wpmath0000008_2

CLEO_(particle_detector).html

  1. B ¯ \overline{B}
  2. B ¯ \overline{B}
  3. p ¯ \overline{p}
  4. p ¯ \overline{p}
  5. B ¯ \overline{B}
  6. B ¯ \overline{B}
  7. p ¯ \overline{p}
  8. n ¯ \overline{n}
  9. D ¯ \overline{D}
  10. ν ¯ \overline{ν}
  11. n ¯ \overline{n}
  12. ν ¯ \overline{ν}
  13. K ¯ \overline{K}
  14. K ¯ \overline{K}
  15. ν ¯ \overline{ν}
  16. X ¯ \overline{X}
  17. ν ¯ \overline{ν}

Clifford_torus.html

  1. S 1 = { ( cos θ , sin θ ) | 0 θ < 2 π } . S^{1}=\{(\cos{\theta},\sin{\theta})\,|\,0\leq\theta<2\pi\}.
  2. S 1 = { ( cos ϕ , sin ϕ ) | 0 ϕ < 2 π } . S^{1}=\{(\cos{\phi},\sin{\phi})\,|\,0\leq\phi<2\pi\}.
  3. S 1 × S 1 = { ( cos θ , sin θ , cos ϕ , sin ϕ ) | 0 θ < 2 π , 0 ϕ < 2 π } . S^{1}\times S^{1}=\{(\cos{\theta},\sin{\theta},\cos{\phi},\sin{\phi})\,|\,0% \leq\theta<2\pi,0\leq\phi<2\pi\}.
  4. x 1 2 + y 1 2 = 1 = x 2 2 + y 2 2 . x_{1}^{2}+y_{1}^{2}=1=x_{2}^{2}+y_{2}^{2}.\,
  5. S 1 = { e i θ | 0 θ < 2 π } S^{1}=\{e^{i\theta}\,|\,0\leq\theta<2\pi\}
  6. S 1 = { e i ϕ | 0 ϕ < 2 π } . S^{1}=\{e^{i\phi}\,|\,0\leq\phi<2\pi\}.
  7. S 1 × S 1 = { ( e i θ , e i ϕ ) | 0 θ < 2 π , 0 ϕ < 2 π } . S^{1}\times S^{1}=\{(e^{i\theta},e^{i\phi})\,|\,0\leq\theta<2\pi,0\leq\phi<2% \pi\}.
  8. | z 1 | 2 = 1 = | z 2 | 2 . \left|z_{1}\right|^{2}=1=\left|z_{2}\right|^{2}.
  9. | e i θ | 2 + | e i ϕ | 2 = 2 . \sqrt{\left|e^{i\theta}\right|^{2}+\left|e^{i\phi}\right|^{2}}=\sqrt{2}.

Clos_network.html

  1. β i j = B A = ( i m - j ) ( m j ) = i ! j ! ( i + j - m ) ! m ! \beta_{ij}=\frac{B}{A}=\frac{\left(\begin{array}[]{c}i\\ m-j\end{array}\right)}{\left(\begin{array}[]{c}m\\ j\end{array}\right)}=\frac{i!j!}{(i+j-m)!m!}
  2. P B = i = 0 u j = 0 u f i g j β i j P_{B}=\sum_{i=0}^{u}\sum_{j=0}^{u}f_{i}g_{j}\beta_{ij}
  3. P B = ( u ! ) 2 ( 2 - q ) 2 u - m q m m ! ( 2 u - m ) ! P_{B}=\frac{(u!)^{2}(2-q)^{2u-m}q^{m}}{m!(2u-m)!}

Closed-loop_pole.html

  1. 𝐆 c = K \,\textbf{G}_{c}=K
  2. 𝐆 c = K \,\textbf{G}_{c}=K
  3. 𝐆 ( s ) \,\textbf{G}(s)
  4. 𝐇 ( s ) \,\textbf{H}(s)
  5. 𝐇 ( s ) = 1 \,\textbf{H}(s)=1
  6. 𝐆 c 𝐆 = K 𝐆 \,\textbf{G}_{c}\,\textbf{G}=K\,\textbf{G}
  7. 𝐆 c 𝐆 𝐇 = K 𝐆 𝐇 \,\textbf{G}_{c}\,\textbf{G}\,\textbf{H}=K\,\textbf{G}\,\textbf{H}
  8. 𝐓 ( s ) = K 𝐆 1 + K 𝐆 𝐇 \,\textbf{T}(s)=\frac{K\,\textbf{G}}{1+K\,\textbf{G}\,\textbf{H}}
  9. 1 + K 𝐆 𝐇 = 0 {1+K\,\textbf{G}\,\textbf{H}}=0
  10. 𝐆 ( s ) \,\textbf{G}(s)
  11. 𝐊 ( s ) \,\textbf{K}(s)
  12. d e t ( 𝐈 + 𝐆 ( s ) 𝐊 ( s ) ) = 0 det(\,\textbf{I}+\,\textbf{G}(s)\,\textbf{K}(s))=0

CMA-ES.html

  1. x x
  2. f ( x ) f(x)
  3. f f
  4. n \mathbb{R}^{n}
  5. f f
  6. λ \lambda
  7. m m
  8. σ \sigma
  9. C = I C=I
  10. p σ = 0 p_{\sigma}=0
  11. p c = 0 p_{c}=0
  12. i i
  13. { 1... λ } \{1...\lambda\}
  14. λ \lambda
  15. x i x_{i}
  16. m m
  17. σ 2 C \sigma^{2}C
  18. f i f_{i}
  19. x i x_{i}
  20. x 1... λ x_{1...\lambda}
  21. x s ( 1 ) s ( λ ) x_{s(1)...s(\lambda)}
  22. s ( i ) s(i)
  23. f 1... λ f_{1...\lambda}
  24. i i
  25. m m^{\prime}
  26. m m
  27. m - m m-m^{\prime}
  28. x i - m x_{i}-m^{\prime}
  29. m m
  30. ( x 1 , , (x_{1},...,
  31. x λ ) x_{\lambda})
  32. p σ p_{\sigma}
  33. ( p σ , (p_{\sigma},
  34. σ - 1 C - 1 / 2 ( m - m ) ) \sigma^{-1}C^{-1/2}(m-m^{\prime}))
  35. p c p_{c}
  36. ( p c , (p_{c},
  37. σ - 1 ( m - m ) , \sigma^{-1}(m-m^{\prime}),
  38. | | p σ | | ) ||p_{\sigma}||)
  39. C C
  40. ( C , (C,
  41. p c , p_{c},
  42. ( x 1 - m ) / σ , , {(x_{1}-m^{\prime})}/{\sigma},...,
  43. ( x λ - m ) / σ ) {(x_{\lambda}-m^{\prime})}/{\sigma})
  44. σ \sigma
  45. ( σ , (\sigma,
  46. | | p σ | | ) ||p_{\sigma}||)
  47. m m
  48. x 1 x_{1}
  49. n n
  50. k k
  51. m k n m_{k}\in\mathbb{R}^{n}
  52. σ k > 0 \sigma_{k}>0
  53. C k C_{k}
  54. n × n n\times n
  55. C 0 = I C_{0}=I
  56. p σ n , p c n p_{\sigma}\in\mathbb{R}^{n},p_{c}\in\mathbb{R}^{n}
  57. λ > 1 \lambda>1
  58. x i n x_{i}\in\mathbb{R}^{n}
  59. 𝒩 ( m k , σ k 2 C k ) \textstyle\mathcal{N}(m_{k},\sigma_{k}^{2}C_{k})
  60. i = 1 , , λ i=1,...,\lambda
  61. x i \displaystyle x_{i}
  62. m k m_{k}
  63. x i x_{i}
  64. f : n f:\mathbb{R}^{n}\to\mathbb{R}
  65. f f
  66. { x i : λ | i = 1 λ } = { x i | i = 1 λ } and f ( x 1 : λ ) f ( x μ : λ ) f ( x μ + 1 : λ ) , \{x_{i:\lambda}\;|\;i=1\dots\lambda\}=\{x_{i}\;|\;i=1\dots\lambda\}\;\;\,\text% {and}\;\;f(x_{1:\lambda})\leq\dots\leq f(x_{\mu:\lambda})\leq f(x_{\mu+1:% \lambda})\dots,
  67. m k + 1 \displaystyle m_{k+1}
  68. w 1 w 2 w μ > 0 w_{1}\geq w_{2}\geq\dots\geq w_{\mu}>0
  69. μ λ / 2 \mu\leq\lambda/2
  70. μ w := 1 / i = 1 μ w i 2 λ / 4 \textstyle\mu_{w}:=1/\sum_{i=1}^{\mu}w_{i}^{2}\approx\lambda/4
  71. i : λ i:\lambda
  72. σ k \sigma_{k}
  73. p σ p_{\sigma}
  74. p σ ( 1 - c σ ) discount factor p σ + 1 - ( 1 - c σ ) 2 complements for discounted variance μ w C k - 1 / 2 m k + 1 - m k displacement of m σ k distributed as 𝒩 ( 0 , I ) under neutral selection p_{\sigma}\leftarrow\underbrace{(1-c_{\sigma})}_{\!\!\!\!\!\,\text{discount % factor}\!\!\!\!\!}\,p_{\sigma}+\overbrace{\sqrt{1-(1-c_{\sigma})^{2}}}^{\!\!\!% \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\,\text{complements % for discounted variance}\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!% \!\!\!\!}\underbrace{\sqrt{\mu_{w}}\,C_{k}^{\;-1/2}\,\frac{\overbrace{m_{k+1}-% m_{k}}^{\!\!\!\,\text{displacement of}\;m\!\!\!}}{\sigma_{k}}}_{\!\!\!\!\!\!\!% \!\!\!\!\!\!\!\!\!\!\!\,\text{distributed as}\;\mathcal{N}(0,I)\;\,\text{under% neutral selection}\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!}
  75. σ k + 1 = σ k × exp ( c σ d σ ( p σ E 𝒩 ( 0 , I ) - 1 ) unbiased about 0 under neutral selection ) \sigma_{k+1}=\sigma_{k}\times\exp\bigg(\frac{c_{\sigma}}{d_{\sigma}}% \underbrace{\left(\frac{\|p_{\sigma}\|}{E\|\mathcal{N}(0,I)\|}-1\right)}_{\!\!% \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\,\text{% unbiased about 0 under neutral selection}\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!% \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!}\bigg)
  76. c σ - 1 n / 3 c_{\sigma}^{-1}\approx n/3
  77. p σ p_{\sigma}
  78. μ w = ( i = 1 μ w i 2 ) - 1 \mu_{w}=\left(\sum_{i=1}^{\mu}w_{i}^{2}\right)^{-1}
  79. 1 μ w μ 1\leq\mu_{w}\leq\mu
  80. w i w_{i}
  81. C k - 1 / 2 = C k - 1 = C k - 1 C_{k}^{\;-1/2}=\sqrt{C_{k}}^{\;-1}=\sqrt{C_{k}^{\;-1}}
  82. C k C_{k}
  83. d σ d_{\sigma}
  84. d σ = d_{\sigma}=\infty
  85. c σ = 0 c_{\sigma}=0
  86. σ k \sigma_{k}
  87. p σ \|p_{\sigma}\|
  88. E 𝒩 ( 0 , I ) \displaystyle E\|\mathcal{N}(0,I)\|
  89. C k - 1 C_{k}^{-1}
  90. ( m k + 2 - m k + 1 σ k + 1 ) T C k - 1 m k + 1 - m k σ k 0 \textstyle\left(\frac{m_{k+2}-m_{k+1}}{\sigma_{k+1}}\right)^{T}\!C_{k}^{-1}% \frac{m_{k+1}-m_{k}}{\sigma_{k}}\approx 0
  91. p c ( 1 - c c ) discount factor p c + 𝟏 [ 0 , α n ] ( p σ ) indicator function 1 - ( 1 - c c ) 2 complements for discounted variance μ w m k + 1 - m k σ k distributed as 𝒩 ( 0 , C k ) under neutral selection p_{c}\leftarrow\underbrace{(1-c_{c})}_{\!\!\!\!\!\,\text{discount factor}\!\!% \!\!\!}\,p_{c}+\underbrace{\mathbf{1}_{[0,\alpha\sqrt{n}]}(\|p_{\sigma}\|)}_{% \,\text{indicator function}}\overbrace{\sqrt{1-(1-c_{c})^{2}}}^{\!\!\!\!\!\!\!% \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\,\text{complements for % discounted variance}\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!% \!\!}\underbrace{\sqrt{\mu_{w}}\,\frac{m_{k+1}-m_{k}}{\sigma_{k}}}_{\!\!\!\!\!% \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\,\text{% distributed as}\;\mathcal{N}(0,C_{k})\;\,\text{under neutral selection}\!\!\!% \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!}
  92. C k + 1 = ( 1 - c 1 - c μ + c s ) discount factor C k + c 1 p c p c T rank one matrix + c μ i = 1 μ w i x i : λ - m k σ k ( x i : λ - m k σ k ) T rank min ( μ , n ) matrix C_{k+1}=\underbrace{(1-c_{1}-c_{\mu}+c_{s})}_{\!\!\!\!\!\,\text{discount % factor}\!\!\!\!\!}\,C_{k}+c_{1}\underbrace{p_{c}p_{c}^{T}}_{\!\!\!\!\!\!\!\!\!% \!\!\!\!\!\!\!\,\text{rank one matrix}\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!}+\,c_{% \mu}\underbrace{\sum_{i=1}^{\mu}w_{i}\frac{x_{i:\lambda}-m_{k}}{\sigma_{k}}% \left(\frac{x_{i:\lambda}-m_{k}}{\sigma_{k}}\right)^{T}}_{\,\text{rank}\;\min(% \mu,n)\;\,\text{matrix}}
  93. T T
  94. c c - 1 n / 4 c_{c}^{-1}\approx n/4
  95. p c p_{c}
  96. α 1.5 \alpha\approx 1.5
  97. 𝟏 [ 0 , α n ] ( p σ ) \mathbf{1}_{[0,\alpha\sqrt{n}]}(\|p_{\sigma}\|)
  98. p σ [ 0 , α n ] \|p_{\sigma}\|\in[0,\alpha\sqrt{n}]
  99. p σ α n \|p_{\sigma}\|\leq\alpha\sqrt{n}
  100. c s = ( 1 - 𝟏 [ 0 , α n ] ( p σ ) 2 ) c 1 c c ( 2 - c c ) c_{s}=(1-\mathbf{1}_{[0,\alpha\sqrt{n}]}(\|p_{\sigma}\|)^{2})\,c_{1}c_{c}(2-c_% {c})
  101. c 1 2 / n 2 c_{1}\approx 2/n^{2}
  102. c μ μ w / n 2 c_{\mu}\approx\mu_{w}/n^{2}
  103. μ \mu
  104. 1 - c 1 1-c_{1}
  105. p c p_{c}
  106. ( x i : λ - m k ) / σ k (x_{i:\lambda}-m_{k})/\sigma_{k}
  107. 𝒩 ( 0 , C k + 1 ) \mathcal{N}(0,C_{k+1})
  108. λ \lambda
  109. λ = 10 \lambda=10
  110. λ = 10 n \lambda=10n
  111. μ w λ / 4 \mu_{w}\approx\lambda/4
  112. λ \lambda
  113. λ \lambda
  114. μ \mu
  115. λ \lambda

Code_(set_theory).html

  1. x H 1 x\in H_{\aleph_{1}}\,
  2. E ω × ω E\subset\omega\times\omega
  3. \in
  4. H 1 H_{\aleph_{1}}
  5. H 1 H_{\aleph_{1}}
  6. H 1 L ( R ) . H_{\aleph_{1}}\subset L(R)\,.

Coherent_risk_measure.html

  1. ϱ \varrho
  2. X X
  3. \mathcal{L}
  4. ϱ : \varrho:\mathcal{L}
  5. \R { + } \R\cup\{+\infty\}
  6. \mathcal{L}
  7. ϱ ( 0 ) = 0 \varrho(0)=0
  8. If Z 1 , Z 2 and Z 1 Z 2 a . s . , then ϱ ( Z 1 ) ϱ ( Z 2 ) \mathrm{If}\;Z_{1},Z_{2}\in\mathcal{L}\;\mathrm{and}\;Z_{1}\leq Z_{2}\;\mathrm% {a.s.},\;\mathrm{then}\;\varrho(Z_{1})\geq\varrho(Z_{2})
  9. Z 2 Z_{2}
  10. Z 1 Z_{1}
  11. Z 2 Z_{2}
  12. Z 1 Z_{1}
  13. Z 1 Z_{1}
  14. Z 2 Z_{2}
  15. If Z 1 , Z 2 , then ϱ ( Z 1 + Z 2 ) ϱ ( Z 1 ) + ϱ ( Z 2 ) \mathrm{If}\;Z_{1},Z_{2}\in\mathcal{L},\;\mathrm{then}\;\varrho(Z_{1}+Z_{2})% \leq\varrho(Z_{1})+\varrho(Z_{2})
  16. If α 0 and Z , then ϱ ( α Z ) = α ϱ ( Z ) \mathrm{If}\;\alpha\geq 0\;\mathrm{and}\;Z\in\mathcal{L},\;\mathrm{then}\;% \varrho(\alpha Z)=\alpha\varrho(Z)
  17. A A
  18. a a
  19. Z Z\in\mathcal{L}
  20. ϱ ( Z + A ) = ϱ ( Z ) - a \varrho(Z+A)=\varrho(Z)-a
  21. A A
  22. a a
  23. Z Z
  24. a = ϱ ( Z ) a=\varrho(Z)
  25. ϱ ( Z + A ) = 0 \varrho(Z+A)=0
  26. If Z 1 , Z 2 and λ [ 0 , 1 ] then ϱ ( λ Z 1 + ( 1 - λ ) Z 2 ) λ ϱ ( Z 1 ) + ( 1 - λ ) ϱ ( Z 2 ) \,\text{If }Z_{1},Z_{2}\in\mathcal{L}\,\text{ and }\lambda\in[0,1]\,\text{ % then }\varrho(\lambda Z_{1}+(1-\lambda)Z_{2})\leq\lambda\varrho(Z_{1})+(1-% \lambda)\varrho(Z_{2})
  27. g : [ 0 , 1 ] [ 0 , 1 ] g\colon[0,1]\rightarrow[0,1]
  28. g ( 0 ) = 0 g(0)=0
  29. g ( 1 ) = 1 g(1)=1
  30. g ~ ( x ) = 1 - g ( 1 - x ) \tilde{g}(x)=1-g(1-x)
  31. ( Ω , , ) (\Omega,\mathcal{F},\mathbb{P})
  32. X X
  33. g g
  34. \mathbb{Q}
  35. A A\in\mathcal{F}
  36. ( A ) = g ( ( X A ) ) . \mathbb{Q}(A)=g(\mathbb{P}(X\in A)).
  37. ϱ ( X ) = 0 + g ( F ¯ X ( x ) ) d x \varrho(X)=\int_{0}^{+\infty}g\left(\bar{F}_{X}(x)\right)dx
  38. g g
  39. g g
  40. g ( x ) = 𝟏 x 1 - α g(x)=\mathbf{1}_{x\geq 1-\alpha}
  41. g g
  42. g ( x ) = min ( x α , 1 ) g(x)=\min(\frac{x}{\alpha},1)
  43. g g
  44. ( λ ( t ) = f ( t ) F ¯ ( t ) ) \scriptstyle\left(\lambda(t)=\frac{f(t)}{\bar{F}(t)}\right)
  45. ξ \xi
  46. g α ( x ) = x ξ g_{\alpha}(x)=x^{\xi}
  47. g g
  48. ξ < 1 2 \scriptstyle\xi<\frac{1}{2}
  49. g α ( x ) = Φ [ Φ - 1 ( x ) - Φ - 1 ( α ) ] g_{\alpha}(x)=\Phi\left[\Phi^{-1}(x)-\Phi^{-1}(\alpha)\right]
  50. g g
  51. d \mathbb{R}^{d}
  52. n d n\leq d
  53. R : L d p 𝔽 M R:L_{d}^{p}\rightarrow\mathbb{F}_{M}
  54. 𝔽 M = { D M : D = c l ( D + K M ) } \mathbb{F}_{M}=\{D\subseteq M:D=cl(D+K_{M})\}
  55. K M = K M K_{M}=K\cap M
  56. K K
  57. M M
  58. m m
  59. R R
  60. K M R ( 0 ) and R ( 0 ) - int K M = K_{M}\subseteq R(0)\;\mathrm{and}\;R(0)\cap-\mathrm{int}K_{M}=\emptyset
  61. X L d p , u M : R ( X + u 1 ) = R ( X ) - u \forall X\in L_{d}^{p},\forall u\in M:R(X+u1)=R(X)-u
  62. X 2 - X 1 L d p ( K ) R ( X 2 ) R ( X 1 ) \forall X_{2}-X_{1}\in L_{d}^{p}(K)\Rightarrow R(X_{2})\supseteq R(X_{1})
  63. ϱ \varrho
  64. ϱ ( X ) = sup Q ( P ) { E Q [ - X ] - α ( Q ) } \varrho(X)=\sup_{Q\in\mathcal{M}(P)}\{E^{Q}[-X]-\alpha(Q)\}
  65. α \alpha
  66. ( P ) \mathcal{M}(P)
  67. ( P ) = { Q P } \mathcal{M}(P)=\{Q\ll P\}
  68. ϱ ( X ) = sup Q 𝒬 E Q [ - X ] \varrho(X)=\sup_{Q\in\mathcal{Q}}E^{Q}[-X]
  69. 𝒬 ( P ) \mathcal{Q}\subseteq\mathcal{M}(P)

Cohn's_irreducibility_criterion.html

  1. [ x ] \mathbb{Z}[x]
  2. p p
  3. p = a m 10 m + a m - 1 10 m - 1 + + a 1 10 + a 0 p=a_{m}10^{m}+a_{m-1}10^{m-1}+\cdots+a_{1}10+a_{0}
  4. 0 a i 9 0\leq a_{i}\leq 9
  5. f ( x ) = a m x m + a m - 1 x m - 1 + + a 1 x + a 0 f(x)=a_{m}x^{m}+a_{m-1}x^{m-1}+\cdots+a_{1}x+a_{0}
  6. [ x ] \mathbb{Z}[x]
  7. b 2 b\geq 2
  8. p ( x ) = a k x k + a k - 1 x k - 1 + + a 1 x + a 0 p(x)=a_{k}x^{k}+a_{k-1}x^{k-1}+\cdots+a_{1}x+a_{0}
  9. 0 a i b - 1 0\leq a_{i}\leq b-1
  10. p ( b ) p(b)
  11. p ( x ) p(x)
  12. [ x ] \mathbb{Z}[x]

Coleman–Weinberg_potential.html

  1. L = - 1 4 ( F μ ν ) 2 + ( D μ ϕ ) 2 - m 2 ϕ 2 - λ 6 ϕ 4 L=-\frac{1}{4}(F_{\mu\nu})^{2}+(D_{\mu}\phi)^{2}-m^{2}\phi^{2}-\frac{\lambda}{% 6}\phi^{4}
  2. F μ ν = μ A ν - ν A μ F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}
  3. D μ = μ - ( e / c ) A μ D_{\mu}=\partial_{\mu}-(e/\hbar c)A_{\mu}
  4. e e
  5. λ \lambda
  6. m 2 < 0 m^{2}<0
  7. m 2 > 0 m^{2}>0
  8. ϕ \phi
  9. m 2 = 0 m^{2}=0
  10. ϕ \phi
  11. m 2 m^{2}
  12. κ λ / e 2 \kappa\equiv\lambda/e^{2}
  13. κ = 1 / 2 \kappa=1/\sqrt{2}
  14. κ \kappa
  15. κ = 0.76 / 2 \kappa=0.76/\sqrt{2}
  16. κ = 1 / 2 \kappa=1/\sqrt{2}

Collision_frequency.html

  1. Z = N A σ A B 8 k B T π μ A B Z=N_{A}\sigma_{AB}\sqrt{\frac{8k_{B}T}{\pi\mu_{AB}}}

Collocation_method.html

  1. y ( t ) = f ( t , y ( t ) ) , y ( t 0 ) = y 0 , y^{\prime}(t)=f(t,y(t)),\quad y(t_{0})=y_{0},
  2. p ( t 0 ) = y 0 , p(t_{0})=y_{0},\,
  3. p ( t 0 ) = f ( t 0 , p ( t 0 ) ) , p^{\prime}(t_{0})=f(t_{0},p(t_{0})),\,
  4. p ( t 0 + h ) = f ( t 0 + h , p ( t 0 + h ) ) . p^{\prime}(t_{0}+h)=f(t_{0}+h,p(t_{0}+h)).\,
  5. p ( t ) = α ( t - t 0 ) 2 + β ( t - t 0 ) + γ p(t)=\alpha(t-t_{0})^{2}+\beta(t-t_{0})+\gamma\,
  6. α = 1 2 h ( f ( t 0 + h , p ( t 0 + h ) ) - f ( t 0 , p ( t 0 ) ) ) , β = f ( t 0 , p ( t 0 ) ) , γ = y 0 . \begin{aligned}\displaystyle\alpha&\displaystyle=\frac{1}{2h}\Big(f(t_{0}+h,p(% t_{0}+h))-f(t_{0},p(t_{0}))\Big),\\ \displaystyle\beta&\displaystyle=f(t_{0},p(t_{0})),\\ \displaystyle\gamma&\displaystyle=y_{0}.\end{aligned}
  7. y 1 = p ( t 0 + h ) = y 0 + 1 2 h ( f ( t 0 + h , y 1 ) + f ( t 0 , y 0 ) ) , y_{1}=p(t_{0}+h)=y_{0}+\frac{1}{2}h\Big(f(t_{0}+h,y_{1})+f(t_{0},y_{0})\Big),\,
  8. y ( t ) = y ( t 0 ) + t 0 t f ( τ , y ( τ ) ) d τ , y(t)=y(t_{0})+\int_{t_{0}}^{t}f(\tau,y(\tau))\,\textrm{d}\tau,\,

Cologarithm.html

  1. colog b x = log b ( 1 x ) = log b 1 - log b x = - log b x \operatorname{colog}_{b}\ x=\log_{b}\left(\frac{1}{x}\right)=\log_{b}1-\log_{b% }x=-\log_{b}x\,

Combinatorial_explosion.html

  1. l = n ( n - 1 ) 2 = ( n 2 ) l=\frac{n(n-1)}{2}={n\choose 2}

Common_pilot_channel.html

  1. 0 < n < 256 0<n<256

Commutation_(neurophysiology).html

  1. a × b b × a a\times b\neq b\times a

Commutation_matrix.html

  1. \otimes
  2. \otimes
  3. i = 1 r \sum_{i=1}^{r}
  4. j = 1 m \sum_{j=1}^{m}
  5. \otimes
  6. 𝐌 = [ a b c d ] \mathbf{M}=\begin{bmatrix}a&b\\ c&d\\ \end{bmatrix}
  7. v e c ( 𝐌 ) = [ a c b d ] vec(\mathbf{M})=\begin{bmatrix}a\\ c\\ b\\ d\\ \end{bmatrix}
  8. [ 1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1 ] [ a c b d ] = [ a b c d ] = v e c ( 𝐌 T ) \begin{bmatrix}1&0&0&0\\ 0&0&1&0\\ 0&1&0&0\\ 0&0&0&1\\ \end{bmatrix}\cdot\begin{bmatrix}a\\ c\\ b\\ d\\ \end{bmatrix}=\begin{bmatrix}a\\ b\\ c\\ d\\ \end{bmatrix}=vec(\mathbf{M}^{T})

Commutative_non-associative_magmas.html

  1. M := { r , p , s } M:=\{r,p,s\}
  2. : M × M M \cdot:M\times M\to M
  3. x , y M x,y\in M
  4. x y x\neq y
  5. x x
  6. y y
  7. x y = y x = x x\cdot y=y\cdot x=x
  8. x x = x x\cdot x=x
  9. x x
  10. r p = p r = p r\cdot p=p\cdot r=p
  11. s s = s s\cdot s=s
  12. r p s r r p r p p p s s r s s \begin{array}[]{c|ccc}\cdot&r&p&s\\ \hline r&r&p&r\\ p&p&p&s\\ s&r&s&s\end{array}
  13. ( M , ) (M,\cdot)
  14. r ( p s ) = r s = r s = p s = ( r p ) s r\cdot(p\cdot s)=r\cdot s=r\neq s=p\cdot s=(r\cdot p)\cdot s
  15. r ( p s ) ( r p ) s r\cdot(p\cdot s)\neq(r\cdot p)\cdot s
  16. x y = ( x + y ) / 2 x\oplus y=(x+y)/2
  17. - 4 ( 0 + 4 ) = - 4 + 2 = - 1 + 1 = - 2 + 4 = ( - 4 0 ) + 4 -4\oplus(0\oplus+4)=-4\oplus+2=-1\neq+1=-2\oplus+4=(-4\oplus 0)\oplus+4
  18. K K
  19. A A
  20. K K
  21. ( x , y , z ) = x r + y p + z s (x,y,z)=xr+yp+zs
  22. x , y , z K x,y,z\in K
  23. r , p , s r,p,s
  24. { ( 1 , 0 , 0 ) , ( 0 , 1 , 0 ) , ( 0 , 0 , 1 ) } \{(1,0,0),(0,1,0),(0,0,1)\}
  25. { r , p , s } \{r,p,s\}
  26. A A
  27. A A
  28. M M
  29. K K
  30. K M K^{M}
  31. M M

