wpmath0000015_8

Stereotype_space.html

  1. X X
  2. \mathbb{C}
  3. i : X X , i ( x ) ( f ) = f ( x ) , x X , f X i:X\to X^{\star\star},\quad i(x)(f)=f(x),\quad x\in X,\quad f\in X^{\star}
  4. X X^{\star}
  5. f : X f:X\to\mathbb{C}
  6. X X^{\star\star}
  7. X X^{\star}
  8. X X
  9. X X
  10. D D
  11. X X
  12. X X
  13. X X
  14. X X
  15. X X
  16. X X
  17. X X^{\star}
  18. X X
  19. X X^{\star}
  20. X X
  21. X X
  22. \Longleftrightarrow
  23. X X
  24. \Longleftrightarrow
  25. X X^{\star}
  26. X X
  27. \Longleftrightarrow
  28. X X
  29. \Longleftrightarrow
  30. X X^{\star}
  31. X X
  32. \Longleftrightarrow
  33. X X^{\star}
  34. X X
  35. \Longleftrightarrow
  36. X X^{\star}
  37. T T
  38. T T
  39. X X
  40. \Longleftrightarrow
  41. X X^{\star}
  42. X X
  43. X X
  44. \Longleftrightarrow
  45. X X
  46. \Longleftrightarrow
  47. X X^{\star}
  48. X X
  49. \Longleftrightarrow
  50. X X^{\star}
  51. T T
  52. X X
  53. \Longleftrightarrow
  54. X X^{\star}
  55. L n L_{n}
  56. n = 1 L n = { 0 } \bigcap_{n=1}^{\infty}L_{n}=\{0\}
  57. X X
  58. \Longleftrightarrow
  59. X X^{\star}
  60. X X
  61. \Longleftrightarrow
  62. X X^{\star}
  63. \Longleftrightarrow
  64. X X^{\star}
  65. X X
  66. \Longleftrightarrow
  67. X X^{\star}
  68. L L
  69. L K L\cap K
  70. K X K\subseteq X^{\star}
  71. X X
  72. \Longleftrightarrow
  73. X X^{\star}
  74. B B
  75. B K B\cap K
  76. K X K\subseteq X^{\star}
  77. X X
  78. X X
  79. X : X X \triangledown_{X}:X\to X^{\triangledown}
  80. X X^{\triangledown}
  81. X X
  82. X X
  83. X : X X \triangledown_{X}:X\to X^{\triangledown}
  84. φ : X Y \varphi:X\to Y
  85. φ : X Y \varphi^{\triangledown}:X^{\triangledown}\to Y^{\triangledown}
  86. Y φ = φ X \triangledown_{Y}\circ\varphi=\varphi^{\triangledown}\circ\triangledown_{X}
  87. X X
  88. X X
  89. X X X\mapsto X^{\triangledown}
  90. X X
  91. X X
  92. X X
  93. X : X X \vartriangle_{X}:X^{\vartriangle}\to X
  94. X X^{\vartriangle}
  95. X X
  96. X X
  97. X : X X \vartriangle_{X}:X^{\vartriangle}\to X
  98. φ : X Y \varphi:X\to Y
  99. φ : X Y \varphi^{\vartriangle}:X^{\vartriangle}\to Y^{\vartriangle}
  100. φ X = Y φ \varphi\circ\vartriangle_{X}=\vartriangle_{Y}\circ\varphi^{\vartriangle}
  101. X X
  102. X X
  103. X X X\mapsto X^{\vartriangle}
  104. X X
  105. X X
  106. X X
  107. X X^{\vartriangle}
  108. X X
  109. X X^{\triangledown}
  110. X X
  111. X X^{\vartriangle\triangledown}
  112. X X^{\triangledown\vartriangle}
  113. X X X\to X^{\star}
  114. φ : X Y \varphi:X\to Y
  115. φ = σ β π \varphi=\sigma\circ\beta\circ\pi
  116. π \pi
  117. β \beta
  118. σ \sigma
  119. X X
  120. Y Y
  121. Hom ( X , Y ) \,\text{Hom}(X,Y)
  122. X X
  123. Y Y
  124. L ( X , Y ) \,\text{L}(X,Y)
  125. φ : X Y \varphi:X\to Y
  126. Hom ( X , Y ) \,\text{Hom}(X,Y)
  127. X Y := Hom ( X , Y ) , X\circledast Y:=\,\text{Hom}(X,Y^{\star})^{\star},
  128. X Y := Hom ( X , Y ) . X\odot Y:=\,\text{Hom}(X^{\star},Y).
  129. X X X , \mathbb{C}\circledast X\cong X\cong X\circledast\mathbb{C},
  130. X X X , \mathbb{C}\odot X\cong X\cong X\odot\mathbb{C},
  131. X Y Y X , X\circledast Y\cong Y\circledast X,
  132. X Y Y X , X\odot Y\cong Y\odot X,
  133. ( X Y ) Z X ( Y Z ) , (X\circledast Y)\circledast Z\cong X\circledast(Y\circledast Z),
  134. ( X Y ) Z X ( Y Z ) , (X\odot Y)\odot Z\cong X\odot(Y\odot Z),
  135. ( X Y ) Y X , (X\circledast Y)^{\star}\cong Y^{\star}\odot X^{\star},
  136. ( X Y ) Y X , (X\odot Y)^{\star}\cong Y^{\star}\circledast X^{\star},
  137. Hom ( X Y , Z ) Hom ( X , Hom ( Y , Z ) ) , \,\text{Hom}(X\circledast Y,Z)\cong\,\text{Hom}(X,\,\text{Hom}(Y,Z)),
  138. Hom ( X , Y Z ) Hom ( X , Y ) Z \,\text{Hom}(X,Y\odot Z)\cong\,\text{Hom}(X,Y)\odot Z
  139. \odot
  140. \circledast
  141. Hom \,\text{Hom}
  142. X X
  143. φ : X X \varphi:X\to X
  144. Hom ( X , X ) \,\text{Hom}(X,X)
  145. X X
  146. Y Y
  147. Hom ( X , Y ) \,\text{Hom}(X,Y)
  148. X Y X\circledast Y
  149. X Y X\odot Y
  150. X X
  151. X X^{\star}
  152. Hom ( X , X ) \,\text{Hom}(X,X)
  153. A A
  154. A A
  155. A A
  156. A A
  157. \mathbb{R}
  158. D X D\subseteq X
  159. A X A\subseteq X
  160. F X F\subseteq X
  161. A D + F A\subseteq D+F
  162. X X
  163. f : X f:X\to\mathbb{C}
  164. S X S\subseteq X
  165. X X
  166. X X
  167. B X B\subseteq X
  168. X X
  169. S X S\subseteq X
  170. U U
  171. X X
  172. B S = U B\cap S=U
  173. X X
  174. X X^{\star}
  175. Q X Q\subseteq X^{\star}
  176. X X
  177. X X
  178. U U^{\circ}
  179. U X U\subseteq X
  180. X X
  181. X X^{\star}
  182. Q X Q\subseteq X^{\star}
  183. X X
  184. X X
  185. U U^{\circ}
  186. U X U\subseteq X
  187. X X^{\vartriangle\triangledown}
  188. X X^{\triangledown\vartriangle}

Steroid-transporting_ATPase.html

  1. \rightleftharpoons

Steroid_15beta-monooxygenase.html

  1. \rightleftharpoons

Stochastic_cellular_automaton.html

  1. E = k G S k E=\prod_{k\in G}S_{k}
  2. G G
  3. \mathbb{Z}
  4. S k S_{k}
  5. S k = { - 1 , + 1 } S_{k}=\{-1,+1\}
  6. S k = { 0 , 1 } S_{k}=\{0,1\}
  7. P ( d σ | η ) = k G p k ( d σ k | η ) P(d\sigma|\eta)=\otimes_{k\in G}p_{k}(d\sigma_{k}|\eta)
  8. η E \eta\in E
  9. p k ( d σ k | η ) p_{k}(d\sigma_{k}|\eta)
  10. S k S_{k}
  11. p k ( d σ k | η ) = p k ( d σ k | η V k ) p_{k}(d\sigma_{k}|\eta)=p_{k}(d\sigma_{k}|\eta_{V_{k}})
  12. η V k = ( η j ) j V k \eta_{V_{k}}=(\eta_{j})_{j\in V_{k}}
  13. V k {V_{k}}

Stochastic_Eulerian_Lagrangian_method.html

  1. ρ d u d t = μ Δ u - p + Λ [ Υ ( V - Γ u ) ] + λ + f t h m ( x , t ) \rho\frac{d{u}}{d{t}}=\mu\,\Delta u-\nabla p+\Lambda[\Upsilon(V-\Gamma{u})]+% \lambda+f_{thm}(x,t)
  2. m d V d t = - Υ ( V - Γ u ) - Φ [ X ] + ξ + F t h m m\frac{d{V}}{d{t}}=-\Upsilon(V-\Gamma{u})-\nabla\Phi[X]+\xi+F_{thm}
  3. d X d t = V . \frac{d{X}}{d{t}}=V.
  4. u = 0. \nabla\cdot u=0.\,
  5. Γ , Λ \Gamma,\Lambda
  6. X , V X,V
  7. Φ \Phi
  8. f t h m , F t h m f_{thm},F_{thm}
  9. λ , ξ \lambda,\xi
  10. Υ \Upsilon
  11. Γ , Λ \Gamma,\Lambda
  12. Γ = Λ T . \Gamma=\Lambda^{T}.
  13. f t h m ( s ) f t h m T ( t ) = - ( 2 k B T ) ( μ Δ - Λ Υ Γ ) δ ( t - s ) . \langle f_{thm}(s)f^{T}_{thm}(t)\rangle=-\left(2k_{B}{T}\right)\left(\mu\Delta% -\Lambda\Upsilon\Gamma\right)\delta(t-s).
  14. F t h m ( s ) F t h m T ( t ) = 2 k B T Υ δ ( t - s ) . \langle F_{thm}(s)F^{T}_{thm}(t)\rangle=2k_{B}{T}\Upsilon\delta(t-s).
  15. f t h m ( s ) F t h m T ( t ) = - 2 k B T Λ Υ δ ( t - s ) . \langle f_{thm}(s)F^{T}_{thm}(t)\rangle=-2k_{B}{T}\Lambda\Upsilon\delta(t-s).

Stochastic_forensics.html

  1. 2 8 10 12 2^{8^{10^{12}}}

Strain_rate_tensor.html

  1. v v
  2. 3 × \mathbb{R}^{3}\times\mathbb{R}
  3. v ( p , t ) v(p,t)
  4. p p
  5. t t
  6. v ( p + r , t ) v(p+r,t)
  7. p p
  8. r r
  9. v ( p + r , t ) = v ( p , t ) + ( v ) ( p , t ) ( r ) + (higher order terms) , v(p+r,t)=v(p,t)+(\nabla v)(p,t)(r)+\,\text{(higher order terms)},
  10. v \nabla v
  11. r r
  12. v \nabla v
  13. v = J = [ 1 v 1 2 v 1 3 v 1 1 v 2 2 v 2 3 v 2 1 v 3 2 v 3 3 v 3 ] . \nabla v=J=\begin{bmatrix}\displaystyle{\partial_{1}v_{1}}&\displaystyle{% \partial_{2}v_{1}}&\displaystyle{\partial_{3}v_{1}}\\ \displaystyle{\partial_{1}v_{2}}&\displaystyle{\partial_{2}v_{2}}&% \displaystyle{\partial_{3}v_{2}}\\ \displaystyle{\partial_{1}v_{3}}&\displaystyle{\partial_{2}v_{3}}&% \displaystyle{\partial_{3}v_{3}}\end{bmatrix}.
  14. v i v_{i}
  15. v v
  16. i i
  17. j f \partial_{j}f
  18. f f
  19. x j x_{j}
  20. J J
  21. p p
  22. t t
  23. p p
  24. v i ( p + r , t ) = v i ( p , t ) + j J i j ( p , t ) r j = v i ( p , t ) + j j v i ( p , t ) r j ; v_{i}(p+r,t)=v_{i}(p,t)+\sum_{j}J_{ij}(p,t)r_{j}=v_{i}(p,t)+\sum_{j}\partial_{% j}v_{i}(p,t)r_{j};
  25. v ( p + r , t ) = v ( p , t ) + J ( p , t ) r v(p+r,t)=v(p,t)+J(p,t)r
  26. v v
  27. r r
  28. J = v J=\nabla v
  29. E E
  30. R R
  31. E = 1 2 ( J + J 𝖳 ) R = 1 2 ( J - J 𝖳 ) E=\frac{1}{2}\left(J+J^{\mathsf{T}}\right)\quad\quad\quad R=\frac{1}{2}\left(J% -J^{\mathsf{T}}\right)
  32. E i j = 1 2 ( j v i + i v j ) R i j = 1 2 ( j v i - i v j ) E_{ij}=\frac{1}{2}(\partial_{j}v_{i}+\partial_{i}v_{j})\quad\quad\quad R_{ij}=% \frac{1}{2}(\partial_{j}v_{i}-\partial_{i}v_{j})
  33. v ( p + r , t ) v ( p , t ) + E ( p , t ) ( r ) + R ( p , t ) ( r ) , v(p+r,t)\approx v(p,t)+E(p,t)(r)+R(p,t)(r),
  34. v i ( p + r , t ) = v i ( p , t ) + j E i j ( p , t ) r j + j R i j ( p , t ) r j = v i ( p , t ) + 1 2 j ( j v i ( p , t ) + i v j ( p , t ) ) r j + 1 2 j ( j v i ( p , t ) - i v j ( p , t ) ) r j \begin{array}[]{lcl}v_{i}(p+r,t)&=&v_{i}(p,t)+\sum_{j}E_{ij}(p,t)r_{j}+\sum_{j% }R_{ij}(p,t)r_{j}\\ &=&v_{i}(p,t)+\frac{1}{2}\sum_{j}\left(\partial_{j}v_{i}(p,t)+\partial_{i}v_{j% }(p,t)\right)r_{j}+\frac{1}{2}\sum_{j}\left(\partial_{j}v_{i}(p,t)-\partial_{i% }v_{j}(p,t)\right)r_{j}\end{array}
  35. R R
  36. p p
  37. ω \omega
  38. ω = 1 2 × v = 1 2 [ 2 v 3 - 3 v 2 3 v 1 - 1 v 3 1 v 2 - 2 v 1 ] . \omega=\frac{1}{2}\nabla\times v=\frac{1}{2}\begin{bmatrix}\partial_{2}v_{3}-% \partial_{3}v_{2}\\ \partial_{3}v_{1}-\partial_{1}v_{3}\\ \partial_{1}v_{2}-\partial_{2}v_{1}\end{bmatrix}.
  39. × v \nabla\times v
  40. R R
  41. E E
  42. E E
  43. E ( p , t ) ( r ) = D ( p , t ) ( r ) + S ( p , t ) ( r ) . E(p,t)(r)=D(p,t)(r)+S(p,t)(r).
  44. E i j = 1 3 ( k k v k ) δ i j rate-of-expansion tensor D i j + ( 1 2 ( i v j + j v i ) - 1 3 ( k k v k ) δ i j ) rate-of-shear tensor S i j , E_{ij}=\underbrace{\frac{1}{3}(\sum_{k}\partial_{k}v_{k})\delta_{ij}}_{\,\text% {rate-of-expansion tensor}D_{ij}}+\underbrace{\left(\frac{1}{2}\left(\partial_% {i}v_{j}+\partial_{j}v_{i}\right)-\frac{1}{3}(\sum_{k}\partial_{k}v_{k})\delta% _{ij}\right)}_{\,\text{rate-of-shear tensor}S_{ij}},
  45. δ \delta
  46. δ i j \delta_{ij}
  47. i = j i=j
  48. i j i\neq j
  49. v = 1 v 1 + 2 v 2 + 3 v 3 ; \nabla\cdot v=\partial_{1}v_{1}+\partial_{2}v_{2}+\partial_{3}v_{3};
  50. v v

Strategic_Network_Formation.html

  1. N = N=
  2. n n
  3. i i
  4. j j
  5. i j ij
  6. i j ij
  7. j i ji
  8. g g
  9. g g
  10. i , j i,j
  11. i , j i,j
  12. \mathbb{N}
  13. N N
  14. G G
  15. u i u_{i}
  16. G ( N ) G(N)
  17. \rightarrow
  18. \mathbb{R}
  19. u i u_{i}
  20. g g
  21. g g
  22. i j ij
  23. g g
  24. u i u_{i}
  25. g g
  26. \geq
  27. u i u_{i}
  28. g g
  29. i j ij
  30. u j u_{j}
  31. g g
  32. \geq
  33. u j u_{j}
  34. g g
  35. i j ij
  36. i j ij
  37. g g
  38. u i u_{i}
  39. g + i j g+ij
  40. u i u_{i}
  41. g g
  42. u j u_{j}
  43. g + i j g+ij
  44. g g
  45. g g
  46. u 1 u_{1}
  47. u n u_{n}
  48. i u i ( g ) i u i ( g ) \textstyle\sum_{i}u_{i}(g)\geq\textstyle\sum_{i}u_{i}(g^{\prime})
  49. g G ( N ) g^{\prime}\in G(N)
  50. g g
  51. u 1 u_{1}
  52. u n u_{n}
  53. g G g^{\prime}\in G
  54. u i u_{i}
  55. g g^{\prime}
  56. \geq
  57. u i u_{i}
  58. g g
  59. i i
  60. i i
  61. b b
  62. n - 1 n-1
  63. \rightarrow
  64. \mathbb{R}
  65. u i ( g ) = j i : j N n - 1 ( g ) b ( l i j ( g ) ) - d i ( g ) c u_{i}(g)=\sum_{j\neq i:j\in N^{n-1}(g)}b(l_{ij}(g))-d_{i}(g)c
  66. l i j ( g ) l_{ij}(g)
  67. i i
  68. j j

Streamer_discharge.html

  1. E E
  2. N N
  3. 1 / N 1/N
  4. E / N E/N

Streptothricin_hydrolase.html

  1. \rightleftharpoons

Stress_resultants.html

  1. 𝐅 1 = A ( σ 11 𝐞 1 + σ 12 𝐞 2 + σ 13 𝐞 3 ) d A \mathbf{F}_{1}=\int_{A}(\sigma_{11}\mathbf{e}_{1}+\sigma_{12}\mathbf{e}_{2}+% \sigma_{13}\mathbf{e}_{3})\,dA
  2. 𝐅 1 = : N 11 𝐞 1 + V 2 𝐞 2 + V 3 𝐞 3 \mathbf{F}_{1}=:N_{11}\mathbf{e}_{1}+V_{2}\mathbf{e}_{2}+V_{3}\mathbf{e}_{3}
  3. N 11 = - b / 2 b / 2 - t / 2 t / 2 σ 11 d x 3 d x 2 . N_{11}=\int_{-b/2}^{b/2}\int_{-t/2}^{t/2}\sigma_{11}\,dx_{3}\,dx_{2}\,.
  4. [ V 2 V 3 ] = - b / 2 b / 2 - t / 2 t / 2 [ σ 12 σ 13 ] d x 3 d x 2 . \begin{bmatrix}V_{2}\\ V_{3}\end{bmatrix}=\int_{-b/2}^{b/2}\int_{-t/2}^{t/2}\begin{bmatrix}\sigma_{12% }\\ \sigma_{13}\end{bmatrix}\,dx_{3}\,dx_{2}\,.
  5. 𝐌 1 = A 𝐫 × ( σ 11 𝐞 1 + σ 12 𝐞 2 + σ 13 𝐞 3 ) d A where 𝐫 = x 2 𝐞 2 + x 3 𝐞 3 . \mathbf{M}_{1}=\int_{A}\mathbf{r}\times(\sigma_{11}\mathbf{e}_{1}+\sigma_{12}% \mathbf{e}_{2}+\sigma_{13}\mathbf{e}_{3})\,dA\quad\,\text{where}\quad\mathbf{r% }=x_{2}\,\mathbf{e}_{2}+x_{3}\,\mathbf{e}_{3}\,.
  6. 𝐌 1 = A ( - x 2 σ 11 𝐞 3 + x 2 σ 13 𝐞 1 + x 3 σ 11 𝐞 2 - x 3 σ 12 𝐞 1 ) d A = : M 11 𝐞 1 + M 12 𝐞 2 + M 13 𝐞 3 . \mathbf{M}_{1}=\int_{A}\left(-x_{2}\sigma_{11}\mathbf{e}_{3}+x_{2}\sigma_{13}% \mathbf{e}_{1}+x_{3}\sigma_{11}\mathbf{e}_{2}-x_{3}\sigma_{12}\mathbf{e}_{1}% \right)dA=:M_{11}\,\mathbf{e}_{1}+M_{12}\,\mathbf{e}_{2}+M_{13}\,\mathbf{e}_{3% }\,.
  7. [ M 11 M 12 M 13 ] := - b / 2 b / 2 - t / 2 t / 2 [ x 2 σ 13 - x 3 σ 12 x 3 σ 11 - x 2 σ 11 ] d x 3 d x 2 . \begin{bmatrix}M_{11}\\ M_{12}\\ M_{13}\end{bmatrix}:=\int_{-b/2}^{b/2}\int_{-t/2}^{t/2}\begin{bmatrix}x_{2}% \sigma_{13}-x_{3}\sigma_{12}\\ x_{3}\sigma_{11}\\ -x_{2}\sigma_{11}\end{bmatrix}\,dx_{3}\,dx_{2}\,.
  8. 𝐅 1 = - t / 2 t / 2 ( σ 11 𝐞 1 + σ 12 𝐞 2 + σ 13 𝐞 3 ) d x 3 and 𝐅 2 = - t / 2 t / 2 ( σ 12 𝐞 1 + σ 22 𝐞 2 + σ 23 𝐞 3 ) d x 3 \mathbf{F}_{1}=\int_{-t/2}^{t/2}(\sigma_{11}\mathbf{e}_{1}+\sigma_{12}\mathbf{% e}_{2}+\sigma_{13}\mathbf{e}_{3})\,dx_{3}\quad\,\text{and}\quad\mathbf{F}_{2}=% \int_{-t/2}^{t/2}(\sigma_{12}\mathbf{e}_{1}+\sigma_{22}\mathbf{e}_{2}+\sigma_{% 23}\mathbf{e}_{3})\,dx_{3}
  9. 𝐅 1 = N 11 𝐞 1 + N 12 𝐞 2 + V 1 𝐞 3 and 𝐅 2 = N 12 𝐞 1 + N 22 𝐞 2 + V 2 𝐞 3 \mathbf{F}_{1}=N_{11}\mathbf{e}_{1}+N_{12}\mathbf{e}_{2}+V_{1}\mathbf{e}_{3}% \quad\,\text{and}\quad\mathbf{F}_{2}=N_{12}\mathbf{e}_{1}+N_{22}\mathbf{e}_{2}% +V_{2}\mathbf{e}_{3}
  10. [ N 11 N 22 N 12 ] := - t / 2 t / 2 [ σ 11 σ 22 σ 12 ] d x 3 \begin{bmatrix}N_{11}\\ N_{22}\\ N_{12}\end{bmatrix}:=\int_{-t/2}^{t/2}\begin{bmatrix}\sigma_{11}\\ \sigma_{22}\\ \sigma_{12}\end{bmatrix}\,dx_{3}
  11. [ V 1 V 2 ] = - t / 2 t / 2 [ σ 13 σ 23 ] d x 3 . \begin{bmatrix}V_{1}\\ V_{2}\end{bmatrix}=\int_{-t/2}^{t/2}\begin{bmatrix}\sigma_{13}\\ \sigma_{23}\end{bmatrix}\,dx_{3}\,.
  12. 𝐌 1 = - t / 2 t / 2 𝐫 × ( σ 11 𝐞 1 + σ 12 𝐞 2 + σ 13 𝐞 3 ) d x 3 and 𝐌 2 = - t / 2 t / 2 𝐫 × ( σ 12 𝐞 1 + σ 22 𝐞 2 + σ 23 𝐞 3 ) d x 3 \mathbf{M}_{1}=\int_{-t/2}^{t/2}\mathbf{r}\times(\sigma_{11}\mathbf{e}_{1}+% \sigma_{12}\mathbf{e}_{2}+\sigma_{13}\mathbf{e}_{3})\,dx_{3}\quad\,\text{and}% \quad\mathbf{M}_{2}=\int_{-t/2}^{t/2}\mathbf{r}\times(\sigma_{12}\mathbf{e}_{1% }+\sigma_{22}\mathbf{e}_{2}+\sigma_{23}\mathbf{e}_{3})\,dx_{3}
  13. 𝐌 1 = - t / 2 t / 2 [ - x 3 σ 12 𝐞 1 + x 3 σ 11 𝐞 2 ] d x 3 and 𝐌 2 = - t / 2 t / 2 [ - x 3 σ 22 𝐞 1 + x 3 σ 12 𝐞 2 ] d x 3 \mathbf{M}_{1}=\int_{-t/2}^{t/2}[-x_{3}\sigma_{12}\mathbf{e}_{1}+x_{3}\sigma_{% 11}\mathbf{e}_{2}]\,dx_{3}\quad\,\text{and}\quad\mathbf{M}_{2}=\int_{-t/2}^{t/% 2}[-x_{3}\sigma_{22}\mathbf{e}_{1}+x_{3}\sigma_{12}\mathbf{e}_{2}]\,dx_{3}
  14. 𝐌 1 = : - M 12 𝐞 1 + M 11 𝐞 2 and 𝐌 2 = : - M 22 𝐞 1 + M 12 𝐞 2 . \mathbf{M}_{1}=:-M_{12}\mathbf{e}_{1}+M_{11}\mathbf{e}_{2}\quad\,\text{and}% \quad\mathbf{M}_{2}=:-M_{22}\mathbf{e}_{1}+M_{12}\mathbf{e}_{2}\,.
  15. [ M 11 M 22 M 12 ] := - t / 2 t / 2 x 3 [ σ 11 σ 22 σ 12 ] d x 3 . \begin{bmatrix}M_{11}\\ M_{22}\\ M_{12}\end{bmatrix}:=\int_{-t/2}^{t/2}x_{3}\,\begin{bmatrix}\sigma_{11}\\ \sigma_{22}\\ \sigma_{12}\end{bmatrix}\,dx_{3}\,.

Strictosidine_beta-glucosidase.html

  1. \rightleftharpoons

Strip_photography.html

  1. ( x , y , t ) (x,y,t)
  2. ( t , y , x ) . (t,y,x).
  3. ( x , y , t ) (x,y,t)
  4. ( t max - t , y , x ) , (t\text{max}-t,y,x),
  5. ( x , t ) (x,t)
  6. ( x , y , t ) (x,y,t)
  7. ( x , t ) (x,t)
  8. ( t , y , x ) (t,y,x)

Strong_Subadditivity_of_Quantum_Entropy.html

  1. \mathcal{H}
  2. ( ) \mathcal{B}(\mathcal{H})
  3. \mathcal{H}
  4. 12 = 1 2 \mathcal{H}^{12}=\mathcal{H}^{1}\otimes\mathcal{H}^{2}
  5. Tr {\rm Tr}
  6. ρ 12 \rho^{12}
  7. 12 \mathcal{H}^{12}
  8. ρ \rho
  9. S ( ρ ) := - Tr ( ρ log ρ ) S(\rho):=-{\rm Tr}(\rho\log\rho)
  10. ρ \rho
  11. σ \sigma
  12. S ( ρ | | σ ) = Tr ( ρ log ρ - ρ log σ ) 0 S(\rho||\sigma)={\rm Tr}(\rho\log\rho-\rho\log\sigma)\geq 0
  13. g g
  14. 0 λ 1 0\leq\lambda\leq 1
  15. g ( λ A 1 + ( 1 - λ ) A 2 , λ B 1 + ( 1 - λ ) B 2 ) λ g ( A 1 , B 1 ) + ( 1 - λ ) g ( A 2 , B 2 ) . g(\lambda A_{1}+(1-\lambda)A_{2},\lambda B_{1}+(1-\lambda)B_{2})\geq\lambda g(% A_{1},B_{1})+(1-\lambda)g(A_{2},B_{2}).
  16. 12 \mathcal{H}^{12}
  17. ρ 12 \rho^{12}
  18. S ( ρ 12 ) S ( ρ 1 ) + S ( ρ 2 ) S(\rho^{12})\leq S(\rho^{1})+S(\rho^{2})
  19. S ( ρ 12 | ρ 1 ) = S ( ρ 12 ) - S ( ρ 1 ) S(\rho^{12}|\rho^{1})=S(\rho^{12})-S(\rho^{1})
  20. S ( ρ 12 | ρ 2 ) = S ( ρ 12 ) - S ( ρ 2 ) S(\rho^{12}|\rho^{2})=S(\rho^{12})-S(\rho^{2})
  21. S ( ρ 12 ) S(\rho^{12})
  22. S ( ρ 1 ) = S ( ρ 12 ) > 0 S(\rho^{1})=S(\rho^{12})>0
  23. S ( ρ 12 ) S(\rho^{12})
  24. S ( ρ 12 ) - S ( ρ 1 ) 0 S(\rho^{12})-S(\rho^{1})\geq 0
  25. S ( ρ 12 ) | S ( ρ 1 ) - S ( ρ 2 ) | S(\rho^{12})\geq|S(\rho^{1})-S(\rho^{2})|
  26. = 1 2 3 . \mathcal{H}=\mathcal{H}^{1}\otimes\mathcal{H}^{2}\otimes\mathcal{H}^{3}.
  27. ρ 123 \rho^{123}
  28. \mathcal{H}
  29. ρ 12 \rho^{12}
  30. 1 2 \mathcal{H}^{1}\otimes\mathcal{H}^{2}
  31. ρ 12 = Tr 3 ρ 123 \rho^{12}={\rm Tr}_{\mathcal{H}^{3}}\rho^{123}
  32. ρ 23 \rho^{23}
  33. ρ 13 \rho^{13}
  34. ρ 1 \rho^{1}
  35. ρ 2 \rho^{2}
  36. ρ 3 \rho^{3}
  37. ρ 123 \rho^{123}
  38. S ( ρ 123 ) + S ( ρ 2 ) S ( ρ 12 ) + S ( ρ 23 ) S(\rho^{123})+S(\rho^{2})\leq S(\rho^{12})+S(\rho^{23})
  39. S ( ρ 12 ) = - Tr 12 ρ 12 log ρ 12 S(\rho^{12})=-{\rm Tr}_{\mathcal{H}^{12}}\rho^{12}\log\rho^{12}
  40. ρ A B C \rho^{ABC}
  41. S ( A B C ) S ( A B ) S(A\mid BC)\leq S(A\mid B)
  42. I ( A : B C ) I ( A : B ) I(A:BC)\geq I(A:B)
  43. S ( ρ 12 ) + S ( ρ 23 ) - S ( ρ 123 ) - S ( ρ 2 ) 2 max { S ( ρ 1 ) - S ( ρ 12 ) , S ( ρ 2 ) - S ( ρ 12 ) , 0 } S(\rho^{12})+S(\rho^{23})-S(\rho^{123})-S(\rho^{2})\geq 2\max\{S(\rho^{1})-S(% \rho^{12}),S(\rho^{2})-S(\rho^{12}),0\}
  44. 2 2
  45. p p
  46. ρ \rho
  47. K K
  48. I p ( ρ , K ) = 1 2 Tr [ ρ p , K * ] [ ρ 1 - p , K ] , I_{p}(\rho,K)=\frac{1}{2}{\rm Tr}[\rho^{p},K^{*}][\rho^{1-p},K],
  49. [ A , B ] = A B - B A [A,B]=AB-BA
  50. K * K^{*}
  51. K K
  52. 0 p 1 0\leq p\leq 1
  53. p p
  54. ρ \rho
  55. 0 p 1 0\leq p\leq 1
  56. - 1 2 Tr ρ K K * -\tfrac{1}{2}{\rm Tr}\rho KK^{*}
  57. T r ρ p K * ρ 1 - p K Tr\rho^{p}K^{*}\rho^{1-p}K
  58. 0 p 1 0\leq p\leq 1
  59. p = 1 2 p=\tfrac{1}{2}
  60. 0 p 1 0\leq p\leq 1
  61. A A
  62. B , B,
  63. 0 r 1 0\leq r\leq 1
  64. p + r 1 p+r\leq 1
  65. p p
  66. p = 1 2 p=\tfrac{1}{2}
  67. S ( ρ 1 ) + S ( ρ 3 ) - S ( ρ 12 ) - S ( ρ 23 ) 0. S(\rho^{1})+S(\rho^{3})-S(\rho^{12})-S(\rho^{23})\leq 0.
  68. S ( ρ 12 | ρ 1 ) S(\rho^{12}|\rho^{1})
  69. S ( ρ 23 | ρ 3 ) S(\rho^{23}|\rho^{3})
  70. ρ 12 S ( ρ 1 ) - S ( ρ 12 ) \rho^{12}\mapsto S(\rho^{1})-S(\rho^{12})
  71. K = 1 K=1
  72. r = 1 - p , A = ρ r=1-p,A=\rho
  73. B = σ B=\sigma
  74. p p
  75. p = 0 p=0
  76. ρ = k λ k ρ k \rho=\sum_{k}\lambda_{k}\rho_{k}
  77. σ = k λ k σ k \sigma=\sum_{k}\lambda_{k}\sigma_{k}
  78. λ k 0 \lambda_{k}\geq 0
  79. k λ k = 1 \sum_{k}\lambda_{k}=1
  80. T T
  81. ( 12 ) ( 12 ) \mathcal{B}(\mathcal{H}^{12})\rightarrow\mathcal{B}(\mathcal{H}^{12})
  82. T = 1 1 T r 2 T=1_{\mathcal{H}^{1}}\otimes Tr_{\mathcal{H}^{2}}
  83. T T
  84. T ( ρ 12 ) = N - 1 j = 1 N ( 1 1 U j ) ρ 12 ( 1 1 U j * ) , T(\rho^{12})=N^{-1}\sum_{j=1}^{N}(1_{\mathcal{H}^{1}}\otimes U_{j})\rho^{12}(1% _{\mathcal{H}^{1}}\otimes U_{j}^{*}),
  85. N N
  86. 2 \mathcal{H}^{2}
  87. 1 \mathcal{H}^{1}
  88. 12 \mathcal{H}^{12}
  89. 2 \mathcal{H}^{2}
  90. 3 \mathcal{H}^{3}
  91. ρ = ρ 123 , σ = ρ 1 ρ 23 , T = 1 12 T r 3 \rho=\rho^{123},\sigma=\rho^{1}\otimes\rho^{23},T=1_{\mathcal{H}^{12}}\otimes Tr% _{\mathcal{H}^{3}}
  92. S ( ρ 12 | | ρ 1 ρ 2 ) S ( ρ 123 | | ρ 1 ρ 23 ) . S(\rho^{12}||\rho^{1}\otimes\rho^{2})\leq S(\rho^{123}||\rho^{1}\otimes\rho^{2% 3}).
  93. S ( ρ 123 | | ρ 1 ρ 23 ) - S ( ρ 12 | | ρ 1 ρ 2 ) = S ( ρ 12 ) + S ( ρ 23 ) - S ( ρ 123 ) - S ( ρ 2 ) 0 , S(\rho^{123}||\rho^{1}\otimes\rho^{23})-S(\rho^{12}||\rho^{1}\otimes\rho^{2})=% S(\rho^{12})+S(\rho^{23})-S(\rho^{123})-S(\rho^{2})\geq 0,
  94. T T
  95. \Rightarrow
  96. \Rightarrow
  97. \Rightarrow
  98. \Rightarrow
  99. ρ 123 \rho_{123}
  100. ρ 12 S ( ρ 1 ) - S ( ρ 12 ) \rho_{12}\mapsto S(\rho_{1})-S(\rho_{12})
  101. \Rightarrow
  102. ρ 12 \rho_{12}
  103. σ 12 \sigma_{12}
  104. λ k ρ k \lambda_{k}\rho_{k}
  105. λ k σ k \lambda_{k}\sigma_{k}
  106. ρ := ρ 2 = k λ k ρ k \rho:=\rho_{2}=\sum_{k}\lambda_{k}\rho_{k}
  107. \Rightarrow
  108. ρ 123 \rho_{123}
  109. ρ 1234 \rho_{1234}
  110. S ( ρ 4 ) + S ( ρ 2 ) S ( ρ 12 ) + S ( ρ 14 ) . S(\rho_{4})+S(\rho_{2})\leq S(\rho_{12})+S(\rho_{14}).
  111. ρ 124 \rho_{124}
  112. S ( ρ 2 ) = S ( ρ 14 ) S(\rho_{2})=S(\rho_{14})
  113. S ( ρ 4 ) = S ( ρ 12 ) S(\rho_{4})=S(\rho_{12})
  114. ρ \rho
  115. σ \sigma
  116. \mathcal{H}
  117. T : ( ) ( 𝒦 ) T:\mathcal{B}(\mathcal{H})\rightarrow\mathcal{B}(\mathcal{K})
  118. S ( T ρ | | T σ ) = S ( ρ | | σ ) , S(T\rho||T\sigma)=S(\rho||\sigma),
  119. T ^ \hat{T}
  120. T ^ T σ = σ , \hat{T}T\sigma=\sigma,
  121. T ^ T ρ = ρ . \hat{T}T\rho=\rho.
  122. T ^ \hat{T}
  123. T ^ ω = σ 1 / 2 T * ( ( T σ ) - 1 / 2 ω ( T σ ) - 1 / 2 ) σ 1 / 2 , \hat{T}\omega=\sigma^{1/2}T^{*}\Bigl((T\sigma)^{-1/2}\omega(T\sigma)^{-1/2}% \Bigr)\sigma^{1/2},
  124. T * T^{*}
  125. T T
  126. t = 0 t=0
  127. ρ \rho
  128. σ \sigma
  129. \mathcal{H}
  130. T : ( ) ( 𝒦 ) T:\mathcal{B}(\mathcal{H})\rightarrow\mathcal{B}(\mathcal{K})
  131. S ( T ρ | | T σ ) = S ( ρ | | σ ) , S(T\rho||T\sigma)=S(\rho||\sigma),
  132. T * ( T ( ρ ) i t T ( σ ) i t ) = ρ i t σ - i t T^{*}(T(\rho)^{it}T(\sigma)^{it})=\rho^{it}\sigma^{-it}
  133. t t
  134. log ρ - log σ = T * ( log T ( ρ ) - log T ( σ ) ) . \log\rho-\log\sigma=T^{*}\Bigl(\log T(\rho)-\log T(\sigma)\Bigr).
  135. T * T^{*}
  136. T T
  137. ρ A B C \rho^{ABC}
  138. A B C \mathcal{H}^{A}\otimes\mathcal{H}^{B}\otimes\mathcal{H}^{C}
  139. B = j B j L B j R \mathcal{H}^{B}=\bigoplus_{j}\mathcal{H}^{B^{L}_{j}}\otimes\mathcal{H}^{B^{R}_% {j}}
  140. ρ A B C = j q j ρ A B j L ρ B j R C , \rho^{ABC}=\bigoplus_{j}q_{j}\rho^{AB^{L}_{j}}\otimes\rho^{B^{R}_{j}C},
  141. ρ A B j L \rho^{AB^{L}_{j}}
  142. A B j L \mathcal{H}^{A}\otimes\mathcal{H}^{B^{L}_{j}}
  143. ρ B j R C \rho^{B^{R}_{j}C}
  144. B j R C \mathcal{H}^{B^{R}_{j}}\otimes\mathcal{H}^{C}
  145. { q j } \{q_{j}\}
  146. ρ 123 \rho^{123}
  147. 1 2 3 \mathcal{H}^{1}\otimes\mathcal{H}^{2}\otimes\mathcal{H}^{3}
  148. T r 12 ( ρ 123 ( - log ( ρ 12 ) - log ( ρ 23 ) + log ( ρ 2 ) + log ( ρ 123 ) ) ) 0. Tr_{12}\Bigl(\rho^{123}(-\log(\rho^{12})-\log(\rho^{23})+\log(\rho^{2})+\log(% \rho^{123}))\Bigr)\geq 0.

