wpmath0000013_1

Brazilian_real_(old).html

  1. S \mathrm{S}\!\!\!\|
  2. S \mathrm{S}\!\!\!\|
  3. 5 / 8 {5}/{8}

Brinkmann_graph.html

  1. ( x - 4 ) ( x - 2 ) ( x + 2 ) ( x 3 - x 2 - 2 x + 1 ) 2 (x-4)(x-2)(x+2)(x^{3}-x^{2}-2x+1)^{2}
  2. ( x 6 + 3 x 5 - 8 x 4 - 21 x 3 + 27 x 2 + 38 x - 41 ) 2 (x^{6}+3x^{5}-8x^{4}-21x^{3}+27x^{2}+38x-41)^{2}

Bromley_equation.html

  1. log γ ± = - A γ I 1 / 2 1 + I 1 / 2 + β b . \log\gamma_{\pm}=\frac{-A_{\gamma}I^{1/2}}{1+I^{1/2}}+\beta b.
  2. γ ± \gamma_{\pm}
  3. 1 z + z - log γ ± \frac{1}{z_{+}z_{-}}\log\gamma_{\pm}
  4. M p z + X q z - M^{z_{+}}_{p}X^{z_{-}}_{q}
  5. log γ ± = - A γ | z + z - | I 1 / 2 1 + ρ I 1 / 2 + ( 0.06 + 0.6 B | z + z - | ) I ( 1 + 1.5 | z + z - | I ) 2 + B I \log\gamma_{\pm}=\frac{-A_{\gamma}|z_{+}z_{-}|I^{1/2}}{1+\rho I^{1/2}}+\frac{(% 0.06+0.6B|z_{+}z_{-}|)I}{\left(1+\frac{1.5}{|z_{+}z_{-}|}I\right)^{2}}+BI
  6. B = B + + B - + δ + δ - B=B_{+}+B_{-}+\delta_{+}\delta_{-}

Brownian_excursion.html

  1. e e
  2. e e
  3. τ - \tau_{-}
  4. τ + \tau_{+}
  5. W W
  6. { e ( t ) : 0 t 1 } = d { | W ( ( 1 - t ) τ - + t τ + ) | τ + - τ - : 0 t 1 } . \{e(t):\ {0\leq t\leq 1}\}\ \stackrel{d}{=}\ \left\{\frac{|W((1-t)\tau_{-}+t% \tau_{+})|}{\sqrt{\tau_{+}-\tau_{-}}}:\ 0\leq t\leq 1\right\}.
  7. τ m \tau_{m}
  8. W 0 W_{0}
  9. { e ( t ) : 0 t 1 } = d { W 0 ( τ m + t mod 1 ) - W 0 ( τ m ) : 0 t 1 } . \{e(t):\ {0\leq t\leq 1}\}\ \stackrel{d}{=}\ \left\{W_{0}(\tau_{m}+t\,\text{ % mod }1)-W_{0}(\tau_{m}):\ 0\leq t\leq 1\right\}.
  10. e e
  11. M + sup 0 t 1 e ( t ) = d sup 0 t 1 W 0 ( t ) - inf 0 t 1 W 0 ( t ) , M_{+}\equiv\sup_{0\leq t\leq 1}e(t)\ \stackrel{d}{=}\ \sup_{0\leq t\leq 1}W_{0% }(t)-\inf_{0\leq t\leq 1}W_{0}(t),
  12. 0 1 e ( t ) d t = d 0 1 W 0 ( t ) d t - inf 0 t 1 W 0 ( t ) . \int_{0}^{1}e(t)\,dt\ \stackrel{d}{=}\ \int_{0}^{1}W_{0}(t)\,dt-\inf_{0\leq t% \leq 1}W_{0}(t).
  13. E M + = π / 2 1.25331 , EM_{+}=\sqrt{\pi/2}\approx 1.25331\ldots,\,
  14. E M + 2 1.64493 , V a r ( M + ) 0.0741337 . EM_{+}^{2}\approx 1.64493\ldots\ ,\ \ Var(M_{+})\approx 0.0741337\ldots.
  15. 0 1 e ( t ) d t \int_{0}^{1}e(t)\,dt
  16. W W
  17. W W
  18. A + 0 1 e ( t ) d t A_{+}\equiv\int_{0}^{1}e(t)\,dt
  19. A + A_{+}
  20. f A + ( x ) = 2 6 x 2 j = 1 v j 2 / 3 e - v j U ( - 5 6 , 4 3 ; v j ) with v j = 2 | a j | 3 / 27 x 2 f_{A_{+}}(x)=\frac{2\sqrt{6}}{x^{2}}\sum_{j=1}^{\infty}v_{j}^{2/3}e^{-v_{j}}U% \left(-\frac{5}{6},\frac{4}{3};v_{j}\right)\ \ \mbox{with}~{}\ \ v_{j}=2|a_{j}% |^{3}/27x^{2}
  21. a j a_{j}
  22. U U
  23. f A + ( x ) 72 6 π x 2 e - 6 x 2 as x , f_{A_{+}}(x)\sim\frac{72\sqrt{6}}{\sqrt{\pi}}x^{2}e^{-6x^{2}}\ \ \mbox{as}~{}% \ \ x\rightarrow\infty,
  24. P ( A + > x ) 6 6 π x e - 6 x 2 as x . P(A_{+}>x)\sim\frac{6\sqrt{6}}{\sqrt{\pi}}xe^{-6x^{2}}\ \ \mbox{as}~{}\ \ x% \rightarrow\infty.
  25. A + A_{+}
  26. E ( A + ) = 1 2 π 2 , E ( A + 2 ) = 5 12 .416666 , V a r ( A + ) = 5 12 - π 8 .0239675 . E(A_{+})=\frac{1}{2}\sqrt{\frac{\pi}{2}},\ \ E(A_{+}^{2})=\frac{5}{12}\approx.% 416666\ldots,\ \ Var(A_{+})=\frac{5}{12}-\frac{\pi}{8}\approx.0239675\ldots\ .

Brownian_model_of_financial_markets.html

  1. N + 1 N+1
  2. N N
  3. = ( r , 𝐛 , δ , σ , A , 𝐒 ( 0 ) ) \mathcal{M}=(r,\mathbf{b},\mathbf{\delta},\mathbf{\sigma},A,\mathbf{S}(0))
  4. ( Ω , , P ) (\Omega,\mathcal{F},P)
  5. [ 0 , T ] [0,T]
  6. D D
  7. 𝐖 ( t ) = ( W 1 ( t ) W D ( t ) ) , 0 t T \mathbf{W}(t)=(W_{1}(t)\ldots W_{D}(t))^{\prime},\;0\leq t\leq T
  8. { ( t ) ; 0 t T } \{\mathcal{F}(t);\;0\leq t\leq T\}
  9. r ( t ) L 1 [ 0 , T ] r(t)\in L_{1}[0,T]
  10. 𝐛 : [ 0 , T ] × N L 2 [ 0 , T ] \mathbf{b}:[0,T]\times\mathbb{R}^{N}\rightarrow\mathbb{R}\in L_{2}[0,T]
  11. δ : [ 0 , T ] × N L 2 [ 0 , T ] \mathbf{\delta}:[0,T]\times\mathbb{R}^{N}\rightarrow\mathbb{R}\in L_{2}[0,T]
  12. σ : [ 0 , T ] × N × D \mathbf{\sigma}:[0,T]\times\mathbb{R}^{N\times D}\rightarrow\mathbb{R}
  13. n = 1 N d = 1 D 0 T σ n , d 2 ( s ) d s < \sum_{n=1}^{N}\sum_{d=1}^{D}\int_{0}^{T}\sigma_{n,d}^{2}(s)ds<\infty
  14. A ( t ) A(t)
  15. 𝐒 ( 0 ) = ( S 0 ( 0 ) , S N ( 0 ) ) \mathbf{S}(0)=(S_{0}(0),\ldots S_{N}(0))^{\prime}
  16. ( Ω , , p ) (\Omega,\mathcal{F},p)
  17. 𝐖 ( t ) = ( W 1 ( t ) W D ( t ) ) , 0 t T \mathbf{W}(t)=(W_{1}(t)\ldots W_{D}(t))^{\prime},\;0\leq t\leq T
  18. 𝐖 ( t ) σ ( { 𝐖 ( s ) ; 0 s t } ) , t [ 0 , T ] . \mathcal{F}^{\mathbf{W}}(t)\triangleq\sigma\left(\{\mathbf{W}(s);\;0\leq s\leq t% \}\right),\quad\forall t\in[0,T].
  19. 𝒩 \mathcal{N}
  20. P P
  21. 𝐖 ( t ) \mathcal{F}^{\mathbf{W}}(t)
  22. ( t ) σ ( 𝐖 ( t ) 𝒩 ) , t [ 0 , T ] \mathcal{F}(t)\triangleq\sigma\left(\mathcal{F}^{\mathbf{W}}(t)\cup\mathcal{N}% \right),\quad\forall t\in[0,T]
  23. { 𝐖 ( t ) ; 0 t T } \{\mathcal{F}^{\mathbf{W}}(t);\;0\leq t\leq T\}
  24. { ( t ) ; 0 t T } \{\mathcal{F}(t);\;0\leq t\leq T\}
  25. ( t ) = σ ( 0 s < t ( s ) ) , \mathcal{F}(t)=\sigma\left(\bigcup_{0\leq s<t}\mathcal{F}(s)\right),
  26. ( t ) = t < s T ( s ) , \mathcal{F}(t)=\bigcap_{t<s\leq T}\mathcal{F}(s),
  27. S 0 ( t ) > 0 S_{0}(t)>0
  28. t t
  29. S 0 ( 0 ) = 1 S_{0}(0)=1
  30. { ( t ) ; 0 t T } \{\mathcal{F}(t);\;0\leq t\leq T\}
  31. S 0 a ( t ) S^{a}_{0}(t)
  32. S 0 s ( t ) S^{s}_{0}(t)
  33. r ( t ) 1 S 0 ( t ) d d t S 0 a ( t ) , r(t)\triangleq\frac{1}{S_{0}(t)}\frac{d}{dt}S^{a}_{0}(t),
  34. A ( t ) 0 t 1 S 0 ( s ) d S 0 s ( s ) , A(t)\triangleq\int_{0}^{t}\frac{1}{S_{0}(s)}dS^{s}_{0}(s),
  35. d S 0 ( t ) = S 0 ( t ) [ r ( t ) d t + d A ( t ) ] , 0 t T , dS_{0}(t)=S_{0}(t)[r(t)dt+dA(t)],\quad\forall 0\leq t\leq T,
  36. S 0 ( t ) = exp ( 0 t r ( s ) d s + A ( t ) ) , 0 t T . S_{0}(t)=\exp\left(\int_{0}^{t}r(s)ds+A(t)\right),\quad\forall 0\leq t\leq T.
  37. S 0 ( t ) S_{0}(t)
  38. A ( ) = 0 A(\cdot)=0
  39. r ( t ) r(t)
  40. ( t ) \mathcal{F}(t)
  41. S 1 ( t ) S N ( t ) S_{1}(t)\ldots S_{N}(t)
  42. N N
  43. d S n ( t ) = S n ( t ) [ b n ( t ) d t + d A ( t ) + d = 1 D σ n , d ( t ) d W d ( t ) ] , 0 t T , n = 1 N . dS_{n}(t)=S_{n}(t)\left[b_{n}(t)dt+dA(t)+\sum_{d=1}^{D}\sigma_{n,d}(t)dW_{d}(t% )\right],\quad\forall 0\leq t\leq T,\quad n=1\ldots N.
  44. σ n , d ( t ) , d = 1 D \sigma_{n,d}(t),\;d=1\ldots D
  45. n n
  46. b n ( t ) b_{n}(t)
  47. A ( t ) A(t)
  48. S n ( t ) = S n ( 0 ) exp ( 0 t d = 1 D σ n , d ( s ) d W d ( s ) + 0 t [ b n ( s ) - 1 2 d = 1 D σ n , d 2 ( s ) ] d s + A ( t ) ) , 0 t T , n = 1 N , S_{n}(t)=S_{n}(0)\exp\left(\int_{0}^{t}\sum_{d=1}^{D}\sigma_{n,d}(s)dW_{d}(s)+% \int_{0}^{t}\left[b_{n}(s)-\frac{1}{2}\sum_{d=1}^{D}\sigma^{2}_{n,d}(s)\right]% ds+A(t)\right),\quad\forall 0\leq t\leq T,\quad n=1\ldots N,
  49. S n ( t ) S 0 ( t ) = S n ( 0 ) exp ( 0 t d = 1 D σ n , d ( s ) d W d ( s ) + 0 t [ b n ( s ) - 1 2 d = 1 D σ n , d 2 ( s ) ] d s ) ) , 0 t T , n = 1 N . \frac{S_{n}(t)}{S_{0}(t)}=S_{n}(0)\exp\left(\int_{0}^{t}\sum_{d=1}^{D}\sigma_{% n,d}(s)dW_{d}(s)+\int_{0}^{t}\left[b_{n}(s)-\frac{1}{2}\sum_{d=1}^{D}\sigma^{2% }_{n,d}(s)\right]ds)\right),\quad\forall 0\leq t\leq T,\quad n=1\ldots N.
  50. A ( t ) A(t)
  51. δ n ( t ) \delta_{n}(t)
  52. t t
  53. Y n ( t ) Y_{n}(t)
  54. d Y n ( t ) = S n ( t ) [ b n ( t ) d t + d A ( t ) + d = 1 D σ n , d ( t ) d W d ( t ) + δ n ( t ) ] , 0 t T , n = 1 N . dY_{n}(t)=S_{n}(t)\left[b_{n}(t)dt+dA(t)+\sum_{d=1}^{D}\sigma_{n,d}(t)dW_{d}(t% )+\delta_{n}(t)\right],\quad\forall 0\leq t\leq T,\quad n=1\ldots N.
  55. = ( r , 𝐛 , δ , σ , A , 𝐒 ( 0 ) ) \mathcal{M}=(r,\mathbf{b},\mathbf{\delta},\mathbf{\sigma},A,\mathbf{S}(0))
  56. ( π 0 , π 1 , π N ) (\pi_{0},\pi_{1},\ldots\pi_{N})
  57. ( t ) \mathcal{F}(t)
  58. N + 1 \mathbb{R}^{N+1}
  59. 0 T | n = 0 N π n ( t ) | [ | r ( t ) | d t + d A ( t ) ] < \int_{0}^{T}|\sum_{n=0}^{N}\pi_{n}(t)|\left[|r(t)|dt+dA(t)\right]<\infty
  60. 0 T | n = 1 N π n ( t ) [ b n ( t ) + δ n ( t ) - r ( t ) ] | d t < \int_{0}^{T}|\sum_{n=1}^{N}\pi_{n}(t)[b_{n}(t)+\mathbf{\delta}_{n}(t)-r(t)]|dt<\infty
  61. 0 T d = 1 D | n = 1 N σ n , d ( t ) π n ( t ) | 2 d t < \int_{0}^{T}\sum_{d=1}^{D}|\sum_{n=1}^{N}\mathbf{\sigma}_{n,d}(t)\pi_{n}(t)|^{% 2}dt<\infty
  62. G ( t ) 0 t [ n = 0 N π n ( t ) ] ( r ( s ) d s + d A ( s ) ) + 0 t [ n = 1 N π n ( t ) ( b n ( t ) + δ n ( t ) - r ( t ) ) ] d t + 0 t d = 1 D n = 1 N σ n , d ( t ) π n ( t ) d W d ( s ) 0 t T G(t)\triangleq\int_{0}^{t}\left[\sum_{n=0}^{N}\pi_{n}(t)\right]\left(r(s)ds+dA% (s)\right)+\int_{0}^{t}\left[\sum_{n=1}^{N}\pi_{n}(t)\left(b_{n}(t)+\mathbf{% \delta}_{n}(t)-r(t)\right)\right]dt+\int_{0}^{t}\sum_{d=1}^{D}\sum_{n=1}^{N}% \mathbf{\sigma}_{n,d}(t)\pi_{n}(t)dW_{d}(s)\quad 0\leq t\leq T
  63. G ( t ) = n = 0 N π n ( t ) G(t)=\sum_{n=0}^{N}\pi_{n}(t)
  64. π 0 \pi_{0}
  65. π = ( π 1 , π N ) \pi=(\pi_{1},\ldots\pi_{N})
  66. π \pi
  67. π 0 < 0 \pi_{0}<0
  68. π n < 0 \pi_{n}<0
  69. b n ( t ) + δ n ( t ) - r ( t ) b_{n}(t)+\mathbf{\delta}_{n}(t)-r(t)
  70. G ( t ) G(t)
  71. n n
  72. 0 = t 0 < t 1 < < t M = T 0=t_{0}<t_{1}<\ldots<t_{M}=T
  73. ν n ( t m ) \nu_{n}(t_{m})
  74. n = 0 N n=0\ldots N
  75. [ t m , t m + 1 m = 0 M - 1 [t_{m},t_{m+1}\;m=0\ldots M-1
  76. ν n ( t m ) \nu_{n}(t_{m})
  77. ( t m ) \mathcal{F}(t_{m})
  78. G ( 0 ) = 0 , G(0)=0,
  79. G ( t m + 1 ) - G ( t m ) = n = 0 N ν n ( t m ) [ Y n ( t m + 1 ) - Y n ( t m ) ] , m = 0 M - 1 , G(t{m+1})-G(t_{m})=\sum_{n=0}^{N}\nu_{n}(t_{m})[Y_{n}(t_{m+1})-Y_{n}(t_{m})],% \quad m=0\ldots M-1,
  80. G ( m ) G(m)
  81. [ 0 , t m ] [0,t_{m}]
  82. n = 0 N ν n ( t m ) S n ( t m ) \sum_{n=0}^{N}\nu_{n}(t_{m})S_{n}(t_{m})
  83. π n ( t ) ν n ( t ) \pi_{n}(t)\triangleq\nu_{n}(t)
  84. Y ( t ) Y(t)
  85. π n ( t ) \pi_{n}(t)
  86. n n
  87. t t
  88. \mathcal{M}
  89. Γ ( t ) 0 t T \Gamma(t)\;0\leq t\leq T
  90. [ 0 , t ] [0,t]
  91. N + 1 N+1
  92. X ( t ) X(t)
  93. X ( t ) G ( t ) + Γ ( t ) X(t)\triangleq G(t)+\Gamma(t)
  94. 0 t T 0\leq t\leq T
  95. Γ ( t ) \Gamma(t)
  96. X ( t ) = n = 0 N π n ( t ) . X(t)=\sum_{n=0}^{N}\pi_{n}(t).
  97. d X ( t ) = d Γ ( t ) + X ( t ) [ r ( t ) d t + d A ( t ) ] + n = 1 N [ π n ( t ) ( b n ( t ) + δ n ( t ) - r ( t ) ) ] + d = 1 D [ n = 1 N π n ( t ) σ n , d ( t ) ] d W d ( t ) dX(t)=d\Gamma(t)+X(t)\left[r(t)dt+dA(t)\right]+\sum_{n=1}^{N}\left[\pi_{n}(t)% \left(b_{n}(t)+\delta_{n}(t)-r(t)\right)\right]+\sum_{d=1}^{D}\left[\sum_{n=1}% ^{N}\pi_{n}(t)\sigma_{n,d}(t)\right]dW_{d}(t)
  98. π 0 \pi_{0}
  99. π n , n = 1 N \pi_{n},\;n=1\ldots N
  100. \mathcal{M}
  101. π ( t ) \pi(t)
  102. G ( T ) 0 G(T)\geq 0
  103. P [ G ( T ) > 0 ] > 0 P[G(T)>0]>0
  104. \mathcal{M}
  105. \mathcal{M}
  106. ( t ) \mathcal{F}(t)
  107. θ : [ 0 , T ] × D \theta:[0,T]\times\mathbb{R}^{D}\rightarrow\mathbb{R}
  108. t [ 0 , T ] t\in[0,T]
  109. b n ( t ) + δ n ( t ) - r ( t ) = d = 1 D σ n , d ( t ) θ d ( t ) b_{n}(t)+\mathbf{\delta}_{n}(t)-r(t)=\sum_{d=1}^{D}\sigma_{n,d}(t)\theta_{d}(t)
  110. θ \theta
  111. n n
  112. σ n , \sigma_{n,\cdot}
  113. θ ( t ) \theta(t)
  114. 0 T d = 1 D | θ d ( t ) | 2 d t < \int_{0}^{T}\sum_{d=1}^{D}|\theta_{d}(t)|^{2}dt<\infty
  115. 𝔼 [ exp { - 0 T d = 1 D θ d ( t ) d W d ( t ) - 1 2 0 T d = 1 D | θ d ( t ) | 2 d t } ] = 1 \mathbb{E}\left[\exp\left\{-\int_{0}^{T}\sum_{d=1}^{D}\theta_{d}(t)dW_{d}(t)-% \frac{1}{2}\int_{0}^{T}\sum_{d=1}^{D}|\theta_{d}(t)|^{2}dt\right\}\right]=1
  116. \mathcal{M}
  117. n n
  118. σ n , d = 0 , d = 1 D \sigma_{n,d}=0,\;d=1\ldots D
  119. δ n ( t ) = 0 \delta_{n}(t)=0
  120. b n ( t ) = r ( t ) b_{n}(t)=r(t)
  121. S n ( t ) = S n ( 0 ) S 0 ( t ) S_{n}(t)=S_{n}(0)S_{0}(t)
  122. \mathcal{M}
  123. N N
  124. D D
  125. 𝐖 ( t ) \mathbf{W}(t)
  126. θ \theta
  127. 0 T d = 1 D | θ d ( t ) | 2 d t < \int_{0}^{T}\sum_{d=1}^{D}|\theta_{d}(t)|^{2}dt<\infty
  128. Z 0 ( t ) = exp { - 0 t d = 1 D θ d ( t ) d W d ( t ) - 1 2 0 t d = 1 D | θ d ( t ) | 2 d t } Z_{0}(t)=\exp\left\{-\int_{0}^{t}\sum_{d=1}^{D}\theta_{d}(t)dW_{d}(t)-\frac{1}% {2}\int_{0}^{t}\sum_{d=1}^{D}|\theta_{d}(t)|^{2}dt\right\}
  129. N N
  130. D D
  131. N - D N-D
  132. ( σ n , 1 σ n , D ) (\sigma_{n,1}\ldots\sigma_{n,D})
  133. D D
  134. σ \sigma
  135. D D
  136. N N
  137. D D
  138. P 0 P_{0}
  139. ( T ) \mathcal{F}(T)
  140. P 0 ( A ) 𝔼 [ Z 0 ( T ) 𝟏 A ] , A ( T ) P_{0}(A)\triangleq\mathbb{E}[Z_{0}(T)\mathbf{1}_{A}],\quad\forall A\in\mathcal% {F}(T)
  141. P P
  142. P 0 P_{0}
  143. 𝐖 0 ( t ) 𝐖 ( t ) + 0 t θ ( s ) d s \mathbf{W}_{0}(t)\triangleq\mathbf{W}(t)+\int_{0}^{t}\theta(s)ds
  144. D D
  145. { ( t ) ; 0 t T } \{\mathcal{F}(t);\;0\leq t\leq T\}
  146. P 0 P_{0}
  147. \mathcal{M}
  148. B B
  149. ( T ) \mathcal{F}(T)
  150. P 0 [ B S 0 ( T ) > - ] = 1 P_{0}\left[\frac{B}{S_{0}(T)}>-\infty\right]=1
  151. x 𝔼 0 [ B S 0 ( T ) ] < x\triangleq\mathbb{E}_{0}\left[\frac{B}{S_{0}(T)}\right]<\infty
  152. \mathcal{M}
  153. B B
  154. x x
  155. ( π n ( t ) ; n = 1 N ) (\pi_{n}(t);\;n=1\ldots N)
  156. X ( t ) X(t)
  157. X ( t ) = B X(t)=B
  158. B B
  159. T T
  160. t = 0 t=0
  161. x = sup ω B ( ω ) x=\sup_{\omega}B(\omega)
  162. x x
  163. T T
  164. B B
  165. \mathcal{M}
  166. N = D N=D
  167. N × D N\times D
  168. σ ( t ) \sigma(t)
  169. t [ 0 , T ] t\in[0,T]

Brusselator.html

  1. A X A\rightarrow X
  2. 2 X + Y 3 X 2X+Y\rightarrow 3X
  3. B + X Y + D B+X\rightarrow Y+D
  4. X E X\rightarrow E
  5. d d t { X } = { A } + { X } 2 { Y } - { B } { X } - { X } {d\over dt}\left\{X\right\}=\left\{A\right\}+\left\{X\right\}^{2}\left\{Y% \right\}-\left\{B\right\}\left\{X\right\}-\left\{X\right\}\,
  6. d d t { Y } = { B } { X } - { X } 2 { Y } {d\over dt}\left\{Y\right\}=\left\{B\right\}\left\{X\right\}-\left\{X\right\}^% {2}\left\{Y\right\}\,
  7. { X } = A \left\{X\right\}=A\,
  8. { Y } = B A \left\{Y\right\}={B\over A}\,
  9. B > 1 + A 2 B>1+A^{2}\,
  10. ( K B r O 3 ) (KBrO_{3})
  11. ( C H 2 ( C O O H ) 2 ) (CH_{2}(COOH)_{2})
  12. ( M n S O 4 ) (MnSO_{4})
  13. ( H 2 S O 4 ) (H_{2}SO_{4})

Bubble_chart.html

  1. v < 0 v<0
  2. v v
  3. v v
  4. - v -v

Bubble_sort.html

  1. \to
  2. \to
  3. \to
  4. \to
  5. \to
  6. \to
  7. \to
  8. \to
  9. \to
  10. \to
  11. \to
  12. \to

Buckingham_potential.html

  1. Φ 12 ( r ) \Phi_{12}(r)
  2. r r
  3. Φ 12 ( r ) = A exp ( - B r ) - C r 6 \Phi_{12}(r)=A\exp\left(-Br\right)-\frac{C}{r^{6}}
  4. A A
  5. B B
  6. C C
  7. r r
  8. r r
  9. 0
  10. r - 6 r^{-6}
  11. r r
  12. Φ 12 ( r ) = A exp ( - B r ) - C r 6 + q 1 q 2 4 π ε 0 r \Phi_{12}(r)=A\exp\left(-Br\right)-\frac{C}{r^{6}}+\frac{q_{1}q_{2}}{4\pi% \varepsilon_{0}r}

Bull_graph.html

  1. ( x - 2 ) ( x - 1 ) 3 x (x-2)(x-1)^{3}x
  2. ( x - 2 ) ( x - 1 ) 3 x (x-2)(x-1)^{3}x
  3. - x ( x 2 - x - 3 ) ( x 2 + x - 1 ) -x(x^{2}-x-3)(x^{2}+x-1)
  4. x 4 + x 3 + x 2 y x^{4}+x^{3}+x^{2}y

Burmester's_theory.html

  1. 𝐖 i = [ T i ] 𝐰 = [ A i ] 𝐰 + 𝐝 i , i = 1 , , 5. \mathbf{W}^{i}=[T_{i}]\mathbf{w}=[A_{i}]\mathbf{w}+\mathbf{d}_{i},\quad i=1,% \ldots,5.
  2. ( 𝐖 i - 𝐆 ) ( 𝐖 i - 𝐆 ) = R 2 , i = 1 , , 5. (\mathbf{W}^{i}-\mathbf{G})\cdot(\mathbf{W}^{i}-\mathbf{G})=R^{2},\quad i=1,% \ldots,5.
  3. ( 𝐖 i - 𝐆 ) ( 𝐖 i - 𝐆 ) - ( 𝐖 1 - 𝐆 ) ( 𝐖 1 - 𝐆 ) = 0 , i = 2 , , 5. (\mathbf{W}^{i}-\mathbf{G})\cdot(\mathbf{W}^{i}-\mathbf{G})-(\mathbf{W}^{1}-% \mathbf{G})\cdot(\mathbf{W}^{1}-\mathbf{G})=0,\quad i=2,\ldots,5.
  4. ( 𝐖 i - 𝐖 1 ) ( 𝐖 i + 𝐖 1 2 - 𝐆 ) = 0 , i = 2 , , 5 , (\mathbf{W}^{i}-\mathbf{W}^{1})\cdot(\frac{\mathbf{W}^{i}+\mathbf{W}^{1}}{2}-% \mathbf{G})=0,\quad i=2,\ldots,5,
  5. 𝐆 i = [ A ( θ i ) ] 𝐠 , 𝐖 i = [ A ( ψ i ) ] 𝐰 + 𝐂 , i = 1 , , 5 , \mathbf{G}^{i}=[A(\theta_{i})]\mathbf{g},\quad\mathbf{W}^{i}=[A(\psi_{i})]% \mathbf{w}+\mathbf{C},\quad i=1,\ldots,5,
  6. ( 𝐖 i - 𝐆 i ) ( 𝐖 i - 𝐆 i ) = R 2 , i = 1 , , 5. (\mathbf{W}^{i}-\mathbf{G}^{i})\cdot(\mathbf{W}^{i}-\mathbf{G}^{i})=R^{2},% \quad i=1,\ldots,5.
  7. ( 𝐖 i - 𝐆 i ) ( 𝐖 i - 𝐆 i ) - ( 𝐖 1 - 𝐆 1 ) ( 𝐖 1 - 𝐆 1 ) = 0 , i = 2 , , 5. (\mathbf{W}^{i}-\mathbf{G}^{i})\cdot(\mathbf{W}^{i}-\mathbf{G}^{i})-(\mathbf{W% }^{1}-\mathbf{G}^{1})\cdot(\mathbf{W}^{1}-\mathbf{G}^{1})=0,\quad i=2,\ldots,5.

Butcher_group.html

  1. α ( t ) = | t | ! t ! | S t | , \displaystyle\alpha(t)={|t|!\over t!|S_{t}|},
  2. [ t 1 , , t n ] ! = | [ t 1 , , t n ] | t 1 ! t n ! [t_{1},\dots,t_{n}]!=|[t_{1},\dots,t_{n}]|\cdot t_{1}!\cdots t_{n}!
  3. ! = 1. \bullet!=1.
  4. d x ( s ) d s = f ( x ( s ) ) , x ( 0 ) = x 0 , \displaystyle{dx(s)\over ds}=f(x(s)),\,\,x(0)=x_{0},
  5. δ i = f i , δ [ t 1 , , t n ] i = j 1 , , j n = 1 N ( δ t 1 j 1 δ t n j n ) j 1 j n f i . \delta_{\bullet}^{i}=f^{i},\,\,\,\delta^{i}_{[t_{1},\dots,t_{n}]}=\sum_{j_{1},% \dots,j_{n}=1}^{N}(\delta^{j_{1}}_{t_{1}}\cdots\delta^{j_{n}}_{t_{n}})\partial% _{j_{1}}\cdots\partial_{j_{n}}f^{i}.
  6. d m x d s m = | t | = m α ( t ) δ t , {d^{m}x\over ds^{m}}=\sum_{|t|=m}\alpha(t)\delta_{t},
  7. x ( s ) = x 0 + t s | t | | t | ! α ( t ) δ t ( 0 ) . \displaystyle x(s)=x_{0}+\sum_{t}{s^{|t|}\over|t|!}\alpha(t)\delta_{t}(0).
  8. x ( 4 ) = f ′′′ f 3 + 3 f ′′ f f 2 + f f ′′ f 2 + ( f ) 3 f , x^{(4)}=f^{\prime\prime\prime}f^{3}+3f^{\prime\prime}f^{\prime}f^{2}+f^{\prime% }f^{\prime\prime}f^{2}+(f^{\prime})^{3}f,
  9. x ( 4 ) = f ′′′ ( f , f , f ) + 3 f ′′ ( f , f ( f ) ) + f ( f ′′ ( f , f ) ) + f ( f ( f ( f ) ) ) , x^{(4)}=f^{\prime\prime\prime}(f,f,f)+3f^{\prime\prime}(f,f^{\prime}(f))+f^{% \prime}(f^{\prime\prime}(f,f))+f^{\prime}(f^{\prime}(f^{\prime}(f))),
  10. Δ : H H H \Delta:H\rightarrow H\otimes H
  11. Δ ( t ) = t I + I t + s t s [ t \ s ] , \Delta(t)=t\otimes I+I\otimes t+\sum_{s\subset t}s\otimes[t\backslash s],
  12. [ t \ s ] [t\backslash s]
  13. S ( t ) = - t - s t ( - 1 ) n ( t \ s ) S ( [ t \ s ] ) s , S ( ) = - . S(t)=-t-\sum_{s\subset t}(-1)^{n(t\backslash s)}S([t\backslash s])s,\,\,\,S(% \bullet)=-\bullet.
  14. φ 1 φ 2 ( t ) = ( φ 1 φ 2 ) Δ ( t ) . \varphi_{1}\star\varphi_{2}(t)=(\varphi_{1}\otimes\varphi_{2})\Delta(t).
  15. φ - 1 ( t ) = φ ( S t ) \varphi^{-1}(t)=\varphi(St)
  16. d x ( s ) d s = f ( x ( s ) ) , x ( 0 ) = x 0 , {dx(s)\over ds}=f(x(s)),\,\,\,x(0)=x_{0},
  17. A = ( a i j ) A=(a_{ij})
  18. b = ( b i ) b=(b_{i})
  19. X i = x n - 1 + h j = 1 m a i j f ( X j ) X_{i}=x_{n-1}+h\sum_{j=1}^{m}a_{ij}f(X_{j})
  20. x n = x n - 1 + h j = 1 m b j f ( x j ) . x_{n}=x_{n-1}+h\sum_{j=1}^{m}b_{j}f(x_{j}).
  21. X i ( s ) = x 0 + s j = 1 m a i j f ( X j ( s ) ) , x ( s ) = x 0 + s j = 1 m b j f ( X j ( s ) ) X_{i}(s)=x_{0}+s\sum_{j=1}^{m}a_{ij}f(X_{j}(s)),\,\,\,x(s)=x_{0}+s\sum_{j=1}^{% m}b_{j}f(X_{j}(s))
  22. X i ( s ) = x 0 + t s | t | | t | ! α ( t ) t ! j = 1 m a i j φ j ( t ) δ t ( 0 ) , x ( s ) = x 0 + t s | t | | t | ! α ( t ) t ! φ ( t ) δ t ( 0 ) , X_{i}(s)=x_{0}+\sum_{t}{s^{|t|}\over|t|!}\alpha(t)t!\sum_{j=1}^{m}a_{ij}% \varphi_{j}(t)\delta_{t}(0),\,\,\,\,x(s)=x_{0}+\sum_{t}{s^{|t|}\over|t|!}% \alpha(t)t!\varphi(t)\delta_{t}(0),
  23. φ j ( ) = 1. φ i ( [ t 1 , , t k ] ) = j 1 , , j k a i j 1 a i j k φ j 1 ( t 1 ) φ j k ( t k ) \varphi_{j}(\bullet)=1.\,\,\,\varphi_{i}([t_{1},\cdots,t_{k}])=\sum_{j_{1},% \dots,j_{k}}a_{ij_{1}}\dots a_{ij_{k}}\varphi_{j_{1}}(t_{1})\dots\varphi_{j_{k% }}(t_{k})
  24. φ ( t ) = j = 1 m b j φ j ( t ) . \varphi(t)=\sum_{j=1}^{m}b_{j}\varphi_{j}(t).
  25. Φ ( t ) = 1 t ! . \Phi(t)={1\over t!}.
  26. φ φ \varphi\star\varphi^{\prime}
  27. ( A 0 0 A ) , ( b , b ) . \begin{pmatrix}A&0\\ 0&A^{\prime}\\ \end{pmatrix},\,\,(b,b^{\prime}).
  28. φ f = 1 + t s | t | | t | ! α ( t ) t ! φ ( t ) δ t ( 0 ) , \varphi\circ f=1+\sum_{t}{s^{|t|}\over|t|!}\alpha(t)t!\varphi(t)\delta_{t}(0),
  29. φ 1 ( φ 2 f ) = ( φ 1 φ 2 ) f . \varphi_{1}\circ(\varphi_{2}\circ f)=(\varphi_{1}\star\varphi_{2})\circ f.
  30. 𝔤 \mathfrak{g}
  31. 𝔤 \mathfrak{g}
  32. θ ( a b ) = ε ( a ) θ ( b ) + θ ( a ) ε ( b ) , \theta(ab)=\varepsilon(a)\theta(b)+\theta(a)\varepsilon(b),
  33. [ θ 1 , θ 2 ] ( t ) = ( θ 1 θ 2 - θ 2 θ 1 ) Δ ( t ) . [\theta_{1},\theta_{2}](t)=(\theta_{1}\otimes\theta_{2}-\theta_{2}\otimes% \theta_{1})\Delta(t).
  34. 𝔤 \mathfrak{g}
  35. θ t ( t ) = δ t t , \theta_{t}(t^{\prime})=\delta_{tt^{\prime}},
  36. R ( f g ) + R ( f ) R ( g ) = R ( f R ( g ) ) + R ( R ( f ) g ) R(fg)+R(f)R(g)=R(fR(g))+R(R(f)g)\,
  37. R ( n a n z n ) = n < 0 a n z n . \displaystyle R(\sum_{n}a_{n}z^{n})=\sum_{n<0}a_{n}z^{n}.
  38. P ( x ) = x - ε ( x ) 1. \displaystyle P(x)=x-\varepsilon(x)1.
  39. m ( S id ) Δ ( x ) = ε ( x ) 1 m\circ(S\otimes{\rm id})\Delta(x)=\varepsilon(x)1
  40. S = - m ( S P ) Δ , S=-m\circ(S\otimes P)\Delta,
  41. Φ S R \Phi_{S}^{R}
  42. Φ S R \Phi_{S}^{R}
  43. Φ S R = - m ( S Φ S R P ) Δ . \Phi_{S}^{R}=-m(S\otimes\Phi_{S}^{R}\circ P)\Delta.
  44. Φ S R \Phi_{S}^{R}
  45. γ ( z ) = γ - ( z ) - 1 γ + ( z ) , \displaystyle\gamma(z)=\gamma_{-}(z)^{-1}\gamma_{+}(z),
  46. { } \cup\{\infty\}
  47. μ γ μ - = 0 , \partial_{\mu}\gamma_{\mu-}=0,
  48. λ w ( t ) = w | t | t \lambda_{w}(t)=w^{|t|}t
  49. F t = lim z = 0 γ - ( z ) λ t z ( γ - ( z ) - 1 ) \displaystyle F_{t}=\lim_{z=0}\gamma_{-}(z)\lambda_{tz}(\gamma_{-}(z)^{-1})
  50. β = t F t | t = 0 . \beta=\partial_{t}F_{t}|_{t=0}.
  51. Φ ( [ t 1 , , t n ] ) = Φ ( t 1 ) Φ ( t n ) | y | 2 + q μ 2 ( | y | 2 ) - z ( c 2 - 1 ) d D y , \displaystyle\Phi([t_{1},\dots,t_{n}])=\int{\Phi(t_{1})\cdots\Phi(t_{n})\over|% y|^{2}+q_{\mu}^{2}}(|y|^{2})^{-z({c\over 2}-1)}\,d^{D}y,
  52. ( | y | 2 ) - u | y | 2 + q μ 2 d D y = π D / 2 ( q μ 2 ) - z - u Γ ( - u + D / 2 ) Γ ( 1 + u - D / 2 ) Γ ( D / 2 ) . \displaystyle\int{(|y|^{2})^{-u}\over|y|^{2}+q_{\mu}^{2}}\,d^{D}y=\pi^{D/2}(q_% {\mu}^{2})^{-z-u}{\Gamma(-u+D/2)\Gamma(1+u-D/2)\over\Gamma(D/2)}.
  53. Φ ( ) = π D / 2 ( q μ 2 ) - z c / 2 Γ ( 1 + c z ) c z . \displaystyle\Phi(\bullet)=\pi^{D/2}(q_{\mu}^{2})^{-zc/2}{\Gamma(1+cz)\over cz}.
  54. Φ S R ( t ) \Phi_{S}^{R}(t)
  55. log q μ 2 \log q_{\mu}^{2}