Compact_closed_category.html

  1. ( 𝐂 , , I ) (\mathbf{C},\otimes,I)
  2. A C A\in C
  3. A * A^{*}
  4. A * A^{*}
  5. η A : I A * A \eta_{A}:I\to A^{*}\otimes A
  6. ε A : A A * I \varepsilon_{A}:A\otimes A^{*}\to I
  7. λ A ( ε A A ) α A , A * , A - 1 ( A η A ) ρ A - 1 = id A \lambda_{A}\circ(\varepsilon_{A}\otimes A)\circ\alpha_{A,A^{*},A}^{-1}\circ(A% \otimes\eta_{A})\circ\rho_{A}^{-1}=\mathrm{id}_{A}
  8. ρ A * ( A * ε A ) α A * , A , A * ( η A A * ) λ A * - 1 = id A * , \rho_{A^{*}}\circ(A^{*}\otimes\varepsilon_{A})\circ\alpha_{A^{*},A,A^{*}}\circ% (\eta_{A}\otimes A^{*})\circ\lambda_{A^{*}}^{-1}=\mathrm{id}_{A^{*}},
  9. λ , ρ \lambda,\rho
  10. ( 𝐂 , , I ) (\mathbf{C},\otimes,I)
  11. id A \mathrm{id}_{A}
  12. A A I A η A ( A * A ) ( A A * ) A ϵ A I A A A\xrightarrow{\cong}A\otimes I\xrightarrow{A\otimes\eta}A\otimes(A^{*}\otimes A% )\xrightarrow{\cong}(A\otimes A^{*})\otimes A\xrightarrow{\epsilon\otimes A}I% \otimes A\xrightarrow{\cong}A
  13. id A * \mathrm{id}_{A^{*}}
  14. A * I A * η A * ( A * A ) A * A * ( A A * ) A * ϵ A * I A * A^{*}\xrightarrow{\cong}I\otimes A^{*}\xrightarrow{\eta\otimes A^{*}}(A^{*}% \otimes A)\otimes A^{*}\xrightarrow{\cong}A^{*}\otimes(A\otimes A^{*})% \xrightarrow{A^{*}\otimes\epsilon}A^{*}\otimes I\xrightarrow{\cong}A^{*}
  15. ( 𝐂 , , I ) (\mathbf{C},\otimes,I)
  16. A * A^{*}
  17. A l A^{l}
  18. A r A^{r}
  19. η A l : I A A l \eta^{l}_{A}:I\to A\otimes A^{l}
  20. η A r : I A r A \eta^{r}_{A}:I\to A^{r}\otimes A
  21. ε A l : A l A I \varepsilon^{l}_{A}:A^{l}\otimes A\to I
  22. ε A r : A A r I \varepsilon^{r}_{A}:A\otimes A^{r}\to I
  23. A A I η r A ( A r A ) ( A A r ) A ϵ r I A A A\to A\otimes I\xrightarrow{\eta^{r}}A\otimes(A^{r}\otimes A)\to(A\otimes A^{r% })\otimes A\xrightarrow{\epsilon^{r}}I\otimes A\to A
  24. A I A η l ( A A l ) A A ( A l A ) ϵ l A I A A\to I\otimes A\xrightarrow{\eta^{l}}(A\otimes A^{l})\otimes A\to A\otimes(A^{% l}\otimes A)\xrightarrow{\epsilon^{l}}A\otimes I\to A
  25. A r I A r η r ( A r A ) A r A r ( A A r ) ϵ r A r I A r A^{r}\to I\otimes A^{r}\xrightarrow{\eta^{r}}(A^{r}\otimes A)\otimes A^{r}\to A% ^{r}\otimes(A\otimes A^{r})\xrightarrow{\epsilon^{r}}A^{r}\otimes I\to A^{r}
  26. A l A l I η l A l ( A A l ) ( A l A ) A l ϵ l I A l A l A^{l}\to A^{l}\otimes I\xrightarrow{\eta^{l}}A^{l}\otimes(A\otimes A^{l})\to(A% ^{l}\otimes A)\otimes A^{l}\xrightarrow{\epsilon^{l}}I\otimes A^{l}\to A^{l}
  27. f : A C B C f:A\otimes C\to B\otimes C
  28. Tr A , B C ( f ) = ρ B ( i d B ε C ) α B , C , C * ( f C * ) α A , C , C * - 1 ( i d A η C * ) ρ A - 1 : A B \mathrm{Tr_{A,B}^{C}}(f)=\rho_{B}\circ(id_{B}\otimes\varepsilon_{C})\circ% \alpha_{B,C,C^{*}}\circ(f\otimes C^{*})\circ\alpha_{A,C,C^{*}}^{-1}\circ(id_{A% }\otimes\eta_{C^{*}})\circ\rho_{A}^{-1}:A\to B
  29. A A I A η C * A ( C C * ) ( A C ) C * f C * ( B C ) C * B ( C C * ) B ε C B I B . A\xrightarrow{\cong}A\otimes I\xrightarrow{A\otimes\eta_{C^{*}}}A\otimes(C% \otimes C^{*})\xrightarrow{\cong}(A\otimes C)\otimes C^{*}\xrightarrow{\;\;f% \otimes C^{*}\;\;}(B\otimes C)\otimes C^{*}\xrightarrow{\cong}B\otimes(C% \otimes C^{*})\xrightarrow{B\otimes\varepsilon_{C}}B\otimes I\xrightarrow{% \cong}B.
  30. A * A^{*}
  31. A A
  32. f r ( n ) = sup { m | f ( m ) n } f^{r}(n)=\sup\{m\in\mathbb{N}|f(m)\leq n\}
  33. f l ( n ) = inf { m | n f ( m ) } f^{l}(n)=\inf\{m\in\mathbb{N}|n\leq f(m)\}
  34. id f f l (left unit) \mbox{id}\leq f\circ f^{l}\qquad\mbox{(left unit)}~{}
  35. id f r f (right unit) \mbox{id}\leq f^{r}\circ f\qquad\mbox{(right unit)}~{}
  36. f l f id (left counit) f^{l}\circ f\leq\mbox{id}\qquad\mbox{(left counit)}~{}
  37. f f r id (right counit) f\circ f^{r}\leq\mbox{id}\qquad\mbox{(right counit)}~{}
  38. f = f id f ( f r f ) = ( f f r ) f id f = f . f=f\circ\mbox{id}\leq f\circ(f^{r}\circ f)=(f\circ f^{r})\circ f\leq\mbox{id}% \circ f=f.
  39. \to
  40. \leq
  41. \otimes
  42. \circ

Compact_convergence.html

  1. ( X , 𝒯 ) (X,\mathcal{T})
  2. ( Y , d Y ) (Y,d_{Y})
  3. f n : X Y f_{n}:X\to Y
  4. n , n\in\mathbb{N},
  5. n n\to\infty
  6. f : X Y f:X\to Y
  7. K X K\subseteq X
  8. ( f n ) | K f | K (f_{n})|_{K}\to f|_{K}
  9. K K
  10. n n\to\infty
  11. K X K\subseteq X
  12. lim n sup x K d Y ( f n ( x ) , f ( x ) ) = 0. \lim_{n\to\infty}\sup_{x\in K}d_{Y}\left(f_{n}(x),f(x)\right)=0.
  13. X = ( 0 , 1 ) X=(0,1)\subset\mathbb{R}
  14. Y = Y=\mathbb{R}
  15. f n ( x ) := x n f_{n}(x):=x^{n}
  16. f n f_{n}
  17. X = ( 0 , 1 ] X=(0,1]
  18. Y = \R Y=\R
  19. f n ( x ) = x n f_{n}(x)=x^{n}
  20. f n f_{n}
  21. ( 0 , 1 ) (0,1)
  22. 1 1
  23. f n f f_{n}\to f
  24. f n f f_{n}\to f
  25. ( X , 𝒯 ) (X,\mathcal{T})
  26. f n f f_{n}\to f
  27. f n f f_{n}\to f
  28. ( X , 𝒯 ) (X,\mathcal{T})
  29. f n f f_{n}\to f
  30. f n f f_{n}\to f
  31. ( X , 𝒯 ) (X,\mathcal{T})
  32. f n f f_{n}\to f
  33. f n f_{n}
  34. f f

Compact_operator_on_Hilbert_space.html

  1. P m T x - T x ( 1 m + 1 ) 2 x . \left\|P_{m}Tx-Tx\right\|\leq\left(\frac{1}{m+1}\right)^{2}\|x\|.
  2. T x , y = x , T y , x , y H . \langle Tx,y\rangle=\langle x,Ty\rangle,\quad x,y\in H.
  3. f = y * T y = λ y * y \nabla f=\nabla\;y^{*}Ty=\lambda\cdot\nabla\;y^{*}y
  4. g ( x ) = T x , x x 2 , 0 x 𝐂 n . g(x)=\frac{\langle Tx,x\rangle}{\|x\|^{2}},\qquad 0\neq x\in\mathbf{C}^{n}.
  5. { h : 𝐑 𝐑 h ( t ) = g ( y + t z ) \begin{cases}h:\mathbf{R}\to\mathbf{R}\\ h(t)=g(y+tz)\end{cases}
  6. h ( 0 ) = lim t 0 h ( t ) - h ( 0 ) t - 0 = lim t 0 g ( y + t z ) - g ( y ) t = lim t 0 1 t ( T ( y + t z ) , y + t z y + t z 2 - T y , y y 2 ) = lim t 0 1 t ( T ( y + t z ) , y + t z - T y , y y 2 ) = 1 y 2 lim t 0 T ( y + t z ) , y + t z - T y , y t = 1 y 2 ( d d t T ( y + t z ) , y + t z y + t z , y + t z ) ( 0 ) = 0. \begin{aligned}\displaystyle h^{\prime}(0)&\displaystyle=\lim_{t\to 0}\frac{h(% t)-h(0)}{t-0}\\ &\displaystyle=\lim_{t\to 0}\frac{g(y+tz)-g(y)}{t}\\ &\displaystyle=\lim_{t\to 0}\frac{1}{t}\left(\frac{\langle T(y+tz),y+tz\rangle% }{\|y+tz\|^{2}}-\frac{\langle Ty,y\rangle}{\|y\|^{2}}\right)\\ &\displaystyle=\lim_{t\to 0}\frac{1}{t}\left(\frac{\langle T(y+tz),y+tz\rangle% -\langle Ty,y\rangle}{\|y\|^{2}}\right)\\ &\displaystyle=\frac{1}{\|y\|^{2}}\lim_{t\to 0}\frac{\langle T(y+tz),y+tz% \rangle-\langle Ty,y\rangle}{t}\\ &\displaystyle=\frac{1}{\|y\|^{2}}\left(\frac{d}{dt}\frac{\langle T(y+tz),y+tz% \rangle}{\langle y+tz,y+tz\rangle}\right)(0)\\ &\displaystyle=0.\end{aligned}
  7. m = T y , y y , y m=\frac{\langle Ty,y\rangle}{\langle y,y\rangle}
  8. ( T y - m y , z ) = 0. \Re\left(\langle Ty-my,z\rangle\right)=0.
  9. m ( T ) := sup { | T x , x | : x H , x 1 } , m(T):=\sup\bigl\{|\langle Tx,x\rangle|:x\in H,\,\|x\|\leq 1\bigr\},
  10. { f : H 𝐑 f ( x ) = T x , x \begin{cases}f:H\to\mathbf{R}\\ f(x)=\langle Tx,x\rangle\end{cases}
  11. Φ ( f ) ( e n ) = f ( λ n ) e n \Phi(f)(e_{n})=f(\lambda_{n})e_{n}
  12. Φ ( f ) = sup λ n σ ( T ) | f ( λ n ) | = f C ( σ ( T ) ) . \|\Phi(f)\|=\sup_{\lambda_{n}\in\sigma(T)}|f(\lambda_{n})|=\|f\|_{C(\sigma(T))}.
  13. Hom ( H , H ) \mathcal{F}\subseteq\operatorname{Hom}(H,H)
  14. ( q Q , T ) ( σ 𝐂 ) ( T - σ ) q = 0 (\forall{q\in Q,T\in\mathcal{F}})~{}(\exists{\sigma\in\mathbf{C}})~{}(T-\sigma% )q=0
  15. \mathcal{F}
  16. \mathcal{F}
  17. \mathcal{F}
  18. s S s\in S
  19. T T\in\mathcal{F}
  20. 0 α σ ( T S ) 0\neq\alpha\in\sigma(T\upharpoonright S)
  21. S := ker ( T S - α ) S^{\prime}:=\ker(T\upharpoonright S-\alpha)
  22. \mathcal{F}
  23. T T^{\prime}\in\mathcal{F}
  24. x ker ( T S - α ) x\in\ker(T\upharpoonright S-\alpha)
  25. ( T - α ) ( T x ) = ( T ( T x ) - α T x ) = 0 (T-\alpha)(T^{\prime}x)=(T^{\prime}(T~{}x)-\alpha T^{\prime}x)=0
  26. dim S < dim S \dim~{}S^{\prime}<\dim~{}S
  27. S S S^{\prime}\subseteq S
  28. \mathcal{F}
  29. \mathcal{F}
  30. 𝐏 = { A H : A is an orthonormal set of common eigenvectors for } , \mathbf{P}=\{A\subseteq H:A\,\text{ is an orthonormal set of common % eigenvectors for }\mathcal{F}\},
  31. S = Q S={\langle Q\rangle}^{\bot}
  32. \mathcal{F}
  33. \mathcal{F}
  34. T 0 T_{0}\in\mathcal{F}
  35. H = λ σ ( T 0 ) ker ( T 0 - σ ) ¯ , H=\overline{\bigoplus_{\lambda\in\sigma(T_{0})}\ker(T_{0}-\sigma)},
  36. σ ( T 0 ) \sigma(T_{0})
  37. \mathcal{F}
  38. ker ( T 0 - σ ) \ker(T_{0}-\sigma)
  39. Q := σ σ ( T 0 ) Q σ Q:=\bigcup_{\sigma\in\sigma(T_{0})}Q_{\sigma}
  40. Hom ( H , H ) \mathcal{F}\subseteq\operatorname{Hom}(H,H)
  41. T 0 T_{0}\in\mathcal{F}
  42. P Hom ( H , H ) × P\in\operatorname{Hom}(H,H)^{\times}
  43. P - 1 T 0 P P^{-1}~{}T_{0}~{}P
  44. P - 1 T 0 P P^{-1}T_{0}P
  45. ker ( P - 1 T 0 ( P - α ) ) , ker ( P - 1 T 0 ( P - α ) ) \ker\left(P^{-1}~{}T_{0}(P-\alpha)\right),\quad\ker\left(P^{-1}~{}T_{0}(P-% \alpha)\right)^{\bot}
  46. P - 1 P P^{-1}\mathcal{F}P
  47. P - 1 P P^{-1}\mathcal{F}P
  48. P ( Q 1 Q 2 ) P(Q_{1}\cup Q_{2})
  49. \mathcal{F}
  50. H = L 2 ( G ) H=L^{2}(G)
  51. { G × H H ( g f ) ( x ) = f ( g - 1 x ) \begin{cases}G\times H\to H\\ (gf)(x)=f(g^{-1}x)\end{cases}
  52. G U ( H ) G\subseteq U(H)
  53. R = T + T * 2 , J = T - T * 2 i . R=\frac{T+T^{*}}{2},\quad J=\frac{T-T^{*}}{2i}.
  54. ( M f ) ( x ) = x f ( x ) , f H , x [ 0 , 1 ] (Mf)(x)=xf(x),\quad f\in H,\,\,x\in[0,1]
  55. ( T K f ) ( x ) = 0 1 K ( x , y ) f ( y ) d y . (T_{K}f)(x)=\int_{0}^{1}K(x,y)f(y)\,\mathrm{d}y.
  56. K ( y , x ) = K ( x , y ) ¯ , x , y [ 0 , 1 ] . K(y,x)=\overline{K(x,y)},\quad x,y\in[0,1].
  57. λ n 2 < , K ( x , y ) λ n φ n ( x ) φ n ( y ) ¯ , \sum\lambda_{n}^{2}<\infty,\ \ K(x,y)\sim\sum\lambda_{n}\varphi_{n}(x)% \overline{\varphi_{n}(y)},

Comparison_triangle.html

  1. M k 2 M_{k}^{2}
  2. k k
  3. M 0 2 M_{0}^{2}
  4. M 1 2 M_{1}^{2}
  5. M - 1 2 M_{-1}^{2}
  6. X X
  7. T T
  8. X X
  9. p p
  10. q q
  11. r r
  12. T * T*
  13. M k 2 M_{k}^{2}
  14. T T
  15. M k 2 M_{k}^{2}
  16. p p^{\prime}
  17. q q^{\prime}
  18. r r^{\prime}
  19. d ( p , q ) = d ( p , q ) d(p,q)=d(p^{\prime},q^{\prime})
  20. d ( p , r ) = d ( p , r ) d(p,r)=d(p^{\prime},r^{\prime})
  21. d ( r , q ) = d ( r , q ) d(r,q)=d(r^{\prime},q^{\prime})
  22. T * T*
  23. p p^{\prime}
  24. q q
  25. r r
  26. p p
  27. q q
  28. r r
  29. p p

Completely_distributive_lattice.html

  1. j J k K j x j , k = f F j J x j , f ( j ) \bigwedge_{j\in J}\bigvee_{k\in K_{j}}x_{j,k}=\bigvee_{f\in F}\bigwedge_{j\in J% }x_{j,f(j)}
  2. { Y Y S } = { Z Z S # } \displaystyle\bigwedge\{\bigvee Y\mid Y\in S\}=\bigvee\{\bigwedge Z\mid Z\in S% ^{\#}\}
  3. ϕ : C L \phi:C\rightarrow L
  4. f : C M f:C\rightarrow M
  5. f ϕ * : L M f^{*}_{\phi}:L\rightarrow M
  6. f = f ϕ * ϕ f=f^{*}_{\phi}\circ\phi
  7. ( 𝒫 ( X ) , ) (\mathcal{P}(X),\subseteq)

Completely_positive_map.html

  1. A A
  2. B B
  3. ϕ : A B \phi:A\to B
  4. ϕ \phi
  5. a 0 ϕ ( a ) 0 a\geq 0\implies\phi(a)\geq 0
  6. ϕ : A B \phi:A\to B
  7. id ϕ : k × k A k × k B \textrm{id}\otimes\phi:\mathbb{C}^{k\times k}\otimes A\to\mathbb{C}^{k\times k% }\otimes B
  8. k × k A \mathbb{C}^{k\times k}\otimes A
  9. A k × k A^{k\times k}
  10. k × k k\times k
  11. A A
  12. id ϕ \textrm{id}\otimes\phi
  13. ( a 11 a 1 k a k 1 a k k ) ( ϕ ( a 11 ) ϕ ( a 1 k ) ϕ ( a k 1 ) ϕ ( a k k ) ) . \begin{pmatrix}a_{11}&\cdots&a_{1k}\\ \vdots&\ddots&\vdots\\ a_{k1}&\cdots&a_{kk}\end{pmatrix}\mapsto\begin{pmatrix}\phi(a_{11})&\cdots&% \phi(a_{1k})\\ \vdots&\ddots&\vdots\\ \phi(a_{k1})&\cdots&\phi(a_{kk})\end{pmatrix}.
  14. ϕ \phi
  15. id k × k Φ \textrm{id}_{\mathbb{C}^{k\times k}}\otimes\Phi
  16. ϕ \phi
  17. ϕ \phi
  18. a 1 a 2 ϕ ( a 1 ) ϕ ( a 2 ) a_{1}\leq a_{2}\implies\phi(a_{1})\leq\phi(a_{2})
  19. a 1 , a 2 A s a a_{1},a_{2}\in A_{sa}
  20. - a A 1 A a a A 1 A -\|a\|_{A}1_{A}\leq a\leq\|a\|_{A}1_{A}
  21. ϕ ( 1 A ) B \|\phi(1_{A})\|_{B}
  22. \to\mathbb{C}
  23. A A
  24. V : H 1 H 2 V:H_{1}\to H_{2}
  25. L ( H 1 ) L ( H 2 ) , A V A V L(H_{1})\to L(H_{2}),A\mapsto VAV^{\ast}
  26. ϕ : A \phi:A\to\mathbb{C}
  27. C ( X ) C ( Y ) C(X)\to C(Y)
  28. n × n \mathbb{C}^{n\times n}
  29. C 2 × 2 2 × 2 C^{2\times 2}\otimes\mathbb{C}^{2\times 2}
  30. [ ( 1 0 0 0 ) ( 0 1 0 0 ) ( 0 0 1 0 ) ( 0 0 0 1 ) ] = [ 1 0 0 1 0 0 0 0 0 0 0 0 1 0 0 1 ] . \begin{bmatrix}\begin{pmatrix}1&0\\ 0&0\end{pmatrix}&\begin{pmatrix}0&1\\ 0&0\end{pmatrix}\\ \begin{pmatrix}0&0\\ 1&0\end{pmatrix}&\begin{pmatrix}0&0\\ 0&1\end{pmatrix}\end{bmatrix}=\begin{bmatrix}1&0&0&1\\ 0&0&0&0\\ 0&0&0&0\\ 1&0&0&1\\ \end{bmatrix}.
  31. I 2 T I_{2}\otimes T
  32. [ ( 1 0 0 0 ) T ( 0 1 0 0 ) T ( 0 0 1 0 ) T ( 0 0 0 1 ) T ] = [ 1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1 ] , \begin{bmatrix}\begin{pmatrix}1&0\\ 0&0\end{pmatrix}^{T}&\begin{pmatrix}0&1\\ 0&0\end{pmatrix}^{T}\\ \begin{pmatrix}0&0\\ 1&0\end{pmatrix}^{T}&\begin{pmatrix}0&0\\ 0&1\end{pmatrix}^{T}\end{bmatrix}=\begin{bmatrix}1&0&0&0\\ 0&0&1&0\\ 0&1&0&0\\ 0&0&0&1\\ \end{bmatrix},
  33. \circ

Complex_Hadamard_matrix.html

  1. N × N N\times N
  2. H H
  3. | H j k | = 1 for j , k = 1 , 2 , , N |H_{jk}|=1{\quad\rm for\quad}j,k=1,2,\dots,N
  4. H H = N 𝕀 HH^{\dagger}=N\;{\mathbb{I}}
  5. {\dagger}
  6. 𝕀 {\mathbb{I}}
  7. H H
  8. 1 N \frac{1}{\sqrt{N}}
  9. 1 N \frac{1}{\sqrt{N}}
  10. N \sqrt{N}
  11. [ F N ] j k := exp [ ( 2 π i ( j - 1 ) ( k - 1 ) / N ] for j , k = 1 , 2 , , N [F_{N}]_{jk}:=\exp[(2\pi i(j-1)(k-1)/N]{\quad\rm for\quad}j,k=1,2,\dots,N
  12. H 1 H 2 H_{1}\simeq H_{2}
  13. D 1 , D 2 D_{1},D_{2}
  14. P 1 , P 2 P_{1},P_{2}
  15. H 1 = D 1 P 1 H 2 P 2 D 2 . H_{1}=D_{1}P_{1}H_{2}P_{2}D_{2}.
  16. N = 2 , 3 N=2,3
  17. 5 5
  18. F N F_{N}
  19. N = 4 N=4
  20. F 4 ( 1 ) ( a ) := [ 1 1 1 1 1 i e i a - 1 - i e i a 1 - 1 1 - 1 1 - i e i a - 1 i e i a ] with a [ 0 , π ) . F_{4}^{(1)}(a):=\begin{bmatrix}1&1&1&1\\ 1&ie^{ia}&-1&-ie^{ia}\\ 1&-1&1&-1\\ 1&-ie^{ia}&-1&ie^{ia}\end{bmatrix}{\quad\rm with\quad}a\in[0,\pi).
  21. N = 6 N=6
  22. F 6 F_{6}
  23. D 6 ( t ) D_{6}(t)
  24. B 6 ( θ ) B_{6}(\theta)
  25. C 6 C_{6}
  26. X 6 ( α ) X_{6}(\alpha)
  27. M 6 ( x ) M_{6}(x)
  28. K 6 ( x , y ) K_{6}(x,y)
  29. K 6 ( x , y , z ) K_{6}(x,y,z)
  30. G 6 G_{6}
  31. K 6 ( x , y , z ) K_{6}(x,y,z)
  32. S 6 H ( 3 , 6 ) S_{6}\in H(3,6)
  33. K 6 ( x , y , z ) , G 6 , S 6 K_{6}(x,y,z),G_{6},S_{6}
  34. N = 6 N=6

Complex_reflection_group.html

  1. S ( m , n ) . S(m,n).
  2. \ell
  3. d 1 d 2 d d_{1}\leq d_{2}\leq\ldots\leq d_{\ell}
  4. i = 1 ( q + d i - 1 ) = w W q dim ( V w ) . \prod_{i=1}^{\ell}(q+d_{i}-1)=\sum_{w\in W}q^{\dim(V^{w})}.
  5. \ell
  6. d 1 * d 2 * d * d^{*}_{1}\geq d^{*}_{2}\geq\ldots\geq d^{*}_{\ell}
  7. i = 1 ( q - d i * - 1 ) = w W det ( w ) q dim ( V w ) . \prod_{i=1}^{\ell}(q-d^{*}_{i}-1)=\sum_{w\in W}\det(w)q^{\dim(V^{w})}.
  8. \ell
  9. \ell
  10. + 1 \ell+1
  11. \ell
  12. d i + d i * = d d_{i}+d^{*}_{i}=d_{\ell}
  13. 1 i 1\leq i\leq\ell
  14. h h
  15. h := d h:=d_{\ell}

Composition_of_relations.html

  1. R X × Y R\subseteq X\times Y
  2. S Y × Z S\subseteq Y\times Z
  3. S R S\circ R
  4. S R = { ( x , z ) X × Z y Y : ( x , y ) R ( y , z ) S } . S\circ R=\{(x,z)\in X\times Z\mid\exists y\in Y:(x,y)\in R\land(y,z)\in S\}.
  5. S R X × Z S\circ R\subseteq X\times Z
  6. ( x , z ) S R (x,z)\in S\circ R
  7. y Y y\in Y
  8. x R y S z x\,R\,y\,S\,z
  9. ( x , y ) R (x,y)\in R
  10. ( y , z ) S (y,z)\in S
  11. l \circ_{l}
  12. r \circ_{r}
  13. \circ
  14. R X × Y R\subseteq X\times Y
  15. R : X Y R\colon X\to Y

Compton_edge.html

  1. 1 E - 1 E = 1 m e c 2 ( 1 - cos θ ) \frac{1}{E^{\prime}}-\frac{1}{E}=\frac{1}{m_{\,\text{e}}c^{2}}\left(1-\cos% \theta\right)
  2. E = E 1 + ( 1 - cos θ ) E m e c 2 E^{\prime}=\frac{E}{1+\frac{(1-\cos\theta)E}{m_{\,\text{e}}c^{2}}}
  3. m e m_{\,\text{e}}
  4. θ \theta
  5. θ \theta
  6. θ \theta
  7. E T = E - E E_{T}=E-E^{\prime}
  8. E Compton = E T ( max ) = 2 E 2 m e c 2 + 2 E E_{\,\text{Compton}}=E_{T}(\,\text{max})=\frac{2E^{2}}{m_{\,\text{e}}c^{2}+2E}

Computation_history.html

  1. M M
  2. M M
  3. ( S , I ) (S,I)
  4. ( T , J ) (T,J)
  5. I = a J I=aJ
  6. a a
  7. M M
  8. S S
  9. T T
  10. a a
  11. ϵ \epsilon
  12. M M
  13. 0011010101 q 00110101010... ...0011010101q00110101010...
  14. q q
  15. q q
  16. M M
  17. w w
  18. q 0 w 0 w 1 q_{0}w_{0}w_{1}...
  19. q 0 q_{0}
  20. q a q_{a}
  21. q r q_{r}
  22. c i + 1 c_{i+1}
  23. c i c_{i}
  24. c i c_{i}
  25. c i + 1 c_{i+1}
  26. c i + 1 c_{i+1}
  27. M M
  28. w w
  29. c 0 , c 1 , , c n c_{0},c_{1},...,c_{n}
  30. C 0 # C 1 r # C 2 # C 3 r # # C n C_{0}\#C^{r}_{1}\#C_{2}\#C^{r}_{3}\#...\#C_{n}
  31. C i C_{i}
  32. c i c_{i}
  33. M M
  34. M M
  35. A L L P D A ALL_{PDA}
  36. D D
  37. M M
  38. w w
  39. P P
  40. D ( P ) D(P)
  41. M M
  42. w w
  43. A T M A_{TM}

Conceptual_clustering.html

  1. C 1 C_{1}
  2. C 1 C_{1}
  3. 1 / 4 = 0.25 1/4=0.25
  4. 3 / 4 = 0.75 3/4=0.75
  5. C 1 C_{1}
  6. p ( x | C 1 ) = ( 0.25 , 0.75 , 0.75 ) p(x|C_{1})=(0.25,0.75,0.75)
  7. C 0 C_{0}
  8. C 1 C_{1}
  9. C 2 C_{2}
  10. C 0 C_{0}
  11. C 2 C_{2}
  12. C 3 C_{3}
  13. C 4 C_{4}
  14. C 5 C_{5}
  15. α \alpha
  16. α = 0.3 \alpha=0.3
  17. α \alpha
  18. α = 0.5 \alpha=0.5

Conchoid_of_Dürer.html

  1. 2 y 2 ( x 2 + y 2 ) - 2 b y 2 ( x + y ) + ( b 2 - 3 a 2 ) y 2 - a 2 x 2 + 2 a 2 b ( x + y ) + a 2 ( a 2 - b 2 ) = 0. 2y^{2}(x^{2}+y^{2})-2by^{2}(x+y)+(b^{2}-3a^{2})y^{2}-a^{2}x^{2}+2a^{2}b(x+y)+a% ^{2}(a^{2}-b^{2})=0.\,
  2. y = ± a / 2 y=\pm a/\sqrt{2}
  3. y = ± x / 2 y=\pm x/\sqrt{2}
  4. x 2 + y 2 = a 2 x^{2}+y^{2}=a^{2}

Concordant_pair.html

  1. sgn ( X 2 - X 1 ) = sgn ( Y 2 - Y 1 ) , \operatorname{sgn}(X_{2}-X_{1})\ =\operatorname{sgn}(Y_{2}-Y_{1}),
  2. sgn x = { - 1 : x < 0 0 : x = 0 1 : x > 0 \operatorname{sgn}x=\left\{\begin{matrix}-1&:&x<0\\ 0&:&x=0\\ 1&:&x>0\end{matrix}\right.
  3. sgn ( X 2 - X 1 ) = - sgn ( Y 2 - Y 1 ) . \operatorname{sgn}(X_{2}-X_{1})\ =-\operatorname{sgn}(Y_{2}-Y_{1}).

