wpmath0000013_4

Eutactic_star.html

  1. k = 1 s ( 𝐚 k 𝐯 ) 2 = ζ | 𝐯 | 2 \sum_{k=1}^{s}(\mathbf{a}_{k}\cdot\mathbf{v})^{2}=\zeta|\mathbf{v}|^{2}
  2. ζ = k = 1 s | 𝐚 k | 2 / n . \zeta=\sum_{k=1}^{s}|\mathbf{a}_{k}|^{2}/n.
  3. T 𝐱 = k = 1 s 𝐚 k ( 𝐚 k 𝐱 ) , T\mathbf{x}=\sum_{k=1}^{s}\mathbf{a}_{k}(\mathbf{a}_{k}\cdot\mathbf{x}),
  4. s \mathbb{R}^{s}

Evasive_Boolean_function.html

  1. f ( x , y , z ) ~{}f(x,y,z)~{}
  2. = ~{}=~{}
  3. ( x and y ) ~{}(x\and y)~{}
  4. ~{}~{}
  5. ( ¬ x and z ) ~{}(\neg x\and z)~{}
  6. = ~{}=~{}
  7. ~{}~{}
  8. and \and
  9. $\or$
  10. ¬ \neg
  11. ( ¬ x = false ) ( ( ¬ x and z ) = false ) (\neg x=\,\text{false})~{}~{}\Rightarrow~{}~{}((\neg x\and z)=\,\text{false})
  12. f ( x , y , z ) ~{}f(x,y,z)~{}
  13. = ( x y z ) ~{}=(x\wedge y\wedge z)~{}

Evidential_decision_theory.html

  1. V V
  2. A A
  3. V ( A ) = j P ( O j | A ) D ( O j ) , V(A)=\sum\limits_{j}P(O_{j}|A)D(O_{j}),
  4. D ( O j ) D(O_{j})
  5. O j O_{j}
  6. P ( O j | A ) P(O_{j}|A)
  7. O j O_{j}
  8. A A

EWMA_chart.html

  1. x ¯ i \bar{x}_{i}
  2. z i = λ x ¯ i + ( 1 - λ ) z i - 1 z_{i}=\lambda\bar{x}_{i}+\left(1-\lambda\right)z_{i-1}
  3. T ± L S n λ 2 - λ [ 1 - ( 1 - λ ) 2 i ] T\pm L\frac{S}{\sqrt{n}}\sqrt{\frac{\lambda}{2-\lambda}[1-\left(1-\lambda% \right)^{2i}]}
  4. ± L σ ^ n \pm L\frac{\hat{\sigma}}{\sqrt{n}}
  5. x ¯ \bar{x}
  6. x ¯ \bar{x}

Exceptional_isomorphism.html

  1. L 2 ( 4 ) L 2 ( 5 ) A 5 , L_{2}(4)\cong L_{2}(5)\cong A_{5},
  2. L 2 ( 7 ) L 3 ( 2 ) , L_{2}(7)\cong L_{3}(2),
  3. L 2 ( 9 ) A 6 , L_{2}(9)\cong A_{6},
  4. L 4 ( 2 ) A 8 , L_{4}(2)\cong A_{8},
  5. PSU 4 ( 2 ) PSp 4 ( 3 ) , \operatorname{PSU}_{4}(2)\cong\operatorname{PSp}_{4}(3),
  6. 𝐅 5 \mathbf{F}_{5}
  7. PSL ( 2 , 5 ) A 5 I , \operatorname{PSL}(2,5)\cong A_{5}\cong I,
  8. PGL ( 2 , 5 ) S 5 , \operatorname{PGL}(2,5)\cong S_{5},
  9. SL ( 2 , 5 ) 2 A 5 2 I , \operatorname{SL}(2,5)\cong 2\cdot A_{5}\cong 2I,
  10. L 2 ( 4 ) L 2 ( 5 ) A 5 , L_{2}(4)\cong L_{2}(5)\cong A_{5},
  11. L 2 ( 9 ) S p 4 ( 2 ) A 6 , L_{2}(9)\cong Sp_{4}(2)^{\prime}\cong A_{6},
  12. S p 4 ( 2 ) S 6 , Sp_{4}(2)\cong S_{6},
  13. L 4 ( 2 ) O 6 ( + , 2 ) A 8 , L_{4}(2)\cong O_{6}(+,2)^{\prime}\cong A_{8},
  14. O 6 ( + , 2 ) S 8 . O_{6}(+,2)\cong S_{8}.
  15. S n S_{n}
  16. n n
  17. A 8 A_{8}
  18. S 8 S_{8}
  19. S L 4 / μ 2 S O 6 SL_{4}/\mu_{2}\cong SO_{6}
  20. C 2 { ± 1 } O ( 1 ) Spin ( 1 ) * C_{2}\cong\{\pm 1\}\cong\operatorname{O}(1)\cong\operatorname{Spin}(1)\cong% \mathbb{Z}^{*}
  21. S 0 O ( 1 ) S^{0}\cong\operatorname{O}(1)
  22. S 1 SO ( 2 ) U ( 1 ) Spin ( 2 ) S^{1}\cong\operatorname{SO}(2)\cong\operatorname{U}(1)\cong\operatorname{Spin}% (2)
  23. S 3 Spin ( 3 ) SU ( 2 ) Sp ( 1 ) S^{3}\cong\operatorname{Spin}(3)\cong\operatorname{SU}(2)\cong\operatorname{Sp% }(1)
  24. 𝔰 𝔩 2 𝔰 𝔬 3 𝔰 𝔭 1 \mathfrak{sl}_{2}\cong\mathfrak{so}_{3}\cong\mathfrak{sp}_{1}
  25. 𝔰 𝔬 5 𝔰 𝔭 2 \mathfrak{so}_{5}\cong\mathfrak{sp}_{2}
  26. 𝔰 𝔬 4 𝔰 𝔩 2 𝔰 𝔩 2 \mathfrak{so}_{4}\cong\mathfrak{sl}_{2}\oplus\mathfrak{sl}_{2}
  27. 𝔰 𝔩 4 𝔰 𝔬 6 \mathfrak{sl}_{4}\cong\mathfrak{so}_{6}

Exchange_current_density.html

  1. j 0 = F k 0 ( C o x y 1 - β C r e d β ) j_{0}=Fk_{0}(C_{oxy}^{1-\beta}C_{red}^{\beta})

Expected_marginal_seat_revenue.html

  1. r 1 r_{1}
  2. r 2 r_{2}
  3. r 1 r_{1}
  4. r 2 r_{2}
  5. r n r_{n}
  6. F j ( x ) F_{j}(x)
  7. y y
  8. P ( D k > y k j + 1 ) = r j + 1 r k P(D_{k}>y_{k}^{j+1})=\frac{r_{j+1}}{r_{k}}
  9. y j y_{j}
  10. y j = k = 1 j y k j + 1 y_{j}=\sum_{k=1}^{j}y_{k}^{j+1}
  11. y j + 1 y_{j+1}
  12. P ( D k > y k j + 1 ) = r j + 1 r P(D_{k}>y_{k}^{j+1})=\frac{r_{j+1}}{r}
  13. j + 1 j+1
  14. y y
  15. S j = k = 1 j D k S_{j}=\sum_{k=1}^{j}D_{k}
  16. r ¯ j = k = 1 j r k D k k = 1 j D k \overline{r}_{j}=\frac{\sum_{k=1}^{j}r_{k}\cdot D_{k}}{\sum_{k=1}^{j}D_{k}}
  17. P ( S j > y j ) = r j + 1 r ¯ j P(S_{j}>y_{j})=\frac{r_{j+1}}{\overline{r}_{j}}
  18. y j y_{j}^{\star}
  19. F j ( x ) F_{j}(x)
  20. y j = μ j + z α σ j y_{j}=\mu_{j}+z_{\alpha}\cdot\sigma_{j}
  21. μ j = k = 1 j μ k \mu_{j}=\sum_{k=1}^{j}\mu_{k}
  22. σ j 2 = k = 1 j σ k 2 \sigma_{j}^{2}=\sum_{k=1}^{j}\sigma_{k}^{2}
  23. z α z_{\alpha}
  24. z α = ϕ - 1 ( 1 - r j + 1 r ¯ j ) z_{\alpha}=\phi^{-1}(1-\frac{r_{j+1}}{\overline{r}_{j}})

