wpmath0000006_5

GENERIC_formalism.html

  1. d x d t = L ( x ) δ E δ x ( x ) + M ( x ) δ S δ x ( x ) . \frac{dx}{dt}=L(x)\cdot\frac{\delta E}{\delta x}(x)+M(x)\cdot\frac{\delta S}{% \delta x}(x).
  2. x x
  3. x x
  4. x x
  5. S S\rightarrow\mathbb{R}
  6. S S
  7. x = ( U , V , T ( r ) ) x=(U,V,T(\vec{r}))
  8. Σ 3 \Sigma\subset\mathbb{R}^{3}
  9. S = { 1 , 2 } Σ S=\{1,2\}\cup\Sigma
  10. E ( x ) E(x)
  11. S ( x ) S(x)
  12. n \mathbb{R}^{n}
  13. \mathbb{R}
  14. x x
  15. δ E / δ x \delta E/\delta x
  16. δ S / δ x \delta S/\delta x
  17. E E
  18. S S
  19. S S\rightarrow\mathbb{R}
  20. L ( x ) L(x)
  21. M ( x ) M(x)
  22. L ( x ) δ S δ x ( x ) = 0 L(x)\cdot\frac{\delta S}{\delta x}(x)=0
  23. M ( x ) δ E δ x ( x ) = 0 M(x)\cdot\frac{\delta E}{\delta x}(x)=0
  24. L L
  25. M M

Genetic_distance.html

  1. X X
  2. Y Y
  3. L L
  4. X u X_{u}
  5. u u
  6. l l
  7. D = - ln l u X u Y u ( l u X u 2 ) ( l u Y u 2 ) \begin{aligned}\displaystyle D=-\ln\frac{\sum\limits_{l}\sum\limits_{u}X_{u}Y_% {u}}{\sqrt{(\sum\limits_{l}\sum\limits_{u}X_{u}^{2})(\sum\limits_{l}\sum% \limits_{u}Y_{u}^{2})}}\end{aligned}
  8. j X j_{X}
  9. X X
  10. j Y j_{Y}
  11. Y Y
  12. j X Y j_{XY}
  13. X X
  14. Y Y
  15. J X J_{X}
  16. J Y J_{Y}
  17. J X Y J_{XY}
  18. j X j_{X}
  19. j Y j_{Y}
  20. j X Y j_{XY}
  21. J X = l u X u 2 L \begin{aligned}\displaystyle J_{X}=\sum\limits_{l}\sum\limits_{u}\frac{{X_{u}}% ^{2}}{L}\end{aligned}
  22. J Y = l u Y u 2 L \begin{aligned}\displaystyle J_{Y}=\sum\limits_{l}\sum\limits_{u}\frac{{Y_{u}}% ^{2}}{L}\end{aligned}
  23. J X Y = l u X u Y u L \begin{aligned}\displaystyle J_{XY}=\sum\limits_{l}\sum\limits_{u}\frac{X_{u}Y% _{u}}{L}\end{aligned}
  24. L L
  25. D = - ln J X Y J X J Y \begin{aligned}\displaystyle D=-\ln{\frac{J_{XY}}{\sqrt{J_{X}J_{Y}}}}\end{aligned}
  26. D C H = 2 π 2 ( 1 - l u X u Y u ) \begin{aligned}\displaystyle D_{CH}=\frac{2}{\pi}\sqrt{2(1-\sum\limits_{l}\sum% \limits_{u}\sqrt{X_{u}Y_{u})}}\end{aligned}
  27. 2 π \frac{2}{\pi}
  28. Θ \Theta
  29. Θ w = l u ( X u - Y u ) 2 2 l ( 1 - u X u Y u ) \begin{aligned}\displaystyle\Theta_{w}=\sqrt{\frac{\sum\limits_{l}\sum\limits_% {u}(X_{u}-Y_{u})^{2}}{2\sum\limits_{l}(1-\sum\limits_{u}X_{u}Y_{u})}}\end{aligned}
  30. D A = 1 - l u X u Y u / L \begin{aligned}\displaystyle D_{A}=1-\sum\limits_{l}\sum\limits_{u}\sqrt{X_{u}% Y_{u}}/{L}\end{aligned}
  31. D E U = u ( X u - Y u ) 2 \begin{aligned}\displaystyle D_{EU}=\sqrt{\sum\limits_{u}(X_{u}-Y_{u})^{2}}% \end{aligned}
  32. μ X \mu_{X}
  33. μ Y \mu_{Y}
  34. ( δ μ ) 2 = l ( μ X - μ Y ) 2 / L \begin{aligned}\displaystyle(\delta\mu)^{2}=\sum\limits_{l}(\mu_{X}-\mu_{Y})^{% 2}/{L}\end{aligned}
  35. D m = ( J X + J Y ) 2 - J X Y \begin{aligned}\displaystyle D_{m}=\frac{(J_{X}+J_{Y})}{2}-J_{XY}\end{aligned}
  36. D R = 1 L u ( X u - Y u ) 2 2 \begin{aligned}\displaystyle D_{R}=\frac{1}{L}\sqrt{\frac{\sum\limits_{u}(X_{u% }-Y_{u})^{2}}{2}}\end{aligned}

Genome_size.html

  1. number of base pairs = mass in pg × 0.978 × 10 9 \,\text{number of base pairs}=\,\text{mass in pg}\times 0.978\times 10^{9}
  2. 1 pg = 978 Mb 1\,\text{pg}=978\,\text{Mb}

Geodesic_deviation.html

  1. T μ = x μ ( s , t ) t . T^{\mu}=\frac{\partial x^{\mu}(s,t)}{\partial t}.
  2. X μ = x μ ( s , t ) s . X^{\mu}=\frac{\partial x^{\mu}(s,t)}{\partial s}.
  3. A μ = T α α ( T β β X μ ) . A^{\mu}=T^{\alpha}\nabla_{\alpha}(T^{\beta}\nabla_{\beta}X^{\mu}).
  4. A μ = R μ ν ρ σ T ν T ρ X σ . A^{\mu}={R^{\mu}}_{\nu\rho\sigma}T^{\nu}T^{\rho}X^{\sigma}.
  5. T α α T^{\alpha}\nabla_{\alpha}
  6. D / d t D/dt
  7. D 2 X μ d t 2 = R μ ν ρ σ T ν T ρ X σ . \frac{D^{2}X^{\mu}}{dt^{2}}={R^{\mu}}_{\nu\rho\sigma}T^{\nu}T^{\rho}X^{\sigma}.

Geodesics_in_general_relativity.html

  1. d 2 x μ d s 2 = - Γ μ d x α d s α β d x β d s . {d^{2}x^{\mu}\over ds^{2}}=-\Gamma^{\mu}{}_{\alpha\beta}{dx^{\alpha}\over ds}{% dx^{\beta}\over ds}\ .
  2. Γ μ α β \Gamma^{\mu}{}_{\alpha\beta}
  3. t x 0 t\equiv x^{0}
  4. d 2 x μ d t 2 = - Γ μ d x α d t α β d x β d t + Γ 0 d x α d t α β d x β d t d x μ d t . {d^{2}x^{\mu}\over dt^{2}}=-\Gamma^{\mu}{}_{\alpha\beta}{dx^{\alpha}\over dt}{% dx^{\beta}\over dt}+\Gamma^{0}{}_{\alpha\beta}{dx^{\alpha}\over dt}{dx^{\beta}% \over dt}{dx^{\mu}\over dt}\ .
  5. d 2 x n d t 2 = - Γ n . 00 {d^{2}x^{n}\over dt^{2}}=-\Gamma^{n}{}_{00}.
  6. X μ X^{\mu}
  7. T X 0 T\equiv X^{0}
  8. d 2 X μ d T 2 = 0. {d^{2}X^{\mu}\over dT^{2}}=0.
  9. d X μ d T = d x ν d T X μ x ν {dX^{\mu}\over dT}={dx^{\nu}\over dT}{\partial X^{\mu}\over\partial x^{\nu}}
  10. d 2 X μ d T 2 = d 2 x ν d T 2 X μ x ν + d x ν d T d x α d T 2 X μ x ν x α {d^{2}X^{\mu}\over dT^{2}}={d^{2}x^{\nu}\over dT^{2}}{\partial X^{\mu}\over% \partial x^{\nu}}+{dx^{\nu}\over dT}{dx^{\alpha}\over dT}{\partial^{2}X^{\mu}% \over\partial x^{\nu}\partial x^{\alpha}}
  11. d 2 x ν d T 2 X μ x ν = - d x ν d T d x α d T 2 X μ x ν x α {d^{2}x^{\nu}\over dT^{2}}{\partial X^{\mu}\over\partial x^{\nu}}=-{dx^{\nu}% \over dT}{dx^{\alpha}\over dT}{\partial^{2}X^{\mu}\over\partial x^{\nu}% \partial x^{\alpha}}
  12. x λ X μ {\partial x^{\lambda}\over\partial X^{\mu}}
  13. d 2 x λ d T 2 = - d x ν d T d x α d T [ 2 X μ x ν x α x λ X μ ] {d^{2}x^{\lambda}\over dT^{2}}=-{dx^{\nu}\over dT}{dx^{\alpha}\over dT}\left[{% \partial^{2}X^{\mu}\over\partial x^{\nu}\partial x^{\alpha}}{\partial x^{% \lambda}\over\partial X^{\mu}}\right]
  14. t x 0 t\equiv x^{0}
  15. d 2 x λ d t 2 = - d x ν d t d x α d t [ 2 X μ x ν x α x λ X μ ] + d x ν d t d x α d t d x λ d t [ 2 X μ x ν x α x 0 X μ ] {d^{2}x^{\lambda}\over dt^{2}}=-{dx^{\nu}\over dt}{dx^{\alpha}\over dt}\left[{% \partial^{2}X^{\mu}\over\partial x^{\nu}\partial x^{\alpha}}{\partial x^{% \lambda}\over\partial X^{\mu}}\right]+{dx^{\nu}\over dt}{dx^{\alpha}\over dt}{% dx^{\lambda}\over dt}\left[{\partial^{2}X^{\mu}\over\partial x^{\nu}\partial x% ^{\alpha}}{\partial x^{0}\over\partial X^{\mu}}\right]
  16. S = d s S=\int ds
  17. d s = - g μ ν ( x ) d x μ d x ν ds=\sqrt{-g_{\mu\nu}(x)dx^{\mu}dx^{\nu}}
  18. λ \lambda
  19. S = - g μ ν d x μ d λ d x ν d λ d λ S=\int\sqrt{-g_{\mu\nu}\frac{dx^{\mu}}{d\lambda}\frac{dx^{\nu}}{d\lambda}}d\lambda
  20. x μ x^{\mu}
  21. 0 = δ S = δ ( - g μ ν d x μ d λ d x ν d λ ) d λ = δ ( - g μ ν d x μ d λ d x ν d λ ) 2 - g μ ν d x μ d λ d x ν d λ d λ 0=\delta S=\int\delta\left(\sqrt{-g_{\mu\nu}\frac{dx^{\mu}}{d\lambda}\frac{dx^% {\nu}}{d\lambda}}\right)d\lambda=\int\frac{\delta\left(-g_{\mu\nu}\frac{dx^{% \mu}}{d\lambda}\frac{dx^{\nu}}{d\lambda}\right)}{2\sqrt{-g_{\mu\nu}\frac{dx^{% \mu}}{d\lambda}\frac{dx^{\nu}}{d\lambda}}}d\lambda
  22. τ \tau
  23. 0 = δ ( g μ ν d x μ d τ d x ν d τ ) d τ 0=\int\delta\left(g_{\mu\nu}\frac{dx^{\mu}}{d\tau}\frac{dx^{\nu}}{d\tau}\right% )d\tau
  24. 0 = ( d x μ d τ d x ν d τ δ g μ ν + g μ ν d δ x μ d τ d x ν d τ + g μ ν d x μ d τ d δ x ν d τ ) d τ = ( d x μ d τ d x ν d τ α g μ ν δ x α + 2 g μ ν d δ x μ d τ d x ν d τ ) d τ 0=\int\left(\frac{dx^{\mu}}{d\tau}\frac{dx^{\nu}}{d\tau}\delta g_{\mu\nu}+g_{% \mu\nu}\frac{d\delta x^{\mu}}{d\tau}\frac{dx^{\nu}}{d\tau}+g_{\mu\nu}\frac{dx^% {\mu}}{d\tau}\frac{d\delta x^{\nu}}{d\tau}\right)d\tau=\int\left(\frac{dx^{\mu% }}{d\tau}\frac{dx^{\nu}}{d\tau}\partial_{\alpha}g_{\mu\nu}\delta x^{\alpha}+2g% _{\mu\nu}\frac{d\delta x^{\mu}}{d\tau}\frac{dx^{\nu}}{d\tau}\right)d\tau
  25. 0 = d τ ( d x μ d τ d x ν d τ α g μ ν δ x α - 2 δ x μ d d τ ( g μ ν d x ν d τ ) ) = d τ ( d x μ d τ d x ν d τ α g μ ν δ x α - 2 δ x μ α g μ ν d x α d τ d x ν d τ - 2 δ x μ g μ ν d 2 x ν d τ 2 ) 0=\int d\tau\left(\frac{dx^{\mu}}{d\tau}\frac{dx^{\nu}}{d\tau}\partial_{\alpha% }g_{\mu\nu}\delta x^{\alpha}-2\delta x^{\mu}\frac{d}{d\tau}\left(g_{\mu\nu}% \frac{dx^{\nu}}{d\tau}\right)\right)=\int d\tau\left(\frac{dx^{\mu}}{d\tau}% \frac{dx^{\nu}}{d\tau}\partial_{\alpha}g_{\mu\nu}\delta x^{\alpha}-2\delta x^{% \mu}\partial_{\alpha}g_{\mu\nu}\frac{dx^{\alpha}}{d\tau}\frac{dx^{\nu}}{d\tau}% -2\delta x^{\mu}g_{\mu\nu}\frac{d^{2}x^{\nu}}{d\tau^{2}}\right)
  26. 0 = d τ δ x μ ( - 2 g μ ν d 2 x ν d τ 2 + d x α d τ d x ν d τ μ g α ν - 2 d x α d τ d x ν d τ α g μ ν ) 0=\int d\tau\delta x^{\mu}\left(-2g_{\mu\nu}\frac{d^{2}x^{\nu}}{d\tau^{2}}+% \frac{dx^{\alpha}}{d\tau}\frac{dx^{\nu}}{d\tau}\partial_{\mu}g_{\alpha\nu}-2% \frac{dx^{\alpha}}{d\tau}\frac{dx^{\nu}}{d\tau}\partial_{\alpha}g_{\mu\nu}\right)
  27. 0 = d τ δ x μ ( - 2 g μ ν d 2 x ν d τ 2 + d x α d τ d x ν d τ μ g α ν - d x α d τ d x ν d τ α g μ ν - d x ν d τ d x α d τ ν g μ α ) 0=\int d\tau\delta x^{\mu}\left(-2g_{\mu\nu}\frac{d^{2}x^{\nu}}{d\tau^{2}}+% \frac{dx^{\alpha}}{d\tau}\frac{dx^{\nu}}{d\tau}\partial_{\mu}g_{\alpha\nu}-% \frac{dx^{\alpha}}{d\tau}\frac{dx^{\nu}}{d\tau}\partial_{\alpha}g_{\mu\nu}-% \frac{dx^{\nu}}{d\tau}\frac{dx^{\alpha}}{d\tau}\partial_{\nu}g_{\mu\alpha}\right)
  28. - 1 2 -\frac{1}{2}
  29. 0 = d τ δ x μ ( g μ ν d 2 x ν d τ 2 + 1 2 d x α d τ d x ν d τ ( α g μ ν + ν g μ α - μ g α ν ) ) 0=\int d\tau\delta x^{\mu}\left(g_{\mu\nu}\frac{d^{2}x^{\nu}}{d\tau^{2}}+\frac% {1}{2}\frac{dx^{\alpha}}{d\tau}\frac{dx^{\nu}}{d\tau}\left(\partial_{\alpha}g_% {\mu\nu}+\partial_{\nu}g_{\mu\alpha}-\partial_{\mu}g_{\alpha\nu}\right)\right)
  30. g μ ν d 2 x ν d τ 2 + 1 2 d x α d τ d x ν d τ ( α g μ ν + ν g μ α - μ g α ν ) = 0 g_{\mu\nu}\frac{d^{2}x^{\nu}}{d\tau^{2}}+\frac{1}{2}\frac{dx^{\alpha}}{d\tau}% \frac{dx^{\nu}}{d\tau}\left(\partial_{\alpha}g_{\mu\nu}+\partial_{\nu}g_{\mu% \alpha}-\partial_{\mu}g_{\alpha\nu}\right)=0
  31. g μ β g^{\mu\beta}
  32. d 2 x β d τ 2 + 1 2 g μ β ( α g μ ν + ν g μ α - μ g α ν ) d x α d τ d x ν d τ = 0 \frac{d^{2}x^{\beta}}{d\tau^{2}}+\frac{1}{2}g^{\mu\beta}\left(\partial_{\alpha% }g_{\mu\nu}+\partial_{\nu}g_{\mu\alpha}-\partial_{\mu}g_{\alpha\nu}\right)% \frac{dx^{\alpha}}{d\tau}\frac{dx^{\nu}}{d\tau}=0
  33. d 2 x β d τ 2 + Γ α ν β d x α d τ d x ν d τ = 0 \frac{d^{2}x^{\beta}}{d\tau^{2}}+\Gamma^{\beta}_{\alpha\nu}\frac{dx^{\alpha}}{% d\tau}\frac{dx^{\nu}}{d\tau}=0
  34. Γ α ν β = 1 2 g μ β ( α g μ ν + ν g μ α - μ g α ν ) \Gamma^{\beta}_{\alpha\nu}=\frac{1}{2}g^{\mu\beta}\left(\partial_{\alpha}g_{% \mu\nu}+\partial_{\nu}g_{\mu\alpha}-\partial_{\mu}g_{\alpha\nu}\right)
  35. d 2 X μ d s 2 = q m F μ β d X α d s η α β . {d^{2}X^{\mu}\over ds^{2}}={q\over m}{F^{\mu\beta}}{dX^{\alpha}\over ds}{\eta_% {\alpha\beta}}.
  36. η α β d X α d s d X β d s = - 1. {\eta_{\alpha\beta}}{dX^{\alpha}\over ds}{dX^{\beta}\over ds}=-1.
  37. η = ( - 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 ) \eta=\begin{pmatrix}-1&0&0&0\\ 0&1&0&0\\ 0&0&1&0\\ 0&0&0&1\end{pmatrix}
  38. d 2 x μ d s 2 = - Γ μ d x α d s α β d x β d s + q m F μ β d x α d s g α β . {d^{2}x^{\mu}\over ds^{2}}=-\Gamma^{\mu}{}_{\alpha\beta}{dx^{\alpha}\over ds}{% dx^{\beta}\over ds}\ +{q\over m}{F^{\mu\beta}}{dx^{\alpha}\over ds}{g_{\alpha% \beta}}.
  39. g α β d x α d s d x β d s = - 1. {g_{\alpha\beta}}{dx^{\alpha}\over ds}{dx^{\beta}\over ds}=-1.
  40. Γ λ = α β 1 2 g λ τ ( g τ α x β + g τ β x α - g α β x τ ) \Gamma^{\lambda}{}_{\alpha\beta}=\frac{1}{2}g^{\lambda\tau}\left(\frac{% \partial g_{\tau\alpha}}{\partial x^{\beta}}+\frac{\partial g_{\tau\beta}}{% \partial x^{\alpha}}-\frac{\partial g_{\alpha\beta}}{\partial x^{\tau}}\right)
  41. l = | g μ ν x ˙ μ x ˙ ν | d s . l=\int\sqrt{\left|g_{\mu\nu}\dot{x}^{\mu}\dot{x}^{\nu}\right|}\,ds\ .
  42. d d s x ˙ α | g μ ν x ˙ μ x ˙ ν | = x α | g μ ν x ˙ μ x ˙ ν | , {d\over ds}{\partial\over\partial\dot{x}^{\alpha}}\sqrt{\left|g_{\mu\nu}\dot{x% }^{\mu}\dot{x}^{\nu}\right|}={\partial\over\partial x^{\alpha}}\sqrt{\left|g_{% \mu\nu}\dot{x}^{\mu}\dot{x}^{\nu}\right|}\ ,
  43. 2 ( Γ λ x ˙ μ μ ν x ˙ ν + x ¨ λ ) = U λ d d s ln | U ν U ν | , 2(\Gamma^{\lambda}{}_{\mu\nu}\dot{x}^{\mu}\dot{x}^{\nu}+\ddot{x}^{\lambda})=U^% {\lambda}{d\over ds}\ln|U_{\nu}U^{\nu}|\ ,
  44. U μ = x ˙ μ . U^{\mu}=\dot{x}^{\mu}.
  45. l = d τ = d τ d ϕ d ϕ = < m t p l > ( d τ ) 2 ( d ϕ ) 2 d ϕ = - g μ ν d x μ d x ν d ϕ d ϕ d ϕ = f d ϕ l=\int d\tau=\int{d\tau\over d\phi}\,d\phi=\int\sqrt{<}mtpl>{{(d\tau)^{2}\over% (d\phi)^{2}}}\,d\phi=\int\sqrt{{-g_{\mu\nu}dx^{\mu}dx^{\nu}\over d\phi\,d\phi}% }\,d\phi=\int f\,d\phi
  46. f = - g μ ν x ˙ μ x ˙ ν f=\sqrt{-g_{\mu\nu}\dot{x}^{\mu}\dot{x}^{\nu}}
  47. d d τ f x ˙ λ = f x λ {d\over d\tau}{\partial f\over\partial\dot{x}^{\lambda}}={\partial f\over% \partial x^{\lambda}}
  48. d d τ - g μ ν x ˙ μ x ˙ ν x ˙ λ = - g μ ν x ˙ μ x ˙ ν x λ {d\over d\tau}{\partial\sqrt{-g_{\mu\nu}\dot{x}^{\mu}\dot{x}^{\nu}}\over% \partial\dot{x}^{\lambda}}={\partial\sqrt{-g_{\mu\nu}\dot{x}^{\mu}\dot{x}^{\nu% }}\over\partial x^{\lambda}}
  49. d d τ ( - g μ ν x ˙ μ x ˙ λ x ˙ ν - g μ ν x ˙ μ x ˙ ν x ˙ λ 2 - g μ ν x ˙ μ x ˙ ν ) = - g μ ν , λ x ˙ μ x ˙ ν 2 - g μ ν x ˙ μ x ˙ ν ( 1 ) {d\over d\tau}\left({-g_{\mu\nu}{\partial\dot{x}^{\mu}\over\partial\dot{x}^{% \lambda}}\dot{x}^{\nu}-g_{\mu\nu}\dot{x}^{\mu}{\partial\dot{x}^{\nu}\over% \partial\dot{x}^{\lambda}}\over 2\sqrt{-g_{\mu\nu}\dot{x}^{\mu}\dot{x}^{\nu}}}% \right)={-g_{\mu\nu,\lambda}\dot{x}^{\mu}\dot{x}^{\nu}\over 2\sqrt{-g_{\mu\nu}% \dot{x}^{\mu}\dot{x}^{\nu}}}\qquad\qquad(1)
  50. d d τ ( g μ ν δ μ x ˙ ν λ + g μ ν x ˙ μ δ ν λ 2 - g μ ν x ˙ μ x ˙ ν ) = g μ ν , λ x ˙ μ x ˙ ν 2 - g μ ν x ˙ μ x ˙ ν ( 2 ) {d\over d\tau}\left({g_{\mu\nu}\delta^{\mu}{}_{\lambda}\dot{x}^{\nu}+g_{\mu\nu% }\dot{x}^{\mu}\delta^{\nu}{}_{\lambda}\over 2\sqrt{-g_{\mu\nu}\dot{x}^{\mu}% \dot{x}^{\nu}}}\right)={g_{\mu\nu,\lambda}\dot{x}^{\mu}\dot{x}^{\nu}\over 2% \sqrt{-g_{\mu\nu}\dot{x}^{\mu}\dot{x}^{\nu}}}\qquad\qquad(2)
  51. d d τ ( g λ ν x ˙ ν + g μ λ x ˙ μ - g μ ν x ˙ μ x ˙ ν ) = g μ ν , λ x ˙ μ x ˙ ν - g μ ν x ˙ μ x ˙ ν ( 3 ) {d\over d\tau}\left({g_{\lambda\nu}\dot{x}^{\nu}+g_{\mu\lambda}\dot{x}^{\mu}% \over\sqrt{-g_{\mu\nu}\dot{x}^{\mu}\dot{x}^{\nu}}}\right)={g_{\mu\nu,\lambda}% \dot{x}^{\mu}\dot{x}^{\nu}\over\sqrt{-g_{\mu\nu}\dot{x}^{\mu}\dot{x}^{\nu}}}% \qquad\qquad(3)
  52. - g μ ν x ˙ μ x ˙ ν d d τ ( g λ ν x ˙ ν + g μ λ x ˙ μ ) - ( g λ ν x ˙ ν + g μ λ x ˙ μ ) d d τ - g μ ν x ˙ μ x ˙ ν - g μ ν x ˙ μ x ˙ ν = g μ ν , λ x ˙ μ x ˙ ν - g μ ν x ˙ μ x ˙ ν ( 4 ) {\sqrt{-g_{\mu\nu}\dot{x}^{\mu}\dot{x}^{\nu}}{d\over d\tau}(g_{\lambda\nu}\dot% {x}^{\nu}+g_{\mu\lambda}\dot{x}^{\mu})-(g_{\lambda\nu}\dot{x}^{\nu}+g_{\mu% \lambda}\dot{x}^{\mu}){d\over d\tau}\sqrt{-g_{\mu\nu}\dot{x}^{\mu}\dot{x}^{\nu% }}\over-g_{\mu\nu}\dot{x}^{\mu}\dot{x}^{\nu}}={g_{\mu\nu,\lambda}\dot{x}^{\mu}% \dot{x}^{\nu}\over\sqrt{-g_{\mu\nu}\dot{x}^{\mu}\dot{x}^{\nu}}}\qquad\qquad(4)
  53. ( - g μ ν x ˙ μ x ˙ ν ) d d τ ( g λ ν x ˙ ν + g μ λ x ˙ μ ) + 1 2 ( g λ ν x ˙ ν + g μ λ x ˙ μ ) d d τ ( g μ ν x ˙ μ x ˙ ν ) - g μ ν x ˙ μ x ˙ ν = g μ ν , λ x ˙ μ x ˙ ν ( 5 ) {(-g_{\mu\nu}\dot{x}^{\mu}\dot{x}^{\nu}){d\over d\tau}(g_{\lambda\nu}\dot{x}^{% \nu}+g_{\mu\lambda}\dot{x}^{\mu})+{1\over 2}(g_{\lambda\nu}\dot{x}^{\nu}+g_{% \mu\lambda}\dot{x}^{\mu}){d\over d\tau}(g_{\mu\nu}\dot{x}^{\mu}\dot{x}^{\nu})% \over-g_{\mu\nu}\dot{x}^{\mu}\dot{x}^{\nu}}=g_{\mu\nu,\lambda}\dot{x}^{\mu}% \dot{x}^{\nu}\qquad\qquad(5)
  54. ( g μ ν x ˙ μ x ˙ ν ) ( g λ ν , μ x ˙ ν x ˙ μ + g μ λ , ν x ˙ μ x ˙ ν + g λ ν x ¨ ν + g λ μ x ¨ μ ) (g_{\mu\nu}\dot{x}^{\mu}\dot{x}^{\nu})(g_{\lambda\nu,\mu}\dot{x}^{\nu}\dot{x}^% {\mu}+g_{\mu\lambda,\nu}\dot{x}^{\mu}\dot{x}^{\nu}+g_{\lambda\nu}\ddot{x}^{\nu% }+g_{\lambda\mu}\ddot{x}^{\mu})
  55. = ( g μ ν , λ x ˙ μ x ˙ ν ) ( g α β x ˙ α x ˙ β ) + 1 2 ( g λ ν x ˙ ν + g λ μ x ˙ μ ) d d τ ( g μ ν x ˙ μ x ˙ ν ) ( 6 ) =(g_{\mu\nu,\lambda}\dot{x}^{\mu}\dot{x}^{\nu})(g_{\alpha\beta}\dot{x}^{\alpha% }\dot{x}^{\beta})+{1\over 2}(g_{\lambda\nu}\dot{x}^{\nu}+g_{\lambda\mu}\dot{x}% ^{\mu}){d\over d\tau}(g_{\mu\nu}\dot{x}^{\mu}\dot{x}^{\nu})\qquad\qquad(6)
  56. g λ ν , μ x ˙ μ x ˙ ν + g λ μ , ν x ˙ μ x ˙ ν - g μ ν , λ x ˙ μ x ˙ ν + 2 g λ μ x ¨ μ = x ˙ λ d d τ ( g μ ν x ˙ μ x ˙ ν ) g α β x ˙ α x ˙ β ( 7 ) g_{\lambda\nu,\mu}\dot{x}^{\mu}\dot{x}^{\nu}+g_{\lambda\mu,\nu}\dot{x}^{\mu}% \dot{x}^{\nu}-g_{\mu\nu,\lambda}\dot{x}^{\mu}\dot{x}^{\nu}+2g_{\lambda\mu}% \ddot{x}^{\mu}={\dot{x}_{\lambda}{d\over d\tau}(g_{\mu\nu}\dot{x}^{\mu}\dot{x}% ^{\nu})\over g_{\alpha\beta}\dot{x}^{\alpha}\dot{x}^{\beta}}\qquad\qquad(7)
  57. 2 ( Γ λ μ ν x ˙ μ x ˙ ν + x ¨ λ ) = x ˙ λ d d τ ( x ˙ ν x ˙ ν ) x ˙ β x ˙ β = U λ d d τ ( U ν U ν ) U β U β = U λ d d τ ln | U ν U ν | ( 8 ) 2(\Gamma_{\lambda\mu\nu}\dot{x}^{\mu}\dot{x}^{\nu}+\ddot{x}_{\lambda})={\dot{x% }_{\lambda}{d\over d\tau}(\dot{x}_{\nu}\dot{x}^{\nu})\over\dot{x}_{\beta}\dot{% x}^{\beta}}={U_{\lambda}{d\over d\tau}(U_{\nu}U^{\nu})\over U_{\beta}U^{\beta}% }=U_{\lambda}{d\over d\tau}\ln|U_{\nu}U^{\nu}|\qquad\qquad(8)
  58. U ν U ν U_{\nu}U^{\nu}
  59. Γ λ x ˙ μ μ ν x ˙ ν + x ¨ λ = 0 . \Gamma^{\lambda}{}_{\mu\nu}\dot{x}^{\mu}\dot{x}^{\nu}+\ddot{x}^{\lambda}=0\ .

Geometric_integrator.html

  1. m = 1 m=1
  2. = 1 \ell=1
  3. g = 1 g=1
  4. q ( t ) q(t)
  5. p ( t ) p(t)
  6. H ( q , p ) = T ( p ) + U ( q ) = 1 2 p 2 - cos q , H(q,p)=T(p)+U(q)=\frac{1}{2}p^{2}-\cos q,
  7. ( q ˙ , p ˙ ) = ( p , - sin q ) . (\dot{q},\dot{p})=(p,-\sin q).\,
  8. Q Q
  9. q q
  10. 𝕊 1 \mathbb{S}^{1}
  11. ( q , p ) (q,p)
  12. 𝕊 1 × \mathbb{S}^{1}\times\mathbb{R}
  13. ( q , p ) 2 (q,p)\in\mathbb{R}^{2}
  14. ( q , p ) (q,p)
  15. z ( t ) = ( q ( t ) , p ( t ) ) T z(t)=(q(t),p(t))^{\mathrm{T}}
  16. f ( z ) = ( p , - sin q ) T f(z)=(p,-\sin q)^{\mathrm{T}}
  17. h h
  18. k k
  19. z k := z ( k h ) z_{k}:=z(kh)
  20. z k + 1 = z k + h f ( z k ) z_{k+1}=z_{k}+hf(z_{k})\,
  21. z k + 1 = z k + h f ( z k + 1 ) z_{k+1}=z_{k}+hf(z_{k+1})\,
  22. z k + 1 = z k + h f ( q k , p k + 1 ) z_{k+1}=z_{k}+hf(q_{k},p_{k+1})\,
  23. z k + 1 = z k + h f ( ( z k + 1 + z k ) / 2 ) z_{k+1}=z_{k}+hf((z_{k+1}+z_{k})/2)\,
  24. p p
  25. H H
  26. p 2 / 2 - cos q p^{2}/2-\cos q
  27. 2 \mathbb{R}^{2}
  28. h = 0.2 h=0.2
  29. h = 0.3 h=0.3
  30. ϕ t \phi_{t}
  31. det ϕ t ( q 0 , p 0 ) = 1 \det\frac{\partial\phi_{t}}{\partial(q_{0},p_{0})}=1
  32. t t
  33. Φ eE , h : z k z k + 1 \Phi_{{\mathrm{eE}},h}:z_{k}\mapsto z_{k+1}
  34. det ( q 0 , p 0 ) Φ eE , h ( z 0 ) = | 1 h - h cos q 0 1 | = 1 + h 2 cos q 0 . \det\frac{\partial}{\partial(q_{0},p_{0})}\Phi_{{\mathrm{eE}},h}(z_{0})=\begin% {vmatrix}1&h\\ -h\cos q_{0}&1\end{vmatrix}=1+h^{2}\cos q_{0}.
  35. det ( q 0 , p 0 ) Φ iE , h ( z 0 ) = ( 1 + h 2 cos q 1 ) - 1 . \det\frac{\partial}{\partial(q_{0},p_{0})}\Phi_{{\mathrm{iE}},h}(z_{0})=(1+h^{% 2}\cos q_{1})^{-1}.
  36. ( 1 - h 0 1 ) ( q 0 , p 0 ) Φ sE , h ( z 0 ) = ( 1 0 - h cos q 0 1 ) , \begin{pmatrix}1&-h\\ 0&1\end{pmatrix}\frac{\partial}{\partial(q_{0},p_{0})}\Phi_{{\mathrm{sE}},h}(z% _{0})=\begin{pmatrix}1&0\\ -h\cos q_{0}&1\end{pmatrix},
  37. det ( Φ sE , h / ( q 0 , p 0 ) ) = 1 \det(\partial\Phi_{{\mathrm{sE}},h}/\partial(q_{0},p_{0}))=1

Georgi–Jarlskog_mass_relation.html

  1. m e 1 3 m d m_{e}\approx\frac{1}{3}m_{d}
  2. m μ 3 m s m_{\mu}\approx 3m_{s}
  3. m τ m b m_{\tau}\approx m_{b}
  4. m d m s 9 m e m μ \frac{m_{d}}{m_{s}}\approx 9\frac{m_{e}}{m_{\mu}}
  5. m s m b 1 3 m μ m τ \frac{m_{s}}{m_{b}}\approx\frac{1}{3}\frac{m_{\mu}}{m_{\tau}}
  6. M S ¯ \overline{MS}
  7. M S ¯ \overline{MS}
  8. M S ¯ \overline{MS}
  9. M S ¯ \overline{MS}
  10. M S ¯ \overline{MS}
  11. m e 1 3 m d m_{e}\approx\frac{1}{3}m_{d}
  12. 0.511 M e V 1.5 M e V 0.511MeV\approx 1.5MeV
  13. m μ 3 m s m_{\mu}\approx 3m_{s}
  14. 105.7 M e V 261 M e V 105.7MeV\approx 261MeV
  15. m τ m b m_{\tau}\approx m_{b}
  16. 1.78 G e V 4.24 G e V 1.78GeV\approx 4.24GeV
  17. m d m s 9 m e m μ \frac{m_{d}}{m_{s}}\approx 9\frac{m_{e}}{m_{\mu}}
  18. 0.051 0.0435 0.051\approx 0.0435
  19. m s m b 1 3 m μ m τ \frac{m_{s}}{m_{b}}\approx\frac{1}{3}\frac{m_{\mu}}{m_{\tau}}
  20. 0.021 0.0198 0.021\approx 0.0198

