wpmath0000010_1

Diphosphomevalonate_decarboxylase.html

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Diphthine_synthase.html

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Dirac_bracket.html

  1. q q
  2. m m
  3. x x
  4. y y
  5. z z
  6. B B
  7. L = 1 2 m v 2 + q c A v - V ( r ) , L=\tfrac{1}{2}m\vec{v}^{2}+\frac{q}{c}\vec{A}\cdot\vec{v}-V(\vec{r}),
  8. c c
  9. V V
  10. x x
  11. y y
  12. A = B 2 ( x y ^ - y x ^ ) \vec{A}=\frac{B}{2}(x\hat{y}-y\hat{x})
  13. L = m 2 ( x ˙ 2 + y ˙ 2 ) + q B 2 c ( x y ˙ - y x ˙ ) - V ( x , y ) , L=\frac{m}{2}(\dot{x}^{2}+\dot{y}^{2})+\frac{qB}{2c}(x\dot{y}-y\dot{x})-V(x,y)% ~{},
  14. m x ¨ = - V x + q B c y ˙ m\ddot{x}=-\frac{\partial V}{\partial x}+\frac{qB}{c}\dot{y}
  15. m y ¨ = - V y - q B c x ˙ . m\ddot{y}=-\frac{\partial V}{\partial y}-\frac{qB}{c}\dot{x}.
  16. L = q B 2 c ( x y ˙ - y x ˙ ) - V ( x , y ) , L=\frac{qB}{2c}(x\dot{y}-y\dot{x})-V(x,y)~{},
  17. y ˙ = c q B V x \dot{y}=\frac{c}{qB}\frac{\partial V}{\partial x}
  18. x ˙ = - c q B V y . \dot{x}=-\frac{c}{qB}\frac{\partial V}{\partial y}~{}.
  19. p x = L x ˙ = - q B 2 c y p_{x}=\frac{\partial L}{\partial\dot{x}}=-\frac{qB}{2c}y
  20. p y = L y ˙ = q B 2 c x , p_{y}=\frac{\partial L}{\partial\dot{y}}=\frac{qB}{2c}x~{},
  21. H ( x , y , p x , p y ) = x ˙ p x + y ˙ p y - L = V ( x , y ) . H(x,y,p_{x},p_{y})=\dot{x}p_{x}+\dot{y}p_{y}-L=V(x,y).
  22. y y
  23. f f
  24. g g
  25. f g f≈g
  26. f f
  27. g g
  28. f = g f=g
  29. φ φ
  30. H H
  31. H * = H + j c j ϕ j H , H^{*}=H+\sum_{j}c_{j}\phi_{j}\approx H,
  32. δ H * δ H δH*≈δH
  33. δ H = H q δ q + H p δ p q ˙ δ p - p ˙ δ q , \delta H=\frac{\partial H}{\partial q}\delta q+\frac{\partial H}{\partial p}% \delta p\approx\dot{q}\delta p-\dot{p}\delta q~{},
  34. ( H q + p ˙ ) δ q + ( H p - q ˙ ) δ p = 0 , \left(\frac{\partial H}{\partial q}+\dot{p}\right)\delta q+\left(\frac{% \partial H}{\partial p}-\dot{q}\right)\delta p=0~{},
  35. δ q δq
  36. δ p δp
  37. n A n δ q n + n B n δ p n = 0 , \sum_{n}A_{n}\delta q_{n}+\sum_{n}B_{n}\delta p_{n}=0,
  38. ϕ j 0 \phi_{j}\approx 0
  39. A n = m u m ϕ m q n A_{n}=\sum_{m}u_{m}\frac{\partial\phi_{m}}{\partial q_{n}}
  40. B n = m u m ϕ m p n , B_{n}=\sum_{m}u_{m}\frac{\partial\phi_{m}}{\partial p_{n}},
  41. p ˙ j = - H q j - k u k ϕ k q j \dot{p}_{j}=-\frac{\partial H}{\partial q_{j}}-\sum_{k}u_{k}\frac{\partial\phi% _{k}}{\partial q_{j}}
  42. q ˙ j = H p j + k u k ϕ k p j \dot{q}_{j}=\frac{\partial H}{\partial p_{j}}+\sum_{k}u_{k}\frac{\partial\phi_% {k}}{\partial p_{j}}
  43. ϕ j ( q , p ) = 0 , \phi_{j}(q,p)=0,
  44. f f
  45. f ˙ { f , H * } P B { f , H } P B + k u k { f , ϕ k } P B , \dot{f}\approx\{f,H^{*}\}_{PB}\approx\{f,H\}_{PB}+\sum_{k}u_{k}\{f,\phi_{k}\}_% {PB},
  46. ϕ j ˙ { ϕ j , H } P B + k u k { ϕ j , ϕ k } P B 0. \dot{\phi_{j}}\approx\{\phi_{j},H\}_{PB}+\sum_{k}u_{k}\{\phi_{j},\phi_{k}\}_{% PB}\approx 0.
  47. L = q L=q
  48. φ φ
  49. { ϕ j , H } P B + k u k { ϕ j , ϕ k } P B 0. \{\phi_{j},H\}_{PB}+\sum_{k}u_{k}\{\phi_{j},\phi_{k}\}_{PB}\approx 0.
  50. u k = U k + V k , u_{k}=U_{k}+V_{k},
  51. U k U_{k}
  52. k V k { ϕ j , ϕ k } P B 0. \sum_{k}V_{k}\{\phi_{j},\phi_{k}\}_{PB}\approx 0.
  53. a a
  54. u k U k + a v a V k a , u_{k}\approx U_{k}+\sum_{a}v_{a}V^{a}_{k},
  55. H T = H + k U k ϕ k + a , k v a V k a ϕ k H_{T}=H+\sum_{k}U_{k}\phi_{k}+\sum_{a,k}v_{a}V^{a}_{k}\phi_{k}
  56. H = H + k U k ϕ k . H^{\prime}=H+\sum_{k}U_{k}\phi_{k}.
  57. f f
  58. f ˙ { f , H T } P B . \dot{f}\approx\{f,H_{T}\}_{PB}.
  59. f ( q , p ) f(q,p)
  60. { f , ϕ j } P B 0 , \{f,\phi_{j}\}_{PB}\approx 0,
  61. j j
  62. φ φ
  63. φ φ
  64. φ φ
  65. c c
  66. { ϕ 1 , ϕ 2 } P B = c . \{\phi_{1},\phi_{2}\}_{PB}=c~{}.
  67. i ħ
  68. [ ϕ ^ 1 , ϕ ^ 2 ] = i c , [\hat{\phi}_{1},\hat{\phi}_{2}]=i\hbar~{}c,
  69. φ φ
  70. φ φ
  71. M a b = { ϕ ~ a , ϕ ~ b } P B . M_{ab}=\{\tilde{\phi}_{a},\tilde{\phi}_{b}\}_{PB}.
  72. f f
  73. g g
  74. a b ab
  75. M M
  76. M M
  77. i ħ
  78. H = V ( x , y ) H=V(x,y)
  79. ϕ 1 = p x + q B 2 c y , ϕ 2 = p y - q B 2 c x . \phi_{1}=p_{x}+\tfrac{qB}{2c}y,\qquad\phi_{2}=p_{y}-\tfrac{qB}{2c}x.
  80. H * = V ( x , y ) + u 1 ( p x + q B 2 c y ) + u 2 ( p y - q B 2 c x ) . H^{*}=V(x,y)+u_{1}\left(p_{x}+\tfrac{qB}{2c}y\right)+u_{2}\left(p_{y}-\tfrac{% qB}{2c}x\right).
  81. { ϕ j , H * } P B \{\phi_{j},H^{*}\}_{PB}
  82. { ϕ 1 , H } P B + j u j { ϕ 1 , ϕ j } P B = - V x + u 2 q B c 0 \{\phi_{1},H\}_{PB}+\sum_{j}u_{j}\{\phi_{1},\phi_{j}\}_{PB}=-\frac{\partial V}% {\partial x}+u_{2}\frac{qB}{c}\approx 0
  83. { ϕ 2 , H } P B + j u j { ϕ 2 , ϕ j } P B = - V y - u 1 q B c 0. \{\phi_{2},H\}_{PB}+\sum_{j}u_{j}\{\phi_{2},\phi_{j}\}_{PB}=-\frac{\partial V}% {\partial y}-u_{1}\frac{qB}{c}\approx 0.
  84. x ˙ = { x , H } P B + u 1 { x , ϕ 1 } P B + u 2 { x , ϕ 2 } = - c q B V y \dot{x}=\{x,H\}_{PB}+u_{1}\{x,\phi_{1}\}_{PB}+u_{2}\{x,\phi_{2}\}=-\frac{c}{qB% }\frac{\partial V}{\partial y}
  85. y ˙ = c q B V x \dot{y}=\frac{c}{qB}\frac{\partial V}{\partial x}
  86. p ˙ x = - 1 2 V x \dot{p}_{x}=-\frac{1}{2}\frac{\partial V}{\partial x}
  87. p ˙ y = - 1 2 V y , \dot{p}_{y}=-\frac{1}{2}\frac{\partial V}{\partial y},
  88. φ φ
  89. φ φ
  90. { ϕ 1 , ϕ 2 } P B = - { ϕ 2 , ϕ 1 } P B = q B c , \{\phi_{1},\phi_{2}\}_{PB}=-\{\phi_{2},\phi_{1}\}_{PB}=\frac{qB}{c},
  91. M = q B c ( 0 1 - 1 0 ) , M=\frac{qB}{c}\left(\begin{matrix}0&1\\ -1&0\end{matrix}\right),
  92. M - 1 = c q B ( 0 - 1 1 0 ) M a b - 1 = - c q B 0 ϵ a b , M^{-1}=\frac{c}{qB}\left(\begin{matrix}0&-1\\ 1&0\end{matrix}\right)\quad\Rightarrow\quad M^{-1}_{ab}=-\frac{c}{qB_{0}}% \epsilon_{ab},
  93. ϵ a b \epsilon_{ab}
  94. { f , g } D B = { f , g } P B + c ϵ a b q B { f , ϕ a } P B { ϕ b , g } P B . \{f,g\}_{DB}=\{f,g\}_{PB}+\frac{c\epsilon_{ab}}{qB}\{f,\phi_{a}\}_{PB}\{\phi_{% b},g\}_{PB}.
  95. { x , y } D B = - c q B \{x,y\}_{DB}=-\tfrac{c}{qB}
  96. { x , p x } D B = { y , p y } D B = 1 2 \{x,p_{x}\}_{DB}=\{y,p_{y}\}_{DB}=\frac{1}{2}
  97. { p x , p y } D B = - q B 4 c . \{p_{x},p_{y}\}_{DB}=-\tfrac{qB}{4c}.
  98. [ x ^ , y ^ ] = - i c q B [\hat{x},\hat{y}]=-i\tfrac{\hbar c}{qB}
  99. [ x ^ , p ^ x ] = [ y ^ , p ^ y ] = i 2 [\hat{x},\hat{p}_{x}]=[\hat{y},\hat{p}_{y}]=i\frac{\hbar}{2}
  100. [ p ^ x , p ^ y ] = - i q B 4 c . [\hat{p}_{x},\hat{p}_{y}]=-i\tfrac{\hbar qB}{4c}~{}.
  101. x x
  102. y y
  103. n n
  104. { x i , x j } D B = 0 , \{x_{i},x_{j}\}_{DB}=0,
  105. { x i , p j } D B = δ i j - x i x j , \{x_{i},p_{j}\}_{DB}=\delta_{ij}-x_{i}x_{j},
  106. { p i , p j } D B = x j p i - x i p j . \{p_{i},p_{j}\}_{DB}=x_{j}p_{i}-x_{i}p_{j}~{}.
  107. n n
  108. n n
  109. x x
  110. p p
  111. n n
  112. x x
  113. z z
  114. x x
  115. L = 1 2 z ˙ 2 1 - z 2 , L=\frac{1}{2}\frac{{\dot{z}}^{2}}{1-z^{2}}~{},
  116. z ¨ = - z z ˙ 2 1 - z 2 = - z 2 E , {\ddot{z}}=-z\frac{{\dot{z}}^{2}}{1-z^{2}}=-z2E~{},
  117. H H
  118. p p
  119. E E
  120. x ˙ i = { x i , H } D B = p i , {\dot{x}}^{i}=\{x^{i},H\}_{DB}=p^{i}~{},
  121. p ˙ i = { p i , H } D B = x i p 2 , {\dot{p}}^{i}=\{p^{i},H\}_{DB}=x^{i}~{}p^{2}~{},
  122. x ¨ i = - x i 2 E . {\ddot{x}}^{i}=-x^{i}2E~{}.

Direct_integration_of_a_beam.html

  1. w ( x ) w(x)
  2. V ( x ) = - w ( x ) d x V(x)=-\int w(x)\,dx
  3. M ( x ) = V ( x ) d x M(x)=\int V(x)\,dx
  4. - [ w ( x ) d x ] d x -\int[\int w(x)\ \,dx]dx
  5. θ \theta
  6. θ ( x ) = 1 E I M ( x ) d x \theta(x)=\frac{1}{EI}\int M(x)\,dx
  7. ν \nu
  8. ν ( x ) = θ ( x ) d x \nu(x)=\int\theta(x)dx
  9. 𝐰 ( x ) = 10 ( k N / m ) \mathbf{w}(x)=10(kN/m)
  10. 𝐕 ( x ) = - w ( x ) d x = - 10 x + C 1 ( k N ) \mathbf{V}(x)=-\int w(x)dx=-10x+C_{1}(kN)
  11. C 1 C_{1}
  12. 𝐕 ( x ) = - 10 x + 75 ( k N ) \mathbf{V}(x)=-10x+75(kN)
  13. 𝐌 ( x ) = V ( x ) = - 5 x 2 + 75 x ( k N m ) \mathbf{M}(x)=\int V(x)=-5x^{2}+75x(kN\cdot m)
  14. C 2 = 0 C_{2}=0
  15. \cdot
  16. \cdot
  17. θ ( x ) = M ( x ) E I = - 5 3 x 3 + 75 2 x 2 + C 3 ( m m ) \mathbf{\theta}(x)=\int\frac{M(x)}{EI}=-\frac{5}{3}x^{3}+\frac{75}{2}x^{2}+C_{% 3}(\frac{m}{m})
  18. ν ( x ) = θ ( x ) = - 5 12 x 4 + 75 6 x 3 + C 3 x + C 4 ( m ) \mathbf{\nu}(x)=\int\theta(x)=-\frac{5}{12}x^{4}+\frac{75}{6}x^{3}+C_{3}x+C_{4% }(m)
  19. v v
  20. C 3 C_{3}
  21. C 4 C_{4}
  22. θ ( x ) = M ( x ) E I = - 5 3 x 3 + 75 2 x 2 - 1406.25 ( m m ) \mathbf{\theta}(x)=\int\frac{M(x)}{EI}=-\frac{5}{3}x^{3}+\frac{75}{2}x^{2}-140% 6.25(\frac{m}{m})
  23. ν ( x ) = θ ( x ) = - 5 12 x 4 + 75 6 x 3 - 1406.25 x ( m ) \mathbf{\nu}(x)=\int\theta(x)=-\frac{5}{12}x^{4}+\frac{75}{6}x^{3}-1406.25x(m)
  24. θ \theta

Discontinuous_Galerkin_method.html

  1. ρ \rho
  2. Ω \Omega
  3. ρ t + 𝐉 = 0 , \frac{\partial\rho}{\partial t}+\nabla\cdot\mathbf{J}=0,
  4. 𝐉 \mathbf{J}
  5. ρ \rho
  6. Ω \Omega
  7. Ω h \Omega_{h}
  8. S h p ( Ω h ) = { v | Ω e i P p ( Ω e i ) , Ω e i Ω h } S_{h}^{p}(\Omega_{h})=\{v_{|\Omega_{e_{i}}}\in P^{p}(\Omega_{e_{i}}),\ \ % \forall\Omega_{e_{i}}\in\Omega_{h}\}
  9. P p ( Ω e i ) P^{p}(\Omega_{e_{i}})
  10. p p
  11. Ω e i \Omega_{e_{i}}
  12. i i
  13. N j P p N_{j}\in P^{p}
  14. ρ h = j = 1 d o f s ρ j i ( t ) N j i ( s y m b o l x ) , \forallsymbol x Ω e i . \rho_{h}=\sum_{j=1}^{dofs}\rho_{j}^{i}(t)N_{j}^{i}(symbol{x}),\quad% \forallsymbol{x}\in\Omega_{e_{i}}.
  15. ϕ h ( s y m b o l x ) = j = 1 d o f s ϕ j i N j i ( s y m b o l x ) , s y m b o l x Ω e i , \phi_{h}(symbol{x})=\sum_{j=1}^{dofs}\phi_{j}^{i}N_{j}^{i}(symbol{x}),\quad% \forall symbol{x}\in\Omega_{e_{i}},
  16. ϕ h \phi_{h}
  17. d d t Ω e i ρ h ϕ h d s y m b o l x + Ω e i ϕ h 𝐉 h \cdotsymbol n d s y m b o l x = Ω e i 𝐉 h ϕ h d s y m b o l x . \frac{d}{dt}\int_{\Omega_{e_{i}}}\rho_{h}\phi_{h}dsymbol{x}+\int_{\partial% \Omega_{e_{i}}}\phi_{h}\mathbf{J}_{h}\cdotsymbol{n}dsymbol{x}=\int_{\Omega_{e_% {i}}}\mathbf{J}_{h}\cdot\nabla\phi_{h}dsymbol{x}.

Divinyl_chlorophyllide_a_8-vinyl-reductase.html

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Dodecenoyl-CoA_isomerase.html

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Double_layer_(interfacial).html

  1. σ d = - 8 ε 0 ε m C R T sinh F Ψ d 2 R T \sigma^{d}=-\sqrt{{8\varepsilon_{0}}{\varepsilon_{m}}CRT}\sinh\frac{F\Psi^{d}}% {2RT}
  2. Ψ ( r ) = Ψ d a r exp ( - κ ( r - a ) ) {\Psi}(r)={\Psi^{d}}\frac{a}{r}\exp({-\kappa}(r-a))
  3. κ a 1 {\kappa}a>>1
  4. κ a < 1 {\kappa}a<1
  5. C = d σ d Ψ C=\frac{d\sigma}{d\Psi}

Double_layer_(plasma).html

  1. e ϕ D L / k B T e 0.1 e\phi_{DL}/k_{B}T_{e}\approx 0.1
  2. 2 e V k B T e 20 e V 2eV\leq k_{B}T_{e}\leq 20eV
  3. x ( n e v e ) = 0 , \frac{\partial}{\partial x}(n_{e}v_{e})=0,
  4. m e v e v e x = - e E . m_{e}v_{e}\frac{\partial v_{e}}{\partial x}=-eE.
  5. E = - x ( m e j e 2 2 n e 2 e 3 ) E=-\frac{\partial}{\partial x}\left(\frac{m_{e}j_{e}^{2}}{2n_{e}^{2}e^{3}}\right)
  6. j e = - n e e v e j_{e}=-n_{e}ev_{e}
  7. F = e n i E = - x ( m e n e v e 2 ) F=en_{i}E=-\frac{\partial}{\partial x}(m_{e}n_{e}v_{e}^{2})
  8. k B T i < m e v e 2 k_{B}T_{i}<m_{e}v_{e}^{2}
  9. v e ( x ) v_{e}(x)
  10. v i ( x ) v_{i}(x)
  11. v α 2 ( x ) / 2 m + q α ϕ ( x ) v_{\alpha}^{2}(x)/2m+q_{\alpha}\phi(x)
  12. j α = q n α ( x ) v α ( x ) j_{\alpha}=qn_{\alpha}(x)v_{\alpha}(x)
  13. | v α ( ϕ α ) | = v α , 0 2 + 2 e ϕ α m α , |v_{\alpha}(\phi_{\alpha})|=\sqrt{v_{\alpha,0}^{2}+\frac{2e\phi_{\alpha}}{m_{% \alpha}}},
  14. n α ( ϕ α ) = n 0 v α , 0 v α - 1 ( ϕ α ) , n_{\alpha}(\phi_{\alpha})=n_{0}v_{\alpha,0}v_{\alpha}^{-1}(\phi_{\alpha}),
  15. ϕ e = ϕ \phi_{e}=\phi
  16. ϕ i = ϕ D L - ϕ \phi_{i}=\phi_{DL}-\phi
  17. n 0 n_{0}
  18. v α , 0 v_{\alpha,0}
  19. ϕ ( x = 0 ) = 0 \phi(x=0)=0
  20. ϕ ( x = d ) = ϕ D L \phi(x=d)=\phi_{DL}
  21. d d
  22. ρ e = j e v e , ρ i = j i v i . \rho_{e}=\frac{j_{e}}{v_{e}},\rho_{i}=\frac{j_{i}}{v_{i}}.
  23. - 1 4 π 2 ϕ x 2 = j i v i , 0 2 + 2 e ( ϕ D L - ϕ ) / m i - j e v e , 0 2 + 2 e ϕ / m e . -\frac{1}{4\pi}\frac{\partial^{2}\phi}{\partial x^{2}}=\frac{j_{i}}{\sqrt{v_{i% ,0}^{2}+2e(\phi_{DL}-\phi)/m_{i}}}-\frac{j_{e}}{\sqrt{v_{e,0}^{2}+2e\phi/m_{e}% }}.
  24. d ϕ / d x d\phi/dx
  25. x x
  26. ϕ \phi
  27. ( d ϕ / d x ) 2 (d\phi/dx)^{2}
  28. j e j i = m i m e . \frac{j_{e}}{j_{i}}=\sqrt{\frac{m_{i}}{m_{e}}}.
  29. 1836 \sqrt{1836}
  30. j d 2 = ( j e + j i ) d 2 = ( 1 + m e m i ) 4 ε 0 C 0 9 2 e m e ϕ D L 1.5 , jd^{2}=(j_{e}+j_{i})d^{2}=\left(1+\sqrt{\frac{m_{e}}{m_{i}}}\right)\frac{4% \varepsilon_{0}C_{0}}{9}\sqrt{\frac{2e}{m_{e}}}\phi_{DL}^{1.5},
  31. C 0 C_{0}
  32. C 0 = 2 - 1.5 ( 4 2 E ( sin π / 8 ) - ( 1 + 2 2 K ( sin π / 8 ) ) 2 1.86518. C_{0}=2^{-1.5}(4\sqrt{2}E(\sin\pi/8)-(1+2\sqrt{2}K(\sin\pi/8))^{2}\approx 1.86% 518.
  33. n α , R = n α , 0 e - e ϕ α k B T α . n_{\alpha,R}=n_{\alpha,0}e^{-\frac{e\phi_{\alpha}}{k_{B}T_{\alpha}}}.
  34. m e v e 2 > k B T i . m_{e}v_{e}^{2}>k_{B}T_{i}.
  35. f α ( x , t ; v ) f_{\alpha}(\vec{x},t;\vec{v})
  36. α \alpha
  37. v \vec{v}
  38. x \vec{x}
  39. t t
  40. α \alpha
  41. t f α + v x f α + q α E m α v f α = 0 , \frac{\partial}{\partial t}f_{\alpha}+\vec{v}\cdot\frac{\partial}{\partial\vec% {x}}f_{\alpha}+\frac{q_{\alpha}\vec{E}}{m_{\alpha}}\cdot\frac{\partial}{% \partial\vec{v}}f_{\alpha}=0,
  42. E = - 2 ϕ x 2 = 4 π ρ . \nabla\cdot\vec{E}=-\frac{\partial^{2}\phi}{\partial x^{2}}=4\pi\rho.
  43. q α q_{\alpha}
  44. m α m_{\alpha}
  45. E ( x , t ) \vec{E}(\vec{x},t)
  46. ϕ ( x , t ) \phi(\vec{x},t)
  47. ρ \rho

Drazin_inverse.html

  1. A k + 1 A D = A k , A D A A D = A D , A A D = A D A . A^{k+1}A^{D}=A^{k},\quad A^{D}AA^{D}=A^{D},\quad AA^{D}=A^{D}A.
  2. A - 1 A^{-1}
  3. A D = A - 1 A^{D}=A^{-1}
  4. A D = 0. A^{D}=0.
  5. A i + 1 := A i + A i ( I - A A i ) ; A_{i+1}:=A_{i}+A_{i}\left(I-AA_{i}\right);
  6. A i + j = A i k = 0 2 j - 1 ( I - A A i ) k . A_{i+j}=A_{i}\sum_{k=0}^{2^{j}-1}(I-AA_{i})^{k}.
  7. A 0 := α A A_{0}:=\alpha A
  8. A 0 A_{0}
  9. A 0 A = A A 0 A_{0}A=AA_{0}
  10. A 0 - A 0 A A 0 < A 0 \|A_{0}-A_{0}AA_{0}\|<\|A_{0}\|
  11. A i A D . A_{i}\rightarrow A^{D}.

DTDP-4-dehydro-6-deoxyglucose_reductase.html

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DTDP-4-dehydrorhamnose_3,5-epimerase.html

  1. \rightleftharpoons

DTDP-4-dehydrorhamnose_reductase.html

  1. \rightleftharpoons

DTDP-6-deoxy-L-talose_4-dehydrogenase.html

  1. \rightleftharpoons

DTDP-galactose_6-dehydrogenase.html

  1. \rightleftharpoons

Dual_norm.html

  1. X X
  2. F F
  3. F = F={\mathbb{C}}
  4. F = F={\mathbb{R}}
  5. \|\cdot\|
  6. X X^{\prime}
  7. X * X^{*}
  8. X X
  9. F F
  10. f : X F f:X\to F
  11. \|\cdot\|^{\prime}
  12. f f
  13. f = sup { | f ( x ) | : x X , x 1 } = sup { | f ( x ) | x : x X , x 0 } . \|f\|^{\prime}=\sup\{|f(x)|:x\in X,\|x\|\leq 1\}=\sup\left\{\frac{|f(x)|}{\|x% \|}:x\in X,x\neq 0\right\}.
  14. X X^{\prime}
  15. X X^{\prime}
  16. [ 1 , ] [1,\infty]
  17. 1 / p + 1 / q = 1 1/p+1/q=1
  18. x T Q x \sqrt{x^{\mathrm{T}}Qx}
  19. y T Q - 1 y \sqrt{y^{\mathrm{T}}Q^{-1}y}
  20. Q Q
  21. A F = i = 1 m j = 1 n | a i j | 2 = trace ( A * A ) = i = 1 min { m , n } σ i 2 \|A\|_{\,\text{F}}=\sqrt{\sum_{i=1}^{m}\sum_{j=1}^{n}|a_{ij}|^{2}}=\sqrt{% \operatorname{trace}(A^{{}^{*}}A)}=\sqrt{\sum_{i=1}^{\min\{m,\,n\}}\sigma_{i}^% {2}}
  22. B F \|B\|_{\,\text{F}}
  23. A 2 = σ m a x ( A ) \|A\|_{2}=\sigma_{max}(A)
  24. i σ i ( B ) \sum_{i}\sigma_{i}(B)

Dukhin_number.html

  1. κ σ \kappa^{\sigma}
  2. D u = κ σ K m a \ Du=\frac{\kappa^{\sigma}}{{K_{m}}a}
  3. D u = 2 ( 1 + 3 m / z 2 ) κ a ( cosh z F ζ 2 R T - 1 ) \ Du=\frac{2(1+3m/z^{2})}{{\kappa}a}\left(\mathrm{cosh}\frac{zF\zeta}{2RT}-1\right)
  4. m = 2 ε 0 ε m R 2 T 2 3 η F 2 D m=\frac{2\varepsilon_{0}\varepsilon_{m}R^{2}T^{2}}{3\eta F^{2}D}

Ecdysone_20-monooxygenase.html

  1. \rightleftharpoons

Ecdysone_oxidase.html

  1. \rightleftharpoons

EHP_spectral_sequence.html

  1. S n ( 2 ) Ω S n + 1 ( 2 ) Ω S 2 n + 1 ( 2 ) S^{n}(2)\rightarrow\Omega S^{n+1}(2)\rightarrow\Omega S^{2n+1}(2)
  2. S ^ 2 n ( p ) Ω S 2 n + 1 ( p ) Ω S 2 p n + 1 ( p ) \widehat{S}^{2n}(p)\rightarrow\Omega S^{2n+1}(p)\rightarrow\Omega S^{2pn+1}(p)
  3. S 2 n - 1 ( p ) Ω S ^ 2 n ( p ) Ω S 2 p n - 1 ( p ) S^{2n-1}(p)\rightarrow\Omega\widehat{S}^{2n}(p)\rightarrow\Omega S^{2pn-1}(p)
  4. S ^ 2 n \widehat{S}^{2n}
  5. Ω S 2 n + 1 \Omega S^{2n+1}
  6. S ^ 2 n \widehat{S}^{2n}
  7. S 2 n S^{2n}

