wpmath0000004_0

1024_(number).html

  1. 2 10 2^{10}
  2. 32 2 32^{2}
  3. a 97 a\geq 97
  4. 2 1000 10 300 \displaystyle\frac{2^{1000}}{10^{300}}

107_Camilla.html

  1. × 10 1 5 \times 10^{1}5

129_(number).html

  1. 11 2 + 2 2 + 2 2 11^{2}+2^{2}+2^{2}
  2. 10 2 + 5 2 + 2 2 10^{2}+5^{2}+2^{2}
  3. 8 2 + 8 2 + 1 2 8^{2}+8^{2}+1^{2}
  4. 8 2 + 7 2 + 4 2 8^{2}+7^{2}+4^{2}

12AX7.html

  1. A v A_{v}
  2. A v A_{v}
  3. A v A_{v}
  4. A v A_{v}
  5. A v A_{v}
  6. A v A_{v}
  7. A v A_{v}
  8. A v A_{v}
  9. A v A_{v}
  10. A v A_{v}
  11. A v A_{v}
  12. A v A_{v}

130_Elektra.html

  1. × 10 1 8 \times 10^{1}8

131_(number).html

  1. 3 n - 1 3n-1

132_(number).html

  1. 12 + 13 + 21 + 23 + 31 + 32 = 132 12+13+21+23+31+32=132

135_(number).html

  1. 135 = ( 1 + 3 + 5 ) ( 1 × 3 × 5 ) 135=(1+3+5)(1\times 3\times 5)
  2. 135 = 1 1 + 3 2 + 5 3 135=1^{1}+3^{2}+5^{3}
  3. 135 = 11 n 2 + 11 n + 3 135=11n^{2}+11n+3
  4. n = 3 n=3
  5. ζ ( 2 ) \zeta(2)

145_(number).html

  1. 145 = 12 2 + 1 2 = 8 2 + 9 2 145=12^{2}+1^{2}=8^{2}+9^{2}
  2. 145 = 1 ! + 4 ! + 5 ! 145=1!+4!+5!

157_(number).html

  1. p p + 1 p + 1 \frac{p^{p}+1}{p+1}

163_(number).html

  1. π 2 9 163 3.1411 \pi\approx{2^{9}\over 163}\approx 3.1411
  2. e 163 3 4 5 2.7166 e\approx{163\over 3\cdot 4\cdot 5}\approx 2.7166\dots
  3. ( - a ) \mathbb{Q}(\sqrt{-a})
  4. a = 163 a=163
  5. a = 1 , 2 , 3 , 7 , 11 , 19 , 43 , 67 a=1,2,3,7,11,19,43,67
  6. f ( n ) = n 2 + n + 41 f(n)=n^{2}+n+41
  7. n n
  8. n < 10 7 n<10^{7}
  9. f ( n ) = 0 f(n)=0
  10. n = ( - 1 + - 163 ) / 2 n=(-1+\sqrt{-163})/2
  11. 163 \sqrt{163}
  12. e π 163 e^{\pi\sqrt{163}}

167_(number).html

  1. 3 n - 1 3n-1

2.4_Metre.html

  1. R = L + 2 d - F + S 2.37 R=\dfrac{L+2d-F+\sqrt{S}}{2.37}

2–3_heap.html

  1. O ( l o g ( n ) ) O(log(n))

2–3_tree.html

  1. T T
  2. T T
  3. T T
  4. T T
  5. r r
  6. a a
  7. r r
  8. L L
  9. R R
  10. L L
  11. R R
  12. a a
  13. L L
  14. a a
  15. R R
  16. T T
  17. r r
  18. a a
  19. b b
  20. a < b a<b
  21. r r
  22. L L
  23. M M
  24. R R
  25. L L
  26. M M
  27. R R
  28. a a
  29. L L
  30. M M
  31. b b
  32. M M
  33. R R
  34. T T
  35. d d
  36. T T
  37. d d
  38. T T
  39. r r
  40. T T
  41. r r
  42. d d
  43. r r
  44. d d
  45. T T
  46. d d
  47. T T
  48. d d
  49. r r
  50. L L
  51. R R
  52. e e
  53. r r
  54. d d
  55. e e
  56. d d
  57. T T
  58. d < e d<e
  59. T T
  60. L L
  61. d > e d>e
  62. T T
  63. R R
  64. r r
  65. L L
  66. M M
  67. R R
  68. a a
  69. b b
  70. r r
  71. a < b a<b
  72. d d
  73. a a
  74. b b
  75. d d
  76. T T
  77. d < a d<a
  78. T T
  79. L L
  80. a < d < b a<d<b
  81. T T
  82. M M
  83. d > b d>b
  84. T T
  85. R R

3-manifold.html

  1. 𝐓 3 = S 1 × S 1 × S 1 . \mathbf{T}^{3}=S^{1}\times S^{1}\times S^{1}.
  2. π / 3 \pi/3
  3. f : ( D 2 , D 2 ) ( M , M ) f\colon(D^{2},\partial D^{2})\to(M,\partial M)\,
  4. f | D 2 f|\partial D^{2}
  5. M \partial M
  6. M M
  7. π 2 ( M ) \pi_{2}(M)
  8. π 2 ( M ) \pi_{2}(M)
  9. S 2 M S^{2}\to M
  10. M ( u 1 , u 2 , , u n ) M(u_{1},u_{2},\dots,u_{n})
  11. E i E_{i}
  12. M ( u 1 , u 2 , , u n ) M(u_{1},u_{2},\dots,u_{n})
  13. p i 2 + q i 2 p_{i}^{2}+q_{i}^{2}\rightarrow\infty
  14. p i / q i p_{i}/q_{i}
  15. u i u_{i}
  16. ω ω \omega^{\omega}
  17. f : S T f:S\rightarrow T
  18. f : π 1 ( S ) π 1 ( T ) f^{\star}:\pi_{1}(S)\rightarrow\pi_{1}(T)
  19. α S \alpha\subset S
  20. f | a f|_{a}

311_(number).html

  1. 3 n - 1 3n-1
  2. 4 n - 1 4n-1

365_(number).html

  1. 365 = 13 2 + 14 2 365=13^{2}+14^{2}
  2. 365 = 10 2 + 11 2 + 12 2 365=10^{2}+11^{2}+12^{2}

44_Nysa.html

  1. σ ^ O C \hat{\sigma}_{OC}

5_yen_coin.html

  1. Zn 65 {}^{65}{\rm Zn}

70_Ophiuchi.html

  1. M V = 4.83 \scriptstyle M_{V_{\odot}}=4.83
  2. L V L V = 10 0.4 ( M V - M V ) \scriptstyle\frac{L_{V_{\ast}}}{L_{V_{\odot}}}=10^{0.4\left(M_{V_{\odot}}-M_{V% _{\ast}}\right)}

A_Dynamical_Theory_of_the_Electromagnetic_Field.html

  1. 𝐉 t o t = \mathbf{J}_{tot}=
  2. 𝐉 \mathbf{J}
  3. + 𝐃 t +\frac{\partial\mathbf{D}}{\partial t}
  4. μ 𝐇 = × 𝐀 \mu\mathbf{H}=\nabla\times\mathbf{A}
  5. × 𝐇 = 𝐉 t o t \nabla\times\mathbf{H}=\mathbf{J}_{tot}
  6. 𝐟 = μ ( 𝐯 × 𝐇 ) - 𝐀 t - ϕ \mathbf{f}=\mu(\mathbf{v}\times\mathbf{H})-\frac{\partial\mathbf{A}}{\partial t% }-\nabla\phi
  7. 𝐟 = 1 ϵ 𝐃 \mathbf{f}=\frac{1}{\epsilon}\mathbf{D}
  8. 𝐟 = 1 σ 𝐉 \mathbf{f}=\frac{1}{\sigma}\mathbf{J}
  9. 𝐃 = - ρ \nabla\cdot\mathbf{D}=-\rho
  10. 𝐉 = - ρ t \nabla\cdot\mathbf{J}=-\frac{\partial\rho}{\partial t}
  11. 𝐇 \mathbf{H}
  12. 𝐉 \mathbf{J}
  13. 𝐉 t o t \mathbf{J}_{tot}
  14. 𝐃 \mathbf{D}
  15. ρ \rho
  16. 𝐀 \mathbf{A}
  17. 𝐟 \mathbf{f}
  18. ϕ \phi
  19. σ \sigma
  20. μ ( 𝐯 × 𝐇 ) \mu(\mathbf{v}\times\mathbf{H})
  21. 𝐯 \mathbf{v}
  22. μ ( 𝐯 × 𝐇 ) \mu(\mathbf{v}\times\mathbf{H})
  23. 𝐇 = 0 \nabla\cdot\mathbf{H}=0
  24. × 𝐇 = ε o 𝐄 t \nabla\times\mathbf{H}=\varepsilon_{o}\frac{\partial\mathbf{E}}{\partial t}
  25. × × 𝐄 = - μ o t × 𝐇 = - μ o ε o 2 𝐄 t 2 \nabla\times\nabla\times\mathbf{E}=-\mu_{o}\frac{\partial}{\partial t}\nabla% \times\mathbf{H}=-\mu_{o}\varepsilon_{o}\frac{\partial^{2}\mathbf{E}}{\partial t% ^{2}}
  26. × × 𝐇 = ε o t × 𝐄 = - μ o ε o 2 𝐇 t 2 \nabla\times\nabla\times\mathbf{H}=\varepsilon_{o}\frac{\partial}{\partial t}% \nabla\times\mathbf{E}=-\mu_{o}\varepsilon_{o}\frac{\partial^{2}\mathbf{H}}{% \partial t^{2}}
  27. × ( × 𝐕 ) = ( 𝐕 ) - 2 𝐕 \nabla\times\left(\nabla\times\mathbf{V}\right)=\nabla\left(\nabla\cdot\mathbf% {V}\right)-\nabla^{2}\mathbf{V}
  28. 𝐕 \mathbf{V}
  29. 2 𝐄 t 2 - c 2 2 𝐄 = 0 {\partial^{2}\mathbf{E}\over\partial t^{2}}\ -\ c^{2}\cdot\nabla^{2}\mathbf{E}% \ \ =\ \ 0
  30. 2 𝐇 t 2 - c 2 2 𝐇 = 0 {\partial^{2}\mathbf{H}\over\partial t^{2}}\ -\ c^{2}\cdot\nabla^{2}\mathbf{H}% \ \ =\ \ 0
  31. c = 1 μ o ε o = 2.99792458 × 10 8 c={1\over\sqrt{\mu_{o}\varepsilon_{o}}}=2.99792458\times 10^{8}

Abbe_sine_condition.html

  1. sin u sin U = sin u sin U \frac{\sin u^{\prime}}{\sin U^{\prime}}=\frac{\sin u}{\sin U}
  2. T ( x o , y o ) = T ( k x , k y ) e j ( k x x o + k y y o ) d k x d k y . T(x_{o},y_{o})=\iint T(k_{x},k_{y})~{}e^{j(k_{x}x_{o}+k_{y}y_{o})}~{}dk_{x}\,% dk_{y}.
  3. x i = M x o x_{i}=Mx_{o}\,
  4. y i = M y o y_{i}=My_{o}\,
  5. T ( x o , y o ) = T ( k x , k y ) e j ( ( k x / M ) ( M x o ) + ( k y / M ) ( M y o ) ) d k x d k y T(x_{o},y_{o})=\iint T(k_{x},k_{y})~{}e^{j((k_{x}/M)(Mx_{o})+(k_{y}/M)(My_{o})% )}~{}dk_{x}\,dk_{y}
  6. T ( x i , y i ) = T ( k x , k y ) e j ( ( k x / M ) x i + ( k y / M ) y i ) d k x d k y T(x_{i},y_{i})=\iint T(k_{x},k_{y})~{}e^{j((k_{x}/M)x_{i}+(k_{y}/M)y_{i})}~{}% dk_{x}\,dk_{y}
  7. k x i = k x M k^{i}_{x}=\frac{k_{x}}{M}
  8. k y i = k y M k^{i}_{y}=\frac{k_{y}}{M}
  9. T ( x i , y i ) = M 2 T ( M k x i , M k y i ) e j ( k x i x i + k y i y i ) d k x i d k y i T(x_{i},y_{i})=M^{2}\iint T(Mk^{i}_{x},Mk^{i}_{y})~{}e^{j(k^{i}_{x}x_{i}+k^{i}% _{y}y_{i})}dk^{i}_{x}\,dk^{i}_{y}
  10. k x = k sin θ cos φ k_{x}=k\sin\theta\cos\varphi\,
  11. k y = k sin θ sin φ k_{y}=k\sin\theta\sin\varphi\,
  12. ϕ = 0 \phi=0
  13. k i sin θ i = k sin θ M . k^{i}\sin\theta^{i}=k\frac{\sin\theta}{M}.

Abelian_integral.html

  1. z 0 z R ( x , w ) d x , \int_{z_{0}}^{z}R\left(x,w\right)dx,
  2. R ( x , w ) R\left(x,w\right)
  3. x x
  4. w w
  5. F ( x , w ) = 0 , F\left(x,w\right)=0,\,
  6. F ( x , w ) F\left(x,w\right)
  7. w w
  8. F ( x , w ) ϕ n ( x ) w n + + ϕ 1 ( x ) w + ϕ 0 ( x ) , F\left(x,w\right)\equiv\phi_{n}\left(x\right)w^{n}+\cdots+\phi_{1}\left(x% \right)w+\phi_{0}\left(x\right),\,
  9. ϕ j ( x ) \phi_{j}\left(x\right)
  10. j = 0 , 1 , , n j=0,1,\ldots,n
  11. x x
  12. z z
  13. F ( x , w ) = w 2 - P ( x ) , F\left(x,w\right)=w^{2}-P\left(x\right),\,
  14. P ( x ) P\left(x\right)
  15. P ( x ) P\left(x\right)
  16. S S
  17. ω \omega
  18. S S
  19. P 0 P_{0}
  20. S S
  21. P 0 P ω \int_{P_{0}}^{P}\omega
  22. f ( P ) f\left(P\right)
  23. C C
  24. S S
  25. P 0 P_{0}
  26. P P
  27. S S
  28. C C
  29. C C
  30. S S
  31. f f
  32. S S
  33. A \sqrt{A}
  34. A A
  35. > 4 >4
  36. J ( S ) J\left(S\right)
  37. P 0 P_{0}
  38. S J ( S ) S\to J\left(S\right)\,
  39. S J ( S ) S\to J\left(S\right)

Abelian_variety_of_CM-type.html

  1. E n d ( A ) End_{\mathbb{Q}}(A)\,

Abiogenic_petroleum_origin.html

  1. CH 4 + 1 2 O 2 2 H 2 + CO \mathrm{CH_{4}+\begin{matrix}\frac{1}{2}\end{matrix}O_{2}\rightarrow 2H_{2}+CO}
  2. ( 2 n + 1 ) H 2 + nCO C n H 2 n + 2 + nH 2 O \mathrm{(2n+1)H_{2}+nCO\rightarrow C_{n}H_{2n+2}+nH_{2}O}
  3. 3 F e 2 SiO 4 + 2 H 2 O 2 F e 3 O 4 + 3 S i O 2 + 2 H 2 \mathrm{3Fe_{2}SiO_{4}+2H_{2}O\rightarrow 2Fe_{3}O_{4}+3SiO_{2}+2H_{2}}
  4. 3 M g 2 SiO 4 + SiO 2 + 4 H 2 O 2 M g 3 Si 2 O 5 ( OH ) 4 \mathrm{3Mg_{2}SiO_{4}+SiO_{2}+4H_{2}O\rightarrow 2Mg_{3}Si_{2}O_{5}(OH)_{4}}
  5. ( Fe , Mg ) 2 SiO 4 + nH 2 O + CO 2 Mg 3 Si 2 O 5 ( OH ) 4 + Fe 3 O 4 + CH 4 \mathrm{(Fe,Mg)_{2}SiO_{4}+nH_{2}O+CO_{2}\rightarrow Mg_{3}Si_{2}O_{5}(OH)_{4}% +Fe_{3}O_{4}+CH_{4}}
  6. 18 M g 2 SiO 4 + 6 F e 2 SiO 4 + 26 H 2 O + CO 2 \mathrm{18Mg_{2}SiO_{4}+6Fe_{2}SiO_{4}+26H_{2}O+CO_{2}}
  7. 12 M g 3 Si 2 O 5 ( OH ) 4 + 4 F e 3 O 4 + CH 4 \mathrm{12Mg_{3}Si_{2}O_{5}(OH)_{4}+4Fe_{3}O_{4}+CH_{4}}
  8. ( Fe , Mg ) 2 SiO 4 + nH 2 O + CO 2 Mg 3 Si 2 O 5 ( OH ) 4 + Fe 3 O 4 + MgCO 3 + SiO 2 \mathrm{(Fe,Mg)_{2}SiO_{4}+nH_{2}O+CO_{2}\rightarrow Mg_{3}Si_{2}O_{5}(OH)_{4}% +Fe_{3}O_{4}+MgCO_{3}+SiO_{2}}
  9. nCH 4 + nFe 3 O 4 + nH 2 O C 2 H 6 + Fe 2 O 3 + HCO 3 + H + \mathrm{nCH_{4}+nFe_{3}O_{4}+nH_{2}O\rightarrow C_{2}H_{6}+Fe_{2}O_{3}+HCO_{3}% +H^{+}}
  10. 4 H 2 + CaCO 3 CH 4 + CaO + 2 H 2 O \mathrm{4H_{2}+CaCO_{3}\rightarrow CH_{4}+CaO+2H_{2}O}

Abraham–Minkowski_controversy.html

  1. p M = n h ν c ; p_{\mathrm{M}}=\frac{nh\nu}{c};
  2. p A = h ν n c , p_{\mathrm{A}}=\frac{h\nu}{nc},
  3. 𝐠 M = 𝐃 × 𝐁 ; \mathbf{g}_{\mathrm{M}}=\mathbf{D}\times\mathbf{B};
  4. 𝐠 A = 1 c 2 𝐄 × 𝐇 , \mathbf{g}_{\mathrm{A}}=\frac{1}{\mathrm{c}^{2}}\mathbf{E}\times\mathbf{H},

Abstract_simplicial_complex.html

  1. Δ Δ
  2. X X
  3. Δ Δ
  4. Y X Y⊂X
  5. Y Y
  6. Δ Δ
  7. Δ Δ
  8. Y Y
  9. X X
  10. Y X Y⊂X
  11. Δ Δ
  12. Δ Δ
  13. Δ Δ
  14. V ( Δ ) = Δ V(Δ)=∪Δ
  15. Δ Δ
  16. Δ Δ
  17. Δ Δ
  18. X X
  19. Δ Δ
  20. d i m ( X ) = | X | 1 dim(X)=|X|−1
  21. d i m ( Δ ) dim(Δ)
  22. Δ Δ
  23. Δ Δ
  24. Δ Δ
  25. d i m ( Δ ) dim(Δ)
  26. d i m ( Δ ) dim(Δ)
  27. Δ Δ
  28. Δ Δ
  29. L Δ L⊂Δ
  30. Δ Δ
  31. Δ Δ
  32. Δ Δ
  33. Δ Δ
  34. Δ Δ
  35. Δ Δ
  36. Δ Δ
  37. Y Y
  38. Δ Δ
  39. Δ / Y Δ/Y
  40. Δ Δ
  41. Δ / Y := { X Δ X Y = , X Y Δ } . \Delta/Y:=\{X\in\Delta\mid X\cap Y=\varnothing,\,X\cup Y\in\Delta\}.
  42. Δ Δ
  43. Δ Δ
  44. Γ Γ
  45. f f
  46. Δ Δ
  47. X X
  48. Δ Δ
  49. f ( X ) f(X)
  50. Γ Γ
  51. t : S 0 , 11 t:S→0,11
  52. s S t s = 1 \sum_{s\in S}t_{s}=1
  53. { s S : t s > 0 } Δ \{s\in S:t_{s}>0\}\in\Delta
  54. 𝒦 \mathcal{K}
  55. K K
  56. K K
  57. 𝒦 \mathcal{K}
  58. X K X∈K
  59. X X
  60. Y X Y⊂X
  61. F ( Y ) F ( X ) F(Y)→F(X)
  62. X K F ( X ) \coprod_{X\in K}{F(X)}
  63. y F ( Y ) y∈F(Y)
  64. F ( Y ) F ( X ) F(Y)→F(X)
  65. Y X Y⊂X
  66. 𝐑 < s u p > N \mathbf{R}<sup>N

Abū_al-Wafā'_Būzjānī.html

  1. sin ( α ± β ) = sin α cos β ± cos α sin β \sin(\alpha\pm\beta)=\sin\alpha\cos\beta\pm\cos\alpha\sin\beta
  2. sin A sin a = sin B sin b = sin C sin c \frac{\sin A}{\sin a}=\frac{\sin B}{\sin b}=\frac{\sin C}{\sin c}

Accelerator_effect.html

  1. I t = μ v i = 1 ( 1 - μ ) i ( Y t - i - Y t - i - 1 ) I_{t}=\mu v\sum_{i=1}^{\infty}\left(1-\mu\right)^{i}\left(Y_{t-i}-Y_{t-i-1}\right)
  2. Y t = m I t = 1 1 - M P C I t Y_{t}=mI_{t}=\frac{1}{1-MPC}I_{t}
  3. I t K t * \frac{I_{t}}{K_{t}^{*}}
  4. I n = x ( K d - K - 1 ) I_{n}=x(K^{d}-K_{-1})
  5. I t = μ v i = 1 ( 1 - μ ) i ( Y t - i - Y t - i - 1 ) I_{t}=\mu v\sum_{i=1}^{\infty}\left(1-\mu\right)^{i}\left(Y_{t-i}-Y_{t-i-1}\right)

Accounting_method.html

  1. m 2 \frac{m}{2}
  2. m 2 \frac{m}{2}
  3. 2 m 2m
  4. 2 m m / 2 = 4 \frac{2m}{m/2}=4

Accrued_interest.html

  1. I A = T × P × R I_{A}=T\times P\times R
  2. I A I_{A}
  3. T T
  4. P P
  5. R R
  6. T T
  7. T = D P D Y T=\frac{D_{P}}{D_{Y}}
  8. D P D_{P}
  9. D Y D_{Y}

Accumulation_function.html

  1. A ( t ) = k a ( t ) A(t)=k\cdot a(t)
  2. a ( t ) = 1 + t i a(t)=1+t\cdot i
  3. a ( t ) = ( 1 + i ) t a(t)=(1+i)^{t}
  4. a ( t ) = ( 1 - d t ) - 1 a(t)=(1-d\cdot t)^{-1}
  5. a ( t ) = ( 1 - d ) - t a(t)=(1-d)^{-t}
  6. δ t = a ( t ) a ( t ) \delta_{t}=\frac{a^{\prime}(t)}{a(t)}\,
  7. a ( t ) = e 0 t δ u d u a(t)=e^{\int_{0}^{t}\delta_{u}\,du}
  8. a ( t ) = e t δ a(t)=e^{t\delta}
  9. δ \delta
  10. r ( t ) = e δ t - 1 r(t)=e^{\delta_{t}}-1

Acid–base_titration.html

  1. pH = - log K a F \mathrm{pH}=-\log\sqrt{K_{a}F}
  2. pH = p K a + log ( n O H - a d d e d n H A i n i t i a l - n O H - a d d e d ) \mathrm{pH}=pK_{a}+\log(\frac{n_{OH^{-}added}}{n_{HAinitial}-n_{OH^{-}added}})
  3. n O H - a d d e d = n H A i n i t i a l - n O H - a d d e d {n_{OH^{-}added}}={n_{HAinitial}-n_{OH^{-}added}}
  4. pH = 14 + log C a C b K w ( C a + C b ) K a \mathrm{pH}=14+\log\sqrt{\frac{C_{a}C_{b}K_{w}}{(C_{a}+C_{b})K_{a}}}
  5. pH = 14 + log C b V b - C a V a ( V a + V b ) \mathrm{pH}=14+\log\frac{C_{b}V_{b}-C_{a}V_{a}}{(V_{a}+V_{b})}
  6. ϕ = C b V b C a V a = α A - - [ H + ] - [ O H - ] C a 1 + [ H + ] - [ O H - ] C b \phi=\frac{C_{b}V_{b}}{C_{a}V_{a}}=\frac{\alpha_{A^{-}}-\frac{[H^{+}]-[OH^{-}]% }{C_{a}}}{1+\frac{[H^{+}]-[OH^{-}]}{C_{b}}}
  7. α A - = K a [ H + ] + K a \alpha_{A^{-}}=\frac{K_{a}}{[H^{+}]+K_{a}}

Actuarial_notation.html

  1. i \,i
  2. i = 0.12 \,i=0.12
  3. i ( m ) \,i^{(m)}
  4. m m
  5. m m
  6. m m
  7. i ( 2 ) \,i^{(2)}
  8. i ( 2 ) / 2 \,i^{(2)}/2
  9. ( 1.0583 ) 2 = 1.12 \,(1.0583)^{2}=1.12
  10. i ( 2 ) / 2 = 0.0583 \,i^{(2)}/2=0.0583
  11. i ( 2 ) = 0.1166 \,i^{(2)}=0.1166
  12. i ( m ) \,i^{(m)}
  13. i = 0.12 \,i=0.12
  14. i ( 12 ) = 0.1139 \,i^{(12)}=0.1139
  15. ( 1 + 0.1139 12 ) 12 = 1.12 \,\left(1+\frac{0.1139}{12}\right)^{12}=1.12
  16. v \,v
  17. v = ( 1 + i ) - 1 1 - i + i 2 \,v={(1+i)}^{-1}\approx 1-i+i^{2}
  18. 1 × v \,1\times v
  19. 25 × v 5 \,25\times v^{5}
  20. d \,d
  21. d = i 1 + i i - i 2 d=\frac{i}{1+i}\approx i-i^{2}
  22. d \,d
  23. ( 1 - d ) = v = ( 1 + i ) - 1 \,(1-d)=v={(1+i)}^{-1}
  24. n \,n
  25. ( 1 - d ) n \,{(1-d)}^{n}
  26. ( 1 + i ) n \,{(1+i)}^{n}
  27. n \,n
  28. d ( m ) \,d^{(m)}
  29. m \,m
  30. i ( m ) \,i^{(m)}
  31. m m
  32. δ \,\delta
  33. m m
  34. δ = lim m i ( m ) \,\delta=\lim_{m\to\infty}i^{(m)}
  35. i \,i
  36. δ \,\delta
  37. d \,d
  38. ( 1 + i ) = ( 1 + i ( m ) m ) m = e δ = ( 1 - d ( m ) m ) - m = ( 1 - d ) - 1 \,(1+i)=\left(1+\frac{i^{(m)}}{m}\right)^{m}=e^{\delta}=\left(1-\frac{d^{(m)}}% {m}\right)^{-m}=(1-d)^{-1}
  39. i > i ( 2 ) > i ( 3 ) > > δ > > d ( 3 ) > d ( 2 ) > d \,i>i^{(2)}>i^{(3)}>\cdots>\delta>\cdots>d^{(3)}>d^{(2)}>d
  40. l x \,l_{x}
  41. x x
  42. l 0 \,l_{0}
  43. l x \,l_{x}
  44. ω \omega
  45. l n \,l_{n}
  46. n ω \,n\geq\omega
  47. d x \,d_{x}
  48. x x
  49. x + 1 x+1
  50. d x \,d_{x}
  51. d x = l x - l x + 1 \,d_{x}=l_{x}-l_{x+1}
  52. q x \,q_{x}
  53. x x
  54. x + 1 x+1
  55. q x = d x / l x \,q_{x}=d_{x}/l_{x}
  56. p x \,p_{x}
  57. x x
  58. x + 1 x+1
  59. p x = l x + 1 / l x \,p_{x}=l_{x+1}/l_{x}
  60. x x
  61. x + 1 x+1
  62. p x + q x = 1 \,p_{x}+q_{x}=1
  63. d x n = d x + d x + 1 + + d x + n - 1 = l x - l x + n \,{}_{n}d_{x}=d_{x}+d_{x+1}+\cdots+d_{x+n-1}=l_{x}-l_{x+n}
  64. x x
  65. x + n x+n
  66. q x n \,{}_{n}q_{x}
  67. x x
  68. x + n x+n
  69. q x n = d x n / l x \,{}_{n}q_{x}={}_{n}d_{x}/l_{x}
  70. p x n \,{}_{n}p_{x}
  71. x x
  72. x + n x+n
  73. p x n = l x + n / l x \,{}_{n}p_{x}=l_{x+n}/l_{x}
  74. e x \,e_{x}
  75. x x
  76. e x = t = 1 p x t \,e_{x}=\sum_{t=1}^{\infty}\ {}_{t}p_{x}
  77. l x + t \,l_{x+t}
  78. l x \,l_{x}
  79. l x + 1 \,l_{x+1}
  80. l x + t = ( 1 - t ) l x + t l x + 1 \,l_{x+t}=(1-t)l_{x}+tl_{x+1}
  81. a \,a
  82. a n | ¯ i a_{\overline{n|}i}
  83. n n
  84. a n | ¯ i = v + v 2 + + v n = 1 - v n i \,a_{\overline{n|}i}=v+v^{2}+\cdots+v^{n}=\frac{1-v^{n}}{i}
  85. a ¨ n | ¯ i \ddot{a}_{\overline{n|}i}
  86. n n
  87. a ¨ n | ¯ i = 1 + v + + v n - 1 = 1 - v n d \ddot{a}_{\overline{n|}i}=1+v+\cdots+v^{n-1}=\frac{1-v^{n}}{d}
  88. s n | ¯ i \,s_{\overline{n|}i}
  89. s ¨ n | ¯ i \ddot{s}_{\overline{n|}i}
  90. ( m ) \,(m)
  91. m m
  92. n n
  93. m m
  94. a n | ¯ i ( m ) = 1 - v n i ( m ) a_{\overline{n|}i}^{(m)}=\frac{1-v^{n}}{i^{(m)}}
  95. a ¨ n | ¯ i ( m ) = 1 - v n d ( m ) \ddot{a}_{\overline{n|}i}^{(m)}=\frac{1-v^{n}}{d^{(m)}}
  96. a ¯ n | ¯ i \overline{a}_{\overline{n|}i}
  97. a n | ¯ i ( m ) \,a_{\overline{n|}i}^{(m)}
  98. m m
  99. a ¯ n | ¯ i = 1 - v n δ \overline{a}_{\overline{n|}i}=\frac{1-v^{n}}{\delta}
  100. a n | ¯ i < a n | ¯ i ( m ) < a ¯ n | ¯ i < a ¨ n | ¯ i ( m ) < a ¨ n | ¯ i a_{\overline{n|}i}<a_{\overline{n|}i}^{(m)}<\overline{a}_{\overline{n|}i}<% \ddot{a}_{\overline{n|}i}^{(m)}<\ddot{a}_{\overline{n|}i}
  101. i i
  102. d d
  103. δ \delta
  104. a 65 \,a_{65}
  105. a 10 | ¯ a_{\overline{10|}}
  106. a 65 : 10 | ¯ a_{65:\overline{10|}}
  107. a 65 ( 12 ) a_{65}^{(12)}
  108. a ¨ 65 {\ddot{a}}_{65}
  109. a x : n | ¯ i ( m ) a_{x:\overline{n|}i}^{(m)}
  110. x x
  111. n n
  112. m m
  113. i i
  114. A \,A
  115. A ( 12 ) A^{(12)}
  116. A x \,A_{x}
  117. A x ( 12 ) \,A_{x}^{(12)}
  118. A ¯ x \,\overline{A}_{x}
  119. P Δ x ( x ) = P ( x < X < x + Δ x X > x ) = F X ( x + Δ x ) - F X ( x ) ( 1 - F X ( x ) ) P_{\Delta x}(x)=P(x<X<x+\Delta\;x\mid\;X>x)=\frac{F_{X}(x+\Delta\;x)-F_{X}(x)}% {(1-F_{X}(x))}
  120. μ ( x ) = F X ( x ) 1 - F X ( x ) \mu\,(x)=\frac{F^{\prime}_{X}(x)}{1-F_{X}(x)}

Adaptive_Huffman_coding.html

  1. W ( A ) > W ( B ) > W ( C ) W(A)>W(B)>W(C)

Adaptive_system.html

  1. S S
  2. E E
  3. S S
  4. P ( S S | E ) P(S\rightarrow S^{\prime}|E)
  5. E E
  6. S S
  7. E E
  8. P ( S S | E ) > P ( S S ) P(S\rightarrow S^{\prime}|E)>P(S\rightarrow S^{\prime})
  9. S S
  10. t t
  11. E E
  12. S S
  13. S S
  14. ( t ) (t\rightarrow\infty)
  15. S S
  16. ( S S ) (S\rightarrow S^{\prime})
  17. t 0 t_{0}
  18. E E
  19. E E
  20. P t 0 ( S S | E ) > P t 0 ( S S ) > 0 P_{t_{0}}(S\rightarrow S^{\prime}|E)>P_{t_{0}}(S\rightarrow S^{\prime})>0
  21. lim t P t ( S S | E ) = P t ( S S ) \lim_{t\rightarrow\infty}P_{t}(S\rightarrow S^{\prime}|E)=P_{t}(S\rightarrow S% ^{\prime})
  22. t t
  23. h h
  24. P t + h ( S S | E ) - P t + h ( S S ) < P t ( S S | E ) - P t ( S S ) P_{t+h}(S\rightarrow S^{\prime}|E)-P_{t+h}(S\rightarrow S^{\prime})<P_{t}(S% \rightarrow S^{\prime}|E)-P_{t}(S\rightarrow S^{\prime})

Addition-chain_exponentiation.html

  1. a 15 = a × ( a × [ a × a 2 ] 2 ) 2 a^{15}=a\times(a\times[a\times a^{2}]^{2})^{2}\!
  2. a 15 = a 3 × ( [ a 3 ] 2 ) 2 a^{15}=a^{3}\times([a^{3}]^{2})^{2}\!
  3. a - 31 = a / ( ( ( ( a 2 ) 2 ) 2 ) 2 ) 2 a^{-31}=a/((((a^{2})^{2})^{2})^{2})^{2}\!

