wpmath0000014_2

Clifton–Pohl_torus.html

  1. M = 2 - { 0 } \mathrm{M}=\mathbb{R}^{2}-\{0\}
  2. g = d x d y + d y d x x 2 + y 2 g=\frac{dx\otimes dy+dy\otimes dx}{x^{2}+y^{2}}
  3. M M
  4. λ ( x , y ) = ( 2 x , 2 y ) \lambda(x,y)=(2x,2y)
  5. Γ \Gamma
  6. λ \lambda
  7. Γ \Gamma
  8. M M
  9. T = M / Γ T=M/\Gamma
  10. σ ( t ) = ( 1 1 - t , 0 ) \sigma(t)=\left(\frac{1}{1-t},0\right)
  11. t = 1 t=1
  12. M M
  13. T T
  14. T T
  15. σ ( t ) = ( tan t , 1 ) \sigma(t)=(\tan t,1)
  16. M M
  17. T T

Closure_phase.html

  1. θ \theta
  2. x 1 = a 1 c o s ( θ ) x_{1}=a_{1}cos(\theta)
  3. x 2 = a 2 c o s ( θ ) x_{2}=a_{2}cos(\theta)
  4. x 3 = ( a 1 + a 2 ) c o s ( θ ) x_{3}=(a_{1}+a_{2})cos(\theta)
  5. θ 0 \theta_{0}
  6. x ( θ ) - x ( θ 0 ) x(\theta)-x(\theta_{0})
  7. P P
  8. ϕ 1 \phi_{1}
  9. ϕ 2 \phi_{2}
  10. ϕ 3 \phi_{3}
  11. ψ 1 \psi_{1}
  12. ψ 2 \psi_{2}
  13. ψ 3 \psi_{3}
  14. ψ 1 = ϕ 1 + e B - e C \psi_{1}=\phi_{1}+e_{B}-e_{C}
  15. ψ 2 = ϕ 2 - e B \psi_{2}=\phi_{2}-e_{B}
  16. ψ 3 = ϕ 3 - e C \psi_{3}=\phi_{3}-e_{C}
  17. O = ψ 1 + ψ 2 - ψ 3 O=\psi_{1}+\psi_{2}-\psi_{3}
  18. O = ϕ 1 + ϕ 2 - ϕ 3 O=\phi_{1}+\phi_{2}-\phi_{3}
  19. O O
  20. O O
  21. | P ( x ) | |P(x)|
  22. P ( x ) P(x)
  23. P ( x 1 ) P ( x 2 ) P * ( x 1 + x 2 ) . P(x_{1})P(x_{2})P^{*}(x_{1}+x_{2}).
  24. B 123 = C 12 C 23 C 13 * B_{123}=C_{12}C_{23}C_{13}^{*}
  25. O = a r g ( B 123 ) O=arg(B_{123})
  26. P ( x ) P(x)
  27. | P ( x ) | |P(x)|
  28. P ( x 1 + x 2 ) P(x_{1}+x_{2})
  29. P ( x 1 ) P(x_{1})
  30. P ( x 2 ) P(x_{2})
  31. P ( x ) P(x)
  32. x x
  33. P ( x ) P(x)

CLs_upper_limits_(particle_physics).html

  1. θ [ 0 , ) \theta\in[0,\infty)
  2. 1 - α 1-\alpha^{\prime}
  3. θ u p ( X ) \theta_{up}(X)
  4. ( θ u p ( X ) < θ | θ ) ( θ u p ( X ) < θ | 0 ) α for all θ . \frac{\mathbb{P}(\theta_{up}(X)<\theta|\theta)}{\mathbb{P}(\theta_{up}(X)<% \theta|0)}\leq\alpha^{\prime}\,\text{ for all }\theta.
  5. ( θ u p ( X ) θ | θ ) \mathbb{P}(\theta_{up}(X)\geq\theta|\theta)
  6. 1 - α 1-\alpha^{\prime}
  7. H 0 : θ = θ 0 H_{0}:\theta=\theta_{0}
  8. H 1 : θ = 0 H_{1}:\theta=0
  9. θ 0 \theta_{0}
  10. α \alpha
  11. θ 0 \theta_{0}
  12. θ u p ( X ) < θ 0 \theta_{up}(X)<\theta_{0}
  13. 1 - β 1-\beta
  14. H 0 H_{0}
  15. α / ( 1 - β ) \alpha/(1-\beta)
  16. α \alpha^{\prime}
  17. θ 0 \theta_{0}
  18. α \alpha^{\prime}
  19. θ 0 \theta_{0}
  20. θ = 0 \theta=0
  21. q θ ( X ) q_{\theta}(X)
  22. θ \theta
  23. ( q θ ( X ) q θ * | θ ) ( q θ ( X ) q θ * | 0 ) = α . \frac{\mathbb{P}(q_{\theta}(X)\geq q_{\theta}^{*}|\theta)}{\mathbb{P}(q_{% \theta}(X)\geq q_{\theta}^{*}|0)}=\alpha^{\prime}.
  24. q θ * q_{\theta}^{*}
  25. n n
  26. s s
  27. b b
  28. n Poiss ( s + b ) n\sim\,\text{Poiss}(s+b)
  29. b b
  30. s s
  31. s s
  32. n * n^{*}
  33. s s
  34. ( n n * | s + b ) α \mathbb{P}(n\leq n^{*}|s+b)\leq\alpha
  35. 1 - α 1-\alpha
  36. b = 3 b=3
  37. n * = 0 n^{*}=0
  38. s + b 3 s+b\geq 3
  39. s 0 s\geq 0
  40. s s
  41. s s
  42. n n * n\leq n^{*}
  43. n b n * n_{b}\leq n^{*}
  44. n b n_{b}
  45. n b n_{b}
  46. n b n_{b}
  47. n b n_{b}
  48. s s
  49. ( n n * | n b n * , s + b ) = ( n n * | s + b ) ( n b n * | s + b ) = ( n n * | s + b ) ( n n * | b ) . \mathbb{P}(n\leq n^{*}|n_{b}\leq n^{*},s+b)=\frac{\mathbb{P}(n\leq n^{*}|s+b)}% {\mathbb{P}(n_{b}\leq n^{*}|s+b)}=\frac{\mathbb{P}(n\leq n^{*}|s+b)}{\mathbb{P% }(n\leq n^{*}|b)}.
  50. q ( X ) q(X)
  51. p θ = ( q ( X ) > q * | θ ) p_{\theta}=\mathbb{P}(q(X)>q^{*}|\theta)
  52. q * q*
  53. p θ p_{\theta}
  54. θ \theta
  55. θ \theta
  56. q * q*
  57. p θ ( q ( X ) > q * | 0 ) p 0 * p_{\theta}\leq\mathbb{P}(q(X)>q^{*}|0)\equiv p_{0}^{*}
  58. n b n_{b}
  59. ( q ( X ) q * | p θ p 0 * , θ ) = ( q ( X ) q * | θ ) ( p θ p 0 * | θ ) = ( q ( X ) q * | θ ) p 0 * = ( q ( X ) q * | θ ) ( q ( X ) > q * | 0 ) . \mathbb{P}(q(X)\geq q^{*}|p_{\theta}\leq p_{0}^{*},\theta)=\frac{\mathbb{P}(q(% X)\geq q^{*}|\theta)}{\mathbb{P}(p_{\theta}\leq p_{0}^{*}|\theta)}=\frac{% \mathbb{P}(q(X)\geq q^{*}|\theta)}{p_{0}^{*}}=\frac{\mathbb{P}(q(X)\geq q^{*}|% \theta)}{\mathbb{P}(q(X)>q^{*}|0)}.
  60. α / ( 1 - β ) \alpha/(1-\beta)
  61. α \alpha
  62. α \alpha
  63. ( 1 - β ) (1-\beta)
  64. 1 - α 1-\alpha
  65. ( β ) (\beta)
  66. H 1 , ( H 2 ) H_{1},(H_{2})
  67. H 1 H_{1}
  68. α / ( 1 - β ) \alpha/(1-\beta)
  69. α / ( 1 - β ) \alpha/(1-\beta)
  70. H 2 H_{2}
  71. H 1 H_{1}
  72. α \alpha
  73. H 1 H_{1}
  74. β \beta
  75. H 2 H_{2}
  76. 1 - α 1-\alpha^{\prime}
  77. θ u p = θ ^ + σ Φ - 1 ( 1 - α Φ ( θ ^ / σ ) ) , \theta_{up}=\hat{\theta}+\sigma\Phi^{-1}(1-\alpha^{\prime}\Phi(\hat{\theta}/% \sigma)),
  78. Φ \Phi
  79. θ ^ \hat{\theta}
  80. θ \theta
  81. σ \sigma
  82. σ \sigma
  83. θ \theta

Co-Büchi_automaton.html

  1. w w
  2. F F
  3. w w
  4. F F
  5. 𝒜 = ( Q , Σ , δ , q 0 , F ) \mathcal{A}=(Q,\Sigma,\delta,q_{0},F)
  6. Q Q
  7. Q Q
  8. 𝒜 \mathcal{A}
  9. Σ \Sigma
  10. 𝒜 \mathcal{A}
  11. δ : Q × Σ Q \delta:Q\times\Sigma\rightarrow Q
  12. 𝒜 \mathcal{A}
  13. q 0 q_{0}
  14. Q Q
  15. F Q F\subseteq Q
  16. 𝒜 \mathcal{A}
  17. w w
  18. ρ ( w ) \rho(w)
  19. ρ ( w ) \rho(w)
  20. F F
  21. δ \delta
  22. Δ \Delta
  23. q 0 q_{0}
  24. Q 0 Q_{0}
  25. i j : j i ρ ( w j ) F . \exists i\forall j:\;j\geq i\quad\rho(w_{j})\in F.
  26. i j : j i ρ ( w j ) F . \forall i\exists j:\;j\geq i\quad\rho(w_{j})\in F.

Cochran's_C_test.html

  1. C j = S j 2 i = 1 N S i 2 C_{j}=\frac{S_{j}^{2}}{\displaystyle\sum_{i=1}^{N}S_{i}^{2}}
  2. C UL ( α , n , N ) = [ 1 + N - 1 F c ( α / N , ( n - 1 ) , ( N - 1 ) ( n - 1 ) ) ] - 1 . C_{\text{UL}}(\alpha,n,N)=\left[1+\frac{N-1}{F_{\text{c}}(\alpha/N,(n-1),(N-1)% (n-1))}\right]^{-1}.

Codon_Adaptation_Index.html

  1. CAI = exp ( 1 / L l = 1 L log ( w i ( l ) ) ) \,\text{CAI}=\exp\left(1/L\sum_{l=1}^{L}{\log\left(w_{i}(l)\right)}\right)
  2. w i = f i max ( f j ) i , j [ synonymous codons for amino acid ] w_{i}=\frac{f_{i}}{\max(f_{j})}\qquad i,j\in[\,\text{synonymous codons for % amino acid}]

Coenergy.html

  1. d W i n p u t = d W s t o r e d ( d W m e c h a n i c a l = 0 ) dW_{input}=dW_{stored}~{}~{}~{}~{}~{}~{}~{}~{}(dW_{mechanical}=0)\;
  2. λ \lambda
  3. d W i n p u t = e i d t dW_{input}=e~{}i~{}dt\;
  4. d W s t o r e d = i d λ dW_{stored}=i~{}d\lambda\;
  5. d W s t o r e d = e i d t = i d λ dW_{stored}=e~{}i~{}dt=i~{}d\lambda\;
  6. i - λ i-\lambda
  7. λ 1 \lambda_{1}
  8. λ 2 \lambda_{2}
  9. Δ W s t o r e d = λ 1 λ 2 i ( λ ) d λ \Delta W_{stored}=\int_{\lambda_{1}}^{\lambda_{2}}i(\lambda)~{}d\lambda\;
  10. λ \lambda
  11. E n e r g y = a r e a O A B O = W s t o r e d = 0 λ i ( λ ) d λ Energy=area~{}OABO=W_{stored}=\int_{0}^{\lambda}i(\lambda)~{}d\lambda\;
  12. C o e n e r g y = a r e a O A C O = W s t o r e d = 0 i λ ( i ) d i Coenergy=area~{}OACO=W^{\prime}_{stored}=\int_{0}^{i}\lambda(i)~{}di\;
  13. a r e a O C A B O = a r e a O A B O + a r e a O A C O area~{}OCABO=area~{}OABO+area~{}OACO\;
  14. λ \lambda
  15. E n e r g y ( W ) + C o e n e r g y ( W ) = i λ Energy(W)+Coenergy(W^{\prime})=i\lambda\;
  16. L = λ i L=\frac{\lambda}{i}
  17. W e n e r g y = 1 2 λ 2 L = 1 2 L i 2 W_{energy}=\frac{1}{2}\frac{\lambda^{2}}{L}=\frac{1}{2}~{}L~{}i^{2}
  18. λ \lambda
  19. W ( λ , x ) e n e r g y = 1 2 λ 2 L ( x ) W(\lambda,x)_{energy}=\frac{1}{2}~{}\frac{\lambda^{2}}{L(x)}
  20. W ( i , x ) c o e n e r g y = 1 2 L ( x ) i 2 W^{\prime}(i,x)_{coenergy}=\frac{1}{2}~{}L(x)~{}i^{2}

Cofree_coalgebra.html

  1. T k V = V k = V V V , T^{k}V=V^{\otimes k}=V\otimes V\otimes\cdots\otimes V,
  2. T ( V ) = k T k V = 𝔽 V ( V V ) ( V V V ) . T(V)=\bigoplus_{k\in\mathbb{N}}T^{k}V=\mathbb{F}\oplus V\oplus(V\otimes V)% \oplus(V\otimes V\otimes V)\oplus\cdots.
  3. Δ ( v 1 v k ) := j = 0 k ( v 1 v j ) ( v j + 1 v k ) \Delta(v_{1}\otimes\dots\otimes v_{k}):=\sum_{j=0}^{k}(v_{1}\otimes\dots% \otimes v_{j})\otimes(v_{j+1}\otimes\dots\otimes v_{k})
  4. T k V × T k V * 𝔽 T^{k}V\times T^{k}V^{*}\to\mathbb{F}
  5. Δ ( f ) ( a b ) = f ( a b ) . \Delta(f)(a\otimes b)=f(ab).
  6. T ^ ( V ) × T ( V * ) 𝔽 , \hat{T}(V)\times T(V^{*})\to\mathbb{F},
  7. T ^ ( V ) = k T k V \hat{T}(V)=\prod_{k\in\mathbb{N}}T^{k}V
  8. Δ ^ : T ^ ( V ) T ^ ( V ) ^ T ^ ( V ) \hat{\Delta}\colon\hat{T}(V)\to\hat{T}(V)\hat{\otimes}\hat{T}(V)
  9. T ^ ( V ) ^ T ^ ( V ) = j , k T j V T k V , \hat{T}(V)\hat{\otimes}\hat{T}(V)=\prod_{j,k\in\mathbb{N}}T^{j}V\otimes T^{k}V,
  10. T ^ ( V ) T ^ ( V ) = { X T ^ ( V ) ^ T ^ ( V ) : k , f j , g j T ^ ( V ) s.t. X = j = 0 k ( f j g j ) } . \hat{T}(V)\otimes\hat{T}(V)=\{X\in\hat{T}(V)\hat{\otimes}\hat{T}(V):\exists\,k% \in\mathbb{N},f_{j},g_{j}\in\hat{T}(V)\,\text{ s.t. }X={\textstyle\sum}_{j=0}^% {k}(f_{j}\otimes g_{j})\}.
  11. T ( V ) C T ^ ( V ) and Δ ^ ( C ) C C T ^ ( V ) ^ T ^ ( V ) , T(V)\subseteq C\subseteq\hat{T}(V)\,\text{ and }\hat{\Delta}(C)\subseteq C% \otimes C\subseteq\hat{T}(V)\hat{\otimes}\hat{T}(V),
  12. C ( V ) = { f T ^ ( V ) : Δ ^ ( f ) T ^ ( V ) T ^ ( V ) } . C(V)=\{f\in\hat{T}(V):\hat{\Delta}(f)\in\hat{T}(V)\otimes\hat{T}(V)\}.
  13. C ( V ) = { I 0 T ^ ( V ) : I T ( V * ) , codim I < } C(V)=\bigcup\{I^{0}\subseteq\hat{T}(V):I\triangleleft T(V^{*}),\,\mathrm{codim% }\,I<\infty\}
  14. T ^ ( V ) = k T k V \hat{T}(V)=\prod_{k\in\mathbb{N}}T^{k}V
  15. j 𝒩 a j τ j \sum_{j\in\mathcal{N}}a_{j}\tau^{j}
  16. Δ ( τ k ) = i + j = k τ i τ j \Delta(\tau^{k})=\sum_{i+j=k}\tau^{i}\otimes\tau^{j}
  17. ( j 𝒩 a j τ j ) ( k = 0 N b k t k ) = k = 0 N a k b k . \biggl(\sum_{j\in\mathcal{N}}a_{j}\tau^{j}\biggr)\biggl(\sum_{k=0}^{N}b_{k}t^{% k}\biggr)=\sum_{k=0}^{N}a_{k}b_{k}.
  18. τ j - N p ( τ - 1 ) = τ j τ N p ( τ - 1 ) \frac{\tau^{j-N}}{p(\tau^{-1})}=\frac{\tau^{j}}{\tau^{N}p(\tau^{-1})}

Cohen's_cryptosystem.html

  1. p p
  2. k k
  3. w 0 , , w k - 1 w_{0},\ldots,w_{k-1}
  4. k k
  5. u 0 , , u k - 1 u_{0},\ldots,u_{k-1}
  6. - B -B
  7. B B
  8. B B
  9. A = p 2 k A=\lfloor\frac{p}{2k}\rfloor
  10. k k
  11. v 0 , , v k - 1 v_{0},\ldots,v_{k-1}
  12. 0
  13. A A
  14. w i = ( u i p + v i ) w_{i}=(u_{i}p+v_{i})
  15. m m
  16. k 2 \frac{k}{2}
  17. c = ( - 1 ) m w i c=(-1)^{m}\sum w_{i}
  18. h = c mod p = ( - 1 ) m v i h=c\mod p=(-1)^{m}\sum v_{i}
  19. m = 0 m=0
  20. 0 < h < p / 2 0<h<p/2
  21. m = 1 m=1
  22. p > h > p / 2 p>h>p/2

Coherent_states_in_mathematical_physics.html

  1. \mathfrak{H}\,
  2. X X
  3. d ν d\nu
  4. X X
  5. x x
  6. X X
  7. | x |x\rangle
  8. \mathfrak{H}
  9. x | x x\mapsto|x\rangle
  10. | ϕ |\phi\rangle
  11. \mathfrak{H}
  12. Ψ ( x ) = x | ϕ \Psi(x)=\langle x|\phi\rangle
  13. X X
  14. X | x x | d ν ( x ) = I \int_{X}|x\rangle\langle x|\;d\nu(x)=I_{\mathfrak{H}}
  15. \mathfrak{H}
  16. | ϕ , | ψ |\phi\rangle,|\psi\rangle
  17. \mathfrak{H}
  18. X ϕ | x x | ψ d ν ( x ) = ϕ | ψ . \int_{X}\langle\phi|x\rangle\langle x|\psi\rangle\;d\nu(x)=\langle\phi|\psi% \rangle\;.
  19. | x |x\rangle
  20. X X\equiv\mathbb{C}
  21. x α x\equiv\alpha
  22. d ν ( x ) 1 π d 2 α . d\nu(x)\equiv\frac{1}{\pi}d^{2}\alpha.
  23. | x |x\rangle
  24. {\mathfrak{H}}\,
  25. Ψ ( x ) = x | ψ \Psi(x)=\langle x|\psi\rangle
  26. | ψ |\psi\rangle
  27. {\mathfrak{H}}
  28. | ψ |\psi\rangle
  29. | ψ = X Ψ ( x ) | x d ν ( x ) , |\psi\rangle=\int_{X}\Psi(x)|x\rangle\;d\nu(x)\;,
  30. Ψ ( x ) = x | ψ \Psi(x)=\langle x|\psi\rangle
  31. Ψ \Psi
  32. X X
  33. X K ( x , y ) Ψ ( y ) d ν ( y ) = Ψ ( x ) , \int_{X}K(x,y)\Psi(y)\;d\nu(y)=\Psi(x)\,,
  34. K ( x , y ) = x | y K(x,y)=\langle x|y\rangle
  35. K ( x , y ) = K ( y , x ) ¯ , K ( x , x ) > 0 , \quad K(x,y)=\overline{K(y,x)}\;,\qquad K(x,x)>0\;,
  36. X K ( x , z ) K ( z , y ) d ν ( z ) = K ( x , y ) . \int_{X}K(x,z)\;K(z,y)\;d\nu(z)=K(x,y)\;.
  37. ε 1 ε 2 ε n \varepsilon_{1}\leq\varepsilon_{2}\leq\ldots\leq\varepsilon_{n}\leq\ldots
  38. ε 1 0 \varepsilon_{1}\neq 0
  39. ε n ! = ε 1 ε 2 ε n \varepsilon_{n}!=\varepsilon_{1}\varepsilon_{2}\ldots\varepsilon_{n}
  40. ε 0 ! = 1 \varepsilon_{0}!=1
  41. | α = 𝒩 ( | α | 2 ) - 1 2 n = 0 α n ε n ! | n . |\alpha\rangle={\mathcal{N}}(|\alpha|^{2})^{-\frac{1}{2}}\;\sum_{n=0}^{\infty}% \frac{\alpha^{n}}{\sqrt{\varepsilon_{n}!}}|n\rangle\,.
  42. 𝒩 ( | α | 2 ) {\mathcal{N}}(|\alpha|^{2})
  43. α | α = 1 \langle\alpha|\alpha\rangle=1
  44. 𝒟 | α α | 𝒩 ( | α | 2 ) d ν ( α , α ¯ ) = I , \int_{\mathcal{D}}|\alpha\rangle\langle\alpha|\;{\mathcal{N}}(|\alpha|^{2})\;d% \nu(\alpha,\overline{\alpha})=I\;,
  45. 𝒟 \mathcal{D}
  46. L L
  47. n = 0 α n ε n ! \sum_{n=0}^{\infty}\frac{\alpha^{n}}{\sqrt{\varepsilon_{n}!}}
  48. L = L=\infty
  49. d ν d\nu
  50. d θ d λ ( r ) d\theta\;d\lambda(r)
  51. α = r e i θ \alpha=re^{i\theta}
  52. d λ d\lambda
  53. ε n ! \varepsilon_{n}!
  54. | ϕ |\phi\rangle
  55. Φ ( α ) = ϕ | α \Phi(\alpha)=\langle\phi|\alpha\rangle
  56. Φ ( α ) = 𝒩 ( | α | 2 ) - 1 2 f ( α ) \Phi(\alpha)={\mathcal{N}}(|\alpha|^{2})^{-\frac{1}{2}}f(\alpha)
  57. f f\,
  58. 𝒟 \mathcal{D}
  59. K ( α ¯ , α ) = α | α = [ 𝒩 ( | α | 2 ) 𝒩 ( | α | 2 ) ] - 1 2 n = 0 ( α ¯ α ) n ε n ! . K(\overline{\alpha},\alpha^{\prime})=\langle\alpha|\alpha^{\prime}\rangle=% \left[{\mathcal{N}}(|\alpha|^{2}){\mathcal{N}}(|\alpha^{\prime}|^{2})\right]^{% -\frac{1}{2}}\sum_{n=0}^{\infty}\frac{(\overline{\alpha}\alpha^{\prime})^{n}}{% \varepsilon_{n}!}\;.
  60. A A
  61. | α |\alpha\rangle
  62. A | α = α | α , A|\alpha\rangle=\alpha|\alpha\rangle\;,
  63. A A^{\dagger}
  64. | n |n\rangle
  65. A | n = ε n | n - 1 , A | n = ε n + 1 | n + 1 . A|n\rangle=\sqrt{\varepsilon_{n}}|n-1\rangle\;,\qquad A^{\dagger}|n\rangle=% \sqrt{\varepsilon_{n+1}}|n+1\rangle\;.
  66. ε n \varepsilon_{n}
  67. I I
  68. A A
  69. A A^{\dagger}
  70. A A
  71. H = n = 0 E n | n n | H=\sum_{n=0}^{\infty}E_{n}|n\rangle\langle n|
  72. E 0 = 0 E_{0}=0
  73. E n E_{n}
  74. | n |n\rangle
  75. {\mathfrak{H}}
  76. E n = ω ε n E_{n}=\omega\varepsilon_{n}
  77. { ε n } \{\varepsilon_{n}\}
  78. 0 = ε 0 < ε 1 < ε 2 < 0=\varepsilon_{0}<\varepsilon_{1}<\varepsilon_{2}<\ldots\;
  79. J 0 J\geq 0
  80. γ \gamma\in\mathbb{R}
  81. | J , γ = 𝒩 ( J ) - 1 2 k = 0 J n / 2 e - i ε n γ ε n ! | n , |J,\gamma\rangle=\mathcal{N}(J)^{-\frac{1}{2}}\sum_{k=0}^{\infty}\,\frac{J^{n/% 2}e^{-i\varepsilon_{n}\gamma}}{\sqrt{\varepsilon_{n}!}}|n\rangle\;,
  82. 𝒩 \mathcal{N}
  83. J J
  84. e - i H t | J , γ = | J , γ + ω t , e^{-iHt}|J,\gamma\rangle=|J,\gamma+\omega t\rangle\;,
  85. J , γ | H | J , γ = ω J . \langle J,\gamma|H|J,\gamma\rangle_{\mathfrak{H}}=\omega J\;.
  86. {\mathfrak{H}}
  87. D ( α ) D(\alpha)\;
  88. X , P X,\,P
  89. I I
  90. G G
  91. U U
  92. {\mathfrak{H}}\,
  93. U ( g ) , g G U(g),\;g\in G
  94. | ψ |\psi\rangle
  95. {\mathfrak{H}}\;
  96. c ( ψ ) = G | ψ | U ( g ) ψ | 2 d μ ( g ) c(\psi)=\int_{G}|\langle\psi|U(g)\psi\rangle|^{2}\;d\mu(g)
  97. d μ d\mu
  98. G G
  99. | ψ |\psi\rangle
  100. c ( ψ ) < c(\psi)<\infty
  101. {\mathfrak{H}}
  102. G G
  103. {\mathfrak{H}}\;
  104. U U
  105. | ψ |\psi\rangle
  106. | g = 1 c ( ψ ) U ( g ) | ψ , for all g G . |g\rangle=\frac{1}{\sqrt{c(\psi)}}\;U(g)|\psi\rangle,\mbox{ for all }~{}g\in G.
  107. G | g g | d μ ( g ) = I \int_{G}|g\rangle\langle g|\;d\mu(g)=I_{\mathfrak{H}}
  108. {\mathfrak{H}}
  109. | g |g\rangle
  110. F ( g ) = g | ϕ F(g)=\langle g|\phi\rangle
  111. | ϕ |\phi\rangle
  112. {\mathfrak{H}}\;
  113. d μ d\mu
  114. G G
  115. L 2 ( G , d μ ) L^{2}(G,d\mu)
  116. ϕ F \phi\mapsto F
  117. {\mathfrak{H}}\;
  118. L 2 ( G , d μ ) L^{2}(G,d\mu)
  119. G G
  120. L 2 ( G , d μ ) L^{2}(G,d\mu)
  121. G Aff G_{\,\text{Aff}}
  122. × \times
  123. g = ( a b 0 1 ) , g=\begin{pmatrix}a&b\\ 0&1\end{pmatrix}\;,
  124. a a
  125. b b
  126. a 0 a\neq 0
  127. g = ( b , a ) g=(b,a)
  128. \mathbb{R}
  129. ( b , a ) x = b + a x (b,a)\cdot x=b+ax
  130. d μ ( b , a ) = a - 2 d b d a d\mu(b,a)=a^{-2}\;db\;da
  131. a - 1 d b d a a^{-1}\;db\;da
  132. L 2 ( , d x ) L^{2}(\mathbb{R},dx)
  133. L 2 ( , d x ) L^{2}(\mathbb{R},dx)
  134. φ ( x ) \varphi(x)
  135. x x
  136. U ( b , a ) U(b,a)
  137. ( U ( b , a ) φ ) ( x ) = 1 | a | φ ( x - b a ) = 1 | a | φ ( ( b , a ) - 1 x ) . (U(b,a)\varphi)(x)=\frac{1}{\sqrt{|a|}}\;\varphi\left(\frac{x-b}{a}\right)=% \frac{1}{\sqrt{|a|}}\;\varphi\left((b,a)^{-1}\cdot x\right)\;.
  138. ψ \psi
  139. L 2 ( , d x ) L^{2}(\mathbb{R},dx)
  140. ψ ^ \widehat{\psi}
  141. | ψ ^ ( k ) | 2 | k | d k < , \int_{\mathbb{R}}\frac{|\widehat{\psi}(k)|^{2}}{|k|}\;dk<\infty\;,
  142. c ( ψ ) = G Aff | ψ | U ( b , a ) ψ | 2 d b d a a 2 < . c(\psi)=\int_{G\text{Aff}}|\langle\psi|U(b,a)\psi\rangle|^{2}\;\frac{db\;da}{a% ^{2}}<\infty\;.
  143. | b , a = 1 c ( ψ ) U ( b , a ) ψ , ( b , a ) G Aff |b,a\rangle=\frac{1}{\sqrt{c(\psi)}}\;U(b,a)\psi\;,\qquad(b,a)\in G_{\,\text{% Aff}}
  144. G Aff | b , a b , a | d b d a a 2 = I \int_{G\text{Aff}}|b,a\rangle\langle b,a|\;\frac{db\;da}{a^{2}}=I
  145. L 2 ( , d x ) L^{2}(\mathbb{R},dx)
  146. | b , a |b,a\rangle
  147. | φ |\varphi\rangle
  148. L 2 ( , d x ) L^{2}(\mathbb{R},dx)
  149. F ( b , a ) = b , a | φ , F(b,a)=\langle b,a|\varphi\rangle\;,
  150. φ \varphi
  151. G Aff G_{\,\text{Aff}}\;
  152. 2 \mathbb{R}^{2}
  153. | 0 |0\rangle
  154. G G
  155. U U
  156. G G
  157. {\mathfrak{H}}
  158. | ψ |\psi\rangle
  159. {\mathfrak{H}}
  160. H H
  161. G G
  162. h h
  163. U ( h ) ψ = e i ω ( h ) ψ , U(h)\mid\psi\rangle=e^{i\omega(h)}\mid\psi\rangle\,,
  164. ω \omega
  165. h h
  166. X = G / H X=G/H
  167. x x
  168. X X
  169. g ( x ) G g(x)\in G
  170. x x
  171. | x = U ( g ( x ) ) | ψ . |x\rangle=U(g(x))|\psi\rangle\in{\mathfrak{H}}.
  172. g ( x ) g(x)
  173. g ( x ) g(x)
  174. g ( x ) G g(x)^{\prime}\in G
  175. x x
  176. g ( x ) = g ( x ) h g(x)^{\prime}=g(x)h
  177. h H h\in H
  178. U ( g ( x ) ) | ψ = e i ω ( h ) | x U(g(x)^{\prime})|\psi\rangle=e^{i\omega(h)}|x\rangle
  179. | x |x\rangle
  180. U ( g ( x ) ) | ψ U(g(x)^{\prime})|\psi\rangle
  181. | x x | |x\rangle\langle x|
  182. | x |x\rangle
  183. U U
  184. x x
  185. G / H G/H
  186. {\mathfrak{H}}
  187. X X
  188. G G
  189. B B
  190. B = X | x x | d μ ( x ) , B=\int_{X}|x\rangle\langle x|\;d\mu(x)\;,
  191. U U
  192. | ψ |\psi\rangle
  193. H H
  194. σ : G / H G \sigma:G/H\to G
  195. G / H | x x | d μ ( x ) = T , \int_{G/H}|x\rangle\langle x|\;d\mu(x)=T\;,
  196. | x = U ( σ ( x ) ) | ψ |x\rangle=U(\sigma(x))|\psi\rangle
  197. T T\;
  198. d μ d\mu
  199. X = G / H X=G/H
  200. | ψ |\psi\rangle
  201. H H
  202. T T
  203. n K \mathbb{R}^{n}\rtimes K
  204. K K
  205. G L ( n , ) GL(n,\mathbb{R})
  206. T T
  207. | ϕ |\phi\rangle
  208. {\mathfrak{H}}\;
  209. | x |x\rangle
  210. | ϕ = X Ψ ( x ) | x d μ ( x ) , Ψ ( x ) = x | T - 1 ϕ , |\phi\rangle=\int_{X}\Psi(x)|x\rangle\;d\mu(x)\;,\qquad\Psi(x)=\langle x|T^{-1% }\phi\rangle\;,
  211. n n
  212. | z |z\rangle
  213. \mathbb{C}
  214. + \mathbb{R}^{+}
  215. X X
  216. μ \mu
  217. L 2 ( X , d μ ) L^{2}(X,d\mu)
  218. μ \mu
  219. L 2 ( X , d μ ) L^{2}(X,d\mu)
  220. 𝒪 = { ϕ n , n = 0 , 1 , } \mathcal{O}=\{\phi_{n}\,,\,n=0,1,\dots\}
  221. ϕ m | ϕ n = X ϕ m ( x ) ¯ ϕ n ( x ) d μ ( x ) = δ m n . \langle\phi_{m}|\phi_{n}\rangle=\int_{X}\overline{\phi_{m}(x)}\,\phi_{n}(x)\,d% \mu(x)=\delta_{mn}\,.
  222. 0 < 𝒩 ( x ) := n | ϕ n ( x ) | 2 < a . e . . 0<\mathcal{N}(x):=\sum_{n}|\phi_{n}(x)|^{2}<\infty\,\quad\mathrm{a.e.}\,.
  223. \mathfrak{H}
  224. { | e n , n = 0 , 1 , } \{|e_{n}\rangle\,,\,n=0,1,\dots\}
  225. 𝒪 \mathcal{O}
  226. = { | x , x X } \mathcal{F}_{\mathfrak{H}}=\{|x\rangle\,,\,x\in X\}
  227. \mathfrak{H}
  228. | x = 1 𝒩 ( x ) n ϕ n ( x ) ¯ | e n , |x\rangle=\frac{1}{\sqrt{\mathcal{N}(x)}}\sum_{n}\overline{\phi_{n}(x)}\,|e_{n% }\rangle\,,
  229. \mathfrak{H}
  230. X d μ ( x ) 𝒩 ( x ) | x x | = I . \int_{X}d\mu(x)\,\mathcal{N}(x)\,|x\rangle\langle x|=I_{\mathfrak{H}}\,.
  231. X X
  232. X x f ( x ) X\ni x\mapsto f(x)
  233. \mathfrak{H}
  234. f ( x ) A f := X μ ( d x ) 𝒩 ( x ) f ( x ) | x x | . f(x)\mapsto A_{f}:=\int_{X}\mu(dx)\,\mathcal{N}(x)\,f(x)\,|x\rangle\langle x|\,.
  235. A f A_{f}
  236. f ( x ) f(x)
  237. f ( x ) f(x)
  238. f ( x ) f(x)
  239. A f A_{f}
  240. \mathcal{F}_{\mathfrak{H}}
  241. A f A_{f}
  242. f ˇ ( x ) := x | A f | x = X μ ( d x ) 𝒩 ( x ) f ( x ) | x | x | 2 . \check{f}(x):=\langle x|A_{f}|x\rangle=\int_{X}\mu(dx^{\prime})\,\mathcal{N}(x% ^{\prime})\,f(x^{\prime})\,|\langle x|x^{\prime}\rangle|^{2}\,.
  243. X X
  244. x x
  245. n | ϕ n ( x ) | 2 𝒩 ( x ) . n\mapsto\frac{|\phi_{n}(x)|^{2}}{{\mathcal{N}}(x)}.
  246. A A
  247. \mathfrak{H}
  248. A = n a n | e n e n | A=\sum_{n}a_{n}|e_{n}\rangle\langle e_{n}|
  249. n n
  250. ( X , μ ) (X,\mu)
  251. X x | ϕ n ( x ) | 2 . X\ni x\mapsto|\phi_{n}(x)|^{2}\,.
  252. | x |x\rangle
  253. X X
  254. A A
  255. ϕ n ( x ) \phi_{n}(x)
  256. | e n |e_{n}\rangle
  257. S U q ( 2 ) SU_{q}(2)
  258. q q