Gershgorin_circle_theorem.html

  1. A A
  2. n × n n\times n
  3. a i j a_{ij}\,
  4. i { 1 , , n } i\in\{1,\dots,n\}
  5. R i = j i | a i j | R_{i}=\sum_{j\neq{i}}\left|a_{ij}\right|
  6. i i
  7. D ( a i i , R i ) D(a_{ii},R_{i})
  8. a i i a_{ii}
  9. R i R_{i}
  10. A A
  11. D ( a i i , R i ) D(a_{ii},R_{i})
  12. λ \lambda
  13. A A
  14. A x = λ x Ax=\lambda x
  15. j a i j x j = λ x i i { 1 , , n } . \sum_{j}a_{ij}x_{j}=\lambda x_{i}\quad\forall i\in\{1,\ldots,n\}.
  16. j i a i j x j = λ x i - a i i x i . \sum_{j\neq i}a_{ij}x_{j}=\lambda x_{i}-a_{ii}x_{i}.
  17. | λ - a i i | = | j i a i j x j x i | j i | a i j x j x i | j i | a i j | = R i |\lambda-a_{ii}|=\left|\frac{\sum_{j\neq i}a_{ij}x_{j}}{x_{i}}\right|\leq\sum_% {j\neq i}\left|\frac{a_{ij}x_{j}}{x_{i}}\right|\leq\sum_{j\neq i}|a_{ij}|=R_{i}
  18. | x j x i | 1 for j i . \left|\frac{x_{j}}{x_{i}}\right|\leq 1\quad\,\text{for }j\neq i.
  19. A = ( 0 1 4 0 ) A=\begin{pmatrix}0&1\\ 4&0\end{pmatrix}
  20. A = ( 1 - 2 1 - 1 ) A=\begin{pmatrix}1&-2\\ 1&-1\end{pmatrix}
  21. B ( t ) = ( 1 - t ) D + t A . B(t)=(1-t)D+tA.\,
  22. t t
  23. t t
  24. D = B ( 0 ) D=B(0)
  25. B ( t ) B(t)
  26. B ( t ) B(t)
  27. d > 0 d>0
  28. B ( t ) B(t)
  29. B ( t ) B(t)
  30. λ ( t ) \lambda(t)
  31. B ( t ) B(t)
  32. d ( t ) d(t)
  33. d ( 0 ) d d(0)\geq d
  34. λ ( 1 ) \lambda(1)
  35. d ( 1 ) = 0 d(1)=0
  36. 0 < t 0 < 1 0<t_{0}<1
  37. 0 < d ( t 0 ) < d 0<d(t_{0})<d
  38. λ ( t 0 ) \lambda(t_{0})
  39. λ ( 1 ) \lambda(1)
  40. A = [ 10 - 1 0 1 0.2 8 0.2 0.2 1 1 2 1 - 1 - 1 - 1 - 11 ] . A=\begin{bmatrix}10&-1&0&1\\ 0.2&8&0.2&0.2\\ 1&1&2&1\\ -1&-1&-1&-11\\ \end{bmatrix}.
  41. j i | a i j | = R i \sum_{j\neq i}|a_{ij}|=R_{i}
  42. D ( 10 , 2 ) , D ( 8 , 0.6 ) , D ( 2 , 3 ) , and D ( - 11 , 3 ) . D(10,2),\;D(8,0.6),\;D(2,3),\;\,\text{and}\;D(-11,3).
  43. D ( 2 , 1.2 ) D(2,1.2)
  44. D ( - 11 , 2.2 ) D(-11,2.2)

Giant_component.html

  1. n n
  2. p p
  3. p 1 - ϵ n p\leq\frac{1-\epsilon}{n}
  4. ϵ > 0 \epsilon>0
  5. O ( l o g n ) O(logn)
  6. p 1 + ϵ n p\geq\frac{1+\epsilon}{n}
  7. O ( l o g n ) O(logn)
  8. p = 1 n p=\frac{1}{n}
  9. n 2 / 3 n^{2/3}
  10. n / 2 n/2
  11. t t
  12. t t
  13. n / 2 n/2
  14. 4 t - 2 n 4t-2n
  15. Θ ( n log n ) \Theta(n\log n)

Gibbons–Hawking–York_boundary_term.html

  1. \mathcal{M}
  2. \partial\mathcal{M}
  3. 𝒮 EH + 𝒮 GHY = 1 16 π d 4 x g R + 1 8 π d 3 x h K , \mathcal{S}_{\mathrm{EH}}+\mathcal{S}_{\mathrm{GHY}}=\frac{1}{16\pi}\int_{% \mathcal{M}}\mathrm{d}^{4}x\,\sqrt{g}R+\frac{1}{8\pi}\int_{\partial\mathcal{M}% }\mathrm{d}^{3}x\,\sqrt{h}K,
  4. 𝒮 EH \mathcal{S}_{\mathrm{EH}}
  5. 𝒮 GHY \mathcal{S}_{\mathrm{GHY}}
  6. h α β h_{\alpha\beta}
  7. K K
  8. g α β g_{\alpha\beta}
  9. h α β h_{\alpha\beta}
  10. h α β h_{\alpha\beta}

Gibbs'_inequality.html

  1. P = { p 1 , , p n } P=\{p_{1},\ldots,p_{n}\}
  2. Q = { q 1 , , q n } Q=\{q_{1},\ldots,q_{n}\}
  3. - i = 1 n p i log 2 p i - i = 1 n p i log 2 q i -\sum_{i=1}^{n}p_{i}\log_{2}p_{i}\leq-\sum_{i=1}^{n}p_{i}\log_{2}q_{i}
  4. p i = q i p_{i}=q_{i}\,
  5. D KL ( P Q ) i = 1 n p i log 2 p i q i 0. D_{\mathrm{KL}}(P\|Q)\equiv\sum_{i=1}^{n}p_{i}\log_{2}\frac{p_{i}}{q_{i}}\geq 0.
  6. log 2 a = ln a ln 2 \log_{2}a=\frac{\ln a}{\ln 2}
  7. ln x x - 1 \ln x\leq x-1
  8. I I
  9. i i
  10. - i I p i ln q i p i - i I p i ( q i p i - 1 ) -\sum_{i\in I}p_{i}\ln\frac{q_{i}}{p_{i}}\geq-\sum_{i\in I}p_{i}\left(\frac{q_% {i}}{p_{i}}-1\right)
  11. = - i I q i + i I p i =-\sum_{i\in I}q_{i}+\sum_{i\in I}p_{i}
  12. = - i I q i + 1 0. =-\sum_{i\in I}q_{i}+1\geq 0.
  13. - i I p i ln q i - i I p i ln p i -\sum_{i\in I}p_{i}\ln q_{i}\geq-\sum_{i\in I}p_{i}\ln p_{i}
  14. - i = 1 n p i ln q i - i = 1 n p i ln p i -\sum_{i=1}^{n}p_{i}\ln q_{i}\geq-\sum_{i=1}^{n}p_{i}\ln p_{i}
  15. q i p i = 1 \frac{q_{i}}{p_{i}}=1
  16. i I i\in I
  17. ln q i p i = q i p i - 1 \ln\frac{q_{i}}{p_{i}}=\frac{q_{i}}{p_{i}}-1
  18. i I q i = 1 \sum_{i\in I}q_{i}=1
  19. p i = q i p_{i}=q_{i}
  20. P P
  21. H ( p 1 , , p n ) log n . H(p_{1},\ldots,p_{n})\leq\log n.
  22. q i = 1 / n q_{i}=1/n

Gibbs_algorithm.html

  1. H = i - p i ln p i H=\sum_{i}-p_{i}\ln p_{i}\,

Gibbs_measure.html

  1. P ( X = x ) = 1 Z ( β ) exp ( - β E ( x ) ) . P(X=x)=\frac{1}{Z(\beta)}\exp(-\beta E(x)).
  2. E ( x ) E(x)
  3. E ( x ) E(x)
  4. β β
  5. Z ( β ) Z(β)
  6. P ( σ k = s σ j , j k ) P(\sigma_{k}=s\mid\sigma_{j},\,j\neq k)
  7. P ( σ k = s σ j , j k ) = P ( σ k = s σ j , j \isin N k ) P(\sigma_{k}=s\mid\sigma_{j},\,j\neq k)=P(\sigma_{k}=s\mid\sigma_{j},\,j\isin N% _{k})
  8. k k
  9. k k
  10. 𝕃 \mathbb{L}
  11. ( S , 𝒮 , λ ) (S,\mathcal{S},\lambda)
  12. ( Ω , ) (\Omega,\mathcal{F})
  13. Ω = S 𝕃 \Omega=S^{\mathbb{L}}
  14. = 𝒮 𝕃 \mathcal{F}=\mathcal{S}^{\mathbb{L}}
  15. ω Ω ω∈Ω
  16. Λ 𝕃 \Lambda\subset\mathbb{L}
  17. ω ω
  18. Λ Λ
  19. ω Λ = ( ω ( t ) ) t Λ \omega_{\Lambda}=(\omega(t))_{t\in\Lambda}
  20. Λ 1 Λ 2 = \Lambda_{1}\cap\Lambda_{2}=\emptyset
  21. Λ 1 Λ 2 = 𝕃 \Lambda_{1}\cup\Lambda_{2}=\mathbb{L}
  22. ω Λ 1 ω Λ 2 \omega_{\Lambda_{1}}\omega_{\Lambda_{2}}
  23. ω Λ 1 \omega_{\Lambda_{1}}
  24. ω Λ 2 \omega_{\Lambda_{2}}
  25. \mathcal{L}
  26. 𝕃 \mathbb{L}
  27. Λ 𝕃 \Lambda\subset\mathbb{L}
  28. Λ \mathcal{F}_{\Lambda}
  29. σ σ
  30. ( σ ( t ) ) t Λ (\sigma(t))_{t\in\Lambda}
  31. σ ( t ) ( ω ) = ω ( t ) \sigma(t)(\omega)=\omega(t)
  32. σ σ
  33. σ σ
  34. Φ = ( Φ A ) A \Phi=(\Phi_{A})_{A\in\mathcal{L}}
  35. A , Φ A A\in\mathcal{L},\Phi_{A}
  36. A \mathcal{F}_{A}
  37. Λ \Lambda\in\mathcal{L}
  38. ω Ω ω∈Ω
  39. H Λ Φ ( ω ) = A , A Λ Φ A ( ω ) . H_{\Lambda}^{\Phi}(\omega)=\sum_{A\in\mathcal{L},A\cap\Lambda\neq\emptyset}% \Phi_{A}(\omega).
  40. Λ \Lambda\in\mathcal{L}
  41. ω ¯ \bar{\omega}
  42. Φ Φ
  43. H Λ Φ ( ω ω ¯ ) = H Λ Φ ( ω Λ ω ¯ Λ c ) H_{\Lambda}^{\Phi}(\omega\mid\bar{\omega})=H_{\Lambda}^{\Phi}\left(\omega_{% \Lambda}\bar{\omega}_{\Lambda^{c}}\right)
  44. Λ c = 𝕃 Λ \Lambda^{c}=\mathbb{L}\setminus\Lambda
  45. Λ \Lambda\in\mathcal{L}
  46. ω ¯ \bar{\omega}
  47. β > 0 β>0
  48. Φ Φ
  49. λ λ
  50. Z Λ Φ ( ω ¯ ) = λ Λ ( d ω ) exp ( - β H Λ Φ ( ω ω ¯ ) ) , Z_{\Lambda}^{\Phi}(\bar{\omega})=\int\lambda^{\Lambda}(\mathrm{d}\omega)\exp(-% \beta H_{\Lambda}^{\Phi}(\omega\mid\bar{\omega})),
  51. λ Λ ( d ω ) = t Λ λ ( d ω ( t ) ) , \lambda^{\Lambda}(\mathrm{d}\omega)=\prod_{t\in\Lambda}\lambda(\mathrm{d}% \omega(t)),
  52. Φ Φ
  53. λ λ
  54. Z Λ Φ ( ω ¯ ) Z_{\Lambda}^{\Phi}(\bar{\omega})
  55. Λ , ω ¯ Ω \Lambda\in\mathcal{L},\bar{\omega}\in\Omega
  56. β > 0 β>0
  57. μ μ
  58. ( Ω , ) (\Omega,\mathcal{F})
  59. λ λ
  60. Φ Φ
  61. μ ( d ω ¯ ) Z Λ Φ ( ω ¯ ) - 1 λ Λ ( d ω ) exp ( - β H Λ Φ ( ω ω ¯ ) ) 1 A ( ω Λ ω ¯ Λ c ) = μ ( A ) , \int\mu(\mathrm{d}\bar{\omega})Z_{\Lambda}^{\Phi}(\bar{\omega})^{-1}\int% \lambda^{\Lambda}(\mathrm{d}\omega)\exp(-\beta H_{\Lambda}^{\Phi}(\omega\mid% \bar{\omega}))1_{A}(\omega_{\Lambda}\bar{\omega}_{\Lambda^{c}})=\mu(A),
  62. A A\in\mathcal{F}
  63. Λ \Lambda\in\mathcal{L}
  64. J J
  65. h h
  66. 𝕃 = 𝐙 d \mathbb{L}=\mathbf{Z}^{d}
  67. Φ A ( ω ) = { - J ω ( t 1 ) ω ( t 2 ) if A = { t 1 , t 2 } with t 2 - t 1 1 = 1 - h ω ( t ) if A = { t } 0 otherwise \Phi_{A}(\omega)=\begin{cases}-J\,\omega(t_{1})\omega(t_{2})&\,\text{if }A=\{t% _{1},t_{2}\}\,\text{ with }\|t_{2}-t_{1}\|_{1}=1\\ -h\,\omega(t)&\,\text{if }A=\{t\}\\ 0&\,\text{otherwise}\end{cases}

Gibbs_state.html

  1. L L\;
  2. ρ 0 \rho_{0}\;
  3. ρ ( t ) = e L t [ ρ 0 ] \rho(t)=e^{Lt}[\rho_{0}]\;
  4. ρ \rho_{\infty}\;
  5. L [ ρ ] = 0 L[\rho_{\infty}]=0

Giovanni_Vacca.html

  1. x 0 = i , x n + 1 = x n + | x n | 2 , lim n x n = 2 π . x_{0}=i,\quad x_{n+1}=\frac{x_{n}+|x_{n}|}{2},\qquad\lim_{n\to\infty}x_{n}=% \frac{2}{\pi}.
  2. γ = k = 2 ( - 1 ) k log 2 k k = 1 2 - 1 3 + 2 ( 1 4 - 1 5 + 1 6 - 1 7 ) + 3 ( 1 8 - - 1 15 ) + ζ ( 2 ) + γ = k = 2 ( 1 k 2 - 1 k ) = k = 2 k - k 2 k k 2 = 1 2 + 2 3 + 1 2 2 ( 1 5 + 2 6 + 3 7 + 4 8 ) + 1 3 2 ( 1 10 + + 6 15 ) + \begin{array}[]{rcl}\gamma&=&\sum_{k=2}^{\infty}(-1)^{k}\frac{\left\lfloor\log% _{2}k\right\rfloor}{k}=\tfrac{1}{2}-\tfrac{1}{3}+2\left(\tfrac{1}{4}-\tfrac{1}% {5}+\tfrac{1}{6}-\tfrac{1}{7}\right)+3\left(\tfrac{1}{8}-\cdots-\tfrac{1}{15}% \right)+\cdots\\ \zeta(2)+\gamma&=&\sum_{k=2}^{\infty}\left(\frac{1}{\lfloor\sqrt{k}\rfloor^{2}% }-\frac{1}{k}\right)=\sum_{k=2}^{\infty}\frac{k-\lfloor\sqrt{k}\rfloor^{2}}{k% \lfloor\sqrt{k}\rfloor^{2}}=\tfrac{1}{2}+\tfrac{2}{3}+\tfrac{1}{2^{2}}\left(% \tfrac{1}{5}+\tfrac{2}{6}+\tfrac{3}{7}+\tfrac{4}{8}\right)+\tfrac{1}{3^{2}}% \left(\tfrac{1}{10}+\cdots+\tfrac{6}{15}\right)+\cdots\end{array}
  3. γ \gamma

Giuga_number.html

  1. p i | ( n p i - 1 ) p_{i}|({n\over p_{i}}-1)
  2. p i 2 | ( n - p i ) p_{i}^{2}|(n-p_{i})
  3. n B φ ( n ) - 1 ( mod n ) nB_{\varphi(n)}\equiv-1\;\;(\mathop{{\rm mod}}n)
  4. φ ( n ) \varphi(n)
  5. i = 1 n - 1 i φ ( n ) - 1 ( mod n ) \sum_{i=1}^{n-1}i^{\varphi(n)}\equiv-1\;\;(\mathop{{\rm mod}}n)
  6. p | n 1 p - p | n 1 p . \sum_{p|n}\frac{1}{p}-\prod_{p|n}\frac{1}{p}\in\mathbb{N}.
  7. p | n 1 p - p | n 1 p = 1. \sum_{p|n}\frac{1}{p}-\prod_{p|n}\frac{1}{p}=1.
  8. p 2 p^{2}
  9. n n
  10. n p - 1 = m - 1 {n\over p}-1=m-1
  11. m = n / p m=n/p
  12. p p
  13. m - 1 m-1
  14. p p
  15. n n
  16. n = p 1 p 2 n=p_{1}p_{2}
  17. p 1 < p 2 p_{1}<p_{2}
  18. n p 2 - 1 = p 1 - 1 < p 2 {n\over p_{2}}-1=p_{1}-1<p_{2}
  19. p 2 p_{2}
  20. n p 2 - 1 {n\over p_{2}}-1
  21. n n
  22. n = i p i n=\prod_{i}{p_{i}}
  23. n = i n p i n^{\prime}=\sum_{i}\frac{n}{p_{i}}

Giulio_Carlo_de'_Toschi_di_Fagnano.html

  1. π = 2 i log 1 - i 1 + i \pi=2i\log{1-i\over 1+i}
  2. - 1 \sqrt{-1}

Giuseppe_Melfi.html

  1. m - 2 , m , m + 2 m-2,m,m+2
  2. k = 0 n σ 1 ( 3 k + 1 ) σ 1 ( 3 n - 3 k + 1 ) = 1 9 σ 3 ( 3 n + 2 ) , \sum_{k=0}^{n}\sigma_{1}(3k+1)\sigma_{1}(3n-3k+1)=\frac{1}{9}\sigma_{3}(3n+2),
  3. σ k ( n ) \sigma_{k}(n)
  4. k k
  5. n n

Glaisher–Kinkelin_constant.html

  1. A 1.2824271291 A\approx 1.2824271291\dots
  2. A A
  3. A = lim n K ( n + 1 ) n n 2 / 2 + n / 2 + 1 / 12 e - n 2 / 4 A=\lim_{n\rightarrow\infty}\frac{K(n+1)}{n^{n^{2}/2+n/2+1/12}e^{-n^{2}/4}}
  4. K ( n ) = k = 1 n - 1 k k K(n)=\prod_{k=1}^{n-1}k^{k}
  5. π \pi
  6. 2 π = lim n n ! e - n n n + 1 2 \sqrt{2\pi}=\lim_{n\to\infty}\frac{n!}{e^{-n}n^{n+\frac{1}{2}}}
  7. π \pi
  8. k = 1 n k \prod_{k=1}^{n}k
  9. k = 1 n k k \prod_{k=1}^{n}k^{k}
  10. G ( n ) = k = 1 n - 2 k ! = [ Γ ( n ) ] n - 1 K ( n ) G(n)=\prod_{k=1}^{n-2}k!=\frac{\left[\Gamma(n)\right]^{n-1}}{K(n)}
  11. Γ ( n ) \Gamma(n)
  12. A = lim n ( 2 π ) n / 2 n n 2 / 2 - 1 / 12 e - 3 n 2 / 4 + 1 / 12 G ( n + 1 ) A=\lim_{n\rightarrow\infty}\frac{(2\pi)^{n/2}n^{n^{2}/2-1/12}e^{-3n^{2}/4+1/12% }}{G(n+1)}
  13. ζ ( - 1 ) = 1 12 - ln A \zeta^{\prime}(-1)=\frac{1}{12}-\ln A
  14. k = 2 ln k k 2 = - ζ ( 2 ) = π 2 6 [ 12 ln A - γ - ln ( 2 π ) ] \sum_{k=2}^{\infty}\frac{\ln k}{k^{2}}=-\zeta^{\prime}(2)=\frac{\pi^{2}}{6}% \left[12\ln A-\gamma-\ln(2\pi)\right]
  15. γ \gamma
  16. k = 1 k 1 k 2 = ( A 12 2 π e γ ) π 2 6 \prod_{k=1}^{\infty}k^{\frac{1}{k^{2}}}=\left(\frac{A^{12}}{2\pi e^{\gamma}}% \right)^{\frac{\pi^{2}}{6}}
  17. 0 1 / 2 ln Γ ( x ) d x = 3 2 ln A + 5 24 ln 2 + 1 4 ln π \int_{0}^{1/2}\ln\Gamma(x)dx=\frac{3}{2}\ln A+\frac{5}{24}\ln 2+\frac{1}{4}\ln\pi
  18. 0 x ln x e 2 π x - 1 d x = 1 2 ζ ( - 1 ) = 1 24 - 1 2 ln A \int_{0}^{\infty}\frac{x\ln x}{e^{2\pi x}-1}dx=\frac{1}{2}\zeta^{\prime}(-1)=% \frac{1}{24}-\frac{1}{2}\ln A
  19. ln A = 1 8 - 1 2 n = 0 1 n + 1 k = 0 n ( - 1 ) k ( n k ) ( k + 1 ) 2 ln ( k + 1 ) \ln A=\frac{1}{8}-\frac{1}{2}\sum_{n=0}^{\infty}\frac{1}{n+1}\sum_{k=0}^{n}% \left(-1\right)^{k}{\left({{n}\atop{k}}\right)}\left(k+1\right)^{2}\ln(k+1)

Glassy_carbon.html

  1. GCE \rm\stackrel{GCE}{\rightleftharpoons}
  2. \cdot
  3. \rightleftharpoons

Gliese_581.html

  1. M V = 4.83 \begin{smallmatrix}M_{V_{\odot}}=4.83\end{smallmatrix}
  2. L V L V = 10 0.4 ( M V - M V ) \begin{smallmatrix}\frac{L_{V_{\ast}}}{L_{V_{\odot}}}=10^{0.4\left(M_{V_{\odot% }}-M_{V_{\ast}}\right)}\end{smallmatrix}

Global_symmetry.html

  1. U ( 1 ) = e i q θ U(1)=e^{iq\theta}
  2. θ \theta
  3. D = ψ ¯ ( i γ μ μ - m ) ψ \mathcal{L}_{D}=\bar{\psi}\left(i\gamma^{\mu}\partial_{\mu}-m\right)\psi
  4. ψ e i q θ ψ \psi\rightarrow e^{iq\theta}\psi
  5. ψ ¯ e - i q θ ψ ¯ \bar{\psi}\rightarrow e^{-iq\theta}\bar{\psi}
  6. ¯ = e - i q θ ψ ¯ ( i γ μ μ - m ) e i q θ ψ = e - i q θ e i q θ ψ ¯ ( i γ μ μ - m ) ψ = \mathcal{L}\rightarrow\bar{\mathcal{L}}=e^{-iq\theta}\bar{\psi}\left(i\gamma^{% \mu}\partial_{\mu}-m\right)e^{iq\theta}\psi=e^{-iq\theta}e^{iq\theta}\bar{\psi% }\left(i\gamma^{\mu}\partial_{\mu}-m\right)\psi=\mathcal{L}

Gloss_(optics).html

  1. R s R_{s}
  2. I 0 I_{0}
  3. i i
  4. I r I_{r}
  5. m m
  6. R s = I r I 0 R_{s}=\frac{I_{r}}{I_{0}}
  7. R s = 1 2 [ ( cos i - m 2 - sin 2 i cos i + m 2 - sin 2 i ) 2 + ( m 2 cos i - m 2 - sin 2 i m 2 cos i + m 2 - sin 2 i ) 2 ] R_{s}=\frac{1}{2}\left[\left(\frac{\cos i-\sqrt{m^{2}-\sin^{2}i}}{\cos i+\sqrt% {m^{2}-\sin^{2}i}}\right)^{2}+\left(\frac{m^{2}\cos i-\sqrt{m^{2}-\sin^{2}i}}{% m^{2}\cos i+\sqrt{m^{2}-\sin^{2}i}}\right)^{2}\right]
  8. i i
  9. h h
  10. Δ r = 2 h cos i \Delta r=2h\cos i\;
  11. λ \lambda
  12. Δ ϕ = 4 π h cos i λ \Delta\phi=\frac{4\pi h\cos i}{\lambda}\;
  13. Δ ϕ \Delta\phi\;
  14. Δ ϕ = π \Delta\phi=\pi\;
  15. Δ ϕ < π 2 \Delta\phi<\frac{\pi}{2}
  16. h < λ 8 cos i h<\frac{\lambda}{8\cos i}\;

Glossary_of_category_theory.html

  1. g = h g=h
  2. g f = h f g\circ f=h\circ f
  3. g f = g g\circ f=g
  4. f h = h f\circ h=h
  5. g f g\circ f
  6. f g f\circ g
  7. f g f\circ g
  8. g = h g=h
  9. f g = f h f\circ g=f\circ h

Glucose-6-phosphate_dehydrogenase.html

  1. \rightleftharpoons

Glycogenin.html

  1. \rightleftharpoons

Glyoxylic_acid.html

  1. \overrightarrow{\leftarrow}
  2. \overrightarrow{\leftarrow}

Goertzel_algorithm.html

  1. x ( n ) x(n)
  2. f f
  3. s ( n ) s(n)
  4. s ( n ) = x ( n ) + 2 cos ( 2 π f ) s ( n - 1 ) - s ( n - 2 ) s(n)=x(n)+2\cos(2\pi f)s(n-1)-s(n-2)
  5. s ( n ) s(n)
  6. y ( n ) y(n)
  7. y ( n ) = s ( n ) - e - 2 π i f s ( n - 1 ) y(n)=s(n)-e^{-2\pi if}s(n-1)
  8. x ( n ) x(n)
  9. n < 0 n<0
  10. x ( 0 ) x(0)
  11. s ( - 2 ) = s ( - 1 ) = 0 s(-2)=s(-1)=0
  12. f f
  13. S ( z ) X ( z ) = 1 1 - 2 cos ( 2 π f ) z - 1 + z - 2 = 1 ( 1 - e + 2 π i f z - 1 ) ( 1 - e - 2 π i f z - 1 ) . \frac{S(z)}{X(z)}=\frac{1}{1-2\cos(2\pi f)z^{-1}+z^{-2}}=\frac{1}{(1-e^{+2\pi if% }z^{-1})(1-e^{-2\pi if}z^{-1})}.
  14. Y ( z ) S ( z ) = 1 - e - 2 π i f z - 1 \frac{Y(z)}{S(z)}=1-e^{-2\pi if}z^{-1}
  15. S ( z ) X ( z ) Y ( z ) S ( z ) = Y ( z ) X ( z ) = ( 1 - e - 2 π i f z - 1 ) ( 1 - e + 2 π i f z - 1 ) ( 1 - e - 2 π i f z - 1 ) = 1 1 - e + 2 π i f z - 1 . \frac{S(z)}{X(z)}\frac{Y(z)}{S(z)}=\frac{Y(z)}{X(z)}=\frac{(1-e^{-2\pi if}z^{-% 1})}{(1-e^{+2\pi if}z^{-1})(1-e^{-2\pi if}z^{-1})}=\frac{1}{1-e^{+2\pi if}z^{-% 1}}.
  16. n = 0 n=0
  17. y ( n ) = x ( n ) + e + 2 π i f y ( n - 1 ) = k = - n x ( k ) e + 2 π i f ( n - k ) = e + 2 π i f n k = 0 n x ( k ) e - 2 π i f k \begin{aligned}\displaystyle y(n)&\displaystyle=x(n)+e^{+2\pi if}y(n-1)\\ &\displaystyle=\sum_{k=-\infty}^{n}x(k)e^{+2\pi if(n-k)}\\ &\displaystyle=e^{+2\pi ifn}\sum_{k=0}^{n}x(k)e^{-2\pi ifk}\end{aligned}
  18. e + 2 π i f e^{+2\pi if}
  19. e - 2 π i f e^{-2\pi if}
  20. n = N n=N
  21. N N
  22. f = K N f=\frac{K}{N}
  23. K K
  24. K ϵ { 0 , 1 , 2 , , N - 1 } K\epsilon\{0,1,2,...,N-1\}
  25. e + 2 π i K = 1 e^{+2\pi iK}=1
  26. y ( N ) = k = 0 N x ( k ) e - 2 π i k K N y(N)=\sum_{k=0}^{N}x(k)e^{-2\pi i\frac{kK}{N}}
  27. X ( K ) X(K)
  28. x x
  29. x ( N ) = 0 x(N)=0
  30. x ( N ) x(N)
  31. x ( N ) = 0 x(N)=0
  32. s ( N ) = 2 cos ( 2 π f ) s ( N - 1 ) - s ( N - 2 ) s(N)=2\cos(2\pi f)s(N-1)-s(N-2)
  33. x ( N - 1 ) x(N-1)
  34. s ( N ) s(N)
  35. s ( N - 1 ) s(N-1)
  36. s ( N - 2 ) s(N-2)
  37. s ( N ) s(N)
  38. s ( N - 1 ) s(N-1)
  39. y ( N ) = s ( N ) - e - 2 π i K N s ( N - 1 ) = ( 2 cos ( 2 π f ) s ( N - 1 ) - s ( N - 2 ) ) - e - 2 π i K N s ( N - 1 ) = e 2 π i K N s ( N - 1 ) - s ( N - 2 ) \begin{aligned}\displaystyle y(N)&\displaystyle=s(N)-e^{-2\pi i\frac{K}{N}}s(N% -1)\\ &\displaystyle=(2\cos(2\pi f)s(N-1)-s(N-2))-e^{-2\pi i\frac{K}{N}}s(N-1)\\ &\displaystyle=e^{2\pi i\frac{K}{N}}s(N-1)-s(N-2)\end{aligned}
  40. N - 1 N-1
  41. y ( N ) y(N)
  42. s ( 0 ) , s ( 1 ) , s(0),s(1),
  43. s ( N ) s(N)
  44. x ( N ) = 0 x(N)=0
  45. s ( N - 1 ) s(N-1)
  46. s ( N ) s(N)
  47. s ( N - 1 ) s(N-1)
  48. y ( N ) y(N)
  49. y ( N - 1 ) y(N-1)
  50. X ( K ) X ( K ) = y ( N ) y ( N ) = y ( N - 1 ) y ( N - 1 ) = s ( N - 1 ) 2 + s ( N - 2 ) 2 - 2 c o s ( 2 π K N ) s ( N - 1 ) s ( N - 2 ) \begin{aligned}\displaystyle X(K)X^{\prime}(K)&\displaystyle=y(N)\ y^{\prime}(% N)=y(N-1)\ y^{\prime}(N-1)\\ &\displaystyle=s(N-1)^{2}+s(N-2)^{2}-2cos(2\pi\frac{K}{N})\ s(N-1)\ s(N-2)\end% {aligned}
  51. x ( N - 1 ) x(N-1)
  52. c r = cos ( 2 π K N ) c i = sin ( 2 π K N ) y ( N ) = c r s ( N - 1 ) - s ( N - 2 ) + i c i s ( N - 1 ) \begin{aligned}\displaystyle c_{r}&\displaystyle=\cos(2\pi\frac{K}{N})\\ \displaystyle c_{i}&\displaystyle=\sin(2\pi\frac{K}{N})\\ \displaystyle y(N)&\displaystyle=c_{r}s(N-1)-s(N-2)+ic_{i}s(N-1)\end{aligned}
  53. X ( K ) X(K)
  54. ϕ = tan - 1 ( X ( K ) ) ( X ( K ) ) \phi=\tan^{-1}\frac{\Im(X(K))}{\Re(X(K))}
  55. y r ( n ) y_{r}(n)
  56. y i ( n ) y_{i}(n)
  57. y ( n ) = y r ( n ) + i y i ( n ) y(n)=y_{r}(n)+i\ y_{i}(n)
  58. M M
  59. M M
  60. N N
  61. K K
  62. O ( K N M ) O(KNM)
  63. X ( f ) X(f)
  64. N N
  65. 2 N 2N
  66. 4 N 4\ N
  67. 2 N + 4 2N+4
  68. 4 N + 4 4N+4
  69. M M
  70. N N
  71. O ( K N log N ) O(KN\log N)
  72. 2 log 2 N 2\log_{2}N
  73. 3 log 2 N 3\log_{2}N
  74. N N
  75. M M
  76. log N \log N
  77. K K
  78. M M
  79. log 2 N \log_{2}N
  80. N 2 N_{2}
  81. M 5 N 2 6 N log 2 N 2 M\leq\frac{5N_{2}}{6N}\log_{2}N_{2}

Goff–Gratch_equation.html

  1. log e * = \log\ e^{*}\ =
  2. - 7.90298 ( T st / T - 1 ) + 5.02808 log ( T st / T ) -7.90298(T_{\mathrm{st}}/T-1)\ +\ 5.02808\ \log(T_{\mathrm{st}}/T)
  3. - 1.3816 × 10 - 7 ( 10 11.344 ( 1 - T / T st ) - 1 ) -\ 1.3816\times 10^{-7}(10^{11.344(1-T/T_{\mathrm{st}})}-1)
  4. + 8.1328 × 10 - 3 ( 10 - 3.49149 ( T st / T - 1 ) - 1 ) + log e st * +\ 8.1328\times 10^{-3}(10^{-3.49149(T_{\mathrm{st}}/T-1)}-1)\ +\ \log\ e^{*}_% {\mathrm{st}}
  5. log e i * = \log\ e^{*}_{i}\ =
  6. - 9.09718 ( T 0 / T - 1 ) - 3.56654 log ( T 0 / T ) -9.09718(T_{0}/T-1)\ -\ 3.56654\ \log(T_{0}/T)
  7. + 0.876793 ( 1 - T / T 0 ) + log e i 0 * +\ 0.876793(1-T/T_{0})+\ \log\ e^{*}_{i0}

Going_up_and_going_down.html

  1. 𝔭 \mathfrak{p}
  2. 𝔮 \mathfrak{q}
  3. 𝔮 A = 𝔭 \mathfrak{q}\cap A=\mathfrak{p}
  4. 𝔮 A \mathfrak{q}\cap A
  5. 𝔭 \mathfrak{p}
  6. 𝔮 \mathfrak{q}
  7. 𝔮 \mathfrak{q}
  8. 𝔭 \mathfrak{p}
  9. 𝔭 1 𝔭 2 𝔭 n \mathfrak{p}_{1}\subseteq\mathfrak{p}_{2}\subseteq\cdots\subseteq\mathfrak{p}_% {n}
  10. 𝔮 1 𝔮 2 𝔮 m \mathfrak{q}_{1}\subseteq\mathfrak{q}_{2}\subseteq\cdots\subseteq\mathfrak{q}_% {m}
  11. 𝔭 i \mathfrak{p}_{i}
  12. 𝔮 1 𝔮 2 𝔮 n \mathfrak{q}_{1}\subseteq\mathfrak{q}_{2}\subseteq\cdots\subseteq\mathfrak{q}_% {n}
  13. 𝔮 i \mathfrak{q}_{i}
  14. 𝔭 i \mathfrak{p}_{i}
  15. 𝔭 1 𝔭 2 𝔭 n \mathfrak{p}_{1}\supseteq\mathfrak{p}_{2}\supseteq\cdots\supseteq\mathfrak{p}_% {n}
  16. 𝔮 1 𝔮 2 𝔮 m \mathfrak{q}_{1}\supseteq\mathfrak{q}_{2}\supseteq\cdots\supseteq\mathfrak{q}_% {m}
  17. 𝔭 i \mathfrak{p}_{i}
  18. 𝔮 1 𝔮 2 𝔮 n \mathfrak{q}_{1}\supseteq\mathfrak{q}_{2}\supseteq\cdots\supseteq\mathfrak{q}_% {n}
  19. 𝔮 i \mathfrak{q}_{i}
  20. 𝔭 i \mathfrak{p}_{i}

Goldwasser–Micali_cryptosystem.html

  1. x p ( p - 1 ) / 2 1 ( mod p ) x_{p}^{(p-1)/2}\equiv 1\;\;(\mathop{{\rm mod}}p)
  2. x q ( q - 1 ) / 2 1 ( mod q ) x_{q}^{(q-1)/2}\equiv 1\;\;(\mathop{{\rm mod}}q)
  3. ( x p ) = ( x q ) = - 1 \left(\frac{x}{p}\right)=\left(\frac{x}{q}\right)=-1
  4. ( x N ) \left(\frac{x}{N}\right)
  5. m i m_{i}
  6. y i y_{i}
  7. gcd ( y i , N ) = 1 \gcd(y_{i},N)=1
  8. c i = y i 2 x m i ( mod N ) c_{i}=y_{i}^{2}x^{m_{i}}\;\;(\mathop{{\rm mod}}N)
  9. m 0 m 1 m_{0}\oplus m_{1}

