wpmath0000009_4

Exponential_polynomial.html

  1. i = 1 m f i ( X ) exp ( w i X ) , \sum_{i=1}^{m}f_{i}(X)\exp(w_{i}X)\ ,

Exponential_utility.html

  1. u ( c ) = { ( 1 - e - a c ) / a a 0 c a = 0 u(c)=\begin{cases}(1-e^{-ac})/a&a\neq 0\\ c&a=0\\ \end{cases}
  2. c c
  3. a a
  4. a > 0 a>0
  5. a = 0 a=0
  6. a < 0 a<0
  7. u ( c ) = 1 - e - a c u(c)=1-e^{-ac}
  8. u ( c ) = ( 1 - e - a c ) / a u(c)=(1-e^{-ac})/a
  9. u ( c ) = - e - a c / a u(c)=-e^{-ac}/a
  10. - u ′′ ( c ) u ( c ) = a . \frac{-u^{\prime\prime}(c)}{u^{\prime}(c)}=a.
  11. ϵ \epsilon
  12. ϵ \epsilon
  13. E ( u ( c ) ) = E [ 1 - e - a ( c ( x ) + ϵ ) ] , \,\text{E}(u(c))=\,\text{E}[1-e^{-a(c(x)+\epsilon)}],
  14. ϵ N ( μ , σ 2 ) , \epsilon\sim N(\mu,\sigma^{2}),\!
  15. E [ e - a ϵ ] = e - a μ + a 2 2 σ 2 . \,\text{E}[e^{-a\epsilon}]=e^{-a\mu+\frac{a^{2}}{2}\sigma^{2}}.
  16. E ( u ( c ) ) = E [ 1 - e - a ( c ( x ) + ϵ ) ] = E [ 1 - e - a c ( x ) e - a ϵ ] = 1 - e - a c ( x ) E [ e - a ϵ ] = 1 - e - a c ( x ) e - a μ + a 2 2 σ 2 . \,\text{E}(u(c))=\,\text{E}[1-e^{-a(c(x)+\epsilon)}]=\,\text{E}[1-e^{-ac(x)}e^% {-a\epsilon}]=1-e^{-ac(x)}\,\text{E}[e^{-a\epsilon}]=1-e^{-ac(x)}e^{-a\mu+% \frac{a^{2}}{2}\sigma^{2}}.
  17. E [ - e - a W ] \,\text{E}[-e^{-aW}]
  18. W = x r + ( W 0 - x k ) r f W=x^{\prime}r+(W_{0}-x^{\prime}k)\cdot r_{f}
  19. W 0 W_{0}
  20. W 0 - x k W_{0}-x^{\prime}k
  21. E [ - e - a W ] = - E [ e - a [ x r + ( W 0 - x k ) r f ] ] = - e - a [ ( W 0 - x k ) r f ] E e - a x r = - e - a [ ( W 0 - x k ) r f ] e - a x μ + a 2 2 σ 2 \,\text{E}[-e^{-aW}]=-\,\text{E}[e^{-a[x^{\prime}r+(W_{0}-x^{\prime}k)\cdot r_% {f}]}]=-e^{-a[(W_{0}-x^{\prime}k)r_{f}]}Ee^{-a\cdot x^{\prime}r}=-e^{-a[(W_{0}% -x^{\prime}k)r_{f}]}e^{-a\cdot x^{\prime}\mu+\frac{a^{2}}{2}\sigma^{2}}
  22. μ \mu
  23. σ 2 \sigma^{2}
  24. e a r f ( x k ) - a x μ + a 2 2 σ 2 , e^{ar_{f}(x^{\prime}k)-a\cdot x^{\prime}\mu+\frac{a^{2}}{2}\sigma^{2}},
  25. x ( μ - r f k ) - a 2 σ 2 . x^{\prime}(\mu-r_{f}\cdot k)-\frac{a}{2}\sigma^{2}.
  26. σ 2 \sigma^{2}
  27. x V x x^{\prime}Vx
  28. x ( μ - r f k ) - a 2 x V x . x^{\prime}(\mu-r_{f}\cdot k)-\frac{a}{2}\cdot x^{\prime}Vx.
  29. x * = 1 a V - 1 ( μ - r f k ) . x^{*}=\frac{1}{a}V^{-1}(\mu-r_{f}\cdot k).

Extension_by_definitions.html

  1. \emptyset
  2. \emptyset
  3. x ( x ) \forall x(x\notin\emptyset)
  4. \emptyset
  5. T T
  6. ϕ ( x 1 , , x n ) \phi(x_{1},\dots,x_{n})
  7. T T
  8. x 1 x_{1}
  9. x n x_{n}
  10. ϕ ( x 1 , , x n ) \phi(x_{1},\dots,x_{n})
  11. T T^{\prime}\,
  12. T T
  13. n n
  14. R R
  15. R R
  16. x 1 x n ( R ( x 1 , , x n ) ϕ ( x 1 , , x n ) ) \forall x_{1}\dots\forall x_{n}(R(x_{1},\dots,x_{n})\leftrightarrow\phi(x_{1},% \dots,x_{n}))
  17. R R
  18. ψ \psi
  19. T T^{\prime}\,
  20. ψ \psi^{\ast}
  21. T T
  22. ψ \psi
  23. R ( t 1 , , t n ) R(t_{1},\dots,t_{n})
  24. ϕ ( t 1 , , t n ) \phi(t_{1},\dots,t_{n})
  25. ϕ \phi
  26. t i t_{i}
  27. ϕ ( t 1 , , t n ) \phi(t_{1},\dots,t_{n})
  28. ψ ψ \psi\leftrightarrow\psi^{\ast}
  29. T T^{\prime}\,
  30. T T^{\prime}\,
  31. T T
  32. T T^{\prime}\,
  33. T T
  34. R R
  35. ψ \psi^{\ast}
  36. ψ \psi
  37. T T
  38. ψ \psi^{\ast}
  39. ψ \psi
  40. R R
  41. T T
  42. ϕ ( y , x 1 , , x n ) \phi(y,x_{1},\dots,x_{n})
  43. T T
  44. y y
  45. x 1 x_{1}
  46. x n x_{n}
  47. ϕ ( y , x 1 , , x n ) \phi(y,x_{1},\dots,x_{n})
  48. x 1 x n ! y ϕ ( y , x 1 , , x n ) \forall x_{1}\dots\forall x_{n}\exists!y\phi(y,x_{1},\dots,x_{n})
  49. T T
  50. x 1 x_{1}
  51. x n x_{n}
  52. ϕ ( y , x 1 , , x n ) \phi(y,x_{1},\dots,x_{n})
  53. T T^{\prime}\,
  54. T T
  55. n n
  56. f f
  57. f f
  58. x 1 x n ϕ ( f ( x 1 , , x n ) , x 1 , , x n ) \forall x_{1}\dots\forall x_{n}\phi(f(x_{1},\dots,x_{n}),x_{1},\dots,x_{n})
  59. f f
  60. ψ \psi
  61. T T^{\prime}\,
  62. ψ \psi^{\ast}
  63. T T
  64. f f
  65. ψ \psi
  66. ψ \psi^{\ast}
  67. ψ \psi
  68. f ( t 1 , , t n ) f(t_{1},\dots,t_{n})
  69. ψ \psi
  70. f f
  71. t i t_{i}
  72. χ \chi
  73. ψ \psi
  74. z z
  75. f f
  76. χ \chi
  77. ψ \psi
  78. χ \chi^{\ast}
  79. ψ \psi^{\ast}
  80. z ( ϕ ( z , t 1 , , t n ) χ ) \forall z(\phi(z,t_{1},\dots,t_{n})\rightarrow\chi^{\ast})
  81. ϕ \phi
  82. t i t_{i}
  83. ϕ ( z , t 1 , , t n ) \phi(z,t_{1},\dots,t_{n})
  84. ψ \psi
  85. ψ \psi^{\ast}
  86. χ \chi
  87. χ \chi^{\ast}
  88. ψ ψ \psi\leftrightarrow\psi^{\ast}
  89. T T^{\prime}\,
  90. T T^{\prime}\,
  91. T T
  92. ψ \psi^{\ast}
  93. ψ \psi
  94. T T
  95. ψ \psi^{\ast}
  96. ψ \psi
  97. f f
  98. T T^{\prime}\,
  99. T T
  100. T T
  101. T T^{\prime}\,
  102. T T
  103. ψ \psi
  104. T T^{\prime}\,
  105. ψ \psi^{\ast}
  106. T T
  107. ψ \psi
  108. T T
  109. ψ ψ \psi\leftrightarrow\psi^{\ast}
  110. T T^{\prime}\,
  111. T T^{\prime}\,
  112. T T^{\prime}\,
  113. = =
  114. \in
  115. \subseteq
  116. \emptyset
  117. T T
  118. \cdot
  119. x ( x e = x and e x = x ) \forall x(x\cdot e=x\,\text{ and }e\cdot x=x)
  120. T T^{\prime}\,
  121. T T^{\prime}\,
  122. T ′′ T^{\prime\prime}\,
  123. T T^{\prime}\,
  124. f f
  125. x ( x f ( x ) = e and f ( x ) x = e ) \forall x(x\cdot f(x)=e\,\text{ and }f(x)\cdot x=e)
  126. f ( x ) f(x)
  127. x - 1 x^{-1}

Extensions_of_symmetric_operators.html

  1. z z - i z + i z\mapsto\frac{z-i}{z+i}
  2. U A = ( A - i ) ( A + i ) - 1 U_{A}=(A-i)(A+i)^{-1}\,
  3. A U A A\mapsto U_{A}
  4. A = - i ( U + 1 ) ( U - 1 ) - 1 , A=-i(U+1)(U-1)^{-1},\,
  5. U ~ \tilde{U}
  6. A ~ = - i ( U ~ + 1 ) ( U ~ - 1 ) - 1 \tilde{A}=-i(\tilde{U}+1)(\tilde{U}-1)^{-1}
  7. R a n ( - i 2 ( U ~ - 1 ) ) = R a n ( U ~ - 1 ) Ran(-\frac{i}{2}(\tilde{U}-1))=Ran(\tilde{U}-1)
  8. K + = R a n ( A + i ) K_{+}=Ran(A+i)^{\perp}
  9. K - = R a n ( A - i ) . K_{-}=Ran(A-i)^{\perp}.
  10. A f = i d d x f . Af=i\frac{d}{dx}f.
  11. K + = s p a n { ϕ + = a e x } K_{+}=span\{\phi_{+}=a\cdot e^{x}\}
  12. K - = s p a n { ϕ - = a e - x } K_{-}=span\{\phi_{-}=a\cdot e^{-x}\}
  13. | α | = 1 {|α|=1}
  14. D o m ( A α ) = { f + β ( α ϕ - - ϕ + ) | f D o m ( A ) , β } . Dom(A_{\alpha})=\{f+\beta(\alpha\phi_{-}-\phi_{+})|f\in Dom(A),\;\beta\in% \mathbb{C}\}.
  15. | f ( 0 ) f ( 1 ) | = | e α - 1 α - e | = 1. \left|\frac{f(0)}{f(1)}\right|=\left|\frac{e\alpha-1}{\alpha-e}\right|=1.
  16. ( A + 1 ) - 1 ( B + 1 ) - 1 (A+1)^{-1}\geq(B+1)^{-1}
  17. D Γ = ( 1 - Γ * Γ ) 1 2 and D Γ * = ( 1 - Γ Γ * ) 1 2 . D_{\Gamma}=(1-\Gamma^{*}\Gamma)^{\frac{1}{2}}\quad\mbox{and}~{}\quad D_{\Gamma% ^{*}}=(1-\Gamma\Gamma^{*})^{\frac{1}{2}}.
  18. 𝒟 Γ = R a n ( D Γ ) and 𝒟 Γ * = R a n ( D Γ * ) . \mathcal{D}_{\Gamma}=Ran(D_{\Gamma})\quad\mbox{and}~{}\quad\mathcal{D}_{\Gamma% ^{*}}=Ran(D_{\Gamma^{*}}).
  19. Γ = [ Γ 1 D Γ 1 * Γ 2 Γ 3 D Γ 1 - Γ 3 Γ 1 * Γ 2 + D Γ 3 * Γ 4 D Γ 2 ] \Gamma=\begin{bmatrix}\Gamma_{1}&D_{\Gamma_{1}^{*}}\Gamma_{2}\\ \Gamma_{3}D_{\Gamma_{1}}&-\Gamma_{3}\Gamma_{1}^{*}\Gamma_{2}+D_{\Gamma_{3}^{*}% }\Gamma_{4}D_{\Gamma_{2}}\end{bmatrix}
  20. | a - 1 a + 1 | 1. \left|\frac{a-1}{a+1}\right|\leq 1.
  21. C A : R a n ( A + 1 ) R a n ( A - 1 ) C_{A}:Ran(A+1)\rightarrow Ran(A-1)\subset\mathcal{H}
  22. C A ( A + 1 ) x = ( A - 1 ) x . i.e. C A = ( A - 1 ) ( A + 1 ) - 1 . C_{A}(A+1)x=(A-1)x.\quad\mbox{i.e.}~{}\quad C_{A}=(A-1)(A+1)^{-1}.\,
  23. C A = [ Γ 1 Γ 3 D Γ 1 ] : R a n ( A + 1 ) R a n ( A + 1 ) R a n ( A + 1 ) . C_{A}=\begin{bmatrix}\Gamma_{1}\\ \Gamma_{3}D_{\Gamma_{1}}\end{bmatrix}:Ran(A+1)\rightarrow\begin{matrix}Ran(A+1% )\\ \oplus\\ Ran(A+1)^{\perp}\end{matrix}.
  24. A = ( 1 + C A ) ( 1 - C A ) - 1 A=(1+C_{A})(1-C_{A})^{-1}\,
  25. C ~ \tilde{C}
  26. A ~ = ( 1 + C ~ ) ( 1 - C ~ ) - 1 \tilde{A}=(1+\tilde{C})(1-\tilde{C})^{-1}
  27. R a n ( 1 - C ~ ) Ran(1-\tilde{C})
  28. C ~ ( Γ 4 ) = [ Γ 1 D Γ 1 Γ 3 * Γ 3 D Γ 1 - Γ 3 Γ 1 Γ 3 * + D Γ 3 * Γ 4 D Γ 3 * ] . \tilde{C}(\Gamma_{4})=\begin{bmatrix}\Gamma_{1}&D_{\Gamma_{1}}\Gamma_{3}^{*}\\ \Gamma_{3}D_{\Gamma_{1}}&-\Gamma_{3}\Gamma_{1}\Gamma_{3}^{*}+D_{\Gamma_{3}^{*}% }\Gamma_{4}D_{\Gamma_{3}^{*}}\end{bmatrix}.
  29. 𝒟 Γ 3 * \mathcal{D}_{\Gamma_{3}^{*}}
  30. C ~ ( - 1 ) and C ~ ( 1 ) \tilde{C}(-1)\quad\mbox{and}~{}\quad\tilde{C}(1)
  31. A 0 and A A_{0}\quad\mbox{and}~{}\quad A_{\infty}
  32. A 0 B A A_{0}\leq B\leq A_{\infty}

Extouch_triangle.html

  1. T A = 0 : csc 2 ( B / 2 ) : csc 2 ( C / 2 ) T_{A}=0:\csc^{2}{\left(B/2\right)}:\csc^{2}{\left(C/2\right)}
  2. T B = csc 2 ( A / 2 ) : 0 : csc 2 ( C / 2 ) T_{B}=\csc^{2}{\left(A/2\right)}:0:\csc^{2}{\left(C/2\right)}
  3. T C = csc 2 ( A / 2 ) : csc 2 ( B / 2 ) : 0 T_{C}=\csc^{2}{\left(A/2\right)}:\csc^{2}{\left(B/2\right)}:0
  4. T A = 0 : a - b + c b : a + b - c c T_{A}=0:\frac{a-b+c}{b}:\frac{a+b-c}{c}
  5. T B = - a + b + c a : 0 : a + b - c c T_{B}=\frac{-a+b+c}{a}:0:\frac{a+b-c}{c}
  6. T C = - a + b + c a : a - b + c b : 0 T_{C}=\frac{-a+b+c}{a}:\frac{a-b+c}{b}:0
  7. A T A_{T}
  8. A T = A 2 r 2 s a b c A_{T}=A\frac{2r^{2}s}{abc}
  9. A A
  10. r r
  11. s s
  12. a a
  13. b b
  14. c c

Extra_element_theorem.html

  1. H ( s ) = H ( s ) 1 + Z n ( s ) Z ( s ) 1 + Z d ( s ) Z ( s ) H(s)=H_{\infty}(s)\frac{1+\frac{Z_{n}(s)}{Z(s)}}{1+\frac{Z_{d}(s)}{Z(s)}}
  2. H ( s ) = H 0 ( s ) 1 + Z ( s ) Z n ( s ) 1 + Z ( s ) Z d ( s ) H(s)=H_{0}(s)\frac{1+\frac{Z(s)}{Z_{n}(s)}}{1+\frac{Z(s)}{Z_{d}(s)}}
  3. H ( s ) H(s)
  4. Z ( s ) Z(s)
  5. Z n ( s ) = v n ( s ) / i n ( s ) Z_{n}(s)=v_{n}(s)/i_{n}(s)
  6. Z i n = Z i n 1 + Z e 0 Z 1 + Z e Z Z_{in}=Z^{\infty}_{in}\cdot\frac{1+\frac{Z^{0}_{e}}{Z}}{1+\frac{Z^{\infty}_{e}% }{Z}}
  7. Z Z
  8. Z i n Z^{\infty}_{in}
  9. Z e 0 Z^{0}_{e}
  10. Z e Z^{\infty}_{e}
  11. Z i n Z_{in}
  12. Z = 1 s Z=\frac{1}{s}
  13. Z i n = 2 1 + 1 = 5 3 Z^{\infty}_{in}=2\|1+1=\frac{5}{3}
  14. Z e 0 = 1 ( 1 + 1 1 ) = 3 5 Z^{0}_{e}=1\|(1+1\|1)=\frac{3}{5}
  15. Z e = 2 1 + 1 = 5 3 Z^{\infty}_{e}=2\|1+1=\frac{5}{3}
  16. Z i n = 5 3 1 + 3 5 s 1 + 5 3 s = 5 + 3 s 3 + 5 s Z_{in}=\frac{5}{3}\cdot\frac{1+\frac{3}{5}s}{1+\frac{5}{3}s}=\frac{5+3s}{3+5s}

Face_diagonal.html

  1. a 2 + b 2 , \sqrt{a^{2}+b^{2}},
  2. a 2 + c 2 , \sqrt{a^{2}+c^{2}},
  3. b 2 + c 2 . \sqrt{b^{2}+c^{2}}.
  4. a 2 a\sqrt{2}

Facility_location_problem.html

  1. O ( n k ) O(n^{\sqrt{k}})
  2. O ( 2 O ( k log k / ε 2 ) d n ) O(2^{O(k\log k/\varepsilon^{2})}dn)
  3. O ( k n ) O(k^{n})

Factor_shares.html

  1. Y = F ( K , L ) Y=F(K,L)\,
  2. π = m a x { K , L } F ( K , L ) * P - r K - w L \pi=max_{\{K,L\}}F(K,L)*P-rK-wL\,
  3. r K + w L rK+wL\,
  4. w = D L [ F ( K , L ) ] w=D_{L}[F(K,L)]\,
  5. r = D K [ F ( K , L ) ] r=D_{K}[F(K,L)]\,
  6. w L = D L [ F ( K , L ) ] * L wL=D_{L}[F(K,L)]*L\,
  7. r K = D K [ F ( K , L ) ] * K rK=D_{K}[F(K,L)]*K\,
  8. w L / Y = D L [ F ( K , L ) ] * L / F ( K , L ) wL/Y=D_{L}[F(K,L)]*L/F(K,L)\,
  9. r K / Y = D K [ F ( K , L ) ] * K / F ( K , L ) rK/Y=D_{K}[F(K,L)]*K/F(K,L)\,

Fair_computational_tree_logic.html

  1. G F P i \bigwedge GFP_{i}
  2. ( G F R G F C ) \bigwedge(GFR\longrightarrow GFC)
  3. π = s o , s 1 \pi=s_{o},s_{1}\dots
  4. ϕ \phi
  5. ϕ \phi
  6. ϕ \phi
  7. ϕ \phi

Fanno_flow.html

  1. d M 2 M 2 = γ M 2 1 - M 2 ( 1 + γ - 1 2 M 2 ) 4 f D h d x \ \frac{dM^{2}}{M^{2}}=\frac{\gamma M^{2}}{1-M^{2}}\left(1+\frac{\gamma-1}{2}M% ^{2}\right)\frac{4f}{D_{h}}dx
  2. 4 f L * D h = ( 1 - M 2 γ M 2 ) + ( γ + 1 2 γ ) ln [ M 2 ( 2 γ + 1 ) ( 1 + γ - 1 2 M 2 ) ] \ \frac{4fL^{*}}{D_{h}}=\left(\frac{1-M^{2}}{\gamma M^{2}}\right)+\left(\frac{% \gamma+1}{2\gamma}\right)\ln\left[\frac{M^{2}}{\left(\frac{2}{\gamma+1}\right)% \left(1+\frac{\gamma-1}{2}M^{2}\right)}\right]
  3. Δ S = Δ s c p = ln [ M γ - 1 γ ( [ 2 γ + 1 ] [ 1 + γ - 1 2 M 2 ] ) - ( γ + 1 ) 2 γ ] \ \Delta S=\frac{\Delta s}{c_{p}}=\ln\left[M^{\frac{\gamma-1}{\gamma}}\left(% \left[\frac{2}{\gamma+1}\right]\left[1+\frac{\gamma-1}{2}M^{2}\right]\right)^{% \frac{-(\gamma+1)}{2\gamma}}\right]
  4. H = h h 0 = c p T c p T 0 = T T 0 \ H=\frac{h}{h_{0}}=\frac{c_{p}T}{c_{p}T_{0}}=\frac{T}{T_{0}}
  5. Δ S = Δ s c p = ln [ ( 1 H - 1 ) γ - 1 2 γ ( 2 γ - 1 ) γ - 1 2 γ ( γ + 1 2 ) γ + 1 2 γ ( H ) γ + 1 2 γ ] \ \Delta S=\frac{\Delta s}{c_{p}}=\ln\left[\left(\frac{1}{H}-1\right)^{\frac{% \gamma-1}{2\gamma}}\left(\frac{2}{\gamma-1}\right)^{\frac{\gamma-1}{2\gamma}}% \left(\frac{\gamma+1}{2}\right)^{\frac{\gamma+1}{2\gamma}}\left(H\right)^{% \frac{\gamma+1}{2\gamma}}\right]
  6. A = A * = constant T 0 = T 0 * = constant m ˙ = m ˙ * = constant \begin{aligned}\displaystyle A&\displaystyle=A^{*}=\mbox{constant}\\ \displaystyle T_{0}&\displaystyle=T_{0}^{*}=\mbox{constant}\\ \displaystyle\dot{m}&\displaystyle=\dot{m}^{*}=\mbox{constant}\end{aligned}
  7. p p * = 1 M 1 ( 2 γ + 1 ) ( 1 + γ - 1 2 M 2 ) ρ ρ * = 1 M ( 2 γ + 1 ) ( 1 + γ - 1 2 M 2 ) T T * = 1 ( 2 γ + 1 ) ( 1 + γ - 1 2 M 2 ) V V * = M 1 ( 2 γ + 1 ) ( 1 + γ - 1 2 M 2 ) p 0 p 0 * = 1 M [ ( 2 γ + 1 ) ( 1 + γ - 1 2 M 2 ) ] γ + 1 2 ( γ - 1 ) \begin{aligned}\displaystyle\frac{p}{p^{*}}&\displaystyle=\frac{1}{M}\frac{1}{% \sqrt{\left(\frac{2}{\gamma+1}\right)\left(1+\frac{\gamma-1}{2}M^{2}\right)}}% \\ \displaystyle\frac{\rho}{\rho^{*}}&\displaystyle=\frac{1}{M}\sqrt{\left(\frac{% 2}{\gamma+1}\right)\left(1+\frac{\gamma-1}{2}M^{2}\right)}\\ \displaystyle\frac{T}{T^{*}}&\displaystyle=\frac{1}{\left(\frac{2}{\gamma+1}% \right)\left(1+\frac{\gamma-1}{2}M^{2}\right)}\\ \displaystyle\frac{V}{V^{*}}&\displaystyle=M\frac{1}{\sqrt{\left(\frac{2}{% \gamma+1}\right)\left(1+\frac{\gamma-1}{2}M^{2}\right)}}\\ \displaystyle\frac{p_{0}}{p_{0}^{*}}&\displaystyle=\frac{1}{M}\left[\left(% \frac{2}{\gamma+1}\right)\left(1+\frac{\gamma-1}{2}M^{2}\right)\right]^{\frac{% \gamma+1}{2\left(\gamma-1\right)}}\end{aligned}
  8. Δ S F = s - s i c p = ln [ ( M M i ) γ - 1 γ ( 1 + γ - 1 2 M i 2 1 + γ - 1 2 M 2 ) γ + 1 2 γ ] Δ S R = s - s i c p = ln [ ( M M i ) 2 ( 1 + γ M i 2 1 + γ M 2 ) γ + 1 γ ] \begin{aligned}\displaystyle\Delta S_{F}&\displaystyle=\frac{s-s_{i}}{c_{p}}=% \ln\left[\left(\frac{M}{M_{i}}\right)^{\frac{\gamma-1}{\gamma}}\left(\frac{1+% \frac{\gamma-1}{2}M_{i}^{2}}{1+\frac{\gamma-1}{2}M^{2}}\right)^{\frac{\gamma+1% }{2\gamma}}\right]\\ \displaystyle\Delta S_{R}&\displaystyle=\frac{s-s_{i}}{c_{p}}=\ln\left[\left(% \frac{M}{M_{i}}\right)^{2}\left(\frac{1+\gamma M_{i}^{2}}{1+\gamma M^{2}}% \right)^{\frac{\gamma+1}{\gamma}}\right]\end{aligned}
  9. ( 1 + γ - 1 2 M i 2 ) [ M i 2 ( 1 + γ M i 2 ) 2 ] = ( 1 + γ - 1 2 M 2 ) [ M 2 ( 1 + γ M 2 ) 2 ] \ \left(1+\frac{\gamma-1}{2}M_{i}^{2}\right)\left[\frac{M_{i}^{2}}{\left(1+% \gamma M_{i}^{2}\right)^{2}}\right]=\left(1+\frac{\gamma-1}{2}M^{2}\right)% \left[\frac{M^{2}}{\left(1+\gamma M^{2}\right)^{2}}\right]

Faraday's_laws_of_electrolysis.html

  1. m = ( Q F ) ( M z ) m\ =\ \left({Q\over F}\right)\left({M\over z}\right)
  2. Q = I t Q=It
  3. m = ( I t F ) ( M z ) m\ =\ \left({It\over F}\right)\left({M\over z}\right)
  4. n = ( I t F ) ( 1 z ) n\ =\ \left({It\over F}\right)\left({1\over z}\right)
  5. τ \tau
  6. τ \tau
  7. Q = 0 t I ( τ ) d τ Q=\int_{0}^{t}I(\tau)\ d\tau

Faro_shuffle.html

  1. S 2 n S_{2n}
  2. ( 1 2 3 4 1 n + 1 2 n + 2 ) \begin{pmatrix}1&2&3&4&\cdots\\ 1&n+1&2&n+2&\cdots\end{pmatrix}
  3. k { k 2 k odd n + k 2 k even k\mapsto\begin{cases}\left\lceil\frac{k}{2}\right\rceil&k\ \,\text{odd}\\ n+\frac{k}{2}&k\ \,\text{even}\end{cases}
  4. ( k , n ) (k,n)
  5. S k n S_{kn}
  6. ( 2 , n ) (2,n)
  7. ρ n \rho_{n}
  8. ( 2 , n - 1 ) (2,n-1)
  9. n n
  10. ρ n \rho_{n}
  11. sgn ( ρ n ) = ( - 1 ) n + 1 sgn ( ρ n - 1 ) . \mbox{sgn}~{}(\rho_{n})=(-1)^{n+1}\mbox{sgn}~{}(\rho_{n-1}).
  12. sgn ( ρ n ) = ( - 1 ) n / 2 = { + 1 n 0 , 1 ( mod 4 ) - 1 n 2 , 3 ( mod 4 ) \mbox{sgn}~{}(\rho_{n})=(-1)^{\lfloor n/2\rfloor}=\begin{cases}+1&n\equiv 0,1% \;\;(\mathop{{\rm mod}}4)\\ -1&n\equiv 2,3\;\;(\mathop{{\rm mod}}4)\end{cases}
  13. ρ 0 \rho_{0}
  14. ρ 1 \rho_{1}
  15. ρ 2 \rho_{2}
  16. ( 23 ) S 4 (23)\in S_{4}

Farouk_Kamoun.html

  1. N N
  2. l n ( N ) ln(N)
  3. e l n ( N ) e\cdot ln(N)

Fast_marching_method.html

  1. F ( x ) | T ( x ) | = 1. F(x)|\nabla T(x)|=1.
  2. T T
  3. F ( x ) F(x)
  4. x x
  5. x x
  6. F ( x ) | S T ( x ) | = 1 , for the surface S , and x S . F(x)|\nabla_{S}T(x)|=1,\,\,\mbox{for the surface}~{}\,\,S,\,\mbox{and}~{}\,\,x% \in S.

Fatou's_theorem.html

  1. f f
  2. D 2 = { z : | z | < 1 } D^{2}=\{z:|z|<1\}
  3. r r
  4. f r : S 1 f_{r}:S^{1}\rightarrow\mathbb{C}
  5. f r ( e i θ ) = f ( r e i θ ) f_{r}(e^{i\theta})=f(re^{i\theta})
  6. S 1 := { e i θ : θ [ 0 , 2 π ] } = { z : | z | = 1 } S^{1}:=\{e^{i\theta}:\theta\in[0,2\pi]\}=\{z\in\mathbb{C}:|z|=1\}
  7. f f
  8. f r f_{r}
  9. r 1 r\rightarrow 1
  10. L p L^{p}
  11. f r f_{r}
  12. f : D 2 f:D^{2}\rightarrow\mathbb{C}
  13. sup 0 < r < 1 f r L p ( S 1 ) < . \sup_{0<r<1}\lVert f_{r}\rVert_{L^{p}(S^{1})}<\infty.
  14. f r f_{r}
  15. f 1 L p ( S 1 ) f_{1}\in L^{p}(S^{1})
  16. L p L^{p}
  17. f r - f 1 L p ( S 1 ) 0 \lVert f_{r}-f_{1}\rVert_{L^{p}(S^{1})}\rightarrow 0
  18. | f r ( e i θ ) - f 1 ( e i θ ) | 0 |f_{r}(e^{i\theta})-f_{1}(e^{i\theta})|\rightarrow 0
  19. θ [ 0 , 2 π ] \theta\in[0,2\pi]
  20. f ( r e i θ ) f 1 ( e i θ ) f(re^{i\theta})\rightarrow f_{1}(e^{i\theta})
  21. θ \theta
  22. γ : [ 0 , 1 ) D 2 \gamma:[0,1)\rightarrow D^{2}
  23. e i θ e^{i\theta}
  24. f f
  25. f 1 ( e i θ ) f_{1}(e^{i\theta})
  26. γ ( t ) = t e i θ \gamma(t)=te^{i\theta}
  27. γ \gamma
  28. γ \gamma
  29. γ : [ 0 , 1 ) D 2 \gamma:[0,1)\rightarrow D^{2}
  30. lim t 1 γ ( t ) = e i θ S 1 \lim_{t\rightarrow 1}\gamma(t)=e^{i\theta}\in S^{1}
  31. Γ α = { z : arg z [ π - α , π + α ] } \Gamma_{\alpha}=\{z:\arg z\in[\pi-\alpha,\pi+\alpha]\}
  32. Γ α ( θ ) = D 2 e i θ ( Γ α + 1 ) . \Gamma_{\alpha}(\theta)=D^{2}\cap e^{i\theta}(\Gamma_{\alpha}+1).
  33. Γ α ( θ ) \Gamma_{\alpha}(\theta)
  34. 2 α 2\alpha
  35. e i θ e^{i\theta}
  36. γ \gamma
  37. e i θ e^{i\theta}
  38. α ( 0 , π 2 ) \alpha\in(0,\frac{\pi}{2})
  39. γ \gamma
  40. Γ α \Gamma_{\alpha}
  41. lim t 1 γ ( t ) = e i θ \lim_{t\rightarrow 1}\gamma(t)=e^{i\theta}
  42. f H p ( D 2 ) f\in H^{p}(D^{2})
  43. θ [ 0 , 2 π ] \theta\in[0,2\pi]
  44. lim t 1 f ( γ ( t ) ) = f 1 ( e i θ ) \lim_{t\rightarrow 1}f(\gamma(t))=f_{1}(e^{i\theta})
  45. γ \gamma
  46. e i θ e^{i\theta}
  47. f 1 f_{1}

Favard_constant.html

  1. K r = 4 π k = 0 [ ( - 1 ) k 2 k + 1 ] r + 1 . K_{r}=\frac{4}{\pi}\sum\limits_{k=0}^{\infty}\left[\frac{(-1)^{k}}{2k+1}\right% ]^{r+1}.

Feasible_region.html

  1. x 1 2 + x 2 4 x_{1}^{2}+x_{2}^{4}
  2. x 1 x_{1}
  3. x 2 , x_{2},
  4. 1 x 1 10 1\leq x_{1}\leq 10
  5. 5 x 2 12. 5\leq x_{2}\leq 12.\,
  6. x 1 2 + x 2 4 . x_{1}^{2}+x_{2}^{4}.

Fedor_Bogomolov.html

  1. H 2 ( M ) H^{2}(M)
  2. c 1 2 > c 2 c_{1}^{2}>c_{2}
  3. V I I 0 VII_{0}
  4. b 2 = 0 b_{2}=0
  5. b 2 = 1 b_{2}=1
  6. V I I 0 VII_{0}
  7. b 2 = 0 b_{2}=0
  8. b 2 = 1 b_{2}=1
  9. V I I 0 VII_{0}
  10. b 2 = 0 b_{2}=0
  11. b 2 = 1 b_{2}=1

Feldman–Mahalanobis_model.html

  1. Y t = Y 0 { 1 + α 0 λ k β k + λ c β c λ k β k [ ( 1 + λ k β k ) t - 1 ] } Y_{t}=Y_{0}\left\{1+\alpha_{0}\frac{\lambda_{k}\beta_{k}+\lambda_{c}\beta_{c}}% {\lambda_{k}\beta_{k}}\left[(1+\lambda_{k}\beta_{k})^{t}-1\right]\right\}
  2. λ k \lambda_{k}
  3. λ c \lambda_{c}
  4. λ k \lambda_{k}
  5. λ c \lambda_{c}

Felici's_law.html

  1. q ( t ) = 1 R [ Φ ( 0 ) - Φ ( t ) ] q(t)={1\over\mathrm{R}}[\Phi(0)-\Phi(t)]
  2. q ( t ) = 0 t i ( τ ) d τ = 1 R 0 t f e m ( τ ) d τ = 1 R [ Φ ( 0 ) - Φ ( t ) ] q(t)=\int_{0}^{t}i(\tau)d\tau={1\over\mathrm{R}}\int_{0}^{t}{f_{em}}(\tau)d% \tau={1\over\mathrm{R}}[\Phi(0)-\Phi(t)]
  3. f e m ( t ) = - d Φ ( t ) d t {f_{em}}(t)=-{d\Phi(t)\over dt}

Feller's_coin-tossing_constants.html

  1. lim n p ( n , k ) α k n + 1 = β k \lim_{n\rightarrow\infty}p(n,k)\alpha_{k}^{n+1}=\beta_{k}\,
  2. x k + 1 = 2 k + 1 ( x - 1 ) x^{k+1}=2^{k+1}(x-1)\,
  3. β k = 2 - α k k + 1 - k α k . \beta_{k}={2-\alpha_{k}\over k+1-k\alpha_{k}}.
  4. α k \alpha_{k}
  5. β k \beta_{k}
  6. k = 2 k=2
  7. 5 - 1 = 2 φ - 2 = 2 / φ \sqrt{5}-1=2\varphi-2=2/\varphi
  8. 1 - 1 / 5 1-1/\sqrt{5}
  9. k k
  10. 9 64 \tfrac{9}{64}

Fermi_coordinates.html

  1. γ \gamma
  2. M M
  3. p p
  4. γ \gamma
  5. ( t , x 2 , , x n ) (t,x^{2},\ldots,x^{n})
  6. p p
  7. ( t , 0 , , 0 ) (t,0,\ldots,0)
  8. p p
  9. γ \gamma
  10. γ \gamma
  11. p p
  12. p p

Feynman_Long_Division_Puzzles.html

  1. . . A . . A . ) . . . . A . . ¯ . . A A ¯ . . . A . . A ¯ . . . . . A . . ¯ . . . . . . . . ¯ 0 \begin{matrix}\qquad\qquad\quad\,\!\!.\;\,.\;\,A\;\,.\\ .\,A\,.\overline{)\;\,.\;\,.\;\,.\;\,.\;\,A\;\,.\;\,.}\\ \;\!\underline{.\;.\,A\;A}\\ \quad\;\,.\;\,.\;\,.\;\,A\\ \qquad\underline{.\;\,.\;\,A}\\ \qquad\;\;\,.\;\,\,.\;\,\,.\;\,\,.\\ \qquad\quad\!\underline{.\;\,A\;\,.\;\,.}\\ \qquad\qquad\quad\!.\;\;.\;\;.\;\;.\\ \qquad\qquad\quad\!\underline{.\;\;.\;\;.\;\;.}\\ \qquad\qquad\qquad\quad\;0\end{matrix}

Fiber_derivative.html

  1. Q Q
  2. L L
  3. T Q TQ
  4. T * Q T^{*}Q
  5. 𝔽 L : T Q T * Q \mathbb{F}L:TQ\rightarrow T^{*}Q
  6. 𝔽 L ( v ) w = d d s | s = 0 L ( v + s w ) \mathbb{F}L(v)\cdot w=\frac{d}{ds}|_{s=0}L(v+sw)
  7. v v
  8. w w

Field_desorption.html

  1. [ M + H ] + [M+H]^{+}\,

File:Beaty_Middle_School.jpg.html

  1. 5 - - 6 + + h ; [ d ] g t u h [ t r u k j = t y u i p ? 9 = 8 / g l [ r e 5--6++h;[d]gtuh[trukj\-=tyuip?9=8~{}/gl[re

File:Binomial_distribution_pdf.png.html

  1. k { 0 , , n } k\in\{0,\dots,n\}\!

File:Breathing_DS_reaction_diffusion.gif.html

  1. t u = d u 2 Δ u + λ u - u 3 - κ 3 v + κ 1 , τ t v = d v 2 Δ v + u - v \begin{array}[]{rl}\partial_{t}u&=d_{u}^{2}\Delta u+\lambda u-u^{3}-\kappa_{3}% v+\kappa_{1},\\ \tau\partial_{t}v&=d_{v}^{2}\Delta v+u-v\end{array}

File:Data_from_National_Vital_Statistics_Report_tPx.png.html

  1. l x \,l_{x}
  2. p x t \,{}_{t}p_{x}
  3. x x
  4. t t
  5. x x
  6. t t

File:DS_collision_ginzburg_landau.gif.html

  1. t q = ( d r + i d i ) Δ q + l r q + ( c r + i c i ) | q | 2 q + ( q r + i q i ) | q | 4 q . \partial_{t}q=(d_{r}+id_{i})\Delta q+l_{r}q+(c_{r}+ic_{i})|q|^{2}q+(q_{r}+iq_{% i})|q|^{4}q.

