wpmath0000012_4

F(R)_gravity.html

  1. S [ g ] = 1 2 κ R - g d 4 x S[g]=\int{1\over 2\kappa}R\sqrt{-g}\,\mathrm{d}^{4}x
  2. S [ g ] = 1 2 κ f ( R ) - g d 4 x S[g]=\int{1\over 2\kappa}f(R)\sqrt{-g}\,\mathrm{d}^{4}x
  3. δ - g = - 1 2 - g g μ ν δ g μ ν \delta\sqrt{-g}=-\frac{1}{2}\sqrt{-g}g_{\mu\nu}\delta g^{\mu\nu}
  4. R = g μ ν R μ ν . R=g^{\mu\nu}R_{\mu\nu}.\!
  5. δ R = R μ ν δ g μ ν + g μ ν δ R μ ν = R μ ν δ g μ ν + g μ ν ( ρ δ Γ ν μ ρ - ν δ Γ ρ μ ρ ) \begin{aligned}\displaystyle\delta R&\displaystyle=R_{\mu\nu}\delta g^{\mu\nu}% +g^{\mu\nu}\delta R_{\mu\nu}\\ &\displaystyle=R_{\mu\nu}\delta g^{\mu\nu}+g^{\mu\nu}(\nabla_{\rho}\delta% \Gamma^{\rho}_{\nu\mu}-\nabla_{\nu}\delta\Gamma^{\rho}_{\rho\mu})\end{aligned}
  6. δ Γ μ ν λ = 1 2 g λ a ( μ δ g a ν + ν δ g a μ - a δ g μ ν ) . \delta\Gamma^{\lambda}_{\mu\nu}=\frac{1}{2}g^{\lambda a}\left(\nabla_{\mu}% \delta g_{a\nu}+\nabla_{\nu}\delta g_{a\mu}-\nabla_{a}\delta g_{\mu\nu}\right).
  7. δ R = R μ ν δ g μ ν + g μ ν δ g μ ν - μ ν δ g μ ν \delta R=R_{\mu\nu}\delta g^{\mu\nu}+g_{\mu\nu}\Box\delta g^{\mu\nu}-\nabla_{% \mu}\nabla_{\nu}\delta g^{\mu\nu}
  8. δ S [ g ] \displaystyle\delta S[g]
  9. δ S [ g ] \displaystyle\delta S[g]
  10. F ( R ) R μ ν - 1 2 f ( R ) g μ ν + [ g μ ν - μ ν ] F ( R ) = κ T μ ν , F(R)R_{\mu\nu}-\frac{1}{2}f(R)g_{\mu\nu}+\left[g_{\mu\nu}\Box-\nabla_{\mu}% \nabla_{\nu}\right]F(R)=\kappa T_{\mu\nu},
  11. T μ ν = - 2 - g δ ( - g L m ) δ g μ ν , T_{\mu\nu}=-\frac{2}{\sqrt{-g}}\frac{\delta(\sqrt{-g}L_{\mathrm{m}})}{\delta g% ^{\mu\nu}},
  12. 3 F H 2 = ρ < m t p l > m + ρ rad + 1 2 ( F R - f ) - 3 H F ˙ 3FH^{2}=\rho_{<}mtpl>{{\rm m}}+\rho_{{\rm rad}}+\frac{1}{2}(FR-f)-3H{\dot{F}}
  13. - 2 F H ˙ = ρ < m t p l > m + 4 3 ρ rad + F ¨ - H F ˙ , -2F\dot{H}=\rho_{<}mtpl>{{\rm m}}+\frac{4}{3}\rho_{{\rm rad}}+\ddot{F}-H\dot{F},
  14. H = a ˙ a , H=\frac{\dot{a}}{a},
  15. ρ ˙ < m t p l > m + 3 H ρ m = 0 ; \dot{\rho}_{<}mtpl>{{\rm m}}+3H\rho_{{\rm m}}=0;
  16. ρ ˙ < m t p l > rad + 4 H ρ rad = 0. \dot{\rho}_{<}mtpl>{{\rm rad}}+4H\rho_{{\rm rad}}=0.
  17. d s 2 = - ( 1 + 2 Φ ) d t 2 + α 2 ( 1 - 2 Ψ ) δ i j d x i d x j \mathrm{d}s^{2}=-(1+2\Phi)\mathrm{d}t^{2}+\alpha^{2}(1-2\Psi)\delta_{ij}% \mathrm{d}x^{i}\mathrm{d}x^{j}\,
  18. Φ = - 4 π G eff a 2 k 2 δ ρ m \Phi=-4\pi G_{\mathrm{eff}}\frac{a^{2}}{k^{2}}\delta\rho_{\mathrm{m}}
  19. G eff = 1 8 π F 1 + 4 k 2 a 2 R m 1 + 3 k 2 a 2 R m , G_{\mathrm{eff}}=\frac{1}{8\pi F}\frac{1+4\frac{k^{2}}{a^{2}R}m}{1+3\frac{k^{2% }}{a^{2}R}m},
  20. m R F , R F . m\equiv\frac{RF_{,R}}{F}.
  21. Φ f ( R ) and d V d Φ 2 f ( R ) - R f ( R ) 3 , \Phi\rightarrow f^{\prime}(R)~{}~{}~{}~{}~{}\textrm{and}~{}~{}~{}~{}\frac{dV}{% d\Phi}\rightarrow\frac{2f(R)-Rf^{\prime}(R)}{3},
  22. Φ = d V d Φ \Box\Phi=\frac{\mathrm{d}V}{\mathrm{d}\Phi}
  23. g μ ν = η μ ν + h μ ν g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}\,
  24. Φ = Φ 0 + δ Φ \Phi=\Phi_{0}+\delta\Phi\,
  25. h μ ν ( t , z ; ω ) = A + ( ω ) exp ( - i ω ( t - z ) ) e μ ν + + A × ( ω ) exp ( - i ω ( t - z ) ) e μ ν × + h f ( v g t - z ; ω ) η μ ν h_{\mu\nu}(t,z;\omega)=A^{+}(\omega)\exp(-i\omega(t-z))e^{+}_{\mu\nu}+A^{% \times}(\omega)\exp(-i\omega(t-z))e^{\times}_{\mu\nu}+h_{f}(v_{\mathrm{g}}t-z;% \omega)\eta_{\mu\nu}
  26. h f δ Φ Φ 0 , h_{f}\equiv\frac{\delta\Phi}{\Phi_{0}},
  27. V ( Φ ) = R V^{\prime}(\Phi)=R
  28. Φ ( R μ ν - 1 2 g μ ν R ) + ( g μ ν - μ ν ) Φ + 1 2 g μ ν V ( Φ ) = κ T μ ν \Phi\left(R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R\right)+\left(g_{\mu\nu}\Box-\nabla% _{\mu}\nabla_{\nu}\right)\Phi+\frac{1}{2}g_{\mu\nu}V(\Phi)=\kappa T_{\mu\nu}
  29. g ~ μ ν = Φ g μ ν , \tilde{g}_{\mu\nu}=\Phi g_{\mu\nu},
  30. R = Φ - 1 [ R ~ + 3 ~ Φ Φ - 9 2 ( ~ Φ Φ ) 2 ] R=\Phi^{-1}\left[\tilde{R}+\frac{3\tilde{\Box}\Phi}{\Phi}-\frac{9}{2}\left(% \frac{\tilde{\nabla}\Phi}{\Phi}\right)^{2}\right]
  31. S = d 4 x - g ~ 1 2 κ [ R ~ - 3 2 ( ~ Φ Φ ) 2 - V ( Φ ) Φ 2 ] S=\int d^{4}x\sqrt{-\tilde{g}}\frac{1}{2\kappa}\left[\tilde{R}-\frac{3}{2}% \left(\frac{\tilde{\nabla}\Phi}{\Phi}\right)^{2}-\frac{V(\Phi)}{\Phi^{2}}\right]
  32. Φ ~ = 3 ln Φ \tilde{\Phi}=\sqrt{3}\ln{\Phi}
  33. S = d 4 x - g ~ 1 2 κ [ R ~ - 1 2 ( ~ Φ ~ ) 2 - V ~ ( Φ ~ ) ] S=\int\mathrm{d}^{4}x\sqrt{-\tilde{g}}\frac{1}{2\kappa}\left[\tilde{R}-\frac{1% }{2}\left(\tilde{\nabla}\tilde{\Phi}\right)^{2}-\tilde{V}(\tilde{\Phi})\right]
  34. V ~ ( Φ ~ ) = e - 2 / 3 Φ ~ V ( e Φ ~ / 3 ) . \tilde{V}(\tilde{\Phi})=e^{-2/\sqrt{3}\;\tilde{\Phi}}V(e^{\tilde{\Phi}/\sqrt{3% }}).
  35. f ( R ) = a 0 + a 1 R + a 2 R 2 + f(R)=a_{0}+a_{1}R+a_{2}R^{2}+\ldots
  36. d D x - g f ( R , R μ ν R μ ν , R μ ν ρ σ R μ ν ρ σ ) \int\mathrm{d}^{D}x\sqrt{-g}\,f(R,R^{\mu\nu}R_{\mu\nu},R^{\mu\nu\rho\sigma}R_{% \mu\nu\rho\sigma})

F._J._Duarte.html

  1. | ψ = 1 2 ( | x 1 | y 2 - | y 1 | x 2 ) \left|\psi\right\rangle={1\over\sqrt{2}}(\left|x\right\rangle_{1}\left|y\right% \rangle_{2}-\left|y\right\rangle_{1}\left|x\right\rangle_{2})

Faddeeva_function.html

  1. w ( z ) := e - z 2 erfc ( - i z ) = erfcx ( - i z ) = e - z 2 ( 1 + 2 i π 0 z e t 2 d t ) . w(z):=e^{-z^{2}}\operatorname{erfc}(-iz)=\operatorname{erfcx}(-iz)=e^{-z^{2}}% \left(1+\frac{2i}{\sqrt{\pi}}\int_{0}^{z}e^{t^{2}}\,\text{d}t\right).
  2. Z ( z ) = i π w ( z ) Z(z)=i\sqrt{\pi}w(z)
  3. w ( x + i y ) = V ( x , y ) + i L ( x , y ) w(x+iy)=V(x,y)+iL(x,y)
  4. w ( z ) = exp ( - z 2 ) ( 1 + 2 i / π 0 z exp ( t 2 ) d t ) w(z)=\exp(-z^{2})(1+2i/\sqrt{\pi}\int_{0}^{z}\exp(t^{2})\,\text{d}t)

Fama–French_three-factor_model.html

  1. r = R f + β 3 ( K m - R f ) + b s 𝑆𝑀𝐵 + b v 𝐻𝑀𝐿 + α r=R_{f}+\beta_{3}(K_{m}-R_{f})+b_{s}\cdot\mathit{SMB}+b_{v}\cdot\mathit{HML}+\alpha

Farrell–Jones_conjecture.html

  1. K n ( R G ) K_{n}(RG)
  2. L n ( R G ) L_{n}(RG)
  3. K n ( R G ) K_{n}(RG)
  4. L n ( R G ) L_{n}(RG)
  5. C * C^{*}
  6. K n t o p ( C * r ( G ) ) K^{top}_{n}(C^{r}_{*}(G))
  7. R R
  8. K R * ? , L R * ? KR^{?}_{*},LR^{?}_{*}
  9. K R n G ( { } ) K n ( R [ G ] ) KR_{n}^{G}(\{\cdot\})\cong K_{n}(R[G])
  10. L R n G ( { } ) L n ( R [ G ] ) . LR_{n}^{G}(\{\cdot\})\cong L_{n}(R[G]).
  11. R [ G ] R[G]
  12. p : E V C Y C ( G ) { } p:E_{VCYC}(G)\rightarrow\{\cdot\}
  13. K R * G ( p ) : K R * G ( E V C Y C ( G ) ) K R * G ( { } ) K * ( R [ G ] ) . KR_{*}^{G}(p):KR_{*}^{G}(E_{VCYC}(G))\rightarrow KR_{*}^{G}(\{\cdot\})\cong K_% {*}(R[G]).
  14. E V C Y C ( G ) E_{VCYC}(G)
  15. R [ G ] R[G]
  16. G G
  17. X X
  18. = X - 1 X 0 X 1 X \emptyset=X^{-1}\subset X^{0}\subset X^{1}\subset\ldots\subset X
  19. G G
  20. K R n G ( j I i G / H j × S i - 1 ) K R n G ( j I i G / H j × D i ) K R n G ( X i - 1 ) K R n G ( X i ) KR_{n}^{G}(\coprod_{j\in I_{i}}G/H_{j}\times S^{i-1})\rightarrow KR_{n}^{G}(% \coprod_{j\in I_{i}}G/H_{j}\times D^{i})\oplus KR_{n}^{G}(X^{i-1})\rightarrow KR% _{n}^{G}(X^{i})
  21. K R n - 1 G ( j I i G / H j × S i - 1 ) K R n - 1 G ( j I i G / H j × D i ) K R n - 1 G ( X i - 1 ) \rightarrow KR_{n-1}^{G}(\coprod_{j\in I_{i}}G/H_{j}\times S^{i-1})\rightarrow KR% _{n-1}^{G}(\coprod_{j\in I_{i}}G/H_{j}\times D^{i})\oplus KR_{n-1}^{G}(X^{i-1})
  22. j I i K n ( R [ H j ] ) j I i K n - 1 ( R H j ) j I i K n ( R H j ) K R n G ( X i - 1 ) K R n G ( X i ) \bigoplus_{j\in I_{i}}K_{n}(R[H_{j}])\oplus\bigoplus_{j\in I_{i}}K_{n-1}(RH_{j% })\rightarrow\bigoplus_{j\in I_{i}}K_{n}(RH_{j})\oplus KR_{n}^{G}(X^{i-1})% \rightarrow KR_{n}^{G}(X^{i})
  23. j I i K n - 1 ( R H j ) j I i K n - 2 ( R H j ) j I i K n - 1 ( R H j ) K R n - 1 G ( X i - 1 ) \rightarrow\bigoplus_{j\in I_{i}}K_{n-1}(RH_{j})\oplus\bigoplus_{j\in I_{i}}K_% {n-2}(RH_{j})\rightarrow\bigoplus_{j\in I_{i}}K_{n-1}(RH_{j})\oplus KR^{G}_{n-% 1}(X^{i-1})
  24. E V C Y C ( G ) E_{VCYC}(G)
  25. \Z \Z
  26. E F I N ( \Z ) E_{FIN}(\Z)
  27. \R \R
  28. \Z \Z
  29. K n \Z ( \R ) = K n ( S 1 ) = K n ( p t ) K n - 1 ( p t ) = K n ( R ) K n - 1 ( R ) . K_{n}^{\Z}(\R)=K_{n}(S^{1})=K_{n}(pt)\oplus K_{n-1}(pt)=K_{n}(R)\oplus K_{n-1}% (R).
  30. K n \Z ( p t ) = K n ( R [ \Z ] ) K n ( R ) K n - 1 ( R ) N K n ( R ) N K n ( R ) . K_{n}^{\Z}(pt)=K_{n}(R[\Z])\cong K_{n}(R)\oplus K_{n-1}(R)\oplus NK_{n}(R)% \oplus NK_{n}(R).
  31. K n ( R ) K n - 1 ( R ) K n ( R ) K n - 1 ( R ) N K n ( R ) N K n ( R ) K_{n}(R)\oplus K_{n-1}(R)\hookrightarrow K_{n}(R)\oplus K_{n-1}(R)\oplus NK_{n% }(R)\oplus NK_{n}(R)
  32. N K n ( R ) = 0 NK_{n}(R)=0
  33. R R
  34. H * ? H^{?}_{*}
  35. F F
  36. E F ( G ) { } E_{F}(G)\rightarrow\{\cdot\}
  37. H * G ( E F ( G ) ) H * G ( { } ) H_{*}^{G}(E_{F}(G))\rightarrow H_{*}^{G}(\{\cdot\})
  38. α : H G \alpha:H\rightarrow G
  39. α * F := { H H | α ( H ) F } \alpha^{*}F:=\{H^{\prime}\leq H|\alpha(H)\in F\}
  40. H H
  41. α * F \alpha^{*}F
  42. F F F\subset F^{\prime}
  43. G G
  44. H F H\in F^{\prime}
  45. F | H := { H F | H H } F|_{H}:=\{H^{\prime}\in F|H^{\prime}\subset H\}
  46. G G
  47. F F
  48. F F^{\prime}
  49. α : H G \alpha:H\rightarrow G
  50. α * F \alpha^{*}F
  51. α \alpha
  52. α * V C Y C \alpha^{*}VCYC
  53. α \alpha
  54. H * G ( E V C Y C ( G ) , L R - ) H * G ( { } , L R - ) = L * - ( R G ) H^{G}_{*}(E_{VCYC}(G),L^{\langle-\infty\rangle}_{R})\rightarrow H^{G}_{*}(\{% \cdot\},L^{\langle-\infty\rangle}_{R})=L^{\langle-\infty\rangle}_{*}(RG)
  55. H * G ( E F I N ( G ) , K t o p ) H * G ( { } , K t o p ) = K n ( C r * ( G ) ) H^{G}_{*}(E_{FIN}(G),K^{top})\rightarrow H^{G}_{*}(\{\cdot\},K^{top})=K_{n}(C^% {*}_{r}(G))
  56. G G
  57. H * G ( E F I N ( G ) , K l 1 t o p ) H * G ( { } , K l 1 t o p ) = K * ( l 1 ( G ) ) H^{G}_{*}(E_{FIN}(G),K^{top}_{l^{1}})\rightarrow H^{G}_{*}(\{\cdot\},K^{top}_{% l^{1}})=K_{*}(l^{1}(G))
  58. l 1 ( G ) C r ( G ) l^{1}(G)\rightarrow C_{r}(G)
  59. K * ( l 1 ( G ) ) K * ( C r ( G ) ) K_{*}(l^{1}(G))\rightarrow K_{*}(C_{r}(G))
  60. H * G ( E F I N ( G ) , K l 1 t o p ) = H * G ( E F I N ( G ) , K t o p ) H * G ( { } , K t o p ) = K * ( C r ( G ) ) H^{G}_{*}(E_{FIN}(G),K^{top}_{l^{1}})=H^{G}_{*}(E_{FIN}(G),K^{top})\rightarrow H% ^{G}_{*}(\{\cdot\},K^{top})=K_{*}(C_{r}(G))
  61. R R
  62. G G
  63. R [ G ] R[G]
  64. 0 , 1 0,1
  65. p p
  66. R [ G ] R[G]
  67. p p
  68. K 0 ( R [ G ] ) K_{0}(R[G])

Fast_inverse_square_root.html

  1. s y m b o l v = v 1 2 + v 2 2 + v 3 2 \|symbol{v}\|=\sqrt{v_{1}^{2}+v_{2}^{2}+v_{3}^{2}}
  2. s y m b o l v ^ = s y m b o l v / s y m b o l v symbol{\hat{v}}=symbol{v}/\|symbol{v}\|
  3. s y m b o l v 2 \|symbol{v}\|^{2}
  4. v 1 2 + v 2 2 + v 3 2 v_{1}^{2}+v_{2}^{2}+v_{3}^{2}
  5. s y m b o l v ^ = s y m b o l v / s y m b o l v 2 symbol{\hat{v}}=symbol{v}/\sqrt{\|symbol{v}\|^{2}}
  6. x - 1 / 2 x^{-1/2}
  7. x = 0.15625 x=0.15625
  8. 1 / [ u r a d i c a l , u x ] 2.52982 1/[u^{\prime}radical^{\prime},u^{\prime}x^{\prime}]≈2.52982
  9. y = 2.61486 y=2.61486
  10. y = 2.52549 y=2.52549
  11. 1 / [ u r a d i c a l , u x ] 1/[u^{\prime}radical^{\prime},u^{\prime}x^{\prime}]
  12. x x
  13. x x
  14. x x
  15. x \displaystyle x
  16. x > 0 x>0
  17. x Align l t ; 0 x&lt;0
  18. B = 127 B=127
  19. x x
  20. x = + 2 - 3 ( 1 + 0.25 ) x=+2^{-3}(1+0.25)
  21. S = 0 S=0
  22. 1 / [ u r a d i c a l , u x ] 1/[u^{\prime}radical^{\prime},u^{\prime}x^{\prime}]
  23. b b
  24. x x
  25. x = 2 e x ( 1 + m x ) x=2^{e_{x}}(1+m_{x})
  26. log 2 ( x ) = e x + log 2 ( 1 + m x ) \log_{2}(x)=e_{x}+\log_{2}(1+m_{x})
  27. log 2 ( 1 + m x ) m x + σ \log_{2}(1+m_{x})\approx m_{x}+\sigma
  28. σ σ
  29. σ = 0 σ=0
  30. σ 0.0430357 σ≈0.0430357
  31. log 2 ( x ) e x + m x + σ . \log_{2}(x)\approx e_{x}+m_{x}+\sigma.
  32. x x
  33. I x = E x L + M x = L ( e x + B + m x ) L log 2 ( x ) + L ( B - σ ) . \begin{aligned}\displaystyle I_{x}&\displaystyle=E_{x}L+M_{x}\\ &\displaystyle=L(e_{x}+B+m_{x})\\ &\displaystyle\approx L\log_{2}(x)+L(B-\sigma).\end{aligned}
  34. log 2 ( x ) I x L - ( B - σ ) . \log_{2}(x)\approx\frac{I_{x}}{L}-(B-\sigma).
  35. y = 1 / [ u r a d i c a l , u x ] y=1/[u^{\prime}radical^{\prime},u^{\prime}x^{\prime}]
  36. log 2 ( y ) = - 1 2 log 2 ( x ) \log_{2}(y)=-\frac{1}{2}\log_{2}(x)
  37. x x
  38. y y
  39. I y 3 2 L ( B - σ ) - 1 2 I x I_{y}\approx\frac{3}{2}L(B-\sigma)-\frac{1}{2}I_{x}
  40. 3 2 L ( B - σ ) = 0x5f3759df \frac{3}{2}L(B-\sigma)=\,\text{0x5f3759df}
  41. σ 0.0450466 σ≈0.0450466
  42. y = 1 x y=\frac{1}{\sqrt{x}}
  43. f ( y ) = 1 y 2 - x = 0 f(y)=\frac{1}{y^{2}}-x=0
  44. y n + 1 = y n - f ( y n ) f ( y n ) y_{n+1}=y_{n}-\frac{f(y_{n})}{f^{\prime}(y_{n})}
  45. y n \,y_{n}
  46. y n + 1 = y n ( 3 - x y n 2 ) 2 y_{n+1}=\frac{y_{n}(3-xy_{n}^{2})}{2}
  47. f ( y ) = 1 y 2 - x f(y)=\frac{1}{y^{2}}-x
  48. f ( y ) = - 2 y 3 f^{\prime}(y)=\frac{-2}{y^{3}}
  49. y n + 1 = y n ( 1.5 - x y n 2 2 ) = y n ( 3 - x y n 2 ) 2 \,y_{n+1}=y_{n}(1.5-\frac{xy_{n}^{2}}{2})=\frac{y_{n}(3-xy_{n}^{2})}{2}
  50. y n + 1 y_{n+1}
  51. E < s u b > x E<sub>x
  52. M < s u b > x M<sub>x
  53. S < s u b > x = 0 S<sub>x=0

Fast_Walsh–Hadamard_transform.html

  1. N 2 N^{2}
  2. N log N N\log N
  3. N N
  4. N / 2 N/2
  5. 2 N × 2 N 2N\times 2N
  6. H N H_{N}
  7. H N = 1 2 ( H N - 1 H N - 1 H N - 1 - H N - 1 ) . H_{N}=\frac{1}{\sqrt{2}}\begin{pmatrix}H_{N-1}&H_{N-1}\\ H_{N-1}&-H_{N-1}\end{pmatrix}.
  8. 1 / 2 1/\sqrt{2}

Faustmann's_formula.html

  1. P V = p f ( T ) exp ( - r T ) ( 1 + exp ( - r T ) + exp ( - 2 r T ) + ) = p f ( T ) exp ( r T ) - 1 . PV=pf(T)\exp(-rT)\cdot{(1+\exp(-rT)+\exp(-2rT)+\cdots)}=\frac{pf(T)}{\exp(rT)-% 1}.

Faxén's_law.html

  1. 𝐔 \mathbf{U}
  2. 𝛀 \mathbf{\Omega}
  3. 𝐅 = 6 π μ a [ ( 1 + a 2 6 2 ) 𝐮 - ( 𝐔 - 𝐮 ) ] , \mathbf{F}=6\pi\mu a\left[\left(1+\frac{a^{2}}{6}\nabla^{2}\right)\mathbf{u}^{% \prime}-(\mathbf{U}-\mathbf{u}^{\infty})\right],
  4. 𝐅 \mathbf{F}
  5. μ \mu
  6. a a
  7. 𝐔 \mathbf{U}
  8. 𝐮 \mathbf{u}^{\prime}
  9. 𝐮 \mathbf{u}^{\infty}
  10. 𝐔 - 𝐮 = 𝐮 + b 0 𝐅 + a 2 6 2 𝐮 , \mathbf{U}-\mathbf{u}^{\infty}=\mathbf{u}^{\prime}+b_{0}\mathbf{F}+\frac{a^{2}% }{6}\nabla^{2}\mathbf{u}^{\prime},
  11. b 0 = - 1 6 π μ a b_{0}=-\frac{1}{6\pi\mu a}
  12. 𝐓 = 8 π μ a 3 [ 1 2 ( s y m b o l × 𝐮 ) - ( 𝛀 - 𝛀 ) ] , \mathbf{T}=8\pi\mu a^{3}\left[\frac{1}{2}\left(symbol{\nabla}\times\mathbf{u}^% {\prime}\right)-(\mathbf{\Omega}-\mathbf{\Omega}^{\infty})\right],
  13. 𝐓 \mathbf{T}
  14. 𝛀 \mathbf{\Omega}
  15. 𝛀 \mathbf{\Omega}^{\infty}
  16. s y m b o l 𝖲 = 10 3 π μ a 3 [ - 2 s y m b o l 𝖤 + ( 1 + 1 10 a 2 2 ) ( s y m b o l 𝐮 + ( s y m b o l 𝐮 ) T ) ] , symbol{\mathsf{S}}=\frac{10}{3}\pi\mu a^{3}\left[-2symbol{\mathsf{E}}^{\infty}% +\left(1+\frac{1}{10}a^{2}\nabla^{2}\right)\left(symbol{\nabla}\mathbf{u}^{% \prime}+(symbol{\nabla}\mathbf{u}^{\prime})^{\mathrm{T}}\right)\right],
  17. s y m b o l 𝖲 symbol{\mathsf{S}}
  18. s y m b o l 𝐮 symbol{\nabla}\mathbf{u}^{\prime}
  19. T {}^{\mathrm{T}}
  20. 1 2 [ s y m b o l 𝐮 + ( s y m b o l 𝐮 ) T ] \frac{1}{2}\left[symbol{\nabla}\mathbf{u}^{\prime}+(symbol{\nabla}\mathbf{u}^{% \prime})^{\mathrm{T}}\right]
  21. s y m b o l 𝖤 = 1 2 [ s y m b o l 𝐮 + ( s y m b o l 𝐮 ) T ] symbol{\mathsf{E}}^{\infty}=\frac{1}{2}\left[symbol{\nabla}\mathbf{u}^{\infty}% +(symbol{\nabla}\mathbf{u}^{\infty})^{\mathrm{T}}\right]
  22. s y m b o l 𝖤 symbol{\mathsf{E}}

Fayet–Iliopoulos_D-term.html

  1. S F I = ξ d 4 θ V S_{FI}=\xi\int d^{4}\theta\,V
  2. ξ \xi

FBI_transform.html

  1. ( f ) ( t ) = ( 2 π ) - n / 2 𝐑 n f ( x ) e - i x t d x . ({\mathcal{F}}f)(t)=(2\pi)^{-n/2}\int_{{\mathbf{R}}^{n}}f(x)e^{-ix\cdot t}\,dx.
  2. ( a f ) ( t , y ) = ( 2 π ) - n / 2 𝐑 n f ( x ) e - a | x - y | 2 / 2 e - i x t d x . ({\mathcal{F}}_{a}f)(t,y)=(2\pi)^{-n/2}\int_{{\mathbf{R}}^{n}}f(x)e^{-a|x-y|^{% 2}/2}e^{-ix\cdot t}\,dx.
  3. f ( x ) = 2 f ( - x ) f(x)={\mathcal{F}}^{2}f(-x)
  4. f ( x ) = ( 2 π ) - n / 2 𝐑 n e i t x ( f ) ( t ) d t f(x)=(2\pi)^{-n/2}\int_{{\mathbf{R}}^{n}}e^{it\cdot x}{\mathcal{F}}(f)(t)\,dt
  5. f ( x ) = ( 2 π ) - n / 2 𝐑 n e i t x e a | x - y | 2 / 2 a ( f ) ( t , y ) d t f(x)=(2\pi)^{-n/2}\int_{{\mathbf{R}}^{n}}e^{it\cdot x}e^{a|x-y|^{2}/2}{% \mathcal{F}}_{a}(f)(t,y)\,dt
  6. | ( | ξ | f ) ( ξ , y ) | C e - ε | ξ | , |({\mathcal{F}}_{|\xi|}f)(\xi,y)|\leq Ce^{-\varepsilon|\xi|},
  7. | ( | ξ | f ) ( ξ , y ) | C e - ε | ξ | , |({\mathcal{F}}_{|\xi|}f)(\xi,y)|\leq Ce^{-\varepsilon|\xi|},
  8. P = | α | m a α ( x ) D α , P=\sum_{|\alpha|\leq m}a_{\alpha}(x)D^{\alpha},
  9. σ P ( x , ξ ) = | α | = m a α ( x ) ξ α , \sigma_{P}(x,\xi)=\sum_{|\alpha|=m}a_{\alpha}(x)\xi^{\alpha},
  10. char P = { ( x , ξ ) : ξ 0 , σ P ( x , ξ ) = 0 } , {\rm char}\,P=\{(x,\xi):\xi\neq 0,\,\sigma_{P}(x,\xi)=0\},
  11. W F A ( P f ) W F A ( f ) WF_{A}(Pf)\subseteq WF_{A}(f)
  12. W F A ( f ) W F A ( P f ) char P . WF_{A}(f)\subseteq WF_{A}(Pf)\cup{\rm char}\,P.