Stufe_(algebra).html

  1. \infty
  2. s ( F ) s(F)\neq\infty
  3. s ( F ) = 2 k s(F)=2^{k}
  4. k k\in\mathbb{N}
  5. k k\in\mathbb{N}
  6. 2 k s ( F ) < 2 k + 1 2^{k}\leq s(F)<2^{k+1}
  7. n = 2 k n=2^{k}
  8. s = s ( F ) s=s(F)
  9. e 1 , , e s F { 0 } e_{1},\ldots,e_{s}\in F\setminus\{0\}
  10. 0 = 1 + e 1 2 + + e n - 1 2 = : a + e n 2 + + e s 2 = : b . 0=\underbrace{1+e_{1}^{2}+\cdots+e_{n-1}^{2}}_{=:a}+\underbrace{e_{n}^{2}+% \cdots+e_{s}^{2}}_{=:b}\;.
  11. a a
  12. b b
  13. n n
  14. a 0 a\neq 0
  15. s ( F ) < 2 k s(F)<2^{k}
  16. k k
  17. a b ab
  18. n n
  19. a b = c 1 2 + + c n 2 ab=c_{1}^{2}+\cdots+c_{n}^{2}
  20. c i F c_{i}\in F
  21. a + b = 0 a+b=0
  22. - a 2 = a b -a^{2}=ab
  23. - 1 = a b a 2 = ( c 1 a ) 2 + + ( c n a ) 2 , -1=\frac{ab}{a^{2}}=\left(\frac{c_{1}}{a}\right)^{2}+\cdots+\left(\frac{c_{n}}% {a}\right)^{2}\;,
  24. s ( F ) = n = 2 k s(F)=n=2^{k}
  25. s ( F ) 2 s(F)\leq 2
  26. F F
  27. p = char ( F ) p=\operatorname{char}(F)
  28. 𝔽 p \mathbb{F}_{p}
  29. p = 2 p=2
  30. - 1 = 1 = 1 2 -1=1=1^{2}
  31. s ( F ) = 1 s(F)=1
  32. p > 2 p>2
  33. S = { x 2 x 𝔽 p } S=\{x^{2}\mid x\in\mathbb{F}_{p}\}
  34. S { 0 } S\setminus\{0\}
  35. 2 2
  36. 𝔽 p × \mathbb{F}_{p}^{\times}
  37. p - 1 p-1
  38. S S
  39. p + 1 2 \tfrac{p+1}{2}
  40. - 1 - S -1-S
  41. 𝔽 p \mathbb{F}_{p}
  42. p p
  43. S S
  44. - 1 - S -1-S
  45. x , y 𝔽 p x,y\in\mathbb{F}_{p}
  46. S x 2 = - 1 - y 2 - 1 - S S\ni x^{2}=-1-y^{2}\in-1-S
  47. - 1 = x 2 + y 2 -1=x^{2}+y^{2}

Subadditive_set_function.html

  1. Ω \Omega
  2. f : 2 Ω f\colon 2^{\Omega}\rightarrow\mathbb{R}
  3. 2 Ω 2^{\Omega}
  4. Ω \Omega
  5. S S
  6. T T
  7. Ω \Omega
  8. f ( S ) + f ( T ) f ( S T ) f(S)+f(T)\geq f(S\cup T)
  9. T 1 , , T m Ω T_{1},\ldots,T_{m}\subseteq\Omega
  10. i = 1 m T i = Ω \cup_{i=1}^{m}T_{i}=\Omega
  11. f f
  12. f ( S ) f(S)
  13. t t
  14. T i 1 , , T i t T_{i_{1}},\ldots,T_{i_{t}}
  15. S j = 1 t T i j S\subseteq\cup_{j=1}^{t}T_{i_{j}}
  16. f f
  17. i { 1 , , m } i\in\{1,\ldots,m\}
  18. a i : Ω + a_{i}\colon\Omega\to\mathbb{R}_{+}
  19. f ( S ) = max i ( x S a i ( x ) ) f(S)=\max_{i}\left(\sum_{x\in S}a_{i}(x)\right)
  20. f f
  21. S Ω S\subseteq\Omega
  22. X 1 , , X n Ω X_{1},\ldots,X_{n}\subseteq\Omega
  23. α 1 , , α n [ 0 , 1 ] \alpha_{1},\ldots,\alpha_{n}\in[0,1]
  24. 1 S i = 1 n α i 1 X i 1_{S}\leq\sum_{i=1}^{n}\alpha_{i}1_{X_{i}}
  25. f ( S ) i = 1 n α i f ( X i ) f(S)\leq\sum_{i=1}^{n}\alpha_{i}f(X_{i})

Subordinator_(mathematics).html

  1. W ( t ) W(t)
  2. θ t \theta t
  3. Γ ( t ; 1 , ν ) \Gamma(t;1,\nu)
  4. X V G ( t ; σ , ν , θ ) := θ Γ ( t ; 1 , ν ) + σ W ( Γ ( t ; 1 , ν ) ) . X^{VG}(t;\sigma,\nu,\theta)\;:=\;\theta\,\Gamma(t;1,\nu)+\sigma\,W(\Gamma(t;1,% \nu)).

Subtract_with_carry.html

  1. x ( i ) = ( x ( i - S ) - x ( i - R ) - c y ( i - 1 ) ) mod M x(i)=(x(i-S)-x(i-R)-cy(i-1))\ \bmod\ M
  2. c y ( i ) cy(i)
  3. x ( i - S ) - x ( i - R ) - c y ( i - 1 ) < 0 x(i-S)-x(i-R)-cy(i-1)<0
  4. c y ( i ) = 0 cy(i)=0

Succinate-semialdehyde_dehydrogenase_(acylating).html

  1. \rightleftharpoons

Succinate-semialdehyde_dehydrogenase_(NADP+).html

  1. \rightleftharpoons

Sugeno_integral.html

  1. ( X , Ω ) (X,\Omega)
  2. h : X [ 0 , 1 ] h:X\to[0,1]
  3. Ω \Omega
  4. A X A\subseteq X
  5. h h
  6. g g
  7. A h ( x ) g = sup E X [ min ( min x E h ( x ) , g ( A E ) ) ] = sup α [ 0 , 1 ] [ min ( α , g ( A F α ) ) ] \int_{A}h(x)\circ g={\sup_{E\subseteq X}}\left[\min\left(\min_{x\in E}h(x),g(A% \cap E)\right)\right]={\sup_{\alpha\in[0,1]}}\left[\min\left(\alpha,g(A\cap F_% {\alpha})\right)\right]
  8. F α = { x | h ( x ) α } F_{\alpha}=\left\{x|h(x)\geq\alpha\right\}
  9. A ~ \tilde{A}
  10. h h
  11. g g
  12. A h ( x ) g = X [ h A ( x ) h ( x ) ] g \int_{A}h(x)\circ g=\int_{X}\left[h_{A}(x)\wedge h(x)\right]\circ g
  13. h A ( x ) h_{A}(x)
  14. A ~ \tilde{A}

Sulfide-cytochrome-c_reductase_(flavocytochrome_c).html

  1. \rightleftharpoons

Sulfide:quinone_reductase.html

  1. \rightleftharpoons

Sulfoacetaldehyde_dehydrogenase.html

  1. \rightleftharpoons

Sulfoacetaldehyde_dehydrogenase_(acylating).html

  1. \rightleftharpoons

Sulfoacetaldehyde_reductase.html

  1. \rightleftharpoons

Sulfopropanediol_3-dehydrogenase.html

  1. \rightleftharpoons

Sulfur_carrier_protein_ThiS_adenylyltransferase.html

  1. \rightleftharpoons

Sulfur_dioxygenase.html

  1. \rightleftharpoons
  2. \rightleftharpoons
  3. \rightleftharpoons

Sum_of_perpetuities_method.html

  1. P P
  2. E E
  3. G G
  4. K K
  5. D D
  6. P = ( E * G K 2 ) + ( D K ) P=(\frac{E*G}{K^{2}})+(\frac{D}{K})
  7. G G
  8. P = ( E * G K 2 ) + ( D K ) P=(\frac{E*G}{K^{2}})+(\frac{D}{K})
  9. P = ( E * G 0.10 2 ) + 0 P=(\frac{E*G}{0.10^{2}})+0
  10. P = E * G * 100 P=E*G*100
  11. X X
  12. R R
  13. X X
  14. R R
  15. E = X + D E=X+D
  16. G = ( X E ) * R G=(\frac{X}{E})*R
  17. P = E K = D K P=\frac{E}{K}=\frac{D}{K}
  18. P = E K = ( X + D ) K = X K + D K P=\frac{E}{K}=\frac{(X+D)}{K}=\frac{X}{K}+\frac{D}{K}
  19. P V x PVx
  20. P = P V x K + D K P=\frac{PVx}{K}+\frac{D}{K}
  21. P V x PVx
  22. X X
  23. X X
  24. P V x PVx
  25. R R
  26. X * R X*R
  27. X X
  28. P V x PVx
  29. P V x = X * R K PVx=\frac{X*R}{K}
  30. X * R K \frac{X*R}{K}
  31. P V x PVx
  32. P = X * R K K + D K P=\frac{\frac{X*R}{K}}{K}+\frac{D}{K}
  33. X * R X*R
  34. E * G E*G
  35. E * G E*G
  36. P = E * G K 2 + D K P=\frac{E*G}{K^{2}}+\frac{D}{K}

Superelliptic_curve.html

  1. y m = f ( x ) , y^{m}=f(x),
  2. C 1 C\to\mathbb{P}^{1}
  3. m 2 m\geq 2
  4. m m
  5. m m
  6. k k
  7. y m = f ( x ) y^{m}=f(x)
  8. f k [ x ] f\in k[x]
  9. m m
  10. < m <m
  11. C C
  12. k k
  13. C ( k ) C(k)
  14. k k
  15. C C
  16. k k
  17. k ( C ) / k ( x ) k(C)/k(x)
  18. C : y m = f ( x ) C:y^{m}=f(x)
  19. k k
  20. B k B^{\prime}\subset k
  21. f f
  22. k k
  23. B = { B if m divides deg ( f ) , B { } otherwise. B=\begin{cases}B^{\prime}&\,\text{ if }m\,\text{ divides }\deg(f),\\ B^{\prime}\cup\{\infty\}&\,\text{ otherwise.}\end{cases}
  24. B 1 ( k ) B\subset\mathbb{P}^{1}(k)
  25. C 1 C\to\mathbb{P}^{1}
  26. x x
  27. α B \alpha\in B
  28. r α r_{\alpha}
  29. α \alpha
  30. f f
  31. 1 r α < m 1\leq r_{\alpha}<m
  32. e α = m ( m , r α ) e_{\alpha}=\frac{m}{(m,r_{\alpha})}
  33. e ( P α , i ) e(P_{\alpha,i})
  34. ( m , r α ) (m,r_{\alpha})
  35. P α , i P_{\alpha,i}
  36. α 𝔸 1 ( k ) 1 ( k ) \alpha\in\mathbb{A}^{1}(k)\subset\mathbb{P}^{1}(k)
  37. α k \alpha\in k
  38. 0 r < m 0\leq r_{\infty}<m
  39. s = min { t | m t deg ( f ) } , s=\min\{t\in\mathbb{Z}|mt\geq\deg(f)\},
  40. r = m s - deg ( f ) r_{\infty}=ms-\deg(f)
  41. ( m , r ) = ( m , deg ( f ) ) (m,r_{\infty})=(m,\deg(f))
  42. e = m ( m , r ) e_{\infty}=\frac{m}{(m,r_{\infty})}
  43. e ( P , i ) e(P_{\infty,i})
  44. ( m , r ) (m,r_{\infty})
  45. P , i P_{\infty,i}
  46. \infty
  47. m m
  48. deg ( f ) \deg(f)
  49. C C
  50. m m
  51. r α r_{\alpha}
  52. g = 1 2 ( m ( | B | - 2 ) - α B ( m , r α ) ) + 1. g=\frac{1}{2}\left(m(|B|-2)-\sum_{\alpha\in B}(m,r_{\alpha})\right)+1.

Supermetric.html

  1. P M P\to M
  2. K K
  3. K K
  4. H H
  5. h h
  6. P / H M P/H\to M
  7. P = F M P=FM
  8. T M TM
  9. M M
  10. G L ( n , ) GL(n,\mathbb{R})
  11. O ( 1 , 3 ) O(1,3)
  12. F M / O ( 1 , 3 ) FM/O(1,3)
  13. M M
  14. G G
  15. P ^ M ^ \widehat{P}\to\widehat{M}
  16. K ^ \widehat{K}
  17. H ^ \widehat{H}
  18. K ^ \widehat{K}
  19. K ^ K ^ / H ^ \widehat{K}\to\widehat{K}/\widehat{H}
  20. P ^ \widehat{P}
  21. H ^ \widehat{H}
  22. P ^ / H ^ M ^ \widehat{P}/\widehat{H}\to\widehat{M}
  23. K ^ / H ^ \widehat{K}/\widehat{H}
  24. G G
  25. K ^ K ^ / H ^ \widehat{K}\to\widehat{K}/\widehat{H}
  26. K ^ \widehat{K}
  27. H ^ \widehat{H}
  28. P ^ = F M ^ \widehat{P}=F\widehat{M}
  29. M ^ \widehat{M}
  30. ( n , 2 m ) (n,2m)
  31. K ^ = G L ^ ( n | 2 m ; Λ ) \widehat{K}=\widehat{GL}(n|2m;\Lambda)
  32. H ^ = O S ^ p ( n | m ; Λ ) \widehat{H}=\widehat{OS}p(n|m;\Lambda)
  33. F M ^ / H ^ M ^ F\widehat{M}/\widehat{H}\to\widehat{M}
  34. M ^ \widehat{M}
  35. B n | 2 m B^{n|2m}

Superradiant_phase_transition.html

  1. N N
  2. H ^ JC = ω a ^ a ^ + ω σ ^ z 2 + Ω 2 ( a ^ σ ^ + + a ^ σ ^ - + a ^ σ ^ - + a ^ σ ^ + ) , \hat{H}_{\,\text{JC}}=\hbar\omega\hat{a}^{\dagger}\hat{a}+\hbar\omega\frac{% \hat{\sigma}_{z}}{2}+\frac{\hbar\Omega}{2}\left(\hat{a}\hat{\sigma}_{+}+\hat{a% }^{\dagger}\hat{\sigma}_{-}+\hat{a}\hat{\sigma}_{-}+\hat{a}^{\dagger}\hat{% \sigma}_{+}\right),
  3. σ ^ - b ^ \hat{\sigma}_{-}\approx\hat{b}
  4. σ ^ + b ^ \hat{\sigma}_{+}\approx\hat{b}^{\dagger}
  5. σ ^ z 2 b ^ b ^ \hat{\sigma}_{z}\approx 2\hat{b}^{\dagger}\hat{b}
  6. H ^ JC = ω a ^ a ^ + ω b ^ b ^ + Ω 2 ( a ^ b ^ + a ^ b ^ + a ^ b ^ + a ^ b ^ ) , \hat{H}_{\,\text{JC}}=\hbar\omega\hat{a}^{\dagger}\hat{a}+\hbar\omega\hat{b}^{% \dagger}\hat{b}+\frac{\hbar\Omega}{2}\left(\hat{a}\hat{b}^{\dagger}+\hat{a}^{% \dagger}\hat{b}+\hat{a}\hat{b}+\hat{a}^{\dagger}\hat{b}^{\dagger}\right),
  7. H ^ JC = Ω + A + ^ A + ^ + Ω - A - ^ A - ^ + C \hat{H}_{\,\text{JC}}=\Omega_{+}\hat{A_{+}}^{\dagger}\hat{A_{+}}+\Omega_{-}% \hat{A_{-}}^{\dagger}\hat{A_{-}}+C
  8. A ± ^ = c ± 1 a ^ + c ± 2 a ^ + c ± 3 b ^ + c ± 4 b ^ \hat{A_{\pm}}=c_{\pm 1}\hat{a}+c_{\pm 2}\hat{a}^{\dagger}+c_{\pm 3}\hat{b}+c_{% \pm 4}\hat{b}^{\dagger}
  9. [ A ± ^ , H ^ JC ] = Ω ± A [\hat{A_{\pm}},\hat{H}_{\,\text{JC}}]=\Omega_{\pm}A
  10. Ω ± = ω 1 ± Ω ω \Omega_{\pm}=\omega\sqrt{1\pm\frac{\Omega}{\omega}}
  11. Ω > ω \Omega>\omega
  12. H ^ JC = ω a ^ a ^ + ω σ ^ z 2 + Ω 2 ( a ^ σ ^ + + a ^ σ ^ - ) , \hat{H}_{\,\text{JC}}=\hbar\omega\hat{a}^{\dagger}\hat{a}+\hbar\omega\frac{% \hat{\sigma}_{z}}{2}+\frac{\hbar\Omega}{2}\left(\hat{a}\hat{\sigma}_{+}+\hat{a% }^{\dagger}\hat{\sigma}_{-}\right),
  13. E ± ( n ) = ω ( n + 1 2 ) ± 1 2 Ω ( n ) , E_{\pm}(n)=\hbar\omega\left(n+\frac{1}{2}\right)\pm\frac{1}{2}\hbar\Omega(n),
  14. Ω ( n ) = Ω n + 1 \Omega(n)=\Omega\sqrt{n+1}
  15. Ω n \Omega_{n}
  16. Z = ± , n e - β E ± ( n ) ± e - β E ± ( n ) d n = e Φ ( n ) d n Z=\sum_{\pm,n}\mathrm{e}^{-\beta E_{\pm}(n)}\approx\sum_{\pm}\int\mathrm{e}^{-% \beta E_{\pm}(n)}dn=\int\mathrm{e}^{\Phi(n)}dn
  17. Φ ( n ) n = 0 \frac{\partial\Phi(n)}{\partial n}=0
  18. Φ ( n ) = - β ω ( n + 1 2 ) + log 2 cosh Ω ( n ) β 2 \Phi(n)=-\beta\hbar\omega\left(n+\frac{1}{2}\right)+\log 2\cosh\frac{\hbar% \Omega(n)\beta}{2}
  19. tanh Ω ( n ) β 2 = 4 ω Ω n + 1 \tanh\frac{\hbar\Omega(n)\beta}{2}=4\frac{\omega}{\Omega}\sqrt{n+1}
  20. Ω > 4 ω \Omega>4\omega
  21. n n
  22. 1 / β 1/\beta

Surface_and_bulk_erosion.html

  1. t d i f f u s i o n = < x > 2 π 4 D t_{diffusion}=\frac{<x>^{2}\pi}{4D}
  2. t e r o s i o n = l n < x > - l n M N a ( N - 1 ) * p 3 k t_{erosion}=\frac{ln<x>-ln\sqrt[3]{M\over N_{a}(N-1)*p}}{k}
  3. E r o s i o n N u m b e r = t d i f f u s i o n t e r o s i o n = < x > 2 π * k 4 D * ( l n < x > - l n M N a ( N - 1 ) * p 3 ) ErosionNumber=\frac{t_{diffusion}}{t_{erosion}}=\frac{<x>^{2}\pi*k}{4D*(ln<x>-% ln\sqrt[3]{M\over N_{a}(N-1)*p})}

Surface_chemistry_of_cooking.html

  1. F = Q 1 Q 2 4 π D 2 ε 0 ε r {F}=\frac{Q_{1}Q_{2}}{4\pi\mathrm{D}^{2}\varepsilon_{0}\varepsilon_{r}}
  2. Q {Q}
  3. D {D}
  4. ε 0 {\varepsilon_{0}}
  5. ε r {\varepsilon_{r}}
  6. D 2 D^{2}
  7. ε r \varepsilon_{r}
  8. V = M 1 2 α 2 + M 2 2 α 1 - ( 4 π ε 0 ε r ) 2 r 6 {V}=\frac{M_{1}^{2}\alpha_{2}+M_{2}^{2}\alpha_{1}}{-\left(4\pi\varepsilon_{0}% \varepsilon_{r}\right)^{2}\mathrm{r}^{6}}
  9. V = - 2 ( M 1 M 2 ) 2 3 ( 4 π ε 0 ε r ) r 6 k b T {V}=\frac{-2\left(M_{1}M_{2}\right)^{2}}{3\left(4\pi\varepsilon_{0}\varepsilon% _{r}\right)\mathrm{r}^{6}\mathrm{k}_{b}\mathrm{T}}
  10. V = - 3 ( α 1 α 2 ) 2 2 ( 4 π ε 0 ε r ) 2 r 6 ( hV 1 V 2 V 1 + V 2 ) {V}=\frac{-3\left(\alpha_{1}\alpha_{2}\right)^{2}}{2\left(4\pi\varepsilon_{0}% \varepsilon_{r}\right)^{2}\mathrm{r}^{6}}\left(\frac{\mathrm{h}\mathrm{V}_{1}% \mathrm{V}_{2}}{\mathrm{V}_{1}+\mathrm{V}_{2}}\right)
  11. M 1 M_{1}
  12. M 2 M_{2}
  13. α 1 \alpha_{1}
  14. α 2 \alpha_{2}
  15. ε 0 {\varepsilon_{0}}
  16. ε r {\varepsilon_{r}}
  17. k b \mathrm{k}_{b}
  18. T T
  19. r r
  20. h V hV
  21. r r
  22. r r

Surface_Chemistry_of_Microvasculature.html

  1. . J = - D C x . \bigg.J=-D\frac{\partial C}{\partial x}\bigg.
  2. . R e = R m e m A . \bigg.R_{e}=\frac{R_{m}em}{A}\bigg.
  3. S V R = 80 × ( M A P - M R A P ) C O SVR=80\times\frac{\left(MAP-MRAP\right)}{CO}
  4. J v = K f [ ( P c - P i ) - σ ( π c - π i ) ] J_{v}=K_{f}\left[\left(P_{c}-P_{i}\right)-\sigma\left(\pi_{c}-\pi_{i}\right)\right]
  5. v = - ϵ o ϵ r ζ E μ \vec{v}=-\frac{\epsilon_{o}\epsilon_{r}\zeta\vec{E}}{\mu}
  6. v \vec{v}
  7. E \vec{E}

Surjunctive_group.html

  1. G G
  2. S S
  3. f : S G S G f:S^{G}\to S^{G}

Surprisal_analysis.html

  1. P ( n ) P(n)
  2. n n
  3. P ( n ) = P 0 ( n ) exp [ - α λ α G α ( n ) ] P(n)=P^{0}(n)\exp\left[-\sum_{\alpha}\lambda_{\alpha}G_{\alpha}(n)\right]
  4. P 0 ( n ) P^{0}(n)
  5. n n
  6. n n
  7. surprisal = def - ln P ( n ) P 0 ( n ) = α λ α G α ( n ) \begin{aligned}\displaystyle\,\text{surprisal}&\displaystyle\stackrel{\,\text{% def}}{=}-\ln\frac{P(n)}{P^{0}(n)}\\ &\displaystyle=\sum_{\alpha}\lambda_{\alpha}G_{\alpha}(n)\end{aligned}
  8. α \alpha
  9. λ α \lambda_{\alpha}
  10. G α ( n ) G_{\alpha}(n)
  11. α \alpha
  12. n n
  13. P 0 ( n ) P^{0}(n)

Surprise_(networks).html

  1. S = - log j = p min ( M , n ) ( M j ) ( F - M n - j ) ( F n ) S=-\log\sum_{j=p}^{\min(M,n)}\frac{{\left({{M}\atop{j}}\right)}{\left({{F-M}% \atop{n-j}}\right)}}{{\left({{F}\atop{n}}\right)}}
  2. F = k ( k - 1 ) 2 F=\frac{k(k-1)}{2}
  3. M = i = 1 C k i ( k i - 1 ) 2 M=\sum_{i=1}^{C}\frac{k_{i}(k_{i}-1)}{2}

Sylvester_matroid.html

  1. n n
  2. n n
  3. U n 2 U{}^{2}_{n}
  4. r r
  5. 2 r - 1 2^{r}-1

Symmetric_cone.html

  1. C * = { X : ( X , Y ) > 0 for Y C ¯ } . \displaystyle{C^{*}=\{X:(X,Y)>0\,\,\mathrm{for}\,\,Y\in\overline{C}\}.}
  2. Aut C = { g GL ( V ) | g C = C } . \displaystyle{\mathrm{Aut}\,C=\{g\in\mathrm{GL}(V)|gC=C\}.}
  3. E × E E , a , b a b = b a , \displaystyle{E\times E\rightarrow E,\,\,\,a,b\mapsto ab=ba,}
  4. L ( a ) L ( a 2 ) = L ( a 2 ) L ( a ) . \displaystyle{L(a)L(a^{2})=L(a^{2})L(a).}
  5. a m a n = a m + n , \displaystyle{a^{m}a^{n}=a^{m+n},}
  6. 2 L ( a b ) L ( a ) + L ( a 2 ) L ( b ) = 2 L ( a ) L ( b ) L ( a ) + L ( a 2 b ) . \displaystyle{2L(ab)L(a)+L(a^{2})L(b)=2L(a)L(b)L(a)+L(a^{2}b).}
  7. a 2 a m - 1 = a m - 1 ( a 2 ) = L ( a m - 1 ) L ( a ) a = L ( a ) L ( a m - 1 ) a = L ( a ) a m = a m + 1 . \displaystyle{a^{2}a^{m-1}=a^{m-1}(a^{2})=L(a^{m-1})L(a)a=L(a)L(a^{m-1})a=L(a)% a^{m}=a^{m+1}.}
  8. L ( a m + 1 ) = 2 L ( a m ) L ( a ) + L ( a 2 ) L ( a m - 1 ) - 2 L ( a ) 2 L ( a m - 1 ) , \displaystyle{L(a^{m+1})=2L(a^{m})L(a)+L(a^{2})L(a^{m-1})-2L(a)^{2}L(a^{m-1}),}
  9. L ( a m + 1 ) a n = 2 L ( a ) L ( a m ) a n + L ( a 2 ) L ( a m - 1 ) a n - 2 L ( a ) 2 L ( a m - 1 ) a n = a m + n + 1 . \displaystyle{L(a^{m+1})a^{n}=2L(a)L(a^{m})a^{n}+L(a^{2})L(a^{m-1})a^{n}-2L(a)% ^{2}L(a^{m-1})a^{n}=a^{m+n+1}.}
  10. a = λ i e i , \displaystyle{a=\sum\lambda_{i}e_{i},}
  11. T = λ i P i \displaystyle{T=\sum\lambda_{i}P_{i}}
  12. F k ( a ) = det 0 m , n < k ( a m , a n ) . \displaystyle{F_{k}(a)=\det_{0\leq m,n<k}(a^{m},a^{n}).}
  13. 2 L ( e ) 3 - 3 L ( e ) 2 + L ( e ) = 0. \displaystyle{2L(e)^{3}-3L(e)^{2}+L(e)=0.}
  14. E = E 0 ( e ) E 1 / 2 ( e ) E 1 ( e ) , \displaystyle{E=E_{0}(e)\oplus E_{1/2}(e)\oplus E_{1}(e),}
  15. U = 8 L ( e ) 2 - 8 L ( e ) + I , \displaystyle{U=8L(e)^{2}-8L(e)+I,}
  16. σ ( x ) = U x \displaystyle{\sigma(x)=Ux}
  17. τ ( a , b ) = Tr L ( a b ) . \displaystyle{\tau(a,b)=\mathrm{Tr}\,L(ab).}
  18. τ ( a , a ) = λ i 2 Tr L ( e i ) > 0. \displaystyle{\tau(a,a)=\sum\lambda_{i}^{2}\mathrm{Tr}\,L(e_{i})>0.}
  19. L ( a ( b c ) - ( a b ) c ) = [ [ L ( a ) , L ( b ) ] , L ( c ) ] . \displaystyle{L(a(bc)-(ab)c)=[[L(a),L(b)],L(c)].}
  20. τ ( a , b c ) = τ ( b a , c ) , \displaystyle{\tau(a,bc)=\tau(ba,c),}
  21. e a = a e = L ( a ) P ( 1 ) = P ( L ( a ) 1 ) = P ( a ) , \displaystyle{ea=ae=L(a)P(1)=P(L(a)1)=P(a),}
  22. E = E i . \displaystyle{E=\oplus E_{i}.}
  23. ( a b ) * = b * a * (a b)*=b*a*
  24. λ ( a * ) = λ ( a ) * λ(a*)=λ(a)*
  25. ρ ( a * ) = ρ ( a ) * ρ(a*)=ρ(a)*
  26. R e ( a b ) = R e ( b a ) Re(a b)=Re(b a)
  27. R e x = ( x + x * ) / 2 = ( x , 1 ) 1 Rex=(x+x*)/2=(x, 1)1
  28. R e ( a b ) c = R e a ( b c ) Re(a b)c=Rea(b c)
  29. A A
  30. Y 1 = ( 0 0 0 0 0 y 1 0 y 1 * 0 ) , Y 2 = ( 0 0 y 2 * 0 0 0 y 2 0 0 ) , Y 3 = ( 0 y 3 0 y 3 * 0 0 0 0 0 ) . \displaystyle{Y_{1}=\begin{pmatrix}0&0&0\\ 0&0&y_{1}\\ 0&y_{1}^{*}&0\end{pmatrix},\,\,\,Y_{2}=\begin{pmatrix}0&0&y_{2}^{*}\\ 0&0&0\\ y_{2}&0&0\end{pmatrix},\,\,\,Y_{3}=\begin{pmatrix}0&y_{3}&0\\ y_{3}^{*}&0&0\\ 0&0&0\end{pmatrix}.}
  31. π i j ( x y ) = π i j ( x ) π i j ( y ) , π i j ( 1 ) = I . \displaystyle{\pi_{ij}(xy)=\pi_{ij}(x)\pi_{ij}(y),\,\,\,\pi_{ij}(1)=I.}
  32. Q ( a ) = 2 L ( a ) 2 - L ( a 2 ) . \displaystyle{Q(a)=2L(a)^{2}-L(a^{2}).}
  33. ( Q ( a ) - 1 a ) a = a Q ( a ) - 1 a = L ( a ) Q ( a ) - 1 a = Q ( a ) - 1 a 2 = 1. \displaystyle{(Q(a)^{-1}a)a=aQ(a)^{-1}a=L(a)Q(a)^{-1}a=Q(a)^{-1}a^{2}=1.}
  34. Q ( a ) L ( a - 1 ) = L ( a ) . \displaystyle{Q(a)L(a^{-1})=L(a).}
  35. c = D c ( Q ( a ) a - 1 ) = 2 [ ( L ( a ) L ( c ) + L ( c ) L ( a ) - L ( a c ) ) a - 1 ] + Q ( a ) D c ( a - 1 ) = 2 c + Q ( a ) D c ( a - 1 ) . \displaystyle{c=D_{c}(Q(a)a^{-1})=2[(L(a)L(c)+L(c)L(a)-L(ac))a^{-1}]+Q(a)D_{c}% (a^{-1})=2c+Q(a)D_{c}(a^{-1}).}
  36. ( Q ( a ) b ) ( Q ( a - 1 ) b - 1 ) = 1. \displaystyle{(Q(a)b)(Q(a^{-1})b^{-1})=1.}
  37. ( Q ( a ) b ) - 1 = Q ( a - 1 ) b - 1 . \displaystyle{(Q(a)b)^{-1}=Q(a^{-1})b^{-1}.}
  38. - Q ( Q ( a ) b ) - 1 Q ( a ) c = - Q ( a ) - 1 Q ( b ) - 1 c . \displaystyle{-Q(Q(a)b)^{-1}Q(a)c=-Q(a)^{-1}Q(b)^{-1}c.}
  39. 𝔤 \mathfrak{g}
  40. 𝔤 \mathfrak{g}
  41. 𝔤 = 𝔨 𝔭 , \displaystyle{\mathfrak{g}=\mathfrak{k}\oplus\mathfrak{p},}
  42. 𝔨 \mathfrak{k}
  43. 𝔭 \mathfrak{p}
  44. 𝔭 \mathfrak{p}
  45. 𝔭 \mathfrak{p}
  46. 𝔭 \mathfrak{p}
  47. 𝔭 \mathfrak{p}
  48. 𝔭 \mathfrak{p}
  49. 𝔭 \mathfrak{p}
  50. 𝔭 \mathfrak{p}
  51. 𝔭 \mathfrak{p}
  52. 𝔨 \mathfrak{k}
  53. 𝔨 \mathfrak{k}
  54. [ a , b 2 , c ] = 2 [ a , b , c ] b . \displaystyle{[a,b^{2},c]=2[a,b,c]b.}
  55. ( [ b 2 , a , b ] , c ) = ( b 2 ( b a ) - b ( b 2 a ) , c ) = - ( b 2 , [ a , b , c ] ) \displaystyle{([b^{2},a,b],c)=(b^{2}(ba)-b(b^{2}a),c)=-(b^{2},[a,b,c])}
  56. ( [ b 2 , a , b ] , c ) = ( b , [ a , b 2 , c ] ) . \displaystyle{([b^{2},a,b],c)=(b,[a,b^{2},c]).}
  57. ( [ b 2 , a , b ] , c ) = 0 , \displaystyle{([b^{2},a,b],c)=0,}
  58. Q ( e a ) = e 2 L ( a ) . \displaystyle{Q(e^{a})=e^{2L(a)}.}
  59. e 2 a = Q ( e a ) 1 = e 2 L ( a ) 1 = e X 1 , \displaystyle{e^{2a}=Q(e^{a})1=e^{2L(a)}1=e^{X}\cdot 1,}
  60. 𝔭 \mathfrak{p}
  61. 𝔤 \mathfrak{g}
  62. 𝔨 \mathfrak{k}
  63. 𝔭 \mathfrak{p}
  64. 𝔨 \mathfrak{k}
  65. 𝔨 \mathfrak{k}
  66. 𝔨 \mathfrak{k}
  67. 𝔭 \mathfrak{p}
  68. 𝔞 \mathfrak{a}
  69. 𝔭 \mathfrak{p}
  70. 𝔭 \mathfrak{p}
  71. 𝔞 \mathfrak{a}
  72. 𝔭 \mathfrak{p}
  73. 𝔞 \mathfrak{a}
  74. E = i j E i j , \displaystyle{E=\bigoplus_{i\leq j}E_{ij},}
  75. 𝔞 \mathfrak{a}
  76. S = { g G | g E i j ( p , q ) ( i , j ) E p q } , \displaystyle{S=\{g\in G|gE_{ij}\subseteq\bigoplus_{(p,q)\geq(i,j)}E_{pq}\},}
  77. G = K A N . \displaystyle{G=KAN.}
  78. 𝔤 i j = { X 𝔤 : [ L ( a ) , X ] = 1 2 ( α i - α j ) X , for a = α i e i } . \displaystyle{\mathfrak{g}_{ij}=\{X\in\mathfrak{g}:[L(a),X]={1\over 2}(\alpha_% {i}-\alpha_{j})X,\,\,\,\mathrm{for}\,\,\,a=\sum\alpha_{i}e_{i}\}.}
  79. 𝔫 = i < j 𝔤 i j , 𝔤 i j = { L ( a ) + 2 [ L ( a ) , L ( e i ) ] : a E i j } . \displaystyle{\mathfrak{n}=\bigoplus_{i<j}\mathfrak{g}_{ij},\,\,\,\,\mathfrak{% g}_{ij}=\{L(a)+2[L(a),L(e_{i})]:a\in E_{ij}\}.}
  80. 𝔫 \mathfrak{n}
  81. [ L ( a ) , L ( a 2 ) ] = 0 \displaystyle{[L(a),L(a^{2})]=0}
  82. Q ( Q ( a ) b ) = Q ( a ) Q ( b ) Q ( a ) , Q ( a m ) = Q ( a ) m ( m 0 ) . \displaystyle{Q(Q(a)b)=Q(a)Q(b)Q(a),\,\,\,Q(a^{m})=Q(a)^{m}\,\,(m\geq 0).}
  83. Q ( a ) - 1 a = a - 1 , Q ( a - 1 ) = Q ( a ) - 1 . \displaystyle{Q(a)^{-1}a=a^{-1},\,\,\,Q(a^{-1})=Q(a)^{-1}.}
  84. ( Q ( a ) - 1 a ) a = a Q ( a ) - 1 a = L ( a ) Q ( a ) - 1 a = Q ( a ) - 1 a 2 = 1 , \displaystyle{(Q(a)^{-1}a)a=aQ(a)^{-1}a=L(a)Q(a)^{-1}a=Q(a)^{-1}a^{2}=1,}
  85. ( Q ( a ) b ) - 1 = Q ( a - 1 ) b - 1 . \displaystyle{(Q(a)b)^{-1}=Q(a^{-1})b^{-1}.}
  86. c - 1 = Q ( c ) - 1 c = Q ( a ) - 1 Q ( b ) - 1 b = Q ( a ) - 1 b - 1 . \displaystyle{c^{-1}=Q(c)^{-1}c=Q(a)^{-1}Q(b)^{-1}b=Q(a)^{-1}b^{-1}.}
  87. Q ( e a ) = e 2 L ( a ) \displaystyle{Q(e^{a})=e^{2L(a)}}
  88. Q ( g a ) = g Q ( a ) g t . \displaystyle{Q(ga)=gQ(a)g^{t}.}
  89. 𝔨 𝔭 \mathfrak{k}\oplus\mathfrak{p}
  90. 𝔨 i 𝔭 \mathfrak{k}\oplus i\mathfrak{p}
  91. a * = a , { a , a * , a } = a 3 . \displaystyle{\|a^{*}\|=\|a\|,\,\,\,\|\{a,a^{*},a\}\|=\|a\|^{3}.}
  92. a b a b . \displaystyle{\|ab\|\leq\|a\|\cdot\|b\|.}
  93. e a = a e = L ( a ) P ( 1 ) = P ( L ( a ) 1 ) = P ( a ) , \displaystyle{ea=ae=L(a)P(1)=P(L(a)1)=P(a),}
  94. T = E + i C T=E+iC
  95. S U ( 1 , 1 ) SU(1,1)
  96. S L ( 2 , 𝐑 ) SL(2,\mathbf{R})
  97. S L ( 2 , 𝐂 ) SL(2,\mathbf{C})
  98. S U ( 2 ) SU(2)
  99. E E
  100. A = E < s u b > 𝐂 = E + i E A=E<sub>\mathbf{C}=E+iE
  101. T = E + i C T=E+iC
  102. E E
  103. g = ( α β γ δ ) , \displaystyle{g=\begin{pmatrix}\alpha&\beta\\ \gamma&\delta\end{pmatrix},}
  104. g ( z ) = ( α z + β ) ( γ z + δ ) - 1 . \displaystyle{g(z)=(\alpha z+\beta)(\gamma z+\delta)^{-1}.}
  105. J = ( 0 1 - 1 0 ) . \displaystyle{J=\begin{pmatrix}0&1\\ -1&0\end{pmatrix}.}
  106. J = ( 1 0 - 1 1 ) ( 1 1 0 1 ) ( 1 0 - 1 1 ) . \displaystyle{J=\begin{pmatrix}1&0\\ -1&1\end{pmatrix}\begin{pmatrix}1&1\\ 0&1\end{pmatrix}\begin{pmatrix}1&0\\ -1&1\end{pmatrix}.}
  107. 𝐒𝐋 ( 2 , k ) = 𝐁 𝐁 J 𝐁 , \displaystyle{\mathbf{SL}(2,k)=\mathbf{B}\cup\mathbf{B}\cdot J\cdot\mathbf{B},}
  108. S L ( 2 , k ) SL(2,k)
  109. 𝐒𝐋 ( 2 , k ) = 𝐁 𝐁 J 𝐔 , \displaystyle{\mathbf{SL}(2,k)=\mathbf{B}\cup\mathbf{B}\cdot J\cdot\mathbf{U},}
  110. T ( β ) = ( 1 β 0 1 ) . \displaystyle{T(\beta)=\begin{pmatrix}1&\beta\\ 0&1\end{pmatrix}.}
  111. ( α 0 0 α - 1 ) = J T ( α - 1 ) J T ( α ) J T ( α - 1 ) . \displaystyle{\begin{pmatrix}\alpha&0\\ 0&\alpha^{-1}\end{pmatrix}=JT(\alpha^{-1})JT(\alpha)JT(\alpha^{-1}).}
  112. S L ( 2 , k ) SL(2,k)
  113. T ( β ) T(β)
  114. β T ( β ) β↦T(β)
  115. D ( 1 ) = J D(−1)=J
  116. D ( α ) D(α)
  117. S L ( 2 , k ) SL(2,k)
  118. T ( β ) T(β)
  119. m 0 m≥0
  120. S L ( 2 , k ) SL(2,k)
  121. S L ( 2 , k ) SL(2,k)
  122. Φ / Δ Φ/Δ
  123. D ( α ) D(α)
  124. 𝐁 𝐁 J 𝐁 \mathbf{B}∪\mathbf{B}J\mathbf{B}
  125. T ( β ) T(β)
  126. D ( α ) D(α)
  127. S L ( 2 , k ) SL(2,k)
  128. ± I ±I
  129. S L ( 2 , k ) SL(2,k)
  130. 𝐊 \mathbf{K}
  131. 𝐁 \mathbf{B}
  132. 𝐊 \mathbf{K}
  133. 𝐁 \mathbf{B}
  134. 𝐊𝐁 = S L ( 2 , k ) \mathbf{KB}=SL(2,k)
  135. 𝐊 \mathbf{K}
  136. S L ( 2 , k ) SL(2,k)
  137. J = k b J=kb
  138. k k
  139. 𝐊 \mathbf{K}
  140. b b
  141. 𝐁 \mathbf{B}
  142. 𝐔 \mathbf{U}
  143. 𝐋 \mathbf{L}
  144. S L ( 2 , k ) = 𝐊𝐔 SL(2,k)=\mathbf{KU}
  145. S L ( 2 , k ) / 𝐊 𝐔 / 𝐔 𝐊 SL(2,k)/\mathbf{K}≅\mathbf{U}/\mathbf{U}∩\mathbf{K}
  146. 𝐊 = S L ( 2 , k ) \mathbf{K}=SL(2,k)
  147. 𝐂 a a \mathbf{C}aa
  148. 𝐂 a a \mathbf{C}aa
  149. p ( t ) p(t)
  150. p ( a ) p(a)
  151. 𝐂 a a \mathbf{C}aa
  152. A A
  153. 𝐂 a a \mathbf{C}aa
  154. p p
  155. a a
  156. a a
  157. a a
  158. F F
  159. G G
  160. G G
  161. F G F∘G
  162. a a
  163. F F
  164. G ( a ) G(a)
  165. F ( G ( a ) ) = ( F G ) ( a ) F(G(a))=(F∘G)(a)
  166. 𝐂 a a \mathbf{C}aa
  167. A A
  168. e e
  169. E E
  170. E = E 1 ( e ) E 1 / 2 ( e ) E 0 ( e ) , A = A 1 ( e ) A 1 / 2 ( e ) A 0 ( e ) . \displaystyle{E=E_{1}(e)\oplus E_{1/2}(e)\oplus E_{0}(e),\,\,\,\,A=A_{1}(e)% \oplus A_{1/2}(e)\oplus A_{0}(e).}
  171. S L ( 2 , 𝐂 ) SL(2,\mathbf{C})
  172. g ( z e x 1 / 2 x 0 ) = α z + β γ z + δ e ( γ z + δ ) - 1 x 1 / 2 x 0 - ( γ z + δ ) - 1 P 0 ( x 1 / 2 2 ) . \displaystyle{g(ze\oplus x_{1/2}\oplus x_{0})={\alpha z+\beta\over\gamma z+% \delta}\cdot e\oplus(\gamma z+\delta)^{-1}x_{1/2}\oplus x_{0}-(\gamma z+\delta% )^{-1}P_{0}(x_{1/2}^{2}).}
  173. S L ( 2 , 𝐂 ) SL(2,\mathbf{C})
  174. S L ( 2 , 𝐂 ) SL(2,\mathbf{C})
  175. A A
  176. C ( z ) = i 1 + z 1 - z = - i + 2 i 1 - z \displaystyle{C(z)=i{1+z\over 1-z}=-i+{2i\over 1-z}}
  177. P ( w ) = w - i w + i = 1 - 2 i w + i . \displaystyle{P(w)={w-i\over w+i}=1-{2i\over w+i}.}
  178. T P ( T ) T↦P(T)
  179. D D
  180. T T
  181. T T
  182. T + i I T+iI
  183. ( T + i I ) x 2 = ( T - i I ) x 2 + 4 ( Im ( T ) x , x ) . \displaystyle{\|(T+iI)x\|^{2}=\|(T-iI)x\|^{2}+4(\mathrm{Im}(T)x,x).}
  184. y = ( T + i I ) x y=(T+iI)x
  185. y 2 = P ( T ) y 2 + 4 ( Im ( T ) x , x ) . \displaystyle{\|y\|^{2}=\|P(T)y\|^{2}+4(\mathrm{Im}(T)x,x).}
  186. I - P ( T ) * P ( T ) = 4 ( T * - i I ) - 1 [ Im T ] ( T + i I ) - 1 \displaystyle{I-P(T)^{*}P(T)=4(T^{*}-iI)^{-1}[\mathrm{Im}\,T](T+iI)^{-1}}
  187. Im C ( U ) = ( 2 i ) - 1 [ C ( U ) - C ( U ) * ] = ( 1 - U * ) - 1 [ I - U * U ] ( I - U ) - 1 . \displaystyle{\mathrm{Im}\,C(U)=(2i)^{-1}[C(U)-C(U)^{*}]=(1-U^{*})^{-1}[I-U^{*% }U](I-U)^{-1}.}
  188. Q ( 1 - u * ) Q ( C ( u ) + C ( u * ) ) Q ( 1 - u ) = - 4 B ( u * , u ) \displaystyle{Q(1-u^{*})Q(C(u)+C(u^{*}))Q(1-u)=-4B(u^{*},u)}
  189. 4 Q ( Im a ) = Q ( a * - i ) B ( P ( a ) * , P ( a ) ) Q ( a + i ) , \displaystyle{4Q(\mathrm{Im}\,a)=Q(a^{*}-i)B(P(a)^{*},P(a))Q(a+i),}
  190. B ( x , y ) B(x,y)
  191. B ( x , y ) = I 2 R ( x , y ) + Q ( x ) Q ( y ) B(x,y)=I−2R(x,y)+Q(x)Q(y)
  192. R ( x , y ) = L L ( x ) , L ( y ) + L ( x y ) R(x,y)=LL(x),L(y)+L(xy)
  193. 1 u 1−u
  194. a a
  195. C C
  196. L ( a ) L(a)
  197. a a
  198. a + i a+i
  199. E E
  200. T T
  201. D D
  202. Q ( x ) Q(x)
  203. x x
  204. a a
  205. b b
  206. Q ( a ) Q ( a - 1 + b - 1 ) Q ( b ) = Q ( a + b ) . \displaystyle{Q(a)Q(a^{-1}+b^{-1})Q(b)=Q(a+b).}
  207. R ( a , b ) = 2 Q ( a ) Q ( a - 1 , b ) = 2 Q ( a , b - 1 ) Q ( b ) \displaystyle{R(a,b)=2Q(a)Q(a^{-1},b)=2Q(a,b^{-1})Q(b)}
  208. B ( a , b ) = Q ( a ) Q ( a - 1 - b ) = Q ( b - 1 - a ) Q ( b ) . \displaystyle{B(a,b)=Q(a)Q(a^{-1}-b)=Q(b^{-1}-a)Q(b).}
  209. b b
  210. a = 1 x a=1−x
  211. b = 1 y b=1−y
  212. Q ( 1 - x ) Q ( C ( x ) + C ( y ) ) Q ( 1 - y ) = - 4 B ( x , y ) . \displaystyle{Q(1-x)Q(C(x)+C(y))Q(1-y)=-4B(x,y).}
  213. C ( x ) + C ( y ) = - 2 i ( 1 - a - 1 - b - 1 ) , \displaystyle{C(x)+C(y)=-2i(1-a^{-1}-b^{-1}),}
  214. Q ( a ) Q ( 1 - a - 1 - b - 1 ) Q ( b ) = B ( 1 - a , 1 - b ) . \displaystyle{Q(a)Q(1-a^{-1}-b^{-1})Q(b)=B(1-a,1-b).}
  215. Q ( a ) Q ( b ) + Q ( a + b ) 2 L ( a ) Q ( b ) 2 Q ( a ) L ( b ) Q(a)Q(b)+Q(a+b)−2L(a)Q(b)−2Q(a)L(b)
  216. 2 L ( a ) L ( b ) + 2 L ( b ) L ( a ) 2 L ( a b ) 2 L ( a ) Q ( b ) 2 Q ( a ) L ( b ) + Q ( a ) Q ( b ) + Q ( a ) + Q ( b ) 2L(a)L(b)+2L(b)L(a)−2L(ab)−2L(a)Q(b)−2Q(a)L(b)+Q(a)Q(b)+Q(a)+Q(b)
  217. ½ Q Q ( a + b ) Q ( a ) Q ( b ) = L ( a ) L ( b ) + L ( b ) L ( a ) L ( a b ) ½QQ(a+b)−Q(a)−Q(b)=L(a)L(b)+L(b)L(a)−L(ab)
  218. a a
  219. D D
  220. a 1 a−1
  221. D D
  222. a λ a−λ
  223. a λ a−λ
  224. g a ga
  225. g g
  226. S U ( 1 , 1 ) SU(1,1)
  227. D D
  228. g g
  229. D D
  230. D D
  231. S S
  232. g g
  233. D D
  234. E E
  235. g u gu
  236. g u gu
  237. S S
  238. S U ( 1 , 1 ) SU(1,1)
  239. D D
  240. D D
  241. D D
  242. z z z↦−z
  243. f f
  244. D D
  245. f ( 0 ) = 0 f(0)=0
  246. I I
  247. f f
  248. f f
  249. i i
  250. ψ ψ
  251. A * A*
  252. z z
  253. D D
  254. w w
  255. n n
  256. g g
  257. D D
  258. I I
  259. D D
  260. S S
  261. g g
  262. U ( n ) × U ( n ) U(n)×U(n)
  263. U ( n , n ) U(n,n)
  264. U ( n ) × U ( n ) U(n)×U(n)
  265. U ( n , n ) U(n,n)
  266. W W
  267. K K
  268. U ( n , n ) U(n,n)
  269. K K
  270. K K
  271. n n
  272. K K
  273. n n
  274. U ( n , n ) U(n,n)
  275. g g
  276. U ( n , n ) U(n,n)
  277. g W gW
  278. W W
  279. G T = K T A T K T . \displaystyle{G_{T}=K_{T}A_{T}K_{T}.}
  280. z z
  281. D D
  282. u u
  283. L ( a , b ) = ad [ a , σ ( b ) ] \displaystyle{L(a,b)=\mathrm{ad}\,[a,\sigma(b)]}
  284. 𝔤 - 1 \mathfrak{g}_{-1}
  285. 𝔤 0 \mathfrak{g}_{0}
  286. 𝔥 \mathfrak{h}
  287. 𝔤 ± 1 \mathfrak{g}_{\pm 1}
  288. [ ( a 1 , T 1 , b 1 ) , ( a 2 , T 2 , b 2 ) ] = ( T 1 a 2 - T 2 a 1 , [ T 1 , T 2 ] + L ( a 1 , b 2 ) - L ( a 2 , b 1 ) , T 2 * b 1 - T 1 * b 2 ) \displaystyle{[(a_{1},T_{1},b_{1}),(a_{2},T_{2},b_{2})]=(T_{1}a_{2}-T_{2}a_{1}% ,[T_{1},T_{2}]+L(a_{1},b_{2})-L(a_{2},b_{1}),T_{2}^{*}b_{1}-T_{1}^{*}b_{2})}
  289. σ ( a , T , b ) = ( b , - T * , a ) . \displaystyle{\sigma(a,T,b)=(b,-T^{*},a).}
  290. B ( ( a 1 , T 1 , b 1 ) , ( a 2 , T 2 , b 2 ) ) = ( a 1 , b 2 ) + ( b 1 , a 2 ) + β ( T 1 , T 2 ) , \displaystyle{B((a_{1},T_{1},b_{1}),(a_{2},T_{2},b_{2}))=(a_{1},b_{2})+(b_{1},% a_{2})+\beta(T_{1},T_{2}),}
  291. β ( L ( a , b ) , L ( c , d ) ) = ( L ( a , b ) c , d ) = ( L ( c , d ) a , b ) . \displaystyle{\beta(L(a,b),L(c,d))=(L(a,b)c,d)=(L(c,d)a,b).}
  292. 𝔤 \mathfrak{g}
  293. 𝔤 - 1 \mathfrak{g}_{-1}
  294. 𝔤 0 \mathfrak{g}_{0}
  295. 𝔤 ± 1 \mathfrak{g}_{\pm 1}
  296. 𝔰 𝔩 2 \mathfrak{sl}_{2}
  297. L 0 ( a , b ) c = L ( a , b ) c \displaystyle{L_{0}(a,b)c=L(a,b)c}