Collapsible_flow.html

  1. a l p h a alpha

Collisionality.html

  1. ν * \nu^{*}
  2. ν * = ν ei m i k B T i ϵ - 3 / 2 q R , \nu^{*}=\nu_{\mathrm{ei}}\,\sqrt{\frac{m_{\mathrm{i}}}{k_{\mathrm{B}}T_{% \mathrm{i}}}}\,\epsilon^{-3/2}\,qR,
  3. ν ei \nu_{\mathrm{ei}}
  4. R R
  5. ϵ \epsilon
  6. q q
  7. m i m_{\mathrm{i}}
  8. T i T_{\mathrm{i}}
  9. k B k_{\mathrm{B}}

Collocation_(remote_sensing).html

  1. v = g k h ( 1 - R a / R s ) c D v=\sqrt{\frac{gkh(1-R_{a}/R_{s})}{c_{D}}}

Color_index.html

  1. T = 4600 ( 1 < m t p l > 0.92 ( B - V ) + 1.7 + 1 0.92 ( B - V ) + 0.62 ) T=4600\left({\frac{1}{<}mtpl>{{0.92(B-V)+1.7}}+\frac{1}{{0.92(B-V)+0.62}}}\right)
  2. E B - V = ( B - V ) Observed - ( B - V ) Intrinsic E_{B-V}=(B-V)_{\textrm{Observed}}-(B-V)_{\textrm{Intrinsic}}

Color_layout_descriptor.html

  1. B p q = α p α q m = 0 M - 1 n = 0 N - 1 A m n cos π ( 2 m + 1 ) p 2 M cos π ( 2 n + 1 ) q 2 N , 0 p M - 1 , 0 q N - 1 B_{pq}=\alpha_{p}\alpha_{q}\sum_{m=0}^{M-1}\sum_{n=0}^{N-1}A_{mn}\cos\frac{\pi% (2m+1)p}{2M}\cos\frac{\pi(2n+1)q}{2N},\qquad 0\leq p\leq M-1,\;0\leq q\leq N-1
  2. α p = { 1 M , p = 0 2 M , 1 p M - 1 α q = { 1 N , q = 0 2 N , 1 q N - 1 \alpha_{p}=\begin{cases}\frac{1}{\sqrt{M}},&p=0\\ \sqrt{\frac{2}{M}},&1\leq p\leq M-1\end{cases}\qquad\alpha_{q}=\begin{cases}% \frac{1}{\sqrt{N}},&q=0\\ \sqrt{\frac{2}{N}},&1\leq q\leq N-1\end{cases}
  3. D = i w y i ( D Y i - D Y i ) 2 + i w b i ( D C b i - D C b i ) 2 + i w r i ( D C r i - D C r i ) 2 D=\sqrt{\sum_{i}w_{yi}(DY_{i}-DY_{i}^{\prime})^{2}}+\sqrt{\sum_{i}w_{bi}(DCb_{% i}-DCb_{i}^{\prime})^{2}}+\sqrt{\sum_{i}w_{ri}(DCr_{i}-DCr_{i}^{\prime})^{2}}

Color_normalization.html

  1. ( α R , β G , γ B ) ( α R α n i R , β G β n i G , γ B γ n i B ) \left(\alpha R,\beta G,\gamma B\right)\rightarrow\left(\frac{\alpha R}{\frac{% \alpha}{n}\sum_{i}R},\frac{\beta G}{\frac{\beta}{n}\sum_{i}G},\frac{\gamma B}{% \frac{\gamma}{n}\sum_{i}B}\right)
  2. y = f ( x ) = 0 x p x ( u ) d u y=f(x)=\int\limits_{0}^{x}p_{x}(u)du
  3. y = g ( z ) = 0 z p z ( u ) d u y^{\prime}=g(z)=\int\limits_{0}^{z}p_{z}(u)du
  4. z = g - 1 ( y ) z=g^{-1}(y^{\prime})
  5. z = g - 1 ( y ) = g - 1 ( y ) = g - 1 ( f ( x ) ) z=g^{-1}(y^{\prime})=g^{-1}(y)=g^{-1}(f(x))
  6. f ( t ) = [ f i j ( t ) ] i = 1... N , j = 1... M f^{(t)}=[f_{ij}^{(t)}]_{i=1...N,j=1...M}
  7. f i j ( t ) = ( r i j ( t ) , g i j ( t ) , b i j ( t ) ) T . f_{ij}^{(t)}=(r_{ij}^{(t)},g_{ij}^{(t)},b_{ij}^{(t)})^{T}.
  8. S i j := r i j ( t ) + g i j ( t ) + b i j ( t ) S_{ij}:=r_{ij}^{(t)}+g_{ij}^{(t)}+b_{ij}^{(t)}
  9. r i j ( t + 1 ) = r i j ( t ) S i j , g i j ( t + 1 ) = g i j ( t ) S i j r_{ij}^{(t+1)}=\frac{r_{ij}^{(t)}}{S_{ij}},g_{ij}^{(t+1)}=\frac{g_{ij}^{(t)}}{% S_{ij}}
  10. b i j ( t + 1 ) = b i j ( t ) S i j . b_{ij}^{(t+1)}=\frac{b_{ij}^{(t)}}{S_{ij}}.
  11. r = 3 N M i = 1 N j = 1 M r i j ( t ) r^{\prime}=\frac{3}{NM}\sum_{i=1}^{N}\sum_{j=1}^{M}r_{ij}^{(t)}
  12. r i j ( t + 2 ) = r i j ( t + 1 ) r . r_{ij}^{(t+2)}=\frac{r_{ij}^{(t+1)}}{r^{\prime}}.

Combined_bisulfite_restriction_analysis.html

  1. % Methylation = 100 × Digested Fragments Undigested Fragments + Digested Fragments \%\;\,\text{Methylation}=100\times\frac{\,\text{Digested Fragments}}{\,\text{% Undigested Fragments + Digested Fragments}}

Commensurator.html

  1. comm G ( H ) = { g G : g H g - 1 H has finite index in both H and g H g - 1 } . \mathrm{comm}_{G}(H)=\{g\in G:gHg^{-1}\cap H\,\text{ has finite index in both % }H\,\text{ and }gHg^{-1}\}.

Comparison_of_optimization_software.html

  1. \to

Comparison_of_orbital_rocket_engines.html

  1. 1 , 340 , 000 N ( 1 , 686 kg ) ( 9.807 m / s 2 ) = 81.04 \frac{1,340,000\ \mathrm{N}}{(1,686\ \mathrm{kg})(9.807\ \mathrm{m/s^{2}})}=81% .04
  2. 64 , 800 N ( 165 kg ) ( 9.807 m / s 2 ) = 40.05 \frac{64,800\ \mathrm{N}}{(165\ \mathrm{kg})(9.807\ \mathrm{m/s^{2}})}=40.05
  3. 3 , 826 , 554 N ( 5 , 480 kg ) ( 9.807 m / s 2 ) = 71.2 \frac{3,826,554\ \mathrm{N}}{(5,480\ \mathrm{kg})(9.807\ \mathrm{m/s^{2}})}=71.2
  4. 1 , 922 , 103 N ( 2 , 200 kg ) ( 9.807 m / s 2 ) = 89.09 \frac{1,922,103\ \mathrm{N}}{(2,200\ \mathrm{kg})(9.807\ \mathrm{m/s^{2}})}=89% .09
  5. 294 , 300 N ( 480 kg ) ( 9.807 m / s 2 ) = 62.5 \frac{294,300\ \mathrm{N}}{(480\ \mathrm{kg})(9.807\ \mathrm{m/s^{2}})}=62.5
  6. 653 , 889 N ( 155 ) ( 9.807 m / s 2 ) = 430 kg \frac{653,889\ \mathrm{N}}{(155)(9.807\ \mathrm{m/s^{2}})}=430\ \mathrm{kg}
  7. 7 , 256 , 921 N ( 9 , 300 kg ) ( 9.807 m / s 2 ) = 79.57 \frac{7,256,921\ \mathrm{N}}{(9,300\ \mathrm{kg})(9.807\ \mathrm{m/s^{2}})}=79% .57
  8. 839 , 449 N ( 1 , 090 kg ) ( 9.807 m / s 2 ) = 78.53 \frac{839,449\ \mathrm{N}}{(1,090\ \mathrm{kg})(9.807\ \mathrm{m/s^{2}})}=78.53
  9. 792 , 377 N ( 1 , 075 kg ) ( 9.807 m / s 2 ) = 75.16 \frac{792,377\ \mathrm{N}}{(1,075\ \mathrm{kg})(9.807\ \mathrm{m/s^{2}})}=75.16
  10. 778 , 648 N ( 1 , 100 kg ) ( 9.807 m / s 2 ) = 72.18 \frac{778,648\ \mathrm{N}}{(1,100\ \mathrm{kg})(9.807\ \mathrm{m/s^{2}})}=72.18
  11. 818 , 855 N ( 1 , 100 kg ) ( 9.807 m / s 2 ) = 75.91 \frac{818,855\ \mathrm{N}}{(1,100\ \mathrm{kg})(9.807\ \mathrm{m/s^{2}})}=75.91
  12. 137 , 200 N ( 285 kg ) ( 9.807 m / s 2 ) = 49.1 \frac{137,200\ \mathrm{N}}{(285\ \mathrm{kg})(9.807\ \mathrm{m/s^{2}})}=49.1
  13. 6 , 770 , 000 N ( 8 , 391 kg ) ( 9.807 m / s 2 ) = 82.27 \frac{6,770,000\ \mathrm{N}}{(8,391\ \mathrm{kg})(9.807\ \mathrm{m/s^{2}})}=82% .27
  14. 1 , 860 , 000 N ( 3 , 526 kg ) ( 9.807 m / s 2 ) = 53.79 \frac{1,860,000\ \mathrm{N}}{(3,526\ \mathrm{kg})(9.807\ \mathrm{m/s^{2}})}=53% .79
  15. 1 , 310 , 000 N ( 2 , 430 kg ) ( 9.807 m / s 2 ) = 54.97 \frac{1,310,000\ \mathrm{N}}{(2,430\ \mathrm{kg})(9.807\ \mathrm{m/s^{2}})}=54% .97
  16. 73 , 550 N ( 445 kg ) ( 9.807 m / s 2 ) = 16.85 \frac{73,550\ \mathrm{N}}{(445\ \mathrm{kg})(9.807\ \mathrm{m/s^{2}})}=16.85
  17. 4 , 158 , 020 N ( 3 , 600 kg ) ( 9.807 m / s 2 ) = 117.77 \frac{4,158,020\ \mathrm{N}}{(3,600\ \mathrm{kg})(9.807\ \mathrm{m/s^{2}})}=11% 7.77
  18. 1 , 671 , 053 N ( 1 , 070 kg ) ( 9.807 m / s 2 ) = 159.25 \frac{1,671,053\ \mathrm{N}}{(1,070\ \mathrm{kg})(9.807\ \mathrm{m/s^{2}})}=15% 9.25
  19. 1 , 961 , 330 N ( 1 , 900 kg ) ( 9.807 m / s 2 ) = 105.26 \frac{1,961,330\ \mathrm{N}}{(1,900\ \mathrm{kg})(9.807\ \mathrm{m/s^{2}})}=10% 5.26
  20. 30 , 000 N ( 111 kg ) ( 9.807 m / s 2 ) = 27.6 \frac{30,000\ \mathrm{N}}{(111\ \mathrm{kg})(9.807\ \mathrm{m/s^{2}})}=27.6
  21. 55 , 400 N ( 138 kg ) ( 9.807 m / s 2 ) = 40.9 \frac{55,400\ \mathrm{N}}{(138\ \mathrm{kg})(9.807\ \mathrm{m/s^{2}})}=40.9

Compartmental_modelling_of_dendrites.html

  1. a i a_{i}
  2. L i L_{i}
  3. V i V_{i}
  4. c i c_{i}
  5. r M i r_{Mi}
  6. I e l e c t r o d e i I^{i}_{electrode}
  7. r L r_{L}
  8. i c a p i + i i o n i = i l o n g i + i e l e c t r o d e i i^{i}_{cap}+i^{i}_{ion}=i^{i}_{long}+i^{i}_{electrode}
  9. i c a p i i^{i}_{cap}
  10. i i o n i i^{i}_{ion}
  11. i c a p i = c i d V i d t i^{i}_{cap}=c_{i}\frac{dV_{i}}{dt}
  12. i i o n i = V i r M i i^{i}_{ion}=\frac{V_{i}}{r_{Mi}}
  13. i l o n g i i^{i}_{long}
  14. R l o n g = r L L 1 2 π a 1 2 + r L L 2 2 π a 2 2 R_{long}=\frac{r_{L}L_{1}}{2\pi a_{1}^{2}}+\frac{r_{L}L_{2}}{2\pi a_{2}^{2}}
  15. i l o n g 1 = g 1 , 2 ( V 2 - V 1 ) i^{1}_{long}=g_{1,2}(V_{2}-V_{1})
  16. i l o n g 2 = g 2 , 1 ( V 1 - V 2 ) i^{2}_{long}=g_{2,1}(V_{1}-V_{2})
  17. g 1 , 2 g_{1,2}
  18. g 2 , 1 g_{2,1}
  19. g 1 , 2 = a 1 a 2 2 r L L 1 ( a 2 2 L 1 + a 1 2 L 2 ) g_{1,2}=\frac{a_{1}a_{2}^{2}}{r_{L}L_{1}(a_{2}^{2}L_{1}+a_{1}^{2}L_{2})}
  20. g 2 , 1 = a 2 a 1 2 r L L 1 ( a 1 2 L 2 + a 2 2 L 1 ) g_{2,1}=\frac{a_{2}a_{1}^{2}}{r_{L}L_{1}(a_{1}^{2}L_{2}+a_{2}^{2}L_{1})}
  21. i e l e c t r o d e I = I e l e c t r o d e i A i i_{electrode}^{I}=\frac{I_{electrode}^{i}}{A_{i}}
  22. A i = 2 π a i L i A_{i}=2\pi a_{i}L_{i}
  23. c 1 d V 1 d t + V 1 r M 1 = g 1 , 2 ( V 2 - V 1 ) + I e l e c t r o d e 1 A 1 c_{1}\frac{dV_{1}}{dt}+\frac{V_{1}}{r_{M1}}=g_{1,2}(V_{2}-V_{1})+\frac{I_{% electrode}^{1}}{A_{1}}
  24. c 2 d V 2 d t + V 2 r M 2 = g 2 , 1 ( V 1 - V 2 ) + I e l e c t r o d e 2 A 2 c_{2}\frac{dV_{2}}{dt}+\frac{V_{2}}{r_{M2}}=g_{2,1}(V_{1}-V_{2})+\frac{I_{% electrode}^{2}}{A_{2}}
  25. r 1 = 1 / g 1 , 2 r_{1}=1/g_{1,2}
  26. r 2 = 1 / g 2 , 1 r_{2}=1/g_{2,1}
  27. c 1 d V 1 d t + V 1 r M 1 = V 2 - V 1 r 1 + I e l e c t r o d e 1 A 1 c_{1}\frac{dV_{1}}{dt}+\frac{V_{1}}{r_{M1}}=\frac{V_{2}-V_{1}}{r_{1}}+\frac{I_% {electrode}^{1}}{A_{1}}
  28. c 2 d V 2 d t + V 2 r M 2 = V 1 - V 2 r 2 + I e l e c t r o d e 2 A 2 c_{2}\frac{dV_{2}}{dt}+\frac{V_{2}}{r_{M2}}=\frac{V_{1}-V_{2}}{r_{2}}+\frac{I_% {electrode}^{2}}{A_{2}}
  29. r 1 = r 2 r r_{1}=r_{2}\equiv r
  30. r M = r M 1 = r M 2 r_{M}=r_{M1}=r_{M2}
  31. V 1 i 1 = r M ( r + r M ) r + 2 r M \frac{V_{1}}{i_{1}}=\frac{r_{M}(r+r_{M})}{r+2r_{M}}
  32. R i n p u t , c o u p l e d R i n p u t , u n c o u p l e d = 1 - r M r + 2 r M \frac{R_{input,coupled}}{R_{input,uncoupled}}=1-\frac{r_{M}}{r+2r_{M}}
  33. C j d V j d t = - V j R j + k c o n n e c t e d j V k - V j R j k + I j C_{j}\frac{dV_{j}}{dt}=-\frac{V_{j}}{R_{j}}+\sum_{kconnectedj}{}\frac{V_{k}-V_% {j}}{R_{jk}}+I_{j}

Complete-linkage_clustering.html

  1. D ( X , Y ) D(X,Y)
  2. X X
  3. Y Y
  4. D ( X , Y ) = max x X , y Y d ( x , y ) D(X,Y)=\max_{x\in X,y\in Y}d(x,y)
  5. d ( x , y ) d(x,y)
  6. x X x\in X
  7. y Y y\in Y
  8. X X
  9. Y Y
  10. N × N N\times N
  11. 𝒪 ( n 3 ) \mathcal{O}(n^{3})
  12. 𝒪 ( n 2 ) \mathcal{O}(n^{2})

Complete_sequence.html

  1. s n = m = 0 n a m s_{n}=\sum_{m=0}^{n}a_{m}
  2. a 0 = 1 a_{0}=1\,
  3. s k - 1 a k - 1 k 1 s_{k-1}\geq a_{k}-1\,\forall\,k\geq 1
  4. a 0 = 1 a_{0}=1\,
  5. 2 a k a k + 1 k 1 2a_{k}\geq a_{k+1}\,\forall\,k\geq 1

Completeness_(logic).html

  1. 𝒮 φ 𝒮 φ . \models_{\mathcal{S}}\varphi\ \to\ \vdash_{\mathcal{S}}\varphi.
  2. 𝒮 \mathcal{S}
  3. Γ 𝒮 φ Γ 𝒮 φ . \Gamma\models_{\mathcal{S}}\varphi\ \to\ \Gamma\vdash_{\mathcal{S}}\varphi.
  4. 𝒮 \mathcal{S}
  5. Γ 𝒮 Γ 𝒮 . \Gamma\models_{\mathcal{S}}\bot\to\ \Gamma\vdash_{\mathcal{S}}\bot.
  6. Γ \Gamma
  7. φ \varphi
  8. Γ \Gamma
  9. Γ \Gamma
  10. φ \varphi
  11. φ \varphi
  12. Γ \Gamma
  13. { a } a b \{a\}\models a\lor b
  14. a b a\lor b
  15. { a } \{a\}
  16. { a , ¬ ( a b ) } \{a,\lnot(a\lor b)\}\vdash\bot
  17. 𝒮 \mathcal{S}
  18. 𝒮 \mathcal{S}

Completeness_of_the_real_numbers.html

  1. S = { x 𝐐 | x 2 < 2 } . S=\{x\in\mathbf{Q}|x^{2}<2\}.
  2. 2 \sqrt{2}
  3. R = { x 𝐐 | x 2 2 x > 0 } . R=\{x\in\mathbf{Q}|x^{2}\geq 2\wedge x>0\}.
  4. 2 \sqrt{2}
  5. 3 , 3.1 , 3.14 , 3.142 , 3.1416 , 3,\quad 3.1,\quad 3.14,\quad 3.142,\quad 3.1416,\quad\ldots
  6. I 1 I 2 I 3 I_{1}\;\supseteq\;I_{2}\;\supseteq\;I_{3}\;\supseteq\;\cdots
  7. [ 3 , 4 ] [ 3.1 , 3.2 ] [ 3.14 , 3.15 ] [ 3.141 , 3.142 ] [3,4]\;\supseteq\;[3.1,3.2]\;\supseteq\;[3.14,3.15]\;\supseteq\;[3.141,3.142]% \;\supseteq\;\cdots

Complexity_function.html

  1. 1 p u ( n ) k n , 1\leq p_{u}(n)\leq k^{n}\ ,
  2. a ( b b ) * a a(bb)^{*}a
  3. L { x y k z : x , y , z F , k } . L\subseteq\{xy^{k}z:x,y,z\in F,\ k\in\mathbb{N}\}\ .
  4. H top ( u ) = lim n log p u ( n ) n log k . H_{\mathrm{top}}(u)=\lim_{n\rightarrow\infty}\frac{\log p_{u}(n)}{n\log k}\ .

Compound_prism.html

  1. θ 0 \theta_{0}
  2. δ \delta
  3. δ = θ 0 - θ 4 \delta=\theta_{0}-\theta_{4}
  4. δ \delta
  5. θ 1 = θ 0 - β 1 θ 3 = θ 2 - α 2 θ 1 = arcsin ( 1 n 1 sin θ 1 ) θ 3 = arcsin ( n 2 sin θ 3 ) θ 2 = θ 1 - α 1 θ 4 = θ 3 + 1 2 α 2 θ 2 = arcsin ( n 1 n 2 sin θ 2 ) \begin{aligned}\displaystyle\theta_{1}&\displaystyle=\theta_{0}-\beta_{1}&% \displaystyle\theta_{3}&\displaystyle=\theta^{\prime}_{2}-\alpha_{2}\\ \displaystyle\theta^{\prime}_{1}&\displaystyle=\arcsin(\tfrac{1}{n_{1}}\,\sin% \theta_{1})&\displaystyle\theta^{\prime}_{3}&\displaystyle=\arcsin(n_{2}\,\sin% \theta_{3})\\ \displaystyle\theta_{2}&\displaystyle=\theta^{\prime}_{1}-\alpha_{1}&% \displaystyle\theta_{4}&\displaystyle=\theta^{\prime}_{3}+\tfrac{1}{2}\alpha_{% 2}\\ \displaystyle\theta^{\prime}_{2}&\displaystyle=\arcsin(\tfrac{n_{1}}{n_{2}}\,% \sin\theta_{2})\end{aligned}
  6. n 1 ( λ ) n_{1}(\lambda)
  7. n 2 ( λ ) n_{2}(\lambda)
  8. α 1 \alpha_{1}
  9. α 2 \alpha_{2}
  10. θ 0 \theta_{0}
  11. α i \alpha_{i}
  12. θ 0 \theta_{0}
  13. α \alpha
  14. sin θ θ \sin\theta\approx\theta
  15. arcsin ( x ) x \,\text{arcsin}(x)\approx x
  16. δ \delta
  17. δ ( λ ) = [ n 1 ( λ - 1 ] α 1 + [ n 2 ( λ ) - 1 ] α 2 . \delta(\lambda)=\big[n_{1}(\lambda-1\big]\alpha_{1}+\big[n_{2}(\lambda)-1\big]% \alpha_{2}\ .
  18. Δ = δ 1 ( λ ¯ ) V 1 + δ 2 ( λ ¯ ) V 2 , \Delta=\frac{\delta_{1}(\bar{\lambda})}{V_{1}}+\frac{\delta_{2}(\bar{\lambda})% }{V_{2}}\ ,
  19. δ i \delta_{i}
  20. V i V_{i}
  21. i i
  22. V i = ( n ¯ - 1 ) / ( n F - n C ) V_{i}=(\bar{n}-1)/(n_{F}-n_{C})
  23. λ ¯ \bar{\lambda}
  24. δ ¯ = 0 \bar{\delta}=0
  25. δ \delta
  26. Δ \Delta
  27. δ 1 ( λ ¯ ) = - δ 2 ( λ ¯ ) = - Δ ( 1 V 2 - 1 V 1 ) - 1 , \delta_{1}(\bar{\lambda})=-\delta_{2}(\bar{\lambda})=-\Delta\Big(\frac{1}{V_{2% }}-\frac{1}{V_{1}}\Big)^{-1}\ ,
  28. α 1 \alpha_{1}
  29. α 2 \alpha_{2}
  30. α 1 \displaystyle\alpha_{1}
  31. δ ( λ ¯ ) = 2 δ 1 ( λ ¯ ) + δ 2 ( λ ¯ ) = 2 ( n ¯ 1 - 1 ) α 1 + ( n ¯ 2 - 1 ) α 2 , Δ = 2 δ 1 ( λ ¯ ) V 1 + δ 2 ( λ ¯ ) V 2 . \begin{aligned}\displaystyle\delta(\bar{\lambda})&\displaystyle=2\delta_{1}(% \bar{\lambda})+\delta_{2}(\bar{\lambda})=2\big(\bar{n}_{1}-1)\alpha_{1}+\big(% \bar{n}_{2}-1)\alpha_{2}\ ,\\ \displaystyle\Delta&\displaystyle=2\frac{\delta_{1}(\bar{\lambda})}{V_{1}}+% \frac{\delta_{2}(\bar{\lambda})}{V_{2}}\ .\end{aligned}
  32. α 1 = Δ 2 ( n ¯ 1 - 1 ) ( 1 V 1 - 1 V 2 ) - 1 , α 2 = Δ n ¯ 2 - 1 ( 1 V 2 - 1 V 1 ) - 1 . \begin{aligned}\displaystyle\alpha_{1}&\displaystyle=\frac{\Delta}{2(\bar{n}_{% 1}-1)}\Big(\frac{1}{V_{1}}-\frac{1}{V_{2}}\Big)^{-1}\ ,\\ \displaystyle\alpha_{2}&\displaystyle=\frac{\Delta}{\bar{n}_{2}-1}\Big(\frac{1% }{V_{2}}-\frac{1}{V_{1}}\Big)^{-1}\ .\end{aligned}
  33. δ \delta
  34. θ 1 \displaystyle\theta_{1}
  35. δ = θ 0 - θ 5 \delta=\theta_{0}-\theta_{5}
  36. θ 1 = θ 0 + α 1 + 1 2 α 2 θ 3 = arcsin ( n 2 n 3 sin θ 3 ) θ 1 = arcsin ( 1 n 1 sin θ 1 ) θ 4 = θ 3 - α 3 θ 2 = θ 1 - α 1 θ 4 = arcsin ( n 3 sin θ 4 ) θ 2 = arcsin ( n 1 n 2 sin θ 2 ) θ 5 = θ 4 + α 3 + 1 2 α 2 θ 3 = θ 2 - α 2 \begin{aligned}\displaystyle\theta_{1}&\displaystyle=\theta_{0}+\alpha_{1}+% \tfrac{1}{2}\alpha_{2}&\displaystyle\theta^{\prime}_{3}&\displaystyle=\arcsin(% \tfrac{n_{2}}{n_{3}}\,\sin\theta_{3})\\ \displaystyle\theta^{\prime}_{1}&\displaystyle=\arcsin(\tfrac{1}{n_{1}}\,\sin% \theta_{1})&\displaystyle\theta_{4}&\displaystyle=\theta^{\prime}_{3}-\alpha_{% 3}\\ \displaystyle\theta_{2}&\displaystyle=\theta^{\prime}_{1}-\alpha_{1}&% \displaystyle\theta^{\prime}_{4}&\displaystyle=\arcsin(n_{3}\,\sin\theta_{4})% \\ \displaystyle\theta^{\prime}_{2}&\displaystyle=\arcsin(\tfrac{n_{1}}{n_{2}}\,% \sin\theta_{2})&\displaystyle\theta_{5}&\displaystyle=\theta^{\prime}_{4}+% \alpha_{3}+\tfrac{1}{2}\alpha_{2}\\ \displaystyle\theta_{3}&\displaystyle=\theta^{\prime}_{2}-\alpha_{2}\end{aligned}
  37. δ = θ 0 - θ 5 \delta=\theta_{0}-\theta_{5}

Compound_probability_distribution.html

  1. F F
  2. p H ( x | α ) = θ p F ( x | θ ) p G ( θ | α ) d θ p_{H}(x|\alpha)={\displaystyle\int\limits_{\theta}p_{F}(x|\theta)\,p_{G}(% \theta|\alpha)\operatorname{d}\!\theta}
  3. p H ( 𝐱 | s y m b o l α ) = s y m b o l θ p F ( 𝐱 | s y m b o l θ ) p G ( s y m b o l θ | s y m b o l α ) d s y m b o l θ p_{H}(\mathbf{x}|symbol\alpha)={\displaystyle\int\limits_{s}ymbol\theta p_{F}(% \mathbf{x}|symbol\theta)\,p_{G}(symbol\theta|symbol\alpha)\operatorname{d}\!% symbol\theta}
  4. H H
  5. F F
  6. H H
  7. F F
  8. H H
  9. G G
  10. F F
  11. n n
  12. α \alpha
  13. β \beta
  14. α \alpha
  15. β \beta
  16. α \alpha
  17. β \beta
  18. N N

Compounds_of_berkelium.html

  1. 3 ¯ \overline{3}
  2. 2 BkO 2 + H 2 Bk 2 O 3 + H 2 O \mathrm{2\ BkO_{2}\ +\ H_{2}\ \longrightarrow\ Bk_{2}O_{3}\ +\ H_{2}O}
  3. 3 ¯ \overline{3}

Computation_of_radiowave_attenuation_in_the_atmosphere.html

  1. | C K | = R sin u = r sin θ . |CK|=R\sin u=r\sin\theta.\,
  2. n 1 sin Φ = n 2 sin u , n_{1}\sin\Phi=n_{2}\sin u,\,
  3. n 1 R sin Φ = n 2 r sin θ = I N V n_{1}R\sin\Phi=n_{2}r\sin\theta=INV\,
  4. 𝐫 ¨ = n grad n \ddot{\mathbf{r}}=n\,\mathrm{grad}\,n