Feature-oriented_programming.html

  1. Δ \Delta
  2. Δ \Delta
  3. Δ \Delta
  4. Δ \Delta
  5. Δ \Delta
  6. Δ \Delta
  7. Δ \Delta
  8. Δ \Delta
  9. Δ \Delta
  10. Δ \Delta
  11. Δ \Delta
  12. Δ \Delta
  13. Δ \Delta
  14. Δ \Delta
  15. Δ \Delta
  16. Δ \Delta
  17. Δ \Delta
  18. Δ \Delta
  19. Δ \Delta
  20. Δ \Delta
  21. Δ \Delta
  22. Δ \Delta
  23. Δ \Delta
  24. Δ \Delta
  25. Δ \Delta
  26. Δ \Delta
  27. Δ \Delta
  28. ( 4 2 ) {\textstyle\left({{4}\atop{2}}\right)}

Feferman–Schütte_ordinal.html

  1. ψ ( Ω Ω ) \psi(\Omega^{\Omega})
  2. θ ( Ω ) \theta(\Omega)
  3. ϕ Ω ( 0 ) \phi_{\Omega}(0)

Fenchel's_Law.html

  1. r = a W - 0.25 r={aW^{-0.25}}

Fermat's_Last_Theorem.html

  1. \mathbb{N}
  2. 1 , 2 , 3 , 1,2,3,\dots
  3. \mathbb{Z}
  4. 0 , ± 1 , ± 2 , 0,\pm 1,\pm 2,\dots
  5. \mathbb{Q}
  6. a / b a/b
  7. a a
  8. b b
  9. \mathbb{Z}
  10. b 0 b\not=0
  11. x n + y n = z n x^{n}+y^{n}=z^{n}
  12. x , y x,y
  13. z z
  14. x n + y n = z n x^{n}+y^{n}=z^{n}
  15. n 3 n\geq 3
  16. x , y , z x,y,z\in\mathbb{N}
  17. \mathbb{Z}
  18. x n + y n = z n x^{n}+y^{n}=z^{n}
  19. n 3 n\geq 3
  20. x , y , z x,y,z\in\mathbb{Z}
  21. n n
  22. n n
  23. x , y , z x,y,z
  24. x , y , z x,y,z
  25. - x , - y , - z -x,-y,-z
  26. \mathbb{N}
  27. x x
  28. z z
  29. y y
  30. z z
  31. x , z x,z
  32. y y
  33. ( - z ) n + y n = ( - x ) n (-z)^{n}+y^{n}=(-x)^{n}
  34. \mathbb{N}
  35. x x
  36. y y
  37. x x
  38. y y
  39. z z
  40. ( - x ) n + z n = y n (-x)^{n}+z^{n}=y^{n}
  41. \mathbb{N}
  42. y y
  43. \mathbb{Z}
  44. \mathbb{N}
  45. x n + y n = z n x^{n}+y^{n}=z^{n}
  46. n 3 n\geq 3
  47. x , y , z x,y,z\in\mathbb{Q}
  48. x , y x,y
  49. z z
  50. n n
  51. \mathbb{Q}
  52. \mathbb{Z}
  53. \mathbb{N}
  54. x n + y n = 1 x^{n}+y^{n}=1
  55. n 3 n\geq 3
  56. x , y x,y\in\mathbb{Q}
  57. x , y , z x,y,z\in\mathbb{Q}
  58. x n + y n = z n x^{n}+y^{n}=z^{n}
  59. ( x / z ) n + ( y / z ) n = 1 (x/z)^{n}+(y/z)^{n}=1
  60. x / z , y / z x/z,y/z\in\mathbb{Q}
  61. \mathbb{Q}
  62. \mathbb{Z}
  63. a 2 + b 2 = c 2 . a^{2}+b^{2}=c^{2}.
  64. A = x + y A=x+y
  65. B = x 2 + y 2 . B=x^{2}+y^{2}.
  66. x 4 - y 4 = z 2 x^{4}-y^{4}=z^{2}
  67. a 1 / m + b 1 / m = c 1 / m a^{1/m}+b^{1/m}=c^{1/m}
  68. a = r s m a=rs^{m}
  69. b = r t m b=rt^{m}
  70. c = r ( s + t ) m c=r(s+t)^{m}
  71. a n / m + b n / m = c n / m a^{n/m}+b^{n/m}=c^{n/m}
  72. a 1 / m a^{1/m}
  73. b 1 / m , b^{1/m},
  74. c 1 / m c^{1/m}
  75. a - 1 + b - 1 = c - 1 a^{-1}+b^{-1}=c^{-1}
  76. a = m n + m 2 , a=mn+m^{2},
  77. b = m n + n 2 , b=mn+n^{2},
  78. c = m n c=mn
  79. a - 2 + b - 2 = d - 2 a^{-2}+b^{-2}=d^{-2}
  80. a = ( v 2 - u 2 ) ( v 2 + u 2 ) , a=(v^{2}-u^{2})(v^{2}+u^{2}),\,
  81. b = 2 u v ( v 2 + u 2 ) , b=2uv(v^{2}+u^{2}),\,
  82. d = 2 u v ( v 2 - u 2 ) , d=2uv(v^{2}-u^{2}),\,
  83. c = ( v 2 + u 2 ) 2 , c=(v^{2}+u^{2})^{2},\,
  84. ( b c ) | n | + ( a c ) | n | = ( a b ) | n | (bc)^{|n|}+(ac)^{|n|}=(ab)^{|n|}
  85. ( a x ) n + ( b y ) n = ( c z ) n \left(\frac{a}{x}\right)^{n}+\left(\frac{b}{y}\right)^{n}=\left(\frac{c}{z}% \right)^{n}
  86. ( a y z ) n + ( b x z ) n = ( c x y ) n (ayz)^{n}+(bxz)^{n}=(cxy)^{n}
  87. ( ( j r + 1 ) s ) r + ( j ( j r + 1 ) s ) ) r = ( j r + 1 ) r s + 1 . \left((j^{r}+1)^{s}\right)^{r}+\left(j(j^{r}+1)^{s})\right)^{r}=(j^{r}+1)^{rs+% 1}.

Fermi_contact_interaction.html

  1. A = - 8 3 π s y m b o l μ n s y m b o l μ e | Ψ ( 0 ) | 2 (c.g.i) A=-\frac{8}{3}\pi\left\langle symbol{\mu}_{n}\cdot symbol{\mu}_{e}\right% \rangle|\Psi(0)|^{2}\qquad\mbox{(c.g.i)}~{}
  2. A = - 2 3 μ 0 s y m b o l μ n s y m b o l μ e | Ψ ( 0 ) | 2 (S.I.) A=-\frac{2}{3}\mu_{0}\left\langle symbol{\mu}_{n}\cdot symbol{\mu}_{e}\right% \rangle|\Psi(0)|^{2}\qquad\mbox{(S.I.)}~{}
  3. S ( r ) B ( r ) d 3 r = - 2 3 μ 0 s y m b o l μ \int_{S(r)}B(r)d^{3}r=-\frac{2}{3}\mu_{0}symbol{\mu}

Fernique's_theorem.html

  1. ( μ ) ( A ) = μ ( - 1 ( A ) ) , (\ell_{\ast}\mu)(A)=\mu(\ell^{-1}(A)),
  2. X exp ( α x 2 ) d μ ( x ) < + . \int_{X}\exp(\alpha\|x\|^{2})\,\mathrm{d}\mu(x)<+\infty.
  3. 𝔼 [ G k ] = X x k d μ ( x ) < + . \mathbb{E}[\|G\|^{k}]=\int_{X}\|x\|^{k}\,\mathrm{d}\mu(x)<+\infty.

Fiber_volume_ratio.html

  1. V f = v f v c V_{f}=\frac{v_{f}}{v_{c}}\!
  2. V f V_{f}
  3. v f v_{f}
  4. v c v_{c}
  5. E = ( 1 - V f ) E m + V f E f E=(1-{V_{f}}){E_{m}}+{V_{f}}{E_{f}}
  6. V f V_{f}
  7. E m E_{m}
  8. E f E_{f}
  9. V f = ( p i 2 3 ) ( r R ) 2 {V_{f}}=({\frac{pi}{2\sqrt{3}}}){(\frac{r}{R})^{2}}
  10. V f = ( p i 4 ) ( r R ) 2 {V_{f}}=({\frac{pi}{4}}){(\frac{r}{R})^{2}}
  11. r r
  12. 2 R 2R
  13. V f , m a x V_{f,max}
  14. V f , m a x V_{f,max}
  15. V v = 1 - V f - V m = v v v c V_{v}=1-V_{f}-V_{m}=\frac{v_{v}}{v_{c}}\!
  16. V v V_{v}
  17. V f V_{f}
  18. V m V_{m}
  19. v v v_{v}
  20. v c v_{c}
  21. V v = ( p c t - p c m ) p c t V_{v}=\frac{({p_{ct}}-{p_{cm}})}{p_{ct}}\!
  22. V v V_{v}
  23. p c t p_{ct}
  24. p c m p_{cm}
  25. p = ( W a p L - W L p a ) ( W a - W L ) p=\frac{(W_{apL}-W_{Lpa})}{(W_{a}-W_{L})}\!
  26. p p
  27. W a p L W_{apL}
  28. W L p a W_{Lpa}
  29. W a W_{a}
  30. W L W_{L}

Field_flattener_lens.html

  1. δ x \delta_{x}
  2. t ( δ x ) = ( n n - 1 ) δ x t(\delta_{x})=\left(\frac{n}{n-1}\right)\delta_{x}
  3. δ x ( y ) \delta_{x}(y)
  4. R p R_{p}
  5. R f = ( n - 1 n ) R p . R_{f}=\left(\frac{n-1}{n}\right)R_{p}.

Field_flow_fractionation.html

  1. c = c 0 e - x l c=c_{0}e^{\frac{-x}{l}}
  2. l = k T F l=\frac{kT}{F}
  3. t r = L V t_{r}=\frac{L}{V}

Fieller's_theorem.html

  1. μ a \mu_{a}
  2. μ b \mu_{b}
  3. ν 11 σ 2 \nu_{11}\sigma^{2}
  4. ν 22 σ 2 \nu_{22}\sigma^{2}
  5. ν 12 σ 2 \nu_{12}\sigma^{2}
  6. ν 11 , ν 12 , ν 22 \nu_{11},\nu_{12},\nu_{22}
  7. a / b a/b
  8. ( m L , m U ) = 1 ( 1 - g ) [ a b - g ν 12 ν 22 t r , α s b ν 11 - 2 a b ν 12 + a 2 b 2 ν 22 - g ( ν 11 - ν 12 2 ν 22 ) ] (m_{L},m_{U})=\frac{1}{(1-g)}\left[\frac{a}{b}-\frac{g\nu_{12}}{\nu_{22}}\mp% \frac{t_{r,\alpha}s}{b}\sqrt{\nu_{11}-2\frac{a}{b}\nu_{12}+\frac{a^{2}}{b^{2}}% \nu_{22}-g\left(\nu_{11}-\frac{\nu_{12}^{2}}{\nu_{22}}\right)}\right]
  9. g = t r , α 2 s 2 ν 22 b 2 . g=\frac{t^{2}_{r,\alpha}s^{2}\nu_{22}}{b^{2}}.
  10. s 2 s^{2}
  11. σ 2 \sigma^{2}
  12. t r , α t_{r,\alpha}
  13. α \alpha
  14. Var ( a b ) = ( a b ) 2 ( Var ( a ) a 2 + Var ( b ) b 2 ) . \operatorname{Var}\left(\frac{a}{b}\right)=\left(\frac{a}{b}\right)^{2}\left(% \frac{\operatorname{Var}(a)}{a^{2}}+\frac{\operatorname{Var}(b)}{b^{2}}\right).
  15. Var ( f ) = ( f x ) 2 Var ( x ) + ( f y ) 2 Var ( y ) + 2 f x . f y Cov ( x , y ) \operatorname{Var}(f)=\left(\frac{\partial f}{\partial x}\right)^{2}% \operatorname{Var}(x)+\left(\frac{\partial f}{\partial y}\right)^{2}% \operatorname{Var}(y)+2\frac{\partial f}{\partial x}.\frac{\partial f}{% \partial y}\operatorname{Cov}(x,y)
  16. Var ( log e a b ) = Var ( a ) a 2 + Var ( b ) b 2 \operatorname{Var}\left(\log_{e}\frac{a}{b}\right)=\frac{\operatorname{Var}(a)% }{a^{2}}+\frac{\operatorname{Var}(b)}{b^{2}}
  17. Var ( log 10 a b ) = ( log 10 e ) 2 ( Var ( a ) a 2 + Var ( b ) b 2 ) . \operatorname{Var}\left(\log_{10}\frac{a}{b}\right)=\left(\log_{10}e\right)^{2% }\left(\frac{\operatorname{Var}(a)}{a^{2}}+\frac{\operatorname{Var}(b)}{b^{2}}% \right).
  18. g = ( t * b ) 2 Var ( b ) . g=\left(\frac{t^{*}}{b}\right)^{2}\operatorname{Var}(b).
  19. g 1 g\geq 1
  20. g < 1 g<1
  21. Var ( a b ) = ( a b ( 1 - g ) ) 2 ( ( 1 - g ) Var ( a ) a 2 + Var ( b ) b 2 ) . \operatorname{Var}\left(\frac{a}{b}\right)=\left(\frac{a}{b(1-g)}\right)^{2}% \left((1-g)\frac{\operatorname{Var}(a)}{a^{2}}+\frac{\operatorname{Var}(b)}{b^% {2}}\right).

File:CrystalBallFunction.gif.html

  1. x ¯ = 0 , σ = 1 , N = 1 \bar{x}=0,\sigma=1,N=1
  2. α = 10 \alpha=10
  3. α = 1 \alpha=1
  4. α = 0.1 \alpha=0.1

File:LF-Initial.png.html

  1. u ( x , 0 ) = { 1 0.4 x 0.6 0 otherwise u(x,0)=\left\{\begin{array}[]{cl}1&0.4\leq x\leq 0.6\\ 0&\,\text{ otherwise }\end{array}\right.
  2. a = 1 a=1
  3. t = 1 t=1

File:LF-Solution.png.html

  1. t = 1 t=1
  2. a = 1 a=1
  3. t = 1 t=1

File:Square_numbers_end_on_3.jpg.html

  1. S q u a r e n u m b e r s e n d o n 3 Squarenumbersendon3

File:Square_numbers_end_on_7.jpg.html

  1. S q u a r e n u m b e r s e n d o n 7 Squarenumbersendon7

File:Square_numbers_end_on_9.jpg.html

  1. S q u a r e n u m b e r s e n d o n 9 Squarenumbersendon9

File:TetrationConvergence3D.png.html

  1. W ( - ln ( z ) ) - ln ( z ) \frac{W(-\ln(z))}{-\ln(z)}

File_sequence.html

  1. ...
  2. ...

Final_value_theorem.html

  1. lim t f ( t ) \lim_{t\to\infty}f(t)
  2. lim t f ( t ) = lim s 0 s F ( s ) \lim_{t\to\infty}f(t)=\lim_{s\to 0}{sF(s)}
  3. F ( s ) F(s)
  4. f ( t ) f(t)
  5. lim k f [ k ] = lim z 1 ( z - 1 ) F ( z ) \lim_{k\to\infty}f[k]=\lim_{z\to 1}{(z-1)F(z)}
  6. F ( z ) F(z)
  7. f [ k ] f[k]
  8. lim s 0 0 d f ( t ) d t e - s t d t = lim s 0 [ s F ( s ) - f ( 0 ) ] \lim_{s\to 0}\int_{0}^{\infty}\frac{df(t)}{dt}e^{-st}dt=\lim_{s\to 0}[sF(s)-f(% 0)]
  9. 0 lim s 0 d f ( t ) d t e - s t d t = 0 d f ( t ) = f ( ) - f ( 0 ) \int_{0}^{\infty}\lim_{s\to 0}\frac{df(t)}{dt}e^{-st}dt=\int_{0}^{\infty}df(t)% =f(\infty)-f(0)
  10. f ( ) = lim s 0 [ s F ( s ) ] f(\infty)=\lim_{s\to 0}[sF(s)]
  11. H ( s ) = 6 s + 2 , H(s)=\frac{6}{s+2},
  12. lim t h ( t ) = lim s 0 6 s s + 2 = 0. \lim_{t\to\infty}h(t)=\lim_{s\to 0}\frac{6s}{s+2}=0.
  13. G ( s ) = 1 s 6 s + 2 G(s)=\frac{1}{s}\frac{6}{s+2}
  14. lim t g ( t ) = lim s 0 s s 6 s + 2 = 6 2 = 3 \lim_{t\to\infty}g(t)=\lim_{s\to 0}\frac{s}{s}\frac{6}{s+2}=\frac{6}{2}=3
  15. H ( s ) = 9 s 2 + 9 , H(s)=\frac{9}{s^{2}+9},
  16. H ( s ) H(s)
  17. H ( s ) H(s)
  18. 0 + j 3 0+j3
  19. 0 - j 3 0-j3

Financial_models_with_long-tailed_distributions_and_volatility_clustering.html

  1. α \alpha
  2. α \alpha
  3. p p
  4. p > α p>\alpha
  5. Y Y
  6. n = 1 , 2 , n=1,2,\dots
  7. Y n , 1 , Y n , 2 , , Y n , n Y_{n,1},Y_{n,2},\dots,Y_{n,n}\,
  8. Y = d k = 1 n Y n , k , Y\stackrel{\mathrm{d}}{=}\sum_{k=1}^{n}Y_{n,k},\,
  9. = d \stackrel{\mathrm{d}}{=}
  10. ν \nu
  11. \mathbb{R}
  12. ν ( 0 ) = 0 \nu({0})=0
  13. ( 1 | x 2 | ) ν ( d x ) < . \int_{\mathbb{R}}(1\wedge|x^{2}|)\,\nu(dx)<\infty.
  14. Y Y
  15. ϕ Y ( u ) = E [ e i u Y ] \phi_{Y}(u)=E[e^{iuY}]
  16. ϕ Y ( u ) = exp ( i γ u - 1 2 σ 2 u 2 + - ( e i u x - 1 - i u x 1 | x | 1 ) ν ( d x ) ) , σ 0 , γ \phi_{Y}(u)=\exp\left(i\gamma u-\frac{1}{2}\sigma^{2}u^{2}+\int_{-\infty}^{% \infty}(e^{iux}-1-iux1_{|x|\leq 1})\,\nu(dx)\right),\sigma\geq 0,~{}~{}\gamma% \in\mathbb{R}
  17. σ 0 \sigma\geq 0
  18. γ \gamma\in\mathbb{R}
  19. ν \nu
  20. ( σ 2 , ν , γ ) (\sigma^{2},\nu,\gamma)
  21. Y Y
  22. ( σ 2 , ν , γ ) (\sigma^{2},\nu,\gamma)
  23. Y Y
  24. ϕ Y \phi_{Y}
  25. X X
  26. α \alpha
  27. n 2 n\geq 2
  28. C n C_{n}
  29. D n D_{n}
  30. X 1 + + X n = d C n X + D n , X_{1}+\cdots+X_{n}\stackrel{\mathrm{d}}{=}C_{n}X+D_{n},\,
  31. X 1 , X 2 , , X n X_{1},X_{2},\dots,X_{n}
  32. X X
  33. C n = n 1 / α C_{n}=n^{1/\alpha}
  34. 0 < α 2 0<\alpha\leq 2
  35. X X
  36. α \alpha
  37. α \alpha
  38. X X
  39. α \alpha
  40. ϕ X \phi_{X}
  41. X X
  42. ϕ X ( u ) = { exp ( i μ u - σ α | u | α ( 1 - i β sgn ( u ) tan ( π α 2 ) ) ) if α ( 0 , 1 ) ( 1 , 2 ) exp ( i μ u - σ | u | ( 1 + i β sgn ( u ) ( 2 π ) ln ( | u | ) ) ) if α = 1 exp ( i μ u - 1 2 σ 2 u 2 ) if α = 2 \phi_{X}(u)=\begin{cases}\exp\left(i\mu u-\sigma^{\alpha}|u|^{\alpha}\left(1-i% \beta\operatorname{sgn}(u)\tan\left(\frac{\pi\alpha}{2}\right)\right)\right)&% \,\text{if }\alpha\in(0,1)\cup(1,2)\\ \exp\left(i\mu u-\sigma|u|\left(1+i\beta\operatorname{sgn}(u)\left(\frac{2}{% \pi}\right)\ln(|u|)\right)\right)&\,\text{if }\alpha=1\\ \exp\left(i\mu u-\frac{1}{2}\sigma^{2}u^{2}\right)&\,\text{if }\alpha=2\end{cases}
  43. μ \mu\in\mathbb{R}
  44. σ > 0 \sigma>0
  45. β [ - 1 , 1 ] \beta\in[-1,1]
  46. ( C 1 , C 2 , λ + , λ - , α ) (C_{1},C_{2},\lambda_{+},\lambda_{-},\alpha)
  47. ( σ 2 , ν , γ ) (\sigma^{2},\nu,\gamma)
  48. σ = 0 \sigma=0
  49. γ \gamma\in\mathbb{R}
  50. ν ( d x ) = ( C 1 e - λ + x x 1 + α 1 x > 0 + C 2 e - λ - | x | | x | 1 + α 1 x < 0 ) d x , \nu(dx)=\left(\frac{C_{1}e^{-\lambda_{+}x}}{x^{1+\alpha}}1_{x>0}+\frac{C_{2}e^% {-\lambda_{-}|x|}}{|x|^{1+\alpha}}1_{x<0}\right)\,dx,
  51. C 1 , C 2 , λ + , λ - > 0 C_{1},C_{2},\lambda_{+},\lambda_{-}>0
  52. α < 2 \alpha<2
  53. C 1 = C 2 = C > 0 C_{1}=C_{2}=C>0
  54. ϕ C T S \phi_{CTS}
  55. ϕ C T S ( u ) = exp ( i u μ + C 1 Γ ( - α ) ( ( λ + - i u ) α - λ + α ) + C 2 Γ ( - α ) ( ( λ - + i u ) α - λ - α ) ) , \phi_{CTS}(u)=\exp\left(iu\mu+C_{1}\Gamma(-\alpha)((\lambda_{+}-iu)^{\alpha}-% \lambda_{+}^{\alpha})+C_{2}\Gamma(-\alpha)((\lambda_{-}+iu)^{\alpha}-\lambda_{% -}^{\alpha})\right),
  56. μ \mu\in\mathbb{R}
  57. ϕ C T S \phi_{CTS}
  58. { z : Im ( z ) ( - λ - , λ + ) } \{z\in\mathbb{C}:\operatorname{Im}(z)\in(-\lambda_{-},\lambda_{+})\}
  59. ( C , λ + , λ - , α ) (C,\lambda_{+},\lambda_{-},\alpha)
  60. ( σ 2 , ν , γ ) (\sigma^{2},\nu,\gamma)
  61. σ = 0 \sigma=0
  62. γ \gamma\in\mathbb{R}
  63. ν ( d x ) = C ( q α ( λ + | x | ) x α + 1 1 x > 0 + q α ( λ - | x | ) | x | α + 1 1 x < 0 ) d x , \nu(dx)=C\left(\frac{q_{\alpha}(\lambda_{+}|x|)}{x^{\alpha+1}}1_{x>0}+\frac{q_% {\alpha}(\lambda_{-}|x|)}{|x|^{\alpha+1}}1_{x<0}\right)\,dx,
  64. C , λ + , λ - > 0 , α < 2 C,\lambda_{+},\lambda_{-}>0,\alpha<2
  65. q α ( x ) = x α + 1 2 K α + 1 2 ( x ) . q_{\alpha}(x)=x^{\frac{\alpha+1}{2}}K_{\frac{\alpha+1}{2}}(x).
  66. K p ( x ) K_{p}(x)
  67. ϵ t ~{}\epsilon_{t}~{}
  68. ϵ t = σ t z t ~{}\epsilon_{t}=\sigma_{t}z_{t}~{}
  69. z t i i d N ( 0 , 1 ) z_{t}\sim iid~{}N(0,1)
  70. σ t 2 \sigma_{t}^{2}
  71. σ t 2 = α 0 + α 1 ϵ t - 1 2 + + α q ϵ t - q 2 = α 0 + i = 1 q α i ϵ t - i 2 \sigma_{t}^{2}=\alpha_{0}+\alpha_{1}\epsilon_{t-1}^{2}+\cdots+\alpha_{q}% \epsilon_{t-q}^{2}=\alpha_{0}+\sum_{i=1}^{q}\alpha_{i}\epsilon_{t-i}^{2}
  72. α 0 > 0 ~{}\alpha_{0}>0~{}
  73. α i 0 , i > 0 \alpha_{i}\geq 0,~{}i>0
  74. z t i i d N ( 0 , 1 ) z_{t}\sim iid~{}N(0,1)
  75. α \alpha