Experimental_uncertainty_analysis.html

  1. T = 2 π < m t p l > L g [ 1 + 1 4 sin 2 ( θ 2 ) ] 𝐄𝐪 ( 𝟏 ) T\,=\,2\,\pi\,\sqrt{<}mtpl>{{L\over g}}\,\,\left[{1\,\,\,+\,\,\,{1\over 4}\sin% ^{2}\left({{\theta\over 2}}\right)\,}\right]{\mathbf{\,\,\,\,\,\,\,\,\,Eq(1)}}
  2. g ^ = 4 π 2 L T 2 [ 1 + 1 4 sin 2 ( < m t p l > θ 2 ) ] 2 𝐄𝐪 ( 𝟐 ) \hat{g}\,=\,{{4\,\pi^{2}L}\over{T^{2}}}\,\,\left[{\,1\,\,\,+\,\,\,{1\over 4}% \sin^{2}\left(<mtpl>{{\theta\over 2}}\right)\,}\right]^{2}{\mathbf{\,\,\,\,\,% \,\,\,\,\,\,\,Eq(2)}}
  3. Δ g ^ = g ^ ( L + Δ L , T + Δ T , θ + Δ θ ) - g ^ ( L , T , θ ) 𝐄𝐪 ( 𝟑 ) \Delta\hat{g}\,\,\,=\,\,\,\,\hat{g}\left({L+\Delta L,\,\,\,T+\Delta T,\,\,\,% \theta+\Delta\theta}\right)\,\,\,-\,\,\,\hat{g}\left({L,\,\,T,\,\,\theta}% \right){\mathbf{\,\,\,\,\,\,\,\,\,Eq(3)}}
  4. Δ g ^ = g ^ ( 0.495 , 1.443 , 30 ) - g ^ ( 0.500 , 1.443 , 30 ) = - 0.098 m / s 2 \Delta\hat{g}\,\,\,=\,\,\,\hat{g}\left({0.495,\,\,\,1.443,\,\,\,30}\right)\,\,% \,-\,\,\,\hat{g}\left({0.500,\,\,1.443,\,\,30}\right)\,\,\,=\,\,\,-0.098{\rm\,% \,\,m/s^{2}}
  5. Δ g ^ g ^ = g ^ ( L + Δ L , T + Δ T , θ + Δ θ ) - g ^ ( L , T , θ ) g ^ ( L , T , θ ) 𝐄𝐪 ( 𝟒 ) {{\Delta\hat{g}}\over{\hat{g}}}\,\,\,=\,\,\,\,{{\hat{g}\left({L+\Delta L,\,\,% \,T+\Delta T,\,\,\,\theta+\Delta\theta}\right)\,\,\,-\,\,\,\hat{g}\left({L,\,% \,T,\,\,\theta}\right)}\over{\hat{g}\left({L,\,\,T,\,\,\theta}\right)}}{% \mathbf{\,\,\,\,\,\,\,\,Eq(4)}}
  6. d z = z x 1 d x 1 + z x 2 d x 2 + z x 3 d x 3 + = i = 1 p z x i d x i 𝐄𝐪 ( 𝟓 ) dz={{\partial z}\over{\partial x_{1}}}dx_{1}\,\,\,+\,\,\,{{\partial z}\over{% \partial x_{2}}}dx_{2}\,\,\,+\,\,\,{{\partial z}\over{\partial x_{3}}}dx_{3}\,% \,\,+\,\,\,\cdots\,\,\,\,\,=\,\,\,\sum\limits_{i\,\,=\,\,1}^{p}{\,{{\partial z% }\over{\partial x_{i}}}dx_{i}}{\mathbf{\,\,\,\,\,\,\,\,\,\,\,\,\,Eq(5)}}
  7. Δ z z x 1 Δ x 1 + z x 2 Δ x 2 + z x 3 Δ x 3 + = i = 1 p z x i Δ x i 𝐄𝐪 ( 𝟔 ) \Delta z\approx{{\partial z}\over{\partial x_{1}}}\Delta x_{1}\,\,\,+\,\,\,{{% \partial z}\over{\partial x_{2}}}\Delta x_{2}\,\,\,+\,\,\,{{\partial z}\over{% \partial x_{3}}}\Delta x_{3}\,\,\,+\,\,\,\cdots\,\,\,\,\,=\,\,\,\sum\limits_{i% \,\,=\,\,1}^{p}{\,{{\partial z}\over{\partial x_{i}}}\Delta x_{i}}{\mathbf{\,% \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,Eq(6)}}
  8. Δ z z 1 z i = 1 p z x i Δ x i 𝐄𝐪 ( 𝟕 ) {{\Delta z}\over z}\,\,\,\approx\,\,\,{1\over z}\,\,\sum\limits_{i\,\,=\,\,1}^% {p}{\,{{\partial z}\over{\partial x_{i}}}\Delta x_{i}}{\mathbf{\,\,\,\,\,\,\,% \,\,\,\,\,\,\,\,\,\,\,\,Eq(7)}}
  9. Δ z ( z x 1 z x 2 z x 3 z x p ) ( Δ x 1 Δ x 2 Δ x 3 Δ x p ) 𝐄𝐪 ( 𝟖 ) \Delta z\,\,\approx\,\,\begin{pmatrix}{\partial z\over\partial x_{1}}&{% \partial z\over\partial x_{2}}&{\partial z\over\partial x_{3}}&\cdots&{% \partial z\over\partial x_{p}}\end{pmatrix}\begin{pmatrix}{\Delta x_{1}}\\ {\Delta x_{2}}\\ {\Delta x_{3}}\\ {\vdots}\\ {\Delta x_{p}}\end{pmatrix}{\mathbf{\,\,\,\,\,\,\,\,\,\,\,\,\,\,Eq(8)}}
  10. Δ g ^ g ^ L Δ L + g ^ T Δ T + g ^ θ Δ θ 𝐄𝐪 ( 𝟗 ) \Delta\hat{g}\,\,\approx\,\,{{\partial\hat{g}}\over{\partial L}}\Delta L\,\,\,% +\,\,\,{{\partial\hat{g}}\over{\partial T}}\Delta T\,\,\,+\,\,\,{{\partial\hat% {g}}\over{\partial\theta}}\Delta\theta{\mathbf{\,\,\,\,\,\,\,\,\,\,\,Eq(9)}}
  11. α ( θ ) [ 1 + 1 4 sin 2 ( < m t p l > θ 2 ) ] 2 \alpha(\theta)\,\,\equiv\,\,\left[{\,1\,\,\,+\,\,\,{1\over 4}\sin^{2}\left(<% mtpl>{{\theta\over 2}}\right)\,}\right]^{2}
  12. g ^ = 4 π 2 L T 2 α ( θ ) g ^ L = 4 π 2 T 2 α ( θ ) g ^ T = - 8 L π 2 T 3 α ( θ ) g ^ θ = L π 2 T 2 α ( θ ) sin ( θ ) 𝐄𝐪 ( 𝟏𝟎 ) \begin{aligned}\displaystyle\hat{g}&\displaystyle={{4\pi^{2}L}\over{T^{2}}}% \alpha(\theta)\\ \\ \displaystyle{{\partial\hat{g}}\over{\partial L}}&\displaystyle=\,\,\,{{4\,\pi% ^{2}}\over{T^{2}}}\alpha(\theta)\\ \\ \displaystyle{{\partial\hat{g}}\over{\partial T}}&\displaystyle=\,\,{{-8\,L\,% \pi^{2}}\over{T^{3}}}\alpha(\theta)\\ \\ \displaystyle{{\partial\hat{g}}\over{\partial\theta}}&\displaystyle=\,\,{{L\,% \pi^{2}}\over{T^{2}}}\,\,\sqrt{\alpha(\theta)}\,\,\sin(\theta)\\ \\ \displaystyle{\mathbf{\,\,\,\,Eq(10)}}\end{aligned}
  13. Δ g ^ [ 4 π 2 T 2 α ( θ ) ] Δ L + [ - 8 L π 2 T 3 α ( θ ) ] Δ T + [ L π 2 T 2 α ( θ ) sin ( θ ) ] Δ θ 𝐄𝐪 ( 𝟏𝟏 ) \Delta\hat{g}\,\,\,\approx\,\,\,\left[{{{4\,\pi^{2}}\over{T^{2}}}\alpha(\theta% )}\right]\,\Delta L\,\,\,\,\,+\,\,\,\,\,\,\left[{{{-8\,L\,\pi^{2}}\over{T^{3}}% }\alpha(\theta)}\right]\Delta T\,\,\,+\,\,\,\,\left[{{{L\,\pi^{2}}\over{T^{2}}% }\,\,\sqrt{\alpha(\theta)}\,\,\sin(\theta)}\right]\Delta\theta{\mathbf{\,\,\,% \,\,\,\,\,Eq(11)}}
  14. Δ g ^ g ^ 1 g ^ g ^ L Δ L + 1 g ^ g ^ T Δ T + 1 g ^ g ^ θ Δ θ {{\Delta\hat{g}}\over{\hat{g}}}\,\,\,\,\approx\,\,\,\,{1\over{\hat{g}}}\,\,{{% \partial\hat{g}}\over{\partial L}}\Delta L\,\,\,+\,\,\,\,{1\over{\hat{g}}}\,\,% {{\partial\hat{g}}\over{\partial T}}\Delta T\,\,\,+\,\,\,\,{1\over{\hat{g}}}\,% \,{{\partial\hat{g}}\over{\partial\theta}}\Delta\theta
  15. Δ g ^ g ^ [ 4 π 2 T 2 α ( θ ) 4 π 2 L T 2 α ( θ ) ] Δ L + [ - 8 L π 2 T 3 α ( θ ) 4 π 2 L T 2 α ( θ ) ] Δ T + [ L π 2 T 2 α ( θ ) sin ( θ ) 4 π 2 L T 2 α ( θ ) ] Δ θ {{\Delta\hat{g}}\over{\hat{g}}}\,\,\,\approx\,\,\,\left[{{{{{4\,\pi^{2}}\over{% T^{2}}}\alpha(\theta)}\over{{{4\,\pi^{2}L}\over{T^{2}}}\alpha(\theta)}}}\right% ]\,\Delta L\,\,\,\,\,+\,\,\,\,\,\,\left[{{{{{-8\,L\,\pi^{2}}\over{T^{3}}}% \alpha(\theta)}\over{{{4\,\pi^{2}L}\over{T^{2}}}\alpha(\theta)}}}\right]\Delta T% \,\,\,+\,\,\,\,\left[{{{{{L\,\pi^{2}}\over{T^{2}}}\,\,\sqrt{\alpha(\theta)}\,% \,\sin(\theta)}\over{{{4\,\pi^{2}L}\over{T^{2}}}\alpha(\theta)}}}\right]\Delta\theta
  16. Δ g ^ g ^ Δ L L - 2 Δ T T + sin ( θ ) 4 α ( θ ) Δ θ {{\Delta\hat{g}}\over{\hat{g}}}\,\,\,\approx\,\,\,{{\Delta L}\over L}\,\,\,-\,% \,\,2\,\,{{\Delta T}\over T}\,\,\,+\,\,\,{{\sin(\theta)}\over{4\,\sqrt{\alpha(% \theta)}}}\Delta\theta
  17. sin ( θ ) 4 [ 1 + 1 4 sin 2 ( < m t p l > θ 2 ) ] θ 4 θ 4 Δ θ = θ 2 4 Δ θ θ {{\sin(\theta)}\over{4\left[{1\,\,\,+\,\,\,{1\over 4}\sin^{2}\left(<mtpl>{{% \theta\over 2}}\right)}\right]}}\,\,\,\approx\,\,\,{\theta\over 4}\,\,\,\,\,\,% \,\,\,\,\,\Rightarrow\,\,\,\,\,\,\,\,{\theta\over 4}\,\,\Delta\theta\,\,\,=\,% \,\,{{\theta^{2}}\over 4}{{\Delta\theta}\over\theta}
  18. Δ g ^ g ^ Δ L L - 2 Δ T T + ( < m t p l > θ 2 ) 2 Δ θ θ 𝐄𝐪 ( 𝟏𝟐 ) {{\Delta\hat{g}}\over{\hat{g}}}\,\,\,\approx\,\,\,{{\Delta L}\over L}\,\,\,\,% \,-\,\,\,2\,\,{{\Delta T}\over T}\,\,\,\,\,+\,\,\,\,\,\left(<mtpl>{{\theta% \over 2}}\right)^{2}{{\Delta\theta}\over\theta}{\mathbf{\,\,\,\,\,\,\,\,\,\,\,% Eq(12)}}
  19. z = f ( x 1 x 2 x 3 x p ) z\,\,\,=\,\,\,f\left({x_{1}\,\,\,x_{2}\,\,\,x_{3}\,\,...\,\,\,x_{p}}\right)
  20. μ z = E [ z ] σ z 2 = E [ ( z - μ z ) 2 ] \mu_{z}\,\,=\,\,\,{\rm E}\,[z]\,\,\,\,\,\,\,\,\,\,\,\,\,\,\sigma_{z}^{2}\,\,\,% =\,\,\,{\rm E}\,\left[{\left({z\,\,-\,\,\mu_{z}}\right)^{2}}\right]
  21. PDF z 1 2 z 1 2 π σ [ exp ( - ( z - μ ) 2 2 σ 2 ) + exp ( - ( - z - μ ) 2 2 σ 2 ) ] {\rm PDF}_{z}\,\,\,\sim\,\,\,{1\over{2\sqrt{z}}}\,\,\,{1\over{\sqrt{2\pi}\,\,% \sigma}}\left[{\exp\left({-\,\,{{\left({\sqrt{z}-\mu}\right)^{2}}\over{2\,% \sigma^{2}}}}\right)\,\,\,+\,\,\,\exp\left({-\,\,{{\left({-\sqrt{z}-\mu}\right% )^{2}}\over{2\,\sigma^{2}}}}\right)}\right]
  22. μ z = 0 z PDF z d z σ z 2 = 0 ( z - μ z ) 2 PDF z d z \mu_{z}\,\,=\,\,\,\int_{\,\,0}^{\,\,\infty}{z\,{\rm PDF}_{z}}\,dz\,\,\,\,\,\,% \,\,\,\,\,\,\sigma_{z}^{2}\,\,=\,\,\int_{\,\,0}^{\,\,\infty}{\left({z-\mu_{z}}% \right)^{2}\,}{\rm PDF}_{z}\,dz
  23. μ z = μ 2 + σ 2 σ z 2 = 2 σ 2 ( 2 μ 2 + σ 2 ) \mu_{z}=\mu^{2}+\,\,\sigma^{2}\,\,\,\,\,\,\,\,\,\,\,\,\,\sigma_{z}^{2}\,\,=\,% \,2\,\sigma^{2}\left({2\mu^{2}+\,\,\sigma^{2}}\right)
  24. σ z 2 i = 1 p j = 1 p ( z x i ) ( z x j ) σ i , j 𝐄𝐪 ( 𝟏𝟑 ) \sigma_{z}^{2}\,\,\,\approx\,\,\,\sum\limits_{i\,=\,1}^{p}{\,\sum\limits_{j\,=% \,1}^{p}{\left({{{\partial z}\over{\partial x_{i}}}}\right)}}\left({{{\partial z% }\over{\partial x_{j}}}}\right)\sigma_{i,j}{\mathbf{\,\,\,\,\,\,\,\,\,\,\,\,Eq% (13)}}
  25. μ z z ( μ 1 , μ 2 ) + 1 2 { 2 z x 1 2 σ 1 2 + 2 z x 2 2 σ 2 2 } + z 2 x 1 x 2 σ 12 𝐄𝐪 ( 𝟏𝟒 ) \mu_{z}\,\,\approx\,\,z\left({\mu_{1},\mu_{2}}\right)\,\,\,+\,\,\,{1\over 2}% \left\{{\,{{\partial^{2}z}\over{\partial x_{1}^{2}}}\sigma_{1}^{2}\,\,\,+\,\,% \,{{\partial^{2}z}\over{\partial x_{2}^{2}}}\sigma_{2}^{2}}\right\}\,\,\,+\,\,% \,{{\partial z^{2}}\over{\partial x_{1}\,\partial x_{2}}}\sigma_{12}{\mathbf{% \,\,\,\,\,\,\,\,\,Eq(14)}}
  26. μ z μ 2 + 1 2 σ 2 2 z x 2 = μ 2 + 1 2 σ 2 [ 2 ] = μ 2 + σ 2 \mu_{z}\,\,\approx\,\,\mu^{2}\,\,+\,\,\,{1\over 2}\,\,\sigma^{2}\,\,{{\partial% ^{2}z}\over{\partial x^{2}}}\,\,\,=\,\,\,\mu^{2}+\,\,\,{1\over 2}\,\,\sigma^{2% }\,\,\left[2\right]\,\,\,\,=\,\,\,\mu^{2}+\,\sigma^{2}
  27. σ z 2 ( z x ) 2 σ 2 = 4 x 2 σ 2 4 ( μ 2 ) σ 2 = 4 μ 2 σ 2 \sigma_{z}^{2}\approx\left({{{\partial z}\over{\partial x}}}\right)^{2}\sigma^% {2}\,\,=\,\,4\,x^{2}\,\sigma^{2}\,\,\,\,\,\Rightarrow\,\,\,\,\,4\left({\mu^{2}% }\right)\sigma^{2}=\,\,\,\,4\,\mu^{2}\sigma^{2}
  28. s y m b o l γ T ( z x 1 z x 2 z x 3 z x p ) symbol{\gamma}^{T}\equiv\begin{pmatrix}{\partial z\over\partial x_{1}}&{% \partial z\over\partial x_{2}}&{\partial z\over\partial x_{3}}&\cdots&{% \partial z\over\partial x_{p}}\end{pmatrix}
  29. 𝐂 ( σ 1 2 σ 12 σ 13 σ 1 p σ 21 σ 2 2 σ 23 σ 2 p σ 31 σ 32 σ 3 2 σ 3 p σ p 1 σ p 2 σ p 3 σ p 2 ) \mathbf{C}\,\,\equiv\,\begin{pmatrix}{\sigma_{1}^{2}}&{\sigma_{12}}&{\sigma_{1% 3}}&\cdots&{\sigma_{1p}}\\ {\sigma_{21}}&{\sigma_{2}^{2}}&{\sigma_{23}}&\cdots&{\sigma_{2p}}\\ {\sigma_{31}}&{\sigma_{32}}&{\sigma_{3}^{2}}&\cdots&{\sigma_{3p}}\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ {\sigma_{p1}}&{\sigma_{p2}}&{\sigma_{p3}}&\cdots&{\sigma_{p}^{2}}\end{pmatrix}
  30. σ z 2 s y m b o l γ T 𝐂 s y m b o l γ 𝐄𝐪 ( 𝟏𝟓 ) \sigma_{z}^{2}\,\,\approx\,\,symbol{\gamma}^{T}\,\mathbf{C}\,\,symbol{\gamma}{% \mathbf{\,\,\,\,\,\,\,\,\,\,\,\,\,\,Eq(15)}}
  31. σ z 2 ( z x 1 ) ( z x 1 ) σ 11 + ( z x 2 ) ( z x 2 ) σ 22 + ( z x 1 ) ( z x 2 ) σ 12 + ( z x 2 ) ( z x 1 ) σ 21 \sigma_{z}^{2}\,\,\,\approx\,\,\,\left({{{\partial z}\over{\partial x_{1}}}}% \right)\left({{{\partial z}\over{\partial x_{1}}}}\right)\sigma_{11}\,\,\,+\,% \,\,\left({{{\partial z}\over{\partial x_{2}}}}\right)\left({{{\partial z}% \over{\partial x_{2}}}}\right)\sigma_{22}\,\,\,+\,\,\,\left({{{\partial z}% \over{\partial x_{1}}}}\right)\left({{{\partial z}\over{\partial x_{2}}}}% \right)\sigma_{12}\,\,\,+\,\,\,\,\left({{{\partial z}\over{\partial x_{2}}}}% \right)\left({{{\partial z}\over{\partial x_{1}}}}\right)\sigma_{21}
  32. σ z 2 ( z x 1 ) 2 σ 1 2 + ( z x 2 ) 2 σ 2 2 + 2 ( z x 1 ) ( z x 2 ) σ 12 \sigma_{z}^{2}\,\,\,\approx\,\,\,\left({{{\partial z}\over{\partial x_{1}}}}% \right)^{2}\sigma_{1}^{2}\,\,\,+\,\,\,\left({{{\partial z}\over{\partial x_{2}% }}}\right)^{2}\sigma_{2}^{2}\,\,\,+\,\,\,2\left({{{\partial z}\over{\partial x% _{1}}}}\right)\left({{{\partial z}\over{\partial x_{2}}}}\right)\,\,\sigma_{12}
  33. z = x 2 y z x = 2 x y z y = x 2 z\,\,=\,\,x^{2}\,y\,\,\,\,\,\,\,\,\,\,\,{{\partial z}\over{\partial x}}\,\,=\,% \,2x\,y\,\,\,\,\,\,\,\,\,{{\partial z}\over{\partial y}}\,\,=\,\,x^{2}
  34. σ z 2 ( 2 x y ) 2 σ x 2 + ( x 2 ) 2 σ y 2 + 2 ( 2 x y ) ( x 2 ) σ x , y \sigma_{z}^{2}\,\,\,\approx\,\,\,\left({2\,x\,y}\right)^{2}\sigma_{x}^{2}\,\,% \,+\,\,\,\left({x^{2}}\right)^{2}\sigma_{y}^{2}\,\,\,+\,\,\,2\left({2\,x\,y}% \right)\left({x^{2}}\right)\sigma_{x,y}
  35. σ z 2 z 2 ( 2 x y ) 2 ( x 2 y ) 2 σ x 2 + ( x 2 ) 2 ( x 2 y ) 2 σ y 2 + 2 ( 2 x y ) ( x 2 ) ( x 2 y ) 2 σ x , y {{\sigma_{z}^{2}\,}\over{z^{2}}}\,\,\approx\,\,\,{{\left({2xy}\right)^{2}}% \over{\left({x^{2}y}\right)^{2}}}\sigma_{x}^{2}\,\,\,+\,\,\,{{\left({x^{2}}% \right)^{2}}\over{\left({x^{2}y}\right)^{2}}}\sigma_{y}^{2}\,\,\,+\,\,\,{{2% \left({2xy}\right)\left({x^{2}}\right)}\over{\left({x^{2}y}\right)^{2}}}\sigma% _{x,y}
  36. σ z 2 z 2 ( 2 σ x x ) 2 + ( σ y y ) 2 + 4 ( σ x , y x y ) {{\sigma_{z}^{2}}\over{z^{2}}}\,\,\approx\,\,\,\left({{{2\sigma_{x}}\over x}}% \right)^{2}\,\,+\,\,\,\,\left({{{\sigma_{y}}\over y}}\right)^{2}\,+\,\,\,4% \left({{{\sigma_{x,y}}\over{x\,y}}}\right)
  37. σ ^ z z ¯ ( 2 σ ^ x x ¯ ) 2 + ( σ ^ y y ¯ ) 2 + 4 ( σ ^ x , y x ¯ y ¯ ) \hat{\sigma}_{z}\,\,\approx\,\,\bar{z}\,\,\sqrt{\,\,\left({{{2\hat{\sigma}_{x}% }\over{\bar{x}}}}\right)^{2}\,\,+\,\,\,\,\left({{{\hat{\sigma}_{y}}\over{\bar{% y}}}}\right)^{2}\,+\,\,\,4\left({{{\hat{\sigma}_{x,y}}\over{\bar{x}\,\bar{y}}}% }\right)}
  38. σ ^ i = k = 1 n ( x k - x ¯ i ) 2 n - 1 σ ^ i , j = k = 1 n ( x k - x ¯ i ) ( x k - x ¯ j ) n - 1 \hat{\sigma}_{i}\,\,\,=\,\,\,\sqrt{{{\,\,\sum\limits_{k=1}^{n}{\left({x_{k}-% \bar{x}_{i}}\right)^{2}}}\over{n-1}}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\hat{% \sigma}_{i,j}\,\,\,=\,\,\,\sqrt{{{\,\,\sum\limits_{k=1}^{n}{\left({x_{k}-\bar{% x}_{i}}\right)\left({x_{k}-\bar{x}_{j}}\right)}}\over{n-1}}}
  39. g ^ = k T 2 \hat{g}=\,\,{k\over{T^{2}}}
  40. g ^ T = - 2 k T 3 2 g ^ T 2 = - 2 k - 3 T 4 = 6 k T 4 {{\partial\hat{g}}\over{\partial T}}\,\,=\,\,\,{{-2k}\over{T^{3}}}\,\,\,\,\,\,% \,\,\,\,\,{{\partial^{2}\hat{g}}\over{\partial T^{2}}}\,\,\,=\,\,\,-2\,k{{-3}% \over{T^{4}}}\,\,\,=\,\,{{6\,k}\over{T^{4}}}
  41. E [ g ^ ] = k μ T 2 + 1 2 ( 6 k μ T 4 ) σ T 2 𝐄𝐪 ( 𝟏𝟔 ) {\rm E}[\hat{g}]\,\,\,=\,\,\,{k\over{\mu_{T}^{2}}}\,\,\,+\,\,\,{1\over 2}\left% ({{{6\,k}\over{\mu_{T}^{4}}}}\right)\sigma_{T}^{2}{\mathbf{\,\,\,\,\,\,\,\,\,% \,\,\,\,\,Eq(16)}}
  42. σ g ^ 2 ( g ^ L g ^ T g ^ θ ) ( σ L 2 0 0 0 σ T 2 0 0 0 σ θ 2 ) ( g ^ L g ^ T g ^ θ ) = ( g ^ L ) 2 σ L 2 + ( g ^ T ) 2 σ T 2 + ( g ^ θ ) 2 σ θ 2 𝐄𝐪 ( 𝟏𝟕 ) \sigma_{\hat{g}}^{2}\,\,\,\approx\,\,\,\,\begin{pmatrix}{{\partial\hat{g}}% \over{\partial L}}&{{\partial\hat{g}}\over{\partial T}}&{{\partial\hat{g}}% \over{\partial\theta}}\end{pmatrix}\begin{pmatrix}{\sigma_{L}^{2}}&0&0\\ 0&{\sigma_{T}^{2}}&0\\ 0&0&{\sigma_{\theta}^{2}}\end{pmatrix}\begin{pmatrix}{{{\partial\hat{g}}\over{% \partial L}}}\\ {{{\partial\hat{g}}\over{\partial T}}}\\ {{{\partial\hat{g}}\over{\partial\theta}}}\end{pmatrix}\,=\,\left({{{\partial% \hat{g}}\over{\partial L}}}\right)^{2}\sigma_{L}^{2}\,\,\,+\,\,\,\left({{{% \partial\hat{g}}\over{\partial T}}}\right)^{2}\sigma_{T}^{2}\,\,\,+\,\,\,\left% ({{{\partial\hat{g}}\over{\partial\theta}}}\right)^{2}\sigma_{\theta}^{2}{% \mathbf{\,\,\,\,\,\,\,\,Eq(17)}}
  43. σ g ^ 2 ( g ^ T ) 2 σ T 2 = ( - 8 L π 2 T 3 α ( θ ) ) 2 σ T 2 \sigma_{\hat{g}}^{2}\,\,\,\approx\,\,\,\left({{{\partial\hat{g}}\over{\partial T% }}}\right)^{2}\sigma_{T}^{2}\,\,\,\,=\,\,\,\left({{{-8L\,\pi^{2}}\over{T^{3}}}% \alpha(\theta)}\right)^{2}\sigma_{T}^{2}
  44. σ g ^ 2 ( - 8 × 0.5 × π 2 1.443 3 1.0338 ) 2 0.03 2 = 0.166 \sigma_{\hat{g}}^{2}\,\,\,\approx\,\,\,\left({{{-8\times 0.5\times\pi^{2}}% \over{1.443^{3}}}1.0338}\right)^{2}0.03^{2}\,\,=\,\,0.166
  45. RE g ^ σ g ^ μ g ^ = 0.166 9.8 = 0.042 {\rm RE}_{\hat{g}}\equiv\,\,\,{{\sigma_{\hat{g}}}\over{\mu_{\hat{g}}}}\,\,\,=% \,\,\,{{\sqrt{0.166}}\over{9.8}}\,\,\,=\,\,0.042
  46. σ g ^ 2 g ^ 2 1 g ^ 2 ( g ^ L ) 2 σ L 2 + 1 g ^ 2 ( g ^ T ) 2 σ T 2 + 1 g ^ 2 ( g ^ θ ) 2 σ θ 2 {{\sigma_{\hat{g}}^{2}\,}\over{\hat{g}^{2}}}\,\,\,\approx\,\,\,{1\over{\hat{g}% ^{2}}}\,\left({{{\partial\hat{g}}\over{\partial L}}}\right)^{2}\sigma_{L}^{2}% \,\,\,+\,\,\,\,{1\over{\hat{g}^{2}}}\,\left({{{\partial\hat{g}}\over{\partial T% }}}\right)^{2}\sigma_{T}^{2}\,\,\,+\,\,\,\,{1\over{\hat{g}^{2}}}\,\left({{{% \partial\hat{g}}\over{\partial\theta}}}\right)^{2}\sigma_{\theta}^{2}
  47. σ g ^ 2 g ^ 2 σ L 2 L 2 + 4 σ T 2 T 2 + ( < m t p l > θ 2 ) 4 σ θ 2 θ 2 {{\sigma_{\hat{g}}^{2}\,}\over{\hat{g}^{2}}}\,\,\,\approx\,\,\,{{\sigma_{L}^{2% }\,}\over{L^{2}}}\,\,\,+\,\,\,\,4{{\sigma_{T}^{2}}\over{T^{2}}}\,\,\,+\,\,\,\,% \left(<mtpl>{{\theta\over 2}}\right)^{4}{{\sigma_{\theta}^{2}}\over{\theta^{2}}}
  48. R E g ^ = σ g g ^ ( σ L L ) 2 + 4 ( σ T T ) 2 + ( < m t p l > θ 2 ) 4 ( σ θ θ ) 2 𝐄𝐪 ( 𝟏𝟖 ) RE_{\hat{g}}\,\,=\,\,{{\sigma_{g}\,}\over{\hat{g}}}\,\,\,\approx\,\,\,\sqrt{\,% \,\left({{{\sigma_{L}}\over L}}\right)^{2}\,\,\,+\,\,\,\,4\left({{{\sigma_{T}}% \over T}}\right)^{2}\,\,+\,\,\,\,\left(<mtpl>{{\theta\over 2}}\right)^{4}\left% ({{{\sigma_{\theta}}\over\theta}}\right)^{2}\,}{\mathbf{\,\,\,\,\,\,\,\,\,\,\,% \,\,Eq(18)}}
  49. RE g ^ 2 σ T T = 2 0.03 1.443 = 0.042 {\rm RE}_{\hat{g}}\,\,\,\approx\,\,\,2\,\,{{\sigma_{T}}\over T}\,\,\,=\,\,\,2{% {0.03}\over{1.443}}\,\,\,=\,\,\,0.042
  50. RE g ^ ( < m t p l > θ 2 ) 2 σ θ θ = ( 0.524 2 ) 2 0.0873 0.524 0.0114 {\rm RE}_{\hat{g}}\,\,\,\approx\,\,\,\left(<mtpl>{{\theta\over 2}}\right)^{2}{% {\sigma_{\theta}}\over\theta}\,\,\,=\,\,\,\left({{{0.524}\over 2}}\right)^{2}{% {0.0873}\over{0.524}}\,\,\,\approx\,\,\,0.0114
  51. 2 g ^ θ 2 = k 32 [ 9 cos ( μ θ ) - cos ( 2 μ θ ) ] {{\partial^{2}\hat{g}}\over{\partial\theta^{2}}}\,\,\,=\,\,\,{k\over{32}}\left% [{9\cos\left({\mu_{\theta}}\right)\,\,\,-\,\,\,\cos\left({2\mu_{\theta}}\right% )}\right]
  52. E [ g ^ ] k α ( μ θ ) + 1 2 k 32 [ 9 cos ( μ θ ) - cos ( 2 μ θ ) ] σ θ 2 {\rm E}[\hat{g}]\,\,\,\approx\,\,\,\,k\alpha\left({\mu_{\theta}}\right)\,\,\,+% \,\,\,{1\over 2}\,\,{k\over{32}}\left[{9\cos\left({\mu_{\theta}}\right)\,\,\,-% \,\,\,\cos\left({2\mu_{\theta}}\right)}\right]\sigma_{\theta}^{2}
  53. β 3 k μ T 2 ( σ T μ T ) 2 30 ( σ T μ T ) 2 𝐄𝐪 ( 𝟏𝟗 ) \beta\,\,\,\approx\,\,\,{{3\,k}\over{\mu_{T}^{2}}}\,\left({{{\sigma_{T}}\over{% \mu_{T}}}}\right)^{2}\,\,\,\approx\,\,\,30\,\,\left({{{\sigma_{T}}\over{\mu_{T% }}}}\right)^{2}{\mathbf{\,\,\,\,\,\,\,\,\,\,\,Eq(19)}}
  54. σ g ^ 2 ( g ^ T ) 2 σ T 2 = ( - 8 L π 2 T 3 α ( θ ) ) 2 σ T 2 ( - 8 L ¯ π 2 T ¯ 3 α ( θ ¯ ) ) 2 σ T 2 n T \sigma_{\hat{g}}^{2}\,\,\,\approx\,\,\,\left({{{\partial\hat{g}}\over{\partial T% }}}\right)^{2}\sigma_{T}^{2}\,\,\,\,=\,\,\,\left({{{-8L\,\pi^{2}}\over{T^{3}}}% \alpha(\theta)}\right)^{2}\sigma_{T}^{2}\,\,\,\,\,\,\,\Rightarrow\,\,\,\,\,% \left({{{-8\bar{L}\,\pi^{2}}\over{\bar{T}^{3}}}\alpha(\bar{\theta})}\right)^{2% }{{\sigma_{T}^{2}}\over{n_{T}}}
  55. σ z 2 i = 1 p ( z x i ) x ¯ i 2 σ i 2 n i \sigma_{z}^{2}\,\,\,\approx\,\,\,\sum\limits_{i\,\,=\,\,1}^{p}{\,\left({{{% \partial z}\over{\partial x_{i}}}}\right)_{\bar{x}_{i}}^{2}}\,\,{{\sigma_{i}^{% 2}}\over{n_{i}}}
  56. β 3 k μ T 2 ( σ T μ T ) 2 30 ( s T n T T ¯ ) 2 \beta\,\,\,\approx\,\,\,{{3\,k}\over{\mu_{T}^{2}}}\,\left({{{\sigma_{T}}\over{% \mu_{T}}}}\right)^{2}\,\,\,\approx\,\,\,30\,\,\left({{{s_{T}}\over{n_{T}\,\bar% {T}}}}\right)^{2}
  57. β 1 30 ( s T n T T ¯ ) 2 β 2 30 ( s T T ¯ ) 2 \beta_{\,\,1}\,\,\,\approx\,\,\,\,30\,\,\left({{{s_{T}}\over{n_{T}\,\bar{T}}}}% \right)^{2}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\beta_{\,\,2}\,\,\,\approx\,\,30% \,\,\left({{{s_{T}}\over{\bar{T}}}}\right)^{2}
  58. σ g ^ 2 ( - 8 L ¯ π 2 T ¯ 3 α ( θ ¯ ) ) 2 σ T 2 n T \sigma_{\hat{g}}^{2}\,\,\,\approx\,\,\,\left({{{-8\bar{L}\,\pi^{2}}\over{\bar{% T}^{3}}}\alpha(\bar{\theta})}\right)^{2}{{\sigma_{T}^{2}}\over{n_{T}}}
  59. R E g ^ = σ ^ g g ^ ( s L n L L ¯ ) 2 + 4 ( s T n T T ¯ ) 2 + ( θ ¯ 2 ) 4 ( s θ n θ θ ¯ ) 2 RE_{\hat{g}}\,\,=\,\,{{\hat{\sigma}_{g}\,}\over{\hat{g}}}\,\,\,\approx\,\,\,% \sqrt{\,\,\left({{{s_{L}}\over{n_{L}\,\bar{L}}}}\right)^{2}\,\,\,+\,\,\,\,4% \left({{{s_{T}}\over{n_{T}\,\bar{T}}}}\right)^{2}\,\,+\,\,\,\,\left({{{\bar{% \theta}}\over 2}}\right)^{4}\left({{{s_{\theta}}\over{n_{\theta}\,\bar{\theta}% }}}\right)^{2}\,}
  60. MSe = σ 2 + β 2 {\rm MSe}\,\,\,=\,\,\,\sigma^{2}+\,\,\,\beta^{2}
  61. E [ z ] = z PDF z d z Var [ z ] = ( z - E [ z ] ) 2 PDF z d z {\rm E}[z]\,\,\,=\,\,\,\int{z\,\,{\rm PDF}_{z}}\,\,dz\,\,\,\,\,\,\,\,\,\,\,\,% \,{\rm Var}[z]\,\,=\,\,\int{\left({z-{\rm E}[z]}\right)^{2}\,\,{\rm PDF}_{z}}% \,\,dz
  62. Var [ z ] E [ ( z - E [ z ] ) 2 ] {\rm Var}[z]\,\,\,\equiv\,\,{\rm E}\left[{\left({\,z\,\,-\,\,{\rm E}[z]\,}% \right)^{2}}\right]
  63. z ( x 1 x 2 x p ) z ( x ¯ 1 x ¯ 2 x ¯ p ) + i = 1 p z x i | x ¯ i ( x i - x ¯ i ) + 1 2 i = 1 p j = 1 p 2 z x i x j | x ¯ i , x ¯ j ( x i - x ¯ i ) ( x j - x ¯ j ) \begin{aligned}\displaystyle z\left({x_{1}\,\,\,x_{2}\,\cdots\,\,\,x_{p}}% \right)\,\,\,\approx\,\,\,z\left({\bar{x}_{1}\,\,\,\bar{x}_{2}\,\,\cdots\,\,\,% \bar{x}_{p}}\right)\,\,\,\,+\,\,\,\,\sum\limits_{i\,=\,1}^{p}{\left.{{{% \partial z}\over{\partial x_{i}}}}\right|}_{\bar{x}_{i}}\left({x_{i}-\bar{x}_{% i}}\right)\\ \displaystyle\,\,\,\,\,\,\,\,\,\,\,\,+\,\,\,\,{1\over 2}\sum\limits_{i\,=\,1}^% {p}{\sum\limits_{j\,=\,1}^{p}{\left.{{{\partial^{2}z}\over{\partial x_{i}% \partial x_{j}}}}\right|}}_{\bar{x}_{i},\bar{x}_{j}}\left({x_{i}-\bar{x}_{i}}% \right)\left({x_{j}-\bar{x}_{j}}\right)\end{aligned}
  64. z ( x 1 x 2 ) z ( x ¯ 1 x ¯ 2 ) + z x 1 ( x 1 - x ¯ 1 ) + z x 2 ( x 2 - x ¯ 2 ) \displaystyle z\left({x_{1}\,\,x_{2}}\right)\,\,\,\,\approx\,\,\,z\left({\bar{% x}_{1}\,\,\bar{x}_{2}}\right)\,\,\,+\,\,\,\,{{\partial z}\over{\partial x_{1}}% }\left({x_{1}-\,\,\bar{x}_{1}}\right)\,\,\,+\,\,\,{{\partial z}\over{\partial x% _{2}}}\left({x_{2}-\,\,\bar{x}_{2}}\right)
  65. z ( x 1 x 2 ) z ( x ¯ 1 x ¯ 2 ) + z x 1 ( x 1 - x ¯ 1 ) + z x 2 ( x 2 - x ¯ 2 ) + 2 z x 1 x 2 ( x 1 - x ¯ 1 ) ( x 2 - x ¯ 2 ) + 1 2 2 z x 1 2 ( x 1 - x ¯ 1 ) 2 + 1 2 2 z x 2 2 ( x 2 - x ¯ 2 ) 2 \begin{aligned}&\displaystyle z\left({x_{1}\,\,x_{2}}\right)\,\,\,\,\approx\,% \,\,z\left({\bar{x}_{1}\,\,\bar{x}_{2}}\right)\,\,\,+\,\,\,\,{{\partial z}% \over{\partial x_{1}}}\left({x_{1}-\,\,\bar{x}_{1}}\right)\,\,\,+\,\,\,{{% \partial z}\over{\partial x_{2}}}\left({x_{2}-\,\,\bar{x}_{2}}\right)\,\,\,+\,% \,\,{{\partial^{2}z}\over{\partial x_{1}\partial x_{2}}}\left({x_{1}-\,\,\bar{% x}_{1}}\right)\left({x_{2}-\,\,\bar{x}_{2}}\right)\\ &\displaystyle\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,% \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,+\,\,\,{1\over 2}\,\,{{\partial^{2}z}\over{% \partial x_{1}^{2}}}\left({x_{1}-\,\,\bar{x}_{1}}\right)^{2}\,\,\,+\,\,\,\,{1% \over 2}\,\,{{\partial^{2}z}\over{\partial x_{2}^{2}}}\left({x_{2}-\,\,\bar{x}% _{2}}\right)^{2}\end{aligned}
  66. E [ z ( x ¯ 1 x ¯ 2 ) ] = z ( μ 1 μ 2 ) E [ z x 1 ( x 1 - x ¯ 1 ) ] = z x 1 E [ ( x 1 - x ¯ 1 ) ] = 0 {\rm E}\left[{z\left({\bar{x}_{1}\,\,\,\bar{x}_{2}}\right)}\right]\,\,\,=\,\,% \,z\left({\mu_{1}\,\,\,\mu_{2}\,}\right)\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\rm E% }\left[{{{\partial z}\over{\partial x_{1}}}\left({x_{1}-\,\,\bar{x}_{1}}\right% )}\right]\,\,\,\,\,=\,\,\,\,\,{{\partial z}\over{\partial x_{1}}}{\rm E}\left[% {\left({x_{1}-\,\,\bar{x}_{1}}\right)}\right]\,\,\,=\,\,0
  67. E [ x i - x ¯ i ] = E [ x i ] - E [ x ¯ i ] = μ i - μ i = 0 {\rm E}\left[{x_{i}-\bar{x}_{i}}\right]\,\,\,=\,\,\,{\rm E}\left[{x_{i}}\right% ]\,\,\,-\,\,\,{\rm E}\left[{\bar{x}_{i}}\right]\,\,\,=\,\,\,\mu_{i}-\,\,\mu_{i% }\,\,\,=\,\,\,0
  68. E [ 1 2 2 z x 1 2 ( x 1 - x ¯ 1 ) 2 ] = 1 2 2 z x 1 2 E [ ( x 1 - x ¯ 1 ) 2 ] = 1 2 2 z x 1 2 σ 1 2 {\rm E}\left[{{1\over 2}{{\partial^{2}z}\over{\partial x_{1}^{2}}}\left({x_{1}% -\,\,\bar{x}_{1}}\right)^{2}}\right]\,\,\,=\,\,\,{1\over 2}\,{{\partial^{2}z}% \over{\partial x_{1}^{2}}}\,{\rm E}\left[{\left({x_{1}-\,\,\bar{x}_{1}}\right)% ^{2}}\right]\,\,\,=\,\,\,{1\over 2}\,{{\partial^{2}z}\over{\partial x_{1}^{2}}% }\sigma_{1}^{2}
  69. E [ 2 z x 1 x 2 ( x 1 - x ¯ 1 ) ( x 2 - x ¯ 2 ) ] = 2 z x 1 x 2 E [ ( x 1 - x ¯ 1 ) ( x 2 - x ¯ 2 ) ] = 2 z x 1 x 2 σ 1 , 2 {\rm E}\left[{{{\partial^{2}z}\over{\partial x_{1}\partial x_{2}}}\left({x_{1}% -\,\,\bar{x}_{1}}\right)\left({x_{2}-\,\,\bar{x}_{2}}\right)}\right]\,\,\,=\,% \,\,{{\partial^{2}z}\over{\partial x_{1}\partial x_{2}}}\,{\rm E}\left[{\left(% {x_{1}-\,\,\bar{x}_{1}}\right)\left({x_{2}-\,\,\bar{x}_{2}}\right)}\right]\,\,% \,=\,\,\,{{\partial^{2}z}\over{\partial x_{1}\partial x_{2}}}\sigma_{1,2}
  70. E [ z ] z ( μ 1 μ 2 ) + 1 2 { 2 z x 1 2 σ 1 2 + 2 z x 2 2 σ 2 2 } + 2 z x 1 x 2 σ 1 , 2 {\rm E}[z]\approx\,\,\,z\left({\mu_{1}\,\,\mu_{2}}\right)\,\,\,+\,\,\,{1\over 2% }\left\{{{{\partial^{2}z}\over{\partial x_{1}^{2}}}\,\,\sigma_{1}^{2}\,\,+\,\,% \,{{\partial^{2}z}\over{\partial x_{2}^{2}}}\,\,\sigma_{2}^{2}}\right\}\,\,\,+% \,\,\,{{\partial^{2}z}\over{\partial x_{1}\partial x_{2}}}\,\,\sigma_{1,2}
  71. ( z - E [ z ] ) 2 [ { < m t p l > z x 1 ( x 1 - x ¯ 1 ) + z x 2 ( x 2 - x ¯ 2 ) } + 2 z x 1 x 2 [ ( x 1 - x ¯ 1 ) ( x 2 - x ¯ 2 ) - σ 1 , 2 ] + 1 2 2 z x 1 2 [ ( x 1 - x ¯ 1 ) 2 - σ 1 2 ] + 1 2 2 z x 2 2 [ ( x 2 - x ¯ 2 ) 2 - σ 2 2 ] ] 2 \begin{array}[]{l}\left({z-{\rm E}[z]}\right)^{2}\approx\,\,\,\left[\begin{% array}[]{l}\left\{{\frac{<}{m}tpl>{{\partial z}}{{\partial x_{1}}}\left({x_{1}% -\,\,\bar{x}_{1}}\right)\,\,+\,\,\,\frac{{\partial z}}{{\partial x_{2}}}\left(% {x_{2}-\,\,\bar{x}_{2}}\right)}\right\}\,\,\,+\\ \,\,\,\frac{{\partial^{2}z}}{{\partial x_{1}\partial x_{2}}}\left[{\left({x_{1% }-\,\,\bar{x}_{1}}\right)\left({x_{2}-\,\,\bar{x}_{2}}\right)\,\,-\,\,\sigma_{% 1,2}}\right]\,\,\,+\\ \,\,\,\frac{1}{2}\frac{{\partial^{2}z}}{{\partial x_{1}^{2}}}\left[{\left({x_{% 1}-\,\,\bar{x}_{1}}\right)^{2}-\,\,\sigma_{1}^{2}}\right]\,\,\,+\,\,\,\frac{1}% {2}\frac{{\partial^{2}z}}{{\partial x_{2}^{2}}}\left[{\left({x_{2}-\,\,\bar{x}% _{2}}\right)^{2}-\,\,\sigma_{2}^{2}}\right]\\ \end{array}\right]^{2}\\ \\ \end{array}
  72. ( z - E [ z ] ) 2 ( < m t p l > z x 1 ) 2 ( x 1 - x ¯ 1 ) 2 + ( z x 2 ) 2 ( x 2 - x ¯ 2 ) 2 + 2 ( z x 1 ) ( z x 2 ) ( x 1 - x ¯ 1 ) ( x 2 - x ¯ 2 ) \left({z\,\,-\,\,{\rm E}[z]}\right)^{2}\approx\,\,\,\left({\frac{<}{m}tpl>{{% \partial z}}{{\partial x_{1}}}}\right)^{2}\left({x_{1}-\bar{x}_{1}}\right)^{2}% \,\,+\,\,\,\,\left({\frac{{\partial z}}{{\partial x_{2}}}}\right)^{2}\left({x_% {2}-\bar{x}_{2}}\right)^{2}\,\,+\,\,\,2\left({\frac{{\partial z}}{{\partial x_% {1}}}}\right)\left({\frac{{\partial z}}{{\partial x_{2}}}}\right)\left({x_{1}-% \bar{x}_{1}}\right)\left({x_{2}-\bar{x}_{2}}\right)
  73. Var [ z ] E [ ( z - E [ z ] ) 2 ] ( < m t p l > z x 1 ) 2 E [ ( x 1 - x ¯ 1 ) 2 ] + ( z x 2 ) 2 E [ ( x 2 - x ¯ 2 ) 2 ] + 2 ( z x 1 ) ( z x 2 ) E [ ( x 1 - x ¯ 1 ) ( x 2 - x ¯ 2 ) ] \begin{array}[]{l}{\rm Var}[z]\,\,\equiv\,{\rm E}\left[{\left({z\,\,-\,\,{\rm E% }[z]}\right)^{2}}\right]\,\,\,\approx\,\,\,\left({\frac{<}{m}tpl>{{\partial z}% }{{\partial x_{1}}}}\right)^{2}{\rm E}\left[{\left({x_{1}-\bar{x}_{1}}\right)^% {2}}\right]\,\,+\\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left({\frac{{\partial z}}{{\partial x% _{2}}}}\right)^{2}{\rm E}\left[{\left({x_{2}-\bar{x}_{2}}\right)^{2}}\right]\,% \,\,\,\,+\,\,\,\,2\left({\frac{{\partial z}}{{\partial x_{1}}}}\right)\left({% \frac{{\partial z}}{{\partial x_{2}}}}\right){\rm E}\left[{\left({x_{1}-\bar{x% }_{1}}\right)\left({x_{2}-\bar{x}_{2}}\right)}\right]\\ \end{array}
  74. Var [ z ] ( < m t p l > z x 1 ) 2 σ 1 2 + ( z x 2 ) 2 σ 2 2 + 2 ( z x 1 ) ( z x 2 ) σ 1 , 2 {\rm Var}[z]\,\,\,\,\approx\,\,\,\left({\frac{<}{m}tpl>{{\partial z}}{{% \partial x_{1}}}}\right)^{2}\sigma_{1}^{2}\,\,\,+\,\,\,\,\left({\frac{{% \partial z}}{{\partial x_{2}}}}\right)^{2}\sigma_{2}^{2}\,\,\,+\,\,\,\,2\left(% {\frac{{\partial z}}{{\partial x_{1}}}}\right)\left({\frac{{\partial z}}{{% \partial x_{2}}}}\right)\sigma_{1,2}
  75. Var [ z ] i = 1 p j = 1 p ( < m t p l > z x i ) ( z x j ) σ i , j {\rm Var}[z]\,\,\,\approx\,\,\,\sum\limits_{i=1}^{p}{\sum\limits_{j=1}^{p}{% \left({\frac{<}{m}tpl>{{\partial z}}{{\partial x_{i}}}}\right)}}\left({\frac{{% \partial z}}{{\partial x_{j}}}}\right)\sigma_{i,j}
  76. E [ z ] z ( x ¯ 1 x ¯ 2 ) + 1 2 { < m t p l > 2 z x 1 2 | x ¯ 1 σ 1 2 n 1 + 2 z x 2 2 | x ¯ 2 σ 2 2 n 2 } + 2 z x 1 x 2 | x ¯ 1 , x ¯ 2 σ 1 , 2 n 1 , 2 {\rm E}[z]\approx\,\,\,z\left({\bar{x}_{1}\,\,\bar{x}_{2}}\right)\,\,\,+\,\,\,% \frac{1}{2}\left\{{\left.{\frac{<}{m}tpl>{{\partial^{2}z}}{{\partial x_{1}^{2}% }}}\right|_{\bar{x}_{1}}\,\,{\sigma_{1}^{2}\over n_{1}}\,\,\,\,+\,\,\,\,\,% \left.{\frac{{\partial^{2}z}}{{\partial x_{2}^{2}}}}\right|_{\bar{x}_{2}}\,{% \sigma_{2}^{2}\over n_{2}}}\right\}\,\,\,+\,\,\,\,\left.{\frac{{\partial^{2}z}% }{{\partial x_{1}\partial x_{2}}}}\right|_{\bar{x}_{1},\bar{x}_{2}}\,\,{\sigma% _{1,2}\over n_{1,2}}
  77. Var [ z ] i = 1 p j = 1 p ( < m t p l > z x i ) x ¯ i ( z x j ) x ¯ j σ i , j n i , j {\rm Var}[z]\,\,\,\approx\,\,\,\sum\limits_{i=1}^{p}{\,\sum\limits_{j=1}^{p}{% \,\left({\frac{<}{m}tpl>{{\partial z}}{{\partial x_{i}}}}\right)_{\bar{x}_{i}}% }}\left({\frac{{\partial z}}{{\partial x_{j}}}}\right)_{\bar{x}_{j}}{\sigma_{i% ,j}\over n_{i,j}}
  78. z = a x r x N ( μ , σ 2 ) a , r constants z\,\,=\,\,a\,x^{r}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,% \,x\,\,\sim\,\,N\left({\mu,\,\,\sigma^{2}}\right)\,\,\,\,\,\,\,a,r\,\,{\rm constants% }\,\,\,\,
  79. Δ z a r μ r - 1 Δ x \Delta z\,\,\approx\,\,a\,r\,\mu^{r-1}\,\Delta x
  80. Δ z z r Δ x μ {{\Delta z}\over z}\,\,\,\approx\,\,\,r\,\,{{\Delta x}\over\mu}
  81. E [ z ] = μ z a μ r + 1 2 a r ( r - 1 ) μ r - 2 σ 2 n {\rm E}[z]\,\,\,=\,\,\,\mu_{z}\approx\,\,a\mu^{r}\,\,\,+\,\,\,{1\over 2}a\,r% \left({r-1}\right)\,\,\mu^{r-2}\,\,{{\sigma^{2}}\over n}
  82. β a r ( r - 1 ) μ r 2 n ( < m t p l > σ μ ) 2 \beta\,\,\approx\,\,{{a\,r\left({r-1}\right)\mu^{r}}\over{2\,n}}\left(<mtpl>{{% \sigma\over\mu}}\right)^{2}
  83. β z r ( r - 1 ) 2 n ( < m t p l > σ μ ) 2 {\beta\over z}\,\,\,\approx\,\,\,{{r\left({r-1}\right)}\over{2\,n}}\,\,\left(<% mtpl>{{\sigma\over\mu}}\right)^{2}
  84. σ z 2 ( a r μ r - 1 ) 2 σ 2 n = ( a r μ r ) 2 n ( < m t p l > σ μ ) 2 \sigma_{z}^{2}\approx\,\,\,\left({a\,r\,\mu^{r-1}}\right)^{2}{{\sigma^{2}}% \over n}\,\,\,=\,\,\,\,{{\left({a\,r\,\mu^{r}}\right)^{2}}\over n}\left(<mtpl>% {{\sigma\over\mu}}\right)^{2}
  85. σ z z a 2 r 2 μ 2 r - 2 σ 2 n a 2 μ 2 r = r n ( < m t p l > σ μ ) {{\sigma_{z}}\over z}\,\,\,\,\,\approx\,\,\,\,\sqrt{{{a^{2}\,r^{2}\,\mu^{2r-2}% \,\,{{\sigma^{2}}\over n}}\over{a^{2}\,\mu^{2r}}}}\,\,\,\,\,=\,\,\,\,\,{r\over% {\sqrt{n}}}\left(<mtpl>{{\sigma\over\mu}}\right)
  86. z = a e b x x N ( μ , σ 2 ) a , b constants z\,\,=\,\,a\,e^{b\,x}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,% \,\,\,x\,\,\sim\,\,N\left({\mu,\,\,\sigma^{2}}\right)\,\,\,\,\,\,\,a,b\,\,{\rm constants}
  87. Δ z a b e b μ Δ x \Delta z\,\,\approx\,\,a\,b\,e^{b\,\mu}\,\Delta x
  88. < m t p l > Δ z z b Δ x \frac{<}{m}tpl>{{\Delta z}}{z}\,\,\,\approx\,\,\,b\,\Delta x
  89. E [ z ] = μ z a e b μ + 1 2 a b 2 e b μ < m t p l > σ 2 n {\rm E}[z]\,\,\,=\,\,\,\mu_{z}\approx\,\,ae^{b\,\mu}\,\,\,+\,\,\,\frac{1}{2}\,% \,a\,b^{2}e^{b\,\mu}\,\,\frac{<}{m}tpl>{{\sigma^{2}}}{n}
  90. β 1 2 a b 2 e b μ < m t p l > σ 2 n \beta\,\,\approx\,\,\frac{1}{2}\,\,a\,b^{2}e^{b\,\mu}\,\,\frac{<}{m}tpl>{{% \sigma^{2}}}{n}
  91. β z < m t p l > b 2 σ 2 2 n \frac{\beta}{z}\,\,\,\approx\,\,\,\frac{<}{m}tpl>{{b^{2}\sigma^{2}}}{{2\,n}}
  92. σ z 2 ( a b e b μ ) 2 < m t p l > σ 2 n \sigma_{z}^{2}\approx\,\,\,\left({a\,b\,e^{b\mu}}\right)^{2}\,\,\frac{<}{m}tpl% >{{\sigma^{2}}}{n}
  93. < m t p l > σ z z ( a b e b μ ) 2 σ 2 n a 2 e 2 b μ = b σ n \frac{<}{m}tpl>{{\sigma_{z}}}{z}\,\,\,\,\,\approx\,\,\,\,\sqrt{\frac{{\left({a% \,b\,e^{b\mu}}\right)^{2}\,\,\frac{{\sigma^{2}}}{n}}}{{a^{2}e^{2b\mu}}}}\,\,\,% \,=\,\,\,\,b\,\,\frac{\sigma}{n}
  94. z = a ln ( b x ) x N ( μ , σ 2 ) a , b constants z\,\,=\,\,a\,\ln\,(b\,x)\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,% \,\,x\,\,\sim\,\,N\left({\mu,\,\,\sigma^{2}}\right)\,\,\,\,\,\,\,a,b\,\,{\rm constants}
  95. Δ z a < m t p l > Δ x μ \Delta z\,\,\approx\,\,a\,\,\frac{<}{m}tpl>{{\Delta x}}{\mu}
  96. < m t p l > Δ z z Δ x μ ln ( b μ ) \frac{<}{m}tpl>{{\Delta z}}{z}\,\,\,\approx\,\,\,\frac{{\Delta x}}{{\mu\,\,\ln% (b\,\mu)}}
  97. E [ z ] = μ z a ln ( b μ ) - 1 2 a < m t p l > μ 2 σ 2 n {\rm E}[z]\,\,\,=\,\,\,\mu_{z}\approx\,\,a\ln(b\mu)\,\,\,-\,\,\,\,\frac{1}{2}% \,\,\frac{a}{<}mtpl>{{\mu^{2}}}\,\,\frac{{\sigma^{2}}}{n}
  98. β - a < m t p l > 2 n ( σ μ ) 2 \beta\,\,\approx\,\,\,-\,\,\,\frac{a}{<}mtpl>{{2\,n}}\left({\frac{\sigma}{\mu}% }\right)^{2}
  99. β z - 1 < m t p l > 2 n ln ( b μ ) ( σ μ ) 2 \frac{\beta}{z}\,\,\,\approx\,\,\,-\,\,\frac{1}{<}mtpl>{{2\,n\,\,\ln(b\mu)}}% \left({\frac{\sigma}{\mu}}\right)^{2}
  100. σ z 2 < m t p l > a 2 n ( σ μ ) 2 σ z a n ( σ μ ) \sigma_{z}^{2}\approx\,\,\,\frac{<}{m}tpl>{{a^{2}\,}}{n}\,\left({\frac{\sigma}% {\mu}}\right)^{2}\,\,\,\,\,\,\,\Rightarrow\,\,\,\,\,\,\sigma_{z}\approx\,\,\,% \frac{a}{{\sqrt{n}}}\,\,\left({\frac{\sigma}{\mu}}\right)
  101. < m t p l > σ z z 1 n ln ( b μ ) ( σ μ ) \frac{<}{m}tpl>{{\sigma_{z}}}{z}\,\,\,\,\,\approx\,\,\,\,\,\frac{1}{{\sqrt{n}% \,\,\,\ln\,(b\mu)}}\,\,\left({\frac{\sigma}{\mu}}\right)
  102. z = a x 1 + b x 2 [ x 1 x 2 ] B V N ( μ 1 , μ 2 , σ 1 2 , σ 2 2 , σ 1 , 2 ) a , b constants z\,\,=\,\,a\,x_{1}\,+\,\,b\,x_{2}\,\,\,\,\,\,\,\,\,\,\left[{x_{1}\,\,x_{2}}% \right]\,\,\sim\,\,BVN\left({\mu_{1},\,\,\mu_{2},\,\,\sigma_{1}^{2},\,\,\sigma% _{2}^{2},\,\,\sigma_{1,2}}\right)\,\,\,\,\,\,\,a,b\,\,{\rm constants}
  103. Δ z a Δ x 1 + b Δ x 2 \Delta z\,\,\approx\,\,a\,\Delta x_{1}+\,\,b\,\Delta x_{2}
  104. < m t p l > Δ z z a Δ x 1 + b Δ x 2 a μ 1 + b μ 2 \frac{<}{m}tpl>{{\Delta z}}{z}\,\,\,\approx\,\,\,\frac{{a\,\Delta x_{1}+\,\,\,% b\,\Delta x_{2}}}{{a\mu_{1}+\,\,b\mu_{2}}}
  105. E [ z ] = μ z a μ 1 + b μ 2 {\rm E}[z]\,\,\,=\,\,\,\mu_{z}\,\,\approx\,\,\,\,a\mu_{1}+\,\,b\mu_{2}
  106. β = 0 \beta\,\,=\,\,\,0
  107. β z = 0 \frac{\beta}{z}\,\,\,=\,\,\,0
  108. σ z 2 1 n [ a 2 σ 1 2 + b 2 σ 2 2 + 2 a b σ 1 , 2 ] \sigma_{z}^{2}\approx\,\,\,\frac{1}{n}\,\,\left[{a^{2}\sigma_{1}^{2}\,\,+\,\,% \,b^{2}\sigma_{2}^{2}\,\,\,+\,\,\,2\,a\,b\,\sigma_{1,2}}\right]
  109. z = a x 1 α x 2 β [ x 1 x 2 ] B V N ( μ 1 , μ 2 , σ 1 2 , σ 2 2 , σ 1 , 2 ) α , β constants z\,\,=\,\,a\,\,x_{1}^{\alpha}\,x_{2}^{\beta}\,\,\,\,\,\,\,\,\,\,\left[{x_{1}\,% \,x_{2}}\right]\,\,\sim\,\,BVN\left({\mu_{1},\,\,\mu_{2},\,\,\sigma_{1}^{2},\,% \,\sigma_{2}^{2},\,\,\sigma_{1,2}}\right)\,\,\,\,\,\,\,\alpha,\beta\,\,{\rm constants}
  110. Δ z ( a α μ 1 α - 1 μ 2 β ) Δ x 1 + ( a β μ 1 α μ 2 β - 1 ) Δ x 2 \Delta z\,\,\approx\,\,\left({a\,\alpha\,\mu_{1}^{\alpha-1}\,\mu_{2}^{\beta}}% \right)\Delta x_{1}\,\,\,+\,\,\,\,\left({a\,\beta\,\mu_{1}^{\alpha}\mu_{2}^{% \beta-1}}\right)\Delta x_{2}
  111. < m t p l > Δ z z α Δ x 1 μ 1 + β Δ x 2 μ 2 \frac{<}{m}tpl>{{\Delta z}}{z}\,\,\,\approx\,\,\,\alpha\frac{{\Delta x_{1}}}{{% \mu_{1}}}\,\,\,+\,\,\,\beta\frac{{\Delta x_{2}}}{{\mu_{2}}}
  112. E [ z ] = μ z a μ 1 α μ 2 β + a 2 n [ ( α ( α - 1 ) μ 1 α - 2 μ 2 β ) σ 1 2 + ( β ( β - 1 ) μ 1 α μ 2 β - 2 ) σ 2 2 + ( 2 α β μ 1 α - 1 μ 2 β - 1 ) σ 1 , 2 ] {\rm E}[z]\,\,\,=\,\,\,\mu_{z}\,\,\approx\,\,\,\,a\mu_{1}^{\alpha}\mu_{2}^{% \beta}\,\,+\,\,\,\frac{a}{2n}\left[\begin{array}[]{l}\left({\alpha\left({% \alpha-1}\right)\mu_{1}^{\alpha-2}\mu_{2}^{\beta}}\right)\sigma_{1}^{2}+\\ \left({\beta\left({\beta-1}\right)\mu_{1}^{\alpha}\mu_{2}^{\beta-2}}\right)% \sigma_{2}^{2}+\\ \left({2\,\alpha\,\beta\,\mu_{1}^{\alpha-1}\,\mu_{2}^{\beta-1}}\right)\sigma_{% 1,2}\\ \end{array}\right]
  113. β a 2 n [ ( α ( α - 1 ) μ 1 α - 2 μ 2 β ) σ 1 2 + ( β ( β - 1 ) μ 1 α μ 2 β - 2 ) σ 2 2 + ( 2 α β μ 1 α - 1 μ 2 β - 1 ) σ 1 , 2 ] \beta\,\,\,\,\approx\,\,\,\,\frac{a}{2n}\left[\begin{array}[]{l}\left({\alpha% \left({\alpha-1}\right)\mu_{1}^{\alpha-2}\mu_{2}^{\beta}}\right)\sigma_{1}^{2}% +\\ \left({\beta\left({\beta-1}\right)\mu_{1}^{\alpha}\mu_{2}^{\beta-2}}\right)% \sigma_{2}^{2}+\\ \left({2\,\alpha\,\beta\,\mu_{1}^{\alpha-1}\,\mu_{2}^{\beta-1}}\right)\sigma_{% 1,2}\\ \end{array}\right]
  114. β z = 1 < m t p l > 2 n [ α ( α - 1 ) ( σ 1 μ 1 ) 2 + β ( β - 1 ) ( σ 2 μ 2 ) 2 + 2 α β ( σ 1 , 2 μ 1 μ 2 ) ] \frac{\beta}{z}\,\,\,=\,\,\,\frac{1}{<}mtpl>{{2n}}\left[{\alpha\left({\alpha-1% }\right)\left({\frac{{\sigma_{1}}}{{\mu_{1}}}}\right)^{2}+\,\,\,\beta\left({% \beta-1}\right)\left({\frac{{\sigma_{2}}}{{\mu_{2}}}}\right)^{2}\,\,+\,\,\,\,2% \,\alpha\,\beta\left({\frac{{\sigma_{1,2}}}{{\mu_{1}\,\mu_{2}}}}\right)\,}\right]
  115. σ z 2 a 2 n [ ( α μ 1 α - 1 μ 2 β ) 2 σ 1 2 + ( β μ 1 α μ 2 β - 1 ) 2 σ 2 2 + ( 2 α β μ 1 2 α - 1 μ 2 2 β - 1 ) σ 1 , 2 ] \sigma_{z}^{2}\approx\,\,\,\frac{a^{2}}{n}\,\,\left[{\left(\alpha\,\mu_{1}^{% \alpha-1}\mu_{2}^{\beta}\right)^{2}\sigma_{1}^{2}\,\,\,+\,\,\,\left(\beta\,\mu% _{1}^{\alpha}\mu_{2}^{\beta-1}\right)^{2}\sigma_{2}^{2}\,\,\,+\,\,\,\left(2% \alpha\,\beta\,\mu_{1}^{2\alpha-1}\mu_{2}^{2\beta-1}\right)\sigma_{1,2}}\right]
  116. σ z z α 2 n ( σ 1 μ 1 ) 2 + β 2 n ( σ 2 μ 2 ) 2 + 2 α β n ( σ 1 , 2 μ 1 μ 2 ) \frac{\sigma_{z}}{z}\,\,\,\approx\,\,\,\sqrt{\,\,\frac{\alpha^{2}}{n}\left(% \frac{\sigma_{1}}{\mu_{1}}\right)^{2}+\,\,\,\,\frac{\beta^{2}}{n}\left(\frac{% \sigma_{2}}{\mu_{2}}\right)^{2}\,\,+\,\,\,\frac{2\,\alpha\,\beta}{n}\left(% \frac{\sigma_{1,2}}{\mu_{1}\,\mu_{2}}\right)}

Exponential-logarithmic_distribution.html

  1. 1 - ln ( 1 - ( 1 - p ) e - β x ) ln p 1-\frac{\ln(1-(1-p)e^{-\beta x})}{\ln p}
  2. - polylog ( 2 , 1 - p ) β ln p -\frac{\,\text{polylog}(2,1-p)}{\beta\ln p}
  3. ln ( 1 + p ) β \frac{\ln(1+\sqrt{p})}{\beta}
  4. - 2 polylog ( 3 , 1 - p ) β 2 ln p -\frac{2\,\text{polylog}(3,1-p)}{\beta^{2}\ln p}
  5. - polylog 2 ( 2 , 1 - p ) β 2 ln 2 p -\frac{\,\text{polylog}^{2}(2,1-p)}{\beta^{2}\ln^{2}p}
  6. - β ( 1 - p ) ln p ( β - t ) hypergeom 2 , 1 -\frac{\beta(1-p)}{\ln p(\beta-t)}\,\text{hypergeom}_{2,1}
  7. ( [ 1 , β - t β ] , [ 2 β - t β ] , 1 - p ) ([1,\frac{\beta-t}{\beta}],[\frac{2\beta-t}{\beta}],1-p)
  8. p ( 0 , 1 ) p\in(0,1)
  9. β > 0 \beta>0
  10. f ( x ; p , β ) := ( 1 - ln p ) β ( 1 - p ) e - β x 1 - ( 1 - p ) e - β x f(x;p,\beta):=\left(\frac{1}{-\ln p}\right)\frac{\beta(1-p)e^{-\beta x}}{1-(1-% p)e^{-\beta x}}
  11. p ( 0 , 1 ) p\in(0,1)
  12. β > 0 \beta>0
  13. x x
  14. x x\rightarrow\infty
  15. β ( 1 - p ) - p ln p \frac{\beta(1-p)}{-p\ln p}
  16. β \beta
  17. p 1 p\rightarrow 1
  18. F ( x ; p , β ) = 1 - ln ( 1 - ( 1 - p ) e - β x ) ln p , F(x;p,\beta)=1-\frac{\ln(1-(1-p)e^{-\beta x})}{\ln p},
  19. x median = ln ( 1 + p ) β x\text{median}=\frac{\ln(1+\sqrt{p})}{\beta}
  20. X X
  21. M X ( t ) = E ( e t X ) = - β ( 1 - p ) ln p ( β - t ) F 2 , 1 ( [ 1 , β - t β ] , [ 2 β - t β ] , 1 - p ) , M_{X}(t)=E(e^{tX})=-\frac{\beta(1-p)}{\ln p(\beta-t)}F_{2,1}\left(\left[1,% \frac{\beta-t}{\beta}\right],\left[\frac{2\beta-t}{\beta}\right],1-p\right),
  22. F 2 , 1 F_{2,1}
  23. F N , D ( n , d , z ) F_{N,D}({n,d},z)
  24. F N , D ( n , d , z ) := k = 0 z k i = 1 p Γ ( n i + k ) Γ - 1 ( n i ) Γ ( k + 1 ) i = 1 q Γ ( d i + k ) Γ - 1 ( d i ) F_{N,D}(n,d,z):=\sum_{k=0}^{\infty}\frac{z^{k}\prod_{i=1}^{p}\Gamma(n_{i}+k)% \Gamma^{-1}(n_{i})}{\Gamma(k+1)\prod_{i=1}^{q}\Gamma(d_{i}+k)\Gamma^{-1}(d_{i})}
  25. n = [ n 1 , n 2 , , n N ] n=[n_{1},n_{2},\dots,n_{N}]
  26. d = [ d 1 , d 2 , , d D ] {d}=[d_{1},d_{2},\dots,d_{D}]
  27. X X
  28. M X ( t ) M_{X}(t)
  29. r r\in\mathbb{N}
  30. E ( X r ; p , β ) = - r ! Li r + 1 ( 1 - p ) β r ln p , E(X^{r};p,\beta)=-r!\frac{\operatorname{Li}_{r+1}(1-p)}{\beta^{r}\ln p},
  31. Li a ( z ) \operatorname{Li}_{a}(z)
  32. Li a ( z ) = k = 1 z k k a . \operatorname{Li}_{a}(z)=\sum_{k=1}^{\infty}\frac{z^{k}}{k^{a}}.
  33. E ( X ) = - Li 2 ( 1 - p ) β ln p , E(X)=-\frac{\operatorname{Li}_{2}(1-p)}{\beta\ln p},
  34. Var ( X ) = - 2 Li 3 ( 1 - p ) β 2 ln p - ( Li 2 ( 1 - p ) β ln p ) 2 . \operatorname{Var}(X)=-\frac{2\operatorname{Li}_{3}(1-p)}{\beta^{2}\ln p}-% \left(\frac{\operatorname{Li}_{2}(1-p)}{\beta\ln p}\right)^{2}.
  35. s ( x ) = ln ( 1 - ( 1 - p ) e - β x ) ln p , s(x)=\frac{\ln(1-(1-p)e^{-\beta x})}{\ln p},
  36. h ( x ) = - β ( 1 - p ) e - β x ( 1 - ( 1 - p ) e - β x ) ln ( 1 - ( 1 - p ) e - β x ) . h(x)=\frac{-\beta(1-p)e^{-\beta x}}{(1-(1-p)e^{-\beta x})\ln(1-(1-p)e^{-\beta x% })}.
  37. m ( x 0 ; p , β ) = E ( X - x 0 | X x 0 ; β , p ) = - Li 2 ( 1 - ( 1 - p ) e - β x 0 ) β ln ( 1 - ( 1 - p ) e - β x 0 ) m(x_{0};p,\beta)=E(X-x_{0}|X\geq x_{0};\beta,p)=-\frac{\operatorname{Li}_{2}(1% -(1-p)e^{-\beta x_{0}})}{\beta\ln(1-(1-p)e^{-\beta x_{0}})}
  38. Li 2 \operatorname{Li}_{2}
  39. X = 1 β ln ( 1 - p 1 - p U ) . X=\frac{1}{\beta}\ln\left(\frac{1-p}{1-p^{U}}\right).
  40. β ( h + 1 ) = n ( i = 1 n x i 1 - ( 1 - p ( h ) ) e - β ( h ) x i ) - 1 , \beta^{(h+1)}=n\left(\sum_{i=1}^{n}\frac{x_{i}}{1-(1-p^{(h)})e^{-\beta^{(h)}x_% {i}}}\right)^{-1},
  41. p ( h + 1 ) = - n ( 1 - p ( h + 1 ) ) ln ( p ( h + 1 ) ) i = 1 n { 1 - ( 1 - p ( h ) ) e - β ( h ) x i } - 1 . p^{(h+1)}=\frac{-n(1-p^{(h+1)})}{\ln(p^{(h+1)})\sum_{i=1}^{n}\{1-(1-p^{(h)})e^% {-\beta^{(h)}x_{i}}\}^{-1}}.