File:DS_interpenetration_swift_hohenberg.gif.html

  1. t q = ( s r + i s i ) Δ 2 q + ( d r + i d i ) Δ q + l r q + ( c r + i c i ) | q | 2 q + ( q r + i q i ) | q | 4 q . \partial_{t}q=(s_{r}+is_{i})\Delta^{2}q+(d_{r}+id_{i})\Delta q+l_{r}q+(c_{r}+% ic_{i})|q|^{2}q+(q_{r}+iq_{i})|q|^{4}q.

File:Fbg2.GIF.html

  1. P B ( λ ) P_{B}(\lambda)
  2. λ \lambda
  3. λ B \lambda_{B}
  4. Δ λ B \Delta\lambda_{B}
  5. P B ( λ B ) P_{B}(\lambda_{B})

File:Fbg3.GIF.html

  1. Λ \Lambda
  2. Δ λ \Delta\lambda

File:Fbg4.GIF.html

  1. π \pi

File:Feynman'sDiagram.jpg.html

  1. γ μ \gamma_{\mu}
  2. γ μ \gamma_{\mu}
  3. 1 / q 2 1/q^{2}

File:FillingFactor.png.html

  1. 1 - exp ( - F π r 2 S α L ) ~{}1-\exp\left(-F\frac{\pi r^{2}}{S}\alpha L\right)~{}

File:Jacobi_lebesgue.svg.html

  1. α = - 0.5 \alpha=-0.5
  2. α = - 0.5 \alpha=-0.5
  3. α = - 0.5 \alpha=-0.5

File:Partially_built_tableau.svg.html

  1. { a b , ¬ a c , ¬ c ¬ b } \{a\wedge b,\neg a\vee c,\neg c\vee\neg b\}

File:PlotDelta.gif.html

  1. π ( x ) \pi(x)
  2. π 0 ( x ) \pi_{0}(x)
  3. li ( x ) \mathrm{li}(x)
  4. π ( x ) \pi(x)
  5. li ( x ) \mathrm{li}(x)

File:Pythagorean_triangle.png.html

  1. d x 2 + d y 2 = d l 2 dx^{2}+dy^{2}=\,dl^{2}

File:QuantumReflection.png.html

  1. r = 1 / ( 1 + k w ) 4 ~{}r=1/(1+kw)^{4}~{}

File:RPDE_detail.gif.html

  1. ϵ \epsilon
  2. T T
  3. P ( T ) P(T)
  4. H n o r m H_{norm}

File:RPDE_real.gif.html

  1. H n o r m H_{norm}

File:Singular_perturbation_convergence.jpg.html

  1. ϵ y ′′ + ( 1 + ϵ ) y + y = 0 , \epsilon y^{\prime\prime}+(1+\epsilon)y^{\prime}+y=0,\,
  2. y ( 0 ) = 0 y(0)=0
  3. y ( 1 ) = 1 y(1)=1
  4. ϵ \epsilon

File:Uniform_curl.JPG.html

  1. F ( x , y ) = y s y m b o l x ^ - x s y m b o l y ^ F(x,y)=ysymbol{\hat{x}}-xsymbol{\hat{y}}

File:Viral_Reproduction_Chart.png.html

  1. = = S u m m a r y = = ==Summary==

Filled_Julia_set.html

  1. K ( f ) \ K(f)
  2. f \ f
  3. K ( f ) \ K(f)
  4. f \ f
  5. z z\,
  6. f \ f
  7. K ( f ) = def { z : f ( k ) ( z ) ↛ a s k } \ K(f)\ \overset{\underset{\mathrm{def}}{}}{=}\ \{z\in\mathbb{C}:f^{(k)}(z)% \not\to\infty\ as\ k\to\infty\}
  8. \mathbb{C}
  9. f ( k ) ( z ) \ f^{(k)}(z)
  10. k \ k
  11. f f\,
  12. f f\,
  13. K ( f ) = A f ( ) K(f)=\mathbb{C}\setminus A_{f}(\infty)
  14. A f ( ) = F A_{f}(\infty)=F_{\infty}
  15. K ( f ) = F C . K(f)=F_{\infty}^{C}.
  16. J ( f ) = K ( f ) = A f ( ) J(f)\,=\partial K(f)=\partial A_{f}(\infty)
  17. A f ( ) A_{f}(\infty)
  18. f f
  19. A f ( ) = def { z : f ( k ) ( z ) a s k } . A_{f}(\infty)\ \overset{\underset{\mathrm{def}}{}}{=}\ \{z\in\mathbb{C}:f^{(k)% }(z)\to\infty\ as\ k\to\infty\}.
  20. f f
  21. f ( z ) = z 2 + c f(z)=z^{2}+c
  22. f c f_{c}
  23. c c
  24. S c S_{c}\,
  25. K \ K\,
  26. β \beta\,
  27. - β -\beta\,
  28. S c = [ - β , β ] S_{c}=\left[-\beta,\beta\right]\,
  29. K \ K\,
  30. K K\,
  31. z c r = 0 z_{cr}=0\,
  32. β \beta\,
  33. 0 K \mathcal{R}^{K}_{0}
  34. - β -\beta\,
  35. 1 / 2 K \mathcal{R}^{K}_{1/2}
  36. - β -\beta\,
  37. β \beta\,
  38. K K\,
  39. K K\,
  40. 0
  41. R R\,
  42. R = def R 1 / 2 S c R 0 R\ \overset{\underset{\mathrm{def}}{}}{=}\ R_{1/2}\ \cup\ S_{c}\ \cup\ R_{0}\,

Filling_area_conjecture.html

  1. π \pi
  2. π \pi
  3. π \pi
  4. S 2 \R 3 S^{2}\subset\R^{3}\,\!
  5. π \pi
  6. π \pi
  7. π \pi
  8. π \pi

Filling_factor.html

  1. F ~{}F~{}
  2. 1 - exp ( - F π r 2 S α L ) , 1-\exp\left(-F\frac{\pi r^{2}}{S}\alpha L\right),
  3. S ~{}S~{}
  4. r ~{}r~{}
  5. α ~{}\alpha~{}
  6. L ~{}L~{}
  7. F ~{}F~{}
  8. 0 < F < 1 ~{}0<F<1~{}

Film_temperature.html

  1. T f T_{f}
  2. T w T_{w}
  3. T T_{\infty}
  4. T f = T w + T 2 T_{f}=\frac{T_{w}+T_{\infty}}{2}

Filtering_problem_(stochastic_processes).html

  1. d Y t = b ( t , Y t ) d t + σ ( t , Y t ) d B t , \mathrm{d}Y_{t}=b(t,Y_{t})\,\mathrm{d}t+\sigma(t,Y_{t})\,\mathrm{d}B_{t},
  2. H t = c ( t , Y t ) + γ ( t , Y t ) noise . H_{t}=c(t,Y_{t})+\gamma(t,Y_{t})\cdot\mbox{noise}~{}.
  3. Z t = 0 t H s d s , Z_{t}=\int_{0}^{t}H_{s}\,\mathrm{d}s,
  4. d Z t = c ( t , Y t ) d t + γ ( t , Y t ) d W t , \mathrm{d}Z_{t}=c(t,Y_{t})\,\mathrm{d}t+\gamma(t,Y_{t})\,\mathrm{d}W_{t},
  5. | c ( t , x ) | + | γ ( t , x ) | C ( 1 + | x | ) \big|c(t,x)\big|+\big|\gamma(t,x)\big|\leq C\big(1+|x|\big)
  6. K = K ( Z , t ) = L 2 ( Ω , G t , 𝐏 ; 𝐑 n ) . K=K(Z,t)=L^{2}(\Omega,G_{t},\mathbf{P};\mathbf{R}^{n}).
  7. 𝐄 [ | Y t - Y ^ t | 2 ] = inf Y K 𝐄 [ | Y t - Y ^ | 2 ] . (M) \mathbf{E}\left[\big|Y_{t}-\hat{Y}_{t}\big|^{2}\right]=\inf_{Y\in K}\mathbf{E}% \left[\big|Y_{t}-\hat{Y}\big|^{2}\right].\qquad\mbox{(M)}~{}
  8. Y ^ t = P K ( Z , t ) ( X t ) , \hat{Y}_{t}=P_{K(Z,t)}\big(X_{t}\big),
  9. P K : L 2 ( Ω , Σ , 𝐏 ; 𝐑 n ) L 2 ( Ω , F , 𝐏 ; 𝐑 n ) P_{K}:L^{2}(\Omega,\Sigma,\mathbf{P};\mathbf{R}^{n})\to L^{2}(\Omega,F,\mathbf% {P};\mathbf{R}^{n})
  10. P K ( X ) = 𝐄 [ X | F ] . P_{K}(X)=\mathbf{E}\big[X\big|F\big].
  11. Y ^ t = P K ( Z , t ) ( X t ) = 𝐄 [ X t | G t ] . \hat{Y}_{t}=P_{K(Z,t)}\big(X_{t}\big)=\mathbf{E}\big[X_{t}\big|G_{t}\big].

Financial_result.html

  1. = Interest income \displaystyle=\mbox{ Interest income}

Finite-dimensional_von_Neumann_algebra.html

  1. Z ( 𝐌 ) = i Z ( 𝐌 ) P i Z(\mathbf{M})=\oplus_{i}Z(\mathbf{M})P_{i}
  2. 𝐌 = i 𝐌 P i . {\mathbf{M}}=\oplus_{i}{\mathbf{M}}P_{i}.

Finite-rank_operator.html

  1. M = α u v * , where u = v = 1 and α 0. M=\alpha\cdot uv^{*},\quad\mbox{where}~{}\quad\|u\|=\|v\|=1\quad\mbox{and}~{}% \quad\alpha\geq 0.
  2. T h = α h , v u for all h H , Th=\alpha\langle h,v\rangle u\quad\mbox{for all}~{}\quad h\in H,
  3. T h = i = 1 n α i h , v i u i for all h H , Th=\sum_{i=1}^{n}\alpha_{i}\langle h,v_{i}\rangle u_{i}\quad\mbox{for all}~{}% \quad h\in H,
  4. S h , k = S g , k T S h , f , S_{h,k}=S_{g,k}TS_{h,f},\,
  5. T : U V T:U\to V
  6. T h = i = 1 n α i h , v i u i for all h U , Th=\sum_{i=1}^{n}\alpha_{i}\langle h,v_{i}\rangle u_{i}\quad\mbox{for all}~{}% \quad h\in U,
  7. u i V u_{i}\in V
  8. v i U v_{i}\in U^{\prime}
  9. U U

First-order_reaction.html

  1. d C d t = - k C \frac{dC}{dt}=-kC
  2. C = C i e - k t C=C_{i}e^{-kt}
  3. k = k 0 e - E a R T k=k_{0}e^{\frac{-E_{a}}{RT}}
  4. t t
  5. C C
  6. C i C_{i}
  7. m o l m - 3 molm^{-3}
  8. k 0 k_{0}
  9. E a E_{a}
  10. k k
  11. R R

Fisher's_noncentral_hypergeometric_distribution.html

  1. A = ω - 1 A=\omega-1
  2. B = m 1 + n - N - ( m 1 + n + 2 ) ω B=m_{1}+n-N-(m_{1}+n+2)\omega
  3. C = ( m 1 + 1 ) ( n + 1 ) ω C=(m_{1}+1)(n+1)\omega
  4. P 2 P 0 - ( P 1 P 0 ) 2 \frac{P_{2}}{P_{0}}-\left(\frac{P_{1}}{P_{0}}\right)^{2}
  5. μ - 2 c b - b 2 - 4 a c \mu\approx\frac{-2c}{b-\sqrt{b^{2}-4ac}}\,
  6. a = ω - 1 a=\omega-1
  7. b = m 1 + n - N - ( m 1 + n ) ω b=m_{1}+n-N-(m_{1}+n)\omega
  8. c = m 1 n ω c=m_{1}n\omega
  9. σ 2 N N - 1 / ( 1 μ + 1 m 1 - μ + 1 n - μ + 1 μ + m 2 - n ) \sigma^{2}\approx\frac{N}{N-1}\bigg/\left(\frac{1}{\mu}+\frac{1}{m_{1}-\mu}+% \frac{1}{n-\mu}+\frac{1}{\mu+m_{2}-n}\right)
  10. fnchypg ( x ; n , m 1 , N , ω ) = fnchypg ( n - x ; n , m 2 , N , 1 / ω ) . \operatorname{fnchypg}(x;n,m_{1},N,\omega)=\operatorname{fnchypg}(n-x;n,m_{2},% N,1/\omega)\,.
  11. fnchypg ( x ; n , m 1 , N , ω ) = fnchypg ( x ; m 1 , n , N , ω ) . \operatorname{fnchypg}(x;n,m_{1},N,\omega)=\operatorname{fnchypg}(x;m_{1},n,N,% \omega)\,.
  12. fnchypg ( x ; n , m 1 , N , ω ) = fnchypg ( m 1 - x ; N - n , m 1 , N , 1 / ω ) . \operatorname{fnchypg}(x;n,m_{1},N,\omega)=\operatorname{fnchypg}(m_{1}-x;N-n,% m_{1},N,1/\omega)\,.
  13. fnchypg ( x ; n , m 1 , N , ω ) = fnchypg ( x - 1 ; n , m 1 , N , ω ) ( m 1 - x + 1 ) ( n - x + 1 ) x ( m 2 - n + x ) ω . \operatorname{fnchypg}(x;n,m_{1},N,\omega)=\operatorname{fnchypg}(x-1;n,m_{1},% N,\omega)\frac{(m_{1}-x+1)(n-x+1)}{x(m_{2}-n+x)}\omega\,.
  14. n n
  15. n tot n_{\,\text{tot}}
  16. n succ n_{\,\text{succ}}
  17. w w
  18. { w f ( x ) ( x - n ) ( n succ - x ) - ( x + 1 ) f ( x + 1 ) ( n + n succ - n tot - x - 1 ) = 0 , f ( 0 ) = 1 F 1 2 ( - n , - n succ ; - n - n succ + n tot + 1 ; w ) } \left\{\begin{array}[]{l}wf(x)(x-n)(n\text{succ}-x)-(x+1)f(x+1)(n+n\text{succ}% -n\text{tot}-x-1)=0,\\ f(0)=\frac{1}{\,{}_{2}F_{1}(-n,-n\text{succ};-n-n\text{succ}+n\text{tot}+1;w)}% \end{array}\right\}
  19. μ i = m i r ω i r ω i + 1 \mu_{i}=\frac{m_{i}r\omega_{i}}{r\omega_{i}+1}
  20. i = 1 c μ i = n \sum_{i=1}^{c}\mu_{i}=n\,
  21. mfnchypg ( 𝐱 ; n , 𝐦 , s y m b o l ω ) = mfnchypg ( 𝐱 ; n , 𝐦 , r s y m b o l ω ) \operatorname{mfnchypg}(\mathbf{x};n,\mathbf{m},symbol{\omega})=\operatorname{% mfnchypg}(\mathbf{x};n,\mathbf{m},rsymbol{\omega})\,\,
  22. r + . r\in\mathbb{R}_{+}.
  23. mfnchypg ( 𝐱 ; n , 𝐦 , ( ω 1 , , ω c - 1 , ω c - 1 ) ) \displaystyle{}\operatorname{mfnchypg}\left(\mathbf{x};n,\mathbf{m},(\omega_{1% },\ldots,\omega_{c-1},\omega_{c-1})\right)
  24. hypg ( x ; n , m , N ) \operatorname{hypg}(x;n,m,N)

Fisher-Kolmogorov_equation.html

  1. u t = α k u ( 1 - u q ) + 2 u x 2 . \frac{\partial u}{\partial t}=\frac{\alpha}{k}u(1-u^{q})+\frac{\partial^{2}u}{% \partial x^{2}}.\,

Fitness_model_(network_theory).html

  1. Π i = η i k i j η j k j . \Pi_{i}=\frac{\eta_{i}k_{i}}{\sum_{j}\eta_{j}k_{j}}.
  2. k i k_{i}
  3. k i t = m Π i = m η i k i j η j k j \frac{\partial k_{i}}{\partial t}=m\Pi_{i}=m\frac{\eta_{i}k_{i}}{\sum_{j}\eta_% {j}k_{j}}
  4. k η ( t , t i ) = m ( t t i ) β ( η i ) k_{\eta}(t,t_{i})=m\left(\frac{t}{t_{i}}\right)^{\beta(\eta_{i})}
  5. β ( η ) = η C and C = ρ ( η ) η 1 - β ( η ) d η . \beta(\eta)=\frac{\eta}{C}\mbox{ and }~{}C=\int\rho(\eta)\frac{\eta}{1-\beta(% \eta)}\,d\eta.
  6. k ( t ) k(t)
  7. P ( k ) ρ ( η ) C η ( m / k ) C / η + 1 d η P(k)\sim\int\rho(\eta)\frac{C}{\eta}(m/k)^{C/\eta+1}\,d\eta

FitzHugh–Nagumo_model.html

  1. I ext I_{\,\text{ext}}
  2. v v
  3. w w
  4. v v
  5. w w
  6. v ˙ = v - v 3 3 - w + I ext \dot{v}=v-\frac{v^{3}}{3}-w+I_{\rm ext}
  7. τ w ˙ = v + a - b w . \tau\dot{w}=v+a-bw.
  8. a = b = 0 a=b=0

Fixation_(population_genetics).html

  1. p p
  2. μ \mu
  3. 2 N μ 2N\mu
  4. 2 N μ × 1 2 N = μ 2N\mu\times\frac{1}{2N}=\mu

Fixed-income_attribution.html

  1. P ( y , t ) P\left({y,t}\right)
  2. δ P = < m t p l > P t δ t + P y δ y + 1 2 2 P y 2 δ y 2 + O ( δ t 2 , δ y 3 ) \delta P=\frac{<}{m}tpl>{{\partial P}}{{\partial t}}\delta t+\frac{{\partial P% }}{{\partial y}}\delta y+\frac{1}{2}\frac{{\partial^{2}P}}{{\partial y^{2}}}% \delta y^{2}+O\left({\delta t^{2},\delta y^{3}}\right)
  3. δ r = < m t p l > δ P P \delta r=\frac{<}{m}tpl>{{\delta P}}{P}
  4. δ r = y δ t - M D δ y + 1 2 C δ y 2 + O ( δ t 2 , δ y 3 ) \delta r=y\cdot\delta t-MD\cdot\delta y+\frac{1}{2}C\cdot\delta y^{2}+O\left({% \delta t^{2},\delta y^{3}}\right)
  5. M D = - 1 P < m t p l > P y MD=-\frac{1}{P}\frac{<}{m}tpl>{{\partial P}}{{\partial y}}
  6. C = 1 P < m t p l > 2 P y 2 C=\frac{1}{P}\frac{<}{m}tpl>{{\partial^{2}P}}{{\partial y^{2}}}
  7. M D MD
  8. C C
  9. y ( m ) = a 0 + a 1 m + a 2 m 2 y\left(m\right)=a_{0}+a_{1}m+a_{2}m^{2}
  10. m m
  11. a 0 , a 1 , a 2 a_{0},a_{1},a_{2}
  12. y ( m ) y\left(m\right)
  13. m m
  14. y ( m ) = β 0 + β 1 [ 1 - exp ( - m / τ ) ] m / τ + β 2 ( [ 1 - exp ( - m / τ ) ] m / τ - exp ( - m / τ ) ) y\left(m\right)=\beta_{0}+\beta_{1}\frac{{\left[{1-\exp\left({-m/\tau}\right)}% \right]}}{m/\tau}+\beta_{2}{\left(\frac{{\left[{1-\exp\left({-m/\tau}\right)}% \right]}}{m/\tau}-\exp\left({-m/\tau}\right)\right)}
  15. y ( m ) y\left(m\right)
  16. m m
  17. β 0 \beta_{0}
  18. β 1 \beta_{1}
  19. β 2 \beta_{2}
  20. τ \tau
  21. β 0 \beta_{0}
  22. β 1 \beta_{1}
  23. β 2 \beta_{2}
  24. τ \tau
  25. τ \tau
  26. β 2 \beta_{2}
  27. + β 3 ( [ 1 - exp ( - m / τ 2 ) ] m / τ 2 - exp ( - m / τ 2 ) ) +\beta_{3}{\left(\frac{{\left[{1-\exp\left({-m/\tau_{2}}\right)}\right]}}{m/% \tau_{2}}-\exp\left({-m/\tau_{2}}\right)\right)}
  28. β 2 \beta_{2}
  29. τ \tau
  30. a 0 a_{0}
  31. r y i e l d = y δ t r_{yield}=y\cdot\delta t
  32. y y
  33. δ t \delta t
  34. D e D_{e}
  35. D e = - P ( y + δ y ) - P ( y - δ y ) < m t p l > 2 P ( y ) δ y D_{e}=-\frac{{P\left({y+\delta y}\right)-P\left({y-\delta y}\right)}}{<}mtpl>{% {2\cdot P\left(y\right)\cdot\delta y}}
  36. P ( y ) P\left(y\right)
  37. y y
  38. δ r y i e l d = i = 1 m K R D i δ y i \delta r_{yield}=\sum\limits_{i=1}^{m}{KRD_{i}\cdot\delta y_{i}}
  39. δ r y i e l d s t e e p e n i n g = i = 1 m K R D i δ y i s t e e p e n i n g \delta r_{yield}^{steepening}=\sum\limits_{i=1}^{m}{KRD_{i}\cdot\delta y_{i}^{% steepening}}

Fixed-point_property.html

  1. f : A A f:A\to A
  2. f : X X f:X\to X
  3. r : X A r:X\to A
  4. f : A A f:A\to A
  5. i f r : X X i\circ f\circ r:X\to X
  6. i : A X i:A\to X
  7. x A x\in A
  8. f r ( x ) = x f\circ r(x)=x
  9. x A x\in A
  10. r ( x ) = x r(x)=x
  11. f ( x ) = x . f(x)=x.

Fizeau_experiment.html

  1. w + = c n + v , w_{+}=\frac{c}{n}+v\ ,
  2. w - = c n - v , w_{-}=\frac{c}{n}-v\ ,
  3. w + = c n + v ( 1 - 1 n 2 ) . w_{+}=\frac{c}{n}+v\left(1-\frac{1}{n^{2}}\right)\ .
  4. f = ( 1 - 1 n 2 ) . f=\left(1-\frac{1}{n^{2}}\right)\ .
  5. w + = c n + v ( 1 - 1 n 2 - λ n d n d λ ) w_{+}=\frac{c}{n}+v\left(1-\frac{1}{n^{2}}-\frac{\lambda}{n}\!\cdot\!\frac{% \mathrm{d}n}{\mathrm{d}\lambda}\right)
  6. t 1 = A B c + v + D E c n - v t_{1}=\frac{AB}{c+v}+\frac{DE}{\frac{c}{n}-v}
  7. t 2 = A B c - v + D E c n + v t_{2}=\frac{AB}{c-v}+\frac{DE}{\frac{c}{n}+v}
  8. t = t - v x c 2 t^{\prime}=t-\frac{vx}{c^{2}}
  9. V L A B = c n + v 1 + c n v c 2 = c n + v 1 + v c n V_{LAB}=\frac{\frac{c}{n}+v}{1+\frac{\frac{c}{n}v}{c^{2}}}=\frac{\frac{c}{n}+v% }{1+\frac{v}{cn}}
  10. V L A B - c n = c n + v 1 + v c n - c n = c n + v - c n ( 1 + v c n ) 1 + v c n V_{LAB}-\frac{c}{n}=\frac{\frac{c}{n}+v}{1+\frac{v}{cn}}-\frac{c}{n}=\frac{% \frac{c}{n}+v-\frac{c}{n}(1+\frac{v}{cn})}{1+\frac{v}{cn}}
  11. = v ( 1 - 1 n 2 ) 1 + v c n v ( 1 - 1 n 2 ) =\frac{v\left(1-\frac{1}{n^{2}}\right)}{1+\frac{v}{cn}}\approx v\left(1-\frac{% 1}{n^{2}}\right)

Flare_(countermeasure).html

  1. e e

Flatness_(systems_theory).html

  1. 𝐱 ˙ ( t ) = 𝐟 ( 𝐱 ( t ) , 𝐮 ( t ) ) , 𝐱 ( 0 ) = 𝐱 0 , 𝐮 ( t ) R m , 𝐱 ( t ) R n , Rank 𝐟 ( 𝐱 , 𝐮 ) 𝐮 = m \dot{\mathbf{x}}(t)=\mathbf{f}(\mathbf{x}(t),\mathbf{u}(t)),\quad\mathbf{x}(0)% =\mathbf{x}_{0},\quad\mathbf{u}(t)\in R^{m},\quad\mathbf{x}(t)\in R^{n},\,% \text{Rank}\frac{\partial\mathbf{f}(\mathbf{x},\mathbf{u})}{\partial\mathbf{u}% }=m
  2. 𝐲 ( t ) = ( y 1 ( t ) , , y m ( t ) ) \mathbf{y}(t)=(y_{1}(t),...,y_{m}(t))
  3. y i , i = 1 , , m y_{i},i=1,...,m
  4. x i , i = 1 , , n x_{i},i=1,...,n
  5. u i , i = 1 , , m u_{i},i=1,...,m
  6. u i ( k ) , k = 1 , , α i u_{i}^{(k)},k=1,...,\alpha_{i}
  7. 𝐲 = Φ ( 𝐱 , 𝐮 , 𝐮 ˙ , , 𝐮 ( α ) ) \mathbf{y}=\Phi(\mathbf{x},\mathbf{u},\dot{\mathbf{u}},...,\mathbf{u}^{(\alpha% )})
  8. x i , i = 1 , , n x_{i},i=1,...,n
  9. u i , i = 1 , , m u_{i},i=1,...,m
  10. y i , i = 1 , , m y_{i},i=1,...,m
  11. y i ( k ) , i = 1 , , m y_{i}^{(k)},i=1,...,m
  12. 𝐲 \mathbf{y}
  13. ϕ ( 𝐲 , 𝐲 ˙ , 𝐲 ( γ ) ) = 𝟎 \phi(\mathbf{y},\dot{\mathbf{y}},\mathbf{y}^{(\gamma)})=\mathbf{0}
  14. 𝐱 ˙ ( t ) = 𝐀𝐱 ( t ) + 𝐁𝐮 ( t ) , 𝐱 ( 0 ) = 𝐱 0 \dot{\mathbf{x}}(t)=\mathbf{A}\mathbf{x}(t)+\mathbf{B}\mathbf{u}(t),\quad% \mathbf{x}(0)=\mathbf{x}_{0}
  15. 𝐱 , 𝐮 \mathbf{x},\mathbf{u}

Flexible_polyhedron.html

  1. n 5 n\geq 5

Florimond_de_Beaune.html

  1. d y d x = α y - x \frac{\operatorname{d}y}{\operatorname{d}x}=\frac{\alpha}{y-x}

Flory_convention.html

  1. l i l_{i}
  2. θ i \theta_{i}
  3. ϕ i \phi_{i}

Flow_velocity.html

  1. 𝐮 = 𝐮 ( 𝐱 , t ) \mathbf{u}=\mathbf{u}(\mathbf{x},t)
  2. 𝐱 \mathbf{x}\,
  3. t t\,
  4. q = || 𝐮 || q=||\mathbf{u}||
  5. 𝐮 \mathbf{u}
  6. 𝐮 t = 0. \frac{\partial\mathbf{u}}{\partial t}=0.
  7. 𝐮 \mathbf{u}
  8. 𝐮 = 0. \nabla\cdot\mathbf{u}=0.
  9. 𝐮 \mathbf{u}
  10. 𝐮 \mathbf{u}
  11. × 𝐮 = 0. \nabla\times\mathbf{u}=0.
  12. 𝐮 \mathbf{u}
  13. Φ , \Phi,
  14. 𝐮 = Φ . \mathbf{u}=\nabla\Phi.
  15. Δ Φ = 0. \Delta\Phi=0.
  16. ω \omega
  17. ω = × 𝐮 . \omega=\nabla\times\mathbf{u}.
  18. ϕ \phi
  19. 𝐮 = ϕ \mathbf{u}=\nabla\mathbf{\phi}
  20. ϕ \phi

Fluid-attenuated_inversion_recovery.html

  1. TI = ln ( 2 ) T 1 , \textrm{TI}=\ln(2)\cdot T_{1},\,

Fluorescence_cross-correlation_spectroscopy.html

  1. G \ G
  2. R \ R
  3. G G R ( τ ) = 1 + δ I G ( t ) δ I R ( t + τ ) I G ( t ) I R ( t ) = I G ( t ) I R ( t + τ ) I G ( t ) I R ( t ) \ G_{GR}(\tau)=1+\frac{\langle\delta I_{G}(t)\delta I_{R}(t+\tau)\rangle}{% \langle I_{G}(t)\rangle\langle I_{R}(t)\rangle}=\frac{\langle I_{G}(t)I_{R}(t+% \tau)\rangle}{\langle I_{G}(t)\rangle\langle I_{R}(t)\rangle}
  4. δ I G \ \delta I_{G}
  5. t \ t
  6. δ I R \ \delta I_{R}
  7. τ \ \tau
  8. V e f f , R G \ V_{eff,RG}
  9. V e f f , R G = π 3 / 2 ( ω x y , G 2 + ω x y , R 2 ) ( ω z , G 2 + ω z , R 2 ) 1 / 2 / 2 3 / 2 \ V_{eff,RG}=\pi^{3/2}(\omega_{xy,G}^{2}+\omega_{xy,R}^{2})(\omega_{z,G}^{2}+% \omega_{z,R}^{2})^{1/2}/2^{3/2}
  10. ω x y , G 2 \ \omega_{xy,G}^{2}
  11. ω x y , R 2 \ \omega_{xy,R}^{2}
  12. ω z , G \ \omega_{z,G}
  13. ω z , R \ \omega_{z,R}
  14. τ D , G R \ \tau_{D,GR}
  15. τ D , G R = ω x y , G 2 + ω x y , R 2 8 D G R \ \tau_{D,GR}=\frac{\omega_{xy,G}^{2}+\omega_{xy,R}^{2}}{8D_{GR}}
  16. D G R \ D_{GR}
  17. G G ( τ ) = 1 + ( < C G > D i f f k ( τ ) + < C G R > D i f f k ( τ ) ) V e f f , G R ( < C G > + < C G R > ) 2 \ G_{G}(\tau)=1+\frac{(<C_{G}>Diff_{k}(\tau)+<C_{GR}>Diff_{k}(\tau))}{V_{eff,% GR}(<C_{G}>+<C_{GR}>)^{2}}
  18. G R ( τ ) = 1 + ( < C R > D i f f k ( τ ) + < C G R > D i f f k ( τ ) ) V e f f , G R ( < C R > + < C G R > ) 2 \ G_{R}(\tau)=1+\frac{(<C_{R}>Diff_{k}(\tau)+<C_{GR}>Diff_{k}(\tau))}{V_{eff,% GR}(<C_{R}>+<C_{GR}>)^{2}}
  19. G G R ( τ ) = 1 + < C G R > D i f f G R ( τ ) V e f f ( < C G > + < C G R > ) ( < C R > + < C G R > ) \ G_{GR}(\tau)=1+\frac{<C_{GR}>Diff_{GR}(\tau)}{V_{eff}(<C_{G}>+<C_{GR}>)(<C_{% R}>+<C_{GR}>)}
  20. D i f f k ( τ ) = 1 ( 1 + τ τ D , i ) ( 1 + a - 2 ( τ τ D , i ) 1 / 2 \ Diff_{k}(\tau)=\frac{1}{(1+\frac{\tau}{\tau_{D,i}})(1+a^{-2}(\frac{\tau}{% \tau_{D,i}})^{1/2}}

Flyback_diode.html

  1. V L = - d Φ B d t = - L d I d t V_{L}=-{d\Phi_{B}\over dt}=-L{dI\over dt}
  2. V R 2 = R 2 I V_{R_{2}}=R_{2}\cdot I
  3. V C C V_{CC}
  4. R 1 R_{1}
  5. V L = V R 2 V_{L}=V_{R_{2}}
  6. - L d I d t = R 2 I -L{dI\over dt}=R_{2}\cdot I
  7. I ( t ) = I 0 e - R 2 L t I(t)=I_{0}\cdot e^{-{R_{2}\over L}t}
  8. I > 0 I>0
  9. V D = c o n s t V_{D}=const
  10. V L = V R 1 + V D V_{L}=V_{R_{1}}+V_{D}
  11. - L d I d t = R 1 I + V D -L{dI\over dt}=R_{1}\cdot I+V_{D}
  12. I ( t ) = ( I 0 + 1 R 1 V D ) e - R 1 L t - 1 R 1 V D I(t)=(I_{0}+{1\over R_{1}}V_{D})\cdot e^{-{R_{1}\over L}t}-{1\over R_{1}}V_{D}
  13. t t
  14. I ( t ) = 0 I(t)=0
  15. t = - L R l n ( V D V D + I 0 R 1 ) t={-L\over R}\cdot ln{\left({V_{D}\over{V_{D}+I_{0}{R_{1}}}}\right)}

Formal_ethics.html

  1. A ¯ \underline{A}
  2. A A
  3. A ¯ \underline{A}
  4. D u x Dux
  5. D u ¯ x D\underline{u}x
  6. ¬ A ¯ \neg\underline{A}
  7. M M
  8. M A ¯ M\underline{A}
  9. ¬ M ¬ A ¯ \neg M\neg\underline{A}
  10. M M
  11. R R
  12. O O
  13. R R
  14. P P
  15. O O
  16. R R
  17. O ( smoke ashtray ) O(\mathrm{smoke}\to\mathrm{ashtray})
  18. smoke O ( ashtray ) \mathrm{smoke}\to O(\mathrm{ashtray})
  19. O ( smoke ashtray ¯ ) O(\mathrm{smoke}\to\underline{\mathrm{ashtray}})
  20. u : A u:A
  21. u : A ¯ u:\underline{A}
  22. i : A ¯ i:\underline{A}
  23. A A
  24. u ¯ : A \underline{u}:A
  25. u ¯ : A ¯ \underline{u}:\underline{A}
  26. G G
  27. O ( u ¯ : G ) O(\underline{u}:G)
  28. ( x ) O ( x ¯ : G ) (x)O(\underline{x}:G)
  29. \square
  30. \diamond
  31. 𝑐 \underset{c}{\square}
  32. 𝑐 \underset{c}{\diamond}
  33. \blacksquare
  34. \square
  35. ( O A ¯ A ¯ ) \blacksquare(O\underline{A}\to\underline{A})
  36. ( O A ¯ A ¯ ) \square(O\underline{A}\to\underline{A})
  37. F A ¯ F\underline{A}
  38. W W
  39. W A ¯ W\underline{A}
  40. G G
  41. G A ¯ G\underline{A}
  42. I I
  43. I A ¯ I\underline{A}
  44. B B
  45. B A ¯ B\underline{A}
  46. G A ¯ G\ast\underline{A}
  47. G G
  48. A ¯ \underline{A}
  49. G A ¯ G\ast\underline{A}
  50. P P
  51. U U
  52. R R
  53. E E
  54. G G
  55. G G
  56. G G
  57. T T
  58. T A ¯ T\underline{A}

Fortune's_algorithm.html

  1. * ( z ) *(z)
  2. * ( z ) = ( z x , z y + d ( z ) ) *(z)=(z_{x},z_{y}+d(z))
  3. d ( z ) d(z)
  4. z z
  5. T T
  6. R p R_{p}
  7. p p
  8. C p q C_{pq}
  9. p p
  10. q q
  11. p 1 , p 2 , , p m p_{1},p_{2},...,p_{m}
  12. y y
  13. x x
  14. Q S - p 1 , p 2 , , p m Q\leftarrow S-{p_{1},p_{2},...,p_{m}}
  15. C p 1 , p 2 0 , C p 2 , p 3 0 , C p m - 1 , p m 0 C_{p_{1},p_{2}}^{0},C_{p_{2},p_{3}}^{0},...C_{p_{m-1},p_{m}}^{0}
  16. T * ( R p 1 ) , C p 1 , p 2 0 , * ( R p 2 ) , C p 2 , p 3 0 , , * ( R p m - 1 ) , C p m - 1 , p m 0 , * ( R p m ) T\leftarrow*(R_{p_{1}}),C_{p_{1},p_{2}}^{0},*(R_{p_{2}}),C_{p_{2},p_{3}}^{0},.% ..,*(R_{p_{m-1}}),C_{p_{m-1},p_{m}}^{0},*(R_{p_{m}})
  17. Q Q
  18. p p
  19. Q Q
  20. p p
  21. p p
  22. * ( V ) *(V)
  23. * ( R q ) *(R_{q})
  24. T T
  25. p p
  26. C r q C_{rq}
  27. C q s C_{qs}
  28. C p q - C_{pq}^{-}
  29. C p q + C_{pq}^{+}
  30. p p
  31. * ( R q ) *(R_{q})
  32. * ( R q ) , C p q - , * ( R p ) , C p q + , * ( R q ) *(R_{q}),C_{pq}^{-},*(R_{p}),C_{pq}^{+},*(R_{q})
  33. T T
  34. Q Q
  35. C r q C_{rq}
  36. C q s C_{qs}
  37. Q Q
  38. C r q C_{rq}
  39. C p q - C_{pq}^{-}
  40. Q Q
  41. C p q + C_{pq}^{+}
  42. C q s C_{qs}
  43. p p
  44. * ( V ) *(V)
  45. p p
  46. C q r C_{qr}
  47. C r s C_{rs}
  48. C u q C_{uq}
  49. C q r C_{qr}
  50. C s v C_{sv}
  51. C r s C_{rs}
  52. T T
  53. C q s 0 C_{qs}^{0}
  54. q y = s y q_{y}=s_{y}
  55. C q s + C_{qs}^{+}
  56. p p
  57. q q
  58. s s
  59. C q s - C_{qs}^{-}
  60. C q r , * ( R r ) , C r s C_{qr},*(R_{r}),C_{rs}
  61. C q s C_{qs}
  62. T T
  63. Q Q
  64. C u q C_{uq}
  65. C q r C_{qr}
  66. Q Q
  67. C r s C_{rs}
  68. C s v C_{sv}
  69. Q Q
  70. C u q C_{uq}
  71. C q s C_{qs}
  72. Q Q
  73. C q s C_{qs}
  74. C s v C_{sv}
  75. p p
  76. C q r C_{qr}
  77. C r s C_{rs}
  78. C q s C_{qs}
  79. C q r C_{qr}
  80. C r s C_{rs}
  81. T T