Finite_element_method.html

  1. P1 : { u ′′ ( x ) = f ( x ) in ( 0 , 1 ) , u ( 0 ) = u ( 1 ) = 0 , \mbox{ P1 }~{}:\begin{cases}u^{\prime\prime}(x)=f(x)\mbox{ in }~{}(0,1),\\ u(0)=u(1)=0,\end{cases}
  2. f f
  3. u u
  4. x x
  5. u ′′ u^{\prime\prime}
  6. u u
  7. x x
  8. P2 : { u x x ( x , y ) + u y y ( x , y ) = f ( x , y ) in Ω , u = 0 on Ω , \mbox{P2 }~{}:\begin{cases}u_{xx}(x,y)+u_{yy}(x,y)=f(x,y)&\mbox{ in }~{}\Omega% ,\\ u=0&\mbox{ on }~{}\partial\Omega,\end{cases}
  9. Ω \Omega
  10. ( x , y ) (x,y)
  11. Ω \partial\Omega
  12. u x x u_{xx}
  13. u y y u_{yy}
  14. x x
  15. y y
  16. u + u ′′ = f u+u^{\prime\prime}=f
  17. u u
  18. v v
  19. v = 0 v=0
  20. x = 0 x=0
  21. x = 1 x=1
  22. 0 1 f ( x ) v ( x ) d x = 0 1 u ′′ ( x ) v ( x ) d x . \int_{0}^{1}f(x)v(x)\,dx=\int_{0}^{1}u^{\prime\prime}(x)v(x)\,dx.
  23. u u
  24. u ( 0 ) = u ( 1 ) = 0 u(0)=u(1)=0
  25. v ( x ) v(x)
  26. u u
  27. u u
  28. 0 1 f ( x ) v ( x ) d x = 0 1 u ′′ ( x ) v ( x ) d x = u ( x ) v ( x ) | 0 1 - 0 1 u ( x ) v ( x ) d x = - 0 1 u ( x ) v ( x ) d x - ϕ ( u , v ) , \begin{aligned}\displaystyle\int_{0}^{1}f(x)v(x)\,dx&\displaystyle=\int_{0}^{1% }u^{\prime\prime}(x)v(x)\,dx\\ &\displaystyle=u^{\prime}(x)v(x)|_{0}^{1}-\int_{0}^{1}u^{\prime}(x)v^{\prime}(% x)\,dx\\ &\displaystyle=-\int_{0}^{1}u^{\prime}(x)v^{\prime}(x)\,dx\equiv-\phi(u,v),% \end{aligned}
  29. v ( 0 ) = v ( 1 ) = 0 v(0)=v(1)=0
  30. u u
  31. v v
  32. Ω f v d s = - Ω u v d s - ϕ ( u , v ) , \int_{\Omega}fv\,ds=-\int_{\Omega}\nabla u\cdot\nabla v\,ds\equiv-\phi(u,v),
  33. \nabla
  34. \cdot
  35. ϕ \,\!\phi
  36. H 0 1 ( Ω ) H_{0}^{1}(\Omega)
  37. Ω \Omega
  38. Ω \partial\Omega
  39. v H 0 1 ( Ω ) v\in H_{0}^{1}(\Omega)
  40. H 0 1 ( 0 , 1 ) H_{0}^{1}(0,1)
  41. ( 0 , 1 ) (0,1)
  42. 0
  43. x = 0 x=0
  44. x = 1 x=1
  45. ϕ \!\,\phi
  46. H 0 1 ( 0 , 1 ) H_{0}^{1}(0,1)
  47. 0 1 f ( x ) v ( x ) d x \int_{0}^{1}f(x)v(x)dx
  48. L 2 ( 0 , 1 ) L^{2}(0,1)
  49. u u
  50. H 0 1 ( 0 , 1 ) H_{0}^{1}(0,1)
  51. f f
  52. H 0 1 , H_{0}^{1},
  53. u H 0 1 u\in H_{0}^{1}
  54. v H 0 1 , - ϕ ( u , v ) = f v \forall v\in H_{0}^{1},\;-\phi(u,v)=\int fv
  55. u V u\in V
  56. v V , - ϕ ( u , v ) = f v \forall v\in V,\;-\phi(u,v)=\int fv
  57. V V
  58. H 0 1 H_{0}^{1}
  59. V V
  60. V V
  61. ( 0 , 1 ) (0,1)
  62. n n
  63. x x
  64. 0 = x 0 < x 1 < < x n < x n + 1 = 1 0=x_{0}<x_{1}<\cdots<x_{n}<x_{n+1}=1
  65. V V
  66. V = { v : [ 0 , 1 ] : v is continuous, v | [ x k , x k + 1 ] is linear for k = 0 , , n , and v ( 0 ) = v ( 1 ) = 0 } V=\{v:[0,1]\rightarrow\mathbb{R}\;:v\mbox{ is continuous, }~{}v|_{[x_{k},x_{k+% 1}]}\mbox{ is linear for }~{}k=0,\dots,n\mbox{, and }~{}v(0)=v(1)=0\}
  67. x 0 = 0 x_{0}=0
  68. x n + 1 = 1 x_{n+1}=1
  69. V V
  70. v V v\in V
  71. x = x k x=x_{k}
  72. k = 1 , , n k=1,\ldots,n
  73. x x
  74. V V
  75. Ω \Omega
  76. Ω \Omega
  77. V V
  78. V h V_{h}
  79. V V
  80. h > 0 h>0
  81. V V
  82. h h
  83. V h V_{h}
  84. V V
  85. x k x_{k}
  86. v k v_{k}
  87. V V
  88. 1 1
  89. x k x_{k}
  90. x j , j k x_{j},\;j\neq k
  91. v k ( x ) = { x - x k - 1 x k - x k - 1 if x [ x k - 1 , x k ] , x k + 1 - x x k + 1 - x k if x [ x k , x k + 1 ] , 0 otherwise , v_{k}(x)=\begin{cases}{x-x_{k-1}\over x_{k}\,-x_{k-1}}&\mbox{ if }~{}x\in[x_{k% -1},x_{k}],\\ {x_{k+1}\,-x\over x_{k+1}\,-x_{k}}&\mbox{ if }~{}x\in[x_{k},x_{k+1}],\\ 0&\mbox{ otherwise}~{},\end{cases}
  92. k = 1 , , n k=1,\dots,n
  93. v k v_{k}
  94. x k x_{k}
  95. Ω \Omega
  96. v k v_{k}
  97. V V
  98. 1 1
  99. x k x_{k}
  100. x j , j k x_{j},\;j\neq k
  101. v j , v k = 0 1 v j v k d x \langle v_{j},v_{k}\rangle=\int_{0}^{1}v_{j}v_{k}\,dx
  102. ϕ ( v j , v k ) = 0 1 v j v k d x \phi(v_{j},v_{k})=\int_{0}^{1}v_{j}^{\prime}v_{k}^{\prime}\,dx
  103. j , k j,k
  104. v j , v k \langle v_{j},v_{k}\rangle
  105. ( j , k ) (j,k)
  106. v k v_{k}
  107. [ x k - 1 , x k + 1 ] [x_{k-1},x_{k+1}]
  108. v j , v k \langle v_{j},v_{k}\rangle
  109. ϕ ( v j , v k ) \phi(v_{j},v_{k})
  110. | j - k | > 1 |j-k|>1
  111. x j x_{j}
  112. x k x_{k}
  113. Ω v j v k d s \int_{\Omega}v_{j}v_{k}\,ds
  114. Ω v j v k d s \int_{\Omega}\nabla v_{j}\cdot\nabla v_{k}\,ds
  115. u ( x ) = k = 1 n u k v k ( x ) u(x)=\sum_{k=1}^{n}u_{k}v_{k}(x)
  116. f ( x ) = k = 1 n f k v k ( x ) f(x)=\sum_{k=1}^{n}f_{k}v_{k}(x)
  117. v ( x ) = v j ( x ) v(x)=v_{j}(x)
  118. j = 1 , , n j=1,\dots,n
  119. - k = 1 n u k ϕ ( v k , v j ) = k = 1 n f k v k v j d x -\sum_{k=1}^{n}u_{k}\phi(v_{k},v_{j})=\sum_{k=1}^{n}f_{k}\int v_{k}v_{j}dx
  120. j = 1 , , n . j=1,\dots,n.
  121. 𝐮 \mathbf{u}
  122. 𝐟 \mathbf{f}
  123. ( u 1 , , u n ) t (u_{1},\dots,u_{n})^{t}
  124. ( f 1 , , f n ) t (f_{1},\dots,f_{n})^{t}
  125. L = ( L i j ) L=(L_{ij})
  126. M = ( M i j ) M=(M_{ij})
  127. L i j = ϕ ( v i , v j ) L_{ij}=\phi(v_{i},v_{j})
  128. M i j = v i v j d x M_{ij}=\int v_{i}v_{j}dx
  129. - L 𝐮 = M 𝐟 . -L\mathbf{u}=M\mathbf{f}.
  130. f ( x ) = k = 1 n f k v k ( x ) f(x)=\sum_{k=1}^{n}f_{k}v_{k}(x)
  131. f ( x ) f(x)
  132. v ( x ) = v j ( x ) v(x)=v_{j}(x)
  133. j = 1 , , n j=1,\dots,n
  134. M M
  135. - L 𝐮 = 𝐛 -L\mathbf{u}=\mathbf{b}
  136. 𝐛 = ( b 1 , , b n ) t \mathbf{b}=(b_{1},\dots,b_{n})^{t}
  137. b j = f v j d x b_{j}=\int fv_{j}dx
  138. j = 1 , , n j=1,\dots,n
  139. L L
  140. M M
  141. v k v_{k}
  142. 𝐮 \mathbf{u}
  143. L L
  144. L L
  145. L L
  146. M M
  147. Ω \Omega
  148. u x x x x + u y y y y = f u_{xxxx}+u_{yyyy}=f
  149. C 1 C^{1}
  150. V V
  151. H 0 1 H_{0}^{1}
  152. V V
  153. H 0 1 H_{0}^{1}
  154. h h
  155. C h p Ch^{p}
  156. C < C<\infty
  157. p > 0 p>0
  158. d d
  159. p = d + 1 p=d+1
  160. n \mathbb{R}^{n}

First-hitting-time_model.html

  1. A A
  2. A A
  3. p ( x , t x 0 ) t = D 2 p ( x , t x 0 ) x 2 , \frac{\partial p(x,t\mid x_{0})}{\partial t}=D\frac{\partial^{2}p(x,t\mid x_{0% })}{\partial x^{2}},
  4. p ( x , t = 0 x 0 ) = δ ( x - x 0 ) p(x,t={0}\mid x_{0})=\delta(x-x_{0})
  5. x ( t ) x(t)
  6. x 0 x_{0}
  7. D D
  8. m 2 s - 1 m^{2}s^{-1}
  9. x ( t ) x(t)
  10. p ( x , t ; x 0 ) = 1 4 π D t exp ( - ( x - x 0 ) 2 4 D t ) . p(x,t;x_{0})=\frac{1}{\sqrt{4\pi Dt}}\exp\left(-\frac{(x-x_{0})^{2}}{4Dt}% \right).
  11. x ( t ) x(t)
  12. FWHM t . \rm{FWHM}\sim\sqrt{t}.
  13. L L
  14. t t
  15. L ( t ) - L ( x , t ) p ( x , t ) d x , \langle L(t)\rangle\equiv\int^{\infty}_{-\infty}L(x,t)p(x,t)dx,
  16. x c x_{c}
  17. t t
  18. p ( x c , t ) = 0 p(x_{c},t)=0
  19. x c x_{c}
  20. p ( x , t ; x 0 , x c ) = 1 4 π D t ( exp ( - ( x - x 0 ) 2 4 D t ) - exp ( - ( x - ( 2 x c - x 0 ) ) 2 4 D t ) ) , p(x,t;x_{0},x_{c})=\frac{1}{\sqrt{4\pi Dt}}\left(\exp\left(-\frac{(x-x_{0})^{2% }}{4Dt}\right)-\exp\left(-\frac{(x-(2x_{c}-x_{0}))^{2}}{4Dt}\right)\right),
  21. x < x c x<x_{c}
  22. x < x c x<x_{c}
  23. t t
  24. S ( t ) - x c p ( x , t ; x 0 , x c ) d x = erf ( x c - x 0 2 D t ) , S(t)\equiv\int_{-\infty}^{x_{c}}p(x,t;x_{0},x_{c})dx=\operatorname{erf}\left(% \frac{x_{c}-x_{0}}{2\sqrt{Dt}}\right),
  25. erf \operatorname{erf}
  26. t t
  27. t + d t t+dt
  28. f ( t ) d t = S ( t ) - S ( t + d t ) f(t)dt=S(t)-S(t+dt)
  29. f ( t ) = - S ( t ) t . f(t)=-\frac{\partial S(t)}{\partial t}.
  30. f ( t ) | x c - x 0 | 4 π D t 3 exp ( - ( x c - x 0 ) 2 4 D t ) . f(t)\equiv\frac{|x_{c}-x_{0}|}{\sqrt{4\pi Dt^{3}}}\exp\left(-\frac{(x_{c}-x_{0% })^{2}}{4Dt}\right).
  31. t ( x c - x 0 ) 2 4 D t\gg\frac{(x_{c}-x_{0})^{2}}{4D}
  32. f ( t ) = Δ x 4 π D t 3 t - 3 / 2 , f(t)=\frac{\Delta x}{\sqrt{4\pi Dt^{3}}}\sim t^{-3/2},
  33. Δ x | x c - x 0 | \Delta x\equiv|x_{c}-x_{0}|
  34. f / t = 0 \partial f/\partial t=0
  35. τ ty = Δ x 2 6 D . \tau_{\rm{ty}}=\frac{\Delta x^{2}}{6D}.
  36. { X ( t ) } \{X(t)\}\,\,
  37. { X ( t ) , t 0 } \{X(t),t\geq 0\,\}\,
  38. μ {\mu}\,\,
  39. σ 2 {\sigma^{2}}\,\,
  40. X ( 0 ) = x 0 > 0 X(0)=x_{0}>0\,

First_variation_of_area_formula.html

  1. Σ ( t ) \Sigma(t)
  2. d d t d A = H d A , \frac{d}{dt}\,dA=H\,dA,
  3. Σ ( t ) \Sigma(t)
  4. Σ ( t ) \Sigma(t)
  5. D α e β D_{\alpha}\vec{e}_{\beta}
  6. e β \vec{e}_{\beta}

Fishburn–Shepp_inequality.html

  1. P ( x < y ) P ( x < z ) < P ( ( x < y ) ( x < z ) ) P(x<y)P(x<z)<P((x<y)\wedge(x<z))\,

Fisher_consistency.html

  1. θ ^ = T ( F ^ n ) , \hat{\theta}=T(\hat{F}_{n})\,,
  2. T ( F θ ) = θ . T(F_{\theta})=\theta\,.
  3. T ( lim n F ^ n ) = θ . T\left(\lim_{n\rightarrow\infty}\hat{F}_{n}\right)=\theta.\,
  4. n - 1 i = 1 n j = 1 m I ( X i = Z j ) Z j , n^{-1}\sum_{i=1}^{n}\sum_{j=1}^{m}I(X_{i}=Z_{j})Z_{j},
  5. n - 1 i = 1 n j = 1 m p j Z j = n - 1 i = 1 n μ = μ , n^{-1}\sum_{i=1}^{n}\sum_{j=1}^{m}p_{j}Z_{j}=n^{-1}\sum_{i=1}^{n}\mu=\mu,
  6. E [ d ln L d b ] = 0 at b = b 0 , E\left[\frac{d\ln L}{db}\right]=0\,\text{ at }b=b_{0},\,

Fisher–Tippett–Gnedenko_theorem.html

  1. X 1 , X 2 , X n X_{1},X_{2}\ldots,X_{n}\ldots
  2. M n = max { X 1 , , X n } M_{n}=\max\{X_{1},\ldots,X_{n}\}
  3. ( a n , b n ) (a_{n},b_{n})
  4. a n > 0 a_{n}>0
  5. lim n P ( M n - b n a n x ) = F ( x ) \lim_{n\to\infty}P\left(\frac{M_{n}-b_{n}}{a_{n}}\leq x\right)=F(x)
  6. F F
  7. F F
  8. a n = G - 1 ( 1 - 1 n ) . a_{n}=G^{-1}\left(1-\frac{1}{n}\right).
  9. ω = sup { G < 1 } < + \omega=\sup\{G<1\}<+\infty
  10. 1 - G ( ω + t x ) 1 - G ( ω - t ) t 0 + ( - x ) θ , x < 0 \frac{1-G(\omega+tx)}{1-G(\omega-t)}\xrightarrow[t\to 0^{+}]{}(-x)^{\theta},% \quad x<0
  11. a n = ω - G - 1 ( 1 - 1 n ) . a_{n}=\omega-G^{-1}\left(1-\frac{1}{n}\right).

FKG_inequality.html

  1. X X
  2. μ ( x y ) μ ( x y ) μ ( x ) μ ( y ) \mu(x\wedge y)\mu(x\vee y)\geq\mu(x)\mu(y)
  3. X X
  4. X X
  5. ( x X f ( x ) g ( x ) μ ( x ) ) ( x X μ ( x ) ) ( x X f ( x ) μ ( x ) ) ( x X g ( x ) μ ( x ) ) . \left(\sum_{x\in X}f(x)g(x)\mu(x)\right)\left(\sum_{x\in X}\mu(x)\right)\geq% \left(\sum_{x\in X}f(x)\mu(x)\right)\left(\sum_{x\in X}g(x)\mu(x)\right).
  6. X X
  7. X X
  8. a 1 a 2 a n a_{1}\leq a_{2}\leq\cdots\leq a_{n}
  9. b 1 b 2 b n b_{1}\leq b_{2}\leq\cdots\leq b_{n}
  10. a 1 b 1 + + a n b n n a 1 + + a n n b 1 + + b n n . \frac{a_{1}b_{1}+\cdots+a_{n}b_{n}}{n}\geq\frac{a_{1}+\cdots+a_{n}}{n}\;\frac{% b_{1}+\cdots+b_{n}}{n}.
  11. \R \R
  12. \R f ( x ) g ( x ) d μ ( x ) \R f ( x ) d μ ( x ) \R g ( x ) d μ ( x ) , \int_{\R}f(x)g(x)\,d\mu(x)\geq\int_{\R}f(x)\,d\mu(x)\,\int_{\R}g(x)\,d\mu(x),
  13. \R \R [ f ( x ) - f ( y ) ] [ g ( x ) - g ( y ) ] d μ ( x ) d μ ( y ) 0. \int_{\R}\int_{\R}[f(x)-f(y)][g(x)-g(y)]\,d\mu(x)\,d\mu(y)\geq 0.
  14. X = X 1 × × X n X=X_{1}\times\cdots\times X_{n}
  15. μ = μ 1 μ n \mu=\mu_{1}\otimes\cdots\otimes\mu_{n}
  16. \R \R
  17. p p
  18. 1 - p 1-p
  19. a b a\leftrightarrow b
  20. c d c\leftrightarrow d
  21. Pr ( a b , c d ) Pr ( a b ) Pr ( c d ) \Pr(a\leftrightarrow b,\ c\leftrightarrow d)\geq\Pr(a\leftrightarrow b)\Pr(c% \leftrightarrow d)
  22. n × n n\times n
  23. S S
  24. { - 1 , + 1 } \{-1,+1\}
  25. Γ \Gamma
  26. S Γ S^{\Gamma}
  27. S S
  28. Φ \Phi
  29. Φ Λ : S Λ \R { } , \Phi_{\Lambda}:S^{\Lambda}\longrightarrow\R\cup\{\infty\},
  30. Λ Γ \Lambda\subset\Gamma
  31. Φ Λ \Phi_{\Lambda}
  32. H Λ ( ϕ ) := Δ Λ Φ Δ ( ϕ ) . H_{\Lambda}(\phi):=\sum_{\Delta\cap\Lambda\not=\emptyset}\Phi_{\Delta}(\phi).
  33. ϕ \phi
  34. Γ \Gamma
  35. S = { - 1 , + 1 } S=\{-1,+1\}
  36. β [ 0 , ] \beta\in[0,\infty]
  37. Φ Λ ( ϕ ) = { β 1 { ϕ ( x ) ϕ ( y ) } if Λ = { x , y } is a pair of adjacent vertices of Γ ; 0 otherwise. \Phi_{\Lambda}(\phi)=\begin{cases}\beta 1_{\{\phi(x)\not=\phi(y)\}}&\,\text{ % if }\Lambda=\{x,y\}\,\text{ is a pair of adjacent vertices of }\Gamma;\\ 0&\,\text{ otherwise.}\end{cases}
  38. Γ \Gamma
  39. β \beta
  40. f i = x X f ( x ) μ i ( x ) x X μ i ( x ) \langle f\rangle_{i}=\frac{\sum_{x\in X}f(x)\mu_{i}(x)}{\sum_{x\in X}\mu_{i}(x)}
  41. X X
  42. f 1 f 2 , \langle f\rangle_{1}\geq\langle f\rangle_{2},
  43. μ 2 ( x y ) μ 1 ( x y ) μ 1 ( x ) μ 2 ( y ) \mu_{2}(x\wedge y)\mu_{1}(x\vee y)\geq\mu_{1}(x)\mu_{2}(y)
  44. X X
  45. f g μ g μ = f 1 f 2 = f μ , \frac{\langle fg\rangle_{\mu}}{\langle g\rangle_{\mu}}=\langle f\rangle_{1}% \geq\langle f\rangle_{2}=\langle f\rangle_{\mu},
  46. X X
  47. \R V \R^{V}
  48. V V
  49. v V v\in V
  50. ϕ ( w ) ψ ( w ) \phi(w)\geq\psi(w)
  51. w v w\not=v
  52. { ϕ ( w ) : w v } \{\phi(w):w\not=v\}
  53. { ψ ( w ) : w v } \{\psi(w):w\not=v\}
  54. V V
  55. f 1 f 2 \langle f\rangle_{1}\geq\langle f\rangle_{2}