Symmetric_decreasing_rearrangement.html

  1. A A
  2. A A
  3. A * A^{*}
  4. A * = { x 𝐑 n : ω n | x | n < | A | } , A^{*}=\{x\in\mathbf{R}^{n}:\,\omega_{n}\cdot|x|^{n}<|A|\},
  5. ω n \omega_{n}
  6. | A | |A|
  7. A A
  8. A A
  9. f f
  10. f * ( x ) = 0 𝕀 { y : f ( y ) > t } * ( x ) d t . f^{*}(x)=\int_{0}^{\infty}\mathbb{I}_{\{y:f(y)>t\}^{*}}(x)\,dt.
  11. f * ( x ) f^{*}(x)
  12. { y : f ( y ) > t } \{y:f(y)>t\}
  13. g ( x ) = 0 𝕀 { y : g ( y ) > t } ( x ) d t , g(x)=\int_{0}^{\infty}\mathbb{I}_{\{y:g(y)>t\}}(x)\,dt,
  14. g g
  15. 𝕀 A * = 𝕀 A * \mathbb{I}_{A}^{*}=\mathbb{I}_{A^{*}}
  16. f * f^{*}
  17. f f
  18. | { x : f * ( x ) > t } | = | { x : f ( x ) > t } | . |\{x:f^{*}(x)>t\}|=|\{x:f(x)>t\}|.
  19. f f
  20. L p L^{p}
  21. f L p = f * L p . \|f\|_{L^{p}}=\|f^{*}\|_{L^{p}}.
  22. f g f * g * . \int fg\leq\int f^{*}g^{*}.
  23. 1 p < 1\leq p<\infty
  24. f W 1 , p f\in W^{1,p}
  25. f * p f p . \|\nabla f^{*}\|_{p}\leq\|\nabla f\|_{p}.
  26. L p L^{p}
  27. f g f * g * f\leq g\Rightarrow f^{*}\leq g^{*}
  28. f - g L p f * - g * L p . \|f-g\|_{L^{p}}\geq\|f^{*}-g^{*}\|_{L^{p}}.
  29. p = 1 p=1

Symmetric_probability_distribution.html

  1. x 0 x_{0}
  2. f ( x 0 - δ ) = f ( x 0 + δ ) f(x_{0}-\delta)=f(x_{0}+\delta)
  3. δ , \delta,
  4. x 0 x_{0}
  5. x 0 x_{0}
  6. x 0 x_{0}
  7. x x
  8. ( x - x 0 ) 2 k (x-x_{0})^{2k}
  9. k k
  10. x = x 0 - δ x=x_{0}-\delta
  11. x = x 0 + δ x=x_{0}+\delta
  12. x 0 x_{0}
  13. | x - x 0 | |x-x_{0}|
  14. x 0 . x_{0}.

Symmetrizable_compact_operator.html

  1. S K = K * S . \displaystyle{SK=K^{*}S.}
  2. ( x , y ) S = ( S x , y ) . \displaystyle{(x,y)_{S}=(Sx,y)}.
  3. S K n = ( K * ) n S . \displaystyle{SK^{n}=(K^{*})^{n}S.}
  4. K x S | 2 n K 2 n x S . \displaystyle{\|Kx\|_{S}|^{2^{n}}\leq\|K^{2^{n}}x\|_{S}.}
  5. K x S lim sup n K ( S x 2 ) 1 / 2 n = K \displaystyle{\|Kx\|_{S}\leq\limsup_{n\rightarrow\infty}\|K\|(\|S\|\|x\|^{2})^% {1/2^{n}}=\|K\|}
  6. A ε = ( R + ε I ) - 1 S K ( R + ε I ) - 1 . \displaystyle{A_{\varepsilon}=(R+\varepsilon I)^{-1}SK(R+\varepsilon I)^{-1}.}
  7. A ε 2 2 = ( R 2 ( R + ε I ) - 2 K , K R 2 ( R + ε I ) - 2 ) 2 K 2 2 , \displaystyle{\|A_{\varepsilon}\|_{2}^{2}=(R^{2}(R+\varepsilon I)^{-2}K,KR^{2}% (R+\varepsilon I)^{-2})_{2}\leq\|K\|_{2}^{2},}
  8. R K = A R . \displaystyle{RK=AR.}
  9. U K S U * = A . \displaystyle{UK_{S}U^{*}=A.}
  10. tr K 2 = λ n 2 , det ( I - z K 2 ) = n = 1 ( 1 - z λ n 2 ) . \displaystyle{\mathrm{tr}\,K^{2}=\sum\lambda_{n}^{2},\,\,\,\det(I-zK^{2})=% \prod_{n=1}^{\infty}(1-z\lambda_{n}^{2}).}
  11. tr ( P N K * P N ) 2 = m = 1 N λ m 2 dim H m , det [ P N - z ( P N K * P N ) 2 ] = m = 1 N ( 1 - z λ m 2 ) dim H m . \displaystyle{\mathrm{tr}\,(P_{N}K^{*}P_{N})^{2}=\sum_{m=1}^{N}\lambda_{m}^{2}% \cdot\mathrm{dim}\,H_{m},\,\,\,\mathrm{det}\,[P_{N}-z(P_{N}K^{*}P_{N})^{2}]=% \prod_{m=1}^{N}(1-z\lambda_{m}^{2})^{\mathrm{dim}\,H_{m}}.}

Symmetry_protected_topological_order.html

  1. H d + 1 [ G , U ( 1 ) ] H^{d+1}[G,U(1)]
  2. k = 1 d H k [ G , i T O d + 1 - k ] \oplus_{k=1}^{d}H^{k}[G,iTO^{d+1-k}]
  3. i T O d + 1 iTO^{d+1}
  4. H d + 1 [ G , U ( 1 ) ] k = 1 d H k [ G , i T O d + 1 - k ] H^{d+1}[G,U(1)]\oplus_{k=1}^{d}H^{k}[G,iTO^{d+1-k}]
  5. Z 2 T Z_{2}^{T}
  6. 0
  7. 0
  8. Z Z
  9. 0
  10. Z 2 Z_{2}
  11. i T O d + 1 iTO^{d+1}
  12. U ( 1 ) Z 2 T U(1)\rtimes Z_{2}^{T}
  13. Z 2 Z_{2}
  14. Z 2 Z_{2}
  15. 2 Z 2 + Z 2 2Z_{2}+Z_{2}
  16. Z Z 2 + Z Z\oplus Z_{2}+Z
  17. Z 2 T Z_{2}^{T}
  18. Z 2 Z_{2}
  19. 0
  20. Z 2 + Z 2 Z_{2}+Z_{2}
  21. 0
  22. Z n Z_{n}
  23. 0
  24. Z n Z_{n}
  25. 0
  26. Z n + Z n Z_{n}+Z_{n}
  27. U ( 1 ) U(1)
  28. 0
  29. Z Z
  30. 0
  31. Z + Z Z+Z
  32. S O ( 3 ) SO(3)
  33. Z 2 Z_{2}
  34. Z Z
  35. 0
  36. Z 2 Z_{2}
  37. S O ( 3 ) × Z 2 T SO(3)\times Z_{2}^{T}
  38. 2 Z 2 2Z_{2}
  39. Z 2 Z_{2}
  40. 3 Z 2 + Z 2 3Z_{2}+Z_{2}
  41. 2 Z 2 2Z_{2}
  42. Z 2 × Z 2 × Z 2 T Z_{2}\times Z_{2}\times Z_{2}^{T}
  43. 4 Z 2 4Z_{2}
  44. 6 Z 2 6Z_{2}
  45. 9 Z 2 + Z 2 9Z_{2}+Z_{2}
  46. 12 Z 2 + 2 Z 2 12Z_{2}+2Z_{2}
  47. H d + 1 [ G , U ( 1 ) ] H^{d+1}[G,U(1)]
  48. k = 1 d H k [ G , i T O d + 1 - k ] \oplus_{k=1}^{d}H^{k}[G,iTO^{d+1-k}]
  49. ( G H , G Ψ , H 2 [ G Ψ , U ( 1 ) ] ) (G_{H},G_{\Psi},H^{2}[G_{\Psi},U(1)])
  50. G H G_{H}
  51. G Ψ G_{\Psi}
  52. H 2 [ G Ψ , U ( 1 ) ] H^{2}[G_{\Psi},U(1)]
  53. G Ψ G_{\Psi}
  54. H 2 [ G , U ( 1 ) ] H^{2}[G,U(1)]
  55. G G
  56. G Ψ = G H G_{\Psi}=G_{H}
  57. G H G_{H}

Synchysite.html

  1. 3 2 / m 3\;2/m
  2. R 3 ¯ m R\;\overline{3}\;m
  3. 2 / m 2/m
  4. C 2 / c C\;2/c
  5. 2 / m 2/m
  6. C 2 / c C\;2/c
  7. 2 / m 2/m
  8. C 2 / c C\;2/c

System_of_bilinear_equations.html

  1. y T A i x = g i y^{T}A_{i}x=g_{i}
  2. i = 1 , 2 , , r i=1,2,\ldots,r
  3. r r
  4. A i A_{i}
  5. g i g_{i}
  6. a x 1 x 2 + b x 1 y 2 + c x 2 y 1 + d y 1 y 2 = α e x 1 x 2 + f x 1 y 2 + g x 2 y 1 + h y 1 y 2 = β \begin{aligned}\displaystyle ax_{1}x_{2}+bx_{1}y_{2}+cx_{2}y_{1}+dy_{1}y_{2}&% \displaystyle=&\displaystyle\alpha\\ \displaystyle ex_{1}x_{2}+fx_{1}y_{2}+gx_{2}y_{1}+hy_{1}y_{2}&\displaystyle=&% \displaystyle\beta\end{aligned}
  7. [ a b c d e f g h ] [ x 1 x 2 x 1 y 2 y 1 x 2 y 1 y 2 ] = [ α β ] \begin{bmatrix}a&b&c&d\\ e&f&g&h\end{bmatrix}\begin{bmatrix}x_{1}x_{2}\\ x_{1}y_{2}\\ y_{1}x_{2}\\ y_{1}y_{2}\end{bmatrix}=\begin{bmatrix}\alpha\\ \beta\end{bmatrix}
  8. m a t ( [ x 1 x 2 x 1 y 2 y 1 x 2 y 1 y 2 ] ) = [ x 1 x 2 x 1 y 2 y 1 x 2 y 1 y 2 ] = [ x 1 y 1 ] [ x 2 y 2 ] mat(\begin{bmatrix}x_{1}x_{2}\\ x_{1}y_{2}\\ y_{1}x_{2}\\ y_{1}y_{2}\end{bmatrix})=\begin{bmatrix}x_{1}x_{2}&x_{1}y_{2}\\ y_{1}x_{2}&y_{1}y_{2}\end{bmatrix}=\begin{bmatrix}x_{1}\\ y_{1}\end{bmatrix}\begin{bmatrix}x_{2}&y_{2}\end{bmatrix}
  9. m × n m\times n
  10. A A
  11. U U
  12. V V
  13. SL ( ) m \mbox{SL}~{}_{m}(\mathbb{Z})
  14. SL ( ) n \mbox{SL}~{}_{n}(\mathbb{Z})
  15. U A V = D UAV=D
  16. D D
  17. D = [ d 1 0 0 0 0 d 2 0 0 d s 0 0 0 0 0 ] m × n D=\begin{bmatrix}d_{1}&0&0&\ldots&0\\ 0&d_{2}&0&\ldots&0\\ \vdots&&&d_{s}&0&\\ 0&0&0&\ldots&0\\ \vdots&\vdots&\vdots&\vdots&\vdots\end{bmatrix}_{m\times n}
  18. d i > 0 d_{i}>0
  19. d i | d i + 1 d_{i}|d_{i+1}
  20. i = 1 , 2 , , s - 1 i=1,2,\ldots,s-1
  21. A 𝐱 = 𝐛 A\,\textbf{x}=\,\textbf{b}
  22. D 𝐲 = 𝐜 D\,\textbf{y}=\,\textbf{c}
  23. V 𝐲 = 𝐱 V\,\textbf{y}=\,\textbf{x}
  24. 𝐜 = U 𝐛 \,\textbf{c}=U\,\textbf{b}
  25. D 𝐲 = 𝐜 D\,\textbf{y}=\,\textbf{c}
  26. D D
  27. D 𝐲 = 𝐜 D\,\textbf{y}=\,\textbf{c}
  28. 𝐱 = V 𝐲 \,\textbf{x}=V\,\textbf{y}
  29. D 𝐲 = 𝐜 D\,\textbf{y}=\,\textbf{c}
  30. 𝐲 = [ a 1 b 1 s t ] \,\textbf{y}=\begin{bmatrix}a_{1}\\ b_{1}\\ s\\ t\end{bmatrix}
  31. s , t s,t\in\mathbb{Z}
  32. D 𝐲 = 𝐜 D\,\textbf{y}=\,\textbf{c}
  33. A 𝐱 = 𝐛 A\,\textbf{x}=\,\textbf{b}
  34. V 𝐲 V\,\textbf{y}
  35. V V
  36. V = [ a 11 a 12 a 13 a 14 a 21 a 22 a 23 a 24 a 31 a 32 a 33 a 34 a 41 a 42 a 43 a 44 ] = [ A 1 B 1 C 1 D 1 ] V=\begin{bmatrix}a_{11}&a_{12}&a_{13}&a_{14}\\ a_{21}&a_{22}&a_{23}&a_{24}\\ a_{31}&a_{32}&a_{33}&a_{34}\\ a_{41}&a_{42}&a_{43}&a_{44}\end{bmatrix}=\begin{bmatrix}A_{1}&B_{1}\\ C_{1}&D_{1}\end{bmatrix}
  37. 𝐱 \,\textbf{x}
  38. M = m a t ( 𝐱 ) = [ a 11 a 1 + a 12 b 1 + a 13 s + a 14 t a 31 a 1 + a 32 b 1 + a 33 s + a 34 t a 21 a 1 + a 22 b 1 + a 23 s + a 24 t a 41 a 1 + a 42 b 1 + a 43 s + a 44 t ] M=mat(\,\textbf{x})=\begin{bmatrix}a_{11}a_{1}+a_{12}b_{1}+a_{13}s+a_{14}t&a_{% 31}a_{1}+a_{32}b_{1}+a_{33}s+a_{34}t\\ a_{21}a_{1}+a_{22}b_{1}+a_{23}s+a_{24}t&a_{41}a_{1}+a_{42}b_{1}+a_{43}s+a_{44}% t\end{bmatrix}
  39. M M

Szyszkowski_equation.html

  1. σ m = σ w ( 1 - 0.411 log ( 1 + x a ) ) \sigma_{m}=\sigma_{w}\left(1-0.411\log\left(1+\frac{x}{a}\right)\right)
  2. a \displaystyle a
  3. γ = γ 0 - R T ω ln ( 1 + K a d c ) \gamma=\gamma_{0}-\frac{RT}{\omega}\ln(1+K_{ad}c)

Šidák_correction.html

  1. H 1 , , H m H_{1},...,H_{m}
  2. H n u l l : H_{null}:
  3. H i H_{i}
  4. H a l t e r n a t i v e : H_{alternative}:
  5. H i H_{i}
  6. α \alpha
  7. H n u l l H_{null}
  8. α \alpha
  9. H i H_{i}
  10. t i t_{i}
  11. t i t_{i}
  12. H n u l l H_{null}
  13. α S I D = 1 - ( 1 - α ) 1 m \alpha_{SID}=1-(1-\alpha)^{\frac{1}{m}}
  14. H n u l l H_{null}
  15. α S I D \alpha_{SID}
  16. α 1 \alpha_{1}
  17. 1 - ( 1 - α 1 ) n 1-(1-\alpha_{1})^{n}
  18. α \alpha
  19. α 1 \alpha_{1}
  20. α 1 = 1 - ( 1 - α ) 1 / n . \alpha_{1}=1-(1-\alpha)^{1/n}.
  21. 1 - 0.95 0.025 1-\sqrt{0.95}\approx 0.025
  22. n 1 n\geq 1
  23. α / n 1 - ( 1 - α ) 1 / n \alpha/n\leq 1-(1-\alpha)^{1/n}

T-coloring.html

  1. G = ( V , E ) G=(V,E)
  2. c : V ( G ) c:V(G)\rightarrow\mathbb{N}
  3. ( u , w ) E ( G ) | c ( u ) - c ( w ) | T (u,w)\in E(G)\Rightarrow\left|c(u)-c(w)\right|\notin T
  4. c ¯ \overline{c}
  5. c ¯ ( v ) = s + 1 - c ( v ) \overline{c}(v)=s+1-c(v)
  6. χ T ( G ) \chi_{T}(G)
  7. χ T ( G ) = χ ( G ) \chi_{T}(G)=\chi(G)
  8. χ T ( G ) χ ( G ) \chi_{T}(G)\geq\chi(G)
  9. χ ( G ) = k \chi(G)=k
  10. r = m a x ( T ) r=max(T)
  11. c : V ( G ) c:V(G)\rightarrow\mathbb{N}
  12. d : V ( G ) d:V(G)\rightarrow\mathbb{N}
  13. d ( v ) = ( r + 1 ) c ( v ) d(v)=(r+1)c(v)
  14. | d ( u ) - d ( w ) | = | ( r + 1 ) c ( u ) - ( r + 1 ) c ( w ) | \left|d(u)-d(w)\right|=\left|(r+1)c(u)-(r+1)c(w)\right|
  15. = ( r + 1 ) | c ( u ) - c ( w ) | r + 1 =(r+1)\left|c(u)-c(w)\right|\geq r+1
  16. | d ( u ) - d ( w ) | T \left|d(u)-d(w)\right|\notin T
  17. χ T ( G ) k = χ ( G ) \chi_{T}(G)\leq k=\chi(G)
  18. χ T ( G ) = χ ( G ) \chi_{T}(G)=\chi(G)
  19. \in
  20. ω \omega
  21. ω \omega
  22. \leq
  23. \leq
  24. \leq
  25. χ \chi
  26. χ T ( G ) = χ ( G ) \chi_{T}(G)=\chi(G)
  27. \leq
  28. χ \chi
  29. χ T ( G ) = χ ( G ) \chi_{T}(G)=\chi(G)

Table_of_congruences.html

  1. 2 n - 1 1 ( mod n ) 2^{n-1}\equiv 1\;\;(\mathop{{\rm mod}}n)\,\!
  2. 2 p - 1 1 ( mod p 2 ) 2^{p-1}\equiv 1\;\;(\mathop{{\rm mod}}p^{2})\,\!
  3. F n - ( < m t p l > n 5 ) 0 ( mod n ) F_{n-\left(\frac{<}{m}tpl>{{n}}{{5}}\right)}\equiv 0\;\;(\mathop{{\rm mod}}n)
  4. F p - ( < m t p l > p 5 ) 0 ( mod p 2 ) F_{p-\left(\frac{<}{m}tpl>{{p}}{{5}}\right)}\equiv 0\;\;(\mathop{{\rm mod}}p^{% 2})
  5. ( 2 n - 1 n - 1 ) 1 ( mod n 3 ) {2n-1\choose n-1}\equiv 1\;\;(\mathop{{\rm mod}}n^{3})
  6. ( 2 p - 1 p - 1 ) 1 ( mod p 4 ) , {2p-1\choose p-1}\equiv 1\;\;(\mathop{{\rm mod}}p^{4}),
  7. ( n - 1 ) ! - 1 ( mod n ) (n-1)!\ \equiv\ -1\;\;(\mathop{{\rm mod}}n)
  8. ( p - 1 ) ! - 1 ( mod p 2 ) (p-1)!\ \equiv\ -1\;\;(\mathop{{\rm mod}}p^{2})

Tangent_space_to_a_functor.html

  1. k [ ϵ ] / ( ϵ ) 2 k[\epsilon]/(\epsilon)^{2}
  2. ( 𝔪 X , p / 𝔪 X , p 2 ) * (\mathfrak{m}_{X,p}/\mathfrak{m}_{X,p}^{2})^{*}
  3. 𝒪 p k [ ϵ ] / ( ϵ ) 2 \mathcal{O}_{p}\to k[\epsilon]/(\epsilon)^{2}
  4. δ p v : u u ( p ) + ϵ v ( u ) , v 𝒪 p * . \delta_{p}^{v}:u\mapsto u(p)+\epsilon v(u),\quad v\in\mathcal{O}_{p}^{*}.
  5. p F ( k ) p\in F(k)
  6. π : F ( k [ ϵ ] / ( ϵ ) 2 ) F ( k ) \pi:F(k[\epsilon]/(\epsilon)^{2})\to F(k)
  7. F = Hom Spec k ( Spec - , X ) F=\operatorname{Hom}_{\operatorname{Spec}k}(\operatorname{Spec}-,X)
  8. π - 1 ( p ) \pi^{-1}(p)
  9. T X = X ( k [ ϵ ] / ( ϵ ) 2 ) T_{X}=X(k[\epsilon]/(\epsilon)^{2})
  10. f : X Y f:X\to Y
  11. f # ( δ p v ) = δ f ( p ) d f p ( v ) f^{\#}(\delta_{p}^{v})=\delta_{f(p)}^{df_{p}(v)}
  12. T X T Y T_{X}\to T_{Y}

Tangential_polygon.html

  1. x 1 + x 2 = a 1 , x 2 + x 3 = a 2 , , x n + x 1 = a n x_{1}+x_{2}=a_{1},\quad x_{2}+x_{3}=a_{2},\quad\ldots,\quad x_{n}+x_{1}=a_{n}
  2. a 1 , , a n a_{1},\dots,a_{n}
  3. r = K s = 2 K i = 1 n a i r=\frac{K}{s}=\frac{2K}{\sum_{i=1}^{n}a_{i}}

Tate_algebra.html

  1. k [ [ t 1 , , t n ] ] k[[t_{1},...,t_{n}]]
  2. a I t I \sum a_{I}t^{I}
  3. | a I | 0 |a_{I}|\to 0
  4. I I\to\infty
  5. f = a I t I f=\sum a_{I}t^{I}
  6. f = max I | a I | \|f\|=\max_{I}|a_{I}|

Tate_curve.html

  1. [ [ q ] ] \mathbb{Z}[[q]]
  2. y 2 + x y = x 3 + a 4 x + a 6 y^{2}+xy=x^{3}+a_{4}x+a_{6}
  3. - a 4 = 5 n n 3 q n 1 - q n = 5 q + 45 q 2 + 140 q 3 + -a_{4}=5\sum_{n}\frac{n^{3}q^{n}}{1-q^{n}}=5q+45q^{2}+140q^{3}+\cdots
  4. - a 6 = n 7 n 5 + 5 n 3 12 × q n 1 - q n = q + 23 q 2 + 154 q 3 + -a_{6}=\sum_{n}\frac{7n^{5}+5n^{3}}{12}\times\frac{q^{n}}{1-q^{n}}=q+23q^{2}+1% 54q^{3}+\cdots
  5. x ( w ) = - y ( w ) - y ( w - 1 ) x(w)=-y(w)-y(w^{-1})
  6. y ( w ) = m Z ( t m w ) 2 ( 1 - t m w ) 3 + m 1 t m w ( 1 - t m w ) 2 y(w)=\sum_{m\in Z}\frac{(t^{m}w)^{2}}{(1-t^{m}w)^{3}}+\sum_{m\geq 1}\frac{t^{m% }w}{(1-t^{m}w)^{2}}

Taurochenodeoxycholate_6α-hydroxylase.html

  1. \rightleftharpoons
  2. \rightleftharpoons

Tax_amortization_benefit.html

  1. F M V = V B A B * T A B f a c t o r FMV\,=\,{VBAB*TAB_{factor}}
  2. T A B f a c t o r = 1 [ 1 - t n * ( 1 k - 1 ( k * ( 1 + k ) n ) ) ] TAB_{factor}\,=\,{1\over[1-{t\over n}*({1\over k}-{1\over(k*(1+k)^{n})})]}

Taxoid_14beta-hydroxylase.html

  1. \rightleftharpoons

Taxoid_7beta-hydroxylase.html

  1. \rightleftharpoons

Telephone_number_(mathematics).html

  1. T ( n ) = T ( n - 1 ) + ( n - 1 ) T ( n - 2 ) , T(n)=T(n-1)+(n-1)T(n-2),
  2. T ( n ) = k = 0 n / 2 ( n 2 k ) ( 2 k - 1 ) ! ! = k = 0 n / 2 n ! 2 k ( n - 2 k ) ! k ! . T(n)=\sum_{k=0}^{\lfloor n/2\rfloor}{\left({{n}\atop{2k}}\right)}(2k-1)!!=\sum% _{k=0}^{\lfloor n/2\rfloor}\frac{n!}{2^{k}(n-2k)!k!}.
  3. k k
  4. ( n 2 k ) {\left({{n}\atop{2k}}\right)}
  5. 2 k 2k
  6. ( 2 k - 1 ) ! ! = ( 2 k ) ! / ( 2 k k ! ) (2k-1)!!=(2k)!/(2^{k}k!)
  7. 2 k 2k
  8. T ( n ) ( n e ) n / 2 e n ( 4 e ) 1 / 4 . T(n)\sim\left(\frac{n}{e}\right)^{n/2}\frac{e^{\sqrt{n}}}{(4e)^{1/4}}\,.
  9. n = 0 T ( n ) x n n ! = exp ( x 2 2 + x ) . \sum_{n=0}^{\infty}\frac{T(n)x^{n}}{n!}=\exp\left(\frac{x^{2}}{2}+x\right).
  10. T ( n ) = H e n ( i ) i n . T(n)=\frac{\mathop{He}_{n}(i)}{i^{n}}.
  11. 2 n / 4 + O ( 1 ) 2^{n/4+O(1)}
  12. T ( 4 k ) T(4k)
  13. T ( 4 k + 1 ) T(4k+1)
  14. k k
  15. T ( 4 k + 2 ) T(4k+2)
  16. k + 1 k+1
  17. T ( 4 k + 3 ) T(4k+3)
  18. k + 2 k+2