Eigendecomposition_of_a_matrix.html

  1. 𝐀𝐯 = λ 𝐯 \mathbf{A}\mathbf{v}=\lambda\mathbf{v}
  2. p ( λ ) := det ( 𝐀 - λ 𝐈 ) = 0. p\left(\lambda\right):=\det\left(\mathbf{A}-\lambda\mathbf{I}\right)=0.\!
  3. p ( λ ) = ( λ - λ 1 ) n 1 ( λ - λ 2 ) n 2 ( λ - λ k ) n k = 0. p\left(\lambda\right)=(\lambda-\lambda_{1})^{n_{1}}(\lambda-\lambda_{2})^{n_{2% }}\cdots(\lambda-\lambda_{k})^{n_{k}}=0.\!
  4. i = 1 N λ n i = N . \sum\limits_{i=1}^{N_{\lambda}}{n_{i}}=N.
  5. ( 𝐀 - λ i 𝐈 ) 𝐯 = 0. \left(\mathbf{A}-\lambda_{i}\mathbf{I}\right)\mathbf{v}=0.\!
  6. i = 1 N λ m i = N 𝐯 . \sum\limits_{i=1}^{N_{\lambda}}{m_{i}}=N_{\mathbf{v}}.
  7. q i ( i = 1 , , N ) . q_{i}\,\,(i=1,\dots,N).
  8. 𝐀 = 𝐐 𝚲 𝐐 - 1 \mathbf{A}=\mathbf{Q}\mathbf{\Lambda}\mathbf{Q}^{-1}
  9. q i q_{i}
  10. Λ i i = λ i \Lambda_{ii}=\lambda_{i}
  11. ( 1 1 0 1 ) \begin{pmatrix}1&1\\ 0&1\\ \end{pmatrix}
  12. q i ( i = 1 , , N ) q_{i}\,\,(i=1,\dots,N)
  13. v i ( i = 1 , , N ) , v_{i}\,\,(i=1,\dots,N),
  14. 𝐀 = [ 1 0 1 3 ] \mathbf{A}=\begin{bmatrix}1&0\\ 1&3\\ \end{bmatrix}
  15. 𝐁 = [ a b c d ] 2 × 2 \mathbf{B}=\begin{bmatrix}a&b\\ c&d\\ \end{bmatrix}\in\mathbb{R}^{2\times 2}
  16. [ a b c d ] - 1 [ 1 0 1 3 ] [ a b c d ] = [ x 0 0 y ] \begin{bmatrix}a&b\\ c&d\\ \end{bmatrix}^{-1}\begin{bmatrix}1&0\\ 1&3\\ \end{bmatrix}\begin{bmatrix}a&b\\ c&d\\ \end{bmatrix}=\begin{bmatrix}x&0\\ 0&y\\ \end{bmatrix}
  17. [ x 0 0 y ] \begin{bmatrix}x&0\\ 0&y\\ \end{bmatrix}
  18. 𝐁 \mathbf{B}
  19. [ 1 0 1 3 ] [ a b c d ] = [ a b c d ] [ x 0 0 y ] \begin{bmatrix}1&0\\ 1&3\\ \end{bmatrix}\begin{bmatrix}a&b\\ c&d\\ \end{bmatrix}=\begin{bmatrix}a&b\\ c&d\\ \end{bmatrix}\begin{bmatrix}x&0\\ 0&y\\ \end{bmatrix}
  20. { [ 1 0 1 3 ] [ a c ] = [ a x c x ] [ 1 0 1 3 ] [ b d ] = [ b y d y ] \begin{cases}\begin{bmatrix}1&0\\ 1&3\end{bmatrix}\begin{bmatrix}a\\ c\end{bmatrix}=\begin{bmatrix}ax\\ cx\end{bmatrix}\\ \begin{bmatrix}1&0\\ 1&3\end{bmatrix}\begin{bmatrix}b\\ d\end{bmatrix}=\begin{bmatrix}by\\ dy\end{bmatrix}\end{cases}
  21. x x
  22. y y
  23. { [ 1 0 1 3 ] [ a c ] = x [ a c ] [ 1 0 1 3 ] [ b d ] = y [ b d ] \begin{cases}\begin{bmatrix}1&0\\ 1&3\end{bmatrix}\begin{bmatrix}a\\ c\end{bmatrix}=x\begin{bmatrix}a\\ c\end{bmatrix}\\ \begin{bmatrix}1&0\\ 1&3\end{bmatrix}\begin{bmatrix}b\\ d\end{bmatrix}=y\begin{bmatrix}b\\ d\end{bmatrix}\end{cases}
  24. a = [ a c ] , b = [ b d ] \overrightarrow{a}=\begin{bmatrix}a\\ c\end{bmatrix},\overrightarrow{b}=\begin{bmatrix}b\\ d\end{bmatrix}
  25. { A a = x a A b = y b \begin{cases}A\overrightarrow{a}=x\overrightarrow{a}\\ A\overrightarrow{b}=y\overrightarrow{b}\end{cases}
  26. 𝐀𝐮 = λ 𝐮 \mathbf{A}\mathbf{u}=\lambda\mathbf{u}
  27. λ \lambda
  28. x x
  29. y y
  30. 𝐮 \mathbf{u}
  31. a \overrightarrow{a}
  32. b \overrightarrow{b}
  33. λ 𝐮 \lambda\mathbf{u}
  34. 𝐮 \mathbf{u}
  35. ( 𝐀 - λ 𝐈 ) 𝐮 = 0 (\mathbf{A}-\lambda\mathbf{I})\mathbf{u}=0
  36. 𝐁 \mathbf{B}
  37. 𝐮 \mathbf{u}
  38. ( 𝐀 - λ 𝐈 ) = 𝟎 (\mathbf{A}-\lambda\mathbf{I})=\mathbf{0}
  39. ( 𝐀 - λ 𝐈 ) (\mathbf{A}-\lambda\mathbf{I})
  40. [ 1 - λ 0 1 3 - λ ] = 0 \begin{bmatrix}1-\lambda&0\\ 1&3-\lambda\end{bmatrix}=0
  41. ( 1 - λ ) ( 3 - λ ) = 0 (1-\lambda)(3-\lambda)=0
  42. 𝐀 \mathbf{A}
  43. λ = 1 \lambda=1
  44. λ = 3 \lambda=3
  45. 𝐀 \mathbf{A}
  46. [ 1 0 0 3 ] \begin{bmatrix}1&0\\ 0&3\end{bmatrix}
  47. { [ 1 0 1 3 ] [ a c ] = 1 [ a c ] [ 1 0 1 3 ] [ b d ] = 3 [ b d ] \begin{cases}\begin{bmatrix}1&0\\ 1&3\end{bmatrix}\begin{bmatrix}a\\ c\end{bmatrix}=1\begin{bmatrix}a\\ c\end{bmatrix}\\ \begin{bmatrix}1&0\\ 1&3\end{bmatrix}\begin{bmatrix}b\\ d\end{bmatrix}=3\begin{bmatrix}b\\ d\end{bmatrix}\end{cases}
  48. a = - 2 c , a a=-2c,a\in\mathbb{R}
  49. b = 0 , d b=0,d\in\mathbb{R}
  50. 𝐁 \mathbf{B}
  51. 𝐀 \mathbf{A}
  52. [ - 2 c 0 c d ] , [ c , d ] \begin{bmatrix}-2c&0\\ c&d\end{bmatrix},[c,d]\in\mathbb{R}
  53. [ - 2 c 0 c d ] - 1 [ 1 0 1 3 ] [ - 2 c 0 c d ] = [ 1 0 0 3 ] , [ c , d ] \begin{bmatrix}-2c&0\\ c&d\\ \end{bmatrix}^{-1}\begin{bmatrix}1&0\\ 1&3\\ \end{bmatrix}\begin{bmatrix}-2c&0\\ c&d\\ \end{bmatrix}=\begin{bmatrix}1&0\\ 0&3\\ \end{bmatrix},[c,d]\in\mathbb{R}
  54. 𝐀 - 1 = 𝐐 𝚲 - 1 𝐐 - 1 \mathbf{A}^{-1}=\mathbf{Q}\mathbf{\Lambda}^{-1}\mathbf{Q}^{-1}
  55. [ Λ - 1 ] i i = 1 λ i \left[\Lambda^{-1}\right]_{ii}=\frac{1}{\lambda_{i}}
  56. min | 2 λ s | \min|\nabla^{2}\lambda_{s}|
  57. f ( x ) = a 0 + a 1 x + a 2 x 2 + f(x)=a_{0}+a_{1}x+a_{2}x^{2}+\cdots
  58. f ( 𝐀 ) = 𝐐 f ( 𝚲 ) 𝐐 - 1 f\left(\mathbf{A}\right)=\mathbf{Q}f\left(\mathbf{\Lambda}\right)\mathbf{Q}^{-1}
  59. [ f ( 𝚲 ) ] i i = f ( λ i ) \left[f\left(\mathbf{\Lambda}\right)\right]_{ii}=f\left(\lambda_{i}\right)
  60. 𝐀 - 1 = 𝐐 𝚲 - 1 𝐐 - 1 \mathbf{A}^{-1}=\mathbf{Q}\mathbf{\Lambda}^{-1}\mathbf{Q}^{-1}
  61. [ f ( 𝚲 ) ] i i = f ( λ i ) \left[f\left(\mathbf{\Lambda}\right)\right]_{ii}=f\left(\lambda_{i}\right)
  62. 𝐀 2 = ( 𝐐 𝚲 𝐐 - 1 ) ( 𝐐 𝚲 𝐐 - 1 ) = 𝐐 𝚲 ( 𝐐 - 1 𝐐 ) 𝚲 𝐐 - 1 = 𝐐 𝚲 2 𝐐 - 1 \mathbf{A}^{2}=(\mathbf{Q}\mathbf{\Lambda}\mathbf{Q}^{-1})(\mathbf{Q}\mathbf{% \Lambda}\mathbf{Q}^{-1})=\mathbf{Q}\mathbf{\Lambda}(\mathbf{Q}^{-1}\mathbf{Q})% \mathbf{\Lambda}\mathbf{Q}^{-1}=\mathbf{Q}\mathbf{\Lambda}^{2}\mathbf{Q}^{-1}
  63. 𝐀 n = 𝐐 𝚲 n 𝐐 - 1 \mathbf{A}^{n}=\mathbf{Q}\mathbf{\Lambda}^{n}\mathbf{Q}^{-1}
  64. A * A = A A * A^{*}A=AA^{*}
  65. 𝐀 = 𝐔 𝚲 𝐔 * \mathbf{A}=\mathbf{U}\mathbf{\Lambda}\mathbf{U}^{*}
  66. A = A * A=A^{*}
  67. 𝐀 = 𝐐 𝚲 𝐐 T \mathbf{A}=\mathbf{Q}\mathbf{\Lambda}\mathbf{Q}^{T}
  68. det ( 𝐀 ) = i = 1 N λ λ i n i \det\left(\mathbf{A}\right)=\prod\limits_{i=1}^{N_{\lambda}}{\lambda_{i}^{n_{i% }}}\!
  69. tr ( 𝐀 ) = i = 1 N λ n i λ i \operatorname{tr}\left(\mathbf{A}\right)=\sum\limits_{i=1}^{N_{\lambda}}{{n_{i% }}\lambda_{i}}\!
  70. A v = λ v Av=\lambda v
  71. e i θ v e^{i\theta}v
  72. - v -v
  73. N 𝐯 = N N_{\mathbf{v}}=N\,
  74. 𝐀 \mathbf{A}
  75. λ i 0 i \lambda_{i}\neq 0\;\forall\,i
  76. λ i 0 i \lambda_{i}\neq 0\;\forall\,i
  77. N 𝐯 = N N_{\mathbf{v}}=N
  78. 𝐀 - 1 = 𝐐 𝚲 - 1 𝐐 - 1 \mathbf{A}^{-1}=\mathbf{Q}\mathbf{\Lambda}^{-1}\mathbf{Q}^{-1}
  79. v v
  80. A v A v , A 2 v A 2 v , A 3 v A 3 v , \frac{Av}{\|Av\|},\frac{A^{2}v}{\|A^{2}v\|},\frac{A^{3}v}{\|A^{3}v\|},\dots
  81. ( 𝐀 - λ i 𝐈 ) 𝐯 i , j = 0 \left(\mathbf{A}-\lambda_{i}\mathbf{I}\right)\mathbf{v}_{i,j}=0\!
  82. A v = λ v * . Av=\lambda v^{*}.\,
  83. A 𝐯 = λ B 𝐯 A\mathbf{v}=\lambda B\mathbf{v}\quad\quad
  84. det ( A - λ B ) = 0. \det(A-\lambda B)=0.\,
  85. n n\in\mathbb{N}
  86. { 𝐯 1 , , 𝐯 n } \{\mathbf{v}_{1}\ ,\dots,\mathbf{v}_{n}\}
  87. i { 1 , , n } i\in\{1,\dots,n\}
  88. A 𝐯 i = λ i B 𝐯 i A\mathbf{v}_{i}=\lambda_{i}B\mathbf{v}_{i}\quad
  89. λ i 𝔽 \lambda_{i}\in\mathbb{F}
  90. P = ( | | 𝐯 1 𝐯 n | | ) ( ( 𝐯 1 ) 1 ( 𝐯 n ) 1 ( 𝐯 1 ) n ( 𝐯 n ) n ) P=\begin{pmatrix}|&&|\\ \mathbf{v}_{1}&\cdots&\mathbf{v}_{n}\\ |&&|\\ \end{pmatrix}\equiv\begin{pmatrix}(\mathbf{v}_{1})_{1}&\cdots&(\mathbf{v}_{n})% _{1}\\ \vdots&&\vdots\\ (\mathbf{v}_{1})_{n}&\cdots&(\mathbf{v}_{n})_{n}\\ \end{pmatrix}
  91. ( D ) i j = { λ i , if i = j 0 , else (D)_{ij}=\begin{cases}\lambda_{i},&\,\text{if }i=j\\ 0,&\,\text{else}\end{cases}
  92. 𝐀 = 𝐁𝐏𝐃𝐏 - 1 \mathbf{A}=\mathbf{B}\mathbf{P}\mathbf{D}\mathbf{P}^{-1}
  93. 𝐀𝐏 = 𝐀 ( | | 𝐯 1 𝐯 n | | ) = ( | | A 𝐯 1 A 𝐯 n | | ) = ( | | λ 1 B 𝐯 1 λ n B 𝐯 n | | ) = ( | | B 𝐯 1 B 𝐯 n | | ) 𝐃 = 𝐁𝐏𝐃 \mathbf{A}\mathbf{P}=\mathbf{A}\begin{pmatrix}|&&|\\ \mathbf{v}_{1}&\cdots&\mathbf{v}_{n}\\ |&&|\\ \end{pmatrix}=\begin{pmatrix}|&&|\\ A\mathbf{v}_{1}&\cdots&A\mathbf{v}_{n}\\ |&&|\\ \end{pmatrix}=\begin{pmatrix}|&&|\\ \lambda_{1}B\mathbf{v}_{1}&\cdots&\lambda_{n}B\mathbf{v}_{n}\\ |&&|\\ \end{pmatrix}=\begin{pmatrix}|&&|\\ B\mathbf{v}_{1}&\cdots&B\mathbf{v}_{n}\\ |&&|\\ \end{pmatrix}\mathbf{D}=\mathbf{B}\mathbf{P}\mathbf{D}
  94. A λ B A−λB
  95. λ λ
  96. B - 1 A 𝐯 = λ 𝐯 B^{-1}A\mathbf{v}=\lambda\mathbf{v}\quad\quad
  97. B - 1 A B^{-1}A
  98. 𝐯 1 * B 𝐯 2 = 0 \mathbf{v}_{1}^{*}B\mathbf{v}_{2}=0

Eight-point_algorithm.html

  1. P P
  2. O L P ¯ \overline{O_{L}P}
  3. O R P ¯ \overline{O_{R}P}
  4. O R O L ¯ \overline{O_{R}O_{L}}
  5. X L X_{L}
  6. P P
  7. X R X_{R}
  8. P P
  9. R , T R,T
  10. X R = R ( X L - T ) X_{R}=R(X_{L}-T)
  11. P P
  12. T X L T\wedge X_{L}
  13. T T
  14. X L X_{L}
  15. X L T T X L - T T T X L = ( X L - T ) T T X L = 0 X_{L}^{T}T\wedge X_{L}-T^{T}T\wedge X_{L}=(X_{L}-T)^{T}T\wedge X_{L}=0
  16. I = R T R I=R^{T}R
  17. ( X L - T ) T R T R T X L = 0 (X_{L}-T)^{T}R^{T}RT\wedge X_{L}=0
  18. ( X L - T ) T R T (X_{L}-T)^{T}R^{T}
  19. X R X_{R}
  20. X R T R T X L = X R T R S X L = X R T E X L = 0 X_{R}^{T}RT\wedge X_{L}=X_{R}^{T}RSX_{L}=X_{R}^{T}EX_{L}=0
  21. T T\wedge
  22. S S
  23. R T T = R S R^{T}T\wedge=RS
  24. E E
  25. O L p L ¯ , O R p R ¯ \overline{O_{L}p_{L}},\overline{O_{R}p_{R}}
  26. O L P ¯ , O R p ¯ \overline{O_{L}P},\overline{O_{R}p}
  27. y , y y,y^{\prime}
  28. P P
  29. y T 𝐄 y = 0 y^{\prime T}\mathbf{E}y=0
  30. 𝐄 \mathbf{E}
  31. 𝐄 \mathbf{E}
  32. 𝐄 \mathbf{E}
  33. ( 𝐲 ) T 𝐄 𝐲 = 0 (\mathbf{y}^{\prime})^{T}\,\mathbf{E}\,\mathbf{y}=0
  34. 𝐲 , 𝐲 \mathbf{y},\mathbf{y}^{\prime}
  35. 𝐄 \mathbf{E}
  36. 𝐄 \mathbf{E}
  37. 𝐲 = ( y 1 y 2 1 ) \mathbf{y}=\begin{pmatrix}y_{1}\\ y_{2}\\ 1\end{pmatrix}
  38. 𝐲 = ( y 1 y 2 1 ) \mathbf{y}^{\prime}=\begin{pmatrix}y^{\prime}_{1}\\ y^{\prime}_{2}\\ 1\end{pmatrix}
  39. 𝐄 = ( e 11 e 12 e 13 e 21 e 22 e 23 e 31 e 32 e 33 ) \mathbf{E}=\begin{pmatrix}e_{11}&e_{12}&e_{13}\\ e_{21}&e_{22}&e_{23}\\ e_{31}&e_{32}&e_{33}\end{pmatrix}
  40. y 1 y 1 e 11 + y 1 y 2 e 12 + y 1 e 13 + y 2 y 1 e 21 + y 2 y 2 e 22 + y 2 e 23 + y 1 e 31 + y 2 e 32 + e 33 = 0 y^{\prime}_{1}y_{1}e_{11}+y^{\prime}_{1}y_{2}e_{12}+y^{\prime}_{1}e_{13}+y^{% \prime}_{2}y_{1}e_{21}+y^{\prime}_{2}y_{2}e_{22}+y^{\prime}_{2}e_{23}+y_{1}e_{% 31}+y_{2}e_{32}+e_{33}=0\,
  41. 𝐞 𝐲 ~ = 0 \mathbf{e}\cdot\tilde{\mathbf{y}}=0
  42. 𝐲 ~ = ( y 1 y 1 y 1 y 2 y 1 y 2 y 1 y 2 y 2 y 2 y 1 y 2 1 ) \tilde{\mathbf{y}}=\begin{pmatrix}y^{\prime}_{1}y_{1}\\ y^{\prime}_{1}y_{2}\\ y^{\prime}_{1}\\ y^{\prime}_{2}y_{1}\\ y^{\prime}_{2}y_{2}\\ y^{\prime}_{2}\\ y_{1}\\ y_{2}\\ 1\end{pmatrix}
  43. 𝐞 = ( e 11 e 12 e 13 e 21 e 22 e 23 e 31 e 32 e 33 ) \mathbf{e}=\begin{pmatrix}e_{11}\\ e_{12}\\ e_{13}\\ e_{21}\\ e_{22}\\ e_{23}\\ e_{31}\\ e_{32}\\ e_{33}\end{pmatrix}
  44. 𝐞 \mathbf{e}
  45. 𝐲 ~ \tilde{\mathbf{y}}
  46. 3 × 3 3\times 3
  47. 𝐲 𝐲 T \mathbf{y}^{\prime}\,\mathbf{y}^{T}
  48. 𝐲 ~ \tilde{\mathbf{y}}
  49. 𝐏 k \mathbf{P}_{k}
  50. 𝐲 ~ k \tilde{\mathbf{y}}_{k}
  51. 𝐞 𝐲 ~ k = 0 \mathbf{e}\cdot\tilde{\mathbf{y}}_{k}=0
  52. 𝐞 \mathbf{e}
  53. 𝐲 ~ k \tilde{\mathbf{y}}_{k}
  54. 𝐞 \mathbf{e}
  55. 𝐲 ~ k \tilde{\mathbf{y}}_{k}
  56. 𝐘 \mathbf{Y}
  57. 𝐞 T 𝐘 = 𝟎 \mathbf{e}^{T}\,\mathbf{Y}=\mathbf{0}
  58. 𝐞 \mathbf{e}
  59. 𝐞 \mathbf{e}
  60. 𝐘 \mathbf{Y}
  61. 𝐲 ~ k \tilde{\mathbf{y}}_{k}
  62. 𝐘 \mathbf{Y}
  63. 𝐞 \mathbf{e}
  64. 𝐄 \mathbf{E}
  65. 𝐘 \mathbf{Y}
  66. 𝐞 \mathbf{e}
  67. 𝐞 T 𝐘 \|\mathbf{e}^{T}\,\mathbf{Y}\|
  68. 𝐞 = 1 \|\mathbf{e}\|=1
  69. 𝐞 \mathbf{e}
  70. 𝐘 \mathbf{Y}
  71. 𝐞 \mathbf{e}
  72. 3 × 3 3\times 3
  73. 𝐄 est \mathbf{E}_{\rm est}
  74. 𝐄 \mathbf{E}^{\prime}
  75. 𝐄 - 𝐄 est \|\mathbf{E}^{\prime}-\mathbf{E}_{\rm est}\|
  76. 𝐄 est \mathbf{E}_{\rm est}
  77. 𝐄 est \mathbf{E}_{\rm est}
  78. 𝐄 est = 𝐔 𝐒 𝐕 T \mathbf{E}_{\rm est}=\mathbf{U}\,\mathbf{S}\,\mathbf{V}^{T}
  79. 𝐔 , 𝐕 \mathbf{U},\mathbf{V}
  80. 𝐒 \mathbf{S}
  81. 𝐄 est \mathbf{E}_{\rm est}
  82. 𝐒 \mathbf{S}
  83. 𝐒 = ( 1 0 0 0 1 0 0 0 0 ) \mathbf{S}^{\prime}=\begin{pmatrix}1&0&0\\ 0&1&0\\ 0&0&0\end{pmatrix}
  84. 𝐄 \mathbf{E}^{\prime}
  85. 𝐄 = 𝐔 𝐒 𝐕 T \mathbf{E}^{\prime}=\mathbf{U}\,\mathbf{S}^{\prime}\,\mathbf{V}^{T}
  86. 𝐄 \mathbf{E}^{\prime}
  87. 𝐅 \mathbf{F}
  88. 𝐅 \mathbf{F}
  89. ( 𝐲 ) T 𝐅 𝐲 = 0 (\mathbf{y}^{\prime})^{T}\,\mathbf{F}\,\mathbf{y}=0
  90. 𝐲 , 𝐲 \mathbf{y},\mathbf{y}^{\prime}
  91. 𝐘 \mathbf{Y}
  92. 𝐟 T 𝐘 = 𝟎 \mathbf{f}^{T}\,\mathbf{Y}=\mathbf{0}
  93. 𝐟 \mathbf{f}
  94. 𝐅 \mathbf{F}
  95. 𝐅 \mathbf{F}
  96. 𝐘 \mathbf{Y}
  97. 𝐘 \mathbf{Y}
  98. 𝐘 \mathbf{Y}
  99. 3 \mathbb{R}^{3}
  100. ( y 1 , y 2 ) (y_{1},y_{2})\,
  101. 𝐲 = ( y 1 y 2 1 ) \mathbf{y}=\begin{pmatrix}y_{1}\\ y_{2}\\ 1\end{pmatrix}
  102. y 1 , y 2 y_{1},y_{2}\,
  103. 𝐲 \mathbf{y}
  104. 𝐘 \mathbf{Y}
  105. ( 700 , 700 ) ± ( 100 , 100 ) (700,700)\pm(100,100)\,
  106. 𝐲 \mathbf{y}
  107. 𝐘 \mathbf{Y}
  108. 2 \sqrt{2}
  109. 𝐲 ¯ , 𝐲 ¯ \mathbf{\bar{y}},\mathbf{\bar{y}}^{\prime}
  110. 𝐲 ¯ = 𝐓 𝐲 \mathbf{\bar{y}}=\mathbf{T}\,\mathbf{y}
  111. 𝐲 ¯ = 𝐓 𝐲 \mathbf{\bar{y}}^{\prime}=\mathbf{T}^{\prime}\,\mathbf{y}^{\prime}
  112. 𝐓 , 𝐓 \mathbf{T},\mathbf{T}^{\prime}
  113. 0 = ( 𝐲 ¯ ) T ( ( 𝐓 ) T ) - 1 𝐅 𝐓 - 1 𝐲 ¯ = ( 𝐲 ¯ ) T 𝐅 ¯ 𝐲 ¯ 0=(\mathbf{\bar{y}}^{\prime})^{T}\,((\mathbf{T}^{\prime})^{T})^{-1}\,\mathbf{F% }\,\mathbf{T}^{-1}\,\mathbf{\bar{y}}=(\mathbf{\bar{y}}^{\prime})^{T}\,\mathbf{% \bar{F}}\,\mathbf{\bar{y}}
  114. 𝐅 ¯ = ( ( 𝐓 ) T ) - 1 𝐅 𝐓 - 1 \mathbf{\bar{F}}=((\mathbf{T}^{\prime})^{T})^{-1}\,\mathbf{F}\,\mathbf{T}^{-1}
  115. 𝐲 ¯ , 𝐲 ¯ \mathbf{\bar{y}},\mathbf{\bar{y}}^{\prime}
  116. 𝐅 ¯ \mathbf{\bar{F}}
  117. 𝐘 ¯ \mathbf{\bar{Y}}
  118. 𝐘 \mathbf{Y}
  119. 𝐟 ¯ \mathbf{\bar{f}}
  120. 𝐘 ¯ 𝐟 ¯ \mathbf{\bar{Y}}\,\mathbf{\bar{f}}
  121. 𝐟 \mathbf{f}
  122. 𝐘 \mathbf{Y}
  123. 𝐟 ¯ \mathbf{\bar{f}}
  124. 𝐅 ¯ \mathbf{\bar{F}}
  125. 𝐅 \mathbf{F}
  126. 𝐅 = ( 𝐓 ) T 𝐅 ¯ 𝐓 \mathbf{F}=(\mathbf{T}^{\prime})^{T}\,\mathbf{\bar{F}}\,\mathbf{T}
  127. 𝐄 \mathbf{E}
  128. 𝐄 \mathbf{E}
  129. 𝐄 \mathbf{E}

Elastic_instability.html

  1. θ \theta
  2. M F = F L sin θ M_{F}=FL\sin\theta
  3. F L sin θ = k θ θ FL\sin\theta=k_{\theta}\theta
  4. k θ k_{\theta}
  5. θ \theta
  6. F L ( θ - 1 6 θ 3 ) k θ θ FL\Bigg(\theta-\frac{1}{6}\theta^{3}\Bigg)\approx k_{\theta}\theta
  7. θ = 0 \theta=0
  8. θ ± 6 ( 1 - k θ F L ) \theta\approx\pm\sqrt{6\Bigg(1-\frac{k_{\theta}}{FL}\Bigg)}
  9. F L < k θ FL<k_{\theta}
  10. θ = 0 \theta=0
  11. k θ / L k_{\theta}/L
  12. E spring = k θ θ d θ = 1 2 k θ θ 2 E_{\mathrm{spring}}=\int k_{\theta}\theta\mathrm{d}\theta=\frac{1}{2}k_{\theta% }\theta^{2}
  13. L ( 1 - cos θ ) L(1-\cos\theta)
  14. E force = F d x = F L ( 1 - cos θ ) E_{\mathrm{force}}=\int{F\mathrm{d}x=FL(1-\cos\theta)}
  15. E spring = E force E_{\mathrm{spring}}=E_{\mathrm{force}}
  16. F = k θ / L F=k_{\theta}/L
  17. θ = 0 \theta=0
  18. θ \theta
  19. Δ θ \Delta\theta
  20. M ( θ ) = F L sin θ - k θ θ M(\theta)=FL\sin\theta-k_{\theta}\theta
  21. θ \theta
  22. M ( θ + Δ θ ) = M + Δ M = F L ( sin θ + Δ θ cos θ ) - k θ ( θ + Δ θ ) M(\theta+\Delta\theta)=M+\Delta M=FL(\sin\theta+\Delta\theta\cos\theta)-k_{% \theta}(\theta+\Delta\theta)
  23. Δ M = Δ θ ( F L cos θ - k θ ) \Delta M=\Delta\theta(FL\cos\theta-k_{\theta})
  24. F L sin θ = k θ θ FL\sin\theta=k_{\theta}\theta
  25. θ \theta
  26. Δ θ > 0 \Delta\theta>0
  27. Δ M < 0 \Delta M<0
  28. Δ M Δ θ = d M d θ = F L cos θ - k θ < 0 \frac{\Delta M}{\Delta\theta}=\frac{\mathrm{d}M}{\mathrm{d}\theta}=FL\cos% \theta-k_{\theta}<0
  29. θ = 0 \theta=0
  30. F L < k θ FL<k_{\theta}
  31. | θ | > 2 ( 1 - k θ F L ) |\theta|>\sqrt{2\Bigg(1-\frac{k_{\theta}}{FL}\Bigg)}
  32. F L > k θ FL>k_{\theta}
  33. F L ( sin θ 1 + sin θ 2 ) = k θ θ 1 FL(\sin\theta_{1}+\sin\theta_{2})=k_{\theta}\theta_{1}
  34. F L sin θ 2 = k θ ( θ 2 - θ 1 ) FL\sin\theta_{2}=k_{\theta}(\theta_{2}-\theta_{1})
  35. θ 1 \theta_{1}
  36. θ 2 \theta_{2}
  37. ( F L - k θ F L k θ F L - k θ ) ( θ 1 θ 2 ) = ( 0 0 ) \begin{pmatrix}FL-k_{\theta}&FL\\ k_{\theta}&FL-k_{\theta}\end{pmatrix}\begin{pmatrix}\theta_{1}\\ \theta_{2}\end{pmatrix}=\begin{pmatrix}0\\ 0\end{pmatrix}
  38. F L k θ = 3 2 5 2 { 0.382 2.618 \frac{FL}{k_{\theta}}=\frac{3}{2}\mp\frac{\sqrt{5}}{2}\approx\left\{\begin{% matrix}0.382\\ 2.618\end{matrix}\right.
  39. θ 1 \theta_{1}
  40. θ 2 θ 1 | θ 1 0 = k θ F L - 1 { 1.618 for F L / k θ 0.382 - 0.618 for F L / k θ 2.618 \frac{\theta_{2}}{\theta_{1}}\Big|_{\theta_{1}\neq 0}=\frac{k_{\theta}}{FL}-1% \approx\left\{\begin{matrix}1.618&\,\text{for }FL/k_{\theta}\approx 0.382\\ -0.618&\,\text{for }FL/k_{\theta}\approx 2.618\end{matrix}\right.

Electroacoustic_phenomena.html

  1. C V I ( E S A ) = A ϕ μ d ρ p - ρ m ρ m \ CVI(ESA)=A\phi\mu_{d}\frac{\rho_{p}-\rho_{m}}{\rho_{m}}
  2. C V I ( E S A ) = A ϕ ε 0 ε m ζ K s η K m ρ p - ρ s ρ s \ CVI(ESA)=A\phi\frac{\varepsilon_{0}\varepsilon_{m}\zeta K_{s}}{\eta K_{m}}% \frac{\rho_{p}-\rho_{s}}{\rho_{s}}
  3. κ a 1 {\kappa}a>>1
  4. D u 1 Du<<1
  5. κ a < 1 {\kappa}a<1
  6. C V I = A 2 σ a 3 η ϕ 1 - ϕ ρ p - ρ s ρ s \ CVI=A\frac{2{\sigma}a}{3\eta}\frac{\phi}{1-\phi}\frac{\rho_{p}-\rho_{s}}{% \rho_{s}}

Electrophoretic_light_scattering.html

  1. υ D \upsilon_{D}\,
  2. υ \upsilon\,
  3. g ( τ ) g(\tau)\,
  4. g ( τ ) = A + B exp ( - Γ τ ) cos ( 2 π υ o ) + C exp ( - 2 Γ τ ) ( 1 ) g(\tau)=A+B\exp(-\Gamma\tau)\cos(2\pi\upsilon_{o})+C\exp(-2\Gamma\tau)\,\qquad% (1)
  5. Γ \Gamma\,
  6. υ o \upsilon_{o}\,
  7. υ o = | υ s - υ r | = | ( υ i + υ D ) - ( υ i + υ M ) | ( 2 ) \upsilon_{o}=|\upsilon_{s}-\upsilon_{r}|=|(\upsilon_{i}+\upsilon_{D})-(% \upsilon_{i}+\upsilon_{M})|\qquad(2)
  8. υ s \upsilon_{s}\,
  9. υ r \upsilon_{r}\,
  10. υ i \upsilon_{i}\,
  11. υ M \upsilon_{M}\,
  12. g ( τ ) g(\tau)\,
  13. ± Δ υ \pm\Delta\upsilon\,
  14. Γ / 2 π \Gamma/2\pi\,
  15. υ = 0 \upsilon=0\,
  16. υ D = V q 2 π ( 3 ) \upsilon_{D}=\frac{Vq}{2\pi}\qquad(3)
  17. Γ = D | q | 2 ( 4 ) \Gamma=D|q|^{2}\qquad(4)
  18. υ D \upsilon_{D}\,
  19. q \ q\,
  20. D \ D\,
  21. q \ q\,
  22. | q | = 4 π n λ 0 sin ( θ 2 ) ( 5 ) \ |q|=\frac{4\pi n}{\lambda_{0}}\sin\left(\frac{\theta}{2}\right)\qquad(5)
  23. V \ V\,
  24. E \ E\,
  25. μ o b s \ \mu_{obs}\,
  26. V = μ o b s E ( 6 ) \ \vec{V}=\mu_{obs}\vec{E}\qquad(6)
  27. υ D = μ o b s n E λ 0 sin θ ( 7 ) \upsilon_{D}=\mu_{obs}\frac{nE}{\lambda_{0}}\sin\theta\qquad(7)
  28. E \ E\,
  29. n \ n\,
  30. λ 0 \ \lambda_{0}\,
  31. θ \ \theta\,
  32. v D \ v_{D}\,
  33. | υ M | > | υ D | \ |\upsilon_{M}|>|\upsilon_{D}|\,
  34. υ p = υ o = ± ( υ D - | υ M | ) ( 8 ) \upsilon_{p}=\upsilon_{o}=\pm(\upsilon_{D}-|\upsilon_{M}|)\qquad(8)
  35. υ M \upsilon_{M}\,
  36. d H \ d_{H}\,
  37. d H = k B T 3 π η D ( 10 ) \ d_{H}=\frac{k_{B}T}{3\pi\eta D}\qquad(10)
  38. k B \ k_{B}\,
  39. T \ T\,
  40. η \ \eta\,
  41. U a ( z ) = A U 0 ( z / b ) 2 + Δ U 0 ( z / b ) + ( 1 - A ) U 0 + U p ( 11 ) \ U_{a}(z)=AU_{0}(z/b)^{2}+\Delta U_{0}(z/b)+(1-A)U_{0}+U_{p}\qquad(11)
  42. z = \ z=\,
  43. U a ( z ) = \ U_{a}(z)=\,
  44. U p = \ U_{p}=\,
  45. z / b = \ z/b=\,
  46. U 0 = \ U_{0}=\,
  47. Δ U 0 = \Delta U_{0}=\,
  48. A = 1 ( 2 / 3 ) - ( 0.420166 / k ) ( 12 ) \ A=\frac{1}{(2/3)-(0.420166/k)}\qquad(12)
  49. k = a / b \ k=a/b\,

End_correction.html

  1. Δ L \Delta L
  2. e e
  3. Δ L \Delta L
  4. Δ L \Delta L
  5. Δ L = 0.6 r = 0.3 D \Delta L=0.6\cdot r=0.3\cdot D
  6. r r
  7. D D
  8. Δ L = 1.2 r = 0.6 D \Delta L=1.2\cdot r=0.6\cdot D

Enoyl-(acyl-carrier-protein)_reductase_(NADPH,_A-specific).html

  1. \rightleftharpoons

Enoyl-(acyl-carrier-protein)_reductase_(NADPH,_B-specific).html

  1. \rightleftharpoons

Enrique_Loedel_Palumbo.html

  1. tanh a = v / c \scriptstyle\tanh\ a\ =\ v/c
  2. tanh b = w / c \scriptstyle\tanh\ b\ =\ w/c
  3. z z e - m j z\mapsto ze^{-mj}
  4. e a j e ( a - b ) j / 2 \scriptstyle e^{aj}\mapsto e^{(a-b)j/2}
  5. e b j e ( b - a ) j / 2 . \scriptstyle e^{bj}\mapsto e^{(b-a)j/2}.