Conditioned_disjunction.html

  1. [ p , q , r ] ( q p ) and ( ¬ q r ) [p,q,r]~{}\leftrightarrow~{}(q\rightarrow p)\and(\neg q\rightarrow r)
  2. ( q and p ) ( ¬ q and r ) (q\and p)(\neg q\and r)

Cone_(category_theory).html

  1. ψ X : N F ( X ) \psi_{X}\colon N\to F(X)\,
  2. ψ X : F ( X ) N \psi_{X}\colon F(X)\to N\,

Conference_graph.html

  1. - 1 ± v 2 , \frac{-1\pm\sqrt{v}}{2},

Conference_matrix.html

  1. ( 0 + 1 + 1 + 1 + 1 + 1 + 1 0 + 1 - 1 - 1 + 1 + 1 + 1 0 + 1 - 1 - 1 + 1 - 1 + 1 0 + 1 - 1 + 1 - 1 - 1 + 1 0 + 1 + 1 + 1 - 1 - 1 + 1 0 ) \begin{pmatrix}0&+1&+1&+1&+1&+1\\ +1&0&+1&-1&-1&+1\\ +1&+1&0&+1&-1&-1\\ +1&-1&+1&0&+1&-1\\ +1&-1&-1&+1&0&+1\\ +1&+1&-1&-1&+1&0\end{pmatrix}
  2. ( 1 0 1 1 0 - 1 - 1 1 1 - 1 0 - 1 1 1 - 1 0 ) \begin{pmatrix}1&0&1&1\\ 0&-1&-1&1\\ 1&-1&0&-1\\ 1&1&-1&0\end{pmatrix}

Configuration_(geometry).html

  1. p p
  2. γ γ
  3. π π
  4. p γ = π p\gamma=\ell\pi\,
  5. p = p=ℓ
  6. γ = π γ=π
  7. n n
  8. p = l p=l
  9. n = 7 n=7
  10. n 16 n≥16
  11. n n
  12. Π Π
  13. n n
  14. Π Π
  15. P P
  16. Π Π
  17. P P
  18. P P
  19. l l
  20. P P
  21. l l
  22. l l
  23. P P
  24. n n
  25. k 3 k≥3
  26. k k
  27. j j
  28. j j
  29. ( j k ) (j≠k)
  30. j j
  31. | | N 00 N 01 N 02 N 10 N 11 N 12 N 20 N 21 N 22 | | \begin{vmatrix}\begin{vmatrix}N_{00}&N_{01}&N_{02}\\ N_{10}&N_{11}&N_{12}\\ N_{20}&N_{21}&N_{22}\end{vmatrix}\end{vmatrix}
  32. n n
  33. 0 j < n 0≤j<n

Conformational_entropy.html

  1. S = - R Σ i p i l n ( p i ) S=-R\Sigma_{i}p_{i}ln(p_{i})
  2. R R
  3. p i p_{i}
  4. i i

Conical_function.html

  1. P - ( 1 / 2 ) + i λ μ ( x ) P^{\mu}_{-(1/2)+i\lambda}(x)
  2. Q - ( 1 / 2 ) + i λ μ ( x ) . Q^{\mu}_{-(1/2)+i\lambda}(x).
  3. P - ( 1 / 2 ) + i λ μ ( x ) P^{\mu}_{-(1/2)+i\lambda}(x)
  4. K μ ( x ) K^{\mu}(x)
  5. K μ ( x ) K^{\mu}(x)

Conjugate_(algebra).html

  1. a 2 - b 2 a^{2}-b^{2}
  2. ( a + b ) ( a - b ) (a+b)(a-b)

Conoid.html

  1. x = v cos u + l f ( u ) , y = v sin u + m f ( u ) , z = n f ( u ) x=v\cos u+lf(u),y=v\sin u+mf(u),z=nf(u)\,

Consensus_(computer_science).html

  1. v v
  2. v v
  3. v v
  4. v v
  5. v v
  6. v v
  7. v v
  8. v v
  9. v v
  10. v v

Consequentia_mirabilis.html

  1. ( ¬ A A ) A (\neg A\rightarrow A)\rightarrow A
  2. ( ¬ ¬ A A ) A (\neg\neg AA)\rightarrow A

Constant-weight_code.html

  1. n n
  2. d d
  3. w w
  4. A ( n , d , w ) A(n,d,w)
  5. d d
  6. w w
  7. l o g 2 N log_{2}N
  8. N N
  9. n = N , d = 2 , w = 1 n=N,~{}d=2,~{}w=1
  10. A ( n , d , w ) = n A(n,d,w)=n
  11. 2 k / 2 2^{k}/2
  12. 2 k 2^{k}
  13. A ( n , d , w ) A(n,d,w)
  14. A ( n , d , w ) A(n,d,w)

Constant_of_motion.html

  1. A A
  2. d A d t = A t + { A , H } \frac{dA}{dt}=\frac{\partial A}{\partial t}+\{A,H\}
  3. A A
  4. B B
  5. { A , B } \{A,B\}
  6. d d t ψ | Q | ψ = - 1 i ψ | [ H , Q ] | ψ + ψ | d Q d t | ψ \frac{d}{dt}\langle\psi|Q|\psi\rangle=\frac{-1}{i\hbar}\langle\psi|\left[H,Q% \right]|\psi\rangle+\langle\psi|\frac{dQ}{dt}|\psi\rangle\,
  7. [ H , Q ] = H Q - Q H [H,Q]=HQ-QH\,
  8. Q = Q ( x , p , t ) Q=Q(x,p,t)\,
  9. i ψ t = H ψ . i\hbar\frac{\partial\psi}{\partial t}=H\psi.\,
  10. d d t Q \frac{d}{dt}\langle Q\rangle\,
  11. = d d t ψ | Q | ψ =\frac{d}{dt}\langle\psi|Q|\psi\rangle\,
  12. = d ψ d t | Q | ψ + ψ | d Q d t | ψ + ψ | Q | d ψ d t =\langle\frac{d\psi}{dt}|Q|\psi\rangle+\langle\psi|\frac{dQ}{dt}|\psi\rangle+% \langle\psi|Q|\frac{d\psi}{dt}\rangle\,
  13. = - 1 i H ψ | Q | ψ + ψ | d Q d t | ψ + 1 i ψ | Q | H ψ =\frac{-1}{i\hbar}\langle H\psi|Q|\psi\rangle+\langle\psi|\frac{dQ}{dt}|\psi% \rangle+\frac{1}{i\hbar}\langle\psi|Q|H\psi\rangle\,
  14. = - 1 i ψ | H Q | ψ + ψ | d Q d t | ψ + 1 i ψ | Q H | ψ =\frac{-1}{i\hbar}\langle\psi|HQ|\psi\rangle+\langle\psi|\frac{dQ}{dt}|\psi% \rangle+\frac{1}{i\hbar}\langle\psi|QH|\psi\rangle\,
  15. = - 1 i ψ | [ H , Q ] | ψ + ψ | d Q d t | ψ =\frac{-1}{i\hbar}\langle\psi|\left[H,Q\right]|\psi\rangle+\langle\psi|\frac{% dQ}{dt}|\psi\rangle\,
  16. d d t ψ | Q | ψ = - 1 i ψ | [ H , Q ] | ψ + ψ | d Q d t | ψ \frac{d}{dt}\langle\psi|Q|\psi\rangle=\frac{-1}{i\hbar}\langle\psi|\left[H,Q% \right]|\psi\rangle+\langle\psi|\frac{dQ}{dt}|\psi\rangle\,
  17. [ H , Q ] = 0 \left[H,Q\right]=0
  18. d d t Q = 0 \frac{d}{dt}\langle Q\rangle=0
  19. ψ \psi
  20. [ H , Q ] 0 \left[H,Q\right]\neq 0
  21. d d t Q = 0 \frac{d}{dt}\langle Q\rangle=0
  22. d d t Q \frac{d}{dt}\langle Q\rangle\,
  23. = - 1 i ψ | [ H , Q ] | ψ =\frac{-1}{i\hbar}\langle\psi|\left[H,Q\right]|\psi\rangle\,
  24. = - 1 i ψ | H Q - Q H | ψ =\frac{-1}{i\hbar}\langle\psi|HQ-QH|\psi\rangle\,
  25. H | ψ = E | ψ H|\psi\rangle=E|\psi\rangle\,
  26. d d t Q \frac{d}{dt}\langle Q\rangle\,
  27. = - 1 i ( E ψ | Q | ψ - E ψ | Q | ψ ) =\frac{-1}{i\hbar}\left(E\langle\psi|Q|\psi\rangle-E\langle\psi|Q|\psi\rangle% \right)\,
  28. = 0 =0
  29. 𝐋 = 𝐱 × 𝐯 \mathbf{L}=\mathbf{x}\times\mathbf{v}
  30. H ( 𝐱 , 𝐯 ) = 1 2 v 2 + Φ H(\mathbf{x},\mathbf{v})=\frac{1}{2}v^{2}+\Phi
  31. C ( x , v , t ) = x - v t C(x,v,t)=x-vt

Constraint_(mathematics).html

  1. min f ( x ) = x 1 2 + x 2 4 subject to: x 1 1 and x 2 = 1 \min f(x)=x_{1}^{2}+x_{2}^{4}\,\text{ subject to: }x_{1}\geq 1\,\text{ and }x_% {2}=1
  2. x x
  3. f ( x ) f(x)
  4. x = ( 1 , 1 ) x=(1,1)
  5. f ( x ) f(x)

Constraint_algebra.html

  1. E = ρ \nabla\cdot\vec{E}=\rho
  2. ( E ( x ) - ρ ( x ) ) | ψ = 0. (\nabla\cdot\vec{E}(x)-\rho(x))|\psi\rangle=0.

Contact_number.html

  1. β \beta
  2. α \alpha
  3. β \beta
  4. α \alpha

Contact_order.html

  1. C O = 1 L N N Δ S i , j CO={1\over{L\cdot N}}\sum^{N}\Delta S_{i,j}

Contact_process_(mathematics).html

  1. { 0 , 1 } S \{0,1\}^{S}
  2. S S
  3. d {}^{d}
  4. η \eta
  5. x x
  6. S S
  7. η ( x ) = 1 \eta(x)=1
  8. η ( x ) = 0 \eta(x)=0
  9. { 0 , , κ } S \{0,\ldots,\kappa\}^{S}
  10. x x
  11. 1 0 at rate 1 , 1\rightarrow 0\quad\mbox{at rate }~{}1,
  12. 0 1 at rate λ y : y x η ( y ) , 0\rightarrow 1\quad\mbox{at rate }~{}\lambda\sum_{y:y\sim x}\eta(y),
  13. S S
  14. x x
  15. S S
  16. λ c \lambda_{c}
  17. λ \lambda
  18. λ > λ c \lambda>\lambda_{c}
  19. λ < λ c \lambda<\lambda_{c}
  20. η ( x ) \eta(x)
  21. ξ t ( x ) : { 0 , 1 } \xi_{t}(x):\mathbb{Z}\to\{0,1\}
  22. x x
  23. t t
  24. r r
  25. θ \theta
  26. θ c \theta_{c}
  27. θ > θ c \theta>\theta_{c}
  28. θ ( 1 / 2 , θ c ) \theta\in(1/2,\theta_{c})
  29. ξ 0 ( x ) = 1 \xi_{0}(x)=1

Contact_resistance.html

  1. r c = { V J } V = 0 r_{c}=\left\{\frac{\partial V}{\partial J}\right\}_{V=0}
  2. Ω cm 2 \Omega\cdot\,\text{cm}^{2}
  3. Ω \Omega
  4. R t o t R_{tot}
  5. R t o t = R c + R c h = R c + L W C μ ( V g s - V d s ) R_{tot}=R_{c}+R_{ch}=R_{c}+\frac{L}{WC\mu(V_{gs}-V_{ds})}
  6. R c R_{c}
  7. R c h R_{ch}
  8. L / W L/W
  9. C C
  10. μ \mu
  11. V g s V_{gs}
  12. V d s V_{ds}
  13. ( 2 * π ) / k F (2*\pi)/k_{F}
  14. k F k_{F}
  15. 2 e 2 / h 2e^{2}/h
  16. e e
  17. h h

Continuous_group_action.html

  1. G × X X , ( g , x ) g x G\times X\to X,\quad(g,x)\mapsto g\cdot x
  2. f : H G f:H\to G
  3. h x = f ( h ) x h\cdot x=f(h)x
  4. G 1 G\to 1
  5. X H X^{H}
  6. h x = x hx=x
  7. F ( X , Y ) F(X,Y)
  8. ( g f ) ( x ) = g f ( g - 1 x ) (g\cdot f)(x)=gf(g^{-1}x)
  9. F ( X , Y ) G F(X,Y)^{G}
  10. f ( g x ) = g f ( x ) f(gx)=gf(x)
  11. F G ( X , Y ) = F ( X , Y ) G F_{G}(X,Y)=F(X,Y)^{G}
  12. F G ( G / H , X ) = X H F_{G}(G/H,X)=X^{H}

Continuous_knapsack_problem.html

  1. i x i W \sum_{i}x_{i}\leq W
  2. i x i v i \sum_{i}x_{i}v_{i}

Continuously_embedded.html

  1. i : X Y : x x i:X\hookrightarrow Y:x\mapsto x
  2. x Y C x X \|x\|_{Y}\leq C\|x\|_{X}
  3. i : 𝐑 𝐑 2 : x ( x , 0 ) i:\mathbf{R}\to\mathbf{R}^{2}:x\mapsto(x,0)
  4. p * = n p n - p . p^{*}=\frac{np}{n-p}.
  5. X = Y = C 0 ( [ 0 , 1 ] ; 𝐑 ) , X=Y=C^{0}([0,1];\mathbf{R}),
  6. f n ( x ) = { - n 2 x + n , 0 x 1 n ; 0 , otherwise. f_{n}(x)=\begin{cases}-n^{2}x+n,&0\leq x\leq\tfrac{1}{n};\\ 0,&\mbox{otherwise.}\end{cases}
  7. f n L 1 = 0 1 | f n ( x ) | d x = 1 2 . \|f_{n}\|_{L^{1}}=\int_{0}^{1}|f_{n}(x)|\,\mathrm{d}x=\frac{1}{2}.

Continuum_function.html

  1. κ 2 κ \kappa\mapsto 2^{\kappa}

Contrast_(statistics).html

  1. θ 1 , , θ t \theta_{1},\ldots,\theta_{t}
  2. a 1 , , a t a_{1},\ldots,a_{t}
  3. i = 1 t a i θ i \sum_{i=1}^{t}a_{i}\theta_{i}
  4. i = 1 t a i θ i \sum_{i=1}^{t}a_{i}\theta_{i}
  5. i = 1 t b i θ i \sum_{i=1}^{t}b_{i}\theta_{i}
  6. μ 1 , μ 2 , μ 3 , μ 4 \mu_{1},\mu_{2},\mu_{3},\mu_{4}
  7. μ 1 \mu_{1}
  8. μ 2 \mu_{2}
  9. μ 3 \mu_{3}
  10. μ 4 \mu_{4}
  11. L = c 1 X ¯ 1 + c 2 X ¯ 2 + + c k X ¯ k = c j X ¯ j L=c_{1}\bar{X}_{1}+c_{2}\bar{X}_{2}+\cdots+c_{k}\bar{X}_{k}=\sum c_{j}\bar{X}_% {j}
  12. X ¯ \bar{X}
  13. n ( c j X ¯ j ) 2 c j 2 \tfrac{n(\sum c_{j}\bar{X}_{j})^{2}}{\sum c_{j}^{2}}

Contrast_(vision).html

  1. Luminance difference Average luminance . \frac{\mbox{Luminance difference}~{}}{\mbox{Average luminance}~{}}.
  2. I - I b I b , \frac{I-I_{\mathrm{b}}}{I_{\mathrm{b}}},
  3. I I
  4. I b I_{\mathrm{b}}
  5. I max - I min I max + I min , \frac{I_{\mathrm{max}}-I_{\mathrm{min}}}{I_{\mathrm{max}}+I_{\mathrm{min}}},
  6. I max I_{\mathrm{max}}
  7. I min I_{\mathrm{min}}
  8. 1 M N i = 0 N - 1 j = 0 M - 1 ( I i j - I ¯ ) 2 , \sqrt{\frac{1}{MN}\sum_{i=0}^{N-1}\sum_{j=0}^{M-1}(I_{ij}-\bar{I})^{2}},
  9. I i j I_{ij}
  10. i i
  11. j j
  12. M M
  13. N N
  14. I ¯ \bar{I}
  15. I I
  16. [ 0 , 1 ] [0,1]

Contributions_of_Leonhard_Euler_to_mathematics.html

  1. π \pi
  2. - 1 \sqrt{-1}
  3. φ \varphi
  4. e i φ = cos φ + i sin φ . e^{i\varphi}=\cos\varphi+i\sin\varphi.\,
  5. e i π + 1 = 0 . e^{i\pi}+1=0\,.
  6. π \pi
  7. log ( x ) = log ( - x ) \log(x)=\log(-x)
  8. x x
  9. 2 log ( - x ) = log ( ( - x ) 2 ) = log ( x 2 ) = 2 log ( x ) 2\log(-x)=\log((-x)^{2})=\log(x^{2})=2\log(x)
  10. i π i\pi
  11. e = n = 0 1 n ! = lim n ( 1 0 ! + 1 1 ! + 1 2 ! + + 1 n ! ) . e=\sum_{n=0}^{\infty}{1\over n!}=\lim_{n\to\infty}\left(\frac{1}{0!}+\frac{1}{% 1!}+\frac{1}{2!}+\cdots+\frac{1}{n!}\right).
  12. arctan z = n = 0 ( - 1 ) n z 2 n + 1 2 n + 1 . \arctan z=\sum_{n=0}^{\infty}\frac{(-1)^{n}z^{2n+1}}{2n+1}.
  13. lim n ( 1 1 2 + 1 2 2 + 1 3 2 + + 1 n 2 ) = π 2 6 . \lim_{n\to\infty}\left(\frac{1}{1^{2}}+\frac{1}{2^{2}}+\frac{1}{3^{2}}+\cdots+% \frac{1}{n^{2}}\right)=\frac{\pi^{2}}{6}.
  14. γ = lim n ( 1 + 1 2 + 1 3 + 1 4 + + 1 n - ln ( n ) ) . \gamma=\lim_{n\rightarrow\infty}\left(1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+% \cdots+\frac{1}{n}-\ln(n)\right).

Control-Lyapunov_function.html

  1. V ( x ) V(x)
  2. x 0 x\neq 0
  3. x = 0 x=0
  4. u ( x , t ) u(x,t)
  5. x ˙ = f ( x , u ) \dot{x}=f(x,u)
  6. x 𝐑 n x\in\mathbf{R}^{n}
  7. u 𝐑 m u\in\mathbf{R}^{m}
  8. x = 0 x=0
  9. D 𝐑 n D\subset\mathbf{R}^{n}
  10. V : D 𝐑 V:D\rightarrow\mathbf{R}
  11. V ( x ) V(x)
  12. x = 0 x=0
  13. x 0 , u V ˙ ( x , u ) = V ( x ) f ( x , u ) < 0. \forall x\neq 0,\exists u\qquad\dot{V}(x,u)=\nabla V(x)\cdot f(x,u)<0.
  14. u * ( x ) = arg min u V ( x ) f ( x , u ) u^{*}(x)=\arg\min_{u}\nabla V(x)\cdot f(x,u)
  15. m ( 1 + q 2 ) q ¨ + b q ˙ + K 0 q + K 1 q 3 = u m(1+q^{2})\ddot{q}+b\dot{q}+K_{0}q+K_{1}q^{3}=u
  16. q d q_{d}
  17. q q
  18. e = q d - q e=q_{d}-q
  19. r r
  20. r = e ˙ + α e r=\dot{e}+\alpha e
  21. V = 1 2 r 2 V=\frac{1}{2}r^{2}
  22. q 0 q\neq 0
  23. q ˙ 0 \dot{q}\neq 0
  24. V V
  25. V ˙ = r r ˙ \dot{V}=r\dot{r}
  26. V ˙ = ( e ˙ + α e ) ( e ¨ + α e ˙ ) \dot{V}=(\dot{e}+\alpha e)(\ddot{e}+\alpha\dot{e})
  27. V ˙ = - κ V \dot{V}=-\kappa V
  28. V V
  29. V ˙ \dot{V}
  30. ( e ¨ + α e ˙ ) = ( q ¨ d - q ¨ + α e ˙ ) (\ddot{e}+\alpha\dot{e})=(\ddot{q}_{d}-\ddot{q}+\alpha\dot{e})
  31. ( q ¨ d - q ¨ + α e ˙ ) = - κ 2 ( e ˙ + α e ) (\ddot{q}_{d}-\ddot{q}+\alpha\dot{e})=-\frac{\kappa}{2}(\dot{e}+\alpha e)
  32. q ¨ \ddot{q}
  33. ( q ¨ d - u - K 0 q - K 1 q 3 - b q ˙ m ( 1 + q 2 ) + α e ˙ ) = - κ 2 ( e ˙ + α e ) (\ddot{q}_{d}-\frac{u-K_{0}q-K_{1}q^{3}-b\dot{q}}{m(1+q^{2})}+\alpha\dot{e})=-% \frac{\kappa}{2}(\dot{e}+\alpha e)
  34. u u
  35. u = m ( 1 + q 2 ) ( q ¨ d + α e ˙ + κ 2 r ) + K 0 q + K 1 q 3 + b q ˙ u=m(1+q^{2})(\ddot{q}_{d}+\alpha\dot{e}+\frac{\kappa}{2}r)+K_{0}q+K_{1}q^{3}+b% \dot{q}
  36. κ \kappa
  37. α \alpha
  38. V ˙ = - κ V \dot{V}=-\kappa V
  39. V = V ( 0 ) e - κ t V=V(0)e^{-\kappa t}
  40. V = 1 2 ( e ˙ + α e ) 2 V=\frac{1}{2}(\dot{e}+\alpha e)^{2}
  41. V V
  42. e e
  43. r r ˙ = - κ 2 r 2 r\dot{r}=-\frac{\kappa}{2}r^{2}
  44. r ˙ = - κ 2 r \dot{r}=-\frac{\kappa}{2}r
  45. r = r ( 0 ) e - κ 2 t r=r(0)e^{-\frac{\kappa}{2}t}
  46. e ˙ + α e = ( e ˙ ( 0 ) + α e ( 0 ) ) e - κ 2 t \dot{e}+\alpha e=(\dot{e}(0)+\alpha e(0))e^{-\frac{\kappa}{2}t}

Control_point_(mathematics).html

  1. d d
  2. i {}_{i}
  3. i \sum_{i}
  4. ϕ i i {}_{i}\phi_{i}
  5. d d
  6. ϕ i \phi_{i}
  7. ϕ i \phi_{i}

Control_reconfiguration.html

  1. { 𝐱 ˙ = 𝐀𝐱 + 𝐁𝐮 𝐲 = 𝐂𝐱 \begin{cases}\dot{\mathbf{x}}&=\mathbf{A}\mathbf{x}+\mathbf{B}\mathbf{u}\\ \mathbf{y}&=\mathbf{C}\mathbf{x}\end{cases}
  2. { 𝐱 ˙ f = 𝐀 f 𝐱 f + 𝐁 f 𝐮 𝐲 f = 𝐂 f 𝐱 f \begin{cases}\dot{\mathbf{x}}_{f}&=\mathbf{A}_{f}\mathbf{x}_{f}+\mathbf{B}_{f}% \mathbf{u}\\ \mathbf{y}_{f}&=\mathbf{C}_{f}\mathbf{x}_{f}\end{cases}
  3. f f
  4. 𝐁 f \mathbf{B}_{f}
  5. 𝐂 f \mathbf{C}_{f}
  6. 𝐀 f \mathbf{A}_{f}
  7. 𝐮 , 𝐲 \mathbf{u},\mathbf{y}
  8. 𝐰 \mathbf{w}
  9. 𝐟 \mathbf{f}
  10. { 𝐱 ˙ f = 𝐀𝐱 f + 𝐁𝐮 + 𝐄𝐟 𝐲 f = 𝐂 f 𝐱 f + 𝐅𝐟 \begin{cases}\dot{\mathbf{x}}_{f}&=\mathbf{A}\mathbf{x}_{f}+\mathbf{B}\mathbf{% u}+\mathbf{E}\mathbf{f}\\ \mathbf{y}_{f}&=\mathbf{C}_{f}\mathbf{x}_{f}+\mathbf{F}\mathbf{f}\end{cases}
  11. 𝐀 ¯ = 𝐀 - 𝐁𝐊 \bar{\mathbf{A}}=\mathbf{A}-\mathbf{B}\mathbf{K}
  12. 𝐊 f \mathbf{K}_{f}
  13. 𝐀 ¯ \bar{\mathbf{A}}

Controllability_Gramian.html

  1. x ˙ = A x + B u \dot{x}=Ax+Bu
  2. A A
  3. W c W_{c}
  4. A W c + W c A T = - B B T AW_{c}+W_{c}A^{T}=-BB^{T}
  5. ( A , B ) (A,B)
  6. W c W_{c}
  7. W c = 0 e A τ B B T e A T τ d τ W_{c}=\int\limits_{0}^{\infty}e^{A\tau}BB^{T}e^{A^{T}\tau}\;d\tau
  8. W c ( t 0 , t 1 ) = t 0 t 1 e A ( t 0 - τ ) B B T e A T ( t 0 - τ ) d τ W_{c}(t_{0},t_{1})=\int_{t_{0}}^{t_{1}}e^{A(t_{0}-\tau)}BB^{T}e^{A^{T}(t_{0}-% \tau)}\;d\tau
  9. ( A , B ) (A,B)
  10. W c ( t 0 , t 1 ) W_{c}(t_{0},t_{1})
  11. t 1 > t 0 t_{1}>t_{0}
  12. W c W_{c}
  13. x ˙ ( t ) = A ( t ) x ( t ) + B ( t ) u ( t ) \dot{x}(t)=A(t)x(t)+B(t)u(t)
  14. y ( t ) = C ( t ) x ( t ) + D ( t ) u ( t ) y(t)=C(t)x(t)+D(t)u(t)
  15. [ t 0 , t 1 ] [t_{0},t_{1}]
  16. W c ( t 0 , t 1 ) W_{c}(t_{0},t_{1})
  17. W c ( t 0 , t 1 ) = t 0 t 1 Φ ( t 0 , τ ) B ( τ ) B T ( τ ) Φ T ( t 0 , τ ) d τ W_{c}(t_{0},t_{1})=\int\limits_{t_{0}}^{t_{1}}\Phi(t_{0},\tau)B(\tau)B^{T}(% \tau)\Phi^{T}(t_{0},\tau)\;d\tau

Convergence_in_measure.html

  1. f , f n ( n ) : X f,f_{n}\ (n\in\mathbb{N}):X\to\mathbb{R}
  2. lim n μ ( { x X : | f ( x ) - f n ( x ) | ε } ) = 0 \lim_{n\to\infty}\mu(\{x\in X:|f(x)-f_{n}(x)|\geq\varepsilon\})=0
  3. F Σ F\in\Sigma
  4. μ ( F ) < \mu(F)<\infty
  5. lim n μ ( { x F : | f ( x ) - f n ( x ) | ε } ) = 0 \lim_{n\to\infty}\mu(\{x\in F:|f(x)-f_{n}(x)|\geq\varepsilon\})=0
  6. \in
  7. μ ( X ) < \mu(X)<\infty
  8. X = X=\mathbb{R}
  9. f n = χ [ n , ) f_{n}=\chi_{[n,\infty)}
  10. f n = χ [ j 2 k , j + 1 2 k ] f_{n}=\chi_{[\frac{j}{2^{k}},\frac{j+1}{2^{k}}]}
  11. k = log 2 n k=\lfloor\log_{2}n\rfloor
  12. j = n - 2 k j=n-2^{k}
  13. χ [ 0 , 1 ] , χ [ 0 , 1 2 ] , χ [ 1 2 , 1 ] , χ [ 0 , 1 4 ] , χ [ 1 4 , 1 2 ] \chi_{\left[0,1\right]},\;\chi_{\left[0,\frac{1}{2}\right]},\;\chi_{\left[% \frac{1}{2},1\right]},\;\chi_{\left[0,\frac{1}{4}\right]},\;\chi_{\left[\frac{% 1}{4},\frac{1}{2}\right]}
  14. f n = n χ [ 0 , 1 n ] f_{n}=n\chi_{\left[0,\frac{1}{n}\right]}
  15. p 1 p\geq 1
  16. { ρ F : F Σ , μ ( F ) < } , \{\rho_{F}:F\in\Sigma,\ \mu(F)<\infty\},
  17. ρ F ( f , g ) = F min { | f - g | , 1 } d μ \rho_{F}(f,g)=\int_{F}\min\{|f-g|,1\}\,d\mu
  18. G X G\subset X
  19. ε > 0 \varepsilon>0
  20. μ ( G F ) < ε . \mu(G\setminus F)<\varepsilon.
  21. μ ( X ) < \mu(X)<\infty
  22. ρ X \rho_{X}
  23. μ \mu
  24. d ( f , g ) := inf δ > 0 μ ( { | f - g | δ } ) + δ d(f,g):=\inf\limits_{\delta>0}\mu(\{|f-g|\geq\delta\})+\delta

