T_{3})=\frac{T_{3}}{T_{2}},
  15. δ Q T = 0 \oint\frac{\delta Q}{T}=0
  16. L δ Q T \int_{L}\frac{\delta Q}{T}
  17. d S = δ Q T dS=\frac{\delta Q}{T}\!
  18. - Δ S + δ Q T = δ Q T < 0 -\Delta S+\int\frac{\delta Q}{T}=\oint\frac{\delta Q}{T}<0
  19. Δ S δ Q T \Delta S\geq\int\frac{\delta Q}{T}\,\!
  20. δ Q = 0 \delta Q=0
  21. Δ S 0 \Delta S\geq 0
  22. d S tot = d S + d S R 0 dS_{\mathrm{tot}}=dS+dS_{R}\geq 0
  23. d U = δ q - δ w + d ( μ i R N i ) dU=\delta q-\delta w+d(\sum\mu_{iR}N_{i})\,
  24. δ q = T R ( - d S R ) T R d S \delta q=T_{R}(-dS_{R})\leq T_{R}dS
  25. δ w - d U + T R d S + μ i R d N i \delta w\leq-dU+T_{R}dS+\sum\mu_{iR}dN_{i}\,
  26. δ w u - d ( U - T R S + p R V - μ i R N i ) \delta w_{u}\leq-d(U-T_{R}S+p_{R}V-\sum\mu_{iR}N_{i})\,
  27. E = U - T R S + p R V - μ i R N i E=U-T_{R}S+p_{R}V-\sum\mu_{iR}N_{i}
  28. d E + δ w u 0 dE+\delta w_{u}\leq 0\,
  29. d S t o t 0 dS_{tot}\geq 0
  30. d E + δ w u 0 dE+\delta w_{u}\leq 0
  31. δ Q T = - N \int\frac{\delta Q}{T}=-N
  32. d S d t 0 \frac{dS}{dt}\geq 0
  33. d S d t = S ˙ i \frac{dS}{dt}=\dot{S}_{i}
  34. S ˙ i 0 \dot{S}_{i}\geq 0
  35. S ˙ i \dot{S}_{i}
  36. T a T_{a}
  37. P d i s s = T a S ˙ i P_{diss}=T_{a}\dot{S}_{i}
  38. d S d t = Q ˙ T + S ˙ i \frac{dS}{dt}=\frac{\dot{Q}}{T}+\dot{S}_{i}
  39. S ˙ i 0 \dot{S}_{i}\geq 0
  40. Q ˙ \dot{Q}
  41. T T
  42. d S d t = Q ˙ T + S ˙ + S ˙ i \frac{dS}{dt}=\frac{\dot{Q}}{T}+\dot{S}+\dot{S}_{i}
  43. S ˙ i 0 \dot{S}_{i}\geq 0
  44. S ˙ \dot{S}
  45. E E
  46. S = k B ln [ Ω ( E ) ] S=k_{\mathrm{B}}\ln\left[\Omega\left(E\right)\right]\,
  47. Ω ( E ) \Omega\left(E\right)
  48. E E
  49. E + δ E E+\delta E
  50. δ E \delta E
  51. δ E \delta E
  52. δ E \delta E
  53. Ω \Omega
  54. Ω \Omega
  55. Ω \Omega
  56. Ω \Omega
  57. 1 / Ω 1/\Omega
  58. H H
  59. d S = δ Q T dS=\frac{\delta Q}{T}
  60. 1 k B T β d ln [ Ω ( E ) ] d E \frac{1}{k_{\mathrm{B}}T}\equiv\beta\equiv\frac{d\ln\left[\Omega\left(E\right)% \right]}{dE}
  61. X d x Xdx
  62. E r E_{r}
  63. X = - d E r d x X=-\frac{dE_{r}}{dx}
  64. δ E \delta E
  65. X = - d E r d x X=-\left\langle\frac{dE_{r}}{dx}\right\rangle\,
  66. Ω ( E ) \Omega\left(E\right)
  67. d E r d x \frac{dE_{r}}{dx}
  68. Y Y
  69. Y + δ Y Y+\delta Y
  70. Ω Y ( E ) \Omega_{Y}\left(E\right)
  71. Ω ( E ) = Y Ω Y ( E ) \Omega\left(E\right)=\sum_{Y}\Omega_{Y}\left(E\right)\,
  72. X = - 1 Ω ( E ) Y Y Ω Y ( E ) X=-\frac{1}{\Omega\left(E\right)}\sum_{Y}Y\Omega_{Y}\left(E\right)\,
  73. Ω ( E ) \Omega\left(E\right)
  74. E E
  75. E + δ E E+\delta E
  76. d E r d x \frac{dE_{r}}{dx}
  77. Y Y
  78. Y + δ Y Y+\delta Y
  79. N Y ( E ) = Ω Y ( E ) δ E Y d x N_{Y}\left(E\right)=\frac{\Omega_{Y}\left(E\right)}{\delta E}Ydx\,
  80. Y d x δ E Ydx\leq\delta E
  81. E E
  82. E + δ E E+\delta E
  83. Ω \Omega
  84. E + δ E E+\delta E
  85. E + δ E E+\delta E
  86. N Y ( E + δ E ) N_{Y}\left(E+\delta E\right)
  87. N Y ( E ) - N Y ( E + δ E ) N_{Y}\left(E\right)-N_{Y}\left(E+\delta E\right)\,
  88. Ω \Omega
  89. δ E \delta E
  90. E + δ E E+\delta E
  91. N Y ( E ) N_{Y}\left(E\right)
  92. N Y ( E + δ E ) N_{Y}\left(E+\delta E\right)
  93. ( Ω x ) E = - Y Y ( Ω Y E ) x = ( ( Ω X ) E ) x \left(\frac{\partial\Omega}{\partial x}\right)_{E}=-\sum_{Y}Y\left(\frac{% \partial\Omega_{Y}}{\partial E}\right)_{x}=\left(\frac{\partial\left(\Omega X% \right)}{\partial E}\right)_{x}\,
  94. Ω \Omega
  95. ( ln ( Ω ) x ) E = β X + ( X E ) x \left(\frac{\partial\ln\left(\Omega\right)}{\partial x}\right)_{E}=\beta X+% \left(\frac{\partial X}{\partial E}\right)_{x}\,
  96. ( S x ) E = X T \left(\frac{\partial S}{\partial x}\right)_{E}=\frac{X}{T}\,
  97. ( S E ) x = 1 T \left(\frac{\partial S}{\partial E}\right)_{x}=\frac{1}{T}\,
  98. d S = ( S E ) x d E + ( S x ) E d x = d E T + X T d x = δ Q T dS=\left(\frac{\partial S}{\partial E}\right)_{x}dE+\left(\frac{\partial S}{% \partial x}\right)_{E}dx=\frac{dE}{T}+\frac{X}{T}dx=\frac{\delta Q}{T}\,
  99. P j = exp ( - E j k B T ) Z P_{j}=\frac{\exp\left(-\frac{E_{j}}{k_{\mathrm{B}}T}\right)}{Z}
  100. S = - k B j P j ln ( P j ) S=-k_{\mathrm{B}}\sum_{j}P_{j}\ln\left(P_{j}\right)
  101. d S = - k B j ln ( P j ) d P j dS=-k_{\mathrm{B}}\sum_{j}\ln\left(P_{j}\right)dP_{j}
  102. P j P_{j}
  103. d S = 1 T j E j d P j = 1 T j d ( E j P j ) - 1 T j P j d E j = d E + δ W T = δ Q T dS=\frac{1}{T}\sum_{j}E_{j}dP_{j}=\frac{1}{T}\sum_{j}d\left(E_{j}P_{j}\right)-% \frac{1}{T}\sum_{j}P_{j}dE_{j}=\frac{dE+\delta W}{T}=\frac{\delta Q}{T}
  104. exp ( S / k ) \sim\exp\left(S/k\right)

Sectional_curvature.html

  1. K ( u , v ) = R ( u , v ) v , u u , u v , v - u , v 2 K(u,v)={\langle R(u,v)v,u\rangle\over\langle u,u\rangle\langle v,v\rangle-% \langle u,v\rangle^{2}}
  2. K ( u , v ) = R ( u , v ) v , u . K(u,v)=\langle R(u,v)v,u\rangle.
  3. d ( z , m ) 2 1 2 d ( z , x ) 2 + 1 2 d ( z , y ) 2 - 1 4 d ( x , y ) 2 d(z,m)^{2}\geq\tfrac{1}{2}d(z,x)^{2}+\tfrac{1}{2}d(z,y)^{2}-\tfrac{1}{4}d(x,y)% ^{2}
  4. d ( z , m ) 2 1 2 d ( z , x ) 2 + 1 2 d ( z , y ) 2 - 1 4 d ( x , y ) 2 . d(z,m)^{2}\leq\tfrac{1}{2}d(z,x)^{2}+\tfrac{1}{2}d(z,y)^{2}-\tfrac{1}{4}d(x,y)% ^{2}.
  5. f p ( x ) = dist 2 ( p , x ) f_{p}(x)=\operatorname{dist}^{2}(p,x)
  6. f p ( x ) = dist 2 ( p , x ) f_{p}(x)=\operatorname{dist}^{2}(p,x)
  7. π i ( M ) \pi_{i}(M)
  8. 2 \mathbb{Z}_{2}
  9. 𝕊 2 × 𝕊 2 \mathbb{S}^{2}\times\mathbb{S}^{2}
  10. ( M , g ) (M,g)
  11. M / G M/G
  12. B 7 = S O ( 5 ) / S O ( 3 ) B^{7}=SO(5)/SO(3)
  13. B 13 = S U ( 5 ) / S p ( 2 ) 𝕊 1 B^{13}=SU(5)/Sp(2)\cdot\mathbb{S}^{1}
  14. W 6 = S U ( 3 ) / T 2 W^{6}=SU(3)/T^{2}
  15. W 12 = S p ( 3 ) / S p ( 1 ) 3 W^{12}=Sp(3)/Sp(1)^{3}
  16. W 24 = F 4 / S p i n ( 8 ) W^{24}=F_{4}/Spin(8)
  17. W p , q 7 = S U ( 3 ) / diag ( z p , z q , z ¯ p + q ) W^{7}_{p,q}=SU(3)/\operatorname{diag}(z^{p},z^{q},\overline{z}^{p+q})
  18. E k , l = diag ( z k 1 , z k 2 , z k 3 ) \ S U ( 3 ) / diag ( z l 1 , z l 2 , z l 3 ) - 1 . E_{k,l}=\operatorname{diag}(z^{k_{1}},z^{k_{2}},z^{k_{3}})\backslash SU(3)/% \operatorname{diag}(z^{l_{1}},z^{l_{2}},z^{l_{3}})^{-1}.
  19. B p 13 = diag ( z 1 p 1 , , z 1 p 5 ) \ U ( 5 ) / diag ( z 2 A , 1 ) - 1 B^{13}_{p}=\operatorname{diag}(z_{1}^{p_{1}},\dots,z_{1}^{p_{5}})\backslash U(% 5)/\operatorname{diag}(z_{2}A,1)^{-1}
  20. A S p ( 2 ) S U ( 4 ) A\in Sp(2)\subset SU(4)

Security_protocol_notation.html

  1. A B : { X } K A , B A\rightarrow B:\{X\}_{K_{A,B}}
  2. B A : { N B } K A B\rightarrow A:\{N_{B}\}_{K_{A}}

Self-adjoint.html

  1. x * = x x^{*}=x
  2. x * = y x^{*}=y
  3. y * = x * * = x y^{*}=x^{**}=x
  4. f f
  5. f = f f=f^{\dagger}
  6. f : A A f\colon A\to A

Self-adjoint_operator.html

  1. , \langle\cdot,\cdot\rangle
  2. A v , w = v , A w \langle Av,w\rangle=\langle v,Aw\rangle
  3. H ψ = V ψ - 2 2 m 2 ψ , H\psi=V\psi-\frac{\hbar^{2}}{2m}\nabla^{2}\psi,
  4. A x y = x A y \langle Ax\mid y\rangle=\langle x\mid Ay\rangle
  5. A x y = x A y \langle Ax\mid y\rangle=\langle x\mid Ay\rangle
  6. A = - d 2 d x 2 A=-\frac{d^{2}}{dx^{2}}
  7. f n ( x ) = sin ( n π x ) n = 1 , 2 , f_{n}(x)=\sin(n\pi x)\qquad n=1,2,\ldots
  8. y x A y y\mapsto\langle x\mid Ay\rangle
  9. x A y = z y y dom A \langle x\mid Ay\rangle=\langle z\mid y\rangle\qquad\forall y\in\operatorname{% dom}A
  10. G ( A ) = { ( ξ , A ξ ) : ξ dom ( A ) } H H . \operatorname{G}(A)=\{(\xi,A\xi):\xi\in\operatorname{dom}(A)\}\subseteq H% \oplus H.
  11. { H H H H J : ( ξ , η ) ( - η , ξ ) \begin{cases}H\oplus H\to H\oplus H\\ \operatorname{J}:(\xi,\eta)\mapsto(-\eta,\xi)\end{cases}
  12. G ( A * ) = ( J G ( A ) ) = { ( x , y ) H H : ( x , y ) | ( - A ξ , ξ ) = 0 ξ dom ( A ) } \operatorname{G}(A^{*})=(\operatorname{J}\operatorname{G}(A))^{\perp}=\{(x,y)% \in H\oplus H:\langle(x,y)|(-A\xi,\xi)\rangle=0\;\;\forall\xi\in\operatorname{% dom}(A)\}
  13. A f ( x ) = x f ( x ) Af(x)=xf(x)
  14. B U ξ = U A ξ , ξ dom A . BU\xi=UA\xi,\qquad\forall\xi\in\operatorname{dom}A.
  15. [ T ψ ] ( x ) = f ( x ) ψ ( x ) [T\psi](x)=f(x)\psi(x)
  16. E T ( λ ) = 𝟏 ( - , λ ] ( T ) \operatorname{E}_{T}(\lambda)=\mathbf{1}_{(-\infty,\lambda]}(T)
  17. 𝟏 ( - , λ ] \mathbf{1}_{(-\infty,\lambda]}
  18. ( - , λ ] (-\infty,\lambda]
  19. T = - + λ d E T ( λ ) . T=\int_{-\infty}^{+\infty}\lambda d\operatorname{E}_{T}(\lambda).
  20. f ( H ) = d E | Ψ E f ( E ) Ψ E | f(H)=\int dE\left|\Psi_{E}\rangle f(E)\langle\Psi_{E}\right|
  21. H | Ψ E = E | Ψ E H|\Psi_{E}\rangle=E|\Psi_{E}\rangle
  22. | Ψ E Ψ E | |\Psi_{E}\rangle\langle\Psi_{E}|
  23. | Ψ |\Psi\rangle
  24. | Ψ |\Psi\rangle
  25. I = d E | Ψ E Ψ E | I=\int dE|\Psi_{E}\rangle\langle\Psi_{E}|
  26. H eff = H - i Γ H_{\,\text{eff}}=H-i\Gamma
  27. - i Γ -i\Gamma
  28. H eff * | Ψ E * = E * | Ψ E * H^{*}_{\,\text{eff}}|\Psi_{E}^{*}\rangle=E^{*}|\Psi_{E}^{*}\rangle
  29. f ( H eff ) = d E | Ψ E f ( E ) Ψ E * | f(H_{\,\text{eff}})=\int dE|\Psi_{E}\rangle f(E)\langle\Psi_{E}^{*}|
  30. W ( A ) : ran ( A + i ) ran ( A - i ) \operatorname{W}(A):\operatorname{ran}(A+i)\to\operatorname{ran}(A-i)
  31. W ( A ) ( A x + i x ) = A x - i x , x dom ( A ) . \operatorname{W}(A)(Ax+ix)=Ax-ix,\qquad x\in\operatorname{dom}(A).
  32. S ( U ) : ran ( 1 - U ) ran ( 1 + U ) \operatorname{S}(U):\operatorname{ran}(1-U)\to\operatorname{ran}(1+U)
  33. S ( U ) ( x - U x ) = i ( x + U x ) x dom ( U ) . \operatorname{S}(U)(x-Ux)=i(x+Ux)\qquad x\in\operatorname{dom}(U).
  34. n + ( V ) = dim dom ( V ) n_{+}(V)=\operatorname{dim}\ \operatorname{dom}(V)^{\perp}
  35. n - ( V ) = dim ran ( V ) n_{-}(V)=\operatorname{dim}\ \operatorname{ran}(V)^{\perp}
  36. V ( x ) = - ( 1 + | x | ) α V(x)=-(1+|x|)^{\alpha}
  37. α > 2 α>2
  38. p 2 p^{2}
  39. N ± = ran ( A ± i ) , N_{\pm}=\operatorname{ran}(A\pm i)^{\perp},
  40. N ± = ker ( A * i ) , N_{\pm}=\operatorname{ker}(A^{*}\mp i),
  41. dom ( A * ) = dom ( A ¯ ) N + N - , \operatorname{dom}(A^{*})=\operatorname{dom}(\overline{A})\oplus N_{+}\oplus N% _{-},
  42. ξ | η graph = ξ | η + A * ξ | A * η \langle\xi|\eta\rangle_{\mathrm{graph}}=\langle\xi|\eta\rangle+\langle A^{*}% \xi|A^{*}\eta\rangle
  43. D : ϕ 1 i ϕ D:\phi\mapsto\frac{1}{i}\phi^{\prime}
  44. u = i u u^{\prime}=iu
  45. u = - i u u^{\prime}=-iu
  46. N ± = { u L 2 ( M ) : P dist u = ± i u } N_{\pm}=\left\{u\in L^{2}(M):P_{\operatorname{dist}}u=\pm iu\right\}
  47. P ( x ) = α c α x α P(\vec{x})=\sum_{\alpha}c_{\alpha}x^{\alpha}
  48. α = ( α 1 , α 2 , , α n ) \alpha=(\alpha_{1},\alpha_{2},\ldots,\alpha_{n})
  49. x α = x 1 α 1 x 2 α 2 x n α n . x^{\alpha}=x_{1}^{\alpha_{1}}x_{2}^{\alpha_{2}}\cdots x_{n}^{\alpha_{n}}.
  50. D α = 1 i | α | x 1 α 1 x 2 α 2 x n α n . D^{\alpha}=\frac{1}{i^{|\alpha|}}\partial_{x_{1}}^{\alpha_{1}}\partial_{x_{2}}% ^{\alpha_{2}}\cdots\partial_{x_{n}}^{\alpha_{n}}.
  51. P ( D ) ϕ = α c α D α ϕ P(\operatorname{D})\phi=\sum_{\alpha}c_{\alpha}\operatorname{D}^{\alpha}\phi
  52. P ϕ ( x ) = α a α ( x ) [ D α ϕ ] ( x ) P\phi(x)=\sum_{\alpha}a_{\alpha}(x)[D^{\alpha}\phi](x)
  53. C 0 ( M ) C 0 ( M ) . C_{0}^{\infty}(M)\to C_{0}^{\infty}(M).
  54. P * form ϕ = α D α ( a α ¯ ϕ ) P^{\mathrm{*form}}\phi=\sum_{\alpha}D^{\alpha}(\overline{a_{\alpha}}\phi)
  55. dom P * = { u L 2 ( M ) : P * form u L 2 ( M ) } . \operatorname{dom}P^{*}=\left\{u\in L^{2}(M):P^{\mathrm{*form}}u\in L^{2}(M)% \right\}.
  56. L μ 2 ( 𝐑 , 𝐇 n ) = { ψ : 𝐑 𝐇 n : ψ measurable and 𝐑 ψ ( t ) 2 d μ ( t ) < } L^{2}_{\mu}(\mathbf{R},\mathbf{H}_{n})=\{\psi:\mathbf{R}\to\mathbf{H}_{n}:\psi% \mbox{ measurable and }~{}\int_{\mathbf{R}}\|\psi(t)\|^{2}d\mu(t)<\infty\}
  57. 𝐑 | λ | 2 ψ ( λ ) 2 d μ ( λ ) < . \int_{\mathbf{R}}|\lambda|^{2}\ \|\psi(\lambda)\|^{2}\,d\mu(\lambda)<\infty.
  58. { μ } 1 ω \{\mu_{\ell}\}_{1\leq\ell\leq\omega}
  59. 1 ω L μ 2 ( 𝐑 , 𝐇 ) . \bigoplus_{1\leq\ell\leq\omega}L^{2}_{\mu_{\ell}}\left(\mathbf{R},\mathbf{H}_{% \ell}\right).
  60. 𝐑 H x d μ ( x ) . \int_{\mathbf{R}}^{\oplus}H_{x}d\mu(x).
  61. Δ = i = 1 n x i 2 . \Delta=\sum_{i=1}^{n}\partial_{x_{i}}^{2}.
  62. - Δ + | x | 2 . -\Delta+|x|^{2}.

Self-clocking_signal.html

  1. M ( t ) M(t)
  2. y ( t ) = M ( t ) cos ( ω c t ) , y(t)=M(t)\cdot\cos(\omega_{c}t),

Self-organizing_map.html

  1. s s
  2. λ \lambda
  3. t t
  4. 𝐃 \mathbf{D}
  5. 𝐃 ( 𝐭 ) \mathbf{D(t)}
  6. v v
  7. 𝐖 𝐯 \mathbf{W_{v}}
  8. u u
  9. Θ ( u , v , s ) \Theta(u,v,s)
  10. α ( s ) \alpha(s)
  11. 𝐃 ( 𝐭 ) \mathbf{D(t)}
  12. s < λ s<\lambda
  13. s < λ s<\lambda

Sellmeier_equation.html

  1. n 2 ( λ ) = 1 + B 1 λ 2 λ 2 - C 1 + B 2 λ 2 λ 2 - C 2 + B 3 λ 2 λ 2 - C 3 , n^{2}(\lambda)=1+\frac{B_{1}\lambda^{2}}{\lambda^{2}-C_{1}}+\frac{B_{2}\lambda% ^{2}}{\lambda^{2}-C_{2}}+\frac{B_{3}\lambda^{2}}{\lambda^{2}-C_{3}},
  2. n 2 ( λ ) = 1 + i B i λ 2 λ 2 - C i , n^{2}(\lambda)=1+\sum_{i}\frac{B_{i}\lambda^{2}}{\lambda^{2}-C_{i}},
  3. n 1 + i B i ε r , \begin{matrix}n\approx\sqrt{1+\sum_{i}B_{i}}\approx\sqrt{\varepsilon_{r}}\end{% matrix},
  4. n 2 ( λ ) = A + B 1 λ 2 λ 2 - C 1 + B 2 λ 2 λ 2 - C 2 . n^{2}(\lambda)=A+\frac{B_{1}\lambda^{2}}{\lambda^{2}-C_{1}}+\frac{B_{2}\lambda% ^{2}}{\lambda^{2}-C_{2}}.

Sensible_heat.html

  1. Δ T \Delta T
  2. Q s e n s i b l e = m c Δ T . Q_{sensible}=mc\Delta T\,.

Sequent.html

  1. A 1 , , A m B 1 , , B n . A_{1},\,\dots,A_{m}\,\vdash\,B_{1},\,\dots,B_{n}.
  2. Γ Σ \Gamma\vdash\Sigma
  3. \vdash
  4. Γ , α Σ Γ α Γ Σ \frac{\Gamma,\alpha\vdash\Sigma\qquad\Gamma\vdash\alpha}{\Gamma\vdash\Sigma}
  5. Γ , α \Gamma,\alpha
  6. Σ \Sigma
  7. Γ \Gamma
  8. α \alpha
  9. Γ \Gamma
  10. Σ \Sigma
  11. Γ Σ \Gamma\vdash\Sigma
  12. Γ \Gamma\vdash
  13. Σ \vdash\Sigma
  14. Γ Σ \Gamma\vdash\Sigma
  15. Γ \Gamma
  16. Σ \Sigma
  17. Σ \Sigma
  18. \vdash

Sequent_calculus.html

  1. B B\,
  2. B B
  3. A 1 , A 2 , , A n B A_{1},A_{2},\ldots,A_{n}\vdash B
  4. A i A_{i}
  5. B B
  6. n 0 n\geq 0
  7. A i A_{i}
  8. \vdash
  9. B B
  10. B \vdash B
  11. A i A_{i}
  12. A 1 A_{1}
  13. A 2 A_{2}
  14. B B
  15. A 1 , , A n B A_{1},\ldots,A_{n}\vdash B
  16. ( A 1 A n ) B \vdash(A_{1}\land\cdots\land A_{n})\rightarrow B
  17. A 1 , , A n B 1 , , B k , A_{1},\ldots,A_{n}\vdash B_{1},\ldots,B_{k},
  18. A i A_{i}
  19. B i B_{i}
  20. n n
  21. k k
  22. B B
  23. B \vdash B
  24. A i A_{i}
  25. B i B_{i}
  26. A 1 , , A n B 1 , , B k A_{1},\ldots,A_{n}\vdash B_{1},\ldots,B_{k}
  27. ( A 1 A n ) ( B 1 B k ) \vdash(A_{1}\land\cdots\land A_{n})\rightarrow(B_{1}\lor\cdots\lor B_{k})
  28. ¬ A 1 ¬ A 2 ¬ A n B 1 B 2 B k \vdash\neg A_{1}\lor\neg A_{2}\lor\cdots\lor\neg A_{n}\lor B_{1}\lor B_{2}\lor% \cdots\lor B_{k}
  29. ¬ ( A 1 A 2 A n ¬ B 1 ¬ B 2 ¬ B k ) \vdash\neg(A_{1}\land A_{2}\land\cdots\land A_{n}\land\neg B_{1}\land\neg B_{2% }\land\cdots\land\neg B_{k})
  30. \vdash
  31. A A
  32. B B
  33. Γ , Δ , Σ \Gamma,\Delta,\Sigma
  34. Π \Pi
  35. \vdash
  36. \vdash
  37. t t
  38. x x
  39. y y
  40. \forall
  41. \exists
  42. A [ t / x ] A[t/x]
  43. t t
  44. x x
  45. A A
  46. t t
  47. x x
  48. A A
  49. t t
  50. A [ t / x ] A[t/x]
  51. W L WL
  52. W R WR
  53. C L CL
  54. C R CR
  55. P L PL
  56. P R PR
  57. A A ( I ) \cfrac{\qquad}{A\vdash A}\quad(I)
  58. Γ Δ , A A , Σ Π Γ , Σ Δ , Π ( 𝐶𝑢𝑡 ) \cfrac{\Gamma\vdash\Delta,A\qquad A,\Sigma\vdash\Pi}{\Gamma,\Sigma\vdash\Delta% ,\Pi}\quad(\mathit{Cut})
  59. Γ , A Δ Γ , A and B Δ ( and L 1 ) \cfrac{\Gamma,A\vdash\Delta}{\Gamma,A\and B\vdash\Delta}\quad({\and}L_{1})
  60. Γ A , Δ Γ A B , Δ ( R 1 ) \cfrac{\Gamma\vdash A,\Delta}{\Gamma\vdash AB,\Delta}\quad({}R_{1})
  61. Γ , B Δ Γ , A and B Δ ( and L 2 ) \cfrac{\Gamma,B\vdash\Delta}{\Gamma,A\and B\vdash\Delta}\quad({\and}L_{2})
  62. Γ B , Δ Γ A B , Δ ( R 2 ) \cfrac{\Gamma\vdash B,\Delta}{\Gamma\vdash AB,\Delta}\quad({}R_{2})
  63. Γ , A Δ Σ , B Π Γ , Σ , A B Δ , Π ( L ) \cfrac{\Gamma,A\vdash\Delta\qquad\Sigma,B\vdash\Pi}{\Gamma,\Sigma,AB\vdash% \Delta,\Pi}\quad({}L)
  64. Γ A , Δ Σ B , Π Γ , Σ A and B , Δ , Π ( and R ) \cfrac{\Gamma\vdash A,\Delta\qquad\Sigma\vdash B,\Pi}{\Gamma,\Sigma\vdash A% \and B,\Delta,\Pi}\quad({\and}R)
  65. Γ A , Δ Σ , B Π Γ , Σ , A B Δ , Π ( L ) \cfrac{\Gamma\vdash A,\Delta\qquad\Sigma,B\vdash\Pi}{\Gamma,\Sigma,A% \rightarrow B\vdash\Delta,\Pi}\quad({\rightarrow}L)
  66. Γ , A B , Δ Γ A B , Δ ( R ) \cfrac{\Gamma,A\vdash B,\Delta}{\Gamma\vdash A\rightarrow B,\Delta}\quad({% \rightarrow}R)
  67. Γ A , Δ Γ , ¬ A Δ ( ¬ L ) \cfrac{\Gamma\vdash A,\Delta}{\Gamma,\lnot A\vdash\Delta}\quad({\lnot}L)
  68. Γ , A Δ Γ ¬ A , Δ ( ¬ R ) \cfrac{\Gamma,A\vdash\Delta}{\Gamma\vdash\lnot A,\Delta}\quad({\lnot}R)
  69. Γ , A [ t / x ] Δ Γ , x A Δ ( L ) \cfrac{\Gamma,A[t/x]\vdash\Delta}{\Gamma,\forall xA\vdash\Delta}\quad({\forall% }L)
  70. Γ A [ y / x ] , Δ Γ x A , Δ ( R ) \cfrac{\Gamma\vdash A[y/x],\Delta}{\Gamma\vdash\forall xA,\Delta}\quad({% \forall}R)
  71. Γ , A [ y / x ] Δ Γ , x A Δ ( L ) \cfrac{\Gamma,A[y/x]\vdash\Delta}{\Gamma,\exists xA\vdash\Delta}\quad({\exists% }L)
  72. Γ A [ t / x ] , Δ Γ x A , Δ ( R ) \cfrac{\Gamma\vdash A[t/x],\Delta}{\Gamma\vdash\exists xA,\Delta}\quad({% \exists}R)
  73. Γ Δ Γ , A Δ ( 𝑊𝐿 ) \cfrac{\Gamma\vdash\Delta}{\Gamma,A\vdash\Delta}\quad(\mathit{WL})
  74. Γ Δ Γ A , Δ ( 𝑊𝑅 ) \cfrac{\Gamma\vdash\Delta}{\Gamma\vdash A,\Delta}\quad(\mathit{WR})
  75. Γ , A , A Δ Γ , A Δ ( 𝐶𝐿 ) \cfrac{\Gamma,A,A\vdash\Delta}{\Gamma,A\vdash\Delta}\quad(\mathit{CL})
  76. Γ A , A , Δ Γ A , Δ ( 𝐶𝑅 ) \cfrac{\Gamma\vdash A,A,\Delta}{\Gamma\vdash A,\Delta}\quad(\mathit{CR})
  77. Γ 1 , A , B , Γ 2 Δ Γ 1 , B , A , Γ 2 Δ ( 𝑃𝐿 ) \cfrac{\Gamma_{1},A,B,\Gamma_{2}\vdash\Delta}{\Gamma_{1},B,A,\Gamma_{2}\vdash% \Delta}\quad(\mathit{PL})
  78. Γ Δ 1 , A , B , Δ 2 Γ Δ 1 , B , A , Δ 2 ( 𝑃𝑅 ) \cfrac{\Gamma\vdash\Delta_{1},A,B,\Delta_{2}}{\Gamma\vdash\Delta_{1},B,A,% \Delta_{2}}\quad(\mathit{PR})
  79. ( R ) ({\forall}R)
  80. ( L ) ({\exists}L)
  81. y y
  82. Γ \Gamma
  83. Δ \Delta
  84. y y
  85. \vdash
  86. ( and L 1 ) ({\and}L_{1})
  87. Δ \Delta
  88. A A
  89. Δ \Delta
  90. A and B A\and B
  91. ( ¬ R ) ({\neg}R)
  92. Γ \Gamma
  93. Δ \Delta
  94. Δ \Delta
  95. ¬ A {\neg}A
  96. ( R ) ({\forall}R)
  97. x A \forall{x}A
  98. A [ y / x ] A[y/x]
  99. A [ y / x ] A[y/x]
  100. ( and R ) ({\and}R)
  101. A and B A\and B
  102. Γ \Gamma
  103. Σ \Sigma
  104. Γ \Gamma
  105. Σ \Sigma
  106. Γ \Gamma
  107. Σ \Sigma
  108. ↚ \not\leftarrow
  109. A ¬ A \vdash A\lnot A
  110. ( I ) (I)
  111. A A A\vdash A
  112. ( ¬ R ) (\lnot R)
  113. ¬ A , A \vdash\lnot A,A
  114. ( R 2 ) (R_{2})
  115. A ¬ A , A \vdash A\lnot A,A
  116. ( P R ) (PR)
  117. A , A ¬ A \vdash A,A\lnot A
  118. ( R 1 ) (R_{1})
  119. A ¬ A , A ¬ A \vdash A\lnot A,A\lnot A
  120. ( C R ) (CR)
  121. A ¬ A \vdash A\lnot A
  122. ( R ) (\forall R)
  123. ( L ) (\exists L)
  124. ( I ) (I)
  125. p ( x , y ) p ( x , y ) p(x,y)\vdash p(x,y)
  126. ( L ) (\forall L)
  127. x ( p ( x , y ) ) p ( x , y ) \forall x\left(p(x,y)\right)\vdash p(x,y)
  128. ( R ) (\exists R)
  129. x ( p ( x , y ) ) y ( p ( x , y ) ) \forall x\left(p(x,y)\right)\vdash\exists y\left(p(x,y)\right)
  130. ( L ) (\exists L)
  131. y ( x ( p ( x , y ) ) ) y ( p ( x , y ) ) \exists y\left(\forall x\left(p(x,y)\right)\right)\vdash\exists y\left(p(x,y)\right)
  132. ( R ) (\forall R)
  133. y ( x ( p ( x , y ) ) ) x ( y ( p ( x , y ) ) ) \exists y\left(\forall x\left(p(x,y)\right)\right)\vdash\forall x\left(\exists y% \left(p(x,y)\right)\right)
  134. ( ( A ( B C ) ) ( ( ( B ¬ A ) and ¬ C ) ¬ A ) ) \left(\left(A\rightarrow\left(BC\right)\right)\rightarrow\left(\left(\left(B% \rightarrow\lnot A\right)\and\lnot C\right)\rightarrow\lnot A\right)\right)
  135. ( I ) (I)
  136. A A A\vdash A
  137. ( ¬ R ) (\lnot R)
  138. ¬ A , A \vdash\lnot A,A
  139. ( P R ) (PR)
  140. A , ¬ A \vdash A,\lnot A
  141. ( I ) (I)
  142. B B B\vdash B
  143. ( I ) (I)
  144. C C C\vdash C
  145. ( L ) (L)
  146. B C B , C BC\vdash B,C
  147. ( P R ) (PR)
  148. B C C , B BC\vdash C,B
  149. ( ¬ L ) (\lnot L)
  150. B C , ¬ C B BC,\lnot C\vdash B
  151. ( I ) (I)
  152. ¬ A ¬ A \lnot A\vdash\lnot A
  153. ( L ) (\rightarrow L)
  154. ( B C ) , ¬ C , ( B ¬ A ) ¬ A \left(BC\right),\lnot C,\left(B\rightarrow\lnot A\right)\vdash\lnot A
  155. ( and L 1 ) (\and L_{1})
  156. ( B C ) , ¬ C , ( ( B ¬ A ) and ¬ C ) ¬ A \left(BC\right),\lnot C,\left(\left(B\rightarrow\lnot A\right)\and\lnot C% \right)\vdash\lnot A
  157. ( P L ) (PL)
  158. ( B C ) , ( ( B ¬ A ) and ¬ C ) , ¬ C ¬ A \left(BC\right),\left(\left(B\rightarrow\lnot A\right)\and\lnot C\right),\lnot C% \vdash\lnot A
  159. ( and L 2 ) (\and L_{2})
  160. ( B C ) , ( ( B ¬ A ) and ¬ C ) , ( ( B ¬ A ) and ¬ C ) ¬ A \left(BC\right),\left(\left(B\rightarrow\lnot A\right)\and\lnot C\right),\left% (\left(B\rightarrow\lnot A\right)\and\lnot C\right)\vdash\lnot A
  161. ( C L ) (CL)
  162. ( B C ) , ( ( B ¬ A ) and ¬ C ) ¬ A \left(BC\right),\left(\left(B\rightarrow\lnot A\right)\and\lnot C\right)\vdash\lnot A
  163. ( P L ) (PL)
  164. ( ( B ¬ A ) and ¬ C ) , ( B C ) ¬ A \left(\left(B\rightarrow\lnot A\right)\and\lnot C\right),\left(BC\right)\vdash\lnot A
  165. ( L ) (\rightarrow L)
  166. ( ( B ¬ A ) and ¬ C ) , ( A ( B C ) ) ¬ A , ¬ A \left(\left(B\rightarrow\lnot A\right)\and\lnot C\right),\left(A\rightarrow% \left(BC\right)\right)\vdash\lnot A,\lnot A
  167. ( C R ) (CR)
  168. ( ( B ¬ A ) and ¬ C ) , ( A ( B C ) ) ¬ A \left(\left(B\rightarrow\lnot A\right)\and\lnot C\right),\left(A\rightarrow% \left(BC\right)\right)\vdash\lnot A
  169. ( P L ) (PL)
  170. ( A ( B C ) ) , ( ( B ¬ A ) and ¬ C ) ¬ A \left(A\rightarrow\left(BC\right)\right),\left(\left(B\rightarrow\lnot A\right% )\and\lnot C\right)\vdash\lnot A
  171. ( R ) (\rightarrow R)
  172. ( A ( B C ) ) ( ( ( B ¬ A ) and ¬ C ) ¬ A ) \left(A\rightarrow\left(BC\right)\right)\vdash\left(\left(\left(B\rightarrow% \lnot A\right)\and\lnot C\right)\rightarrow\lnot A\right)
  173. ( R ) (\rightarrow R)
  174. ( ( A ( B C ) ) ( ( ( B ¬ A ) and ¬ C ) ¬ A ) ) \vdash\left(\left(A\rightarrow\left(BC\right)\right)\rightarrow\left(\left(% \left(B\rightarrow\lnot A\right)\and\lnot C\right)\rightarrow\lnot A\right)\right)
  175. A A\,
  176. Γ \Gamma\,
  177. ( Γ A ) (\Gamma\vDash A)
  178. Γ A \Gamma\vdash A
  179. Γ , A A , Δ \Gamma,A\vdash A,\Delta
  180. A A
  181. A A
  182. ( and R ) , ( L ) ({\and}R),({}L)
  183. ( L ) ({\rightarrow}L)
  184. Γ \Gamma
  185. Σ \Sigma
  186. \bot
  187. \cfrac{}{\bot\vdash\quad}
  188. Γ , Δ \cfrac{}{\Gamma,\bot\vdash\Delta}
  189. \bot
  190. ¬ A A \neg A\iff A\to\bot
  191. ( L ) ({}L)
  192. Γ , A C Σ , B C Γ , Σ , A B C ( L ) \cfrac{\Gamma,A\vdash C\qquad\Sigma,B\vdash C}{\Gamma,\Sigma,AB\vdash C}\quad(% {}L)
  193. ( R ) ({\to}R)
  194. ( ¬ R ) (\neg R)
  195. \bot
  196. Γ , A B C Γ ( A B ) C \cfrac{\Gamma,A\vdash BC}{\Gamma\vdash(A\to B)C}
  197. ( A ( B C ) ) ( ( A B ) C ) (A\to(BC))\to((A\to B)C)