Goodness_of_fit.html

  1. χ 2 = ( O - E ) 2 σ 2 \chi^{2}=\sum{\frac{(O-E)^{2}}{\sigma^{2}}}
  2. σ 2 \sigma^{2}
  3. χ red 2 = χ 2 ν = 1 ν ( O - E ) 2 σ 2 \chi_{\mathrm{red}}^{2}=\frac{\chi^{2}}{\nu}=\frac{1}{\nu}\sum{\frac{(O-E)^{2}% }{\sigma^{2}}}
  4. ν \nu
  5. N - n - 1 N-n-1
  6. N N
  7. n n
  8. χ red 2 1 \chi_{\mathrm{red}}^{2}\gg 1
  9. χ red 2 > 1 \chi_{\mathrm{red}}^{2}>1
  10. χ red 2 = 1 \chi_{\mathrm{red}}^{2}=1
  11. χ red 2 < 1 \chi_{\mathrm{red}}^{2}<1
  12. χ 2 = i = 1 n ( O i - E i ) E i 2 \chi^{2}=\sum_{i=1}^{n}{\frac{(O_{i}-E_{i})}{E_{i}}^{2}}
  13. E i = ( F ( Y u ) - F ( Y l ) ) N E_{i}\,=\,\bigg(F(Y_{u})\,-\,F(Y_{l})\bigg)\,N
  14. χ 2 = ( 44 - 50 ) 2 50 + ( 56 - 50 ) 2 50 = 1.44 \chi^{2}={(44-50)^{2}\over 50}+{(56-50)^{2}\over 50}=1.44
  15. χ 2 = i = 1 k ( N i - n p i ) 2 n p i = all cells ( O - E ) 2 E . \chi^{2}=\sum_{i=1}^{k}{\frac{(N_{i}-np_{i})^{2}}{np_{i}}}=\sum_{\mathrm{all\ % cells}}{\frac{(\mathrm{O}-\mathrm{E})^{2}}{\mathrm{E}}}.
  16. N i = n \sum N_{i}=n

Gorman_polar_form.html

  1. i i
  2. e i ( p , u i ) e^{i}\left(p,u^{i}\right)
  3. u u
  4. e i ( p , u i ) = f i ( p ) + u i g ( p ) e^{i}\left(p,u^{i}\right)=f^{i}(p)+u^{i}g(p)
  5. f i ( p ) f^{i}\left(p\right)
  6. g ( p ) g\left(p\right)
  7. p p
  8. e i ( p , u i ) e^{i}\left(p,u^{i}\right)
  9. f i ( p ) f^{i}\left(p\right)
  10. g ( p ) g\left(p\right)
  11. f i ( p ) f^{i}\left(p\right)
  12. i i
  13. g ( p ) g\left(p\right)
  14. e i ( p , u i ) - f i ( p ) e^{i}\left(p,u^{i}\right)-f^{i}(p)
  15. u ¯ \bar{u}
  16. g ( p ) g\left(p\right)
  17. v i ( p , m i ) = m i - f i ( p ) g ( p ) v^{i}\left(p,m^{i}\right)=\frac{m^{i}-f^{i}(p)}{g(p)}
  18. m m
  19. e i ( p , u i ) e^{i}\left(p,u^{i}\right)
  20. m i m^{i}
  21. p p
  22. i i
  23. n n
  24. x n i ( p , m i ) = - v i ( p , m i ) p n v i ( p , m i ) m i = f i ( p ) p n + g ( p ) p n m - f i ( p ) g ( p ) x^{i}_{n}(p,m^{i})=-\frac{\frac{\partial v^{i}(p,m^{i})}{\partial p_{n}}}{% \frac{\partial v^{i}(p,m^{i})}{\partial m^{i}}}=\frac{\partial f^{i}(p)}{% \partial p_{n}}+\frac{\partial g(p)}{\partial p_{n}}\cdot\frac{m-f^{i}(p)}{g(p)}
  25. m m
  26. x n i ( p , m i ) m = g ( p ) p n g ( p ) \frac{\partial x^{i}_{n}(p,m^{i})}{\partial m}=\frac{\frac{\partial g(p)}{% \partial p_{n}}}{g(p)}
  27. f i ( p ) f^{i}(p)
  28. g ( p ) g(p)

Gossip_(video_game).html

  1. Δ x l , s = x l , o x s , o k 1 \Delta x_{l,s}=\frac{x_{l,o}x^{\prime}_{s,o}}{k_{1}}
  2. Δ x l , o = x l , s x s , o k 2 \Delta x_{l,o}=\frac{x_{l,s}x^{\prime}_{s,o}}{k_{2}}

GOST_(hash_function).html

  1. 𝒻 0 j \mathcal{f}0\mathcal{g}^{j}
  2. 𝒿 M 𝒿 \mathcal{j}M\mathcal{j}
  3. 𝓀 \mathcal{k}
  4. + +
  5. \oplus
  6. M M
  7. m n , m n - 1 , m n - 2 , , m 1 m_{n},m_{n-1},m_{n-2},...,m_{1}
  8. m n m_{n}
  9. H o u t = f ( H i n , m ) H_{out}\ =\ f(H_{in},m)
  10. H o u t H_{out}
  11. H i n H_{in}
  12. m m
  13. f f
  14. H i + 1 = f ( H i , m i ) \!H_{i+1}=f(H_{i},m_{i})
  15. H 1 H_{1}
  16. 0 256 0^{256}
  17. H n + 1 H_{n+1}
  18. H n + 2 = f ( H n + 1 , L ) H_{n+2}\ =\ f(H_{n+1},\ L)
  19. 2 256 2^{256}
  20. h = f ( H n + 2 , K ) h\ =\ f(H_{n+2},\ K)
  21. m 1 + m 2 + m 3 + + m n m_{1}+m_{2}+m_{3}+...+m_{n}
  22. h h
  23. h := i n i t i a l h\ :=initial
  24. Σ := 0 \Sigma\ :=\ 0
  25. L := 0 L\ :=\ 0
  26. | M | > 256 |M|>256
  27. h := f ( h , m i ) h\ :=\ f(h,\ m_{i})
  28. L := L + 256 L\ :=\ L\ +\ 256
  29. Σ := Σ + m i \Sigma\ :=\ \Sigma\ +\ m_{i}
  30. L := L + 𝒿 m n 𝒿 L\ :=\ L\ +\ \mathcal{j}\ m_{n}\ \mathcal{j}
  31. m n := 0 256 - 𝒿 m n 𝒿 𝓀 m n m_{n}\ :=\ {0}^{256\ -\ \mathcal{j}m_{n}\mathcal{j}}\mathcal{k}m_{n}
  32. Σ := Σ + m n \Sigma\ :=\ \Sigma\ +\ m_{n}
  33. h := f ( h , m n ) h\ :=\ f(h,\ m_{n})
  34. h := f ( h , L ) h\ :=\ f(h,\ L)
  35. h := f ( h , Σ ) h\ :=\ f(h,\ \Sigma)
  36. h h
  37. f f
  38. H o u t = f ( H i n , m ) H_{out}\ =\ f(H_{in},\ m)
  39. K 1 , K 2 , K 3 , K 4 K_{1},\ K_{2},\ K_{3},\ K_{4}
  40. H i n \ H_{in}
  41. K 1 , K 2 , K 3 , K 4 K_{1},\ K_{2},\ K_{3},\ K_{4}
  42. A ( Y ) = A ( y 4 𝓀 y 3 𝓀 y 2 𝓀 y 1 ) = ( y 1 y 2 ) 𝓀 y 4 𝓀 y 3 𝓀 y 2 A(Y)=A(y_{4}\ \mathcal{k}\ y_{3}\ \mathcal{k}\ y_{2}\ \mathcal{k}\ y_{1})=(y_{% 1}\oplus y_{2})\ \mathcal{k}\ y_{4}\ \mathcal{k}\ y_{3}\ \mathcal{k}\ y_{2}
  43. y 1 , y 2 , y 3 , y 4 y_{1},\ y_{2},\ y_{3},\ y_{4}
  44. P ( Y ) = P ( y 32 𝓀 y 31 𝓀 𝓀 y 1 ) = y φ ( 32 ) 𝓀 y φ ( 31 ) 𝓀 𝓀 y φ ( 1 ) P(Y)=P(y_{32}\mathcal{k}y_{31}\mathcal{k}\dots\mathcal{k}y_{1})=y_{\varphi(32)% }\mathcal{k}y_{\varphi(31)}\mathcal{k}\dots\mathcal{k}y_{\varphi(1)}
  45. φ ( i + 1 + 4 ( k - 1 ) ) = 8 i + k , i = 0 , , 3 , k = 1 , , 8 \varphi(i+1+4(k-1))=8i+k,\ i=0,\dots,3,\ k=1,\dots,8
  46. y 32 , y 31 , , y 1 y_{32},\ y_{31},\ \dots,\ y_{1}
  47. U := H i n , V := m , W := U V , K 1 = P ( W ) U\ :=\ H_{in},\quad V\ :=\ m,\quad W\ :=\ U\ \oplus\ V,\quad K_{1}\ =\ P(W)
  48. U := A ( U ) C j , V := A ( A ( V ) ) , W := U V , K j = P ( W ) U:=A(U)\oplus C_{j},\quad V:=A(A(V)),\quad W:=U\oplus V,\quad K_{j}=P(W)
  49. H i n H_{in}
  50. K 1 , K 2 , K 3 , K 4 K_{1},K_{2},K_{3},K_{4}
  51. H i n H_{in}
  52. H i n = h 4 𝓀 h 3 𝓀 h 2 𝓀 h 1 H_{in}=h_{4}\mathcal{k}h_{3}\mathcal{k}h_{2}\mathcal{k}h_{1}
  53. s 1 = E ( h 1 , K 1 ) s_{1}=E(h_{1},K_{1})
  54. s 2 = E ( h 2 , K 2 ) s_{2}=E(h_{2},K_{2})
  55. s 3 = E ( h 3 , K 3 ) s_{3}=E(h_{3},K_{3})
  56. s 4 = E ( h 4 , K 4 ) s_{4}=E(h_{4},K_{4})
  57. S = s 4 𝓀 s 3 𝓀 s 2 𝓀 s 1 S=s_{4}\mathcal{k}s_{3}\mathcal{k}s_{2}\mathcal{k}s_{1}
  58. H i n H_{in}
  59. H o u t H_{out}
  60. ψ ( Y ) = ψ ( y 16 𝓀 y 15 𝓀 𝓀 y 2 𝓀 y 1 ) = ( y 1 y 2 y 3 y 4 y 13 y 16 ) 𝓀 y 16 𝓀 y 15 𝓀 𝓀 y 3 𝓀 y 2 \psi(Y)=\psi(y_{16}\mathcal{k}y_{15}\mathcal{k}...\mathcal{k}y_{2}\mathcal{k}y% _{1})=(y_{1}\oplus y_{2}\oplus y_{3}\oplus y_{4}\oplus y_{13}\oplus y_{16})% \mathcal{k}y_{16}\mathcal{k}y_{15}\mathcal{k}...\mathcal{k}y_{3}\mathcal{k}y_{2}
  61. y 16 , y 15 , , y 2 , y 1 y_{16},y_{15},...,y_{2},y_{1}
  62. H o u t = ψ 61 ( H i n ψ ( m ψ 12 ( S ) ) ) H_{out}={\psi}^{61}(H_{in}\oplus\psi(m\oplus{\psi}^{12}(S)))
  63. ψ i {\psi}^{i}
  64. ψ \psi
  65. H 1 H_{1}
  66. H 1 H_{1}
  67. E E

Gouy_balance.html

  1. 1 2 \tfrac{1}{2}

Gödel_metric.html

  1. d s 2 = 1 2 ω 2 [ - ( d t + e x d z ) 2 + d x 2 + d y 2 + 1 2 e 2 x d z 2 ] , - < t , x , y , z < , ds^{2}=\frac{1}{2\omega^{2}}[-(dt+e^{x}dz)^{2}+dx^{2}+dy^{2}+\tfrac{1}{2}e^{2x% }dz^{2}],\qquad\qquad-\infty<t,x,y,z<\infty,
  2. ω \omega
  3. x , y , z x,y,z
  4. e 0 = 2 ω t \vec{e}_{0}=\sqrt{2}\omega\,\partial_{t}
  5. e 1 = 2 ω x \vec{e}_{1}=\sqrt{2}\omega\,\partial_{x}
  6. e 2 = 2 ω y \vec{e}_{2}=\sqrt{2}\omega\,\partial_{y}
  7. e 3 = 2 ω ( exp ( - x ) z - t ) . \vec{e}_{3}=2\omega\,\left(\exp(-x)\,\partial_{z}-\,\partial_{t}\right).
  8. e 0 \vec{e}_{0}
  9. e 2 \vec{e}_{2}
  10. - ω -\omega
  11. f 0 = e 0 \vec{f}_{0}=\vec{e}_{0}
  12. f 1 = cos ( ω t ) e 1 - sin ( ω t ) e 3 \vec{f}_{1}=\cos(\omega t)\,\vec{e}_{1}-\sin(\omega t)\,\vec{e}_{3}
  13. f 2 = e 2 \vec{f}_{2}=\vec{e}_{2}
  14. f 3 = sin ( ω t ) e 1 + cos ( ω t ) e 3 . \vec{f}_{3}=\sin(\omega t)\,\vec{e}_{1}+\cos(\omega t)\,\vec{e}_{3}.
  15. G a ^ b ^ = ω 2 diag ( - 1 , 1 , 1 , 1 ) + 2 ω 2 diag ( 1 , 0 , 0 , 0 ) . G^{\hat{a}\hat{b}}=\omega^{2}\,\operatorname{diag}(-1,1,1,1)+2\omega^{2}\,% \operatorname{diag}(1,0,0,0).
  16. ω 2 \omega^{2}
  17. ω 2 \omega^{2}
  18. t \partial_{t}
  19. y , z \partial_{y},\;\partial_{z}
  20. x - z z \partial_{x}-z\,\partial_{z}
  21. - 2 exp ( - x ) t + z x + ( exp ( - 2 x ) - z 2 / 2 ) z . -2\exp(-x)\,\partial_{t}+z\,\partial_{x}+\left(\exp(-2x)-z^{2}/2\right)\,% \partial_{z}.
  22. t , y , z t,y,z
  23. x x
  24. x = x 0 x=x_{0}
  25. y = y 0 y=y_{0}
  26. t = t 0 t=t_{0}
  27. u = e 0 \vec{u}=\vec{e}_{0}
  28. E [ u ] m ^ n ^ = ω 2 diag ( 1 , 0 , 1 ) . {E\left[\vec{u}\right]}_{\hat{m}\hat{n}}=\omega^{2}\,\operatorname{diag}(1,0,1).
  29. y \partial_{y}
  30. B [ u ] m ^ n ^ = 0. {B\left[\vec{u}\right]}_{\hat{m}\hat{n}}=0.
  31. R a b c d R a b c d = 12 ω 4 , R a b c d R a b c d = 0. R_{abcd}\,R^{abcd}=12\omega^{4},\;R_{abcd}{{}^{\star}R}^{abcd}=0.
  32. e 0 e 0 = 0 \nabla_{\vec{e}_{0}}\vec{e}_{0}=0
  33. - ω e 2 -\omega\vec{e}_{2}
  34. e 0 \vec{e}_{0}
  35. y \partial_{y}
  36. e j \vec{e}_{j}
  37. f j \vec{f}_{j}
  38. ω \omega
  39. e 2 \vec{e}_{2}
  40. e 0 = t , e 1 = z , e 2 = r , e 3 = 1 b ( r ) ( - a ( r ) t + ϕ ) \vec{e}_{0}=\partial_{t},\;\vec{e}_{1}=\partial_{z},\;\vec{e}_{2}=\partial_{r}% ,\,\vec{e}_{3}=\frac{1}{b(r)}\,\left(-a(r)\,\partial_{t}+\partial_{\phi}\right)
  41. e 0 \vec{e}_{0}
  42. G i ^ j ^ = μ diag ( 1 , 0 , 0 , 0 ) + p diag ( 0 , 1 , 1 , 1 ) G^{\hat{i}\hat{j}}=\mu\,\operatorname{diag}(1,0,0,0)+p\,\operatorname{diag}(0,% 1,1,1)
  43. b ′′′ = b ′′ b b , ( a ) 2 = 2 b ′′ b b^{\prime\prime\prime}=\frac{b^{\prime\prime}\,b^{\prime}}{b},\;\left(a^{% \prime}\right)^{2}=2\,b^{\prime\prime}\,b
  44. μ = p \mu=p
  45. μ = ω 2 \mu=\omega^{2}
  46. b ( r ) = sinh ( 2 ω r ) 2 ω , a ( r ) = cosh ( 2 ω r ) ω + c b(r)=\frac{\sinh(\sqrt{2}\omega\,r)}{\sqrt{2}\omega},\;a(r)=\frac{\cosh(\sqrt{% 2}\omega r)}{\omega}+c
  47. e 3 = 1 r ϕ + O ( 1 r 2 ) \vec{e}_{3}=\frac{1}{r}\,\partial_{\phi}+O\left(\frac{1}{r^{2}}\right)
  48. c = - 1 / ω c=-1/\omega
  49. e 0 = t , e 1 = z , e 2 = r , e 3 = 2 ω sinh ( 2 ω r ) ϕ - 2 sinh ( 2 ω r ) 1 + cosh ( 2 ω r ) t \vec{e}_{0}=\partial_{t},\;\vec{e}_{1}=\partial_{z},\;\vec{e}_{2}=\partial_{r}% ,\;\vec{e}_{3}=\frac{\sqrt{2}\omega}{\sinh(\sqrt{2}\omega r)}\,\partial_{\phi}% -\frac{\sqrt{2}\sinh(\sqrt{2}\omega r)}{1+\cosh(\sqrt{2}\omega r)}\,\partial_{t}
  50. ϕ \partial_{\phi}
  51. r = r c r=r_{c}
  52. r c = arccosh ( 3 ) 2 ω r_{c}=\frac{\operatorname{arccosh}(3)}{\sqrt{2}\omega}
  53. e 3 = ω 2 ϕ - t , \vec{e}_{3}=\frac{\omega}{2}\,\partial_{\phi}-\partial_{t},
  54. ω 2 ϕ = e 3 + e 0 \frac{\omega}{2}\,\partial_{\phi}=\vec{e}_{3}+\vec{e}_{0}
  55. r = r c r=r_{c}
  56. z z
  57. z z
  58. r = 0 r=0
  59. r = r c r=r_{c}
  60. ϕ \partial_{\phi}
  61. r = r c r=r_{c}
  62. r r c r\rightarrow r_{c}
  63. r = r c r=r_{c}

Graded_poset.html

  1. 0 < 1 < 2 < 0<1<2<\cdots
  2. W 0 , W 1 , W 2 , W_{0},W_{1},W_{2},...
  3. W i W_{i}
  4. 𝒫 ( S ) \mathcal{P}(S)
  5. S S
  6. x < z 1 < y x<z_{1}<y
  7. x < w 1 < w 2 < y x<w_{1}<w_{2}<y
  8. 𝒩 { } \mathcal{N}\cup\{\infty\}
  9. \infty

Grad–Shafranov_equation.html

  1. ψ \psi
  2. Δ * ψ = - μ 0 R 2 d p d ψ - 1 2 d F 2 d ψ , \Delta^{*}\psi=-\mu_{0}R^{2}\frac{dp}{d\psi}-\frac{1}{2}\frac{dF^{2}}{d\psi},
  3. μ 0 \mu_{0}
  4. p ( ψ ) p(\psi)
  5. F ( ψ ) = R B ϕ F(\psi)=RB_{\phi}
  6. B = 1 R ψ × e ^ ϕ + F R e ^ ϕ , \vec{B}=\frac{1}{R}\nabla\psi\times\hat{e}_{\phi}+\frac{F}{R}\hat{e}_{\phi},
  7. μ 0 J = 1 R d F d ψ ψ × e ^ ϕ - 1 R Δ * ψ e ^ ϕ . \mu_{0}\vec{J}=\frac{1}{R}\frac{dF}{d\psi}\nabla\psi\times\hat{e}_{\phi}-\frac% {1}{R}\Delta^{*}\psi\hat{e}_{\phi}.
  8. Δ * \Delta^{*}
  9. Δ * ψ R 2 ( 1 R 2 ψ ) = R R ( 1 R ψ R ) + 2 ψ Z 2 \Delta^{*}\psi\equiv R^{2}\vec{\nabla}\cdot\left(\frac{1}{R^{2}}\vec{\nabla}% \psi\right)=R\frac{\partial}{\partial R}\left(\frac{1}{R}\frac{\partial\psi}{% \partial R}\right)+\frac{\partial^{2}\psi}{\partial Z^{2}}
  10. F ( ψ ) F(\psi)
  11. p ( ψ ) p(\psi)
  12. z z
  13. / z = 0 \partial/\partial z=0
  14. B = ( A / y , - A / x , B z ( x , y ) ) , {B}=(\partial A/\partial y,-\partial A/\partial x,B_{z}(x,y)),
  15. B = A × z ^ + B z z ^ , {B}=\nabla A\times\hat{{z}}+B_{z}\hat{{z}},
  16. A ( x , y ) z ^ A(x,y)\hat{{z}}
  17. A \nabla A
  18. ψ \psi
  19. p = j × B , \nabla p={j}\times{B},
  20. p \nabla p
  21. / z \partial/\partial z
  22. j × B = 0 {j}_{\perp}\times{B}_{\perp}=0
  23. j {j}_{\perp}
  24. B {B}_{\perp}
  25. j × B = j z ( z ^ × B ) + j × z ^ B z , {j}\times{B}=j_{z}(\hat{{z}}\times{B_{\perp}})+{j_{\perp}}\times\hat{{z}}B_{z},
  26. \perp
  27. z z
  28. z z
  29. j z = - 2 A / μ 0 . j_{z}=-\nabla^{2}A/\mu_{0}.
  30. B = A × z ^ {B}_{\perp}=\nabla A\times\hat{{z}}
  31. j = ( 1 / μ 0 ) B z × z ^ {j}_{\perp}=(1/\mu_{0})\nabla B_{z}\times\hat{{z}}
  32. B {B}_{\perp}
  33. B z \nabla B_{z}
  34. B {B}_{\perp}
  35. B z B_{z}
  36. p p
  37. z ^ × B = A - ( z ^ A ) z ^ = A \hat{{z}}\times{B}_{\perp}=\nabla A-({\hat{z}}\cdot\nabla A){\hat{z}}=\nabla A
  38. j × B z z ^ = - ( 1 / μ 0 ) B z B z + ( B z / μ 0 ) ( z ^ B z ) z ^ = - ( 1 / μ 0 ) B z B z . {j}_{\perp}\times B_{z}{\hat{z}}=-(1/\mu_{0})B_{z}\nabla B_{z}+(B_{z}/\mu_{0})% ({\hat{z}}\cdot\nabla B_{z}){\hat{z}}=-(1/\mu_{0})B_{z}\nabla B_{z}.
  39. p \nabla p
  40. p = - [ ( 1 / μ 0 ) 2 A ] A - ( 1 / μ 0 ) B z B z . \nabla p=-\left[(1/\mu_{0})\nabla^{2}A\right]\nabla A-(1/\mu_{0})B_{z}\nabla B% _{z}.
  41. p p
  42. B z B_{z}
  43. A A
  44. p = ( d p / d A ) A \nabla p=(dp/dA)\nabla A
  45. B z = ( d B z / d A ) A \nabla B_{z}=(dB_{z}/dA)\nabla A
  46. A \nabla A
  47. 2 A = - μ 0 d d A ( p + B z 2 2 μ 0 ) . \nabla^{2}A=-\mu_{0}\frac{d}{dA}\left(p+\frac{B_{z}^{2}}{2\mu_{0}}\right).

Grammar-based_code.html

  1. x = x 1 x n x=x_{1}\cdots x_{n}
  2. x x
  3. G G
  4. G G

Graph_pebbling.html

  1. π ( K n ) = n \scriptstyle\pi(K_{n})\,=\,n
  2. K n K_{n}
  3. π ( P n ) = 2 n - 1 \scriptstyle\pi(P_{n})\,=\,2^{n-1}
  4. P n P_{n}
  5. π ( W n ) = n \scriptstyle\pi(W_{n})\,=\,n
  6. W n W_{n}
  7. s ( v ) = u V ( G ) 2 d ( u , v ) s(v)=\sum_{u\in V(G)}2^{d(u,v)}
  8. γ ( K n ) = 2 n - 1 \scriptstyle\gamma(K_{n})\,=\,2n-1
  9. K n \scriptstyle K_{n}
  10. γ ( P n ) = 2 n - 1 \scriptstyle\gamma(P_{n})\,=\,2^{n}-1
  11. P n \scriptstyle P_{n}
  12. γ ( W n ) = 4 n - 9 \scriptstyle\gamma(W_{n})\,=\,4n-9
  13. W n \scriptstyle W_{n}

Grassmann_number.html

  1. θ i \theta_{i}
  2. x x
  3. θ i θ j = - θ j θ i θ i x = x θ i . \theta_{i}\theta_{j}=-\theta_{j}\theta_{i}\qquad\theta_{i}x=x\theta_{i}.
  4. ( θ i ) 2 = 0 , (\theta_{i})^{2}=0,\,
  5. θ i θ i = - θ i θ i . \theta_{i}\theta_{i}=-\theta_{i}\theta_{i}.
  6. [ a f ( θ ) + b g ( θ ) ] d θ = a f ( θ ) d θ + b g ( θ ) d θ \int\,[af(\theta)+bg(\theta)]\,d\theta=a\int\,f(\theta)\,d\theta+b\int\,g(% \theta)\,d\theta
  7. [ θ f ( θ ) ] d θ = 0. \int\left[\frac{\partial}{\partial\theta}f(\theta)\right]\,d\theta=0.
  8. 1 d θ = 0 \int\,1\,d\theta=0
  9. θ d θ = 1. \int\,\theta\,d\theta=1.
  10. exp [ θ T A η ] d θ d η = det A \int\exp\left[\theta^{T}A\eta\right]\,d\theta\,d\eta=\det A
  11. θ 1 \theta_{1}
  12. θ 2 \theta_{2}
  13. θ 1 = [ 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 ] θ 2 = [ 0 0 0 0 0 0 0 0 1 0 0 0 0 - 1 0 0 ] θ 1 θ 2 = - θ 2 θ 1 = [ 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 ] . \theta_{1}=\begin{bmatrix}0&0&0&0\\ 1&0&0&0\\ 0&0&0&0\\ 0&0&1&0\end{bmatrix}\qquad\theta_{2}=\begin{bmatrix}0&0&0&0\\ 0&0&0&0\\ 1&0&0&0\\ 0&-1&0&0\end{bmatrix}\qquad\theta_{1}\theta_{2}=-\theta_{2}\theta_{1}=\begin{% bmatrix}0&0&0&0\\ 0&0&0&0\\ 0&0&0&0\\ 1&0&0&0\end{bmatrix}.
  14. θ i 1 θ i 2 θ i N + θ i N θ i 1 θ i 2 + = 0 \theta_{i_{1}}\theta_{i_{2}}\cdots\theta_{i_{N}}+\theta_{i_{N}}\theta_{i_{1}}% \theta_{i_{2}}\cdots+\cdots=0
  15. ( θ i ) N = 0 (\theta_{i})^{N}=0\,
  16. θ = [ 0 1 0 0 0 1 0 0 0 ] \theta=\begin{bmatrix}0&1&0\\ 0&0&1\\ 0&0&0\end{bmatrix}\qquad
  17. θ 3 = 0 \theta^{3}=0
  18. θ 1 ( θ 2 ) 2 + θ 2 θ 1 θ 2 + ( θ 2 ) 2 θ 1 = 0 \theta_{1}(\theta_{2})^{2}+\theta_{2}\theta_{1}\theta_{2}+(\theta_{2})^{2}% \theta_{1}=0
  19. θ 1 ( θ 2 ) 2 = - 1 2 θ 2 θ 1 θ 2 = ( θ 2 ) 2 θ 1 \theta_{1}(\theta_{2})^{2}=-\frac{1}{2}\theta_{2}\theta_{1}\theta_{2}=(\theta_% {2})^{2}\theta_{1}
  20. ( θ 1 ) 2 ( θ 2 ) 2 = ( θ 2 ) 2 ( θ 1 ) 2 = θ 1 ( θ 2 ) 2 θ 1 = θ 2 ( θ 1 ) 2 θ 2 = - 1 2 θ 1 θ 2 θ 1 θ 2 = - 1 2 θ 2 θ 1 θ 2 θ 1 , (\theta_{1})^{2}(\theta_{2})^{2}=(\theta_{2})^{2}(\theta_{1})^{2}=\theta_{1}(% \theta_{2})^{2}\theta_{1}=\theta_{2}(\theta_{1})^{2}\theta_{2}=-\frac{1}{2}% \theta_{1}\theta_{2}\theta_{1}\theta_{2}=-\frac{1}{2}\theta_{2}\theta_{1}% \theta_{2}\theta_{1},
  21. ( A a b c θ a η b ψ c ) 4 = det ( A ) ( θ 1 ) 2 ( θ 2 ) 2 ( η 1 ) 2 ( η 2 ) 2 ( ψ 1 ) 2 ( ψ 2 ) 2 . (A^{abc}\theta_{a}\eta_{b}\psi_{c})^{4}=\det(A)(\theta_{1})^{2}(\theta_{2})^{2% }(\eta_{1})^{2}(\eta_{2})^{2}(\psi_{1})^{2}(\psi_{2})^{2}.

Gravitational_acceleration.html

  1. F = G m 1 m 2 r 2 F=G\frac{m_{1}m_{2}}{r^{2}}
  2. m 1 m_{1}
  3. m 2 m_{2}
  4. G G
  5. r r
  6. 𝐠 = - G M r 2 𝐫 ^ \mathbf{g}=-{GM\over r^{2}}\mathbf{\hat{r}}
  7. M M
  8. 𝐫 ^ \mathbf{\hat{r}}
  9. 𝐅 = m 𝐠 \mathbf{F}=m\mathbf{g}
  10. 𝐅 \mathbf{F}
  11. m m
  12. 𝐠 \mathbf{g}
  13. 𝐠 \mathbf{g}
  14. g = g=
  15. g g
  16. g = g 45 - 1 2 ( g poles - g equator ) cos ( 2 l a t π 180 ) g=g_{45}-\tfrac{1}{2}(g_{\mathrm{poles}}-g_{\mathrm{equator}})\cos\left(2\,lat% \,\frac{\pi}{180}\right)
  17. g poles g_{\mathrm{poles}}
  18. g 45 g_{45}
  19. g equator g_{\mathrm{equator}}
  20. J 2 J_{2}
  21. 𝐠 = - G M r 2 𝐫 ^ \mathbf{g}=-{GM\over r^{2}}\mathbf{\hat{r}}
  22. 𝐫 ^ \scriptstyle\mathbf{\hat{r}}

Gravitational_instanton.html

  1. σ 1 = sin ψ d θ - cos ψ sin θ d ϕ \sigma_{1}=\sin\psi\,d\theta-\cos\psi\sin\theta\,d\phi
  2. σ 2 = cos ψ d θ + sin ψ sin θ d ϕ \sigma_{2}=\cos\psi\,d\theta+\sin\psi\sin\theta\,d\phi
  3. σ 3 = d ψ + cos θ d ϕ . \sigma_{3}=d\psi+\cos\theta\,d\phi.
  4. d s 2 = 1 4 r + n r - n d r 2 + r - n r + n n 2 σ 3 2 + 1 4 ( r 2 - n 2 ) ( σ 1 2 + σ 2 2 ) ds^{2}=\frac{1}{4}\frac{r+n}{r-n}dr^{2}+\frac{r-n}{r+n}n^{2}{\sigma_{3}}^{2}+% \frac{1}{4}(r^{2}-n^{2})({\sigma_{1}}^{2}+{\sigma_{2}}^{2})
  5. d s 2 = ( 1 - a r 4 ) - 1 d r 2 + r 2 4 ( 1 - a r 4 ) σ 3 2 + r 2 4 ( σ 1 2 + σ 2 2 ) . ds^{2}=\left(1-\frac{a}{r^{4}}\right)^{-1}dr^{2}+\frac{r^{2}}{4}\left(1-\frac{% a}{r^{4}}\right){\sigma_{3}}^{2}+\frac{r^{2}}{4}(\sigma_{1}^{2}+\sigma_{2}^{2}).
  6. r a 1 / 4 r\geq a^{1/4}
  7. r a 1 / 4 r\rightarrow a^{1/4}
  8. θ = 0 , π \theta=0,\pi
  9. a = 0 a=0
  10. ψ \psi
  11. 4 π 4\pi
  12. a 0 a\neq 0
  13. ψ \psi
  14. 2 π 2\pi
  15. r r\rightarrow\infty
  16. d s 2 = d r 2 + r 2 4 σ 3 2 + r 2 4 ( σ 1 2 + σ 2 2 ) ds^{2}=dr^{2}+\frac{r^{2}}{4}\sigma_{3}^{2}+\frac{r^{2}}{4}(\sigma_{1}^{2}+% \sigma_{2}^{2})
  17. a 0 a\neq 0
  18. ψ \psi
  19. ψ ψ + 2 π \psi\,{\sim}\,\psi+2\pi
  20. d s 2 = 1 V ( 𝐱 ) ( d ψ + s y m b o l ω d 𝐱 ) 2 + V ( 𝐱 ) d 𝐱 d 𝐱 , ds^{2}=\frac{1}{V(\mathbf{x})}(d\psi+symbol{\omega}\cdot d\mathbf{x})^{2}+V(% \mathbf{x})d\mathbf{x}\cdot d\mathbf{x},
  21. V = ± × s y m b o l ω , V = i = 1 2 1 | 𝐱 - 𝐱 i | . \nabla V=\pm\nabla\times symbol{\omega},\quad V=\sum_{i=1}^{2}\frac{1}{|% \mathbf{x}-\mathbf{x}_{i}|}.
  22. V = 1 | 𝐱 | V=\frac{1}{|\mathbf{x}|}
  23. ρ = r 2 / 4 \rho=r^{2}/4
  24. ρ \rho
  25. θ \theta
  26. ϕ \phi
  27. 𝐱 \mathbf{x}
  28. 𝐱 = ( ρ sin θ cos ϕ , ρ sin θ sin ϕ , ρ cos θ ) \mathbf{x}=(\rho\sin\theta\cos\phi,\rho\sin\theta\sin\phi,\rho\cos\theta)
  29. ψ \psi
  30. ψ ψ + 4 π . \psi\ {\sim}\ \psi+4\pi.
  31. V = i = 1 n 1 | 𝐱 - 𝐱 i | . \quad V=\sum_{i=1}^{n}\frac{1}{|\mathbf{x}-\mathbf{x}_{i}|}.
  32. 𝐱 i \mathbf{x}_{i}
  33. r r\rightarrow\infty
  34. 𝐱 i \mathbf{x}_{i}
  35. θ \theta
  36. ϕ \phi
  37. r r / n r\rightarrow r/\sqrt{n}
  38. d s 2 = d r 2 + r 2 4 ( d ψ n + cos θ d ϕ ) 2 + r 2 4 [ ( σ 1 L ) 2 + ( σ 2 L ) 2 ] . ds^{2}=dr^{2}+\frac{r^{2}}{4}\left({d\psi\over n}+\cos\theta\,d\phi\right)^{2}% +\frac{r^{2}}{4}[(\sigma_{1}^{L})^{2}+(\sigma_{2}^{L})^{2}].
  39. ψ \psi
  40. ψ / n \psi/n
  41. 4 π / n 4\pi/n
  42. 4 π 4\pi
  43. ψ ψ + 4 π k / n \psi\ {\sim}\ \psi+4\pi k/n
  44. e 2 π i k / n e^{2\pi ik/n}
  45. d s 2 = 1 V ( 𝐱 ) ( d τ + s y m b o l ω d 𝐱 ) 2 + V ( 𝐱 ) d 𝐱 d 𝐱 , ds^{2}=\frac{1}{V(\mathbf{x})}(d\tau+symbol{\omega}\cdot d\mathbf{x})^{2}+V(% \mathbf{x})d\mathbf{x}\cdot d\mathbf{x},
  46. V = ± × s y m b o l ω , V = ε + 2 M i = 1 k 1 | 𝐱 - 𝐱 i | . \nabla V=\pm\nabla\times symbol{\omega},\quad V=\varepsilon+2M\sum_{i=1}^{k}% \frac{1}{|\mathbf{x}-\mathbf{x}_{i}|}.
  47. ϵ = 1 \epsilon=1
  48. ϵ = 0 \epsilon=0
  49. k = 1 k=1
  50. ϵ = 0 \epsilon=0
  51. k = 2 k=2

Gravity_well.html

  1. Φ ( 𝐱 ) = - G M | 𝐱 | . \Phi(\mathbf{x})=-\frac{GM}{|\mathbf{x}|}.
  2. k 2 h = - g ρ k\nabla^{2}h=-g\rho
  3. 2 Φ = - 4 π G ρ \nabla^{2}\Phi=-4\pi G\rho

Gray_graph.html

  1. [ - 25 , 7 , - 7 , 13 , - 13 , 25 ] 9 . [-25,7,-7,13,-13,25]^{9}.
  2. ( x - 3 ) x 16 ( x + 3 ) ( x 2 - 6 ) 6 ( x 2 - 3 ) 12 . (x-3)x^{16}(x+3)(x^{2}-6)^{6}(x^{2}-3)^{12}.