Flamant_solution.html

  1. σ r r = 2 C 1 cos θ r + 2 C 3 sin θ r σ r θ = 0 σ θ θ = 0 \begin{aligned}\displaystyle\sigma_{rr}&\displaystyle=\frac{2C_{1}\cos\theta}{% r}+\frac{2C_{3}\sin\theta}{r}\\ \displaystyle\sigma_{r\theta}&\displaystyle=0\\ \displaystyle\sigma_{\theta\theta}&\displaystyle=0\end{aligned}
  2. C 1 , C 3 C_{1},C_{3}
  3. α , β \alpha,\beta
  4. F 1 + 2 α β ( C 1 cos θ + C 3 sin θ ) cos θ d θ = 0 F 2 + 2 α β ( C 1 cos θ + C 3 sin θ ) sin θ d θ = 0 \begin{aligned}\displaystyle F_{1}&\displaystyle+2\int_{\alpha}^{\beta}(C_{1}% \cos\theta+C_{3}\sin\theta)\,\cos\theta\,d\theta=0\\ \displaystyle F_{2}&\displaystyle+2\int_{\alpha}^{\beta}(C_{1}\cos\theta+C_{3}% \sin\theta)\,\sin\theta\,d\theta=0\end{aligned}
  5. F 1 , F 2 F_{1},F_{2}
  6. σ = f ( r ) g ( θ ) \sigma=f(r)g(\theta)
  7. ( 1 / r ) (1/r)
  8. α = - π \alpha=-\pi
  9. β = 0 \beta=0
  10. C 1 = - F 1 π , C 3 = - F 2 π C_{1}=-\frac{F_{1}}{\pi},\quad C_{3}=-\frac{F_{2}}{\pi}
  11. σ r r = - 2 π r ( F 1 cos θ + F 2 sin θ ) σ r θ = 0 σ θ θ = 0 \begin{aligned}\displaystyle\sigma_{rr}&\displaystyle=-\frac{2}{\pi\,r}(F_{1}% \cos\theta+F_{2}\sin\theta)\\ \displaystyle\sigma_{r\theta}&\displaystyle=0\\ \displaystyle\sigma_{\theta\theta}&\displaystyle=0\end{aligned}
  12. u r = - 1 4 π μ [ F 1 { ( κ - 1 ) θ sin θ - cos θ + ( κ + 1 ) ln r cos θ } + F 2 { ( κ - 1 ) θ cos θ + sin θ - ( κ + 1 ) ln r sin θ } ] u θ = - 1 4 π μ [ F 1 { ( κ - 1 ) θ cos θ - sin θ - ( κ + 1 ) ln r sin θ } - F 2 { ( κ - 1 ) θ sin θ + cos θ + ( κ + 1 ) ln r cos θ } ] \begin{aligned}\displaystyle u_{r}&\displaystyle=-\cfrac{1}{4\pi\mu}\left[F_{1% }\{(\kappa-1)\theta\sin\theta-\cos\theta+(\kappa+1)\ln r\cos\theta\}+\right.\\ &\displaystyle\qquad\qquad\left.F_{2}\{(\kappa-1)\theta\cos\theta+\sin\theta-(% \kappa+1)\ln r\sin\theta\}\right]\\ \displaystyle u_{\theta}&\displaystyle=-\cfrac{1}{4\pi\mu}\left[F_{1}\{(\kappa% -1)\theta\cos\theta-\sin\theta-(\kappa+1)\ln r\sin\theta\}-\right.\\ &\displaystyle\qquad\qquad\left.F_{2}\{(\kappa-1)\theta\sin\theta+\cos\theta+(% \kappa+1)\ln r\cos\theta\}\right]\end{aligned}
  13. ln r \ln r
  14. x 1 , x 2 x_{1},x_{2}
  15. u 1 = F 1 ( κ + 1 ) ln | x 1 | 4 π μ + F 2 ( κ + 1 ) sign ( x 1 ) 8 μ u 2 = F 2 ( κ + 1 ) ln | x 1 | 4 π μ + F 1 ( κ + 1 ) sign ( x 1 ) 8 μ \begin{aligned}\displaystyle u_{1}&\displaystyle=\frac{F_{1}(\kappa+1)\ln|x_{1% }|}{4\pi\mu}+\frac{F_{2}(\kappa+1)\,\text{sign}(x_{1})}{8\mu}\\ \displaystyle u_{2}&\displaystyle=\frac{F_{2}(\kappa+1)\ln|x_{1}|}{4\pi\mu}+% \frac{F_{1}(\kappa+1)\,\text{sign}(x_{1})}{8\mu}\end{aligned}
  16. κ = { 3 - 4 ν plane strain 3 - ν 1 + ν plane stress \kappa=\begin{cases}3-4\nu&\qquad\,\text{plane strain}\\ \cfrac{3-\nu}{1+\nu}&\qquad\,\text{plane stress}\end{cases}
  17. ν \nu
  18. μ \mu
  19. sign ( x ) = { + 1 x > 0 - 1 x < 0 \,\text{sign}(x)=\begin{cases}+1&x>0\\ -1&x<0\end{cases}
  20. ( 1 / r ) (1/r)
  21. 1 / r 1/r
  22. φ = C 1 r θ sin θ + C 2 r ln r cos θ + C 3 r θ cos θ + C 4 r ln r sin θ \varphi=C_{1}r\theta\sin\theta+C_{2}r\ln r\cos\theta+C_{3}r\theta\cos\theta+C_% {4}r\ln r\sin\theta
  23. σ r r = C 1 ( 2 cos θ r ) + C 2 ( cos θ r ) + C 3 ( 2 sin θ r ) + C 4 ( sin θ r ) σ r θ = C 2 ( sin θ r ) + C 4 ( - cos θ r ) σ θ θ = C 2 ( cos θ r ) + C 4 ( sin θ r ) \begin{aligned}\displaystyle\sigma_{rr}&\displaystyle=C_{1}\left(\frac{2\cos% \theta}{r}\right)+C_{2}\left(\frac{\cos\theta}{r}\right)+C_{3}\left(\frac{2% \sin\theta}{r}\right)+C_{4}\left(\frac{\sin\theta}{r}\right)\\ \displaystyle\sigma_{r\theta}&\displaystyle=C_{2}\left(\frac{\sin\theta}{r}% \right)+C_{4}\left(\frac{-\cos\theta}{r}\right)\\ \displaystyle\sigma_{\theta\theta}&\displaystyle=C_{2}\left(\frac{\cos\theta}{% r}\right)+C_{4}\left(\frac{\sin\theta}{r}\right)\end{aligned}
  24. C 1 , C 2 , C 3 , C 4 C_{1},C_{2},C_{3},C_{4}
  25. a a\,
  26. 𝐧 = 𝐞 r \mathbf{n}=\mathbf{e}_{r}
  27. ( 𝐞 r , 𝐞 θ ) (\mathbf{e}_{r},\mathbf{e}_{\theta})
  28. 𝐭 = s y m b o l σ 𝐧 t r = σ r r , t θ = σ r θ . \mathbf{t}=symbol{\sigma}\cdot\mathbf{n}\quad\implies t_{r}=\sigma_{rr},~{}t_{% \theta}=\sigma_{r\theta}~{}.
  29. f 1 = F 1 + α β [ σ r r ( a , θ ) cos θ - σ r θ ( a , θ ) sin θ ] a d θ = 0 f 2 = F 2 + α β [ σ r r ( a , θ ) sin θ + σ r θ ( a , θ ) cos θ ] a d θ = 0 m 3 = α β [ a σ r θ ( a , θ ) ] a d θ = 0 \begin{aligned}\displaystyle\sum f_{1}&\displaystyle=F_{1}+\int_{\alpha}^{% \beta}\left[\sigma_{rr}(a,\theta)~{}\cos\theta-\sigma_{r\theta}(a,\theta)~{}% \sin\theta\right]~{}a~{}d\theta=0\\ \displaystyle\sum f_{2}&\displaystyle=F_{2}+\int_{\alpha}^{\beta}\left[\sigma_% {rr}(a,\theta)~{}\sin\theta+\sigma_{r\theta}(a,\theta)~{}\cos\theta\right]~{}a% ~{}d\theta=0\\ \displaystyle\sum m_{3}&\displaystyle=\int_{\alpha}^{\beta}\left[a~{}\sigma_{r% \theta}(a,\theta)\right]~{}a~{}d\theta=0\end{aligned}
  30. a a\,
  31. θ = α \theta=\alpha
  32. θ = β \theta=\beta
  33. σ r θ = σ θ θ = 0 at θ = α , θ = β \sigma_{r\theta}=\sigma_{\theta\theta}=0\qquad\,\text{at}~{}~{}\theta=\alpha,% \theta=\beta
  34. r = 0 r=0
  35. σ r θ = 0 \sigma_{r\theta}=0
  36. F 1 + α β σ r r ( a , θ ) a cos θ d θ = 0 F 2 + α β σ r r ( a , θ ) a sin θ d θ = 0 \begin{aligned}\displaystyle F_{1}&\displaystyle+\int_{\alpha}^{\beta}\sigma_{% rr}(a,\theta)~{}a~{}\cos\theta~{}d\theta=0\\ \displaystyle F_{2}&\displaystyle+\int_{\alpha}^{\beta}\sigma_{rr}(a,\theta)~{% }a~{}\sin\theta~{}d\theta=0\end{aligned}
  37. σ θ θ = 0 \sigma_{\theta\theta}=0
  38. θ = α , θ = β \theta=\alpha,\theta=\beta
  39. r = 0 r=0
  40. σ θ θ = 0 \sigma_{\theta\theta}=0
  41. σ r θ = 0 \sigma_{r\theta}=0
  42. C 2 = C 4 = 0 C_{2}=C_{4}=0
  43. σ r r = 2 C 1 cos θ r + 2 C 3 sin θ r ; σ r θ = 0 ; σ θ θ = 0 \sigma_{rr}=\frac{2C_{1}\cos\theta}{r}+\frac{2C_{3}\sin\theta}{r}~{};~{}~{}% \sigma_{r\theta}=0~{};~{}~{}\sigma_{\theta\theta}=0
  44. σ r r \sigma_{rr}
  45. σ r r \sigma_{rr}
  46. C 1 , C 3 C_{1},C_{3}
  47. F 1 \displaystyle F_{1}
  48. α = - π \alpha=-\pi
  49. β = 0 \beta=0
  50. F 2 F_{2}
  51. F 1 F_{1}
  52. F 1 + 2 - π 0 ( C 1 cos θ + C 3 sin θ ) cos θ d θ = 0 F 1 + C 1 π = 0 F 2 + 2 - π 0 ( C 1 cos θ + C 3 sin θ ) sin θ d θ = 0 F 2 + C 3 π = 0 \begin{aligned}\displaystyle F_{1}&\displaystyle+2\int_{-\pi}^{0}(C_{1}\cos% \theta+C_{3}\sin\theta)~{}\cos\theta~{}d\theta=0\qquad\implies F_{1}+C_{1}\pi=% 0\\ \displaystyle F_{2}&\displaystyle+2\int_{-\pi}^{0}(C_{1}\cos\theta+C_{3}\sin% \theta)~{}\sin\theta~{}d\theta=0\qquad\implies F_{2}+C_{3}\pi=0\end{aligned}
  53. C 1 = - F 1 π ; C 3 = - F 2 π . C_{1}=-\cfrac{F_{1}}{\pi}~{};~{}~{}C_{3}=-\cfrac{F_{2}}{\pi}~{}.
  54. σ r r = - 2 π r ( F 1 cos θ + F 2 sin θ ) ; σ r θ = 0 ; σ θ θ = 0 \sigma_{rr}=-\frac{2}{\pi r}(F_{1}\cos\theta+F_{2}\sin\theta)~{};~{}~{}\sigma_% {r\theta}=0~{};~{}~{}\sigma_{\theta\theta}=0
  55. u r = - 1 4 π μ [ F 1 { ( κ - 1 ) θ sin θ - cos θ + ( κ + 1 ) ln r cos θ } + F 2 { ( κ - 1 ) θ cos θ + sin θ - ( κ + 1 ) ln r sin θ } ] u θ = - 1 4 π μ [ F 1 { ( κ - 1 ) θ cos θ - sin θ - ( κ + 1 ) ln r sin θ } - F 2 { ( κ - 1 ) θ sin θ + cos θ + ( κ + 1 ) ln r cos θ } ] \begin{aligned}\displaystyle u_{r}&\displaystyle=-\cfrac{1}{4\pi\mu}\left[F_{1% }\{(\kappa-1)\theta\sin\theta-\cos\theta+(\kappa+1)\ln r\cos\theta\}+\right.\\ &\displaystyle\qquad\qquad\left.F_{2}\{(\kappa-1)\theta\cos\theta+\sin\theta-(% \kappa+1)\ln r\sin\theta\}\right]\\ \displaystyle u_{\theta}&\displaystyle=-\cfrac{1}{4\pi\mu}\left[F_{1}\{(\kappa% -1)\theta\cos\theta-\sin\theta-(\kappa+1)\ln r\sin\theta\}-\right.\\ &\displaystyle\qquad\qquad\left.F_{2}\{(\kappa-1)\theta\sin\theta+\cos\theta+(% \kappa+1)\ln r\cos\theta\}\right]\end{aligned}
  56. x 1 x_{1}
  57. θ = 0 \theta=0
  58. x 1 x_{1}
  59. θ = π \theta=\pi
  60. r = | x 1 | r=|x_{1}|
  61. θ = 0 \theta=0
  62. u r = u 1 = F 1 4 π μ [ 1 - ( κ + 1 ) ln | x 1 | ] u θ = u 2 = F 2 4 π μ [ 1 + ( κ + 1 ) ln | x 1 | ] \begin{aligned}\displaystyle u_{r}=u_{1}&\displaystyle=\cfrac{F_{1}}{4\pi\mu}% \left[1-(\kappa+1)\ln|x_{1}|\right]\\ \displaystyle u_{\theta}=u_{2}&\displaystyle=\cfrac{F_{2}}{4\pi\mu}\left[1+(% \kappa+1)\ln|x_{1}|\right]\end{aligned}
  63. θ = π \theta=\pi
  64. u r = - u 1 = - F 1 4 π μ [ 1 - ( κ + 1 ) ln | x 1 | ] + F 2 4 μ ( κ - 1 ) u θ = - u 2 = F 1 4 μ ( κ - 1 ) - F 2 4 π μ [ 1 + ( κ + 1 ) ln | x 1 | ] \begin{aligned}\displaystyle u_{r}=-u_{1}&\displaystyle=-\cfrac{F_{1}}{4\pi\mu% }\left[1-(\kappa+1)\ln|x_{1}|\right]+\cfrac{F_{2}}{4\mu}(\kappa-1)\\ \displaystyle u_{\theta}=-u_{2}&\displaystyle=\cfrac{F_{1}}{4\mu}(\kappa-1)-% \cfrac{F_{2}}{4\pi\mu}\left[1+(\kappa+1)\ln|x_{1}|\right]\end{aligned}
  65. u 1 = F 2 8 μ ( κ - 1 ) ; u 2 = F 1 8 μ ( κ - 1 ) u_{1}=\cfrac{F_{2}}{8\mu}(\kappa-1)~{};~{}~{}u_{2}=\cfrac{F_{1}}{8\mu}(\kappa-1)
  66. u 1 = F 1 4 π μ ; u 2 = F 2 4 π μ . u_{1}=\cfrac{F_{1}}{4\pi\mu}~{};~{}~{}u_{2}=\cfrac{F_{2}}{4\pi\mu}~{}.
  67. u 1 = F 1 4 π μ ( κ + 1 ) ln | x 1 | + F 2 8 μ ( κ - 1 ) sign ( x 1 ) u 2 = F 2 4 π μ ( κ + 1 ) ln | x 1 | + F 1 8 μ ( κ - 1 ) sign ( x 1 ) \begin{aligned}\displaystyle u_{1}&\displaystyle=\cfrac{F_{1}}{4\pi\mu}(\kappa% +1)\ln|x_{1}|+\cfrac{F_{2}}{8\mu}(\kappa-1)\,\text{sign}(x_{1})\\ \displaystyle u_{2}&\displaystyle=\cfrac{F_{2}}{4\pi\mu}(\kappa+1)\ln|x_{1}|+% \cfrac{F_{1}}{8\mu}(\kappa-1)\,\text{sign}(x_{1})\end{aligned}
  68. sign ( x ) = { + 1 x > 0 - 1 x < 0 \,\text{sign}(x)=\begin{cases}+1&x>0\\ -1&x<0\end{cases}

Flippin–Lodge_angle.html

  1. α B D \alpha_{BD}
  2. α F L \alpha_{FL}
  3. α B D \alpha_{BD}
  4. α F L \alpha_{FL}
  5. α F L \alpha_{FL}
  6. α B D \alpha_{BD}
  7. α B D \alpha_{BD}
  8. α F L \alpha_{FL}
  9. α F L \alpha_{FL}
  10. α F L \alpha_{FL}
  11. α F L \alpha_{FL}
  12. α B D \alpha_{BD}
  13. α \alpha
  14. α F L \alpha_{FL}
  15. α \alpha
  16. α B D \alpha_{BD}
  17. α F L \alpha_{FL}
  18. α B D \alpha_{BD}
  19. α F L \alpha_{FL}
  20. α B D \alpha_{BD}
  21. α F L \alpha_{FL}
  22. α F L \alpha_{FL}
  23. α F L \alpha_{FL}
  24. α B D \alpha_{BD}
  25. α F L \alpha_{FL}
  26. α F L \alpha_{FL}
  27. α F L \alpha_{FL}
  28. α B D \alpha_{BD}
  29. α B D \alpha_{BD}
  30. α F L \alpha_{FL}
  31. α F L \alpha_{FL}
  32. α F L \alpha_{FL}
  33. α F L \alpha_{FL}
  34. α F L \alpha_{FL}
  35. α F L \alpha_{FL}
  36. α F L \alpha_{FL}
  37. α B D \alpha_{BD}
  38. α B D \alpha_{BD}
  39. α B D \alpha_{BD}
  40. α F L \alpha_{FL}
  41. α F L \alpha_{FL}
  42. α F L \alpha_{FL}
  43. α F L \alpha_{FL}

Fluorine_perchlorate.html

  1. F 2 + HClO 4 FClO 4 + HF \rm\ F_{2}+HClO_{4}\rightarrow FClO_{4}+HF
  2. NF 4 ClO 4 Δ NF 3 + FClO 4 \rm\ NF_{4}ClO_{4}\xrightarrow{\Delta}NF_{3}+FClO_{4}

Focus_recovery_based_on_the_linear_canonical_transform.html

  1. U ( x 0 , y 0 ) = 1 j λ U ( x 1 , y 1 ) e j k r 01 r 01 cos θ d x 1 d y 1 U(x_{0},y_{0})=\frac{1}{j\lambda}\int\!\int U(x_{1},y_{1})\frac{e^{jkr_{01}}}{% r_{01}}\cos\theta dx_{1}dy_{1}
  2. r 01 r_{01}
  3. z z
  4. r 01 z \frac{r_{01}}{z}
  5. r 01 r_{01}
  6. [ ( x 0 - x 1 ) 2 + ( y 0 - y 1 ) 2 + z 2 ] 1 / 2 [(x_{0}-x_{1})^{2}+(y_{0}-y_{1})^{2}+z^{2}]^{1/2}
  7. U ( x 0 , y 0 ) = 1 j λ z U ( x 1 , y 1 ) exp ( j k z [ 1 + ( x 0 - x 1 z ) 2 + ( y 0 - y 1 z ) 2 ] 1 / 2 ) 1 + ( x 0 - x 1 z ) 2 + ( y 0 - y 1 z ) 2 d x 1 d y 1 U(x_{0},y_{0})=\frac{1}{j\lambda z}\int\!\int U(x_{1},y_{1})\frac{\exp(jkz[1+(% \frac{x_{0}-x_{1}}{z})^{2}+(\frac{y_{0}-y_{1}}{z})^{2}]^{1/2})}{1+(\frac{x_{0}% -x_{1}}{z})^{2}+(\frac{y_{0}-y_{1}}{z})^{2}}dx_{1}dy_{1}
  8. L M ( f ( u ) ) = L M ( u , u ) f ( u ) d u L_{M}(f(u))=\int L_{M}(u,u^{\prime})f(u^{\prime})du^{\prime}
  9. L M ( u , u ) = { 1 b e - j π / 4 e [ j π ( d b u 2 ) - 2 1 b u u + a b u 2 ] , if b 0 d e j 2 c d u 2 δ ( u - d u ) , if b = 0 L_{M}(u,u^{\prime})=\begin{cases}\sqrt{\frac{1}{b}}e^{-j\pi/4}e^{[j\pi(\frac{d% }{b}u^{2})-2\frac{1}{b}uu^{\prime}+\frac{a}{b}u^{\prime 2}]},&\mbox{if }~{}b% \neq 0\\ \sqrt{d}e^{\frac{j}{2}cdu^{2}}\delta(u^{\prime}-du),&\mbox{if }~{}b=0\end{cases}
  10. [ 1 - z 1 f λ z 0 - λ z 0 z 1 f + λ z 1 - 1 λ f 1 - z 0 f ] \begin{bmatrix}1-\frac{z_{1}}{f}&\lambda z_{0}-\frac{\lambda z_{0}z_{1}}{f}+% \lambda z_{1}\\ -\frac{1}{\lambda f}&1-\frac{z_{0}}{f}\end{bmatrix}

Foldy–Wouthuysen_transformation.html

  1. U = e β α p ^ θ = cos θ + β α p ^ sin θ = e γ p ^ θ = cos θ + γ p ^ sin θ U=e^{\beta\mathbf{\alpha}\cdot\hat{p}\theta}=\cos\theta+\beta\mathbf{\alpha}% \cdot\hat{p}\sin\theta=e^{\mathbf{\gamma}\cdot\hat{p}\theta}=\cos\theta+% \mathbf{\gamma}\cdot\hat{p}\sin\theta
  2. p i ^ p i / | p | \hat{p^{i}}\equiv p^{i}/|p|
  3. U - 1 = e - β α p ^ θ = cos θ - β α p ^ sin θ , U^{-1}=e^{-\beta\mathbf{\alpha}\cdot\hat{p}\theta}=\cos\theta-\beta\mathbf{% \alpha}\cdot\hat{p}\sin\theta\,,
  4. H ^ 0 α p + β m \hat{H}_{0}\equiv\alpha\cdot p+\beta m
  5. H ^ 0 H ^ 0 U H ^ 0 U - 1 = U ( α p + β m ) U - 1 = ( cos θ + β α p ^ sin θ ) ( α p + β m ) ( cos θ - β α p ^ sin θ ) \hat{H}_{0}\to\hat{H}^{\prime}_{0}\equiv U\hat{H}_{0}U^{-1}=U(\alpha\cdot p+% \beta m)U^{-1}=(\cos\theta+\beta\mathbf{\alpha}\cdot\hat{p}\sin\theta)(\alpha% \cdot p+\beta m)(\cos\theta-\beta\mathbf{\alpha}\cdot\hat{p}\sin\theta)
  6. H ^ 0 = ( α p + β m ) ( cos θ - β α p ^ sin θ ) 2 = ( α p + β m ) e - 2 β α p ^ θ = ( α p + β m ) ( cos 2 θ - β α p ^ sin 2 θ ) \hat{H}^{\prime}_{0}=(\alpha\cdot p+\beta m)(\cos\theta-\beta\mathbf{\alpha}% \cdot\hat{p}\sin\theta)^{2}=(\alpha\cdot p+\beta m)e^{-2\beta\mathbf{\alpha}% \cdot\hat{p}\theta}=(\alpha\cdot p+\beta m)(\cos 2\theta-\beta\mathbf{\alpha}% \cdot\hat{p}\sin 2\theta)
  7. H ^ 0 = α p ( cos 2 θ - m | p | sin 2 θ ) + β ( m cos 2 θ + | p | sin 2 θ ) \hat{H}^{\prime}_{0}=\alpha\cdot p\left(\cos 2\theta-\frac{m}{|p|}\sin 2\theta% \right)+\beta(m\cos 2\theta+|p|\sin 2\theta)
  8. H ^ 0 \hat{H}^{\prime}_{0}
  9. H ^ 0 = β ( m cos 2 θ + | p | sin 2 θ ) \hat{H}^{\prime}_{0}=\beta(m\cos 2\theta+|p|\sin 2\theta)
  10. sin 2 θ = | p | / m 2 + | p | 2 \sin 2\theta=|p|/\sqrt{m^{2}+|p|^{2}}
  11. cos 2 θ = m / m 2 + | p | 2 \cos 2\theta=m/\sqrt{m^{2}+|p|^{2}}
  12. H ^ 0 = β m 2 + | p | 2 \hat{H}^{\prime}_{0}=\beta\sqrt{m^{2}+|p|^{2}}
  13. p 0 = m 2 + | p | 2 p^{0}=\sqrt{m^{2}+{|p|}^{2}}
  14. H ^ 0 = β E \hat{H}^{\prime}_{0}=\beta E
  15. O O U O U - 1 = ( ± I ) ( O ) ( ± I ) = O O\to O^{\prime}\equiv UOU^{-1}=(\pm I)(O)(\pm I)=O
  16. H ^ 0 α p + β m \hat{H}_{0}\equiv\alpha\cdot p+\beta m
  17. H ^ 0 = H ^ 0 = β m \hat{H}_{0}=\hat{H}^{\prime}_{0}=\beta m
  18. H ^ 0 \hat{H}_{0}
  19. i i
  20. v i ^ i [ H ^ 0 , x i ] \hat{v_{i}}\equiv i[\hat{H}_{0},x_{i}]
  21. m = γ 0 H ^ 0 + γ j p j m=\gamma^{0}\hat{H}_{0}+\gamma^{j}p_{j}
  22. i i
  23. 0 = [ m , x i ] = [ ( γ 0 H ^ 0 + γ j p j ) , x i ] = [ γ 0 H ^ 0 , x i ] + i γ i 0=[m,x_{i}]=[(\gamma^{0}\hat{H}_{0}+\gamma^{j}p_{j}),x_{i}]=[\gamma^{0}\hat{H}% _{0},x_{i}]+i\gamma_{i}
  24. [ x i , p j ] = - i η i j [x_{i},p_{j}]=-i\eta_{ij}
  25. γ 0 \gamma^{0}
  26. d x i ^ d t = v i ^ i [ H ^ 0 , x i ] = α i \frac{\hat{dx_{i}}}{dt}=\hat{v_{i}}\equiv i[\hat{H}_{0},x_{i}]=\alpha_{i}
  27. i [ H ^ 0 , v ^ i ] 0 i[\hat{H}_{0},\hat{v}_{i}]\neq 0
  28. v i ^ i [ H ^ 0 , x i ] \hat{v_{i}}^{\prime}\equiv i[\hat{H}^{\prime}_{0},x_{i}]
  29. H ^ 0 = β p 0 \hat{H}^{\prime}_{0}=\beta p_{0}
  30. v i ^ i [ H ^ 0 , x i ] = i β [ p 0 , x i ] \hat{v_{i}}^{\prime}\equiv i[\hat{H}^{\prime}_{0},x_{i}]=i\beta[p_{0},x_{i}]
  31. [ p 0 , x i ] [p_{0},x_{i}]
  32. p 0 = m 2 + | p | 2 p^{0}=\sqrt{m^{2}+|p|^{2}}
  33. [ x i , p j ] = - i η i j [x_{i},p_{j}]=-i\eta_{ij}
  34. [ p 0 , x i ] [p_{0},x_{i}]
  35. i [ p 0 , x i ] = p i p 0 = v i i[p_{0},x_{i}]=\frac{p_{i}}{p^{0}}=v_{i}
  36. [ x i , p j ] = - i η i j [x_{i},p_{j}]=-i\eta_{ij}
  37. d x i ^ d t = v i ^ i [ H ^ 0 , x i ] = β p i p 0 = β v i \frac{\hat{dx_{i}}^{\prime}}{dt}=\hat{v_{i}}^{\prime}\equiv i[\hat{H}^{\prime}% _{0},x_{i}]=\beta\frac{p_{i}}{p^{0}}=\beta v_{i}
  38. i [ H ^ 0 , v ^ i ] = i [ β p 0 , β v i ] = 0 i[\hat{H}^{\prime}_{0},\hat{v}_{i}^{\prime}]=i[\beta p_{0},\beta v_{i}]=0
  39. v i ^ i [ H ^ 0 , x i ] = α i \hat{v_{i}}\equiv i[\hat{H}_{0},x_{i}]=\alpha_{i}
  40. v i ^ i [ H ^ 0 , x i ] = β p i p 0 = 0 \hat{v_{i}}^{\prime}\equiv i[\hat{H}^{\prime}_{0},x_{i}]=\beta\frac{p_{i}}{p^{% 0}}=0
  41. v i ^ = v i ^ \hat{v_{i}}^{\prime}=\hat{v_{i}}
  42. H ^ 0 \hat{H}_{0}
  43. ( c α p + β m o c 2 + I V ) Φ = E Φ (c\vec{\alpha}\cdot\vec{p}+\beta m_{o}c^{2}+IV)\Phi=E\Phi
  44. H ^ H ^ 0 + λ H ^ 1 \hat{H}\equiv\hat{H}_{0}+\lambda\hat{H}_{1}
  45. H ^ 0 = c α p + β m o c 2 + 1 2 ( I + β ) V \hat{H}_{0}=c\vec{\alpha}\cdot\vec{p}+\beta m_{o}c^{2}+\frac{1}{2}(I+\beta)V
  46. H ^ 1 = 1 α 2 H ^ 1 = 1 2 α 2 ( I - β ) V \hat{H}_{1}=\frac{1}{\alpha^{2}}\hat{H^{\prime}}_{1}=\frac{1}{2\alpha^{2}}(I-% \beta)V
  47. ϕ 0 = A ( Ψ 1 0 Ψ 2 0 ) \phi^{0}=A\left(\begin{array}[]{c}\Psi_{1}^{0}\\ \Psi_{2}^{0}\end{array}\right)
  48. H ^ 0 \hat{H}_{0}
  49. Ψ 2 0 = c σ p ( E 0 + m 0 c 2 ) Ψ 1 0 \Psi_{2}^{0}=c~{}\frac{\vec{\sigma}\cdot\vec{p}}{(E_{0}+m_{0}c^{2})}\Psi_{1}^{0}
  50. c σ p Ψ 2 0 + V Ψ 1 0 = ϵ 0 Ψ 1 0 c\sigma\cdot\vec{p}~{}\Psi_{2}^{0}+V\Psi_{1}^{0}=\epsilon_{0}\Psi_{1}^{0}
  51. ϵ 0 = E 0 - m 0 c 2 \epsilon_{0}=E_{0}-m_{0}c^{2}
  52. Ψ 2 0 \Psi_{2}^{0}
  53. { p 2 2 m 0 + ( 1 + ϵ 0 2 m 0 c 2 ) ( V - ϵ 0 ) } Ψ 1 0 0. \left\{\frac{p^{2}}{2m_{0}}+\left(1+\frac{\epsilon_{0}}{2m_{0}c^{2}}\right)(V-% \epsilon_{0})\right\}\Psi_{1}^{0}\equiv 0.
  54. E = ( 1 + ϵ 0 2 m 0 c 2 ) ϵ 0 and V = ( 1 + ϵ 0 2 m 0 c 2 ) V E^{\prime}=\left(1+\frac{\epsilon_{0}}{2m_{0}c^{2}}\right)\epsilon_{0}\quad% \mbox{and}~{}\quad V^{\prime}=\left(1+\frac{\epsilon_{0}}{2m_{0}c^{2}}\right)V
  55. H ^ = - ( n 2 ( r ) - p ^ 2 ) 1 / 2 \widehat{H}=-\left(n^{2}(r)-\widehat{p}_{\perp}^{2}\right)^{1/2}
  56. ( p ^ 2 / n 0 2 ) \left({\widehat{p}_{\perp}^{2}}/{n_{0}^{2}}\right)

Fontana_bridge.html

  1. Z P Z_{P}
  2. Z S Z_{S}
  3. I S I_{S}
  4. Z S Z_{S}
  5. Z S Z_{S}
  6. Z 1 Z 3 = Z P Z 2 Z_{1}Z_{3}=Z_{P}Z_{2}
  7. I S = V I N Z P + Z 1 Z P Z 1 I_{S}=V_{IN}\frac{Z_{P}+Z_{1}}{Z_{P}Z_{1}}

Forecast_skill.html

  1. 𝑀𝑆𝐸 = t = 1 N E t 2 N \ \mathit{MSE}=\frac{\sum_{t=1}^{N}{E_{t}^{2}}}{N}
  2. 𝑆𝑆 = 1 - 𝑀𝑆𝐸 forecast 𝑀𝑆𝐸 ref \ \mathit{SS}=1-\frac{\mathit{MSE}\text{forecast}}{\mathit{MSE}\text{ref}}

Forking_lemma.html

  1. frk acc ( acc q - 1 h ) . \,\text{frk}\geq\,\text{acc}\cdot\left(\frac{\,\text{acc}}{q}-\frac{1}{h}% \right).