Convergence_of_measures.html

  1. μ n \mu_{n}
  2. f d μ n f d μ \int f\,d\mu_{n}\to\int f\,d\mu
  3. f f
  4. f f
  5. μ n ( A ) μ ( A ) \mu_{n}(A)\to\mu(A)
  6. A A
  7. f d μ n f d μ \int f\,d\mu_{n}\to\int f\,d\mu
  8. f f
  9. f f
  10. ϵ > 0 \epsilon>0
  11. | μ n ( A ) - μ ( A ) | < ϵ |\mu_{n}(A)-\mu(A)|<\epsilon
  12. A A
  13. ( X , ) (X,\mathcal{F})
  14. μ - ν T V = sup f { X f d μ - X f d ν } . \left\|\mu-\nu\right\|_{TV}=\sup_{f}\left\{\int_{X}fd\mu-\int_{X}fd\nu\right\}.
  15. μ - ν T V = 2 sup A | μ ( A ) - ν ( A ) | . \left\|\mu-\nu\right\|_{TV}=2\cdot\sup_{A\in\mathcal{F}}|\mu(A)-\nu(A)|.
  16. 2 + μ - ν T V 4 {2+\|\mu-\nu\|_{TV}\over 4}
  17. μ n - μ T V < ϵ \|\mu_{n}-\mu\|_{TV}<\epsilon
  18. ( X , ) (X,\mathcal{F})
  19. lim n μ n ( A ) = μ ( A ) \lim_{n\to\infty}\mu_{n}(A)=\mu(A)
  20. A A\in\mathcal{F}
  21. P n P P_{n}\Rightarrow P
  22. { U ϕ , x , δ | ϕ : S 𝐑 is bounded and continuous, x 𝐑 and δ > 0 } , \left\{U_{\phi,x,\delta}\left|\begin{array}[]{c}\phi\colon S\to\mathbf{R}\,% \text{ is bounded and continuous,}\\ x\in\mathbf{R}\,\text{ and }\delta>0\end{array}\right.\right\},
  23. U ϕ , x , δ := { μ s y m b o l P ( S ) | | S ϕ d μ - x | < δ } . U_{\phi,x,\delta}:=\left\{\mu\in symbol{P}(S)\left|\left|\int_{S}\phi\mathrm{d% }\mu-x\right|<\delta\right.\right\}.
  24. P n P P_{n}\Rightarrow P
  25. P n P P_{n}\rightharpoonup P
  26. P n 𝒟 P . P_{n}\xrightarrow{\mathcal{D}}P.
  27. ( Ω , , ) (\Omega,\mathcal{F},\mathbb{P})

Convex_preferences.html

  1. \geq
  2. \succeq
  3. x , y , z X x,y,z\in X
  4. y x y\succeq x
  5. z x z\succeq x
  6. θ y + ( 1 - θ ) z x \theta y+(1-\theta)z\succeq x
  7. θ [ 0 , 1 ] \theta\in[0,1]
  8. P P
  9. x , y , z X x,y,z\in X
  10. y x y\succeq x
  11. z x z\succeq x
  12. y z y\neq z
  13. θ y + ( 1 - θ ) z x \theta y+(1-\theta)z\succ x
  14. θ ( 0 , 1 ) ; \theta\in(0,1);
  15. \succ

Coordinate-induced_basis.html

  1. x a x^{a}
  2. e a e_{a}
  3. e a = x a e_{a}=\frac{\partial}{\partial x^{a}}
  4. ω a \omega^{a}
  5. ω a = d x a . \omega^{a}=dx^{a}.\,

Correlogram.html

  1. r h r_{h}\,
  2. h h\,
  3. Y = constant + error Y=\mathrm{constant}+\mathrm{error}
  4. s Y ¯ = s / N s_{\bar{Y}}=s/\sqrt{N}
  5. s Y ¯ = s / N s_{\bar{Y}}=s/\sqrt{N}
  6. Y = constant + error Y=\mathrm{constant}+\mathrm{error}
  7. r h = c h / c 0 r_{h}=c_{h}/c_{0}\,
  8. c h = 1 N t = 1 N - h ( Y t - Y ¯ ) ( Y t + h - Y ¯ ) c_{h}=\frac{1}{N}\sum_{t=1}^{N-h}\left(Y_{t}-\bar{Y}\right)\left(Y_{t+h}-\bar{% Y}\right)
  9. c 0 = 1 N t = 1 N ( Y t - Y ¯ ) 2 c_{0}=\frac{1}{N}\sum_{t=1}^{N}\left(Y_{t}-\bar{Y}\right)^{2}
  10. c h = 1 N - h t = 1 N - h ( Y t - Y ¯ ) ( Y t + h - Y ¯ ) c_{h}=\frac{1}{N-h}\sum_{t=1}^{N-h}\left(Y_{t}-\bar{Y}\right)\left(Y_{t+h}-% \bar{Y}\right)
  11. α \alpha\,
  12. B = ± z 1 - α / 2 S E ( r h ) B=\pm z_{1-\alpha/2}SE(r_{h})\,
  13. r h r_{h}\,
  14. h h\,
  15. α \alpha\,
  16. S E ( r 1 ) = 1 N SE(r_{1})=\frac{1}{\sqrt{N}}
  17. S E ( r h ) = 1 + 2 i = 1 h - 1 r i 2 N SE(r_{h})=\sqrt{\frac{1+2\sum_{i=1}^{h-1}r^{2}_{i}}{N}}
  18. h > 1. h>1.\,
  19. ± z 1 - α / 2 N \pm\frac{z_{1-\alpha/2}}{\sqrt{N}}
  20. ± z 1 - α / 2 1 N ( 1 + 2 i = 1 k r i 2 ) \pm z_{1-\alpha/2}\sqrt{\frac{1}{N}\left(1+2\sum_{i=1}^{k}r_{i}^{2}\right)}

COST_Hata_model.html

  1. L = 46.3 + 33.9 log f - 13.82 log h B - a ( h R ) + [ 44.9 - 6.55 log h B ] log d + C L\;=\;46.3\;+\;33.9\log f\;-\;13.82\log h_{B}\;-\;a(h_{R})\;+\;[44.9\;-\;6.55% \log h_{B}]\log d\;+\;C
  2. a ( h R ) = ( 1.1 log f - 0.7 ) h R - ( 1.56 log f - 0.8 ) a(h_{R})\;=\;(1.1\log f\;-\;0.7)h_{R}\;-\;(1.56\log f\;-\;0.8)
  3. C = { 0 d B for medium cities and suburban areas 3 d B for metropolitan areas C\;=\;\begin{cases}0\;dB\mbox{ for medium cities and suburban areas}\\ 3\;dB\mbox{ for metropolitan areas}\end{cases}

Cotlar–Stein_lemma.html

  1. L 2 L^{2}
  2. E , F E,\,F
  3. T j T_{j}
  4. j 1 j\geq 1
  5. T j T_{j}
  6. E E
  7. F F
  8. a j k = T j T k , b j k = T j T k . a_{jk}=\|T_{j}T_{k}^{\ast}\|,\qquad b_{jk}=\|T_{j}^{\ast}T_{k}\|.
  9. T j : E F T_{j}:\;E\to F
  10. j 1 , j\geq 1,
  11. A = sup j k a j k < , B = sup j k b j k < . A=\sup_{j}\sum_{k}\sqrt{a_{jk}}<\infty,\qquad B=\sup_{j}\sum_{k}\sqrt{b_{jk}}<\infty.
  12. T j T_{j}
  13. j T j \sum_{j}T_{j}
  14. j T j A B . \|\sum_{j}T_{j}\|\leq\sqrt{AB}.
  15. i , j | ( R i v , R j v ) | ( max i j R i * R j 1 2 ) ( max i j R i R j * 1 2 ) v 2 . \displaystyle{\sum_{i,j}|(R_{i}v,R_{j}v)|\leq\left(\max_{i}\sum_{j}\|R_{i}^{*}% R_{j}\|^{1\over 2}\right)\left(\max_{i}\sum_{j}\|R_{i}R_{j}^{*}\|^{1\over 2}% \right)\|v\|^{2}.}
  16. i , j | ( T i v , T j v ) | A B v 2 . \displaystyle{\sum_{i,j}|(T_{i}v,T_{j}v)|\leq AB\|v\|^{2}.}
  17. i = 1 n T i v 2 A B v 2 , \displaystyle{\|\sum_{i=1}^{n}T_{i}v\|^{2}\leq AB\|v\|^{2},}
  18. i = m n T i v 2 i , j m | ( T i v , T j v ) | . \displaystyle{\|\sum_{i=m}^{n}T_{i}v\|^{2}\leq\sum_{i,j\geq m}|(T_{i}v,T_{j}v)% |.}
  19. s n = i = 1 n T i v \displaystyle{s_{n}=\sum_{i=1}^{n}T_{i}v}
  20. R = a i j R i * R j \displaystyle{R=\sum a_{ij}R_{i}^{*}R_{j}}
  21. ( R v , v ) = | ( R v , v ) | = | ( R i v , R j v ) | . \displaystyle{(Rv,v)=|(Rv,v)|=\sum|(R_{i}v,R_{j}v)|.}
  22. R 2 m = ( R * R ) m R i 1 * R i 2 R i 3 * R i 4 R i 2 m ( R i 1 * R i 1 * R i 2 R i 2 R i 3 * R i 2 m - 1 * R i 2 m R i 2 m ) 1 2 . \displaystyle{\|R\|^{2m}=\|(R^{*}R)^{m}\|\leq\sum\|R_{i_{1}}^{*}R_{i_{2}}R_{i_% {3}}^{*}R_{i_{4}}\cdots R_{i_{2m}}\|\leq\sum\left(\|R_{i_{1}}^{*}\|\|R_{i_{1}}% ^{*}R_{i_{2}}\|\|R_{i_{2}}R_{i_{3}}^{*}\|\cdots\|R_{i_{2m-1}}^{*}R_{i_{2m}}\|% \|R_{i_{2m}}\|\right)^{1\over 2}.}
  23. R 2 m n max R i ( max i j R i * R j 1 2 ) 2 m ( max i j R i R j * 1 2 ) 2 m - 1 . \displaystyle{\|R\|^{2m}\leq n\cdot\max\|R_{i}\|\left(\max_{i}\sum_{j}\|R_{i}^% {*}R_{j}\|^{1\over 2}\right)^{2m}\left(\max_{i}\sum_{j}\|R_{i}R_{j}^{*}\|^{1% \over 2}\right)^{2m-1}.}
  24. R ( max i j R i * R j 1 2 ) ( max i j R i R j * 1 2 ) , \displaystyle{\|R\|\leq\left(\max_{i}\sum_{j}\|R_{i}^{*}R_{j}\|^{1\over 2}% \right)\left(\max_{i}\sum_{j}\|R_{i}R_{j}^{*}\|^{1\over 2}\right),}
  25. A = sup x X T ( x ) * T ( y ) 1 2 d μ ( y ) , B = sup x X T ( y ) T ( x ) * 1 2 d μ ( y ) , \displaystyle{A=\sup_{x}\int_{X}\|T(x)^{*}T(y)\|^{1\over 2}\,d\mu(y),\,\,\,B=% \sup_{x}\int_{X}\|T(y)T(x)^{*}\|^{1\over 2}\,d\mu(y),}
  26. X T ( x ) v d μ ( x ) A B v . \displaystyle{\|\int_{X}T(x)v\,d\mu(x)\|\leq\sqrt{AB}\cdot\|v\|.}
  27. T = [ 1 0 0 0 1 0 0 0 1 ] T=\left[\begin{array}[]{cccc}1&0&0&\vdots\\ 0&1&0&\vdots\\ 0&0&1&\vdots\\ \cdots&\cdots&\cdots&\ddots\end{array}\right]
  28. T 1 = [ 1 0 0 0 0 0 0 0 0 ] , T 2 = [ 0 0 0 0 1 0 0 0 0 ] , T 3 = [ 0 0 0 0 0 0 0 0 1 ] , . \qquad T_{1}=\left[\begin{array}[]{cccc}1&0&0&\vdots\\ 0&0&0&\vdots\\ 0&0&0&\vdots\\ \cdots&\cdots&\cdots&\ddots\end{array}\right],\qquad T_{2}=\left[\begin{array}% []{cccc}0&0&0&\vdots\\ 0&1&0&\vdots\\ 0&0&0&\vdots\\ \cdots&\cdots&\cdots&\ddots\end{array}\right],\qquad T_{3}=\left[\begin{array}% []{cccc}0&0&0&\vdots\\ 0&0&0&\vdots\\ 0&0&1&\vdots\\ \cdots&\cdots&\cdots&\ddots\end{array}\right],\qquad\dots.
  29. T j = 1 \|T_{j}\|=1
  30. j j
  31. j T j \sum_{j\in\mathbb{N}}T_{j}
  32. T j T k = 0 \|T_{j}T_{k}^{\ast}\|=0
  33. T j T k = 0 \|T_{j}^{\ast}T_{k}\|=0
  34. j k j\neq k
  35. T = j T j T=\sum_{j\in\mathbb{N}}T_{j}

Cotorsion_group.html

  1. C C
  2. E x t ( G , C ) = 0 Ext(G,C)=0
  3. G G
  4. G G

Countably_generated_space.html

  1. V U V\cap U

Couple_(mechanics).html

  1. τ = F d \tau=Fd\,
  2. τ \tau
  3. e ^ \hat{e}
  4. τ = 𝐝 × 𝐅 . \mathbf{\tau}=\mathbf{d}\times\mathbf{F}.
  5. M = 𝐫 1 × 𝐅 1 + 𝐫 2 × 𝐅 2 + M=\mathbf{r}_{1}\times\mathbf{F}_{1}+\mathbf{r}_{2}\times\mathbf{F}_{2}+\cdots
  6. M = ( 𝐫 1 + 𝐫 ) × 𝐅 1 + ( 𝐫 2 + 𝐫 ) × 𝐅 2 + M^{\prime}=(\mathbf{r}_{1}+\mathbf{r})\times\mathbf{F}_{1}+(\mathbf{r}_{2}+% \mathbf{r})\times\mathbf{F}_{2}+\cdots
  7. M = ( 𝐫 1 × 𝐅 1 + 𝐫 2 × 𝐅 2 + ) + 𝐫 × ( 𝐅 1 + 𝐅 2 + ) . M^{\prime}=\left(\mathbf{r}_{1}\times\mathbf{F}_{1}+\mathbf{r}_{2}\times% \mathbf{F}_{2}+\cdots\right)+\mathbf{r}\times\left(\mathbf{F}_{1}+\mathbf{F}_{% 2}+\cdots\right).
  8. 𝐅 1 + 𝐅 2 + = 0. \mathbf{F}_{1}+\mathbf{F}_{2}+\cdots=0.
  9. M = 𝐫 1 × 𝐅 1 + 𝐫 2 × 𝐅 2 + = M M^{\prime}=\mathbf{r}_{1}\times\mathbf{F}_{1}+\mathbf{r}_{2}\times\mathbf{F}_{% 2}+\cdots=M

Courant_bracket.html

  1. [ X + ξ , Y + η ] = [ X , Y ] + X η - Y ξ - 1 2 d ( i ( X ) η - i ( Y ) ξ ) [X+\xi,Y+\eta]=[X,Y]+\mathcal{L}_{X}\eta-\mathcal{L}_{Y}\xi-\frac{1}{2}d(i(X)% \eta-i(Y)\xi)
  2. X \mathcal{L}_{X}
  3. X + ξ X + ξ + i ( X ) α X+\xi\mapsto X+\xi+i(X)\alpha
  4. 𝐓 * {\mathbf{T}}^{*}
  5. 𝐓 𝐓 * {\mathbf{T}}\oplus{\mathbf{T}}^{*}
  6. 𝐓 𝐓 * {\mathbf{T}}\oplus{\mathbf{T}}^{*}
  7. 𝐓 𝐓 * {\mathbf{T}}\oplus{\mathbf{T}}^{*}
  8. X + ξ , Y + η = 1 2 ( ξ ( Y ) + η ( X ) ) . \langle X+\xi,Y+\eta\rangle=\frac{1}{2}(\xi(Y)+\eta(X)).
  9. 𝐓 𝐓 * {\mathbf{T}}\oplus{\mathbf{T}}^{*}
  10. 𝐓 𝐓 * {\mathbf{T}}\oplus{\mathbf{T}}^{*}
  11. 𝐓 𝐓 * {\mathbf{T}}\oplus{\mathbf{T}}^{*}
  12. 𝐓 𝐓 * {\mathbf{T}}\oplus{\mathbf{T}}^{*}
  13. \oplus
  14. \otimes
  15. [ A , B ] D = [ A , B ] + d A , B [A,B]_{D}=[A,B]+d\langle A,B\rangle
  16. [ A , [ B , C ] D ] D = [ [ A , B ] D , C ] D + [ B , [ A , C ] D ] D . [A,[B,C]_{D}]_{D}=[[A,B]_{D},C]_{D}+[B,[A,C]_{D}]_{D}.
  17. [ X + f , Y + g ] = [ X , Y ] + X g - Y f [X+f,Y+g]=[X,Y]+Xg-Yf
  18. [ X + f , Y + g ] = [ X + f θ , Y + g θ ] L i e [X+f,Y+g]=[X+f\frac{\partial}{\partial\theta},Y+g\frac{\partial}{\partial% \theta}]_{Lie}

Coxeter_graph.html

  1. ( x - 3 ) ( x - 2 ) 8 ( x + 1 ) 7 ( x 2 + 2 x - 1 ) 6 (x-3)(x-2)^{8}(x+1)^{7}(x^{2}+2x-1)^{6}

Coxeter–Dynkin_diagram.html

  1. [ 2 a 12 a 21 2 ] \left[\begin{matrix}2&a_{12}\\ a_{21}&2\end{matrix}\right]
  2. [ 2 0 0 2 ] \left[\begin{smallmatrix}2&0\\ 0&2\end{smallmatrix}\right]
  3. [ 2 - 1 - 1 2 ] \left[\begin{smallmatrix}2&-1\\ -1&2\end{smallmatrix}\right]
  4. [ 2 - 2 - 2 2 ] \left[\begin{smallmatrix}2&-\sqrt{2}\\ -\sqrt{2}&2\end{smallmatrix}\right]
  5. [ 2 - ϕ - ϕ 2 ] \left[\begin{smallmatrix}2&-\phi\\ -\phi&2\end{smallmatrix}\right]
  6. 4 sin 2 ( π / 5 ) 4\sin^{2}(\pi/5)
  7. ( 5 - 5 ) / 2 (5-\sqrt{5})/2
  8. [ 2 - 3 - 3 2 ] \left[\begin{smallmatrix}2&-\sqrt{3}\\ -\sqrt{3}&2\end{smallmatrix}\right]
  9. [ 2 - 2 cos ( π / 8 ) - 2 cos ( π / 8 ) 2 ] \left[\begin{smallmatrix}2&-2\cos(\pi/8)\\ -2\cos(\pi/8)&2\end{smallmatrix}\right]
  10. 2 - 2 2-\sqrt{2}
  11. [ 2 - 2 cos ( π / 10 ) - 2 cos ( π / 10 ) 2 ] \left[\begin{smallmatrix}2&-2\cos(\pi/10)\\ -2\cos(\pi/10)&2\end{smallmatrix}\right]
  12. 4 sin 2 ( π / 10 ) 4\sin^{2}(\pi/10)
  13. ( 3 - 5 ) / 2 (3-\sqrt{5})/2
  14. [ 2 - 2 cos ( π / 12 ) - 2 cos ( π / 12 ) 2 ] \left[\begin{smallmatrix}2&-2\cos(\pi/12)\\ -2\cos(\pi/12)&2\end{smallmatrix}\right]
  15. 2 - 3 2-\sqrt{3}
  16. [ 2 - 2 cos ( π / p ) - 2 cos ( π / p ) 2 ] \left[\begin{smallmatrix}2&-2\cos(\pi/p)\\ -2\cos(\pi/p)&2\end{smallmatrix}\right]
  17. 4 sin 2 ( π / p ) 4\sin^{2}(\pi/p)
  18. I ~ 1 {\tilde{I}}_{1}
  19. A ~ 1 {\tilde{A}}_{1}
  20. [ 2 - 2 - 2 2 ] \left[\begin{smallmatrix}2&-2\\ -2&2\end{smallmatrix}\right]
  21. [ 2 - 2 - 2 2 ] \left[\begin{smallmatrix}2&-2\\ -2&2\end{smallmatrix}\right]
  22. [ 2 - 2 c o s h ( 2 λ ) - 2 c o s h ( 2 λ ) 2 ] \left[\begin{smallmatrix}2&-2cosh(2\lambda)\\ -2cosh(2\lambda)&2\end{smallmatrix}\right]
  23. - 4 sinh 2 ( 2 λ ) 0 -4\sinh^{2}(2\lambda)\leq 0
  24. I ~ 1 {\tilde{I}}_{1}
  25. I ~ 1 {\tilde{I}}_{1}
  26. C ~ 2 {\tilde{C}}_{2}
  27. G ~ 2 {\tilde{G}}_{2}
  28. A ~ 2 {\tilde{A}}_{2}
  29. G ~ 2 {\tilde{G}}_{2}
  30. C ~ 3 {\tilde{C}}_{3}
  31. B ~ 3 {\tilde{B}}_{3}
  32. A ~ 3 {\tilde{A}}_{3}
  33. A 1 + {A}_{1+}
  34. B C 2 + {BC}_{2+}
  35. D 2 + {D}_{2+}
  36. E 3 - 8 {E}_{3-8}
  37. F 3 - 4 {F}_{3-4}
  38. G 2 {G}_{2}
  39. H 2 - 4 {H}_{2-4}
  40. I 2 ( p ) {I}_{2}(p)
  41. A ~ n - 1 {\tilde{A}}_{n-1}
  42. C ~ n - 1 {\tilde{C}}_{n-1}
  43. B ~ n - 1 {\tilde{B}}_{n-1}
  44. D ~ n - 1 {\tilde{D}}_{n-1}
  45. E ~ 6 {\tilde{E}}_{6}
  46. E ~ 7 {\tilde{E}}_{7}
  47. E ~ 8 {\tilde{E}}_{8}
  48. F ~ 4 {\tilde{F}}_{4}
  49. G ~ 2 {\tilde{G}}_{2}
  50. I ~ 1 {\tilde{I}}_{1}
  51. A ~ 1 {\tilde{A}}_{1}
  52. C ~ 1 {\tilde{C}}_{1}
  53. A ~ 1 {\tilde{A}}_{1}
  54. A ~ 1 2 {\tilde{A}}_{1}^{2}
  55. A ~ 1 G ~ 2 {\tilde{A}}_{1}{\tilde{G}}_{2}
  56. A ~ 1 + {\tilde{A}}_{1+}
  57. B ~ 3 + {\tilde{B}}_{3+}
  58. C ~ 1 + {\tilde{C}}_{1+}
  59. D ~ 4 + {\tilde{D}}_{4+}
  60. E ~ n {\tilde{E}}_{n}
  61. F ~ 4 {\tilde{F}}_{4}
  62. G ~ 2 {\tilde{G}}_{2}
  63. A ~ 1 {\tilde{A}}_{1}
  64. C ~ 1 {\tilde{C}}_{1}
  65. A ~ 2 {\tilde{A}}_{2}
  66. C ~ 2 {\tilde{C}}_{2}
  67. G ~ 2 {\tilde{G}}_{2}
  68. A ~ 3 {\tilde{A}}_{3}
  69. B ~ 3 {\tilde{B}}_{3}
  70. C ~ 3 {\tilde{C}}_{3}
  71. D ~ 3 {\tilde{D}}_{3}
  72. A ~ 3 {\tilde{A}}_{3}
  73. A ~ 4 {\tilde{A}}_{4}
  74. B ~ 4 {\tilde{B}}_{4}
  75. C ~ 4 {\tilde{C}}_{4}
  76. D ~ 4 {\tilde{D}}_{4}
  77. F ~ 4 {\tilde{F}}_{4}
  78. A ~ 5 {\tilde{A}}_{5}
  79. B ~ 5 {\tilde{B}}_{5}
  80. C ~ 5 {\tilde{C}}_{5}
  81. D ~ 5 {\tilde{D}}_{5}
  82. A ~ 6 {\tilde{A}}_{6}
  83. B ~ 6 {\tilde{B}}_{6}
  84. C ~ 6 {\tilde{C}}_{6}
  85. D ~ 6 {\tilde{D}}_{6}
  86. E ~ 6 {\tilde{E}}_{6}
  87. A ~ 7 {\tilde{A}}_{7}
  88. B ~ 7 {\tilde{B}}_{7}
  89. C ~ 7 {\tilde{C}}_{7}
  90. D ~ 7 {\tilde{D}}_{7}
  91. E ~ 7 {\tilde{E}}_{7}
  92. A ~ 8 {\tilde{A}}_{8}
  93. B ~ 8 {\tilde{B}}_{8}
  94. C ~ 8 {\tilde{C}}_{8}
  95. D ~ 8 {\tilde{D}}_{8}
  96. E ~ 8 {\tilde{E}}_{8}
  97. A ~ 9 {\tilde{A}}_{9}
  98. B ~ 9 {\tilde{B}}_{9}
  99. C ~ 9 {\tilde{C}}_{9}
  100. D ~ 9 {\tilde{D}}_{9}
  101. 1 p + 1 q + 1 r < 1. \frac{1}{p}+\frac{1}{q}+\frac{1}{r}<1.
  102. B H ¯ 3 {\bar{BH}}_{3}
  103. K ¯ 3 {\bar{K}}_{3}
  104. J ¯ 3 {\bar{J}}_{3}
  105. D H ¯ 3 {\bar{DH}}_{3}
  106. A B ^ 3 {\widehat{AB}}_{3}
  107. A H ^ 3 {\widehat{AH}}_{3}
  108. B B ^ 3 {\widehat{BB}}_{3}
  109. B H ^ 3 {\widehat{BH}}_{3}
  110. H H ^ 3 {\widehat{HH}}_{3}
  111. H ¯ 4 {\bar{H}}_{4}
  112. B H ¯ 4 {\bar{BH}}_{4}
  113. K ¯ 4 {\bar{K}}_{4}
  114. D H ¯ 4 {\bar{DH}}_{4}
  115. A F ^ 4 {\widehat{AF}}_{4}
  116. B R ^ 3 {\widehat{BR}}_{3}
  117. C R ^ 3 {\widehat{CR}}_{3}
  118. R R ^ 3 {\widehat{RR}}_{3}
  119. A V ^ 3 {\widehat{AV}}_{3}
  120. B V ^ 3 {\widehat{BV}}_{3}
  121. H V ^ 3 {\widehat{HV}}_{3}
  122. V V ^ 3 {\widehat{VV}}_{3}
  123. P ¯ 3 {\bar{P}}_{3}
  124. B P ¯ 3 {\bar{BP}}_{3}
  125. H P ¯ 3 {\bar{HP}}_{3}
  126. V P ¯ 3 {\bar{VP}}_{3}
  127. D V ¯ 3 {\bar{DV}}_{3}
  128. O ¯ 3 {\bar{O}}_{3}
  129. M ¯ 3 {\bar{M}}_{3}
  130. R ¯ 3 {\bar{R}}_{3}
  131. N ¯ 3 {\bar{N}}_{3}
  132. V ¯ 3 {\bar{V}}_{3}
  133. B V ¯ 3 {\bar{BV}}_{3}
  134. H V ¯ 3 {\bar{HV}}_{3}
  135. Y ¯ 3 {\bar{Y}}_{3}
  136. Z ¯ 3 {\bar{Z}}_{3}
  137. D P ¯ 3 {\bar{DP}}_{3}
  138. P P ¯ 3 {\bar{PP}}_{3}
  139. P ¯ 4 {\bar{P}}_{4}
  140. B P ¯ 4 {\bar{BP}}_{4}
  141. F R ^ 4 {\widehat{FR}}_{4}
  142. D P ¯ 4 {\bar{DP}}_{4}
  143. N ¯ 4 {\bar{N}}_{4}
  144. O ¯ 4 {\bar{O}}_{4}
  145. S ¯ 4 {\bar{S}}_{4}
  146. R ¯ 4 {\bar{R}}_{4}
  147. M ¯ 4 {\bar{M}}_{4}
  148. P ¯ 5 {\bar{P}}_{5}
  149. A U ^ 5 {\widehat{AU}}_{5}
  150. A R ^ 5 {\widehat{AR}}_{5}
  151. S ¯ 5 {\bar{S}}_{5}
  152. O ¯ 5 {\bar{O}}_{5}
  153. N ¯ 5 {\bar{N}}_{5}
  154. U ¯ 5 {\bar{U}}_{5}
  155. X ¯ 5 {\bar{X}}_{5}
  156. R ¯ 5 {\bar{R}}_{5}
  157. Q ¯ 5 {\bar{Q}}_{5}
  158. M ¯ 5 {\bar{M}}_{5}
  159. L ¯ 5 {\bar{L}}_{5}
  160. P ¯ 6 {\bar{P}}_{6}
  161. Q ¯ 6 {\bar{Q}}_{6}
  162. S ¯ 6 {\bar{S}}_{6}
  163. P ¯ 7 {\bar{P}}_{7}
  164. Q ¯ 7 {\bar{Q}}_{7}
  165. S ¯ 7 {\bar{S}}_{7}
  166. T ¯ 7 {\bar{T}}_{7}
  167. P ¯ 8 {\bar{P}}_{8}
  168. Q ¯ 8 {\bar{Q}}_{8}
  169. S ¯ 8 {\bar{S}}_{8}
  170. T ¯ 8 {\bar{T}}_{8}
  171. Q ¯ 9 {\bar{Q}}_{9}
  172. S ¯ 9 {\bar{S}}_{9}
  173. T ¯ 9 {\bar{T}}_{9}
  174. E ~ 8 {\tilde{E}}_{8}
  175. E ~ 7 {\tilde{E}}_{7}
  176. E ~ 6 {\tilde{E}}_{6}
  177. F ~ 4 {\tilde{F}}_{4}
  178. G ~ 2 {\tilde{G}}_{2}
  179. A 2 A_{2}
  180. C 2 C_{2}
  181. G 2 G_{2}
  182. A 3 A_{3}
  183. B 3 B_{3}
  184. C 3 C_{3}
  185. H 4 H_{4}
  186. A ~ 2 {\tilde{A}}_{2}
  187. C ~ 2 {\tilde{C}}_{2}
  188. G ~ 2 {\tilde{G}}_{2}
  189. P ¯ 3 {\bar{P}}_{3}
  190. R ¯ 3 {\bar{R}}_{3}
  191. V ¯ 3 {\bar{V}}_{3}
  192. A ~ 3 {\tilde{A}}_{3}
  193. B ~ 3 {\tilde{B}}_{3}
  194. C ~ 3 {\tilde{C}}_{3}
  195. P ¯ 4 {\bar{P}}_{4}
  196. S ¯ 4 {\bar{S}}_{4}
  197. R ¯ 4 {\bar{R}}_{4}
  198. H ¯ 4 {\bar{H}}_{4}
  199. A 4 A_{4}
  200. B 4 B_{4}
  201. C 4 C_{4}
  202. D 4 D_{4}
  203. F 4 F_{4}
  204. A 5 A_{5}
  205. B 5 B_{5}
  206. D 5 D_{5}
  207. A ~ 4 {\tilde{A}}_{4}
  208. B ~ 4 {\tilde{B}}_{4}
  209. C ~ 4 {\tilde{C}}_{4}
  210. D ~ 4 {\tilde{D}}_{4}
  211. F ~ 4 {\tilde{F}}_{4}
  212. P ¯ 5 {\bar{P}}_{5}
  213. S ¯ 5 {\bar{S}}_{5}
  214. R ¯ 5 {\bar{R}}_{5}
  215. Q ¯ 5 {\bar{Q}}_{5}
  216. U ¯ 5 {\bar{U}}_{5}
  217. A ~ 5 {\tilde{A}}_{5}
  218. B ~ 5 {\tilde{B}}_{5}
  219. D ~ 5 {\tilde{D}}_{5}
  220. P ¯ 6 {\bar{P}}_{6}
  221. S ¯ 6 {\bar{S}}_{6}
  222. Q ¯ 6 {\bar{Q}}_{6}
  223. A 6 A_{6}
  224. B 6 B_{6}
  225. D 6 D_{6}
  226. E 6 E_{6}
  227. A 7 A_{7}
  228. B 7 B_{7}
  229. D 7 D_{7}
  230. E 7 E_{7}
  231. E 8 E_{8}
  232. A ~ 6 {\tilde{A}}_{6}
  233. B ~ 6 {\tilde{B}}_{6}
  234. D ~ 6 {\tilde{D}}_{6}
  235. E ~ 6 {\tilde{E}}_{6}
  236. P ¯ 7 {\bar{P}}_{7}
  237. S ¯ 7 {\bar{S}}_{7}
  238. Q ¯ 7 {\bar{Q}}_{7}
  239. T ¯ 7 {\bar{T}}_{7}
  240. A ~ 7 {\tilde{A}}_{7}
  241. B ~ 7 {\tilde{B}}_{7}
  242. D ~ 7 {\tilde{D}}_{7}
  243. E ~ 7 {\tilde{E}}_{7}
  244. P ¯ 8 {\bar{P}}_{8}
  245. S ¯ 8 {\bar{S}}_{8}
  246. Q ¯ 8 {\bar{Q}}_{8}
  247. T ¯ 8 {\bar{T}}_{8}
  248. E ~ 8 {\tilde{E}}_{8}
  249. T ¯ 9 {\bar{T}}_{9}
  250. A ~ n - 1 {\tilde{A}}_{n-1}
  251. A ~ k n - 1 {\tilde{A}}_{kn-1}
  252. B ~ n {\tilde{B}}_{n}
  253. D ~ 2 n + 1 {\tilde{D}}_{2n+1}
  254. D ~ n + 1 {\tilde{D}}_{n+1}
  255. D ~ 2 n {\tilde{D}}_{2n}
  256. C ~ n {\tilde{C}}_{n}
  257. B ~ n + 1 {\tilde{B}}_{n+1}
  258. C ~ 2 n {\tilde{C}}_{2n}
  259. C ~ 2 n + 1 {\tilde{C}}_{2n+1}
  260. C ~ n {\tilde{C}}_{n}
  261. A ~ 2 n + 1 {\tilde{A}}_{2n+1}
  262. A ~ 2 n {\tilde{A}}_{2n}
  263. A ~ 2 n - 1 {\tilde{A}}_{2n-1}
  264. C ~ n {\tilde{C}}_{n}
  265. D ~ n + 2 {\tilde{D}}_{n+2}
  266. C ~ 2 {\tilde{C}}_{2}
  267. D ~ 5 {\tilde{D}}_{5}
  268. F ~ 4 {\tilde{F}}_{4}
  269. E ~ 6 {\tilde{E}}_{6}
  270. E ~ 7 {\tilde{E}}_{7}
  271. G ~ 2 {\tilde{G}}_{2}
  272. D ~ 6 {\tilde{D}}_{6}
  273. E ~ 7 {\tilde{E}}_{7}
  274. B ~ 3 {\tilde{B}}_{3}
  275. F ~ 4 {\tilde{F}}_{4}
  276. D ~ 4 {\tilde{D}}_{4}
  277. E ~ 6 {\tilde{E}}_{6}