Gresley_conjugated_valve_gear.html

  1. sin θ \scriptstyle\sin\;\theta
  2. sin ( θ + 120 ) \scriptstyle\sin(\theta+120^{\circ})
  3. sin ( θ - 120 ) \scriptstyle\sin(\theta-120^{\circ})
  4. - 1 2 sin θ \scriptstyle-\tfrac{1}{2}\sin\theta
  5. sin ( θ + 120 ) \scriptstyle\sin(\theta+120^{\circ})
  6. sin ( θ - 120 ) \scriptstyle\sin(\theta-120^{\circ})
  7. θ \scriptstyle\;\!\theta

Grill_(cryptology).html

  1. L L
  2. M M
  3. N N
  4. N N
  5. 26 ! = 403 , 291 , 461 , 126 , 605 , 635 , 584 , 000 , 000 26!=403,291,461,126,605,635,584,000,000
  6. 26 ! 2 13 13 ! = 7 , 905 , 853 , 580 , 025 \frac{26!}{2^{13}\,13!}=7,905,853,580,025
  7. 26 ! 2 6 6 ! 14 ! = 100 , 391 , 791 , 500 \frac{26!}{2^{6}\,6!\,14!}=100,391,791,500
  8. A = ( a b c d e f g h i j k l m n o p q r s t u v w x y z ) 𝚙𝚝𝚔𝚡𝚛𝚣𝚚𝚜𝚠𝚖𝚌𝚘𝚓𝚢𝚕𝚊𝚐𝚎𝚑𝚋𝚟𝚞𝚒𝚍𝚗𝚏 = (ap)(bt)(ck)(dx)(er)(fz)(gq)(hs)(iw)(jm)(lo)(ny)(uv) B = ( a b c d e f g h i j k l m n o p q r s t u v w x y z ) 𝚞𝚔𝚣𝚖𝚢𝚡𝚛𝚜𝚗𝚚𝚋𝚠𝚍𝚒𝚙𝚘𝚓𝚐𝚑𝚟𝚊𝚝𝚕𝚏𝚎𝚌 = (au)(bk)(cz)(dm)(ey)(fx)(gr)(hs)(in)(jq)(lw)(op)(tv) C = ( a b c d e f g h i j k l m n o p q r s t u v w x y z ) 𝚞𝚢𝚖𝚜𝚣𝚗𝚚𝚠𝚘𝚟𝚝𝚙𝚌𝚏𝚒𝚕𝚐𝚡𝚍𝚔𝚊𝚓𝚑𝚛𝚋𝚎 = (au)(by)(cm)(ds)(ez)(fn)(gq)(hw)(io)(jv)(kt)(lp)(rx) D = ( a b c d e f g h i j k l m n o p q r s t u v w x y z ) 𝚓𝚠𝚟𝚛𝚘𝚜𝚞𝚢𝚣𝚊𝚝𝚚𝚡𝚙𝚎𝚗𝚕𝚍𝚏𝚔𝚐𝚌𝚋𝚖𝚑𝚒 = (aj)(bw)(cv)(dr)(eo)(fs)(gu)(hy)(iz)(kt)(lq)(mx)(np) E = ( a b c d e f g h i j k l m n o p q r s t u v w x y z ) 𝚓𝚡𝚟𝚚𝚕𝚝𝚗𝚢𝚙𝚊𝚜𝚎𝚞𝚐𝚣𝚒𝚍𝚠𝚔𝚏𝚖𝚌𝚛𝚋𝚑𝚘 = (aj)(bx)(cv)(dq)(el)(ft)(gn)(hy)(ip)(ks)(mu)(oz)(rw) F = ( a b c d e f g h i j k l m n o p q r s t u v w x y z ) 𝚗𝚟𝚢𝚔𝚣𝚞𝚝𝚜𝚕𝚡𝚍𝚒𝚘𝚊𝚖𝚠𝚛𝚚𝚑𝚐𝚏𝚋𝚙𝚓𝚌𝚎 = (an)(bv)(cy)(dk)(ez)(fu)(gt)(hs)(il)(jx)(mo)(pw)(qr) \begin{aligned}\displaystyle A=&\displaystyle{\left({{\texttt{}}\atop{% abcdefghijklmnopqrstuvwxyz}}\right)}\texttt{ptkxrzqswmcojylagehbvuidnf}&% \displaystyle=\texttt{(ap)(bt)(ck)(dx)(er)(fz)(gq)(hs)(iw)(jm)(lo)(ny)(uv)}\\ \displaystyle B=&\displaystyle{\left({{\texttt{}}\atop{% abcdefghijklmnopqrstuvwxyz}}\right)}\texttt{ukzmyxrsnqbwdipojghvatlfec}&% \displaystyle=\texttt{(au)(bk)(cz)(dm)(ey)(fx)(gr)(hs)(in)(jq)(lw)(op)(tv)}\\ \displaystyle C=&\displaystyle{\left({{\texttt{}}\atop{% abcdefghijklmnopqrstuvwxyz}}\right)}\texttt{uymsznqwovtpcfilgxdkajhrbe}&% \displaystyle=\texttt{(au)(by)(cm)(ds)(ez)(fn)(gq)(hw)(io)(jv)(kt)(lp)(rx)}\\ \displaystyle D=&\displaystyle{\left({{\texttt{}}\atop{% abcdefghijklmnopqrstuvwxyz}}\right)}\texttt{jwvrosuyzatqxpenldfkgcbmhi}&% \displaystyle=\texttt{(aj)(bw)(cv)(dr)(eo)(fs)(gu)(hy)(iz)(kt)(lq)(mx)(np)}\\ \displaystyle E=&\displaystyle{\left({{\texttt{}}\atop{% abcdefghijklmnopqrstuvwxyz}}\right)}\texttt{jxvqltnypaseugzidwkfmcrbho}&% \displaystyle=\texttt{(aj)(bx)(cv)(dq)(el)(ft)(gn)(hy)(ip)(ks)(mu)(oz)(rw)}\\ \displaystyle F=&\displaystyle{\left({{\texttt{}}\atop{% abcdefghijklmnopqrstuvwxyz}}\right)}\texttt{nvykzutslxdioamwrqhgfbpjce}&% \displaystyle=\texttt{(an)(bv)(cy)(dk)(ez)(fu)(gt)(hs)(il)(jx)(mo)(pw)(qr)}\\ \end{aligned}
  9. p 1 = c 1 A - 1 = p 4 = c 4 D - 1 p 2 = c 2 B - 1 = p 5 = c 5 E - 1 p 3 = c 3 C - 1 = p 6 = c 6 F - 1 \begin{aligned}\displaystyle p_{1}&\displaystyle=c_{1}A^{-1}&\displaystyle=p_{% 4}&\displaystyle=c_{4}D^{-1}\\ \displaystyle p_{2}&\displaystyle=c_{2}B^{-1}&\displaystyle=p_{5}&% \displaystyle=c_{5}E^{-1}\\ \displaystyle p_{3}&\displaystyle=c_{3}C^{-1}&\displaystyle=p_{6}&% \displaystyle=c_{6}F^{-1}\\ \end{aligned}
  10. p 1 D \displaystyle p_{1}D
  11. c 1 A - 1 D \displaystyle c_{1}A^{-1}D
  12. A A
  13. D D
  14. A A = I AA=I
  15. I I
  16. c 1 A D \displaystyle c_{1}AD
  17. A D \displaystyle AD
  18. A D AD
  19. A D \displaystyle AD
  20. C = (ez)... F = (ez)... \begin{aligned}\displaystyle C&\displaystyle=\texttt{(ez)...}\\ \displaystyle F&\displaystyle=\texttt{(ez)...}\\ \end{aligned}
  21. A = (er)... D = (eo)... \begin{aligned}\displaystyle A&\displaystyle=\texttt{(er)...}\\ \displaystyle D&\displaystyle=\texttt{(eo)...}\\ \end{aligned}
  22. A = (er)(dx)... D = (eo)... \begin{aligned}\displaystyle A&\displaystyle=\texttt{(er)(dx)...}\\ \displaystyle D&\displaystyle=\texttt{(eo)...}\\ \end{aligned}
  23. A = (er)(dx)(jm)(ap)(ny)(hs)(fz)(iw)(bt)(ck)(uv)(gq)(lo) D = (eo)(lq)(gu)(cv)(kt)(bw)(iz)(fs)(hy)(np)(ag)(mx)(dr) \begin{aligned}\displaystyle A&\displaystyle=\texttt{(er)(dx)(jm)(ap)(ny)(hs)(% fz)(iw)(bt)(ck)(uv)(gq)(lo)}\\ \displaystyle D&\displaystyle=\texttt{(eo)(lq)(gu)(cv)(kt)(bw)(iz)(fs)(hy)(np)% (ag)(mx)(dr)}\\ \end{aligned}
  24. A = (ap)(bt)(ck)(dx)(er)(fz)(gq)(hs)(iw)(jm)(lo)(ny)(uv) D = (aj)(bw)(cv)(dr)(eo)(fs)(gu)(hy)(iz)(kt)(lq)(mx)(np) \begin{aligned}\displaystyle A&\displaystyle=\texttt{(ap)(bt)(ck)(dx)(er)(fz)(% gq)(hs)(iw)(jm)(lo)(ny)(uv)}\\ \displaystyle D&\displaystyle=\texttt{(aj)(bw)(cv)(dr)(eo)(fs)(gu)(hy)(iz)(kt)% (lq)(mx)(np)}\\ \end{aligned}
  25. B = (ey)(hs)(kb)(xf)(tv)(cz)(op)(in)(gr)(wl)... E = (le)(rw)(ng)(pi)(zo)(vc)(ft)(bx)(sk)(yh)... \begin{aligned}\displaystyle B&\displaystyle=\texttt{(ey)(hs)(kb)(xf)(tv)(cz)(% op)(in)(gr)(wl)...}\\ \displaystyle E&\displaystyle=\texttt{(le)(rw)(ng)(pi)(zo)(vc)(ft)(bx)(sk)(yh)% ...}\\ \end{aligned}
  26. S S
  27. H H
  28. L M N LMN
  29. R R
  30. P P
  31. P P
  32. A = S H ( P 1 N P - 1 ) L M R M - 1 L - 1 ( P 1 N - 1 P - 1 ) H - 1 S - 1 B = S H ( P 2 N P - 2 ) L M R M - 1 L - 1 ( P 2 N - 1 P - 2 ) H - 1 S - 1 C = S H ( P 3 N P - 3 ) L M R M - 1 L - 1 ( P 3 N - 1 P - 3 ) H - 1 S - 1 D = S H ( P 4 N P - 4 ) L M R M - 1 L - 1 ( P 4 N - 1 P - 4 ) H - 1 S - 1 E = S H ( P 5 N P - 5 ) L M R M - 1 L - 1 ( P 5 N - 1 P - 5 ) H - 1 S - 1 F = S H ( P 6 N P - 6 ) L M R M - 1 L - 1 ( P 6 N - 1 P - 6 ) H - 1 S - 1 \begin{aligned}\displaystyle A&\displaystyle=SH(P^{1}NP^{-1})LMRM^{-1}L^{-1}(P% ^{1}N^{-1}P^{-1})H^{-1}S^{-1}\\ \displaystyle B&\displaystyle=SH(P^{2}NP^{-2})LMRM^{-1}L^{-1}(P^{2}N^{-1}P^{-2% })H^{-1}S^{-1}\\ \displaystyle C&\displaystyle=SH(P^{3}NP^{-3})LMRM^{-1}L^{-1}(P^{3}N^{-1}P^{-3% })H^{-1}S^{-1}\\ \displaystyle D&\displaystyle=SH(P^{4}NP^{-4})LMRM^{-1}L^{-1}(P^{4}N^{-1}P^{-4% })H^{-1}S^{-1}\\ \displaystyle E&\displaystyle=SH(P^{5}NP^{-5})LMRM^{-1}L^{-1}(P^{5}N^{-1}P^{-5% })H^{-1}S^{-1}\\ \displaystyle F&\displaystyle=SH(P^{6}NP^{-6})LMRM^{-1}L^{-1}(P^{6}N^{-1}P^{-6% })H^{-1}S^{-1}\\ \end{aligned}
  33. Q Q
  34. Q = L M R M - 1 L - 1 Q=LMRM^{-1}L^{-1}
  35. A \displaystyle A
  36. H H
  37. H = ( q w e r t z u i o a s d f g h j k p y x c v b n m l a b c d e f g h i j k l m n o p q r s t u v w x y z ) H={\left({{qwertzuioasdfghjkpyxcvbnml}\atop{abcdefghijklmnopqrstuvwxyz}}\right)}
  38. H H
  39. H = ( a b c d e f g h i j k l m n o p q r s t u v w x y z a b c d e f g h i j k l m n o p q r s t u v w x y z ) H={\left({{abcdefghijklmnopqrstuvwxyz}\atop{abcdefghijklmnopqrstuvwxyz}}\right)}
  40. S S
  41. H - 1 S - 1 A S H = ( P 1 N P - 1 ) Q ( P 1 N - 1 P - 1 ) H - 1 S - 1 B S H = ( P 2 N P - 2 ) Q ( P 2 N - 1 P - 2 ) H - 1 S - 1 C S H = ( P 3 N P - 3 ) Q ( P 3 N - 1 P - 3 ) H - 1 S - 1 D S H = ( P 4 N P - 4 ) Q ( P 4 N - 1 P - 4 ) H - 1 S - 1 E S H = ( P 5 N P - 5 ) Q ( P 5 N - 1 P - 5 ) H - 1 S - 1 F S H = ( P 6 N P - 6 ) Q ( P 6 N - 1 P - 6 ) \begin{aligned}\displaystyle H^{-1}S^{-1}ASH&\displaystyle=(P^{1}NP^{-1})Q(P^{% 1}N^{-1}P^{-1})\\ \displaystyle H^{-1}S^{-1}BSH&\displaystyle=(P^{2}NP^{-2})Q(P^{2}N^{-1}P^{-2})% \\ \displaystyle H^{-1}S^{-1}CSH&\displaystyle=(P^{3}NP^{-3})Q(P^{3}N^{-1}P^{-3})% \\ \displaystyle H^{-1}S^{-1}DSH&\displaystyle=(P^{4}NP^{-4})Q(P^{4}N^{-1}P^{-4})% \\ \displaystyle H^{-1}S^{-1}ESH&\displaystyle=(P^{5}NP^{-5})Q(P^{5}N^{-1}P^{-5})% \\ \displaystyle H^{-1}S^{-1}FSH&\displaystyle=(P^{6}NP^{-6})Q(P^{6}N^{-1}P^{-6})% \\ \end{aligned}
  42. P P
  43. U \displaystyle U
  44. U V \displaystyle UV
  45. V W = N P - 1 N - 1 ( U V ) N P 1 N - 1 W X = N P - 1 N - 1 ( V W ) N P 1 N - 1 X Y = N P - 1 N - 1 ( W X ) N P 1 N - 1 Y Z = N P - 1 N - 1 ( X Y ) N P 1 N - 1 \begin{aligned}\displaystyle VW&\displaystyle=NP^{-1}N^{-1}(UV)NP^{1}N^{-1}\\ \displaystyle WX&\displaystyle=NP^{-1}N^{-1}(VW)NP^{1}N^{-1}\\ \displaystyle XY&\displaystyle=NP^{-1}N^{-1}(WX)NP^{1}N^{-1}\\ \displaystyle YZ&\displaystyle=NP^{-1}N^{-1}(XY)NP^{1}N^{-1}\\ \end{aligned}
  46. U V W X Y Z UVWXYZ
  47. U V UV
  48. V W VW
  49. U V UV
  50. V W VW
  51. N N
  52. P P
  53. S S
  54. Q Q
  55. ( P 1 N - 1 P - 1 ) S - 1 A S ( P 1 N P - 1 ) \displaystyle(P^{1}N^{-1}P^{-1})S^{-1}AS(P^{1}NP^{-1})
  56. Q Q
  57. N N
  58. N N
  59. Q Q
  60. S S
  61. Q Q
  62. P 0 N P - 0 𝚋𝚍𝚏𝚑𝚓𝚕𝚌𝚙𝚛𝚝𝚡𝚟𝚣𝚗𝚢𝚎𝚒𝚠𝚐𝚊𝚔𝚖𝚞𝚜𝚚𝚘 P 1 N P - 1 𝚌𝚎𝚐𝚒𝚔𝚋𝚘𝚚𝚜𝚠𝚞𝚢𝚖𝚡𝚍𝚑𝚟𝚏𝚣𝚓𝚕𝚝𝚛𝚙𝚗𝚊 P 2 N P - 2 𝚍𝚏𝚑𝚓𝚊𝚗𝚙𝚛𝚟𝚝𝚡𝚕𝚠𝚌𝚐𝚞𝚎𝚢𝚒𝚔𝚜𝚚𝚘𝚖𝚣𝚋 P 25 N P - 25 𝚙𝚌𝚎𝚐𝚒𝚔𝚖𝚍𝚚𝚜𝚞𝚢𝚠𝚊𝚘𝚣𝚏𝚓𝚡𝚑𝚋𝚕𝚗𝚟𝚝𝚛 P 0 N P - 0 𝚋𝚍𝚏𝚑𝚓𝚕𝚌𝚙𝚛𝚝𝚡𝚟𝚣𝚗𝚢𝚎𝚒𝚠𝚐𝚊𝚔𝚖𝚞𝚜𝚚𝚘 P 1 N P - 1 𝚌𝚎𝚐𝚒𝚔𝚋𝚘𝚚𝚜𝚠𝚞𝚢𝚖𝚡𝚍𝚑𝚟𝚏𝚣𝚓𝚕𝚝𝚛𝚙𝚗𝚊 P 2 N P - 2 𝚍𝚏𝚑𝚓𝚊𝚗𝚙𝚛𝚟𝚝𝚡𝚕𝚠𝚌𝚐𝚞𝚎𝚢𝚒𝚔𝚜𝚚𝚘𝚖𝚣𝚋 P 3 N P - 3 𝚎𝚐𝚒𝚣𝚖𝚘𝚚𝚞𝚜𝚠𝚔𝚟𝚋𝚏𝚝𝚍𝚡𝚑𝚓𝚛𝚙𝚗𝚕𝚢𝚊𝚌 P 4 N P - 4 𝚏𝚑𝚢𝚕𝚗𝚙𝚝𝚛𝚟𝚓𝚞𝚊𝚎𝚜𝚌𝚠𝚐𝚒𝚚𝚘𝚖𝚔𝚡𝚣𝚋𝚍 \begin{aligned}\displaystyle P^{0}&\displaystyle NP^{-0}&\displaystyle\ % \texttt{bdfhjlcprtxvznyeiwgakmusqo}\\ \displaystyle P^{1}&\displaystyle NP^{-1}&\displaystyle\ \texttt{% cegikboqswuymxdhvfzjltrpna}\\ \displaystyle P^{2}&\displaystyle NP^{-2}&\displaystyle\ \texttt{% dfhjanprvtxlwcgueyiksqomzb}\\ &\displaystyle...&\displaystyle...\\ \displaystyle P^{25}&\displaystyle NP^{-25}&\displaystyle\ \texttt{% pcegikmdqsuywaozfjxhblnvtr}\\ \displaystyle P^{0}&\displaystyle NP^{-0}&\displaystyle\ \texttt{% bdfhjlcprtxvznyeiwgakmusqo}\\ \displaystyle P^{1}&\displaystyle NP^{-1}&\displaystyle\ \texttt{% cegikboqswuymxdhvfzjltrpna}\\ \displaystyle P^{2}&\displaystyle NP^{-2}&\displaystyle\ \texttt{% dfhjanprvtxlwcgueyiksqomzb}\\ \displaystyle P^{3}&\displaystyle NP^{-3}&\displaystyle\ \texttt{% egizmoquswkvbftdxhjrpnlyac}\\ \displaystyle P^{4}&\displaystyle NP^{-4}&\displaystyle\ \texttt{% fhylnptrvjuaescwgiqomkxzbd}\\ \end{aligned}
  63. A A
  64. F F
  65. N N
  66. Q Q
  67. Q Q
  68. A A
  69. Q Q
  70. Q Q
  71. A B C D E F ABCDEF
  72. D D
  73. Q Q
  74. D D
  75. A A
  76. Q Q
  77. E E
  78. Q Q
  79. Q Q
  80. A B C D E F ABCDEF
  81. Q Q
  82. Q Q
  83. Q Q
  84. Q Q
  85. S S
  86. Q Q
  87. N N
  88. S S
  89. Q Q
  90. S S
  91. Q Q
  92. Q Q
  93. N N
  94. S S
  95. A A
  96. Q Q
  97. B C D E F BCDEF
  98. Q Q
  99. Q Q
  100. B B
  101. Q Q
  102. D D
  103. Q Q
  104. S S
  105. Q Q
  106. S S
  107. E E
  108. Q Q
  109. F F
  110. Q Q
  111. S S
  112. Q Q
  113. S S
  114. Q Q
  115. Q Q
  116. N N
  117. S S
  118. Q Q
  119. L L
  120. M M
  121. 26 < s u p > 2 = 676 26<sup>2=676

Gromov–Witten_invariant.html

  1. ¯ g , n \overline{\mathcal{M}}_{g,n}
  2. ¯ g , n ( X , A ) \overline{\mathcal{M}}_{g,n}(X,A)
  3. ¯ g , n ( X , A ) \overline{\mathcal{M}}_{g,n}(X,A)
  4. ( C , x 1 , , x n , f ) (C,x_{1},\cdots,x_{n},f)
  5. d := 2 c 1 X ( A ) + ( 2 k - 6 ) ( 1 - g ) + 2 n . d:=2c_{1}^{X}(A)+(2k-6)(1-g)+2n.
  6. st ( C , x 1 , , x n ) ¯ g , n ( X , A ) \mathrm{st}(C,x_{1},\cdots,x_{n})\in\overline{\mathcal{M}}_{g,n}(X,A)
  7. Y := ¯ g , n × X n , Y:=\overline{\mathcal{M}}_{g,n}\times X^{n},
  8. { ev : ¯ g , n ( X , A ) Y ev ( C , x 1 , , x n , f ) = ( st ( C , x 1 , , x n ) , f ( x 1 ) , , f ( x n ) ) . \begin{cases}\mathrm{ev}:\overline{\mathcal{M}}_{g,n}(X,A)\to Y\\ \mathrm{ev}(C,x_{1},\cdots,x_{n},f)=\left(\mathrm{st}(C,x_{1},\cdots,x_{n}),f(% x_{1}),\cdots,f(x_{n})\right).\end{cases}
  9. G W g , n X , A H d ( Y , 𝐐 ) . GW_{g,n}^{X,A}\in H_{d}(Y,\mathbf{Q}).
  10. ¯ g , n \overline{\mathcal{M}}_{g,n}
  11. G W g , n X , A ( β , α 1 , , α n ) := G W g , n X , A β α 1 α n H 0 ( Y , 𝐐 ) , GW_{g,n}^{X,A}(\beta,\alpha_{1},\ldots,\alpha_{n}):=GW_{g,n}^{X,A}\cdot\beta% \cdot\alpha_{1}\cdot\cdots\cdot\alpha_{n}\in H_{0}(Y,\mathbf{Q}),
  12. \cdot
  13. ¯ j , J \bar{\partial}_{j,J}

Groombridge_34.html

  1. M V = 10.32 \scriptstyle M_{V_{\ast}}=10.32
  2. M V = 13.29 \scriptstyle M_{V_{\ast}}=13.29
  3. M V = 4.83 \scriptstyle M_{V_{\odot}}=4.83
  4. L V L V = 10 0.4 ( M V - M V ) \scriptstyle\frac{L_{V_{\ast}}}{L_{V_{\odot}}}=10^{0.4\left(M_{V_{\odot}}-M_{V% _{\ast}}\right)}

Gross–Neveu_model.html

  1. = ψ ¯ a ( i / - m ) ψ a + g 2 2 N [ ψ ¯ a ψ a ] 2 \mathcal{L}=\bar{\psi}_{a}\left(i\partial\!\!\!/-m\right)\psi^{a}+\frac{g^{2}}% {2N}\left[\bar{\psi}_{a}\psi^{a}\right]^{2}
  2. ψ ¯ a ψ a \overline{\psi}_{a}\psi^{a}
  3. ϕ 4 \phi^{4}
  4. = ψ ¯ a ( i / - m ) ψ a + g 2 2 N ( [ ψ ¯ a ψ a ] 2 - [ ψ ¯ a γ 5 ψ a ] 2 ) \mathcal{L}=\bar{\psi}_{a}\left(i\partial\!\!\!/-m\right)\psi^{a}+\frac{g^{2}}% {2N}(\left[\bar{\psi}_{a}\psi^{a}\right]^{2}-\left[\bar{\psi}_{a}\gamma_{5}% \psi^{a}\right]^{2})
  5. ψ γ 5 ψ \psi\rightarrow\gamma_{5}\psi
  6. ψ e i θ γ 5 ψ \psi\rightarrow e^{i\theta\gamma_{5}}\psi

Grothendieck_spectral_sequence.html

  1. G F G\circ F
  2. F : 𝒜 F:\mathcal{A}\to\mathcal{B}
  3. G : 𝒞 G:\mathcal{B}\to\mathcal{C}
  4. F F
  5. G G
  6. \mathcal{B}
  7. A A
  8. 𝒜 \mathcal{A}
  9. E 2 p q = ( R p G R q F ) ( A ) R p + q ( G F ) ( A ) . E_{2}^{pq}=({\rm R}^{p}G\circ{\rm R}^{q}F)(A)\Longrightarrow{\rm R}^{p+q}(G% \circ F)(A).
  10. X X
  11. Y Y
  12. 𝒜 = 𝐀𝐛 ( X ) \mathcal{A}=\mathbf{Ab}(X)
  13. = 𝐀𝐛 ( Y ) \mathcal{B}=\mathbf{Ab}(Y)
  14. 𝒞 = 𝐀𝐛 \mathcal{C}=\mathbf{Ab}
  15. f : X Y f:X\to Y
  16. f * : 𝐀𝐛 ( X ) 𝐀𝐛 ( Y ) f_{*}:\mathbf{Ab}(X)\to\mathbf{Ab}(Y)
  17. Γ X : 𝐀𝐛 ( X ) 𝐀𝐛 \Gamma_{X}:\mathbf{Ab}(X)\to\mathbf{Ab}
  18. Γ Y : 𝐀𝐛 ( Y ) 𝐀𝐛 . \Gamma_{Y}:\mathbf{Ab}(Y)\to\mathbf{Ab}.
  19. Γ Y f * = Γ X \Gamma_{Y}\circ f_{*}=\Gamma_{X}
  20. f * f_{*}
  21. Γ Y \Gamma_{Y}
  22. f - 1 f^{-1}
  23. H p ( Y , R q f * ) H p + q ( X , ) H^{p}(Y,{\rm R}^{q}f_{*}\mathcal{F})\implies H^{p+q}(X,\mathcal{F})
  24. \mathcal{F}
  25. X X
  26. ( X , 𝒪 ) (X,\mathcal{O})
  27. E 2 p , q = H p ( X ; x t 𝒪 q ( F , G ) ) Ext 𝒪 p + q ( F , G ) . E^{p,q}_{2}=\operatorname{H}^{p}(X;\mathcal{E}xt^{q}_{\mathcal{O}}(F,G))% \Rightarrow\operatorname{Ext}^{p+q}_{\mathcal{O}}(F,G).
  28. R p Γ ( X , - ) = H p ( X , - ) R^{p}\Gamma(X,-)=\operatorname{H}^{p}(X,-)
  29. R q o m 𝒪 ( F , - ) = x t 𝒪 q ( F , - ) R^{q}\mathcal{H}om_{\mathcal{O}}(F,-)=\mathcal{E}xt^{q}_{\mathcal{O}}(F,-)
  30. R n Γ ( X , o m 𝒪 ( F , - ) ) = Ext 𝒪 n ( F , - ) R^{n}\Gamma(X,\mathcal{H}om_{\mathcal{O}}(F,-))=\operatorname{Ext}^{n}_{% \mathcal{O}}(F,-)
  31. o m 𝒪 ( F , - ) \mathcal{H}om_{\mathcal{O}}(F,-)
  32. 𝒪 \mathcal{O}
  33. Γ ( X , - ) \Gamma(X,-)
  34. Z n , B n + 1 Z^{n},B^{n+1}
  35. d : K n K n + 1 d:K^{n}\to K^{n+1}
  36. 0 Z n K n 𝑑 B n + 1 0 0\to Z^{n}\to K^{n}\overset{d}{\to}B^{n+1}\to 0
  37. B n + 1 B^{n+1}
  38. 0 B n Z n H n ( K ) 0. 0\to B^{n}\to Z^{n}\to H^{n}(K^{\bullet})\to 0.
  39. 0 G ( B n ) G ( Z n ) G ( H n ( K ) ) 0. 0\to G(B^{n})\to G(Z^{n})\to G(H^{n}(K^{\bullet}))\to 0.
  40. 0 G ( Z n ) G ( K n ) G ( d ) G ( B n + 1 ) 0 0\to G(Z^{n})\to G(K^{n})\overset{G(d)}{\to}G(B^{n+1})\to 0
  41. \square
  42. A 0 A 1 A^{0}\to A^{1}\to\cdots
  43. ϕ p \phi^{p}
  44. F ( A p ) F ( A p + 1 ) F(A^{p})\to F(A^{p+1})
  45. 0 ker ϕ p F ( A p ) ϕ p im ϕ p 0. 0\to\operatorname{ker}\phi^{p}\to F(A^{p})\overset{\phi^{p}}{\to}\operatorname% {im}\phi^{p}\to 0.
  46. J 0 J 1 J^{0}\to J^{1}\to\cdots
  47. K 0 K 1 K^{0}\to K^{1}\to\cdots
  48. I p , = J K I^{p,\bullet}=J\oplus K
  49. F ( A p ) F(A^{p})
  50. 0 F ( A ) I , 0 I , 1 . 0\to F(A^{\bullet})\to I^{\bullet,0}\to I^{\bullet,1}\to\cdots.
  51. I 0 , q I 1 , q I^{0,q}\to I^{1,q}\to\cdots
  52. E 0 p , q = G ( I p , ) E_{0}^{p,q}=G(I^{p,\bullet})
  53. E 1 p , q ′′ = H q ( G ( I p , ) ) = R q G ( F ( A p ) ) {}^{\prime\prime}E_{1}^{p,q}=H^{q}(G(I^{p,\bullet}))=R^{q}G(F(A^{p}))
  54. F ( A p ) F(A^{p})
  55. E 2 n ′′ = R n ( G F ) ( A ) {}^{\prime\prime}E_{2}^{n}=R^{n}(G\circ F)(A)
  56. E 2 ′′ = E ′′ {}^{\prime\prime}E_{2}={}^{\prime\prime}E_{\infty}
  57. E 1 p , q = H q ( G ( I , p ) ) = G ( H q ( I , p ) ) . {}^{\prime}E^{p,q}_{1}=H^{q}(G(I^{\bullet,p}))=G(H^{q}(I^{\bullet,p})).
  58. H q ( I , 0 ) H q ( I , 1 ) H^{q}(I^{\bullet,0})\to H^{q}(I^{\bullet,1})\to\cdots
  59. H q ( F ( A ) ) = R q F ( A ) H^{q}(F(A^{\bullet}))=R^{q}F(A)
  60. E 2 p , q = R p G ( R q F ( A ) ) . {}^{\prime}E^{p,q}_{2}=R^{p}G(R^{q}F(A)).
  61. E r {}^{\prime}E_{r}
  62. E r ′′ {}^{\prime\prime}E_{r}
  63. \square

Grothendieck–Katz_p-curvature_conjecture.html

  1. d v / d z = A ( z ) v dv/dz=A(z)v

Group_code.html

  1. n n
  2. G n G^{n}
  3. G G
  4. C C
  5. G n G^{n}
  6. | G | k \left|G\right|^{k}
  7. n - k n-k
  8. k k
  9. G = ( ( 00 11 ) ( 01 01 ) ( 11 01 ) ( 00 11 ) ( 11 11 ) ( 00 00 ) ) G=\begin{pmatrix}\begin{pmatrix}00\\ 11\end{pmatrix}\begin{pmatrix}01\\ 01\end{pmatrix}\begin{pmatrix}11\\ 01\end{pmatrix}\\ \begin{pmatrix}00\\ 11\end{pmatrix}\begin{pmatrix}11\\ 11\end{pmatrix}\begin{pmatrix}00\\ 00\end{pmatrix}\end{pmatrix}
  10. 2 × 2 2\times 2
  11. g 1 m 1 g 2 m 2 g r m r g_{1}^{m_{1}}g_{2}^{m_{2}}...g_{r}^{m_{r}}
  12. g 1 , g r g_{1},...g_{r}
  13. G G

Haber's_rule.html

  1. C × t = k C\times t=k
  2. C C
  3. t t
  4. k k
  5. C C
  6. t t
  7. C C
  8. t t

Hadamard_space.html

  1. d ( z , m ) 2 + d ( x , y ) 2 4 d ( z , x ) 2 + d ( z , y ) 2 2 . d(z,m)^{2}+{d(x,y)^{2}\over 4}\leq{d(z,x)^{2}+d(z,y)^{2}\over 2}.
  2. d ( x , m ) = d ( y , m ) = d ( x , y ) / 2 d(x,m)=d(y,m)=d(x,y)/2
  3. m = ( x + y ) / 2 m=(x+y)/2
  4. Γ \Gamma
  5. Γ \Gamma

Hagedorn_temperature.html

  1. lim T T H - T r [ e - β H ] = \lim_{T\rightarrow T_{H}^{-}}Tr[e^{-\beta H}]=\infty
  2. lim T T H - E = lim T T H - T r [ H e - β H ] T r [ e - β H ] = \lim_{T\rightarrow T_{H}^{-}}E=\lim_{T\rightarrow T_{H}^{-}}\frac{Tr[He^{-% \beta H}]}{Tr[e^{-\beta H}]}=\infty

Hahn_decomposition_theorem.html

  1. μ + ( E ) := μ ( E P ) \mu^{+}(E):=\mu(E\cap P)\,
  2. μ - ( E ) := - μ ( E N ) \mu^{-}(E):=-\mu(E\cap N)\,
  3. μ + ( E ) = sup B Σ , B E μ ( B ) \mu^{+}(E)=\sup_{B\in\Sigma,B\subset E}\mu(B)
  4. μ - ( E ) = - inf B Σ , B E μ ( B ) \mu^{-}(E)=-\inf_{B\in\Sigma,B\subset E}\mu(B)
  5. ν + μ + and ν - μ - . \nu^{+}\geq\mu^{+}\,\text{ and }\nu^{-}\geq\mu^{-}.
  6. t n = sup { μ ( B ) : B Σ , B A n } t_{n}=\sup\{\mu(B):B\in\Sigma,\,B\subset A_{n}\}
  7. μ ( B n ) min { 1 , t n / 2 } . \mu(B_{n})\geq\min\{1,t_{n}/2\}.
  8. A = D n = 0 B n . A=D\setminus\bigcup_{n=0}^{\infty}B_{n}.
  9. μ ( A ) = μ ( D ) - n = 0 μ ( B n ) μ ( D ) - n = 0 min { 1 , t n / 2 } . \mu(A)=\mu(D)-\sum_{n=0}^{\infty}\mu(B_{n})\leq\mu(D)-\sum_{n=0}^{\infty}\min% \{1,t_{n}/2\}.
  10. s n := inf { μ ( D ) : D Σ , D X N n } . s_{n}:=\inf\{\mu(D):D\in\Sigma,\,D\subset X\setminus N_{n}\}.
  11. μ ( D n ) max { s n / 2 , - 1 } 0. \mu(D_{n})\leq\max\{s_{n}/2,-1\}\leq 0.
  12. N = n = 0 A n . N=\bigcup_{n=0}^{\infty}A_{n}.
  13. μ ( B ) = n = 0 μ ( B A n ) \mu(B)=\sum_{n=0}^{\infty}\mu(B\cap A_{n})
  14. μ ( N ) = n = 0 μ ( A n ) n = 0 max { s n / 2 , - 1 } = - , \mu(N)=\sum_{n=0}^{\infty}\mu(A_{n})\leq\sum_{n=0}^{\infty}\max\{s_{n}/2,-1\}=% -\infty,
  15. ( N , P ) (N^{\prime},P^{\prime})
  16. X X
  17. P N P\cap N^{\prime}
  18. N P N\cap P^{\prime}
  19. P P = N N = ( P N ) ( N P ) , P\,\triangle\,P^{\prime}=N\,\triangle\,N^{\prime}=(P\cap N^{\prime})\cup(N\cap P% ^{\prime}),