Computer_for_operations_with_functions.html

  1. A A
  2. α \alpha
  3. A = α 0 α 1 α k α n A=\alpha_{0}\alpha_{1}\dots\alpha_{k}\dots\alpha_{n}
  4. x x
  5. F ( x ) F(x)
  6. F ( x ) = ( α 22 α 2 k α 11 α 12 α 1 k α 00 α 01 α 02 α 0 k ) F(x)=\begin{pmatrix}&&\cdots\\ &&\alpha_{22}\cdots\alpha_{2k}\cdots\\ &\alpha_{11}&\alpha_{12}\cdots\alpha_{1k}\cdots\\ \alpha_{00}&\alpha_{01}&\alpha_{02}\cdots\alpha_{0k}\cdots\end{pmatrix}
  7. A A
  8. A = k = 0 n α k ρ k A=\sum_{k=0}^{n}\alpha_{k}\rho^{k}
  9. ρ \rho
  10. F ( x ) = k = 0 n m = 0 k α m k R k y k - m ( 1 - y ) m F(x)=\sum_{k=0}^{n}\sum_{m=0}^{k}\alpha_{mk}R^{k}y^{k-m}(1-y)^{m}
  11. R R
  12. α \alpha
  13. y y
  14. x x
  15. α k α k + 1 \alpha_{k}\longrightarrow\alpha_{k+1}
  16. ( α k + 1 , m + 1 α k , m α k + 1 , m ) \begin{pmatrix}&\alpha_{k+1,m+1}\\ \ \nearrow&\\ \alpha_{k,m}\longrightarrow&\alpha_{k+1,m}\end{pmatrix}
  17. T K R TK_{R}
  18. α m k \alpha_{mk}
  19. D R = { - r 1 , - r 1 + 1 , , - 1 , 0 , 1 , , r 2 - 1 , r 2 } D_{R}=\{-r_{1},-r_{1}+1,\dots,-1,0,1,\dots,r_{2}-1,r_{2}\}
  20. r 1 , r 2 0 r_{1},\;r_{2}\geq 0
  21. R = r 1 + r 2 + 1 R=r_{1}+r_{2}+1
  22. T K 3 TK_{3}
  23. α m k ( - 1 , 0 , 1 ) \alpha_{mk}\in(-1,0,1)
  24. T K 4 TK_{4}
  25. α m k ( - 2 , - 1 , 0 , 1 ) \alpha_{mk}\in(-2,-1,0,1)
  26. ( 0 a R 0 ) = ( a 0 a ) , ( a 0 0 ) = ( 0 a R - a ) , ( 0 0 a ) = ( - a a R 0 ) \begin{pmatrix}&0\\ \ \nearrow&\\ aR\longrightarrow&0\end{pmatrix}=\begin{pmatrix}&a\\ \ \nearrow&\\ 0\longrightarrow&a\end{pmatrix},\quad\begin{pmatrix}&a\\ \ \nearrow&\\ 0\longrightarrow&0\end{pmatrix}=\begin{pmatrix}&0\\ \ \nearrow&\\ aR\longrightarrow&-a\end{pmatrix},\quad\begin{pmatrix}&0\\ \ \nearrow&\\ 0\longrightarrow&a\end{pmatrix}=\begin{pmatrix}&-a\\ \ \nearrow&\\ aR\longrightarrow&0\end{pmatrix}
  27. a a
  28. T K R TK_{R}
  29. T K R ( α ) = α TK_{R}(\alpha)=\alpha
  30. T K R TK_{R}
  31. y k y^{k}
  32. T K R ( y 2 ) = ( 0 0 1 ) TK_{R}(y^{2})=(0\ 0\ 1)
  33. ( m k ) (mk)
  34. S m k S_{mk}
  35. α m k , β m k \alpha_{mk},\ \beta_{mk}
  36. p m , k - 1 , p m - 1 , k - 1 p_{m,k-1},\ p_{m-1,k-1}
  37. S m k = α m k + β m k + p m , k - 1 + p m - 1 , k - 1 S_{mk}=\alpha_{mk}+\beta_{mk}+p_{m,k-1}+p_{m-1,k-1}
  38. S m k = σ m k + R p m k S_{mk}=\sigma_{mk}+Rp_{mk}
  39. σ m k D R \sigma_{mk}\in D_{R}
  40. σ m k \sigma_{mk}
  41. ( m k ) (mk)
  42. p m k p_{mk}
  43. ( m , k + 1 ) (m,k+1)
  44. ( m + 1 , k + 1 ) (m+1,k+1)
  45. α m k D R \alpha_{mk}\in D_{R}
  46. β m k D R \beta_{mk}\in D_{R}
  47. S m k = σ m k + R p m k S_{mk}=\sigma_{mk}+Rp_{mk}
  48. R > 2. R>2.
  49. R = 3 R=3
  50. ( m k ) (mk)
  51. S m k S_{mk}
  52. S m k = α m k - β m k + p m , k - 1 + p m - 1 , k - 1 S_{mk}=\alpha_{mk}-\beta_{mk}+p_{m,k-1}+p_{m-1,k-1}
  53. ( a 0 0 ) = ( 0 a R - a ) \begin{pmatrix}&a\\ \ \nearrow&\\ 0\longrightarrow&0\end{pmatrix}=\begin{pmatrix}&0\\ \ \nearrow&\\ aR\longrightarrow&-a\end{pmatrix}
  54. ( m k ) (mk)
  55. S m k = α m k / R - p m + 1 , k / R + p m + 1 , k + 1 S_{mk}=\alpha_{mk}/R-p_{m+1,k}/R+p_{m+1,k+1}
  56. S m k = σ m k + p m k / R S_{mk}=\sigma_{mk}+p_{mk}/R
  57. σ m k D R \sigma_{mk}\in D_{R}
  58. σ m k \sigma_{mk}
  59. ( m k ) (mk)
  60. p m k p_{mk}
  61. ( m - 1 , k - 1 ) (m-1,k-1)
  62. ( m - 1 , k ) (m-1,k)
  63. S m k = σ m k + p m k / R S_{mk}=\sigma_{mk}+p_{mk}/R
  64. R > 2. R>2.
  65. R = 3 R=3
  66. T K R TK_{R}^{\prime}
  67. ( m k ) (mk)
  68. T K R ′′ TK_{R}^{\prime\prime}
  69. ( m k ) (mk)
  70. T K R TK_{R}^{\prime}
  71. T K R TK_{R}^{\prime}
  72. T K R ′′ TK_{R}^{\prime\prime}
  73. ( m k ) (mk)
  74. T K R TK_{R}^{\prime}
  75. T K R TK_{R}^{\prime}
  76. F ( x ) F(x)
  77. F ( x ) x = y x F ( x ) y \frac{\partial F(x)}{\partial x}=\frac{\partial y}{\partial x}\frac{\partial F% (x)}{\partial y}
  78. F ( x ) F(x)
  79. F ( x ) y \frac{\partial F(x)}{\partial y}
  80. y x \frac{\partial y}{\partial x}
  81. F ( x ) y \frac{\partial F(x)}{\partial y}
  82. x ( 0 0 α m k 0 0 0 ) = ( ( k - m ) α m k 0 ( k - 2 m ) α m k 0 0 ( - m ) α m k ) \frac{\partial}{\partial x}\begin{pmatrix}&&0\\ &0&\alpha_{mk}\\ 0&0&0\end{pmatrix}=\begin{pmatrix}&&(k-m)\alpha_{mk}\\ &0&(k-2m)\alpha_{mk}\\ 0&0&(-m)\alpha_{mk}\end{pmatrix}
  83. F ( x ) = k = 0 n A k y k F(x)=\sum_{k=0}^{n}A_{k}y^{k}
  84. A k A_{k}
  85. y k y^{k}
  86. α m k R k y k ( 1 - y ) m \alpha_{mk}R^{k}y^{k}(1-y)^{m}
  87. F ( x , v ) = k = 0 n m 1 = 0 k m 2 = 0 k α m 1 , m 2 , k R k y k - m 1 ( 1 - y ) m 1 z k - m 2 ( 1 - z ) m 2 F(x,v)=\sum_{k=0}^{n}\sum_{m1=0}^{k}\sum_{m2=0}^{k}\alpha_{m1,m2,k}R^{k}y^{k-m% 1}(1-y)^{m1}z^{k-m2}(1-z)^{m2}
  88. R R
  89. α m 1 , m 2 , k \alpha_{m1,m2,k}
  90. y ( x ) , z ( v ) y(x),~{}z(v)
  91. x , v x,~{}v
  92. α m 1 , m 2 , k \alpha_{m1,m2,k}
  93. m 1 , m 2 , k {m1,m2,k}
  94. P K R PK_{R}
  95. α m 1 , m 2 , k \alpha_{m1,m2,k}
  96. D R D_{R}
  97. P K R PK_{R}
  98. R 7 R\geq 7
  99. F ( x 1 , , x i , , x a ) = k = 0 n m 1 = 0 k m a = 0 k ( α m 1 , , m a , k R k i = 1 a ( y i k - m i ( 1 - y i ) m i ) ) F(x_{1},\ldots,x_{i},\ldots,x_{a})=\sum_{k=0}^{n}\sum_{m_{1}=0}^{k}\ldots\sum_% {m_{a}=0}^{k}(\alpha_{m_{1},\ldots,m_{a},k}R^{k}\prod^{a}_{i=1}(y_{i}^{k-m_{i}% }(1-y_{i})^{m_{i}}))
  100. R R
  101. α m 1 , , m a , k \alpha_{m_{1},\ldots,m_{a},k}
  102. y i ( x i ) y_{i}(x_{i})
  103. x i x_{i}
  104. α m 1 , m 2 , m 3 , k \alpha_{m1,m2,m3,k}
  105. m 1 , m 2 , m 3 , k {m1,m2,m3,k}
  106. G P K R GPK_{R}
  107. α m 1 , , m a , k \alpha_{m_{1},\ldots,m_{a},k}
  108. D R D_{R}
  109. G P K R GPK_{R}
  110. 2 a 2^{a}
  111. R ( 2 a - 1 - 1 ) R\geq(2^{a-1}-1)

Computer_representation_of_surfaces.html

  1. 𝐱 = 𝐱 ( u , v ) , \mathbf{x}=\mathbf{x}(u,v),

Concentration_inequality.html

  1. Pr ( | X | a ) E ( | X | ) a . \Pr(|X|\geq a)\leq\frac{\textrm{E}(|X|)}{a}.
  2. Φ \Phi
  3. Pr ( X a ) = Pr ( Φ ( X ) Φ ( a ) ) E ( Φ ( X ) ) Φ ( a ) . \Pr(X\geq a)=\Pr(\Phi(X)\geq\Phi(a))\leq\frac{\textrm{E}(\Phi(X))}{\Phi(a)}.
  4. Φ = x 2 \Phi=x^{2}
  5. Pr ( | X - E ( X ) | a ) Var ( X ) a 2 , \Pr(|X-\textrm{E}(X)|\geq a)\leq\frac{\textrm{Var}(X)}{a^{2}},
  6. Var ( X ) = E [ ( X - E ( X ) ) 2 ] . \operatorname{Var}(X)=\operatorname{E}[(X-\operatorname{E}(X))^{2}].
  7. n n
  8. p p
  9. k k
  10. n n
  11. f ( k ; n , p ) = Pr ( K = k ) = ( n k ) p k ( 1 - p ) n - k . f(k;n,p)=\Pr(K=k)={n\choose k}p^{k}(1-p)^{n-k}.
  12. S n = i = 1 n X i S_{n}=\sum_{i=1}^{n}X_{i}
  13. X i X_{i}
  14. p p
  15. S n S_{n}
  16. n n
  17. p p
  18. n n\to\infty
  19. S n S_{n}
  20. n p np
  21. n p ( 1 - p ) np(1-p)
  22. lim n Pr [ a σ < S n - n p < b σ ] = a b 1 2 π e - x 2 / 2 d x \lim_{n\to\infty}\Pr[a\sigma<S_{n}-np<b\sigma]=\int_{a}^{b}\frac{1}{\sqrt{2\pi% }}e^{-x^{2}/2}dx
  23. p = λ n p=\frac{\lambda}{n}
  24. λ \lambda
  25. B ( n , p ) B(n,p)
  26. P ( λ ) P(\lambda)
  27. X i X_{i}
  28. X i E ( X i ) - a i - M X_{i}\geq E(X_{i})-a_{i}-M
  29. 1 i n 1\leq i\leq n
  30. X = i = 1 n X i X=\sum_{i=1}^{n}X_{i}
  31. Pr [ X E ( X ) - λ ] e - λ 2 2 ( V a r ( X ) + i = 1 n a i 2 + M λ / 3 ) \Pr[X\leq E(X)-\lambda]\leq e^{-\frac{\lambda^{2}}{2(Var(X)+\sum_{i=1}^{n}a_{i% }^{2}+M\lambda/3)}}
  32. X i X_{i}
  33. X i E ( X i ) + a i + M X_{i}\leq E(X_{i})+a_{i}+M
  34. Pr [ X E ( X ) + λ ] e - λ 2 2 ( V a r ( X ) + i = 1 n a i 2 + M λ / 3 ) \Pr[X\geq E(X)+\lambda]\leq e^{-\frac{\lambda^{2}}{2(Var(X)+\sum_{i=1}^{n}a_{i% }^{2}+M\lambda/3)}}
  35. X i X_{i}
  36. | X i | 1 |X_{i}|\leq 1
  37. σ 2 \sigma^{2}
  38. X i X_{i}
  39. Pr [ | X | k σ ] 2 e - k 2 / 4 n \Pr[|X|\geq k\sigma]\leq 2e^{-k^{2}/4n}
  40. 0 k 2 σ < c o d e > < / c o d e > 0\leq k\leq 2\sigma\par <code> </code>
  41. X 1 , , X n X_{1},\dots,X_{n}\!
  42. X i X_{i}
  43. 1 i n 1\leq i\leq n
  44. Pr ( X i [ a i , b i ] ) = 1. \Pr(X_{i}\in[a_{i},b_{i}])=1.\!
  45. X ¯ = X 1 + + X n n \overline{X}=\frac{X_{1}+\cdots+X_{n}}{n}
  46. Pr ( X ¯ - E [ X ¯ ] t ) exp ( - 2 t 2 n 2 i = 1 n ( b i - a i ) 2 ) , \Pr(\overline{X}-\mathrm{E}[\overline{X}]\geq t)\leq\exp\left(-\frac{2t^{2}n^{% 2}}{\sum_{i=1}^{n}(b_{i}-a_{i})^{2}}\right),\!
  47. Pr ( | X ¯ - E [ X ¯ ] | t ) 2 exp ( - 2 t 2 n 2 i = 1 n ( b i - a i ) 2 ) , \Pr(|\overline{X}-\mathrm{E}[\overline{X}]|\geq t)\leq 2\exp\left(-\frac{2t^{2% }n^{2}}{\sum_{i=1}^{n}(b_{i}-a_{i})^{2}}\right),\!
  48. i i
  49. σ 2 = 1 n i = 1 n Var ( X i ) . \sigma^{2}=\frac{1}{n}\sum_{i=1}^{n}\operatorname{Var}(X_{i}).
  50. t 0 t≥0
  51. Pr ( i = 1 n X i > t ) exp ( - n σ 2 a 2 h ( a t n σ 2 ) ) , \Pr\left(\sum_{i=1}^{n}X_{i}>t\right)\leq\exp\left(-\frac{n\sigma^{2}}{a^{2}}h% \left(\frac{at}{n\sigma^{2}}\right)\right),
  52. h ( u ) = ( 1 + u ) l o g ( 1 + u ) u h(u)=(1+u)log(1+u)–u
  53. ε \varepsilon
  54. 𝐏 { | 1 n i = 1 n X i | > ε } 2 exp { - n ε 2 2 ( 1 + ε / 3 ) } . \mathbf{P}\left\{\left|\;\frac{1}{n}\sum_{i=1}^{n}X_{i}\;\right|>\varepsilon% \right\}\leq 2\exp\left\{-\frac{n\varepsilon^{2}}{2(1+\varepsilon/3)}\right\}.
  55. X 1 X n X_{1}\dots X_{n}
  56. X 1 X n X_{1}^{\prime}\dots X_{n}^{\prime}
  57. X i X_{i}^{\prime}
  58. X i X_{i}
  59. i i
  60. X = ( X 1 , , X n ) , X ( i ) = ( X 1 , , X i - 1 , X i , X i + 1 , , X n ) . X=(X_{1},\dots,X_{n}),X^{(i)}=(X_{1},\dots,X_{i-1},X_{i}^{\prime},X_{i+1},% \dots,X_{n}).
  61. Var ( f ( X ) ) 1 2 i = 1 n E [ ( f ( X ) - f ( X ( i ) ) ) 2 ] . \mathrm{Var}(f(X))\leq\frac{1}{2}\sum_{i=1}^{n}E[(f(X)-f(X^{(i)}))^{2}].

Concomitant_(statistics).html

  1. Y [ r : n ] Y_{[r:n]}
  2. 1 r n 1\leq r\leq n
  3. f Y [ r : n ] ( y ) = - f Y X ( y | x ) f X r : n ( x ) d x f_{Y_{[r:n]}}(y)=\int_{-\infty}^{\infty}f_{Y\mid X}(y|x)f_{X_{r:n}}(x)\,% \mathrm{d}x
  4. ( X i , Y i ) (X_{i},Y_{i})
  5. 1 r 1 < < r k n 1\leq r_{1}<\cdots<r_{k}\leq n
  6. ( Y [ r 1 : n ] , , Y [ r k : n ] ) \left(Y_{[r_{1}:n]},\cdots,Y_{[r_{k}:n]}\right)
  7. f Y [ r 1 : n ] , , Y [ r k : n ] ( y 1 , , y k ) = - - x k - x 2 i = 1 k f Y X ( y i | x i ) f X r 1 : n , , X r k : n ( x 1 , , x k ) d x 1 d x k f_{Y_{[r_{1}:n]},\cdots,Y_{[r_{k}:n]}}(y_{1},\cdots,y_{k})=\int_{-\infty}^{% \infty}\int_{-\infty}^{x_{k}}\cdots\int_{-\infty}^{x_{2}}\prod^{k}_{i=1}f_{Y% \mid X}(y_{i}|x_{i})f_{X_{r_{1}:n},\cdots,X_{r_{k}:n}}(x_{1},\cdots,x_{k})% \mathrm{d}x_{1}\cdots\mathrm{d}x_{k}
  8. ( Y [ r 1 : n ] , , Y [ r k : n ] ) \left(Y_{[r_{1}:n]},\cdots,Y_{[r_{k}:n]}\right)
  9. X r 1 : n = x 1 , , X r k : n = x k X_{r_{1}:n}=x_{1},\cdots,X_{r_{k}:n}=x_{k}
  10. x 1 x k x_{1}\leq\cdots\leq x_{k}
  11. f Y [ r 1 : n ] , , Y [ r k : n ] X r 1 : n X r k : n ( y 1 , , y k | x 1 , , x k ) = i = 1 k f Y X ( y i | x i ) f_{Y_{[r_{1}:n]},\cdots,Y_{[r_{k}:n]}\mid X_{r_{1}:n}\cdots X_{r_{k}:n}}(y_{1}% ,\cdots,y_{k}|x_{1},\cdots,x_{k})=\prod^{k}_{i=1}f_{Y\mid X}(y_{i}|x_{i})

Conductivity_near_the_percolation_threshold.html

  1. σ \sigma
  2. ϵ \epsilon
  3. p p
  4. 1 - p 1-p
  5. p c p_{c}
  6. p > p c p>p_{c}
  7. p c p_{c}
  8. ν \nu
  9. β \beta
  10. ξ \xi
  11. ξ ( p ) ( p c - p ) - ν \xi(p)\propto(p_{c}-p)^{-\nu}
  12. P ( p ) ( p - p c ) β P(p)\propto(p-p_{c})^{\beta}
  13. σ m \sigma_{m}
  14. σ d \sigma_{d}
  15. σ d = 0 \sigma_{d}=0
  16. σ D C ( p ) σ m ( p - p c ) t \sigma_{DC}(p)\propto\sigma_{m}(p-p_{c})^{t}
  17. p > p c p>p_{c}
  18. σ m = \sigma_{m}=\infty
  19. σ D C ( p ) σ d ( p c - p ) - s \sigma_{DC}(p)\propto\sigma_{d}(p_{c}-p)^{-s}
  20. p < p c p<p_{c}
  21. σ ( p ) σ m | Δ p | t Φ ± ( h | Δ p | - s - t ) \sigma(p)\propto\sigma_{m}|\Delta p|^{t}\Phi_{\pm}\left(h|\Delta p|^{-s-t}\right)
  22. Δ p p - p c \Delta p\equiv p-p_{c}
  23. h σ d σ m h\equiv\frac{\sigma_{d}}{\sigma_{m}}
  24. σ D C ( p c ) σ m ( σ d σ m ) u \sigma_{DC}(p_{c})\propto\sigma_{m}\left(\frac{\sigma_{d}}{\sigma_{m}}\right)^% {u}
  25. u = t t + s u=\frac{t}{t+s}
  26. ϵ 1 ( ω = 0 , p ) = ϵ d | p - p c | s \epsilon_{1}(\omega=0,p)=\frac{\epsilon_{d}}{|p-p_{c}|^{s}}
  27. σ m = 1 / R \sigma_{m}=1/R
  28. σ d = i C ω \sigma_{d}=iC\omega
  29. ω \omega
  30. σ ( p , ω ) 1 R | Δ p | t Φ ± ( i ω ω 0 | Δ p | - ( s + t ) ) \sigma(p,\omega)\propto\frac{1}{R}|\Delta p|^{t}\Phi_{\pm}\left(\frac{i\omega}% {\omega_{0}}|\Delta p|^{-(s+t)}\right)
  31. τ * = 1 ω 0 | Δ p | - ( s + t ) \tau^{*}=\frac{1}{\omega_{0}}|\Delta p|^{-(s+t)}

Confidence_distribution.html

  1. H Φ ( μ ) H_{\Phi}(\mu)
  2. H t ( μ ) H_{t}(\mu)
  3. H Φ ( μ ) = Φ ( n ( μ - X ¯ ) σ ) , and H t ( μ ) = F t n - 1 ( n ( μ - X ¯ ) s ) , H_{\Phi}(\mu)=\Phi\left(\frac{\sqrt{n}(\mu-\bar{X})}{\sigma}\right),\quad\,% \text{and}\quad H_{t}(\mu)=F_{t_{n-1}}\left(\frac{\sqrt{n}(\mu-\bar{X})}{s}% \right),
  4. F t n - 1 F_{t_{n-1}}
  5. t n - 1 t_{n-1}
  6. H A ( μ ) = Φ ( n ( μ - X ¯ ) s ) H_{A}(\mu)=\Phi\left(\frac{\sqrt{n}(\mu-\bar{X})}{s}\right)
  7. H t ( μ ) H_{t}(\mu)
  8. H A ( μ ) H_{A}(\mu)
  9. N ( X ¯ , σ 2 ) N(\bar{X},\sigma^{2})
  10. N ( X ¯ , s 2 ) N(\bar{X},s^{2})
  11. μ \mu
  12. H Φ ( μ ) H_{\Phi}(\mu)
  13. H t ( μ ) H_{t}(\mu)
  14. H A ( μ ) H_{A}(\mu)
  15. H χ 2 ( θ ) = 1 - F χ n - 1 2 ( ( n - 1 ) s 2 / θ ) H_{\chi^{2}}(\theta)=1-F_{\chi^{2}_{n-1}}((n-1)s^{2}/\theta)
  16. F χ n - 1 2 F_{\chi^{2}_{n-1}}
  17. χ n - 1 2 \chi^{2}_{n-1}
  18. H Φ ( μ ) = Φ ( n ( μ - X ¯ ) σ ) H_{\Phi}(\mu)=\Phi\left(\frac{\sqrt{n}(\mu-\bar{X})}{\sigma}\right)
  19. H t ( μ ) = F t n - 1 ( n ( μ - X ¯ ) s ) H_{t}(\mu)=F_{t_{n-1}}\left(\frac{\sqrt{n}(\mu-\bar{X})}{s}\right)
  20. z = 1 2 ln 1 + r 1 - r z={1\over 2}\ln{1+r\over 1-r}
  21. N ( 1 2 ln 1 + ρ 1 - ρ , 1 n - 3 ) N({1\over 2}\ln{{1+\rho}\over{1-\rho}},{1\over n-3})
  22. H n ( ρ ) = 1 - Φ ( n - 3 ( 1 2 ln 1 + r 1 - r - 1 2 ln 1 + ρ 1 - ρ ) ) H_{n}(\rho)=1-\Phi\left(\sqrt{n-3}\left({1\over 2}\ln{1+r\over 1-r}-{1\over 2}% \ln{{1+\rho}\over{1-\rho}}\right)\right)
  23. ( - , H n - 1 ( 1 - α ) ] , [ H n - 1 ( α ) , ) (-\infty,H_{n}^{-1}(1-\alpha)],[H_{n}^{-1}(\alpha),\infty)
  24. [ H n - 1 ( α / 2 ) , H n - 1 ( 1 - α / 2 ) ] [H_{n}^{-1}(\alpha/2),H_{n}^{-1}(1-\alpha/2)]
  25. [ H n - 1 ( α 1 ) , H n - 1 ( 1 - α 2 ) ] [H_{n}^{-1}(\alpha_{1}),H_{n}^{-1}(1-\alpha_{2})]
  26. H n ( θ ) H_{n}(\theta)
  27. H n ( θ ) = β H_{n}(\theta)=\beta
  28. θ ¯ n = - t d H n ( t ) \bar{\theta}_{n}=\int_{-\infty}^{\infty}t\,dH_{n}(t)
  29. θ ^ n = arg max θ h n ( θ ) , h n ( θ ) = H n ( θ ) . \widehat{\theta}_{n}=\arg\max_{\theta}h_{n}(\theta),h_{n}(\theta)=H^{\prime}_{% n}(\theta).
  30. p s ( C ) = H n ( C ) = C d H ( θ ) . p_{s}(C)=H_{n}(C)=\int_{C}dH(\theta).

Configuration_graph.html

  1. s s
  2. n n
  3. ( n + 1 ) s (n+1)s
  4. n + 1 n+1

Conformal_cyclic_cosmology.html

  1. g a b g_{ab}
  2. Ω \Omega

Conformal_geometric_algebra.html

  1. 𝐱 \mathbf{x}
  2. n o 2 = 0 n o n = - 1 n o 𝐱 = 0 n 2 = 0 n o n = e - e + n 𝐱 = 0 \begin{array}[]{lllll}{n_{o}}^{2}&=0\qquad n\text{o}\cdot n_{\infty}&=-1&n% \text{o}\cdot\mathbf{x}&=0\\ {n_{\infty}}^{2}&=0\qquad n\text{o}\wedge n_{\infty}&=e_{-}e_{+}&n_{\infty}% \cdot\mathbf{x}&=0\end{array}
  3. r r
  4. r r
  5. r r
  6. F : 𝐱 n o + 𝐱 + 1 2 𝐱 2 n F:\mathbf{x}\mapsto n\text{o}+\mathbf{x}+\tfrac{1}{2}\mathbf{x}^{2}n_{\infty}
  7. 𝐱 \mathbf{x}
  8. 1 1
  9. X X
  10. X 𝒫 n n o ( X - X n ) X\mapsto\mathcal{P}^{\perp}_{n_{\infty}\wedge n_{o}}\left(\frac{X}{-X\cdot n_{% \infty}}\right)
  11. 𝐱 \mathbf{x}
  12. 𝐱 = 0 \mathbf{x}=0
  13. A A
  14. X 2 = 0 X^{2}=0
  15. X A = 0 X\wedge A=0
  16. A A
  17. A A
  18. X A X∧A
  19. A A
  20. A A
  21. g ( 𝐚 ) . g ( 𝐛 ) = - 1 2 𝐚 - 𝐛 2 g(\mathbf{a}).g(\mathbf{b})=-\frac{1}{2}\|\mathbf{a}-\mathbf{b}\|^{2}
  22. 𝐒 = g ( 𝐚 ) - 1 2 ρ 2 𝐞 \mathbf{S}=g(\mathbf{a})-\frac{1}{2}\rho^{2}\mathbf{e}_{\infty}
  23. - 1 2 ( 𝐚 - 𝐱 ) 2 + 1 2 ρ 2 = 0 -\frac{1}{2}(\mathbf{a}-\mathbf{x})^{2}+\frac{1}{2}\rho^{2}=0
  24. 𝐏 = 𝐚 ^ + α 𝐞 \mathbf{P}=\hat{\mathbf{a}}+\alpha\mathbf{e}_{\infty}
  25. 𝐚 ^ \hat{\mathbf{a}}
  26. 𝐏 = g ( 𝐚 ) - g ( 𝐛 ) \mathbf{P}=g(\mathbf{a})-g(\mathbf{b})
  27. g ( 𝐱 ) = 𝐏 β 𝐏 α g ( 𝐱 ) 𝐏 α 𝐏 β g(\mathbf{x}^{\prime})=\mathbf{P}_{\beta}\mathbf{P}_{\alpha}\;g(\mathbf{x})\;% \mathbf{P}_{\alpha}\mathbf{P}_{\beta}
  28. 𝐏 α = 𝐚 ^ + α 𝐞 \mathbf{P}_{\alpha}=\hat{\mathbf{a}}+\alpha\mathbf{e}_{\infty}
  29. 𝐏 β = 𝐚 ^ + β 𝐞 \mathbf{P}_{\beta}=\hat{\mathbf{a}}+\beta\mathbf{e}_{\infty}
  30. 𝐱 = 𝐱 + 2 ( β - α ) 𝐚 ^ \mathbf{x}^{\prime}=\mathbf{x}+2(\beta-\alpha)\hat{\mathbf{a}}
  31. g ( 𝐱 ) = 𝐛 ^ 𝐚 ^ g ( 𝐱 ) 𝐚 ^ 𝐛 ^ g(\mathbf{x}^{\prime})=\hat{\mathbf{b}}\hat{\mathbf{a}}\;g(\mathbf{x})\;\hat{% \mathbf{a}}\hat{\mathbf{b}}
  32. 𝐓𝐑 𝐓 ~ \mathbf{TR{\tilde{T}}}
  33. g ( 𝒢 x ) = 𝐓𝐑 𝐓 ~ g ( 𝐱 ) 𝐓 𝐑 ~ 𝐓 ~ g(\mathcal{G}x)=\mathbf{TR{\tilde{T}}}\;g(\mathbf{x})\;\mathbf{T\tilde{R}% \tilde{T}}
  34. 𝐌 = 𝐓 𝟐 𝐓 𝟏 𝐑 𝐓 𝟏 ~ \mathbf{M}=\mathbf{T_{2}T_{1}R{\tilde{T_{1}}}}
  35. 𝐌 = 𝐓 𝐑 \mathbf{M}=\mathbf{T^{\prime}R^{\prime}}

Conformal_welding.html

  1. F ( r , θ ) = r exp [ i ψ ( r ) g ( θ ) + i ( 1 - ψ ( r ) ) θ ] , \displaystyle{F(r,\theta)=r\exp[i\psi(r)g(\theta)+i(1-\psi(r))\theta],}
  2. f ( e i θ ) = e i g ( θ ) , \displaystyle{f(e^{i\theta})=e^{ig(\theta)},}
  3. μ < 1 , μ ( z ) = F z ¯ / F z . \displaystyle{\|\mu\|_{\infty}<1,\,\,\,\mu(z)=F_{\overline{z}}/F_{z}.}
  4. G z ¯ = μ G z . \displaystyle{G_{\overline{z}}=\mu G_{z}.}
  5. f = f 1 - 1 f 2 . \displaystyle{f=f_{1}^{-1}\circ f_{2}.}
  6. f - ( z ) = a 0 + a 1 z + a 2 z 2 + , f + ( z ) = z + b 1 z - 1 + b 2 z - 2 + , \displaystyle{f_{-}(z)=a_{0}+a_{1}z+a_{2}z^{2}+\cdots,\,\,\,\,\,f_{+}(z)=z+b_{% 1}z^{-1}+b_{2}z^{-2}+\cdots,}
  7. f - ( e i θ ) = f + ( f ( e i θ ) ) . \displaystyle{f_{-}(e^{i\theta})=f_{+}(f(e^{i\theta})).}
  8. T F = - F + 2 e i θ \displaystyle{TF=-F+2e^{i\theta}}
  9. T F f = F f . \displaystyle{TF\circ f=F\circ f.}
  10. T ( F f ) f - 1 - T ( F ) = 2 F - 2 e i θ . \displaystyle{T(F\circ f)\circ f^{-1}-T(F)=2F-2e^{i\theta}.}
  11. K f = V ( f ) P V ( f ) - 1 - P \displaystyle{K_{f}=V(f)PV(f)^{-1}-P}
  12. ( I - K f ) F = e i θ . \displaystyle{(I-K_{f})F=e^{i\theta}.}