Forward_kinematics.html

  1. [ T ] = [ Z 1 ] [ X 1 ] [ Z 2 ] [ X 2 ] [ X n - 1 ] [ Z n ] , [T]=[Z_{1}][X_{1}][Z_{2}][X_{2}]\ldots[X_{n-1}][Z_{n}],\!
  2. [ Z i ] = Trans Z i ( d i ) Rot Z i ( θ i ) , [Z_{i}]=\operatorname{Trans}_{Z_{i}}(d_{i})\operatorname{Rot}_{Z_{i}}(\theta_{% i}),
  3. [ X i ] = Trans X i ( a i , i + 1 ) Rot X i ( α i , i + 1 ) . [X_{i}]=\operatorname{Trans}_{X_{i}}(a_{i,i+1})\operatorname{Rot}_{X_{i}}(% \alpha_{i,i+1}).
  4. T i i - 1 = [ Z i ] [ X i ] = Trans Z i ( d i ) Rot Z i ( θ i ) Trans X i ( a i , i + 1 ) Rot X i ( α i , i + 1 ) , {}^{i-1}T_{i}=[Z_{i}][X_{i}]=\operatorname{Trans}_{Z_{i}}(d_{i})\operatorname{% Rot}_{Z_{i}}(\theta_{i})\operatorname{Trans}_{X_{i}}(a_{i,i+1})\operatorname{% Rot}_{X_{i}}(\alpha_{i,i+1}),
  5. [ T ] = T n 0 = i = 1 n T i i - 1 ( θ i ) , [T]={}^{0}T_{n}=\prod_{i=1}^{n}{}^{i-1}T_{i}(\theta_{i}),
  6. T i i - 1 ( θ i ) {}^{i-1}T_{i}(\theta_{i})
  7. i i
  8. i - 1 i-1
  9. Trans Z i ( d i ) = [ 1 0 0 0 0 1 0 0 0 0 1 d i 0 0 0 1 ] , Rot Z i ( θ i ) = [ cos θ i - sin θ i 0 0 sin θ i cos θ i 0 0 0 0 1 0 0 0 0 1 ] . \operatorname{Trans}_{Z_{i}}(d_{i})=\begin{bmatrix}1&0&0&0\\ 0&1&0&0\\ 0&0&1&d_{i}\\ 0&0&0&1\end{bmatrix},\quad\operatorname{Rot}_{Z_{i}}(\theta_{i})=\begin{% bmatrix}\cos\theta_{i}&-\sin\theta_{i}&0&0\\ \sin\theta_{i}&\cos\theta_{i}&0&0\\ 0&0&1&0\\ 0&0&0&1\end{bmatrix}.
  10. Trans X i ( a i , i + 1 ) = [ 1 0 0 a i , i + 1 0 1 0 0 0 0 1 0 0 0 0 1 ] , Rot X i ( α i , i + 1 ) = [ 1 0 0 0 0 cos α i , i + 1 - sin α i , i + 1 0 0 sin α i , i + 1 cos α i , i + 1 0 0 0 0 1 ] . \operatorname{Trans}_{X_{i}}(a_{i,i+1})=\begin{bmatrix}1&0&0&a_{i,i+1}\\ 0&1&0&0\\ 0&0&1&0\\ 0&0&0&1\end{bmatrix},\quad\operatorname{Rot}_{X_{i}}(\alpha_{i,i+1})=\begin{% bmatrix}1&0&0&0\\ 0&\cos\alpha_{i,i+1}&-\sin\alpha_{i,i+1}&0\\ 0&\sin\alpha_{i,i+1}&\cos\alpha_{i,i+1}&0\\ 0&0&0&1\end{bmatrix}.
  11. i - 1 T i = [ cos θ i - sin θ i cos α i , i + 1 sin θ i sin α i , i + 1 a i , i + 1 cos θ i sin θ i cos θ i cos α i , i + 1 - cos θ i sin α i , i + 1 a i , i + 1 sin θ i 0 sin α i , i + 1 cos α i , i + 1 d i 0 0 0 1 ] , \operatorname{}^{i-1}T_{i}=\begin{bmatrix}\cos\theta_{i}&-\sin\theta_{i}\cos% \alpha_{i,i+1}&\sin\theta_{i}\sin\alpha_{i,i+1}&a_{i,i+1}\cos\theta_{i}\\ \sin\theta_{i}&\cos\theta_{i}\cos\alpha_{i,i+1}&-\cos\theta_{i}\sin\alpha_{i,i% +1}&a_{i,i+1}\sin\theta_{i}\\ 0&\sin\alpha_{i,i+1}&\cos\alpha_{i,i+1}&d_{i}\\ 0&0&0&1\end{bmatrix},

Forward_measure.html

  1. B ( T ) = exp ( 0 T r ( u ) d u ) B(T)=\exp\left(\int_{0}^{T}r(u)\,du\right)
  2. D ( T ) = 1 / B ( T ) = exp ( - 0 T r ( u ) d u ) D(T)=1/B(T)=\exp\left(-\int_{0}^{T}r(u)\,du\right)
  3. Q * Q_{*}
  4. Q T Q_{T}
  5. d Q T d Q * = 1 B ( T ) E Q * [ 1 / B ( T ) ] = D ( T ) E Q * [ D ( T ) ] . \frac{dQ_{T}}{dQ_{*}}=\frac{1}{B(T)E_{Q_{*}}[1/B(T)]}=\frac{D(T)}{E_{Q_{*}}[D(% T)]}.
  6. P ( t , T ) = E Q * [ B ( t ) B ( T ) | ( t ) ] = E Q * [ D ( T ) D ( t ) | ( t ) ] P(t,T)=E_{Q_{*}}\left[\frac{B(t)}{B(T)}|\mathcal{F}(t)\right]=E_{Q_{*}}\left[% \frac{D(T)}{D(t)}|\mathcal{F}(t)\right]
  7. ( t ) \mathcal{F}(t)
  8. d Q T d Q * = B ( 0 ) P ( T , T ) B ( T ) P ( 0 , T ) \frac{dQ_{T}}{dQ_{*}}=\frac{B(0)P(T,T)}{B(T)P(0,T)}
  9. S ( t ) D ( t ) = E Q * [ D ( T ) S ( T ) | ( t ) ] . S(t)D(t)=E_{Q_{*}}[D(T)S(T)|\mathcal{F}(t)].\,
  10. F S ( t , T ) = S ( t ) P ( t , T ) F_{S}(t,T)=\frac{S(t)}{P(t,T)}
  11. F S ( T , T ) = S ( T ) F_{S}(T,T)=S(T)
  12. F S ( t , T ) = E Q * [ D ( T ) S ( T ) | ( t ) ] D ( t ) P ( t , T ) = E Q T [ F S ( T , T ) | ( t ) ] E Q * [ D ( T ) | ( t ) ] D ( t ) P ( t , T ) F_{S}(t,T)=\frac{E_{Q_{*}}[D(T)S(T)|\mathcal{F}(t)]}{D(t)P(t,T)}=E_{Q_{T}}[F_{% S}(T,T)|\mathcal{F}(t)]\frac{E_{Q_{*}}[D(T)|\mathcal{F}(t)]}{D(t)P(t,T)}
  13. F S ( t , T ) = E Q T [ F S ( T , T ) | ( t ) ] . F_{S}(t,T)=E_{Q_{T}}[F_{S}(T,T)|\mathcal{F}(t)].\,

Forward–backward_algorithm.html

  1. o 1 : t := o 1 , , o t o_{1:t}:=o_{1},\dots,o_{t}
  2. X k { X 1 , , X t } X_{k}\in\{X_{1},\dots,X_{t}\}
  3. P ( X k | o 1 : t ) P(X_{k}\ |\ o_{1:t})
  4. k { 1 , , t } k\in\{1,\dots,t\}
  5. k k
  6. P ( X k | o 1 : k ) P(X_{k}\ |\ o_{1:k})
  7. k k
  8. P ( o k + 1 : t | X k ) P(o_{k+1:t}\ |\ X_{k})
  9. P ( X k | o 1 : t ) = P ( X k | o 1 : k , o k + 1 : t ) P ( o k + 1 : t | X k ) P ( X k | o 1 : k ) P(X_{k}\ |\ o_{1:t})=P(X_{k}\ |\ o_{1:k},o_{k+1:t})\propto P(o_{k+1:t}\ |\ X_{% k})P(X_{k}\ |\ o_{1:k})
  10. o k + 1 : t o_{k+1:t}
  11. o 1 : k o_{1:k}
  12. X k X_{k}
  13. 𝐏 ( X t X t - 1 ) \mathbf{P}(X_{t}\mid X_{t-1})
  14. X t X_{t}
  15. 𝐓 \mathbf{T}
  16. 𝐓 = ( 0.7 0.3 0.3 0.7 ) \mathbf{T}=\begin{pmatrix}0.7&0.3\\ 0.3&0.7\end{pmatrix}
  17. 𝐁 = ( 0.9 0.1 0.2 0.8 ) \mathbf{B}=\begin{pmatrix}0.9&0.1\\ 0.2&0.8\end{pmatrix}
  18. π \mathbf{\pi}
  19. 𝐏 ( O = j ) = i π i b i , j \mathbf{P}(O=j)=\sum_{i}\pi_{i}b_{i,j}
  20. π \mathbf{\pi}
  21. 𝐎 𝐣 = diag ( b * , o j ) \mathbf{O_{j}}=\mathrm{diag}(b_{*,o_{j}})
  22. 𝐎 𝟏 = ( 0.9 0.0 0.0 0.2 ) \mathbf{O_{1}}=\begin{pmatrix}0.9&0.0\\ 0.0&0.2\end{pmatrix}
  23. 𝐟 𝟎 : 𝟏 = π 𝐎 𝟏 \mathbf{f_{0:1}}=\mathbf{\pi}\mathbf{O_{1}}
  24. 𝐟 𝟎 : 𝐭 = 𝐟 𝟎 : 𝐭 - 𝟏 𝐓𝐎 𝐭 \mathbf{f_{0:t}}=\mathbf{f_{0:t-1}}\mathbf{T}\mathbf{O_{t}}
  25. 𝐟 𝟎 : 𝐭 ( i ) = 𝐏 ( o 1 , o 2 , , o t , X t = x i | π ) \mathbf{f_{0:t}}(i)=\mathbf{P}(o_{1},o_{2},\dots,o_{t},X_{t}=x_{i}|\mathbf{\pi})
  26. 𝐟 ^ 𝟎 : 𝐭 = c t - 1 𝐟 ^ 𝟎 : 𝐭 - 𝟏 𝐓𝐎 𝐭 \mathbf{\hat{f}_{0:t}}=c_{t}^{-1}\ \mathbf{\hat{f}_{0:t-1}}\mathbf{T}\mathbf{O% _{t}}
  27. 𝐟 ^ 𝟎 : 𝐭 - 𝟏 \mathbf{\hat{f}_{0:t-1}}
  28. c t c_{t}
  29. 𝐏 ( o 1 , o 2 , , o t | π ) = s = 1 t c s \mathbf{P}(o_{1},o_{2},\dots,o_{t}|\mathbf{\pi})=\prod_{s=1}^{t}c_{s}
  30. 𝐟 ^ 𝟎 : 𝐭 ( i ) = 𝐟 𝟎 : 𝐭 ( i ) s = 1 t c s = 𝐏 ( o 1 , o 2 , , o t , X t = x i | π ) 𝐏 ( o 1 , o 2 , , o t | π ) = 𝐏 ( X t = x i | o 1 , o 2 , , o t , π ) \mathbf{\hat{f}_{0:t}}(i)=\frac{\mathbf{f_{0:t}}(i)}{\prod_{s=1}^{t}c_{s}}=% \frac{\mathbf{P}(o_{1},o_{2},\dots,o_{t},X_{t}=x_{i}|\mathbf{\pi})}{\mathbf{P}% (o_{1},o_{2},\dots,o_{t}|\mathbf{\pi})}=\mathbf{P}(X_{t}=x_{i}|o_{1},o_{2},% \dots,o_{t},\mathbf{\pi})
  31. 𝐛 𝐭 : 𝐓 ( i ) = 𝐏 ( o t + 1 , o t + 2 , , o T | X t = x i ) \mathbf{b_{t:T}}(i)=\mathbf{P}(o_{t+1},o_{t+2},\dots,o_{T}|X_{t}=x_{i})
  32. X t = x i X_{t}=x_{i}
  33. 𝐛 𝐓 : 𝐓 = [ 1 1 1 ] T \mathbf{b_{T:T}}=[1\ 1\ 1\ \dots]^{T}
  34. 𝐛 𝐭 - 𝟏 : 𝐓 = 𝐓𝐎 𝐭 𝐛 𝐭 : 𝐓 \mathbf{b_{t-1:T}}=\mathbf{T}\mathbf{O_{t}}\mathbf{b_{t:T}}
  35. c t c_{t}
  36. 𝐛 𝐓 : 𝐓 \mathbf{b_{T:T}}
  37. 𝐛 ^ 𝐭 - 𝟏 : 𝐓 = c t - 1 𝐓𝐎 𝐭 𝐛 ^ 𝐭 : 𝐓 \mathbf{\hat{b}_{t-1:T}}=c_{t}^{-1}\mathbf{T}\mathbf{O_{t}}\mathbf{\hat{b}_{t:% T}}
  38. 𝐛 ^ 𝐭 : 𝐓 \mathbf{\hat{b}_{t:T}}
  39. 𝐛 ^ 𝐭 : 𝐓 ( i ) = 𝐛 𝐭 : 𝐓 ( i ) s = t + 1 T c s \mathbf{\hat{b}_{t:T}}(i)=\frac{\mathbf{b_{t:T}}(i)}{\prod_{s=t+1}^{T}c_{s}}
  40. γ 𝐭 ( i ) = 𝐏 ( X t = x i | o 1 , o 2 , , o T , π ) = 𝐏 ( o 1 , o 2 , , o T , X t = x i | π ) 𝐏 ( o 1 , o 2 , , o T | π ) = 𝐟 𝟎 : 𝐭 ( i ) 𝐛 𝐭 : 𝐓 ( i ) s = 1 T c s = 𝐟 ^ 𝟎 : 𝐭 ( i ) 𝐛 ^ 𝐭 : 𝐓 ( i ) \mathbf{\gamma_{t}}(i)=\mathbf{P}(X_{t}=x_{i}|o_{1},o_{2},\dots,o_{T},\mathbf{% \pi})=\frac{\mathbf{P}(o_{1},o_{2},\dots,o_{T},X_{t}=x_{i}|\mathbf{\pi})}{% \mathbf{P}(o_{1},o_{2},\dots,o_{T}|\mathbf{\pi})}=\frac{\mathbf{f_{0:t}}(i)% \cdot\mathbf{b_{t:T}}(i)}{\prod_{s=1}^{T}c_{s}}=\mathbf{\hat{f}_{0:t}}(i)\cdot% \mathbf{\hat{b}_{t:T}}(i)
  41. 𝐟 𝟎 : 𝐭 ( i ) 𝐛 𝐭 : 𝐓 ( i ) \mathbf{f_{0:t}}(i)\cdot\mathbf{b_{t:T}}(i)
  42. x i x_{i}
  43. X t = x i X_{t}=x_{i}
  44. γ 𝐭 ( i ) \mathbf{\gamma_{t}}(i)
  45. 𝐏 ( X t = x i , X t + 1 = x j ) 𝐏 ( X t = x i ) 𝐏 ( X t + 1 = x j ) \mathbf{P}(X_{t}=x_{i},X_{t+1}=x_{j})\neq\mathbf{P}(X_{t}=x_{i})\mathbf{P}(X_{% t+1}=x_{j})
  46. 𝐓 = ( 0.7 0.3 0.3 0.7 ) \mathbf{T}=\begin{pmatrix}0.7&0.3\\ 0.3&0.7\end{pmatrix}
  47. 𝐁 = ( 0.9 0.1 0.2 0.8 ) \mathbf{B}=\begin{pmatrix}0.9&0.1\\ 0.2&0.8\end{pmatrix}
  48. 𝐎 𝟏 = ( 0.9 0.0 0.0 0.2 ) 𝐎 𝟐 = ( 0.9 0.0 0.0 0.2 ) 𝐎 𝟑 = ( 0.1 0.0 0.0 0.8 ) 𝐎 𝟒 = ( 0.9 0.0 0.0 0.2 ) 𝐎 𝟓 = ( 0.9 0.0 0.0 0.2 ) \mathbf{O_{1}}=\begin{pmatrix}0.9&0.0\\ 0.0&0.2\end{pmatrix}~{}~{}\mathbf{O_{2}}=\begin{pmatrix}0.9&0.0\\ 0.0&0.2\end{pmatrix}~{}~{}\mathbf{O_{3}}=\begin{pmatrix}0.1&0.0\\ 0.0&0.8\end{pmatrix}~{}~{}\mathbf{O_{4}}=\begin{pmatrix}0.9&0.0\\ 0.0&0.2\end{pmatrix}~{}~{}\mathbf{O_{5}}=\begin{pmatrix}0.9&0.0\\ 0.0&0.2\end{pmatrix}
  49. 𝐎 𝟑 \mathbf{O_{3}}
  50. 𝐟 𝟎 : 𝟎 = ( 0.5 0.5 ) \mathbf{f_{0:0}}=\begin{pmatrix}0.5&0.5\end{pmatrix}
  51. ( 𝐟 ^ 𝟎 : 𝐭 ) T = c - 1 𝐎 𝐭 ( 𝐓 ) T ( 𝐟 ^ 𝟎 : 𝐭 - 𝟏 ) T (\mathbf{\hat{f}_{0:t}})^{T}=c^{-1}\mathbf{O_{t}}(\mathbf{T})^{T}(\mathbf{\hat% {f}_{0:t-1}})^{T}
  52. 𝐟 ^ 𝟎 : 𝐭 = c - 1 𝐟 ^ 𝟎 : 𝐭 - 𝟏 𝐓𝐎 𝐭 \mathbf{\hat{f}_{0:t}}=c^{-1}\mathbf{\hat{f}_{0:t-1}}\mathbf{T}\mathbf{O_{t}}
  53. ( 𝐟 ^ 𝟎 : 𝟏 ) T = c 1 - 1 ( 0.9 0.0 0.0 0.2 ) ( 0.7 0.3 0.3 0.7 ) ( 0.5000 0.5000 ) = c 1 - 1 ( 0.4500 0.1000 ) = ( 0.8182 0.1818 ) (\mathbf{\hat{f}_{0:1}})^{T}=c_{1}^{-1}\begin{pmatrix}0.9&0.0\\ 0.0&0.2\end{pmatrix}\begin{pmatrix}0.7&0.3\\ 0.3&0.7\end{pmatrix}\begin{pmatrix}0.5000\\ 0.5000\end{pmatrix}=c_{1}^{-1}\begin{pmatrix}0.4500\\ 0.1000\end{pmatrix}=\begin{pmatrix}0.8182\\ 0.1818\end{pmatrix}
  54. ( 𝐟 ^ 𝟎 : 𝟐 ) T = c 2 - 1 ( 0.9 0.0 0.0 0.2 ) ( 0.7 0.3 0.3 0.7 ) ( 0.8182 0.1818 ) = c 2 - 1 ( 0.5645 0.0745 ) = ( 0.8834 0.1166 ) (\mathbf{\hat{f}_{0:2}})^{T}=c_{2}^{-1}\begin{pmatrix}0.9&0.0\\ 0.0&0.2\end{pmatrix}\begin{pmatrix}0.7&0.3\\ 0.3&0.7\end{pmatrix}\begin{pmatrix}0.8182\\ 0.1818\end{pmatrix}=c_{2}^{-1}\begin{pmatrix}0.5645\\ 0.0745\end{pmatrix}=\begin{pmatrix}0.8834\\ 0.1166\end{pmatrix}
  55. ( 𝐟 ^ 𝟎 : 𝟑 ) T = c 3 - 1 ( 0.1 0.0 0.0 0.8 ) ( 0.7 0.3 0.3 0.7 ) ( 0.8834 0.1166 ) = c 3 - 1 ( 0.0653 0.2772 ) = ( 0.1907 0.8093 ) (\mathbf{\hat{f}_{0:3}})^{T}=c_{3}^{-1}\begin{pmatrix}0.1&0.0\\ 0.0&0.8\end{pmatrix}\begin{pmatrix}0.7&0.3\\ 0.3&0.7\end{pmatrix}\begin{pmatrix}0.8834\\ 0.1166\end{pmatrix}=c_{3}^{-1}\begin{pmatrix}0.0653\\ 0.2772\end{pmatrix}=\begin{pmatrix}0.1907\\ 0.8093\end{pmatrix}
  56. ( 𝐟 ^ 𝟎 : 𝟒 ) T = c 4 - 1 ( 0.9 0.0 0.0 0.2 ) ( 0.7 0.3 0.3 0.7 ) ( 0.1907 0.8093 ) = c 4 - 1 ( 0.3386 0.1247 ) = ( 0.7308 0.2692 ) (\mathbf{\hat{f}_{0:4}})^{T}=c_{4}^{-1}\begin{pmatrix}0.9&0.0\\ 0.0&0.2\end{pmatrix}\begin{pmatrix}0.7&0.3\\ 0.3&0.7\end{pmatrix}\begin{pmatrix}0.1907\\ 0.8093\end{pmatrix}=c_{4}^{-1}\begin{pmatrix}0.3386\\ 0.1247\end{pmatrix}=\begin{pmatrix}0.7308\\ 0.2692\end{pmatrix}
  57. ( 𝐟 ^ 𝟎 : 𝟓 ) T = c 5 - 1 ( 0.9 0.0 0.0 0.2 ) ( 0.7 0.3 0.3 0.7 ) ( 0.7308 0.2692 ) = c 5 - 1 ( 0.5331 0.0815 ) = ( 0.8673 0.1327 ) (\mathbf{\hat{f}_{0:5}})^{T}=c_{5}^{-1}\begin{pmatrix}0.9&0.0\\ 0.0&0.2\end{pmatrix}\begin{pmatrix}0.7&0.3\\ 0.3&0.7\end{pmatrix}\begin{pmatrix}0.7308\\ 0.2692\end{pmatrix}=c_{5}^{-1}\begin{pmatrix}0.5331\\ 0.0815\end{pmatrix}=\begin{pmatrix}0.8673\\ 0.1327\end{pmatrix}
  58. 𝐛 𝟓 : 𝟓 = ( 1.0 1.0 ) \mathbf{b_{5:5}}=\begin{pmatrix}1.0\\ 1.0\end{pmatrix}
  59. 𝐛 ^ 𝟒 : 𝟓 = α ( 0.7 0.3 0.3 0.7 ) ( 0.9 0.0 0.0 0.2 ) ( 1.0000 1.0000 ) = α ( 0.6900 0.4100 ) = ( 0.6273 0.3727 ) \mathbf{\hat{b}_{4:5}}=\alpha\begin{pmatrix}0.7&0.3\\ 0.3&0.7\end{pmatrix}\begin{pmatrix}0.9&0.0\\ 0.0&0.2\end{pmatrix}\begin{pmatrix}1.0000\\ 1.0000\end{pmatrix}=\alpha\begin{pmatrix}0.6900\\ 0.4100\end{pmatrix}=\begin{pmatrix}0.6273\\ 0.3727\end{pmatrix}
  60. 𝐛 ^ 𝟑 : 𝟓 = α ( 0.7 0.3 0.3 0.7 ) ( 0.9 0.0 0.0 0.2 ) ( 0.6273 0.3727 ) = α ( 0.4175 0.2215 ) = ( 0.6533 0.3467 ) \mathbf{\hat{b}_{3:5}}=\alpha\begin{pmatrix}0.7&0.3\\ 0.3&0.7\end{pmatrix}\begin{pmatrix}0.9&0.0\\ 0.0&0.2\end{pmatrix}\begin{pmatrix}0.6273\\ 0.3727\end{pmatrix}=\alpha\begin{pmatrix}0.4175\\ 0.2215\end{pmatrix}=\begin{pmatrix}0.6533\\ 0.3467\end{pmatrix}
  61. 𝐛 ^ 𝟐 : 𝟓 = α ( 0.7 0.3 0.3 0.7 ) ( 0.1 0.0 0.0 0.8 ) ( 0.6533 0.3467 ) = α ( 0.1289 0.2138 ) = ( 0.3763 0.6237 ) \mathbf{\hat{b}_{2:5}}=\alpha\begin{pmatrix}0.7&0.3\\ 0.3&0.7\end{pmatrix}\begin{pmatrix}0.1&0.0\\ 0.0&0.8\end{pmatrix}\begin{pmatrix}0.6533\\ 0.3467\end{pmatrix}=\alpha\begin{pmatrix}0.1289\\ 0.2138\end{pmatrix}=\begin{pmatrix}0.3763\\ 0.6237\end{pmatrix}
  62. 𝐛 ^ 𝟏 : 𝟓 = α ( 0.7 0.3 0.3 0.7 ) ( 0.9 0.0 0.0 0.2 ) ( 0.3763 0.6237 ) = α ( 0.2745 0.1889 ) = ( 0.5923 0.4077 ) \mathbf{\hat{b}_{1:5}}=\alpha\begin{pmatrix}0.7&0.3\\ 0.3&0.7\end{pmatrix}\begin{pmatrix}0.9&0.0\\ 0.0&0.2\end{pmatrix}\begin{pmatrix}0.3763\\ 0.6237\end{pmatrix}=\alpha\begin{pmatrix}0.2745\\ 0.1889\end{pmatrix}=\begin{pmatrix}0.5923\\ 0.4077\end{pmatrix}
  63. 𝐛 ^ 𝟎 : 𝟓 = α ( 0.7 0.3 0.3 0.7 ) ( 0.9 0.0 0.0 0.2 ) ( 0.5923 0.4077 ) = α ( 0.3976 0.2170 ) = ( 0.6469 0.3531 ) \mathbf{\hat{b}_{0:5}}=\alpha\begin{pmatrix}0.7&0.3\\ 0.3&0.7\end{pmatrix}\begin{pmatrix}0.9&0.0\\ 0.0&0.2\end{pmatrix}\begin{pmatrix}0.5923\\ 0.4077\end{pmatrix}=\alpha\begin{pmatrix}0.3976\\ 0.2170\end{pmatrix}=\begin{pmatrix}0.6469\\ 0.3531\end{pmatrix}
  64. c t c_{t}
  65. ( γ 𝟎 ) T = α ( 0.5000 0.5000 ) ( 0.6469 0.3531 ) = α ( 0.3235 0.1765 ) = ( 0.6469 0.3531 ) (\mathbf{\gamma_{0}})^{T}=\alpha\begin{pmatrix}0.5000\\ 0.5000\end{pmatrix}\circ\begin{pmatrix}0.6469\\ 0.3531\end{pmatrix}=\alpha\begin{pmatrix}0.3235\\ 0.1765\end{pmatrix}=\begin{pmatrix}0.6469\\ 0.3531\end{pmatrix}
  66. ( γ 𝟏 ) T = α ( 0.8182 0.1818 ) ( 0.5923 0.4077 ) = α ( 0.4846 0.0741 ) = ( 0.8673 0.1327 ) (\mathbf{\gamma_{1}})^{T}=\alpha\begin{pmatrix}0.8182\\ 0.1818\end{pmatrix}\circ\begin{pmatrix}0.5923\\ 0.4077\end{pmatrix}=\alpha\begin{pmatrix}0.4846\\ 0.0741\end{pmatrix}=\begin{pmatrix}0.8673\\ 0.1327\end{pmatrix}
  67. ( γ 𝟐 ) T = α ( 0.8834 0.1166 ) ( 0.3763 0.6237 ) = α ( 0.3324 0.0728 ) = ( 0.8204 0.1796 ) (\mathbf{\gamma_{2}})^{T}=\alpha\begin{pmatrix}0.8834\\ 0.1166\end{pmatrix}\circ\begin{pmatrix}0.3763\\ 0.6237\end{pmatrix}=\alpha\begin{pmatrix}0.3324\\ 0.0728\end{pmatrix}=\begin{pmatrix}0.8204\\ 0.1796\end{pmatrix}
  68. ( γ 𝟑 ) T = α ( 0.1907 0.8093 ) ( 0.6533 0.3467 ) = α ( 0.1246 0.2806 ) = ( 0.3075 0.6925 ) (\mathbf{\gamma_{3}})^{T}=\alpha\begin{pmatrix}0.1907\\ 0.8093\end{pmatrix}\circ\begin{pmatrix}0.6533\\ 0.3467\end{pmatrix}=\alpha\begin{pmatrix}0.1246\\ 0.2806\end{pmatrix}=\begin{pmatrix}0.3075\\ 0.6925\end{pmatrix}
  69. ( γ 𝟒 ) T = α ( 0.7308 0.2692 ) ( 0.6273 0.3727 ) = α ( 0.4584 0.1003 ) = ( 0.8204 0.1796 ) (\mathbf{\gamma_{4}})^{T}=\alpha\begin{pmatrix}0.7308\\ 0.2692\end{pmatrix}\circ\begin{pmatrix}0.6273\\ 0.3727\end{pmatrix}=\alpha\begin{pmatrix}0.4584\\ 0.1003\end{pmatrix}=\begin{pmatrix}0.8204\\ 0.1796\end{pmatrix}
  70. ( γ 𝟓 ) T = α ( 0.8673 0.1327 ) ( 1.0000 1.0000 ) = α ( 0.8673 0.1327 ) = ( 0.8673 0.1327 ) (\mathbf{\gamma_{5}})^{T}=\alpha\begin{pmatrix}0.8673\\ 0.1327\end{pmatrix}\circ\begin{pmatrix}1.0000\\ 1.0000\end{pmatrix}=\alpha\begin{pmatrix}0.8673\\ 0.1327\end{pmatrix}=\begin{pmatrix}0.8673\\ 0.1327\end{pmatrix}
  71. γ 𝟎 \mathbf{\gamma_{0}}
  72. 𝐛 ^ 𝟎 : 𝟓 \mathbf{\hat{b}_{0:5}}
  73. γ 𝟓 \mathbf{\gamma_{5}}
  74. 𝐟 ^ 𝟎 : 𝟓 \mathbf{\hat{f}_{0:5}}
  75. 𝐟 ^ 𝟎 : 𝟓 \mathbf{\hat{f}_{0:5}}
  76. 𝐛 ^ 𝟎 : 𝟓 \mathbf{\hat{b}_{0:5}}
  77. γ 𝟎 \mathbf{\gamma_{0}}
  78. 𝐛 ^ 𝟎 : 𝟓 \mathbf{\hat{b}_{0:5}}
  79. 𝐛 ^ 𝟎 : 𝟓 \mathbf{\hat{b}_{0:5}}
  80. γ 𝟓 \mathbf{\gamma_{5}}
  81. N T N^{T}
  82. O ( T N T ) O(T\cdot N^{T})
  83. T T
  84. N N
  85. O ( N 2 T ) O(N^{2}T)\,
  86. O ( N 2 T log T ) O(N^{2}T\log T)\,
  87. O ( N 2 log T ) O(N^{2}\log T)\,
  88. O ( N 2 T ) O(N^{2}T)\,
  89. O ( N 2 log T ) O(N^{2}\log T)\,
  90. 𝐟 𝟎 : 𝐭 + 𝟏 \mathbf{f_{0:t+1}}

Foster_graph.html

  1. ( x - 3 ) ( x - 2 ) 9 ( x - 1 ) 18 x 10 ( x + 1 ) 18 ( x + 2 ) 9 ( x + 3 ) ( x 2 - 6 ) 12 (x-3)(x-2)^{9}(x-1)^{18}x^{10}(x+1)^{18}(x+2)^{9}(x+3)(x^{2}-6)^{12}

Fourier_algebra.html

  1. L 1 ( G ^ ) L_{1}(\widehat{\mathit{G}})
  2. ( G ^ ) (\widehat{\mathit{G}})
  3. M ( G ^ ) M(\widehat{\mathit{G}})
  4. ( G ^ ) (\widehat{\mathit{G}})
  5. B ( G ) B(\mathit{G})
  6. A ( G ) A(\mathit{G})
  7. G \mathit{G}
  8. M ( G ^ ) M(\widehat{\mathit{G}})
  9. G ^ \widehat{G}
  10. L 1 ( G ^ ) L_{1}(\widehat{\mathit{G}})
  11. G ^ \widehat{G}
  12. G ^ \widehat{\mathit{G}}
  13. G \mathit{G}
  14. μ \mu
  15. G ^ \widehat{\mathit{G}}
  16. μ ^ \widehat{\mu}
  17. G \mathit{G}
  18. μ ^ ( x ) = G ^ X ( x ) ¯ d μ ( X ) , x G \widehat{\mu}(x)=\int_{\widehat{G}}\overline{X(x)}\,d\mu(X),\quad x\in G
  19. B ( G ) B(\mathit{G})
  20. M ( G ^ ) M(\widehat{\mathit{G}})
  21. L 1 ( G ^ ) L_{1}(\widehat{\mathit{G}})
  22. M ( G ^ ) M(\widehat{\mathit{G}})
  23. L 1 ( G ^ ) L_{1}(\widehat{\mathit{G}})
  24. A ( G ) A(\mathit{G})
  25. G \mathit{G}
  26. G ^ \widehat{G}
  27. B ( G ) B(\mathit{G})
  28. G \mathit{G}
  29. G \mathit{G}

Fourier_integral_operator.html

  1. ( T f ) ( x ) = n e 2 π i Φ ( x , ξ ) a ( x , ξ ) f ^ ( ξ ) d ξ (Tf)(x)=\int_{\mathbb{R}^{n}}e^{2\pi i\Phi(x,\xi)}a(x,\xi)\hat{f}(\xi)\,d\xi
  2. f ^ \hat{f}
  3. det ( 2 Φ x i ξ j ) 0 \det\left(\frac{\partial^{2}\Phi}{\partial x_{i}\,\partial\xi_{j}}\right)\neq 0
  4. 1 c 2 2 u t 2 ( t , x ) = Δ u ( t , x ) for ( t , x ) + × n , \frac{1}{c^{2}}\frac{\partial^{2}u}{\partial t^{2}}(t,x)=\Delta u(t,x)\quad% \mathrm{for}\quad(t,x)\in\mathbb{R}^{+}\times\mathbb{R}^{n},
  5. u ( 0 , x ) = 0 , u t ( 0 , x ) = f ( x ) , for f 𝒮 ( n ) . u(0,x)=0,\quad\frac{\partial u}{\partial t}(0,x)=f(x),\quad\mathrm{for}\quad f% \in\mathcal{S}^{\prime}(\mathbb{R}^{n}).
  6. u ( t , x ) = 1 ( 2 π ) n e i ( x , ξ + c t | ξ | ) 2 i | ξ | f ^ ( ξ ) d ξ - 1 ( 2 π ) n e i ( x , ξ - c t | ξ | ) 2 i | ξ | f ^ ( ξ ) d ξ . u(t,x)=\frac{1}{(2\pi)^{n}}\int\frac{e^{i(\langle x,\xi\rangle+ct|\xi|)}}{2i|% \xi|}\hat{f}(\xi)\,d\xi-\frac{1}{(2\pi)^{n}}\int\frac{e^{i(\langle x,\xi% \rangle-ct|\xi|)}}{2i|\xi|}\hat{f}(\xi)\,d\xi.