Hahn–Kolmogorov_theorem.html

  1. Σ 0 \Sigma_{0}
  2. X . X.
  3. μ 0 : Σ 0 [ 0 , ] \mu_{0}\colon\Sigma_{0}\to[0,\infty]
  4. μ 0 ( n = 1 N A n ) = n = 1 N μ 0 ( A n ) \mu_{0}\left(\bigcup_{n=1}^{N}A_{n}\right)=\sum_{n=1}^{N}\mu_{0}(A_{n})
  5. A 1 , A 2 , , A N A_{1},A_{2},\dots,A_{N}
  6. Σ 0 \Sigma_{0}
  7. μ 0 ( n = 1 A n ) = n = 1 μ 0 ( A n ) \mu_{0}\left(\bigcup_{n=1}^{\infty}A_{n}\right)=\sum_{n=1}^{\infty}\mu_{0}(A_{% n})
  8. { A n : n } \{A_{n}:n\in\mathbb{N}\}
  9. Σ 0 \Sigma_{0}
  10. n = 1 A n Σ 0 \cup_{n=1}^{\infty}A_{n}\in\Sigma_{0}
  11. μ 0 \mu_{0}
  12. μ 0 \mu_{0}
  13. Σ \Sigma
  14. Σ 0 \Sigma_{0}
  15. μ : Σ [ 0 , ] \mu\colon\Sigma\to[0,\infty]
  16. Σ 0 \Sigma_{0}
  17. μ 0 . \mu_{0}.
  18. μ 0 \mu_{0}
  19. σ \sigma
  20. μ 0 \mu_{0}
  21. σ \sigma
  22. σ \sigma
  23. \mathbb{Q}
  24. [ a , b ) [a,b)
  25. a , b a,b\in\mathbb{Q}
  26. X X
  27. [ 0 , 1 ) \mathbb{Q}\cap[0,1)
  28. Σ 0 \Sigma_{0}
  29. [ 0 , 1 ) \mathbb{Q}\cap[0,1)
  30. Σ 0 \Sigma_{0}
  31. Σ 0 \Sigma_{0}
  32. μ 0 \mu_{0}
  33. # \#
  34. Σ 0 \Sigma_{0}
  35. μ 0 \mu_{0}
  36. σ \sigma
  37. Σ 0 \Sigma_{0}
  38. Σ 0 \Sigma_{0}
  39. A Σ 0 A\in\Sigma_{0}
  40. μ 0 ( A ) = + \mu_{0}(A)=+\infty
  41. Σ \Sigma
  42. σ \sigma
  43. Σ 0 \Sigma_{0}
  44. Σ \Sigma
  45. σ \sigma
  46. X X
  47. # \#
  48. 2 # 2\#
  49. Σ \Sigma
  50. μ 0 \mu_{0}
  51. μ 0 \mu_{0}
  52. μ 0 \mu_{0}
  53. σ \sigma

Half_range_Fourier_series.html

  1. [ 0 , L ] [0,L]
  2. [ - L , L ] [-L,L]
  3. f ( x ) , x [ 0 , L ] f(x),x\in[0,L]
  4. [ - L , 0 ] [-L,0]
  5. f ( x ) f(x)
  6. f ( x ) = cos ( x ) f(x)=\cos(x)
  7. 0 < x < π 0<x<\pi
  8. a n = 0 n a_{n}=0\ \quad\forall n
  9. b n = 2 π 0 π cos ( x ) sin ( n x ) d x = 2 n ( ( - 1 ) n + 1 ) π ( n 2 - 1 ) n 2 b_{n}=\frac{2}{\pi}\int_{0}^{\pi}\cos(x)\sin(nx)\,\mathrm{d}x=\frac{2n((-1)^{n% }+1)}{\pi(n^{2}-1)}\quad\forall n\geq 2
  10. b n = 0 b_{n}=0
  11. b n = 4 n π ( n 2 - 1 ) b_{n}={4n\over\pi(n^{2}-1)}
  12. b 2 k = 8 k π ( 4 k 2 - 1 ) b_{2k}={8k\over\pi(4k^{2}-1)}
  13. b 1 = 0 b_{1}=0
  14. cos ( x ) = 8 π n = 1 n ( 4 n 2 - 1 ) sin ( 2 n x ) \cos(x)={{8\over\pi}\sum_{n=1}^{\infty}{n\over(4n^{2}-1)}\sin(2nx)}

Hamiltonian_fluid_mechanics.html

  1. [ φ ( x ) , ρ ( y ) ] = δ d ( x - y ) [\varphi(\vec{x}),\rho(\vec{y})]=\delta^{d}(\vec{x}-\vec{y})
  2. = d d x ( 1 2 ρ ( φ ) 2 + e ( ρ ) ) , \mathcal{H}=\int\mathrm{d}^{d}x\left(\frac{1}{2}\rho(\nabla\varphi)^{2}+e(\rho% )\right),
  3. e ′′ = 1 ρ p , e^{\prime\prime}=\frac{1}{\rho}p^{\prime},
  4. ρ t \displaystyle\frac{\partial\rho}{\partial t}
  5. u = def φ \vec{u}\ \stackrel{\mathrm{def}}{=}\ \nabla\varphi
  6. u t + ( u ) u = - e ′′ ρ = - 1 ρ p \frac{\partial\vec{u}}{\partial t}+(\vec{u}\cdot\nabla)\vec{u}=-e^{\prime% \prime}\nabla\rho=-\frac{1}{\rho}\nabla{p}
  7. × u = 0 . \nabla\times\vec{u}=\vec{0}.

Hamiltonian_lattice_gauge_theory.html

  1. 𝔤 \mathfrak{g}
  2. T r [ E ( e ) 2 ] Tr[E(e)^{2}]

Handle_decompositions_of_3-manifolds.html

  1. S 2 × ~ S 1 \scriptstyle S^{2}\tilde{\times}S^{1}

Hans_Hellmut_Kirst.html

  1. * *
  2. * *

Happy_ending_problem.html

  1. f ( N ) = 1 + 2 N - 2 for all N 3. f(N)=1+2^{N-2}\quad\mbox{for all }~{}N\geq 3.
  2. f ( N ) 1 + 2 N - 2 , f(N)\geq 1+2^{N-2},
  3. f ( N ) ( 2 N - 5 N - 2 ) + 1 = O ( 4 N N ) . f(N)\leq{2N-5\choose N-2}+1=O\left(\frac{4^{N}}{\sqrt{N}}\right).

Harmonic_balance.html

  1. V V
  2. I linear I\text{linear}
  3. V V
  4. I nonlinear I\text{nonlinear}
  5. V V
  6. ϵ = I linear + I nonlinear = 0 \epsilon=I\text{linear}+I\text{nonlinear}=0
  7. V V
  8. ϵ \epsilon
  9. d ϵ d V \tfrac{d\epsilon}{dV}
  10. ϵ \epsilon

Harnack's_inequality.html

  1. R - r R + r f ( x 0 ) f ( x ) R + r R - r f ( x 0 ) . {R-r\over R+r}f(x_{0})\leq f(x)\leq{R+r\over R-r}f(x_{0}).
  2. Ω \Omega
  3. 𝐑 n \mathbf{R}^{n}
  4. ω \omega
  5. ω ¯ Ω \bar{\omega}\subset\Omega
  6. C C
  7. u ( x ) u(x)
  8. C C
  9. u u
  10. Ω \Omega
  11. ω \omega
  12. f ( x ) = 1 ω n - 1 | y - x 0 | = R R 2 - r 2 R | x - y | n f ( y ) d y , \displaystyle{f(x)={1\over\omega_{n-1}}\int_{|y-x_{0}|=R}{R^{2}-r^{2}\over R|x% -y|^{n}}\cdot f(y)\,dy,}
  13. R - r | x - y | R + r , \displaystyle{R-r\leq|x-y|\leq R+r,}
  14. R - r R ( R + r ) n - 1 R 2 - r 2 R | x - y | n R + r R ( R - r ) n - 1 . \displaystyle{{R-r\over R(R+r)^{n-1}}\leq{R^{2}-r^{2}\over R|x-y|^{n}}\leq{R+r% \over R(R-r)^{n-1}}.}
  15. f ( x 0 ) = 1 R n - 1 ω n - 1 | y - x 0 | = R f ( y ) d y . \displaystyle{f(x_{0})={1\over R^{n-1}\omega_{n-1}}\int_{|y-x_{0}|=R}f(y)\,dy.}
  16. sup u C ( inf u + || f || ) \sup u\leq C(\inf u+||f||)
  17. \mathcal{M}
  18. n \mathbb{R}^{n}
  19. u = i , j = 1 n a i j ( t , x ) 2 u x i x j + i = 1 n b i ( t , x ) u x i + c ( t , x ) u \mathcal{L}u=\sum_{i,j=1}^{n}a_{ij}(t,x)\frac{\partial^{2}u}{\partial x_{i}\,% \partial x_{j}}+\sum_{i=1}^{n}b_{i}(t,x)\frac{\partial u}{\partial x_{i}}+c(t,% x)u
  20. ( a i j ) (a_{ij})
  21. u ( t , x ) C 2 ( ( 0 , T ) × ) u(t,x)\in C^{2}((0,T)\times\mathcal{M})
  22. u t - u 0 \frac{\partial u}{\partial t}-\mathcal{L}u\geq 0
  23. ( 0 , T ) × (0,T)\times\mathcal{M}
  24. u ( t , x ) 0 \quad u(t,x)\geq 0
  25. ( 0 , T ) × . \quad(0,T)\times\mathcal{M}.
  26. K K
  27. \mathcal{M}
  28. τ ( 0 , T ) \tau\in(0,T)
  29. C > 0 \quad C>0
  30. K K
  31. τ \tau
  32. \mathcal{L}
  33. t ( τ , T ) \quad t\in(\tau,T)
  34. sup K u ( t - τ , ) C inf K u ( t , ) . \sup_{K}u(t-\tau,\cdot)\leq C\inf_{K}u(t,\cdot).\,

Hartogs'_theorem.html

  1. F : 𝐂 n 𝐂 F:{\,\textbf{C}}^{n}\to{\,\textbf{C}}
  2. f : 𝐑 n 𝐑 f\colon{\,\textbf{R}}^{n}\to{\,\textbf{R}}
  3. f f
  4. f ( x , y ) = x y x 2 + y 2 . f(x,y)=\frac{xy}{x^{2}+y^{2}}.
  5. f ( 0 , 0 ) = 0 f(0,0)=0
  6. x x
  7. y y
  8. x = y x=y
  9. x = - y x=-y
  10. f f

Hartogs_number.html

  1. α = { β Ord i : β X } \alpha=\{\beta\in\textrm{Ord}\mid\exists i:\beta\hookrightarrow X\}
  2. \cong

Harvard_step_test.html

  1. t e t_{e}
  2. h b h_{b}
  3. t e * 100 h b * 2 \frac{t_{e}*100}{h_{b}*2}

Hauptvermutung.html

  1. κ ( M ) H 4 ( M ; / 2 ) \kappa(M)\in H^{4}(M;\mathbb{Z}/2\mathbb{Z})

Hazy_Sighted_Link_State_Routing_Protocol.html

  1. O ( N 1.5 ) O(N^{1.5})
  2. O ( N 2 ) O(N^{2})

Heap_leaching.html

  1. A u + ( s ) + 2 C N - ( a q ) A u ( C N ) 2 - ( a q ) Au^{+}(s)+2CN^{-}(aq)\rightarrow Au(CN)_{2}^{-}(aq)
  2. 2 A u ( C N ) 2 - ( a q ) + Z n ( s ) Z n ( C N ) 4 - ( a q ) + 2 A u ( s ) 2Au(CN)_{2}^{-}(aq)+Zn(s)\rightarrow Zn(CN)_{4}^{-}(aq)+2Au(s)

Heath–Jarrow–Morton_framework.html

  1. f ( t , T ) \textstyle f(t,T)
  2. t T \textstyle t\leq T
  3. T \textstyle T
  4. t \textstyle t
  5. P ( t , T ) = e - t T f ( t , s ) d s P(t,T)=e^{-\int_{t}^{T}f(t,s)ds}
  6. P ( t , T ) \textstyle P(t,T)
  7. t \textstyle t
  8. T t \textstyle T\geq t
  9. β ( t ) = e 0 t f ( u , u ) d u \beta(t)=e^{\int_{0}^{t}f(u,u)du}
  10. f ( t , t ) r ( t ) \textstyle f(t,t)\triangleq r(t)
  11. f ( t , s ) \textstyle f(t,s)
  12. \textstyle\mathbb{Q}
  13. d f ( t , s ) = μ ( t , s ) d t + s y m b o l Σ ( t , s ) d W t df(t,s)=\mu(t,s)dt+symbol\Sigma(t,s)dW_{t}
  14. W t \textstyle W_{t}
  15. d \textstyle d
  16. μ ( u , s ) \textstyle\mu(u,s)
  17. s y m b o l Σ ( u , s ) \textstyle symbol\Sigma(u,s)
  18. u \textstyle\mathcal{F}_{u}
  19. f \textstyle f
  20. P ( t , s ) \textstyle P(t,s)
  21. Y t log P ( t , s ) = - t s f ( t , u ) d u Y_{t}\triangleq\log P(t,s)=-\int_{t}^{s}f(t,u)du
  22. Y t \textstyle Y_{t}
  23. d Y t \displaystyle dY_{t}
  24. μ ( t , s ) * = t s μ ( t , u ) d u \textstyle\mu(t,s)^{*}=\int_{t}^{s}\mu(t,u)du
  25. s y m b o l Σ ( t , s ) * = t s s y m b o l Σ ( t , u ) d u \textstyle symbol\Sigma(t,s)^{*}=\int_{t}^{s}symbol\Sigma(t,u)du
  26. Y t \textstyle Y_{t}
  27. d Y t = ( r t - μ ( t , s ) * ) d t - s y m b o l Σ ( t , s ) * d W t dY_{t}=\left(r_{t}-\mu(t,s)^{*}\right)dt-symbol\Sigma(t,s)^{*}dW_{t}
  28. P ( t , T ) \textstyle P(t,T)
  29. d P ( t , s ) P ( t , s ) = ( r t - μ ( t , s ) * + 1 2 s y m b o l Σ ( t , s ) * s y m b o l Σ ( t , s ) * T ) d t - s y m b o l Σ ( t , s ) * d W t \frac{dP(t,s)}{P(t,s)}=\left(r_{t}-\mu(t,s)^{*}+\frac{1}{2}symbol\Sigma(t,s)^{% *}symbol\Sigma(t,s)^{*T}\right)dt-symbol\Sigma(t,s)^{*}dW_{t}
  30. P ( t , s ) β ( t ) \textstyle\frac{P(t,s)}{\beta(t)}
  31. \textstyle\mathbb{Q}
  32. μ ( t , s ) * = 1 2 s y m b o l Σ ( t , s ) * s y m b o l Σ ( t , s ) * T \textstyle\mu(t,s)^{*}=\frac{1}{2}symbol\Sigma(t,s)^{*}symbol\Sigma(t,s)^{*T}
  33. s \textstyle s
  34. μ ( t , u ) = s y m b o l Σ ( t , u ) t u s y m b o l Σ ( t , s ) T d s \mu(t,u)=symbol\Sigma(t,u)\int_{t}^{u}symbol\Sigma(t,s)^{T}ds
  35. f \textstyle f
  36. d f ( t , u ) = ( s y m b o l Σ ( t , u ) t u s y m b o l Σ ( t , s ) T d s ) d t + s y m b o l Σ ( t , u ) d W t df(t,u)=\left(symbol\Sigma(t,u)\int_{t}^{u}symbol\Sigma(t,s)^{T}ds\right)dt+% symbol\Sigma(t,u)dW_{t}
  37. s y m b o l Σ \textstyle symbol\Sigma

Heawood_number.html

  1. H ( S ) = 7 + 49 - 24 e ( S ) 2 H(S)=\left\lfloor\frac{7+\sqrt{49-24e(S)}}{2}\right\rfloor
  2. e ( S ) e(S)
  3. H ( S ) H(S)
  4. 6 6
  5. 6 6
  6. H ( S ) H(S)
  7. S S
  8. S S
  9. 7 7

Hecke_character.html

  1. ( I , m ) = 1 χ ( I ) N ( I ) - s = L ( s , χ ) \sum_{(I,m)=1}\chi(I)N(I)^{-s}=L(s,\chi)\,

Heisenberg's_microscope.html

  1. x x
  2. ε \varepsilon
  3. λ \lambda
  4. Δ x = λ sin ε / 2 . \Delta x=\frac{\lambda}{\sin\varepsilon/2}.
  5. h / λ h/\lambda
  6. h h
  7. x x
  8. Δ p x 2 h λ sin ε / 2. \Delta p_{x}\approx 2\frac{h}{\lambda}\sin\varepsilon/2.
  9. Δ x \Delta x
  10. Δ p x \Delta p_{x}
  11. Δ x Δ p x ( λ sin ε / 2 ) ( 2 h λ sin ε / 2 ) = 2 h \Delta x\Delta p_{x}\approx\left(\frac{\lambda}{\sin\varepsilon/2}\right)\left% (2\frac{h}{\lambda}\sin\varepsilon/2\right)=2h

Helicity_(particle_physics).html

  1. h = J p ^ = L p ^ + S p ^ = S p ^ , p ^ = p | p | , h=\vec{J}\cdot\hat{p}=\vec{L}\cdot\hat{p}+\vec{S}\cdot\hat{p}=\vec{S}\cdot\hat% {p},\qquad\hat{p}=\frac{\vec{p}}{\left|\vec{p}\right|},
  2. S S
  3. S S
  4. S S
  5. S S
  6. S S
  7. S S
  8. e < s u p > i h θ e<sup>ihθ

Helmholtz_resonance.html

  1. ω H = γ A 2 m P 0 V 0 \omega_{H}=\sqrt{\gamma\frac{A^{2}}{m}\frac{P_{0}}{V_{0}}}
  2. γ \gamma
  3. A A
  4. m m
  5. P 0 P_{0}
  6. V 0 V_{0}
  7. A = V n L e q A=\frac{V_{n}}{L_{eq}}
  8. L e q L_{eq}
  9. L e q = L n + 0.6 D L_{eq}=L_{n}+0.6D
  10. L n L_{n}
  11. D D
  12. V n V_{n}
  13. ω H = γ A m V n L e q P 0 V 0 \omega_{H}=\sqrt{\gamma\frac{A}{m}\frac{V_{n}}{L_{eq}}\frac{P_{0}}{V_{0}}}
  14. ρ {\rho}
  15. V n m = 1 ρ \frac{V_{n}}{m}=\frac{1}{\rho}
  16. ω H = γ P 0 ρ A V 0 L e q \omega_{H}=\sqrt{\gamma\frac{P_{0}}{\rho}\frac{A}{V_{0}L_{eq}}}
  17. f H = ω H 2 π f_{H}=\frac{\omega_{H}}{2\pi}
  18. v = γ P 0 ρ v=\sqrt{\gamma\frac{P_{0}}{\rho}}
  19. f H = v 2 π A V 0 L e q f_{H}=\frac{v}{2\pi}\sqrt{\frac{A}{V_{0}L_{eq}}}

Helmholtz_theorem_(classical_mechanics).html

  1. H ( x , p ; V ) = K ( p ) + φ ( x ; V ) H(x,p;V)=K(p)+\varphi(x;V)
  2. K = p 2 2 m K=\frac{p^{2}}{2m}
  3. φ ( x ; V ) \varphi(x;V)
  4. V V
  5. t \left\langle\cdot\right\rangle_{t}
  6. E = K + φ , E=K+\varphi,
  7. T = 2 K t , T=2\left\langle K\right\rangle_{t},
  8. P = - φ V t , P=\left\langle-\frac{\partial\varphi}{\partial V}\right\rangle_{t},
  9. S ( E , V ) = log 2 m ( E - φ ( x , V ) ) d x . S(E,V)=\log\oint\sqrt{2m\left(E-\varphi\left(x,V\right)\right)}\,dx.
  10. d S = d E + P d V T . dS=\frac{dE+PdV}{T}.
  11. T T
  12. S S
  13. d x 2 m ( E - φ ( x , V ) ) \oint dx\sqrt{2m\left(E-\varphi\left(x,V\right)\right)}

Hemicontinuity.html

  1. G r ( Γ ) = { ( a , b ) A × B : b Γ ( a ) } Gr(\Gamma)=\{(a,b)\in A\times B:b\in\Gamma(a)\}
  2. A × B A\times B
  3. a A a\in A
  4. a n A \forall a_{n}\in A
  5. b B \forall b\in B
  6. b n Γ ( a n ) \forall b_{n}\in\Gamma(a_{n})
  7. lim n a n = a , lim n b n = b b Γ ( a ) \lim_{n\to\infty}a_{n}=a,\;\lim_{n\to\infty}b_{n}=b\implies b\in\Gamma(a)
  8. V S V\cap S\neq\emptyset
  9. a m A , a m a , b Γ ( a ) , a m k \forall a_{m}\in A,\,a_{m}\rightarrow a,\forall b\in\Gamma(a),\exists a_{m_{k}}
  10. a m , b k Γ ( a m k ) , b k b a_{m},\,\exists b_{k}\in\Gamma(a_{m_{k}}),\,b_{k}\rightarrow b

Henri_Tresca.html

  1. σ t r e s c a = σ 1 - σ 3 > σ m a x \ \sigma_{tresca}=\sigma_{1}-\sigma_{3}>\sigma_{max}

Herbrand's_theorem.html

  1. ( y 1 , , y n ) F ( y 1 , , y n ) (\exists y_{1},\ldots,y_{n})F(y_{1},\ldots,y_{n})
  2. F ( y 1 , , y n ) F(y_{1},\ldots,y_{n})
  3. ( y 1 , , y n ) F ( y 1 , , y n ) (\exists y_{1},\ldots,y_{n})F(y_{1},\ldots,y_{n})
  4. t i j t_{ij}
  5. 1 i k 1\leq i\leq k
  6. 1 j n 1\leq j\leq n
  7. F ( t 11 , , t 1 n ) F ( t k 1 , , t k n ) F(t_{11},\ldots,t_{1n})\vee\ldots\vee F(t_{k1},\ldots,t_{kn})
  8. F ( t 11 , , t 1 n ) F ( t k 1 , , t k n ) F(t_{11},\ldots,t_{1n})\vee\ldots\vee F(t_{k1},\ldots,t_{kn})
  9. ( y 1 , , y n ) F ( y 1 , , y n ) . (\exists y_{1},\ldots,y_{n})F(y_{1},\ldots,y_{n}).
  10. A A
  11. A A
  12. ( y 1 , , y n ) F ( y 1 , , y n ) (\exists y_{1},\ldots,y_{n})F(y_{1},\ldots,y_{n})
  13. ( y 1 , , y n ) F ( y 1 , , y n ) \vdash(\exists y_{1},\ldots,y_{n})F(y_{1},\ldots,y_{n})
  14. F ( y 1 , , y n ) F(y_{1},\ldots,y_{n})
  15. F ( t 11 , , t 1 n ) , , F ( t k 1 , , t k n ) \vdash F(t_{11},\ldots,t_{1n}),\ldots,F(t_{k1},\ldots,t_{kn})

Hermite_interpolation.html

  1. ( x 0 , y 0 ) , ( x 1 , y 1 ) , , ( x n - 1 , y n - 1 ) , ( x 0 , y 0 ) , ( x 1 , y 1 ) , , ( x n - 1 , y n - 1 ) , ( x 0 , y 0 ( m ) ) , ( x 1 , y 1 ( m ) ) , , ( x n - 1 , y n - 1 ( m ) ) \begin{matrix}(x_{0},y_{0}),&(x_{1},y_{1}),&\ldots,&(x_{n-1},y_{n-1}),\\ (x_{0},y_{0}^{\prime}),&(x_{1},y_{1}^{\prime}),&\ldots,&(x_{n-1},y_{n-1}^{% \prime}),\\ \vdots&\vdots&&\vdots\\ (x_{0},y_{0}^{(m)}),&(x_{1},y_{1}^{(m)}),&\ldots,&(x_{n-1},y_{n-1}^{(m)})\end{matrix}
  2. m = 1 m=1
  3. n + 1 n+1
  4. x 0 , x 1 , x 2 , , x n x_{0},x_{1},x_{2},\ldots,x_{n}
  5. f ( x 0 ) , f ( x 1 ) , , f ( x n ) f(x_{0}),f(x_{1}),\ldots,f(x_{n})
  6. f ( x 0 ) , f ( x 1 ) , , f ( x n ) f^{\prime}(x_{0}),f^{\prime}(x_{1}),\ldots,f^{\prime}(x_{n})
  7. f f
  8. z 0 , z 1 , , z 2 n + 1 z_{0},z_{1},\ldots,z_{2n+1}
  9. z 2 i = z 2 i + 1 = x i . z_{2i}=z_{2i+1}=x_{i}.
  10. z 0 , z 1 , , z 2 n + 1 z_{0},z_{1},\ldots,z_{2n+1}
  11. z i = z i + 1 f [ z i , z i + 1 ] = f ( z i + 1 ) - f ( z i ) z i + 1 - z i = 0 0 z_{i}=z_{i+1}\implies f[z_{i},z_{i+1}]=\frac{f(z_{i+1})-f(z_{i})}{z_{i+1}-z_{i% }}=\frac{0}{0}
  12. f ( z i ) f^{\prime}(z_{i})
  13. x i x_{i}
  14. z 0 , z 1 , , z N z_{0},z_{1},\ldots,z_{N}
  15. x i x_{i}
  16. j = 2 , 3 , , k j=2,3,\ldots,k
  17. f ( j ) ( x i ) j ! . \frac{f^{(j)}(x_{i})}{j!}.
  18. f [ x i , x i , x i ] = f ′′ ( x i ) 2 f[x_{i},x_{i},x_{i}]=\frac{f^{\prime\prime}(x_{i})}{2}
  19. f [ x i , x i , x i , x i ] = f ( 3 ) ( x i ) 6 f[x_{i},x_{i},x_{i},x_{i}]=\frac{f^{(3)}(x_{i})}{6}
  20. f ( x ) = x 8 + 1 f(x)=x^{8}+1
  21. x { - 1 , 0 , 1 } x\in\{-1,0,1\}
  22. { z i } = { - 1 , - 1 , - 1 , 0 , 0 , 0 , 1 , 1 , 1 } \{z_{i}\}=\{-1,-1,-1,0,0,0,1,1,1\}
  23. z 0 = - 1 f [ z 0 ] = 2 f ( z 0 ) 1 = - 8 z 1 = - 1 f [ z 1 ] = 2 f ′′ ( z 1 ) 2 = 28 f ( z 1 ) 1 = - 8 f [ z 3 , z 2 , z 1 , z 0 ] = - 21 z 2 = - 1 f [ z 2 ] = 2 f [ z 3 , z 2 , z 1 ] = 7 15 f [ z 3 , z 2 ] = - 1 f [ z 4 , z 3 , z 2 , z 1 ] = - 6 - 10 z 3 = 0 f [ z 3 ] = 1 f [ z 4 , z 3 , z 2 ] = 1 5 4 f ( z 3 ) 1 = 0 f [ z 5 , z 4 , z 3 , z 2 ] = - 1 - 2 - 1 z 4 = 0 f [ z 4 ] = 1 f ′′ ( z 4 ) 2 = 0 1 2 1 f ( z 4 ) 1 = 0 f [ z 6 , z 5 , z 4 , z 3 ] = 1 2 1 z 5 = 0 f [ z 5 ] = 1 f [ z 6 , z 5 , z 4 ] = 1 5 4 f [ z 6 , z 5 ] = 1 f [ z 7 , z 6 , z 5 , z 4 ] = 6 10 z 6 = 1 f [ z 6 ] = 2 f [ z 7 , z 6 , z 5 ] = 7 15 f ( z 7 ) 1 = 8 f [ z 8 , z 7 , z 6 , z 5 ] = 21 z 7 = 1 f [ z 7 ] = 2 f ′′ ( z 7 ) 2 = 28 f ( z 8 ) 1 = 8 z 8 = 1 f [ z 8 ] = 2 \begin{matrix}z_{0}=-1&f[z_{0}]=2&&&&&&&&\\ &&\frac{f^{\prime}(z_{0})}{1}=-8&&&&&&&\\ z_{1}=-1&f[z_{1}]=2&&\frac{f^{\prime\prime}(z_{1})}{2}=28&&&&&&\\ &&\frac{f^{\prime}(z_{1})}{1}=-8&&f[z_{3},z_{2},z_{1},z_{0}]=-21&&&&&\\ z_{2}=-1&f[z_{2}]=2&&f[z_{3},z_{2},z_{1}]=7&&15&&&&\\ &&f[z_{3},z_{2}]=-1&&f[z_{4},z_{3},z_{2},z_{1}]=-6&&-10&&&\\ z_{3}=0&f[z_{3}]=1&&f[z_{4},z_{3},z_{2}]=1&&5&&4&&\\ &&\frac{f^{\prime}(z_{3})}{1}=0&&f[z_{5},z_{4},z_{3},z_{2}]=-1&&-2&&-1&\\ z_{4}=0&f[z_{4}]=1&&\frac{f^{\prime\prime}(z_{4})}{2}=0&&1&&2&&1\\ &&\frac{f^{\prime}(z_{4})}{1}=0&&f[z_{6},z_{5},z_{4},z_{3}]=1&&2&&1&\\ z_{5}=0&f[z_{5}]=1&&f[z_{6},z_{5},z_{4}]=1&&5&&4&&\\ &&f[z_{6},z_{5}]=1&&f[z_{7},z_{6},z_{5},z_{4}]=6&&10&&&\\ z_{6}=1&f[z_{6}]=2&&f[z_{7},z_{6},z_{5}]=7&&15&&&&\\ &&\frac{f^{\prime}(z_{7})}{1}=8&&f[z_{8},z_{7},z_{6},z_{5}]=21&&&&&\\ z_{7}=1&f[z_{7}]=2&&\frac{f^{\prime\prime}(z_{7})}{2}=28&&&&&&\\ &&\frac{f^{\prime}(z_{8})}{1}=8&&&&&&&\\ z_{8}=1&f[z_{8}]=2&&&&&&&&\\ \end{matrix}
  24. P ( x ) = 2 - 8 ( x + 1 ) + 28 ( x + 1 ) 2 - 21 ( x + 1 ) 3 + 15 x ( x + 1 ) 3 - 10 x 2 ( x + 1 ) 3 + 4 x 3 ( x + 1 ) 3 - 1 x 3 ( x + 1 ) 3 ( x - 1 ) + x 3 ( x + 1 ) 3 ( x - 1 ) 2 = 2 - 8 + 28 - 21 - 8 x + 56 x - 63 x + 15 x + 28 x 2 - 63 x 2 + 45 x 2 - 10 x 2 - 21 x 3 + 45 x 3 - 30 x 3 + 4 x 3 + x 3 + x 3 + 15 x 4 - 30 x 4 + 12 x 4 + 2 x 4 + x 4 - 10 x 5 + 12 x 5 - 2 x 5 + 4 x 5 - 2 x 5 - 2 x 5 - x 6 + x 6 - x 7 + x 7 + x 8 = x 8 + 1. \begin{aligned}\displaystyle P(x)&\displaystyle=2-8(x+1)+28(x+1)^{2}-21(x+1)^{% 3}+15x(x+1)^{3}-10x^{2}(x+1)^{3}\\ &\displaystyle\quad{}+4x^{3}(x+1)^{3}-1x^{3}(x+1)^{3}(x-1)+x^{3}(x+1)^{3}(x-1)% ^{2}\\ &\displaystyle=2-8+28-21-8x+56x-63x+15x+28x^{2}-63x^{2}+45x^{2}-10x^{2}-21x^{3% }\\ &\displaystyle\quad{}+45x^{3}-30x^{3}+4x^{3}+x^{3}+x^{3}+15x^{4}-30x^{4}+12x^{% 4}+2x^{4}+x^{4}\\ &\displaystyle\quad{}-10x^{5}+12x^{5}-2x^{5}+4x^{5}-2x^{5}-2x^{5}-x^{6}+x^{6}-% x^{7}+x^{7}+x^{8}\\ &\displaystyle=x^{8}+1.\end{aligned}
  25. i = 0 k - 1 ( x - z i ) \prod_{i=0}^{k-1}(x-z_{i})
  26. x [ x 0 , x n ] x\in[x_{0},x_{n}]
  27. f ( x ) - H ( x ) = f ( K ) ( c ) K ! i ( x - x i ) k i f(x)-H(x)=\frac{f^{(K)}(c)}{K!}\prod_{i}(x-x_{i})^{k_{i}}
  28. [ x 0 , x N ] [x_{0},x_{N}]
  29. k i k_{i}
  30. x i x_{i}

Hermitian_function.html

  1. f ( - x ) = f ( x ) ¯ f(-x)=\overline{f(x)}
  2. x x
  3. f f
  4. f f
  5. f ( - x 1 , - x 2 ) = f ( x 1 , x 2 ) ¯ f(-x_{1},-x_{2})=\overline{f(x_{1},x_{2})}
  6. ( x 1 , x 2 ) (x_{1},x_{2})
  7. f f
  8. f f
  9. f f
  10. f f
  11. f f
  12. f f
  13. f f
  14. f f
  15. f g = f * g f\star g=f*g
  16. \star
  17. * *
  18. f g = g f f\star g=g\star f

Hermitian_manifold.html

  1. h Γ ( E E ¯ ) * h\in\Gamma(E\otimes\bar{E})^{*}
  2. h p ( η , ζ ¯ ) = h p ( ζ , η ¯ ) ¯ h_{p}(\eta,\bar{\zeta})=\overline{h_{p}(\zeta,\bar{\eta})}
  3. h p ( ζ , ζ ¯ ) > 0 h_{p}(\zeta,\bar{\zeta})>0
  4. h = h α β ¯ d z α d z ¯ β h=h_{\alpha\bar{\beta}}\,dz^{\alpha}\otimes d\bar{z}^{\beta}
  5. h α β ¯ h_{\alpha\bar{\beta}}
  6. g = 1 2 ( h + h ¯ ) . g={1\over 2}(h+\bar{h}).
  7. g = 1 2 h α β ¯ ( d z α d z ¯ β + d z ¯ β d z α ) . g={1\over 2}h_{\alpha\bar{\beta}}\,(dz^{\alpha}\otimes d\bar{z}^{\beta}+d\bar{% z}^{\beta}\otimes dz^{\alpha}).
  8. ω = i 2 ( h - h ¯ ) . \omega={i\over 2}(h-\bar{h}).
  9. ω = i 2 h α β ¯ d z α d z ¯ β . \omega={i\over 2}h_{\alpha\bar{\beta}}\,dz^{\alpha}\wedge d\bar{z}^{\beta}.
  10. ω ( u , v ) = g ( J u , v ) g ( u , v ) = ω ( u , J v ) \begin{aligned}\displaystyle\omega(u,v)&\displaystyle=g(Ju,v)\\ \displaystyle g(u,v)&\displaystyle=\omega(u,Jv)\end{aligned}
  11. h = g - i ω . h=g-i\omega.\,
  12. h ( J u , J v ) = h ( u , v ) g ( J u , J v ) = g ( u , v ) ω ( J u , J v ) = ω ( u , v ) \begin{aligned}\displaystyle h(Ju,Jv)&\displaystyle=h(u,v)\\ \displaystyle g(Ju,Jv)&\displaystyle=g(u,v)\\ \displaystyle\omega(Ju,Jv)&\displaystyle=\omega(u,v)\end{aligned}
  13. g ( u , v ) = 1 2 ( g ( u , v ) + g ( J u , J v ) ) . g^{\prime}(u,v)={1\over 2}\left(g(u,v)+g(Ju,Jv)\right).
  14. vol M = ω n n ! Ω n , n ( M ) \mathrm{vol}_{M}=\frac{\omega^{n}}{n!}\in\Omega^{n,n}(M)
  15. vol M = ( i 2 ) n det ( h α β ¯ ) d z 1 d z ¯ 1 d z n d z ¯ n . \mathrm{vol}_{M}=\left(\frac{i}{2}\right)^{n}\det(h_{\alpha\bar{\beta}})\,dz^{% 1}\wedge d\bar{z}^{1}\wedge\cdots\wedge dz^{n}\wedge d\bar{z}^{n}.
  16. d ω = 0 . d\omega=0\,.