Büchi's_problem.html

  1. { x 2 2 - 2 x 1 2 + x 0 2 = 2 x 3 2 - 2 x 2 2 + x 1 2 = 2 x M - 1 2 - 2 x M - 2 2 + x M - 3 2 = 2 \begin{cases}x_{2}^{2}-2x_{1}^{2}+x_{0}^{2}=2\\ x_{3}^{2}-2x_{2}^{2}+x_{1}^{2}=2\\ {}\quad\vdots\\ x_{M-1}^{2}-2x_{M-2}^{2}+x_{M-3}^{2}=2\end{cases}
  2. x n 2 = ( x 0 + n ) 2 . x_{n}^{2}=(x_{0}+n)^{2}.
  3. σ = ( x n 2 ) n = 0 , , M - 1 \sigma=(x_{n}^{2})_{n=0,\dots,M-1}
  4. Δ ( 1 ) ( σ ) = ( x n + 1 2 - x n 2 ) n = 0 , , M - 2 \Delta^{(1)}(\sigma)=(x_{n+1}^{2}-x_{n}^{2})_{n=0,\dots,M-2}
  5. σ \sigma
  6. Δ ( 2 ) ( σ ) = ( ( x n + 2 2 - x n + 1 2 ) - ( x n + 1 2 - x n 2 ) ) n = 0 , , M - 3 = ( x n + 2 2 - 2 x n + 1 2 + x n 2 ) n = 0 , , M - 3 . \Delta^{(2)}(\sigma)=((x_{n+2}^{2}-x_{n+1}^{2})-(x_{n+1}^{2}-x_{n}^{2}))_{n=0,% \dots,M-3}=(x_{n+2}^{2}-2x_{n+1}^{2}+x_{n}^{2})_{n=0,\dots,M-3}.
  7. Δ ( 2 ) ( σ ) = ( 2 ) n = 0 , , M - 3 \Delta^{(2)}(\sigma)=(2)_{n=0,\dots,M-3}
  8. σ \sigma
  9. ( ) ( x + 2 ) 2 - 2 ( x + 1 ) 2 + x 2 = 2. (\star)\qquad(x+2)^{2}-2(x+1)^{2}+x^{2}=2.
  10. x 2 2 - 2 x 1 2 + x 0 2 = 2 x_{2}^{2}-2x_{1}^{2}+x_{0}^{2}=2
  11. x 2 2 = ( x 0 + 2 ) 2 x_{2}^{2}=(x_{0}+2)^{2}
  12. x 1 2 = ( x 0 + 1 ) 2 x_{1}^{2}=(x_{0}+1)^{2}
  13. ( ) (\star)
  14. ( x + 3 ) 2 - 2 ( x + 2 ) 2 + ( x + 1 ) 2 = 2 (x+3)^{2}-2(x+2)^{2}+(x+1)^{2}=2
  15. { x 2 2 - 2 x 1 2 + x 0 2 = 2 x 3 2 - 2 x 2 2 + x 1 2 = 2 \begin{cases}x_{2}^{2}-2x_{1}^{2}+x_{0}^{2}=2\\ x_{3}^{2}-2x_{2}^{2}+x_{1}^{2}=2\end{cases}
  16. x n 2 = ( x 0 + n ) 2 x_{n}^{2}=(x_{0}+n)^{2}

Cable_fairing.html

  1. a a
  2. a a
  3. a a

Caffeine_synthase.html

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

Calabi–Eckmann_manifold.html

  1. n \ 0 × m \ 0 {\mathbb{C}}^{n}\backslash 0\times{\mathbb{C}}^{m}\backslash 0
  2. {\mathbb{C}}
  3. t , ( x , y ) n \ 0 × m \ 0 | t ( x , y ) = ( e t x , e α t y ) t\in{\mathbb{C}},\ (x,y)\in{\mathbb{C}}^{n}\backslash 0\times{\mathbb{C}}^{m}% \backslash 0\ \ |\ \ t(x,y)=(e^{t}x,e^{\alpha t}y)
  4. α \ \alpha\in{\mathbb{C}}\backslash{\mathbb{R}}
  5. G L ( n , ) × G L ( m , ) GL(n,{\mathbb{C}})\times GL(m,{\mathbb{C}})
  6. H 2 ( M ) = 0 H^{2}(M)=0
  7. n \ 0 × m \ 0 P n - 1 × P m - 1 {\mathbb{C}}^{n}\backslash 0\times{\mathbb{C}}^{m}\backslash 0\mapsto{\mathbb{% C}}P^{n-1}\times{\mathbb{C}}P^{m-1}
  8. P n - 1 × P m - 1 {\mathbb{C}}P^{n-1}\times{\mathbb{C}}P^{m-1}
  9. \mathbb{C}
  10. + α {\mathbb{Z}}+\alpha\cdot{\mathbb{Z}}

Calculus_of_functors.html

  1. Emb ( M , N ) Imm ( M , N ) \mathrm{Emb}(M,N)\to\mathrm{Imm}(M,N)
  2. F T k + 1 F T k F F\to T_{k+1}F\to T_{k}F
  3. F T k F , F\to T_{k}F,
  4. F T k + 1 F T k F T 1 F T 0 F , F\to\cdots\to T_{k+1}F\to T_{k}F\to\cdots T_{1}F\to T_{0}F,
  5. ( k - 1 ) (k-1)
  6. T k F T_{k}F
  7. F T k F F\to T_{k}F

CALICE.html

  1. e + e - e^{+}e^{-}
  2. e + e - e^{+}e^{-}
  3. m 3 m^{3}
  4. c m cm
  5. c m 2 cm^{2}
  6. c m cm
  7. c m 2 cm^{2}
  8. c m 2 cm^{2}
  9. c m 2 cm^{2}
  10. λ \lambda
  11. 1 - λ 1-\lambda
  12. 5 - λ 5-\lambda

Calkin–Wilf_tree.html

  1. q i + 1 = 1 2 q i - q i + 1 q_{i+1}=\frac{1}{2\lfloor q_{i}\rfloor-q_{i}+1}
  2. q i q_{i}
  3. q 1 = 1 q_{1}=1
  4. q i \lfloor q_{i}\rfloor
  5. q i q_{i}
  6. q i q_{i}
  7. 10000111001 2 10000111001_{2}
  8. q 1081 = 53 / 37 q_{1081}=53/37
  9. 11111000110 2 11111000110_{2}
  10. q 1990 = 37 / 53 q_{1990}=37/53
  11. q i q_{i}
  12. q i = 3 / 4 q_{i}=3/4
  13. 1110 2 1110_{2}
  14. q i = 4 / 3 q_{i}=4/3
  15. 1001 2 1001_{2}
  16. ( n - r r ) , 0 2 r < n , \scriptstyle{n-r\choose r},\ 0\leq 2r<n,

Cambridge_equation.html

  1. P Y P\cdot Y
  2. M 𝑑 = 𝑘 P Y M^{\,\textit{d}}=\,\textit{k}\cdot P\cdot Y
  3. M 𝑑 = M M^{\,\textit{d}}=M
  4. Y Y
  5. M 1 k = P Y M\cdot\frac{1}{k}=P\cdot Y

Camera_auto-calibration.html

  1. P i P^{i}
  2. X j X_{j}
  3. H H
  4. { P j H , H - 1 X J } \left\{P^{j}H,H^{-1}X_{J}\right\}
  5. K i K_{i}
  6. P M i P i H = K i ( R i | t i ) P_{M}^{i}\equiv P^{i}H=K_{i}\left(R_{i}|t_{i}\right)

Canonical_signed_digit.html

  1. + 2 5 - 2 3 - 2 0 +2^{5}-2^{3}-2^{0}
  2. 32 - 8 - 1 = 23 32-8-1=23
  3. ( 7 = 0 × 2 3 + 1 × 2 2 + 1 × 2 1 + 1 × 2 0 = 4 + 2 + 1 ) (7=0\times 2^{3}+1\times 2^{2}+1\times 2^{1}+1\times 2^{0}=4+2+1)
  4. ( 7 = + 1 × 2 3 + 0 × 2 2 + 0 × 2 1 - 1 × 2 0 = 8 - 1 ) (7=+1\times 2^{3}+0\times 2^{2}+0\times 2^{1}-1\times 2^{0}=8-1)

Canonical_singularity.html

  1. K X = f * ( K Y ) + i a i E i \displaystyle K_{X}=f^{*}(K_{Y})+\sum_{i}a_{i}E_{i}
  2. ( X , Δ ) (X,\Delta)
  3. Δ \Delta
  4. K X + Δ K_{X}+\Delta
  5. \mathbb{Q}
  6. ( X , Δ ) > 0 (X,\Delta)>0
  7. ( X , Δ ) 0 (X,\Delta)\geq 0
  8. ( X , Δ ) > - 1 (X,\Delta)>-1
  9. Δ 0 \lfloor\Delta\rfloor\leq 0
  10. ( X , Δ ) > - 1 (X,\Delta)>-1
  11. ( X , Δ ) - 1 (X,\Delta)\geq-1

Capacitated_minimum_spanning_tree.html

  1. r r
  2. c c
  3. r r
  4. c c
  5. c c
  6. G = ( V , E ) G=(V,E)
  7. n = | G | n=|G|
  8. r G r\in G
  9. a i a_{i}
  10. G G
  11. c i j c_{ij}
  12. a i a_{i}
  13. a j a_{j}
  14. C = c i j C={c_{ij}}
  15. r r
  16. i = 0 n c r i \displaystyle\sum_{i=0}^{n}c_{ri}
  17. a j a_{j}
  18. G - r G-{r}
  19. t ( a i ) = g i - c i j t(a_{i})=g_{i}-c_{ij}
  20. t ( a i ) t(a_{i})
  21. g i g_{i}
  22. i i
  23. a j a_{j}
  24. c i j c_{ij}
  25. c 1 r c_{1r}
  26. c 2 r c_{2r}
  27. c n r c_{nr}
  28. a i a_{i}
  29. a i a_{i}
  30. t ( a i ) t(a_{i})
  31. g i g_{i}
  32. c i j c_{ij}
  33. t ( a i ) t(a_{i})
  34. t ( a i ) t(a_{i})
  35. c k j c_{kj}

Capacitive_deionization.html

  1. Δ G = R * T * Φ v , f r e s h * ( C f e e d - C f r e s h ) [ l n α 1 - α - l n β 1 - α ] \Delta G=R*T*\Phi_{v,fresh}*(C_{feed}-C_{fresh})\left[\frac{ln\alpha}{1-\alpha% }-\frac{ln\beta}{1-\alpha}\right]

Capacitive_displacement_sensor.html

  1. C = ε 0 K A d C=\dfrac{\varepsilon_{0}KA}{d}
  2. C 1 d C\propto\dfrac{1}{d}

Capelli's_identity.html

  1. 𝔤 𝔩 n \mathfrak{gl}_{n}
  2. E i j = a = 1 n x i a x j a . E_{ij}=\sum_{a=1}^{n}x_{ia}\frac{\partial}{\partial x_{ja}}.
  3. | E 11 + n - 1 E 1 , n - 1 E 1 n E n - 1 , 1 E n - 1 , n - 1 + 1 E n - 1 , n E n 1 E n , n - 1 E n n + 0 | = | x 11 x 1 n x n 1 x n n | | x 11 x 1 n x n 1 x n n | . \begin{vmatrix}E_{11}+n-1&\cdots&E_{1,n-1}&E_{1n}\\ \vdots&\ddots&\vdots&\vdots\\ E_{n-1,1}&\cdots&E_{n-1,n-1}+1&E_{n-1,n}\\ E_{n1}&\cdots&E_{n,n-1}&E_{nn}+0\end{vmatrix}=\begin{vmatrix}x_{11}&\cdots&x_{% 1n}\\ \vdots&\ddots&\vdots\\ x_{n1}&\cdots&x_{nn}\end{vmatrix}\begin{vmatrix}\frac{\partial}{\partial x_{11% }}&\cdots&\frac{\partial}{\partial x_{1n}}\\ \vdots&\ddots&\vdots\\ \frac{\partial}{\partial x_{n1}}&\cdots&\frac{\partial}{\partial x_{nn}}\end{% vmatrix}.
  4. det ( A ) = σ S n sgn ( σ ) A σ ( 1 ) , 1 A σ ( 2 ) , 2 A σ ( n ) , n , \det(A)=\sum_{\sigma\in S_{n}}\operatorname{sgn}(\sigma)A_{\sigma(1),1}A_{% \sigma(2),2}\cdots A_{\sigma(n),n},
  5. E = X D t , E=XD^{t},
  6. E , X , D E,X,D
  7. x i j \frac{\partial}{\partial x_{ij}}
  8. det ( E ) = det ( X ) det ( D t ) \det(E)=\det(X)\det(D^{t})
  9. ( n - i ) δ i j (n-i)\delta_{ij}
  10. det ( A B ) = det ( A ) det ( B ) \det(AB)=\det(A)\det(B)
  11. n n
  12. m m
  13. x i j x_{ij}
  14. i = 1 , , n , j = 1 , , m i=1,\dots,n,\ j=1,\dots,m
  15. E i j E_{ij}
  16. E i j = a = 1 m x i a x j a . E_{ij}=\sum_{a=1}^{m}x_{ia}\frac{\partial}{\partial x_{ja}}.
  17. a a
  18. 1 1
  19. m m
  20. [ E i j , E k l ] = δ j k E i l - δ i l E k j . [E_{ij},E_{kl}]=\delta_{jk}E_{il}-\delta_{il}E_{kj}.~{}~{}~{}~{}~{}~{}~{}~{}~{}
  21. [ a , b ] [a,b]
  22. a b - b a ab-ba
  23. e i j e_{ij}
  24. ( i , j ) (i,j)
  25. e i j e_{ij}
  26. π : e i j E i j \pi:e_{ij}\mapsto E_{ij}
  27. 𝔤 𝔩 n \mathfrak{gl}_{n}
  28. x i j x_{ij}
  29. E i j = x i x j . E_{ij}=x_{i}\frac{\partial}{\partial x_{j}}.
  30. E i j x k = δ j k x i . E_{ij}x_{k}=\delta_{jk}x_{i}.~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}
  31. E i j E_{ij}
  32. e i j e_{ij}
  33. n \mathbb{C}^{n}
  34. 𝔤 𝔩 n \mathfrak{gl}_{n}
  35. n \mathbb{C}^{n}
  36. E i j E_{ij}
  37. 𝔤 𝔩 n \mathfrak{gl}_{n}
  38. S k n S^{k}\mathbb{C}^{n}
  39. n \mathbb{C}^{n}
  40. x 1 k x^{k}_{1}
  41. E i j x 1 k = 0 E_{ij}x^{k}_{1}=0
  42. 𝔤 𝔩 n \mathfrak{gl}_{n}
  43. E i j = ψ i ψ j E_{ij}=\psi_{i}\frac{\partial}{\partial\psi_{j}}
  44. ψ i \psi_{i}
  45. Λ k n \Lambda^{k}\mathbb{C}^{n}
  46. n \mathbb{C}^{n}
  47. 𝔤 𝔩 n \mathfrak{gl}_{n}
  48. det ( E + ( n - i ) δ i j ) = 0 , n > 1 \det(E+(n-i)\delta_{ij})=0,\qquad n>1
  49. E i j c = x i p j E^{c}_{ij}=x_{i}p_{j}
  50. x i , p j x_{i},p_{j}
  51. E c E^{c}
  52. E E
  53. det ( E c ) = 0 \det(E^{c})=0
  54. E E
  55. ( n - i ) δ i j (n-i)\delta_{ij}
  56. det ( t + E + ( n - i ) δ i j ) = t [ n ] + Tr ( E ) t [ n - 1 ] , \det(t+E+(n-i)\delta_{ij})=t^{[n]}+\mathrm{Tr}(E)t^{[n-1]},~{}~{}~{}~{}
  57. t [ k ] = t ( t + 1 ) ( t + k - 1 ) t^{[k]}=t(t+1)\cdots(t+k-1)
  58. | t + E 11 + 1 E 12 E 21 t + E 22 | = | t + x 1 1 + 1 x 1 2 x 2 1 t + x 2 2 | = ( t + x 1 1 + 1 ) ( t + x 2 2 ) - x 2 1 x 1 2 = t ( t + 1 ) + t ( x 1 1 + x 2 2 ) + x 1 1 x 2 2 + x 2 2 - x 2 1 x 1 2 \begin{aligned}&\displaystyle\begin{vmatrix}t+E_{11}+1&E_{12}\\ E_{21}&t+E_{22}\end{vmatrix}=\begin{vmatrix}t+x_{1}\partial_{1}+1&x_{1}% \partial_{2}\\ x_{2}\partial_{1}&t+x_{2}\partial_{2}\end{vmatrix}\\ &\displaystyle=(t+x_{1}\partial_{1}+1)(t+x_{2}\partial_{2})-x_{2}\partial_{1}x% _{1}\partial_{2}\\ &\displaystyle=t(t+1)+t(x_{1}\partial_{1}+x_{2}\partial_{2})+x_{1}\partial_{1}% x_{2}\partial_{2}+x_{2}\partial_{2}-x_{2}\partial_{1}x_{1}\partial_{2}\end{aligned}
  59. 1 x 1 = x 1 1 + 1 , 1 x 2 = x 2 1 , x 1 x 2 = x 2 x 1 \partial_{1}x_{1}=x_{1}\partial_{1}+1,\partial_{1}x_{2}=x_{2}\partial_{1},x_{1% }x_{2}=x_{2}x_{1}\,
  60. t ( t + 1 ) + t ( x 1 1 + x 2 2 ) + x 2 x 1 1 2 + x 2 2 - x 2 x 1 1 2 - x 2 2 \displaystyle{}\quad t(t+1)+t(x_{1}\partial_{1}+x_{2}\partial_{2})+x_{2}x_{1}% \partial_{1}\partial_{2}+x_{2}\partial_{2}-x_{2}x_{1}\partial_{1}\partial_{2}-% x_{2}\partial_{2}
  61. U ( 𝔤 𝔩 n ) U(\mathfrak{gl}_{n})
  62. [ E i j , det ( E + ( n - i ) δ i j ) ] = 0 [E_{ij},\det(E+(n-i)\delta_{ij})]=0
  63. [ E i j , E k l ] = δ j k E i l - δ i l E k j [E_{ij},E_{kl}]=\delta_{jk}E_{il}-\delta_{il}E_{kj}
  64. det ( t + E + ( n - i ) δ i j ) = t [ n ] + k = n - 1 , , 0 t [ k ] C k , \det(t+E+(n-i)\delta_{ij})=t^{[n]}+\sum_{k=n-1,\dots,0}t^{[k]}C_{k},~{}~{}~{}~% {}~{}
  65. t [ k ] = t ( t + 1 ) ( t + k - 1 ) , t^{[k]}=t(t+1)\cdots(t+k-1),
  66. C k = I = ( i 1 < i 2 < < i k ) det ( E + ( k - i ) δ i j ) I I , C_{k}=\sum_{I=(i_{1}<i_{2}<\cdots<i_{k})}\det(E+(k-i)\delta_{ij})_{II},
  67. + ( k - i ) δ i j +(k-i)\delta_{ij}
  68. U ( 𝔤 𝔩 n ) U(\mathfrak{gl}_{n})
  69. [ E i j , E k l ] = δ j k E i l - δ i l E k j [E_{ij},E_{kl}]=\delta_{jk}E_{il}-\delta_{il}E_{kj}
  70. U ( 𝔤 𝔩 n ) U(\mathfrak{gl}_{n})
  71. U ( 𝔤 𝔩 n ) U(\mathfrak{gl}_{n})
  72. | t + E 11 + 1 E 12 E 21 t + E 22 | = ( t + E 11 + 1 ) ( t + E 22 ) - E 21 E 12 = t ( t + 1 ) + t ( E 11 + E 22 ) + E 11 E 22 - E 21 E 12 + E 22 . \begin{aligned}\displaystyle{}\quad\begin{vmatrix}t+E_{11}+1&E_{12}\\ E_{21}&t+E_{22}\end{vmatrix}&\displaystyle=(t+E_{11}+1)(t+E_{22})-E_{21}E_{12}% \\ &\displaystyle=t(t+1)+t(E_{11}+E_{22})+E_{11}E_{22}-E_{21}E_{12}+E_{22}.\end{aligned}
  73. ( E 11 + E 22 ) (E_{11}+E_{22})
  74. E i j E_{ij}
  75. E i j E_{ij}
  76. E 12 E_{12}
  77. [ E 12 , E 11 E 22 - E 21 E 12 + E 22 ] [E_{12},E_{11}E_{22}-E_{21}E_{12}+E_{22}]
  78. = [ E 12 , E 11 ] E 22 + E 11 [ E 12 , E 22 ] - [ E 12 , E 21 ] E 12 - E 21 [ E 12 , E 12 ] + [ E 12 , E 22 ] =[E_{12},E_{11}]E_{22}+E_{11}[E_{12},E_{22}]-[E_{12},E_{21}]E_{12}-E_{21}[E_{1% 2},E_{12}]+[E_{12},E_{22}]
  79. = - E 12 E 22 + E 11 E 12 - ( E 11 - E 22 ) E 12 - 0 + E 12 =-E_{12}E_{22}+E_{11}E_{12}-(E_{11}-E_{22})E_{12}-0+E_{12}
  80. = - E 12 E 22 + E 22 E 12 + E 12 = - E 12 + E 12 = 0. =-E_{12}E_{22}+E_{22}E_{12}+E_{12}=-E_{12}+E_{12}=0.
  81. E 11 E 22 - E 21 E 12 E_{11}E_{22}-E_{21}E_{12}
  82. E 12 E_{12}
  83. + E 22 +E_{22}
  84. E i j = a = 1 m x i a x j a , E_{ij}=\sum_{a=1}^{m}x_{ia}\frac{\partial}{\partial x_{ja}},
  85. E = X D t E=XD^{t}
  86. E E
  87. n × n n\times n
  88. E i j E_{ij}
  89. X X
  90. n × m n\times m
  91. x i j x_{ij}
  92. D D
  93. n × m n\times m
  94. x i j \frac{\partial}{\partial x_{ij}}
  95. E ( E + ( n - i ) δ i j ) E\rightarrow(E+(n-i)\delta_{ij})
  96. det ( E + ( n - i ) δ i j ) = I = ( 1 i 1 < i 2 < < i n m ) det ( X I ) det ( D I t ) \det(E+(n-i)\delta_{ij})=\sum_{I=(1\leq i_{1}<i_{2}<\cdots<i_{n}\leq m)}\det(X% _{I})\det(D^{t}_{I})
  97. det ( E + ( n - i ) δ i j ) = 0 \det(E+(n-i)\delta_{ij})=0
  98. det ( E + ( s - i ) δ i j ) K L = I = ( 1 i 1 < i 2 < < i s m ) det ( X K I ) det ( D I L t ) \det(E+(s-i)\delta_{ij})_{KL}=\sum_{I=(1\leq i_{1}<i_{2}<\cdots<i_{s}\leq m)}% \det(X_{KI})\det(D^{t}_{IL})
  99. M K L M_{KL}
  100. det ( t + E + ( n - i ) δ i j ) = t [ n ] + k = n - 1 , , 0 t [ k ] I , J det ( X I J ) det ( D J I t ) , \det(t+E+(n-i)\delta_{ij})=t^{[n]}+\sum_{k=n-1,\dots,0}t^{[k]}\sum_{I,J}\det(X% _{IJ})\det(D^{t}_{JI}),
  101. I = ( 1 i 1 < < i k n ) , I=(1\leq i_{1}<\cdots<i_{k}\leq n),
  102. J = ( 1 j 1 < < j k n ) J=(1\leq j_{1}<\cdots<j_{k}\leq n)
  103. + ( n - i ) δ i j +(n-i)\delta_{ij}
  104. E i j E_{ij}
  105. E i j x k l = x i l δ j k E_{ij}x_{kl}=x_{il}\delta_{jk}
  106. n n \mathbb{C}^{n}\oplus\cdots\oplus\mathbb{C}^{n}
  107. x i l x_{il}
  108. n n = n m . \mathbb{C}^{n}\oplus\cdots\oplus\mathbb{C}^{n}=\mathbb{C}^{n}\otimes\mathbb{C}% ^{m}.
  109. E i j dual = a = 1 n x a i x a j . E_{ij}\text{dual}=\sum_{a=1}^{n}x_{ai}\frac{\partial}{\partial x_{aj}}.
  110. E i j E_{ij}
  111. i j i\leftrightarrow j
  112. E i j dual E_{ij}\text{dual}
  113. 𝔤 𝔩 m \mathfrak{gl}_{m}
  114. E i j dual E_{ij}\text{dual}
  115. E k l E_{kl}
  116. G L n × G L m GL_{n}\times GL_{m}
  117. n m \mathbb{C}^{n}\otimes\mathbb{C}^{m}
  118. 𝔤 𝔩 n × 𝔤 𝔩 m \mathfrak{gl}_{n}\times\mathfrak{gl}_{m}
  119. E i j E_{ij}~{}~{}~{}~{}
  120. E i j dual E_{ij}\text{dual}
  121. E i j E_{ij}~{}~{}~{}~{}
  122. E i j dual E_{ij}\text{dual}
  123. G L n GL_{n}
  124. G L m GL_{m}
  125. [ x i j ] = S ( n m ) = D ρ n D ρ m D . \mathbb{C}[x_{ij}]=S(\mathbb{C}^{n}\otimes\mathbb{C}^{m})=\sum_{D}\rho_{n}^{D}% \otimes\rho_{m}^{D^{\prime}}.
  126. ρ D \rho^{D}
  127. D {D}
  128. D {D^{\prime}}
  129. G L n × G L m GL_{n}\times GL_{m}
  130. 𝔤 𝔩 n \mathfrak{gl}_{n}
  131. X = | x 11 x 12 x 13 x 1 n x 12 x 22 x 23 x 2 n x 13 x 23 x 33 x 3 n x 1 n x 2 n x 3 n x n n | , D = | 2 x 11 x 12 x 13 x 1 n x 12 2 x 22 x 23 x 2 n x 13 x 23 2 x 33 x 3 n x 1 n x 2 n x 3 n 2 x n n | X=\begin{vmatrix}x_{11}&x_{12}&x_{13}&\cdots&x_{1n}\\ x_{12}&x_{22}&x_{23}&\cdots&x_{2n}\\ x_{13}&x_{23}&x_{33}&\cdots&x_{3n}\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ x_{1n}&x_{2n}&x_{3n}&\cdots&x_{nn}\end{vmatrix},D=\begin{vmatrix}2\frac{% \partial}{\partial x_{11}}&\frac{\partial}{\partial x_{12}}&\frac{\partial}{% \partial x_{13}}&\cdots&\frac{\partial}{\partial x_{1n}}\\ \frac{\partial}{\partial x_{12}}&2\frac{\partial}{\partial x_{22}}&\frac{% \partial}{\partial x_{23}}&\cdots&\frac{\partial}{\partial x_{2n}}\\ \frac{\partial}{\partial x_{13}}&\frac{\partial}{\partial x_{23}}&2\frac{% \partial}{\partial x_{33}}&\cdots&\frac{\partial}{\partial x_{3n}}\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ \frac{\partial}{\partial x_{1n}}&\frac{\partial}{\partial x_{2n}}&\frac{% \partial}{\partial x_{3n}}&\cdots&2\frac{\partial}{\partial x_{nn}}\end{vmatrix}
  132. det ( X D + ( n - i ) δ i j ) = det ( X ) det ( D ) \det(XD+(n-i)\delta_{ij})=\det(X)\det(D)\,
  133. X = | 0 x 12 x 13 x 1 n - x 12 0 x 23 x 2 n - x 13 - x 23 0 x 3 n - x 1 n - x 2 n - x 3 n 0 | , D = | 0 x 12 x 13 x 1 n - x 12 0 x 23 x 2 n - x 13 - x 23 0 x 3 n - x 1 n - x 2 n - x 3 n 0 | . X=\begin{vmatrix}0&x_{12}&x_{13}&\cdots&x_{1n}\\ -x_{12}&0&x_{23}&\cdots&x_{2n}\\ -x_{13}&-x_{23}&0&\cdots&x_{3n}\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ -x_{1n}&-x_{2n}&-x_{3n}&\cdots&0\end{vmatrix},D=\begin{vmatrix}0&\frac{% \partial}{\partial x_{12}}&\frac{\partial}{\partial x_{13}}&\cdots&\frac{% \partial}{\partial x_{1n}}\\ -\frac{\partial}{\partial x_{12}}&0&\frac{\partial}{\partial x_{23}}&\cdots&% \frac{\partial}{\partial x_{2n}}\\ -\frac{\partial}{\partial x_{13}}&-\frac{\partial}{\partial x_{23}}&0&\cdots&% \frac{\partial}{\partial x_{3n}}\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ -\frac{\partial}{\partial x_{1n}}&-\frac{\partial}{\partial x_{2n}}&-\frac{% \partial}{\partial x_{3n}}&\cdots&0\end{vmatrix}.
  134. det ( X D + ( n - i ) δ i j ) = det ( X ) det ( D ) . \det(XD+(n-i)\delta_{ij})=\det(X)\det(D).\,
  135. [ M i j , Y k l ] = - δ j k Q i l [M_{ij},Y_{kl}]=-\delta_{jk}Q_{il}~{}~{}~{}~{}~{}
  136. det ( M Y + Q diag ( n - 1 , n - 2 , , 1 , 0 ) ) = det ( M ) det ( Y ) . \det(MY+Q\,\mathrm{diag}(n-1,n-2,\dots,1,0))=\det(M)\det(Y).~{}~{}~{}~{}~{}~{}% ~{}
  137. x i j + f i j ( x 11 , , x k l , ) \frac{\partial}{\partial x_{ij}}+f_{ij}(x_{11},\dots,x_{kl},\dots)
  138. x i j x_{ij}
  139. i j \partial_{ij}
  140. det ( z - A - X 1 z - B D t ) \det\left(\frac{\partial}{\partial_{z}}-A-X\frac{1}{z-B}D^{t}\right)
  141. = det calculate as if all commute Put all x and z on the left, while all derivations on the right ={\det}\text{calculate as if all commute}_{\,\text{Put all }x\,\text{ and }z\,% \text{ on the left, while all derivations on the right}}
  142. ( z - A - X 1 z - B D t ) \left(\frac{\partial}{\partial_{z}}-A-X\frac{1}{z-B}D^{t}\right)
  143. L ( z ) = A + X 1 z - B D t L(z)=A+X\frac{1}{z-B}D^{t}
  144. det ( z - L ( z ) ) = i = 0 n H i ( z ) ( z ) i . \det\left(\frac{\partial}{\partial_{z}}-L(z)\right)=\sum_{i=0}^{n}H_{i}(z)% \left(\frac{\partial}{\partial_{z}}\right)^{i}.
  145. [ H i ( z ) , H j ( w ) ] = 0 , [H_{i}(z),H_{j}(w)]=0,~{}~{}~{}~{}~{}~{}~{}~{}
  146. perm ( X t D - ( n - i ) δ i j ) = perm Calculate as if all commute Put all x on the left, with all derivations on the right ( X t D ) . \mathrm{perm}(X^{t}D-(n-i)\delta_{ij})=\mathrm{perm}\text{Calculate as if all % commute}_{\,\text{Put all }x\,\text{ on the left, with all derivations on the % right}}(X^{t}D).