Series_and_parallel_circuits.html

  1. I = I 1 = I 2 = = I n I=I_{1}=I_{2}=\dots=I_{n}
  2. R total = R 1 + R 2 + + R n R_{\mathrm{total}}=R_{1}+R_{2}+\cdots+R_{n}
  3. 1 G total = 1 G 1 + 1 G 2 + + 1 G n \frac{1}{G_{\mathrm{total}}}=\frac{1}{G_{1}}+\frac{1}{G_{2}}+\cdots+\frac{1}{G% _{n}}
  4. G t o t a l = G 1 G 2 G 1 + G 2 . G_{total}=\frac{G_{1}G_{2}}{G_{1}+G_{2}}.
  5. L total = L 1 + L 2 + + L n L_{\mathrm{total}}=L_{1}+L_{2}+\cdots+L_{n}
  6. M 12 M_{12}
  7. M 13 M_{13}
  8. M 23 M_{23}
  9. M 21 M_{21}
  10. M 31 M_{31}
  11. M 32 M_{32}
  12. M 11 M_{11}
  13. M 22 M_{22}
  14. M 33 M_{33}
  15. L total = ( M 11 + M 22 + M 33 ) + ( M 12 + M 13 + M 23 ) + ( M 21 + M 31 + M 32 ) L_{\mathrm{total}}=(M_{11}+M_{22}+M_{33})+(M_{12}+M_{13}+M_{23})+(M_{21}+M_{31% }+M_{32})
  16. M i j M_{ij}
  17. M j i M_{ji}
  18. 1 C total = 1 C 1 + 1 C 2 + + 1 C n \frac{1}{C_{\mathrm{total}}}=\frac{1}{C_{1}}+\frac{1}{C_{2}}+\cdots+\frac{1}{C% _{n}}
  19. V = V 1 = V 2 = = V n V=V_{1}=V_{2}=\ldots=V_{n}
  20. I total = V ( 1 R 1 + 1 R 2 + + 1 R n ) I_{\mathrm{total}}=V\left(\frac{1}{R_{1}}+\frac{1}{R_{2}}+\cdots+\frac{1}{R_{n% }}\right)
  21. R i R_{i}
  22. 1 R total = 1 R 1 + 1 R 2 + + 1 R n \frac{1}{R_{\mathrm{total}}}=\frac{1}{R_{1}}+\frac{1}{R_{2}}+\cdots+\frac{1}{R% _{n}}
  23. R total = R 1 R 2 R 1 + R 2 . R_{\mathrm{total}}=\frac{R_{1}R_{2}}{R_{1}+R_{2}}.
  24. 1 R total = 1 R × N \frac{1}{R_{\mathrm{total}}}=\frac{1}{R}\times N
  25. R total = R N {R_{\mathrm{total}}}=\frac{R}{N}
  26. R i R_{i}
  27. I i = V R i I_{i}=\frac{V}{R_{i}}\,
  28. I 1 I 2 = R 2 R 1 \frac{I_{1}}{I_{2}}=\frac{R_{2}}{R_{1}}
  29. G G
  30. G total = G 1 + G 2 + + G n {G_{\mathrm{total}}}={G_{1}}+{G_{2}}+\cdots+{G_{n}}
  31. 1 L total = 1 L 1 + 1 L 2 + + 1 L n \frac{1}{L_{\mathrm{total}}}=\frac{1}{L_{1}}+\frac{1}{L_{2}}+\cdots+\frac{1}{L% _{n}}
  32. 1 L total = L 1 + L 2 - 2 M L 1 L 2 - M 2 \frac{1}{L_{\mathrm{total}}}=\frac{L_{1}+L_{2}-2M}{L_{1}L_{2}-M^{2}}
  33. L 1 = L 2 L_{1}=L_{2}
  34. L t o t a l = L + M 2 L_{total}=\frac{L+M}{2}
  35. M M
  36. M 12 M_{12}
  37. M 13 M_{13}
  38. M 23 M_{23}
  39. L L
  40. v i = j L i , j d i j d t v_{i}=\sum_{j}L_{i,j}\frac{di_{j}}{dt}
  41. C total = C 1 + C 2 + + C n C_{\mathrm{total}}=C_{1}+C_{2}+\cdots+C_{n}
  42. G 1 G_{1}
  43. G 2 G_{2}
  44. I E q = I 1 + I 2 . I_{Eq}=I_{1}+I_{2}.\ \,
  45. G E q V = G 1 V + G 2 V G_{Eq}V=G_{1}V+G_{2}V\ \,
  46. G E q = G 1 + G 2 . G_{Eq}=G_{1}+G_{2}.\ \,
  47. G 1 G_{1}
  48. G 2 G_{2}
  49. V E q = V 1 + V 2 . V_{Eq}=V_{1}+V_{2}.\ \,
  50. I G E q = I G 1 + I G 2 \frac{I}{G_{Eq}}=\frac{I}{G_{1}}+\frac{I}{G_{2}}
  51. 1 G E q = 1 G 1 + 1 G 2 . \frac{1}{G_{Eq}}=\frac{1}{G_{1}}+\frac{1}{G_{2}}.
  52. G E q = G 1 G 2 G 1 + G 2 . G_{Eq}=\frac{G_{1}G_{2}}{G_{1}+G_{2}}.
  53. R eq = R 1 R 2 = R 1 R 2 R 1 + R 2 R_{\mathrm{eq}}=R_{1}\|R_{2}={R_{1}R_{2}\over R_{1}+R_{2}}
  54. R 1 R 2 R 3 R_{1}\|R_{2}\|R_{3}
  55. R 1 R 2 R 3 R 1 R 2 + R 1 R 3 + R 2 R 3 \frac{R_{1}R_{2}R_{3}}{R_{1}R_{2}+R_{1}R_{3}+R_{2}R_{3}}

Set-builder_notation.html

  1. { 7 , 3 , 15 , 31 } \{7,3,15,31\}
  2. { a , c , b } \{a,c,b\}
  3. { 1 , 2 , 3 , , 100 } \{1,2,3,\ldots,100\}
  4. { 0 , 1 , 2 , } \{0,1,2,\ldots\}
  5. { \{
  6. } \}
  7. { x | Φ ( x ) } \{x~{}|~{}\Phi(x)\}
  8. { x : Φ ( x ) } \{x\,:\,\Phi(x)\}
  9. { x | x 𝐑 x = x 2 } \{x~{}|~{}x\in\mathbf{R}\land x=x^{2}\}\,\!
  10. { 0 , 1 } \{0,1\}
  11. { x | x 𝐑 x > 0 } \{x~{}|~{}x\in\mathbf{R}\land x>0\}
  12. { ( x , y ) | 0 < y < f ( x ) } \{(x,y)~{}|~{}0<y<f(x)\}
  13. { k | n : ( n 𝐍 k = 2 n ) } \{k~{}|~{}\exists n:(n\in\mathbf{N}\land k=2n)\}
  14. { a | p , q : ( p , q 𝐙 q 0 a q = p ) } \{a~{}|~{}\exists p,q:(p,q\in\mathbf{Z}\land q\not=0\land aq=p)\}
  15. 𝐍 m = { x | x 𝐙 x m } = { m , m + 1 , m + 2 , } , \mathbf{N}_{m}=\{x~{}|~{}x\in\mathbf{Z}\land x\geq m\}=\{m,m+1,m+2,\ldots\},
  16. x 1 = m , x 2 = m + 1 x_{1}=m,\ x_{2}=m+1
  17. x 1 = m + 2 x_{1}=m+2
  18. 𝐍 3 = { x | x 𝐙 x 3 } = { 3 , 4 , 5 , } . \mathbf{N}_{3}=\{x~{}|~{}x\in\mathbf{Z}\land x\geq 3\}=\{3,4,5,\ldots\}.
  19. and \and
  20. $\or$
  21. and \and
  22. { 2 n : n N } \{2n:n\in N\}
  23. { p / q : p , q Z , q 0 } \{p/q:p,q\in Z,q\not=0\}
  24. { 2 t + 1 | t 𝐙 } \{2t+1~{}|~{}t\in\mathbf{Z}\}
  25. { ( t , 2 t + 1 ) | t 𝐙 } \{(t,2t+1)~{}|~{}t\in\mathbf{Z}\}
  26. { 2 x + 1 = 5 | x 𝐍 } \{2x+1=5~{}|~{}x\in\mathbf{N}\}
  27. { t r u e , f a l s e } \{true,false\}
  28. 2 x + 1 = 5 2x+1=5
  29. { x R | x 𝐂 and n 𝐍 and x = π n } \{x\in R~{}|~{}x\in\mathbf{C}\and n\in\mathbf{N}\and x=\pi n\}
  30. { t r u e } \{true\}
  31. { 2 t + 1 | t 𝐙 } = { u | ( u - 1 ) / 2 𝐙 } \{2t+1~{}|~{}t\in\mathbf{Z}\}=\{u~{}|~{}(u-1)/2\in\mathbf{Z}\}
  32. { x 𝐑 | x = x 2 } \{x\in\mathbf{R}~{}|~{}x=x^{2}\}\,\!
  33. { 0 , 1 } \{0,1\}
  34. { x | x 𝐑 and x = x 2 } \{x~{}|~{}x\in\mathbf{R}\and x=x^{2}\}\,\!
  35. { 0 , 1 } \{0,1\}
  36. { x | x = x 2 } \{x~{}|~{}x=x^{2}\}\,\!
  37. { 0 , 1 } \{0,1\}
  38. { x | x 𝐑 and x 2 = 1 } = { x | x 𝐑 and | x | = 1 } \{x|x\in\mathbf{R}\and x^{2}=1\}=\{x|x\in\mathbf{R}\and\ |x|=1\}
  39. ( x 𝐑 and x 2 = 1 ) ( x 𝐑 and | x | = 1 ) (x\in\mathbf{R}\and x^{2}=1)\Leftrightarrow(x\in\mathbf{R}\and\ |x|=1)
  40. A = { x | P ( x ) } A=\{x|P(x)\}
  41. B = { x | Q ( x ) } B=\{x|Q(x)\}
  42. t : P ( t ) Q ( t ) \forall t:P(t)\Leftrightarrow Q(t)
  43. ( x : P ( x ) Q ( x ) ) ( { x | P ( x ) } = { x | Q ( x ) } ) (\forall_{x}:P(x)\Leftrightarrow Q(x))\Leftrightarrow(\{x|P(x)\}=\{x|Q(x)\})
  44. R = { S | S S } R=\{S~{}|~{}S\notin S\}
  45. { x : A | P ( x ) } \{x:A\ |\ P(x)\}
  46. ( x : A ) (x:A)
  47. ( x A ) (x\in A)
  48. { x : A | P ( x ) F ( x ) } \{x:A\ |\ P(x)\bullet F(x)\}
  49. { l | l L } \{l\ |\ l\in L\}
  50. { ( k , x ) | k K x X P ( x ) } \{(k,x)\ |\ k\in K\wedge x\in X\wedge P(x)\}

Set_(abstract_data_type).html

  1. A A
  2. 2 A 2^{A}
  3. 𝒫 ( A ) \mathcal{P}(A)
  4. F F
  5. S S
  6. F ( x ) = { 1 , if x S 0 , if x S F(x)=\begin{cases}1,&\mbox{if }~{}x\in S\\ 0,&\mbox{if }~{}x\not\in S\end{cases}

Settling.html

  1. w = 2 ( ρ p - ρ f ) g r 2 9 μ w=\frac{2(\rho_{p}-\rho_{f})gr^{2}}{9\mu}
  2. R e 0.1 Re\leq 0.1
  3. R e 0.5 Re\leq 0.5
  4. R e 1.0 Re\leq 1.0
  5. C d C_{d}
  6. C d = F d 1 2 ρ f U 2 A C_{d}=\frac{F_{d}}{\frac{1}{2}\rho_{f}U^{2}A}
  7. w = 2.46 ( ( ρ p - ρ f ) g r ρ f ) 1 2 . w=2.46\left(\frac{(\rho_{p}-\rho_{f})gr}{\rho_{f}}\right)^{\frac{1}{2}}.
  8. 0.2 R e 1000 0.2\leq Re\leq 1000
  9. F ρ f U 2 A = 12 Re ( 1 + 0.15 Re 0.687 ) . \frac{F}{\rho_{f}U^{2}A}=\frac{12}{\mathrm{Re}}\left(1+0.15\mathrm{Re}^{0.687}% \right).

Seventh_chord.html

  1. 7 {}^{7}
  2. 5 6 {}^{6}_{5}
  3. 3 4 {}^{4}_{3}

Sexagesimal.html

  1. 8 , 34 , 17 ¯ \overline{8,34,17}
  2. 5 , 27 , 16 , 21 , 49 ¯ \overline{5,27,16,21,49}
  3. 4 , 36 , 55 , 23 ¯ \overline{4,36,55,23}
  4. 17 , 8 , 34 ¯ \overline{17,8,34}
  5. 3 , 31 , 45 , 52 , 56 , 28 , 14 , 7 ¯ \overline{3,31,45,52,56,28,14,7}
  6. 3 , 9 , 28 , 25 , 15 , 47 , 22 , 6 , 18 , 56 , 50 , 31 , 34 , 44 , 12 , 37 , 53 , 41 ¯ \overline{3,9,28,25,15,47,22,6,18,56,50,31,34,44,12,37,53,41}
  7. 1 ¯ \overline{1}
  8. 0 , 59 ¯ \overline{0,59}
  9. 1 ; 24 , 51 , 10 = 1 + 24 60 + 51 60 2 + 10 60 3 = 30547 21600 1.41421296 1;24,51,10=1+\frac{24}{60}+\frac{51}{60^{2}}+\frac{10}{60^{3}}=\frac{30547}{21% 600}\approx 1.41421296\ldots
  10. 2 1.41421356 \sqrt{2}\approx 1.41421356\ldots

Seyfert_galaxy.html

  1. F g r a v = G M B H m p r 2 F_{grav}=\frac{GM_{BH}m_{p}}{r^{2}}
  2. F r a d = d p d t = 1 c d E d t = 1 c σ t L 4 π r 2 F_{rad}=\frac{dp}{dt}=\frac{1}{c}\frac{dE}{dt}=\frac{1}{c}\sigma_{t}\frac{L}{4% \pi r^{2}}
  3. F r a d = F g r a v L < L E d d i n g t o n F_{rad}=F_{grav}\rightarrow L<L_{Eddington}
  4. = 4 π c G M B H m p σ t =\frac{4\pi cGM_{BH}m_{p}}{\sigma_{t}}
  5. = 1.3 × 10 38 M B H M s o l a r e r g / s e c =1.3\times 10^{38}\frac{M_{BH}}{M_{solar}}\,erg/sec
  6. = 30000 M B H M s o l a r L s o l a r =30000\frac{M_{BH}}{M_{solar}}L_{solar}

Shannon–Hartley_theorem.html

  1. C = B log 2 ( 1 + S N ) C=B\log_{2}\left(1+\frac{S}{N}\right)
  2. f p 2 B f_{p}\leq 2B\,
  3. M = 1 + A Δ V . M=1+{A\over\Delta V}.
  4. R = f p log 2 ( M ) , R=f_{p}\log_{2}(M),\,
  5. R 2 B log 2 ( M ) . R\leq 2B\log_{2}(M).
  6. R < C R<C\,
  7. R > C R>C\,
  8. 2 B log 2 ( M ) = B log 2 ( 1 + S N ) 2B\log_{2}(M)=B\log_{2}\left(1+\frac{S}{N}\right)
  9. M = 1 + S N . M=\sqrt{1+\frac{S}{N}}.
  10. C = 0 B log 2 ( 1 + S ( f ) N ( f ) ) d f C=\int_{0}^{B}\log_{2}\left(1+\frac{S(f)}{N(f)}\right)df
  11. C 0.332 B SNR ( in dB ) C\approx 0.332\cdot B\cdot\mathrm{SNR\ (in\ dB)}
  12. SNR ( in dB ) = 10 log 10 S N . \mathrm{SNR\ (in\ dB)}=10\log_{10}{S\over N}.
  13. C 1.44 B S N . C\approx 1.44\cdot B\cdot{S\over N}.
  14. N 0 N_{0}
  15. B N 0 B\cdot N_{0}
  16. C 1.44 S N 0 C\approx 1.44\cdot{S\over N_{0}}

Shape.html

  1. S ( u , v , w ) = u - w u - v S(u,v,w)=\frac{u-w}{u-v}
  2. z a z + b , a 0 , z\mapsto az+b,\quad a\neq 0,
  3. 1 - p = 1 - ( u - w ) / ( u - v ) = ( w - v ) / ( u - v ) = ( v - w ) / ( v - u ) = S ( v , u , w ) . 1-p=1-(u-w)/(u-v)=(w-v)/(u-v)=(v-w)/(v-u)=S(v,u,w).
  4. p - 1 = S ( u , w , v ) . p^{-1}=S(u,w,v).
  5. S ( v , w , u ) = ( 1 - p ) - 1 . S(v,w,u)=(1-p)^{-1}.
  6. p ( 1 - p ) - 1 = S ( u , v , w ) S ( v , w , u ) = ( u - w ) / ( v - w ) = S ( w , v , u ) . p(1-p)^{-1}=S(u,v,w)S(v,w,u)=(u-w)/(v-w)=S(w,v,u).
  7. p = ( 1 - q ) - 1 , p=(1-q)^{-1},
  8. p = r ( 1 - q - 1 ) p=r(1-q^{-1})
  9. ( z 1 , z 2 , z n ) (z_{1},z_{2},...z_{n})
  10. S ( z j , z j + 1 , z j + 2 ) , j = 1 , , n - 2. S(z_{j},z_{j+1},z_{j+2}),\ j=1,...,n-2.

Shape_of_the_universe.html

  1. Ω Ω
  2. Ω = 1 Ω=1
  3. Ω > 1 Ω>1
  4. Ω Align l t ; 1 Ω&lt;1
  5. Ω Ω
  6. Ω m a s s 0.315 ± 0.018 \Omega_{mass}\approx 0.315\pm 0.018
  7. Ω r e l a t i v i s t i c 9.24 * 10 - 5 \Omega_{relativistic}\approx 9.24*10^{-5}
  8. Ω Λ 0.6817 ± 0.0018 \Omega_{\Lambda}\approx 0.6817\pm 0.0018
  9. Ω t o t a l = Ω m a s s + Ω r e l a t i v i s t i c + Ω Λ = 1.00 ± 0.02 \Omega_{total}=\Omega_{mass}+\Omega_{relativistic}+\Omega_{\Lambda}=1.00\pm 0.02
  10. ρ c r i t i c a l = 9.47 * 10 - 27 k g / m 3 \rho_{critical}=9.47*10^{-27}kg/m^{3}
  11. Ω Ω
  12. Ω t o t a l 1.00 ± 0.12 \Omega_{total}\approx 1.00\pm 0.12
  13. d d
  14. d d
  15. d d
  16. d d
  17. d d
  18. t > 0 t>0
  19. t = 0 t=0
  20. c t ct
  21. Ω = 0 Ω=0

Shapley_value.html

  1. v v
  2. v : 2 n v\;:\;2^{n}\to\mathbb{R}
  3. v ( ) = 0 v(\emptyset)=0
  4. \emptyset
  5. v v
  6. v v
  7. S S
  8. ( v , N ) (v,N)
  9. ϕ i ( v ) = S N { i } | S | ! ( n - | S | - 1 ) ! n ! ( v ( S { i } ) - v ( S ) ) \phi_{i}(v)=\sum_{S\subseteq N\setminus\{i\}}\frac{|S|!\;(n-|S|-1)!}{n!}(v(S% \cup\{i\})-v(S))
  10. ϕ i ( v ) = 1 | N | ! R [ v ( P i R { i } ) - v ( P i R ) ] \phi_{i}(v)=\frac{1}{|N|!}\sum_{R}\left[v(P_{i}^{R}\cup\left\{i\right\})-v(P_{% i}^{R})\right]\,\!
  11. | N | ! |N|!
  12. R R\,\!
  13. P i R P_{i}^{R}\,\!
  14. N N\,\!
  15. i i\,\!
  16. R R\,\!
  17. k p 2 \frac{kp}{2}
  18. p 2 \frac{p}{2}
  19. N = { 1 , 2 , 3 } N=\{1,2,3\}\,\!
  20. v ( S ) = { 1 , if S { { 1 , 3 } , { 2 , 3 } , { 1 , 2 , 3 } } 0 , otherwise v(S)=\begin{cases}1,&\,\text{if }S\in\left\{\{1,3\},\{2,3\},\{1,2,3\}\right\}% \\ 0,&\,\text{otherwise}\\ \end{cases}
  21. ϕ i ( v ) = 1 | N | ! R [ v ( P i R { i } ) - v ( P i R ) ] \phi_{i}(v)=\frac{1}{|N|!}\sum_{R}\left[v(P_{i}^{R}\cup\left\{i\right\})-v(P_{% i}^{R})\right]\,\!
  22. R R\,\!
  23. P i R P_{i}^{R}\,\!
  24. N N\,\!
  25. i i\,\!
  26. R R\,\!
  27. R R\,\!
  28. M C 1 MC_{1}
  29. 1 , 2 , 3 {1,2,3}\,\!
  30. v ( { 1 } ) - v ( ) = 0 - 0 = 0 v(\{1\})-v(\varnothing)=0-0=0\,\!
  31. 1 , 3 , 2 {1,3,2}\,\!
  32. v ( { 1 } ) - v ( ) = 0 - 0 = 0 v(\{1\})-v(\varnothing)=0-0=0\,\!
  33. 2 , 1 , 3 {2,1,3}\,\!
  34. v ( { 1 , 2 } ) - v ( { 2 } ) = 0 - 0 = 0 v(\{1,2\})-v(\{2\})=0-0=0\,\!
  35. 2 , 3 , 1 {2,3,1}\,\!
  36. v ( { 1 , 2 , 3 } ) - v ( { 2 , 3 } ) = 1 - 1 = 0 v(\{1,2,3\})-v(\{2,3\})=1-1=0\,\!
  37. 3 , 1 , 2 {3,1,2}\,\!
  38. v ( { 1 , 3 } ) - v ( { 3 } ) = 1 - 0 = 1 v(\{1,3\})-v(\{3\})=1-0=1\,\!
  39. 3 , 2 , 1 {3,2,1}\,\!
  40. v ( { 1 , 2 , 3 } ) - v ( { 2 , 3 } ) = 1 - 1 = 0 v(\{1,2,3\})-v(\{2,3\})=1-1=0\,\!
  41. ϕ 1 ( v ) = ( 1 ) ( 1 6 ) = 1 6 \phi_{1}(v)=(1)\!\left(\frac{1}{6}\right)=\frac{1}{6}\,\!
  42. ϕ 2 ( v ) = ϕ 1 ( v ) = 1 6 \phi_{2}(v)=\phi_{1}(v)=\frac{1}{6}\,\!
  43. ϕ 3 ( v ) = 4 6 = 2 3 . \phi_{3}(v)=\frac{4}{6}=\frac{2}{3}.\,
  44. i N ϕ i ( v ) = v ( N ) \sum_{i\in N}\phi_{i}(v)=v(N)
  45. v ( S { i } ) = v ( S { j } ) v(S\cup\{i\})=v(S\cup\{j\})
  46. ϕ i ( v + w ) = ϕ i ( v ) + ϕ i ( w ) \phi_{i}(v+w)=\phi_{i}(v)+\phi_{i}(w)
  47. ϕ i ( a v ) = a ϕ i ( v ) \phi_{i}(av)=a\phi_{i}(v)
  48. ϕ i ( v ) \phi_{i}(v)
  49. i i
  50. v v
  51. v ( S { i } ) = v ( S ) v(S\cup\{i\})=v(S)
  52. S S
  53. d s ds
  54. v v
  55. c c
  56. I I
  57. I = [ 0 , 1 ] I=[0,1]
  58. ( S v ) ( d s ) = 0 1 ( v ( t I + d s ) - v ( t I ) ) d t . (Sv)(ds)=\int_{0}^{1}(v(tI+ds)-v(tI))dt.
  59. ( S v ) ( d s ) (Sv)(ds)
  60. d s ds
  61. t I tI
  62. I I
  63. t t
  64. t I + d s tI+ds
  65. d s ds
  66. t I tI
  67. v v
  68. I I
  69. μ \mu
  70. v ( c ) = f ( μ ( c ) ) v(c)=f(\mu(c))
  71. ϕ \phi
  72. μ ( c ) = 1 c ( u ) ϕ ( u ) d u , \mu(c)=\int 1_{c}(u)\phi(u)du,
  73. 1 c ( ) 1_{c}()
  74. c c
  75. μ ( t I ) = t μ ( I ) \mu(tI)=t\mu(I)
  76. t t
  77. v ( t I + d s ) = f ( t μ ( I ) ) + f ( t μ ( I ) ) μ ( d s ) . v(tI+ds)=f(t\mu(I))+f^{\prime}(t\mu(I))\mu(ds).
  78. ( S v ) ( d s ) = 0 1 f t μ ( I ) ( μ ( d s ) ) d t (Sv)(ds)=\int_{0}^{1}f^{\prime}_{t\mu(I)}(\mu(ds))dt
  79. μ \mu
  80. μ \mu
  81. μ ( t I ) = t μ ( I ) \mu(tI)=t\mu(I)
  82. f f
  83. ( S v ) ( d s ) = lim ϵ 0 , ϵ > 0 1 ϵ 0 1 - ϵ ( f ( t + ϵ μ ( d s ) ) - f ( t ) ) d t (Sv)(ds)=\lim_{\epsilon\to 0,\epsilon>0}\frac{1}{\epsilon}\int_{0}^{1-\epsilon% }(f(t+\epsilon\mu(ds))-f(t))dt

Sharkovskii's_theorem.html

  1. I I\subset\mathbb{R}
  2. f : I I f:I\to I
  3. { 2 k k } \{2^{k}\ \mid\ k\in\mathbb{N}\}
  4. f : x ( 1 - x ) - 1 f:x\mapsto(1-x)^{-1}