Conjectural_variation.html

  1. ϕ * = - 1 \phi^{*}=-1
  2. P = 1 - x - y P=1-x-y
  3. x P = x ( 1 - x - y ) = x - x 2 - x y xP=x(1-x-y)=x-x^{2}-xy
  4. Π = x - x 2 - x y \Pi=x-x^{2}-xy
  5. d Π d x = ( 1 - 2 x - y ) - x d y d x = 0 \frac{d\Pi}{dx}=(1-2x-y)-x\frac{dy}{dx}=0
  6. d y d x = ϕ \frac{dy}{dx}=\phi
  7. x = R ( y , ϕ ) = 1 - y 2 + ϕ x=R(y,\phi)=\frac{1-y}{2+\phi}
  8. ϕ = 0 \phi=0
  9. x = 1 - x 2 + ϕ x=\frac{1-x}{2+\phi}
  10. x * = y * = 1 3 + ϕ x^{*}=y^{*}=\frac{1}{3+\phi}
  11. P * = 1 + ϕ 3 + ϕ P^{*}=\frac{1+\phi}{3+\phi}
  12. ϕ = 0 \phi=0
  13. P * = 1 3 P^{*}=\frac{1}{3}
  14. ϕ = - 1 \phi=-1
  15. ϕ = + 1 \phi=+1
  16. P * = 1 2 P^{*}=\frac{1}{2}
  17. ϕ \phi
  18. Π = ( x - x 2 - x y ) - a . x 2 2 \Pi=(x-x^{2}-xy)-\frac{a.x^{2}}{2}
  19. d Π d x = ( 1 - 2 x - y ) - x d y d x - a x = 0 \frac{d\Pi}{dx}=(1-2x-y)-x\frac{dy}{dx}-ax=0
  20. x = R ( y , ϕ ) = 1 - y 2 + a + ϕ x=R(y,\phi)=\frac{1-y}{2+a+\phi}
  21. R y = - 1 2 + a + ϕ R_{y}=-\frac{1}{2+a+\phi}
  22. ϕ = 0 \phi=0
  23. ϕ \phi
  24. R y R_{y}
  25. ϕ = - 1 2 + a + ϕ \phi=-\frac{1}{2+a+\phi}
  26. ϕ * = - ( 1 + a 2 ) + 4 a + a 2 4 \phi^{*}=-(1+\frac{a}{2})+\sqrt{\frac{4a+a^{2}}{4}}
  27. ϕ * = - 1 \phi^{*}=-1

Conjugate_beam_method.html

  1. E q .1 d V d x = w Eq.1\;\frac{dV}{dx}=w
  2. E q .3 d 2 M d x 2 = w Eq.3\;\frac{d^{2}M}{dx^{2}}=w
  3. E q .2 d θ d x = M E I Eq.2\;\frac{d\theta}{dx}=\frac{M}{EI}
  4. E q .4 d 2 v d x 2 = M E I Eq.4\;\frac{d^{2}v}{dx^{2}}=\frac{M}{EI}
  5. V = w d x V=\int w\,dx
  6. M = [ w d x ] d x M=\int\left[\int w\,dx\right]dx
  7. θ = ( M E I ) d x \theta=\int\left(\frac{M}{EI}\right)dx
  8. v = [ ( M E I ) d x ] d x v=\int\left[\int\left(\frac{M}{EI}\right)dx\right]dx
  9. v = 0 v=0
  10. θ = 0 \theta=0
  11. M ¯ = 0 \overline{M}=0
  12. Q ¯ = 0 \overline{Q}=0
  13. v 0 v\not=0
  14. θ 0 \theta\not=0
  15. M ¯ 0 \overline{M}\not=0
  16. Q ¯ 0 \overline{Q}\not=0
  17. v = 0 v=0
  18. θ 0 \theta\not=0
  19. M ¯ = 0 \overline{M}=0
  20. Q ¯ 0 \overline{Q}\not=0
  21. v = 0 v=0
  22. θ \theta
  23. M ¯ = 0 \overline{M}=0
  24. Q ¯ \overline{Q}
  25. v v
  26. θ \theta
  27. M ¯ \overline{M}
  28. Q ¯ \overline{Q}

Conjugate_depth.html

  1. M = y 1 2 2 + q 2 g y 1 = y 2 2 2 + q 2 g y 2 . M=\frac{y_{1}^{2}}{2}+\frac{q^{2}}{gy_{1}}=\frac{y_{2}^{2}}{2}+\frac{q^{2}}{gy% _{2}}.
  2. q 2 g ( 1 y 1 - 1 y 2 ) = 1 2 ( y z 2 - y 1 2 ) . \frac{q^{2}}{g}\left(\frac{1}{y_{1}}-\frac{1}{y_{2}}\right)=\frac{1}{2}\left(y% _{z}^{2}-y_{1}^{2}\right).
  3. q 2 g ( y 2 - y 1 y 1 y 2 ) = 1 2 ( y 2 - y 1 ) ( y 2 + y 1 ) . \frac{q^{2}}{g}\left(\frac{y_{2}-y_{1}}{y_{1}y_{2}}\right)=\frac{1}{2}(y_{2}-y% _{1})(y_{2}+y_{1}).
  4. q 2 g ( 1 y 1 y 2 ) = 1 2 ( y 2 + y 1 ) where q 1 2 = y 1 2 v 1 2 = y 2 2 v 2 2 . \frac{q^{2}}{g}\left(\frac{1}{y_{1}y_{2}}\right)=\frac{1}{2}(y_{2}+y_{1})% \qquad\,\text{where }q_{1}^{2}=y_{1}^{2}v_{1}^{2}=y_{2}^{2}v_{2}^{2}.
  5. v 1 2 g ( 1 y 1 y 2 ) = 1 2 y 1 2 ( y 2 + y 1 ) recall F r 1 2 = v 1 2 g y 1 . \frac{v_{1}^{2}}{g}\left(\frac{1}{y_{1}y_{2}}\right)=\frac{1}{2y_{1}^{2}}(y_{2% }+y_{1})\qquad\,\text{recall }Fr_{1}^{2}=\frac{v_{1}^{2}}{gy_{1}}.
  6. F r 2 2 = y 2 2 2 y 1 2 + y 2 2 y 1 . Fr_{2}^{2}=\frac{y_{2}^{2}}{2y_{1}^{2}}+\frac{y_{2}}{2y_{1}}.
  7. F r 1 2 = x 2 2 + x 2 0 = x 2 2 + x 2 - F r 1 2 . Fr_{1}^{2}=\frac{x^{2}}{2}+\frac{x}{2}\Rightarrow 0=\frac{x^{2}}{2}+\frac{x}{2% }-Fr_{1}^{2}.
  8. 4 2 \tfrac{\sqrt{4}}{2}
  9. x = - 1 2 ± ( 1 / 2 ) 2 - 4 ( 1 / 2 ) ( F r 1 2 ) 2 ( 1 / 2 ) = - 1 2 ( 1 / 4 ) - 8 ( F r 1 2 ) . x=\frac{-\tfrac{1}{2}\pm\sqrt{(1/2)^{2}-4(1/2)(Fr_{1}^{2})}}{2(1/2)}=-\frac{1}% {2}\sqrt{(1/4)-8(Fr_{1}^{2})}.
  10. y 2 y 1 = 1 2 ( 1 + 8 F r 1 2 - 1 ) . \frac{y_{2}}{y_{1}}=\frac{1}{2}\left(\sqrt{1+8Fr_{1}^{2}}-1\right).

Conjugate_residual_method.html

  1. 𝐀𝐱 = 𝐛 \mathbf{A}\mathbf{x}=\mathbf{b}
  2. 𝐱 0 \mathbf{x}_{0}
  3. 𝐱 0 := Some initial guess \displaystyle\mathbf{x}_{0}:=\,\text{Some initial guess}
  4. 𝐱 k \mathbf{x}_{k}
  5. α k \alpha_{k}
  6. β k \beta_{k}
  7. 𝐀𝐩 k \mathbf{Ap}_{k}
  8. 𝐱 0 := Some initial guess \displaystyle\mathbf{x}_{0}:=\,\text{Some initial guess}
  9. 𝐌 - 1 \mathbf{M}^{-1}

Consistent_pricing_process.html

  1. ( Ω , , { t } t = 0 T , P ) (\Omega,\mathcal{F},\{\mathcal{F}_{t}\}_{t=0}^{T},P)
  2. t t
  3. i t h i^{th}
  4. i t h i^{th}
  5. Z = ( Z t ) t = 0 T Z=(Z_{t})_{t=0}^{T}
  6. d \mathbb{R}^{d}
  7. P P
  8. Z t K t + \ { 0 } Z_{t}\in K_{t}^{+}\backslash\{0\}
  9. t t
  10. K t K_{t}
  11. t t
  12. Z Z
  13. Q Q

Constant_elasticity_of_variance_model.html

  1. d S t = μ S t d t + σ S t γ d W t dS_{t}=\mu S_{t}dt+\sigma S_{t}^{\gamma}dW_{t}
  2. σ , γ \sigma,\;\gamma
  3. σ 0 , γ 0 \sigma\geq 0,\;\gamma\geq 0
  4. γ \gamma
  5. γ < 1 \gamma<1
  6. γ > 1 \gamma>1

Constrained_conditional_model.html

  1. { ϕ i ( x , y ) } \{\phi_{i}(x,y)\}
  2. { C i ( x , y ) } \{C_{i}(x,y)\}
  3. x X x\in X
  4. y Y y\in Y
  5. ρ \rho
  6. a r g m a x y i w i ϕ i ( x , y ) - ρ i C i ( x , y ) argmax_{y}\sum_{i}w_{i}\phi_{i}(x,y)-\sum\rho_{i}C_{i}(x,y)
  7. C i C C_{i}\in C
  8. ( x , y ) (x,y)
  9. ρ \rho

Constructible_topology.html

  1. Spec ( A ) \operatorname{Spec}(A)
  2. A A
  3. Spec ( B ) \operatorname{Spec}(B)
  4. Spec ( A ) \operatorname{Spec}(A)
  5. Spec ( B ) Spec ( A ) \operatorname{Spec}(B)\to\operatorname{Spec}(A)
  6. Spec ( A ) \operatorname{Spec}(A)
  7. A / nil ( A ) A/\operatorname{nil}(A)
  8. nil ( A ) \operatorname{nil}(A)\,

Constructing_skill_trees.html

  1. R t R_{t}
  2. t T t\in T
  3. p ( q Q ) p(q\in Q)
  4. j + 1 j+1
  5. t t
  6. q q
  7. P ( j , t , q ) P(j,t,q)
  8. P ( j , t , q ) P(j,t,q)
  9. I n v e r s e G a m m a ( v 2 , u 2 ) InverseGamma(\frac{v}{2},\frac{u}{2})
  10. N o r m a l ( 0 , σ 2 δ ) Normal(0,\sigma^{2}\delta)
  11. P ( j , t , q ) P(j,t,q)
  12. P ( j , t , q ) = π - n 2 δ m | ( A + D ) - 1 | 1 2 u v 2 ( y + u ) u + v 2 Γ ( n + v 2 ) Γ ( v 2 ) P(j,t,q)=\frac{\pi^{-\frac{n}{2}}}{\delta^{m}}\left|(A+D)^{-1}\right|^{\frac{1% }{2}}\frac{u^{\frac{v}{2}}}{(y+u)^{\frac{u+v}{2}}}\frac{\Gamma(\frac{n+v}{2})}% {\Gamma({\frac{v}{2}})}
  13. P t ( j , q ) P_{t}(j,q)
  14. P j M A P P^{MAP}_{j}
  15. P t ( j , q ) = ( 1 - G ( t - j - 1 ) ) P ( j , t , q ) p ( q ) P j M A P P_{t}(j,q)=(1-G(t-j-1))P(j,t,q)p(q)P^{MAP}_{j}
  16. P j M A P = max i , q P j ( i , q ) g ( j - i ) 1 - G ( j - i - 1 ) , j < t P^{MAP}_{j}=\max_{i,q}\frac{P_{j}(i,q)g(j-i)}{1-G(j-i-1)},\forall j<t
  17. A = i = j t Φ ( x i ) Φ ( x i ) T A=\sum^{t}_{i=j}\Phi(x_{i})\Phi(x_{i})^{T}
  18. Φ ( x i ) \Phi(x_{i})
  19. x i x_{i}
  20. y = ( i = j t R i 2 ) - b T ( A + D ) - 1 b y=(\sum^{t}_{i=j}R^{2}_{i})-b^{T}(A+D)^{-1}b
  21. b = i = j t R i Φ ( x i ) b=\sum^{t}_{i=j}R_{i}\Phi(x_{i})
  22. R i = j = i T γ j - i r j R_{i}=\sum^{T}_{j=i}\gamma^{j-i}r_{j}
  23. Γ \Gamma
  24. n = t - j n=t-j
  25. m m
  26. D D
  27. δ - 1 \delta^{-1}
  28. l l
  29. g ( l ) = ( 1 - p ) l - 1 p g(l)=(1-p)^{l-1}p
  30. G ( l ) = ( 1 - ( 1 - p ) l ) G(l)=(1-(1-p)^{l})
  31. p = 1 k p=\frac{1}{k}
  32. k : k:
  33. O ( N L ) O(NL)
  34. O ( N c ) O(Nc)
  35. N N
  36. L L
  37. P ( j , t , q ) P(j,t,q)
  38. O ( c ) O(c)
  39. P ( j , t , q ) P(j,t,q)
  40. p p a r t i c l e s p\in particles
  41. max p \max_{p}
  42. \cup
  43. q Q q\in Q
  44. \cup
  45. p P p\in P
  46. Φ t \Phi_{t}
  47. Φ \Phi
  48. Φ t Φ t T \Phi_{t}\Phi_{t}^{T}
  49. γ \gamma
  50. Φ t \Phi_{t}
  51. r t r_{t}
  52. γ 2 \gamma^{2}
  53. r t 2 r_{t}^{2}
  54. γ r t \gamma r_{t}
  55. γ \gamma
  56. r t r_{t}
  57. γ \gamma

Constructive_function_theory.html

  1. max 0 x 2 π | f ( x ) - P n ( x ) | C ( f ) n α , \max_{0\leq x\leq 2\pi}|f(x)-P_{n}(x)|\leq\frac{C(f)}{n^{\alpha}},

Contextual_image_classification.html

  1. x 0 x_{0}
  2. x 0 x_{0}
  3. N ( x 0 ) N(x_{0})
  4. f ( x i ) f(x_{i})
  5. ξ = ( f ( x 0 ) , f ( x 1 ) , , f ( x k ) ) \xi=\left(f(x_{0}),f(x_{1}),\ldots,f(x_{k})\right)
  6. x i N ( x 0 ) ; i = 1 , , k x_{i}\in N(x_{0});\quad i=1,\ldots,k
  7. N ( x 0 ) N(x_{0})
  8. η = ( θ 0 , θ 1 , , θ k ) \eta=\left(\theta_{0},\theta_{1},\ldots,\theta_{k}\right)
  9. θ i { ω 0 , ω 1 , , ω k } \theta_{i}\in\left\{\omega_{0},\omega_{1},\ldots,\omega_{k}\right\}
  10. ω s \omega_{s}
  11. N ( x 0 ) N(x_{0})
  12. x 0 x_{0}
  13. η ^ = ( θ 1 , θ 2 , , θ k ) \hat{\eta}=\left(\theta_{1},\theta_{2},\ldots,\theta_{k}\right)
  14. x 0 x_{0}
  15. 3 × 3 3\times 3
  16. x 0 x_{0}
  17. x 0 x_{0}
  18. ω r \omega_{r}
  19. x 0 x_{0}
  20. ω r \omega_{r}
  21. θ 0 = ω r if P ( ω r f ( x 0 ) ) = max s = 1 , 2 , , R P ( ω s f ( x 0 ) ) \theta_{0}=\omega_{r}\quad\,\text{ if }\quad P(\omega_{r}\mid f(x_{0}))=\max_{% s=1,2,\ldots,R}P(\omega_{s}\mid f(x_{0}))
  22. x 1 x_{1}
  23. x 0 x_{0}
  24. θ 0 = ω r if P ( ω r ξ ) = max s = 1 , 2 , , R P ( ω s ξ ) \theta_{0}=\omega_{r}\quad\,\text{ if }\quad P(\omega_{r}\mid\xi)=\max_{s=1,2,% \ldots,R}P(\omega_{s}\mid\xi)
  25. P ( ω s ξ ) P(\omega_{s}\mid\xi)
  26. P ( ω s ξ ) = p ( ξ ω s ) P ( ω s ) p ( ξ ) P(\omega_{s}\mid\xi)=\frac{p(\xi\mid\omega_{s})P(\omega_{s})}{p\left(\xi\right)}
  27. x i x_{i}
  28. ξ \xi
  29. p ( ξ ω s ) p(\xi\mid\omega_{s})
  30. P ( ω s ) P(\omega_{s})
  31. P ( ω r ξ ) P(\omega_{r}\mid\xi)
  32. θ 0 \theta_{0}
  33. m × n m\times n
  34. 2 m × n 2^{m\times n}

Continuous_dual_Hahn_polynomials.html

  1. S n ( x 2 ; a , b , c ) = F 2 3 ( - n , a + i x , a - i x ; a + b , a + c ; 1 ) . S_{n}(x^{2};a,b,c)={}_{3}F_{2}(-n,a+ix,a-ix;a+b,a+c;1).
  2. a ¯ \overline{a}
  3. b ¯ \overline{b}

Continuous_geometry.html

  1. ( α A a α ) b = α ( a α b ) (\bigwedge_{\alpha\in A}a_{\alpha})b=\bigwedge_{\alpha}(a_{\alpha}b)
  2. P G ( F ) P G ( F 2 ) P G ( F 4 ) P G ( F 8 ) PG(F)\subset PG(F^{2})\subset PG(F^{4})\subset PG(F^{8})\cdots

Continuous_Hahn_polynomials.html

  1. p n ( x ; a , b , c , d ) = i n ( a + c ) n ( a + d ) n n ! F 2 3 ( - n , n + a + b + c + d - 1 , a + i x ; a + c , a + d ; 1 ) p_{n}(x;a,b,c,d)=i^{n}\frac{(a+c)_{n}(a+d)_{n}}{n!}{}_{3}F_{2}(-n,n+a+b+c+d-1,% a+ix;a+c,a+d;1)

Continuous_q-Hahn_polynomials.html

  1. p n ( x ; a , b , c | q ) = a - n e - i n u ( a b e 2 i u , a c , a d ; q ) n * 4 Φ 3 ( q - n , a b c d q n - 1 , a e i ( t + 2 u ) , a e - i t ; a b e 2 i u , a c , a d ; q ; q ) p_{n}(x;a,b,c|q)=a^{-n}e^{-inu}(abe^{2iu},ac,ad;q)_{n}*_{4}\Phi_{3}(q^{-n},% abcdq^{n-1},ae^{i{(t+2u)}},ae^{-it};abe^{2iu},ac,ad;q;q)
  2. x = c o s ( t + u ) x=cos(t+u)

Continuous_q-Hermite_polynomials.html

  1. $\displaystyle$
  2. 2 x H n ( x | q ) = H n + 1 ( x | q ) + ( 1 - q n ) H n - 1 ( x | q ) 2xH_{n}(x|q)=H_{n+1}(x|q)+(1-q^{n})H_{n-1}(x|q)
  3. H 0 ( x | q ) = 1 , H - 1 ( x | q ) = 0 \textstyle H_{0}(x|q)=1,H_{-1}(x|q)=0
  4. H 0 ( x | q ) = 1 H_{0}(x|q)=1
  5. H 1 ( x | q ) = 2 x H_{1}(x|q)=2x
  6. H 2 ( x | q ) = 4 x 2 - ( 1 - q n ) H_{2}(x|q)=4x^{2}-(1-q^{n})
  7. H 3 ( x | q ) = 8 x 3 - 4 x ( 1 - q n ) H_{3}(x|q)=8x^{3}-4x(1-q^{n})
  8. H 4 ( x | q ) = 16 x 4 - 12 x 2 ( 1 - q n ) + ( 1 - q n ) 2 H_{4}(x|q)=16x^{4}-12x^{2}(1-q^{n})+(1-q^{n})^{2}
  9. H 5 ( x | q ) = 32 x 5 - 32 x 3 ( 1 - q n ) + 6 x ( 1 - q n ) 2 H_{5}(x|q)=32x^{5}-32x^{3}(1-q^{n})+6x(1-q^{n})^{2}
  10. n = 0 H n ( x | q ) t n ( q ; q ) n = 1 ( t e i θ , t e - i θ ; q ) \displaystyle\sum_{n=0}^{\infty}H_{n}(x|q)\frac{t^{n}}{(q;q)_{n}}=\frac{1}{% \left(te^{i\theta},te^{-i\theta};q\right)_{\infty}}
  11. x = cos θ \textstyle x=\cos\theta

Continuous_q-Laguerre_polynomials.html

  1. P n ( α ) ( x | q ) = ( q α + 1 ; q ) n ( q ; q ) n P_{n}^{(\alpha)}(x|q)=\frac{(q^{\alpha}+1;q)_{n}}{(q;q)_{n}}
  2. Φ 2 3 ( q - n , q α / 2 + 1 / 4 e i θ , q α / 2 + 1 / 4 * e - i θ ; q α + 1 , 0 | q , q ) {}_{3}\Phi_{2}(q^{-n},q^{\alpha/2+1/4}e^{i\theta},q^{\alpha/2+1/4}*e^{-i\theta% };q^{\alpha+1},0|q,q)

Contraction_hierarchies.html

  1. < s = u 0 , u 1 , , u p = v , , v q = t > , p , q <s=u_{0},u_{1},...,u_{p}=v,...,v_{q}=t>,p,q\in\mathbb{N}
  2. u i < u i + 1 , i , i < p u_{i}<u_{i+1},i\in\mathbb{N},i<p
  3. u j > u j + 1 , j , p j < q u_{j}>u_{j+1},j\in\mathbb{N},p\leq j<q
  4. u k u_{k}
  5. u k - 1 u_{k-1}
  6. u k + 1 u_{k+1}

Contrast_set_learning.html

  1. χ 2 \chi^{2}

Converse_theorem.html

  1. L ( s ) = a n n s L(s)=\sum\frac{a_{n}}{n^{s}}
  2. L χ ( s ) = χ ( n ) a n n s L_{\chi}(s)=\sum\frac{\chi(n)a_{n}}{n^{s}}

Convexity_(finance).html

  1. E [ f ( X ) ] f ( E [ X ] ) . E[f(X)]\geq f(E[X]).
  2. Θ = - Γ , \Theta=-\Gamma,

Convexity_in_economics.html

  1. [ 0 , 1 ] [0,1]
  2. S S
  3. n \mathbb{R}^{n}
  4. S S
  5. S S
  6. S S
  7. n , \mathbb{R}^{n},
  8. x x
  9. S , S,
  10. x . x.
  11. S S
  12. S , S,
  13. S S
  14. S S

Convolution_of_probability_distributions.html

  1. i = 1 2 Bernoulli ( p ) Binomial ( 2 , p ) . \sum_{i=1}^{2}\mathrm{Bernoulli}(p)\sim\mathrm{Binomial}(2,p).
  2. X i Bernoulli ( p ) , 0 < p < 1 , 1 i 2 X_{i}\sim\mathrm{Bernoulli}(p),\quad 0<p<1,\quad 1\leq i\leq 2
  3. Y = i = 1 2 X i . Y=\sum_{i=1}^{2}X_{i}.
  4. Z Binomial ( 2 , p ) . Z\sim\mathrm{Binomial}(2,p)\,\!.
  5. X 1 and X 2 X_{1}\,\text{ and }X_{2}
  6. [ Y = n ] = [ i = 1 2 X i = n ] = m [ X 1 = m ] × [ X 2 = n - m ] = m [ ( 1 m ) p m ( 1 - p ) 1 - m ] [ ( 1 n - m ) p n - m ( 1 - p ) 1 - n + m ] = p n ( 1 - p ) 2 - n m ( 1 m ) ( 1 n - m ) = p n ( 1 - p ) 2 - n [ ( 1 0 ) ( 1 n ) + ( 1 1 ) ( 1 n - 1 ) ] = ( 2 n ) p n ( 1 - p ) 2 - n = [ Z = n ] . \begin{aligned}\displaystyle\mathbb{P}[Y=n]&\displaystyle=\mathbb{P}\left[\sum% _{i=1}^{2}X_{i}=n\right]\\ &\displaystyle=\sum_{m\in\mathbb{Z}}\mathbb{P}[X_{1}=m]\times\mathbb{P}[X_{2}=% n-m]\\ &\displaystyle=\sum_{m\in\mathbb{Z}}\left[{\left({{1}\atop{m}}\right)}p^{m}% \left(1-p\right)^{1-m}\right]\left[{\left({{1}\atop{n-m}}\right)}p^{n-m}\left(% 1-p\right)^{1-n+m}\right]\\ &\displaystyle=p^{n}\left(1-p\right)^{2-n}\sum_{m\in\mathbb{Z}}{\left({{1}% \atop{m}}\right)}{\left({{1}\atop{n-m}}\right)}\\ &\displaystyle=p^{n}\left(1-p\right)^{2-n}\left[{\left({{1}\atop{0}}\right)}{% \left({{1}\atop{n}}\right)}+{\left({{1}\atop{1}}\right)}{\left({{1}\atop{n-1}}% \right)}\right]\\ &\displaystyle={\left({{2}\atop{n}}\right)}p^{n}\left(1-p\right)^{2-n}=\mathbb% {P}[Z=n].\end{aligned}
  7. ( n k ) = 0 {\textstyle\left({{n}\atop{k}}\right)}=0
  8. X k X_{k}
  9. Z Z
  10. φ X k ( t ) = 1 - p + p e i t φ Z ( t ) = ( 1 - p + p e i t ) 2 \varphi_{X_{k}}(t)=1-p+pe^{it}\qquad\varphi_{Z}(t)=\left(1-p+pe^{it}\right)^{2}
  11. φ Y ( t ) = E ( e i t k = 1 2 X k ) = E ( k = 1 2 e i t X k ) = k = 1 2 E ( e i t X k ) = k = 1 2 ( 1 - p + p e i t ) = ( 1 - p + p e i t ) 2 = φ Z ( t ) \begin{aligned}\displaystyle\varphi_{Y}(t)&\displaystyle=\operatorname{E}\left% (e^{it\sum_{k=1}^{2}X_{k}}\right)=\operatorname{E}\left(\prod_{k=1}^{2}e^{itX_% {k}}\right)\\ &\displaystyle=\prod_{k=1}^{2}\operatorname{E}\left(e^{itX_{k}}\right)=\prod_{% k=1}^{2}\left(1-p+pe^{it}\right)\\ &\displaystyle=\left(1-p+pe^{it}\right)^{2}=\varphi_{Z}(t)\end{aligned}
  12. X k X_{k}
  13. Y Y
  14. Z Z

Cooling_capacity.html

  1. Q ˙ = m ˙ C p Δ T \dot{Q}=\dot{m}C_{p}\Delta T
  2. Q ˙ \dot{Q}
  3. m ˙ \dot{m}
  4. C p C_{p}
  5. Δ T \Delta T

Coorbit_theory.html

  1. π \pi
  2. 𝒢 \mathcal{G}
  3. \mathcal{H}
  4. f f\in\mathcal{H}
  5. g g\in\mathcal{H}
  6. V g f ( x ) = f , π ( x ) g V_{g}f(x)=\langle f,\pi(x)g\rangle
  7. V g f = V g f * V g g V_{g}f=V_{g}f*V_{g}g
  8. Y Y
  9. V g f Y V_{g}f\in Y

Coordinate_descent.html

  1. F ( 𝐱 ) F(\mathbf{x})
  2. 𝐞 1 , 𝐞 2 , , 𝐞 n \mathbf{e}_{1},\mathbf{e}_{2},\dots,\mathbf{e}_{n}
  3. 𝐱 k \mathbf{x}^{k}
  4. i i
  5. 𝐱 k + 1 \mathbf{x}^{k+1}
  6. 𝐱 i k + 1 = arg min y f ( x 1 k + 1 , , x i - 1 k + 1 , y , x i + 1 k , , x n k ) ; \mathbf{x}^{k+1}_{i}=\underset{y\in\mathbb{R}}{\operatorname{arg\,min}}\;f(x^{% k+1}_{1},...,x^{k+1}_{i-1},y,x^{k}_{i+1},...,x^{k}_{n});
  7. 𝐱 0 \mathbf{x}^{0}
  8. F F
  9. 𝐱 0 , 𝐱 1 , 𝐱 2 , \mathbf{x}^{0},\mathbf{x}^{1},\mathbf{x}^{2},\dots
  10. F ( 𝐱 0 ) F ( 𝐱 1 ) F ( 𝐱 2 ) , F(\mathbf{x}^{0})\geq F(\mathbf{x}^{1})\geq F(\mathbf{x}^{2})\geq\cdots,