Formal_grammar.html

  1. S a S b S\rightarrow aSb
  2. S b a S\rightarrow ba
  3. S a S b a a S b b a a b a b b S\Rightarrow aSb\Rightarrow aaSbb\Rightarrow aababb
  4. { a n b a b n | n 0 } = { b a , a b a b , a a b a b b , a a a b a b b b , } \{a^{n}bab^{n}|n\geq 0\}=\{ba,abab,aababb,aaababbb,\ldots\}
  5. a k a^{k}
  6. a a
  7. k k
  8. n n
  9. Σ \Sigma
  10. ( Σ N ) * N ( Σ N ) * ( Σ N ) * (\Sigma\cup N)^{*}N(\Sigma\cup N)^{*}\rightarrow(\Sigma\cup N)^{*}
  11. * {*}
  12. \cup
  13. Λ \Lambda
  14. ϵ \epsilon
  15. S N S\in N
  16. ( N , Σ , P , S ) (N,\Sigma,P,S)
  17. G = ( N , Σ , P , S ) G=(N,\Sigma,P,S)
  18. G \Rightarrow_{G}
  19. ( Σ N ) * (\Sigma\cup N)^{*}
  20. x G y iff u , v , p , q ( Σ N ) * : ( x = u p v ) ( p q P ) ( y = u q v ) x\Rightarrow_{G}y\mbox{ iff }~{}\exists u,v,p,q\in(\Sigma\cup N)^{*}:(x=upv)% \wedge(p\rightarrow q\in P)\wedge(y=uqv)
  21. G * {\Rightarrow_{G}}^{*}
  22. G \Rightarrow_{G}
  23. ( Σ N ) * (\Sigma\cup N)^{*}
  24. S S
  25. { w ( Σ N ) * S G * w } \{w\in(\Sigma\cup N)^{*}\mid S{\Rightarrow_{G}}^{*}w\}
  26. Σ * \Sigma^{*}
  27. G G
  28. s y m b o l L ( G ) symbol{L}(G)
  29. S S
  30. { w Σ * S G * w } \{w\in\Sigma^{*}\mid S{\Rightarrow_{G}}^{*}w\}
  31. G = ( N , Σ , P , S ) G=(N,\Sigma,P,S)
  32. ( N Σ , P ) (N\cup\Sigma,P)
  33. S S
  34. G G
  35. N = { S , B } N=\left\{S,B\right\}
  36. Σ = { a , b , c } \Sigma=\left\{a,b,c\right\}
  37. S S
  38. P P
  39. S a B S c S\rightarrow aBSc
  40. S a b c S\rightarrow abc
  41. B a a B Ba\rightarrow aB
  42. B b b b Bb\rightarrow bb
  43. L ( G ) = { a n b n c n | n 1 } L(G)=\left\{a^{n}b^{n}c^{n}|n\geq 1\right\}
  44. a n a^{n}
  45. a a
  46. a a
  47. b b
  48. c c
  49. L ( G ) L(G)
  50. s y m b o l S 2 s y m b o l a b c symbol{S}\Rightarrow_{2}symbol{abc}
  51. s y m b o l S 1 s y m b o l a B S c 2 a B s y m b o l a b c c 3 a s y m b o l a B b c c 4 a a s y m b o l b b c c symbol{S}\Rightarrow_{1}symbol{aBSc}\Rightarrow_{2}aBsymbol{abc}c\Rightarrow_{% 3}asymbol{aB}bcc\Rightarrow_{4}aasymbol{bb}cc
  52. s y m b o l S 1 s y m b o l a B S c 1 a B s y m b o l a B S c c 2 a B a B s y m b o l a b c c c 3 a s y m b o l a B B a b c c c 3 a a B s y m b o l a B b c c c symbol{S}\Rightarrow_{1}symbol{aBSc}\Rightarrow_{1}aBsymbol{aBSc}c\Rightarrow_% {2}aBaBsymbol{abc}cc\Rightarrow_{3}asymbol{aB}Babccc\Rightarrow_{3}aaBsymbol{% aB}bccc
  53. 3 a a s y m b o l a B B b c c c 4 a a a B s y m b o l b b c c c 4 a a a s y m b o l b b b c c c \Rightarrow_{3}aasymbol{aB}Bbccc\Rightarrow_{4}aaaBsymbol{bb}ccc\Rightarrow_{4% }aaasymbol{bb}bccc
  54. P i Q P\Rightarrow_{i}Q
  55. L ( G ) = { a n b n c n | n 1 } L(G)=\left\{a^{n}b^{n}c^{n}|n\geq 1\right\}
  56. { a n b n | n 1 } \left\{a^{n}b^{n}|n\geq 1\right\}
  57. a a
  58. b b
  59. G 2 G_{2}
  60. N = { S } N=\left\{S\right\}
  61. Σ = { a , b } \Sigma=\left\{a,b\right\}
  62. S S
  63. S a S b S\rightarrow aSb
  64. S a b S\rightarrow ab
  65. O ( n 3 ) O(n^{3})
  66. O ( n 3 ) O(n^{3})
  67. n n
  68. { a n b n | n 1 } \left\{a^{n}b^{n}|n\geq 1\right\}
  69. { a n b m | m , n 1 } \left\{a^{n}b^{m}\,|\,m,n\geq 1\right\}
  70. a a
  71. b b
  72. G 3 G_{3}
  73. N = { S , A , B } N=\left\{S,A,B\right\}
  74. Σ = { a , b } \Sigma=\left\{a,b\right\}
  75. S S
  76. S a A S\rightarrow aA
  77. A a A A\rightarrow aA
  78. A b B A\rightarrow bB
  79. B b B B\rightarrow bB
  80. B ϵ B\rightarrow\epsilon

Formal_scheme.html

  1. 𝒥 \mathcal{J}
  2. 𝒥 n V \mathcal{J}^{n}\subseteq V
  3. 𝒥 \mathcal{J}
  4. 𝒥 \mathcal{J}
  5. A / 𝒥 A/\mathcal{J}
  6. 𝒥 λ \mathcal{J}_{\lambda}
  7. A / 𝒥 λ A/\mathcal{J}_{\lambda}
  8. lim λ 𝒪 Spec A / 𝒥 λ \underleftarrow{\lim}_{\lambda}\mathcal{O}_{\,\text{Spec}A/\mathcal{J}_{% \lambda}}
  9. 𝒪 Spf A ( D f ) = A f ^ \mathcal{O}_{\,\text{Spf}A}(D_{f})=\widehat{A_{f}}
  10. A f ^ \widehat{A_{f}}
  11. ( 𝔛 , 𝒪 𝔛 ) (\mathfrak{X},\mathcal{O}_{\mathfrak{X}})
  12. 𝔛 \mathfrak{X}
  13. f : 𝔛 𝔜 f:\mathfrak{X}\to\mathfrak{Y}
  14. f # : Γ ( U , 𝒪 𝔜 ) Γ ( f - 1 ( U ) , 𝒪 𝔛 ) f^{\#}:\Gamma(U,\mathcal{O}_{\mathfrak{Y}})\to\Gamma(f^{-1}(U),\mathcal{O}_{% \mathfrak{X}})
  15. 𝔛 \mathfrak{X}
  16. 𝔜 \mathfrak{Y}
  17. \mathcal{I}
  18. f * ( ) 𝒪 𝔛 f^{*}(\mathcal{I})\mathcal{O}_{\mathfrak{X}}
  19. 𝔛 \mathfrak{X}

Formulas_for_generating_Pythagorean_triples.html

  1. 1 , 3 , 5 , 7 , 9 , 11 , 1,3,5,7,9,11,\ldots
  2. n n
  3. n 2 n^{2}
  4. k k
  5. n n
  6. n = ( k + 1 ) / 2 n=(k+1)/2
  7. k k
  8. k = a 2 k=a^{2}
  9. n n
  10. b 2 b^{2}
  11. n - 1 n-1
  12. c 2 c^{2}
  13. n n
  14. a 2 + b 2 = c 2 a^{2}+b^{2}=c^{2}
  15. k = 9 = 3 2 = a 2 k=9=3^{2}=a^{2}
  16. 5 = n = ( a 2 + 1 ) / 2 5=n=(a^{2}+1)/2
  17. b 2 = 4 2 b^{2}=4^{2}
  18. n = 5 n=5
  19. c 2 = 5 2 c^{2}=5^{2}
  20. a 2 + b 2 = c 2 a^{2}+b^{2}=c^{2}
  21. 1 1 3 , 2 2 5 , 3 3 7 , 4 4 9 , 1\tfrac{1}{3},\,\text{ }2\tfrac{2}{5},\,\text{ }3\tfrac{3}{7},\,\text{ }4% \tfrac{4}{9},\ldots
  22. 3 , 5 , 7 , 9 , 3,\,\text{ }5,\,\text{ }7,\,\text{ }9,
  23. 3 3 7 3\tfrac{3}{7}
  24. 24 7 \tfrac{24}{7}
  25. 1 1 3 yields [ 3 , 4 , 5 ] , 2 2 5 yields [ 5 , 12 , 13 ] , 3 3 7 yields [ 7 , 24 , 25 ] , 4 4 9 yields [ 9 , 40 , 41 ] , 1\tfrac{1}{3}\,\text{ }\xrightarrow{\,\text{yields}}\,\text{ }[3,4,5],\,\text{% 2}\tfrac{2}{5}\,\text{ }\xrightarrow{\,\text{yields}}\,\text{ }[5,12,13],\,% \text{ 3}\tfrac{3}{7}\,\text{ }\xrightarrow{\,\text{yields}}\,\text{ }[7,24,25% ],\,\text{ 4}\tfrac{4}{9}\,\text{ }\xrightarrow{\,\text{yields}}\,\text{ }[9,4% 0,41],\,\text{ }\ldots
  26. 1 7 8 , 2 11 12 , 3 15 16 , 4 19 20 , 1\tfrac{7}{8},\,\text{ }2\tfrac{11}{12},\,\text{ }3\tfrac{15}{16},\,\text{ }4% \tfrac{19}{20},\ldots
  27. n + 4 n + 3 4 n + 4 n+\tfrac{4n+3}{4n+4}
  28. 1 7 8 yields [ 15 , 8 , 17 ] , 2 11 12 yields [ 35 , 12 , 37 ] , 3 15 16 yields [ 63 , 16 , 65 ] , 4 19 20 yields [ 99 , 20 , 101 ] , 1\tfrac{7}{8}\xrightarrow{\,\text{yields}}[15,8,17],2\tfrac{11}{12}% \xrightarrow{\,\text{yields}}[35,12,37],3\tfrac{15}{16}\xrightarrow{\,\text{% yields}}[63,16,65],4\tfrac{19}{20}\xrightarrow{\,\text{yields}}[99,20,101],\ldots
  29. P l a t o : c - b = 1 , P y t h a g o r a s : c - a = 2 , F e r m a t : | a - b | = 1 Plato:c-b=1,\quad\quad Pythagoras:c-a=2,\quad\quad Fermat:\left|a-b\right|=1
  30. x 2 + y 2 = z 2 x^{2}+y^{2}=z^{2}
  31. r 2 = 2 s t r^{2}=2st
  32. x = r + s , y = r + t , z = r + s + t . x=r+s\,,\,y=r+t\,,\,z=r+s+t.
  33. r r
  34. r 2 2 \tfrac{r^{2}}{2}
  35. r 2 2 = 18 \tfrac{r^{2}}{2}=18
  36. ( a n , b n , c n ) = ( a n - 1 + b n - 1 + c n - 1 , F 2 n - 1 - b n - 1 , F 2 n ) (a_{n},b_{n},c_{n})=(a_{n-1}+b_{n-1}+c_{n-1},\,F_{2n-1}-b_{n-1},\,F_{2n})
  37. ( 2 h n + 1 h n + 2 , h n h n + 3 , 2 h n + 1 h n + 2 + h n 2 ) (2h_{n+1}h_{n+2},h_{n}h_{n+3},2h_{n+1}h_{n+2}+h_{n}^{2})
  38. [ q q ] \left[{\begin{array}[]{*{20}c}q&{q^{\prime}}\\ \bullet&\bullet\end{array}}\right]
  39. q + q = p q + p = p [ q q p p ] \begin{array}[]{*{20}c}q^{\prime}+q=p\\ q+p=p^{\prime}\end{array}\to\left[{\begin{array}[]{*{20}c}q&q^{\prime}\\ p&p^{\prime}\end{array}}\right]
  40. a = 2 q p b = q p c = p p - q q = q p + q p radii ( r 1 = q q , r 2 = q p , r 3 = q p , r 4 = p p ) A = q q p p P = r 1 + r 2 + r 3 + r 4 \begin{array}[]{l}a=2qp\\ b=q^{\prime}p^{\prime}\\ c=pp^{\prime}-qq^{\prime}=qp^{\prime}+q^{\prime}p\\ \\ \,\text{radii}\to(r_{1}=qq^{\prime},r_{2}=qp^{\prime},r_{3}=q^{\prime}p,r_{4}=% pp^{\prime})\\ A=qq^{\prime}pp^{\prime}\\ P=r_{1}+r_{2}+r_{3}+r_{4}\end{array}
  41. [ 2 9 ] [ 2 9 11 13 ] \left[{\begin{array}[]{*{20}c}2&9\\ \bullet&\bullet\end{array}}\right]\to\left[{\begin{array}[]{*{20}c}2&9\\ 11&13\end{array}}\right]
  42. a = 2 ( 22 ) = 44 b = 117 c = ( 143 - 18 ) = ( 26 + 99 ) = 125 radii ( r 1 = 18 , r 2 = 26 , r 3 = 99 , r 4 = 143 ) A = ( 18 ) ( 143 ) = 2574 P = ( 18 + 26 + 99 + 143 ) = 286 \begin{array}[]{l}a=2(22)=44\\ b=117\\ c=(143-18)=(26+99)=125\\ \\ \,\text{radii}\to(r_{1}=18,\quad r_{2}=26,\quad r_{3}=99,\quad r_{4}=143)\\ A=(18)(143)=2574\\ P=(18+26+99+143)=286\end{array}
  43. ( k 1 + k 2 + k 3 + k 4 ) 2 = 2 ( k 1 2 + k 2 2 + k 3 2 + k 4 2 ) , \left(k_{1}+k_{2}+k_{3}+k_{4}\right)^{2}=2\left(k_{1}^{2}+k_{2}^{2}+k_{3}^{2}+% k_{4}^{2}\right),
  44. [ x y ] \left[{\begin{array}[]{*{20}{c}}\bullet&x\\ \bullet&y\end{array}}\right]
  45. [ x y ] , [ x y ] , [ y x ] \left[{\begin{array}[]{*{20}{c}}\bullet&x\\ y&\bullet\end{array}}\right],\left[{\begin{array}[]{*{20}{c}}x&y\\ \bullet&\bullet\end{array}}\right],\left[{\begin{array}[]{*{20}{c}}y&x\\ \bullet&\bullet\end{array}}\right]
  46. [ 1 1 2 3 ] parent \left[{\begin{array}[]{*{20}{c}}1&1\\ 2&3\end{array}}\right]\leftarrow\,\text{parent}
  47. [ 2 1 3 5 ] , [ 1 3 4 5 ] , [ 3 1 4 7 ] children \left[{\begin{array}[]{*{20}{c}}2&1\\ 3&5\end{array}}\right],\left[{\begin{array}[]{*{20}{c}}1&3\\ 4&5\end{array}}\right],\left[{\begin{array}[]{*{20}{c}}3&1\\ 4&7\end{array}}\right]\leftarrow\,\text{children}
  48. m 1 = ( 4 x + m ) m_{1}=(4x+m)
  49. n 1 = ( x + n ) n_{1}=(x+n)
  50. Side A \displaystyle\,\text{Side }A
  51. [ a , b , c ] [\,\text{ }a,\,\text{ }b,\,\text{ }c]
  52. a a
  53. [ a 1 , b 1 , c 1 ] , [ a 2 , b 2 , c 2 ] , [ a 3 , b 3 , c 3 ] [{{a}_{1}},{{b}_{1}},{{c}_{1}}],\,\text{ }[{{a}_{2}},{{b}_{2}},{{c}_{2}}],\,% \text{ }[{{a}_{3}},{{b}_{3}},{{c}_{3}}]
  54. [ a , b , c ] [\,\text{ }a,\,\text{ }b,\,\text{ }c]
  55. [ a , b , c ] [\,\text{ }a,\,\text{ }b,\,\text{ }c]
  56. [ 3 , 4 , 5 ] [\,\text{ }3,\,\text{ }4,\,\text{ }5]
  57. [ a , b , c ] [\,\text{ }a,\,\text{ }b,\,\text{ }c]
  58. [ - 1 2 2 - 2 1 2 - 2 2 3 ] 𝐴 [ a b c ] = [ a 1 b 1 c 1 ] , [ 1 2 2 2 1 2 2 2 3 ] 𝐵 [ a b c ] = [ a 2 b 2 c 2 ] , [ 1 - 2 2 2 - 1 2 2 - 2 3 ] 𝐶 [ a b c ] = [ a 3 b 3 c 3 ] \overset{A}{\mathop{\left[\begin{matrix}-1&2&2\\ -2&1&2\\ -2&2&3\\ \end{matrix}\right]}}\left[\begin{matrix}a\\ b\\ c\\ \end{matrix}\right]=\left[\begin{matrix}a_{1}\\ b_{1}\\ c_{1}\\ \end{matrix}\right],\quad\,\text{ }\overset{B}{\mathop{\left[\begin{matrix}1&2% &2\\ 2&1&2\\ 2&2&3\\ \end{matrix}\right]}}\left[\begin{matrix}a\\ b\\ c\\ \end{matrix}\right]=\left[\begin{matrix}a_{2}\\ b_{2}\\ c_{2}\end{matrix}\right],\quad\,\text{ }\overset{C}{\mathop{\left[\begin{% matrix}1&-2&2\\ 2&-1&2\\ 2&-2&3\end{matrix}\right]}}\left[\begin{matrix}a\\ b\\ c\end{matrix}\right]=\left[\begin{matrix}a_{3}\\ b_{3}\\ c_{3}\end{matrix}\right]
  59. - a + 2 b + 2 c = a 1 - 2 a + b + 2 c = b 1 - 2 a + 2 b + 3 c = c 1 [ a 1 , b 1 , c 1 ] \displaystyle\begin{matrix}-a+2b+2c=a_{1}&-2a+b+2c=b_{1}&-2a+2b+3c=c_{1}&\quad% \to\left[\,\text{ }a_{1},\,\text{ }b_{1},\,\text{ }c_{1}\right]\\ \end{matrix}
  60. [ 2 1 - 1 - 2 2 2 - 2 1 3 ] A [ a b c ] = [ a 1 b 1 c 1 ] , [ 2 1 1 2 - 2 2 2 - 1 3 ] B [ a b c ] = [ a 2 b 2 c 2 ] , [ 2 - 1 1 2 2 2 2 1 3 ] C [ a b c ] = [ a 3 b 3 c 3 ] \overset{{{A}^{\prime}}}{\mathop{\left[\begin{matrix}2&1&-1\\ -2&2&2\\ -2&1&3\end{matrix}\right]}}\left[\begin{matrix}a\\ b\\ c\end{matrix}\right]=\left[\begin{matrix}a_{1}\\ b_{1}\\ c_{1}\end{matrix}\right],\quad\,\text{ }\overset{{{B}^{\prime}}}{\mathop{\left% [\begin{matrix}2&1&1\\ 2&-2&2\\ 2&-1&3\end{matrix}\right]}}\left[\begin{matrix}a\\ b\\ c\\ \end{matrix}\right]=\left[\begin{matrix}a_{2}\\ b_{2}\\ c_{2}\end{matrix}\right],\quad\,\text{ }\overset{{{C}^{\prime}}}{\mathop{\left% [\begin{matrix}2&-1&1\\ 2&2&2\\ 2&1&3\\ \end{matrix}\right]}}\left[\begin{matrix}a\\ b\\ c\\ \end{matrix}\right]=\left[\begin{matrix}a_{3}\\ b_{3}\\ c_{3}\end{matrix}\right]
  61. + 2 a + b - c = a 1 - 2 a + 2 b + 2 c = b 1 - 2 a + b + 3 c = c 1 [ a 1 , b 1 , c 1 ] \displaystyle\begin{matrix}+2a+b-c=a_{1}&-2a+2b+2c=b_{1}&-2a+b+3c=c_{1}&\quad% \to\left[\,\text{ }a_{1},\,\text{ }b_{1},\,\text{ }c_{1}\right]\end{matrix}
  62. [ 5 , 12 , 13 ] A B C [ 45 , 28 , 53 ] [ 55,48,73 ] [ 7,24,25 ] [ 5 , 12 , 13 ] A B C [ 9 , 40 , 41 ] [ 35,12,37 ] [ 11,60,61 ] \begin{matrix}&\left[\,\text{5},12,13\right]&\\ A&B&C\\ \left[45,28,53\right]&\left[\,\text{55,48,73}\right]&\left[\,\text{7,24,25}% \right]\end{matrix}\quad\quad\quad\quad\quad\quad\begin{matrix}&\left[\,\text{% 5},12,13\right]&\\ {{A}^{\prime}}&{{B}^{\prime}}&{{C}^{\prime}}\\ \left[9,40,41\right]&\left[\,\text{35,12,37}\right]&\left[\,\text{11,60,61}% \right]\end{matrix}
  63. b + 1 = c b+1=c
  64. ( 3 , 4 , 5 ) \left(3,4,5\right)
  65. 6 × ( 1 2 ) 6\times(1^{2})
  66. 5 2 \tfrac{5}{2}
  67. ( 5 , 12 , 13 ) \left(5,12,13\right)
  68. 6 × ( 1 2 + 2 2 ) 6\times(1^{2}+2^{2})
  69. 13 2 \tfrac{13}{2}
  70. ( 7 , 24 , 25 ) \left(7,24,25\right)
  71. 6 × ( 1 2 + 2 2 + 3 2 ) 6\times(1^{2}+2^{2}+3^{2})
  72. 25 2 \tfrac{25}{2}
  73. ( a , a 2 - 1 2 , a 2 + 1 2 ) \left(a,\tfrac{a^{2}-1}{2},\tfrac{a^{2}+1}{2}\right)
  74. 6 × [ 1 2 + 2 2 + + ( a - 1 2 ) 2 ] 6\times\left[1^{2}+2^{2}+\cdots+\left(\tfrac{a-1}{2}\right)^{2}\right]
  75. ( a - 1 2 ) \left(\tfrac{a-1}{2}\right)
  76. c 2 \tfrac{c}{2}
  77. k > h 2 d : k>\frac{h\sqrt{2}}{d}:
  78. ( h + d k , d k + ( d k ) 2 2 h , h + d k + ( d k ) 2 2 h ) . (h+dk,dk+\frac{(dk)^{2}}{2h},h+dk+\frac{(dk)^{2}}{2h}).

FOSD_origami.html

  1. U W = U × W = [ U W 11 U W 12 U W 1 n U W m 1 U W m 2 U W m n ] UW=U\times W=\begin{bmatrix}UW_{11}&UW_{12}&\cdots&UW_{1n}\\ \vdots&\vdots&\ddots&\vdots\\ UW_{m1}&UW_{m2}&\cdots&UW_{mn}\end{bmatrix}
  2. T L = T × L = [ P B P G P S H B H G H S D B D G D S J B J G J S ] TL=T\times L=\begin{bmatrix}PB&PG&PS\\ HB&HG&HS\\ DB&DG&DS\\ JB&JG&JS\end{bmatrix}
  3. F = U 1 U 2 U 4 = i = 1 , 2 , 4 U i F=U_{1}\cdot U_{2}\cdot U_{4}=\sum_{i=1,2,4}U_{i}
  4. F = W 1 W 3 = j = 1 , 3 W i F=W_{1}\cdot W_{3}=\sum_{j=1,3}W_{i}
  5. F = U W 11 U W 21 U W 33 = i = 1 , 2 , 3 j = 1 , 3 U W i , j = j = 1 , 3 i = 1 , 2 , 3 U W i , j F=UW_{11}\cdot UW_{21}\cdot...\cdot UW_{33}=\sum_{i=1,2,3}\sum_{j=1,3}UW_{i,j}% =\sum_{j=1,3}\sum_{i=1,2,3}UW_{i,j}
  6. A = G 1 × × G n = k = 1 n G k A=G_{1}\times...\times G_{n}=\prod_{k=1}^{n}G_{k}
  7. H = i 1 i 2 i n G i 1 , i 2 i n H=\sum_{i_{1}}\sum_{i_{2}}...\sum_{i_{n}}G_{i_{1},i_{2}...i_{n}}
  8. E = i = 2 , 0 j = 2 , 0 T L i , j = j = 2 , 0 i = 2 , 0 T L i , j E=\sum_{i=2,0}\sum_{j=2,0}TL_{i,j}=\sum_{j=2,0}\sum_{i=2,0}TL_{i,j}

Foster's_theorem.html

  1. V ( i ) - i S V(i)\geq-\infty\,\text{ }\forall\,\text{ }i\in S
  2. j S p i j V ( j ) < \sum_{j\in S}p_{ij}V(j)<{\infty}
  3. i F i\in F
  4. j S p i j V ( j ) V ( i ) - ϵ \sum_{j\in S}p_{ij}V(j)\leq V(i)-\epsilon
  5. i F i\notin F

Foster–Greer–Thorbecke_indices.html

  1. F G T α = 1 N i = 1 H ( z - y i z ) α FGT_{\alpha}=\frac{1}{N}\sum_{i=1}^{H}(\frac{z-y_{i}}{z})^{\alpha}
  2. α \alpha
  3. F G T 0 = H N FGT_{0}=\frac{H}{N}
  4. F G T 1 = 1 N i = 1 H ( z - y i z ) FGT_{1}=\frac{1}{N}\sum_{i=1}^{H}(\frac{z-y_{i}}{z})
  5. F G T 2 = 1 N i = 1 H ( z - y i z ) 2 FGT_{2}=\frac{1}{N}\sum_{i=1}^{H}(\frac{z-y_{i}}{z})^{2}
  6. F G T 2 = H μ 2 + ( 1 - μ 2 ) C v 2 FGT_{2}=H\mu^{2}+(1-\mu^{2})C_{v}^{2}
  7. μ = 1 H i = 1 H ( z - y i z ) \mu=\frac{1}{H}\sum_{i=1}^{H}(\frac{z-y_{i}}{z})

Foundations_of_geometry.html

  1. \angle
  2. \angle
  3. \angle
  4. \angle
  5. \angle
  6. \angle
  7. \angle
  8. \ell
  9. \ell
  10. \ell
  11. \ell
  12. \ell
  13. \ell

Fourier_division.html

  1. c a = c 1 , c 2 , c 3 , c 4 , c 5 a 1 , a 2 , a 3 , a 4 , a 5 = b 1 , b 2 , b 3 , b 4 , b 5 = b \frac{c}{a}=\frac{c_{1},c_{2},c_{3},c_{4},c_{5}\dots}{a_{1},a_{2},a_{3},a_{4},% a_{5}\dots}=b_{1},b_{2},b_{3},b_{4},b_{5}\dots=b
  2. b 1 = c 1 , c 2 a 1 with remainder r 1 b_{1}=\frac{c_{1},c_{2}}{a_{1}}\mbox{ with remainder }~{}r_{1}
  3. b 2 = r 1 , c 3 - b 1 × a 2 a 1 with remainder r 2 b_{2}=\frac{r_{1},c_{3}-b_{1}\times a_{2}}{a_{1}}\mbox{ with remainder }~{}r_{2}
  4. b 3 = r 2 , c 4 - b 2 × a 2 - b 1 × a 3 a 1 with remainder r 3 b_{3}=\frac{r_{2},c_{4}-b_{2}\times a_{2}-b_{1}\times a_{3}}{a_{1}}\mbox{ with% remainder }~{}r_{3}
  5. b 4 = r 3 , c 5 - b 3 × a 2 - b 2 × a 3 - b 1 × a 4 a 1 with remainder r 4 b_{4}=\frac{r_{3},c_{5}-b_{3}\times a_{2}-b_{2}\times a_{3}-b_{1}\times a_{4}}% {a_{1}}\mbox{ with remainder }~{}r_{4}\dots
  6. b i = r i - 1 , c i + 1 - j = 2 i b i - j + 1 × a j a 1 with remainder r i b_{i}=\frac{r_{i-1},c_{i+1}-\textstyle\sum_{j=2}^{i}b_{i-j+1}\times a_{j}}{a_{% 1}}\mbox{ with remainder }~{}r_{i}
  7. b 1 , b_{1},
  8. B 1 = b 1 B_{1}=b_{1}
  9. B i = 100 b i - 1 + b i B_{i}=100b_{i-1}+b_{i}
  10. 1 π = 10 , 00 , 00 31 , 41 , 59 = b 1 , b 2 , b 3 = b \frac{1}{\pi}=\frac{10,00,00\dots}{31,41,59\dots}=b_{1},b_{2},b_{3}\dots=b
  11. b 1 = 10 , 00 31 = 32 with remainder 8 b_{1}=\frac{10,00}{31}=32\mbox{ with remainder }~{}8
  12. b 2 = 8 , 00 - 32 × 41 31 = - 512 31 = - 17 with remainder 15 b_{2}=\frac{8,00-32\times 41}{31}=\frac{-512}{31}=-17\mbox{ with remainder }~{% }15
  13. b 3 = 15 , 00 + 17 × 41 - 32 × 59 31 = 309 31 = 10 with remainder - 1. b_{3}=\frac{15,00+17\times 41-32\times 59}{31}=\frac{309}{31}=10\mbox{ with % remainder }~{}-1.