Enzyme-thiol_transhydrogenase_(glutathione-disulfide).html

  1. \rightleftharpoons

Ephedrine_dehydrogenase.html

  1. \rightleftharpoons

Epstein_frame.html

  1. P c = N 1 N 2 P m - ( 1 , 111 | U 2 ¯ | ) 2 R i P_{c}=\frac{N_{1}}{N_{2}}\cdot P_{m}-\frac{\left(1,111\cdot|\bar{U_{2}}|\right% )^{2}}{R_{i}}
  2. N 1 N_{1}~{}
  3. N 2 N_{2}~{}
  4. P m P_{m}~{}
  5. R i R_{i}~{}
  6. | U 2 ¯ | |\bar{U_{2}}|
  7. P s = P c 4 l m l m P_{s}=\frac{P_{c}\cdot 4\cdot l}{m\cdot l_{m}}
  8. l l~{}
  9. l m l_{m}~{}
  10. m m~{}

Erdős–Mordell_inequality.html

  1. π \pi
  2. c r a x + b y . cr\geq ax+by.
  3. c ( r + z ) 2 a x + b y + c z 2 . \frac{c(r+z)}{2}\geq\frac{ax+by+cz}{2}.
  4. r ( a / c ) y + ( b / c ) x , r\geq(a/c)y+(b/c)x,
  5. q ( a / b ) z + ( c / b ) x , q\geq(a/b)z+(c/b)x,
  6. p ( b / a ) z + ( c / a ) y . p\geq(b/a)z+(c/a)y.
  7. p + q + r ( b c + c b ) x + ( a c + c a ) y + ( a b + b a ) z . p+q+r\geq\left(\frac{b}{c}+\frac{c}{b}\right)x+\left(\frac{a}{c}+\frac{c}{a}% \right)y+\left(\frac{a}{b}+\frac{b}{a}\right)z.

Erythro-3-hydroxyaspartate_ammonia-lyase.html

  1. \rightleftharpoons

Erythrose-4-phosphate_dehydrogenase.html

  1. \rightleftharpoons

Erythrulose_reductase.html

  1. \rightleftharpoons

Estradiol_17alpha-dehydrogenase.html

  1. \rightleftharpoons

Estradiol_17beta-dehydrogenase.html

  1. \rightleftharpoons

Estradiol_6beta-monooxygenase.html

  1. \rightleftharpoons

Estrone_sulfotransferase.html

  1. \rightleftharpoons

Ethanolamine_ammonia-lyase.html

  1. \rightleftharpoons

Ethanolamine_oxidase.html

  1. \rightleftharpoons

Ethylbenzene_hydroxylase.html

  1. \rightleftharpoons

Euclidean_field.html

  1. ¯ \mathbb{R}\cap\mathbb{\overline{Q}}

Evolutionary_invasion_analysis.html

  1. S r ( m ) S_{r}(m)
  2. S r ( r ) = 0 S_{r}(r)=0
  3. m = r m=r
  4. S r ( r ) S_{r}^{\prime}(r)
  5. S r ( m ) S r ( r ) + S r ( r ) ( m - r ) S_{r}(m)\approx S_{r}(r)+S_{r}^{\prime}(r)(m-r)
  6. m r m\approx r
  7. m m
  8. S r ( m ) S_{r}(m)
  9. r r
  10. m m
  11. S r ( m ) S_{r}(m)
  12. S r ( m ) S_{r}(m)
  13. m = r m=r
  14. r r
  15. S r ( r ) S_{r}^{\prime}(r)
  16. S r ( r ) S_{r}^{\prime}(r)
  17. r * r^{*}
  18. S r * ( r * ) = 0 S_{r^{*}}^{\prime}(r^{*})=0
  19. r * r^{*}
  20. S r * ( r * ) = 0 S^{\prime}_{r^{*}}(r^{*})=0
  21. r * r^{*}
  22. r * r^{*}
  23. S r * ′′ ( r * ) < 0 S_{r^{*}}^{\prime\prime}(r^{*})<0
  24. r * r^{*}
  25. r * r^{*}
  26. S r ( r ) S_{r}^{\prime}(r)
  27. r * r^{*}
  28. r < r * r<r^{*}
  29. r > r * r>r^{*}
  30. S r ( r ) S_{r}^{\prime}(r)
  31. r r
  32. r * r^{*}
  33. d d r S r ( r ) | r = r * < 0. \frac{d}{dr}S_{r}^{\prime}(r)\Big|_{r=r^{*}}<0.
  34. r 1 r_{1}
  35. r 2 r_{2}
  36. S r 1 ( r 2 ) > 0 S_{r_{1}}(r_{2})>0
  37. S r 2 ( r 1 ) > 0 , S_{r_{2}}(r_{1})>0,
  38. r 1 r_{1}
  39. r 2 r_{2}
  40. S r 1 , r 2 ( m ) S_{r_{1},r_{2}}(m)
  41. r 1 r_{1}
  42. r 2 r_{2}
  43. r 1 r_{1}
  44. r 2 r_{2}
  45. S r 1 , r 2 ( r 1 ) S_{r_{1},r_{2}}^{\prime}(r_{1})
  46. S r 1 , r 2 ( r 2 ) S_{r_{1},r_{2}}^{\prime}(r_{2})
  47. r 1 r_{1}
  48. r 2 r_{2}
  49. S r ( m ) S r 1 , r 2 ( m ) S_{r}(m)\approx S_{r_{1},r_{2}}(m)
  50. S r 1 , r 2 ( r 1 ) = S r 1 , r 2 ( r 2 ) = 0 , S_{r_{1},r_{2}}(r_{1})=S_{r_{1},r_{2}}(r_{2})=0,

Exceptional_divisor.html

  1. f : X Y f:X\rightarrow Y
  2. X X
  3. f f
  4. f : X Y f:X\rightarrow Y
  5. X X
  6. Y Y
  7. Z X Z\subset X
  8. f ( Z ) f(Z)
  9. Y Y
  10. f f
  11. i Z i D i v ( X ) , \sum_{i}Z_{i}\in Div(X),
  12. f f
  13. X X
  14. σ : X ~ X \sigma:\tilde{X}\rightarrow X
  15. W X W\subset X
  16. W W

Extension_of_scalars.html

  1. S S
  2. R R
  3. f : R S f:R\to S
  4. f : R S f:R\to S
  5. M M
  6. R R
  7. M S = S R M {}_{S}M=S\otimes_{R}M
  8. S S
  9. R R
  10. f f
  11. S S
  12. s ( s r ) = ( s s ) r s\cdot(s^{\prime}\cdot r)=(s\cdot s^{\prime})\cdot r
  13. s , s S s,s^{\prime}\in S
  14. r R r\in R
  15. S S
  16. ( S , R ) (S,R)
  17. M S {}_{S}M
  18. S S
  19. s ( s m ) = s s m s\cdot(s^{\prime}\otimes m)=ss^{\prime}\otimes m
  20. s , s S s,s^{\prime}\in S
  21. m M m\in M
  22. M M
  23. ( S , R ) (S,R)
  24. x 2 + 1 , x^{2}+1,
  25. R R
  26. S S
  27. M M
  28. M S {}_{S}M
  29. R R
  30. u : M N u:M\to N
  31. S S
  32. u S : S M S N u_{S}:_{S}M\to_{S}N
  33. u S = id S u u_{S}=\,\text{id}_{S}\otimes u
  34. R R
  35. M M
  36. S S
  37. N N
  38. u Hom R ( M , N ) u\in\,\text{Hom}_{R}(M,N)
  39. N N
  40. R R
  41. F u : S M N Fu:_{S}M\to N
  42. M S = S R M id S u S R N N {}_{S}M=S\otimes_{R}M\xrightarrow{\,\text{id}_{S}\otimes u}S\otimes_{R}N\to N
  43. s n s n s\otimes n\mapsto sn
  44. F u Fu
  45. S S
  46. F : Hom R ( M , N ) Hom S ( S M , N ) F:\,\text{Hom}_{R}(M,N)\to\,\text{Hom}_{S}(_{S}M,N)
  47. R R
  48. S S
  49. G : Hom S ( S M , N ) Hom R ( M , N ) G:\,\text{Hom}_{S}(_{S}M,N)\to\,\text{Hom}_{R}(M,N)
  50. v Hom S ( S M , N ) v\in\,\text{Hom}_{S}(_{S}M,N)
  51. G v Gv
  52. M R R M f id M S R M 𝑣 N M\to R\otimes_{R}M\xrightarrow{f\otimes\,\text{id}_{M}}S\otimes_{R}M% \xrightarrow{v}N
  53. m 1 m m\mapsto 1\otimes m
  54. Hom S ( S M , N ) \,\text{Hom}_{S}(_{S}M,N)
  55. Hom R ( M , N ) \,\text{Hom}_{R}(M,N)
  56. f f

Extensional_viscosity.html

  1. η e = σ n ε ˙ \eta_{e}=\frac{\sigma_{n}}{\dot{\varepsilon}}\,\!
  2. η e \eta_{e}\,\!
  3. σ n \sigma_{n}\,\!
  4. σ n = F A \sigma_{n}=\frac{F}{A}\,\!
  5. ε ˙ \dot{\varepsilon}\,\!
  6. ε ˙ = 1 L d L d t \dot{\varepsilon}=\frac{1}{L}\frac{dL}{dt}\,\!
  7. η e = 3 η \eta_{e}=3\eta\,\!

External_(mathematics).html

  1. f : R × S S f:R\times S\rightarrow S
  2. f : S × R S f:S\times R\rightarrow S
  3. f : R × R S f:R\times R\rightarrow S
  4. f : Q × R S f:Q\times R\rightarrow S
  5. z q : × z^{q}:\mathbb{Z}\times\mathbb{Q}\rightarrow\mathbb{C}
  6. ( - 1 ) 1 / 2 = i {(-1)}^{1/2}=i
  7. ( ) : S × 𝐒𝐞𝐭 𝔹 (\in):S\times\mathbf{Set}\rightarrow\mathbb{B}
  8. 𝐒𝐞𝐭 \mathbf{Set}
  9. f : R × R S f:R\times R\rightarrow S
  10. ( ) : R × R 𝔹 (\leq):R\times R\rightarrow\mathbb{B}
  11. f : R × S S f:R\times S\rightarrow S
  12. f : S × S S f:S\times S\rightarrow S
  13. ( S , × ) (S,\times)
  14. r × s S r\times s\in S
  15. s S , r R s\in S,r\in R
  16. ( S , × ) (S,\times)
  17. ( R , ) (R,\cdot)
  18. ( r 1 r 2 ) × s = r 1 × ( r 2 × s ) (r_{1}\cdot r_{2})\times s=r_{1}\times(r_{2}\times s)
  19. s S , r 1 , r 2 R s\in S,r_{1},r_{2}\in R
  20. ( S , × ) (S,\times)
  21. ( R , ) (R,\cdot)
  22. 1 R 1\in R
  23. 1 × s = s 1\times s=s
  24. s S s\in S
  25. \otimes
  26. ( S , , ) (S,\oplus,\otimes)
  27. ( R , + , ) (R,+,\cdot)
  28. r ( s 1 s 2 ) = ( r s 1 ) ( r s 2 ) r\otimes(s_{1}\oplus s_{2})=(r\otimes s_{1})\oplus(r\otimes s_{2})
  29. s 1 , s 2 S , r R s_{1},s_{2}\in S,r\in R
  30. ( r 1 + r 2 ) s = ( r 1 s ) ( r 2 s ) (r_{1}+r_{2})\otimes s=(r_{1}\otimes s)\oplus(r_{2}\otimes s)
  31. s S , r 1 , r 2 R s\in S,r_{1},r_{2}\in R
  32. ( S , , ) (S,\oplus,\otimes)
  33. ( R , + , ) (R,+,\cdot)
  34. ( S , ) (S,\oplus)
  35. ( S , ) (S,\otimes)
  36. \otimes
  37. ( S , , ) (S,\oplus,\otimes)
  38. ( R , + , ) (R,+,\cdot)
  39. ( S , , ) (S,\oplus,\otimes)
  40. ( R , + , ) (R,+,\cdot)
  41. ( S , ) (S,\oplus)
  42. ( S , ) (S,\otimes)
  43. \otimes
  44. ( S , , ) (S,\oplus,\otimes)
  45. ( R , + , ) (R,+,\cdot)
  46. ( , ) (\mathbb{C},\uparrow)
  47. ( , ) (\mathbb{C},\cdot)
  48. ( , , ) (\mathbb{N},\cdot,\uparrow)
  49. ( , + , ) (\mathbb{N},+,\cdot)
  50. ( T , S , Φ ) (T,S,\Phi)
  51. ( S , Φ ) (S,\Phi)
  52. ( T , + ) (T,{+})

FAD-AMP_lyase_(cyclizing).html

  1. \rightleftharpoons

Farnesol_2-isomerase.html

  1. \rightleftharpoons

Farnesol_dehydrogenase.html

  1. \rightleftharpoons

Fatty-acid_O-methyltransferase.html

  1. \rightleftharpoons

Fatty-acid_peroxidase.html

  1. \rightleftharpoons

Fåhræus–Lindqvist_effect.html

  1. Q = π R 4 Δ P 8 μ e L \ Q=\frac{\pi R^{4}\Delta P}{8\mu_{e}L}
  2. Q Q
  3. Δ P \Delta P
  4. L L
  5. μ e \mu_{e}
  6. R R
  7. π \pi
  8. μ e \mu_{e}
  9. μ e \mu_{e}
  10. μ e \mu_{e}
  11. H R = tube hematocrit feed reservoir hematocrit \mathrm{H_{R}}={\mbox{tube hematocrit}~{}\over\mbox{feed reservoir hematocrit}% ~{}}

Feit–Thompson_conjecture.html

  1. p q - 1 p - 1 divides q p - 1 q - 1 . \frac{p^{q}-1}{p-1}\,\text{ divides }\frac{q^{p}-1}{q-1}.

Fermat's_Last_Theorem_in_fiction.html

  1. 1782 12 + 1841 12 = 1922 12 1782^{12}+1841^{12}=1922^{12}
  2. 1922 12 1922^{12}
  3. 3987 12 + 4365 12 = 4472 12 3987^{12}+4365^{12}=4472^{12}
  4. a n + b n = c n a^{n}+b^{n}=c^{n}

Ferredoxin_hydrogenase.html

  1. \rightleftharpoons

Ferredoxin—NAD(+)_reductase.html

  1. \rightleftharpoons

Ferredoxin—NADP(+)_reductase.html

  1. \rightleftharpoons

Ferredoxin—nitrate_reductase.html

  1. \rightleftharpoons

Ferredoxin—nitrite_reductase.html

  1. \rightleftharpoons

Ferric-chelate_reductase.html

  1. \rightleftharpoons

Feynman_checkerboard.html

  1. x / ϵ c x/\epsilon c
  2. t / ϵ t/\epsilon
  3. ϵ \epsilon\,
  4. m m\,
  5. ϵ c \epsilon c\,
  6. c c\,
  7. - i ϵ m c 2 / -i\epsilon mc^{2}/\hbar
  8. \hbar\,

Field_with_one_element.html

  1. [ n ] q := q n - 1 q - 1 = 1 + q + q 2 + + q n - 1 . [n]_{q}:=\frac{q^{n}-1}{q-1}=1+q+q^{2}+\dots+q^{n-1}.
  2. [ n ] q ! := [ 1 ] q [ 2 ] q [ n ] q [n]_{q}!:=[1]_{q}[2]_{q}\dots[n]_{q}
  3. n ! m ! ( n - m ) ! \frac{n!}{m!(n-m)!}
  4. [ n ] q ! [ m ] q ! [ n - m ] q ! \frac{[n]_{q}!}{[m]_{q}![n-m]_{q}!}
  5. 𝐅 1 n = μ n . \mathbf{F}_{1^{n}}=\mu_{n}.
  6. 𝔽 1 \mathbb{F}_{1}

File:GeometricSegment.png.html

  1. 1 2 + 1 4 + 1 8 + 1 16 + = 1 \frac{1}{2}\,+\,\frac{1}{4}\,+\,\frac{1}{8}\,+\,\frac{1}{16}\,+\,\cdots\;=\;1

File:LogDirichletDensity-alpha_0.1_to_alpha_1.9.gif.html

  1. log ( f ( x 1 , , x K - 1 ; α 1 , , α K ) ) = log ( 1 B ( α ) i = 1 K x i α i - 1 ) \log(f(x_{1},\dots,x_{K-1};\alpha_{1},\dots,\alpha_{K}))=\log(\frac{1}{\mathrm% {B}(\alpha)}\prod_{i=1}^{K}x_{i}^{\alpha_{i}-1})
  2. K = 3 K=3
  3. x 1 , x 2 x_{1},x_{2}
  4. x 3 = 1 - x 1 - x 2 x_{3}=1-x_{1}-x_{2}
  5. α 1 = α 2 = α 3 = α \alpha_{1}=\alpha_{2}=\alpha_{3}=\alpha
  6. α \alpha

File:Twin_5.png.html

  1. v = c 3 , γ = 3 2 v=\frac{c}{\sqrt{3}},\ \gamma=\sqrt{\frac{3}{2}}

Filling_radius.html

  1. FillRad ( C 2 ) = R . \mathrm{FillRad}(C\subset\mathbb{R}^{2})=R.
  2. ε \varepsilon
  3. U ε C 2 . U_{\varepsilon}C\subset\mathbb{R}^{2}.
  4. ε > 0 \varepsilon>0
  5. ε \varepsilon
  6. U ε C U_{\varepsilon}C
  7. FillRad ( C 2 ) \mathrm{FillRad}(C\subset\mathbb{R}^{2})
  8. ε > 0 \varepsilon>0
  9. U ε C U_{\varepsilon}C
  10. U ε X E U_{\varepsilon}X\subset E
  11. \mathbb{Z}
  12. 2 \mathbb{Z}_{2}
  13. H n ( X ; A ) A H_{n}(X;A)\simeq A
  14. FillRad ( X E ) = inf { ε > 0 ι ε ( [ X ] ) = 0 H n ( U ε X ) } , \mathrm{FillRad}(X\subset E)=\inf\left\{\varepsilon>0\mid\iota_{\varepsilon}([% X])=0\in H_{n}(U_{\varepsilon}X)\right\},
  15. ι ε \iota_{\varepsilon}
  16. L ( X ) L^{\infty}(X)
  17. \|\cdot\|
  18. x X x\in X
  19. f x L ( X ) f_{x}\in L^{\infty}(X)
  20. f x ( y ) = d ( x , y ) f_{x}(y)=d(x,y)
  21. y X y\in X
  22. d ( x , y ) = f x - f y , d(x,y)=\|f_{x}-f_{y}\|,
  23. π \pi
  24. E = L ( X ) E=L^{\infty}(X)
  25. FillRad ( X ) = FillRad ( X L ( X ) ) . \mathrm{FillRad}(X)=\mathrm{FillRad}\left(X\subset L^{\infty}(X)\right).
  26. FillRad M InjRad M 2 ( dim M + 2 ) . \mathrm{FillRad}M\geq\frac{\mathrm{InjRad}M}{2(\dim M+2)}.

Financial_accelerator.html

  1. X = C + B X=C+B
  2. B P A B\leq PA
  3. X C + P A X\leq C+PA

First_Hurwitz_triplet.html

  1. K K
  2. [ ρ ] \mathbb{Q}[\rho]
  3. ρ \rho
  4. [ η ] \mathbb{Z}[\eta]
  5. η = 2 cos ( 2 π 7 ) \eta=2\cos(\tfrac{2\pi}{7})
  6. D D
  7. ( η , η ) K (\eta,\eta)_{K}
  8. τ = 1 + η + η 2 \tau=1+\eta+\eta^{2}
  9. j = 1 2 ( 1 + η i + τ j ) j^{\prime}=\tfrac{1}{2}(1+\eta i+\tau j)
  10. 𝒬 Hur = [ η ] [ i , j , j ] \mathcal{Q}_{\mathrm{Hur}}=\mathbb{Z}[\eta][i,j,j^{\prime}]
  11. 𝒬 Hur \mathcal{Q}_{\mathrm{Hur}}
  12. D D
  13. [ η ] \mathbb{Z}[\eta]
  14. 13 = η ( η + 2 ) ( 2 η - 1 ) ( 3 - 2 η ) ( η + 3 ) , 13=\eta(\eta+2)(2\eta-1)(3-2\eta)(\eta+3),
  15. η ( η + 2 ) \eta(\eta+2)
  16. 𝒬 Hur 1 ( I ) = { x 𝒬 Hur 1 : x 1 ( mod I 𝒬 Hur ) } , \mathcal{Q}^{1}_{\mathrm{Hur}}(I)=\{x\in\mathcal{Q}_{\mathrm{Hur}}^{1}:x\equiv 1% \;\;(\mathop{{\rm mod}}I\mathcal{Q}_{\mathrm{Hur}})\},
  17. 𝒬 Hur \mathcal{Q}_{\mathrm{Hur}}
  18. I 𝒬 H u r I\mathcal{Q}_{\mathrm{H}ur}
  19. χ ( Σ ) = 1 2 π Σ K ( u ) d A , \chi(\Sigma)=\frac{1}{2\pi}\int_{\Sigma}K(u)\,dA,
  20. χ ( Σ ) \chi(\Sigma)
  21. K ( u ) K(u)
  22. g = 14 g=14
  23. χ ( Σ ) = - 26 \chi(\Sigma)=-26
  24. K ( u ) = - 1 , K(u)=-1,
  25. 52 π 52\pi
  26. 4 3 log ( g ( Σ ) ) , \frac{4}{3}\log(g(\Sigma)),
  27. 𝒬 H u r 1 ( I ) \mathcal{Q}^{1}_{Hur}(I)
  28. 3 - 2 η O K 3-2\eta\vartriangleleft O_{K}
  29. - 4 η 2 - 8 η - 3 -4\eta^{2}-8\eta-3
  30. η + 3 O K \eta+3\vartriangleleft O_{K}
  31. 5 η 2 + 11 η + 3 5\eta^{2}+11\eta+3
  32. 2 η - 1 O K 2\eta-1\vartriangleleft O_{K}
  33. - 7 η 2 - 14 η - 3 -7\eta^{2}-14\eta-3

Fixed_end_moment.html

  1. q q
  2. d x dx
  3. x x
  4. q d x qdx
  5. M right fixed = 0 L q d x x 2 ( L - x ) L 2 = q L 2 12 M_{\mathrm{right}}^{\mathrm{fixed}}=\int_{0}^{L}\frac{qdx\,x^{2}(L-x)}{L^{2}}=% \frac{qL^{2}}{12}
  6. M left fixed = 0 L { - q d x x ( L - x ) 2 L 2 } = - q L 2 12 M_{\mathrm{left}}^{\mathrm{fixed}}=\int_{0}^{L}\left\{-\frac{qdx\,x(L-x)^{2}}{% L^{2}}\right\}=-\frac{qL^{2}}{12}
  7. q d x qdx
  8. q 0 q_{0}
  9. M right fixed = 0 L q 0 x L d x x 2 ( L - x ) L 2 = q 0 L 2 20 M_{\mathrm{right}}^{\mathrm{fixed}}=\int_{0}^{L}q_{0}\frac{x}{L}dx\frac{x^{2}(% L-x)}{L^{2}}=\frac{q_{0}L^{2}}{20}
  10. M left fixed = 0 L { - q 0 x L d x x ( L - x ) 2 L 2 } = - q 0 L 2 30 M_{\mathrm{left}}^{\mathrm{fixed}}=\int_{0}^{L}\left\{-q_{0}\frac{x}{L}dx\frac% {x(L-x)^{2}}{L^{2}}\right\}=-\frac{q_{0}L^{2}}{30}

Flashsort.html

  1. O ( n ) O(n)
  2. m m
  3. A i A_{i}
  4. K ( A i ) = 1 + INT ( ( m - 1 ) A i - A min A max - A min ) K(A_{i})=1+\textrm{INT}\left((m-1)\frac{A_{i}-A_{\textrm{min}}}{A_{\textrm{max% }}-A_{\textrm{min}}}\right)
  5. A A
  6. A A
  7. L L
  8. A A
  9. L L
  10. L L
  11. A A
  12. A A
  13. L L
  14. A A
  15. i i
  16. A 0 A_{0}
  17. A i-1 A_{\textrm{i-1}}
  18. i i
  19. A i A_{i}
  20. A i A_{i}
  21. A A
  22. A i A_{i}
  23. A i A_{i}
  24. L L
  25. O ( 1 ) O(1)
  26. m m
  27. n n
  28. m O ( 1 ) = O ( m ) = O ( n ) m\cdot O(1)=O(m)=O(n)
  29. O ( n 2 ) O(n^{2})
  30. m m
  31. m m
  32. m m
  33. m = 0.42 n m=0.42n
  34. m = 0.1 n m=0.1n
  35. n n
  36. n > 80 n>80
  37. n = 10000 n=10000
  38. O ( n ) O(n)

Flat_(geometry).html

  1. n n
  2. n n
  3. n 1 n−1
  4. n 1 n−1
  5. x x
  6. y y
  7. 3 x + 5 y = 8. 3x+5y=8.
  8. x x
  9. y y
  10. z z
  11. n n
  12. k k
  13. n k n−k
  14. x = 2 + 3 t , y = - 1 + t z = 3 2 - 4 t x=2+3t,\;\;\;\;y=-1+t\;\;\;\;z=\frac{3}{2}-4t
  15. x = 5 + 2 t 1 - 3 t 2 , y = - 4 + t 1 + 2 t 2 z = 5 t 1 - 3 t 2 . x=5+2t_{1}-3t_{2},\;\;\;\;y=-4+t_{1}+2t_{2}\;\;\;\;z=5t_{1}-3t_{2}.\,\!
  16. k k
  17. t < s u b > 1 , , t k t<sub>1, … ,t_{k}
  18. n n
  19. n n

Flat_manifold.html

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

Flavanone_3-dioxygenase.html

  1. \rightleftharpoons

Flavanone_4-reductase.html

  1. \rightleftharpoons

Flavin_reductase.html

  1. \rightleftharpoons

Flavone_synthase.html

  1. \rightleftharpoons

Flavonoid_3'-monooxygenase.html

  1. \rightleftharpoons

Flavonol_3-sulfotransferase.html

  1. \rightleftharpoons

Flavonol_synthase.html

  1. \rightleftharpoons

Flexural_modulus.html

  1. E bend = L 3 F 4 w h 3 d E_{\mathrm{bend}}=\frac{L^{3}F}{4wh^{3}d}
  2. d = L 3 F 48 I E d=\frac{L^{3}F}{48IE}
  3. I = 1 12 w h 3 I=\frac{1}{12}wh^{3}
  4. E bend = E E_{\mathrm{bend}}=E

Flip_(mathematics).html

  1. X X
  2. X = X 1 X 2 X n X=X_{1}\rightarrow X_{2}\rightarrow\cdots\rightarrow X_{n}
  3. K X i K_{X_{i}}
  4. K X n K_{X_{n}}
  5. X i X_{i}
  6. K X i K_{X_{i}}
  7. K X i C K_{X_{i}}\cdot C
  8. C C
  9. X i X_{i}
  10. X i X_{i}
  11. f : X i X i + f:X_{i}\rightarrow X_{i}^{+}
  12. X i X_{i}
  13. X i + 1 = X i + X_{i+1}=X_{i}^{+}
  14. m f * ( 𝒪 X ( m K ) ) \oplus_{m}f_{*}(\mathcal{O}_{X}(mK))
  15. f + : X + = P r o j ( m f * ( 𝒪 X ( m K ) ) ) Y f^{+}:X^{+}=Proj(\oplus_{m}f_{*}(\mathcal{O}_{X}(mK)))\to Y

Fluoren-9-ol_dehydrogenase.html

  1. \rightleftharpoons

Fluoroacetaldehyde_dehydrogenase.html

  1. \rightleftharpoons

FMN_reductase.html

  1. \rightleftharpoons

Foldback_(power_supply_design).html

  1. P dissipation in regulator = ( V in - V out ) × I out P_{\rm dissipation\,in\,regulator}=(V_{\rm in}-V_{\rm out})\times I_{\rm out}

Folded_spectrum_method.html

  1. ε \varepsilon
  2. Ψ \Psi
  3. Ψ i + 1 = Ψ i - α ( H - ε 𝟏 ) 2 Ψ i \Psi_{i+1}=\Psi_{i}-\alpha(H-\varepsilon\mathbf{1})^{2}\Psi_{i}
  4. 0 < α < 1 0<\alpha<1
  5. 𝟏 \mathbf{1}
  6. H : G H G H 2 . H:\;G\sim H\rightarrow G\sim H^{2}.