Address_space_layout_randomization.html

  1. E s E_{s}
  2. E m E_{m}
  3. E x E_{x}
  4. E h E_{h}
  5. A s A_{s}
  6. A m A_{m}
  7. A x A_{x}
  8. A h A_{h}
  9. α \alpha
  10. N N
  11. N = E s - A s + E m - A m + E x - A x + E h - A h N=E_{s}-A_{s}+E_{m}-A_{m}+E_{x}-A_{x}+E_{h}-A_{h}\,
  12. α \alpha
  13. α \alpha\,
  14. N N
  15. g ( α ) = isolated guessing; address space is re-randomized after each attempt g\left(\alpha\,\right)=\mbox{isolated guessing; address space is re-randomized% after each attempt}~{}\,
  16. g ( α ) = 1 - ( 1 - 2 - N ) α : 0 α g\left(\alpha\,\right)=1-{\left(1-{2^{-N}}\right)^{\alpha}\,}:0\leq\,\alpha\,
  17. b ( α ) = systematic brute forcing on copies of the program with the same address space b\left(\alpha\,\right)=\mbox{systematic brute forcing on copies of the program% with the same address space}~{}
  18. b ( α ) = α < m t p l > 2 N : 0 α 2 N b\left(\alpha\,\right)=\frac{\alpha\,}{<}mtpl>{{2^{N}}}:0\leq\,\alpha\,\leq\,{% 2^{N}}
  19. 2 N 2^{N}
  20. l = 1 l=1
  21. l l
  22. β \beta\,
  23. E m = log 2 ( l ) : β = 1 , l 1 E_{m}=\log_{2}\left(l\right):\beta\,=1,l\geq\,1
  24. E m = i = l l - ( β - 1 ) log 2 ( i ) : β 1 , l 1 E_{m}=\sum_{i=l}^{l-\left(\beta\,-1\right)}\log_{2}\left(i\right):\beta\,\geq% \,1,l\geq\,1
  25. l l
  26. β = 1 \beta\,=1
  27. log 2 ( n ) \log_{2}\!\left(n\right)
  28. n n

ADE_classification.html

  1. π / 2 = 90 \pi/2=90^{\circ}
  2. 2 π / 3 = 120 2\pi/3=120^{\circ}
  3. A n , D n , E 6 , E 7 , E 8 . A_{n},\,D_{n},\,E_{6},\,E_{7},\,E_{8}.
  4. B n B_{n}
  5. C n C_{n}
  6. F 4 F_{4}
  7. G 2 G_{2}
  8. n 4 n\geq 4
  9. D n . D_{n}.
  10. D 3 A 3 , E 4 A 4 , E 5 D 5 , D_{3}\cong A_{3},E_{4}\cong A_{4},E_{5}\cong D_{5},
  11. A n A_{n}
  12. 𝔰 𝔩 n + 1 ( 𝐂 ) , \mathfrak{sl}_{n+1}(\mathbf{C}),
  13. D n D_{n}
  14. 𝔰 𝔬 2 n ( 𝐂 ) , \mathfrak{so}_{2n}(\mathbf{C}),
  15. E 6 , E 7 , E 8 E_{6},E_{7},E_{8}
  16. A n A_{n}
  17. 𝔰 𝔲 n + 1 , \mathfrak{su}_{n+1},
  18. S U ( n + 1 ) ; SU(n+1);
  19. D n D_{n}
  20. 𝔰 𝔬 2 n ( 𝐑 ) , \mathfrak{so}_{2n}(\mathbf{R}),
  21. P S O ( 2 n ) PSO(2n)
  22. E 6 , E 7 , E 8 E_{6},E_{7},E_{8}
  23. S U ( 2 ) SU(2)
  24. A ~ n , D ~ n , E ~ k , \tilde{A}_{n},\tilde{D}_{n},\tilde{E}_{k},
  25. E 6 , E 7 , E 8 , E_{6},E_{7},E_{8},
  26. A 3 , B C 3 , A_{3},BC_{3},
  27. H 3 . H_{3}.
  28. 𝐂 2 \mathbf{C}^{2}
  29. Δ ϕ = ϕ . \Delta\phi=\phi.
  30. Δ - I . \Delta-I.
  31. Δ ϕ = ϕ - 2. \Delta\phi=\phi-2.
  32. E ~ 6 , E ~ 7 , E ~ 8 \tilde{E}_{6},\tilde{E}_{7},\tilde{E}_{8}
  33. S 3 , S 2 , S 1 , S_{3},S_{2},S_{1},
  34. G ~ 2 , F ~ 4 , E ~ 8 \tilde{G}_{2},\tilde{F}_{4},\tilde{E}_{8}
  35. E ~ 8 \tilde{E}_{8}
  36. E ~ 7 \tilde{E}_{7}
  37. E ~ 6 \tilde{E}_{6}
  38. A 4 , S 4 , A 5 A_{4},S_{4},A_{5}
  39. A 4 × Z 5 , A_{4}\times Z_{5},
  40. S 4 × Z 7 , S_{4}\times Z_{7},
  41. A 5 × Z 11 . A_{5}\times Z_{11}.

Adelic_algebraic_group.html

  1. G ( A ) G(A)
  2. G = G L 1 G=GL_{1}
  3. G L 1 GL_{1}

Adhesion.html

  1. E = 3 h ν - 3 4 h ν α 2 R 6 E=3h\nu-\frac{3}{4}\frac{h\nu\alpha^{2}}{R^{6}}
  2. P a r e a = - A 24 π z 3 \frac{P}{area}=-\frac{A}{24\pi z^{3}}

Adiabatic_theorem.html

  1. t 0 \scriptstyle{t_{0}}
  2. H ^ ( t 0 ) \scriptstyle{\hat{H}(t_{0})}
  3. H ^ ( t 0 ) \scriptstyle{\hat{H}(t_{0})}
  4. ψ ( x , t 0 ) \scriptstyle{\psi(x,t_{0})}
  5. H ^ ( t 1 ) \scriptstyle{\hat{H}(t_{1})}
  6. t 1 \scriptstyle{t_{1}}
  7. ψ ( x , t 1 ) \scriptstyle{\psi(x,t_{1})}
  8. τ = t 1 - t 0 \scriptstyle{\tau=t_{1}-t_{0}}
  9. τ \scriptstyle{\tau\rightarrow\infty}
  10. ψ ( x , t 1 ) \scriptstyle{\psi(x,t_{1})}
  11. H ^ ( t 1 ) \scriptstyle{\hat{H}(t_{1})}
  12. | ψ ( x , t 1 ) | 2 | ψ ( x , t 0 ) | 2 |\psi(x,t_{1})|^{2}\neq|\psi(x,t_{0})|^{2}
  13. ψ ( x , t 0 ) \scriptstyle{\psi(x,t_{0})}
  14. τ \scriptstyle{\tau}
  15. ψ ( x , t 0 ) \scriptstyle{\psi(x,t_{0})}
  16. τ i n t = 2 π / E 0 \scriptstyle{\tau_{int}=2\pi\hbar/E_{0}}
  17. E 0 \scriptstyle{E_{0}}
  18. ψ ( x , t 0 ) \scriptstyle{\psi(x,t_{0})}
  19. τ 0 \scriptstyle{\tau\rightarrow 0}
  20. | ψ ( x , t 1 ) | 2 = | ψ ( x , t 0 ) | 2 |\psi(x,t_{1})|^{2}=|\psi(x,t_{0})|^{2}\quad
  21. H ^ \scriptstyle{\hat{H}}
  22. H ^ ( t 1 ) \scriptstyle{\hat{H}(t_{1})}
  23. ψ ( t 0 ) \scriptstyle{\psi(t_{0})}
  24. | ψ ( t ) | 2 \scriptstyle{|\psi(t)|^{2}}
  25. k \scriptstyle{k}
  26. k \scriptstyle{k}
  27. ( d k d t 0 ) \scriptstyle{\left(\frac{dk}{dt}\rightarrow 0\right)}
  28. t \scriptstyle{t}
  29. ψ ( t ) \scriptstyle{\psi(t)}
  30. H ^ ( t ) \scriptstyle{\hat{H}(t)}
  31. H ^ ( 0 ) \scriptstyle{\hat{H}(0)}
  32. n = 0 \scriptstyle{n=0}
  33. ( d k d t ) \scriptstyle{\left(\frac{dk}{dt}\rightarrow\infty\right)}
  34. ( | ψ ( t ) | 2 = | ψ ( 0 ) | 2 ) \scriptstyle{\left(|\psi(t)|^{2}=|\psi(0)|^{2}\right)}
  35. H ^ ( t ) \scriptstyle{\hat{H}(t)}
  36. H ^ ( t ) \scriptstyle{\hat{H}(t)}
  37. | 1 \scriptstyle{|1\rangle}
  38. | 2 \scriptstyle{|2\rangle}
  39. | Ψ = c 1 ( t ) | 1 + c 2 ( t ) | 2 . |\Psi\rangle=c_{1}(t)|1\rangle+c_{2}(t)|2\rangle.
  40. ω 0 \scriptstyle{\hbar\omega_{0}}
  41. | 1 \scriptstyle{|1\rangle}
  42. | 2 \scriptstyle{|2\rangle}
  43. 𝐇 = ( μ B ( t ) - ω 0 / 2 a a * ω 0 / 2 - μ B ( t ) ) \mathbf{H}=\begin{pmatrix}\mu B(t)-\hbar\omega_{0}/2&a\\ a^{*}&\hbar\omega_{0}/2-\mu B(t)\end{pmatrix}
  44. μ \scriptstyle{\mu}
  45. a \scriptstyle{a}
  46. E 1 ( t ) \scriptstyle{E_{1}(t)}
  47. E 2 ( t ) \scriptstyle{E_{2}(t)}
  48. 𝐇 \scriptstyle{\mathbf{H}}
  49. 𝐇 \scriptstyle{\mathbf{H}}
  50. | ϕ 1 ( t ) \scriptstyle{|\phi_{1}(t)\rangle}
  51. | ϕ 2 ( t ) \scriptstyle{|\phi_{2}(t)\rangle}
  52. ε 1 ( t ) = - 1 2 4 a 2 + ( ω 0 - 2 μ B ( t ) ) 2 ε 2 ( t ) = + 1 2 4 a 2 + ( ω 0 - 2 μ B ( t ) ) 2 . \begin{aligned}\displaystyle\varepsilon_{1}(t)&\displaystyle=-\frac{1}{2}\sqrt% {4a^{2}+(\hbar\omega_{0}-2\mu B(t))^{2}}\\ \displaystyle\varepsilon_{2}(t)&\displaystyle=+\frac{1}{2}\sqrt{4a^{2}+(\hbar% \omega_{0}-2\mu B(t))^{2}}.\\ \end{aligned}
  53. ε 1 ( t ) \scriptstyle{\varepsilon_{1}(t)}
  54. ε 2 ( t ) \scriptstyle{\varepsilon_{2}(t)}
  55. E 1 ( t ) \scriptstyle{E_{1}(t)}
  56. E 2 ( t ) \scriptstyle{E_{2}(t)}
  57. | 1 \scriptstyle{|1\rangle}
  58. | 2 \scriptstyle{|2\rangle}
  59. | ϕ 1 ( t 0 ) \scriptstyle{|\phi_{1}(t_{0})\rangle}
  60. ( d B d t 0 ) \scriptstyle{\left(\frac{dB}{dt}\rightarrow 0\right)}
  61. | ϕ 1 ( t ) \scriptstyle{|\phi_{1}(t)\rangle}
  62. ( d B d t ) \scriptstyle{\left(\frac{dB}{dt}\rightarrow\infty\right)}
  63. | ϕ 2 ( t 1 ) \scriptstyle{|\phi_{2}(t_{1})\rangle}
  64. ( 0 < d B d t < ) \scriptstyle{\left(0<\frac{dB}{dt}<\infty\right)}
  65. i t Ψ ( x , t ) = H ^ Ψ ( x , t ) i\hbar{\partial\over\partial t}\Psi(x,t)=\hat{H}\Psi(x,t)
  66. H ^ ψ n ( x ) = E n ψ n ( x ) \hat{H}\psi_{n}(x)=E_{n}\psi_{n}(x)
  67. Ψ ( x , t ) = n c n Ψ n ( x , t ) = n c n ψ n ( x ) e - i E n t / \Psi(x,t)=\sum_{n}c_{n}\Psi_{n}(x,t)=\sum_{n}c_{n}\psi_{n}(x)e^{-iE_{n}t/\hbar}
  68. Ψ n ( x , t ) = ψ n ( x ) e - i E n t / \ \Psi_{n}(x,t)=\psi_{n}(x)e^{-iE_{n}t/\hbar}
  69. ( - E n t / ) \scriptstyle{(-E_{n}t/\hbar)}
  70. ψ n ( t ) | ψ m ( t ) = δ n m \langle\psi_{n}(t)|\psi_{m}(t)\rangle=\delta_{nm}
  71. Ψ ( t ) \scriptstyle{\Psi(t)}
  72. Ψ ( t ) = n c n ( t ) ψ n ( t ) e i θ n ( t ) \Psi(t)=\sum_{n}c_{n}(t)\psi_{n}(t)e^{i\theta_{n}(t)}
  73. θ n ( t ) = - 1 0 t E n ( t ) d t \scriptstyle{\theta_{n}(t)=-\frac{1}{\hbar}\int\limits_{0}^{t}E_{n}(t^{\prime}% )dt^{\prime}}
  74. θ n ( t ) \scriptstyle{\theta_{n}(t)}
  75. i n ( c n ˙ ψ n + c n ψ n ˙ + i c n ψ n θ n ˙ ) e i θ n = n c n H ^ ψ n e i θ n i\hbar\sum_{n}(\dot{c_{n}}\psi_{n}+c_{n}\dot{\psi_{n}}+ic_{n}\psi_{n}\dot{% \theta_{n}})e^{i\theta_{n}}=\sum_{n}c_{n}\hat{H}\psi_{n}e^{i\theta_{n}}
  76. θ n ˙ \scriptstyle{\dot{\theta_{n}}}
  77. - E / \scriptstyle{-E/\hbar}
  78. n c n ˙ ψ n e i θ n = - n c n ψ n ˙ e i θ n \sum_{n}\dot{c_{n}}\psi_{n}e^{i\theta_{n}}=-\sum_{n}c_{n}\dot{\psi_{n}}e^{i% \theta_{n}}
  79. ψ m | \scriptstyle{\langle\psi_{m}|}
  80. ψ m | ψ n \scriptstyle{\langle\psi_{m}|\psi_{n}\rangle}
  81. δ n m \delta_{nm}
  82. c ˙ m ( t ) = - n c n ψ m | ψ n ˙ e i ( θ n - θ m ) \dot{c}_{m}(t)=-\sum_{n}c_{n}\langle\psi_{m}|\dot{\psi_{n}}\rangle e^{i(\theta% _{n}-\theta_{m})}
  83. ψ m | ψ n ˙ \scriptstyle{\langle\psi_{m}|\dot{\psi_{n}}\rangle}
  84. c ˙ m ( t ) = - c m ψ m | ψ m ˙ - n m c n ψ m | H ^ ˙ | ψ n E n - E m e i ( θ n - θ m ) \dot{c}_{m}(t)=-c_{m}\langle\psi_{m}|\dot{\psi_{m}}\rangle-\sum_{n\neq m}c_{n}% \frac{\langle\psi_{m}|\dot{\hat{H}}|\psi_{n}\rangle}{E_{n}-E_{m}}e^{i(\theta_{% n}-\theta_{m})}
  85. H ^ ˙ \dot{\hat{H}}
  86. c ˙ m ( t ) = - c m ψ m | ψ m ˙ \dot{c}_{m}(t)=-c_{m}\langle\psi_{m}|\dot{\psi_{m}}\rangle
  87. c m ( t ) = c m ( 0 ) exp [ - 0 t ψ m ( t ) | ψ m ˙ ( t ) d t ] = c m ( 0 ) e i γ m ( t ) , c_{m}(t)=c_{m}(0)\exp[-\textstyle\int\limits_{0}^{t}\langle\psi_{m}(t^{\prime}% )|\dot{\psi_{m}}(t^{\prime})\rangle dt^{\prime}]=c_{m}(0)e^{i\gamma_{m}(t)},
  88. γ m ( t ) = i 0 t ψ m ( t ) | ψ m ˙ ( t ) d t \scriptstyle{\gamma_{m}(t)=i\int_{0}^{t}\langle\psi_{m}(t^{\prime})|\dot{\psi_% {m}}(t^{\prime})\rangle dt^{\prime}}
  89. ψ m ( t ) | ψ m ˙ ( t ) \scriptstyle{\langle\psi_{m}(t^{\prime})|\dot{\psi_{m}}(t^{\prime})\rangle}
  90. ψ m ( t ) | ψ m ( t ) = 1 \scriptstyle{\langle\psi_{m}(t^{\prime})|\psi_{m}(t^{\prime})\rangle=1}
  91. c m ( t ) \scriptstyle{c_{m}(t)}
  92. Ψ n ( t ) = ψ n ( t ) e i θ n ( t ) e i γ n ( t ) . \Psi_{n}(t)=\psi_{n}(t)e^{i\theta_{n}(t)}e^{i\gamma_{n}(t)}.
  93. γ n ( t ) \scriptstyle{\gamma_{n}(t)}
  94. γ n ( t ) \scriptstyle{\gamma_{n}(t)}
  95. t \scriptstyle{t}
  96. | ψ ( t ) = n c n A ( t ) e - i E n t / | ϕ n |\psi(t)\rangle=\sum_{n}c^{A}_{n}(t)e^{-iE_{n}t/\hbar}|\phi_{n}\rangle
  97. ψ ( x , t ) = x | ψ ( t ) \psi(x,t)=\langle x|\psi(t)\rangle
  98. τ \scriptstyle{\tau}
  99. H ^ 0 \scriptstyle{\hat{H}_{0}}
  100. t 0 \scriptstyle{t_{0}}
  101. H ^ 1 \scriptstyle{\hat{H}_{1}}
  102. t 1 \scriptstyle{t_{1}}
  103. τ = t 1 - t 0 \scriptstyle{\tau=t_{1}-t_{0}}
  104. U ^ ( t , t 0 ) = 1 - i t 0 t H ^ ( t ) U ^ ( t , t 0 ) d t \hat{U}(t,t_{0})=1-\frac{i}{\hbar}\int_{t_{0}}^{t}\hat{H}(t^{\prime})\hat{U}(t% ^{\prime},t_{0})dt^{\prime}
  105. i t U ^ ( t , t 0 ) = H ^ ( t ) U ^ ( t , t 0 ) i\hbar\frac{\partial}{\partial t}\hat{U}(t,t_{0})=\hat{H}(t)\hat{U}(t,t_{0})
  106. U ^ ( t 0 , t 0 ) = 1 \scriptstyle{\hat{U}(t_{0},t_{0})=1}
  107. t 0 \scriptstyle{t_{0}}
  108. t \scriptstyle{t}
  109. | ψ ( t ) = U ^ ( t , t 0 ) | ψ ( t 0 ) . |\psi(t)\rangle=\hat{U}(t,t_{0})|\psi(t_{0})\rangle.
  110. U ^ ( t 1 , t 0 ) \scriptstyle{\hat{U}(t_{1},t_{0})}
  111. τ \scriptstyle{\tau}
  112. | 0 | ψ ( t 0 ) \scriptstyle{|0\rangle\equiv|\psi(t_{0})\rangle}
  113. ζ = 0 | U ^ ( t 1 , t 0 ) U ^ ( t 1 , t 0 ) | 0 - 0 | U ^ ( t 1 , t 0 ) | 0 0 | U ^ ( t 1 , t 0 ) | 0 \zeta=\langle 0|\hat{U}^{\dagger}(t_{1},t_{0})\hat{U}(t_{1},t_{0})|0\rangle-% \langle 0|\hat{U}^{\dagger}(t_{1},t_{0})|0\rangle\langle 0|\hat{U}(t_{1},t_{0}% )|0\rangle
  114. U ^ ( t 1 , t 0 ) \scriptstyle{\hat{U}(t_{1},t_{0})}
  115. U ^ ( t 1 , t 0 ) = 1 + 1 i t 0 t 1 H ^ ( t ) d t + 1 ( i ) 2 t 0 t 1 d t t 0 t d t ′′ H ^ ( t ) H ^ ( t ′′ ) + \hat{U}(t_{1},t_{0})=1+{1\over i\hbar}\int_{t_{0}}^{t_{1}}\hat{H}(t)dt+{1\over% (i\hbar)^{2}}\int_{t_{0}}^{t_{1}}dt^{\prime}\int_{t_{0}}^{t^{\prime}}dt^{% \prime\prime}\hat{H}(t^{\prime})\hat{H}(t^{\prime\prime})+\ldots
  116. ζ \scriptstyle{\zeta}
  117. 1 τ t 0 t 1 H ^ ( t ) d t H ¯ {1\over\tau}\int_{t_{0}}^{t_{1}}\hat{H}(t)dt\equiv\bar{H}
  118. t 0 t 1 \scriptstyle{t_{0}\rightarrow t_{1}}
  119. ζ = 0 | ( 1 + i τ H ¯ ) ( 1 - i τ H ¯ ) | 0 - 0 | ( 1 + i τ H ¯ ) | 0 0 | ( 1 - i τ H ¯ ) | 0 \zeta=\langle 0|(1+\frac{i}{\hbar}\tau\bar{H})(1-{i\over\hbar}\tau\bar{H})|0% \rangle-\langle 0|(1+{i\over\hbar}\tau\bar{H})|0\rangle\langle 0|(1-{i\over% \hbar}\tau\bar{H})|0\rangle
  120. ζ = τ 2 2 ( 0 | H ¯ 2 | 0 - 0 | H ¯ | 0 0 | H ¯ | 0 ) \zeta=\frac{\tau^{2}}{\hbar^{2}}\left(\langle 0|\bar{H}^{2}|0\rangle-\langle 0% |\bar{H}|0\rangle\langle 0|\bar{H}|0\rangle\right)
  121. ζ = τ 2 Δ H ¯ 2 2 \zeta=\frac{\tau^{2}\Delta\bar{H}^{2}}{\hbar^{2}}
  122. Δ H ¯ \scriptstyle{\Delta\bar{H}}
  123. ζ 1 \scriptstyle{\zeta\ll 1}
  124. τ Δ H ¯ \tau\gg{\hbar\over\Delta\bar{H}}
  125. τ 0 \scriptstyle{\tau\rightarrow 0}
  126. lim τ 0 U ^ ( t 1 , t 0 ) = 1 \lim_{\tau\rightarrow 0}\hat{U}(t_{1},t_{0})=1
  127. | x | ψ ( t 1 ) | 2 = | x | ψ ( t 0 ) | 2 |\langle x|\psi(t_{1})\rangle|^{2}=|\langle x|\psi(t_{0})\rangle|^{2}\quad
  128. P D = 1 - ζ P_{D}=1-\zeta\quad
  129. τ \scriptstyle{\tau\rightarrow\infty}
  130. | x | ψ ( t 1 ) | 2 | x | ψ ( t 0 ) | 2 |\langle x|\psi(t_{1})\rangle|^{2}\neq|\langle x|\psi(t_{0})\rangle|^{2}
  131. H ^ ( t 0 ) \scriptstyle{\hat{H}(t_{0})}
  132. τ \scriptstyle{\tau}
  133. H ^ ( t 1 ) \scriptstyle{\hat{H}(t_{1})}
  134. P A = ζ P_{A}=\zeta\quad
  135. v L Z = t | E 2 - E 1 | q | E 2 - E 1 | d q d t v_{LZ}={\frac{\partial}{\partial t}|E_{2}-E_{1}|\over\frac{\partial}{\partial q% }|E_{2}-E_{1}|}\approx\frac{dq}{dt}
  136. q \scriptstyle{q}
  137. E 1 \scriptstyle{E_{1}}
  138. E 2 \scriptstyle{E_{2}}
  139. v L Z \scriptstyle{v_{LZ}}
  140. P D \scriptstyle{P_{D}}
  141. P D = e - 2 π Γ Γ = a 2 / | t ( E 2 - E 1 ) | = a 2 / | d q d t q ( E 2 - E 1 ) | = a 2 | α | \begin{aligned}\displaystyle P_{D}&\displaystyle=e^{-2\pi\Gamma}\\ \displaystyle\Gamma&\displaystyle={a^{2}/\hbar\over\left|\frac{\partial}{% \partial t}(E_{2}-E_{1})\right|}={a^{2}/\hbar\over\left|\frac{dq}{dt}\frac{% \partial}{\partial q}(E_{2}-E_{1})\right|}\\ &\displaystyle={a^{2}\over\hbar|\alpha|}\\ \end{aligned}
  142. i c ¯ ˙ A ( t ) = 𝐇 A ( t ) c ¯ A ( t ) i\hbar\dot{\underline{c}}^{A}(t)=\mathbf{H}_{A}(t)\underline{c}^{A}(t)
  143. c ¯ A ( t ) \scriptstyle{\underline{c}^{A}(t)}
  144. 𝐇 A ( t ) \scriptstyle{\mathbf{H}_{A}(t)}
  145. P D = | c 2 A ( t 1 ) | 2 P_{D}=|c^{A}_{2}(t_{1})|^{2}\quad
  146. | c 1 A ( t 0 ) | 2 = 1 \scriptstyle{|c^{A}_{1}(t_{0})|^{2}=1}