Coppersmith's_Attack.html

  1. ( N , e ) (N,e)
  2. e d 1 mod ( p - 1 ) ( q - 1 ) ed\equiv 1\bmod\ (p-1)(q-1)
  3. d p d mod ( p - 1 ) d_{p}\equiv d\bmod(p-1)
  4. d q d mod ( q - 1 ) d_{q}\equiv d\bmod(q-1)
  5. C M e mod N C\equiv M^{e}\bmod N
  6. d d
  7. C d M mod N C^{d}\equiv M\bmod N
  8. e e
  9. e e
  10. ( 2 16 + 1 ) (2^{16}+1)
  11. F 0 , F 2 F_{0},F_{2}
  12. F 4 F_{4}
  13. ( F x = 2 2 x + 1 ) (F_{x}=2^{2^{x}}+1)
  14. e e
  15. gcd ( e , p - 1 ) = 1 \gcd(e,p-1)=1
  16. gcd ( e , q - 1 ) = 1 \gcd(e,q-1)=1
  17. p p
  18. q q
  19. p mod e 1 p\,\bmod\,e\neq 1
  20. gcd ( p - 1 , e ) = 1 \gcd(p-1,e)=1
  21. m m
  22. e = 2 16 + 1 e=2^{16}+1
  23. e e
  24. e e
  25. f [ x ] f\in{\mathbb{Z}}[x]
  26. d d
  27. X = N 1 d - ϵ X=N^{\frac{1}{d}-\epsilon}
  28. 1 d > ϵ > 0 \frac{1}{d}>\epsilon>0
  29. N , f \left\langle N,f\right\rangle
  30. x 0 < X x_{0}<X
  31. f ( x 0 ) = 0 mod N f(x_{0})=0\,\bmod\,N
  32. ( w ) (w)
  33. w = min ( 1 ϵ , log 2 N ) w={\rm min}(\frac{1}{\epsilon},\log_{2}N)
  34. f f
  35. N N
  36. X = N 1 d X=N^{\frac{1}{d}}
  37. X X
  38. N N
  39. M M
  40. P 1 ; P 2 ; ; P k P_{1};P_{2};\dots;P_{k}
  41. e e
  42. e = 3 e=3
  43. N i , e i \left\langle N_{i},e_{i}\right\rangle
  44. k 3 k\geq 3
  45. M M
  46. C 1 , C 2 C_{1},C_{2}
  47. C 3 C_{3}
  48. C i M 3 mod N i C_{i}\equiv M^{3}\,\bmod\,N_{i}
  49. gcd ( N i , N j ) = 1 \gcd(N_{i},N_{j})=1
  50. i , j i,j
  51. N i N_{i}
  52. gcd ( N i , N j ) \gcd(N_{i},N_{j})
  53. C N 1 N 2 N 3 * C\in\mathbb{Z}^{*}_{N_{1}N_{2}N_{3}}
  54. C i C mod N i C_{i}\equiv C\,\bmod\,N_{i}
  55. C M 3 mod N 1 N 2 N 3 C\equiv M^{3}\,\bmod\,N_{1}N_{2}N_{3}
  56. M < N i M<N_{i}
  57. i i
  58. M 3 < N 1 N 2 N 3 M^{3}<N_{1}N_{2}N_{3}
  59. C = M 3 C=M^{3}
  60. C C
  61. M M
  62. e e
  63. e e
  64. M M
  65. C i = f i ( M ) e C_{i}=f_{i}(M)^{e}
  66. 1 i k 1\leq i\leq k
  67. f i f_{i}
  68. M M
  69. M M
  70. m m
  71. M i = i 2 m + M M_{i}=i2^{m}+M
  72. M i M_{i}
  73. g 1 ( M ) = 0 g_{1}(M)=0
  74. N i N_{i}
  75. N 1 , , N k N_{1},\dots,N_{k}
  76. N min = min i { N i } N_{\rm min}={\rm min}_{i}\{N_{i}\}
  77. g i ( x ) / N i [ x ] g_{i}(x)\in\mathbb{Z}/N_{i}\left[x\right]
  78. q q
  79. M < N min M<N_{\rm min}
  80. g i ( M ) = 0 g_{i}(M)=0
  81. N i N_{i}
  82. i { 1 , , k } i\in\left\{1,\dots,k\right\}
  83. k > q k>q
  84. N i , g i ( x ) \left\langle N_{i},g_{i}\left(x\right)\right\rangle
  85. i i
  86. M M
  87. N i N_{i}
  88. T i T_{i}
  89. T i 1 mod N i ( i s 1 ) T_{i}\equiv 1\bmod N_{i}(is_{1})
  90. T i 0 mod N j T_{i}\equiv 0\bmod\ N_{j}
  91. i j i\neq j
  92. g ( x ) = i T i g i ( x ) g(x)=\sum i\cdot T_{i}\cdot g_{i}(x)
  93. g ( M ) 0 mod N i g(M)\equiv 0\bmod\ \prod N_{i}
  94. T i T_{i}
  95. g ( x ) g\left(x\right)
  96. g ( x ) g\left(x\right)
  97. q q
  98. x 0 x_{0}
  99. g ( x 0 ) 0 mod N i g(x_{0})\equiv 0\bmod\prod N_{i}
  100. | x 0 | < ( N i ) 1 q \left|x_{0}\right|<\left(\prod N_{i}\right)^{\frac{1}{q}}
  101. M < N min < ( N 1 ) 1 k < ( N 1 ) 1 q M<N_{\rm min}<(\prod N_{1})^{\frac{1}{k}}<(\prod N_{1})^{\frac{1}{q}}
  102. M M
  103. f i ( x ) f_{i}\left(x\right)
  104. N i = ( f i ( x ) ) e i - C i mod N i N_{i}=\left(f_{i}\left(x\right)\right)^{e_{i}}-C_{i}\bmod\ N_{i}
  105. g i = ( f i ( x ) ) e i - C i mod N i g_{i}=\left(f_{i}\left(x\right)\right)^{e_{i}}-C_{i}\bmod N_{i}
  106. k > max i ( e i deg f i ) k>{\rm max}_{i}(e_{i}\cdot\deg f_{i})
  107. k = O ( q 2 ) k=O(q^{2})
  108. q = max i ( e i . deg f i ) q={\rm max}_{i}(e_{i}.\deg f_{i})
  109. e = 3 e=3
  110. N N
  111. N ; e i \left\langle N;e_{i}\right\rangle
  112. M 1 ; M 2 N M_{1};M_{2}\in\mathbb{Z}_{N}
  113. M 1 f ( M 2 ) mod N M_{1}\equiv f(M_{2})\,\bmod\,N
  114. f N [ x ] f\in\mathbb{Z}_{N}[x]
  115. M 1 M_{1}
  116. M 2 M_{2}
  117. C 1 ; C 2 C_{1};C_{2}
  118. M 1 ; M 2 M_{1};M_{2}
  119. C 1 ; C 2 C_{1};C_{2}
  120. e = 3 e=3
  121. N , e \left\langle N,e\right\rangle
  122. M 1 M 2 N * M_{1}\neq M_{2}\in\mathbb{Z}^{*}_{N}
  123. M 1 f ( M 2 ) mod N M_{1}\equiv f(M_{2})\,\bmod\,N
  124. f = a x + b N [ x ] f=ax+b\in\mathbb{Z}_{N}[x]
  125. b 0 b\neq 0
  126. N , e , C 1 , C 2 , f \left\langle N,e,C_{1},C_{2},f\right\rangle
  127. M 1 , M 2 M_{1},M_{2}
  128. log 2 N \log_{2}N
  129. e e
  130. e = 3 e=3
  131. e e
  132. log 2 N \log_{2}N
  133. C 1 = M 1 e mod N C_{1}=M_{1}^{e}\,\bmod\,N
  134. M 2 M_{2}
  135. g 1 ( x ) = f ( x ) e - C 1 N [ x ] g_{1}(x)=f(x)^{e}-C_{1}\in\mathbb{Z}_{N}[x]
  136. M 2 M_{2}
  137. g 2 ( x ) = x e - C 2 N [ x ] g_{2}(x)=x^{e}-C_{2}\in\mathbb{Z}_{N}[x]
  138. x - M 2 x-M_{2}
  139. g 1 g_{1}
  140. g 2 g_{2}
  141. M 2 M_{2}
  142. e e
  143. log 2 N \log_{2}N
  144. e = 3 e=3
  145. M M
  146. M M
  147. M M
  148. M M
  149. N , e \left\langle N,e\right\rangle
  150. N N
  151. n n
  152. m = n e 2 m=\lfloor\frac{n}{e^{2}}\rfloor
  153. M N * M\in\mathbb{Z}^{*}_{N}
  154. n - m n-m
  155. M 1 = 2 m M + r 1 M_{1}=2^{m}M+r_{1}
  156. M 2 = 2 m M + r 2 M_{2}=2^{m}M+r_{2}
  157. r 1 r_{1}
  158. r 2 r_{2}
  159. 0 r 1 , r 2 < 2 m 0\leq r_{1},r_{2}<2^{m}
  160. N , e \left\langle N,e\right\rangle
  161. C 1 , C 2 C_{1},C_{2}
  162. M 1 , M 2 M_{1},M_{2}
  163. r 1 r_{1}
  164. r 2 r_{2}
  165. M M
  166. g 1 ( x , y ) = x e - C 1 g_{1}(x,y)=x^{e}-C_{1}
  167. g 2 ( x , y ) = ( x + y ) e - C 2 g_{2}(x,y)=(x+y)^{e}-C_{2}
  168. y = r 2 - r 1 y=r_{2}-r_{1}
  169. x = M 1 x=M_{1}
  170. = r 2 - r 1 \vartriangle=r_{2}-r_{1}
  171. h ( y ) = res x ( g 1 , g 2 ) N [ y ] h(y)={\rm res}_{x}(g_{1},g_{2})\in\mathbb{Z}_{N}[y]
  172. | | < 2 m < N 1 e 2 \left|\vartriangle\right|<2^{m}<N^{\frac{1}{e^{2}}}
  173. \vartriangle
  174. h h
  175. N N
  176. \vartriangle
  177. M 2 M_{2}
  178. M M

Core_recovery_parameters.html

  1. T C R = ( l sum of pieces l tot core run ) × 100 TCR=\left(\frac{l_{\mathrm{sum~{}of~{}pieces}}}{l_{\mathrm{tot~{}core~{}run}}}% \right)\times 100
  2. l sum of pieces l_{\mathrm{sum~{}of~{}pieces}}
  3. l tot core run l_{\mathrm{tot~{}core~{}run}}
  4. S C R = ( l sum of solid core pieces l tot core run ) × 100 SCR=\left(\frac{l_{\mathrm{sum~{}of~{}solid~{}core~{}pieces}}}{l_{\mathrm{tot~% {}core~{}run}}}\right)\times 100
  5. l sum of solid core pieces l_{\mathrm{sum~{}of~{}solid~{}core~{}pieces}}
  6. l tot core run l_{\mathrm{tot~{}core~{}run}}
  7. R Q D = ( l sum of 100 l tot core run ) × 100 RQD=\left(\frac{l_{\mathrm{sum~{}of~{}100}}}{l_{\mathrm{tot~{}core~{}run}}}% \right)\times 100
  8. l sum of 100 l_{\mathrm{sum~{}of~{}100}}
  9. l tot core run l_{\mathrm{tot~{}core~{}run}}

Coronal_radiative_losses.html

  1. L r = n e n l C l k h ν l k + L r e c + L b r e m s L_{r}=n_{e}\sum n_{l}C_{lk}h\nu_{lk}+L_{rec}+L_{brems}
  2. n e n_{e}
  3. n k n_{k}
  4. h h
  5. ν \nu
  6. h ν h\nu
  7. C l k C_{lk}
  8. L r e c L_{rec}
  9. L b r e m s L_{brems}
  10. n u n_{u}
  11. n l n_{l}
  12. n l n e C l u = n u A u l n_{l}n_{e}C_{lu}=n_{u}A_{ul}
  13. A l u A_{lu}
  14. L r e c L_{rec}
  15. L b r e m s L_{brems}
  16. L r = n e n H P ( T ) W m - 3 L_{r}=n_{e}n_{H}P(T)~{}~{}{W~{}m^{-3}}
  17. P ( T ) P(T)
  18. n e = n H n_{e}=n_{H}
  19. P ( T ) P(T)
  20. P ( T ) 10 - 21.85 ( 10 4.3 < T < 10 4.6 K ) P(T)\approx 10^{-21.85}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}(10^{4.3}<T<10^{4.6}K)
  21. P ( T ) 10 - 31 T 2 ( 10 4.6 < T < 10 4.9 K ) P(T)\approx 10^{-31}~{}T^{2}~{}~{}~{}~{}~{}~{}~{}~{}~{}(10^{4.6}<T<10^{4.9}K)
  22. P ( T ) 10 - 21.2 ( 10 4.9 < T < 10 5.4 K ) P(T)\approx 10^{-21.2}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}(10^{4.9}<T<10^{5.4}K)
  23. P ( T ) 10 - 10.4 T - 2 ( 10 5.4 < T < 10 5.75 K ) P(T)\approx 10^{-10.4}~{}T^{-2}~{}~{}~{}~{}~{}~{}(10^{5.4}<T<10^{5.75}K)
  24. P ( T ) 10 - 21.94 ( 10 5.75 < T < 10 6.3 K ) P(T)\approx 10^{-21.94}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}(10^{5.75}<T<10^{6.3}K)
  25. P ( T ) 10 - 17.73 T - 2 / 3 ( 10 6.3 < T < 10 7 K ) P(T)\approx 10^{-17.73}~{}T^{-2/3}~{}~{}~{}(10^{6.3}<T<10^{7}K)

Counterion_condensation.html

  1. Γ = λ B / l c h a r g e > 1 \Gamma=\lambda_{B}/l_{charge}>1
  2. λ B \lambda_{B}
  3. l c h a r g e l_{charge}
  4. λ B \lambda_{B}\approx
  5. Γ \Gamma\approx

Count–min_sketch.html

  1. i i
  2. 0 i n 0≤i≤n
  3. n n
  4. w w
  5. d d
  6. w w
  7. d d
  8. d d
  9. w w
  10. d d
  11. w = e / ε w=⌈e/ε⌉
  12. d = l n 1 / δ d=⌈ln1/δ⌉
  13. ε ε
  14. δ δ
  15. i i
  16. j j
  17. j j
  18. k k
  19. i i
  20. i i
  21. a ^ i = min j count [ j , h j ( i ) ] \hat{a}_{i}=\min_{j}\mathrm{count}[j,h_{j}(i)]
  22. count \mathrm{count}
  23. a ^ i a i + ϵ j n | a j | \hat{a}_{i}\leq a_{i}+\epsilon\sum_{j}^{n}|a_{j}|
  24. 1 - δ 1-\delta
  25. i i
  26. count a \mathrm{count}_{a}
  27. count b \mathrm{count}_{b}
  28. c c
  29. i i
  30. a ^ i = min j count [ j , h j ( i ) ] \hat{a}_{i}=\min_{j}\mathrm{count}[j,h_{j}(i)]
  31. count [ j , h j ( i ) ] max { count [ j , h j ( i ) ] , a i ^ + c } \mathrm{count}[j,h_{j}(i)]\leftarrow\max\{\mathrm{count}[j,h_{j}(i)],\hat{a_{i% }}+c\}
  32. j j

Coupled_mode_theory.html

  1. P ω ( z ) = m n P m ω ( z ) = 1 4 m n N m ω | a m ω ( z ) | 2 P^{\omega}(z)=\sum_{m}^{n}P^{\omega}_{m}(z)=\frac{1}{4}\sum_{m}^{n}N^{\omega}_% {m}\left|a^{\omega}_{m}(z)\right|^{2}\,\!
  2. N m N_{m}
  3. a m a_{m}

Covariance_operator.html

  1. , \langle\cdot,\cdot\rangle
  2. Cov ( x , y ) = H x , z y , z d 𝐏 ( z ) \mathrm{Cov}(x,y)=\int_{H}\langle x,z\rangle\langle y,z\rangle\,\mathrm{d}% \mathbf{P}(z)
  3. Cov ( x , y ) = C x , y \mathrm{Cov}(x,y)=\langle Cx,y\rangle
  4. Cov ( x , y ) = B x , z y , z d 𝐏 ( z ) \mathrm{Cov}(x,y)=\int_{B}\langle x,z\rangle\langle y,z\rangle\,\mathrm{d}% \mathbf{P}(z)
  5. x , z \langle x,z\rangle
  6. Cov ( x , y ) = z ( x ) z ( y ) d 𝐏 ( z ) = E ( z ( x ) z ( y ) ) \mathrm{Cov}(x,y)=\int z(x)z(y)\,\mathrm{d}\mathbf{P}(z)=E(z(x)z(y))
  7. u u ( x ) u\mapsto u(x)

Coxeter_complex.html

  1. ( W , S ) (W,S)
  2. M = ( m ( s , t ) ) s , t S M=(m(s,t))_{s,t\in S}
  3. ( e s ) s S (e_{s})_{s\in S}
  4. B ( e s , e t ) = - cos ( π m ( s , t ) ) B(e_{s},e_{t})=-\cos\left(\frac{\pi}{m(s,t)}\right)
  5. s ( v ) = v - 2 B ( e s , v ) B ( e s , e s ) e s s(v)=v-2\frac{B(e_{s},v)}{B(e_{s},e_{s})}e_{s}
  6. e s e_{s}
  7. e s e_{s}^{\vee}
  8. e s , v = 2 B ( e s , v ) B ( e s , e s ) , \langle e_{s}^{\vee},v\rangle=2\frac{B(e_{s},v)}{B(e_{s},e_{s})},
  9. s ( f ) = f - f , e s e s , s(f)=f-\langle f,e_{s}\rangle e_{s}^{\vee},\,
  10. s S s\in S
  11. H s = { f V * : f , e s = 0 } H_{s}=\{f\in V^{*}:\langle f,e_{s}\rangle=0\}
  12. 𝒞 = { f V * : f , e s > 0 s S } \mathcal{C}=\{f\in V^{*}:\langle f,e_{s}\rangle>0\ \forall s\in S\}
  13. H s H_{s}
  14. 𝒞 \mathcal{C}
  15. w 𝒞 w\mathcal{C}
  16. w W w\in W
  17. 𝒞 \mathcal{C}
  18. X = w W w 𝒞 ¯ X=\bigcup_{w\in W}w\overline{\mathcal{C}}
  19. 𝒞 \mathcal{C}
  20. Σ ( W , S ) \Sigma(W,S)
  21. Σ ( W , S ) = ( X { 0 } ) / + \Sigma(W,S)=(X\setminus\{0\})/\mathbb{R}^{+}
  22. D n D_{n}
  23. I 2 ( n ) \mathrm{I}_{2}(n)
  24. s , t | s 2 , t 2 , ( s t ) n \left\langle s,t\,\left|\,s^{2},t^{2},(st)^{n}\right\rangle\right.
  25. I 2 ( n ) \mathrm{I}_{2}(n)
  26. V = 2 V=\mathbb{R}^{2}
  27. I 2 ( 3 ) = A 2 \mathrm{I}_{2}(3)=\mathrm{A}_{2}
  28. D D_{\infty}
  29. x = 0 x=0
  30. x = 1 2 x={1\over 2}
  31. s , t | s 2 , t 2 \left\langle s,t\,\left|\,s^{2},t^{2}\right\rangle\right.
  32. w W J wW_{J}
  33. W J = J W_{J}=\langle J\rangle
  34. W S = W W_{S}=W
  35. W = { 1 } W_{\emptyset}=\{1\}
  36. Σ ( W , S ) \Sigma(W,S)
  37. { 1 , 2 , , n } \{1,2,\ldots,n\}
  38. ( W , S ) (W,S)
  39. | S | - 1 |S|-1
  40. ( | S | - 1 ) (|S|-1)

Coxeter_notation.html

  1. I ~ 1 {\tilde{I}}_{1}
  2. A ~ 2 {\tilde{A}}_{2}
  3. C ~ 2 {\tilde{C}}_{2}
  4. G ~ 2 {\tilde{G}}_{2}
  5. A ~ 3 {\tilde{A}}_{3}
  6. B ~ 3 {\tilde{B}}_{3}
  7. C ~ 3 {\tilde{C}}_{3}
  8. A ~ 4 {\tilde{A}}_{4}
  9. B ~ 4 {\tilde{B}}_{4}
  10. C ~ 4 {\tilde{C}}_{4}
  11. D ~ 4 {\tilde{D}}_{4}
  12. F ~ 4 {\tilde{F}}_{4}
  13. A ~ n {\tilde{A}}_{n}
  14. B ~ n {\tilde{B}}_{n}
  15. C ~ n {\tilde{C}}_{n}
  16. D ~ n {\tilde{D}}_{n}
  17. E ~ 6 {\tilde{E}}_{6}
  18. E ~ 7 {\tilde{E}}_{7}
  19. E ~ 8 {\tilde{E}}_{8}
  20. B H ¯ 3 {\bar{BH}}_{3}
  21. K ¯ 3 {\bar{K}}_{3}
  22. J ¯ 3 {\bar{J}}_{3}
  23. D H ¯ 3 {\bar{DH}}_{3}
  24. A B ^ 3 {\widehat{AB}}_{3}
  25. A H ^ 3 {\widehat{AH}}_{3}
  26. B B ^ 3 {\widehat{BB}}_{3}
  27. B H ^ 3 {\widehat{BH}}_{3}
  28. H H ^ 3 {\widehat{HH}}_{3}
  29. H ¯ 4 {\bar{H}}_{4}
  30. B H ¯ 4 {\bar{BH}}_{4}
  31. K ¯ 4 {\bar{K}}_{4}
  32. D H ¯ 4 {\bar{DH}}_{4}
  33. A F ^ 4 {\widehat{AF}}_{4}
  34. A ~ 3 {\tilde{A}}_{3}
  35. A ~ n {\tilde{A}}_{n}
  36. D 3 D_{3}
  37. E ~ 6 {\tilde{E}}_{6}
  38. D ~ 4 {\tilde{D}}_{4}
  39. A ~ n {\tilde{A}}_{n}
  40. D 3 D_{3}
  41. E ~ 6 {\tilde{E}}_{6}
  42. D ~ 4 {\tilde{D}}_{4}
  43. L ¯ 5 {\bar{L}}_{5}
  44. 3 ¯ \overline{3}
  45. 3 ¯ \overline{3}
  46. [ 1 0 0 - 1 ] \left[\begin{smallmatrix}1&0\\ 0&-1\\ \end{smallmatrix}\right]
  47. [ cos 2 π / p sin 2 π / p sin 2 π / p - cos 2 π / p ] \left[\begin{smallmatrix}\cos 2\pi/p&\sin 2\pi/p\\ \sin 2\pi/p&-\cos 2\pi/p\\ \end{smallmatrix}\right]
  48. [ cos 2 π / p sin 2 π / p - sin 2 π / p cos 2 π / p ] \left[\begin{smallmatrix}\cos 2\pi/p&\sin 2\pi/p\\ -\sin 2\pi/p&\cos 2\pi/p\\ \end{smallmatrix}\right]
  49. [ 1 0 0 0 - 1 0 0 0 1 ] \left[\begin{smallmatrix}1&0&0\\ 0&-1&0\\ 0&0&1\\ \end{smallmatrix}\right]
  50. [ 0 1 0 1 0 0 0 0 1 ] \left[\begin{smallmatrix}0&1&0\\ 1&0&0\\ 0&0&1\\ \end{smallmatrix}\right]
  51. [ - 1 0 2 0 1 0 0 0 1 ] \left[\begin{smallmatrix}-1&0&2\\ 0&1&0\\ 0&0&1\\ \end{smallmatrix}\right]
  52. [ 0 1 0 - 1 0 0 0 0 1 ] \left[\begin{smallmatrix}0&1&0\\ -1&0&0\\ 0&0&1\\ \end{smallmatrix}\right]
  53. [ 0 1 0 - 1 0 2 0 0 1 ] \left[\begin{smallmatrix}0&1&0\\ -1&0&2\\ 0&0&1\\ \end{smallmatrix}\right]
  54. [ - 1 0 2 0 - 1 0 0 0 1 ] \left[\begin{smallmatrix}-1&0&2\\ 0&-1&0\\ 0&0&1\\ \end{smallmatrix}\right]
  55. [ 0 1 0 1 0 - 2 0 0 1 ] \left[\begin{smallmatrix}0&1&0\\ 1&0&-2\\ 0&0&1\\ \end{smallmatrix}\right]
  56. 1 ¯ \overline{1}
  57. 2 ¯ \overline{2}
  58. 2 ¯ \overline{2}
  59. 3 ¯ \overline{3}
  60. 4 ¯ \overline{4}
  61. 5 ¯ \overline{5}
  62. 6 ¯ \overline{6}
  63. n ¯ \overline{n}
  64. 2 ¯ \overline{2}
  65. 3 ¯ \overline{3}
  66. 4 ¯ \overline{4}
  67. 5 ¯ \overline{5}
  68. 6 ¯ \overline{6}
  69. n ¯ \overline{n}
  70. 4 ¯ \overline{4}
  71. 3 ¯ \overline{3}
  72. 8 ¯ \overline{8}
  73. 5 ¯ \overline{5}
  74. 12 ¯ \overline{12}
  75. 2 n ¯ \overline{2n}
  76. n ¯ \overline{n}
  77. 22 ¯ \overline{22}
  78. 1 ¯ \overline{1}
  79. 42 ¯ \overline{42}
  80. 62 ¯ \overline{62}
  81. 82 ¯ \overline{82}
  82. 102 ¯ \overline{102}
  83. 122 ¯ \overline{122}
  84. 2 n 2 ¯ \overline{2n2}
  85. 2 ¯ \overline{2}
  86. 2 ¯ \overline{2}
  87. 3 ¯ \overline{3}
  88. 2 ¯ \overline{2}
  89. 4 ¯ \overline{4}
  90. 2 ¯ \overline{2}
  91. 5 ¯ \overline{5}
  92. 2 ¯ \overline{2}
  93. 6 ¯ \overline{6}
  94. 2 ¯ \overline{2}
  95. n ¯ \overline{n}
  96. 2 ¯ \overline{2}
  97. 6 ¯ \overline{6}
  98. 10 ¯ \overline{10}
  99. 2 n ¯ \overline{2n}
  100. 2 ¯ \overline{2}
  101. 2 ¯ \overline{2}
  102. 2 ¯ \overline{2}
  103. 2 ¯ \overline{2}
  104. 2 ¯ \overline{2}
  105. 2 ¯ \overline{2}
  106. 3 ¯ \overline{3}
  107. 3 ¯ \overline{3}
  108. 3 ¯ \overline{3}
  109. 3 ¯ \overline{3}
  110. 4 ¯ \overline{4}
  111. 4 ¯ \overline{4}
  112. 3 ¯ \overline{3}
  113. 3 ¯ \overline{3}
  114. 5 ¯ \overline{5}
  115. 3 ¯ \overline{3}
  116. 5 ¯ \overline{5}
  117. 3 ¯ \overline{3}
  118. 1 ¯ \overline{1}
  119. 2 ¯ \overline{2}
  120. 2 ¯ \overline{2}
  121. 4 ¯ \overline{4}
  122. 3 ¯ \overline{3}
  123. 6 ¯ \overline{6}
  124. 2 ¯ n \overline{2}{n}
  125. n ¯ \overline{n}
  126. 2 ¯ n \overline{2}{n}
  127. n ¯ \overline{n}
  128. 2 ¯ n \overline{2}{n}
  129. 2 ¯ n \overline{2}{n}
  130. 2 ¯ n \overline{2}{n}
  131. n ¯ \overline{n}
  132. 2 ¯ n \overline{2}{n}
  133. n ¯ \overline{n}
  134. 2 ¯ n \overline{2}{n}
  135. n ¯ \overline{n}
  136. 2 ¯ n \overline{2}{n}
  137. 2 ¯ n \overline{2}{n}
  138. 2 ¯ n \overline{2}{n}
  139. 1 ¯ \overline{1}
  140. 2 ¯ \overline{2}

Cramér's_V.html

  1. A A
  2. B B
  3. i = 1 , , r ; j = 1 , , k i=1,\ldots,r;j=1,\ldots,k
  4. n i j = n_{ij}=
  5. ( A i , B j ) (A_{i},B_{j})
  6. χ 2 = i , j ( n i j - n i . n . j n ) 2 n i . n . j n \chi^{2}=\sum_{i,j}\frac{(n_{ij}-\frac{n_{i.}n_{.j}}{n})^{2}}{\frac{n_{i.}n_{.% j}}{n}}
  7. V = φ 2 min ( k - 1 , r - 1 ) = χ 2 / n min ( k - 1 , r - 1 ) V=\sqrt{\frac{\varphi^{2}}{\min(k-1,r-1)}}=\sqrt{\frac{\chi^{2}/n}{\min(k-1,r-% 1)}}
  8. φ 2 \varphi^{2}
  9. χ 2 \chi^{2}
  10. n n
  11. k k
  12. r r

Credal_set.html

  1. X X
  2. P ( X ) P(X)
  3. X X
  4. K ( X ) K(X)
  5. X X
  6. K ( X ) K(X)
  7. ext [ K ( X ) ] \mathrm{ext}[K(X)]
  8. f f
  9. X X
  10. K ( X ) K(X)
  11. E ¯ [ f ] = min P ( X ) K ( X ) x f ( x ) P ( x ) . \underline{E}[f]=\min_{P(X)\in K(X)}\sum_{x}f(x)P(x).

Crocco's_theorem.html

  1. D 𝐮 D t = T s - h \frac{D\mathbf{u}}{Dt}=T\nabla\,s-\nabla\,h
  2. 𝐮 × s y m b o l ω = v p 0 \mathbf{u}\times symbol\omega=v\nabla p_{0}
  3. T d s d n = d h 0 d n + u ω T\frac{ds}{dn}=\frac{dh_{0}}{dn}+u\omega
  4. 𝐮 t + ( u 2 2 + h ) = u × s y m b o l ω + T s + 𝐠 , \frac{\partial\mathbf{u}}{\partial t}+\nabla\left(\frac{u^{2}}{2}+h\right)=u% \times symbol\omega+T\nabla s+\mathbf{g},
  5. 𝐮 \mathbf{u}
  6. ω \omega
  7. v v
  8. p 0 p_{0}
  9. T T
  10. s s
  11. h h
  12. 𝐠 \mathbf{g}
  13. n n
  14. rot 𝐮 × 𝐮 = T grad S \scriptstyle\mathrm{rot}\,\mathbf{u}\times\mathbf{u}=T\mathrm{grad}\,S

Cross-spectrum.html

  1. ( X t , Y t ) (X_{t},Y_{t})
  2. γ x x \gamma_{xx}
  3. γ y y \gamma_{yy}
  4. γ x y \gamma_{xy}
  5. Γ x y \Gamma_{xy}
  6. γ x y \gamma_{xy}
  7. Γ x y ( f ) = { γ x y } ( f ) = τ = - γ x y ( τ ) e - 2 π i τ f , \Gamma_{xy}(f)=\mathcal{F}\{\gamma_{xy}\}(f)=\sum_{\tau=-\infty}^{\infty}\,% \gamma_{xy}(\tau)\,e^{-2\,\pi\,i\,\tau\,f},
  8. γ x y ( τ ) = E [ ( X t - μ x ) ( Y t + τ - μ y ) ] \gamma_{xy}(\tau)=\operatorname{E}[(X_{t}-\mu_{x})(Y_{t+\tau}-\mu_{y})]
  9. Γ x y ( f ) = Λ x y ( f ) + i Ψ x y ( f ) , \Gamma_{xy}(f)=\Lambda_{xy}(f)+i\Psi_{xy}(f),
  10. Γ x y ( f ) = A x y ( f ) e i ϕ x y ( f ) . \Gamma_{xy}(f)=A_{xy}(f)\,e^{i\phi_{xy}(f)}.
  11. A x y A_{xy}
  12. A x y ( f ) = ( Λ x y ( f ) 2 + Ψ x y ( f ) 2 ) 1 2 , A_{xy}(f)=(\Lambda_{xy}(f)^{2}+\Psi_{xy}(f)^{2})^{\frac{1}{2}},
  13. Φ x y \Phi_{xy}
  14. { tan - 1 ( Ψ x y ( f ) / Λ x y ( f ) ) if Ψ x y ( f ) 0 Λ x y ( f ) 0 0 if Ψ x y ( f ) = 0 and Λ x y ( f ) > 0 ± π if Ψ x y ( f ) = 0 and Λ x y ( f ) < 0 π / 2 if Ψ x y ( f ) > 0 and Λ x y ( f ) = 0 - π / 2 if Ψ x y ( f ) < 0 and Λ x y ( f ) = 0 \begin{cases}\tan^{-1}(\Psi_{xy}(f)/\Lambda_{xy}(f))&\,\text{if }\Psi_{xy}(f)% \neq 0\wedge\Lambda_{xy}(f)\neq 0\\ 0&\,\text{if }\Psi_{xy}(f)=0\,\text{ and }\Lambda_{xy}(f)>0\\ \pm\pi&\,\text{if }\Psi_{xy}(f)=0\,\text{ and }\Lambda_{xy}(f)<0\\ \pi/2&\,\text{if }\Psi_{xy}(f)>0\,\text{ and }\Lambda_{xy}(f)=0\\ -\pi/2&\,\text{if }\Psi_{xy}(f)<0\,\text{ and }\Lambda_{xy}(f)=0\\ \end{cases}
  15. κ x y ( f ) = A x y 2 Γ x x ( f ) Γ y y ( f ) , \kappa_{xy}(f)=\frac{A_{xy}^{2}}{\Gamma_{xx}(f)\Gamma_{yy}(f)},

Cryptanalysis_of_the_Lorenz_cipher.html

  1. ψ \psi
  2. ψ \psi
  3. ψ \psi
  4. ψ \psi
  5. ψ \psi
  6. μ \mu
  7. μ \mu
  8. χ \chi
  9. χ \chi
  10. χ \chi
  11. χ \chi
  12. χ \chi
  13. χ \chi
  14. ψ \psi
  15. μ \mu
  16. μ \mu
  17. μ \mu
  18. χ \chi
  19. ψ \psi
  20. ψ \psi^{\prime}
  21. χ \chi
  22. ψ \psi
  23. χ \chi
  24. ψ \psi
  25. ψ \psi
  26. χ \chi
  27. ψ \psi
  28. ψ \psi
  29. ψ \psi
  30. μ \mu
  31. χ \chi
  32. ψ \psi
  33. χ \chi
  34. ψ \psi
  35. χ \chi
  36. ψ \psi
  37. χ \chi
  38. ψ \psi
  39. χ \chi
  40. ψ \psi
  41. ψ \psi
  42. χ \chi
  43. χ \chi
  44. χ \chi
  45. ψ \psi
  46. χ \chi
  47. χ \chi
  48. χ \chi
  49. χ \chi
  50. χ \chi
  51. ψ \psi
  52. χ \chi
  53. ψ \psi
  54. χ \chi
  55. ψ \psi
  56. χ \chi
  57. χ \chi
  58. ψ \psi
  59. ψ \psi
  60. χ \chi
  61. χ \chi
  62. ψ \psi
  63. ψ \psi
  64. χ \chi
  65. χ \chi
  66. ψ \psi
  67. ψ \psi
  68. χ \chi
  69. χ \chi
  70. χ \chi
  71. χ \chi
  72. χ \chi
  73. χ \chi
  74. χ \chi
  75. χ \chi
  76. χ \chi
  77. χ \chi
  78. χ \chi
  79. χ \chi
  80. χ \chi
  81. χ \chi
  82. χ \chi
  83. χ \chi
  84. χ \chi
  85. χ \chi
  86. χ \chi
  87. χ \chi
  88. χ \chi
  89. χ \chi
  90. χ \chi
  91. χ \chi
  92. χ \chi
  93. χ \chi
  94. χ \chi
  95. χ \chi
  96. χ \chi
  97. χ \chi
  98. χ \chi
  99. χ \chi
  100. χ \chi

CSS_code.html

  1. C 1 C_{1}
  2. C 2 C_{2}
  3. [ n , k 1 ] [n,k_{1}]
  4. [ n , k 2 ] [n,k_{2}]
  5. C 2 C 1 C_{2}\subset C_{1}
  6. C 1 , C 2 C_{1},C_{2}^{\perp}
  7. 2 t + 1 \geq 2t+1
  8. C 2 C_{2}^{\perp}
  9. C 2 C_{2}
  10. CSS ( C 1 , C 2 ) \,\text{CSS}(C_{1},C_{2})
  11. C 1 C_{1}
  12. C 2 C_{2}
  13. [ n , k 1 - k 2 , d ] [n,k_{1}-k_{2},d]
  14. d 2 t + 1 d\geq 2t+1
  15. x C 1 : < m t p l > | x + C 2 := x\in C_{1}:<mtpl>{{|}}x+C_{2}\rangle:=
  16. 1 / < m t p l > | C 2 | 1/\sqrt{<mtpl>{{|}}C_{2}{{|}}}
  17. y C 2 < m t p l > | x + y \sum_{y\in C_{2}}<mtpl>{{|}}x+y\rangle
  18. + +
  19. CSS ( C 1 , C 2 ) \,\text{CSS}(C_{1},C_{2})
  20. { < m t p l > | x + C 2 x C 1 } \{<mtpl>{{|}}x+C_{2}\rangle\mid x\in C_{1}\}

CT_Value.html

  1. ln ( N N 0 ) = Λ C W C n t \ln(\frac{N}{N_{0}})=\Lambda_{CW}C^{n}t\!
  2. ( N N 0 ) (\frac{N}{N_{0}})\!
  3. Λ C W \Lambda_{CW}\!
  4. C C\!
  5. n n\!
  6. t t\!