Forbidden_graph_characterization.html

  1. \mathcal{F}
  2. C 4 , C 5 , C 4 ¯ ( = K 2 + K 2 ) C_{4},C_{5},\bar{C_{4}}(=K_{2}+K_{2})

Forcing_(recursion_theory).html

  1. \Vdash
  2. 2 ω 2^{\omega}
  3. 2 < ω 2^{<\omega}
  4. P P
  5. P P
  6. P \succ_{P}
  7. 0 P 0_{P}
  8. p p
  9. q q
  10. q P p q\succ_{P}p
  11. p q p\mid q
  12. p , q p,q
  13. p p
  14. q q
  15. r r
  16. p P r p\succ_{P}r
  17. q P r q\succ_{P}r
  18. p p
  19. q q
  20. F F
  21. P P
  22. p , q F p q p,q\in F\implies p\nmid q
  23. p F q P p q F p\in F\land q\succ_{P}p\implies q\in F
  24. F F
  25. F F
  26. F F^{\prime}
  27. F F
  28. C C
  29. 2 < ω 2^{<\omega}
  30. ( τ C σ σ τ (\tau\succ_{C}\sigma\iff\sigma\supset\tau
  31. C \succ_{C}
  32. P \succ_{P}
  33. P \prec_{P}
  34. σ \sigma
  35. τ \tau
  36. σ \sigma
  37. τ \tau
  38. τ C σ \tau\succ_{C}\sigma
  39. σ \sigma
  40. σ P ψ \sigma\Vdash_{P}\psi
  41. σ \sigma
  42. ψ \psi
  43. P P
  44. ψ \psi
  45. ψ \psi
  46. P P

Formaldehyde_dehydrogenase.html

  1. \rightleftharpoons

Formaldehyde_dismutase.html

  1. \rightleftharpoons

Formate_dehydrogenase_(cytochrome).html

  1. \rightleftharpoons

Formate_dehydrogenase_(cytochrome-c-553).html

  1. \rightleftharpoons

Formate_dehydrogenase_(NADP+).html

  1. \rightleftharpoons

Formimidoyltetrahydrofolate_cyclodeaminase.html

  1. \rightleftharpoons

Formyl-CoA_transferase.html

  1. \rightleftharpoons

Formylmethanofuran_dehydrogenase.html

  1. \rightleftharpoons

Formyltetrahydrofolate_dehydrogenase.html

  1. \rightleftharpoons

Fractional_vortices.html

  1. | Ψ | e i ϕ |\Psi|e^{i\phi}
  2. 2 π 2\pi
  3. ϕ \phi
  4. | Ψ | e i ϕ |\Psi|e^{i\phi}
  5. ϕ {\phi}
  6. 2 π 2\pi
  7. 2 π × 𝑖𝑛𝑡𝑒𝑔𝑒𝑟 2\pi\times\mathit{integer}
  8. π \pi
  9. 2 π N 2\pi N
  10. N = ± 1 , ± 2 , N=\pm 1,\pm 2,...
  11. κ π κ≠π
  12. ψ = ± φ ψ=±φ
  13. φ = c o s ( - 1 / g ) φ=cos(-1/g)
  14. ψ ( x ) ψ(x)
  15. - φ
  16. + φ
  17. ψ ( x ) ψ(x)
  18. + φ
  19. - φ + 2 π -φ+2π
  20. 2 π - 2 φ 2π-2φ
  21. π \pi
  22. π \pi
  23. 2 π 2\pi
  24. 2 π 2\pi
  25. U ( 1 ) × U ( 1 ) U(1)\times U(1)
  26. 2 π 2\pi

FRACTRAN.html

  1. ( 17 91 , 78 85 , 19 51 , 23 38 , 29 33 , 77 29 , 95 23 , 77 19 , 1 17 , 11 13 , 13 11 , 15 14 , 15 2 , 55 1 ) \left(\frac{17}{91},\frac{78}{85},\frac{19}{51},\frac{23}{38},\frac{29}{33},% \frac{77}{29},\frac{95}{23},\frac{77}{19},\frac{1}{17},\frac{11}{13},\frac{13}% {11},\frac{15}{14},\frac{15}{2},\frac{55}{1}\right)
  2. 2 2 = 4 , 2 3 = 8 , 2 5 = 32 , 2 7 = 128 , 2 11 = 2048 , 2 13 = 8192 , 2 17 = 131072 , 2 19 = 524288 , 2^{2}=4,\,2^{3}=8,\,2^{5}=32,\,2^{7}=128,\,2^{11}=2048,\,2^{13}=8192,\,2^{17}=% 131072,\,2^{19}=524288,\,\dots
  3. 60 = 2 2 × 3 1 × 5 1 60=2^{2}\times 3^{1}\times 5^{1}
  4. f 1 = 21 20 = 3 × 7 2 2 × 5 1 f_{1}=\frac{21}{20}=\frac{3\times 7}{2^{2}\times 5^{1}}
  5. v 2 2 v_{2}\geq 2
  6. v 5 1 v_{5}\geq 1
  7. 60 f 1 = 2 2 × 3 1 × 5 1 3 × 7 2 2 × 5 1 = 3 2 × 7 1 60\cdot f_{1}=2^{2}\times 3^{1}\times 5^{1}\cdot\frac{3\times 7}{2^{2}\times 5% ^{1}}=3^{2}\times 7^{1}
  8. ( 3 2 ) \left(\frac{3}{2}\right)
  9. 3 2 \frac{3}{2}
  10. 2 a 3 b 2^{a}3^{b}
  11. 2 a - 1 3 b + 1 2^{a-1}3^{b+1}
  12. 2 a - 2 3 b + 2 2^{a-2}3^{b+2}
  13. a a
  14. 3 2 \frac{3}{2}
  15. 3 a + b 3^{a+b}
  16. 2 a 3 b 2^{a}3^{b}
  17. 5 a b 5^{ab}
  18. 3 7 \frac{3}{7}
  19. 11 2 \frac{11}{2}
  20. 1 3 \frac{1}{3}
  21. 5 7 13 3 11 , 11 13 \frac{5\cdot 7\cdot 13}{3\cdot 11},\frac{11}{13}
  22. 1 11 \frac{1}{11}
  23. ( 455 33 , 11 13 , 1 11 , 3 7 , 11 2 , 1 3 ) \left(\frac{455}{33},\frac{11}{13},\frac{1}{11},\frac{3}{7},\frac{11}{2},\frac% {1}{3}\right)
  24. 2 3 × 3 2 = 72 2^{3}\times 3^{2}=72
  25. 5 6 5^{6}
  26. 7 13 2 3 11 , 11 13 \frac{7\cdot 13}{2\cdot 3\cdot 11},\frac{11}{13}
  27. 1 3 11 \frac{1}{3\cdot 11}
  28. 5 17 11 \frac{5\cdot 17}{11}
  29. 3 19 7 17 , 17 19 \frac{3\cdot 19}{7\cdot 17},\frac{17}{19}
  30. 11 17 \frac{11}{17}
  31. 1 3 \frac{1}{3}
  32. ( 91 66 , 11 13 , 1 33 , 85 11 , 57 119 , 17 19 , 11 17 , 1 3 ) \left(\frac{91}{66},\frac{11}{13},\frac{1}{33},\frac{85}{11},\frac{57}{119},% \frac{17}{19},\frac{11}{17},\frac{1}{3}\right)
  33. ( 3 11 2 2 5 , 5 11 , 13 2 5 , 1 5 , 2 3 , 2 5 7 , 7 2 ) \left(\frac{3\cdot 11}{2^{2}\cdot 5},\frac{5}{11},\frac{13}{2\cdot 5},\frac{1}% {5},\frac{2}{3},\frac{2\cdot 5}{7},\frac{7}{2}\right)
  34. 3 11 2 2 5 , 5 11 \frac{3\cdot 11}{2^{2}\cdot 5},\frac{5}{11}
  35. 13 2 5 \frac{13}{2\cdot 5}
  36. 1 5 \frac{1}{5}
  37. 2 3 \frac{2}{3}
  38. 2 5 7 \frac{2\cdot 5}{7}
  39. 7 2 \frac{7}{2}

Francis_Birch_(geophysicist).html

  1. M ¯ \bar{M}
  2. ρ \rho
  3. V p = a ( M ¯ ) + b ρ V_{p}=a(\bar{M})+b\rho

Freidlin–Wentzell_theorem.html

  1. { d X t ε = b ( X t ε ) d t + ε d B t ; X 0 ε = 0 ; \begin{cases}\mathrm{d}X_{t}^{\varepsilon}=b(X_{t}^{\varepsilon})\,\mathrm{d}t% +\sqrt{\varepsilon}\,\mathrm{d}B_{t};\\ X_{0}^{\varepsilon}=0;\end{cases}
  2. I ( ω ) = 1 2 0 T | ω ˙ t - b ( ω t ) | 2 d t I(\omega)=\frac{1}{2}\int_{0}^{T}|\dot{\omega}_{t}-b(\omega_{t})|^{2}\,\mathrm% {d}t
  3. lim sup ε 0 ε log 𝐏 [ X ε F ] - inf ω F I ( ω ) \limsup_{\varepsilon\downarrow 0}\varepsilon\log\mathbf{P}\big[X^{\varepsilon}% \in F\big]\leq-\inf_{\omega\in F}I(\omega)
  4. lim inf ε 0 ε log 𝐏 [ X ε G ] - inf ω G I ( ω ) . \liminf_{\varepsilon\downarrow 0}\varepsilon\log\mathbf{P}\big[X^{\varepsilon}% \in G\big]\geq-\inf_{\omega\in G}I(\omega).

Freiheitssatz.html

  1. G = x 1 , , x n | r = 1 G=\langle x_{1},\dots,x_{n}|r=1\rangle

Freudenthal_magic_square.html

  1. L = ( 𝔡 𝔢 𝔯 ( A ) 𝔡 𝔢 𝔯 ( J 3 ( B ) ) ) ( A 0 J 3 ( B ) 0 ) L=\left(\mathfrak{der}(A)\oplus\mathfrak{der}(J_{3}(B))\right)\oplus\left(A_{0% }\otimes J_{3}(B)_{0}\right)
  2. 𝔡 𝔢 𝔯 \mathfrak{der}
  3. 𝔡 𝔢 𝔯 ( A ) 𝔡 𝔢 𝔯 ( J 3 ( B ) ) \mathfrak{der}(A)\oplus\mathfrak{der}(J_{3}(B))
  4. A 0 J 3 ( B ) 0 A_{0}\otimes J_{3}(B)_{0}
  5. A 0 J 3 ( B ) 0 A_{0}\otimes J_{3}(B)_{0}
  6. 𝔰 𝔭 1 \mathfrak{sp}_{1}
  7. 𝔤 2 \mathfrak{g}_{2}
  8. 𝔰 𝔬 3 \mathfrak{so}_{3}
  9. 𝔰 𝔲 3 \mathfrak{su}_{3}
  10. 𝔰 𝔭 3 \mathfrak{sp}_{3}
  11. 𝔣 4 \mathfrak{f}_{4}
  12. 𝔰 𝔲 3 \mathfrak{su}_{3}
  13. 𝔰 𝔲 3 𝔰 𝔲 3 \mathfrak{su}_{3}\oplus\mathfrak{su}_{3}
  14. 𝔰 𝔲 6 \mathfrak{su}_{6}
  15. 𝔢 6 \mathfrak{e}_{6}
  16. 𝔰 𝔭 1 \mathfrak{sp}_{1}
  17. 𝔰 𝔭 3 \mathfrak{sp}_{3}
  18. 𝔰 𝔲 6 \mathfrak{su}_{6}
  19. 𝔰 𝔬 12 \mathfrak{so}_{12}
  20. 𝔢 7 \mathfrak{e}_{7}
  21. 𝔤 2 \mathfrak{g}_{2}
  22. 𝔣 4 \mathfrak{f}_{4}
  23. 𝔢 6 \mathfrak{e}_{6}
  24. 𝔢 7 \mathfrak{e}_{7}
  25. 𝔢 8 \mathfrak{e}_{8}
  26. 𝔡 𝔢 𝔯 ( J 3 ( B ) ) \mathfrak{der}(J_{3}(B))
  27. 𝔰 𝔞 3 ( A B ) \mathfrak{sa}_{3}(A\otimes B)
  28. 𝔡 𝔢 𝔯 ( A ) 𝔡 𝔢 𝔯 ( B ) 𝔰 𝔞 3 ( A B ) . \mathfrak{der}(A)\oplus\mathfrak{der}(B)\oplus\mathfrak{sa}_{3}(A\otimes B).
  29. 𝔰 𝔞 3 ( A B ) \mathfrak{sa}_{3}(A\otimes B)
  30. 𝔰 𝔞 3 ( A B ) \mathfrak{sa}_{3}(A\otimes B)
  31. 𝔰 𝔞 3 ( A B ) \mathfrak{sa}_{3}(A\otimes B)
  32. 𝔡 𝔢 𝔯 ( A ) 𝔡 𝔢 𝔯 ( B ) \mathfrak{der}(A)\oplus\mathfrak{der}(B)
  33. A 1 × A 2 × A 3 𝐑 A_{1}\times A_{2}\times A_{3}\to\mathbf{R}
  34. 𝔡 𝔢 𝔯 ( A ) \mathfrak{der}(A)
  35. 𝔰 𝔭 1 \mathfrak{sp}_{1}
  36. 𝔤 2 \mathfrak{g}_{2}
  37. 𝔱 𝔯 𝔦 ( A ) \mathfrak{tri}(A)
  38. 𝔲 1 𝔲 1 \mathfrak{u}_{1}\oplus\mathfrak{u}_{1}
  39. 𝔰 𝔭 1 𝔰 𝔭 1 𝔰 𝔭 1 \mathfrak{sp}_{1}\oplus\mathfrak{sp}_{1}\oplus\mathfrak{sp}_{1}
  40. 𝔰 𝔬 8 \mathfrak{so}_{8}
  41. 𝔱 𝔯 𝔦 ( A ) 𝔱 𝔯 𝔦 ( B ) ( A 1 B 1 ) ( A 2 B 2 ) ( A 3 B 3 ) . \mathfrak{tri}(A)\oplus\mathfrak{tri}(B)\oplus(A_{1}\otimes B_{1})\oplus(A_{2}% \otimes B_{2})\oplus(A_{3}\otimes B_{3}).
  42. 𝔢 8 𝔰 𝔬 8 𝔰 𝔬 ^ 8 ( V V ^ ) ( S + S ^ + ) ( S - S ^ - ) \mathfrak{e}_{8}\cong\mathfrak{so}_{8}\oplus\widehat{\mathfrak{so}}_{8}\oplus(% V\otimes\widehat{V})\oplus(S_{+}\otimes\widehat{S}_{+})\oplus(S_{-}\otimes% \widehat{S}_{-})
  43. 𝔰 𝔬 8 \mathfrak{so}_{8}
  44. 𝔰 𝔬 16 \mathfrak{so}_{16}
  45. 𝔣 4 𝔰 𝔬 9 Δ 16 \mathfrak{f}_{4}\cong\mathfrak{so}_{9}\oplus\Delta^{16}
  46. 𝔢 6 ( 𝔰 𝔬 10 𝔲 1 ) Δ 32 \mathfrak{e}_{6}\cong(\mathfrak{so}_{10}\oplus\mathfrak{u}_{1})\oplus\Delta^{32}
  47. 𝔢 7 ( 𝔰 𝔬 12 𝔰 𝔭 1 ) Δ + 64 \mathfrak{e}_{7}\cong(\mathfrak{so}_{12}\oplus\mathfrak{sp}_{1})\oplus\Delta_{% +}^{64}
  48. 𝔢 8 𝔰 𝔬 16 Δ + 128 . \mathfrak{e}_{8}\cong\mathfrak{so}_{16}\oplus\Delta_{+}^{128}.
  49. 𝔰 𝔬 3 \mathfrak{so}_{3}
  50. 𝔰 𝔩 3 ( 𝐑 ) \mathfrak{sl}_{3}(\mathbf{R})
  51. 𝔰 𝔭 6 ( 𝐑 ) \mathfrak{sp}_{6}(\mathbf{R})
  52. 𝔣 4 ( 4 ) \mathfrak{f}_{4(4)}
  53. 𝔰 𝔩 3 ( 𝐑 ) \mathfrak{sl}_{3}(\mathbf{R})
  54. 𝔰 𝔩 3 ( 𝐑 ) 𝔰 𝔩 3 ( 𝐑 ) \mathfrak{sl}_{3}(\mathbf{R})\oplus\mathfrak{sl}_{3}(\mathbf{R})
  55. 𝔰 𝔩 6 ( 𝐑 ) \mathfrak{sl}_{6}(\mathbf{R})
  56. 𝔢 6 ( 6 ) \mathfrak{e}_{6(6)}
  57. 𝔰 𝔭 6 ( 𝐑 ) \mathfrak{sp}_{6}(\mathbf{R})
  58. 𝔰 𝔩 6 ( 𝐑 ) \mathfrak{sl}_{6}(\mathbf{R})
  59. 𝔰 𝔬 6 , 6 \mathfrak{so}_{6,6}
  60. 𝔢 7 ( 7 ) \mathfrak{e}_{7(7)}
  61. 𝔣 4 ( 4 ) \mathfrak{f}_{4(4)}
  62. 𝔢 6 ( 6 ) \mathfrak{e}_{6(6)}
  63. 𝔢 7 ( 7 ) \mathfrak{e}_{7(7)}
  64. 𝔢 8 ( 8 ) \mathfrak{e}_{8(8)}
  65. 𝔣 4 ( 4 ) \mathfrak{f}_{4(4)}
  66. 𝔢 6 ( 6 ) \mathfrak{e}_{6(6)}
  67. 𝔢 7 ( 7 ) \mathfrak{e}_{7(7)}
  68. 𝔢 8 ( 8 ) \mathfrak{e}_{8(8)}
  69. 𝔰 𝔭 3 𝔰 𝔭 1 \mathfrak{sp}_{3}\oplus\mathfrak{sp}_{1}
  70. 𝔰 𝔭 4 \mathfrak{sp}_{4}
  71. 𝔰 𝔲 8 \mathfrak{su}_{8}
  72. 𝔰 𝔬 16 \mathfrak{so}_{16}
  73. 𝔰 𝔬 3 \mathfrak{so}_{3}
  74. 𝔰 𝔲 3 \mathfrak{su}_{3}
  75. 𝔰 𝔭 3 \mathfrak{sp}_{3}
  76. 𝔣 4 \mathfrak{f}_{4}
  77. 𝔰 𝔩 3 ( 𝐑 ) \mathfrak{sl}_{3}(\mathbf{R})
  78. 𝔰 𝔩 3 ( 𝐂 ) \mathfrak{sl}_{3}(\mathbf{C})
  79. 𝔰 𝔩 3 ( 𝐇 ) \mathfrak{sl}_{3}(\mathbf{H})
  80. 𝔢 6 ( - 26 ) \mathfrak{e}_{6(-26)}
  81. 𝔰 𝔭 6 ( 𝐑 ) \mathfrak{sp}_{6}(\mathbf{R})
  82. 𝔰 𝔲 3 , 3 \mathfrak{su}_{3,3}
  83. 𝔰 𝔬 6 * ( 𝐇 ) \mathfrak{so}^{*}_{6}(\mathbf{H})
  84. 𝔢 7 ( - 25 ) \mathfrak{e}_{7(-25)}
  85. 𝔣 4 ( 4 ) \mathfrak{f}_{4(4)}
  86. 𝔢 6 ( 2 ) \mathfrak{e}_{6(2)}
  87. 𝔢 7 ( - 5 ) \mathfrak{e}_{7(-5)}
  88. 𝔢 8 ( - 24 ) \mathfrak{e}_{8(-24)}
  89. 𝔢 6 ( 2 ) \mathfrak{e}_{6(2)}
  90. 𝔢 6 ( - 26 ) \mathfrak{e}_{6(-26)}
  91. 𝔢 7 ( - 5 ) \mathfrak{e}_{7(-5)}
  92. 𝔢 7 ( - 25 ) \mathfrak{e}_{7(-25)}
  93. 𝔢 8 ( - 24 ) \mathfrak{e}_{8(-24)}
  94. 𝔰 𝔲 6 𝔰 𝔭 1 \mathfrak{su}_{6}\oplus\mathfrak{sp}_{1}
  95. 𝔣 4 \mathfrak{f}_{4}
  96. 𝔰 𝔲 12 𝔰 𝔭 1 \mathfrak{su}_{12}\oplus\mathfrak{sp}_{1}
  97. 𝔢 6 𝔲 1 \mathfrak{e}_{6}\oplus\mathfrak{u}_{1}
  98. 𝔢 7 𝔰 𝔭 1 \mathfrak{e}_{7}\oplus\mathfrak{sp}_{1}
  99. 𝔰 𝔬 3 ( 𝐊 ) \mathfrak{so}_{3}(\mathbf{K})
  100. 𝔰 𝔩 3 ( 𝐊 ) \mathfrak{sl}_{3}(\mathbf{K})
  101. 𝔰 𝔭 6 ( 𝐊 ) \mathfrak{sp}_{6}(\mathbf{K})
  102. 𝔣 4 ( 𝐊 ) \mathfrak{f}_{4}(\mathbf{K})
  103. 𝔰 𝔩 3 ( 𝐊 ) \mathfrak{sl}_{3}(\mathbf{K})
  104. 𝔰 𝔩 3 ( 𝐊 ) 𝔰 𝔩 3 ( 𝐊 ) \mathfrak{sl}_{3}(\mathbf{K})\oplus\mathfrak{sl}_{3}(\mathbf{K})
  105. 𝔰 𝔩 6 ( 𝐊 ) \mathfrak{sl}_{6}(\mathbf{K})
  106. 𝔢 6 ( 𝐊 ) \mathfrak{e}_{6}(\mathbf{K})
  107. 𝔰 𝔭 6 ( 𝐊 ) \mathfrak{sp}_{6}(\mathbf{K})
  108. 𝔰 𝔩 6 ( 𝐊 ) \mathfrak{sl}_{6}(\mathbf{K})
  109. 𝔰 𝔬 12 ( 𝐊 ) \mathfrak{so}_{12}(\mathbf{K})
  110. 𝔢 7 ( 𝐊 ) \mathfrak{e}_{7}(\mathbf{K})
  111. 𝔣 4 ( 𝐊 ) \mathfrak{f}_{4}(\mathbf{K})
  112. 𝔢 6 ( 𝐊 ) \mathfrak{e}_{6}(\mathbf{K})
  113. 𝔢 7 ( 𝐊 ) \mathfrak{e}_{7}(\mathbf{K})
  114. 𝔢 8 ( 𝐊 ) \mathfrak{e}_{8}(\mathbf{K})
  115. 𝔰 𝔬 n ( 𝐊 ) \mathfrak{so}_{n}(\mathbf{K})
  116. 𝔰 𝔩 n ( 𝐊 ) or 𝔰 𝔲 n \mathfrak{sl}_{n}(\mathbf{K})\,\text{ or }\mathfrak{su}_{n}
  117. 𝔰 𝔭 2 n ( 𝐊 ) or 𝔰 𝔭 n \mathfrak{sp}_{2n}(\mathbf{K})\,\text{ or }\mathfrak{sp}_{n}
  118. 𝔰 𝔩 n ( 𝐊 ) or 𝔰 𝔲 n \mathfrak{sl}_{n}(\mathbf{K})\,\text{ or }\mathfrak{su}_{n}
  119. 𝔰 𝔩 n ( 𝐊 ) 𝔰 𝔩 n ( 𝐊 ) or 𝔰 𝔲 n 𝔰 𝔲 n \mathfrak{sl}_{n}(\mathbf{K})\oplus\mathfrak{sl}_{n}(\mathbf{K})\,\text{ or }% \mathfrak{su}_{n}\oplus\mathfrak{su}_{n}
  120. 𝔰 𝔩 2 n ( 𝐊 ) or 𝔰 𝔲 2 n \mathfrak{sl}_{2n}(\mathbf{K})\,\text{ or }\mathfrak{su}_{2n}
  121. 𝔰 𝔭 2 n ( 𝐊 ) or 𝔰 𝔭 n \mathfrak{sp}_{2n}(\mathbf{K})\,\text{ or }\mathfrak{sp}_{n}
  122. 𝔰 𝔩 2 n ( 𝐊 ) or 𝔰 𝔲 2 n \mathfrak{sl}_{2n}(\mathbf{K})\,\text{ or }\mathfrak{su}_{2n}
  123. 𝔰 𝔬 4 n ( 𝐊 ) \mathfrak{so}_{4n}(\mathbf{K})
  124. 𝔰 𝔬 2 \mathfrak{so}_{2}
  125. 𝔰 𝔬 3 \mathfrak{so}_{3}
  126. 𝔰 𝔬 5 \mathfrak{so}_{5}
  127. 𝔰 𝔬 9 \mathfrak{so}_{9}
  128. 𝔰 𝔬 3 \mathfrak{so}_{3}
  129. 𝔰 𝔬 4 \mathfrak{so}_{4}
  130. 𝔰 𝔬 6 \mathfrak{so}_{6}
  131. 𝔰 𝔬 10 \mathfrak{so}_{10}
  132. 𝔰 𝔬 5 \mathfrak{so}_{5}
  133. 𝔰 𝔬 6 \mathfrak{so}_{6}
  134. 𝔰 𝔬 8 \mathfrak{so}_{8}
  135. 𝔰 𝔬 12 \mathfrak{so}_{12}
  136. 𝔰 𝔬 9 \mathfrak{so}_{9}
  137. 𝔰 𝔬 10 \mathfrak{so}_{10}
  138. 𝔰 𝔬 12 \mathfrak{so}_{12}
  139. 𝔰 𝔬 16 \mathfrak{so}_{16}
  140. ( 𝐀 𝐁 ) n , (\mathbf{A}\otimes\mathbf{B})^{n},

Fréchet_distribution.html

  1. e - ( x - m s ) - α e^{-(\frac{x-m}{s})^{-\alpha}}
  2. { m + s Γ ( 1 - 1 α ) for α > 1 otherwise \begin{cases}\ m+s\Gamma\left(1-\frac{1}{\alpha}\right)&\,\text{for }\alpha>1% \\ \ \infty&\,\text{otherwise}\end{cases}
  3. m + s log e ( 2 ) α m+\frac{s}{\sqrt[\alpha]{\log_{e}(2)}}
  4. m + s ( α 1 + α ) 1 / α m+s\left(\frac{\alpha}{1+\alpha}\right)^{1/\alpha}
  5. { s 2 ( Γ ( 1 - 2 α ) - ( Γ ( 1 - 1 α ) ) 2 ) for α > 2 otherwise \begin{cases}\ s^{2}\left(\Gamma\left(1-\frac{2}{\alpha}\right)-\left(\Gamma% \left(1-\frac{1}{\alpha}\right)\right)^{2}\right)&\,\text{for }\alpha>2\\ \ \infty&\,\text{otherwise}\end{cases}
  6. { Γ ( 1 - 3 α ) - 3 Γ ( 1 - 2 α ) Γ ( 1 - 1 α ) + 2 Γ 3 ( 1 - 1 α ) ( Γ ( 1 - 2 α ) - Γ 2 ( 1 - 1 α ) ) 3 for α > 3 otherwise \begin{cases}\ \frac{\Gamma\left(1-\frac{3}{\alpha}\right)-3\Gamma\left(1-% \frac{2}{\alpha}\right)\Gamma\left(1-\frac{1}{\alpha}\right)+2\Gamma^{3}\left(% 1-\frac{1}{\alpha}\right)}{\sqrt{\left(\Gamma\left(1-\frac{2}{\alpha}\right)-% \Gamma^{2}\left(1-\frac{1}{\alpha}\right)\right)^{3}}}&\,\text{for }\alpha>3\\ \ \infty&\,\text{otherwise}\end{cases}
  7. { - 6 + Γ ( 1 - 4 α ) - 4 Γ ( 1 - 3 α ) Γ ( 1 - 1 α ) + 3 Γ 2 ( 1 - 2 α ) [ Γ ( 1 - 2 α ) - Γ 2 ( 1 - 1 α ) ] 2 for α > 4 otherwise \begin{cases}\ -6+\frac{\Gamma\left(1-\frac{4}{\alpha}\right)-4\Gamma\left(1-% \frac{3}{\alpha}\right)\Gamma\left(1-\frac{1}{\alpha}\right)+3\Gamma^{2}\left(% 1-\frac{2}{\alpha}\right)}{\left[\Gamma\left(1-\frac{2}{\alpha}\right)-\Gamma^% {2}\left(1-\frac{1}{\alpha}\right)\right]^{2}}&\,\text{for }\alpha>4\\ \ \infty&\,\text{otherwise}\end{cases}
  8. 1 + γ α + γ + ln ( s α ) 1+\frac{\gamma}{\alpha}+\gamma+\ln\left(\frac{s}{\alpha}\right)
  9. γ \gamma
  10. k k
  11. α > k \alpha>k
  12. Pr ( X x ) = e - x - α if x > 0. \Pr(X\leq x)=e^{-x^{-\alpha}}\,\text{ if }x>0.
  13. Pr ( X x ) = e - ( x - m s ) - α if x > m . \Pr(X\leq x)=e^{-\left(\frac{x-m}{s}\right)^{-\alpha}}\,\text{ if }x>m.
  14. α \alpha
  15. μ k = 0 x k f ( x ) d x = 0 t - k α e - t d t , \mu_{k}=\int_{0}^{\infty}x^{k}f(x)dx=\int_{0}^{\infty}t^{-\frac{k}{\alpha}}e^{% -t}\,dt,
  16. t = x - α t=x^{-\alpha}
  17. k < α k<\alpha
  18. μ k = Γ ( 1 - k α ) \mu_{k}=\Gamma\left(1-\frac{k}{\alpha}\right)
  19. Γ ( z ) \Gamma\left(z\right)
  20. α > 1 \alpha>1
  21. E [ X ] = Γ ( 1 - 1 α ) E[X]=\Gamma(1-\tfrac{1}{\alpha})
  22. α > 2 \alpha>2
  23. Var ( X ) = Γ ( 1 - 2 α ) - ( Γ ( 1 - 1 α ) ) 2 . \,\text{Var}(X)=\Gamma(1-\tfrac{2}{\alpha})-\big(\Gamma(1-\tfrac{1}{\alpha})% \big)^{2}.
  24. q y q_{y}
  25. y y
  26. q y = F - 1 ( y ) = ( - log e y ) - 1 α q_{y}=F^{-1}(y)=\left(-\log_{e}y\right)^{-\frac{1}{\alpha}}
  27. q 1 / 2 = ( log e 2 ) - 1 α . q_{1/2}=(\log_{e}2)^{-\frac{1}{\alpha}}.
  28. ( α α + 1 ) 1 α . \left(\frac{\alpha}{\alpha+1}\right)^{\frac{1}{\alpha}}.
  29. q 1 = m + s log ( 4 ) α q_{1}=m+\frac{s}{\sqrt[\alpha]{\log(4)}}
  30. q 3 = m + s log ( 4 3 ) α . q_{3}=m+\frac{s}{\sqrt[\alpha]{\log(\frac{4}{3})}}.
  31. F ( m e a n ) = exp ( - Γ - α ( 1 - 1 α ) ) F(mean)=\exp\left(-\Gamma^{-\alpha}\left(1-\frac{1}{\alpha}\right)\right)
  32. F ( m o d e ) = exp ( - α + 1 α ) . F(mode)=\exp\left(-\frac{\alpha+1}{\alpha}\right).
  33. Z i = - 1 / l o g F i ( X i ) Z_{i}=-1/logF_{i}(X_{i})
  34. ( R , W ) = ( Z 1 + Z 2 , Z 1 / ( Z 1 + Z 2 ) ) (R,W)=(Z_{1}+Z_{2},Z_{1}/(Z_{1}+Z_{2}))
  35. R 1 R>>1
  36. W W
  37. X U ( 0 , 1 ) X\sim U(0,1)\,
  38. m + s ( - log ( X ) ) - 1 / α Frechet ( α , s , m ) m+s(-\log(X))^{-1/\alpha}\sim\textrm{Frechet}(\alpha,s,m)\,
  39. X Frechet ( α , s , m ) X\sim\textrm{Frechet}(\alpha,s,m)\,
  40. k X + b Frechet ( α , k s , k m + b ) kX+b\sim\textrm{Frechet}(\alpha,ks,km+b)\,
  41. X i Frechet ( α , s , m ) X_{i}\sim\textrm{Frechet}(\alpha,s,m)\,
  42. Y = max { X 1 , , X n } Y=\max\{\,X_{1},\ldots,X_{n}\,\}\,
  43. Y Frechet ( α , n 1 α s , m ) Y\sim\textrm{Frechet}(\alpha,n^{\tfrac{1}{\alpha}}s,m)\,
  44. X Frechet ( α , s , m = 0 ) X\sim\textrm{Frechet}(\alpha,s,m=0)\,
  45. X - 1 Weibull ( k = α , λ = s - 1 ) X^{-1}\sim\textrm{Weibull}(k=\alpha,\lambda=s^{-1})\,

Fructose-6-phosphate_phosphoketolase.html

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Fructose_5-dehydrogenase.html

  1. \rightleftharpoons

Fructose_5-dehydrogenase_(NADP+).html

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Fructuronate_reductase.html

  1. \rightleftharpoons

Fucosterol-epoxide_lyase.html

  1. \rightleftharpoons

Fumarate_reductase_(NADH).html

  1. \rightleftharpoons

Functional-theoretic_algebra.html

  1. x y = L 1 ( x ) y + L 2 ( y ) x - L 1 ( x ) L 2 ( y ) e . x\cdot y=L_{1}(x)y+L_{2}(y)x-L_{1}(x)L_{2}(y)e.
  2. x y = L ( x ) y + L ( y ) x - L ( x ) L ( y ) e . x\cdot y=L(x)y+L(y)x-L(x)L(y)e.
  3. ( f + g ) ( x ) = f ( x ) + g ( x ) (f+g)(x)=f(x)+g(x)\,
  4. ( α f ) ( x ) = α f ( x ) . (\alpha f)(x)=\alpha f(x).\,
  5. f g = L 1 ( f ) g + L 2 ( g ) f - L 1 ( f ) L 2 ( g ) e = f ( a ) g + g ( b ) f - f ( a ) g ( b ) e . f\cdot g=L_{1}(f)g+L_{2}(g)f-L_{1}(f)L_{2}(g)e=f(a)g+g(b)f-f(a)g(b)e.
  6. ( f g ) ( a ) = f ( a ) g ( a ) and ( f g ) ( b ) = f ( b ) g ( b ) . (f\cdot g)(a)=f(a)g(a)\mbox{ and }~{}(f\cdot g)(b)=f(b)g(b).
  7. e ( t ) = 1 , [ 0 , 1 ] e(t)=1,\forall\in[0,1]
  8. α β = α ( 0 ) β + β ( 1 ) α - α ( 0 ) β ( 1 ) e {\alpha}\cdot{\beta}={\alpha}(0){\beta}+{\beta}(1){\alpha}-{\alpha}(0){\beta}(% 1)e
  9. f ( t ) = 1 - t + i t and g ( t ) = cos ( 2 π t ) + i sin ( 2 π t ) f(t)=1-t+it\mbox{ and }~{}g(t)=\cos(2\pi t)+i\sin(2\pi t)
  10. g ( 0 ) = g ( 1 ) = 1 g(0)=g(1)=1
  11. f ( 0 ) = 1 f(0)=1
  12. f ( 1 ) = i f(1)=i
  13. f g and g f f\cdot g\mbox{ and }~{}g\cdot f
  14. ( f g ) ( t ) = [ - t + cos ( 2 π t ) ] + i [ t + sin ( 2 π t ) ] (f\cdot g)(t)=[-t+\cos(2\pi t)]+i[t+\sin(2\pi t)]
  15. ( g f ) ( t ) = [ 1 - t - sin ( 2 π t ) ] + i [ t - 1 + cos ( 2 π t ) ] (g\cdot f)(t)=[1-t-\sin(2\pi t)]+i[t-1+\cos(2\pi t)]
  16. f g g f f\cdot g\neq g\cdot f
  17. f ( 0 ) g ( 0 ) = 1 and ends at f ( 1 ) g ( 1 ) = i . f(0)g(0)=1\mbox{ and ends at }~{}f(1)g(1)=i.