Sheaf_(mathematics).html

  1. \mathcal{F}
  2. F ( U ) i F ( U i ) i , j F ( U i U j ) . F(U)\rightarrow\prod_{i}F(U_{i}){{{}\atop\longrightarrow}\atop{\longrightarrow% \atop{}}}\prod_{i,j}F(U_{i}\cap U_{j}).
  3. res U i , U : F ( U ) F ( U i ) \operatorname{res}_{U_{i},U}\colon F(U)\rightarrow F(U_{i})
  4. res U i U j , U i : F ( U i ) F ( U i U j ) \operatorname{res}_{U_{i}\cap U_{j},U_{i}}\colon F(U_{i})\rightarrow F(U_{i}% \cap U_{j})
  5. res U i U j , U j : F ( U j ) F ( U i U j ) . \operatorname{res}_{U_{i}\cap U_{j},U_{j}}\colon F(U_{j})\rightarrow F(U_{i}% \cap U_{j}).
  6. 𝒪 M j \mathcal{O}^{j}_{M}
  7. 𝒪 M \mathcal{O}_{M}
  8. 𝒪 X × \mathcal{O}_{X}^{\times}
  9. 𝒟 \mathcal{DB}
  10. 𝒟 M \mathcal{D}_{M}
  11. f ~ : a F G \tilde{f}:aF\rightarrow G
  12. f = f ~ i f=\tilde{f}i
  13. s ¯ \bar{s}
  14. s ¯ \bar{s}
  15. U F ( U ) / K ( U ) U\mapsto F(U)/K(U)
  16. 0 K F Q 0. 0\to K\to F\to Q\to 0.
  17. o m ( F , G ) \mathcal{H}om(F,G)
  18. U Hom ( F | U , G | U ) U\mapsto\operatorname{Hom}(F|_{U},G|_{U})
  19. F | U F|_{U}
  20. ( F | U ) ( V ) = F ( V ) (F|_{U})(V)=F(V)
  21. U F ( U ) G ( U ) U\mapsto F(U)\otimes G(U)
  22. 𝐙 ¯ \underline{\mathbf{Z}}
  23. \mathcal{F}
  24. f * f_{*}
  25. f ! f_{!}
  26. 𝒢 \mathcal{G}
  27. f - 1 f^{-1}
  28. f ! f^{!}
  29. f ! f^{!}
  30. f - 1 f^{-1}
  31. f * f_{*}
  32. R f ! Rf_{!}
  33. f ! f^{!}
  34. f * f^{*}
  35. f - 1 f^{-1}
  36. x \mathcal{F}_{x}
  37. \mathcal{F}
  38. x = lim U x ( U ) , \mathcal{F}_{x}=\underrightarrow{\lim}_{U\ni x}\mathcal{F}(U),
  39. x := i - 1 ( { x } ) , \mathcal{F}_{x}:=i^{-1}\mathcal{F}(\{x\}),
  40. ( X , 𝒪 X ) (X,\mathcal{O}_{X})
  41. 𝒪 X \mathcal{O}_{X}
  42. 𝐑 ¯ \underline{\mathbf{R}}
  43. 𝐂 ¯ \underline{\mathbf{C}}
  44. ( X , 𝒪 X ) (X,\mathcal{O}_{X})
  45. \mathcal{M}
  46. ( U ) \mathcal{M}(U)
  47. 𝒪 X ( U ) \mathcal{O}_{X}(U)
  48. ( U ) ( V ) \mathcal{M}(U)\to\mathcal{M}(V)
  49. 𝒪 X ( U ) \mathcal{O}_{X}(U)
  50. 𝒪 X \mathcal{O}_{X}
  51. 𝐙 ¯ \underline{\mathbf{Z}}
  52. \mathcal{M}
  53. 𝒪 X n | U | U \mathcal{O}_{X}^{n}|_{U}\to\mathcal{M}|_{U}
  54. \mathcal{M}
  55. 𝒪 X m | U 𝒪 X n | U \mathcal{O}_{X}^{m}|_{U}\to\mathcal{O}_{X}^{n}|_{U}
  56. 𝒪 X m | U 𝒪 X n | U \mathcal{O}_{X}^{m}|_{U}\to\mathcal{O}_{X}^{n}|_{U}\to\mathcal{M}
  57. 𝒪 X n | U \mathcal{O}_{X}^{n}|_{U}\to\mathcal{M}
  58. \mathcal{M}
  59. ϕ : 𝒪 X n \phi:\mathcal{O}_{X}^{n}\to\mathcal{M}
  60. 𝒪 X \mathcal{O}_{X}
  61. 𝒪 X \mathcal{O}_{X}
  62. 𝒪 X \mathcal{O}_{X}
  63. 𝒪 X \mathcal{O}_{X}
  64. ( f - 1 F ) ( Y ) Hom 𝐓𝐨𝐩 / X ( f , π ) (f^{-1}F)(Y)\cong\operatorname{Hom}_{\mathbf{Top}/X}(f,\pi)
  65. Γ ( U , - ) \Gamma(U,-)
  66. Γ ( U , - ) \Gamma(U,-)
  67. Γ ( U , - ) \Gamma(U,-)
  68. H i ( U , - ) H^{i}(U,-)
  69. 𝐑 ¯ \underline{\mathbf{R}}
  70. 𝐑 ¯ \underline{\mathbf{R}}
  71. H 1 H^{1}

Sheffer_sequence.html

  1. Q p n ( x ) = n p n - 1 ( x ) . Qp_{n}(x)=np_{n-1}(x)\,.
  2. p n ( x ) = k = 0 n a n , k x k and q n ( x ) = k = 0 n b n , k x k . p_{n}(x)=\sum_{k=0}^{n}a_{n,k}x^{k}\ \mbox{and}~{}\ q_{n}(x)=\sum_{k=0}^{n}b_{% n,k}x^{k}.
  3. p q p\circ q
  4. ( p n q ) ( x ) = k = 0 n a n , k q k ( x ) = 0 k n a n , k b k , x (p_{n}\circ q)(x)=\sum_{k=0}^{n}a_{n,k}q_{k}(x)=\sum_{0\leq k\leq\ell\leq n}a_% {n,k}b_{k,\ell}x^{\ell}
  5. e n ( x ) = x n = k = 0 n δ n , k x k . e_{n}(x)=x^{n}=\sum_{k=0}^{n}\delta_{n,k}x^{k}.
  6. p n ( x + y ) = k = 0 n ( n k ) p k ( x ) p n - k ( y ) . p_{n}(x+y)=\sum_{k=0}^{n}{n\choose k}p_{k}(x)p_{n-k}(y).
  7. p 0 ( x ) = 1 p_{0}(x)=1\,
  8. p n ( 0 ) = 0 for n 1. p_{n}(0)=0\mbox{ for }~{}n\geq 1.\,
  9. s n ( x + y ) = k = 0 n ( n k ) p k ( x ) s n - k ( y ) . s_{n}(x+y)=\sum_{k=0}^{n}{n\choose k}p_{k}(x)s_{n-k}(y).
  10. s n ( x + y ) = k = 0 n ( n k ) x k s n - k ( y ) . s_{n}(x+y)=\sum_{k=0}^{n}{n\choose k}x^{k}s_{n-k}(y).
  11. n = 0 p n ( x ) n ! t n = A ( t ) exp ( x B ( t ) ) \sum_{n=0}^{\infty}\frac{p_{n}(x)}{n!}t^{n}=A(t)\exp(xB(t))\,

Shellsort.html

  1. a 1 a 2 a 3 a 4 a 5 a 6 a 7 a 8 a 9 a 10 a 11 a 12 input data: 62 83 18 53 07 17 95 86 47 69 25 28 after 5-sorting: 17 28 18 47 07 25 83 86 53 69 62 95 after 3-sorting: 17 07 18 47 28 25 69 62 53 83 86 95 after 1-sorting: 07 17 18 25 28 47 53 62 69 83 86 95 \begin{array}[]{rcccccccccccc}&a_{1}&a_{2}&a_{3}&a_{4}&a_{5}&a_{6}&a_{7}&a_{8}% &a_{9}&a_{10}&a_{11}&a_{12}\\ \hbox{input data:}&62&83&18&53&07&17&95&86&47&69&25&28\\ \hbox{after 5-sorting:}&17&28&18&47&07&25&83&86&53&69&62&95\\ \hbox{after 3-sorting:}&17&07&18&47&28&25&69&62&53&83&86&95\\ \hbox{after 1-sorting:}&07&17&18&25&28&47&53&62&69&83&86&95\\ \end{array}
  2. N / 2 k \lfloor N/2^{k}\rfloor
  3. N 2 , N 4 , , 1 \left\lfloor\frac{N}{2}\right\rfloor,\left\lfloor\frac{N}{4}\right\rfloor,% \ldots,1
  4. Θ ( N 2 ) \Theta(N^{2})
  5. 2 N / 2 k + 1 + 1 2\lfloor N/2^{k+1}\rfloor+1
  6. 2 N 4 + 1 , , 3 , 1 2\left\lfloor\frac{N}{4}\right\rfloor+1,\ldots,3,1
  7. Θ ( N 3 / 2 ) \Theta(N^{3/2})
  8. 2 k - 1 2^{k}-1
  9. 1 , 3 , 7 , 15 , 31 , 63 , 1,3,7,15,31,63,\ldots
  10. Θ ( N 3 / 2 ) \Theta(N^{3/2})
  11. 2 k + 1 2^{k}+1
  12. 1 , 3 , 5 , 9 , 17 , 33 , 65 , 1,3,5,9,17,33,65,\ldots
  13. Θ ( N 3 / 2 ) \Theta(N^{3/2})
  14. 2 p 3 q 2^{p}3^{q}
  15. 1 , 2 , 3 , 4 , 6 , 8 , 9 , 12 , 1,2,3,4,6,8,9,12,\ldots
  16. Θ ( N log 2 N ) \Theta(N\log^{2}N)
  17. ( 3 k - 1 ) / 2 (3^{k}-1)/2
  18. N / 3 \lceil N/3\rceil
  19. 1 , 4 , 13 , 40 , 121 , 1,4,13,40,121,\ldots
  20. Θ ( N 3 / 2 ) \Theta(N^{3/2})
  21. 0 q < r q ( r 2 + r ) / 2 - k a q , where \prod\limits_{\scriptscriptstyle 0\leq q<r\atop\scriptscriptstyle q\neq(r^{2}+% r)/2-k}a_{q},\hbox{where}
  22. r = 2 k + 2 k , r=\left\lfloor\sqrt{2k+\sqrt{2k}}\right\rfloor,
  23. a q = min { n : n ( 5 / 2 ) q + 1 , a_{q}=\min\{n\in\mathbb{N}\colon n\geq(5/2)^{q+1},
  24. p : 0 p < q gcd ( a p , n ) = 1 } \forall p\colon 0\leq p<q\Rightarrow\gcd(a_{p},n)=1\}
  25. 1 , 3 , 7 , 21 , 48 , 112 , 1,3,7,21,48,112,\ldots
  26. O ( N 1 + 8 ln ( 5 / 2 ) / ln N ) O(N^{1+\sqrt{8\ln(5/2)/\ln N}})
  27. 4 k + 3 2 k - 1 + 1 4^{k}+3\cdot 2^{k-1}+1
  28. 1 , 8 , 23 , 77 , 281 , 1,8,23,77,281,\ldots
  29. O ( N 4 / 3 ) O(N^{4/3})
  30. 9 ( 4 k - 1 - 2 k 2 ) + 1 , 4 k + 1 - 6 2 k + 1 2 + 1 9(4^{k-1}-2^{\frac{k}{2}})+1,4^{k+1}-6\cdot 2^{\frac{k+1}{2}}+1
  31. 1 , 5 , 19 , 41 , 109 , 1,5,19,41,109,\ldots
  32. O ( N 4 / 3 ) O(N^{4/3})
  33. h k = max { 5 h k - 1 / 11 , 1 } , h 0 = N h_{k}=\max\left\{\left\lfloor 5h_{k-1}/11\right\rfloor,1\right\},h_{0}=N
  34. 5 N 11 , 5 11 5 N 11 , , 1 \left\lfloor\frac{5N}{11}\right\rfloor,\left\lfloor\frac{5}{11}\left\lfloor% \frac{5N}{11}\right\rfloor\right\rfloor,\ldots,1
  35. 9 k - 4 k 5 4 k - 1 \left\lceil\frac{9^{k}-4^{k}}{5\cdot 4^{k-1}}\right\rceil
  36. 1 , 4 , 9 , 20 , 46 , 103 , 1,4,9,20,46,103,\ldots
  37. 1 , 4 , 10 , 23 , 57 , 132 , 301 , 701 1,4,10,23,57,132,301,701
  38. h k = 2.25 h k - 1 h_{k}=\lfloor 2.25h_{k-1}\rfloor
  39. h k = h k h_{k}=\lceil h^{\prime}_{k}\rceil
  40. h k = 2.25 h k - 1 + 1 h^{\prime}_{k}=2.25h^{\prime}_{k-1}+1
  41. h 1 = 1 h^{\prime}_{1}=1
  42. 0.5349 N N - 0.4387 N - 0.097 N + O ( 1 ) 0.5349N\sqrt{N}-0.4387N-0.097\sqrt{N}+O(1)
  43. 2 N 2 / h + π N 3 h 2N^{2}/h+\sqrt{\pi N^{3}h}
  44. N 2 4 c h + O ( N ) \frac{N^{2}}{4ch}+O(N)
  45. 1 8 g π c h ( h - 1 ) N 3 / 2 + O ( h N ) \frac{1}{8g}\sqrt{\frac{\pi}{ch}}(h-1)N^{3/2}+O(hN)
  46. ψ ( h , g ) N + 1 8 π c ( c - 1 ) N 3 / 2 + O ( ( c - 1 ) g h 1 / 2 N ) + O ( c 2 g 3 h 2 ) \psi(h,g)N+\frac{1}{8}\sqrt{\frac{\pi}{c}}(c-1)N^{3/2}+O((c-1)gh^{1/2}N)+O(c^{% 2}g^{3}h^{2})
  47. π h 128 g + O ( g - 1 / 2 h 1 / 2 ) + O ( g h - 1 / 2 ) \sqrt{\frac{\pi h}{128}}g+O(g^{-1/2}h^{1/2})+O(gh^{-1/2})
  48. h k = 2.25 h k - 1 h_{k}=\lfloor 2.25h_{k-1}\rfloor

Shutter_speed.html

  1. 2 \scriptstyle\sqrt{2}
  2. E = F 360 S E=\frac{F\cdot 360^{\circ}}{S}
  3. S = F 360 E S=\frac{F\cdot 360^{\circ}}{E}

Sierpinski_number.html

  1. { k 2 n + 1 : n } . \left\{\,k2^{n}+1:n\in\mathbb{N}\,\right\}.
  2. 2 - n + k 2 - n \frac{2^{-n}+k}{2^{-n}}
  3. - 2 - n - ( - k ) 2 - n -\frac{2^{-n}-(-k)}{2^{-n}}
  4. Ξ ( k , n ) = a b s ( n u m e r a t o r ( k 2 n + 1 ) ) \Xi(k,n)=abs(numerator(k2^{n}+1))
  5. Ξ ( k , n ) \Xi(k,n)

Sieve_of_Eratosthenes.html

  1. n \sqrt{n}
  2. n n
  3. i i
  4. i ²
  5. O ( n l o g l o g n ) O(nloglogn)
  6. n n
  7. n n
  8. A A
  9. n n
  10. Δ n Δ≤\sqrt{n}
  11. n \sqrt{n}
  12. Δ Δ
  13. n + 1 \sqrt{n}+1
  14. n n
  15. Δ Δ
  16. p n p≤\sqrt{n}
  17. Δ Δ
  18. n \sqrt{n}
  19. O ( n ) O(\sqrt{n})
  20. n n
  21. n \sqrt{n}
  22. n p n 1 p n\sum_{p\leq n}\frac{1}{p}
  23. ln ln n + M \ln\ln n+M
  24. n n
  25. n \sqrt{n}
  26. x x
  27. 1 2 x 2 ln x \frac{1}{2}x^{2}\ln x
  28. x x
  29. x ln x \frac{x}{\ln x}
  30. n n
  31. 1 2 n 2 ln n \frac{1}{2}\frac{n^{2}}{\ln n}
  32. n \sqrt{n}
  33. 1 ln n \frac{1}{\ln n}
  34. n n
  35. 2 π ( n 1 2 ) 2\pi(n^{\frac{1}{2}})
  36. π \pi
  37. 4 n 1 2 ln n \frac{4n^{\frac{1}{2}}}{\ln n}
  38. n n
  39. 4 n ln n \frac{4}{\sqrt{n}\ln n}
  40. ln ln n - 1 ln n ( 1 - 4 n ) + M - ln 2 \ln\ln n-\frac{1}{\ln n}\left(1-\frac{4}{\sqrt{n}}\right)+M-\ln 2
  41. p x 1 p \sum_{p\leq x}\frac{1}{p}
  42. x x
  43. p x p \prod_{p\leq x}p
  44. p x p - 1 p \prod_{p\leq x}\frac{p-1}{p}
  45. p - 1 p \frac{p-1}{p}
  46. x x
  47. n n
  48. x x
  49. p x p - 1 p ( ln ln n - 1 ln n ( 1 - 4 n ) + M - ln 2 - p x 1 p ) \prod_{p\leq x}\frac{p-1}{p}\left(\ln\ln n-\frac{1}{\ln n}\left(1-\frac{4}{% \sqrt{n}}\right)+M-\ln 2-\sum_{p\leq x}\frac{1}{p}\right)
  50. O ( n log log n ) O(n\log\log n)
  51. log log n \log\log n
  52. O ( n ) O(n)
  53. O ( n ( log n ) ( log log n ) ) O(n(\log n)(\log\log n))
  54. O ( n ) O(n)
  55. O ( n log log n ) O(n\log\log n)
  56. O ( n log n ) O(\frac{\sqrt{n}}{\log n})
  57. O ( n ) O(n)
  58. O ( n 1 / 2 log log n / log n ) O(n^{1/2}\log\log n/\log n)
  59. O ( n log n ) O(\frac{n}{\log n})
  60. O ( n log n ) O(\frac{\sqrt{n}}{\log n})
  61. 3 · 3 = 9 3·3=9
  62. 3 · 5 = 15 3·5=15
  63. 3 · 7 = 21 3·7=21
  64. 3 · 𝟗 = 27 3·\mathbf{9}=27
  65. 3 · 𝟏𝟓 = 45 3·\mathbf{15}=45

Sievert.html

  1. H T = R W R D T , R H_{T}=\sum_{R}W_{R}\cdot D_{T,R}

Sigmoid_function.html

  1. S ( t ) = 1 1 + e - t . S(t)=\frac{1}{1+e^{-t}}.
  2. t ± t\rightarrow\pm\infty
  3. d d t S ( t ) = c 1 S ( t ) ( c 2 - S ( t ) ) \tfrac{d}{dt}S(t)=c_{1}S(t)\left(c_{2}-S(t)\right)
  4. c 3 c_{3}
  5. f ( x ) = x 1 + x 2 f(x)=\tfrac{x}{\sqrt{1+x^{2}}}

Sign_convention.html

  1. ( 1 0 0 0 0 - 1 0 0 0 0 - 1 0 0 0 0 - 1 ) \begin{pmatrix}1&0&0&0\\ 0&-1&0&0\\ 0&0&-1&0\\ 0&0&0&-1\end{pmatrix}
  2. ( - 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 ) \begin{pmatrix}-1&0&0&0\\ 0&1&0&0\\ 0&0&1&0\\ 0&0&0&1\end{pmatrix}
  3. R a b = R c a c b R_{ab}\,=R^{c}{}_{acb}
  4. R a b = R c a b c R_{ab}\,=R^{c}{}_{abc}
  5. R a b = R a c b c R_{ab}\,={R_{acb}}^{c}
  6. ± \pm
  7. F a b \,F_{ab}
  8. e - i ω t \,e^{-i\omega t}
  9. e + j ω t \,e^{+j\omega t}

Silicon_bandgap_temperature_sensor.html

  1. V B E = V G 0 ( 1 - T T 0 ) + V B E 0 ( T T 0 ) + ( n K T q ) ln ( T 0 T ) + ( K T q ) ln ( I C I C 0 ) V_{BE}=V_{G0}\left(1-{\frac{T}{T_{0}}}\right)+V_{BE0}\left(\frac{T}{T_{0}}% \right)+\left(\frac{nKT}{q}\right)\ln\left(\frac{T_{0}}{T}\right)+\left(\frac{% KT}{q}\right)\ln\left(\frac{I_{C}}{I_{C0}}\right)\,
  2. Δ V B E = K T q ln ( I C 1 I C 2 ) \Delta V_{BE}=\frac{KT}{q}\cdot\ln\left(\frac{I_{C1}}{I_{C2}}\right)\,
  3. Δ V B E = K T q ln ( N ) \Delta V_{BE}=\frac{KT}{q}\cdot\ln\left(N\right)\,

Simple_path.html

  1. \R \R
  2. \R n \R^{n}

Simplicial_complex.html

  1. 𝒦 \mathcal{K}
  2. 𝒦 \mathcal{K}
  3. 𝒦 \mathcal{K}
  4. σ 1 , σ 2 𝒦 \sigma_{1},\sigma_{2}\in\mathcal{K}
  5. σ 1 \sigma_{1}
  6. σ 2 \sigma_{2}
  7. 𝒦 \mathcal{K}
  8. 𝒦 \mathcal{K}
  9. 𝒦 \mathcal{K}
  10. σ 𝒦 \sigma\in\mathcal{K}
  11. ( f 0 , f 1 , f 2 , , f d + 1 ) (f_{0},f_{1},f_{2},...,f_{d+1})
  12. x 3 + 6 x 2 + 12 x + 8 x^{3}+6x^{2}+12x+8
  13. x 4 + 18 x 3 + 23 x 2 + 8 x + 1 x^{4}+18x^{3}+23x^{2}+8x+1
  14. F Δ ( x - 1 ) = h 0 x d + 1 + h 1 x d + h 2 x d - 1 + + h d x + h d + 1 F_{\Delta}(x-1)=h_{0}x^{d+1}+h_{1}x^{d}+h_{2}x^{d-1}+...+h_{d}x+h_{d+1}
  15. ( h 0 , h 1 , h 2 , , h d + 1 ) . (h_{0},h_{1},h_{2},...,h_{d+1}).
  16. F ( x - 1 ) = ( x - 1 ) 3 + 6 ( x - 1 ) 2 + 12 ( x - 1 ) + 8 = x 3 + 3 x 2 + 3 x + 1. F(x-1)=(x-1)^{3}+6(x-1)^{2}+12(x-1)+8=x^{3}+3x^{2}+3x+1.

Simpson's_rule.html

  1. a b f ( x ) d x b - a 6 [ f ( a ) + 4 f ( a + b 2 ) + f ( b ) ] . \int_{a}^{b}f(x)\,dx\approx\tfrac{b-a}{6}\left[f(a)+4f\left(\tfrac{a+b}{2}% \right)+f(b)\right].
  2. f ( x ) f(x)
  3. P ( x ) P(x)
  4. f ( x ) f(x)
  5. P ( x ) = f ( a ) ( x - m ) ( x - b ) ( a - m ) ( a - b ) + f ( m ) ( x - a ) ( x - b ) ( m - a ) ( m - b ) + f ( b ) ( x - a ) ( x - m ) ( b - a ) ( b - m ) . P(x)=f(a)\tfrac{(x-m)(x-b)}{(a-m)(a-b)}+f(m)\tfrac{(x-a)(x-b)}{(m-a)(m-b)}+f(b% )\tfrac{(x-a)(x-m)}{(b-a)(b-m)}.
  6. a b P ( x ) d x = b - a 6 [ f ( a ) + 4 f ( a + b 2 ) + f ( b ) ] . \int_{a}^{b}P(x)\,dx=\tfrac{b-a}{6}\left[f(a)+4f\left(\tfrac{a+b}{2}\right)+f(% b)\right].
  7. a = - 1 a=-1
  8. b = 1 b=1
  9. M = ( b - a ) f ( a + b 2 ) M=(b-a)f\left(\tfrac{a+b}{2}\right)
  10. T = 1 2 ( b - a ) ( f ( a ) + f ( b ) ) . T=\tfrac{1}{2}(b-a)(f(a)+f(b)).
  11. - 1 24 ( b - a ) 3 f ′′ ( a ) + O ( ( b - a ) 4 ) and 1 12 ( b - a ) 3 f ′′ ( a ) + O ( ( b - a ) 4 ) , -\tfrac{1}{24}(b-a)^{3}f^{\prime\prime}(a)+O((b-a)^{4})\quad\,\text{and}\quad% \tfrac{1}{12}(b-a)^{3}f^{\prime\prime}(a)+O((b-a)^{4}),
  12. O ( ( b - a ) 4 ) O((b-a)^{4})
  13. ( b - a ) 4 (b-a)^{4}
  14. O ( ( b - a ) 4 ) O((b-a)^{4})
  15. 2 M + T 3 . \tfrac{2M+T}{3}.
  16. 1 b - a a b f ( x ) d x α f ( a ) + β f ( a + b 2 ) + γ f ( b ) . \tfrac{1}{b-a}\int_{a}^{b}f(x)\,dx\approx\alpha f(a)+\beta f\left(\tfrac{a+b}{% 2}\right)+\gamma f(b).
  17. 1 90 ( b - a 2 ) 5 | f ( 4 ) ( ξ ) | , \tfrac{1}{90}\left(\tfrac{b-a}{2}\right)^{5}\left|f^{(4)}(\xi)\right|,
  18. ξ \xi
  19. a a
  20. b b
  21. ( b - a ) 5 (b-a)^{5}
  22. ( b - a ) 4 (b-a)^{4}
  23. ξ \xi
  24. [ a , b ] [a,b]
  25. [ a , b ] [a,b]
  26. [ a , b ] [a,b]
  27. [ a , b ] [a,b]
  28. n n
  29. n n
  30. a b f ( x ) d x h 3 [ f ( x 0 ) + 2 j = 1 n / 2 - 1 f ( x 2 j ) + 4 j = 1 n / 2 f ( x 2 j - 1 ) + f ( x n ) ] , \int_{a}^{b}f(x)\,dx\approx\tfrac{h}{3}\bigg[f(x_{0})+2\sum_{j=1}^{n/2-1}f(x_{% 2j})+4\sum_{j=1}^{n/2}f(x_{2j-1})+f(x_{n})\bigg],
  31. x j = a + j h x_{j}=a+jh
  32. j = 0 , 1 , , n - 1 , n j=0,1,...,n-1,n
  33. h = ( b - a ) / n h=(b-a)/n
  34. x 0 = a x_{0}=a
  35. x n = b x_{n}=b
  36. n = 2 n=2
  37. a b f ( x ) d x h 3 [ f ( x 0 ) + 4 f ( x 1 ) + 2 f ( x 2 ) + 4 f ( x 3 ) + 2 f ( x 4 ) + + 4 f ( x n - 1 ) + f ( x n ) ] = h 3 j = 1 n / 2 [ f ( x 2 j - 2 ) + 4 f ( x 2 j - 1 ) + f ( x 2 j ) ] . \int_{a}^{b}f(x)\,dx\approx\tfrac{h}{3}\bigg[f(x_{0})+4f(x_{1})+2f(x_{2})+4f(x% _{3})+2f(x_{4})+\cdots+4f(x_{n-1})+f(x_{n})\bigg]=\tfrac{h}{3}\sum_{j=1}^{n/2}% \bigg[f(x_{2j-2})+4f(x_{2j-1})+f(x_{2j})\bigg].
  38. h 4 180 ( b - a ) max ξ [ a , b ] | f ( 4 ) ( ξ ) | , \tfrac{h^{4}}{180}(b-a)\max_{\xi\in[a,b]}|f^{(4)}(\xi)|,
  39. h h
  40. h = ( b - a ) / n . h=(b-a)/n.
  41. [ a , b ] [a,b]
  42. a b f ( x ) d x h 48 [ 17 f ( x 0 ) + 59 f ( x 1 ) + 43 f ( x 2 ) + 49 f ( x 3 ) + 48 i = 4 n - 4 f ( x i ) + 49 f ( x n - 3 ) + 43 f ( x n - 2 ) + 59 f ( x n - 1 ) + 17 f ( x n ) ] . \int_{a}^{b}f(x)\,dx\approx\tfrac{h}{48}\bigg[17f(x_{0})+59f(x_{1})+43f(x_{2})% +49f(x_{3})+48\sum_{i=4}^{n-4}f(x_{i})+49f(x_{n-3})+43f(x_{n-2})+59f(x_{n-1})+% 17f(x_{n})\bigg].
  43. a b f ( x ) d x 3 h 8 [ f ( a ) + 3 f ( 2 a + b 3 ) + 3 f ( a + 2 b 3 ) + f ( b ) ] = ( b - a ) 8 [ f ( a ) + 3 f ( 2 a + b 3 ) + 3 f ( a + 2 b 3 ) + f ( b ) ] , \int_{a}^{b}f(x)\,dx\approx\tfrac{3h}{8}\left[f(a)+3f\left(\tfrac{2a+b}{3}% \right)+3f\left(\tfrac{a+2b}{3}\right)+f(b)\right]=\tfrac{(b-a)}{8}\left[f(a)+% 3f\left(\tfrac{2a+b}{3}\right)+3f\left(\tfrac{a+2b}{3}\right)+f(b)\right]\,,
  44. | ( b - a ) 5 6480 f ( 4 ) ( ξ ) | , \left|\tfrac{(b-a)^{5}}{6480}f^{(4)}(\xi)\right|,
  45. ξ \xi
  46. a a
  47. b b
  48. h = ( b - a ) / n , x i = a + i h , h=(b-a)/n,\quad\quad x_{i}=a+ih,
  49. a b f ( x ) d x 3 h 8 [ f ( x 0 ) + 3 f ( x 1 ) + 3 f ( x 2 ) + 2 f ( x 3 ) + 3 f ( x 4 ) + 3 f ( x 5 ) + 2 f ( x 6 ) + + f ( x n ) ] . \int_{a}^{b}f(x)\,dx\approx\tfrac{3h}{8}\left[f(x_{0})+3f(x_{1})+3f(x_{2})+2f(% x_{3})+3f(x_{4})+3f(x_{5})+2f(x_{6})+...+f(x_{n})\right].
  50. n n

Simulated_annealing.html

  1. s s
  2. s s^{\prime}
  3. P ( e , e , T ) P(e,e^{\prime},T)
  4. e = E ( s ) e=E(s)
  5. e = E ( s ) e^{\prime}=E(s^{\prime})
  6. T T
  7. P P
  8. e e^{\prime}
  9. e e
  10. T T
  11. P ( e , e , T ) P(e,e^{\prime},T)
  12. e > e e^{\prime}>e
  13. T T
  14. T = 0 T=0
  15. P ( e , e , T ) P(e,e^{\prime},T)
  16. e < e e^{\prime}<e
  17. P P
  18. e - e e^{\prime}-e
  19. T T
  20. s s
  21. T T
  22. s s
  23. T T
  24. T T
  25. T = 0 T=0
  26. n e i g h b o u r ( s ) neighbour(s)
  27. s s
  28. r a n d o m ( 0 , 1 ) random(0,1)
  29. 0 , 1 ) ) 0,1))
  30. t e m p e r a t u r e ( r ) temperature(r)
  31. r r
  32. s = s < s u b > 0 s=s<sub>0
  33. n = 20 n=20
  34. n ! n!
  35. n ( n - 1 ) / 2 = 190 n(n-1)/2=190
  36. i = 1 n - 1 i \sum_{i=1}^{n-1}i
  37. ( s , s ) (s,s^{\prime})
  38. s s^{\prime}
  39. s s
  40. P ( e , e , T ) P(e,e^{\prime},T)
  41. P ( e , e , T ) P(e,e^{\prime},T)
  42. e < e e^{\prime}<e
  43. exp ( - ( e - e ) / T ) \exp(-(e^{\prime}-e)/T)
  44. T T
  45. s s^{\prime}
  46. n - 1 n-1
  47. n ( n - 1 ) / 2 n(n-1)/2
  48. s s^{\prime}
  49. P ( E ( s ) , E ( s ) , T ) P(E(s),E(s^{\prime}),T)
  50. P P
  51. E ( s ) - E ( s ) E(s^{\prime})-E(s)
  52. T T
  53. T T
  54. A A
  55. B B
  56. A A
  57. A A
  58. B B
  59. A A
  60. B B
  61. A A
  62. B B