Cubic_harmonic.html

  1. l ( l + 1 ) l(l+1)
  2. n n
  3. l l
  4. ψ n l m ( r ) = R n l ( r ) Y l m ( θ , φ ) \psi_{nlm}({r})=R_{nl}(r)Y_{l}^{m}(\theta,\varphi)
  5. R n l ( r ) R_{nl}(r)
  6. Y l m ( θ , φ ) Y_{l}^{m}(\theta,\varphi)
  7. Y l m ( θ , φ ) Y_{l}^{m}(\theta,\varphi)
  8. ψ n l c ( r ) = R n l ( r ) X l c ( r ) \psi_{nlc}({r})=R_{nl}(r)X_{lc}({r})
  9. X l c ( r ) X_{lc}({r})
  10. l l
  11. l ( l + 1 ) l(l+1)
  12. l ( l + 1 ) l(l+1)
  13. X l c ( r ) = Y l 0 X_{lc}({r})=Y_{l}^{0}
  14. X l c ( r ) = 1 i n c 2 ( Y l m - Y l - m ) X_{lc^{\prime}}({r})=\frac{1}{i^{n_{c^{\prime}}}\sqrt{2}}\left(Y_{l}^{m}-Y_{l}% ^{-m}\right)
  15. X l c ′′ ( r ) = 1 i n c ′′ 2 ( Y l m + Y l - m ) X_{lc^{\prime\prime}}({r})=\frac{1}{i^{n_{c^{\prime\prime}}}\sqrt{2}}\left(Y_{% l}^{m}+Y_{l}^{-m}\right)
  16. ψ n 00 ( r ) = R n 0 ( r ) Y 0 0 \psi_{n00}({r})=R_{n0}(r)Y_{0}^{0}
  17. s = X 00 = Y 0 0 = 1 4 π s=X_{00}=Y_{0}^{0}=\frac{1}{\sqrt{4\pi}}
  18. p z = N 1 c z r = Y 1 0 p_{z}=N_{1}^{c}\frac{z}{r}=Y_{1}^{0}
  19. p x = N 1 c x r = 1 2 ( Y 1 - 1 - Y 1 1 ) p_{x}=N_{1}^{c}\frac{x}{r}=\frac{1}{\sqrt{2}}\left(Y_{1}^{-1}-Y_{1}^{1}\right)
  20. p y = N 1 c y r = i 2 ( Y 1 - 1 + Y 1 1 ) p_{y}=N_{1}^{c}\frac{y}{r}=\frac{i}{\sqrt{2}}\left(Y_{1}^{-1}+Y_{1}^{1}\right)
  21. N 1 c = ( 3 4 π ) 1 / 2 N_{1}^{c}=\left(\frac{3}{4\pi}\right)^{1/2}
  22. ψ n 2 c ( r ) = R n 2 ( r ) X 2 c ( r ) \psi_{n2c}({r})=R_{n2}(r)X_{2c}({r})
  23. X 2 c ( r ) X_{2c}({r})
  24. d z 2 = N 2 c 3 z 2 - r 2 2 r 2 3 = Y 2 0 d_{z^{2}}=N_{2}^{c}\frac{3z^{2}-r^{2}}{2r^{2}\sqrt{3}}=Y_{2}^{0}
  25. d x z = N 2 c x z r 2 = 1 2 ( Y 2 - 1 - Y 2 1 ) d_{xz}=N_{2}^{c}\frac{xz}{r^{2}}=\frac{1}{\sqrt{2}}\left(Y_{2}^{-1}-Y_{2}^{1}\right)
  26. d y z = N 2 c y z r 2 = i 2 ( Y 2 - 1 + Y 2 1 ) d_{yz}=N_{2}^{c}\frac{yz}{r^{2}}=\frac{i}{\sqrt{2}}\left(Y_{2}^{-1}+Y_{2}^{1}\right)
  27. d x y = N 2 c x y r 2 = i 2 ( Y 2 - 2 - Y 2 2 ) d_{xy}=N_{2}^{c}\frac{xy}{r^{2}}=\frac{i}{\sqrt{2}}\left(Y_{2}^{-2}-Y_{2}^{2}\right)
  28. d x 2 - y 2 = N 2 c x 2 - y 2 2 r 2 = 1 2 ( Y 2 - 2 + Y 2 2 ) d_{x^{2}-y^{2}}=N_{2}^{c}\frac{x^{2}-y^{2}}{2r^{2}}=\frac{1}{\sqrt{2}}\left(Y_% {2}^{-2}+Y_{2}^{2}\right)
  29. N 2 c = ( 15 4 π ) 1 / 2 N_{2}^{c}=\left(\frac{15}{4\pi}\right)^{1/2}
  30. ψ n 3 c ( r ) = R n 3 ( r ) X 3 c ( r ) \psi_{n3c}({r})=R_{n3}(r)X_{3c}({r})
  31. X 3 c ( r ) X_{3c}({r})
  32. f z 3 = N 3 c z ( 2 z 2 - 3 x 2 - 3 y 2 ) 2 r 3 15 = Y 3 0 f_{z^{3}}=N_{3}^{c}\frac{z(2z^{2}-3x^{2}-3y^{2})}{2r^{3}\sqrt{15}}=Y_{3}^{0}
  33. f x z 2 = N 3 c x ( 4 z 2 - x 2 - y 2 ) 2 r 3 5 = 1 2 ( Y 3 - 1 - Y 3 1 ) f_{xz^{2}}=N_{3}^{c}\frac{x(4z^{2}-x^{2}-y^{2})}{2r^{3}\sqrt{5}}=\frac{1}{% \sqrt{2}}\left(Y_{3}^{-1}-Y_{3}^{1}\right)
  34. f y z 2 = N 3 c y ( 4 z 2 - x 2 - y 2 ) r 3 5 = i 2 ( Y 3 - 1 + Y 3 1 ) f_{yz^{2}}=N_{3}^{c}\frac{y(4z^{2}-x^{2}-y^{2})}{r^{3}\sqrt{5}}=\frac{i}{\sqrt% {2}}\left(Y_{3}^{-1}+Y_{3}^{1}\right)
  35. f x y z = N 3 c x y z r 3 = i 2 ( Y 3 - 2 - Y 3 2 ) f_{xyz}=N_{3}^{c}\frac{xyz}{r^{3}}=\frac{i}{\sqrt{2}}\left(Y_{3}^{-2}-Y_{3}^{2% }\right)
  36. f z ( x 2 - y 2 ) = N 3 c z ( x 2 - y 2 ) 2 r 3 = 1 2 ( Y 3 - 2 + Y 3 2 ) f_{z(x^{2}-y^{2})}=N_{3}^{c}\frac{z\left(x^{2}-y^{2}\right)}{2r^{3}}=\frac{1}{% \sqrt{2}}\left(Y_{3}^{-2}+Y_{3}^{2}\right)
  37. f x ( x 2 - 3 y 2 ) = N 3 c x ( x 2 - 3 y 2 ) 2 r 3 3 = 1 2 ( Y 3 - 3 - Y 3 3 ) f_{x(x^{2}-3y^{2})}=N_{3}^{c}\frac{x\left(x^{2}-3y^{2}\right)}{2r^{3}\sqrt{3}}% =\frac{1}{\sqrt{2}}\left(Y_{3}^{-3}-Y_{3}^{3}\right)
  38. f y ( 3 x 2 - y 2 ) = N 3 c y ( 3 x 2 - y 2 ) 2 r 3 3 = i 2 ( Y 3 - 3 + Y 3 3 ) f_{y(3x^{2}-y^{2})}=N_{3}^{c}\frac{y\left(3x^{2}-y^{2}\right)}{2r^{3}\sqrt{3}}% =\frac{i}{\sqrt{2}}\left(Y_{3}^{-3}+Y_{3}^{3}\right)
  39. N 3 c = ( 105 4 π ) 1 / 2 N_{3}^{c}=\left(\frac{105}{4\pi}\right)^{1/2}

Curie's_law.html

  1. 𝐌 = C 𝐁 T , \mathbf{M}=C\cdot\frac{\mathbf{B}}{T},
  2. 𝐌 \mathbf{M}
  3. 𝐁 \mathbf{B}
  4. T T
  5. C C
  6. μ \vec{\mu}
  7. E = - μ B . E=-\vec{\mu}\cdot\vec{B}.
  8. μ \mu
  9. - μ -\mu
  10. E 0 = - μ B E_{0}=-\mu B
  11. E 1 = μ B . E_{1}=\mu B.
  12. μ \mu
  13. μ = μ P ( μ ) + ( - μ ) P ( - μ ) = 1 Z ( μ e μ B β - μ e - μ B β ) = 2 μ Z sinh ( μ B β ) , \left\langle\mu\right\rangle=\mu P\left(\mu\right)+(-\mu)P\left(-\mu\right)={1% \over Z}\left(\mu e^{\mu B\beta}-\mu e^{-\mu B\beta}\right)={2\mu\over Z}\sinh% (\mu B\beta),
  14. Z Z
  15. Z = n = 0 , 1 e - E n β = e μ B β + e - μ B β = 2 cosh ( μ B β ) . Z=\sum_{n=0,1}e^{-E_{n}\beta}=e^{\mu B\beta}+e^{-\mu B\beta}=2\cosh\left(\mu B% \beta\right).
  16. μ = μ tanh ( μ B β ) . \left\langle\mu\right\rangle=\mu\tanh\left(\mu B\beta\right).
  17. M = N μ = N μ tanh ( μ B k T ) M=N\left\langle\mu\right\rangle=N\mu\tanh\left({\mu B\over kT}\right)
  18. T T
  19. B B
  20. ( μ B k T ) 1 \left({\mu B\over kT}\right)\ll 1
  21. | x | 1 |x|\ll 1
  22. tanh x x \tanh x\approx x
  23. 𝐌 ( T ) = N μ 2 k 𝐁 T , \mathbf{M}(T\rightarrow\infty)={N\mu^{2}\over k}{\mathbf{B}\over T},
  24. C = N μ 2 / k C=N\mu^{2}/k
  25. M M
  26. N μ N\mu
  27. C = μ B 2 3 k B N g 2 J ( J + 1 ) C=\frac{\mu_{B}^{2}}{3k_{B}}Ng^{2}J(J+1)
  28. J J
  29. g g
  30. μ = g J μ B \mu=gJ\mu_{B}
  31. E = - μ B cos θ , E=-\mu B\cos\theta,
  32. θ \theta
  33. z z
  34. Z = 0 2 π d ϕ 0 π d θ sin θ exp ( μ B β cos θ ) . Z=\int_{0}^{2\pi}d\phi\int_{0}^{\pi}d\theta\sin\theta\exp(\mu B\beta\cos\theta).
  35. ϕ \phi
  36. y = cos θ y=\cos\theta
  37. Z = 2 π - 1 1 d y exp ( μ B β y ) = 2 π exp ( μ B β ) - exp ( - μ B β ) μ B β = 4 π sinh ( μ B β ) μ B β . Z=2\pi\int_{-1}^{1}dy\exp(\mu B\beta y)=2\pi{\exp(\mu B\beta)-\exp(-\mu B\beta% )\over\mu B\beta}={4\pi\sinh(\mu B\beta)\over\mu B\beta.}
  38. z z
  39. ϕ \phi
  40. μ z = 1 Z 0 2 π d ϕ 0 π d θ sin θ exp ( μ B β cos θ ) [ μ cos θ ] . \left\langle\mu_{z}\right\rangle={1\over Z}\int_{0}^{2\pi}d\phi\int_{0}^{\pi}d% \theta\sin\theta\exp(\mu B\beta\cos\theta)\left[\mu\cos\theta\right].
  41. Z Z
  42. μ z = 1 Z B β Z . \left\langle\mu_{z}\right\rangle={1\over ZB}\partial_{\beta}Z.
  43. μ z = μ L ( μ B β ) , \left\langle\mu_{z}\right\rangle=\mu L(\mu B\beta),
  44. L L
  45. L ( x ) = coth x - 1 x . L(x)=\coth x-{1\over x}.
  46. x x
  47. L ( x ) x / 3 L(x)\approx x/3
  48. 1 1

Cyanase.html

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

Cycles_of_Time.html

  1. \rightarrow\infty

Cyclohedron.html

  1. S 1 × W n S^{1}\times W_{n}

Cyclotomic_field.html

  1. 𝐐 \mathbf{Q}
  2. n n
  3. n > 2 n>2
  4. n n
  5. n n
  6. Φ n ( x ) = gcd ( k , n ) = 1 1 k n ( x - e 2 i π k n ) \Phi_{n}(x)=\prod_{\stackrel{1\leq k\leq n}{\gcd(k,n)=1}}\left(x-e^{2i\pi\frac% {k}{n}}\right)
  7. φ ( n ) φ(n)
  8. φ φ
  9. a a
  10. n n
  11. a a
  12. n n
  13. n n
  14. n n
  15. n n
  16. p p
  17. p p
  18. p p
  19. φ ( p ) = 2 k \varphi(p)=2^{k}
  20. n = 3 n=3
  21. n = 6 n=6
  22. ζ < s u b > 3 = ζ<sub>3=

Cyclotruncated_5-simplex_honeycomb.html

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

Cyclotruncated_simplectic_honeycomb.html

  1. A ~ n {\tilde{A}}_{n}
  2. A ~ n {\tilde{A}}_{n}
  3. A ~ 1 {\tilde{A}}_{1}
  4. A ~ 2 {\tilde{A}}_{2}
  5. A ~ 3 {\tilde{A}}_{3}
  6. A ~ 4 {\tilde{A}}_{4}
  7. A ~ 5 {\tilde{A}}_{5}
  8. A ~ 6 {\tilde{A}}_{6}
  9. A ~ 7 {\tilde{A}}_{7}
  10. A ~ 8 {\tilde{A}}_{8}

Cyrus–Beck_algorithm.html

  1. p ( t ) \displaystyle p(t)
  2. 0 t 1 0\leq t\leq 1
  3. n ( p ( t ) - p E ) n\cdot(p(t)-p_{E})

Dagum_distribution.html

  1. ( 1 + ( x b ) - a ) - p {\left(1+{\left(\frac{x}{b}\right)}^{-a}\right)}^{-p}
  2. { - b a Γ ( - 1 a ) Γ ( 1 a + p ) Γ ( p ) if a > 1 Indeterminate otherwise \begin{cases}-\frac{b}{a}\frac{\Gamma\left(-\tfrac{1}{a}\right)\Gamma\left(% \tfrac{1}{a}+p\right)}{\Gamma(p)}&\,\text{if}\ a>1\\ \,\text{Indeterminate}&\,\text{otherwise}\end{cases}
  3. b ( - 1 + 2 1 p ) - 1 a b{\left(-1+2^{\tfrac{1}{p}}\right)}^{-\tfrac{1}{a}}
  4. b ( a p - 1 a + 1 ) 1 a b{\left(\frac{ap-1}{a+1}\right)}^{\tfrac{1}{a}}
  5. { - b 2 a 2 ( 2 a Γ ( - 2 a ) Γ ( 2 a + p ) Γ ( p ) + ( Γ ( - 1 a ) Γ ( 1 a + p ) Γ ( p ) ) 2 ) if a > 2 Indeterminate otherwise \begin{cases}-\frac{b^{2}}{a^{2}}\left(2a\frac{\Gamma\left(-\tfrac{2}{a}\right% )\,\Gamma\left(\tfrac{2}{a}+p\right)}{\Gamma\left(p\right)}+\left(\frac{\Gamma% \left(-\tfrac{1}{a}\right)\Gamma\left(\tfrac{1}{a}+p\right)}{\Gamma\left(p% \right)}\right)^{2}\right)&\,\text{if}\ a>2\\ \,\text{Indeterminate}&\,\text{otherwise}\end{cases}
  6. F ( x ; a , b , p ) = ( 1 + ( x b ) - a ) - p F(x;a,b,p)={\left(1+{\left(\frac{x}{b}\right)}^{-a}\right)}^{-p}
  7. x > 0 x>0
  8. a , b , p > 0. a,b,p>0.
  9. f ( x ; a , b , p ) = a p x ( ( x b ) a p ( ( x b ) a + 1 ) p + 1 ) . f(x;a,b,p)=\frac{ap}{x}\left(\frac{(\tfrac{x}{b})^{ap}}{\left((\tfrac{x}{b})^{% a}+1\right)^{p+1}}\right).
  10. X D ( a , b , p ) 1 X S M ( a , 1 b , p ) X\sim D(a,b,p)\iff\frac{1}{X}\sim SM(a,\tfrac{1}{b},p)
  11. F ( x ; a , b , p , δ ) = δ + ( 1 - δ ) ( 1 + ( x b ) - a ) - p . F(x;a,b,p,\delta)=\delta+(1-\delta){\left(1+{\left(\frac{x}{b}\right)}^{-a}% \right)}^{-p}.

DAHP_synthase.html

  1. \rightleftharpoons

Damped_sine_wave.html

  1. y ( t ) = e - t cos ( 2 π t ) y(t)=e^{-t}\cdot\cos(2\pi t)
  2. y ( t ) = A e - λ t ( cos ( ω t + ϕ ) + sin ( ω t + ϕ ) ) y(t)=A\cdot e^{-\lambda t}\cdot(\cos(\omega t+\phi)+\sin(\omega t+\phi))
  3. y ( t ) y(t)
  4. A A
  5. λ \lambda
  6. ϕ \phi
  7. ω \omega
  8. y ( t ) = A e - λ t ( cos ( ω t + ϕ ) ) y(t)=A\cdot e^{-\lambda t}\cdot(\cos(\omega t+\phi))
  9. ϕ \phi
  10. τ \tau
  11. f - 1 f^{-1}
  12. f f
  13. ω / ( 2 π ) \omega/(2\pi)
  14. τ - 1 \tau^{-1}
  15. t - 1 t^{-1}
  16. ln ( 2 ) / λ \ln(2)/\lambda
  17. 0.693 / λ 0.693/\lambda

Damping_matrix.html

  1. f D i = c i 1 u 1 ˙ + c i 2 u 2 ˙ + + c i n u n ˙ = j = 1 n c i , j u j ˙ f_{Di}=c_{i1}\dot{u_{1}}+c_{i2}\dot{u_{2}}+\cdots+c_{in}\dot{u_{n}}=\sum_{j=1}% ^{n}c_{i,j}\dot{u_{j}}\,
  2. F D = C U ˙ F_{D}=C\dot{U}\,
  3. C = ( c i , j ) 1 i n , 1 j m C=(c_{i,j})_{1\leq i\leq n,1\leq j\leq m}\,

Darwin_Lagrangian.html

  1. v 2 c 2 \frac{v^{2}}{c^{2}}
  2. L = L f + L int , L=L\text{f}+L\text{int},
  3. L f = 1 2 m 1 v 1 2 + 1 8 c 2 m 1 v 1 4 + 1 2 m 2 v 2 2 + 1 8 c 2 m 2 v 2 4 , L\text{f}=\frac{1}{2}m_{1}v_{1}^{2}+\frac{1}{8c^{2}}m_{1}v_{1}^{4}+\frac{1}{2}% m_{2}v_{2}^{2}+\frac{1}{8c^{2}}m_{2}v_{2}^{4},
  4. L int = L C + L D , L\text{int}=L\text{C}+L\text{D},
  5. L C = - q 1 q 2 r , L\text{C}=-\frac{q_{1}q_{2}}{r},
  6. L D = q 1 q 2 r 1 2 c 2 𝐯 1 [ 𝟏 + 𝐫 ^ 𝐫 ^ ] 𝐯 2 . L\text{D}=\frac{q_{1}q_{2}}{r}\frac{1}{2c^{2}}\mathbf{v}_{1}\cdot\left[\mathbf% {1}+\mathbf{\hat{r}}\mathbf{\hat{r}}\right]\cdot\mathbf{v}_{2}.
  7. 𝐫 ^ \hat{\mathbf{r}}
  8. L int = - q Φ + q c 𝐮 𝐀 , L\text{int}=-q\Phi+{q\over c}\mathbf{u}\cdot\mathbf{A},
  9. 2 𝐀 - 1 c 2 2 𝐀 t 2 = - 4 π c 𝐉 t \nabla^{2}\mathbf{A}-{1\over c^{2}}{\partial^{2}\mathbf{A}\over\partial t^{2}}% =-{4\pi\over c}\mathbf{J}_{t}
  10. 𝐉 = q 2 𝐯 2 δ ( 𝐫 - 𝐫 2 ) , \mathbf{J}=q_{2}\mathbf{v}_{2}\delta\left(\mathbf{r}-\mathbf{r}_{2}\right),
  11. 𝐉 ( 𝐤 ) d 3 r exp ( - i 𝐤 𝐫 ) 𝐉 ( 𝐫 ) = q 2 𝐯 2 exp ( - i 𝐤 𝐫 2 ) . \mathbf{J}\left(\mathbf{k}\right)\equiv\int d^{3}r\exp\left(-i\mathbf{k}\cdot% \mathbf{r}\right)\mathbf{J}\left(\mathbf{r}\right)=q_{2}\mathbf{v}_{2}\exp% \left(-i\mathbf{k}\cdot\mathbf{r}_{2}\right).
  12. 𝐉 t ( 𝐤 ) = q 2 [ 𝟏 - 𝐤 ^ 𝐤 ^ ] 𝐯 2 exp ( - i 𝐤 𝐫 2 ) . \mathbf{J}_{t}\left(\mathbf{k}\right)=q_{2}\left[\mathbf{1}-\mathbf{\hat{k}}% \mathbf{\hat{k}}\right]\cdot\mathbf{v}_{2}\exp\left(-i\mathbf{k}\cdot\mathbf{r% }_{2}\right).
  13. 𝐤 𝐉 t ( 𝐤 ) = 0 , \mathbf{k}\cdot\mathbf{J}_{t}\left(\mathbf{k}\right)=0,
  14. 𝐉 t ( 𝐤 ) \mathbf{J}_{t}\left(\mathbf{k}\right)
  15. 𝐀 ( 𝐤 ) = 4 π c q 2 k 2 [ 𝟏 - 𝐤 ^ 𝐤 ^ ] 𝐯 2 exp ( - i 𝐤 𝐫 2 ) \mathbf{A}\left(\mathbf{k}\right)={4\pi\over c}{q_{2}\over k^{2}}\left[\mathbf% {1}-\mathbf{\hat{k}}\mathbf{\hat{k}}\right]\cdot\mathbf{v}_{2}\exp\left(-i% \mathbf{k}\cdot\mathbf{r}_{2}\right)
  16. 𝐀 ( 𝐫 ) = d 3 k ( 2 π ) 3 𝐀 ( 𝐤 ) exp ( i 𝐤 𝐫 1 ) = q 2 2 c 1 r [ 𝟏 + 𝐫 ^ 𝐫 ^ ] 𝐯 2 \mathbf{A}\left(\mathbf{r}\right)=\int{d^{3}k\over\left(2\pi\right)^{3}}\;% \mathbf{A}\left(\mathbf{k}\right)\;{\exp\left(i\mathbf{k}\cdot\mathbf{r}_{1}% \right)}={q_{2}\over 2c}{1\over r}\left[\mathbf{1}+\mathbf{\hat{r}}\mathbf{% \hat{r}}\right]\cdot\mathbf{v}_{2}
  17. 𝐫 = 𝐫 1 - 𝐫 2 \mathbf{r}=\mathbf{r}_{1}-\mathbf{r}_{2}
  18. L D = q 1 q 2 r 1 2 c 2 𝐯 1 [ 𝟏 + 𝐫 ^ 𝐫 ^ ] 𝐯 2 L_{D}={q_{1}q_{2}\over r}{1\over 2c^{2}}\mathbf{v}_{1}\cdot\left[\mathbf{1}+% \mathbf{\hat{r}}\mathbf{\hat{r}}\right]\cdot\mathbf{v}_{2}
  19. d d t 𝐯 1 L ( 𝐫 1 , 𝐯 1 ) = 1 L ( 𝐫 1 , 𝐯 1 ) {d\over dt}{\partial\over\partial\mathbf{v}_{1}}L\left(\mathbf{r}_{1},\mathbf{% v}_{1}\right)=\nabla_{1}L\left(\mathbf{r}_{1},\mathbf{v}_{1}\right)
  20. d 𝐩 1 d t = 1 L ( 𝐫 1 , 𝐯 1 ) {d\mathbf{p}_{1}\over dt}=\nabla_{1}L\left(\mathbf{r}_{1},\mathbf{v}_{1}\right)
  21. d d t [ ( 1 + 1 2 v 1 2 c 2 ) m 1 𝐯 1 ] = 0 {d\over dt}\left[\left(1+{1\over 2}{v_{1}^{2}\over c^{2}}\right)m_{1}\mathbf{v% }_{1}\right]=0
  22. 𝐩 1 = ( 1 + 1 2 v 1 2 c 2 ) m 1 𝐯 1 \mathbf{p}_{1}=\left(1+{1\over 2}{v_{1}^{2}\over c^{2}}\right)m_{1}\mathbf{v}_% {1}
  23. d d t [ ( 1 + 1 2 v 1 2 c 2 ) m 1 𝐯 1 + q 1 c 𝐀 ( 𝐫 1 ) ] = - q 1 q 2 r + [ q 1 q 2 r 1 2 c 2 𝐯 1 [ 𝟏 + 𝐫 ^ 𝐫 ^ ] 𝐯 2 ] {d\over dt}\left[\left(1+{1\over 2}{v_{1}^{2}\over c^{2}}\right)m_{1}\mathbf{v% }_{1}+{q_{1}\over c}\mathbf{A}\left(\mathbf{r}_{1}\right)\right]=-\nabla{q_{1}% q_{2}\over r}+\nabla\left[{q_{1}q_{2}\over r}{1\over 2c^{2}}\mathbf{v}_{1}% \cdot\left[\mathbf{1}+\mathbf{\hat{r}}\mathbf{\hat{r}}\right]\cdot\mathbf{v}_{% 2}\right]
  24. d 𝐩 1 d t = q 1 q 2 r 2 𝐫 ^ + q 1 q 2 r 2 1 2 c 2 { 𝐯 1 ( 𝐫 ^ 𝐯 2 ) + 𝐯 2 ( 𝐫 ^ 𝐯 1 ) - 𝐫 ^ [ 𝐯 1 ( 𝟏 + 3 𝐫 ^ 𝐫 ^ ) 𝐯 2 ] } {d\mathbf{p}_{1}\over dt}={q_{1}q_{2}\over r^{2}}{\hat{\mathbf{r}}}+{q_{1}q_{2% }\over r^{2}}{1\over 2c^{2}}\left\{\mathbf{v}_{1}\left({{\hat{\mathbf{r}}}% \cdot\mathbf{v}_{2}}\right)+\mathbf{v}_{2}\left({{\hat{\mathbf{r}}}\cdot% \mathbf{v}_{1}}\right)-{\hat{\mathbf{r}}}\left[\mathbf{v}_{1}\cdot\left(% \mathbf{1}+3{\hat{\mathbf{r}}}{\hat{\mathbf{r}}}\right)\cdot\mathbf{v}_{2}% \right]\right\}
  25. 𝐩 1 = ( 1 + 1 2 v 1 2 c 2 ) m 1 𝐯 1 + q 1 c 𝐀 ( 𝐫 1 ) \mathbf{p}_{1}=\left(1+{1\over 2}{v_{1}^{2}\over c^{2}}\right)m_{1}\mathbf{v}_% {1}+{q_{1}\over c}\mathbf{A}\left(\mathbf{r}_{1}\right)
  26. 𝐀 ( 𝐫 1 ) = q 2 2 c 1 r [ 𝟏 + 𝐫 ^ 𝐫 ^ ] 𝐯 2 \mathbf{A}\left(\mathbf{r}_{1}\right)={q_{2}\over 2c}{1\over r}\left[\mathbf{1% }+\mathbf{\hat{r}}\mathbf{\hat{r}}\right]\cdot\mathbf{v}_{2}
  27. 𝐫 = 𝐫 1 - 𝐫 2 \mathbf{r}=\mathbf{r}_{1}-\mathbf{r}_{2}
  28. H = 𝐩 1 𝐯 1 + 𝐩 2 𝐯 2 - L . H=\mathbf{p}_{1}\cdot\mathbf{v}_{1}+\mathbf{p}_{2}\cdot\mathbf{v}_{2}-L.
  29. H ( 𝐫 1 , 𝐩 1 , 𝐫 2 , 𝐩 2 ) = ( 1 - 1 4 p 1 2 m 1 2 c 2 ) p 1 2 2 m 1 + ( 1 - 1 4 p 2 2 m 2 2 c 2 ) p 2 2 2 m 2 + q 1 q 2 r - q 1 q 2 r 1 2 m 1 m 2 c 2 𝐩 1 [ 𝟏 + 𝐫 ^ 𝐫 ^ ] 𝐩 2 . H\left(\mathbf{r}_{1},\mathbf{p}_{1},\mathbf{r}_{2},\mathbf{p}_{2}\right)=% \left(1-{1\over 4}{p_{1}^{2}\over m_{1}^{2}c^{2}}\right){p_{1}^{2}\over 2m_{1}% }\;+\;\left(1-{1\over 4}{p_{2}^{2}\over m_{2}^{2}c^{2}}\right){p_{2}^{2}\over 2% m_{2}}\;+\;{q_{1}q_{2}\over r}\;-\;{q_{1}q_{2}\over r}{1\over 2m_{1}m_{2}c^{2}% }\mathbf{p}_{1}\cdot\left[\mathbf{1}+\mathbf{\hat{r}}\mathbf{\hat{r}}\right]% \cdot\mathbf{p}_{2}.
  30. 𝐯 1 = H 𝐩 1 \mathbf{v}_{1}={\partial H\over\partial\mathbf{p}_{1}}
  31. d 𝐩 1 d t = - 1 H {d\mathbf{p}_{1}\over dt}=-\nabla_{1}H
  32. 𝐯 1 = ( 1 - 1 2 p 1 2 m 1 2 c 2 ) 𝐩 1 m 1 - q 1 q 2 2 m 1 m 2 c 2 1 r [ 𝟏 + 𝐫 ^ 𝐫 ^ ] 𝐩 2 \mathbf{v}_{1}=\left(1-{1\over 2}{p_{1}^{2}\over m_{1}^{2}c^{2}}\right){% \mathbf{p}_{1}\over m_{1}}-{q_{1}q_{2}\over 2m_{1}m_{2}c^{2}}{1\over r}\left[% \mathbf{1}+\mathbf{\hat{r}}\mathbf{\hat{r}}\right]\cdot\mathbf{p}_{2}
  33. d 𝐩 1 d t = q 1 q 2 r 2 𝐫 ^ + q 1 q 2 r 2 1 2 m 1 m 2 c 2 { 𝐩 1 ( 𝐫 ^ 𝐩 2 ) + 𝐩 2 ( 𝐫 ^ 𝐩 1 ) - 𝐫 ^ [ 𝐩 1 ( 𝟏 + 3 𝐫 ^ 𝐫 ^ ) 𝐩 2 ] } {d\mathbf{p}_{1}\over dt}={q_{1}q_{2}\over r^{2}}{\hat{\mathbf{r}}}\;+\;{q_{1}% q_{2}\over r^{2}}{1\over 2m_{1}m_{2}c^{2}}\left\{\mathbf{p}_{1}\left({{\hat{% \mathbf{r}}}\cdot\mathbf{p}_{2}}\right)+\mathbf{p}_{2}\left({{\hat{\mathbf{r}}% }\cdot\mathbf{p}_{1}}\right)-{\hat{\mathbf{r}}}\left[\mathbf{p}_{1}\cdot\left(% \mathbf{1}+3{\hat{\mathbf{r}}}{\hat{\mathbf{r}}}\right)\cdot\mathbf{p}_{2}% \right]\right\}