Admissible_rule.html

  1. { , , , } \{\to,\land,\lor,\bot\}
  2. { , , } \{\to,\bot,\Box\}
  3. σ f ( A 1 , , A n ) = f ( σ A 1 , , σ A n ) \sigma f(A_{1},\dots,A_{n})=f(\sigma A_{1},\dots,\sigma A_{n})
  4. \vdash
  5. \vdash
  6. A \varnothing\vdash A
  7. L \vdash_{L}
  8. L \vdash_{L}
  9. A 1 , , A n B or A 1 , , A n / B , \frac{A_{1},\dots,A_{n}}{B}\qquad\,\text{or}\qquad A_{1},\dots,A_{n}/B,
  10. σ A 1 , , σ A n / σ B \sigma A_{1},\dots,\sigma A_{n}/\sigma B
  11. \vdash
  12. Γ B \Gamma\vdash B
  13. Γ | B \Gamma\,|\!\!\!\sim B
  14. | |\!\!\!\sim
  15. = | {\vdash}={\,|\!\!\!\sim}
  16. A 1 , , A n / B A_{1},\dots,A_{n}/B
  17. A 1 A n / B A_{1}\land\dots\land A_{n}/B
  18. \top
  19. \bot
  20. ( 𝐾𝑃𝑅 ) ¬ p q r ( ¬ p q ) ( ¬ p r ) (\mathit{KPR})\qquad\frac{\neg p\to q\lor r}{(\neg p\to q)\lor(\neg p\to r)}
  21. ( ¬ p q r ) ( ¬ p q ) ( ¬ p r ) (\neg p\to q\lor r)\to(\neg p\to q)\lor(\neg p\to r)
  22. p p \frac{\Box p}{p}
  23. p ¬ p \frac{\Diamond p\land\Diamond\neg p}{\bot}
  24. ( 𝐿𝑅 ) p p p (\mathit{LR})\qquad\frac{\Box p\to p}{p}
  25. Π 1 0 \Pi^{0}_{1}
  26. i = 0 n ( j = 0 k ¬ i , j 0 p j j = 0 k ¬ i , j 1 p j ) p 0 , \frac{\bigvee_{i=0}^{n}\bigl(\bigwedge_{j=0}^{k}\neg_{i,j}^{0}p_{j}\land% \bigwedge_{j=0}^{k}\neg_{i,j}^{1}\Box p_{j}\bigr)}{p_{0}},
  27. ¬ i , j u \neg_{i,j}^{u}
  28. ¬ \neg
  29. i = 0 n φ i / p 0 \textstyle\bigvee_{i=0}^{n}\varphi_{i}/p_{0}
  30. φ i \varphi_{i}
  31. { ¬ i , j 0 p j , ¬ i , j 1 p j j k } \{\neg_{i,j}^{0}p_{j},\neg_{i,j}^{1}\Box p_{j}\mid j\leq k\}
  32. { φ i i n } \{\varphi_{i}\mid i\leq n\}
  33. M = W , R , M=\langle W,R,{\Vdash}\rangle
  34. φ i p j p j φ i , \varphi_{i}\Vdash p_{j}\iff p_{j}\in\varphi_{i},
  35. φ i R φ i j k ( p j φ i { p j , p j } φ i ) . \varphi_{i}\,R\,\varphi_{i^{\prime}}\iff\forall j\leq k\,(\Box p_{j}\in\varphi% _{i}\Rightarrow\{p_{j},\Box p_{j}\}\subseteq\varphi_{i^{\prime}}).
  36. i = 0 n φ i / p 0 \textstyle\bigvee_{i=0}^{n}\varphi_{i}/p_{0}
  37. W { φ i i n } W\subseteq\{\varphi_{i}\mid i\leq n\}
  38. φ i p 0 \varphi_{i}\nVdash p_{0}
  39. i n , i\leq n,
  40. φ i φ i \varphi_{i}\Vdash\varphi_{i}
  41. i n , i\leq n,
  42. α , β W \alpha,\beta\in W
  43. α p j \alpha\Vdash\Box p_{j}
  44. φ p j p j \varphi\Vdash p_{j}\land\Box p_{j}
  45. φ D \varphi\in D
  46. α p j \alpha\Vdash\Box p_{j}
  47. α p j \alpha\Vdash p_{j}
  48. φ p j p j \varphi\Vdash p_{j}\land\Box p_{j}
  49. φ D \varphi\in D
  50. A | I P C B A\,|\!\!\!\sim_{IPC}B
  51. T ( A ) | G r z T ( B ) . T(A)\,|\!\!\!\sim_{Grz}T(B).
  52. L σ p υ τ p \vdash_{L}\sigma p\leftrightarrow\upsilon\tau p
  53. A L B σ B A\vdash_{L}B\leftrightarrow\sigma B
  54. P L A P\vdash_{L}A
  55. P Π ( A ) , P\in\Pi(A),
  56. P | L B P\,|\!\!\!\sim_{L}B
  57. P L B P\vdash_{L}B
  58. A | L B A\,|\!\!\!\sim_{L}B
  59. P Π ( A ) ( P L B ) . \forall P\in\Pi(A)\,(P\vdash_{L}B).
  60. | L |\!\!\!\sim_{L}
  61. L \vdash_{L}
  62. L \vdash_{L}
  63. p ¬ p . \frac{\Diamond p\land\Diamond\neg p}{\bot}.
  64. ( i = 1 n ( p i q i ) p n + 1 p n + 2 ) r j = 1 n + 2 ( i = 1 n ( p i q i ) p j ) r , n 1 \frac{\displaystyle\Bigl(\bigwedge_{i=1}^{n}(p_{i}\to q_{i})\to p_{n+1}\lor p_% {n+2}\Bigr)\lor r}{\displaystyle\bigvee_{j=1}^{n+2}\Bigl(\bigwedge_{i=1}^{n}(p% _{i}\to q_{i})\to p_{j}\Bigr)\lor r},\qquad n\geq 1
  65. ( q i = 1 n p i ) r i = 1 n ( q q p i ) r , n 0 \frac{\displaystyle\Box\Bigl(\Box q\to\bigvee_{i=1}^{n}\Box p_{i}\Bigr)\lor% \Box r}{\displaystyle\bigvee_{i=1}^{n}\Box(q\land\Box q\to p_{i})\lor r},% \qquad n\geq 0
  66. \bot
  67. ( ( q q ) i = 1 n p i ) r i = 1 n ( q p i ) r , n 0 \frac{\displaystyle\Box\Bigl(\Box(q\to\Box q)\to\bigvee_{i=1}^{n}\Box p_{i}% \Bigr)\lor\Box r}{\displaystyle\bigvee_{i=1}^{n}\Box(\Box q\to p_{i})\lor r},% \qquad n\geq 0
  68. F = W , R F=\langle W,R\rangle
  69. \Vdash
  70. x W ( x A ) \forall x\in W\,(x\Vdash A)
  71. A Γ A\in\Gamma
  72. x W ( x B ) \forall x\in W\,(x\Vdash B)
  73. ( p q ) p r ( ( p q ) p ) ( ( p q ) r ) \frac{(p\to q)\to p\lor r}{((p\to q)\to p)\lor((p\to q)\to r)}
  74. A ( p 1 , , p n , s 1 , , s k ) B ( p 1 , , p n , s 1 , , s k ) , \frac{A(p_{1},\dots,p_{n},s_{1},\dots,s_{k})}{B(p_{1},\dots,p_{n},s_{1},\dots,% s_{k})},
  75. A 1 , , A n B 1 , , B m or A 1 , , A n / B 1 , , B m . \frac{A_{1},\dots,A_{n}}{B_{1},\dots,B_{m}}\qquad\,\text{or}\qquad A_{1},\dots% ,A_{n}/B_{1},\dots,B_{m}.
  76. , \frac{\;\bot\;}{},
  77. p q p , q . \frac{p\lor q}{p,q}.
  78. A 1 , , A n B 1 B m . \frac{A_{1},\dots,A_{n}}{\Box B_{1}\lor\dots\lor\Box B_{m}}.
  79. Γ A , Δ Π , A Λ Γ , Π Δ , Λ . \frac{\Gamma\vdash A,\Delta\qquad\Pi,A\vdash\Lambda}{\Gamma,\Pi\vdash\Delta,% \Lambda}.
  80. Γ Δ \Gamma\vdash\Delta
  81. Γ Δ \bigwedge\Gamma\to\bigvee\Delta

Adobe_RGB_color_space.html

  1. X = X a - X K X W - X K X W Y W X=\frac{X_{a}-X_{K}}{X_{W}-X_{K}}\frac{X_{W}}{Y_{W}}
  2. Y = Y a - Y K Y W - Y K Y=\frac{Y_{a}-Y_{K}}{Y_{W}-Y_{K}}
  3. Z = Z a - Z K Z W - Z K Z W Y W Z=\frac{Z_{a}-Z_{K}}{Z_{W}-Z_{K}}\frac{Z_{W}}{Y_{W}}
  4. [ R G B ] = [ 2.04159 - 0.56501 - 0.34473 - 0.96924 1.87597 0.04156 0.01344 - 0.11836 1.01517 ] [ X Y Z ] \begin{bmatrix}R\\ G\\ B\end{bmatrix}=\begin{bmatrix}2.04159&-0.56501&-0.34473\\ -0.96924&1.87597&0.04156\\ 0.01344&-0.11836&1.01517\end{bmatrix}\begin{bmatrix}X\\ Y\\ Z\end{bmatrix}
  5. [ X Y Z ] = [ 0.57667 0.18556 0.18823 0.29734 0.62736 0.07529 0.02703 0.07069 0.99134 ] [ R G B ] \begin{bmatrix}X\\ Y\\ Z\end{bmatrix}=\begin{bmatrix}0.57667&0.18556&0.18823\\ 0.29734&0.62736&0.07529\\ 0.02703&0.07069&0.99134\end{bmatrix}\begin{bmatrix}R\\ G\\ B\end{bmatrix}
  6. [ R G B ] = [ 1.96253 - 0.61068 - 0.34137 - 0.97876 1.91615 0.03342 0.02869 - 0.14067 1.34926 ] [ X Y Z ] \begin{bmatrix}R\\ G\\ B\end{bmatrix}=\begin{bmatrix}1.96253&-0.61068&-0.34137\\ -0.97876&1.91615&0.03342\\ 0.02869&-0.14067&1.34926\end{bmatrix}\begin{bmatrix}X\\ Y\\ Z\end{bmatrix}
  7. R = R 1 2.19921875 , R^{\prime}=R^{\frac{1}{2.19921875}},
  8. G = G 1 2.19921875 , G^{\prime}=G^{\frac{1}{2.19921875}},
  9. B = B 1 2.19921875 B^{\prime}=B^{\frac{1}{2.19921875}}
  10. [ X Y Z ] = [ 0.60974 0.20528 0.14919 0.31111 0.62567 0.06322 0.01947 0.06087 0.74457 ] [ R G B ] \begin{bmatrix}X\\ Y\\ Z\end{bmatrix}=\begin{bmatrix}0.60974&0.20528&0.14919\\ 0.31111&0.62567&0.06322\\ 0.01947&0.06087&0.74457\end{bmatrix}\begin{bmatrix}R\\ G\\ B\end{bmatrix}

AdS::CFT_correspondence.html

  1. A d S 5 × S 5 AdS_{5}\times S^{5}
  2. A d S 5 AdS_{5}
  3. S 5 S^{5}
  4. A d S 7 × S 4 AdS_{7}\times S^{4}
  5. A d S 4 × S 7 AdS_{4}\times S^{7}
  6. 10 - 11 10^{-11}
  7. η \eta
  8. s s
  9. η s 4 π k \frac{\eta}{s}\approx\frac{\hbar}{4\pi k}
  10. \hbar
  11. k k
  12. η / s \eta/s
  13. q ^ \widehat{q}
  14. q ^ \widehat{q}
  15. N N
  16. N N

Affine_logic.html

  1. A B A\rightarrow B
  2. A - B A{-\!\circ}B\otimes\top

Affine_plane_(incidence_geometry).html

  1. n n
  2. n n
  3. n + 1 n+1
  4. n n
  5. n n
  6. n n
  7. n n
  8. n + 1 n+1
  9. n n
  10. n + 1 n+1
  11. n n
  12. n 1 n−1
  13. l l
  14. Π Π
  15. l l
  16. l l
  17. Π Π
  18. V V
  19. 2 n 2n
  20. F F
  21. V V
  22. S S
  23. n n
  24. V V
  25. V V
  26. S S
  27. A A
  28. V V
  29. v + U v+U
  30. v v
  31. V V
  32. U U
  33. S S
  34. A A
  35. x x + w x→x+w
  36. w w
  37. k k
  38. k k
  39. n n
  40. n k nk
  41. n n
  42. n n
  43. k k
  44. k k
  45. ( n + 1 ) (n+1)
  46. n n
  47. n n
  48. k k
  49. n n
  50. k 2 k−2
  51. n n
  52. F F
  53. Σ Σ
  54. n n
  55. Σ Σ
  56. | Σ | = k |Σ|=k
  57. k k
  58. F F
  59. Σ Σ
  60. | F | |F|
  61. M M
  62. F F
  63. M M
  64. F F
  65. C C
  66. Hull ( C ) = C C , \operatorname{Hull}(C)=C\cap C^{\perp},
  67. C < s u p > C<sup>⊥

Affirming_a_disjunct.html

  1. p q p\vee q
  2. p p
  3. {}\vdash{}
  4. q q
  5. {}\vdash{}

Age_of_the_universe.html

  1. H 0 H_{0}
  2. t 0 = 1 H 0 F ( Ω r , Ω m , Ω Λ , ) t_{0}=\frac{1}{H_{0}}F(\Omega_{r},\Omega_{m},\Omega_{\Lambda},\dots)
  3. H 0 H_{0}
  4. H 0 H_{0}
  5. 1 / H 0 1/H_{0}
  6. t 0 t_{0}
  7. k m / M p c · s {km}/{Mpc·s}
  8. H 0 H_{0}

Agoh–Giuga_conjecture.html

  1. p B p - 1 - 1 ( mod p ) . pB_{p-1}\equiv-1\;\;(\mathop{{\rm mod}}p).
  2. 1 p - 1 + 2 p - 1 + + ( p - 1 ) p - 1 - 1 ( mod p ) 1^{p-1}+2^{p-1}+\cdots+(p-1)^{p-1}\equiv-1\;\;(\mathop{{\rm mod}}p)
  3. i = 1 p - 1 i p - 1 - 1 ( mod p ) . \sum_{i=1}^{p-1}i^{p-1}\equiv-1\;\;(\mathop{{\rm mod}}p).
  4. a p - 1 1 ( mod p ) a^{p-1}\equiv 1\;\;(\mathop{{\rm mod}}p)
  5. a = 1 , 2 , , p - 1 a=1,2,\dots,p-1
  6. p - 1 - 1 ( mod p ) . p-1\equiv-1\;\;(\mathop{{\rm mod}}p).
  7. ( p - 1 ) ! - 1 ( mod p ) , (p-1)!\equiv-1\;\;(\mathop{{\rm mod}}p),
  8. i = 1 p - 1 i - 1 ( mod p ) . \prod_{i=1}^{p-1}i\equiv-1\;\;(\mathop{{\rm mod}}p).
  9. i = 1 p - 1 i p - 1 ( - 1 ) p - 1 1 ( mod p ) , \prod_{i=1}^{p-1}i^{p-1}\equiv(-1)^{p-1}\equiv 1\;\;(\mathop{{\rm mod}}p),
  10. i = 1 p - 1 i p - 1 ( - 1 ) p - 1 1 ( mod p ) . \prod_{i=1}^{p-1}i^{p-1}\equiv(-1)^{p-1}\equiv 1\;\;(\mathop{{\rm mod}}p).
  11. i = 1 p - 1 i p - 1 - 1 ( mod p ) \sum_{i=1}^{p-1}i^{p-1}\equiv-1\;\;(\mathop{{\rm mod}}p)
  12. i = 1 p - 1 i p - 1 1 ( mod p ) . \prod_{i=1}^{p-1}i^{p-1}\equiv 1\;\;(\mathop{{\rm mod}}p).

Air_mass_(astronomy).html

  1. h h
  2. z z
  3. h = 90 - z . h=90^{\circ}-z\,.
  4. X X
  5. z z
  6. X = sec z . X=\sec\,z\,.
  7. X = sec z X=\sec\,z
  8. X = sec z t [ 1 - 0.0012 ( sec 2 z t - 1 ) ] , X=\sec\,z_{\mathrm{t}}\,\left[1-0.0012\,(\sec^{2}z_{\mathrm{t}}-1)\right]\,,
  9. z t z_{\mathrm{t}}
  10. sec z - 1 \sec\,z-1
  11. X = sec z - 0.0018167 ( sec z - 1 ) - 0.002875 ( sec z - 1 ) 2 - 0.0008083 ( sec z - 1 ) 3 , X=\sec\,z\,-\,0.0018167\,(\sec\,z\,-\,1)\,-\,0.002875\,(\sec\,z\,-\,1)^{2}\,-% \,0.0008083\,(\sec\,z\,-\,1)^{3}\,,
  12. X = ( cos z + 0.025 e - 11 cos z ) - 1 , X=\left(\cos\,z+0.025e^{-11\cos\,z}\right)^{-1}\,,
  13. γ \gamma
  14. X = 1 sin γ + 0.50572 ( γ + 6.07995 ) - 1.6364 ; X=\frac{1}{\sin\,\gamma+0.50572\,(\gamma+6.07995^{\circ})^{-1.6364}}\;;
  15. X = 1 cos z + 0.50572 ( 6.07995 + 90 - z ) - 1.6364 , X=\frac{1}{\cos\,z+0.50572\,(6.07995^{\circ}+90-z)^{-1.6364}}\,,
  16. z z
  17. X = 1.002432 cos 2 z t + 0.148386 cos z t + 0.0096467 cos 3 z t + 0.149864 cos 2 z t + 0.0102963 cos z t + 0.000303978 , X=\frac{1.002432\,\cos^{2}z_{\mathrm{t}}+0.148386\,\cos\,z_{\mathrm{t}}+0.0096% 467}{\cos^{3}z_{\mathrm{t}}+0.149864\,\cos^{2}z_{\mathrm{t}}+0.0102963\,\cos\,% z_{\mathrm{t}}+0.000303978}\,,
  18. z t z_{\mathrm{t}}
  19. X = 1 sin ( h + 244 / ( 165 + 47 h 1.1 ) ) , X=\frac{1}{\sin(h+{244}/(165+47h^{1.1}))}\,,
  20. h h
  21. ( 90 - z ) (90^{\circ}-z)
  22. s s
  23. z z
  24. y atm y_{\mathrm{atm}}
  25. s = R E 2 cos 2 z + 2 R E y atm + y atm 2 - R E cos z , s=\sqrt{R_{\mathrm{E}}^{2}\cos^{2}z+2R_{\mathrm{E}}y_{\mathrm{atm}}+y_{\mathrm% {atm}}^{2}}-R_{\mathrm{E}}\cos\,z\,,
  26. s = ( R E + y atm ) 2 - R E 2 sin 2 z - R E cos z , s=\sqrt{\left(R_{\mathrm{E}}+y_{\mathrm{atm}}\right)^{2}-R_{\mathrm{E}}^{2}% \sin^{2}z}-R_{\mathrm{E}}\cos\,z\,,
  27. R E R_{\mathrm{E}}
  28. y atm y_{\mathrm{atm}}
  29. X = s y atm = R E y atm cos 2 z + 2 y atm R E + ( y atm R E ) 2 - R E y atm cos z . X=\frac{s}{y_{\mathrm{atm}}}=\frac{R_{\mathrm{E}}}{y_{\mathrm{atm}}}\sqrt{\cos% ^{2}z+2\frac{y_{\mathrm{atm}}}{R_{\mathrm{E}}}+\left(\frac{y_{\mathrm{atm}}}{R% _{\mathrm{E}}}\right)^{2}}-\frac{R_{\mathrm{E}}}{y_{\mathrm{atm}}}\cos\,z\,.
  30. y atm = k T 0 m g , y_{\mathrm{atm}}=\frac{kT_{0}}{mg}\,,
  31. k k
  32. T 0 T_{0}
  33. m m
  34. g g
  35. T 0 T_{0}
  36. m m
  37. g g
  38. y atm y_{\mathrm{atm}}
  39. X horiz = 1 + 2 R E y atm 38.87 . X_{\mathrm{horiz}}=\sqrt{1+2\frac{R_{\mathrm{E}}}{y_{\mathrm{atm}}}}\approx 38% .87\,.
  40. R E y atm = X 2 - 1 2 ( 1 - X cos z ) ; \frac{R_{\mathrm{E}}}{y_{\mathrm{atm}}}=\frac{X^{2}-1}{2\left(1-X\cos z\right)% }\,;
  41. z z
  42. R E / y atm R_{\mathrm{E}}/y_{\mathrm{atm}}
  43. X horiz X_{\mathrm{horiz}}
  44. R E R_{\mathrm{E}}
  45. y atm y_{\mathrm{atm}}
  46. σ \sigma
  47. σ = ρ d s . \sigma=\int\rho\,\mathrm{d}s\,.
  48. σ = 0 y atm ρ ( R E + y ) d y R E 2 cos 2 z + 2 R E y + y 2 . \sigma=\int_{0}^{y_{\mathrm{atm}}}\frac{\rho\,\left(R_{\mathrm{E}}+y\right)% \mathrm{d}y}{\sqrt{R_{\mathrm{E}}^{2}\cos^{2}z+2R_{\mathrm{E}}y+y^{2}}}\,.
  49. X = σ σ zen . X=\frac{\sigma}{\sigma_{\mathrm{zen}}}\,.
  50. σ zen \sigma_{\mathrm{zen}}
  51. ρ = ρ 0 e - y / H , \rho=\rho_{0}e^{-y/H}\,,
  52. ρ 0 \rho_{0}
  53. H H
  54. X π R 2 H exp ( R cos 2 z 2 H ) erfc ( R cos 2 z 2 H ) . X\approx\sqrt{\frac{\pi R}{2H}}\exp{\left(\frac{R\cos^{2}z}{2H}\right)}\,% \mathrm{erfc}\left(\sqrt{\frac{R\cos^{2}z}{2H}}\right)\,.
  55. R = 7 / 6 R E , R=7/6\,R_{\mathrm{E}}\,,
  56. R E R_{\mathrm{E}}
  57. X horiz π R 2 H . X_{\mathrm{horiz}}\approx\sqrt{\frac{\pi R}{2H}}\,.
  58. X horiz 37.20 . X_{\mathrm{horiz}}\approx 37.20\,.
  59. T = T 0 - α y , T=T_{0}-\alpha y\,,
  60. T 0 T_{0}
  61. α \alpha
  62. ρ = ρ 0 ( 1 - α T 0 y ) 1 / ( κ - 1 ) , \rho=\rho_{0}\left(1-\frac{\alpha}{T}_{0}y\right)^{1/(\kappa-1)}\,,
  63. κ \kappa
  64. σ = r obs r atm ρ d r 1 - ( n obs n r obs r ) 2 sin 2 z , \sigma=\int_{r_{\mathrm{obs}}}^{r_{\mathrm{atm}}}\frac{\rho\,\mathrm{d}r}{% \sqrt{1-\left(\frac{n_{\mathrm{obs}}}{n}\frac{r_{\mathrm{obs}}}{r}\right)^{2}% \sin^{2}z}}\,,
  65. n obs n_{\mathrm{obs}}
  66. y obs y_{\mathrm{obs}}
  67. n n
  68. y y
  69. r obs = R E + y obs r_{\mathrm{obs}}=R_{\mathrm{E}}+y_{\mathrm{obs}}
  70. r = R E + y r=R_{\mathrm{E}}+y
  71. y y
  72. r atm = R E + y atm r_{\mathrm{atm}}=R_{\mathrm{E}}+y_{\mathrm{atm}}
  73. y atm y_{\mathrm{atm}}
  74. n - 1 n obs - 1 = ρ ρ obs . \frac{n-1}{n_{\mathrm{obs}}-1}=\frac{\rho}{\rho_{\mathrm{obs}}}\,.
  75. σ = r obs r atm ρ d r 1 - ( n obs 1 + ( n obs - 1 ) ρ / ρ obs ) 2 ( r obs r ) 2 sin 2 z . \sigma=\int_{r_{\mathrm{obs}}}^{r_{\mathrm{atm}}}\frac{\rho\,\mathrm{d}r}{% \sqrt{1-\left(\frac{n_{\mathrm{obs}}}{1+(n_{\mathrm{obs}}-1)\rho/\rho_{\mathrm% {obs}}}\right)^{2}\left(\frac{r_{\mathrm{obs}}}{r}\right)^{2}\sin^{2}z}}\,.
  76. n obs - 1 n_{\mathrm{obs}}-1
  77. ( n obs - 1 ) 2 (n_{\mathrm{obs}}-1)^{2}
  78. σ = r obs r atm ρ d r 1 - [ 1 + 2 ( n obs - 1 ) ( 1 - ρ ρ obs ) ] ( r obs r ) 2 sin 2 z . \sigma=\int_{r_{\mathrm{obs}}}^{r_{\mathrm{atm}}}\frac{\rho\,\mathrm{d}r}{% \sqrt{1-\left[1+2(n_{\mathrm{obs}}-1)(1-\frac{\rho}{\rho_{\mathrm{obs}}})% \right]\left(\frac{r_{\mathrm{obs}}}{r}\right)^{2}\sin^{2}z}}\,.
  79. y obs y_{\mathrm{obs}}
  80. y atm y_{\mathrm{atm}}
  81. z z
  82. s s
  83. R E R_{\mathrm{E}}
  84. ( R E + y a t m ) 2 = s 2 + ( R E + y o b s ) 2 - 2 ( R E + y o b s ) s cos ( 180 - z ) = s 2 + ( R E + y o b s ) 2 + 2 ( R E + y o b s ) s cos z \begin{aligned}\displaystyle\left(R_{E}+y_{atm}\right)^{2}&\displaystyle=s^{2}% +\left(R_{E}+y_{obs}\right)^{2}-2\left(R_{E}+y_{obs}\right)s\cos\left(180^{% \circ}-z\right)\\ &\displaystyle=s^{2}+\left(R_{E}+y_{obs}\right)^{2}+2\left(R_{E}+y_{obs}\right% )s\cos z\end{aligned}
  85. s 2 + 2 ( R E + y obs ) s cos z - 2 R E y atm - y atm 2 + 2 R E y obs + y obs 2 = 0 . {{s}^{2}}+2\left({{R}_{\,\text{E}}}+{{y}_{\,\text{obs}}}\right)s\cos z-2{{R}_{% \,\text{E}}}{{y}_{\,\text{atm}}}-y_{\,\text{atm}}^{2}+2{{R}_{\,\text{E}}}{{y}_% {\,\text{obs}}}+y_{\,\text{obs}}^{2}=0\,.
  86. s = ± ( R E + y obs ) 2 cos 2 z + 2 R E ( y atm - y obs ) + y atm 2 - y obs 2 - ( R E + y obs ) cos z . s=\pm\sqrt{{{\left({{R}_{\,\text{E}}}+{{y}_{\,\text{obs}}}\right)}^{2}}{{\cos}% ^{2}}z+2{{R}_{\,\text{E}}}\left({{y}_{\,\text{atm }}}-{{y}_{\,\text{obs}}}% \right)+y_{\,\text{atm}}^{2}-y_{\,\text{obs}}^{2}}-({{R}_{\,\text{E}}}+{{y}_{% \,\text{obs}}})\cos z\,.
  87. y atm y_{\mathrm{atm}}
  88. X = ( R E + y obs y atm ) 2 cos 2 z + 2 R E y atm 2 ( y atm - y obs ) - ( y obs y atm ) 2 + 1 - R E + y obs y atm cos z . X=\sqrt{{{\left(\frac{{{R}_{\,\text{E}}}+{{y}_{\,\text{obs}}}}{{{y}_{\,\text{% atm}}}}\right)}^{2}}{{\cos}^{2}}z+\frac{2{{R}_{\,\text{E}}}}{y_{\,\text{atm}}^% {2}}\left({{y}_{\,\text{atm}}}-{{y}_{\,\text{obs}}}\right)-{{\left(\frac{{{y}_% {\,\text{obs}}}}{{{y}_{\,\text{atm}}}}\right)}^{2}}+1}-\frac{{{R}_{\,\text{E}}% }+{{y}_{\,\text{obs}}}}{{{y}_{\,\text{atm}}}}\cos z\,.
  89. r ^ = R E / y atm \hat{r}=R_{\mathrm{E}}/y_{\mathrm{atm}}
  90. y ^ = y obs / y atm \hat{y}=y_{\mathrm{obs}}/y_{\mathrm{atm}}
  91. X = ( r ^ + y ^ ) 2 cos 2 z + 2 r ^ ( 1 - y ^ ) - y ^ 2 + 1 - ( r ^ + y ^ ) cos z . X=\sqrt{{{(\hat{r}+\hat{y})}^{2}}{{\cos}^{2}}z+2\hat{r}(1-\hat{y})-\hat{y}^{2}% +1}\;-\;(\hat{r}+\hat{y})\cos z\,.
  92. X = ( R E y atm ) 2 cos 2 z + 2 R E y atm + 1 - R E y atm cos z . X=\sqrt{{{\left(\frac{{{R}_{\,\text{E}}}}{{{y}_{\,\text{atm}}}}\right)}^{2}}{{% \cos}^{2}}z+\frac{2{{R}_{\,\text{E}}}}{{{y}_{\,\text{atm}}}}+1}-\frac{{{R}_{\,% \text{E}}}}{{{y}_{\,\text{atm}}}}\cos z\,.
  93. cos γ = R E + y obs - h R E + y obs , \cos\gamma=\frac{{{R}_{\,\text{E}}}+{{y}_{\,\text{obs}}}-h}{{{R}_{\,\text{E}}}% +{{y}_{\,\text{obs}}}}\,,
  94. h h
  95. γ \gamma
  96. z max z_{\mathrm{max}}
  97. γ = z max - 90 , \gamma={{z}_{\,\text{max}}}-90{}^{\circ}\,,
  98. cos γ = cos ( z max - 90 ) = sin z max . \cos\gamma=\cos\left({{z}_{\,\text{max}}}-90{}^{\circ}\right)=\sin{{z}_{\,% \text{max}}}\,.
  99. sin z max = R E + y obs - h R E + y obs . \sin{{z}_{\,\text{max}}}=\frac{{{R}_{\,\text{E}}}+{{y}_{\,\text{obs}}}-h}{{{R}% _{\,\text{E}}}+{{y}_{\,\text{obs}}}}\,.
  100. h h
  101. z max z_{\mathrm{max}}
  102. z max = 180 - sin - 1 R E + y obs - h R E + y obs . {{z}_{\,\text{max}}}=180{}^{\circ}-{{\sin}^{-1}}\frac{{{R}_{\,\text{E}}}+{{y}_% {\,\text{obs}}}-h}{{{R}_{\,\text{E}}}+{{y}_{\,\text{obs}}}}\,.
  103. y obs = h y_{\mathrm{obs}}=h
  104. z max = 180 - sin - 1 R E R E + h . {{z}_{\,\text{max}}}=180{}^{\circ}-{{\sin}^{-1}}\frac{{{R}_{\,\text{E}}}}{{{R}% _{\,\text{E}}}+h}\,.

Akaike_information_criterion.html

  1. AIC = 2 k - 2 ln ( L ) \mathrm{AIC}=2k-2\ln(L)
  2. AICc = AIC + 2 k ( k + 1 ) n - k - 1 \mathrm{AICc}=\mathrm{AIC}+\frac{2k(k+1)}{n-k-1}
  3. x 1 x 1 2 π σ 2 exp ( - ( ln x - μ ) 2 2 σ 2 ) x\mapsto\,\frac{1}{x}\frac{1}{\sqrt{2\pi\sigma^{2}}}\,\exp\left(-\frac{\left(% \ln x-\mu\right)^{2}}{2\sigma^{2}}\right)
  4. ln ( μ , σ 2 ) = - n 2 ln ( 2 π ) - n 2 ln σ 2 - 1 2 σ 2 i = 1 n ( x i - μ ) 2 \ln\mathcal{L}(\mu,\sigma^{2})=-\frac{n}{2}\ln(2\pi)-\frac{n}{2}\ln\sigma^{2}-% \frac{1}{2\sigma^{2}}\sum_{i=1}^{n}(x_{i}-\mu)^{2}
  5. L = i = 1 n ( 1 2 π σ i ^ 2 ) 1 / 2 exp ( - i = 1 n ( y i - f ( x i ; θ ^ ) ) 2 2 σ i ^ 2 ) L=\prod_{i=1}^{n}\left(\frac{1}{2\pi\hat{\sigma_{i}}^{2}}\right)^{1/2}\exp% \left(-\sum_{i=1}^{n}\frac{(y_{i}-f(x_{i};\hat{\theta}))^{2}}{2\hat{\sigma_{i}% }^{2}}\right)
  6. ln ( L ) = ln ( i = 1 n ( 1 2 π σ i ^ 2 ) 1 / 2 ) - 1 2 i = 1 n ( y i - f ( x i ; θ ^ ) ) 2 σ i ^ 2 \therefore\,\ln(L)=\ln\left(\prod_{i=1}^{n}\left(\frac{1}{2\pi\hat{\sigma_{i}}% ^{2}}\right)^{1/2}\right)-\frac{1}{2}\sum_{i=1}^{n}\frac{(y_{i}-f(x_{i};\hat{% \theta}))^{2}}{\hat{\sigma_{i}}^{2}}
  7. ln ( L ) = C - χ 2 / 2 \therefore\,\ln(L)=C-\chi^{2}/2\,
  8. RSS = i = 1 n ( y i - f ( x i ; θ ^ ) ) 2 \textstyle\mathrm{RSS}=\sum_{i=1}^{n}(y_{i}-f(x_{i};\hat{\theta}))^{2}

Al-Battani.html

  1. tan a = sin a cos a \tan a=\frac{\sin a}{\cos a}
  2. sec a = 1 + tan 2 a \sec a=\sqrt{1+\tan^{2}a}
  3. sin x = a 1 + a 2 \sin x=\frac{a}{\sqrt{1+a^{2}}}
  4. b sin ( A ) = a sin ( 90 - A ) b\sin(A)=a\sin(90^{\circ}-A)

Alexander_Macfarlane.html

  1. h α A = cosh A + sinh A α π / 2 . h\alpha^{A}=\cosh A+\sinh A\ \alpha^{\pi/2}.
  2. α π / 2 \alpha^{\pi/2}

Alexander_polynomial.html

  1. H 1 ( X ) H_{1}(X)
  2. H 1 ( X ) H_{1}(X)
  3. [ t , t - 1 ] \mathbb{Z}[t,t^{-1}]
  4. ± t n \pm t^{n}
  5. Δ K ( t ) \Delta_{K}(t)
  6. ± t n \pm t^{n}
  7. π 1 ( S 3 \ K ) \pi_{1}(S^{3}\backslash K)
  8. Δ K ( t ) \Delta_{K}(t)
  9. Δ K ( t - 1 ) = Δ K ( t ) \Delta_{K}(t^{-1})=\Delta_{K}(t)
  10. H 1 X ¯ Hom [ t , t - 1 ] ( H 1 X , G ) \overline{H_{1}X}\simeq\mathrm{Hom}_{\mathbb{Z}[t,t^{-1}]}(H_{1}X,G)
  11. G G
  12. [ t , t - 1 ] \mathbb{Z}[t,t^{-1}]
  13. [ t , t - 1 ] \mathbb{Z}[t,t^{-1}]
  14. [ t , t - 1 ] \mathbb{Z}[t,t^{-1}]
  15. H 1 X ¯ \overline{H_{1}X}
  16. [ t , t - 1 ] \mathbb{Z}[t,t^{-1}]
  17. H 1 X H_{1}X
  18. H 1 X H_{1}X
  19. t t
  20. t - 1 t^{-1}
  21. Δ K ( 1 ) = ± 1 \Delta_{K}(1)=\pm 1
  22. t t
  23. M M
  24. r a n k ( H 1 M ) = 1 rank(H_{1}M)=1
  25. Δ M ( t ) \Delta_{M}(t)
  26. Δ M ( 1 ) \Delta_{M}(1)
  27. H 1 M H_{1}M
  28. Δ K ( t ) = 1 \Delta_{K}(t)=1
  29. Δ K ( t ) = f ( t ) f ( t - 1 ) \Delta_{K}(t)=f(t)f(t^{-1})
  30. f ( t ) f(t)
  31. ± 1 \pm 1
  32. S C K S 1 S\to C_{K}\to S^{1}
  33. C K C_{K}
  34. g : S S g:S\to S
  35. Δ K ( t ) = D e t ( t I - g * ) \Delta_{K}(t)=Det(tI-g_{*})
  36. g * : H 1 S H 1 S g_{*}:H_{1}S\to H_{1}S
  37. K K
  38. K K^{\prime}
  39. f : S 1 × D 2 S 3 f:S^{1}\times D^{2}\to S^{3}
  40. K = f ( K ) K=f(K^{\prime})
  41. S 1 × D 2 S 3 S^{1}\times D^{2}\subset S^{3}
  42. Δ K ( t ) = Δ f ( S 1 × { 0 } ) ( t a ) Δ K ( t ) \Delta_{K}(t)=\Delta_{f(S^{1}\times\{0\})}(t^{a})\Delta_{K^{\prime}}(t)
  43. a a\in\mathbb{Z}
  44. K S 1 × D 2 K^{\prime}\subset S^{1}\times D^{2}
  45. H 1 ( S 1 × D 2 ) = H_{1}(S^{1}\times D^{2})=\mathbb{Z}
  46. Δ K 1 # K 2 ( t ) = Δ K 1 ( t ) Δ K 2 ( t ) \Delta_{K_{1}\#K_{2}}(t)=\Delta_{K_{1}}(t)\Delta_{K_{2}}(t)
  47. K K
  48. Δ K ( t ) = ± 1 \Delta_{K}(t)=\pm 1
  49. ( z ) \nabla(z)
  50. L + , L - , L 0 L_{+},L_{-},L_{0}
  51. ( O ) = 1 \nabla(O)=1
  52. ( L + ) - ( L - ) = z ( L 0 ) \nabla(L_{+})-\nabla(L_{-})=z\nabla(L_{0})
  53. Δ L ( t 2 ) = L ( t - t - 1 ) \Delta_{L}(t^{2})=\nabla_{L}(t-t^{-1})
  54. Δ L \Delta_{L}
  55. ± t n / 2 \pm t^{n/2}
  56. Δ ( L + ) - Δ ( L - ) = ( t 1 / 2 - t - 1 / 2 ) Δ ( L 0 ) \Delta(L_{+})-\Delta(L_{-})=(t^{1/2}-t^{-1/2})\Delta(L_{0})
  57. K K

Alexander–Spanier_cohomology.html

  1. d f ( x 0 , , x p ) = i ( - 1 ) i f ( x 0 , , x i - 1 , x i + 1 , , x p ) . df(x_{0},\ldots,x_{p})=\sum_{i}(-1)^{i}f(x_{0},\ldots,x_{i-1},x_{i+1},\ldots,x% _{p}).