Capital_allocation_line.html

  1. CAL : E ( r C ) = r F + σ C E ( r P ) - r F σ P \mathrm{CAL}:E(r_{C})=r_{F}+\sigma_{C}\frac{E(r_{P})-r_{F}}{\sigma_{P}}

Capitol_Hill_Babysitting_Co-op.html

  1. S g + S m = 0 S_{g}+S_{m}=0
  2. S g S_{g}
  3. S m S_{m}
  4. S m = - S g S_{m}=-S_{g}
  5. S m S_{m}
  6. S g = 0 S_{g}=0
  7. S m = 0 S_{m}=0
  8. S m = - S g = - S_{m}=-S_{g}=-

Carbon_nanotube_nanomotor.html

  1. C = ε A d L C={{\varepsilon A}\over d}\propto L
  2. E L 0 E\propto L^{0}
  3. V = E * L = E * L L V=E*L=E*L\propto L
  4. Q = C V L 2 Q=CV\propto L^{2}
  5. F = A * E 2 L 2 F=A*E^{2}\propto L^{2}
  6. F = Q 1 * Q 2 d 2 L 2 * L 2 L 2 L 2 F={{Q_{1}*Q_{2}}\over d^{2}}\propto{{L^{2}*L^{2}}\over L^{2}}\propto L^{2}
  7. S 1 = V 0 s i n ( ω t ) S_{1}=V_{0}sin(\omega t)
  8. S 2 = V 0 s i n ( ω t - π ) S_{2}=V_{0}sin(\omega t-\pi)
  9. S 3 = V 0 s i n ( 2 ω t + π / 2 ) S_{3}=V_{0}sin(2\omega t+{\pi/2})
  10. R = - V 0 R=-V_{0}
  11. ω \omega
  12. k k
  13. T T
  14. Γ = ω 2 π e - Δ E k T \Gamma={\omega\over 2\pi}e^{{-\Delta E}\over kT}
  15. Γ 1 H z \Gamma\approx 1Hz
  16. ω = Δ E m a 0 2 \omega=\sqrt{\Delta E\over{ma^{2}_{0}}}
  17. a 0 2 a^{2}_{0}

Carbonaceous_biochemical_oxygen_demand.html

  1. I 0 / I = 1 + K S V [ O 2 ] I_{0}/I~{}=~{}1~{}+~{}K_{SV}~{}[O_{2}]
  2. I = L u m i n e s c e n c e i n t h e p r e s e n c e o f o x y g e n I~{}=~{}Luminescence~{}in~{}the~{}presence~{}of~{}oxygen
  3. I 0 = L u m i n e s c e n c e i n t h e a b s e n c e o f o x y g e n I_{0}~{}=~{}Luminescence~{}in~{}the~{}absence~{}of~{}oxygen
  4. K S V = S t e r n - V o l m e r c o n s t a n t f o r o x y g e n q u e n c h i n g K_{SV}~{}=~{}Stern-Volmer~{}constant~{}for~{}oxygen~{}quenching
  5. [ O 2 ] = D i s s o l v e d o x y g e n c o n c e n t r a t i o n [O_{2}]~{}=~{}Dissolved~{}oxygen~{}concentration

Carey_Foster_bridge.html

  1. σ σ
  2. X - Y = σ ( l 2 - l 1 ) {X-Y=\sigma(l_{2}-l_{1})}\,
  3. α α
  4. β β
  5. σ σ
  6. P Q = X + σ ( l 1 + α ) Y + σ ( 100 - l 1 + β ) {P\over Q}={{X+\sigma(l_{1}+\alpha)}\over{Y+\sigma(100-l_{1}+\beta)}}
  7. P Q + 1 = X + Y + σ ( 100 + α + β ) Y + σ ( 100 - l 1 + β ) {{P\over Q}+1}={{X+Y+\sigma(100+\alpha+\beta)}\over{Y+\sigma(100-l_{1}+\beta)}}
  8. P Q = Y + σ ( l 2 + α ) X + σ ( 100 - l 2 + β ) {P\over Q}={{Y+\sigma(l_{2}+\alpha)}\over{X+\sigma(100-l_{2}+\beta)}}
  9. P Q + 1 = X + Y + σ ( 100 + α + β ) X + σ ( 100 - l 2 + β ) {{P\over Q}+1}={{X+Y+\sigma(100+\alpha+\beta)}\over{X+\sigma(100-l_{2}+\beta)}}
  10. Y + σ ( 100 - l 1 + β ) = X + σ ( 100 - l 2 + β ) \displaystyle Y+\sigma(100-l_{1}+\beta)=X+\sigma(100-l_{2}+\beta)
  11. l < s u b > 1 l<sub>1

Carleson's_theorem.html

  1. f ^ ( n ) \hat{f}(n)
  2. lim N | n | N f ^ ( n ) e i n x = f ( x ) \lim_{N\rightarrow\infty}\sum_{|n|\leq N}\hat{f}(n)e^{inx}=f(x)
  3. f ^ ( ξ ) \hat{f}(\xi)
  4. lim R | ξ | R f ^ ( ξ ) e 2 π i x ξ d ξ = f ( x ) \lim_{R\rightarrow\infty}\int_{|\xi|\leq R}\hat{f}(\xi)e^{2\pi ix\xi}\,d\xi=f(x)
  5. s n ( x ) = o ( log ( n ) 1 / p ) almost everywhere , s_{n}(x)=o(\log(n)^{1/p})\,\text{ almost everywhere},\,
  6. C f ( x ) = sup N | - N N f ^ ( y ) e 2 π i x y d y | Cf(x)=\sup_{N}\left|\int_{-N}^{N}\hat{f}(y)e^{2\pi ixy}\,dy\right|

Carlo_Severini.html

  1. { u x = f ( x , y , u , u y ) ( x , y ) + × [ a , b ] u ( 0 , y ) = U ( y ) y [ a , b ] , \left\{\begin{array}[]{lc}\frac{\partial u}{\partial x}=f\left(x,y,u,\frac{% \partial u}{\partial y}\right)&(x,y)\in\mathbb{R}^{+}\times[a,b]\\ u(0,y)=U(y)&y\in[a,b]\Subset\mathbb{R}\end{array}\right.,
  2. U U
  3. [ a , b ] [a,b]
  4. f f
  5. { ( x , y , z , p ) = ( 0 , y , U ( y ) , U ( y ) ) ; y [ a , b ] } \scriptstyle\{(x,y,z,p)=(0,y,U(y),U^{\prime}(y));y\in[a,b]\}
  6. f f
  7. C ( 1 , 1 ) C^{(1,1)}

Cartan's_lemma_(potential_theory).html

  1. u ( z ) = 1 2 π 𝐂 log | z - ζ | d μ ( ζ ) u(z)=\frac{1}{2\pi}\int_{\mathbf{C}}\log|z-\zeta|\,d\mu(\zeta)
  2. i r i < 5 H \sum_{i}r_{i}<5H\,
  3. u ( z ) n 2 π log H e u(z)\geq\frac{n}{2\pi}\log\frac{H}{e}

Carter_constant.html

  1. C = p θ 2 + cos 2 θ ( a 2 ( m 2 - E 2 ) + ( L sin θ ) 2 ) C=p_{\theta}^{2}+\cos^{2}\theta\Bigg(a^{2}(m^{2}-E^{2})+\left(\frac{L}{\sin% \theta}\right)^{2}\Bigg)
  2. p θ p_{\theta}
  3. E E
  4. L L
  5. m m
  6. a a
  7. C C
  8. C C
  9. K = C + ( L - a E ) 2 K=C+(L-aE)^{2}
  10. C C
  11. K K
  12. K K
  13. K K
  14. C = K μ ν u μ u ν C=K^{\mu\nu}u_{\mu}u_{\nu}
  15. u u
  16. K μ ν = 2 Σ l ( μ n ν ) + r 2 g μ ν K^{\mu\nu}=2\Sigma\ l^{(\mu}n^{\nu)}+r^{2}g^{\mu\nu}
  17. g μ ν g^{\mu\nu}
  18. l μ l^{\mu}
  19. n ν n^{\nu}
  20. l μ = ( r 2 + a 2 Δ , 1 , 0 , a Δ ) l^{\mu}=\left(\frac{r^{2}+a^{2}}{\Delta},1,0,\frac{a}{\Delta}\right)
  21. n ν = ( r 2 + a 2 2 Σ , - Δ 2 Σ , 0 , a 2 Σ ) n^{\nu}=\left(\frac{r^{2}+a^{2}}{2\Sigma},-\frac{\Delta}{2\Sigma},0,\frac{a}{2% \Sigma}\right)
  22. E E
  23. L L
  24. m m
  25. C = p θ 2 + ( L sin θ ) 2 C=p_{\theta}^{2}+\left(\frac{L}{\sin\theta}\right)^{2}
  26. θ = π / 2 \theta=\pi/2
  27. p θ = 0 p_{\theta}=0
  28. C = L 2 C=L^{2}

Cartesian_product.html

  1. A × B = { ( a , b ) a A and b B } . A\times B=\{\,(a,b)\mid a\in A\ \mbox{ and }~{}\ b\in B\,\}.
  2. ( x , y ) = { { x } , { x , y } } (x,y)=\{\{x\},\{x,y\}\}
  3. X × Y 𝒫 ( 𝒫 ( X Y ) ) X\times Y\subseteq\mathcal{P}(\mathcal{P}(X\cup Y))
  4. 𝒫 \mathcal{P}
  5. A × B B × A , A\times B\neq B\times A,
  6. ( A × B ) × C A × ( B × C ) (A\times B)\times C\neq A\times(B\times C)
  7. ( A B ) × ( C D ) = ( A × C ) ( B × D ) (A\cap B)\times(C\cap D)=(A\times C)\cap(B\times D)
  8. ( A B ) × ( C D ) ( A × C ) ( B × D ) (A\cup B)\times(C\cup D)\neq(A\times C)\cup(B\times D)
  9. ( A × C ) ( B × D ) = [ ( A B ) × C ] [ ( A B ) × ( C D ) ] [ ( B A ) × D ] (A\times C)\cup(B\times D)=[(A\setminus B)\times C]\cup[(A\cap B)\times(C\cup D% )]\cup[(B\setminus A)\times D]
  10. ( A × C ) ( B × D ) = [ A × ( C D ) ] [ ( A B ) × C ] (A\times C)\setminus(B\times D)=[A\times(C\setminus D)]\cup[(A\setminus B)% \times C]
  11. A × ( B C ) = ( A × B ) ( A × C ) , A\times(B\cap C)=(A\times B)\cap(A\times C),
  12. A × ( B C ) = ( A × B ) ( A × C ) , A\times(B\cup C)=(A\times B)\cup(A\times C),
  13. A × ( B C ) = ( A × B ) ( A × C ) , A\times(B\setminus C)=(A\times B)\setminus(A\times C),
  14. ( A × B ) c = ( A c × B c ) ( A c × B ) ( A × B c ) . (A\times B)^{c}=(A^{c}\times B^{c})\cup(A^{c}\times B)\cup(A\times B^{c}).
  15. if A B then A × C B × C , \,\text{if }A\subseteq B\,\text{ then }A\times C\subseteq B\times C,
  16. if both A , B then A × B C × D A C and B D . \,\text{if both }A,B\neq\emptyset\,\text{ then }A\times B\subseteq C\times D% \iff A\subseteq C\and B\subseteq D.
  17. X n = X × X × × X n = { ( x 1 , , x n ) | x i X for all i = 1 , , n } . X^{n}=\underbrace{X\times X\times\cdots\times X}_{n}=\{(x_{1},\ldots,x_{n})\ |% \ x_{i}\in X\,\text{ for all }i=1,\ldots,n\}.
  18. X 1 × × X n = { ( x 1 , , x n ) : x i X i } . X_{1}\times\cdots\times X_{n}=\{(x_{1},\ldots,x_{n}):x_{i}\in X_{i}\}.
  19. { X i | i I } \left\{X_{i}\,|\,i\in I\right\}
  20. i I X i = { f : I i I X i | ( i ) ( f ( i ) X i ) } , \prod_{i\in I}X_{i}=\left\{f:I\to\bigcup_{i\in I}X_{i}\ \Big|\ (\forall i)(f(i% )\in X_{i})\right\},
  21. π j : i I X i X j , \pi_{j}:\prod_{i\in I}X_{i}\to X_{j},
  22. π j ( f ) = f ( j ) \pi_{j}(f)=f(j)
  23. \mathbb{N}
  24. n = 1 = × × \prod_{n=1}^{\infty}\mathbb{R}=\mathbb{R}\times\mathbb{R}\times\cdots
  25. ω \mathbb{R}^{\omega}
  26. \mathbb{R}^{\mathbb{N}}
  27. i I X i = i I X \prod_{i\in I}X_{i}=\prod_{i\in I}X
  28. ( f × g ) ( a , b ) = ( f ( a ) , g ( b ) ) . (f\times g)(a,b)=(f(a),g(b)).

Casio_FX-502P_series.html

  1. 1 × n × ( n - 1 ) × × 2 × 1 = n ! 1\times n\times(n-1)\times\cdots\times 2\times 1=n!

Casio_FX-602P_series.html

  1. 1 × n × ( n - 1 ) × × 2 × 1 = n ! 1\times n\times(n-1)\times\cdots\times 2\times 1=n!

Casio_FX-603P.html

  1. 1 × n × ( n - 1 ) × × 2 × 1 = n ! 1\times n\times(n-1)\times\cdots\times 2\times 1=n!

Castelnuovo–Mumford_regularity.html

  1. H i ( 𝐏 n , F ( r - i ) ) = 0 H^{i}(\mathbf{P}^{n},F(r-i))=0\,
  2. F j F 0 M 0 \cdots\rightarrow F_{j}\rightarrow\cdots\rightarrow F_{0}\rightarrow M\rightarrow 0

Catalan_surface.html

  1. x = f ( u ) + v i ( u ) , y = g ( u ) + v j ( u ) , z = h ( u ) + v k ( u ) x=f(u)+vi(u),\quad y=g(u)+vj(u),\quad z=h(u)+vk(u)\,

Catchment_hydrology.html

  1. I - O = d S / d t I-O=dS/dt
  2. I I
  3. O O
  4. d S / d t dS/dt
  5. P - R - E T = d S / d t P-R-ET=dS/dt
  6. P - R = E T P-R=ET

Caterpillar_tree.html

  1. 2 n - 4 + 2 ( n - 4 ) / 2 . 2^{n-4}+2^{\lfloor(n-4)/2\rfloor}.

Causal_decision_theory.html

  1. U U
  2. A A
  3. U ( A ) = j P ( A > O j ) D ( O j ) , U(A)=\sum\limits_{j}P(A>O_{j})D(O_{j}),
  4. D ( O j ) D(O_{j})
  5. O j O_{j}
  6. P ( A > O j ) P(A>O_{j})
  7. A A
  8. O j O_{j}
  9. P ( A > O j ) P(A>O_{j})
  10. P ( O j | A ) P(O_{j}|A)
  11. A A
  12. A > O j A>O_{j}
  13. P ( A > O j ) = P ( O j | A ) P(A>O_{j})=P(O_{j}|A)
  14. P ( A > O j ) P(A>O_{j})
  15. w w
  16. w A w_{A}
  17. A A
  18. A A
  19. A A

Cavalieri's_principle.html

  1. 1 3 ( base × height ) \frac{1}{3}\left(\,\text{base}\times\,\text{height}\right)
  2. 4 3 π r 3 \frac{4}{3}\pi r^{3}
  3. r r
  4. r r
  5. r r
  6. r r
  7. y y
  8. π ( r 2 - y 2 ) \pi\left(r^{2}-y^{2}\right)
  9. π ( r 2 - y 2 ) \pi\left(r^{2}-y^{2}\right)
  10. 1 3 \frac{1}{3}
  11. 2 3 \frac{2}{3}
  12. 2 3 \frac{2}{3}
  13. base × height = π r 2 r = π r 3 \,\text{base}\times\,\text{height}=\pi r^{2}\cdot r=\pi r^{3}
  14. ( 2 3 ) π r 3 \left(\frac{2}{3}\right)\pi r^{3}
  15. ( 4 3 ) π r 3 \left(\frac{4}{3}\right)\pi r^{3}

Cayley's_ruled_cubic_surface.html

  1. 3 z - 3 x y + x 3 = 0. 3z-3xy+x^{3}=0.

Cayley's_Ω_process.html

  1. Ω = | x 11 x 1 n x n 1 x n n | . \Omega=\begin{vmatrix}\frac{\partial}{\partial x_{11}}&\cdots&\frac{\partial}{% \partial x_{1n}}\\ \vdots&\ddots&\vdots\\ \frac{\partial}{\partial x_{n1}}&\cdots&\frac{\partial}{\partial x_{nn}}\end{% vmatrix}.
  2. 2 f g x 1 y 2 - 2 f g x 2 y 1 \frac{\partial^{2}fg}{\partial x_{1}\partial y_{2}}-\frac{\partial^{2}fg}{% \partial x_{2}\partial y_{1}}

Cellular_approximation_theorem.html

  1. f ( X n ) Y n f(X^{n})\subseteq Y^{n}
  2. e k Y e^{k}\subseteq Y
  3. k n k\leq n
  4. X n - 1 e n X^{n-1}\cup e^{n}
  5. X n - 1 e n X^{n-1}\cup e^{n}
  6. f ( e n ) f(e^{n})
  7. n < k n<k
  8. π n ( S k ) = 0 \pi_{n}(S^{k})=0\,
  9. S n S^{n}\,
  10. S k S^{k}\,
  11. S n S^{n}\,
  12. S k S^{k}\,
  13. S n S^{n}\,
  14. S k S^{k}\,
  15. π n ( S k ) = 0 \pi_{n}(S^{k})=0\,
  16. A X A\subseteq X\,
  17. X - A X-A
  18. i n i\leq n\,
  19. ( D i , D i ) (D^{i},\partial D^{i})\,
  20. π i ( X , A ) \pi_{i}(X,A)\,
  21. ( X , X n ) (X,X^{n})\,
  22. ( X , X n ) (X,X^{n})\,
  23. π i ( X n ) \pi_{i}(X^{n})\,
  24. π i ( X ) \pi_{i}(X)\,
  25. i < n i<n\,
  26. π n ( X n ) \pi_{n}(X^{n})\,
  27. π n ( X ) \pi_{n}(X)\,

Centre_wavelength.html

  1. λ c = 1 P t o t a l p ( λ ) λ d λ \lambda_{c}=\frac{1}{P_{total}}\int p(\lambda)\lambda\,d\lambda
  2. P t o t a l = p ( λ ) d λ P_{total}=\int p(\lambda)d\lambda
  3. p ( λ ) p(\lambda)
  4. p ( λ ) p(\lambda)