Fourier–Motzkin_elimination.html

  1. S S
  2. n n
  3. r r
  4. x 1 x_{1}
  5. x r x_{r}
  6. x r x_{r}
  7. x r x_{r}
  8. x r b i - k = 1 r - 1 a i k x k x_{r}\geq b_{i}-\sum_{k=1}^{r-1}a_{ik}x_{k}
  9. x r A j ( x 1 , , x r - 1 ) x_{r}\geq A_{j}(x_{1},\dots,x_{r-1})
  10. j j
  11. n A n_{A}
  12. n A n_{A}
  13. x r b i - k = 1 r - 1 a i k x k x_{r}\leq b_{i}-\sum_{k=1}^{r-1}a_{ik}x_{k}
  14. x r B j ( x 1 , , x r - 1 ) x_{r}\leq B_{j}(x_{1},\dots,x_{r-1})
  15. j j
  16. n B n_{B}
  17. n B n_{B}
  18. x r x_{r}
  19. ϕ \phi
  20. max ( A 1 ( x 1 , , x r - 1 ) , , A n A ( x 1 , , x r - 1 ) ) x r min ( B 1 ( x 1 , , x r - 1 ) , , B n B ( x 1 , , x r - 1 ) ) ϕ \max(A_{1}(x_{1},\dots,x_{r-1}),\dots,A_{n_{A}}(x_{1},\dots,x_{r-1}))\leq x_{r% }\leq\min(B_{1}(x_{1},\dots,x_{r-1}),\dots,B_{n_{B}}(x_{1},\dots,x_{r-1}))\wedge\phi
  21. x r S \exists x_{r}~{}S
  22. max ( A 1 ( x 1 , , x r - 1 ) , , A n A ( x 1 , , x r - 1 ) ) min ( B 1 ( x 1 , , x r - 1 ) , , B n B ( x 1 , , x r - 1 ) ) ϕ \max(A_{1}(x_{1},\dots,x_{r-1}),\dots,A_{n_{A}}(x_{1},\dots,x_{r-1}))\leq\min(% B_{1}(x_{1},\dots,x_{r-1}),\dots,B_{n_{B}}(x_{1},\dots,x_{r-1}))\wedge\phi
  23. max ( A 1 ( x 1 , , x r - 1 ) , , A n A ( x 1 , , x r - 1 ) ) min ( B 1 ( x 1 , , x r - 1 ) , , B n B ( x 1 , , x r - 1 ) ) \max(A_{1}(x_{1},\dots,x_{r-1}),\dots,A_{n_{A}}(x_{1},\dots,x_{r-1}))\leq\min(% B_{1}(x_{1},\dots,x_{r-1}),\dots,B_{n_{B}}(x_{1},\dots,x_{r-1}))
  24. n A n B n_{A}n_{B}
  25. A i ( x 1 , , x r - 1 ) B j ( x 1 , , x r - 1 ) A_{i}(x_{1},\dots,x_{r-1})\leq B_{j}(x_{1},\dots,x_{r-1})
  26. 1 i n A 1\leq i\leq n_{A}
  27. 1 j n B 1\leq j\leq n_{B}
  28. x r x_{r}
  29. ( n - n A - n B ) + n A n B (n-n_{A}-n_{B})+n_{A}n_{B}
  30. n A = n B = n / 2 n_{A}=n_{B}=n/2
  31. n 2 / 4 n^{2}/4
  32. n n
  33. n 2 / 4 n^{2}/4
  34. d d
  35. 4 ( n / 4 ) 2 d 4(n/4)^{2^{d}}

Fourier–Mukai_transform.html

  1. X X
  2. X ^ \hat{X}
  3. 𝒫 \mathcal{P}
  4. X × X ^ , X\times\hat{X},
  5. p p
  6. p ^ \hat{p}
  7. R 𝒮 : D ( X ) R p ^ ( p 𝒫 ) D ( X ^ ) R\mathcal{S}:\mathcal{F}\in D(X)\mapsto R\hat{p}_{\ast}(p^{\ast}\mathcal{F}% \otimes\mathcal{P})\in D(\hat{X})
  8. R 𝒮 ^ : D ( X ^ ) D ( X ) . R\widehat{\mathcal{S}}:D(\hat{X})\to D(X).\,
  9. R 𝒮 R 𝒮 ^ = ( - 1 ) [ - g ] R\mathcal{S}\circ R\widehat{\mathcal{S}}=(-1)^{\ast}[-g]
  10. R 𝒮 ( 𝒢 ) = R 𝒮 ( ) R 𝒮 ( 𝒢 ) R\mathcal{S}(\mathcal{F}\ast\mathcal{G})=R\mathcal{S}(\mathcal{F})\otimes R% \mathcal{S}(\mathcal{G})
  11. R 𝒮 ( 𝒢 ) = R 𝒮 ( ) R 𝒮 ( 𝒢 ) [ g ] R\mathcal{S}(\mathcal{F}\otimes\mathcal{G})=R\mathcal{S}(\mathcal{F})\ast R% \mathcal{S}(\mathcal{G})[g]

Fox_derivative.html

  1. g i \frac{\partial}{\partial g_{i}}
  2. g i ( g j ) = δ i j \frac{\partial}{\partial g_{i}}(g_{j})=\delta_{ij}
  3. δ i j \delta_{ij}
  4. g i ( e ) = 0 \frac{\partial}{\partial g_{i}}(e)=0
  5. g i ( u v ) = g i ( u ) + u g i ( v ) \frac{\partial}{\partial g_{i}}(uv)=\frac{\partial}{\partial g_{i}}(u)+u\frac{% \partial}{\partial g_{i}}(v)
  6. g i ( u - 1 ) = - u - 1 g i ( u ) \frac{\partial}{\partial g_{i}}(u^{-1})=-u^{-1}\frac{\partial}{\partial g_{i}}% (u)

Fractal_dimension_on_networks.html

  1. N N
  2. l ln N \left\langle l\right\rangle\sim\ln{N}
  3. l l
  4. N e l / l 0 N\sim e^{\left\langle l\right\rangle/l_{0}}
  5. l 0 l_{0}
  6. N B N_{B}
  7. l B l_{B}
  8. d B d_{B}
  9. N B l B - d B N_{B}\sim l_{B}^{-d_{B}}
  10. M B ( l B ) \left\langle M_{B}\left(l_{B}\right)\right\rangle
  11. l B l_{B}
  12. M B ( l B ) l B d B \left\langle M_{B}\left(l_{B}\right)\right\rangle\sim l_{B}^{d_{B}}
  13. N N
  14. M B ( l B ) \left\langle M_{B}\left(l_{B}\right)\right\rangle
  15. d B d_{B}
  16. l l
  17. l l
  18. d f d_{f}
  19. M C l d f \left\langle M_{C}\right\rangle\sim l^{d_{f}}
  20. M C \left\langle M_{C}\right\rangle
  21. M B ( l B ) l B d B \left\langle M_{B}\left(l_{B}\right)\right\rangle\sim l_{B}^{d_{B}}
  22. d B = 4.1 , 3.4 , 2.0 , and 1.8 d_{B}=4.1,\mbox{ }~{}3.4,\mbox{ }~{}2.0,\mbox{ and }~{}1.8

Fraction_of_variance_unexplained.html

  1. y ^ i = f ( x i ) \widehat{y}_{i}=f(x_{i})
  2. x i x_{i}
  3. FVU \displaystyle\,\text{FVU}
  4. S S err = i = 1 N ( y i - y i ^ ) 2 S S tot = i = 1 N ( y i - y ¯ ) 2 S S reg = i = 1 N ( y i ^ - y ¯ ) 2 and y ¯ = 1 N y i i = 1 N . \begin{aligned}\displaystyle SS_{\rm err}&\displaystyle=\sum_{i=1}^{N}\;(y_{i}% -\widehat{y_{i}})^{2}\\ \displaystyle SS_{\rm tot}&\displaystyle=\sum_{i=1}^{N}\;(y_{i}-\bar{y})^{2}\\ \displaystyle SS_{\rm reg}&\displaystyle=\sum_{i=1}^{N}\;(\widehat{y_{i}}-\bar% {y})^{2}\,\text{ and}\\ \displaystyle\bar{y}&\displaystyle=\frac{1}{N}\sum{}_{i=1}^{N}\;y_{i}.\end{aligned}
  5. FVU = MSE ( f ) var [ Y ] , \,\text{FVU}=\frac{\,\text{MSE}(f)}{\,\text{var}[Y]},
  6. f ( x i ) = y ¯ f(x_{i})=\bar{y}
  7. y ^ i = y i \hat{y}_{i}=y_{i}

Fraňková–Helly_selection_theorem.html

  1. ( f n ( k ) ) ( f n ) BV ( [ 0 , T ] ; X ) \left(f_{n(k)}\right)\subseteq(f_{n})\subset\mathrm{BV}([0,T];X)
  2. λ ( f n ( k ) ( t ) ) λ ( f ( t ) ) in as k . \lambda\left(f_{n(k)}(t)\right)\to\lambda(f(t))\mbox{ in }~{}\mathbb{R}\mbox{ % as }~{}k\to\infty.
  3. f n ( t ) = sin ( n t ) . f_{n}(t)=\sin(nt).
  4. f n - u n < ε \|f_{n}-u_{n}\|_{\infty}<\varepsilon
  5. | u n ( 0 ) | + Var ( u n ) L ε , |u_{n}(0)|+\mathrm{Var}(u_{n})\leq L_{\varepsilon},
  6. sup Π j = 1 m | u ( t j ) - u ( t j - 1 ) | \sup_{\Pi}\sum_{j=1}^{m}|u(t_{j})-u(t_{j-1})|
  7. Π = { 0 = t 0 < t 1 < < t m = T , m 𝐍 } \Pi=\{0=t_{0}<t_{1}<\dots<t_{m}=T,m\in\mathbf{N}\}
  8. ( f n ( k ) ) ( f n ) Reg ( [ 0 , T ] ; X ) \left(f_{n(k)}\right)\subseteq(f_{n})\subset\mathrm{Reg}([0,T];X)
  9. λ ( f n ( k ) ( t ) ) λ ( f ( t ) ) in as k . \lambda\left(f_{n(k)}(t)\right)\to\lambda(f(t))\mbox{ in }~{}\mathbb{R}\mbox{ % as }~{}k\to\infty.

Fraser_Filter.html

  1. f ( i ) = f i f(i)=f_{i}
  2. a v e r a g e 12 = f 1 + f 2 2 average_{12}=\frac{f_{1}+f_{2}}{2}
  3. a v e r a g e 34 = f 3 + f 4 2 average_{34}=\frac{f_{3}+f_{4}}{2}
  4. Δ x \Delta x
  5. a v e r a g e 12 - a v e r a g e 34 2 Δ x = ( f 1 + f 2 ) - ( f 3 + f 4 ) 4 Δ x \frac{average_{12}-average_{34}}{2\Delta x}=\frac{(f_{1}+f_{2})-(f_{3}+f_{4})}% {4\Delta x}
  6. 4 Δ x 4\Delta x
  7. ( f 1 + f 2 ) - ( f 3 + f 4 ) (f_{1}+f_{2})-(f_{3}+f_{4})

Fred_Galvin.html

  1. 2 ω 1 < ( 2 1 ) + 2^{\aleph_{\omega_{1}}}<\aleph_{(2^{\aleph_{1}})^{+}}
  2. ω 1 ( α ) k 2 \omega_{1}\to(\alpha)^{2}_{k}
  3. α < ω 1 , k < ω \alpha<\omega_{1},k<\omega
  4. 1 ↛ [ 1 ] 4 2 \aleph_{1}\not\to[\aleph_{1}]^{2}_{4}
  5. 2 0 ↛ [ 2 0 ] 0 2 2^{\aleph_{0}}\not\to[2^{\aleph_{0}}]^{2}_{\aleph_{0}}
  6. η [ η ] 3 2 \eta\to[\eta]^{2}_{3}

Free-by-cyclic_group.html

  1. G G
  2. F F
  3. G / F G/F
  4. G G
  5. F F
  6. G G

Free_androgen_index.html

  1. F A I = 100 × ( T o t a l T e s t o s t e r o n e S H B G ) FAI=100\times(\frac{Total\ Testosterone}{SHBG})

Free_energy_perturbation.html

  1. Δ F ( A B ) = F B - F A = - k B T ln exp ( - E B - E A k B T ) A \Delta F(A\rightarrow B)=F_{B}-F_{A}=-k_{B}T\ln\left\langle\exp\left(-\frac{E_% {B}-E_{A}}{k_{B}T}\right)\right\rangle_{A}

Free_lattice.html

  1. f ~ : F X L \tilde{f}:FX\to L
  2. f = f ~ η f=\tilde{f}\circ\eta
  3. f ~ \tilde{f}
  4. S F X { f ( s ) | s S } S\in FX\mapsto\bigvee\left\{f(s)|s\in S\right\}
  5. \bigvee
  6. f ~ \tilde{f}
  7. sup N : ( f : N F X ) \operatorname{sup}_{N}:(f:N\to FX)
  8. sup N \operatorname{sup}_{N}
  9. p n p_{n}
  10. p α p_{\alpha}
  11. p α = sup { p β | β < α } p_{\alpha}=\operatorname{sup}\{p_{\beta}|\beta<\alpha\}
  12. α \alpha
  13. p α + 1 p_{\alpha+1}
  14. p α p_{\alpha}

Free_recoil.html

  1. E t g u = 0.5 [ ( m p v p ) + ( m c v c ) 1000 ] 2 / m g u E_{tgu}=0.5\cdot[\tfrac{(m_{p}\cdot v_{p})+(m_{c}\cdot v_{c})}{1000}]^{2}/m_{gu}
  2. v g u = ( m p v p ) + ( m c v c ) 1000 m g u v_{gu}=\tfrac{(m_{p}\cdot v_{p})+(m_{c}\cdot v_{c})}{1000\cdot m_{gu}}
  3. E t g u = 0.5 m g u v g u 2 E_{tgu}=0.5\cdot m_{gu}\cdot v_{gu}^{2}\,
  4. E t g u = 0.5 [ ( m p v p ) + ( m c v c ) 1000 ] 2 / m g u E_{tgu}=0.5\cdot[\tfrac{(m_{p}\cdot v_{p})+(m_{c}\cdot v_{c})}{1000}]^{2}/m_{gu}
  5. E t g u = 0.5 [ ( 9.1 823 ) + ( 2.75 1585 ) 1000 ] 2 / 4.54 = E_{tgu}=0.5\cdot[\tfrac{(9.1\cdot 823)+(2.75\cdot 1585)}{1000}]^{2}/4.54=
  6. E t g u = 0.5 [ ( 7489.3 ) + ( 4358.75 ) 1000 ] 2 / 4.54 = E_{tgu}=0.5\cdot[\tfrac{(7489.3)+(4358.75)}{1000}]^{2}/4.54=
  7. E t g u = 0.5 [ 11848.05 1000 ] 2 / 4.54 = E_{tgu}=0.5\cdot[\tfrac{11848.05}{1000}]^{2}/4.54=
  8. E t g u = 0.5 11.848 2 / 4.54 = E_{tgu}=0.5\cdot 11.848^{2}/4.54=\,
  9. E t g u = 0.5 140.367 / 4.54 = E_{tgu}=0.5\cdot 140.367/4.54=\,
  10. E t g u = 70.188 / 4.54 = E_{tgu}=70.188/4.54=\,
  11. E t g u = 15.46 J E_{tgu}=15.46J\,
  12. v g u = ( m p v p ) + ( m c v c ) 7000 / m g u v_{gu}=\tfrac{(m_{p}\cdot v_{p})+(m_{c}\cdot v_{c})}{7000}/m_{gu}
  13. E t g u = m g u v g u 2 2 g c E_{tgu}=\tfrac{m_{gu}\cdot v_{gu}^{2}}{2g_{c}}\,

Frequency_addition_source_of_optical_radiation.html

  1. λ 1 \lambda_{1}
  2. λ 2 \lambda_{2}
  3. λ = ( 1 λ 1 + 1 λ 2 ) - 1 . \lambda=\left(\frac{1}{\lambda_{1}}+\frac{1}{\lambda_{2}}\right)^{-1}.

Frequency_scaling.html

  1. Runtime = Instructions Program × Cycles Instruction × Time Cycle \mathrm{Runtime}=\frac{\mathrm{Instructions}}{\mathrm{Program}}\times\frac{% \mathrm{Cycles}}{\mathrm{Instruction}}\times\frac{\mathrm{Time}}{\mathrm{Cycle}}
  2. P = C × V 2 × F P=C\times V^{2}\times F

Frettenheim.html

  1. \vdots

Frictionless_plane.html

  1. F f = μ k F N , F_{\mathrm{f}}=\mu_{\mathrm{k}}F_{\mathrm{N}},
  2. F f F_{\mathrm{f}}
  3. F N F_{\mathrm{N}}
  4. μ k \mu_{\mathrm{k}}

Friedel's_law.html

  1. f ( x ) f(x)
  2. F ( k ) = - + f ( x ) e i k x d x F(k)=\int^{+\infty}_{-\infty}f(x)e^{ik\cdot x}dx
  3. F ( k ) = F * ( - k ) F(k)=F^{*}(-k)\,
  4. F * F^{*}
  5. F F
  6. ( k , - k ) (k,-k)
  7. | F | 2 |F|^{2}
  8. | F ( k ) | 2 = | F ( - k ) | 2 |F(k)|^{2}=|F(-k)|^{2}\,
  9. ϕ \phi
  10. F F
  11. ϕ ( k ) = - ϕ ( - k ) \phi(k)=-\phi(-k)\,

Friedlander–Iwaniec_theorem.html

  1. a 2 + b 4 a^{2}+b^{4}
  2. a 2 + b 4 a^{2}+b^{4}
  3. X X
  4. X 3 / 4 X^{3/4}
  5. a 2 + 1 a^{2}+1

Frobenius_covariant.html

  1. A A
  2. A A
  3. f ( A ) f(A)
  4. A A
  5. A A
  6. k k
  7. A i j = 1 j i k 1 λ i - λ j ( A - λ j I ) . A_{i}\equiv\prod_{j=1\atop j\neq i}^{k}\frac{1}{\lambda_{i}-\lambda_{j}}(A-% \lambda_{j}I).
  8. A A
  9. S S
  10. D D
  11. A A
  12. i i
  13. A A
  14. i i
  15. S S
  16. i i
  17. A A
  18. i i
  19. S S
  20. A A
  21. A = [ 1 3 4 2 ] . A=\begin{bmatrix}1&3\\ 4&2\end{bmatrix}.
  22. A = [ 3 1 / 7 4 - 1 / 7 ] [ 5 0 0 - 2 ] [ 3 1 / 7 4 - 1 / 7 ] - 1 = [ 3 1 / 7 4 - 1 / 7 ] [ 5 0 0 - 2 ] [ 1 / 7 1 / 7 4 - 3 ] . A=\begin{bmatrix}3&1/7\\ 4&-1/7\end{bmatrix}\begin{bmatrix}5&0\\ 0&-2\end{bmatrix}\begin{bmatrix}3&1/7\\ 4&-1/7\end{bmatrix}^{-1}=\begin{bmatrix}3&1/7\\ 4&-1/7\end{bmatrix}\begin{bmatrix}5&0\\ 0&-2\end{bmatrix}\begin{bmatrix}1/7&1/7\\ 4&-3\end{bmatrix}.
  23. A 1 = c 1 r 1 = [ 3 4 ] [ 1 / 7 1 / 7 ] = [ 3 / 7 3 / 7 4 / 7 4 / 7 ] = A 1 2 A 2 = c 2 r 2 = [ 1 / 7 - 1 / 7 ] [ 4 - 3 ] = [ 4 / 7 - 3 / 7 - 4 / 7 3 / 7 ] = A 2 2 , \begin{array}[]{rl}A_{1}&=c_{1}r_{1}=\begin{bmatrix}3\\ 4\end{bmatrix}\begin{bmatrix}1/7&1/7\end{bmatrix}=\begin{bmatrix}3/7&3/7\\ 4/7&4/7\end{bmatrix}=A_{1}^{2}\\ A_{2}&=c_{2}r_{2}=\begin{bmatrix}1/7\\ -1/7\end{bmatrix}\begin{bmatrix}4&-3\end{bmatrix}=\begin{bmatrix}4/7&-3/7\\ -4/7&3/7\end{bmatrix}=A_{2}^{2}~{},\end{array}
  24. A 1 A 2 = 0 , A 1 + A 2 = I . A_{1}A_{2}=0,\qquad A_{1}+A_{2}=I~{}.
  25. t r A < s u b > 1 = t r A 2 = 1 trA<sub>1=trA_{2}=1

Frobenius_matrix.html

  1. A = ( 1 0 0 0 0 1 0 0 0 a 32 1 0 0 a n 2 0 1 ) A=\begin{pmatrix}1&0&0&\cdots&0\\ 0&1&0&\cdots&0\\ 0&a_{32}&1&\cdots&0\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ 0&a_{n2}&0&\cdots&1\end{pmatrix}
  2. A - 1 = ( 1 0 0 0 0 1 0 0 0 - a 32 1 0 0 - a n 2 0 1 ) A^{-1}=\begin{pmatrix}1&0&0&\cdots&0\\ 0&1&0&\cdots&0\\ 0&-a_{32}&1&\cdots&0\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ 0&-a_{n2}&0&\cdots&1\end{pmatrix}

Frontogenesis.html

  1. F = 1 | θ | θ x { 1 C p ( p p ) κ [ x ( d Q d t ) ] - ( u x θ x ) - ( v x θ y ) - ( w x θ z ) } + θ y { 1 C p ( p p ) κ [ y ( d Q d t ) ] - ( u y θ x ) - ( v y θ y ) - ( w y θ z ) } + θ z { p κ C p [ z ( p - κ d Q d t ) ] - ( u z θ x ) - ( v z θ y ) - ( w z θ z ) } \begin{aligned}\displaystyle F=\frac{1}{|\nabla\theta|}\cdot\frac{\partial% \theta}{\partial x}\left\{\frac{1}{C_{p}}\left(\frac{p_{\circ}}{p}\right)^{% \kappa}\left[\frac{\partial}{\partial x}\left(\frac{dQ}{dt}\right)\right]-% \left(\frac{\partial u}{\partial x}\frac{\partial\theta}{\partial x}\right)-% \left(\frac{\partial v}{\partial x}\frac{\partial\theta}{\partial y}\right)-% \left(\frac{\partial w}{\partial x}\frac{\partial\theta}{\partial z}\right)% \right\}\\ \displaystyle+\frac{\partial\theta}{\partial y}\left\{\frac{1}{C_{p}}\left(% \frac{p_{\circ}}{p}\right)^{\kappa}\left[\frac{\partial}{\partial y}\left(% \frac{dQ}{dt}\right)\right]-\left(\frac{\partial u}{\partial y}\frac{\partial% \theta}{\partial x}\right)-\left(\frac{\partial v}{\partial y}\frac{\partial% \theta}{\partial y}\right)-\left(\frac{\partial w}{\partial y}\frac{\partial% \theta}{\partial z}\right)\right\}\\ \displaystyle+\frac{\partial\theta}{\partial z}\left\{\frac{p_{\circ}^{\kappa}% }{C_{p}}\left[\frac{\partial}{\partial z}\left(p^{-\kappa}\frac{dQ}{dt}\right)% \right]-\left(\frac{\partial u}{\partial z}\frac{\partial\theta}{\partial x}% \right)-\left(\frac{\partial v}{\partial z}\frac{\partial\theta}{\partial y}% \right)-\left(\frac{\partial w}{\partial z}\frac{\partial\theta}{\partial z}% \right)\right\}\end{aligned}
  2. x x
  3. 1 C p ( p p ) κ [ x ( d Q d t ) ] \frac{1}{C_{p}}\left(\frac{p_{\circ}}{p}\right)^{\kappa}\left[\frac{\partial}{% \partial x}\left(\frac{dQ}{dt}\right)\right]
  4. y y
  5. 1 C p ( p p ) κ [ y ( d Q d t ) ] \frac{1}{C_{p}}\left(\frac{p_{\circ}}{p}\right)^{\kappa}\left[\frac{\partial}{% \partial y}\left(\frac{dQ}{dt}\right)\right]
  6. z z
  7. p κ C p [ z ( p - κ d Q d t ) ] \frac{p_{\circ}^{\kappa}}{C_{p}}\left[\frac{\partial}{\partial z}\left(p^{-% \kappa}\frac{dQ}{dt}\right)\right]
  8. x x
  9. - ( u x θ x ) - ( v x θ y ) -\left(\frac{\partial u}{\partial x}\frac{\partial\theta}{\partial x}\right)-% \left(\frac{\partial v}{\partial x}\frac{\partial\theta}{\partial y}\right)
  10. y y
  11. - ( u y θ x ) - ( v y θ y ) -\left(\frac{\partial u}{\partial y}\frac{\partial\theta}{\partial x}\right)-% \left(\frac{\partial v}{\partial y}\frac{\partial\theta}{\partial y}\right)
  12. z z
  13. - ( u z θ x ) - ( v z θ y ) -\left(\frac{\partial u}{\partial z}\frac{\partial\theta}{\partial x}\right)-% \left(\frac{\partial v}{\partial z}\frac{\partial\theta}{\partial y}\right)
  14. x x
  15. y y
  16. - ( w x θ z ) -\left(\frac{\partial w}{\partial x}\frac{\partial\theta}{\partial z}\right)
  17. - ( w y θ z ) -\left(\frac{\partial w}{\partial y}\frac{\partial\theta}{\partial z}\right)
  18. - ( w z θ z ) -\left(\frac{\partial w}{\partial z}\frac{\partial\theta}{\partial z}\right)

Froude–Krylov_force.html

  1. F F K = - S w p n d s , \vec{F}_{FK}=-\iint_{S_{w}}p~{}\vec{n}~{}ds,
  2. F F K \vec{F}_{FK}
  3. S w S_{w}
  4. p p
  5. n \vec{n}
  6. F F K = A p d y n . F_{FK}=A\cdot p_{dyn}.
  7. p d y n p_{dyn}
  8. p d y n = ρ g H / 2 p_{dyn}=\rho\cdot g\cdot H/2
  9. ρ \rho
  10. g g
  11. H H

Fuel_fraction.html

  1. ζ = Δ W W 1 \ \zeta=\frac{\Delta W}{W_{1}}

Fujiki_class_C.html

  1. X M X\subset M
  2. X M X\subset M
  3. M ω d i m M > 0. \int_{M}\omega^{{dim_{\mathbb{C}}M}}>0.
  4. [ ω ] H 2 ( M ) [\omega]\in H^{2}(M)
  5. c 1 ( L ) = [ ω ] c_{1}(L)=[\omega]
  6. H 0 ( L N ) {\mathbb{P}}H^{0}(L^{N})

Full_flight_simulator.html

  1. φ arctan ( l / R ) \varphi\approx\arctan\,(l/R)
  2. φ 17 \varphi\approx 17^{\circ}

Fully_differential_amplifier.html

  1. V i d = V i n + - V i n - V_{id}=V_{in+}-V_{in-}
  2. V o d = V o u t + - V o u t - = V i d × G a i n V_{od}=V_{out+}-V_{out-}=V_{id}\times Gain
  3. V o c = ( V o u t + ) + ( V o u t - ) 2 V_{oc}=\frac{(V_{out+})+(V_{out-})}{2}

Fundamental_diagram_of_traffic_flow.html

  1. μ ( n ) \mu(n)
  2. μ 1 \mu_{1}
  3. η \eta
  4. q ¯ \bar{q}
  5. q ¯ = k = 1 n d i ( B ) n T L \bar{q}=\frac{\sum_{k=1}^{n}d_{i}(B)}{nTL}
  6. k ¯ \bar{k}
  7. k ¯ = k = 1 n t i ( B ) n T L \bar{k}=\frac{\sum_{k=1}^{n}t_{i}(B)}{nTL}
  8. v ¯ \bar{v}
  9. v ¯ = q ¯ k ¯ \bar{v}=\frac{\bar{q}}{\bar{k}}
  10. n = k ¯ k = 1 n l i = k ¯ L n=\bar{k}\sum_{k=1}^{n}l_{i}=\bar{k}L
  11. L L
  12. d d
  13. τ \tau
  14. τ = d v ¯ = n d M F D ( n ) L \tau=\frac{d}{\bar{v}}=\frac{nd}{MFD(n)L}

Fundamental_matrix_(linear_differential_equation).html

  1. 𝐱 ˙ ( t ) = A ( t ) 𝐱 ( t ) \dot{\mathbf{x}}(t)=A(t)\mathbf{x}(t)
  2. Ψ ( t ) \Psi(t)
  3. 𝐱 = Ψ ( t ) 𝐜 \mathbf{x}=\Psi(t)\mathbf{c}
  4. 𝐜 \mathbf{c}
  5. Ψ \Psi
  6. 𝐱 ˙ ( t ) = A ( t ) 𝐱 ( t ) \dot{\mathbf{x}}(t)=A(t)\mathbf{x}(t)
  7. Ψ ˙ ( t ) = A ( t ) Ψ ( t ) \dot{\Psi}(t)=A(t)\Psi(t)
  8. Ψ \Psi
  9. t t

Futile_cycle.html

  1. A T P + H 2 O A D P + P i + h e a t ATP+H_{2}O{\rightleftharpoons}ADP+P_{i}+heat

Fxgrep.html

  1. k k
  2. k k

G-index.html

  1. g 2 < m t p l > i g c i g^{2}\leq\sum_{<}mtpl>{{i\leq g}}c_{i}
  2. g 1 g < m t p l > i g c i g\leq\frac{1}{g}\sum_{<}mtpl>{{i\leq g}}c_{i}

G-network.html

  1. Λ i \scriptstyle{\Lambda_{i}}
  2. λ i \scriptstyle{\lambda_{i}}
  3. p i j + \scriptstyle{p_{ij}^{+}}
  4. p i j - \scriptstyle{p_{ij}^{-}}
  5. d i \scriptstyle{d_{i}}
  6. ρ i = λ i + μ i + λ i - \rho_{i}=\frac{\lambda^{+}_{i}}{\mu_{i}+\lambda^{-}_{i}}
  7. λ i + , λ i - \scriptstyle{\lambda^{+}_{i},\lambda^{-}_{i}}
  8. i = 1 , , m \scriptstyle{i=1,\ldots,m}
  9. λ i + = j ρ j μ j p j i + + Λ i \lambda^{+}_{i}=\sum_{j}\rho_{j}\mu_{j}p^{+}_{ji}+\Lambda_{i}\,
  10. λ i - = j ρ j μ j p j i - + λ i . \lambda^{-}_{i}=\sum_{j}\rho_{j}\mu_{j}p^{-}_{ji}+\lambda_{i}.\,
  11. ( λ i + , λ i - ) \scriptstyle{(\lambda^{+}_{i},\lambda^{-}_{i})}
  12. π ( n 1 , n 2 , , n m ) = i = 1 m ( 1 - ρ i ) ρ i n i . \pi(n_{1},n_{2},\ldots,n_{m})=\prod_{i=1}^{m}(1-\rho_{i})\rho_{i}^{n_{i}}.
  13. π \pi
  14. W ( s ) = μ ( 1 - ρ ) λ + s + λ + μ ( 1 - ρ ) - [ s + λ + μ ( 1 - ρ ) ] 2 - 4 λ + λ - λ - - λ + - μ ( 1 - ρ ) - s + [ s + λ + μ ( 1 - ρ ) ] 2 - 4 λ + λ - W^{\ast}(s)=\frac{\mu(1-\rho)}{\lambda^{+}}\frac{s+\lambda+\mu(1-\rho)-\sqrt{[% s+\lambda+\mu(1-\rho)]^{2}-4\lambda^{+}\lambda^{-}}}{\lambda^{-}-\lambda^{+}-% \mu(1-\rho)-s+\sqrt{[s+\lambda+\mu(1-\rho)]^{2}-4\lambda^{+}\lambda^{-}}}

Gain_(information_retrieval).html

  1. r = T P + F N T P + T N + F P + F N = 𝑃𝑜𝑠𝑖𝑡𝑖𝑣𝑒𝑠 N r=\frac{TP+FN}{TP+TN+FP+FN}=\frac{\,\textit{Positives}}{N}
  2. G = 𝑝𝑟𝑒𝑐𝑖𝑠𝑖𝑜𝑛 r G=\frac{\,\textit{precision}}{r}
  3. A c c = T P + T N T P + T N + F P + F N = 𝐶𝑜𝑟𝑟𝑒𝑐𝑡𝑠 N Acc=\frac{TP+TN}{TP+TN+FP+FN}=\frac{\,\textit{Corrects}}{N}
  4. r = ( 𝑃𝑜𝑠𝑖𝑡𝑖𝑣𝑒𝑠 N ) 2 + ( 𝑁𝑒𝑔𝑎𝑡𝑖𝑣𝑒𝑠 N ) 2 = f ( 𝑃𝑜𝑠𝑖𝑡𝑖𝑣𝑒𝑠 ) 2 + f ( 𝑁𝑒𝑔𝑎𝑡𝑖𝑣𝑒𝑠 ) 2 r=\left(\frac{\,\textit{Positives}}{N}\right)^{2}+\left(\frac{\,\textit{% Negatives}}{N}\right)^{2}=f(\,\textit{Positives})^{2}+f(\,\textit{Negatives})^% {2}
  5. G = 𝐴𝑐𝑐 r G=\frac{\,\textit{Acc}}{r}

Gain_(laser).html

  1. P ~{}P~{}
  2. G = < m t p l > dd z ln ( P ) = d P / d z P G=\frac{<}{m}tpl>{{\rm d}}{{\rm d}z}\ln(P)=\frac{{\rm d}P/{\rm d}z}{P}
  3. z ~{}z~{}
  4. 2 i k E z = Δ E + 2 ν E + i G E 2ik\frac{\partial E}{\partial z}=\Delta_{\perp}E+2\nu E+iGE
  5. ν ~{}\nu~{}
  6. E ~{}E~{}
  7. E phys ~{}E_{\rm phys}~{}
  8. E phys = Re ( e E exp ( i k z - i ω t ) ) ~{}E_{\rm phys}={\rm Re}\left(\vec{e}E\exp(ikz-i\omega t)\right)~{}
  9. e ~{}\vec{e}~{}
  10. k ~{}k~{}
  11. ω ~{}\omega~{}
  12. Δ perp = ( 2 x 2 + 2 y 2 ) ~{}\Delta_{\rm perp}=\left(\frac{\partial^{2}}{\partial x^{2}}+\frac{\partial^% {2}}{\partial y^{2}}\right)~{}
  13. Re ~{}\rm Re~{}
  14. N 1 ~{}N_{1}~{}
  15. N 2 ~{}N_{2}~{}
  16. G = σ e N 2 - σ a N 1 ~{}G=\sigma_{\rm e}N_{2}-\sigma_{\rm a}N_{1}~{}
  17. σ e ~{}\sigma_{\rm e}~{}
  18. σ a ~{}\sigma_{\rm a}~{}
  19. g = G d z g=\int G{\rm d}z
  20. K ~{}K~{}
  21. P out ~{}P_{\rm out}
  22. P in ~{}P_{\rm in}
  23. K = P out / P in ~{}K=P_{\rm out}/P_{\rm in}
  24. K = exp ( G d z ) ~{}K=\exp\left(\int G{\rm d}z\right)~{}
  25. K ~{}K~{}

Gaisser–Hillas_function.html

  1. N ( X ) N(X)
  2. X X
  3. N ( X ) = N max ( X - X 0 X max - X 0 ) X max - X 0 λ exp ( X max - X λ ) , N(X)=N\text{max}\left(\frac{X-X_{0}}{X\text{max}-X_{0}}\right)^{\frac{X\text{% max}-X_{0}}{\lambda}}\exp\left(\frac{X\text{max}-X}{\lambda}\right),
  4. N max N\text{max}
  5. X max X\text{max}
  6. X 0 X_{0}
  7. λ \lambda
  8. n = N N max n=\frac{N}{N\text{max}}
  9. x = X - X 0 λ x=\frac{X-X_{0}}{\lambda}
  10. m = X max - X 0 λ m=\frac{X\text{max}-X_{0}}{\lambda}
  11. n ( x ) = [ x m ] m exp ( m - x ) = x m e - x m m e - m = exp [ m ( ln x - ln m ) - ( x - m ) ] . n(x)=\left[\frac{x}{m}\right]^{m}\exp(m-x)=\frac{x^{m}\,e^{-x}}{m^{m}\,e^{-m}}% =\exp[m(\ln x-\ln m)-(x-m)].

Gamma_diversity.html

  1. γ = α × β \gamma=\alpha\times\beta
  2. D γ q = 1 i = 1 S p i p i q - 1 q - 1 {}^{q}\!D_{\gamma}={1\over{\sqrt[q-1]{{\sum_{i=1}^{S}p_{i}p_{i}^{q-1}}}}}
  3. p i p_{i}

Gammatone_filter.html

  1. g ( t ) = a t n - 1 e - 2 π b t cos ( 2 π f t + ϕ ) , g(t)=at^{n-1}e^{-2\pi bt}\cos(2\pi ft+\phi),\,
  2. f f
  3. ϕ \phi
  4. a a
  5. n n
  6. b b
  7. t t