Data_matrix_(multivariate_statistics).html

  1. 𝐘 𝐗𝐁 \mathbf{Y}\approx\mathbf{X}\mathbf{B}

Datar–Mathews_method_for_real_option_valuation.html

  1. C 0 = E 0 [ max ( S T e - μ t - X T e - r t , 0 ) ] C_{0}=E_{0}\left[\max\left(S_{T}e^{-\mu t}-X_{T}e^{-rt},0\right)\right]
  2. Real option value = average [ max ( operating profit - launch costs ) , 0 ) ] \,\text{Real option value}=\,\text{average}\left[\max\left(\,\text{operating % profit}-\,\text{launch costs}\right),0)\right]
  3. S 0 = S T e - μ T and σ 0 = ln ( 1 + ( σ T S T ) 2 ) T . S_{0}=S_{T}e^{-\mu T}\,\text{ and }\sigma_{0}=\frac{\sqrt{\ln\left(1+\left(% \frac{\sigma_{T}}{S_{T}}\right)^{2}\right)}}{\sqrt{T}}.
  4. X < s u b > t 0 = X T e r T X<sub>t_{0}=X_{T}e^{ −rT}

Davis_distribution.html

  1. Γ ( n ) \Gamma(n)
  2. ζ ( n ) \zeta(n)
  3. { μ + b ζ ( n - 1 ) ( n - 1 ) ζ ( n ) if n > 2 Indeterminate otherwise \begin{cases}\mu+\frac{b\zeta(n-1)}{(n-1)\zeta(n)}&\,\text{if}\ n>2\\ \,\text{Indeterminate}&\,\text{otherwise}\end{cases}
  4. { b 2 ( - ( n - 2 ) ζ ( n - 1 ) 2 + ( n - 1 ) ζ ( n - 2 ) ζ ( n ) ) ( n - 2 ) ( n - 1 ) 2 ζ ( n ) 2 if n > 3 Indeterminate otherwise \begin{cases}\frac{b^{2}\left(-(n-2){\zeta(n-1)}^{2}+(n-1)\zeta(n-2)\zeta(n)% \right)}{(n-2){(n-1)}^{2}{\zeta(n)}^{2}}&\,\text{if}\ n>3\\ \,\text{Indeterminate}&\,\text{otherwise}\end{cases}
  5. f ( x ; μ , b , n ) = b n ( x - μ ) - 1 - n ( e b x - μ - 1 ) Γ ( n ) ζ ( n ) f(x;\mu,b,n)=\frac{b^{n}{(x-\mu)}^{-1-n}}{\left(e^{\frac{b}{x-\mu}}-1\right)% \Gamma(n)\zeta(n)}
  6. Γ ( n ) \Gamma(n)
  7. ζ ( n ) \zeta(n)
  8. f ( μ ) = 0 f(\mu)=0\,
  9. μ > 0 \mu>0\,
  10. f ( x ) A ( x - μ ) - α - 1 . f(x)\sim A{(x-\mu)}^{-\alpha-1}\,.
  11. X Davis ( b = 1 , n = 4 , μ = 0 ) X\sim\mathrm{Davis}(b=1,n=4,\mu=0)\,
  12. 1 X Planck \tfrac{1}{X}\sim\mathrm{Planck}

DE-9IM.html

  1. DE9IM ( a , b ) = [ dim ( I ( a ) I ( b ) ) dim ( I ( a ) B ( b ) ) dim ( I ( a ) E ( b ) ) dim ( B ( a ) I ( b ) ) dim ( B ( a ) B ( b ) ) dim ( B ( a ) E ( b ) ) dim ( E ( a ) I ( b ) ) dim ( E ( a ) B ( b ) ) dim ( E ( a ) E ( b ) ) ] \operatorname{DE9IM}(a,b)=\begin{bmatrix}\dim(I(a)\cap I(b))&\dim(I(a)\cap B(b% ))&\dim(I(a)\cap E(b))\\ \dim(B(a)\cap I(b))&\dim(B(a)\cap B(b))&\dim(B(a)\cap E(b))\\ \dim(E(a)\cap I(b))&\dim(E(a)\cap B(b))&\dim(E(a)\cap E(b))\end{bmatrix}
  2. bin ( DE9IM ( a , b ) ) = 9 I M ( a , b ) = [ a o b o a o b a o b e a b o a b a b e a e b o a e b a e b e ] \operatorname{bin}(\operatorname{DE9IM}(a,b))=\operatorname{9IM}(a,b)=\begin{% bmatrix}a^{o}\cap b^{o}\neq\emptyset&a^{o}\cap\partial{b}\neq\emptyset&a^{o}% \cap b^{e}\neq\emptyset\\ \partial{a}\cap b^{o}\neq\emptyset&\partial{a}\cap\partial{b}\neq\emptyset&% \partial{a}\cap b^{e}\neq\emptyset\\ a^{e}\cap b^{o}\neq\emptyset&a^{e}\cap\partial{b}\neq\emptyset&a^{e}\cap b^{e}% \neq\emptyset\end{bmatrix}
  3. [ I I I B I E B I B B B E E I E B E E ] \begin{bmatrix}II&IB&IE\\ BI&BB&BE\\ EI&EB&EE\end{bmatrix}
  4. [ T * F * * F F F * ] \Bigl[\begin{smallmatrix}\mathrm{T}&\mathrm{*}&\mathrm{F}\\ \mathrm{*}&\mathrm{*}&\mathrm{F}\\ \mathrm{F}&\mathrm{F}&\mathrm{*}\end{smallmatrix}\Bigr]
  5. [ F F * F F * * * * ] \Bigl[\begin{smallmatrix}\mathrm{F}&\mathrm{F}&\mathrm{*}\\ \mathrm{F}&\mathrm{F}&\mathrm{*}\\ \mathrm{*}&\mathrm{*}&\mathrm{*}\end{smallmatrix}\Bigr]
  6. [ F T * * * * * * * ] \Bigl[\begin{smallmatrix}\mathrm{F}&\mathrm{T}&\mathrm{*}\\ \mathrm{*}&\mathrm{*}&\mathrm{*}\\ \mathrm{*}&\mathrm{*}&\mathrm{*}\end{smallmatrix}\Bigr]
  7. [ F * * T * * * * * ] \Bigl[\begin{smallmatrix}\mathrm{F}&\mathrm{*}&\mathrm{*}\\ \mathrm{T}&\mathrm{*}&\mathrm{*}\\ \mathrm{*}&\mathrm{*}&\mathrm{*}\end{smallmatrix}\Bigr]
  8. [ F * * * T * * * * ] \Bigl[\begin{smallmatrix}\mathrm{F}&\mathrm{*}&\mathrm{*}\\ \mathrm{*}&\mathrm{T}&\mathrm{*}\\ \mathrm{*}&\mathrm{*}&\mathrm{*}\end{smallmatrix}\Bigr]
  9. [ T * * * * * F F * ] \Bigl[\begin{smallmatrix}\mathrm{T}&\mathrm{*}&\mathrm{*}\\ \mathrm{*}&\mathrm{*}&\mathrm{*}\\ \mathrm{F}&\mathrm{F}&\mathrm{*}\end{smallmatrix}\Bigr]
  10. [ T * * * * * F F * ] \Bigl[\begin{smallmatrix}\mathrm{T}&\mathrm{*}&\mathrm{*}\\ \mathrm{*}&\mathrm{*}&\mathrm{*}\\ \mathrm{F}&\mathrm{F}&\mathrm{*}\end{smallmatrix}\Bigr]
  11. [ * T * * * * F F * ] \Bigl[\begin{smallmatrix}\mathrm{*}&\mathrm{T}&\mathrm{*}\\ \mathrm{*}&\mathrm{*}&\mathrm{*}\\ \mathrm{F}&\mathrm{F}&\mathrm{*}\end{smallmatrix}\Bigr]
  12. [ * * * T * * F F * ] \Bigl[\begin{smallmatrix}\mathrm{*}&\mathrm{*}&\mathrm{*}\\ \mathrm{T}&\mathrm{*}&\mathrm{*}\\ \mathrm{F}&\mathrm{F}&\mathrm{*}\end{smallmatrix}\Bigr]
  13. [ * * * * T * F F * ] \Bigl[\begin{smallmatrix}\mathrm{*}&\mathrm{*}&\mathrm{*}\\ \mathrm{*}&\mathrm{T}&\mathrm{*}\\ \mathrm{F}&\mathrm{F}&\mathrm{*}\end{smallmatrix}\Bigr]
  14. [ T * * * * * * * * ] \Bigl[\begin{smallmatrix}\mathrm{T}&\mathrm{*}&\mathrm{*}\\ \mathrm{*}&\mathrm{*}&\mathrm{*}\\ \mathrm{*}&\mathrm{*}&\mathrm{*}\end{smallmatrix}\Bigr]
  15. [ * T * * * * * * * ] \Bigl[\begin{smallmatrix}\mathrm{*}&\mathrm{T}&\mathrm{*}\\ \mathrm{*}&\mathrm{*}&\mathrm{*}\\ \mathrm{*}&\mathrm{*}&\mathrm{*}\end{smallmatrix}\Bigr]
  16. [ * * * T * * * * * ] \Bigl[\begin{smallmatrix}\mathrm{*}&\mathrm{*}&\mathrm{*}\\ \mathrm{T}&\mathrm{*}&\mathrm{*}\\ \mathrm{*}&\mathrm{*}&\mathrm{*}\end{smallmatrix}\Bigr]
  17. [ * * * * T * * * * ] \Bigl[\begin{smallmatrix}\mathrm{*}&\mathrm{*}&\mathrm{*}\\ \mathrm{*}&\mathrm{T}&\mathrm{*}\\ \mathrm{*}&\mathrm{*}&\mathrm{*}\end{smallmatrix}\Bigr]
  18. [ T * F * * F * * * ] \Bigl[\begin{smallmatrix}\mathrm{T}&\mathrm{*}&\mathrm{F}\\ \mathrm{*}&\mathrm{*}&\mathrm{F}\\ \mathrm{*}&\mathrm{*}&\mathrm{*}\end{smallmatrix}\Bigr]
  19. [ T * F * * F * * * ] \Bigl[\begin{smallmatrix}\mathrm{T}&\mathrm{*}&\mathrm{F}\\ \mathrm{*}&\mathrm{*}&\mathrm{F}\\ \mathrm{*}&\mathrm{*}&\mathrm{*}\end{smallmatrix}\Bigr]
  20. [ * T F * * F * * * ] \Bigl[\begin{smallmatrix}\mathrm{*}&\mathrm{T}&\mathrm{F}\\ \mathrm{*}&\mathrm{*}&\mathrm{F}\\ \mathrm{*}&\mathrm{*}&\mathrm{*}\end{smallmatrix}\Bigr]
  21. [ * * F T * F * * * ] \Bigl[\begin{smallmatrix}\mathrm{*}&\mathrm{*}&\mathrm{F}\\ \mathrm{T}&\mathrm{*}&\mathrm{F}\\ \mathrm{*}&\mathrm{*}&\mathrm{*}\end{smallmatrix}\Bigr]
  22. [ * * F * T F * * * ] \Bigl[\begin{smallmatrix}\mathrm{*}&\mathrm{*}&\mathrm{F}\\ \mathrm{*}&\mathrm{T}&\mathrm{F}\\ \mathrm{*}&\mathrm{*}&\mathrm{*}\end{smallmatrix}\Bigr]
  23. [ T * T * * * * * * ] \Bigl[\begin{smallmatrix}\mathrm{T}&\mathrm{*}&\mathrm{T}\\ \mathrm{*}&\mathrm{*}&\mathrm{*}\\ \mathrm{*}&\mathrm{*}&\mathrm{*}\end{smallmatrix}\Bigr]
  24. [ T * * * * * T * * ] \Bigl[\begin{smallmatrix}\mathrm{T}&\mathrm{*}&\mathrm{*}\\ \mathrm{*}&\mathrm{*}&\mathrm{*}\\ \mathrm{T}&\mathrm{*}&\mathrm{*}\end{smallmatrix}\Bigr]
  25. [ 0 * * * * * * * * ] \Bigl[\begin{smallmatrix}\mathrm{0}&\mathrm{*}&\mathrm{*}\\ \mathrm{*}&\mathrm{*}&\mathrm{*}\\ \mathrm{*}&\mathrm{*}&\mathrm{*}\end{smallmatrix}\Bigr]
  26. [ T * T * * * T * * ] \Bigl[\begin{smallmatrix}\mathrm{T}&\mathrm{*}&\mathrm{T}\\ \mathrm{*}&\mathrm{*}&\mathrm{*}\\ \mathrm{T}&\mathrm{*}&\mathrm{*}\end{smallmatrix}\Bigr]
  27. [ 1 * T * * * T * * ] \Bigl[\begin{smallmatrix}\mathrm{1}&\mathrm{*}&\mathrm{T}\\ \mathrm{*}&\mathrm{*}&\mathrm{*}\\ \mathrm{T}&\mathrm{*}&\mathrm{*}\end{smallmatrix}\Bigr]

De_analysi_per_aequationes_numero_terminorum_infinitas.html

  1. log ( 1 + x ) \log(1+x)

De_Donder–Weyl_theory.html

  1. p a i / x i = - H / y a \partial p^{i}_{a}/\partial x^{i}=-\partial H/\partial y^{a}
  2. y a / x i = H / p a i \partial y^{a}/\partial x^{i}=\partial H/\partial p^{i}_{a}
  3. L = L ( y a , i y a , x i ) L=L(y^{a},\partial_{i}y^{a},x^{i})
  4. p a i = L / ( i y a ) p^{i}_{a}=\partial L/\partial(\partial_{i}y^{a})
  5. H = p a i i y a - L H=p^{i}_{a}\partial_{i}y^{a}-L
  6. p a i / x i = - H / y a , y a / x i = H / p a i \partial p^{i}_{a}/\partial x^{i}=-\partial H/\partial y^{a}\,,\,\partial y^{a% }/\partial x^{i}=\partial H/\partial p^{i}_{a}

De_Moivre's_law.html

  1. ω \omega
  2. S ( x ) = 1 - x ω , 0 x < ω . S(x)=1-\frac{x}{\omega},\qquad 0\leq x<\omega.
  3. p x t {}_{t}p_{x}
  4. p x t = S ( x + t ) S ( x ) = ω - ( x + t ) ω - x , 0 t < ω - x , {}_{t}p_{x}=\frac{S(x+t)}{S(x)}=\frac{\omega-(x+t)}{\omega-x},\qquad 0\leq t<% \omega-x,
  5. ( 0 , ω - x ) (0,\,\omega-x)
  6. q x t {}_{t}q_{x}
  7. q x t = S ( x ) - S ( x + t ) S ( x ) = t ω - x . {}_{t}q_{x}=\frac{S(x)-S(x+t)}{S(x)}=\frac{t}{\omega-x}.
  8. μ ( x ) = - S ( x ) / S ( x ) = f ( x ) / S ( x ) , \mu(x)=-S^{\prime}(x)/S(x)=f(x)/S(x),
  9. μ ( x + t ) = 1 ω - ( x + t ) , 0 t < ω - x , \mu(x+t)=\frac{1}{\omega-(x+t)},\qquad 0\leq t<\omega-x,
  10. x \ell_{x}
  11. 0 \ell_{0}
  12. x + t = ( 1 - t ) x + t x + 1 , 0 < t < 1 , \ell_{x+t}=(1-t)\ell_{x}+t\ell_{x+1},\qquad 0<t<1,
  13. S ( x + t ) = ( 1 - t ) S ( x ) + t S ( x + 1 ) , 0 < t < 1. S(x+t)=(1-t)S(x)+tS(x+1),\qquad 0<t<1.
  14. q x t {}_{t}q_{x}
  15. t q x tq_{x}
  16. μ ( x + t ) = q x 1 - t q x \mu(x+t)=\frac{q_{x}}{1-tq_{x}}
  17. 0 < t < 1 \,0<t<1
  18. p x t {}_{t}p_{x}

Decoupling_(cosmology).html

  1. Γ \Gamma
  2. H H
  3. λ \lambda
  4. H - 1 H^{-1}
  5. Γ = c λ = n e σ e c \Gamma=\frac{c}{\lambda}=n_{e}\sigma_{e}c
  6. n e n_{e}
  7. σ e \sigma_{e}
  8. c c
  9. H a - 3 / 2 H\varpropto a^{-{3/2}}
  10. a a
  11. Γ \Gamma
  12. a a
  13. H H
  14. H H
  15. Γ \Gamma
  16. Γ = H \Gamma=H
  17. z = 1100 z=1100
  18. ± \pm

Deep_learning.html

  1. w i j ( t + 1 ) = w i j ( t ) + η C w i j w_{ij}(t+1)=w_{ij}(t)+\eta\frac{\partial C}{\partial w_{ij}}
  2. η \eta
  3. C C
  4. p j = exp ( x j ) k exp ( x k ) p_{j}=\frac{\exp(x_{j})}{\sum_{k}\exp(x_{k})}
  5. p j p_{j}
  6. x j x_{j}
  7. x k x_{k}
  8. j j
  9. k k
  10. C = - j d j log ( p j ) C=-\sum_{j}d_{j}\log(p_{j})
  11. d j d_{j}
  12. j j
  13. p j p_{j}
  14. j j
  15. 2 \ell_{2}
  16. 1 \ell_{1}
  17. Δ w i j ( t + 1 ) = w i j ( t ) + η log ( p ( v ) ) w i j \Delta w_{ij}(t+1)=w_{ij}(t)+\eta\frac{\partial\log(p(v))}{\partial w_{ij}}
  18. p ( v ) p(v)
  19. p ( v ) = 1 Z h e - E ( v , h ) p(v)=\frac{1}{Z}\sum_{h}e^{-E(v,h)}
  20. Z Z
  21. E ( v , h ) E(v,h)
  22. log ( p ( v ) ) w i j \frac{\partial\log(p(v))}{\partial w_{ij}}
  23. v i h j data - v i h j model \langle v_{i}h_{j}\rangle\text{data}-\langle v_{i}h_{j}\rangle\text{model}
  24. p \langle\cdots\rangle_{p}
  25. p p
  26. v i h j model \langle v_{i}h_{j}\rangle\text{model}
  27. n n
  28. n = 1 n=1
  29. n n
  30. v i h j model \langle v_{i}h_{j}\rangle\text{model}
  31. p ( h j = 1 𝐕 ) = σ ( b j + i v i w i j ) p(h_{j}=1\mid\,\textbf{V})=\sigma(b_{j}+\sum_{i}v_{i}w_{ij})
  32. σ \sigma
  33. b j b_{j}
  34. h j h_{j}
  35. p ( v i = 1 𝐇 ) = σ ( a i + j h j w i j ) p(v_{i}=1\mid\,\textbf{H})=\sigma(a_{i}+\sum_{j}h_{j}w_{ij})
  36. a i a_{i}
  37. v i v_{i}
  38. Δ w i j v i h j data - v i h j reconstruction \Delta w_{ij}\propto\langle v_{i}h_{j}\rangle\text{data}-\langle v_{i}h_{j}% \rangle\text{reconstruction}
  39. s y m b o l ν { 0 , 1 } D symbol{\nu}\in\{0,1\}^{D}
  40. s y m b o l h ( 1 ) { 0 , 1 } F 1 , s y m b o l h ( 2 ) { 0 , 1 } F 2 , , s y m b o l h ( L ) { 0 , 1 } F L symbol{h}^{(1)}\in\{0,1\}^{F_{1}},symbol{h}^{(2)}\in\{0,1\}^{F_{2}},\ldots,% symbol{h}^{(L)}\in\{0,1\}^{F_{L}}
  41. ν \nu
  42. p ( s y m b o l ν ) = 1 Z h e i j W i j ( 1 ) ν i h j ( 1 ) + j l W j l ( 2 ) h j ( 1 ) h l ( 2 ) + l m W l m ( 3 ) h l ( 2 ) h m ( 3 ) , p(symbol{\nu})=\frac{1}{Z}\sum_{h}e^{\sum_{ij}W_{ij}^{(1)}\nu_{i}h_{j}^{(1)}+% \sum_{jl}W_{jl}^{(2)}h_{j}^{(1)}h_{l}^{(2)}+\sum_{lm}W_{lm}^{(3)}h_{l}^{(2)}h_% {m}^{(3)}},
  43. s y m b o l h = { s y m b o l h ( 1 ) , s y m b o l h ( 2 ) , s y m b o l h ( 3 ) } symbol{h}=\{symbol{h}^{(1)},symbol{h}^{(2)},symbol{h}^{(3)}\}
  44. θ = { s y m b o l W ( 1 ) , s y m b o l W ( 2 ) , s y m b o l W ( 3 ) } \theta=\{symbol{W}^{(1)},symbol{W}^{(2)},symbol{W}^{(3)}\}
  45. s y m b o l W ( 2 ) = 0 symbol{W}^{(2)}=0
  46. s y m b o l W ( 3 ) = 0 symbol{W}^{(3)}=0
  47. f θ f_{\theta}
  48. θ = { s y m b o l W , b } \theta=\{symbol{W},b\}
  49. s y m b o l W symbol{W}
  50. g θ g_{\theta}
  51. s y m b o l x symbol{x}
  52. s y m b o l x ~ \tilde{symbol{x}}
  53. q D ( s y m b o l x ~ | s y m b o l x ) q_{D}(\tilde{symbol{x}}|symbol{x})
  54. s y m b o l x ~ \tilde{symbol{x}}
  55. s y m b o l y = f θ ( s y m b o l x ~ ) = s ( s y m b o l W s y m b o l x ~ + b ) symbol{y}=f_{\theta}(\tilde{symbol{x}})=s(symbol{W}\tilde{symbol{x}}+b)
  56. s y m b o l z = g θ ( s y m b o l y ) symbol{z}=g_{\theta}(symbol{y})
  57. s y m b o l x symbol{x}
  58. L H ( s y m b o l x , s y m b o l z ) L_{H}(symbol{x},symbol{z})
  59. f θ f_{\theta}
  60. s y m b o l H = σ ( s y m b o l W T s y m b o l X ) symbol{H}=\sigma(symbol{W}^{T}symbol{X})
  61. min U T f = || s y m b o l U T s y m b o l H - s y m b o l T || F 2 , \min_{U^{T}}f=||symbol{U}^{T}symbol{H}-symbol{T}||^{2}_{F},
  62. s y m b o l ν symbol{\nu}
  63. p ( s y m b o l ν , ψ ) = 1 Z h e i j W i j ( 1 ) ν i h j 1 + j l W j l ( 2 ) h j 1 h l 2 + l m W l m ( 3 ) h l 2 h m 3 , p(symbol{\nu},\psi)=\frac{1}{Z}\sum_{h}e^{\sum_{ij}W_{ij}^{(1)}\nu_{i}h_{j}^{1% }+\sum_{jl}W_{jl}^{(2)}h_{j}^{1}h_{l}^{2}+\sum_{lm}W_{lm}^{(3)}h_{l}^{2}h_{m}^% {3}},
  64. s y m b o l h = { s y m b o l h ( 1 ) , s y m b o l h ( 2 ) , s y m b o l h ( 3 ) } symbol{h}=\{symbol{h}^{(1)},symbol{h}^{(2)},symbol{h}^{(3)}\}
  65. ψ = { s y m b o l W ( 1 ) , s y m b o l W ( 2 ) , s y m b o l W ( 3 ) } \psi=\{symbol{W}^{(1)},symbol{W}^{(2)},symbol{W}^{(3)}\}
  66. P ( ν , h 1 , h 2 , h 3 ) P(\nu,h^{1},h^{2},h^{3})
  67. P ( ν , h 1 , h 2 | h 3 ) P(\nu,h^{1},h^{2}|h^{3})
  68. P ( h 3 ) P(h^{3})
  69. P ( ν , h 1 , h 2 | h 3 ) P(\nu,h^{1},h^{2}|h^{3})
  70. h 3 h^{3}
  71. P ( ν , h 1 , h 2 | h 3 ) = 1 Z ( ψ , h 3 ) e i j W i j ( 1 ) ν i h j 1 + j l W j l ( 2 ) h j 1 h l 2 + l m W l m ( 3 ) h l 2 h m 3 . P(\nu,h^{1},h^{2}|h^{3})=\frac{1}{Z(\psi,h^{3})}e^{\sum_{ij}W_{ij}^{(1)}\nu_{i% }h_{j}^{1}+\sum_{jl}W_{jl}^{(2)}h_{j}^{1}h_{l}^{2}+\sum_{lm}W_{lm}^{(3)}h_{l}^% {2}h_{m}^{3}}.
  72. l + 1 l+1
  73. l l
  74. n l n_{l}
  75. l l
  76. n l n_{l}
  77. m l { 1 , , n l } m_{l}\in\{1,\ldots,n_{l}\}
  78. m l m_{l}
  79. m l m_{l}

Deep_sequencing.html

  1. N × L / G N\times L/G

Defective_coloring.html

  1. g 0 g\geq 0
  2. k = k ( g ) k=k(g)
  3. g g
  4. χ d ( G ) \chi_{d}(G)
  5. C 5 C_{5}
  6. χ 0 ( C 5 ) = χ ( C 5 ) = 3 \chi_{0}(C_{5})=\chi(C_{5})=3
  7. χ 1 ( C 5 ) = 3 \chi_{1}(C_{5})=3
  8. χ 2 ( C n ) = 1 ; n \chi_{2}(C_{n})=1;\forall n\in\mathbb{N}
  9. G G
  10. v 0 v_{0}
  11. G G
  12. V i V_{i}
  13. G G
  14. i i
  15. v 0 v_{0}
  16. G i G_{i}
  17. V i \langle V_{i}\rangle
  18. V i V_{i}
  19. G i G_{i}
  20. K 1 , 3 K_{1,3}
  21. V 0 V 1 V i - 1 V_{0}\cup V_{1}\cup...\cup V_{i-1}
  22. G G^{\prime}
  23. V 0 V 1 V i - 1 \langle V_{0}\cup V_{1}\cup...\cup V_{i-1}\rangle
  24. G G^{\prime}
  25. V i V_{i}
  26. V i - 1 V_{i-1}
  27. G G^{\prime}
  28. V 0 V 1 V i - 1 \langle V_{0}\cup V_{1}\cup...\cup V_{i-1}\rangle
  29. G G^{\prime}
  30. V i V_{i}
  31. G G^{\prime}
  32. K 2 , 3 K_{2,3}
  33. K 2 , 3 K_{2,3}
  34. G i G_{i}
  35. G G
  36. V i V_{i}
  37. V i - 1 V_{i-1}
  38. V i + 1 , i 1 V_{i+1},i\geqslant 1
  39. V i V_{i}
  40. i i
  41. G G
  42. G G
  43. G i G_{i}
  44. G i G_{i}
  45. V i V_{i}
  46. i i
  47. i i
  48. G G

Defining_equation_(physics).html

  1. S 𝐁 d 𝐥 = μ 0 S ( 𝐉 + ϵ 0 𝐄 t ) d 𝐀 \oint_{S}\mathbf{B}\cdot{\rm d}\mathbf{l}=\mu_{0}\oint_{S}\left(\mathbf{J}+% \epsilon_{0}\frac{\partial\mathbf{E}}{\partial t}\right)\cdot{\rm d}\mathbf{A}\,\!
  2. 𝐁 = μ 0 𝐇 , \mathbf{B}=\mu_{0}\mathbf{H},\,\!
  3. I = S 𝐉 d 𝐀 , I=\oint_{S}\mathbf{J}\cdot{\rm d}\mathbf{A},\,\!
  4. 𝐉 d = ϵ 0 𝐄 t \mathbf{J}_{\rm d}=\epsilon_{0}\frac{\partial\mathbf{E}}{\partial t}\,\!
  5. I d = S 𝐉 d d 𝐀 , I_{d}=\oint_{S}\mathbf{J}_{\rm d}\cdot{\rm d}\mathbf{A},\,\!
  6. S 𝐁 d 𝐥 = μ 0 S 𝐉 d 𝐀 + μ 0 S 𝐉 d d 𝐀 , \oint_{S}\mathbf{B}\cdot{\rm d}\mathbf{l}=\mu_{0}\oint_{S}\mathbf{J}\cdot{\rm d% }\mathbf{A}+\mu_{0}\oint_{S}\mathbf{J}_{\rm d}\cdot{\rm d}\mathbf{A},\,\!
  7. S 𝐇 d 𝐥 = I + I d , \oint_{S}\mathbf{H}\cdot{\rm d}\mathbf{l}=I+I_{d},\,\!
  8. p = L r p=\frac{L}{r}\,\!
  9. J = I A J=\frac{I}{A}\,\!
  10. L = p r L=pr\,\!
  11. I = J A I=JA\,\!
  12. 𝐉 𝐧 ^ = I A \mathbf{J}\cdot\mathbf{\hat{n}}=\frac{I}{A}\,\!
  13. L = p r , L=pr,\,\!
  14. L = p r sin θ . L=pr\sin\theta.\,\!
  15. 𝐋 = 𝐫 × 𝐩 . \mathbf{L}=\mathbf{r}\times\mathbf{p}.\,\!
  16. 𝐉 𝐧 ^ A = I , \mathbf{J}\cdot\mathbf{\hat{n}}A=I,\,\!
  17. 𝐉 𝐀 = I , \mathbf{J}\cdot\mathbf{A}=I,\,\!
  18. J = lim A 0 I A = d I d A J=\lim_{A\rightarrow 0}\frac{I}{A}=\frac{\mathrm{d}I}{\mathrm{d}A}\,\!
  19. I = lim A i 0 i J A i = S J d A I=\lim_{A_{i}\rightarrow 0}\sum_{i}JA_{i}=\int_{S}J{\mathrm{d}A}\,\!
  20. d I = J d A , \mathrm{d}I=J{\mathrm{d}A},\,\!
  21. I = S J d A . I=\int_{S}J{\mathrm{d}A}.\,\!
  22. 𝐉 𝐧 ^ = d I d A \mathbf{J}\cdot\mathbf{\hat{n}}=\frac{\mathrm{d}I}{\mathrm{d}A}\,\!
  23. I = S 𝐉 d 𝐀 I=\int_{S}\mathbf{J}\cdot\mathrm{d}\mathbf{A}\,\!
  24. J i n i = d I d A J_{i}n_{i}=\frac{\mathrm{d}I}{\mathrm{d}A}\,\!
  25. 𝐋 = 𝐫 × 𝐩 \mathbf{L}=\mathbf{r}\times\mathbf{p}\,\!
  26. 𝐚 = 𝐛 × 𝐜 , a i = ϵ i j k b j c k , \mathbf{a}=\mathbf{b}\times\mathbf{c},\quad a_{i}=\epsilon_{ijk}b_{j}c_{k},\,\!
  27. L i = ϵ i j k r j p k . L_{i}=\epsilon_{ijk}r_{j}p_{k}.\,\!
  28. J i n i d A = d I J_{i}n_{i}\mathrm{d}A=\mathrm{d}I\,\!
  29. S J i d A i = I \int_{S}J_{i}\mathrm{d}A_{i}=I\,\!
  30. 𝐅 = q ( 𝐯 × 𝐁 ) = ( I d t ) ( d 𝐫 d t × 𝐁 ) = ( I d t d 𝐫 d t ) × 𝐁 = I ( d 𝐫 ) × 𝐁 = I ( 𝐥 × 𝐁 ) , \begin{aligned}\displaystyle\mathbf{F}&\displaystyle=q\left(\mathbf{v}\times% \mathbf{B}\right)\\ &\displaystyle=\left(\int I\mathrm{d}t\right)\left(\frac{\mathrm{d}\mathbf{r}}% {\mathrm{d}t}\times\mathbf{B}\right)\\ &\displaystyle=\left(\int I\mathrm{d}t\frac{\mathrm{d}\mathbf{r}}{\mathrm{d}t}% \right)\times\mathbf{B}\\ &\displaystyle=I\left(\int\mathrm{d}\mathbf{r}\right)\times\mathbf{B}\\ &\displaystyle=I\left(\mathbf{l}\times\mathbf{B}\right),\end{aligned}\,\!
  31. 𝐥 = d 𝐫 \mathbf{l}=\int\mathrm{d}\mathbf{r}\,\!
  32. 𝐀 = A 𝐧 ^ \mathbf{A}=A\mathbf{\hat{n}}\,\!
  33. 𝐧 ^ \mathbf{\hat{n}}\,\!
  34. 𝐁 𝐧 ^ = d Φ B d A , \mathbf{B}\cdot\mathbf{\hat{n}}=\frac{\mathrm{d}\Phi_{B}}{\mathrm{d}A},\,\!
  35. 𝐁 𝐧 ^ d A = d Φ B , \mathbf{B}\cdot\mathbf{\hat{n}}\mathrm{d}A=\mathrm{d}\Phi_{B},\,\!
  36. Φ B = S 𝐁 d 𝐀 . \Phi_{B}=\int_{S}\mathbf{B}\cdot\mathrm{d}\mathbf{A}.\,\!
  37. L = N d Φ B d I , L=N\frac{\mathrm{d}\Phi_{B}}{\mathrm{d}I},\,\!
  38. V = - L d I d t . V=-L\frac{\mathrm{d}I}{\mathrm{d}t}.\,\!
  39. V = - N d Φ B d t , V=-N\frac{\mathrm{d}\Phi_{B}}{\mathrm{d}t},\,\!
  40. V d t = - N d Φ B , V{\mathrm{d}t}=-N\mathrm{d}\Phi_{B},\,\!
  41. L = - V d t d I L=-V\frac{{\mathrm{d}t}}{\mathrm{d}I}\,\!
  42. V = - L d I d t V=-L\frac{\mathrm{d}I}{\mathrm{d}t}\,\!
  43. V = d W d q , I = d q d t , V=\frac{\mathrm{d}W}{\mathrm{d}q},\quad I=\frac{\mathrm{d}q}{\mathrm{d}t},\,\!
  44. 𝐅 = q [ 𝐄 + ( 𝐯 × 𝐁 ) ] , \mathbf{F}=q\left[\mathbf{E}+\left(\mathbf{v}\times\mathbf{B}\right)\right],\,\!
  45. 𝐄 = 𝐅 / q . \mathbf{E}=\mathbf{F}/q.\,\!
  46. i N 1 𝐩 i = j N 2 𝐩 j \sum_{i}^{N_{1}}\mathbf{p}_{\rm i}=\sum_{j}^{N_{2}}\mathbf{p}_{\rm j}\,\!
  47. 𝐩 = m 𝐯 \mathbf{p}=m\mathbf{v}\,\!
  48. 𝐩 i = m i 𝐯 i \mathbf{p}_{\rm i}=m_{i}\mathbf{v}_{\rm i}\,\!
  49. 𝐩 j = m j 𝐯 j \mathbf{p}_{\rm j}=m_{j}\mathbf{v}_{\rm j}\,\!
  50. i N 1 m i 𝐯 i = j N 2 m i 𝐯 i . \sum_{i}^{N_{1}}m_{i}\mathbf{v}_{\rm i}=\sum_{j}^{N_{2}}m_{i}\mathbf{v}_{\rm i% }.\,\!
  51. m = γ m 0 m=\gamma m_{0}\,\!
  52. 𝐩 = m 𝐯 𝐩 = γ m 0 𝐯 \mathbf{p}=m\mathbf{v}\rightarrow\mathbf{p}=\gamma m_{0}\mathbf{v}
  53. E = m c 2 E = γ m 0 c 2 E=mc^{2}\rightarrow E=\gamma m_{0}c^{2}
  54. T = m 2 𝐯 𝐯 T = γ m 0 2 𝐯 𝐯 T=\frac{m}{2}\mathbf{v}\cdot\mathbf{v}\nrightarrow T=\frac{\gamma m_{0}}{2}% \mathbf{v}\cdot\mathbf{v}
  55. 𝐅 = m 𝐚 𝐅 = γ m 0 𝐚 \mathbf{F}=m\mathbf{a}\nrightarrow\mathbf{F}=\gamma m_{0}\mathbf{a}
  56. 𝐅 = - m ( 𝐯 𝐯 ) 𝐫 ^ | 𝐫 | = q ( 𝐯 × 𝐁 ) , \mathbf{F}=-\frac{m\left(\mathbf{v}\cdot{\mathbf{v}}\right)\mathbf{\hat{r}}}{% \left|\mathbf{r}\right|}=q\left(\mathbf{v}\times\mathbf{B}\right),\,\!
  57. m | 𝐯 | 2 | 𝐫 | = q | 𝐯 | | 𝐁 | sin θ , \frac{m\left|\mathbf{v}\right|^{2}}{\left|\mathbf{r}\right|}=q\left|\mathbf{v}% \right|\left|\mathbf{B}\right|\sin\theta,\,\!
  58. m | 𝐯 | | 𝐫 | = q | 𝐁 | sin θ , \frac{m\left|\mathbf{v}\right|}{\left|\mathbf{r}\right|}=q\left|\mathbf{B}% \right|\sin\theta,\,\!
  59. | 𝐁 | | 𝐫 | = m | 𝐯 | q sin θ , \left|\mathbf{B}\right|\left|\mathbf{r}\right|=\frac{m\left|\mathbf{v}\right|}% {q\sin\theta},\,\!