Fundamental_discriminant.html

  1. S = { - 8 , - 4 , 8 , - 3 , 5 , - 7 , - 11 , 13 , 17 , - 19 , } S=\{-8,-4,8,-3,5,-7,-11,13,17,-19,\;\ldots\}

Furstenberg's_proof_of_the_infinitude_of_primes.html

  1. S ( a , b ) = { a n + b n } = a + b . S(a,b)=\{an+b\mid n\in\mathbb{Z}\}=a\mathbb{Z}+b.\,
  2. S ( a , b ) = j = 1 a - 1 S ( a , b + j ) . S(a,b)=\mathbb{Z}\setminus\bigcup_{j=1}^{a-1}S(a,b+j).
  3. { - 1 , + 1 } = p prime S ( p , 0 ) . \mathbb{Z}\setminus\{-1,+1\}=\bigcup_{p\mathrm{\,prime}}S(p,0).

Furylfuramide_isomerase.html

  1. \rightleftharpoons

GABA_transaminase.html

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  2. \rightleftharpoons
  3. \rightleftharpoons

Galactitol-1-phosphate_5-dehydrogenase.html

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Galactitol_2-dehydrogenase.html

  1. \rightleftharpoons

Galactose-6-phosphate_isomerase.html

  1. \rightleftharpoons

Galactose_1-dehydrogenase.html

  1. \rightleftharpoons

Galactose_1-dehydrogenase_(NADP+).html

  1. \rightleftharpoons

Galactosylceramide_sulfotransferase.html

  1. \rightleftharpoons

Gallate_decarboxylase.html

  1. \rightleftharpoons

Gamma-butyrobetaine_dioxygenase.html

  1. \rightleftharpoons

Gamma-guanidinobutyraldehyde_dehydrogenase.html

  1. \rightleftharpoons

Gamma-ray_burst_emission_mechanisms.html

  1. N ( E ) = { E α exp ( - E < m t p l > E 0 ) , if E ( α - β ) E 0 [ ( α - β ) E 0 ] ( α - β ) E β exp ( β - α ) , if E > ( α - β ) E 0 N(E)=\begin{cases}{E^{\alpha}\exp\left({-\frac{E}{<}mtpl>{{E_{0}}}}\right)},&% \mbox{if }~{}E\leq(\alpha-\beta)E_{0}\mbox{ }\\ {\left[{\left({\alpha-\beta}\right)E_{0}}\right]^{\left({\alpha-\beta}\right)}% E^{\beta}\exp\left({\beta-\alpha}\right)},&\mbox{if }~{}E>(\alpha-\beta)E_{0}% \mbox{ }\end{cases}
  2. ν a \nu_{a}
  3. ν m \nu_{m}
  4. ν c \nu_{c}
  5. ν m > ν c \nu_{m}>\nu_{c}
  6. F ν { ν 2 , ν < ν a ν 1 / 3 , ν a < ν < ν c ν - 1 / 2 , ν c < ν < ν m ν - p / 2 , ν m < ν F_{\nu}\propto\begin{cases}{\nu^{2}},&\nu<\nu_{a}\\ {\nu^{1/3}},&\nu_{a}<\nu<\nu_{c}\\ {\nu^{-1/2}},&\nu_{c}<\nu<\nu_{m}\\ {\nu^{-p/2}},&\nu_{m}<\nu\end{cases}
  7. ν m < ν c \nu_{m}<\nu_{c}
  8. F ν { ν 2 , ν < ν a ν 1 / 3 , ν a < ν < ν m ν - ( p - 1 ) / 2 , ν m < ν < ν c ν - p / 2 , ν c < ν F_{\nu}\propto\begin{cases}{\nu^{2}},&\nu<\nu_{a}\\ {\nu^{1/3}},&\nu_{a}<\nu<\nu_{m}\\ {\nu^{-(p-1)/2}},&\nu_{m}<\nu<\nu_{c}\\ {\nu^{-p/2}},&\nu_{c}<\nu\end{cases}
  9. ν c t 1 / 2 \nu_{c}\propto t^{1/2}
  10. ν m t - 3 / 2 \nu_{m}\propto t^{-3/2}
  11. F ν , m a x = c o n s t F_{\nu,max}=const
  12. F ν , m a x F_{\nu,max}
  13. ν c \nu_{c}
  14. ν m \nu_{m}
  15. ν m \nu_{m}
  16. ν c \nu_{c}

Gamma-ray_burst_progenitors.html

  1. E s p i n = 1 2 I Ω H 2 E_{spin}=\frac{1}{2}I\Omega_{H}^{2}
  2. I = 4 M 3 ( cos ( λ / 2 ) / cos ( λ / 4 ) ) 2 I=4M^{3}(\cos(\lambda/2)/\cos(\lambda/4))^{2}
  3. Ω H = ( 1 / 2 M ) tan ( λ / 2 ) \Omega_{H}=(1/2M)\tan(\lambda/2)
  4. sin λ = a / M \sin\lambda=a/M
  5. a a
  6. M M
  7. M M

Gate_orbit.html

  1. r = 2 μ v 2 r={2\mu\over v_{\infty}^{2}}
  2. r r\,
  3. μ \mu\,
  4. v v_{\infty}\,
  5. v 2 v_{\infty}^{2}
  6. C 3 C3

GDP-4-dehydro-6-deoxy-D-mannose_reductase.html

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GDP-4-dehydro-D-rhamnose_reductase.html

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GDP-6-deoxy-D-talose_4-dehydrogenase.html

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GDP-L-fucose_synthase.html

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GDP-mannose_3,5-epimerase.html

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GDP-mannose_6-dehydrogenase.html

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Geissoschizine_dehydrogenase.html

  1. \rightleftharpoons

General_matrix_notation_of_a_VAR(p).html

  1. y t = c + A 1 y t - 1 + A 2 y t - 2 + + A p y t - p + e t , y_{t}=c+A_{1}y_{t-1}+A_{2}y_{t-2}+\cdots+A_{p}y_{t-p}+e_{t},\,
  2. y i y_{i}
  3. A i A_{i}
  4. [ y 1 , t y 2 , t y k , t ] = [ c 1 c 2 c k ] + [ a 1 , 1 1 a 1 , 2 1 a 1 , k 1 a 2 , 1 1 a 2 , 2 1 a 2 , k 1 a k , 1 1 a k , 2 1 a k , k 1 ] [ y 1 , t - 1 y 2 , t - 1 y k , t - 1 ] + + [ a 1 , 1 p a 1 , 2 p a 1 , k p a 2 , 1 p a 2 , 2 p a 2 , k p a k , 1 p a k , 2 p a k , k p ] [ y 1 , t - p y 2 , t - p y k , t - p ] + [ e 1 , t e 2 , t e k , t ] \begin{bmatrix}y_{1,t}\\ y_{2,t}\\ \vdots\\ y_{k,t}\end{bmatrix}=\begin{bmatrix}c_{1}\\ c_{2}\\ \vdots\\ c_{k}\end{bmatrix}+\begin{bmatrix}a_{1,1}^{1}&a_{1,2}^{1}&\cdots&a_{1,k}^{1}\\ a_{2,1}^{1}&a_{2,2}^{1}&\cdots&a_{2,k}^{1}\\ \vdots&\vdots&\ddots&\vdots\\ a_{k,1}^{1}&a_{k,2}^{1}&\cdots&a_{k,k}^{1}\end{bmatrix}\begin{bmatrix}y_{1,t-1% }\\ y_{2,t-1}\\ \vdots\\ y_{k,t-1}\end{bmatrix}+\cdots+\begin{bmatrix}a_{1,1}^{p}&a_{1,2}^{p}&\cdots&a_% {1,k}^{p}\\ a_{2,1}^{p}&a_{2,2}^{p}&\cdots&a_{2,k}^{p}\\ \vdots&\vdots&\ddots&\vdots\\ a_{k,1}^{p}&a_{k,2}^{p}&\cdots&a_{k,k}^{p}\end{bmatrix}\begin{bmatrix}y_{1,t-p% }\\ y_{2,t-p}\\ \vdots\\ y_{k,t-p}\end{bmatrix}+\begin{bmatrix}e_{1,t}\\ e_{2,t}\\ \vdots\\ e_{k,t}\end{bmatrix}
  5. y 1 , t = c 1 + a 1 , 1 1 y 1 , t - 1 + a 1 , 2 1 y 2 , t - 1 + + a 1 , k 1 y k , t - 1 + + a 1 , 1 p y 1 , t - p + a 1 , 2 p y 2 , t - p + + a 1 , k p y k , t - p + e 1 , t y_{1,t}=c_{1}+a_{1,1}^{1}y_{1,t-1}+a_{1,2}^{1}y_{2,t-1}+\cdots+a_{1,k}^{1}y_{k% ,t-1}+\cdots+a_{1,1}^{p}y_{1,t-p}+a_{1,2}^{p}y_{2,t-p}+\cdots+a_{1,k}^{p}y_{k,% t-p}+e_{1,t}\,
  6. y 2 , t = c 2 + a 2 , 1 1 y 1 , t - 1 + a 2 , 2 1 y 2 , t - 1 + + a 2 , k 1 y k , t - 1 + + a 2 , 1 p y 1 , t - p + a 2 , 2 p y 2 , t - p + + a 2 , k p y k , t - p + e 2 , t y_{2,t}=c_{2}+a_{2,1}^{1}y_{1,t-1}+a_{2,2}^{1}y_{2,t-1}+\cdots+a_{2,k}^{1}y_{k% ,t-1}+\cdots+a_{2,1}^{p}y_{1,t-p}+a_{2,2}^{p}y_{2,t-p}+\cdots+a_{2,k}^{p}y_{k,% t-p}+e_{2,t}\,
  7. \qquad\vdots
  8. y k , t = c k + a k , 1 1 y 1 , t - 1 + a k , 2 1 y 2 , t - 1 + + a k , k 1 y k , t - 1 + + a k , 1 p y 1 , t - p + a k , 2 p y 2 , t - p + + a k , k p y k , t - p + e k , t y_{k,t}=c_{k}+a_{k,1}^{1}y_{1,t-1}+a_{k,2}^{1}y_{2,t-1}+\cdots+a_{k,k}^{1}y_{k% ,t-1}+\cdots+a_{k,1}^{p}y_{1,t-p}+a_{k,2}^{p}y_{2,t-p}+\cdots+a_{k,k}^{p}y_{k,% t-p}+e_{k,t}\,
  9. y 0 y_{0}
  10. y T y_{T}
  11. Y = B Z + U Y=BZ+U\,
  12. Y = [ y p y p + 1 y T ] = [ y 1 , p y 1 , p + 1 y 1 , T y 2 , p y 2 , p + 1 y 2 , T y k , p y k , p + 1 y k , T ] Y=\begin{bmatrix}y_{p}&y_{p+1}&\cdots&y_{T}\end{bmatrix}=\begin{bmatrix}y_{1,p% }&y_{1,p+1}&\cdots&y_{1,T}\\ y_{2,p}&y_{2,p+1}&\cdots&y_{2,T}\\ \vdots&\vdots&\vdots&\vdots\\ y_{k,p}&y_{k,p+1}&\cdots&y_{k,T}\end{bmatrix}
  13. B = [ c A 1 A 2 A p ] = [ c 1 a 1 , 1 1 a 1 , 2 1 a 1 , k 1 a 1 , 1 p a 1 , 2 p a 1 , k p c 2 a 2 , 1 1 a 2 , 2 1 a 2 , k 1 a 2 , 1 p a 2 , 2 p a 2 , k p c k a k , 1 1 a k , 2 1 a k , k 1 a k , 1 p a k , 2 p a k , k p ] B=\begin{bmatrix}c&A_{1}&A_{2}&\cdots&A_{p}\end{bmatrix}=\begin{bmatrix}c_{1}&% a_{1,1}^{1}&a_{1,2}^{1}&\cdots&a_{1,k}^{1}&\cdots&a_{1,1}^{p}&a_{1,2}^{p}&% \cdots&a_{1,k}^{p}\\ c_{2}&a_{2,1}^{1}&a_{2,2}^{1}&\cdots&a_{2,k}^{1}&\cdots&a_{2,1}^{p}&a_{2,2}^{p% }&\cdots&a_{2,k}^{p}\\ \vdots&\vdots&\vdots&\ddots&\vdots&\cdots&\vdots&\vdots&\ddots&\vdots\\ c_{k}&a_{k,1}^{1}&a_{k,2}^{1}&\cdots&a_{k,k}^{1}&\cdots&a_{k,1}^{p}&a_{k,2}^{p% }&\cdots&a_{k,k}^{p}\end{bmatrix}
  14. Z = [ 1 1 1 y p - 1 y p y T - 1 y p - 2 y p - 1 y T - 2 y 0 y 1 y T - p ] = [ 1 1 1 y 1 , p - 1 y 1 , p y 1 , T - 1 y 2 , p - 1 y 2 , p y 2 , T - 1 y k , p - 1 y k , p y k , T - 1 y 1 , p - 2 y 1 , p - 1 y 1 , T - 2 y 2 , p - 2 y 2 , p - 1 y 2 , T - 2 y k , p - 2 y k , p - 1 y k , T - 2 y 1 , 0 y 1 , 1 y 1 , T - p y 2 , 0 y 2 , 1 y 2 , T - p y k , 0 y k , 1 y k , T - p ] Z=\begin{bmatrix}1&1&\cdots&1\\ y_{p-1}&y_{p}&\cdots&y_{T-1}\\ y_{p-2}&y_{p-1}&\cdots&y_{T-2}\\ \vdots&\vdots&\ddots&\vdots\\ y_{0}&y_{1}&\cdots&y_{T-p}\end{bmatrix}=\begin{bmatrix}1&1&\cdots&1\\ y_{1,p-1}&y_{1,p}&\cdots&y_{1,T-1}\\ y_{2,p-1}&y_{2,p}&\cdots&y_{2,T-1}\\ \vdots&\vdots&\ddots&\vdots\\ y_{k,p-1}&y_{k,p}&\cdots&y_{k,T-1}\\ y_{1,p-2}&y_{1,p-1}&\cdots&y_{1,T-2}\\ y_{2,p-2}&y_{2,p-1}&\cdots&y_{2,T-2}\\ \vdots&\vdots&\ddots&\vdots\\ y_{k,p-2}&y_{k,p-1}&\cdots&y_{k,T-2}\\ \vdots&\vdots&\ddots&\vdots\\ y_{1,0}&y_{1,1}&\cdots&y_{1,T-p}\\ y_{2,0}&y_{2,1}&\cdots&y_{2,T-p}\\ \vdots&\vdots&\ddots&\vdots\\ y_{k,0}&y_{k,1}&\cdots&y_{k,T-p}\end{bmatrix}
  15. U = [ e p e p + 1 e T ] = [ e 1 , p e 1 , p + 1 e 1 , T e 2 , p e 2 , p + 1 e 2 , T e k , p e k , p + 1 e k , T ] . U=\begin{bmatrix}e_{p}&e_{p+1}&\cdots&e_{T}\end{bmatrix}=\begin{bmatrix}e_{1,p% }&e_{1,p+1}&\cdots&e_{1,T}\\ e_{2,p}&e_{2,p+1}&\cdots&e_{2,T}\\ \vdots&\vdots&\ddots&\vdots\\ e_{k,p}&e_{k,p+1}&\cdots&e_{k,T}\end{bmatrix}.
  16. Y B Z Y\approx BZ

Generalised_metric.html

  1. \scriptstyle\mathbb{R}
  2. ( F , + , , < ) (F,+,\cdot,<)
  3. M M
  4. d : M × M F + { 0 } d:M\times M\to F^{+}\cup\{0\}
  5. M M
  6. d ( x , y ) = 0 x = y d(x,y)=0\Leftrightarrow x=y
  7. d ( x , y ) = d ( y , x ) d(x,y)=d(y,x)
  8. d ( x , y ) + d ( y , z ) d ( x , z ) d(x,y)+d(y,z)\leq d(x,z)
  9. B ( x , δ ) := { y M : d ( x , y ) < δ } B(x,\delta)\;:=\{y\in M\;:d(x,y)<\delta\}
  10. M M
  11. F F
  12. F F
  13. M M
  14. x G x\in G
  15. G G
  16. B ( x , δ ) B(x,\delta)
  17. x B ( x , δ ) G x\in B(x,\delta)\subseteq G
  18. μ ( x , G ) = B ( x , δ / 2 ) \mu(x,G)=B(x,\delta/2)
  19. G , G G,G
  20. μ ( x , G ) := B ( x , 1 / 2 n ( x , G ) ) \mu(x,G):=B(x,1/2n(x,G))
  21. n ( x , G ) := min { n : B ( x , 1 / n ) G } n(x,G):=\min\{n\in\mathbb{N}:B(x,1/n)\subseteq G\}
  22. x G x\in G
  23. A ( x , G ) := { a F : n , B ( x , n a ) G } A(x,G):=\{a\in F\colon\forall n\in\mathbb{N},B(x,n\cdot a)\subseteq G\}
  24. F \mathbb{N}_{F}
  25. ξ F \xi\in F
  26. n : n 1 ξ \forall n\in\mathbb{N}\colon n\cdot 1\leq\xi
  27. a = k ( 2 ξ ) - 1 a=k\cdot(2\xi)^{-1}
  28. a a
  29. μ ( x , G ) = { B ( x , a ) : a A ( x , G ) } \mu(x,G)=\bigcup\{B(x,a)\colon a\in A(x,G)\}
  30. μ ( x , G ) G \mu(x,G)\subseteq G
  31. μ ( x , G ) μ ( y , H ) \mu(x,G)\cap\mu(y,H)
  32. a A ( x , G ) : d ( x , z ) < a ; b A ( y , H ) : d ( z , y ) < b \exists a\in A(x,G)\colon d(x,z)<a;\;\;\exists b\in A(y,H)\colon d(z,y)<b
  33. d ( x , y ) d ( x , z ) + d ( z , y ) < 2 max { a , b } d(x,y)\leq d(x,z)+d(z,y)<2\cdot\max\{a,b\}
  34. μ ( x , G ) G \mu(x,G)\subseteq G
  35. μ ( y , H ) H \mu(y,H)\subseteq H

Generalized_forces.html

  1. δ W = i = 1 n 𝐅 i δ 𝐫 i \delta W=\sum_{i=1}^{n}\mathbf{F}_{i}\cdot\delta\mathbf{r}_{i}
  2. δ 𝐫 i = j = 1 m 𝐫 i q j δ q j , i = 1 , , n , \delta\mathbf{r}_{i}=\sum_{j=1}^{m}\frac{\partial\mathbf{r}_{i}}{\partial q_{j% }}\delta q_{j},\quad i=1,\ldots,n,
  3. δ W = 𝐅 1 j = 1 m 𝐫 1 q j δ q j + + 𝐅 n j = 1 m 𝐫 n q j δ q j . \delta W=\mathbf{F}_{1}\cdot\sum_{j=1}^{m}\frac{\partial\mathbf{r}_{1}}{% \partial q_{j}}\delta q_{j}+\ldots+\mathbf{F}_{n}\cdot\sum_{j=1}^{m}\frac{% \partial\mathbf{r}_{n}}{\partial q_{j}}\delta q_{j}.
  4. δ W = i = 1 n 𝐅 i 𝐫 i q 1 δ q 1 + + i = 1 n 𝐅 i 𝐫 i q m δ q m . \delta W=\sum_{i=1}^{n}\mathbf{F}_{i}\cdot\frac{\partial\mathbf{r}_{i}}{% \partial q_{1}}\delta q_{1}+\ldots+\sum_{i=1}^{n}\mathbf{F}_{i}\cdot\frac{% \partial\mathbf{r}_{i}}{\partial q_{m}}\delta q_{m}.
  5. δ W = Q 1 δ q 1 + + Q m δ q m , \delta W=Q_{1}\delta q_{1}+\ldots+Q_{m}\delta q_{m},
  6. Q j = i = 1 n 𝐅 i 𝐫 i q j , j = 1 , , m , Q_{j}=\sum_{i=1}^{n}\mathbf{F}_{i}\cdot\frac{\partial\mathbf{r}_{i}}{\partial q% _{j}},\quad j=1,\ldots,m,
  7. δ 𝐫 i = j = 1 m 𝐕 i q ˙ j δ q j , i = 1 , , n . \delta\mathbf{r}_{i}=\sum_{j=1}^{m}\frac{\partial\mathbf{V}_{i}}{\partial\dot{% q}_{j}}\delta q_{j},\quad i=1,\ldots,n.
  8. Q j = i = 1 n 𝐅 i 𝐕 i q ˙ j , j = 1 , , m . Q_{j}=\sum_{i=1}^{n}\mathbf{F}_{i}\cdot\frac{\partial\mathbf{V}_{i}}{\partial% \dot{q}_{j}},\quad j=1,\ldots,m.
  9. 𝐅 i * = - m i 𝐀 i , i = 1 , , n , \mathbf{F}_{i}^{*}=-m_{i}\mathbf{A}_{i},\quad i=1,\ldots,n,
  10. Q j * = i = 1 n 𝐅 i * 𝐕 i q ˙ j , j = 1 , , m . Q^{*}_{j}=\sum_{i=1}^{n}\mathbf{F}^{*}_{i}\cdot\frac{\partial\mathbf{V}_{i}}{% \partial\dot{q}_{j}},\quad j=1,\ldots,m.
  11. δ W = ( Q 1 + Q 1 * ) δ q 1 + + ( Q m + Q m * ) δ q m . \delta W=(Q_{1}+Q^{*}_{1})\delta q_{1}+\ldots+(Q_{m}+Q^{*}_{m})\delta q_{m}.

Generic_property.html

  1. f : M N f\colon M\to N

Gentisate_1,2-dioxygenase.html

  1. \rightleftharpoons

Gentisate_decarboxylase.html

  1. \rightleftharpoons

Geodesic_convexity.html

  1. f γ : [ 0 , T ] f\circ\gamma:[0,T]\to\mathbb{R}

Georg_Nöbeling.html

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

Geraniol_dehydrogenase.html

  1. \rightleftharpoons

Giacinto_Morera.html

  1. C C
  2. D D
  3. f f
  4. C f ( z ) d z = 0 \oint_{C}f(z)\,\mathrm{d}z=0
  5. C C
  6. D D
  7. f f

Gibberellin-44_dioxygenase.html

  1. \rightleftharpoons

Gibberellin_2beta-dioxygenase.html

  1. \rightleftharpoons

Gibberellin_3beta-dioxygenase.html

  1. \rightleftharpoons

Giordano_Riccati.html

  1. E steel E brass = 2.06 \frac{E_{\mbox{steel}~{}}}{E_{\mbox{brass}~{}}}=2.06

Glossary_of_chemistry_terms.html

  1. k k

Gluconate_2-dehydrogenase.html

  1. \rightleftharpoons

Gluconate_2-dehydrogenase_(acceptor).html

  1. \rightleftharpoons

Gluconate_5-dehydrogenase.html

  1. \rightleftharpoons

Glucosaminate_ammonia-lyase.html

  1. \rightleftharpoons

Glucose-6-phosphate_1-epimerase.html

  1. \rightleftharpoons

Glucose-fructose_oxidoreductase.html

  1. \rightleftharpoons

Glucose_1-dehydrogenase.html

  1. \rightleftharpoons

Glucose_1-dehydrogenase_(NAD+).html

  1. \rightleftharpoons

Glucose_1-dehydrogenase_(NADP+).html

  1. \rightleftharpoons

Glucose_dehydrogenase_(acceptor).html

  1. \rightleftharpoons

Glucoside_3-dehydrogenase.html

  1. \rightleftharpoons

Glucuronate_isomerase.html

  1. \rightleftharpoons

Glucuronate_reductase.html

  1. \rightleftharpoons

Glucuronolactone_reductase.html

  1. \rightleftharpoons

Glucuronoxylan_4-O-methyltransferase.html

  1. \rightleftharpoons

Glutaconate_CoA-transferase.html

  1. \rightleftharpoons

Glutaconyl-CoA_decarboxylase.html

  1. \rightleftharpoons

Glutamate-1-semialdehyde_2,1-aminomutase.html

  1. \rightleftharpoons

Glutamate-5-semialdehyde_dehydrogenase.html

  1. \rightleftharpoons

Glutamate_racemase.html

  1. \rightleftharpoons

Glutamate_synthase_(ferredoxin).html

  1. \rightleftharpoons

Glutamate_synthase_(NADH).html

  1. \rightleftharpoons

Glutamate_synthase_(NADPH).html

  1. \rightleftharpoons

Glutamyl-tRNA_reductase.html

  1. \rightleftharpoons

Glutarate-semialdehyde_dehydrogenase.html

  1. \rightleftharpoons

Glutathione_dehydrogenase_(ascorbate).html

  1. \rightleftharpoons

Glutathione_oxidase.html

  1. \rightleftharpoons

Glutathione—CoA-glutathione_transhydrogenase.html

  1. \rightleftharpoons

Glutathione—cystine_transhydrogenase.html

  1. \rightleftharpoons

Glutathione—homocystine_transhydrogenase.html

  1. \rightleftharpoons

Glyceollin_synthase.html

  1. \rightleftharpoons

Glyceraldehyde-3-phosphate_dehydrogenase_(ferredoxin).html

  1. \rightleftharpoons

Glyceraldehyde-3-phosphate_dehydrogenase_(NAD(P)+).html

  1. \rightleftharpoons

Glyceraldehyde-3-phosphate_dehydrogenase_(NADP+)_(phosphorylating).html

  1. \rightleftharpoons

Glyceraldehyde-3-phosphate_dehydrogenase_(phosphorylating).html

  1. \rightleftharpoons

Glycerate_dehydrogenase.html

  1. \rightleftharpoons

Glycerol-3-phosphate_1-dehydrogenase_(NADP+).html

  1. \rightleftharpoons

Glycerol-3-phosphate_dehydrogenase_(NAD(P)+).html

  1. \rightleftharpoons

Glycerol-3-phosphate_dehydrogenase_(NAD+).html

  1. \rightleftharpoons

Glycerol-3-phosphate_oxidase.html

  1. \rightleftharpoons

Glycerol_2-dehydrogenase_(NADP+).html

  1. \rightleftharpoons

Glycerol_dehydrogenase.html

  1. \rightleftharpoons

Glycerol_dehydrogenase_(acceptor).html

  1. \rightleftharpoons

Glycerol_dehydrogenase_(NADP+).html

  1. \rightleftharpoons

Glycine_dehydrogenase.html

  1. \rightleftharpoons

Glycine_dehydrogenase_(cyanide-forming).html

  1. \rightleftharpoons

Glycine_dehydrogenase_(cytochrome).html

  1. \rightleftharpoons

Glycine_dehydrogenase_(decarboxylating).html

  1. \rightleftharpoons

Glycine_formimidoyltransferase.html

  1. \rightleftharpoons

Glycine_hydroxymethyltransferase.html

  1. \rightleftharpoons

Glycine_N-methyltransferase.html

  1. \rightleftharpoons

Glycine_reductase.html

  1. \rightleftharpoons

Glycochenodeoxycholate_sulfotransferase.html

  1. \rightleftharpoons

Glycolaldehyde_dehydrogenase.html

  1. \rightleftharpoons

Glycolate_dehydrogenase.html

  1. \rightleftharpoons

Glycosylphosphatidylinositol_diacylglycerol-lyase.html

  1. \rightleftharpoons

Glyoxylate_dehydrogenase_(acylating).html

  1. \rightleftharpoons

Glyoxylate_oxidase.html

  1. \rightleftharpoons

Glyoxylate_reductase_(NADP+).html

  1. \rightleftharpoons

Goku_(disambiguation).html

  1. 10 48 10^{48}

Golomb–Dickman_constant.html

  1. λ = 0.62432998854355087099293638310083724 . \lambda=0.62432998854355087099293638310083724\dots.
  2. λ = lim n a n n . \lambda=\lim_{n\to\infty}\frac{a_{n}}{n}.
  3. λ n \lambda n
  4. λ = lim n 1 n k = 2 n log ( P 1 ( k ) ) log ( k ) , \lambda=\lim_{n\to\infty}\frac{1}{n}\sum_{k=2}^{n}\frac{\log(P_{1}(k))}{\log(k% )},
  5. P 1 ( k ) P_{1}(k)
  6. λ d \lambda d
  7. λ \lambda
  8. λ = lim n Prob { P 2 ( n ) P 1 ( n ) } \lambda=\lim_{n\to\infty}\,\text{Prob}\left\{P_{2}(n)\leq\sqrt{P_{1}(n)}\right\}
  9. P 2 ( n ) P_{2}(n)
  10. λ \lambda
  11. λ = 0 e - t - E 1 ( t ) d t \lambda=\int_{0}^{\infty}e^{-t-E_{1}(t)}dt
  12. E 1 ( t ) E_{1}(t)
  13. λ = 0 ρ ( t ) t + 2 d t \lambda=\int_{0}^{\infty}\frac{\rho(t)}{t+2}dt
  14. λ = 0 ρ ( t ) ( t + 1 ) 2 d t \lambda=\int_{0}^{\infty}\frac{\rho(t)}{(t+1)^{2}}dt
  15. ρ ( t ) \rho(t)

Gravity_turn.html

  1. F \vec{F}
  2. m d v d t = F - m g k ^ . m\frac{d\vec{v}}{dt}=\vec{F}-mg\hat{k}\;.
  3. k ^ \hat{k}
  4. m m
  5. v \vec{v}
  6. v \vec{v}
  7. v ˙ = g ( n - cos β ) , v β ˙ = g sin β . \begin{aligned}\displaystyle\dot{v}&\displaystyle=g(n-\cos{\beta})\;,\\ \displaystyle v\dot{\beta}&\displaystyle=g\sin{\beta}\;.\\ \end{aligned}
  8. n = F / m g n=F/mg
  9. β = arccos ( τ 1 k ^ ) \beta=\arccos{(\vec{\tau_{1}}\cdot\hat{k})}
  10. n n

Great_ellipse.html

  1. 10 000 km 10\,000\,\mathrm{km}
  2. a a
  3. b b
  4. f = ( a - b ) / a f=(a-b)/a
  5. e = f ( 2 - f ) e=\sqrt{f(2-f)}
  6. e = e / ( 1 - f ) e^{\prime}=e/(1-f)
  7. A A
  8. ϕ 1 \phi_{1}
  9. λ 1 \lambda_{1}
  10. B B
  11. ϕ 2 \phi_{2}
  12. λ 2 \lambda_{2}
  13. A A
  14. B B
  15. s 12 s_{12}
  16. α 1 \alpha_{1}
  17. α 2 \alpha_{2}
  18. a a
  19. ϕ \phi
  20. β \beta
  21. ϕ \phi
  22. θ \theta
  23. a 2 / b a^{2}/b
  24. ϕ \phi
  25. A A
  26. B B
  27. ( ϕ 1 , λ 1 ) (\phi_{1},\lambda_{1})
  28. ( ϕ 2 , λ 2 ) (\phi_{2},\lambda_{2})
  29. ϕ \phi
  30. β \beta
  31. a tan β = b tan ϕ . a\tan\beta=b\tan\phi.
  32. λ \lambda
  33. α \alpha
  34. γ \gamma
  35. tan α = tan γ 1 - e 2 cos 2 β , tan γ = tan α 1 + e 2 cos 2 ϕ , \begin{aligned}\displaystyle\tan\alpha&\displaystyle=\frac{\tan\gamma}{\sqrt{1% -e^{2}\cos^{2}\beta}},\\ \displaystyle\tan\gamma&\displaystyle=\frac{\tan\alpha}{\sqrt{1+e^{\prime 2}% \cos^{2}\phi}},\end{aligned}
  36. α \alpha
  37. γ \gamma
  38. a a
  39. σ \sigma
  40. a a
  41. a 1 - e 2 cos 2 γ 0 a\sqrt{1-e^{2}\cos^{2}\gamma_{0}}
  42. γ 0 \gamma_{0}
  43. σ \sigma
  44. α \alpha
  45. λ \lambda
  46. ω \omega
  47. b 1 + e 2 cos 2 α 0 b\sqrt{1+e^{\prime 2}\cos^{2}\alpha_{0}}
  48. b b
  49. s 12 s_{12}
  50. α 1 \alpha_{1}
  51. α 2 \alpha_{2}
  52. A A
  53. B B
  54. β 1 \beta_{1}
  55. β 2 \beta_{2}
  56. ( β 1 , λ 1 ) (\beta_{1},\lambda_{1})
  57. ( β 2 , λ 2 ) (\beta_{2},\lambda_{2})
  58. γ \gamma
  59. α \alpha
  60. γ 0 \gamma_{0}
  61. γ 1 \gamma_{1}
  62. γ 2 \gamma_{2}
  63. A A
  64. B B
  65. α 1 \alpha_{1}
  66. α 2 \alpha_{2}
  67. γ 1 \gamma_{1}
  68. γ 2 \gamma_{2}
  69. γ 0 \gamma_{0}
  70. σ 01 \sigma_{01}
  71. σ 02 \sigma_{02}
  72. A A
  73. B B
  74. s 12 s_{12}
  75. σ 01 \sigma_{01}
  76. σ 02 \sigma_{02}
  77. β \beta
  78. B B
  79. A A
  80. α 1 \alpha_{1}
  81. s 12 s_{12}

Gromov_product.html

  1. ( y , z ) x = 1 2 ( d ( x , y ) + d ( x , z ) - d ( y , z ) ) . (y,z)_{x}=\frac{1}{2}\big(d(x,y)+d(x,z)-d(y,z)\big).
  2. d ( x , y ) = a + b , d ( x , z ) = a + c , d ( y , z ) = b + c d(x,y)=a+b,\ d(x,z)=a+c,\ d(y,z)=b+c
  3. ( y , z ) x = a , ( x , z ) y = b , ( x , y ) z = c (y,z)_{x}=a,\ (x,z)_{y}=b,\ (x,y)_{z}=c
  4. d ( x , y ) = ( x , z ) y + ( y , z ) x , d(x,y)=(x,z)_{y}+(y,z)_{x},
  5. 0 ( y , z ) x min { d ( y , x ) , d ( z , x ) } , 0\leq(y,z)_{x}\leq\min\big\{d(y,x),d(z,x)\big\},
  6. | ( y , z ) p - ( y , z ) q | d ( p , q ) , \big|(y,z)_{p}-(y,z)_{q}\big|\leq d(p,q),
  7. | ( x , y ) p - ( x , z ) p | d ( y , z ) . \big|(x,y)_{p}-(x,z)_{p}\big|\leq d(y,z).
  8. x x_{\infty}
  9. y y_{\infty}
  10. lim x x y y ( x , y ) p \lim_{x\to x_{\infty}\atop y\to y_{\infty}}(x,y)_{p}
  11. ( x , y ) p = log csc ( θ / 2 ) , (x_{\infty},y_{\infty})_{p}=\log\csc(\theta/2),
  12. θ \theta
  13. p x px_{\infty}
  14. p y py_{\infty}
  15. ( x , z ) p min { ( x , y ) p , ( y , z ) p } - δ . (x,z)_{p}\geq\min\big\{(x,y)_{p},(y,z)_{p}\big\}-\delta.