Alfvén_wave.html

  1. ϵ \epsilon\,
  2. ϵ = 1 + 1 B 2 c 2 μ 0 ρ \epsilon=1+\frac{1}{B^{2}}c^{2}\mu_{0}\rho
  3. B B\,
  4. c c\,
  5. μ 0 \mu_{0}\,
  6. ρ = Σ n s m s \rho=\Sigma n_{s}m_{s}\,
  7. s s\,
  8. v = c ϵ = c 1 + 1 B 2 c 2 μ 0 ρ v=\frac{c}{\sqrt{\epsilon}}=\frac{c}{\sqrt{1+\frac{1}{B^{2}}c^{2}\mu_{0}\rho}}
  9. v = v A 1 + 1 c 2 v A 2 v=\frac{v_{A}}{\sqrt{1+\frac{1}{c^{2}}v_{A}^{2}}}
  10. v A = B μ 0 ρ v_{A}=\frac{B}{\sqrt{\mu_{0}\rho}}
  11. v A c v_{A}\ll c
  12. v v A v\approx v_{A}
  13. v A c v_{A}\gg c
  14. v c v\approx c
  15. v A = B μ 0 n i m i v_{A}=\frac{B}{\sqrt{\mu_{0}n_{i}m_{i}}}~{}~{}
  16. v A = B n i m i v_{A}=\frac{B}{\sqrt{n_{i}m_{i}}}~{}~{}
  17. v A ( 2.18 × 10 11 cm/s ) ( m i / m p ) - 1 / 2 ( n i / cm - 3 ) - 1 / 2 ( B / gauss ) v_{A}\approx(2.18\times 10^{11}\,\mbox{cm/s}~{})\,(m_{i}/m_{p})^{-1/2}\,(n_{i}% /{\rm cm}^{-3})^{-1/2}\,(B/{\rm gauss})
  18. n i n_{i}\,
  19. m i m_{i}\,
  20. τ A \tau_{A}
  21. τ A = a v A \tau_{A}=\frac{a}{v_{A}}
  22. a a
  23. a a
  24. v = c 1 + e + P 2 P m v=\frac{c}{\sqrt{1+\frac{e+P}{2P_{m}}}}
  25. e e\,
  26. P P\,
  27. P m = 1 2 μ 0 B 2 P_{m}=\frac{1}{2\mu_{0}}B^{2}\,
  28. P e ρ c 2 P\ll e\approx\rho c^{2}

Algebra_of_sets.html

  1. \cup
  2. \cap
  3. A B = B A A\cup B=B\cup A\,\!
  4. A B = B A A\cap B=B\cap A\,\!
  5. ( A B ) C = A ( B C ) (A\cup B)\cup C=A\cup(B\cup C)\,\!
  6. ( A B ) C = A ( B C ) (A\cap B)\cap C=A\cap(B\cap C)\,\!
  7. A ( B C ) = ( A B ) ( A C ) A\cup(B\cap C)=(A\cup B)\cap(A\cup C)\,\!
  8. A ( B C ) = ( A B ) ( A C ) A\cap(B\cup C)=(A\cap B)\cup(A\cap C)\,\!
  9. U U
  10. A = A A\cup\varnothing=A\,\!
  11. A U = A A\cap U=A\,\!
  12. A A C = U A\cup A^{C}=U\,\!
  13. A A C = A\cap A^{C}=\varnothing\,\!
  14. ( A C ) C = A (A^{C})^{C}=A
  15. A A = A A\cup A=A\,\!
  16. A A = A A\cap A=A\,\!
  17. A U = U A\cup U=U\,\!
  18. A = A\cap\varnothing=\varnothing\,\!
  19. A ( A B ) = A A\cup(A\cap B)=A\,\!
  20. A ( A B ) = A A\cap(A\cup B)=A\,\!
  21. A A A\cup A\,\!
  22. = ( A A ) U =(A\cup A)\cap U\,\!
  23. = ( A A ) ( A A C ) =(A\cup A)\cap(A\cup A^{C})\,\!
  24. = A ( A A C ) =A\cup(A\cap A^{C})\,\!
  25. = A =A\cup\varnothing\,\!
  26. = A =A\,\!
  27. A A A\cap A\,\!
  28. = ( A A ) =(A\cap A)\cup\varnothing
  29. = ( A A ) ( A A C ) =(A\cap A)\cup(A\cap A^{C})\,\!
  30. = A ( A A C ) =A\cap(A\cup A^{C})\,\!
  31. = A U =A\cap U\,\!
  32. = A =A\,\!
  33. A B = A ( A B ) A\cap B\,\!=A\smallsetminus(A\smallsetminus B)
  34. ( A B ) C = A C B C (A\cup B)^{C}=A^{C}\cap B^{C}\,\!
  35. ( A B ) C = A C B C (A\cap B)^{C}=A^{C}\cup B^{C}\,\!
  36. ( A C ) C = A {(A^{C})}^{C}=A\,\!
  37. C = U \varnothing^{C}=U
  38. U C = U^{C}=\varnothing
  39. A B = U A\cup B=U\,\!
  40. A B = A\cap B=\varnothing\,\!
  41. B = A C B=A^{C}\,\!
  42. A A A\subseteq A\,\!
  43. A B A\subseteq B\,\!
  44. B A B\subseteq A\,\!
  45. A = B A=B\,\!
  46. A B A\subseteq B\,\!
  47. B C B\subseteq C\,\!
  48. A C A\subseteq C\,\!
  49. A S \varnothing\subseteq A\subseteq S\,\!
  50. A A B A\subseteq A\cup B\,\!
  51. A C A\subseteq C\,\!
  52. B C B\subseteq C\,\!
  53. A B C A\cup B\subseteq C\,\!
  54. A B A A\cap B\subseteq A\,\!
  55. C A C\subseteq A\,\!
  56. C B C\subseteq B\,\!
  57. C A B C\subseteq A\cap B\,\!
  58. A B A\subseteq B\,\!
  59. A B A\subseteq B\,\!
  60. A B = A A\cap B=A\,\!
  61. A B = B A\cup B=B\,\!
  62. A B = A\smallsetminus B=\varnothing
  63. B C A C B^{C}\subseteq A^{C}
  64. C ( A B ) = ( C A ) ( C B ) C\setminus(A\cap B)=(C\setminus A)\cup(C\setminus B)\,\!
  65. C ( A B ) = ( C A ) ( C B ) C\setminus(A\cup B)=(C\setminus A)\cap(C\setminus B)\,\!
  66. C ( B A ) = ( A C ) ( C B ) C\setminus(B\setminus A)=(A\cap C)\cup(C\setminus B)\,\!
  67. ( B A ) C = ( B C ) A = B ( C A ) (B\setminus A)\cap C=(B\cap C)\setminus A=B\cap(C\setminus A)\,\!
  68. ( B A ) C = ( B C ) ( A C ) (B\setminus A)\cup C=(B\cup C)\setminus(A\setminus C)\,\!
  69. A A = A\setminus A=\varnothing\,\!
  70. A = \varnothing\setminus A=\varnothing\,\!
  71. A = A A\setminus\varnothing=A\,\!
  72. B A = A C B B\setminus A=A^{C}\cap B\,\!
  73. ( B A ) C = A B C (B\setminus A)^{C}=A\cup B^{C}\,\!
  74. U A = A C U\setminus A=A^{C}\,\!
  75. A U = A\setminus U=\varnothing\,\!

Algebra_representation.html

  1. = [ i ] / ( i 2 + 1 ) , \mathbb{C}=\mathbb{R}[i]/(i^{2}+1),
  2. End ( V ) \mathbb{C}\to\mathrm{End}(V)
  3. i i
  4. i i
  5. k k
  6. k k
  7. K [ T 1 , , T k ] , K[T_{1},\dots,T_{k}],
  8. K [ x 1 , , x k ] K[x_{1},\dots,x_{k}]
  9. x i T i . x_{i}\mapsto T_{i}.
  10. K [ T ] K[T]
  11. λ : A R \lambda\colon\ A\to R
  12. R [ T ] , R[T],
  13. R [ T ] R R[T]\to R
  14. A × M M A\times M\to M
  15. m M m\in M
  16. a m = λ ( a ) m am=\lambda(a)m
  17. a A , a\in A,
  18. λ \lambda
  19. 𝒜 \mathcal{A}
  20. v v
  21. T T
  22. U U
  23. T U v = U T v TUv=UTv
  24. 𝐅 [ T 1 , , T k ] \mathbf{F}[T_{1},\dots,T_{k}]
  25. k k
  26. λ = ( λ 1 , , λ k ) \lambda=(\lambda_{1},\dots,\lambda_{k})
  27. k k
  28. k k
  29. k k

Algebraic_code-excited_linear_prediction.html

  1. 1 B ( z ) = 1 1 - g p z - T \frac{1}{B(z)}=\frac{1}{1-g_{p}z^{-T}}
  2. 1 A ( z ) = 1 1 + i = 1 P a i z - i \frac{1}{A(z)}=\frac{1}{1+\sum_{i=1}^{P}a_{i}z^{-i}}

Algebraic_function.html

  1. f ( x ) = 1 / x , f ( x ) = x , f ( x ) = 1 + x 3 x 3 / 7 - 7 x 1 / 3 f(x)=1/x,f(x)=\sqrt{x},f(x)=\frac{\sqrt{1+x^{3}}}{x^{3/7}-\sqrt{7}x^{1/3}}
  2. f ( x ) 5 + f ( x ) 4 + x = 0 f(x)^{5}+f(x)^{4}+x=0
  3. y = f ( x ) y=f(x)
  4. a n ( x ) y n + a n - 1 ( x ) y n - 1 + + a 0 ( x ) = 0 a_{n}(x)y^{n}+a_{n-1}(x)y^{n-1}+\cdots+a_{0}(x)=0
  5. S = S=\mathbb{Q}
  6. \mathbb{Q}
  7. exp ( x ) , tan ( x ) , ln ( x ) , Γ ( x ) \exp(x),\tan(x),\ln(x),\Gamma(x)
  8. f ( x ) = cos ( arcsin ( x ) ) = 1 - x 2 f(x)=\cos(\arcsin(x))=\sqrt{1-x^{2}}
  9. y 2 + x 2 = 1. y^{2}+x^{2}=1.\,
  10. y = ± 1 - x 2 . y=\pm\sqrt{1-x^{2}}.\,
  11. p ( y , x 1 , x 2 , , x m ) = 0. p(y,x_{1},x_{2},\dots,x_{m})=0.\,
  12. y = p ( x ) y=p(x)
  13. y - p ( x ) = 0. y-p(x)=0.\,
  14. y = p ( x ) q ( x ) y=\frac{p(x)}{q(x)}
  15. q ( x ) y - p ( x ) = 0. q(x)y-p(x)=0.
  16. y = p ( x ) n y=\sqrt[n]{p(x)}
  17. y n - p ( x ) = 0. y^{n}-p(x)=0.
  18. a n ( x ) y n + + a 0 ( x ) = 0 , a_{n}(x)y^{n}+\cdots+a_{0}(x)=0,
  19. b m ( y ) x m + b m - 1 ( y ) x m - 1 + + b 0 ( y ) = 0. b_{m}(y)x^{m}+b_{m-1}(y)x^{m-1}+\cdots+b_{0}(y)=0.
  20. x = ± y x=\pm\sqrt{y}
  21. y = - 2 x - 108 + 12 81 - 12 x 3 3 + - 108 + 12 81 - 12 x 3 3 6 . y=-\frac{2x}{\sqrt[3]{-108+12\sqrt{81-12x^{3}}}}+\frac{\sqrt[3]{-108+12\sqrt{8% 1-12x^{3}}}}{6}.
  22. x 3 4 3 , x\leq\frac{3}{\sqrt[3]{4}},
  23. x > 3 4 3 , x>\frac{3}{\sqrt[3]{4}},
  24. 1 2 π i Δ i p y ( x 0 , y ) p ( x 0 , y ) d y = 1. \frac{1}{2\pi i}\oint_{\partial\Delta_{i}}\frac{p_{y}(x_{0},y)}{p(x_{0},y)}\,% dy=1.
  25. f i ( x ) = 1 2 π i Δ i y p y ( x , y ) p ( x , y ) d y f_{i}(x)=\frac{1}{2\pi i}\oint_{\partial\Delta_{i}}y\frac{p_{y}(x,y)}{p(x,y)}% \,dy
  26. p ( x , y ) = a n ( x ) ( y - f 1 ( x ) ) ( y - f 2 ( x ) ) ( y - f n ( x ) ) p(x,y)=a_{n}(x)(y-f_{1}(x))(y-f_{2}(x))\cdots(y-f_{n}(x))

Algebraic_normal_form.html

  1. a b ( a b ) ( a b c ) a\veebar b\veebar\left(a\wedge b\right)\veebar\left(a\wedge b\wedge c\right)
  2. f ( x 1 , x 2 , , x n ) = f(x_{1},x_{2},\ldots,x_{n})=\!
  3. a 0 a_{0}\oplus\!
  4. a 1 x 1 a 2 x 2 a n x n a_{1}x_{1}\oplus a_{2}x_{2}\oplus\cdots\oplus a_{n}x_{n}\oplus\!
  5. a 1 , 2 x 1 x 2 a n - 1 , n x n - 1 x n a_{1,2}x_{1}x_{2}\oplus\cdots\oplus a_{n-1,n}x_{n-1}x_{n}\oplus\!
  6. \cdots\oplus\!
  7. a 1 , 2 , , n x 1 x 2 x n a_{1,2,\ldots,n}x_{1}x_{2}\ldots x_{n}\!
  8. a 0 , a 1 , , a 1 , 2 , , n { 0 , 1 } * a_{0},a_{1},\ldots,a_{1,2,\ldots,n}\in\{0,1\}^{*}
  9. f f
  10. f ( x ) = 0 f(x)=0
  11. f ( x ) = 1 f(x)=1
  12. f ( x ) = x f(x)=x
  13. f ( x ) = 1 x f(x)=1\oplus x
  14. f ( x 1 , x 2 , , x n ) = g ( x 2 , , x n ) x 1 h ( x 2 , , x n ) f(x_{1},x_{2},\ldots,x_{n})=g(x_{2},\ldots,x_{n})\oplus x_{1}h(x_{2},\ldots,x_% {n})
  15. g ( x 2 , , x n ) = f ( 0 , x 2 , , x n ) g(x_{2},\ldots,x_{n})=f(0,x_{2},\ldots,x_{n})
  16. h ( x 2 , , x n ) = f ( 0 , x 2 , , x n ) f ( 1 , x 2 , , x n ) h(x_{2},\ldots,x_{n})=f(0,x_{2},\ldots,x_{n})\oplus f(1,x_{2},\ldots,x_{n})
  17. x 1 = 0 x_{1}=0
  18. x 1 h = 0 x_{1}h=0
  19. f ( 0 , ) = f ( 0 , ) f(0,\ldots)=f(0,\ldots)
  20. x 1 = 1 x_{1}=1
  21. x 1 h = h x_{1}h=h
  22. f ( 1 , ) = f ( 0 , ) f ( 0 , ) f ( 1 , ) f(1,\ldots)=f(0,\ldots)\oplus f(0,\ldots)\oplus f(1,\ldots)
  23. g g
  24. h h
  25. f f
  26. f ( x , y ) = x y f(x,y)=x\lor y
  27. f ( x , y ) = f ( 0 , y ) x ( f ( 0 , y ) f ( 1 , y ) ) f(x,y)=f(0,y)\oplus x(f(0,y)\oplus f(1,y))
  28. f ( 0 , y ) = 0 y = y f(0,y)=0\lor y=y
  29. f ( 1 , y ) = 1 y = 1 f(1,y)=1\lor y=1
  30. f ( x , y ) = y x ( y 1 ) f(x,y)=y\oplus x(y\oplus 1)
  31. f ( x , y ) = y x y x = x y x y f(x,y)=y\oplus xy\oplus x=x\oplus y\oplus xy

Algebraic_surface.html

  1. p a p_{a}
  2. p g p_{g}
  3. 𝒟 ( S ) \mathcal{D}(S)
  4. 𝒟 ( S ) × 𝒟 ( S ) : ( X , Y ) X Y \mathcal{D}(S)\times\mathcal{D}(S)\rightarrow\mathbb{Z}:(X,Y)\mapsto X\cdot Y
  5. 𝒟 0 ( S ) := { D 𝒟 ( S ) | D X = 0 , for all X 𝒟 ( S ) } \mathcal{D}_{0}(S):=\{D\in\mathcal{D}(S)|D\cdot X=0,\,\text{for all }X\in% \mathcal{D}(S)\}
  6. 𝒟 / 𝒟 0 ( S ) := N u m ( S ) \mathcal{D}/\mathcal{D}_{0}(S):=Num(S)
  7. N u m ( S ) × N u m ( S ) = ( D ¯ , E ¯ ) D E Num(S)\times Num(S)\mapsto\mathbb{Z}=(\bar{D},\bar{E})\mapsto D\cdot E
  8. N u m ( S ) Num(S)
  9. D ¯ \bar{D}
  10. D ¯ \bar{D}
  11. { H } := { D N u m ( S ) | D H = 0 } . \{H\}^{\perp}:=\{D\in Num(S)|D\cdot H=0\}.
  12. D { { H } | D 0 } , D D < 0 D\in\{\{H\}^{\perp}|D\neq 0\},D\cdot D<0
  13. { H } \{H\}^{\perp}
  14. { H } \{H\}^{\perp}

Alkaline_fuel_cell.html

  1. 2 H 2 + 4 O H - 4 H 2 O + 4 e - \mathrm{2H}_{2}+\mathrm{4OH}^{-}\longrightarrow\mathrm{4H}_{2}\mathrm{O}+% \mathrm{4e}^{-}
  2. O 2 + 2 H 2 O + 4 e - 4 O H - \mathrm{O}_{2}+\mathrm{2H}_{2}\mathrm{O}+\mathrm{4e}^{-}\longrightarrow\mathrm% {4OH}^{-}

All_one_polynomial.html

  1. A O P m ( x ) = i = 0 m x i AOP_{m}(x)=\sum_{i=0}^{m}x^{i}
  2. A O P m ( x ) = x m + x m - 1 + + x + 1 AOP_{m}(x)=x^{m}+x^{m-1}+\cdots+x+1
  3. A O P m ( x ) = x m + 1 - 1 x - 1 AOP_{m}(x)={x^{m+1}-1\over{x-1}}
  4. \mathbb{Q}

Allen_Hatcher.html

  1. 2 \scriptstyle 2
  2. Diff ( S 3 ) O ( 4 ) \scriptstyle{\mathrm{Diff}}(S^{3})\simeq{\mathrm{O}}(4)

Almost_flat_manifold.html

  1. ε > 0 \varepsilon>0
  2. g ε g_{\varepsilon}
  3. diam ( M , g ε ) 1 \mbox{diam}~{}(M,g_{\varepsilon})\leq 1
  4. g ε g_{\varepsilon}
  5. ε \varepsilon
  6. K g ε K_{g_{\varepsilon}}
  7. | K g ϵ | < ε |K_{g_{\epsilon}}|<\varepsilon
  8. ε n > 0 \varepsilon_{n}>0
  9. ε n \varepsilon_{n}
  10. 1 \leq 1

Alternating_finite_automaton.html

  1. ( q , a , q 1 q 2 ) (q,a,q_{1}\vee q_{2})
  2. q 1 q_{1}
  3. q 2 q_{2}
  4. ( q , a , q 1 q 2 ) (q,a,q_{1}\wedge q_{2})
  5. q 1 q_{1}
  6. q 2 q_{2}
  7. 2 k 2^{k}
  8. { { q 1 } , { q 2 , q 3 } } \{\{q_{1}\},\{q_{2},q_{3}\}\}
  9. q 1 ( q 2 q 3 ) q_{1}\vee(q_{2}\wedge q_{3})
  10. { { } } \{\{\}\}
  11. \varnothing
  12. ( S ( ) , S ( ) , Σ , δ , P 0 , F ) (S(\exists),S(\forall),\Sigma,\delta,P_{0},F)
  13. S ( ) S(\exists)
  14. S ( ) S(\vee)
  15. S ( ) S(\forall)
  16. S ( ) S(\wedge)
  17. Σ \ \Sigma
  18. δ \ \delta
  19. ( S ( ) S ( ) ) × ( Σ { ε } ) 2 S ( ) S ( ) (S(\exists)\cup S(\forall))\times(\Sigma\cup\{\varepsilon\})\to 2^{S(\exists)% \cup S(\forall)}
  20. P 0 \ P_{0}
  21. P 0 S ( ) S ( ) P_{0}\in S(\exists)\cup S(\forall)
  22. F \ F
  23. F S ( ) S ( ) F\subseteq S(\exists)\cup S(\forall)

Alternating_Turing_machine.html

  1. M = ( Q , Γ , δ , q 0 , g ) M=(Q,\Gamma,\delta,q_{0},g)
  2. Q Q
  3. Γ \Gamma
  4. δ : Q × Γ 𝒫 ( Q × Γ × { L , R } ) \delta:Q\times\Gamma\rightarrow\mathcal{P}(Q\times\Gamma\times\{L,R\})
  5. q 0 Q q_{0}\in Q
  6. g : Q { , , a c c e p t , r e j e c t } g:Q\rightarrow\{\wedge,\vee,accept,reject\}
  7. q Q q\in Q
  8. g ( q ) = a c c e p t g(q)=accept
  9. g ( q ) = r e j e c t g(q)=reject
  10. g ( q ) = g(q)=\wedge
  11. g ( q ) = g(q)=\vee
  12. q 0 q_{0}
  13. t ( n ) t(n)
  14. n n
  15. t ( n ) t(n)
  16. s ( n ) s(n)
  17. s ( n ) s(n)
  18. c t ( n ) c\cdot t(n)
  19. c > 0 c>0
  20. ATIME ( t ( n ) ) {\rm ATIME}(t(n))
  21. c s ( n ) c\cdot s(n)
  22. ASPACE ( s ( n ) ) {\rm ASPACE}(s(n))
  23. n 2 n^{2}
  24. n n
  25. AP = k > 0 ATIME ( n k ) {\rm AP}=\bigcup_{k>0}{\rm ATIME}(n^{k})
  26. APSPACE = k > 0 ASPACE ( n k ) {\rm APSPACE}=\bigcup_{k>0}{\rm ASPACE}(n^{k})
  27. AEXPTIME = k > 0 ATIME ( k n ) {\rm AEXPTIME}=\bigcup_{k>0}{\rm ATIME}(k^{n})
  28. ASPACE ( f ( n ) ) = c > 0 DTIME ( 2 c f ( n ) ) = DTIME ( 2 O ( f ( n ) ) ) \mathrm{ASPACE}(f(n))=\bigcup_{c>0}{\rm DTIME}(2^{cf(n)})={\rm DTIME}(2^{O(f(n% ))})
  29. ATIME ( g ( n ) ) DSPACE ( g ( n ) ) \mathrm{ATIME}(g(n))\subseteq{\rm DSPACE}(g(n))
  30. NSPACE ( g ( n ) ) c > 0 ATIME ( c × g ( n ) 2 ) , \mathrm{NSPACE}(g(n))\subseteq\bigcup_{c>0}{\rm ATIME}(c\times g(n)^{2}),
  31. f ( n ) log ( n ) f(n)\geq\log(n)
  32. g ( n ) log ( n ) g(n)\geq\log(n)
  33. ATIME ( C , j ) = Σ j TIME ( C ) {\rm ATIME}(C,j)=\Sigma_{j}{\rm TIME}(C)
  34. f C f\in C
  35. j - 1 j-1
  36. j j
  37. TIME ( C ) {\rm TIME}(C)
  38. coATIME ( C , j ) = Π j TIME ( C ) {\rm coATIME}(C,j)=\Pi_{j}{\rm TIME}(C)
  39. ATIME ( f , j ) {\rm ATIME}(f,j)
  40. ASPACE ( C , j ) = Σ SPACE ( C ) {\rm ASPACE}(C,j)=\Sigma_{\rm SPACE}(C)
  41. j j
  42. k j k\geq j
  43. j j
  44. SPACE ( f ) {\rm SPACE}(f)
  45. f = Ω ( log ) f=\Omega(\log)
  46. Σ k p \Sigma_{k}^{p}
  47. Π k p \Pi_{k}^{p}
  48. Σ k P \Sigma_{k}\rm{P}
  49. Π k P \Pi_{k}\rm{P}

Amber_(color).html

  1. y 0.429 y\leq 0.429
  2. y 0.398 y\geq 0.398
  3. z 0.007 z\leq 0.007
  4. y x - 0.120 y\leq x-0.120
  5. y 0.390 y\geq 0.390
  6. y 0.790 - 0.670 x y\geq 0.790-0.670x

Ambiguity_function.html

  1. χ ( τ , f ) \chi(\tau,f)
  2. s ( t ) s(t)
  3. χ ( τ , f ) = - s ( t ) s * ( t - τ ) e i 2 π f t d t \chi(\tau,f)=\int_{-\infty}^{\infty}s(t)s^{*}(t-\tau)e^{i2\pi ft}\,dt
  4. * {}^{*}
  5. i i
  6. f = 0 f=0
  7. s ( t ) s(t)
  8. χ ( 0 , f ) \chi(0,f)
  9. χ ( τ , 0 ) \chi(\tau,0)
  10. χ ( τ , f D ) \chi(\tau,f_{D})
  11. s L 2 ( R ) s\in L^{2}(R)
  12. W B s s ( τ , α ) = | α | - s ( t ) s * ( α ( t - τ ) ) d t WB_{ss}(\tau,\alpha)=\sqrt{|{\alpha}|}\int_{-\infty}^{\infty}s(t)s^{*}(\alpha(% t-\tau))\,dt
  13. α {\alpha}
  14. α = c - v c + v \alpha=\frac{c-v}{c+v}
  15. α \alpha
  16. α - 1 \alpha^{-1}
  17. χ ( τ , f ) = δ ( τ ) δ ( f ) \chi(\tau,f)=\delta(\tau)\delta(f)\,
  18. s ( t ) s(t)
  19. δ ( τ ) δ ( f ) \delta(\tau)\delta(f)
  20. | χ ( τ , f ) | 2 | χ ( 0 , 0 ) | 2 |\chi(\tau,f)|^{2}\leq|\chi(0,0)|^{2}
  21. χ ( τ , f ) = exp [ j 2 π τ f ] χ * ( - τ , - f ) \chi(\tau,f)=\exp[j2\pi\tau f]\chi^{*}(-\tau,-f)\,
  22. - - | χ ( τ , f ) | 2 d τ d f = | χ ( 0 , 0 ) | 2 = E 2 \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}|\chi(\tau,f)|^{2}\,d\tau\,df=|% \chi(0,0)|^{2}=E^{2}
  23. If s ( t ) | χ ( τ , f ) | then s ( t ) exp [ j 2 π k t ] | χ ( τ , f + k t ) | \,\text{If }s(t)\rightarrow|\chi(\tau,f)|\,\text{ then }s(t)\exp[j2\pi kt]{% \rightarrow}|\chi(\tau,f+kt)|\,
  24. S ( f ) S * ( f ) = - χ ( τ , 0 ) e - j 2 π τ f d τ S(f)S^{*}(f)=\int_{-\infty}^{\infty}\chi(\tau,0)e^{-j2\pi\tau f}\,d\tau
  25. τ \tau
  26. A A
  27. A ( u ( t ) - u ( t - τ ) ) A(u(t)-u(t-\tau))\,
  28. u ( t ) u(t)
  29. τ A 2 \tau A^{2}
  30. 2 τ 2\tau

Ambiguous_grammar.html

  1. { a n b m c m d n | n , m > 0 } \{a^{n}b^{m}c^{m}d^{n}|n,m>0\}
  2. { a n b n c m d m | n , m > 0 } \{a^{n}b^{n}c^{m}d^{m}|n,m>0\}
  3. { a n b n c n d n | n > 0 } \{a^{n}b^{n}c^{n}d^{n}|n>0\}

Amenable_group.html

  1. G f d μ \int_{G}f\,d\mu
  2. ν ( A ) = g G μ ( A g - 1 ) d μ - . \nu(A)=\int_{g\in G}\mu\left(Ag^{-1}\right)\,d\mu^{-}.