Hessian_form_of_an_elliptic_curve.html

  1. K K
  2. E E
  3. K K
  4. Y 2 + a 1 X Y + a 3 Y = X 3 Y^{2}+a_{1}XY+a_{3}Y=X^{3}
  5. Δ = ( a 3 3 ( a 1 3 - 27 a 3 ) ) = a 3 3 δ . \Delta=(a_{3}^{3}(a_{1}^{3}-27a_{3}))=a_{3}^{3}\delta.
  6. P = ( 0 , 0 ) P=(0,0)
  7. P = ( 0 , 0 ) P=(0,0)
  8. E E
  9. P P
  10. Y = 0 Y=0
  11. E E
  12. P P
  13. P P
  14. E E
  15. K K
  16. P = ( 0 , 0 ) P=(0,0)
  17. P P
  18. Y = 0 Y=0
  19. Y 2 + a 1 X Y + a 3 Y = X 3 Y^{2}+a_{1}XY+a_{3}Y=X^{3}
  20. a 1 , a 3 K a_{1},a_{3}\in K
  21. μ \mu
  22. T 3 - δ T 2 + δ 2 3 T + a 3 δ 2 = 0. T^{3}-\delta T^{2}+{\delta^{2}\over 3}T+a_{3}\delta^{2}=0.
  23. μ = δ - a 1 δ 2 / 3 3 . \mu={\delta-a_{1}\delta^{2/3}\over 3}.
  24. K K
  25. q 2 q\equiv 2
  26. K K
  27. μ \mu
  28. D = 3 ( μ - δ ) μ D=\frac{3(\mu-\delta)}{\mu}
  29. C C
  30. x 3 + y 3 + z 3 = D x y z x^{3}+y^{3}+z^{3}=Dxyz
  31. C C
  32. x 3 + y 3 + 1 = D x y x^{3}+y^{3}+1=Dxy
  33. x = X Z x=\frac{X}{Z}
  34. y = Y Z y=\frac{Y}{Z}
  35. D 3 1 D^{3}\neq 1
  36. v 2 = u 3 - 27 D ( D 3 + 8 ) u + 54 ( D 6 - 20 D 3 - 8 ) , v^{2}=u^{3}-27D(D^{3}+8)u+54(D^{6}-20D^{3}-8),\,
  37. ( x , y ) = ( η ( u + 9 D 2 ) , - 1 + η ( 3 D 3 - D x - 12 ) ) (x,y)=(\eta(u+9D^{2}),-1+\eta(3D^{3}-Dx-12))\,
  38. ( u , v ) = ( - 9 D 2 + ε x , 3 ε ( y - 1 ) ) (u,v)=(-9D^{2}+\varepsilon x,3\varepsilon(y-1))\,
  39. η = 6 ( D 3 - 1 ) ( v + 9 D 3 - 3 D u - 36 ) ( u + 9 D 2 ) 3 + ( 3 D d - D u - 12 ) 3 \eta=\frac{6(D^{3}-1)(v+9D^{3}-3Du-36)}{(u+9D^{2})^{3}+(3Dd-Du-12)^{3}}
  40. ε = 12 ( D 3 - 1 ) D x + y + 1 \varepsilon=\frac{12(D^{3}-1)}{Dx+y+1}
  41. θ \theta
  42. θ \theta
  43. θ \theta
  44. y = - x + ( x 1 + y 1 ) y=-x+(x_{1}+y_{1})
  45. P P
  46. θ \theta
  47. x 2 - ( x 1 + y 1 ) x + x 1 y 1 = θ x_{2}-(x_{1}+y_{1})\cdot x+x_{1}\cdot y_{1}=\theta
  48. x 1 + y 1 + D x_{1}+y_{1}+D
  49. D 3 D^{3}
  50. - P -P
  51. y 1 y_{1}
  52. - P -P
  53. x 1 x_{1}
  54. - P = ( y 1 , x 1 ) -P=(y_{1},x_{1})
  55. - P = ( Y 1 : X 1 : Z 1 ) -P=(Y_{1}:X_{1}:Z_{1})
  56. P = ( X 1 : Y 1 : Z 1 ) P=(X_{1}:Y_{1}:Z_{1})
  57. [ 2 ] P = ( Y 1 ( X 1 3 - Z 1 3 ) : X 1 ( Z 1 3 - Y 1 3 ) : Z 1 ( Y 1 3 - X 1 3 ) ) [2]P=(Y_{1}\cdot(X_{1}^{3}-Z_{1}^{3}):X_{1}\cdot(Z_{1}^{3}-Y_{1}^{3}):Z_{1}% \cdot(Y_{1}^{3}-X_{1}^{3}))
  58. P = ( X 1 : Y 1 : Z 1 ) P=(X_{1}:Y_{1}:Z_{1})
  59. Q = ( X 2 : Y 2 : Z 2 ) Q=(X_{2}:Y_{2}:Z_{2})
  60. R R
  61. R = P + Q R=P+Q
  62. R = ( Y 1 X 2 Z 2 - Y 2 X 1 Z 1 : X 1 Y 2 Z 2 - X 2 Y 1 Z 1 : Z 1 X 2 Y 2 - Z 2 X 1 Y 1 ) R=(Y_{1}\cdot X_{2}\cdot Z_{2}-Y_{2}\cdot X_{1}\cdot Z_{1}:X_{1}\cdot Y_{2}% \cdot Z_{2}-X_{2}\cdot Y_{1}\cdot Z_{1}:Z_{1}\cdot X_{2}\cdot Y_{2}-Z_{2}\cdot X% _{1}\cdot Y_{1})
  63. P = ( X 1 : Y 1 : Z 1 ) P=(X_{1}:Y_{1}:Z_{1})
  64. Q = ( X 2 : Y 2 : Z 2 ) Q=(X_{2}:Y_{2}:Z_{2})
  65. θ \theta
  66. P = ( - 1 : - 1 : 1 ) P=(-1:-1:1)
  67. 2 P = ( X : Y : Z ) 2P=(X:Y:Z)
  68. 2 P = ( - 4 : - 2 : 0 ) 2P=(-4:-2:0)
  69. x x
  70. y y
  71. X , Y , Z , X X , Y Y , Z Z , X Y , Y Z , X Z X,Y,Z,XX,YY,ZZ,XY,YZ,XZ
  72. x = X / Z x=X/Z
  73. y = Y / Z y=Y/Z
  74. X X = X X XX=X\cdot X
  75. Y Y = Y Y YY=Y\cdot Y
  76. Z Z = Z Z ZZ=Z\cdot Z
  77. X Y = 2 X Y XY=2\cdot X\cdot Y
  78. Y Z = 2 Y Z YZ=2\cdot Y\cdot Z
  79. X Z = 2 X Z XZ=2\cdot X\cdot Z

Heun_function.html

  1. d 2 w d z 2 + [ γ z + δ z - 1 + ϵ z - a ] d w d z + α β z - q z ( z - 1 ) ( z - a ) w = 0. \frac{d^{2}w}{dz^{2}}+\left[\frac{\gamma}{z}+\frac{\delta}{z-1}+\frac{\epsilon% }{z-a}\right]\frac{dw}{dz}+\frac{\alpha\beta z-q}{z(z-1)(z-a)}w=0.
  2. ϵ = α + β - γ - δ + 1 \epsilon=\alpha+\beta-\gamma-\delta+1

Hexagonal_bipyramid.html

  1. n 7. n\geq 7.

Hexagonal_lattice.html

  1. 1 3 3 \frac{1}{3}\sqrt{3}

Hexagonal_prism.html

  1. a a
  2. h h
  3. V = 3 3 2 a 2 × h V=\frac{3\sqrt{3}}{2}a^{2}\times h

Hidden_Field_Equations.html

  1. 𝔽 q \mathbb{F}_{q}
  2. 𝔽 q n \mathbb{F}_{q^{n}}
  3. 𝔽 q m \mathbb{F}_{q^{m}}
  4. 𝔽 q \mathbb{F}_{q}
  5. m m
  6. n n
  7. 𝔽 q \mathbb{F}_{q}
  8. 𝔽 q n 𝔽 q m \mathbb{F}_{q^{n}}\to\mathbb{F}_{q^{m}}
  9. 𝔽 q n \mathbb{F}_{q^{n}}
  10. 𝔽 q \mathbb{F}_{q}
  11. 𝔽 q \mathbb{F}_{q}
  12. q q
  13. K K
  14. β 1 , , β n \beta_{1},...,\beta_{n}
  15. K K
  16. 𝔽 q \mathbb{F}_{q}
  17. 0 < h < q n 0<h<q^{n}
  18. h = q θ + 1 h=q^{\theta}+1
  19. θ \theta
  20. ( h , q n - 1 ) = 1 (h,q^{n}-1)=1
  21. u 𝔽 q n u\in\mathbb{F}_{q^{n}}
  22. u u
  23. u = ( u 1 , , u n ) u=(u_{1},...,u_{n})
  24. v 𝔽 q n v\in\mathbb{F}_{q^{n}}
  25. v = u q θ u ( 1 ) v=u^{q^{\theta}}u\ \ \ \ (1)
  26. ( h , q n - 1 ) = 1 (h,q^{n}-1)=1
  27. u u h u\to u^{h}
  28. K K
  29. u u h u\to u^{h^{\prime}}
  30. h h^{\prime}
  31. h mod q n - 1 h\ \bmod q^{n}-1
  32. n × n n\times n
  33. S = { S i j } S=\{S_{ij}\}
  34. T = { T i j } T=\{T_{ij}\}
  35. 𝔽 q \mathbb{F}_{q}
  36. c = ( c 1 , , c n ) c=(c_{1},...,c_{n})
  37. d = ( d 1 , , d n ) d=(d_{1},...,d_{n})
  38. n n
  39. 𝔽 q \mathbb{F}_{q}
  40. x x
  41. y y
  42. u = S x + c v = T y + d ( 2 ) u=Sx+c\ \ \ \ v=Ty+d\ \ \ \ (2)
  43. A ( k ) = a i j ( k ) A^{(k)}={a_{ij}^{(k)}}
  44. β 1 , , β n \beta_{1},...,\beta_{n}
  45. β i q k = j = 1 n a i j k β j , a i j k 𝔽 q \beta_{i}^{q^{k}}=\sum_{j=1}^{n}a_{ij}^{k}\beta_{j},\ \ a_{ij}^{k}\in\mathbb{F% }_{q}
  46. 1 i , k n 1\leq i,k\leq n
  47. β i β j = l = 1 n m i j l β l , m i j l 𝔽 q \beta_{i}\beta_{j}=\sum_{l=1}^{n}m_{ijl}\beta_{l},\ \ m_{ijl}\in\mathbb{F}_{q}
  48. 1 i , j n 1\leq i,j\leq n
  49. n n
  50. v i v_{i}
  51. u j u_{j}
  52. β i \beta_{i}
  53. u j , v i u_{j},v_{i}
  54. x k , y l x_{k},y_{l}
  55. n n
  56. y l y_{l}
  57. x k x_{k}
  58. n n
  59. y l y_{l}
  60. x k x_{k}
  61. P P
  62. x x
  63. 𝔽 q n \mathbb{F}_{q^{n}}
  64. q = 2 q=2
  65. 𝔽 q n \mathbb{F}_{q^{n}}
  66. P ( x ) = y P(x)=y
  67. P P
  68. n n
  69. ( p 1 , , p n ) (p_{1},...,p_{n})
  70. ( p 1 , , p n ) (p_{1},...,p_{n})
  71. 𝔽 q n \mathbb{F}_{q^{n}}
  72. 𝔽 q \mathbb{F}_{q}
  73. S S
  74. T T
  75. ( S , P , T ) (S,P,T)
  76. P P
  77. 𝔽 q n \mathbb{F}_{q^{n}}
  78. ( p 1 , , p n ) (p_{1},...,p_{n})
  79. ( S , P , T ) (S,P,T)
  80. input x x = ( x 1 , , x n ) secret : S x secret : P y secret : T output y \,\text{input}x\to x=(x_{1},...,x_{n})\overset{\,\text{secret}:S}{\to}x^{% \prime}\overset{\,\text{secret}:P}{\to}y^{\prime}\overset{\,\text{secret}:T}{% \to}\,\text{output}y
  81. P P
  82. d d
  83. 𝔽 q n \mathbb{F}_{q^{n}}
  84. 𝔽 q n [ x ] \mathbb{F}_{q^{n}}[x]
  85. P P
  86. 𝔽 q \mathbb{F}_{q}
  87. P P
  88. x q s i + q t i x^{q^{s_{i}}+q^{t_{i}}}
  89. q q
  90. P P
  91. P ( x ) = c i x q s i + q t i P(x)=\sum c_{i}x^{q^{s_{i}}+q^{t_{i}}}
  92. d d
  93. d d
  94. P P
  95. d d
  96. P P
  97. P - 1 P^{-1}
  98. d 2 ( ln d ) O ( 1 ) n 2 𝔽 q d^{2}(\ln d)^{O(1)}n^{2}\mathbb{F}_{q}
  99. n n
  100. ( p 1 , , p n ) (p_{1},...,p_{n})
  101. 𝔽 q \mathbb{F}_{q}
  102. M M
  103. 𝔽 q n 𝔽 q n \mathbb{F}_{q^{n}}\to\mathbb{F}_{q}^{n}
  104. M M
  105. ( x 1 , , x n ) 𝔽 q n (x_{1},...,x_{n})\in\mathbb{F}_{q}^{n}
  106. M M
  107. p i p_{i}
  108. ( x 1 , , x n ) (x_{1},...,x_{n})
  109. ( p 1 ( x 1 , , x n ) , p 2 ( x 1 , , x n ) , , p n ( x 1 , , x n ) ) 𝔽 q n (p_{1}(x_{1},...,x_{n}),p_{2}(x_{1},...,x_{n}),...,p_{n}(x_{1},...,x_{n}))\in% \mathbb{F}_{q}^{n}
  110. S , T , P S,T,P
  111. p i p_{i}
  112. S S
  113. x x^{\prime}
  114. x x^{\prime}
  115. 𝔽 q n 𝔽 q n \mathbb{F}{q^{n}}\to\mathbb{F}_{q^{n}}
  116. P P
  117. 𝔽 q n \mathbb{F}_{q^{n}}
  118. y 𝔽 q n y^{\prime}\in\mathbb{F}_{q^{n}}
  119. y y^{\prime}
  120. ( y 1 , , y n ) (y_{1}^{\prime},...,y_{n}^{\prime})
  121. T T
  122. y 𝔽 q n y\in\mathbb{F}_{q^{n}}
  123. ( y 1 , , y n ) 𝔽 q n (y_{1},...,y_{n})\in\mathbb{F}_{q}^{n}
  124. y y
  125. ( S , P , T ) (S,P,T)
  126. S S
  127. T T
  128. P ( x ) = y P(x^{\prime})=y^{\prime}
  129. P P
  130. X = ( x 1 , , x d ) 𝔽 q n X^{\prime}=(x_{1}^{\prime},...,x_{d}^{\prime})\in\mathbb{F}_{q^{n}}
  131. P P
  132. r r
  133. M M
  134. M M
  135. X X^{\prime}
  136. M + r x secret : S x secret : P y secret : T y M\overset{+r}{\to}x\overset{\,\text{secret}:S}{\to}x^{\prime}\overset{\,\text{% secret}:P}{\to}y^{\prime}\overset{\,\text{secret}:T}{\to}y
  137. f f
  138. 𝔽 q n \mathbb{F}_{q^{n}}

High-κ_dielectric.html

  1. C = κ ε 0 A t C=\frac{\kappa\varepsilon_{0}A}{t}
  2. I D , S a t = W L μ C i n v ( V G - V t h ) 2 2 I_{D,Sat}=\frac{W}{L}\mu\,C_{inv}\frac{(V_{G}-V_{th})^{2}}{2}

Higher-order_abstract_syntax.html

  1. λ y . e \lambda y.e

Highly_optimized_tolerance.html

  1. X X
  2. x i x_{i}
  3. p i p_{i}
  4. r i r_{i}
  5. x i = r i - β x_{i}=r_{i}^{-\beta}
  6. β \beta
  7. L = i = 0 N - 1 p i x i L=\sum_{i=0}^{N-1}p_{i}x_{i}
  8. i = 0 N - 1 r i = κ \sum_{i=0}^{N-1}r_{i}=\kappa
  9. p i x i - ( 1 + 1 / β ) p_{i}\propto x_{i}^{-(1+1/\beta)}
  10. x i x_{i}
  11. r i r_{i}

Hilbert's_eighteenth_problem.html

  1. n n
  2. n n

Hilbert's_nineteenth_problem.html

  1. F ( p , q , z ; x , y ) d x d y = Minimum [ z x = p ; z y = q ] {\iint F(p,q,z;x,y)dxdy}=\,\text{Minimum}\qquad\left[\frac{\partial z}{% \partial x}=p\quad;\quad\frac{\partial z}{\partial y}=q\right]
  2. 2 F 2 p 2 F 2 q - ( 2 F p q ) 2 > 0 \frac{\partial^{2}F}{\partial^{2}p}\cdot\frac{\partial^{2}F}{\partial^{2}q}-% \left(\frac{\partial^{2}F}{{\partial p}{\partial q}}\right)^{2}>0
  3. F F
  4. p , q , z , x p,q,z,x
  5. y y
  6. F F
  7. C 3 {C}_{3}
  8. C 3 {C}_{3}
  9. D i ( a i j ( x ) D j u ) = 0 D_{i}(a^{ij}(x)D_{j}u)=0
  10. U L ( D w ) d x \int_{U}L(Dw)\mathrm{d}x
  11. Σ i ( L p i ( D w ) ) x i = 0 \Sigma_{i}(L_{p_{i}}(Dw))_{x_{i}}=0
  12. Σ i ( L p i p j ( D w ) w x j x k ) x i = 0 \Sigma_{i}(L_{p_{i}p_{j}}(Dw)w_{x_{j}x_{k}})_{x_{i}}=0
  13. D i ( a i j ( x ) D j u ) = 0 D_{i}(a^{ij}(x)D_{j}u)=0
  14. a i j = L p i p j ( D w ) a^{ij}=L_{p_{i}p_{j}}(Dw)
  15. D i ( a i j ( x ) D j u ) = D t ( u ) D_{i}(a^{ij}(x)D_{j}u)=D_{t}(u)
  16. D i ( a i j ( x ) D j u ) = 0 D_{i}(a^{ij}(x)D_{j}u)=0
  17. K K
  18. - 1 / 2 -1/2
  19. F F
  20. C 2 {C}_{2}

Hilbert's_seventeenth_problem.html

  1. f ( x , y , z ) = z 6 + x 4 y 2 + x 2 y 4 - 3 x 2 y 2 z 2 f(x,y,z)=z^{6}+x^{4}y^{2}+x^{2}y^{4}-3x^{2}y^{2}z^{2}\,
  2. l 1 l_{1}
  3. v ( n , d ) , v(n,d),\,
  4. v ( n , d ) v(n,d)
  5. v ( n , d ) 2 n , v(n,d)\leq 2^{n},\,

Hilbert's_thirteenth_problem.html

  1. x 7 + a x 3 + b x 2 + c x + 1 = 0 x^{7}+ax^{3}+bx^{2}+cx+1=0

Hilbert's_twelfth_problem.html

  1. 𝐐 ( i , 1 + 2 i 4 ) / 𝐐 ( i ) \mathbf{Q}(i,\sqrt[4]{1+2i})/\mathbf{Q}(i)

Hilbert_class_field.html

  1. K = K=\mathbb{Q}
  2. K = ( - 15 ) K=\mathbb{Q}(\sqrt{-15})
  3. L = ( - 3 , 5 ) L=\mathbb{Q}(\sqrt{-3},\sqrt{5})
  4. L L
  5. ( 3 , i ) \mathbb{Q}(\sqrt{3},i)
  6. ( 3 ) \mathbb{Q}(\sqrt{3})

Hilbert_manifold.html

  1. g ( v , w ) ( p ) := v , w H for v , w T p H , g(v,w)(p):=\langle v,w\rangle_{H}\,\text{ for }v,w\in\mathrm{T}_{p}H,

Hilbert_modular_form.html

  1. \mathcal{H}
  2. σ 1 , , σ m \sigma_{1},\dots,\sigma_{m}
  3. G L 2 ( F ) GL_{2}(F)
  4. G L 2 ( ) m . GL_{2}(\mathbb{R})^{m}.
  5. 𝒪 F \mathcal{O}_{F}
  6. G L 2 + ( 𝒪 F ) GL_{2}^{+}(\mathcal{O}_{F})
  7. z = ( z 1 , , z m ) m z=(z_{1},\dots,z_{m})\in\mathcal{H}^{m}
  8. G L 2 + ( 𝒪 F ) GL_{2}^{+}(\mathcal{O}_{F})
  9. γ z = ( σ 1 ( γ ) z 1 , , σ m ( γ ) z m ) \gamma\cdot z=(\sigma_{1}(\gamma)z_{1},\dots,\sigma_{m}(\gamma)z_{m})
  10. g = ( a b c d ) G L 2 ( ) g=\begin{pmatrix}a&b\\ c&d\end{pmatrix}\in GL_{2}(\mathbb{R})
  11. j ( g , z ) = det ( g ) - 1 / 2 ( c z + d ) j(g,z)=\det(g)^{-1/2}(cz+d)
  12. ( k 1 , , k m ) (k_{1},\dots,k_{m})
  13. m \mathcal{H}^{m}
  14. γ G L 2 + ( 𝒪 F ) \gamma\in GL_{2}^{+}(\mathcal{O}_{F})
  15. f ( γ z ) = i = 1 m j ( σ i ( γ ) , z i ) k i f ( z ) . f(\gamma z)=\prod_{i=1}^{m}j(\sigma_{i}(\gamma),z_{i})^{k_{i}}f(z).

Hill_equation_(biochemistry).html

  1. θ = [ L ] n K d + [ L ] n = [ L ] n ( K A ) n + [ L ] n = 1 ( K A [ L ] ) n + 1 \theta={[L]^{n}\over K_{d}+[L]^{n}}={[L]^{n}\over(K_{A})^{n}+[L]^{n}}={1\over(% {K_{A}\over[L]})^{n}+1}
  2. θ \theta
  3. [ L ] [L]
  4. K d K_{d}
  5. K A K_{A}
  6. K D K_{D}
  7. n n
  8. log ( θ 1 - θ ) = n log [ L ] - log K A . \log\left({\theta\over 1-\theta}\right)=n\log{[L]}-\log{K_{A}}.
  9. n > 1 n>1
  10. n < 1 n<1
  11. n = 1 n=1

Hippopede.html

  1. ( x 2 + y 2 ) 2 = c x 2 + d y 2 (x^{2}+y^{2})^{2}=cx^{2}+dy^{2}
  2. r 2 = 4 b ( a - b sin 2 θ ) r^{2}=4b(a-b\sin^{2}\theta)\,
  3. ( x 2 + y 2 ) 2 + 4 b ( b - a ) ( x 2 + y 2 ) = 4 b 2 x 2 (x^{2}+y^{2})^{2}+4b(b-a)(x^{2}+y^{2})=4b^{2}x^{2}

Hirzebruch–Riemann–Roch_theorem.html

  1. χ ( X , E ) = i = 0 dim X ( - 1 ) i dim H i ( X , E ) \chi(X,E)=\sum_{i=0}^{\dim_{\mathbb{C}}X}(-1)^{i}\dim_{\mathbb{C}}H^{i}(X,E)
  2. χ ( X , E ) = ch n - j ( E ) T j j ! \chi(X,E)=\sum\operatorname{ch}_{n-j}(E)\frac{T_{j}}{j!}
  3. ch ( E ) = exp ( x i ) \operatorname{ch}(E)=\sum\exp(x_{i})
  4. C ( E ) = C j ( E ) = ( 1 + x i ) . C(E)=\sum C_{j}(E)=\prod(1+x_{i}).
  5. χ ( X , E ) = X ch ( E ) td ( X ) \chi(X,E)=\int_{X}\operatorname{ch}(E)\operatorname{td}(X)
  6. h 0 ( 𝒪 ( D ) ) - h 1 ( 𝒪 ( D ) ) = c 1 ( 𝒪 ( D ) ) + c 1 ( T ( X ) ) / 2 h^{0}(\mathcal{O}(D))-h^{1}(\mathcal{O}(D))=c_{1}(\mathcal{O}(D))+c_{1}(T(X))/% 2\ \
  7. ( D ) - ( K - D ) = deg ( D ) + 1 - g . \ell(D)-\ell(K-D)=\,\text{deg}(D)+1-g.
  8. h 0 ( V ) - h 1 ( V ) = c 1 ( V ) + rank ( V ) ( 1 - g ) . h^{0}(V)-h^{1}(V)=c_{1}(V)+\,\text{rank}(V)(1-g).
  9. χ ( D ) = χ ( 𝒪 ) + ( ( D . D ) - ( D . K ) ) / 2. \chi(D)=\chi(\mathcal{O})+((D.D)-(D.K))/2.

History_of_thermodynamics.html

  1. S = k log W S=k\log W\,

Hit_rate.html

  1. Sales Won Sales Won + Sales Lost + Sales Abandoned . \frac{\sum{\mbox{Sales Won}~{}}}{\sum{\mbox{Sales Won}~{}}+\sum{\mbox{Sales % Lost}~{}}+\sum{\mbox{Sales Abandoned}~{}}}.

HNN_extension.html

  1. G * α = S , t | R , t h t - 1 = α ( h ) , h H . G*_{\alpha}=\left\langle S,t\Big|R,tht^{-1}=\alpha(h),\forall h\in H\right\rangle.
  2. w = g 0 t ε 1 g 1 t ε 2 g n - 1 t ε n g n , g i G , ε i = ± 1. w=g_{0}t^{\varepsilon_{1}}g_{1}t^{\varepsilon_{2}}\cdots g_{n-1}t^{\varepsilon% _{n}}g_{n},\qquad g_{i}\in G,\varepsilon_{i}=\pm 1.

Hodgkin–Huxley_model.html

  1. I c = C m d V m d t I_{c}=C_{m}\frac{{\mathrm{d}}V_{m}}{{\mathrm{d}}t}
  2. I i = g i ( V m - V i ) I_{i}={g_{i}}(V_{m}-V_{i})\;
  3. V i V_{i}
  4. I = C m d V m d t + g K ( V m - V K ) + g N a ( V m - V N a ) + g l ( V m - V l ) , I=C_{m}\frac{{\mathrm{d}}V_{m}}{{\mathrm{d}}t}+g_{K}(V_{m}-V_{K})+g_{Na}(V_{m}% -V_{Na})+g_{l}(V_{m}-V_{l}),
  5. I = C m d V m d t + g ¯ K n 4 ( V m - V K ) + g ¯ Na m 3 h ( V m - V N a ) + g ¯ l ( V m - V l ) , I=C_{m}\frac{{\mathrm{d}}V_{m}}{{\mathrm{d}}t}+\bar{g}\text{K}n^{4}(V_{m}-V_{K% })+\bar{g}\text{Na}m^{3}h(V_{m}-V_{Na})+\bar{g}_{l}(V_{m}-V_{l}),
  6. d n d t = α n ( V m ) ( 1 - n ) - β n ( V m ) n \frac{dn}{dt}=\alpha_{n}(V_{m})(1-n)-\beta_{n}(V_{m})n
  7. d m d t = α m ( V m ) ( 1 - m ) - β m ( V m ) m \frac{dm}{dt}=\alpha_{m}(V_{m})(1-m)-\beta_{m}(V_{m})m
  8. d h d t = α h ( V m ) ( 1 - h ) - β h ( V m ) h \frac{dh}{dt}=\alpha_{h}(V_{m})(1-h)-\beta_{h}(V_{m})h
  9. α i \alpha_{i}
  10. β i \beta_{i}
  11. g ¯ n \bar{g}_{n}
  12. p = ( n , m , h ) p=(n,m,h)
  13. α p \alpha_{p}
  14. β p \beta_{p}
  15. α p ( V m ) = p ( V m ) / τ p \alpha_{p}(V_{m})=p_{\infty}(V_{m})/\tau_{p}
  16. β p ( V m ) = ( 1 - p ( V m ) ) / τ p . \beta_{p}(V_{m})=(1-p_{\infty}(V_{m}))/\tau_{p}.
  17. p p_{\infty}
  18. ( 1 - p ) (1-p_{\infty})
  19. V m V_{m}
  20. α \alpha
  21. β \beta
  22. α n ( V m ) = 0.01 ( V m - 10 ) exp ( V m - 10 10 ) - 1 α m ( V m ) = 0.1 ( V m - 25 ) exp ( V m - 25 10 ) - 1 α h ( V m ) = 0.07 exp ( V m 20 ) β n ( V m ) = 0.125 exp ( V m 80 ) β m ( V m ) = 4 exp ( V m 18 ) β h ( V m ) = 1 exp ( V m - 30 10 ) + 1 \begin{array}[]{lll}\alpha_{n}(V_{m})=\frac{0.01(V_{m}-10)}{\exp\big(\frac{V_{% m}-10}{10}\big)-1}&\alpha_{m}(V_{m})=\frac{0.1(V_{m}-25)}{\exp\big(\frac{V_{m}% -25}{10}\big)-1}&\alpha_{h}(V_{m})=0.07\exp\bigg(\frac{V_{m}}{20}\bigg)\\ \beta_{n}(V_{m})=0.125\exp\bigg(\frac{V_{m}}{80}\bigg)&\beta_{m}(V_{m})=4\exp% \bigg(\frac{V_{m}}{18}\bigg)&\beta_{h}(V_{m})=\frac{1}{\exp\big(\frac{V_{m}-30% }{10}\big)+1}\end{array}
  23. α \alpha
  24. β \beta
  25. A p ( V m - B p ) exp ( V m - B p C p ) - D p \frac{A_{p}(V_{m}-B_{p})}{\exp\big(\frac{V_{m}-B_{p}}{C_{p}}\big)-D_{p}}
  26. m ( t ) = m 0 - [ ( m 0 - m ) ( 1 - e - t / τ m ) ] m(t)=m_{0}-[(m_{0}-m_{\infty})(1-e^{-t/\tau_{m}})]\,
  27. h ( t ) = h 0 - [ ( h 0 - h ) ( 1 - e - t / τ h ) ] h(t)=h_{0}-[(h_{0}-h_{\infty})(1-e^{-t/\tau_{h}})]\,
  28. n ( t ) = n 0 - [ ( n 0 - n ) ( 1 - e - t / τ n ) ] n(t)=n_{0}-[(n_{0}-n_{\infty})(1-e^{-t/\tau_{n}})]\,
  29. V m V_{m}
  30. I Na ( t ) = g ¯ Na m ( V m ) 3 h ( V m ) ( V m - E Na ) , I_{\mathrm{Na}}(t)=\bar{g}_{\mathrm{Na}}m(V_{m})^{3}h(V_{m})(V_{m}-E_{\mathrm{% Na}}),
  31. I K ( t ) = g ¯ K n ( V m ) 4 ( V m - E K ) . I_{\mathrm{K}}(t)=\bar{g}_{\mathrm{K}}n(V_{m})^{4}(V_{m}-E_{\mathrm{K}}).
  32. I = a 2 R 2 V x 2 , I=\frac{a}{2R}\frac{\partial^{2}V}{\partial x^{2}},
  33. g i g_{i}
  34. I I

Hofstadter's_butterfly.html

  1. λ = 1 \lambda=1

Hole_argument.html

  1. \to
  2. e μ I ( x ) e_{\mu}^{I}(x)
  3. g μ ν ( x ) g_{\mu\nu}(x)
  4. d s 2 = g μ ν ( x ) d x μ d x ν ds^{2}=g_{\mu\nu}(x)dx^{\mu}dx^{\nu}
  5. x x
  6. y y
  7. x x
  8. y y
  9. x x
  10. x x
  11. y y
  12. y y
  13. t = 0 t=0
  14. t = 0 t=0
  15. x x
  16. y y

Hollow-cathode_lamp.html

  1. A + h ν A * A+h\nu\rightarrow A^{*}
  2. A * + e - A + + 2 e - A^{*}+e^{-}\rightarrow A^{+}+2e^{-}
  3. A A
  4. h ν h\nu
  5. A * A^{*}
  6. e - e^{-}

Holomorph_(mathematics).html

  1. G G
  2. G G
  3. Hol ( G ) \operatorname{Hol}(G)
  4. Aut ( G ) \operatorname{Aut}(G)
  5. G G
  6. Hol ( G ) = G Aut ( G ) \operatorname{Hol}(G)=G\rtimes\operatorname{Aut}(G)
  7. ( g , α ) ( h , β ) = ( g α ( h ) , α β ) . (g,\alpha)(h,\beta)=(g\alpha(h),\alpha\beta).
  8. G ϕ A G\rtimes_{\phi}A
  9. G G
  10. A A
  11. ϕ : A Aut ( G ) \phi:A\rightarrow\operatorname{Aut}(G)
  12. ( g , a ) ( h , b ) = ( g ϕ ( a ) ( h ) , a b ) (g,a)(h,b)=(g\phi(a)(h),ab)
  13. ϕ ( a ) Aut ( G ) \phi(a)\in\operatorname{Aut}(G)
  14. ϕ ( a ) ( h ) G \phi(a)(h)\in G
  15. A = Aut ( G ) A=\operatorname{Aut}(G)
  16. ϕ \phi
  17. ϕ \phi
  18. G = C 3 = x = { 1 , x , x 2 } G=C_{3}=\langle x\rangle=\{1,x,x^{2}\}
  19. Aut ( G ) = σ = { 1 , σ } \operatorname{Aut}(G)=\langle\sigma\rangle=\{1,\sigma\}
  20. σ ( x ) = x 2 \sigma(x)=x^{2}
  21. Hol ( G ) = { ( x i , σ j ) } \operatorname{Hol}(G)=\{(x^{i},\sigma^{j})\}
  22. ( x i 1 , σ j 1 ) ( x i 2 , σ j 2 ) = ( x i 1 + i 2 2 j 1 , σ j 1 + j 2 ) (x^{i_{1}},\sigma^{j_{1}})(x^{i_{2}},\sigma^{j_{2}})=(x^{i_{1}+i_{2}2^{{}^{j_{% 1}}}},\sigma^{j_{1}+j_{2}})
  23. x x
  24. σ \sigma
  25. ( x , σ ) ( x 2 , σ ) = ( x 1 + 2 2 , σ 2 ) = ( x 2 , 1 ) (x,\sigma)(x^{2},\sigma)=(x^{1+2\cdot 2},\sigma^{2})=(x^{2},1)
  26. ( x 2 , σ ) ( x , σ ) = ( x , 1 ) (x^{2},\sigma)(x,\sigma)=(x,1)
  27. Hol ( C 3 ) \operatorname{Hol}(C_{3})
  28. S 3 S_{3}
  29. Inn ( G ) Im ( g λ ( g ) ρ ( g ) ) \operatorname{Inn}(G)\cong\operatorname{Im}(g\mapsto\lambda(g)\rho(g))