Fourier_sine_and_cosine_series.html

  1. [ 0 , L ] [0,L]
  2. n = 1 c n sin n π x L \sum_{n=1}^{\infty}c_{n}\sin\frac{n\pi x}{L}
  3. c n = 2 L 0 L f ( x ) sin n π x L d x , n c_{n}=\frac{2}{L}\int_{0}^{L}f(x)\sin\frac{n\pi x}{L}\,dx,n\in\mathbb{N}
  4. f ( 0 ) = f ( L ) = 0 f(0)=f(L)=0
  5. [ 0 , L ] [0,L]
  6. 2 L 2L
  7. c 0 2 + n = 1 c n cos n π x L \frac{c_{0}}{2}+\sum_{n=1}^{\infty}c_{n}\cos\frac{n\pi x}{L}
  8. c n = 2 L 0 L f ( x ) cos n π x L d x , n 0 c_{n}=\frac{2}{L}\int_{0}^{L}f(x)\cos\frac{n\pi x}{L}\,dx,n\in\mathbb{N}_{0}
  9. [ 0 , L ] [0,L]
  10. 2 L 2L

Fourier_transform_on_finite_groups.html

  1. f : G f:G\rightarrow\mathbb{C}\,
  2. ϱ : G G L ( d ϱ , ) \varrho:G\rightarrow GL(d_{\varrho},\mathbb{C})\,
  3. G G\,
  4. f ^ ( ϱ ) = a G f ( a ) ϱ ( a ) . \widehat{f}(\varrho)=\sum_{a\in G}f(a)\varrho(a).
  5. ϱ \varrho\,
  6. G G\,
  7. f ^ ( ϱ ) \widehat{f}(\varrho)\,
  8. d ϱ × d ϱ d_{\varrho}\times d_{\varrho}\,
  9. d ϱ d_{\varrho}\,
  10. ϱ \varrho\,
  11. ϱ i \varrho_{i}\,
  12. G G
  13. ϱ i \varrho_{i}
  14. G G
  15. | G | |G|
  16. i d ϱ i 2 = | G | \sum_{i}d_{\varrho_{i}}^{2}=|G|
  17. a a\,
  18. G G\,
  19. f ( a ) = 1 | G | i d ϱ i Tr ( ϱ i ( a - 1 ) f ^ ( ϱ i ) ) . f(a)=\frac{1}{|G|}\sum_{i}d_{\varrho_{i}}\,\text{Tr}\left(\varrho_{i}(a^{-1})% \widehat{f}(\varrho_{i})\right).
  20. f , g : G f,g:G\rightarrow\mathbb{C}\,
  21. ( f g ) ( a ) = b G f ( a b - 1 ) g ( b ) . (f\ast g)(a)=\sum_{b\in G}f(ab^{-1})g(b).
  22. ϱ \varrho\,
  23. G G\,
  24. f g ^ ( ϱ ) = f ^ ( ϱ ) g ^ ( ϱ ) . \widehat{f\ast g}(\varrho)=\widehat{f}(\varrho)\widehat{g}(\varrho).
  25. f , g : G f,g:G\rightarrow\mathbb{C}\,
  26. a G f ( a - 1 ) g ( a ) = 1 | G | i d ϱ i Tr ( f ^ ( ϱ i ) g ^ ( ϱ i ) ) , \sum_{a\in G}f(a^{-1})g(a)=\frac{1}{|G|}\sum_{i}d_{\varrho_{i}}\,\text{Tr}% \left(\widehat{f}(\varrho_{i})\widehat{g}(\varrho_{i})\right),
  27. ϱ i \varrho_{i}\,
  28. G . G.\,
  29. f ^ ( s ) = a G f ( a ) χ s ¯ ( a ) . \widehat{f}(s)=\sum_{a\in G}f(a)\bar{\chi_{s}}(a).
  30. f , χ s \langle f,\chi_{s}\rangle
  31. G G\,
  32. \mathbb{C}\,
  33. f , g = a G f ( a ) g ¯ ( a ) . \langle f,g\rangle=\sum_{a\in G}f(a)\bar{g}(a).
  34. f ( a ) = 1 | G | s G f ^ ( s ) χ s ( a ) . f(a)=\frac{1}{|G|}\sum_{s\in G}\widehat{f}(s)\chi_{s}(a).
  35. δ a , 0 , \delta_{a,0},\,
  36. δ i , j \delta_{i,j}\,

Fourt–Woodlock_equation.html

  1. V = ( H H T R T U ) + ( H H T R M R R R R U ) V=(HH\cdot TR\cdot TU)+(HH\cdot TR\cdot MR\cdot RR\cdot RU)

Fractional_coordinates.html

  1. [ x y z ] = [ a b cos ( γ ) c cos ( β ) 0 b sin ( γ ) c cos ( α ) - cos ( β ) cos ( γ ) sin ( γ ) 0 0 c v sin ( γ ) ] [ a ^ b ^ c ^ ] \begin{bmatrix}x\\ y\\ z\\ \end{bmatrix}=\begin{bmatrix}a&b\cos(\gamma)&c\cos(\beta)\\ 0&b\sin(\gamma)&c\frac{\cos(\alpha)-\cos(\beta)\cos(\gamma)}{\sin(\gamma)}\\ 0&0&c\frac{v}{\sin(\gamma)}\\ \end{bmatrix}\begin{bmatrix}\hat{a}\\ \hat{b}\\ \hat{c}\\ \end{bmatrix}
  2. v v
  3. v = 1 - cos 2 ( α ) - cos 2 ( β ) - cos 2 ( γ ) + 2 cos ( α ) cos ( β ) cos ( γ ) v=\sqrt{1-\cos^{2}(\alpha)-\cos^{2}(\beta)-\cos^{2}(\gamma)+2\cos(\alpha)\cos(% \beta)\cos(\gamma)}
  4. x = a x f r a c + c z f r a c cos ( β ) x=a\,x_{frac}+c\,z_{frac}\,\cos(\beta)
  5. y = b y f r a c y=b\,y_{frac}
  6. z = c v z f r a c = c z f r a c sin ( β ) z=c\,v\,z_{frac}=c\,z_{frac}\,\sin(\beta)
  7. [ 𝐚 ^ 𝐛 ^ 𝐜 ^ ] = [ 𝟏 𝐚 - cos ( γ ) 𝐚 sin ( γ ) cos ( α ) cos ( γ ) - cos ( β ) 𝐚𝐯 sin ( γ ) 𝟎 𝟏 𝐛 sin ( γ ) cos ( β ) cos ( γ ) - cos ( α ) 𝐛𝐯 sin ( γ ) 𝟎 𝟎 sin ( γ ) 𝐜𝐯 ] [ x y z ] \mathbf{\begin{bmatrix}\hat{a}\\ \hat{b}\\ \hat{c}\\ \end{bmatrix}=\begin{bmatrix}\frac{1}{a}&-\frac{\cos(\gamma)}{a\sin(\gamma)}&% \frac{\cos(\alpha)\cos(\gamma)-\cos(\beta)}{av\sin(\gamma)}\\ 0&\frac{1}{b\sin(\gamma)}&\frac{\cos(\beta)\cos(\gamma)-\cos(\alpha)}{bv\sin(% \gamma)}\\ 0&0&\frac{\sin(\gamma)}{cv}\\ \end{bmatrix}}\begin{bmatrix}x\\ y\\ z\\ \end{bmatrix}

Fractional_quantum_mechanics.html

  1. i ψ ( 𝐫 , t ) t = D α ( - 2 Δ ) α / 2 ψ ( 𝐫 , t ) + V ( 𝐫 , t ) ψ ( 𝐫 , t ) i\hbar\frac{\partial\psi(\mathbf{r},t)}{\partial t}=D_{\alpha}(-\hbar^{2}% \Delta)^{\alpha/2}\psi(\mathbf{r},t)+V(\mathbf{r},t)\psi(\mathbf{r},t)\,
  2. ( - 2 Δ ) α / 2 ψ ( 𝐫 , t ) = 1 ( 2 π ) 3 d 3 p e i 𝐩 𝐫 / | 𝐩 | α φ ( 𝐩 , t ) , (-\hbar^{2}\Delta)^{\alpha/2}\psi(\mathbf{r},t)=\frac{1}{(2\pi\hbar)^{3}}\int d% ^{3}pe^{i\mathbf{p}\cdot\mathbf{r}/\hbar}|\mathbf{p}|^{\alpha}\varphi(\mathbf{% p},t),
  3. ψ ( 𝐫 , t ) \psi(\mathbf{r},t)
  4. φ ( 𝐩 , t ) \varphi(\mathbf{p},t)
  5. ψ ( 𝐫 , t ) = 1 ( 2 π ) 3 d 3 p e i 𝐩 𝐫 / φ ( 𝐩 , t ) , φ ( 𝐩 , t ) = d 3 r e - i 𝐩 𝐫 / ψ ( 𝐫 , t ) . \psi(\mathbf{r},t)=\frac{1}{(2\pi\hbar)^{3}}\int d^{3}pe^{i\mathbf{p}\cdot% \mathbf{r}/\hbar}\varphi(\mathbf{p},t),\qquad\varphi(\mathbf{p},t)=\int d^{3}% re^{-i\mathbf{p}\cdot\mathbf{r}/\hbar}\psi(\mathbf{r},t).

Free_convolution.html

  1. μ \mu
  2. ν \nu
  3. X X
  4. μ \mu
  5. Y Y
  6. ν \nu
  7. X X
  8. Y Y
  9. μ ν \mu\boxplus\nu
  10. X + Y X+Y
  11. A A
  12. B B
  13. n n
  14. n n
  15. A A
  16. B B
  17. μ \mu
  18. ν \nu
  19. n n
  20. A + B A+B
  21. μ ν \mu\boxplus\nu
  22. μ ν \mu\boxplus\nu
  23. μ \mu
  24. ν \nu
  25. c c
  26. c \boxplus_{c}
  27. c [ 0 , 1 ] c\in[0,1]
  28. A A
  29. B B
  30. n n
  31. p p
  32. A A
  33. B B
  34. μ \mu
  35. ν \nu
  36. n n
  37. p p
  38. n / p n/p
  39. c c
  40. A + B A+B
  41. μ c ν \mu\boxplus_{c}\nu
  42. μ c ν \mu\boxplus_{c}\nu
  43. c c
  44. μ \mu
  45. ν \nu
  46. μ \mu
  47. ν \nu
  48. [ 0 , + ) [0,+\infty)
  49. X X
  50. μ \mu
  51. Y Y
  52. ν \nu
  53. X X
  54. Y Y
  55. μ ν \mu\boxtimes\nu
  56. X 1 / 2 Y X 1 / 2 X^{1/2}YX^{1/2}
  57. Y 1 / 2 X Y 1 / 2 Y^{1/2}XY^{1/2}
  58. A A
  59. B B
  60. n n
  61. n n
  62. A A
  63. B B
  64. μ \mu
  65. ν \nu
  66. n n
  67. A B AB
  68. μ ν \mu\boxtimes\nu
  69. μ , ν \mu,\nu
  70. { z : | z | = 1 } \{z:|z|=1\}

Free_loop.html

  1. X X
  2. X X
  3. S 1 S^{1}
  4. X X
  5. f g f\sim g
  6. ψ : S 1 S 1 \psi:S^{1}\rightarrow S^{1}
  7. g = f ψ g=f\circ\psi
  8. L X LX

Free_spectral_range.html

  1. Δ ν \Delta\nu
  2. Δ λ \Delta\lambda
  3. Δ ν = c 2 n g l \Delta\nu=\frac{c}{2n_{g}l}
  4. l l
  5. n g n_{g}
  6. c c
  7. Δ λ = λ 2 2 n g l \Delta\lambda=\frac{\lambda^{2}}{2n_{g}l}
  8. λ \lambda
  9. λ \lambda
  10. ( λ + Δ λ ) (\lambda+\Delta\lambda)
  11. Δ λ = λ m \Delta\lambda={\lambda\over m}
  12. Δ λ = λ 0 2 2 n l cos θ + λ 0 λ 0 2 2 n l cos θ \Delta\lambda=\frac{\lambda_{0}^{2}}{2nl\cos\theta+\lambda_{0}}\approx\frac{% \lambda_{0}^{2}}{2nl\cos\theta}
  13. θ \theta
  14. Δ f c 2 n l cos θ \Delta f\approx\frac{c}{2nl\cos\theta}
  15. = Δ λ δ λ = π 2 arcsin ( 1 / F ) , \mathcal{F}=\frac{\Delta\lambda}{\delta\lambda}=\frac{\pi}{2\arcsin(1/\sqrt{F}% )},
  16. F = 4 R < m t p l > ( 1 - R ) 2 F=\frac{4R}{<}mtpl>{{(1-R)^{2}}}
  17. π F 2 = π R 1 / 2 ( 1 - R ) . \mathcal{F}\approx\frac{\pi\sqrt{F}}{2}=\frac{\pi R^{1/2}}{(1-R)}.

Freshman's_dream.html

  1. ( 1 + 4 ) 2 = 5 2 = 25 (1+4)^{2}=5^{2}=25
  2. 1 2 + 4 2 = 17 1^{2}+4^{2}=17
  3. x 2 + y 2 \sqrt{x^{2}+y^{2}}
  4. x 2 + y 2 = | x | + | y | \sqrt{x^{2}}+\sqrt{y^{2}}=|x|+|y|
  5. 9 + 16 = 25 = 5 \sqrt{9+16}=\sqrt{25}=5
  6. 1 / 2 {1}/{2}
  7. ( p n ) = p ! n ! ( p - n ) ! . {\left({{p}\atop{n}}\right)}=\frac{p!}{n!(p-n)!}.
  8. n [ x ] \mathbb{Z}_{n}[x]

Fréchet_distance.html

  1. S S
  2. A A
  3. S S
  4. S S
  5. A : [ 0 , 1 ] S A:[0,1]\rightarrow S
  6. α \alpha
  7. [ 0 , 1 ] [0,1]
  8. α : [ 0 , 1 ] [ 0 , 1 ] \alpha:[0,1]\rightarrow[0,1]
  9. A A
  10. B B
  11. S S
  12. A A
  13. B B
  14. α \alpha
  15. β \beta
  16. [ 0 , 1 ] [0,1]
  17. t [ 0 , 1 ] t\in[0,1]
  18. S S
  19. A ( α ( t ) ) A(\alpha(t))
  20. B ( β ( t ) ) B(\beta(t))
  21. F ( A , B ) F(A,B)
  22. F ( A , B ) = inf α , β max t [ 0 , 1 ] { d ( A ( α ( t ) ) , B ( β ( t ) ) ) } F(A,B)=\inf_{\alpha,\beta}\,\,\max_{t\in[0,1]}\,\,\Bigg\{d\Big(A(\alpha(t)),\,% B(\beta(t))\Big)\Bigg\}
  23. d d
  24. S S
  25. t t
  26. A ( α ( t ) ) A(\alpha(t))
  27. B ( β ( t ) ) B(\beta(t))
  28. t t
  29. t t
  30. A ( α ( t ) ) A(\alpha(t))
  31. B ( β ( t ) ) B(\beta(t))
  32. [ 0 , 1 ] [0,1]
  33. α \alpha
  34. β \beta
  35. O ( m n log ( m n ) ) O(mn\cdot\log(mn))
  36. D ε ( A , B ) := { ( α , β ) [ 0 , 1 ] 2 d ( A ( α ) , B ( β ) ) ε } D_{\varepsilon}(A,B):=\{\,(\alpha,\beta)\in[0,1]^{2}\mid d(A(\alpha),B(\beta))% \leq\varepsilon\,\}
  37. F ( A , B ) F(A,B)
  38. D ε ( A , B ) D_{\varepsilon}(A,B)
  39. r 1 r_{1}
  40. r 2 r_{2}
  41. | r 1 - r 2 | . |r_{1}-r_{2}|.
  42. r 1 + r 2 r_{1}+r_{2}
  43. | r 1 - r 2 | |r_{1}-r_{2}|