Grothendieck_construction.html

  1. F : C CAT F\colon C\rightarrow\,\text{CAT}
  2. F F
  3. Γ ( F ) \Gamma(F)
  4. C F C\int F
  5. ( c , x ) (c,x)
  6. c o b j ( C ) c\in obj(C)
  7. x o b j ( F ( c ) ) x\in obj(F(c))
  8. Hom Γ ( F ) ( ( c 1 , x 1 ) , ( c 2 , x 2 ) ) \,\text{Hom}_{\Gamma(F)}((c_{1},x_{1}),(c_{2},x_{2}))
  9. ( f , x ) (f,x)
  10. f : c 1 c 2 f\colon c_{1}\rightarrow c_{2}
  11. m o r ( C ) mor(C)
  12. x : F ( f ) ( x 1 ) x 2 x\colon F(f)(x_{1})\rightarrow x_{2}
  13. m o r ( F ( c 2 ) ) mor(F(c_{2}))
  14. ( f , x ) ( f , x ) = ( f f , x F ( f ) ( x ) ) (f,x)\cdot(f^{\prime},x^{\prime})=(ff^{\prime},x\cdot F(f)(x^{\prime}))

Group_Hopf_algebra.html

  1. x = g G a g g x=\sum_{g\in G}a_{g}g
  2. f : G k , f\colon G\to k,
  3. ( x , f ) = g G a g f ( g ) , (x,f)=\sum_{g\in G}a_{g}f(g),
  4. Δ ( x ) = x x ; \Delta(x)=x\otimes x;
  5. ϵ ( x ) = 1 k ; \epsilon(x)=1_{k};
  6. S ( x ) = x - 1 . S(x)=x^{-1}.
  7. 𝒢 ( k G ) \mathcal{G}(kG)
  8. a k G a\in kG
  9. Δ ( a ) = a a \Delta(a)=a\otimes a
  10. ϵ ( a ) = 1 \epsilon(a)=1
  11. α : G × X X \alpha\colon G\times X\to X
  12. ϕ α : G Aut ( F ( X ) ) \phi_{\alpha}\colon G\to\mathrm{Aut}(F(X))
  13. C 0 ( X ) C_{0}(X)
  14. ϕ α \phi_{\alpha}
  15. ϕ α ( g ) = α g * \phi_{\alpha}(g)=\alpha^{*}_{g}
  16. α g * \alpha^{*}_{g}
  17. α g * ( f ) x = f ( α ( g , x ) ) \alpha^{*}_{g}(f)x=f(\alpha(g,x))
  18. g G , f F ( X ) g\in G,f\in F(X)
  19. x X x\in X
  20. λ : k G F ( X ) F ( X ) \lambda\colon kG\otimes F(X)\to F(X)
  21. λ ( ( c 1 g 1 + c 2 g 2 + ) f ) ( x ) = c 1 f ( g 1 x ) + c 2 f ( g 2 x ) + \lambda((c_{1}g_{1}+c_{2}g_{2}+\cdots)\otimes f)(x)=c_{1}f(g_{1}\cdot x)+c_{2}% f(g_{2}\cdot x)+\cdots
  22. c 1 , c 2 , k c_{1},c_{2},\ldots\in k
  23. g 1 , g 2 , g_{1},g_{2},\ldots
  24. g i x := α ( g i , x ) g_{i}\cdot x:=\alpha(g_{i},x)
  25. λ \lambda
  26. h 1 A = ϵ ( h ) 1 A h\cdot 1_{A}=\epsilon(h)1_{A}
  27. h ( a b ) = ( h ( 1 ) a ) ( h ( 2 ) b ) h\cdot(ab)=(h_{(1)}\cdot a)(h_{(2)}\cdot b)
  28. a , b A a,b\in A
  29. h H h\in H
  30. Δ ( h ) = h ( 1 ) h ( 2 ) \Delta(h)=h_{(1)}\otimes h_{(2)}
  31. λ \lambda
  32. F ( X ) F(X)
  33. A # H A\#H
  34. A H A\otimes H
  35. ( a h ) ( b k ) := a ( h ( 1 ) b ) h ( 2 ) k (a\otimes h)(b\otimes k):=a(h_{(1)}\cdot b)\otimes h_{(2)}k
  36. a # h a\#h
  37. a h a\otimes h
  38. ( a # g 1 ) ( b # g 2 ) = a ( g 1 b ) # g 1 g 2 (a\#g_{1})(b\#g_{2})=a(g_{1}\cdot b)\#g_{1}g_{2}
  39. A # k G A\#kG
  40. A # G A\#G
  41. A H A\rtimes H
  42. C * C^{*}

Group_method_of_data_handling.html

  1. Y ( x 1 , , x n ) = a 0 + i = 1 m a i f i Y(x_{1},\dots,x_{n})=a_{0}+\sum\limits_{i=1}^{m}a_{i}f_{i}
  2. Y ( x 1 , , x n ) = a 0 + i = 1 n a i x i + i = 1 n j = i n a i j x i x j + i = 1 n j = i n k = j n a i j k x i x j x k + Y(x_{1},\dots,x_{n})=a_{0}+\sum\limits_{i=1}^{n}{a_{i}}x_{i}+\sum\limits_{i=1}% ^{n}{\sum\limits_{j=i}^{n}{a_{ij}}}x_{i}x_{j}+\sum\limits_{i=1}^{n}{\sum% \limits_{j=i}^{n}{\sum\limits_{k=j}^{n}{a_{ijk}}}}x_{i}x_{j}x_{k}+\cdots

Guanidinoacetate_N-methyltransferase.html

  1. \rightleftharpoons

Hadwiger–Finsler_inequality.html

  1. a 2 + b 2 + c 2 ( a - b ) 2 + ( b - c ) 2 + ( c - a ) 2 + 4 3 T (HF) . a^{2}+b^{2}+c^{2}\geq(a-b)^{2}+(b-c)^{2}+(c-a)^{2}+4\sqrt{3}T\quad\mbox{(HF)}~% {}.
  2. a 2 + b 2 + c 2 4 3 T (W) . a^{2}+b^{2}+c^{2}\geq 4\sqrt{3}T\quad\mbox{(W)}~{}.

Haloacetate_dehalogenase.html

  1. \rightleftharpoons

Haloalkane_dehalogenase.html

  1. \rightleftharpoons

Hanes–Woolf_plot.html

  1. [ S ] v = [ S ] V max + K m V max {[S]\over v}={[S]\over V_{\max}}+{K_{m}\over V_{\max}}
  2. v = V max [ S ] K m + [ S ] v={{V_{\max}[S]}\over{K_{m}+[S]}}
  3. [ S ] v = [ S ] ( K m + [ S ] ) V max [ S ] = K m + [ S ] V max {[S]\over v}={{[S](K_{m}+[S])}\over{V_{\max}[S]}}={{K_{m}+[S]}\over{V_{\max}}}
  4. [ S ] v = 1 V max [ S ] + K m V max {[S]\over v}={1\over V_{\max}}[S]+{K_{m}\over V_{\max}}

HAZMAT_Class_5_Oxidizing_agents_and_organic_peroxides.html

  1. i = 1 k n i c i m i \sum_{i=1}^{k}\frac{n_{i}c_{i}}{m_{i}}
  2. n i n_{i}
  3. i t h i^{th}
  4. c i c_{i}
  5. i t h i^{th}
  6. m i m_{i}
  7. i t h i^{th}

Heavy_baryon_chiral_perturbation_theory.html

  1. 𝒪 ( 1 ) \mathcal{O}(1)
  2. m B - n m_{B}^{-n}
  3. m B m_{B}

Heine's_identity.html

  1. 1 z - cos ψ = 2 π m = - Q m - 1 2 ( z ) e i m ψ \frac{1}{\sqrt{z-\cos\psi}}=\frac{\sqrt{2}}{\pi}\sum_{m=-\infty}^{\infty}Q_{m-% \frac{1}{2}}(z)e^{im\psi}
  2. Q m - 1 2 Q_{m-\frac{1}{2}}
  3. ( z - cos ψ ) n - 1 2 = 2 π ( z 2 - 1 ) n 2 Γ ( 1 2 - n ) m = - Γ ( m - n + 1 2 ) Γ ( m + n + 1 2 ) Q m - 1 2 n ( z ) e i m ψ , (z-\cos\psi)^{n-\frac{1}{2}}=\sqrt{\frac{2}{\pi}}\frac{(z^{2}-1)^{\frac{n}{2}}% }{\Gamma(\frac{1}{2}-n)}\sum_{m=-\infty}^{\infty}\frac{\Gamma(m-n+\frac{1}{2})% }{\Gamma(m+n+\frac{1}{2})}Q_{m-\frac{1}{2}}^{n}(z)e^{im\psi},
  4. Γ \scriptstyle\,\Gamma

Helicity_basis.html

  1. ξ λ \xi_{\lambda}
  2. σ p ^ ξ λ ( p ^ ) = λ ξ λ ( p ^ ) \sigma\cdot\hat{p}\xi_{\lambda}(\hat{p})=\lambda\xi_{\lambda}(\hat{p})\,
  3. σ \sigma\,
  4. p ^ \hat{p}\,
  5. λ = ± 1 \lambda=\pm 1\,
  6. p ^ \hat{p}\,
  7. ξ λ \xi_{\lambda}\,
  8. p μ = ( E , | p | sin θ cos ϕ , | p | sin θ sin ϕ , | p | cos θ ) p^{\mu}=\left(E,|\vec{p}|\sin{\theta}\cos{\phi},|\vec{p}|\sin{\theta}\sin{\phi% },|\vec{p}|\cos{\theta}\right)\,
  9. ξ + 1 ( p ) = 1 2 | p | ( | p | + p z ) ( | p | + p z p x + i p y ) = ( cos θ 2 e i ϕ sin θ 2 ) \xi_{+1}(\vec{p})=\frac{1}{\sqrt{2|\vec{p}|(|\vec{p}|+p_{z})}}\begin{pmatrix}|% \vec{p}|+p_{z}\\ p_{x}+ip_{y}\end{pmatrix}=\begin{pmatrix}\cos{\frac{\theta}{2}}\\ e^{i\phi}\sin{\frac{\theta}{2}}\end{pmatrix}\,
  10. ξ - 1 ( p ) = 1 2 | p | ( | p | + p z ) ( - p x + i p y | p | + p z ) = ( - e - i ϕ sin θ 2 cos θ 2 ) \xi_{-1}(\vec{p})=\frac{1}{\sqrt{2|\vec{p}|(|\vec{p}|+p_{z})}}\begin{pmatrix}-% p_{x}+ip_{y}\\ |\vec{p}|+p_{z}\end{pmatrix}=\begin{pmatrix}-e^{-i\phi}\sin{\frac{\theta}{2}}% \\ \cos{\frac{\theta}{2}}\end{pmatrix}\,
  11. z ^ = ± p ^ \hat{z}=\pm\hat{p}\,
  12. p ^ = + z ^ \hat{p}=+\hat{z}\,
  13. ξ + 1 ( z ^ ) = ( 1 0 ) \xi_{+1}(\hat{z})=\begin{pmatrix}1\\ 0\end{pmatrix}\,
  14. ξ - 1 ( z ^ ) = ( 0 1 ) \xi_{-1}(\hat{z})=\begin{pmatrix}0\\ 1\end{pmatrix}\,
  15. p ^ = - z ^ \hat{p}=-\hat{z}\,
  16. ξ + 1 ( - z ^ ) = ( 0 1 ) \xi_{+1}(-\hat{z})=\begin{pmatrix}0\\ 1\end{pmatrix}\,
  17. ξ - 1 ( - z ^ ) = ( - 1 0 ) \xi_{-1}(-\hat{z})=\begin{pmatrix}-1\\ 0\end{pmatrix}\,
  18. ψ \psi\,
  19. ψ ( x ) = d 3 p ( 2 π ) 3 2 E λ ± 1 ( a ^ p λ u λ ( p ) e - i p x + b ^ p λ v λ ( p ) e i p x ) \psi(x)=\int{\frac{d^{3}p}{(2\pi)^{3}\sqrt{2E}}\sum_{\lambda\pm 1}{\left(\hat{% a}_{p}^{\lambda}u_{\lambda}(p)e^{-ip\cdot x}+\hat{b}_{p}^{\lambda}v_{\lambda}(% p)e^{ip\cdot x}\right)}}\,
  20. a ^ p λ \hat{a}_{p}^{\lambda}\,
  21. b ^ p λ \hat{b}_{p}^{\lambda}\,
  22. u λ ( p ) u_{\lambda}(p)\,
  23. v λ ( p ) v_{\lambda}(p)\,
  24. u λ ( p ) = ( u - 1 u + 1 ) = ( E - λ | p | χ λ ( p ^ ) E + λ | p | χ λ ( p ^ ) ) u_{\lambda}(p)=\begin{pmatrix}u_{-1}\\ u_{+1}\end{pmatrix}=\begin{pmatrix}\sqrt{E-\lambda|\vec{p}|}\chi_{\lambda}(% \hat{p})\\ \sqrt{E+\lambda|\vec{p}|}\chi_{\lambda}(\hat{p})\end{pmatrix}\,
  25. v λ ( p ) = ( v - 1 v + 1 ) = ( - λ E + λ | p | χ - λ ( p ^ ) λ E - λ | p | χ - λ ( p ^ ) ) v_{\lambda}(p)=\begin{pmatrix}v_{-1}\\ v_{+1}\end{pmatrix}=\begin{pmatrix}-\lambda\sqrt{E+\lambda|\vec{p}|}\chi_{-% \lambda}(\hat{p})\\ \lambda\sqrt{E-\lambda|\vec{p}|}\chi_{-\lambda}(\hat{p})\end{pmatrix}\,
  26. ψ ( x ) = d 3 p ( 2 π ) 3 2 E λ = 0 3 ( a ^ p , λ ϵ λ ( p ) e - i p x + a ^ p , λ ϵ λ * ( p ) e i p x ) \psi(x)=\int{\frac{d^{3}p}{(2\pi)^{3}\sqrt{2E}}\sum_{\lambda=0}^{3}{\left(\hat% {a}_{p,\lambda}\epsilon_{\lambda}(p)e^{-ip\cdot x}+\hat{a}_{p,\lambda}^{% \dagger}\epsilon^{*}_{\lambda}(p)e^{ip\cdot x}\right)}}\,
  27. q μ = ( E , q x , q y , q z ) q^{\mu}=(E,q_{x},q_{y},q_{z})\,
  28. ϵ μ ( q , x ) \epsilon^{\mu}(q,x)\,
  29. = 1 | q | q T ( 0 , q x q z , q y q z , - q T 2 ) =\frac{1}{|\vec{q}|q_{T}}\left(0,q_{x}q_{z},q_{y}q_{z},-q_{T}^{2}\right)\,
  30. ϵ μ ( q , y ) \epsilon^{\mu}(q,y)\,
  31. = 1 q T ( 0 , - q y , q x , 0 ) =\frac{1}{q_{T}}\left(0,-q_{y},q_{x},0\right)\,
  32. ϵ μ ( q , z ) \epsilon^{\mu}(q,z)\,
  33. = E m | q | ( | q | 2 E , q x , q y , q z ) =\frac{E}{m|\vec{q}|}\left(\frac{|\vec{q}|^{2}}{E},q_{x},q_{y},q_{z}\right)\,
  34. q T = q x 2 + q y 2 q_{T}=\sqrt{q_{x}^{2}+q_{y}^{2}}\,
  35. E = | q | 2 + m 2 E=\sqrt{|\vec{q}|^{2}+m^{2}}\,

Helium_atom_scattering.html

  1. V ( z ) = D { e x p [ - 2 α ( z - z m ) ] - 2 e x p [ - α ( z - z m > ) ] } + 2 β D e x p [ 2 α ( z - z m ) ] ξ ( x , y ) V(z)=D\big\{exp\left[-2\alpha(z-z_{m})\right]-2exp\left[-\alpha(z-z_{m}>)% \right]\big\}+2\beta Dexp\left[2\alpha(z-z_{m})\right]\xi(x,y)

Helix_angle.html

  1. Helix angle = arctan ( 2 π r m l ) \mbox{Helix angle}~{}=\arctan\left(\frac{2\pi r_{m}}{l}\right)
  2. α = 45 o - ϕ 2 \alpha=45^{o}-\frac{\phi}{2}
  3. η m a x = 1 - sin ϕ 1 + sin ϕ \eta_{max}=\frac{1-\sin{\phi}}{1+\sin{\phi}}
  4. α \alpha\,
  5. ϕ \phi\,
  6. η m a x \eta_{max}

Hellinger_distance.html

  1. H 2 ( P , Q ) = 1 2 ( d P d λ - d Q d λ ) 2 d λ . H^{2}(P,Q)=\frac{1}{2}\displaystyle\int\left(\sqrt{\frac{dP}{d\lambda}}-\sqrt{% \frac{dQ}{d\lambda}}\right)^{2}d\lambda.
  2. H 2 ( P , Q ) = 1 2 ( d P - d Q ) 2 . H^{2}(P,Q)=\frac{1}{2}\int\left(\sqrt{dP}-\sqrt{dQ}\right)^{2}.
  3. H 2 ( P , Q ) = 1 2 ( f ( x ) - g ( x ) ) 2 d x = 1 - f ( x ) g ( x ) d x , H^{2}(P,Q)=\frac{1}{2}\int\left(\sqrt{f(x)}-\sqrt{g(x)}\right)^{2}\,dx=1-\int% \sqrt{f(x)g(x)}\,dx,
  4. 0 H ( P , Q ) 1. 0\leq H(P,Q)\leq 1.
  5. P = ( p 1 , , p k ) P=(p_{1},\ldots,p_{k})
  6. Q = ( q 1 , , q k ) Q=(q_{1},\ldots,q_{k})
  7. H ( P , Q ) = 1 2 i = 1 k ( p i - q i ) 2 , H(P,Q)=\frac{1}{\sqrt{2}}\;\sqrt{\sum_{i=1}^{k}(\sqrt{p_{i}}-\sqrt{q_{i}})^{2}},
  8. H ( P , Q ) = 1 2 P - Q 2 . H(P,Q)=\frac{1}{\sqrt{2}}\;\bigl\|\sqrt{P}-\sqrt{Q}\bigr\|_{2}.
  9. H ( P , Q ) H(P,Q)
  10. δ ( P , Q ) \delta(P,Q)
  11. H 2 ( P , Q ) δ ( P , Q ) 2 H ( P , Q ) . H^{2}(P,Q)\leq\delta(P,Q)\leq\sqrt{2}H(P,Q)\,.
  12. B C ( P , Q ) BC(P,Q)
  13. H ( P , Q ) = 1 - B C ( P , Q ) . H(P,Q)=\sqrt{1-BC(P,Q)}.
  14. P 𝒩 ( μ 1 , σ 1 2 ) \scriptstyle P\,\sim\,\mathcal{N}(\mu_{1},\sigma_{1}^{2})
  15. Q 𝒩 ( μ 2 , σ 2 2 ) \scriptstyle Q\,\sim\,\mathcal{N}(\mu_{2},\sigma_{2}^{2})
  16. H 2 ( P , Q ) = 1 - 2 σ 1 σ 2 σ 1 2 + σ 2 2 e - 1 4 ( μ 1 - μ 2 ) 2 σ 1 2 + σ 2 2 . H^{2}(P,Q)=1-\sqrt{\frac{2\sigma_{1}\sigma_{2}}{\sigma_{1}^{2}+\sigma_{2}^{2}}% }\,e^{-\frac{1}{4}\frac{(\mu_{1}-\mu_{2})^{2}}{\sigma_{1}^{2}+\sigma_{2}^{2}}}.
  17. P Exp ( α ) \scriptstyle P\,\sim\,\rm{Exp}(\alpha)
  18. Q Exp ( β ) \scriptstyle Q\,\sim\,\rm{Exp}(\beta)
  19. H 2 ( P , Q ) = 1 - 2 α β α + β . H^{2}(P,Q)=1-\frac{2\sqrt{\alpha\beta}}{\alpha+\beta}.
  20. P W ( k , α ) \scriptstyle P\,\sim\,\rm{W}(k,\alpha)
  21. Q W ( k , β ) \scriptstyle Q\,\sim\,\rm{W}(k,\beta)
  22. k k
  23. α , β \alpha\,,\beta
  24. H 2 ( P , Q ) = 1 - 2 ( α β ) k / 2 α k + β k . H^{2}(P,Q)=1-\frac{2(\alpha\beta)^{k/2}}{\alpha^{k}+\beta^{k}}.
  25. α \alpha
  26. β \beta
  27. P Poisson ( α ) \scriptstyle P\,\sim\,\rm{Poisson}(\alpha)
  28. Q Poisson ( β ) \scriptstyle Q\,\sim\,\rm{Poisson}(\beta)
  29. H 2 ( P , Q ) = 1 - e - 1 2 ( α - β ) 2 . H^{2}(P,Q)=1-e^{-\frac{1}{2}(\sqrt{\alpha}-\sqrt{\beta})^{2}}.
  30. P Beta ( a 1 , b 1 ) \scriptstyle P\,\sim\,\,\text{Beta}(a_{1},b_{1})
  31. Q Beta ( a 2 , b 2 ) \scriptstyle Q\,\sim\,\,\text{Beta}(a_{2},b_{2})
  32. H 2 ( P , Q ) = 1 - B ( a 1 + a 2 2 , b 1 + b 2 2 ) B ( a 1 , b 1 ) B ( a 2 , b 2 ) H^{2}(P,Q)=1-\frac{B\left(\frac{a_{1}+a_{2}}{2},\frac{b_{1}+b_{2}}{2}\right)}{% \sqrt{B(a_{1},b_{1})B(a_{2},b_{2})}}
  33. B B

Hennessy–Milner_logic.html

  1. Φ : := t t | f f | Φ 1 Φ 2 | Φ 1 Φ 2 | [ L ] Φ | L Φ \Phi::=tt\,\,\,|\,\,\,ff\,\,\,|\,\,\,\Phi_{1}\land\Phi_{2}\,\,\,|\,\,\,\Phi_{1% }\lor\Phi_{2}\,\,\,|\,\,\,[L]\Phi\,\,\,|\,\,\,\langle L\rangle\Phi
  2. t t tt
  3. f f ff
  4. [ L ] Φ \scriptstyle{[L]\Phi}
  5. L Φ \scriptstyle{\langle L\rangle\Phi}

Hepoxilin-epoxide_hydrolase.html

  1. \rightleftharpoons

Hereditary_property.html

  1. 𝒢 \mathcal{G}
  2. { } \{\varnothing\}
  3. \varnothing
  4. { , { } } \{\varnothing,\{\varnothing\}\}
  5. Φ ( x ) \Phi(x)
  6. Φ ( y ) \Phi(y)
  7. Φ ( y ) \Phi(y)
  8. x tc ( x ) { y | Φ ( y ) } x\cup\mathop{\rm tc}(x)\subseteq\{y|\Phi(y)\}
  9. Φ ( y ) \Phi(y)
  10. { y | Φ ( y ) } \{y|\Phi(y)\}
  11. Φ ( x ) \Phi(x)
  12. H κ H_{\kappa}\!
  13. H ( κ ) H(\kappa)\!
  14. H ( ω ) H(\omega)
  15. H ( ω 1 ) H(\omega_{1})
  16. ω 1 \omega_{1}

Heteroscedasticity-consistent_standard_errors.html

  1. u i ^ \scriptstyle\widehat{u_{i}}
  2. Y = X β + U , Y=X^{\prime}\beta+U,\,
  3. β ^ O L S = ( 𝕏 𝕏 ) - 1 𝕏 𝕐 . \widehat{\beta}_{OLS}=(\mathbb{X}^{\prime}\mathbb{X})^{-1}\mathbb{X}^{\prime}% \mathbb{Y}.\,
  4. 𝕏 \mathbb{X}
  5. X i X_{i}^{\prime}
  6. v O L S [ β ^ O L S ] = s 2 ( 𝕏 𝕏 ) - 1 , s 2 = i u ^ i 2 n - k v_{OLS}[\hat{\beta}_{OLS}]=s^{2}(\mathbb{X}^{\prime}\mathbb{X})^{-1},s^{2}=% \frac{\sum_{i}\hat{u}_{i}^{2}}{n-k}
  7. u ^ i \hat{u}_{i}
  8. E [ u u ] = σ 2 I n E[uu^{\prime}]=\sigma^{2}I_{n}
  9. V [ β ^ O L S ] = V [ ( 𝕏 𝕏 ) - 1 𝕏 𝕐 ] = ( 𝕏 𝕏 ) - 1 𝕏 Σ 𝕏 ( 𝕏 𝕏 ) - 1 V[\hat{\beta}_{OLS}]=V[(\mathbb{X}^{\prime}\mathbb{X})^{-1}\mathbb{X}^{\prime}% \mathbb{Y}]=(\mathbb{X}^{\prime}\mathbb{X})^{-1}\mathbb{X}^{\prime}\Sigma% \mathbb{X}(\mathbb{X}^{\prime}\mathbb{X})^{-1}
  10. Σ = V [ u ] \Sigma=V[u]
  11. v O L S [ β ^ O L S ] v_{OLS}[\hat{\beta}_{OLS}]
  12. u i u_{i}
  13. Σ = diag ( σ 1 2 , , σ n 2 ) \Sigma=\operatorname{diag}(\sigma_{1}^{2},\ldots,\sigma_{n}^{2})
  14. σ ^ i 2 = u ^ i 2 \hat{\sigma}_{i}^{2}=\hat{u}_{i}^{2}
  15. v H C E [ β ^ O L S ] \displaystyle v_{HCE}[\hat{\beta}_{OLS}]
  16. 𝕏 \mathbb{X}
  17. X i X_{i}^{\prime}
  18. Ω ^ n \hat{\Omega}_{n}
  19. n \sqrt{n}
  20. n ( β ^ n - β ) 𝑑 N ( 0 , Ω ) , \sqrt{n}(\hat{\beta}_{n}-\beta)\xrightarrow{d}N(0,\Omega),
  21. Ω = E [ X X ] - 1 V a r [ X u ] E [ X X ] - 1 , \Omega=E[XX^{\prime}]^{-1}Var[Xu]E[XX^{\prime}]^{-1},
  22. Ω ^ n \displaystyle\hat{\Omega}_{n}
  23. Ω ^ n = n v H C E [ β ^ O L S ] \hat{\Omega}_{n}=n\cdot v_{HCE}[\hat{\beta}_{OLS}]
  24. V a r ^ [ X u ] = 1 n i X i X i u ^ i 2 = 1 n 𝕏 diag ( u ^ 1 2 , , u ^ n 2 ) 𝕏 \widehat{Var}[Xu]=\frac{1}{n}\sum_{i}X_{i}X_{i}^{\prime}\hat{u}_{i}^{2}=\frac{% 1}{n}\mathbb{X}^{\prime}\operatorname{diag}(\hat{u}_{1}^{2},\ldots,\hat{u}_{n}% ^{2})\mathbb{X}

Heuristic_(computer_science).html

  1. h ( v i , v g ) h(v_{i},v_{g})
  2. d ( v i , v g ) d^{\star}(v_{i},v_{g})
  3. v g v_{g}
  4. G G
  5. n n
  6. v 0 , v 1 , , v n v_{0},v_{1},\cdots,v_{n}
  7. h ( v i , v g ) d ( v i , v g ) h(v_{i},v_{g})\leq d^{\star}(v_{i},v_{g})
  8. ( v i , v g ) (v_{i},v_{g})
  9. i , g [ 0 , 1 , , n ] {i,g}\in[0,1,...,n]
  10. G G
  11. v i v_{i}
  12. v j v_{j}
  13. i , j g {i,j}\neq g

Heuristic_function.html

  1. h ( n ) h(n)
  2. g ( n ) + h ( n ) g(n)+h(n)
  3. g ( n ) g(n)
  4. h ( n ) h(n)
  5. h ( n ) h(n)
  6. b b
  7. d d
  8. b d b^{d}
  9. b b
  10. b b^{\prime}
  11. h 1 ( n ) < h 2 ( n ) h_{1}(n)<h_{2}(n)
  12. h 1 ( n ) h_{1}(n)
  13. h 2 ( n ) h_{2}(n)
  14. n n
  15. h 1 h 2 h_{1}\leq h_{2}
  16. h ( n ) h ( n ) h(n)\leq h^{\prime}(n)
  17. h 1 h 2 h_{1}\leq h_{2}
  18. h 2 h_{2}
  19. h 1 h_{1}
  20. h 1 ( n ) , h 2 ( n ) , , h i ( n ) h_{1}(n),h_{2}(n),...,h_{i}(n)
  21. h ( n ) = max { h 1 ( n ) , h 2 ( n ) , , h i ( n ) } h(n)=\max\{h_{1}(n),h_{2}(n),...,h_{i}(n)\}
  22. h ( n ) h(n)
  23. h ( n ) - h ( n ) c ( n , n ) h(n)-h(n^{\prime})\leq c(n,n^{\prime})
  24. n n^{\prime}
  25. c ( n , n ) c(n,n^{\prime})
  26. n n
  27. n n^{\prime}
  28. h ( g o a l ) = 0 h(goal)=0
  29. h ( n ) - h ( n ) c relaxed ( n , n ) c ( n , n ) h(n)-h(n^{\prime})\leq c_{\mathrm{relaxed}}(n,n^{\prime})\leq c(n,n^{\prime})
  30. h 1 h_{1}
  31. h 1 h_{1}
  32. h 1 h_{1}
  33. h 2 h_{2}
  34. h 2 h_{2}
  35. h 2 h_{2}
  36. h 1 h_{1}
  37. h 2 h_{2}
  38. h 2 h_{2}
  39. h 1 h_{1}
  40. h 2 h_{2}
  41. h 1 h_{1}