American_Invitational_Mathematics_Examination.html

  1. ( ( 3 ! ) ! ) ! 3 ! = k n ! , \frac{((3!)!)!}{3!}=k\cdot n!,
  2. k k
  3. n n
  4. n n
  5. k + n . k+n.
  6. k k
  7. 36 36
  8. 300 300
  9. 596 596
  10. k k
  11. a a
  12. b b
  13. c c
  14. P ( z ) = z 3 + q z + r P(z)=z^{3}+qz+r
  15. | a | 2 + | b | 2 + | c | 2 = 250 |a|^{2}+|b|^{2}+|c|^{2}=250
  16. a a
  17. b b
  18. c c
  19. h h
  20. h 2 h^{2}

Ample_line_bundle.html

  1. M M
  2. f : X Y f\colon X\to Y
  3. \mathcal{F}
  4. 𝒪 Y \mathcal{O}_{Y}
  5. f : X N , f\colon X\to\mathbb{P}^{N},
  6. = 𝒪 ( 1 ) Pic ( N ) \mathcal{F}=\mathcal{O}(1)\in\mathrm{Pic}(\mathbb{P}^{N})
  7. a i F ( X ) a_{i}\in F(X)
  8. f : X n , x [ a 0 ( x ) : : a n ( x ) ] , f\colon X\rightarrow\mathbb{P}^{n},\ x\mapsto[a_{0}(x):\cdots:a_{n}(x)],
  9. A = Γ ( F , X ) A=\Gamma(F,X)
  10. ρ = ρ X , U \rho=\rho_{X,U}
  11. i * ( 𝒪 ( 1 ) ) L . i^{*}(\mathcal{O}(1))\cong L.
  12. H i ( X , F ) H^{i}(X,F)
  13. \mathcal{L}
  14. n 0 n\gg 0
  15. j : X N j:X\to\mathbb{P}^{N}
  16. n = j * ( 𝒪 ( 1 ) ) \mathcal{L}^{\otimes n}=j^{*}(\mathcal{O}(1))
  17. n \mathcal{L}^{\otimes n}
  18. D D
  19. D D
  20. n D nD
  21. D D
  22. L L
  23. M M
  24. \mathcal{L}
  25. X X
  26. \mathcal{L}
  27. m \mathcal{L}^{\otimes m}
  28. \mathcal{F}
  29. m \mathcal{F}\otimes\mathcal{L}^{\otimes m}
  30. X X
  31. \mathcal{F}
  32. H i ( X , m ) , i 1 H^{i}(X,\mathcal{F}\otimes\mathcal{L}^{\otimes m}),\ i\geq 1
  33. F F
  34. 𝒪 ( 1 ) \mathcal{O}(1)
  35. ( F ) \mathbb{P}(F)
  36. \mathcal{L}
  37. \mathcal{L}
  38. k = 0 Γ ( X , k ) \bigoplus_{k=0}^{\infty}\Gamma(X,\mathcal{L}^{\otimes k})
  39. X Γ ( X , k ) X\to\mathbb{P}\Gamma(X,\mathcal{L}^{\otimes k})
  40. k 0 k\gg 0

Amplitude-shift_keying.html

  1. v i = 2 A L - 1 i - A ; i = 0 , 1 , , L - 1 v_{i}=\frac{2A}{L-1}i-A;\quad i=0,1,\dots,L-1
  2. Δ = 2 A L - 1 \Delta=\frac{2A}{L-1}
  3. s ( t ) = n = - v [ n ] h t ( t - n T s ) s(t)=\sum_{n=-\infty}^{\infty}v[n]\cdot h_{t}(t-nT_{s})
  4. z ( t ) = n r ( t ) + n = - v [ n ] g ( t - n T s ) z(t)=n_{r}(t)+\sum_{n=-\infty}^{\infty}v[n]\cdot g(t-nT_{s})
  5. n r ( t ) = n ( t ) * h r ( f ) n_{r}(t)=n(t)*h_{r}(f)
  6. g ( t ) = h t ( t ) * h c ( f ) * h r ( t ) g(t)=h_{t}(t)*h_{c}(f)*h_{r}(t)
  7. z [ k ] = n r [ k ] + v [ k ] g [ 0 ] + n k v [ n ] g [ k - n ] z[k]=n_{r}[k]+v[k]g[0]+\sum_{n\neq k}v[n]g[k-n]
  8. z [ k ] = n r [ k ] + v [ k ] g [ 0 ] z[k]=n_{r}[k]+v[k]g[0]
  9. σ N 2 = - + Φ N ( f ) | H r ( f ) | 2 d f \sigma_{N}^{2}=\int_{-\infty}^{+\infty}\Phi_{N}(f)\cdot|H_{r}(f)|^{2}df
  10. Φ N ( f ) \Phi_{N}(f)
  11. P e = P e | H 0 P H 0 + P e | H 1 P H 1 + + P e | H L - 1 P H L - 1 P_{e}=P_{e|H_{0}}\cdot P_{H_{0}}+P_{e|H_{1}}\cdot P_{H_{1}}+\cdots+P_{e|H_{L-1% }}\cdot P_{H_{L-1}}
  12. P e | H 0 P_{e|H_{0}}
  13. P H 0 P_{H_{0}}
  14. P H i = 1 L P_{H_{i}}=\frac{1}{L}
  15. P e = 2 ( 1 - 1 L ) P + P_{e}=2\left(1-\frac{1}{L}\right)P^{+}
  16. P + = A g ( 0 ) L - 1 1 2 π σ N e - x 2 2 σ N 2 d x = 1 2 erfc ( A g ( 0 ) 2 ( L - 1 ) σ N ) P^{+}=\int_{\frac{Ag(0)}{L-1}}^{\infty}\frac{1}{\sqrt{2\pi}\sigma_{N}}e^{-% \frac{x^{2}}{2\sigma_{N}^{2}}}dx=\frac{1}{2}\operatorname{erfc}\left(\frac{Ag(% 0)}{\sqrt{2}(L-1)\sigma_{N}}\right)

Analytic_capacity.html

  1. γ ( K ) = sup { | f ( ) | ; f ( 𝐂 K ) , f 1 , f ( ) = 0 } \gamma(K)=\sup\{|f^{\prime}(\infty)|;\ f\in\mathcal{H}^{\infty}(\mathbf{C}% \setminus K),\ \|f\|_{\infty}\leq 1,\ f(\infty)=0\}
  2. ( U ) \mathcal{H}^{\infty}(U)
  3. f ( ) := lim z z ( f ( z ) - f ( ) ) f^{\prime}(\infty):=\lim_{z\to\infty}z\left(f(z)-f(\infty)\right)
  4. f ( ) := lim z f ( z ) f(\infty):=\lim_{z\to\infty}f(z)
  5. f ( ) lim z f ( z ) f^{\prime}(\infty)\neq\lim_{z\to\infty}f^{\prime}(z)
  6. f ( 𝐂 K ) f\in\mathcal{H}^{\infty}(\mathbf{C}\setminus K)
  7. f 1 \|f\|\leq 1
  8. Q n j Q_{n}^{j}
  9. Q n j Q_{n}^{j}
  10. Q n - 1 k Q_{n-1}^{k}
  11. H 1 ( K ) = 2 H^{1}(K)=\sqrt{2}
  12. γ ( K ) = 0 K is purely unrectifiable \gamma(K)=0\ \Leftrightarrow\ K\ \,\text{ is purely unrectifiable}

Analytic_combinatorics.html

  1. Z ( G ) ( f ( z ) , f ( z 2 ) , , f ( z n ) ) . Z(G)(f(z),f(z^{2}),\ldots,f(z^{n})).\,
  2. g ( z ) n | G | . \frac{g(z)^{n}}{|G|}.
  3. E 2 E_{2}
  4. E 3 E_{3}
  5. X 2 / E 2 + X 3 / E 3 X^{2}/E_{2}\;+\;X^{3}/E_{3}
  6. X n / G X^{n}/G
  7. X n = X × × X X^{n}=X\times\ldots\times X
  8. X / C 1 + X 2 / C 2 + X 3 / C 3 + X 4 / C 4 + . X/C_{1}\;+\;X^{2}/C_{2}\;+\;X^{3}/C_{3}\;+\;X^{4}/C_{4}\;+\cdots.
  9. Cl ( S n ) \operatorname{Cl}(S_{n})
  10. S n S_{n}
  11. 𝒞 [ 𝔄 ] \mathcal{C}\in\mathbb{N}[\mathfrak{A}]
  12. 𝒞 = n 1 G Cl ( S n ) c G ( X n / G ) \mathcal{C}=\sum_{n\geq 1}\sum_{G\in\operatorname{Cl}(S_{n})}c_{G}(X^{n}/G)
  13. 𝔄 \mathfrak{A}
  14. { Cl ( S n ) } n 1 \{\operatorname{Cl}(S_{n})\}_{n\geq 1}
  15. c G . c_{G}\in\mathbb{N}.
  16. E 2 + E 3 and C 1 + C 2 + C 3 + . E_{2}+E_{3}\mbox{ and }~{}C_{1}+C_{2}+C_{3}+\cdots.
  17. 𝒞 [ 𝔄 ] \mathcal{C}\in\mathbb{N}[\mathfrak{A}]
  18. F ( z ) F(z)
  19. 𝒞 ( X ) \mathcal{C}(X)
  20. f ( z ) f(z)
  21. G ( z ) G(z)
  22. 𝒞 ( X ) \mathcal{C}(X)
  23. g ( z ) g(z)
  24. F ( z ) = n 1 G Cl ( S n ) c G Z ( G ) ( f ( z ) , f ( z 2 ) , , f ( z n ) ) F(z)=\sum_{n\geq 1}\sum_{G\in\operatorname{Cl}(S_{n})}c_{G}Z(G)(f(z),f(z^{2}),% \ldots,f(z^{n}))
  25. G ( z ) = n 1 ( G Cl ( S n ) c G | G | ) g ( z ) n . G(z)=\sum_{n\geq 1}\left(\sum_{G\in\operatorname{Cl}(S_{n})}\frac{c_{G}}{|G|}% \right)g(z)^{n}.
  26. G ( z ) G(z)
  27. 𝒞 [ 𝔄 ] \mathcal{C}\in\mathbb{Z}[\mathfrak{A}]
  28. 𝒞 [ 𝔄 ] . \mathcal{C}\in\mathbb{Q}[\mathfrak{A}].
  29. g ( z ) g(z)
  30. 𝔖 \mathfrak{S}
  31. 1 + E 1 + E 2 + E 3 + 1+E_{1}+E_{2}+E_{3}+\cdots\,
  32. F ( z ) = 1 + n 1 Z ( E n ) ( f ( z ) , f ( z 2 ) , , f ( z n ) ) = 1 + n 1 f ( z ) n = 1 1 - f ( z ) F(z)=1+\sum_{n\geq 1}Z(E_{n})(f(z),f(z^{2}),\ldots,f(z^{n}))=1+\sum_{n\geq 1}f% (z)^{n}=\frac{1}{1-f(z)}
  33. G ( z ) = 1 + n 1 ( 1 | E n | ) g ( z ) n = 1 1 - g ( z ) . G(z)=1+\sum_{n\geq 1}\left(\frac{1}{|E_{n}|}\right)g(z)^{n}=\frac{1}{1-g(z)}.
  34. \mathfrak{C}
  35. C 1 + C 2 + C 3 + C_{1}+C_{2}+C_{3}+\cdots\,
  36. F ( z ) = n 1 Z ( C n ) ( f ( z ) , f ( z 2 ) , , f ( z n ) ) = n 1 1 n d | n φ ( d ) f ( z d ) n / d F(z)=\sum_{n\geq 1}Z(C_{n})(f(z),f(z^{2}),\ldots,f(z^{n}))=\sum_{n\geq 1}\frac% {1}{n}\sum_{d|n}\varphi(d)f(z^{d})^{n/d}
  37. F ( z ) = k 1 φ ( k ) m 1 1 k m f ( z k ) m = k 1 φ ( k ) k log 1 1 - f ( z k ) F(z)=\sum_{k\geq 1}\varphi(k)\sum_{m\geq 1}\frac{1}{km}f(z^{k})^{m}=\sum_{k% \geq 1}\frac{\varphi(k)}{k}\log\frac{1}{1-f(z^{k})}
  38. G ( z ) = n 1 ( 1 | C n | ) g ( z ) n = log 1 1 - g ( z ) . G(z)=\sum_{n\geq 1}\left(\frac{1}{|C_{n}|}\right)g(z)^{n}=\log\frac{1}{1-g(z)}.
  39. 𝔓 \mathfrak{P}
  40. even \mathfrak{C}_{\operatorname{even}}
  41. C 2 + C 4 + C 6 + C_{2}+C_{4}+C_{6}+\cdots\,
  42. G ( z ) = n 1 ( 1 | C 2 n | ) g ( z ) 2 n = 1 2 log 1 1 - g ( z ) 2 . G(z)=\sum_{n\geq 1}\left(\frac{1}{|C_{2n}|}\right)g(z)^{2n}=\frac{1}{2}\log% \frac{1}{1-g(z)^{2}}.
  43. odd \mathfrak{C}_{\operatorname{odd}}
  44. C 1 + C 3 + C 5 + C_{1}+C_{3}+C_{5}+\cdots
  45. G ( z ) = log 1 1 - g ( z ) - 1 2 log 1 1 - g ( z ) 2 = 1 2 log 1 + g ( z ) 1 - g ( z ) . G(z)=\log\frac{1}{1-g(z)}-\frac{1}{2}\log\frac{1}{1-g(z)^{2}}=\frac{1}{2}\log% \frac{1+g(z)}{1-g(z)}.
  46. 𝔐 / 𝔓 \mathfrak{M}/\mathfrak{P}
  47. 1 + S 1 + S 2 + S 3 + 1+S_{1}+S_{2}+S_{3}+\cdots\,
  48. M ( f ( z ) , y ) = n 0 y n Z ( S n ) ( f ( z ) , f ( z 2 ) , , f ( z n ) ) M(f(z),y)=\sum_{n\geq 0}y^{n}Z(S_{n})(f(z),f(z^{2}),\ldots,f(z^{n}))
  49. 𝔐 ( f ( z ) ) = M ( f ( z ) , 1 ) . \mathfrak{M}(f(z))=M(f(z),1).\,
  50. M ( f ( z ) , 1 ) M(f(z),1)
  51. F ( z ) = exp ( l 1 f ( z l ) l ) . F(z)=\exp\left(\sum_{l\geq 1}\frac{f(z^{l})}{l}\right).
  52. G ( z ) = 1 + n 1 ( 1 | S n | ) g ( z ) n = n 0 g ( z ) n n ! = exp g ( z ) . G(z)=1+\sum_{n\geq 1}\left(\frac{1}{|S_{n}|}\right)g(z)^{n}=\sum_{n\geq 0}% \frac{g(z)^{n}}{n!}=\exp g(z).
  53. 𝔓 \mathfrak{P}
  54. 𝔐 \mathfrak{M}
  55. \mathcal{E}
  56. ϵ \epsilon
  57. 𝒵 \mathcal{Z}
  58. 𝒜 \mathcal{A}
  59. A ( z ) A(z)
  60. \mathcal{E}
  61. 𝒵 \mathcal{Z}
  62. E ( z ) = 1 E(z)=1
  63. Z ( z ) = z Z(z)=z
  64. \mathcal{B}
  65. 𝒞 \mathcal{C}
  66. 𝒜 = 𝒞 \mathcal{A}=\mathcal{B}\cup\mathcal{C}
  67. A ( z ) = B ( z ) + C ( z ) A(z)=B(z)+C(z)
  68. 𝒜 \mathcal{A}
  69. \mathcal{B}
  70. \circ
  71. 𝒜 \mathcal{A}
  72. \bullet
  73. \mathcal{B}
  74. 𝒜 + = ( 𝒜 × { } ) ( × { } ) \mathcal{A}+\mathcal{B}=(\mathcal{A}\times\{\circ\})\cup(\mathcal{B}\times\{% \bullet\})
  75. A n A_{n}
  76. A ( x ) = n = 0 A n x n A(x)=\sum_{n=0}^{\infty}A_{n}x^{n}
  77. 𝒜 \mathcal{A}
  78. \mathcal{B}
  79. a 𝒜 a\in\mathcal{A}
  80. b b\in\mathcal{B}
  81. | ( a , b ) | = | a | + | b | |(a,b)|=|a|+|b|
  82. 𝒜 × \mathcal{A}\times\mathcal{B}
  83. k = 0 n A k B n - k . \sum_{k=0}^{n}A_{k}B_{n-k}.
  84. 𝒜 = × 𝒞 \mathcal{A}=\mathcal{B}\times\mathcal{C}
  85. A ( z ) = B ( z ) C ( z ) . A(z)=B(z)\cdot C(z).
  86. 𝒜 = 𝔊 { } \mathcal{A}=\mathfrak{G}\{\mathcal{B}\}
  87. 𝔊 { } = + + ( × ) + ( × × ) + . \mathfrak{G}\{\mathcal{B}\}=\mathcal{E}+\mathcal{B}+(\mathcal{B}\times\mathcal% {B})+(\mathcal{B}\times\mathcal{B}\times\mathcal{B})+\cdots.
  88. \mathcal{B}
  89. A ( z ) = 1 + B ( z ) + B ( z ) 2 + B ( z ) 3 + = 1 1 - B ( z ) . A(z)=1+B(z)+B(z)^{2}+B(z)^{3}+\cdots=\frac{1}{1-B(z)}.
  90. 𝒜 = 𝔓 { } \mathcal{A}=\mathfrak{P}\{\mathcal{B}\}
  91. 𝔓 { } = β ( + { β } ) , \mathfrak{P}\{\mathcal{B}\}=\prod_{\beta\in\mathcal{B}}(\mathcal{E}+\{\beta\}),
  92. A ( z ) = β ( 1 + z | β | ) = n = 1 ( 1 + z n ) B n = exp ( ln n = 1 ( 1 + z n ) B n ) = exp ( n = 1 B n ln ( 1 + z n ) ) = exp ( n = 1 B n k = 1 ( - 1 ) k - 1 z n k k ) = exp ( k = 1 ( - 1 ) k - 1 k n = 1 B n z n k ) = exp ( k = 1 ( - 1 ) k - 1 B ( z k ) k ) , \begin{aligned}\displaystyle A(z)&\displaystyle{}=\prod_{\beta\in\mathcal{B}}(% 1+z^{|\beta|})\\ &\displaystyle{}=\prod_{n=1}^{\infty}(1+z^{n})^{B_{n}}\\ &\displaystyle{}=\exp\left(\ln\prod_{n=1}^{\infty}(1+z^{n})^{B_{n}}\right)\\ &\displaystyle{}=\exp\left(\sum_{n=1}^{\infty}B_{n}\ln(1+z^{n})\right)\\ &\displaystyle{}=\exp\left(\sum_{n=1}^{\infty}B_{n}\cdot\sum_{k=1}^{\infty}% \frac{(-1)^{k-1}z^{nk}}{k}\right)\\ &\displaystyle{}=\exp\left(\sum_{k=1}^{\infty}\frac{(-1)^{k-1}}{k}\cdot\sum_{n% =1}^{\infty}B_{n}z^{nk}\right)\\ &\displaystyle{}=\exp\left(\sum_{k=1}^{\infty}\frac{(-1)^{k-1}B(z^{k})}{k}% \right),\end{aligned}
  93. ln ( 1 + u ) = k = 1 ( - 1 ) k - 1 u k k \ln(1+u)=\sum_{k=1}^{\infty}\frac{(-1)^{k-1}u^{k}}{k}
  94. 𝒜 = 𝔐 { } \mathcal{A}=\mathfrak{M}\{\mathcal{B}\}
  95. 𝔐 { } = β 𝔊 { β } . \mathfrak{M}\{\mathcal{B}\}=\prod_{\beta\in\mathcal{B}}\mathfrak{G}\{\beta\}.
  96. A ( z ) = β ( 1 - z | β | ) - 1 = n = 1 ( 1 - z n ) - B n = exp ( ln n = 1 ( 1 - z n ) - B n ) = exp ( n = 1 - B n ln ( 1 - z n ) ) = exp ( k = 1 B ( z k ) k ) , \begin{aligned}\displaystyle A(z)&\displaystyle{}=\prod_{\beta\in\mathcal{B}}(% 1-z^{|\beta|})^{-1}\\ &\displaystyle{}=\prod_{n=1}^{\infty}(1-z^{n})^{-B_{n}}\\ &\displaystyle{}=\exp\left(\ln\prod_{n=1}^{\infty}(1-z^{n})^{-B_{n}}\right)\\ &\displaystyle{}=\exp\left(\sum_{n=1}^{\infty}-B_{n}\ln(1-z^{n})\right)\\ &\displaystyle{}=\exp\left(\sum_{k=1}^{\infty}\frac{B(z^{k})}{k}\right),\end{aligned}
  97. ln ( 1 - z n ) \ln(1-z^{n})
  98. \mathcal{B}
  99. { } \mathfrak{C}\{\mathcal{B}\}
  100. Θ \Theta\mathcal{B}
  101. \mathcal{B}
  102. 𝒞 \mathcal{B}\circ\mathcal{C}
  103. \mathcal{B}
  104. 𝒞 \mathcal{C}
  105. 𝒜 = { } \mathcal{A}=\mathfrak{C}\{\mathcal{B}\}
  106. A ( z ) = k = 1 ϕ ( k ) k ln 1 1 - B ( z k ) A(z)=\sum_{k=1}^{\infty}\frac{\phi(k)}{k}\ln\frac{1}{1-B(z^{k})}
  107. ϕ ( k ) \phi(k)\,
  108. 𝒜 = Θ \mathcal{A}=\Theta\mathcal{B}
  109. A ( z ) = z d d z B ( z ) A(z)=z\frac{d}{dz}B(z)
  110. 𝒜 = 𝒞 \mathcal{A}=\mathcal{B}\circ\mathcal{C}
  111. A ( z ) = B ( C ( z ) ) A(z)=B(C(z))\,
  112. 𝒢 = 𝒵 × 𝔊 { 𝒢 } . \mathcal{G}=\mathcal{Z}\times\mathfrak{G}\{\mathcal{G}\}.
  113. G ( z ) = z 1 - G ( z ) G(z)=\frac{z}{1-G(z)}
  114. 1 - G ( z ) 1-G(z)
  115. G ( z ) - G ( z ) 2 = z G(z)-G(z)^{2}=z
  116. G ( z ) = 1 - 1 - 4 z 2 . G(z)=\frac{1-\sqrt{1-4z}}{2}.
  117. \mathcal{I}
  118. = 𝒵 × 𝔊 { 𝒵 } \mathcal{I}=\mathcal{Z}\times\mathfrak{G}\{\mathcal{Z}\}
  119. \mathcal{I}
  120. I ( z ) = z 1 - z . I(z)=\frac{z}{1-z}.
  121. 𝒫 \mathcal{P}
  122. 𝒫 = 𝔐 { } . \mathcal{P}=\mathfrak{M}\{\mathcal{I}\}.
  123. 𝒫 \mathcal{P}
  124. P ( z ) = exp ( I ( z ) + 1 2 I ( z 2 ) + 1 3 I ( z 3 ) + ) . P(z)=\exp\left(I(z)+\frac{1}{2}I(z^{2})+\frac{1}{3}I(z^{3})+\cdots\right).
  125. P ( z ) P(z)
  126. [ 1 n ] [1\ldots n]
  127. A n A_{n}
  128. A ( x ) = n = 0 A n x n n ! . A(x)=\sum_{n=0}^{\infty}A_{n}\frac{x^{n}}{n!}.
  129. 𝒜 . \mathcal{A}\star\mathcal{B}.
  130. β \beta\in\mathcal{B}
  131. γ 𝒞 \gamma\in\mathcal{C}
  132. β \beta
  133. γ \gamma
  134. ρ \rho
  135. α \alpha
  136. ρ ( α ) \rho(\alpha)
  137. α \alpha
  138. β \beta
  139. α β = { ( α , β ) : ( α , β ) is well-labelled, ρ ( α ) = α , ρ ( β ) = β } . \alpha\star\beta=\{(\alpha^{\prime},\beta^{\prime}):(\alpha^{\prime},\beta^{% \prime})\mbox{ is well-labelled, }~{}\rho(\alpha^{\prime})=\alpha,\rho(\beta^{% \prime})=\beta\}.
  140. 𝒜 \mathcal{A}
  141. \mathcal{B}
  142. 𝒜 = α 𝒜 , β ( α β ) . \mathcal{A}\star\mathcal{B}=\bigcup_{\alpha\in\mathcal{A},\beta\in\mathcal{B}}% (\alpha\star\beta).
  143. k k
  144. n - k n-k
  145. ( n k ) {n\choose k}
  146. n n
  147. k = 0 n ( n k ) A k B n - k . \sum_{k=0}^{n}{n\choose k}A_{k}B_{n-k}.
  148. A ( z ) B ( z ) . A(z)\cdot B(z).\,
  149. 𝒜 = 𝔊 { } \mathcal{A}=\mathfrak{G}\{\mathcal{B}\}
  150. 𝔊 { } = + + ( ) + ( ) + \mathfrak{G}\{\mathcal{B}\}=\mathcal{E}+\mathcal{B}+(\mathcal{B}\star\mathcal{% B})+(\mathcal{B}\star\mathcal{B}\star\mathcal{B})+\cdots
  151. A ( z ) = 1 1 - B ( z ) A(z)=\frac{1}{1-B(z)}
  152. k k
  153. k ! k!
  154. 𝒜 = 𝔓 { } \mathcal{A}=\mathfrak{P}\{\mathcal{B}\}
  155. A ( z ) = k = 0 B ( z ) k k ! = exp ( B ( z ) ) A(z)=\sum_{k=0}^{\infty}\frac{B(z)^{k}}{k!}=\exp(B(z))
  156. k k
  157. k k
  158. 𝒜 = { } \mathcal{A}=\mathfrak{C}\{\mathcal{B}\}
  159. A ( z ) = k = 0 B ( z ) k k = ln ( 1 1 - B ( z ) ) . A(z)=\sum_{k=0}^{\infty}\frac{B(z)^{k}}{k}=\ln\left(\frac{1}{1-B(z)}\right).
  160. 𝒜 m i n = 𝒞 \mathcal{A}_{min}=\mathcal{B}^{\square}\star\mathcal{C}
  161. \mathcal{B}
  162. 𝒜 m a x = 𝒞 \mathcal{A}_{max}=\mathcal{B}^{\blacksquare}\star\mathcal{C}
  163. A m i n ( z ) = A m a x ( z ) = 0 z B ( t ) C ( t ) d t . A_{min}(z)=A_{max}(z)=\int^{z}_{0}B^{\prime}(t)C(t)dt.
  164. A m i n ( t ) = A m a x ( t ) = B ( t ) C ( t ) . A_{min}^{\prime}(t)=A_{max}^{\prime}(t)=B^{\prime}(t)C(t).
  165. \mathcal{L}
  166. = 𝒵 S E T ( ) . \mathcal{L}=\mathcal{Z}^{\square}\star SET(\mathcal{L}).
  167. even , odd , 𝔓 even , and 𝔓 odd \mathfrak{C}_{\operatorname{even}},\mathfrak{C}_{\operatorname{odd}},\mathfrak% {P}_{\operatorname{even}},\mbox{ and }~{}\mathfrak{P}_{\operatorname{odd}}
  168. 𝔓 ( 𝔓 1 ( 𝒵 ) ) . \mathfrak{P}(\mathfrak{P}_{\geq 1}(\mathcal{Z})).
  169. 𝔓 ( ( 𝒵 ) ) \mathfrak{P}(\mathfrak{C}(\mathcal{Z}))

Analytic_signal.html

  1. s ( t ) s(t)
  2. S ( f ) S(f)
  3. f = 0 f=0
  4. S ( - f ) = S ( f ) * , S(-f)=S(f)^{*},
  5. S ( f ) S(f)
  6. S a ( f ) \displaystyle S_{\mathrm{a}}(f)
  7. u ( f ) \operatorname{u}(f)
  8. sgn ( f ) \operatorname{sgn}(f)
  9. S ( f ) S(f)
  10. S ( f ) S(f)
  11. S ( f ) \displaystyle S(f)
  12. s ( t ) s(t)
  13. S a ( f ) S_{\mathrm{a}}(f)
  14. s a ( t ) \displaystyle s_{\mathrm{a}}(t)
  15. s ^ ( t ) = def [ s ( t ) ] \hat{s}(t)\stackrel{\mathrm{def}}{{}={}}\operatorname{\mathcal{H}}[s(t)]
  16. s ( t ) s(t)
  17. * *
  18. j j
  19. s ( t ) = cos ( ω t ) , s(t)=\cos(\omega t),
  20. ω > 0. \omega>0.
  21. s ^ ( t ) = cos ( ω t - π / 2 ) = sin ( ω t ) , \hat{s}(t)=\cos(\omega t-\pi/2)=\sin(\omega t),
  22. s a ( t ) = s ( t ) + j s ^ ( t ) = cos ( ω t ) + j sin ( ω t ) = e j ω t . s_{\mathrm{a}}(t)=s(t)+j\hat{s}(t)=\cos(\omega t)+j\sin(\omega t)=e^{j\omega t}.
  23. cos ( ω t ) = 1 2 ( e j ω t + e j ( - ω ) t ) . \cos(\omega t)=\tfrac{1}{2}(e^{j\omega t}+e^{j(-\omega)t}).
  24. s ( t ) = cos ( ω t + θ ) = 1 2 ( e j ( ω t + θ ) + e - j ( ω t + θ ) ) s(t)=\cos(\omega t+\theta)=\tfrac{1}{2}(e^{j(\omega t+\theta)}+e^{-j(\omega t+% \theta)})
  25. s a ( t ) = { e j ( ω t + θ ) = e j | ω | t e j θ , if ω > 0 , e - j ( ω t + θ ) = e j | ω | t e - j θ , if ω < 0. s_{\mathrm{a}}(t)=\begin{cases}e^{j(\omega t+\theta)}\ \ =\ e^{j|\omega|t}% \cdot e^{j\theta},&\,\text{if}\ \omega>0,\\ e^{-j(\omega t+\theta)}=\ e^{j|\omega|t}\cdot e^{-j\theta},&\,\text{if}\ % \omega<0.\end{cases}
  26. s a ( t ) s_{\mathrm{a}}(t)
  27. s ( t ) s(t)
  28. s ( t ) s(t)
  29. s ( t ) = e - j ω t s(t)=e^{-j\omega t}
  30. ω > 0 \omega>0
  31. s ^ ( t ) = j e - j ω t , \hat{s}(t)=je^{-j\omega t},
  32. s a ( t ) = e - j ω t + j 2 e - j ω t = e - j ω t - e - j ω t = 0. s_{\mathrm{a}}(t)=e^{-j\omega t}+j^{2}e^{-j\omega t}=e^{-j\omega t}-e^{-j% \omega t}=0.
  33. s ( t ) = Re [ s a ( t ) ] s(t)=\operatorname{Re}[s_{\mathrm{a}}(t)]
  34. Im [ s a ( t ) ] \operatorname{Im}[s_{\mathrm{a}}(t)]
  35. s a * ( t ) s_{\mathrm{a}}^{*}(t)
  36. Re [ s a * ( t ) ] \operatorname{Re}[s_{\mathrm{a}}^{*}(t)]
  37. s a ( t ) = s m ( t ) e j ϕ ( t ) , s_{\mathrm{a}}(t)=s_{\mathrm{m}}(t)e^{j\phi(t)},
  38. s m ( t ) = def | s a ( t ) | s_{\mathrm{m}}(t)\stackrel{\mathrm{def}}{{}={}}|s_{\mathrm{a}}(t)|
  39. ϕ ( t ) = def arg [ s a ( t ) ] \phi(t)\stackrel{\mathrm{def}}{{}={}}\arg\!\left[s_{\mathrm{a}}(t)\right]
  40. s ( t ) s(t)
  41. s m ( t ) s_{\mathrm{m}}(t)
  42. ω ( t ) = def d ϕ d t ( t ) . \omega(t)\stackrel{\mathrm{def}}{{}={}}\frac{d\phi}{dt}(t).
  43. f ( t ) = def 1 2 π ω ( t ) . f(t)\stackrel{\mathrm{def}}{{}={}}\frac{1}{2\pi}\omega(t).
  44. s a ¯ ( t ) = def s a ( t ) e - j ω 0 t = s m ( t ) e j ( ϕ ( t ) - ω 0 t ) , \underline{s_{\mathrm{a}}}(t)\stackrel{\mathrm{def}}{{}={}}s_{\mathrm{a}}(t)e^% {-j\omega_{0}t}=s_{\mathrm{m}}(t)e^{j(\phi(t)-\omega_{0}t)},
  45. ω 0 \omega_{0}
  46. ω 0 \omega_{0}
  47. s ( t ) s(t)
  48. ω 0 \omega_{0}
  49. ω 0 \omega_{0}
  50. ω 0 \omega_{0}
  51. s a ( t ) , s_{\mathrm{a}}(t),
  52. s a ¯ ( t ) \underline{s_{\mathrm{a}}}(t)
  53. ω 0 \omega_{0}
  54. ω 0 \omega_{0}
  55. 0 + ( ω - ω 0 ) 2 | S a ( ω ) | 2 d ω . \int_{0}^{+\infty}(\omega-\omega_{0})^{2}|S_{\mathrm{a}}(\omega)|^{2}\,d\omega.
  56. ω 0 \omega_{0}
  57. ϕ ( t ) \phi(t)
  58. - + [ ω ( t ) - ω 0 ] 2 | s a ( t ) | 2 d t \int_{-\infty}^{+\infty}[\omega(t)-\omega_{0}]^{2}|s_{\mathrm{a}}(t)|^{2}\,dt
  59. θ \theta
  60. - + [ ϕ ( t ) - ( ω 0 t + θ ) ] 2 d t . \int_{-\infty}^{+\infty}[\phi(t)-(\omega_{0}t+\theta)]^{2}\,dt.
  61. s y m b o l u ^ symbol\hat{u}
  62. s y m b o l ξ symbol\xi
  63. s y m b o l ξ s y m b o l u ^ < 0 symbol\xi\cdot symbol\hat{u}<0
  64. s y m b o l u ^ symbol\hat{u}
  65. s y m b o l u ^ symbol\hat{u}