Cox–Ingersoll–Ross_model.html

  1. d r t = a ( b - r t ) d t + σ r t d W t dr_{t}=a(b-r_{t})\,dt+\sigma\sqrt{r_{t}}\,dW_{t}
  2. W t W_{t}
  3. a a
  4. b b
  5. σ \sigma\,
  6. a a
  7. b b
  8. σ \sigma\,
  9. a ( b - r t ) a(b-r_{t})
  10. b b
  11. a a
  12. σ r t \sigma\sqrt{r_{t}}
  13. a a
  14. b b
  15. 2 a b σ 2 2ab\geq\sigma^{2}\,
  16. r t + T = Y 2 c r_{t+T}=\frac{Y}{2c}
  17. c = 2 a ( 1 - e - a T ) σ 2 c=\frac{2a}{(1-e^{-aT})\sigma^{2}}
  18. 4 a b σ 2 \frac{4ab}{\sigma^{2}}
  19. 2 c r t e - a T 2cr_{t}e^{-aT}
  20. f ( r t + T ; r t , a , b , σ ) = c e - u - v ( v u ) q / 2 I q ( 2 u v ) f(r_{t+T};r_{t},a,b,\sigma)=c\,e^{-u-v}\left(\frac{v}{u}\right)^{q/2}I_{q}(2% \sqrt{uv})
  21. q = 2 a b σ 2 - 1 q=\frac{2ab}{\sigma^{2}}-1
  22. u = c r t e - a T u=cr_{t}e^{-aT}
  23. v = c r t + T v=cr_{t+T}
  24. r r_{\infty}
  25. f ( r ; a , b , σ ) = ω ν Γ ( ν ) r ν - 1 e - ω r f(r_{\infty};a,b,\sigma)=\frac{\omega^{\nu}}{\Gamma(\nu)}r_{\infty}^{\nu-1}e^{% -\omega r_{\infty}}
  26. ω = 2 a / σ 2 \omega=2a/\sigma^{2}
  27. ν = 2 a b / σ 2 \nu=2ab/\sigma^{2}
  28. σ r t \sigma\sqrt{r_{t}}
  29. r 0 r_{0}
  30. 2 a b σ 2 2ab\geq\sigma^{2}
  31. E [ r t | r 0 ] = r 0 e - a t + b ( 1 - e - a t ) E[r_{t}|r_{0}]=r_{0}e^{-at}+b(1-e^{-at})
  32. b b
  33. V a r [ r t | r 0 ] = r 0 σ 2 a ( e - a t - e - 2 a t ) + b σ 2 2 a ( 1 - e - a t ) 2 Var[r_{t}|r_{0}]=r_{0}\frac{\sigma^{2}}{a}(e^{-at}-e^{-2at})+\frac{b\sigma^{2}% }{2a}(1-e^{-at})^{2}
  34. r t + Δ t - r t = a ( b - r t ) Δ t + σ r t ϵ t r_{t+\Delta t}-r_{t}=a(b-r_{t})\,\Delta t+\sigma\,\sqrt{r}_{t}\epsilon_{t}
  35. r t + Δ t - r t r t = a b Δ t r t - a r t Δ t + σ ϵ t \frac{r_{t+\Delta t}-r_{t}}{\sqrt{r}_{t}}=\frac{ab\Delta t}{\sqrt{r}_{t}}-a% \sqrt{r}_{t}\Delta t+\sigma\,\epsilon_{t}
  36. P ( t , T ) = A ( t , T ) exp ( - B ( t , T ) r t ) P(t,T)=A(t,T)\exp(-B(t,T)r_{t})\!
  37. A ( t , T ) = ( 2 h exp ( ( a + h ) ( T - t ) / 2 ) 2 h + ( a + h ) ( exp ( ( T - t ) h ) - 1 ) ) 2 a b / σ 2 A(t,T)=\left(\frac{2h\exp((a+h)(T-t)/2)}{2h+(a+h)(\exp((T-t)h)-1)}\right)^{2ab% /\sigma^{2}}
  38. B ( t , T ) = 2 ( exp ( ( T - t ) h ) - 1 ) 2 h + ( a + h ) ( exp ( ( T - t ) h ) - 1 ) B(t,T)=\frac{2(\exp((T-t)h)-1)}{2h+(a+h)(\exp((T-t)h)-1)}
  39. h = a 2 + 2 σ 2 h=\sqrt{a^{2}+2\sigma^{2}}

Cramér–von_Mises_criterion.html

  1. F * F^{*}
  2. F n F_{n}
  3. ω 2 = - [ F n ( x ) - F * ( x ) ] 2 d F * ( x ) \omega^{2}=\int_{-\infty}^{\infty}[F_{n}(x)-F^{*}(x)]^{2}\,\mathrm{d}F^{*}(x)
  4. F * F^{*}
  5. F n F_{n}
  6. x 1 , x 2 , , x n x_{1},x_{2},\cdots,x_{n}
  7. T = n ω 2 = 1 12 n + i = 1 n [ 2 i - 1 2 n - F ( x i ) ] 2 . T=n\omega^{2}=\frac{1}{12n}+\sum_{i=1}^{n}\left[\frac{2i-1}{2n}-F(x_{i})\right% ]^{2}.
  8. F F
  9. U 2 = T - n ( F ¯ - 1 2 ) 2 , U^{2}=T-n(\bar{F}-\tfrac{1}{2})^{2},
  10. F ¯ = 1 n F ( x i ) . \bar{F}=\frac{1}{n}\sum F(x_{i}).
  11. x 1 , x 2 , , x N x_{1},x_{2},\cdots,x_{N}
  12. y 1 , y 2 , , y M y_{1},y_{2},\cdots,y_{M}
  13. r 1 , r 2 , , r N r_{1},r_{2},\cdots,r_{N}
  14. s 1 , s 2 , , s M s_{1},s_{2},\cdots,s_{M}
  15. T = N ω 2 = U N M ( N + M ) - 4 M N - 1 6 ( M + N ) T=N\omega^{2}=\frac{U}{NM(N+M)}-\frac{4MN-1}{6(M+N)}
  16. U = N i = 1 N ( r i - i ) 2 + M j = 1 M ( s j - j ) 2 U=N\sum_{i=1}^{N}(r_{i}-i)^{2}+M\sum_{j=1}^{M}(s_{j}-j)^{2}
  17. x x
  18. y y
  19. r r
  20. x i x_{i}
  21. i i
  22. x 1 , x N x_{1},...x_{N}
  23. x i x_{i}
  24. x j x_{j}
  25. ( i + j ) / 2 (i+j)/2
  26. ( r i - i ) 2 (r_{i}-i)^{2}
  27. ( s j - j ) 2 (s_{j}-j)^{2}
  28. r i r_{i}
  29. i i
  30. s j s_{j}
  31. j j

Craps_principle.html

  1. E 1 E_{1}
  2. E 2 E_{2}
  3. E 1 E_{1}
  4. E 1 E_{1}
  5. E 2 E_{2}
  6. P [ E 1 E 1 E 2 ] = P [ E 1 ] P [ E 1 ] + P [ E 2 ] \operatorname{P}\left[E_{1}\mid E_{1}\cup E_{2}\right]=\frac{\operatorname{P}[% E_{1}]}{\operatorname{P}[E_{1}]+\operatorname{P}[E_{2}]}
  7. E 1 E_{1}
  8. E 2 E_{2}
  9. E 1 E_{1}
  10. E 2 E_{2}
  11. P [ E 1 E 2 ] = P [ E 1 ] + P [ E 2 ] \operatorname{P}[E_{1}\cup E_{2}]=\operatorname{P}[E_{1}]+\operatorname{P}[E_{% 2}]
  12. E 1 ( E 1 E 2 ) = E 1 E_{1}\cap(E_{1}\cup E_{2})=E_{1}
  13. P [ E 1 ( E 1 E 2 ) ] = P [ E 1 E 1 E 2 ] P [ E 1 E 2 ] \operatorname{P}[E_{1}\cap(E_{1}\cup E_{2})]=\operatorname{P}\left[E_{1}\mid E% _{1}\cup E_{2}\right]\operatorname{P}\left[E_{1}\cup E_{2}\right]
  14. E 1 : player 1 wins E_{1}:\mathrm{player\ 1\ wins}
  15. E 2 : player 2 wins E_{2}:\mathrm{player\ 2\ wins}
  16. P [ E 1 ] \operatorname{P}[E_{1}]
  17. P [ E 2 ] \operatorname{P}[E_{2}]
  18. E 1 : the original roll (called ’the point’) is rolled (a win) E_{1}:\textrm{ the\ original\ roll\ (called\ 'the\ point')\ is\ rolled\ (a\ % win) }
  19. E 2 : a 7 is rolled (a loss) E_{2}:\textrm{ a\ 7\ is\ rolled\ (a\ loss) }
  20. E 1 E_{1}
  21. E 2 E_{2}
  22. 3 / 36 3 / 36 + 6 / 36 = 1 3 \frac{3/36}{3/36+6/36}=\frac{1}{3}
  23. i = 0 P [ first i rolls are ties, ( i + 1 ) th roll is ’the point’ ] \sum_{i=0}^{\infty}\operatorname{P}[\textrm{first\ }i\textrm{\ rolls\ are\ % ties,\ }(i+1)^{\textrm{th}}\textrm{\ roll\ is\ 'the\ point'}]
  24. i i
  25. P [ first i rolls are ties, ( i + 1 ) th roll is ’the point’ ] = ( 1 - P [ E 1 ] - P [ E 2 ] ) i P [ E 1 ] \operatorname{P}[\textrm{first\ }i\textrm{\ rolls\ are\ ties,\ }(i+1)^{\textrm% {th}}\textrm{\ roll\ is\ 'the\ point'}]=(1-\operatorname{P}[E_{1}]-% \operatorname{P}[E_{2}])^{i}\operatorname{P}[E_{1}]
  26. i = 0 ( 1 - P [ E 1 ] - P [ E 2 ] ) i P [ E 1 ] = P [ E 1 ] i = 0 ( 1 - P [ E 1 ] - P [ E 2 ] ) i \sum_{i=0}^{\infty}(1-\operatorname{P}[E_{1}]-\operatorname{P}[E_{2}])^{i}% \operatorname{P}[E_{1}]=\operatorname{P}[E_{1}]\sum_{i=0}^{\infty}(1-% \operatorname{P}[E_{1}]-\operatorname{P}[E_{2}])^{i}
  27. = P [ E 1 ] 1 - ( 1 - P [ E 1 ] - P [ E 2 ] ) = P [ E 1 ] P [ E 1 ] + P [ E 2 ] =\frac{\operatorname{P}[E_{1}]}{1-(1-\operatorname{P}[E_{1}]-\operatorname{P}[% E_{2}])}=\frac{\operatorname{P}[E_{1}]}{\operatorname{P}[E_{1}]+\operatorname{% P}[E_{2}]}

Critical_speed.html

  1. ω 1 g i = 1 n w i y i i = 1 n w i y i 2 \omega_{1}\approx\sqrt{\frac{g\sum_{i=1}^{n}{w_{i}y_{i}}}{\sum_{i=1}^{n}{w_{i}% y_{i}^{2}}}}
  2. w i w_{i}
  3. y i y_{i}
  4. ω 1 g y m a x \omega_{1}\approx\sqrt{\frac{g}{y_{max}}}
  5. y m a x y_{max}
  6. 60 2 π \frac{60}{2\pi}

Crooks_fluctuation_theorem.html

  1. x ( t ) x(t)
  2. x ~ ( t ) \tilde{x}(t)
  3. P [ x ( t ) ] P ~ [ x ~ ( t ) ] = e σ [ x ( t ) ] \frac{P[x(t)]}{\tilde{P}[\tilde{x}(t)]}=e^{\sigma[x(t)]}
  4. λ \lambda
  5. λ = 0 \lambda=0
  6. λ = 1 \lambda=1
  7. λ \lambda
  8. P ( A B ) P(A\rightarrow B)
  9. A A
  10. λ = 0 \lambda=0
  11. λ = 1 \lambda=1
  12. P ( A B ) P(A\leftarrow B)
  13. B B
  14. λ = 1 \lambda=1
  15. A A
  16. λ = 0 \lambda=0
  17. β = ( k B T ) - 1 \beta=(k_{B}T)^{-1}
  18. k B k_{B}
  19. T T
  20. W A B W_{AB}
  21. A A
  22. B B
  23. Δ F = F ( B ) - F ( A ) \Delta F=F(B)-F(A)
  24. A A
  25. B B
  26. λ = 0 \lambda=0
  27. λ = 1 \lambda=1
  28. P ( A B ) P ( A B ) = exp [ β ( W A B - Δ F ) ] . \frac{P(A\rightarrow B)}{P(A\leftarrow B)}=\exp[\beta(W_{A\rightarrow B}-% \Delta F)].
  29. W A B - Δ F W_{A\rightarrow B}-\Delta F
  30. W d W_{d}
  31. P ( A B ) P(A\rightarrow B)
  32. P ( A B ) P(A\leftarrow B)
  33. W A B = Δ F W_{A\rightarrow B}=\Delta F
  34. W d W_{d}
  35. W A B = - W A B W_{A\rightarrow B}=-W_{A\leftarrow B}
  36. P A B ( W ) = P A B ( - W ) exp [ β ( W - Δ F ) ] . P_{A\rightarrow B}(W)=P_{A\leftarrow B}(-W)~{}\exp[\beta(W-\Delta F)].
  37. W = Δ F W=\Delta F

Cross-entropy_method.html

  1. = 𝔼 𝐮 [ H ( 𝐗 ) ] = H ( 𝐱 ) f ( 𝐱 ; 𝐮 ) d 𝐱 \ell=\mathbb{E}_{\mathbf{u}}[H(\mathbf{X})]=\int H(\mathbf{x})\,f(\mathbf{x};% \mathbf{u})\,\textrm{d}\mathbf{x}
  2. H H
  3. f ( 𝐱 ; 𝐮 ) f(\mathbf{x};\mathbf{u})
  4. ^ = 1 N i = 1 N H ( 𝐗 i ) f ( 𝐗 i ; 𝐮 ) g ( 𝐗 i ) \hat{\ell}=\frac{1}{N}\sum_{i=1}^{N}H(\mathbf{X}_{i})\frac{f(\mathbf{X}_{i};% \mathbf{u})}{g(\mathbf{X}_{i})}
  5. 𝐗 1 , , 𝐗 N \mathbf{X}_{1},\dots,\mathbf{X}_{N}
  6. g g\,
  7. H H
  8. g * ( 𝐱 ) = H ( 𝐱 ) f ( 𝐱 ; 𝐮 ) / g^{*}(\mathbf{x})=H(\mathbf{x})f(\mathbf{x};\mathbf{u})/\ell
  9. \ell
  10. g * g^{*}
  11. 𝐯 ( 0 ) \mathbf{v}^{(0)}
  12. 𝐗 1 , , 𝐗 N \mathbf{X}_{1},\dots,\mathbf{X}_{N}
  13. f ( ; 𝐯 ( t - 1 ) ) f(\cdot;\mathbf{v}^{(t-1)})
  14. 𝐯 ( t ) \mathbf{v}^{(t)}
  15. 𝐯 ( t ) = argmax 𝐯 1 N i = 1 N H ( 𝐗 i ) f ( 𝐗 i ; 𝐮 ) f ( 𝐗 i ; 𝐯 ( t - 1 ) ) log f ( 𝐗 i ; 𝐯 ) \mathbf{v}^{(t)}=\mathop{\textrm{argmax}}_{\mathbf{v}}\frac{1}{N}\sum_{i=1}^{N% }H(\mathbf{X}_{i})\frac{f(\mathbf{X}_{i};\mathbf{u})}{f(\mathbf{X}_{i};\mathbf% {v}^{(t-1)})}\log f(\mathbf{X}_{i};\mathbf{v})
  16. f f\,
  17. f f\,
  18. H ( 𝐗 ) = I { 𝐱 A } H(\mathbf{X})=\mathrm{I}_{\{\mathbf{x}\in A\}}
  19. f ( 𝐗 i ; 𝐮 ) = f ( 𝐗 i ; 𝐯 ( t - 1 ) ) f(\mathbf{X}_{i};\mathbf{u})=f(\mathbf{X}_{i};\mathbf{v}^{(t-1)})
  20. 𝐯 ( t ) \mathbf{v}^{(t)}
  21. 𝐗 k A \mathbf{X}_{k}\in A
  22. S ( x ) S(x)
  23. S ( x ) = e - ( x - 2 ) 2 + 0.8 e - ( x + 2 ) 2 S(x)=\textrm{e}^{-(x-2)^{2}}+0.8\,\textrm{e}^{-(x+2)^{2}}
  24. s y m b o l θ ( S ( X ) γ ) \mathbb{P}_{symbol{\theta}}(S(X)\geq\gamma)
  25. γ \gamma\,
  26. { f ( ; s y m b o l θ ) } \left\{f(\cdot;symbol{\theta})\right\}
  27. μ t \mu_{t}\,
  28. σ t 2 \sigma_{t}^{2}
  29. s y m b o l θ = ( μ , σ 2 ) symbol{\theta}=(\mu,\sigma^{2})
  30. γ \gamma\,
  31. s y m b o l θ symbol{\theta}
  32. D KL ( I { S ( x ) γ } f s y m b o l θ ) D_{\mathrm{KL}}(\textrm{I}_{\{S(x)\geq\gamma\}}\|f_{symbol{\theta}})
  33. γ \geq\gamma

Cross-flow_filtration.html

  1. J = Δ P ( R m + R c ) μ J=\frac{\Delta P}{(R_{m}+R_{c})\mu}
  2. J J
  3. Δ P \Delta P
  4. R m R_{m}
  5. R c R_{c}
  6. μ \mu
  7. R m R_{m}
  8. R c R_{c}

Crossed_ladders_problem.html

  1. 1 A + 1 B = 1 h , \tfrac{1}{A}+\tfrac{1}{B}=\tfrac{1}{h},
  2. h h\,
  3. w 1 w_{1}\,
  4. w 2 w_{2}\,
  5. a a\,
  6. w w\,
  7. w w\,
  8. w 1 w_{1}\,
  9. b b\,
  10. w w\,
  11. w w\,
  12. w 2 w_{2}\,
  13. B w = h w 1 and A w = h w 2 where w 1 > 0 , w 2 > 0 , \frac{B}{w}=\frac{h}{w_{1}}\quad\,\text{and}\quad\frac{A}{w}=\frac{h}{w_{2}}% \quad\,\text{where}\quad w_{1}>0,w_{2}>0,
  14. w 1 + w 2 = w w_{1}+w_{2}=w
  15. 1 A + 1 B = 1 h . \frac{1}{A}+\frac{1}{B}=\frac{1}{h}.
  16. A 2 + w 2 = b 2 A^{2}+w^{2}=b^{2}
  17. B 2 + w 2 = a 2 B^{2}+w^{2}=a^{2}
  18. 1 A + 1 B = 1 h \tfrac{1}{A}+\tfrac{1}{B}=\tfrac{1}{h}
  19. A 4 - 2 h A 3 + ( h - A ) 2 ( a 2 - b 2 ) = 0 , A^{4}-2hA^{3}+(h-A)^{2}(a^{2}-b^{2})=0,
  20. B 4 - 2 h B 3 + ( h - B ) 2 ( b 2 - a 2 ) = 0. B^{4}-2hB^{3}+(h-B)^{2}(b^{2}-a^{2})=0.
  21. 1 A + 1 B = 1 h , \tfrac{1}{A}+\tfrac{1}{B}=\tfrac{1}{h},
  22. (Eq. 1) A B = h ( A + B ) \,\text{(Eq. 1)}\quad AB=h(A+B)\,
  23. w 2 + B 2 = a 2 w^{2}+B^{2}=a^{2}
  24. w 2 + A 2 = b 2 w^{2}+A^{2}=b^{2}
  25. a 2 - B 2 = b 2 - A 2 a^{2}-B^{2}=b^{2}-A^{2}
  26. (Eq. 2) a 2 - b 2 = ( B + A ) ( B - A ) \,\text{(Eq. 2)}\quad a^{2}-b^{2}=(B+A)(B-A)\,
  27. ( a 2 - b 2 ) 2 = ( B + A ) 2 ( B - A ) 2 , (a^{2}-b^{2})^{2}=(B+A)^{2}(B-A)^{2},\,
  28. ( a 2 - b 2 ) 2 = ( B + A ) 2 ( B 2 - 2 A B + A 2 ) (a^{2}-b^{2})^{2}=(B+A)^{2}(B^{2}-2AB+A^{2})
  29. ( a 2 - b 2 ) 2 = ( A + B ) 2 ( A 2 + B 2 - 2 A B ) (a^{2}-b^{2})^{2}=(A+B)^{2}(A^{2}+B^{2}-2AB)
  30. ( a 2 - b 2 ) 2 = ( A + B ) 2 ( A 2 + B 2 + 2 A B - 4 A B ) , (a^{2}-b^{2})^{2}=(A+B)^{2}(A^{2}+B^{2}+2AB-4AB),\,
  31. ( a 2 - b 2 ) 2 = ( A + B ) 2 ( ( A 2 + 2 A B + B 2 ) - 4 A B ) , (a^{2}-b^{2})^{2}=(A+B)^{2}((A^{2}+2AB+B^{2})-4AB),\,
  32. ( a 2 - b 2 ) 2 = ( A + B ) 2 ( ( A + B ) 2 - 4 A B ) , (a^{2}-b^{2})^{2}=(A+B)^{2}((A+B)^{2}-4AB),
  33. ( a 2 - b 2 ) 2 = ( A + B ) 2 ( ( A + B ) 2 - 4 h ( A + B ) ) , (a^{2}-b^{2})^{2}=(A+B)^{2}((A+B)^{2}-4h(A+B)),\,
  34. ( a 2 - b 2 ) 2 = ( A + B ) 2 ( A + B ) ( ( A + B ) - 4 h ) (a^{2}-b^{2})^{2}=(A+B)^{2}(A+B)((A+B)-4h)\,
  35. (Eq. 3) ( a 2 - b 2 ) 2 = ( A + B ) 3 ( A + B - 4 h ) \,\text{(Eq. 3)}\quad(a^{2}-b^{2})^{2}=(A+B)^{3}(A+B-4h)\,
  36. x = A + B a 2 - b 2 , x=\frac{A+B}{\sqrt{a^{2}-b^{2}}},\,
  37. c = 4 h a 2 - b 2 . c=\frac{4h}{\sqrt{a^{2}-b^{2}}}.\,
  38. x 3 ( x - c ) - 1 = 0. x^{3}(x-c)-1=0.\,
  39. A + B = A + A 2 + ( a 2 - b 2 ) A+B=A+\sqrt{A^{2}+(a^{2}-b^{2})}
  40. w = b 2 - A 2 . w=\sqrt{b^{2}-A^{2}}.
  41. h = a b a + b . h=\frac{ab}{a+b}.
  42. 1 A + 1 B = 1 h \tfrac{1}{A}+\tfrac{1}{B}=\tfrac{1}{h}
  43. h = A B A + B , h=\tfrac{AB}{A+B},
  44. a 2 - b 2 a^{2}-b^{2}
  45. x 4 - 2 h x 3 + D x 2 - 2 h D x + h 2 D = 0 x^{4}-2hx^{3}+Dx^{2}-2hDx+h^{2}D=0
  46. a 2 - b 2 a^{2}-b^{2}
  47. b 2 - a 2 b^{2}-a^{2}
  48. 1 A + 1 B = 1 h . \tfrac{1}{A}+\tfrac{1}{B}=\tfrac{1}{h}.