Ceyuan_haijing.html

  1. ( b 14 - a 14 ) + ( b 15 - a 15 ) (b_{14}-a_{14})+(b_{15}-a_{15})
  2. r = 120 r=120
  3. a 1 = 320 a_{1}=320
  4. b 1 = 640 b_{1}=640
  5. c i c_{i}
  6. a i a_{i}
  7. b i b_{i}
  8. a i + b i a_{i}+b_{i}
  9. b i - a i b_{i}-a_{i}
  10. a i + c i a_{i}+c_{i}
  11. c i - a i c_{i}-a_{i}
  12. b i + c i b_{i}+c_{i}
  13. c i - b i c_{i}-b_{i}
  14. c i + ( b i - a i ) c_{i}+(b_{i}-a_{i})
  15. c i - ( b i - a i ) c_{i}-(b_{i}-a_{i})
  16. a i + b i + c i a_{i}+b_{i}+c_{i}
  17. a i + b i a_{i}+b_{i}
  18. a 6 = a 7 a_{6}=a_{7}
  19. b 6 = b 7 b_{6}=b_{7}
  20. c 6 = c 7 c_{6}=c_{7}
  21. a 8 = a 9 a_{8}=a_{9}
  22. b 8 = b 9 b_{8}=b_{9}
  23. c 8 = c 9 c_{8}=c_{9}
  24. ( c 1 - a 1 ) * ( c 1 * b 1 ) (c_{1}-a_{1})*(c_{1}*b_{1})
  25. 1 2 1\over 2
  26. ( d 1 ) 2 (d_{1})^{2}
  27. a 10 * b 11 a_{10}*b_{11}
  28. 1 2 1\over 2
  29. ( d 1 ) 2 (d_{1})^{2}
  30. a 13 * b 1 a_{13}*b_{1}
  31. 1 2 1\over 2
  32. ( d 1 ) 2 (d_{1})^{2}
  33. a 1 * b 13 a_{1}*b_{13}
  34. 1 2 1\over 2
  35. ( d 1 ) 2 (d_{1})^{2}
  36. b 2 * b 15 b_{2}*b_{15}
  37. [ - F o r m u l a E r r o r - ] [-FormulaError-]
  38. ( r 1 ) 2 (r_{1})^{2}
  39. a 14 * a 3 a_{14}*a_{3}
  40. ( r 1 ) 2 (r_{1})^{2}
  41. a 5 * b 4 a_{5}*b_{4}
  42. ( d 1 ) 2 (d_{1})^{2}
  43. a 8 * b 6 a_{8}*b_{6}
  44. a 9 * b 7 a_{9}*b_{7}
  45. = ( r 1 ) 2 =(r_{1})^{2}
  46. ( b 14 * c 14 ) * ( a 15 + c 15 ) (b_{14}*c_{14})*(a_{15}+c_{15})
  47. ( r 1 ) 2 (r_{1})^{2}
  48. c 6 * c 8 c_{6}*c_{8}
  49. c 7 * c 9 ) c_{7}*c_{9})
  50. a 13 * b 13 a_{13}*b_{13}
  51. [ - F o r m u l a E r r o r - ] [-FormulaError-]
  52. a 2 + b 2 + c 2 = b 1 + c 1 a_{2}+b_{2}+c_{2}=b_{1}+c_{1}
  53. a 3 + b 3 + c 3 = a 1 + c 1 a_{3}+b_{3}+c_{3}=a_{1}+c_{1}
  54. a 4 + b 4 + c 4 = 2 b 1 a_{4}+b_{4}+c_{4}=2b_{1}
  55. a 5 + b 5 + c 5 = 2 a 1 a_{5}+b_{5}+c_{5}=2a_{1}
  56. a 6 + b 6 + c 6 = b 1 a_{6}+b_{6}+c_{6}=b_{1}
  57. a 7 + b 7 + c 7 = b 1 a_{7}+b_{7}+c_{7}=b_{1}
  58. a 8 + b 8 + c 8 = a 1 a_{8}+b_{8}+c_{8}=a_{1}
  59. a 9 + b 9 + c 9 = a 1 a_{9}+b_{9}+c_{9}=a_{1}
  60. a 10 + b 10 + c 10 = b 1 + c 1 - a 1 a_{10}+b_{10}+c_{10}=b_{1}+c_{1}-a_{1}
  61. a 11 + b 11 + c 11 = c 1 - b 1 + a 1 a_{11}+b_{11}+c_{11}=c_{1}-b_{1}+a_{1}
  62. a 12 + b 12 + c 12 = c 1 a_{12}+b_{12}+c_{12}=c_{1}
  63. a 13 + b 13 + c 13 = a 1 + b 1 - c 1 a_{13}+b_{13}+c_{13}=a_{1}+b_{1}-c_{1}
  64. a 14 + b 14 + c 14 = c 1 - a 1 a_{14}+b_{14}+c_{14}=c_{1}-a_{1}
  65. a 15 + b 15 + c 15 = c 1 - c 1 a_{15}+b_{15}+c_{15}=c_{1}-c_{1}
  66. ( b 7 - a 7 ) + ( b 8 - a 8 ) + ( b 14 - a 14 ) + ( b 15 - a 15 ) = 2 * ( b 12 - a 12 ) (b_{7}-a_{7})+(b_{8}-a_{8})+(b_{14}-a_{14})+(b_{15}-a_{15})=2*(b_{12}-a_{12})
  67. a 8 + ( b 7 - a 7 ) + ( b 8 - a 8 ) = b 7 a_{8}+(b_{7}-a_{7})+(b_{8}-a_{8})=b_{7}
  68. d = 2 a 1 × b 1 a 1 + b 1 + c 1 d={2a_{1}\times b_{1}\over a_{1}+b_{1}+c_{1}}
  69. = 2 * 320 * 600 320 + 600 + ( 320 2 + 600 2 ) = 240 ={2*320*600\over 320+600+\sqrt{(}320^{2}+600^{2})}=240
  70. a 2 a_{2}
  71. b 2 b_{2}
  72. 2 a 2 × b 2 a 2 + b 2 + c 2 = d {2a_{2}\times b_{2}\over a_{2}+b_{2}+c_{2}}=d
  73. = 2 * 256 * 480 256 + 600 + ( 256 + 600 2 ) = 240 ={2*256*480\over 256+600+\sqrt{(}256^{+}600^{2})}=240
  74. 2 a 3 × b 3 a 3 + b 3 + c 3 = d {2a_{3}\times b_{3}\over a_{3}+b_{3}+c_{3}}=d
  75. 2 a 12 × b 12 c 12 = d {2a_{12}\times b_{12}\over c_{12}}=d
  76. 2 a × b a + b = d {2a\times b\over a+b}=d
  77. 2 a 10 × b 10 b 10 - a 10 + c 10 = d {2a_{10}\times b_{10}\over b_{10}-a_{10}+c_{10}}=d
  78. 2 a 11 × b 11 b 11 - a 11 + c 11 = d {2a_{11}\times b_{11}\over b_{11}-a_{11}+c_{11}}=d
  79. 2 a 13 × b 13 b 13 + a 13 - c 13 = d {2a_{13}\times b_{13}\over b_{13}+a_{13}-c_{13}}=d
  80. 2 a 14 × b 14 c 14 - a 14 = d {2a_{14}\times b_{14}\over c_{14}-a_{14}}=d
  81. 2 a 15 × b 15 c 15 - b 15 = d {2a_{15}\times b_{15}\over c_{15}-b_{15}}=d
  82. 480 - x 480-x
  83. 200 - x 200-x
  84. x 2 - 680 x + 96000 x^{2}-680x+96000
  85. x 2 - 680 x + 96000 = 2 x 2 x^{2}-680x+96000=2x^{2}
  86. - x 2 - 680 x + 96000 = 0 -x^{2}-680x+96000=0
  87. r = 120 r=120
  88. b 2 b_{2}
  89. a 10 a_{10}
  90. b 11 b_{11}
  91. a 11 a_{11}
  92. b 10 b_{10}
  93. a 15 a_{15}
  94. b 14 b_{14}
  95. a 11 a_{11}
  96. b 10 b_{10}
  97. b 2 b_{2}
  98. c 4 c_{4}
  99. b 2 b_{2}
  100. a 11 a_{11}
  101. x 2 + a 11 x - 2 b 2 a 11 = 0 x^{2}+a_{11}x-2b_{2}a_{11}=0
  102. b 2 b_{2}
  103. b 11 b_{11}
  104. x 2 + b 2 x - b 2 b 11 = 0 x^{2}+b_{2}x-b_{2}b_{11}=0
  105. b 2 b_{2}
  106. a 15 a_{15}
  107. x 3 + a 15 x 2 - 4 a 15 b 2 2 = 0 x^{3}+a_{15}x^{2}-4a_{15}b_{2}^{2}=0
  108. b 2 b_{2}
  109. a 14 a_{14}
  110. x 3 - ( b 2 - 2 a 14 ) x 2 + a 14 2 * x + a 14 2 * b 2 = 0 x^{3}-(b_{2}-2a_{14})x^{2}+a_{14}^{2}*x+a_{14}^{2}*b_{2}=0
  111. b 2 b_{2}
  112. a 10 a_{10}
  113. x 2 + ( b 2 - ( b 2 - c 10 ) ) x + b 2 ( b 2 - c 10 ) = 0 x^{2}+(b_{2}-(b_{2}-c_{10}))x+b_{2}(b_{2}-c_{10})=0
  114. b 2 b_{2}
  115. c 2 c_{2}
  116. ( ( 1 / 2 ) * c 2 - ( 1 / 2 ) * b 2 + b 2 ) * x 2 - ( 1 / 2 ) * ( c 2 - b 2 ) b 2 2 = 0 ((1/2)*c_{2}-(1/2)*b_{2}+b_{2})*x^{2}-(1/2)*(c_{2}-b_{2})b_{2}^{2}=0
  117. b 2 b_{2}
  118. c 1 c_{1}
  119. 2 x 2 + ( ( c 1 + b 2 ) + ( c 1 - b 2 ) ) x - ( ( c 1 + b 2 ) ( c 1 - b 2 ) - ( c 1 - b 2 ) 2 ) ) = 0 2x^{2}+((c_{1}+b_{2})+(c_{1}-b_{2}))x-((c_{1}+b_{2})(c_{1}-b_{2})-(c_{1}-b_{2}% )^{2}))=0
  120. b 2 b_{2}
  121. c 6 c_{6}
  122. 2 x 2 - 2 ( b 2 - 2 ( b 2 - c 5 ) ) b 2 = 0 2x^{2}-2(b_{2}-2(b_{2}-c_{5}))b_{2}=0
  123. b 2 b_{2}
  124. b 14 b_{14}
  125. x 2 - 2 b 2 x + ( ( b 2 - b 14 ) 2 - b 14 2 = 0 x^{2}-2b_{2}x+((b_{2}-b_{14})^{2}-b_{14}^{2}=0
  126. b 2 b_{2}
  127. a 10 a_{10}
  128. ( 2 b 2 - a 10 ) x - b 2 a 10 = 0 (2b_{2}-a_{10})x-b_{2}a_{10}=0
  129. b 2 b_{2}
  130. c 15 c_{15}
  131. b 15 b_{15}
  132. x 2 + ( b 2 + c 15 ) x - b 2 c 15 = 0 x^{2}+(b_{2}+c_{15})x-b_{2}c_{15}=0
  133. b 2 b_{2}
  134. c 14 c_{14}
  135. a 14 a_{14}
  136. x 4 - 2 ( b 2 - c 14 ) x 3 + ( b 2 - c 14 ) 2 x 2 + 2 b 2 c 14 2 x - ( 2 ( b 2 - c 14 ) - b 2 ) ) b 2 c 14 2 = 0 x^{4}-2(b_{2}-c_{14})x^{3}+(b_{2}-c_{14})^{2}x^{2}+2b_{2}c_{14}^{2}x-(2(b_{2}-% c_{14})-b_{2}))b_{2}c_{14}^{2}=0
  137. b 2 b_{2}
  138. c 6 c_{6}
  139. r = ( ( 2 c 6 - b 2 ) b 2 ) r=\sqrt{(}(2c_{6}-b_{2})b_{2})
  140. b 2 b_{2}
  141. c 8 c_{8}
  142. - x 3 - c 8 x 2 - b 2 2 x + c 8 b 2 2 = 0 -x^{3}-c_{8}x^{2}-b_{2}^{2}x+c_{8}b_{2}^{2}=0
  143. b 2 b_{2}
  144. b 14 + c 14 b_{14}+c_{14}
  145. b 2 b_{2}
  146. a 15 + c 15 a_{15}+c_{15}
  147. a 3 a_{3}
  148. c 5 c_{5}
  149. b 10 b_{10}
  150. a 10 a_{10}
  151. b 14 b_{14}
  152. b 15 b_{15}
  153. c 11 c_{11}
  154. c 13 c_{13}
  155. c 1 c_{1}
  156. c 9 c_{9}
  157. a 15 a_{15}
  158. b 11 b_{11}
  159. c 14 c_{14}
  160. c 15 c_{15}
  161. c 9 c_{9}
  162. c 7 c_{7}
  163. a 15 + c 15 a_{15}+c_{15}
  164. b 14 + c 14 b_{14}+c_{14}
  165. b 1 b_{1}
  166. b 14 b_{14}
  167. a 14 a_{14}
  168. a 15 a_{15}
  169. b 15 b_{15}
  170. b 11 b_{11}
  171. a 11 a_{11}
  172. c 10 c_{10}
  173. c 4 c_{4}
  174. c 2 c_{2}
  175. c 1 c_{1}
  176. c 6 c_{6}
  177. c 9 - a 11 c_{9}-a_{11}
  178. c 15 c_{15}
  179. c 14 c_{14}
  180. c 9 c_{9}
  181. c 12 c_{12}
  182. a 15 + b 14 a_{15}+b_{14}
  183. c 13 c_{13}
  184. a 1 a_{1}
  185. a 1 + c 3 a_{1}+c_{3}
  186. a 1 a_{1}
  187. a 1 a_{1}
  188. a 1 a_{1}
  189. a 1 a_{1}
  190. a 1 a_{1}
  191. a 1 a_{1}
  192. a 1 a_{1}
  193. a 1 a_{1}
  194. a 1 a_{1}
  195. a 1 a_{1}
  196. a 1 a_{1}
  197. a 1 + c 3 a_{1}+c_{3}
  198. a 1 a_{1}
  199. a 1 a_{1}
  200. a 1 a_{1}
  201. a 1 a_{1}
  202. a 1 a_{1}
  203. a 1 a_{1}
  204. a 15 a_{15}
  205. b 15 b_{15}
  206. b 14 b_{14}
  207. a 14 a_{14}
  208. a 10 a_{10}
  209. b 10 b_{10}
  210. c 11 c_{11}
  211. c 5 c_{5}
  212. c 3 c_{3}
  213. c 1 c_{1}
  214. c 9 c_{9}
  215. b 10 - c 6 b_{10}-c_{6}
  216. c 14 c_{14}
  217. c 15 c_{15}
  218. c 6 c_{6}
  219. c 12 c_{12}
  220. a 15 + b 14 a_{15}+b_{14}
  221. a 13 a_{13}
  222. a 14 a_{14}
  223. b 15 b_{15}
  224. a 15 a_{15}
  225. b 14 b_{14}
  226. a 14 a_{14}
  227. a 15 a_{15}
  228. b 14 b_{14}
  229. b 15 b_{15}
  230. a 14 a_{14}
  231. c 8 c_{8}
  232. b 15 b_{15}
  233. c 7 c_{7}
  234. b 15 b_{15}
  235. c 13 c_{13}
  236. a 14 a_{14}
  237. c 13 c_{13}
  238. d - a 14 d-a_{14}
  239. d - b 15 d-b_{15}
  240. d - b 14 d-b_{14}
  241. d - a 15 d-a_{15}
  242. c 12 c_{12}
  243. a 15 + b 14 a_{15}+b_{14}
  244. a 15 + b 14 a_{15}+b_{14}
  245. c 13 c_{13}
  246. b 15 + c 13 b_{15}+c_{13}
  247. c 13 - b 15 c_{13}-b_{15}
  248. a 14 + b 15 + c 13 a_{14}+b_{15}+c_{13}
  249. a 14 + b 15 + c 13 - a 14 a_{14}+b_{15}+c_{13}-a_{14}
  250. c 14 c_{14}
  251. d - b 15 d-b_{15}
  252. c 5 c_{5}
  253. d - a 14 d-a_{14}
  254. a 1 - a 14 a_{1}-a_{14}
  255. b 1 - b 15 b_{1}-b_{15}
  256. a 1 + a 14 a_{1}+a_{14}
  257. b 1 - b 15 b_{1}-b_{15}
  258. a 14 + b 14 a_{14}+b_{14}
  259. a 15 + b 15 a_{15}+b_{15}
  260. c 12 c_{12}
  261. a 14 + b 14 a_{14}+b_{14}
  262. a 15 + b 15 a_{15}+b_{15}
  263. c 13 c_{13}
  264. ( c 12 - a 12 ) + ( c 12 - b 12 ) (c_{12}-a_{12})+(c_{12}-b_{12})
  265. d 14 + d 15 d_{14}+d_{15}
  266. c 15 c_{15}
  267. c 14 c_{14}
  268. b 14 + c 14 b_{14}+c_{14}
  269. c 15 + b 15 c_{15}+b_{15}
  270. a 15 + c 15 a_{15}+c_{15}
  271. a 14 + c 14 a_{14}+c_{14}
  272. a 1 + c 1 a_{1}+c_{1}
  273. a 15 + c 15 a_{15}+c_{15}
  274. [ - F o r m u l a E r r o r - ] [-FormulaError-]
  275. a 1 + c 1 a_{1}+c_{1}
  276. a 14 + c 14 a_{14}+c_{14}
  277. [ - F o r m u l a E r r o r - ] [-FormulaError-]
  278. b 1 + c 1 b_{1}+c_{1}
  279. b 15 + c 15 b_{15}+c_{15}
  280. [ - F o r m u l a E r r o r - ] [-FormulaError-]
  281. b 1 + c 1 b_{1}+c_{1}
  282. b 14 + c 14 b_{14}+c_{14}
  283. [ - F o r m u l a E r r o r - ] [-FormulaError-]
  284. b 14 + c 14 b_{14}+c_{14}
  285. a 15 + c 15 a_{15}+c_{15}
  286. b 14 + a 15 - c 13 b_{14}+a_{15}-c_{13}
  287. ( b 8 - a 8 ) + ( b 2 - a 2 ) (b_{8}-a_{8})+(b_{2}-a_{2})
  288. b 14 + a 15 - c 13 b_{14}+a_{15}-c_{13}
  289. [ - F o r m u l a E r r o r - ] [-FormulaError-]
  290. b 7 - a 8 b_{7}-a_{8}
  291. ( b 14 - a 14 ) + ( b 15 - a 15 ) (b_{14}-a_{14})+(b_{15}-a_{15})
  292. c 12 - d c_{12}-d
  293. ( b 7 - a 7 ) + ( b 8 - a 8 ) (b_{7}-a_{7})+(b_{8}-a_{8})
  294. ( b 14 - a 14 ) + ( b 15 - a 15 ) (b_{14}-a_{14})+(b_{15}-a_{15})
  295. [ - F o r m u l a E r r o r - ] [-FormulaError-]
  296. a 14 + b 14 a_{14}+b_{14}
  297. a 15 + b 15 a_{15}+b_{15}
  298. [ - F o r m u l a E r r o r - ] [-FormulaError-]
  299. a 14 + a 15 a_{14}+a_{15}
  300. b 14 + b 15 b_{14}+b_{15}
  301. [ - F o r m u l a E r r o r - ] [-FormulaError-]
  302. ( b 7 - a 7 ) + ( b 8 - a 8 ) = 161 (b_{7}-a_{7})+(b_{8}-a_{8})=161
  303. ( b 14 - a 14 ) + ( b 15 - a 15 ) = 77 (b_{14}-a_{14})+(b_{15}-a_{15})=77
  304. ( b 7 - a 7 ) + ( b 8 - a 8 ) + ( b 14 - a 14 ) + ( b 15 - a 15 ) 2 (b_{7}-a_{7})+(b_{8}-a_{8})+(b_{14}-a_{14})+(b_{15}-a_{15})\over 2
  305. = ( b 12 - a 12 ) =(b_{12}-a_{12})
  306. b 12 - a 12 = b_{12}-a_{12}=
  307. 161 + 77 2 161+77\over 2
  308. = 119 =119
  309. x = a 8 x=a_{8}
  310. x + 161 x+161
  311. x + ( b 7 - a 7 ) + ( b 8 - a 8 ) = a 8 + ( b 7 - a 7 ) + ( b 8 - a 8 ) x+(b_{7}-a_{7})+(b_{8}-a_{8})=a_{8}+(b_{7}-a_{7})+(b_{8}-a_{8})
  312. = b 7 =b_{7}
  313. a 8 + b 7 = c 12 a_{8}+b_{7}=c_{12}
  314. c 12 = x + b 7 = 2 * x + ( b 7 - a 7 ) + ( b 8 - a 8 ) = 2 * x + 161 c_{12}=x+b_{7}=2*x+(b_{7}-a_{7})+(b_{8}-a_{8})=2*x+161
  315. c 12 2 = ( x + b 7 ) 2 = ( 2 * x + 161 ) 2 = 4 * x 2 + 644 * x + 25921 c_{12}^{2}=(x+b_{7})^{2}=(2*x+161)^{2}=4*x^{2}+644*x+25921
  316. c 12 2 - ( b 12 - a 12 ) 2 c_{12}^{2}-(b_{12}-a_{12})^{2}
  317. = 4 * x 2 + 644 * x + 25921 - =4*x^{2}+644*x+25921-
  318. ( ( b 7 - a 7 ) + ( b 8 - a 8 ) + ( b 14 - a 14 ) + ( b 15 - a 15 ) ) 2 4 ((b_{7}-a_{7})+(b_{8}-a_{8})+(b_{14}-a_{14})+(b_{15}-a_{15}))^{2}\over 4
  319. = 4 * x 2 + 644 * x + 11760 = d =4*x^{2}+644*x+11760=d
  320. d 2 = ( 4 * x 2 + 644 * x + 11760 ) 2 = 16 * x 4 + 5152 * x 3 + 508816 * x 2 + 15146880 * x + 138297600 d^{2}=(4*x^{2}+644*x+11760)^{2}=16*x^{4}+5152*x^{3}+508816*x^{2}+15146880*x+13% 8297600
  321. 4 * x 4*x
  322. 4 * x * b 7 = 4 * x * ( x + ( b 7 - a 7 ) + ( b 8 - a 8 ) ) = 4 * x * ( x + 161 ) = 4 * x 2 + 644 * x 4*x*b_{7}=4*x*(x+(b_{7}-a_{7})+(b_{8}-a_{8}))=4*x*(x+161)=4*x^{2}+644*x
  323. d 2 = 4 * x * b 7 * c 12 2 = d^{2}=4*x*b_{7}*c_{12}^{2}=
  324. ( 4 * x 2 + 644 * x ) * ( 4 * x 2 + 644 * x + 25921 ) = (4*x^{2}+644*x)*(4*x^{2}+644*x+25921)=
  325. 16 * x 4 + 5152 * x 3 + 518420 * x 2 + 16693124 16*x^{4}+5152*x^{3}+518420*x^{2}+16693124
  326. d 2 d^{2}
  327. 16 * x 4 + 5152 * x 3 + 518420 * x 2 + 16693124 = 16*x^{4}+5152*x^{3}+518420*x^{2}+16693124=
  328. 16 * x 4 + 5152 * x 3 + 508816 * x 2 + 15146880 * x + 138297600 16*x^{4}+5152*x^{3}+508816*x^{2}+15146880*x+138297600
  329. 9604 * x 2 + 1546244 * x - 138297600 = 0 9604*x^{2}+1546244*x-138297600=0
  330. x = a 8 = 64 x=a_{8}=64
  331. a 12 + b 12 + c 12 a_{12}+b_{12}+c_{12}
  332. b 12 - a 12 b_{12}-a_{12}
  333. c 1 c_{1}
  334. b 1 - a 1 b_{1}-a_{1}
  335. c 1 c_{1}
  336. a 10 + b 11 a_{10}+b_{11}
  337. c 1 c_{1}
  338. a 2 + b 3 a_{2}+b_{3}
  339. a 1 + b 1 a_{1}+b_{1}
  340. a 2 a_{2}
  341. b 3 b_{3}
  342. a 1 + b 1 a_{1}+b_{1}
  343. c 13 + b 13 - a 13 c_{13}+b_{13}-a_{13}
  344. c 13 - b 13 + a 13 c_{13}-b_{13}+a_{13}
  345. a 1 + b 1 a_{1}+b_{1}
  346. a 11 + b 11 a_{11}+b_{11}
  347. a 10 + b 10 a_{10}+b_{10}
  348. a 1 + b 1 a_{1}+b_{1}
  349. c 10 - a 10 c_{10}-a_{10}
  350. c 11 - b 11 c_{11}-b_{11}
  351. a 1 + b 1 a_{1}+b_{1}
  352. c 6 + c 8 c_{6}+c_{8}
  353. c 6 - c 8 c_{6}-c_{8}
  354. a 1 + b 1 a_{1}+b_{1}
  355. c 10 c_{10}
  356. c 11 c_{11}
  357. a 1 + b 1 a_{1}+b_{1}
  358. c 4 c_{4}
  359. c 5 c_{5}
  360. a 1 + b 1 a_{1}+b_{1}
  361. c 2 c_{2}
  362. c 3 c_{3}
  363. a 1 + b 1 + c 1 a_{1}+b_{1}+c_{1}
  364. c 1 - b 1 c_{1}-b_{1}
  365. a 1 + b 1 + c 1 a_{1}+b_{1}+c_{1}
  366. c 1 - a 1 c_{1}-a_{1}
  367. a 1 + b 1 + c 1 a_{1}+b_{1}+c_{1}
  368. b 1 - a 1 b_{1}-a_{1}
  369. a 1 + b 1 + c 1 a_{1}+b_{1}+c_{1}
  370. ( c 1 - b 1 ) + ( c 1 - a 1 ) (c_{1}-b_{1})+(c_{1}-a_{1})
  371. a 1 + b 1 + c 1 a_{1}+b_{1}+c_{1}
  372. ( c 1 - b 1 ) + ( b 1 - a 1 ) + ( c 1 - a 1 ) (c_{1}-b_{1})+(b_{1}-a_{1})+(c_{1}-a_{1})
  373. a 1 + b 1 + c 1 a_{1}+b_{1}+c_{1}
  374. d 14 + d 15 d_{14}+d_{15}
  375. a 1 + b 1 + c 1 a_{1}+b_{1}+c_{1}
  376. c 12 c_{12}
  377. a 1 + b 1 + c 1 a_{1}+b_{1}+c_{1}
  378. c 13 c_{13}
  379. c 2 c_{2}
  380. c 3 c_{3}
  381. [ - F o r m u l a E r r o r - ] [-FormulaError-]
  382. c 5 c_{5}
  383. c 4 c_{4}
  384. [ - F o r m u l a E r r o r - ] [-FormulaError-]
  385. b 11 b_{11}
  386. c 4 c_{4}
  387. [ - F o r m u l a E r r o r - ] [-FormulaError-]
  388. a 10 a_{10}
  389. c 3 c_{3}
  390. [ - F o r m u l a E r r o r - ] [-FormulaError-]
  391. a 10 a_{10}
  392. b 11 b_{11}
  393. [ - F o r m u l a E r r o r - ] [-FormulaError-]
  394. b 7 b_{7}
  395. a 8 a_{8}
  396. [ - F o r m u l a E r r o r - ] [-FormulaError-]
  397. b 1 - b 11 b_{1}-b_{11}
  398. a 1 - a 10 a_{1}-a_{10}
  399. b 10 - a 10 b_{10}-a_{10}
  400. b 11 - a 11 b_{11}-a{11}
  401. [ - F o r m u l a E r r o r - ] [-FormulaError-]
  402. c 13 c_{13}
  403. a 10 - b 11 a_{10}-b_{11}
  404. [ - F o r m u l a E r r o r - ] [-FormulaError-]
  405. a 12 + b 12 a_{12}+b_{12}
  406. a 13 + b 13 a_{13}+b_{13}
  407. [ - F o r m u l a E r r o r - ] [-FormulaError-]
  408. c 1 c_{1}
  409. b 1 a 1 b_{1}\over a_{1}
  410. [ - F o r m u l a E r r o r - ] [-FormulaError-]
  411. d 10 - d 11 d_{10}-d_{11}
  412. d 12 - d 13 d_{12}-d_{13}
  413. [ - F o r m u l a E r r o r - ] [-FormulaError-]
  414. c 12 - [ c 10 - ( b 10 - a 10 ) ] c_{12}-[c_{10}-(b_{10}-a_{10})]
  415. c 11 + ( b 11 - a 11 ) - c 13 c_{11}+(b_{11}-a_{11})-c_{13}
  416. b 12 - a 12 b_{12}-a_{12}
  417. c 8 - ( c 1 - b 1 ) c_{8}-(c_{1}-b_{1})
  418. ( c 1 - a 1 ) - c 7 (c_{1}-a_{1})-c_{7}
  419. [ - F o r m u l a E r r o r - ] [-FormulaError-]
  420. a 1 + c 1 a_{1}+c_{1}
  421. ( c 1 - a 1 ) + ( c 1 - b 1 ) (c_{1}-a_{1})+(c_{1}-b_{1})
  422. [ - F o r m u l a E r r o r - ] [-FormulaError-]
  423. a 12 + b 12 + c 12 a_{12}+b_{12}+c_{12}
  424. ( a 13 + b 13 ) - c 13 (a_{13}+b_{13})-c_{13}
  425. [ - F o r m u l a E r r o r - ] [-FormulaError-]
  426. b 1 + c 1 b_{1}+c_{1}
  427. a 1 a_{1}
  428. 8 15 8\over 15
  429. b 1 b_{1}
  430. a 1 + c 1 a_{1}+c_{1}
  431. a 1 a_{1}
  432. 8 15 8\over 15
  433. b 1 b_{1}
  434. a 1 = ( 1 - 5 / 9 ) * 3 d a_{1}=(1-5/9)*3d
  435. b 1 - a 1 b_{1}-a_{1}
  436. [ - F o r m u l a E r r o r - ] [-FormulaError-]
  437. a 3 = ( 5 / 6 ) * d a_{3}=(5/6)*d
  438. b 2 - a 3 b_{2}-a_{3}
  439. ( 15 / 16 ) b 1 = d (15/16)b_{1}=d
  440. a 1 + b 1 a_{1}+b_{1}
  441. [ - F o r m u l a E r r o r - ] [-FormulaError-]
  442. a 12 = ( 8 / 15 ) * b 12 a_{12}=(8/15)*b_{12}
  443. c 12 - b 12 c_{12}-b_{12}
  444. c 12 - a 12 c_{12}-a_{12}
  445. c 1 c_{1}
  446. d = ( 1 / 2 ) b 2 d=(1/2)b_{2}
  447. a 3 = ( 5 / 6 ) d a_{3}=(5/6)d
  448. b 2 + a 3 + c 2 b_{2}+a_{3}+c_{2}
  449. b 2 = ( 12 / 17 ) c 1 b_{2}=(12/17)c_{1}
  450. a 3 = ( 5 / 17 ) c 1 a_{3}=(5/17)c_{1}
  451. a 3 + ( 5 / 6 ) b 2 a_{3}+(5/6)b_{2}
  452. b 2 + ( 3 / 5 ) a 3 b_{2}+(3/5)a_{3}
  453. [ - F o r m u l a E r r o r - ] [-FormulaError-]
  454. a 11 + ( 1 / 3 ) b 10 a_{11}+(1/3)b_{10}
  455. b 10 - ( 3 / 4 ) a 11 b_{10}-(3/4)a_{11}
  456. [ - F o r m u l a E r r o r - ] [-FormulaError-]
  457. b 1 - d = ( 3 / 5 ) b 1 b_{1}-d=(3/5)b_{1}
  458. a 1 - d = ( 1 / 4 ) a 1 a_{1}-d=(1/4)a_{1}
  459. ( b 1 - d ) - ( a 1 - d ) (b_{1}-d)-(a_{1}-d)
  460. b 1 - d = ( 3 / 5 ) b 1 b_{1}-d=(3/5)b_{1}
  461. a 1 - d = ( 1 / 4 ) a 1 a_{1}-d=(1/4)a_{1}
  462. ( 1 / 5 ) b 1 - ( 1 / 4 ) a 1 (1/5)b_{1}-(1/4)a_{1}
  463. b 14 = ( 1 - ( 15 / 24 ) b 10 ) b_{14}=(1-(15/24)b_{10})
  464. a 15 = ( 1 - ( 4 / 5 ) ) a 11 a_{15}=(1-(4/5))a_{11}
  465. b 14 - a 15 b_{14}-a_{15}
  466. b 10 - a 11 b_{10}-a_{11}
  467. a 1 + b 1 + c 1 a_{1}+b_{1}+c_{1}
  468. ( b 1 / a 1 ) = 8 ( 1 / 3 ) (b_{1}/a_{1})=8(1/3)
  469. ( a 1 / b 15 ) = 10 ( 2 / 3 ) (a_{1}/b_{15})=10(2/3)
  470. a 14 - a 13 a_{14}-a_{13}
  471. b 13 - b 15 b_{13}-b_{15}
  472. a 9 * b 7 = r 2 a_{9}*b_{7}=r^{2}
  473. a 8 = a 9 a_{8}=a_{9}
  474. a 8 * b 7 = r 2 a_{8}*b_{7}=r^{2}
  475. a 8 a_{8}
  476. b 7 b_{7}

Chain_rule_(disambiguation).html

  1. d y d x = d y d u d u d x . \frac{\mathrm{d}y}{\mathrm{d}x}=\frac{\mathrm{d}y}{\mathrm{d}u}\cdot\frac{% \mathrm{d}u}{\mathrm{d}x}.
  2. ( x y ) z ( y z ) x ( z x ) y = - 1. \left(\frac{\partial x}{\partial y}\right)_{z}\left(\frac{\partial y}{\partial z% }\right)_{x}\left(\frac{\partial z}{\partial x}\right)_{y}=-1.
  3. P ( X 1 = x 1 , , X n = x n ) = i = 1 n P ( X i = x i X i + 1 = x i + 1 , , X n = x n ) \mathrm{P}(X_{1}=x_{1},\ldots,X_{n}=x_{n})=\prod_{i=1}^{n}\mathrm{P}(X_{i}=x_{% i}\mid X_{i+1}=x_{i+1},\ldots,X_{n}=x_{n})
  4. K ( X , Y ) = K ( X ) + K ( Y | X ) + O ( log ( K ( X , Y ) ) ) K(X,Y)=K(X)+K(Y|X)+O(\log(K(X,Y)))
  5. H ( X , Y ) = H ( X ) + H ( Y | X ) H(X,Y)=H(X)+H(Y|X)

Chain_rule_(probability).html

  1. A 1 , , A n A_{1},\ldots,A_{n}
  2. P ( A n , , A 1 ) = P ( A n | A n - 1 , , A 1 ) P ( A n - 1 , , A 1 ) \mathrm{P}(A_{n},\ldots,A_{1})=\mathrm{P}(A_{n}|A_{n-1},\ldots,A_{1})\cdot% \mathrm{P}(A_{n-1},\ldots,A_{1})
  3. P ( k = 1 n A k ) = k = 1 n P ( A k | j = 1 k - 1 A j ) \mathrm{P}\left(\bigcap_{k=1}^{n}A_{k}\right)=\prod_{k=1}^{n}\mathrm{P}\left(A% _{k}\,\Bigg|\,\bigcap_{j=1}^{k-1}A_{j}\right)
  4. P ( A 4 , A 3 , A 2 , A 1 ) = P ( A 4 A 3 , A 2 , A 1 ) P ( A 3 A 2 , A 1 ) P ( A 2 A 1 ) P ( A 1 ) \mathrm{P}(A_{4},A_{3},A_{2},A_{1})=\mathrm{P}(A_{4}\mid A_{3},A_{2},A_{1})% \cdot\mathrm{P}(A_{3}\mid A_{2},A_{1})\cdot\mathrm{P}(A_{2}\mid A_{1})\cdot% \mathrm{P}(A_{1})
  5. P ( A , B ) = P ( B A ) P ( A ) = 2 / 3 × 1 / 2 = 1 / 3 \mathrm{P}(A,B)=\mathrm{P}(B\mid A)\mathrm{P}(A)=2/3\times 1/2=1/3

Chaman_Fault.html

  1. M w M_{\mathrm{w}}
  2. M w M_{\mathrm{w}}

Chang's_conjecture.html

  1. ( ω 2 , ω 1 ) ( ω 1 , ω ) (\omega_{2},\omega_{1})\twoheadrightarrow(\omega_{1},\omega)
  2. ( ω 3 , ω 2 ) ( ω 2 , ω 1 ) (\omega_{3},\omega_{2})\twoheadrightarrow(\omega_{2},\omega_{1})

Chaotic_mixing.html

  1. d x d t = v ( x , t ) \frac{d\vec{x}}{dt}=\vec{v}(\vec{x},t)
  2. v = ( u , v , w ) \vec{v}=(u,v,w)
  3. x = ( x , y , z ) \vec{x}=(x,y,z)
  4. x x
  5. y y
  6. x 1 \vec{x}_{1}
  7. x 2 \vec{x}_{2}
  8. δ x = x 2 - x 1 \delta\vec{x}=\vec{x}_{2}-\vec{x}_{1}
  9. v \vec{v}
  10. t + δ t t+\delta t
  11. d d t ( x + δ x ) v + v δ x \frac{\mathrm{d}}{\mathrm{d}t}(\vec{x}+\delta\vec{x})\approx\vec{v}+\nabla\vec% {v}\cdot\delta\vec{x}
  12. δ x ( t + δ t ) δ x ( t ) + δ t ( δ x ) v \delta x(t+\delta t)\approx\delta x(t)+\delta t(\delta x\cdot\nabla)\vec{v}
  13. d d t δ x v δ x \frac{\mathrm{d}}{\mathrm{d}t}\delta\vec{x}\approx\nabla\vec{v}\cdot\delta\vec% {x}
  14. δ x \delta\vec{x}
  15. v \nabla\vec{v}
  16. d d t s y m b o l H v s y m b o l H , s y m b o l H ( t = 0 ) = s y m b o l I \frac{\mathrm{d}}{\mathrm{d}t}symbol{H}\equiv\nabla\vec{v}\cdot symbol{H},% \qquad symbol{H}(t=0)=symbol{I}
  17. δ x ( t ) s y m b o l H δ x 0 \delta\vec{x}(t)\approx symbol{H}\cdot\delta\vec{x}_{0}
  18. s y m b o l H T s y m b o l H δ x 0 i = h i δ x 0 i symbol{H^{T}}\cdot symbol{H}\cdot\delta\vec{x}_{0i}=h_{i}\cdot\delta\vec{x}_{0i}
  19. λ i ( x , t ) 1 2 t ln h i ( x , t ) \lambda_{i}(\vec{x},t)\equiv\frac{1}{2t}\ln{h_{i}(\vec{x},t)}
  20. λ i ( x , t ) λ i + 1 ( x , t ) \lambda_{i}(\vec{x},t)\geq\lambda_{i+1}(\vec{x},t)
  21. δ x 0 i \delta\vec{x}_{0i}
  22. { δ x 0 i } \{\delta\vec{x}_{0i}\}
  23. { h i ( x , t ) } \{\sqrt{h_{i}(\vec{x},t)}\}
  24. h 1 ( x , t ) \sqrt{h_{1}(\vec{x},t)}
  25. h N ( x , t ) \sqrt{h_{N}(\vec{x},t)}
  26. λ = lim t λ 1 ( x , t ) \lambda=\lim_{t\to\infty}\lambda_{1}(\vec{x},t)
  27. | δ x | | δ x 0 | e λ 1 t . |\delta\vec{x}|\approx|\delta\vec{x}_{0}|e^{\lambda_{1}t}.
  28. λ = lim t λ 1 ( x , t ) \lambda=\lim_{t\to\infty}\lambda_{1}(\vec{x},t)
  29. < λ > t r a j e c t o r i e s <\lambda>_{trajectories}
  30. d q d t = - q v \frac{\mathrm{d}\nabla q}{\mathrm{d}t}=-\nabla q\cdot\nabla\vec{v}
  31. s y m b o l H symbol{H}^{\prime}
  32. d s y m b o l H d t = - s y m b o l H v s y m b o l H ( t = 0 ) = s y m b o l I \frac{\mathrm{d}symbol{H^{\prime}}}{\mathrm{d}t}=-\nabla symbol{H^{\prime}}% \cdot\nabla\vec{v}\qquad symbol{H^{\prime}}(t=0)=symbol{I}
  33. s y m b o l H = s y m b o l H - 1 symbol{H^{\prime}}=symbol{H}^{-1}
  34. { h i } \{h_{i}^{\prime}\}
  35. s y m b o l H symbol{H}^{\prime}
  36. h i = 1 / h i h_{i}^{\prime}=1/h_{i}
  37. { h i } \{h_{i}^{\prime}\}
  38. d L d t = | v d s | \frac{\mathrm{d}L}{\mathrm{d}t}=\int|\nabla\vec{v}\cdot\mathrm{d}\vec{s}|
  39. d s \mathrm{d}\vec{s}
  40. L L 0 exp ( λ ¯ 1 t ) L\approx L_{0}\exp(\bar{\lambda}_{1}t)
  41. s y m b o l M : x i ( t i ) \displaystyle symbol{M}\colon\vec{x_{i}}(t_{i})
  42. s y m b o l M symbol{M}
  43. c c
  44. . q t = D q - v q . \big.\frac{\partial q}{\partial t}=\nabla\cdot D\nabla q-\vec{v}\cdot\nabla q.
  45. v \vec{v}
  46. w B = D λ w_{B}=\sqrt{\frac{D}{\lambda}}
  47. λ \lambda

Chaplygin_gas.html

  1. p = - A / ρ α p=-A/\rho^{\alpha}
  2. p p
  3. ρ \rho
  4. α = 1 \alpha=1
  5. A A
  6. α \alpha
  7. 0 < α 1 0<\alpha\leq 1

Chapman–Enskog_theory.html

  1. f t + 𝐯 f 𝐱 + 𝐅 f 𝐩 = B ( f f ) \frac{\partial f}{\partial t}+\mathbf{v}\cdot\frac{\partial f}{\partial\mathbf% {x}}+\mathbf{F}\cdot\frac{\partial f}{\partial\mathbf{p}}=B\left(f\otimes f\right)
  2. ε \varepsilon
  3. f = f 0 + ε f 1 + ε 2 f 2 + = n = 0 ε n f n f=f_{0}+\varepsilon f_{1}+\varepsilon^{2}f_{2}+...=\sum_{n=0}^{\infty}% \varepsilon^{n}f_{n}
  4. f 0 f_{0}
  5. f n t + 𝐯 f n 𝐱 + 𝐅 f n 𝐩 = i = 0 n B ( f i f n - i ) \frac{\partial f_{n}}{\partial t}+\mathbf{v}\cdot\frac{\partial f_{n}}{% \partial\mathbf{x}}+\mathbf{F}\cdot\frac{\partial f_{n}}{\partial\mathbf{p}}=% \sum_{i=0}^{n}B\left(f_{i}\otimes f_{n-i}\right)

Characteristic_admittance.html

  1. Y 0 = G + j ω C R + j ω L Y_{0}=\sqrt{\frac{G+j\omega C}{R+j\omega L}}
  2. R R
  3. L L
  4. G G
  5. C C
  6. j j
  7. ω \omega
  8. I + V + = Y 0 = - I - V - \frac{I^{+}}{V^{+}}=Y_{0}=-\frac{I^{-}}{V^{-}}
  9. + +
  10. - -

Charge_transfer_coefficient.html

  1. α c ν = - ( R T n F ) ( l n ( | I r e d | ) E ) T , p , c i , i n t e r f a c e \frac{\alpha_{c}}{\nu}=-\left(\frac{RT}{nF}\right)\left(\frac{\partial ln(|I_{% red}|)}{\partial E}\right)_{T,p,c_{i,interface}}
  2. α a ν = ( R T n F ) ( l n ( | I o x | ) E ) T , p , c i , i n t e r f a c e \frac{\alpha_{a}}{\nu}=\left(\frac{RT}{nF}\right)\left(\frac{\partial ln(|I_{% ox}|)}{\partial E}\right)_{T,p,c_{i,interface}}
  3. ν \nu
  4. R R
  5. T T
  6. n n
  7. F F
  8. E E
  9. I I
  10. β c + β a = 1 \beta_{c}+\beta_{a}=1

Châtelet_surface.html

  1. y 2 - a z 2 = P ( x ) , y^{2}-az^{2}=P(x),\,

Cheng's_eigenvalue_comparison_theorem.html

  1. K M k . K_{M}\leq k.
  2. λ 1 ( B N ( k ) ( r ) ) λ 1 ( B M ( p , r ) ) . \lambda_{1}\left(B_{N(k)}(r)\right)\leq\lambda_{1}\left(B_{M}(p,r)\right).
  3. Ric ( X , X ) k ( n - 1 ) | X | 2 . \operatorname{Ric}(X,X)\geq k(n-1)|X|^{2}.
  4. λ 1 ( B N ( k ) ( r ) ) λ 1 ( B M ( p , r ) ) . \lambda_{1}\left(B_{N(k)}(r)\right)\geq\lambda_{1}\left(B_{M}(p,r)\right).

Cherenkov_radiation.html

  1. v p v\text{p}
  2. c / n < v p < c c/n<v\text{p}<c
  3. c c
  4. n n
  5. 0.75 c < v p < c 0.75c<v\text{p}<c
  6. n = 1.33 n=1.33
  7. β = v p / c \beta=v\text{p}/c
  8. v em = c / n v\text{em}=c/n
  9. x p = v p t = β c t x\text{p}=v\text{p}t=\beta\,ct
  10. x em = v em t = c n t . x\text{em}=v\text{em}t=\frac{c}{n}t.
  11. cos θ = 1 n β . \cos\theta=\frac{1}{n\beta}.
  12. cos θ = 1 / ( n β ) \cos\theta=1/(n\beta)
  13. n n
  14. v 0 = c / n v_{0}=c/n
  15. c c
  16. n n

Chess_rating_system.html

  1. R n e w = R o l d + K 2 ( W - L + 1 2 i D i C ) R_{new}=R_{old}+\frac{K}{2}\left(W-L+\frac{1}{2}\frac{\sum_{i}D_{i}}{C}\right)
  2. 2635 + 10 2 ( 10.5 - 4.5 - 1 2 1620 200 ) = 2644.75 2635+\frac{10}{2}\left(10.5-4.5-\frac{1}{2}\frac{1620}{200}\right)=2644.75

Chinese_monoid.html

  1. a * ( b a ) * b * ( c a ) * ( c b ) * c * a^{*}\ (ba)^{*}b^{*}\ (ca)^{*}(cb)^{*}c^{*}\cdots
  2. n ( n + 1 ) 2 \frac{n(n+1)}{2}

Chlorobenzene_(data_page).html

  1. γ \gamma

Choice_model_simulation.html

  1. U i = α P i + β D i + ε i U_{i}=\alpha P_{i}+\beta D_{i}+\varepsilon_{i}\,
  2. U a = α P a + β D a + ε a U_{a}=\alpha P_{a}+\beta D_{a}+\varepsilon_{a}\,
  3. ε \varepsilon
  4. P i \displaystyle P_{i}
  5. P i = Pr ( ε i - ε a > V a - V i ) P_{i}=\Pr(\varepsilon_{i}-\varepsilon_{a}>V_{a}-V_{i})
  6. ε \varepsilon
  7. ε i - ε a \varepsilon_{i}-\varepsilon_{a}
  8. ε \varepsilon
  9. P i P_{i}
  10. P n j = Pr ( U n j > U n i ) P_{nj}=\Pr(U_{nj}>U_{ni})
  11. P n j P_{nj}
  12. U n 1 U n J U_{n1}...U_{nJ}
  13. U i = C i + α P i + β D i + ε i U_{i}=C_{i}+\alpha P_{i}+\beta D_{i}+\varepsilon_{i}\,
  14. U a = C a + α P a + β D a + ε a U_{a}=C_{a}+\alpha P_{a}+\beta D_{a}+\varepsilon_{a}\,
  15. P i = Pr ( ε i - ε a > ( C a - C i ) + α P a - α P i + β D a - β D i ) P_{i}=\Pr(\varepsilon_{i}-\varepsilon_{a}>(C_{a}-C_{i})+\alpha P_{a}-\alpha P_% {i}+\beta D_{a}-\beta D_{i})
  16. C a a n d C i C_{a}andC_{i}
  17. C i C_{i}
  18. U i = α P i + β D i + ε i U_{i}=\alpha P_{i}+\beta D_{i}+\varepsilon_{i}\,
  19. U a = ( C a - C i ) + α P a + β D a + ε a U_{a}=(C_{a}-C_{i})+\alpha P_{a}+\beta D_{a}+\varepsilon_{a}\,
  20. C i C_{i}
  21. U i = α P i + β D i + γ Y + ε i U_{i}=\alpha P_{i}+\beta D_{i}+\gamma Y+\varepsilon_{i}\,
  22. U i = α P i / Y + β D i + ε i U_{i}=\alpha P_{i}/Y+\beta D_{i}+\varepsilon_{i}\,

Chooser_option.html

  1. t 1 t_{1}
  2. t 2 t_{2}
  3. m a x ( S - K , 0 ) max(S-K,0)
  4. m a x ( K - S , 0 ) max(K-S,0)
  5. K K
  6. S S
  7. K K
  8. t 2 t_{2}
  9. K e - r ( t 2 - t 1 ) Ke^{-r(t_{2}-t_{1})}
  10. t 1 t_{1}

Christoffel–Darboux_formula.html

  1. j = 0 n f j ( x ) f j ( y ) h j = k n h n k n + 1 f n ( y ) f n + 1 ( x ) - f n + 1 ( y ) f n ( x ) x - y \sum_{j=0}^{n}\frac{f_{j}(x)f_{j}(y)}{h_{j}}=\frac{k_{n}}{h_{n}k_{n+1}}\frac{f% _{n}(y)f_{n+1}(x)-f_{n+1}(y)f_{n}(x)}{x-y}

Chvátal_graph.html

  1. χ ( G ) Δ + ω + 1 2 . \chi(G)\leq\left\lceil\frac{\Delta+\omega+1}{2}\right\rceil.
  2. ( x - 4 ) ( x - 1 ) 4 x 2 ( x + 1 ) ( x + 3 ) 2 ( x 2 + x - 4 ) (x-4)(x-1)^{4}x^{2}(x+1)(x+3)^{2}(x^{2}+x-4)