Analytization_trick.html

  1. f ( z ) = g ( z ¯ , z ) f(z)=g(\bar{z},z)
  2. z 1 g | z 1 = z ¯ ; z 2 = z \left.\frac{\partial}{\partial z_{1}}g\right|_{z_{1}=\bar{z};z_{2}=z}
  3. z 2 g | z 1 = z ¯ ; z 2 = z \left.\frac{\partial}{\partial z_{2}}g\right|_{z_{1}=\bar{z};z_{2}=z}
  4. z ¯ f ( z ¯ , z ) \frac{\partial}{\partial\bar{z}}f(\bar{z},z)
  5. z f ( z ¯ , z ) \frac{\partial}{\partial z}f(\bar{z},z)
  6. z ¯ \bar{z}
  7. f ( z ¯ , z ) f(\bar{z},z)

Anchoring.html

  1. 1 × 2 × 3 × 4 × 5 × 6 × 7 × 8 1\times 2\times 3\times 4\times 5\times 6\times 7\times 8
  2. 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 8\times 7\times 6\times 5\times 4\times 3\times 2\times 1

Ancient_Egyptian_mathematics.html

  1. 1 2 , 1 3 \tfrac{1}{2},\tfrac{1}{3}
  2. 2 3 \tfrac{2}{3}
  3. 3 4 \tfrac{3}{4}
  4. 1 n \tfrac{1}{n}
  5. 2 n \tfrac{2}{n}
  6. 1 n \tfrac{1}{n}
  7. 1 / 2 1/2
  8. 1 / 3 1/3
  9. 2 / 3 2/3
  10. 1 / 4 1/4
  11. 1 / 5 1/5
  12. ( 2 / 3 + 1 / 10 + 1 / 2190 ) (2/3+1/10+1/2190)
  13. 8 + 2 / 3 + 1 / 10 + 1 / 2190 8+2/3+1/10+1/2190
  14. 3 / 2 × x + 4 = 10. 3/2\times x+4=10.

Ancillary_statistic.html

  1. X ¯ n = X 1 + + X n n \overline{X}_{n}=\frac{X_{1}+\,\cdots\,+X_{n}}{n}
  2. σ ^ 2 := ( X i - X ¯ ) 2 n \hat{\sigma}^{2}:=\,\frac{\sum\left(X_{i}-\overline{X}\right)^{2}}{n}
  3. X ¯ \overline{X}

Anders_Johan_Lexell.html

  1. d x V d x \scriptstyle dx\int{Vdx}
  2. d x d x V d x \scriptstyle dx\int{dx\int{Vdx}}
  3. d x d x d x V d x \scriptstyle dx\int{dx\int{dx\int{Vdx}}}
  4. x = y ϕ ( x ) + ψ ( x ) x=y\phi(x^{\prime})+\psi(x^{\prime})
  5. n n
  6. n n

Angel_problem.html

  1. d 1 < d 2 < d 3 < d_{1}<d_{2}<d_{3}<\cdots
  2. d d
  3. π = i = 1 ( σ i γ i ) \pi=\cup^{\infty}_{i=1}(\sigma_{i}\cup\gamma_{i})
  4. σ = i = 1 σ i \sigma=\cup^{\infty}_{i=1}\sigma_{i}
  5. γ i {\gamma_{i}}
  6. i : | γ i | i \forall i:|\gamma_{i}|\leq i
  7. | γ i | |\gamma_{i}|
  8. γ i \gamma_{i}
  9. σ i \sigma_{i}
  10. σ i \sigma_{i}
  11. σ i + 1 \sigma_{i+1}
  12. σ \sigma

Angular_defect.html

  1. π \pi\,
  2. 4 π 4\pi\,
  3. 2 π 3 {2\pi\over 3}
  4. 4 π 4\pi\,
  5. π 2 {\pi\over 2}
  6. 4 π 4\pi\,
  7. π 3 {\pi\over 3}
  8. 4 π 4\pi\,
  9. π 5 {\pi\over 5}
  10. 4 π 4\pi\,

Angular_diameter.html

  1. δ = 2 arctan ( d 2 D ) , \delta=2\arctan\left(\frac{d}{2D}\right),
  2. δ \delta
  3. d d
  4. D D
  5. D d D\gg d
  6. δ d / D \delta\approx d/D
  7. d act , d_{\mathrm{act}},
  8. D D
  9. δ = 2 arcsin ( d act 2 D ) \delta=2\arcsin\left(\frac{d_{\mathrm{act}}}{2D}\right)
  10. x 1 x\ll 1
  11. arcsin x arctan x x \arcsin x\approx\arctan x\approx x
  12. π \pi
  13. π \pi
  14. δ \delta
  15. 1 / 2 {1}/{2}
  16. d 2 D tan ( δ 2 ) d\equiv 2D\tan\left(\frac{\delta}{2}\right)

Anisometropia.html

  1. Magnification = 1 ( 1 - ( t n ) P ) 1 ( 1 - h F ) \textrm{Magnification}=\frac{1}{(1-(\frac{t}{n})P)}\cdot\frac{1}{(1-hF)}

Annual_percentage_rate.html

  1. l = 1 M S l ( 1 + APR / 100 ) - t l = k = 1 N A k ( 1 + APR / 100 ) - t k \sum_{l=1}^{M}S_{l}(1+\mathrm{APR}/100)^{-t_{l}}=\sum_{k=1}^{N}A_{k}(1+\mathrm% {APR}/100)^{-t_{k}}
  2. S - A = R ( 1 + APR / 100 ) - t N + k = 1 N A k ( 1 + APR / 100 ) - t k S-A=R(1+\mathrm{APR}/100)^{-t_{N}}+\sum_{k=1}^{N}A_{k}(1+\mathrm{APR}/100)^{-t% _{k}}
  3. p = P 0 r ( 1 + r ) n ( 1 + r ) n - 1 p=\frac{P_{0}\cdot r\cdot(1+r)^{n}}{(1+r)^{n}-1}
  4. N ( C r + F r ) 2 , \frac{N(Cr+Fr)}{2}\,,
  5. ( C + F ) r 2 . \frac{(C+F)r}{2}\,.
  6. EAR = ( 1 + APR n ) n - 1 \mathrm{EAR}=(1+\tfrac{\mathrm{APR}}{n})^{n}-1
  7. ( 1 + 0.129949 12 ) 12 - 1 (1+\tfrac{0.129949}{12})^{12}-1

Annualized_failure_rate.html

  1. A F R = 1 - e x p ( - 8760 / M T B F ) . AFR=1-exp(-8760/MTBF).
  2. A F R = 1 M T B F / 8760 100 AFR={1\over{MTBF/8760}}\cdot 100

Antecedent_(logic).html

  1. ϕ \phi
  2. ψ \psi
  3. ϕ \phi
  4. ψ \psi

Antihomomorphism.html

  1. ϕ : X Y op \phi\colon X\to Y^{\,\text{op}}
  2. Y op Y^{\,\text{op}}
  3. \cdot
  4. Y op Y^{\,\text{op}}
  5. * *
  6. x * y := y x x*y:=y\cdot x
  7. Y op Y^{\,\text{op}}
  8. ϕ : X op Y \phi\colon X^{\,\text{op}}\to Y
  9. X op X^{\,\text{op}}

Antimagic_square.html

  1. 1 , , m 1,\ldots,m
  2. m n 2 m\leq n^{2}

Antimatroid.html

  1. G = { U S S F } G=\{U\setminus S\mid S\in F\}
  2. A B = { S T S A T B } . A\vee B=\{S\cup T\mid S\in A\wedge T\in B\}.

Antisymmetric_tensor.html

  1. T i j k = - T j i k = T j k i = - T k j i = T k i j = - T i k j T_{ijk\dots}=-T_{jik\dots}=T_{jki\dots}=-T_{kji\dots}=T_{kij\dots}=-T_{ikj\dots}
  2. U i j k U_{ijk\dots}
  3. U ( i j ) k = 1 2 ( U i j k + U j i k ) U_{(ij)k\dots}=\frac{1}{2}(U_{ijk\dots}+U_{jik\dots})
  4. U [ i j ] k = 1 2 ( U i j k - U j i k ) U_{[ij]k\dots}=\frac{1}{2}(U_{ijk\dots}-U_{jik\dots})
  5. U i j k = U ( i j ) k + U [ i j ] k . U_{ijk\dots}=U_{(ij)k\dots}+U_{[ij]k\dots}.
  6. M [ a b ] = 1 2 ! ( M a b - M b a ) , M_{[ab]}=\frac{1}{2!}(M_{ab}-M_{ba}),
  7. T [ a b c ] = 1 3 ! ( T a b c - T a c b + T b c a - T b a c + T c a b - T c b a ) . T_{[abc]}=\frac{1}{3!}(T_{abc}-T_{acb}+T_{bca}-T_{bac}+T_{cab}-T_{cba}).
  8. M [ a b ] = 1 2 ! δ a b c d M c d , M_{[ab]}=\frac{1}{2!}\,\delta_{ab}^{cd}M_{cd},
  9. T [ a b c ] = 1 3 ! δ a b c d e f T d e f . T_{[abc]}=\frac{1}{3!}\,\delta_{abc}^{def}T_{def}.
  10. S [ a 1 a p ] = 1 p ! δ a 1 a p b 1 b p S b 1 b p . S_{[a_{1}\dots a_{p}]}=\frac{1}{p!}\delta_{a_{1}\dots a_{p}}^{b_{1}\dots b_{p}% }S_{b_{1}\dots b_{p}}.
  11. δ a b c d \delta_{ab\dots}^{cd\dots}
  12. F μ ν F_{\mu\nu}

Anton_Zeilinger.html

  1. C 60 C_{60}
  2. C 70 C_{70}

Apache_Point_Observatory_Lunar_Laser-ranging_Operation.html

  1. 2 {}^{2}

Aperiodic_tiling.html

  1. T \R d T\in\R^{d}
  2. { T + x : x d } \{T+x\,:\,x\in\mathbb{R}^{d}\}
  3. ε \varepsilon
  4. 1 / ε 1/\varepsilon
  5. ε \varepsilon
  6. 1 , 2 , 4 , , 2 n , 1,2,4,\ldots,2^{n},\ldots

Apollonian_gasket.html

  1. a 2 + b 2 + c 2 + d 2 = 2 a b + 2 a c + 2 a d + 2 b c + 2 b d + 2 c d , a^{2}+b^{2}+c^{2}+d^{2}=2ab+2ac+2ad+2bc+2bd+2cd,\,
  2. 2 3 - 3 \scriptstyle 2\sqrt{3}-3
  3. a ( n ) = 4 a ( n 1 ) a ( n 2 ) a(n)=4a(n−1)−a(n−2)
  4. 3 + 2 3.732050807 \scriptstyle\sqrt{3}+2\approx 3.732050807\dots

Approximate_identity.html

  1. { e λ : λ Λ } \{\,e_{\lambda}:\lambda\in\Lambda\,\}
  2. { a e λ : λ Λ } \{\,ae_{\lambda}:\lambda\in\Lambda\,\}
  3. { e λ : λ Λ } \{\,e_{\lambda}:\lambda\in\Lambda\,\}
  4. { e λ a : λ Λ } \{\,e_{\lambda}a:\lambda\in\Lambda\,\}

Approximation_error.html

  1. ϵ = | v - v approx | , \epsilon=|v-v\text{approx}|\ ,
  2. v 0 , v\neq 0,
  3. η = ϵ | v | = | v - v approx v | = | 1 - v approx v | , \eta=\frac{\epsilon}{|v|}=\left|\frac{v-v\text{approx}}{v}\right|=\left|1-% \frac{v\text{approx}}{v}\right|,
  4. δ = 100 × η = 100 × ϵ | v | = 100 × | v - v approx v | . \delta=100\times\eta=100\times\frac{\epsilon}{|v|}=100\times\left|\frac{v-v% \text{approx}}{v}\right|.
  5. v v
  6. v approx v_{\,\text{approx}}
  7. × 10 - 3 \times 10^{-}3

Approximation_theory.html

  1. P ( x ) - f ( x ) \mid P(x)-f(x)\mid
  2. + ε +\varepsilon
  3. - ε -\varepsilon
  4. ε \varepsilon
  5. + ε +\varepsilon
  6. - ε -\varepsilon
  7. | Q ( x i ) - f ( x i ) | < | P ( x i ) - f ( x i ) | . |Q(x_{i})-f(x_{i})|<|P(x_{i})-f(x_{i})|.
  8. Q ( x i ) - f ( x i ) | Q ( x i ) - f ( x i ) | < | P ( x i ) - f ( x i ) | = P ( x i ) - f ( x i ) , Q(x_{i})-f(x_{i})\leq|Q(x_{i})-f(x_{i})|<|P(x_{i})-f(x_{i})|=P(x_{i})-f(x_{i}),
  9. f ( x i ) - Q ( x i ) | Q ( x i ) - f ( x i ) | < | P ( x i ) - f ( x i ) | = f ( x i ) - P ( x i ) . f(x_{i})-Q(x_{i})\leq|Q(x_{i})-f(x_{i})|<|P(x_{i})-f(x_{i})|=f(x_{i})-P(x_{i}).
  10. f ( x ) i = 0 c i T i ( x ) f(x)\sim\sum_{i=0}^{\infty}c_{i}T_{i}(x)
  11. T N T_{N}
  12. T N T_{N}
  13. T N + 1 T_{N+1}
  14. T N + 1 T_{N+1}
  15. T N T_{N}
  16. x 1 x_{1}
  17. x 2 x_{2}
  18. x N + 2 x_{N+2}
  19. x 1 x_{1}
  20. x N + 2 x_{N+2}
  21. P ( x 1 ) - f ( x 1 ) = + ε P(x_{1})-f(x_{1})=+\varepsilon\,
  22. P ( x 2 ) - f ( x 2 ) = - ε P(x_{2})-f(x_{2})=-\varepsilon\,
  23. P ( x 3 ) - f ( x 3 ) = + ε P(x_{3})-f(x_{3})=+\varepsilon\,
  24. \vdots
  25. P ( x N + 2 ) - f ( x N + 2 ) = ± ε . P(x_{N+2})-f(x_{N+2})=\pm\varepsilon.\,
  26. P 0 + P 1 x 1 + P 2 x 1 2 + P 3 x 1 3 + + P N x 1 N - f ( x 1 ) = + ε P_{0}+P_{1}x_{1}+P_{2}x_{1}^{2}+P_{3}x_{1}^{3}+\dots+P_{N}x_{1}^{N}-f(x_{1})=+\varepsilon\,
  27. P 0 + P 1 x 2 + P 2 x 2 2 + P 3 x 2 3 + + P N x 2 N - f ( x 2 ) = - ε P_{0}+P_{1}x_{2}+P_{2}x_{2}^{2}+P_{3}x_{2}^{3}+\dots+P_{N}x_{2}^{N}-f(x_{2})=-\varepsilon\,
  28. \vdots
  29. x 1 x_{1}
  30. x N + 2 x_{N+2}
  31. f ( x 1 ) f(x_{1})
  32. f ( x N + 2 ) f(x_{N+2})
  33. P 0 P_{0}
  34. P 1 P_{1}
  35. P N P_{N}
  36. ε \varepsilon
  37. x 1 x_{1}
  38. x N + 2 x_{N+2}
  39. ε \varepsilon
  40. e x e^{x}
  41. ε \varepsilon
  42. ± ε \pm\varepsilon
  43. f ( x ) f(x)\,
  44. f ( x ) f^{\prime}(x)\,
  45. f ′′ ( x ) f^{\prime\prime}(x)\,
  46. 10 - 15 10^{-15}
  47. 10 - 30 10^{-30}
  48. T N + 1 T_{N+1}

Aquifer_test.html

  1. s = Q 4 π T W ( u ) u = r 2 S 4 T t \begin{aligned}\displaystyle s&\displaystyle=\frac{Q}{4\pi T}W(u)\\ \displaystyle u&\displaystyle=\frac{r^{2}S}{4Tt}\end{aligned}
  2. W ( u ) = - 0.577216 - ln ( u ) + u - u 2 2 × 2 ! + u 3 3 × 3 ! - u 4 4 × 4 ! + \begin{aligned}\displaystyle W(u)=-0.577216-\ln(u)+u-\frac{u^{2}}{2\times 2!}+% \frac{u^{3}}{3\times 3!}-\frac{u^{4}}{4\times 4!}+\cdots\end{aligned}
  3. h - h 0 = Q 2 π T ln ( R r ) h-h_{0}=\frac{Q}{2\pi T}\ln\left(\frac{R}{r}\right)

Arbelos.html

  1. A A
  2. B B
  3. C C
  4. B C BC
  5. A B AB
  6. A C AC
  7. H H
  8. B C BC
  9. A A
  10. H A HA
  11. r 2 + h 2 = x 2 r^{2}+h^{2}=x^{2}
  12. ( 1 - r ) 2 + h 2 = y 2 (1-r)^{2}+h^{2}=y^{2}
  13. x 2 + y 2 = 1 x^{2}+y^{2}=1
  14. y 2 = ( 1 - r ) 2 + x 2 - r 2 y^{2}=(1-r)^{2}+x^{2}-r^{2}
  15. y 2 = 1 - 2 r + x 2 y^{2}=1-2r+x^{2}
  16. x = r x=\sqrt{r}
  17. y = 1 - r y=\sqrt{1-r}
  18. h = r - r 2 h=\sqrt{r-r^{2}}
  19. 1 2 r - r 2 . \frac{1}{2}\sqrt{r-r^{2}}.
  20. A c i r c l e = π ( 1 2 r - r 2 ) 2 A_{circle}=\pi\left(\frac{1}{2}\sqrt{r-r^{2}}\right)^{2}
  21. A c i r c l e = π r 4 - π r 2 4 A_{circle}=\frac{\pi r}{4}-\frac{\pi r^{2}}{4}
  22. A a r b e l o s = π 8 - ( π 2 ( r 2 ) 2 + π 2 ( 1 - r 2 ) 2 ) A_{arbelos}=\frac{\pi}{8}-\left(\frac{\pi}{2}\left(\frac{r}{2}\right)^{2}+% \frac{\pi}{2}\left(\frac{1-r}{2}\right)^{2}\right)
  23. A a r b e l o s = π - π r 2 - π + 2 π r - π r 2 8 A_{arbelos}=\frac{\pi-\pi r^{2}-\pi+2\pi r-\pi r^{2}}{8}
  24. A a r b e l o s = π r 4 - π r 2 4 = A c i r c l e A_{arbelos}=\frac{\pi r}{4}-\frac{\pi r^{2}}{4}=A_{circle}
  25. D D
  26. E E
  27. B H BH
  28. C H CH
  29. A B AB
  30. A C AC
  31. A D H E ADHE
  32. B D A BDA
  33. B H C BHC
  34. A E C AEC
  35. A D H E ADHE
  36. D E DE
  37. B A BA
  38. D D
  39. A C AC
  40. E E
  41. A H AH

Arbitrage_pricing_theory.html

  1. r j = a j + b j 1 F 1 + b j 2 F 2 + + b j n F n + ϵ j r_{j}=a_{j}+b_{j1}F_{1}+b_{j2}F_{2}+\cdots+b_{jn}F_{n}+\epsilon_{j}
  2. a j a_{j}
  3. j j
  4. F k F_{k}
  5. b j k b_{jk}
  6. j j
  7. k k
  8. ϵ j \epsilon_{j}
  9. E ( r j ) = r f + b j 1 R P 1 + b j 2 R P 2 + + b j n R P n E\left(r_{j}\right)=r_{f}+b_{j1}RP_{1}+b_{j2}RP_{2}+\cdots+b_{jn}RP_{n}
  10. R P k RP_{k}
  11. r f r_{f}

Arbitrarily_large.html

  1. n x x > n f ( x ) 0 \forall n\in\mathbb{R}\exists x\in\mathbb{R}x>n\land f(x)\geq 0
  2. n x x > n f ( x ) 0 \exists n\in\mathbb{R}\forall x\in\mathbb{R}x>n\Rightarrow f(x)\geq 0

Area_density.html

  1. ρ A = m A \rho_{A}=\frac{m}{A}
  2. ρ A = ρ l \rho_{A}=\rho\cdot l
  3. ρ < s u b > A ρ<sub>A
  4. ρ \rho
  5. σ = ρ d s . \sigma=\int\rho\;\operatorname{d}s.
  6. σ = ρ d z \sigma=\int\rho\;\operatorname{d}z
  7. z z
  8. ρ A \rho_{A}
  9. ρ ¯ \bar{\rho}
  10. ρ ¯ = ρ A Δ z , \bar{\rho}=\frac{\rho_{A}}{\Delta z},
  11. Δ z = 1 d z \Delta z=\int 1\;\operatorname{d}z
  12. ρ ¯ \bar{\rho}
  13. ρ A \rho_{A}
  14. Δ z \Delta z
  15. N = n d s . N=\int n\;\operatorname{d}s.
  16. P = σ ρ 0 P=\frac{\sigma}{\rho_{0}}

Areal_velocity.html

  1. vector area of parallelogram A B C D = r ( t ) × r ( t + Δ t ) . \,\text{vector area of parallelogram }ABCD=\vec{r}(t)\times\vec{r}(t+\Delta t).
  2. vector area of triangle A B C = r ( t ) × r ( t + Δ t ) 2 . \,\text{vector area of triangle }ABC=\frac{\vec{r}(t)\times\vec{r}(t+\Delta t)% }{2}.
  3. areal velocity \displaystyle\,\text{areal velocity}
  4. r ( t ) \vec{r}\,^{\prime}(t)
  5. v ( t ) \vec{v}(t)
  6. d A d t = r × v 2 . \frac{d\vec{A}}{dt}=\frac{\vec{r}\times\vec{v}}{2}.
  7. L = r × m v , \vec{L}=\vec{r}\times m\vec{v},

Areostationary_orbit.html

  1. v = ω r . v=\omega r\,\text{.}

Argument_of_periapsis.html

  1. ω = arccos 𝐧 𝐞 | 𝐧 | | 𝐞 | \omega=\arccos{{\mathbf{n}\cdot\mathbf{e}}\over{\mathbf{\left|n\right|}\mathbf% {\left|e\right|}}}
  2. e z < 0 e_{z}<0\,
  3. ω = 2 π - ω \omega=2\pi-\omega\,
  4. 𝐧 \mathbf{n}
  5. 𝐧 \mathbf{n}
  6. 𝐞 \mathbf{e}
  7. ω = arctan 2 ( e y , e x ) \omega=\arctan 2({e_{y}},{e_{x}})
  8. ( 𝐫 × 𝐯 ) z < 0 (\mathbf{r}\times\mathbf{v})_{z}<0
  9. ω = 2 π - ω \omega=2\pi-\omega\,
  10. e x e_{x}\,
  11. e y e_{y}\,
  12. 𝐞 . \mathbf{e}.\,

Argument_principle.html

  1. C f ( z ) f ( z ) d z = 2 π i ( 4 - 5 ) \oint_{C}{f^{\prime}(z)\over f(z)}\,dz=2\pi i(4-5)
  2. C f ( z ) f ( z ) d z = 2 π i ( N - P ) \oint_{C}{f^{\prime}(z)\over f(z)}\,dz=2\pi i(N-P)
  3. C f ( z ) f ( z ) d z = 2 π i ( a n ( C , a ) - b n ( C , b ) ) \oint_{C}\frac{f^{\prime}(z)}{f(z)}\,dz=2\pi i\left(\sum_{a}n(C,a)-\sum_{b}n(C% ,b)\right)
  4. C f ( z ) f ( z ) d z \oint_{C}\frac{f^{\prime}(z)}{f(z)}\,dz
  5. d d z log ( f ( z ) ) = f ( z ) f ( z ) \frac{d}{dz}\log(f(z))=\frac{f^{\prime}(z)}{f(z)}
  6. C f ( z ) f ( z ) d z = f ( C ) 1 w d w \oint_{C}\frac{f^{\prime}(z)}{f(z)}\,dz=\oint_{f(C)}\frac{1}{w}\,dw
  7. f ( z ) = k ( z - z N ) k - 1 g ( z ) + ( z - z N ) k g ( z ) f^{\prime}(z)=k(z-z_{N})^{k-1}g(z)+(z-z_{N})^{k}g^{\prime}(z)\,\!
  8. f ( z ) f ( z ) = k z - z N + g ( z ) g ( z ) . {f^{\prime}(z)\over f(z)}={k\over z-z_{N}}+{g^{\prime}(z)\over g(z)}.
  9. f ( z ) = - m ( z - z P ) - m - 1 h ( z ) + ( z - z P ) - m h ( z ) . f^{\prime}(z)=-m(z-z_{P})^{-m-1}h(z)+(z-z_{P})^{-m}h^{\prime}(z)\,\!.
  10. f ( z ) f ( z ) = - m z - z P + h ( z ) h ( z ) {f^{\prime}(z)\over f(z)}={-m\over z-z_{P}}+{h^{\prime}(z)\over h(z)}
  11. 1 2 π i C f ( z ) f ( z ) d z {1\over 2\pi i}\oint_{C}{f^{\prime}(z)\over f(z)}\,dz
  12. ξ ( s ) \xi(s)
  13. 1 2 π i C g ( z ) f ( z ) f ( z ) d z = a n ( C , a ) g ( a ) - b n ( C , b ) g ( b ) . \frac{1}{2\pi i}\oint_{C}g(z)\frac{f^{\prime}(z)}{f(z)}\,dz=\sum_{a}n(C,a)g(a)% -\sum_{b}n(C,b)g(b).
  14. 1 2 π i C z k f ( z ) f ( z ) d z = z 1 k + z 2 k + + z p k , \frac{1}{2\pi i}\oint_{C}z^{k}\frac{f^{\prime}(z)}{f(z)}\,dz=z_{1}^{k}+z_{2}^{% k}+\dots+z_{p}^{k},
  15. C f ( z ) g ( z ) g ( z ) d z \oint_{C}f(z){g^{\prime}(z)\over g(z)}\,dz
  16. n = 0 f ( n ) - 0 f ( x ) d x = f ( 0 ) / 2 + i 0 f ( i t ) - f ( - i t ) e 2 π t - 1 d t \sum_{n=0}^{\infty}f(n)-\int_{0}^{\infty}f(x)\,dx=f(0)/2+i\int_{0}^{\infty}% \frac{f(it)-f(-it)}{e^{2\pi t}-1}\,dt
  17. Ω \Omega
  18. C f ( z ) f ( z ) g ( z ) d z = 2 π i ( a g ( a ) n ( C , a ) - b g ( b ) n ( C , b ) ) \oint_{C}{f^{\prime}(z)\over f(z)}g(z)\,dz=2\pi i\left(\sum_{a}g(a)n(C,a)-\sum% _{b}g(b)n(C,b)\right)

Arnoldi_iteration.html

  1. A A
  2. λ 1 \lambda_{1}
  3. A n - 1 b A^{n-1}b
  4. K n = [ b A b A 2 b A n - 1 b ] . K_{n}=\begin{bmatrix}b&Ab&A^{2}b&\cdots&A^{n-1}b\end{bmatrix}.
  5. 𝒦 n \mathcal{K}_{n}
  6. n n
  7. A n - 1 b A^{n-1}b
  8. 𝒦 n \mathcal{K}_{n}
  9. q k A q k - 1 q_{k}\leftarrow Aq_{k-1}\,
  10. h j , k - 1 q j * q k h_{j,k-1}\leftarrow q_{j}^{*}q_{k}\,
  11. q k q k - h j , k - 1 q j q_{k}\leftarrow q_{k}-h_{j,k-1}q_{j}\,
  12. h k , k - 1 q k h_{k,k-1}\leftarrow\|q_{k}\|\,
  13. q k q k h k , k - 1 q_{k}\leftarrow\frac{q_{k}}{h_{k,k-1}}\,
  14. q k q_{k}
  15. q 1 , , q k - 1 q_{1},\dots,q_{k-1}
  16. H n = [ h 1 , 1 h 1 , 2 h 1 , 3 h 1 , n h 2 , 1 h 2 , 2 h 2 , 3 h 2 , n 0 h 3 , 2 h 3 , 3 h 3 , n 0 0 h n , n - 1 h n , n ] . H_{n}=\begin{bmatrix}h_{1,1}&h_{1,2}&h_{1,3}&\cdots&h_{1,n}\\ h_{2,1}&h_{2,2}&h_{2,3}&\cdots&h_{2,n}\\ 0&h_{3,2}&h_{3,3}&\cdots&h_{3,n}\\ \vdots&\ddots&\ddots&\ddots&\vdots\\ 0&\cdots&0&h_{n,n-1}&h_{n,n}\end{bmatrix}.
  17. H n = Q n * A Q n . H_{n}=Q_{n}^{*}AQ_{n}.\,
  18. 𝒦 n \mathcal{K}_{n}
  19. A Q n = Q n + 1 H ~ n AQ_{n}=Q_{n+1}\tilde{H}_{n}
  20. H ~ n = [ h 1 , 1 h 1 , 2 h 1 , 3 h 1 , n h 2 , 1 h 2 , 2 h 2 , 3 h 2 , n 0 h 3 , 2 h 3 , 3 h 3 , n 0 h n , n - 1 h n , n 0 0 h n + 1 , n ] \tilde{H}_{n}=\begin{bmatrix}h_{1,1}&h_{1,2}&h_{1,3}&\cdots&h_{1,n}\\ h_{2,1}&h_{2,2}&h_{2,3}&\cdots&h_{2,n}\\ 0&h_{3,2}&h_{3,3}&\cdots&h_{3,n}\\ \vdots&\ddots&\ddots&\ddots&\vdots\\ \vdots&&0&h_{n,n-1}&h_{n,n}\\ 0&\cdots&\cdots&0&h_{n+1,n}\end{bmatrix}