Exponential_map_(discrete_dynamical_systems).html

  1. E c : z e z + c E_{c}:z\to e^{z}+c\,
  2. E λ : z λ * e z E_{\lambda}:z\to\lambda*e^{z}
  3. λ * e z . = e z + l n ( λ ) \lambda*e^{z}.=e^{z+ln(\lambda)}
  4. E λ : z e z + l n ( λ ) E_{\lambda}:z\to e^{z}+ln(\lambda)
  5. z = z + l n ( λ ) z=z+ln(\lambda)

Exponential_mechanism_(differential_privacy).html

  1. n n\,\!
  2. 𝒟 \mathcal{D}\,\!
  3. \mathcal{R}\,\!
  4. D D\,\!
  5. R R\,\!
  6. 𝒟 \mathcal{D}\,\!
  7. \mathcal{R}\,\!
  8. μ \mu\,\!
  9. \mathcal{R}\,\!
  10. q : 𝒟 n × q:\mathcal{D}^{n}\times\mathcal{R}\rightarrow\mathbb{R}\,\!
  11. ( d , r ) (d,r)\,\!
  12. d 𝒟 n d\in\mathcal{D}^{n}\,\!
  13. r r\in\mathcal{R}\,\!
  14. ( d , r ) (d,r)\,\!
  15. d 𝒟 n d\in\mathcal{D}^{n}\,\!
  16. r r\in\mathcal{R}\,\!
  17. q ( d , r ) q(d,r)\,\!
  18. q ϵ ( d ) \mathcal{E}_{q}^{\epsilon}(d)\,\!
  19. q : ( 𝒟 n × ) q:(\mathcal{D}^{n}\times\mathcal{R})\rightarrow\mathbb{R}\,\!
  20. μ \mu\,\!
  21. \mathcal{R}\,\!
  22. q ϵ ( d ) := \mathcal{E}_{q}^{\epsilon}(d):=\,\!
  23. r r\,\!
  24. e ϵ q ( d , r ) × μ ( r ) e^{\epsilon q(d,r)}\times\mu(r)\,\!
  25. d 𝒟 n , r R d\in\mathcal{D}^{n},r\in R\,\!
  26. r r\,\!
  27. q ( d , r ) q(d,r)\,\!
  28. μ \mu\,\!
  29. r r\,\!
  30. q ( d , r ) q(d,r)\,\!
  31. q ϵ ( d ) \mathcal{E}_{q}^{\epsilon}(d)\,\!
  32. r e ϵ q ( d , r ) × μ ( r ) \int_{r}e^{\epsilon q(d,r)}\times\mu(r)\,\!
  33. q ϵ ( d ) \mathcal{E}_{q}^{\epsilon}(d)\,\!
  34. ( 2 ϵ Δ q ) (2\epsilon\Delta q)\,\!
  35. q ϵ ( d ) \mathcal{E}_{q}^{\epsilon}(d)\,\!
  36. r r\,\!
  37. e ϵ q ( d , r ) μ ( r ) e ϵ q ( d , r ) μ ( r ) d r \frac{e^{\epsilon q(d,r)}\mu(r)}{\int e^{\epsilon q(d,r)}\mu(r)dr}\,\!
  38. d d\,\!
  39. q q\,\!
  40. Δ q \Delta q\,\!
  41. e ϵ Δ q e^{\epsilon\Delta q}\,\!
  42. e - ϵ Δ q e^{-\epsilon\Delta q}\,\!
  43. d d\,\!
  44. exp ( 2 ϵ Δ q ) \exp(2\epsilon\Delta q)\,\!
  45. r r\,\!
  46. q ϵ ( d ) \mathcal{E}_{q}^{\epsilon}(d)\,\!
  47. q ( d , r ) q(d,r)\,\!
  48. max r q ( d , r ) \max_{r}q(d,r)\,\!
  49. O P T OPT\,\!
  50. O P T OPT\,\!
  51. μ \mu
  52. r r\,\!
  53. q q\,\!
  54. S t = { r : q ( d , r ) > O P T - t } S_{t}=\{r:q(d,r)>OPT-t\}\,\!
  55. S ¯ 2 t = { r : q ( d , r ) O P T - 2 t } \bar{S}_{2t}=\{r:q(d,r)\leq OPT-2t\}\,\!
  56. p ( S ¯ 2 t ) p(\bar{S}_{2t})\,\!
  57. exp ( - ϵ t ) / μ ( S t ) \exp(-\epsilon t)/\mu(S_{t})\,\!
  58. R R\,\!
  59. p ( S ¯ 2 t ) p(\bar{S}_{2t})\,\!
  60. p ( S ¯ 2 t ) / p ( S t ) p(\bar{S}_{2t})/p(S_{t})\,\!
  61. p ( S ¯ 2 t ) p ( S t ) = S ¯ 2 t exp ( ϵ q ( d , r ) ) μ ( r ) d r S t exp ( ϵ q ( d , r ) ) μ ( r ) d r exp ( - ϵ t ) μ ( S ¯ 2 t ) μ ( S t ) . \frac{p(\bar{S}_{2t})}{p(S_{t})}=\frac{\int_{\bar{S}_{2t}}\exp(\epsilon q(d,r)% )\mu(r)\,dr}{\int_{S_{t}}\exp(\epsilon q(d,r))\mu(r)\,dr}\leq\exp(-\epsilon t)% \frac{\mu(\bar{S}_{2t})}{\mu(S_{t})}.
  62. μ ( S ¯ 2 t ) \mu(\bar{S}_{2t})\,\!
  63. t ln ( O P T t μ ( S t ) ) / ϵ t\geq\ln(\frac{OPT}{t\mu(S_{t})})/\epsilon\,\!
  64. E [ q ( d , q ϵ ( d ) ) ] O P T - 3 t E[q(d,\mathcal{E}_{q}^{\epsilon}(d))]\geq OPT-3t\,\!
  65. O P T - 2 t OPT-2t\,\!
  66. 1 - exp ( - ϵ t ) / μ ( S t ) 1-\exp(-\epsilon t)/\mu(S_{t})\,\!
  67. t ln ( O P T t μ ( S t ) ) / ϵ t\geq\ln(\frac{OPT}{t\mu(S_{t})})/\epsilon\,\!
  68. t t\,\!
  69. 1 - t / O P T 1-t/OPT\,\!
  70. O P T - 2 t OPT-2t\,\!
  71. μ ( A ) \mu(A)\,\!
  72. A A\subseteq\mathcal{R}\,\!
  73. μ ( ) \mu(\mathcal{R})\,\!
  74. Q Q\,\!
  75. D 1 , D 2 𝒟 n D_{1},D_{2}\in\mathcal{D}^{n}\,\!
  76. G S Q = max D 1 , D 2 : d ( D 1 , D 2 ) = 1 | ( Q ( D 1 ) - Q ( D 2 ) ) | GS_{Q}=\max_{D_{1},D_{2}:d(D_{1},D_{2})=1}|(Q(D_{1})-Q(D_{2}))|\,\!
  77. Q φ Q_{\varphi}\,\!
  78. φ \varphi\,\!
  79. Q φ = | { x D : φ ( x ) } | | D | Q_{\varphi}=\frac{|\{x\in D:\varphi(x)\}|}{|D|}\,\!
  80. G S Q φ 1 / n GS_{Q_{\varphi}}\leq 1/n\,\!
  81. φ \varphi\,\!
  82. 𝒜 \mathcal{A}\,\!
  83. ( α , δ ) (\alpha,\delta)\,\!
  84. H H\,\!
  85. 1 - δ 1-\delta\,\!
  86. h H \forall h\in H\,\!
  87. D D\,\!
  88. D ^ = 𝒜 ( D ) \widehat{D}=\mathcal{A}(D)\,\!
  89. | Q h ( D ^ ) - Q h ( D ) | α |Q_{h}(\widehat{D})-Q_{h}(D)|\leq\alpha\,\!
  90. Q h Q_{h}\,\!
  91. D D\,\!
  92. D ^ \widehat{D}\,\!
  93. D D\,\!
  94. n n\,\!
  95. k k\,\!
  96. ( x 1 , x 2 , , x k ) (x_{1},x_{2},\dots,x_{k})\,\!
  97. x i { 0 , 1 } x_{i}\in\{0,1\}\,\!
  98. π 1 x 1 + π 2 x 2 + + π k - 1 x k - 1 x k \pi_{1}x_{1}+\pi_{2}x_{2}+\cdots+\pi_{k-1}x_{k-1}\geq x_{k}\,\!
  99. π 1 , π 2 , , π k - 1 \pi_{1},\pi_{2},\dots,\pi_{k-1}\,\!
  100. D ^ \widehat{D}\,\!
  101. D D\,\!
  102. ϵ \epsilon\,\!
  103. H H\,\!
  104. D { 0 , 1 } k D\subset\{0,1\}^{k}\,\!
  105. | D | O ( k V C D I M ( H ) log ( 1 / α ) α 3 ϵ + log ( 1 / δ ) α ϵ ) |D|\geq O\left(\frac{k\cdot VCDIM(H)\log(1/\alpha)}{\alpha^{3}\epsilon}+\frac{% \log(1/\delta)}{\alpha\epsilon}\right)\,\!
  106. ( α , δ ) (\alpha,\delta)\,\!
  107. D ^ \widehat{D}\,\!
  108. ϵ \epsilon\,\!
  109. α \alpha\,\!
  110. O ~ ( V C D I M ( H ) / α 2 ) \tilde{O}(VCDIM(H)/\alpha^{2})\,\!
  111. D D\,\!
  112. D ^ \widehat{D}\,\!
  113. = O ( V C D I M ( H ) log ( 1 / α ) ) / α 2 =O(VCDIM(H)\log(1/\alpha))/\alpha^{2}\,\!
  114. max h H | Q h ( D ) - Q h ( D ^ ) | α / 2 \max_{h\in H}|Q_{h}(D)-Q_{h}(\widehat{D})|\leq\alpha/2\,\!
  115. Pr [ | Q h ( D ) - Q h ( D ^ ) | α / 2 \Pr[|Q_{h}(D)-Q_{h}(\widehat{D})|\geq\alpha/2\,\!
  116. h H ] 2 ( e m V C D I M ( H ) ) V C D I M ( H ) e - α 2 m 8 h\in H]\leq 2(\frac{em}{VCDIM(H)})^{VCDIM(H)}\cdot e^{-\frac{\alpha^{2}m}{8}}\,\!
  117. D ^ \widehat{D}\,\!
  118. m λ ( V C D I M ( H ) log ( m / V C D I M ( H ) ) / α 2 ) m\geq\lambda(VCDIM(H)\log(m/VCDIM(H))/\alpha^{2})\,\!
  119. λ \lambda\,\!
  120. O ~ ( V C D I M ( H ) / α 2 ) \tilde{O}(VCDIM(H)/\alpha^{2})\,\!
  121. m m\,\!
  122. m λ ( V C D I M ( H ) log ( 1 / α ) / α 2 ) m\geq\lambda(VCDIM(H)\log(1/\alpha)/\alpha^{2})\,\!
  123. q : ( ( { 0 , 1 } k ) n × ( { 0 , 1 } k ) m ) q:((\{0,1\}^{k})^{n}\times(\{0,1\}^{k})^{m})\rightarrow\mathbb{R}\,\!
  124. D D\,\!
  125. D ^ \widehat{D}\,\!
  126. e q ( D , D ^ ) ϵ n / 2 e^{q(D,\widehat{D})\epsilon n/2}\,\!
  127. ( ϵ n G S q ) (\epsilon nGS_{q})\,\!
  128. ( q ( D ) , q ( D ^ ) ) = - max h H | Q h ( D ) - Q h ( D ^ ) | (q(D),q(\widehat{D}))=-\max_{h\in H}|Q_{h}(D)-Q_{h}(\widehat{D})|\,\!
  129. ( α , δ ) (\alpha,\delta)\,\!
  130. D ^ \widehat{D}\,\!
  131. q ( D , D ^ ) - α q(D,\widehat{D})\geq-\alpha\,\!
  132. 1 - δ 1-\delta\,\!
  133. 2 k m 2^{km}\,\!
  134. q ( D , D ^ ) - α q(D,\widehat{D})\leq-\alpha\,\!
  135. e - ϵ α n / 2 e^{-\epsilon\alpha n/2}\,\!
  136. D ^ \widehat{D}\,\!
  137. 2 k m e - ϵ α n / 2 2^{km}e^{-\epsilon\alpha n/2}\,\!
  138. D ^ ( { 0 , 1 } k ) m \widehat{D}\in(\{0,1\}^{k})^{m}\,\!
  139. q ( D , D ^ ) - α / 2 q(D,\widehat{D})\geq-\alpha/2\,\!
  140. e - α ϵ n / 4 e^{-\alpha\epsilon n/4}\,\!
  141. A := A:=\,\!
  142. D ^ \widehat{D}\,\!
  143. q ( D , D ^ ) - α / 2 q(D,\widehat{D})\geq-\alpha/2\,\!
  144. B := B:=\,\!
  145. D ^ \widehat{D}\,\!
  146. q ( D , D ^ ) - α q(D,\widehat{D})\leq-\alpha\,\!
  147. Pr [ A ] Pr [ B ] e - α ϵ n / 4 2 k m e - α ϵ n / 2 = e α ϵ n / 4 2 k m . \therefore\frac{\Pr[A]}{\Pr[B]}\geq\frac{e^{-\alpha\epsilon n/4}}{2^{km}e^{-% \alpha\epsilon n/2}}=\frac{e^{\alpha\epsilon n/4}}{2^{km}}.\,\!
  148. 1 / δ ( 1 - δ ) / δ 1/\delta\geq(1-\delta)/\delta\,\!
  149. n 4 ϵ α ( k m + l n 1 δ ) O ( d V C D I M ( H ) log ( 1 / α ) α 3 ϵ + log ( 1 / δ ) α ϵ ) n\geq\frac{4}{\epsilon\alpha}\left(km+ln\frac{1}{\delta}\right)\geq O\left(% \frac{d\cdot VCDIM(H)\log(1/\alpha)}{\alpha^{3}\epsilon}+\frac{\log(1/\delta)}% {\alpha\epsilon}\right)\,\!

Exponentiated_Weibull_distribution.html

  1. F ( x ; k , λ ; α ) = [ 1 - e - ( x / λ ) k ] α F(x;k,\lambda;\alpha)=\left[1-e^{-(x/\lambda)^{k}}\right]^{\alpha}\,
  2. f ( x ; k , λ ; α ) = α k λ [ x λ ] k - 1 [ 1 - e - ( x / λ ) k ] α - 1 e - ( x / λ ) k f(x;k,\lambda;\alpha)=\alpha\frac{k}{\lambda}\left[\frac{x}{\lambda}\right]^{k% -1}\left[1-e^{-(x/\lambda)^{k}}\right]^{\alpha-1}e^{-(x/\lambda)^{k}}\,

Extended_Boolean_model.html

  1. w x , j = f x , j * I d f x m a x i I d f i w_{x,j}=f_{x,j}*\frac{Idf_{x}}{max_{i}Idf_{i}}
  2. 𝐯 d j = [ w 1 , j , w 2 , j , , w i , j ] \mathbf{v}_{d_{j}}=[w_{1,j},w_{2,j},\ldots,w_{i,j}]
  3. s i m ( q o r , d ) = w 1 2 + w 2 2 2 sim(q_{or},d)=\sqrt{\frac{w_{1}^{2}+w_{2}^{2}}{2}}
  4. s i m ( q a n d , d ) = 1 - ( 1 - w 1 ) 2 + ( 1 - w 2 ) 2 2 sim(q_{and},d)=1-\sqrt{\frac{(1-w_{1})^{2}+(1-w_{2})^{2}}{2}}
  5. 1 p 1≤p≤∞
  6. q o r = k 1 p k 2 p . p k t q_{or}=k_{1}\lor^{p}k_{2}\lor^{p}....\lor^{p}k_{t}
  7. q o r q_{or}
  8. d j d_{j}
  9. s i m ( q o r , d j ) = w 1 p + w 2 p + . + w t p t p sim(q_{or},d_{j})=\sqrt[p]{\frac{w_{1}^{p}+w_{2}^{p}+....+w_{t}^{p}}{t}}
  10. q a n d = k 1 p k 2 p . p k t q_{and}=k_{1}\land^{p}k_{2}\land^{p}....\land^{p}k_{t}
  11. q a n d q_{and}
  12. d j d_{j}
  13. s i m ( q a n d , d j ) = 1 - ( 1 - w 1 ) p + ( 1 - w 2 ) p + . + ( 1 - w t ) p t p sim(q_{and},d_{j})=1-\sqrt[p]{\frac{(1-w_{1})^{p}+(1-w_{2})^{p}+....+(1-w_{t})% ^{p}}{t}}
  14. q q
  15. d d
  16. s i m ( q , d ) = ( 1 - ( ( 1 - w 1 ) p + ( 1 - w 2 ) p 2 p ) ) p + w 3 p 2 p sim(q,d)=\sqrt[p]{\frac{(1-\sqrt[p]{(\frac{(1-w_{1})^{p}+(1-w_{2})^{p}}{2}}))^% {p}+w_{3}^{p}}{2}}

Extender_(set_theory).html

  1. E = { E a | a [ λ ] < ω } E=\{E_{a}|a\in[\lambda]^{<\omega}\}
  2. α κ \alpha\in\kappa
  3. { s [ κ ] | a | : α s } \{s\in[\kappa]^{|a|}:\alpha\in s\}
  4. { s [ κ ] | a | : f ( s ) max s } E a \{s\in[\kappa]^{|a|}:f(s)\in\max s\}\in E_{a}
  5. b a , { t κ | b | : ( f π b a ) ( t ) t } E b b\supseteq a,\ \{t\in\kappa^{|b|}:(f\circ\pi_{ba})(t)\in t\}\in E_{b}
  6. b = { α 1 , , α n } b=\{\alpha_{1},\dots,\alpha_{n}\}
  7. α 1 < < α n < λ \alpha_{1}<\dots<\alpha_{n}<\lambda
  8. a = { α i 1 , , α i m } a=\{\alpha_{i_{1}},\dots,\alpha_{i_{m}}\}
  9. π b a : { ξ 1 , , ξ n } { ξ i 1 , , ξ i m } ( ξ 1 < < ξ n ) \pi_{ba}:\{\xi_{1},\dots,\xi_{n}\}\mapsto\{\xi_{i_{1}},\dots,\xi_{i_{m}}\}\ (% \xi_{1}<\dots<\xi_{n})
  10. X E a { s : π b a ( s ) X } E b X\in E_{a}\Leftrightarrow\{s:\pi_{ba}(s)\in X\}\in E_{b}
  11. E = { E a | a [ λ ] < ω } E=\{E_{a}|a\in[\lambda]^{<\omega}\}
  12. for a [ λ ] < ω , X [ κ ] < ω : X E a a j ( X ) . \,\text{for }a\in[\lambda]^{<\omega},X\subseteq[\kappa]^{<\omega}:\quad X\in E% _{a}\Leftrightarrow a\in j(X).

Extremal_orders_of_an_arithmetic_function.html

  1. lim inf n f ( n ) m ( n ) = 1 \liminf_{n\to\infty}\frac{f(n)}{m(n)}=1
  2. lim sup n f ( n ) M ( n ) = 1 \limsup_{n\to\infty}\frac{f(n)}{M(n)}=1
  3. lim inf n σ ( n ) n = 1 \liminf_{n\to\infty}\frac{\sigma(n)}{n}=1
  4. lim sup n σ ( n ) n ln ln n = e γ , \limsup_{n\to\infty}\frac{\sigma(n)}{n\ln\ln n}=e^{\gamma},
  5. lim inf n ϕ ( n ) n = 1 \liminf_{n\to\infty}\frac{\phi(n)}{n}=1
  6. lim inf n ϕ ( n ) ln ln n n = e - γ , \liminf_{n\to\infty}\frac{\phi(n)\ln\ln n}{n}=e^{-\gamma},

Extremal_principles_in_non-equilibrium_thermodynamics.html

  1. X t h X_{th}
  2. X t h = - 1 T T X_{th}=-\frac{1}{T}\nabla T
  3. T T

Extremum_estimator.html

  1. θ ^ \scriptstyle\hat{\theta}
  2. Q ^ n \scriptstyle\hat{Q}_{n}
  3. θ ^ = arg max θ Θ Q ^ n ( θ ) , \hat{\theta}=\underset{\theta\in\Theta}{\operatorname{arg\;max}}\ \widehat{Q}_% {n}(\theta),
  4. Q ^ n ( θ ^ ) max θ Θ Q ^ n ( θ ) - o p ( 1 ) , \widehat{Q}_{n}(\hat{\theta})\geq\max_{\theta\in\Theta}\,\widehat{Q}_{n}(% \theta)-o_{p}(1),
  5. θ ^ \scriptstyle\hat{\theta}
  6. θ ^ \scriptscriptstyle\hat{\theta}
  7. Q ^ n ( θ ) \scriptstyle\hat{Q}_{n}(\theta)
  8. θ ^ \scriptstyle\hat{\theta}
  9. Q ^ n ( θ ) \scriptstyle\hat{Q}_{n}(\theta)
  10. sup θ Θ | Q ^ n ( θ ) - Q 0 ( θ ) | 𝑝 0. \sup_{\theta\in\Theta}\big|\hat{Q}_{n}(\theta)-Q_{0}(\theta)\big|\ % \xrightarrow{p}\ 0.

Étale_algebra.html

  1. K K
  2. \mathfrak{R}
  3. K K
  4. \mathfrak{R}
  5. K K ¯ K ¯ × × K ¯ \mathfrak{R}\otimes_{K}\bar{K}\cong\bar{K}\times\cdots\times\bar{K}
  6. Spec Spec K \mathrm{Spec}\,\mathfrak{R}\to\mathrm{Spec}\,K

Étale_group_scheme.html

  1. G G
  2. K K
  3. \mathfrak{R}
  4. K K ¯ \mathfrak{R}\otimes_{K}\bar{K}
  5. K ¯ × × K ¯ \bar{K}\times...\times\bar{K}

F-test_of_equality_of_variances.html

  1. X ¯ = 1 n i = 1 n X i and Y ¯ = 1 m i = 1 m Y i \overline{X}=\frac{1}{n}\sum_{i=1}^{n}X_{i}\,\text{ and }\overline{Y}=\frac{1}% {m}\sum_{i=1}^{m}Y_{i}
  2. S X 2 = 1 n - 1 i = 1 n ( X i - X ¯ ) 2 and S Y 2 = 1 m - 1 i = 1 m ( Y i - Y ¯ ) 2 S_{X}^{2}=\frac{1}{n-1}\sum_{i=1}^{n}\left(X_{i}-\overline{X}\right)^{2}\,% \text{ and }S_{Y}^{2}=\frac{1}{m-1}\sum_{i=1}^{m}\left(Y_{i}-\overline{Y}% \right)^{2}
  3. F = S X 2 S Y 2 F=\frac{S_{X}^{2}}{S_{Y}^{2}}

F26A_graph.html

  1. D 26 = a , b | a 2 = b 13 = 1 , a b a = b - 1 . D_{26}=\langle a,b|a^{2}=b^{13}=1,aba=b^{-1}\rangle.
  2. ( x - 3 ) ( x + 3 ) ( x 4 - 5 x 2 + 3 ) 6 . (x-3)(x+3)(x^{4}-5x^{2}+3)^{6}.\,

Factor_(chord).html

  1. 3 6 {}^{6}_{3}
  2. 2 4 {}^{4}_{2}

Factor_payments_(economics).html

  1. w w
  2. l l
  3. w l wl
  4. l l
  5. w l wl
  6. y y
  7. f ( l ) f(l)
  8. P r o f i t = P y Profit=Py
  9. w l wl
  10. P P
  11. f ( l ) f(l)
  12. w l wl
  13. P . M P L P.MPL
  14. w w
  15. P . M P L = w P.MPL=w
  16. M P L = w / P MPL=w/P
  17. w / P w/P
  18. M P K = R / P MPK=R/P
  19. P r o f i t = P Y - W L - R K Profit=PY-WL-RK
  20. P . Y P.Y
  21. W . L W.L
  22. R . K R.K
  23. Y = F ( L , K ) Y=F(L,K)
  24. P r o f i t = P F ( K , L ) - W L R K Profit=PF(K,L)-WL_{R}K

False_positive_rate.html

  1. F P F P + T N \frac{FP}{FP+TN}
  2. H i H_{i}
  3. V V
  4. S S
  5. R R
  6. U U
  7. T T
  8. m - R m-R
  9. m 0 m_{0}
  10. m - m 0 m-m_{0}
  11. m m
  12. m 0 m_{0}
  13. m - m 0 m-m_{0}
  14. V V
  15. S S
  16. T T
  17. U U
  18. R R
  19. S S
  20. T T
  21. U U
  22. V V
  23. V / m 0 V/m_{0}
  24. V m 0 V\leq m_{0}
  25. E ( V / m 0 ) E(V/m_{0})
  26. V / m 0 V/m_{0}
  27. FWER = Pr ( V 1 ) \mathrm{FWER}=\Pr(V\geq 1)\,
  28. E ( V / m 0 ) E(V/m_{0})
  29. E ( V / R ) E(V/R)

Farinograph.html

  1. B U n - B U n - x B U n \tfrac{BU^{n}-BU^{n-x}}{BU^{n}}

Fast-growing_hierarchy.html

  1. f α + 1 ( n ) = f α n ( n ) , f_{\alpha+1}(n)=f_{\alpha}^{n}(n),\,
  2. f α ( n ) = f α [ n ] ( n ) f_{\alpha}(n)=f_{\alpha[n]}(n)\,\!
  3. n \mathcal{E}^{n}
  4. Π 1 1 \Pi^{1}_{1}
  5. n \mathcal{E}^{n}

Fast_Syndrome_Based_Hash.html

  1. ϕ \phi
  2. n , r , w {n,r,w}
  3. n > w n>w
  4. w log ( n / w ) > r w\log(n/w)>r
  5. s = w log ( n / w ) s=w\log(n/w)
  6. r r
  7. n , r , w , s n,r,w,s
  8. log ( n / w ) \log(n/w)
  9. log \log
  10. w log ( n / w ) > r w\cdot\log(n/w)>r
  11. ϕ \phi
  12. r × n r\times n
  13. H H
  14. n n
  15. w log ( n / w ) w\log(n/w)
  16. ( 𝐅 2 ) n (\mathbf{F}_{2})^{n}
  17. n n
  18. r r
  19. w w
  20. w w
  21. n n
  22. n n
  23. w w
  24. w w
  25. n n
  26. [ ( i - 1 ) w , i w ) [(i-1)w,iw)
  27. 0 < i < n / w + 1 0<i<n/w+1
  28. ( n / w ) w (n/w)^{w}
  29. w w
  30. n n
  31. log ( ( n / w ) w ) = w log ( n / w ) = s \log((n/w)^{w})=w\log(n/w)=s
  32. s s
  33. w w
  34. n n
  35. s s
  36. n n
  37. w w
  38. H H
  39. r r
  40. H H
  41. H i H_{i}
  42. r × n / w r\times n/w
  43. w log ( n / w ) w\log(n/w)
  44. w w
  45. n / w n/w
  46. n n
  47. w w
  48. n / w n/w
  49. s s
  50. s s
  51. w w
  52. s 1 , , s w s_{1},\dots,s_{w}
  53. n / w n/w
  54. H i H_{i}
  55. r r
  56. r r
  57. s = 010011 s=010011
  58. 4 × 12 4\times 12
  59. H = ( 1 0 1 1 0 1 0 0 1 0 1 1 0 1 0 0 0 1 1 1 0 1 0 0 0 1 1 1 0 1 0 0 1 0 1 0 1 1 0 0 1 0 1 1 0 0 0 1 ) H=\left(\begin{array}[]{llllcllllcllll}1&0&1&1&&0&1&0&0&&1&0&1&1\\ 0&1&0&0&&0&1&1&1&&0&1&0&0\\ 0&1&1&1&&0&1&0&0&&1&0&1&0\\ 1&1&0&0&&1&0&1&1&&0&0&0&1\end{array}\right)
  60. w = 3 w=3
  61. H 1 H_{1}
  62. H 2 H_{2}
  63. H 3 H_{3}
  64. s s
  65. w = 3 w=3
  66. log 2 ( 12 / 3 ) = 2 \log_{2}(12/3)=2
  67. s 1 = 01 s_{1}=01
  68. s 2 = 00 s_{2}=00
  69. s 3 = 11 s_{3}=11
  70. s i s_{i}
  71. s 1 = 1 s_{1}=1
  72. s 2 = 0 s_{2}=0
  73. s 3 = 3 s_{3}=3
  74. H 1 H_{1}
  75. H 2 H_{2}
  76. r = 0111 0001 1001 = 1111 r=0111\oplus 0001\oplus 1001=1111
  77. ϕ \phi
  78. w w
  79. H i H_{i}
  80. r × ( n / w ) r\times(n/w)
  81. S S
  82. r r
  83. w w
  84. H i H_{i}
  85. S S
  86. S S
  87. w w
  88. H i H_{i}
  89. r × ( n / w ) r\times(n/w)
  90. S S
  91. r r
  92. w w^{\prime}
  93. H i H_{i}
  94. ( 0 < w < 2 w ) (0<w^{\prime}<2w)
  95. w w
  96. H w = S Hw=S
  97. w w^{\prime}
  98. H w = 0 Hw^{\prime}=0
  99. n n
  100. w w
  101. n n
  102. [ ( i - 1 ) w , i w ) [(i-1)w,iw)
  103. m 1 m 2 m_{1}\neq m_{2}
  104. w 1 w_{1}
  105. w 2 w_{2}
  106. H w 1 = H w 2 Hw_{1}=Hw_{2}
  107. H ( w 1 + w 2 ) = H w 1 + H w 2 = 2 H w 1 = 0 H(w_{1}+w_{2})=Hw_{1}+Hw_{2}=2Hw_{1}=0
  108. ( w 1 + w 2 ) (w_{1}+w_{2})
  109. H H
  110. w = 3 w=3
  111. r = 1111 r=1111
  112. r r
  113. s 1 = 1 s_{1}=1
  114. s 2 = 0 s_{2}=0
  115. s 3 = 3 s_{3}=3
  116. r r
  117. H 1 H_{1}
  118. H 2 H_{2}
  119. H 3 H_{3}
  120. H 3 H_{3}
  121. H H
  122. H H

FASTQ_format.html

  1. Q sanger = - 10 log 10 p Q\text{sanger}=-10\,\log_{10}p
  2. Q solexa-prior to v.1.3 = - 10 log 10 p 1 - p Q\text{solexa-prior to v.1.3}=-10\,\log_{10}\frac{p}{1-p}
  3. Q phred = - 10 log 10 e Q\text{phred}=-10\log\text{10}e

Feature_model.html

  1. f 1 f_{1}
  2. f 2 f_{2}
  3. f 1 f 2 f_{1}\Leftrightarrow f_{2}
  4. r r
  5. r r
  6. f 1 f_{1}
  7. f f
  8. f 1 f f_{1}\Rightarrow f
  9. f 1 f_{1}
  10. f f
  11. f 1 f f_{1}\Leftrightarrow f
  12. f 1 , , f n f_{1},\dots,f_{n}
  13. f f
  14. ( f 1 f n f ) i < j ¬ ( f i f j ) \left(f_{1}\lor\dots\lor f_{n}\Leftrightarrow f\right)\land\bigwedge_{i<j}% \lnot(f_{i}\land f_{j})
  15. f 1 , , f n f_{1},\dots,f_{n}
  16. f f
  17. f 1 f n f f_{1}\lor\dots\lor f_{n}\Leftrightarrow f
  18. f 1 f_{1}
  19. f 2 f_{2}
  20. ¬ ( f 1 f 2 ) \lnot(f_{1}\land f_{2})
  21. f 1 f_{1}
  22. f 2 f_{2}
  23. f 1 f 2 f_{1}\Rightarrow f_{2}

Fedosov_manifold.html

  1. Γ j k i = 0 \Gamma^{i}_{jk}=0

FEE_method.html

  1. E E
  2. e x . e^{x}.
  3. y = f ( x ) y=f(x)
  4. s f ( n ) = O ( M ( n ) log 2 n ) . s_{f}(n)=O(M(n)\log^{2}n).\,
  5. s f ( n ) s_{f}(n)
  6. f ( x ) f(x)
  7. n n
  8. M ( n ) M(n)
  9. n n
  10. π , \pi,
  11. γ , \gamma,
  12. s f ( n ) = O ( M ( n ) log 2 n ) . s_{f}(n)=O(M(n)\log^{2}n).\,
  13. π , \pi,
  14. π 4 = arctan 1 2 + arctan 1 3 , \frac{\pi}{4}=\arctan\frac{1}{2}+\arctan\frac{1}{3},
  15. arctan 1 2 = 1 1 2 - 1 3 2 3 + + ( - 1 ) r - 1 ( 2 r - 1 ) 2 2 r - 1 + R 1 , \arctan\frac{1}{2}=\frac{1}{1\cdot 2}-\frac{1}{3\cdot 2^{3}}+\cdots+\frac{(-1)% ^{r-1}}{(2r-1)2^{2r-1}}+R_{1},
  16. arctan 1 3 = 1 1 3 - 1 3 3 3 + + ( - 1 ) r - 1 ( 2 r - 1 ) 3 2 r - 1 + R 2 , \arctan\frac{1}{3}=\frac{1}{1\cdot 3}-\frac{1}{3\cdot 3^{3}}+\cdots+\frac{(-1)% ^{r-1}}{(2r-1)3^{2r-1}}+R_{2},
  17. R 1 , R_{1},
  18. R 2 , R_{2},
  19. | R 1 | 4 5 1 2 r + 1 1 2 2 r + 1 ; |R_{1}|\leq\frac{4}{5}\frac{1}{2r+1}\frac{1}{2^{2r+1}};
  20. | R 2 | 9 10 1 2 r + 1 1 3 2 r + 1 ; |R_{2}|\leq\frac{9}{10}\frac{1}{2r+1}\frac{1}{3^{2r+1}};
  21. r = n , r=n,\,
  22. 4 ( | R 1 | + | R 2 | ) < 2 - n . 4(|R_{1}|+|R_{2}|)\ <\ 2^{-n}.
  23. π \pi
  24. s π = O ( M ( n ) log 2 n ) . s_{\pi}=O(M(n)\log^{2}n).\,
  25. n n
  26. m = 6 n , k = n , k 1 , m=6n,\quad k=n,\quad k\geq 1,\,
  27. γ = - log n r = 0 12 n ( - 1 ) r n r + 1 ( r + 1 ) ! + r = 0 12 n ( - 1 ) r n r + 1 ( r + 1 ) ! ( r + 1 ) + O ( 2 - n ) . \gamma=-\log n\sum_{r=0}^{12n}\frac{(-1)^{r}n^{r+1}}{(r+1)!}+\sum_{r=0}^{12n}% \frac{(-1)^{r}n^{r+1}}{(r+1)!(r+1)}+O(2^{-n}).
  28. s γ = O ( M ( n ) log 2 n ) . s_{\gamma}=O(M(n)\log^{2}n).\,
  29. γ \gamma
  30. f 1 = f 1 ( z ) = j = 0 a ( j ) b ( j ) z j , f_{1}=f_{1}(z)=\sum_{j=0}^{\infty}\frac{a(j)}{b(j)}z^{j},
  31. f 2 = f 2 ( z ) = j = 0 a ( j ) b ( j ) z j j ! , f_{2}=f_{2}(z)=\sum_{j=0}^{\infty}\frac{a(j)}{b(j)}\frac{z^{j}}{j!},
  32. a ( j ) , b ( j ) a(j),\quad b(j)
  33. | a ( j ) | + | b ( j ) | ( C j ) K ; | z | < 1 ; K |a(j)|+|b(j)|\leq(Cj)^{K};\quad|z|\ <\ 1;\quad K
  34. C C
  35. z z
  36. s f 1 ( n ) = O ( M ( n ) log 2 n ) , s_{f_{1}}(n)=O\left(M(n)\log^{2}n\right),\,
  37. s f 2 ( n ) = O ( M ( n ) log n ) . s_{f_{2}}(n)=O\left(M(n)\log n\right).
  38. e e
  39. m = 2 k , k 1 m=2^{k},\quad k\geq 1
  40. e , e,
  41. e = 1 + 1 1 ! + 1 2 ! + + 1 ( m - 1 ) ! + R m . e=1+\frac{1}{1!}+\frac{1}{2!}+\cdots+\frac{1}{(m-1)!}+R_{m}.
  42. m m
  43. R m R_{m}
  44. R m 2 - n - 1 R_{m}\leq 2^{-n-1}
  45. m 4 n log n . m\geq\frac{4n}{\log n}.
  46. m = 2 k m=2^{k}
  47. k k
  48. 2 k 4 n log n > 2 k - 1 . 2^{k}\geq\frac{4n}{\log n}>2^{k-1}.
  49. S = 1 + 1 1 ! + 1 2 ! + + 1 ( m - 1 ) ! = j = 0 m - 1 1 ( m - 1 - j ) ! , S=1+\frac{1}{1!}+\frac{1}{2!}+\cdots+\frac{1}{(m-1)!}=\sum_{j=0}^{m-1}\frac{1}% {(m-1-j)!},
  50. k k
  51. S S
  52. S \displaystyle S
  53. m , m - 2 , m - 4 , . m,m-2,m-4,\dots.\,
  54. S S
  55. S = S ( 1 ) = j = 0 m 1 - 1 1 ( m - 1 - 2 j ) ! α m 1 - j ( 1 ) , S=S(1)=\sum_{j=0}^{m_{1}-1}\frac{1}{(m-1-2j)!}\alpha_{m_{1}-j}(1),
  56. m 1 = m 2 , m = 2 k , k 1. m_{1}=\frac{m}{2},m=2^{k},k\geq 1.
  57. m 2 \frac{m}{2}
  58. α m 1 - j ( 1 ) = m - 2 j , j = 0 , 1 , , m 1 - 1 , \alpha_{m_{1}-j}(1)=m-2j,\quad j=0,1,\dots,m_{1}-1,
  59. S S
  60. i i
  61. i + 1 i+1
  62. i + 1 k i+1\leq k
  63. S = S ( i + 1 ) = j = 0 m i + 1 - 1 1 ( m - 1 - 2 i + 1 j ) ! α m i + 1 - j ( i + 1 ) , S=S(i+1)=\sum_{j=0}^{m_{i+1}-1}\frac{1}{(m-1-2^{i+1}j)!}\alpha_{m_{i+1}-j}(i+1),
  64. m i + 1 = m i 2 = m 2 i + 1 , m_{i+1}=\frac{m_{i}}{2}=\frac{m}{2^{i+1}},
  65. m 2 i + 1 \frac{m}{2^{i+1}}
  66. α m i + 1 - j ( i + 1 ) = α m i - 2 j ( i ) + α m i - ( 2 j + 1 ) ( i ) ( m - 1 - 2 i + 1 j ) ! ( m - 1 - 2 i - 2 i + 1 j ) ! , \alpha_{m_{i+1}-j}(i+1)=\alpha_{m_{i}-2j}(i)+\alpha_{m_{i}-(2j+1)}(i)\frac{(m-% 1-2^{i+1}j)!}{(m-1-2^{i}-2^{i+1}j)!},
  67. j = 0 , 1 , , m i + 1 - 1 , m = 2 k , k i + 1. j=0,1,\dots,\quad m_{i+1}-1,\quad m=2^{k},\quad k\geq i+1.
  68. ( m - 1 - 2 i + 1 j ) ! ( m - 1 - 2 i - 2 i + 1 j ) ! \frac{(m-1-2^{i+1}j)!}{(m-1-2^{i}-2^{i+1}j)!}
  69. 2 i 2^{i}
  70. k k
  71. α 1 ( k ) , \alpha_{1}(k),
  72. ( m - 1 ) ! , (m-1)!,
  73. α 1 ( k ) \alpha_{1}(k)
  74. ( m - 1 ) ! , (m-1)!,
  75. n n
  76. S , S,
  77. e e
  78. n n
  79. O ( M ( m ) log 2 m ) = O ( M ( n ) log n ) . O\left(M(m)\log^{2}m\right)=O\left(M(n)\log n\right).\,
  80. ζ ( 3 ) \zeta(3)
  81. ζ ( s ) \zeta(s)
  82. s s
  83. L L

Feed-Drum.html

  1. z ( ρ , ϕ , τ ) = R ( ρ ) * Φ ( ϕ ) * c o s ( ω * τ ) z(\rho,\phi,\tau)=R(\rho)*\Phi(\phi)*cos(\omega*\tau)
  2. R ( ρ ) = J n ( m , k ρ ) R(\rho)=Jn(m,k\rho)
  3. Φ ( ϕ ) = A * c o s ( m * ϕ + ϕ 0 ) \Phi(\phi)=A*cos(m*\phi+\phi^{0})
  4. J n ( m , x ) Jn(m,x)
  5. m * ϕ 0 m*\phi^{0}
  6. R ( a ) = J n ( m , k * a ) = 0 R(a)=Jn(m,k*a)=0
  7. a a
  8. k m , n = R m , n / a k_{m,n}=R_{m,n}/a
  9. R m , n R_{m,n}
  10. m , J n ( m , x ) m,Jn(m,x)
  11. z m , n ( ρ , ϕ , τ ) = A m , n * J n ( m , k m , n ρ ) * c o s ( m * ϕ + ϕ m , n 0 ) z_{m,n}(\rho,\phi,\tau)=A_{m,n}*Jn(m,k_{m,n}\rho)*cos(m*\phi+\phi_{m,n}^{0})
  12. ω m , n = k m , n * c \omega_{m,n}=k_{m,n}*c
  13. c = T / σ c=\sqrt{T/\sigma}
  14. T T
  15. σ \sigma
  16. c c
  17. v 1 v_{1}
  18. ( 0 , 1 ) (0,1)
  19. R 0 , 1 = 2.405 R_{0,1}=2.405
  20. c = 2 * π * v 1 R 0 , 1 * a c=\tfrac{2*\pi*v_{1}}{R_{0,1}}*a
  21. a = 0.51 m a=0.51m
  22. v 1 = 30 H z v_{1}=30Hz
  23. c = 40 m / s c=40m/s
  24. π \pi
  25. m m\rightarrow\infty
  26. m m
  27. n n
  28. c c
  29. ϕ 0 \phi_{0}
  30. m = 0 m=0
  31. m 1 m\geq 1

Feedback_with_Carry_Shift_Registers.html

  1. N > 1 N>1
  2. r r
  3. ( a ; z ) = ( a 0 , a 1 , , a r - 1 ; z ) (a;z)=(a_{0},a_{1},\dots,a_{r-1};z)
  4. a i a_{i}
  5. { 0 , 1 , , N - 1 } = S \{0,1,\dots,N-1\}=S
  6. z z
  7. q 1 , , q n q_{1},\dots,q_{n}
  8. s = q r a 0 + q r - 1 a 1 + + q 1 a r - 1 + z s=q_{r}a_{0}+q_{r-1}a_{1}+\dots+q_{1}a_{r-1}+z
  9. s = a r + N z s=a_{r}+Nz^{\prime}
  10. a r a_{r}
  11. S S
  12. ( a 1 , a 2 , , a r ; z ) (a_{1},a_{2},\dots,a_{r};z^{\prime})
  13. S S
  14. q = q r N r + + q 1 N 1 - 1 q=q_{r}N^{r}+\dots+q_{1}N^{1}-1
  15. a = a 0 + a 1 N + a 2 N 2 + a=a_{0}+a_{1}N+a_{2}N^{2}+\dots
  16. u u
  17. a = u / q a=u/q
  18. u u
  19. - q -q
  20. 0
  21. g g
  22. N mod q N\mod q
  23. a i = ( A g i mod q ) mod N a_{i}=(Ag_{i}\mod q)\mod N
  24. A A
  25. q - 1 q-1
  26. N = 2 N=2
  27. 2 L 2L

Fenwick_tree.html

  1. O ( log n ) O(\log n)
  2. n n
  3. O ( log n ) O(\log n)
  4. O ( n log n ) O(n\log n)
  5. O ( n ) O(n)
  6. O ( log n ) O(\log n)
  7. O ( 2 d log d n ) O(2^{d}\log^{d}n)

Fermat_quintic_threefold.html

  1. V 5 + W 5 + X 5 + Y 5 + Z 5 = 0. V^{5}+W^{5}+X^{5}+Y^{5}+Z^{5}=0.\,

Fermat_quotient.html

  1. q p ( a ) = a p - 1 - 1 p . q_{p}(a)=\frac{a^{p-1}-1}{p}.
  2. δ p ( a ) = a - a p p \delta_{p}(a)=\frac{a-a^{p}}{p}
  3. q p ( 1 ) 0 ( mod p ) q_{p}(1)\equiv 0\;\;(\mathop{{\rm mod}}p)
  4. q p ( - a ) q p ( a ) ( mod p ) q_{p}(-a)\equiv q_{p}(a)\;\;(\mathop{{\rm mod}}p)
  5. q p ( a b ) q p ( a ) + q p ( b ) ( mod p ) q_{p}(ab)\equiv q_{p}(a)+q_{p}(b)\;\;(\mathop{{\rm mod}}p)
  6. q p ( a r ) r q p ( a ) ( mod p ) q_{p}(a^{r})\equiv rq_{p}(a)\;\;(\mathop{{\rm mod}}p)
  7. q p ( p - a ) q p ( a ) + 1 a ( mod p ) q_{p}(p-a)\equiv q_{p}(a)+\frac{1}{a}\;\;(\mathop{{\rm mod}}p)
  8. q p ( p + a ) q p ( a ) - 1 a ( mod p ) q_{p}(p+a)\equiv q_{p}(a)-\frac{1}{a}\;\;(\mathop{{\rm mod}}p)
  9. q p ( p - 1 ) 1 ( mod p ) q_{p}(p-1)\equiv 1\;\;(\mathop{{\rm mod}}p)
  10. q p ( p + 1 ) - 1 ( mod p ) q_{p}(p+1)\equiv-1\;\;(\mathop{{\rm mod}}p)
  11. q p ( 1 / a ) - q p ( a ) ( mod p ) q_{p}(1/a)\equiv-q_{p}(a)\;\;(\mathop{{\rm mod}}p)
  12. q p ( a / b ) q p ( a ) - q p ( b ) ( mod p ) q_{p}(a/b)\equiv q_{p}(a)-q_{p}(b)\;\;(\mathop{{\rm mod}}p)
  13. q p ( a + n p ) q p ( a ) - n 1 a ( mod p ) . q_{p}(a+np)\equiv q_{p}(a)-n\cdot\frac{1}{a}\;\;(\mathop{{\rm mod}}p).
  14. q p ( a + n p 2 ) q p ( a ) ( mod p ) . q_{p}(a+np^{2})\equiv q_{p}(a)\;\;(\mathop{{\rm mod}}p).
  15. - 2 q p ( 2 ) k = 1 p - 1 2 1 k ( mod p ) . -2q_{p}(2)\equiv\sum_{k=1}^{\frac{p-1}{2}}\frac{1}{k}\;\;(\mathop{{\rm mod}}p).
  16. - 3 q p ( 2 ) k = 1 p 4 1 k ( mod p ) . -3q_{p}(2)\equiv\sum_{k=1}^{\lfloor\frac{p}{4}\rfloor}\frac{1}{k}\;\;(\mathop{% {\rm mod}}p).
  17. 4 q p ( 2 ) k = p 10 + 1 2 p 10 1 k + k = 3 p 10 + 1 4 p 10 1 k ( mod p ) . 4q_{p}(2)\equiv\sum_{k=\lfloor\frac{p}{10}\rfloor+1}^{\lfloor\frac{2p}{10}% \rfloor}\frac{1}{k}+\sum_{k=\lfloor\frac{3p}{10}\rfloor+1}^{\lfloor\frac{4p}{1% 0}\rfloor}\frac{1}{k}\;\;(\mathop{{\rm mod}}p).
  18. 2 q p ( 2 ) k = p 6 + 1 p 3 1 k ( mod p ) . 2q_{p}(2)\equiv\sum_{k=\lfloor\frac{p}{6}\rfloor+1}^{\lfloor\frac{p}{3}\rfloor% }\frac{1}{k}\;\;(\mathop{{\rm mod}}p).
  19. - 3 q p ( 3 ) 2 k = 1 p 3 1 k ( mod p ) . -3q_{p}(3)\equiv 2\sum_{k=1}^{\lfloor\frac{p}{3}\rfloor}\frac{1}{k}\;\;(% \mathop{{\rm mod}}p).
  20. - 5 q p ( 5 ) 4 k = 1 p 5 1 k + 2 k = p 5 + 1 2 p 5 1 k ( mod p ) . -5q_{p}(5)\equiv 4\sum_{k=1}^{\lfloor\frac{p}{5}\rfloor}\frac{1}{k}+2\sum_{k=% \lfloor\frac{p}{5}\rfloor+1}^{\lfloor\frac{2p}{5}\rfloor}\frac{1}{k}\;\;(% \mathop{{\rm mod}}p).