Gauss's_continued_fraction.html

  1. f 0 , f 1 , f 2 , f_{0},f_{1},f_{2},\dots
  2. f i - 1 - f i = k i z f i + 1 f_{i-1}-f_{i}=k_{i}\,z\,f_{i+1}
  3. i > 0 i>0
  4. k i k_{i}
  5. f i - 1 f i = 1 + k i z f i + 1 < m t p l > f i , \frac{f_{i-1}}{f_{i}}=1+k_{i}z\frac{f_{i+1}}{<}mtpl>{{f_{i}}},\,
  6. f i f i - 1 = 1 1 + k i z f i + 1 < m t p l > f i \frac{f_{i}}{f_{i-1}}=\frac{1}{1+k_{i}z\frac{f_{i+1}}{<}mtpl>{{f_{i}}}}
  7. g i = f i / f i - 1 g_{i}=f_{i}/f_{i-1}
  8. g i = 1 1 + k i z g i + 1 g_{i}=\frac{1}{1+k_{i}zg_{i+1}}
  9. g 1 = f 1 f 0 = 1 1 + k 1 z g 2 = 1 1 + k 1 z 1 + k 2 z g 3 = 1 1 + k 1 z 1 + k 2 z 1 + k 3 z g 4 = g_{1}=\frac{f_{1}}{f_{0}}=\cfrac{1}{1+k_{1}zg_{2}}=\cfrac{1}{1+\cfrac{k_{1}z}{% 1+k_{2}zg_{3}}}=\cfrac{1}{1+\cfrac{k_{1}z}{1+\cfrac{k_{2}z}{1+k_{3}zg_{4}}}}=\dots
  10. f 1 f 0 = 1 1 + k 1 z 1 + k 2 z 1 + k 3 z 1 + \frac{f_{1}}{f_{0}}=\cfrac{1}{1+\cfrac{k_{1}z}{1+\cfrac{k_{2}z}{1+\cfrac{k_{3}% z}{1+{}\ddots}}}}
  11. f i f_{i}
  12. F 1 0 {}_{0}F_{1}
  13. F 1 1 {}_{1}F_{1}
  14. F 1 2 {}_{2}F_{1}
  15. f i - 1 - f i = k i z f i + 1 f_{i-1}-f_{i}=k_{i}zf_{i+1}
  16. F 1 0 ( ; a ; z ) = 1 + 1 a 1 ! z + 1 a ( a + 1 ) 2 ! z 2 + 1 a ( a + 1 ) ( a + 2 ) 3 ! z 3 + \,{}_{0}F_{1}(;a;z)=1+\frac{1}{a\,1!}z+\frac{1}{a(a+1)\,2!}z^{2}+\frac{1}{a(a+% 1)(a+2)\,3!}z^{3}+\cdots
  17. F 1 0 ( ; a - 1 ; z ) - 0 F 1 ( ; a ; z ) = z a ( a - 1 ) 0 F 1 ( ; a + 1 ; z ) \,{}_{0}F_{1}(;a-1;z)-\,_{0}F_{1}(;a;z)=\frac{z}{a(a-1)}\,_{0}F_{1}(;a+1;z)
  18. f i = F 1 0 ( ; a + i ; z ) , k i = 1 ( a + i ) ( a + i - 1 ) f_{i}={}_{0}F_{1}(;a+i;z),\,k_{i}=\tfrac{1}{(a+i)(a+i-1)}
  19. F 1 0 ( a + 1 ; z ) F 1 0 ( a ; z ) = 1 1 + 1 a ( a + 1 ) z 1 + 1 ( a + 1 ) ( a + 2 ) z 1 + 1 ( a + 2 ) ( a + 3 ) z 1 + \frac{\,{}_{0}F_{1}(a+1;z)}{\,{}_{0}F_{1}(a;z)}=\cfrac{1}{1+\cfrac{\frac{1}{a(% a+1)}z}{1+\cfrac{\frac{1}{(a+1)(a+2)}z}{1+\cfrac{\frac{1}{(a+2)(a+3)}z}{1+{}% \ddots}}}}
  20. F 1 0 ( a + 1 ; z ) a 0 F 1 ( a ; z ) = 1 a + z ( a + 1 ) + z ( a + 2 ) + z ( a + 3 ) + \frac{\,{}_{0}F_{1}(a+1;z)}{a\,_{0}F_{1}(a;z)}=\cfrac{1}{a+\cfrac{z}{(a+1)+% \cfrac{z}{(a+2)+\cfrac{z}{(a+3)+{}\ddots}}}}
  21. F 1 1 ( a ; b ; z ) = 1 + a b 1 ! z + a ( a + 1 ) b ( b + 1 ) 2 ! z 2 + a ( a + 1 ) ( a + 2 ) b ( b + 1 ) ( b + 2 ) 3 ! z 3 + {}_{1}F_{1}(a;b;z)=1+\frac{a}{b\,1!}z+\frac{a(a+1)}{b(b+1)\,2!}z^{2}+\frac{a(a% +1)(a+2)}{b(b+1)(b+2)\,3!}z^{3}+\dots
  22. F 1 1 ( a ; b - 1 ; z ) - 1 F 1 ( a + 1 ; b ; z ) = ( a - b + 1 ) z b ( b - 1 ) 1 F 1 ( a + 1 ; b + 1 ; z ) \,{}_{1}F_{1}(a;b-1;z)-\,_{1}F_{1}(a+1;b;z)=\frac{(a-b+1)z}{b(b-1)}\,_{1}F_{1}% (a+1;b+1;z)
  23. F 1 1 ( a ; b - 1 ; z ) - 1 F 1 ( a ; b ; z ) = a z b ( b - 1 ) 1 F 1 ( a + 1 ; b + 1 ; z ) \,{}_{1}F_{1}(a;b-1;z)-\,_{1}F_{1}(a;b;z)=\frac{az}{b(b-1)}\,_{1}F_{1}(a+1;b+1% ;z)
  24. f 0 ( z ) = 1 F 1 ( a ; b ; z ) f_{0}(z)=\,_{1}F_{1}(a;b;z)
  25. f 1 ( z ) = 1 F 1 ( a + 1 ; b + 1 ; z ) f_{1}(z)=\,_{1}F_{1}(a+1;b+1;z)
  26. f 2 ( z ) = 1 F 1 ( a + 1 ; b + 2 ; z ) f_{2}(z)=\,_{1}F_{1}(a+1;b+2;z)
  27. f 3 ( z ) = 1 F 1 ( a + 2 ; b + 3 ; z ) f_{3}(z)=\,_{1}F_{1}(a+2;b+3;z)
  28. f 4 ( z ) = 1 F 1 ( a + 2 ; b + 4 ; z ) f_{4}(z)=\,_{1}F_{1}(a+2;b+4;z)
  29. f i - 1 - f i = k i z f i + 1 f_{i-1}-f_{i}=k_{i}zf_{i+1}
  30. k 1 = a - b b ( b + 1 ) , k 2 = a + 1 ( b + 1 ) ( b + 2 ) , k 3 = a - b - 1 ( b + 2 ) ( b + 3 ) , k 4 = a + 2 ( b + 3 ) ( b + 4 ) k_{1}=\tfrac{a-b}{b(b+1)},k_{2}=\tfrac{a+1}{(b+1)(b+2)},k_{3}=\tfrac{a-b-1}{(b% +2)(b+3)},k_{4}=\tfrac{a+2}{(b+3)(b+4)}
  31. F 1 1 ( a + 1 ; b + 1 ; z ) F 1 1 ( a ; b ; z ) = 1 1 + a - b b ( b + 1 ) z 1 + a + 1 ( b + 1 ) ( b + 2 ) z 1 + a - b - 1 ( b + 2 ) ( b + 3 ) z 1 + a + 2 ( b + 3 ) ( b + 4 ) z 1 + \frac{{}_{1}F_{1}(a+1;b+1;z)}{{}_{1}F_{1}(a;b;z)}=\cfrac{1}{1+\cfrac{\frac{a-b% }{b(b+1)}z}{1+\cfrac{\frac{a+1}{(b+1)(b+2)}z}{1+\cfrac{\frac{a-b-1}{(b+2)(b+3)% }z}{1+\cfrac{\frac{a+2}{(b+3)(b+4)}z}{1+{}\ddots}}}}}
  32. F 1 1 ( a + 1 ; b + 1 ; z ) b F 1 1 ( a ; b ; z ) = 1 b + ( a - b ) z ( b + 1 ) + ( a + 1 ) z ( b + 2 ) + ( a - b - 1 ) z ( b + 3 ) + ( a + 2 ) z ( b + 4 ) + \frac{{}_{1}F_{1}(a+1;b+1;z)}{b{}_{1}F_{1}(a;b;z)}=\cfrac{1}{b+\cfrac{(a-b)z}{% (b+1)+\cfrac{(a+1)z}{(b+2)+\cfrac{(a-b-1)z}{(b+3)+\cfrac{(a+2)z}{(b+4)+{}% \ddots}}}}}
  33. F 1 1 ( a ; b + 1 ; z ) F 1 1 ( a ; b ; z ) = 1 1 + a b ( b + 1 ) z 1 + a - b - 1 ( b + 1 ) ( b + 2 ) z 1 + a + 1 ( b + 2 ) ( b + 3 ) z 1 + a - b - 2 ( b + 3 ) ( b + 4 ) z 1 + \frac{{}_{1}F_{1}(a;b+1;z)}{{}_{1}F_{1}(a;b;z)}=\cfrac{1}{1+\cfrac{\frac{a}{b(% b+1)}z}{1+\cfrac{\frac{a-b-1}{(b+1)(b+2)}z}{1+\cfrac{\frac{a+1}{(b+2)(b+3)}z}{% 1+\cfrac{\frac{a-b-2}{(b+3)(b+4)}z}{1+{}\ddots}}}}}
  34. F 1 1 ( a ; b + 1 ; z ) b F 1 1 ( a ; b ; z ) = 1 b + a z ( b + 1 ) + ( a - b - 1 ) z ( b + 2 ) + ( a + 1 ) z ( b + 3 ) + ( a - b - 2 ) z ( b + 4 ) + \frac{{}_{1}F_{1}(a;b+1;z)}{b{}_{1}F_{1}(a;b;z)}=\cfrac{1}{b+\cfrac{az}{(b+1)+% \cfrac{(a-b-1)z}{(b+2)+\cfrac{(a+1)z}{(b+3)+\cfrac{(a-b-2)z}{(b+4)+{}\ddots}}}}}
  35. F 1 1 ( 0 ; b ; z ) = 1 {}_{1}F_{1}(0;b;z)=1
  36. F 1 1 ( 1 ; b ; z ) = 1 1 + - z b + z ( b + 1 ) + - b z ( b + 2 ) + 2 z ( b + 3 ) + - ( b + 1 ) z ( b + 4 ) + {}_{1}F_{1}(1;b;z)=\cfrac{1}{1+\cfrac{-z}{b+\cfrac{z}{(b+1)+\cfrac{-bz}{(b+2)+% \cfrac{2z}{(b+3)+\cfrac{-(b+1)z}{(b+4)+{}\ddots}}}}}}
  37. F 1 2 ( a , b ; c ; z ) = 1 + a b c 1 ! z + a ( a + 1 ) b ( b + 1 ) c ( c + 1 ) 2 ! z 2 + a ( a + 1 ) ( a + 2 ) b ( b + 1 ) ( b + 2 ) c ( c + 1 ) ( c + 2 ) 3 ! z 3 + {}_{2}F_{1}(a,b;c;z)=1+\frac{ab}{c\,1!}z+\frac{a(a+1)b(b+1)}{c(c+1)\,2!}z^{2}+% \frac{a(a+1)(a+2)b(b+1)(b+2)}{c(c+1)(c+2)\,3!}z^{3}+\dots\,
  38. F 1 2 ( a , b ; c - 1 ; z ) - 2 F 1 ( a + 1 , b ; c ; z ) = ( a - c + 1 ) b z c ( c - 1 ) 2 F 1 ( a + 1 , b + 1 ; c + 1 ; z ) \,{}_{2}F_{1}(a,b;c-1;z)-\,_{2}F_{1}(a+1,b;c;z)=\frac{(a-c+1)bz}{c(c-1)}\,_{2}% F_{1}(a+1,b+1;c+1;z)
  39. F 1 2 ( a , b ; c - 1 ; z ) - 2 F 1 ( a , b + 1 ; c ; z ) = ( b - c + 1 ) a z c ( c - 1 ) 2 F 1 ( a + 1 , b + 1 ; c + 1 ; z ) \,{}_{2}F_{1}(a,b;c-1;z)-\,_{2}F_{1}(a,b+1;c;z)=\frac{(b-c+1)az}{c(c-1)}\,_{2}% F_{1}(a+1,b+1;c+1;z)
  40. f 0 ( z ) = 2 F 1 ( a , b ; c ; z ) f_{0}(z)=\,_{2}F_{1}(a,b;c;z)
  41. f 1 ( z ) = 2 F 1 ( a + 1 , b ; c + 1 ; z ) f_{1}(z)=\,_{2}F_{1}(a+1,b;c+1;z)
  42. f 2 ( z ) = 2 F 1 ( a + 1 , b + 1 ; c + 2 ; z ) f_{2}(z)=\,_{2}F_{1}(a+1,b+1;c+2;z)
  43. f 3 ( z ) = 2 F 1 ( a + 2 , b + 1 ; c + 3 ; z ) f_{3}(z)=\,_{2}F_{1}(a+2,b+1;c+3;z)
  44. f 4 ( z ) = 2 F 1 ( a + 2 , b + 2 ; c + 4 ; z ) f_{4}(z)=\,_{2}F_{1}(a+2,b+2;c+4;z)
  45. f i - 1 - f i = k i z f i + 1 f_{i-1}-f_{i}=k_{i}zf_{i+1}
  46. k 1 = ( a - c ) b c ( c + 1 ) , k 2 = ( b - c - 1 ) ( a + 1 ) ( c + 1 ) ( c + 2 ) , k 3 = ( a - c - 1 ) ( b + 1 ) ( c + 2 ) ( c + 3 ) , k 4 = ( b - c - 2 ) ( a + 2 ) ( c + 3 ) ( c + 4 ) k_{1}=\tfrac{(a-c)b}{c(c+1)},k_{2}=\tfrac{(b-c-1)(a+1)}{(c+1)(c+2)},k_{3}=% \tfrac{(a-c-1)(b+1)}{(c+2)(c+3)},k_{4}=\tfrac{(b-c-2)(a+2)}{(c+3)(c+4)}
  47. F 1 2 ( a + 1 , b ; c + 1 ; z ) F 1 2 ( a , b ; c ; z ) = 1 1 + ( a - c ) b c ( c + 1 ) z 1 + ( b - c - 1 ) ( a + 1 ) ( c + 1 ) ( c + 2 ) z 1 + ( a - c - 1 ) ( b + 1 ) ( c + 2 ) ( c + 3 ) z 1 + ( b - c - 2 ) ( a + 2 ) ( c + 3 ) ( c + 4 ) z 1 + \frac{{}_{2}F_{1}(a+1,b;c+1;z)}{{}_{2}F_{1}(a,b;c;z)}=\cfrac{1}{1+\cfrac{\frac% {(a-c)b}{c(c+1)}z}{1+\cfrac{\frac{(b-c-1)(a+1)}{(c+1)(c+2)}z}{1+\cfrac{\frac{(% a-c-1)(b+1)}{(c+2)(c+3)}z}{1+\cfrac{\frac{(b-c-2)(a+2)}{(c+3)(c+4)}z}{1+{}% \ddots}}}}}
  48. F 1 2 ( a + 1 , b ; c + 1 ; z ) c F 1 2 ( a , b ; c ; z ) = 1 c + ( a - c ) b z ( c + 1 ) + ( b - c - 1 ) ( a + 1 ) z ( c + 2 ) + ( a - c - 1 ) ( b + 1 ) z ( c + 3 ) + ( b - c - 2 ) ( a + 2 ) z ( c + 4 ) + \frac{{}_{2}F_{1}(a+1,b;c+1;z)}{c{}_{2}F_{1}(a,b;c;z)}=\cfrac{1}{c+\cfrac{(a-c% )bz}{(c+1)+\cfrac{(b-c-1)(a+1)z}{(c+2)+\cfrac{(a-c-1)(b+1)z}{(c+3)+\cfrac{(b-c% -2)(a+2)z}{(c+4)+{}\ddots}}}}}
  49. F 1 2 ( 0 , b ; c ; z ) = 1 {}_{2}F_{1}(0,b;c;z)=1
  50. F 1 2 ( 1 , b ; c ; z ) = 1 1 + - b z c + ( b - c ) z ( c + 1 ) + - c ( b + 1 ) z ( c + 2 ) + 2 ( b - c - 1 ) z ( c + 3 ) + - ( c + 1 ) ( b + 2 ) z ( c + 4 ) + {}_{2}F_{1}(1,b;c;z)=\cfrac{1}{1+\cfrac{-bz}{c+\cfrac{(b-c)z}{(c+1)+\cfrac{-c(% b+1)z}{(c+2)+\cfrac{2(b-c-1)z}{(c+3)+\cfrac{-(c+1)(b+2)z}{(c+4)+{}\ddots}}}}}}
  51. F 1 0 {}_{0}F_{1}
  52. F 1 1 {}_{1}F_{1}
  53. F 1 2 {}_{2}F_{1}
  54. + 1 +1
  55. + 1 +1
  56. + 1 +1
  57. cosh ( z ) = 0 F 1 ( 1 2 ; z 2 4 ) , \cosh(z)=\,_{0}F_{1}({\tfrac{1}{2}};{\tfrac{z^{2}}{4}}),
  58. sinh ( z ) = z 0 F 1 ( 3 2 ; z 2 4 ) , \sinh(z)=z\,_{0}F_{1}({\tfrac{3}{2}};{\tfrac{z^{2}}{4}}),
  59. tanh ( z ) = z 0 F 1 ( 3 2 ; z 2 4 ) F 1 0 ( 1 2 ; z 2 4 ) = z / 2 1 2 + z 2 4 3 2 + z 2 4 5 2 + z 2 4 7 2 + = z 1 + z 2 3 + z 2 5 + z 2 7 + . \tanh(z)=\frac{z\,_{0}F_{1}({\tfrac{3}{2}};{\tfrac{z^{2}}{4}})}{\,{}_{0}F_{1}(% {\tfrac{1}{2}};{\tfrac{z^{2}}{4}})}=\cfrac{z/2}{\tfrac{1}{2}+\cfrac{\tfrac{z^{% 2}}{4}}{\tfrac{3}{2}+\cfrac{\tfrac{z^{2}}{4}}{\tfrac{5}{2}+\cfrac{\tfrac{z^{2}% }{4}}{\tfrac{7}{2}+{}\ddots}}}}=\cfrac{z}{1+\cfrac{z^{2}}{3+\cfrac{z^{2}}{5+% \cfrac{z^{2}}{7+{}\ddots}}}}.
  60. tan ( z ) = z 1 - z 2 3 - z 2 5 - z 2 7 - . \tan(z)=\cfrac{z}{1-\cfrac{z^{2}}{3-\cfrac{z^{2}}{5-\cfrac{z^{2}}{7-{}\ddots}}% }}.
  61. J ν J_{\nu}
  62. J ν ( z ) = ( 1 2 z ) ν Γ ( ν + 1 ) 0 F 1 ( ; ν + 1 ; - z 2 4 ) , J_{\nu}(z)=\frac{(\tfrac{1}{2}z)^{\nu}}{\Gamma(\nu+1)}\,_{0}F_{1}(;\nu+1;-% \frac{z^{2}}{4}),
  63. J ν ( z ) J ν - 1 ( z ) = z 2 ν - z 2 2 ( ν + 1 ) - z 2 2 ( ν + 2 ) - z 2 2 ( ν + 3 ) - . \frac{J_{\nu}(z)}{J_{\nu-1}(z)}=\cfrac{z}{2\nu-\cfrac{z^{2}}{2(\nu+1)-\cfrac{z% ^{2}}{2(\nu+2)-\cfrac{z^{2}}{2(\nu+3)-{}\ddots}}}}.
  64. e z = F 1 1 ( 1 ; 1 ; z ) e^{z}={}_{1}F_{1}(1;1;z)
  65. 1 / e z = e - z 1/e^{z}=e^{-z}
  66. e z = 1 1 + - z 1 + z 2 + - z 3 + 2 z 4 + - 2 z 5 + e^{z}=\cfrac{1}{1+\cfrac{-z}{1+\cfrac{z}{2+\cfrac{-z}{3+\cfrac{2z}{4+\cfrac{-2% z}{5+{}\ddots}}}}}}
  67. e z = 1 + z 1 + - z 2 + z 3 + - 2 z 4 + 2 z 5 + e^{z}=1+\cfrac{z}{1+\cfrac{-z}{2+\cfrac{z}{3+\cfrac{-2z}{4+\cfrac{2z}{5+{}% \ddots}}}}}
  68. e = 2 + 1 1 + 1 2 + 1 1 + 1 1 + 1 4 + e=2+\cfrac{1}{1+\cfrac{1}{2+\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{4+{}\ddots}}}}}
  69. erf ( z ) = 2 π 0 z e - t 2 d t , \operatorname{erf}(z)=\frac{2}{\sqrt{\pi}}\int_{0}^{z}e^{-t^{2}}dt,
  70. erf ( z ) = 2 z π e 1 - z 2 F 1 ( 1 ; 3 2 ; z 2 ) . \operatorname{erf}(z)=\frac{2z}{\sqrt{\pi}}e^{-z^{2}}\,_{1}F_{1}(1;{% \scriptstyle\frac{3}{2}};z^{2}).
  71. π 2 e z 2 erf ( z ) = z 1 - z 2 3 2 + z 2 5 2 - 3 2 z 2 7 2 + 2 z 2 9 2 - 5 2 z 2 11 2 + 3 z 2 13 2 - 7 2 z 2 15 2 + - . \frac{\sqrt{\pi}}{2}e^{z^{2}}\operatorname{erf}(z)=\cfrac{z}{1-\cfrac{z^{2}}{% \frac{3}{2}+\cfrac{z^{2}}{\frac{5}{2}-\cfrac{\frac{3}{2}z^{2}}{\frac{7}{2}+% \cfrac{2z^{2}}{\frac{9}{2}-\cfrac{\frac{5}{2}z^{2}}{\frac{11}{2}+\cfrac{3z^{2}% }{\frac{13}{2}-\cfrac{\frac{7}{2}z^{2}}{\frac{15}{2}+-\ddots}}}}}}}}.
  72. ( 1 - z ) - b = F 0 1 ( b ; ; z ) = 2 F 1 ( 1 , b ; 1 ; z ) (1-z)^{-b}={}_{1}F_{0}(b;;z)=\,_{2}F_{1}(1,b;1;z)
  73. ( 1 - z ) - b = 1 1 + - b z 1 + ( b - 1 ) z 2 + - ( b + 1 ) z 3 + 2 ( b - 2 ) z 4 + (1-z)^{-b}=\cfrac{1}{1+\cfrac{-bz}{1+\cfrac{(b-1)z}{2+\cfrac{-(b+1)z}{3+\cfrac% {2(b-2)z}{4+{}\ddots}}}}}
  74. arctan z = z F ( 1 2 , 1 ; 3 2 ; - z 2 ) . \arctan z=zF({\scriptstyle\frac{1}{2}},1;{\scriptstyle\frac{3}{2}};-z^{2}).
  75. arctan z = z 1 + ( 1 z ) 2 3 + ( 2 z ) 2 5 + ( 3 z ) 2 7 + ( 4 z ) 2 9 + , \arctan z=\cfrac{z}{1+\cfrac{(1z)^{2}}{3+\cfrac{(2z)^{2}}{5+\cfrac{(3z)^{2}}{7% +\cfrac{(4z)^{2}}{9+\ddots}}}}},
  76. π 4 = 1 1 + 1 2 2 + 3 2 2 + 5 2 2 + = 1 - 1 3 + 1 5 - 1 7 + - \frac{\pi}{4}=\cfrac{1}{1+\cfrac{1^{2}}{2+\cfrac{3^{2}}{2+\cfrac{5^{2}}{2+% \ddots}}}}=1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+-\dots

Gaussian_free_field.html

  1. φ \varphi
  2. H ( φ ) = 1 2 ( x , y ) P ( x , y ) ( φ ( x ) - φ ( y ) ) 2 . H(\varphi)=\frac{1}{2}\sum_{(x,y)}P(x,y)\big(\varphi(x)-\varphi(y)\big)^{2}.
  3. exp ( - H ( φ ) ) \exp(-H(\varphi))
  4. \R V U \R^{V\setminus U}
  5. 𝔼 [ φ ( x ) ] \mathbb{E}[\varphi(x)]
  6. Cov [ φ ( x ) , φ ( y ) ] \mathrm{Cov}[\varphi(x),\varphi(y)]
  7. f , g := Ω ( D f ( x ) , D g ( x ) ) d x \langle f,g\rangle:=\int_{\Omega}(Df(x),Dg(x))\,dx
  8. Ω \partial\Omega
  9. D f ( x ) Df\,(x)
  10. x Ω x\in\Omega
  11. H 1 ( Ω ) H^{1}(\Omega)
  12. φ \varphi
  13. Ω \Omega
  14. H 1 ( Ω ) H^{1}(\Omega)
  15. f H 1 ( Ω ) f\in H^{1}(\Omega)
  16. φ , f \langle\varphi,f\rangle
  17. Cov [ φ , f , φ , g ] = f , g \mathrm{Cov}[\langle\varphi,f\rangle,\langle\varphi,g\rangle]=\langle f,g\rangle
  18. f , g H 1 ( Ω ) f,g\in H^{1}(\Omega)
  19. ψ 1 , ψ 2 , \psi_{1},\psi_{2},\dots
  20. H 1 ( Ω ) H^{1}(\Omega)
  21. φ := k = 1 ξ k ψ k , \varphi:=\sum_{k=1}^{\infty}\xi_{k}\psi_{k},
  22. ξ k \xi_{k}
  23. H 1 ( Ω ) H^{1}(\Omega)
  24. f H 1 ( Ω ) f\in H^{1}(\Omega)
  25. f = k = 1 c k ψ k , with k = 1 c k 2 < , f=\sum_{k=1}^{\infty}c_{k}\psi_{k},\,\text{ with }\sum_{k=1}^{\infty}c_{k}^{2}% <\infty,
  26. φ , f := k = 1 ξ k c k \langle\varphi,f\rangle:=\sum_{k=1}^{\infty}\xi_{k}c_{k}
  27. φ \varphi
  28. H 1 ( Ω ) H^{1}(\Omega)
  29. Ω \Omega
  30. n = 1 n=1
  31. H 1 [ 0 , 1 ] H^{1}[0,1]
  32. ψ k ( t ) := 0 t φ k ( s ) d s , \psi_{k}(t):=\int_{0}^{t}\varphi_{k}(s)\,ds\,,
  33. ( φ k ) (\varphi_{k})
  34. L 2 [ 0 , 1 ] , L^{2}[0,1]\,,
  35. φ ( t ) := k = 1 ξ k ψ k ( t ) \varphi(t):=\sum_{k=1}^{\infty}\xi_{k}\psi_{k}(t)
  36. φ k \varphi_{k}
  37. ( φ k ) (\varphi_{k})
  38. n 2 n\geq 2

Gaussian_isoperimetric_inequality.html

  1. A \scriptstyle A
  2. 𝐑 n \scriptstyle\mathbf{R}^{n}
  3. A ε = { x 𝐑 n | dist ( x , A ) ε } A_{\varepsilon}=\left\{x\in\mathbf{R}^{n}\,|\,\,\text{dist}(x,A)\leq% \varepsilon\right\}
  4. lim inf ε + 0 ε - 1 { γ n ( A ε ) - γ n ( A ) } φ ( Φ - 1 ( γ n ( A ) ) ) , \liminf_{\varepsilon\to+0}\varepsilon^{-1}\left\{\gamma^{n}(A_{\varepsilon})-% \gamma^{n}(A)\right\}\geq\varphi(\Phi^{-1}(\gamma^{n}(A))),
  5. φ ( t ) = exp ( - t 2 / 2 ) 2 π and Φ ( t ) = - t φ ( s ) d s . \varphi(t)=\frac{\exp(-t^{2}/2)}{\sqrt{2\pi}}\quad{\rm and}\quad\Phi(t)=\int_{% -\infty}^{t}\varphi(s)\,ds.

Gaussian_network_model.html

  1. V G N M = γ 2 [ i , j N ( Δ R j - Δ R i ) 2 ] = γ 2 [ i , j N Δ R i Γ i j Δ R j ] V_{GNM}=\frac{\gamma}{2}\left[\sum_{i,j}^{N}(\Delta R_{j}-\Delta R_{i})^{2}% \right]=\frac{\gamma}{2}\left[\sum_{i,j}^{N}\Delta R_{i}\Gamma_{ij}\Delta R_{j% }\right]
  2. Γ i j = { - 1 , if i j and R i j r c 0 , if i j and R i j > r c - j , j i N Γ i j , if i = j \Gamma_{ij}=\left\{\begin{matrix}-1,&\mbox{if }~{}i\neq j&\mbox{and }~{}R_{ij}% \leq r_{c}\\ 0,&\mbox{if }~{}i\neq j&\mbox{and }~{}R_{ij}>r_{c}\\ -\sum_{j,j\neq i}^{N}\Gamma_{ij},&\mbox{if }~{}i=j\end{matrix}\right.
  3. V G N M = γ 2 [ Δ X T Γ Δ X + Δ Y T Γ Δ Y + Δ Z T Γ Δ Z ] V_{GNM}=\frac{\gamma}{2}[\Delta X^{T}\Gamma\Delta X+\Delta Y^{T}\Gamma\Delta Y% +\Delta Z^{T}\Gamma\Delta Z]
  4. P ( Δ R ) = P ( Δ X , Δ Y , Δ Z ) = p ( Δ X ) p ( Δ Y ) p ( Δ Z ) P(\Delta R)=P(\Delta X,\Delta Y,\Delta Z)=p(\Delta X)p(\Delta Y)p(\Delta Z)
  5. p ( Δ X ) exp { - γ 2 k B T Δ X T Γ Δ X } = exp { - 1 2 ( Δ X T ( k B T γ Γ - 1 ) - 1 Δ X ) } p(\Delta X)\propto\exp\left\{-\frac{\gamma}{2k_{B}T}\Delta X^{T}\Gamma\Delta X% \right\}=\exp\left\{-\frac{1}{2}\left(\Delta X^{T}\left(\frac{k_{B}T}{\gamma}% \Gamma^{-1}\right)^{-1}\Delta X\right)\right\}
  6. W ( x , μ , Σ ) = 1 ( 2 π ) N | Σ | exp { - 1 2 ( x - μ ) T Σ - 1 ( x - μ ) } W(x,\mu,\Sigma)=\frac{1}{\sqrt{(2\pi)^{N}|\Sigma|}}\exp\left\{-\frac{1}{2}(x-% \mu)^{T}\Sigma^{-1}(x-\mu)\right\}
  7. ( 2 π ) N | Σ | \sqrt{(2\pi)^{N}|\Sigma|}
  8. p ( Δ X ) = 1 ( 2 π ) N k B T γ | Γ - 1 | exp { - 1 2 ( Δ X T ( k B T γ Γ - 1 ) - 1 Δ X ) } p(\Delta X)=\frac{1}{\sqrt{(2\pi)^{N}\frac{k_{B}T}{\gamma}|\Gamma^{-1}|}}\exp% \left\{-\frac{1}{2}\left(\Delta X^{T}\left(\frac{k_{B}T}{\gamma}\Gamma^{-1}% \right)^{-1}\Delta X\right)\right\}
  9. Z X = 0 exp { - 1 2 ( Δ X T ( k B T γ Γ - 1 ) - 1 Δ X ) } d Δ X Z_{X}=\int_{0}^{\infty}\exp\left\{-\frac{1}{2}\left(\Delta X^{T}\left(\frac{k_% {B}T}{\gamma}\Gamma^{-1}\right)^{-1}\Delta X\right)\right\}d\Delta X
  10. k B T γ Γ - 1 \frac{k_{B}T}{\gamma}\Gamma^{-1}
  11. P ( Δ R ) = p ( Δ X ) p ( Δ Y ) p ( Δ Z ) = 1 ( 2 π ) 3 N | k B T γ Γ - 1 | 3 exp { - 3 2 ( Δ X T ( k B T γ Γ - 1 ) - 1 Δ X ) } P(\Delta R)=p(\Delta X)p(\Delta Y)p(\Delta Z)=\frac{1}{\sqrt{(2\pi)^{3N}|\frac% {k_{B}T}{\gamma}\Gamma^{-1}|^{3}}}\exp\left\{-\frac{3}{2}\left(\Delta X^{T}% \left(\frac{k_{B}T}{\gamma}\Gamma^{-1}\right)^{-1}\Delta X\right)\right\}
  12. Z G N M = Z X Z Y Z Z = 1 ( 2 π ) 3 N | k B T γ Γ - 1 | 3 Z_{GNM}=Z_{X}Z_{Y}Z_{Z}=\frac{1}{\sqrt{(2\pi)^{3N}|\frac{k_{B}T}{\gamma}\Gamma% ^{-1}|^{3}}}
  13. < Δ X Δ X T Δ X Δ X T p ( Δ X ) d Δ X = k B T γ Γ - 1 <\Delta X\cdot\Delta X^{T}>=\int\Delta X\cdot\Delta X^{T}p(\Delta X)d\Delta X=% \frac{k_{B}T}{\gamma}\Gamma^{-1}
  14. < Δ X Δ X T < Δ Y Δ Y T < Δ Z Δ Z T 1 3 < Δ R Δ R T Align g t ; <\Delta X\cdot\Delta X^{T}>=<\Delta Y\cdot\Delta Y^{T}>=<\Delta Z\cdot\Delta Z% ^{T}>=\frac{1}{3}<\Delta R\cdot\Delta R^{T}&gt;
  15. < Δ R i 2 3 k B T γ ( Γ - 1 ) i i <\Delta R_{i}^{2}>=\frac{3k_{B}T}{\gamma}(\Gamma^{-1})_{ii}
  16. < Δ R i Δ R j 3 k B T γ ( Γ - 1 ) i j <\Delta R_{i}\cdot\Delta R_{j}>=\frac{3k_{B}T}{\gamma}(\Gamma^{-1})_{ij}
  17. < Δ R i Δ R j 3 k B T γ [ U Λ - 1 U T ] i j = 3 k B T γ k = 1 N - 1 λ k - 1 [ u k u k T ] i j <\Delta R_{i}\cdot\Delta R_{j}>=\frac{3k_{B}T}{\gamma}[U\Lambda^{-1}U^{T}]_{ij% }=\frac{3k_{B}T}{\gamma}\sum_{k=1}^{N-1}\lambda_{k}^{-1}[u_{k}u_{k}^{T}]_{ij}
  18. [ Δ R i Δ R j ] k = 3 k B T γ λ k - 1 [ u k ] i [ u k ] j [\Delta R_{i}\cdot\Delta R_{j}]_{k}=\frac{3k_{B}T}{\gamma}\lambda_{k}^{-1}[u_{% k}]_{i}[u_{k}]_{j}
  19. B i = 8 π 2 3 < Δ R i Δ R i 8 π 2 k B T γ ( Γ - 1 ) i i B_{i}=\frac{8\pi^{2}}{3}<\Delta R_{i}\cdot\Delta R_{i}>=\frac{8\pi^{2}k_{B}T}{% \gamma}(\Gamma^{-1})_{ii}

Gauss–Codazzi_equations.html

  1. 0 T x M T x P | M T x M 0. 0\rightarrow T_{x}M\rightarrow T_{x}P|_{M}\rightarrow T_{x}^{\perp}M% \rightarrow 0.
  2. T P | M = T M T M . TP|_{M}=TM\oplus T^{\perp}M.
  3. X Y = ( X Y ) + ( X Y ) . \nabla^{\prime}_{X}Y=\top(\nabla^{\prime}_{X}Y)+\bot(\nabla^{\prime}_{X}Y).
  4. X Y = ( X Y ) , α ( X , Y ) = ( X Y ) . \nabla_{X}Y=\top(\nabla^{\prime}_{X}Y),\quad\alpha(X,Y)=\bot(\nabla^{\prime}_{% X}Y).
  5. R ( X , Y ) Z , W = R ( X , Y ) Z , W + α ( X , Z ) , α ( Y , W ) - α ( Y , Z ) , α ( X , W ) \langle R^{\prime}(X,Y)Z,W\rangle=\langle R(X,Y)Z,W\rangle+\langle\alpha(X,Z),% \alpha(Y,W)\rangle-\langle\alpha(Y,Z),\alpha(X,W)\rangle
  6. X ξ = ( X ξ ) + ( X ξ ) = - A ξ ( X ) + D X ( ξ ) . \nabla^{\prime}_{X}\xi=\top(\nabla^{\prime}_{X}\xi)+\bot(\nabla^{\prime}_{X}% \xi)=-A_{\xi}(X)+D_{X}(\xi).
  7. A ξ X , Y = α ( X , Y ) , ξ \langle A_{\xi}X,Y\rangle=\langle\alpha(X,Y),\xi\rangle
  8. ( ~ X α ) ( Y , Z ) = D X ( α ( Y , Z ) ) - α ( X Y , Z ) - α ( Y , X Z ) . (\tilde{\nabla}_{X}\alpha)(Y,Z)=D_{X}\left(\alpha(Y,Z)\right)-\alpha(\nabla_{X% }Y,Z)-\alpha(Y,\nabla_{X}Z).
  9. ( R ( X , Y ) Z ) = ( ~ X α ) ( Y , Z ) - ( ~ Y α ) ( X , Z ) . \bot\left(R^{\prime}(X,Y)Z\right)=(\tilde{\nabla}_{X}\alpha)(Y,Z)-(\tilde{% \nabla}_{Y}\alpha)(X,Z).
  10. L v - M u = L Γ 1 + 12 M ( Γ 2 - 12 Γ 1 ) 11 - N Γ 2 11 L_{v}-M_{u}=L\Gamma^{1}{}_{12}+M(\Gamma^{2}{}_{12}-\Gamma^{1}{}_{11})-N\Gamma^% {2}{}_{11}
  11. M v - N u = L Γ 1 + 22 M ( Γ 2 - 22 Γ 1 ) 12 - N Γ 2 12 M_{v}-N_{u}=L\Gamma^{1}{}_{22}+M(\Gamma^{2}{}_{22}-\Gamma^{1}{}_{12})-N\Gamma^% {2}{}_{12}
  12. 𝐫 ( u , v ) = ( x ( u , v ) , y ( u , v ) , z ( u , v ) ) \mathbf{r}(u,v)=(x(u,v),y(u,v),z(u,v))
  13. r u u = Γ 1 r u 11 + Γ 2 r v 11 + L n {r}_{uu}=\Gamma^{1}{}_{11}{r}_{u}+\Gamma^{2}{}_{11}{r}_{v}+L{n}
  14. r u v = Γ 1 r u 12 + Γ 2 r v 12 + M n {r}_{uv}=\Gamma^{1}{}_{12}{r}_{u}+\Gamma^{2}{}_{12}{r}_{v}+M{n}
  15. r v v = Γ 1 r u 22 + Γ 2 r v 22 + N n {r}_{vv}=\Gamma^{1}{}_{22}{r}_{u}+\Gamma^{2}{}_{22}{r}_{v}+N{n}
  16. ( r u u ) v = ( r u v ) u \left({r}_{uu}\right)_{v}=\left({r}_{uv}\right)_{u}
  17. ( Γ 1 ) 11 v r u + Γ 1 r u v 11 + ( Γ 2 ) 11 v r v + Γ 2 r v v 11 + L v n + L n v \left(\Gamma^{1}{}_{11}\right)_{v}{r}_{u}+\Gamma^{1}{}_{11}{r}_{uv}+\left(% \Gamma^{2}{}_{11}\right)_{v}{r}_{v}+\Gamma^{2}{}_{11}{r}_{vv}+L_{v}{n}+L{n}_{v}
  18. = ( Γ 1 ) 12 u r u + Γ 1 r u u 12 + ( Γ 12 2 ) u r v + Γ 2 r u v 12 + M u n + M n u =\left(\Gamma^{1}{}_{12}\right)_{u}{r}_{u}+\Gamma^{1}{}_{12}{r}_{uu}+\left(% \Gamma_{12}^{2}\right)_{u}{r}_{v}+\Gamma^{2}{}_{12}{r}_{uv}+M_{u}{n}+M{n}_{u}
  19. M Γ 1 + 11 N Γ 2 + 11 L v = L Γ 1 + 12 M Γ 2 + 12 M u M\Gamma^{1}{}_{11}+N\Gamma^{2}{}_{11}+L_{v}=L\Gamma^{1}{}_{12}+M\Gamma^{2}{}_{% 12}+M_{u}
  20. e 1 , e 2 , , e k e_{1},e_{2},\ldots,e_{k}
  21. α ( X , Y ) = j = 1 k α j ( X , Y ) e j \alpha(X,Y)=\sum_{j=1}^{k}\alpha_{j}(X,Y)e_{j}
  22. E 1 , E 2 , , E m E_{1},E_{2},\ldots,E_{m}
  23. H j = i = 1 m α j ( E i , E i ) H_{j}=\sum_{i=1}^{m}\alpha_{j}(E_{i},E_{i})
  24. k = 1 k=1
  25. H j H_{j}
  26. 1 / m 1/m
  27. R ( X , Y ) Z , W = R ( X , Y ) Z , W + j = 1 k α j ( X , Z ) α j ( Y , W ) - α j ( Y , Z ) α j ( X , W ) \langle R^{\prime}(X,Y)Z,W\rangle=\langle R(X,Y)Z,W\rangle+\sum_{j=1}^{k}% \alpha_{j}(X,Z)\alpha_{j}(Y,W)-\alpha_{j}(Y,Z)\alpha_{j}(X,W)
  28. Y , Z Y,Z
  29. R i c ( X , W ) = R i c ( X , W ) + j = 1 k R ( X , e j ) e j , W + ( i = 1 m α j ( X , E i ) α j ( E i , W ) ) - H j α j ( X , W ) Ric^{\prime}(X,W)=Ric(X,W)+\sum_{j=1}^{k}\langle R^{\prime}(X,e_{j})e_{j},W% \rangle+\left(\sum_{i=1}^{m}\alpha_{j}(X,E_{i})\alpha_{j}(E_{i},W)\right)-H_{j% }\alpha_{j}(X,W)
  30. X , W X,W
  31. R i c ( X , W ) = R i c ( X , W ) + R ( X , n ) n , W + ( i = 1 m h ( X , E i ) h ( E i , W ) ) - H h ( X , W ) Ric^{\prime}(X,W)=Ric(X,W)+\langle R^{\prime}(X,n)n,W\rangle+\left(\sum_{i=1}^% {m}h(X,E_{i})h(E_{i},W)\right)-Hh(X,W)
  32. n = e 1 n=e_{1}
  33. h = α 1 h=\alpha_{1}
  34. H = H 1 H=H_{1}
  35. R = R + 2 R i c ( n , n ) + h 2 - H 2 R^{\prime}=R+2Ric^{\prime}(n,n)+\|h\|^{2}-H^{2}
  36. R R^{\prime}
  37. R R
  38. h 2 = i , j = 1 m h ( E i , E j ) 2 \|h\|^{2}=\sum_{i,j=1}^{m}h(E_{i},E_{j})^{2}
  39. k > 1 k>1
  40. x 1 2 + x 2 2 + + x m + k + 1 2 = 1 x_{1}^{2}+x_{2}^{2}+\cdots+x_{m+k+1}^{2}=1
  41. x j + λ x j = 0 \triangle x_{j}+\lambda x_{j}=0
  42. j j
  43. m + k + 1 m+k+1
  44. = i = 1 m E i E i \triangle=\sum_{i=1}^{m}\nabla_{E_{i}}\nabla_{E_{i}}
  45. λ > 0 \lambda>0

Gauss–Lucas_theorem.html

  1. x 1 < x 2 < < x n x_{1}<x_{2}<\cdots<x_{n}\,
  2. [ x 1 , x n ] [x_{1},x_{n}]\,
  3. p n x n + p n - 1 x n - 1 + p 0 p_{n}x^{n}+p_{n-1}x^{n-1}+\cdots p_{0}
  4. - p n - 1 n p n -\frac{p_{n-1}}{n\cdot p_{n}}
  5. P ( z ) = α i = 1 n ( z - a i ) P(z)=\alpha\prod_{i=1}^{n}(z-a_{i})
  6. a 1 , a 2 , , a n a_{1},a_{2},\ldots,a_{n}
  7. α \alpha
  8. P ( z ) 0 P(z)\neq 0
  9. P ( z ) P ( z ) = i = 1 n 1 z - a i . \frac{P^{\prime}(z)}{P(z)}=\sum_{i=1}^{n}\frac{1}{z-a_{i}}.
  10. P P^{\prime}
  11. P ( z ) 0 P(z)\neq 0
  12. i = 1 n 1 z - a i = 0 \sum_{i=1}^{n}\frac{1}{z-a_{i}}=0
  13. i = 1 n z ¯ - a i ¯ | z - a i | 2 = 0. \ \sum_{i=1}^{n}\frac{\overline{z}-\overline{a_{i}}}{|z-a_{i}|^{2}}=0.
  14. ( i = 1 n 1 | z - a i | 2 ) z ¯ = ( i = 1 n 1 | z - a i | 2 a i ¯ ) . \left(\sum_{i=1}^{n}\frac{1}{|z-a_{i}|^{2}}\right)\overline{z}=\left(\sum_{i=1% }^{n}\frac{1}{|z-a_{i}|^{2}}\overline{a_{i}}\right).
  15. z z
  16. a i a_{i}
  17. P ( z ) = P ( z ) = 0 P(z)=P^{\prime}(z)=0
  18. z = 1 z + 0 a i z=1\cdot z+0\cdot a_{i}
  19. P P

General_set_theory.html

  1. x y [ z [ z x z y ] x = y ] . \forall x\forall y[\forall z[z\in x\leftrightarrow z\in y]\rightarrow x=y].
  2. ϕ \phi\!
  3. ϕ \phi\!
  4. ϕ ( x ) \phi(x)\!
  5. z y x [ x y ( x z ϕ ( x ) ) ] . \forall z\exists y\forall x[x\in y\leftrightarrow(x\in z\land\phi(x))].
  6. x y \exist w z [ z w ( z x z = y ) ] . \forall x\forall y\exist w\forall z[z\in w\leftrightarrow(z\in xz=y)].
  7. S ( x ) = x { x } S(x)=x\cup\{x\}
  8. , S ( ) , S ( S ( ) ) , , \varnothing,\,S(\varnothing),\,S(S(\varnothing)),\,\ldots,
  9. 1 \aleph_{1}

Generalised_circle.html

  1. Γ ( γ , r ) = { z : the distance between z and γ is r } \Gamma(\gamma,r)=\{z:\mathrm{\ the\ distance\ between\ }z\mathrm{\ and\ }% \gamma\mathrm{\ is\ }r\}
  2. | z - γ | = r {\left|z-\gamma\right|}=r
  3. | z - γ | 2 = r 2 {\left|z-\gamma\right|}^{2}=r^{2}
  4. ( z - γ ) ( z - γ ) ¯ = r 2 (z-\gamma)\overline{(z-\gamma)}=r^{2}
  5. z z ¯ - z γ ¯ - z ¯ γ + γ γ ¯ = r 2 z\bar{z}-z\bar{\gamma}-\bar{z}\gamma+\gamma\bar{\gamma}=r^{2}
  6. z z ¯ - z γ ¯ - z ¯ γ + γ γ ¯ - r 2 = 0. z\bar{z}-z\bar{\gamma}-\bar{z}\gamma+\gamma\bar{\gamma}-r^{2}=0.
  7. A z z ¯ + B z + C z ¯ + D = 0 Az\bar{z}+Bz+C\bar{z}+D=0
  8. A z z ¯ + B z + C z ¯ + D = 0 Az\bar{z}+Bz+C\bar{z}+D=0
  9. = ( A B C D ) = . \mathfrak{C}=\begin{pmatrix}A&B\\ C&D\end{pmatrix}=\mathfrak{C}^{\dagger}.
  10. \mathfrak{C}
  11. \mathfrak{H}
  12. ( - 1 ) T ( - 1 ) * . \mathfrak{C}\mapsto({\mathfrak{H}}^{-1})^{T}{\mathfrak{C}}({\mathfrak{H}}^{-1}% )^{*}.