Arrangement_of_hyperplanes.html

  1. p A ( y ) := B ( - 1 ) | B | y dim f ( B ) , p_{A}(y):=\sum_{B}(-1)^{|B|}y^{\dim f(B)},
  2. w A ( x , y ) := B x n - dim f ( B ) C ( - 1 ) | C - B | y dim f ( C ) , w_{A}(x,y):=\sum_{B}x^{n-\dim f(B)}\sum_{C}(-1)^{|C-B|}y^{\dim f(C)},
  3. H A K e H \bigoplus_{H\in A}Ke_{H}
  4. \partial
  5. e H 1 e H p e_{H_{1}}\wedge\cdots\wedge e_{H_{p}}
  6. H 1 , , H p H_{1},\dots,H_{p}
  7. H 1 H p H_{1}\cap\cdots\cap H_{p}
  8. 1 2 π i d α α . \frac{1}{2\pi i}\frac{d\alpha}{\alpha}.
  9. α \alpha

Arrangement_of_lines.html

  1. 9.5 n - 1 \lfloor 9.5n\rfloor-1

Arthur_Amos_Noyes.html

  1. d W d t = D A ( C s - C ) L \frac{dW}{dt}=\frac{DA(C_{s}-C)}{L}
  2. d W d t \frac{dW}{dt}
  3. C s C_{s}

Arthur–Merlin_protocol.html

  1. B P P \exists BPP
  2. z { 0 , 1 } q ( n ) Pr y { 0 , 1 } p ( n ) ( M ( x , y , z ) = 1 ) 2 / 3 , \exists z\in\{0,1\}^{q(n)}\,\Pr\nolimits_{y\in\{0,1\}^{p(n)}}(M(x,y,z)=1)\geq 2% /3,
  3. z { 0 , 1 } q ( n ) Pr y { 0 , 1 } p ( n ) ( M ( x , y , z ) = 0 ) 2 / 3. \forall z\in\{0,1\}^{q(n)}\,\Pr\nolimits_{y\in\{0,1\}^{p(n)}}(M(x,y,z)=0)\geq 2% /3.
  4. z { 0 , 1 } q ( n ) Pr y { 0 , 1 } p ( n ) ( M ( x , y , z ) = 1 ) 1 / 3. \forall z\in\{0,1\}^{q(n)}\,\Pr\nolimits_{y\in\{0,1\}^{p(n)}}(M(x,y,z)=1)\leq 1% /3.
  5. Pr y { 0 , 1 } p ( n ) ( z { 0 , 1 } q ( n ) M ( x , y , z ) = 1 ) 2 / 3 , \Pr\nolimits_{y\in\{0,1\}^{p(n)}}(\exists z\in\{0,1\}^{q(n)}\,M(x,y,z)=1)\geq 2% /3,
  6. Pr y { 0 , 1 } p ( n ) ( z { 0 , 1 } q ( n ) M ( x , y , z ) = 0 ) 2 / 3. \Pr\nolimits_{y\in\{0,1\}^{p(n)}}(\forall z\in\{0,1\}^{q(n)}\,M(x,y,z)=0)\geq 2% /3.
  7. Pr y { 0 , 1 } p ( n ) ( z { 0 , 1 } q ( n ) M ( x , y , z ) = 1 ) 1 / 3. \Pr\nolimits_{y\in\{0,1\}^{p(n)}}(\exists z\in\{0,1\}^{q(n)}\,M(x,y,z)=1)\leq 1% /3.
  8. . B P P \exists.BPP
  9. . B P P \exists.BPP
  10. f i f_{i}

Artificial_chemistry.html

  1. \subset

Artificial_gravity.html

  1. g = R × ( π × rpm 30 ) 2 9.81 g=\frac{R\times(\frac{\pi\times\mathrm{rpm}}{30})^{2}}{9.81}
  2. R = 9.81 g ( π × rpm 30 ) 2 R=\frac{9.81g}{(\frac{\pi\times\mathrm{rpm}}{30})^{2}}
  3. π \pi\approx

Ashtekar_variables.html

  1. q a b ( x ) q_{ab}(x)
  2. K a b ( x ) K^{ab}(x)
  3. E i a E^{a}_{i}
  4. i = 1 , 2 , 3 i=1,2,3
  5. δ i j = q a b E i a E j b \delta_{ij}=q_{ab}E_{i}^{a}E_{j}^{b}
  6. E i a E_{i}^{a}
  7. a , b , c a,b,c
  8. i , j , k i,j,k
  9. δ i j \delta_{ij}
  10. E a i E^{i}_{a}
  11. E a i = q a b E i b E^{i}_{a}=q_{ab}E^{b}_{i}
  12. δ i j = q a b E a i E b j \delta^{ij}=q^{ab}E^{i}_{a}E^{j}_{b}
  13. q a b q^{ab}
  14. q a b q_{ab}
  15. q a b E a i E b j q^{ab}E^{i}_{a}E^{j}_{b}
  16. E i a E b i = δ b a E_{i}^{a}E^{i}_{b}=\delta_{b}^{a}
  17. δ i j = q a b E j b E i a \delta_{ij}=q_{ab}E_{j}^{b}E_{i}^{a}
  18. E c i E^{i}_{c}
  19. E a j E_{a}^{j}
  20. E i a E b i = δ b a E_{i}^{a}E^{i}_{b}=\delta_{b}^{a}
  21. q a b = i = 1 3 δ i j E i a E j b = i = 1 3 E i a E i b , q^{ab}=\sum_{i=1}^{3}\delta_{ij}E_{i}^{a}E_{j}^{b}=\sum_{i=1}^{3}E_{i}^{a}E_{i% }^{b},
  22. q a b q^{ab}
  23. E i a E_{i}^{a}
  24. ( det ( q ) ) q a b = i = 1 3 E ~ i a E ~ i b , (\mathrm{det}(q))q^{ab}=\sum_{i=1}^{3}\tilde{E}_{i}^{a}\tilde{E}_{i}^{b},
  25. E ~ i a \tilde{E}_{i}^{a}
  26. E ~ i a = d e t ( q ) E i a \tilde{E}_{i}^{a}=\sqrt{det(q)}E_{i}^{a}
  27. E ~ i a \tilde{E}_{i}^{a}
  28. E ~ i a \tilde{E}_{i}^{a}
  29. E i a E_{i}^{a}
  30. E ~ i a \tilde{E}_{i}^{a}
  31. i i
  32. S U ( 2 ) SU(2)
  33. V i b V_{i}^{b}
  34. D a V i b = a V i b - Γ a i j V j b + Γ a c b V i c D_{a}V_{i}^{b}=\partial_{a}V_{i}^{b}-\Gamma_{a\;\;i}^{\;\;j}V_{j}^{b}+\Gamma^{% b}_{ac}V_{i}^{c}
  35. Γ a c b \Gamma^{b}_{ac}
  36. Γ a i j \Gamma_{a\;\;i}^{\;\;j}
  37. A a i = Γ a i + β K a i A_{a}^{i}=\Gamma_{a}^{i}+\beta K_{a}^{i}
  38. Γ a i = Γ a j k ϵ j k i \Gamma_{a}^{i}=\Gamma_{ajk}\epsilon^{jki}
  39. K a i = K a b E ~ b i / d e t ( q ) K_{a}^{i}=K_{ab}\tilde{E}^{bi}/\sqrt{det(q)}
  40. A b j A^{j}_{b}
  41. { E ~ i a ( x ) , A b j ( y ) } = 8 π G Newton β δ b a δ i j δ 3 ( x - y ) \{\tilde{E}_{i}^{a}(x),A^{j}_{b}(y)\}=8\pi G_{\mathrm{Newton}}\beta\delta^{a}_% {b}\delta^{j}_{i}\delta^{3}(x-y)
  42. β \beta
  43. G Newton G_{\mathrm{Newton}}
  44. β = - i \beta=-i
  45. A a i A_{a}^{i}
  46. H ~ = d e t ( q ) H \tilde{H}=\sqrt{det(q)}H
  47. β \beta

Associated_Legendre_polynomials.html

  1. ( 1 - x 2 ) d 2 d x 2 P m ( x ) - 2 x d d x P m ( x ) + [ ( + 1 ) - m 2 1 - x 2 ] P m ( x ) = 0 (1-x^{2})\frac{d^{2}}{dx^{2}}P_{\ell}^{m}(x)-2x\frac{d}{dx}P_{\ell}^{m}(x)+% \left[\ell(\ell+1)-\frac{m^{2}}{1-x^{2}}\right]P_{\ell}^{m}(x)=0
  2. d d x [ ( 1 - x 2 ) d d x P m ( x ) ] + [ ( + 1 ) - m 2 1 - x 2 ] P m ( x ) = 0 \frac{d}{dx}\left[(1-x^{2})\frac{d}{dx}P_{\ell}^{m}(x)\right]+\left[\ell(\ell+% 1)-\frac{m^{2}}{1-x^{2}}\right]P_{\ell}^{m}(x)=0
  3. P m ( x ) P_{\ell}^{m}(x)
  4. P m ( x ) = ( - 1 ) m ( 1 - x 2 ) m / 2 d m d x m ( P ( x ) ) P_{\ell}^{m}(x)=(-1)^{m}\ (1-x^{2})^{m/2}\ \frac{d^{m}}{dx^{m}}\left(P_{\ell}(% x)\right)\,
  5. ( 1 - x 2 ) d 2 d x 2 P ( x ) - 2 x d d x P ( x ) + ( + 1 ) P ( x ) = 0. (1-x^{2})\frac{d^{2}}{dx^{2}}P_{\ell}(x)-2x\frac{d}{dx}P_{\ell}(x)+\ell(\ell+1% )P_{\ell}(x)=0.
  6. P ( x ) = 1 2 ! d d x [ ( x 2 - 1 ) ] , P_{\ell}(x)=\frac{1}{2^{\ell}\,\ell!}\ \frac{d^{\ell}}{dx^{\ell}}\left[(x^{2}-% 1)^{\ell}\right],
  7. P m ( x ) = ( - 1 ) m 2 ! ( 1 - x 2 ) m / 2 d + m d x + m ( x 2 - 1 ) . P_{\ell}^{m}(x)=\frac{(-1)^{m}}{2^{\ell}\ell!}(1-x^{2})^{m/2}\ \frac{d^{\ell+m% }}{dx^{\ell+m}}(x^{2}-1)^{\ell}.
  8. d - m d x - m ( x 2 - 1 ) = c l m ( 1 - x 2 ) m d + m d x + m ( x 2 - 1 ) , \frac{d^{\ell-m}}{dx^{\ell-m}}(x^{2}-1)^{\ell}=c_{lm}(1-x^{2})^{m}\frac{d^{% \ell+m}}{dx^{\ell+m}}(x^{2}-1)^{\ell},
  9. c l m = ( - 1 ) m ( - m ) ! ( + m ) ! , c_{lm}=(-1)^{m}\frac{(\ell-m)!}{(\ell+m)!},
  10. P - m ( x ) = ( - 1 ) m ( - m ) ! ( + m ) ! P m ( x ) . P^{-m}_{\ell}(x)=(-1)^{m}\frac{(\ell-m)!}{(\ell+m)!}P^{m}_{\ell}(x).
  11. P - m ( x ) = ( - 1 ) m P m ( x ) P_{\ell-m}(x)=(-1)^{m}P_{\ell}^{m}(x)
  12. - 1 1 P k m P m d x = 2 ( + m ) ! ( 2 + 1 ) ( - m ) ! δ k , \int_{-1}^{1}P_{k}^{m}P_{\ell}^{m}dx=\frac{2(\ell+m)!}{(2\ell+1)(\ell-m)!}\ % \delta_{k,\ell}
  13. - 1 1 P m P n 1 - x 2 d x = { 0 if m n ( + m ) ! m ( - m ) ! if m = n 0 if m = n = 0 \int_{-1}^{1}\frac{P_{\ell}^{m}P_{\ell}^{n}}{1-x^{2}}dx=\begin{cases}0&\mbox{% if }~{}m\neq n\\ \frac{(\ell+m)!}{m(\ell-m)!}&\mbox{if }~{}m=n\neq 0\\ \infty&\mbox{if }~{}m=n=0\end{cases}
  14. P - m = ( - 1 ) m ( - m ) ! ( + m ) ! P m P_{\ell}^{-m}=(-1)^{m}\frac{(\ell-m)!}{(\ell+m)!}P_{\ell}^{m}
  15. If m > then P m = 0. \textrm{If}\quad{\mid}m{\mid}>\ell\,\quad\mathrm{then}\quad P_{\ell}^{m}=0.\,
  16. P - m = P - 1 m , ( = 1 , 2 , ) P_{-\ell}^{m}=P_{\ell-1}^{m},\ (\ell=1,\,2,\,...)
  17. P 0 0 ( x ) = 1 P_{0}^{0}(x)=1
  18. P 1 - 1 ( x ) = - 1 2 P 1 1 ( x ) P_{1}^{-1}(x)=-\begin{matrix}\frac{1}{2}\end{matrix}P_{1}^{1}(x)
  19. P 1 0 ( x ) = x P_{1}^{0}(x)=x
  20. P 1 1 ( x ) = - ( 1 - x 2 ) 1 / 2 P_{1}^{1}(x)=-(1-x^{2})^{1/2}
  21. P 2 - 2 ( x ) = 1 24 P 2 2 ( x ) P_{2}^{-2}(x)=\begin{matrix}\frac{1}{24}\end{matrix}P_{2}^{2}(x)
  22. P 2 - 1 ( x ) = - 1 6 P 2 1 ( x ) P_{2}^{-1}(x)=-\begin{matrix}\frac{1}{6}\end{matrix}P_{2}^{1}(x)
  23. P 2 0 ( x ) = 1 2 ( 3 x 2 - 1 ) P_{2}^{0}(x)=\begin{matrix}\frac{1}{2}\end{matrix}(3x^{2}-1)
  24. P 2 1 ( x ) = - 3 x ( 1 - x 2 ) 1 / 2 P_{2}^{1}(x)=-3x(1-x^{2})^{1/2}
  25. P 2 2 ( x ) = 3 ( 1 - x 2 ) P_{2}^{2}(x)=3(1-x^{2})
  26. P 3 - 3 ( x ) = - 1 720 P 3 3 ( x ) P_{3}^{-3}(x)=-\begin{matrix}\frac{1}{720}\end{matrix}P_{3}^{3}(x)
  27. P 3 - 2 ( x ) = 1 120 P 3 2 ( x ) P_{3}^{-2}(x)=\begin{matrix}\frac{1}{120}\end{matrix}P_{3}^{2}(x)
  28. P 3 - 1 ( x ) = - 1 12 P 3 1 ( x ) P_{3}^{-1}(x)=-\begin{matrix}\frac{1}{12}\end{matrix}P_{3}^{1}(x)
  29. P 3 0 ( x ) = 1 2 ( 5 x 3 - 3 x ) P_{3}^{0}(x)=\begin{matrix}\frac{1}{2}\end{matrix}(5x^{3}-3x)
  30. P 3 1 ( x ) = - 3 2 ( 5 x 2 - 1 ) ( 1 - x 2 ) 1 / 2 P_{3}^{1}(x)=-\begin{matrix}\frac{3}{2}\end{matrix}(5x^{2}-1)(1-x^{2})^{1/2}
  31. P 3 2 ( x ) = 15 x ( 1 - x 2 ) P_{3}^{2}(x)=15x(1-x^{2})
  32. P 3 3 ( x ) = - 15 ( 1 - x 2 ) 3 / 2 P_{3}^{3}(x)=-15(1-x^{2})^{3/2}
  33. P 4 - 4 ( x ) = 1 40320 P 4 4 ( x ) P_{4}^{-4}(x)=\begin{matrix}\frac{1}{40320}\end{matrix}P_{4}^{4}(x)
  34. P 4 - 3 ( x ) = - 1 5040 P 4 3 ( x ) P_{4}^{-3}(x)=-\begin{matrix}\frac{1}{5040}\end{matrix}P_{4}^{3}(x)
  35. P 4 - 2 ( x ) = 1 360 P 4 2 ( x ) P_{4}^{-2}(x)=\begin{matrix}\frac{1}{360}\end{matrix}P_{4}^{2}(x)
  36. P 4 - 1 ( x ) = - 1 20 P 4 1 ( x ) P_{4}^{-1}(x)=-\begin{matrix}\frac{1}{20}\end{matrix}P_{4}^{1}(x)
  37. P 4 0 ( x ) = 1 8 ( 35 x 4 - 30 x 2 + 3 ) P_{4}^{0}(x)=\begin{matrix}\frac{1}{8}\end{matrix}(35x^{4}-30x^{2}+3)
  38. P 4 1 ( x ) = - 5 2 ( 7 x 3 - 3 x ) ( 1 - x 2 ) 1 / 2 P_{4}^{1}(x)=-\begin{matrix}\frac{5}{2}\end{matrix}(7x^{3}-3x)(1-x^{2})^{1/2}
  39. P 4 2 ( x ) = 15 2 ( 7 x 2 - 1 ) ( 1 - x 2 ) P_{4}^{2}(x)=\begin{matrix}\frac{15}{2}\end{matrix}(7x^{2}-1)(1-x^{2})
  40. P 4 3 ( x ) = - 105 x ( 1 - x 2 ) 3 / 2 P_{4}^{3}(x)=-105x(1-x^{2})^{3/2}
  41. P 4 4 ( x ) = 105 ( 1 - x 2 ) 2 P_{4}^{4}(x)=105(1-x^{2})^{2}
  42. ( - m + 1 ) P + 1 m ( x ) = ( 2 + 1 ) x P m ( x ) - ( + m ) P - 1 m ( x ) (\ell-m+1)P_{\ell+1}^{m}(x)=(2\ell+1)xP_{\ell}^{m}(x)-(\ell+m)P_{\ell-1}^{m}(x)
  43. 2 m x P m ( x ) = - 1 - x 2 [ P m + 1 ( x ) + ( + m ) ( - m + 1 ) P m - 1 ( x ) ] 2mxP_{\ell}^{m}(x)=-\sqrt{1-x^{2}}\left[P_{\ell}^{m+1}(x)+(\ell+m)(\ell-m+1)P_% {\ell}^{m-1}(x)\right]
  44. 1 1 - x 2 P m ( x ) = - 1 2 m [ P - 1 m + 1 ( x ) + ( + m - 1 ) ( + m ) P - 1 m - 1 ( x ) ] \frac{1}{\sqrt{1-x^{2}}}P_{\ell}^{m}(x)=\frac{-1}{2m}\left[P_{\ell-1}^{m+1}(x)% +(\ell+m-1)(\ell+m)P_{\ell-1}^{m-1}(x)\right]
  45. 1 1 - x 2 P m ( x ) = - 1 2 m [ P + 1 m + 1 ( x ) + ( - m + 1 ) ( - m + 2 ) P + 1 m - 1 ( x ) ] \frac{1}{\sqrt{1-x^{2}}}P_{\ell}^{m}(x)=\frac{-1}{2m}\left[P_{\ell+1}^{m+1}(x)% +(\ell-m+1)(\ell-m+2)P_{\ell+1}^{m-1}(x)\right]
  46. 1 - x 2 P m ( x ) = 1 2 + 1 [ ( - m + 1 ) ( - m + 2 ) P + 1 m - 1 ( x ) - ( + m - 1 ) ( + m ) P - 1 m - 1 ( x ) ] \sqrt{1-x^{2}}P_{\ell}^{m}(x)=\frac{1}{2\ell+1}\left[(\ell-m+1)(\ell-m+2)P_{% \ell+1}^{m-1}(x)-(\ell+m-1)(\ell+m)P_{\ell-1}^{m-1}(x)\right]
  47. 1 - x 2 P m ( x ) = 1 2 + 1 [ - P + 1 m + 1 ( x ) + P - 1 m + 1 ( x ) ] \sqrt{1-x^{2}}P_{\ell}^{m}(x)=\frac{1}{2\ell+1}\left[-P_{\ell+1}^{m+1}(x)+P_{% \ell-1}^{m+1}(x)\right]
  48. 1 - x 2 P m + 1 ( x ) = ( - m ) x P m ( x ) - ( + m ) P - 1 m ( x ) \sqrt{1-x^{2}}P_{\ell}^{m+1}(x)=(\ell-m)xP_{\ell}^{m}(x)-(\ell+m)P_{\ell-1}^{m% }(x)
  49. 1 - x 2 P m + 1 ( x ) = ( - m + 1 ) P + 1 m ( x ) - ( + m + 1 ) x P m ( x ) \sqrt{1-x^{2}}P_{\ell}^{m+1}(x)=(\ell-m+1)P_{\ell+1}^{m}(x)-(\ell+m+1)xP_{\ell% }^{m}(x)
  50. 1 - x 2 d d x P m ( x ) = 1 2 [ ( + m ) ( - m + 1 ) P m - 1 ( x ) - P m + 1 ( x ) ] \sqrt{1-x^{2}}\frac{d}{dx}{P_{\ell}^{m}}(x)=\frac{1}{2}\left[(\ell+m)(\ell-m+1% )P_{\ell}^{m-1}(x)-P_{\ell}^{m+1}(x)\right]
  51. ( 1 - x 2 ) d d x P m ( x ) = 1 2 + 1 [ ( + 1 ) ( + m ) P - 1 m ( x ) - ( - m + 1 ) P + 1 m ( x ) ] (1-x^{2})\frac{d}{dx}{P_{\ell}^{m}}(x)=\frac{1}{2\ell+1}\left[(\ell+1)(\ell+m)% P_{\ell-1}^{m}(x)-\ell(\ell-m+1)P_{\ell+1}^{m}(x)\right]
  52. ( x 2 - 1 ) d d x P m ( x ) = x P m ( x ) - ( + m ) P - 1 m ( x ) (x^{2}-1)\frac{d}{dx}{P_{\ell}^{m}}(x)={\ell}xP_{\ell}^{m}(x)-(\ell+m)P_{\ell-% 1}^{m}(x)
  53. ( x 2 - 1 ) d d x P m ( x ) = 1 - x 2 P m + 1 ( x ) + m x P m ( x ) (x^{2}-1)\frac{d}{dx}{P_{\ell}^{m}}(x)=\sqrt{1-x^{2}}P_{\ell}^{m+1}(x)+mxP_{% \ell}^{m}(x)
  54. ( x 2 - 1 ) d d x P m ( x ) = - ( + m ) ( - m + 1 ) 1 - x 2 P m - 1 ( x ) - m x P m ( x ) (x^{2}-1)\frac{d}{dx}{P_{\ell}^{m}}(x)=-(\ell+m)(\ell-m+1)\sqrt{1-x^{2}}P_{% \ell}^{m-1}(x)-mxP_{\ell}^{m}(x)
  55. P + 1 + 1 ( x ) = - ( 2 + 1 ) 1 - x 2 P ( x ) P_{\ell+1}^{\ell+1}(x)=-(2\ell+1)\sqrt{1-x^{2}}P_{\ell}^{\ell}(x)
  56. P ( x ) = ( - 1 ) l ( 2 - 1 ) ! ! ( 1 - x 2 ) ( l / 2 ) P_{\ell}^{\ell}(x)=(-1)^{l}(2\ell-1)!!(1-x^{2})^{(l/2)}
  57. P + 1 ( x ) = x ( 2 + 1 ) P ( x ) P_{\ell+1}^{\ell}(x)=x(2\ell+1)P_{\ell}^{\ell}(x)
  58. 1 2 - 1 1 P l u ( x ) P m v ( x ) P n w ( x ) d x = \frac{1}{2}\int_{-1}^{1}P_{l}^{u}(x)P_{m}^{v}(x)P_{n}^{w}(x)dx=
  59. ( - 1 ) s - m - w ( m + v ) ! ( n + w ) ! ( 2 s - 2 n ) ! s ! ( m - v ) ! ( s - l ) ! ( s - m ) ! ( s - n ) ! ( 2 s + 1 ) ! (-1)^{s-m-w}\frac{(m+v)!(n+w)!(2s-2n)!s!}{(m-v)!(s-l)!(s-m)!(s-n)!(2s+1)!}
  60. × t = p q ( - 1 ) t ( l + u + t ) ! ( m + n - u - t ) ! t ! ( l - u - t ) ! ( m - n + u + t ) ! ( n - w - t ) ! \times\ \sum_{t=p}^{q}(-1)^{t}\frac{(l+u+t)!(m+n-u-t)!}{t!(l-u-t)!(m-n+u+t)!(n% -w-t)!}
  61. l , m , n 0 l,m,n\geq 0
  62. u , v , w 0 u,v,w\geq 0
  63. u u
  64. u = v + w u=v+w
  65. m n m\geq n
  66. 2 s = l + m + n \ 2s=l+m+n
  67. p = max ( 0 , n - m - u ) \ p=\max(0,\,n-m-u)
  68. q = min ( m + n - u , l - u , n - w ) \ q=\min(m+n-u,\,l-u,\,n-w)
  69. s s
  70. m + n l m - n m+n\geq l\geq m-n
  71. P λ μ ( z ) = 1 Γ ( 1 - μ ) [ 1 + z 1 - z ] 2 μ / 2 F 1 ( - λ , λ + 1 ; 1 - μ ; 1 - z 2 ) P_{\lambda}^{\mu}(z)=\frac{1}{\Gamma(1-\mu)}\left[\frac{1+z}{1-z}\right]^{\mu/% 2}\,_{2}F_{1}(-\lambda,\lambda+1;1-\mu;\frac{1-z}{2})
  72. Γ \Gamma
  73. F 1 2 {}_{2}F_{1}
  74. F 1 2 ( α , β ; γ ; z ) = Γ ( γ ) Γ ( α ) Γ ( β ) n = 0 Γ ( n + α ) Γ ( n + β ) Γ ( n + γ ) n ! z n , \,{}_{2}F_{1}(\alpha,\beta;\gamma;z)=\frac{\Gamma(\gamma)}{\Gamma(\alpha)% \Gamma(\beta)}\sum_{n=0}^{\infty}\frac{\Gamma(n+\alpha)\Gamma(n+\beta)}{\Gamma% (n+\gamma)\ n!}z^{n},
  75. ( 1 - z 2 ) y ′′ - 2 z y + ( λ [ λ + 1 ] - μ 2 1 - z 2 ) y = 0. (1-z^{2})\,y^{\prime\prime}-2zy^{\prime}+\left(\lambda[\lambda+1]-\frac{\mu^{2% }}{1-z^{2}}\right)\,y=0.\,
  76. Q λ μ ( z ) Q_{\lambda}^{\mu}(z)
  77. Q λ μ ( z ) = π Γ ( λ + μ + 1 ) 2 λ + 1 Γ ( λ + 3 / 2 ) 1 z λ + μ + 1 ( 1 - z 2 ) 2 μ / 2 F 1 ( λ + μ + 1 2 , λ + μ + 2 2 ; λ + 3 2 ; 1 z 2 ) Q_{\lambda}^{\mu}(z)=\frac{\sqrt{\pi}\ \Gamma(\lambda+\mu+1)}{2^{\lambda+1}% \Gamma(\lambda+3/2)}\frac{1}{z^{\lambda+\mu+1}}(1-z^{2})^{\mu/2}\,_{2}F_{1}% \left(\frac{\lambda+\mu+1}{2},\frac{\lambda+\mu+2}{2};\lambda+\frac{3}{2};% \frac{1}{z^{2}}\right)
  78. P λ μ ( z ) P_{\lambda}^{\mu}(z)
  79. Q λ μ ( z ) Q_{\lambda}^{\mu}(z)
  80. x = cos θ x=\cos\theta
  81. P m ( cos θ ) = ( - 1 ) m ( sin θ ) m d m d ( cos θ ) m ( P ( cos θ ) ) P_{\ell}^{m}(\cos\theta)=(-1)^{m}(\sin\theta)^{m}\ \frac{d^{m}}{d(\cos\theta)^% {m}}\left(P_{\ell}(\cos\theta)\right)\,
  82. P 0 0 ( cos θ ) \displaystyle P_{0}^{0}(\cos\theta)
  83. P m ( cos θ ) P_{\ell}^{m}(\cos\theta)
  84. [ 0 , π ] [0,\pi]
  85. sin θ \sin\theta
  86. 0 π P k m ( cos θ ) P m ( cos θ ) sin θ d θ = 2 ( + m ) ! ( 2 + 1 ) ( - m ) ! δ k , \int_{0}^{\pi}P_{k}^{m}(\cos\theta)P_{\ell}^{m}(\cos\theta)\,\sin\theta\,d% \theta=\frac{2(\ell+m)!}{(2\ell+1)(\ell-m)!}\ \delta_{k,\ell}
  87. 0 π P m ( cos θ ) P n ( cos θ ) csc θ d θ = { 0 if m n ( + m ) ! m ( - m ) ! if m = n 0 if m = n = 0 \int_{0}^{\pi}P_{\ell}^{m}(\cos\theta)P_{\ell}^{n}(\cos\theta)\csc\theta\,d% \theta=\begin{cases}0&\,\text{if }m\neq n\\ \frac{(\ell+m)!}{m(\ell-m)!}&\,\text{if }m=n\neq 0\\ \infty&\,\text{if }m=n=0\end{cases}
  88. P m ( cos θ ) P_{\ell}^{m}(\cos\theta)
  89. d 2 y d θ 2 + cot θ d y d θ + [ λ - m 2 sin 2 θ ] y = 0 \frac{d^{2}y}{d\theta^{2}}+\cot\theta\frac{dy}{d\theta}+\left[\lambda-\frac{m^% {2}}{\sin^{2}\theta}\right]\,y=0\,
  90. \geq
  91. λ = ( + 1 ) \lambda=\ell(\ell+1)\,
  92. P m ( cos θ ) P_{\ell}^{m}(\cos\theta)
  93. θ \theta
  94. ϕ \phi
  95. 2 ψ + λ ψ = 0 \nabla^{2}\psi+\lambda\psi=0
  96. 2 ψ = 2 ψ θ 2 + cot θ ψ θ + csc 2 θ 2 ψ ϕ 2 . \nabla^{2}\psi=\frac{\partial^{2}\psi}{\partial\theta^{2}}+\cot\theta\frac{% \partial\psi}{\partial\theta}+\csc^{2}\theta\frac{\partial^{2}\psi}{\partial% \phi^{2}}.
  97. 2 ψ θ 2 + cot θ ψ θ + csc 2 θ 2 ψ ϕ 2 + λ ψ = 0 \frac{\partial^{2}\psi}{\partial\theta^{2}}+\cot\theta\frac{\partial\psi}{% \partial\theta}+\csc^{2}\theta\frac{\partial^{2}\psi}{\partial\phi^{2}}+% \lambda\psi=0
  98. sin ( m ϕ ) \sin(m\phi)
  99. cos ( m ϕ ) \cos(m\phi)
  100. d 2 y d θ 2 + cot θ d y d θ + [ λ - m 2 sin 2 θ ] y = 0 \frac{d^{2}y}{d\theta^{2}}+\cot\theta\frac{dy}{d\theta}+\left[\lambda-\frac{m^% {2}}{\sin^{2}\theta}\right]\,y=0\,
  101. P m ( cos θ ) P_{\ell}^{m}(\cos\theta)
  102. m \ell{\geq}m
  103. λ = ( + 1 ) \lambda=\ell(\ell+1)
  104. 2 ψ + λ ψ = 0 \nabla^{2}\psi+\lambda\psi=0
  105. λ = ( + 1 ) \lambda=\ell(\ell+1)
  106. P m ( cos θ ) cos ( m ϕ ) 0 m P_{\ell}^{m}(\cos\theta)\ \cos(m\phi)\ \ \ \ 0\leq m\leq\ell
  107. P m ( cos θ ) sin ( m ϕ ) 0 < m . P_{\ell}^{m}(\cos\theta)\ \sin(m\phi)\ \ \ \ 0<m\leq\ell.
  108. Y , m ( θ , ϕ ) = ( 2 + 1 ) ( - m ) ! 4 π ( + m ) ! P m ( cos θ ) e i m ϕ - m . Y_{\ell,m}(\theta,\phi)=\sqrt{\frac{(2\ell+1)(\ell-m)!}{4\pi(\ell+m)!}}\ P_{% \ell}^{m}(\cos\theta)\ e^{im\phi}\qquad-\ell\leq m\leq\ell.
  109. Y , m ( θ , ϕ ) Y_{\ell,m}(\theta,\phi)
  110. Y , m * ( θ , ϕ ) = ( - 1 ) m Y , - m ( θ , ϕ ) . Y_{\ell,m}^{*}(\theta,\phi)=(-1)^{m}Y_{\ell,-m}(\theta,\phi).
  111. 2 ψ ( θ , ϕ ) + λ ψ ( θ , ϕ ) = 0 , \nabla^{2}\psi(\theta,\phi)+\lambda\psi(\theta,\phi)=0,

Associator.html

  1. R R
  2. [ , , ] : R × R × R R [\cdot,\cdot,\cdot]:R\times R\times R\to R
  3. [ x , y , z ] = ( x y ) z - x ( y z ) . [x,y,z]=(xy)z-x(yz).\,
  4. R R
  5. w [ x , y , z ] + [ w , x , y ] z = [ w x , y , z ] - [ w , x y , z ] + [ w , x , y z ] . w[x,y,z]+[w,x,y]z=[wx,y,z]-[w,xy,z]+[w,x,yz].\,
  6. R R
  7. R R
  8. [ n , R , R ] = [ R , n , R ] = [ R , R , n ] = { 0 } . [n,R,R]=[R,n,R]=[R,R,n]=\{0\}\ .
  9. ( [ n , R , R ] , [ R , n , R ] , [ R , R , n ] ) ([n,R,R],[R,n,R],[R,R,n])
  10. { 0 } \{0\}
  11. : Q × Q Q \cdot:Q\times Q\to Q
  12. a x = b a\cdot x=b
  13. y a = b y\cdot a=b
  14. ( , , ) : Q × Q × Q Q (\cdot,\cdot,\cdot):Q\times Q\times Q\to Q
  15. ( a b ) c = ( a ( b c ) ) ( a , b , c ) (a\cdot b)\cdot c=(a\cdot(b\cdot c))\cdot(a,b,c)
  16. a x , y , z : ( x y ) z x ( y z ) . a_{x,y,z}:(xy)z\mapsto x(yz).