Cipolla's_algorithm.html

  1. x 2 n ( mod p ) . x^{2}\equiv n\;\;(\mathop{{\rm mod}}p).
  2. x , n 𝐅 p x,n\in\mathbf{F}_{p}
  3. p p
  4. 𝐅 p \mathbf{F}_{p}
  5. p p
  6. { 0 , 1 , , p - 1 } \{0,1,\dots,p-1\}
  7. p p
  8. n 𝐅 p n\in\mathbf{F}_{p}
  9. x 𝐅 p x\in\mathbf{F}_{p}
  10. x 2 = n . x^{2}=n.
  11. a 𝐅 p a\in\mathbf{F}_{p}
  12. a 2 - n a^{2}-n
  13. a a
  14. a a
  15. ( a 2 - n | p ) (a^{2}-n|p)
  16. a a
  17. a a
  18. ( p - 1 ) / 2 p (p-1)/2p
  19. p p
  20. 1 / 2 1/2
  21. x = ( a + a 2 - n ) ( p + 1 ) / 2 x=\left(a+\sqrt{a^{2}-n}\right)^{(p+1)/2}
  22. 𝐅 p 2 = 𝐅 p ( a 2 - n ) \mathbf{F}_{p^{2}}=\mathbf{F}_{p}(\sqrt{a^{2}-n})
  23. x 2 = n . x^{2}=n.
  24. x 2 = n x^{2}=n
  25. ( - x ) 2 = n (-x)^{2}=n
  26. x - x x\neq-x
  27. 𝐅 13 \mathbf{F}_{13}
  28. 𝐅 13 2 \mathbf{F}_{13^{2}}
  29. x 2 = 10. x^{2}=10.
  30. 10 10
  31. 𝐅 13 \mathbf{F}_{13}
  32. ( 10 | 13 ) (10|13)
  33. ( 10 | 13 ) 10 6 1 mod 13. (10|13)\equiv 10^{6}\equiv 1\bmod 13.
  34. a 2 - n a^{2}-n
  35. a = 2 a=2
  36. a 2 - n a^{2}-n
  37. ( 7 | 13 ) (7|13)
  38. 7 6 = 343 2 5 2 25 - 1 mod 13. 7^{6}=343^{2}\equiv 5^{2}\equiv 25\equiv-1\bmod 13.
  39. a = 2 a=2
  40. x = ( a + a 2 - n ) ( p + 1 ) / 2 = ( 2 + - 6 ) 7 . x=\left(a+\sqrt{a^{2}-n}\right)^{(p+1)/2}=\left(2+\sqrt{-6}\right)^{7}.
  41. ( 2 + - 6 ) 2 = 4 + 4 - 6 - 6 = - 2 + 4 - 6 . \left(2+\sqrt{-6}\right)^{2}=4+4\sqrt{-6}-6=-2+4\sqrt{-6}.
  42. ( 2 + - 6 ) 4 = ( - 2 + 4 - 6 ) 2 = - 1 - 3 - 6 . \left(2+\sqrt{-6}\right)^{4}=\left(-2+4\sqrt{-6}\right)^{2}=-1-3\sqrt{-6}.
  43. ( 2 + - 6 ) 6 = ( - 2 + 4 - 6 ) ( - 1 - 3 - 6 ) = 9 + 2 - 6 . \left(2+\sqrt{-6}\right)^{6}=\left(-2+4\sqrt{-6}\right)\left(-1-3\sqrt{-6}% \right)=9+2\sqrt{-6}.
  44. ( 2 + - 6 ) 7 = ( 9 + 2 - 6 ) ( 2 + - 6 ) = 6. \left(2+\sqrt{-6}\right)^{7}=\left(9+2\sqrt{-6}\right)\left(2+\sqrt{-6}\right)% =6.
  45. x = 6 x=6
  46. x = - 6 = 7. x=-6=7.
  47. 6 2 = 36 = 10 \ 6^{2}=36=10
  48. 7 2 = 49 = 10. 7^{2}=49=10.
  49. 𝐅 p 2 = 𝐅 p ( a 2 - n ) = { x + y a 2 - n : x , y 𝐅 p } \mathbf{F}_{p^{2}}=\mathbf{F}_{p}(\sqrt{a^{2}-n})=\{x+y\sqrt{a^{2}-n}:x,y\in% \mathbf{F}_{p}\}
  50. ω \omega
  51. a 2 - n \sqrt{a^{2}-n}
  52. a 2 - n a^{2}-n
  53. 𝐅 p \mathbf{F}_{p}
  54. ω \omega
  55. ( x 1 + y 1 ω ) + ( x 2 + y 2 ω ) = ( x 1 + x 2 ) + ( y 1 + y 2 ) ω \left(x_{1}+y_{1}\omega\right)+\left(x_{2}+y_{2}\omega\right)=\left(x_{1}+x_{2% }\right)+\left(y_{1}+y_{2}\right)\omega
  56. ω 2 = a 2 - n \omega^{2}=a^{2}-n
  57. ( x 1 + y 1 ω ) ( x 2 + y 2 ω ) = x 1 x 2 + x 1 y 2 ω + y 1 x 2 ω + y 1 y 2 ω 2 = ( x 1 x 2 + y 1 y 2 ( a 2 - n ) ) + ( x 1 y 2 + y 1 x 2 ) ω \left(x_{1}+y_{1}\omega\right)\left(x_{2}+y_{2}\omega\right)=x_{1}x_{2}+x_{1}y% _{2}\omega+y_{1}x_{2}\omega+y_{1}y_{2}\omega^{2}=\left(x_{1}x_{2}+y_{1}y_{2}% \left(a^{2}-n\right)\right)+\left(x_{1}y_{2}+y_{1}x_{2}\right)\omega
  58. 𝐅 p 2 \mathbf{F}_{p^{2}}
  59. ω \omega
  60. 0
  61. 0 + 0 ω 0+0\omega
  62. α 𝐅 p 2 \alpha\in\mathbf{F}_{p^{2}}
  63. α + 0 = ( x + y ω ) + ( 0 + 0 ω ) = ( x + 0 ) + ( y + 0 ) ω = x + y ω = α \alpha+0=(x+y\omega)+(0+0\omega)=(x+0)+(y+0)\omega=x+y\omega=\alpha
  64. 1 1
  65. 1 + 0 ω 1+0\omega
  66. α 1 = ( x + y ω ) ( 1 + 0 ω ) = ( x 1 + 0 0 ( n 2 - a ) ) + ( x 0 + 1 x ) ω = x + y ω = α \alpha\cdot 1=(x+y\omega)(1+0\omega)=\left(x\cdot 1+0\cdot 0\left(n^{2}-a% \right)\right)+(x\cdot 0+1\cdot x)\omega=x+y\omega=\alpha
  67. 𝐅 p 2 \mathbf{F}_{p^{2}}
  68. x + y ω x+y\omega
  69. - x - y ω -x-y\omega
  70. 𝐅 p 2 \mathbf{F}_{p^{2}}
  71. - x , - y 𝐅 p -x,-y\in\mathbf{F}_{p}
  72. α \alpha
  73. α = x 1 + y 1 ω \alpha=x_{1}+y_{1}\omega
  74. α - 1 = x 2 + y 2 ω \alpha^{-1}=x_{2}+y_{2}\omega
  75. ( x 1 + y 1 ω ) ( x 2 + y 2 ω ) = ( x 1 x 2 + y 1 y 2 ( n 2 - a ) ) + ( x 1 y 2 + y 1 x 2 ) ω = 1 (x_{1}+y_{1}\omega)(x_{2}+y_{2}\omega)=\left(x_{1}x_{2}+y_{1}y_{2}\left(n^{2}-% a\right)\right)+\left(x_{1}y_{2}+y_{1}x_{2}\right)\omega=1
  76. x 1 x 2 + y 1 y 2 ( n 2 - a ) = 1 x_{1}x_{2}+y_{1}y_{2}(n^{2}-a)=1
  77. x 1 y 2 + y 1 x 2 = 0 x_{1}y_{2}+y_{1}x_{2}=0
  78. x 2 x_{2}
  79. y 2 y_{2}
  80. x 2 = - y 1 - 1 x 1 ( y 1 ( n 2 - a ) - x 1 2 y 1 - 1 ) - 1 x_{2}=-y_{1}^{-1}x_{1}\left(y_{1}\left(n^{2}-a\right)-x_{1}^{2}y_{1}^{-1}% \right)^{-1}
  81. y 2 = ( y 1 ( n 2 - a ) - x 1 2 y 1 - 1 ) - 1 y_{2}=\left(y_{1}\left(n^{2}-a\right)-x_{1}^{2}y_{1}^{-1}\right)^{-1}
  82. x 2 x_{2}
  83. y 2 y_{2}
  84. 𝐅 p \mathbf{F}_{p}
  85. 𝐅 p 2 \mathbf{F}_{p^{2}}
  86. x + y ω 𝐅 p 2 : ( x + y ω ) p = x - y ω x+y\omega\in\mathbf{F}_{p^{2}}:(x+y\omega)^{p}=x-y\omega
  87. ω 2 = a 2 - n \omega^{2}=a^{2}-n
  88. 𝐅 p \mathbf{F}_{p}
  89. ω p - 1 = ( ω 2 ) p - 1 2 = - 1 \omega^{p-1}=\left(\omega^{2}\right)^{\frac{p-1}{2}}=-1
  90. ω p = - ω \omega^{p}=-\omega
  91. x p = x x^{p}=x
  92. x 𝐅 p x\in\mathbf{F}_{p}
  93. ( a + b ) p = a p + b p \left(a+b\right)^{p}=a^{p}+b^{p}
  94. ( x + y ω ) p = x p + y p ω p = x - y ω (x+y\omega)^{p}=x^{p}+y^{p}\omega^{p}=x-y\omega
  95. x 0 = ( a + ω ) p + 1 2 𝐅 p 2 x_{0}=\left(a+\omega\right)^{\frac{p+1}{2}}\in\mathbf{F}_{p^{2}}
  96. x 0 2 = n 𝐅 p x_{0}^{2}=n\in\mathbf{F}_{p}
  97. x 0 2 = ( a + ω ) p + 1 = ( a + ω ) ( a + ω ) p = ( a + ω ) ( a - ω ) = a 2 - ω 2 = a 2 - ( a 2 - n ) = n x_{0}^{2}=\left(a+\omega\right)^{p+1}=(a+\omega)(a+\omega)^{p}=(a+\omega)(a-% \omega)=a^{2}-\omega^{2}=a^{2}-\left(a^{2}-n\right)=n
  98. 𝐅 p 2 \mathbf{F}_{p^{2}}
  99. x 0 𝐅 p 2 x_{0}\in\mathbf{F}_{p^{2}}
  100. x 2 - n x^{2}-n
  101. 𝐅 p \mathbf{F}_{p}
  102. 𝐅 p 2 \mathbf{F}_{p^{2}}
  103. x 0 x_{0}
  104. - x 0 -x_{0}
  105. x 2 - n x^{2}-n
  106. 𝐅 p 2 \mathbf{F}_{p^{2}}
  107. x 0 , - x 0 𝐅 p x_{0},-x_{0}\in\mathbf{F}_{p}
  108. 4 m + 2 k - 4 4m+2k-4
  109. 4 m - 2 4m-2
  110. 𝐅 p 2 \mathbf{F}_{p^{2}}
  111. ( x + y ω ) 2 = ( x 2 + y 2 ω 2 ) + ( ( x + y ) 2 - x 2 - y 2 ) ω , (x+y\omega)^{2}=\left(x^{2}+y^{2}\omega^{2}\right)+\left(\left(x+y\right)^{2}-% x^{2}-y^{2}\right)\omega,
  112. ω 2 = a 2 - n \omega^{2}=a^{2}-n
  113. ( x + y ω ) 2 ( c + ω ) = ( c d - b ( x + d ) ) + ( d 2 - b y ) ω , \left(x+y\omega\right)^{2}\left(c+\omega\right)=\left(cd-b\left(x+d\right)% \right)+\left(d^{2}-by\right)\omega,
  114. d = ( x + y c ) d=(x+yc)
  115. b = n y b=ny
  116. p 1 ( mod 4 ) , p\equiv 1\;\;(\mathop{{\rm mod}}4),
  117. p 3 ( mod 4 ) p\equiv 3\;\;(\mathop{{\rm mod}}4)
  118. x ± n p + 1 4 x\equiv\pm n^{\frac{p+1}{4}}
  119. ( p + 1 ) / 2 (p+1)/2
  120. m - 1 m-1
  121. ( p + 1 ) / 2 (p+1)/2
  122. ( a + ω ) \left(a+\omega\right)
  123. n - k - 1 n-k-1
  124. k - 1 k-1
  125. S ( S - 1 ) > 8 m + 20 S(S-1)>8m+20
  126. 2 S 2^{S}
  127. p - 1 p-1

Circular_ensemble.html

  1. U R = ( 0 - 1 1 0 0 - 1 1 0 0 - 1 1 0 ) U T ( 0 1 - 1 0 0 1 - 1 0 0 1 - 1 0 ) . U^{R}=\left(\begin{array}[]{ccccccc}0&-1&&&&&\\ 1&0&&&&&\\ &&0&-1&&&\\ &&1&0&&&\\ &&&&\ddots&&\\ &&&&&0&-1\\ &&&&&1&0\end{array}\right)U^{T}\left(\begin{array}[]{ccccccc}0&1&&&&&\\ -1&0&&&&&\\ &&0&1&&&\\ &&-1&0&&&\\ &&&&\ddots&&\\ &&&&&0&1\\ &&&&&-1&0\end{array}\right)~{}.
  2. e i θ k e^{i\theta_{k}}
  3. p ( θ 1 , , θ n ) = 1 Z n , β 1 k < j n | e i θ k - e i θ j | β , p(\theta_{1},\cdots,\theta_{n})=\frac{1}{Z_{n,\beta}}\prod_{1\leq k<j\leq n}|e% ^{i\theta_{k}}-e^{i\theta_{j}}|^{\beta}~{},
  4. Z n , β = ( 2 π ) n Γ ( β n / 2 + 1 ) ( Γ ( β / 2 + 1 ) ) n . Z_{n,\beta}=(2\pi)^{n}\frac{\Gamma(\beta n/2+1)}{\left(\Gamma(\beta/2+1)\right% )^{n}}~{}.
  5. e i θ k e^{i\theta_{k}}
  6. e - i θ k e^{-i\theta_{k}}
  7. p ( θ 1 , , θ m ) = C 1 k < j m ( cos θ k - cos θ j ) 2 , p(\theta_{1},\cdots,\theta_{m})=C\prod_{1\leq k<j\leq m}(\cos\theta_{k}-\cos% \theta_{j})^{2}~{},
  8. p ( θ 1 , , θ m ) = C 1 i m ( 1 - σ cos θ i ) 1 k < j m ( cos θ k - cos θ j ) 2 . p(\theta_{1},\cdots,\theta_{m})=C\prod_{1\leq i\leq m}(1-\sigma\cos\theta_{i})% \prod_{1\leq k<j\leq m}(\cos\theta_{k}-\cos\theta_{j})^{2}~{}.
  9. p ( θ 1 , , θ m ) = C 1 i m ( 1 - cos 2 θ i ) 1 k < j m ( cos θ k - cos θ j ) 2 . p(\theta_{1},\cdots,\theta_{m})=C\prod_{1\leq i\leq m}(1-\cos^{2}\theta_{i})% \prod_{1\leq k<j\leq m}(\cos\theta_{k}-\cos\theta_{j})^{2}~{}.

Circular_surface.html

  1. f ( t , θ ) := γ ( t ) + r ( t ) u ( t ) cos θ + r ( t ) v ( t ) sin θ , f(t,\theta):=\gamma(t)+r(t){u}(t)\cos\theta+r(t){v}(t)\sin\theta,\,

Circular_uniform_distribution.html

  1. f U C ( θ ) = 1 2 π . f_{UC}(\theta)=\frac{1}{2\pi}.
  2. z = e i θ z=e^{i\theta}
  3. m 0 m_{0}
  4. z n = δ n \langle z^{n}\rangle=\delta_{n}
  5. δ n \delta_{n}
  6. R = | z n | = 0 R=|\langle z^{n}\rangle|=0\,
  7. z n = e i θ n z_{n}=e^{i\theta_{n}}
  8. z ¯ = 1 N n = 1 N z n = C ¯ + i S ¯ = R ¯ e i θ ¯ \overline{z}=\frac{1}{N}\sum_{n=1}^{N}z_{n}=\overline{C}+i\overline{S}=% \overline{R}e^{i\overline{\theta}}
  9. C ¯ = 1 N n = 1 N cos ( θ n ) S ¯ = 1 N n = 1 N sin ( θ n ) \overline{C}=\frac{1}{N}\sum_{n=1}^{N}\cos(\theta_{n})\qquad\qquad\overline{S}% =\frac{1}{N}\sum_{n=1}^{N}\sin(\theta_{n})
  10. R ¯ 2 = | z ¯ | 2 = C ¯ 2 + S ¯ 2 \overline{R}^{2}=|\overline{z}|^{2}=\overline{C}^{2}+\overline{S}^{2}
  11. θ ¯ = Arg ( z ¯ ) . \overline{\theta}=\mathrm{Arg}(\overline{z}).\,
  12. 1 ( 2 π ) N Γ n = 1 N d θ n = P ( R ¯ ) P ( θ ¯ ) d R ¯ d θ ¯ \frac{1}{(2\pi)^{N}}\int_{\Gamma}\prod_{n=1}^{N}d\theta_{n}=P(\overline{R})P(% \overline{\theta})\,d\overline{R}\,d\overline{\theta}
  13. Γ \Gamma\,
  14. 2 π 2\pi
  15. R ¯ \overline{R}
  16. θ ¯ \overline{\theta}
  17. C ¯ \overline{C}
  18. S ¯ \overline{S}
  19. P ( θ ¯ ) P(\overline{\theta})
  20. P ( θ ¯ ) = 1 2 π P(\overline{\theta})=\frac{1}{2\pi}
  21. R ¯ \overline{R}
  22. P N ( R ¯ ) = N 2 R ¯ 0 J 0 ( N R ¯ t ) J 0 ( t ) N t d t P_{N}(\overline{R})=N^{2}\overline{R}\int_{0}^{\infty}J_{0}(N\overline{R}\,t)J% _{0}(t)^{N}t\,dt
  23. J 0 J_{0}
  24. P 2 ( R ¯ ) = 2 π 1 - R ¯ 2 . P_{2}(\overline{R})=\frac{2}{\pi\sqrt{1-\overline{R}^{2}}}.
  25. P ( u ) d u = 1 π d u 1 - u 2 P(u)du=\frac{1}{\pi}\,\frac{du}{\sqrt{1-u^{2}}}
  26. u = cos θ n u=\cos\theta_{n}\,
  27. sin θ n \sin\theta_{n}\,
  28. C ¯ \overline{C}\,
  29. S ¯ \overline{S}\,
  30. 1 / 2 N 1/2N
  31. R ¯ \overline{R}\,
  32. 1 / 2 N 1/2N
  33. lim N P N ( R ¯ ) = 2 N R ¯ e - N R ¯ 2 . \lim_{N\rightarrow\infty}P_{N}(\overline{R})=2N\overline{R}\,e^{-N\overline{R}% ^{2}}.
  34. H U = - Γ 1 2 π ln ( 1 2 π ) d θ = ln ( 2 π ) H_{U}=-\int_{\Gamma}\frac{1}{2\pi}\ln\left(\frac{1}{2\pi}\right)\,d\theta=\ln(% 2\pi)
  35. Γ \Gamma
  36. 2 π 2\pi

Clapeyron's_theorem_(elasticity).html

  1. 1 2 F ( L 1 - L 0 ) . \frac{1}{2}F(L_{1}-L_{0}).

Clasper_(mathematics).html

  1. G = 𝐀 𝐁 G=\mathbf{A}\cup\mathbf{B}
  2. M M
  3. 𝐀 \mathbf{A}
  4. 𝐁 \mathbf{B}
  5. G G
  6. G G
  7. C n C_{n}
  8. C n C_{n}
  9. n n
  10. L M L\subset M
  11. C 1 C_{1}
  12. C 2 C_{2}
  13. C n C_{n}
  14. {}^{\prime}
  15. k k
  16. K K
  17. {}^{\prime}
  18. k k
  19. K K
  20. {}^{\prime}
  21. C k C_{k}

Classification_of_obesity.html

  1. BMI = kilograms meters 2 \mathrm{BMI}=\frac{\mathrm{kilograms}}{\mathrm{meters}^{2}}
  2. BMI = pounds 703 inches 2 \mathrm{BMI}=\mathrm{pounds}\frac{703}{\mathrm{inches}^{2}}
  3. body fat percentage = 1.2 × BMI + 0.23 × age - 5.4 - 10.8 × gender \,\text{body fat percentage}=1.2\times\,\text{BMI}+0.23\times\,\text{age}-5.4-% 10.8\times\,\text{gender}

Clausius–Duhem_inequality.html

  1. d d t ( Ω ρ η dV ) Ω ρ η ( u n - 𝐯 𝐧 ) dA - Ω 𝐪 𝐧 T dA + Ω ρ s T dV . \cfrac{d}{dt}\left(\int_{\Omega}\rho~{}\eta~{}\,\text{dV}\right)\geq\int_{% \partial\Omega}\rho~{}\eta~{}(u_{n}-\mathbf{v}\cdot\mathbf{n})~{}\,\text{dA}-% \int_{\partial\Omega}\cfrac{\mathbf{q}\cdot\mathbf{n}}{T}~{}\,\text{dA}+\int_{% \Omega}\cfrac{\rho~{}s}{T}~{}\,\text{dV}.
  2. t t\,
  3. Ω \Omega\,
  4. Ω \partial\Omega\,
  5. ρ \rho\,
  6. η \eta\,
  7. u n u_{n}\,
  8. Ω \partial\Omega\,
  9. 𝐯 \mathbf{v}
  10. Ω \Omega\,
  11. 𝐧 \mathbf{n}
  12. 𝐪 \mathbf{q}
  13. s s\,
  14. T T\,
  15. 𝐱 \mathbf{x}
  16. t t\,
  17. ρ η ˙ - s y m b o l ( 𝐪 T ) + ρ s T \rho~{}\dot{\eta}\geq-symbol{\nabla}\cdot\left(\cfrac{\mathbf{q}}{T}\right)+% \cfrac{\rho~{}s}{T}
  18. η ˙ \dot{\eta}
  19. η \eta\,
  20. s y m b o l ( 𝐚 ) symbol{\nabla}\cdot(\mathbf{a})
  21. 𝐚 \mathbf{a}
  22. Ω \Omega
  23. u n = 0 u_{n}=0
  24. Ω t ( ρ η ) dV - Ω ρ η ( 𝐯 𝐧 ) dA - Ω 𝐪 𝐧 T dA + Ω ρ s T dV . \int_{\Omega}\frac{\partial}{\partial t}(\rho~{}\eta)~{}\,\text{dV}\geq-\int_{% \partial\Omega}\rho~{}\eta~{}(\mathbf{v}\cdot\mathbf{n})~{}\,\text{dA}-\int_{% \partial\Omega}\cfrac{\mathbf{q}\cdot\mathbf{n}}{T}~{}\,\text{dA}+\int_{\Omega% }\cfrac{\rho~{}s}{T}~{}\,\text{dV}.
  25. Ω t ( ρ η ) dV - Ω s y m b o l ( ρ η 𝐯 ) dV - Ω s y m b o l ( 𝐪 T ) dV + Ω ρ s T dV . \int_{\Omega}\frac{\partial}{\partial t}(\rho~{}\eta)~{}\,\text{dV}\geq-\int_{% \Omega}symbol{\nabla}\cdot(\rho~{}\eta~{}\mathbf{v})~{}\,\text{dV}-\int_{% \Omega}symbol{\nabla}\cdot\left(\cfrac{\mathbf{q}}{T}\right)~{}\,\text{dV}+% \int_{\Omega}\cfrac{\rho~{}s}{T}~{}\,\text{dV}.
  26. Ω \Omega
  27. t ( ρ η ) - s y m b o l ( ρ η 𝐯 ) - s y m b o l ( 𝐪 T ) + ρ s T . \frac{\partial}{\partial t}(\rho~{}\eta)\geq-symbol{\nabla}\cdot(\rho~{}\eta~{% }\mathbf{v})-symbol{\nabla}\cdot\left(\cfrac{\mathbf{q}}{T}\right)+\cfrac{\rho% ~{}s}{T}.
  28. ρ t η + ρ η t - s y m b o l ( ρ η ) 𝐯 - ρ η ( s y m b o l 𝐯 ) - s y m b o l ( 𝐪 T ) + ρ s T \frac{\partial\rho}{\partial t}~{}\eta+\rho~{}\frac{\partial\eta}{\partial t}% \geq-symbol{\nabla}(\rho_{\eta})\cdot\mathbf{v}-\rho~{}\eta~{}(symbol{\nabla}% \cdot\mathbf{v})-symbol{\nabla}\cdot\left(\cfrac{\mathbf{q}}{T}\right)+\cfrac{% \rho~{}s}{T}
  29. ρ t η + ρ η t - η s y m b o l ρ 𝐯 - ρ s y m b o l η 𝐯 - ρ η ( s y m b o l 𝐯 ) - s y m b o l ( 𝐪 T ) + ρ s T \frac{\partial\rho}{\partial t}~{}\eta+\rho~{}\frac{\partial\eta}{\partial t}% \geq-\eta~{}symbol{\nabla}\rho\cdot\mathbf{v}-\rho~{}symbol{\nabla}\eta\cdot% \mathbf{v}-\rho~{}\eta~{}(symbol{\nabla}\cdot\mathbf{v})-symbol{\nabla}\cdot% \left(\cfrac{\mathbf{q}}{T}\right)+\cfrac{\rho~{}s}{T}
  30. ( ρ t + s y m b o l ρ 𝐯 + ρ s y m b o l 𝐯 ) η + ρ ( η t + s y m b o l η 𝐯 ) - s y m b o l ( 𝐪 T ) + ρ s T . \left(\frac{\partial\rho}{\partial t}+symbol{\nabla}\rho\cdot\mathbf{v}+\rho~{% }symbol{\nabla}\cdot\mathbf{v}\right)~{}\eta+\rho~{}\left(\frac{\partial\eta}{% \partial t}+symbol{\nabla}\eta\cdot\mathbf{v}\right)\geq-symbol{\nabla}\cdot% \left(\cfrac{\mathbf{q}}{T}\right)+\cfrac{\rho~{}s}{T}.
  31. ρ \rho
  32. η \eta
  33. ρ ˙ = ρ t + s y m b o l ρ 𝐯 ; η ˙ = η t + s y m b o l η 𝐯 . \dot{\rho}=\frac{\partial\rho}{\partial t}+symbol{\nabla}\rho\cdot\mathbf{v}~{% };~{}~{}\dot{\eta}=\frac{\partial\eta}{\partial t}+symbol{\nabla}\eta\cdot% \mathbf{v}.
  34. ( ρ ˙ + ρ s y m b o l 𝐯 ) η + ρ η ˙ - s y m b o l ( 𝐪 T ) + ρ s T . \left(\dot{\rho}+\rho~{}symbol{\nabla}\cdot\mathbf{v}\right)~{}\eta+\rho~{}% \dot{\eta}\geq-symbol{\nabla}\cdot\left(\cfrac{\mathbf{q}}{T}\right)+\cfrac{% \rho~{}s}{T}.
  35. ρ ˙ + ρ s y m b o l 𝐯 = 0 \dot{\rho}+\rho~{}symbol{\nabla}\cdot\mathbf{v}=0
  36. ρ η ˙ - s y m b o l ( 𝐪 T ) + ρ s T . {\rho~{}\dot{\eta}\geq-symbol{\nabla}\cdot\left(\cfrac{\mathbf{q}}{T}\right)+% \cfrac{\rho~{}s}{T}.}
  37. ρ ( e ˙ - T η ˙ ) - s y m b o l σ : s y m b o l 𝐯 - 𝐪 \cdotsymbol T T \rho~{}(\dot{e}-T~{}\dot{\eta})-symbol{\sigma}:symbol{\nabla}\mathbf{v}\leq-% \cfrac{\mathbf{q}\cdotsymbol{\nabla}T}{T}
  38. e ˙ \dot{e}
  39. e e\,
  40. s y m b o l σ symbol{\sigma}
  41. s y m b o l 𝐯 symbol{\nabla}\mathbf{v}
  42. s y m b o l ( φ 𝐯 ) = φ s y m b o l 𝐯 + 𝐯 \cdotsymbol φ symbol{\nabla}\cdot(\varphi~{}\mathbf{v})=\varphi~{}symbol{\nabla}\cdot\mathbf% {v}+\mathbf{v}\cdotsymbol{\nabla}\varphi
  43. ρ η ˙ - s y m b o l ( 𝐪 T ) + ρ s T or ρ η ˙ - 1 T s y m b o l 𝐪 - 𝐪 \cdotsymbol ( 1 T ) + ρ s T . \rho~{}\dot{\eta}\geq-symbol{\nabla}\cdot\left(\cfrac{\mathbf{q}}{T}\right)+% \cfrac{\rho~{}s}{T}\qquad\,\text{or}\qquad\rho~{}\dot{\eta}\geq-\cfrac{1}{T}~{% }symbol{\nabla}\cdot\mathbf{q}-\mathbf{q}\cdotsymbol{\nabla}\left(\cfrac{1}{T}% \right)+\cfrac{\rho~{}s}{T}.
  44. 𝐞 j \mathbf{e}_{j}
  45. s y m b o l ( 1 T ) = x j ( T - 1 ) 𝐞 j = - ( T - 2 ) T x j 𝐞 j = - 1 T 2 s y m b o l T . symbol{\nabla}\left(\cfrac{1}{T}\right)=\frac{\partial}{\partial x_{j}}\left(T% ^{-1}\right)~{}\mathbf{e}_{j}=-\left(T^{-2}\right)~{}\frac{\partial T}{% \partial x_{j}}~{}\mathbf{e}_{j}=-\cfrac{1}{T^{2}}~{}symbol{\nabla}T.
  46. ρ η ˙ - 1 T s y m b o l 𝐪 + 1 T 2 𝐪 \cdotsymbol T + ρ s T or ρ η ˙ - 1 T ( s y m b o l 𝐪 - ρ s ) + 1 T 2 𝐪 \cdotsymbol T . \rho~{}\dot{\eta}\geq-\cfrac{1}{T}~{}symbol{\nabla}\cdot\mathbf{q}+\cfrac{1}{T% ^{2}}~{}\mathbf{q}\cdotsymbol{\nabla}T+\cfrac{\rho~{}s}{T}\qquad\,\text{or}% \qquad\rho~{}\dot{\eta}\geq-\cfrac{1}{T}\left(symbol{\nabla}\cdot\mathbf{q}-% \rho~{}s\right)+\cfrac{1}{T^{2}}~{}\mathbf{q}\cdotsymbol{\nabla}T.
  47. ρ e ˙ - s y m b o l σ : s y m b o l 𝐯 + s y m b o l 𝐪 - ρ s = 0 ρ e ˙ - s y m b o l σ : s y m b o l 𝐯 = - ( s y m b o l 𝐪 - ρ s ) . \rho~{}\dot{e}-symbol{\sigma}:symbol{\nabla}\mathbf{v}+symbol{\nabla}\cdot% \mathbf{q}-\rho~{}s=0\qquad\implies\qquad\rho~{}\dot{e}-symbol{\sigma}:symbol{% \nabla}\mathbf{v}=-(symbol{\nabla}\cdot\mathbf{q}-\rho~{}s).
  48. ρ η ˙ 1 T ( ρ e ˙ - s y m b o l σ : s y m b o l 𝐯 ) + 1 T 2 𝐪 \cdotsymbol T ρ η ˙ T ρ e ˙ - s y m b o l σ : s y m b o l 𝐯 + 𝐪 \cdotsymbol T T . \rho~{}\dot{\eta}\geq\cfrac{1}{T}\left(\rho~{}\dot{e}-symbol{\sigma}:symbol{% \nabla}\mathbf{v}\right)+\cfrac{1}{T^{2}}~{}\mathbf{q}\cdotsymbol{\nabla}T% \qquad\implies\qquad\rho~{}\dot{\eta}~{}T\geq\rho~{}\dot{e}-symbol{\sigma}:% symbol{\nabla}\mathbf{v}+\cfrac{\mathbf{q}\cdotsymbol{\nabla}T}{T}.
  49. ρ ( e ˙ - T η ˙ ) - s y m b o l σ : s y m b o l 𝐯 - 𝐪 \cdotsymbol T T {\rho~{}(\dot{e}-T~{}\dot{\eta})-symbol{\sigma}:symbol{\nabla}\mathbf{v}\leq-% \cfrac{\mathbf{q}\cdotsymbol{\nabla}T}{T}\qquad\qquad\square}
  50. 𝒟 := ρ ( T η ˙ - e ˙ ) + s y m b o l σ : s y m b o l 𝐯 - 𝐪 \cdotsymbol T T 0 \mathcal{D}:=\rho~{}(T~{}\dot{\eta}-\dot{e})+symbol{\sigma}:symbol{\nabla}% \mathbf{v}-\cfrac{\mathbf{q}\cdotsymbol{\nabla}T}{T}\geq 0

Clebsch_graph.html

  1. ( v , k , λ , μ ) = ( 16 , 5 , 0 , 2 ) (v,k,\lambda,\mu)=(16,5,0,2)
  2. ( x + 3 ) 5 ( x - 1 ) 10 ( x - 5 ) (x+3)^{5}(x-1)^{10}(x-5)
  3. D 5 D_{5}

Clebsch_surface.html

  1. x 0 + x 1 + x 2 + x 3 + x 4 = 0 \displaystyle x_{0}+x_{1}+x_{2}+x_{3}+x_{4}=0
  2. x 0 3 + x 1 3 + x 2 3 + x 3 3 + x 4 3 = 0. \displaystyle x_{0}^{3}+x_{1}^{3}+x_{2}^{3}+x_{3}^{3}+x_{4}^{3}=0.
  3. x 1 3 + x 2 3 + x 3 3 + x 4 3 = ( x 1 + x 2 + x 3 + x 4 ) 3 \displaystyle x_{1}^{3}+x_{2}^{3}+x_{3}^{3}+x_{4}^{3}=(x_{1}+x_{2}+x_{3}+x_{4}% )^{3}