Tellurite_methyltransferase.html

  1. \rightleftharpoons

Template:1_3k_polytopes.html

  1. E ~ 7 {\tilde{E}}_{7}
  2. T ¯ 8 {\bar{T}}_{8}

Template:1_k2_polytopes.html

  1. E ~ 8 {\tilde{E}}_{8}
  2. T ¯ 8 {\bar{T}}_{8}

Template:2_2k_polytopes.html

  1. E ~ 6 {\tilde{E}}_{6}

Template:2_k1_polytopes.html

  1. E ~ 8 {\tilde{E}}_{8}
  2. T ¯ 8 {\bar{T}}_{8}

Template:3_k1_polytopes.html

  1. E ~ 7 {\tilde{E}}_{7}
  2. T ¯ 8 {\bar{T}}_{8}

Template:A5_honeycombs.html

  1. A ~ 5 {\tilde{A}}_{5}
  2. A ~ 5 {\tilde{A}}_{5}

Template:A6_honeycombs.html

  1. A ~ 6 {\tilde{A}}_{6}

Template:A7_honeycombs.html

  1. A ~ 7 {\tilde{A}}_{7}

Template:Alpha::Fe.html

  1. [ α F e ] \begin{smallmatrix}\left[\frac{\alpha}{Fe}\right]\end{smallmatrix}

Template:Bra-ket::doc.html

  1. | ξ |ξ\rangle
  2. | ψ |ψ\rangle
  3. ψ ξ \langle ψξ\rangle
  4. ξ ψ \langle ξψ\rangle
  5. ψ ξ \langle ψξ\rangle

Template:Bra::doc.html

  1. Ψ | \langle Ψ|
  2. Ψ | \langle Ψ|

Template:Braket::doc.html

  1. [ u b r a k e t , u k e t , u 3 c 8 ] [u^{\prime}braket^{\prime},u^{\prime}ket^{\prime},u^{\prime}\u{0}3c8^{\prime}]
  2. [ u b r a k e t , u b r a , u 3 c 8 ] = [ u b r a k e t , u k e t , u 3 c 8 ] < s u p > [u^{\prime}braket^{\prime},u^{\prime}bra^{\prime},u^{\prime}\u{0}3c8^{\prime}]% =[u^{\prime}braket^{\prime},u^{\prime}ket^{\prime},u^{\prime}\u{0}3c8^{\prime}% ]<sup>†

Template:Calculation_results.html

  1. summand + summand augend + addend } = \scriptstyle\left.\begin{matrix}\scriptstyle\,\text{summand}+\,\text{summand}% \\ \scriptstyle\,\text{augend}+\,\text{addend}\end{matrix}\right\}=
  2. sum \scriptstyle\,\text{sum}
  3. minuend - subtrahend = \scriptstyle\,\text{minuend}-\,\text{subtrahend}=
  4. difference \scriptstyle\,\text{difference}
  5. multiplicand × multiplicand multiplicand × multiplier } = \scriptstyle\left.\begin{matrix}\scriptstyle\,\text{multiplicand}\times\,\text% {multiplicand}\\ \scriptstyle\,\text{multiplicand}\times\,\text{multiplier}\end{matrix}\right\}=
  6. product \scriptstyle\,\text{product}
  7. dividend divisor numerator denominator } = \scriptstyle\left.\begin{matrix}\scriptstyle\frac{\scriptstyle\,\text{dividend% }}{\scriptstyle\,\text{divisor}}\\ \scriptstyle\frac{\scriptstyle\,\text{numerator}}{\scriptstyle\,\text{% denominator}}\end{matrix}\right\}=
  8. quotient \scriptstyle\,\text{quotient}
  9. dividend mod divisor = \scriptstyle\,\text{dividend}\bmod\,\text{divisor}=
  10. remainder \scriptstyle\,\text{remainder}
  11. baseexponent = \scriptstyle\,\text{base}\text{exponent}=
  12. power \scriptstyle\,\text{power}
  13. radicand degree = \scriptstyle\sqrt[\,\text{degree}]{\scriptstyle\,\text{radicand}}=
  14. root \scriptstyle\,\text{root}
  15. log base ( antilogarithm ) = \scriptstyle\log\text{base}(\,\text{antilogarithm})=
  16. logarithm \scriptstyle\,\text{logarithm}

Template:Did_you_know_nominations::Lanczos_tensor.html

  1. \Box

Template:Fe::H.html

  1. [ F e H ] \begin{smallmatrix}\left[\frac{Fe}{H}\right]\end{smallmatrix}

Template:Honeycombs.html

  1. A ~ n - 1 {\tilde{A}}_{n-1}
  2. C ~ n - 1 {\tilde{C}}_{n-1}
  3. B ~ n - 1 {\tilde{B}}_{n-1}
  4. D ~ n - 1 {\tilde{D}}_{n-1}
  5. G ~ 2 {\tilde{G}}_{2}
  6. F ~ 4 {\tilde{F}}_{4}
  7. E ~ n - 1 {\tilde{E}}_{n-1}

Template:Image_label_marker::doc.html

  1. x m a r k e r = x m a r k e r _ o r i g x o r i g x s c a l e d - m a r k e r _ s i z e 2 + x a d j u s t x_{marker}={x_{marker\_orig}\over x_{orig}}\cdot x_{scaled}-{marker\_size\over 2% }+x_{adjust}
  2. y m a r k e r = y m a r k e r _ o r i g y o r i g y s c a l e d - m a r k e r _ s i z e 2 + y a d j u s t y_{marker}={y_{marker\_orig}\over y_{orig}}\cdot y_{scaled}-{marker\_size\over 2% }+y_{adjust}
  3. y s c a l e d = x s c a l e d x o r i g y o r i g y_{scaled}={x_{scaled}\over x_{orig}}\cdot y_{orig}
  4. x t e x t = x m a r k e r _ o r i g x o r i g x s c a l e d + m a r k e r _ s i z e 2 + x a d j u s t + x t e x t a d j u s t x_{text}={x_{marker\_orig}\over x_{orig}}\cdot x_{scaled}+{marker\_size\over 2% }+x_{adjust}+x_{textadjust}
  5. y t e x t = y m a r k e r _ o r i g y o r i g y s c a l e d - m a r k e r _ s i z e 2 + y a d j u s t + y t e x t a d j u s t y_{text}={y_{marker\_orig}\over y_{orig}}\cdot y_{scaled}-{marker\_size\over 2% }+y_{adjust}+y_{textadjust}

Template:Just_chromatic_scale.html

  1. 1 1 \frac{1}{1}
  2. 27 25 \frac{27}{25}
  3. 25 24 \frac{25}{24}
  4. 9 8 \frac{9}{8}
  5. 6 5 \frac{6}{5}
  6. 75 64 \frac{75}{64}
  7. 5 4 \frac{5}{4}
  8. 4 3 \frac{4}{3}
  9. 36 25 \frac{36}{25}
  10. 25 18 \frac{25}{18}
  11. 3 2 \frac{3}{2}
  12. 8 5 \frac{8}{5}
  13. 25 16 \frac{25}{16}
  14. 5 3 \frac{5}{3}
  15. 9 5 \frac{9}{5}
  16. 125 72 \frac{125}{72}
  17. 15 8 \frac{15}{8}

Template:Just_tuning.html

  1. 1 1 \frac{1}{1}
  2. 3 2 \frac{3}{2}
  3. 1 1 \frac{1}{1}
  4. 4 3 \frac{4}{3}
  5. 1 1 \frac{1}{1}
  6. 6 5 \frac{6}{5}
  7. 5 4 \frac{5}{4}
  8. 3 2 \frac{3}{2}
  9. 1 1 \frac{1}{1}
  10. 5 3 \frac{5}{3}
  11. 8 5 \frac{8}{5}
  12. 4 3 \frac{4}{3}
  13. 1 1 \frac{1}{1}
  14. 7 6 \frac{7}{6}
  15. 6 5 \frac{6}{5}
  16. 5 4 \frac{5}{4}
  17. 7 5 \frac{7}{5}
  18. 3 2 \frac{3}{2}
  19. 7 4 \frac{7}{4}
  20. 1 1 \frac{1}{1}
  21. 12 7 \frac{12}{7}
  22. 5 3 \frac{5}{3}
  23. 8 5 \frac{8}{5}
  24. 10 7 \frac{10}{7}
  25. 4 3 \frac{4}{3}
  26. 8 7 \frac{8}{7}
  27. 1 1 \frac{1}{1}
  28. 12 11 \frac{12}{11}
  29. 11 10 \frac{11}{10}
  30. 10 9 \frac{10}{9}
  31. 9 8 \frac{9}{8}
  32. 7 6 \frac{7}{6}
  33. 6 5 \frac{6}{5}
  34. 11 9 \frac{11}{9}
  35. 5 4 \frac{5}{4}
  36. 14 11 \frac{14}{11}
  37. 11 8 \frac{11}{8}
  38. 7 5 \frac{7}{5}
  39. 3 2 \frac{3}{2}
  40. 14 9 \frac{14}{9}
  41. 7 4 \frac{7}{4}
  42. 1 1 \frac{1}{1}
  43. 11 6 \frac{11}{6}
  44. 20 11 \frac{20}{11}
  45. 9 5 \frac{9}{5}
  46. 16 9 \frac{16}{9}
  47. 12 7 \frac{12}{7}
  48. 5 3 \frac{5}{3}
  49. 18 11 \frac{18}{11}
  50. 8 5 \frac{8}{5}
  51. 11 7 \frac{11}{7}
  52. 16 11 \frac{16}{11}
  53. 10 7 \frac{10}{7}
  54. 4 3 \frac{4}{3}
  55. 9 7 \frac{9}{7}
  56. 8 7 \frac{8}{7}

Template:K_21_polytopes.html

  1. E ~ 8 {\tilde{E}}_{8}
  2. T ¯ 8 {\bar{T}}_{8}

Template:K_22_polytopes.html

  1. E ~ 6 {\tilde{E}}_{6}
  2. T ¯ 7 {\bar{T}}_{7}

Template:K_31_polytopes.html

  1. E ~ 7 {\tilde{E}}_{7}
  2. T ¯ 8 {\bar{T}}_{8}

Template:Ket::doc.html

  1. | Ψ |Ψ\rangle
  2. | Ψ |Ψ\rangle
  3. i ħ d d t | ψ ( t ) = Ĥ | ψ ( t ) iħ\frac{d}{dt}|ψ(t)\rangle=Ĥ|ψ(t)\rangle

Template:Mtag.html

  1. < m t p l > 1 {<mtpl>{{1}}}
  2. < m t p l > 2 {<mtpl>{{2}}}
  3. < m t p l > 3 {<mtpl>{{3}}}
  4. < m t p l > 4 {<mtpl>{{4}}}
  5. < m t p l > 5 {<mtpl>{{5}}}
  6. < m t p l > 6 {<mtpl>{{6}}}
  7. < m t p l > 7 {<mtpl>{{7}}}
  8. < m t p l > 8 {<mtpl>{{8}}}
  9. < m t p l > 9 {<mtpl>{{9}}}
  10. < m t p l > 10 {<mtpl>{{10}}}
  11. < m t p l > 11 {<mtpl>{{11}}}

Template:PAGR::doc.html

  1. P A G R = [ ( P 2 P 1 ) 1 t 2 - t 1 - 1 ] × 100 % PAGR=\left[\left(\frac{P_{2}}{P_{1}}\right)^{\frac{1}{t_{2}-t_{1}}}-1\right]% \times 100\%
  2. P 1 P_{1}
  3. P 2 P_{2}
  4. t 1 t_{1}
  5. t 2 t_{2}

Template:PGR::doc.html

  1. P G R = ln ( P ( t 2 ) ) - ln ( P ( t 1 ) ) ( t 2 - t 1 ) × 100 % PGR=\frac{\ln(P(t_{2}))-\ln(P(t_{1}))}{(t_{2}-t_{1})}\times 100\%

Template:Positionskarte::Lineare_Kegelprojektion.html

  1. n = sin ( latitude × π / 180 ) n=\sin(\rm{latitude}\times\pi/180)

Template:Preradicals.html

  1. σ \sigma
  2. τ \tau
  3. σ τ \sigma\leq\tau
  4. σ M τ M \sigma M\leq\tau M

Template:Pythagorean_chromatic_scale.html

  1. 1 1 \frac{1}{1}
  2. 256 243 \frac{256}{243}
  3. 2187 2048 \frac{2187}{2048}
  4. 9 8 \frac{9}{8}
  5. 32 27 \frac{32}{27}
  6. 19683 16384 \frac{19683}{16384}
  7. 81 64 \frac{81}{64}
  8. 4 3 \frac{4}{3}
  9. 1024 729 \frac{1024}{729}
  10. 729 512 \frac{729}{512}
  11. 3 2 \frac{3}{2}
  12. 128 81 \frac{128}{81}
  13. 6561 4096 \frac{6561}{4096}
  14. 27 16 \frac{27}{16}
  15. 16 9 \frac{16}{9}
  16. 59049 32768 \frac{59049}{32768}
  17. 243 128 \frac{243}{128}

Template:Pythagorean_tuning.html

  1. 1 1 \frac{1}{1}
  2. 3 2 \frac{3}{2}
  3. 9 8 \frac{9}{8}
  4. 27 16 \frac{27}{16}
  5. 81 64 \frac{81}{64}
  6. 243 128 \frac{243}{128}
  7. 729 512 \frac{729}{512}
  8. 1 1 \frac{1}{1}
  9. 4 3 \frac{4}{3}
  10. 16 9 \frac{16}{9}
  11. 32 27 \frac{32}{27}
  12. 128 81 \frac{128}{81}
  13. 256 243 \frac{256}{243}
  14. 1024 729 \frac{1024}{729}

Template:Vec::doc.html

  1. A \vec{A}
  2. d [ u v e c , u A , u 032 ] / d d{[u^{\prime}vec^{\prime},u^{\prime}A^{\prime},u^{\prime}\u{2}032^{\prime}]}/{d}
  3. d [ u v e c , u A ] / d d{[u^{\prime}vec^{\prime},u^{\prime}A^{\prime}]}/{d}
  4. d [ u v e c , u A , u 032 ] d d\frac{[u^{\prime}vec^{\prime},u^{\prime}A^{\prime},u^{\prime}\u{2}032^{\prime% }]}{d}
  5. d [ u v e c , u A ] d d\frac{[u^{\prime}vec^{\prime},u^{\prime}A^{\prime}]}{d}
  6. [ u v e c , u A ] [u^{\prime}vec^{\prime},u^{\prime}A^{\prime}]
  7. [ u v e c , u A , u o v e r ] [u^{\prime}vec^{\prime},u^{\prime}A^{\prime},u^{\prime}over^{\prime}]
  8. [ u v e c , u A , u u n d e r ] [u^{\prime}vec^{\prime},u^{\prime}A^{\prime},u^{\prime}under^{\prime}]
  9. [ u v e c , u A , u l e f t ] [u^{\prime}vec^{\prime},u^{\prime}A^{\prime},u^{\prime}left^{\prime}]
  10. [ u v e c , u A , u l e f t , u o v e r ] [u^{\prime}vec^{\prime},u^{\prime}A^{\prime},u^{\prime}left^{\prime},u^{\prime% }over^{\prime}]
  11. [ u v e c , u A , u l e f t , u u n d e r ] [u^{\prime}vec^{\prime},u^{\prime}A^{\prime},u^{\prime}left^{\prime},u^{\prime% }under^{\prime}]
  12. [ u v e c , u A , u r i g h t ] [u^{\prime}vec^{\prime},u^{\prime}A^{\prime},u^{\prime}right^{\prime}]
  13. [ u v e c , u A , u r i g h t , u o v e r ] [u^{\prime}vec^{\prime},u^{\prime}A^{\prime},u^{\prime}right^{\prime},u^{% \prime}over^{\prime}]
  14. [ u v e c , u A , u r i g h t , u u n d e r ] [u^{\prime}vec^{\prime},u^{\prime}A^{\prime},u^{\prime}right^{\prime},u^{% \prime}under^{\prime}]
  15. [ u v e c , u A , u d o u b l e ] [u^{\prime}vec^{\prime},u^{\prime}A^{\prime},u^{\prime}double^{\prime}]
  16. [ u v e c , u A , u d o u b l e , u o v e r ] [u^{\prime}vec^{\prime},u^{\prime}A^{\prime},u^{\prime}double^{\prime},u^{% \prime}over^{\prime}]
  17. [ u v e c , u A , u d o u b l e , u u n d e r ] [u^{\prime}vec^{\prime},u^{\prime}A^{\prime},u^{\prime}double^{\prime},u^{% \prime}under^{\prime}]
  18. [ u v e c , u 3 c 9 ] = [ u v e c , u 207 ] × [ u v e c , u u ] [u^{\prime}vec^{\prime},u^{\prime}\u{0}3c9^{\prime}]=[u^{\prime}vec^{\prime},u% ^{\prime}\u{2}207^{\prime}]×[u^{\prime}vec^{\prime},u^{\prime}u^{\prime}]
  19. [ u v e c , u v ] = [ u v e c , u 3 c 9 ] × [ u v e c , u r ] [u^{\prime}vec^{\prime},u^{\prime}v^{\prime}]=[u^{\prime}vec^{\prime},u^{% \prime}\u{0}3c9^{\prime}]×[u^{\prime}vec^{\prime},u^{\prime}r^{\prime}]
  20. d [ u v e c , u A , u 032 ] / d = d [ u v e c , u A ] / d [ u v e c , u 3 c 9 ] × [ u v e c , u A ] d{[u^{\prime}vec^{\prime},u^{\prime}A^{\prime},u^{\prime}\u{2}032^{\prime}]}/{% d}=d{[u^{\prime}vec^{\prime},u^{\prime}A^{\prime}]}/{d}−[u^{\prime}vec^{\prime% },u^{\prime}\u{0}3c9^{\prime}]×[u^{\prime}vec^{\prime},u^{\prime}A^{\prime}]
  21. d [ u v e c , u A , u 032 ] d = d [ u v e c , u A ] d [ u v e c , u 3 c 9 ] × [ u v e c , u A ] d\frac{[u^{\prime}vec^{\prime},u^{\prime}A^{\prime},u^{\prime}\u{2}032^{\prime% }]}{d}=d\frac{[u^{\prime}vec^{\prime},u^{\prime}A^{\prime}]}{d}−[u^{\prime}vec% ^{\prime},u^{\prime}\u{0}3c9^{\prime}]×[u^{\prime}vec^{\prime},u^{\prime}A^{% \prime}]

Temporal_motivation_theory.html

  1. Motivation = Expectancy × Value 1 + Impulsiveness × Delay \mathrm{Motivation}=\frac{\mbox{Expectancy × Value}~{}}{\mbox{1 + % Impulsiveness × Delay}~{}}
  2. M o t i v a t i o n Motivation
  3. E x p e c t a n c y Expectancy
  4. V a l u e Value
  5. I m p u l s i v e n e s s Impulsiveness
  6. D e l a y Delay