Crosswordese.html

  1. I , II , III , IV , V , VI , VII , VIII , IX , and X . \mathrm{I,\;II,\;III,\;IV,\;V,\;VI,\;VII,\;VIII,\;IX,and\;X.}
  2. I V ¯ \overline{IV}
  3. V ¯ \overline{V}
  4. V I ¯ \overline{VI}
  5. V I I ¯ \overline{VII}
  6. V I I I ¯ \overline{VIII}
  7. I X ¯ \overline{IX}
  8. X ¯ \overline{X}
  9. X X ¯ \overline{XX}
  10. X X X ¯ \overline{XXX}
  11. X L ¯ \overline{XL}
  12. L ¯ \overline{L}
  13. L X ¯ \overline{LX}
  14. L X X ¯ \overline{LXX}
  15. L X X X ¯ \overline{LXXX}
  16. X C ¯ \overline{XC}
  17. C ¯ \overline{C}
  18. C C ¯ \overline{CC}
  19. C C C ¯ \overline{CCC}
  20. C D ¯ \overline{CD}
  21. D ¯ \overline{D}
  22. D C ¯ \overline{DC}
  23. D C C ¯ \overline{DCC}
  24. D C C C ¯ \overline{DCCC}
  25. C M ¯ \overline{CM}

CTL*.html

  1. Φ : := p ( ¬ Φ ) ( Φ and Φ ) ( Φ Φ ) ( Φ Φ ) ( Φ Φ ) A ϕ E ϕ \Phi::=\bot\mid\top\mid p\mid(\neg\Phi)\mid(\Phi\and\Phi)\mid(\Phi\Phi)\mid(% \Phi\Rightarrow\Phi)\mid(\Phi\Leftrightarrow\Phi)\mid A\phi\mid E\phi
  2. ϕ : := Φ ( ¬ ϕ ) ( ϕ and ϕ ) ( ϕ ϕ ) ( ϕ ϕ ) ( ϕ ϕ ) X ϕ F ϕ G ϕ [ ϕ U ϕ ] \phi::=\Phi\mid(\neg\phi)\mid(\phi\and\phi)\mid(\phi\phi)\mid(\phi\Rightarrow% \phi)\mid(\phi\Leftrightarrow\phi)\mid X\phi\mid F\phi\mid G\phi\mid[\phi U\phi]
  3. p p
  4. Φ \Phi
  5. ϕ \phi
  6. Φ Φ \Phi\Phi
  7. X , F , G , U X,F,G,U
  8. A ϕ = ¬ E ¬ ϕ A\phi=\neg E\neg\phi
  9. E X ( p ) and A F G ( p ) EX(p)\and AFG(p)
  10. A F G ( p ) \ AFG(p)
  11. E X ( p ) \ EX(p)
  12. A G ( p ) \ AG(p)
  13. A \ A
  14. s s
  15. Φ \Phi
  16. s Φ s\models\Phi
  17. ( ( , s ) ) ( ( , s ) ⊧̸ ) \Big((\mathcal{M},s)\models\top\Big)\land\Big((\mathcal{M},s)\not\models\bot\Big)
  18. ( ( , s ) p ) ( p L ( s ) ) \Big((\mathcal{M},s)\models p\Big)\Leftrightarrow\Big(p\in L(s)\Big)
  19. ( ( , s ) ¬ Φ ) ( ( , s ) ⊧̸ Φ ) \Big((\mathcal{M},s)\models\neg\Phi\Big)\Leftrightarrow\Big((\mathcal{M},s)% \not\models\Phi\Big)
  20. ( ( , s ) Φ 1 Φ 2 ) ( ( ( , s ) Φ 1 ) ( ( , s ) Φ 2 ) ) \Big((\mathcal{M},s)\models\Phi_{1}\land\Phi_{2}\Big)\Leftrightarrow\Big(\big(% (\mathcal{M},s)\models\Phi_{1}\big)\land\big((\mathcal{M},s)\models\Phi_{2}% \big)\Big)
  21. ( ( , s ) Φ 1 Φ 2 ) ( ( ( , s ) Φ 1 ) ( ( , s ) Φ 2 ) ) \Big((\mathcal{M},s)\models\Phi_{1}\lor\Phi_{2}\Big)\Leftrightarrow\Big(\big((% \mathcal{M},s)\models\Phi_{1}\big)\lor\big((\mathcal{M},s)\models\Phi_{2}\big)\Big)
  22. ( ( , s ) Φ 1 Φ 2 ) ( ( ( , s ) ⊧̸ Φ 1 ) ( ( , s ) Φ 2 ) ) \Big((\mathcal{M},s)\models\Phi_{1}\Rightarrow\Phi_{2}\Big)\Leftrightarrow\Big% (\big((\mathcal{M},s)\not\models\Phi_{1}\big)\lor\big((\mathcal{M},s)\models% \Phi_{2}\big)\Big)
  23. ( ( , s ) Φ 1 Φ 2 ) ( ( ( ( , s ) Φ 1 ) ( ( , s ) Φ 2 ) ) ( ¬ ( ( , s ) Φ 1 ) ¬ ( ( , s ) Φ 2 ) ) ) \bigg((\mathcal{M},s)\models\Phi_{1}\Leftrightarrow\Phi_{2}\bigg)% \Leftrightarrow\bigg(\Big(\big((\mathcal{M},s)\models\Phi_{1}\big)\land\big((% \mathcal{M},s)\models\Phi_{2}\big)\Big)\lor\Big(\neg\big((\mathcal{M},s)% \models\Phi_{1}\big)\land\neg\big((\mathcal{M},s)\models\Phi_{2}\big)\Big)\bigg)
  24. ( ( , s ) A ϕ ) ( π ϕ \Big((\mathcal{M},s)\models A\phi\Big)\Leftrightarrow\Big(\pi\models\phi
  25. π \ \pi
  26. s ) s\Big)
  27. ( ( , s ) E ϕ ) ( π ϕ \Big((\mathcal{M},s)\models E\phi\Big)\Leftrightarrow\Big(\pi\models\phi
  28. π \ \pi
  29. s ) s\Big)
  30. π ϕ \pi\models\phi
  31. ϕ \ \phi
  32. π = s 0 s 1 \pi=s_{0}\to s_{1}\to\cdots
  33. π [ n ] \ \pi[n]
  34. s n s n + 1 s_{n}\to s_{n+1}\to\cdots
  35. ( π Φ ) ( ( , s 0 ) Φ ) \Big(\pi\models\Phi\Big)\Leftrightarrow\Big((\mathcal{M},s_{0})\models\Phi\Big)
  36. ( π ¬ ϕ ) ( π ⊧̸ ϕ ) \Big(\pi\models\neg\phi\Big)\Leftrightarrow\Big(\pi\not\models\phi\Big)
  37. ( π ϕ 1 ϕ 2 ) ( ( π ϕ 1 ) ( π ϕ 2 ) ) \Big(\pi\models\phi_{1}\land\phi_{2}\Big)\Leftrightarrow\Big(\big(\pi\models% \phi_{1}\big)\land\big(\pi\models\phi_{2}\big)\Big)
  38. ( π ϕ 1 ϕ 2 ) ( ( π ϕ 1 ) ( π ϕ 2 ) ) \Big(\pi\models\phi_{1}\lor\phi_{2}\Big)\Leftrightarrow\Big(\big(\pi\models% \phi_{1}\big)\lor\big(\pi\models\phi_{2}\big)\Big)
  39. ( π ϕ 1 ϕ 2 ) ( ( π ⊧̸ ϕ 1 ) ( π ϕ 2 ) ) \Big(\pi\models\phi_{1}\Rightarrow\phi_{2}\Big)\Leftrightarrow\Big(\big(\pi% \not\models\phi_{1}\big)\lor\big(\pi\models\phi_{2}\big)\Big)
  40. ( π ϕ 1 ϕ 2 ) ( ( ( π ϕ 1 ) ( π ϕ 2 ) ) ( ¬ ( π ϕ 1 ) ¬ ( π ϕ 2 ) ) ) \bigg(\pi\models\phi_{1}\Leftrightarrow\phi_{2}\bigg)\Leftrightarrow\bigg(\Big% (\big(\pi\models\phi_{1}\big)\land\big(\pi\models\phi_{2}\big)\Big)\lor\Big(% \neg\big(\pi\models\phi_{1}\big)\land\neg\big(\pi\models\phi_{2}\big)\Big)\bigg)
  41. ( π X ϕ ) ( π [ 1 ] ϕ ) \Big(\pi\models X\phi\Big)\Leftrightarrow\Big(\pi[1]\models\phi\Big)
  42. ( π F ϕ ) ( n 0 : π [ n ] ϕ ) \Big(\pi\models F\phi\Big)\Leftrightarrow\Big(\exists n\geqslant 0:\pi[n]% \models\phi\Big)
  43. ( π G ϕ ) ( n 0 : π [ n ] ϕ ) \Big(\pi\models G\phi\Big)\Leftrightarrow\Big(\forall n\geqslant 0:\pi[n]% \models\phi\Big)
  44. ( π [ ϕ 1 U ϕ 2 ] ) ( n 0 : ( π [ n ] ϕ 2 0 k < n : π [ k ] ϕ 1 ) ) \Big(\pi\models[\phi_{1}U\phi_{2}]\Big)\Leftrightarrow\Big(\exists n\geqslant 0% :\big(\pi[n]\models\phi_{2}\land\forall 0\leqslant k<n:~{}\pi[k]\models\phi_{1% }\big)\Big)

Cubic_form.html

  1. a x 3 + 3 b x 2 y + 3 c x y 2 + d y 3 ax^{3}+3bx^{2}y+3cxy^{2}+dy^{3}

Cubic_reciprocity.html

  1. x q = x 3 n + 2 x ( mod q ) and x q - 1 = x 3 n + 1 1 ( mod q ) , so x^{q}=x^{3n+2}\equiv x\;\;(\mathop{{\rm mod}}q)\;\mbox{ and }~{}\;x^{q-1}=x^{3% n+1}\equiv 1\;\;(\mathop{{\rm mod}}q),\mbox{ so }~{}
  2. x = 1 x x q x q - 1 = x 3 n + 2 x 3 n + 1 = x 6 n + 3 = ( x 2 n + 1 ) 3 ( mod q ) x=1\cdot x\equiv x^{q}x^{q-1}=x^{3n+2}x^{3n+1}=x^{6n+3}=(x^{2n+1})^{3}\;\;(% \mathop{{\rm mod}}q)
  3. 4 p = ( 2 m - n ) 2 + 3 n 2 = ( 2 n - m ) 2 + 3 m 2 = ( m + n ) 2 + 3 ( m - n ) 2 , \begin{aligned}\displaystyle 4p&\displaystyle=(2m-n)^{2}+3n^{2}\\ &\displaystyle=(2n-m)^{2}+3m^{2}\\ &\displaystyle=(m+n)^{2}+3(m-n)^{2},\end{aligned}
  4. p = 1 4 ( L 2 + 27 M 2 ) , p=\tfrac{1}{4}\left(L^{2}+27M^{2}\right),
  5. [ m n ] 3 = { + 1 if m is a cubic residue ( mod n ) - 1 if m is a cubic nonresidue ( mod n ) \left[\frac{m}{n}\right]_{3}=\begin{cases}&+1\mbox{ if }~{}m\mbox{ is a cubic % residue }~{}\;\;(\mathop{{\rm mod}}n)\\ &-1\mbox{ if }~{}m\mbox{ is a cubic nonresidue }~{}\;\;(\mathop{{\rm mod}}n)% \end{cases}
  6. [ 2 p ] 3 = 1 if and only if 3 | b [ 3 p ] 3 = 1 if and only if 9 | b ; or 9 | ( a ± b ) [ 5 p ] 3 = 1 if and only if 15 | b ; or 3 | b and 5 | a ; or 15 | ( a ± b ) ; or 15 | ( 2 a ± b ) [ 6 p ] 3 = 1 if and only if 9 | b ; or 9 | ( a ± 2 b ) \begin{aligned}\displaystyle\left[\frac{2}{p}\right]_{3}=1&\displaystyle\mbox{% if and only if }~{}3|b\\ \displaystyle\left[\frac{3}{p}\right]_{3}=1&\displaystyle\mbox{ if and only if% }~{}9|b;\mbox{ or }~{}9|(a\pm b)\\ \displaystyle\left[\frac{5}{p}\right]_{3}=1&\displaystyle\mbox{ if and only if% }~{}15|b;\mbox{ or }~{}3|b\mbox{ and }~{}5|a;\mbox{ or }~{}15|(a\pm b);\mbox{% or }~{}15|(2a\pm b)\\ \displaystyle\left[\frac{6}{p}\right]_{3}=1&\displaystyle\mbox{ if and only if% }~{}9|b;\mbox{ or }~{}9|(a\pm 2b)\\ \end{aligned}
  7. If [ 7 p ] 3 = 1 , then \mbox{ If }~{}\left[\frac{7}{p}\right]_{3}=1,\mbox{ then }~{}
  8. ( 3 | b and 7 | a ) , or 21 | ( b ± a ) , or 7 | ( 4 b ± a ) , or 21 | b , or 7 | ( b ± 2 a ) . (3|b\mbox{ and }~{}7|a),\mbox{ or }~{}21|(b\pm a),\mbox{ or }~{}7|(4b\pm a),% \mbox{ or }~{}21|b,\mbox{ or }~{}7|(b\pm 2a).
  9. p = 3 n + 1 = 1 4 ( L 2 + 27 M 2 ) , p=3n+1=\tfrac{1}{4}\left(L^{2}+27M^{2}\right),
  10. L ( n ! ) 3 1 ( mod p ) , L(n!)^{3}\equiv 1\;\;(\mathop{{\rm mod}}p),
  11. [ L p ] 3 = [ M p ] 3 = 1 \left[\frac{L}{p}\right]_{3}=\left[\frac{M}{p}\right]_{3}=1
  12. p = 1 4 ( L 2 + 27 M 2 ) , p=\tfrac{1}{4}\left(L^{2}+27M^{2}\right),
  13. [ q p ] 3 = 1 if and only if [ L + 3 M x 2 p q ] 3 = 1 if and only if [ ( L + 3 M x L - 3 M x ) q ] 3 = 1. \left[\frac{q}{p}\right]_{3}=1\mbox{ if and only if }~{}\left[\frac{\frac{L+3% Mx}{2}p}{q}\right]_{3}=1\mbox{ if and only if }~{}\left[\frac{(\frac{L+3Mx}{L-% 3Mx})}{q}\right]_{3}=1.
  14. q = 1 4 ( L 2 + 27 M 2 ) , q=\tfrac{1}{4}\left(L^{\prime 2}+27M^{\prime 2}\right),
  15. x ± L 3 M ( mod q ) x\equiv\pm\frac{L^{\prime}}{3M^{\prime}}\;\;(\mathop{{\rm mod}}q)
  16. [ q p ] 3 = 1 if and only if [ ( L M + L M L M - L M ) q ] 3 = 1. \left[\frac{q}{p}\right]_{3}=1\mbox{ if and only if }~{}\left[\frac{(\frac{LM^% {\prime}+L^{\prime}M}{LM^{\prime}-L^{\prime}M})}{q}\right]_{3}=1.
  17. ( p q ) 3 ( q p ) 3 = ( ( L M + L M 2 M ) p ) 3 2 . \left(\frac{p}{q}\right)_{3}\left(\frac{q}{p}\right)_{3}=\left(\frac{(\frac{L^% {\prime}M+LM^{\prime}}{2M})}{p}\right)_{3}^{2}.
  18. ( p q ) 3 \left(\frac{p}{q}\right)_{3}
  19. p p
  20. q q
  21. p = π π ¯ p=\pi\overline{\pi}
  22. q = ρ ρ ¯ q=\rho\overline{\rho}
  23. π , π ¯ , ρ , ρ ¯ \pi,\overline{\pi},\rho,\overline{\rho}
  24. 2 ( m o d 3 ) ≡2(mod3)
  25. ρ N / 2 M ( m o d π ) ρ≡N/2M(modπ)
  26. π N / 2 M ( m o d ρ ) π≡N/2M′(modρ)
  27. N = L M + L M N=L′M+LM′
  28. L L 1 ( m o d 3 ) L≡L′≡1(mod3)
  29. ( p q ) 3 \left(\frac{p}{q}\right)_{3}
  30. ρ ρ
  31. L i ( p , q ) = Li(p,q)=
  32. ( p q ) 3 \left(\frac{p}{q}\right)_{3}
  33. p = α α ¯ p=\alpha\overline{\alpha}
  34. α \alpha
  35. α ¯ \overline{\alpha}
  36. π π
  37. π π
  38. ρ N / 2 M ( m o d π ) ρ≡N/2M(modπ)
  39. p = 13 = ( 1 + 3 ω ) ( 4 3 ω ) p=13=(−1+3ω)(−4−3ω)
  40. q = 79 = ( 7 + 3 ω ) ( 10 3 ω ) q=79=(−7+3ω)(−10−3ω)
  41. L = 5 L=−5
  42. M = 1 M=1
  43. L = 17 L′=−17
  44. M = 1 M′=1
  45. π = 4 3 ω π=−4−3ω
  46. ρ = 7 + 3 ω ρ=−7+3ω
  47. π = 1 + 3 ω π=−1+3ω
  48. ρ = 10 3 ω ρ=−10−3ω
  49. q and p = 1 4 ( L 2 + 27 M 2 ) q\mbox{ and }~{}p=\tfrac{1}{4}\left(L^{2}+27M^{2}\right)
  50. [ q p ] 3 = 1 if and only if { q | L M or L ± 9 r 2 u + 1 M ( mod q ) , where u 0 , 1 , - 1 2 , - 1 3 ( mod q ) and 3 u + 1 r 2 ( 3 u - 3 ) ( mod q ) \left[\frac{q}{p}\right]_{3}=1\mbox{ if and only if }~{}\begin{cases}q|LM\mbox% { or }\\ L\equiv\pm\frac{9r}{2u+1}M\;\;(\mathop{{\rm mod}}q),\;\;\;\mbox{ where }\\ \;\;\;\;\;u\not\equiv 0,1,-\frac{1}{2},-\frac{1}{3}\;\;(\mathop{{\rm mod}}q)\;% \;\;\mbox{ and }\\ \;\;\;\;\;3u+1\equiv r^{2}(3u-3)\;\;(\mathop{{\rm mod}}q)\end{cases}
  51. [ 2 p ] 3 = 1 \displaystyle\left[\frac{2}{p}\right]_{3}=1
  52. [ 2 p ] 3 = 1 if and only if M 0 ( mod 2 ) \left[\frac{2}{p}\right]_{3}=1\mbox{ if and only if }~{}\qquad\quad M\equiv 0% \;\;(\mathop{{\rm mod}}2)
  53. p q = 1 4 ( L 2 + 27 M 2 ) . pq=\tfrac{1}{4}\left(L^{2}+27M^{2}\right).
  54. [ L p ] 3 [ L q ] 3 = 1 if and only if [ q p ] 3 [ p q ] 3 = 1 \left[\frac{L}{p}\right]_{3}\left[\frac{L}{q}\right]_{3}=1\;\;\mbox{ if and % only if }~{}\;\;\left[\frac{q}{p}\right]_{3}\left[\frac{p}{q}\right]_{3}=1
  55. ω = - 1 + i 3 2 = e 2 π i 3 \omega=\frac{-1+i\sqrt{3}}{2}=e^{\frac{2\pi i}{3}}
  56. ω 3 = ω ω 2 = ω ω ¯ = 1 , ω ¯ = ω 2 \omega^{3}=\omega\omega^{2}=\omega\overline{\omega}=1,\;\;\ \overline{\omega}=% \omega^{2}
  57. ω 2 ¯ = ω \overline{\omega^{2}}=\omega
  58. ω - ω ¯ = i 3 . \omega-\overline{\omega}=i\sqrt{3}.
  59. λ = a + b ω ω λ = - b + ( a - b ) ω ω 2 λ = ( b - a ) - a ω - λ = - a - b ω - ω λ = b + ( b - a ) ω - ω 2 λ = ( a - b ) + a ω \begin{aligned}\displaystyle\lambda&\displaystyle=a+b\omega\\ \displaystyle\omega\lambda&\displaystyle=-b+(a-b)\omega\\ \displaystyle\omega^{2}\lambda&\displaystyle=(b-a)-a\omega\\ \displaystyle-\lambda&\displaystyle=-a-b\omega\\ \displaystyle-\omega\lambda&\displaystyle=b+(b-a)\omega\\ \displaystyle-\omega^{2}\lambda&\displaystyle=(a-b)+a\omega\end{aligned}
  60. N λ = λ λ ¯ = a 2 - a b + b 2 . \mathrm{N}\lambda=\lambda\overline{\lambda}=a^{2}-ab+b^{2}.
  61. p = N π = N π ¯ = π π ¯ p=\mathrm{N}\pi=\mathrm{N}\overline{\pi}=\pi\overline{\pi}
  62. λ = ± ω μ ( 1 - ω ) ν π 1 α 1 π 2 α 2 π 3 α 3 \lambda=\pm\omega^{\mu}(1-\omega)^{\nu}\pi_{1}^{\alpha_{1}}\pi_{2}^{\alpha_{2}% }\pi_{3}^{\alpha_{3}}\dots
  63. α N π - 1 1 ( mod π ) \alpha^{\mathrm{N}\pi-1}\equiv 1\;\;(\mathop{{\rm mod}}\pi)
  64. α N π - 1 3 \alpha^{\frac{\mathrm{N}\pi-1}{3}}
  65. α N π - 1 3 ω k ( mod π ) \alpha^{\frac{\mathrm{N}\pi-1}{3}}\equiv\omega^{k}\;\;(\mathop{{\rm mod}}\pi)
  66. ( α π ) 3 = ω k α N π - 1 3 ( mod π ) . \left(\frac{\alpha}{\pi}\right)_{3}=\omega^{k}\equiv\alpha^{\frac{\mathrm{N}% \pi-1}{3}}\;\;(\mathop{{\rm mod}}\pi).
  67. x 3 α ( mod π ) x^{3}\equiv\alpha\;\;(\mathop{{\rm mod}}\pi)
  68. ( α π ) 3 = 1. \left(\frac{\alpha}{\pi}\right)_{3}=1.
  69. ( α β π ) 3 = ( α π ) 3 ( β π ) 3 \Bigg(\frac{\alpha\beta}{\pi}\Bigg)_{3}=\Bigg(\frac{\alpha}{\pi}\Bigg)_{3}% \Bigg(\frac{\beta}{\pi}\Bigg)_{3}
  70. ( α π ) 3 ¯ = ( α ¯ π ¯ ) 3 \overline{\Bigg(\frac{\alpha}{\pi}\Bigg)_{3}}=\Bigg(\frac{\overline{\alpha}}{% \overline{\pi}}\Bigg)_{3}
  71. ( α π ) 3 = ( α θ ) 3 \Bigg(\frac{\alpha}{\pi}\Bigg)_{3}=\Bigg(\frac{\alpha}{\theta}\Bigg)_{3}
  72. ( α π ) 3 = ( β π ) 3 \Bigg(\frac{\alpha}{\pi}\Bigg)_{3}=\Bigg(\frac{\beta}{\pi}\Bigg)_{3}
  73. ( α λ ) 3 = ( α π 1 ) 3 α 1 ( α π 2 ) 3 α 2 \left(\frac{\alpha}{\lambda}\right)_{3}=\left(\frac{\alpha}{\pi_{1}}\right)_{3% }^{\alpha_{1}}\left(\frac{\alpha}{\pi_{2}}\right)_{3}^{\alpha_{2}}\dots
  74. λ = π 1 α 1 π 2 α 2 π 3 α 3 \lambda=\pi_{1}^{\alpha_{1}}\pi_{2}^{\alpha_{2}}\pi_{3}^{\alpha_{3}}\dots
  75. ( a b ) 3 = 1. \left(\frac{a}{b}\right)_{3}=1.
  76. ( α β ) 3 = ( β α ) 3 . \Bigg(\frac{\alpha}{\beta}\Bigg)_{3}=\Bigg(\frac{\beta}{\alpha}\Bigg)_{3}.
  77. ( ω α ) 3 = ω 1 - a - b 3 = ω - m - n , ( 1 - ω α ) 3 = ω a - 1 3 = ω m , ( 3 α ) 3 = ω b 3 = ω n . \Bigg(\frac{\omega}{\alpha}\Bigg)_{3}=\omega^{\frac{1-a-b}{3}}=\omega^{-m-n},% \;\;\;\Bigg(\frac{1-\omega}{\alpha}\Bigg)_{3}=\omega^{\frac{a-1}{3}}=\omega^{m% },\;\;\;\Bigg(\frac{3}{\alpha}\Bigg)_{3}=\omega^{\frac{b}{3}}=\omega^{n}.

Cumulative_incidence.html

  1. C I ( t ) = 1 - e - I R ( t ) D . CI(t)=1-e^{-IR(t)\cdot D}\,.

Cumulative_prospect_theory.html

  1. p p
  2. U ( p ) := - 0 v ( x ) d d x ( w ( F ( x ) ) ) d x + 0 + v ( x ) d d x ( - w ( 1 - F ( x ) ) ) d x , U(p):=\int_{-\infty}^{0}v(x)\frac{d}{dx}(w(F(x)))\,dx+\int_{0}^{+\infty}v(x)% \frac{d}{dx}(-w(1-F(x)))\,dx,
  3. v v
  4. w w
  5. F ( x ) := - x d p F(x):=\int_{-\infty}^{x}\,dp
  6. x x

Current_density.html

  1. J = lim A 0 I ( A ) A J=\lim\limits_{A\rightarrow 0}\frac{I(A)}{A}
  2. q = t 1 t 2 S J n ^ d A d t q=\int_{t_{1}}^{t_{2}}\iint_{S}{J}\cdot{\hat{n}}{\rm d}A{\rm d}t
  3. n ^ {\hat{n}}
  4. A = A n ^ {A}=A{\hat{n}}
  5. n ^ {\hat{n}}
  6. J n ^ = J cos θ {J}\cdot{\hat{n}}=J\cos\theta
  7. 𝐉 ( 𝐫 , t ) = ρ ( 𝐫 , t ) 𝐯 d ( 𝐫 , t ) \mathbf{J}(\mathbf{r},t)=\rho(\mathbf{r},t)\;\mathbf{v}\text{d}(\mathbf{r},t)\,
  8. ρ ( 𝐫 , t ) = q n ( 𝐫 , t ) \rho(\mathbf{r},t)=qn(\mathbf{r},t)
  9. 𝐉 = σ 𝐄 \mathbf{J}=\sigma\mathbf{E}\,
  10. 𝐉 ( 𝐫 , t ) = - t d t d 3 𝐫 σ ( 𝐫 - 𝐫 , t - t ) 𝐄 ( 𝐫 , t ) \mathbf{J}(\mathbf{r},t)=\int_{-\infty}^{t}\mathrm{d}t^{\prime}\int\mathrm{d}^% {3}\mathbf{r}^{\prime}\;\sigma(\mathbf{r}-\mathbf{r}^{\prime},t-t^{\prime})\;% \mathbf{E}(\mathbf{r}^{\prime},\ t^{\prime})\,
  11. 𝐉 ( 𝐤 , ω ) = σ ( 𝐤 , ω ) 𝐄 ( 𝐤 , ω ) \mathbf{J}(\mathbf{k},\omega)=\sigma(\mathbf{k},\omega)\;\mathbf{E}(\mathbf{k}% ,\omega)\,
  12. 𝐉 P = 𝐏 t \mathbf{J}_{\mathrm{P}}=\frac{\partial\mathbf{P}}{\partial t}
  13. 𝐉 M = × 𝐌 \mathbf{J}_{\mathrm{M}}=\nabla\times\mathbf{M}
  14. 𝐉 b = 𝐉 P + 𝐉 M \mathbf{J}_{\mathrm{b}}=\mathbf{J}_{\mathrm{P}}+\mathbf{J}_{\mathrm{M}}
  15. 𝐉 = 𝐉 f + 𝐉 b \mathbf{J}=\mathbf{J}_{\mathrm{f}}+\mathbf{J}_{\mathrm{b}}
  16. 𝐉 D = 𝐃 t \mathbf{J}_{\mathrm{D}}=\frac{\partial\mathbf{D}}{\partial t}
  17. S 𝐉 d 𝐀 = - d d t V ρ d V = - V ρ t d V \int_{S}{\mathbf{J}\cdot\mathrm{d}\mathbf{A}}=-\frac{\mathrm{d}}{\mathrm{d}t}% \int_{V}{\rho\;\mathrm{d}V}=-\int_{V}{\frac{\partial\rho}{\partial t}\;\mathrm% {d}V}
  18. S 𝐉 d 𝐀 = V 𝐉 d V \int_{S}{\mathbf{J}\cdot\mathrm{d}\mathbf{A}}=\int_{V}{\mathbf{\nabla}\cdot% \mathbf{J}\;\mathrm{d}V}
  19. V 𝐉 d V = - V ρ t d V \int_{V}{\mathbf{\nabla}\cdot\mathbf{J}\;\mathrm{d}V}\ =-\int_{V}{\frac{% \partial\rho}{\partial t}\;\mathrm{d}V}
  20. 𝐉 = - ρ t \nabla\cdot\mathbf{J}=-\frac{\partial\rho}{\partial t}

Current_differencing_buffered_amplifier.html

  1. V p = V n = 0 Vp=Vn=0
  2. I z = I p - I n Iz=Ip-In
  3. V w = V z Vw=Vz

Curved_mirror.html

  1. S > F , S = F , S < F S>F,\ S=F,\ S<F
  2. S < F S<F
  3. S = F S=F
  4. F < S < 2 F F<S<2F
  5. S = 2 F S=2F
  6. S > 2 F S>2F
  7. d o d_{\mathrm{o}}
  8. d i d_{\mathrm{i}}
  9. f f
  10. 1 d o + 1 d i = 1 f \frac{1}{d_{\mathrm{o}}}+\frac{1}{d_{\mathrm{i}}}=\frac{1}{f}
  11. d o d_{\mathrm{o}}
  12. d i d_{\mathrm{i}}
  13. 1 / d o 1/d_{\mathrm{o}}
  14. 1 / d i 1/d_{\mathrm{i}}
  15. 1 / f 1/f
  16. 1 / d o 1/d_{\mathrm{o}}
  17. 1 / d i 1/d_{\mathrm{i}}
  18. m h i h o = - d i d o m\equiv\frac{h_{\mathrm{i}}}{h_{\mathrm{o}}}=-\frac{d_{\mathrm{i}}}{d_{\mathrm% {o}}}
  19. C C
  20. - 1 f -\frac{1}{f}
  21. f f
  22. arccos ( - r R ) \arccos\left(-\frac{r}{R}\right)

Cuspidal_representation.html

  1. L 2 L^{2}
  2. GL 2 \operatorname{GL}_{2}
  3. Z ( 𝐀 ) G ( K ) \ G ( 𝐀 ) | f ( g ) | 2 d g < \int_{Z(\mathbf{A})G(K)\backslash G(\mathbf{A})}|f(g)|^{2}\,dg<\infty
  4. U ( K ) \ U ( 𝐀 ) f ( u g ) d u = 0 \int_{U(K)\backslash U(\mathbf{A})}f(ug)\,du=0
  5. ( g f ) ( x ) = f ( x g ) . (g\cdot f)(x)=f(xg).
  6. L 0 2 ( G ( K ) \ G ( 𝐀 ) , ω ) = ^ ( π , V π ) m π V π L^{2}_{0}(G(K)\backslash G(\mathbf{A}),\omega)=\hat{\bigoplus}_{(\pi,V_{\pi})}% m_{\pi}V_{\pi}

CVSS.html

  1. E x p l o i t a b i l i t y = < m t p l > 20 × A c c e s s V e c t o r × A c c e s s C o m p l e x i t y × A u t h e n t i c a t i o n \begin{aligned}\displaystyle Exploitability={<mtpl>{{20}}}\times{{{% AccessVector}}}\times{{{AccessComplexity}}}\times{{{Authentication}}}\\ \end{aligned}
  2. I m p a c t = < m t p l > 10.41 × ( 1 - ( 1 - C o n f I m p a c t ) × ( 1 - I n t e g I m p a c t ) × ( 1 - A v a i l I m p a c t ) ) \begin{aligned}\displaystyle Impact={<mtpl>{{10.41}}}\times(1-(1-ConfImpact)% \times(1-IntegImpact)\times(1-AvailImpact))\\ \end{aligned}
  3. f ( I m p a c t ) = { 0 , if I m p a c t = 0 1.176 , otherwise f(Impact)=\begin{cases}0,&\,\text{if }Impact\,\text{ = 0}\\ 1.176,&\,\text{otherwise }\end{cases}
  4. B a s e S c o r e = < m t p l > r o u n d T o 1 D e c i m a l ( ( ( 0.6 × I m p a c t ) + ( 0.4 × E x p l o i t a b i l i t y ) - 1.5 ) × f ( I m p a c t ) ) \begin{aligned}\displaystyle BaseScore={<mtpl>{{roundTo1Decimal}}}(((0.6\times Impact% )+(0.4\times Exploitability)-1.5)\times f(Impact))\end{aligned}

Cyclic_cellular_automaton.html

  1. n n
  2. 0
  3. n - 1 n-1
  4. n - 1 n-1

Cyclotomic_unit.html

  1. p b = a g b = g a \prod_{pb=a}g_{b}=g_{a}

Cyclotron_resonance.html

  1. m v 2 r = q B v \frac{mv^{2}}{r}=qBv
  2. f = v 2 π r f=\frac{v}{2\pi r}
  3. f = q B 2 π m f=\frac{qB}{2\pi m}
  4. ω = 2 π f = q B m \omega=2\pi f=\frac{qB}{m}

Cylinder_set_measure.html

  1. 𝒜 ( E ) \mathcal{A}(E)
  2. 𝒜 ( E ) := { T Lin ( E ; F T ) | T surjective and dim F T < + } . \mathcal{A}(E):=\{T\in\mathrm{Lin}(E;F_{T})|T\mbox{ surjective and }~{}\dim_{% \mathbb{R}}F_{T}<+\infty\}.
  3. { μ T | T 𝒜 ( E ) } . \{\mu_{T}|T\in\mathcal{A}(E)\}.
  4. μ T = ( π S T ) * ( μ S ) . \mu_{T}=\left(\pi_{ST}\right)_{*}(\mu_{S}).
  5. μ T = ( π S T ) * ( μ S ) \mu_{T}=\left(\pi_{ST}\right)_{*}(\mu_{S})
  6. T \mathcal{B}_{T}
  7. Cyl ( E ) := { T - 1 ( B ) | B T , T 𝒜 ( E ) } . \mathrm{Cyl}(E):=\{T^{-1}(B)|B\in\mathcal{B}_{T},T\in\mathcal{A}(E)\}.
  8. T \mathcal{B}_{T}
  9. Borel ( E ) = σ ( Cyl ( E ) ) . \mathrm{Borel}(E)=\sigma\left(\mathrm{Cyl}(E)\right).
  10. { μ T | T 𝒜 ( E ) } \{\mu_{T}|T\in\mathcal{A}(E)\}
  11. γ T H := i * ( γ dim F T ) , \gamma_{T}^{H}:=i_{*}\left(\gamma^{\dim F_{T}}\right),
  12. S H S . S\subseteq H\subseteq S^{\prime}.