Clifford_analysis.html

  1. d + * d * d+*d*
  2. C 0 ( 𝐑 n ) C_{0}^{\infty}(\mathbf{R}^{n})
  3. D = j = 1 n e j x j D=\sum_{j=1}^{n}e_{j}\frac{\partial}{\partial x_{j}}
  4. D 2 = - Δ n D^{2}=-\Delta_{n}
  5. G ( x - y ) := 1 ω n x - y x - y n G(x-y):=\frac{1}{\omega_{n}}\frac{x-y}{\|x-y\|^{n}}
  6. D 1 ( n - 2 ) ω n x - y n - 2 = G ( x - y ) D\frac{1}{(n-2)\omega_{n}\|x-y\|^{n-2}}=G(x-y)
  7. 1 ( n - 2 ) ω n x - y n - 2 \frac{1}{(n-2)\;\omega_{n}\;\|x-y\|^{n-2}}
  8. x + i y \frac{\partial}{\partial x}+i\frac{\partial}{\partial y}
  9. - x x 2 𝐑 n . -\frac{x}{\|x\|^{2}}\in\mathbf{R}^{n}.
  10. D n - 1 = j = 1 n - 1 x j D_{n-1}=\sum_{j=1}^{n-1}\frac{\partial}{\partial x_{j}}
  11. ζ = ζ 1 e 1 + + ζ n - 1 e n - 1 . \zeta=\zeta_{1}e_{1}+\ldots+\zeta_{n-1}e_{n-1}.
  12. ± 1 2 + G ( x - y ) | 𝐑 n - 1 \pm\tfrac{1}{2}+G(x-y)|_{\mathbf{R}^{n-1}}
  13. 1 2 ( 1 ± i ζ ζ ) . \frac{1}{2}\left(1\pm i\frac{\zeta}{\|\zeta\|}\right).
  14. G | 𝐑 n = j = 1 n - 1 e j R j G|_{\mathbf{R}^{n}}=\sum_{j=1}^{n-1}e_{j}R_{j}
  15. x j x n . \frac{x_{j}}{\|x\|^{n}}.
  16. G | 𝐑 n G|_{\mathbf{R}^{n}}
  17. i ζ ζ \frac{i\zeta}{\|\zeta\|}
  18. j = 1 n - 1 R j 2 = 1. \sum_{j=1}^{n-1}R_{j}^{2}=1.
  19. G | 𝐑 n G|_{\mathbf{R}^{n}}
  20. j = 0 ( x n e n - 1 D n - 1 ) j g ( x ) . \sum_{j=0}^{\infty}\left(x_{n}e_{n}^{-1}D_{n-1}\right)^{j}g(x).
  21. e - i x , ζ ( 1 2 ( 1 ± i ζ ζ ) ) e^{-i\langle x,\zeta\rangle}\left(\tfrac{1}{2}\left(1\pm i\frac{\zeta}{\|\zeta% \|}\right)\right)
  22. e - i x , ζ e^{-i\langle x,\zeta\rangle}
  23. D S = x ( Γ n + n 2 ) D_{S}=x(\Gamma_{n}+\frac{n}{2})
  24. 1 i < j n + 1 e i e j ( x i x j - x j x i ) \sum\nolimits_{1\leq i<j\leq n+1}e_{i}e_{j}\left(x_{i}\frac{\partial}{\partial x% _{j}}-x_{j}\frac{\partial}{\partial x_{i}}\right)
  25. y = C ( x ) = ( e n + 1 x + 1 ) ( x + e n + 1 ) - 1 , x 𝐑 n . y=C(x)=(e_{n+1}x+1)(x+e_{n+1})^{-1},\qquad x\in\mathbf{R}^{n}.
  26. x = ( - e n + 1 + 1 ) ( y - e n + 1 ) - 1 . x=(-e_{n+1}+1)(y-e_{n+1})^{-1}.
  27. J ( C - 1 , y ) f ( C - 1 ( y ) ) J\left(C^{-1},y\right)f\left(C^{-1}(y)\right)
  28. J ( C - 1 , y ) = y - e n + 1 y - e n + 1 n . J(C^{-1},y)=\frac{y-e_{n+1}}{\|y-e_{n+1}\|^{n}}.
  29. D S ( D S - x ) = S , D_{S}(D_{S}-x)=\triangle_{S},
  30. S = - L B + 1 4 n ( n - 2 ) \triangle_{S}=-\triangle_{LB}+\tfrac{1}{4}n(n-2)
  31. L B \triangle_{LB}
  32. S \triangle_{S}
  33. D s ( D S - x ) ( D S - x ) ( D S - 2 x ) D_{s}(D_{S}-x)(D_{S}-x)(D_{S}-2x)
  34. - S ( S + 2 ) , -\triangle_{S}(\triangle_{S}+2),
  35. n 2 \triangle_{n}^{2}
  36. a x + b c x + d , \frac{ax+b}{cx+d},
  37. y = M ( x ) + a x + b c x + d y=M(x)+\frac{ax+b}{cx+d}
  38. J ( M , x ) f ( M ( x ) ) J(M,x)f(M(x))
  39. J ( M , x ) = c x + d ~ c x + d n J(M,x)=\frac{\widetilde{cx+d}}{\|cx+d\|^{n}}
  40. a x + b c x + d = - a x - b - c x - d \frac{ax+b}{cx+d}=\frac{-ax-b}{-cx-d}
  41. D s ( x ) = j = 1 n e j ( x ) Γ ~ e j ( x ) s ( x ) Ds(x)=\sum_{j=1}^{n}e_{j}(x)\tilde{\Gamma}_{e_{j}(x)}s(x)
  42. Γ ~ \widetilde{\Gamma}
  43. D 2 = Γ * Γ + τ 4 D^{2}=\Gamma^{*}\Gamma+\tfrac{\tau}{4}
  44. C ( x , y ) := D - 1 * δ y , x y M , C(x,y):=D^{-1}*\delta_{y},\qquad x\neq y\in M,
  45. M f = D f + n - 2 x n Q ( f ) Mf=Df+\frac{n-2}{x_{n}}Q(f)
  46. M 2 f = - n P ( f ) + n - 2 x n P ( f ) x n - ( n Q ( f ) - n - 2 x n Q ( f ) x n + n - 2 x n 2 Q ( f ) ) e n M^{2}f=-\triangle_{n}P(f)+\frac{n-2}{x_{n}}\frac{\partial P(f)}{\partial x_{n}% }-\left(\triangle_{n}Q(f)-\frac{n-2}{x_{n}}\frac{\partial Q(f)}{\partial x_{n}% }+\frac{n-2}{x_{n}^{2}}Q(f)\right)e_{n}
  47. n - n - 2 x n x n \triangle_{n}-\frac{n-2}{x_{n}}\frac{\partial}{\partial x_{n}}
  48. h k ( x ) = p k ( x ) + x p k - 1 ( x ) h_{k}(x)=p_{k}(x)+xp_{k-1}(x)
  49. D u u p k - 1 ( u ) = ( - n - 2 k + 2 ) p k - 1 . D_{u}up_{k-1}(u)=(-n-2k+2)p_{k-1}.
  50. R k = ( I + 1 n + 2 k - 2 u D u ) D x . R_{k}=\left(I+\frac{1}{n+2k-2}uD_{u}\right)D_{x}.

Clifford_parallel.html

  1. { e a r : 0 a < π } \{e^{ar}:\ 0\leq a<\pi\}
  2. { u e a r : 0 a < π } , \{ue^{ar}:\ 0\leq a<\pi\},
  3. { e a r u : 0 a < π } . \{e^{ar}u:\ 0\leq a<\pi\}.
  4. { e a r e b r : 0 a , b < π } . \{e^{ar}e^{br}:\ 0\leq a,b<\pi\}.

Clifford_semigroup.html

  1. x x - 1 = x - 1 x xx^{-1}=x^{-1}x
  2. x y = y x x - 1 y = y x - 1 xy=yx\leftrightarrow x^{-1}y=yx^{-1}

Closed_range_theorem.html

  1. X X
  2. Y Y
  3. T : D ( T ) Y T\colon D(T)\to Y
  4. D ( T ) D(T)
  5. X X
  6. T T^{\prime}
  7. T T
  8. R ( T ) R(T)
  9. T T
  10. Y Y
  11. R ( T ) R(T^{\prime})
  12. T T^{\prime}
  13. X X^{\prime}
  14. X X
  15. R ( T ) = N ( T ) = { y Y | x * , y = 0 for all x * N ( T ) } R(T)=N(T^{\prime})^{\perp}=\{y\in Y|\langle x^{*},y\rangle=0\quad{\,\text{for % all}}\quad x^{*}\in N(T^{\prime})\}
  16. R ( T ) = N ( T ) = { x * X | x * , y = 0 for all y N ( T ) } R(T^{\prime})=N(T)^{\perp}=\{x^{*}\in X^{\prime}|\langle x^{*},y\rangle=0\quad% {\,\text{for all}}\quad y\in N(T)\}
  17. T T
  18. R ( T ) = Y R(T)=Y
  19. T T^{\prime}
  20. R ( T ) = X R(T^{\prime})=X^{\prime}
  21. T T

Cloud_drop_effective_radius.html

  1. r e = 0 π r 3 n ( r ) d r 0 π r 2 n ( r ) d r r_{e}=\dfrac{\int\limits_{0}^{\infty}\pi\cdot r^{3}\cdot n(r)\,dr}{\int\limits% _{0}^{\infty}\pi\cdot r^{2}\cdot n(r)\,dr}

Cluster-weighted_modeling.html

  1. p ( y , x ) = 1 n w j p j ( y , x ) , p(y,x)=\sum_{1}^{n}w_{j}p_{j}(y,x),
  2. p j ( y , x ) = p j ( y | x ) p j ( x ) , p_{j}(y,x)=p_{j}(y|x)p_{j}(x),

Cluster_algebra.html

  1. w y = t , b t , y > 0 t b t , y + t , b t , y < 0 t - b t , y wy=\prod_{t,b_{t,y}>0}t^{b_{t,y}}+\prod_{t,b_{t,y}<0}t^{-b_{t,y}}
  2. x n - 1 x n + 1 = 1 + x n \displaystyle x_{n-1}x_{n+1}=1+x_{n}
  3. x 1 , x 2 , x 3 = 1 + x 2 x 1 , x 4 = 1 + x 3 x 2 = 1 + x 1 + x 2 x 1 x 2 , x_{1},x_{2},x_{3}=\frac{1+x_{2}}{x_{1}},x_{4}=\frac{1+x_{3}}{x_{2}}=\frac{1+x_% {1}+x_{2}}{x_{1}x_{2}},
  4. x 5 = 1 + x 4 x 3 = 1 + x 1 x 2 , x 6 = 1 + x 5 x 4 = x 1 , x 7 = 1 + x 6 x 5 = x 2 , x_{5}=\frac{1+x_{4}}{x_{3}}=\frac{1+x_{1}}{x_{2}},x_{6}=\frac{1+x_{5}}{x_{4}}=% x_{1},x_{7}=\frac{1+x_{6}}{x_{5}}=x_{2},\ldots
  5. { x 1 , x 2 , x 3 } , \left\{x_{1},x_{2},x_{3}\right\},
  6. { 1 + x 2 x 1 , x 2 , x 3 } , \left\{\frac{1+x_{2}}{x_{1}},x_{2},x_{3}\right\},
  7. { x 1 , x 1 + x 3 x 2 , x 3 } , \left\{x_{1},\frac{x_{1}+x_{3}}{x_{2}},x_{3}\right\},
  8. { x 1 , x 2 , 1 + x 2 x 3 } , \left\{x_{1},x_{2},\frac{1+x_{2}}{x_{3}}\right\},
  9. { 1 + x 2 x 1 , x 1 + ( 1 + x 2 ) x 3 x 1 x 2 , x 3 } , \left\{\frac{1+x_{2}}{x_{1}},\frac{x_{1}+(1+x_{2})x_{3}}{x_{1}x_{2}},x_{3}% \right\},
  10. { 1 + x 2 x 1 , x 2 , 1 + x 2 x 3 } , \left\{\frac{1+x_{2}}{x_{1}},x_{2},\frac{1+x_{2}}{x_{3}}\right\},
  11. { x 1 + ( 1 + x 2 ) x 3 x 1 x 2 , x 1 + x 3 x 2 , x 3 } , \left\{\frac{x_{1}+(1+x_{2})x_{3}}{x_{1}x_{2}},\frac{x_{1}+x_{3}}{x_{2}},x_{3}% \right\},
  12. { x 1 , x 1 + x 3 x 2 , ( 1 + x 2 ) x 1 + x 3 x 2 x 3 } , \left\{x_{1},\frac{x_{1}+x_{3}}{x_{2}},\frac{(1+x_{2})x_{1}+x_{3}}{x_{2}x_{3}}% \right\},
  13. { x 1 , ( 1 + x 2 ) x 1 + x 3 x 2 x 3 , 1 + x 2 x 3 } , \left\{x_{1},\frac{(1+x_{2})x_{1}+x_{3}}{x_{2}x_{3}},\frac{1+x_{2}}{x_{3}}% \right\},
  14. { 1 + x 2 x 1 , x 1 + ( 1 + x 2 ) x 3 x 1 x 2 , ( 1 + x 2 ) x 1 + ( 1 + x 2 ) x 3 x 1 x 2 x 3 } , \left\{\frac{1+x_{2}}{x_{1}},\frac{x_{1}+(1+x_{2})x_{3}}{x_{1}x_{2}},\frac{(1+% x_{2})x_{1}+(1+x_{2})x_{3}}{x_{1}x_{2}x_{3}}\right\},
  15. { 1 + x 2 x 1 , ( 1 + x 2 ) x 1 + ( 1 + x 2 ) x 3 x 1 x 2 x 3 , 1 + x 2 x 3 } , \left\{\frac{1+x_{2}}{x_{1}},\frac{(1+x_{2})x_{1}+(1+x_{2})x_{3}}{x_{1}x_{2}x_% {3}},\frac{1+x_{2}}{x_{3}}\right\},
  16. { x 1 + ( 1 + x 2 ) x 3 x 1 x 2 , x 1 + x 3 x 2 , ( 1 + x 2 ) x 1 + ( 1 + x 2 ) x 3 x 1 x 2 x 3 } , \left\{\frac{x_{1}+(1+x_{2})x_{3}}{x_{1}x_{2}},\frac{x_{1}+x_{3}}{x_{2}},\frac% {(1+x_{2})x_{1}+(1+x_{2})x_{3}}{x_{1}x_{2}x_{3}}\right\},
  17. { ( 1 + x 2 ) x 1 + ( 1 + x 2 ) x 3 x 1 x 2 x 3 , x 1 + x 3 x 2 , ( 1 + x 2 ) x 1 + x 3 x 2 x 3 } , \left\{\frac{(1+x_{2})x_{1}+(1+x_{2})x_{3}}{x_{1}x_{2}x_{3}},\frac{x_{1}+x_{3}% }{x_{2}},\frac{(1+x_{2})x_{1}+x_{3}}{x_{2}x_{3}}\right\},
  18. { ( 1 + x 2 ) x 1 + ( 1 + x 2 ) x 3 x 1 x 2 x 3 , ( 1 + x 2 ) x 1 + x 3 x 2 x 3 , 1 + x 2 x 3 } . \left\{\frac{(1+x_{2})x_{1}+(1+x_{2})x_{3}}{x_{1}x_{2}x_{3}},\frac{(1+x_{2})x_% {1}+x_{3}}{x_{2}x_{3}},\frac{1+x_{2}}{x_{3}}\right\}.
  19. 1 + x 2 x 1 , x 1 + x 3 x 2 , 1 + x 2 x 3 , x 1 + ( 1 + x 2 ) x 3 x 1 x 2 , ( 1 + x 2 ) x 1 + x 3 x 2 x 3 , ( 1 + x 2 ) x 1 + ( 1 + x 2 ) x 3 x 1 x 2 x 3 \frac{1+x_{2}}{x_{1}},\frac{x_{1}+x_{3}}{x_{2}},\frac{1+x_{2}}{x_{3}},\frac{x_% {1}+(1+x_{2})x_{3}}{x_{1}x_{2}},\frac{(1+x_{2})x_{1}+x_{3}}{x_{2}x_{3}},\frac{% (1+x_{2})x_{1}+(1+x_{2})x_{3}}{x_{1}x_{2}x_{3}}

Cluster_labeling.html

  1. X X
  2. Y Y
  3. I ( X , Y ) = x X y Y p ( x , y ) l o g 2 ( p ( x , y ) p 1 ( x ) p 2 ( y ) ) I(X,Y)=\sum_{x\in X}{\sum_{y\in Y}{p(x,y)log_{2}\left(\frac{p(x,y)}{p_{1}(x)p_% {2}(y)}\right)}}
  4. I ( C , T ) = c 0 , 1 t 0 , 1 p ( C = c , T = t ) l o g 2 ( p ( C = c , T = t ) p ( C = c ) p ( T = t ) ) I(C,T)=\sum_{c\in{0,1}}{\sum_{t\in{0,1}}{p(C=c,T=t)log_{2}\left(\frac{p(C=c,T=% t)}{p(C=c)p(T=t)}\right)}}
  5. X 2 = a A b B ( O a , b - E a , b ) 2 E a , b X^{2}=\sum_{a\in A}{\sum_{b\in B}{\frac{(O_{a,b}-E_{a,b})^{2}}{E_{a,b}}}}
  6. X 2 = a 0 , 1 b 0 , 1 ( O a , b - E a , b ) 2 E a , b X^{2}=\sum_{a\in{0,1}}{\sum_{b\in{0,1}}{\frac{(O_{a,b}-E_{a,b})^{2}}{E_{a,b}}}}
  7. T k T_{k}
  8. S k S_{k}
  9. i i
  10. j j
  11. T k s i m T_{k}^{sim}
  12. T k T_{k}
  13. T k s i m = T k T k = ( t s i m i j ) T_{k}^{sim}=T_{k}^{\prime}T_{k}=(t_{{sim}_{ij}})
  14. t ~ i \tilde{t}_{i}
  15. t ~ j \tilde{t}_{j}
  16. t s i m i j t_{{sim}_{ij}}
  17. i i
  18. j j
  19. T k s i m T_{k}^{sim}

Clustering_high-dimensional_data.html

  1. lim d d i s t max - d i s t min d i s t min = 0 \lim_{d\to\infty}\frac{dist_{\max}-dist_{\min}}{dist_{\min}}=0
  2. c a c_{a}
  3. { x } \{x\}
  4. c b c_{b}
  5. c c c_{c}
  6. c d c_{d}
  7. { y } \{y\}
  8. c c c_{c}
  9. x x
  10. c a b c_{ab}
  11. c a d c_{ad}
  12. 2 d 2^{d}
  13. d d
  14. c c c_{c}
  15. x x
  16. y y

Cmin.html

  1. C p m i n = S F D k a V d ( k a - k ) × { e - k τ 1 - e - k τ - e - k a τ 1 - e - k a τ } C_{pmin}=\frac{SFDk_{a}}{V_{d}(k_{a}-k)}\times\{\frac{e^{-k\tau}}{1-e^{-k\tau}% }-\frac{e^{-k_{a}\tau}}{1-e^{-k_{a}\tau}}\}