Tensors_in_curvilinear_coordinates.html

  1. 𝐯 = v k 𝐛 k \mathbf{v}=v^{k}\,\mathbf{b}_{k}
  2. 𝐛 i 𝐛 j = δ j i \mathbf{b}^{i}\cdot\mathbf{b}_{j}=\delta^{i}_{j}
  3. 𝐯 = v k 𝐛 k \mathbf{v}=v_{k}~{}\mathbf{b}^{k}
  4. 𝐯 \mathbf{v}
  5. s y m b o l S = S i j 𝐛 i 𝐛 j = S j i 𝐛 i 𝐛 j = S i j 𝐛 i 𝐛 j = S i j 𝐛 i 𝐛 j symbol{S}=S^{ij}~{}\mathbf{b}_{i}\otimes\mathbf{b}_{j}=S^{i}_{~{}j}~{}\mathbf{% b}_{i}\otimes\mathbf{b}^{j}=S_{i}^{~{}j}~{}\mathbf{b}^{i}\otimes\mathbf{b}_{j}% =S_{ij}~{}\mathbf{b}^{i}\otimes\mathbf{b}^{j}
  6. g i j = 𝐛 i 𝐛 j = g j i ; g i j = 𝐛 i 𝐛 j = g j i g_{ij}=\mathbf{b}_{i}\cdot\mathbf{b}_{j}=g_{ji}~{};~{}~{}g^{ij}=\mathbf{b}^{i}% \cdot\mathbf{b}^{j}=g^{ji}
  7. v i = g i k v k ; v i = g i k v k ; 𝐛 i = g i j 𝐛 j ; 𝐛 i = g i j 𝐛 j v^{i}=g^{ik}~{}v_{k}~{};~{}~{}v_{i}=g_{ik}~{}v^{k}~{};~{}~{}\mathbf{b}^{i}=g^{% ij}~{}\mathbf{b}_{j}~{};~{}~{}\mathbf{b}_{i}=g_{ij}~{}\mathbf{b}^{j}
  8. 𝐯 𝐛 i = v k 𝐛 k 𝐛 i = v k δ k i = v i \mathbf{v}\cdot\mathbf{b}^{i}=v^{k}~{}\mathbf{b}_{k}\cdot\mathbf{b}^{i}=v^{k}~% {}\delta^{i}_{k}=v^{i}
  9. 𝐯 𝐛 i = v k 𝐛 k 𝐛 i = v k δ i k = v i \mathbf{v}\cdot\mathbf{b}_{i}=v_{k}~{}\mathbf{b}^{k}\cdot\mathbf{b}_{i}=v_{k}~% {}\delta_{i}^{k}=v_{i}
  10. 𝐯 𝐛 i = v k 𝐛 k 𝐛 i = g k i v k \mathbf{v}\cdot\mathbf{b}_{i}=v^{k}~{}\mathbf{b}_{k}\cdot\mathbf{b}_{i}=g_{ki}% ~{}v^{k}
  11. 𝐯 𝐛 i = v k 𝐛 k 𝐛 i = g k i v k \mathbf{v}\cdot\mathbf{b}^{i}=v_{k}~{}\mathbf{b}^{k}\cdot\mathbf{b}^{i}=g^{ki}% ~{}v_{k}
  12. S i j = g i k S k j = g j k S k i = g i k g j l S k l S^{ij}=g^{ik}~{}S_{k}^{~{}j}=g^{jk}~{}S^{i}_{~{}k}=g^{ik}~{}g^{jl}~{}S_{kl}
  13. s y m b o l = ε i j k 𝐞 i 𝐞 j 𝐞 k symbol{\mathcal{E}}=\varepsilon_{ijk}~{}\mathbf{e}^{i}\otimes\mathbf{e}^{j}% \otimes\mathbf{e}^{k}
  14. s y m b o l = i j k 𝐛 i 𝐛 j 𝐛 k = i j k 𝐛 i 𝐛 j 𝐛 k symbol{\mathcal{E}}=\mathcal{E}_{ijk}~{}\mathbf{b}^{i}\otimes\mathbf{b}^{j}% \otimes\mathbf{b}^{k}=\mathcal{E}^{ijk}~{}\mathbf{b}_{i}\otimes\mathbf{b}_{j}% \otimes\mathbf{b}_{k}
  15. i j k = [ 𝐛 i , 𝐛 j , 𝐛 k ] = ( 𝐛 i × 𝐛 j ) 𝐛 k ; i j k = [ 𝐛 i , 𝐛 j , 𝐛 k ] \mathcal{E}_{ijk}=\left[\mathbf{b}_{i},\mathbf{b}_{j},\mathbf{b}_{k}\right]=(% \mathbf{b}_{i}\times\mathbf{b}_{j})\cdot\mathbf{b}_{k}~{};~{}~{}\mathcal{E}^{% ijk}=\left[\mathbf{b}^{i},\mathbf{b}^{j},\mathbf{b}^{k}\right]
  16. 𝐛 i × 𝐛 j = J ε i j p 𝐛 p = g ε i j p 𝐛 p \mathbf{b}_{i}\times\mathbf{b}_{j}=J~{}\varepsilon_{ijp}~{}\mathbf{b}^{p}=% \sqrt{g}~{}\varepsilon_{ijp}~{}\mathbf{b}^{p}
  17. i j k = J ε i j k = g ε i j k \mathcal{E}_{ijk}=J~{}\varepsilon_{ijk}=\sqrt{g}~{}\varepsilon_{ijk}
  18. i j k = 1 J ε i j k = 1 g ε i j k \mathcal{E}^{ijk}=\cfrac{1}{J}~{}\varepsilon^{ijk}=\cfrac{1}{\sqrt{g}}~{}% \varepsilon^{ijk}
  19. 𝖨 𝐯 = 𝐯 \mathsf{I}\cdot\mathbf{v}=\mathbf{v}
  20. 𝖨 = g i j 𝐛 i 𝐛 j = g i j 𝐛 i 𝐛 j = 𝐛 i 𝐛 i = 𝐛 i 𝐛 i \mathsf{I}=g^{ij}~{}\mathbf{b}_{i}\otimes\mathbf{b}_{j}=g_{ij}~{}\mathbf{b}^{i% }\otimes\mathbf{b}^{j}=\mathbf{b}_{i}\otimes\mathbf{b}^{i}=\mathbf{b}^{i}% \otimes\mathbf{b}_{i}
  21. 𝐮 𝐯 = u i v i = u i v i = g i j u i v j = g i j u i v j \mathbf{u}\cdot\mathbf{v}=u^{i}~{}v_{i}=u_{i}~{}v^{i}=g_{ij}~{}u^{i}~{}v^{j}=g% ^{ij}~{}u_{i}~{}v_{j}
  22. 𝐮 × 𝐯 = ε i j k u j v k 𝐞 i \mathbf{u}\times\mathbf{v}=\varepsilon_{ijk}~{}{u}_{j}~{}{v}_{k}~{}\mathbf{e}_% {i}
  23. 𝐮 × 𝐯 = [ ( 𝐛 m × 𝐛 n ) 𝐛 s ] u m v n 𝐛 s = s m n u m v n 𝐛 s \mathbf{u}\times\mathbf{v}=[(\mathbf{b}_{m}\times\mathbf{b}_{n})\cdot\mathbf{b% }_{s}]~{}u^{m}~{}v^{n}~{}\mathbf{b}^{s}=\mathcal{E}_{smn}~{}u^{m}~{}v^{n}~{}% \mathbf{b}^{s}
  24. i j k \mathcal{E}_{ijk}
  25. 𝐮 × 𝐯 = ε i j k u ^ j v ^ k 𝐞 i \mathbf{u}\times\mathbf{v}=\varepsilon_{ijk}~{}\hat{u}_{j}~{}\hat{v}_{k}~{}% \mathbf{e}_{i}
  26. 𝐞 i \mathbf{e}_{i}
  27. 𝐞 p × 𝐞 q = ε i p q 𝐞 i \mathbf{e}_{p}\times\mathbf{e}_{q}=\varepsilon_{ipq}~{}\mathbf{e}_{i}
  28. 𝐛 m × 𝐛 n = 𝐱 q m × 𝐱 q n = ( x p 𝐞 p ) q m × ( x q 𝐞 q ) q n = x p q m x q q n 𝐞 p × 𝐞 q = ε i p q x p q m x q q n 𝐞 i \begin{aligned}\displaystyle\mathbf{b}_{m}\times\mathbf{b}_{n}&\displaystyle=% \frac{\partial\mathbf{x}}{\partial q^{m}}\times\frac{\partial\mathbf{x}}{% \partial q^{n}}=\frac{\partial(x_{p}~{}\mathbf{e}_{p})}{\partial q^{m}}\times% \frac{\partial(x_{q}~{}\mathbf{e}_{q})}{\partial q^{n}}\\ &\displaystyle=\frac{\partial x_{p}}{\partial q^{m}}~{}\frac{\partial x_{q}}{% \partial q^{n}}~{}\mathbf{e}_{p}\times\mathbf{e}_{q}=\varepsilon_{ipq}~{}\frac% {\partial x_{p}}{\partial q^{m}}~{}\frac{\partial x_{q}}{\partial q^{n}}~{}% \mathbf{e}_{i}\end{aligned}
  29. ( 𝐛 m × 𝐛 n ) 𝐛 s = ε i p q x p q m x q q n x i q s (\mathbf{b}_{m}\times\mathbf{b}_{n})\cdot\mathbf{b}_{s}=\varepsilon_{ipq}~{}% \frac{\partial x_{p}}{\partial q^{m}}~{}\frac{\partial x_{q}}{\partial q^{n}}~% {}\frac{\partial x_{i}}{\partial q^{s}}
  30. u ^ j = x j q m u m ; v ^ k = x k q n v n ; 𝐞 i = x i q s 𝐛 s \hat{u}_{j}=\frac{\partial x_{j}}{\partial q^{m}}~{}u^{m}~{};~{}~{}\hat{v}_{k}% =\frac{\partial x_{k}}{\partial q^{n}}~{}v^{n}~{};~{}~{}\mathbf{e}_{i}=\frac{% \partial x_{i}}{\partial q^{s}}~{}\mathbf{b}^{s}
  31. 𝐮 × 𝐯 = ε i j k u ^ j v ^ k 𝐞 i = ε i j k x j q m x k q n x i q s u m v n 𝐛 s = [ ( 𝐛 m × 𝐛 n ) 𝐛 s ] u m v n 𝐛 s = s m n u m v n 𝐛 s \begin{aligned}\displaystyle\mathbf{u}\times\mathbf{v}&\displaystyle=% \varepsilon_{ijk}~{}\hat{u}_{j}~{}\hat{v}_{k}~{}\mathbf{e}_{i}=\varepsilon_{% ijk}~{}\frac{\partial x_{j}}{\partial q^{m}}~{}\frac{\partial x_{k}}{\partial q% ^{n}}~{}\frac{\partial x_{i}}{\partial q^{s}}~{}u^{m}~{}v^{n}~{}\mathbf{b}^{s}% \\ &\displaystyle=[(\mathbf{b}_{m}\times\mathbf{b}_{n})\cdot\mathbf{b}_{s}]~{}u^{% m}~{}v^{n}~{}\mathbf{b}^{s}=\mathcal{E}_{smn}~{}u^{m}~{}v^{n}~{}\mathbf{b}^{s}% \end{aligned}
  32. 𝖨 \mathsf{I}
  33. 𝖨 𝐯 = 𝐯 \mathsf{I}\cdot\mathbf{v}=\mathbf{v}
  34. 𝖨 = g i j 𝐛 i 𝐛 j = g i j 𝐛 i 𝐛 j = 𝐛 i 𝐛 i = 𝐛 i 𝐛 i \mathsf{I}=g^{ij}\mathbf{b}_{i}\otimes\mathbf{b}_{j}=g_{ij}\mathbf{b}^{i}% \otimes\mathbf{b}^{j}=\mathbf{b}_{i}\otimes\mathbf{b}^{i}=\mathbf{b}^{i}% \otimes\mathbf{b}_{i}
  35. 𝐯 = s y m b o l S 𝐮 \mathbf{v}=symbol{S}\cdot\mathbf{u}
  36. v i 𝐛 i = S i j u j 𝐛 i = S j i u j 𝐛 i ; v i 𝐛 i = S i j u i 𝐛 i = S i j u j 𝐛 i v^{i}\mathbf{b}_{i}=S^{ij}u_{j}\mathbf{b}_{i}=S^{i}_{j}u^{j}\mathbf{b}_{i};% \qquad v_{i}\mathbf{b}^{i}=S_{ij}u^{i}\mathbf{b}^{i}=S_{i}^{j}u_{j}\mathbf{b}^% {i}
  37. s y m b o l U = s y m b o l S \cdotsymbol T symbol{U}=symbol{S}\cdotsymbol{T}
  38. U i j 𝐛 i 𝐛 j = S i k T . j k 𝐛 i 𝐛 j = S i . k T k j 𝐛 i 𝐛 j U_{ij}\mathbf{b}^{i}\otimes\mathbf{b}^{j}=S_{ik}T^{k}_{.j}\mathbf{b}^{i}% \otimes\mathbf{b}^{j}=S_{i}^{.k}T_{kj}\mathbf{b}^{i}\otimes\mathbf{b}^{j}
  39. s y m b o l U = S i j T . n m g j m 𝐛 i 𝐛 n = S . m i T . n m 𝐛 i 𝐛 n = S i j T j n 𝐛 i 𝐛 n symbol{U}=S^{ij}T^{m}_{.n}g_{jm}\mathbf{b}_{i}\otimes\mathbf{b}^{n}=S^{i}_{.m}% T^{m}_{.n}\mathbf{b}_{i}\otimes\mathbf{b}^{n}=S^{ij}T_{jn}\mathbf{b}_{i}% \otimes\mathbf{b}^{n}
  40. s y m b o l S symbol{S}
  41. [ s y m b o l S 𝐮 , s y m b o l S 𝐯 , s y m b o l S 𝐰 ] = \detsymbol S [ 𝐮 , 𝐯 , 𝐰 ] \left[symbol{S}\cdot\mathbf{u},symbol{S}\cdot\mathbf{v},symbol{S}\cdot\mathbf{% w}\right]=\detsymbol{S}\left[\mathbf{u},\mathbf{v},\mathbf{w}\right]
  42. 𝐮 , 𝐯 , 𝐰 \mathbf{u},\mathbf{v},\mathbf{w}
  43. [ 𝐮 , 𝐯 , 𝐰 ] := 𝐮 ( 𝐯 × 𝐰 ) . \left[\mathbf{u},\mathbf{v},\mathbf{w}\right]:=\mathbf{u}\cdot(\mathbf{v}% \times\mathbf{w}).
  44. 𝐛 i = s y m b o l F 𝐞 i \mathbf{b}_{i}=symbol{F}\cdot\mathbf{e}_{i}
  45. 𝐛 i 𝐞 i = ( s y m b o l F 𝐞 i ) 𝐞 i = s y m b o l F ( 𝐞 i 𝐞 i ) = s y m b o l F . \mathbf{b}_{i}\otimes\mathbf{e}_{i}=(symbol{F}\cdot\mathbf{e}_{i})\otimes% \mathbf{e}_{i}=symbol{F}\cdot(\mathbf{e}_{i}\otimes\mathbf{e}_{i})=symbol{F}~{}.
  46. 𝐛 i = s y m b o l F - T 𝐞 i ; g i j = [ s y m b o l F - 1 \cdotsymbol F - T ] i j ; g i j = [ g i j ] - 1 = [ s y m b o l F T \cdotsymbol F ] i j \mathbf{b}^{i}=symbol{F}^{-\rm{T}}\cdot\mathbf{e}^{i}~{};~{}~{}g^{ij}=[symbol{% F}^{-\rm{1}}\cdotsymbol{F}^{-\rm{T}}]_{ij}~{};~{}~{}g_{ij}=[g^{ij}]^{-1}=[% symbol{F}^{\rm{T}}\cdotsymbol{F}]_{ij}
  47. J := \detsymbol F J:=\detsymbol{F}
  48. [ 𝐛 1 , 𝐛 2 , 𝐛 3 ] = \detsymbol F [ 𝐞 1 , 𝐞 2 , 𝐞 3 ] . \left[\mathbf{b}_{1},\mathbf{b}_{2},\mathbf{b}_{3}\right]=\detsymbol{F}\left[% \mathbf{e}_{1},\mathbf{e}_{2},\mathbf{e}_{3}\right]~{}.
  49. [ 𝐞 1 , 𝐞 2 , 𝐞 3 ] = 1 \left[\mathbf{e}_{1},\mathbf{e}_{2},\mathbf{e}_{3}\right]=1
  50. J = \detsymbol F = [ 𝐛 1 , 𝐛 2 , 𝐛 3 ] = 𝐛 1 ( 𝐛 2 × 𝐛 3 ) J=\detsymbol{F}=\left[\mathbf{b}_{1},\mathbf{b}_{2},\mathbf{b}_{3}\right]=% \mathbf{b}_{1}\cdot(\mathbf{b}_{2}\times\mathbf{b}_{3})
  51. g := det [ g i j ] g:=\det[g_{ij}]\,
  52. g = det [ s y m b o l F T ] det [ s y m b o l F ] = J J = J 2 g=\det[symbol{F}^{\rm{T}}]\cdot\det[symbol{F}]=J\cdot J=J^{2}
  53. det [ g i j ] = 1 J 2 \det[g^{ij}]=\cfrac{1}{J^{2}}
  54. [ g i j ] = [ g i j ] - 1 [g^{ij}]=[g_{ij}]^{-1}
  55. g g i j = 2 J J g i j = g g i j \cfrac{\partial g}{\partial g_{ij}}=2~{}J~{}\cfrac{\partial J}{\partial g_{ij}% }=g~{}g^{ij}
  56. 𝐛 i 𝐛 j = δ j i 𝐛 1 𝐛 1 = 1 , 𝐛 1 𝐛 2 = 𝐛 1 𝐛 3 = 0 𝐛 1 = A ( 𝐛 2 × 𝐛 3 ) \mathbf{b}^{i}\cdot\mathbf{b}_{j}=\delta^{i}_{j}\quad\Rightarrow\quad\mathbf{b% }^{1}\cdot\mathbf{b}_{1}=1,~{}\mathbf{b}^{1}\cdot\mathbf{b}_{2}=\mathbf{b}^{1}% \cdot\mathbf{b}_{3}=0\quad\Rightarrow\quad\mathbf{b}^{1}=A~{}(\mathbf{b}_{2}% \times\mathbf{b}_{3})
  57. 𝐛 1 𝐛 1 = A 𝐛 1 ( 𝐛 2 × 𝐛 3 ) = A J = 1 A = 1 J \mathbf{b}^{1}\cdot\mathbf{b}_{1}=A~{}\mathbf{b}_{1}\cdot(\mathbf{b}_{2}\times% \mathbf{b}_{3})=AJ=1\quad\Rightarrow\quad A=\cfrac{1}{J}
  58. 𝐛 1 = 1 J ( 𝐛 2 × 𝐛 3 ) ; 𝐛 2 = 1 J ( 𝐛 3 × 𝐛 1 ) ; 𝐛 3 = 1 J ( 𝐛 1 × 𝐛 2 ) \mathbf{b}^{1}=\cfrac{1}{J}(\mathbf{b}_{2}\times\mathbf{b}_{3})~{};~{}~{}% \mathbf{b}^{2}=\cfrac{1}{J}(\mathbf{b}_{3}\times\mathbf{b}_{1})~{};~{}~{}% \mathbf{b}^{3}=\cfrac{1}{J}(\mathbf{b}_{1}\times\mathbf{b}_{2})
  59. ε i j k 𝐛 k = 1 J ( 𝐛 i × 𝐛 j ) = 1 g ( 𝐛 i × 𝐛 j ) \varepsilon_{ijk}~{}\mathbf{b}^{k}=\cfrac{1}{J}(\mathbf{b}_{i}\times\mathbf{b}% _{j})=\cfrac{1}{\sqrt{g}}(\mathbf{b}_{i}\times\mathbf{b}_{j})
  60. ε i j k \varepsilon_{ijk}\,
  61. 𝐛 i = 𝐱 q i = 𝐱 x j x j q i = 𝐞 j x j q i \mathbf{b}_{i}=\cfrac{\partial\mathbf{x}}{\partial q^{i}}=\cfrac{\partial% \mathbf{x}}{\partial x_{j}}~{}\cfrac{\partial x_{j}}{\partial q^{i}}=\mathbf{e% }_{j}~{}\cfrac{\partial x_{j}}{\partial q^{i}}
  62. 𝐞 i = 𝐛 j q j x i \mathbf{e}_{i}=\mathbf{b}_{j}~{}\cfrac{\partial q^{j}}{\partial x_{i}}
  63. 𝐞 k 𝐛 i = x k q i x k q i 𝐛 i = 𝐞 k ( 𝐛 i 𝐛 i ) = 𝐞 k \mathbf{e}^{k}\cdot\mathbf{b}_{i}=\frac{\partial x_{k}}{\partial q^{i}}\quad% \Rightarrow\quad\frac{\partial x_{k}}{\partial q^{i}}~{}\mathbf{b}^{i}=\mathbf% {e}^{k}\cdot(\mathbf{b}_{i}\otimes\mathbf{b}^{i})=\mathbf{e}^{k}
  64. 𝐛 k = q k x i 𝐞 i \mathbf{b}^{k}=\frac{\partial q^{k}}{\partial x_{i}}~{}\mathbf{e}^{i}
  65. ( q 1 , q 2 , q 3 ) (q^{1},q^{2},q^{3})
  66. 𝐱 = s y m b o l φ ( q 1 , q 2 , q 3 ) ; q i = ψ i ( 𝐱 ) = [ s y m b o l φ - 1 ( 𝐱 ) ] i \mathbf{x}=symbol{\varphi}(q^{1},q^{2},q^{3})~{};~{}~{}q^{i}=\psi^{i}(\mathbf{% x})=[symbol{\varphi}^{-1}(\mathbf{x})]^{i}
  67. 𝐱 i ( α ) = s y m b o l φ ( α , q j , q k ) , i j k \mathbf{x}_{i}(\alpha)=symbol{\varphi}(\alpha,q^{j},q^{k})~{},~{}~{}i\neq j\neq k
  68. d 𝐱 i d α 𝐱 q i \cfrac{\rm{d}\mathbf{x}_{i}}{\rm{d}\alpha}\equiv\cfrac{\partial\mathbf{x}}{% \partial q^{i}}
  69. f ( 𝐱 ) = f [ s y m b o l φ ( q 1 , q 2 , q 3 ) ] = f φ ( q 1 , q 2 , q 3 ) f(\mathbf{x})=f[symbol{\varphi}(q^{1},q^{2},q^{3})]=f_{\varphi}(q^{1},q^{2},q^% {3})
  70. [ s y m b o l f ( 𝐱 ) ] 𝐜 = d d α f ( 𝐱 + α 𝐜 ) | α = 0 [symbol{\nabla}f(\mathbf{x})]\cdot\mathbf{c}=\cfrac{\rm{d}}{\rm{d}\alpha}f(% \mathbf{x}+\alpha\mathbf{c})\biggr|_{\alpha=0}
  71. q i + α c i = ψ i ( 𝐱 + α 𝐜 ) q^{i}+\alpha~{}c^{i}=\psi^{i}(\mathbf{x}+\alpha~{}\mathbf{c})
  72. [ s y m b o l f ( 𝐱 ) ] 𝐜 = d d α f φ ( q 1 + α c 1 , q 2 + α c 2 , q 3 + α c 3 ) | α = 0 = f φ q i c i = f q i c i [symbol{\nabla}f(\mathbf{x})]\cdot\mathbf{c}=\cfrac{\rm{d}}{\rm{d}\alpha}f_{% \varphi}(q^{1}+\alpha~{}c^{1},q^{2}+\alpha~{}c^{2},q^{3}+\alpha~{}c^{3})\biggr% |_{\alpha=0}=\cfrac{\partial f_{\varphi}}{\partial q^{i}}~{}c^{i}=\cfrac{% \partial f}{\partial q^{i}}~{}c^{i}
  73. f ( 𝐱 ) = ψ i ( 𝐱 ) f(\mathbf{x})=\psi^{i}(\mathbf{x})
  74. q i = ψ i ( 𝐱 ) q^{i}=\psi^{i}(\mathbf{x})
  75. [ s y m b o l ψ i ( 𝐱 ) ] 𝐜 = ψ i q j c j = c i [symbol{\nabla}\psi^{i}(\mathbf{x})]\cdot\mathbf{c}=\cfrac{\partial\psi^{i}}{% \partial q^{j}}~{}c^{j}=c^{i}
  76. [ s y m b o l f ( 𝐱 ) ] 𝐜 = f q i c i = ( f q i 𝐛 i ) ( c i 𝐛 i ) s y m b o l f ( 𝐱 ) = f q i 𝐛 i [symbol{\nabla}f(\mathbf{x})]\cdot\mathbf{c}=\cfrac{\partial f}{\partial q^{i}% }~{}c^{i}=\left(\cfrac{\partial f}{\partial q^{i}}~{}\mathbf{b}^{i}\right)% \left(c^{i}~{}\mathbf{b}_{i}\right)\quad\Rightarrow\quad symbol{\nabla}f(% \mathbf{x})=\cfrac{\partial f}{\partial q^{i}}~{}\mathbf{b}^{i}
  77. [ s y m b o l 𝐟 ( 𝐱 ) ] 𝐜 = 𝐟 q i c i [symbol{\nabla}\mathbf{f}(\mathbf{x})]\cdot\mathbf{c}=\cfrac{\partial\mathbf{f% }}{\partial q^{i}}~{}c^{i}
  78. 𝐜 = 𝐱 q i c i = 𝐛 i ( 𝐱 ) c i ; 𝐛 i ( 𝐱 ) := 𝐱 q i \mathbf{c}=\cfrac{\partial\mathbf{x}}{\partial q^{i}}~{}c^{i}=\mathbf{b}_{i}(% \mathbf{x})~{}c^{i}~{};~{}~{}\mathbf{b}_{i}(\mathbf{x}):=\cfrac{\partial% \mathbf{x}}{\partial q^{i}}
  79. s y m b o l 𝐟 ( 𝐱 ) = 𝐟 q i 𝐛 i symbol{\nabla}\mathbf{f}(\mathbf{x})=\cfrac{\partial\mathbf{f}}{\partial q^{i}% }\otimes\mathbf{b}^{i}
  80. 𝐛 i = s y m b o l ψ i \mathbf{b}^{i}=symbol{\nabla}\psi^{i}
  81. 𝐛 i , j = 𝐛 i q j := Γ i j k 𝐛 k 𝐛 i , j 𝐛 l = Γ i j l \mathbf{b}_{i,j}=\frac{\partial\mathbf{b}_{i}}{\partial q^{j}}:=\Gamma_{ijk}~{% }\mathbf{b}^{k}\quad\Rightarrow\quad\mathbf{b}_{i,j}\cdot\mathbf{b}_{l}=\Gamma% _{ijl}
  82. g i j , k = ( 𝐛 i 𝐛 j ) , k = 𝐛 i , k 𝐛 j + 𝐛 i 𝐛 j , k = Γ i k j + Γ j k i g i k , j = ( 𝐛 i 𝐛 k ) , j = 𝐛 i , j 𝐛 k + 𝐛 i 𝐛 k , j = Γ i j k + Γ k j i g j k , i = ( 𝐛 j 𝐛 k ) , i = 𝐛 j , i 𝐛 k + 𝐛 j 𝐛 k , i = Γ j i k + Γ k i j \begin{aligned}\displaystyle g_{ij,k}&\displaystyle=(\mathbf{b}_{i}\cdot% \mathbf{b}_{j})_{,k}=\mathbf{b}_{i,k}\cdot\mathbf{b}_{j}+\mathbf{b}_{i}\cdot% \mathbf{b}_{j,k}=\Gamma_{ikj}+\Gamma_{jki}\\ \displaystyle g_{ik,j}&\displaystyle=(\mathbf{b}_{i}\cdot\mathbf{b}_{k})_{,j}=% \mathbf{b}_{i,j}\cdot\mathbf{b}_{k}+\mathbf{b}_{i}\cdot\mathbf{b}_{k,j}=\Gamma% _{ijk}+\Gamma_{kji}\\ \displaystyle g_{jk,i}&\displaystyle=(\mathbf{b}_{j}\cdot\mathbf{b}_{k})_{,i}=% \mathbf{b}_{j,i}\cdot\mathbf{b}_{k}+\mathbf{b}_{j}\cdot\mathbf{b}_{k,i}=\Gamma% _{jik}+\Gamma_{kij}\end{aligned}
  83. Γ i j k = 1 2 ( g i k , j + g j k , i - g i j , k ) = 1 2 [ ( 𝐛 i 𝐛 k ) , j + ( 𝐛 j 𝐛 k ) , i - ( 𝐛 i 𝐛 j ) , k ] \Gamma_{ijk}=\frac{1}{2}(g_{ik,j}+g_{jk,i}-g_{ij,k})=\frac{1}{2}[(\mathbf{b}_{% i}\cdot\mathbf{b}_{k})_{,j}+(\mathbf{b}_{j}\cdot\mathbf{b}_{k})_{,i}-(\mathbf{% b}_{i}\cdot\mathbf{b}_{j})_{,k}]
  84. Γ i j k = Γ j i k \Gamma_{ij}^{k}=\Gamma_{ji}^{k}
  85. 𝐛 i q j = Γ i j k 𝐛 k \cfrac{\partial\mathbf{b}_{i}}{\partial q^{j}}=\Gamma_{ij}^{k}~{}\mathbf{b}_{k}
  86. Γ i j k = 𝐛 i q j 𝐛 k = - 𝐛 i 𝐛 k q j \Gamma_{ij}^{k}=\cfrac{\partial\mathbf{b}_{i}}{\partial q^{j}}\cdot\mathbf{b}^% {k}=-\mathbf{b}_{i}\cdot\cfrac{\partial\mathbf{b}^{k}}{\partial q^{j}}
  87. 𝐛 i q j = - Γ j k i 𝐛 k ; s y m b o l 𝐛 i = Γ i j k 𝐛 k 𝐛 j ; s y m b o l 𝐛 i = - Γ j k i 𝐛 k 𝐛 j \cfrac{\partial\mathbf{b}^{i}}{\partial q^{j}}=-\Gamma^{i}_{jk}~{}\mathbf{b}^{% k}~{};~{}~{}symbol{\nabla}\mathbf{b}_{i}=\Gamma_{ij}^{k}~{}\mathbf{b}_{k}% \otimes\mathbf{b}^{j}~{};~{}~{}symbol{\nabla}\mathbf{b}^{i}=-\Gamma_{jk}^{i}~{% }\mathbf{b}^{k}\otimes\mathbf{b}^{j}
  88. Γ i j k = g k m 2 ( g m i q j + g m j q i - g i j q m ) \Gamma^{k}_{ij}=\frac{g^{km}}{2}\left(\frac{\partial g_{mi}}{\partial q^{j}}+% \frac{\partial g_{mj}}{\partial q^{i}}-\frac{\partial g_{ij}}{\partial q^{m}}\right)
  89. s y m b o l 𝐯 = [ v i q k + Γ l k i v l ] 𝐛 i 𝐛 k = [ v i q k - Γ k i l v l ] 𝐛 i 𝐛 k \begin{aligned}\displaystyle symbol{\nabla}\mathbf{v}&\displaystyle=\left[% \cfrac{\partial v^{i}}{\partial q^{k}}+\Gamma^{i}_{lk}~{}v^{l}\right]~{}% \mathbf{b}_{i}\otimes\mathbf{b}^{k}\\ &\displaystyle=\left[\cfrac{\partial v_{i}}{\partial q^{k}}-\Gamma^{l}_{ki}~{}% v_{l}\right]~{}\mathbf{b}^{i}\otimes\mathbf{b}^{k}\end{aligned}
  90. 𝐯 = v i 𝐛 i = v ^ i 𝐛 ^ i \mathbf{v}=v_{i}~{}\mathbf{b}^{i}=\hat{v}_{i}~{}\hat{\mathbf{b}}^{i}
  91. v i v_{i}\,
  92. v ^ i \hat{v}_{i}
  93. 𝐛 ^ i = 𝐛 i g i i \hat{\mathbf{b}}^{i}=\cfrac{\mathbf{b}^{i}}{\sqrt{g^{ii}}}
  94. s y m b o l s y m b o l S = s y m b o l S q i 𝐛 i symbol{\nabla}symbol{S}=\cfrac{\partial symbol{S}}{\partial q^{i}}\otimes% \mathbf{b}^{i}
  95. s y m b o l s y m b o l S = q k [ S i j 𝐛 i 𝐛 j ] 𝐛 k = [ S i j q k - Γ k i l S l j - Γ k j l S i l ] 𝐛 i 𝐛 j 𝐛 k symbol{\nabla}symbol{S}=\cfrac{\partial}{\partial q^{k}}[S_{ij}~{}\mathbf{b}^{% i}\otimes\mathbf{b}^{j}]\otimes\mathbf{b}^{k}=\left[\cfrac{\partial S_{ij}}{% \partial q^{k}}-\Gamma^{l}_{ki}~{}S_{lj}-\Gamma^{l}_{kj}~{}S_{il}\right]~{}% \mathbf{b}^{i}\otimes\mathbf{b}^{j}\otimes\mathbf{b}^{k}
  96. s y m b o l s y m b o l S = [ S i j q k + Γ k l i S l j + Γ k l j S i l ] 𝐛 i 𝐛 j 𝐛 k = [ S j i q k + Γ k l i S j l - Γ k j l S l i ] 𝐛 i 𝐛 j 𝐛 k = [ S i j q k - Γ i k l S l j + Γ k l j S i l ] 𝐛 i 𝐛 j 𝐛 k \begin{aligned}\displaystyle symbol{\nabla}symbol{S}&\displaystyle=\left[% \cfrac{\partial S^{ij}}{\partial q^{k}}+\Gamma^{i}_{kl}~{}S^{lj}+\Gamma^{j}_{% kl}~{}S^{il}\right]~{}\mathbf{b}_{i}\otimes\mathbf{b}_{j}\otimes\mathbf{b}^{k}% \\ &\displaystyle=\left[\cfrac{\partial S^{i}_{~{}j}}{\partial q^{k}}+\Gamma^{i}_% {kl}~{}S^{l}_{~{}j}-\Gamma^{l}_{kj}~{}S^{i}_{~{}l}\right]~{}\mathbf{b}_{i}% \otimes\mathbf{b}^{j}\otimes\mathbf{b}^{k}\\ &\displaystyle=\left[\cfrac{\partial S_{i}^{~{}j}}{\partial q^{k}}-\Gamma^{l}_% {ik}~{}S_{l}^{~{}j}+\Gamma^{j}_{kl}~{}S_{i}^{~{}l}\right]~{}\mathbf{b}^{i}% \otimes\mathbf{b}_{j}\otimes\mathbf{b}^{k}\end{aligned}
  97. s y m b o l S = S i j 𝐛 i 𝐛 j = S ^ i j 𝐛 ^ i 𝐛 ^ j symbol{S}=S_{ij}~{}\mathbf{b}^{i}\otimes\mathbf{b}^{j}=\hat{S}_{ij}~{}\hat{% \mathbf{b}}^{i}\otimes\hat{\mathbf{b}}^{j}
  98. S ^ i j = S i j g i i g j j \hat{S}_{ij}=S_{ij}~{}\sqrt{g^{ii}~{}g^{jj}}
  99. 𝐯 \mathbf{v}
  100. div 𝐯 = s y m b o l 𝐯 = tr ( s y m b o l 𝐯 ) \operatorname{div}~{}\mathbf{v}=symbol{\nabla}\cdot\mathbf{v}=\,\text{tr}(% symbol{\nabla}\mathbf{v})
  101. s y m b o l 𝐯 = v i q i + Γ i i v = [ v i q j - Γ j i v ] g i j symbol{\nabla}\cdot\mathbf{v}=\cfrac{\partial v^{i}}{\partial q^{i}}+\Gamma^{i% }_{\ell i}~{}v^{\ell}=\left[\cfrac{\partial v_{i}}{\partial q^{j}}-\Gamma^{% \ell}_{ji}~{}v_{\ell}\right]~{}g^{ij}
  102. s y m b o l 𝐯 = v i q i + Γ i i v symbol{\nabla}\cdot\mathbf{v}=\frac{\partial v^{i}}{\partial q^{i}}+\Gamma_{% \ell i}^{i}~{}v^{\ell}
  103. Γ i i = Γ i i = g m i 2 [ g i m q + g m q i - g i l q m ] \Gamma_{\ell i}^{i}=\Gamma_{i\ell}^{i}=\cfrac{g^{mi}}{2}\left[\frac{\partial g% _{im}}{\partial q^{\ell}}+\frac{\partial g_{\ell m}}{\partial q^{i}}-\frac{% \partial g_{il}}{\partial q^{m}}\right]
  104. s y m b o l g symbol{g}
  105. g m i g m q i = g m i g i q m g^{mi}~{}\frac{\partial g_{\ell m}}{\partial q^{i}}=g^{mi}~{}\frac{\partial g_% {i\ell}}{\partial q^{m}}
  106. s y m b o l 𝐯 = v i q i + g m i 2 g i m q v symbol{\nabla}\cdot\mathbf{v}=\frac{\partial v^{i}}{\partial q^{i}}+\cfrac{g^{% mi}}{2}~{}\frac{\partial g_{im}}{\partial q^{\ell}}~{}v^{\ell}
  107. [ g i j ] - 1 = [ g i j ] [g_{ij}]^{-1}=[g^{ij}]
  108. [ g i j ] = [ g i j ] - 1 = A i j g ; g := det ( [ g i j ] ) = \detsymbol g [g^{ij}]=[g_{ij}]^{-1}=\cfrac{A^{ij}}{g}~{};~{}~{}g:=\det([g_{ij}])=\detsymbol% {g}
  109. g = det ( [ g i j ] ) = i g i j A i j g g i j = A i j g=\det([g_{ij}])=\sum_{i}g_{ij}~{}A^{ij}\quad\Rightarrow\quad\frac{\partial g}% {\partial g_{ij}}=A^{ij}
  110. [ g i j ] = 1 g g g i j [g^{ij}]=\cfrac{1}{g}~{}\frac{\partial g}{\partial g_{ij}}
  111. s y m b o l 𝐯 = v i q i + 1 2 g g g m i g i m q v = v i q i + 1 2 g g q v symbol{\nabla}\cdot\mathbf{v}=\frac{\partial v^{i}}{\partial q^{i}}+\cfrac{1}{% 2g}~{}\frac{\partial g}{\partial g_{mi}}~{}\frac{\partial g_{im}}{\partial q^{% \ell}}~{}v^{\ell}=\frac{\partial v^{i}}{\partial q^{i}}+\cfrac{1}{2g}~{}\frac{% \partial g}{\partial q^{\ell}}~{}v^{\ell}
  112. s y m b o l 𝐯 = 1 g q i ( v i g ) symbol{\nabla}\cdot\mathbf{v}=\cfrac{1}{\sqrt{g}}~{}\frac{\partial}{\partial q% ^{i}}(v^{i}~{}\sqrt{g})
  113. ( s y m b o l \cdotsymbol S ) 𝐚 = s y m b o l ( s y m b o l S 𝐚 ) (symbol{\nabla}\cdotsymbol{S})\cdot\mathbf{a}=symbol{\nabla}\cdot(symbol{S}% \cdot\mathbf{a})
  114. s y m b o l \cdotsymbol S = [ S i j q k - Γ k i l S l j - Γ k j l S i l ] g i k 𝐛 j = [ S i j q i + Γ i l i S l j + Γ i l j S i l ] 𝐛 j = [ S j i q i + Γ i l i S j l - Γ i j l S l i ] 𝐛 j = [ S i j q k - Γ i k l S l j + Γ k l j S i l ] g i k 𝐛 j \begin{aligned}\displaystyle symbol{\nabla}\cdotsymbol{S}&\displaystyle=\left[% \cfrac{\partial S_{ij}}{\partial q^{k}}-\Gamma^{l}_{ki}~{}S_{lj}-\Gamma^{l}_{% kj}~{}S_{il}\right]~{}g^{ik}~{}\mathbf{b}^{j}\\ &\displaystyle=\left[\cfrac{\partial S^{ij}}{\partial q^{i}}+\Gamma^{i}_{il}~{% }S^{lj}+\Gamma^{j}_{il}~{}S^{il}\right]~{}\mathbf{b}_{j}\\ &\displaystyle=\left[\cfrac{\partial S^{i}_{~{}j}}{\partial q^{i}}+\Gamma^{i}_% {il}~{}S^{l}_{~{}j}-\Gamma^{l}_{ij}~{}S^{i}_{~{}l}\right]~{}\mathbf{b}^{j}\\ &\displaystyle=\left[\cfrac{\partial S_{i}^{~{}j}}{\partial q^{k}}-\Gamma^{l}_% {ik}~{}S_{l}^{~{}j}+\Gamma^{j}_{kl}~{}S_{i}^{~{}l}\right]~{}g^{ik}~{}\mathbf{b% }_{j}\end{aligned}
  115. 2 φ := s y m b o l ( s y m b o l φ ) \nabla^{2}\varphi:=symbol{\nabla}\cdot(symbol{\nabla}\varphi)
  116. 2 φ = 1 g q i ( [ s y m b o l φ ] i g ) \nabla^{2}\varphi=\cfrac{1}{\sqrt{g}}~{}\frac{\partial}{\partial q^{i}}([% symbol{\nabla}\varphi]^{i}~{}\sqrt{g})
  117. s y m b o l φ = φ q l 𝐛 l = g l i φ q l 𝐛 i [ s y m b o l φ ] i = g l i φ q l symbol{\nabla}\varphi=\frac{\partial\varphi}{\partial q^{l}}~{}\mathbf{b}^{l}=% g^{li}~{}\frac{\partial\varphi}{\partial q^{l}}~{}\mathbf{b}_{i}\quad% \Rightarrow\quad[symbol{\nabla}\varphi]^{i}=g^{li}~{}\frac{\partial\varphi}{% \partial q^{l}}
  118. 2 φ = 1 g q i ( g l i φ q l g ) \nabla^{2}\varphi=\cfrac{1}{\sqrt{g}}~{}\frac{\partial}{\partial q^{i}}\left(g% ^{li}~{}\frac{\partial\varphi}{\partial q^{l}}~{}\sqrt{g}\right)
  119. s y m b o l × 𝐯 = r s t v s | r 𝐛 t symbol{\nabla}\times\mathbf{v}=\mathcal{E}^{rst}v_{s|r}~{}\mathbf{b}_{t}
  120. v s | r = v s , r - Γ s r i v i v_{s|r}=v_{s,r}-\Gamma^{i}_{sr}~{}v_{i}
  121. 𝐛 i 𝐛 j = { g i i if i = j 0 if i j , \mathbf{b}_{i}\cdot\mathbf{b}_{j}=\begin{cases}g_{ii}&\,\text{if }i=j\\ 0&\,\text{if }i\neq j,\end{cases}
  122. 𝐛 i 𝐛 j = { g i i if i = j 0 if i j , \mathbf{b}^{i}\cdot\mathbf{b}^{j}=\begin{cases}g^{ii}&\,\text{if }i=j\\ 0&\,\text{if }i\neq j,\end{cases}
  123. g i i = g i i - 1 g^{ii}=g_{ii}^{-1}
  124. 𝐛 i , 𝐛 j \mathbf{b}_{i},\mathbf{b}_{j}
  125. 𝐱 = i = 1 3 x i 𝐞 i \mathbf{x}=\sum_{i=1}^{3}x_{i}~{}\mathbf{e}_{i}
  126. d 𝐱 = i = 1 3 j = 1 3 ( x i q j 𝐞 i ) d q j \mathrm{d}\mathbf{x}=\sum_{i=1}^{3}\sum_{j=1}^{3}\left(\cfrac{\partial x_{i}}{% \partial q^{j}}~{}\mathbf{e}_{i}\right)\mathrm{d}q^{j}
  127. d 𝐱 d 𝐱 = i = 1 3 j = 1 3 k = 1 3 x i q j x i q k d q j d q k \mathrm{d}\mathbf{x}\cdot\mathrm{d}\mathbf{x}=\sum_{i=1}^{3}\sum_{j=1}^{3}\sum% _{k=1}^{3}\cfrac{\partial x_{i}}{\partial q^{j}}~{}\cfrac{\partial x_{i}}{% \partial q^{k}}~{}\mathrm{d}q^{j}~{}\mathrm{d}q^{k}
  128. g i j ( q i , q j ) = k = 1 3 x k q i x k q j = 𝐛 i 𝐛 j g_{ij}(q^{i},q^{j})=\sum_{k=1}^{3}\cfrac{\partial x_{k}}{\partial q^{i}}~{}% \cfrac{\partial x_{k}}{\partial q^{j}}=\mathbf{b}_{i}\cdot\mathbf{b}_{j}
  129. g i j = 𝐱 q i 𝐱 q j = ( k h k i 𝐞 k ) ( m h m j 𝐞 m ) = k h k i h k j g_{ij}=\cfrac{\partial\mathbf{x}}{\partial q^{i}}\cdot\cfrac{\partial\mathbf{x% }}{\partial q^{j}}=\left(\sum_{k}h_{ki}~{}\mathbf{e}_{k}\right)\cdot\left(\sum% _{m}h_{mj}~{}\mathbf{e}_{m}\right)=\sum_{k}h_{ki}~{}h_{kj}
  130. 𝐛 i 𝐛 i = g i i = k h k i 2 = : h i 2 | 𝐱 q i | = | 𝐛 i | = g i i = h i \mathbf{b}_{i}\cdot\mathbf{b}_{i}=g_{ii}=\sum_{k}h_{ki}^{2}=:h_{i}^{2}\quad% \Rightarrow\quad\left|\cfrac{\partial\mathbf{x}}{\partial q^{i}}\right|=\left|% \mathbf{b}_{i}\right|=\sqrt{g_{ii}}=h_{i}
  131. ( x , y ) = ( r cos θ , r sin θ ) (x,y)=(r\cos\theta,r\sin\theta)\,\!
  132. n n
  133. C f d s = a b f ( 𝐱 ( t ) ) | 𝐱 t | d t \int_{C}f\,ds=\int_{a}^{b}f(\mathbf{x}(t))\left|{\partial\mathbf{x}\over% \partial t}\right|\;dt
  134. | 𝐱 t | = | i = 1 3 𝐱 q i q i t | \left|{\partial\mathbf{x}\over\partial t}\right|=\left|\sum_{i=1}^{3}{\partial% \mathbf{x}\over\partial q^{i}}{\partial q^{i}\over\partial t}\right|
  135. 𝐱 q i = k h k i 𝐞 k {\partial\mathbf{x}\over\partial q^{i}}=\sum_{k}h_{ki}~{}\mathbf{e}_{k}
  136. | 𝐱 t | \displaystyle\left|{\partial\mathbf{x}\over\partial t}\right|
  137. g i j = 0 g_{ij}=0\,
  138. i j i\neq j
  139. | 𝐱 t | = i g i i ( q i t ) 2 = i h i 2 ( q i t ) 2 \left|{\partial\mathbf{x}\over\partial t}\right|=\sqrt{\sum_{i}g_{ii}~{}\left(% \cfrac{\partial q^{i}}{\partial t}\right)^{2}}=\sqrt{\sum_{i}h_{i}^{2}~{}\left% (\cfrac{\partial q^{i}}{\partial t}\right)^{2}}
  140. S f d S = T f ( 𝐱 ( s , t ) ) | 𝐱 s × 𝐱 t | d s d t \int_{S}f\,dS=\iint_{T}f(\mathbf{x}(s,t))\left|{\partial\mathbf{x}\over% \partial s}\times{\partial\mathbf{x}\over\partial t}\right|\,ds\,dt
  141. | 𝐱 s × 𝐱 t | = | ( i 𝐱 q i q i s ) × ( j 𝐱 q j q j t ) | \left|{\partial\mathbf{x}\over\partial s}\times{\partial\mathbf{x}\over% \partial t}\right|=\left|\left(\sum_{i}{\partial\mathbf{x}\over\partial q^{i}}% {\partial q^{i}\over\partial s}\right)\times\left(\sum_{j}{\partial\mathbf{x}% \over\partial q^{j}}{\partial q^{j}\over\partial t}\right)\right|
  142. 𝐱 q i q i s = k ( i = 1 3 h k i q i s ) 𝐞 k ; 𝐱 q j q j t = m ( j = 1 3 h m j q j t ) 𝐞 m {\partial\mathbf{x}\over\partial q^{i}}{\partial q^{i}\over\partial s}=\sum_{k% }\left(\sum_{i=1}^{3}h_{ki}~{}{\partial q^{i}\over\partial s}\right)\mathbf{e}% _{k}~{};~{}~{}{\partial\mathbf{x}\over\partial q^{j}}{\partial q^{j}\over% \partial t}=\sum_{m}\left(\sum_{j=1}^{3}h_{mj}~{}{\partial q^{j}\over\partial t% }\right)\mathbf{e}_{m}
  143. | 𝐱 s × 𝐱 t | \displaystyle\left|{\partial\mathbf{x}\over\partial s}\times{\partial\mathbf{x% }\over\partial t}\right|
  144. \mathcal{E}
  145. | 𝐞 1 𝐞 2 𝐞 3 i h 1 i q i s i h 2 i q i s i h 3 i q i s j h 1 j q j t j h 2 j q j t j h 3 j q j t | \begin{vmatrix}\mathbf{e}_{1}&\mathbf{e}_{2}&\mathbf{e}_{3}\\ &&\\ \sum_{i}h_{1i}{\partial q^{i}\over\partial s}&\sum_{i}h_{2i}{\partial q^{i}% \over\partial s}&\sum_{i}h_{3i}{\partial q^{i}\over\partial s}\\ &&\\ \sum_{j}h_{1j}{\partial q^{j}\over\partial t}&\sum_{j}h_{2j}{\partial q^{j}% \over\partial t}&\sum_{j}h_{3j}{\partial q^{j}\over\partial t}\end{vmatrix}
  146. 𝐛 i = k g i k 𝐛 k ; g i i = 1 g i i = 1 h i 2 \mathbf{b}^{i}=\sum_{k}g^{ik}~{}\mathbf{b}_{k}~{};~{}~{}g^{ii}=\cfrac{1}{g_{ii% }}=\cfrac{1}{h_{i}^{2}}
  147. φ = i φ q i 𝐛 i = i j φ q i g i j 𝐛 j = i 1 h i 2 f q i 𝐛 i ; 𝐯 = i 1 h i 2 𝐯 q i 𝐛 i \nabla\varphi=\sum_{i}{\partial\varphi\over\partial q^{i}}~{}\mathbf{b}^{i}=% \sum_{i}\sum_{j}{\partial\varphi\over\partial q^{i}}~{}g^{ij}~{}\mathbf{b}_{j}% =\sum_{i}\cfrac{1}{h_{i}^{2}}~{}{\partial f\over\partial q^{i}}~{}\mathbf{b}_{% i}~{};~{}~{}\nabla\mathbf{v}=\sum_{i}\cfrac{1}{h_{i}^{2}}~{}{\partial\mathbf{v% }\over\partial q^{i}}\otimes\mathbf{b}_{i}
  148. g = g 11 g 22 g 33 = h 1 2 h 2 2 h 3 2 g = h 1 h 2 h 3 g=g_{11}~{}g_{22}~{}g_{33}=h_{1}^{2}~{}h_{2}^{2}~{}h_{3}^{2}\quad\Rightarrow% \quad\sqrt{g}=h_{1}h_{2}h_{3}
  149. s y m b o l 𝐯 = 1 h 1 h 2 h 3 q i ( h 1 h 2 h 3 v i ) symbol{\nabla}\cdot\mathbf{v}=\cfrac{1}{h_{1}h_{2}h_{3}}~{}\frac{\partial}{% \partial q^{i}}(h_{1}h_{2}h_{3}~{}v^{i})
  150. v i = g i k v k v 1 = g 11 v 1 = v 1 h 1 2 ; v 2 = g 22 v 2 = v 2 h 2 2 ; v 3 = g 33 v 3 = v 3 h 3 2 v^{i}=g^{ik}~{}v_{k}\quad\Rightarrow v^{1}=g^{11}~{}v_{1}=\cfrac{v_{1}}{h_{1}^% {2}}~{};~{}~{}v^{2}=g^{22}~{}v_{2}=\cfrac{v_{2}}{h_{2}^{2}}~{};~{}~{}v^{3}=g^{% 33}~{}v_{3}=\cfrac{v_{3}}{h_{3}^{2}}
  151. s y m b o l 𝐯 = 1 h 1 h 2 h 3 i q i ( h 1 h 2 h 3 h i 2 v i ) symbol{\nabla}\cdot\mathbf{v}=\cfrac{1}{h_{1}h_{2}h_{3}}~{}\sum_{i}\frac{% \partial}{\partial q^{i}}\left(\cfrac{h_{1}h_{2}h_{3}}{h_{i}^{2}}~{}v_{i}\right)
  152. g l i φ q l = { g 11 φ q 1 , g 22 φ q 2 , g 33 φ q 3 } = { 1 h 1 2 φ q 1 , 1 h 2 2 φ q 2 , 1 h 3 2 φ q 3 } g^{li}~{}\frac{\partial\varphi}{\partial q^{l}}=\left\{g^{11}~{}\frac{\partial% \varphi}{\partial q^{1}},g^{22}~{}\frac{\partial\varphi}{\partial q^{2}},g^{33% }~{}\frac{\partial\varphi}{\partial q^{3}}\right\}=\left\{\cfrac{1}{h_{1}^{2}}% ~{}\frac{\partial\varphi}{\partial q^{1}},\cfrac{1}{h_{2}^{2}}~{}\frac{% \partial\varphi}{\partial q^{2}},\cfrac{1}{h_{3}^{2}}~{}\frac{\partial\varphi}% {\partial q^{3}}\right\}
  153. 2 φ = 1 h 1 h 2 h 3 i q i ( h 1 h 2 h 3 h i 2 φ q i ) \nabla^{2}\varphi=\cfrac{1}{h_{1}h_{2}h_{3}}~{}\sum_{i}\frac{\partial}{% \partial q^{i}}\left(\cfrac{h_{1}h_{2}h_{3}}{h_{i}^{2}}~{}\frac{\partial% \varphi}{\partial q^{i}}\right)
  154. × 𝐯 = 1 h 1 h 2 h 3 i = 1 n 𝐞 i j k ε i j k h i ( h k v k ) q j \nabla\times\mathbf{v}=\frac{1}{h_{1}h_{2}h_{3}}\sum_{i=1}^{n}\mathbf{e}_{i}% \sum_{jk}\varepsilon_{ijk}h_{i}\frac{\partial(h_{k}v_{k})}{\partial q^{j}}
  155. ( x 1 , x 2 , x 3 ) = 𝐱 = s y m b o l φ ( q 1 , q 2 , q 3 ) = s y m b o l φ ( r , θ , z ) = { r cos θ , r sin θ , z } (x_{1},x_{2},x_{3})=\mathbf{x}=symbol{\varphi}(q^{1},q^{2},q^{3})=symbol{% \varphi}(r,\theta,z)=\{r\cos\theta,r\sin\theta,z\}
  156. { ψ 1 ( 𝐱 ) , ψ 2 ( 𝐱 ) , ψ 3 ( 𝐱 ) } = ( q 1 , q 2 , q 3 ) ( r , θ , z ) = { x 1 2 + x 2 2 , tan - 1 ( x 2 / x 1 ) , x 3 } \{\psi^{1}(\mathbf{x}),\psi^{2}(\mathbf{x}),\psi^{3}(\mathbf{x})\}=(q^{1},q^{2% },q^{3})\equiv(r,\theta,z)=\{\sqrt{x_{1}^{2}+x_{2}^{2}},\tan^{-1}(x_{2}/x_{1})% ,x_{3}\}
  157. 0 < r < , 0 < θ < 2 π , - < z < 0<r<\infty~{},~{}~{}0<\theta<2\pi~{},~{}~{}-\infty<z<\infty
  158. 𝐛 1 = 𝐞 r = 𝐛 1 𝐛 2 = r 𝐞 θ = r 2 𝐛 2 𝐛 3 = 𝐞 z = 𝐛 3 \begin{aligned}\displaystyle\mathbf{b}_{1}&\displaystyle=\mathbf{e}_{r}=% \mathbf{b}^{1}\\ \displaystyle\mathbf{b}_{2}&\displaystyle=r~{}\mathbf{e}_{\theta}=r^{2}~{}% \mathbf{b}^{2}\\ \displaystyle\mathbf{b}_{3}&\displaystyle=\mathbf{e}_{z}=\mathbf{b}^{3}\end{aligned}
  159. 𝐞 r , 𝐞 θ , 𝐞 z \mathbf{e}_{r},\mathbf{e}_{\theta},\mathbf{e}_{z}
  160. r , θ , z r,\theta,z
  161. g i j = g i j = 0 ( i j ) ; g 11 = 1 , g 22 = 1 r , g 33 = 1 g^{ij}=g_{ij}=0(i\neq j)~{};~{}~{}\sqrt{g^{11}}=1,~{}\sqrt{g^{22}}=\cfrac{1}{r% },~{}\sqrt{g^{33}}=1
  162. Γ 12 2 = Γ 21 2 = 1 r ; Γ 22 1 = - r \Gamma_{12}^{2}=\Gamma_{21}^{2}=\cfrac{1}{r}~{};~{}~{}\Gamma_{22}^{1}=-r
  163. 𝐛 ^ 1 = 𝐞 r ; 𝐛 ^ 2 = 𝐞 θ ; 𝐛 ^ 3 = 𝐞 z \hat{\mathbf{b}}^{1}=\mathbf{e}_{r}~{};~{}~{}\hat{\mathbf{b}}^{2}=\mathbf{e}_{% \theta}~{};~{}~{}\hat{\mathbf{b}}^{3}=\mathbf{e}_{z}
  164. ( v ^ 1 , v ^ 2 , v ^ 3 ) = ( v 1 , v 2 / r , v 3 ) = : ( v r , v θ , v z ) (\hat{v}_{1},\hat{v}_{2},\hat{v}_{3})=(v_{1},v_{2}/r,v_{3})=:(v_{r},v_{\theta}% ,v_{z})
  165. s y m b o l f = f r 𝐞 r + 1 r f θ 𝐞 θ + f z 𝐞 z symbol{\nabla}f=\cfrac{\partial f}{\partial r}~{}\mathbf{e}_{r}+\cfrac{1}{r}~{% }\cfrac{\partial f}{\partial\theta}~{}\mathbf{e}_{\theta}+\cfrac{\partial f}{% \partial z}~{}\mathbf{e}_{z}
  166. s y m b o l 𝐯 = v r r 𝐞 r 𝐞 r + 1 r ( v r θ - v θ ) 𝐞 r 𝐞 θ + v r z 𝐞 r 𝐞 z + v θ r 𝐞 θ 𝐞 r + 1 r ( v θ θ + v r ) 𝐞 θ 𝐞 θ + v θ z 𝐞 θ 𝐞 z + v z r 𝐞 z 𝐞 r + 1 r v z θ 𝐞 z 𝐞 θ + v z z 𝐞 z 𝐞 z \begin{aligned}\displaystyle symbol{\nabla}\mathbf{v}&\displaystyle=\cfrac{% \partial v_{r}}{\partial r}~{}\mathbf{e}_{r}\otimes\mathbf{e}_{r}+\cfrac{1}{r}% \left(\cfrac{\partial v_{r}}{\partial\theta}-v_{\theta}\right)~{}\mathbf{e}_{r% }\otimes\mathbf{e}_{\theta}+\cfrac{\partial v_{r}}{\partial z}~{}\mathbf{e}_{r% }\otimes\mathbf{e}_{z}\\ &\displaystyle+\cfrac{\partial v_{\theta}}{\partial r}~{}\mathbf{e}_{\theta}% \otimes\mathbf{e}_{r}+\cfrac{1}{r}\left(\cfrac{\partial v_{\theta}}{\partial% \theta}+v_{r}\right)~{}\mathbf{e}_{\theta}\otimes\mathbf{e}_{\theta}+\cfrac{% \partial v_{\theta}}{\partial z}~{}\mathbf{e}_{\theta}\otimes\mathbf{e}_{z}\\ &\displaystyle+\cfrac{\partial v_{z}}{\partial r}~{}\mathbf{e}_{z}\otimes% \mathbf{e}_{r}+\cfrac{1}{r}\cfrac{\partial v_{z}}{\partial\theta}~{}\mathbf{e}% _{z}\otimes\mathbf{e}_{\theta}+\cfrac{\partial v_{z}}{\partial z}~{}\mathbf{e}% _{z}\otimes\mathbf{e}_{z}\end{aligned}
  167. s y m b o l 𝐯 = v r r + 1 r ( v θ θ + v r ) + v z z \begin{aligned}\displaystyle symbol{\nabla}\cdot\mathbf{v}&\displaystyle=% \cfrac{\partial v_{r}}{\partial r}+\cfrac{1}{r}\left(\cfrac{\partial v_{\theta% }}{\partial\theta}+v_{r}\right)+\cfrac{\partial v_{z}}{\partial z}\end{aligned}
  168. s y m b o l 2 f = s y m b o l \cdotsymbol f symbol{\nabla}^{2}f=symbol{\nabla}\cdotsymbol{\nabla}f
  169. 𝐯 = s y m b o l f = [ v r v θ v z ] = [ f r 1 r f θ f z ] \mathbf{v}=symbol{\nabla}f=\left[v_{r}~{}~{}v_{\theta}~{}~{}v_{z}\right]=\left% [\cfrac{\partial f}{\partial r}~{}~{}\cfrac{1}{r}\cfrac{\partial f}{\partial% \theta}~{}~{}\cfrac{\partial f}{\partial z}\right]
  170. s y m b o l 𝐯 = s y m b o l 2 f = 2 f r 2 + 1 r ( 1 r 2 f θ 2 + f r ) + 2 f z 2 = 1 r [ r ( r f r ) ] + 1 r 2 2 f θ 2 + 2 f z 2 symbol{\nabla}\cdot\mathbf{v}=symbol{\nabla}^{2}f=\cfrac{\partial^{2}f}{% \partial r^{2}}+\cfrac{1}{r}\left(\cfrac{1}{r}\cfrac{\partial^{2}f}{\partial% \theta^{2}}+\cfrac{\partial f}{\partial r}\right)+\cfrac{\partial^{2}f}{% \partial z^{2}}=\cfrac{1}{r}\left[\cfrac{\partial}{\partial r}\left(r\cfrac{% \partial f}{\partial r}\right)\right]+\cfrac{1}{r^{2}}\cfrac{\partial^{2}f}{% \partial\theta^{2}}+\cfrac{\partial^{2}f}{\partial z^{2}}
  171. S ^ 11 = S 11 = : S r r ; S ^ 12 = S 12 r = : S r θ ; S ^ 13 = S 13 = : S r z S ^ 21 = S 11 r = : S θ r ; S ^ 22 = S 22 r 2 = : S θ θ ; S ^ 23 = S 23 r = : S θ z S ^ 31 = S 31 = : S z r ; S ^ 32 = S 32 r = : S z θ ; S ^ 33 = S 33 = : S z z \begin{aligned}\displaystyle\hat{S}_{11}&\displaystyle=S_{11}=:S_{rr}~{};~{}~{% }\hat{S}_{12}=\cfrac{S_{12}}{r}=:S_{r\theta}~{};~{}~{}\hat{S}_{13}&% \displaystyle=S_{13}=:S_{rz}\\ \displaystyle\hat{S}_{21}&\displaystyle=\cfrac{S_{11}}{r}=:S_{\theta r}~{};~{}% ~{}\hat{S}_{22}=\cfrac{S_{22}}{r^{2}}=:S_{\theta\theta}~{};~{}~{}\hat{S}_{23}&% \displaystyle=\cfrac{S_{23}}{r}=:S_{\theta z}\\ \displaystyle\hat{S}_{31}&\displaystyle=S_{31}=:S_{zr}~{};~{}~{}\hat{S}_{32}=% \cfrac{S_{32}}{r}=:S_{z\theta}~{};~{}~{}\hat{S}_{33}&\displaystyle=S_{33}=:S_{% zz}\end{aligned}
  172. s y m b o l s y m b o l S = S r r r 𝐞 r 𝐞 r 𝐞 r + 1 r [ S r r θ - ( S θ r + S r θ ) ] 𝐞 r 𝐞 r 𝐞 θ + S r r z 𝐞 r 𝐞 r 𝐞 z + S r θ r 𝐞 r 𝐞 θ 𝐞 r + 1 r [ S r θ θ + ( S r r - S θ θ ) ] 𝐞 r 𝐞 θ 𝐞 θ + S r θ z 𝐞 r 𝐞 θ 𝐞 z + S r z r 𝐞 r 𝐞 z 𝐞 r + 1 r [ S r z θ - S θ z ] 𝐞 r 𝐞 z 𝐞 θ + S r z z 𝐞 r 𝐞 z 𝐞 z + S θ r r 𝐞 θ 𝐞 r 𝐞 r + 1 r [ S θ r θ + ( S r r - S θ θ ) ] 𝐞 θ 𝐞 r 𝐞 θ + S θ r z 𝐞 θ 𝐞 r 𝐞 z + S θ θ r 𝐞 θ 𝐞 θ 𝐞 r + 1 r [ S θ θ θ + ( S r θ + S θ r ) ] 𝐞 θ 𝐞 θ 𝐞 θ + S θ θ z 𝐞 θ 𝐞 θ 𝐞 z + S θ z r 𝐞 θ 𝐞 z 𝐞 r + 1 r [ S θ z θ + S r z ] 𝐞 θ 𝐞 z 𝐞 θ + S θ z z 𝐞 θ 𝐞 z 𝐞 z + S z r r 𝐞 z 𝐞 r 𝐞 r + 1 r [ S z r θ - S z θ ] 𝐞 z 𝐞 r 𝐞 θ + S z r z 𝐞 z 𝐞 r 𝐞 z + S z θ r 𝐞 z 𝐞 θ 𝐞 r + 1 r [ S z θ θ + S z r ] 𝐞 z 𝐞 θ 𝐞 θ + S z θ z 𝐞 z 𝐞 θ 𝐞 z + S z z r 𝐞 z 𝐞 z 𝐞 r + 1 r S z z θ 𝐞 z 𝐞 z 𝐞 θ + S z z z 𝐞 z 𝐞 z 𝐞 z \begin{aligned}\displaystyle symbol{\nabla}symbol{S}&\displaystyle=\frac{% \partial S_{rr}}{\partial r}~{}\mathbf{e}_{r}\otimes\mathbf{e}_{r}\otimes% \mathbf{e}_{r}+\cfrac{1}{r}\left[\frac{\partial S_{rr}}{\partial\theta}-(S_{% \theta r}+S_{r\theta})\right]~{}\mathbf{e}_{r}\otimes\mathbf{e}_{r}\otimes% \mathbf{e}_{\theta}+\frac{\partial S_{rr}}{\partial z}~{}\mathbf{e}_{r}\otimes% \mathbf{e}_{r}\otimes\mathbf{e}_{z}\\ &\displaystyle+\frac{\partial S_{r\theta}}{\partial r}~{}\mathbf{e}_{r}\otimes% \mathbf{e}_{\theta}\otimes\mathbf{e}_{r}+\cfrac{1}{r}\left[\frac{\partial S_{r% \theta}}{\partial\theta}+(S_{rr}-S_{\theta\theta})\right]~{}\mathbf{e}_{r}% \otimes\mathbf{e}_{\theta}\otimes\mathbf{e}_{\theta}+\frac{\partial S_{r\theta% }}{\partial z}~{}\mathbf{e}_{r}\otimes\mathbf{e}_{\theta}\otimes\mathbf{e}_{z}% \\ &\displaystyle+\frac{\partial S_{rz}}{\partial r}~{}\mathbf{e}_{r}\otimes% \mathbf{e}_{z}\otimes\mathbf{e}_{r}+\cfrac{1}{r}\left[\frac{\partial S_{rz}}{% \partial\theta}-S_{\theta z}\right]~{}\mathbf{e}_{r}\otimes\mathbf{e}_{z}% \otimes\mathbf{e}_{\theta}+\frac{\partial S_{rz}}{\partial z}~{}\mathbf{e}_{r}% \otimes\mathbf{e}_{z}\otimes\mathbf{e}_{z}\\ &\displaystyle+\frac{\partial S_{\theta r}}{\partial r}~{}\mathbf{e}_{\theta}% \otimes\mathbf{e}_{r}\otimes\mathbf{e}_{r}+\cfrac{1}{r}\left[\frac{\partial S_% {\theta r}}{\partial\theta}+(S_{rr}-S_{\theta\theta})\right]~{}\mathbf{e}_{% \theta}\otimes\mathbf{e}_{r}\otimes\mathbf{e}_{\theta}+\frac{\partial S_{% \theta r}}{\partial z}~{}\mathbf{e}_{\theta}\otimes\mathbf{e}_{r}\otimes% \mathbf{e}_{z}\\ &\displaystyle+\frac{\partial S_{\theta\theta}}{\partial r}~{}\mathbf{e}_{% \theta}\otimes\mathbf{e}_{\theta}\otimes\mathbf{e}_{r}+\cfrac{1}{r}\left[\frac% {\partial S_{\theta\theta}}{\partial\theta}+(S_{r\theta}+S_{\theta r})\right]~% {}\mathbf{e}_{\theta}\otimes\mathbf{e}_{\theta}\otimes\mathbf{e}_{\theta}+% \frac{\partial S_{\theta\theta}}{\partial z}~{}\mathbf{e}_{\theta}\otimes% \mathbf{e}_{\theta}\otimes\mathbf{e}_{z}\\ &\displaystyle+\frac{\partial S_{\theta z}}{\partial r}~{}\mathbf{e}_{\theta}% \otimes\mathbf{e}_{z}\otimes\mathbf{e}_{r}+\cfrac{1}{r}\left[\frac{\partial S_% {\theta z}}{\partial\theta}+S_{rz}\right]~{}\mathbf{e}_{\theta}\otimes\mathbf{% e}_{z}\otimes\mathbf{e}_{\theta}+\frac{\partial S_{\theta z}}{\partial z}~{}% \mathbf{e}_{\theta}\otimes\mathbf{e}_{z}\otimes\mathbf{e}_{z}\\ &\displaystyle+\frac{\partial S_{zr}}{\partial r}~{}\mathbf{e}_{z}\otimes% \mathbf{e}_{r}\otimes\mathbf{e}_{r}+\cfrac{1}{r}\left[\frac{\partial S_{zr}}{% \partial\theta}-S_{z\theta}\right]~{}\mathbf{e}_{z}\otimes\mathbf{e}_{r}% \otimes\mathbf{e}_{\theta}+\frac{\partial S_{zr}}{\partial z}~{}\mathbf{e}_{z}% \otimes\mathbf{e}_{r}\otimes\mathbf{e}_{z}\\ &\displaystyle+\frac{\partial S_{z\theta}}{\partial r}~{}\mathbf{e}_{z}\otimes% \mathbf{e}_{\theta}\otimes\mathbf{e}_{r}+\cfrac{1}{r}\left[\frac{\partial S_{z% \theta}}{\partial\theta}+S_{zr}\right]~{}\mathbf{e}_{z}\otimes\mathbf{e}_{% \theta}\otimes\mathbf{e}_{\theta}+\frac{\partial S_{z\theta}}{\partial z}~{}% \mathbf{e}_{z}\otimes\mathbf{e}_{\theta}\otimes\mathbf{e}_{z}\\ &\displaystyle+\frac{\partial S_{zz}}{\partial r}~{}\mathbf{e}_{z}\otimes% \mathbf{e}_{z}\otimes\mathbf{e}_{r}+\cfrac{1}{r}~{}\frac{\partial S_{zz}}{% \partial\theta}~{}\mathbf{e}_{z}\otimes\mathbf{e}_{z}\otimes\mathbf{e}_{\theta% }+\frac{\partial S_{zz}}{\partial z}~{}\mathbf{e}_{z}\otimes\mathbf{e}_{z}% \otimes\mathbf{e}_{z}\end{aligned}
  173. s y m b o l s y m b o l S = S r r r 𝐞 r + S r θ r 𝐞 θ + S r z r 𝐞 z + 1 r [ S θ r θ + ( S r r - S θ θ ) ] 𝐞 r + 1 r [ S θ θ θ + ( S r θ + S θ r ) ] 𝐞 θ + 1 r [ S θ z θ + S r z ] 𝐞 z + S z r z 𝐞 r + S z θ z 𝐞 θ + S z z z 𝐞 z \begin{aligned}\displaystyle symbol{\nabla}\cdot symbol{S}&\displaystyle=\frac% {\partial S_{rr}}{\partial r}~{}\mathbf{e}_{r}+\frac{\partial S_{r\theta}}{% \partial r}~{}\mathbf{e}_{\theta}+\frac{\partial S_{rz}}{\partial r}~{}\mathbf% {e}_{z}\\ &\displaystyle+\cfrac{1}{r}\left[\frac{\partial S_{\theta r}}{\partial\theta}+% (S_{rr}-S_{\theta\theta})\right]~{}\mathbf{e}_{r}+\cfrac{1}{r}\left[\frac{% \partial S_{\theta\theta}}{\partial\theta}+(S_{r\theta}+S_{\theta r})\right]~{% }\mathbf{e}_{\theta}+\cfrac{1}{r}\left[\frac{\partial S_{\theta z}}{\partial% \theta}+S_{rz}\right]~{}\mathbf{e}_{z}\\ &\displaystyle+\frac{\partial S_{zr}}{\partial z}~{}\mathbf{e}_{r}+\frac{% \partial S_{z\theta}}{\partial z}~{}\mathbf{e}_{\theta}+\frac{\partial S_{zz}}% {\partial z}~{}\mathbf{e}_{z}\end{aligned}