Generalised_Hough_transform.html

  1. R ϕ = T s { T θ [ k = 1 N R s k ( ϕ ) ] } R_{\phi}=T_{s}\left\{T_{\theta}\left[\bigcup_{k=1}^{N}R_{s_{k}}(\phi)\right]\right\}
  2. H ( y ) = i = 1 N h i ( y - y i ) H(y)=\sum_{i=1}^{N}h_{i}(y-y_{i})
  3. x = x c + x o r x c = x - x x=x_{c}+x^{\prime}\ \ or\ \ x_{c}=x-x^{\prime}
  4. y = y c + y o r y c = y - y y=y_{c}+y^{\prime}\ \ or\ \ y_{c}=y-y^{\prime}
  5. c o s ( π - α ) = y / r o r y = r c o s ( π - α ) = - r / s i n ( α ) cos(\pi-\alpha)=y^{\prime}/r\ \ or\ \ y^{\prime}=rcos(\pi-\alpha)=-r/sin(\alpha)
  6. s i n ( π - α ) = x / r o r x = r s i n ( π - α ) = - r / c o s ( α ) sin(\pi-\alpha)=x^{\prime}/r\ \ or\ \ x^{\prime}=rsin(\pi-\alpha)=-r/cos(\alpha)
  7. x c = x + r c o s ( α ) x_{c}=x+rcos(\alpha)
  8. y c = y + r s i n ( α ) y_{c}=y+rsin(\alpha)

Generalized_assignment_problem.html

  1. a 1 a_{1}
  2. a n a_{n}
  3. b 1 b_{1}
  4. b m b_{m}
  5. b i b_{i}
  6. t i t_{i}
  7. b i b_{i}
  8. a j a_{j}
  9. p i j p_{ij}
  10. w i j w_{ij}
  11. b i b_{i}
  12. t i t_{i}
  13. i = 1 m j = 1 n p i j x i j . \sum_{i=1}^{m}\sum_{j=1}^{n}p_{ij}x_{ij}.
  14. j = 1 n w i j x i j t i i = 1 , , m \sum_{j=1}^{n}w_{ij}x_{ij}\leq t_{i}\qquad i=1,\ldots,m
  15. i = 1 m x i j = 1 j = 1 , , n \sum_{i=1}^{m}x_{ij}=1\qquad j=1,\ldots,n
  16. x i j { 0 , 1 } i = 1 , , m , j = 1 , , n x_{ij}\in\{0,1\}\qquad i=1,\ldots,m,\quad j=1,\ldots,n
  17. e / ( e - 1 ) - ε e/(e-1)-\varepsilon
  18. ε \varepsilon
  19. α \alpha
  20. α + 1 \alpha+1
  21. j j
  22. b j b_{j}
  23. b j b_{j}
  24. x i x_{i}
  25. b j b_{j}
  26. p i j p_{ij}
  27. x i x_{i}
  28. p i j p_{ij}
  29. p i k p_{ik}
  30. x i x_{i}
  31. b k b_{k}
  32. T T
  33. T [ i ] = j T[i]=j
  34. x i x_{i}
  35. b j b_{j}
  36. T [ i ] = - 1 T[i]=-1
  37. x i x_{i}
  38. j j
  39. P j P_{j}
  40. P j [ i ] = p i j P_{j}[i]=p_{ij}
  41. x i x_{i}
  42. T [ i ] = - 1 T[i]=-1
  43. P j [ i ] = p i j - p i k P_{j}[i]=p_{ij}-p_{ik}
  44. x i x_{i}
  45. b k b_{k}
  46. T [ i ] = k T[i]=k
  47. T [ i ] = - 1 T[i]=-1
  48. i = 1 n i=1\ldots n
  49. j = 1... m j=1...m
  50. b j b_{j}
  51. P j P_{j}
  52. S j S_{j}
  53. T T
  54. S j S_{j}
  55. T [ i ] = j T[i]=j
  56. i S j i\in S_{j}

Generalized_Dirichlet_distribution.html

  1. p 1 , , p k - 1 p_{1},\ldots,p_{k-1}
  2. [ i = 1 k - 1 B ( a i , b i ) ] - 1 p k b k - 1 - 1 i = 1 k - 1 [ p i a i - 1 ( j = i k p j ) b i - 1 - ( a i + b i ) ] \left[\prod_{i=1}^{k-1}B(a_{i},b_{i})\right]^{-1}p_{k}^{b_{k-1}-1}\prod_{i=1}^% {k-1}\left[p_{i}^{a_{i}-1}\left(\sum_{j=i}^{k}p_{j}\right)^{b_{i-1}-(a_{i}+b_{% i})}\right]
  3. p k = 1 - i = 1 k - 1 p i p_{k}=1-\sum_{i=1}^{k-1}p_{i}
  4. B ( x , y ) B(x,y)
  5. b i - 1 = a i + b i b_{i-1}=a_{i}+b_{i}
  6. 2 i k - 1 2\leqslant i\leqslant k-1
  7. b 0 b_{0}
  8. p 1 , p 2 , p 3 p_{1},p_{2},p_{3}
  9. [ i = 1 3 B ( a i , b i ) ] - 1 p 1 a 1 - 1 p 2 a 2 - 1 p 3 a 3 - 1 p 4 b 3 - 1 ( p 2 + p 3 + p 4 ) b 1 - ( a 2 + b 2 ) ( p 3 + p 4 ) b 2 - ( a 3 + b 3 ) \left[\prod_{i=1}^{3}B(a_{i},b_{i})\right]^{-1}p_{1}^{a_{1}-1}p_{2}^{a_{2}-1}p% _{3}^{a_{3}-1}p_{4}^{b_{3}-1}\left(p_{2}+p_{3}+p_{4}\right)^{b_{1}-\left(a_{2}% +b_{2}\right)}\left(p_{3}+p_{4}\right)^{b_{2}-\left(a_{3}+b_{3}\right)}
  10. p 1 + p 2 + p 3 < 1 p_{1}+p_{2}+p_{3}<1
  11. p 4 = 1 - p 1 - p 2 - p 3 p_{4}=1-p_{1}-p_{2}-p_{3}
  12. z 1 , , z k - 1 z_{1},\ldots,z_{k-1}
  13. z 1 = p 1 , z 2 = p 2 / ( 1 - p 1 ) , z 3 = p 3 / ( 1 - ( p 1 + p 2 ) ) , , z i = p i / ( 1 - p 1 + + p i - 1 ) z_{1}=p_{1},z_{2}=p_{2}/\left(1-p_{1}\right),z_{3}=p_{3}/\left(1-(p_{1}+p_{2})% \right),\ldots,z_{i}=p_{i}/\left(1-p_{1}+\cdots+p_{i-1}\right)
  14. p 1 , , p k p_{1},\ldots,p_{k}
  15. z i z_{i}
  16. a i , b i a_{i},b_{i}
  17. i = 1 , , k - 1 i=1,\ldots,k-1
  18. x 1 + + x k 1 x_{1}+\cdots+x_{k}\leqslant 1
  19. i = 1 k x i α i - 1 ( 1 - x 1 - - x i ) γ i B ( α i , β i ) \prod_{i=1}^{k}\frac{x_{i}^{\alpha_{i}-1}\left(1-x_{1}-\ldots-x_{i}\right)^{% \gamma_{i}}}{B(\alpha_{i},\beta_{i})}
  20. γ j = β j - α j + 1 - β j + 1 \gamma_{j}=\beta_{j}-\alpha_{j+1}-\beta_{j+1}
  21. 1 j k - 1 1\leqslant j\leqslant k-1
  22. γ k = β k - 1 \gamma_{k}=\beta_{k}-1
  23. k k
  24. x k + 1 = 1 - i = 1 k x i x_{k+1}=1-\sum_{i=1}^{k}x_{i}
  25. k - 1 k-1
  26. x k = 1 - i = 1 k - 1 x i x_{k}=1-\sum_{i=1}^{k-1}x_{i}
  27. X = ( X 1 , , X k ) G D k ( α 1 , , α k ; β 1 , , β k ) X=\left(X_{1},\ldots,X_{k}\right)\sim GD_{k}\left(\alpha_{1},\ldots,\alpha_{k}% ;\beta_{1},\ldots,\beta_{k}\right)
  28. E [ X 1 r 1 X 2 r 2 X k r k ] = j = 1 k Γ ( α j + β j ) Γ ( α j + r j ) Γ ( β j + δ j ) Γ ( α j ) Γ ( β j ) Γ ( α j + β j + r j + δ j ) E\left[X_{1}^{r_{1}}X_{2}^{r_{2}}\cdots X_{k}^{r_{k}}\right]=\prod_{j=1}^{k}% \frac{\Gamma\left(\alpha_{j}+\beta_{j}\right)\Gamma\left(\alpha_{j}+r_{j}% \right)\Gamma\left(\beta_{j}+\delta_{j}\right)}{\Gamma\left(\alpha_{j}\right)% \Gamma\left(\beta_{j}\right)\Gamma\left(\alpha_{j}+\beta_{j}+r_{j}+\delta_{j}% \right)}
  29. δ j = r j + 1 + r j + 2 + + r k \delta_{j}=r_{j+1}+r_{j+2}+\cdots+r_{k}
  30. j = 1 , 2 , , k - 1 j=1,2,\cdots,k-1
  31. δ k = 0 \delta_{k}=0
  32. E ( X j ) = α j α j + β j m = 1 j - 1 β m α m + β m . E\left(X_{j}\right)=\frac{\alpha_{j}}{\alpha_{j}+\beta_{j}}\prod_{m=1}^{j-1}% \frac{\beta_{m}}{\alpha_{m}+\beta_{m}}.
  33. b i - 1 = a i + b i b_{i-1}=a_{i}+b_{i}
  34. 2 i k 2\leqslant i\leqslant k
  35. X = ( X 1 , , X k ) G D k ( α 1 , , α k ; β 1 , , β k ) X=\left(X_{1},\ldots,X_{k}\right)\sim GD_{k}\left(\alpha_{1},\ldots,\alpha_{k}% ;\beta_{1},\ldots,\beta_{k}\right)
  36. Y | X Y|X
  37. n n
  38. Y = ( Y 1 , , Y k ) Y=\left(Y_{1},\ldots,Y_{k}\right)
  39. Y j = y j Y_{j}=y_{j}
  40. 1 j k 1\leqslant j\leqslant k
  41. y k + 1 = n - i = 1 k y i y_{k+1}=n-\sum_{i=1}^{k}y_{i}
  42. X | Y X|Y
  43. X | Y G D k ( α 1 , , α k ; β 1 , , β k ) X|Y\sim GD_{k}\left({\alpha^{\prime}}_{1},\ldots,{\alpha^{\prime}}_{k};{\beta^% {\prime}}_{1},\ldots,{\beta^{\prime}}_{k}\right)
  44. α j = α j + y j {\alpha^{\prime}}_{j}=\alpha_{j}+y_{j}
  45. β j = β j + i = j + 1 k + 1 y i {\beta^{\prime}}_{j}=\beta_{j}+\sum_{i=j+1}^{k+1}y_{i}
  46. 1 k . 1\leqslant k.
  47. k + 1 k+1
  48. X = ( X 1 , , X k ) X=(X_{1},\ldots,X_{k})
  49. j j
  50. X D ( α 1 , , α k , α k + 1 ) X\sim D(\alpha_{1},\ldots,\alpha_{k},\alpha_{k+1})
  51. X X
  52. α i \alpha_{i}
  53. k + 1 k+1
  54. α i \alpha_{i}
  55. i i
  56. i i
  57. α i \alpha_{i}
  58. 1 \geq 1
  59. j j
  60. j j
  61. n n
  62. X X
  63. X G D ( α 1 , , α k ; β 1 , , β k ) X\sim GD(\alpha_{1},\ldots,\alpha_{k};\beta_{1},\ldots,\beta_{k})
  64. k + 1 k+1
  65. j = 1 , , k j=1,\ldots,k
  66. α j \alpha_{j}
  67. j j
  68. β j \beta_{j}
  69. j j
  70. k + 1 k+1
  71. k + 1 k+1
  72. n n
  73. α \alpha
  74. β \beta

Generalized_Pareto_distribution.html

  1. μ \mu
  2. σ \sigma
  3. ξ \xi
  4. κ = - ξ \kappa=-\xi\,
  5. F ξ ( z ) = { 1 - ( 1 + ξ z ) - 1 / ξ for ξ 0 , 1 - e - z for ξ = 0. F_{\xi}(z)=\begin{cases}1-\left(1+\xi z\right)^{-1/\xi}&\,\text{for }\xi\neq 0% ,\\ 1-e^{-z}&\,\text{for }\xi=0.\end{cases}
  6. z 0 z\geq 0
  7. ξ 0 \xi\geq 0
  8. 0 z - 1 / ξ 0\leq z\leq-1/\xi
  9. ξ < 0 \xi<0
  10. f ξ ( z ) = { ( ξ z + 1 ) - ξ + 1 ξ for ξ 0 , e - z for ξ = 0. f_{\xi}(z)=\begin{cases}(\xi z+1)^{-\frac{\xi+1}{\xi}}&\,\text{for }\xi\neq 0,% \\ e^{-z}&\,\text{for }\xi=0.\end{cases}
  11. { ( ξ z + 1 ) f ξ ( z ) + ( ξ + 1 ) f ξ ( z ) = 0 , f ξ ( 0 ) = 1 } \left\{\begin{array}[]{l}(\xi z+1)f_{\xi}^{\prime}(z)+(\xi+1)f_{\xi}(z)=0,\\ f_{\xi}(0)=1\end{array}\right\}
  12. x - μ σ \frac{x-\mu}{\sigma}
  13. F ( ξ , μ , σ ) ( x ) = { 1 - ( 1 + ξ ( x - μ ) σ ) - 1 / ξ for ξ 0 , 1 - exp ( - x - μ σ ) for ξ = 0. F_{(\xi,\mu,\sigma)}(x)=\begin{cases}1-\left(1+\frac{\xi(x-\mu)}{\sigma}\right% )^{-1/\xi}&\,\text{for }\xi\neq 0,\\ 1-\exp\left(-\frac{x-\mu}{\sigma}\right)&\,\text{for }\xi=0.\end{cases}
  14. x μ x\geqslant\mu
  15. ξ 0 \xi\geqslant 0\,
  16. μ x μ - σ / ξ \mu\leqslant x\leqslant\mu-\sigma/\xi
  17. ξ < 0 \xi<0
  18. μ \mu\in\mathbb{R}
  19. σ > 0 \sigma>0
  20. ξ \xi\in\mathbb{R}
  21. f ( ξ , μ , σ ) ( x ) = 1 σ ( 1 + ξ ( x - μ ) σ ) ( - 1 ξ - 1 ) f_{(\xi,\mu,\sigma)}(x)=\frac{1}{\sigma}\left(1+\frac{\xi(x-\mu)}{\sigma}% \right)^{\left(-\frac{1}{\xi}-1\right)}
  22. f ( ξ , μ , σ ) ( x ) = σ 1 ξ ( σ + ξ ( x - μ ) ) 1 ξ + 1 f_{(\xi,\mu,\sigma)}(x)=\frac{\sigma^{\frac{1}{\xi}}}{\left(\sigma+\xi(x-\mu)% \right)^{\frac{1}{\xi}+1}}
  23. x μ x\geqslant\mu
  24. ξ 0 \xi\geqslant 0
  25. μ x μ - σ / ξ \mu\leqslant x\leqslant\mu-\sigma/\xi
  26. ξ < 0 \xi<0
  27. { f ( x ) ( - μ ξ + σ + ξ x ) + ( ξ + 1 ) f ( x ) = 0 , f ( 0 ) = ( 1 - μ ξ σ ) - 1 ξ - 1 σ } \left\{\begin{array}[]{l}f^{\prime}(x)(-\mu\xi+\sigma+\xi x)+(\xi+1)f(x)=0,\\ f(0)=\frac{\left(1-\frac{\mu\xi}{\sigma}\right)^{-\frac{1}{\xi}-1}}{\sigma}% \end{array}\right\}
  28. ξ \xi
  29. μ \mu
  30. ξ > 0 \xi>0
  31. μ = σ / ξ \mu=\sigma/\xi
  32. x m = σ / ξ x_{m}=\sigma/\xi
  33. α = 1 / ξ \alpha=1/\xi
  34. X = μ + σ ( U - ξ - 1 ) ξ GPD ( μ , σ , ξ 0 ) X=\mu+\frac{\sigma(U^{-\xi}-1)}{\xi}\sim\mbox{GPD}~{}(\mu,\sigma,\xi\neq 0)
  35. X = μ - σ ln ( U ) GPD ( μ , σ , ξ = 0 ) . X=\mu-\sigma\ln(U)\sim\mbox{GPD}~{}(\mu,\sigma,\xi=0).

Generalized_Pochhammer_symbol.html

  1. α > 0 \alpha>0
  2. κ = ( κ 1 , κ 2 , , κ m ) \kappa=(\kappa_{1},\kappa_{2},\ldots,\kappa_{m})
  3. ( a ) κ ( α ) = i = 1 m j = 1 κ i ( a - i - 1 α + j - 1 ) . (a)^{(\alpha)}_{\kappa}=\prod_{i=1}^{m}\prod_{j=1}^{\kappa_{i}}\left(a-\frac{i% -1}{\alpha}+j-1\right).

Generation_time.html

  1. T = log R 0 r T=\frac{\log R_{0}}{r}
  2. T \textstyle T
  3. T = 0 x e - r x ( x ) m ( x ) d x T=\int_{0}^{\infty}xe^{-rx}\ell(x)m(x)\,\mathrm{d}x
  4. T = λ 𝐯𝐰 𝐯𝐅𝐰 = 1 e λ ( f i j ) T=\frac{\lambda\mathbf{vw}}{\mathbf{vFw}}=\frac{1}{\sum e_{\lambda}(f_{ij})}
  5. e λ ( f i j ) = f i j λ λ f i j \textstyle e_{\lambda}(f_{ij})=\frac{f_{ij}}{\lambda}\frac{\partial\lambda}{% \partial f_{ij}}
  6. T = x = 0 x ( x ) m ( x ) d x x = 0 ( x ) m ( x ) d x T=\frac{\int_{x=0}^{\infty}x\ell(x)m(x)\,\mathrm{d}x}{\int_{x=0}^{\infty}\ell(% x)m(x)\,\mathrm{d}x}

Generator_(category_theory).html

  1. 𝒞 \mathcal{C}
  2. { G i O b ( 𝒞 ) | i I } \{G_{i}\in Ob(\mathcal{C})|i\in I\}
  3. f , g : X Y f,g:X\rightarrow Y
  4. 𝒞 \mathcal{C}
  5. f g f\neq g
  6. h : G i X h:G_{i}\rightarrow X
  7. f h g h f\circ h\neq g\circ h
  8. 𝐙 \mathbf{Z}
  9. x X x\in X
  10. f ( x ) g ( x ) f(x)\neq g(x)
  11. 𝐙 X , \mathbf{Z}\rightarrow X,
  12. n n x n\mapsto n\cdot x

Geodetic_effect.html

  1. d s 2 = d t 2 ( 1 - 2 m r ) - d r 2 ( 1 - 2 m r ) - 1 - r 2 ( d θ 2 + sin 2 θ d ϕ 2 ) , ds^{2}=dt^{2}\left(1-\frac{2m}{r}\right)-dr^{2}\left(1-\frac{2m}{r}\right)^{-1% }-r^{2}(d\theta^{2}+\sin^{2}\theta\,d\phi^{\prime 2}),
  2. ω \omega
  3. d ϕ = d ϕ - ω d t . d\phi=d\phi^{\prime}-\omega\,dt.
  4. θ \theta
  5. d s 2 = ( 1 - 2 m r - r 2 β ω 2 ) ( d t - r 2 β ω 1 - 2 m / r - r 2 β ω 2 d ϕ ) 2 - d r 2 ( 1 - 2 m r ) - 1 - r 2 β - 2 m r β 1 - 2 m / r - r 2 β ω 2 d ϕ 2 ds^{2}=\left(1-\frac{2m}{r}-r^{2}\beta\omega^{2}\right)\left(dt-\frac{r^{2}% \beta\omega}{1-2m/r-r^{2}\beta\omega^{2}}\,d\phi\right)^{2}-dr^{2}\left(1-% \frac{2m}{r}\right)^{-1}-\frac{r^{2}\beta-2mr\beta}{1-2m/r-r^{2}\beta\omega^{2% }}\,d\phi^{2}
  6. β = sin 2 ( θ ) \beta=\sin^{2}(\theta)
  7. d s 2 = e 2 Φ ( d t - w i d x i ) 2 - k i j d x i d x j . ds^{2}=e^{2\Phi}\left(dt-w_{i}\,dx^{i}\right)^{2}-k_{ij}\,dx^{i}\,dx^{j}.
  8. Ω = 2 4 e Φ [ k i k k j l ( ω i , j - ω j , i ) ( ω k , l - ω l , k ) ] 1 / 2 = β ω ( r - 3 m ) r - 2 m - β ω 2 r 3 = β ω . \Omega=\frac{\sqrt{2}}{4}e^{\Phi}[k^{ik}k^{jl}(\omega_{i,j}-\omega_{j,i})(% \omega_{k,l}-\omega_{l,k})]^{1/2}=\frac{\sqrt{\beta}\omega(r-3m)}{r-2m-\beta% \omega^{2}r^{3}}=\sqrt{\beta}\omega.
  9. Φ , i = 0 \Phi,_{i}=0
  10. Φ , i = 2 m / r 2 - 2 r β ω 2 2 ( 1 - 2 m / r - r 2 β ω 2 ) = 0. \Phi,_{i}=\frac{2m/r^{2}-2r\beta\omega^{2}}{2(1-2m/r-r^{2}\beta\omega^{2})}=0.
  11. ω 2 = m r 3 β . \omega^{2}=\frac{m}{r^{3}\beta}.
  12. Δ τ = ( 1 - 2 m r - r 2 β ω 2 ) 1 / 2 d t = ( 1 - 3 m r ) 1 / 2 d t . \Delta\tau=\left(1-\frac{2m}{r}-r^{2}\beta\omega^{2}\right)^{1/2}\,dt=\left(1-% \frac{3m}{r}\right)^{1/2}\,dt.
  13. α = Ω Δ τ \alpha^{\prime}=\Omega\Delta\tau
  14. α = α + 2 π = - 2 π β ( ( 1 - 3 m r ) 1 / 2 - 1 ) . \alpha=\alpha^{\prime}+2\pi=-2\pi\sqrt{\beta}\Bigg(\left(1-\frac{3m}{r}\right)% ^{1/2}-1\Bigg).
  15. α 3 π m r β = 3 π m r sin ( θ ) . \alpha\approx\frac{3\pi m}{r}\sqrt{\beta}=\frac{3\pi m}{r}\sin(\theta).

Geometric_measure_theory.html

  1. 3 \mathbb{R}^{3}
  2. vol ( ( 1 - λ ) K + λ L ) 1 / n ( 1 - λ ) vol ( K ) 1 / n + λ vol ( L ) 1 / n , \mathrm{vol}\big((1-\lambda)K+\lambda L\big)^{1/n}\geq(1-\lambda)\mathrm{vol}(% K)^{1/n}+\lambda\,\mathrm{vol}(L)^{1/n},

Geometric_median.html

  1. x 1 , x 2 , , x m x_{1},x_{2},\dots,x_{m}\,
  2. x i n x_{i}\in\mathbb{R}^{n}
  3. = arg min y n i = 1 m x i - y 2 =\underset{y\in\mathbb{R}^{n}}{\operatorname{arg\,min}}\sum_{i=1}^{m}\left\|x_% {i}-y\right\|_{2}
  4. y y
  5. y y
  6. x i x_{i}
  7. y i + 1 = ( j = 1 m x j x j - y i ) / ( j = 1 m 1 x j - y i ) . \left.y_{i+1}=\left(\sum_{j=1}^{m}\frac{x_{j}}{\|x_{j}-y_{i}\|}\right)\right/% \left(\sum_{j=1}^{m}\frac{1}{\|x_{j}-y_{i}\|}\right).
  8. 0 = j = 1 m x j - y x j - y . 0=\sum_{j=1}^{m}\frac{x_{j}-y}{\left\|x_{j}-y\right\|}.
  9. y = ( j = 1 m x j x j - y ) / ( j = 1 m 1 x j - y ) , \left.y=\left(\sum_{j=1}^{m}\frac{x_{j}}{\|x_{j}-y\|}\right)\right/\left(\sum_% {j=1}^{m}\frac{1}{\|x_{j}-y\|}\right),
  10. 0 = j = 1 m u j 0=\sum_{j=1}^{m}u_{j}
  11. u j = x j - y x j - y u_{j}=\frac{x_{j}-y}{\left\|x_{j}-y\right\|}
  12. u j 1. \|u_{j}\|\leq 1.
  13. 1 j m , x j y x j - y x j - y | { j 1 j m , x j = y } | . \sum_{1\leq j\leq m,x_{j}\neq y}\frac{x_{j}-y}{\left\|x_{j}-y\right\|}\leq% \left|\{\,j\mid 1\leq j\leq m,x_{j}=y\,\}\right|.
  14. M M
  15. d ( , ) d(\cdot,\cdot)
  16. w 1 , , w n w_{1},\ldots,w_{n}
  17. n n
  18. x 1 , , x n x_{1},\ldots,x_{n}
  19. n n
  20. M M
  21. m m
  22. m = arg min x M i = 1 n w i d ( x , x i ) m=\underset{x\in M}{\operatorname{arg\,min}}\sum_{i=1}^{n}w_{i}d(x,x_{i})
  23. m m

Geometric_programming.html

  1. f 0 ( x ) \ f_{0}(x)
  2. f i ( x ) 1 , i = 1 , , m f_{i}(x)\leq 1,\quad i=1,\dots,m
  3. h i ( x ) = 1 , i = 1 , , p h_{i}(x)=1,\quad i=1,\dots,p
  4. f 0 , , f m f_{0},\dots,f_{m}
  5. h 1 , , h p h_{1},\dots,h_{p}
  6. f : + + n f:\mathbb{R}_{++}^{n}\to\mathbb{R}
  7. f ( x ) = c x 1 a 1 x 2 a 2 x n a n f(x)=cx_{1}^{a_{1}}x_{2}^{a_{2}}\cdots x_{n}^{a_{n}}
  8. c > 0 c>0
  9. a i a_{i}\in\mathbb{R}
  10. y i = log ( x i ) y_{i}=\log(x_{i})
  11. f ( x ) = c x 1 a 1 x n a n e a T y + b f(x)=cx_{1}^{a_{1}}\cdots x_{n}^{a_{n}}\mapsto e^{a^{T}y+b}
  12. b = log ( c ) b=\log(c)
  13. f f
  14. f ( x ) = k = 1 K c k x 1 a 1 k x n a n k f(x)=\sum_{k=1}^{K}c_{k}x_{1}^{a_{1k}}\cdots x_{n}^{a_{nk}}
  15. f ( x ) = k = 1 K e a k T y + b k f(x)=\sum_{k=1}^{K}e^{a_{k}^{T}y+b_{k}}
  16. a k = ( a 1 k , , a n k ) a_{k}=(a_{1k},\dots,a_{nk})
  17. b k = log ( c k ) b_{k}=\log(c_{k})

Geometric_spanner.html

  1. O ( n ) O(n)
  2. O ( M S T ) O(MST)
  3. O ( 1 ) O(1)
  4. O ( n log n ) O(n\log n)
  5. Θ \Theta
  6. Θ \Theta
  7. Θ \Theta
  8. O ( n log n ) O(n\log n)
  9. O ( n 3 log n ) O(n^{3}\log n)
  10. O ( n 2 ) O(n^{2})
  11. O ( n 2 log n ) O(n^{2}\log n)
  12. O ( n 2 ) O(n^{2})
  13. O ( n 2 log 2 n ) O(n^{2}\log^{2}n)
  14. 1 0 \scriptstyle\sqrt{1}0
  15. ( 4 3 / 9 ) π 2.418 \scriptstyle{(4\sqrt{3}/9)\pi}\approx 2.418
  16. π / 2 \scriptstyle{{\pi}/2}

GF_method.html

  1. 2 V s , t = 1 3 N - 6 F s t S s S t . 2V\approx\sum_{s,t=1}^{3N-6}F_{st}S_{s}\,S_{t}.
  2. 2 T = s , t = 1 3 N - 6 g s t ( 𝐪 ) S ˙ s S ˙ t , 2T=\sum_{s,t=1}^{3N-6}g_{st}(\mathbf{q})\dot{S}_{s}\dot{S}_{t},
  3. S s S ˙ t S_{s}\,\dot{S}_{t}
  4. 𝐋 T 𝐅𝐋 = s y m b o l Φ and 𝐋 T 𝐆 - 1 𝐋 = 𝐄 , \mathbf{L}^{\mathrm{T}}\mathbf{F}\mathbf{L}=symbol{\Phi}\quad\mathrm{and}\quad% \mathbf{L}^{\mathrm{T}}\mathbf{G}^{-1}\mathbf{L}=\mathbf{E},
  5. 𝐆𝐅𝐋 = 𝐋 s y m b o l Φ , \mathbf{G}\mathbf{F}\mathbf{L}=\mathbf{L}symbol{\Phi},
  6. s y m b o l Φ = diag ( f 1 , , f 3 N - 6 ) symbol{\Phi}=\operatorname{diag}(f_{1},\ldots,f_{3N-6})
  7. 4 π 2 ν i 2 4{\pi}^{2}{\nu}_{i}^{2}
  8. ν i {\nu}_{i}
  9. 𝐄 \mathbf{E}\,
  10. Q k = t = 1 3 N - 6 ( 𝐋 - 1 ) k t S t , k = 1 , , 3 N - 6. Q_{k}=\sum_{t=1}^{3N-6}(\mathbf{L}^{-1})_{kt}S_{t},\quad k=1,\ldots,3N-6.\,
  11. 𝐬 = col ( S 1 , , S 3 N - 6 ) and 𝐐 = col ( Q 1 , , Q 3 N - 6 ) , \mathbf{s}=\operatorname{col}(S_{1},\ldots,S_{3N-6})\quad\mathrm{and}\quad% \mathbf{Q}=\operatorname{col}(Q_{1},\ldots,Q_{3N-6}),
  12. 𝐬 = 𝐋𝐐 . \mathbf{s}=\mathbf{L}\mathbf{Q}.
  13. 2 E = 𝐬 ˙ T 𝐆 - 1 𝐬 ˙ + 𝐬 T 𝐅𝐬 2E=\dot{\mathbf{s}}^{\mathrm{T}}\mathbf{G}^{-1}\dot{\mathbf{s}}+\mathbf{s}^{% \mathrm{T}}\mathbf{F}\mathbf{s}
  14. = 𝐐 ˙ T ( 𝐋 T 𝐆 - 1 𝐋 ) 𝐐 ˙ + 𝐐 T ( 𝐋 T 𝐅𝐋 ) 𝐐 =\dot{\mathbf{Q}}^{\mathrm{T}}\;\left(\mathbf{L}^{\mathrm{T}}\mathbf{G}^{-1}% \mathbf{L}\right)\;\dot{\mathbf{Q}}+\mathbf{Q}^{\mathrm{T}}\left(\mathbf{L}^{% \mathrm{T}}\mathbf{F}\mathbf{L}\right)\;\mathbf{Q}
  15. = 𝐐 ˙ T 𝐐 ˙ + 𝐐 T s y m b o l Φ 𝐐 = t = 1 3 N - 6 ( Q ˙ t 2 + f t Q t 2 ) . =\dot{\mathbf{Q}}^{\mathrm{T}}\dot{\mathbf{Q}}+\mathbf{Q}^{\mathrm{T}}symbol{% \Phi}\mathbf{Q}=\sum_{t=1}^{3N-6}\big(\dot{Q}_{t}^{2}+f_{t}Q_{t}^{2}\big).
  16. L = 1 2 t = 1 3 N - 6 ( Q ˙ t 2 - f t Q t 2 ) . L=\frac{1}{2}\sum_{t=1}^{3N-6}\big(\dot{Q}_{t}^{2}-f_{t}Q_{t}^{2}\big).
  17. Q ¨ t + f t Q t = 0 \ddot{Q}_{t}+f_{t}\,Q_{t}=0
  18. 𝐝 A 𝐑 A - 𝐑 A 0 \mathbf{d}_{A}\equiv\mathbf{R}_{A}-\mathbf{R}_{A}^{0}
  19. S t = A = 1 N i = 1 3 s A i t d A i = A = 1 N 𝐬 A t 𝐝 A , for t = 1 , , 3 N - 6 , S_{t}=\sum_{A=1}^{N}\sum_{i=1}^{3}s^{t}_{Ai}\,d_{Ai}=\sum_{A=1}^{N}\mathbf{s}^% {t}_{A}\cdot\mathbf{d}_{A},\quad\mathrm{for}\quad t=1,\ldots,3N-6,
  20. s A i t s^{t}_{Ai}
  21. 𝐬 = 𝐁𝐝 . \mathbf{s}=\mathbf{B}\mathbf{d}.
  22. s A i t s^{t}_{Ai}
  23. 𝐐 = 𝐋 - 1 𝐬 = 𝐋 - 1 𝐁𝐝 𝐃𝐝 . \mathbf{Q}=\mathbf{L}^{-1}\mathbf{s}=\mathbf{L}^{-1}\mathbf{B}\mathbf{d}\equiv% \mathbf{D}\mathbf{d}.
  24. Q k = A = 1 N i = 1 3 D A i k d A i for k = 1 , , 3 N - 6. Q_{k}=\sum_{A=1}^{N}\sum_{i=1}^{3}D^{k}_{Ai}\,d_{Ai}\quad\mathrm{for}\quad k=1% ,\ldots,3N-6.
  25. A = 1 N 𝐬 A t = 0 and A = 1 N 𝐑 A 0 × 𝐬 A t = 0 , t = 1 , , 3 N - 6. \sum_{A=1}^{N}\mathbf{s}^{t}_{A}=0\quad\mathrm{and}\quad\sum_{A=1}^{N}\mathbf{% R}^{0}_{A}\times\mathbf{s}^{t}_{A}=0,\quad t=1,\ldots,3N-6.
  26. A = 1 N M A 𝐝 A = 0 and A = 1 N M A 𝐑 A 0 × 𝐝 A = 0. \sum_{A=1}^{N}M_{A}\;\mathbf{d}_{A}=0\quad\mathrm{and}\quad\sum_{A=1}^{N}M_{A}% \;\mathbf{R}^{0}_{A}\times\mathbf{d}_{A}=0.