Simultaneous_equations_model.html

  1. y i t = y - i , t γ i + x i t β i + u i t , i = 1 , , m , y_{it}=y_{-i,t}^{\prime}\gamma_{i}+x_{it}^{\prime}\;\!\beta_{i}+u_{it},\quad i% =1,\ldots,m,
  2. y i = Y - i γ i + X i β i + u i , i = 1 , , m , y_{i}=Y_{-i}\gamma_{i}+X_{i}\beta_{i}+u_{i},\quad i=1,\ldots,m,
  3. Y Γ = X B + U . Y\Gamma=XB+U.\,
  4. Y = X B Γ - 1 + U Γ - 1 = X Π + V . Y=XB\Gamma^{-1}+U\Gamma^{-1}=X\Pi+V.\,
  5. Π ^ \scriptstyle\hat{\Pi}
  6. 1 T X X \scriptstyle\frac{1}{T}X^{\prime}\!X
  7. Y ^ - i \scriptstyle\hat{Y}_{\!-i}
  8. Y ^ - i \scriptstyle\hat{Y}_{\!-i}
  9. y i = ( Y - i X i ) ( γ i β i ) + u i Z i δ i + u i , y_{i}=\begin{pmatrix}Y_{-i}&X_{i}\end{pmatrix}\begin{pmatrix}\gamma_{i}\\ \beta_{i}\end{pmatrix}+u_{i}\equiv Z_{i}\delta_{i}+u_{i},
  10. δ ^ i = ( Z ^ i Z ^ i ) - 1 Z ^ i y i = ( Z i P Z i ) - 1 Z i P y i , \hat{\delta}_{i}=\big(\hat{Z}^{\prime}_{i}\hat{Z}_{i}\big)^{-1}\hat{Z}^{\prime% }_{i}y_{i}=\big(Z^{\prime}_{i}PZ_{i}\big)^{-1}Z^{\prime}_{i}Py_{i},
  11. y i = Y - i γ i + X i β i + u i Z i δ i + u i y_{i}=Y_{-i}\gamma_{i}+X_{i}\beta_{i}+u_{i}\equiv Z_{i}\delta_{i}+u_{i}
  12. Y - i = X Π + U - 1 Y_{-i}=X\Pi+U_{-1}
  13. Y - i Y_{-i}
  14. X - i X_{-i}
  15. X X
  16. Z Z
  17. δ ^ i = ( Z i ( I - λ M ) Z i ) - 1 Z i ( I - λ M ) y i , \hat{\delta}_{i}=\Big(Z^{\prime}_{i}(I-\lambda M)Z_{i}\Big)^{\!-1}Z^{\prime}_{% i}(I-\lambda M)y_{i},
  18. ( [ y i Y - i ] M i [ y i Y - i ] ) ( [ y i Y - i ] M [ y i Y - i ] ) - 1 \Big(\begin{bmatrix}y_{i}\\ Y_{-i}\end{bmatrix}M_{i}\begin{bmatrix}y_{i}&Y_{-i}\end{bmatrix}\Big)\Big(% \begin{bmatrix}y_{i}\\ Y_{-i}\end{bmatrix}M\begin{bmatrix}y_{i}&Y_{-i}\end{bmatrix}\Big)^{\!-1}
  19. | [ y i Y - i ] M i [ y i Y - i ] - λ [ y i Y - i ] M [ y i Y - i ] | = 0 \Big|\begin{bmatrix}y_{i}&Y_{-i}\end{bmatrix}^{\prime}M_{i}\begin{bmatrix}y_{i% }&Y_{-i}\end{bmatrix}-\lambda\begin{bmatrix}y_{i}&Y_{-i}\end{bmatrix}^{\prime}% M\begin{bmatrix}y_{i}&Y_{-i}\end{bmatrix}\Big|=0
  20. δ ^ = ( Z ( I - κ M ) Z ) - 1 Z ( I - κ M ) y , \hat{\delta}=\Big(Z^{\prime}(I-\kappa M)Z\Big)^{\!-1}Z^{\prime}(I-\kappa M)y,
  21. δ = [ β i γ i ] \delta=\begin{bmatrix}\beta_{i}&\gamma_{i}\end{bmatrix}
  22. Z = [ X i Y - i ] Z=\begin{bmatrix}X_{i}&Y_{-i}\end{bmatrix}
  23. I - κ M = I - M = P I-\kappa M=I-M=P

Sinc_filter.html

  1. H ( f ) = rect ( f 2 B ) H(f)=\mathrm{rect}\left(\frac{f}{2B}\right)
  2. B B\,
  3. h ( t ) = - 1 { H ( f ) } \displaystyle h(t)=\mathcal{F}^{-1}\{H(f)\}
  4. h L P F ( t ) = 2 B L sinc ( 2 B L t ) h_{LPF}(t)=2B_{L}\,\mathrm{sinc}\left(2B_{L}t\right)
  5. H L P F ( f ) = rect ( f 2 B L ) . H_{LPF}(f)=\mathrm{rect}\left(\frac{f}{2B_{L}}\right).
  6. h B P F ( t ) = 2 B H sinc ( 2 B H t ) - 2 B L sinc ( 2 B L t ) h_{BPF}(t)=2B_{H}\,\mathrm{sinc}\left(2B_{H}t\right)-2B_{L}\,\mathrm{sinc}% \left(2B_{L}t\right)
  7. H B P F ( f ) = rect ( f 2 B H ) - rect ( f 2 B L ) . H_{BPF}(f)=\mathrm{rect}\left(\frac{f}{2B_{H}}\right)-\mathrm{rect}\left(\frac% {f}{2B_{L}}\right).
  8. h H P F ( t ) = δ ( t ) - 2 B H sinc ( 2 B H t ) h_{HPF}(t)=\delta(t)-2B_{H}\,\mathrm{sinc}\left(2B_{H}t\right)
  9. H H P F ( f ) = 1 - rect ( f 2 B H ) . H_{HPF}(f)=1-\mathrm{rect}\left(\frac{f}{2B_{H}}\right).
  10. π \pi

Sine-Gordon_equation.html

  1. φ t t - φ x x + sin φ = 0. \,\varphi_{tt}-\varphi_{xx}+\sin\varphi=0.
  2. u = x + t 2 , v = x - t 2 , u=\frac{x+t}{2},\quad v=\frac{x-t}{2},
  3. φ u v = sin φ . \varphi_{uv}=\sin\varphi.\,
  4. d s 2 = d u 2 + 2 cos φ d u d v + d v 2 , ds^{2}=du^{2}+2\cos\varphi\,du\,dv+dv^{2},\,
  5. φ \varphi
  6. φ t t - φ x x + φ = 0. \varphi_{tt}-\varphi_{xx}+\varphi\ =0.\,
  7. SG ( φ ) = 1 2 ( φ t 2 - φ x 2 ) - 1 + cos φ . \mathcal{L}\text{SG}(\varphi)=\frac{1}{2}(\varphi_{t}^{2}-\varphi_{x}^{2})-1+% \cos\varphi.
  8. cos ( φ ) = n = 0 ( - φ 2 ) n ( 2 n ) ! , \cos(\varphi)=\sum_{n=0}^{\infty}\frac{(-\varphi^{2})^{n}}{(2n)!},
  9. SG ( φ ) = 1 2 ( φ t 2 - φ x 2 ) - φ 2 2 + n = 2 ( - φ 2 ) n ( 2 n ) ! = KG ( φ ) + n = 2 ( - φ 2 ) n ( 2 n ) ! . \begin{aligned}\displaystyle\mathcal{L}\text{SG}(\varphi)&\displaystyle=\frac{% 1}{2}(\varphi_{t}^{2}-\varphi_{x}^{2})-\frac{\varphi^{2}}{2}+\sum_{n=2}^{% \infty}\frac{(-\varphi^{2})^{n}}{(2n)!}\\ &\displaystyle=\mathcal{L}\text{KG}(\varphi)+\sum_{n=2}^{\infty}\frac{(-% \varphi^{2})^{n}}{(2n)!}.\end{aligned}
  10. φ soliton ( x , t ) := 4 arctan e m γ ( x - v t ) + δ \varphi\text{soliton}(x,t):=4\arctan e^{m\gamma(x-vt)+\delta}\,
  11. γ 2 = 1 1 - v 2 . \gamma^{2}=\frac{1}{1-v^{2}}.
  12. φ t t - φ x x + m 2 sin φ = 0. \,\varphi_{tt}-\varphi_{xx}+m^{2}\sin\varphi=0.
  13. γ \gamma
  14. φ \varphi
  15. φ = 0 \varphi=0
  16. φ = 2 π \varphi=2\pi
  17. φ = 0 ( mod 2 π ) \varphi=0(\textrm{mod}2\pi)
  18. γ \gamma
  19. φ u = φ u + 2 β sin ( φ + φ 2 ) , {\varphi^{\prime}}_{u}=\varphi_{u}+2\beta\sin\left(\frac{\varphi^{\prime}+% \varphi}{2}\right),
  20. φ v = - φ v + 2 β sin ( φ - φ 2 ) with φ = φ 0 = 0 {\varphi^{\prime}}_{v}=-\varphi_{v}+\frac{2}{\beta}\sin\left(\frac{\varphi^{% \prime}-\varphi}{2}\right)\,\text{ with }\varphi=\varphi_{0}=0
  21. ϑ K = - 1 \vartheta_{\textrm{K}}=-1
  22. ϑ AK = + 1 \vartheta_{\textrm{AK}}=+1
  23. Δ B \Delta_{\textrm{B}}
  24. Δ B = 2 arctanh ( 1 - ω 2 ) ( 1 - v K 2 ) 1 - ω 2 \Delta_{B}=\frac{2\textrm{arctanh}\sqrt{(1-\omega^{2})(1-v\text{K}^{2})}}{% \sqrt{1-\omega^{2}}}
  25. v K v\text{K}
  26. ω \omega
  27. x 0 x_{0}
  28. x 0 + Δ B x_{0}+\Delta\text{B}
  29. φ x x - φ t t = sinh φ . \varphi_{xx}-\varphi_{tt}=\sinh\varphi.\,
  30. = 1 2 ( φ t 2 - φ x 2 ) - cosh φ . \mathcal{L}={1\over 2}(\varphi_{t}^{2}-\varphi_{x}^{2})-\cosh\varphi.\,
  31. φ x x + φ y y = sin φ , \varphi_{xx}+\varphi_{yy}=\sin\varphi,\,
  32. φ \varphi

Singular_value_decomposition.html

  1. m × n m×n
  2. 𝐌 \mathbf{M}
  3. 𝐔 \mathbf{U}
  4. m × m m×m
  5. 𝚺 \mathbf{Σ}
  6. m × n m×n
  7. 𝐕 \mathbf{V}
  8. 𝐕 \mathbf{V}
  9. 𝐕 \mathbf{V}
  10. n × n n×n
  11. 𝚺 \mathbf{Σ}
  12. 𝐌 \mathbf{M}
  13. m m
  14. 𝐔 \mathbf{U}
  15. n n
  16. 𝐕 \mathbf{V}
  17. 𝐌 \mathbf{M}
  18. 𝐌 \mathbf{M}
  19. 𝐌 \mathbf{M}
  20. 𝐌 \mathbf{M}
  21. 𝚺 \mathbf{Σ}
  22. 𝐌 \mathbf{M}
  23. m × n m×n
  24. K K
  25. 𝐌 = 𝐔 s y m b o l Σ 𝐕 * \mathbf{M}=\mathbf{U}symbol{\Sigma}\mathbf{V}^{*}
  26. 𝐔 \mathbf{U}
  27. m × m m×m
  28. K K
  29. K = 𝐑 K=\mathbf{R}
  30. 𝚺 \mathbf{Σ}
  31. m × n m×n
  32. n × n n×n
  33. n × n n×n
  34. 𝐕 \mathbf{V}
  35. 𝐌 \mathbf{M}
  36. 𝚺 \mathbf{Σ}
  37. 𝐌 \mathbf{M}
  38. 𝚺 \mathbf{Σ}
  39. 𝐌 \mathbf{M}
  40. 𝐔 \mathbf{U}
  41. 𝐕 \mathbf{V}
  42. 𝐔 \mathbf{U}
  43. 𝐌 \mathbf{M}
  44. m × m m×m
  45. 𝚺 \mathbf{Σ}
  46. m × m m×m
  47. 𝚺 \mathbf{Σ}
  48. n n
  49. n × n n×n
  50. n n
  51. 𝐔 \mathbf{U}
  52. 𝐕 \mathbf{V}
  53. 4 × 5 4×5
  54. 𝐌 = [ 1 0 0 0 2 0 0 3 0 0 0 0 0 0 0 0 4 0 0 0 ] \mathbf{M}=\begin{bmatrix}1&0&0&0&2\\ 0&0&3&0&0\\ 0&0&0&0&0\\ 0&4&0&0&0\end{bmatrix}
  55. 𝐔 = [ 0 0 1 0 0 1 0 0 0 0 0 - 1 1 0 0 0 ] s y m b o l Σ = [ 4 0 0 0 0 0 3 0 0 0 0 0 5 0 0 0 0 0 0 0 ] 𝐕 * = [ 0 1 0 0 0 0 0 1 0 0 0.2 0 0 0 0.8 0 0 0 1 0 - 0.8 0 0 0 0.2 ] \begin{aligned}\displaystyle\mathbf{U}&\displaystyle=\begin{bmatrix}0&0&1&0\\ 0&1&0&0\\ 0&0&0&-1\\ 1&0&0&0\end{bmatrix}\\ \displaystyle symbol{\Sigma}&\displaystyle=\begin{bmatrix}4&0&0&0&0\\ 0&3&0&0&0\\ 0&0&\sqrt{5}&0&0\\ 0&0&0&0&0\end{bmatrix}\\ \displaystyle\mathbf{V}^{*}&\displaystyle=\begin{bmatrix}0&1&0&0&0\\ 0&0&1&0&0\\ \sqrt{0.2}&0&0&0&\sqrt{0.8}\\ 0&0&0&1&0\\ -\sqrt{0.8}&0&0&0&\sqrt{0.2}\end{bmatrix}\end{aligned}
  56. 𝚺 \mathbf{Σ}
  57. 𝐔 \mathbf{U}
  58. 𝐔 \mathbf{U}
  59. 𝐔𝐔 𝐓 = [ 0 0 1 0 0 1 0 0 0 0 0 - 1 1 0 0 0 ] [ 0 0 0 1 0 1 0 0 1 0 0 0 0 0 - 1 0 ] = [ 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 ] = 𝐈 4 𝐕𝐕 𝐓 = [ 0 0 0.2 0 - 0.8 1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0.8 0 0.2 ] [ 0 1 0 0 0 0 0 1 0 0 0.2 0 0 0 0.8 0 0 0 1 0 - 0.8 0 0 0 0.2 ] = [ 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 ] = 𝐈 5 \begin{aligned}\displaystyle\mathbf{U}\mathbf{U^{T}}&\displaystyle=\begin{% bmatrix}0&0&1&0\\ 0&1&0&0\\ 0&0&0&-1\\ 1&0&0&0\end{bmatrix}\cdot\begin{bmatrix}0&0&0&1\\ 0&1&0&0\\ 1&0&0&0\\ 0&0&-1&0\end{bmatrix}=\begin{bmatrix}1&0&0&0\\ 0&1&0&0\\ 0&0&1&0\\ 0&0&0&1\end{bmatrix}=\mathbf{I}_{4}\\ \displaystyle\mathbf{V}\mathbf{V^{T}}&\displaystyle=\begin{bmatrix}0&0&\sqrt{0% .2}&0&-\sqrt{0.8}\\ 1&0&0&0&0\\ 0&1&0&0&0\\ 0&0&0&1&0\\ 0&0&\sqrt{0.8}&0&\sqrt{0.2}\end{bmatrix}\cdot\begin{bmatrix}0&1&0&0&0\\ 0&0&1&0&0\\ \sqrt{0.2}&0&0&0&\sqrt{0.8}\\ 0&0&0&1&0\\ -\sqrt{0.8}&0&0&0&\sqrt{0.2}\end{bmatrix}=\begin{bmatrix}1&0&0&0&0\\ 0&1&0&0&0\\ 0&0&1&0&0\\ 0&0&0&1&0\\ 0&0&0&0&1\end{bmatrix}=\mathbf{I}_{5}\end{aligned}
  60. V V
  61. 𝐕 * = [ 0 1 0 0 0 0 0 1 0 0 0.2 0 0 0 0.8 0.4 0 0 0.5 - 0.1 - 0.4 0 0 0.5 0.1 ] \mathbf{V}^{*}=\begin{bmatrix}0&1&0&0&0\\ 0&0&1&0&0\\ \sqrt{0.2}&0&0&0&\sqrt{0.8}\\ \sqrt{0.4}&0&0&\sqrt{0.5}&-\sqrt{0.1}\\ -\sqrt{0.4}&0&0&\sqrt{0.5}&\sqrt{0.1}\end{bmatrix}
  62. σ σ
  63. 𝐌 \mathbf{M}
  64. 𝐌 v = σ u and 𝐌 * u = σ v \mathbf{M}\vec{v}=\sigma\vec{u}\,\,\text{ and }\mathbf{M}^{*}\vec{u}=\sigma% \vec{v}
  65. σ σ
  66. 𝐌 = 𝐔 s y m b o l Σ 𝐕 * \mathbf{M}=\mathbf{U}symbol{\Sigma}\mathbf{V}^{*}
  67. 𝚺 \mathbf{Σ}
  68. 𝐌 \mathbf{M}
  69. 𝐔 \mathbf{U}
  70. 𝐕 \mathbf{V}
  71. m × n m×n
  72. 𝐌 \mathbf{M}
  73. p = m i n ( m , n ) p=min(m,n)
  74. 𝐔 \mathbf{U}
  75. 𝐌 \mathbf{M}
  76. 𝐕 \mathbf{V}
  77. 𝐌 \mathbf{M}
  78. 𝐌 \mathbf{M}
  79. 𝐔 \mathbf{U}
  80. 𝐕 \mathbf{V}
  81. 𝐌 \mathbf{M}
  82. 𝐌 \mathbf{M}
  83. 𝐌 + = 𝐕 s y m b o l Σ + 𝐔 * \mathbf{M}^{+}=\mathbf{V}symbol{\Sigma}^{+}\mathbf{U}^{*}
  84. 𝚺 \mathbf{Σ}
  85. 𝐀𝐱 = 𝟎 \mathbf{Ax}=\mathbf{0}
  86. 𝐀 \mathbf{A}
  87. 𝐱 \mathbf{x}
  88. 𝐀 \mathbf{A}
  89. 𝐱 \mathbf{x}
  90. 𝐱 \mathbf{x}
  91. 𝐀 \mathbf{A}
  92. 𝐀 \mathbf{A}
  93. 𝐱 \mathbf{x}
  94. 𝐀 \mathbf{A}
  95. 𝐀 \mathbf{A}
  96. 𝐱 \mathbf{x}
  97. 𝐱 \mathbf{x}
  98. 𝐱 \mathbf{x}
  99. 𝐀 \mathbf{A}
  100. 𝐱 \mathbf{x}
  101. 𝐀𝐱 \mathbf{Ax}
  102. [ u ! ! ] 𝐱 [ u ! ! ] = 1 [u^{\prime}!!^{\prime}]\mathbf{x}[u^{\prime}!!^{\prime}]=1
  103. 𝐀 \mathbf{A}
  104. 𝐌 \mathbf{M}
  105. 𝐌 \mathbf{M}
  106. 𝐌 \mathbf{M}
  107. 𝐕 \mathbf{V}
  108. 𝐌 \mathbf{M}
  109. 𝐌 \mathbf{M}
  110. 𝐌 \mathbf{M}
  111. 𝚺 \mathbf{Σ}
  112. 𝐌 \mathbf{M}
  113. 𝐌 ~ \tilde{\mathbf{M}}
  114. r r
  115. 𝐌 \mathbf{M}
  116. 𝐌 ~ \tilde{\mathbf{M}}
  117. rank ( 𝐌 ~ ) = r \operatorname{rank}\left(\tilde{\mathbf{M}}\right)=r
  118. 𝐌 \mathbf{M}
  119. 𝐌 ~ = 𝐔 s y m b o l Σ ~ 𝐕 * \tilde{\mathbf{M}}=\mathbf{U}\tilde{symbol{\Sigma}}\mathbf{V}^{*}
  120. s y m b o l Σ ~ \tilde{symbol{\Sigma}}
  121. 𝚺 \mathbf{Σ}
  122. r r
  123. 𝐀 \mathbf{A}
  124. 𝐀 = 𝐮 𝐯 \mathbf{A}=\mathbf{u}⊗\mathbf{v}
  125. A i j = u i v j A_{ij}=u_{i}v_{j}
  126. 𝐌 \mathbf{M}
  127. 𝐌 = i 𝐀 i = i σ i 𝐔 i 𝐕 i \mathbf{M}=\sum_{i}\mathbf{A}_{i}=\sum_{i}\sigma_{i}\mathbf{U}_{i}\otimes% \mathbf{V}_{i}^{\dagger}
  128. i i
  129. 𝐔 \mathbf{U}
  130. 𝐕 \mathbf{V}
  131. α = σ 1 2 i σ i 2 , \alpha=\frac{\sigma_{1}^{2}}{\sum_{i}\sigma_{i}^{2}},
  132. 𝐀 \mathbf{A}
  133. 𝐎 \mathbf{O}
  134. 𝐀 \mathbf{A}
  135. 𝐎 𝐀 \mathbf{O}−\mathbf{A}
  136. 𝐈 \mathbf{I}
  137. 𝐎 \mathbf{O}
  138. 𝐀 \mathbf{A}
  139. 𝐁 \mathbf{B}
  140. 𝐎 = arg min s y m b o l Ω 𝐀 s y m b o l Ω - 𝐁 F subject to s y m b o l Ω T s y m b o l Ω = 𝐈 \mathbf{O}=\arg\min_{s}ymbol{\Omega}\|\mathbf{A}symbol{\Omega}-\mathbf{B}\|_{F% }\quad\mathrm{subject\ to}\quad symbol{\Omega}^{T}symbol{\Omega}=\mathbf{I}
  141. F \|\cdot\|_{F}
  142. 𝚺 \mathbf{Σ}
  143. m × n m×n
  144. 𝐌 \mathbf{M}
  145. 𝐌 * 𝐌 \displaystyle\mathbf{M}^{*}\mathbf{M}
  146. 𝐕 \mathbf{V}
  147. 𝐔 \mathbf{U}
  148. 𝚺 \mathbf{Σ}
  149. 𝐌 \mathbf{M}
  150. 𝐔 \mathbf{U}
  151. 𝐃 \mathbf{D}
  152. 𝐌 \mathbf{M}
  153. 𝐌 \mathbf{M}
  154. 𝐔 \mathbf{U}
  155. 𝐃 \mathbf{D}
  156. 𝚺 \mathbf{Σ}
  157. 𝐔 \mathbf{U}
  158. 𝐕 \mathbf{V}
  159. 𝐌 \mathbf{M}
  160. λ λ
  161. 𝐌 \mathbf{M}
  162. 𝐌 u = λ u \mathbf{M}u=λu
  163. 𝐌 \mathbf{M}
  164. 𝐌 \mathbf{M}
  165. n × n n×n
  166. { f : 𝐑 n 𝐑 f ( x ) = x T 𝐌 x \begin{cases}f:\mathbf{R}^{n}\to\mathbf{R}\\ f(x)=x^{T}\mathbf{M}x\end{cases}
  167. f = x T 𝐌 x = λ x T x \nabla f=\nabla x^{T}\mathbf{M}x=\lambda\cdot\nabla x^{T}x
  168. 𝐌 u = λ u \mathbf{M}u=λu
  169. 𝐌 \mathbf{M}
  170. λ λ
  171. 𝐌 \mathbf{M}
  172. 2 n 2n
  173. 𝐌 \mathbf{M}
  174. 𝐌 \mathbf{M}
  175. m × n m×n
  176. n × n n×n
  177. 𝐕 \mathbf{V}
  178. 𝐕 * 𝐌 * 𝐌𝐕 = [ 𝐃 0 0 0 ] \mathbf{V}^{*}\mathbf{M}^{*}\mathbf{M}\mathbf{V}=\begin{bmatrix}\mathbf{D}&0\\ 0&0\end{bmatrix}
  179. 𝐃 \mathbf{D}
  180. 𝐕 \mathbf{V}
  181. [ 𝐕 1 * 𝐕 2 * ] 𝐌 * 𝐌 [ 𝐕 1 𝐕 2 ] = [ 𝐕 1 * 𝐌 * 𝐌𝐕 1 𝐕 1 * 𝐌 * 𝐌𝐕 2 𝐕 2 * 𝐌 * 𝐌𝐕 1 𝐕 2 * 𝐌 * 𝐌𝐕 2 ] = [ 𝐃 0 0 0 ] \begin{bmatrix}\mathbf{V}_{1}^{*}\\ \mathbf{V}_{2}^{*}\end{bmatrix}\mathbf{M}^{*}\mathbf{M}\begin{bmatrix}\mathbf{% V}_{1}&\mathbf{V}_{2}\end{bmatrix}=\begin{bmatrix}\mathbf{V}_{1}^{*}\mathbf{M}% ^{*}\mathbf{M}\mathbf{V}_{1}&\mathbf{V}_{1}^{*}\mathbf{M}^{*}\mathbf{M}\mathbf% {V}_{2}\\ \mathbf{V}_{2}^{*}\mathbf{M}^{*}\mathbf{M}\mathbf{V}_{1}&\mathbf{V}_{2}^{*}% \mathbf{M}^{*}\mathbf{M}\mathbf{V}_{2}\end{bmatrix}=\begin{bmatrix}\mathbf{D}&% 0\\ 0&0\end{bmatrix}
  182. 𝐕 1 * 𝐌 * 𝐌𝐕 1 = 𝐃 , 𝐕 2 * 𝐌 * 𝐌𝐕 2 = 0. \mathbf{V}_{1}^{*}\mathbf{M}^{*}\mathbf{M}\mathbf{V}_{1}=\mathbf{D},\qquad% \mathbf{V}_{2}^{*}\mathbf{M}^{*}\mathbf{M}\mathbf{V}_{2}=\mathbf{0}.
  183. 𝐕 \mathbf{V}
  184. 𝐕 1 * 𝐕 1 \displaystyle\mathbf{V}_{1}^{*}\mathbf{V}_{1}
  185. 𝐔 1 = 𝐌𝐕 1 𝐃 - 1 2 \mathbf{U}_{1}=\mathbf{M}\mathbf{V}_{1}\mathbf{D}^{-\frac{1}{2}}
  186. 𝐔 1 𝐃 1 2 𝐕 1 * = 𝐌𝐕 1 𝐃 - 1 2 𝐃 1 2 𝐕 1 * = 𝐌 \mathbf{U}_{1}\mathbf{D}^{\frac{1}{2}}\mathbf{V}_{1}^{*}=\mathbf{M}\mathbf{V}_% {1}\mathbf{D}^{-\frac{1}{2}}\mathbf{D}^{\frac{1}{2}}\mathbf{V}_{1}^{*}=\mathbf% {M}
  187. 𝐃 \mathbf{D}
  188. m m
  189. n n
  190. 𝐔 1 * 𝐔 1 = 𝐃 - 1 2 𝐕 1 * 𝐌 * 𝐌𝐕 1 𝐃 - 1 2 = 𝐃 - 1 2 𝐃𝐃 - 1 2 = 𝐈 𝟏 \mathbf{U}_{1}^{*}\mathbf{U}_{1}=\mathbf{D}^{-\frac{1}{2}}\mathbf{V}_{1}^{*}% \mathbf{M}^{*}\mathbf{M}\mathbf{V}_{1}\mathbf{D}^{-\frac{1}{2}}=\mathbf{D}^{-% \frac{1}{2}}\mathbf{D}\mathbf{D}^{-\frac{1}{2}}=\mathbf{I_{1}}
  191. 𝐔 = [ 𝐔 1 𝐔 2 ] \mathbf{U}=\begin{bmatrix}\mathbf{U}_{1}&\mathbf{U}_{2}\end{bmatrix}
  192. s y m b o l Σ = [ [ 𝐃 1 2 0 0 0 ] 0 ] symbol{\Sigma}=\begin{bmatrix}\begin{bmatrix}\mathbf{D}^{\frac{1}{2}}&0\\ 0&0\end{bmatrix}\\ 0\end{bmatrix}
  193. [ 𝐔 1 𝐔 2 ] [ [ D 1 2 0 0 0 ] 0 ] [ 𝐕 1 𝐕 2 ] * = [ 𝐔 1 𝐔 2 ] [ 𝐃 1 2 𝐕 1 * 0 ] = 𝐔 1 𝐃 1 2 𝐕 1 * = 𝐌 \begin{bmatrix}\mathbf{U}_{1}&\mathbf{U}_{2}\end{bmatrix}\begin{bmatrix}\begin% {bmatrix}\mathbf{}D^{\frac{1}{2}}&0\\ 0&0\end{bmatrix}\\ 0\end{bmatrix}\begin{bmatrix}\mathbf{V}_{1}&\mathbf{V}_{2}\end{bmatrix}^{*}=% \begin{bmatrix}\mathbf{U}_{1}&\mathbf{U}_{2}\end{bmatrix}\begin{bmatrix}% \mathbf{D}^{\frac{1}{2}}\mathbf{V}_{1}^{*}\\ 0\end{bmatrix}=\mathbf{U}_{1}\mathbf{D}^{\frac{1}{2}}\mathbf{V}_{1}^{*}=% \mathbf{M}
  194. 𝐌 = 𝐔 s y m b o l Σ 𝐕 * \mathbf{M}=\mathbf{U}symbol{\Sigma}\mathbf{V}^{*}
  195. 𝐮 \mathbf{u}
  196. 𝐯 \mathbf{v}
  197. 𝐮 \mathbf{u}
  198. 𝐯 \mathbf{v}
  199. 𝐌 \mathbf{M}
  200. m × n m×n
  201. σ ( 𝐮 , 𝐯 ) = 𝐮 T 𝐌𝐯 , 𝐮 S m - 1 , 𝐯 S n - 1 . \sigma(\mathbf{u},\mathbf{v})=\mathbf{u}^{T}\mathbf{M}\mathbf{v},\qquad\mathbf% {u}\in S^{m-1},\mathbf{v}\in S^{n-1}.
  202. σ σ
  203. σ σ
  204. σ ( 𝐮 , 𝐯 ) σ(\mathbf{u},\mathbf{v})
  205. 𝐌 \mathbf{M}
  206. σ = 𝐮 T 𝐌𝐯 - λ 1 𝐮 T 𝐮 - λ 2 𝐯 T 𝐯 \nabla\sigma=\nabla\mathbf{u}^{T}\mathbf{M}\mathbf{v}-\lambda_{1}\cdot\nabla% \mathbf{u}^{T}\mathbf{u}-\lambda_{2}\cdot\nabla\mathbf{v}^{T}\mathbf{v}
  207. 𝐌𝐯 1 = 2 λ 1 𝐮 1 + 0 𝐌 T 𝐮 1 = 0 + 2 λ 2 𝐯 1 \begin{aligned}\displaystyle\mathbf{M}\mathbf{v}_{1}&\displaystyle=2\lambda_{1% }\mathbf{u}_{1}+0\\ \displaystyle\mathbf{M}^{T}\mathbf{u}_{1}&\displaystyle=0+2\lambda_{2}\mathbf{% v}_{1}\end{aligned}
  208. 𝐮 1 T \mathbf{u}_{1}^{T}
  209. 𝐯 1 T \mathbf{v}_{1}^{T}
  210. [ u ! ! ] 𝐮 [ u ! ! ] = [ u ! ! ] 𝐯 [ u ! ! ] = 1 [u^{\prime}!!^{\prime}]\mathbf{u}[u^{\prime}!!^{\prime}]=[u^{\prime}!!^{\prime% }]\mathbf{v}[u^{\prime}!!^{\prime}]=1
  211. σ 1 = 2 λ 1 = 2 λ 2 . \sigma_{1}=2\lambda_{1}=2\lambda_{2}.
  212. 𝐌𝐯 1 = σ 1 𝐮 1 𝐌 T 𝐮 1 = σ 1 𝐯 1 \begin{aligned}\displaystyle\mathbf{M}\mathbf{v}_{1}&\displaystyle=\sigma_{1}% \mathbf{u}_{1}\\ \displaystyle\mathbf{M}^{T}\mathbf{u}_{1}&\displaystyle=\sigma_{1}\mathbf{v}_{% 1}\end{aligned}
  213. σ ( 𝐮 , 𝐯 ) σ(\mathbf{u},\mathbf{v})
  214. 𝐮 , 𝐯 \mathbf{u},\mathbf{v}
  215. 𝐔 \mathbf{U}
  216. 𝐕 \mathbf{V}
  217. 𝐔 \mathbf{U}
  218. 𝐕 \mathbf{V}
  219. { T : K n K m x 𝐌 x \begin{cases}T:K^{n}\to K^{m}\\ x\mapsto\mathbf{M}x\end{cases}
  220. T ( 𝐕 i ) = σ i 𝐔 i , i = 1 , , min ( m , n ) , T(\mathbf{V}_{i})=\sigma_{i}\mathbf{U}_{i},\qquad i=1,\cdots,\min(m,n),
  221. i i
  222. 𝚺 \mathbf{Σ}
  223. i > m i n ( m , n ) i>min(m,n)
  224. T T
  225. i i
  226. i i
  227. T T
  228. S S
  229. T T
  230. n = m n=m
  231. T T
  232. T ( S ) T(S)
  233. T T
  234. 𝐃 \mathbf{D}
  235. T ( S ) T(S)
  236. T ( S ) T(S)
  237. 𝐔 \mathbf{U}
  238. T ( S ) T(S)
  239. T T
  240. 𝐌 \mathbf{M}
  241. 𝐌 \mathbf{M}
  242. ( 𝐎 𝐌 𝐌 * 𝐎 ) . \begin{pmatrix}\mathbf{O}&\mathbf{M}\\ \mathbf{M}^{*}&\mathbf{O}\end{pmatrix}.
  243. 𝐌 𝐐𝐑 \mathbf{M}⇒\mathbf{Q}\mathbf{R}
  244. 𝐑 \mathbf{R}
  245. 𝐌 𝐋 \mathbf{M}⇐\mathbf{L}
  246. 𝐌 = z 0 𝐈 + z 1 σ 1 + z 2 σ 2 + z 3 σ 3 \mathbf{M}=z_{0}\mathbf{I}+z_{1}\sigma_{1}+z_{2}\sigma_{2}+z_{3}\sigma_{3}
  247. z i z_{i}\in\mathbb{C}
  248. 𝐈 \mathbf{I}
  249. σ i \sigma_{i}
  250. σ ± = | z 0 | 2 + | z 1 | 2 + | z 2 | 2 + | z 3 | 2 ± ( | z 0 | 2 + | z 1 | 2 + | z 2 | 2 + | z 3 | 2 ) 2 - | z 0 2 - z 1 2 - z 2 2 - z 3 2 | 2 = | z 0 | 2 + | z 1 | 2 + | z 2 | 2 + | z 3 | 2 ± 2 ( Re z 0 z 1 * ) 2 + ( Re z 0 z 2 * ) 2 + ( Re z 0 z 3 * ) 2 + ( Im z 1 z 2 * ) 2 + ( Im z 2 z 3 * ) 2 + ( Im z 3 z 1 * ) 2 \begin{aligned}\displaystyle\sigma_{\pm}&\displaystyle=\sqrt{|z_{0}|^{2}+|z_{1% }|^{2}+|z_{2}|^{2}+|z_{3}|^{2}\pm\sqrt{(|z_{0}|^{2}+|z_{1}|^{2}+|z_{2}|^{2}+|z% _{3}|^{2})^{2}-|z_{0}^{2}-z_{1}^{2}-z_{2}^{2}-z_{3}^{2}|^{2}}}\\ &\displaystyle=\sqrt{|z_{0}|^{2}+|z_{1}|^{2}+|z_{2}|^{2}+|z_{3}|^{2}\pm 2\sqrt% {(\mathrm{Re}z_{0}z_{1}^{*})^{2}+(\mathrm{Re}z_{0}z_{2}^{*})^{2}+(\mathrm{Re}z% _{0}z_{3}^{*})^{2}+(\mathrm{Im}z_{1}z_{2}^{*})^{2}+(\mathrm{Im}z_{2}z_{3}^{*})% ^{2}+(\mathrm{Im}z_{3}z_{1}^{*})^{2}}}\end{aligned}
  251. 𝐌 = 𝐔 n s y m b o l Σ n 𝐕 * \mathbf{M}=\mathbf{U}_{n}symbol{\Sigma}_{n}\mathbf{V}^{*}
  252. 𝐌 = 𝐔 r s y m b o l Σ r 𝐕 r * \mathbf{M}=\mathbf{U}_{r}symbol{\Sigma}_{r}\mathbf{V}_{r}^{*}
  253. 𝐌 ~ = 𝐔 t s y m b o l Σ t 𝐕 t * \tilde{\mathbf{M}}=\mathbf{U}_{t}symbol{\Sigma}_{t}\mathbf{V}_{t}^{*}
  254. 𝐌 ~ \tilde{\mathbf{M}}
  255. 𝐌 = 𝐌 * 𝐌 1 2 \|\mathbf{M}\|=\|\mathbf{M}^{*}\mathbf{M}\|^{\frac{1}{2}}
  256. n × n n×n
  257. 𝐌 , 𝐍 = trace ( 𝐍 * 𝐌 ) . \langle\mathbf{M},\mathbf{N}\rangle=\operatorname{trace}\left(\mathbf{N}^{*}% \mathbf{M}\right).
  258. 𝐌 = 𝐌 , 𝐌 = trace ( 𝐌 * 𝐌 ) . \|\mathbf{M}\|=\sqrt{\langle\mathbf{M},\mathbf{M}\rangle}=\sqrt{\operatorname{% trace}\left(\mathbf{M}^{*}\mathbf{M}\right)}.
  259. 𝐌 = i σ i 2 \|\mathbf{M}\|=\sqrt{\sum_{i}\sigma_{i}^{2}}
  260. 𝐌 \mathbf{M}
  261. 𝐌 \mathbf{M}
  262. i j | m i j | 2 . \sqrt{\sum_{ij}|m_{ij}|^{2}}.
  263. 𝐌 = 𝐔 T f 𝐕 * \mathbf{M}=\mathbf{U}T_{f}\mathbf{V}^{*}
  264. T f T_{f}
  265. [ U 1 U 2 ] \begin{bmatrix}U_{1}\\ U_{2}\end{bmatrix}
  266. 𝐌 = 𝐔𝐕 * 𝐕 T f 𝐕 * \mathbf{M}=\mathbf{U}\mathbf{V}^{*}\cdot\mathbf{V}T_{f}\mathbf{V}^{*}
  267. T T
  268. λ λ
  269. 𝐌 \mathbf{M}
  270. ψ H ψ∈H
  271. 𝐌 ψ = 𝐔 T f 𝐕 * ψ = i 𝐔 T f 𝐕 * ψ , 𝐔 e i 𝐔 e i = i σ i ψ , 𝐕 e i 𝐔 e i \mathbf{M}\psi=\mathbf{U}T_{f}\mathbf{V}^{*}\psi=\sum_{i}\left\langle\mathbf{U% }T_{f}\mathbf{V}^{*}\psi,\mathbf{U}e_{i}\right\rangle\mathbf{U}e_{i}=\sum_{i}% \sigma_{i}\left\langle\psi,\mathbf{V}e_{i}\right\rangle\mathbf{U}e_{i}
  272. H H
  273. 𝐌 \mathbf{M}
  274. 𝐌 \mathbf{M}
  275. 𝐌 \mathbf{M}
  276. σ k \sigma_{k}