Ternary_commutator.html

  1. [ a , b , c ] = a b c - a c b - b a c + b c a + c a b - c b a . [a,b,c]=abc-acb-bac+bca+cab-cba.\,

Test_functions_for_optimization.html

  1. f ( x , y ) = - 20 exp ( - 0.2 0.5 ( x 2 + y 2 ) ) f(x,y)=-20\exp\left(-0.2\sqrt{0.5\left(x^{2}+y^{2}\right)}\right)
  2. - exp ( 0.5 ( cos ( 2 π x ) + cos ( 2 π y ) ) ) + e + 20 -\exp\left(0.5\left(\cos\left(2\pi x\right)+\cos\left(2\pi y\right)\right)% \right)+e+20
  3. f ( 0 , 0 ) = 0 f(0,0)=0
  4. - 5 x , y 5 -5\leq x,y\leq 5
  5. f ( s y m b o l x ) = i = 1 n x i 2 f(symbol{x})=\sum_{i=1}^{n}x_{i}^{2}
  6. f ( x 1 , , x n ) = f ( 0 , , 0 ) = 0 f(x_{1},\dots,x_{n})=f(0,\dots,0)=0
  7. - x i -\infty\leq x_{i}\leq\infty
  8. 1 i n 1\leq i\leq n
  9. f ( s y m b o l x ) = i = 1 n - 1 [ 100 ( x i + 1 - x i 2 ) 2 + ( x i - 1 ) 2 ] f(symbol{x})=\sum_{i=1}^{n-1}\left[100\left(x_{i+1}-x_{i}^{2}\right)^{2}+\left% (x_{i}-1\right)^{2}\right]
  10. Min = { n = 2 f ( 1 , 1 ) = 0 , n = 3 f ( 1 , 1 , 1 ) = 0 , n > 3 f ( 1 , , 1 ( n ) times ) = 0 \,\text{Min}=\begin{cases}n=2&\rightarrow\quad f(1,1)=0,\\ n=3&\rightarrow\quad f(1,1,1)=0,\\ n>3&\rightarrow\quad f\left(\underbrace{1,\dots,1}_{(n)\,\text{ times}}\right)% =0\\ \end{cases}
  11. - x i -\infty\leq x_{i}\leq\infty
  12. 1 i n 1\leq i\leq n
  13. f ( x , y ) = ( 1.5 - x + x y ) 2 + ( 2.25 - x + x y 2 ) 2 f(x,y)=\left(1.5-x+xy\right)^{2}+\left(2.25-x+xy^{2}\right)^{2}
  14. + ( 2.625 - x + x y 3 ) 2 +\left(2.625-x+xy^{3}\right)^{2}
  15. f ( 3 , 0.5 ) = 0 f(3,0.5)=0
  16. - 4.5 x , y 4.5 -4.5\leq x,y\leq 4.5
  17. f ( x , y ) = ( 1 + ( x + y + 1 ) 2 ( 19 - 14 x + 3 x 2 - 14 y + 6 x y + 3 y 2 ) ) f(x,y)=\left(1+\left(x+y+1\right)^{2}\left(19-14x+3x^{2}-14y+6xy+3y^{2}\right)\right)
  18. ( 30 + ( 2 x - 3 y ) 2 ( 18 - 32 x + 12 x 2 + 48 y - 36 x y + 27 y 2 ) ) \left(30+\left(2x-3y\right)^{2}\left(18-32x+12x^{2}+48y-36xy+27y^{2}\right)\right)
  19. f ( 0 , - 1 ) = 3 f(0,-1)=3
  20. - 2 x , y 2 -2\leq x,y\leq 2
  21. f ( x , y ) = ( x + 2 y - 7 ) 2 + ( 2 x + y - 5 ) 2 f(x,y)=\left(x+2y-7\right)^{2}+\left(2x+y-5\right)^{2}
  22. f ( 1 , 3 ) = 0 f(1,3)=0
  23. - 10 x , y 10 -10\leq x,y\leq 10
  24. f ( x , y ) = 100 | y - 0.01 x 2 | + 0.01 | x + 10 | . f(x,y)=100\sqrt{\left|y-0.01x^{2}\right|}+0.01\left|x+10\right|.\quad
  25. f ( - 10 , 1 ) = 0 f(-10,1)=0
  26. - 15 x - 5 -15\leq x\leq-5
  27. - 3 y 3 -3\leq y\leq 3
  28. f ( x , y ) = 0.26 ( x 2 + y 2 ) - 0.48 x y f(x,y)=0.26\left(x^{2}+y^{2}\right)-0.48xy
  29. f ( 0 , 0 ) = 0 f(0,0)=0
  30. - 10 x , y 10 -10\leq x,y\leq 10
  31. f ( x , y ) = sin 2 ( 3 π x ) + ( x - 1 ) 2 ( 1 + sin 2 ( 3 π y ) ) f(x,y)=\sin^{2}\left(3\pi x\right)+\left(x-1\right)^{2}\left(1+\sin^{2}\left(3% \pi y\right)\right)
  32. + ( y - 1 ) 2 ( 1 + sin 2 ( 2 π y ) ) +\left(y-1\right)^{2}\left(1+\sin^{2}\left(2\pi y\right)\right)
  33. f ( 1 , 1 ) = 0 f(1,1)=0
  34. - 10 x , y 10 -10\leq x,y\leq 10
  35. f ( x , y ) = 2 x 2 - 1.05 x 4 + x 6 6 + x y + y 2 f(x,y)=2x^{2}-1.05x^{4}+\frac{x^{6}}{6}+xy+y^{2}
  36. f ( 0 , 0 ) = 0 f(0,0)=0
  37. - 5 x , y 5 -5\leq x,y\leq 5
  38. f ( x , y ) = - cos ( x ) cos ( y ) exp ( - ( ( x - π ) 2 + ( y - π ) 2 ) ) f(x,y)=-\cos\left(x\right)\cos\left(y\right)\exp\left(-\left(\left(x-\pi\right% )^{2}+\left(y-\pi\right)^{2}\right)\right)
  39. f ( π , π ) = - 1 f(\pi,\pi)=-1
  40. - 100 x , y 100 -100\leq x,y\leq 100
  41. f ( x , y ) = - 0.0001 ( | sin ( x ) sin ( y ) exp ( | 100 - x 2 + y 2 π | ) | + 1 ) 0.1 f(x,y)=-0.0001\left(\left|\sin\left(x\right)\sin\left(y\right)\exp\left(\left|% 100-\frac{\sqrt{x^{2}+y^{2}}}{\pi}\right|\right)\right|+1\right)^{0.1}
  42. Min = { f ( 1.34941 , - 1.34941 ) = - 2.06261 f ( 1.34941 , 1.34941 ) = - 2.06261 f ( - 1.34941 , 1.34941 ) = - 2.06261 f ( - 1.34941 , - 1.34941 ) = - 2.06261 \,\text{Min}=\begin{cases}f\left(1.34941,-1.34941\right)&=-2.06261\\ f\left(1.34941,1.34941\right)&=-2.06261\\ f\left(-1.34941,1.34941\right)&=-2.06261\\ f\left(-1.34941,-1.34941\right)&=-2.06261\\ \end{cases}
  43. - 10 x , y 10 -10\leq x,y\leq 10
  44. f ( x , y ) = - ( y + 47 ) sin ( | y + x 2 + 47 | ) - x sin ( | x - ( y + 47 ) | ) f(x,y)=-\left(y+47\right)\sin\left(\sqrt{\left|y+\frac{x}{2}+47\right|}\right)% -x\sin\left(\sqrt{\left|x-\left(y+47\right)\right|}\right)
  45. f ( 512 , 404.2319 ) = - 959.6407 f(512,404.2319)=-959.6407
  46. - 512 x , y 512 -512\leq x,y\leq 512
  47. f ( x , y ) = - | sin ( x ) cos ( y ) exp ( | 1 - x 2 + y 2 π | ) | f(x,y)=-\left|\sin\left(x\right)\cos\left(y\right)\exp\left(\left|1-\frac{% \sqrt{x^{2}+y^{2}}}{\pi}\right|\right)\right|
  48. Min = { f ( 8.05502 , 9.66459 ) = - 19.2085 f ( - 8.05502 , 9.66459 ) = - 19.2085 f ( 8.05502 , - 9.66459 ) = - 19.2085 f ( - 8.05502 , - 9.66459 ) = - 19.2085 \,\text{Min}=\begin{cases}f\left(8.05502,9.66459\right)&=-19.2085\\ f\left(-8.05502,9.66459\right)&=-19.2085\\ f\left(8.05502,-9.66459\right)&=-19.2085\\ f\left(-8.05502,-9.66459\right)&=-19.2085\end{cases}
  49. - 10 x , y 10 -10\leq x,y\leq 10
  50. f ( x , y ) = sin ( x + y ) + ( x - y ) 2 - 1.5 x + 2.5 y + 1 f(x,y)=\sin\left(x+y\right)+\left(x-y\right)^{2}-1.5x+2.5y+1
  51. f ( - 0.54719 , - 1.54719 ) = - 1.9133 f(-0.54719,-1.54719)=-1.9133
  52. - 1.5 x 4 -1.5\leq x\leq 4
  53. - 3 y 4 -3\leq y\leq 4
  54. f ( x , y ) = 0.5 + sin 2 ( x 2 - y 2 ) - 0.5 ( 1 + 0.001 ( x 2 + y 2 ) ) 2 f(x,y)=0.5+\frac{\sin^{2}\left(x^{2}-y^{2}\right)-0.5}{\left(1+0.001\left(x^{2% }+y^{2}\right)\right)^{2}}
  55. f ( 0 , 0 ) = 0 f(0,0)=0
  56. - 100 x , y 100 -100\leq x,y\leq 100
  57. f ( x , y ) = 0.5 + cos 2 ( sin ( | x 2 - y 2 | ) ) - 0.5 ( 1 + 0.001 ( x 2 + y 2 ) ) 2 f(x,y)=0.5+\frac{\cos^{2}\left(\sin\left(\left|x^{2}-y^{2}\right|\right)\right% )-0.5}{\left(1+0.001\left(x^{2}+y^{2}\right)\right)^{2}}
  58. f ( 0 , 1.25313 ) = 0.292579 f(0,1.25313)=0.292579
  59. - 100 x , y 100 -100\leq x,y\leq 100
  60. f ( s y m b o l x ) = i = 1 n x i 4 - 16 x i 2 + 5 x i 2 f(symbol{x})=\frac{\sum_{i=1}^{n}x_{i}^{4}-16x_{i}^{2}+5x_{i}}{2}
  61. - 39.16617 n < f ( - 2.903534 , , - 2.903534 ( n ) times ) < - 39.16616 n -39.16617n<f\left(\underbrace{-2.903534,\ldots,-2.903534}_{(n)\,\text{ times}}% \right)<-39.16616n
  62. - 5 x i 5 -5\leq x_{i}\leq 5
  63. 1 i n 1\leq i\leq n
  64. f ( x , y ) = 0.1 x y f(x,y)=0.1xy
  65. subjected to: x 2 + y 2 ( r T + r S cos ( n arctan x y ) ) 2 \,\text{subjected to: }x^{2}+y^{2}\leq\left(r_{T}+r_{S}\cos\left(n\arctan\frac% {x}{y}\right)\right)^{2}
  66. where: r T = 1 , r S = 0.2 and n = 8 \,\text{where: }r_{T}=1,r_{S}=0.2\,\text{ and }n=8
  67. f ( ± 0.84852813 , 0.84852813 ) = - 0.072 f(\pm 0.84852813,\mp 0.84852813)=-0.072
  68. - 1.25 x , y 1.25 -1.25\leq x,y\leq 1.25
  69. Minimize = { f 1 ( x , y ) = 4 x 2 + 4 y 2 f 2 ( x , y ) = ( x - 5 ) 2 + ( y - 5 ) 2 \,\text{Minimize}=\begin{cases}f_{1}\left(x,y\right)&=4x^{2}+4y^{2}\\ f_{2}\left(x,y\right)&=\left(x-5\right)^{2}+\left(y-5\right)^{2}\\ \end{cases}
  70. s.t. = { g 1 ( x , y ) = ( x - 5 ) 2 + y 2 25 g 2 ( x , y ) = ( x - 8 ) 2 + ( y + 3 ) 2 7.7 \,\text{s.t.}=\begin{cases}g_{1}\left(x,y\right)&=\left(x-5\right)^{2}+y^{2}% \leq 25\\ g_{2}\left(x,y\right)&=\left(x-8\right)^{2}+\left(y+3\right)^{2}\geq 7.7\\ \end{cases}
  71. 0 x 5 0\leq x\leq 5
  72. 0 y 3 0\leq y\leq 3
  73. Minimize = { f 1 ( x , y ) = 2 + ( x - 2 ) 2 + ( y - 1 ) 2 f 2 ( x , y ) = 9 x - ( y - 1 ) 2 \,\text{Minimize}=\begin{cases}f_{1}\left(x,y\right)&=2+\left(x-2\right)^{2}+% \left(y-1\right)^{2}\\ f_{2}\left(x,y\right)&=9x-\left(y-1\right)^{2}\\ \end{cases}
  74. s.t. = { g 1 ( x , y ) = x 2 + y 2 225 g 2 ( x , y ) = x - 3 y + 10 0 \,\text{s.t.}=\begin{cases}g_{1}\left(x,y\right)&=x^{2}+y^{2}\leq 225\\ g_{2}\left(x,y\right)&=x-3y+10\leq 0\\ \end{cases}
  75. - 20 x , y 20 -20\leq x,y\leq 20
  76. Minimize = { f 1 ( s y m b o l x ) = 1 - exp ( - i = 1 n ( x i - 1 n ) 2 ) f 2 ( s y m b o l x ) = 1 - exp ( - i = 1 n ( x i + 1 n ) 2 ) \,\text{Minimize}=\begin{cases}f_{1}\left(symbol{x}\right)&=1-\exp\left(-\sum_% {i=1}^{n}\left(x_{i}-\frac{1}{\sqrt{n}}\right)^{2}\right)\\ f_{2}\left(symbol{x}\right)&=1-\exp\left(-\sum_{i=1}^{n}\left(x_{i}+\frac{1}{% \sqrt{n}}\right)^{2}\right)\\ \end{cases}
  77. - 4 x i 4 -4\leq x_{i}\leq 4
  78. 1 i n 1\leq i\leq n
  79. Minimize = { f 1 ( x , y ) = x 2 - y f 2 ( x , y ) = - 0.5 x - y - 1 \,\text{Minimize}=\begin{cases}f_{1}\left(x,y\right)&=x^{2}-y\\ f_{2}\left(x,y\right)&=-0.5x-y-1\\ \end{cases}
  80. s.t. = { g 1 ( x , y ) = 6.5 - x 6 - y 0 g 2 ( x , y ) = 7.5 - 0.5 x - y 0 g 3 ( x , y ) = 30 - 5 x - y 0 \,\text{s.t.}=\begin{cases}g_{1}\left(x,y\right)&=6.5-\frac{x}{6}-y\geq 0\\ g_{2}\left(x,y\right)&=7.5-0.5x-y\geq 0\\ g_{3}\left(x,y\right)&=30-5x-y\geq 0\\ \end{cases}
  81. - 7 x , y 4 -7\leq x,y\leq 4
  82. Minimize = { f 1 ( s y m b o l x ) = i = 1 2 [ - 10 exp ( - 0.2 x i 2 + x i + 1 2 ) ] f 2 ( s y m b o l x ) = i = 1 3 [ | x i | 0.8 + 5 sin ( x i 3 ) ] \,\text{Minimize}=\begin{cases}f_{1}\left(symbol{x}\right)&=\sum_{i=1}^{2}% \left[-10\exp\left(-0.2\sqrt{x_{i}^{2}+x_{i+1}^{2}}\right)\right]\\ &\\ f_{2}\left(symbol{x}\right)&=\sum_{i=1}^{3}\left[\left|x_{i}\right|^{0.8}+5% \sin\left(x_{i}^{3}\right)\right]\\ \end{cases}
  83. - 5 x i 5 -5\leq x_{i}\leq 5
  84. 1 i 3 1\leq i\leq 3
  85. Minimize = { f 1 ( x ) = x 2 f 2 ( x ) = ( x - 2 ) 2 \,\text{Minimize}=\begin{cases}f_{1}\left(x\right)&=x^{2}\\ f_{2}\left(x\right)&=\left(x-2\right)^{2}\\ \end{cases}
  86. - A x A -A\leq x\leq A
  87. A A
  88. 10 10
  89. 10 5 10^{5}
  90. A A
  91. Minimize = { f 1 ( x ) = { - x , if x 1 x - 2 , if 1 < x 3 4 - x , if 3 < x 4 x - 4 , if x > 4 f 2 ( x ) = ( x - 5 ) 2 \,\text{Minimize}=\begin{cases}f_{1}\left(x\right)&=\begin{cases}-x,&\,\text{% if }x\leq 1\\ x-2,&\,\text{if }1<x\leq 3\\ 4-x,&\,\text{if }3<x\leq 4\\ x-4,&\,\text{if }x>4\\ \end{cases}\\ f_{2}\left(x\right)&=\left(x-5\right)^{2}\\ \end{cases}
  92. - 5 x 10 -5\leq x\leq 10
  93. Minimize = { f 1 ( x , y ) = [ 1 + ( A 1 - B 1 ( x , y ) ) 2 + ( A 2 - B 2 ( x , y ) ) 2 ] f 2 ( x , y ) = ( x + 3 ) 2 + ( y + 1 ) 2 \,\text{Minimize}=\begin{cases}f_{1}\left(x,y\right)&=\left[1+\left(A_{1}-B_{1% }\left(x,y\right)\right)^{2}+\left(A_{2}-B_{2}\left(x,y\right)\right)^{2}% \right]\\ f_{2}\left(x,y\right)&=\left(x+3\right)^{2}+\left(y+1\right)^{2}\\ \end{cases}
  94. where = { A 1 = 0.5 sin ( 1 ) - 2 cos ( 1 ) + sin ( 2 ) - 1.5 cos ( 2 ) A 2 = 1.5 sin ( 1 ) - cos ( 1 ) + 2 sin ( 2 ) - 0.5 cos ( 2 ) B 1 ( x , y ) = 0.5 sin ( x ) - 2 cos ( x ) + sin ( y ) - 1.5 cos ( y ) B 2 ( x , y ) = 1.5 sin ( x ) - cos ( x ) + 2 sin ( y ) - 0.5 cos ( y ) \,\text{where}=\begin{cases}A_{1}&=0.5\sin\left(1\right)-2\cos\left(1\right)+% \sin\left(2\right)-1.5\cos\left(2\right)\\ A_{2}&=1.5\sin\left(1\right)-\cos\left(1\right)+2\sin\left(2\right)-0.5\cos% \left(2\right)\\ B_{1}\left(x,y\right)&=0.5\sin\left(x\right)-2\cos\left(x\right)+\sin\left(y% \right)-1.5\cos\left(y\right)\\ B_{2}\left(x,y\right)&=1.5\sin\left(x\right)-\cos\left(x\right)+2\sin\left(y% \right)-0.5\cos\left(y\right)\end{cases}
  95. - π x , y π -\pi\leq x,y\leq\pi
  96. Minimize = { f 1 ( s y m b o l x ) = x 1 f 2 ( s y m b o l x ) = g ( s y m b o l x ) h ( f 1 ( s y m b o l x ) , g ( s y m b o l x ) ) g ( s y m b o l x ) = 1 + 9 29 i = 2 30 x i h ( f 1 ( s y m b o l x ) , g ( s y m b o l x ) ) = 1 - f 1 ( s y m b o l x ) g ( s y m b o l x ) \,\text{Minimize}=\begin{cases}f_{1}\left(symbol{x}\right)&=x_{1}\\ f_{2}\left(symbol{x}\right)&=g\left(symbol{x}\right)h\left(f_{1}\left(symbol{x% }\right),g\left(symbol{x}\right)\right)\\ g\left(symbol{x}\right)&=1+\frac{9}{29}\sum_{i=2}^{30}x_{i}\\ h\left(f_{1}\left(symbol{x}\right),g\left(symbol{x}\right)\right)&=1-\sqrt{% \frac{f_{1}\left(symbol{x}\right)}{g\left(symbol{x}\right)}}\\ \end{cases}
  97. 0 x i 1 0\leq x_{i}\leq 1
  98. 1 i 30 1\leq i\leq 30
  99. Minimize = { f 1 ( s y m b o l x ) = x 1 f 2 ( s y m b o l x ) = g ( s y m b o l x ) h ( f 1 ( s y m b o l x ) , g ( s y m b o l x ) ) g ( s y m b o l x ) = 1 + 9 29 i = 2 30 x i h ( f 1 ( s y m b o l x ) , g ( s y m b o l x ) ) = 1 - ( f 1 ( s y m b o l x ) g ( s y m b o l x ) ) 2 \,\text{Minimize}=\begin{cases}f_{1}\left(symbol{x}\right)&=x_{1}\\ f_{2}\left(symbol{x}\right)&=g\left(symbol{x}\right)h\left(f_{1}\left(symbol{x% }\right),g\left(symbol{x}\right)\right)\\ g\left(symbol{x}\right)&=1+\frac{9}{29}\sum_{i=2}^{30}x_{i}\\ h\left(f_{1}\left(symbol{x}\right),g\left(symbol{x}\right)\right)&=1-\left(% \frac{f_{1}\left(symbol{x}\right)}{g\left(symbol{x}\right)}\right)^{2}\\ \end{cases}
  100. 0 x i 1 0\leq x_{i}\leq 1
  101. 1 i 30 1\leq i\leq 30
  102. Minimize = { f 1 ( s y m b o l x ) = x 1 f 2 ( s y m b o l x ) = g ( s y m b o l x ) h ( f 1 ( s y m b o l x ) , g ( s y m b o l x ) ) g ( s y m b o l x ) = 1 + 9 29 i = 2 30 x i h ( f 1 ( s y m b o l x ) , g ( s y m b o l x ) ) = 1 - f 1 ( s y m b o l x ) g ( s y m b o l x ) - ( f 1 ( s y m b o l x ) g ( s y m b o l x ) ) sin ( 10 π f 1 ( s y m b o l x ) ) \,\text{Minimize}=\begin{cases}f_{1}\left(symbol{x}\right)&=x_{1}\\ f_{2}\left(symbol{x}\right)&=g\left(symbol{x}\right)h\left(f_{1}\left(symbol{x% }\right),g\left(symbol{x}\right)\right)\\ g\left(symbol{x}\right)&=1+\frac{9}{29}\sum_{i=2}^{30}x_{i}\\ h\left(f_{1}\left(symbol{x}\right),g\left(symbol{x}\right)\right)&=1-\sqrt{% \frac{f_{1}\left(symbol{x}\right)}{g\left(symbol{x}\right)}}-\left(\frac{f_{1}% \left(symbol{x}\right)}{g\left(symbol{x}\right)}\right)\sin\left(10\pi f_{1}% \left(symbol{x}\right)\right)\end{cases}
  103. 0 x i 1 0\leq x_{i}\leq 1
  104. 1 i 30 1\leq i\leq 30
  105. Minimize = { f 1 ( s y m b o l x ) = x 1 f 2 ( s y m b o l x ) = g ( s y m b o l x ) h ( f 1 ( s y m b o l x ) , g ( s y m b o l x ) ) g ( s y m b o l x ) = 91 + i = 2 10 ( x i 2 - 10 cos ( 4 π x i ) ) h ( f 1 ( s y m b o l x ) , g ( s y m b o l x ) ) = 1 - f 1 ( s y m b o l x ) g ( s y m b o l x ) \,\text{Minimize}=\begin{cases}f_{1}\left(symbol{x}\right)&=x_{1}\\ f_{2}\left(symbol{x}\right)&=g\left(symbol{x}\right)h\left(f_{1}\left(symbol{x% }\right),g\left(symbol{x}\right)\right)\\ g\left(symbol{x}\right)&=91+\sum_{i=2}^{10}\left(x_{i}^{2}-10\cos\left(4\pi x_% {i}\right)\right)\\ h\left(f_{1}\left(symbol{x}\right),g\left(symbol{x}\right)\right)&=1-\sqrt{% \frac{f_{1}\left(symbol{x}\right)}{g\left(symbol{x}\right)}}\end{cases}
  106. 0 x 1 1 0\leq x_{1}\leq 1
  107. - 5 x i 5 -5\leq x_{i}\leq 5
  108. 2 i 10 2\leq i\leq 10
  109. Minimize = { f 1 ( s y m b o l x ) = 1 - exp ( - 4 x 1 ) sin 6 ( 6 π x 1 ) f 2 ( s y m b o l x ) = g ( s y m b o l x ) h ( f 1 ( s y m b o l x ) , g ( s y m b o l x ) ) g ( s y m b o l x ) = 1 + 9 [ i = 2 10 x i 9 ] 0.25 h ( f 1 ( s y m b o l x ) , g ( s y m b o l x ) ) = 1 - ( f 1 ( s y m b o l x ) g ( s y m b o l x ) ) 2 \,\text{Minimize}=\begin{cases}f_{1}\left(symbol{x}\right)&=1-\exp\left(-4x_{1% }\right)\sin^{6}\left(6\pi x_{1}\right)\\ f_{2}\left(symbol{x}\right)&=g\left(symbol{x}\right)h\left(f_{1}\left(symbol{x% }\right),g\left(symbol{x}\right)\right)\\ g\left(symbol{x}\right)&=1+9\left[\frac{\sum_{i=2}^{10}x_{i}}{9}\right]^{0.25}% \\ h\left(f_{1}\left(symbol{x}\right),g\left(symbol{x}\right)\right)&=1-\left(% \frac{f_{1}\left(symbol{x}\right)}{g\left(symbol{x}\right)}\right)^{2}\\ \end{cases}
  110. 0 x i 1 0\leq x_{i}\leq 1
  111. 1 i 10 1\leq i\leq 10
  112. Minimize = { f 1 ( x , y ) = 0.5 ( x 2 + y 2 ) + sin ( x 2 + y 2 ) f 2 ( x , y ) = ( 3 x - 2 y + 4 ) 2 8 + ( x - y + 1 ) 2 27 + 15 f 3 ( x , y ) = 1 x 2 + y 2 + 1 - 1.1 exp ( - ( x 2 + y 2 ) ) \,\text{Minimize}=\begin{cases}f_{1}\left(x,y\right)&=0.5\left(x^{2}+y^{2}% \right)+\sin\left(x^{2}+y^{2}\right)\\ f_{2}\left(x,y\right)&=\frac{\left(3x-2y+4\right)^{2}}{8}+\frac{\left(x-y+1% \right)^{2}}{27}+15\\ f_{3}\left(x,y\right)&=\frac{1}{x^{2}+y^{2}+1}-1.1\exp\left(-\left(x^{2}+y^{2}% \right)\right)\\ \end{cases}
  113. - 3 x , y 3 -3\leq x,y\leq 3
  114. Minimize = { f 1 ( s y m b o l x ) = - 25 ( x 1 - 2 ) 2 - ( x 2 - 2 ) 2 - ( x 3 - 1 ) 2 - ( x 4 - 4 ) 2 - ( x 5 - 1 ) 2 f 2 ( s y m b o l x ) = i = 1 6 x i 2 \,\text{Minimize}=\begin{cases}f_{1}\left(symbol{x}\right)&=-25\left(x_{1}-2% \right)^{2}-\left(x_{2}-2\right)^{2}-\left(x_{3}-1\right)^{2}-\left(x_{4}-4% \right)^{2}-\left(x_{5}-1\right)^{2}\\ f_{2}\left(symbol{x}\right)&=\sum_{i=1}^{6}x_{i}^{2}\\ \end{cases}
  115. s.t. = { g 1 ( s y m b o l x ) = x 1 + x 2 - 2 0 g 2 ( s y m b o l x ) = 6 - x 1 - x 2 0 g 3 ( s y m b o l x ) = 2 - x 2 + x 1 0 g 4 ( s y m b o l x ) = 2 - x 1 + 3 x 2 0 g 5 ( s y m b o l x ) = 4 - ( x 3 - 3 ) 2 - x 4 0 g 6 ( s y m b o l x ) = ( x 5 - 3 ) 2 + x 6 - 4 0 \,\text{s.t.}=\begin{cases}g_{1}\left(symbol{x}\right)&=x_{1}+x_{2}-2\geq 0\\ g_{2}\left(symbol{x}\right)&=6-x_{1}-x_{2}\geq 0\\ g_{3}\left(symbol{x}\right)&=2-x_{2}+x_{1}\geq 0\\ g_{4}\left(symbol{x}\right)&=2-x_{1}+3x_{2}\geq 0\\ g_{5}\left(symbol{x}\right)&=4-\left(x_{3}-3\right)^{2}-x_{4}\geq 0\\ g_{6}\left(symbol{x}\right)&=\left(x_{5}-3\right)^{2}+x_{6}-4\geq 0\end{cases}
  116. 0 x 1 , x 2 , x 6 10 0\leq x_{1},x_{2},x_{6}\leq 10
  117. 1 x 3 , x 5 5 1\leq x_{3},x_{5}\leq 5
  118. 0 x 4 6 0\leq x_{4}\leq 6
  119. Minimize = { f 1 ( x , y ) = x f 2 ( x , y ) = ( 1 + y ) exp ( - x 1 + y ) \,\text{Minimize}=\begin{cases}f_{1}\left(x,y\right)&=x\\ f_{2}\left(x,y\right)&=\left(1+y\right)\exp\left(-\frac{x}{1+y}\right)\end{cases}
  120. s.t. = { g 1 ( x , y ) = f 2 ( x , y ) 0.858 exp ( - 0.541 f 1 ( x , y ) ) 1 g 1 ( x , y ) = f 2 ( x , y ) 0.728 exp ( - 0.295 f 1 ( x , y ) ) 1 \,\text{s.t.}=\begin{cases}g_{1}\left(x,y\right)&=\frac{f_{2}\left(x,y\right)}% {0.858\exp\left(-0.541f_{1}\left(x,y\right)\right)}\geq 1\\ g_{1}\left(x,y\right)&=\frac{f_{2}\left(x,y\right)}{0.728\exp\left(-0.295f_{1}% \left(x,y\right)\right)}\geq 1\end{cases}
  121. 0 x , y 1 0\leq x,y\leq 1
  122. Minimize = { f 1 ( x , y ) = x f 2 ( x , y ) = 1 + y x \,\text{Minimize}=\begin{cases}f_{1}\left(x,y\right)&=x\\ f_{2}\left(x,y\right)&=\frac{1+y}{x}\\ \end{cases}
  123. s.t. = { g 1 ( x , y ) = y + 9 x 6 g 1 ( x , y ) = - y + 9 x 1 \,\text{s.t.}=\begin{cases}g_{1}\left(x,y\right)&=y+9x\geq 6\\ g_{1}\left(x,y\right)&=-y+9x\geq 1\\ \end{cases}
  124. 0.1 x 1 0.1\leq x\leq 1
  125. 0 y 5 0\leq y\leq 5