Cylindric_algebra.html

  1. α \alpha
  2. α \alpha
  3. ( A , + , , - , 0 , 1 , c κ , d κ λ ) κ , λ < α (A,+,\cdot,-,0,1,c_{\kappa},d_{\kappa\lambda})_{\kappa,\lambda<\alpha}
  4. ( A , + , , - , 0 , 1 ) (A,+,\cdot,-,0,1)
  5. c κ c_{\kappa}
  6. A A
  7. κ \kappa
  8. d κ λ d_{\kappa\lambda}
  9. A A
  10. κ \kappa
  11. λ \lambda
  12. c κ 0 = 0 c_{\kappa}0=0
  13. x c κ x x\leq c_{\kappa}x
  14. c κ ( x c κ y ) = c κ x c κ y c_{\kappa}(x\cdot c_{\kappa}y)=c_{\kappa}x\cdot c_{\kappa}y
  15. c κ c λ x = c λ c κ x c_{\kappa}c_{\lambda}x=c_{\lambda}c_{\kappa}x
  16. d κ κ = 1 d_{\kappa\kappa}=1
  17. κ { λ , μ } \kappa\notin\{\lambda,\mu\}
  18. d λ μ = c κ ( d λ κ d κ μ ) d_{\lambda\mu}=c_{\kappa}(d_{\lambda\kappa}\cdot d_{\kappa\mu})
  19. κ λ \kappa\neq\lambda
  20. c κ ( d κ λ x ) c κ ( d κ λ - x ) = 0 c_{\kappa}(d_{\kappa\lambda}\cdot x)\cdot c_{\kappa}(d_{\kappa\lambda}\cdot-x)=0
  21. c κ x c_{\kappa}x
  22. κ \kappa
  23. x x
  24. d κ λ d_{\kappa\lambda}
  25. κ \kappa
  26. λ \lambda
  27. κ . 𝑓𝑎𝑙𝑠𝑒 𝑓𝑎𝑙𝑠𝑒 \exists\kappa.\mathit{false}\Leftrightarrow\mathit{false}
  28. x κ . x x\Rightarrow\exists\kappa.x
  29. κ . ( x κ . y ) ( κ . x ) ( κ . y ) \exists\kappa.(x\wedge\exists\kappa.y)\Leftrightarrow(\exists\kappa.x)\wedge(% \exists\kappa.y)
  30. κ λ . x λ κ . x \exists\kappa\exists\lambda.x\Leftrightarrow\exists\lambda\exists\kappa.x
  31. κ = κ 𝑡𝑟𝑢𝑒 \kappa=\kappa\Leftrightarrow\mathit{true}
  32. κ \kappa
  33. λ \lambda
  34. μ \mu
  35. λ = μ κ . ( λ = κ κ = μ ) \lambda=\mu\Leftrightarrow\exists\kappa.(\lambda=\kappa\wedge\kappa=\mu)
  36. κ \kappa
  37. λ \lambda
  38. κ . ( κ = λ x ) κ . ( κ = λ ¬ x ) 𝑓𝑎𝑙𝑠𝑒 \exists\kappa.(\kappa=\lambda\wedge x)\wedge\exists\kappa.(\kappa=\lambda% \wedge\neg x)\Leftrightarrow\mathit{false}

D'Alembert–Euler_condition.html

  1. 𝐱 ¨ = D 2 𝐱 D t \ddot{\mathbf{x}}=\frac{D^{2}\mathbf{x}}{Dt}
  2. curl 𝐱 = 0. \mathrm{curl}\ \mathbf{x}=\mathbf{0}.\,

D-arginase.html

  1. \rightleftharpoons

Dagger_(disambiguation).html

  1. M M^{\dagger}
  2. : o p \dagger\colon\mathbb{C}^{op}\rightarrow\mathbb{C}
  3. A B A\dagger B

Dagger_category.html

  1. 𝒞 \mathcal{C}
  2. : 𝒞 o p 𝒞 \dagger\colon\mathcal{C}^{op}\rightarrow\mathcal{C}
  3. 𝒞 o p \mathcal{C}^{op}
  4. f : A B f\colon A\to B
  5. 𝒞 \mathcal{C}
  6. f : B A f^{\dagger}\colon B\to A
  7. f : A B f\colon A\to B
  8. g : B C g\colon B\to C
  9. id A = id A : A A \mathrm{id}_{A}=\mathrm{id}_{A}^{\dagger}\colon A\rightarrow A
  10. ( g f ) = f g : C A (g\circ f)^{\dagger}=f^{\dagger}\circ g^{\dagger}\colon C\rightarrow A
  11. f = f : A B f^{\dagger\dagger}=f\colon A\rightarrow B\,
  12. R : X Y R:X\rightarrow Y
  13. R : Y X R^{\dagger}:Y\rightarrow X
  14. R R
  15. f : A B f:A\rightarrow B
  16. f : B A f^{\dagger}:B\rightarrow A
  17. 𝒞 \mathcal{C}
  18. f f
  19. f = f - 1 f^{\dagger}=f^{-1}
  20. f = f f=f^{\dagger}
  21. f : A A f\colon A\to A

Dagger_compact_category.html

  1. 𝐂 \mathbf{C}
  2. A A
  3. 𝐂 \mathbf{C}
  4. 𝐂 𝐂 𝐂 \mathbf{C}\otimes\mathbf{C}\to\mathbf{C}
  5. σ A , B : A B B A \sigma_{A,B}:A\otimes B\simeq B\otimes A
  6. A 𝐂 A\in\mathbf{C}
  7. A * A^{*}
  8. η A : I A * A \eta_{A}:I\to A^{*}\otimes A
  9. ε A : A A * I \varepsilon_{A}:A\otimes A^{*}\to I
  10. : 𝐂 o p 𝐂 \dagger\colon\mathbf{C}^{op}\rightarrow\mathbf{C}
  11. σ A , A * ε A = η A \sigma_{A,A^{*}}\circ\varepsilon^{\dagger}_{A}=\eta_{A}
  12. ( A , δ , ε ) (A,\delta,\varepsilon)
  13. ( 1 A ε ) δ = 1 A = ( ε 1 A ) δ (1_{A}\otimes\varepsilon)\circ\delta=1_{A}=(\varepsilon\otimes 1_{A})\circ\delta
  14. ( 1 A δ ) δ = ( δ 1 A ) δ (1_{A}\otimes\delta)\circ\delta=(\delta\otimes 1_{A})\circ\delta
  15. σ A , A δ = δ \sigma_{A,A}\circ\delta=\delta
  16. δ δ = 1 A \delta^{\dagger}\circ\delta=1_{A}
  17. ( δ 1 A ) ( 1 A δ ) = δ δ (\delta^{\dagger}\otimes 1_{A})\circ(1_{A}\otimes\delta)=\delta\circ\delta^{\dagger}
  18. | j |j\rangle
  19. δ : H \displaystyle\delta:H
  20. ε : H | j 1 \begin{aligned}\displaystyle\varepsilon:H&\displaystyle\to\mathbb{C}\\ \displaystyle|j\rangle&\displaystyle\mapsto 1\\ \end{aligned}
  21. | j |j\rangle
  22. ( A , δ , ε ) (A,\delta,\varepsilon)
  23. ( A , δ , ε ) (A,\delta,\varepsilon)
  24. ψ \psi
  25. δ ψ = ψ ψ \delta\circ\psi=\psi\otimes\psi
  26. ψ \psi
  27. ( A , δ , ε ) (A,\delta,\varepsilon)
  28. δ ( ψ ¯ ψ ) = ε \delta^{\dagger}\circ(\overline{\psi}\otimes\psi)=\varepsilon^{\dagger}
  29. ψ \psi
  30. ( A , δ X , ε X ) (A,\delta_{X},\varepsilon_{X})
  31. ( A , δ Z , ε Z ) (A,\delta_{Z},\varepsilon_{Z})
  32. δ Z δ X = ε Z ε X \delta^{\dagger}_{Z}\circ\delta_{X}=\varepsilon_{Z}\circ\varepsilon_{X}^{\dagger}
  33. δ ( ψ 1 A ) \delta^{\dagger}\circ(\psi\otimes 1_{A})
  34. ψ \psi
  35. ( A , δ , ε ) (A,\delta,\varepsilon)

Dagger_symmetric_monoidal_category.html

  1. , , I \langle\mathbb{C},\otimes,I\rangle
  2. \mathbb{C}
  3. \mathbb{C}
  4. f : A B f:A\rightarrow B
  5. g : C D g:C\rightarrow D
  6. A , B A,B
  7. C C
  8. O b ( ) Ob(\mathbb{C})
  9. ( f g ) = f g : B D A C (f\otimes g)^{\dagger}=f^{\dagger}\otimes g^{\dagger}:B\otimes D\rightarrow A\otimes C
  10. α A , B , C = α A , B , C - 1 : ( A B ) C A ( B C ) \alpha^{\dagger}_{A,B,C}=\alpha^{-1}_{A,B,C}:(A\otimes B)\otimes C\rightarrow A% \otimes(B\otimes C)
  11. ρ A = ρ A - 1 : A A I \rho^{\dagger}_{A}=\rho^{-1}_{A}:A\rightarrow A\otimes I
  12. λ A = λ A - 1 : A I A \lambda^{\dagger}_{A}=\lambda^{-1}_{A}:A\rightarrow I\otimes A
  13. σ A , B = σ A , B - 1 : B A A B \sigma^{\dagger}_{A,B}=\sigma^{-1}_{A,B}:B\otimes A\rightarrow A\otimes B
  14. α , λ , ρ \alpha,\lambda,\rho
  15. σ \sigma

Damage_per_second.html

  1. 75 / ( .25 + .75 ) 75/(.25+.75)
  2. 75 / 1 75/1
  3. 75 75
  4. 75 / .25 = 300 75/.25=300
  5. 100 / ( .50 + .75 ) = 80 100/(.50+.75)=80
  6. 100 / .50 = 200 100/.50=200
  7. 75 + 100 = 175 75+100=175
  8. 5 × 0.25 = 1.25 5\times 0.25=1.25
  9. 175 / 1.25 = 140 175/1.25=140

Damgård–Jurik_cryptosystem.html

  1. n s + 1 n^{s+1}
  2. n n
  3. s s
  4. s = 1 s=1
  5. φ ( n s + 1 ) \varphi(n^{s+1})
  6. Z n s + 1 * Z^{*}_{n^{s+1}}
  7. n s n^{s}
  8. Z n s + 1 * Z^{*}_{n^{s+1}}
  9. G × H G\times H
  10. G G
  11. n s n^{s}
  12. H H
  13. Z n * Z^{*}_{n}
  14. G / H G/H
  15. H H
  16. n = p q n=pq
  17. λ = lcm ( p - 1 , q - 1 ) \lambda=\operatorname{lcm}(p-1,q-1)
  18. g n s + 1 * g\in\mathbb{Z}^{*}_{n^{s+1}}
  19. g = ( 1 + n ) j x mod n s + 1 g=(1+n)^{j}x\mod n^{s+1}
  20. j j
  21. n n
  22. x H x\in H
  23. d d
  24. d mod n n * d\mod n\in\mathbb{Z}^{*}_{n}
  25. d = 0 mod λ d=0\mod\lambda
  26. d d
  27. λ \lambda
  28. ( n , g ) (n,g)
  29. d d
  30. m m
  31. m n s m\in\mathbb{Z}_{n^{s}}
  32. r r
  33. r n s + 1 * r\in\mathbb{Z}^{*}_{n^{s+1}}
  34. c = g m r n s mod n s + 1 c=g^{m}\cdot r^{n^{s}}\mod n^{s+1}
  35. c n s + 1 * c\in\mathbb{Z}^{*}_{n^{s+1}}
  36. c d m o d n s + 1 c^{d}\;mod\;n^{s+1}
  37. c d = ( g m r n s ) d = ( ( 1 + n ) j m x m r n s ) d = ( 1 + n ) j m d m o d n s ( x m r n s ) d m o d λ = ( 1 + n ) j m d m o d n s c^{d}=(g^{m}r^{n^{s}})^{d}=((1+n)^{jm}x^{m}r^{n^{s}})^{d}=(1+n)^{jmd\;mod\;n^{% s}}(x^{m}r^{n^{s}})^{d\;mod\;\lambda}=(1+n)^{jmd\;mod\;n^{s}}
  38. j m d jmd
  39. j d jd
  40. m = ( j m d ) ( j d ) - 1 m o d n s m=(jmd)\cdot(jd)^{-1}\;mod\;n^{s}
  41. g = n + 1 g=n+1
  42. d = 1 m o d n s d=1\;mod\;n^{s}
  43. d = 0 m o d λ d=0\;mod\;\lambda
  44. c d = ( 1 + n ) m m o d n s + 1 c^{d}=(1+n)^{m}\;mod\;n^{s+1}

Dangerously_irrelevant_operator.html

  1. ϕ \phi
  2. V ( ϕ ) = - a ϕ α + b ϕ β V\left(\phi\right)=-a\phi^{\alpha}+b\phi^{\beta}
  3. β \beta
  4. α \alpha
  5. ϕ \phi
  6. d - β c d-\beta c
  7. ϕ = ( a α b β ) 1 β - α = ( a α β ) 1 β - α b - 1 β - α \langle\phi\rangle=\left(\frac{a\alpha}{b\beta}\right)^{\frac{1}{\beta-\alpha}% }=\left(\frac{a\alpha}{\beta}\right)^{\frac{1}{\beta-\alpha}}b^{-\frac{1}{% \beta-\alpha}}
  8. ϕ = 0 \phi=0

Darboux_derivative.html

  1. G G
  2. 𝔤 \mathfrak{g}
  3. ω G \omega_{G}
  4. 𝔤 \mathfrak{g}
  5. 1 1
  6. G G
  7. ω G ( X g ) = ( T g L g ) - 1 X g \omega_{G}(X_{g})=(T_{g}L_{g})^{-1}X_{g}
  8. g G g\in G
  9. X g T g G X_{g}\in T_{g}G
  10. L g L_{g}
  11. g G g\in G
  12. T g L g T_{g}L_{g}
  13. g g
  14. f : M G f:M\to G
  15. M M
  16. G G
  17. f f
  18. 𝔤 \mathfrak{g}
  19. 1 1
  20. ω f := f * ω G , \omega_{f}:=f^{*}\omega_{G},
  21. ω G \omega_{G}
  22. f f
  23. f f
  24. ω f \omega_{f}
  25. f f^{\prime}
  26. f : f:\mathbb{R}\to\mathbb{R}
  27. \mathbb{R}
  28. \mathbb{R}
  29. 0
  30. \mathbb{R}
  31. \mathbb{R}
  32. x x\in\mathbb{R}
  33. v T x ( T f ( x ) L f ( x ) ) - 1 ( T x f ) v T 0 . v\in T_{x}\mathbb{R}\mapsto(T_{f(x)}L_{f(x)})^{-1}\circ(T_{x}f)v\in T_{0}% \mathbb{R}.
  34. t \frac{\partial}{\partial t}
  35. \mathbb{R}
  36. f ( x ) f^{\prime}(x)
  37. M M
  38. f , g : M G f,g:M\to G
  39. ω f \omega_{f}
  40. ω f = ω g \omega_{f}=\omega_{g}
  41. C G C\in G
  42. f ( x ) = C g ( x ) f(x)=C\cdot g(x)
  43. x M x\in M
  44. C C
  45. d ω + 1 2 [ ω , ω ] = 0. d\omega+\frac{1}{2}[\omega,\omega]=0.
  46. X X
  47. Y Y
  48. G G
  49. x G x\in G
  50. ( d ω ) x ( X x , Y x ) + [ ω x ( X x ) , ω x ( Y x ) ] = 0. (d\omega)_{x}(X_{x},Y_{x})+[\omega_{x}(X_{x}),\omega_{x}(Y_{x})]=0.
  51. 1 1
  52. 𝔤 \mathfrak{g}
  53. 1 1
  54. ω \omega
  55. M M
  56. p M p\in M
  57. U U
  58. f : U G f:U\to G
  59. ω f = ω | U , \omega_{f}=\omega|_{U},
  60. ω \omega
  61. M M
  62. M M
  63. G G

DART_ion_source.html

  1. M * + S S + + M + e - M^{*}+S\to S^{+\bullet}+M+e^{-}\,
  2. H e ( 2 3 S ) + H 2 O H 2 O + + H e ( 1 1 S ) + e - He(2^{3}S)+H_{2}O\to H_{2}O^{+\bullet}+He(1^{1}S)+e^{-}
  3. H 2 O + + H 2 O H 3 O + + O H H_{2}O^{+\bullet}+H_{2}O\to H_{3}O^{+}+OH^{\bullet}
  4. H 3 O + + n H 2 O [ ( H 2 O ) n H ] + H_{3}O^{+}+nH_{2}O\to\left[(H_{2}O\right)_{n}H]^{+}
  5. [ ( H 2 O ) n H ] + + S S H + + n H 2 O \left[(H_{2}O\right)_{n}H]^{+}+S\to SH^{+}+nH_{2}O

Darwin_(unit).html

  1. r r
  2. r = ln X 2 - ln X 1 Δ t r=\frac{{\rm ln}X_{2}-{\rm ln}X_{1}}{\Delta t}
  3. X 1 X_{1}
  4. X 2 X_{2}
  5. Δ t \Delta t
  6. r = ln X 2 X 1 Δ t r=\frac{{\rm ln}\frac{X_{2}}{X_{1}}}{\Delta t}
  7. M a - 1 Ma^{-1}

Data_synchronization.html

  1. S A S B = ( S A - S B ) ( S B - S A ) S_{A}\oplus S_{B}=(S_{A}-S_{B})\cup(S_{B}-S_{A})
  2. S A S_{A}
  3. S B S_{B}
  4. σ A \sigma_{A}
  5. σ B \sigma_{B}
  6. σ A \sigma_{A}
  7. σ B \sigma_{B}

Daya_Bay_Reactor_Neutrino_Experiment.html

  1. sin 2 ( 2 θ 13 ) = 0.092 ± 0.016 ( stat ) ± 0.005 ( syst ) . \sin^{2}(2\theta_{13})=0.092\pm 0.016\,\mathrm{(stat)}\pm 0.005\,\mathrm{(syst% )}.
  2. sin 2 ( 2 θ 13 ) = 0.090 - 0.009 + 0.008 \sin^{2}(2\theta_{13})=0.090^{+0.008}_{-0.009}
  3. sin 2 ( 2 θ 13 ) = 0.083 ± 0.018 \sin^{2}(2\theta_{13})=0.083\pm 0.018
  4. sin 2 ( θ 13 ) = 0.084 ± 0.005 , | Δ m e e 2 | = 2.44 - 11 + 10 × 10 - 3 eV 2 \sin^{2}(\theta_{13})=0.084\pm 0.005,\quad|\Delta m^{2}_{ee}|=2.44^{+10}_{-11}% \times 10^{-3}{\rm eV}^{2}

De_Branges_space.html

  1. \mathbb{C}
  2. \mathbb{C}
  3. | E ( z ) | > | E ( z ¯ ) | |E(z)|>|E(\bar{z})|
  4. + = { z | Im ( z ) > 0 } \mathbb{C}^{+}=\{z\in\mathbb{C}|{\rm Im}(z)>0\}
  5. F / E , F # / E H 2 ( + ) F/E,F^{\#}/E\in H_{2}(\mathbb{C}^{+})
  6. + = { z | Im ( z ) > 0 } \mathbb{C}^{+}=\{z\in\mathbb{C}|{\rm Im(z)}>0\}
  7. F # ( z ) = F ( z ¯ ) ¯ F^{\#}(z)=\overline{F(\bar{z})}
  8. H 2 ( + ) H_{2}(\mathbb{C}^{+})
  9. | ( F / E ) ( λ ) | 2 d λ < \int_{\mathbb{R}}|(F/E)(\lambda)|^{2}d\lambda<\infty
  10. | ( F / E ) ( z ) | , | ( F # / E ) ( z ) | C F ( Im ( z ) ) ( - 1 / 2 ) , z + |(F/E)(z)|,|(F^{\#}/E)(z)|\leq C_{F}(\operatorname{Im}(z))^{(-1/2)},\forall z% \in\mathbb{C}^{+}
  11. [ F , G ] = 1 π F ( λ ) ¯ G ( λ ) d λ | E ( λ ) | 2 . [F,G]=\frac{1}{\pi}\int_{\mathbb{R}}\overline{F(\lambda)}G(\lambda)\frac{d% \lambda}{|E(\lambda)|^{2}}.

De_Gua's_theorem.html

  1. A A B C 2 = A \color b l u e A B O 2 + A \color g r e e n A C O 2 + A \color r e d B C O 2 A_{ABC}^{2}=A_{\color{blue}ABO}^{2}+A_{\color{green}ACO}^{2}+A_{\color{red}BCO% }^{2}
  2. n \mathbb{R}^{n}
  3. k n k\leq n
  4. I { 1 , , n } I\subseteq\{1,\ldots,n\}
  5. C I C_{I}
  6. e i 1 , , e i k e_{i_{1}},\ldots,e_{i_{k}}
  7. I = { i 1 , , i k } I=\{i_{1},\ldots,i_{k}\}
  8. e 1 , , e n e_{1},\ldots,e_{n}
  9. n \mathbb{R}^{n}
  10. vol ( C ) k 2 = I vol ( C I ) k 2 , \mbox{vol}~{}_{k}^{2}(C)=\sum_{I}\mbox{vol}~{}_{k}^{2}(C_{I}),
  11. vol ( C ) k \mbox{vol}~{}_{k}(C)
  12. I { 1 , , n } I\subseteq\{1,\ldots,n\}
  13. n \mathbb{R}^{n}

De_Haas–van_Alphen_effect.html

  1. 1 / B 1/B
  2. S S
  3. Δ ( 1 B ) = 2 π e S \Delta\left(\frac{1}{B}\right)=\frac{2\pi e}{\hbar S}

Dead-end_elimination.html

  1. E E
  2. N N
  3. r k r_{k}
  4. k th \mathrm{k^{th}}
  5. E T O T = k E k ( r k ) + k l E k l ( r k , r l ) E_{TOT}=\sum_{k}E_{k}(r_{k})+\sum_{k\neq l}E_{kl}(r_{k},r_{l})\,
  6. E k ( r k ) E_{k}(r_{k})
  7. r k r_{k}
  8. E k l ( r k , r l ) E_{kl}(r_{k},r_{l})
  9. r k , r j r_{k},r_{j}
  10. E k k ( r k A , r k A ) E_{kk}(r_{k}^{A},r_{k}^{A})
  11. r k A r_{k}^{A}
  12. k k
  13. r k B r_{k}^{B}
  14. E k ( r k A ) + l = 1 N min X E k l ( r k A , r l X ) > E k ( r k B ) + l = 1 N max X E k l ( r k B , r l X ) E_{k}(r_{k}^{A})+\sum_{l=1}^{N}\min_{X}E_{kl}(r_{k}^{A},r_{l}^{X})>E_{k}(r_{k}% ^{B})+\sum_{l=1}^{N}\max_{X}E_{kl}(r_{k}^{B},r_{l}^{X})
  15. min X E k l ( r k A , r l X ) \min_{X}E_{kl}(r_{k}^{A},r_{l}^{X})
  16. r k A r_{k}^{A}
  17. k k
  18. l l
  19. max X E k l ( r k B , r l X ) \max_{X}E_{kl}(r_{k}^{B},r_{l}^{X})
  20. r k B r_{k}^{B}
  21. k k
  22. l l
  23. U k l A B U_{kl}^{AB}
  24. A A
  25. B B
  26. k k
  27. l l
  28. U k l A B = def E k ( r k A ) + E l ( r l B ) + E k l ( r k A , r l B ) U_{kl}^{AB}\ \stackrel{\mathrm{def}}{=}\ E_{k}(r_{k}^{A})+E_{l}(r_{l}^{B})+E_{% kl}(r_{k}^{A},r_{l}^{B})
  29. A A
  30. B B
  31. k k
  32. l l
  33. C C
  34. D D
  35. U k l A B + i = 1 N min X ( E k i ( r k A , r i X ) + E l j ( r l B , r j X ) ) > U k l C D + i = 1 N max X ( E k i ( r k C , r i X ) + E l j ( r l D , r j X ) ) U_{kl}^{AB}+\sum_{i=1}^{N}\min_{X}\left(E_{ki}(r_{k}^{A},r_{i}^{X})+E_{lj}(r_{% l}^{B},r_{j}^{X})\right)>U_{kl}^{CD}+\sum_{i=1}^{N}\max_{X}\left(E_{ki}(r_{k}^% {C},r_{i}^{X})+E_{lj}(r_{l}^{D},r_{j}^{X})\right)
  36. A C A\neq C
  37. B D B\neq D
  38. k l k\neq l
  39. N N
  40. N N
  41. p p
  42. p p
  43. N p Np
  44. r k r_{k}
  45. r l r_{l}
  46. p p
  47. p × p p\times p
  48. N 2 p 2 N^{2}p^{2}
  49. O ( p N ) O(p^{N})
  50. r k r_{k}
  51. k k
  52. E k ( r k A ) - E k ( r k B ) + l = 1 N min X ( E k l ( r k A , r l X ) - E k l ( r k B , r l X ) ) > 0 E_{k}(r_{k}^{A})-E_{k}(r_{k}^{B})+\sum_{l=1}^{N}\min_{X}\left(E_{kl}(r_{k}^{A}% ,r_{l}^{X})-E_{kl}(r_{k}^{B},r_{l}^{X})\right)>0
  53. r k A r_{k}^{A}
  54. r k r_{k}
  55. r k A r_{k}^{A}
  56. r k A r_{k}^{A}