Hexadecanal_dehydrogenase_(acylating).html

  1. \rightleftharpoons

Hexadecanol_dehydrogenase.html

  1. \rightleftharpoons

Hexaprenyldihydroxybenzoate_methyltransferase.html

  1. \rightleftharpoons

Hexose_oxidase.html

  1. \rightleftharpoons

Histidinol_dehydrogenase.html

  1. \rightleftharpoons

History_of_Young_Physicists'_Tournament_in_Russia.html

  1. ν \nu
  2. ω \omega
  3. ν \nu

Hofstadter_sequence.html

  1. R ( 1 ) = 1 ; S ( 1 ) = 2 R ( n ) = R ( n - 1 ) + S ( n - 1 ) , n > 1. \begin{aligned}\displaystyle R(1)&\displaystyle=1~{};\ S(1)=2\\ \displaystyle R(n)&\displaystyle=R(n-1)+S(n-1),\quad n>1.\end{aligned}
  2. G ( 0 ) = 0 G ( n ) = n - G ( G ( n - 1 ) ) , n > 0. \begin{aligned}\displaystyle G(0)&\displaystyle=0\\ \displaystyle G(n)&\displaystyle=n-G(G(n-1)),\quad n>0.\end{aligned}
  3. H ( 0 ) = 0 H ( n ) = n - H ( H ( H ( n - 1 ) ) ) , n > 0. \begin{aligned}\displaystyle H(0)&\displaystyle=0\\ \displaystyle H(n)&\displaystyle=n-H(H(H(n-1))),\quad n>0.\end{aligned}
  4. F ( 0 ) = 1 ; M ( 0 ) = 0 F ( n ) = n - M ( F ( n - 1 ) ) , n > 0 M ( n ) = n - F ( M ( n - 1 ) ) , n > 0. \begin{aligned}\displaystyle F(0)&\displaystyle=1~{};\ M(0)=0\\ \displaystyle F(n)&\displaystyle=n-M(F(n-1)),\quad n>0\\ \displaystyle M(n)&\displaystyle=n-F(M(n-1)),\quad n>0.\end{aligned}
  5. Q ( 1 ) = Q ( 2 ) = 1 , Q ( n ) = Q ( n - Q ( n - 1 ) ) + Q ( n - Q ( n - 2 ) ) , n > 2. \begin{aligned}\displaystyle Q(1)&\displaystyle=Q(2)=1,\\ \displaystyle Q(n)&\displaystyle=Q(n-Q(n-1))+Q(n-Q(n-2)),\quad n>2.\end{aligned}
  6. Q r , s ( n ) = { 1 , 1 n s , Q r , s ( n - Q r , s ( n - r ) ) + Q r , s ( n - Q r , s ( n - s ) ) , n > s , Q_{r,s}(n)=\begin{cases}1,\quad 1\leq n\leq s,\\ Q_{r,s}(n-Q_{r,s}(n-r))+Q_{r,s}(n-Q_{r,s}(n-s)),\quad n>s,\end{cases}
  7. F i , j ( n ) = { 1 , n = 1 , 2 , F i , j ( n - i - F i , j ( n - 1 ) ) + F i , j ( n - j - F i , j ( n - 2 ) ) , n > 2. F_{i,j}(n)=\begin{cases}1,\quad n=1,2,\\ F_{i,j}(n-i-F_{i,j}(n-1))+F_{i,j}(n-j-F_{i,j}(n-2)),\quad n>2.\end{cases}
  8. a ( 1 ) = a ( 2 ) = 1 , a ( n ) = a ( a ( n - 1 ) ) + a ( n - a ( n - 1 ) ) , n > 2. \begin{aligned}\displaystyle a(1)&\displaystyle=a(2)=1,\\ \displaystyle a(n)&\displaystyle=a(a(n-1))+a(n-a(n-1)),\quad n>2.\end{aligned}

Holocytochrome-c_synthase.html

  1. \rightleftharpoons

Homocysteine_desulfhydrase.html

  1. \rightleftharpoons

Homocysteine_S-methyltransferase.html

  1. \rightleftharpoons

Homoisocitrate_dehydrogenase.html

  1. \rightleftharpoons

Homoserine_dehydrogenase.html

  1. \rightleftharpoons

Householder's_method.html

  1. x n + 1 = x n + d ( 1 / f ) ( d - 1 ) ( x n ) ( 1 / f ) ( d ) ( x n ) x_{n+1}=x_{n}+d\;\frac{\left(1/f\right)^{(d-1)}(x_{n})}{\left(1/f\right)^{(d)}% (x_{n})}
  2. | x n + 1 - a | K | x n - a | d + 1 |x_{n+1}-a|\leq K\cdot{|x_{n}-a|}^{d+1}
  3. K > 0. K>0.\!
  4. d + 1 d + 1 \sqrt[d+1]{d+1}
  5. 2 3 1.2599 \sqrt[3]{2}\approx 1.2599
  6. 3 6 1.2009 \sqrt[6]{3}\approx 1.2009
  7. f ( a ) 0 f^{\prime}(a)\neq 0
  8. 1 f ( x ) = 1 f ( x ) - f ( a ) = x - a f ( x ) - f ( a ) - 1 a ( 1 - x / a ) - 1 a f ( a ) k = 0 ( x a ) k . \frac{1}{f(x)}=\frac{1}{f(x)-f(a)}=\frac{x-a}{f(x)-f(a)}\cdot\frac{-1}{a(1-x/a% )}\approx\frac{-1}{af^{\prime}(a)}\cdot\sum_{k=0}^{\infty}\left(\frac{x}{a}% \right)^{k}.
  9. f ( a ) 0 f^{\prime}(a)\neq 0
  10. C a - d C\,a^{-d}
  11. a b + ( 1 / f ) ( d - 1 ) ( b ) ( d - 1 ) ! d ! ( 1 / f ) ( d ) ( b ) = b + d ( 1 / f ) ( d - 1 ) ( b ) ( 1 / f ) ( d ) ( b ) . a\approx b+\frac{(1/f)^{(d-1)}(b)}{(d-1)!}\;\frac{d!}{(1/f)^{(d)}(b)}=b+d\;% \frac{(1/f)^{(d-1)}(b)}{(1/f)^{(d)}(b)}.
  12. ( 1 / f ) ( x ) = d = 0 ( 1 / f ) ( d ) ( b ) d ! ( x - b ) d . (1/f)(x)=\sum_{d=0}^{\infty}\frac{(1/f)^{(d)}(b)}{d!}(x-b)^{d}.
  13. a - b = lim d ( 1 / f ) ( d - 1 ) ( b ) ( d - 1 ) ! ( 1 / f ) ( d ) ( b ) d ! = d ( 1 / f ) ( d - 1 ) ( b ) ( 1 / f ) ( d ) ( b ) . a-b=\lim_{d\rightarrow\infty}\frac{\frac{(1/f)^{(d-1)}(b)}{(d-1)!}}{\frac{(1/f% )^{(d)}(b)}{d!}}=d\frac{(1/f)^{(d-1)}(b)}{(1/f)^{(d)}(b)}.
  14. x n + 1 = x n + 1 ( 1 / f ) ( x n ) ( 1 / f ) ( 1 ) ( x n ) = x n + 1 f ( x n ) ( - f ( x n ) f ( x n ) 2 ) - 1 = x n - f ( x n ) f ( x n ) . \begin{array}[]{rl}x_{n+1}=&x_{n}+1\,\frac{\left(1/f\right)(x_{n})}{\left(1/f% \right)^{(1)}(x_{n})}\\ =&x_{n}+\frac{1}{f(x_{n})}\cdot\left(\frac{-f^{\prime}(x_{n})}{f(x_{n})^{2}}% \right)^{-1}\\ =&x_{n}-\frac{f(x_{n})}{f^{\prime}(x_{n})}.\end{array}
  15. ( 1 / f ) ( x ) = - f ( x ) f ( x ) 2 \textstyle(1/f)^{\prime}(x)=-\frac{f^{\prime}(x)}{f(x)^{2}}
  16. ( 1 / f ) ′′ ( x ) = - f ′′ ( x ) f ( x ) 2 + 2 f ( x ) 2 f ( x ) 3 \textstyle\ (1/f)^{\prime\prime}(x)=-\frac{f^{\prime\prime}(x)}{f(x)^{2}}+2% \frac{f^{\prime}(x)^{2}}{f(x)^{3}}
  17. x n + 1 = x n + 2 ( 1 / f ) ( x n ) ( 1 / f ) ′′ ( x n ) = x n + - 2 f ( x n ) f ( x n ) - f ( x n ) f ′′ ( x n ) + 2 f ( x n ) 2 = x n - f ( x n ) f ( x n ) f ( x n ) 2 - 1 2 f ( x n ) f ′′ ( x n ) = x n + h n 1 1 + 1 2 ( f ′′ / f ) ( x n ) h n . \begin{array}[]{rl}x_{n+1}=&x_{n}+2\,\frac{\left(1/f\right)^{\prime}(x_{n})}{% \left(1/f\right)^{\prime\prime}(x_{n})}\\ =&x_{n}+\frac{-2f(x_{n})\,f^{\prime}(x_{n})}{-f(x_{n})f^{\prime\prime}(x_{n})+% 2f^{\prime}(x_{n})^{2}}\\ =&x_{n}-\frac{f(x_{n})f^{\prime}(x_{n})}{f^{\prime}(x_{n})^{2}-\tfrac{1}{2}f(x% _{n})f^{\prime\prime}(x_{n})}\\ =&x_{n}+h_{n}\;\frac{1}{1+\frac{1}{2}(f^{\prime\prime}/f^{\prime})(x_{n})\,h_{% n}}.\end{array}
  18. h n = - f ( x n ) f ( x n ) h_{n}=-\tfrac{f(x_{n})}{f^{\prime}(x_{n})}
  19. x n x_{n}
  20. ( 1 / f ) ′′′ ( x ) = - f ′′′ ( x ) f ( x ) 2 + 6 f ( x ) f ′′ ( x ) f ( x ) 3 - 6 f ( x ) 3 f ( x ) 4 \textstyle(1/f)^{\prime\prime\prime}(x)=-\frac{f^{\prime\prime\prime}(x)}{f(x)% ^{2}}+6\frac{f^{\prime}(x)\,f^{\prime\prime}(x)}{f(x)^{3}}-6\frac{f^{\prime}(x% )^{3}}{f(x)^{4}}
  21. x n + 1 = x n + 3 ( 1 / f ) ′′ ( x n ) ( 1 / f ) ′′′ ( x n ) = x n - 6 f ( x n ) f ( x n ) 2 - 3 f ( x n ) 2 f ′′ ( x n ) 6 f ( x ) 3 - 6 f ( x n ) f ( x n ) f ′′ ( x ) + f ( x n ) 2 f ′′′ ( x n ) = x n + h n 1 + 1 2 ( f ′′ / f ) ( x n ) h n 1 + ( f ′′ / f ) ( x n ) h n + 1 6 ( f ′′′ / f ) ( x n ) h n 2 \begin{array}[]{rl}x_{n+1}=&x_{n}+3\,\frac{\left(1/f\right)^{\prime\prime}(x_{% n})}{\left(1/f\right)^{\prime\prime\prime}(x_{n})}\\ =&x_{n}-\frac{6f(x_{n})\,f^{\prime}(x_{n})^{2}-3f(x_{n})^{2}f^{\prime\prime}(x% _{n})}{6f^{\prime}(x)^{3}-6f(x_{n})f^{\prime}(x_{n})\,f^{\prime\prime}(x)+f(x_% {n})^{2}\,f^{\prime\prime\prime}(x_{n})}\\ =&x_{n}+h_{n}\frac{1+\frac{1}{2}(f^{\prime\prime}/f^{\prime})(x_{n})\,h_{n}}{1% +(f^{\prime\prime}/f^{\prime})(x_{n})\,h_{n}+\frac{1}{6}(f^{\prime\prime\prime% }/f^{\prime})(x_{n})\,h_{n}^{2}}\end{array}
  22. y 3 - 2 y - 5 = 0 y^{3}-2y-5=0
  23. 0 = f ( x ) = - 1 + 10 x + 6 x 2 + x 3 0=f(x)=-1+10x+6x^{2}+x^{3}
  24. 1 / f ( x ) = - 1 - 10 x - 106 x 2 - 1121 x 3 - 11856 x 4 - 125392 x 5 - 1326177 x 6 - 14025978 x 7 - 148342234 x 8 - 1568904385 x 9 - 16593123232 x 10 + O ( x 11 ) \begin{array}[]{rl}1/f(x)=&-1-10\,x-106\,x^{2}-1121\,x^{3}-11856\,x^{4}-125392% \,x^{5}\\ &-1326177\,x^{6}-14025978\,x^{7}-148342234\,x^{8}-1568904385\,x^{9}\\ &-16593123232\,x^{10}+O(x^{11})\end{array}
  25. x 1 = 0.0 + 106 / 1121 = 0.09455842997324 x_{1}=0.0+106/1121=0.09455842997324
  26. x 2 , x 3 , x 4 x_{2},x_{3},x_{4}
  27. f = - 1 + 10 x + 6 x 2 + x 3 f=-1+10x+6x^{2}+x^{3}
  28. f = 10 + 12 x + 3 x 2 f^{\prime}=10+12x+3x^{2}
  29. f ′′ = 12 + 6 x f^{\prime\prime}=12+6x
  30. f ′′′ = 6 f^{\prime\prime\prime}=6
  31. x i + 1 = x i - f ( x i ) / f ( x i ) x_{i+1}=x_{i}-f(x_{i})/f^{\prime}(x_{i})
  32. x i + 1 = x i + ( - 2 f f ) / ( 2 f 2 - f f ′′ ) x_{i+1}=x_{i}+(-2ff^{\prime})/(2{f^{\prime}}^{2}-ff^{\prime\prime})
  33. x i + 1 = x i - 6 f f 2 - 3 f 2 f ′′ 6 f 3 - 6 f f f ′′ + f 2 f ′′′ x_{i+1}=x_{i}-\frac{6f{f^{\prime}}^{2}-3f^{2}f^{\prime\prime}}{6{f^{\prime}}^{% 3}-6ff^{\prime}f^{\prime\prime}+f^{2}f^{\prime\prime\prime}}
  34. f ( x + h ) = a 0 + h b 0 + b 1 h + + b d - 1 h d - 1 + O ( h d + 1 ) . f(x+h)=\frac{a_{0}+h}{b_{0}+b_{1}h+\cdots+b_{d-1}h^{d-1}}+O(h^{d+1}).
  35. h = - a 0 h=-a_{0}
  36. b d = 0 b_{d}=0
  37. ( 1 / f ) ( x + h ) = ( 1 / f ) ( x ) + ( 1 / f ) ( x ) h + + ( 1 / f ) ( d - 1 ) ( x ) h d - 1 ( d - 1 ) ! + ( 1 / f ) ( d ) ( x ) h d d ! + O ( h d + 1 ) (1/f)(x+h)=(1/f)(x)+(1/f)^{\prime}(x)h+\cdots+(1/f)^{(d-1)}(x)\frac{h^{d-1}}{(% d-1)!}+(1/f)^{(d)}(x)\frac{h^{d}}{d!}+O(h^{d+1})
  38. a 0 + h a_{0}+h
  39. h d h^{d}
  40. 0 = b d = a 0 ( 1 / f ) ( d ) ( x ) 1 d ! + ( 1 / f ) ( d - 1 ) ( x ) 1 ( d - 1 ) ! 0=b_{d}=a_{0}(1/f)^{(d)}(x)\frac{1}{d!}+(1/f)^{(d-1)}(x)\frac{1}{(d-1)!}
  41. h = - a 0 h=-a_{0}
  42. h = - a 0 = 1 ( d - 1 ) ! ( 1 / f ) ( d - 1 ) ( x ) 1 d ! ( 1 / f ) ( d ) ( x ) = d ( 1 / f ) ( d - 1 ) ( x ) ( 1 / f ) ( d ) ( x ) \begin{array}[]{rl}h=&-a_{0}=\frac{\frac{1}{(d-1)!}(1/f)^{(d-1)}(x)}{\frac{1}{% d!}(1/f)^{(d)}(x)}\\ =&d\,\frac{(1/f)^{(d-1)}(x)}{(1/f)^{(d)}(x)}\end{array}
  43. x n + 1 = x n + d ( 1 / f ) ( d - 1 ) ( x n ) ( 1 / f ) ( d ) ( x n ) x_{n+1}=x_{n}+d\;\frac{\left(1/f\right)^{(d-1)}(x_{n})}{\left(1/f\right)^{(d)}% (x_{n})}

Hörmander's_condition.html

  1. [ V , W ] ( x ) = D V ( x ) W ( x ) - D W ( x ) V ( x ) , [V,W](x)=\mathrm{D}V(x)W(x)-\mathrm{D}W(x)V(x),
  2. A j 0 ( x ) , [ A j 0 ( x ) , A j 1 ( x ) ] , [ [ A j 0 ( x ) , A j 1 ( x ) ] , A j 2 ( x ) ] , 0 j 0 , j 1 , , j n n \begin{aligned}&\displaystyle A_{j_{0}}(x)~{},\\ &\displaystyle[A_{j_{0}}(x),A_{j_{1}}(x)]~{},\\ &\displaystyle[[A_{j_{0}}(x),A_{j_{1}}(x)],A_{j_{2}}(x)]~{},\\ &\displaystyle\quad\vdots\end{aligned}\qquad 0\leq j_{0},j_{1},\ldots,j_{n}\leq n
  3. j 0 j_{0}
  4. d x = V 0 ( x ) d t + i = 1 m V i ( x ) d W i dx=V_{0}(x)\;dt+\sum_{i=1}^{m}V_{i}(x)\circ dW_{i}
  5. F = 1 2 i = 1 n A i 2 + A 0 . F=\frac{1}{2}\sum_{i=1}^{n}A_{i}^{2}+A_{0}.
  6. { u t ( t , x ) = F u ( t , x ) , t > 0 , x 𝐑 d ; u ( t , ) f , as t 0 ; \begin{cases}\dfrac{\partial u}{\partial t}(t,x)=Fu(t,x),&t>0,x\in\mathbf{R}^{% d};\\ u(t,\cdot)\to f,&\mbox{ as }~{}t\to 0;\end{cases}
  7. u ( t , x ) = 𝐑 d p ( t , x , y ) f ( y ) d y u(t,x)=\int_{\mathbf{R}^{d}}p(t,x,y)f(y)\,\mathrm{d}y
  8. A i = j = 1 d a j i x j , A_{i}=\sum_{j=1}^{d}a_{ji}\frac{\partial}{\partial x_{j}},

HPO_formalism.html

  1. \mathcal{H}
  2. \mathcal{H}
  3. P \,P
  4. P ^ \hat{P}
  5. \mathcal{H}
  6. α \,\alpha
  7. α t i \alpha_{t_{i}}
  8. t 1 < t 2 < < t n t_{1}<t_{2}<\ldots<t_{n}
  9. α \,\alpha
  10. ( α 1 , α 2 , , α n ) (\alpha_{1},\alpha_{2},\ldots,\alpha_{n})
  11. α t 1 \alpha_{t_{1}}
  12. t 1 t_{1}
  13. α t 2 \alpha_{t_{2}}
  14. t 2 t_{2}
  15. \ldots
  16. α t n \alpha_{t_{n}}
  17. t n t_{n}
  18. α \,\alpha
  19. β \,\beta
  20. α , β \,\alpha,\beta
  21. α = ( α 1 , α 2 , , α n ) \alpha=(\alpha_{1},\alpha_{2},\ldots,\alpha_{n})
  22. α ^ := α ^ t 1 α ^ t 2 α ^ t n \hat{\alpha}:=\hat{\alpha}_{t_{1}}\otimes\hat{\alpha}_{t_{2}}\otimes\ldots% \otimes\hat{\alpha}_{t_{n}}
  23. α ^ t i \hat{\alpha}_{t_{i}}
  24. \mathcal{H}
  25. α t i \alpha_{t_{i}}
  26. t i t_{i}
  27. α ^ \hat{\alpha}
  28. H = H=\mathcal{H}\otimes\mathcal{H}\otimes\ldots\otimes\mathcal{H}
  29. H H
  30. α ^ \hat{\alpha}
  31. H H
  32. α \,\alpha
  33. β \,\beta
  34. t i \,t_{i}
  35. α \,\alpha
  36. β \,\beta
  37. β \,\beta
  38. t i \,t_{i}
  39. α , β \,\alpha,\beta
  40. α \,\alpha
  41. β \,\beta
  42. α ^ β ^ = β ^ α ^ \hat{\alpha}\hat{\beta}=\hat{\beta}\hat{\alpha}
  43. α \alpha
  44. β \beta
  45. α \,\alpha
  46. β \,\beta
  47. α β ^ := α ^ β ^ \widehat{\alpha\wedge\beta}:=\hat{\alpha}\hat{\beta}
  48. ( = β ^ α ^ ) (=\hat{\beta}\hat{\alpha})
  49. α \alpha
  50. β \beta
  51. α \,\alpha
  52. β \,\beta
  53. α β ^ := α ^ + β ^ - α ^ β ^ \widehat{\alpha\vee\beta}:=\hat{\alpha}+\hat{\beta}-\hat{\alpha}\hat{\beta}
  54. P ^ \hat{P}
  55. ¬ P ^ := 𝕀 - P ^ \neg\hat{P}:=\mathbb{I}-\hat{P}
  56. 𝕀 \mathbb{I}
  57. ¬ α \neg\alpha
  58. α \alpha
  59. ¬ α ^ := 𝕀 - α ^ \widehat{\neg\alpha}:=\mathbb{I}-\hat{\alpha}
  60. 𝕀 \mathbb{I}
  61. α = ( α 1 , α 2 ) \,\alpha=(\alpha_{1},\alpha_{2})
  62. ¬ α \neg\alpha
  63. ¬ α ^ = 𝕀 𝕀 - α ^ 1 α ^ 2 \widehat{\neg\alpha}=\mathbb{I}\otimes\mathbb{I}-\hat{\alpha}_{1}\otimes\hat{% \alpha}_{2}
  64. = ( 𝕀 - α ^ 1 ) α ^ 2 + α ^ 1 ( 𝕀 - α ^ 2 ) + ( 𝕀 - α ^ 1 ) ( 𝕀 - α ^ 2 ) =(\mathbb{I}-\hat{\alpha}_{1})\otimes\hat{\alpha}_{2}+\hat{\alpha}_{1}\otimes(% \mathbb{I}-\hat{\alpha}_{2})+(\mathbb{I}-\hat{\alpha}_{1})\otimes(\mathbb{I}-% \hat{\alpha}_{2})
  65. ( 𝕀 - α ^ 1 ) α ^ 2 (\mathbb{I}-\hat{\alpha}_{1})\otimes\hat{\alpha}_{2}
  66. α ^ 1 ( 𝕀 - α ^ 2 ) \hat{\alpha}_{1}\otimes(\mathbb{I}-\hat{\alpha}_{2})
  67. ( 𝕀 - α ^ 1 ) ( 𝕀 - α ^ 2 ) (\mathbb{I}-\hat{\alpha}_{1})\otimes(\mathbb{I}-\hat{\alpha}_{2})
  68. α 1 \,\alpha_{1}
  69. α 2 \,\alpha_{2}
  70. α 1 \,\alpha_{1}
  71. α 2 \,\alpha_{2}
  72. α 1 \,\alpha_{1}
  73. α 2 \,\alpha_{2}
  74. α 1 \,\alpha_{1}
  75. α 2 \,\alpha_{2}
  76. ¬ α ^ \widehat{\neg\alpha}
  77. ¬ α \neg\alpha

Hudson's_equation.html

  1. W = γ r H 3 K D Δ 3 cot θ W=\frac{\gamma_{r}H^{3}}{K_{D}\Delta^{3}\cot\theta}
  2. γ r \gamma_{r}
  3. H s Δ D n 50 = ( K D c o t θ ) 1 / 3 1.27 \frac{H_{s}}{\Delta D_{n50}}=\frac{(K_{D}cot\theta)^{1/3}}{1.27}

Hundred-dollar,_Hundred-digit_Challenge_problems.html

  1. lim ε 0 ε 1 x - 1 cos ( x - 1 log x ) d x \lim_{\varepsilon\to 0}\int_{\varepsilon}^{1}x^{-1}\cos\left(x^{-1}\log x% \right)\,dx
  2. a 11 = 1 , a 12 = 1 / 2 , a 21 = 1 / 3 , a 13 = 1 / 4 , a 22 = 1 / 5 , a 31 = 1 / 6 , a_{11}=1,a_{12}=1/2,a_{21}=1/3,a_{13}=1/4,a_{22}=1/5,a_{31}=1/6,\dots
  3. 2 \ell^{2}
  4. || A || ||A||
  5. exp ( sin ( 50 x ) ) + sin ( 60 e y ) + sin ( 70 sin x ) + sin ( sin ( 80 y ) ) - sin ( 10 ( x + y ) ) + 1 / 4 ( x 2 + y 2 ) \exp\left(\sin\left(50x\right)\right)+\sin\left(60e^{y}\right)+\sin\left(70% \sin x\right)+\sin\left(\sin\left(80y\right)\right)-\sin\left(10\left(x+y% \right)\right)+1/4\left(x^{2}+y^{2}\right)
  6. f ( z ) = 1 / Γ ( z ) f(z)=1/\Gamma(z)
  7. Γ ( z ) \Gamma(z)
  8. p ( z ) p(z)
  9. f ( z ) f(z)
  10. | | . | | ||.||_{\infty}
  11. || f - p || ||f-p||_{\infty}
  12. ( 0 , 0 ) (0,0)
  13. 1 / 4 1/4
  14. 1 / 4 + ε 1/4+\varepsilon
  15. 1 / 4 - ε 1/4-\varepsilon
  16. 1 / 2 1/2
  17. ε \varepsilon
  18. a i j a_{ij}
  19. | i - j | = 1 , 2 , 4 , 8 , , 16384 |i-j|=1,2,4,8,\dots,16384
  20. A - 1 A^{-1}
  21. [ - 1 , 1 ] × [ - 1 , 1 ] [-1,1]\times[-1,1]
  22. u = 0 u=0
  23. t = 0 t=0
  24. u = 5 u=5
  25. u = 0 u=0
  26. u t = Δ u u_{t}=\Delta u
  27. u = 1 u=1
  28. I ( α ) = 0 2 [ 2 + sin ( 10 α ) ] x α sin ( α / ( 2 - x ) ) d x I(\alpha)=\int_{0}^{2}\left[2+\sin\left(10\alpha\right)\right]x^{\alpha}\sin% \left(\alpha/\left(2-x\right)\right)\,dx

Hybrid-pi_model.html

  1. v be v_{\mathrm{be}}
  2. v ce v_{\mathrm{ce}}
  3. i b i_{\mathrm{b}}
  4. i c i_{\mathrm{c}}
  5. g m = i c v be | v ce = 0 = I C V T g_{m}=\frac{i_{\mathrm{c}}}{v_{\mathrm{be}}}\Bigg|_{v_{\mathrm{ce}}=0}=\frac{I% _{\mathrm{C}}}{V_{\mathrm{T}}}
  6. I C I_{\mathrm{C}}\,
  7. V T = k T q V_{\mathrm{T}}=\begin{matrix}\frac{kT}{q}\end{matrix}
  8. k k
  9. q q
  10. T T
  11. V T V_{\mathrm{T}}
  12. r π = v be i b | v ce = 0 = β 0 g m = V T I B r_{\pi}=\frac{v_{\mathrm{be}}}{i_{\mathrm{b}}}\Bigg|_{v_{\mathrm{ce}}=0}=\frac% {\beta_{0}}{g_{m}}=\frac{V_{\mathrm{T}}}{I_{\mathrm{B}}}\,
  13. β 0 = I C I B \beta_{0}=\frac{I_{\mathrm{C}}}{I_{\mathrm{B}}}\,
  14. I B I_{\mathrm{B}}
  15. r O = v ce i c | v be = 0 = V A + V CE I C V A I C r_{\mathrm{O}}=\frac{v_{\mathrm{ce}}}{i_{\mathrm{c}}}\Bigg|_{v_{\mathrm{be}}=0% }=\frac{V_{\mathrm{A}}+V_{\mathrm{CE}}}{I_{\mathrm{C}}}\approx\frac{V_{\mathrm% {A}}}{I_{\mathrm{C}}}
  16. V A V_{\mathrm{A}}
  17. g ce = 1 r O g_{\mathrm{ce}}=\frac{1}{r_{\mathrm{O}}}
  18. r e = 1 g m r_{\mathrm{e}}=\frac{1}{g_{m}}
  19. g m = i d v gs | v ds = 0 g_{m}=\frac{i_{\mathrm{d}}}{v_{\mathrm{gs}}}\Bigg|_{v_{\mathrm{ds}}=0}
  20. I D I_{\mathrm{D}}
  21. g m = 2 I D V GS - V th \ g_{m}=\frac{2I_{\mathrm{D}}}{V_{\mathrm{GS}}-V_{\mathrm{th}}}
  22. I D I_{\mathrm{D}}
  23. V th V_{\mathrm{th}}
  24. V GS V_{\mathrm{GS}}
  25. V ov = V GS - V th \ V_{\mathrm{ov}}=V_{\mathrm{GS}}-V_{\mathrm{th}}
  26. r O = v ds i d | v gs = 0 r_{\mathrm{O}}=\frac{v_{\mathrm{ds}}}{i_{\mathrm{d}}}\Bigg|_{v_{\mathrm{gs}}=0}
  27. r O = 1 / λ + V DS I D V E L I D r_{\mathrm{O}}=\frac{1/\lambda+V_{\mathrm{DS}}}{I_{\mathrm{D}}}\approx\frac{V_% {E}L}{I_{\mathrm{D}}}
  28. λ = 1 V E L \lambda=\frac{1}{V_{E}L}
  29. g ds = 1 r O g_{\mathrm{ds}}=\frac{1}{r_{\mathrm{O}}}

Hydrogen:quinone_oxidoreductase.html

  1. \rightleftharpoons

Hydrogen_dehydrogenase.html

  1. \rightleftharpoons

Hydrogen_dehydrogenase_(NADP+).html

  1. \rightleftharpoons

Hydrogenase_(acceptor).html

  1. \rightleftharpoons

Hydrogensulfite_reductase.html

  1. \rightleftharpoons

Hydroxyacid-oxoacid_transhydrogenase.html

  1. \rightleftharpoons

Hydroxycyclohexanecarboxylate_dehydrogenase.html

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Hydroxyglutamate_decarboxylase.html

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Hydroxylamine_oxidase.html

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Hydroxylamine_reductase.html

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Hydroxylamine_reductase_(NADH).html

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Hydroxymalonate_dehydrogenase.html

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Hydroxymandelonitrile_lyase.html

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Hydroxymethylglutaryl-CoA_reductase_(NADPH).html

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Hydroxynitrilase.html

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Hydroxyphenylacetonitrile_2-monooxygenase.html

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Hydroxyphenylpyruvate_reductase.html

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Hydroxyphytanate_oxidase.html

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Hydroxypyruvate_decarboxylase.html

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Hydroxypyruvate_isomerase.html

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Hydroxypyruvate_reductase.html

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Hydroxyquinol_1,2-dioxygenase.html

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Hyoscyamine_(6S)-dioxygenase.html

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Hyponitrite_reductase.html

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Hypotaurine_dehydrogenase.html