Holomorphic_functional_calculus.html

  1. p ( z ) = i = 0 m a i z i p(z)=\sum_{i=0}^{m}a_{i}z^{i}
  2. p ( T ) = i = 0 m a i T i p(T)=\sum_{i=0}^{m}a_{i}T^{i}
  3. f ( z ) = i = 0 a i z i , f(z)=\sum_{i=0}^{\infty}a_{i}z^{i},
  4. f ( T ) = i = 0 a i T i . f(T)=\sum_{i=0}^{\infty}a_{i}T^{i}.
  5. f ( T ) = e T = I + T + T 2 2 ! + T 3 3 ! + . f(T)=e^{T}=I+T+\frac{T^{2}}{2!}+\frac{T^{3}}{3!}+\cdots.
  6. ln ( z + 1 ) = z - z 2 2 + z 3 3 - , \ln(z+1)=z-\frac{z^{2}}{2}+\frac{z^{3}}{3}-\cdots,
  7. f ( z ) = 1 ( z - 2 ) ( z - 5 ) f(z)=\frac{1}{(z-2)(z-5)}
  8. f ( T ) = ( T - 2 I ) - 1 ( T - 5 I ) - 1 . f(T)=(T-2I)^{-1}(T-5I)^{-1}.\,
  9. f ( z ) = 1 2 π i Γ f ( ζ ) ζ - z d ζ f(z)=\frac{1}{2\pi i}\int\nolimits_{\Gamma}\frac{f(\zeta)}{\zeta-z}\,d\zeta
  10. f ( T ) = 1 2 π i Γ f ( ζ ) ζ - T d ζ , f(T)=\frac{1}{2\pi i}\int_{\Gamma}\frac{f(\zeta)}{\zeta-T}\,d\zeta,
  11. f ( T ) = 1 2 π i Γ f ( ζ ) ζ - T d ζ , f(T)=\frac{1}{2\pi i}\int\nolimits_{\Gamma}\frac{f(\zeta)}{\zeta-T}\,d\zeta,
  12. Γ g = i γ i g . \int_{\Gamma}g=\sum\nolimits_{i}\int_{\gamma_{i}}g.
  13. 1 2 π i Γ f ( ζ ) ζ - T d ζ \frac{1}{2\pi i}\int_{\Gamma}\frac{f(\zeta)}{\zeta-T}\,d\zeta
  14. 1 2 π i Γ d ζ ζ - z . \frac{1}{2\pi i}\int_{\Gamma}\frac{d\zeta}{\zeta-z}.
  15. ( z 1 - T ) - 1 - ( z 2 - T ) - 1 = ( z 1 - T ) - 1 ( z 2 - z 1 ) ( z 2 - T ) - 1 . (z_{1}-T)^{-1}-(z_{2}-T)^{-1}=(z_{1}-T)^{-1}(z_{2}-z_{1})(z_{2}-T)^{-1}.\,
  16. ( z 1 - T ) - 1 ( z 2 - T ) - 1 = ( z 1 - T ) - 1 - ( z 2 - T ) - 1 ( z 2 - z 1 ) . (z_{1}-T)^{-1}(z_{2}-T)^{-1}=\frac{(z_{1}-T)^{-1}-(z_{2}-T)^{-1}}{(z_{2}-z_{1}% )}.
  17. 1 z 2 - T = 1 z 1 - T 1 1 - z 1 - z 2 z 1 - T \frac{1}{z_{2}-T}=\frac{1}{z_{1}-T}\cdot\frac{1}{1-\frac{z_{1}-z_{2}}{z_{1}-T}}
  18. ( z 1 - T ) - 1 n 0 ( ( z 1 - z 2 ) ( z 1 - T ) - 1 ) n (z_{1}-T)^{-1}\sum_{n\geq 0}\left((z_{1}-z_{2})(z_{1}-T)^{-1}\right)^{n}
  19. | z 1 - z 2 | < 1 ( z 1 - T ) - 1 . |z_{1}-z_{2}|<\frac{1}{\left\|(z_{1}-T)^{-1}\right\|}.
  20. 1 z - T = 1 z 1 1 - T z \frac{1}{z-T}=\frac{1}{z}\cdot\frac{1}{1-\frac{T}{z}}
  21. 1 z n 0 ( T z ) n . \frac{1}{z}\sum_{n\geq 0}\left(\frac{T}{z}\right)^{n}.
  22. T z < 1 , i.e. | z | > T . \left\|\frac{T}{z}\right\|<1,\;\,\text{i.e.}\;|z|>\|T\|.
  23. n ( Γ , a ) = i n ( γ i , a ) , n(\Gamma,a)=\sum\nolimits_{i}n(\gamma_{i},a),
  24. Γ g L ( X ) , \int_{\Gamma}g\in L(X),
  25. ϕ ( Γ g ) = Γ ϕ ( g ) . \phi\left(\int_{\Gamma}g\right)=\int_{\Gamma}\phi(g).
  26. Γ ϕ ( g ) = 0. \int_{\Gamma}\phi(g)=0.
  27. Γ f ( ζ ) ζ - T d ζ = Ω f ( ζ ) ζ - T d ζ . \int_{\Gamma}\frac{f(\zeta)}{\zeta-T}\,d\zeta=\int_{\Omega}\frac{f(\zeta)}{% \zeta-T}\,d\zeta.
  28. Ω f ( ζ ) ζ - T d ζ = - Ω f ( ζ ) ζ - T d ζ . \int_{\Omega}\frac{f(\zeta)}{\zeta-T}\,d\zeta=-\int_{\Omega^{\prime}}\frac{f(% \zeta)}{\zeta-T}\,d\zeta.
  29. ζ f ( ζ ) ζ - T \zeta\rightarrow\frac{f(\zeta)}{\zeta-T}
  30. Γ Ω f ( ζ ) ζ - T d ζ = 0 \int_{\Gamma\cup\Omega^{\prime}}\frac{f(\zeta)}{\zeta-T}\,d\zeta=0
  31. Γ f ( ζ ) ζ - T d ζ + Ω f ( ζ ) ζ - T d ζ = Γ f ( ζ ) ζ - T d ζ - Ω f ( ζ ) ζ - T d ζ = 0. \int_{\Gamma}\frac{f(\zeta)}{\zeta-T}\,d\zeta+\int_{\Omega^{\prime}}\frac{f(% \zeta)}{\zeta-T}\,d\zeta=\int_{\Gamma}\frac{f(\zeta)}{\zeta-T}\,d\zeta-\int_{% \Omega}\frac{f(\zeta)}{\zeta-T}\,d\zeta=0.
  32. 1 2 π i Γ ζ k ζ - T d ζ = T k \frac{1}{2\pi i}\int_{\Gamma}\frac{\zeta^{k}}{\zeta-T}d\zeta=T^{k}
  33. ( z - T ) - 1 = 1 z n 0 ( T z ) n . (z-T)^{-1}=\frac{1}{z}\sum_{n\geq 0}\left(\frac{T}{z}\right)^{n}.
  34. f ( T ) = 1 2 π i Γ ( n 0 T n ζ n + 1 - k ) d ζ f(T)=\frac{1}{2\pi i}\int_{\Gamma}\left(\sum_{n\geq 0}\frac{T^{n}}{\zeta^{n+1-% k}}\right)d\zeta
  35. n 0 T n 1 2 π i ( Γ d ζ ζ n + 1 - k ) = n 0 T n δ n k = T k . \sum_{n\geq 0}T^{n}\cdot\frac{1}{2\pi i}\left(\int_{\Gamma}\frac{d\zeta}{\zeta% ^{n+1-k}}\right)=\sum_{n\geq 0}T^{n}\cdot\delta_{nk}=T^{k}.
  36. f 1 ( T ) f 2 ( T ) = ( f 1 f 2 ) ( T ) . f_{1}(T)f_{2}(T)=(f_{1}\cdot f_{2})(T).\,
  37. f 1 ( T ) f 2 ( T ) = ( 1 2 π i Γ 1 f 1 ( ζ ) ζ - T d ζ ) ( 1 2 π i Γ 2 f 2 ( ω ) ω - T d ω ) = 1 ( 2 π i ) 2 Γ 1 Γ 2 f 1 ( ζ ) f 2 ( ω ) ( ζ - T ) ( ω - T ) d ω d ζ = 1 ( 2 π i ) 2 Γ 1 Γ 2 f 1 ( ζ ) f 2 ( ω ) ( ( ζ - T ) - 1 - ( ω - T ) - 1 ω - ζ ) d ω d ζ First Resolvent Formula = 1 ( 2 π i ) 2 { ( Γ 1 f 1 ( ζ ) ζ - T [ Γ 2 f 2 ( ω ) ω - ζ d ω ] d ζ ) - ( Γ 2 f 2 ( ω ) ω - T [ Γ 1 f 1 ( ζ ) ω - ζ d ζ ] d ω ) } = 1 ( 2 π i ) 2 Γ 1 f 1 ( ζ ) ζ - T [ Γ 2 f 2 ( ω ) ω - ζ d ω ] d ζ \begin{aligned}\displaystyle f_{1}(T)f_{2}(T)&\displaystyle=\left(\frac{1}{2% \pi i}\int_{\Gamma_{1}}\frac{f_{1}(\zeta)}{\zeta-T}d\zeta\right)\left(\frac{1}% {2\pi i}\int_{\Gamma_{2}}\frac{f_{2}(\omega)}{\omega-T}\,d\omega\right)\\ &\displaystyle=\frac{1}{(2\pi i)^{2}}\int_{\Gamma_{1}}\int_{\Gamma_{2}}\frac{f% _{1}(\zeta)f_{2}(\omega)}{(\zeta-T)(\omega-T)}\;d\omega\,d\zeta\\ &\displaystyle=\frac{1}{(2\pi i)^{2}}\int_{\Gamma_{1}}\int_{\Gamma_{2}}f_{1}(% \zeta)f_{2}(\omega)\left(\frac{(\zeta-T)^{-1}-(\omega-T)^{-1}}{\omega-\zeta}% \right)d\omega\,d\zeta&&\displaystyle\,\text{First Resolvent Formula}\\ &\displaystyle=\frac{1}{(2\pi i)^{2}}\left\{\left(\int_{\Gamma_{1}}\frac{f_{1}% (\zeta)}{\zeta-T}\left[\int_{\Gamma_{2}}\frac{f_{2}(\omega)}{\omega-\zeta}d% \omega\right]d\zeta\right)-\left(\int_{\Gamma_{2}}\frac{f_{2}(\omega)}{\omega-% T}\left[\int_{\Gamma_{1}}\frac{f_{1}(\zeta)}{\omega-\zeta}d\zeta\right]d\omega% \right)\right\}\\ &\displaystyle=\frac{1}{(2\pi i)^{2}}\int_{\Gamma_{1}}\frac{f_{1}(\zeta)}{% \zeta-T}\left[\int_{\Gamma_{2}}\frac{f_{2}(\omega)}{\omega-\zeta}d\omega\right% ]d\zeta\end{aligned}
  38. f 1 ( T ) f 2 ( T ) \displaystyle f_{1}(T)f_{2}(T)
  39. f k ( T ) - f l ( T ) = 1 2 π Γ ( f k - f l ) ( ζ ) ζ - T d ζ 1 2 π Γ | ( f k - f l ) ( ζ ) | ( ζ - T ) - 1 d ζ \begin{aligned}\displaystyle\left\|f_{k}(T)-f_{l}(T)\right\|&\displaystyle=% \frac{1}{2\pi}\left\|\int_{\Gamma}\frac{(f_{k}-f_{l})(\zeta)}{\zeta-T}d\zeta% \right\|\\ &\displaystyle\leq\frac{1}{2\pi}\int_{\Gamma}\left|(f_{k}-f_{l})(\zeta)\right|% \cdot\left\|(\zeta-T)^{-1}\right\|d\zeta\end{aligned}
  40. f ( z ) - f ( μ ) = ( z - μ ) g ( z ) . f(z)-f(\mu)=(z-\mu)g(z).\,
  41. g ( z ) = 1 f ( z ) - μ . g(z)=\frac{1}{f(z)-\mu}.
  42. e ( T ) = 1 2 π i Γ e ( z ) z - T d z e(T)=\frac{1}{2\pi i}\int_{\Gamma}\frac{e(z)}{z-T}\,dz
  43. P ( K ; T ) = 1 2 π i Γ d z z - T P(K;T)=\frac{1}{2\pi i}\int\nolimits_{\Gamma}\frac{dz}{z-T}
  44. σ ( T ) = i = 1 m F i . \sigma(T)=\bigcup_{i=1}^{m}F_{i}.
  45. i e i ( T ) = I , \sum_{i}e_{i}(T)=I,\,
  46. X = i X i . X=\sum_{i}X_{i}.\,
  47. T = i T i . T=\sum_{i}T_{i}.\,
  48. X = i X i . X^{\prime}=\bigoplus_{i}X_{i}.
  49. i x i = i x i , \left\|\bigoplus_{i}x_{i}\right\|=\sum_{i}\|x_{i}\|,
  50. R ( i x i ) = i x i R\left(\bigoplus_{i}x_{i}\right)=\sum_{i}x_{i}
  51. R T R - 1 = i T i . RTR^{-1}=\bigoplus_{i}T_{i}.
  52. i T i \bigoplus_{i}T_{i}

Holomorphically_convex_hull.html

  1. G n G\subset{\mathbb{C}}^{n}
  2. G G
  3. n n
  4. 𝒪 ( G ) {\mathcal{O}}(G)
  5. G . G.
  6. K G K\subset G
  7. K K
  8. K ^ G := { z G | | f ( z ) | sup w K | f ( w ) | for all f 𝒪 ( G ) } . \hat{K}_{G}:=\{z\in G\big|\left|f(z)\right|\leq\sup_{w\in K}\left|f(w)\right|% \mbox{ for all }~{}f\in{\mathcal{O}}(G)\}.
  9. G G
  10. K G K\subset G
  11. G G
  12. K ^ G \hat{K}_{G}
  13. G G
  14. n = 1 n=1
  15. G G
  16. K ^ G \hat{K}_{G}
  17. K K
  18. G K G G\setminus K\subset G

Hom_functor.html

  1. \definecolor g r a y R G B 249 , 249 , 249 \pagecolor g r a y g f g \definecolor{gray}{RGB}{249,249,249}\pagecolor{gray}g\mapsto f\circ g
  2. \definecolor g r a y R G B 249 , 249 , 249 \pagecolor g r a y g g h \definecolor{gray}{RGB}{249,249,249}\pagecolor{gray}g\mapsto g\circ h
  3. [ - - ] : C o p × C C \left[-\ -\right]:C^{op}\times C\to C
  4. : C o p × C C \Rightarrow:C^{op}\times C\to C
  5. hom ( - , - ) : C o p × C C \,\text{hom}(-,-):C^{op}\times C\to C
  6. U : C 𝐒𝐞𝐭 U:C\to\,\textbf{Set}
  7. U hom ( - , - ) Hom ( - , - ) U\circ\,\text{hom}(-,-)\simeq\,\text{Hom}(-,-)
  8. \simeq
  9. Hom ( I , hom ( - , - ) ) Hom ( - , - ) \,\text{Hom}(I,\,\text{hom}(-,-))\simeq\,\text{Hom}(-,-)
  10. Hom ( X , Y Z ) Hom ( X Y , Z ) \,\text{Hom}(X,Y\Rightarrow Z)\simeq\,\text{Hom}(X\otimes Y,Z)
  11. \otimes
  12. Y Z Y\Rightarrow Z
  13. \otimes
  14. × \times
  15. Y Z Y\Rightarrow Z
  16. Z Y Z^{Y}
  17. id C : C C \,\text{id}_{C}\colon C\nrightarrow C
  18. hom ( X , - ) : C C \,\text{hom}(X,-):C\to C
  19. hom ( - , X ) : C op C \,\text{hom}(-,X):C\text{op}\to C
  20. \otimes

Homoeoid.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. a , b , c a,b,c
  3. 0 m 1 0\leq m\leq 1
  4. x 2 a 2 + y 2 b 2 + z 2 c 2 = m 2 \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}+\frac{z^{2}}{c^{2}}=m^{2}
  5. m 1 m\to 1

Homogeneity_(physics).html

  1. E k = 1 2 m v 2 ; E = m c 2 ; E = p v ; E = h c / λ E_{k}=\frac{1}{2}mv^{2};~{}~{}E=mc^{2};~{}~{}E=pv;~{}~{}E=hc/\lambda

Homogeneity_and_heterogeneity.html

  1. h i = ( c i - c batch ) m i c batch m aver . h_{i}=\frac{(c_{i}-c\text{batch})m_{i}}{c\text{batch}m\text{aver}}.
  2. h i h_{i}
  3. i i
  4. c i c_{i}
  5. i i
  6. c batch c\text{batch}
  7. m i m_{i}
  8. i i
  9. m aver m\text{aver}

Homography.html

  1. ( x 0 , , x n ) (x_{0},\ldots,x_{n})
  2. f : V W f:V\rightarrow W
  3. [ x 0 : : x n ] [x_{0}:\cdots:x_{n}]
  4. [ y 0 : : y n ] [y_{0}:\cdots:y_{n}]
  5. y 0 = a 0 , 0 x 0 + + a 0 , n x n y n = a n , 0 x 0 + + a n , n x n . \begin{aligned}\displaystyle y_{0}&\displaystyle=a_{0,0}x_{0}+\dots+a_{0,n}x_{% n}\\ &\displaystyle\vdots\\ \displaystyle y_{n}&\displaystyle=a_{n,0}x_{0}+\dots+a_{n,n}x_{n}.\end{aligned}
  6. y 1 = a 1 , 0 + a 1 , 1 x 1 + + a 1 , n x n a 0 , 0 + a 0 , 1 x 1 + + a 0 , n x n y n = a n , 0 + a n , 1 x 1 + + a n , n x n a 0 , 0 + a 0 , 1 x 1 + + a 0 , n x n \begin{aligned}\displaystyle y_{1}&\displaystyle=\frac{a_{1,0}+a_{1,1}x_{1}+% \dots+a_{1,n}x_{n}}{a_{0,0}+a_{0,1}x_{1}+\dots+a_{0,n}x_{n}}\\ &\displaystyle\vdots\\ \displaystyle y_{n}&\displaystyle=\frac{a_{n,0}+a_{n,1}x_{1}+\dots+a_{n,n}x_{n% }}{a_{0,0}+a_{0,1}x_{1}+\dots+a_{0,n}x_{n}}\end{aligned}
  7. z a z + b c z + d , where a d - b c 0 , z\mapsto\frac{az+b}{cz+d},\,\text{ where }ad-bc\neq 0,
  8. p : V { 0 } P ( V ) p:V\setminus\{0\}\to P(V)
  9. ( p ( e 0 ) , , p ( e n + 1 ) ) \left(p(e_{0}),\ldots,p(e_{n+1})\right)
  10. λ 0 e 0 + + λ n + 1 e n + 1 = 0 . \lambda_{0}e_{0}+\cdots+\lambda_{n+1}e_{n+1}=0\,.
  11. e i e_{i}
  12. λ i e i \lambda_{i}e_{i}
  13. i n i\leq n
  14. e n + 1 e_{n+1}
  15. - λ n + 1 e n + 1 -\lambda_{n+1}e_{n+1}
  16. ( p ( e 0 ) , , p ( e n ) , p ( e 0 + + e n ) ) (p(e_{0}),\ldots,p(e_{n}),p(e_{0}+\cdots+e_{n}))
  17. ( e 0 , , e n ) . (e_{0},\ldots,e_{n}).
  18. e i e_{i}
  19. \ell
  20. = α ( ) \ell^{\prime}=\alpha(\ell)
  21. R = R=\ell\cap\ell^{\prime}
  22. \ell
  23. \ell^{\prime}
  24. \ell
  25. S = A B M , S=AB\cap M,
  26. B = S A O B . B^{\prime}=SA^{\prime}\cap OB.
  27. a , b a,b
  28. c c
  29. F F
  30. h h
  31. F F∪∞
  32. a a
  33. b b
  34. c c
  35. a a
  36. b b
  37. c c
  38. d d
  39. a a , b ; c , d aa,b;c,d
  40. h ( d ) h(d)
  41. F F∪∞
  42. d d
  43. k k : 1 kk:1
  44. ( a , b , c ) (a,b,c)
  45. a a , b ; c , d = k aa,b;c,d=k
  46. U ( z , 1 ) ( a c b d ) = U ( z a + b , z c + d ) . U(z,1)\begin{pmatrix}a&c\\ b&d\end{pmatrix}=U(za+b,\ zc+d).
  47. z z a + b z c + d , z\mapsto\frac{za+b}{zc+d}\ ,
  48. U ( z a + b , z c + d ) U ( ( z c + d ) - 1 ( z a + b ) , 1 ) . U(za+b,\ zc+d)\sim U((zc+d)^{-1}(za+b),\ 1).
  49. h = ( 1 1 0 1 ) h=\begin{pmatrix}1&1\\ 0&1\end{pmatrix}
  50. h n = ( 1 n 0 1 ) = ( 1 0 0 1 ) . h^{n}=\begin{pmatrix}1&n\\ 0&1\end{pmatrix}=\begin{pmatrix}1&0\\ 0&1\end{pmatrix}.
  51. ( a b c d ) , \scriptstyle\begin{pmatrix}a&b\\ c&d\end{pmatrix},
  52. a x + b c x + d \scriptstyle\frac{ax+b}{cx+d}

Homothetic_vector_field.html

  1. X g a b = 2 c g a b \mathcal{L}_{X}g_{ab}=2cg_{ab}

Horrocks–Mumford_bundle.html

  1. 2 F \wedge^{2}F

Hosohedron.html

  1. N 2 = 4 n 2 m + 2 n - m n N_{2}=\frac{4n}{2m+2n-mn}

Hungarian_algorithm.html

  1. O ( n 4 ) O(n^{4})
  2. O ( n 3 ) O(n^{3})
  3. y : ( S T ) y:(S\cup T)\mapsto\mathbb{R}
  4. y ( i ) + y ( j ) c ( i , j ) y(i)+y(j)\leq c(i,j)
  5. i S , j T i\in S,j\in T
  6. v S T y ( v ) \sum_{v\in S\cup T}y(v)
  7. y ( i ) + y ( j ) = c ( i , j ) y(i)+y(j)=c(i,j)
  8. G y G_{y}
  9. G y G_{y}
  10. G y G_{y}
  11. G y \overrightarrow{G_{y}}
  12. R S S R_{S}\subseteq S
  13. R T T R_{T}\subseteq T
  14. R S R_{S}
  15. R T R_{T}
  16. Z Z
  17. G y \overrightarrow{G_{y}}
  18. R S R_{S}
  19. R T Z R_{T}\cap Z
  20. G y \overrightarrow{G_{y}}
  21. R S R_{S}
  22. R T R_{T}
  23. R T Z R_{T}\cap Z
  24. Δ := min { c ( i , j ) - y ( i ) - y ( j ) : i Z S , j T Z } \Delta:=\min\{c(i,j)-y(i)-y(j):i\in Z\cap S,j\in T\setminus Z\}
  25. Δ \Delta
  26. Z S Z\cap S
  27. T Z T\setminus Z
  28. Δ \Delta
  29. Z S Z\cap S
  30. Δ \Delta
  31. Z T Z\cap T
  32. G y G_{y}
  33. Δ \Delta
  34. R S R_{S}
  35. O ( n 4 ) O(n^{4})
  36. O ( n 2 ) O(n^{2})
  37. n n
  38. [ a 1 a 2 a 3 a 4 b 1 b 2 b 3 b 4 c 1 c 2 c 3 c 4 d 1 d 2 d 3 d 4 ] \begin{bmatrix}a1&a2&a3&a4\\ b1&b2&b3&b4\\ c1&c2&c3&c4\\ d1&d2&d3&d4\end{bmatrix}
  39. [ a 2 a 3 a 4 c 2 c 3 c 4 ] \begin{bmatrix}a2&a3&a4\\ c2&c3&c4\end{bmatrix}
  40. O ( n 3 ) O(n^{3})
  41. O ( n 3 ) O(n^{3})
  42. O ( n 3 ) O(n^{3})
  43. O ( n 3 ) O(n^{3})
  44. O ( n 3 ) O(n^{3})

Huntington–Hill_method.html

  1. D = n ( n + 1 ) \scriptstyle D=\sqrt{n(n+1)}

Hybrid_automaton.html

  1. H H
  2. X = { x 1 , , x n } X=\{x_{1},...,x_{n}\}
  3. n n
  4. H H
  5. X ˙ \dot{X}
  6. { x ˙ 1 , , x ˙ n } \{\dot{x}_{1},...,\dot{x}_{n}\}
  7. X X^{\prime}
  8. { x 1 , , x n } \{x^{\prime}_{1},...,x^{\prime}_{n}\}
  9. ( V , E ) (V,E)
  10. V V
  11. E E
  12. v V v\in V
  13. ( v ) (v)
  14. X X
  15. ( v ) (v)
  16. X X
  17. ( v ) (v)
  18. X X ˙ X\cup\dot{X}
  19. e E e\in E
  20. ( e ) (e)
  21. X X X\cup X^{\prime}
  22. Σ \Sigma
  23. E Σ E\rightarrow\Sigma

Hydrogen_cycle.html

  1. H 2 + OH H + H 2 O \mathrm{H_{2}+OH\longrightarrow H+H_{2}O}
  2. CH 4 + OH CH 3 + H 2 O \mathrm{CH_{4}+OH\longrightarrow CH_{3}+H_{2}O}

Hydrogen_spectral_series.html

  1. n n
  2. n n′
  3. n n
  4. n n′
  5. 1 λ = R ( 1 n 2 - 1 n 2 ) {1\over\lambda}=R\left({1\over{n^{\prime}}^{2}}-{1\over n^{2}}\right)
  6. n n
  7. n n′
  8. R R
  9. 7 {}^{7}
  10. 1 {}^{−1}
  11. n n
  12. n n′
  13. n n
  14. n n
  15. n n
  16. n n
  17. n n
  18. n n

Hyperbolic_group.html

  1. δ \delta
  2. [ x , y ] B δ ( [ y , z ] [ z , x ] ) , [x,y]\subseteq B_{\delta}([y,z]\cup[z,x]),
  3. [ y , z ] B δ ( [ z , x ] [ x , y ] ) , [y,z]\subseteq B_{\delta}([z,x]\cup[x,y]),
  4. [ z , x ] B δ ( [ x , y ] [ y , z ] ) . [z,x]\subseteq B_{\delta}([x,y]\cup[y,z]).
  5. δ \delta
  6. Δ ( l , m , n ) \Delta(l,m,n)

Hyperbolic_growth.html

  1. 1 / x 1/x
  2. x 0 x\to 0
  3. x 0 x_{0}
  4. x 0 x_{0}
  5. x ( t ) = 1 t c - t x(t)=\frac{1}{t_{c}-t}
  6. t c t_{c}
  7. t t c t\to t_{c}
  8. x ( t ) = K t c - t x(t)=\frac{K}{t_{c}-t}
  9. K K
  10. d x d t = K ( t c - t ) 2 = x 2 K \frac{dx}{dt}=\frac{K}{(t_{c}-t)^{2}}=\frac{x^{2}}{K}
  11. x ( t ) = K ( t c - t ) 2 . x(t)=\frac{K}{(t_{c}-t)^{2}}.

Hyperbolic_set.html

  1. T Λ M = E s E u T_{\Lambda}M=E^{s}\oplus E^{u}
  2. ( D f ) x E x s = E f ( x ) s (Df)_{x}E^{s}_{x}=E^{s}_{f(x)}
  3. ( D f ) x E x u = E f ( x ) u (Df)_{x}E^{u}_{x}=E^{u}_{f(x)}
  4. x Λ x\in\Lambda
  5. D f n v c λ n v \|Df^{n}v\|\leq c\lambda^{n}\|v\|
  6. v E s v\in E^{s}
  7. n > 0 n>0
  8. D f - n v c λ n v \|Df^{-n}v\|\leq c\lambda^{n}\|v\|
  9. v E u v\in E^{u}
  10. n > 0 n>0

Hypergeometric_function.html

  1. F 1 2 ( a , b ; c ; z ) = n = 0 ( a ) n ( b ) n ( c ) n z n n ! . {}_{2}F_{1}(a,b;c;z)=\sum_{n=0}^{\infty}\frac{(a)_{n}(b)_{n}}{(c)_{n}}\frac{z^% {n}}{n!}.
  2. ( q ) n = { 1 n = 0 q ( q + 1 ) ( q + n - 1 ) n > 0 (q)_{n}=\begin{cases}1&n=0\\ q(q+1)\cdots(q+n-1)&n>0\end{cases}
  3. F 1 2 ( - m , b ; c ; z ) = n = 0 m ( - 1 ) n ( m n ) ( b ) n ( c ) n z n . {}_{2}F_{1}(-m,b;c;z)=\sum_{n=0}^{m}(-1)^{n}{\left({{m}\atop{n}}\right)}\frac{% (b)_{n}}{(c)_{n}}z^{n}.
  4. c m c→−m
  5. m m
  6. Γ ( c ) Γ(c)
  7. lim c - m F 1 2 ( a , b ; c ; z ) Γ ( c ) = ( a ) m + 1 ( b ) m + 1 ( m + 1 ) ! z m + 1 F 1 2 ( a + m + 1 , b + m + 1 ; m + 2 ; z ) \lim_{c\to-m}\frac{{}_{2}F_{1}(a,b;c;z)}{\Gamma(c)}=\frac{(a)_{m+1}(b)_{m+1}}{% (m+1)!}z^{m+1}{}_{2}F_{1}(a+m+1,b+m+1;m+2;z)
  8. F ( z ) F(z)
  9. ln ( 1 + z ) = z 2 F 1 ( 1 , 1 ; 2 ; - z ) \ln(1+z)=z\ _{2}F_{1}(1,1;2;-z)
  10. ( 1 - z ) - a = 2 F 1 ( a , 1 ; 1 ; z ) (1-z)^{-a}=\ _{2}F_{1}(a,1;1;z)
  11. arcsin ( z ) = z 2 F 1 ( 1 2 , 1 2 ; 3 2 ; z 2 ) \arcsin(z)=z\ _{2}F_{1}\left(\tfrac{1}{2},\tfrac{1}{2};\tfrac{3}{2};z^{2}\right)
  12. M ( a , c , z ) = lim b F 1 2 ( a , b ; c ; b - 1 z ) M(a,c,z)=\lim_{b\to\infty}{}_{2}F_{1}(a,b;c;b^{-1}z)
  13. F 1 2 ( a , 1 - a ; c ; z ) = Γ ( c ) z 1 - c 2 ( 1 - z ) c - 1 2 P - a 1 - c ( 1 - 2 z ) {}_{2}F_{1}(a,1-a;c;z)=\Gamma(c)z^{\tfrac{1-c}{2}}(1-z)^{\tfrac{c-1}{2}}P_{-a}% ^{1-c}(1-2z)
  14. F 1 2 ( - n , α + 1 + β + n ; α + 1 ; x ) = n ! ( α + 1 ) n P n ( α , β ) ( 1 - 2 x ) {}_{2}F_{1}(-n,\alpha+1+\beta+n;\alpha+1;x)=\frac{n!}{(\alpha+1)_{n}}P^{(% \alpha,\beta)}_{n}(1-2x)
  15. τ = i F 1 2 ( 1 2 , 1 2 ; 1 ; 1 - z ) F 1 2 ( 1 2 , 1 2 ; 1 ; z ) \tau={\rm{i}}\frac{{}_{2}F_{1}\left(\frac{1}{2},\frac{1}{2};1;1-z\right)}{{}_{% 2}F_{1}\left(\frac{1}{2},\frac{1}{2};1;z\right)}
  16. z = κ 2 ( τ ) = θ 2 ( τ ) 4 θ 3 ( τ ) 4 z=\kappa^{2}(\tau)=\frac{\theta_{2}(\tau)^{4}}{\theta_{3}(\tau)^{4}}
  17. B x ( p , q ) = x p p F 1 2 ( p , 1 - q ; p + 1 ; x ) B_{x}(p,q)=\tfrac{x^{p}}{p}{}_{2}F_{1}(p,1-q;p+1;x)
  18. K ( k ) = π 2 2 F 1 ( 1 2 , 1 2 ; 1 ; k 2 ) K(k)=\tfrac{\pi}{2}\,_{2}F_{1}\left(\tfrac{1}{2},\tfrac{1}{2};1;k^{2}\right)
  19. E ( k ) = π 2 2 F 1 ( - 1 2 , 1 2 ; 1 ; k 2 ) E(k)=\tfrac{\pi}{2}\,_{2}F_{1}\left(-\tfrac{1}{2},\tfrac{1}{2};1;k^{2}\right)
  20. z ( 1 - z ) d 2 w d z 2 + [ c - ( a + b + 1 ) z ] d w d z - a b w = 0. z(1-z)\frac{d^{2}w}{dz^{2}}+\left[c-(a+b+1)z\right]\frac{dw}{dz}-ab\,w=0.
  21. F 1 2 ( a , b ; c ; z ) \,{}_{2}F_{1}(a,b;c;z)
  22. z 2 1 - c F 1 ( 1 + a - c , 1 + b - c ; 2 - c ; z ) z^{1-c}\,_{2}F_{1}(1+a-c,1+b-c;2-c;z)
  23. z m F ( a + m , b + m ; 1 + m ; z ) . z^{m}F(a+m,b+m;1+m;z).
  24. F 1 2 ( a , b ; 1 + a + b - c ; 1 - z ) \,{}_{2}F_{1}(a,b;1+a+b-c;1-z)
  25. ( 1 - z ) 2 c - a - b F 1 ( c - a , c - b ; 1 + c - a - b ; 1 - z ) (1-z)^{c-a-b}\;_{2}F_{1}(c-a,c-b;1+c-a-b;1-z)
  26. z 2 - a F 1 ( a , 1 + a - c ; 1 + a - b ; z - 1 ) z^{-a}\,_{2}F_{1}\left(a,1+a-c;1+a-b;z^{-1}\right)
  27. z 2 - b F 1 ( b , 1 + b - c ; 1 + b - a ; z - 1 ) . z^{-b}\,_{2}F_{1}\left(b,1+b-c;1+b-a;z^{-1}\right).
  28. ( 1 - z ) - a F ( a , c - b ; c ; z z - 1 ) (1-z)^{-a}F\left(a,c-b;c;\tfrac{z}{z-1}\right)
  29. F ( a , b ; 1 + a + b - c ; 1 - z ) F(a,b;1+a+b-c;1-z)
  30. ( 1 - z ) - b F ( c - a , b ; c ; z z - 1 ) (1-z)^{-b}F\left(c-a,b;c;\tfrac{z}{z-1}\right)
  31. F 1 2 ( a , b ; c ; z ) = ( 1 - z ) c - a - b F 1 2 ( c - a , c - b ; c ; z ) Euler transformation {}_{2}F_{1}(a,b;c;z)=(1-z)^{c-a-b}\,{}_{2}F_{1}(c-a,c-b;c;z)\ \ \ \,\text{% Euler transformation}
  32. F 1 2 ( a , b ; c ; z ) = ( 1 - z ) - a F 1 2 ( a , c - b ; c ; z z - 1 ) Pfaff transformation {}_{2}F_{1}(a,b;c;z)=(1-z)^{-a}\,{}_{2}F_{1}(a,c-b;c;\tfrac{z}{z-1})\ \ \ \,% \text{Pfaff transformation}
  33. F 1 2 ( a , b ; c ; z ) = ( 1 - z ) - b F 1 2 ( c - a , b ; c ; z z - 1 ) Pfaff transformation {}_{2}F_{1}(a,b;c;z)=(1-z)^{-b}\,{}_{2}F_{1}(c-a,b;c;\tfrac{z}{z-1})\ \ \ \,% \text{Pfaff transformation}
  34. d 2 u d z 2 + Q ( z ) u ( z ) = 0 \frac{d^{2}u}{dz^{2}}+Q(z)u(z)=0
  35. Q = z 2 [ 1 - ( a - b ) 2 ] + z [ 2 c ( a + b + 1 ) - 4 a b ] + c ( 2 - c ) 4 z 2 ( 1 - z ) 2 Q=\frac{z^{2}[1-(a-b)^{2}]+z[2c(a+b+1)-4ab]+c(2-c)}{4z^{2}(1-z)^{2}}
  36. d d z log v ( z ) = - c - z ( a + b + 1 ) 2 z ( 1 - z ) = - c 2 z - 1 + a + b - c 2 ( z - 1 ) \frac{d}{dz}\log v(z)=-\frac{c-z(a+b+1)}{2z(1-z)}=-\frac{c}{2z}-\frac{1+a+b-c}% {2(z-1)}
  37. v ( z ) = z - c / 2 ( 1 - z ) ( c - a - b - 1 ) / 2 . v(z)=z^{-c/2}(1-z)^{(c-a-b-1)/2}.
  38. s k ( z ) = ϕ k ( 1 ) ( z ) ϕ k ( 0 ) ( z ) s_{k}(z)=\frac{\phi_{k}^{(1)}(z)}{\phi_{k}^{(0)}(z)}
  39. D k ( λ , μ , ν ; z ) = s k ( z ) D_{k}(\lambda,\mu,\nu;z)=s_{k}(z)
  40. s 0 ( z ) = z λ ( 1 + 𝒪 ( z ) ) s_{0}(z)=z^{\lambda}(1+\mathcal{O}(z))
  41. s 1 ( z ) = ( 1 - z ) μ ( 1 + 𝒪 ( 1 - z ) ) s_{1}(z)=(1-z)^{\mu}(1+\mathcal{O}(1-z))
  42. s ( z ) = z ν ( 1 + 𝒪 ( 1 z ) ) . s_{\infty}(z)=z^{\nu}(1+\mathcal{O}(\tfrac{1}{z})).
  43. π 1 ( 𝐂 { 0 , 1 } , z 0 ) G L ( 2 , 𝐂 ) \pi_{1}(\mathbf{C}\setminus\{0,1\},z_{0})\to GL(2,\mathbf{C})
  44. 1 / k + 1 / l + 1 / m > 1 1/k+1/l+1/m>1
  45. B ( b , c - b ) 2 F 1 ( a , b ; c ; z ) = 0 1 x b - 1 ( 1 - x ) c - b - 1 ( 1 - z x ) - a d x \real ( c ) > \real ( b ) > 0 , B(b,c-b)\,_{2}F_{1}(a,b;c;z)=\int_{0}^{1}x^{b-1}(1-x)^{c-b-1}(1-zx)^{-a}\,dx% \qquad\real(c)>\real(b)>0,
  46. 1 2 π i - i i Γ ( a + s ) Γ ( b + s ) Γ ( - s ) Γ ( c + s ) ( - z ) s d s \frac{1}{2\pi i}\int_{-i\infty}^{i\infty}\frac{\Gamma(a+s)\Gamma(b+s)\Gamma(-s% )}{\Gamma(c+s)}(-z)^{s}\,ds
  47. Γ ( a ) Γ ( b ) Γ ( c ) 2 F 1 ( a , b ; c ; z ) , \frac{\Gamma(a)\Gamma(b)}{\Gamma(c)}\,_{2}F_{1}(a,b;c;z),
  48. F 1 2 ( a ± 1 , b ; c ; z ) , F 1 2 ( a , b ± 1 ; c ; z ) , F 1 2 ( a , b ; c ± 1 ; z ) {}_{2}F_{1}(a\pm 1,b;c;z),\quad{}_{2}F_{1}(a,b\pm 1;c;z),\quad{}_{2}F_{1}(a,b;% c\pm 1;z)
  49. a , b , c a,b,c
  50. z z
  51. ( 6 2 ) = 15 \begin{pmatrix}6\\ 2\end{pmatrix}=15
  52. z d F d z = z a b c F ( a + , b + , c + ) = a ( F ( a + ) - F ) = b ( F ( b + ) - F ) = ( c - 1 ) ( F ( c - ) - F ) = ( c - a ) F ( a - ) + ( a - c + b z ) F 1 - z = ( c - b ) F ( b - ) + ( b - c + a z ) F 1 - z = z ( c - a ) ( c - b ) F ( c + ) + c ( a + b - c ) F c ( 1 - z ) \begin{aligned}\displaystyle z\frac{dF}{dz}&\displaystyle=z\frac{ab}{c}F(a+,b+% ,c+)\\ &\displaystyle=a(F(a+)-F)\\ &\displaystyle=b(F(b+)-F)\\ &\displaystyle=(c-1)(F(c-)-F)\\ &\displaystyle=\frac{(c-a)F(a-)+(a-c+bz)F}{1-z}\\ &\displaystyle=\frac{(c-b)F(b-)+(b-c+az)F}{1-z}\\ &\displaystyle=z\frac{(c-a)(c-b)F(c+)+c(a+b-c)F}{c(1-z)}\end{aligned}
  53. 𝐂 ( z ) \mathbf{C}(z)
  54. F 1 2 ( a + m , b + n ; c + l ; z ) , {}_{2}F_{1}(a+m,b+n;c+l;z),
  55. F 1 2 ( a + 1 , b ; c + 1 ; z ) F 1 2 ( a , b ; c ; z ) = 1 1 + ( a - c ) b c ( c + 1 ) z 1 + ( b - c - 1 ) ( a + 1 ) ( c + 1 ) ( c + 2 ) z 1 + ( a - c - 1 ) ( b + 1 ) ( c + 2 ) ( c + 3 ) z 1 + ( b - c - 2 ) ( a + 2 ) ( c + 3 ) ( c + 4 ) z 1 + \frac{{}_{2}F_{1}(a+1,b;c+1;z)}{{}_{2}F_{1}(a,b;c;z)}=\cfrac{1}{1+\cfrac{\frac% {(a-c)b}{c(c+1)}z}{1+\cfrac{\frac{(b-c-1)(a+1)}{(c+1)(c+2)}z}{1+\cfrac{\frac{(% a-c-1)(b+1)}{(c+2)(c+3)}z}{1+\cfrac{\frac{(b-c-2)(a+2)}{(c+3)(c+4)}z}{1+{}% \ddots}}}}}
  56. F 1 2 ( a , b ; c ; z ) = ( 1 - z ) c - a - b F 1 2 ( c - a , c - b ; c ; z ) . {}_{2}F_{1}(a,b;c;z)=(1-z)^{c-a-b}{}_{2}F_{1}(c-a,c-b;c;z).
  57. F 1 2 ( a , b ; c ; z ) = ( 1 - z ) - b F 1 2 ( b , c - a ; c ; z z - 1 ) {}_{2}F_{1}(a,b;c;z)=(1-z)^{-b}{}_{2}F_{1}\left(b,c-a;c;\tfrac{z}{z-1}\right)
  58. F 1 2 ( a , b ; c ; z ) = ( 1 - z ) - a F 1 2 ( a , c - b ; c ; z z - 1 ) {}_{2}F_{1}(a,b;c;z)=(1-z)^{-a}{}_{2}F_{1}\left(a,c-b;c;\tfrac{z}{z-1}\right)
  59. F 1 2 ( a , b ; 2 b ; z ) = ( 1 - z ) - a 2 F 1 2 ( 1 2 a , b - 1 2 a ; b + 1 2 ; z 2 4 z - 4 ) {}_{2}F_{1}(a,b;2b;z)=(1-z)^{-\frac{a}{2}}{}_{2}F_{1}\left(\tfrac{1}{2}a,b-% \tfrac{1}{2}a;b+\tfrac{1}{2};\frac{z^{2}}{4z-4}\right)
  60. F 1 2 ( 3 2 a , 1 2 ( 3 a - 1 ) ; a + 1 2 ; - z 2 3 ) = ( 1 + z ) 1 - 3 a F 1 2 ( a - 1 3 , a , 2 a , 2 z ( 3 + z 2 ) ( 1 + z ) - 3 ) {}_{2}F_{1}\left(\tfrac{3}{2}a,\tfrac{1}{2}(3a-1);a+\tfrac{1}{2};-\tfrac{z^{2}% }{3}\right)=(1+z)^{1-3a}\,{}_{2}F_{1}\left(a-\tfrac{1}{3},a,2a,2z(3+z^{2})(1+z% )^{-3}\right)
  61. F 1 2 ( 1 4 , 3 8 ; 7 8 ; z ) ( z 4 - 60 z 3 + 134 z 2 - 60 z + 1 ) 1 / 16 = F 1 2 ( 1 48 , 17 48 ; 7 8 ; - 432 z ( z - 1 ) 2 ( z + 1 ) 8 ( z 4 - 60 z 3 + 134 z 2 - 60 z + 1 ) 3 ) . {}_{2}F_{1}\left(\tfrac{1}{4},\tfrac{3}{8};\tfrac{7}{8};z\right)(z^{4}-60z^{3}% +134z^{2}-60z+1)^{1/16}={}_{2}F_{1}\left(\tfrac{1}{48},\tfrac{17}{48};\tfrac{7% }{8};\tfrac{-432z(z-1)^{2}(z+1)^{8}}{(z^{4}-60z^{3}+134z^{2}-60z+1)^{3}}\right).
  62. F 1 2 ( a , b ; c ; 1 ) = Γ ( c ) Γ ( c - a - b ) Γ ( c - a ) Γ ( c - b ) , ( c ) > ( a + b ) {}_{2}F_{1}(a,b;c;1)=\frac{\Gamma(c)\Gamma(c-a-b)}{\Gamma(c-a)\Gamma(c-b)},% \qquad\Re(c)>\Re(a+b)
  63. a = - m a=-m
  64. F 1 2 ( - m , b ; c ; 1 ) = ( c - b ) ( m ) ( c ) ( m ) {}_{2}F_{1}(-m,b;c;1)=\frac{(c-b)_{(m)}}{(c)_{(m)}}
  65. F 1 2 ( a , b ; 1 + a - b ; - 1 ) = Γ ( 1 + a - b ) Γ ( 1 + 1 2 a ) Γ ( 1 + a ) Γ ( 1 + 1 2 a - b ) {}_{2}F_{1}(a,b;1+a-b;-1)=\frac{\Gamma(1+a-b)\Gamma(1+\tfrac{1}{2}a)}{\Gamma(1% +a)\Gamma(1+\tfrac{1}{2}a-b)}
  66. F 1 2 ( a , b ; 1 + a - b ; z ) = ( 1 - z ) 2 - a F 1 ( a 2 , 1 + a 2 - b ; 1 + a - b ; - 4 z ( 1 - z ) 2 ) = ( 1 + z ) 2 - a F 1 ( a 2 , a + 1 2 ; 1 + a - b ; 4 z ( 1 + z ) 2 ) \begin{aligned}{}_{2}F_{1}(a,b;1+a-b;z)&\displaystyle=(1-z)^{-a}\;_{2}F_{1}% \left(\frac{a}{2},\frac{1+a}{2}-b;1+a-b;-\frac{4z}{(1-z)^{2}}\right)\\ &\displaystyle=(1+z)^{-a}\,_{2}F_{1}\left(\frac{a}{2},\frac{a+1}{2};1+a-b;% \frac{4z}{(1+z)^{2}}\right)\end{aligned}
  67. F 1 2 ( a , b ; 1 2 ( 1 + a + b ) ; 1 2 ) = Γ ( 1 2 ) Γ ( 1 2 ( 1 + a + b ) ) Γ ( 1 2 ( 1 + a ) ) Γ ( 1 2 ( 1 + b ) ) . {}_{2}F_{1}\left(a,b;\tfrac{1}{2}\left(1+a+b\right);\tfrac{1}{2}\right)=\frac{% \Gamma(\tfrac{1}{2})\Gamma(\tfrac{1}{2}\left(1+a+b\right))}{\Gamma(\tfrac{1}{2% }\left(1+a)\right)\Gamma(\tfrac{1}{2}\left(1+b\right))}.
  68. F 1 2 ( a , 1 - a ; c ; 1 2 ) = Γ ( 1 2 c ) Γ ( 1 2 ( 1 + c ) ) Γ ( 1 2 ( c + a ) ) Γ ( 1 2 ( 1 + c - a ) ) . {}_{2}F_{1}\left(a,1-a;c;\tfrac{1}{2}\right)=\frac{\Gamma(\tfrac{1}{2}c)\Gamma% (\tfrac{1}{2}\left(1+c\right))}{\Gamma(\tfrac{1}{2}\left(c+a\right))\Gamma(% \tfrac{1}{2}\left(1+c-a\right))}.
  69. F 1 2 ( a , - a ; 1 2 ; x 2 4 ( x - 1 ) ) = ( 1 - x ) a + ( 1 - x ) - a 2 , {}_{2}F_{1}\left(a,-a;\tfrac{1}{2};\tfrac{x^{2}}{4(x-1)}\right)=\frac{(1-x)^{a% }+(1-x)^{-a}}{2},
  70. T a ( cos x ) = F 1 2 ( a , - a ; 1 2 ; 1 2 ( 1 - cos x ) ) = cos ( a x ) T_{a}(\cos x)={}_{2}F_{1}\left(a,-a;\tfrac{1}{2};\tfrac{1}{2}(1-\cos x)\right)% =\cos(ax)