Fermat–Catalan_conjecture.html

  1. 1 m + 1 n + 1 k 1. \frac{1}{m}+\frac{1}{n}+\frac{1}{k}\leq 1.
  2. 1 m + 2 3 = 3 2 1^{m}+2^{3}=3^{2}\;
  3. 2 5 + 7 2 = 3 4 2^{5}+7^{2}=3^{4}\;
  4. 13 2 + 7 3 = 2 9 13^{2}+7^{3}=2^{9}\;
  5. 2 7 + 17 3 = 71 2 2^{7}+17^{3}=71^{2}\;
  6. 3 5 + 11 4 = 122 2 3^{5}+11^{4}=122^{2}\;
  7. 33 8 + 1549034 2 = 15613 3 33^{8}+1549034^{2}=15613^{3}\;
  8. 1414 3 + 2213459 2 = 65 7 1414^{3}+2213459^{2}=65^{7}\;
  9. 9262 3 + 15312283 2 = 113 7 9262^{3}+15312283^{2}=113^{7}\;
  10. 17 7 + 76271 3 = 21063928 2 17^{7}+76271^{3}=21063928^{2}\;
  11. 43 8 + 96222 3 = 30042907 2 43^{8}+96222^{3}=30042907^{2}\;

Ferromagnetic_material_properties.html

  1. μ i = B μ 0 * H \mu_{i}=\frac{B}{\mu_{0}*H}
  2. μ Δ = Δ B μ 0 * Δ H \mu_{\Delta}=\frac{\Delta B}{\mu_{0}*\Delta H}
  3. μ a = B μ 0 * H \mu_{a}=\frac{B}{\mu_{0}*H}
  4. μ Δ \mu_{\Delta}
  5. ρ \rho
  6. Ω * m \Omega*m
  7. α F = μ θ - μ r e f μ r e f 2 * ( θ - θ r e f ) \alpha_{F}=\frac{\mu_{\theta}-\mu_{ref}}{\mu_{ref}^{2}*(\theta-\theta_{ref})}
  8. α F = μ θ - μ r e f μ θ * μ r e f * ( θ - θ r e f ) \alpha_{F}=\frac{\mu_{\theta}-\mu_{ref}}{\mu_{\theta}*\mu_{ref}*(\theta-\theta% _{ref})}
  9. 2 * P i * f * L 2*Pi*f*L
  10. μ e = μ i \mu_{e}=\mu_{i}
  11. t a n δ μ i * 10 6 {\frac{tan\delta}{\mu_{i}}}*10^{6}
  12. D F = μ 1 - μ 2 ( μ 1 ) 2 * l o g ( t 2 / t 1 ) D_{F}=\frac{\mu_{1}-\mu_{2}}{(\mu_{1})^{2}*log(t2/t1)}
  13. μ 1 , μ 2 \mu_{1},\mu_{2}
  14. η B \eta_{B}
  15. β F \beta_{F}
  16. V e = C 1 3 C 2 2 V_{e}=\frac{{C_{1}}^{3}}{{C_{2}}^{2}}
  17. μ e = C 1 Σ l μ A \mu_{e}=\frac{C_{1}}{\Sigma\frac{l}{\mu A}}
  18. α \alpha
  19. α 2 * A L = 1000000 \alpha^{2}*A_{L}=1000000

FFAG_accelerator.html

  1. B r = 0 , B θ = 0 , B z = a r k f ( ψ ) B_{r}=0,\quad B_{\theta}=0,\quad B_{z}=ar^{k}~{}f(\psi)
  2. ψ = N [ tan ζ ln ( r / r 0 ) - θ ] \psi=N~{}[\tan~{}\zeta~{}\ln(r/r_{0})~{}-~{}\theta]
  3. k k
  4. N N
  5. ζ \zeta
  6. r r
  7. f ( ψ ) f(\psi)
  8. k 1 k>>1

Fiat–Shamir_heuristic.html

  1. x x
  2. y = g x y=g^{x}
  3. g g
  4. v \Z q v\in\Z_{q}
  5. t = g v t=g^{v}
  6. t t
  7. c \Z q * c\in\Z^{*}_{q}
  8. r = v - c x r=v-cx
  9. r r
  10. t g r y c t\equiv g^{r}y^{c}
  11. g r y c = g v - c x g x c = g v = t g^{r}y^{c}=g^{v-cx}g^{xc}=g^{v}=t
  12. x x
  13. y = g x y=g^{x}
  14. g g
  15. v \Z q v\in\Z_{q}
  16. t = g v t=g^{v}
  17. c = H ( g , y , t ) c=H(g,y,t)
  18. H ( ) H()
  19. r = v - c x r=v-cx
  20. ( t , r ) (t,r)
  21. r r
  22. g g
  23. q - 1 q-1
  24. t g r y c t\equiv g^{r}y^{c}

Fiber_pull-out.html

  1. W d = π d 2 σ f 2 l d 24 E f W_{d}=\frac{\pi\;d^{2}\;\sigma_{f}^{2}\;l_{d}}{24\;E_{f}}
  2. d d
  3. σ f 2 \sigma_{f}^{2}
  4. l d l_{d}
  5. E f E_{f}

Fibonacci_retracement.html

  1. F 100 % = ( 1 + 5 2 ) 0 = 1 F_{100\%}=\left(\frac{1+\sqrt{5}}{2}\right)^{0}=1\,
  2. F 61.8 % = ( 1 + 5 2 ) - 1 0.618034 F_{61.8\%}=\left({\frac{1+\sqrt{5}}{2}}\right)^{-1}\approx 0.618034\,
  3. F 38.2 % = ( 1 + 5 2 ) - 2 0.381966 F_{38.2\%}=\left({\frac{1+\sqrt{5}}{2}}\right)^{-2}\approx 0.381966\,
  4. F 23.6 % = ( 1 + 5 2 ) - 3 0.236068 F_{23.6\%}=\left({\frac{1+\sqrt{5}}{2}}\right)^{-3}\approx 0.236068\,
  5. F 0 % = ( 1 + 5 2 ) - = 0 F_{0\%}=\left({\frac{1+\sqrt{5}}{2}}\right)^{-\infty}=0\,
  6. F 76.4 % = 1 - ( 1 + 5 2 ) - 3 0.763932 F_{76.4\%}=1-\left({\frac{1+\sqrt{5}}{2}}\right)^{-3}\approx 0.763932\,
  7. F 78.6 % = ( 1 + 5 2 ) - 1 2 0.786151 F_{78.6\%}=\left({\frac{1+\sqrt{5}}{2}}\right)^{-\frac{1}{2}}\approx 0.786151\,
  8. F 50 % = 1 2 = 0.500000 F_{50\%}=\frac{1}{2}=0.500000\,

Fibonomial_coefficient.html

  1. ( n k ) F = F n F n - 1 F n - k + 1 F k F k - 1 F 1 = n ! F k ! F ( n - k ) ! F {\left({{n}\atop{k}}\right)}_{F}=\frac{F_{n}F_{n-1}\cdots F_{n-k+1}}{F_{k}F_{k% -1}\cdots F_{1}}=\frac{n!_{F}}{k!_{F}(n-k)!_{F}}
  2. ( n 0 ) F = ( n n ) F = 1 {\left({{n}\atop{0}}\right)}_{F}={\left({{n}\atop{n}}\right)}_{F}=1
  3. ( n 1 ) F = ( n n - 1 ) F = F n {\left({{n}\atop{1}}\right)}_{F}={\left({{n}\atop{n-1}}\right)}_{F}=F_{n}
  4. ( n 2 ) F = ( n n - 2 ) F = F n F n - 1 F 2 F 1 = F n F n - 1 , {\left({{n}\atop{2}}\right)}_{F}={\left({{n}\atop{n-2}}\right)}_{F}=\frac{F_{n% }F_{n-1}}{F_{2}F_{1}}=F_{n}F_{n-1},
  5. ( n 3 ) F = ( n n - 3 ) F = F n F n - 1 F n - 2 F 3 F 2 F 1 = F n F n - 1 F n - 2 / 2 , {\left({{n}\atop{3}}\right)}_{F}={\left({{n}\atop{n-3}}\right)}_{F}=\frac{F_{n% }F_{n-1}F_{n-2}}{F_{3}F_{2}F_{1}}=F_{n}F_{n-1}F_{n-2}/2,
  6. ( n k ) F = ( n n - k ) F . {\left({{n}\atop{k}}\right)}_{F}={\left({{n}\atop{n-k}}\right)}_{F}.
  7. n = 0 n=0
  8. n = 1 n=1
  9. n = 2 n=2
  10. n = 3 n=3
  11. n = 4 n=4
  12. n = 5 n=5
  13. n = 6 n=6
  14. n = 7 n=7
  15. ( n k ) F = F n - k + 1 ( n - 1 k - 1 ) F + F k - 1 ( n - 1 k ) F {\left({{n}\atop{k}}\right)}_{F}=F_{n-k+1}{\left({{n-1}\atop{k-1}}\right)}_{F}% +F_{k-1}{\left({{n-1}\atop{k}}\right)}_{F}

Fibonorial.html

  1. n ! F := i = 1 n F i , n 1 , and 0 ! F := 1 , {n!}_{F}:=\prod_{i=1}^{n}F_{i},\quad n\geq 1,\,\text{ and }0!_{F}:=1,

FibroTest.html

  1. z = 4.467 × log 10 [ α 2 m a c r o g l o b u l i n ( g / L ) ] - 1.357 × log 10 [ H a p t o g l o b i n ( g / L ) ] + 1.017 × log 10 [ G G T ( I U / L ) ] + 0.0281 × [ A g e ( y e a r s ) ] z=4.467\times\log_{10}[\alpha 2macroglobulin(g/L)]-1.357\times\log_{10}[% Haptoglobin(g/L)]+1.017\times\log_{10}[GGT(IU/L)]+0.0281\times[Age(years)]
  2. + 1.737 × log 10 [ B i l i r u b i n ( μ m o l / L ) ] - 1.184 × [ A p o A 1 ( g / L ) ] + 0.301 × S e x ( f e m a l e = 0 , m a l e = 1 ) - 5.54 +1.737\times\log_{10}[Bilirubin(\mu mol/L)]-1.184\times[ApoA1(g/L)]+0.301% \times Sex(female=0,male=1)-5.54

Field_(physics).html

  1. 𝐠 ( 𝐫 ) = 𝐅 ( 𝐫 ) m . \mathbf{g}(\mathbf{r})=\frac{\mathbf{F}(\mathbf{r})}{m}.
  2. 𝐅 ( 𝐫 ) = - G M m r 2 𝐫 ^ , \mathbf{F}(\mathbf{r})=-\frac{GMm}{r^{2}}\hat{\mathbf{r}},
  3. 𝐫 ^ \hat{\mathbf{r}}
  4. 𝐠 ( 𝐫 ) = 𝐅 ( 𝐫 ) m = - G M r 2 𝐫 ^ . \mathbf{g}(\mathbf{r})=\frac{\mathbf{F}(\mathbf{r})}{m}=-\frac{GM}{r^{2}}\hat{% \mathbf{r}}.
  5. 𝐠 ( 𝐫 ) = - Φ ( 𝐫 ) . \mathbf{g}(\mathbf{r})=-\nabla\Phi(\mathbf{r}).
  6. 𝐄 = 1 4 π ϵ 0 q r 2 𝐫 ^ . \mathbf{E}=\frac{1}{4\pi\epsilon_{0}}\frac{q}{r^{2}}\hat{\mathbf{r}}.
  7. 𝐄 ( 𝐫 ) = - V ( 𝐫 ) . \mathbf{E}(\mathbf{r})=-\nabla V(\mathbf{r}).
  8. 𝐅 ( 𝐫 ) = q 𝐯 × 𝐁 ( 𝐫 ) , \mathbf{F}(\mathbf{r})=q\mathbf{v}\times\mathbf{B}(\mathbf{r}),
  9. 𝐁 ( 𝐫 ) = μ 0 I 4 π d s y m b o l × d 𝐫 ^ r 2 . \mathbf{B}(\mathbf{r})=\frac{\mu_{0}I}{4\pi}\int\frac{dsymbol{\ell}\times d% \hat{\mathbf{r}}}{r^{2}}.
  10. 𝐁 ( 𝐫 ) = s y m b o l × 𝐀 ( 𝐫 ) \mathbf{B}(\mathbf{r})=symbol{\nabla}\times\mathbf{A}(\mathbf{r})
  11. 𝐄 = - s y m b o l V - 𝐀 t \mathbf{E}=-symbol{\nabla}V-\frac{\partial\mathbf{A}}{\partial t}
  12. 𝐁 = s y m b o l × 𝐀 . \mathbf{B}=symbol{\nabla}\times\mathbf{A}.
  13. ± \pm\infty

File:Cml2e.gif.html

  1. x n + 1 = r x n ( 1 - x n ) x_{n+1}=rx_{n}(1-x_{n})
  2. r = 3.5 , 3.51 , , 3.9 r=3.5,3.51,...,3.9
  3. 3.58 3.58

File:Natural_boundary_example.gif.html

  1. n = 0 z 2 n \sum_{n=0}^{\infty}z^{2^{n}}

File:Proton_decay4.svg.html

  1. T ¯ ( 3 ¯ , 1 ) 1 3 \bar{T}(\bar{3},1)_{\frac{1}{3}}
  2. S U ( 5 ) SU(5)

File:Sech_squared_a_arctanh.png.html

  1. f ( x ; μ , s ) = a 4 s cosh 2 [ a sinh - 1 ( x - μ 2 s ) ] ( x - μ ) 2 4 s 2 + 1 f(x;\mu,s)=\frac{a}{4s\cosh^{2}\left[a\sinh^{-1}\left(\frac{x-\mu}{2s}\right)% \right]\sqrt{\frac{(x-\mu)^{2}}{4s^{2}}+1}}
  2. a = 1.6 , s σ / 2.25 = 2 σ / 3 a=1.6,\,s\approx\sigma/\sqrt{2.25}=2\sigma/3

File:Shine1019.jpg.html

  1. p i n k pink

Filter_(signal_processing).html

  1. H ( s ) \ H(s)
  2. Y ( s ) \ Y(s)
  3. X ( s ) \ X(s)
  4. s \ s
  5. H ( s ) = Y ( s ) X ( s ) \ H(s)=\frac{Y(s)}{X(s)}
  6. s = σ + j ω \ s=\sigma+j\omega
  7. s \ s
  8. s \ s
  9. s \ s
  10. { d f d t } = s { f ( t ) } - f ( 0 ) , \mathcal{L}\left\{\frac{df}{dt}\right\}=s\cdot\mathcal{L}\left\{f(t)\right\}-f% (0),
  11. { d f d t } = s { f ( t ) } \mathcal{L}\left\{\frac{df}{dt}\right\}=s\cdot\mathcal{L}\left\{f(t)\right\}

Filtered-popping_recursive_transition_network.html

  1. ε \varepsilon
  2. ( Q , K , Σ , δ , κ , Q I , F ) (Q,K,\Sigma,\delta,\kappa,Q_{I},F)
  3. Q Q
  4. K K
  5. Σ \Sigma
  6. δ : Q × ( Σ { ε } Q ) Q \delta:Q\times(\Sigma\cup\{\varepsilon\}\cup Q)\to Q
  7. ε \varepsilon
  8. κ : Q K \kappa:Q\to K
  9. Q I Q Q_{I}\subseteq Q
  10. F Q F\subseteq Q
  11. q s q_{s}
  12. q t q_{t}
  13. ε \varepsilon
  14. δ ( q s , ε ) q t \delta(q_{s},\varepsilon)\to q_{t}
  15. δ ( q s , σ ) q t \delta(q_{s},\sigma)\to q_{t}
  16. σ \sigma
  17. δ ( q s , q c ) q t \delta(q_{s},q_{c})\to q_{t}
  18. q c q_{c}
  19. q t q_{t}
  20. δ ( q s , q c ) q t \delta(q_{s},q_{c})\to q_{t}
  21. q s q_{s}
  22. q c q_{c}
  23. q t q_{t}
  24. ( q f , q r ) F × Q (q_{f},q_{r})\in F\times Q
  25. q f q_{f}
  26. q r q_{r}
  27. q r q_{r}
  28. q r q_{r}
  29. κ ( q f ) = κ ( q r ) \kappa(q_{f})=\kappa(q_{r})
  30. κ \kappa
  31. σ 1 σ l \sigma_{1}\ldots\sigma_{l}
  32. p p
  33. q q
  34. q q^{\prime}
  35. p p
  36. σ κ ( q ) + 1 σ κ ( q ) \sigma_{\kappa(q)+1}\ldots\sigma_{\kappa(q^{\prime})}

Financial_independence.html

  1. Y e a r s U n t i l F I = Y e a r l y E x p e n s e s W i t h d r a w a l R a t e - N e t W o r t h Y e a r l y E a r n i n g s A f t e r T a x S a v i n g s R a t e Years\;Until\;FI=\frac{\frac{Yearly\;Expenses}{Withdrawal\;Rate}-Net\;Worth}{% Yearly\;Earnings\;After\;Tax\cdot Savings\;Rate}
  2. $ 10 , 000 4 % - $ 5000 $ 30 , 000 75 % = 10.9 y e a r s \frac{\frac{\$10,000}{4\%}-\$5000}{\$30,000\cdot 75\%}=10.9\;years

Finite_difference_coefficient.html

  1. f ′′′ ( x 0 ) - 1 2 f ( x - 2 ) + f ( x - 1 ) - f ( x + 1 ) + 1 2 f ( x + 2 ) h x 3 + O ( h x 2 ) \displaystyle f^{\prime\prime\prime}(x_{0})\approx\displaystyle\frac{-\frac{1}% {2}f(x_{-2})+f(x_{-1})-f(x_{+1})+\frac{1}{2}f(x_{+2})}{h^{3}_{x}}+O\left(h_{x}% ^{2}\right)
  2. h x h_{x}
  3. f ( x 0 ) - 11 6 f ( x 0 ) + 3 f ( x + 1 ) - 3 2 f ( x + 2 ) + 1 3 f ( x + 3 ) h x + O ( h x 3 ) , \displaystyle f^{\prime}(x_{0})\approx\displaystyle\frac{-\frac{11}{6}f(x_{0})% +3f(x_{+1})-\frac{3}{2}f(x_{+2})+\frac{1}{3}f(x_{+3})}{h_{x}}+O\left(h_{x}^{3}% \right),
  4. f ′′ ( x 0 ) 2 f ( x 0 ) - 5 f ( x + 1 ) + 4 f ( x + 2 ) - f ( x + 3 ) h x 2 + O ( h x 2 ) , \displaystyle f^{\prime\prime}(x_{0})\approx\displaystyle\frac{2f(x_{0})-5f(x_% {+1})+4f(x_{+2})-f(x_{+3})}{h_{x}^{2}}+O\left(h_{x}^{2}\right),
  5. f ( x 0 ) 11 6 f ( x 0 ) - 3 f ( x - 1 ) + 3 2 f ( x - 2 ) - 1 3 f ( x - 3 ) h x + O ( h x 3 ) , \displaystyle f^{\prime}(x_{0})\approx\displaystyle\frac{\frac{11}{6}f(x_{0})-% 3f(x_{-1})+\frac{3}{2}f(x_{-2})-\frac{1}{3}f(x_{-3})}{h_{x}}+O\left(h_{x}^{3}% \right),
  6. f ′′ ( x 0 ) 2 f ( x 0 ) - 5 f ( x - 1 ) + 4 f ( x - 2 ) - f ( x - 3 ) h x 2 + O ( h x 2 ) , \displaystyle f^{\prime\prime}(x_{0})\approx\displaystyle\frac{2f(x_{0})-5f(x_% {-1})+4f(x_{-2})-f(x_{-3})}{h_{x}^{2}}+O\left(h_{x}^{2}\right),

Finite_ring.html

  1. / m \mathbb{Z}/m\mathbb{Z}
  2. M n ( 𝔽 q ) M_{n}(\mathbb{F}_{q})
  3. M n ( 𝔽 q ) M_{n}(\mathbb{F}_{q})

Firefly_algorithm.html

  1. f ( 𝐱 ) , 𝐱 = ( x 1 , x 2 , , x d ) f(\mathbf{x}),\quad\mathbf{x}=(x_{1},x_{2},...,x_{d})
  2. 𝐱 i ( i = 1 , 2 , , n ) \mathbf{x}_{i}\quad(i=1,2,\dots,n)
  3. I I
  4. f ( 𝐱 ) f(\mathbf{x})
  5. I f ( 𝐱 ) I\propto f(\mathbf{x})
  6. I = f ( 𝐱 ) I=f(\mathbf{x})
  7. γ \gamma
  8. exp ( - γ r ) \exp(-\gamma\;r)
  9. 𝐱 i \mathbf{x}_{i}
  10. 𝐱 j \mathbf{x}_{j}
  11. 𝐱 i t + 1 = 𝐱 i t + β exp [ - γ r i j 2 ] ( 𝐱 j t - 𝐱 i t ) + α t s y m b o l ϵ t \mathbf{x}_{i}^{t+1}=\mathbf{x}_{i}^{t}+\beta\exp[-\gamma r_{ij}^{2}](\mathbf{% x}_{j}^{t}-\mathbf{x}_{i}^{t})+\alpha_{t}symbol{\epsilon}_{t}
  12. α t \alpha_{t}
  13. s y m b o l ϵ t symbol{\epsilon}_{t}
  14. γ 0 \gamma\rightarrow 0
  15. I j I_{j}
  16. g * g^{*}
  17. γ \gamma
  18. β \beta
  19. γ \gamma
  20. γ = 1 / L \gamma=1/\sqrt{L}
  21. L L
  22. γ \gamma
  23. α t \alpha_{t}
  24. α t 0.01 L α t \alpha_{t}\leftarrow 0.01L\alpha_{t}
  25. α = α 0 δ t \alpha=\alpha_{0}\delta^{t}
  26. 0 < δ < 1 ( e . g . δ = 0.97 ) 0<\delta<1(e.g.\delta=0.97)
  27. α t \alpha_{t}
  28. α t \alpha_{t}

Firehose_instability.html

  1. h ( x , t ) h(x,t)
  2. z z
  3. x x
  4. u u
  5. a z = ( t + u x ) 2 h = 2 h t 2 + 2 u 2 h t x + u 2 2 h x 2 , a_{z}=\left({\partial\over\partial t}+u{\partial\over\partial x}\right)^{2}h={% \partial^{2}h\over\partial t^{2}}+2u{\partial^{2}h\over\partial t\partial x}+u% ^{2}{\partial^{2}h\over\partial x^{2}},\,
  6. x x
  7. F x F_{x}
  8. 2 h t 2 + σ u 2 2 h x 2 - F z ( x , t ) = 0 , {\partial^{2}h\over\partial t^{2}}+\sigma_{u}^{2}{\partial^{2}h\over\partial x% ^{2}}-F_{z}(x,t)=0,\,
  9. σ u \sigma_{u}
  10. h ( x , t ) = H exp [ i ( k x - ω t ) ] h(x,t)=H\exp\left[i\left(kx-\omega t\right)\right]
  11. F z ( x , t ) = - G Σ - d y - [ h ( x , t ) - h ( x , t ) ] [ ( x - x ) 2 + ( y - y ) 2 ] 3 / 2 d x = - 2 π G Σ k h ( x , t ) F_{z}(x,t)=-G\Sigma\int_{-\infty}^{\infty}dy^{\prime}\int_{-\infty}^{\infty}{% \left[h(x,t)-h(x^{\prime},t)\right]\over\left[(x-x^{\prime})^{2}+(y-y^{\prime}% )^{2}\right]^{3/2}}dx^{\prime}=-2\pi G\Sigma kh(x,t)
  12. Σ \Sigma
  13. ω 2 = 2 π G Σ k - σ u 2 k 2 . \omega^{2}=2\pi G\Sigma k-\sigma_{u}^{2}k^{2}.
  14. λ = 2 π / k > λ J = σ u 2 / G Σ \lambda=2\pi/k>\lambda_{J}=\sigma_{u}^{2}/G\Sigma
  15. λ < λ J \lambda<\lambda_{J}
  16. κ z \kappa_{z}
  17. k u ku
  18. k u > κ z ku>\kappa_{z}
  19. κ z \kappa_{z}
  20. Ω z \Omega_{z}
  21. Ω z \Omega_{z}
  22. 2 Ω x > Ω z 2\Omega_{x}>\Omega_{z}\,
  23. δ z e i m ϕ \delta z\propto e^{im\phi}
  24. m Ω > κ z m\Omega>\kappa_{z}\,
  25. Ω \Omega

First_break_picking.html

  1. d i = r i + 1 - r i , i = 1 , 2 , ( n - 1 ) \displaystyle d_{i}=r_{i+1}-r_{i},i=1,2,...(n-1)
  2. e r = j = i - n e i x j 2 / j = i i + n e x j 2 \displaystyle er=\sum_{j=i-ne}^{i}x^{2}_{j}/\sum_{j=i}^{i+ne}x^{2}_{j}
  3. e r 3 i = ( a b s ( x i ) * e r ) 3 \displaystyle er3_{i}=(abs(x_{i})^{*}er)^{3}
  4. B T A ( t ) ¯ = i = i m | u ( t - i ) | m \displaystyle\overline{BTA(t)}=\sum_{i=i}^{m}\frac{|u(t-i)|}{m}
  5. H 1 ( t ) = E m ( t - p ) + α E s d ( t - p ) \displaystyle H_{1}(t)=E_{m}(t-p)+\alpha E_{sd}(t-p)

Five_points_determine_a_conic.html

  1. 𝐏 5 ; \mathbf{P}^{5};
  2. ( x , y ) , (x,y),
  3. A x 2 + B x y + C y 2 + D x + E y + F = 0 Ax^{2}+Bxy+Cy^{2}+Dx+Ey+F=0
  4. ( A , B , C , D , E , F ) ; (A,B,C,D,E,F);
  5. D x + E y + F = 0 Dx+Ey+F=0
  6. 𝐏 5 \mathbf{P}^{5}
  7. [ x 2 : x y : y 2 : x z : y z : z 2 ] , [x^{2}:xy:y^{2}:xz:yz:z^{2}],
  8. 𝐏 5 \mathbf{P}^{5}
  9. [ A : B : C : D : E : F ] [A:B:C:D:E:F]
  10. 𝐏 5 \mathbf{P}^{5}
  11. f ( x ) f(x)
  12. [ 1 : 0 : 0 ] [1:0:0]
  13. [ 0 : 1 : 0 ] [0:1:0]
  14. X A , X B , X C XA,XB,XC
  15. Y A , Y B , Y C YA,YB,YC
  16. m = n - 1 m=n-1
  17. k = ( ( n + 1 d ) ) - 1 = ( n + d d ) - 1 = 1 n ! ( d + 1 ) ( n ) - 1. k=\left(\!\!{n+1\choose d}\!\!\right)-1={n+d\choose d}-1=\frac{1}{n!}(d+1)^{(n% )}-1.
  18. n + 1 n+1
  19. n + 1 n+1
  20. ( ( n + 1 d ) ) , \textstyle{\left(\!\!{n+1\choose d}\!\!\right)},
  21. 1 / n ! 1/n!
  22. n = 2 , n=2,
  23. 1 2 ( d + 1 ) ( d + 2 ) - 1 = 1 2 ( d 2 + 3 d ) \textstyle{\frac{1}{2}}(d+1)(d+2)-1=\textstyle{\frac{1}{2}}(d^{2}+3d)
  24. d = 0 , 1 , 2 , 3 , 4 d=0,1,2,3,4
  25. 0 , 2 , 5 , 9 , 14 0,2,5,9,14

Flag_of_China.html

  1. Δ E a b 2.0 \Delta E_{ab}^{\,\bullet}\leq 2.0

Flat_function.html

  1. f ( x ) lim n k = 0 n f ( k ) ( x 0 ) k ! ( x - x 0 ) k . f(x)\sim\lim_{n\to\infty}\sum_{k=0}^{n}\frac{f^{(k)}(x_{0})}{k!}(x-x_{0})^{k}.
  2. f ( x ) = { e - 1 / x 2 if x 0 0 if x = 0 f(x)=\begin{cases}e^{-1/x^{2}}&\,\text{if }x\neq 0\\ 0&\,\text{if }x=0\end{cases}

Flatness_(electrical_engineering).html

  1. flatness = max ( P out ) - min ( P out ) \,\text{flatness}=\max\left(P\text{out}\right)-\min\left(P\text{out}\right)
  2. F = < f 4 Align g t ; / < f 2 > 2 F=<f^{4}&gt;/<f^{2}>^{2}

Flatness_(mathematics).html

  1. A A
  2. - A -\otimes A

Flexoelectricity.html

  1. P i = d i j k σ j k + μ i j k l ϵ j k x l P_{i}=d_{ijk}\sigma_{jk}+\mu_{ijkl}\frac{\partial\epsilon_{jk}}{\partial x_{l}}
  2. μ i j k l \mu_{ijkl}
  3. d i j k d_{ijk}

FLiBe.html

  1. 2 L i F ( s ) + B e F 2 ( s ) 2 L i ( l ) + + B e F 4 ( l ) - 2 2LiF_{(s)}+BeF_{2(s)}\longrightarrow 2Li^{+}_{(l)}+BeF_{4(l)}^{-2}
  2. B e F 2 ( l ) + 2 H 2 O ( g ) B e ( O H ) 2 ( d ) + 2 H F ( d ) BeF_{2(l)}+2H_{2}O_{(g)}\leftrightharpoons Be(OH)_{2(d)}+2HF_{(d)}
  3. B e F 2 ( l ) + H 2 O ( g ) B e ( O ) ( d ) + 2 H F ( d ) BeF_{2(l)}+H_{2}O_{(g)}\leftrightharpoons Be(O)_{(d)}+2HF_{(d)}
  4. H F ( g ) + e - F - + 1 / 2 H 2 ( g ) HF_{(g)}+e^{-}\longrightarrow F^{-}+1/2H_{2(g)}
  5. N i F 2 ( d ) + 2 e - N i ( c ) + 2 F - NiF_{2(d)}+2e^{-}\longrightarrow Ni_{(c)}+2F^{-}

Florida_pompano.html

  1. W = c L b W=cL^{b}\!\,

Flow_shop_scheduling.html

  1. ( w i ) F i \sum(w_{i})F_{i}
  2. ( w i ) T i \sum(w_{i})T_{i}

Flux_pumping.html

  1. 2 𝐇 = λ - 2 𝐇 \nabla^{2}\mathbf{H}=\lambda^{-2}\mathbf{H}\,
  2. Φ 0 = h 2 e \Phi_{0}=\frac{h}{2e}
  3. ρ = d E d J \rho=\frac{dE}{dJ}\,
  4. 𝐄 = 𝐄 𝟎 * ( J J c ) n \mathbf{E}=\mathbf{E_{0}}*\left(\frac{J}{J_{c}}\right)^{n}\,
  5. ρ ( J ) = E 0 * n * ( J J c ) n - 1 J c \rho(J)=\frac{E_{0}*n*(\frac{J}{J_{c}})^{n-1}}{J_{c}}\,

Flywheel_energy_storage.html

  1. E m = K ( σ ρ ) \frac{E}{m}=K\left(\frac{\sigma}{\rho}\right)
  2. E E
  3. m m
  4. K K
  5. σ \sigma
  6. ρ \rho
  7. K = 1 K=1
  8. K = 0.606 K=0.606
  9. K = 0.333 K=0.333
  10. K = 0.5 K=0.5

FM-index.html

  1. T = T=
  2. M M
  3. L L
  4. I I
  5. F F
  6. L L
  7. L F ( i ) LF(i)
  8. i i
  9. j j
  10. F j j Fjj
  11. L i i Lii
  12. C c c Ccc
  13. O c c ( c , k ) Occ(c,k)
  14. C c c Ccc
  15. c c
  16. O c c ( c , k ) Occ(c,k)
  17. c c
  18. L 1.. k k L1..kk
  19. O c c ( c , k ) Occ(c,k)
  20. C c c Ccc
  21. c c
  22. C c c Ccc
  23. L F ( i ) = C L L i i LF(i)=CLLii
  24. L L
  25. a a
  26. a a
  27. F F
  28. L F ( 9 ) LF(9)
  29. L F ( 9 ) = C a a + O c c ( a , 9 ) = 5 LF(9)=Caa+Occ(a,9)=5
  30. i i
  31. L i i Lii
  32. F i i Fii
  33. L i i = T k k Lii=Tkk
  34. L L F ( i ) ) = T k - 11 LLF(i))=Tk-11
  35. T T
  36. L L
  37. L L
  38. C C
  39. O c c Occ
  40. L L
  41. T T
  42. O c c ( c , k ) Occ(c,k)
  43. P 1.. p p P1..pp
  44. T T
  45. M M
  46. T T
  47. P P
  48. a a
  49. C C a a + 1.. C C a + 11 = 2..66 CCaa+1..CCa+11=2..66
  50. L L
  51. T T
  52. r r
  53. C C r r + O c c ( r , s t a r t - 1 ) + 1.. C C r r + O c c ( r , e n d ) ) = CCrr+Occ(r,start-1)+1..CCrr+Occ(r,end))=
  54. 10 + 0 + 1..10 + 22 = 10+0+1..10+22=
  55. 11..122 11..122
  56. s t a r t start
  57. e n d end
  58. L L
  59. T T
  60. b b
  61. C C b b + O c c ( b , s t a r t - 1 ) + 1.. C C b b + O c c ( b , e n d ) ) = CCbb+Occ(b,start-1)+1..CCbb+Occ(b,end))=
  62. 6 + 0 + 1..6 + 22 = 6+0+1..6+22=
  63. 7..88 7..88
  64. L L
  65. 8 - 7 + 1 = 2 8-7+1=2
  66. T T
  67. O c c ( c , k ) Occ(c,k)
  68. O ( p ) O(p)
  69. L L
  70. i i
  71. T T
  72. l o c a t e ( 7 ) = 8 locate(7)=8
  73. L L
  74. T T
  75. L L
  76. T T
  77. L j j Ljj
  78. l o c a t e ( j ) locate(j)
  79. L F ( i ) LF(i)
  80. P 1.. p p P1..pp
  81. T 1.. u u T1..uu
  82. O ( H k ( T ) + log log u log ϵ u ) O(H_{k}(T)+{{\log\log u}\over{\log^{\epsilon}u}})
  83. k 0 k≥0

Focal_subgroup_theorem.html

  1. p k p^{k}
  2. [ G , G ] [G,G]
  3. p k p^{k}
  4. 𝐄 p ( G ) 𝐀 p ( G ) 𝐎 p ( G ) . \mathbf{E}^{p}(G)\supseteq\mathbf{A}^{p}(G)\supseteq\mathbf{O}^{p}(G).
  5. | |
  6. | |

Focaloid.html

  1. x 2 a 2 + y 2 b 2 + z 2 c 2 = 1 \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}+\frac{z^{2}}{c^{2}}=1
  2. x 2 a 2 + λ + y 2 b 2 + λ + z 2 c 2 + λ = 1. \frac{x^{2}}{a^{2}+\lambda}+\frac{y^{2}}{b^{2}+\lambda}+\frac{z^{2}}{c^{2}+% \lambda}=1.
  3. λ 0 \lambda\to 0
  4. f 1 2 = a 2 - b 2 = ( a 2 + λ ) - ( b 2 + λ ) , f_{1}^{2}=a^{2}-b^{2}=(a^{2}+\lambda)-(b^{2}+\lambda),\,
  5. f 2 2 = a 2 - c 2 = ( a 2 + λ ) - ( c 2 + λ ) , f_{2}^{2}=a^{2}-c^{2}=(a^{2}+\lambda)-(c^{2}+\lambda),\,
  6. f 3 2 = b 2 - c 2 = ( b 2 + λ ) - ( c 2 + λ ) . f_{3}^{2}=b^{2}-c^{2}=(b^{2}+\lambda)-(c^{2}+\lambda).