Astroid.html

  1. x 2 / 3 + y 2 / 3 = a 2 / 3 . x^{2/3}+y^{2/3}=a^{2/3}.\,
  2. x = a cos 3 t = a 4 ( 3 cos t + cos 3 t ) , x=a\cos^{3}t={a\over 4}(3\cos t+\cos 3t),
  3. y = a sin 3 t = a 4 ( 3 sin t - sin 3 t ) . y=a\sin^{3}t={a\over 4}(3\sin t-\sin 3t).
  4. r 2 = a 2 - 3 p 2 , r^{2}=a^{2}-3p^{2},
  5. s = 3 a 4 cos 2 φ , s={3a\over 4}\cos 2\varphi,
  6. R 2 + 4 s 2 = 9 a 2 4 . R^{2}+4s^{2}=\frac{9a^{2}}{4}.
  7. r = a ( cos 2 / 3 θ + sin 2 / 3 θ ) 3 / 2 . r=\frac{a}{(\cos^{2/3}\theta+\sin^{2/3}\theta)^{3/2}}.
  8. ( x 2 + y 2 - a 2 ) 3 + 27 a 2 x 2 y 2 = 0. (x^{2}+y^{2}-a^{2})^{3}+27a^{2}x^{2}y^{2}=0.\,
  9. 3 8 π a 2 \frac{3}{8}\pi a^{2}
  10. 6 a 6a
  11. 32 105 π a 3 \frac{32}{105}\pi a^{3}
  12. 12 5 π a 2 \frac{12}{5}\pi a^{2}
  13. x 2 y 2 = x 2 + y 2 . \textstyle x^{2}y^{2}=x^{2}+y^{2}.

Asymptotic_analysis.html

  1. n n
  2. n n
  3. n n
  4. n n
  5. n n
  6. f f
  7. g g
  8. n n
  9. f g ( as n ) f\sim g\quad(\,\text{as }n\to\infty)
  10. lim n f ( n ) g ( n ) = 1 . \lim_{n\to\infty}\frac{f(n)}{g(n)}=1~{}.
  11. n n
  12. f f
  13. g g
  14. f f
  15. f ( x ) f(x)
  16. f f
  17. f f
  18. f g 1 f\sim g_{1}
  19. f g 1 + g 2 f\sim g_{1}+g_{2}
  20. f g 1 + + g k f\sim g_{1}+\cdots+g_{k}
  21. f - ( g 1 + + g k ) = o ( g k ) . f-(g_{1}+\cdots+g_{k})=o(g_{k})~{}.
  22. ε ε
  23. y ( x ) e S ( x ) . y(x)\sim e^{S(x)}~{}.
  24. c c
  25. a a
  26. x y ′′ + ( c - x ) y - a y = 0 . xy^{\prime\prime}+(c-x)y^{\prime}-ay=0~{}.
  27. x x
  28. y e x y\sim e^{x}\,
  29. y y
  30. x x
  31. y y
  32. x S ′′ + x S 2 + ( c - x ) S - a = 0 xS^{\prime\prime}+xS^{\prime 2}+(c-x)S^{\prime}-a=0\,
  33. S ′′ + S 2 + ( c x - 1 ) S - a x = 0 S^{\prime\prime}+S^{\prime 2}+\left(\frac{c}{x}-1\right)S^{\prime}-\frac{a}{x}% =0\,
  34. y y
  35. S 2 S , S^{\prime 2}\sim S^{\prime}~{},
  36. S ′′ , c x S , a x = o ( S 2 ) , o ( S ) S^{\prime\prime},~{}\frac{c}{x}S^{\prime},~{}\frac{a}{x}=o(S^{\prime 2}),~{}o(% S^{\prime})\,
  37. S 0 2 = S 0 . S_{0}^{\prime 2}=S_{0}^{\prime}~{}.
  38. S 0 S_{0}
  39. S 0 S_{0}
  40. S S
  41. S 0 S_{0}
  42. S 0 = 1 S_{0}^{\prime}=1\,
  43. S 0 2 = 1 S_{0}^{\prime 2}=1\,
  44. S 0 ′′ = 0 = o ( S 0 ) S_{0}^{\prime\prime}=0=o(S_{0}^{\prime})\,
  45. c x S 0 = c x = o ( S 0 ) \frac{c}{x}S_{0}^{\prime}=\frac{c}{x}=o(S_{0}^{\prime})\,
  46. a x = o ( S 0 ) \frac{a}{x}=o(S_{0}^{\prime})\,
  47. S 0 = x S_{0}=x\,
  48. y e x . y\sim e^{x}~{}.
  49. y A x p e λ x r ( 1 + u 1 x + u 2 x 2 ) , y\sim Ax^{p}e^{\lambda x^{r}}\left(1+\frac{u_{1}}{x}+\frac{u_{2}}{x^{2}}\cdots% \right)~{},
  50. x x
  51. S ( x ) S 0 ( x ) + C ( x ) S(x)\equiv S_{0}(x)+C(x)\,
  52. C ′′ + C 2 + C + c x C + c - a x = 0 . C^{\prime\prime}+C^{\prime 2}+C^{\prime}+\frac{c}{x}C^{\prime}+\frac{c-a}{x}=0% ~{}.
  53. C 0 = log x a - c . C_{0}=\log x^{a-c}~{}.
  54. y x a - c e x . y\sim x^{a-c}e^{x}~{}.

Asymptotic_expansion.html

  1. exp ( - c / ϵ ) \sim\exp\left(-c/\epsilon\right)
  2. φ n + 1 ( x ) = o ( φ n ( x ) ) ( x L ) \varphi_{n+1}(x)=o(\varphi_{n}(x))\ (x\rightarrow L)
  3. x L x\rightarrow L
  4. n = 0 N a n φ n ( x ) \sum_{n=0}^{N}a_{n}\varphi_{n}(x)
  5. f ( x ) - n = 0 N - 1 a n φ n ( x ) = O ( φ N ( x ) ) ( x L ) f(x)-\sum_{n=0}^{N-1}a_{n}\varphi_{n}(x)=O(\varphi_{N}(x))\ (x\rightarrow L)
  6. f ( x ) - n = 0 N - 1 a n φ n ( x ) = o ( φ N - 1 ( x ) ) ( x L ) . f(x)-\sum_{n=0}^{N-1}a_{n}\varphi_{n}(x)=o(\varphi_{N-1}(x))\ (x\rightarrow L).
  7. f ( x ) n = 0 a n φ n ( x ) ( x L ) . f(x)\sim\sum_{n=0}^{\infty}a_{n}\varphi_{n}(x)\ (x\rightarrow L).
  8. f f
  9. x x
  10. N N\rightarrow\infty
  11. N N
  12. x L x\rightarrow L
  13. L L
  14. e x x x 2 π x Γ ( x + 1 ) 1 + 1 12 x + 1 288 x 2 - 139 51840 x 3 - ( x ) \frac{e^{x}}{x^{x}\sqrt{2\pi x}}\Gamma(x+1)\sim 1+\frac{1}{12x}+\frac{1}{288x^% {2}}-\frac{139}{51840x^{3}}-\cdots\ (x\rightarrow\infty)
  15. x e x E 1 ( x ) n = 0 ( - 1 ) n n ! x n ( x ) xe^{x}E_{1}(x)\sim\sum_{n=0}^{\infty}\frac{(-1)^{n}n!}{x^{n}}\ (x\rightarrow\infty)
  16. ζ ( s ) n = 1 N - 1 n - s + N 1 - s s - 1 + N - s m = 1 B 2 m s 2 m - 1 ¯ ( 2 m ) ! N 2 m - 1 \zeta(s)\sim\sum_{n=1}^{N-1}n^{-s}+\frac{N^{1-s}}{s-1}+N^{-s}\sum_{m=1}^{% \infty}\frac{B_{2m}s^{\overline{2m-1}}}{(2m)!N^{2m-1}}
  17. B 2 m B_{2m}
  18. s 2 m - 1 ¯ s^{\overline{2m-1}}
  19. N > | s | N>|s|
  20. π x e x 2 erfc ( x ) 1 + n = 1 ( - 1 ) n ( 2 n ) ! n ! ( 2 x ) 2 n ( x ) . \sqrt{\pi}xe^{x^{2}}{\rm erfc}(x)\sim 1+\sum_{n=1}^{\infty}(-1)^{n}\frac{(2n)!% }{n!(2x)^{2n}}\ (x\rightarrow\infty).
  21. 1 1 - w = n = 0 w n . \frac{1}{1-w}=\sum_{n=0}^{\infty}w^{n}.
  22. w 1 w\neq 1
  23. | w | < 1 |w|<1
  24. e - w / t e^{-w/t}
  25. 0 e - w / t 1 - w d w = n = 0 t n + 1 0 e - u u n d u , \int_{0}^{\infty}\frac{e^{-w/t}}{1-w}\,dw=\sum_{n=0}^{\infty}t^{n+1}\int_{0}^{% \infty}e^{-u}u^{n}\,du,
  26. u = w / t u=w/t
  27. e - 1 / t Ei ( 1 t ) = n = 0 n ! t n + 1 . e^{-1/t}\operatorname{Ei}\left(\frac{1}{t}\right)=\sum_{n=0}^{\infty}n!\;t^{n+% 1}.
  28. Ei ( 1 / t ) \operatorname{Ei}(1/t)
  29. x = - 1 / t x=-1/t
  30. Ei ( x ) = - E 1 ( - x ) \operatorname{Ei}(x)=-E_{1}(-x)

Asymptotic_formula.html

  1. lim n P ( n ) F ( n ) = 1. \lim_{n\rightarrow\infty}\frac{P(n)}{F(n)}=1.
  2. P ( n ) F ( n ) P(n)\sim F(n)\,
  3. π ( x ) x log ( x ) . \pi(x)\sim\frac{x}{\log(x)}.
  4. n ! = 1 × 2 × × n n!=1\times 2\times\ldots\times n
  5. n ! 2 π n ( n e ) n . n!\sim\sqrt{2\pi n}\left(\frac{n}{e}\right)^{n}.
  6. P ( n ) 1 4 n 3 e π 2 n / 3 . P(n)\sim\frac{1}{4n\sqrt{3}}e^{\pi\sqrt{2n/3}}.
  7. y ′′ - x y = 0 y^{\prime\prime}-xy=0\,
  8. Ai ( x ) e - 2 3 x 3 / 2 2 π x 1 / 4 . \mathrm{Ai}(x)\sim\frac{e^{-\frac{2}{3}x^{3/2}}}{2\sqrt{\pi}x^{1/4}}.

Asymptotic_freedom.html

  1. x b x x\rightarrow bx
  2. n f n_{f}
  3. β 1 ( α ) = α 2 π ( - 11 N 6 + n f 3 ) \beta_{1}(\alpha)={\alpha^{2}\over\pi}\left(-{11N\over 6}+{n_{f}\over 3}\right)
  4. α \alpha
  5. g 2 / ( 4 π ) g^{2}/(4\pi)
  6. N = 3 , N=3,
  7. β 1 < 0 \beta_{1}<0
  8. n f < 33 2 . n_{f}<{33\over 2}.
  9. σ \sigma

Asymptotic_gain_model.html

  1. G = G ( T T + 1 ) + G 0 ( 1 T + 1 ) , G=G_{\infty}\left(\frac{T}{T+1}\right)+G_{0}\left(\frac{1}{T+1}\right)\ ,
  2. T T
  3. G = G | T , G_{\infty}=G\ \Big|_{T\rightarrow\infty}\ ,
  4. G 0 = G | T 0 . G_{0}=G\ \Big|_{T\rightarrow 0}\ .
  5. G = G T 1 + T = G T 1 + 1 G G T , G=G_{\infty}\frac{T}{1+T}=\frac{G_{\infty}T}{1+\frac{1}{G_{\infty}}G_{\infty}T% }\ ,
  6. A F B = A 1 + β F B A . A_{FB}=\frac{A}{1+{\beta}_{FB}A}\ .
  7. β F B = 1 G , \beta_{FB}=\frac{1}{G_{\infty}}\ ,
  8. A = G T . A=G_{\infty}\ T\ .
  9. T = g m ( R D | | r O ) g m R D , T=g_{m}\left(R_{D}\ ||r_{O}\right)\approx g_{m}R_{D}\ ,
  10. G = v o u t i i n = - R f . G_{\infty}=\frac{v_{out}}{i_{in}}=-R_{f}\ .
  11. G 0 G_{0}
  12. G 0 = v o u t i i n = R D r O R D , G_{0}=\frac{v_{out}}{i_{in}}=R_{D}\|r_{O}\approx R_{D}\ ,
  13. G = v o u t i i n = - R f g m R D 1 + g m R D + R D 1 1 + g m R D . G=\frac{v_{out}}{i_{in}}=-R_{f}\frac{g_{m}R_{D}}{1+g_{m}R_{D}}+R_{D}\frac{1}{1% +g_{m}R_{D}}\ .
  14. i B = - v π 1 + R 2 / R 1 + R f / R 1 ( β + 1 ) R 2 . i_{B}=-v_{\pi}\frac{1+R_{2}/R_{1}+R_{f}/R_{1}}{(\beta+1)R_{2}}\ .
  15. v C = v π ( 1 + R f R 1 ) - i B r π 2 . v_{C}=v_{\pi}\left(1+\frac{R_{f}}{R_{1}}\right)-i_{B}r_{\pi 2}\ .
  16. i T = i B - v C R C . i_{T}=i_{B}-\frac{v_{C}}{R_{C}}\ .
  17. T = - i R i T = - g m v π i T T=-\frac{i_{R}}{i_{T}}=-g_{m}\frac{v_{\pi}}{i_{T}}
  18. = g m R C ( 1 + R f R 1 ) ( 1 + R C + r π 2 ( β + 1 ) R 2 ) + R C + r π 2 ( β + 1 ) R 1 . =\frac{g_{m}R_{C}}{\left(1+\frac{R_{f}}{R_{1}}\right)\left(1+\frac{R_{C}+r_{% \pi 2}}{(\beta+1)R_{2}}\right)+\frac{R_{C}+r_{\pi 2}}{(\beta+1)R_{1}}}\ .
  19. G 0 = β i B i S . G_{0}=\frac{\beta i_{B}}{i_{S}}\ .
  20. ( i S - i R ) R 1 = i R R f + v E , (i_{S}-i_{R})R_{1}=i_{R}R_{f}+v_{E}\ \ ,
  21. i S = i R ( 1 + R f R 1 ) + v E R 1 . i_{S}=i_{R}\left(1+\frac{R_{f}}{R_{1}}\right)+\frac{v_{E}}{R_{1}}\ .
  22. i R = v E R 2 + ( β + 1 ) i B . i_{R}=\frac{v_{E}}{R_{2}}+(\beta+1)i_{B}\ .
  23. G 0 = β ( β + 1 ) ( 1 + R f R 1 ) + ( r π 2 + R C ) [ 1 R 1 + 1 R 2 ( 1 + R f R 1 ) ] G_{0}=\frac{\beta}{(\beta+1)\left(1+\frac{R_{f}}{R_{1}}\right)+(r_{\pi 2}+R_{C% })\left[\frac{1}{R_{1}}+\frac{1}{R_{2}}\left(1+\frac{R_{f}}{R_{1}}\right)% \right]}
  24. G = β i B i S = ( β β + 1 ) ( 1 + R f R 2 ) . G_{\infty}=\frac{\beta i_{B}}{i_{S}}=\left(\frac{\beta}{\beta+1}\right)\left(1% +\frac{R_{f}}{R_{2}}\right)\ .
  25. β F B = 1 G 1 ( 1 + R f R 2 ) = R 2 ( R f + R 2 ) , \beta_{FB}=\frac{1}{G_{\infty}}\approx\frac{1}{(1+\frac{R_{f}}{R_{2}})}=\frac{% R_{2}}{(R_{f}+R_{2})}\ ,
  26. A = G T ( 1 + R f R 2 ) g m R C ( 1 + R f R 1 ) ( 1 + R C + r π 2 ( β + 1 ) R 2 ) + R C + r π 2 ( β + 1 ) R 1 . A=G_{\infty}T\approx\frac{\left(1+\frac{R_{f}}{R_{2}}\right)g_{m}R_{C}}{\left(% 1+\frac{R_{f}}{R_{1}}\right)\left(1+\frac{R_{C}+r_{\pi 2}}{(\beta+1)R_{2}}% \right)+\frac{R_{C}+r_{\pi 2}}{(\beta+1)R_{1}}}\ .
  27. A ρ = G ( R C 2 / / R L ) . A_{\rho}=G\left(R_{C2}//R_{L}\right)\ .
  28. A i = G ( R C 2 R C 2 + R L ) . A_{i}=G\left(\frac{R_{C2}}{R_{C2}+R_{L}}\right)\ .

Atwood_machine.html

  1. m 1 m_{1}
  2. m 2 m_{2}
  3. m 1 m_{1}
  4. m 2 m_{2}
  5. W 1 = m 1 g W_{1}=m_{1}g
  6. W 2 = m 2 g W_{2}=m_{2}g
  7. m 1 g - T = m 1 a \;m_{1}g-T=m_{1}a
  8. T - m 2 g = m 2 a \;T-m_{2}g=m_{2}a
  9. m 1 g - m 2 g = m 1 a + m 2 a \;m_{1}g-m_{2}g=m_{1}a+m_{2}a
  10. a = g m 1 - m 2 m 1 + m 2 a=g{m_{1}-m_{2}\over m_{1}+m_{2}}
  11. d = 1 2 a t 2 d={1\over 2}at^{2}
  12. a = g m 1 - m 2 m 1 + m 2 a=g{m_{1}-m_{2}\over m_{1}+m_{2}}
  13. m 1 a = m 1 g - T m_{1}a=m_{1}g-T
  14. T = 2 g m 1 m 2 m 1 + m 2 = 2 g 1 / m 1 + 1 / m 2 T={2gm_{1}m_{2}\over m_{1}+m_{2}}={2g\over 1/m_{1}+1/m_{2}}
  15. α = a r , \alpha={a\over r},
  16. α \alpha
  17. τ net = ( T 1 - T 2 ) r - τ friction = I α \tau_{\mathrm{net}}=\left(T_{1}-T_{2}\right)r-\tau_{\mathrm{friction}}=I\alpha
  18. a = g ( m 1 - m 2 ) - τ friction r m 1 + m 2 + I r 2 a={{g(m_{1}-m_{2})-{\tau_{\mathrm{friction}}\over r}}\over{m_{1}+m_{2}+{{I}% \over{r^{2}}}}}
  19. T 1 = m 1 g ( 2 m 2 + I r 2 + τ friction r g ) m 1 + m 2 + I r 2 T_{1}={{m_{1}g(2m_{2}+{{I}\over{r^{2}}}+{{\tau_{\mathrm{friction}}}\over{rg}})% }\over{m_{1}+m_{2}+{{I}\over{r^{2}}}}}
  20. T 2 = m 2 g ( 2 m 1 + I r 2 + τ friction r g ) m 1 + m 2 + I r 2 T_{2}={{m_{2}g(2m_{1}+{{I}\over{r^{2}}}+{{\tau_{\mathrm{friction}}}\over{rg}})% }\over{m_{1}+m_{2}+{{I}\over{r^{2}}}}}
  21. a = g ( m 1 - m 2 ) m 1 + m 2 + I r 2 a={{g(m_{1}-m_{2})}\over{m_{1}+m_{2}+{{I}\over{r^{2}}}}}
  22. T 1 = m 1 g ( 2 m 2 + I r 2 ) m 1 + m 2 + I r 2 T_{1}={{m_{1}g(2m_{2}+{{I}\over{r^{2}}})}\over{m_{1}+m_{2}+{{I}\over{r^{2}}}}}
  23. T 2 = m 2 g ( 2 m 1 + I r 2 ) m 1 + m 2 + I r 2 T_{2}={{m_{2}g(2m_{1}+{{I}\over{r^{2}}})}\over{m_{1}+m_{2}+{{I}\over{r^{2}}}}}

Authorship_of_the_Johannine_works.html

  1. 𝔓 \mathfrak{P}

Automatic_differentiation.html

  1. y = g ( h ( x ) ) = g ( w ) y=g(h(x))=g(w)
  2. d y d x = d y d w d w d x \frac{dy}{dx}=\frac{dy}{dw}\frac{dw}{dx}
  3. d w / d x dw/dx
  4. d y / d w dy/dw
  5. y x = y w 1 w 1 x = y w 1 ( w 1 w 2 w 2 x ) = y w 1 ( w 1 w 2 ( w 2 w 3 w 3 x ) ) = \frac{\partial y}{\partial x}=\frac{\partial y}{\partial w_{1}}\frac{\partial w% _{1}}{\partial x}=\frac{\partial y}{\partial w_{1}}\left(\frac{\partial w_{1}}% {\partial w_{2}}\frac{\partial w_{2}}{\partial x}\right)=\frac{\partial y}{% \partial w_{1}}\left(\frac{\partial w_{1}}{\partial w_{2}}\left(\frac{\partial w% _{2}}{\partial w_{3}}\frac{\partial w_{3}}{\partial x}\right)\right)=\cdots
  6. w w
  7. w ˙ = w x \dot{w}=\frac{\partial w}{\partial x}
  8. z \displaystyle z
  9. w ˙ 1 = x 1 x 1 = 1 \displaystyle\dot{w}_{1}=\frac{\partial x_{1}}{\partial x_{1}}=1
  10. w 1 = x 1 w_{1}=x_{1}
  11. w ˙ 1 = 1 \dot{w}_{1}=1
  12. w 2 = x 2 w_{2}=x_{2}
  13. w ˙ 2 = 0 \dot{w}_{2}=0
  14. w 3 = w 1 w 2 w_{3}=w_{1}\cdot w_{2}
  15. w ˙ 3 = w 2 w ˙ 1 + w 1 w ˙ 2 \dot{w}_{3}=w_{2}\cdot\dot{w}_{1}+w_{1}\cdot\dot{w}_{2}
  16. w 4 = sin w 1 w_{4}=\sin w_{1}
  17. w ˙ 4 = cos w 1 w ˙ 1 \dot{w}_{4}=\cos w_{1}\cdot\dot{w}_{1}
  18. w 5 = w 3 + w 4 w_{5}=w_{3}+w_{4}
  19. w ˙ 5 = w ˙ 3 + w ˙ 4 \dot{w}_{5}=\dot{w}_{3}+\dot{w}_{4}
  20. f f
  21. w ˙ 1 = 0 ; w ˙ 2 = 1 \dot{w}_{1}=0;\dot{w}_{2}=1
  22. m n m≫n
  23. n n
  24. m m
  25. y x = y w 1 w 1 x = ( y w 2 w 2 w 1 ) w 1 x = ( ( y w 3 w 3 w 2 ) w 2 w 1 ) w 1 x = \frac{\partial y}{\partial x}=\frac{\partial y}{\partial w_{1}}\frac{\partial w% _{1}}{\partial x}=\left(\frac{\partial y}{\partial w_{2}}\frac{\partial w_{2}}% {\partial w_{1}}\right)\frac{\partial w_{1}}{\partial x}=\left(\left(\frac{% \partial y}{\partial w_{3}}\frac{\partial w_{3}}{\partial w_{2}}\right)\frac{% \partial w_{2}}{\partial w_{1}}\right)\frac{\partial w_{1}}{\partial x}=\cdots
  26. w w
  27. w ¯ = y w \bar{w}=\frac{\partial y}{\partial w}
  28. w ¯ 5 = 1 \bar{w}_{5}=1
  29. w ¯ 4 = w ¯ 5 \bar{w}_{4}=\bar{w}_{5}
  30. w ¯ 3 = w ¯ 5 \bar{w}_{3}=\bar{w}_{5}
  31. w ¯ 2 = w ¯ 3 w 1 \bar{w}_{2}=\bar{w}_{3}\cdot w_{1}
  32. w ¯ 1 = w ¯ 3 w 2 + w ¯ 4 cos w 1 \bar{w}_{1}=\bar{w}_{3}\cdot w_{2}+\bar{w}_{4}\cdot\cos w_{1}
  33. y = f ( x ) y=f(x)
  34. = ȳ f ( x ) x̄=ȳf′(x)
  35. m n m≪n
  36. m m
  37. n n
  38. x \,x
  39. x + x ε x+x^{\prime}\varepsilon
  40. x x^{\prime}
  41. ε \varepsilon
  42. ε 2 = 0 \varepsilon^{2}=0
  43. ( x + x ε ) + ( y + y ε ) = x + y + ( x + y ) ε ( x + x ε ) ( y + y ε ) = x y + x y ε + y x ε + x y ε 2 = x y + ( x y + y x ) ε \begin{aligned}\displaystyle(x+x^{\prime}\varepsilon)+(y+y^{\prime}\varepsilon% )&\displaystyle=x+y+(x^{\prime}+y^{\prime})\varepsilon\\ \displaystyle(x+x^{\prime}\varepsilon)\cdot(y+y^{\prime}\varepsilon)&% \displaystyle=xy+xy^{\prime}\varepsilon+yx^{\prime}\varepsilon+x^{\prime}y^{% \prime}\varepsilon^{2}=xy+(xy^{\prime}+yx^{\prime})\varepsilon\end{aligned}
  44. P ( x ) = p 0 + p 1 x + p 2 x 2 + + p n x n P(x)=p_{0}+p_{1}x+p_{2}x^{2}+\cdots+p_{n}x^{n}
  45. P ( x + x ε ) = p 0 + p 1 ( x + x ε ) + + p n ( x + x ε ) n = p 0 + p 1 x + + p n x n + p 1 x ε + 2 p 2 x x ε + + n p n x n - 1 x ε = P ( x ) + P ( 1 ) ( x ) x ε \begin{aligned}\displaystyle P(x+x^{\prime}\varepsilon)&\displaystyle=p_{0}+p_% {1}(x+x^{\prime}\varepsilon)+\cdots+p_{n}(x+x^{\prime}\varepsilon)^{n}\\ &\displaystyle=p_{0}+p_{1}x+\cdots+p_{n}x^{n}+p_{1}x^{\prime}\varepsilon+2p_{2% }xx^{\prime}\varepsilon+\cdots+np_{n}x^{n-1}x^{\prime}\varepsilon\\ &\displaystyle=P(x)+P^{(1)}(x)x^{\prime}\varepsilon\end{aligned}
  46. P ( 1 ) P^{(1)}
  47. P P
  48. x x^{\prime}
  49. x , x \langle x,x^{\prime}\rangle
  50. u , u + v , v \displaystyle\left\langle u,u^{\prime}\right\rangle+\left\langle v,v^{\prime}\right\rangle
  51. g g
  52. g ( u , u , v , v ) = g ( u , v ) , g u ( u , v ) u + g v ( u , v ) v g(\langle u,u^{\prime}\rangle,\langle v,v^{\prime}\rangle)=\langle g(u,v),g_{u% }(u,v)u^{\prime}+g_{v}(u,v)v^{\prime}\rangle
  53. g u g_{u}
  54. g v g_{v}
  55. g g
  56. u , u \langle u,u^{\prime}\rangle
  57. c c
  58. c , 0 \langle c,0\rangle
  59. f : f:\mathbb{R}\rightarrow\mathbb{R}
  60. x 0 x_{0}
  61. f ( x 0 , 1 ) f(\langle x_{0},1\rangle)
  62. f ( x 0 ) , f ( x 0 ) \langle f(x_{0}),f^{\prime}(x_{0})\rangle
  63. y = f ( x ) x y^{\prime}=\nabla f(x)\cdot x^{\prime}
  64. y m y^{\prime}\in\mathbb{R}^{m}
  65. f : n m f:\mathbb{R}^{n}\rightarrow\mathbb{R}^{m}
  66. x n x\in\mathbb{R}^{n}
  67. x n x^{\prime}\in\mathbb{R}^{n}
  68. ( y 1 , y 1 , , y m , y m ) = f ( x 1 , x 1 , , x n , x n ) (\langle y_{1},y^{\prime}_{1}\rangle,\ldots,\langle y_{m},y^{\prime}_{m}% \rangle)=f(\langle x_{1},x^{\prime}_{1}\rangle,\ldots,\langle x_{n},x^{\prime}% _{n}\rangle)
  69. f \nabla f
  70. n n