Deformed_power.html

  1. S = P 2 + Q 2 + D 2 S=\sqrt{P^{2}+Q^{2}+D^{2}}

Degeneracy_(graph_theory).html

  1. u u
  2. c c
  3. c c
  4. ( c + 1 ) (c+1)

Delaporte_distribution.html

  1. j = 0 k i = 0 j Γ ( α + i ) β i λ j - i e - λ Γ ( α ) i ! ( 1 + β ) α + i ( j - i ) ! \sum_{j=0}^{k}\sum_{i=0}^{j}\frac{\Gamma(\alpha+i)\beta^{i}\lambda^{j-i}e^{-% \lambda}}{\Gamma(\alpha)i!(1+\beta)^{\alpha+i}(j-i)!}
  2. λ + α β \lambda+\alpha\beta
  3. { z , z + 1 { z } : z = ( α - 1 ) β + λ z otherwise \begin{cases}z,z+1&\{z\in\mathbb{Z}\}:\;z=(\alpha-1)\beta+\lambda\\ \lfloor z\rfloor&\textrm{otherwise}\end{cases}
  4. λ + α β ( 1 + β ) \lambda+\alpha\beta(1+\beta)
  5. λ \lambda
  6. α \alpha
  7. β \beta
  8. λ + α β ( 1 + 3 β + 2 β 2 ) ( λ + α β ( 1 + β ) ) 3 2 \frac{\lambda+\alpha\beta(1+3\beta+2\beta^{2})}{\left(\lambda+\alpha\beta(1+% \beta)\right)^{\frac{3}{2}}}
  9. λ + 3 λ 2 + α β ( 1 + 6 λ + 6 λ β + 7 β + 12 β 2 + 6 β 3 + 3 α β + 6 α β 2 + 3 α β 3 ) ( λ + α β ( 1 + β ) ) 2 \frac{\lambda+3\lambda^{2}+\alpha\beta(1+6\lambda+6\lambda\beta+7\beta+12\beta% ^{2}+6\beta^{3}+3\alpha\beta+6\alpha\beta^{2}+3\alpha\beta^{3})}{\left(\lambda% +\alpha\beta(1+\beta)\right)^{2}}

Deligne_cohomology.html

  1. 0 𝐙 ( p ) Ω X 0 Ω X 1 Ω X p - 1 0 0\rightarrow\mathbf{Z}(p)\rightarrow\Omega^{0}_{X}\rightarrow\Omega^{1}_{X}% \rightarrow\cdots\rightarrow\Omega_{X}^{p-1}\rightarrow 0\rightarrow\dots
  2. Ω X * \Omega^{*}_{X}

Demagnetizing_field.html

  1. 𝐇 ( 𝐫 ) \mathbf{H}(\mathbf{r})
  2. μ 0 \mu_{0}
  3. 𝐌 \mathbf{M}
  4. 2 U out = 0. \nabla^{2}U\text{out}=0.
  5. 𝐇 \mathbf{H}
  6. 𝐁 \mathbf{B}
  7. U in \displaystyle U\text{in}
  8. ( / n ) \scriptstyle\left(\partial/\partial n\right)
  9. E = - 1 2 magnet 𝐌 𝐇 d d V E=-\frac{1}{2}\int\text{magnet}\mathbf{M}\cdot\mathbf{H}\text{d}dV
  10. E = magnet 1 𝐌 1 𝐇 d ( 2 ) d V . E=\int\text{magnet 1}\mathbf{M}_{1}\cdot\mathbf{H}\text{d}^{(2)}dV.
  11. magnet 1 𝐌 1 𝐇 d ( 2 ) d V = magnet 2 𝐌 2 𝐇 d ( 1 ) d V . \int\text{magnet 1}\mathbf{M}_{1}\cdot\mathbf{H}\text{d}^{(2)}dV=\int\text{% magnet 2}\mathbf{M}_{2}\cdot\mathbf{H}\text{d}^{(1)}dV.
  12. U ( 𝐫 ) = - 1 4 π volume 𝐌 ( 𝐫 ) | 𝐫 - 𝐫 | d V + 1 4 π surface 𝐧 𝐌 ( 𝐫 ) | 𝐫 - 𝐫 | d S , U(\mathbf{r})=-\frac{1}{4\pi}\int\text{volume}\frac{\nabla^{\prime}\cdot% \mathbf{M\left(r^{\prime}\right)}}{|\mathbf{r}-\mathbf{r}^{\prime}|}dV^{\prime% }+\frac{1}{4\pi}\int\text{surface}\frac{\mathbf{n}\cdot\mathbf{M\left(r^{% \prime}\right)}}{|\mathbf{r}-\mathbf{r}^{\prime}|}dS^{\prime},
  13. 𝐫 \mathbf{r}′
  14. · 𝐌 −∇·\mathbf{M}
  15. 𝐧 · 𝐌 \mathbf{n}·\mathbf{M}
  16. 𝐧 · 𝐌 \mathbf{n}·\mathbf{M}
  17. 𝐇 \mathbf{H}
  18. 𝐇 = 𝐇 0 - γ 4 π 𝐌 0 , \mathbf{H}=\mathbf{H}_{0}-\frac{\gamma}{4\pi}\mathbf{M}_{0},
  19. H k = ( H 0 ) k - γ k 4 π ( M 0 ) k , k = x , y , z . H_{k}=(H_{0})_{k}-\frac{\gamma_{k}}{4\pi}(M_{0})_{k},\qquad k=x,y,z.
  20. < v a r > γ <var>γ

Demazure_module.html

  1. Ch ( F ( w λ ) ) = Δ 1 Δ 2 Δ n e λ \,\text{Ch}(F(w\lambda))=\Delta_{1}\Delta_{2}\cdots\Delta_{n}e^{\lambda}
  2. ρ \rho
  3. w u = w ( u + ρ ) - ρ w\cdot u=w(u+\rho)-\rho
  4. Δ α ( u ) = u - s α u 1 - e - α \Delta_{\alpha}(u)=\frac{u-s_{\alpha}\cdot u}{1-e^{-\alpha}}

Dempwolff_group.html

  1. 2 5 . GL 5 ( 𝔽 2 ) 2^{5\,.}\mathrm{GL}_{5}(\mathbb{F}_{2})
  2. GL 5 ( 𝔽 2 ) \mathrm{GL}_{5}(\mathbb{F}_{2})
  3. 2 5 2^{5}
  4. E 8 E_{8}
  5. E 8 E_{8}
  6. GL n ( 𝔽 q ) \mathrm{GL}_{n}(\mathbb{F}_{q})
  7. 𝔽 q n \mathbb{F}_{q}^{n}
  8. q > 2 q>2
  9. n n
  10. 2 3 . GL 3 ( 𝔽 2 ) 2^{3\,.}\mathrm{GL}_{3}(\mathbb{F}_{2})
  11. G 2 ( 𝔽 3 ) G_{2}(\mathbb{F}_{3})
  12. 2 4 . GL 4 ( 𝔽 2 ) 2^{4\,.}\mathrm{GL}_{4}(\mathbb{F}_{2})
  13. 2 5 . GL 5 ( 𝔽 2 ) 2^{5\,.}\mathrm{GL}_{5}(\mathbb{F}_{2})

Denisyuk_polynomials.html

  1. n = 0 t n M n ( x ) = 1 1 + t exp - x t 1 - t \displaystyle\sum_{n=0}^{\infty}t^{n}M_{n}(x)=\frac{1}{1+t}\exp-\frac{xt}{1-t}

Denjoy–Koksma_inequality.html

  1. k = 0 m - 1 f ( x + k ω ) \sum_{k=0}^{m-1}f(x+k\omega)
  2. | i = 0 q - 1 ϕ f i ( x ) - q T ϕ d μ | < Var ( ϕ ) \left|\sum_{i=0}^{q-1}\phi f^{i}(x)-q\int_{T}\phi\,d\mu\right|<\operatorname{% Var}(\phi)

Denjoy–Wolff_theorem.html

  1. | f ( z ) | δ ( r ) | z | , |f(z)|\leq\delta(r)|z|,
  2. δ ( r ) = M ( r ) r < 1. \delta(r)={M(r)\over r}<1.
  3. | f n ( z ) | δ ( r ) n |f^{n}(z)|\leq\delta(r)^{n}
  4. r k r_{k}
  5. f k ( z ) = r k f ( z ) . f_{k}(z)=r_{k}f(z).
  6. f k ( z ) - z f_{k}(z)-z
  7. g ( z ) = z g(z)=z
  8. f k f_{k}
  9. z k z_{k}
  10. z k w . z_{k}\rightarrow w.
  11. f k f_{k}
  12. f n f^{n}
  13. f n k f^{n_{k}}
  14. f n k + 1 - n k f^{n_{k+1}-n_{k}}
  15. h ( g ( w ) ) = g ( w ) , h(g(w))=g(w),\,
  16. h ( w ) = w h(w)=w
  17. m k = n k + 1 - n k m_{k}=n_{k+1}-n_{k}
  18. f m k - 1 f^{m_{k}-1}
  19. f n f^{n}

Dense_subgraph.html

  1. G G
  2. d G = 1.375 d_{G}=1.375
  3. b , c , d , e b,c,d,e
  4. h h
  5. 1.4 1.4
  6. G = ( E , V ) G=(E,V)
  7. S = ( E S , V S ) S=(E_{S},V_{S})
  8. G G
  9. S S
  10. d S = | E S | | V S | d_{S}={|E_{S}|\over|V_{S}|}
  11. k k
  12. k k
  13. k k
  14. k k
  15. n 1 / 4 + ϵ n^{1/4+\epsilon}
  16. ϵ > 0 \epsilon>0
  17. P = N P P=NP
  18. k k
  19. k k
  20. k k
  21. α \alpha
  22. Θ ( α 2 ) \Theta(\alpha^{2})
  23. k k
  24. k k
  25. k k
  26. k k

Dense_submodule.html

  1. y - 1 N = { r R y r N } y^{-1}N=\{r\in R\mid yr\in N\}\,
  2. x ( y - 1 N ) { 0 } x(y^{-1}N)\neq\{0\}\,
  3. N d M N\subseteq_{d}M\,
  4. Hom R ( M / N , E ( M ) ) = { 0 } \mathrm{Hom}_{R}(M/N,E(M))=\{0\}\,
  5. Ann ( I ) = { 0 } \ell\cdot\mathrm{Ann}(I)=\{0\}\,
  6. E ~ ( M ) = { x E ( M ) f S , f ( M ) = 0 f ( x ) = 0 } \tilde{E}(M)=\{x\in E(M)\mid\forall f\in S,f(M)=0\implies f(x)=0\}\,

Dependability_state_model.html

  1. λ \lambda
  2. μ - 1 \mu^{-1}
  3. 2 λ 2\lambda
  4. λ \lambda
  5. λ \lambda
  6. μ \mu
  7. 𝐀 𝟎 = [ 0 - μ 0 - λ 0 - μ 0 λ 0 ] . \mathbf{A_{0}}=\begin{bmatrix}0&-\mu&0\\ -\lambda&0&-\mu\\ 0&\lambda&0\end{bmatrix}.
  8. 𝐀 𝟏 = [ ( λ ) - μ 0 - λ ( λ + μ ) - μ 0 - λ ( μ ) ] . \mathbf{A_{1}}=\begin{bmatrix}(\lambda)&-\mu&0\\ -\lambda&(\lambda+\mu)&-\mu\\ 0&-\lambda&(\mu)\\ \end{bmatrix}.
  9. n P n = 1. \sum_{n}P_{n}=1.
  10. R ( t ) = e - λ t R(t)=e^{-\lambda t}\,

Dependence_logic.html

  1. = ( t 1 t n ) =\!\!(t_{1}\ldots t_{n})
  2. t 1 t n t_{1}\ldots t_{n}
  3. t n \!t_{n}
  4. t 1 t n - 1 t_{1}\ldots t_{n-1}
  5. R t 1 t n Rt_{1}\ldots t_{n}
  6. R \!R
  7. t 1 t n t_{1}\ldots t_{n}
  8. t 1 = t 2 \!t_{1}=t_{2}
  9. t 1 \!t_{1}
  10. t 2 \!t_{2}
  11. = ( t 1 t n ) =\!\!(t_{1}\ldots t_{n})
  12. n n\in\mathbb{N}
  13. t 1 t n t_{1}\ldots t_{n}
  14. ϕ \!\phi
  15. Free ( ϕ ) \mbox{Free}~{}(\phi)
  16. ϕ \!\phi
  17. Free ( ϕ ) \mbox{Free}~{}(\phi)
  18. ϕ \!\phi
  19. ¬ ϕ \lnot\phi
  20. Free ( ¬ ϕ ) = Free ( ϕ ) \mbox{Free}~{}(\lnot\phi)=\mbox{Free}~{}(\phi)
  21. ϕ \!\phi
  22. ψ \!\psi
  23. ϕ ψ \!\phi\vee\psi
  24. Free ( ϕ ψ ) = Free ( ϕ ) Free ( ψ ) \mbox{Free}~{}(\phi\vee\psi)=\mbox{Free}~{}(\phi)\cup\mbox{Free}~{}(\psi)
  25. ϕ \!\phi
  26. x \!x
  27. x ϕ \!\exists x\phi
  28. Free ( v ϕ ) = Free ( ϕ ) \ { v } \mbox{Free}~{}(\exists v\phi)=\mbox{Free}~{}(\phi)\backslash\{v\}
  29. ϕ \!\phi
  30. Free ( ϕ ) = \mbox{Free}~{}(\phi)=\emptyset
  31. ϕ ψ \!\phi\wedge\psi
  32. ¬ ( ¬ ϕ ¬ ψ ) \!\lnot(\lnot\phi\vee\lnot\psi)
  33. x ϕ \!\forall x\phi
  34. ¬ ( x ( ¬ ϕ ) ) \!\lnot(\exists x(\lnot\phi))
  35. 𝒜 = ( A , σ , I ) \!\mathcal{A}=(A,\sigma,I)
  36. V = { v 1 v n } V=\{v_{1}\ldots v_{n}\}
  37. 𝒜 \!\mathcal{A}
  38. V \!V
  39. 𝒜 \!\mathcal{A}
  40. V \!V
  41. μ \!\mu
  42. V \!V
  43. A \!A
  44. v 1 v n v_{1}\ldots v_{n}
  45. A \!A
  46. X \!X
  47. μ 1 μ 4 \!\mu_{1}\ldots\mu_{4}
  48. { v 1 , v 2 , v 3 } \!\{v_{1},v_{2},v_{3}\}
  49. v 1 \!v_{1}
  50. v 2 \!v_{2}
  51. v 3 \!v_{3}
  52. μ 1 \!\mu_{1}
  53. μ 1 ( v 1 ) \!\mu_{1}(v_{1})
  54. μ 1 ( v 2 ) \!\mu_{1}(v_{2})
  55. μ 1 ( v 3 ) \!\mu_{1}(v_{3})
  56. μ 2 \!\mu_{2}
  57. μ 2 ( v 1 ) \!\mu_{2}(v_{1})
  58. μ 2 ( v 2 ) \!\mu_{2}(v_{2})
  59. μ 2 ( v 3 ) \!\mu_{2}(v_{3})
  60. μ 3 \!\mu_{3}
  61. μ 3 ( v 1 ) \!\mu_{3}(v_{1})
  62. μ 3 ( v 2 ) \!\mu_{3}(v_{2})
  63. μ 3 ( v 3 ) \!\mu_{3}(v_{3})
  64. μ 4 \!\mu_{4}
  65. μ 4 ( v 1 ) \!\mu_{4}(v_{1})
  66. μ 4 ( v 2 ) \!\mu_{4}(v_{2})
  67. μ 4 ( v 3 ) \!\mu_{4}(v_{3})
  68. 𝒯 \!\mathcal{T}
  69. 𝒞 \mathcal{C}
  70. 𝒜 \mathcal{A}
  71. X X
  72. ϕ \!\phi
  73. X \!\!X
  74. ( 𝒜 , X , ϕ ) 𝒯 \!(\mathcal{A},X,\phi)\in\mathcal{T}
  75. X \!X
  76. ϕ \!\phi
  77. 𝒜 \!\mathcal{A}
  78. 𝒜 X + ϕ \!\mathcal{A}\models_{X}^{+}\phi
  79. ( 𝒜 , X , ϕ ) 𝒞 \!(\mathcal{A},X,\phi)\in\mathcal{C}
  80. X \!X
  81. ϕ \!\phi
  82. 𝒜 \!\mathcal{A}
  83. 𝒜 X - ϕ \!\mathcal{A}\models_{X}^{-}\phi
  84. 𝒜 X + ϕ \!\mathcal{A}\models_{X}^{+}\phi
  85. ϕ \!\phi
  86. X \!X
  87. 𝒜 \mathcal{A}
  88. 𝒜 X - ϕ \!\mathcal{A}\models_{X}^{-}\phi
  89. ϕ \!\phi
  90. X \!X
  91. 𝒜 \mathcal{A}
  92. ϕ \!\phi
  93. ϕ \!\phi
  94. 𝒜 \!\mathcal{A}
  95. 𝒜 { } + ϕ \!\mathcal{A}\models_{\{\emptyset\}}^{+}\phi
  96. ϕ \!\phi
  97. 𝒜 \!\mathcal{A}
  98. 𝒜 { } - ϕ \!\mathcal{A}\models_{\{\emptyset\}}^{-}\phi
  99. + \!\models^{+}
  100. - \!\models^{-}
  101. 𝒜 X + R t 1 t n \!\mathcal{A}\models_{X}^{+}Rt_{1}\ldots t_{n}
  102. R \!R
  103. 𝒜 \!\mathcal{A}
  104. t 1 t n \!t_{1}\ldots t_{n}
  105. X \!X
  106. μ X \!\mu\in X
  107. ( t 1 t n ) \!(t_{1}\ldots t_{n})
  108. μ \!\mu
  109. R \!R
  110. 𝒜 \!\mathcal{A}
  111. 𝒜 X + t 1 = t 2 \!\mathcal{A}\models_{X}^{+}t_{1}=t_{2}
  112. t 1 \!t_{1}
  113. t 2 \!t_{2}
  114. X \!X
  115. μ X \!\mu\in X
  116. t 1 \!t_{1}
  117. t 2 \!t_{2}
  118. 𝒜 \!\mathcal{A}
  119. 𝒜 X + = ( t 1 t n ) \!\mathcal{A}\models_{X}^{+}=\!\!(t_{1}\ldots t_{n})
  120. s , s X \!s,s^{\prime}\in X
  121. ( t 1 t n - 1 ) \!(t_{1}\ldots t_{n-1})
  122. t n \!t_{n}
  123. 𝒜 X + ¬ ϕ \!\mathcal{A}\models_{X}^{+}\lnot\phi
  124. 𝒜 X - ϕ \!\mathcal{A}\models_{X}^{-}\phi
  125. 𝒜 X + ϕ ψ \!\mathcal{A}\models_{X}^{+}\phi\vee\psi
  126. Y \!Y
  127. Z \!Z
  128. X = Y Z X=Y\cup Z
  129. 𝒜 Y + ϕ \!\mathcal{A}\models_{Y}^{+}\phi
  130. 𝒜 Z + ψ \!\mathcal{A}\models_{Z}^{+}\psi
  131. 𝒜 X + x ϕ \!\mathcal{A}\models_{X}^{+}\exists x\phi
  132. F \!F
  133. X \!X
  134. 𝒜 \!\mathcal{A}
  135. 𝒜 X [ F / x ] + ϕ \!\mathcal{A}\models_{X[F/x]}^{+}\phi
  136. X [ F / x ] = { s [ F ( s ) / x ] : s X } \!X[F/x]=\{s[F(s)/x]:s\in X\}
  137. 𝒜 X - R t 1 t n \!\mathcal{A}\models_{X}^{-}Rt_{1}\ldots t_{n}
  138. R \!R
  139. 𝒜 \!\mathcal{A}
  140. t 1 t n \!t_{1}\ldots t_{n}
  141. X \!X
  142. μ X \!\mu\in X
  143. ( t 1 t n ) \!(t_{1}\ldots t_{n})
  144. μ \!\mu
  145. R \!R
  146. 𝒜 \!\mathcal{A}
  147. 𝒜 X - t 1 = t 2 \!\mathcal{A}\models_{X}^{-}t_{1}=t_{2}
  148. t 1 \!t_{1}
  149. t 2 \!t_{2}
  150. X \!X
  151. μ X \!\mu\in X
  152. t 1 \!t_{1}
  153. t 2 \!t_{2}
  154. 𝒜 \!\mathcal{A}
  155. 𝒜 X - = ( t 1 t n ) \!\mathcal{A}\models_{X}^{-}=\!\!(t_{1}\ldots t_{n})
  156. X \!X
  157. 𝒜 X - ¬ ϕ \!\mathcal{A}\models_{X}^{-}\lnot\phi
  158. 𝒜 X + ϕ \!\mathcal{A}\models_{X}^{+}\phi
  159. 𝒜 X - ϕ ψ \!\mathcal{A}\models_{X}^{-}\phi\vee\psi
  160. 𝒜 X - ϕ \!\mathcal{A}\models_{X}^{-}\phi
  161. 𝒜 X - ψ \!\mathcal{A}\models_{X}^{-}\psi
  162. 𝒜 X - x ϕ \!\mathcal{A}\models_{X}^{-}\exists x\phi
  163. 𝒜 X [ A / x ] - ϕ \!\mathcal{A}\models_{X[A/x]}^{-}\phi
  164. X [ A / x ] = { s [ m / x ] : s A } \!X[A/x]=\{s[m/x]:s\in A\}
  165. A \!A
  166. 𝒜 \!\mathcal{A}
  167. ϕ \!\phi
  168. 𝒜 \!\mathcal{A}
  169. 𝒜 { } + ϕ \!\mathcal{A}\models_{\{\emptyset\}}^{+}\phi
  170. ϕ \!\phi
  171. 𝒜 \!\mathcal{A}
  172. ϕ \!\phi
  173. 𝒜 X + ϕ \!\mathcal{A}\models_{X}^{+}\phi
  174. μ X \!\mu\in X
  175. ϕ \!\phi
  176. 𝒜 \!\mathcal{A}
  177. z x 1 x 2 y 1 y 2 ( = ( x 1 , y 1 ) = ( x 2 , y 2 ) ( x 1 = x 2 y 1 = y 2 ) y 1 z ) \!\exists z\forall x_{1}\forall x_{2}\exists y_{1}\exists y_{2}(=\!\!(x_{1},y_% {1})\wedge=\!\!(x_{2},y_{2})\wedge(x_{1}=x_{2}\leftrightarrow y_{1}=y_{2})% \wedge y_{1}\not=z)
  178. 𝒜 \mathcal{A}
  179. ϕ \!\phi
  180. R 1 R n ψ ( R 1 R n ) \!\exists R_{1}\ldots\exists R_{n}\psi(R_{1}\ldots R_{n})
  181. ψ ( R 1 R n ) \!\psi(R_{1}\ldots R_{n})
  182. ( Q H x 1 , x 2 , y 1 , y 2 ) ϕ ( x 1 , x 2 , y 1 , y 2 ) ( x 1 y 1 x 2 y 2 ) ϕ ( x 1 , x 2 , y 1 , y 2 ) (Q_{H}x_{1},x_{2},y_{1},y_{2})\phi(x_{1},x_{2},y_{1},y_{2})\equiv\begin{% pmatrix}\forall x_{1}\exists y_{1}\\ \forall x_{2}\exists y_{2}\end{pmatrix}\phi(x_{1},x_{2},y_{1},y_{2})
  183. x 1 y 1 x 2 y 2 ( = ( x 1 , y 1 ) = ( x 2 , y 2 ) ϕ ) \forall x_{1}\exists y_{1}\forall x_{2}\exists y_{2}(=\!\!(x_{1},y_{1})\wedge=% \!\!(x_{2},y_{2})\wedge\phi)
  184. V = { v 1 v n } \!V=\{v_{1}\ldots v_{n}\}
  185. ϕ \!\phi
  186. V \!V
  187. ϕ D \!\phi^{D}
  188. 𝒜 X + ϕ 𝒜 X + ϕ D \mathcal{A}\models_{X}^{+}\phi\Leftrightarrow\mathcal{A}\models_{X}^{+}\phi^{D}
  189. 𝒜 \mathcal{A}
  190. X \!X
  191. V \!V
  192. ψ \!\psi
  193. V \!V
  194. ψ I \!\psi^{I}
  195. 𝒜 X + ψ 𝒜 X + ψ I \mathcal{A}\models_{X}^{+}\psi\Leftrightarrow\mathcal{A}\models_{X}^{+}\psi^{I}
  196. 𝒜 \mathcal{A}
  197. X \!X
  198. V \!V
  199. 𝒜 X ϕ \!\mathcal{A}\models_{X}\phi
  200. Y X \!Y\subseteq X
  201. 𝒜 Y ψ \!\mathcal{A}\models_{Y}\psi
  202. y ( = ( y ) y = x ) \!\exists y(=\!\!(y)\wedge y=x)
  203. X = { ( x : 0 ) , ( x : 1 ) } \!X=\{(x:0),(x:1)\}
  204. ϕ \phi
  205. ψ \psi
  206. ϕ \!\phi
  207. ψ \!\psi
  208. θ \!\theta
  209. ϕ \!\phi
  210. θ \!\theta
  211. θ \!\theta
  212. ψ \!\psi
  213. τ ( x ) \!\tau(x)
  214. ϕ \!\phi
  215. ω \mathcal{M}_{\omega}
  216. ϕ \!{}^{\prime}\phi^{\prime}
  217. ϕ \!\phi
  218. ω { } + ϕ \mathcal{M}_{\omega}\models^{+}_{\{\emptyset\}}\!\phi
  219. ω { } + τ ( ϕ ) . \mathcal{M}_{\omega}\models^{+}_{\{\emptyset\}}\tau(^{\prime}\phi^{\prime}).
  220. Π 2 \Pi_{2}
  221. ϕ \sim\!\!\phi
  222. ϕ ψ \phi\rightarrow\psi
  223. 𝒜 X ϕ ψ \!\mathcal{A}\models_{X}\phi\rightarrow\psi
  224. Y X \!Y\subseteq X
  225. 𝒜 Y ϕ \!\mathcal{A}\models_{Y}\phi
  226. 𝒜 Y ψ \!\mathcal{A}\models_{Y}\psi
  227. t 1 t 3 t 2 \vec{t_{1}}\bot_{\vec{t_{3}}}\vec{t_{2}}
  228. t 1 \vec{t_{1}}
  229. t 2 \vec{t_{2}}
  230. t 3 \vec{t_{3}}
  231. 𝒜 X t 1 t 3 t 2 \mathcal{A}\models_{X}\vec{t_{1}}\bot_{\vec{t_{3}}}\vec{t_{2}}
  232. s , s X s,s^{\prime}\in X
  233. t 3 s = t 3 s \vec{t_{3}}\langle s\rangle=\vec{t_{3}}\langle s^{\prime}\rangle
  234. s ′′ X s^{\prime\prime}\in X
  235. t 3 s ′′ = t 3 s \vec{t_{3}}\langle s^{\prime\prime}\rangle=\vec{t_{3}}\langle s\rangle
  236. t 1 s ′′ = t 1 s \vec{t_{1}}\langle s^{\prime\prime}\rangle=\vec{t_{1}}\langle s\rangle
  237. t 2 s ′′ = t 2 s \vec{t_{2}}\langle s^{\prime\prime}\rangle=\vec{t_{2}}\langle s^{\prime}\rangle
  238. t 1 t 2 \vec{t_{1}}\subseteq\vec{t_{2}}
  239. t 1 t 2 \vec{t_{1}}\mid\vec{t_{2}}
  240. t 1 \vec{t_{1}}
  241. t 2 \vec{t_{2}}
  242. 𝒜 X t 1 t 2 \mathcal{A}\models_{X}\vec{t_{1}}\subseteq\vec{t_{2}}
  243. s X s\in X
  244. s X s^{\prime}\in X
  245. t 1 s = t 2 s \vec{t_{1}}\langle s\rangle=\vec{t_{2}}\langle s^{\prime}\rangle
  246. 𝒜 X t 1 t 2 \mathcal{A}\models_{X}\vec{t_{1}}\mid\vec{t_{2}}
  247. s , s X s,s^{\prime}\in X
  248. t 1 s t 2 s \vec{t_{1}}\langle s\rangle\neq\vec{t_{2}}\langle s^{\prime}\rangle