Cnoidal_wave.html

  1. 7 h / g \scriptstyle 7\sqrt{h/g}
  2. η ( x , t ) = η 2 + H cn 2 ( 2 K ( m ) x - c t λ m ) , \eta(x,t)=\eta_{2}+H\,\operatorname{cn}^{2}\,\left(\begin{array}[]{c|c}% \displaystyle 2\,K(m)\,\frac{x-c\,t}{\lambda}&m\end{array}\right),
  3. U = H λ 2 h 3 = H h ( λ h ) 2 . U=\frac{H\,\lambda^{2}}{h^{3}}=\frac{H}{h}\,\left(\frac{\lambda}{h}\right)^{2}.
  4. t η + g h x η + 3 2 g h η x η + 1 6 h 2 g h x 3 η = 0 , \partial_{t}\eta+\sqrt{gh}\;\partial_{x}\eta+\tfrac{3}{2}\,\sqrt{\frac{g}{h}}% \;\eta\,\partial_{x}\eta+\tfrac{1}{6}\,h^{2}\,\sqrt{gh}\;\partial_{x}^{3}\eta=0,
  5. η ~ = η h , \tilde{\eta}=\frac{\eta}{h},
  6. x ~ = x h \tilde{x}=\frac{x}{h}
  7. t ~ = g h t . \tilde{t}=\sqrt{\frac{g}{h}}\,t.
  8. t ~ η ~ + x ~ η ~ + 3 2 η ~ x ~ η ~ + 1 6 x ~ 3 η ~ = 0 , \partial_{\tilde{t}}\tilde{\eta}+\partial_{\tilde{x}}\tilde{\eta}+\tfrac{3}{2}% \,\tilde{\eta}\,\partial_{\tilde{x}}\tilde{\eta}+\tfrac{1}{6}\,\partial_{% \tilde{x}}^{3}\tilde{\eta}=0,
  9. t ^ ϕ + 6 ϕ x ^ ϕ + x ^ 3 ϕ = 0 \partial_{\hat{t}}\phi+6\,\phi\ \partial_{\hat{x}}\phi+\partial_{\hat{x}}^{3}% \phi=0
  10. t ^ = 1 6 t , \hat{t}=\tfrac{1}{6}\,t,\,
  11. x ^ = x - t \hat{x}=x-t\,
  12. ϕ = 3 2 η , \phi=\tfrac{3}{2}\,\eta,\,
  13. η = η ( ξ ) \eta=\eta(\xi)\,
  14. ξ \xi
  15. ξ = x - c t . \xi=x-c\,t.\,
  16. x η = η \partial_{x}\eta=\eta^{\prime}\,
  17. t η = - c η , \partial_{t}\eta=-c\,\eta^{\prime},\,
  18. 1 6 η ′′′ + ( 1 - c ) η + 3 2 η η = 0. \tfrac{1}{6}\,\eta^{\prime\prime\prime}+\left(1-c\right)\,\eta^{\prime}+\tfrac% {3}{2}\,\eta\,\eta^{\prime}=0.\,
  19. 1 6 η ′′ + ( 1 - c ) η + 3 4 η 2 = 1 4 r , \tfrac{1}{6}\eta^{\prime\prime}+\left(1-c\right)\,\eta+\tfrac{3}{4}\,\eta^{2}=% \tfrac{1}{4}r,\,
  20. 1 3 ( η ) 2 + 2 ( 1 - c ) η 2 + η 3 = r η + s , \tfrac{1}{3}\left(\eta^{\prime}\right)^{2}+2\,\left(1-c\right)\,\eta^{2}+\eta^% {3}=r\,\eta+s,\,
  21. η 1 η 2 η 3 . \eta_{1}\geq\eta_{2}\geq\eta_{3}.\,
  22. η - η 1 \displaystyle\eta-\eta_{1}
  23. 4 3 ( d ψ d ξ ) 2 = ( η 1 - η 3 ) - ( η 1 - η 2 ) sin 2 ψ ( ξ ) , \frac{4}{3}\,\left(\frac{\,\text{d}\psi}{\,\text{d}\xi}\right)^{2}=\left(\eta_% {1}-\eta_{3}\right)-\left(\eta_{1}-\eta_{2}\right)\;\sin^{2}\,\psi(\xi),
  24. 1 Δ d ξ d ψ = ± 1 1 - m sin 2 ψ , \frac{1}{\Delta}\,\frac{\,\text{d}\xi}{\,\text{d}\psi}=\pm\,\frac{1}{\sqrt{1-m% \sin^{2}\,\psi}},
  25. Δ 2 = 4 3 1 η 1 - η 3 \Delta^{2}=\frac{4}{3}\,\frac{1}{\eta_{1}-\eta_{3}}
  26. m = η 1 - η 2 η 1 - η 3 , m=\frac{\eta_{1}-\eta_{2}}{\eta_{1}-\eta_{3}},
  27. cos ψ = cn ( ξ Δ m ) \cos\,\psi=\operatorname{cn}\left(\begin{array}[]{c|c}\displaystyle\frac{\xi}{% \Delta}&m\end{array}\right)
  28. sin ψ = sn ( ξ Δ m ) . \sin\,\psi=\operatorname{sn}\left(\begin{array}[]{c|c}\displaystyle\frac{\xi}{% \Delta}&m\end{array}\right).
  29. η ( ξ ) = η 2 + ( η 1 - η 2 ) cn 2 ( ξ Δ m ) . \eta(\xi)=\eta_{2}+\left(\eta_{1}-\eta_{2}\right)\,\operatorname{cn}^{2}\left(% \begin{array}[]{c|c}\displaystyle\frac{\xi}{\Delta}&m\end{array}\right).
  30. m = H η 1 - η 3 m=\frac{H}{\eta_{1}-\eta_{3}}
  31. Δ 2 m = 4 3 H \frac{\Delta^{2}}{m}=\frac{4}{3\,H}
  32. Δ = 4 3 m H . \Delta=\sqrt{\frac{4}{3}\frac{m}{H}}.
  33. η ( ξ ) = η 2 + H cn 2 ( ξ Δ m ) . \eta(\xi)=\eta_{2}+H\,\operatorname{cn}^{2}\left(\begin{array}[]{c|c}% \displaystyle\frac{\xi}{\Delta}&m\end{array}\right).
  34. 1 2 λ = Δ F ( 1 2 π m ) = Δ K ( m ) , \tfrac{1}{2}\,\lambda=\Delta\,F\left(\begin{array}[]{c|c}\tfrac{1}{2}\,\pi&m% \end{array}\right)=\Delta\,K(m),
  35. λ = 2 Δ K ( m ) = 16 3 m H K ( m ) , \lambda=2\,\Delta\,K(m)=\sqrt{\frac{16}{3}\frac{m}{H}}\;K(m),
  36. 0 = 0 λ η ( ξ ) d ξ = 2 0 1 2 λ [ η 2 + ( η 1 - η 2 ) cn 2 ( ξ Δ m ) ] d ξ = 2 0 1 2 π [ η 2 + ( η 1 - η 2 ) cos 2 ψ ] d ξ d ψ d ψ = 2 Δ 0 1 2 π η 1 - ( η 1 - η 2 ) sin 2 ψ 1 - m sin 2 ψ d ψ = 2 Δ 0 1 2 π η 1 - m ( η 1 - η 3 ) sin 2 ψ 1 - m sin 2 ψ d ψ = 2 Δ 0 1 2 π [ η 3 1 - m sin 2 ψ + ( η 1 - η 3 ) 1 - m sin 2 ψ ] d ψ = 2 Δ [ η 3 K ( m ) + ( η 1 - η 3 ) E ( m ) ] = 2 Δ [ η 3 K ( m ) + H m E ( m ) ] , \begin{aligned}\displaystyle 0&\displaystyle=\int_{0}^{\lambda}\eta(\xi)\;\,% \text{d}\xi=2\,\int_{0}^{\tfrac{1}{2}\lambda}\left[\eta_{2}+\left(\eta_{1}-% \eta_{2}\right)\,\operatorname{cn}^{2}\,\left(\begin{array}[]{c|c}% \displaystyle\frac{\xi}{\Delta}&m\end{array}\right)\right]\;\,\text{d}\xi\\ &\displaystyle=2\,\int_{0}^{\tfrac{1}{2}\pi}\Bigl[\eta_{2}+\left(\eta_{1}-\eta% _{2}\right)\,\cos^{2}\,\psi\Bigr]\,\frac{\,\text{d}\xi}{\,\text{d}\psi}\;\,% \text{d}\psi=2\,\Delta\,\int_{0}^{\tfrac{1}{2}\pi}\frac{\eta_{1}-\left(\eta_{1% }-\eta_{2}\right)\,\sin^{2}\,\psi}{\sqrt{1-m\,\sin^{2}\,\psi}}\;\,\text{d}\psi% \\ &\displaystyle=2\,\Delta\,\int_{0}^{\tfrac{1}{2}\pi}\frac{\eta_{1}-m\,\left(% \eta_{1}-\eta_{3}\right)\,\sin^{2}\,\psi}{\sqrt{1-m\,\sin^{2}\,\psi}}\;\,\text% {d}\psi=2\,\Delta\,\int_{0}^{\tfrac{1}{2}\pi}\left[\frac{\eta_{3}}{\sqrt{1-m\,% \sin^{2}\,\psi}}+\left(\eta_{1}-\eta_{3}\right)\,\sqrt{1-m\,\sin^{2}\,\psi}% \right]\;\,\text{d}\psi\\ &\displaystyle=2\,\Delta\,\Bigl[\eta_{3}\,K(m)+\left(\eta_{1}-\eta_{3}\right)% \,E(m)\Bigr]=2\,\Delta\,\Bigl[\eta_{3}\,K(m)+\frac{H}{m}\,E(m)\Bigr],\end{aligned}
  37. η 3 = - H m E ( m ) K ( m ) , \eta_{3}=-\,\frac{H}{m}\,\frac{E(m)}{K(m)},
  38. η 1 = H m ( 1 - E ( m ) K ( m ) ) \eta_{1}=\frac{H}{m}\,\left(1-\frac{E(m)}{K(m)}\right)
  39. η 2 = H m ( 1 - m - E ( m ) K ( m ) ) . \eta_{2}=\frac{H}{m}\,\left(1-m-\frac{E(m)}{K(m)}\right).
  40. c = 1 + 1 2 ( η 1 + η 2 + η 3 ) = 1 + H m ( 1 - 1 2 m - 3 2 E ( m ) K ( m ) ) . c=1+\tfrac{1}{2}\,\left(\eta_{1}+\eta_{2}+\eta_{3}\right)=1+\frac{H}{m}\,\left% (1-\frac{1}{2}\,m-\frac{3}{2}\,\frac{E(m)}{K(m)}\right).
  41. g h \scriptstyle\sqrt{gh}
  42. η ( x , t ) = η 2 + H cn 2 ( x - c t Δ m ) , \eta(x,t)=\eta_{2}+H\,\operatorname{cn}^{2}\left(\begin{array}[]{c|c}% \displaystyle\frac{x-c\,t}{\Delta}&m\end{array}\right),
  43. η 2 = H m ( 1 - m - E ( m ) K ( m ) ) , \eta_{2}=\frac{H}{m}\,\left(1-m-\frac{E(m)}{K(m)}\right),
  44. Δ = λ 2 K ( m ) = h 4 3 m h H , \Delta=\frac{\lambda}{2\,K(m)}\,=h\,\sqrt{\frac{4}{3}\frac{m\,h}{H}},
  45. λ = h 16 3 m h H K ( m ) , \lambda=h\,\sqrt{\frac{16}{3}\frac{m\,h}{H}}\;K(m),
  46. c = g h [ 1 + H m h ( 1 - 1 2 m - 3 2 E ( m ) K ( m ) ) ] c=\sqrt{gh}\,\left[1+\frac{H}{m\,h}\,\left(1-\frac{1}{2}\,m-\frac{3}{2}\,\frac% {E(m)}{K(m)}\right)\right]
  47. τ = λ c , \tau=\frac{\lambda}{c},
  48. t η + g h x η + 3 2 g h η x η - 1 6 h 2 t x 2 η = 0. \partial_{t}\eta+\sqrt{g\,h}\,\partial_{x}\eta+\tfrac{3}{2}\,\sqrt{\frac{g}{h}% }\,\eta\,\partial_{x}\eta-\tfrac{1}{6}\,h^{2}\,\partial_{t}\,\partial_{x}^{2}% \eta=0.
  49. t η + x η + 3 2 η x η - 1 6 t x 2 η = 0. \partial_{t}\eta+\partial_{x}\eta+\tfrac{3}{2}\,\eta\,\partial_{x}\eta-\tfrac{% 1}{6}\,\partial_{t}\,\partial_{x}^{2}\eta=0.
  50. t ^ φ + x ^ φ + φ x ^ φ - t ^ x ^ 2 φ = 0 \partial_{\hat{t}}\varphi+\partial_{\hat{x}}\varphi+\varphi\,\partial_{\hat{x}% }\varphi-\partial_{\hat{t}}\,\partial_{\hat{x}}^{2}\varphi=0\,
  51. t ^ = 6 t , \hat{t}=\sqrt{6}\,t,
  52. x ^ = 6 x \hat{x}=\sqrt{6}\,x
  53. φ = 3 2 η , \varphi=\tfrac{3}{2}\,\eta,
  54. 1 3 c ( η ) 2 = f ( η ) \tfrac{1}{3}\,c\,\left(\eta^{\prime}\right)^{2}=f(\eta)\,
  55. f ( η ) = - η 3 + 2 ( c - 1 ) η 2 + r η + s . f(\eta)=-\eta^{3}+2\,\left(c-1\right)\,\eta^{2}+r\,\eta+s.\,
  56. c \scriptstyle\sqrt{c}
  57. Δ = 4 3 m c H . \Delta=\sqrt{\frac{4}{3}\frac{m\,c}{H}}.
  58. Δ : \Delta:
  59. λ = 16 3 m c H K ( m ) . \lambda=\sqrt{\frac{16}{3}\,\frac{m\,c}{H}}\;K(m).
  60. η ( x , t ) = η 2 + H cn 2 ( x - c t Δ m ) , η 2 = H m ( 1 - m - E ( m ) K ( m ) ) , Δ = h 4 3 m h H c g h = λ 2 K ( m ) , λ = h 16 3 m h H c g h K ( m ) , c = g h [ 1 + H m h ( 1 - 1 2 m - 3 2 E ( m ) K ( m ) ) ] and τ = λ c . \begin{aligned}\displaystyle\eta(x,t)&\displaystyle=\eta_{2}+H\,\operatorname{% cn}^{2}\left(\begin{array}[]{c|c}\displaystyle\frac{x-c\,t}{\Delta}&m\end{% array}\right),\\ \displaystyle\eta_{2}&\displaystyle=\frac{H}{m}\,\left(1-m-\frac{E(m)}{K(m)}% \right),\\ \displaystyle\Delta&\displaystyle=h\,\sqrt{\frac{4}{3}\,\frac{m\,h}{H}\,\frac{% c}{\sqrt{g\,h}}}&&\displaystyle=\frac{\lambda}{2\,K(m)},\\ \displaystyle\lambda&\displaystyle=h\,\sqrt{\frac{16}{3}\,\frac{m\,h}{H}\,% \frac{c}{\sqrt{gh}}}\;K(m),\\ \displaystyle c&\displaystyle=\sqrt{gh}\,\left[1+\frac{H}{m\,h}\,\left(1-\frac% {1}{2}\,m-\frac{3}{2}\,\frac{E(m)}{K(m)}\right)\right]&&\displaystyle\,\text{% and}\\ \displaystyle\tau&\displaystyle=\frac{\lambda}{c}.\end{aligned}
  61. τ g h = 9.80 , \tau\,\sqrt{\frac{g}{h}}=9.80,
  62. λ = h 16 3 m h H K ( m ) , \lambda=h\,\sqrt{\frac{16}{3}\frac{m\,h}{H}}\;K(m),
  63. c = g h [ 1 + H m h ( 1 - 1 2 m - 3 2 E ( m ) K ( m ) ) ] c=\sqrt{gh}\,\left[1+\frac{H}{m\,h}\,\left(1-\frac{1}{2}\,m-\frac{3}{2}\,\frac% {E(m)}{K(m)}\right)\right]
  64. τ = λ c , \tau=\frac{\lambda}{c},
  65. m = 0.9832 m=0.9832\,
  66. g h \scriptstyle\sqrt{gh}
  67. c g h = 1.0376 , \frac{c}{\sqrt{g\,h}}=1.0376,
  68. U = H λ 2 h 3 = 62 , U=\frac{H\,\lambda^{2}}{h^{3}}=62,
  69. cn ( z | m ) sech ( z ) - 1 4 ( 1 - m ) [ sinh ( z ) cosh ( z ) - z ] tanh ( z ) sech ( z ) , \operatorname{cn}\left(z|m\right)\approx\operatorname{sech}(z)-\tfrac{1}{4}\,(% 1-m)\,\Bigl[\sinh(z)\;\cosh(z)-z\Bigr]\,\tanh(z)\;\operatorname{sech}(z),
  70. sech ( z ) = 1 cosh ( z ) . \operatorname{sech}(z)=\frac{1}{\cosh(z)}.
  71. cn ( z | m ) sech ( z ) , \operatorname{cn}\left(z|m\right)\to\operatorname{sech}(z),
  72. c = g h ( 1 + 1 2 H h ) c=\sqrt{g\,h}\,\left(1+\frac{1}{2}\,\frac{H}{h}\right)
  73. η 2 = 0. \eta_{2}=0.\,
  74. η ( x , t ) = H sech 2 ( x - c t Δ ) . \eta(x,t)=H\,\operatorname{sech}^{2}\left(\frac{x-c\,t}{\Delta}\right).
  75. Δ = h 4 h 3 H \Delta=h\,\sqrt{\frac{4\,h}{3\,H}}
  76. Δ = h 4 h 3 H c g h \Delta=h\,\sqrt{\frac{4\,h}{3\,H}\,\frac{c}{\sqrt{g\,h}}}
  77. cn ( z | m ) = π m K ( m ) n = 0 sech ( ( 2 n + 1 ) π K ( m ) 2 K ( m ) ) cos ( ( 2 n + 1 ) π z 2 K ( m ) ) . \operatorname{cn}(z|m)=\frac{\pi}{\sqrt{m}\,K(m)}\,\sum_{n=0}^{\infty}\,% \operatorname{sech}\left((2n+1)\,\frac{\pi\,K^{\prime}(m)}{2\,K(m)}\right)\;% \cos\left((2n+1)\,\frac{\pi\,z}{2\,K(m)}\right).
  78. K ( m ) = π 2 [ 1 + ( 1 2 ) 2 m + ( 1 3 2 4 ) 2 m 2 + ( 1 3 5 2 4 6 ) 2 m 3 + ] , E ( m ) = π 2 [ 1 - ( 1 2 ) 2 m 1 - ( 1 3 2 4 ) 2 m 2 3 - ( 1 3 5 2 4 6 ) 2 m 3 5 - ] . \begin{aligned}\displaystyle K(m)&\displaystyle=\frac{\pi}{2}\,\left[1+\left(% \frac{1}{2}\right)^{2}\,m+\left(\frac{1\,\cdot\,3}{2\,\cdot\,4}\right)^{2}\,m^% {2}+\left(\frac{1\,\cdot\,3\,\cdot\,5}{2\,\cdot\,4\,\cdot\,6}\right)^{2}\,m^{3% }+\cdots\right],\\ \displaystyle E(m)&\displaystyle=\frac{\pi}{2}\,\left[1-\left(\frac{1}{2}% \right)^{2}\,\frac{m}{1}-\left(\frac{1\,\cdot\,3}{2\,\cdot\,4}\right)^{2}\,% \frac{m^{2}}{3}-\left(\frac{1\,\cdot\,3\,\cdot\,5}{2\,\cdot\,4\,\cdot\,6}% \right)^{2}\,\frac{m^{3}}{5}-\cdots\right].\end{aligned}
  79. sech ( ( 2 n + 1 ) π K ( m ) 2 K ( m ) ) = 2 q n + 1 2 1 + q 2 n + 1 \operatorname{sech}\left((2n+1)\,\frac{\pi\,K^{\prime}(m)}{2\,K(m)}\right)=2\,% \frac{q^{n+\tfrac{1}{2}}}{1+q^{2n+1}}
  80. q = exp ( - π K ( m ) K ( m ) ) . q=\exp\left(-\pi\,\frac{K^{\prime}(m)}{K(m)}\right).
  81. q = m 16 + 8 ( m 16 ) 2 + 84 ( m 16 ) 3 + 992 ( m 16 ) 4 + . q=\frac{m}{16}+8\,\left(\frac{m}{16}\right)^{2}+84\,\left(\frac{m}{16}\right)^% {3}+992\,\left(\frac{m}{16}\right)^{4}+\cdots.
  82. n = 0 n=0\,
  83. π m K ( m ) sech ( π K ( m ) 2 K ( m ) ) = 1 - 1 16 m - 9 16 m 2 + , \frac{\pi}{\sqrt{m}\,K(m)}\,\operatorname{sech}\,\left(\frac{\pi\,K^{\prime}(m% )}{2\,K(m)}\right)=1-\tfrac{1}{16}\,m-\tfrac{9}{16}\,m^{2}+\cdots,
  84. n = 1 n=1\,
  85. π m K ( m ) sech ( 3 π K ( m ) 2 K ( m ) ) = 1 16 m + 1 32 m 2 + , \frac{\pi}{\sqrt{m}\,K(m)}\,\operatorname{sech}\,\left(\frac{3\,\pi\,K^{\prime% }(m)}{2\,K(m)}\right)=\tfrac{1}{16}\,m+\tfrac{1}{32}\,m^{2}+\cdots,
  86. n = 2 n=2\,
  87. π m K ( m ) sech ( 5 π K ( m ) 2 K ( m ) ) = 1 256 m 2 + . \frac{\pi}{\sqrt{m}\,K(m)}\,\operatorname{sech}\,\left(\frac{5\,\pi\,K^{\prime% }(m)}{2\,K(m)}\right)=\tfrac{1}{256}\,m^{2}+\cdots.
  88. cn ( z | m ) \displaystyle\operatorname{cn}\,(z|m)
  89. α π 2 K ( m ) . \alpha\equiv\frac{\pi}{2\,K(m)}.
  90. cn 2 ( z | m ) = ( 1 2 - 1 16 m - 1 32 m 2 + ) + ( 1 2 - 3 512 m 2 + ) cos 2 α z + ( 1 16 m + 1 32 m 2 + ) cos 4 α z + ( 3 512 m 2 + ) cos 6 α z + . \begin{aligned}\displaystyle\operatorname{cn}^{2}\,(z|m)&\displaystyle=\Bigl(% \tfrac{1}{2}-\tfrac{1}{16}\,m-\tfrac{1}{32}\,m^{2}+\cdots&&\displaystyle\Bigr)% \\ &\displaystyle+\;\Bigl(\tfrac{1}{2}-\tfrac{3}{512}\,m^{2}+\cdots&&% \displaystyle\Bigr)\;\cos\,2\,\alpha\,z\\ &\displaystyle+\;\Bigl(\tfrac{1}{16}\,m+\tfrac{1}{32}\,m^{2}+\cdots&&% \displaystyle\Bigr)\;\cos\,4\,\alpha\,z\\ &\displaystyle+\;\Bigl(\tfrac{3}{512}\,m^{2}+\cdots&&\displaystyle\Bigr)\;\cos% \,6\,\alpha\,z\;+\;\cdots.\end{aligned}
  91. θ π ξ K ( m ) ξ Δ = 2 π ξ λ . \theta\equiv\frac{\pi\,\xi}{K(m)}\,\frac{\xi}{\Delta}=2\,\pi\,\frac{\xi}{% \lambda}.
  92. η ( x , t ) = H ( 1 2 - 3 512 m 2 + ) cos θ + H ( 1 16 m + 1 32 m 2 + ) cos 2 θ + H ( 3 512 m 2 + ) cos 3 θ + . \eta(x,t)=\;H\,\Bigl(\tfrac{1}{2}-\tfrac{3}{512}\,m^{2}+\cdots\Bigr)\,\cos\,% \theta\;+\;H\,\Bigl(\tfrac{1}{16}\,m+\tfrac{1}{32}\,m^{2}+\cdots\Bigr)\,\cos\,% 2\theta\;+\;H\,\Bigl(\tfrac{3}{512}\,m^{2}+\cdots\Bigr)\,\cos\,3\theta\;+\;\cdots.
  93. cn ( z | m ) cos ( z ) + 1 4 m ( z - sin ( z ) cos ( z ) ) sin ( z ) + , \operatorname{cn}(z|m)\approx\cos(z)+\tfrac{1}{4}\,m\,\bigl(z-\sin(z)\,\cos(z)% \bigr)\,\sin(z)+\cdots,
  94. η ( x , t ) = 1 2 H cos θ , \eta(x,t)=\tfrac{1}{2}\,H\,\cos\,\theta,
  95. θ = 2 π ξ λ = 2 π x - c t λ . \theta=2\,\pi\,\frac{\xi}{\lambda}=2\,\pi\ \frac{x-c\,t}{\lambda}.
  96. c = g h [ 1 + H m h ( 1 - 1 2 m - 3 2 E ( m ) K ( m ) ) ] . c=\sqrt{gh}\,\left[1+\frac{H}{m\,h}\,\left(1-\frac{1}{2}\,m-\frac{3}{2}\,\frac% {E(m)}{K(m)}\right)\right].
  97. λ = h 16 3 m h H K ( m ) , \lambda=h\,\sqrt{\frac{16}{3}\,\frac{mh}{H}}\,K(m),
  98. λ = h 16 3 m h H c g h K ( m ) , \lambda=h\,\sqrt{\frac{16}{3}\,\frac{mh}{H}\,\frac{c}{\sqrt{g\,h}}}\,K(m),
  99. κ h = 2 π λ h , \kappa\,h=\frac{2\,\pi}{\lambda}\,h,
  100. c = g h [ 1 + ( κ h ) 2 4 3 π 2 K 2 ( m ) ( 1 - 1 2 m - 3 2 E ( m ) K ( m ) ) ] , c=\sqrt{gh}\,\left[1+(\kappa\,h)^{2}\,\frac{4}{3\,\pi^{2}}\,K^{2}(m)\,\left(1-% \frac{1}{2}\,m-\frac{3}{2}\,\frac{E(m)}{K(m)}\right)\right],
  101. c = g h 1 1 - ( κ h ) 2 4 3 π 2 K 2 ( m ) ( 1 - 1 2 m - 3 2 E ( m ) K ( m ) ) . c=\sqrt{gh}\,\frac{1}{\displaystyle 1-(\kappa\,h)^{2}\,\frac{4}{3\,\pi^{2}}\,K% ^{2}(m)\,\left(1-\frac{1}{2}\,m-\frac{3}{2}\,\frac{E(m)}{K(m)}\right)}.
  102. γ = 4 3 π 2 K 2 ( m ) ( 1 - 1 2 m - 3 2 E ( m ) K ( m ) ) = - 1 6 + 1 64 m 2 + 1 64 m 3 + , \gamma=\frac{4}{3\,\pi^{2}}\,K^{2}(m)\,\left(1-\frac{1}{2}\,m-\,\frac{3}{2}\,% \frac{E(m)}{K(m)}\right)=-\tfrac{1}{6}+\tfrac{1}{64}\,m^{2}+\tfrac{1}{64}\,m^{% 3}+\cdots,
  103. 1 / 6 {1}/{6}
  104. c = [ 1 - 1 6 ( κ h ) 2 ] g h , c=\Bigl[1-\tfrac{1}{6}\,\left(\kappa h\right)^{2}\Bigr]\,\sqrt{g\,h},
  105. c = 1 1 + 1 6 ( κ h ) 2 g h , c=\frac{1}{1+\tfrac{1}{6}\,\left(\kappa h\right)^{2}}\,\sqrt{g\,h},
  106. 6 \scriptstyle\sqrt{6}
  107. Q = 0 ζ ( ξ ) z Ψ d z , R = p ρ + 1 2 [ ( ξ Ψ ) 2 + ( z Ψ ) 2 ] + g z and S = 0 ζ ( ξ ) [ p ρ + ( z Ψ ) 2 ] d z . \begin{aligned}\displaystyle Q&\displaystyle=\int_{0}^{\zeta(\xi)}\partial_{z}% \Psi\;\,\text{d}z,\\ \displaystyle R&\displaystyle=\frac{p}{\rho}+\tfrac{1}{2}\,\Bigl[\left(% \partial_{\xi}\Psi\right)^{2}+\left(\partial_{z}\Psi\right)^{2}\Bigr]+g\,z% \qquad\,\text{and}\\ \displaystyle S&\displaystyle=\int_{0}^{\zeta(\xi)}\left[\frac{p}{\rho}+\left(% \partial_{z}\Psi\right)^{2}\right]\;\,\text{d}z.\end{aligned}
  108. Q 0 = v h , Q_{0}=v\,h,
  109. R 0 = 1 2 v 2 + g h R_{0}=\tfrac{1}{2}\,v^{2}+g\,h
  110. S 0 = v 2 h + 1 2 g h 2 . S_{0}=v^{2}\,h+\tfrac{1}{2}\,g\,h^{2}.
  111. h c = Q 2 g 3 , h_{c}=\sqrt[3]{\frac{Q^{2}}{g}},
  112. 1 3 ( ζ ~ ) 2 - ζ 3 + 2 R ~ ζ ~ 2 - 2 S ~ ζ ~ + 1 , \tfrac{1}{3}\,\left(\tilde{\zeta}^{\prime}\right)^{2}\approx-\zeta^{3}+2\,% \tilde{R}\,\tilde{\zeta}^{2}-2\,\tilde{S}\,\tilde{\zeta}+1,
  113. ζ ~ = ζ h c , \tilde{\zeta}=\frac{\zeta}{h_{c}},
  114. ξ ~ = ξ h c , \tilde{\xi}=\frac{\xi}{h_{c}},
  115. R ~ = R g h c , \tilde{R}=\frac{R}{g\,h_{c}},
  116. S ~ = S g h c 2 . \tilde{S}=\frac{S}{g\,h_{c}^{2}}.
  117. S = R ζ - 1 2 g ζ 2 + 0 ζ 1 2 [ ( z Ψ ) 2 - ( ξ Ψ ) 2 ] d z . S=R\,\zeta-\tfrac{1}{2}\,g\,\zeta^{2}+\int_{0}^{\zeta}\tfrac{1}{2}\left[\left(% \partial_{z}\Psi\right)^{2}-\left(\partial_{\xi}\Psi\right)^{2}\right]\;\,% \text{d}z.
  118. Ψ = z u b ( ξ ) - z 3 3 ! u b ′′ ( ξ ) + z 5 5 ! u b iv ( ξ ) + , \Psi=z\,u_{b}(\xi)-\frac{z^{3}}{3!}\,u_{b}^{\prime\prime}(\xi)+\frac{z^{5}}{5!% }\,u_{b}\text{iv}(\xi)+\cdots,
  119. S = R ζ - 1 2 g ζ 2 + 1 2 ζ u b 2 - 1 6 ζ 3 u b u b ′′ - 1 6 ζ 3 ( u b ) 2 + . S=R\,\zeta-\tfrac{1}{2}\,g\,\zeta^{2}+\tfrac{1}{2}\,\zeta\,u_{b}^{2}-\tfrac{1}% {6}\,\zeta^{3}\,u_{b}\,u_{b}^{\prime\prime}-\tfrac{1}{6}\,\zeta^{3}\,\left(u_{% b}^{\prime}\right)^{2}+\cdots.
  120. Q = ζ u b ( ξ ) - 1 6 ζ 3 u b ′′ ξ + . Q=\zeta\,u_{b}(\xi)-\tfrac{1}{6}\,\zeta^{3}\,u_{b}^{\prime\prime}{\xi}+\cdots.
  121. u b = Q ζ + 1 6 ζ 2 u b ′′ + . u_{b}=\frac{Q}{\zeta}+\tfrac{1}{6}\,\zeta^{2}\,u_{b}^{\prime\prime}+\cdots.
  122. u b 2 \displaystyle u_{b}^{2}
  123. S R ζ - 1 2 g ζ 2 + 1 2 Q 2 ζ - 1 6 Q 2 ζ ( ζ ) 2 . S\approx R\,\zeta-\tfrac{1}{2}\,g\,\zeta^{2}+\tfrac{1}{2}\,\frac{Q^{2}}{\zeta}% -\tfrac{1}{6}\,\frac{Q^{2}}{\zeta}\,\left(\zeta^{\prime}\right)^{2}.
  124. E pot = 1 λ 0 λ 1 2 ρ g η 2 ( x , t ) d x E\text{pot}=\frac{1}{\lambda}\,\int_{0}^{\lambda}\tfrac{1}{2}\,\rho\,g\,\eta^{% 2}(x,t)\;\,\text{d}x
  125. t ( 1 2 η 2 ) + x { 1 2 g h η 2 + 1 2 g h η 3 + 1 12 h 2 g h [ x 2 ( η 2 ) - 3 ( x η ) 2 ] } = 0 , \partial_{t}\left(\tfrac{1}{2}\,\eta^{2}\right)+\partial_{x}\left\{\tfrac{1}{2% }\,\sqrt{g\,h}\,\eta^{2}+\tfrac{1}{2}\,\sqrt{\frac{g}{h}}\,\eta^{3}+\tfrac{1}{% 12}\,h^{2}\sqrt{g\,h}\,\left[\partial_{x}^{2}\left(\eta^{2}\right)-3\left(% \partial_{x}\eta\right)^{2}\right]\right\}=0,
  126. 1 / 6 {1}/{6}
  127. 1 λ 0 λ η 2 d x = 1 λ 0 λ { η 2 + H cn 2 ( ξ Δ m ) } 2 d ξ = H 2 λ 0 λ cn 4 ( ξ Δ m ) d ξ - η 2 2 = Δ H 2 λ 0 π cos 4 ψ d ξ d ψ d ψ - η 2 2 = H 2 2 K ( m ) 0 π cos 4 ψ 1 - m sin 2 ψ d ψ - η 2 2 = 1 3 H 2 m 2 [ ( 2 - 5 m + 3 m 2 ) + ( 4 m - 2 ) E ( m ) K ( m ) ] - H 2 m 2 ( 1 - m - E ( m ) K ( m ) ) 2 \begin{aligned}\displaystyle\frac{1}{\lambda}\,\int_{0}^{\lambda}\eta^{2}\;\,% \text{d}x&\displaystyle=\frac{1}{\lambda}\int_{0}^{\lambda}\left\{\eta_{2}+H\,% \operatorname{cn}^{2}\left(\begin{array}[]{c|c}\displaystyle\frac{\xi}{\Delta}% &m\end{array}\right)\right\}^{2}\;\,\text{d}\xi=\frac{H^{2}}{\lambda}\int_{0}^% {\lambda}\operatorname{cn}^{4}\left(\begin{array}[]{c|c}\displaystyle\frac{\xi% }{\Delta}&m\end{array}\right)\;\,\text{d}\xi-\eta_{2}^{2}\\ &\displaystyle=\frac{\Delta\,H^{2}}{\lambda}\int_{0}^{\pi}\cos^{4}\,\psi\,% \frac{\,\text{d}\xi}{\,\text{d}\psi}\;\,\text{d}\psi-\eta_{2}^{2}=\frac{H^{2}}% {2\,K(m)}\int_{0}^{\pi}\frac{\cos^{4}\,\psi}{\sqrt{1-m\,\sin^{2}\,\psi}}\;\,% \text{d}\psi-\eta_{2}^{2}\\ &\displaystyle=\frac{1}{3}\,\frac{H^{2}}{m^{2}}\,\left[\left(2-5\,m+3\,m^{2}% \right)+\left(4\,m-2\right)\,\frac{E(m)}{K(m)}\right]-\frac{H^{2}}{m^{2}}\,% \left(1-m-\frac{E(m)}{K(m)}\right)^{2}\end{aligned}
  128. E pot = 1 2 ρ g H 2 [ - 1 3 m + 2 3 m ( 1 + 1 m ) ( 1 - E ( m ) K ( m ) ) - 1 m 2 ( 1 - E ( m ) K ( m ) ) 2 ] . E\text{pot}=\tfrac{1}{2}\,\rho\,g\,H^{2}\,\left[-\frac{1}{3\,m}+\frac{2}{3\,m}% \,\left(1+\frac{1}{m}\right)\left(1-\frac{E(m)}{K(m)}\right)-\frac{1}{m^{2}}\,% \left(1-\frac{E(m)}{K(m)}\right)^{2}\right].
  129. 1 / 16 {1}/{16}

Coanda_effect_mixer.html

  1. μ \mu

CoBoosting.html

  1. f 1 ( x 1 ) f_{1}(x_{1})
  2. f 2 ( x 2 ) f_{2}(x_{2})
  3. x = ( x 1 , x 2 ) x=(x_{1},x_{2})
  4. f 1 ( x 1 ) = f 2 ( x 2 ) = f ( x ) f_{1}(x_{1})=f_{2}(x_{2})=f(x)
  5. { ( x 1 , i , x 2 , i ) } i = 1 n \{(x_{1,i},x_{2,i})\}_{i=1}^{n}
  6. { y i } i = 1 m \{y_{i}\}_{i=1}^{m}
  7. i , j : g j 0 ( s y m b o l x i ) = 0 \forall i,j:g_{j}^{0}(symbol{x_{i}})=0
  8. t = 1 , , T t=1,...,T
  9. j = 1 , 2 j=1,2
  10. y i ^ = { y i 1 i m s i g n ( g 3 - j t - 1 ( s y m b o l x 3 - j , i ) ) m < i n \hat{y_{i}}=\left\{\begin{array}[]{ll}y_{i}1\leq i\leq m\\ sign(g_{3-j}^{t-1}(symbol{x_{3-j,i}}))m<i\leq n\end{array}\right.
  11. D t j ( i ) = 1 Z t j e - y i ^ g j t - 1 ( s y m b o l x j , i ) D_{t}^{j}(i)=\frac{1}{Z_{t}^{j}}e^{-\hat{y_{i}}g_{j}^{t-1}(symbol{x_{j,i}})}
  12. Z t j = i = 1 n e - y i ^ g j t - 1 ( s y m b o l x j , i ) Z_{t}^{j}=\sum_{i=1}^{n}e^{-\hat{y_{i}}g_{j}^{t-1}(symbol{x_{j,i}})}
  13. h t j h_{t}^{j}
  14. α t \alpha_{t}
  15. i : g j t ( s y m b o l x j , i ) = g j t - 1 ( s y m b o l x j , i ) + α t h t j ( s y m b o l x j , i ) \forall i:g_{j}^{t}(symbol{x_{j,i}})=g_{j}^{t-1}(symbol{x_{j,i}})+\alpha_{t}h_% {t}^{j}(symbol{x_{j,i}})
  16. f ( s y m b o l x ) = s i g n ( j = 1 2 g j T ( s y m b o l x j ) ) f(symbol{x})=sign\left(\sum_{j=1}^{2}g_{j}^{T}(symbol{x_{j}})\right)
  17. 1 m i = 1 m e ( - y i ( t = 1 T α t h t ( s y m b o l x i ) ) ) = t Z t \frac{1}{m}\sum_{i=1}^{m}e^{\left(-y_{i}\left(\sum_{t=1}^{T}\alpha_{t}h_{t}(% symbol{x_{i}})\right)\right)}=\prod_{t}Z_{t}
  18. Z t Z_{t}
  19. D t + 1 D_{t+1}
  20. Z t Z_{t}
  21. D t ( i ) D_{t}(i)
  22. Z t = i : x t x i D t ( i ) + i : x t x i D t ( i ) e - y i α i h t ( s y m b o l x i ) Z_{t}=\sum_{i:x_{t}\notin x_{i}}D_{t}(i)+\sum_{i:x_{t}\in x_{i}}D_{t}(i)e^{-y_% {i}\alpha_{i}h_{t}(symbol{x_{i}})}
  23. x t x_{t}
  24. W 0 = i : h t ( x i ) = 0 D t ( i ) W_{0}=\sum_{i:h_{t}(x_{i})=0}D_{t}(i)
  25. W + = i : h t ( x i ) = y i D t ( i ) W_{+}=\sum_{i:h_{t}(x_{i})=y_{i}}D_{t}(i)
  26. W - = i : h t ( x i ) = - y i D t ( i ) W_{-}=\sum_{i:h_{t}(x_{i})=-y_{i}}D_{t}(i)
  27. Z t Z_{t}
  28. α t \alpha_{t}
  29. α t = 1 2 ln ( W + W - ) \alpha_{t}=\frac{1}{2}\ln\left(\frac{W_{+}}{W_{-}}\right)
  30. W - W_{-}
  31. Z t Z_{t}
  32. Z t = W 0 + 2 W + W - Z_{t}=W_{0}+2\sqrt{W_{+}W_{-}}
  33. 1... m 1...m
  34. m 1 n m_{1}...n
  35. x i = ( x 1 , i , x 2 , i ) x_{i}=(x_{1,i},x_{2,i})
  36. Z C O Z_{CO}
  37. Z t Z_{t}
  38. Z C O = i = 1 m e - y i g 1 ( s y m b o l x 1 , i ) + i = 1 m e - y i g 2 ( s y m b o l x 2 , i ) + i = m + 1 n e - f 2 ( s y m b o l x 2 , i ) g 1 ( s y m b o l x 1 , i ) + i = m + 1 n e - f 1 ( s y m b o l x 1 , i ) g 2 ( s y m b o l x 2 , i ) Z_{CO}=\sum_{i=1}^{m}e^{-y_{i}g_{1}(symbol{x_{1,i}})}+\sum_{i=1}^{m}e^{-y_{i}g% _{2}(symbol{x_{2,i}})}+\sum_{i=m+1}^{n}e^{-f_{2}(symbol{x_{2,i}})g_{1}(symbol{% x_{1,i}})}+\sum_{i=m+1}^{n}e^{-f_{1}(symbol{x_{1,i}})g_{2}(symbol{x_{2,i}})}
  39. g j g_{j}
  40. j t h j^{th}
  41. f j f_{j}
  42. g j g_{j}
  43. g j t - 1 g_{j}^{t-1}
  44. j t h j^{th}
  45. t - 1 t-1
  46. y i ^ = { y i 1 i m s i g n ( g 3 - j t - 1 ( s y m b o l x 3 - j , i ) ) m < i n \hat{y_{i}}=\left\{\begin{array}[]{ll}y_{i}1\leq i\leq m\\ sign(g_{3-j}^{t-1}(symbol{x_{3-j,i}}))m<i\leq n\end{array}\right.
  47. 3 - j 3-j
  48. Z C O Z_{CO}
  49. Z C O = Z C O 1 + Z C O 2 Z_{CO}=Z_{CO}^{1}+Z_{CO}^{2}
  50. Z C O j = i = 1 n e - y i ^ ( g j t - 1 ( s y m b o l x i ) + α t j g t j ( s y m b o l x j , i ) ) Z_{CO}^{j}=\sum_{i=1}^{n}e^{-\hat{y_{i}}(g_{j}^{t-1}(symbol{x_{i}})+\alpha_{t}% ^{j}g_{t}^{j}(symbol{x_{j,i}}))}
  51. j j
  52. t t
  53. D t j ( i ) = 1 Z t j e - y i ^ g j t - 1 ( s y m b o l x j , i ) D_{t}^{j}(i)=\frac{1}{Z_{t}^{j}}e^{-\hat{y_{i}}g_{j}^{t-1}(symbol{x_{j,i}})}
  54. Z C O j Z_{CO}^{j}
  55. Z C O j = i = 1 n D t j e - y i ^ α t j g t j ( s y m b o l x j , i ) Z_{CO}^{j}=\sum_{i=1}^{n}D_{t}^{j}e^{-\hat{y_{i}}\alpha_{t}^{j}g_{t}^{j}(% symbol{x_{j,i}})}
  56. α t j \alpha_{t}^{j}
  57. y i ^ \hat{y_{i}}
  58. D t j D_{t}^{j}
  59. Z C O 1 Z_{CO}^{1}
  60. Z C O 2 Z_{CO}^{2}
  61. Z C O Z_{CO}