Gibbons–Hawking_effect.html

  1. T T
  2. M M
  3. T M - 1 T\propto M^{-1}
  4. T T
  5. H H
  6. T H T\propto H

Gibbs–Duhem_equation.html

  1. i = 1 I N i d μ i = - S d T + V d p \sum_{i=1}^{I}N_{i}\,\mathrm{d}\mu_{i}=-S\,\mathrm{d}T+V\,\mathrm{d}p\,
  2. N i N_{i}\,
  3. i i\,
  4. d μ i \mathrm{d}\mu_{i}\,
  5. S S\,
  6. T T\,
  7. V V\,
  8. p p\,
  9. I - 1 I-1\,
  10. I I\,
  11. G G\,
  12. d G = G p | T , N d p + G T | p , N d T + i = 1 I G N i | p , T , N j i d N i \mathrm{d}G=\left.\frac{\partial G}{\partial p}\right|_{T,N}\,\mathrm{d}p+% \left.\frac{\partial G}{\partial T}\right|_{p,N}\,\mathrm{d}T+\sum_{i=1}^{I}% \left.\frac{\partial G}{\partial N_{i}}\right|_{p,T,N_{j\neq i}}\,\mathrm{d}N_% {i}\,
  13. d G = V d p - S d T + i = 1 I μ i d N i \mathrm{d}G=V\,\mathrm{d}p-S\,\mathrm{d}T+\sum_{i=1}^{I}\mu_{i}\,\mathrm{d}N_{% i}\,
  14. G = i = 1 I μ i N i G=\sum_{i=1}^{I}\mu_{i}N_{i}\,
  15. d G = i = 1 I μ i d N i + i = 1 I N i d μ i \mathrm{d}G=\sum_{i=1}^{I}\mu_{i}\,\mathrm{d}N_{i}+\sum_{i=1}^{I}N_{i}\,% \mathrm{d}\mu_{i}\,
  16. i = 1 I N i d μ i = - S d T + V d p \sum_{i=1}^{I}N_{i}\,\mathrm{d}\mu_{i}=-S\,\mathrm{d}T+V\,\mathrm{d}p\,
  17. I I\,
  18. I + 1 I+1\,
  19. 0 = N 1 d μ 1 + N 2 d μ 2 0=N_{1}\,\mathrm{d}\mu_{1}+N_{2}\,\mathrm{d}\mu_{2}\,
  20. N 1 + N 2 N_{1}+N_{2}\,
  21. γ \gamma
  22. x 1 + x 2 = 1 x_{1}+x_{2}=1\,
  23. 0 = x 1 d ln ( γ 1 ) + x 2 d ln ( γ 2 ) 0=x_{1}\,\mathrm{d}\ln(\gamma_{1})+x_{2}\,\mathrm{d}\ln(\gamma_{2})\,

Girvan–Newman_algorithm.html

  1. i i

Gittins_index.html

  1. ν ( i ) = sup τ > 0 t = 0 τ - 1 β t R [ Z ( t ) ] Z ( 0 ) = i t = 0 τ - 1 β t Z ( 0 ) = i \nu(i)=\sup_{\tau>0}\frac{\left\langle\sum_{t=0}^{\tau-1}\beta^{t}R[Z(t)]% \right\rangle_{Z(0)=i}}{\left\langle\sum_{t=0}^{\tau-1}\beta^{t}\right\rangle_% {Z(0)=i}}
  2. Z ( ) Z(\cdot)
  3. R ( i ) R(i)
  4. i i
  5. β < 1 \beta<1
  6. c \langle\cdot\rangle_{c}
  7. X c x χ x P { X = x | c } \langle X\rangle_{c}\doteq\sum_{x\in\chi}xP\{X=x|c\}
  8. χ \chi
  9. w ( i ) = inf { k : v ( i , k ) = k } w(i)=\inf\{k:v(i,k)=k\}
  10. v ( i , k ) v(i,k)
  11. v ( i , k ) = sup τ > 0 t = 0 τ - 1 β t R [ Z ( t ) ] + β t k Z ( 0 ) = i v(i,k)=\sup_{\tau>0}\left\langle\sum_{t=0}^{\tau-1}\beta^{t}R[Z(t)]+\beta^{t}k% \right\rangle_{Z(0)=i}
  12. ν ( i ) = ( 1 - β ) w ( i ) . \nu(i)=(1-\beta)w(i).
  13. Z ( ) Z(\cdot)
  14. i i
  15. M i M_{i}
  16. i i
  17. M i M_{i}
  18. i i
  19. h ( i ) = sup π t = 0 τ - 1 β t R [ Z π ( t ) ] Z ( 0 ) = i h(i)=\sup_{\pi}\left\langle\sum_{t=0}^{\tau-1}\beta^{t}R[Z^{\pi}(t)]\right% \rangle_{Z(0)=i}
  20. π \pi
  21. M i M_{i}
  22. h ( i ) = w ( i ) h(i)=w(i)
  23. β ( i ) \beta(i)
  24. i i
  25. α ( i ) \alpha(i)
  26. α ( i ) = sup τ > 0 R τ ( i ) Q τ ( i ) \alpha(i)=\sup_{\tau>0}\frac{R^{\tau}(i)}{Q^{\tau}(i)}
  27. R τ ( i ) = t = 0 τ - 1 R [ Z ( t ) ] Z ( 0 ) = i R^{\tau}(i)=\left\langle\sum_{t=0}^{\tau-1}R[Z(t)]\right\rangle_{Z(0)=i}
  28. Q τ ( i ) = t = 0 τ - 1 j = 0 t β [ Z ( j ) ] Z ( 0 ) = i Q^{\tau}(i)=\left\langle\sum_{t=0}^{\tau-1}\prod_{j=0}^{t}\beta[Z(j)]\right% \rangle_{Z(0)=i}
  29. β t \beta^{t}
  30. j = 0 t β [ Z ( j ) ] \prod_{j=0}^{t}\beta[Z(j)]
  31. ν ( i ) \nu(i)
  32. w ( i ) w(i)
  33. h ( i ) h(i)
  34. α ( i ) = h ( i ) = w ( i ) \alpha(i)=h(i)=w(i)
  35. α ( i ) k ν ( i ) , k \alpha(i)\neq k\nu(i),\forall k
  36. α ( i ) \alpha(i)
  37. ν ( i ) \nu(i)

Gliding_flight.html

  1. L D = Δ s Δ h = v forward v down {L\over D}={{\Delta s}\over{\Delta h}}={v_{\,\text{forward}}\over v_{\,\text{% down}}}

Gliese_832.html

  1. B C = - 1.821 \scriptstyle BC=-1.821
  2. M b o l = 8.369 \scriptstyle M_{bol_{\ast}}=8.369
  3. M b o l = 4.73 \scriptstyle M_{bol_{\odot}}=4.73
  4. L b o l L b o l = 10 0.4 ( M b o l - M b o l ) \scriptstyle\frac{L_{bol_{\ast}}}{L_{bol_{\odot}}}=10^{0.4\left(M_{bol_{\odot}% }-M_{bol_{\ast}}\right)}
  5. M V = 4.83 \scriptstyle M_{V_{\odot}}=4.83
  6. L V L V = 10 0.4 ( M V - M V ) \scriptstyle\frac{L_{V_{\ast}}}{L_{V_{\odot}}}=10^{0.4\left(M_{V_{\odot}}-M_{V% _{\ast}}\right)}

Global_language_system.html

  1. Q i Q_{i}
  2. Q i = p i × c i = ( P i N S ) × ( C i M S ) Q_{i}=p_{i}\times c_{i}=\left(\frac{P_{i}}{N^{S}}\right)\times\left(\frac{C_{i% }}{M^{S}}\right)
  3. p i p_{i}
  4. P i P_{i}
  5. N S N^{S}
  6. c i c_{i}
  7. C i C_{i}
  8. M S M^{S}
  9. p i p_{i}
  10. c i c_{i}

Gloss_and_matte_paint.html

  1. P V C = V p i g m e n t V p i g m e n t + V b i n d e r PVC=\frac{V_{pigment}}{V_{pigment}+V_{binder}}

Glycerol_(data_page).html

  1. log e P m m H g = \scriptstyle\log_{e}P_{mmHg}=
  2. log e ( 760 101.325 ) - 21.25867 log e ( T + 273.15 ) - 16726.26 T + 273.15 + 165.5099 + 1.100480 × 10 - 05 ( T + 273.15 ) 2 \scriptstyle\log_{e}(\frac{760}{101.325})-21.25867\log_{e}(T+273.15)-\frac{167% 26.26}{T+273.15}+165.5099+1.100480\times 10^{-05}(T+273.15)^{2}

GMP_synthase_(glutamine—hydrolysing).html

  1. \rightleftharpoons

Goldbach's_comet.html

  1. g ( E ) g(E)
  2. g ( E ) g(E)
  3. E > 2 E>2
  4. g ( 22 ) = 3 g(22)=3
  5. 22 = 11 + 11 = 5 + 17 = 3 + 19 22=11+11=5+17=3+19
  6. E / 2 E/2
  7. g ( E ) g(E)
  8. E / 2 E/2
  9. E / 2 E/2
  10. g ( E ) g a v = Π 2 ( p - 1 ) ( p - 2 ) , ( 1 ) \frac{g(E)}{g_{av}}=\Pi_{2}\prod\frac{(p-1)}{(p-2)}\,,\quad\quad\quad\quad% \quad\quad(1)
  11. E / 2 E/2
  12. Π 2 = p > 2 ( 1 - 1 ( p - 1 ) 2 ) = 0.6601618... \Pi_{2}=\prod_{p>2}\left(1-\frac{1}{(p-1)^{2}}\right)=0.6601618...
  13. E / 2 E/2
  14. Π 2 \Pi_{2}
  15. 1 / g p a v ( E ) 1/\sqrt{g_{pav}(E)}
  16. 2 / g p a v ( E ) \sqrt{2/g_{pav}(E)}
  17. E / 2 E/2
  18. E / 2 E/2
  19. E / 2 E/2
  20. Π p \Pi\,p

Golod–Shafarevich_theorem.html

  1. b j n b j - 1 - i = 2 j b j - i r i . b_{j}\geq nb_{j-1}-\sum_{i=2}^{j}b_{j-i}r_{i}.

Golomb_sequence.html

  1. a ( 1 ) = 1 ; a ( n + 1 ) = 1 + a ( n + 1 - a ( a ( n ) ) ) a(1)=1;a(n+1)=1+a(n+1-a(a(n)))
  2. φ 2 - φ n φ - 1 , \varphi^{2-\varphi}n^{\varphi-1},

Goodman_and_Kruskal's_gamma.html

  1. G = N s - N d N s + N d . G=\frac{N_{s}-N_{d}}{N_{s}+N_{d}}\ .
  2. γ \gamma
  3. γ = P s - P d P s + P d , \gamma=\frac{P_{s}-P_{d}}{P_{s}+P_{d}}\ ,
  4. t G N s + N d n ( 1 - G 2 ) , t\approx G\sqrt{\frac{N_{s}+N_{d}}{n(1-G^{2})}}\ ,
  5. n N s + N d . n\neq N_{s}+N_{d}.\,
  6. a a
  7. b b
  8. a + b a+b
  9. c c
  10. d d
  11. c + d c+d
  12. a + c a+c
  13. b + d b+d
  14. n n
  15. Q = a d - b c a d + b c . Q=\frac{ad-bc}{ad+bc}\ .

Goormaghtigh_conjecture.html

  1. x m - 1 x - 1 = y n - 1 y - 1 \frac{x^{m}-1}{x-1}=\frac{y^{n}-1}{y-1}

Goos–Hänchen_effect.html

  1. 𝐄 ¯ ( x , z , t ) = 𝐄 ¯ T E / T M ( e j 𝐤 1 𝐫 + e j 𝐤 2 𝐫 ) e - j ω t \mathbf{\underline{E}}(x,z,t)=\mathbf{\underline{E}}^{TE/TM}\left(e^{j\mathbf{% k}_{1}\cdot\mathbf{r}}+e^{j\mathbf{k}_{2}\cdot\mathbf{r}}\right)\cdot e^{-j% \omega t}
  2. 𝐤 1 = k ( cos ( θ 0 + Δ θ ) 𝐱 ^ + sin ( θ 0 + Δ θ ) 𝐳 ^ ) \mathbf{k}_{1}=k\left(\cos{\left(\theta_{0}+\Delta\theta\right)}\mathbf{\hat{x% }}+\sin{\left(\theta_{0}+\Delta\theta\right)}\mathbf{\hat{z}}\right)
  3. 𝐤 2 = k ( cos ( θ 0 - Δ θ ) 𝐱 ^ + sin ( θ 0 - Δ θ ) 𝐳 ^ ) \mathbf{k}_{2}=k\left(\cos{\left(\theta_{0}-\Delta\theta\right)}\mathbf{\hat{x% }}+\sin{\left(\theta_{0}-\Delta\theta\right)}\mathbf{\hat{z}}\right)
  4. k = ω c n 1 k=\begin{matrix}\frac{\omega}{c}\end{matrix}n_{1}
  5. 𝐤 0 = k ( cos θ 0 𝐱 ^ + sin θ 0 𝐳 ^ ) \mathbf{k}_{0}=k\left(\cos{\theta_{0}}\mathbf{\hat{x}}+\sin{\theta_{0}}\mathbf% {\hat{z}}\right)
  6. ( y , z ) (y,z)

Gosset–Elte_figures.html

  1. 1 i + 1 + 1 j + 1 + 1 k + 1 > 1 \frac{1}{i+1}+\frac{1}{j+1}+\frac{1}{k+1}>1
  2. E ~ 6 {\tilde{E}}_{6}
  3. E ~ 7 {\tilde{E}}_{7}
  4. E ~ 8 {\tilde{E}}_{8}
  5. T ¯ 7 {\bar{T}}_{7}
  6. T ¯ 8 {\bar{T}}_{8}
  7. T ¯ 9 {\bar{T}}_{9}
  8. Q ~ 4 {\tilde{Q}}_{4}
  9. L ¯ 4 {\bar{L}}_{4}

Gödel_numbering_for_sequences.html

  1. a 0 , a n - 1 \langle a_{0},\dots a_{n-1}\rangle
  2. f ( a , i ) = a i f(a,i)=a_{i}
  3. β \beta
  4. β \beta
  5. a 0 , a n - 1 \langle a_{0},\dots a_{n-1}\rangle
  6. β ( a , i ) = a i \beta(a,i)=a_{i}
  7. π \pi
  8. K ( π ( x , y ) ) = x K\left(\pi\left(x,y\right)\right)=x
  9. L ( π ( x , y ) ) = y L\left(\pi\left(x,y\right)\right)=y
  10. rem ( 5 , 3 ) = 2 \mathrm{rem}(5,3)=2
  11. rem ( 7 , 2 ) = 1 \mathrm{rem}(7,2)=1
  12. β \beta
  13. β ( s , i ) = rem ( K ( s ) , ( i + 1 ) L ( s ) + 1 ) \beta(s,i)=\mathrm{rem}\left(K\left(s\right),\left(i+1\right)\cdot L\left(s% \right)+1\right)
  14. β \beta
  15. β ( π ( x 0 , m ) , i ) = rem ( x 0 , ( i + 1 ) m + 1 ) \beta\left(\pi\left(x_{0},m\right),i\right)=\mathrm{rem}\left(x_{0},\left(i+1% \right)\cdot m+1\right)
  16. i < n \forall i<n
  17. m i = ( i + 1 ) m + 1 m_{i}=(i+1)\cdot m+1
  18. β ( π ( x 0 , m ) , i ) = rem ( x 0 , m i ) \beta\left(\pi\left(x_{0},m\right),i\right)=\mathrm{rem}\left(x_{0},m_{i}\right)
  19. m i m_{i}
  20. β \beta
  21. a 0 , a n - 1 a_{0},\dots a_{n-1}
  22. i n ¯ { 0 } ( i m ) \forall i\in\overline{n}\setminus\left\{0\right\}\left(i\mid m\right)
  23. i < n ( a i < m i ) \forall i<n\left(a_{i}<m_{i}\right)
  24. 1 m n - 1 m 1\mid m\land\dots\land n-1\mid m
  25. β \beta
  26. x ~ \tilde{x}
  27. x a i ( mod m i ) x\equiv a_{i}\;\;(\mathop{{\rm mod}}m_{i})
  28. a i = rem ( x ~ , m i ) a_{i}=\mathrm{rem}(\tilde{x},m_{i})
  29. i < n ( a i < m ) \forall i<n\;(a_{i}<m)
  30. m i m_{i}
  31. i n ¯ { 0 } ( i m ) \forall i\in\overline{n}\setminus\left\{0\right\}\left(i\mid m\right)
  32. i < n , j < n ( i j coprime ( m i , m j ) ) \forall i<n,j<n\;\left(i\neq j\rightarrow\mathrm{coprime}\left(m_{i},m_{j}% \right)\right)
  33. i < n \forall i<n
  34. m i = ( i + 1 ) m + 1 m_{i}=(i+1)\cdot m+1
  35. i < n , j < n ( i j ¬ coprime ( m i , m j ) ) \exists i<n,j<n\;\left(i\neq j\land\lnot\mathrm{coprime}\left(m_{i},m_{j}% \right)\right)
  36. i < n , j < n ( i j p Prime ( p m i p m j ) ) \exists i<n,j<n\;\left(i\neq j\land\exists p\in\mathrm{Prime}\;\left(p\mid m_{% i}\land p\mid m_{j}\right)\right)
  37. i < n , j < n , p Prime ( i j p m i p m j ) \exists i<n,j<n,p\in\mathrm{Prime}\;\left(i\neq j\land p\mid m_{i}\land p\mid m% _{j}\right)
  38. p m i p m j p\mid m_{i}\land p\mid m_{j}
  39. p m i - m j p\mid m_{i}-m_{j}
  40. m k m_{k}
  41. m i - m j = ( i - j ) m m_{i}-m_{j}=(i-j)\cdot m
  42. p ( i - j ) m p\mid(i-j)\cdot m
  43. p i - j p m p\mid i-j\lor p\mid m
  44. i n ¯ { 0 } ( i m ) \forall i\in\overline{n}\setminus\left\{0\right\}\left(i\mid m\right)
  45. i < n j < n i j i<n\land j<n\land i\neq j
  46. i - j n ¯ { 0 } i-j\in\overline{n}\setminus\left\{0\right\}
  47. i - j m i-j\mid m
  48. ( A ( A B ) ) B \left(A\land\left(A\rightarrow B\right)\right)\rightarrow B
  49. i - j m i-j\mid m
  50. p i - j p m p\mid i-j\rightarrow p\mid m
  51. p m p\mid m
  52. p m i p\mid m_{i}
  53. p m p\mid m
  54. p m i - ( i + 1 ) m p\mid m_{i}-\left(i+1\right)\cdot m
  55. m i m_{i}
  56. m i - ( i + 1 ) m = 1 m_{i}-\left(i+1\right)\cdot m=1
  57. p 1 p\mid 1
  58. p Prime \exists p\in\mathrm{Prime}
  59. i < n , j < n ( i j coprime ( m i , m j ) ) \forall i<n,j<n\;\left(i\neq j\rightarrow\mathrm{coprime}\left(m_{i},m_{j}% \right)\right)
  60. x a 0 ( mod m 0 ) x\equiv a_{0}\;\;(\mathop{{\rm mod}}m_{0})
  61. \vdots
  62. x a n - 1 ( mod m n - 1 ) x\equiv a_{n-1}\;\;(\mathop{{\rm mod}}m_{n-1})
  63. i < n ( x a i ( mod m i ) ) \forall i<n\;\left(x\equiv a_{i}\;\;(\mathop{{\rm mod}}m_{i})\right)
  64. i < n ( ) \forall i<n\;\left(\dots\right)
  65. i < n ( ) \forall i<n\;\left(\dots\right)
  66. i < n ( \forall i<n(
  67. x 0 x_{0}
  68. m 0 , m n - 1 m_{0},\dots m_{n-1}
  69. x 0 x_{0}
  70. x 0 a i ( mod m i ) x_{0}\equiv a_{i}\;\;(\mathop{{\rm mod}}m_{i})
  71. rem ( x 0 , m i ) = rem ( a i , m i ) \mathrm{rem}\left(x_{0},m_{i}\right)=\mathrm{rem}\left(a_{i},m_{i}\right)
  72. i < n ( a i < m i ) \forall i<n\;\left(a_{i}<m_{i}\right)
  73. a i < m i a_{i}<m_{i}
  74. rem ( a i , m i ) = a i \mathrm{rem}\left(a_{i},m_{i}\right)=a_{i}
  75. rem ( x 0 , m i ) = a i \mathrm{rem}\left(x_{0},m_{i}\right)=a_{i}
  76. β ( π ( x 0 , m ) , i ) = rem ( x 0 , m i ) \beta\left(\pi\left(x_{0},m\right),i\right)=\mathrm{rem}\left(x_{0},m_{i}\right)
  77. β \beta
  78. β ( π ( x 0 , m ) , i ) = a i \beta\left(\pi\left(x_{0},m\right),i\right)=a_{i}
  79. β \beta
  80. a 0 , , a n - 1 s i < n β ( s , i ) = a i \forall a_{0},\dots,a_{n-1}\;\exists s\;\forall i<n\;\beta(s,i)=a_{i}
  81. a 0 , , a n - 1 \left\langle a_{0},\dots,a_{n-1}\right\rangle
  82. β \beta
  83. a 0 , , a n - 1 s i < n β ( s , i ) = a i \forall a_{0},\dots,a_{n-1}\;\exists s\;\forall i<n\;\beta(s,i)=a_{i}
  84. f : n + 1 f:\mathbb{N}^{n+1}\to\mathbb{N}
  85. f ( a 0 , , a n - 1 , s ) = { 0 if i < n ( β ( s , i ) = a i ) 1 if i < n ( β ( s , i ) a i ) f\left(a_{0},\dots,a_{n-1},s\right)=\begin{cases}0&\mathrm{if}\;\forall i<n\;% \left(\beta(s,i)=a_{i}\right)\\ 1&\mathrm{if}\;\exists i<n\;\left(\beta(s,i)\neq a_{i}\right)\end{cases}
  86. f ( a 0 , , a n - 1 , s ) = 0 i < n ( β ( s , i ) = a i ) f\left(a_{0},\dots,a_{n-1},s\right)=0\leftrightarrow\forall i<n\;\left(\beta(s% ,i)=a_{i}\right)
  87. a 0 , , a n - 1 s ( f ( a 0 , , a n - 1 , s ) = 0 ) \forall a_{0},\dots,a_{n-1}\;\exists s\;\left(f\left(a_{0},\dots,a_{n-1},s% \right)=0\right)
  88. β \beta
  89. g : n g:\mathbb{N}^{n}\to\mathbb{N}
  90. a 0 , , a n - 1 μ a . [ i < n ( β ( a , i ) = a i ) ] \left\langle a_{0},\dots,a_{n-1}\right\rangle\longmapsto\mu a.\left[\forall i<% n\;\left(\beta\left(a,i\right)=a_{i}\right)\right]
  91. g : * g:\mathbb{N}^{*}\to\mathbb{N}
  92. a 0 , , a n - 1 , a n μ a . [ a 0 = n i < n ( β ( a , i + 1 ) = a i ) ] \left\langle a_{0},\dots,a_{n-1},a_{n}\right\rangle\longmapsto\mu a.\left[a_{0% }=n\land\forall i<n\;\left(\beta\left(a,i+1\right)=a_{i}\right)\right]
  93. x n ( mod m 0 ) x\equiv n\;\;(\mathop{{\rm mod}}m_{0})

GPS_signals.html

  1. C i ( t ) = A ( t ) B ( t + D i ) C_{i}(t)=A(t)\oplus B(t+D_{i})
  2. C i C_{i}
  3. i i
  4. A A
  5. x x 10 + x 3 + 1 x\to x^{10}+x^{3}+1
  6. B B
  7. x x 10 + x 9 + x 8 + x 6 + x 3 + 1 x\to x^{10}+x^{9}+x^{8}+x^{6}+x^{3}+1
  8. D i D_{i}
  9. i i
  10. \oplus
  11. S ( t ) = P I X I ( t ) cos ( ω t + ϕ 0 ) - P Q X Q ( t ) sin ( ω t + ϕ 0 ) + P Q X Q ( t ) cos ( ω t + ϕ 0 + π 2 ) , S(t)=\sqrt{P\text{I}}X\text{I}(t)\cos(\omega t+\phi_{0})\underbrace{{}-\sqrt{P% \text{Q}}X\text{Q}(t)\sin(\omega t+\phi_{0})}_{+\sqrt{P\text{Q}}X\text{Q}(t)% \cos\left(\omega t+\phi_{0}+\frac{\pi}{2}\right)},
  12. P I \scriptstyle\ P\text{I}\,
  13. P Q \scriptstyle\ P\text{Q}\,
  14. X I ( t ) \scriptstyle\ X\text{I}(t)\,
  15. X Q ( t ) \scriptstyle\ X\text{Q}(t)\,
  16. ( = ± 1 ) \scriptstyle\ (=\pm 1)\,

Gradient_network.html

  1. = =
  2. h j | j S i ( 1 ) i {h_{j}|j\in S_{i}^{(1)}\cup{i}}
  3. G = G=
  4. G G
  5. ( V , F ) (V,F)
  6. G G
  7. R ( l ) = P R(l)=P
  8. = l =l
  9. R ( l ) = 1 N n = 0 N - 1 C l N - 1 - n [ 1 - p ( 1 - p ) ] N - 1 - n - l [ p ( 1 - p ) n ] l ] R(l)=\frac{1}{N}\sum_{n=0}^{N-1}\mathrm{C}^{N-1-n}_{l}[1-p(1-p)]^{N-1-n-l}[p(1% -p)^{n}]^{l}]
  10. R ( l ) l - 1 R(l)\approx l^{-1}
  11. J = 1 - N receive N send h network = R ( 0 ) J=1-\langle\langle\frac{N\text{receive}}{N\text{send}}\rangle_{h}\rangle\text{% network}=R(0)
  12. J ( N , P ) = 1 - ln N N ln ( 1 1 - P ) [ 1 + O ( 1 N ) ] 1. J(N,P)=1-\frac{\ln N}{N\ln(\frac{1}{1-P})}\left[1+O(\frac{1}{N})\right]% \rightarrow 1.

Graph_cuts_in_computer_vision.html

  1. k k
  2. k > 2 , k>2,
  3. x { R , G , B } N x\in\{R,G,B\}^{N}
  4. S R N S\in R^{N}
  5. S { 0 for background , 1 for foreground/object to be detected } N S\in\{0\,\text{ for background},1\,\text{ for foreground/object to be detected% }\}^{N}
  6. E ( x , S , C , λ ) E(x,S,C,\lambda)
  7. E ( x , S , C , λ ) = E color + E coherence E(x,S,C,\lambda)=E_{\rm color}+E_{\rm coherence}
  8. arg min S E ( x , S , C , λ ) {\arg\min}_{S}E(x,S,C,\lambda)
  9. P r ( x | S ) = K Pr(x|S)=K
  10. ( - E ) (-E)
  11. E E
  12. E color E_{\rm color}
  13. E coherence E_{\rm coherence}
  14. E color E_{\rm color}
  15. E coherence E_{\rm coherence}
  16. 24 n + 14 m 24n+14m
  17. n n
  18. m m

Graph_enumeration.html

  1. C n = 2 ( n 2 ) - 1 n k = 1 n - 1 k ( n k ) 2 ( n - k 2 ) C k . C_{n}=2^{n\choose 2}-\frac{1}{n}\sum_{k=1}^{n-1}k{n\choose k}2^{n-k\choose 2}C% _{k}.
  2. 2 n - 4 + 2 ( n - 4 ) / 2 . 2^{n-4}+2^{\lfloor(n-4)/2\rfloor}.

Graph_partition.html

  1. max i | V i | ( 1 + ε ) | V | k . \max_{i}|V_{i}|\leq(1+\varepsilon)\left\lceil\frac{|V|}{k}\right\rceil.

Graph_product.html

  1. \sim
  2. ≁ \not\sim
  3. u 1 u_{1}
  4. u 2 u_{2}
  5. v 1 v_{1}
  6. v 2 v_{2}
  7. G H G\square H
  8. u 1 u_{1}
  9. v 1 v_{1}
  10. u 2 u_{2}
  11. \sim
  12. v 2 v_{2}
  13. u 1 u_{1}
  14. \sim
  15. v 1 v_{1}
  16. u 2 u_{2}
  17. v 2 v_{2}
  18. G V 1 , E 1 H V 2 , E 2 J ( V 1 V 2 ) , ( E 2 V 1 + E 1 V 2 ) G_{V_{1},E_{1}}\square H_{V_{2},E_{2}}\rightarrow J_{(V_{1}V_{2}),(E_{2}V_{1}+% E_{1}V_{2})}
  19. G × H G\times H
  20. u 1 u_{1}
  21. \sim
  22. v 1 v_{1}
  23. u 2 u_{2}
  24. \sim
  25. v 2 v_{2}
  26. G V 1 , E 1 × H V 2 , E 2 J ( V 1 V 2 ) , ( 2 E 1 E 2 ) G_{V_{1},E_{1}}\times H_{V_{2},E_{2}}\rightarrow J_{(V_{1}V_{2}),(2E_{1}E_{2})}
  27. G H G\cdot H
  28. G [ H ] G[H]
  29. G V 1 , E 1 H V 2 , E 2 J ( V 1 V 2 ) , ( E 2 V 1 + E 1 V 2 2 ) G_{V_{1},E_{1}}\cdot H_{V_{2},E_{2}}\rightarrow J_{(V_{1}V_{2}),(E_{2}V_{1}+E_% {1}V_{2}^{2})}
  30. G H G\boxtimes H
  31. G V 1 , E 1 H V 2 , E 2 J ( V 1 V 2 ) , ( V 1 E 2 + V 2 E 1 + 2 E 1 E 2 ) G_{V_{1},E_{1}}\boxtimes H_{V_{2},E_{2}}\rightarrow J_{(V_{1}V_{2}),(V_{1}E_{2% }+V_{2}E_{1}+2E_{1}E_{2})}
  32. G * H G*H
  33. ( u 1 v 1 and u 2 v 2 ) (u_{1}\sim v_{1}\,\text{ and }u_{2}\sim v_{2})
  34. ( u 1 ≁ v 1 and u 2 ≁ v 2 ) (u_{1}\not\sim v_{1}\,\text{ and }u_{2}\not\sim v_{2})
  35. G V 1 , E 1 H V 2 , E 2 J ( V 1 V 2 ) , ( E 2 V 1 + E 1 ) G_{V_{1},E_{1}}\cdot H_{V_{2},E_{2}}\rightarrow J_{(V_{1}V_{2}),(E_{2}V_{1}+E_% {1})}
  36. G H G\ltimes H
  37. ( u 1 = v 1 ) (u_{1}=v_{1})
  38. ( u 1 v 1 and u 2 ≁ v 2 ) (u_{1}\sim v_{1}\,\text{ and }u_{2}\not\sim v_{2})
  39. K 2 K_{2}
  40. K 2 K 2 K_{2}\square K_{2}
  41. K 2 × K 2 K_{2}\times K_{2}
  42. K 2 K 2 K_{2}\boxtimes K_{2}
  43. K 2 K 2 K_{2}\square K_{2}
  44. K 2 K 2 K_{2}\boxtimes K_{2}
  45. G [ H ] G[H]

Graph_toughness.html

  1. G G
  2. t t
  3. t t
  4. k > 1 k>1
  5. G G
  6. k k
  7. t k tk
  8. 1 1
  9. t t
  10. t t
  11. k k
  12. k + 1 k+1
  13. 1 2 \frac{1}{2}
  14. k k
  15. k k
  16. k k
  17. 1 1
  18. t t
  19. k = 2 k=2
  20. 2 t - 1 2t-1
  21. t t
  22. 2 t 2t
  23. 1 1
  24. 1 1
  25. t t
  26. t t
  27. t = 2 t=2

Gravitational-wave_astronomy.html

  1. 10 - 25 10^{-25}

Gravitational-wave_observatory.html

  1. h 10 - 18 h\approx 10^{-18}
  2. h 5 × 10 - 22 h\approx 5\times 10^{-22}
  3. h 2 × 10 - 13 / 𝐻𝑧 h\sim{2\times 10^{-13}/\sqrt{\mathit{Hz}}}
  4. h 2 × 10 - 17 / 𝐻𝑧 h\sim{2\times 10^{-17}/\sqrt{\mathit{Hz}}}
  5. h 2 × 10 - 20 / 𝐻𝑧 h\sim{2\times 10^{-20}/\sqrt{\mathit{Hz}}}

Gravitational_two-body_problem.html

  1. μ = G ( m 1 + m 2 ) \mu=G(m_{1}+m_{2})
  2. m 1 m_{1}
  3. m 2 m_{2}
  4. u ( θ ) 1 r ( θ ) = μ h 2 ( 1 + e cos ( θ - θ 0 ) ) u(\theta)\equiv\frac{1}{r(\theta)}=\frac{\mu}{h^{2}}(1+e\cos(\theta-\theta_{0}))
  5. m 2 / ( m 1 + m 2 ) m_{2}/(m_{1}+m_{2})
  6. - m 1 / ( m 1 + m 2 ) -m_{1}/(m_{1}+m_{2})
  7. r v 2 = r 3 ω 2 = 4 π 2 r 3 / T 2 = μ rv^{2}=r^{3}\omega^{2}=4\pi^{2}r^{3}/T^{2}=\mu
  8. 4 π 2 a 3 / T 2 = μ 4\pi^{2}a^{3}/T^{2}=\mu
  9. a 3 / T 2 = M a^{3}/T^{2}=M
  10. r v 2 rv^{2}
  11. 2 μ 2\mu
  12. ϵ \epsilon
  13. μ = 2 a ϵ \mu=2a\mid\epsilon\mid
  14. 1 2 2 {1\over 2}\sqrt{2}
  15. 2 3 1.26 \sqrt[3]{2}\approx 1.26
  16. 2 3 \sqrt[3]{2}

Gravity_gradiometry.html

  1. G z z = g z z g z ( z + l 2 ) - g z ( z - l 2 ) l G_{zz}={\partial g_{z}\over\partial z}\approx{g_{z}\left(z+\tfrac{l}{2}\right)% -g_{z}\left(z-\tfrac{l}{2}\right)\over l}
  2. G M z ( r 2 + z 2 ) 3 / 2 × 10 5 [ mGal ] {GM\,z\over\left(r^{2}+z^{2}\right)^{3/2}}\times 10^{5}\;\left[\,\text{mGal}\right]
  3. G M ( r 2 - 2 z 2 ) ( r 2 + z 2 ) 5 / 2 × 10 9 [ E ] {GM\left(r^{2}-2z^{2}\right)\over\left(r^{2}+z^{2}\right)^{5/2}}\times 10^{9}% \;\left[\,\text{E}\right]
  4. G M z 2 × 10 5 {GM\over z^{2}}\times 10^{5}
  5. 2 G M z 3 × 10 9 {2GM\over z^{3}}\times 10^{9}
  6. 1.53 z 1.53\,z
  7. z \approx z
  8. 3.07 z 3.07\,z
  9. 2 z 2\,z

Green_measure.html

  1. d X t = b ( X t ) d t + σ ( X t ) d B t . \mathrm{d}X_{t}=b(X_{t})\,\mathrm{d}t+\sigma(X_{t})\,\mathrm{d}B_{t}.
  2. L X = i b i x i + 1 2 i , j ( σ σ ) i , j 2 x i x j . L_{X}=\sum_{i}b_{i}\frac{\partial}{\partial x_{i}}+\frac{1}{2}\sum_{i,j}\big(% \sigma\sigma^{\top}\big)_{i,j}\frac{\partial^{2}}{\partial x_{i}\,\partial x_{% j}}.
  3. τ D := inf { t 0 | X t D } . \tau_{D}:=\inf\{t\geq 0|X_{t}\not\in D\}.
  4. G ( x , H ) = 𝐄 x [ 0 τ D χ H ( X s ) d s ] , G(x,H)=\mathbf{E}^{x}\left[\int_{0}^{\tau_{D}}\chi_{H}(X_{s})\,\mathrm{d}s% \right],
  5. D f ( y ) G ( x , d y ) = 𝐄 x [ 0 τ D f ( X s ) d s ] \int_{D}f(y)\,G(x,\mathrm{d}y)=\mathbf{E}^{x}\left[\int_{0}^{\tau_{D}}f(X_{s})% \,\mathrm{d}s\right]
  6. G ( x , H ) = H G ( x , y ) d y , G(x,H)=\int_{H}G(x,y)\,\mathrm{d}y,
  7. f ( x ) = 𝐄 x [ f ( X τ D ) ] - D L X f ( y ) G ( x , d y ) . f(x)=\mathbf{E}^{x}\big[f\big(X_{\tau_{D}}\big)\big]-\int_{D}L_{X}f(y)\,G(x,% \mathrm{d}y).
  8. f ( x ) = - D L X f ( y ) G ( x , d y ) . f(x)=-\int_{D}L_{X}f(y)\,G(x,\mathrm{d}y).
  9. 𝐄 x [ f ( X τ D ) ] \mathbf{E}^{x}\big[f\big(X_{\tau_{D}}\big)\big]
  10. = f ( x ) + 𝐄 x [ 0 τ D L X f ( X s ) d s ] =f(x)+\mathbf{E}^{x}\left[\int_{0}^{\tau_{D}}L_{X}f(X_{s})\,\mathrm{d}s\right]
  11. = f ( x ) + D L X f ( y ) G ( x , d y ) . =f(x)+\int_{D}L_{X}f(y)\,G(x,\mathrm{d}y).