Test_vector.html

  1. y = f ( x ) y=f(x)
  2. y y
  3. x x
  4. y = f ( x 1 , x 2 , ) y=f(x_{1},x_{2},...)
  5. Y = C ( X ) Y=C(X)
  6. Y Y
  7. C C
  8. X X
  9. y = L ( u , p ) y=L(u,p)
  10. y { t r u e , f a l s e } y\in\{true,false\}
  11. u , p { S t r i n g } u,p\in\{String\}
  12. t r u e true
  13. f a l s e false
  14. L L
  15. Y = L ( X ) Y=L(X)
  16. X = [ x 1 , x 2 ] = [ u , p ] ; Y = [ y 1 ] X=[x_{1},x_{2}]=[u,p]\;;\;Y=[y_{1}]
  17. X X
  18. Y Y
  19. { X 1 , X 2 , X 3 , } \{X_{1},X_{2},X_{3},...\}
  20. X i X_{i}

Tetradic_Palatini_action.html

  1. g α β = e α I e β J η I J g_{\alpha\beta}=e_{\alpha}^{I}e_{\beta}^{J}\eta_{IJ}
  2. η I J = d i a g ( - 1 , 1 , 1 , 1 ) \eta_{IJ}=diag(-1,1,1,1)
  3. 𝒟 α V I = α V I + ω α I J V J . \mathcal{D}_{\alpha}V_{I}=\partial_{\alpha}V_{I}+\omega_{\alpha I}^{\;\;\;\;J}% V_{J}.
  4. ω α I J \omega_{\alpha I}^{\;\;\;\;J}
  5. η I J \eta_{IJ}
  6. Ω α β I J V J = ( 𝒟 α 𝒟 β - 𝒟 β 𝒟 α ) V I \Omega_{\alpha\beta I}^{\;\;\;\;\;\;J}V_{J}=(\mathcal{D}_{\alpha}\mathcal{D}_{% \beta}-\mathcal{D}_{\beta}\mathcal{D}_{\alpha})V_{I}
  7. Ω α β I J = 2 [ α ω β ] I J + 2 ω [ α I K ω β ] K J {\Omega_{\alpha\beta}}^{IJ}=2\partial_{[\alpha}{\omega_{\beta]}}^{IJ}+2{\omega% _{[\alpha}}^{IK}{\omega_{\beta]K}}^{J}
  8. α e β I = 0 \nabla_{\alpha}e_{\beta}^{I}=0
  9. V β I V_{\beta}^{I}
  10. α V β I = α V β I - Γ α β γ V γ I - Γ α J I V β J . \nabla_{\alpha}V_{\beta}^{I}=\partial_{\alpha}V_{\beta}^{I}-\Gamma_{\alpha% \beta}^{\gamma}V_{\gamma}^{I}-\Gamma_{\alpha\;J}^{\;\;I}V_{\beta}^{J}.
  11. R α β I J R_{\alpha\beta}^{\;\;\;\;IJ}
  12. R α β I J V J = ( α β - β α ) V I R_{\alpha\beta I}^{\;\;\;\;\;\;J}V_{J}=(\nabla_{\alpha}\nabla_{\beta}-\nabla_{% \beta}\nabla_{\alpha})V_{I}
  13. R α β γ δ V δ = ( α β - β α ) V γ R_{\alpha\beta\gamma}^{\;\;\;\;\;\;\delta}V_{\delta}=(\nabla_{\alpha}\nabla_{% \beta}-\nabla_{\beta}\nabla_{\alpha})V_{\gamma}
  14. V γ = V I e γ I V_{\gamma}=V_{I}e^{I}_{\gamma}
  15. R α β γ δ = e γ I R α β I J e J δ , R α β = R α γ I J e β I e J γ a n d R = R α β I J e I α e J β R_{\alpha\beta\gamma}^{\;\;\;\;\;\;\delta}=e_{\gamma}^{I}R_{\alpha\beta I}^{\;% \;\;\;\;\;J}e_{J}^{\delta},\quad R_{\alpha\beta}=R_{\alpha\gamma I}^{\;\;\;\;% \;\;J}e^{I}_{\beta}e^{\gamma}_{J}\;\;and\;\;R=R_{\alpha\beta}^{\;\;\;\;IJ}e_{I% }^{\alpha}e_{J}^{\beta}
  16. e I α e J β Ω α β I J e^{\alpha}_{I}e^{\beta}_{J}\Omega_{\alpha\beta}^{\;\;\;\;IJ}
  17. S H - P = d 4 x e e I α e J β Ω α β I J S_{H-P}=\int d^{4}x\;e\;e^{\alpha}_{I}e^{\beta}_{J}\Omega_{\alpha\beta}^{\;\;% \;\;IJ}
  18. e = - g e=\sqrt{-g}
  19. g g
  20. α e β I = 0. \nabla_{\alpha}e^{I}_{\beta}=0.
  21. C α I J C_{\alpha I}^{\;\;\;J}
  22. C α I J V J = ( D α - α ) V I . C_{\alpha I}^{\;\;\;J}V_{J}=(D_{\alpha}-\nabla_{\alpha})V_{I}.
  23. Ω α β I J - R α β I J = [ α C β ] I J + C [ α I M C β ] M J \Omega_{\alpha\beta}^{\;\;\;\;IJ}-R_{\alpha\beta}^{\;\;\;\;IJ}=\nabla_{[\alpha% }C_{\beta]}^{\;\;IJ}+C_{[\alpha}^{\;\;\;IM}C_{\beta]M}^{\;\;\;\;J}
  24. \nabla
  25. C α I J C_{\alpha}^{\;\;IJ}
  26. ω α I J \omega_{\alpha}^{\;\;\;IJ}
  27. C α I J C_{\alpha}^{\;\;IJ}
  28. S H - P = d 4 x e e I α e J β ( R α β I J + [ α C β ] I J + C [ α I M C β ] M J ) S_{H-P}=\int d^{4}x\;e\;e^{\alpha}_{I}e^{\beta}_{J}(R_{\alpha\beta}^{\;\;\;\;% IJ}+\nabla_{[\alpha}C_{\beta]}^{\;\;IJ}+C_{[\alpha}^{\;\;\;IM}C_{\beta]M}^{\;% \;\;\;J})
  29. C α I J C_{\alpha}^{\;\;IJ}
  30. C α I J C_{\alpha}^{\;\;IJ}
  31. e M [ a e N b ] δ [ I M δ J ] K C b K N = 0 e^{[a}_{M}e^{b]}_{N}\delta^{M}_{[I}\delta^{K}_{J]}C_{bK}^{\;\;\;N}=0
  32. C α I J = 0 C_{\alpha}^{\;\;IJ}=0
  33. e M [ a e N b ] δ [ I M δ J ] K e^{[a}_{M}e^{b]}_{N}\delta^{M}_{[I}\delta^{K}_{J]}
  34. \nabla
  35. D D
  36. D D
  37. Ω \Omega
  38. R R
  39. e = det e α I e=\det e_{\alpha}^{I}
  40. δ det ( a ) = det ( a ) ( a - 1 ) j i δ a i j \delta\det(a)=\det(a)(a^{-1})_{ji}\delta a_{ij}
  41. δ e = e e I α δ e α I \delta e=ee_{I}^{\alpha}\delta e_{\alpha}^{I}
  42. δ ( e α I e I α ) = 0 \delta(e_{\alpha}^{I}e_{I}^{\alpha})=0
  43. δ e = - e e α I δ e I α \delta e=-ee_{\alpha}^{I}\delta e_{I}^{\alpha}
  44. δ S H - P = d 4 x e ( ( δ e I α ) e J β Ω α β I J + e I α ( δ e J β ) Ω α β I J - e γ K ( δ e K γ ) e I α e J β Ω α β I J ) \delta S_{H-P}=\int d^{4}x\;e\;\Big((\delta e^{\alpha}_{I})e^{\beta}_{J}\Omega% _{\alpha\beta}^{\;\;\;\;IJ}+e^{\alpha}_{I}(\delta e^{\beta}_{J})\Omega_{\alpha% \beta}^{\;\;\;\;IJ}-e_{\gamma}^{K}(\delta e_{K}^{\gamma})e^{\alpha}_{I}e^{% \beta}_{J}\Omega_{\alpha\beta}^{\;\;\;\;IJ}\Big)
  45. = 2 d 4 x e ( e J β Ω α β I J - 1 2 e M γ e N δ e α I Ω γ δ M N ) ( δ e I α ) \;\;\;\;\;=2\int d^{4}x\;e\;\Big(e^{\beta}_{J}\Omega_{\alpha\beta}^{\;\;\;\;IJ% }-{1\over 2}e_{M}^{\gamma}e_{N}^{\delta}e_{\alpha}^{I}\Omega_{\gamma\delta}^{% \;\;\;\;MN}\Big)(\delta e_{I}^{\alpha})
  46. Ω α β I J \Omega_{\alpha\beta}^{\;\;\;\;IJ}
  47. R α β I J R_{\alpha\beta}^{\;\;\;\;IJ}
  48. e J γ R α γ I J - 1 2 R γ δ M N e M γ e N δ e α I = 0 e_{J}^{\gamma}R_{\alpha\gamma}^{\;\;\;\;IJ}-{1\over 2}R_{\gamma\delta}^{\;\;\;% \;MN}e_{M}^{\gamma}e_{N}^{\delta}e_{\alpha}^{I}=0
  49. e I β e_{I\beta}
  50. R α β - 1 2 R g α β R_{\alpha\beta}-{1\over 2}Rg_{\alpha\beta}
  51. - 1 2 γ e e I α e J β Ω α β M N [ ω ] ϵ M N I J -{1\over 2\gamma}ee_{I}^{\alpha}e_{J}^{\beta}\Omega_{\alpha\beta}^{\;\;\;\;MN}% [\omega]\epsilon^{IJ}_{\;\;\;MN}
  52. S = d 4 x e e I α e J β P M N I J Ω α β M N S=\int d^{4}x\;e\;e^{\alpha}_{I}e^{\beta}_{J}P^{IJ}_{\;\;\;\;MN}\Omega_{\alpha% \beta}^{\;\;\;\;MN}
  53. P M N I J = δ M [ I δ N J ] - 1 2 γ ϵ M N I J . P^{IJ}_{\;\;\;\;MN}=\delta_{M}^{[I}\delta_{N}^{J]}-{1\over 2\gamma}\epsilon^{% IJ}_{\;\;\;MN}.
  54. γ \gamma
  55. γ = - i \gamma=-i
  56. γ = ± i \gamma=\pm i
  57. γ ± i \gamma\not=\pm i
  58. P M N I J P^{IJ}_{\;\;\;\;MN}
  59. ( P - 1 ) I J M N = γ 2 γ 2 + 1 ( δ I [ M δ J N ] + 1 2 γ ϵ I J M N ) . (P^{-1})_{IJ}^{\;\;\;\;MN}={\gamma^{2}\over\gamma^{2}+1}\Big(\delta_{I}^{[M}% \delta_{J}^{N]}+{1\over 2\gamma}\epsilon_{IJ}^{\;\;\;MN}\Big).
  60. γ = ± i \gamma=\pm i
  61. e M [ a e N b ] δ [ I M δ J ] K e^{[a}_{M}e^{b]}_{N}\delta^{M}_{[I}\delta^{K}_{J]}
  62. C α I J = 0 C_{\alpha}^{\;\;IJ}=0
  63. R α β γ δ R_{\alpha\beta\gamma}^{\;\;\;\;\;\;\delta}
  64. R α β γ δ V δ = ( α β - β α ) V γ . R_{\alpha\beta\gamma}^{\;\;\;\;\;\;\delta}V_{\delta}=(\nabla_{\alpha}\nabla_{% \beta}-\nabla_{\beta}\nabla_{\alpha})V_{\gamma}.
  65. V γ = e γ I V I V_{\gamma}=e_{\gamma}^{I}V_{I}
  66. R α β γ δ V δ = ( α β - β α ) V γ R_{\alpha\beta\gamma}^{\;\;\;\;\;\;\delta}V_{\delta}=(\nabla_{\alpha}\nabla_{% \beta}-\nabla_{\beta}\nabla_{\alpha})V_{\gamma}
  67. = ( α β - β α ) ( e γ I V I ) =(\nabla_{\alpha}\nabla_{\beta}-\nabla_{\beta}\nabla_{\alpha})(e_{\gamma}^{I}V% _{I})
  68. = e γ I ( α β - β α ) V I =e_{\gamma}^{I}(\nabla_{\alpha}\nabla_{\beta}-\nabla_{\beta}\nabla_{\alpha})V_% {I}
  69. = e γ I R α β I J e J δ V δ =e_{\gamma}^{I}R_{\alpha\beta I}^{\;\;\;\;\;\;J}e_{J}^{\delta}V_{\delta}
  70. α e β I = 0 \nabla_{\alpha}e_{\beta}^{I}=0
  71. V δ V_{\delta}
  72. R α β γ δ = e γ I R α β I J e J δ R_{\alpha\beta\gamma}^{\;\;\;\;\;\;\delta}=e_{\gamma}^{I}R_{\alpha\beta I}^{\;% \;\;\;\;\;J}e_{J}^{\delta}
  73. R α β = R α γ β γ = R α γ I J e β I e J γ . R_{\alpha\beta}=R_{\alpha\gamma\beta}^{\;\;\;\;\;\;\;\gamma}=R_{\alpha\gamma I% }^{\;\;\;\;\;\;J}e_{\beta}^{I}e_{J}^{\gamma}.
  74. α \alpha
  75. β \beta
  76. R = R α β I J e I α e J β . R=R_{\alpha\beta}^{\;\;\;\;IJ}e_{I}^{\alpha}e_{J}^{\beta}.
  77. D α V I D_{\alpha}V_{I}
  78. 𝒟 a \mathcal{D}_{a}
  79. V I V_{I}
  80. 𝒟 α 𝒟 β V I = 𝒟 α ( β V I + C β I J V J ) \mathcal{D}_{\alpha}\mathcal{D}_{\beta}V_{I}=\mathcal{D}_{\alpha}(\nabla_{% \beta}V_{I}+C_{\beta I}^{\;\;\;J}V_{J})
  81. = α ( β V I + C β I J V J ) + C α I K ( b V K + C β K J V J ) + Γ ¯ α β γ ( γ V I + C γ I J V J ) =\nabla_{\alpha}(\nabla_{\beta}V_{I}+C_{\beta I}^{\;\;\;J}V_{J})+C_{\alpha I}^% {\;\;\;K}(\nabla_{b}V_{K}+C_{\beta K}^{\;\;\;J}V_{J})+\overline{\Gamma}_{% \alpha\beta}^{\gamma}(\nabla_{\gamma}V_{I}+C_{\gamma I}^{\;\;\;J}V_{J})
  82. Γ ¯ α β γ \overline{\Gamma}_{\alpha\beta}^{\gamma}
  83. α \alpha
  84. β \beta
  85. Ω α β I J V J = ( 𝒟 α 𝒟 β - 𝒟 β 𝒟 α ) V I \Omega_{\alpha\beta I}^{\;\;\;\;\;\;J}V_{J}=(\mathcal{D}_{\alpha}\mathcal{D}_{% \beta}-\mathcal{D}_{\beta}\mathcal{D}_{\alpha})V_{I}
  86. = ( α β - β α ) V I + α ( C β I J V J ) - β ( C α I J V J ) =(\nabla_{\alpha}\nabla_{\beta}-\nabla_{\beta}\nabla_{\alpha})V_{I}+\nabla_{% \alpha}(C_{\beta I}^{\;\;\;J}V_{J})-\nabla_{\beta}(C_{\alpha I}^{\;\;\;J}V_{J})
  87. + C α I K β V K - C β I K α V K + C α I K C β K J V J - C β I K C α K J V J +\;C_{\alpha I}^{\;\;\;K}\nabla_{\beta}V_{K}-C_{\beta I}^{\;\;\;K}\nabla_{% \alpha}V_{K}+C_{\alpha I}^{\;\;\;K}C_{\beta K}^{\;\;\;J}V_{J}-C_{\beta I}^{\;% \;\;K}C_{\alpha K}^{\;\;\;J}V_{J}
  88. = R α β I J V J + ( α C β I J - β C α I J + C α I K C β K J - C β I K C α K J ) V J =R_{\alpha\beta I}^{\;\;\;\;\;J}V_{J}+(\nabla_{\alpha}C_{\beta I}^{\;\;\;J}-% \nabla_{\beta}C_{\alpha I}^{\;\;\;J}+C_{\alpha I}^{\;\;\;K}C_{\beta K}^{\;\;\;% J}-C_{\beta_{I}}^{\;\;\;K}C_{\alpha K}^{\;\;\;J})V_{J}
  89. Ω a b I J - R a b I J = 2 [ a C b ] I J + 2 C [ a I K C b ] K J \Omega_{ab}^{\;\;\;\;IJ}-R_{ab}^{\;\;\;\;IJ}=2\nabla_{[a}C_{b]}^{\;\;\;IJ}+2C_% {[a}^{\;\;\;IK}C_{b]K}^{\;\;\;\;\;J}
  90. C α I J C_{\alpha}^{\;\;IJ}
  91. a \nabla_{a}
  92. η I J = e β I e J β \eta_{IJ}=e_{\beta I}e^{\beta}_{J}
  93. 𝒟 α \mathcal{D}_{\alpha}
  94. 0 = ( 𝒟 α - α ) η I J 0=(\mathcal{D}_{\alpha}-\nabla_{\alpha})\eta_{IJ}
  95. = C α I K η K J + C a J K η I K =C_{\alpha I}^{\;\;\;K}\eta_{KJ}+C_{aJ}^{\;\;\;K}\eta_{IK}
  96. = C α I J + C α J I . =C_{\alpha IJ}+C_{\alpha JI}.
  97. C α I J = C α [ I J ] C_{\alpha IJ}=C_{\alpha[IJ]}
  98. C α I J C_{\alpha I}^{\;\;\;\;J}
  99. δ S E H = δ d 4 x e e M γ e N β C [ γ M K C β ] K N \delta S_{EH}=\delta\int d^{4}x\;e\;e_{M}^{\gamma}e_{N}^{\beta}C_{[\gamma}^{\;% \;\;MK}C_{\beta]K}^{\;\;\;\;N}
  100. = δ d 4 x e e M [ γ e N β ] C γ M K C β K N =\delta\int d^{4}x\;e\;e_{M}^{[\gamma}e_{N}^{\beta]}C_{\gamma}^{\;\;\;MK}C_{% \beta K}^{\;\;\;\;N}
  101. = δ d 4 x e e M [ γ e N β ] C γ M K C β K N =\delta\int d^{4}x\;e\;e^{M[\gamma}e^{\beta]}_{N}C_{\gamma M}^{\;\;\;\;K}C_{% \beta K}^{\;\;\;\;\;N}
  102. = d 4 x e e M [ γ e N β ] ( δ γ α δ M I δ J K C β K N + C γ M K δ β α δ K I δ J N ) δ C α I J =\int d^{4}x\;ee^{M[\gamma}e^{\beta]}_{N}\big(\delta_{\gamma}^{\alpha}\delta^{% I}_{M}\delta^{K}_{J}C_{\beta K}^{\;\;\;\;\;N}+C_{\gamma M}^{\;\;\;\;K}\delta^{% \alpha}_{\beta}\delta^{I}_{K}\delta^{N}_{J}\big)\delta C_{\alpha I}^{\;\;\;\;J}
  103. = d 4 x e ( e I [ α e N β ] C β J N + e M [ β e J α ] C β M I ) δ C α I J =\int d^{4}x\;e(e^{I[\alpha}e^{\beta]}_{N}C_{\beta J}^{\;\;\;\;N}+e^{M[\beta}e% ^{\alpha]}_{J}C_{\beta M}^{\;\;\;\;I})\delta C_{\alpha I}^{\;\;\;\;J}
  104. e I [ α e K β ] C β J K + e K [ β e J α ] C β K I = 0 e_{I}^{[\alpha}e^{\beta]}_{K}C_{\beta J}^{\;\;\;\;K}+e^{K[\beta}e^{\alpha]}_{J% }C_{\beta KI}=0
  105. C β I K e K [ α e J β ] + C β J K e I [ α e K β ] = 0. C_{\beta I}^{\;\;\;K}e^{[\alpha}_{K}e^{\beta]}_{J}+C_{\beta J}^{\;\;\;K}e^{[% \alpha}_{I}e^{\beta]}_{K}=0.
  106. C β K I = - C β I K C_{\beta KI}=-C_{\beta IK}
  107. e M [ α e N β ] δ [ I M δ J ] K C β K N = 0. e^{[\alpha}_{M}e^{\beta]}_{N}\delta^{M}_{[I}\delta^{K}_{J]}C_{\beta K}^{\;\;\;% N}=0.
  108. C α I J C_{\alpha}^{\;\;IJ}
  109. C β I K e K [ α e J β ] + C β J K e I [ α e K β ] = 0 E q .1 C_{\beta I}^{\;\;\;K}e^{[\alpha}_{K}e^{\beta]}_{J}+C_{\beta J}^{\;\;\;K}e^{[% \alpha}_{I}e^{\beta]}_{K}=0\;\;\;Eq.1
  110. C α I J = 0 C_{\alpha I}^{\;\;\;\;J}=0
  111. S α β γ := C α I J e β I e γ J . S_{\alpha\beta\gamma}:=C_{\alpha IJ}e^{I}_{\beta}e^{J}_{\gamma}.
  112. C α I J = C α [ I J ] C_{\alpha IJ}=C_{\alpha[IJ]}
  113. S α β γ = S α [ β γ ] S_{\alpha\beta\gamma}=S_{\alpha[\beta\gamma]}
  114. e α I e γ J e_{\alpha}^{I}e_{\gamma}^{J}
  115. C β J I e γ J e I β = 0. C_{\beta J}^{\;\;\;\;I}e_{\gamma}^{J}e_{I}^{\beta}=0.
  116. S α β γ = C α I J e β I e J γ S_{\alpha\beta}^{\;\;\;\;\gamma}=C_{\alpha I}^{\;\;\;J}e_{\beta}^{I}e_{J}^{\gamma}
  117. S β γ β = 0 S_{\beta\gamma}^{\;\;\;\;\beta}=0
  118. ( C β I J e J β ) e γ I = 0 , (C_{\beta I}^{\;\;\;J}e_{J}^{\beta})e_{\gamma}^{I}=0,
  119. e α I e_{\alpha}^{I}
  120. C β I J e J β = 0. C_{\beta I}^{\;\;\;J}e_{J}^{\beta}=0.
  121. C β I K e K β e J α , C_{\beta I}^{\;\;\;K}e^{\beta}_{K}e^{\alpha}_{J},
  122. C β J K e I α e K β C_{\beta J}^{\;\;\;K}e^{\alpha}_{I}e^{\beta}_{K}
  123. C β I K e K α e J β - C β J K e I β e K α = 0. C_{\beta I}^{\;\;\;K}e^{\alpha}_{K}e^{\beta}_{J}-C_{\beta J}^{\;\;\;K}e^{\beta% }_{I}e^{\alpha}_{K}=0.
  124. e γ I e δ J e^{I}_{\gamma}e^{J}_{\delta}
  125. 0 = ( C β I K e K α e J β - C β J K e I β e K α ) e γ I e δ J 0=(C_{\beta I}^{\;\;\;K}e^{\alpha}_{K}e^{\beta}_{J}-C_{\beta J}^{\;\;\;K}e^{% \beta}_{I}e^{\alpha}_{K})e^{I}_{\gamma}e^{J}_{\delta}
  126. = C β I K e K α e γ I δ δ β - C β J K δ γ β e K α e δ J =C_{\beta I}^{\;\;\;K}e^{\alpha}_{K}e^{I}_{\gamma}\delta_{\delta}^{\beta}-C_{% \beta J}^{\;\;\;K}\delta_{\gamma}^{\beta}e^{\alpha}_{K}e^{J}_{\delta}
  127. = C δ I K e γ I e K α - C γ J K e δ J e K α =C_{\delta I}^{\;\;\;K}e^{I}_{\gamma}e^{\alpha}_{K}-C_{\gamma J}^{\;\;\;K}e^{J% }_{\delta}e^{\alpha}_{K}
  128. S γ δ α = S ( γ δ ) α . S_{\gamma\delta}^{\;\;\;\alpha}=S_{(\gamma\delta)}^{\;\;\;\;\;\;\alpha}.
  129. S α β γ = S α [ β γ ] S_{\alpha\beta\gamma}=S_{\alpha[\beta\gamma]}
  130. S α β γ = S ( α β ) γ S_{\alpha\beta\gamma}=S_{(\alpha\beta)\gamma}
  131. S α β γ = S β α γ = - S β γ α = - S γ β α = S γ α β = S α γ β = - S α β γ S_{\alpha\beta\gamma}=S_{\beta\alpha\gamma}=-S_{\beta\gamma\alpha}=-S_{\gamma% \beta\alpha}=S_{\gamma\alpha\beta}=S_{\alpha\gamma\beta}=-S_{\alpha\beta\gamma}
  132. S α β γ = 0 S_{\alpha\beta\gamma}=0
  133. C α I J e β I e γ J = 0 , C_{\alpha IJ}e_{\beta}^{I}e_{\gamma}^{J}=0,
  134. e α I e_{\alpha}^{I}
  135. C α I J = 0 C_{\alpha IJ}=0

Tetrahydrocannabinolic_acid_synthase.html

  1. \rightleftharpoons

Tetrahymanol_synthase.html

  1. \rightleftharpoons

Tetramethylammonium-corrinoid_protein_Co-methyltransferase.html

  1. \rightleftharpoons

The_Kjartansson_constant_Q_model.html

  1. α ( w ) = a 1 | w | 1 - γ ( 1.1 ) \alpha(w)=a_{1}|w|^{1-\gamma}\quad(1.1)
  2. 1 c ( w ) = 1 c + a 1 | w | - γ c o t ( π γ 2 ) ( 1.2 ) \frac{1}{c(w)}=\frac{1}{c_{\infty}}+a_{1}|w|^{-\gamma}cot(\frac{\pi\gamma}{2})% \quad(1.2)
  3. 1 c ( w ) = a 1 | w | - γ c o t ( π γ 2 ) ( 1.2 ) \frac{1}{c(w)}=a_{1}|w|^{-\gamma}cot(\frac{\pi\gamma}{2})\quad(1.2)

The_Kolsky_basic_model_and_modified_model_for_attenuation_and_dispersion.html

  1. K ( i w ) = k ( w ) + i a ( w ) ( 1.3 ) K(iw)=k(w)+ia(w)\quad(1.3)
  2. c ( w ) = w k ( w ) ( 1.4 ) c(w)=\frac{w}{k(w)}\quad(1.4)
  3. α = | w | ( 2 c r Q r ) ( 1.5 ) \alpha=\frac{|w|}{(2c_{r}Q_{r})}\quad(1.5)
  4. 1 c ( w ) = 1 c r ( 1 - 1 π Q r l n | w w r | ) ( 1.6 ) \frac{1}{c(w)}=\frac{1}{c_{r}}(1-\frac{1}{\pi Q_{r}}ln|\frac{w}{w_{r}}|)\quad(% 1.6)
  5. 1 c ( w ) = 1 c r | w w r | - γ ( 1.7 ) \frac{1}{c(w)}=\frac{1}{c_{r}}|\frac{w}{w_{r}}|^{-\gamma}\quad(1.7)
  6. γ = ( π Q r ) - 1 \gamma=(\pi Q_{r})^{-1}

Thebaine_6-O-demethylase.html

  1. \rightleftharpoons

Theorem_on_formal_functions.html

  1. f : X S f:X\to S
  2. \mathcal{F}
  3. S 0 S_{0}
  4. \mathcal{I}
  5. X ^ , S ^ \widehat{X},\widehat{S}
  6. X 0 = f - 1 ( S 0 ) X_{0}=f^{-1}(S_{0})
  7. S 0 S_{0}
  8. p 0 p\geq 0
  9. ( R p f * ) lim k R p f * k (R^{p}f_{*}\mathcal{F})^{\wedge}\to\underleftarrow{\lim}_{k}R^{p}f_{*}\mathcal% {F}_{k}
  10. 𝒪 S ^ \mathcal{O}_{\widehat{S}}
  11. lim R p f * 𝒪 S 𝒪 S / k + 1 \underleftarrow{\lim}R^{p}f_{*}\mathcal{F}\otimes_{\mathcal{O}_{S}}\mathcal{O}% _{S}/{\mathcal{I}^{k+1}}
  12. k = 𝒪 S ( 𝒪 S / k + 1 ) \mathcal{F}_{k}=\mathcal{F}\otimes_{\mathcal{O}_{S}}(\mathcal{O}_{S}/{\mathcal% {I}}^{k+1})
  13. s S s\in S
  14. ( ( R p f * ) s ) lim H p ( f - 1 ( s ) , 𝒪 S ( 𝒪 s / 𝔪 s k ) ) ((R^{p}f_{*}\mathcal{F})_{s})^{\wedge}\simeq\underleftarrow{\lim}H^{p}(f^{-1}(% s),\mathcal{F}\otimes_{\mathcal{O}_{S}}(\mathcal{O}_{s}/\mathfrak{m}_{s}^{k}))
  15. 𝔪 s \mathfrak{m}_{s}
  16. dim f - 1 ( s ) r \operatorname{dim}f^{-1}(s)\leq r
  17. s S s\in S
  18. R i f * = 0 , i > r . R^{i}f_{*}\mathcal{F}=0,\quad i>r.
  19. s S s\in S
  20. R i f * | U = 0 , i > dim f - 1 ( s ) . R^{i}f_{*}\mathcal{F}|_{U}=0,\quad i>\operatorname{dim}f^{-1}(s).
  21. f * 𝒪 X = 𝒪 S f_{*}\mathcal{O}_{X}=\mathcal{O}_{S}
  22. f - 1 ( s ) f^{-1}(s)
  23. s S s\in S
  24. i : X ^ X , i : S ^ S i^{\prime}:\widehat{X}\to X,i:\widehat{S}\to S
  25. 𝒪 S ^ \mathcal{O}_{\widehat{S}}
  26. i * R q f * R p f ^ * ( i * ) i^{*}R^{q}f_{*}\mathcal{F}\to R^{p}\widehat{f}_{*}(i^{\prime*}\mathcal{F})
  27. f ^ : X ^ S ^ \widehat{f}:\widehat{X}\to\widehat{S}
  28. f : X S f:X\to S
  29. \mathcal{F}
  30. i * i^{\prime*}\mathcal{F}
  31. ^ \widehat{\mathcal{F}}
  32. R q f * R^{q}f_{*}\mathcal{F}
  33. ( R q f * ) R p f ^ * ^ (R^{q}f_{*}\mathcal{F})^{\wedge}\to R^{p}\widehat{f}_{*}\widehat{\mathcal{F}}
  34. f : X n S n f:X_{n}\to S_{n}
  35. X n = ( X 0 , 𝒪 X / 𝒥 n + 1 ) X_{n}=(X_{0},\mathcal{O}_{X}/\mathcal{J}^{n+1})
  36. S n = ( S 0 , 𝒪 S / n + 1 ) S_{n}=(S_{0},\mathcal{O}_{S}/\mathcal{I}^{n+1})
  37. R q f ^ * ^ lim R p f * n R^{q}\widehat{f}_{*}\widehat{\mathcal{F}}\to\underleftarrow{\lim}R^{p}f_{*}% \mathcal{F}_{n}
  38. n \mathcal{F}_{n}

Thermal_conductance_quantum.html

  1. g 0 g_{0}
  2. g 0 = π 2 k B 2 T 3 h ( 9.456 × 10 - 13 W / K 2 ) T g_{0}=\frac{\pi^{2}{k_{B}}^{2}T}{3h}\approx(9.456\times 10^{-13}W/K^{2})T
  3. g 0 g_{0}
  4. g 0 g_{0}

Thermal_history_of_the_Earth.html

  1. Q surf = Q sec,man + Q rad + Q cmb Q\text{surf}=Q\text{sec,man}+Q\text{rad}+Q\text{cmb}
  2. Q cmb = Q sec,core + Q L + Q G Q\text{cmb}=Q\text{sec,core}+Q\text{L}+Q\text{G}
  3. Q surf Q\text{surf}
  4. Q sec,man = M man c man d T man / d t Q\text{sec,man}=M\text{man}c\text{man}dT\text{man}/dt
  5. M man M\text{man}
  6. c man c\text{man}
  7. T man T\text{man}
  8. Q rad Q\text{rad}
  9. Q cmb Q\text{cmb}
  10. Q sec,core = M core c core d T core / d t Q\text{sec,core}=M\text{core}c\text{core}dT\text{core}/dt
  11. Q L Q\text{L}
  12. Q G Q\text{G}
  13. d T man / d t dT\text{man}/dt
  14. d T core / d t dT\text{core}/dt
  15. d T man d t = 3 ( - Q surf - Q cmb ) 4 π ρ m c m ( R 3 - R c 3 ) + Q rad V m ρ m c m \frac{dT\text{man}}{dt}=\frac{3(-Q\text{surf}-Q\text{cmb})}{4\pi\rho\text{m}c% \text{m}(R^{3}-R\text{c}^{3})}+\frac{Q\text{rad}}{V\text{m}\rho\text{m}c\text{% m}}
  16. d T core d t = Q cmb [ A c ( L + E G ) ( R i R c ) 2 ρ i d R i d T cmb η c - R c 3 - R i 3 3 R c 3 ρ c c c ] - 1 \frac{dT\text{core}}{dt}=Q\text{cmb}\left[A\text{c}(L+E_{G})\left(\frac{R_{i}}% {R\text{c}}\right)^{2}\rho_{i}\frac{dR_{i}}{dT\text{cmb}\eta\text{c}}-\frac{R% \text{c}^{3}-R_{i}^{3}}{3R\text{c}^{3}}\rho\text{c}c\text{c}\right]^{-1}
  17. Q cmb = 0 Q\text{cmb}=0
  18. T man T\text{man}
  19. U r = 1 / 3 Ur=1/3
  20. beta = 1 / 3 \,\text{beta}=1/3

Thermodynamics_of_micellization.html

  1. Δ G m i c e l l e = R T × l n ( C M C ) \Delta G_{micelle}=RT\times ln(CMC)
  2. Δ G m i c e l l e \Delta G_{micelle}
  3. R R
  4. T T
  5. C M C CMC
  6. N S M NS\rightleftharpoons M
  7. Δ G m i c e l l e = - R T N l n [ m i c e l l e ] + R T × l n ( C M C ) \Delta G_{micelle}=-\frac{RT}{N}ln[micelle]+RT\times ln(CMC)
  8. Δ G m i c e l l e \Delta G_{micelle}
  9. R R
  10. T T
  11. N N
  12. [ m i c e l l e ] [micelle]
  13. C M C CMC
  14. N N
  15. I I
  16. N S - + N I + M I ( N - i ) - + ( N - i ) I + NS^{-}+NI^{+}\rightleftharpoons MI^{(N-i)-}+(N-i)I^{+}
  17. Δ G m i c e l l e = R T × l n ( x s ) + R T N l n ( x m N ) \Delta G_{micelle}=RT\times ln(x_{s})+\frac{RT}{N}ln(\frac{x_{m}}{N})
  18. Δ G m i c e l l e \Delta G_{micelle}
  19. R R
  20. T T
  21. x s x_{s}
  22. x m x_{m}
  23. Δ G m i c e l l e = Δ G H P + Δ G E L + Δ G I F \Delta G_{micelle}=\Delta G_{HP}+\Delta G_{EL}+\Delta G_{IF}
  24. N S = V c a * L c N_{\,\text{S}}=\frac{V_{c}}{a*L_{c}}

Thermospermine_synthase.html

  1. \rightleftharpoons

Theta_operator.html

  1. θ = z d d z \theta=z{d\over dz}
  2. θ ( z k ) = k z k , k = 0 , 1 , 2 , \theta(z^{k})=kz^{k},\quad k=0,1,2,\dots
  3. θ = k = 1 n x k x k . \theta=\sum_{k=1}^{n}x_{k}\frac{\partial}{\partial x_{k}}.