Sintering.html

  1. E n / E = ( D / d ) 3.4 E_{n}/E=(D/d)^{3.4}
  2. σ G B \sigma_{GB}
  3. γ G B \gamma_{GB}
  4. σ G B d A (work done) = γ G B d A (energy change) \sigma_{GB}dA\,\text{ (work done)}=\gamma_{GB}dA\,\text{ (energy change)}\,\!
  5. σ G B d A (work done) = γ G B d A (energy change) \sigma_{GB}dA\,\text{ (work done)}=\gamma_{GB}dA\,\text{ (energy change)}\,\!
  6. σ G B d A (work done) = d G (energy change) = γ G B d A + A d γ G B \sigma_{GB}dA\,\text{ (work done)}=dG\,\text{ (energy change)}=\gamma_{GB}dA+% Ad\gamma_{GB}\,\!
  7. σ G B = γ G B + A d γ G B d A \sigma_{GB}=\gamma_{GB}+\frac{Ad\gamma_{GB}}{dA}\,\!
  8. σ G B \sigma_{GB}
  9. N m \frac{N}{m}
  10. γ G B \gamma_{GB}
  11. J m 2 \frac{J}{m^{2}}
  12. ( J = N m ) (J=Nm)
  13. G m = G 0 m + K t G^{m}=G_{0}^{m}+Kt
  14. K = K 0 e - Q R T K=K_{0}e^{\frac{-Q}{RT}}
  15. F = π r λ sin ( 2 θ ) F=\pi r\lambda\sin(2\theta)\,\!
  16. f = 4 3 π r 3 N f=\frac{4}{3}\pi r^{3}N\,\!
  17. n = 3 f 2 π r 2 n=\frac{3f}{2\pi r^{2}}\,\!
  18. 2 λ R \frac{2\lambda}{R}
  19. n F m a x = 2 λ D c r i t nF_{max}=\frac{2\lambda}{D_{crit}}\,\!
  20. D c r i t = 4 r 3 f D_{crit}=\frac{4r}{3f}\,\!

Skein_relation.html

  1. F ( L - , L 0 , L + ) = 0 F\Big(L_{-},L_{0},L_{+}\Big)=0
  2. F ( L - ( x ) , L 0 ( x ) , L + ( x ) , x ) = 0 F\Big(L_{-}(x),L_{0}(x),L_{+}(x),x\Big)=0
  3. x x
  4. x \sqrt{x}
  5. P ( unknot ) = 1 P({\rm unknot})=1
  6. ( D - , D 0 , D + ) (D_{-},D_{0},D_{+})
  7. P ( D - ) = ( x - 1 / 2 - x 1 / 2 ) P ( D 0 ) + P ( D + ) P(D_{-})=(x^{-1/2}-x^{1/2})P(D_{0})+P(D_{+})

Skew-symmetric_matrix.html

  1. [ 0 2 - 1 - 2 0 - 4 1 4 0 ] . \begin{bmatrix}0&2&-1\\ -2&0&-4\\ 1&4&0\end{bmatrix}.
  2. Mat = n Skew n Sym , n \mbox{Mat}~{}_{n}=\mbox{Skew}~{}_{n}\oplus\mbox{Sym}~{}_{n},
  3. A = 1 2 ( A - A 𝖳 ) + 1 2 ( A + A 𝖳 ) . A=\frac{1}{2}(A-A^{\mathsf{T}})+\frac{1}{2}(A+A^{\mathsf{T}}).
  4. , \langle\cdot,\cdot\rangle
  5. A x , y = - x , A y x , y n . \langle Ax,y\rangle=-\langle x,Ay\rangle\quad\forall x,y\in\mathbb{R}^{n}.
  6. x , A x = 0 \langle x,Ax\rangle=0
  7. x + y , A ( x + y ) = 0 \langle x+y,A(x+y)\rangle=0
  8. n = 0 s ( n ) n ! x n = ( 1 - x 2 ) - 1 4 exp ( x 2 4 ) . \sum_{n=0}^{\infty}\frac{s(n)}{n!}x^{n}=(1-x^{2})^{-\frac{1}{4}}\exp\left(% \frac{x^{2}}{4}\right).
  9. s ( n ) = π - 1 2 2 3 4 Γ ( 3 / 4 ) ( n / e ) n - 1 4 ( 1 + O ( 1 n ) ) . s(n)=\pi^{-\frac{1}{2}}2^{\frac{3}{4}}\Gamma\left(3/4\right)(n/e)^{n-\frac{1}{% 4}}\left(1+O\big(\frac{1}{n}\big)\right).
  10. Σ = [ 0 λ 1 - λ 1 0 0 0 0 0 λ 2 - λ 2 0 0 0 0 0 λ r - λ r 0 0 0 ] \Sigma=\begin{bmatrix}\begin{matrix}0&\lambda_{1}\\ -\lambda_{1}&0\end{matrix}&0&\cdots&0\\ 0&\begin{matrix}0&\lambda_{2}\\ -\lambda_{2}&0\end{matrix}&&0\\ \vdots&&\ddots&\vdots\\ 0&0&\cdots&\begin{matrix}0&\lambda_{r}\\ -\lambda_{r}&0\end{matrix}\\ &&&&\begin{matrix}0\\ &\ddots\\ &&0\end{matrix}\end{bmatrix}
  11. [ A , B ] = A B - B A . [A,B]=AB-BA.\,
  12. [ A , B ] 𝖳 = B 𝖳 A 𝖳 - A 𝖳 B 𝖳 = B A - A B = - [ A , B ] . [A,B]^{\mathsf{T}}=B^{\mathsf{T}}A^{\mathsf{T}}-A^{\mathsf{T}}B^{\mathsf{T}}=% BA-AB=-[A,B]\,.
  13. R = exp ( A ) = n = 0 A n n ! . R=\exp(A)=\sum_{n=0}^{\infty}\frac{A^{n}}{n!}.
  14. [ a - b b a ] , \begin{bmatrix}a&-b\\ b&\,a\end{bmatrix},
  15. [ cos θ - sin θ sin θ cos θ ] = exp ( θ [ 0 - 1 1 0 ] ) , \begin{bmatrix}\cos\,\theta&-\sin\,\theta\\ \sin\,\theta&\,\cos\,\theta\end{bmatrix}=\exp\left(\theta\begin{bmatrix}0&-1\\ 1&\,0\end{bmatrix}\right),
  16. n / 2 \scriptstyle\lfloor{n/2}\rfloor
  17. v w v\wedge w
  18. v w v * w - w * v , v\wedge w\mapsto v^{*}\otimes w-w^{*}\otimes v,
  19. v * v^{*}
  20. v v

Skewes'_number.html

  1. π ( x ) > li ( x ) , \pi(x)>\operatorname{li}(x),
  2. e 727.95133 e^{727.95133}
  3. e e e e 7.705 < 10 10 10 964 . e^{e^{e^{e^{7.705}}}}<10^{10^{10^{964}}}.
  4. × 10 1 165 \times 10^{1}165
  5. × 10 1 165 \times 10^{1}165
  6. × 10 3 70 \times 10^{3}70
  7. × 10 3 16 \times 10^{3}16
  8. × 10 1 0 \times 10^{1}0
  9. × 10 1 7 \times 10^{1}7
  10. π ( x ) = li ( x ) - li ( x ) 2 - ρ li ( x ρ ) + smaller terms \pi(x)=\operatorname{li}(x)-\frac{\operatorname{li}(\sqrt{x})}{2}-\sum_{\rho}% \operatorname{li}(x^{\rho})+\,\text{smaller terms}
  11. x \sqrt{x}
  12. x \sqrt{x}
  13. li ( x 1 / 2 ) / 2 \mathrm{li}(x^{1/2})/2
  14. li ( x ) \mathrm{li}(x)
  15. p n p^{n}
  16. 1 / n 1/n
  17. li ( x 1 / 2 ) / 2 \mathrm{li}(x^{1/2})/2

Skin_effect.html

  1. J = J S e - d / δ J=J_{\mathrm{S}}\,e^{-{d/\delta}}
  2. δ = 2 ρ ω μ r μ 0 \delta=\sqrt{{2\rho}\over{\omega\mu_{r}\mu_{0}}}
  3. ρ \rho
  4. ω \omega
  5. μ r \mu_{r}
  6. μ 0 \mu_{0}
  7. δ = ( 1 ω ) { ( μ ϵ 2 ) [ ( 1 + ( 1 ρ ω ϵ ) 2 ) 1 / 2 - 1 ] } - 1 / 2 \delta=\left({1\over\omega}\right)\left\{\left({{\mu\epsilon}\over 2}\right)% \left[\left(1+\left({1\over{\rho\omega\epsilon}}\right)^{2}\right)^{1/2}-1% \right]\right\}^{-1/2}
  8. μ \mu
  9. μ r \mu_{r}
  10. μ 0 \mu_{0}
  11. ϵ r \epsilon_{r}
  12. ϵ 0 \epsilon_{0}
  13. ϵ \epsilon
  14. ϵ r \epsilon_{r}
  15. ϵ 0 \epsilon_{0}
  16. ϵ \epsilon
  17. ϵ \epsilon
  18. δ = 2 ρ ω μ 1 + ( ρ ω ϵ ) 2 + ρ ω ϵ \delta=\sqrt{{2\rho}\over{\omega\mu}}\;\;\sqrt{\sqrt{1+\left({\rho\omega% \epsilon}\right)^{2}}+\rho\omega\epsilon}
  19. 1 / ρ ϵ 1/\rho\epsilon
  20. 1 / ρ ϵ 1/\rho\epsilon
  21. δ 2 ρ ϵ μ \delta\approx{2\rho}\sqrt{\epsilon\over\mu}
  22. ρ \rho
  23. R L ρ π ( D - δ ) δ L ρ π D δ R\approx{{L\rho}\over{\pi(D-\delta)\delta}}\approx{{L\rho}\over{\pi D\delta}}
  24. D δ D\gg\delta
  25. D W = 200 mm f / Hz D_{\mathrm{W}}={\frac{200~{}\mathrm{mm}}{\sqrt{f/\mathrm{Hz}}}}
  26. δ = 2 ρ ( 2 π f ) ( μ 0 μ r ) 503 ρ μ r f \delta=\sqrt{{2\rho}\over{(2\pi f)(\mu_{0}\mu_{r})}}\approx 503\,\sqrt{\frac{% \rho}{\mu_{r}f}}
  27. δ = \delta=
  28. μ r = \mu_{r}=
  29. ρ = \rho=
  30. ρ = 1 / σ \rho=1/\sigma
  31. f = f=
  32. μ r = \mu_{r}=
  33. δ = 503 2.44 10 - 8 1 50 = 11.1 mm \delta=503\,\sqrt{\frac{2.44\cdot 10^{-8}}{1\cdot 50}}=11.1\,\mathrm{mm}
  34. 9 = 3 \sqrt{9}=3
  35. μ r \mu_{r}
  36. L cen L\text{cen}\,
  37. r < a r<a\,
  38. L ext L\text{ext}\,
  39. a < r < b a<r<b\,
  40. L shd L\text{shd}\,
  41. b < r < c b<r<c\,
  42. L total = L cen + L shd + L ext L\text{total}=L\text{cen}+L\text{shd}+L\text{ext}\,
  43. L ext L\text{ext}\,
  44. L / D = μ 0 2 π ln ( b a ) L/D=\frac{\mu_{0}}{2\pi}\ln\left(\frac{b}{a}\right)\,
  45. L DC = L cen + L shd + L ext L\text{DC}=L\text{cen}+L\text{shd}+L\text{ext}\,
  46. L = L ext L_{\infty}=L\text{ext}\,
  47. b < r < c b<r<c\,
  48. L ext L\text{ext}
  49. L ( f ) = l 0 + l ( f f m ) b 1 + ( f f m ) b L(f)=\frac{l_{0}+l_{\infty}(\frac{f}{f_{m}})^{b}}{1+(\frac{f}{f_{m}})^{b}}\,

Slinky.html

  1. 1 / 2 {1}/{2}
  2. T = 2 π m k . T=2\pi\sqrt{\frac{m}{k}}.
  3. L = W 2 k . L=\frac{W}{2k}.
  4. p ( n ) = L ( n - 1 ) 2 . p(n)=L(n-1)^{2}.

Small_number.html

  1. 1 360 × 60 × 60 \frac{1}{360\times 60\times 60}
  2. 10 - 10 10 10 10 4.829 * 10 183230 10^{\,\!-10^{10^{10^{10^{4.829*10^{183230}}}}}}

Small_population_size.html

  1. 1 / 2 n - 1 1/2^{n-1}
  2. 1 / 2 n 1/2^{n}

Smoothsort.html

  1. O ( n ) \scriptstyle O(n)

SN1_reaction.html

  1. log ( k k 0 ) = m Y \log{\left(\frac{k}{k_{0}}\right)}=mY\,

Snake_lemma.html

  1. ker a \color G r a y ker b \color G r a y ker c 𝑑 coker a \color G r a y coker b \color G r a y coker c \ker a~{}{\color{Gray}\longrightarrow}~{}\ker b~{}{\color{Gray}\longrightarrow% }~{}\ker c~{}\overset{d}{\longrightarrow}~{}\operatorname{coker}a~{}{\color{% Gray}\longrightarrow}~{}\operatorname{coker}b~{}{\color{Gray}\longrightarrow}~% {}\operatorname{coker}c

Snub_cube.html

  1. s { 4 3 } s\begin{Bmatrix}4\\ 3\end{Bmatrix}
  2. 6 + 8 3 \scriptstyle{6+8\sqrt{3}}
  3. 613 t + 203 9 ( 35 t - 62 ) \sqrt{\tfrac{613t+203}{9(35t-62)}}
  4. 1 3 ( 1 + 19 - 3 33 3 + 19 + 3 33 3 ) 1.83929 \tfrac{1}{3}\scriptstyle{\left(1+\sqrt[3]{19-3\sqrt{33}}+\sqrt[3]{19+3\sqrt{33% }}\right)\approx 1.83929}
  5. 1 t + 1 0.593465 \tfrac{1}{\sqrt{t+1}}\scriptstyle{\approx 0.593465}
  6. 1 2 t + 1 0.842509 \tfrac{1}{2}\scriptstyle{\sqrt{t+1}\approx 0.842509}
  7. α 6 - 4 α 4 + 16 α 2 - 32 = 0 , \alpha^{6}-4\alpha^{4}+16\alpha^{2}-32=0,\,
  8. α = 4 3 - 16 3 β + 2 β 3 1.60972 \alpha=\sqrt{\frac{4}{3}-\frac{16}{3\beta}+\frac{2\beta}{3}}\approx 1.60972
  9. β = 26 + 6 33 3 \beta=\sqrt[3]{26+6\sqrt{33}}

Snub_dodecahedron.html

  1. s { 5 3 } s\begin{Bmatrix}5\\ 3\end{Bmatrix}
  2. ξ = ϕ 2 + 1 2 ϕ - 5 27 3 + ϕ 2 - 1 2 ϕ - 5 27 3 \xi=\sqrt[3]{\frac{\phi}{2}+\frac{1}{2}\sqrt{\phi-\frac{5}{27}}}+\sqrt[3]{% \frac{\phi}{2}-\frac{1}{2}\sqrt{\phi-\frac{5}{27}}}
  3. A = 20 3 + 3 25 + 10 5 55.28674495844515 A=20\sqrt{3}+3\sqrt{25+10\sqrt{5}}\approx 55.28674495844515
  4. V = 12 ξ 2 ( 3 ϕ + 1 ) - ξ ( 36 ϕ + 7 ) - ( 53 ϕ + 6 ) 6 3 - ξ 2 3 37.61664996273336 V=\frac{12\xi^{2}(3\phi+1)-\xi(36\phi+7)-(53\phi+6)}{6\sqrt{3-\xi^{2}}^{3}}% \approx 37.61664996273336

Soap_film.html

  1. V = 2 γ ρ h V=\sqrt{\frac{2\gamma}{\rho h}}
  2. γ \gamma
  3. ρ \rho
  4. h h

Social_credit.html

  1. real cost (production) = M T 1 T 2 d C d t d t T 1 T 2 d P d t d t \,\text{real cost (production)}=M\cdot\cfrac{\int_{T_{1}}^{T_{2}}\frac{dC}{dt}% \,dt}{\int_{T_{1}}^{T_{2}}\frac{dP}{dt}\,dt}
  2. true price ( $ ) = cost ( $ ) consumption ( $ ) + depreciation ( $ ) credit ( $ ) + production ( $ ) \,\text{true price }(\$)=\,\text{cost }(\$)\cdot\dfrac{\,\text{consumption }(% \$)+\,\text{depreciation }(\$)}{\,\text{credit }(\$)+\,\text{production }(\$)}

Social_welfare_function.html

  1. W = i = 1 n Y i W=\sum_{i=1}^{n}Y_{i}
  2. W W
  3. Y i Y_{i}
  4. i i
  5. n n
  6. W = 1 n i = 1 n Y i = Y ¯ W=\frac{1}{n}\sum_{i=1}^{n}Y_{i}=\overline{Y}
  7. W = min ( Y 1 , Y 2 , , Y n ) W=\min(Y_{1},Y_{2},\cdots,Y_{n})
  8. W Gini = Y ¯ ( 1 - G ) W_{\mathrm{Gini}}=\overline{Y}\left(1-G\right)
  9. ( 1 - G ) (1-G)
  10. G G
  11. W Theil - L = Y ¯ e - T L W_{\mathrm{Theil-L}}=\overline{Y}\mathrm{e}^{-T_{L}}
  12. W Theil - T - 1 = Y ¯ e T T W^{-1}_{\mathrm{Theil-T}}=\overline{Y}\mathrm{e}^{T_{T}}

Soft_tissue.html

  1. 𝐒 = 𝐒 ( 𝐄 , 𝐄 ˙ ) 𝐒 = 𝐒 ( 𝐄 ) \mathbf{S}=\mathbf{S}(\mathbf{E},\dot{\mathbf{E}})\quad\rightarrow\quad\mathbf% {S}=\mathbf{S}(\mathbf{E})
  2. W = 1 2 [ q + c ( e Q - 1 ) ] W=\frac{1}{2}\left[q+c\left(e^{Q}-1\right)\right]
  3. q = a i j k l E i j E k l Q = b i j k l E i j E k l q=a_{ijkl}E_{ij}E_{kl}\qquad Q=b_{ijkl}E_{ij}E_{kl}
  4. E i j E_{ij}
  5. a i j k l a_{ijkl}
  6. b i j k l b_{ijkl}
  7. c c
  8. w w
  9. λ i \lambda_{i}
  10. W = 1 2 [ a ( λ 1 2 + λ 2 2 + λ 3 2 - 3 ) + b ( e c ( λ 1 2 + λ 2 2 + λ 3 2 - 3 ) - 1 ) ] W=\frac{1}{2}\left[a(\lambda_{1}^{2}+\lambda_{2}^{2}+\lambda_{3}^{2}-3)+b\left% (e^{c(\lambda_{1}^{2}+\lambda_{2}^{2}+\lambda_{3}^{2}-3)}-1\right)\right]
  11. W = 1 2 a i j k l E i j E k l W=\frac{1}{2}a_{ijkl}E_{ij}E_{kl}
  12. W = 1 2 c ( e b i j k l E i j E k l - 1 ) W=\frac{1}{2}c\left(e^{b_{ijkl}E_{ij}E_{kl}}-1\right)
  13. W = - μ J m 2 ln ( 1 - ( λ 1 2 + λ 2 2 + λ 3 2 - 3 J m ) ) W=-\frac{\mu J_{m}}{2}\ln\left(1-\left(\frac{\lambda_{1}^{2}+\lambda_{2}^{2}+% \lambda_{3}^{2}-3}{J_{m}}\right)\right)
  14. μ > 0 \mu>0
  15. J m > 0 Jm>0
  16. J m Jm
  17. λ m 2 + 2 λ m - J m - 3 = 0 \lambda_{m}^{2}+2\lambda_{m}-J_{m}-3=0

Solar_luminosity.html

  1. L = 4 π k I A 2 L_{\odot}=4\pi kI_{\odot}A^{2}\,

Solar_mass.html

  1. [ u s o l a r m a s s ] = [ u v a l , u 1.98855 , u 0.00025 , u e = 30 , u u = k i l o g r a m ] [u^{\prime}solarmass^{\prime}]=[u^{\prime}val^{\prime},u^{\prime}1.98855^{% \prime},u^{\prime}0.00025^{\prime},u^{\prime}e=30^{\prime},u^{\prime}u=% kilogram^{\prime}]
  2. G G
  3. M = 4 π 2 × ( 1 AU ) 3 G × ( 1 yr ) 2 M_{\odot}=\frac{4\pi^{2}\times(1\,\mathrm{AU})^{3}}{G\times(1\,\mathrm{yr})^{2}}
  4. [ u s o l a r m a s s ] [u^{\prime}solarmass^{\prime}]
  5. [ u v a l , u 27068510 ] M < s u b > L [u^{\prime}val^{\prime},u^{\prime}27068510^{\prime}]M<sub>L

Solenoid.html

  1. B l = μ 0 N I , Bl=\mu_{0}NI,
  2. μ 0 \mu_{0}
  3. N N
  4. I I
  5. B = μ 0 N I l . B=\mu_{0}\frac{NI}{l}.
  6. B = μ 0 μ r N I l . B=\mu_{0}\mu_{\mathrm{r}}\frac{NI}{l}.
  7. B = μ 0 μ eff N I l = μ N I l , B=\mu_{0}\mu_{\mathrm{eff}}\frac{NI}{l}=\mu\frac{NI}{l},
  8. μ eff = μ r 1 + k ( μ r - 1 ) , \mu_{\mathrm{eff}}=\frac{\mu_{r}}{1+k(\mu_{r}-1)},
  9. K = I L ϕ ^ . \vec{K}=\frac{I}{L}\hat{\phi}.
  10. ( ρ , ϕ , z ) (\rho,\phi,z)
  11. A ϕ = μ 0 I 2 π 1 L a ρ [ ζ k ( k 2 + h 2 - h 2 k 2 h 2 k 2 K ( k 2 ) - 1 k 2 E ( k 2 ) + h 2 - 1 h 2 Π ( h 2 , k 2 ) ) ] ζ - ζ + , A_{\phi}=\frac{\mu_{0}I}{2\pi}\frac{1}{L}\sqrt{\frac{a}{\rho}}\left[\zeta k% \left(\frac{k^{2}+h^{2}-h^{2}k^{2}}{h^{2}k^{2}}K(k^{2})-\frac{1}{k^{2}}E(k^{2}% )+\frac{h^{2}-1}{h^{2}}\Pi(h^{2},k^{2})\right)\right]_{\zeta_{-}}^{\zeta_{+}},
  12. ζ ± = z ± L 2 , \zeta_{\pm}=z\pm\frac{L}{2},
  13. h 2 = 4 a ρ ( a + ρ ) 2 , h^{2}=\frac{4a\rho}{(a+\rho)^{2}},
  14. k 2 = 4 a ρ ( a + ρ ) 2 + ζ 2 , k^{2}=\frac{4a\rho}{(a+\rho)^{2}+\zeta^{2}},
  15. K ( m ) = 0 π / 2 1 1 - m sin 2 θ d θ , K(m)=\int_{0}^{\pi/2}{\frac{1}{\sqrt{1-m\sin^{2}\theta}}}d\theta,
  16. E ( m ) = 0 π / 2 1 - m sin 2 θ d θ , E(m)=\int_{0}^{\pi/2}{\sqrt{1-m\sin^{2}\theta}}d\theta,
  17. Π ( n , m ) = 0 π / 2 1 ( 1 - n sin 2 θ ) 1 - m sin 2 θ d θ . \Pi(n,m)=\int_{0}^{\pi/2}{\frac{1}{(1-n\sin^{2}\theta)\sqrt{1-m\sin^{2}\theta}% }}d\theta.
  18. K ( m ) K(m)
  19. E ( m ) E(m)
  20. Π ( n , m ) \Pi(n,m)
  21. B = × A , \vec{B}=\nabla\times\vec{A},
  22. B ρ = μ 0 I 2 π 1 L a ρ [ k 2 - 2 k K ( k 2 ) + 2 k E ( k 2 ) ] ζ - ζ + , B_{\rho}=\frac{\mu_{0}I}{2\pi}\frac{1}{L}\sqrt{\frac{a}{\rho}}\left[\frac{k^{2% }-2}{k}K(k^{2})+\frac{2}{k}E(k^{2})\right]_{\zeta_{-}}^{\zeta_{+}},
  23. B z = μ 0 I 4 π 1 L 1 a ρ [ ζ k ( K ( k 2 ) + a - ρ a + ρ Π ( h 2 , k 2 ) ) ] ζ - ζ + . B_{z}=\frac{\mu_{0}I}{4\pi}\frac{1}{L}\frac{1}{\sqrt{a\rho}}\left[\zeta k\left% (K(k^{2})+\frac{a-\rho}{a+\rho}\Pi(h^{2},k^{2})\right)\right]_{\zeta_{-}}^{% \zeta_{+}}.
  24. B B
  25. B = μ 0 N i l , B=\mu_{0}\frac{Ni}{l},
  26. N N
  27. i i
  28. l l
  29. B B
  30. A A
  31. Φ = μ 0 N i A l . \Phi=\mu_{0}\frac{NiA}{l}.
  32. L = N Φ i , L=\frac{N\Phi}{i},
  33. L = μ 0 N 2 A l . L=\mu_{0}\frac{N^{2}A}{l}.