Fock–Lorentz_symmetry.html

  1. c = 1 ε 0 μ 0 , c=\frac{1}{\sqrt{\varepsilon_{0}\mu_{0}}}\ ,
  2. c ( x ) = 1 χ ( x ) , c(x)=\frac{1}{\sqrt{\chi(x)}}\ ,
  3. χ ( x ) \chi(x)

Fold_change.html

  1. n 100 % n\cdot 100\%
  2. n n
  3. ( n - 1 ) 100 % (n-1)\cdot 100\%

Folkman_graph.html

  1. ( x - 4 ) x 10 ( x + 4 ) ( x 2 - 6 ) 4 (x-4)x^{10}(x+4)(x^{2}-6)^{4}

Forbidden_subgraph_problem.html

  1. K r , n r 3 K_{r},\,n\geq r\geq 3

Fork–join_queue.html

  1. 12 - ρ 8 μ ( 1 - ρ ) \frac{12-\rho}{8\mu(1-\rho)}
  2. ρ = λ / μ \rho=\lambda/\mu
  3. λ \lambda
  4. μ \mu
  5. ( n , k ) (n,k)
  6. k k
  7. n n
  8. ( n , k ) (n,k)
  9. k = n k=n

Forms_of_energy.html

  1. E = T + V E=T+V\,\!
  2. H = T + V , H=T+V,\,\!
  3. L = T - V L=T-V\,\!
  4. E k = 𝐅 d 𝐱 = 𝐯 d 𝐩 E_{\mathrm{k}}=\int\mathbf{F}\cdot d\mathbf{x}=\int\mathbf{v}\cdot d\mathbf{p}
  5. E k = 𝐅 d 𝐱 = m d 𝐯 d t 𝐯 d t = m d 𝐯 𝐯 = m 1 2 d ( 𝐯 𝐯 ) = 1 2 m v 2 E_{\mathrm{k}}=\int\mathbf{F}\cdot d\mathbf{x}=m\int\frac{d\mathbf{v}}{dt}% \cdot\mathbf{v}dt=m\int d\mathbf{v}\cdot\mathbf{v}=m\int\frac{1}{2}d(\mathbf{v% }\cdot\mathbf{v})=\frac{1}{2}mv^{2}\,\!
  6. E k = ( γ - 1 ) m c 2 , E_{\mathrm{k}}=\left(\gamma-1\right)mc^{2},
  7. γ = 1 1 - ( v c ) 2 \gamma=\frac{1}{\sqrt{1-\left(\frac{v}{c}\right)^{2}}}
  8. E 0 = m 0 c 2 E_{0}=m_{0}c^{2}\,\!
  9. Δ U = - Δ W \Delta U=-\Delta W
  10. Δ W = 𝐫 1 𝐫 2 𝐅 d 𝐫 \Delta W=\int_{\mathbf{r}_{1}}^{\mathbf{r}_{2}}\mathbf{F}\cdot\mathrm{d}% \mathbf{r}
  11. 𝐅 = W \mathbf{F}=\nabla W
  12. Δ W = θ 1 θ 2 | s y m b o l τ | d θ \Delta W=\int_{\theta_{1}}^{\theta_{2}}\left|symbol{\tau}\right|\mathrm{d}\theta
  13. 𝐅 = k 𝐱 \mathbf{F}=k\mathbf{x}\,\!
  14. E p , e = 1 2 k x 2 E_{\mathrm{p,e}}=\frac{1}{2}kx^{2}\,\!
  15. d W = γ d S . \mathrm{d}W=\gamma\mathrm{d}S.\,\!
  16. E p , g m g h E_{\mathrm{p,g}}\approx mgh\,\!
  17. E p , g = - G m 1 m 2 r E_{\mathrm{p,g}}=-\frac{Gm_{1}m_{2}}{r}\,\!
  18. E p , g = - Φ m E_{\mathrm{p,g}}=-\Phi m\,\!
  19. Δ q = T 1 T 2 C v d T \Delta q=\int_{T_{1}}^{T_{2}}C_{\mathrm{v}}\mathrm{d}T
  20. U = d f 2 k B T U=\frac{d_{f}}{2}k_{B}T
  21. Δ E = p Δ V \Delta E=p\Delta V\,\!
  22. Δ H = Δ U + p Δ V \Delta H=\Delta U+p\Delta V\,\!
  23. Δ E = T Δ S \Delta E=T\Delta S\,\!
  24. Δ G = Δ H - T Δ S \Delta G=\Delta H-T\Delta S\,\!
  25. R y = m e e 4 8 ε 0 2 h 2 = 1 2 α 2 m e c 2 = 13.605 692 53 ( 30 ) eV R_{y}=\frac{m_{e}e^{4}}{8\varepsilon_{0}^{2}h^{2}}=\frac{1}{2}\alpha^{2}m_{e}c% ^{2}=13.605\;692\;53(30)\ \mathrm{eV}
  26. E p , e = 1 < m t p l > 4 π ϵ 0 Q 1 Q 2 r E_{\mathrm{p,e}}=\frac{1}{<}mtpl>{{4\pi\epsilon_{0}}}{{Q_{1}Q_{2}}\over{r}}
  27. E p , e = ϕ q E_{\mathrm{p,e}}=\phi q\,\!
  28. E p , e = Q 2 2 C = 1 2 C V 2 = 1 2 V Q E_{\mathrm{p,e}}=\frac{Q^{2}}{2C}=\frac{1}{2}CV^{2}=\frac{1}{2}VQ\,\!
  29. E = V Q = V I t = P t = V 2 t R = I 2 R t E=VQ=VIt=Pt=\frac{V^{2}t}{R}={I^{2}}Rt\,\!
  30. E p , m = - 𝐦 𝐁 E_{\mathrm{p,m}}=-\mathbf{m}\cdot\mathbf{B}
  31. E p , m = 1 2 L I 2 E_{\mathrm{p,m}}=\frac{1}{2}LI^{2}
  32. u e = ϵ 0 2 | 𝐄 | 2 , u m = 1 2 μ 0 | 𝐁 | 2 u_{e}=\frac{\epsilon_{0}}{2}\left|\mathbf{E}\right|^{2},\quad u_{m}=\frac{1}{2% \mu_{0}}\left|\mathbf{B}\right|^{2}\,\!
  33. 𝐒 = 1 μ 𝐄 × 𝐁 , \mathbf{S}=\frac{1}{\mu}\mathbf{E}\times\mathbf{B},
  34. E = h ν = h c λ E=h\nu=\frac{hc}{\lambda}\,\!
  35. Δ E = E 2 - E 1 = h c ( ν 2 - ν 1 ) = h c ( 1 λ 2 - 1 λ 1 ) \Delta E=E_{2}-E_{1}=hc\left(\nu_{2}-\nu_{1}\right)=hc\left(\frac{1}{\lambda_{% 2}}-\frac{1}{\lambda_{1}}\right)\,\!

Forward_converter.html

  1. N S / N P \textstyle N_{\mathrm{S}}/N_{\mathrm{P}}
  2. V out = D N S N P V supply V_{\mathrm{out}}=D\cdot\frac{N_{\mathrm{S}}}{N_{\mathrm{P}}}\cdot V_{\mathrm{% supply}}
  3. D D

Forward_volatility.html

  1. t 0 = 0 t_{0}=0
  2. σ 0 , j 2 = 1 j ( σ 0 , 1 2 + σ 1 , 2 2 + + σ j - 2 , j - 1 2 + σ j - 1 , j 2 ) σ j - 1 , j = j σ 0 , j 2 - k = 1 j - 1 σ k - 1 , k 2 , \begin{aligned}\displaystyle\sigma_{0,j}^{2}&\displaystyle=\frac{1}{j}(\sigma_% {0,1}^{2}+\sigma_{1,2}^{2}+\cdots+\sigma_{j-2,j-1}^{2}+\sigma_{j-1,j}^{2})\\ \displaystyle\Rightarrow\sigma_{j-1,j}&\displaystyle=\sqrt{j\sigma_{0,j}^{2}-% \sum_{k=1}^{j-1}\sigma_{k-1,k}^{2}},\end{aligned}
  3. j = 1 , 2 , j=1,2,\ldots
  4. 1 j \frac{1}{j}
  5. σ i , j \sigma_{i,\,j}
  6. [ i , j ] [i,\,j]
  7. σ 0 , j \sigma_{0,\,j}
  8. j j
  9. σ 0 , j 2 = 1 j ( σ 0 , 1 2 + σ 1 , 2 2 + + σ j - 1 , j 2 ) = j - 1 j 1 j - 1 ( σ 0 , 1 2 + σ 1 , 2 2 + + σ j - 2 , j - 1 2 ) + 1 j σ j - 1 , j 2 = j - 1 j σ 0 , j - 1 2 + 1 j σ j - 1 , j 2 σ j - 1 , j = j σ 0 , j 2 - ( j - 1 ) σ 0 , j - 1 2 \begin{aligned}\displaystyle\sigma_{0,j}^{2}&\displaystyle=\frac{1}{j}(\sigma_% {0,1}^{2}+\sigma_{1,2}^{2}+\cdots+\sigma_{j-1,j}^{2})\\ &\displaystyle=\frac{j-1}{j}\cdot\frac{1}{j-1}(\sigma_{0,1}^{2}+\sigma_{1,2}^{% 2}+\cdots+\sigma_{j-2,j-1}^{2})+\frac{1}{j}\sigma_{j-1,j}^{2}\\ &\displaystyle=\frac{j-1}{j}\,\sigma_{0,j-1}^{2}+\frac{1}{j}\sigma_{j-1,j}^{2}% \\ \displaystyle\Rightarrow\sigma_{j-1,j}&\displaystyle=\sqrt{j\sigma_{0,j}^{2}-(% j-1)\sigma_{0,j-1}^{2}}\end{aligned}
  10. t 0 < t < T t_{0}<t<T
  11. t 0 t_{0}
  12. σ t , T = ( T - t 0 ) σ t 0 , T 2 - ( t - t 0 ) σ t 0 , t 2 T - t \sigma_{t,T}=\sqrt{\frac{(T-t_{0})\sigma_{t_{0},T}^{2}-(t-t_{0})\sigma_{t_{0},% t}^{2}}{T-t}}
  13. t 0 = 0 t_{0}=0
  14. σ t , T = T σ 0 , T 2 - t σ 0 , t 2 T - t \sigma_{t,T}=\sqrt{\frac{T\sigma_{0,T}^{2}-t\sigma_{0,t}^{2}}{T-t}}
  15. σ 0 , 0.25 \sigma_{0,\,0.25}
  16. σ 0 , 0.5 \sigma_{0,\,0.5}
  17. t 0 = 0 t_{0}=0
  18. σ 0.25 , 0.5 = 0.5 0.166 2 - 0.25 0.18 2 0.25 = 0.1507 15.1 % \sigma_{0.25,\,0.5}=\sqrt{\frac{0.5\cdot 0.166^{2}-0.25\cdot 0.18^{2}}{0.25}}=% 0.1507\approx 15.1\%

FOSD_Program_Cubes.html

  1. P = F 8 + F 4 + F 2 + F 1 + F 0 P=F_{8}+F_{4}+F_{2}+F_{1}+F_{0}
  2. P = i = ( 0 , 1 , 2 , 4 , 8 ) 𝐅 i P=\sum_{i=(0,1,2,4,8)}\mathbf{F}_{i}
  3. C = M c b + M c t + M w b + M w t C=M_{cb}+M_{ct}+M_{wb}+M_{wt}
  4. P = i 1 = S 1 , i 2 = S 2 , . . i n = S n K i 1 i 2 . . i n P=\sum_{i_{1}=S_{1},i_{2}=S_{2},..i_{n}=S_{n}}K_{i_{1}i_{2}..i_{n}}
  5. C = M c b + M w b + M c t + M w t C=M_{cb}+M_{wb}+M_{ct}+M_{wt}
  6. P = i n = S n , . . i 2 = S 2 , i 1 = S 1 K i 1 i 2 . . i n P=\sum_{i_{n}=S_{n},..i_{2}=S_{2},i_{1}=S_{1}}K_{i_{1}i_{2}..i_{n}}

Fourier_domain_mode_locking.html

  1. L L
  2. f f
  3. 1 / τ = c / L 1/\tau=c/L
  4. f i = i c L f_{i}=i\frac{c}{L}
  5. c c

Fourier_shell_correlation.html

  1. F S C ( r ) = r i r F 1 ( r i ) F 2 ( r i ) r i r | F 1 ( r i ) | 2 r i r | F 2 ( r i ) | 2 2 FSC(r)=\frac{\displaystyle\sum_{r_{i}\in r}{F_{1}(r_{i})\cdot F_{2}(r_{i})^{% \ast}}}{\displaystyle\sqrt[2]{\sum_{r_{i}\in r}{\left|F_{1}(r_{i})\right|^{2}}% \cdot\sum_{r_{i}\in r}{\left|F_{2}(r_{i})\right|^{2}}}}
  2. F 1 F_{1}
  3. F 2 F_{2}^{\ast}
  4. r i r_{i}
  5. r r

Fox_H-function.html

  1. H p , q m , n [ z | ( a 1 , A 1 ) ( a 2 , A 2 ) ( a p , A p ) ( b 1 , B 1 ) ( b 2 , B 2 ) ( b q , B q ) ] = 1 2 π i L ( j = 1 m Γ ( b j + B j s ) ) ( j = 1 n Γ ( 1 - a j - A j s ) ) ( j = m + 1 q Γ ( 1 - b j - B j s ) ) ( j = n + 1 p Γ ( a j + A j s ) ) z - s d s H_{p,q}^{\,m,n}\!\left[z\left|\begin{matrix}(a_{1},A_{1})&(a_{2},A_{2})&\ldots% &(a_{p},A_{p})\\ (b_{1},B_{1})&(b_{2},B_{2})&\ldots&(b_{q},B_{q})\end{matrix}\right.\right]=% \frac{1}{2\pi i}\int_{L}\frac{(\prod_{j=1}^{m}\Gamma(b_{j}+B_{j}s))(\prod_{j=1% }^{n}\Gamma(1-a_{j}-A_{j}s))}{(\prod_{j=m+1}^{q}\Gamma(1-b_{j}-B_{j}s))(\prod_% {j=n+1}^{p}\Gamma(a_{j}+A_{j}s))}z^{-s}\,ds
  2. H p , q m , n [ z | ( a 1 , C ) ( a 2 , C ) ( a p , C ) ( b 1 , C ) ( b 2 , C ) ( b q , C ) ] = 1 C G p , q m , n ( a 1 , , a p b 1 , , b q | z 1 / C ) . H_{p,q}^{\,m,n}\!\left[z\left|\begin{matrix}(a_{1},C)&(a_{2},C)&\ldots&(a_{p},% C)\\ (b_{1},C)&(b_{2},C)&\ldots&(b_{q},C)\end{matrix}\right.\right]=\frac{1}{C}G_{p% ,q}^{\,m,n}\!\left(\left.\begin{matrix}a_{1},\dots,a_{p}\\ b_{1},\dots,b_{q}\end{matrix}\;\right|\,z^{1/C}\right).

Föppl–von_Kármán_equations.html

  1. ( 1 ) E h 3 12 ( 1 - ν 2 ) Δ 2 w - h x β ( σ α β w x α ) = P ( 2 ) σ α β x β = 0 \begin{aligned}\displaystyle(1)&\displaystyle\frac{Eh^{3}}{12(1-\nu^{2})}% \Delta^{2}w-h\frac{\partial}{\partial x_{\beta}}\left(\sigma_{\alpha\beta}% \frac{\partial w}{\partial x_{\alpha}}\right)=P\\ \displaystyle(2)&\displaystyle\frac{\partial\sigma_{\alpha\beta}}{\partial x_{% \beta}}=0\end{aligned}
  2. E E
  3. υ υ
  4. h h
  5. w w
  6. P P
  7. α , β α,β
  8. Δ 2 w := 2 x α x α [ 2 w x β x β ] = 4 w x 1 4 + 4 w x 2 4 + 2 4 w x 1 2 x 2 2 . \Delta^{2}w:=\frac{\partial^{2}}{\partial x_{\alpha}\partial x_{\alpha}}\left[% \frac{\partial^{2}w}{\partial x_{\beta}\partial x_{\beta}}\right]=\frac{% \partial^{4}w}{\partial x_{1}^{4}}+\frac{\partial^{4}w}{\partial x_{2}^{4}}+2% \frac{\partial^{4}w}{\partial x_{1}^{2}\partial x_{2}^{2}}\,.
  9. φ \varphi
  10. σ 11 = 2 φ x 2 2 , σ 22 = 2 φ x 1 2 , σ 12 = - 2 φ x 1 x 2 . \sigma_{11}=\frac{\partial^{2}\varphi}{\partial x_{2}^{2}}~{},~{}~{}\sigma_{22% }=\frac{\partial^{2}\varphi}{\partial x_{1}^{2}}~{},~{}~{}\sigma_{12}=-\frac{% \partial^{2}\varphi}{\partial x_{1}\partial x_{2}}\,.
  11. E h 3 12 ( 1 - ν 2 ) Δ 2 w - h ( 2 φ x 2 2 2 w x 1 2 + 2 φ x 1 2 2 w x 2 2 - 2 2 φ x 1 x 2 2 w x 1 x 2 ) = P \frac{Eh^{3}}{12(1-\nu^{2})}\Delta^{2}w-h\left(\frac{\partial^{2}\varphi}{% \partial x_{2}^{2}}\frac{\partial^{2}w}{\partial x_{1}^{2}}+\frac{\partial^{2}% \varphi}{\partial x_{1}^{2}}\frac{\partial^{2}w}{\partial x_{2}^{2}}-2\frac{% \partial^{2}\varphi}{\partial x_{1}\,\partial x_{2}}\frac{\partial^{2}w}{% \partial x_{1}\,\partial x_{2}}\right)=P
  12. Δ 2 φ + E { 2 w x 1 2 2 w x 2 2 - ( 2 w x 1 x 2 ) 2 } = 0 . \Delta^{2}\varphi+E\left\{\frac{\partial^{2}w}{\partial x_{1}^{2}}\frac{% \partial^{2}w}{\partial x_{2}^{2}}-\left(\frac{\partial^{2}w}{\partial x_{1}\,% \partial x_{2}}\right)^{2}\right\}=0\,.
  13. D Δ 2 w = P D\Delta^{2}\ w=P
  14. D := E h 3 12 ( 1 - ν 2 ) D:=\frac{Eh^{3}}{12(1-\nu^{2})}
  15. 𝐮 \mathbf{u}
  16. u 1 ( x 1 , x 2 , x 3 ) = - x 3 w x 1 , u 2 ( x 1 , x 2 , x 3 ) = - x 3 w x 2 , u 3 ( x 1 , x 2 , x 3 ) = w ( x 1 , x 2 ) u_{1}(x_{1},x_{2},x_{3})=-x_{3}\,\frac{\partial w}{\partial x_{1}}~{},~{}~{}u_% {2}(x_{1},x_{2},x_{3})=-x_{3}\,\frac{\partial w}{\partial x_{2}}~{},~{}~{}u_{3% }(x_{1},x_{2},x_{3})=w(x_{1},x_{2})
  17. E i j := 1 2 [ u i x j + u j x i + u k x i u k x j ] . E_{ij}:=\frac{1}{2}\left[\frac{\partial u_{i}}{\partial x_{j}}+\frac{\partial u% _{j}}{\partial x_{i}}+\frac{\partial u_{k}}{\partial x_{i}}\,\frac{\partial u_% {k}}{\partial x_{j}}\right]\,.
  18. E 11 = u 1 x 1 + 1 2 [ ( u 1 x 1 ) 2 + ( u 2 x 1 ) 2 + ( u 3 x 1 ) 2 ] = - x 3 2 w x 1 2 + 1 2 [ x 3 2 ( 2 w x 1 2 ) 2 + x 3 2 ( 2 w x 1 x 2 ) 2 + ( w x 1 ) 2 ] E 22 = u 2 x 2 + 1 2 [ ( u 1 x 2 ) 2 + ( u 2 x 2 ) 2 + ( u 3 x 2 ) 2 ] = - x 3 2 w x 2 2 + 1 2 [ x 3 2 ( 2 w x 1 x 2 ) 2 + x 3 2 ( 2 w x 2 2 ) 2 + ( w x 2 ) 2 ] E 33 = u 3 x 3 + 1 2 [ ( u 1 x 3 ) 2 + ( u 2 x 3 ) 2 + ( u 3 x 3 ) 2 ] = 1 2 [ ( w x 1 ) 2 + ( w x 2 ) 2 ] E 12 = 1 2 [ u 1 x 2 + u 2 x 1 + u 1 x 1 u 1 x 2 + u 2 x 1 u 2 x 2 + u 3 x 1 u 3 x 2 ] = - x 3 2 w x 1 x 2 + 1 2 [ x 3 2 ( 2 w x 1 2 ) ( 2 w x 1 x 2 ) + x 3 2 ( 2 w x 1 x 2 ) ( 2 w x 2 2 ) + w x 1 w x 2 ] E 23 = 1 2 [ u 2 x 3 + u 3 x 2 + u 1 x 2 u 1 x 3 + u 2 x 2 u 2 x 3 + u 3 x 2 u 3 x 3 ] = 1 2 [ x 3 ( 2 w x 1 x 2 ) ( w x 1 ) + x 3 ( 2 w x 2 2 ) ( w x 2 ) ] E 31 = 1 2 [ u 3 x 1 + u 1 x 3 + u 1 x 3 u 1 x 1 + u 2 x 3 u 2 x 1 + u 3 x 3 u 3 x 1 ] = 1 2 [ x 3 ( w x 1 ) ( 2 w x 1 2 ) + x 3 ( w x 2 ) ( 2 w x 1 x 2 ) ] \begin{aligned}\displaystyle E_{11}&\displaystyle=\frac{\partial u_{1}}{% \partial x_{1}}+\frac{1}{2}\left[\left(\frac{\partial u_{1}}{\partial x_{1}}% \right)^{2}+\left(\frac{\partial u_{2}}{\partial x_{1}}\right)^{2}+\left(\frac% {\partial u_{3}}{\partial x_{1}}\right)^{2}\right]\\ &\displaystyle=-x_{3}\,\frac{\partial^{2}w}{\partial x_{1}^{2}}+\frac{1}{2}% \left[x_{3}^{2}\left(\frac{\partial^{2}w}{\partial x_{1}^{2}}\right)^{2}+x_{3}% ^{2}\left(\frac{\partial^{2}w}{\partial x_{1}\partial x_{2}}\right)^{2}+\left(% \frac{\partial w}{\partial x_{1}}\right)^{2}\right]\\ \displaystyle E_{22}&\displaystyle=\frac{\partial u_{2}}{\partial x_{2}}+\frac% {1}{2}\left[\left(\frac{\partial u_{1}}{\partial x_{2}}\right)^{2}+\left(\frac% {\partial u_{2}}{\partial x_{2}}\right)^{2}+\left(\frac{\partial u_{3}}{% \partial x_{2}}\right)^{2}\right]\\ &\displaystyle=-x_{3}\,\frac{\partial^{2}w}{\partial x_{2}^{2}}+\frac{1}{2}% \left[x_{3}^{2}\left(\frac{\partial^{2}w}{\partial x_{1}\partial x_{2}}\right)% ^{2}+x_{3}^{2}\left(\frac{\partial^{2}w}{\partial x_{2}^{2}}\right)^{2}+\left(% \frac{\partial w}{\partial x_{2}}\right)^{2}\right]\\ \displaystyle E_{33}&\displaystyle=\frac{\partial u_{3}}{\partial x_{3}}+\frac% {1}{2}\left[\left(\frac{\partial u_{1}}{\partial x_{3}}\right)^{2}+\left(\frac% {\partial u_{2}}{\partial x_{3}}\right)^{2}+\left(\frac{\partial u_{3}}{% \partial x_{3}}\right)^{2}\right]\\ &\displaystyle=\frac{1}{2}\left[\left(\frac{\partial w}{\partial x_{1}}\right)% ^{2}+\left(\frac{\partial w}{\partial x_{2}}\right)^{2}\right]\\ \displaystyle E_{12}&\displaystyle=\frac{1}{2}\left[\frac{\partial u_{1}}{% \partial x_{2}}+\frac{\partial u_{2}}{\partial x_{1}}+\frac{\partial u_{1}}{% \partial x_{1}}\,\frac{\partial u_{1}}{\partial x_{2}}+\frac{\partial u_{2}}{% \partial x_{1}}\,\frac{\partial u_{2}}{\partial x_{2}}+\frac{\partial u_{3}}{% \partial x_{1}}\,\frac{\partial u_{3}}{\partial x_{2}}\right]\\ &\displaystyle=-x_{3}\frac{\partial^{2}w}{\partial x_{1}\partial x_{2}}+\frac{% 1}{2}\left[x_{3}^{2}\left(\frac{\partial^{2}w}{\partial x_{1}^{2}}\right)\left% (\frac{\partial^{2}w}{\partial x_{1}\partial x_{2}}\right)+x_{3}^{2}\left(% \frac{\partial^{2}w}{\partial x_{1}\partial x_{2}}\right)\left(\frac{\partial^% {2}w}{\partial x_{2}^{2}}\right)+\frac{\partial w}{\partial x_{1}}\,\frac{% \partial w}{\partial x_{2}}\right]\\ \displaystyle E_{23}&\displaystyle=\frac{1}{2}\left[\frac{\partial u_{2}}{% \partial x_{3}}+\frac{\partial u_{3}}{\partial x_{2}}+\frac{\partial u_{1}}{% \partial x_{2}}\,\frac{\partial u_{1}}{\partial x_{3}}+\frac{\partial u_{2}}{% \partial x_{2}}\,\frac{\partial u_{2}}{\partial x_{3}}+\frac{\partial u_{3}}{% \partial x_{2}}\,\frac{\partial u_{3}}{\partial x_{3}}\right]\\ &\displaystyle=\frac{1}{2}\left[x_{3}\left(\frac{\partial^{2}w}{\partial x_{1}% \partial x_{2}}\right)\left(\frac{\partial w}{\partial x_{1}}\right)+x_{3}% \left(\frac{\partial^{2}w}{\partial x_{2}^{2}}\right)\left(\frac{\partial w}{% \partial x_{2}}\right)\right]\\ \displaystyle E_{31}&\displaystyle=\frac{1}{2}\left[\frac{\partial u_{3}}{% \partial x_{1}}+\frac{\partial u_{1}}{\partial x_{3}}+\frac{\partial u_{1}}{% \partial x_{3}}\,\frac{\partial u_{1}}{\partial x_{1}}+\frac{\partial u_{2}}{% \partial x_{3}}\,\frac{\partial u_{2}}{\partial x_{1}}+\frac{\partial u_{3}}{% \partial x_{3}}\,\frac{\partial u_{3}}{\partial x_{1}}\right]\\ &\displaystyle=\frac{1}{2}\left[x_{3}\left(\frac{\partial w}{\partial x_{1}}% \right)\left(\frac{\partial^{2}w}{\partial x_{1}^{2}}\right)+x_{3}\left(\frac{% \partial w}{\partial x_{2}}\right)\left(\frac{\partial^{2}w}{\partial x_{1}% \partial x_{2}}\right)\right]\end{aligned}
  19. ( w x 1 ) 2 , ( w x 2 ) 2 , w x 1 w x 2 . \left(\frac{\partial w}{\partial x_{1}}\right)^{2}~{},~{}~{}\left(\frac{% \partial w}{\partial x_{2}}\right)^{2}~{},~{}~{}\frac{\partial w}{\partial x_{% 1}}\,\frac{\partial w}{\partial x_{2}}\,.
  20. E 11 = - x 3 2 w x 1 2 + 1 2 ( w x 1 ) 2 E 22 = - x 3 2 w x 2 2 + 1 2 ( w x 2 ) 2 E 12 = - x 3 2 w x 1 x 2 + 1 2 w x 1 w x 2 E 33 = 0 , E 23 = 0 , E 31 = 0 . \begin{aligned}\displaystyle E_{11}&\displaystyle=-x_{3}\,\frac{\partial^{2}w}% {\partial x_{1}^{2}}+\frac{1}{2}\left(\frac{\partial w}{\partial x_{1}}\right)% ^{2}\\ \displaystyle E_{22}&\displaystyle=-x_{3}\,\frac{\partial^{2}w}{\partial x_{2}% ^{2}}+\frac{1}{2}\left(\frac{\partial w}{\partial x_{2}}\right)^{2}\\ \displaystyle E_{12}&\displaystyle=-x_{3}\frac{\partial^{2}w}{\partial x_{1}% \partial x_{2}}+\frac{1}{2}\,\frac{\partial w}{\partial x_{1}}\,\frac{\partial w% }{\partial x_{2}}\\ \displaystyle E_{33}&\displaystyle=0~{},~{}~{}E_{23}=0~{},~{}~{}E_{31}=0\,.% \end{aligned}
  21. [ σ 11 σ 22 σ 12 ] = E ( 1 - ν 2 ) [ 1 ν 0 ν 1 0 0 0 1 - ν ] [ E 11 E 22 E 12 ] \begin{bmatrix}\sigma_{11}\\ \sigma_{22}\\ \sigma_{12}\end{bmatrix}=\cfrac{E}{(1-\nu^{2})}\begin{bmatrix}1&\nu&0\\ \nu&1&0\\ 0&0&1-\nu\end{bmatrix}\begin{bmatrix}E_{11}\\ E_{22}\\ E_{12}\end{bmatrix}
  22. σ 11 = E ( 1 - ν 2 ) [ ( - x 3 2 w x 1 2 + 1 2 ( w x 1 ) 2 ) + ν ( - x 3 2 w x 2 2 + 1 2 ( w x 2 ) 2 ) ] σ 22 = E ( 1 - ν 2 ) [ ν ( - x 3 2 w x 1 2 + 1 2 ( w x 1 ) 2 ) + ( - x 3 2 w x 2 2 + 1 2 ( w x 2 ) 2 ) ] σ 12 = E ( 1 + ν ) [ - x 3 2 w x 1 x 2 + 1 2 w x 1 w x 2 ] . \begin{aligned}\displaystyle\sigma_{11}&\displaystyle=\cfrac{E}{(1-\nu^{2})}% \left[\left(-x_{3}\,\frac{\partial^{2}w}{\partial x_{1}^{2}}+\frac{1}{2}\left(% \frac{\partial w}{\partial x_{1}}\right)^{2}\right)+\nu\left(-x_{3}\,\frac{% \partial^{2}w}{\partial x_{2}^{2}}+\frac{1}{2}\left(\frac{\partial w}{\partial x% _{2}}\right)^{2}\right)\right]\\ \displaystyle\sigma_{22}&\displaystyle=\cfrac{E}{(1-\nu^{2})}\left[\nu\left(-x% _{3}\,\frac{\partial^{2}w}{\partial x_{1}^{2}}+\frac{1}{2}\left(\frac{\partial w% }{\partial x_{1}}\right)^{2}\right)+\left(-x_{3}\,\frac{\partial^{2}w}{% \partial x_{2}^{2}}+\frac{1}{2}\left(\frac{\partial w}{\partial x_{2}}\right)^% {2}\right)\right]\\ \displaystyle\sigma_{12}&\displaystyle=\cfrac{E}{(1+\nu)}\left[-x_{3}\frac{% \partial^{2}w}{\partial x_{1}\partial x_{2}}+\frac{1}{2}\,\frac{\partial w}{% \partial x_{1}}\,\frac{\partial w}{\partial x_{2}}\right]\,.\end{aligned}
  23. N α β := - h / 2 h / 2 σ α β d x 3 , M α β := - h / 2 h / 2 x 3 σ α β d x 3 . N_{\alpha\beta}:=\int_{-h/2}^{h/2}\sigma_{\alpha\beta}\,dx_{3}~{},~{}~{}M_{% \alpha\beta}:=\int_{-h/2}^{h/2}x_{3}\,\sigma_{\alpha\beta}\,dx_{3}\,.
  24. N 11 = E h 2 ( 1 - ν 2 ) [ ( w x 1 ) 2 + ν ( w x 2 ) 2 ] N 22 = E h 2 ( 1 - ν 2 ) [ ν ( w x 1 ) 2 + ( w x 2 ) 2 ] N 12 = E h 2 ( 1 + ν ) w x 1 w x 2 \begin{aligned}\displaystyle N_{11}&\displaystyle=\cfrac{Eh}{2(1-\nu^{2})}% \left[\left(\frac{\partial w}{\partial x_{1}}\right)^{2}+\nu\left(\frac{% \partial w}{\partial x_{2}}\right)^{2}\right]\\ \displaystyle N_{22}&\displaystyle=\cfrac{Eh}{2(1-\nu^{2})}\left[\nu\left(% \frac{\partial w}{\partial x_{1}}\right)^{2}+\left(\frac{\partial w}{\partial x% _{2}}\right)^{2}\right]\\ \displaystyle N_{12}&\displaystyle=\cfrac{Eh}{2(1+\nu)}\,\frac{\partial w}{% \partial x_{1}}\,\frac{\partial w}{\partial x_{2}}\end{aligned}
  25. M 11 = - E h 3 12 ( 1 - ν 2 ) [ 2 w x 1 2 + ν 2 w x 2 2 ] M 22 = - E h 3 12 ( 1 - ν 2 ) [ ν 2 w x 1 2 + 2 w x 2 2 ] M 12 = - E h 3 12 ( 1 + ν ) 2 w x 1 x 2 . \begin{aligned}\displaystyle M_{11}&\displaystyle=-\cfrac{Eh^{3}}{12(1-\nu^{2}% )}\left[\frac{\partial^{2}w}{\partial x_{1}^{2}}+\nu\,\frac{\partial^{2}w}{% \partial x_{2}^{2}}\right]\\ \displaystyle M_{22}&\displaystyle=-\cfrac{Eh^{3}}{12(1-\nu^{2})}\left[\nu\,% \frac{\partial^{2}w}{\partial x_{1}^{2}}+\frac{\partial^{2}w}{\partial x_{2}^{% 2}}\right]\\ \displaystyle M_{12}&\displaystyle=-\cfrac{Eh^{3}}{12(1+\nu)}\,\frac{\partial^% {2}w}{\partial x_{1}\partial x_{2}}\,.\end{aligned}
  26. 2 M 11 x 1 2 + 2 M 22 x 2 2 + 2 2 M 12 x 1 x 2 + x 1 ( N 11 w x 1 + N 12 w x 2 ) + x 2 ( N 12 w x 1 + N 22 w x 2 ) = P N α β x β = 0 . \begin{aligned}&\displaystyle\frac{\partial^{2}M_{11}}{\partial x_{1}^{2}}+% \frac{\partial^{2}M_{22}}{\partial x_{2}^{2}}+2\frac{\partial^{2}M_{12}}{% \partial x_{1}\partial x_{2}}+\frac{\partial}{\partial x_{1}}\left(N_{11}\,% \frac{\partial w}{\partial x_{1}}+N_{12}\,\frac{\partial w}{\partial x_{2}}% \right)+\frac{\partial}{\partial x_{2}}\left(N_{12}\,\frac{\partial w}{% \partial x_{1}}+N_{22}\,\frac{\partial w}{\partial x_{2}}\right)=P\\ &\displaystyle\frac{\partial N_{\alpha\beta}}{\partial x_{\beta}}=0\,.\end{aligned}
  27. Δ Δ
  28. Δ w := 2 w x α x α = 2 w x 1 2 + 2 w x 2 2 \Delta w:=\frac{\partial^{2}w}{\partial x_{\alpha}\partial x_{\alpha}}=\frac{% \partial^{2}w}{\partial x_{1}^{2}}+\frac{\partial^{2}w}{\partial x_{2}^{2}}

Förster_coupling.html

  1. H c = 1 2 i , j , k , l V ijkl a i a j a l a k H_{c}=\frac{1}{2}\sum_{i,j,k,l}V_{\,\text{ijkl}}a_{i}{}^{\dagger}a_{j}{}^{% \dagger}a_{l}a_{k}
  2. V ijkl = e 2 4 π ϵ 0 ϵ r d 3 x d 3 x ϕ i ( x ) * ϕ j ( x ) * 1 | x - x | ϕ k ( x ) ϕ l ( x ) V_{\,\text{ijkl}}=\frac{e^{2}}{4\pi\epsilon_{0}\epsilon_{r}}\int d^{3}x\int d^% {3}x^{\prime}\phi_{i}{}^{*}\left(\overset{\rightharpoonup}{x}\right)\phi_{j}{}% ^{*}\left(\overset{\rightharpoonup}{x}^{\prime}\right)\frac{1}{\left|\overset{% \rightharpoonup}{x}-\overset{\rightharpoonup}{x}^{\prime}\right|}\phi_{k}\left% (\overset{\rightharpoonup}{x}\right)\phi_{l}\left(\overset{\rightharpoonup}{x}% ^{\prime}\right)
  3. ϵ r \epsilon_{r}
  4. | c |c\rangle
  5. | v |v\rangle
  6. H cc = i > j V cc ij a c i a c j a c j a c i H_{\,\text{cc}}=\sum_{i>j}V_{\,\text{cc}}^{\,\text{ij}}a_{c_{i}}{}^{\dagger}a_% {c_{j}}{}^{\dagger}a_{c_{j}}a_{c_{i}}
  7. H cv = i j V cv ij a c i a v j a v j a c i H_{\,\text{cv}}=\sum_{i\neq j}V_{\,\text{cv}}^{\,\text{ij}}a_{c_{i}}{}^{% \dagger}a_{v_{j}}{}^{\dagger}a_{v_{j}}a_{c_{i}}
  8. H F = i j V F ij a c i a v j a c j a v i H_{F}=\sum_{i\neq j}V_{F}^{\,\text{ij}}a_{c_{i}}{}^{\dagger}a_{v_{j}}{}^{% \dagger}a_{c_{j}}a_{v_{i}}
  9. V bs = V cv - V cc V_{\,\text{bs}}=V_{\,\text{cv}}-V_{\,\text{cc}}
  10. V F V_{F}
  11. { | 00 , | 01 , | 10 , | 11 } a s ( = 1 ) \{|00\rangle,|01\rangle,|10\rangle,|11\rangle\}\ as\ (\hbar=1)
  12. H ^ = ( ω 0 0 0 0 0 ω 0 + ω 2 V F 0 0 V F ω 0 + ω 1 0 0 0 0 ω 0 + ω 1 + ω 2 + V XX ) \hat{H}=\left(\begin{array}[]{cccc}\omega_{0}&0&0&0\\ 0&\omega_{0}+\omega_{2}&V_{F}&0\\ 0&V_{F}&\omega_{0}+\omega_{1}&0\\ 0&0&0&\omega_{0}+\omega_{1}+\omega_{2}+V_{\,\text{XX}}\end{array}\right)
  13. V F V_{F}
  14. V XX V_{\,\text{XX}}
  15. ω 0 \omega_{0}
  16. Δ ω ω 1 - ω 2 \Delta\omega\equiv\omega_{1}-\omega_{2}
  17. | 10 |10\rangle
  18. | 01 |01\rangle

Fractional_anisotropy.html

  1. λ 1 , λ 2 , λ 3 \lambda_{1},\lambda_{2},\lambda_{3}
  2. ϵ \epsilon
  3. λ \lambda
  4. FA = 3 2 ( λ 1 - λ ^ ) 2 + ( λ 2 - λ ^ ) 2 + ( λ 3 - λ ^ ) 2 λ 1 2 + λ 2 2 + λ 3 2 \,\text{FA}=\sqrt{\frac{3}{2}}\frac{\sqrt{(\lambda_{1}-\hat{\lambda})^{2}+(% \lambda_{2}-\hat{\lambda})^{2}+(\lambda_{3}-\hat{\lambda})^{2}}}{\sqrt{\lambda% _{1}^{2}+\lambda_{2}^{2}+\lambda_{3}^{2}}}
  5. λ ^ = ( λ 1 + λ 2 + λ 3 ) / 3 \hat{\lambda}=(\lambda_{1}+\lambda_{2}+\lambda_{3})/3
  6. FA = 1 2 ( λ 1 - λ 2 ) 2 + ( λ 2 - λ 3 ) 2 + ( λ 3 - λ 1 ) 2 λ 1 2 + λ 2 2 + λ 3 2 \,\text{FA}=\sqrt{\frac{1}{2}}\frac{\sqrt{(\lambda_{1}-\lambda_{2})^{2}+(% \lambda_{2}-\lambda_{3})^{2}+(\lambda_{3}-\lambda_{1})^{2}}}{\sqrt{\lambda_{1}% ^{2}+\lambda_{2}^{2}+\lambda_{3}^{2}}}
  7. FA = 1 2 ( 3 - 1 t r a c e ( R 2 ) ) \,\text{FA}=\sqrt{\frac{1}{2}(3-\frac{1}{trace(\,\text{R}^{2})})}
  8. R = D t r a c e ( D ) \,\text{R}=\frac{\,\text{D}}{trace(\,\text{D})}