Codex_Palatinus.html

  1. 𝔓 \mathfrak{P}
  2. 𝔓 \mathfrak{P}

Codex_Veronensis.html

  1. 𝔓 \mathfrak{P}
  2. 𝔓 \mathfrak{P}

Coding_theory_approaches_to_nucleic_acid_design.html

  1. 𝒬 = { A , T , C , G } \mathcal{Q}=\{\mathit{A},\mathit{T},\mathit{C},\mathit{G}\}
  2. A ¯ = T , T ¯ = A , C ¯ = G , G ¯ = C \bar{A}=T,\bar{T}=A,\bar{C}=G,\bar{G}=C
  3. q = q 1 q 2 . q n \mathit{q}=\mathit{q}_{1}\mathit{q}_{2}....\mathit{q}_{n}
  4. n \mathit{n}
  5. 𝒬 \mathcal{Q}
  6. 1 i j n 1\leqslant i\leqslant j\leqslant n
  7. q [ i , j ] \mathit{q}_{[i,j]}
  8. q i q i + 1 q j \mathit{q}_{i}\mathit{q}_{i+1}...\mathit{q}_{j}
  9. q \mathit{q}
  10. q R \mathit{q}^{R}
  11. q R C = q ¯ n q ¯ n - 1 q ¯ 1 \mathit{q}^{RC}=\mathit{\bar{q}_{n}}\mathit{\bar{q}_{n-1}}...\mathit{\bar{q}_{% 1}}
  12. q ¯ i \mathit{\bar{q}_{i}}
  13. q i \mathit{q}_{i}
  14. n \mathit{n}
  15. p \mathit{p}
  16. q \mathit{q}
  17. 𝒬 \mathcal{Q}
  18. d H ( p , q ) \mathit{d}_{H}(\mathit{p},\mathit{q})
  19. i \mathit{i}
  20. p i q i \mathit{p}_{i}\neq\mathit{q}_{i}
  21. d H R ( p , q ) = d H ( p , q R ) \mathit{d_{H}}^{R}(\mathit{p},\mathit{q})=\mathit{d}_{H}(\mathit{p},\mathit{q}% ^{R})
  22. d H R C ( p , q ) = d H ( p , q R C ) \mathit{d}_{H}^{RC}(\mathit{p},\mathit{q})=\mathit{d}_{H}(\mathit{p},\mathit{q% }^{RC})
  23. R C RC
  24. w G C ( q ) \mathit{w}_{GC}(\mathit{q})
  25. q = q 1 q 2 . q n \mathit{q}=\mathit{q}_{1}\mathit{q}_{2}....\mathit{q}_{n}
  26. i \mathit{i}
  27. q i { G , C } \mathit{q}_{i}\in\{G,C\}
  28. w \mathit{w}
  29. H H ( n , m \mathit{H}\equiv\mathit{H}(n,\mathbb{C}_{m}
  30. n \mathit{n}
  31. × \times
  32. n \mathit{n}
  33. m \mathit{m}
  34. m \mathbb{C}_{m}
  35. { e - 2 π 𝑖𝑙 / m \{e^{-2\pi\mathit{i}\mathit{l}}/\mathit{m}
  36. l \mathit{l}
  37. m - 1 } \mathit{m}-1\}
  38. 𝐻𝐻 * \mathit{H}\mathit{H}^{*}
  39. 𝑛𝐼 \mathit{n}\mathit{I}
  40. I \mathit{I}
  41. n \mathit{n}
  42. m = p \mathit{m}=\mathit{p}
  43. p \mathit{p}
  44. H ( n , p ) \mathit{H}(\mathit{n},\mathbb{C}_{p})
  45. p | n \mathit{p}|\mathit{n}
  46. E \mathit{E}
  47. ( n , p ) (\mathit{n},\mathbb{Z}_{p})
  48. H ( n , p ) \mathit{H}(\mathit{n},\mathbb{C}_{p})
  49. n × n \mathit{n}\times\mathit{n}
  50. Z p = { 0 , 1 , 2 , , p - 1 } \mathit{Z}_{p}=\{0,1,2,...,\mathit{p}-1\}
  51. ( e - 2 π i l ) (e^{-2\pi\mathit{i}l})
  52. H ( n , p ) \mathit{H}(\mathit{n},\mathbb{C}_{p})
  53. l \mathit{l}
  54. 𝐺𝐹 ( p ) \mathit{GF(p)}
  55. E \mathit{E}
  56. 𝐺𝐹 ( p ) \mathit{GF(p)}
  57. H \mathit{H}
  58. ( n - 1 ) × ( n - 1 ) (\mathit{n}-1)\times(\mathit{n}-1)
  59. H \mathit{H}
  60. H \mathit{H}
  61. E \mathit{E}
  62. p \mathbb{Z}_{p}
  63. n / p \mathit{n}/\mathit{p}
  64. n ( p - 1 ) / p \mathit{n}(\mathit{p}-1)/\mathit{p}
  65. C p = 1 , x , x 2 , , x p - I \mathit{C_{p}}={1,x,x2,...,xp-I}
  66. x \mathit{x}
  67. x = exp ( 2 π j / p ) x=\exp(2\pi j/p)
  68. p p
  69. p p
  70. 2 2
  71. A = ( x a i ) \mathit{A}=(x^{a_{i}})
  72. B = ( x b i ) \mathit{B}=(x^{b_{i}})
  73. C p \mathit{C_{p}}
  74. N = p t \mathit{N}=pt
  75. t \mathit{t}
  76. Q = a i - b i mod p : i = 1 , 2 , , N \mathit{Q}={\mathit{a_{i}}-\mathit{b_{i}}\mod\mathit{p}:i=1,2,...,N}
  77. n q \mathit{n_{q}}
  78. q \mathit{q}
  79. 𝐺𝐹 ( p ) \mathit{GF(p)}
  80. Q \mathit{Q}
  81. Q \mathit{Q}
  82. q \mathit{q}
  83. 𝐺𝐹 ( p ) \mathit{GF(p)}
  84. Q \mathit{Q}
  85. t \mathit{t}
  86. n q = t , q = 0 , 1 , , p - 1 \mathit{n_{q}}=t,q=0,1,...,p-1
  87. C p \mathit{C_{p}}
  88. p \mathit{p}
  89. A , B \mathit{A},\mathit{B}
  90. N = p t \mathit{N}=pt
  91. C p \mathit{C_{p}}
  92. Q \mathit{Q}
  93. Q \mathit{Q}
  94. mod p \mod\mathit{p}
  95. A , B \mathit{A},\mathit{B}
  96. V \mathit{V}
  97. N \mathit{N}
  98. 𝐺𝐹 ( p ) \mathit{GF(p)}
  99. p \mathit{p}
  100. V \mathit{V}
  101. a ( V ) \mathit{a(V)}
  102. N \mathit{N}
  103. N \mathit{N}
  104. N \mathit{N}
  105. V \mathit{V}
  106. V * \mathit{V^{*}}
  107. a ( V * ) \mathit{a(V^{*})}
  108. a ( V ) \mathit{a(V)}
  109. E = E c \mathit{E}=\mathit{E_{c}}
  110. 𝐺𝐹 ( p ) \mathit{GF(p)}
  111. N = p n - 1 \mathit{N=p^{n}-1}
  112. E \mathit{E}
  113. K K
  114. g ( x ) \mathit{g(x)}
  115. 𝐺𝐹 ( p ) \mathit{GF(p)}
  116. x N - 1 \mathit{x^{N}-1}
  117. K K
  118. N N
  119. g ( x ) \mathit{g(x)}
  120. N - n \mathit{N-n}
  121. g ( x ) \mathit{g(x)}
  122. N \mathit{N}
  123. E \mathit{E}
  124. N N
  125. x N - 1 = g ( x ) h ( x ) \mathit{x^{N}-1}=\mathit{g(x)h(x)}
  126. h ( x ) \mathit{h(x)}
  127. [ 0 , g ( x ) ] [0,\mathit{g(x)}]
  128. 𝐺𝐹 ( p ) \mathit{GF(p)}
  129. 0 , 1 , , p - 1 0,1,...,p-1
  130. h ( x ) \mathit{h(x)}
  131. 𝐺𝐹 ( p ) \mathit{GF(p)}
  132. n \mathit{n}
  133. g ( x ) \mathit{g(x)}
  134. N - n \mathit{N-n}
  135. g ( x ) h ( x ) = x N - 1 \mathit{g(x)h(x)}=\mathit{x^{N}}-1
  136. [ 0 , g ( x ) ] [0,\mathit{g(x)}]
  137. 𝐺𝐹 ( p ) \mathit{GF(p)}
  138. h ( x ) | x N - 1 \mathit{h(x)}|\mathit{x^{N}}-1
  139. g ( x ) \mathit{g(x)}
  140. mod \mod
  141. x N - 1 \mathit{x^{N}}-1
  142. K \mathit{K}
  143. N \mathit{N}
  144. H ( p , 𝑝𝑛 ) \mathit{H(p,pn)}
  145. H ( 3 , 9 ) \mathit{H(3,9)}
  146. h ( x ) = x 2 + x + 2 \mathit{h(x)}=\mathit{x^{2}}+\mathit{x}+2
  147. g ( x ) = x 6 + 2 x 5 + 2 x 4 + 2 x 2 + x + 1 \mathit{g(x)}=\mathit{x^{6}}+2\mathit{x^{5}}+2\mathit{x^{4}}+2\mathit{x^{2}}+% \mathit{x}+1
  148. g \mathit{g}
  149. 0 , 1 , 6 {0,1,6}
  150. mod 8 \mod 8
  151. p \mathit{p}
  152. N + 1 = 𝑝𝑛 \mathit{N}+1=\mathit{pn}
  153. g ( x ) \mathit{g}(\mathit{x})
  154. N - n \mathit{N}-\mathit{n}
  155. C = [ c 0 , c 1 , , c N - 1 ] \mathit{C}=[\mathit{c}_{0},\mathit{c}_{1},...,\mathit{c}_{N-1}]
  156. 𝐺𝐹 ( p ) \mathit{GF}(\mathit{p})
  157. C = [ c 0 , c 1 , , c N - 1 ] \mathit{C}=[\mathit{c}_{0},\mathit{c}_{1},...,\mathit{c}_{N-1}]
  158. g ( x ) h ( x ) = 𝑥𝑁 - 1 \mathit{g(x)h(x)=xN-1}
  159. h ( x ) \mathit{h(x)}
  160. n \mathit{n}
  161. K ¯ \mathit{\bar{K}}
  162. N \mathit{N}
  163. K = [ 0 , K ¯ ] \mathit{K}=[0,\mathit{\bar{K}}]
  164. H ( p , p n ) = 𝑥𝐾 \mathit{H(p,p_{n})=xK}
  165. x = e 2 π i / p \mathit{x=e^{2\pi i/p}}
  166. H \mathit{H}
  167. g ( x ) \mathit{g(x)}
  168. x N - 1 \mathit{x^{N-1}}
  169. N - n \mathit{N-n}
  170. E c \mathit{E_{c}}
  171. H \mathit{H}
  172. C \mathit{C}
  173. 𝐺𝐹 ( p ) \mathit{GF(p)}
  174. C \mathit{C}
  175. E c \mathit{E_{c}}
  176. E c \mathit{E_{c}}
  177. C \mathit{C}
  178. E c \mathit{E_{c}}
  179. mod p \mod p
  180. K \mathit{K}
  181. 𝑥𝐾 \mathit{xK}
  182. H \mathit{H}
  183. E \mathit{E}
  184. N = p k - 1 N=\mathit{p}^{k}-1
  185. p \mathbb{Z}_{p}
  186. E \mathit{E}
  187. ( N + 1 ) / p = p k - 1 (N+1)/p=\mathit{p}^{k-1}
  188. ( N + 1 ) ( p - 1 ) / p = ( p - 1 ) p k - 1 (N+1)(p-1)/p=\mathit{(p-1)}\mathit{p}^{k-1}
  189. N \mathit{N}
  190. E \mathit{E}
  191. N \mathit{N}
  192. N \mathit{N}
  193. p \mathbb{Z}_{p}
  194. p \mathbb{Z}_{p}
  195. ( p - 1 ) p k - 1 \mathit{(p-1)}\mathit{p}^{k-1}
  196. N = p k - 1 \mathit{N}=\mathit{p}^{\mathit{k}}-1
  197. p \mathit{p}
  198. k + \mathit{k}\in\mathbb{Z}^{+}
  199. g ( x ) = c 0 + c 1 x + c 2 x 2 + + c N - k x N - k \mathit{g}(\mathit{x})=\mathit{c}_{0}+\mathit{c}_{1}\mathit{x}+\mathit{c}_{2}% \mathit{x}^{2}+...+\mathit{c}_{N-k}\mathit{x}^{N-k}
  200. p \mathbb{Z}_{p}
  201. g ( x ) h ( x ) = x N - 1 \mathit{g}(\mathit{x})\mathit{h}(\mathit{x})=\mathit{x}^{N}-1
  202. p \mathbb{Z}_{p}
  203. h ( x ) p [ x ] \mathit{h}(\mathit{x})\in\mathbb{Z}_{p}[\mathit{x}]
  204. ( c 0 , c 1 , . , c N - k , c N - k + 1 , , c N - 1 ) \mathit{(}{c}_{0},\mathit{c}_{1},....,\mathit{c}_{N-k},\mathit{c}_{N-k+1},...,% \mathit{c}_{N-1})
  205. c i = 0 \mathit{c}_{i}=0
  206. N \mathit{N}
  207. g = ( c 0 , c 1 , , c N - 1 ) \mathit{g}=(\mathit{c}_{0},\mathit{c}_{1},...,\mathit{c}_{N-1})
  208. p \mathbb{Z}_{p}
  209. p \mathbb{Z}_{p}
  210. 𝒬 = { A , T , C , G } \mathcal{Q}=\{\mathit{A},\mathit{T},\mathit{C},\mathit{G}\}
  211. 3 k - 1 \mathit{3}^{k}-1
  212. 3 \mathbb{Z}_{3}
  213. 0
  214. A \mathit{A}
  215. 1 1
  216. T \mathit{T}
  217. 2 2
  218. G \mathit{G}
  219. 𝒟 \mathcal{D}
  220. 3 k - 1 \mathit{3}^{k}-1
  221. 3 k - 1 \mathit{3}^{k-1}
  222. d H = 2.3 k - 1 \mathit{d_{H}}=2.\mathit{3}^{k-1}
  223. G ¯ = C \mathit{\bar{G}}=\mathit{C}
  224. 𝒟 \mathcal{D}
  225. C \mathit{C}
  226. d H R C ( 𝒟 ) 3 k - 1 \mathit{d}_{H}^{RC}(\mathcal{D})\geq 3^{k-1}
  227. k + \mathit{k}\in\mathbb{Z}^{+}
  228. 𝔻 \mathbb{D}
  229. 3 k - 1 \mathit{3}^{k}-1
  230. 3 k - 1 \mathit{3}^{k}-1
  231. 3 k - 1 \mathit{3}^{k-1}
  232. d H R C ( 𝔻 ) 3 k - 1 \mathit{d}_{H}^{RC}(\mathbb{D})\geqslant\mathit{3}^{k-1}
  233. g \mathit{g}
  234. H ( p , p n ) \mathit{H(p,p^{n})}
  235. N + 1 = p n \mathit{N}+1=\mathit{p^{n}}
  236. n = 3 \mathit{n}=3
  237. g ( 1 ) \mathit{g^{(1)}}
  238. ( 22201221202001110211210200 ) (22201221202001110211210200)
  239. g ( 2 ) \mathit{g^{(2)}}
  240. ( 20212210222001012112011100 ) (20212210222001012112011100)
  241. g ( x ) = a 0 + a 1 x + . + a n x n \mathit{g(x)}=a_{0}+a_{1}x+....+a_{n}x^{n}
  242. 0 , 1 , 2 {0,1,2}
  243. A , T , G {A,T,G}
  244. g ( 1 ) \mathit{g^{(1)}}
  245. 0 - A ; 1 - T ; 2 - G 0-A;1-T;2-G
  246. g ( 2 ) \mathit{g^{(2)}}
  247. g ( 1 ) \mathit{g^{(1)}}
  248. 0 - G ; 1 - T ; 2 - A 0-G;1-T;2-A
  249. 𝒬 \mathcal{Q}
  250. A \mathit{A}
  251. 00 00
  252. T \mathit{T}
  253. 01 01
  254. C \mathit{C}
  255. 10 10
  256. G \mathit{G}
  257. 11 11
  258. q \mathit{q}
  259. b ( q ) \mathit{b(q)}
  260. q \mathit{q}
  261. q \mathit{q}
  262. b ( q ) \mathit{b(q)}
  263. b 0 b 1 b 2 b 2 n - 1 \mathit{b}_{0}\mathit{b}_{1}\mathit{b}_{2}...\mathit{b}_{2n-1}
  264. e ( q ) \mathit{e(q)}
  265. b 0 b 2 b 2 n - 2 \mathit{b}_{0}\mathit{b}_{2}...\mathit{b}_{2n-2}
  266. b ( q ) \mathit{b(q)}
  267. o ( q ) \mathit{o(q)}
  268. b 1 b 3 b 5 b 2 n - 1 \mathit{b}_{1}\mathit{b}_{3}\mathit{b}_{5}...\mathit{b}_{2n-1}
  269. b ( q ) \mathit{b(q)}
  270. q \mathit{q}
  271. A C G T C C ACGTCC
  272. b ( q ) \mathit{b(q)}
  273. 001011011010 001011011010
  274. e ( q ) \mathit{e(q)}
  275. 011011 011011
  276. o ( q ) \mathit{o(q)}
  277. 001100 001100
  278. ( 𝒞 ) = { e ( x ) : x 𝒞 } \mathcal{E}(\mathcal{C})=\{e(x):x\in\mathcal{C}\}
  279. 𝒪 ( 𝒞 ) = { o ( x ) : x 𝒞 } \mathcal{O}(\mathcal{C})=\{o(x):x\in\mathcal{C}\}
  280. q \mathit{q}
  281. e ( q ) \mathit{e(q)}
  282. 𝒞 \mathcal{C}
  283. ( 𝒞 ) \mathcal{E}(\mathcal{C})
  284. \mathcal{B}
  285. M M
  286. n \mathit{n}
  287. d 𝑚𝑖𝑛 \mathit{d_{min}}
  288. c \mathit{c}\in\mathcal{B}
  289. c ¯ \mathit{\bar{c}}\in\mathcal{B}
  290. w > 0 \mathit{w}>0
  291. w = { u : w H ( u ) = w } \mathcal{B_{\mathit{w}}}=\{u\in\mathcal{B}:\mathit{w_{H}}(u)=\mathit{w}\}
  292. w H ( . ) \mathit{w_{H}(.)}
  293. w > 0 \mathit{w}>0
  294. n 2 w + d 𝑚𝑖𝑛 / 2 \mathit{n}\geq\mathit{2w}+\lceil\mathit{d_{min}}/2\rceil
  295. 𝒞 w \mathcal{C}_{w}
  296. = { a b ¯ : a , b w } \mathcal{E}=\{a\bar{b}:a,b\in\mathcal{B}_{w}\}
  297. 𝒪 = { a b R C : a , b , a \mathcal{O}=\{ab^{RC}:a,b\in\mathcal{B},a
  298. } \}
  299. a a
  300. 𝒪 \mathcal{O}
  301. a b R C 𝒪 ab^{RC}\in\mathcal{O}
  302. b a R C 𝒪 ba^{RC}\notin\mathcal{O}
  303. 𝒪 \mathcal{O}
  304. w \mathcal{E}_{w}
  305. | w | 2 {\left|\mathcal{B}_{w}\right|}^{2}
  306. 2 n 2n
  307. n n
  308. d H ( w d 𝑚𝑖𝑛 ) \mathit{d_{H}}(\mathcal{E}_{w}\geq\mathit{d_{min}})
  309. d H R ( w d 𝑚𝑖𝑛 ) \mathit{d_{H}}^{R}(\mathcal{E}_{w}\geq\mathit{d_{min}})
  310. w \mathcal{B}_{w}
  311. \mathcal{B}
  312. d H ( a b ¯ , d R C c R ) = d H ( a , d R C ) + d H ( b ¯ , c R ) = d H ( a , d R C ) + d H ( c , b R C ) \mathit{d_{H}}(a\bar{b},d^{RC}c^{R})=\mathit{d_{H}}(a,d^{RC})+\mathit{d_{H}}(% \bar{b},c^{R})=\mathit{d_{H}}(a,d^{RC})+\mathit{d_{H}}(c,b^{RC})
  313. b b
  314. d d
  315. w \mathit{w}
  316. b R C b^{RC}
  317. d R C d^{RC}
  318. n - w \mathit{n-w}
  319. w \mathit{w}
  320. a , b , c , d w a,b,c,d\in\mathcal{B}_{w}
  321. d H ( a b ¯ , d R C c R ) 2 d 𝑚𝑖𝑛 / 2 d 𝑚𝑖𝑛 \mathit{d_{H}}(a\bar{b},d^{RC}c^{R})\geq 2\lceil\mathit{d_{min}}/2\rceil\geq% \mathit{d_{min}}
  322. 𝒪 \mathcal{O}
  323. M ( M - 1 ) / 2 M(M-1)/2
  324. 2 n 2n
  325. d H ( ( O ) ) d 𝑚𝑖𝑛 \mathit{d_{H}}(\mathcal{(}O))\geq\mathit{d_{min}}
  326. ( O ) \mathcal{(}O)
  327. \mathcal{B}
  328. d H 𝑅𝐶 ( ( O ) ) d 𝑚𝑖𝑛 \mathit{d_{H}^{RC}}(\mathcal{(}O))\geq\mathit{d_{min}}
  329. 𝒞 = w = d m i n w 𝑚𝑎𝑥 𝒞 w \mathcal{C}=\bigcup_{\mathit{w}=d_{min}}^{\mathit{w_{max}}}\mathcal{C}_{w}
  330. w 𝑚𝑎𝑥 = ( n - d m i n / 2 ) / 2 \mathit{w_{max}}=(\mathit{n}-\lceil d_{min}/2\rceil)/2
  331. 1 2 M ( M - 1 ) w = d m i n w m a x | A w 2 | \frac{1}{2}M(M-1)\sum_{w=d_{min}}^{w_{max}}\left|\mathit{A_{w}}^{2}\right|
  332. 2 n 2\mathit{n}
  333. d H ( ) d 𝑚𝑖𝑛 \mathit{d_{H}}(\mathcal{B})\geq\mathit{d_{min}}
  334. d H R C ( ) d 𝑚𝑖𝑛 \mathit{d_{H}}^{RC}(\mathcal{B})\geq\mathit{d_{min}}

Coefficient_of_fractional_parentage.html

  1. Ψ 1 N ( j N α J M ) = α 1 J 1 ( j N - 1 α 1 J 1 ; j | } j N α J ) [ Ψ 1 N - 1 ( j N - 1 α 1 J 1 ) ψ N ( j ) ] J M \Psi_{1\ldots N}(j^{N}\alpha JM)=\sum_{\alpha_{1}J_{1}}(j^{N-1}\alpha_{1}J_{1}% ;j|\}j^{N}\alpha J)\left[\Psi_{1\ldots N-1}(j^{N-1}\alpha_{1}J_{1})\otimes\psi% _{N}(j)\right]^{J}_{M}
  2. Ψ 1 N ( j N α J M ) \Psi_{1\ldots N}(j^{N}\alpha JM)
  3. Ψ 1 N - 1 ( j N - 1 α 1 J 1 ) \Psi_{1\ldots N-1}(j^{N-1}\alpha_{1}J_{1})

Collective_action_theory.html

  1. F i * d V g d T = d C d T F_{i}*\frac{dV_{g}}{dT}=\frac{dC}{dT}
  2. d V g d T d C d T = 1 F i = V g V i \frac{\frac{dV_{g}}{dT}}{\frac{dC}{dT}}=\frac{1}{F_{i}}=\frac{V_{g}}{V_{i}}
  3. F i > C V g F_{i}>\frac{C}{V_{g}}
  4. F i = V i V g F_{i}=\frac{V_{i}}{V_{g}}
  5. V i > C V_{i}>C
  6. F i F_{i}
  7. V i V_{i}
  8. V g V_{g}
  9. T T
  10. C C
  11. T T
  12. C = f ( T ) C=f(T)

Collinearity_equation.html

  1. x P , y P , z P x_{P},y_{P},z_{P}
  2. x 0 , y 0 , z 0 x_{0},y_{0},z_{0}
  3. λ \lambda
  4. x - x 0 x-x_{0}
  5. x 0 - x P x_{0}-x_{P}
  6. y - y 0 y-y_{0}
  7. y 0 - y P y_{0}-y_{P}
  8. z 0 = c z_{0}=c
  9. z P - z 0 z_{P}-z_{0}
  10. x - x 0 = - λ ( x P - x 0 ) x-x_{0}=-\lambda(x_{P}-x_{0})
  11. y - y 0 = - λ ( y P - y 0 ) y-y_{0}=-\lambda(y_{P}-y_{0})
  12. c = λ ( z P - z 0 ) , c=\lambda(z_{P}-z_{0}),
  13. λ \lambda
  14. x - x 0 = - c x P - x 0 z P - z 0 x-x_{0}=-c\ \frac{x_{P}-x_{0}}{z_{P}-z_{0}}
  15. y - y 0 = - c y P - y 0 z P - z 0 y-y_{0}=-c\ \frac{y_{P}-y_{0}}{z_{P}-z_{0}}
  16. X 0 , Y 0 , Z 0 X_{0},Y_{0},Z_{0}
  17. ( X - X 0 , Y - Y 0 , Z - Z 0 ) (X-X_{0},Y-Y_{0},Z-Z_{0})
  18. x P - x 0 = R 11 ( X - X 0 ) + R 21 ( Y - Y 0 ) + R 31 ( Z - Z 0 ) x_{P}-x_{0}=R_{11}(X-X_{0})+R_{21}(Y-Y_{0})+R_{31}(Z-Z_{0})
  19. y P - y 0 = R 12 ( X - X 0 ) + R 22 ( Y - Y 0 ) + R 32 ( Z - Z 0 ) y_{P}-y_{0}=R_{12}(X-X_{0})+R_{22}(Y-Y_{0})+R_{32}(Z-Z_{0})
  20. z P - z 0 = R 13 ( X - X 0 ) + R 23 ( Y - Y 0 ) + R 33 ( Z - Z 0 ) z_{P}-z_{0}=R_{13}(X-X_{0})+R_{23}(Y-Y_{0})+R_{33}(Z-Z_{0})
  21. x - x 0 = - c R 11 ( X - X 0 ) + R 21 ( Y - Y 0 ) + R 31 ( Z - Z 0 ) R 13 ( X - X 0 ) + R 23 ( Y - Y 0 ) + R 33 ( Z - Z 0 ) x-x_{0}=-c\ \frac{R_{11}(X-X_{0})+R_{21}(Y-Y_{0})+R_{31}(Z-Z_{0})}{R_{13}(X-X_% {0})+R_{23}(Y-Y_{0})+R_{33}(Z-Z_{0})}
  22. y - y 0 = - c R 12 ( X - X 0 ) + R 22 ( Y - Y 0 ) + R 32 ( Z - Z 0 ) R 13 ( X - X 0 ) + R 23 ( Y - Y 0 ) + R 33 ( Z - Z 0 ) y-y_{0}=-c\ \frac{R_{12}(X-X_{0})+R_{22}(Y-Y_{0})+R_{32}(Z-Z_{0})}{R_{13}(X-X_% {0})+R_{23}(Y-Y_{0})+R_{33}(Z-Z_{0})}

Collision_response.html

  1. e [ 0..1 ] e\in[0..1]
  2. e e
  3. 𝐟 ( t ) 3 \mathbf{f}(t)\in\mathbb{R}^{3}
  4. t t\in\mathbb{R}
  5. m m\in\mathbb{R}
  6. [ t 0 . . t 1 ] [t_{0}..t_{1}]
  7. 𝐩 ( t ) = m 𝐯 ( t ) \mathbf{p}(t)=m\mathbf{v}(t)
  8. 𝐯 ( t ) \mathbf{v}(t)
  9. j 3 j\in\mathbb{R}^{3}
  10. 𝐣 = t 0 t 1 𝐟 d t \mathbf{j}=\int_{t_{0}}^{t_{1}}\mathbf{f}dt
  11. 𝐣 \mathbf{j}
  12. t 1 t 0 | 𝐟 | t_{1}\rightarrow t_{0}\Rightarrow\left|\mathbf{f}\right|\rightarrow\infty
  13. 𝐟 \mathbf{f}
  14. lim t 1 t 0 t 0 t 1 𝐟 d t \lim_{t_{1}\rightarrow t_{0}}\int_{t_{0}}^{t_{1}}\mathbf{f}dt
  15. 𝐣 \mathbf{j}
  16. 𝐟 r ( t ) 3 \mathbf{f}_{r}(t)\in\mathbb{R}^{3}
  17. [ t 0 . . t 1 ] [t_{0}..t_{1}]
  18. 𝐣 r ( t ) 3 \mathbf{j}_{r}(t)\in\mathbb{R}^{3}
  19. 𝐣 r = t 0 t 1 𝐟 r d t \mathbf{j}_{r}=\int_{t_{0}}^{t_{1}}\mathbf{f}_{r}dt
  20. 𝐩 r 3 \mathbf{p}_{r}\in\mathbb{R}^{3}
  21. 𝐣 r \mathbf{j}_{r}
  22. - 𝐣 r -\mathbf{j}_{r}
  23. ± 𝐣 r = ± j r 𝐧 ^ \pm\mathbf{j}_{r}=\pm j_{r}\mathbf{\hat{n}}
  24. j r j_{r}\in\mathbb{R}
  25. 𝐧 ^ \mathbf{\hat{n}}
  26. - 𝐧 ^ -\mathbf{\hat{n}}
  27. 𝐧 ^ \mathbf{\hat{n}}
  28. 𝐣 r \mathbf{j}_{r}
  29. 𝐯 1 = 𝐯 1 - j r m 1 𝐧 ^ \mathbf{v^{\prime}}_{1}=\mathbf{v}_{1}-\frac{j_{r}}{m_{1}}\mathbf{\hat{n}}
  30. 𝐯 2 = 𝐯 2 + j r m 2 𝐧 ^ \mathbf{v^{\prime}}_{2}=\mathbf{v}_{2}+\frac{j_{r}}{m_{2}}\mathbf{\hat{n}}
  31. i i
  32. 𝐯 i 3 \mathbf{v}_{i}\in\mathbb{R}^{3}
  33. 𝐯 i 3 \mathbf{v^{\prime}}_{i}\in\mathbb{R}^{3}
  34. ω 1 = ω 1 - j r 𝐈 1 - 1 ( 𝐫 1 × 𝐧 ^ ) \mathbf{\omega^{\prime}}_{1}=\mathbf{\omega}_{1}-j_{r}\mathbf{I}_{1}^{-1}(% \mathbf{r}_{1}\times\mathbf{\hat{n}})
  35. ω 2 = ω 2 + j r 𝐈 2 - 1 ( 𝐫 2 × 𝐧 ^ ) \mathbf{\omega^{\prime}}_{2}=\mathbf{\omega}_{2}+j_{r}\mathbf{I}_{2}^{-1}(% \mathbf{r}_{2}\times\mathbf{\hat{n}})
  36. i i
  37. ω i 3 {\omega}_{i}\in\mathbb{R}^{3}
  38. ω i 3 {\omega^{\prime}}_{i}\in\mathbb{R}^{3}
  39. 𝐈 i 3 × 3 \mathbf{I}_{i}\in\mathbb{R}^{3\times 3}
  40. 𝐫 i 3 \mathbf{r}_{i}\in\mathbb{R}^{3}
  41. 𝐩 \mathbf{p}
  42. v p 1 , v p 2 3 v_{p1},v_{p2}\in\mathbb{R}^{3}
  43. 𝐯 p i = 𝐯 i + ω i × 𝐫 i \mathbf{v}_{pi}=\mathbf{v}_{i}+\mathbf{\omega}_{i}\times\mathbf{r}_{i}
  44. i = 1 , 2 i=1,2
  45. e e
  46. 𝐯 r = 𝐯 p 2 - 𝐯 p 1 \mathbf{v}_{r}=\mathbf{v}_{p2}-\mathbf{v}_{p1}
  47. 𝐯 r = 𝐯 p 2 - 𝐯 p 1 \mathbf{v^{\prime}}_{r}=\mathbf{v^{\prime}}_{p2}-\mathbf{v^{\prime}}_{p1}
  48. 𝐧 ^ \mathbf{\hat{n}}
  49. 𝐯 r 𝐧 ^ = - e 𝐯 r 𝐧 ^ \mathbf{v^{\prime}}_{r}\cdot\mathbf{\hat{n}}=-e\mathbf{v}_{r}\cdot\mathbf{\hat% {n}}
  50. j r j_{r}
  51. j r = - ( 1 + e ) 𝐯 r 𝐧 ^ m 1 - 1 + m 2 - 1 + ( 𝐈 1 - 1 ( 𝐫 1 × 𝐧 ^ ) × 𝐫 1 + 𝐈 2 - 1 ( 𝐫 2 × 𝐧 ^ ) × 𝐫 2 ) 𝐧 ^ j_{r}=\frac{-(1+e)\mathbf{v}_{r}\cdot\mathbf{\hat{n}}}{{m_{1}}^{-1}+{m_{2}}^{-% 1}+({\mathbf{I}_{1}}^{-1}(\mathbf{r}_{1}\times\mathbf{\hat{n}})\times\mathbf{r% }_{1}+{\mathbf{I}_{2}}^{-1}(\mathbf{r}_{2}\times\mathbf{\hat{n}})\times\mathbf% {r}_{2})\cdot\mathbf{\hat{n}}}
  52. 𝐯 i \mathbf{v^{\prime}}_{i}
  53. ω i \mathbf{\omega^{\prime}}_{i}
  54. j r j_{r}
  55. 𝐯 r \mathbf{v}_{r}
  56. m 1 m_{1}
  57. m 2 m_{2}
  58. 𝐈 1 \mathbf{I}_{1}
  59. 𝐈 2 \mathbf{I}_{2}
  60. 𝐫 1 \mathbf{r}_{1}
  61. 𝐫 2 \mathbf{r}_{2}
  62. 𝐧 ^ \mathbf{\hat{n}}
  63. e e
  64. 𝐣 r \mathbf{j}_{r}
  65. j r j_{r}
  66. 𝐧 ^ \mathbf{\hat{n}}
  67. 𝐣 r = j r 𝐧 ^ \mathbf{j}_{r}=j_{r}\mathbf{\hat{n}}
  68. 𝐯 i \mathbf{v^{\prime}}_{i}
  69. 𝐯 i \mathbf{v}_{i}
  70. m i m_{i}
  71. 𝐣 r \mathbf{j}_{r}
  72. ω i \mathbf{\omega^{\prime}}_{i}
  73. ω i \mathbf{\omega}_{i}
  74. 𝐈 i \mathbf{I}_{i}
  75. 𝐣 r \mathbf{j}_{r}
  76. μ s {\mu}_{s}\in\mathbb{R}
  77. μ d {\mu}_{d}\in\mathbb{R}
  78. μ s > μ d {\mu}_{s}>{\mu}_{d}
  79. f s , f d f_{s},f_{d}\in\mathbb{R}
  80. f r = | 𝐟 r | f_{r}=|\mathbf{f}_{r}|
  81. f s = μ s f r f_{s}={\mu}_{s}f_{r}
  82. f d = μ d f r f_{d}={\mu}_{d}f_{r}
  83. f s f_{s}
  84. f d f_{d}
  85. 𝐧 ^ 3 \mathbf{\hat{n}}\in\mathbb{R}^{3}
  86. 𝐯 r 3 \mathbf{v}_{r}\in\mathbb{R}^{3}
  87. 𝐭 ^ 3 \mathbf{\hat{t}}\in\mathbb{R}^{3}
  88. 𝐧 ^ \mathbf{\hat{n}}
  89. 𝐭 ^ = { 𝐯 r - ( 𝐯 r 𝐧 ^ ) 𝐧 ^ | 𝐯 r - ( 𝐯 r 𝐧 ^ ) 𝐧 ^ | 𝐯 r 𝐧 ^ 0 𝐟 e - ( 𝐟 e 𝐧 ^ ) 𝐧 ^ | 𝐟 e - ( 𝐟 e 𝐧 ^ ) 𝐧 ^ | 𝐯 r 𝐧 ^ = 0 𝐟 e 𝐧 ^ 0 𝟎 𝐯 r 𝐧 ^ = 0 𝐟 e 𝐧 ^ = 0 \mathbf{\hat{t}}=\left\{\begin{matrix}\frac{\mathbf{v}_{r}-(\mathbf{v}_{r}% \cdot\mathbf{\hat{n}})\mathbf{\hat{n}}}{|\mathbf{v}_{r}-(\mathbf{v}_{r}\cdot% \mathbf{\hat{n}})\mathbf{\hat{n}}|}&\mathbf{v}_{r}\cdot\mathbf{\hat{n}}\neq 0&% \\ \frac{\mathbf{f}_{e}-(\mathbf{f}_{e}\cdot\mathbf{\hat{n}})\mathbf{\hat{n}}}{|% \mathbf{f}_{e}-(\mathbf{f}_{e}\cdot\mathbf{\hat{n}})\mathbf{\hat{n}}|}&\mathbf% {v}_{r}\cdot\mathbf{\hat{n}}=0&\mathbf{f}_{e}\cdot\mathbf{\hat{n}}\neq 0\\ \mathbf{0}&\mathbf{v}_{r}\cdot\mathbf{\hat{n}}=0&\mathbf{f}_{e}\cdot\mathbf{% \hat{n}}=0\\ \end{matrix}\right.
  90. 𝐟 e 3 \mathbf{f}_{e}\in\mathbb{R}^{3}
  91. 𝐭 ^ \mathbf{\hat{t}}
  92. 𝐟 f 3 \mathbf{f}_{f}\in\mathbb{R}^{3}
  93. 𝐧 ^ \mathbf{\hat{n}}
  94. 𝐭 ^ \mathbf{\hat{t}}
  95. 𝐟 e 3 \mathbf{f}_{e}\in\mathbb{R}^{3}
  96. 𝐭 ^ \mathbf{\hat{t}}
  97. 𝐟 f 3 \mathbf{f}_{f}\in\mathbb{R}^{3}
  98. 𝐟 f = { - ( 𝐟 e 𝐭 ^ ) 𝐭 ^ 𝐯 r 𝐭 ^ = 0 𝐟 e 𝐭 ^ f s - f d 𝐭 ^ (otherwise) \mathbf{f}_{f}=\left\{\begin{matrix}-(\mathbf{f}_{e}\cdot\mathbf{\hat{t}})% \mathbf{\hat{t}}&\mathbf{v}_{r}\cdot\mathbf{\hat{t}}=0&\mathbf{f}_{e}\cdot% \mathbf{\hat{t}}\leq f_{s}\\ -f_{d}\mathbf{\hat{t}}&\,\text{(otherwise)}\\ \end{matrix}\right.
  99. 𝐣 f 3 \mathbf{j}_{f}\in\mathbb{R}^{3}
  100. 𝐭 ^ \mathbf{\hat{t}}
  101. 𝐣 r 3 \mathbf{j}_{r}\in\mathbb{R}^{3}
  102. [ t 0 . . t 1 ] [t_{0}..t_{1}]
  103. j s = μ s j r j_{s}={\mu}_{s}j_{r}
  104. j d = μ d j r j_{d}={\mu}_{d}j_{r}
  105. j r = | 𝐣 r | j_{r}=|\mathbf{j}_{r}|
  106. 𝐧 ^ \mathbf{\hat{n}}
  107. 𝐭 ^ \mathbf{\hat{t}}
  108. 𝐣 f = { - ( m 𝐯 r 𝐭 ^ ) 𝐭 ^ 𝐯 r 𝐭 ^ = 0 m 𝐯 r 𝐭 ^ j s - j d 𝐭 ^ (otherwise) \mathbf{j}_{f}=\left\{\begin{matrix}-(m\mathbf{v}_{r}\cdot\mathbf{\hat{t}})% \mathbf{\hat{t}}&\mathbf{v}_{r}\cdot\mathbf{\hat{t}}=0&m\mathbf{v}_{r}\cdot% \mathbf{\hat{t}}\leq j_{s}\\ -j_{d}\mathbf{\hat{t}}&\,\text{(otherwise)}\\ \end{matrix}\right.
  109. μ s {\mu}_{s}
  110. μ d {\mu}_{d}

Color-coding.html

  1. k k
  2. k k
  3. k k
  4. G = ( V , E ) G=(V,E)
  5. k k
  6. V ω V^{\omega}
  7. V ω log V V^{\omega}\log V
  8. ω ω
  9. k k
  10. G = ( V , E ) G=(V,E)
  11. G G
  12. k k
  13. O ( V ) O(V)
  14. O ( V l o g V ) O(VlogV)
  15. G = ( V , E ) G=(V,E)
  16. O ( l o g V ) O(logV)
  17. H = ( V H , E H ) H=(V_{H},E_{H})
  18. G = ( V , E ) G=(V,E)
  19. H H
  20. | V H | = O ( log V ) |V_{H}|=O(\log V)
  21. G G
  22. k = | V H | k=|V_{H}|
  23. H H
  24. G G
  25. H H
  26. p p
  27. 1 p \frac{1}{p}
  28. H H
  29. p p
  30. | V H | = O ( log V ) |V_{H}|=O(\log V)
  31. p p
  32. G G
  33. G G
  34. k k
  35. H H
  36. O ( r ) O(r)
  37. H H
  38. G G
  39. O ( r p ) O(\tfrac{r}{p})
  40. k k
  41. G = ( V , E ) G=(V,E)
  42. k ! / k k > e - k k!/k^{k}>e^{-k}
  43. k k k^{k}
  44. k k
  45. k ! k!
  46. O ( V ω ) O(V^{\omega})
  47. G G
  48. e k O ( V ω ) e^{k}\cdot O(V^{\omega})
  49. k k
  50. G G
  51. V V
  52. k 1 k−1
  53. G G
  54. k / 2 k/2
  55. V V
  56. k / 2 - 1 k/2-1
  57. B B
  58. A 1 B A 2 A_{1}BA_{2}
  59. V V
  60. k 1 k−1
  61. t ( k ) 2 k t ( k / 2 ) t(k)\leq 2^{k}\cdot t(k/2)
  62. 2 O ( k ) V ω O ( V ω ) 2^{O(k)}\cdot V^{\omega}\in O(V^{\omega})
  63. G G
  64. G G
  65. H H
  66. G G
  67. H H
  68. k k
  69. F F
  70. F F
  71. k k
  72. S S
  73. | S | = k |S|=k
  74. h h
  75. F F
  76. F F
  77. k k
  78. k k
  79. k k
  80. e k k O ( log k ) log | V | e^{k}k^{O(\log k)}\log|V|
  81. 2 O ( k ) log 2 | V | 2^{O(k)}\log^{2}|V|
  82. k k
  83. k k
  84. 2 n l o g k 2nlogk
  85. 2 l o g k 2logk
  86. k O ( 1 ) log | V | k^{O(1)}\log|V|
  87. k k
  88. 2 O ( k ) 2^{O(k)}
  89. k k
  90. k k
  91. 2 O ( k ) log | V | 2^{O(k)}\log|V|
  92. k k
  93. k k
  94. n k l o g k nklogk
  95. k l o g k klogk
  96. k k
  97. k O ( k ) log | V | k^{O(k)}\log|V|
  98. k = O ( log n ) k=O(\log n)
  99. G G
  100. n n