Solid_angle.html

  1. d Ω = sin θ d θ d φ d\Omega=\sin\theta\,d\theta\,d\varphi
  2. θ \theta
  3. φ \varphi
  4. Ω = S r ^ n ^ d S r 2 = S sin θ d θ d φ \Omega=\iint_{S}\frac{\hat{r}\cdot\hat{n}\,dS}{r^{2}}\ =\iint_{S}\sin\theta\,d% \theta\,d\varphi
  5. r ^ = r / r \hat{r}=\vec{r}/r
  6. r \vec{r}
  7. d S \,dS
  8. n ^ \hat{n}
  9. d S \,dS
  10. r ^ n ^ \hat{r}\cdot\hat{n}
  11. n ^ \hat{n}
  12. d Ω = 4 π ( d S / A ) ( r ^ n ^ ) d\Omega=4\pi(dS/A)\,(\hat{r}\cdot\hat{n})
  13. A = 4 π r 2 A=4\pi r^{2}
  14. 2 θ 2\theta\,\!
  15. Ω = 2 π ( 1 - cos θ ) \Omega=2\pi\left(1-\cos{\theta}\right)
  16. 0 2 π 0 θ sin θ d θ d ϕ = 2 π 0 θ sin θ d θ = 2 π [ - cos θ ] 0 θ = 2 π ( 1 - cos θ ) \int_{0}^{2\pi}\int_{0}^{\theta}\sin\theta^{\prime}\,d\theta^{\prime}\,d\phi=2% \pi\int_{0}^{\theta}\sin\theta^{\prime}\,d\theta^{\prime}\ =2\pi\left[-\cos% \theta^{\prime}\right]_{0}^{\theta}\ =2\pi\left(1-\cos\theta\right)
  17. 2 r sin ( θ 2 ) 2r\sin\left(\frac{\theta}{2}\right)
  18. Ω = 4 π sin 2 ( θ 2 ) = 2 π ( 1 - cos θ ) \Omega=4\pi\sin^{2}\left(\frac{\theta}{2}\right)\ =2\pi\left(1-\cos{\theta}\right)
  19. 4 π - Ω = 2 π ( 1 + cos θ ) 4\pi\ -\Omega=2\pi\left(1+\cos{\theta}\right)
  20. θ \theta\,\!
  21. 2 π ( 1 + cos θ ) 2\pi\left(1+\cos{\theta}\right)
  22. γ \gamma
  23. Ω = 2 ( arccos sin γ sin θ - cos θ arccos tan γ tan θ ) \Omega=2\left(\arccos\frac{\sin\gamma}{\sin\theta}-\cos\theta\arccos\frac{\tan% \gamma}{\tan\theta}\right)
  24. a , b , c \vec{a}\ ,\,\vec{b}\ ,\,\vec{c}
  25. θ a \theta_{a}\,
  26. θ b , θ c \theta_{b},\,\theta_{c}
  27. ϕ a b \phi_{ab}\,
  28. ϕ b c , ϕ a c \phi_{bc},\,\phi_{ac}
  29. Ω \Omega
  30. Ω = ( ϕ a b + ϕ b c + ϕ a c ) - π \Omega=\left(\phi_{ab}+\phi_{bc}+\phi_{ac}\right)\ -\pi
  31. π \pi
  32. i = 1 4 Ω i = 2 i = 1 6 ϕ i - 4 π \sum_{i=1}^{4}\Omega_{i}=2\sum_{i=1}^{6}\phi_{i}\ -4\pi
  33. ϕ i \phi_{i}\,
  34. Ω \Omega
  35. a , b , c \vec{a}\ ,\,\vec{b}\ ,\,\vec{c}
  36. tan ( 1 2 Ω ) = | a b c | a b c + ( a b ) c + ( a c ) b + ( b c ) a \tan\left(\frac{1}{2}\Omega\right)=\frac{\left|\vec{a}\ \vec{b}\ \vec{c}\right% |}{abc+\left(\vec{a}\cdot\vec{b}\right)c+\left(\vec{a}\cdot\vec{c}\right)b+% \left(\vec{b}\cdot\vec{c}\right)a}
  37. | a b c | \left|\vec{a}\ \vec{b}\ \vec{c}\right|
  38. M i 1 = a i M_{i1}=\vec{a}_{i}
  39. a \vec{a}
  40. a \,a
  41. a b \vec{a}\cdot\vec{b}
  42. π \pi
  43. θ a , θ b , θ c \theta_{a},\,\theta_{b},\,\theta_{c}
  44. tan ( 1 4 Ω ) = tan ( θ s 2 ) tan ( θ s - θ a 2 ) tan ( θ s - θ b 2 ) tan ( θ s - θ c 2 ) \tan\left(\frac{1}{4}\Omega\right)=\sqrt{\tan\left(\frac{\theta_{s}}{2}\right)% \tan\left(\frac{\theta_{s}-\theta_{a}}{2}\right)\tan\left(\frac{\theta_{s}-% \theta_{b}}{2}\right)\tan\left(\frac{\theta_{s}-\theta_{c}}{2}\right)}
  45. θ s = θ a + θ b + θ c 2 \theta_{s}=\frac{\theta_{a}+\theta_{b}+\theta_{c}}{2}
  46. a \,a
  47. b \,b
  48. Ω = 4 arcsin ( sin a 2 sin b 2 ) \Omega=4\arcsin\left(\sin{a\over 2}\sin{b\over 2}\right)
  49. Ω = 4 arctan α β 2 d 4 d 2 + α 2 + β 2 \Omega=4\arctan\frac{\alpha\beta}{2d\sqrt{4d^{2}+\alpha^{2}+\beta^{2}}}
  50. Ω = 2 π - 2 n arctan ( tan π n 1 + r 2 h 2 ) \Omega=2\pi-2n\arctan\left(\frac{\tan{\pi\over n}}{\sqrt{1+{r^{2}\over h^{2}}}% }\right)
  51. { s 1 , s 2 , , s n } \{s_{1},s_{2},...,s_{n}\}
  52. Ω = 2 π - arg j = 1 n ( ( s j - 1 s j ) ( s j s j + 1 ) - ( s j - 1 s j + 1 ) + i [ s j - 1 s j s j + 1 ] ) \Omega=2\pi-\arg\prod_{j=1}^{n}\left(\left(s_{j-1}s_{j}\right)\left(s_{j}s_{j+% 1}\right)-\left(s_{j-1}s_{j+1}\right)+i\left[s_{j-1}s_{j}s_{j+1}\right]\right)
  53. i i
  54. s 0 = s n s_{0}=s_{n}
  55. s n + 1 = s 1 s_{n+1}=s_{1}
  56. ( sin ϕ N - sin ϕ S ) ( θ E - θ W ) sr \left(\sin\phi_{N}-\sin\phi_{S}\right)\left(\theta_{E}-\theta_{W}\,\!\right)% \mathrm{sr}
  57. ϕ N \phi_{N}\,\!
  58. ϕ S \phi_{S}\,\!
  59. θ E \theta_{E}\,\!
  60. θ W \theta_{W}\,\!
  61. ϕ N - ϕ S \phi_{N}-\phi_{S}\,\!
  62. θ E - θ W \theta_{E}-\theta_{W}\,\!
  63. × 10 - 3 \times 10^{-}3
  64. × 10 - 3 \times 10^{-}3
  65. 2 θ 2\theta\,\!
  66. Ω = 2 π ( 1 - cos θ ) \Omega=2\pi\left(1-\cos{\theta}\right)
  67. × 10 - 5 \times 10^{-}5
  68. × 10 - 5 \times 10^{-}5
  69. d d
  70. Ω d = 2 π d 2 Γ ( d 2 ) \Omega_{d}=\frac{2\pi^{\frac{d}{2}}}{\Gamma\left(\frac{d}{2}\right)}
  71. Γ \Gamma
  72. d d
  73. Ω d = { 1 ( d 2 - 1 ) ! 2 π d 2 d even ( 1 2 ( d - 1 ) ) ! ( d - 1 ) ! 2 d π 1 2 ( d - 1 ) d odd \Omega_{d}=\begin{cases}\frac{1}{\left(\frac{d}{2}-1\right)!}2\pi^{\frac{d}{2}% }&d\,\text{ even}\\ \frac{\left(\frac{1}{2}\left(d-1\right)\right)!}{(d-1)!}2^{d}\pi^{\frac{1}{2}(% d-1)}&d\,\text{ odd}\end{cases}

Solvable_group.html

  1. G G
  2. { 1 } = G 0 < G 1 < < G k = G \{1\}=G_{0}<G_{1}<\cdots<G_{k}=G
  3. G j - 1 G_{j-1}
  4. G j G_{j}
  5. G j / G j - 1 G_{j}/G_{j-1}
  6. j = 1 , 2 , , k j=1,2,\dots,k
  7. G G ( 1 ) G ( 2 ) , G\triangleright G^{(1)}\triangleright G^{(2)}\triangleright\cdots,
  8. G ( n ) = { 1 } G^{(n)}=\{1\}
  9. p a q b p^{a}q^{b}

Sone.html

  1. N = ( 10 L N - 40 10 ) 0.30103 2 L N - 40 10 N=\left(10^{\frac{L_{N}-40}{10}}\right)^{0.30103}\approx 2^{\frac{L_{N}-40}{10}}
  2. L N = 40 + 10 log 2 ( N ) L_{N}=40+10\log_{2}(N)

Sonic_boom.html

  1. α \alpha
  2. sin ( α ) = v sound v object \sin(\alpha)=\frac{v\text{sound}}{v\text{object}}
  3. v object v sound \frac{v\text{object}}{v\text{sound}}
  4. α \alpha

Sophie_Germain_prime.html

  1. 2 C n ( ln n ) 2 1.32032 n ( ln n ) 2 2C\frac{n}{(\ln n)^{2}}\approx 1.32032\frac{n}{(\ln n)^{2}}
  2. C = p > 2 p ( p - 2 ) ( p - 1 ) 2 0.660161 C=\prod_{p>2}\frac{p(p-2)}{(p-1)^{2}}\approx 0.660161
  3. p > 2 , p>2,
  4. p 1 , p_{1},
  5. p 2 , p_{2},
  6. p 1 < p 2 , p_{1}<p_{2},
  7. ( p - 1 ) p 1 = p 2 + 1. (p-1)p_{1}=p_{2}+1.
  8. p 2 = 2 a p 1 - 1 , p_{2}=2ap_{1}-1,
  9. 2 a + 1 2a+1

Sorites_paradox.html

  1. p ¬ p p\vee\neg p
  2. p p
  3. p p
  4. v v
  5. L L
  6. A t ( x ) At(x)
  7. x x
  8. x x
  9. 2 A t ( x ) 2^{At(x)}
  10. V V
  11. x x
  12. V ( x ) = True V(x)=\,\text{True}
  13. v ( x ) = True v(x)=\,\text{True}
  14. v v
  15. V ( x ) V(x)
  16. v v
  17. v v^{\prime}
  18. v ( x ) = True v(x)=\,\text{True}
  19. v ( x ) = False v^{\prime}(x)=\,\text{False}
  20. L p L\;p
  21. v v
  22. v v^{\prime}
  23. L p L\;p
  24. v ( L p ) = True v(L\;p)=\,\text{True}
  25. v ( L p ) = False v^{\prime}(L\;p)=\,\text{False}
  26. L p L\;p
  27. L p ¬ L p L\;p\lor\lnot L\;p
  28. True \,\text{True}
  29. H 1000 H\;1000
  30. H 1000 ¬ H 1000 H\;1000\lor\lnot H\;1000

Space_charge.html

  1. J = ( 1 - r ~ ) A 0 T 2 exp ( - ϕ k T ) J=(1-\tilde{r})A_{0}T^{2}\exp\left(\frac{-\phi}{kT}\right)
  2. A 0 = 4 π e m e k 2 h 3 1.2 × 10 5 A_{0}=\frac{4\pi em_{e}k^{2}}{h^{3}}\approx 1.2\times 10^{5}
  3. J = I a S = 4 ϵ 0 9 2 e / m e V a 3 / 2 d 2 J=\frac{I_{a}}{S}=\frac{4\epsilon_{0}}{9}\sqrt{2e/m_{e}}\frac{V_{a}^{3/2}}{d^{% 2}}
  4. e e
  5. m e m_{e}
  6. v v
  7. \mathcal{E}
  8. μ \mu
  9. v = μ v=\mu\mathcal{E}
  10. L L
  11. J J
  12. J = 9 ϵ μ V a 2 8 L 3 J=\frac{9{\epsilon}{\mu}{V_{a}}^{2}}{8{L}^{3}}
  13. V a V_{a}
  14. ϵ \epsilon
  15. J = 2 ϵ v V a L 2 J=\frac{2{\epsilon}{v}{V_{a}}}{{L}^{2}}
  16. J J
  17. V a V_{a}
  18. μ \mu
  19. ϵ \epsilon

Spatial_anti-aliasing.html

  1. cos ( 2 j π x ) cos ( 2 k π y ) \ \cos(2j\pi x)\cos(2k\pi y)

Spearman's_rank_correlation_coefficient.html

  1. ρ \rho
  2. r s r_{s}
  3. ρ \rho
  4. τ \tau
  5. X i , Y i X_{i},Y_{i}
  6. x i , y i x_{i},y_{i}
  7. ρ = 1 - 6 d i 2 n ( n 2 - 1 ) . \rho={1-\frac{6\sum d_{i}^{2}}{n(n^{2}-1)}}.
  8. d i = x i - y i d_{i}=x_{i}-y_{i}
  9. X i X_{i}
  10. x i x_{i}
  11. 2 + 3 2 = 2.5 \frac{2+3}{2}=2.5
  12. 2 + 3 2 = 2.5 \frac{2+3}{2}=2.5
  13. σ = 0.6325 n - 1 \sigma=\frac{0.6325}{\sqrt{n-1}}
  14. X i X_{i}
  15. Y i Y_{i}
  16. d i 2 d^{2}_{i}
  17. X i X_{i}
  18. x i x_{i}
  19. Y i Y_{i}
  20. y i y_{i}
  21. d i d_{i}
  22. x i x_{i}
  23. y i y_{i}
  24. d i 2 d^{2}_{i}
  25. d i d_{i}
  26. X i X_{i}
  27. Y i Y_{i}
  28. x i x_{i}
  29. y i y_{i}
  30. d i d_{i}
  31. d i 2 d^{2}_{i}
  32. d i 2 d^{2}_{i}
  33. d i 2 = 194 \sum d_{i}^{2}=194
  34. ρ = 1 - 6 d i 2 n ( n 2 - 1 ) . \rho=1-{\frac{6\sum d_{i}^{2}}{n(n^{2}-1)}}.
  35. ρ = 1 - 6 × 194 10 ( 10 2 - 1 ) \rho=1-{\frac{6\times 194}{10(10^{2}-1)}}
  36. F ( r ) = 1 2 ln 1 + r 1 - r = artanh ( r ) . F(r)={1\over 2}\ln{1+r\over 1-r}=\operatorname{artanh}(r).
  37. z = n - 3 1.06 F ( r ) z=\sqrt{\frac{n-3}{1.06}}F(r)
  38. t = r n - 2 1 - r 2 t=r\sqrt{\frac{n-2}{1-r^{2}}}

Special_linear_group.html

  1. det : GL ( n , F ) F × . \det\colon\operatorname{GL}(n,F)\to F^{\times}.
  2. 𝔰 𝔩 ( n , F ) \mathfrak{sl}(n,F)
  3. Alt ( 3 ) [ GL ( 2 , 𝐅 2 ) , GL ( 2 , 𝐅 2 ) ] < E ( 2 , 𝐅 2 ) = SL ( 2 , 𝐅 2 ) = GL ( 2 , 𝐅 2 ) Sym ( 3 ) , \operatorname{Alt}(3)\cong[\operatorname{GL}(2,\mathbf{F}_{2}),\operatorname{% GL}(2,\mathbf{F}_{2})]<\operatorname{E}(2,\mathbf{F}_{2})=\operatorname{SL}(2,% \mathbf{F}_{2})=\operatorname{GL}(2,\mathbf{F}_{2})\cong\operatorname{Sym}(3),
  4. [ T i j , T j k ] = T i k for i k [ T i j , T k l ] = 𝟏 for i l , j k ( T 12 T 21 - 1 T 12 ) 4 = 𝟏 \begin{aligned}\displaystyle\left[T_{ij},T_{jk}\right]&\displaystyle=T_{ik}&&% \displaystyle\mbox{for }~{}i\neq k\\ \displaystyle\left[T_{ij},T_{kl}\right]&\displaystyle=\mathbf{1}&&% \displaystyle\mbox{for }~{}i\neq l,j\neq k\\ \displaystyle(T_{12}T_{21}^{-1}T_{12})^{4}&\displaystyle=\mathbf{1}\\ \end{aligned}

Special_unitary_group.html

  1. n n
  2. S U ( n ) SU(n)
  3. n × n n×n
  4. U ( n ) U(n)
  5. n × n n×n
  6. U ( n ) U(n)
  7. S U ( n ) U ( n ) G L ( n , 𝐂 ) SU(n)⊂U(n)⊂GL(n,\mathbf{C})
  8. S U ( n ) SU(n)
  9. S U ( 2 ) SU(2)
  10. S U ( 3 ) SU(3)
  11. S U ( 1 ) SU(1)
  12. S U ( 2 ) SU(2)
  13. S U ( 2 ) SU(2)
  14. S O ( 3 ) SO(3)
  15. S U ( 2 ) SU(2)
  16. S U ( n ) SU(n)
  17. S U ( n ) SU(n)
  18. ζ I ζI
  19. ζ ζ
  20. n n
  21. I I
  22. S U ( 2 ) SU(2)
  23. n 1 n−1
  24. S U ( n ) SU(n)
  25. 𝐬𝐮 ( n ) \mathbf{su}(n)
  26. n × n n×n
  27. n × n n×n
  28. i −i
  29. 𝐬𝐮 ( n ) \mathbf{su}(n)
  30. O ^ i j \hat{O}_{ij}
  31. i , j = 1 , 2 , , n i,j=1,2,...,n
  32. [ O ^ i j , O ^ k ] = δ j k O ^ i - δ i O ^ k j \left[\hat{O}_{ij},\hat{O}_{k\ell}\right]=\delta_{jk}\hat{O}_{i\ell}-\delta_{i% \ell}\hat{O}_{kj}
  33. N ^ = i = 1 n O ^ i i \hat{N}=\sum_{i=1}^{n}\hat{O}_{ii}
  34. [ N ^ , O ^ i j ] = 0 , \left[\hat{N},\hat{O}_{ij}\right]=0,
  35. 𝐬𝐮 ( n ) \mathbf{su}(n)
  36. n × n n×n
  37. T a T b = 1 2 n δ a b I n + 1 2 c = 1 n 2 - 1 ( i f a b c + d a b c ) T c T_{a}T_{b}=\frac{1}{2n}\delta_{ab}I_{n}+\frac{1}{2}\sum_{c=1}^{n^{2}-1}{(if_{% abc}+d_{abc})T_{c}}\,
  38. f f
  39. d d
  40. [ T a , T b ] + = 1 n δ a b I n + c = 1 n 2 - 1 d a b c T c \left[T_{a},T_{b}\right]_{+}=\frac{1}{n}\delta_{ab}I_{n}+\sum_{c=1}^{n^{2}-1}{% d_{abc}T_{c}}\,
  41. [ T a , T b ] - = i c = 1 n 2 - 1 f a b c T c . \left[T_{a},T_{b}\right]_{-}=i\sum_{c=1}^{n^{2}-1}{f_{abc}T_{c}}\,.
  42. c , e = 1 n 2 - 1 d a c e d b c e = n 2 - 4 n δ a b \sum_{c,e=1}^{n^{2}-1}d_{ace}d_{bce}=\frac{n^{2}-4}{n}\delta_{ab}\,
  43. ( T a ) j k = - i f a j k . (T_{a})_{jk}=-if_{ajk}.
  44. S U ( 2 ) SU(2)
  45. SU ( 2 ) = { ( α - β ¯ β α ¯ ) : α , β 𝐂 , | α | 2 + | β | 2 = 1 } , \mathrm{SU}(2)=\left\{\begin{pmatrix}\alpha&-\overline{\beta}\\ \beta&\overline{\alpha}\end{pmatrix}:\ \ \alpha,\beta\in\mathbf{C},|\alpha|^{2% }+|\beta|^{2}=1\right\}~{},
  46. φ : 𝐂 2 M ( 2 , 𝐂 ) φ ( α , β ) = ( α - β ¯ β α ¯ ) , \begin{aligned}\displaystyle\varphi:\mathbf{C}^{2}&\displaystyle\to% \operatorname{M}(2,\mathbf{C})\\ \displaystyle\varphi(\alpha,\beta)&\displaystyle=\begin{pmatrix}\alpha&-% \overline{\beta}\\ \beta&\overline{\alpha}\end{pmatrix},\end{aligned}
  47. M ( 2 , 𝐂 ) M(2,\mathbf{C})
  48. M ( 2 , 𝐂 ) M(2,\mathbf{C})
  49. φ φ
  50. φ φ
  51. M ( 2 , 𝐂 ) M(2,\mathbf{C})
  52. S U ( 2 ) SU(2)
  53. S U ( 2 ) SU(2)
  54. S U ( 2 ) SU(2)
  55. 𝔰 𝔲 ( 2 ) = { ( i a - z ¯ z - i a ) : a 𝐑 , z 𝐂 } . \mathfrak{su}(2)=\left\{\begin{pmatrix}ia&-\overline{z}\\ z&-ia\end{pmatrix}:\ a\in\mathbf{R},z\in\mathbf{C}\right\}~{}.
  56. u 1 = ( 0 i i 0 ) u 2 = ( 0 - 1 1 0 ) u 3 = ( i 0 0 - i ) , u_{1}=\begin{pmatrix}0&i\\ i&0\end{pmatrix}\qquad u_{2}=\begin{pmatrix}0&-1\\ 1&0\end{pmatrix}\qquad u_{3}=\begin{pmatrix}i&0\\ 0&-i\end{pmatrix}~{},
  57. [ u 3 , u 1 ] = 2 u 2 , [ u 1 , u 2 ] = 2 u 3 , [ u 2 , u 3 ] = 2 u 1 . [u_{3},u_{1}]=2u_{2},\qquad[u_{1},u_{2}]=2u_{3},\qquad[u_{2},u_{3}]=2u_{1}~{}.
  58. S U ( 2 ) SU(2)
  59. 𝐬𝐮 ( 3 ) \mathbf{su}(3)
  60. T T
  61. T a = λ a 2 . T_{a}=\frac{\lambda_{a}}{2}.\,
  62. λ λ
  63. S U ( 3 ) SU(3)
  64. S U ( 2 ) SU(2)
  65. λ 1 = ( 0 1 0 1 0 0 0 0 0 ) \lambda_{1}=\begin{pmatrix}0&1&0\\ 1&0&0\\ 0&0&0\end{pmatrix}
  66. λ 2 = ( 0 - i 0 i 0 0 0 0 0 ) \lambda_{2}=\begin{pmatrix}0&-i&0\\ i&0&0\\ 0&0&0\end{pmatrix}
  67. λ 3 = ( 1 0 0 0 - 1 0 0 0 0 ) \lambda_{3}=\begin{pmatrix}1&0&0\\ 0&-1&0\\ 0&0&0\end{pmatrix}
  68. λ 4 = ( 0 0 1 0 0 0 1 0 0 ) \lambda_{4}=\begin{pmatrix}0&0&1\\ 0&0&0\\ 1&0&0\end{pmatrix}
  69. λ 5 = ( 0 0 - i 0 0 0 i 0 0 ) \lambda_{5}=\begin{pmatrix}0&0&-i\\ 0&0&0\\ i&0&0\end{pmatrix}
  70. λ 6 = ( 0 0 0 0 0 1 0 1 0 ) \lambda_{6}=\begin{pmatrix}0&0&0\\ 0&0&1\\ 0&1&0\end{pmatrix}
  71. λ 7 = ( 0 0 0 0 0 - i 0 i 0 ) \lambda_{7}=\begin{pmatrix}0&0&0\\ 0&0&-i\\ 0&i&0\end{pmatrix}
  72. λ 8 = 1 3 ( 1 0 0 0 1 0 0 0 - 2 ) . \lambda_{8}=\frac{1}{\sqrt{3}}\begin{pmatrix}1&0&0\\ 0&1&0\\ 0&0&-2\end{pmatrix}.
  73. λ a \lambda_{a}
  74. H H
  75. [ T a , T b ] = i c = 1 8 f a b c T c \left[T_{a},T_{b}\right]=i\sum_{c=1}^{8}{f_{abc}T_{c}}\,
  76. { T a , T b } = 1 3 δ a b + c = 1 8 d a b c T c \{T_{a},T_{b}\}=\frac{1}{3}\delta_{ab}+\sum_{c=1}^{8}{d_{abc}T_{c}}\,
  77. { λ a , λ b } = 4 3 δ a b + 2 c = 1 8 d a b c λ c \{\lambda_{a},\lambda_{b}\}=\frac{4}{3}\delta_{ab}+2\sum_{c=1}^{8}{d_{abc}% \lambda_{c}}
  78. f f
  79. f 123 = 1 f_{123}=1\,
  80. f 147 = - f 156 = f 246 = f 257 = f 345 = - f 367 = 1 2 f_{147}=-f_{156}=f_{246}=f_{257}=f_{345}=-f_{367}=\frac{1}{2}\,
  81. f 458 = f 678 = 3 2 , f_{458}=f_{678}=\frac{\sqrt{3}}{2},\,
  82. f a b c f_{abc}
  83. d d
  84. d 118 = d 228 = d 338 = - d 888 = 1 3 d_{118}=d_{228}=d_{338}=-d_{888}=\frac{1}{\sqrt{3}}\,
  85. d 448 = d 558 = d 668 = d 778 = - 1 2 3 d_{448}=d_{558}=d_{668}=d_{778}=-\frac{1}{2\sqrt{3}}\,
  86. d 146 = d 157 = - d 247 = d 256 = d 344 = d 355 = - d 366 = - d 377 = 1 2 . d_{146}=d_{157}=-d_{247}=d_{256}=d_{344}=d_{355}=-d_{366}=-d_{377}=\frac{1}{2}.\,
  87. H H
  88. exp ( i θ H ) = \exp\left(i\theta H\right)=
  89. [ - 1 3 I sin ( ϕ + 2 π / 3 ) sin ( ϕ - 2 π / 3 ) - 1 2 3 H sin ( ϕ ) - 1 4 H 2 ] exp ( 2 3 i θ sin ϕ ) cos ( ϕ + 2 π / 3 ) cos ( ϕ - 2 π / 3 ) \left[-\tfrac{1}{3}~{}I\sin\left(\phi+2\pi/3\right)\sin\left(\phi-2\pi/3\right% )-\tfrac{1}{2\sqrt{3}}~{}H\sin\left(\phi\right)-\tfrac{1}{4}~{}H^{2}\right]% \frac{\exp\left(\frac{2}{\sqrt{3}}~{}i\theta\sin\phi\right)}{\cos\left(\phi+2% \pi/3\right)\cos\left(\phi-2\pi/3\right)}
  90. + [ - 1 3 I sin ( ϕ ) sin ( ϕ - 2 π / 3 ) - 1 2 3 H sin ( ϕ + 2 π / 3 ) - 1 4 H 2 ] exp ( 2 3 i θ sin ( ϕ + 2 π / 3 ) ) cos ( ϕ ) cos ( ϕ - 2 π / 3 ) +\left[-\tfrac{1}{3}~{}I\sin\left(\phi\right)\sin\left(\phi-2\pi/3\right)-% \tfrac{1}{2\sqrt{3}}~{}H\sin\left(\phi+2\pi/3\right)-\tfrac{1}{4}~{}H^{2}% \right]\frac{\exp\left(\frac{2}{\sqrt{3}}~{}i\theta\sin\left(\phi+2\pi/3\right% )\right)}{\cos\left(\phi\right)\cos\left(\phi-2\pi/3\right)}
  91. + [ - 1 3 I sin ( ϕ ) sin ( ϕ + 2 π / 3 ) - 1 2 3 H sin ( ϕ - 2 π / 3 ) - 1 4 H 2 ] exp ( 2 3 i θ sin ( ϕ - 2 π / 3 ) ) cos ( ϕ ) cos ( ϕ + 2 π / 3 ) +\left[-\tfrac{1}{3}~{}I\sin\left(\phi\right)\sin\left(\phi+2\pi/3\right)-% \tfrac{1}{2\sqrt{3}}~{}H\sin\left(\phi-2\pi/3\right)-\tfrac{1}{4}~{}H^{2}% \right]\frac{\exp\left(\frac{2}{\sqrt{3}}~{}i\theta\sin\left(\phi-2\pi/3\right% )\right)}{\cos\left(\phi\right)\cos\left(\phi+2\pi/3\right)}
  92. ϕ 1 3 ( arccos ( 3 2 3 det H ) - π 2 ) \phi\equiv\tfrac{1}{3}\left(\arccos\left(\tfrac{3}{2}\sqrt{3}\det H\right)-% \tfrac{\pi}{2}\right)
  93. n × n n×n
  94. n × n n×n
  95. 𝐡 \mathbf{h}
  96. i i
  97. i i
  98. n n
  99. n n
  100. S U ( n ) SU(n)
  101. n n
  102. n n
  103. n n
  104. ( 1 , 1 , 0 , , 0 ) (1,−1,0,...,0)
  105. ( 1 , - 1 , 0 , , 0 ) , ( 0 , 1 , - 1 , , 0 ) , ( 0 , 0 , 0 , , 1 , - 1 ) . \begin{aligned}\displaystyle(&\displaystyle 1,-1,0,\dots,0),\\ \displaystyle(&\displaystyle 0,1,-1,\dots,0),\\ &\displaystyle\ldots\\ \displaystyle(&\displaystyle 0,0,0,\dots,1,-1).\end{aligned}
  106. ( 2 - 1 0 0 - 1 2 - 1 0 0 - 1 2 0 0 0 0 2 ) . \begin{pmatrix}2&-1&0&\dots&0\\ -1&2&-1&\dots&0\\ 0&-1&2&\dots&0\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ 0&0&0&\dots&2\end{pmatrix}.
  107. F F
  108. S U ( p , q ; F ) SU(p,q;F)
  109. F F
  110. ( p , q ) (p,q)
  111. p q pq
  112. F F
  113. F F
  114. A A
  115. p q pq
  116. G L ( n , 𝐑 ) GL(n,\mathbf{R})
  117. M SU ( p , q , R ) M\in\mathrm{SU}(p,q,R)
  118. M * A M = A det M = 1. \begin{aligned}\displaystyle M^{*}AM&\displaystyle=A\\ \displaystyle\det M&\displaystyle=1.\end{aligned}
  119. S U ( p , q ) SU(p,q)
  120. 𝐂 \mathbf{C}
  121. A A
  122. A = [ 0 0 i 0 I n - 2 0 - i 0 0 ] . A=\begin{bmatrix}0&0&i\\ 0&I_{n-2}&0\\ -i&0&0\end{bmatrix}.
  123. A A
  124. 𝐂 \mathbf{C}
  125. S U ( 2 , 1 ; 𝐙 i i ) ) SU(2,1;\mathbf{Z}ii))
  126. S L ( 2 , 9 ; 𝐙 ) SL(2,9;\mathbf{Z})
  127. S U ( 1 , 1 ; 𝐂 ) SU(1,1;\mathbf{C})
  128. S L ( 2 , 𝐑 ) SL(2,\mathbf{R})
  129. S U ( n ) SU(n)
  130. SU ( n ) SU ( p ) × SU ( n - p ) × U ( 1 ) \mathrm{SU}(n)\supset\mathrm{SU}(p)\times\mathrm{SU}(n-p)\times\mathrm{U}(1)
  131. U ( 1 ) U(1)
  132. SU ( n ) SO ( n ) , SU ( 2 n ) Sp ( n ) . \begin{aligned}\displaystyle\mathrm{SU}(n)&\displaystyle\supset\mathrm{SO}(n),% \\ \displaystyle\mathrm{SU}(2n)&\displaystyle\supset\mathrm{Sp}(n).\end{aligned}
  133. S U ( n ) SU(n)
  134. U ( 1 ) U(1)
  135. S U ( n ) SU(n)
  136. SO ( 2 n ) SU ( n ) Sp ( n ) SU ( n ) Spin ( 4 ) = SU ( 2 ) × SU ( 2 ) E 6 SU ( 6 ) E 7 SU ( 8 ) G 2 SU ( 3 ) \begin{aligned}\displaystyle\mathrm{SO}(2n)&\displaystyle\supset\mathrm{SU}(n)% \\ \displaystyle\mathrm{Sp}(n)&\displaystyle\supset\mathrm{SU}(n)\\ \displaystyle\mathrm{Spin}(4)&\displaystyle=\mathrm{SU}(2)\times\mathrm{SU}(2)% \\ \displaystyle\mathrm{E}_{6}&\displaystyle\supset\mathrm{SU}(6)\\ \displaystyle\mathrm{E}_{7}&\displaystyle\supset\mathrm{SU}(8)\\ \displaystyle\mathrm{G}_{2}&\displaystyle\supset\mathrm{SU}(3)\end{aligned}
  137. S U ( 2 ) SU(2)
  138. S O ( 3 ) SO(3)
  139. P S U ( n ) PSU(n)
  140. U ( n ) U(n)
  141. S U ( n ) SU(n)
  142. < s u p > n ℂ<sup>n
  143. S U ( 2 ) S O ( 3 ) SU(2)→SO(3)
  144. S p ( n ) Sp(n)
  145. S p ( 2 n , 𝐂 ) Sp(2n,\mathbf{C})
  146. U S p ( 2 n ) USp(2n)
  147. S p ( n ) Sp(n)
  148. 2 n × 2 n 2n×2n