Frobenius_solution_to_the_hypergeometric_equation.html

  1. x ( 1 - x ) y ′′ + { γ - ( 1 + α + β ) x } y - α β y = 0 x(1-x)y^{\prime\prime}+\left\{\gamma-(1+\alpha+\beta)x\right\}y^{\prime}-% \alpha\beta y=0
  2. P 0 ( x ) = - α β , P 1 ( x ) = γ - ( 1 + α + β ) x , P 2 ( x ) = x ( 1 - x ) \begin{aligned}\displaystyle P_{0}(x)&\displaystyle=-\alpha\beta,\\ \displaystyle P_{1}(x)&\displaystyle=\gamma-(1+\alpha+\beta)x,\\ \displaystyle P_{2}(x)&\displaystyle=x(1-x)\end{aligned}
  3. P 2 ( 0 ) = P 2 ( 1 ) = 0. P_{2}(0)=P_{2}(1)=0.
  4. lim x a ( x - a ) P 1 ( x ) P 2 ( x ) \displaystyle\lim_{x\to a}\frac{(x-a)P_{1}(x)}{P_{2}(x)}
  5. y = r = 0 a r x r + c y=\sum_{r=0}^{\infty}a_{r}x^{r+c}
  6. y = r = 0 a r ( r + c ) x r + c - 1 y ′′ = r = 0 a r ( r + c ) ( r + c - 1 ) x r + c - 2 . \begin{aligned}\displaystyle y^{\prime}&\displaystyle=\sum_{r=0}^{\infty}a_{r}% (r+c)x^{r+c-1}\\ \displaystyle y^{\prime\prime}&\displaystyle=\sum_{r=0}^{\infty}a_{r}(r+c)(r+c% -1)x^{r+c-2}.\end{aligned}
  7. x r = 0 a r ( r + c ) ( r + c - 1 ) x r + c - 2 - x 2 r = 0 a r ( r + c ) ( r + c - 1 ) x r + c - 2 + γ r = 0 a r ( r + c ) x r + c - 1 - ( 1 + α + β ) x r = 0 a r ( r + c ) x r + c - 1 - α β r = 0 a r x r + c = 0 x\sum_{r=0}^{\infty}a_{r}(r+c)(r+c-1)x^{r+c-2}-x^{2}\sum_{r=0}^{\infty}a_{r}(r% +c)(r+c-1)x^{r+c-2}+\gamma\sum_{r=0}^{\infty}a_{r}(r+c)x^{r+c-1}-(1+\alpha+% \beta)x\sum_{r=0}^{\infty}a_{r}(r+c)x^{r+c-1}-\alpha\beta\sum_{r=0}^{\infty}a_% {r}x^{r+c}=0
  8. r = 0 a r ( r + c ) ( r + c - 1 ) x r + c - 1 - r = 0 a r ( r + c ) ( r + c - 1 ) x r + c + γ r = 0 a r ( r + c ) x r + c - 1 - ( 1 + α + β ) r = 0 a r ( r + c ) x r + c - α β r = 0 a r x r + c = 0 \sum_{r=0}^{\infty}a_{r}(r+c)(r+c-1)x^{r+c-1}-\sum_{r=0}^{\infty}a_{r}(r+c)(r+% c-1)x^{r+c}+\gamma\sum_{r=0}^{\infty}a_{r}(r+c)x^{r+c-1}-(1+\alpha+\beta)\sum_% {r=0}^{\infty}a_{r}(r+c)x^{r+c}-\alpha\beta\sum_{r=0}^{\infty}a_{r}x^{r+c}=0
  9. r = 0 a r ( r + c ) ( r + c - 1 ) x r + c - 1 - r = 1 a r - 1 ( r + c - 1 ) ( r + c - 2 ) x r + c - 1 + γ r = 0 a r ( r + c ) x r + c - 1 \displaystyle\sum_{r=0}^{\infty}a_{r}(r+c)(r+c-1)x^{r+c-1}-\sum_{r=1}^{\infty}% a_{r-1}(r+c-1)(r+c-2)x^{r+c-1}+\gamma\sum_{r=0}^{\infty}a_{r}(r+c)x^{r+c-1}
  10. a 0 ( c ( c - 1 ) + γ c ) x c - 1 + r = 1 a r ( r + c ) ( r + c - 1 ) x r + c - 1 - r = 1 a r - 1 ( r + c - 1 ) ( r + c - 2 ) x r + c - 1 + γ r = 1 a r ( r + c ) x r + c - 1 - ( 1 + α + β ) r = 1 a r - 1 ( r + c - 1 ) x r + c - 1 - α β r = 1 a r - 1 x r + c - 1 = 0 \begin{aligned}&\displaystyle a_{0}(c(c-1)+\gamma c)x^{c-1}+\sum_{r=1}^{\infty% }a_{r}(r+c)(r+c-1)x^{r+c-1}-\sum_{r=1}^{\infty}a_{r-1}(r+c-1)(r+c-2)x^{r+c-1}% \\ &\displaystyle\qquad+\gamma\sum_{r=1}^{\infty}a_{r}(r+c)x^{r+c-1}-(1+\alpha+% \beta)\sum_{r=1}^{\infty}a_{r-1}(r+c-1)x^{r+c-1}-\alpha\beta\sum_{r=1}^{\infty% }a_{r-1}x^{r+c-1}=0\end{aligned}
  11. a 0 ( c ( c - 1 ) + γ c ) = 0 a_{0}(c(c-1)+\gamma c)=0
  12. c ( c - 1 + γ ) = 0. c(c-1+\gamma)=0.
  13. c 1 = 0 , c 2 = 1 - γ c_{1}=0,c_{2}=1-\gamma
  14. ( ( r + c ) ( r + c - 1 ) + γ ( r + c ) ) a r + ( - ( r + c - 1 ) ( r + c - 2 ) - ( 1 + α + β ) ( r + c - 1 ) - α β ) a r - 1 = 0 ((r+c)(r+c-1)+\gamma(r+c))a_{r}+(-(r+c-1)(r+c-2)-(1+\alpha+\beta)(r+c-1)-% \alpha\beta)a_{r-1}=0
  15. a r = ( r + c - 1 ) ( r + c - 2 ) + ( 1 + α + β ) ( r + c - 1 ) + α β ( r + c ) ( r + c - 1 ) + γ ( r + c ) a r - 1 = ( r + c - 1 ) ( r + c + α + β - 1 ) + α β ( r + c ) ( r + c + γ - 1 ) a r - 1 \begin{aligned}\displaystyle a_{r}&\displaystyle=\frac{(r+c-1)(r+c-2)+(1+% \alpha+\beta)(r+c-1)+\alpha\beta}{(r+c)(r+c-1)+\gamma(r+c)}a_{r-1}\\ &\displaystyle=\frac{(r+c-1)(r+c+\alpha+\beta-1)+\alpha\beta}{(r+c)(r+c+\gamma% -1)}a_{r-1}\end{aligned}
  16. ( r + c - 1 ) ( r + c + α + β - 1 ) + α β = ( r + c - 1 ) ( r + c + α - 1 ) + ( r + c - 1 ) β + α β = ( r + c - 1 ) ( r + c + α - 1 ) + β ( r + c + α - 1 ) \begin{aligned}\displaystyle(r+c-1)(r+c+\alpha+\beta-1)+\alpha\beta&% \displaystyle=(r+c-1)(r+c+\alpha-1)+(r+c-1)\beta+\alpha\beta\\ &\displaystyle=(r+c-1)(r+c+\alpha-1)+\beta(r+c+\alpha-1)\end{aligned}
  17. a r = ( r + c + α - 1 ) ( r + c + β - 1 ) ( r + c ) ( r + c + γ - 1 ) a r - 1 , for r 1. a_{r}=\frac{(r+c+\alpha-1)(r+c+\beta-1)}{(r+c)(r+c+\gamma-1)}a_{r-1},\,\text{ % for }r\geq 1.
  18. a 1 = ( c + α ) ( c + β ) ( c + 1 ) ( c + γ ) a 0 a 2 = ( c + α + 1 ) ( c + β + 1 ) ( c + 2 ) ( c + γ + 1 ) a 1 = ( c + α + 1 ) ( c + α ) ( c + β ) ( c + β + 1 ) ( c + 2 ) ( c + 1 ) ( c + γ ) ( c + γ + 1 ) a 0 = ( c + α ) 2 ( c + β ) 2 ( c + 1 ) 2 ( c + γ ) 2 a 0 a 3 = ( c + α + 2 ) ( c + β + 2 ) ( c + 3 ) ( c + γ + 2 ) a 2 = ( c + α ) 2 ( c + α + 2 ) ( c + β ) 2 ( c + β + 2 ) ( c + 1 ) 2 ( c + 3 ) ( c + γ ) 2 ( c + γ + 2 ) a 0 = ( c + α ) 3 ( c + β ) 3 ( c + 1 ) 3 ( c + γ ) 3 a 0 \begin{aligned}\displaystyle a_{1}&\displaystyle=\frac{(c+\alpha)(c+\beta)}{(c% +1)(c+\gamma)}a_{0}\\ \displaystyle a_{2}&\displaystyle=\frac{(c+\alpha+1)(c+\beta+1)}{(c+2)(c+% \gamma+1)}a_{1}=\frac{(c+\alpha+1)(c+\alpha)(c+\beta)(c+\beta+1)}{(c+2)(c+1)(c% +\gamma)(c+\gamma+1)}a_{0}=\frac{(c+\alpha)_{2}(c+\beta)_{2}}{(c+1)_{2}(c+% \gamma)_{2}}a_{0}\\ \displaystyle a_{3}&\displaystyle=\frac{(c+\alpha+2)(c+\beta+2)}{(c+3)(c+% \gamma+2)}a_{2}=\frac{(c+\alpha)_{2}(c+\alpha+2)(c+\beta)_{2}(c+\beta+2)}{(c+1% )_{2}(c+3)(c+\gamma)_{2}(c+\gamma+2)}a_{0}=\frac{(c+\alpha)_{3}(c+\beta)_{3}}{% (c+1)_{3}(c+\gamma)_{3}}a_{0}\end{aligned}
  19. a r = ( c + α ) r ( c + β ) r ( c + 1 ) r ( c + γ ) r a 0 , for r 0 a_{r}=\frac{(c+\alpha)_{r}(c+\beta)_{r}}{(c+1)_{r}(c+\gamma)_{r}}a_{0},\,\text% { for }r\geq 0
  20. y = a 0 r = 0 ( c + α ) r ( c + β ) r ( c + 1 ) r ( c + γ ) r x r + c . y=a_{0}\sum_{r=0}^{\infty}\frac{(c+\alpha)_{r}(c+\beta)_{r}}{(c+1)_{r}(c+% \gamma)_{r}}x^{r+c}.
  21. y = a 0 r = 0 ( c + α ) r ( c + β ) r ( c + 1 ) r ( c + γ ) r x r + c , y=a_{0}\sum_{r=0}^{\infty}\frac{(c+\alpha)_{r}(c+\beta)_{r}}{(c+1)_{r}(c+% \gamma)_{r}}x^{r+c},
  22. y 1 = a 0 r = 0 ( α ) r ( β ) r ( 1 ) r ( γ ) r x r = a 0 F 1 2 ( α , β ; γ ; x ) y 2 = a 0 r = 0 ( α + 1 - γ ) r ( β + 1 - γ ) r ( 1 - γ + 1 ) r ( 1 - γ + γ ) r x r + 1 - γ = a 0 x 1 - γ r = 0 ( α + 1 - γ ) r ( β + 1 - γ ) r ( 1 ) r ( 2 - γ ) r x r = a 0 x 1 - γ F 1 2 ( α - γ + 1 , β - γ + 1 ; 2 - γ ; x ) \begin{aligned}\displaystyle y_{1}&\displaystyle=a_{0}\sum_{r=0}^{\infty}\frac% {(\alpha)_{r}(\beta)_{r}}{(1)_{r}(\gamma)_{r}}x^{r}=a_{0}\cdot{{}_{2}F_{1}}(% \alpha,\beta;\gamma;x)\\ \displaystyle y_{2}&\displaystyle=a_{0}\sum_{r=0}^{\infty}\frac{(\alpha+1-% \gamma)_{r}(\beta+1-\gamma)_{r}}{(1-\gamma+1)_{r}(1-\gamma+\gamma)_{r}}x^{r+1-% \gamma}\\ &\displaystyle=a_{0}x^{1-\gamma}\sum_{r=0}^{\infty}\frac{(\alpha+1-\gamma)_{r}% (\beta+1-\gamma)_{r}}{(1)_{r}(2-\gamma)_{r}}x^{r}\\ &\displaystyle=a_{0}x^{1-\gamma}{{}_{2}F_{1}}(\alpha-\gamma+1,\beta-\gamma+1;2% -\gamma;x)\end{aligned}
  23. y = A y 1 + B y 2 . y=A^{\prime}y_{1}+B^{\prime}y_{2}.
  24. y = A F 1 2 ( α , β ; γ ; x ) + B x 1 - γ F 1 2 ( α - γ + 1 , β - γ + 1 ; 2 - γ ; x ) y=A{{}_{2}F_{1}}(\alpha,\beta;\gamma;x)+Bx^{1-\gamma}{{}_{2}F_{1}}(\alpha-% \gamma+1,\beta-\gamma+1;2-\gamma;x)\,
  25. y = a 0 r = 0 ( c + α ) r ( c + β ) r ( c + 1 ) r 2 x r + c . y=a_{0}\sum_{r=0}^{\infty}\frac{(c+\alpha)_{r}(c+\beta)_{r}}{(c+1)_{r}^{2}}x^{% r+c}.
  26. y 1 = a 0 r = 0 ( α ) r ( β ) r ( 1 ) r ( 1 ) r x r = a 0 F 1 2 ( α , β ; 1 ; x ) y 2 = y c | c = 0 . \begin{aligned}\displaystyle y_{1}&\displaystyle=a_{0}\sum_{r=0}^{\infty}\frac% {(\alpha)_{r}(\beta)_{r}}{(1)_{r}(1)_{r}}x^{r}=a_{0}{{}_{2}F_{1}}(\alpha,\beta% ;1;x)\\ \displaystyle y_{2}&\displaystyle=\left.\frac{\partial y}{\partial c}\right|_{% c=0}.\end{aligned}
  27. M r = ( c + α ) r ( c + β ) r ( c + 1 ) r 2 . M_{r}=\frac{(c+\alpha)_{r}(c+\beta)_{r}}{(c+1)_{r}^{2}}.
  28. ln ( M r ) = ln ( ( c + α ) r ( c + β ) r ( c + 1 ) r 2 ) = ln ( c + α ) r + ln ( c + β ) r - 2 ln ( c + 1 ) r \ln(M_{r})=\ln\left(\frac{(c+\alpha)_{r}(c+\beta)_{r}}{(c+1)_{r}^{2}}\right)=% \ln(c+\alpha)_{r}+\ln(c+\beta)_{r}-2\ln(c+1)_{r}
  29. ln ( c + α ) r = ln ( ( c + α ) ( c + α + 1 ) ( c + α + r - 1 ) ) = k = 0 r - 1 ln ( c + α + k ) . \ln(c+\alpha)_{r}=\ln\left((c+\alpha)(c+\alpha+1)\cdots(c+\alpha+r-1)\right)=% \sum_{k=0}^{r-1}\ln(c+\alpha+k).
  30. ln ( M r ) = k = 0 r - 1 ln ( c + α + k ) + k = 0 r - 1 ln ( c + β + k ) - 2 k = 0 r - 1 ln ( c + 1 + k ) = k = 0 r - 1 ( ln ( c + α + k ) + ln ( c + β + k ) - 2 ln ( c + 1 + k ) ) \begin{aligned}\displaystyle\ln(M_{r})&\displaystyle=\sum_{k=0}^{r-1}\ln(c+% \alpha+k)+\sum_{k=0}^{r-1}\ln(c+\beta+k)-2\sum_{k=0}^{r-1}\ln(c+1+k)\\ &\displaystyle=\sum_{k=0}^{r-1}\left(\ln(c+\alpha+k)+\ln(c+\beta+k)-2\ln(c+1+k% )\right)\end{aligned}
  31. 1 M r M r c = k = 0 r - 1 ( 1 c + α + k + 1 c + β + k - 2 c + 1 + k ) . \frac{1}{M_{r}}\frac{\partial M_{r}}{\partial c}=\sum_{k=0}^{r-1}\left(\frac{1% }{c+\alpha+k}+\frac{1}{c+\beta+k}-\frac{2}{c+1+k}\right).
  32. M r c = ( c + α ) r ( c + β ) r ( c + 1 ) r 2 k = 0 r - 1 ( 1 c + α + k + 1 c + β + k - 2 c + 1 + k ) . \frac{\partial M_{r}}{\partial c}=\frac{(c+\alpha)_{r}(c+\beta)_{r}}{(c+1)_{r}% ^{2}}\sum_{k=0}^{r-1}\left(\frac{1}{c+\alpha+k}+\frac{1}{c+\beta+k}-\frac{2}{c% +1+k}\right).
  33. y = a 0 x c r = 0 ( c + α ) r ( c + β ) r ( c + 1 ) r 2 x r = a 0 x c r = 0 M r x r . y=a_{0}x^{c}\sum_{r=0}^{\infty}\frac{(c+\alpha)_{r}(c+\beta)_{r}}{(c+1)_{r}^{2% }}x^{r}=a_{0}x^{c}\sum_{r=0}^{\infty}M_{r}x^{r}.
  34. y c = a 0 x c ln ( x ) r = 0 ( c + α ) r ( c + β ) r ( c + 1 ) r 2 x r + a 0 x c r = 0 ( ( c + α ) r ( c + β ) r ( c + 1 ) r 2 { k = 0 r - 1 ( 1 c + α + k + 1 c + β + k - 2 c + 1 + k ) } ) x r = a 0 x c r = 0 ( c + α ) r ( c + β ) r ( c + 1 ) r ) 2 ( ln x + k = 0 r - 1 ( 1 c + α + k + 1 c + β + k - 2 c + 1 + k ) ) x r . \begin{aligned}\displaystyle\frac{\partial y}{\partial c}&\displaystyle=a_{0}x% ^{c}\ln(x)\sum_{r=0}^{\infty}\frac{(c+\alpha)_{r}(c+\beta)_{r}}{(c+1)_{r}^{2}}% x^{r}+a_{0}x^{c}\sum_{r=0}^{\infty}\left(\frac{(c+\alpha)_{r}(c+\beta)_{r}}{(c% +1)_{r}^{2}}\left\{\sum_{k=0}^{r-1}\left(\frac{1}{c+\alpha+k}+\frac{1}{c+\beta% +k}-\frac{2}{c+1+k}\right)\right\}\right)x^{r}\\ &\displaystyle=a_{0}x^{c}\sum_{r=0}^{\infty}\frac{(c+\alpha)_{r}(c+\beta)_{r}}% {(c+1)_{r})^{2}}\left(\ln x+\sum_{k=0}^{r-1}\left(\frac{1}{c+\alpha+k}+\frac{1% }{c+\beta+k}-\frac{2}{c+1+k}\right)\right)x^{r}.\end{aligned}
  35. y 2 = a 0 r = 0 ( α ) r ( β ) r ( 1 ) r 2 ( ln x + k = 0 r - 1 ( 1 α + k + 1 β + k - 2 1 + k ) ) x r . y_{2}=a_{0}\sum_{r=0}^{\infty}\frac{(\alpha)_{r}(\beta)_{r}}{(1)_{r}^{2}}\left% (\ln x+\sum_{k=0}^{r-1}\left(\frac{1}{\alpha+k}+\frac{1}{\beta+k}-\frac{2}{1+k% }\right)\right)x^{r}.
  36. y = C F 1 2 ( α , β ; 1 ; x ) + D r = 0 ( α ) r ( β ) r ( 1 ) r 2 ( ln ( x ) + k = 0 r - 1 ( 1 α + k + 1 β + k - 2 1 + k ) ) x r y=C{{}_{2}F_{1}}(\alpha,\beta;1;x)+D\sum_{r=0}^{\infty}\frac{(\alpha)_{r}(% \beta)_{r}}{(1)_{r}^{2}}\left(\ln(x)+\sum_{k=0}^{r-1}\left(\frac{1}{\alpha+k}+% \frac{1}{\beta+k}-\frac{2}{1+k}\right)\right)x^{r}
  37. a r = ( r + c + α - 1 ) ( r + c + β - 1 ) ( r + c ) ( r + c + γ - 1 ) a r - 1 , a_{r}=\frac{(r+c+\alpha-1)(r+c+\beta-1)}{(r+c)(r+c+\gamma-1)}a_{r-1},
  38. y b = b 0 x c r = 0 c ( c + α ) r ( c + β ) r ( c + 1 ) r ( c + γ ) r x r y_{b}=b_{0}x^{c}\sum_{r=0}^{\infty}\frac{c(c+\alpha)_{r}(c+\beta)_{r}}{(c+1)_{% r}(c+\gamma)_{r}}x^{r}
  39. c ( c + α ) 1 - γ ( c + β ) 1 - γ ( c + 1 ) 1 - γ ( c + γ ) 1 - γ x 1 - γ \frac{c(c+\alpha)_{1-\gamma}(c+\beta)_{1-\gamma}}{(c+1)_{1-\gamma}(c+\gamma)_{% 1-\gamma}}x^{1-\gamma}
  40. ( c + γ ) 1 - γ = ( c + γ ) ( c + γ + 1 ) c . (c+\gamma)_{1-\gamma}=(c+\gamma)(c+\gamma+1)\cdots c.
  41. y 1 = b 0 ( ( α ) 1 - γ ( β ) 1 - γ ( 1 ) 1 - γ ( γ ) - γ x 1 - γ + ( α ) 2 - γ ( β ) 2 - γ ( 1 ) 2 - γ ( γ ) - γ ( 1 ) x 2 - γ + ( α ) 3 - γ ( β ) 3 - γ ( 1 ) 3 - γ ( γ ) - γ ( 1 ) ( 2 ) x 3 - γ + ) = b 0 ( γ ) - γ r = 1 - γ ( α ) r ( β ) r ( 1 ) r ( 1 ) r + γ - 1 x r . \begin{aligned}\displaystyle y_{1}&\displaystyle=b_{0}\left(\frac{(\alpha)_{1-% \gamma}(\beta)_{1-\gamma}}{(1)_{1-\gamma}(\gamma)_{-\gamma}}x^{1-\gamma}+\frac% {(\alpha)_{2-\gamma}(\beta)_{2-\gamma}}{(1)_{2-\gamma}(\gamma)_{-\gamma}(1)}x^% {2-\gamma}+\frac{(\alpha)_{3-\gamma}(\beta)_{3-\gamma}}{(1)_{3-\gamma}(\gamma)% _{-\gamma}(1)(2)}x^{3-\gamma}+\cdots\right)\\ &\displaystyle=\frac{b_{0}}{(\gamma)_{-\gamma}}\sum_{r=1-\gamma}^{\infty}\frac% {(\alpha)_{r}(\beta)_{r}}{(1)_{r}(1)_{r+\gamma-1}}x^{r}.\end{aligned}
  42. y 2 = y b c | c = 1 - γ . y_{2}=\left.\frac{\partial y_{b}}{\partial c}\right|_{c=1-\gamma}.
  43. M r = c ( c + α ) r ( c + β ) r ( c + 1 ) r ( c + γ ) r . M_{r}=\frac{c(c+\alpha)_{r}(c+\beta)_{r}}{(c+1)_{r}(c+\gamma)_{r}}.
  44. M r c = c ( c + α ) r ( c + β ) r ( c + 1 ) r ( c + γ ) r { 1 c + k = 0 r - 1 ( 1 c + α + k + 1 c + β + k - 1 c + 1 + k - 1 c + γ + k ) } . \frac{\partial M_{r}}{\partial c}=\frac{c(c+\alpha)_{r}(c+\beta)_{r}}{(c+1)_{r% }(c+\gamma)_{r}}\left\{\frac{1}{c}+\sum_{k=0}^{r-1}\left(\frac{1}{c+\alpha+k}+% \frac{1}{c+\beta+k}-\frac{1}{c+1+k}-\frac{1}{c+\gamma+k}\right)\right\}.
  45. y b = b 0 r = 0 c ( c + α ) r ( c + β ) r ( c + 1 ) r ( c + γ ) r x r + c = b 0 x c r = 0 M r x r . y_{b}=b_{0}\sum_{r=0}^{\infty}\frac{c(c+\alpha)_{r}(c+\beta)_{r}}{(c+1)_{r}(c+% \gamma)_{r}}x^{r+c}=b_{0}x^{c}\sum_{r=0}^{\infty}M_{r}x^{r}.
  46. y c = b 0 x c ( ln ( x ) r = 0 c ( c + α ) r ( c + β ) r ( c + 1 ) r ( c + γ ) r x r + r = 0 c ( c + α ) r ( c + β ) r ( c + 1 ) r ( c + γ ) r { 1 c + k = 0 r - 1 ( 1 c + α + k + 1 c + β + k - 1 c + 1 + k - 1 c + γ + k ) } x r ) \begin{aligned}\displaystyle\frac{\partial y}{\partial c}&\displaystyle=b_{0}x% ^{c}\left(\ln(x)\sum_{r=0}^{\infty}\frac{c(c+\alpha)_{r}(c+\beta)_{r}}{(c+1)_{% r}(c+\gamma)_{r}}x^{r}+\sum_{r=0}^{\infty}\frac{c(c+\alpha)_{r}(c+\beta)_{r}}{% (c+1)_{r}(c+\gamma)_{r}}\left\{\frac{1}{c}+\sum_{k=0}^{r-1}\left(\frac{1}{c+% \alpha+k}+\frac{1}{c+\beta+k}-\frac{1}{c+1+k}-\frac{1}{c+\gamma+k}\right)% \right\}x^{r}\right)\end{aligned}
  47. y c = b 0 x c r = 0 c ( c + α ) r ( c + β ) r ( c + 1 ) r ( c + γ ) r ( ln ( x ) + 1 c + k = 0 r - 1 ( 1 c + α + k + 1 c + β + k - 1 c + 1 + k - 1 c + γ + k ) ) x r . \frac{\partial y}{\partial c}=b_{0}x^{c}\sum_{r=0}^{\infty}\frac{c(c+\alpha)_{% r}(c+\beta)_{r}}{(c+1)_{r}(c+\gamma)_{r}}\left(\ln(x)+\frac{1}{c}+\sum_{k=0}^{% r-1}\left(\frac{1}{c+\alpha+k}+\frac{1}{c+\beta+k}-\frac{1}{c+1+k}-\frac{1}{c+% \gamma+k}\right)\right)x^{r}.
  48. y = E ( γ ) - γ r = 1 - γ ( α ) r ( β ) r ( 1 ) r ( 1 ) r + γ - 1 x r + F x 1 - γ r = 0 ( 1 - γ ) ( α + 1 - γ ) r ( β + 1 - γ ) r ( 2 - γ ) r ( 1 ) r ( ln ( x ) + + 1 1 - γ + k = 0 r - 1 ( 1 α + k + 1 - γ + 1 β + k + 1 - γ - 1 2 + k - γ - 1 1 + k ) ) x r . \begin{aligned}\displaystyle y&\displaystyle=\frac{E}{(\gamma)_{-\gamma}}\sum_% {r=1-\gamma}^{\infty}\frac{(\alpha)_{r}(\beta)_{r}}{(1)_{r}(1)_{r+\gamma-1}}x^% {r}+Fx^{1-\gamma}\sum_{r=0}^{\infty}\frac{(1-\gamma)(\alpha+1-\gamma)_{r}(% \beta+1-\gamma)_{r}}{(2-\gamma)_{r}(1)_{r}}\Biggl(\ln(x)+\\ &\displaystyle\qquad\qquad+\frac{1}{1-\gamma}+\sum_{k=0}^{r-1}\left(\frac{1}{% \alpha+k+1-\gamma}+\frac{1}{\beta+k+1-\gamma}-\frac{1}{2+k-\gamma}-\frac{1}{1+% k}\right)\Biggr)x^{r}.\end{aligned}
  49. a r = ( r + c + α - 1 ) ( r + c + β - 1 ) ( r + c ) ( r + c + γ - 1 ) a r - 1 , a_{r}=\frac{(r+c+\alpha-1)(r+c+\beta-1)}{(r+c)(r+c+\gamma-1)}a_{r-1},
  50. y b = b 0 x c r = 0 ( c + γ - 1 ) ( c + α ) r ( c + β ) r ( c + 1 ) r ( c + γ ) r x r . y_{b}=b_{0}x^{c}\sum_{r=0}^{\infty}\frac{(c+\gamma-1)(c+\alpha)_{r}(c+\beta)_{% r}}{(c+1)_{r}(c+\gamma)_{r}}x^{r}.
  51. ( c + γ - 1 ) ( c + α ) γ - 1 ( c + β ) γ - 1 ( c + 1 ) γ - 1 ( c + γ ) γ - 1 x γ - 1 \frac{(c+\gamma-1)(c+\alpha)_{\gamma-1}(c+\beta)_{\gamma-1}}{(c+1)_{\gamma-1}(% c+\gamma)_{\gamma-1}}x^{\gamma-1}
  52. ( c + 1 ) γ - 1 = ( c + 1 ) ( c + 2 ) ( c + γ - 1 ) . (c+1)_{\gamma-1}=(c+1)(c+2)\cdots(c+\gamma-1).
  53. y 1 = b 0 x 1 - γ ( ( α + 1 - γ ) γ - 1 ( β + 1 - γ ) γ - 1 ( 2 - γ ) γ - 2 ( 1 ) γ - 1 x γ - 1 + ( α + 1 - γ ) γ ( β + 1 - γ ) γ ( 2 - γ ) γ - 2 ( 1 ) ( 1 ) γ x γ + ) = b 0 ( 2 - γ ) γ - 2 x 1 - γ r = γ - 1 ( α + 1 - γ ) r ( β + 1 - γ ) r ( 1 ) r ( 1 ) r + 1 - γ x r . \begin{aligned}\displaystyle y_{1}&\displaystyle=b_{0}x^{1-\gamma}\left(\frac{% (\alpha+1-\gamma)_{\gamma-1}(\beta+1-\gamma)_{\gamma-1}}{(2-\gamma)_{\gamma-2}% (1)_{\gamma-1}}x^{\gamma-1}+\frac{(\alpha+1-\gamma)_{\gamma}(\beta+1-\gamma)_{% \gamma}}{(2-\gamma)_{\gamma-2}(1)(1)_{\gamma}}x^{\gamma}+\cdots\right)\\ &\displaystyle=\frac{b_{0}}{(2-\gamma)_{\gamma-2}}x^{1-\gamma}\sum_{r=\gamma-1% }^{\infty}\frac{(\alpha+1-\gamma)_{r}(\beta+1-\gamma)_{r}}{(1)_{r}(1)_{r+1-% \gamma}}x^{r}.\end{aligned}
  54. y 2 = y b c | c = 0 y_{2}=\left.\frac{\partial y_{b}}{\partial c}\right|_{c=0}
  55. M r = ( c + γ - 1 ) ( c + α ) r ( c + β ) r ( c + 1 ) r ( c + γ ) r . M_{r}=\frac{(c+\gamma-1)(c+\alpha)_{r}(c+\beta)_{r}}{(c+1)_{r}(c+\gamma)_{r}}.
  56. M r c = ( c + γ - 1 ) ( c + α ) r ( c + β ) r ( c + 1 ) r ( c + γ ) r ( 1 c + γ - 1 + k = 0 r - 1 ( 1 c + α + k + 1 c + β + k - 1 c + 1 + k - 1 c + γ + k ) ) \frac{\partial M_{r}}{\partial c}=\frac{(c+\gamma-1)(c+\alpha)_{r}(c+\beta)_{r% }}{(c+1)_{r}(c+\gamma)_{r}}\left(\frac{1}{c+\gamma-1}+\sum_{k=0}^{r-1}\left(% \frac{1}{c+\alpha+k}+\frac{1}{c+\beta+k}-\frac{1}{c+1+k}-\frac{1}{c+\gamma+k}% \right)\right)
  57. y b = b 0 r = 0 ( c + γ - 1 ) ( c + α ) r ( c + β ) r ( c + 1 ) r ( c + γ ) r x r + c = b 0 x c r = 0 M r x r . y_{b}=b_{0}\sum_{r=0}^{\infty}\frac{(c+\gamma-1)(c+\alpha)_{r}(c+\beta)_{r}}{(% c+1)_{r}(c+\gamma)_{r}}x^{r+c}=b_{0}x^{c}\sum_{r=0}^{\infty}M_{r}x^{r}.
  58. y c = b 0 x c ( ln ( x ) r = 0 ( c + γ - 1 ) ( c + α ) r ( c + β ) r ( c + 1 ) r ( c + γ ) r x r + r = 0 ( c + γ - 1 ) ( c + α ) r ( c + β ) r ( c + 1 ) r ( c + γ ) r { 1 c + γ - 1 + + k = 0 r - 1 ( 1 c + α + k + 1 c + β + k - 1 c + 1 + k - 1 c + γ + k ) } x r ) = b 0 x c r = 0 ( c + γ - 1 ) ( c + α ) r ( c + β ) r ( c + 1 ) r ( c + γ ) r ( ln ( x ) + 1 c + γ - 1 + k = 0 r - 1 { 1 c + α + k + 1 c + β + k - 1 c + 1 + k - 1 c + γ + k } ) x r . \begin{aligned}\displaystyle\frac{\partial y}{\partial c}&\displaystyle=b_{0}x% ^{c}\Bigg(\ln(x)\sum_{r=0}^{\infty}\frac{(c+\gamma-1)(c+\alpha)_{r}(c+\beta)_{% r}}{(c+1)_{r}(c+\gamma)_{r}}x^{r}+\sum_{r=0}^{\infty}\frac{(c+\gamma-1)(c+% \alpha)_{r}(c+\beta)_{r}}{(c+1)_{r}(c+\gamma)_{r}}\Bigg\{\frac{1}{c+\gamma-1}+% \\ &\displaystyle\qquad\qquad+\sum_{k=0}^{r-1}\left(\frac{1}{c+\alpha+k}+\frac{1}% {c+\beta+k}-\frac{1}{c+1+k}-\frac{1}{c+\gamma+k}\right)\Bigg\}x^{r}\Bigg)\\ &\displaystyle=b_{0}x^{c}\sum_{r=0}^{\infty}\frac{(c+\gamma-1)(c+\alpha)_{r}(c% +\beta)_{r}}{(c+1)_{r}(c+\gamma)_{r}}\left(\ln(x)+\frac{1}{c+\gamma-1}+\sum_{k% =0}^{r-1}\left\{\frac{1}{c+\alpha+k}+\frac{1}{c+\beta+k}-\frac{1}{c+1+k}-\frac% {1}{c+\gamma+k}\right\}\right)x^{r}.\end{aligned}