Thiazole_synthase.html

  1. \rightleftharpoons

Thiele_modulus.html

  1. h T = diffusion time reaction time h_{T}=\dfrac{\mbox{diffusion time}~{}}{\mbox{reaction time}~{}}
  2. π r 2 ( - D c d C d x ) x = π r 2 ( - D c d C d x ) x + Δ x + ( 2 π r Δ x ) ( k 1 C ) {\pi}r^{2}\left(-D_{c}\frac{dC}{dx}\right)_{x}={\pi}r^{2}\left(-D_{c}\frac{dC}% {dx}\right)_{x+{\Delta}x}+\left(2{\pi}r{\Delta}x\right)\left(k_{1}C\right)
  3. D c D_{c}
  4. k 1 k_{1}
  5. Δ x {\Delta}x
  6. Δ x {\Delta}x
  7. D c ( d 2 C d x 2 ) = 2 k 1 C r D_{c}\left(\frac{d^{2}C}{dx^{2}}\right)=\frac{2k_{1}C}{r}
  8. C = C o at x = 0 C=C_{o}\,\text{ at }x=0
  9. d C d x = 0 at x = L \frac{dC}{dx}=0\,\text{ at }x=L
  10. d 2 C d ( x / L ) 2 = ( 2 k 1 L 2 r D c ) C \frac{d^{2}C}{d(x/L)^{2}}=\left(\frac{2k_{1}L^{2}}{rD_{c}}\right)C
  11. h T 2 = 2 k 1 L 2 r D c h^{2}_{T}=\frac{2k_{1}L^{2}}{rD_{c}}
  12. h T h_{T}
  13. h 2 2 = 2 L 2 k 2 C o r D c h^{2}_{2}=\frac{2L^{2}k_{2}C_{o}}{rD_{c}}
  14. h o 2 = 2 L 2 k o r D c C o h^{2}_{o}=\frac{2L^{2}k_{o}}{rD_{c}C_{o}}
  15. η = tanh h T h T {\eta}=\frac{\tanh h_{T}}{h_{T}}

Thinning_(morphology).html

  1. E = Z 2 E=Z^{2}
  2. C 1 = { ( 0 , 0 ) , ( - 1 , - 1 ) , ( 0 , - 1 ) , ( 1 , - 1 ) } C_{1}=\{(0,0),(-1,-1),(0,-1),(1,-1)\}
  3. D 1 = { ( - 1 , 1 ) , ( 0 , 1 ) , ( 1 , 1 ) } D_{1}=\{(-1,1),(0,1),(1,1)\}
  4. C 2 = { ( - 1 , 0 ) , ( 0 , 0 ) , ( - 1 , - 1 ) , ( 0 , - 1 ) } C_{2}=\{(-1,0),(0,0),(-1,-1),(0,-1)\}
  5. D 2 = { ( 0 , 1 ) , ( 1 , 1 ) , ( 1 , 0 ) } D_{2}=\{(0,1),(1,1),(1,0)\}
  6. 90 o 90^{o}
  7. 180 o 180^{o}
  8. 270 o 270^{o}
  9. B 1 , , B 8 B_{1},\ldots,B_{8}
  10. X B i = X ( X B i ) X\otimes B_{i}=X\setminus(X\odot B_{i})
  11. \setminus
  12. \odot
  13. A B 1 B 2 B 8 B 1 B 2 A\otimes B_{1}\otimes B_{2}\otimes\ldots\otimes B_{8}\otimes B_{1}\otimes B_{2% }\otimes\ldots

Thomas–Fermi_screening.html

  1. k 0 2 = 4 π e 2 n μ k_{0}^{2}=4\pi e^{2}\frac{\partial n}{\partial\mu}
  2. k 0 2 = 4 π e 2 n / ( k B T ) k_{0}^{2}=4\pi e^{2}n/(k_{B}T)
  3. n ( μ ) n(\mu)
  4. n ( μ ) μ 3 / 2 n(\mu)\propto\mu^{3/2}
  5. n = 2 1 ( 2 π ) 3 4 3 π k F 3 , μ = 2 k F 2 2 m , n μ 3 / 2 . n=2\frac{1}{(2\pi)^{3}}\frac{4}{3}\pi k_{F}^{3}\quad,\quad\mu=\frac{\hbar^{2}k% _{F}^{2}}{2m}\quad,\quad n\propto\mu^{3/2}.
  6. n ( μ ) e μ / k B T n(\mu)\propto e^{\mu/k_{B}T}
  7. ρ induced ( 𝐫 ) = - e [ n ( μ 0 + e ϕ ( 𝐫 ) ) - n ( μ 0 ) ] \rho^{\,\text{induced}}(\mathbf{r})=-e[n(\mu_{0}+e\phi(\mathbf{r}))-n(\mu_{0})]
  8. ρ induced ( 𝐫 ) \rho^{\,\text{induced}}(\mathbf{r})
  9. ϕ \phi
  10. ϕ ( 𝐫 ) = 0 \phi(\mathbf{r})=0
  11. ρ induced ( 𝐫 ) - e 2 n μ ϕ ( 𝐫 ) \rho^{\,\text{induced}}(\mathbf{r})\approx-e^{2}\frac{\partial n}{\partial\mu}% \phi(\mathbf{r})
  12. n / μ \partial n/\partial\mu
  13. ϵ ( 𝐪 ) = 1 + k 0 2 q 2 \epsilon(\mathbf{q})=1+\frac{k_{0}^{2}}{q^{2}}
  14. k 0 = 4 π e 2 n μ k_{0}=\sqrt{4\pi e^{2}\frac{\partial n}{\partial\mu}}
  15. ϕ ( 𝐫 ) \phi(\mathbf{r})
  16. ϕ ( 𝐫 ) = Q r e - k 0 r \phi(\mathbf{r})=\frac{Q}{r}e^{-k_{0}r}

Thyristor_switched_capacitor.html

  1. I t s c = V s v c X t s c I_{tsc}={V_{svc}\over{X_{tsc}}}
  2. X t s c = 1 2 π f C t s c - 2 π f L t s c X_{tsc}={{1\over{2\pi fC_{tsc}}}-2\pi fL_{tsc}}
  3. f t s c = 1 2 π f C t s c L t s c f_{tsc}={{1\over{2\pi fC_{tsc}L_{tsc}}}}

Titanium_biocompatibility.html

  1. Γ \Gamma
  2. Γ = Q ADS M n F \Gamma={Q_{\,\text{ADS}}M\over nF}
  3. v c = 2 π D c d N A v_{\,\text{c}}={2\pi DcdN_{\,\text{A}}}
  4. Γ = B ADS Γ max ( 1 + B ADSc ) \Gamma={B_{\,\text{ADS}}\Gamma_{\,\text{max}}\over(1+B_{\,\text{ADSc}})}
  5. Γ \Gamma
  6. c Γ = 1 B ADS Γ max + c Γ max {c\over\Gamma}={{1\over{B_{\,\text{ADS}}\Gamma_{\,\text{max}}}}+{c\over\Gamma_% {\,\text{max}}}}

Tobler's_hiking_function.html

  1. W = 6 e - 3.5 | d h d x + 0.05 | W=6e^{\displaystyle-3.5\left|\frac{dh}{dx}+0.05\right|}
  2. d h d x = S = tan Θ \frac{dh}{dx}=S=\tan\Theta

Tom_Sanders_(mathematician).html

  1. O ( N ( log log N ) 5 log N ) O\left(\frac{N(\log\log N)^{5}}{\log N}\right)

Toom's_rule.html

  1. t = 0 t=0
  2. ( t > 0 ) (t>0)

Topological_complexity.html

  1. P X = { γ : [ 0 , 1 ] X } PX=\{\gamma:[0,1]\,\to\,X\}
  2. π : P X X × X \pi:PX\to\,X\times X
  3. π ( γ ) = ( γ ( 0 ) , γ ( 1 ) ) \pi(\gamma)=(\gamma(0),\gamma(1))
  4. { U i } i = 1 k \{U_{i}\}_{i=1}^{k}
  5. X × X X\times X
  6. i = 1 , , k i=1,\ldots,k
  7. s i : U i P X . s_{i}:\,U_{i}\to\,PX.
  8. S n S^{n}
  9. S 1 S^{1}
  10. F ( \R m , n ) F(\R^{m},n)
  11. T C ( F ( \R m , n ) ) = { 2 n - 1 for m odd 2 n - 2 for m even . TC(F(\R^{m},n))=\begin{cases}2n-1&\mathrm{for\,\,{\it m}\,\,odd}\\ 2n-2&\mathrm{for\,\,{\it m}\,\,even.}\end{cases}

Topological_degeneracy.html

  1. 2 N d / 2 / 2 2^{N_{d}/2}/2
  2. N d N_{d}

Topological_graph.html

  1. O ( k 4 n ) O(k^{4}n)
  2. O ( k 2 n ) O(k^{2}n)
  3. O ( n log 4 k - 8 n ) O(n\log^{4k-8}n)
  4. c log n log log n c\frac{\log n}{\log\log n}
  5. c n 1 3 cn^{\frac{1}{3}}
  6. n log O ( log k ) n n\log^{O(\log k)}n
  7. O ( pcr ( G ) 7 4 log 3 2 pcr ( G ) ) O(\operatorname{pcr}(G)^{\frac{7}{4}}\log^{\frac{3}{2}}\operatorname{pcr}(G))
  8. n + 1 3 \left\lceil\frac{n+1}{3}\right\rceil
  9. n 2 3 n^{\frac{2}{3}}
  10. d \mathbb{R}^{d}
  11. O ( n d ) O(n^{d})
  12. 2 \mathbb{R}^{2}
  13. O ( n d + 1 - δ ) O(n^{d+1-\delta})
  14. δ = δ ( k , d ) < 1 \delta=\delta(k,d)<1
  15. O ( n d ) O(n^{d})
  16. O ( n d 2 ) O(n^{\lceil\frac{d}{2}\rceil})
  17. d \mathbb{R}^{d}
  18. 3 \mathbb{R}^{3}
  19. 3 \mathbb{R}^{3}
  20. o ( n 2 ) o(n^{2})
  21. c n 2 3 cn^{\frac{2}{3}}

TOPSIS.html

  1. x i j x_{ij}
  2. ( x i j ) m × n (x_{ij})_{m\times n}
  3. ( x i j ) m × n (x_{ij})_{m\times n}
  4. R = ( r i j ) m × n R=(r_{ij})_{m\times n}
  5. r i j = x i j i = 1 m x i j 2 , i = 1 , 2 , , m , j = 1 , 2 , , n r_{ij}=\frac{x_{ij}}{\sqrt{\sum_{i=1}^{m}x_{ij}^{2}}},i=1,2,...,m,j=1,2,...,n
  6. T = ( t i j ) m × n = ( w j r i j ) m × n , i = 1 , 2 , , m T=(t_{ij})_{m\times n}=(w_{j}r_{ij})_{m\times n},i=1,2,...,m
  7. w j = W j / j = 1 n W j , j = 1 , 2 , , n w_{j}=W_{j}/\sum_{j=1}^{n}W_{j},j=1,2,...,n
  8. j = 1 n w j = 1 \sum_{j=1}^{n}w_{j}=1
  9. W j W_{j}
  10. v j , j = 1 , 2 , , n . v_{j},j=1,2,...,n.
  11. ( A w ) (A_{w})
  12. ( A b ) (A_{b})
  13. A w = { m a x ( t i j | i = 1 , 2 , , m ) | j J - , m i n ( t i j | i = 1 , 2 , , m ) | j J + } { t w j | j = 1 , 2 , , n } , A_{w}=\{\langle max(t_{ij}|i=1,2,...,m)|j\in J_{-}\rangle,\langle min(t_{ij}|i% =1,2,...,m)|j\in J_{+}\rangle\}\equiv\{t_{wj}|j=1,2,...,n\},
  14. A b = { m i n ( t i j | i = 1 , 2 , , m ) | j J - , m a x ( t i j | i = 1 , 2 , , m ) | j J + } { t b j | j = 1 , 2 , , n } , A_{b}=\{\langle min(t_{ij}|i=1,2,...,m)|j\in J_{-}\rangle,\langle max(t_{ij}|i% =1,2,...,m)|j\in J_{+}\rangle\}\equiv\{t_{bj}|j=1,2,...,n\},
  15. J + = { j = 1 , 2 , , n | j J_{+}=\{j=1,2,...,n|j
  16. J - = { j = 1 , 2 , , n | j J_{-}=\{j=1,2,...,n|j
  17. i i
  18. A w A_{w}
  19. d i w = j = 1 n ( t i j - t w j ) 2 , i = 1 , 2 , , m d_{iw}=\sqrt{\sum_{j=1}^{n}(t_{ij}-t_{wj})^{2}},i=1,2,...,m
  20. i i
  21. A b A_{b}
  22. d i b = j = 1 n ( t i j - t b j ) 2 , i = 1 , 2 , , m d_{ib}=\sqrt{\sum_{j=1}^{n}(t_{ij}-t_{bj})^{2}},i=1,2,...,m
  23. d i w d_{iw}
  24. d i b d_{ib}
  25. i i
  26. s i w = d i w / ( d i w + d i b ) , 0 s i w 1 , i = 1 , 2 , , m s_{iw}=d_{iw}/(d_{iw}+d_{ib}),0\leq s_{iw}\leq 1,i=1,2,...,m
  27. s i w = 1 s_{iw}=1
  28. s i w = 0 s_{iw}=0
  29. s i w ( i = 1 , 2 , , m ) s_{iw}(i=1,2,...,m)
  30. r i j = x i j i = 1 m x i j 2 , i = 1 , 2 , , m , j = 1 , 2 , , n r_{ij}=\frac{x_{ij}}{\sqrt{\sum_{i=1}^{m}x_{ij}^{2}}},i=1,2,...,m,j=1,2,...,n

Torsion_sheaf.html

  1. \mathcal{F}
  2. Γ ( U , ) \Gamma(U,\mathcal{F})
  3. \mathcal{F}

Torulene_dioxygenase.html

  1. \rightleftharpoons

Total_dual_integrality.html

  1. A x b Ax\leq b
  2. A A
  3. b b
  4. c n c\in\mathbb{Z}^{n}
  5. max c T x A x b , \begin{aligned}&&\displaystyle\max c^{\mathrm{T}}x\\ &&\displaystyle Ax\leq b,\end{aligned}
  6. P P
  7. A x b Ax\leq b
  8. b b
  9. P P
  10. P P
  11. P P
  12. A x b Ax\leq b
  13. b b

Touschek_effect.html

  1. 1 τ = 1 C 1 τ l ( s ) d s \frac{1}{\tau}=\frac{1}{C}\oint\frac{1}{\tau_{l}}(s)\,ds
  2. 1 τ = 1 2 ( 1 τ + + 1 τ - ) \frac{1}{\tau}=\frac{1}{2}\left(\frac{1}{\tau_{+}}+\frac{1}{\tau_{-}}\right)
  3. 1 τ l ( s ) = r 0 2 c N 8 π γ 2 σ x σ y σ z δ acc 3 F ( ε m ) . \frac{1}{\tau_{l}}(s)=\frac{r_{0}^{2}cN}{8\pi\gamma^{2}\sigma_{x}\sigma_{y}% \sigma_{z}\delta_{\mathrm{acc}}^{3}}F(\varepsilon_{m}).
  4. r 0 r_{0}
  5. γ \gamma
  6. σ x , y , z \sigma_{x,y,z}
  7. ε m = ( δ acc γ σ x ) 2 \varepsilon_{m}=\left(\frac{\delta_{\mathrm{acc}}}{\gamma\sigma_{x^{\prime}}}% \right)^{2}
  8. F ( ε ) = ε 2 0 1 ( 2 u - ln ( 1 u ) - 2 ) e - ε u d u F(\varepsilon)=\frac{\sqrt{\varepsilon}}{2}\int_{0}^{1}\left(\frac{2}{u}-\ln% \left(\frac{1}{u}\right)-2\right)e^{-\frac{\varepsilon}{u}}\,du

TP_model_transformation_in_control_theory.html

  1. ( 𝐱 ˙ ( t ) 𝐲 ( t ) ) = 𝐒 ( 𝐩 ( t ) ) ( 𝐱 ( t ) 𝐮 ( t ) ) , \begin{pmatrix}{{\mathbf{{\dot{x}}}}}(t)\\ {\mathbf{y}}(t)\end{pmatrix}={\mathbf{S}}({\mathbf{p}}(t))\begin{pmatrix}{% \mathbf{x}}(t)\\ {\mathbf{u}}(t)\end{pmatrix},
  2. 𝐮 ( t ) {\mathbf{u}}(t)
  3. 𝐲 ( t ) {\mathbf{y}}(t)
  4. 𝐱 ( t ) {\mathbf{x}}(t)
  5. 𝐒 ( 𝐩 ( t ) ) \R L 1 × L 2 {\mathbf{S}}({\mathbf{p}}(t))\in\R^{L_{1}\times L_{2}}
  6. 𝐩 ( t ) Ω {\mathbf{p}}(t)\in\Omega
  7. N N
  8. Ω = [ a 1 , b 1 ] × [ a 2 , b 2 ] × × [ a N , b N ] \R N \Omega=[a_{1},b_{1}]\times[a_{2},b_{2}]\times\cdots\times[a_{N},b_{N}]\subset% \R^{N}
  9. 𝐒 ( 𝐩 ( t ) ) {\mathbf{S}}({\mathbf{p}}(t))
  10. 𝐩 ( t ) {\mathbf{p}}(t)
  11. 𝐱 ( t ) {\mathbf{x}}(t)
  12. ( 𝐱 ˙ ( t ) 𝐲 ( t ) ) = 𝒮 n = 1 N 𝐰 n ( p n ( t ) ) ( 𝐱 ( t ) 𝐮 ( t ) ) , \begin{pmatrix}{{\mathbf{{\dot{x}}}}}(t)\\ {\mathbf{y}}(t)\end{pmatrix}=\mathcal{S}\boxtimes_{n=1}^{N}\mathbf{w}_{n}(p_{n% }(t))\begin{pmatrix}{\mathbf{x}}(t)\\ {\mathbf{u}}(t)\end{pmatrix},
  13. 𝐮 ( t ) {\mathbf{u}}(t)
  14. 𝐲 ( t ) {\mathbf{y}}(t)
  15. 𝐱 ( t ) {\mathbf{x}}(t)
  16. 𝐒 ( 𝐩 ( t ) ) = 𝒮 n = 1 N 𝐰 ( p n ( t ) ) \R L 1 × L 2 {\mathbf{S}}({\mathbf{p}}(t))=\mathcal{S}\boxtimes_{n=1}^{N}\mathbf{w}(p_{n}(t% ))\in\R^{L_{1}\times L_{2}}
  17. 𝐩 ( t ) Ω {\mathbf{p}}(t)\in\Omega
  18. N N
  19. Ω = [ a 1 , b 1 ] × [ a 2 , b 2 ] × × [ a N , b N ] \R N \Omega=[a_{1},b_{1}]\times[a_{2},b_{2}]\times\cdots\times[a_{N},b_{N}]\subset% \R^{N}
  20. w n , i n ( p n ( t ) ) [ 0 , 1 ] w_{n,i_{n}}(p_{n}(t))\in[0,1]
  21. 𝐰 n ( p n ( t ) ) \mathbf{w}_{n}(p_{n}(t))
  22. 𝐒 i 1 , i 2 , , i N \mathbf{S}_{i_{1},i_{2},\ldots,i_{N}}
  23. 𝐒 ( 𝐩 ( t ) ) {\mathbf{S}}({\mathbf{p}}(t))
  24. n : i n = 1 I n w n , i n ( p n ( t ) ) = 1. \forall n:\sum_{i_{n}=1}^{I_{n}}w_{n,i_{n}}(p_{n}(t))=1.
  25. w n , i n ( p n ( t ) ) [ 0 , 1 ] . w_{n,i_{n}}(p_{n}(t))\in[0,1].
  26. 𝐒 ( 𝐩 ( t ) ) \mathbf{S}({\mathbf{p}}(t))
  27. 𝐒 i 1 , i 2 , , i N \mathbf{S}_{i_{1},i_{2},\ldots,i_{N}}
  28. 𝐩 ( t ) Ω \mathbf{p}(t)\in\Omega
  29. 𝐒 ( 𝐩 ( t ) ) = r = 1 R 𝐒 r w r ( 𝐩 ( t ) ) , \mathbf{S}(\mathbf{p}(t))=\sum_{r=1}^{R}\mathbf{S}_{r}w_{r}(\mathbf{p}(t)),
  30. i 1 , i 2 , i N i_{1},i_{2},\ldots i_{N}
  31. 𝐒 ( 𝐩 ( t ) ) \mathbf{S}(\mathbf{p}(t))
  32. 𝐩 ( t ) Ω R N \mathbf{p}(t)\in\Omega\subset R^{N}
  33. 𝐒 ( 𝐩 ( t ) ) = 𝒮 n = 1 N 𝐰 n ( p n ( t ) ) \mathbf{S}(\mathbf{p}(t))=\mathcal{S}\boxtimes_{n=1}^{N}\mathbf{w}_{n}(p_{n}(t))
  34. 𝒮 \mathcal{S}
  35. 𝐰 n ( p n ( t ) ) \mathbf{w}_{n}(p_{n}(t))
  36. n = 1 N n=1\ldots N
  37. 𝐒 ( 𝐩 ( t ) ) 𝒮 n = 1 N 𝐰 n ( p n ( t ) ) , \mathbf{S}(\mathbf{p}(t))\approx\mathcal{S}\boxtimes_{n=1}^{N}\mathbf{w}_{n}(p% _{n}(t)),
  38. u = - 𝐅 ( 𝐩 ( t ) ) 𝐱 ( t ) = - r = 1 R F r w r ( 𝐩 ( t ) ) 𝐱 ( t ) , u=-\mathbf{F}(\mathbf{p}(t))\mathbf{x}(t)=-\sum_{r=1}^{R}F_{r}w_{r}(\mathbf{p}% (t))\mathbf{x}(t),
  39. 𝐅 r \mathbf{F}_{r}
  40. 𝐒 r \mathbf{S}_{r}
  41. 𝐒 r \mathbf{S}_{r}
  42. 𝐅 r \mathbf{F}_{r}
  43. u = - 𝐅 ( 𝐩 ( t ) ) 𝐱 ( t ) = - n = 1 N 𝐰 n ( p n ( t ) ) 𝐱 ( t ) , u=-\mathbf{F}(\mathbf{p}(t))\mathbf{x}(t)=-\mathcal{F}\boxtimes_{n=1}^{N}% \mathbf{w}_{n}(p_{n}(t))\mathbf{x}(t),
  44. 𝐅 i 1 , i 2 , , i N \mathbf{F}_{i_{1},i_{2},\ldots,i_{N}}
  45. \mathcal{F}
  46. 𝐒 i 1 , i 2 , , i N \mathbf{S}_{i_{1},i_{2},\ldots,i_{N}}
  47. 𝒮 \mathcal{S}
  48. 𝒮 \mathcal{S}
  49. 𝐒 ( 𝐩 ( t ) ) \mathbf{S}(\mathbf{p}(t))
  50. Z = Z=\infty
  51. 𝐒 ( 𝐩 ( t ) ) = 𝒮 z n = 1 N 𝐰 z , n ( p n ( t ) ) , \mathbf{S}(\mathbf{p}(t))=\mathcal{S}_{z}\boxtimes_{n=1}^{N}\mathbf{w}_{z,n}(p% _{n}(t)),
  52. z = 1 Z z=1\ldots Z
  53. 𝐒 ( 𝐩 ( t ) ) \mathbf{S}(\mathbf{p}(t))
  54. u z = - 𝐅 z ( 𝐩 ( t ) ) 𝐱 ( t ) = - z n = 1 N 𝐰 z , n ( p n ( t ) ) 𝐱 ( t ) . u_{z}=-\mathbf{F}_{z}(\mathbf{p}(t))\mathbf{x}(t)=-\mathcal{F}_{z}\boxtimes_{n% =1}^{N}\mathbf{w}_{z,n}(p_{n}(t))\mathbf{x}(t).
  55. 𝒮 \mathcal{S}
  56. \mathcal{F}
  57. z = 1 Z z=1\ldots Z
  58. Z Z
  59. 𝐒 ( 𝐩 ( t ) ) \mathbf{S}(\mathbf{p}(t))
  60. 𝐒 ( 𝐩 ( t ) ) = 𝒮 n = 1 N 𝐰 n ( p n ( t ) ) , \mathbf{S}(\mathbf{p}(t))=\mathcal{S}\boxtimes_{n=1}^{N}\mathbf{w}_{n}(p_{n}(t% )),
  61. 𝐒 ( 𝐩 ( t ) ) \mathbf{S}(\mathbf{p}(t))
  62. u = - n = 1 N 𝐰 n ( p n ( t ) ) 𝐱 ( t ) , u=-\mathcal{F}\boxtimes_{n=1}^{N}\mathbf{w}_{n}(p_{n}(t))\mathbf{x}(t),
  63. k = 1 K = k=1\ldots K=\infty
  64. 𝐒 k ( 𝐩 ( t ) ) = 𝒮 n = 1 N 𝐰 k , n ( p n ( t ) ) , \mathbf{S}_{k}(\mathbf{p}(t))=\mathcal{S}\boxtimes_{n=1}^{N}\mathbf{w}_{k,n}(p% _{n}(t)),
  65. u k = - n = 1 N 𝐰 k , n ( p n ( t ) ) 𝐱 ( t ) . u_{k}=-\mathcal{F}\boxtimes_{n=1}^{N}\mathbf{w}_{k,n}(p_{n}(t))\mathbf{x}(t).
  66. L 2 L_{2}

Trace_inequalities.html

  1. n n
  2. n n
  3. n n
  4. n n
  5. f f
  6. I I
  7. f ( A ) f(A)
  8. λ λ
  9. I I
  10. P P
  11. f ( A ) j f ( λ j ) P j , f(A)\equiv\sum_{j}f(\lambda_{j})P_{j}~{},
  12. A = j λ j P j . A=\sum_{j}\lambda_{j}P_{j}.
  13. f : I f:I→ℝ
  14. I I
  15. n n
  16. I I
  17. A B f ( A ) f ( B ) , A\geq B\Rightarrow f(A)\geq f(B),
  18. A B A≥B
  19. A B 0 A−B≥0
  20. f : I f:I\rightarrow\mathbb{R}
  21. n n
  22. I I
  23. 0 < λ < 1 0<\lambda<1
  24. f ( λ A + ( 1 - λ ) B ) λ f ( A ) + ( 1 - λ ) f ( B ) . f(\lambda A+(1-\lambda)B)\leq\lambda f(A)+(1-\lambda)f(B).
  25. λ A + ( 1 - λ ) B \lambda A+(1-\lambda)B
  26. I I
  27. A A
  28. B B
  29. I I
  30. f f
  31. - f -f
  32. f f
  33. g : I × J g:I\times J\rightarrow\mathbb{R}
  34. I , J I,J\subset\mathbb{R}
  35. n n
  36. A 1 , A 2 𝐇 n A_{1},A_{2}\in\mathbf{H}_{n}
  37. I I
  38. B 1 , B 2 𝐇 n B_{1},B_{2}\in\mathbf{H}_{n}
  39. J J
  40. 0 λ 1 0\leq\lambda\leq 1
  41. g ( λ A 1 + ( 1 - λ ) A 2 , λ B 1 + ( 1 - λ ) B 2 ) λ g ( A 1 , B 1 ) + ( 1 - λ ) g ( A 2 , B 2 ) . g(\lambda A_{1}+(1-\lambda)A_{2},\lambda B_{1}+(1-\lambda)B_{2})\leq\lambda g(% A_{1},B_{1})+(1-\lambda)g(A_{2},B_{2}).
  42. g g
  43. g g
  44. g g
  45. f f
  46. A Tr f ( A ) = j f ( λ j ) , A\mapsto{\rm Tr}f(A)=\sum_{j}f(\lambda_{j}),
  47. A A
  48. λ λ
  49. f f
  50. n n
  51. t f ( t ) t\mapsto f(t)
  52. A Tr f ( A ) A\mapsto{\rm Tr}f(A)
  53. t f ( t ) t\mapsto f(t)
  54. A Tr f ( A ) A\mapsto{\rm Tr}f(A)
  55. f f
  56. - 1 p 0 -1\leq p\leq 0
  57. f ( t ) = - t p f(t)=-t^{p}
  58. 0 p 1 0\leq p\leq 1
  59. f ( t ) = t p f(t)=t^{p}
  60. 1 p 2 1\leq p\leq 2
  61. f ( t ) = t p f(t)=t^{p}
  62. f ( t ) = log ( t ) f(t)=\log(t)
  63. f ( t ) = t log ( t ) f(t)=t\log(t)
  64. f f
  65. n n
  66. n n
  67. A A
  68. B B
  69. f f
  70. f f
  71. n n
  72. n n
  73. A A
  74. B B
  75. f f
  76. f f
  77. A A
  78. B B
  79. f ( t ) = t l o g t f(t)=tlogt
  80. C = A B C=A−B
  81. ϕ ( t ) = Tr [ f ( B + t C ) ] . \phi(t)={\rm Tr}[f(B+tC)]~{}.
  82. φ φ
  83. t t
  84. t t
  85. f f
  86. C C
  87. φ φ
  88. ϕ ( t ) - ϕ ( 0 ) t \tfrac{\phi(t)-\phi(0)}{t}
  89. t t
  90. A , B 𝐇 n A,B\in\mathbf{H}_{n}
  91. Tr e A + B Tr e A e B . {\rm Tr}\,e^{A+B}\leq{\rm Tr}\,e^{A}e^{B}.
  92. A , B , C 𝐇 n + A,B,C\in\mathbf{H}_{n}^{+}
  93. Tr e ln A - ln B + ln C 0 d t Tr A ( B + t ) - 1 C ( B + t ) - 1 . {\rm Tr}\,e^{\ln A-\ln B+\ln C}\leq\int_{0}^{\infty}dt\,{\rm Tr}\,A(B+t)^{-1}C% (B+t)^{-1}.
  94. R , F 𝐇 n R,F\in\mathbf{H}_{n}
  95. Tr e F e R Tr e F + R e g . {\rm Tr}\,e^{F}e^{R}\geq{\rm Tr}\,e^{F+R}\geq e^{g}.
  96. f ( x ) = e x p ( x ) , A = R + F , a n d B = R + g I f(x)=exp(x),A=R+F,andB=R+gI
  97. H H
  98. e - H e^{-H}
  99. γ 0 \gamma\geq 0
  100. Tr γ = 1 , {\rm Tr}\,\gamma=1,
  101. Tr γ H + Tr γ ln γ - ln Tr e - H , {\rm Tr}\,\gamma H+{\rm Tr}\,\gamma\ln\gamma\geq-\ln{\rm Tr}\,e^{-H},
  102. γ = exp ( - H ) / Tr exp ( - H ) \gamma={\rm exp}(-H)/{\rm Tr}\,{\rm exp}(-H)
  103. m × n m\times n
  104. K K
  105. q q
  106. r r
  107. 0 q 1 0\leq q\leq 1
  108. 0 r 1 0\leq r\leq 1
  109. q + r 1 q+r\leq 1
  110. 𝐇 m + × 𝐇 n + \mathbf{H}^{+}_{m}\times\mathbf{H}^{+}_{n}
  111. F ( A , B , K ) = Tr ( K * A q K B r ) F(A,B,K)={\rm Tr}(K^{*}A^{q}KB^{r})
  112. ( A , B ) (A,B)
  113. K K
  114. K * K^{*}
  115. K . K.
  116. L 𝐇 n L\in\mathbf{H}_{n}
  117. f ( A ) = Tr exp { L + ln A } f(A)={\rm Tr}\,\exp\{L+\ln A\}
  118. 𝐇 n + + \mathbf{H}_{n}^{++}
  119. m × n m\times n
  120. K K
  121. 1 q 2 1\leq q\leq 2
  122. 0 r 1 0\leq r\leq 1
  123. q - r 1 q-r\geq 1
  124. 𝐇 m + + × 𝐇 n + + \mathbf{H}^{++}_{m}\times\mathbf{H}^{++}_{n}
  125. ( A , B ) Tr ( K * A q K B - r ) (A,B)\mapsto{\rm Tr}(K^{*}A^{q}KB^{-r})
  126. A , B 𝐇 n + + A,B\in\mathbf{H}^{++}_{n}
  127. R ( A B ) := Tr ( A log A ) - Tr ( A log B ) . R(A\|B):={\rm Tr}(A\log A)-{\rm Tr}(A\log B).
  128. ρ \rho
  129. σ \sigma
  130. R ( ρ σ ) = S ( ρ σ ) R(\rho\|\sigma)=S(\rho\|\sigma)
  131. R ( A B ) R(A\|B)
  132. f ( x ) = x log x f(x)=x\log x
  133. R ( A B ) : 𝐇 n + + × 𝐇 n + + 𝐑 R(A\|B):\mathbf{H}^{++}_{n}\times\mathbf{H}^{++}_{n}\rightarrow\mathbf{R}
  134. 0 < p < 1 0<p<1
  135. ( A , B ) T r ( B 1 - p A p ) (A,B)\mapsto Tr(B^{1-p}A^{p})
  136. ( A , B ) 1 p - 1 ( Tr ( B 1 - p A p ) - Tr A ) (A,B)\mapsto\frac{1}{p-1}({\rm Tr}(B^{1-p}A^{p})-{\rm Tr}\,A)
  137. lim p 1 1 p - 1 ( Tr ( B 1 - p A p ) - Tr A ) = R ( A B ) , \lim_{p\rightarrow 1}\frac{1}{p-1}({\rm Tr}(B^{1-p}A^{p})-{\rm Tr}\,A)=R(A\|B),
  138. f f
  139. I I
  140. f ( k A k * X k A k ) k A k * f ( X k ) A k , f\left(\sum_{k}A_{k}^{*}X_{k}A_{k}\right)\leq\sum_{k}A_{k}^{*}f(X_{k})A_{k},
  141. { A k } k \{A_{k}\}_{k}
  142. k A k * A k = 1 \sum_{k}A^{*}_{k}A_{k}=1
  143. { X k } k \{X_{k}\}_{k}
  144. I I
  145. f f
  146. I I
  147. m m
  148. n n
  149. f f
  150. Tr ( f ( k = 1 n A k * X k A k ) ) Tr ( k = 1 n A k * f ( X k ) A k ) , {\rm Tr}\Bigl(f\Bigl(\sum_{k=1}^{n}A_{k}^{*}X_{k}A_{k}\Bigr)\Bigr)\leq{\rm Tr}% \Bigl(\sum_{k=1}^{n}A_{k}^{*}f(X_{k})A_{k}\Bigr),
  151. X X
  152. X X
  153. m m
  154. m m
  155. I I
  156. A A
  157. A A
  158. m m
  159. m m
  160. k = 1 n A k * A k = 1 \sum_{k=1}^{n}A_{k}^{*}A_{k}=1
  161. n n
  162. m m
  163. n n
  164. f f
  165. f f
  166. I I
  167. f f
  168. n n
  169. f ( k = 1 n A k * X k A k ) k = 1 n A k * f ( X k ) A k , f\Bigl(\sum_{k=1}^{n}A_{k}^{*}X_{k}A_{k}\Bigr)\leq\sum_{k=1}^{n}A_{k}^{*}f(X_{% k})A_{k},
  170. ( X 1 , , X n ) (X_{1},\ldots,X_{n})
  171. \mathcal{H}
  172. I I
  173. ( A 1 , , A n ) (A_{1},\ldots,A_{n})
  174. \mathcal{H}
  175. k = 1 n A k * A k = 1 \sum_{k=1}^{n}A^{*}_{k}A_{k}=1
  176. f ( V * X V ) V * f ( X ) V f(V^{*}XV)\leq V^{*}f(X)V
  177. V V
  178. \mathcal{H}
  179. X X
  180. I I
  181. P f ( P X P + λ ( 1 - P ) ) P P f ( X ) P Pf(PXP+\lambda(1-P))P\leq Pf(X)P
  182. P P
  183. \mathcal{H}
  184. X X
  185. I I
  186. λ \lambda
  187. I I
  188. A 0 A\geq 0
  189. B 0 B\geq 0
  190. r 1 , r\geq 1,
  191. Tr ( B 1 / 2 A 1 / 2 B 1 / 2 ) r Tr B r / 2 A r / 2 B r / 2 . {\rm Tr}(B^{1/2}A^{1/2}B^{1/2})^{r}\leq{\rm Tr}B^{r/2}A^{r/2}B^{r/2}.
  192. A 0 A\geq 0
  193. B 0 B\geq 0
  194. q 0 , q\geq 0,
  195. Tr ( B 1 / 2 A B 1 / 2 ) r q Tr ( B r / 2 A r B r / 2 ) q , {\rm Tr}(B^{1/2}AB^{1/2})^{rq}\leq{\rm Tr}(B^{r/2}A^{r}B^{r/2})^{q},
  196. r 1 , r\geq 1,
  197. Tr ( B r / 2 A r B r / 2 ) q Tr ( B 1 / 2 A B 1 / 2 ) r q , {\rm Tr}(B^{r/2}A^{r}B^{r/2})^{q}\leq{\rm Tr}(B^{1/2}AB^{1/2})^{rq},
  198. 0 r 1. 0\leq r\leq 1.
  199. f ( x ) f(x)
  200. L L
  201. R R
  202. [ L , R ] = L R - R L = 0 [L,R]=LR-RL=0
  203. g ( L , R ) := f ( L R - 1 ) R g(L,R):=f(LR^{-1})R
  204. L = λ L 1 + ( 1 - λ ) L 2 L=\lambda L_{1}+(1-\lambda)L_{2}
  205. R = λ R 1 + ( 1 - λ ) R 2 R=\lambda R_{1}+(1-\lambda)R_{2}
  206. [ L i , R i ] = 0 [L_{i},R_{i}]=0
  207. 0 λ 1 0\leq\lambda\leq 1
  208. g ( L , R ) λ g ( L 1 , R 1 ) + ( 1 - λ ) g ( L 2 , R 2 ) . g(L,R)\leq\lambda g(L_{1},R_{1})+(1-\lambda)g(L_{2},R_{2}).
  209. α 1 α 2 α n \alpha_{1}\geq\alpha_{2}\geq\cdots\geq\alpha_{n}
  210. β 1 β 2 β n \beta_{1}\geq\beta_{2}\geq\cdots\geq\beta_{n}
  211. | trace ( A B ) | i = 1 n α i β i . \left|\mathrm{trace}(AB)\right|\leq\sum_{i=1}^{n}\alpha_{i}\beta_{i}.
  212. A A
  213. B B