Spectral_density.html

  1. x ( t ) x(t)
  2. x ( t ) x(t)
  3. x ( t ) x(t)
  4. μ \mu
  5. S ( f ) = S(f)=
  6. f f
  7. [ a , b ) [a,b)
  8. x ( t ) x(t)
  9. [ a , b ) [a,b)
  10. S ( b ) - S ( a ) S(b)-S(a)
  11. S S
  12. x x
  13. S S
  14. S S^{\prime}
  15. f f
  16. f f
  17. x x
  18. x x
  19. x x
  20. X ( t ) X(t)
  21. x x
  22. x x
  23. μ \mu
  24. S S
  25. x x
  26. x 2 x^{2}
  27. X n X_{n}
  28. X t X_{t}
  29. x ( t ) x(t)
  30. x ( t ) x(t)
  31. X X
  32. E \operatorname{E}
  33. Var \operatorname{Var}
  34. x n x_{n}
  35. n = 0 n=0
  36. N - 1 N-1
  37. x n \displaystyle x_{n}
  38. x n x_{n}
  39. 1 N n = 0 N - 1 x n 2 \frac{1}{N}\sum_{n=0}^{N-1}x_{n}^{2}
  40. N N\rightarrow\infty
  41. lim N 1 N n = 0 N - 1 x n 2 . \lim_{N\rightarrow\infty}\frac{1}{N}\sum_{n=0}^{N-1}x_{n}^{2}.
  42. x ( t ) = k A k sin ( 2 π ν k t + ϕ k ) x(t)=\sum_{k}A_{k}\cdot\sin(2\pi\nu_{k}t+\phi_{k})
  43. lim T 1 2 T - T T x ( t ) 2 d t . \lim_{T\rightarrow\infty}\frac{1}{2T}\int_{-T}^{T}x(t)^{2}dt.
  44. sin \sin
  45. 1 / 2 1/\sqrt{2}
  46. A k sin ( 2 π ν k t + ϕ k ) A_{k}\sin(2\pi\nu_{k}t+\phi_{k})
  47. A k 2 / 2 A_{k}^{2}/2
  48. x ( t ) x(t)
  49. ν k \nu_{k}
  50. A k 2 / 2. A_{k}^{2}/2.
  51. x ( t ) x(t)
  52. A k 2 / 2 A_{k}^{2}/2
  53. S ( ν ) S(\nu)
  54. S ( ν ) = k : ν k < ν A k 2 / 2. S(\nu)=\sum_{k:\nu_{k}<\nu}A_{k}^{2}/2.
  55. S S
  56. x x
  57. τ \tau
  58. x ( t ) x(t)
  59. x ( t + τ ) x(t+\tau)
  60. c c
  61. x x
  62. c ( τ ) = lim T 1 2 T - T T x ( t ) x ( t + τ ) d t . c(\tau)=\lim_{T\rightarrow\infty}\frac{1}{2T}\int_{-T}^{T}x(t)x(t+\tau)dt.
  63. τ \tau
  64. c c
  65. c ( 0 ) c(0)
  66. c c
  67. x x
  68. c ( τ ) = k 1 2 A k 2 cos ( 2 π ν k τ ) . c(\tau)=\sum_{k}\frac{1}{2}A_{k}^{2}\cos(2\pi\nu_{k}\tau).
  69. c c
  70. x x
  71. c c
  72. x ( t ) x(t)
  73. - | x ( t ) | 2 d t . \int\limits_{-\infty}^{\infty}|x(t)|^{2}\,dt.
  74. x ^ ( f ) = - e - 2 π i f t x ( t ) d t . \hat{x}(f)=\int\limits_{-\infty}^{\infty}e^{-2\pi ift}x(t)dt.
  75. - | x ( t ) | 2 d t = - | x ^ ( f ) | 2 d f . \int\limits_{-\infty}^{\infty}|x(t)|^{2}\,dt=\int\limits_{-\infty}^{\infty}|% \hat{x}(f)|^{2}\,df.
  76. f f
  77. ω = 2 π f \omega=2\pi f
  78. | x ^ ( f ) | 2 |\hat{x}(f)|^{2}
  79. f f
  80. x ( t ) x(t)
  81. S x x ( f ) = | x ^ ( f ) | 2 S_{xx}(f)=|\hat{x}(f)|^{2}
  82. V ( t ) V(t)
  83. Z Z
  84. t t
  85. V ( t ) 2 / Z V(t)^{2}/Z
  86. V ( t ) 2 / Z V(t)^{2}/Z
  87. S x x ( f ) S_{xx}(f)
  88. f f
  89. Δ f \Delta f
  90. E ( f ) E(f)
  91. f f
  92. E ( f ) / Δ f E(f)/\Delta f
  93. V ( t ) 2 / Z V(t)^{2}/Z
  94. E ( f ) E(f)
  95. E ( f ) / Δ f E(f)/\Delta f
  96. Z Z
  97. x n x_{n}
  98. x n = x ( n Δ t ) x_{n}=x(n\,\Delta t)
  99. S x x ( f ) = ( Δ t ) 2 | n = - x n e - 2 π i f n | 2 = ( Δ t ) 2 x ^ d ( f ) x ^ d * ( f ) , S_{xx}(f)=(\Delta t)^{2}\left|\sum_{n=-\infty}^{\infty}x_{n}e^{-2\pi ifn}% \right|^{2}=(\Delta t)^{2}\hat{x}_{d}(f)\hat{x}_{d}^{*}(f),
  100. x ^ d ( f ) \hat{x}_{d}(f)
  101. x n x_{n}
  102. x ^ d * ( f ) \hat{x}_{d}^{*}(f)
  103. x ^ d ( f ) \hat{x}_{d}(f)
  104. Δ t \Delta t
  105. Δ t 0 \Delta t\rightarrow 0
  106. x ( t ) x(t)
  107. P = lim T 1 2 T - T T x ( t ) 2 d t . P=\lim_{T\rightarrow\infty}\frac{1}{2T}\int_{-T}^{T}x(t)^{2}\,dt.
  108. x ( t ) x(t)
  109. x ^ ( ω ) \hat{x}(\omega)
  110. x ^ ( ω ) x ^ ( ω ) = 2 π f ( ω ) δ ( ω - ω ) \langle\hat{x}(\omega)\hat{x}^{\ast}(\omega^{\prime})\rangle=2\pi\,f(\omega)\,% \delta(\omega-\omega^{\prime})
  111. δ ( ω - ω ) \delta(\omega-\omega^{\prime})
  112. x ^ T ( ω ) \hat{x}_{T}(\omega)
  113. x ^ T ( ω ) = 1 2 π T 0 T x ( t ) e - i ω t d t . \hat{x}_{T}(\omega)=\frac{1}{\sqrt{2\pi T}}\int_{0}^{T}x(t)e^{-i\omega t}\,dt.
  114. x ^ T ( ω ) = x ^ T ( f ) / 2 π \hat{x}_{T}(\omega)=\hat{x}_{T}(f)/\sqrt{2\pi}
  115. S x x ( ω ) = lim T 𝐄 [ | x ^ T ( ω ) | 2 ] . S_{xx}(\omega)=\lim_{T\rightarrow\infty}\mathbf{E}\left[|\hat{x}_{T}(\omega)|^% {2}\right].
  116. 𝐄 [ | x ^ T ( ω ) | 2 ] = 𝐄 [ 1 T 0 T x * ( t ) e i ω t d t 0 T x ( t ) e - i ω t d t ] = 1 T 0 T 0 T 𝐄 [ x * ( t ) x ( t ) ] e i ω ( t - t ) d t d t . \mathbf{E}\left[|\hat{x}_{T}(\omega)|^{2}\right]=\mathbf{E}\left[\frac{1}{T}% \int\limits_{0}^{T}x^{*}(t)e^{i\omega t}\,dt\int\limits_{0}^{T}x(t^{\prime})e^% {-i\omega t^{\prime}}\,dt^{\prime}\right]=\frac{1}{T}\int\limits_{0}^{T}\int% \limits_{0}^{T}\mathbf{E}\left[x^{*}(t)x(t^{\prime})\right]e^{i\omega(t-t^{% \prime})}\,dt\,dt^{\prime}.
  117. S x x ( ω ) S_{xx}(\omega)
  118. γ ( τ ) = X ( t ) X ( t + τ ) \gamma(\tau)=\langle X(t)X(t+\tau)\rangle
  119. γ ( τ ) \gamma(\tau)
  120. S x x ( ω ) = - γ ( τ ) e - i ω τ d τ = γ ^ ( ω ) . S_{xx}(\omega)=\int_{-\infty}^{\infty}\,\gamma(\tau)\,e^{-i\omega\tau}\,d\tau=% \hat{\gamma}(\omega).
  121. γ \gamma
  122. γ \gamma
  123. [ ω 1 , ω 2 ] [\omega_{1},\omega_{2}]
  124. ω 1 ω 2 S x x ( ω ) + S x x ( - ω ) d ω = F ( ω 2 ) - F ( - ω 2 ) \int_{\omega_{1}}^{\omega_{2}}\,S_{xx}(\omega)+S_{xx}(-\omega)\,d\omega=F(% \omega_{2})-F(-\omega_{2})
  125. F F
  126. S x x S_{xx}
  127. x n x_{n}
  128. 1 n N 1\leq n\leq N
  129. x n = x ( n Δ t ) x_{n}=x(n\Delta t)
  130. T = N Δ t T=N\Delta t
  131. S x x ( ω ) = ( Δ t ) 2 T | n = 1 N x n e - i ω n | 2 S_{xx}(\omega)=\frac{(\Delta t)^{2}}{T}\left|\sum_{n=1}^{N}x_{n}e^{-i\omega n}% \right|^{2}
  132. S x x ( - ω ) = S x x ( ω ) S_{xx}(-\omega)=S_{xx}(\omega)
  133. Var ( X n ) = γ 0 = 2 0 S x x ( ω ) d ω . \,\text{Var}(X_{n})=\gamma_{0}=2\int_{0}^{\infty}S_{xx}(\omega)d\omega.
  134. γ \gamma
  135. γ ( τ ) = α 1 γ 1 ( τ ) + α 2 γ 2 ( τ ) \gamma(\tau)=\alpha_{1}\gamma_{1}(\tau)+\alpha_{2}\gamma_{2}(\tau)
  136. f = α 1 S x x , 1 + α 2 S x x , 2 . f=\alpha_{1}S_{xx,1}+\alpha_{2}S_{xx,2}.
  137. F ( ω ) F(\omega)
  138. F ( ω ) = - ω S x x ( ω ) d ω . F(\omega)=\int_{-\infty}^{\omega}S_{xx}(\omega^{\prime})\,d\omega^{\prime}.
  139. x ( t ) x(t)
  140. y ( t ) y(t)
  141. S x x ( ω ) S_{xx}(\omega)
  142. S y y ( ω ) S_{yy}(\omega)
  143. S x y ( ω ) = lim T 𝐄 { [ F x T ( ω ) ] * F y T ( ω ) } . S_{xy}(\omega)=\lim_{T\rightarrow\infty}\mathbf{E}\left\{\left[F_{x}^{T}(% \omega)\right]^{*}F_{y}^{T}(\omega)\right\}.
  144. S x y ( ω ) = - R x y ( t ) e - j ω t d t = - [ - x ( τ ) y ( τ + t ) d τ ] e - j ω t d t , S_{xy}(\omega)=\int_{-\infty}^{\infty}R_{xy}(t)e^{-j\omega t}dt=\int_{-\infty}% ^{\infty}\left[\int_{-\infty}^{\infty}x(\tau)\cdot y(\tau+t)d\tau\right]\,e^{-% j\omega t}dt,
  145. R x y ( t ) R_{xy}(t)
  146. x ( t ) x(t)
  147. y ( t ) y(t)
  148. S x y ( ω ) S_{xy}(\omega)
  149. x ( t ) = y ( t ) x(t)=y(t)
  150. S x y ( ω ) = 1 2 π n = - R x y ( n ) e - j ω n S_{xy}(\omega)=\frac{1}{2\pi}\sum_{n=-\infty}^{\infty}R_{xy}(n)e^{-j\omega n}
  151. f ( t ) f(t)
  152. f ( t ) f(t)
  153. - | f ( t ) | 2 d t = - E S D ( ω ) d ω . \int_{-\infty}^{\infty}\left|f(t)\right|^{2}\,dt=\int_{-\infty}^{\infty}ESD(% \omega)\,d\omega.
  154. R ( 0 ) R(0)
  155. f ( t ) f(t)
  156. μ ( [ a , b ) ) \mu([a,b))
  157. μ \mu

Spectral_line.html

  1. ( Δ E 1 / r 2 ) (\Delta E\sim 1/r^{2})
  2. ( Δ E 1 / r 3 ) (\Delta E\sim 1/r^{3})
  3. ( Δ E 1 / r 4 ) (\Delta E\sim 1/r^{4})
  4. ( Δ E 1 / r 6 ) (\Delta E\sim 1/r^{6})

Spectral_method.html

  1. ( 2 x 2 + 2 y 2 ) f ( x , y ) = g ( x , y ) for all x , y \left(\frac{\partial^{2}}{\partial x^{2}}+\frac{\partial^{2}}{\partial y^{2}}% \right)f(x,y)=g(x,y)\quad\,\text{for all }x,y
  2. f = : a j , k e i ( j x + k y ) f=:\sum a_{j,k}e^{i(jx+ky)}
  3. g = : b j , k e i ( j x + k y ) g=:\sum b_{j,k}e^{i(jx+ky)}
  4. - a j , k ( j 2 + k 2 ) e i ( j x + k y ) = b j , k e i ( j x + k y ) \sum-a_{j,k}(j^{2}+k^{2})e^{i(jx+ky)}=\sum b_{j,k}e^{i(jx+ky)}
  5. a j , k = - b j , k j 2 + k 2 a_{j,k}=-\frac{b_{j,k}}{j^{2}+k^{2}}
  6. h n h^{n}
  7. h := 1 / n h:=1/n
  8. n n
  9. u ( x , 0 ) u(x,0)
  10. x [ 0 , 2 π ) x\in\left[0,2\pi\right)
  11. u 𝒰 u\in\mathcal{U}
  12. t u + u x u = ρ x x u + f x [ 0 , 2 π ) , t > 0 \partial_{t}u+u\partial_{x}u=\rho\partial_{xx}u+f\quad\forall x\in\left[0,2\pi% \right),\forall t>0
  13. t u , v = x ( - 1 2 u 2 + ρ x u ) , v + f , v v 𝒱 , t > 0 \langle\partial_{t}u,v\rangle=\langle\partial_{x}\left(-\frac{1}{2}u^{2}+\rho% \partial_{x}u\right),v\rangle+\langle f,v\rangle\quad\forall v\in\mathcal{V},% \forall t>0
  14. f , g := 0 2 π f ( x ) g ( x ) ¯ d x \langle f,g\rangle:=\int_{0}^{2\pi}f(x)\overline{g(x)}\,dx
  15. t u , v = 1 2 u 2 - ρ x u , x v + f , v v 𝒱 , t > 0. \langle\partial_{t}u,v\rangle=\langle\frac{1}{2}u^{2}-\rho\partial_{x}u,% \partial_{x}v\rangle+\langle f,v\rangle\quad\forall v\in\mathcal{V},\forall t>0.
  16. 𝒰 N := { u : u ( x , t ) = k = - N / 2 N / 2 - 1 u ^ k ( t ) e i k x } \mathcal{U}^{N}:=\left\{u:u(x,t)=\sum_{k=-N/2}^{N/2-1}\hat{u}_{k}(t)e^{ikx}\right\}
  17. 𝒱 N := span { e i k x : k - N / 2 , , N / 2 - 1 } \mathcal{V}^{N}:=\,\text{ span}\left\{e^{ikx}:k\in-N/2,\dots,N/2-1\right\}
  18. u ^ k ( t ) := 1 2 π u ( x , t ) , e i k x \hat{u}_{k}(t):=\frac{1}{2\pi}\langle u(x,t),e^{ikx}\rangle
  19. u 𝒰 N u\in\mathcal{U}^{N}
  20. t u , e i k x = 1 2 u 2 - ρ x u , x e i k x + f , e i k x k { - N / 2 , , N / 2 - 1 } , t > 0. \langle\partial_{t}u,e^{ikx}\rangle=\langle\frac{1}{2}u^{2}-\rho\partial_{x}u,% \partial_{x}e^{ikx}\rangle+\langle f,e^{ikx}\rangle\quad\forall k\in\left\{-N/% 2,\dots,N/2-1\right\},\forall t>0.
  21. e i l x , e i k x = 2 π δ l k \langle e^{ilx},e^{ikx}\rangle=2\pi\delta_{lk}
  22. δ l k \delta_{lk}
  23. k k
  24. t u , e i k x = t l u ^ l e i l x , e i k x = l t u ^ l e i l x , e i k x = 2 π t u ^ k , f , e i k x = l f ^ l e i l x , e i k x = 2 π f ^ k , and 1 2 u 2 - ρ x u , x e i k x = 1 2 ( p u ^ p e i p x ) ( q u ^ q e i q x ) - ρ x l u ^ l e i l x , x e i k x = 1 2 p q u ^ p u ^ q e i ( p + q ) x , i k e i k x - ρ i l l u ^ l e i l x , i k e i k x = - i k 2 p q u ^ p u ^ q e i ( p + q ) x , e i k x - ρ k l l u ^ l e i l x , e i k x = - i π k p + q = k u ^ p u ^ q - 2 π ρ k 2 u ^ k . \begin{aligned}\displaystyle\langle\partial_{t}u,e^{ikx}\rangle&\displaystyle=% \langle\partial_{t}\sum_{l}\hat{u}_{l}e^{ilx},e^{ikx}\rangle=\langle\sum_{l}% \partial_{t}\hat{u}_{l}e^{ilx},e^{ikx}\rangle=2\pi\partial_{t}\hat{u}_{k},\\ \displaystyle\langle f,e^{ikx}\rangle&\displaystyle=\langle\sum_{l}\hat{f}_{l}% e^{ilx},e^{ikx}\rangle=2\pi\hat{f}_{k},\,\text{ and}\\ \displaystyle\langle\frac{1}{2}u^{2}-\rho\partial_{x}u,\partial_{x}e^{ikx}% \rangle&\displaystyle=\langle\frac{1}{2}\left(\sum_{p}\hat{u}_{p}e^{ipx}\right% )\left(\sum_{q}\hat{u}_{q}e^{iqx}\right)-\rho\partial_{x}\sum_{l}\hat{u}_{l}e^% {ilx},\partial_{x}e^{ikx}\rangle\\ &\displaystyle=\langle\frac{1}{2}\sum_{p}\sum_{q}\hat{u}_{p}\hat{u}_{q}e^{i% \left(p+q\right)x},ike^{ikx}\rangle-\langle\rho i\sum_{l}l\hat{u}_{l}e^{ilx},% ike^{ikx}\rangle\\ &\displaystyle=-\frac{ik}{2}\langle\sum_{p}\sum_{q}\hat{u}_{p}\hat{u}_{q}e^{i% \left(p+q\right)x},e^{ikx}\rangle-\rho k\langle\sum_{l}l\hat{u}_{l}e^{ilx},e^{% ikx}\rangle\\ &\displaystyle=-i\pi k\sum_{p+q=k}\hat{u}_{p}\hat{u}_{q}-2\pi\rho{}k^{2}\hat{u% }_{k}.\end{aligned}
  25. k k
  26. 2 π t u ^ k = - i π k p + q = k u ^ p u ^ q - 2 π ρ k 2 u ^ k + 2 π f ^ k k { - N / 2 , , N / 2 - 1 } , t > 0. 2\pi\partial_{t}\hat{u}_{k}=-i\pi k\sum_{p+q=k}\hat{u}_{p}\hat{u}_{q}-2\pi\rho% {}k^{2}\hat{u}_{k}+2\pi\hat{f}_{k}\quad k\in\left\{-N/2,\dots,N/2-1\right\},% \forall t>0.
  27. 2 π 2\pi
  28. t u ^ k = - i k 2 p + q = k u ^ p u ^ q - ρ k 2 u ^ k + f ^ k k { - N / 2 , , N / 2 - 1 } , t > 0. \partial_{t}\hat{u}_{k}=-\frac{ik}{2}\sum_{p+q=k}\hat{u}_{p}\hat{u}_{q}-\rho{}% k^{2}\hat{u}_{k}+\hat{f}_{k}\quad k\in\left\{-N/2,\dots,N/2-1\right\},\forall t% >0.
  29. u ^ k ( 0 ) \hat{u}_{k}(0)
  30. f ^ k ( t ) \hat{f}_{k}(t)
  31. g g
  32. C < C<\infty
  33. C h n Ch^{n}
  34. h h
  35. n n

Spectral_theorem.html

  1. A A
  2. V V
  3. x , y V x,y∈V
  4. A x , y = x , A y . \langle Ax,\,y\rangle=\langle x,\,Ay\rangle.
  5. A A
  6. A A
  7. A A
  8. A A
  9. x = y x=y
  10. A A
  11. x x
  12. A x = λ x Ax=λx
  13. λ λ
  14. λ λ
  15. V V
  16. A A
  17. A A
  18. λ 1 e 1 , e 1 = A ( e 1 ) , e 1 = e 1 , A ( e 1 ) = λ ¯ 1 e 1 , e 1 \lambda_{1}\langle e_{1},e_{1}\rangle=\langle A(e_{1}),e_{1}\rangle=\langle e_% {1},A(e_{1})\rangle=\bar{\lambda}_{1}\langle e_{1},e_{1}\rangle
  19. K K
  20. A A
  21. K K
  22. A A
  23. A A
  24. A A
  25. A A
  26. A A
  27. V λ = { v V : A v = λ v } V_{\lambda}=\{\,v\in V:Av=\lambda v\,\}
  28. λ λ
  29. V V
  30. A A
  31. A = λ 1 P λ 1 + + λ m P λ m . A=\lambda_{1}P_{\lambda_{1}}+\cdots+\lambda_{m}P_{\lambda_{m}}.
  32. A A
  33. A A
  34. A A
  35. U U
  36. T T
  37. A A
  38. T T
  39. A A
  40. U U
  41. A = U D U * , A=UDU^{*},
  42. D D
  43. D D
  44. A A
  45. U U
  46. A A
  47. D D
  48. A A
  49. V V
  50. V V
  51. A A
  52. A A
  53. t t
  54. [ A φ ] ( t ) = t φ ( t ) . [A\varphi](t)=t\varphi(t).\;
  55. A A
  56. H H
  57. ( X , Σ , μ ) (X,Σ,μ)
  58. f f
  59. X X
  60. U * T U = A , U^{*}TU=A,
  61. T T
  62. [ T φ ] ( x ) = f ( x ) φ ( x ) . [T\varphi](x)=f(x)\varphi(x).
  63. T = f \|T\|=\|f\|_{\infty}
  64. f f
  65. A A
  66. A = σ ( A ) λ d E λ . A=\int_{\sigma(A)}\lambda\,dE_{\lambda}.
  67. T T
  68. H H
  69. H H
  70. L < s u p > 2 ( M , μ ) L<sup>2(M,μ)

Spectrogram.html

  1. spectrogram ( t , ω ) = | STFT ( t , ω ) | 2 \mathrm{spectrogram}(t,\omega)=\left|\mathrm{STFT}(t,\omega)\right|^{2}

Spectrum_(functional_analysis).html

  1. ( x 1 , x 2 , ) ( 0 , x 1 , x 2 , ) . (x_{1},x_{2},\dots)\mapsto(0,x_{1},x_{2},\dots).
  2. T T
  3. X X
  4. 𝕂 \mathbb{K}
  5. I I
  6. X X
  7. T T
  8. λ 𝕂 \lambda\in\mathbb{K}
  9. λ I - T \lambda I-T
  10. λ I - T \lambda I-T
  11. λ \lambda
  12. λ I - T \lambda I-T
  13. T T
  14. σ ( T ) \sigma(T)
  15. ρ ( T ) = 𝕂 σ ( T ) \rho(T)=\mathbb{K}\setminus\sigma(T)
  16. λ \lambda
  17. T T
  18. T - λ I T-\lambda I
  19. ( T - λ I ) - 1 (T-\lambda I)^{-1}
  20. T - λ I T-\lambda I
  21. λ \lambda
  22. 2 ( ) \ell^{2}(\mathbb{Z})
  23. v = ( , v - 2 , v - 1 , v 0 , v 1 , v 2 , ) v=(\ldots,v_{-2},v_{-1},v_{0},v_{1},v_{2},\ldots)
  24. i = - + v i 2 \sum_{i=-\infty}^{+\infty}v_{i}^{2}
  25. T T
  26. u = T ( v ) u=T(v)
  27. u i = v i - 1 u_{i}=v_{i-1}
  28. i i
  29. T ( v ) = λ v T(v)=\lambda v
  30. v i v_{i}
  31. λ = 1 \lambda=1
  32. λ 1 \lambda\neq 1
  33. T - λ I T-\lambda I
  34. | λ | = 1 |\lambda|=1
  35. u u
  36. u i = 1 / ( | i | + 1 ) u_{i}=1/(|i|+1)
  37. 2 ( ) \ell^{2}(\mathbb{Z})
  38. v v
  39. 2 ( ) \ell^{2}(\mathbb{Z})
  40. ( T - I ) v = u (T-I)v=u
  41. v i - 1 = u i + v i v_{i-1}=u_{i}+v_{i}
  42. i i
  43. R ( λ ) = ( λ I - T ) - 1 R(\lambda)=(\lambda I-T)^{-1}\,
  44. r ( T ) = sup { | λ | : λ σ ( T ) } . r(T)=\sup\{|\lambda|:\lambda\in\sigma(T)\}.
  45. T T
  46. r ( T ) = lim n T n 1 / n . r(T)=\lim_{n\to\infty}\|T^{n}\|^{1/n}.
  47. T T
  48. T T
  49. T T
  50. λ σ ( T ) \lambda\in\sigma(T)
  51. λ I - T \lambda I-T
  52. λ I - T \lambda I-T
  53. λ \lambda
  54. T T
  55. σ p ( T ) \sigma_{p}(T)
  56. λ I - T \lambda I-T
  57. λ \lambda
  58. T T
  59. T T
  60. σ a p ( T ) \sigma_{ap}(T)
  61. λ σ ( T ) \lambda\in\sigma(T)
  62. λ I - T \lambda I-T
  63. λ \lambda
  64. λ I - T \lambda I-T
  65. λ \lambda
  66. T T
  67. σ r ( T ) \sigma_{r}(T)
  68. σ ( T ) \sigma(T)
  69. x x
  70. T ( x ) = 0 T(x)=0
  71. λ \lambda
  72. T T
  73. λ σ ( T ) \lambda\in\sigma(T)
  74. T T
  75. T T
  76. σ p ( T ) \sigma_{p}(T)
  77. lim n T x n - λ x n = 0 \lim_{n\to\infty}\|Tx_{n}-\lambda x_{n}\|=0
  78. T ( , a - 1 , a ^ 0 , a 1 , ) = ( , a ^ - 1 , a 0 , a 1 , ) T(\cdots,a_{-1},\hat{a}_{0},a_{1},\cdots)=(\cdots,\hat{a}_{-1},a_{0},a_{1},\cdots)
  79. 1 n ( , 0 , 1 , λ - 1 , λ - 2 , , λ 1 - n , 0 , ) \frac{1}{\sqrt{n}}(\dots,0,1,\lambda^{-1},\lambda^{-2},\dots,\lambda^{1-n},0,\dots)
  80. T x n - λ x n = 2 n 0. \|Tx_{n}-\lambda x_{n}\|=\sqrt{\frac{2}{n}}\to 0.
  81. σ c ( T ) = σ a p ( T ) ( σ r ( T ) σ p ( T ) ) \sigma_{c}(T)=\sigma_{ap}(T)\setminus(\sigma_{r}(T)\cup\sigma_{p}(T))
  82. T : D X X T:D\subset X\to X
  83. T - λ I : D X T-\lambda I:D\to X
  84. S : X D S:X\rightarrow D
  85. S ( T - I λ ) = I D , ( T - I λ ) S = I X . S(T-I\lambda)=I_{D},\,(T-I\lambda)S=I_{X}.

Spectrum_analyzer.html

  1. S T = k ( S p a n ) R B W 2 \ ST=\frac{k(Span)}{RBW^{2}}
  2. Δ ν = 1 / T \Delta\nu=1/T
  3. ν s \nu_{s}
  4. ν s / 2 \nu_{s}/2
  5. t sweep = k ( f 2 - f 1 ) RBW × VBW . t_{\mathrm{sweep}}=\frac{k(f_{2}-f_{1})}{\mathrm{RBW}\times\mathrm{VBW}}.