Gromov's_inequality_for_complex_projective_space.html

  1. stsys 2 n n ! vol 2 n ( n ) \mathrm{stsys}_{2}{}^{n}\leq n!\;\mathrm{vol}_{2n}(\mathbb{CP}^{n})
  2. stsys 2 \operatorname{stsys_{2}}
  3. 1 n \mathbb{CP}^{1}\subset\mathbb{CP}^{n}
  4. , , \mathbb{R,C,H}
  5. stsys 2 2 2 v o l 4 ( 2 ) \mathrm{stsys}_{2}{}^{2}\leq 2\mathrm{vol}_{4}(\mathbb{CP}^{2})
  6. 2 \mathbb{RP}^{2}
  7. 2 \mathbb{HP}^{2}
  8. 2 \mathbb{HP}^{2}
  9. stsys 4 / 2 vol 8 \mathrm{stsys}_{4}{}^{2}/\mathrm{vol}_{8}

Gromov's_systolic_inequality_for_essential_manifolds.html

  1. ( sys π 1 ( M ) ) n C n vol ( M ) , \left(\operatorname{sys\pi}_{1}(M)\right)^{n}\leq C_{n}\operatorname{vol}(M),
  2. sys π 1 6 FillRad ( X ) , \mathrm{sys\pi}_{1}\leq 6\;\mathrm{FillRad}(X),
  3. FillRad ( X ) C n vol n ( X ) 1 n , \mathrm{FillRad}(X)\leq C_{n}\mathrm{vol}_{n}{}^{\tfrac{1}{n}}(X),

Gross_Production_Average.html

  1. ( 1.8 ) O B P + S L G 4 \frac{{(1.8)OBP}+SLG}{4}

Grothendieck_inequality.html

  1. | i , j a i j s i t j | 1 \left|\sum_{i,j}a_{ij}s_{i}t_{j}\right|\leq 1
  2. | i , j a i j S i , T j | k \left|\sum_{i,j}a_{ij}\langle S_{i},T_{j}\rangle\right|\leq k
  3. 1.57 π 2 k \R sinh ( π 2 ) 2.3 1.57\approx\frac{\pi}{2}\leq k_{\R}\leq\mathrm{sinh}(\frac{\pi}{2})\approx 2.3
  4. π 2 ln ( 1 + 2 ) \frac{\pi}{2\ln(1+\sqrt{2})}

Ground_dipole.html

  1. P = π 2 f 2 I 2 L 2 2 c 2 h σ P=\frac{\pi^{2}f^{2}I^{2}L^{2}}{2c^{2}h\sigma}\,

Groundwater_discharge.html

  1. Q = d h d l K A Q=\frac{dh}{dl}KA

Guitar_harmonics.html

  1. F 1.. n = log s m m - n F_{1..n}=\log_{s}{m\over{m-n}}
  2. s = 2 1 / 12 s=2^{1/12}

Gullstrand–Painlevé_coordinates.html

  1. d τ 2 = ( 1 - 2 M r ) d t 2 - d r 2 ( 1 - 2 M r ) - r 2 d θ 2 - r 2 sin 2 θ d ϕ 2 d\tau^{2}=\left(1-\frac{2M}{r}\right)\,dt^{2}-\frac{dr^{2}}{\left(1-\frac{2M}{% r}\right)}-r^{2}\,d\theta^{2}-r^{2}\sin^{2}\theta\,d\phi^{2}\,
  2. t r = t - f ( r ) t_{r}=t-f(r)
  3. d τ 2 = ( 1 - 2 M r ) d t r 2 + 2 ( 1 - 2 M r ) f d t r d r - ( 1 1 - 2 M r - ( 1 - 2 M r ) f 2 ) d r 2 - r 2 d θ 2 - r 2 sin 2 θ d ϕ 2 d\tau^{2}=\left(1-\frac{2M}{r}\right)\,dt_{r}^{2}+2\left(1-\frac{2M}{r}\right)% \,f^{\prime}dt_{r}dr-\left({\frac{1}{1-\frac{2M}{r}}}-\left(1-\frac{2M}{r}% \right)\,{f^{\prime}}^{2}\right)dr^{2}-r^{2}\,d\theta^{2}-r^{2}\sin^{2}\theta% \,d\phi^{2}\,
  4. f = d f ( r ) d r f^{\prime}={\frac{df(r)}{dr}}
  5. d r 2 dr^{2}
  6. f = - 1 1 - 2 M r 2 M r f^{\prime}=-{\frac{1}{1-{\frac{2M}{r}}}}\sqrt{\frac{2M}{r}}
  7. d τ 2 = ( 1 - 2 M r ) d t r 2 - 2 2 M r d t r d r - d r 2 - r 2 d θ 2 - r 2 sin 2 θ d ϕ 2 d\tau^{2}=\left(1-\frac{2M}{r}\right)\,dt_{r}^{2}-2\sqrt{\frac{2M}{r}}dt_{r}dr% -dr^{2}-r^{2}\,d\theta^{2}-r^{2}\sin^{2}\theta\,d\phi^{2}\,
  8. t r t_{r}
  9. - r 4 sin ( θ ) 2 -r^{4}\sin(\theta)^{2}
  10. f ( r ) = - 2 M r 1 - 2 M r d r = 2 M ( - 2 y + ln ( y + 1 y - 1 ) ) f(r)=-\int{\frac{\sqrt{\frac{2M}{r}}}{1-{\frac{2M}{r}}}}dr=2M\left(-2y+\ln% \left({\frac{y+1}{y-1}}\right)\right)
  11. y = r 2 M y=\sqrt{\frac{r}{2M}}
  12. d r d t = - ( 1 - 2 M r ) 2 M r . \frac{dr}{dt}=-\left(1-\frac{2M}{r}\right)\sqrt{\frac{2M}{r}}.\,
  13. d r d t r = β = - 2 M r . \frac{dr}{dt_{r}}=\beta=-\sqrt{\frac{2M}{r}}.\,
  14. r < 2 M r<2M\,\!
  15. 0 T r d t = - 2 M 0 ( 2 M r ) - 1 d r . \int_{0}^{T_{r}}\,dt=-\int_{2M}^{0}\left(\sqrt{\frac{2M}{r}}\right)^{-1}\,dr.\,
  16. T r = 4 3 M . T_{r}=\frac{4}{3}M.\,
  17. d τ 2 = d t r 2 [ 1 - 2 M r + 2 2 M r 2 M r - 2 M r 2 ] = d t r 2 d\tau^{2}=dt_{r}^{2}\left[1-{\frac{2M}{r}}+2\sqrt{\frac{2M}{r}}{\sqrt{\frac{2M% }{r}}}-{\sqrt{\frac{2M}{r}}}^{2}\right]=dt_{r}^{2}
  18. t r t_{r}
  19. d τ = 0 d\tau=0\,\!
  20. 0 = ( d r + ( 1 + 2 M r ) d t r ) ( d r - ( 1 - 2 M r ) d t r ) , 0=\left(dr+\left(1+\sqrt{\frac{2M}{r}}\ \right)\,dt_{r}\right)\left(dr-\left(1% -\sqrt{\frac{2M}{r}}\ \right)\,dt_{r}\right),\,
  21. d r d t r = \plusmn 1 - 2 M r . \frac{dr}{dt_{r}}=\plusmn 1-\sqrt{\frac{2M}{r}}.\,
  22. r r\to\infty\,\!
  23. d r d t r = \plusmn 1 \frac{dr}{dt_{r}}=\plusmn 1\,\!
  24. r = 2 M r=2M\,\!
  25. d r d t r = 0 \frac{dr}{dt_{r}}=0\,\!
  26. ( d r d t r ) raindrop ( d r d t r ) light = 2 M r 1 + 2 M r < 1 \frac{\left(\dfrac{dr}{dt_{r}}\right)\text{raindrop}}{\left(\dfrac{dr}{dt_{r}}% \right)\text{light}}=\frac{\sqrt{\dfrac{2M}{r}}}{1+\sqrt{\dfrac{2M}{r}}}<1\,\!
  27. 0 T r d t = - 2 M 0 ( 2 M r + 1 ) - 1 d r . \int_{0}^{T_{r}}\,dt=-\int_{2M}^{0}\left(\sqrt{\frac{2M}{r}}+1\right)^{-1}\,dr% .\,\!
  28. T r = 4 M ln 2 - 2 M 0.77 M . T_{r}=4M\ln 2-2M\approx 0.77M.\,\!
  29. cos s y m b o l Φ r = d r r d t r = 2 M r + cos s y m b o l Φ s 1 + 2 M r cos s y m b o l Φ s , \cos symbol{\Phi}_{r}=\frac{dr_{r}}{dt_{r}}=\frac{\sqrt{\dfrac{2M}{r}}+\cos symbol% {\Phi}_{s}}{1+\sqrt{\dfrac{2M}{r}}\ \cos symbol{\Phi}_{s}},\,
  30. cos s y m b o l Φ s = d r s d t s = ± 1 - ( 1 - 2 M r ) I 2 r 2 , \cos symbol{\Phi}_{s}=\frac{dr_{s}}{dt_{s}}=\pm\sqrt{1-\left(1-\frac{2M}{r}% \right)\frac{\mathit{I}^{2}}{r^{2}}},\,\!
  31. ϕ = I r 0 d r r 2 cos s y m b o l Φ s , \phi=\mathit{I}\int_{r_{0}}^{\infty}\frac{dr}{r^{2}\ \cos symbol{\Phi}_{s}},\,\!
  32. s y m b o l Φ r , s y m b o l Φ s symbol{\Phi}_{r},\ symbol{\Phi}_{s}\,\!
  33. ϕ \ \phi\,\!
  34. I \mathit{I}\,\!
  35. θ = π 2 \theta=\frac{\pi}{2}\,\!
  36. I \mathit{I}\,\!
  37. r 0 r_{0}\,\!
  38. s y m b o l Φ r 0 \ symbol{\Phi}_{r0}\,\!
  39. ϕ \ \phi\,\!
  40. d r dr\,\!
  41. r 0 r_{0}\,\!
  42. r 0 r\rightarrow 0\,\!
  43. cos s y m b o l Φ r r 2 M \cos symbol{\Phi}_{r}\rightarrow\sqrt{\frac{r}{2M}}\,\!

Gunduz_Caginalp.html

  1. f f_{\infty}
  2. f f_{\infty}
  3. Ω \Omega
  4. | Ω | |\Omega|
  5. | Ω | |\partial\Omega|
  6. F ( Ω ) = | Ω | f + | Ω | f x + F(\Omega)=|\Omega|f_{\infty}+|\partial\Omega|f_{x}+...
  7. f x f_{x}
  8. | Ω | |\Omega|
  9. | Ω | |\partial\Omega|
  10. ϕ \phi
  11. ϕ \phi
  12. ε \varepsilon
  13. ϕ \phi
  14. ( ϕ , T ) (\phi,T)
  15. C P T t + l 2 ϕ = K Δ T α ε 2 ϕ t = ε 2 Δ ϕ + 1 2 ( ϕ - ϕ 3 ) + ε [ s ] E 3 σ ( T - T E ) \begin{array}[]{lcl}C_{P}T_{t}+\frac{l}{2}\phi=K\Delta T\\ \alpha\varepsilon^{2}\phi_{t}=\varepsilon^{2}\Delta\phi+\frac{1}{2}(\phi-\phi^% {3})+\frac{\varepsilon[s]_{E}}{3\sigma}(T-T_{E})\end{array}
  16. C P , l , α , σ , [ s ] E C_{P},l,\alpha,\sigma,[s]_{E}
  17. ε \varepsilon
  18. ε \varepsilon
  19. d 0 d_{0}
  20. ε \varepsilon
  21. d 0 d_{0}
  22. d 0 d_{0}
  23. ε \varepsilon
  24. ε 2 \varepsilon^{2}
  25. \ast

Günter_Nimtz.html

  1. n := ϵ r μ r n:=\sqrt{\epsilon_{r}\mu_{r}}

H-vector.html

  1. f ( Δ ) = ( f - 1 , f 0 , , f d - 1 ) . f(\Delta)=(f_{-1},f_{0},\ldots,f_{d-1}).
  2. h k = i = 0 k ( - 1 ) k - i ( d - i k - i ) f i - 1 . h_{k}=\sum_{i=0}^{k}(-1)^{k-i}{\left({{d-i}\atop{k-i}}\right)}f_{i-1}.
  3. h ( Δ ) = ( h 0 , h 1 , , h d ) h(\Delta)=(h_{0},h_{1},\ldots,h_{d})
  4. i = 0 d f i - 1 ( t - 1 ) d - i = k = 0 d h k t d - k . \sum_{i=0}^{d}f_{i-1}(t-1)^{d-i}=\sum_{k=0}^{d}h_{k}t^{d-k}.
  5. P R ( t ) = i = 0 d f i - 1 t i ( 1 - t ) i = h 0 + h 1 t + + h d t d ( 1 - t ) d . P_{R}(t)=\sum_{i=0}^{d}\frac{f_{i-1}t^{i}}{(1-t)^{i}}=\frac{h_{0}+h_{1}t+% \cdots+h_{d}t^{d}}{(1-t)^{d}}.
  6. h k = h d - k . h_{k}=h_{d-k}.
  7. h k = dim IH 2 k ( X , ) h_{k}=\operatorname{dim}_{\mathbb{Q}}\operatorname{IH}^{2k}(X,\mathbb{Q})
  8. | | : P { 0 , 1 , , n } |\cdot|:P\to\{0,1,\ldots,n\}
  9. P S = { x P : | x | S } . P_{S}=\{x\in P:|x|\in S\}.
  10. S α P ( S ) S\mapsto\alpha_{P}(S)
  11. S β P ( S ) , β P ( S ) = T S ( - 1 ) | S | - | T | α P ( S ) S\mapsto\beta_{P}(S),\quad\beta_{P}(S)=\sum_{T\subseteq S}(-1)^{|S|-|T|}\alpha% _{P}(S)
  12. α P ( S ) = T S β P ( T ) . \alpha_{P}(S)=\sum_{T\subseteq S}\beta_{P}(T).
  13. f i - 1 ( Δ ( P ) ) = | S | = i α P ( S ) , h i ( Δ ( P ) ) = | S | = i β P ( S ) . f_{i-1}(\Delta(P))=\sum_{|S|=i}\alpha_{P}(S),\quad h_{i}(\Delta(P))=\sum_{|S|=% i}\beta_{P}(S).
  14. u S = u 1 u n , u i = a for i S , u i = b for i S . u_{S}=u_{1}\cdots u_{n},\quad u_{i}=a\,\text{ for }i\notin S,u_{i}=b\,\text{ % for }i\in S.
  15. Ψ P ( a , b ) = S β P ( S ) u S . \Psi_{P}(a,b)=\sum_{S}\beta_{P}(S)u_{S}.
  16. Ψ P ( a , a + b ) = S α P ( S ) u S . \Psi_{P}(a,a+b)=\sum_{S}\alpha_{P}(S)u_{S}.
  17. Ψ P ( a , b ) = Φ P ( a + b , a b + b a ) . \Psi_{P}(a,b)=\Phi_{P}(a+b,ab+ba).

Haar-like_features.html

  1. sum = I ( C ) + I ( A ) - I ( B ) - I ( D ) . \,\text{sum}=I(C)+I(A)-I(B)-I(D).\,
  2. A , B , C , D A,B,C,D
  3. I I

Hafner–Sarnak–McCurley_constant.html

  1. D ( n ) = k = 1 { 1 - [ 1 - j = 1 n ( 1 - p k - j ) ] 2 } , D(n)=\prod_{k=1}^{\infty}\left\{1-\left[1-\prod_{j=1}^{n}(1-p_{k}^{-j})\right]% ^{2}\right\},

Hall's_universal_group.html

  1. Γ 0 \Gamma_{0}
  2. 3 \geq 3
  3. Γ 1 \Gamma_{1}
  4. S Γ 0 S_{\Gamma_{0}}
  5. Γ 0 \Gamma_{0}
  6. Γ 2 \Gamma_{2}
  7. S Γ 1 = S S Γ 0 S_{\Gamma_{1}}=S_{S_{\Gamma_{0}}}\,
  8. x g x x\mapsto gx\,
  9. Γ 0 Γ 1 Γ 2 . \Gamma_{0}\hookrightarrow\Gamma_{1}\hookrightarrow\Gamma_{2}\hookrightarrow% \cdots.\,
  10. Γ i \Gamma_{i}
  11. Γ i U \Gamma_{i}\subset U
  12. Γ i + 1 = S Γ i \Gamma_{i+1}=S_{\Gamma_{i}}
  13. Γ i \Gamma_{i}
  14. G U G\hookrightarrow U

Halley's_method.html

  1. x n + 1 = x n - 2 f ( x n ) f ( x n ) 2 [ f ( x n ) ] 2 - f ( x n ) f ′′ ( x n ) x_{n+1}=x_{n}-\frac{2f(x_{n})f^{\prime}(x_{n})}{2{[f^{\prime}(x_{n})]}^{2}-f(x% _{n})f^{\prime\prime}(x_{n})}
  2. | x n + 1 - a | K | x n - a | 3 , for some K > 0. |x_{n+1}-a|\leq K\cdot{|x_{n}-a|}^{3},\,\text{ for some }K>0.
  3. f ( x n ) / f ( x n ) f(x_{n})/f^{\prime}(x_{n})
  4. f ′′ ( x n ) / f ( x n ) f^{\prime\prime}(x_{n})/f^{\prime}(x_{n})
  5. x n + 1 = x n - f ( x n ) f ( x n ) [ 1 - f ( x n ) f ( x n ) f ′′ ( x n ) 2 f ( x n ) ] - 1 x_{n+1}=x_{n}-\frac{f(x_{n})}{f^{\prime}(x_{n})}\left[1-\frac{f(x_{n})}{f^{% \prime}(x_{n})}\cdot\frac{f^{\prime\prime}(x_{n})}{2f^{\prime}(x_{n})}\right]^% {-1}
  6. x n + 1 = x n - f ( x n ) f ( x n ) - f ( x n ) f ′′ ( x n ) 2 f ( x n ) x_{n+1}=x_{n}-\frac{f(x_{n})}{f^{\prime}(x_{n})-\frac{f(x_{n})f^{\prime\prime}% (x_{n})}{2f^{\prime}(x_{n})}}
  7. g ( x ) = f ( x ) | f ( x ) | . g(x)=\frac{f(x)}{\sqrt{|f^{\prime}(x)|}}.
  8. x n + 1 = x n - g ( x n ) g ( x n ) x_{n+1}=x_{n}-\frac{g(x_{n})}{g^{\prime}(x_{n})}
  9. g ( x ) = 2 [ f ( x ) ] 2 - f ( x ) f ′′ ( x ) 2 f ( x ) | f ( x ) | , g^{\prime}(x)=\frac{2[f^{\prime}(x)]^{2}-f(x)f^{\prime\prime}(x)}{2f^{\prime}(% x)\sqrt{|f^{\prime}(x)|}},
  10. 0 = f ( a ) = f ( x n ) + f ( x n ) ( a - x n ) + f ′′ ( x n ) 2 ( a - x n ) 2 + f ′′′ ( ξ ) 6 ( a - x n ) 3 0=f(a)=f(x_{n})+f^{\prime}(x_{n})(a-x_{n})+\frac{f^{\prime\prime}(x_{n})}{2}(a% -x_{n})^{2}+\frac{f^{\prime\prime\prime}(\xi)}{6}(a-x_{n})^{3}
  11. 0 = f ( a ) = f ( x n ) + f ( x n ) ( a - x n ) + f ′′ ( η ) 2 ( a - x n ) 2 , 0=f(a)=f(x_{n})+f^{\prime}(x_{n})(a-x_{n})+\frac{f^{\prime\prime}(\eta)}{2}(a-% x_{n})^{2},
  12. 2 f ( x n ) 2f^{\prime}(x_{n})
  13. f ′′ ( x n ) ( a - x n ) f^{\prime\prime}(x_{n})(a-x_{n})
  14. 0 \displaystyle 0
  15. f ( x n ) f ′′ ( x n ) ( a - x n ) 2 f^{\prime}(x_{n})f^{\prime\prime}(x_{n})(a-x_{n})^{2}
  16. 0 = 2 f ( x n ) f ( x n ) + ( 2 [ f ( x n ) ] 2 - f ( x n ) f ′′ ( x n ) ) ( a - x n ) + ( f ( x n ) f ′′′ ( ξ ) 3 - f ′′ ( x n ) f ′′ ( η ) 2 ) ( a - x n ) 3 . 0=2f(x_{n})f^{\prime}(x_{n})+\left(2[f^{\prime}(x_{n})]^{2}-f(x_{n})f^{\prime% \prime}(x_{n})\right)(a-x_{n})+\left(\frac{f^{\prime}(x_{n})f^{\prime\prime% \prime}(\xi)}{3}-\frac{f^{\prime\prime}(x_{n})f^{\prime\prime}(\eta)}{2}\right% )(a-x_{n})^{3}.
  17. 2 [ f ( x n ) ] 2 - f ( x n ) f ′′ ( x n ) 2[f^{\prime}(x_{n})]^{2}-f(x_{n})f^{\prime\prime}(x_{n})
  18. a - x n = - 2 f ( x n ) f ( x n ) 2 [ f ( x n ) ] 2 - f ( x n ) f ′′ ( x n ) - 2 f ( x n ) f ′′′ ( ξ ) - 3 f ′′ ( x n ) f ′′ ( η ) 6 ( 2 [ f ( x n ) ] 2 - f ( x n ) f ′′ ( x n ) ) ( a - x n ) 3 . a-x_{n}=\frac{-2f(x_{n})f^{\prime}(x_{n})}{2[f^{\prime}(x_{n})]^{2}-f(x_{n})f^% {\prime\prime}(x_{n})}-\frac{2f^{\prime}(x_{n})f^{\prime\prime\prime}(\xi)-3f^% {\prime\prime}(x_{n})f^{\prime\prime}(\eta)}{6(2[f^{\prime}(x_{n})]^{2}-f(x_{n% })f^{\prime\prime}(x_{n}))}(a-x_{n})^{3}.
  19. a - x n + 1 = - 2 f ( x n ) f ′′′ ( ξ ) - 3 f ′′ ( x n ) f ′′ ( η ) 12 [ f ( x n ) ] 2 - 6 f ( x n ) f ′′ ( x n ) ( a - x n ) 3 . a-x_{n+1}=-\frac{2f^{\prime}(x_{n})f^{\prime\prime\prime}(\xi)-3f^{\prime% \prime}(x_{n})f^{\prime\prime}(\eta)}{12[f^{\prime}(x_{n})]^{2}-6f(x_{n})f^{% \prime\prime}(x_{n})}(a-x_{n})^{3}.
  20. - 2 f ( a ) f ′′′ ( a ) - 3 f ′′ ( a ) f ′′ ( a ) 12 [ f ( a ) ] 2 . -\frac{2f^{\prime}(a)f^{\prime\prime\prime}(a)-3f^{\prime\prime}(a)f^{\prime% \prime}(a)}{12[f^{\prime}(a)]^{2}}.
  21. | a - x n + 1 | K | a - x n | 3 |a-x_{n+1}|\leq K|a-x_{n}|^{3}
  22. Δ x i + 1 = 3 ( f ′′ ) 2 - 2 f f ′′′ 12 ( f ) 2 ( Δ x i ) 3 + O [ Δ x i ] 4 , Δ x i x i - a . \Delta x_{i+1}=\frac{3(f^{\prime\prime})^{2}-2f^{\prime}f^{\prime\prime\prime}% }{12(f^{\prime})^{2}}(\Delta x_{i})^{3}+O[\Delta x_{i}]^{4},\qquad\Delta x_{i}% \triangleq x_{i}-a.

Hamada's_equation.html

  1. β L = β U [ 1 + ( 1 - T ) ϕ ] ( 1 ) \beta_{L}=\beta_{U}[1+(1-T)\phi]\qquad(1)
  2. ϕ \phi\,\!
  3. β i = c o v ( r i , r M ) σ 2 ( r M ) ( 2 ) \beta_{i}=\frac{cov(r_{i},r_{M})}{\sigma^{2}(r_{M})}\qquad(2)
  4. r E , U = E B I T ( 1 - T ) - Δ I C E U ( 3 ) r_{E,U}=\frac{EBIT(1-T)-\Delta IC}{E_{U}}\qquad(3)
  5. r E , L = E B I T ( 1 - T ) - Δ I C + D e b t n e w - I n t e r e s t E L ( 4 ) r_{E,L}=\frac{EBIT(1-T)-\Delta IC+Debt_{new}-Interest}{E_{L}}\qquad(4)
  6. Δ I C \Delta IC
  7. β U = c o v ( E B I T ( 1 - T ) E U , r M ) σ 2 ( r M ) \beta_{U}=\frac{cov(\frac{EBIT(1-T)}{E_{U}},r_{M})}{\sigma^{2}(r_{M})}
  8. β L = c o v ( E B I T ( 1 - T ) E L , r M ) σ 2 ( r M ) \beta_{L}=\frac{cov(\frac{EBIT(1-T)}{E_{L}},r_{M})}{\sigma^{2}(r_{M})}
  9. E L β L = E U β U β L = E U E L β U E_{L}\beta_{L}=E_{U}\beta_{U}\rightarrow\beta_{L}=\frac{E_{U}}{E_{L}}\beta_{U}
  10. V L = V U = V A = E U = E L | T = 0 + D V_{L}=V_{U}=V_{A}=E_{U}=E_{L|T=0}+D
  11. i D r D T ( 1 + r D ) i = D r D T r D = D T \sum_{i}\frac{Dr_{D}T}{(1+r_{D})^{i}}=\frac{Dr_{D}T}{r_{D}}=DT
  12. V L = V { U , A } + D T = E U + D T = E L | T = 0 + D + D T = E L | T > 0 + D V_{L}=V_{\{U,A\}}+DT=E_{U}+DT=E_{L|T=0}+D+DT=E_{L|T>0}+D
  13. E U = E L | T > 0 + D - D T E_{U}=E_{L|T>0}+D-DT
  14. β L = E L + D - D T E L β U = [ 1 + D E L ( 1 - T ) ] β U = β U [ 1 + ( 1 - T ) ϕ ] \beta_{L}=\frac{E_{L}+D-DT}{E_{L}}\beta_{U}=\left[1+\frac{D}{E_{L}}(1-T)\right% ]\beta_{U}=\beta_{U}[1+(1-T)\phi]

Hamming(7,4).html

  1. p 1 p_{1}
  2. p 2 p_{2}
  3. d 1 d_{1}
  4. p 3 p_{3}
  5. d 2 d_{2}
  6. d 3 d_{3}
  7. d 4 d_{4}
  8. p 1 p_{1}
  9. p 2 p_{2}
  10. p 3 p_{3}
  11. d 1 d_{1}
  12. d 2 d_{2}
  13. d 3 d_{3}
  14. d 4 d_{4}
  15. p 1 p_{1}
  16. p 2 p_{2}
  17. p 3 p_{3}
  18. 𝐆 := ( 1 1 0 1 1 0 1 1 1 0 0 0 0 1 1 1 0 1 0 0 0 0 1 0 0 0 0 1 ) , 𝐇 := ( 1 0 1 0 1 0 1 0 1 1 0 0 1 1 0 0 0 1 1 1 1 ) . \mathbf{G}:=\begin{pmatrix}1&1&0&1\\ 1&0&1&1\\ 1&0&0&0\\ 0&1&1&1\\ 0&1&0&0\\ 0&0&1&0\\ 0&0&0&1\\ \end{pmatrix},\qquad\mathbf{H}:=\begin{pmatrix}1&0&1&0&1&0&1\\ 0&1&1&0&0&1&1\\ 0&0&0&1&1&1&1\\ \end{pmatrix}.
  19. 𝐩 = ( d 1 d 2 d 3 d 4 ) = ( 1 0 1 1 ) \mathbf{p}=\begin{pmatrix}d_{1}\\ d_{2}\\ d_{3}\\ d_{4}\\ \end{pmatrix}=\begin{pmatrix}1\\ 0\\ 1\\ 1\end{pmatrix}
  20. 𝐱 = 𝐆𝐩 = ( 1 1 0 1 1 0 1 1 1 0 0 0 0 1 1 1 0 1 0 0 0 0 1 0 0 0 0 1 ) ( 1 0 1 1 ) = ( 2 3 1 2 0 1 1 ) = ( 0 1 1 0 0 1 1 ) \mathbf{x}=\mathbf{G}\mathbf{p}=\begin{pmatrix}1&1&0&1\\ 1&0&1&1\\ 1&0&0&0\\ 0&1&1&1\\ 0&1&0&0\\ 0&0&1&0\\ 0&0&0&1\\ \end{pmatrix}\begin{pmatrix}1\\ 0\\ 1\\ 1\end{pmatrix}=\begin{pmatrix}2\\ 3\\ 1\\ 2\\ 0\\ 1\\ 1\end{pmatrix}=\begin{pmatrix}0\\ 1\\ 1\\ 0\\ 0\\ 1\\ 1\end{pmatrix}
  21. 𝐫 = 𝐱 \mathbf{r}=\mathbf{x}
  22. 𝐳 = 𝐇𝐫 = ( 1 0 1 0 1 0 1 0 1 1 0 0 1 1 0 0 0 1 1 1 1 ) ( 0 1 1 0 0 1 1 ) = ( 2 4 2 ) = ( 0 0 0 ) \mathbf{z}=\mathbf{H}\mathbf{r}=\begin{pmatrix}1&0&1&0&1&0&1\\ 0&1&1&0&0&1&1\\ 0&0&0&1&1&1&1\\ \end{pmatrix}\begin{pmatrix}0\\ 1\\ 1\\ 0\\ 0\\ 1\\ 1\end{pmatrix}=\begin{pmatrix}2\\ 4\\ 2\end{pmatrix}=\begin{pmatrix}0\\ 0\\ 0\end{pmatrix}
  23. 𝐫 = 𝐱 + 𝐞 i \mathbf{r}=\mathbf{x}+\mathbf{e}_{i}
  24. i t h i_{th}
  25. i t h i^{th}
  26. 𝐞 2 = ( 0 1 0 0 0 0 0 ) \mathbf{e}_{2}=\begin{pmatrix}0\\ 1\\ 0\\ 0\\ 0\\ 0\\ 0\end{pmatrix}
  27. i t h i^{th}
  28. 𝐇𝐫 = 𝐇 ( 𝐱 + 𝐞 i ) = 𝐇𝐱 + 𝐇𝐞 i \mathbf{Hr}=\mathbf{H}\left(\mathbf{x}+\mathbf{e}_{i}\right)=\mathbf{Hx}+% \mathbf{He}_{i}
  29. 𝐇𝐱 + 𝐇𝐞 i = 𝟎 + 𝐇𝐞 i = 𝐇𝐞 i \mathbf{Hx}+\mathbf{He}_{i}=\mathbf{0}+\mathbf{He}_{i}=\mathbf{He}_{i}
  30. i t h i^{th}
  31. 𝐫 = 𝐱 + 𝐞 5 = ( 0 1 1 0 0 1 1 ) + ( 0 0 0 0 1 0 0 ) = ( 0 1 1 0 1 1 1 ) \mathbf{r}=\mathbf{x}+\mathbf{e}_{5}=\begin{pmatrix}0\\ 1\\ 1\\ 0\\ 0\\ 1\\ 1\end{pmatrix}+\begin{pmatrix}0\\ 0\\ 0\\ 0\\ 1\\ 0\\ 0\end{pmatrix}=\begin{pmatrix}0\\ 1\\ 1\\ 0\\ 1\\ 1\\ 1\end{pmatrix}
  32. 𝐳 = 𝐇𝐫 = ( 1 0 1 0 1 0 1 0 1 1 0 0 1 1 0 0 0 1 1 1 1 ) ( 0 1 1 0 1 1 1 ) = ( 3 4 3 ) = ( 1 0 1 ) \mathbf{z}=\mathbf{Hr}=\begin{pmatrix}1&0&1&0&1&0&1\\ 0&1&1&0&0&1&1\\ 0&0&0&1&1&1&1\\ \end{pmatrix}\begin{pmatrix}0\\ 1\\ 1\\ 0\\ 1\\ 1\\ 1\end{pmatrix}=\begin{pmatrix}3\\ 4\\ 3\end{pmatrix}=\begin{pmatrix}1\\ 0\\ 1\end{pmatrix}
  33. 𝐫 corrected = ( 0 1 1 0 1 ¯ 1 1 ) = ( 0 1 1 0 0 1 1 ) \mathbf{r}_{\,\text{corrected}}=\begin{pmatrix}0\\ 1\\ 1\\ 0\\ \overline{1}\\ 1\\ 1\end{pmatrix}=\begin{pmatrix}0\\ 1\\ 1\\ 0\\ 0\\ 1\\ 1\end{pmatrix}
  34. 𝐑 = ( 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 ) \mathbf{R}=\begin{pmatrix}0&0&1&0&0&0&0\\ 0&0&0&0&1&0&0\\ 0&0&0&0&0&1&0\\ 0&0&0&0&0&0&1\\ \end{pmatrix}
  35. 𝐩 𝐫 = ( 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 ) ( 0 1 1 0 0 1 1 ) = ( 1 0 1 1 ) \mathbf{p_{r}}=\begin{pmatrix}0&0&1&0&0&0&0\\ 0&0&0&0&1&0&0\\ 0&0&0&0&0&1&0\\ 0&0&0&0&0&0&1\\ \end{pmatrix}\begin{pmatrix}0\\ 1\\ 1\\ 0\\ 0\\ 1\\ 1\end{pmatrix}=\begin{pmatrix}1\\ 0\\ 1\\ 1\end{pmatrix}
  36. ( \color b l u e d 1 , \color b l u e d 2 , \color b l u e d 3 , \color b l u e d 4 ) ({\color{blue}d_{1}},{\color{blue}d_{2}},{\color{blue}d_{3}},{\color{blue}d_{4% }})
  37. ( \color r e d p 1 , \color r e d p 2 , \color b l u e d 1 , \color r e d p 3 , \color b l u e d 2 , \color b l u e d 3 , \color b l u e d 4 ) ({\color{red}p_{1}},{\color{red}p_{2}},{\color{blue}d_{1}},{\color{red}p_{3}},% {\color{blue}d_{2}},{\color{blue}d_{3}},{\color{blue}d_{4}})
  38. ( \color r e d p 1 , \color r e d p 2 , \color b l u e d 1 , \color r e d p 3 , \color b l u e d 2 , \color b l u e d 3 , \color b l u e d 4 , \color g r e e n p 4 ) ({\color{red}p_{1}},{\color{red}p_{2}},{\color{blue}d_{1}},{\color{red}p_{3}},% {\color{blue}d_{2}},{\color{blue}d_{3}},{\color{blue}d_{4}},{\color{green}p_{4% }})

Handshaking_lemma.html

  1. v V deg ( v ) = 2 | E | \sum_{v\in V}\deg(v)=2|E|

Hannan–Quinn_information_criterion.html

  1. HQC = - 2 L m a x + 2 k log log n , \mathrm{HQC}=-2L_{max}+2k\log\log n,
  2. L m a x L_{max}

Hans_B._Pacejka.html

  1. R ( k ) = d s i n { c a r c t a n [ b ( 1 - e ) k + e a r c t a n ( b k ) ] } R(k)=d\cdot sin\{c\cdot arctan[b(1-e)k+e\cdot arctan(bk)]\}\,

Hapke_parameters.html

  1. ω ¯ 0 \bar{\omega}_{0}
  2. K s / ( K s + K a ) K_{s}/(K_{s}+K_{a})
  3. K s K_{s}
  4. K a K_{a}
  5. h h
  6. B 0 B_{0}
  7. S 0 S_{0}
  8. P 0 P_{0}
  9. θ \theta

Hardy–Littlewood_maximal_function.html

  1. M f ( x ) = sup r > 0 1 | B ( x , r ) | B ( x , r ) | f ( y ) | d y Mf(x)=\sup_{r>0}\frac{1}{|B(x,r)|}\int_{B(x,r)}|f(y)|\,dy
  2. | { M f > λ } | < C d λ f L 1 ( 𝐑 d ) . \left|\{Mf>\lambda\}\right|<\frac{C_{d}}{\lambda}\|f\|_{L^{1}(\mathbf{R}^{d})}.
  3. M f L p ( 𝐑 d ) C p , d f L p ( 𝐑 d ) . \|Mf\|_{L^{p}(\mathbf{R}^{d})}\leq C_{p,d}\|f\|_{L^{p}(\mathbf{R}^{d})}.
  4. \mathcal{F}
  5. \mathcal{F}
  6. \mathcal{F}^{\prime}
  7. B B B 5 B \bigcup_{B\in\mathcal{F}}B\subset\bigcup_{B\in\mathcal{F^{\prime}}}5B
  8. B x | f | d y > t | B x | . \int_{B_{x}}|f|dy>t|B_{x}|.
  9. | { M f > t } | 5 d j | B j | 5 d t | f | d y . |\{Mf>t\}|\leq 5^{d}\sum_{j}|B_{j}|\leq{5^{d}\over t}\int|f|dy.
  10. | { M f > t } | 2 C t | f | > t 2 | f | d x , |\{Mf>t\}|\leq{2C\over t}\int_{|f|>\frac{t}{2}}|f|dx,
  11. M f p p = 0 M f ( x ) p t p - 1 d t d x = p 0 t p - 1 | { M f > t } | d t \|Mf\|_{p}^{p}=\int\int_{0}^{Mf(x)}pt^{p-1}dtdx=p\int_{0}^{\infty}t^{p-1}|\{Mf% >t\}|dt
  12. M f p p p 0 t p - 1 ( 2 C t | f | > t 2 | f | d x ) d t = 2 C p 0 | f | > t 2 t p - 2 | f | d x d t = C p f p p \|Mf\|_{p}^{p}\leq p\int_{0}^{\infty}t^{p-1}\left({2C\over t}\int_{|f|>\frac{t% }{2}}|f|dx\right)dt=2Cp\int_{0}^{\infty}\int_{|f|>\frac{t}{2}}t^{p-2}|f|dxdt=C% _{p}\|f\|_{p}^{p}
  13. C = 5 d C=5^{d}
  14. 3 d 3^{d}
  15. Ω f ( x ) = lim sup r 0 f r ( x ) - lim inf r 0 f r ( x ) \Omega f(x)=\limsup_{r\to 0}f_{r}(x)-\liminf_{r\to 0}f_{r}(x)
  16. f r ( x ) = 1 | B ( x , r ) | B ( x , r ) f ( y ) d y . f_{r}(x)=\frac{1}{|B(x,r)|}\int_{B(x,r)}f(y)dy.
  17. Ω f Ω g + Ω h = Ω g \Omega f\leq\Omega g+\Omega h=\Omega g
  18. | { Ω g > ε } | 2 A ε g 1 \left|\{\Omega g>\varepsilon\}\right|\leq\frac{2A}{\varepsilon}\|g\|_{1}
  19. g 1 0 \|g\|_{1}\to 0
  20. lim r 0 f r ( x ) \lim_{r\to 0}f_{r}(x)
  21. f r - f 1 0 \|f_{r}-f\|_{1}\to 0
  22. f r k f f_{r_{k}}\to f
  23. f * ( x ) = sup x B x 1 | B x | B x | f ( y ) | d y f^{*}(x)=\sup_{x\in B_{x}}\frac{1}{|B_{x}|}\int_{B_{x}}|f(y)|dy
  24. M Δ f ( x ) = sup x Q x 1 | Q x | Q x | f ( y ) | d y M_{\Delta}f(x)=\sup_{x\in Q_{x}}\frac{1}{|Q_{x}|}\int_{Q_{x}}|f(y)|dy

Harish-Chandra_character.html

  1. π ( f ) = G f ( x ) π ( x ) d x \pi(f)=\int_{G}f(x)\pi(x)\,dx
  2. Θ π : f Tr ( π ( f ) ) \Theta_{\pi}:f\mapsto\operatorname{Tr}(\pi(f))

Harmonic_measure.html

  1. R n R^{n}
  2. n 2 n\geq 2
  3. { - Δ H f ( x ) = 0 , x D ; H f ( x ) = f ( x ) , x D . \begin{cases}-\Delta H_{f}(x)=0,&x\in D;\\ H_{f}(x)=f(x),&x\in\partial D.\end{cases}
  4. H f ( x ) = D f ( y ) d ω ( x , D ) ( y ) . H_{f}(x)=\int_{\partial D}f(y)\,\mathrm{d}\omega(x,D)(y).
  5. 0 ω ( x , D ) ( E ) 1 ; 0\leq\omega(x,D)(E)\leq 1;
  6. 1 - ω ( x , D ) ( E ) = ω ( x , D ) ( D E ) ; 1-\omega(x,D)(E)=\omega(x,D)(\partial D\setminus E);
  7. y ω ( y , D ) ( E ) y\mapsto\omega(y,D)(E)
  8. D 2 D\subset\mathbb{R}^{2}
  9. H 1 ( D ) < H^{1}(\partial D)<\infty
  10. E D E\subset\partial D
  11. ω ( X , D ) ( E ) = 0 \omega(X,D)(E)=0
  12. H 1 ( E ) = 0 H^{1}(E)=0
  13. D 2 D\subset\mathbb{R}^{2}
  14. E D E\subset\partial D
  15. H s ( E ) = 0 H^{s}(E)=0
  16. s < 1 s<1
  17. ω ( x , D ) ( E ) = 0 \omega(x,D)(E)=0
  18. D n D\subset\mathbb{R}^{n}
  19. E D E\subset\partial D
  20. ω ( X , D ) ( E ) = 0 \omega(X,D)(E)=0
  21. H n - 1 ( E ) = 0 H^{n-1}(E)=0
  22. 𝔻 = { X 2 : | X | < 1 } \mathbb{D}=\{X\in\mathbb{R}^{2}:|X|<1\}
  23. 𝔻 \mathbb{D}
  24. ω ( 0 , 𝔻 ) ( E ) = | E | / 2 π \omega(0,\mathbb{D})(E)=|E|/2\pi
  25. E S 1 E\subset S^{1}
  26. | E | |E|
  27. E E
  28. 𝔻 \mathbb{D}
  29. X 𝔻 X\in\mathbb{D}
  30. ω ( X , 𝔻 ) ( E ) = E 1 - | X | 2 | X - Q | 2 d H 1 ( Q ) 2 π \omega(X,\mathbb{D})(E)=\int_{E}\frac{1-|X|^{2}}{|X-Q|^{2}}\frac{dH^{1}(Q)}{2\pi}
  31. E S 1 E\subset S^{1}
  32. H 1 H^{1}
  33. d ω ( X , 𝔻 ) / d H 1 d\omega(X,\mathbb{D})/dH^{1}
  34. n 2 n\geq 2
  35. 𝔹 n = { X n : | X | < 1 } \mathbb{B}^{n}=\{X\in\mathbb{R}^{n}:|X|<1\}
  36. X 𝔹 n X\in\mathbb{B}^{n}
  37. ω ( X , 𝔹 n ) ( E ) = E 1 - | X | 2 | X - Q | n d H n - 1 ( Q ) σ n - 1 \omega(X,\mathbb{B}^{n})(E)=\int_{E}\frac{1-|X|^{2}}{|X-Q|^{n}}\frac{dH^{n-1}(% Q)}{\sigma_{n-1}}
  38. E S n - 1 E\subset S^{n-1}
  39. H n - 1 H^{n-1}
  40. S n - 1 S^{n-1}
  41. H n - 1 ( S n - 1 ) = σ n - 1 H^{n-1}(S^{n-1})=\sigma_{n-1}
  42. D 2 D\subset\mathbb{R}^{2}
  43. \in
  44. ω ( X , D ) ( E ) = | ϕ - 1 ( E ) | / 2 π \omega(X,D)(E)=|\phi^{-1}(E)|/2\pi
  45. E D E\subset\partial D
  46. ϕ : 𝔻 D \phi:\mathbb{D}\rightarrow D
  47. ϕ ( 0 ) = X \phi(0)=X
  48. D 2 D\subset\mathbb{R}^{2}
  49. E D E\subset\partial D
  50. E E
  51. H 1 ( E ) = 0 H^{1}(E)=0
  52. ω ( X , D ) ( E ) = 1 \omega(X,D)(E)=1
  53. μ G x ( F ) = 𝐏 x [ X τ G F ] \mu_{G}^{x}(F)=\mathbf{P}^{x}\big[X_{\tau_{G}}\in F\big]