Fractional_Schrödinger_equation.html

  1. ( - 2 Δ ) α / 2 ψ ( 𝐫 , t ) = 1 ( 2 π ) 3 d 3 p e i 𝐩𝐫 | 𝐩 | α φ ( 𝐩 , t ) , (-\hbar^{2}\Delta)^{\alpha/2}\psi(\mathbf{r},t)=\frac{1}{(2\pi\hbar)^{3}}\int d% ^{3}pe^{i\frac{\mathbf{pr}}{\hbar}}|\mathbf{p}|^{\alpha}\varphi(\mathbf{p},t),
  2. ψ ( 𝐫 , t ) \psi(\mathbf{r},t)
  3. φ ( 𝐩 , t ) \varphi(\mathbf{p},t)
  4. ψ ( 𝐫 , t ) = 1 ( 2 π ) 3 d 3 p e i 𝐩 𝐫 / φ ( 𝐩 , t ) , φ ( 𝐩 , t ) = d 3 r e - i 𝐩 𝐫 / ψ ( 𝐫 , t ) . \psi(\mathbf{r},t)=\frac{1}{(2\pi\hbar)^{3}}\int d^{3}pe^{i\mathbf{p}\cdot% \mathbf{r}/\hbar}\varphi(\mathbf{p},t),\qquad\varphi(\mathbf{p},t)=\int d^{3}% re^{-i\mathbf{p}\cdot\mathbf{r}/\hbar}\psi(\mathbf{r},t).
  5. H ^ α \widehat{H}_{\alpha}
  6. H ^ α = D α ( - 2 Δ ) α / 2 + V ( 𝐫 , t ) . \widehat{H}_{\alpha}=D_{\alpha}(-\hbar^{2}\Delta)^{\alpha/2}+V(\mathbf{r},t).
  7. H ^ α \widehat{H}_{\alpha}
  8. H α ( 𝐩 , 𝐫 ) = D α | 𝐩 | α + V ( 𝐫 , t ) , H_{\alpha}(\mathbf{p},\mathbf{r})=D_{\alpha}|\mathbf{p}|^{\alpha}+V(\mathbf{r}% ,t),
  9. H α H_{\alpha}
  10. H α = D α ( - 2 Δ ) α / 2 + V ( 𝐫 ) , H_{\alpha}=D_{\alpha}(-\hbar^{2}\Delta)^{\alpha/2}+V(\mathbf{r}),
  11. ψ ( 𝐫 , t ) = e - ( i / ) E t ϕ ( 𝐫 ) , \psi(\mathbf{r},t)=e^{-(i/\hbar)Et}\phi(\mathbf{r}),
  12. ϕ ( 𝐫 ) \phi(\mathbf{r})
  13. H α ϕ ( 𝐫 ) = E ϕ ( 𝐫 ) , H_{\alpha}\phi(\mathbf{r})=E\phi(\mathbf{r}),
  14. D α ( - 2 Δ ) α / 2 ϕ ( 𝐫 ) + V ( 𝐫 ) ϕ ( 𝐫 ) = E ϕ ( 𝐫 ) . D_{\alpha}(-\hbar^{2}\Delta)^{\alpha/2}\phi(\mathbf{r})+V(\mathbf{r})\phi(% \mathbf{r})=E\phi(\mathbf{r}).
  15. ψ ( 𝐫 , t ) \psi(\mathbf{r},t)
  16. 𝐫 \mathbf{r}
  17. | ψ ( 𝐫 , t ) | 2 . |\psi(\mathbf{r},t)|^{2}.
  18. | ϕ ( 𝐫 ) | 2 |\phi(\mathbf{r})|^{2}
  19. 𝐫 \mathbf{r}
  20. ρ ( 𝐫 , t ) t + 𝐣 ( 𝐫 , t ) + K ( 𝐫 , t ) = 0 , \frac{\partial\rho(\mathbf{r},t)}{\partial t}+\nabla\cdot\mathbf{j}(\mathbf{r}% ,t)+K(\mathbf{r},t)=0,
  21. ρ ( 𝐫 , t ) = ψ ( 𝐫 , t ) ψ ( 𝐫 , t ) \rho(\mathbf{r},t)=\psi^{\ast}(\mathbf{r},t)\psi(\mathbf{r},t)
  22. 𝐣 ( 𝐫 , t ) \mathbf{j}(\mathbf{r},t)
  23. 𝐣 ( 𝐫 , t ) = D α i ( ψ * ( 𝐫 , t ) ( - 2 Δ ) α / 2 - 1 ψ ( 𝐫 , t ) - ψ ( 𝐫 , t ) ( - 2 Δ ) α / 2 - 1 ψ * ( 𝐫 , t ) ) , \mathbf{j}(\mathbf{r},t)=\frac{D_{\alpha}\hbar}{i}\left(\psi^{*}(\mathbf{r},t)% (-\hbar^{2}\Delta)^{\alpha/2-1}\mathbf{\nabla}\psi(\mathbf{r},t)-\psi(\mathbf{% r},t)(-\hbar^{2}\Delta)^{\alpha/2-1}\mathbf{\nabla}\psi^{*}(\mathbf{r},t)% \right),
  24. K ( 𝐫 , t ) = D α i ( ψ ( 𝐫 , t ) ( - 2 Δ ) α / 2 - 1 ψ * ( 𝐫 , t ) - ( ψ * ( 𝐫 , t ) ( - 2 Δ ) α / 2 - 1 ψ ( 𝐫 , t ) ) , \mathit{K}(\mathbf{r},t)=\frac{D_{\alpha}\hbar}{i}\left(\mathbf{\nabla}\psi(% \mathbf{r},t)(-\hbar^{2}\Delta)^{\alpha/2-1}\mathbf{\nabla}\psi^{*}(\mathbf{r}% ,t)-(\mathbf{\nabla}\psi^{*}(\mathbf{r},t)(-\hbar^{2}\Delta)^{\alpha/2-1}% \mathbf{\nabla}\psi(\mathbf{r},t)\right),
  25. = / 𝐫 \mathbf{\nabla=\partial/\partial r}
  26. K ( 𝐫 , t ) \mathit{K}(\mathbf{r},t)
  27. ρ ( 𝐫 , t ) t + 𝐣 ( 𝐫 , t ) = 0. \frac{\partial\rho(\mathbf{r},t)}{\partial t}+\nabla\cdot\mathbf{j}(\mathbf{r}% ,t)=0.
  28. 𝐩 ^ = i 𝐫 \widehat{\mathbf{p}}=\frac{\hbar}{i}\frac{\partial}{\partial\mathbf{r}}
  29. 𝐣 \mathbf{j}
  30. 𝐣 = D α ( ψ ( 𝐩 ^ 2 ) α / 2 - 1 𝐩 ^ ψ + ψ ( 𝐩 ^ 2 ) α / 2 - 1 𝐩 ^ ψ ) . \mathbf{j=}D_{\alpha}\left(\psi(\widehat{\mathbf{p}}^{2})^{\alpha/2-1}\widehat% {\mathbf{p}}\psi^{\ast}+\psi^{\ast}(\widehat{\mathbf{p}}^{\ast 2})^{\alpha/2-1% }\widehat{\mathbf{p}}^{\ast}\psi\right).
  31. 𝐯 ^ \widehat{\mathbf{v}}
  32. 𝐯 ^ = i ( H α 𝐫 ^ - 𝐫 ^ H α ) , \widehat{\mathbf{v}}=\frac{i}{\hbar}(H_{\alpha}\widehat{\mathbf{r}}\mathbf{-}% \widehat{\mathbf{r}}H_{\alpha}),
  33. 𝐯 ^ = α D α | 𝐩 ^ 2 | α / 2 - 1 𝐩 ^ . \widehat{\mathbf{v}}=\alpha D_{\alpha}|\widehat{\mathbf{p}}^{2}|^{\alpha/2-1}% \widehat{\mathbf{p}}\,.
  34. 𝐣 = 1 α ( ψ 𝐯 ^ ψ + ψ 𝐯 ^ ψ ) , 1 < α 2. \mathbf{j=}\frac{1}{\alpha}\left(\psi\widehat{\mathbf{v}}\psi^{\ast}+\psi^{% \ast}\widehat{\mathbf{v}}\psi\right),\qquad 1<\alpha\leq 2.
  35. ψ ( 𝐫 , t ) = α 2 v exp [ i ( 𝐩 𝐫 - E t ) ] , E = D α | 𝐩 | α , 1 < α 2 , \psi(\mathbf{r},t)=\sqrt{\frac{\alpha}{2\mathrm{v}}}\exp\left[\frac{i}{\hbar}(% \mathbf{p}\cdot\mathbf{r}-Et)\right],\qquad E=D_{\alpha}|\mathbf{p}|^{\alpha},% \qquad 1<\alpha\leq 2,
  36. v \mathrm{v}
  37. v = α D α p α - 1 \mathrm{v}=\alpha D_{\alpha}p^{\alpha-1}
  38. 𝐣 = 𝐯 v , 𝐯 = α D α | 𝐩 2 | α 2 - 1 𝐩 , \mathbf{j=}\frac{\mathbf{v}}{\mathrm{v}},\qquad\mathbf{v}=\alpha D_{\alpha}|% \mathbf{p}^{2}|^{\frac{\alpha}{2}-1}\mathbf{p,}
  39. 𝐣 \mathbf{j}
  40. V ( 𝐫 ) V(\mathbf{r})
  41. V ( 𝐫 ) = - Z e 2 | 𝐫 | , V(\mathbf{r})=-\frac{Ze^{2}}{|\mathbf{r}|},
  42. D α ( - 2 Δ ) α / 2 ϕ ( 𝐫 ) - Z e 2 | 𝐫 | ϕ ( 𝐫 ) = E ϕ ( 𝐫 ) . D_{\alpha}(-\hbar^{2}\Delta)^{\alpha/2}\phi(\mathbf{r})-\frac{Ze^{2}}{|\mathbf% {r|}}\phi(\mathbf{r})=E\phi(\mathbf{r}).
  43. α D α ( n a n ) α = Z e 2 a n , \alpha D_{\alpha}\left(\frac{n\hbar}{a_{n}}\right)^{\alpha}=\frac{Ze^{2}}{a_{n% }},
  44. a n = a 0 n α / ( α - 1 ) . a_{n}=a_{0}n^{\alpha/(\alpha-1)}.
  45. a 0 = ( α D α α Z e 2 ) 1 / ( α - 1 ) . a_{0}=\left(\frac{\alpha D_{\alpha}\hbar^{\alpha}}{Ze^{2}}\right)^{1/(\alpha-1% )}.
  46. E n = ( 1 - α ) E 0 n - α / ( α - 1 ) , 1 < α 2 , E_{n}=(1-\alpha)E_{0}n^{-\alpha/(\alpha-1)},\qquad 1<\alpha\leq 2,
  47. E 0 = ( Z e 2 α D α 1 / α ) α / ( α - 1 ) . E_{0}=\left(\frac{Ze^{2}}{\alpha D_{\alpha}^{1/\alpha}\hbar}\right)^{\alpha/(% \alpha-1)}.
  48. ( α - 1 ) E 0 / c (\alpha-1)E_{0}/\hbar c
  49. Ry = m e 4 / 2 3 c \mathrm{Ry}=me^{4}/2\hbar^{3}c
  50. ω \omega
  51. ω = ( 1 - α ) E 0 [ 1 n α α - 1 - 1 m α α - 1 ] \omega=\frac{(1-\alpha)E_{0}}{\hbar}\left[\frac{1}{n^{\frac{\alpha}{\alpha-1}}% }-\frac{1}{m^{\frac{\alpha}{\alpha-1}}}\right]
  52. V ( x ) V({x})
  53. - a x a -a\leq x\leq a
  54. V ( x ) = , x < - a ( i ) V(x)=\infty,\qquad x<-a\qquad\qquad(\mathrm{i})
  55. V ( x ) = 0 , - a x a ( ii ) V(x)=0,\quad-a\leq x\leq a\quad\quad\quad\ (\mathrm{ii})
  56. V ( x ) = , x > a ( iii ) V(x)=\infty,\qquad\ x>a\qquad\qquad\ (\mathrm{iii})
  57. ψ ( x ) \psi(x)
  58. ψ ( x , t ) = ( - i E t ) ϕ ( x ) \psi(x,t)=\left(-i\frac{Et}{\hbar}\right)\phi(x)
  59. ϕ ( x ) \phi(x)
  60. ϕ ( x ) = 0 \phi(x)=0
  61. D α ( ) α ϕ ( x ) = E ϕ ( x ) . D_{\alpha}(\hbar\nabla)^{\alpha}\phi(x)=E\phi(x).
  62. ϕ ( x ) \phi(x)
  63. ϕ ( - a ) = ϕ ( a ) = 0 \phi(-a)=\phi(a)=0
  64. ϕ ( x ) = A exp ( i k x ) + B exp ( - i k x ) . \phi(x)=A\exp(ikx)+B\exp(-ikx).
  65. A = - B exp ( - i 2 k a ) , A=-B\exp(-i2ka),
  66. sin ( 2 k a ) = 0. \sin(2ka)=0.
  67. 2 k a = n π . 2ka=n\pi.
  68. ϕ n even ( - x ) = ϕ n even ( x ) \phi_{n}^{\mathrm{even}}(-x)=\phi_{n}^{\mathrm{even}}(x)
  69. x - x x\rightarrow-x
  70. ϕ even ( x ) \phi^{\mathrm{even}}(x)
  71. ϕ n even ( x ) = 1 a cos [ n π x 2 a ] , n = 1 , 3 , 5 , . \phi_{n}^{\mathrm{even}}(x)=\frac{1}{\sqrt{a}}\cos\left[\frac{n\pi x}{2a}% \right],\quad n=1,3,5,....
  72. ϕ n odd ( - x ) = - ϕ n odd ( x ) \phi_{n}^{\mathrm{odd}}(-x)=-\phi_{n}^{\mathrm{odd}}(x)
  73. x - x x\rightarrow-x
  74. ϕ even ( x ) \phi^{\mathrm{even}}(x)
  75. ϕ n odd ( x ) = 1 a sin [ n π x 2 a ] , n = 2 , 4 , 6 , . \phi_{n}^{\mathrm{odd}}(x)=\frac{1}{\sqrt{a}}\sin\left[\frac{n\pi x}{2a}\right% ],\quad n=2,4,6,....
  76. ϕ even ( x ) \phi^{\mathrm{even}}(x)
  77. ϕ odd ( x ) \phi^{\mathrm{odd}}(x)
  78. - a a d x ϕ m even ( x ) ϕ n even ( x ) = - a a d x ϕ m odd ( x ) ϕ n odd ( x ) = δ m n , \int\limits_{-a}^{a}dx\phi_{m}^{\mathrm{even}}(x)\phi_{n}^{\mathrm{even}}(x)=% \int\limits_{-a}^{a}dx\phi_{m}^{\mathrm{odd}}(x)\phi_{n}^{\mathrm{odd}}(x)=% \delta_{mn},
  79. δ m n \delta_{mn}
  80. - a a d x ϕ m even ( x ) ϕ n odd ( x ) = 0. \int\limits_{-a}^{a}dx\phi_{m}^{\mathrm{even}}(x)\phi_{n}^{\mathrm{odd}}(x)=0.
  81. E n = D α ( π 2 a ) α n α , n = 1 , 2 , 3.... , 1 < α 2. E_{n}=D_{\alpha}\left(\frac{\pi\hbar}{2a}\right)^{\alpha}n^{\alpha},\qquad% \qquad n=1,2,3....,\qquad 1<\alpha\leq 2.
  82. ϕ n even ( x ) \phi_{n}^{\mathrm{even}}(x)
  83. ϕ ground ( x ) ϕ 1 even ( x ) = 1 a cos ( π x 2 a ) , \phi_{\mathrm{ground}}(x)\equiv\phi_{1}^{\mathrm{even}}(x)=\frac{1}{\sqrt{a}}% \cos\left(\frac{\pi x}{2a}\right),
  84. E ground = D α ( π 2 a ) α . E_{\mathrm{ground}}=D_{\alpha}\left(\frac{\pi\hbar}{2a}\right)^{\alpha}.
  85. H α , β H_{\alpha,\beta}
  86. H α , β = D α ( - 2 Δ ) α / 2 + q 2 | 𝐫 | β , 1 < α 2 , 1 < β 2 , H_{\alpha,\beta}=D_{\alpha}(-\hbar^{2}\Delta)^{\alpha/2}+q^{2}|\mathbf{r}|^{% \beta},\quad 1<\alpha\leq 2,\quad 1<\beta\leq 2,
  87. ψ ( 𝐫 , t ) \psi(\mathbf{r},t)
  88. i ψ ( 𝐫 , t ) t = D α ( - 2 Δ ) α / 2 ψ ( 𝐫 , t ) + q 2 | 𝐫 | β ψ ( 𝐫 , t ) i\hbar\frac{\partial\psi(\mathbf{r},t)}{\partial t}=D_{\alpha}(-\hbar^{2}% \Delta)^{\alpha/2}\psi(\mathbf{r},t)+q^{2}|\mathbf{r}|^{\beta}\psi(\mathbf{r},t)
  89. ψ ( 𝐫 , t ) = e - i E t / ϕ ( 𝐫 ) , \psi(\mathbf{r},t)=e^{-iEt/\hbar}\phi(\mathbf{r}),
  90. D α ( - 2 Δ ) α / 2 ϕ ( 𝐫 , t ) + q 2 | 𝐫 | β ϕ ( 𝐫 , t ) = E ϕ ( 𝐫 , t ) . D_{\alpha}(-\hbar^{2}\Delta)^{\alpha/2}\phi(\mathbf{r},t)+q^{2}|\mathbf{r}|^{% \beta}\phi(\mathbf{r},t)=E\phi(\mathbf{r},t).
  91. H α , β H_{\alpha,\beta}
  92. H α = D α | p | α + q 2 | x | β H_{\alpha}=D_{\alpha}|p|^{\alpha}+q^{2}|x|^{\beta}
  93. E = D α | p | α + q 2 | x | β , E=D_{\alpha}|p|^{\alpha}+q^{2}|x|^{\beta},
  94. | p | = ( 1 D α ( E - q 2 | x | β ) ) 1 / α |p|=\left(\frac{1}{D_{\alpha}}(E-q^{2}|x|^{\beta})\right)^{1/\alpha}
  95. p = 0 p=0
  96. | x | ( E / q 2 ) 1 / β |x|\leq(E/q^{2})^{1/\beta}
  97. 2 π ( n + 1 2 ) = p d x = 4 0 x m p d x = 4 0 x m D α - 1 / α ( E - q 2 | x | β ) 1 / α d x , 2\pi\hbar(n+\frac{1}{2})=\oint pdx=4\int\limits_{0}^{x_{m}}pdx=4\int\limits_{0% }^{x_{m}}D_{\alpha}^{-1/\alpha}(E-q^{2}|x|^{\beta})^{1/\alpha}dx,
  98. \oint
  99. x m = ( E / q 2 ) 1 / β x_{m}=(E/q^{2})^{1/\beta}
  100. y = x ( E / q 2 ) - 1 / β y=x(E/q^{2})^{-1/\beta}
  101. 0 x m D α - 1 / α ( E - q 2 | x | β ) 1 / α d x = 1 D α 1 / α q 2 / β E 1 α + 1 β 0 1 d y ( 1 - y β ) 1 / α . \int\limits_{0}^{x_{m}}D_{\alpha}^{-1/\alpha}(E-q^{2}|x|^{\beta})^{1/\alpha}dx% =\frac{1}{D_{\alpha}^{1/\alpha}q^{2/\beta}}E^{\frac{1}{\alpha}+\frac{1}{\beta}% }\int\limits_{0}^{1}dy(1-y^{\beta})^{1/\alpha}.
  102. 0 1 d y ( 1 - y β ) 1 / α = 1 β 0 1 d z z 1 β - 1 ( 1 - z ) 1 α = 1 β B ( 1 β , 1 α + 1 ) . \int\limits_{0}^{1}dy(1-y^{\beta})^{1/\alpha}=\frac{1}{\beta}\int\limits_{0}^{% 1}dzz^{\frac{1}{\beta}-1}(1-z)^{\frac{1}{\alpha}}=\frac{1}{\beta}B\left(\frac{% 1}{\beta},\frac{1}{\alpha}+1\right).
  103. 2 π ( n + 1 2 ) = 4 D α 1 / α q 2 / β E 1 α + 1 β 1 β B ( 1 β , 1 α + 1 ) . 2\pi\hbar(n+\frac{1}{2})=\frac{4}{D_{\alpha}^{1/\alpha}q^{2/\beta}}E^{\frac{1}% {\alpha}+\frac{1}{\beta}}\frac{1}{\beta}B\left(\frac{1}{\beta},\frac{1}{\alpha% }+1\right).
  104. E n = ( π β D α 1 / α q 2 / β 2 B ( 1 β , 1 α + 1 ) ) α β α + β ( n + 1 2 ) α β α + β . E_{n}=\left(\frac{\pi\hbar\beta D_{\alpha}^{1/\alpha}q^{2/\beta}}{2B(\frac{1}{% \beta},\frac{1}{\alpha}+1)}\right)^{\frac{\alpha\beta}{\alpha+\beta}}\left(n+% \frac{1}{2}\right)^{\frac{\alpha\beta}{\alpha+\beta}}.
  105. 1 α + 1 β = 1 \frac{1}{\alpha}+\frac{1}{\beta}=1
  106. 1 < α 2 1<\alpha\leq 2
  107. 1 < β 2 1<\beta\leq 2

Frame-dragging.html

  1. c 2 d τ 2 = ( 1 - r s r ρ 2 ) c 2 d t 2 - ρ 2 Λ 2 d r 2 - ρ 2 d θ 2 c^{2}d\tau^{2}=\left(1-\frac{r_{s}r}{\rho^{2}}\right)c^{2}dt^{2}-\frac{\rho^{2% }}{\Lambda^{2}}dr^{2}-\rho^{2}d\theta^{2}
  2. - ( r 2 + α 2 + r s r α 2 ρ 2 sin 2 θ ) sin 2 θ d ϕ 2 + 2 r s r α c sin 2 θ ρ 2 d ϕ d t -\left(r^{2}+\alpha^{2}+\frac{r_{s}r\alpha^{2}}{\rho^{2}}\sin^{2}\theta\right)% \sin^{2}\theta\ d\phi^{2}+\frac{2r_{s}r\alpha c\sin^{2}\theta}{\rho^{2}}d\phi dt
  3. r s = 2 G M c 2 r_{s}=\frac{2GM}{c^{2}}
  4. α = J M c \alpha=\frac{J}{Mc}
  5. ρ 2 = r 2 + α 2 cos 2 θ \rho^{2}=r^{2}+\alpha^{2}\cos^{2}\theta\,\!
  6. Λ 2 = r 2 - r s r + α 2 \Lambda^{2}=r^{2}-r_{s}r+\alpha^{2}\,\!
  7. c 2 d τ 2 = c 2 d t 2 - ρ 2 r 2 + α 2 d r 2 - ρ 2 d θ 2 - ( r 2 + α 2 ) sin 2 θ d ϕ 2 c^{2}d\tau^{2}=c^{2}dt^{2}-\frac{\rho^{2}}{r^{2}+\alpha^{2}}dr^{2}-\rho^{2}d% \theta^{2}-\left(r^{2}+\alpha^{2}\right)\sin^{2}\theta d\phi^{2}
  8. c 2 d τ 2 = ( g t t - g t ϕ 2 g ϕ ϕ ) d t 2 + g r r d r 2 + g θ θ d θ 2 + g ϕ ϕ ( d ϕ + g t ϕ g ϕ ϕ d t ) 2 c^{2}d\tau^{2}=\left(g_{tt}-\frac{g_{t\phi}^{2}}{g_{\phi\phi}}\right)dt^{2}+g_% {rr}dr^{2}+g_{\theta\theta}d\theta^{2}+g_{\phi\phi}\left(d\phi+\frac{g_{t\phi}% }{g_{\phi\phi}}dt\right)^{2}
  9. Ω = - g t ϕ g ϕ ϕ = r s α r c ρ 2 ( r 2 + α 2 ) + r s α 2 r sin 2 θ \Omega=-\frac{g_{t\phi}}{g_{\phi\phi}}=\frac{r_{s}\alpha rc}{\rho^{2}\left(r^{% 2}+\alpha^{2}\right)+r_{s}\alpha^{2}r\sin^{2}\theta}
  10. Ω = r s α c r 3 + α 2 r + r s α 2 \Omega=\frac{r_{s}\alpha c}{r^{3}+\alpha^{2}r+r_{s}\alpha^{2}}
  11. r i n n e r = r s + r s 2 - 4 α 2 2 r_{inner}=\frac{r_{s}+\sqrt{r_{s}^{2}-4\alpha^{2}}}{2}
  12. r o u t e r = r s + r s 2 - 4 α 2 cos 2 θ 2 r_{outer}=\frac{r_{s}+\sqrt{r_{s}^{2}-4\alpha^{2}\cos^{2}\theta}}{2}
  13. a ¯ = - 2 d 1 ( ω ¯ × v ¯ ) - d 2 [ ω ¯ × ( ω ¯ × r ¯ ) + 2 ( ω ¯ r ¯ ) ω ¯ ] \bar{a}=-2d_{1}\left(\bar{\omega}\times\bar{v}\right)-d_{2}\left[\bar{\omega}% \times\left(\bar{\omega}\times\bar{r}\right)+2\left(\bar{\omega}\bar{r}\right)% \bar{\omega}\right]
  14. d 1 = 4 M G 3 R c 2 d_{1}=\frac{4MG}{3Rc^{2}}
  15. d 2 = 4 M G 15 R c 2 d_{2}=\frac{4MG}{15Rc^{2}}
  16. d 1 = 4 α ( 2 - α ) ( 1 + α ) ( 3 - α ) , α = M G 2 R c 2 d_{1}=\frac{4\alpha(2-\alpha)}{(1+\alpha)(3-\alpha)},\qquad\alpha=\frac{MG}{2% Rc^{2}}

Franklin_graph.html

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

Fraunhofer_distance.html

  1. d = 2 D 2 λ , d={{2D^{2}}\over{\lambda}},
  2. λ {\lambda}

Free_carrier_absorption.html

  1. ρ λ λ 0 = 1 e ( ε λ , k - μ ) β + 1 = f λ , k \rho_{\lambda\lambda}^{0}=\frac{1}{{e^{(\varepsilon_{\lambda,k}-\mu)\beta}+1}}% =f_{\lambda,k}
  2. ε \varepsilon
  3. N λ = λ f λ , k N_{\lambda}=\sum\limits_{\lambda}{f_{\lambda,k}}
  4. ρ c v int ( k , t ) = < m t p l > d ω 2 π d c v ε ( ω ) e i ( ε c , k - ε v , k - ω ) t ( ε c , k - ε v , k - ω - i γ ) ( f v , k - f c , k ) \rho_{cv}^{{\mathop{\rm int}}}(k,t)=\int{\frac{<}{m}tpl>{{d\omega}}{{2\pi}}% \frac{{d_{cv}\varepsilon(\omega)e^{i(\varepsilon_{c,k}-\varepsilon_{v,k}-% \omega)t}}}{{\hbar(\varepsilon_{c,k}-\varepsilon_{v,k}-\omega-i\gamma)}}(f_{v,% k}-f_{c,k})}
  5. P ( t ) = t r [ ρ ( t ) d ] \displaystyle P(t)=tr[\rho(t)d]
  6. χ ( ω ) = - k | d c v | 2 < m t p l > L 3 ( f v , k - f c , k ) ( 1 ( ε v , k - ε c , k + ω + i γ ) - 1 ( ε c , k - ε v , k + ω + i γ ) ) \chi(\omega)=-\sum\limits_{k}{\frac{{\left|{d_{cv}}\right|^{{}_{2}}}}{<}mtpl>{% {L^{3}}}}(f_{v,k}-f_{c,k})\left({\frac{1}{{\hbar(\varepsilon_{v,k}-\varepsilon% _{c,k}+\omega+i\gamma)}}-\frac{1}{{\hbar(\varepsilon_{c,k}-\varepsilon_{v,k}+% \omega+i\gamma)}}}\right)
  7. α ( ω ) = < m t p l > 4 π ω n b c χ ′′ ( ω ) \alpha(\omega)=\frac{<}{m}tpl>{{4\pi\omega}}{{n_{b}c}}\chi^{\prime\prime}(\omega)
  8. = < m t p l > 4 π ω n b c k | d c v | 2 ( f v , k - f c , k ) δ ( ( ε v , k - ε c , k + ω ) ) {\rm{}}=\frac{<}{m}tpl>{{4\pi\omega}}{{n_{b}c}}\sum\limits_{k}{\left|{d_{cv}}% \right|^{2}(f_{v,k}-f_{c,k})\delta(\hbar(\varepsilon_{v,k}-\varepsilon_{c,k}+% \omega))}
  9. α ( ω ) = α 0 d < m t p l > ω E 0 ( ω - E g - E 0 ( d ) E 0 ) ( d - 2 ) / 2 k Θ ( ω - E g - E 0 ( d ) ) A ( ω ) \alpha(\omega)=\alpha_{0}^{d}\frac{<}{m}tpl>{{\hbar\omega}}{{E_{0}}}\left({% \frac{{\hbar\omega-E_{g}-E_{0}^{(d)}}}{{E_{0}}}}\right)^{(d-2)/2}\sum\limits_{% k}{\Theta(\hbar\omega-E_{g}-E_{0}^{(d)})A(\omega)}

Free_independence.html

  1. ( A , ϕ ) (A,\phi)
  2. A A
  3. \mathbb{C}
  4. ϕ : A \phi:A\to\mathbb{C}
  5. μ \mu
  6. A = L ( , μ ) , ϕ ( f ) = f ( t ) d μ ( t ) . A=L^{\infty}(\mathbb{R},\mu),\phi(f)=\int f(t)\,d\mu(t).
  7. A = M N A=M_{N}
  8. N × N N\times N
  9. ϕ = 1 N T r \phi=\frac{1}{N}Tr
  10. A A
  11. ϕ \phi
  12. A A
  13. A = Γ A=\mathbb{C}\Gamma
  14. Γ \Gamma
  15. ϕ \phi
  16. ϕ ( g ) = δ g = e , g Γ \phi(g)=\delta_{g=e},g\in\Gamma
  17. { A i : i I } \{A_{i}:i\in I\}
  18. A A
  19. { A i : i I } \{A_{i}:i\in I\}
  20. ϕ ( x 1 x 2 x n ) = 0 \phi(x_{1}x_{2}\cdots x_{n})=0
  21. ϕ ( x j ) = 0 \phi(x_{j})=0
  22. x j A i ( j ) x_{j}\in A_{i(j)}
  23. i ( 1 ) i ( 2 ) , i ( 2 ) i ( 3 ) , i(1)\neq i(2),i(2)\neq i(3),\dots
  24. X i A X_{i}\in A
  25. i I i\in I
  26. A A
  27. A A
  28. A i A_{i}
  29. 1 1
  30. X i X_{i}
  31. Γ \Gamma
  32. Γ i , i I \Gamma_{i},i\in I
  33. A = Γ A=\mathbb{C}\Gamma
  34. ϕ ( g ) = δ g = e \phi(g)=\delta_{g=e}
  35. A i = Γ i A A_{i}=\mathbb{C}\Gamma_{i}\subset A
  36. A i : i I A_{i}:i\in I
  37. U i ( N ) , i = 1 , 2 U_{i}(N),i=1,2
  38. N × N N\times N
  39. N × N N\times N
  40. U 1 ( N ) , U 2 ( N ) U_{1}(N),U_{2}(N)
  41. N N\to\infty
  42. N N\to\infty

Free_Poisson_distribution.html

  1. α \alpha
  2. λ \lambda
  3. ( ( 1 - λ N ) δ 0 + λ N δ α ) N \left(\left(1-\frac{\lambda}{N}\right)\delta_{0}+\frac{\lambda}{N}\delta_{% \alpha}\right)^{\boxplus N}
  4. X N X_{N}
  5. X N X_{N}
  6. α \alpha
  7. λ N \frac{\lambda}{N}
  8. X 1 , X 2 , X_{1},X_{2},\ldots
  9. N N\to\infty
  10. X 1 + + X N X_{1}+\cdots+X_{N}
  11. λ , α \lambda,\alpha
  12. μ = { ( 1 - λ ) δ 0 + λ ν , if 0 λ 1 ν , if λ > 1 , \mu=\begin{cases}(1-\lambda)\delta_{0}+\lambda\nu,&\,\text{if }0\leq\lambda% \leq 1\\ \nu,&\,\text{if }\lambda>1,\end{cases}
  13. ν = 1 2 π α t 4 λ α 2 - ( t - α ( 1 + λ ) ) 2 d t \nu=\frac{1}{2\pi\alpha t}\sqrt{4\lambda\alpha^{2}-(t-\alpha(1+\lambda))^{2}}% \,dt
  14. [ α ( 1 - λ ) 2 , α ( 1 + λ ) 2 ] [\alpha(1-\sqrt{\lambda})^{2},\alpha(1+\sqrt{\lambda})^{2}]
  15. λ \lambda
  16. R ( z ) = λ α 1 - α z . R(z)=\frac{\lambda\alpha}{1-\alpha z}.
  17. G ( z ) = z + α - λ α - ( z - α ( 1 + λ ) ) 2 - 4 λ α 2 2 α z G(z)=\frac{z+\alpha-\lambda\alpha-\sqrt{(z-\alpha(1+\lambda))^{2}-4\lambda% \alpha^{2}}}{2\alpha z}
  18. S ( z ) = 1 z + λ S(z)=\frac{1}{z+\lambda}
  19. α = 1 \alpha=1

FreeArc.html

  1. score X = time X × 2 10 ( size X size TOP - 1 ) \,\text{score}_{X}=\,\text{time}_{X}\times 2^{10(\frac{\,\text{size}_{X}}{\,% \text{size}_{\,\text{TOP}}}-1)}

FreeON.html

  1. Γ \Gamma

Freestream.html

  1. \infty
  2. V V_{\infty}

Frequency_offset.html

  1. f o s = f c h + f L p 12 f_{os}=f_{ch}+\frac{f_{L}\cdot p}{12}
  2. f o s f_{os}
  3. f c h f_{ch}
  4. p p
  5. - 12 < p < 12 -12<p<12
  6. f L f_{L}
  7. f o s = 203250000 + 15626 8 12 203260417 f_{os}=203250000+\frac{15626\cdot 8}{12}\approx 203260417
  8. f o s = 203250000 - 15626 8 12 203239583 f_{os}=203250000-\frac{15626\cdot 8}{12}\approx 203239583

Fresnel–Arago_laws.html

  1. 𝐄 𝟏 ( 𝐫 , t ) = 𝐄 01 cos ( 𝐤 𝟏 𝐫 - ω t + ϵ 1 ) \mathbf{E_{1}}(\mathbf{r},t)=\mathbf{E}_{01}\cos(\mathbf{k_{1}\cdot r}-\omega t% +\epsilon_{1})
  2. 𝐄 𝟐 ( 𝐫 , t ) = 𝐄 02 cos ( 𝐤 𝟐 𝐫 - ω t + ϵ 2 ) \mathbf{E_{2}}(\mathbf{r},t)=\mathbf{E}_{02}\cos(\mathbf{k_{2}\cdot r}-\omega t% +\epsilon_{2})
  3. I = ϵ v 𝐄 2 T I=\epsilon v\langle\mathbf{E}^{2}\rangle_{T}
  4. I 12 = ϵ v 𝐄 𝟎𝟏 𝐄 𝟎𝟐 cos δ I_{12}=\epsilon v\mathbf{E_{01}\cdot E_{02}}\cos\delta
  5. δ = ( 𝐤 𝟏 𝐫 - 𝐤 𝟐 𝐫 + ϵ 1 - ϵ 2 ) \delta=(\mathbf{k_{1}\cdot r-k_{2}\cdot r}+\epsilon_{1}-\epsilon_{2})
  6. 𝐄 𝟎𝟏 \mathbf{E_{01}}
  7. 𝐄 𝟎𝟐 \mathbf{E_{02}}
  8. I 12 = 0 I_{12}=0
  9. 𝐄 𝟎𝟏 \mathbf{E_{01}}
  10. 𝐄 𝟎𝟐 \mathbf{E_{02}}
  11. cos δ \cos\delta
  12. δ \delta
  13. cos δ T = 0 \langle\cos\delta\rangle_{T}=0
  14. I 12 = 0 I_{12}=0
  15. 𝐄 𝟎𝟏 \mathbf{E_{01}}
  16. 𝐄 𝟎𝟐 \mathbf{E_{02}}

Fréchet_mean.html

  1. m = arg min p M i = 1 N w i d 2 ( p , x i ) m=\mathop{\mathrm{arg\ min}}_{p\in M}\sum_{i=1}^{N}w_{i}d^{2}(p,x_{i})
  2. d ( x , y ) = | log ( x ) - log ( y ) | d(x,y)=|\log(x)-\log(y)|
  3. f : x e x f:x\mapsto e^{x}
  4. x i x_{i}
  5. f f
  6. f - 1 ( x i ) f^{-1}(x_{i})
  7. f ( 1 n i = 1 n f - 1 ( x i ) ) = exp ( 1 n i = 1 n log x i ) = x 1 x n n f\left(\frac{1}{n}\sum_{i=1}^{n}f^{-1}(x_{i})\right)=\exp\left(\frac{1}{n}\sum% _{i=1}^{n}\log x_{i}\right)=\sqrt[n]{x_{1}\cdots x_{n}}
  8. d H ( x , y ) = | 1 x - 1 y | d_{H}(x,y)=\left|\frac{1}{x}-\frac{1}{y}\right|
  9. m m
  10. d m ( x , y ) = | x m - y m | d_{m}(x,y)=|x^{m}-y^{m}|
  11. f f
  12. d f ( x , y ) = | f ( x ) - f ( y ) | d_{f}(x,y)=|f(x)-f(y)|

Fréedericksz_transition.html

  1. 𝐧 ^ = n x 𝐱 ^ + n y 𝐲 ^ \mathbf{\hat{n}}=n_{x}\mathbf{\hat{x}}+n_{y}\mathbf{\hat{y}}
  2. n x = cos θ ( z ) n_{x}=\cos{\theta(z)}
  3. n y = sin θ ( z ) n_{y}=\sin{\theta(z)}
  4. d = 1 2 K 2 ( d θ d z ) 2 \mathcal{F}_{d}=\frac{1}{2}K_{2}\left(\frac{d\theta}{dz}\right)^{2}
  5. U = 1 2 K 2 ( d θ d z ) 2 - 1 2 ϵ 0 Δ χ e E 2 sin 2 θ U=\frac{1}{2}K_{2}\left(\frac{d\theta}{dz}\right)^{2}-\frac{1}{2}\epsilon_{0}% \Delta\chi_{e}E^{2}\sin^{2}{\theta}
  6. F A = 0 d 1 2 K 2 ( d θ d z ) 2 - 1 2 ϵ 0 Δ χ e E 2 sin 2 θ d z F_{A}=\int_{0}^{d}\frac{1}{2}K_{2}\left(\frac{d\theta}{dz}\right)^{2}-\frac{1}% {2}\epsilon_{0}\Delta\chi_{e}E^{2}\sin^{2}{\theta}\,dz\,
  7. ( U θ ) - d d z ( U ( d θ d z ) ) = 0 \left(\frac{\partial U}{\partial\theta}\right)-\frac{d}{dz}\left(\frac{% \partial U}{\partial\left(\frac{d\theta}{dz}\right)}\right)=0
  8. K 2 ( d 2 θ d z 2 ) + ϵ 0 Δ χ e E 2 sin θ cos θ = 0 K_{2}\left(\frac{d^{2}\theta}{dz^{2}}\right)+\epsilon_{0}\Delta\chi_{e}E^{2}% \sin{\theta}\cos{\theta}=0
  9. ζ = z d \zeta=\frac{z}{d}
  10. ξ d = d - 1 K 2 ϵ 0 Δ χ e E 2 \xi_{d}=d^{-1}\sqrt{\frac{K_{2}}{\epsilon_{0}\Delta\chi_{e}E^{2}}}
  11. d d
  12. ξ d 2 ( d 2 θ d ζ 2 ) + sin θ cos θ = 0 \xi_{d}^{2}\left(\frac{d^{2}\theta}{d\zeta^{2}}\right)+\sin{\theta}\cos{\theta% }=0
  13. d θ d ζ \frac{d\theta}{d\zeta}
  14. d θ d ζ ξ d 2 ( d 2 θ d ζ 2 ) + d θ d ζ sin θ cos θ = 1 2 ξ d 2 d d ζ ( ( d θ d ζ ) 2 ) + 1 2 d d ζ ( sin 2 θ ) = 0 \frac{d\theta}{d\zeta}\xi_{d}^{2}\left(\frac{d^{2}\theta}{d\zeta^{2}}\right)+% \frac{d\theta}{d\zeta}\sin{\theta}\cos{\theta}=\frac{1}{2}\xi_{d}^{2}\frac{d}{% d\zeta}\left(\left(\frac{d\theta}{d\zeta}\right)^{2}\right)+\frac{1}{2}\frac{d% }{d\zeta}\left(\sin^{2}{\theta}\right)=0
  15. 1 2 ξ d 2 d d ζ ( ( d θ d ζ ) 2 ) + 1 2 d d ζ ( sin 2 θ ) d ζ = 0 \int\frac{1}{2}\xi_{d}^{2}\frac{d}{d\zeta}\left(\left(\frac{d\theta}{d\zeta}% \right)^{2}\right)+\frac{1}{2}\frac{d}{d\zeta}\left(\sin^{2}{\theta}\right)\,d% \zeta\,=0
  16. d θ d ζ = 1 ξ d sin 2 θ m - sin 2 θ \frac{d\theta}{d\zeta}=\frac{1}{\xi_{d}}\sqrt{\sin^{2}{\theta_{m}}-\sin^{2}{% \theta}}
  17. θ m \theta_{m}
  18. θ \theta
  19. ζ = 1 / 2 \zeta=1/2
  20. k = sin θ m k=\sin{\theta_{m}}
  21. t = sin θ sin θ m t=\frac{\sin{\theta}}{\sin{\theta_{m}}}
  22. t t
  23. 0 1 1 ( 1 - t 2 ) ( 1 - k 2 t 2 ) d t K ( k ) = 1 2 ξ d \int_{0}^{1}\frac{1}{\sqrt{(1-t^{2})(1-k^{2}t^{2})}}\,dt\,\equiv K(k)=\frac{1}% {2\xi_{d}}
  24. K ( 0 ) = π 2 K(0)=\frac{\pi}{2}
  25. E t E_{t}
  26. E t = π d K 2 ϵ 0 Δ χ e E_{t}=\frac{\pi}{d}\sqrt{\frac{K_{2}}{\epsilon_{0}\Delta\chi_{e}}}

Friedel_oscillations.html

  1. ψ k ( r ) = 1 Ω e i k r \psi_{{k}}({r})=\frac{1}{\sqrt{\Omega}}e^{i{k}\cdot{r}}
  2. ρ ( r ) = ρ 0 + δ n cos ( 2 k F | r | + δ ) | r | 3 \rho({r})=\rho_{0}+\delta n\frac{\cos(2k_{F}|{r}|+\delta)}{|{r}|^{3}}

Friedman_translation.html

  1. Δ 0 0 \Delta^{0}_{0}

Friendship_graph.html

  1. ( x - 2 ) n ( x - 1 ) n x (x-2)^{n}(x-1)^{n}x
  2. n 2 4 + f ( k ) , \left\lfloor\frac{n^{2}}{4}\right\rfloor+f(k),