Deal_or_No_Deal_(Netherlands).html

  1. 1 20 \tfrac{1}{20}

Debye–Hückel_theory.html

  1. γ \gamma
  2. a = γ c a=\gamma c
  3. γ ± \gamma_{\pm}
  4. γ ± = ( γ Na + γ Cl - ) 1 / 2 \gamma_{\pm}=\left(\gamma_{\mathrm{Na^{+}}}\gamma_{\mathrm{Cl^{-}}}\right)^{1/2}
  5. γ ± = ( γ A n γ B m ) 1 / ( n + m ) \gamma_{\pm}=\left({\gamma_{A}}^{n}{\gamma_{B}}^{m}\right)^{1/(n+m)}
  6. force = z 1 z 2 e 2 4 π ϵ 0 ϵ r r 2 \,\text{force}=\frac{z_{1}z_{2}e^{2}}{4\pi\epsilon_{0}\epsilon_{r}r^{2}}
  7. 2 ψ j ( r ) = - 1 ϵ 0 ϵ r ρ j ( r ) \nabla^{2}\psi_{j}(r)=-\frac{1}{\epsilon_{0}\epsilon_{r}}\rho_{j}(r)
  8. n i = n i exp ( - z i e ψ j ( r ) k T ) n^{\prime}_{i}=n_{i}\exp\left(\frac{-z_{i}e\psi_{j}(r)}{kT}\right)
  9. 2 ψ j ( r ) = - 1 ϵ 0 ϵ r i { n i ( z i e ) exp ( - z i e ψ j ( r ) k T ) } \nabla^{2}\psi_{j}(r)=-\frac{1}{\epsilon_{0}\epsilon_{r}}\sum_{i}\left\{n_{i}(% z_{i}e)\exp\left(\frac{-z_{i}e\psi_{j}(r)}{kT}\right)\right\}
  10. 2 ψ j ( r ) = κ 2 ψ j ( r ) with κ 2 = e 2 ϵ 0 ϵ r k T i n i z i 2 \nabla^{2}\psi_{j}(r)=\kappa^{2}\psi_{j}(r)\qquad\,\text{with}\qquad\kappa^{2}% =\frac{e^{2}}{\epsilon_{0}\epsilon_{r}kT}\sum_{i}n_{i}z_{i}^{2}
  11. ( r 2 + 2 r r - κ 2 ) ψ j = 0 with solutions ψ j ( r ) = A e - κ r r + A ′′ e κ r r (\partial_{r}^{2}+\frac{2}{r}\partial_{r}-\kappa^{2})\psi_{j}=0\qquad\,\text{% with solutions}\qquad\psi_{j}(r)=A^{\prime}\frac{e^{-\kappa r}}{r}+A^{\prime% \prime}\frac{e^{\kappa r}}{r}
  12. A ′′ = 0 A^{\prime\prime}=0
  13. r = a 0 r=a_{0}
  14. r = 0 r=0
  15. π \pi
  16. A A^{\prime}
  17. ψ j ( r ) = z j e 4 π ε 0 ε r e κ a 0 1 + κ a 0 e - κ r r \psi_{j}(r)=\frac{z_{j}e}{4\pi\varepsilon_{0}\varepsilon_{r}}\frac{e^{\kappa a% _{0}}}{1+\kappa a_{0}}\frac{e^{-\kappa r}}{r}
  18. u j u_{j}
  19. r = 0 r=0
  20. u j = z j e ( ψ j ( a 0 ) - z j e 4 π ε 0 ε r 1 a 0 ) = - z j 2 e 2 4 π ε 0 ε r κ 1 + κ a 0 u_{j}=z_{j}e\Big(\psi_{j}(a_{0})-\frac{z_{j}e}{4\pi\varepsilon_{0}\varepsilon_% {r}}\frac{1}{a_{0}}\Big)=-\frac{z_{j}^{2}e^{2}}{4\pi\varepsilon_{0}\varepsilon% _{r}}\frac{\kappa}{1+\kappa a_{0}}
  21. log γ ± \log\gamma_{\pm}
  22. log 10 γ ± = - A z j 2 I 1 + B a 0 I \log_{10}\gamma_{\pm}=-Az_{j}^{2}\frac{\sqrt{I}}{1+Ba_{0}\sqrt{I}}
  23. A = e 2 B 2.303 × 8 π ϵ 0 ϵ r k T A=\frac{e^{2}B}{2.303\times 8\pi\epsilon_{0}\epsilon_{r}kT}
  24. B = ( 2 e 2 N ϵ 0 ϵ r k T ) 1 / 2 B=\left(\frac{2e^{2}N}{\epsilon_{0}\epsilon_{r}kT}\right)^{1/2}
  25. log γ ± \log\gamma_{\pm}
  26. Λ m = Λ m 0 - K c \Lambda_{m}=\Lambda_{m}^{0}-K\sqrt{c}
  27. Λ m 0 \Lambda_{m}^{0}
  28. Λ m = Λ m 0 - ( A + B Λ m 0 ) c \Lambda_{m}=\Lambda_{m}^{0}-(A+B\Lambda_{m}^{0})\sqrt{c}

Deceleration_parameter.html

  1. q \!q
  2. q = def - a ¨ a a ˙ 2 q\ \stackrel{\mathrm{def}}{=}\ -\frac{\ddot{a}a}{\dot{a}^{2}}
  3. a \!a
  4. a ¨ \ddot{a}
  5. q \!q
  6. 3 a ¨ a = - 4 π G ( ρ + 3 p ) = - 4 π G ( 1 + 3 w ) ρ , 3\frac{\ddot{a}}{a}=-4\pi G(\rho+3p)=-4\pi G(1+3w)\rho,
  7. ρ \!\rho
  8. p \!p
  9. w \!w
  10. q = 1 2 ( 1 + 3 w ) ( 1 + K / ( a H ) 2 ) q=\frac{1}{2}(1+3w)\left(1+K/(aH)^{2}\right)
  11. H \!H
  12. K = 1 , 0 \!K=1,0
  13. - 1 \!-1
  14. H ˙ H 2 = - ( 1 + q ) . \frac{\dot{H}}{H^{2}}=-(1+q).
  15. q - 1 \!q\geq-1
  16. q = 1 2 ( 1 + 3 w ) q=\frac{1}{2}(1+3w)
  17. w \!w
  18. - 1 / 3 \!-1/3
  19. q \!q
  20. w 0 \!w\approx 0

Decomposition_of_time_series.html

  1. T t T_{t}
  2. C t C_{t}
  3. S t S_{t}
  4. I t I_{t}

Dedekind_psi_function.html

  1. ψ ( n ) = n p | n ( 1 + 1 p ) , \psi(n)=n\prod_{p|n}\left(1+\frac{1}{p}\right),
  2. ψ ( n ) n s = ζ ( s ) ζ ( s - 1 ) ζ ( 2 s ) . \sum\frac{\psi(n)}{n^{s}}=\frac{\zeta(s)\zeta(s-1)}{\zeta(2s)}.
  3. ψ = Id * | μ | \psi=\mathrm{Id}*|\mu|
  4. ψ k ( n ) = J 2 k ( n ) J k ( n ) \psi_{k}(n)=\frac{J_{2k}(n)}{J_{k}(n)}
  5. n 1 ψ k ( n ) n s = ζ ( s ) ζ ( s - k ) ζ ( 2 s ) \sum_{n\geq 1}\frac{\psi_{k}(n)}{n^{s}}=\frac{\zeta(s)\zeta(s-k)}{\zeta(2s)}
  6. ψ k ( n ) = n k * μ 2 ( n ) \psi_{k}(n)=n^{k}*\mu^{2}(n)
  7. ϵ 2 = 1 , 0 , 0 , 1 , 0 , 0 , 0 , 0 , 1 , 0 , 0 , 0 , 0 , 0 , 0 , 1 , 0 , 0 , 0 \epsilon_{2}=1,0,0,1,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0\ldots
  8. ϵ 2 ( n ) * ψ k ( n ) = σ k ( n ) \epsilon_{2}(n)*\psi_{k}(n)=\sigma_{k}(n)

Deductive_closure.html

  1. \vdash
  2. \vdash

Defective_matrix.html

  1. J = [ λ 1 λ 1 λ ] , J=\begin{bmatrix}\lambda&1&&\\ &\lambda&\ddots&\\ &&\ddots&1\\ &&&\lambda\end{bmatrix},
  2. v = [ 1 0 0 ] . v=\begin{bmatrix}1\\ 0\\ \vdots\\ 0\end{bmatrix}.
  3. [ 3 1 0 3 ] \begin{bmatrix}3&1\\ 0&3\end{bmatrix}
  4. [ 1 0 ] \begin{bmatrix}1\\ 0\end{bmatrix}

Definable_set.html

  1. \mathcal{L}
  2. \mathcal{M}
  3. \mathcal{L}
  4. M M
  5. X X
  6. M M
  7. m m
  8. A M m A\subseteq M^{m}
  9. \mathcal{M}
  10. X X
  11. φ [ x 1 , , x m , y 1 , , y n ] \varphi[x_{1},\ldots,x_{m},y_{1},\ldots,y_{n}]
  12. b 1 , , b n X b_{1},\ldots,b_{n}\in X
  13. a 1 , , a m M a_{1},\ldots,a_{m}\in M
  14. ( a 1 , , a m ) A (a_{1},\ldots,a_{m})\in A
  15. φ [ a 1 , , a m , b 1 , , b n ] \mathcal{M}\models\varphi[a_{1},\ldots,a_{m},b_{1},\ldots,b_{n}]
  16. A A
  17. \mathcal{M}
  18. \mathcal{M}
  19. \mathcal{M}
  20. \mathcal{M}
  21. a a
  22. \mathcal{M}
  23. { a } \{a\}
  24. \mathcal{M}
  25. 𝒩 = ( , < ) \mathcal{N}=(\mathbb{N},<)
  26. 𝒩 \mathcal{N}
  27. 0
  28. φ ( x ) \varphi(x)
  29. φ = ¬ y ( y < x ) , \varphi=\neg\exists y(y<x),
  30. n > 0 n>0
  31. φ ( x ) \varphi(x)
  32. n n
  33. φ = x 0 x n - 1 ( x 0 < x 1 x n - 1 < x y ( y < x ( y x 0 y x n - 1 ) ) ) \varphi=\exists x_{0}\cdots\exists x_{n-1}(x_{0}<x_{1}\land\cdots\land x_{n-1}% <x\land\forall y(y<x\rightarrow(y\equiv x_{0}\lor\cdots\lor y\equiv x_{n-1})))
  34. 𝒵 = ( , < ) \mathcal{Z}=(\mathbb{Z},<)
  35. 𝒩 = ( , + , , < ) \mathcal{N}=(\mathbb{N},+,\cdot,<)
  36. = ( , 0 , 1 , + , ) \mathcal{R}=(\mathbb{R},0,1,+,\cdot)
  37. φ = y ( y y x ) . \varphi=\exists y(y\cdot y\equiv x).
  38. a \R a\in\R
  39. φ [ a ] \mathcal{R}\models\varphi[a]
  40. \mathcal{R}
  41. φ \varphi
  42. \mathcal{R}
  43. a , b \R a,b\in\R
  44. a b a\leq b
  45. b - a b-a
  46. = ( , 0 , 1 , + , , ) \mathcal{R}^{\leq}=(\mathbb{R},0,1,+,\cdot,\leq)
  47. \mathcal{R}^{\leq}
  48. \mathcal{M}
  49. \mathcal{L}
  50. M M
  51. X M X\subseteq M
  52. A M m A\subseteq M^{m}
  53. \mathcal{M}
  54. X X
  55. π : M M \pi:M\to M
  56. \mathcal{M}
  57. X X
  58. a 1 , , a m M a_{1},\ldots,a_{m}\in M
  59. ( a 1 , , a m ) A (a_{1},\ldots,a_{m})\in A
  60. ( π ( a 1 ) , , π ( a m ) ) A (\pi(a_{1}),\ldots,\pi(a_{m}))\in A
  61. 𝒵 = ( , < ) \mathcal{Z}=(\mathbb{Z},<)
  62. 𝒵 \mathcal{Z}
  63. 𝒵 \mathcal{Z}
  64. 𝒵 \mathcal{Z}
  65. \mathbb{Z}
  66. 𝒵 \mathcal{Z}

Degree_Lintner.html

  1. Lintner = WK + 16 3.5 {}^{\circ}\mbox{Lintner}~{}=\frac{{}^{\circ}\mbox{WK}~{}+16}{3.5}
  2. WK = ( 3.5 × Lintner ) - 16 {}^{\circ}\mbox{WK}~{}=\left(3.5\times{}^{\circ}\mbox{Lintner}~{}\right)-16

Degree_of_a_polynomial.html

  1. 7 x 2 y 3 + 4 x - 9 7x^{2}y^{3}+4x-9
  2. 7 x 2 y 3 + 4 x 1 y 0 - 9 x 0 y 0 7x^{2}y^{3}+4x^{1}y^{0}-9x^{0}y^{0}
  3. ( x + 1 ) 2 - ( x - 1 ) 2 (x+1)^{2}-(x-1)^{2}
  4. ( x + 1 ) 2 - ( x - 1 ) 2 = 4 x (x+1)^{2}-(x-1)^{2}=4x
  5. x 2 + x y + y 2 x^{2}+xy+y^{2}
  6. x 2 + y 2 x^{2}+y^{2}
  7. 3 - 5 x + 2 x 5 - 7 x 9 3-5x+2x^{5}-7x^{9}
  8. ( y - 3 ) ( 2 y + 6 ) ( - 4 y - 21 ) (y-3)(2y+6)(-4y-21)
  9. ( 3 z 8 + z 5 - 4 z 2 + 6 ) + ( - 3 z 8 + 8 z 4 + 2 z 3 + 14 z ) (3z^{8}+z^{5}-4z^{2}+6)+(-3z^{8}+8z^{4}+2z^{3}+14z)
  10. z 8 z^{8}
  11. 3 - 5 x + 2 x 5 - 7 x 9 3-5x+2x^{5}-7x^{9}
  12. - 7 x 9 + 2 x 5 - 5 x + 3 -7x^{9}+2x^{5}-5x+3
  13. ( y - 3 ) ( 2 y + 6 ) ( - 4 y - 21 ) (y-3)(2y+6)(-4y-21)
  14. - 8 y 3 - 42 y 2 + 72 y + 378 -8y^{3}-42y^{2}+72y+378
  15. ( 3 z 8 + z 5 - 4 z 2 + 6 ) + ( - 3 z 8 + 8 z 4 + 2 z 3 + 14 z ) (3z^{8}+z^{5}-4z^{2}+6)+(-3z^{8}+8z^{4}+2z^{3}+14z)
  16. z 5 + 8 z 4 + 2 z 3 - 4 z 2 + 14 z + 6 z^{5}+8z^{4}+2z^{3}-4z^{2}+14z+6
  17. deg ( P + Q ) max ( deg ( P ) , deg ( Q ) ) \deg(P+Q)\leq\max(\deg(P),\deg(Q))
  18. deg ( P - Q ) max ( deg ( P ) , deg ( Q ) ) \deg(P-Q)\leq\max(\deg(P),\deg(Q))
  19. ( x 3 + x ) + ( x 2 + 1 ) = x 3 + x 2 + x + 1 (x^{3}+x)+(x^{2}+1)=x^{3}+x^{2}+x+1
  20. ( x 3 + x ) - ( x 3 + x 2 ) = - x 2 + x (x^{3}+x)-(x^{3}+x^{2})=-x^{2}+x
  21. deg ( c P ) = deg ( P ) \deg(cP)=\deg(P)
  22. 2 ( x 2 + 3 x - 2 ) = 2 x 2 + 6 x - 4 2(x^{2}+3x-2)=2x^{2}+6x-4
  23. x 2 + 3 x - 2 x^{2}+3x-2
  24. 𝐙 / 4 𝐙 \mathbf{Z}/4\mathbf{Z}
  25. deg ( 1 + 2 x ) = 1 \deg(1+2x)=1
  26. deg ( 2 ( 1 + 2 x ) ) = deg ( 2 + 4 x ) = deg ( 2 ) = 0 \deg(2(1+2x))=\deg(2+4x)=\deg(2)=0
  27. deg ( P Q ) = deg ( P ) + deg ( Q ) \deg(PQ)=\deg(P)+\deg(Q)
  28. ( x 3 + x ) ( x 2 + 1 ) = x 5 + 2 x 3 + x (x^{3}+x)(x^{2}+1)=x^{5}+2x^{3}+x
  29. 𝐙 / 4 𝐙 \mathbf{Z}/4\mathbf{Z}
  30. deg ( 2 x ) + deg ( 1 + 2 x ) = 1 + 1 = 2 \deg(2x)+\deg(1+2x)=1+1=2
  31. deg ( 2 x ( 1 + 2 x ) ) = deg ( 2 x ) = 1 \deg(2x(1+2x))=\deg(2x)=1
  32. P P
  33. Q Q
  34. deg ( P Q ) = deg ( P ) deg ( Q ) \deg(P\circ Q)=\deg(P)\deg(Q)
  35. P = ( x 3 + x ) P=(x^{3}+x)
  36. Q = ( x 2 + 1 ) Q=(x^{2}+1)
  37. P Q = P ( x 2 + 1 ) = ( x 2 + 1 ) 3 + ( x 2 + 1 ) = x 6 + 3 x 4 + 4 x 2 + 2 P\circ Q=P\circ(x^{2}+1)=(x^{2}+1)^{3}+(x^{2}+1)=x^{6}+3x^{4}+4x^{2}+2
  38. 𝐙 / 4 𝐙 \mathbf{Z}/4\mathbf{Z}
  39. deg ( 2 x ) deg ( 1 + 2 x ) = 1 1 = 1 \deg(2x)\deg(1+2x)=1\cdot 1=1
  40. deg ( 2 x ( 1 + 2 x ) ) = deg ( 2 + 4 x ) = deg ( 2 ) = 0 \deg(2x\circ(1+2x))=\deg(2+4x)=\deg(2)=0
  41. - -\infty
  42. - -\infty
  43. - -\infty
  44. - -\infty
  45. max ( a , - ) = a , \max(a,-\infty)=a,
  46. a + - = - . a+-\infty=-\infty.
  47. ( x 3 + x ) + ( 0 ) = x 3 + x (x^{3}+x)+(0)=x^{3}+x
  48. 3 max ( 3 , - ) 3\leq\max(3,-\infty)
  49. ( x ) - ( x ) = 0 (x)-(x)=0
  50. - -\infty
  51. - max ( 1 , 1 ) -\infty\leq\max(1,1)
  52. ( 0 ) ( x 2 + 1 ) = 0 (0)(x^{2}+1)=0
  53. - -\infty
  54. - = - + 2 -\infty=-\infty+2
  55. deg f = lim x log | f ( x ) | log x . \deg f=\lim_{x\rightarrow\infty}\frac{\log|f(x)|}{\log x}.
  56. 1 / x \ 1/x
  57. x \sqrt{x}
  58. log x \ \log x
  59. exp x \ \exp x
  60. deg f = lim x x f ( x ) f ( x ) . \deg f=\lim_{x\to\infty}\frac{xf^{\prime}(x)}{f(x)}.
  61. / 4 \mathbb{Z}/4\mathbb{Z}
  62. f ( x ) = a 0 f(x)=a_{0}
  63. a 0 a_{0}
  64. - -\infty

Delta-ring.html

  1. \mathcal{R}
  2. A B A\cup B\in\mathcal{R}
  3. A , B A,B\in\mathcal{R}
  4. A - B A-B\in\mathcal{R}
  5. A , B A,B\in\mathcal{R}
  6. n = 1 A n \bigcap_{n=1}^{\infty}A_{n}\in\mathcal{R}
  7. A n A_{n}\in\mathcal{R}
  8. n n\in\mathbb{N}
  9. \mathcal{R}

Delta_potential.html

  1. - 2 2 m d 2 ψ d x 2 ( x ) + V ( x ) ψ ( x ) = E ψ ( x ) -\frac{\hbar^{2}}{2m}\frac{d^{2}\psi}{dx^{2}}(x)+V(x)\psi(x)=E\psi(x)
  2. V ( x ) = λ δ ( x ) \displaystyle V(x)=\lambda\delta(x)
  3. d 2 ψ d x 2 = - 2 m E 2 ψ ; \textstyle\frac{d^{2}\psi}{dx^{2}}=-\frac{2mE}{\hbar^{2}}\psi;
  4. k = 2 m E k=\frac{\sqrt{2mE}}{\hbar}
  5. ψ ( x ) = { ψ L ( x ) = A r e i k x + A l e - i k x , if x < 0 ; ψ R ( x ) = B r e i k x + B l e - i k x , if x > 0 , \psi(x)=\begin{cases}\psi_{\mathrm{L}}(x)=A_{\mathrm{r}}e^{ikx}+A_{\mathrm{l}}% e^{-ikx},&\,\text{ if }x<0;\\ \psi_{\mathrm{R}}(x)=B_{\mathrm{r}}e^{ikx}+B_{\mathrm{l}}e^{-ikx},&\,\text{ if% }x>0,\end{cases}
  6. ψ ( 0 ) = ψ L ( 0 ) = ψ R ( 0 ) = A r + A l = B r + B l \psi(0)=\psi_{L}(0)=\psi_{R}(0)=A_{r}+A_{l}=B_{r}+B_{l}
  7. - 2 2 m - ϵ + ϵ ψ ′′ ( x ) d x + - ϵ + ϵ V ( x ) ψ ( x ) d x = E - ϵ + ϵ ψ ( x ) d x . -\frac{\hbar^{2}}{2m}\int_{-\epsilon}^{+\epsilon}\psi^{\prime\prime}(x)\,dx+% \int_{-\epsilon}^{+\epsilon}V(x)\psi(x)\,dx=E\int_{-\epsilon}^{+\epsilon}\psi(% x)\,dx.
  8. - 2 2 m [ ψ R ( 0 ) - ψ L ( 0 ) ] + λ ψ ( 0 ) \textstyle-\frac{\hbar^{2}}{2m}[\psi^{\prime}_{R}(0)-\psi^{\prime}_{L}(0)]+% \lambda\psi(0)
  9. - ϵ + ϵ ψ ′′ ( x ) d x = [ ψ ( + ϵ ) - ψ ( - ϵ ) ] \int_{-\epsilon}^{+\epsilon}\psi^{\prime\prime}(x)\,dx=[\psi^{\prime}({+% \epsilon})-\psi^{\prime}({-\epsilon})]
  10. - 2 2 m i k ( - A r + A l + B r - B l ) + λ ( A r + A l ) = 0. -\frac{\hbar^{2}}{2m}ik(-A_{r}+A_{l}+B_{r}-B_{l})+\lambda(A_{r}+A_{l})=0.
  11. { A r + A l - B r - B l = 0 ; - A r + A l + B r - B l = 2 m λ i k 2 ( A r + A l ) . \begin{cases}A_{r}+A_{l}-B_{r}-B_{l}&=0;\\ -A_{r}+A_{l}+B_{r}-B_{l}&=\frac{2m\lambda}{ik\hbar^{2}}(A_{r}+A_{l}).\end{cases}
  12. ψ ( x ) = { ψ L ( x ) = A l e κ x , if x < 0 ; ψ R ( x ) = B r e - κ x , if x > 0. \psi(x)=\begin{cases}\psi_{\,\text{L}}(x)=A_{\,\text{l}}e^{\kappa x},&\,\text{% if }x<0;\\ \psi_{\,\text{R}}(x)=B_{\,\text{r}}e^{-\kappa x},&\,\text{ if }x>0.\end{cases}
  13. { A l = B r = κ ; κ = - m λ 2 ; \begin{cases}A_{l}=B_{r}=\sqrt{\kappa};\\ \kappa=-\frac{m\lambda}{\hbar^{2}};\end{cases}
  14. E = - 2 κ 2 2 m = - m λ 2 2 2 . E=-\frac{\hbar^{2}\kappa^{2}}{2m}=-\frac{m\lambda^{2}}{2\hbar^{2}}.
  15. E > 0 E>0\,\!
  16. m λ 2 2 2 \frac{m\lambda^{2}}{2\hbar^{2}}\,\!
  17. t = 1 1 - m λ i 2 k t=\cfrac{1}{1-\cfrac{m\lambda}{i\hbar^{2}k}}\,\!
  18. r = 1 i 2 k m λ - 1 r=\cfrac{1}{\cfrac{i\hbar^{2}k}{m\lambda}-1}\,\!
  19. R = | r | 2 = 1 1 + 4 k 2 m 2 λ 2 = 1 1 + 2 2 E m λ 2 . R=|r|^{2}=\cfrac{1}{1+\cfrac{\hbar^{4}k^{2}}{m^{2}\lambda^{2}}}=\cfrac{1}{1+% \cfrac{2\hbar^{2}E}{m\lambda^{2}}}.\,\!
  20. T = | t | 2 = 1 - R = 1 1 + m 2 λ 2 4 k 2 = 1 1 + m λ 2 2 2 E T=|t|^{2}=1-R=\cfrac{1}{1+\cfrac{m^{2}\lambda^{2}}{\hbar^{4}k^{2}}}=\cfrac{1}{% 1+\cfrac{m\lambda^{2}}{2\hbar^{2}E}}\,\!
  21. m m\,\!
  22. Ψ ( x , y , z ) = ψ ( x ) ϕ ( y , z ) \Psi(x,y,z)=\psi(x)\phi(y,z)\,\!
  23. - 2 2 m d 2 ψ d x 2 ( x ) + V ( x ) ψ ( x ) = E ψ ( x ) -\frac{\hbar^{2}}{2m}\frac{d^{2}\psi}{dx^{2}}(x)+V(x)\psi(x)=E\psi(x)
  24. V ( x ) = - q [ δ ( x + R 2 ) + λ δ ( x - R 2 ) ] V(x)=-q\left[\delta(x+\frac{R}{2})+\lambda\delta(x-\frac{R}{2})\right]
  25. 0 < R < 0<R<\infty
  26. x = ± R 2 x=\pm{\textstyle\frac{R}{2}}
  27. = m = 1 \hbar=m=1
  28. 0 < λ < 1 0<\lambda<1
  29. ψ ( x ) = A e - d | x + R 2 | + B e - d | x - R 2 | \psi(x)~{}=~{}Ae^{-d\left|x+\frac{R}{2}\right|}+Be^{-d\left|x-\frac{R}{2}% \right|}
  30. | q - d q e - d R q λ e - d R q λ - d | = 0 where E = - d 2 2 . \left|\begin{array}[]{cc}q-d&qe^{-dR}\\ q\lambda e^{-dR}&q\lambda-d\end{array}\right|=0\quad\mbox{where}~{}\quad E=-% \frac{d^{2}}{2}~{}.
  31. d d
  32. d ± ( λ ) = 1 2 q ( λ + 1 ) ± 1 2 { q 2 ( 1 + λ ) 2 - 4 λ q 2 [ 1 - e - 2 d ± ( λ ) R ] } 1 / 2 d_{\pm}(\lambda)~{}=~{}{\textstyle\frac{1}{2}}q(\lambda+1)\pm{\textstyle\frac{% 1}{2}}\left\{q^{2}(1+\lambda)^{2}-4\,\lambda q^{2}[1-e^{-2d_{\pm}(\lambda)R}]% \right\}^{1/2}
  33. d = d ± d=d_{\pm}
  34. λ = 1 \lambda=1
  35. d ± = q [ 1 ± e - d ± R ] d_{\pm}=q[1\pm e^{-d_{\pm}R}]
  36. A = B A=B
  37. A = - B A=-B
  38. H 2 + H_{2}^{+}
  39. d ± = q + W ( ± q R e - q R ) / R d_{\pm}=q~{}+~{}W(\pm qRe^{-qR})/R
  40. d + d_{+}
  41. q R 1 qR\leq 1
  42. d - = 0 d_{-}=0
  43. E = 0 E=0

Demand_set.html

  1. L L
  2. R + l R+l
  3. L L
  4. > p >p
  5. x > p x x>px^{\prime}
  6. x x
  7. x x^{\prime}
  8. e e
  9. p p
  10. D ( > p , p , e ) D(>p,p,e)

Dember_effect.html

  1. ( I 3 ) (I_{3})
  2. ( I 1 ) (I_{1})
  3. ( I 2 ) (I_{2})
  4. I 3 - I 2 - I 1 = I 4 I_{3}-I_{2}-I_{1}=I_{4}\,
  5. I 1 I_{1}
  6. I 2 I_{2}
  7. I 4 I_{4}
  8. I 1 I_{1}
  9. I 2 I_{2}
  10. I 2 I_{2}
  11. I 4 I_{4}
  12. I 1 I_{1}

Demihypercube.html

  1. D n , D_{n},
  2. 2 n - 1 n ! , 2^{n-1}n!,
  3. B C n BC_{n}

Dendrite_(mathematics).html

  1. f ( z ) = z 2 + c f(z)=z^{2}+c
  2. f f

Deneb_in_fiction.html

  1. M \begin{smallmatrix}M_{\odot}\end{smallmatrix}
  2. M \begin{smallmatrix}M_{\odot}\end{smallmatrix}

Dense_order.html

  1. x y x R y ( z x R z z R y ) . \forall x\ \forall y\ xRy\Rightarrow(\exists z\ xRz\land zRy).

Departure_function.html

  1. H ig - H R T = V [ T ( Z T ) V ] d V V + 1 - Z \frac{H^{\mathrm{ig}}-H}{RT}=\int_{V}^{\infty}\left[T\left(\frac{\partial Z}{% \partial T}\right)_{V}\right]\frac{\mathrm{d}V}{V}+1-Z
  2. S ig - S R = V [ T ( Z T ) V - 1 + Z ] d V V - ln Z \frac{S^{\mathrm{ig}}-S}{R}=\int_{V}^{\infty}\left[T\left(\frac{\partial Z}{% \partial T}\right)_{V}-1+Z\right]\frac{\mathrm{d}V}{V}-\ln Z
  3. G ig - G R T = V ( 1 - Z ) d V V + ln Z + 1 - Z \frac{G^{\mathrm{ig}}-G}{RT}=\int_{V}^{\infty}(1-Z)\frac{\mathrm{d}V}{V}+\ln Z% +1-Z
  4. h T , P - h T , P ideal = R T C [ T r ( Z - 1 ) - 2.078 ( 1 + κ ) α ln ( Z + 2.414 B Z - 0.414 B ) ] h_{T,P}-h_{T,P}^{\mathrm{ideal}}=RT_{C}\left[T_{r}(Z-1)-2.078(1+\kappa)\sqrt{% \alpha}\ln\left(\frac{Z+2.414B}{Z-0.414B}\right)\right]
  5. s T , P - s T , P ideal = R [ ln ( Z - B ) - 2.078 κ ( 1 + κ T r - κ ) ln ( Z + 2.414 B Z - 0.414 B ) ] s_{T,P}-s_{T,P}^{\mathrm{ideal}}=R\left[\ln(Z-B)-2.078\kappa\left(\frac{1+% \kappa}{\sqrt{T_{r}}}-\kappa\right)\ln\left(\frac{Z+2.414B}{Z-0.414B}\right)\right]
  6. α \alpha
  7. κ = 0.37464 + 1.54226 ω - 0.26992 ω 2 \kappa=0.37464+1.54226\;\omega-0.26992\;\omega^{2}
  8. B = 0.07780 P r T r B=0.07780\frac{P_{r}}{T_{r}}