Hypsochromic_shift.html

  1. Δ λ = λ observed state1 - λ observed state2 \Delta\lambda=\lambda^{\mathrm{state1}}_{\mathrm{observed}}-\lambda^{\mathrm{% state2}}_{\mathrm{observed}}
  2. λ \lambda
  3. λ observed state1 > λ observed state2 \lambda^{\mathrm{state1}}_{\mathrm{observed}}>\lambda^{\mathrm{state2}}_{% \mathrm{observed}}

Hysteresivity.html

  1. G * ( f ) = G + j G ′′ \ G^{*}(f)=G^{{}^{\prime}}+jG^{{}^{\prime\prime}}
  2. G * ( f ) = G ( 1 + j h ) \ G^{*}(f)=G^{{}^{\prime}}(1+jh)

I-beam.html

  1. σ x x \sigma_{xx}
  2. σ max \sigma_{\mathrm{max}}
  3. M max M_{\mathrm{max}}
  4. S = M max σ max = I c S=\cfrac{M_{\mathrm{max}}}{\sigma_{\mathrm{max}}}=\cfrac{I}{c}
  5. I I
  6. c c
  7. a a
  8. h h
  9. h / 2 h/2
  10. h / 2 h/2
  11. I = a h 2 4 ; S = 0.5 a h I=\cfrac{ah^{2}}{4}~{};~{}~{}S=0.5ah
  12. S 0.35 a h S\approx 0.35ah

Icosahedral_symmetry.html

  1. I : s , t s 2 , t 3 , ( s t ) 5 I:\langle s,t\mid s^{2},t^{3},(st)^{5}\rangle
  2. I h : s , t s 3 ( s t ) - 2 , t 5 ( s t ) - 2 . I_{h}:\langle s,t\mid s^{3}(st)^{-2},t^{5}(st)^{-2}\rangle.
  3. 1 A 5 S 5 Z 2 1 1\to A_{5}\to S_{5}\to Z_{2}\to 1
  4. I h = A 5 × Z 2 I_{h}=A_{5}\times Z_{2}
  5. 1 Z 2 2 I A 5 1 1\to Z_{2}\to 2I\to A_{5}\to 1
  6. A 5 A_{5}
  7. S 5 S_{5}
  8. A 5 A_{5}
  9. I h I_{h}
  10. A 5 A_{5}
  11. 2 I 2I
  12. A 5 A_{5}
  13. S 5 S_{5}
  14. A 5 PSL ( 2 , 5 ) , A_{5}\cong\operatorname{PSL}(2,5),
  15. S 5 PGL ( 2 , 5 ) , S_{5}\cong\operatorname{PGL}(2,5),
  16. 2 I SL ( 2 , 5 ) , 2I\cong\operatorname{SL}(2,5),
  17. 53 ¯ \overline{53}
  18. 2 ¯ \overline{2}
  19. 3 ¯ \overline{3}
  20. 10 ¯ \overline{10}
  21. 3 ¯ \overline{3}
  22. 5 ¯ \overline{5}
  23. 3 ¯ \overline{3}
  24. 1 ¯ \overline{1}
  25. - 1 -1
  26. D 3 × ± 1 D_{3}\times\pm 1
  27. Z 2 × Z 2 Z_{2}\times Z_{2}
  28. Z 2 × Z 2 Z_{2}\times Z_{2}
  29. Z 2 × Z 2 × Z 2 Z_{2}\times Z_{2}\times Z_{2}
  30. D 5 × ± 1 D_{5}\times\pm 1
  31. I A 5 < S 5 I\stackrel{\sim}{\to}A_{5}<S_{5}

Ideal_chain.html

  1. L = N l L=N\,l
  2. R \vec{R}
  3. r 1 , , r N \vec{r}_{1},\ldots,\vec{r}_{N}
  4. R = Σ i = 1 N r i = 0 \langle\vec{R}\rangle=\Sigma_{i=1}^{N}\langle\vec{r}_{i}\rangle=\vec{0}~{}
  5. \langle\rangle
  6. r 1 , , r N \vec{r}_{1},\ldots,\vec{r}_{N}
  7. R \vec{R}
  8. R x , R y , R_{x},R_{y},
  9. R z R_{z}
  10. σ 2 = R x 2 - R x 2 = R x 2 - 0 \sigma^{2}=\langle R_{x}^{2}\rangle-\langle R_{x}\rangle^{2}=\langle R_{x}^{2}% \rangle-0
  11. R x 2 = R y 2 = R z 2 = N l 2 3 \langle R_{x}^{2}\rangle=\langle R_{y}^{2}\rangle=\langle R_{z}^{2}\rangle=N\,% \frac{l^{2}}{3}
  12. R 2 = N l 2 = L l \langle\vec{R^{2}}\rangle=N\,l^{2}=L\,l~{}
  13. P ( R ) = ( 3 2 π N l 2 ) 3 / 2 e - 3 R 2 2 N l 2 P(\vec{R})=\left(\frac{3}{2\pi Nl^{2}}\right)^{3/2}e^{-\frac{3\vec{R}^{2}}{2Nl% ^{2}}}
  14. R 2 = N l = L l \sqrt{\langle\vec{R^{2}}\rangle}=\sqrt{N}\,l=\sqrt{L\,l}~{}
  15. R G = N l 6 \mathit{R}_{G}=\frac{\sqrt{N}\,l}{\sqrt{6}\ }
  16. N l N\,l
  17. P ( R ) P(\vec{R})
  18. R x R_{x}
  19. R y R_{y}
  20. R z R_{z}
  21. N l 2 / 3 N\,l^{2}/3
  22. P ( R ) P(\vec{R})
  23. R x R_{x}
  24. R y R_{y}
  25. R z R_{z}
  26. r 1 , , r N \vec{r}_{1},\ldots,\vec{r}_{N}
  27. R x R_{x}
  28. R y R_{y}
  29. R z R_{z}
  30. R \vec{R}
  31. P ( R ) P(\vec{R})
  32. r i \vec{r}_{i}
  33. R \vec{R}
  34. Ω ( R ) \Omega(\vec{R})
  35. S ( R ) = k B log ( Ω ( R ) ) S(\vec{R})=k_{B}\log(\Omega(\vec{R}))
  36. k B k_{B}
  37. Ω ( R ) \Omega(\vec{R})
  38. P ( R ) P(\vec{R})
  39. P ( R ) P(\vec{R})
  40. Ω ( R ) \Omega(\vec{R})
  41. S ( R ) = k B log ( P ( R ) ) + C s t S(\vec{R})=k_{B}\log(P(\vec{R}))+C_{st}
  42. C s t C_{st}
  43. F \vec{F}
  44. R \vec{R}
  45. R \vec{R}
  46. d R \vec{dR}
  47. 0 = d U = δ W + δ Q 0=dU=\delta W+\delta Q~{}
  48. δ W \delta W
  49. δ Q \delta Q
  50. R \vec{R}
  51. R + d R \vec{R}+\vec{dR}
  52. δ Q = T d S \delta Q=TdS~{}
  53. f o p \vec{f}_{op}
  54. f \vec{f}
  55. δ W = f o p d R = - f d R \delta W=\langle\vec{f}_{op}\rangle\cdot\vec{dR}=-\langle\vec{f}\rangle\cdot% \vec{dR}~{}
  56. f = T d S d R = k B T P ( R ) d P ( R ) d R \langle\vec{f}\rangle=T\frac{dS}{\vec{dR}}=\frac{k_{B}T}{P(\vec{R})}\frac{dP(% \vec{R})}{\vec{dR}}~{}
  57. f = - k B T 3 R N l 2 \langle\vec{f}\rangle=-k_{B}T\frac{3\vec{R}}{Nl^{2}}~{}
  58. f o p \vec{f}_{op}
  59. R \langle\vec{R}\rangle
  60. f = - f o p \vec{f}=-\vec{f}_{op}
  61. f \vec{f}
  62. R \vec{R}
  63. f = - k B T 3 R N l 2 \vec{f}=-k_{B}T\frac{3\langle\vec{R}\rangle}{Nl^{2}}~{}

Idempotent_matrix.html

  1. 2 × 2 2\times 2
  2. 3 × 3 3\times 3
  3. [ 1 0 0 1 ] \begin{bmatrix}1&0\\ 0&1\end{bmatrix}
  4. [ 2 - 2 - 4 - 1 3 4 1 - 2 - 3 ] \begin{bmatrix}2&-2&-4\\ -1&3&4\\ 1&-2&-3\end{bmatrix}
  5. ( a b c d ) \begin{pmatrix}a&b\\ c&d\end{pmatrix}
  6. a = a 2 + b c , a=a^{2}+bc,
  7. b = a b + b d , b=ab+bd,
  8. b ( 1 - a - d ) = 0 b(1-a-d)=0
  9. b = 0 b=0
  10. d = 1 - a , d=1-a,
  11. c = c a + c d , c=ca+cd,
  12. c ( 1 - a - d ) = 0 c(1-a-d)=0
  13. c = 0 c=0
  14. d = 1 - a , d=1-a,
  15. d = b c + d 2 . d=bc+d^{2}.
  16. ( a b b 1 - a ) \begin{pmatrix}a&b\\ b&1-a\end{pmatrix}
  17. a 2 + b 2 = a , a^{2}+b^{2}=a,
  18. a 2 - a + b 2 = 0 , a^{2}-a+b^{2}=0,
  19. ( a - 1 2 ) 2 + b 2 = 1 4 (a-\tfrac{1}{2})^{2}+b^{2}=\tfrac{1}{4}
  20. M = 1 2 ( 1 - cos θ sin θ sin θ 1 + cos θ ) M=\tfrac{1}{2}\begin{pmatrix}1-\cos\theta&\sin\theta\\ \sin\theta&1+\cos\theta\end{pmatrix}
  21. ( a b c 1 - a ) \begin{pmatrix}a&b\\ c&1-a\end{pmatrix}
  22. a 2 + b c = a a^{2}+bc=a
  23. A n = A A^{n}=A
  24. A 1 = A A^{1}=A
  25. A k - 1 = A A^{k-1}=A
  26. A k = A k - 1 A = A A = A A^{k}=A^{k-1}A=AA=A
  27. β \beta
  28. Minimize ( y - X β ) T ( y - X β ) \,\text{Minimize }(y-X\beta)^{T}(y-X\beta)\,
  29. β = ( X T X ) - 1 X T y \beta=(X^{T}X)^{-1}X^{T}y\,
  30. e = y - X β = y - X ( X T X ) - 1 X T y = [ I - X ( X T X ) - 1 X T ] y = M y . e=y-X\beta=y-X(X^{T}X)^{-1}X^{T}y=[I-X(X^{T}X)^{-1}X^{T}]y=My.\,
  31. X ( X T X ) - 1 X T X(X^{T}X)^{-1}X^{T}
  32. e T e = ( M y ) T ( M y ) = y T M T M y = y T M M y = y T M y . e^{T}e=(My)^{T}(My)=y^{T}M^{T}My=y^{T}MMy=y^{T}My.\,
  33. β \beta

Ikeda_map.html

  1. z n + 1 = A + B z n e i ( | z n | 2 + C ) z_{n+1}=A+Bz_{n}e^{i(|z_{n}|^{2}+C)}
  2. z n z_{n}
  3. A A
  4. C C
  5. B 1 B\leq 1
  6. B = 1 B=1
  7. z n + 1 = A + B z n e i K / ( | z n | 2 + 1 ) + C z_{n+1}=A+Bz_{n}e^{iK/(|z_{n}|^{2}+1)+C}
  8. x n + 1 = 1 + u ( x n cos t n - y n sin t n ) , x_{n+1}=1+u(x_{n}\cos t_{n}-y_{n}\sin t_{n}),\,
  9. y n + 1 = u ( x n sin t n + y n cos t n ) , y_{n+1}=u(x_{n}\sin t_{n}+y_{n}\cos t_{n}),\,
  10. t n = 0.4 - 6 1 + x n 2 + y n 2 . t_{n}=0.4-\frac{6}{1+x_{n}^{2}+y_{n}^{2}}.
  11. u 0.6 u\geq 0.6
  12. u u
  13. u u
  14. u = 0.3 u=0.3
  15. u = 0.5 u=0.5
  16. u = 0.7 u=0.7
  17. u = 0.9 u=0.9
  18. u u

Illuminant_D65.html

  1. 1.4388 1.438 \frac{1.4388}{1.438}
  2. 6500 K × 1.4388 1.438 = 6503.6 K 6500\ \,\text{K}\times\frac{1.4388}{1.438}=6503.6\ \,\text{K}

Image-based_modeling_and_rendering.html

  1. ( x , y , z ) (x,y,z)
  2. ( θ , ϕ ) (\theta,\phi)
  3. ( λ ) (\lambda)
  4. ( t ) (t)
  5. P ( x , y , z , θ , ϕ , λ , t ) P(x,y,z,\theta,\phi,\lambda,t)

Image_histogram.html

  1. H × W H\times W
  2. T T
  3. n h = i = 1 min ( T , H W ) ( T i ) ( H W i - 1 ) . n_{h}=\sum_{i=1}^{\min(T,\ H\cdot W)}{T\choose i}\cdot{H\cdot W\choose i-1}.

Image_moment.html

  1. M p q = - - x p y q f ( x , y ) d x d y M_{pq}=\int\limits_{-\infty}^{\infty}\int\limits_{-\infty}^{\infty}x^{p}y^{q}f% (x,y)\,dx\,dy
  2. M i j = x y x i y j I ( x , y ) M_{ij}=\sum_{x}\sum_{y}x^{i}y^{j}I(x,y)\,\!
  3. x y I ( x , y ) \sum_{x}\sum_{y}I(x,y)\,\!
  4. μ p q = - - ( x - x ¯ ) p ( y - y ¯ ) q f ( x , y ) d x d y \mu_{pq}=\int\limits_{-\infty}^{\infty}\int\limits_{-\infty}^{\infty}(x-\bar{x% })^{p}(y-\bar{y})^{q}f(x,y)\,dx\,dy
  5. x ¯ = M 10 M 00 \bar{x}=\frac{M_{10}}{M_{00}}
  6. y ¯ = M 01 M 00 \bar{y}=\frac{M_{01}}{M_{00}}
  7. μ p q = x y ( x - x ¯ ) p ( y - y ¯ ) q f ( x , y ) \mu_{pq}=\sum_{x}\sum_{y}(x-\bar{x})^{p}(y-\bar{y})^{q}f(x,y)
  8. μ 00 = M 00 , \mu_{00}=M_{00},\,\!
  9. μ 01 = 0 , \mu_{01}=0,\,\!
  10. μ 10 = 0 , \mu_{10}=0,\,\!
  11. μ 11 = M 11 - x ¯ M 01 = M 11 - y ¯ M 10 , \mu_{11}=M_{11}-\bar{x}M_{01}=M_{11}-\bar{y}M_{10},
  12. μ 20 = M 20 - x ¯ M 10 , \mu_{20}=M_{20}-\bar{x}M_{10},
  13. μ 02 = M 02 - y ¯ M 01 , \mu_{02}=M_{02}-\bar{y}M_{01},
  14. μ 21 = M 21 - 2 x ¯ M 11 - y ¯ M 20 + 2 x ¯ 2 M 01 , \mu_{21}=M_{21}-2\bar{x}M_{11}-\bar{y}M_{20}+2\bar{x}^{2}M_{01},
  15. μ 12 = M 12 - 2 y ¯ M 11 - x ¯ M 02 + 2 y ¯ 2 M 10 , \mu_{12}=M_{12}-2\bar{y}M_{11}-\bar{x}M_{02}+2\bar{y}^{2}M_{10},
  16. μ 30 = M 30 - 3 x ¯ M 20 + 2 x ¯ 2 M 10 , \mu_{30}=M_{30}-3\bar{x}M_{20}+2\bar{x}^{2}M_{10},
  17. μ 03 = M 03 - 3 y ¯ M 02 + 2 y ¯ 2 M 01 . \mu_{03}=M_{03}-3\bar{y}M_{02}+2\bar{y}^{2}M_{01}.
  18. μ p q = m p n q ( p m ) ( q n ) ( - x ¯ ) ( p - m ) ( - y ¯ ) ( q - n ) M m n \mu_{pq}=\sum_{m}^{p}\sum_{n}^{q}{p\choose m}{q\choose n}(-\bar{x})^{(p-m)}(-% \bar{y})^{(q-n)}M_{mn}
  19. μ 20 = μ 20 / μ 00 = M 20 / M 00 - x ¯ 2 \mu^{\prime}_{20}=\mu_{20}/\mu_{00}=M_{20}/M_{00}-\bar{x}^{2}
  20. μ 02 = μ 02 / μ 00 = M 02 / M 00 - y ¯ 2 \mu^{\prime}_{02}=\mu_{02}/\mu_{00}=M_{02}/M_{00}-\bar{y}^{2}
  21. μ 11 = μ 11 / μ 00 = M 11 / M 00 - x ¯ y ¯ \mu^{\prime}_{11}=\mu_{11}/\mu_{00}=M_{11}/M_{00}-\bar{x}\bar{y}
  22. I ( x , y ) I(x,y)
  23. cov [ I ( x , y ) ] = [ μ 20 μ 11 μ 11 μ 02 ] \operatorname{cov}[I(x,y)]=\begin{bmatrix}\mu^{\prime}_{20}&\mu^{\prime}_{11}% \\ \mu^{\prime}_{11}&\mu^{\prime}_{02}\end{bmatrix}
  24. Θ = 1 2 arctan ( 2 μ 11 μ 20 - μ 02 ) \Theta=\frac{1}{2}\arctan\left(\frac{2\mu^{\prime}_{11}}{\mu^{\prime}_{20}-\mu% ^{\prime}_{02}}\right)
  25. μ 20 - μ 02 0 \mu^{\prime}_{20}-\mu^{\prime}_{02}\neq 0
  26. λ i = μ 20 + μ 02 2 ± 4 μ 11 2 + ( μ 20 - μ 02 ) 2 2 , \lambda_{i}=\frac{\mu^{\prime}_{20}+\mu^{\prime}_{02}}{2}\pm\frac{\sqrt{4{\mu^% {\prime}}_{11}^{2}+({\mu^{\prime}}_{20}-{\mu^{\prime}}_{02})^{2}}}{2},
  27. 1 - λ 2 λ 1 . \sqrt{1-\frac{\lambda_{2}}{\lambda_{1}}}.
  28. η i j = μ i j μ 00 ( 1 + i + j 2 ) \eta_{ij}=\frac{\mu_{ij}}{\mu_{00}^{\left(1+\frac{i+j}{2}\right)}}\,\!
  29. I 1 = η 20 + η 02 I_{1}=\eta_{20}+\eta_{02}
  30. I 2 = ( η 20 - η 02 ) 2 + 4 η 11 2 I_{2}=(\eta_{20}-\eta_{02})^{2}+4\eta_{11}^{2}
  31. I 3 = ( η 30 - 3 η 12 ) 2 + ( 3 η 21 - η 03 ) 2 I_{3}=(\eta_{30}-3\eta_{12})^{2}+(3\eta_{21}-\eta_{03})^{2}
  32. I 4 = ( η 30 + η 12 ) 2 + ( η 21 + η 03 ) 2 I_{4}=(\eta_{30}+\eta_{12})^{2}+(\eta_{21}+\eta_{03})^{2}
  33. I 5 = ( η 30 - 3 η 12 ) ( η 30 + η 12 ) [ ( η 30 + η 12 ) 2 - 3 ( η 21 + η 03 ) 2 ] + ( 3 η 21 - η 03 ) ( η 21 + η 03 ) [ 3 ( η 30 + η 12 ) 2 - ( η 21 + η 03 ) 2 ] I_{5}=(\eta_{30}-3\eta_{12})(\eta_{30}+\eta_{12})[(\eta_{30}+\eta_{12})^{2}-3(% \eta_{21}+\eta_{03})^{2}]+(3\eta_{21}-\eta_{03})(\eta_{21}+\eta_{03})[3(\eta_{% 30}+\eta_{12})^{2}-(\eta_{21}+\eta_{03})^{2}]
  34. I 6 = ( η 20 - η 02 ) [ ( η 30 + η 12 ) 2 - ( η 21 + η 03 ) 2 ] + 4 η 11 ( η 30 + η 12 ) ( η 21 + η 03 ) I_{6}=(\eta_{20}-\eta_{02})[(\eta_{30}+\eta_{12})^{2}-(\eta_{21}+\eta_{03})^{2% }]+4\eta_{11}(\eta_{30}+\eta_{12})(\eta_{21}+\eta_{03})
  35. I 7 = ( 3 η 21 - η 03 ) ( η 30 + η 12 ) [ ( η 30 + η 12 ) 2 - 3 ( η 21 + η 03 ) 2 ] - ( η 30 - 3 η 12 ) ( η 21 + η 03 ) [ 3 ( η 30 + η 12 ) 2 - ( η 21 + η 03 ) 2 ] . I_{7}=(3\eta_{21}-\eta_{03})(\eta_{30}+\eta_{12})[(\eta_{30}+\eta_{12})^{2}-3(% \eta_{21}+\eta_{03})^{2}]-(\eta_{30}-3\eta_{12})(\eta_{21}+\eta_{03})[3(\eta_{% 30}+\eta_{12})^{2}-(\eta_{21}+\eta_{03})^{2}].
  36. I 8 = η 11 [ ( η 30 + η 12 ) 2 - ( η 03 + η 21 ) 2 ] - ( η 20 - η 02 ) ( η 30 + η 12 ) ( η 03 + η 21 ) I_{8}=\eta_{11}[(\eta_{30}+\eta_{12})^{2}-(\eta_{03}+\eta_{21})^{2}]-(\eta_{20% }-\eta_{02})(\eta_{30}+\eta_{12})(\eta_{03}+\eta_{21})

Imaginary_time.html

  1. τ \scriptstyle\tau
  2. π / 2 \scriptstyle\pi/2
  3. τ = i t \scriptstyle\tau\ =\ it
  4. 2 β = 2 / T \scriptstyle 2\beta\ =\ 2/T
  5. F e - i t H I \scriptstyle\langle F\mid e^{-itH}\mid I\rangle
  6. Z = Tr e - β H \scriptstyle Z\ =\ \operatorname{Tr}\ e^{-\beta H}
  7. t = β / i \scriptstyle t\,=\,\beta/i

Index_ellipsoid.html

  1. 8 π W = D 1 2 ε 1 + D 2 2 ε 2 + D 3 2 ε 3 . 8\pi W=\frac{D^{2}_{1}}{\varepsilon_{1}}+\frac{D^{2}_{2}}{\varepsilon_{2}}+% \frac{D^{2}_{3}}{\varepsilon_{3}}.
  2. R i = D i 8 π W , R_{i}=\frac{D_{i}}{\sqrt{8\pi W}},
  3. R 1 2 ε 1 + R 2 2 ε 2 + R 3 2 ε 3 = 1. \frac{R_{1}^{2}}{\varepsilon_{1}}+\frac{R_{2}^{2}}{\varepsilon_{2}}+\frac{R_{3% }^{2}}{\varepsilon_{3}}=1.
  4. s \vec{s}
  5. R s = 0 \vec{R}\cdot\vec{s}=0

Inductive_dimension.html

  1. ind ( ) = Ind ( ) = - 1 \operatorname{ind}(\varnothing)=\operatorname{Ind}(\varnothing)=-1
  2. x \isin X x\isin X
  3. dim \operatorname{dim}
  4. dim X = 0 \operatorname{dim}X=0
  5. Ind X = 0. \operatorname{Ind}X=0.
  6. dim X = Ind X = ind X \operatorname{dim}X=\operatorname{Ind}X=\operatorname{ind}X
  7. Ind X n \operatorname{Ind}X\leq n
  8. A A
  9. X X
  10. f : A S n f:A\to S^{n}
  11. f ¯ : X S n \bar{f}:X\to S^{n}

Inductive_effect.html

  1. - I -I
  2. + I +I

Inelastic_mean_free_path.html

  1. I 0 \textstyle I_{0}
  2. d \textstyle d
  3. I ( d ) = I 0 e - d / λ ( E ) I(d)=I_{0}\ e^{-d\ /\lambda(E)}
  4. I ( d ) \textstyle I(d)
  5. λ ( E ) \textstyle\lambda(E)
  6. 1 / e \textstyle 1/e

Infiltration_(hydrology).html

  1. F = B I + P - E - T - E T - S - I A - R - B O F=B_{I}+P-E-T-ET-S-I_{A}-R-B_{O}
  2. B I B_{I}
  3. B O B_{O}
  4. I A I_{A}
  5. 0 F ( t ) F F + ψ Δ θ d F = 0 t K d t \int_{0}^{F(t)}{F\over F+\psi\,\Delta\theta}\,dF=\int_{0}^{t}K\,dt
  6. ψ {\psi}
  7. θ \theta
  8. K K
  9. F F
  10. F ( t ) = K t + ψ Δ θ ln [ 1 + F ( t ) ψ Δ θ ] . F(t)=Kt+\psi\,\Delta\theta\ln\left[1+{F(t)\over\psi\,\Delta\theta}\right].
  11. F ( t ) F(t)
  12. F F
  13. K t Kt
  14. 2 ψ Δ θ K t \sqrt{2\psi\,\Delta\theta Kt}
  15. h 0 h_{0}
  16. F F
  17. t t
  18. F F
  19. f ( t ) = K [ ψ Δ θ F ( t ) + 1 ] . f(t)=K\left[{\psi\,\Delta\theta\over F(t)}+1\right].
  20. f 0 f_{0}
  21. t t
  22. f c f_{c}
  23. f t = f c + ( f 0 - f c ) e - k t f_{t}=f_{c}+(f_{0}-f_{c})e^{-kt}
  24. f t f_{t}
  25. f 0 f_{0}
  26. f c f_{c}
  27. k k
  28. F t = f c t + ( f 0 - f c ) k ( 1 - e - k t ) F_{t}=f_{c}t+{(f_{0}-f_{c})\over k}(1-e^{-kt})
  29. f ( t ) = a k t a - 1 f(t)=akt^{a-1}\!
  30. a a
  31. k k
  32. f ( t ) = a k t a - 1 + f 0 f(t)=akt^{a-1}+f_{0}\!
  33. F ( t ) = k t a + f 0 t F(t)=kt^{a}+f_{0}t\!
  34. f 0 f_{0}
  35. h 0 h_{0}
  36. - ψ - L -\psi-L
  37. f = K [ h 0 - ( - ψ - L ) L ] f=K\left[{h_{0}-(-\psi-L)\over L}\right]
  38. ψ {\psi}
  39. h 0 h_{0}
  40. K K
  41. f = K [ L + S f + h 0 L ] f=K\left[{L+S_{f}+h_{0}\over L}\right]
  42. f {f}
  43. K K
  44. L L
  45. S f {S_{f}}
  46. - ψ {-\psi}
  47. - ψ f {-\psi_{f}}
  48. h 0 h_{0}