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Hypsicles.html

  1. 10 3 ( 5 - 5 ) \sqrt{\tfrac{10}{3(5-\sqrt{5})}}

Iitaka_dimension.html

  1. R ( X , L ) = d = 0 H 0 ( X , L d ) . R(X,L)=\bigoplus_{d=0}^{\infty}H^{0}(X,L^{\otimes d}).
  2. - -\infty
  3. N ( M ) = { m 1 | P m ( M ) 1 } , N(M)=\{m\geq 1|P_{m}(M)\geq 1\},
  4. m N ( M ) m\in N(M)
  5. Φ m K \Phi_{mK}
  6. Φ m K : M N z ( φ 0 ( z ) : φ 1 ( z ) : : φ N ( z ) ) \begin{aligned}\displaystyle\Phi_{mK}:&\displaystyle M\longrightarrow\ \ \ \ % \ \ \mathbb{P}^{N}\\ &\displaystyle z\ \ \ \mapsto\ \ (\varphi_{0}(z):\varphi_{1}(z):\cdots:\varphi% _{N}(z))\end{aligned}
  7. φ i \varphi_{i}
  8. Φ m K \Phi_{mK}
  9. Φ m K ( M ) \Phi_{mK}(M)
  10. N \mathbb{P}^{N}
  11. m m
  12. Φ m k : M W = Φ m K ( M ) N \Phi_{mk}:M\rightarrow W=\Phi_{mK}(M)\subset\mathbb{P}^{N}
  13. φ : M W \varphi:M\longrightarrow W
  14. Φ m K ( M ) = Φ m K ( W ) \Phi_{mK}(M)=\Phi_{mK}(W)
  15. Φ m 1 K : M W m 1 ( M ) \Phi_{m_{1}K}:M\longrightarrow W_{m_{1}}(M)
  16. Φ m 2 K : M W m 2 ( M ) \Phi_{m_{2}K}:M\longrightarrow W_{m_{2}}(M)
  17. φ : W m 1 W m 2 ( M ) \varphi:W_{m_{1}}\longrightarrow W_{m_{2}}(M)
  18. M * M^{*}
  19. M M
  20. W * W^{*}
  21. W m 1 W_{m_{1}}
  22. W m 1 W_{m_{1}}
  23. Φ : M * W * \Phi:M^{*}\longrightarrow W^{*}
  24. Φ \Phi
  25. Φ \Phi
  26. M w * Φ - 1 ( w ) , w W * M^{*}_{w}\Phi^{-1}(w),\ \ w\in W^{*}

Imidazoleacetate_4-monooxygenase.html

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Impulse_excitation_technique.html

  1. E = 0.9465 ( m f f 2 b ) ( L 3 t 3 ) T E=0.9465\left(\frac{mf^{2}_{f}}{b}\right)\left(\frac{L^{3}}{t^{3}}\right)T
  2. T = 1 + 6.858 ( t L ) 2 T=1+6.858\left(\frac{t}{L}\right)^{2}
  3. G = 4 L m f t 2 b t ( B 1 + A ) G=\frac{4Lmf_{t}^{2}}{bt}\left(\frac{B}{1+A}\right)
  4. B = ( b / t + t / b 4 ( t / b ) - 2.52 ( t / b ) 2 + 0.21 ( t / b ) 6 ) B=\left(\frac{b/t+t/b}{4\left(t/b\right)-2.52\left(t/b\right)^{2}+0.21\left(t/% b\right)^{6}}\right)
  5. A = ( 0.5062 - 0.8776 ( b / t ) + 0.3504 ( b / t ) 2 - 0.0078 ( b / t ) 3 12.03 ( b / t ) + 9.892 ( b / t ) 2 ) A=\left(\frac{0.5062-0.8776\left(b/t\right)+0.3504\left(b/t\right)^{2}-0.0078% \left(b/t\right)^{3}}{12.03\left(b/t\right)+9.892\left(b/t\right)^{2}}\right)
  6. E = 1.6067 ( L 3 d 4 ) m f f 2 T E=1.6067\left(\frac{L^{3}}{d^{4}}\right)mf_{f}^{2}T^{\prime}
  7. T = 1 + 4.939 ( d L ) 2 T^{\prime}=1+4.939\left(\frac{d}{L}\right)^{2}
  8. G = 16 ( L π d 2 ) m f t 2 G=16\left(\frac{L}{\pi d^{2}}\right)mf_{t}^{2}
  9. ν = ( E 2 G ) - 1 \nu=\left(\frac{E}{2G}\right)-1
  10. x ( t ) = A e - δ t sin ( ω t + ϕ ) x\left(t\right)=Ae^{-\delta t}\sin\left(\omega t+\phi\right)

Indanol_dehydrogenase.html

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Index_of_industrial_production.html

  1. I = ( W i R i ) W i I=\frac{\sum(W_{i}R_{i})}{\sum W_{i}}

Indole-3-acetaldehyde_oxidase.html

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Indole-3-acetaldehyde_reductase_(NADH).html

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Indole-3-acetaldehyde_reductase_(NADPH).html

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Indole-3-glycerol-phosphate_lyase.html

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Indole-3-glycerol-phosphate_synthase.html

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Indole_2,3-dioxygenase.html

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Indoleacetaldoxime_dehydratase.html

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Indolelactate_dehydrogenase.html

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Indolepyruvate_C-methyltransferase.html

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Indolepyruvate_decarboxylase.html

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Indolepyruvate_ferredoxin_oxidoreductase.html

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Infinite_divisibility_(probability).html

  1. X 11 X 21 X 22 X 31 X 32 X 33 \begin{array}[]{cccc}X_{11}\\ X_{21}&X_{22}\\ X_{31}&X_{32}&X_{33}\\ \vdots&\vdots&\vdots&\ddots\end{array}
  2. lim n max 1 k n P ( | X n k | > ε ) = 0 for every ε > 0. \lim_{n\to\infty}\,\max_{1\leq k\leq n}\;P(\left|X_{nk}\right|>\varepsilon)=0% \,\text{ for every }\varepsilon>0.
  3. lim n n p n = λ , \lim_{n\rightarrow\infty}np_{n}=\lambda,

Initial_value_formulation_(general_relativity).html

  1. Σ \Sigma
  2. Σ \Sigma
  3. Σ \Sigma

Inositol-3-phosphate_synthase.html

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Inositol_1-methyltransferase.html

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Inositol_2-dehydrogenase.html

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Inositol_3-methyltransferase.html

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Inositol_4-methyltransferase.html

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Integer_relation_algorithm.html

  1. a 1 x 1 + a 2 x 2 + + a n x n = 0. a_{1}x_{1}+a_{2}x_{2}+\cdots+a_{n}x_{n}=0.\,

Integraph.html

  1. Y = F ( x ) = f ( x ) d x , Y=F(x)=\int f(x)dx,
  2. y = f ( x ) . y=f(x).
  3. ( x , y ) (x,y)
  4. ( x , y ) , ( x , 0 ) (x,y),(x,0)
  5. ( x 1 , 0 ) (x−1,0)
  6. X X
  7. y y
  8. ( X , Y ) (X,Y)
  9. ( x , y ) (x,y)
  10. X = ± 1 X=±1
  11. π \pi

Integration_by_parts_operator.html

  1. E D φ ( x ) h ( x ) d μ ( x ) = E φ ( x ) ( A h ) ( x ) d μ ( x ) \int_{E}\mathrm{D}\varphi(x)h(x)\,\mathrm{d}\mu(x)=\int_{E}\varphi(x)(Ah)(x)\,% \mathrm{d}\mu(x)
  2. E * i * H * H 𝑖 E . E^{*}\xrightarrow{i^{*}}H^{*}\cong H\xrightarrow{i}E.
  3. ( A h ) ( x ) = h ( x ) x - trace H D h ( x ) . (Ah)(x)=h(x)x-\mathrm{trace}_{H}\mathrm{D}h(x).
  4. S = { h : C 0 L 0 2 , 1 | h is bounded and non-anticipating } , S=\left\{\left.h\colon C_{0}\to L_{0}^{2,1}\right|h\mbox{ is bounded and non-% anticipating}~{}\right\},
  5. C 0 φ ( x + λ h ( x ) ) d γ ( x ) = C 0 φ ( x ) exp ( λ 0 1 h ˙ s d x s - λ 2 2 0 1 | h ˙ s | 2 d s ) d γ ( x ) . \int_{C_{0}}\varphi(x+\lambda h(x))\,\mathrm{d}\gamma(x)=\int_{C_{0}}\varphi(x% )\exp\left(\lambda\int_{0}^{1}\dot{h}_{s}\cdot\mathrm{d}x_{s}-\frac{\lambda^{2% }}{2}\int_{0}^{1}|\dot{h}_{s}|^{2}\,\mathrm{d}s\right)\,\mathrm{d}\gamma(x).
  6. C 0 D φ ( x ) h ( x ) d γ ( x ) = C 0 φ ( x ) ( A h ) ( x ) d γ ( x ) , \int_{C_{0}}\mathrm{D}\varphi(x)h(x)\,\mathrm{d}\gamma(x)=\int_{C_{0}}\varphi(% x)(Ah)(x)\,\mathrm{d}\gamma(x),
  7. 0 1 h ˙ s d x s . \int_{0}^{1}\dot{h}_{s}\cdot\mathrm{d}x_{s}.

Inversion_(music).html

  1. 4 6 {}^{6}_{4}
  2. T n p I ( x ) = - x + n T^{p}_{n}I(x)=-x+n
  3. T n p I ( x ) = n - x T^{p}_{n}I(x)=n-x
  4. T n I ( x ) = - x + n ( mod 12 ) T_{n}I(x)=-x+n\;\;(\mathop{{\rm mod}}12)\,
  5. 4 6 {}^{6}_{4}
  6. 3 5 {}^{5}_{3}

Inversion_temperature.html

  1. H = 5 2 N k B T + N 2 V ( b k B T - 2 a ) H=\frac{5}{2}Nk_{B}T+\frac{N^{2}}{V}(bk_{B}T-2a)
  2. N N
  3. V V
  4. T T
  5. k B k_{B}
  6. a a
  7. b b
  8. b k B T - 2 a bk_{B}T-2a
  9. T inv = 2 a b k B = 27 4 T c T\text{inv}=\frac{2a}{bk_{B}}=\frac{27}{4}T_{c}
  10. T c T_{c}
  11. T > T inv T>T\text{inv}
  12. T < T inv T<T\text{inv}

Iodophenol_O-methyltransferase.html

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Ion-attachment_mass_spectrometry.html

  1. M + X + + A M X + + A M+X^{+}+A\to MX^{+}+A

Ion_cyclotron_resonance.html

  1. ω = z e B m , \omega=\frac{zeB}{m},
  2. m z = e B 2 π f . \frac{m}{z}=\frac{eB}{2\pi f}.

Iron—cytochrome-c_reductase.html

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Isobutyraldoxime_O-methyltransferase.html

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Isobutyryl-CoA_mutase.html

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Isochorismatase.html

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Isocitrate_epimerase.html

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Isoflavone_2'-hydroxylase.html

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Isoflavone_3'-hydroxylase.html

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Isoflavone_4'-O-methyltransferase.html

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Isoflavone_7-O-methyltransferase.html

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Isoliquiritigenin_2'-O-methyltransferase.html

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Isomaltulose_synthase.html

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Isoorientin_3'-O-methyltransferase.html

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Isopenicillin_N_epimerase.html

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Isopenicillin_N_synthase.html

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Isopentenyl-diphosphate_Delta-isomerase.html

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Isopiperitenol_dehydrogenase.html

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Isopiperitenone_Delta-isomerase.html

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Isopropanol_dehydrogenase_(NADP+).html

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Isoquinoline_1-oxidoreductase.html

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Isovaleryl-CoA_dehydrogenase.html

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IκB_kinase.html

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Jacobi–Anger_expansion.html

  1. e i z cos θ = n = - i n J n ( z ) e i n θ , e^{iz\cos\theta}=\sum_{n=-\infty}^{\infty}i^{n}\,J_{n}(z)\,e^{in\theta},
  2. J n ( z ) J_{n}(z)
  3. n n
  4. i i
  5. i 2 = - 1. i^{2}=-1.
  6. e i z sin θ = n = - J n ( z ) e i n θ . e^{iz\sin\theta}=\sum_{n=-\infty}^{\infty}J_{n}(z)\,e^{in\theta}.
  7. J - n ( z ) = ( - 1 ) n J n ( z ) , J_{-n}(z)=(-1)^{n}\,J_{n}(z),
  8. n n
  9. e i z cos θ = J 0 ( z ) + 2 n = 1 i n J n ( z ) cos ( n θ ) . e^{iz\cos\theta}=J_{0}(z)\,+\,2\,\sum_{n=1}^{\infty}\,i^{n}\,J_{n}(z)\,\cos\,(% n\theta).
  10. cos ( z cos θ ) = J 0 ( z ) + 2 n = 1 ( - 1 ) n J 2 n ( z ) cos ( 2 n θ ) , sin ( z cos θ ) = - 2 n = 1 ( - 1 ) n J 2 n - 1 ( z ) cos [ ( 2 n - 1 ) θ ] , cos ( z sin θ ) = J 0 ( z ) + 2 n = 1 J 2 n ( z ) cos ( 2 n θ ) , sin ( z sin θ ) = 2 n = 1 J 2 n - 1 ( z ) sin [ ( 2 n - 1 ) θ ] . \begin{aligned}\displaystyle\cos(z\cos\theta)&\displaystyle=J_{0}(z)+2\sum_{n=% 1}^{\infty}(-1)^{n}J_{2n}(z)\cos(2n\theta),\\ \displaystyle\sin(z\cos\theta)&\displaystyle=-2\sum_{n=1}^{\infty}(-1)^{n}J_{2% n-1}(z)\cos\left[\left(2n-1\right)\theta\right],\\ \displaystyle\cos(z\sin\theta)&\displaystyle=J_{0}(z)+2\sum_{n=1}^{\infty}J_{2% n}(z)\cos(2n\theta),\\ \displaystyle\sin(z\sin\theta)&\displaystyle=2\sum_{n=1}^{\infty}J_{2n-1}(z)% \sin\left[\left(2n-1\right)\theta\right].\end{aligned}

Jacobsthal_number.html

  1. U n ( P , Q ) U_{n}(P,Q)
  2. J n = { 0 if n = 0 ; 1 if n = 1 ; J n - 1 + 2 J n - 2 if n > 1. J_{n}=\begin{cases}0&\mbox{if }~{}n=0;\\ 1&\mbox{if }~{}n=1;\\ J_{n-1}+2J_{n-2}&\mbox{if }~{}n>1.\\ \end{cases}
  3. J n + 1 = 2 J n + ( - 1 ) n , J_{n+1}=2J_{n}+(-1)^{n}\,,
  4. J n + 1 = 2 n - J n . J_{n+1}=2^{n}-J_{n}.\,
  5. J n = 2 n - ( - 1 ) n 3 . J_{n}=\frac{2^{n}-(-1)^{n}}{3}.
  6. x ( 1 + x ) ( 1 - 2 x ) . \frac{x}{(1+x)(1-2x)}.
  7. V n ( 1 , - 2 ) V_{n}(1,-2)
  8. L n = { 2 if n = 0 ; 1 if n = 1 ; L n - 1 + 2 L n - 2 if n > 1. L_{n}=\begin{cases}2&\mbox{if }~{}n=0;\\ 1&\mbox{if }~{}n=1;\\ L_{n-1}+2L_{n-2}&\mbox{if }~{}n>1.\\ \end{cases}
  9. L n + 1 = 2 L n - 3 ( - 1 ) n . L_{n+1}=2L_{n}-3(-1)^{n}.\,
  10. L n = 2 n + ( - 1 ) n . L_{n}=2^{n}+(-1)^{n}.\,

Jaffard_ring.html

  1. dim R [ T 1 , , T n ] = n + dim R , \dim R[T_{1},\ldots,T_{n}]=n+\dim R,\,
  2. 𝐐 ¯ [ [ T ] ] \overline{\mathbf{Q}}[[T]]

Japanese_theorem_for_cyclic_quadrilaterals.html

  1. A B C D \square ABCD
  2. M 1 , M 2 , M 3 , M 4 M_{1},M_{2},M_{3},M_{4}
  3. A B D , A B C , B C D , A C D \triangle ABD,\triangle ABC,\triangle BCD,\triangle ACD
  4. M 1 , M 2 , M 3 , M 4 M_{1},M_{2},M_{3},M_{4}

Jasmonate_O-methyltransferase.html

  1. \rightleftharpoons

Java_hashCode().html

  1. h ( s ) h(s)
  2. h ( s ) = i = 0 n - 1 s [ i ] 31 n - 1 - i h(s)=\sum_{i=0}^{n-1}s\left[\,i\,\right]\cdot 31^{n-1-i}
  3. s [ i ] s\left[\,i\,\right]
  4. i i
  5. n n

Jensen_hierarchy.html

  1. J α + 1 Pow ( J α ) = Def ( J α ) J_{\alpha+1}\cap\textrm{Pow}(J_{\alpha})=\textrm{Def}(J_{\alpha})
  2. W n α W^{\alpha}_{n}
  3. X α ( n + 1 , e ) = { X ( n , f ) W n + 1 α ( e , f ) } X_{\alpha}(n+1,e)=\{X(n,f)\mid W^{\alpha}_{n+1}(e,f)\}
  4. X α ( 0 , e ) = e X_{\alpha}(0,e)=e
  5. n ω J α , n \bigcup_{n\in\omega}J_{\alpha,n}
  6. J α + 1 Pow ( J α ) = Def ( J α ) , J_{\alpha+1}\cap\,\text{Pow}(J_{\alpha})=\,\text{Def}(J_{\alpha}),

Jet_damping.html

  1. C m q Cm_{q}
  2. C n r Cn_{r}
  3. C m q Cm_{q}
  4. C n r Cn_{r}

JLO_cocycle.html

  1. 𝒜 \mathcal{A}
  2. 𝒜 \mathcal{A}
  3. θ \theta
  4. θ \theta
  5. θ \theta
  6. θ \theta
  7. \mathcal{H}
  8. 𝒜 \mathcal{A}
  9. 2 \mathbb{Z}_{2}
  10. γ \gamma
  11. \mathcal{H}
  12. = 0 1 \mathcal{H}=\mathcal{H}_{0}\oplus\mathcal{H}_{1}
  13. 𝒜 \mathcal{A}
  14. 2 \mathbb{Z}_{2}
  15. a γ = γ a a\gamma=\gamma a
  16. a 𝒜 a\in\mathcal{A}
  17. D D
  18. D D
  19. γ \gamma
  20. D γ = - γ D D\gamma=-\gamma D
  21. a 𝒜 a\in\mathcal{A}
  22. D D
  23. Dom ( D ) \mathrm{Dom}\left(D\right)
  24. [ D , a ] : Dom ( D ) \left[D,a\right]:\mathrm{Dom}\left(D\right)\to\mathcal{H}
  25. tr ( e - t D 2 ) < \mathrm{tr}\left(e^{-tD^{2}}\right)<\infty
  26. t > 0 t>0
  27. θ \theta
  28. M M
  29. 𝒜 = C ( M ) \mathcal{A}=C^{\infty}\left(M\right)
  30. M M
  31. \mathcal{H}
  32. M M
  33. D D
  34. Φ t ( D ) \Phi_{t}\left(D\right)
  35. Φ t ( D ) = ( Φ t 0 ( D ) , Φ t 2 ( D ) , Φ t 4 ( D ) , ) \Phi_{t}\left(D\right)=\left(\Phi_{t}^{0}\left(D\right),\Phi_{t}^{2}\left(D% \right),\Phi_{t}^{4}\left(D\right),\ldots\right)
  36. 𝒜 \mathcal{A}
  37. Φ t 0 ( D ) ( a 0 ) = tr ( γ a 0 e - t D 2 ) , \Phi_{t}^{0}\left(D\right)\left(a_{0}\right)=\mathrm{tr}\left(\gamma a_{0}e^{-% tD^{2}}\right),
  38. Φ t n ( D ) ( a 0 , a 1 , , a n ) = 0 s 1 s n t tr ( γ a 0 e - s 1 D 2 [ D , a 1 ] e - ( s 2 - s 1 ) D 2 [ D , a n ] e - ( t - s n ) D 2 ) d s 1 d s n , \Phi_{t}^{n}\left(D\right)\left(a_{0},a_{1},\ldots,a_{n}\right)=\int_{0\leq s_% {1}\leq\ldots s_{n}\leq t}\mathrm{tr}\left(\gamma a_{0}e^{-s_{1}D^{2}}\left[D,% a_{1}\right]e^{-\left(s_{2}-s_{1}\right)D^{2}}\ldots\left[D,a_{n}\right]e^{-% \left(t-s_{n}\right)D^{2}}\right)ds_{1}\ldots ds_{n},
  39. n = 2 , 4 , n=2,4,\dots
  40. Φ t ( D ) \Phi_{t}\left(D\right)
  41. t t

John_Hilton_Grace.html

  1. a ( z ) = a 0 + ( n 1 ) a 1 z + ( n 2 ) a 2 z 2 + + a n z n a(z)=a_{0}+{\textstyle\left({{n}\atop{1}}\right)}a_{1}z+{\textstyle\left({{n}% \atop{2}}\right)}a_{2}z^{2}+\dots+a_{n}z^{n}
  2. b ( z ) = b 0 + ( n 1 ) b 1 z + ( n 2 ) b 2 z 2 + + b n z n b(z)=b_{0}+{\textstyle\left({{n}\atop{1}}\right)}b_{1}z+{\textstyle\left({{n}% \atop{2}}\right)}b_{2}z^{2}+\dots+b_{n}z^{n}
  3. a 0 b n - ( n 1 ) a 1 b n - 1 + ( n 2 ) a 2 b n - 2 - + ( - 1 ) n a n b 0 = 0 a_{0}b_{n}-{\textstyle\left({{n}\atop{1}}\right)}a_{1}b_{n-1}+{\textstyle\left% ({{n}\atop{2}}\right)}a_{2}b_{n-2}-\cdots+(-1)^{n}a_{n}b_{0}=0
  4. a ( z ) a(z)
  5. b ( z ) b(z)
  6. c ( z ) = a 0 b 0 + ( n 1 ) a 1 b 1 z + ( n 2 ) a 2 b 2 z 2 + + a n b n z n c(z)=a_{0}b_{0}+{\textstyle\left({{n}\atop{1}}\right)}a_{1}b_{1}z+{\textstyle% \left({{n}\atop{2}}\right)}a_{2}b_{2}z^{2}+\cdots+a_{n}b_{n}z^{n}

Juglone_3-monooxygenase.html

  1. \rightleftharpoons

Jury_stability_criterion.html

  1. f ( z ) = a 0 z n + a 1 z n - 1 + a 2 z n - 2 + + a n - 1 z + a n f(z)=a_{0}z^{n}+a_{1}z^{n-1}+a_{2}z^{n-2}+\cdots+a_{n-1}z+a_{n}
  2. a n a 0 \frac{a_{n}}{a_{0}}
  3. a 0 \displaystyle a_{0}
  4. a n a 0 \frac{a_{n}}{a_{0}}
  5. b n - 1 b 0 \frac{b_{n-1}}{b_{0}}
  6. a 0 > 0 {a_{0}}>0
  7. a 0 {a_{0}}
  8. b 0 {b_{0}}
  9. c 0 {c_{0}}

Kachurovskii's_theorem.html

  1. d f ( x ) ( y - x ) f ( y ) - f ( x ) ; \mathrm{d}f(x)(y-x)\leq f(y)-f(x);
  2. ( d f ( x ) - d f ( y ) ) ( x - y ) 0. \big(\mathrm{d}f(x)-\mathrm{d}f(y)\big)(x-y)\geq 0.

Kaempferol_4'-O-methyltransferase.html

  1. \rightleftharpoons

Katz's_back-off_model.html

  1. P b o ( w i w i - n + 1 w i - 1 ) = { d w i - n + 1 w i C ( w i - n + 1 w i - 1 w i ) C ( w i - n + 1 w i - 1 ) if C ( w i - n + 1 w i ) > k α w i - n + 1 w i - 1 P b o ( w i w i - n + 2 w i - 1 ) otherwise \begin{aligned}&\displaystyle P_{bo}(w_{i}\mid w_{i-n+1}\cdots w_{i-1})\\ \displaystyle=&\displaystyle\begin{cases}d_{w_{i-n+1}\cdots w_{i}}\dfrac{C(w_{% i-n+1}\cdots w_{i-1}w_{i})}{C(w_{i-n+1}\cdots w_{i-1})}&\,\text{if }C(w_{i-n+1% }\cdots w_{i})>k\\ \alpha_{w_{i-n+1}\cdots w_{i-1}}P_{bo}(w_{i}\mid w_{i-n+2}\cdots w_{i-1})&\,% \text{otherwise}\end{cases}\end{aligned}
  2. k k
  3. d d
  4. C C
  5. C * C^{*}
  6. d = C * C d=\frac{C^{*}}{C}
  7. α \alpha
  8. β w i - n + 1 w i - 1 = 1 - { w i : C ( w i - n + 1 w i ) > k } d w i - n + 1 w i C ( w i - n + 1 w i - 1 w i ) C ( w i - n + 1 w i - 1 ) \beta_{w_{i-n+1}\cdots w_{i-1}}=1-\sum_{\{w_{i}:C(w_{i-n+1}\cdots w_{i})>k\}}d% _{w_{i-n+1}\cdots w_{i}}\frac{C(w_{i-n+1}\cdots w_{i-1}w_{i})}{C(w_{i-n+1}% \cdots w_{i-1})}
  9. α w i - n + 1 w i - 1 = β w i - n + 1 w i - 1 { w i : C ( w i - n + 1 w i ) k } P b o ( w i w i - n + 2 w i - 1 ) \alpha_{w_{i-n+1}\cdots w_{i-1}}=\frac{\beta_{w_{i-n+1}\cdots w_{i-1}}}{\sum_{% \{w_{i}:C(w_{i-n+1}\cdots w_{i})\leq k\}}P_{bo}(w_{i}\mid w_{i-n+2}\cdots w_{i% -1})}

KdV_hierarchy.html

  1. T T
  2. T ( g ) ( x ) = g ( x + 1 ) T(g)(x)=g(x+1)
  3. 𝒞 \mathcal{C}
  4. T ( g ) ( x ) = g ( x ) T(g)(x)=g(x)
  5. g 𝒞 g\in\mathcal{C}
  6. L g ( ψ ) ( x ) = ψ ′′ ( x ) + g ( x ) ψ ( x ) L_{g}(\psi)(x)=\psi^{\prime\prime}(x)+g(x)\psi(x)
  7. \mathbb{R}
  8. g \mathcal{B}_{g}
  9. ( λ , α ) × * (\lambda,\alpha)\in\mathbb{C}\times\mathbb{C}^{*}
  10. ψ \psi
  11. L g ( ψ ) = λ ψ L_{g}(\psi)=\lambda\psi
  12. T ( ψ ) = α ψ T(\psi)=\alpha\psi
  13. D i : 𝒞 𝒞 D_{i}:\mathcal{C}\to\mathcal{C}
  14. i i
  15. g ( x , t ) g(x,t)
  16. g t ( x ) g_{t}(x)
  17. g ( x , t ) g(x,t)
  18. D i ( g t ) = d d t g t D_{i}(g_{t})=\frac{d}{dt}g_{t}
  19. g \mathcal{B}_{g}
  20. t t

Kepler_triangle.html

  1. φ = 1 + 5 2 \varphi={1+\sqrt{5}\over 2}
  2. 1 : φ : φ 1:\sqrt{\varphi}:\varphi
  3. 1 1
  4. φ \sqrt{\varphi}
  5. φ \varphi
  6. φ \varphi
  7. φ 2 = φ + 1 \varphi^{2}=\varphi+1
  8. ( φ ) 2 = ( φ ) 2 + ( 1 ) 2 . (\varphi)^{2}=(\sqrt{\varphi})^{2}+(1)^{2}.
  9. a , a φ , a φ , a,a\sqrt{\varphi},a\varphi,
  10. 4 a φ 4a\sqrt{\varphi}
  11. a π φ a\pi\varphi
  12. π 4 / φ \pi\approx 4/\sqrt{\varphi}
  13. π 4 / φ \pi\neq 4/\sqrt{\varphi}
  14. π \pi
  15. π \pi
  16. φ \varphi

Keratan_sulfotransferase.html

  1. \rightleftharpoons

Kervaire_invariant.html

  1. L 4 k + 2 , L_{4k+2},
  2. L 4 k L 4 k L^{4k}\cong L_{4k}
  3. L 4 k + 1 . L^{4k+1}.
  4. \to
  5. S 2 m + 1 S^{2m+1}
  6. \to
  7. M 4 m + 2 M^{4m+2}
  8. S 2 m + 1 S^{2m+1}
  9. \to
  10. M 4 m + 2 M^{4m+2}
  11. m 0 , 1 , 3 m\neq 0,1,3
  12. S 4 m + 2 + k S 2 m + 1 + k S^{4m+2+k}\to S^{2m+1+k}
  13. m = 0 , 1 , 3 m=0,1,3
  14. π n + 2 ( S n ) = Z / 2 Z \pi_{n+2}(S^{n})=Z/2Z
  15. S n + 2 S^{n+2}
  16. \to
  17. S n S^{n}
  18. n 2 n\geq 2
  19. S n + 2 S^{n+2}
  20. Θ n / b P n + 1 π n S / J , \Theta_{n}/bP_{n+1}\to\pi_{n}^{S}/J,\,
  21. b P n + 1 bP_{n+1}
  22. π n S \pi_{n}^{S}
  23. b P n + 1 bP_{n+1}
  24. J J
  25. n = 4 k + 3 , n=4k+3,
  26. ( 0 1 - 1 0 ) \begin{pmatrix}0&1\\ -1&0\end{pmatrix}
  27. x y xy
  28. Q ( 1 , 0 ) = Q ( 0 , 1 ) = 0 Q(1,0)=Q(0,1)=0
  29. Q ( 1 , 1 ) = 1 Q(1,1)=1
  30. k 8 k\geq 8
  31. n 254 n\geq 254
  32. 𝐎 P 2 \mathbf{O}P^{2}

Ketol-acid_reductoisomerase.html

  1. \rightleftharpoons

Ketotetrose-phosphate_aldolase.html

  1. \rightleftharpoons

Kinetic_proofreading.html

  1. e - 10 e^{-10}
  2. p = e - Δ F / k T p=e^{-\Delta F/kT}
  3. p 2 p^{2}
  4. p N p^{N}
  5. Δ F \Delta F
  6. N Δ F N\Delta F
  7. e - 10 e^{-10}

KK-theory.html

  1. [ F , ρ ( a ) ] , ( F 2 - 1 ) ρ ( a ) , ( F - F * ) ρ ( a ) [F,\rho(a)],(F^{2}-1)\rho(a),(F-F^{*})\rho(a)
  2. K K ( A , B ) = [ q A , K ( H ) B ] KK(A,B)=[qA,K(H)\otimes B]
  3. K K ( A , B ) × K K ( B , C ) K K ( A , C ) KK(A,B)\times KK(B,C)\to KK(A,C)
  4. 𝖪𝖪 \mathsf{KK}
  5. 𝖪𝖪 \mathsf{KK}
  6. 𝖪𝖪 \mathsf{KK}
  7. 𝖢 * - 𝖺𝗅𝗀 𝖪𝖪 \mathsf{C^{*}\!-\!alg}\to\mathsf{KK}
  8. 𝖪𝖪 \mathsf{KK}
  9. K K ( A , B E ) × K K ( B D , C ) K K ( A D , C E ) . KK(A,B\otimes E)\times KK(B\otimes D,C)\to KK(A\otimes D,C\otimes E).

Kleene's_T_predicate.html

  1. T k ( e , i 1 , , i k , x ) T_{k}(e,i_{1},\ldots,i_{k},x)
  2. f ( n ) U ( μ x T ( e , n , x ) ) f(n)\simeq U(\mu x\,T(e,n,x))
  3. μ x ϕ ( x ) \mu x\,\phi(x)
  4. ϕ ( x ) \phi(x)
  5. \simeq
  6. K = { e : x T ( e , 0 , x ) } K=\{e\mbox{ }~{}:\mbox{ }~{}\exists xT(e,0,x)\}
  7. Σ 1 0 \Sigma^{0}_{1}
  8. K n + 1 = { e , a 1 , , a n : x T ( e , a 1 , , a n , x ) } K_{n+1}=\{\langle e,a_{1},\ldots,a_{n}\rangle:\exists xT(e,a_{1},\ldots,a_{n},% x)\}
  9. Σ 1 0 \Sigma^{0}_{1}
  10. Σ 1 0 \Sigma^{0}_{1}
  11. A k + 1 A\subseteq\mathbb{N}^{k+1}
  12. Σ n 0 \Sigma^{0}_{n}
  13. { a 1 , , a k : x ( a 1 , , a k , x ) A ) } \{\langle a_{1},\ldots,a_{k}\rangle:\forall x(\langle a_{1},\ldots,a_{k},x)\in A)\}
  14. Π n + 1 0 \Pi^{0}_{n+1}