Friendship_paradox.html

  1. G = ( V , E ) G=(V,E)
  2. V V
  3. E E
  4. X X
  5. Y Y
  6. Y Y
  7. X X
  8. v v
  9. d ( v ) d(v)
  10. d ( v ) d(v)
  11. μ μ
  12. μ = v V d ( v ) | V | = 2 | E | | V | . \mu=\frac{\sum_{v\in V}d(v)}{|V|}=\frac{2|E|}{|V|}.
  13. v V d ( v ) 2 2 | E | = μ + σ 2 μ , \frac{\sum_{v\in V}d(v)^{2}}{2|E|}=\mu+\frac{\sigma^{2}}{\mu},
  14. σ 2 {\sigma}^{2}
  15. μ μ
  16. σ 2 {\sigma}^{2}
  17. ( u , v ) (u,v)
  18. u u
  19. v v
  20. v v
  21. d ( v ) d(v)
  22. d ( v ) d(v)
  23. d ( v ) d(v)
  24. u u
  25. v v
  26. u u
  27. v v

Frink_ideal.html

  1. \subseteq
  2. \subseteq
  3. \subseteq

Fritz_John_conditions.html

  1. minimize \displaystyle\,\text{minimize }
  2. g i g_{i}
  3. h j h_{j}
  4. \mathcal{I}
  5. \mathcal{I^{\prime}}
  6. \mathcal{E}
  7. x * x^{*}
  8. f f
  9. λ = [ λ 0 , λ 1 , λ 2 , , λ n ] \lambda=[\lambda_{0},\lambda_{1},\lambda_{2},\dots,\lambda_{n}]
  10. { λ 0 f ( x * ) = i λ i g i ( x * ) + i λ i h i ( x * ) λ i 0 , i { 0 } i ( { 0 , 1 , , n } \ ) ( λ i 0 ) \begin{cases}\lambda_{0}\nabla f(x^{*})=\sum\limits_{i\in\mathcal{I}^{\prime}}% \lambda_{i}\nabla g_{i}(x^{*})+\sum\limits_{i\in\mathcal{E}}\lambda_{i}\nabla h% _{i}(x^{*})\\ \lambda_{i}\geq 0,\ i\in\mathcal{I}^{\prime}\cup\{0\}\\ \exists i\in\left(\{0,1,\ldots,n\}\backslash\mathcal{I}\right)\left(\lambda_{i% }\neq 0\right)\end{cases}
  11. λ 0 > 0 \lambda_{0}>0
  12. g i ( i ) \nabla g_{i}(i\in\mathcal{I}^{\prime})
  13. h i ( i ) \nabla h_{i}(i\in\mathcal{E})
  14. λ 0 > 0 \lambda_{0}>0

Froda's_theorem.html

  1. f f
  2. x x
  3. x 0 x_{0}
  4. f f
  5. x = x 0 x=x_{0}
  6. f ( x + 0 ) := lim h 0 f ( x + h ) f(x+0):=\lim_{h\searrow 0}f(x+h)
  7. f ( x - 0 ) := lim h 0 f ( x - h ) f(x-0):=\lim_{h\searrow 0}f(x-h)
  8. f ( x 0 + 0 ) f(x_{0}+0)
  9. f ( x 0 - 0 ) f(x_{0}-0)
  10. f ( x 0 + 0 ) - f ( x 0 - 0 ) f(x_{0}+0)-f(x_{0}-0)
  11. x 0 x_{0}
  12. x 0 x_{0}
  13. x 0 x_{0}
  14. f f
  15. x 0 x_{0}
  16. x 0 x_{0}
  17. f ( x 0 + 0 ) = f ( x 0 - 0 ) f ( x 0 ) f(x_{0}+0)=f(x_{0}-0)\neq f(x_{0})
  18. I := [ a , b ] I:=[a,b]
  19. f f
  20. I I
  21. f ( a ) f ( a + 0 ) f ( x - 0 ) f ( x + 0 ) f ( b - 0 ) f ( b ) f(a)\leq f(a+0)\leq f(x-0)\leq f(x+0)\leq f(b-0)\leq f(b)
  22. a < x < b a<x<b
  23. α > 0 \alpha>0
  24. x 1 < x 2 < < x n x_{1}<x_{2}<\cdots<x_{n}
  25. n n
  26. I I
  27. f f
  28. α \alpha
  29. f ( x i + 0 ) - f ( x i - 0 ) α , i = 1 , 2 , , n f(x_{i}+0)-f(x_{i}-0)\geq\alpha,\ i=1,2,\ldots,n
  30. f ( x i + 0 ) f ( x i + 1 - 0 ) f(x_{i}+0)\leq f(x_{i+1}-0)
  31. f ( x i + 1 - 0 ) - f ( x i + 0 ) 0 , i = 1 , 2 , , n f(x_{i+1}-0)-f(x_{i}+0)\geq 0,\ i=1,2,\ldots,n
  32. f ( b ) - f ( a ) f ( x n + 0 ) - f ( x 1 - 0 ) = i = 1 n [ f ( x i + 0 ) - f ( x i - 0 ) ] + f(b)-f(a)\geq f(x_{n}+0)-f(x_{1}-0)=\sum_{i=1}^{n}[f(x_{i}+0)-f(x_{i}-0)]+
  33. + i = 1 n - 1 [ f ( x i + 1 - 0 ) - f ( x i + 0 ) ] i = 1 n [ f ( x i + 0 ) - f ( x i - 0 ) ] n α +\sum_{i=1}^{n-1}[f(x_{i+1}-0)-f(x_{i}+0)]\geq\sum_{i=1}^{n}[f(x_{i}+0)-f(x_{i% }-0)]\geq n\alpha
  34. n f ( b ) - f ( a ) α n\leq\frac{f(b)-f(a)}{\alpha}
  35. f ( b ) - f ( a ) < f(b)-f(a)<\infty
  36. α \alpha
  37. S 1 := { x : x I , f ( x + 0 ) - f ( x - 0 ) 1 } S_{1}:=\{x:x\in I,f(x+0)-f(x-0)\geq 1\}
  38. S n := { x : x I , 1 n f ( x + 0 ) - f ( x - 0 ) < 1 n - 1 } , n 2. S_{n}:=\{x:x\in I,\frac{1}{n}\leq f(x+0)-f(x-0)<\frac{1}{n-1}\},\ n\geq 2.
  39. S n S_{n}
  40. S = n = 1 S n S=\cup_{n=1}^{\infty}S_{n}
  41. S i , i = 1 , 2 , S_{i},\ i=1,2,\ldots
  42. S S
  43. f f
  44. I I
  45. I n I_{n}
  46. I = n = 1 I n . I=\cup_{n=1}^{\infty}I_{n}.
  47. I = ( a , b ] , a - I=(a,b],\ a\geq-\infty
  48. I 1 = [ α 1 , b ] , I 2 = [ α 2 , α 1 ] , , I n = [ α n , α n - 1 ] , I_{1}=[\alpha_{1},b],\ I_{2}=[\alpha_{2},\alpha_{1}],\ldots,\ I_{n}=[\alpha_{n% },\alpha_{n-1}],\ldots
  49. { α n } n \{\alpha_{n}\}_{n}
  50. α n a . \alpha_{n}\rightarrow a.
  51. I = [ a , b ) , b + I=[a,b),\ b\leq+\infty
  52. I = ( a , b ) - a < b I=(a,b)\ -\infty\leq a<b\leq\infty
  53. I n I_{n}
  54. I I

FRP_tanks_and_vessels.html

  1. 13 * π 4 * 10 2 π 4 = 36.055 f t {\sqrt{13*{{\pi}\over{4}}*10^{2}}\over{\pi\over 4}}=36.055ft

Frucht_graph.html

  1. ( x - 3 ) ( x - 2 ) x ( x + 1 ) ( x + 2 ) ( x 3 + x 2 - 2 x - 1 ) ( x 4 + x 3 - 6 x 2 - 5 x + 4 ) (x-3)(x-2)x(x+1)(x+2)(x^{3}+x^{2}-2x-1)(x^{4}+x^{3}-6x^{2}-5x+4)

Functional_renormalization_group.html

  1. Γ \Gamma
  2. S S
  3. Γ \Gamma
  4. Γ \Gamma
  5. Γ \Gamma
  6. Γ \Gamma
  7. Γ k \Gamma_{k}
  8. k k
  9. R k R_{k}
  10. Γ k ( 2 ) \Gamma^{(2)}_{k}
  11. R k R_{k}
  12. q k q\lesssim k
  13. Γ k \Gamma_{k}
  14. q k q\gtrsim k
  15. Γ k \Gamma_{k}
  16. k Γ k = 1 2 STr k R k ( Γ k ( 2 ) + R k ) - 1 , \partial_{k}\Gamma_{k}=\frac{1}{2}\,\text{STr}\,\partial_{k}R_{k}\,(\Gamma^{(2% )}_{k}+R_{k})^{-1},
  17. k \partial_{k}
  18. k k
  19. Γ k \Gamma_{k}
  20. Γ k Λ = S \Gamma_{k\to\Lambda}=S
  21. S S
  22. k = Λ k=\Lambda
  23. k 0 k\to 0
  24. Γ = Γ k 0 \Gamma=\Gamma_{k\to 0}
  25. STr \,\text{STr}
  26. Γ k \Gamma_{k}
  27. Γ k ( 2 ) \Gamma^{(2)}_{k}
  28. R k R_{k}
  29. Γ k \Gamma_{k}
  30. { c n } \{c_{n}\}
  31. k = Λ k=\Lambda
  32. Γ k = Λ = S \Gamma_{k=\Lambda}=S
  33. k k
  34. Γ k \Gamma_{k}
  35. R k R_{k}
  36. R k R_{k}
  37. k = 0 k=0
  38. Γ k = 0 = Γ \Gamma_{k=0}=\Gamma
  39. R k R_{k}
  40. Γ k \Gamma_{k}
  41. R k R_{k}
  42. β \beta
  43. S S
  44. Λ \Lambda
  45. Λ \Lambda
  46. k k
  47. 𝒱 [ η , η + ] = - ln Z [ G 0 - 1 η , G 0 - 1 η + ] - η G 0 - 1 η + \mathcal{V}[\eta,\eta^{+}]=-\ln Z[G_{0}^{-1}\eta,G_{0}^{-1}\eta^{+}]-\eta G_{0% }^{-1}\eta^{+}
  48. G 0 G_{0}
  49. Z [ η , η + ] Z[\eta,\eta^{+}]
  50. D D
  51. 𝒲 [ η , η + ] = exp ( - Δ D ) 𝒱 [ η , η + ] \mathcal{W}[\eta,\eta^{+}]=\exp(-\Delta_{D})\mathcal{V}[\eta,\eta^{+}]
  52. Δ = D δ 2 / ( δ η δ η + ) \Delta=D\delta^{2}/(\delta\eta\delta\eta^{+})
  53. Λ \Lambda
  54. < m t p l > Λ V Λ ( ψ ) = - Δ ˙ G 0 , Λ V Λ ( ψ ) + Δ G ˙ 0 , Λ 12 𝒱 Λ ( 1 ) 𝒱 Λ ( 2 ) \frac{\partial}{<}mtpl>{{\partial\Lambda}}{{V}_{\Lambda}}(\psi)=-{\dot{\Delta}% _{{G_{0,\Lambda}}}}{{V}_{\Lambda}}(\psi)+\Delta_{{{\dot{G}}_{0,\Lambda}}}^{12}% \mathcal{V}_{\Lambda}^{(1)}\mathcal{V}_{\Lambda}^{(2)}
  55. Λ 𝒲 Λ = - Δ D ˙ Λ + G ˙ 0 , Λ 𝒲 Λ + e - Δ < m t p l > D Λ 12 Δ G ˙ 0 , Λ 12 𝒲 Λ ( 1 ) 𝒲 Λ ( 2 ) {\partial_{\Lambda}}{\mathcal{W}_{\Lambda}}=-{\Delta_{{{\dot{D}}_{\Lambda}}+{{% \dot{G}}_{0,\Lambda}}}}{\mathcal{W}_{\Lambda}}+{e^{-\Delta_{<}mtpl>{{D_{% \Lambda}}}^{12}}}\Delta_{{{\dot{G}}_{0,\Lambda}}}^{12}\mathcal{W}_{\Lambda}^{(% 1)}\mathcal{W}_{\Lambda}^{(2)}
  56. Δ G ˙ 0 , Λ 12 𝒱 Λ ( 1 ) 𝒱 Λ ( 2 ) = 1 2 ( δ V Λ ( ψ ) < m t p l > δ ψ , G ˙ 0 , Λ δ V Λ ( ψ ) δ ψ ) \Delta_{{{\dot{G}}_{0,\Lambda}}}^{12}\mathcal{V}_{\Lambda}^{(1)}\mathcal{V}_{% \Lambda}^{(2)}=\frac{1}{2}\left({\frac{{\delta{{V}_{\Lambda}}(\psi)}}{<}mtpl>{% {\delta\psi}},{{\dot{G}}_{0,\Lambda}}\frac{{\delta{{V}_{\Lambda}}(\psi)}}{{% \delta\psi}}}\right)
  57. O ( N ) O(N)
  58. d d
  59. d = 3 d=3
  60. d = 2 d=2
  61. N = 2 N=2

Fundamental_lemma_(Langlands_program).html

  1. S O γ H ( 1 K H ) = Δ ( γ H , γ G ) O γ G κ ( 1 K G ) SO_{\gamma_{H}}(1_{K_{H}})=\Delta(\gamma_{H},\gamma_{G})O^{\kappa}_{\gamma_{G}% }(1_{K_{G}})

Fundamental_theorem_of_linear_programming.html

  1. min c T x subject to x P \min c^{T}x\,\text{ subject to }x\in P
  2. P = { x n : A x b } P=\{x\in\mathbb{R}^{n}:Ax\leq b\}
  3. P P
  4. x x^{\ast}
  5. x x^{\ast}
  6. P P
  7. F P F\subset P
  8. x int ( P ) x^{\ast}\in\mathrm{int}(P)
  9. ϵ > 0 \epsilon>0
  10. ϵ \epsilon
  11. x x^{\ast}
  12. P P
  13. B ϵ ( x ) P B_{\epsilon}(x^{\ast})\subset P
  14. x - ϵ 2 c || c || P x^{\ast}-\frac{\epsilon}{2}\frac{c}{||c||}\in P
  15. c T ( x - ϵ 2 c || c || ) = c T x - ϵ 2 c T c || c || = c T x - ϵ 2 || c || < c T x . c^{T}\left(x^{\ast}-\frac{\epsilon}{2}\frac{c}{||c||}\right)=c^{T}x^{\ast}-% \frac{\epsilon}{2}\frac{c^{T}c}{||c||}=c^{T}x^{\ast}-\frac{\epsilon}{2}||c||<c% ^{T}x^{\ast}.
  16. x x^{\ast}
  17. x x^{\ast}
  18. P P
  19. x x^{\ast}
  20. P P
  21. x 1 , , x t x_{1},...,x_{t}
  22. x = i = 1 t λ i x i x^{\ast}=\sum_{i=1}^{t}\lambda_{i}x_{i}
  23. λ i 0 \lambda_{i}\geq 0
  24. i = 1 t λ i = 1 \sum_{i=1}^{t}\lambda_{i}=1
  25. 0 = c T ( ( i = 1 t λ i x i ) - x ) = c T ( i = 1 t λ i ( x i - x ) ) = i = 1 t λ i ( c T x i - c T x ) . 0=c^{T}\left(\left(\sum_{i=1}^{t}\lambda_{i}x_{i}\right)-x^{\ast}\right)=c^{T}% \left(\sum_{i=1}^{t}\lambda_{i}(x_{i}-x^{\ast})\right)=\sum_{i=1}^{t}\lambda_{% i}(c^{T}x_{i}-c^{T}x^{\ast}).
  26. x x^{\ast}
  27. c T x = c T x i c^{T}x^{\ast}=c^{T}x_{i}
  28. x i x_{i}
  29. x i x_{i}
  30. x 1 , , x t x_{1},...,x_{t}

Funk_transform.html

  1. F f ( 𝐱 ) = 𝐮 C ( 𝐱 ) f ( 𝐮 ) d s ( 𝐮 ) Ff(\mathbf{x})=\int_{\mathbf{u}\in C(\mathbf{x})}f(\mathbf{u})\,ds(\mathbf{u})
  2. C ( 𝐱 ) = { 𝐮 S 2 𝐮 𝐱 = 0 } . C(\mathbf{x})=\{\mathbf{u}\in S^{2}\mid\mathbf{u}\cdot\mathbf{x}=0\}.
  3. f L 2 ( S 2 ) f\in L^{2}(S^{2})
  4. f = n = 0 k = - n n f ^ ( n , k ) Y n k . f=\sum_{n=0}^{\infty}\sum_{k=-n}^{n}\hat{f}(n,k)Y_{n}^{k}.
  5. f = n = 0 k = - n n P n ( 0 ) f ^ ( n , k ) Y n k f=\sum_{n=0}^{\infty}\sum_{k=-n}^{n}P_{n}(0)\hat{f}(n,k)Y_{n}^{k}
  6. P 2 n + 1 ( 0 ) = 0 P_{2n+1}(0)=0
  7. P 2 n ( 0 ) = ( - 1 ) n 1 3 5 2 n - 1 2 4 6 2 n P_{2n}(0)=(-1)^{n}\frac{1\cdot 3\cdot 5\cdots 2n-1}{2\cdot 4\cdot 6\cdots 2n}
  8. ( F * f ) ( p , 𝐱 ) = 1 2 π cos p 𝐮 = 1 , 𝐱 𝐮 = sin p f ( 𝐮 ) | d 𝐮 | . (F^{*}f)(p,\mathbf{x})=\frac{1}{2\pi\cos p}\int_{\|\mathbf{u}\|=1,\mathbf{x}% \cdot\mathbf{u}=\sin p}f(\mathbf{u})\,|d\mathbf{u}|.
  9. f ( 𝐱 ) = 1 2 π { d d u 0 u F * ( F f ) ( cos - 1 v , 𝐱 ) v ( u 2 - v 2 ) - 1 / 2 d v } u = 1 . f(\mathbf{x})=\frac{1}{2\pi}\left\{\frac{d}{du}\int_{0}^{u}F^{*}(Ff)(\cos^{-1}% v,\mathbf{x})v(u^{2}-v^{2})^{-1/2}\,dv\right\}_{u=1}.
  10. φ ( 𝐱 , 𝐲 ) = 1 2 π f ( u 𝐱 + v 𝐲 ) ( u d v - v d u ) \varphi(\mathbf{x},\mathbf{y})=\frac{1}{2\pi}\oint f(u\mathbf{x}+v\mathbf{y})(% u\,dv-v\,du)
  11. f ( u 𝐱 + v 𝐲 ) ( u d v - v d u ) f(u\mathbf{x}+v\mathbf{y})(u\,dv-v\,du)
  12. ϕ ( a 𝐱 + b 𝐲 , c 𝐱 + d 𝐲 ) = 1 | a d - b c | ϕ ( 𝐱 , 𝐲 ) , \phi(a\mathbf{x}+b\mathbf{y},c\mathbf{x}+d\mathbf{y})=\frac{1}{|ad-bc|}\phi(% \mathbf{x},\mathbf{y}),
  13. F f ( 𝐱 𝐲 ) = ϕ ( 𝐱 , 𝐲 ) . Ff(\mathbf{x}\wedge\mathbf{y})=\phi(\mathbf{x},\mathbf{y}).

Furuta_pendulum.html

  1. τ 1 \tau_{1}
  2. L 1 L_{1}
  3. L 2 L_{2}
  4. m 1 m_{1}
  5. m 2 m_{2}
  6. l 1 l_{1}
  7. l 2 l_{2}
  8. s y m b o l J 1 symbol{J}_{1}
  9. s y m b o l J 2 symbol{J}_{2}
  10. b 1 b_{1}
  11. b 2 b_{2}
  12. b 1 b_{1}
  13. b 2 b_{2}
  14. θ 1 \theta_{1}
  15. θ 2 \theta_{2}
  16. θ 2 = 0 \theta_{2}=0
  17. τ 1 \tau_{1}
  18. τ 2 \tau_{2}
  19. θ ¨ 1 ( J 1 z z + m 1 l 1 2 + m 2 L 1 2 + ( J 2 y y + m 2 l 2 2 ) sin 2 ( θ 2 ) + J 2 x x cos 2 ( θ 2 ) ) + θ ¨ 2 m 2 L 1 l 2 cos ( θ 2 ) - m 2 L 1 l 2 sin ( θ 2 ) θ ˙ 2 2 + θ ˙ 1 θ ˙ 2 sin ( 2 θ 2 ) ( m 2 l 2 2 + J 2 y y - J 2 x x ) + b 1 θ ˙ 1 = τ 1 \ddot{\theta}_{1}\left(J_{1zz}+m_{1}l_{1}^{2}+m_{2}L_{1}^{2}+(J_{2yy}+m_{2}l_{% 2}^{2})\sin^{2}(\theta_{2})+J_{2xx}\cos^{2}(\theta_{2})\right)+\ddot{\theta}_{% 2}m_{2}L_{1}l_{2}\cos(\theta_{2})-m_{2}L_{1}l_{2}\sin(\theta_{2})\dot{\theta}_% {2}^{2}+\dot{\theta}_{1}\dot{\theta}_{2}\sin(2\theta_{2})(m_{2}l_{2}^{2}+J_{2% yy}-J_{2xx})+b_{1}\dot{\theta}_{1}=\tau_{1}
  20. θ ¨ 1 m 2 L 1 l 2 cos ( θ 2 ) + θ ¨ 2 ( m 2 l 2 2 + J 2 z z ) + 1 / 2 θ ˙ 1 2 sin ( 2 θ 2 ) ( - m 2 l 2 2 - J 2 y y + J 2 x x ) + b 2 θ ˙ 2 + g m 2 l 2 sin ( θ 2 ) = τ 2 \ddot{\theta}_{1}m_{2}L_{1}l_{2}\cos(\theta_{2})+\ddot{\theta}_{2}(m_{2}l_{2}^% {2}+J_{2zz})+1/2\dot{\theta}_{1}^{2}\sin(2\theta_{2})(-m_{2}l_{2}^{2}-J_{2yy}+% J_{2xx})+b_{2}\dot{\theta}_{2}+gm_{2}l_{2}\sin(\theta_{2})=\tau_{2}
  21. s y m b o l J 1 = d i a g [ J 1 x x , J 1 y y , J 1 z z ] = d i a g [ 0 , J 1 , J 1 ] symbol{J}_{1}=diag[J_{1xx},J_{1yy},J_{1zz}]=diag[0,J_{1},J_{1}]
  22. s y m b o l J 2 = d i a g [ J 2 x x , J 2 y y , J 2 z z ] = d i a g [ 0 , J 2 , J 2 ] symbol{J}_{2}=diag[J_{2xx},J_{2yy},J_{2zz}]=diag[0,J_{2},J_{2}]
  23. J 1 ^ = J 1 + m 1 l 1 2 \hat{J_{1}}=J_{1}+m_{1}l_{1}^{2}
  24. J 2 ^ = J 2 + m 2 l 2 2 \hat{J_{2}}=J_{2}+m_{2}l_{2}^{2}
  25. J 0 ^ = J ^ 1 + m 2 L 1 2 = J 1 + m 1 l 1 2 + m 2 L 1 2 \hat{J_{0}}=\hat{J}_{1}+m_{2}L_{1}^{2}=J_{1}+m_{1}l_{1}^{2}+m_{2}L_{1}^{2}
  26. θ ¨ 1 ( J 0 ^ + J 2 ^ sin 2 ( θ 2 ) ) + θ ¨ 2 m 2 L 1 l 2 cos ( θ 2 ) - m 2 L 1 l 2 sin ( θ 2 ) θ ˙ 2 2 + θ ˙ 1 θ ˙ 2 sin ( 2 θ 2 ) J 2 ^ + b 1 θ ˙ 1 = τ 1 \ddot{\theta}_{1}\left(\hat{J_{0}}+\hat{J_{2}}\sin^{2}(\theta_{2})\right)+% \ddot{\theta}_{2}m_{2}L_{1}l_{2}\cos(\theta_{2})-m_{2}L_{1}l_{2}\sin(\theta_{2% })\dot{\theta}_{2}^{2}+\dot{\theta}_{1}\dot{\theta}_{2}\sin(2\theta_{2})\hat{J% _{2}}+b_{1}\dot{\theta}_{1}=\tau_{1}
  27. θ ¨ 1 m 2 L 1 l 2 cos ( θ 2 ) + θ ¨ 2 J 2 ^ - 1 / 2 θ ˙ 1 2 sin ( 2 θ 2 ) J 2 ^ + b 2 θ ˙ 2 + g m 2 l 2 sin ( θ 2 ) = τ 2 \ddot{\theta}_{1}m_{2}L_{1}l_{2}\cos(\theta_{2})+\ddot{\theta}_{2}\hat{J_{2}}-% 1/2\dot{\theta}_{1}^{2}\sin(2\theta_{2})\hat{J_{2}}+b_{2}\dot{\theta}_{2}+gm_{% 2}l_{2}\sin(\theta_{2})=\tau_{2}

Fuzzy_pay-off_method_for_real_option_valuation.html

  1. ROV = A ( Pos ) A ( Pos ) + A ( Neg ) × E [ A + ] \mathrm{ROV}=\frac{A(\mathrm{Pos})}{A(\mathrm{Pos})+A(\mathrm{Neg})}\times E[A% _{+}]

Fuzzy_retrieval.html

  1. d A B = m i n ( d A , d B ) d_{A\cap B}=min(d_{A},d_{B})
  2. d A B = m a x ( d A , d B ) d_{A\cup B}=max(d_{A},d_{B})
  3. S ( D , Q ) = i = 1 n r i - 1 * w d i j = 1 n r j - 1 S(D,Q)=\sum_{i=1}^{n}\frac{r^{i-1}*w_{di}}{\sum_{j=1}^{n}r^{j-1}}

G-module.html

  1. ( g f ) ( x , y ) = f ( ( x , y ) g ) = f ( α x + γ y , β x + δ y ) (g\cdot f)(x,y)=f((x,y)g)=f(\alpha x+\gamma y,\beta x+\delta y)
  2. g = [ α β γ δ ] g=\begin{bmatrix}\alpha&\beta\\ \gamma&\delta\\ \end{bmatrix}

G._B._Pant_University_of_Agriculture_and_Technology.html

  1. C G P A = i = 1 N C i . G P i i = 1 N C i CGPA\,\!={\sum_{i=1}^{N}C_{i}.{GP}_{i}\over\sum_{i=1}^{N}C_{i}}
  2. N N
  3. C i C_{i}
  4. i t h i^{th}
  5. G P i {GP}_{i}
  6. i t h i^{th}

Gabriel's_theorem.html

  1. A n A_{n}
  2. D n D_{n}
  3. E 6 E_{6}
  4. E 7 E_{7}
  5. E 8 E_{8}

Gaetano_Fichera.html

  1. n > 2 n>2
  2. Ω \scriptstyle\partial\Omega
  3. C 1 , α C^{1,\alpha}
  4. H 1 / 2 ( Ω ) \scriptstyle H^{1/2}(\partial\Omega)
  5. f f
  6. n 2 n \scriptstyle\mathbb{C}^{n}\equiv\mathbb{R}^{2n}
  7. n 2 \scriptstyle n\geq 2
  8. 2 4 \scriptstyle\mathbb{C}^{2}\equiv\mathbb{R}^{4}
  9. k k
  10. k k
  11. R 2 n R^{2n}
  12. f f

Gammoid.html

  1. G G
  2. S S
  3. T T
  4. S S
  5. Γ \Gamma
  6. T T
  7. I I
  8. T T
  9. Γ \Gamma
  10. S S
  11. I I
  12. T T
  13. G G
  14. U n r U{}^{r}_{n}
  15. n n
  16. r r
  17. K r , n K_{r,n}
  18. S S
  19. r r
  20. T T
  21. n n
  22. S S
  23. T T
  24. T T
  25. r r
  26. S S
  27. U n r U{}^{r}_{n}
  28. n n
  29. S S
  30. r r
  31. r r
  32. X T X\subset T
  33. G G
  34. S S
  35. T T
  36. S S
  37. X X
  38. Y Y
  39. S S
  40. X X
  41. X X
  42. X X
  43. ( S , T , E ) (S,T,E)
  44. S S
  45. T T
  46. γ \gamma
  47. G G
  48. S S
  49. N v N_{v}
  50. v V ( G ) S v\in V(G)\setminus S
  51. u u
  52. N v N_{v}
  53. v v
  54. v v
  55. | S | |S|
  56. S S
  57. N v N_{v}
  58. u u
  59. u v uv
  60. v v
  61. v v
  62. u v u_{v}
  63. N v N_{v}
  64. u v v u_{v}v
  65. N v N_{v}
  66. v v
  67. U 4 2 U{}^{2}_{4}
  68. n n
  69. U n 2 U{}^{2}_{n}
  70. n - 1 n-1
  71. S S
  72. 2 | S | 2^{|S|}

Gardner's_relation.html

  1. ρ = α V p β \rho=\alpha V_{p}^{\beta}
  2. ρ \rho
  3. V p V_{p}
  4. α \alpha
  5. β \beta
  6. α = 0.23 \alpha=0.23
  7. β = 0.25 \beta=0.25
  8. ρ = 0.23 V p 0.25 . \rho=0.23V_{p}^{0.25}.
  9. V p V_{p}
  10. α = 0.31 \alpha=0.31
  11. ρ = 0.31 V p 0.25 . \rho=0.31V_{p}^{0.25}.
  12. α \alpha
  13. β \beta

Gas_collecting_tube.html

  1. p p
  2. m f u l l m_{full}
  3. m s u c k e d m_{sucked}
  4. m = m f u l l - m s u c k e d m=m_{full}-m_{sucked}
  5. p p
  6. m c o n t a i n i n g l i q u i d m_{containingliquid}
  7. m l i q u i d = m c o n t a i n i n g l i q u i d - m s u c k e d m_{liquid}=m_{containingliquid}-m_{sucked}
  8. ρ l i q u i d \rho_{liquid}
  9. V = m l i q u i d / ρ l i q u i d V=m_{liquid}/\rho_{liquid}
  10. m m
  11. V V
  12. ρ = m V \rho=\frac{m}{V}
  13. ρ g a s \rho_{gas}
  14. ρ l i q u i d \rho_{liquid}
  15. V V
  16. m e v a c t u b e m_{evactube}
  17. ρ \rho
  18. m f u l l g a s m_{fullgas}
  19. m f u l l l i q u i d m_{fullliquid}
  20. ρ g a s = m f u l l g a s - m e v a c t u b e V \rho_{gas}=\frac{m_{fullgas}-m_{evactube}}{V}\,
  21. ρ l i q u i d = m f u l l l i q u i d - m e v a c t u b e V \rho_{liquid}=\frac{m_{fullliquid}-m_{evactube}}{V}
  22. m e v a c t u b e m_{evactube}
  23. V V
  24. V = m f u l l l i q u i d - m f u l l g a s ρ l i q u i d - ρ g a s V=\frac{m_{fullliquid}-m_{fullgas}}{\rho_{liquid}-\rho_{gas}}\,
  25. m e v a c t u b e = ρ g a s m f u l l l i q u i d - ρ l i q u i d m f u l l g a s ρ l i q u i d - ρ g a s m_{evactube}=\frac{\rho_{gas}\cdot m_{fullliquid}-\rho_{liquid}\cdot m_{% fullgas}}{\rho_{liquid}-\rho_{gas}}
  26. m f u l l m_{full}
  27. m = m f u l l - m e v a c t u b e m=m_{full}-m_{evactube}
  28. ρ = m V \rho=\frac{m}{V}
  29. ρ = m V \rho=\frac{m}{V}
  30. M M
  31. M = m R T p V M=\frac{m\cdot R\cdot T}{p\cdot V}\,
  32. R R
  33. T T

Gaseous_detection_device.html

  1. R = I ( r , d ) δ I b = 1 - 1 Q exp [ e V 2 k T ϵ ( 1 - Q ) ] R=\frac{I(r,d)}{\delta I_{b}}=1-\frac{1}{Q}\exp\left[\frac{eV}{2kT\epsilon}(1-% Q)\right]
  2. Q = ( 1 + r 2 d 2 ) 1 / 2 Q=\left(1+\frac{r^{2}}{d^{2}}\right)^{1/2}
  3. I ( r , d ) = δ I b R exp ( α d ) \ I(r,d)=\delta I_{b}R\exp(\alpha d)
  4. I ( r , d ) = η I b ( 1 - 1 Q ) \ I(r,d)=\eta I_{b}\left(1-\frac{1}{Q}\right)
  5. I ( r , d ) = η I b p d S [ r d tan - 1 ( r d ) + ln Q ] \ I(r,d)=\eta I_{b}pdS\left[\frac{r}{d}\tan^{-1}\left(\frac{r}{d}\right)+\ln Q\right]
  6. I ( r , d ) = 1 2 η I b p S 0 d ln [ 1 + ( r h ) 2 ] exp { α ( d - h ) } d h \ I(r,d)=\frac{1}{2}\eta I_{b}pS\int\limits_{0}^{d}\ln\left[1+\left(\frac{r}{h% }\right)^{2}\right]\exp\{\alpha(d-h)\}\,dh
  7. i = e υ d i=\frac{e\upsilon}{d}
  8. G = I t o t δ I b = exp [ A p d exp ( - B p d V ) ] \ G=\frac{I_{tot}}{\delta I_{b}}=\exp\left[Apd\exp\left(-B\frac{pd}{V}\right)\right]

Gather-scatter_(vector_addressing).html

  1. i d x idx
  2. x x
  3. y y
  4. x x
  5. x y | x x\leftarrow y|_{x}
  6. x ( i ) = y ( i d x ( i ) ) x(i)=y(idx(i))
  7. y | x x y|_{x}\leftarrow x
  8. x x
  9. y y
  10. y ( i d x ( i ) ) = x ( i ) y(idx(i))=x(i)

Gauge_symmetry_(mathematics).html

  1. L L
  2. E E
  3. L L
  4. L L
  5. E E
  6. J μ J^{\mu}
  7. J μ = W μ + d ν U ν μ J^{\mu}=W^{\mu}+d_{\nu}U^{\nu\mu}
  8. W μ W^{\mu}
  9. U ν μ U^{\nu\mu}

Gauge_theory.html

  1. V V + C V\rightarrow V+C
  2. 𝐄 = - V \mathbf{E}=-\nabla V
  3. 𝐄 = - V - 𝐀 t 𝐁 = × 𝐀 \begin{aligned}\displaystyle\mathbf{E}&\displaystyle=-\nabla V-\frac{\partial% \mathbf{A}}{\partial t}\\ \displaystyle\mathbf{B}&\displaystyle=\nabla\times\mathbf{A}\end{aligned}
  4. V V + C V\rightarrow V+C
  5. 𝐀 𝐀 + f V V - f t \begin{aligned}\displaystyle\mathbf{A}&\displaystyle\rightarrow\mathbf{A}+% \nabla f\\ \displaystyle V&\displaystyle\rightarrow V-\frac{\partial f}{\partial t}\end{aligned}
  6. φ i \varphi_{i}
  7. 𝒮 = d 4 x i = 1 n [ 1 2 μ φ i μ φ i - 1 2 m 2 φ i 2 ] \mathcal{S}=\int\,\mathrm{d}^{4}x\sum_{i=1}^{n}\left[\frac{1}{2}\partial_{\mu}% \varphi_{i}\partial^{\mu}\varphi_{i}-\frac{1}{2}m^{2}\varphi_{i}^{2}\right]
  8. = 1 2 ( μ Φ ) T μ Φ - 1 2 m 2 Φ T Φ \ \mathcal{L}=\frac{1}{2}(\partial_{\mu}\Phi)^{T}\partial^{\mu}\Phi-\frac{1}{2% }m^{2}\Phi^{T}\Phi
  9. Φ = ( φ 1 , φ 2 , , φ n ) T \ \Phi=(\varphi_{1},\varphi_{2},\ldots,\varphi_{n})^{T}
  10. μ \partial_{\mu}
  11. Φ \Phi
  12. Φ Φ = G Φ \ \Phi\mapsto\Phi^{\prime}=G\Phi
  13. Φ \Phi
  14. Φ \Phi
  15. ( μ Φ ) ( μ Φ ) = G μ Φ \ (\partial_{\mu}\Phi)\mapsto(\partial_{\mu}\Phi)^{\prime}=G\partial_{\mu}\Phi
  16. J μ a = i μ Φ T T a Φ \ J^{a}_{\mu}=i\partial_{\mu}\Phi^{T}T^{a}\Phi
  17. μ ( G Φ ) G ( μ Φ ) \ \partial_{\mu}(G\Phi)\neq G(\partial_{\mu}\Phi)
  18. Φ \Phi
  19. Φ \Phi
  20. ( D μ Φ ) = G D μ Φ \ (D_{\mu}\Phi)^{\prime}=GD_{\mu}\Phi
  21. D μ = μ + i g A μ \ D_{\mu}=\partial_{\mu}+igA_{\mu}
  22. A μ = G A μ G - 1 + i g ( μ G ) G - 1 \ A^{\prime}_{\mu}=GA_{\mu}G^{-1}+\frac{i}{g}(\partial_{\mu}G)G^{-1}
  23. A μ = a A μ a T a \ A_{\mu}=\sum_{a}A_{\mu}^{a}T^{a}
  24. loc = 1 2 ( D μ Φ ) T D μ Φ - 1 2 m 2 Φ T Φ \ \mathcal{L}_{\mathrm{loc}}=\frac{1}{2}(D_{\mu}\Phi)^{T}D^{\mu}\Phi-\frac{1}{% 2}m^{2}\Phi^{T}\Phi
  25. Φ \Phi
  26. A A
  27. int = i g 2 Φ T A μ T μ Φ + i g 2 ( μ Φ ) T A μ Φ - g 2 2 ( A μ Φ ) T A μ Φ \ \mathcal{L}_{\mathrm{int}}=i\frac{g}{2}\Phi^{T}A_{\mu}^{T}\partial^{\mu}\Phi% +i\frac{g}{2}(\partial_{\mu}\Phi)^{T}A^{\mu}\Phi-\frac{g^{2}}{2}(A_{\mu}\Phi)^% {T}A^{\mu}\Phi
  28. gf \mathcal{L}_{\mathrm{gf}}
  29. A ( x ) A(x)
  30. gf = - 1 2 Tr ( F μ ν F μ ν ) \ \mathcal{L}_{\mathrm{gf}}=-\frac{1}{2}\operatorname{Tr}(F^{\mu\nu}F_{\mu\nu})
  31. F μ ν = 1 i g [ D μ , D ν ] \ F_{\mu\nu}=\frac{1}{ig}[D_{\mu},D_{\nu}]
  32. A A
  33. = loc + gf = global + int + gf \ \mathcal{L}=\mathcal{L}_{\mathrm{loc}}+\mathcal{L}_{\mathrm{gf}}=\mathcal{L}% _{\mathrm{global}}+\mathcal{L}_{\mathrm{int}}+\mathcal{L}_{\mathrm{gf}}
  34. 𝒮 = ψ ¯ ( i c γ μ μ - m c 2 ) ψ d 4 x \mathcal{S}=\int\bar{\psi}(i\hbar c\,\gamma^{\mu}\partial_{\mu}-mc^{2})\psi\,% \mathrm{d}^{4}x
  35. ψ e i θ ψ \psi\mapsto e^{i\theta}\psi
  36. D μ = μ - i e A μ \ D_{\mu}=\partial_{\mu}-i\frac{e}{\hbar}A_{\mu}
  37. int = e ψ ¯ ( x ) γ μ ψ ( x ) A μ ( x ) = J μ ( x ) A μ ( x ) \ \mathcal{L}_{\mathrm{int}}=\frac{e}{\hbar}\bar{\psi}(x)\gamma^{\mu}\psi(x)A_% {\mu}(x)=J^{\mu}(x)A_{\mu}(x)
  38. J μ ( x ) J^{\mu}(x)
  39. A μ ( x ) A_{\mu}(x)
  40. QED = ψ ¯ ( i c γ μ D μ - m c 2 ) ψ - 1 4 μ 0 F μ ν F μ ν \ \mathcal{L}_{\mathrm{QED}}=\bar{\psi}(i\hbar c\,\gamma^{\mu}D_{\mu}-mc^{2})% \psi-\frac{1}{4\mu_{0}}F_{\mu\nu}F^{\mu\nu}
  41. F = d A + A A {F}=\mathrm{d}{A}+{A}\wedge{A}
  42. \wedge
  43. A {A}
  44. T a T^{a}
  45. A {A}
  46. A A {A}\wedge{A}
  47. δ ε A = [ ε , A ] - d ε \delta_{\varepsilon}{A}=[\varepsilon,{A}]-\mathrm{d}\varepsilon
  48. [ , ] [\cdot,\cdot]
  49. δ ε X = ε X \delta_{\varepsilon}X=\varepsilon X
  50. δ ε D X = ε D X \delta_{\varepsilon}DX=\varepsilon DX
  51. D X = def d X + A X DX\ \stackrel{\mathrm{def}}{=}\ \mathrm{d}X+{A}X
  52. δ ε F = ε F \delta_{\varepsilon}{F}=\varepsilon{F}
  53. F {F}
  54. 1 4 g 2 Tr [ * F F ] \frac{1}{4g^{2}}\int\operatorname{Tr}[*F\wedge F]
  55. χ ( ρ ) ( 𝒫 { e γ A } ) \chi^{(\rho)}\left(\mathcal{P}\left\{e^{\int_{\gamma}A}\right\}\right)
  56. 𝒫 \mathcal{P}
  57. A μ ( x ) A μ ( x ) = A μ ( x ) + μ f ( x ) A_{\mu}(x)\rightarrow A^{\prime}_{\mu}(x)=A_{\mu}(x)+\partial_{\mu}f(x)
  58. A μ ( x ) = μ f ( x ) A^{\prime}_{\mu}(x)=\partial_{\mu}f(x)
  59. f ( x ) f(x)

Gauss's_inequality.html

  1. Pr ( X - m > k ) { ( 2 τ 3 k ) 2 if k 2 τ 3 1 - k τ 3 if 0 k 2 τ 3 . \Pr(\mid X-m\mid>k)\leq\begin{cases}\left(\frac{2\tau}{3k}\right)^{2}&\,\text{% if }k\geq\frac{2\tau}{\sqrt{3}}\\ 1-\frac{k}{\tau\sqrt{3}}&\,\text{if }0\leq k\leq\frac{2\tau}{\sqrt{3}}.\end{cases}