Speed_of_sound.html

  1. c = K s ρ , c=\sqrt{\frac{K_{s}}{\rho}},
  2. ρ \rho
  3. c c
  4. c = ( p ρ ) s , c=\sqrt{\left(\frac{\partial p}{\partial\rho}\right)_{s}},
  5. p p
  6. ρ \rho
  7. c air = ( 331.3 + 0.606 ϑ ) m / s , c_{\mathrm{air}}=(331{.}3+0{.}606\cdot\vartheta)~{}\mathrm{m/s},
  8. ϑ \vartheta
  9. c air = 331.3 m / s 1 + ϑ 273.15 . c_{\mathrm{air}}=331.3~{}\mathrm{m/s}\sqrt{1+\frac{\vartheta}{273.15}}.
  10. 273.15 \sqrt{273.15}
  11. c air = 20.05 m / s ϑ + 273.15 . c_{\mathrm{air}}=20.05~{}\mathrm{m/s}\sqrt{\vartheta+273.15}.
  12. γ \gamma
  13. γ \gamma
  14. K = γ p , K=\gamma\cdot p,
  15. c = γ p ρ , c=\sqrt{\gamma\cdot{p\over\rho}},
  16. γ \gamma
  17. C p / C v C_{p}/C_{v}
  18. ρ \rho
  19. p p
  20. c ideal = γ p ρ = γ R T M = γ k T m , c_{\mathrm{ideal}}=\sqrt{\gamma\cdot{p\over\rho}}=\sqrt{\gamma\cdot R\cdot T% \over M}=\sqrt{\gamma\cdot k\cdot T\over m},
  21. c ideal c_{\mathrm{ideal}}
  22. R R
  23. k k
  24. γ \gamma
  25. T T
  26. M M
  27. m m
  28. c air c_{\mathrm{air}}
  29. γ \gamma
  30. γ = 1.4000 \ \gamma\,=1.4000
  31. R * = R / M air R_{*}=R/M_{\mathrm{air}}
  32. ϑ = T - 273.15 \vartheta=T-273.15
  33. c ideal = γ R * T = γ R * ( ϑ + 273.15 ) , c_{\mathrm{ideal}}=\sqrt{\gamma\cdot R_{*}\cdot T}=\sqrt{\gamma\cdot R_{*}% \cdot(\vartheta+273.15)},
  34. c ideal = γ R * 273.15 1 + ϑ 273.15 . c_{\mathrm{ideal}}=\sqrt{\gamma\cdot R_{*}\cdot 273.15}\cdot\sqrt{1+\frac{% \vartheta}{273.15}}.
  35. ϑ \vartheta
  36. R = 8.314510 J / ( mol K ) R=8.314510~{}\mathrm{J/(mol\cdot K)}
  37. M air = 0.0289645 kg / mol M_{\mathrm{air}}=0.0289645~{}\mathrm{kg/mol}
  38. γ = 1.4000 \gamma=1.4000
  39. c air = 331.3 m / s 1 + ϑ C 273.15 C . c_{\mathrm{air}}=331.3~{}\mathrm{m/s}\sqrt{1+\frac{\vartheta^{\circ}\mathrm{C}% }{273.15\;^{\circ}\mathrm{C}}}.
  40. c air = 331.3 m / s ( 1 + ϑ C 2 273.15 C ) , c_{\mathrm{air}}=331.3~{}\mathrm{m/s}(1+\frac{\vartheta^{\circ}\mathrm{C}}{2% \cdot 273.15\;^{\circ}\mathrm{C}}),
  41. c air = ( 331.3 + 0.606 C - 1 ϑ ) m / s . c_{\mathrm{air}}=(331{.}3+0{.}606\;^{\circ}\mathrm{C}^{-1}\cdot\vartheta)~{}% \mathrm{m/s}.
  42. U ( h ) = U ( 0 ) h ζ , U(h)=U(0)h^{\zeta},
  43. d U d H = ζ U ( h ) h , \frac{dU}{dH}=\zeta\frac{U(h)}{h},
  44. U ( h ) U(h)
  45. h h
  46. U ( 0 ) U(0)
  47. ζ \zeta
  48. d U d H \frac{dU}{dH}
  49. h h
  50. γ \gamma
  51. c gas , monatomic c gas , diatomic = 5 / 3 7 / 5 = 25 21 = 1.091 {c_{\mathrm{gas,monatomic}}\over c_{\mathrm{gas,diatomic}}}=\sqrt{{{{5/3}\over% {7/5}}}}=\sqrt{25\over 21}=1.091\ldots
  52. c solid , p = K + 4 3 G ρ = E ( 1 - ν ) ρ ( 1 + ν ) ( 1 - 2 ν ) , c_{\mathrm{solid,p}}=\sqrt{\frac{K+\frac{4}{3}G}{\rho}}=\sqrt{\frac{E(1-\nu)}{% \rho(1+\nu)(1-2\nu)}},
  53. c solid , s = G ρ , c_{\mathrm{solid,s}}=\sqrt{\frac{G}{\rho}},
  54. ρ \rho
  55. ν \nu
  56. E = 3 K ( 1 - 2 ν ) E=3K(1-2\nu)
  57. ρ \rho
  58. c solid = E ρ , c_{\mathrm{solid}}=\sqrt{\frac{E}{\rho}},
  59. c fluid = K ρ , c_{\mathrm{fluid}}=\sqrt{\frac{K}{\rho}},
  60. c ( T , S , z ) = a 1 + a 2 T + a 3 T 2 + a 4 T 3 + a 5 ( S - 35 ) + a 6 z + a 7 z 2 + a 8 T ( S - 35 ) + a 9 T z 3 , c(T,S,z)=a_{1}+a_{2}T+a_{3}T^{2}+a_{4}T^{3}+a_{5}(S-35)+a_{6}z+a_{7}z^{2}+a_{8% }T(S-35)+a_{9}Tz^{3},
  61. a 1 \displaystyle a_{1}
  62. c s = ( γ Z k T e / m i ) 1 / 2 = 9.79 × 10 3 ( γ Z T e / μ ) 1 / 2 m / s , c_{s}=(\gamma ZkT_{e}/m_{i})^{1/2}=9.79\times 10^{3}(\gamma ZT_{e}/\mu)^{1/2}~% {}\mathrm{m/s},
  63. m i m_{i}
  64. μ \mu
  65. μ = m i / m p \mu=m_{i}/m_{p}
  66. T e T_{e}
  67. γ \gamma

Sphenic_number.html

  1. n = p q r n=p\cdot q\cdot r
  2. { 1 , p , q , r , p q , p r , q r , n } . \left\{1,\ p,\ q,\ r,\ pq,\ pr,\ qr,\ n\right\}.
  3. Φ n ( x ) \Phi_{n}(x)
  4. ± 1 \pm 1

Spherical_aberration.html

  1. R 1 R_{1}
  2. R 2 R_{2}
  3. R 1 + R 2 R 1 - R 2 = 2 ( n 2 - 1 ) n + 2 ( i + o i - o ) \frac{R_{1}+R_{2}}{R_{1}-R_{2}}=\frac{2\left(n^{2}-1\right)}{n+2}\left(\frac{i% +o}{i-o}\right)

Spherical_harmonics.html

  1. Y m Y_{\ell}^{m}
  2. V ( 𝐱 ) = i m i | 𝐱 i - 𝐱 | . V(\mathbf{x})=\sum_{i}\frac{m_{i}}{|\mathbf{x}_{i}-\mathbf{x}|}.
  3. 1 | 𝐱 1 - 𝐱 | = P 0 ( cos γ ) 1 r 1 + P 1 ( cos γ ) r r 1 2 + P 2 ( cos γ ) r 2 r 1 3 + \frac{1}{|\mathbf{x}_{1}-\mathbf{x}|}=P_{0}(\cos\gamma)\frac{1}{r_{1}}+P_{1}(% \cos\gamma)\frac{r}{r_{1}^{2}}+P_{2}(\cos\gamma)\frac{r^{2}}{r_{1}^{3}}+\cdots
  4. 2 u x 2 + 2 u y 2 + 2 u z 2 = 0. \frac{\partial^{2}u}{\partial x^{2}}+\frac{\partial^{2}u}{\partial y^{2}}+% \frac{\partial^{2}u}{\partial z^{2}}=0.
  5. - i 𝐫 × , -i\hbar\mathbf{r}\times\nabla,
  6. = 0 , , 4 ℓ=0,…,4
  7. m = 0 , , m=0,…,ℓ
  8. Y - m Y_{\ell}^{-m}
  9. 90 / m 90^{\circ}/m
  10. f f
  11. 2 f = 1 r 2 r ( r 2 f r ) + 1 r 2 sin θ θ ( sin θ f θ ) + 1 r 2 sin 2 θ 2 f φ 2 = 0. \nabla^{2}f=\frac{1}{r^{2}}\frac{\partial}{\partial r}\left(r^{2}\frac{% \partial f}{\partial r}\right)+\frac{1}{r^{2}\sin\theta}\frac{\partial}{% \partial\theta}\left(\sin\theta\frac{\partial f}{\partial\theta}\right)+\frac{% 1}{r^{2}\sin^{2}\theta}\frac{\partial^{2}f}{\partial\varphi^{2}}=0.
  12. f ( r , θ , φ ) = R ( r ) Y ( θ , φ ) f(r,θ,φ)=R(r)Y(θ,φ)
  13. 1 R d d r ( r 2 d R d r ) = λ , 1 Y 1 sin θ θ ( sin θ Y θ ) + 1 Y 1 sin 2 θ 2 Y φ 2 = - λ . \frac{1}{R}\frac{d}{dr}\left(r^{2}\frac{dR}{dr}\right)=\lambda,\qquad\frac{1}{% Y}\frac{1}{\sin\theta}\frac{\partial}{\partial\theta}\left(\sin\theta\frac{% \partial Y}{\partial\theta}\right)+\frac{1}{Y}\frac{1}{\sin^{2}\theta}\frac{% \partial^{2}Y}{\partial\varphi^{2}}=-\lambda.
  14. Y Y
  15. Y ( θ , φ ) = Θ ( θ ) Φ ( φ ) Y(θ,φ)=Θ(θ)Φ(φ)
  16. 1 Φ d 2 Φ d φ 2 = - m 2 \frac{1}{\Phi}\frac{d^{2}\Phi}{d\varphi^{2}}=-m^{2}
  17. λ sin 2 θ + sin θ Θ d d θ ( sin θ d Θ d θ ) = m 2 \lambda\sin^{2}\theta+\frac{\sin\theta}{\Theta}\frac{d}{d\theta}\left(\sin% \theta\frac{d\Theta}{d\theta}\right)=m^{2}
  18. m m
  19. m m
  20. Φ Φ
  21. 2 π
  22. m m
  23. Y ( θ , φ ) Y(θ,φ)
  24. θ = 0 , π θ=0,π
  25. Θ Θ
  26. λ λ
  27. λ = ( + 1 ) λ=ℓ(ℓ+1)
  28. | m | ℓ≥|m|
  29. t = c o s θ t=cosθ
  30. R R
  31. B = 0 B=0
  32. Y ( θ , φ ) = Θ ( θ ) Φ ( φ ) Y(θ,φ)=Θ(θ)Φ(φ)
  33. 2 + 1 2ℓ+1
  34. m m
  35. m −ℓ≤m≤ℓ
  36. Y m ( θ , φ ) = N e i m φ P m ( cos θ ) Y_{\ell}^{m}(\theta,\varphi)=Ne^{im\varphi}P_{\ell}^{m}(\cos{\theta})
  37. r 2 2 Y m ( θ , φ ) = - ( + 1 ) Y m ( θ , φ ) . r^{2}\nabla^{2}Y_{\ell}^{m}(\theta,\varphi)=-\ell(\ell+1)Y_{\ell}^{m}(\theta,% \varphi).
  38. m m
  39. N N
  40. θ θ
  41. φ φ
  42. θ θ
  43. 0
  44. π / 2 π/2
  45. π π
  46. φ φ
  47. Y ( θ , φ ) Y(θ,φ)
  48. r 2 2 Y = - ( + 1 ) Y r^{2}\nabla^{2}Y=-\ell(\ell+1)Y
  49. 2 + 1 2ℓ+1
  50. f ( r , θ , φ ) = = 0 m = - f m r Y m ( θ , φ ) , f(r,\theta,\varphi)=\sum_{\ell=0}^{\infty}\sum_{m=-\ell}^{\ell}f_{\ell}^{m}r^{% \ell}Y_{\ell}^{m}(\theta,\varphi),
  51. r < R = 1 lim sup | f m | 1 . r<R=\frac{1}{\limsup_{\ell\to\infty}|f_{\ell}^{m}|^{\frac{1}{\ell}}}.
  52. 𝐋 = - i 𝐱 × = L x 𝐢 + L y 𝐣 + L z 𝐤 . \mathbf{L}=-i\hbar\mathbf{x}\times\nabla=L_{x}\mathbf{i}+L_{y}\mathbf{j}+L_{z}% \mathbf{k}.
  53. ħ ħ
  54. ħ = 1 ħ=1
  55. 𝐋 2 = - r 2 2 + ( r r + 1 ) r r = - 1 sin θ θ sin θ θ - 1 sin 2 θ 2 φ 2 . \begin{aligned}\displaystyle\mathbf{L}^{2}&\displaystyle=-r^{2}\nabla^{2}+% \left(r\frac{\partial}{\partial r}+1\right)r\frac{\partial}{\partial r}\\ &\displaystyle=-\frac{1}{\sin\theta}\frac{\partial}{\partial\theta}\sin\theta% \frac{\partial}{\partial\theta}-\frac{1}{\sin^{2}\theta}\frac{\partial^{2}}{% \partial\varphi^{2}}.\end{aligned}
  56. L z \displaystyle L_{z}
  57. 1 ( 2 π ) 3 / 2 𝐑 3 | f ( x ) | 2 e - | x | 2 / 2 d x < . \frac{1}{(2\pi)^{3/2}}\int_{\mathbf{R}^{3}}|f(x)|^{2}e^{-|x|^{2}/2}\,dx<\infty.
  58. 𝐋 2 Y = λ Y L z Y = m Y \begin{aligned}\displaystyle\mathbf{L}^{2}Y&\displaystyle=\lambda Y\\ \displaystyle L_{z}Y&\displaystyle=mY\end{aligned}
  59. 𝐋 2 = L x 2 + L y 2 + L z 2 \mathbf{L}^{2}=L_{x}^{2}+L_{y}^{2}+L_{z}^{2}
  60. L + = L x + i L y L - = L x - i L y \begin{aligned}\displaystyle L_{+}&\displaystyle=L_{x}+iL_{y}\\ \displaystyle L_{-}&\displaystyle=L_{x}-iL_{y}\end{aligned}
  61. [ L z , L + ] = L + , [ L z , L - ] = - L - , [ L + , L - ] = 2 L z . [L_{z},L_{+}]=L_{+},\quad[L_{z},L_{-}]=-L_{-},\quad[L_{+},L_{-}]=2L_{z}.
  62. L + k Y = 0. L_{+}^{k}Y=0.
  63. L - L + = 𝐋 2 - L z 2 - L z L_{-}L_{+}=\mathbf{L}^{2}-L_{z}^{2}-L_{z}
  64. 0 = L - L + k Y = ( λ - ( m + k ) 2 - ( m + k ) ) Y . 0=L_{-}L_{+}^{k}Y=(\lambda-(m+k)^{2}-(m+k))Y.
  65. P - m = ( - 1 ) m ( - m ) ! ( + m ) ! P m P_{\ell}^{-m}=(-1)^{m}\frac{(\ell-m)!}{(\ell+m)!}P_{\ell}^{m}
  66. Y m ( θ , φ ) = ( 2 + 1 ) 4 π ( - m ) ! ( + m ) ! P m ( cos θ ) e i m φ Y_{\ell}^{m}(\theta,\varphi)=\sqrt{{(2\ell+1)\over 4\pi}{(\ell-m)!\over(\ell+m% )!}}\,P_{\ell}^{m}(\cos{\theta})\,e^{im\varphi}
  67. Y m ( θ , φ ) = ( - 1 ) m ( 2 + 1 ) 4 π ( - m ) ! ( + m ) ! P m ( cos θ ) e i m φ Y_{\ell}^{m}(\theta,\varphi)=(-1)^{m}\sqrt{{(2\ell+1)\over 4\pi}{(\ell-m)!% \over(\ell+m)!}}\,P_{\ell}^{m}(\cos{\theta})\,e^{im\varphi}
  68. θ = 0 π φ = 0 2 π Y m Y m d * Ω = δ δ m m , \int_{\theta=0}^{\pi}\int_{\varphi=0}^{2\pi}Y_{\ell}^{m}\,Y_{\ell^{\prime}}^{m% ^{\prime}}{}^{*}\,d\Omega=\delta_{\ell\ell^{\prime}}\,\delta_{mm^{\prime}},
  69. | Y m | 2 d Ω = 1. \int{|Y_{\ell}^{m}|^{2}d\Omega}=1.
  70. Y m ( θ , φ ) = ( 2 + 1 ) ( - m ) ! ( + m ) ! P m ( cos θ ) e i m φ Y_{\ell}^{m}(\theta,\varphi)=\sqrt{{(2\ell+1)}{(\ell-m)!\over(\ell+m)!}}\,P_{% \ell}^{m}(\cos{\theta})\,e^{im\varphi}
  71. 1 4 π θ = 0 π φ = 0 2 π Y m Y m d * Ω = δ δ m m . {1\over 4\pi}\int_{\theta=0}^{\pi}\int_{\varphi=0}^{2\pi}Y_{\ell}^{m}\,Y_{\ell% ^{\prime}}^{m^{\prime}}{}^{*}d\Omega=\delta_{\ell\ell^{\prime}}\,\delta_{mm^{% \prime}}.
  72. Y m ( θ , φ ) = < m t p l > ( - m ) ! ( + m ) ! Y_{\ell}^{m}(\theta,\varphi)=\sqrt{<}mtpl>{{(\ell-m)!\over(\ell+m)!}}
  73. θ = 0 π φ = 0 2 π Y m Y m d * Ω = 4 π ( 2 + 1 ) δ δ m m . \int_{\theta=0}^{\pi}\int_{\varphi=0}^{2\pi}Y_{\ell}^{m}\,Y_{\ell^{\prime}}^{m% ^{\prime}}{}^{*}d\Omega={4\pi\over(2\ell+1)}\delta_{\ell\ell^{\prime}}\,\delta% _{mm^{\prime}}.
  74. Y m ( θ , φ ) * = ( - 1 ) m Y - m ( θ , φ ) , Y_{\ell}^{m}{}^{*}(\theta,\varphi)=(-1)^{m}Y_{\ell}^{-m}(\theta,\varphi),
  75. * *
  76. Y m \displaystyle Y_{\ell m}
  77. Y m = { 1 2 ( Y | m | - i Y , - | m | ) if m < 0 Y 0 if m = 0 ( - 1 ) m 2 ( Y | m | + i Y , - | m | ) if m > 0. Y_{\ell}^{m}=\begin{cases}\displaystyle{1\over\sqrt{2}}\left(Y_{\ell|m|}-iY_{% \ell,-|m|}\right)&\,\text{if}\ m<0\\ \displaystyle Y_{\ell 0}&\,\text{if}\ m=0\\ \displaystyle{(-1)^{m}\over\sqrt{2}}\left(Y_{\ell|m|}+iY_{\ell,-|m|}\right)&\,% \text{if}\ m>0.\end{cases}
  78. = 4 \ell=4
  79. l = 1 l=1
  80. r ( Y m Y - m ) = [ 2 + 1 4 π ] 1 / 2 Π ¯ m ( z ) ( ( - 1 ) m ( A m + i B m ) ( A m - i B m ) ) , m > 0. r^{\ell}\,\begin{pmatrix}Y_{\ell}^{m}\\ Y_{\ell}^{-m}\end{pmatrix}=\left[\frac{2\ell+1}{4\pi}\right]^{1/2}\bar{\Pi}^{m% }_{\ell}(z)\begin{pmatrix}(-1)^{m}(A_{m}+iB_{m})\\ \qquad(A_{m}-iB_{m})\\ \end{pmatrix},\qquad m>0.
  81. r Y 0 2 + 1 4 π Π ¯ 0 . r^{\ell}\,Y_{\ell}^{0}\equiv\sqrt{\frac{2\ell+1}{4\pi}}\bar{\Pi}^{0}_{\ell}.
  82. A m ( x , y ) = p = 0 m ( m p ) x p y m - p cos ( ( m - p ) π 2 ) , A_{m}(x,y)=\sum_{p=0}^{m}{\left({{m}\atop{p}}\right)}x^{p}y^{m-p}\cos((m-p)% \frac{\pi}{2}),
  83. B m ( x , y ) = p = 0 m ( m p ) x p y m - p sin ( ( m - p ) π 2 ) , B_{m}(x,y)=\sum_{p=0}^{m}{\left({{m}\atop{p}}\right)}x^{p}y^{m-p}\sin((m-p)% \frac{\pi}{2}),
  84. Π ¯ m ( z ) = [ ( - m ) ! ( + m ) ! ] 1 / 2 k = 0 ( - m ) / 2 ( - 1 ) k 2 - ( k ) ( 2 - 2 k ) ( - 2 k ) ! ( - 2 k - m ) ! r 2 k z - 2 k - m . \bar{\Pi}^{m}_{\ell}(z)=\left[\frac{(\ell-m)!}{(\ell+m)!}\right]^{1/2}\sum_{k=% 0}^{\left\lfloor(\ell-m)/2\right\rfloor}(-1)^{k}2^{-\ell}{\left({{\ell}\atop{k% }}\right)}{\left({{2\ell-2k}\atop{\ell}}\right)}\frac{(\ell-2k)!}{(\ell-2k-m)!% }\;r^{2k}\;z^{\ell-2k-m}.
  85. m = 0 m=0
  86. Π ¯ 0 ( z ) = k = 0 / 2 ( - 1 ) k 2 - ( k ) ( 2 - 2 k ) r 2 k z - 2 k . \bar{\Pi}^{0}_{\ell}(z)=\sum_{k=0}^{\left\lfloor\ell/2\right\rfloor}(-1)^{k}2^% {-\ell}{\left({{\ell}\atop{k}}\right)}{\left({{2\ell-2k}\atop{\ell}}\right)}\;% r^{2k}\;z^{\ell-2k}.
  87. Π ¯ m ( z ) \bar{\Pi}^{\ell}_{m}(z)
  88. A m ( x , y ) A_{m}(x,y)\,
  89. B m ( x , y ) B_{m}(x,y)\,
  90. Y 3 1 = - 1 r 3 [ 7 4 π 3 16 ] 1 / 2 ( 5 z 2 - r 2 ) ( x + i y ) = - [ 7 4 π 3 16 ] 1 / 2 ( 5 cos 2 θ - 1 ) ( sin θ e i φ ) Y^{1}_{3}=-\frac{1}{r^{3}}\left[\tfrac{7}{4\pi}\cdot\tfrac{3}{16}\right]^{1/2}% (5z^{2}-r^{2})(x+iy)=-\left[\tfrac{7}{4\pi}\cdot\tfrac{3}{16}\right]^{1/2}(5% \cos^{2}\theta-1)(\sin\theta e^{i\varphi})
  91. Y 4 - 2 = 1 r 4 [ 9 4 π 5 32 ] 1 / 2 ( 7 z 2 - r 2 ) ( x - i y ) 2 = [ 9 4 π 5 32 ] 1 / 2 ( 7 cos 2 θ - 1 ) ( sin 2 θ e - 2 i φ ) Y^{-2}_{4}=\frac{1}{r^{4}}\left[\tfrac{9}{4\pi}\cdot\tfrac{5}{32}\right]^{1/2}% (7z^{2}-r^{2})(x-iy)^{2}=\left[\tfrac{9}{4\pi}\cdot\tfrac{5}{32}\right]^{1/2}(% 7\cos^{2}\theta-1)(\sin^{2}\theta e^{-2i\varphi})
  92. m > 0 m>0
  93. A m A_{m}
  94. m < 0 m<0
  95. B m B_{m}
  96. r ( Y m Y - m ) = [ 2 + 1 4 π ] 1 / 2 Π ¯ m ( z ) ( A m B m ) , m > 0. r^{\ell}\,\begin{pmatrix}Y_{\ell m}\\ Y_{\ell-m}\end{pmatrix}=\left[\frac{2\ell+1}{4\pi}\right]^{1/2}\bar{\Pi}^{m}_{% \ell}(z)\begin{pmatrix}A_{m}\\ B_{m}\\ \end{pmatrix},\qquad m>0.
  97. r Y 0 2 + 1 4 π Π ¯ 0 . r^{\ell}\,Y_{\ell 0}\equiv\sqrt{\frac{2\ell+1}{4\pi}}\bar{\Pi}^{0}_{\ell}.
  98. f ( θ , φ ) = = 0 m = - f m Y m ( θ , φ ) . f(\theta,\varphi)=\sum_{\ell=0}^{\infty}\sum_{m=-\ell}^{\ell}f_{\ell}^{m}\,Y_{% \ell}^{m}(\theta,\varphi).
  99. lim N 0 2 π 0 π | f ( θ , φ ) - = 0 N m = - f m Y m ( θ , φ ) | 2 sin θ d θ d ϕ = 0. \lim_{N\to\infty}\int_{0}^{2\pi}\int_{0}^{\pi}\left|f(\theta,\varphi)-\sum_{% \ell=0}^{N}\sum_{m=-\ell}^{\ell}f_{\ell}^{m}Y_{\ell}^{m}(\theta,\varphi)\right% |^{2}\sin\theta\,d\theta\,d\phi=0.
  100. f m = Ω f ( θ , φ ) Y m * ( θ , φ ) d Ω = 0 2 π d φ 0 π d θ sin θ f ( θ , φ ) Y m * ( θ , φ ) . f_{\ell}^{m}=\int_{\Omega}f(\theta,\varphi)\,Y_{\ell}^{m*}(\theta,\varphi)\,d% \Omega=\int_{0}^{2\pi}d\varphi\int_{0}^{\pi}\,d\theta\,\sin\theta f(\theta,% \varphi)Y_{\ell}^{m*}(\theta,\varphi).
  101. f ( θ , φ ) = = 0 m = - f m Y m ( θ , φ ) . f(\theta,\varphi)=\sum_{\ell=0}^{\infty}\sum_{m=-\ell}^{\ell}f_{\ell m}\,Y_{% \ell m}(\theta,\varphi).
  102. 1 4 π Ω | f ( Ω ) | 2 d Ω = = 0 S f f ( ) , \frac{1}{4\,\pi}\int_{\Omega}|f(\Omega)|^{2}\,d\Omega=\sum_{\ell=0}^{\infty}S_% {f\!f}(\ell),
  103. S f f ( ) = 1 2 + 1 m = - | f m | 2 S_{f\!f}(\ell)=\frac{1}{2\ell+1}\sum_{m=-\ell}^{\ell}|f_{\ell m}|^{2}
  104. 1 4 π Ω f ( Ω ) g ( Ω ) d Ω = = 0 S f g ( ) , \frac{1}{4\,\pi}\int_{\Omega}f(\Omega)\,g^{\ast}(\Omega)\,d\Omega=\sum_{\ell=0% }^{\infty}S_{fg}(\ell),
  105. S f g ( ) = 1 2 + 1 m = - f m g m S_{fg}(\ell)=\frac{1}{2\ell+1}\sum_{m=-\ell}^{\ell}f_{\ell m}g^{\ast}_{\ell m}
  106. S f f ( ) = C β . S_{f\!f}(\ell)=C\,\ell^{\beta}.
  107. = 0 ( 1 + 2 ) s S f f ( ) < , \sum_{\ell=0}^{\infty}(1+\ell^{2})^{s}S_{ff}(\ell)<\infty,
  108. S f f ( ) = O ( - s ) as S_{ff}(\ell)=O(\ell^{-s})\quad\rm{as\ }\ell\to\infty
  109. cos ( θ - θ ) = cos θ cos θ + sin θ sin θ \cos(\theta^{\prime}-\theta)=\cos\theta^{\prime}\cos\theta+\sin\theta\sin% \theta^{\prime}
  110. P ( 𝐱 𝐲 ) = 4 π 2 + 1 m = - Y m * ( θ , φ ) Y m ( θ , φ ) . P_{\ell}(\mathbf{x}\cdot\mathbf{y})=\frac{4\pi}{2\ell+1}\sum_{m=-\ell}^{\ell}Y% _{\ell m}^{*}(\theta^{\prime},\varphi^{\prime})\,Y_{\ell m}(\theta,\varphi).
  111. m = - Y m * ( θ , φ ) Y m ( θ , φ ) = 2 + 1 4 π \sum_{m=-\ell}^{\ell}Y_{\ell m}^{*}(\theta,\varphi)\,Y_{\ell m}(\theta,\varphi% )=\frac{2\ell+1}{4\pi}
  112. Z 𝐱 ( ) Z^{(\ell)}_{\mathbf{x}}
  113. Z 𝐱 ( ) ( 𝐲 ) = j = 1 dim ( 𝐇 ) Y j ( 𝐱 ) ¯ Y j ( 𝐲 ) Z^{(\ell)}_{\mathbf{x}}({\mathbf{y}})=\sum_{j=1}^{\dim(\mathbf{H}_{\ell})}% \overline{Y_{j}({\mathbf{x}})}\,Y_{j}({\mathbf{y}})
  114. Z 𝐱 ( ) ( 𝐲 ) Z^{(\ell)}_{\mathbf{x}}({\mathbf{y}})
  115. Z 𝐱 ( ) ( 𝐲 ) = C ( ( n - 1 ) / 2 ) ( 𝐱 𝐲 ) Z^{(\ell)}_{\mathbf{x}}({\mathbf{y}})=C_{\ell}^{((n-1)/2)}({\mathbf{x}}\cdot{% \mathbf{y}})
  116. dim 𝐇 ω n - 1 = j = 1 dim ( 𝐇 ) | Y j ( 𝐱 ) | 2 \frac{\dim\mathbf{H}_{\ell}}{\omega_{n-1}}=\sum_{j=1}^{\dim(\mathbf{H}_{\ell})% }|Y_{j}({\mathbf{x}})|^{2}
  117. P Ψ ( r ) = Ψ ( - r ) P\Psi(\vec{r})=\Psi(-\vec{r})
  118. { θ , ϕ } \{\theta,\phi\}
  119. { π - θ , π + ϕ } \{\pi-\theta,\pi+\phi\}
  120. Y m ( θ , ϕ ) Y m ( π - θ , π + ϕ ) = ( - 1 ) Y m ( θ , ϕ ) Y_{\ell}^{m}(\theta,\phi)\rightarrow Y_{\ell}^{m}(\pi-\theta,\pi+\phi)=(-1)^{% \ell}Y_{\ell}^{m}(\theta,\phi)
  121. Y m Y_{\ell m}
  122. Re [ Y m ] \,\text{Re}[Y_{\ell m}]
  123. Y m Y_{\ell}^{m}
  124. Re [ Y m ] = 0 \,\text{Re}[Y_{\ell}^{m}]=0
  125. Im [ Y m ] = 0 \,\text{Im}[Y_{\ell}^{m}]=0
  126. Y m Y_{\ell}^{m}
  127. Y m Y_{\ell}^{m}
  128. Y m Y_{\ell}^{m}
  129. Y 0 0 ( θ , φ ) = 1 2 1 π Y_{0}^{0}(\theta,\varphi)={1\over 2}\sqrt{1\over\pi}
  130. Y 1 - 1 ( θ , φ ) = 1 2 3 2 π sin θ e - i φ Y_{1}^{-1}(\theta,\varphi)={1\over 2}\sqrt{3\over 2\pi}\,\sin\theta\,e^{-i\varphi}
  131. Y 1 0 ( θ , φ ) = 1 2 3 π cos θ Y_{1}^{0}(\theta,\varphi)={1\over 2}\sqrt{3\over\pi}\,\cos\theta
  132. Y 1 1 ( θ , φ ) = - 1 2 3 2 π sin θ e i φ Y_{1}^{1}(\theta,\varphi)={-1\over 2}\sqrt{3\over 2\pi}\,\sin\theta\,e^{i\varphi}
  133. Y 2 - 2 ( θ , φ ) = 1 4 15 2 π sin 2 θ e - 2 i φ Y_{2}^{-2}(\theta,\varphi)={1\over 4}\sqrt{15\over 2\pi}\,\sin^{2}\theta\,e^{-% 2i\varphi}
  134. Y 2 - 1 ( θ , φ ) = 1 2 15 2 π sin θ cos θ e - i φ Y_{2}^{-1}(\theta,\varphi)={1\over 2}\sqrt{15\over 2\pi}\,\sin\theta\,\cos% \theta\,e^{-i\varphi}
  135. Y 2 0 ( θ , φ ) = 1 4 5 π ( 3 cos 2 θ - 1 ) Y_{2}^{0}(\theta,\varphi)={1\over 4}\sqrt{5\over\pi}\,(3\cos^{2}\theta-1)
  136. Y 2 1 ( θ , φ ) = - 1 2 15 2 π sin θ cos θ e i φ Y_{2}^{1}(\theta,\varphi)={-1\over 2}\sqrt{15\over 2\pi}\,\sin\theta\,\cos% \theta\,e^{i\varphi}
  137. Y 2 2 ( θ , φ ) = 1 4 15 2 π sin 2 θ e 2 i φ Y_{2}^{2}(\theta,\varphi)={1\over 4}\sqrt{15\over 2\pi}\,\sin^{2}\theta\,e^{2i\varphi}
  138. P ( λ 𝐱 ) = λ P ( 𝐱 ) . P(\lambda\mathbf{x})=\lambda^{\ell}P(\mathbf{x}).
  139. S n - 1 = { 𝐱 𝐑 n | x | = 1 } S^{n-1}=\{\mathbf{x}\in\mathbf{R}^{n}\,\mid\,|x|=1\}
  140. Δ S n - 1 f = - ( + n - 2 ) f . \Delta_{S^{n-1}}f=-\ell(\ell+n-2)f.
  141. 2 = r 1 - n r r n - 1 r + r - 2 Δ S n - 1 . \nabla^{2}=r^{1-n}\frac{\partial}{\partial r}r^{n-1}\frac{\partial}{\partial r% }+r^{-2}\Delta_{S^{n-1}}.
  142. S n - 1 f g ¯ d Ω = 0 \int_{S^{n-1}}f\bar{g}\,d\Omega=0
  143. Δ S n - 1 - 1 \Delta_{S^{n-1}}^{-1}
  144. P ( x ) = P ( x ) + | x | 2 P - 2 + + { | x | P 0 even | x | - 1 P 1 ( x ) odd P(x)=P_{\ell}(x)+|x|^{2}P_{\ell-2}+\cdots+\begin{cases}|x|^{\ell}P_{0}&\ell\rm% {\ even}\\ |x|^{\ell-1}P_{1}(x)&\ell\rm{\ odd}\end{cases}
  145. dim 𝐇 = ( n + - 1 n - 1 ) - ( n + - 3 n - 1 ) . \dim\mathbf{H}_{\ell}={\left({{n+\ell-1}\atop{n-1}}\right)}-{\left({{n+\ell-3}% \atop{n-1}}\right)}.
  146. Δ S n - 1 = sin 2 - n ϕ ϕ sin n - 2 ϕ ϕ + sin - 2 ϕ Δ S n - 2 \Delta_{S^{n-1}}=\sin^{2-n}\phi\frac{\partial}{\partial\phi}\sin^{n-2}\phi% \frac{\partial}{\partial\phi}+\sin^{-2}\phi\Delta_{S^{n-2}}
  147. Y l 1 , l n - 1 ( θ 1 , θ n - 1 ) = 1 2 π e i l 1 θ 1 j = 2 n - 1 P ¯ l j l n - 2 j ( θ j ) Y_{l_{1},\dots l_{n-1}}(\theta_{1},\dots\theta_{n-1})=\frac{1}{\sqrt{2\pi}}e^{% il_{1}\theta_{1}}\prod_{j=2}^{n-1}{}_{j}\bar{P}^{l_{n-2}}_{l_{j}}(\theta_{j})
  148. P ¯ L l j ( θ ) = 2 L + j - 1 2 ( L + l + j - 2 ) ! ( L - l ) ! sin 2 - j 2 ( θ ) P L + j - 2 2 - ( l + j - 2 2 ) ( cos θ ) {}_{j}\bar{P}^{l}_{L}(\theta)=\sqrt{\frac{2L+j-1}{2}\frac{(L+l+j-2)!}{(L-l)!}}% \sin^{\frac{2-j}{2}}(\theta)P^{-(l+\frac{j-2}{2})}_{L+\frac{j-2}{2}}(\cos\theta)
  149. ψ ψ ρ \psi\mapsto\psi\circ\rho
  150. ψ i 1 i \psi_{i_{1}\dots i_{\ell}}
  151. ψ ( x 1 , , x n ) = i 1 i ψ i 1 i x i 1 x i . \psi(x_{1},\dots,x_{n})=\sum_{i_{1}\dots i_{\ell}}\psi_{i_{1}\dots i_{\ell}}x_% {i_{1}}\cdots x_{i_{\ell}}.
  152. ψ i 1 i \psi_{i_{1}\dots i_{\ell}}
  153. A = 0 A=0