Gauss_iterated_map.html

  1. x n + 1 = exp ( - α x n 2 ) + β , x_{n+1}=\exp(-\alpha x^{2}_{n})+\beta,\,
  2. x n x_{n}
  3. α = 4.90 \alpha=4.90
  4. β \beta
  5. α = 6.20 \alpha=6.20
  6. β \beta

Gauss_map_(disambiguation).html

  1. h ( x ) = 1 / x - 1 / x , h(x)=1/x-\lfloor 1/x\rfloor,

Gaussian_q-distribution.html

  1. s q ( x ) = { 0 if x < - ν 1 c ( q ) E q 2 - q 2 x 2 [ 2 ] q if - ν x ν 0 if x > ν . s_{q}(x)=\begin{cases}0&\,\text{if }x<-\nu\\ \frac{1}{c(q)}E_{q^{2}}^{\frac{-q^{2}x^{2}}{[2]_{q}}}&\,\text{if }-\nu\leq x% \leq\nu\\ 0&\mbox{if }~{}x>\nu.\end{cases}
  2. ν = ν ( q ) = 1 1 - q , \nu=\nu(q)=\frac{1}{\sqrt{1-q}},
  3. c ( q ) = 2 ( 1 - q ) 1 / 2 m = 0 ( - 1 ) m q m ( m + 1 ) ( 1 - q 2 m + 1 ) ( 1 - q 2 ) q 2 m . c(q)=2(1-q)^{1/2}\sum_{m=0}^{\infty}\frac{(-1)^{m}q^{m(m+1)}}{(1-q^{2m+1})(1-q% ^{2})_{q^{2}}^{m}}.
  4. t t
  5. [ t ] q = q t - 1 q - 1 . [t]_{q}=\frac{q^{t}-1}{q-1}.
  6. E q x = j = 0 q j ( j - 1 ) / 2 x j [ j ] ! E_{q}^{x}=\sum_{j=0}^{\infty}q^{j(j-1)/2}\frac{x^{j}}{[j]!}
  7. [ n ] q ! = [ n ] q [ n - 1 ] q [ 2 ] q [n]_{q}!=[n]_{q}[n-1]_{q}\cdots[2]_{q}\,
  8. G q ( x ) = { 0 if x < - ν 1 c ( q ) - ν x E q 2 - q 2 t 2 / [ 2 ] d q t if - ν x ν 1 if x > ν G_{q}(x)=\begin{cases}0&\,\text{if }x<-\nu\\ \displaystyle\frac{1}{c(q)}\int_{-\nu}^{x}E_{q^{2}}^{-q^{2}t^{2}/[2]}\,d_{q}t&% \,\text{if }-\nu\leq x\leq\nu\\ 1&\,\text{if }x>\nu\end{cases}
  9. G q ( x ) = { 0 if x < - ν , 1 2 + 1 - q c ( q ) n = 0 q n ( n + 1 ) ( q - 1 ) n ( 1 - q 2 n + 1 ) ( 1 - q 2 ) q 2 n x 2 n + 1 if - ν x ν 1 if x > ν G_{q}(x)=\begin{cases}0&\,\text{if }x<-\nu,\\ \displaystyle\frac{1}{2}+\frac{1-q}{c(q)}\sum_{n=0}^{\infty}\frac{q^{n(n+1)}(q% -1)^{n}}{(1-q^{2n+1})(1-q^{2})_{q^{2}}^{n}}x^{2n+1}&\,\text{if }-\nu\leq x\leq% \nu\\ 1&\,\text{if}\ x>\nu\end{cases}
  10. ( a + b ) q n = i = 0 n - 1 ( a + q i b ) . (a+b)_{q}^{n}=\prod_{i=0}^{n-1}(a+q^{i}b).
  11. 1 c ( q ) - ν ν E q 2 - q 2 x 2 / [ 2 ] x 2 n d q x = [ 2 n - 1 ] ! ! , \frac{1}{c(q)}\int_{-\nu}^{\nu}E_{q^{2}}^{-q^{2}x^{2}/[2]}\,x^{2n}\,d_{q}x=[2n% -1]!!,
  12. 1 c ( q ) - ν ν E q 2 - q 2 x 2 / [ 2 ] x 2 n + 1 d q x = 0 , \frac{1}{c(q)}\int_{-\nu}^{\nu}E_{q^{2}}^{-q^{2}x^{2}/[2]}\,x^{2n+1}\,d_{q}x=0,
  13. [ 2 n - 1 ] [ 2 n - 3 ] [ 1 ] = [ 2 n - 1 ] ! ! . [2n-1][2n-3]\cdots[1]=[2n-1]!!.\,

Gábor_Tardos.html

  1. \mathcal{H}
  2. τ ( ) 2 ν ( ) \tau(\mathcal{H})\leq 2\nu(\mathcal{H})
  3. τ ( ) \tau(\mathcal{H})
  4. \mathcal{H}
  5. ν ( ) \nu(\mathcal{H})
  6. \mathcal{H}

Gell-Mann–Okubo_mass_formula.html

  1. M = a 0 + a 1 S + a 2 [ I ( I + 1 ) - 1 4 S 2 ] , M=a_{0}+a_{1}S+a_{2}\left[I\left(I+1\right)-\frac{1}{4}S^{2}\right],
  2. 1 / 2 {1}/{2}
  3. 1 / 2 {1}/{2}
  4. 1 / 2 {1}/{2}
  5. 1 / 2 {1}/{2}
  6. N + Ξ 2 = 3 Λ + Σ 4 \frac{N+\Xi}{2}=\frac{3\Lambda+\Sigma}{4}\,
  7. N + Ξ 2 = 1128.5 MeV / c 2 \frac{N+\Xi}{2}=1128.5~{}\mathrm{MeV}/c^{2}
  8. 3 Λ + Σ 4 = 1135.25 MeV / c 2 \frac{3\Lambda+\Sigma}{4}=1135.25~{}\mathrm{MeV}/c^{2}
  9. Δ - Σ * = Σ * - Ξ * = Ξ * - Ω = a 1 - 2 a 2 147 MeV / c 2 \Delta-\Sigma^{*}=\Sigma^{*}-\Xi^{*}=\Xi^{*}-\Omega=a_{1}-2a_{2}\approx\,147~{% }\mathrm{MeV}/c^{2}
  10. K - + K ¯ 0 2 + K + + K 0 2 2 = 3 η + π 4 \frac{\frac{K^{-}+\bar{K}^{0}}{2}+\frac{K^{+}+K^{0}}{2}}{2}=\frac{3\eta+\pi}{4}
  11. K - + K ¯ 0 2 + K + + K 0 2 2 = 248 MeV / c 2 \frac{\frac{K^{-}+\bar{K}^{0}}{2}+\frac{K^{+}+K^{0}}{2}}{2}=248~{}\mathrm{MeV}% /c^{2}
  12. 3 η + π 4 = 445 MeV / c 2 \frac{3\eta+\pi}{4}=445~{}\mathrm{MeV}/c^{2}
  13. ( K - + K ¯ 0 2 ) 2 + ( K + + K 0 2 ) 2 2 = 3 η 2 + π 2 4 \frac{\left(\frac{K^{-}+\bar{K}^{0}}{2}\right)^{2}+\left(\frac{K^{+}+K^{0}}{2}% \right)^{2}}{2}=\frac{3\eta^{2}+\pi^{2}}{4}
  14. ( K - + K ¯ 0 2 ) 2 + ( K + + K 0 2 ) 2 2 = 246 × 10 3 MeV 2 / c 4 \frac{\left(\frac{K^{-}+\bar{K}^{0}}{2}\right)^{2}+\left(\frac{K^{+}+K^{0}}{2}% \right)^{2}}{2}=246\times 10^{3}~{}\mathrm{MeV^{2}}/c^{4}
  15. 3 η 2 + π 2 4 = 230 × 10 3 MeV 2 / c 4 \frac{3\eta^{2}+\pi^{2}}{4}=230\times 10^{3}~{}\mathrm{MeV^{2}}/c^{4}

General_Dirichlet_series.html

  1. n = 1 a n e - λ n s , \sum_{n=1}^{\infty}a_{n}e^{-\lambda_{n}s},
  2. a n a_{n}
  3. s s
  4. { λ n } \{\lambda_{n}\}
  5. n = 1 a n n s , \sum_{n=1}^{\infty}\frac{a_{n}}{n^{s}},
  6. λ n = log n \lambda_{n}=\log n
  7. n = 1 a n ( e - s ) n , \sum_{n=1}^{\infty}a_{n}(e^{-s})^{n},
  8. λ n = n \lambda_{n}=n
  9. s 0 = σ 0 + t 0 i s_{0}=\sigma_{0}+t_{0}i
  10. | arg ( s - s 0 ) | θ < π 2 , |\,\text{arg}(s-s_{0})|\leq\theta<\frac{\pi}{2},
  11. s = σ + t i s=\sigma+ti
  12. σ > σ 0 \sigma>\sigma_{0}
  13. σ c \sigma_{c}
  14. σ > σ c \sigma>\sigma_{c}
  15. σ < σ c \sigma<\sigma_{c}
  16. σ c = \sigma_{c}=\infty
  17. σ c = - \sigma_{c}=-\infty
  18. σ c \sigma_{c}
  19. σ c = inf { σ : n = 1 a n e - λ n s converges for any s where Re ( s ) > σ } \sigma_{c}=\inf\{\sigma\in\mathbb{R}:\sum_{n=1}^{\infty}a_{n}e^{-\lambda_{n}s}% \,\text{ converges for any }s\,\text{ where Re}(s)>\sigma\}
  20. σ = σ c \sigma=\sigma_{c}
  21. σ c = { s : Re ( s ) > σ c } . \mathbb{C}_{\sigma_{c}}=\{s\in\mathbb{C}:\,\text{Re}(s)>\sigma_{c}\}.
  22. n = 1 1 n e - n s , \sum_{n=1}^{\infty}\frac{1}{n}e^{-ns},
  23. s = - π i s=-\pi i
  24. s = 0 s=0
  25. σ = 0 \sigma=0
  26. s = 0 s=0
  27. σ c 0 \sigma_{c}\geq 0
  28. a n \sum a_{n}
  29. s = 0 s=0
  30. σ c 0 \sigma_{c}\leq 0
  31. a n \sum a_{n}
  32. σ c \sigma_{c}
  33. a n \sum a_{n}
  34. a k \sum a_{k}
  35. σ c 0 \sigma_{c}\geq 0
  36. σ c \sigma_{c}
  37. σ c = lim sup n log | a 1 + a 2 + + a n | λ n . \sigma_{c}=\limsup_{n\to\infty}\frac{\log|a_{1}+a_{2}+\cdots+a_{n}|}{\lambda_{% n}}.
  38. a k \sum a_{k}
  39. σ c 0 \sigma_{c}\leq 0
  40. σ c \sigma_{c}
  41. σ c = lim sup n log | a n + 1 + a n + 2 + | λ n . \sigma_{c}=\limsup_{n\to\infty}\frac{\log|a_{n+1}+a_{n+2}+\cdots|}{\lambda_{n}}.
  42. n = 1 | a n e - λ n s | , \sum_{n=1}^{\infty}|a_{n}e^{-\lambda_{n}s}|,
  43. s 0 s_{0}
  44. Re ( s ) > Re ( s 0 ) \,\text{Re}(s)>\,\text{Re}(s_{0})
  45. σ a \sigma_{a}
  46. σ > σ a \sigma>\sigma_{a}
  47. σ < σ a \sigma<\sigma_{a}
  48. σ a \sigma_{a}
  49. σ a = inf { σ : n = 1 a n e - λ n s converges absolutely for any s where Re ( s ) > σ } \sigma_{a}=\inf\{\sigma\in\mathbb{R}:\sum_{n=1}^{\infty}a_{n}e^{-\lambda_{n}s}% \,\text{ converges absolutely for any }s\,\text{ where Re}(s)>\sigma\}
  50. σ a \sigma_{a}
  51. | a k | \sum|a_{k}|
  52. σ a \sigma_{a}
  53. σ a = lim sup n log ( | a 1 | + | a 2 | + + | a n | ) λ n . \sigma_{a}=\limsup_{n\to\infty}\frac{\log(|a_{1}|+|a_{2}|+\cdots+|a_{n}|)}{% \lambda_{n}}.
  54. | a k | \sum|a_{k}|
  55. σ a \sigma_{a}
  56. σ a = lim sup n log ( | a n + 1 | + | a n + 2 | + ) λ n . \sigma_{a}=\limsup_{n\to\infty}\frac{\log(|a_{n+1}|+|a_{n+2}|+\cdots)}{\lambda% _{n}}.
  57. 0 σ a - σ c L := lim sup n log n λ n . 0\leq\sigma_{a}-\sigma_{c}\leq L:=\limsup_{n\to\infty}\frac{\log n}{\lambda_{n% }}.
  58. σ c = σ a = lim sup n log | a n | λ n . \sigma_{c}=\sigma_{a}=\limsup_{n\to\infty}\frac{\log|a_{n}|}{\lambda_{n}}.
  59. λ n = log n \lambda_{n}=\log n
  60. f ( s ) = n = 1 a n e - λ n s , f(s)=\sum_{n=1}^{\infty}a_{n}e^{-\lambda_{n}s},
  61. k = 1 , 2 , 3 , k=1,2,3,...
  62. f ( k ) ( s ) = ( - 1 ) k n = 1 a n λ n k e - λ n s . f^{(k)}(s)=(-1)^{k}\sum_{n=1}^{\infty}a_{n}\lambda_{n}^{k}e^{-\lambda_{n}s}.
  63. λ n k \lambda_{n}\in\mathbb{R}^{k}
  64. λ n m \lambda_{n}\in\mathbb{C}^{m}

Generalized_additive_model_for_location,_scale_and_shape.html

  1. y y
  2. y i y_{i}
  3. i = 1 , 2 , , n i=1,2,\dots,n
  4. f ( y i | μ i , σ i , ν i , τ i ) f(y_{i}|\mu_{i},\sigma_{i},\nu_{i},\tau_{i})
  5. ( μ i , σ i , ν i , τ i ) (\mu_{i},\sigma_{i},\nu_{i},\tau_{i})
  6. μ i \mu_{i}
  7. σ i \sigma_{i}
  8. g 1 ( μ ) = η 1 = X 1 β 1 + j = 1 J 1 h j 1 ( x j 1 ) \displaystyle g_{1}(\mu)=\eta_{1}=X_{1}\beta_{1}+\sum_{j=1}^{J_{1}}{h}_{j1}(x_% {j1})
  9. η k \eta_{k}
  10. n n
  11. β k T = ( β 1 k , β 2 k , , β J k k ) \beta^{T}_{k}=(\beta_{1k},\beta_{2k},\ldots,\beta_{J^{\prime}_{k}k})
  12. J k J^{\prime}_{k}
  13. X k X_{k}
  14. n × J k n\times J^{\prime}_{k}
  15. h j k h_{jk}
  16. x j k x_{jk}
  17. j = 1 , 2 , , J k j=1,2,\ldots,J_{k}
  18. k = 1 , 2 , 3 , 4 k=1,2,3,4

Generalized_chi-squared_distribution.html

  1. X = ( z + a ) T A ( z + a ) + c T z = ( x + b ) T D ( x + b ) + d T x + e , X=(z+a)^{\mathrm{T}}A(z+a)+c^{\mathrm{T}}z=(x+b)^{\mathrm{T}}D(x+b)+d^{\mathrm% {T}}x+e,
  2. X = i = 1 r λ i Y i + f Z 0 , X=\sum_{i=1}^{r}\lambda_{i}Y_{i}+fZ_{0},
  3. Z i Z_{i}
  4. σ i 2 \sigma_{i}^{2}
  5. Q ~ = i = 1 k | Z i | 2 \tilde{Q}=\sum_{i=1}^{k}|Z_{i}|^{2}
  6. Z i Z_{i}
  7. μ = σ i 2 \mu=\sigma_{i}^{2}
  8. Q ~ \tilde{Q}
  9. μ / 2 \mu/2
  10. 2 / μ 2/\mu
  11. χ 2 ( 2 k ) \chi^{2}(2k)
  12. σ i 2 \sigma_{i}^{2}
  13. Q ~ \tilde{Q}
  14. f ( x ; k , σ 1 2 , , σ k 2 ) = i = 1 k e - x σ i 2 σ i 2 j = 1 , j i k ( 1 - σ j 2 σ i 2 ) for x 0. f(x;k,\sigma_{1}^{2},\ldots,\sigma_{k}^{2})=\sum_{i=1}^{k}\frac{e^{-\frac{x}{% \sigma_{i}^{2}}}}{\sigma_{i}^{2}\prod_{j=1,j\neq i}^{k}(1-\frac{\sigma_{j}^{2}% }{\sigma_{i}^{2}})}\quad\mbox{for }~{}x\geq 0.
  15. σ i 2 \sigma_{i}^{2}
  16. 𝐫 = ( r 1 , r 2 , , r M ) \mathbf{r}=(r_{1},r_{2},\dots,r_{M})
  17. r m r_{m}
  18. σ m 2 . \sigma^{2}_{m}.
  19. χ 2 \chi^{2}
  20. Q ~ = m = 1 M σ m 2 Q m , Q m χ 2 ( 2 r m ) . \tilde{Q}=\sum_{m=1}^{M}\sigma^{2}_{m}Q_{m},\quad Q_{m}\sim\chi^{2}(2r_{m})\,.
  21. Q ~ \tilde{Q}
  22. f ( x ; 𝐫 , σ 1 2 , σ M 2 ) = m = 1 M 1 σ m 2 r m k = 1 M l = 1 r k Ψ k , l , 𝐫 ( r k - l ) ! ( - x ) r k - l e - x σ k 2 , for x 0 , f(x;\mathbf{r},\sigma^{2}_{1},\dots\sigma^{2}_{M})=\prod_{m=1}^{M}\frac{1}{% \sigma^{2r_{m}}_{m}}\sum_{k=1}^{M}\sum_{l=1}^{r_{k}}\frac{\Psi_{k,l,\mathbf{r}% }}{(r_{k}-l)!}(-x)^{r_{k}-l}e^{-\frac{x}{\sigma^{2}_{k}}}\quad\,\text{, for }x% \geq 0,
  23. Ψ k , l , 𝐫 = ( - 1 ) r k - 1 𝐢 Ω k , l j k ( i j + r j - 1 i j ) ( 1 σ j 2 - 1 σ k 2 ) - ( r j + i j ) , \Psi_{k,l,\mathbf{r}}=(-1)^{r_{k}-1}\sum_{\mathbf{i}\in\Omega_{k,l}}\prod_{j% \neq k}\Big(\!\!\!\begin{array}[]{c}i_{j}+r_{j}-1\\ i_{j}\end{array}\!\!\!\Big)\Big(\frac{1}{\sigma^{2}_{j}}\!-\!\frac{1}{\sigma^{% 2}_{k}}\Big)^{-(r_{j}+i_{j})},
  24. 𝐢 = [ i 1 , , i M ] T \mathbf{i}=[i_{1},\ldots,i_{M}]^{T}
  25. Ω k , l \Omega_{k,l}
  26. l - 1 l-1
  27. i k = 0 i_{k}=0
  28. Ω k , l = { [ i 1 , , i m ] m ; j = 1 M i j = l - 1 , i k = 0 , i j 0 , for all j } . \Omega_{k,l}=\Big\{[i_{1},\ldots,i_{m}]\in\mathbb{Z}^{m};\sum_{j=1}^{M}i_{j}\!% =l-1,i_{k}=0,i_{j}\geq 0\,\,\,\text{, for all }j\Big\}.

Generalized_context-free_grammar.html

  1. f ( x 1 , , x m , y 1 , , y n , ) = γ f(\langle x_{1},...,x_{m}\rangle,\langle y_{1},...,y_{n}\rangle,...)=\gamma
  2. γ \gamma
  3. X f ( Y , Z , ) X\to f(Y,Z,...)
  4. Y Y
  5. Z Z
  6. { w w R : w { a , b } * } \{ww^{R}:w\in\{a,b\}^{*}\}
  7. w R w^{R}
  8. w w
  9. S ϵ | a S a | b S b S\to\epsilon~{}|~{}aSa~{}|~{}bSb
  10. c o n c ( x , y , z ) = x y z conc(\langle x\rangle,\langle y\rangle,\langle z\rangle)=\langle xyz\rangle
  11. S c o n c ( ϵ , ϵ , ϵ ) | c o n c ( a , S , a ) | c o n c ( b , S , b ) S\to conc(\langle\epsilon\rangle,\langle\epsilon\rangle,\langle\epsilon\rangle% )~{}|~{}conc(\langle a\rangle,S,\langle a\rangle)~{}|~{}conc(\langle b\rangle,% S,\langle b\rangle)
  12. S c o n c ( a , S , a ) S\to conc(\langle a\rangle,S,\langle a\rangle)
  13. c o n c ( a , c o n c ( b , S , b ) , a ) conc(\langle a\rangle,conc(\langle b\rangle,S,\langle b\rangle),\langle a\rangle)
  14. c o n c ( a , c o n c ( b , c o n c ( b , S , b ) , b ) , a ) conc(\langle a\rangle,conc(\langle b\rangle,conc(\langle b\rangle,S,\langle b% \rangle),\langle b\rangle),\langle a\rangle)
  15. c o n c ( a , c o n c ( b , c o n c ( b , c o n c ( ϵ , ϵ , ϵ ) , b ) , b ) , a ) conc(\langle a\rangle,conc(\langle b\rangle,conc(\langle b\rangle,conc(\langle% \epsilon\rangle,\langle\epsilon\rangle,\langle\epsilon\rangle),\langle b% \rangle),\langle b\rangle),\langle a\rangle)
  16. c o n c ( a , c o n c ( b , c o n c ( b , ϵ , b ) , b ) , a ) conc(\langle a\rangle,conc(\langle b\rangle,conc(\langle b\rangle,\langle% \epsilon\rangle,\langle b\rangle),\langle b\rangle),\langle a\rangle)
  17. c o n c ( a , c o n c ( b , b b , b ) , a ) conc(\langle a\rangle,conc(\langle b\rangle,\langle bb\rangle,\langle b\rangle% ),\langle a\rangle)
  18. c o n c ( a , b b b b , a ) conc(\langle a\rangle,\langle bbbb\rangle,\langle a\rangle)
  19. a b b b b a \langle abbbba\rangle
  20. f ( x 1 , , x n ) = f(x_{1},...,x_{n})=...
  21. f ( x ) = g ( x , y ) f(x)=g(x,y)
  22. f ( x ) = g ( x , x ) f(x)=g(x,x)
  23. f ( x 1 , , x n ) = f(x_{1},...,x_{n})=...
  24. f ( x , y ) = g ( y , x ) f(x,y)=g(y,x)
  25. f ( x ) = g ( x , y ) f(x)=g(x,y)
  26. f ( x , y ) = g ( x ) f(x,y)=g(x)

Generalized_gamma_distribution.html

  1. a 2 ( Γ ( ( d + 2 ) / p ) Γ ( d / p ) - ( Γ ( ( d + 1 ) / p ) Γ ( d / p ) ) 2 ) a^{2}\left(\frac{\Gamma((d+2)/p)}{\Gamma(d/p)}-\left(\frac{\Gamma((d+1)/p)}{% \Gamma(d/p)}\right)^{2}\right)
  2. ln a Γ ( d / p ) p + d p + ( 1 a - d p ) ψ ( d p ) \ln\frac{a\Gamma(d/p)}{p}+\frac{d}{p}+\left(\frac{1}{a}-\frac{d}{p}\right)\psi% \left(\frac{d}{p}\right)
  3. a > 0 a>0
  4. d > 0 d>0
  5. p > 0 p>0
  6. f ( x ; a , d , p ) = ( p / a d ) x d - 1 e - ( x / a ) p Γ ( d / p ) , f(x;a,d,p)=\frac{(p/a^{d})x^{d-1}e^{-(x/a)^{p}}}{\Gamma(d/p)},
  7. Γ ( ) \Gamma(\cdot)
  8. F ( x ; a , d , p ) = γ ( d / p , ( x / a ) p ) Γ ( d / p ) , F(x;a,d,p)=\frac{\gamma(d/p,(x/a)^{p})}{\Gamma(d/p)},
  9. γ ( ) \gamma(\cdot)
  10. d = p d=p
  11. p = 1 p=1
  12. E ( X r ) = a r Γ ( d + r p ) Γ ( d p ) . \operatorname{E}(X^{r})=a^{r}\frac{\Gamma(\frac{d+r}{p})}{\Gamma(\frac{d}{p})}.
  13. f 1 f_{1}
  14. f 2 f_{2}
  15. D K L ( f 1 f 2 ) = 0 f 1 ( x ; a 1 , d 1 , p 1 ) ln f 1 ( x ; a 1 , d 1 , p 1 ) f 2 ( x ; a 2 , d 2 , p 2 ) d x = ln p 1 a 2 d 2 Γ ( d 2 / p 2 ) p 2 a 1 d 1 Γ ( d 1 / p 1 ) + [ ψ ( d 1 / p 1 ) p 1 + ln a 1 ] ( d 1 - d 2 ) + Γ ( ( d 1 + p 2 ) / p 1 ) Γ ( d 1 / p 1 ) ( a 1 a 2 ) p 2 - d 1 p 1 \begin{aligned}\displaystyle D_{KL}(f_{1}\parallel f_{2})&\displaystyle=\int_{% 0}^{\infty}f_{1}(x;a_{1},d_{1},p_{1})\,\ln\frac{f_{1}(x;a_{1},d_{1},p_{1})}{f_% {2}(x;a_{2},d_{2},p_{2})}\,dx\\ &\displaystyle=\ln\frac{p_{1}\,a_{2}^{d_{2}}\,\Gamma\left(d_{2}/p_{2}\right)}{% p_{2}\,a_{1}^{d_{1}}\,\Gamma\left(d_{1}/p_{1}\right)}+\left[\frac{\psi\left(d_% {1}/p_{1}\right)}{p_{1}}+\ln a_{1}\right](d_{1}-d_{2})+\frac{\Gamma\bigl((d_{1% }+p_{2})/p_{1}\bigr)}{\Gamma\left(d_{1}/p_{1}\right)}\left(\frac{a_{1}}{a_{2}}% \right)^{p_{2}}-\frac{d_{1}}{p_{1}}\end{aligned}
  16. ψ ( ) \psi(\cdot)
  17. μ = ln a + ln d - ln p p \mu=\ln a+\frac{\ln d-\ln p}{p}
  18. σ = 1 p d \sigma=\frac{1}{\sqrt{pd}}
  19. Q = p d Q=\sqrt{\frac{p}{d}}

Generalized_lifting.html

  1. f j [ n ] f_{j}[n]
  2. f j o [ n ] f_{j}^{o}[n]
  3. f j e [ n ] f_{j}^{e}[n]
  4. g j + 1 [ n ] g_{j+1}[n]
  5. g j + 1 [ n ] g_{j+1}[n]
  6. f j e [ n ] f_{j}^{e}[n]
  7. f j + 1 [ n ] f_{j+1}[n]
  8. f j e [ n ] f_{j}^{e}[n]
  9. g j + 1 [ n ] g_{j+1}[n]
  10. f j e [ n ] f_{j}^{e}[n]
  11. f j o [ n ] f_{j}^{o}[n]
  12. g j + 1 [ n ] = P ( f j o [ n ] ; f j e [ n ] ) \textstyle g_{j+1}[n]=P(f_{j}^{o}[n];f_{j}^{e}[n])
  13. f j + 1 [ n ] = U ( f j e [ n ] ; f j + 1 [ n ] ) \textstyle f_{j+1}[n]=U(f_{j}^{e}[n];f_{j+1}[n])

Generalized_Lotka–Volterra_equation.html

  1. x 1 , x 2 , x_{1},x_{2},\dots
  2. n n
  3. 𝐱 \mathbf{x}
  4. d x i d t = x i f i ( 𝐱 ) , \frac{dx_{i}}{dt}=x_{i}f_{i}(\mathbf{x}),
  5. 𝐟 \mathbf{f}
  6. 𝐟 = 𝐫 + A 𝐱 , \mathbf{f}=\mathbf{r}+A\mathbf{x},
  7. 𝐫 \mathbf{r}
  8. 𝐫 \mathbf{r}
  9. r i r_{i}
  10. a i j a_{ij}
  11. a i j a_{ij}
  12. a i j a_{ij}
  13. a j i a_{ji}
  14. a i j a_{ij}
  15. a j i a_{ji}
  16. a i j a_{ij}
  17. a j i a_{ji}
  18. a i i a_{ii}
  19. d x i / d t dx_{i}/dt
  20. x i 0 x_{i}\neq 0
  21. i i
  22. 𝐱 = - A - 1 𝐫 . \mathbf{x}=-A^{-1}\mathbf{r}.
  23. x i x_{i}
  24. 𝐱 = ( 0 , 0 , 0 ) \mathbf{x}=(0,0,\dots 0)
  25. n = 2 n=2

Generalized_normal_distribution.html

  1. β = 2 \textstyle\beta=2
  2. μ \textstyle\mu
  3. α 2 2 \textstyle\frac{\alpha^{2}}{2}
  4. β = 1 \textstyle\beta=1
  5. β \textstyle\beta\rightarrow\infty
  6. ( μ - α , μ + α ) \textstyle(\mu-\alpha,\mu+\alpha)
  7. β < 2 \beta<2
  8. β > 2 \beta>2
  9. β = 2 \textstyle\beta=2
  10. β = \textstyle\beta=\infty
  11. β = 1 \textstyle\beta=1
  12. β = 2 \textstyle\beta=2
  13. β \textstyle\beta
  14. β \textstyle\lfloor\beta\rfloor
  15. β \beta
  16. β 2 \textstyle\beta\geq 2
  17. μ \mu
  18. m 1 m_{1}
  19. β \textstyle\beta
  20. β = β 0 \textstyle\beta=\textstyle\beta_{0}
  21. β 0 = m 1 m 2 , \beta_{0}=\frac{m_{1}}{\sqrt{m_{2}}},
  22. m 1 = 1 N i = 1 N | x i | , m_{1}={1\over N}\sum_{i=1}^{N}|x_{i}|,
  23. m 2 m_{2}
  24. β i + 1 = β i - g ( β i ) g ( β i ) , \beta_{i+1}=\beta_{i}-\frac{g(\beta_{i})}{g^{\prime}(\beta_{i})},
  25. g ( β ) = 1 + ψ ( 1 / β ) β - i = 1 N | x i - μ | β log | x i - μ | i = 1 N | x i - μ | β + log ( β N i = 1 N | x i - μ | β ) β , g(\beta)=1+\frac{\psi(1/\beta)}{\beta}-\frac{\sum_{i=1}^{N}|x_{i}-\mu|^{\beta}% \log|x_{i}-\mu|}{\sum_{i=1}^{N}|x_{i}-\mu|^{\beta}}+\frac{\log(\frac{\beta}{N}% \sum_{i=1}^{N}|x_{i}-\mu|^{\beta})}{\beta},
  26. g ( β ) = - ψ ( 1 / β ) β 2 - ψ ( 1 / β ) β 3 + 1 β 2 - i = 1 N | x i - μ | β ( log | x i - μ | ) 2 i = 1 N | x i - μ | β + ( i = 1 N | x i - μ | β log | x i - μ | ) 2 ( i = 1 N | x i - μ | β ) 2 + i = 1 N | x i - μ | β log | x i - μ | β i = 1 N | x i - μ | β - log ( β N i = 1 N | x i - μ | β ) β 2 , g^{\prime}(\beta)=-\frac{\psi(1/\beta)}{\beta^{2}}-\frac{\psi^{\prime}(1/\beta% )}{\beta^{3}}+\frac{1}{\beta^{2}}-\frac{\sum_{i=1}^{N}|x_{i}-\mu|^{\beta}(\log% |x_{i}-\mu|)^{2}}{\sum_{i=1}^{N}|x_{i}-\mu|^{\beta}}+\frac{(\sum_{i=1}^{N}|x_{% i}-\mu|^{\beta}\log|x_{i}-\mu|)^{2}}{(\sum_{i=1}^{N}|x_{i}-\mu|^{\beta})^{2}}+% \frac{\sum_{i=1}^{N}|x_{i}-\mu|^{\beta}\log|x_{i}-\mu|}{\beta\sum_{i=1}^{N}|x_% {i}-\mu|^{\beta}}-\frac{\log(\frac{\beta}{N}\sum_{i=1}^{N}|x_{i}-\mu|^{\beta})% }{\beta^{2}},
  27. ψ ( ) \psi()
  28. ψ ( ) \psi^{\prime}()
  29. β \textstyle\beta
  30. μ \mu
  31. m i n μ = i = 1 N | x i - μ | β min_{\mu}=\sum_{i=1}^{N}|x_{i}-\mu|^{\beta}
  32. α \textstyle\alpha
  33. α = ( β N i = 1 N | x i - μ | β ) 1 β . \alpha=(\frac{\beta}{N}\sum_{i=1}^{N}|x_{i}-\mu|^{\beta})^{\frac{1}{\beta}}.
  34. n n
  35. β \beta
  36. α \alpha
  37. p ( 𝐱 ) = g ( 𝐱 β ) p(\mathbf{x})=g(\|\mathbf{x}\|_{\beta})
  38. e 4 κ 2 + 2 e 3 κ 2 + 3 e 2 κ 2 - 6 e^{4\kappa^{2}}+2e^{3\kappa^{2}}+3e^{2\kappa^{2}}-6

Generalized_p-value.html

  1. μ \mu
  2. σ 2 \sigma^{2}
  3. X ¯ \overline{X}
  4. S 2 S^{2}
  5. Z = n ( X ¯ - μ ) / σ N ( 0 , 1 ) Z=\sqrt{n}(\overline{X}-\mu)/\sigma\sim N(0,1)
  6. U = n S 2 / σ 2 χ n - 1 2 . U=nS^{2}/\sigma^{2}\sim\chi^{2}_{n-1}.
  7. ρ = μ / σ \rho=\mu/\sigma
  8. R = x ¯ S s σ - X ¯ - μ σ = x ¯ s U n - Z n , R=\frac{\overline{x}S}{s\sigma}-\frac{\overline{X}-\mu}{\sigma}=\frac{% \overline{x}}{s}\frac{\sqrt{U}}{\sqrt{n}}~{}-~{}\frac{Z}{\sqrt{n}},
  9. x ¯ \overline{x}
  10. X ¯ \overline{X}
  11. s s
  12. S S
  13. R R
  14. H A : ρ < ρ 0 H_{A}:\rho<\rho_{0}
  15. p = P r ( R ρ 0 ) p=Pr(R\geq\rho_{0})

Generalized_second-price_auction.html

  1. n n
  2. k < n k<n
  3. α i \alpha_{i}
  4. α 1 α 2 α k . \alpha_{1}\geq\alpha_{2}\geq\cdots\geq\alpha_{k}.\,
  5. n - k n-k
  6. α i = 0 \alpha_{i}=0
  7. i > k i>k
  8. v i v_{i}
  9. b i b_{i}
  10. v i v_{i}
  11. v 1 v 2 v n , v_{1}\geq v_{2}\geq\cdots\geq v_{n},\,
  12. p i p_{i}
  13. i i
  14. j j
  15. u i = α j ( v i - p i ) u_{i}=\alpha_{j}(v_{i}-p_{i})
  16. j α j v π ( j ) \sum_{j}\alpha_{j}v_{\pi(j)}
  17. π ( j ) \pi(j)
  18. j j
  19. i α i p i \sum_{i}\alpha_{i}p_{i}
  20. i i
  21. i i
  22. p i = b i + 1 p_{i}=b_{i+1}
  23. α 1 = 1 \alpha_{1}=1
  24. α 2 = 0.4 \alpha_{2}=0.4
  25. v 1 = 7 v_{1}=7
  26. v 2 = 6 v_{2}=6
  27. v 3 = 1 v_{3}=1
  28. S W = i α i v π ( i ) SW=\sum_{i}\alpha_{i}v_{\pi(i)}
  29. π ( i ) \pi(i)
  30. i i
  31. π \pi
  32. ( b 1 , , b n ) (b_{1},\dots,b_{n})
  33. 1.282 1.282

Generalized_symmetric_group.html

  1. S ( m , n ) := Z m S n S(m,n):=Z_{m}\wr S_{n}
  2. m = 1 , m=1,
  3. S ( 1 , n ) = S n . S(1,n)=S_{n}.
  4. m = 2 , m=2,
  5. Z 2 { ± 1 } Z_{2}\cong\{\pm 1\}
  6. S ( 2 , n ) S(2,n)
  7. S ( m , n ) S(m,n)
  8. Z m μ m . Z_{m}\cong\mu_{m}.
  9. Z m × Z 2 Z_{m}\times Z_{2}
  10. Z 2 m Z_{2m}
  11. Z m Z_{m}
  12. Z m Z_{m}
  13. Z m Z_{m}
  14. Z 2 . Z_{2}.
  15. H 2 ( S ( 2 k + 1 , n ) ) = { 1 n < 4 𝐙 / 2 n 4. H_{2}(S(2k+1,n))=\begin{cases}1&n<4\\ \mathbf{Z}/2&n\geq 4.\end{cases}
  16. H 2 ( S ( 2 k + 2 , n ) ) = { 1 n = 0 , 1 𝐙 / 2 n = 2 ( 𝐙 / 2 ) 2 n = 3 ( 𝐙 / 2 ) 3 n 4. H_{2}(S(2k+2,n))=\begin{cases}1&n=0,1\\ \mathbf{Z}/2&n=2\\ (\mathbf{Z}/2)^{2}&n=3\\ (\mathbf{Z}/2)^{3}&n\geq 4.\end{cases}
  17. H 2 ( S ( 2 k + 1 , n ) ) H 2 ( S ( 1 , n ) ) H_{2}(S(2k+1,n))\approx H_{2}(S(1,n))
  18. H 2 ( S ( 2 k + 2 , n ) ) H 2 ( S ( 2 , n ) ) , H_{2}(S(2k+2,n))\approx H_{2}(S(2,n)),

Generalized_vector_space_model.html

  1. s i m ( d k , q ) = j = 1 n i = 1 n w i , k * w j , q * t i t j i = 1 n w i , k 2 * i = 1 n w i , q 2 sim(d_{k},q)=\frac{\sum_{j=1}^{n}\sum_{i=1}^{n}w_{i,k}*w_{j,q}*t_{i}\cdot t_{j% }}{\sqrt{\sum_{i=1}^{n}w_{i,k}^{2}}*\sqrt{\sum_{i=1}^{n}w_{i,q}^{2}}}
  2. t i t j t_{i}\cdot t_{j}
  3. t i t j t_{i}\cdot t_{j}
  4. t i t j = S R ( ( t i , t j ) , ( s i , s j ) , O ) t_{i}\cdot t_{j}=SR((t_{i},t_{j}),(s_{i},s_{j}),O)
  5. S C M S P E SCM\cdot SPE

Generic-case_complexity.html

  1. σ : I \sigma:I\to\mathbb{N}
  2. B n = { x I σ ( x ) n } B_{n}=\{x\in I\mid\sigma(x)\leq n\}
  3. { μ n } \{\mu_{n}\}
  4. μ n \mu_{n}
  5. B n B_{n}
  6. B n B_{n}
  7. μ n \mu_{n}
  8. B n B_{n}
  9. μ n ( B n ) = 0 \mu_{n}(B_{n})=0
  10. X I X\subset I
  11. ρ ( X ) = lim n μ n ( X B n ) \rho(X)=\lim_{n\to\infty}\mu_{n}(X\cap B_{n})
  12. B n B_{n}
  13. μ n \mu_{n}
  14. ρ ( X ) = lim | X B n | | B n | . \rho(X)=\lim\frac{|X\cap B_{n}|}{|B_{n}|}.
  15. I n = { x I σ ( x ) = n } I_{n}=\{x\in I\mid\sigma(x)=n\}
  16. ρ ( X ) = lim | X I n | | I n | \rho^{\prime}(X)=\lim\frac{|X\cap I_{n}|}{|I_{n}|}
  17. ρ ( X ) \rho(X)
  18. ρ ( X ) \rho^{\prime}(X)
  19. X I X\subseteq I
  20. ρ ( X ) = 1 \rho(X)=1
  21. ρ ( X ) = 0 \rho(X)=0
  22. μ n ( X B n ) \mu_{n}(X\cap B_{n})
  23. f : f:\mathbb{N}\to\mathbb{N}
  24. \mathbb{N}
  25. { 0 , 1 } \{0,1\}
  26. \mathbb{N}
  27. G e n ( f ) G e n ( f 3 ) Gen(f)\subsetneq Gen(f^{3})
  28. 𝒜 \mathcal{A}
  29. T : I T:I\to\mathbb{N}
  30. μ \mu
  31. 𝒜 \mathcal{A}
  32. { 0 , 1 } \{0,1\}
  33. σ ( w ) \sigma(w)
  34. | w | |w|
  35. I n I_{n}
  36. μ n \mu_{n}
  37. I n I_{n}
  38. T ( w ) = 2 2 n T(w)=2^{2^{n}}
  39. σ ( w ) = | w | \sigma(w)=|w|
  40. μ ( w ) = 2 - 2 | w | - 1 \mu(w)=2^{-2|w|-1}
  41. T ( w ) = 2 | w | T(w)=2^{|w|}
  42. f : I + f:I\rightarrow\mathbb{R}^{+}
  43. μ \mu
  44. k 1 k\geq 1
  45. w I n f 1 / k ( w ) μ n ( w ) = O ( n ) \sum_{w\in I_{n}}f^{1/k}(w)\mu_{n}(w)=O(n)
  46. { μ n } \{\mu_{n}\}
  47. μ \mu
  48. μ \mu
  49. μ \mu
  50. f : I + f:I\rightarrow\mathbb{R}^{+}
  51. μ \mu
  52. ρ \rho^{\prime}
  53. 𝒜 \mathcal{A}
  54. \mathcal{B}
  55. { μ n } \{\mu_{n}\}
  56. μ \mu
  57. 𝒜 \mathcal{A}
  58. μ \mu
  59. I n I_{n}