  59. y = G ( 2 - γ ) γ - 2 x 1 - γ r = γ - 1 ( α + 1 - γ ) r ( β + 1 - γ ) r ( 1 ) r ( 1 ) r + 1 - γ x r + + H r = 0 ( 1 - γ ) ( α + 1 - γ ) r ( β + 1 - γ ) r ( 2 - γ ) r ( 1 ) r ( ln ( x ) + 1 γ - 1 + k = 0 r - 1 { 1 α + k + 1 β + k - 1 1 + k - 1 γ + k } ) x r . \begin{aligned}\displaystyle y&\displaystyle=\frac{G}{(2-\gamma)_{\gamma-2}}x^% {1-\gamma}\sum_{r=\gamma-1}^{\infty}\frac{(\alpha+1-\gamma)_{r}(\beta+1-\gamma% )_{r}}{(1)_{r}(1)_{r+1-\gamma}}x^{r}+\\ &\displaystyle\qquad\qquad+H\sum_{r=0}^{\infty}\frac{(1-\gamma)(\alpha+1-% \gamma)_{r}(\beta+1-\gamma)_{r}}{(2-\gamma)_{r}(1)_{r}}\left(\ln(x)+\frac{1}{% \gamma-1}+\sum_{k=0}^{r-1}\left\{\frac{1}{\alpha+k}+\frac{1}{\beta+k}-\frac{1}% {1+k}-\frac{1}{\gamma+k}\right\}\right)x^{r}.\end{aligned}
  60. lim x a ( x - a ) P 1 ( x ) P 2 ( x ) = lim x 1 ( x - 1 ) ( γ - ( 1 + α + β ) x ) x ( 1 - x ) = lim x 1 - ( γ - ( 1 + α + β ) x ) x = 1 + α + β - γ lim x a ( x - a ) 2 P 0 ( x ) P 2 ( x ) = lim x 1 ( x - 1 ) 2 ( - α β ) x ( 1 - x ) = lim x 1 ( x - 1 ) α β x = 0 \begin{aligned}\displaystyle\lim_{x\to a}\frac{(x-a)P_{1}(x)}{P_{2}(x)}&% \displaystyle=\lim_{x\to 1}\frac{(x-1)(\gamma-(1+\alpha+\beta)x)}{x(1-x)}=\lim% _{x\to 1}\frac{-(\gamma-(1+\alpha+\beta)x)}{x}=1+\alpha+\beta-\gamma\\ \displaystyle\lim_{x\to a}\frac{(x-a)^{2}P_{0}(x)}{P_{2}(x)}&\displaystyle=% \lim_{x\to 1}\frac{(x-1)^{2}(-\alpha\beta)}{x(1-x)}=\lim_{x\to 1}\frac{(x-1)% \alpha\beta}{x}=0\end{aligned}
  61. y = r = 0 a r ( x - 1 ) r + c , y=\sum_{r=0}^{\infty}a_{r}(x-1)^{r+c},
  62. x ( 1 - x ) y ′′ + ( γ - ( 1 + α + β ) x ) y - α β y = 0. x(1-x)y^{\prime\prime}+(\gamma-(1+\alpha+\beta)x)y^{\prime}-\alpha\beta y=0.
  63. d y d x = d y d z × d z d x = - d y d z = - y d 2 y d x 2 = d d x ( d y d x ) = d d x ( - d y d z ) = d d z ( - d y d z ) × d z d x = d 2 y d z 2 = y ′′ \begin{aligned}\displaystyle\frac{dy}{dx}&\displaystyle=\frac{dy}{dz}\times% \frac{dz}{dx}=-\frac{dy}{dz}=-y^{\prime}\\ \displaystyle\frac{d^{2}y}{dx^{2}}&\displaystyle=\frac{d}{dx}\left(\frac{dy}{% dx}\right)=\frac{d}{dx}\left(-\frac{dy}{dz}\right)=\frac{d}{dz}\left(-\frac{dy% }{dz}\right)\times\frac{dz}{dx}=\frac{d^{2}y}{dz^{2}}=y^{\prime\prime}\end{aligned}
  64. z ( 1 - z ) y ′′ + ( α + β - γ + 1 - ( 1 + α + β ) z ) y - α β y = 0. z(1-z)y^{\prime\prime}+(\alpha+\beta-\gamma+1-(1+\alpha+\beta)z)y^{\prime}-% \alpha\beta y=0.
  65. y = A { F 1 2 ( α , β ; - Δ + 1 ; 1 - x ) } + B { ( 1 - x ) Δ F 1 2 ( Δ + β , Δ + α ; Δ + 1 ; 1 - x ) } y=A\left\{{{}_{2}F_{1}}(\alpha,\beta;-\Delta+1;1-x)\right\}+B\left\{(1-x)^{% \Delta}{{}_{2}F_{1}}(\Delta+\beta,\Delta+\alpha;\Delta+1;1-x)\right\}
  66. y = C { F 1 2 ( α , β ; 1 ; 1 - x ) } + D { r = 0 ( α ) r ( β ) r ( 1 ) r 2 ( ln ( 1 - x ) + k = 0 r - 1 ( 1 α + k + 1 β + k - 2 1 + k ) ) ( 1 - x ) r } y=C\left\{{{}_{2}F_{1}}(\alpha,\beta;1;1-x)\right\}+D\left\{\sum_{r=0}^{\infty% }\frac{(\alpha)_{r}(\beta)_{r}}{(1)_{r}^{2}}\left(\ln(1-x)+\sum_{k=0}^{r-1}% \left(\frac{1}{\alpha+k}+\frac{1}{\beta+k}-\frac{2}{1+k}\right)\right)(1-x)^{r% }\right\}
  67. y = E { 1 ( - Δ + 1 ) Δ - 1 r = 1 - Δ - α - β ( α ) r ( β ) r ( 1 ) r ( 1 ) r - Δ ( 1 - x ) r } + + F { ( 1 - x ) Δ r = 0 ( Δ ) ( Δ + α ) r ( Δ + β ) r ( Δ + 1 ) r ( 1 ) r ( ln ( 1 - x ) + 1 Δ + k = 0 r - 1 ( 1 Δ + α + k + 1 Δ + β + k - 1 Δ + 1 + k - 1 1 + k ) ) ( 1 - x ) r } \begin{aligned}\displaystyle y&\displaystyle=E\left\{\frac{1}{(-\Delta+1)_{% \Delta-1}}\ \sum_{r=1-\Delta-\alpha-\beta}^{\infty}\frac{(\alpha)_{r}(\beta)_{% r}}{(1)_{r}(1)_{r-\Delta}}(1-x)^{r}\right\}+\\ &\displaystyle\quad+F\left\{(1-x)^{\Delta}\ \sum_{r=0}^{\infty}\frac{(\Delta)(% \Delta+\alpha)_{r}(\Delta+\beta)_{r}}{(\Delta+1)_{r}(1)_{r}}\left(\ln(1-x)+% \frac{1}{\Delta}+\sum_{k=0}^{r-1}\left(\frac{1}{\Delta+\alpha+k}+\frac{1}{% \Delta+\beta+k}-\frac{1}{\Delta+1+k}-\frac{1}{1+k}\right)\right)(1-x)^{r}% \right\}\end{aligned}
  68. x ( 1 - x ) y ′′ + ( γ - ( 1 + α + β ) x ) y - α β y = 0 d y d x = d y d s × d s d x = - s 2 × d y d s = - s 2 y d 2 y d x 2 = d d x ( d y d x ) = d d x ( - s 2 × d y d s ) = d d s ( - s 2 × d y d s ) × d s d x = ( ( - 2 s ) × d y d s + ( - s 2 ) d 2 y d s 2 ) × ( - s 2 ) = 2 s 3 y + s 4 y ′′ \begin{aligned}&\displaystyle x(1-x)y^{\prime\prime}+\left(\gamma-(1+\alpha+% \beta)x\right)y^{\prime}-\alpha\beta y=0\\ &\displaystyle\frac{dy}{dx}=\frac{dy}{ds}\times\frac{ds}{dx}=-s^{2}\times\frac% {dy}{ds}=-s^{2}y^{\prime}\\ &\displaystyle\frac{d^{2}y}{dx^{2}}=\frac{d}{dx}\left(\frac{dy}{dx}\right)=% \frac{d}{dx}\left(-s^{2}\times\frac{dy}{ds}\right)=\frac{d}{ds}\left(-s^{2}% \times\frac{dy}{ds}\right)\times\frac{ds}{dx}=\left((-2s)\times\frac{dy}{ds}+(% -s^{2})\frac{d^{2}y}{ds^{2}}\right)\times(-s^{2})=2s^{3}y^{\prime}+s^{4}y^{% \prime\prime}\end{aligned}
  69. 1 s ( 1 - 1 s ) ( 2 s 3 y + s 4 y ′′ ) + ( γ - ( 1 + α + β ) 1 s ) ( - s 2 y ) - α β y = 0 \frac{1}{s}\left(1-\frac{1}{s}\right)\left(2s^{3}y^{\prime}+s^{4}y^{\prime% \prime}\right)+\left(\gamma-(1+\alpha+\beta)\frac{1}{s}\right)(-s^{2}y^{\prime% })-\alpha\beta y=0
  70. ( s 3 - s 2 ) y ′′ + ( ( 2 - γ ) s 2 + ( α + β - 1 ) s ) y - α β y = 0. \left(s^{3}-s^{2}\right)y^{\prime\prime}+\left((2-\gamma)s^{2}+(\alpha+\beta-1% )s\right)y^{\prime}-\alpha\beta y=0.
  71. P 0 ( s ) = - α β , P 1 ( s ) = ( 2 - γ ) s 2 + ( α + β - 1 ) s , P 2 ( s ) = s 3 - s 2 . \begin{aligned}\displaystyle P_{0}(s)&\displaystyle=-\alpha\beta,\\ \displaystyle P_{1}(s)&\displaystyle=(2-\gamma)s^{2}+(\alpha+\beta-1)s,\\ \displaystyle P_{2}(s)&\displaystyle=s^{3}-s^{2}.\end{aligned}
  72. lim s a ( s - a ) P 1 ( s ) P 2 ( s ) = lim s 0 ( s - 0 ) ( ( 2 - γ ) s 2 + ( α + β - 1 ) s ) s 3 - s 2 = lim s 0 ( 2 - γ ) s 2 + ( α + β - 1 ) s s 2 - s = lim s 0 ( 2 - γ ) s + ( α + β - 1 ) s - 1 = 1 - α - β . lim s a ( s - a ) 2 P 0 ( s ) P 2 ( s ) = lim s 0 ( s - 0 ) 2 ( - α β ) s 3 - s 2 = lim s 0 ( - α β ) s - 1 = α β . \begin{aligned}\displaystyle\lim_{s\to a}\frac{(s-a)P_{1}(s)}{P_{2}(s)}&% \displaystyle=\lim_{s\to 0}\frac{(s-0)((2-\gamma)s^{2}+(\alpha+\beta-1)s)}{s^{% 3}-s^{2}}\\ &\displaystyle=\lim_{s\to 0}\frac{(2-\gamma)s^{2}+(\alpha+\beta-1)s}{s^{2}-s}% \\ &\displaystyle=\lim_{s\to 0}\frac{(2-\gamma)s+(\alpha+\beta-1)}{s-1}=1-\alpha-% \beta.\\ \displaystyle\lim_{s\to a}\frac{(s-a)^{2}P_{0}(s)}{P_{2}(s)}&\displaystyle=% \lim_{s\to 0}\frac{(s-0)^{2}(-\alpha\beta)}{s^{3}-s^{2}}=\lim_{s\to 0}\frac{(-% \alpha\beta)}{s-1}=\alpha\beta.\end{aligned}
  73. y = r = 0 a r s r + c y=\sum_{r=0}^{\infty}{a_{r}s^{r+c}}
  74. y = r = 0 a r ( r + c ) s r + c - 1 y ′′ = r = 0 a r ( r + c ) ( r + c - 1 ) s r + c - 2 \begin{aligned}\displaystyle y^{\prime}&\displaystyle=\sum\limits_{r=0}^{% \infty}{a_{r}(r+c)s^{r+c-1}}\\ \displaystyle y^{\prime\prime}&\displaystyle=\sum\limits_{r=0}^{\infty}{a_{r}(% r+c)(r+c-1)s^{r+c-2}}\end{aligned}
  75. ( s 3 - s 2 ) y ′′ + ( ( 2 - γ ) s 2 + ( α + β - 1 ) s ) y - ( α β ) y = 0 \left(s^{3}-s^{2}\right)y^{\prime\prime}+\left((2-\gamma)s^{2}+(\alpha+\beta-1% )s\right)y^{\prime}-(\alpha\beta)y=0
  76. ( s 3 - s 2 ) r = 0 a r ( r + c ) ( r + c - 1 ) s r + c - 2 + ( ( 2 - γ ) s 2 + ( α + β - 1 ) s ) r = 0 a r ( r + c ) s r + c - 1 - ( α β ) r = 0 a r s r + c = 0 \left(s^{3}-s^{2}\right)\sum_{r=0}^{\infty}{a_{r}(r+c)(r+c-1)s^{r+c-2}}+\left(% (2-\gamma)s^{2}+(\alpha+\beta-1)s\right)\sum_{r=0}^{\infty}{a_{r}(r+c)s^{r+c-1% }}-(\alpha\beta)\sum_{r=0}^{\infty}{a_{r}s^{r+c}}=0
  77. r = 0 a r ( r + c ) ( r + c - 1 ) s r + c + 1 - r = 0 a r ( r + c ) ( r + c - 1 ) x r + c + ( 2 - γ ) r = 0 a r ( r + c ) s r + c + 1 + ( α + β - 1 ) r = 0 a r ( r + c ) s r + c - α β r = 0 a r s r + c = 0. \sum_{r=0}^{\infty}{a_{r}(r+c)(r+c-1)s^{r+c+1}}-\sum_{r=0}^{\infty}{a_{r}(r+c)% (r+c-1)x^{r+c}}+(2-\gamma)\sum_{r=0}^{\infty}{a_{r}(r+c)s^{r+c+1}}+(\alpha+% \beta-1)\sum_{r=0}^{\infty}{a_{r}(r+c)s^{r+c}}-\alpha\beta\sum_{r=0}^{\infty}{% a_{r}s^{r+c}}=0.
  78. r = 1 a r - 1 ( r + c - 1 ) ( r + c - 2 ) s r + c - r = 0 a r ( r + c ) ( r + c - 1 ) s r + c + ( 2 - γ ) r = 1 a r - 1 ( r + c - 1 ) s r + c + + ( α + β - 1 ) r = 0 a r ( r + c ) s r + c - α β r = 0 a r s r + c = 0 \begin{aligned}&\displaystyle\sum_{r=1}^{\infty}{a_{r-1}(r+c-1)(r+c-2)s^{r+c}}% -\sum_{r=0}^{\infty}{a_{r}(r+c)(r+c-1)s^{r+c}}+(2-\gamma)\sum_{r=1}^{\infty}{a% _{r-1}(r+c-1)s^{r+c}}+\\ &\displaystyle\qquad\qquad+(\alpha+\beta-1)\sum_{r=0}^{\infty}{a_{r}(r+c)s^{r+% c}}-\alpha\beta\sum_{r=0}^{\infty}{a_{r}s^{r+c}}=0\end{aligned}
  79. a 0 ( - ( c ) ( c - 1 ) + ( α + β - 1 ) ( c ) - α β ) s c + r = 1 a r - 1 ( r + c - 1 ) ( r + c - 2 ) s r + c - r = 1 a r ( r + c ) ( r + c - 1 ) x r + c + + ( 2 - γ ) r = 1 a r - 1 ( r + c - 1 ) s r + c + ( α + β - 1 ) r = 1 a r ( r + c ) s r + c - α β r = 1 a r s r + c = 0 \begin{aligned}&\displaystyle a_{0}\left(-(c)(c-1)+(\alpha+\beta-1)(c)-\alpha% \beta\right)s^{c}+\sum_{r=1}^{\infty}{a_{r-1}(r+c-1)(r+c-2)s^{r+c}}-\sum_{r=1}% ^{\infty}{a_{r}(r+c)(r+c-1)x^{r+c}}+\\ &\displaystyle\qquad\qquad+(2-\gamma)\sum_{r=1}^{\infty}{a_{r-1}(r+c-1)s^{r+c}% }+(\alpha+\beta-1)\sum_{r=1}^{\infty}{a_{r}(r+c)s^{r+c}}-\alpha\beta\sum_{r=1}% ^{\infty}{a_{r}s^{r+c}}=0\end{aligned}
  80. a 0 ( - ( c ) ( c - 1 ) + ( α + β - 1 ) ( c ) - α β ) = 0 a_{0}\left(-(c)(c-1)+(\alpha+\beta-1)(c)-\alpha\beta\right)=0
  81. ( c ) ( - c + 1 + α + β - 1 ) - α β ) = 0. (c)(-c+1+\alpha+\beta-1)-\alpha\beta)=0.
  82. ( ( r + c - 1 ) ( r + c - 2 ) + ( 2 - γ ) ( r + c - 1 ) ) a r - 1 + ( - ( r + c ) ( r + c - 1 ) + ( α + β - 1 ) ( r + c ) - α β ) a r = 0 \left((r+c-1)(r+c-2)+(2-\gamma)(r+c-1)\right)a_{r-1}+\left(-(r+c)(r+c-1)+(% \alpha+\beta-1)(r+c)-\alpha\beta\right)a_{r}=0
  83. a r = - ( ( r + c - 1 ) ( r + c - 2 ) + ( 2 - γ ) ( r + c - 1 ) ) ( - ( r + c ) ( r + c - 1 ) + ( α + β - 1 ) ( r + c ) - α β ) a r - 1 = ( ( r + c - 1 ) ( r + c - γ ) ) ( ( r + c ) ( r + c - α - β ) + α β ) a r - 1 a_{r}=-\frac{\left((r+c-1)(r+c-2)+(2-\gamma)(r+c-1)\right)}{\left(-(r+c)(r+c-1% )+(\alpha+\beta-1)(r+c)-\alpha\beta\right)}a_{r-1}=\frac{\left((r+c-1)(r+c-% \gamma)\right)}{\left((r+c)(r+c-\alpha-\beta)+\alpha\beta\right)}a_{r-1}
  84. ( r + c ) ( r + c - α - β ) + α β = ( r + c - α ) ( r + c ) - β ( r + c ) + α β = ( r + c - α ) ( r + c ) - β ( r + c - α ) . \begin{aligned}\displaystyle(r+c)(r+c-\alpha-\beta)+\alpha\beta&\displaystyle=% (r+c-\alpha)(r+c)-\beta(r+c)+\alpha\beta\\ &\displaystyle=(r+c-\alpha)(r+c)-\beta(r+c-\alpha).\end{aligned}
  85. a r = ( r + c - 1 ) ( r + c - γ ) ( r + c - α ) ( r + c - β ) a r - 1 , r 1 a_{r}=\frac{(r+c-1)(r+c-\gamma)}{(r+c-\alpha)(r+c-\beta)}a_{r-1},\quad\forall r\geq 1
  86. a 1 = ( c ) ( c + 1 - γ ) ( c + 1 - α ) ( c + 1 - β ) a 0 a 2 = ( c + 1 ) ( c + 2 - γ ) ( c + 2 - α ) ( c + 2 - β ) a 1 = ( c + 1 ) ( c ) ( c + 2 - γ ) ( c + 1 - γ ) ( c + 2 - α ) ( c + 1 - α ) ( c + 2 - β ) ( c + 1 - β ) a 0 = ( c ) 2 ( c + 1 - γ ) 2 ( c + 1 - α ) 2 ( c + 1 - β ) 2 a 0 \begin{aligned}\displaystyle a_{1}&\displaystyle=\frac{(c)(c+1-\gamma)}{(c+1-% \alpha)(c+1-\beta)}a_{0}\\ \displaystyle a_{2}&\displaystyle=\frac{(c+1)(c+2-\gamma)}{(c+2-\alpha)(c+2-% \beta)}a_{1}=\frac{(c+1)(c)(c+2-\gamma)(c+1-\gamma)}{(c+2-\alpha)(c+1-\alpha)(% c+2-\beta)(c+1-\beta)}a_{0}=\frac{(c)_{2}(c+1-\gamma)_{2}}{(c+1-\alpha)_{2}(c+% 1-\beta)_{2}}a_{0}\end{aligned}
  87. a r = ( c ) r ( c + 1 - γ ) r ( c + 1 - α ) r ( c + 1 - β ) r a 0 r 0 a_{r}=\frac{(c)_{r}(c+1-\gamma)_{r}}{(c+1-\alpha)_{r}(c+1-\beta)_{r}}a_{0}% \quad\forall r\geq 0
  88. y = a 0 r = 0 ( c ) r ( c + 1 - γ ) r ( c + 1 - α ) r ( c + 1 - β ) r s r + c y=a_{0}\sum_{r=0}^{\infty}\frac{(c)_{r}(c+1-\gamma)_{r}}{(c+1-\alpha)_{r}(c+1-% \beta)_{r}}s^{r+c}
  89. y = a 0 r = 0 ( c ) r ( c + 1 - γ ) r ( c + 1 - α ) r ( c + 1 - β ) r s r + c , y=a_{0}\sum_{r=0}^{\infty}\frac{(c)_{r}(c+1-\gamma)_{r}}{(c+1-\alpha)_{r}(c+1-% \beta)_{r}}s^{r+c},
  90. y 1 = a 0 r = 0 ( α ) r ( α + 1 - γ ) r ( 1 ) r ( α + 1 - β ) r s r + α = a 0 s α F 1 2 ( α , α + 1 - γ ; α + 1 - β ; s ) y 2 = a 0 r = 0 ( β ) r ( β + 1 - γ ) r ( β + 1 - α ) r ( 1 ) r s r + β = a 0 s β F 1 2 ( β , β + 1 - γ ; β + 1 - α ; s ) \begin{aligned}\displaystyle y_{1}&\displaystyle=a_{0}\sum_{r=0}^{\infty}\frac% {(\alpha)_{r}(\alpha+1-\gamma)_{r}}{(1)_{r}(\alpha+1-\beta)_{r}}s^{r+\alpha}=a% _{0}s^{\alpha}\ {}_{2}F_{1}(\alpha,\alpha+1-\gamma;\alpha+1-\beta;s)\\ \displaystyle y_{2}&\displaystyle=a_{0}\sum_{r=0}^{\infty}\frac{(\beta)_{r}(% \beta+1-\gamma)_{r}}{(\beta+1-\alpha)_{r}(1)_{r}}s^{r+\beta}=a_{0}s^{\beta}\ {% }_{2}F_{1}(\beta,\beta+1-\gamma;\beta+1-\alpha;s)\end{aligned}
  91. y = A { x - α F 1 2 ( α , α + 1 - γ ; α + 1 - β ; x - 1 ) } + B { x - β F 1 2 ( β , β + 1 - γ ; β + 1 - α ; x - 1 ) } y=A\left\{x^{-\alpha}\ {}_{2}F_{1}\left(\alpha,\alpha+1-\gamma;\alpha+1-\beta;% x^{-1}\right)\right\}+B\left\{x^{-\beta}\ {}_{2}F_{1}\left(\beta,\beta+1-% \gamma;\beta+1-\alpha;x^{-1}\right)\right\}
  92. y = a 0 r = 0 ( c ) r ( c + 1 - γ ) r ( ( c + 1 - α ) r ) 2 s r + c y=a_{0}\sum_{r=0}^{\infty}{\frac{(c)_{r}(c+1-\gamma)_{r}}{\left((c+1-\alpha)_{% r}\right)^{2}}s^{r+c}}
  93. y 1 = a 0 r = 0 ( α ) r ( α + 1 - γ ) r ( 1 ) r ( 1 ) r s r + α = a 0 s α F 1 2 ( α , α + 1 - γ ; 1 ; s ) y 2 = y c | c = α \begin{aligned}\displaystyle y_{1}&\displaystyle=a_{0}\sum_{r=0}^{\infty}{% \frac{(\alpha)_{r}(\alpha+1-\gamma)_{r}}{(1)_{r}(1)_{r}}s^{r+\alpha}}=a_{0}s^{% \alpha}\ {}_{2}F_{1}(\alpha,\alpha+1-\gamma;1;s)\\ \displaystyle y_{2}&\displaystyle=\left.\frac{\partial y}{\partial c}\right|_{% c=\alpha}\end{aligned}
  94. M r = ( c ) r ( c + 1 - γ ) r ( ( c + 1 - α ) r ) 2 M_{r}=\frac{(c)_{r}(c+1-\gamma)_{r}}{\left((c+1-\alpha)_{r}\right)^{2}}
  95. M r c = ( c ) r ( c + 1 - γ ) r ( ( c + 1 - α ) r ) 2 k = 0 r - 1 ( 1 c + k + 1 c + 1 - γ + k - 2 c + 1 - α + k ) \frac{\partial M_{r}}{\partial c}=\frac{(c)_{r}(c+1-\gamma)_{r}}{\left((c+1-% \alpha)_{r}\right)^{2}}\sum_{k=0}^{r-1}\left(\frac{1}{c+k}+\frac{1}{c+1-\gamma% +k}-\frac{2}{c+1-\alpha+k}\right)
  96. y = a 0 s c r = 0 ( c ) r ( c + 1 - γ ) r ( ( c + 1 - α ) r ) 2 s r = a 0 s c r = 0 M r s r = a 0 s c ( ln ( s ) r = 0 ( c ) r ( c + 1 - γ ) r ( ( c + 1 - α ) r ) 2 s r + r = 0 ( c ) r ( c + 1 - γ ) r ( ( c + 1 - α ) r ) 2 { k = 0 r - 1 ( 1 c + k + 1 c + 1 - γ + k - 2 c + 1 - α + k ) } s r ) \begin{aligned}\displaystyle y&\displaystyle=a_{0}s^{c}\sum_{r=0}^{\infty}% \frac{(c)_{r}(c+1-\gamma)_{r}}{\left((c+1-\alpha)_{r}\right)^{2}}s^{r}\\ &\displaystyle=a_{0}s^{c}\sum_{r=0}^{\infty}{M_{r}s^{r}}\\ &\displaystyle=a_{0}s^{c}\left(\ln(s)\sum_{r=0}^{\infty}\frac{(c)_{r}(c+1-% \gamma)_{r}}{\left((c+1-\alpha)_{r}\right)^{2}}s^{r}+\sum_{r=0}^{\infty}\frac{% (c)_{r}(c+1-\gamma)_{r}}{\left((c+1-\alpha)_{r}\right)^{2}}\left\{\sum_{k=0}^{% r-1}{\left(\frac{1}{c+k}+\frac{1}{c+1-\gamma+k}-\frac{2}{c+1-\alpha+k}\right)}% \right\}s^{r}\right)\end{aligned}
  97. y c = a 0 s c r = 0 ( c ) r ( c + 1 - γ ) r ( ( c + 1 - α ) r ) 2 ( ln ( s ) + k = 0 r - 1 ( 1 c + k + 1 c + 1 - γ + k - 2 c + 1 - α + k ) ) s r \frac{\partial y}{\partial c}=a_{0}s^{c}\sum_{r=0}^{\infty}\frac{(c)_{r}(c+1-% \gamma)_{r}}{\left((c+1-\alpha)_{r}\right)^{2}}\left(\ln(s)+\sum_{k=0}^{r-1}{% \left(\frac{1}{c+k}+\frac{1}{c+1-\gamma+k}-\frac{2}{c+1-\alpha+k}\right)}% \right)s^{r}
  98. y 2 = y c | c = α = a 0 s α r = 0 ( α ) r ( α + 1 - γ ) r ( 1 ) r ( 1 ) r ( ln ( s ) + k = 0 r - 1 ( 1 α + k + 1 α + 1 - γ + k - 2 1 + k ) ) s r y_{2}=\left.\frac{\partial y}{\partial c}\right|_{c=\alpha}=a_{0}s^{\alpha}% \sum_{r=0}^{\infty}\frac{(\alpha)_{r}(\alpha+1-\gamma)_{r}}{(1)_{r}(1)_{r}}% \left(\ln(s)+\sum_{k=0}^{r-1}\left(\frac{1}{\alpha+k}+\frac{1}{\alpha+1-\gamma% +k}-\frac{2}{1+k}\right)\right)s^{r}
  99. y = C { x - α F 1 2 ( α , α + 1 - γ ; 1 ; x - 1 ) } + D { x - α r = 0 ( α ) r ( α + 1 - γ ) r ( 1 ) r ( 1 ) r ( ln ( x - 1 ) + k = 0 r - 1 ( 1 α + k + 1 α + 1 - γ + k - 2 1 + k ) ) x - r } y=C\left\{x^{-\alpha}{}_{2}F_{1}\left(\alpha,\alpha+1-\gamma;1;x^{-1}\right)% \right\}+D\left\{x^{-\alpha}\sum_{r=0}^{\infty}\frac{(\alpha)_{r}(\alpha+1-% \gamma)_{r}}{(1)_{r}(1)_{r}}\left(\ln\left(x^{-1}\right)+\sum_{k=0}^{r-1}\left% (\frac{1}{\alpha+k}+\frac{1}{\alpha+1-\gamma+k}-\frac{2}{1+k}\right)\right)x^{% -r}\right\}
  100. a r = ( r + c - 1 ) ( r + c - γ ) ( r + c - α ) ( r + c - β ) a r - 1 a_{r}=\frac{(r+c-1)(r+c-\gamma)}{(r+c-\alpha)(r+c-\beta)}a_{r-1}
  101. y b = b 0 r = 0 ( c - β ) ( c ) r ( c + 1 - γ ) r ( c + 1 - α ) r ( c + 1 - β ) r s r + c y_{b}=b_{0}\sum_{r=0}^{\infty}\frac{(c-\beta)(c)_{r}(c+1-\gamma)_{r}}{(c+1-% \alpha)_{r}(c+1-\beta)_{r}}s^{r+c}
  102. ( c - β ) ( c ) α - β ( c + 1 - γ ) α - β ( c + 1 - α ) α - β ( c + 1 - β ) α - β s α - β \frac{(c-\beta)(c)_{\alpha-\beta}(c+1-\gamma)_{\alpha-\beta}}{(c+1-\alpha)_{% \alpha-\beta}(c+1-\beta)_{\alpha-\beta}}s^{\alpha-\beta}
  103. ( c + 1 - α ) α - β = ( c + 1 - α ) ( c + 2 - α ) ( c - β ) . (c+1-\alpha)_{\alpha-\beta}=(c+1-\alpha)(c+2-\alpha)\cdots(c-\beta).
  104. y 1 = b 0 ( ( β ) α - β ( β + 1 - γ ) α - β ( β + 1 - α ) α - β - 1 ( 1 ) α - β s α - β + ( β ) α - β + 1 ( β + 1 - γ ) α - β + 1 ( β + 1 - α ) α - β - 1 ( 1 ) ( 1 ) α - β + 1 s α - β + 1 + ) = b 0 ( β + 1 - α ) α - β - 1 r = α - β ( β ) r ( β + 1 - γ ) r ( 1 ) r ( 1 ) r + β - α s r \begin{aligned}\displaystyle y_{1}&\displaystyle=b_{0}\left(\frac{(\beta)_{% \alpha-\beta}(\beta+1-\gamma)_{\alpha-\beta}}{(\beta+1-\alpha)_{\alpha-\beta-1% }(1)_{\alpha-\beta}}s^{\alpha-\beta}+\frac{(\beta)_{\alpha-\beta+1}(\beta+1-% \gamma)_{\alpha-\beta+1}}{(\beta+1-\alpha)_{\alpha-\beta-1}(1)(1)_{\alpha-% \beta+1}}s^{\alpha-\beta+1}+\cdots\right)\\ &\displaystyle=\frac{b_{0}}{(\beta+1-\alpha)_{\alpha-\beta-1}}\sum_{r=\alpha-% \beta}^{\infty}\frac{(\beta)_{r}(\beta+1-\gamma)_{r}}{(1)_{r}(1)_{r+\beta-% \alpha}}s^{r}\end{aligned}
  105. y 2 = y b c | c = α . y_{2}=\left.\frac{\partial y_{b}}{\partial c}\right|_{c=\alpha}.
  106. M r = ( c - β ) ( c ) r ( c + 1 - γ ) r ( c + 1 - α ) r ( c + 1 - β ) r . M_{r}=\frac{(c-\beta)(c)_{r}(c+1-\gamma)_{r}}{(c+1-\alpha)_{r}(c+1-\beta)_{r}}.
  107. M r c = ( c - β ) ( c ) r ( c + 1 - γ ) r ( c + 1 - α ) r ( c + 1 - β ) r ( 1 c - β + k = 0 r - 1 ( 1 c + k + 1 c + 1 - γ + k - 1 c + 1 - α + k - 1 c + 1 - β + k ) ) \frac{\partial M_{r}}{\partial c}=\frac{(c-\beta)(c)_{r}(c+1-\gamma)_{r}}{(c+1% -\alpha)_{r}(c+1-\beta)_{r}}\left(\frac{1}{c-\beta}+\sum_{k=0}^{r-1}\left(% \frac{1}{c+k}+\frac{1}{c+1-\gamma+k}-\frac{1}{c+1-\alpha+k}-\frac{1}{c+1-\beta% +k}\right)\right)
  108. y b = b 0 r = 0 ( ( c - β ) ( c ) r ( c + 1 - γ ) r ( c + 1 - α ) r ( c + 1 - β ) r s r + c ) = b 0 s c r = 0 M r s r y_{b}=b_{0}\sum_{r=0}^{\infty}{\left(\frac{(c-\beta)(c)_{r}(c+1-\gamma)_{r}}{(% c+1-\alpha)_{r}(c+1-\beta)_{r}}s^{r+c}\right)}=b_{0}s^{c}\sum_{r=0}^{\infty}{M% _{r}s^{r}}
  109. y c = b 0 s c ln ( s ) r = 0 ( c - β ) ( c ) r ( c + 1 - γ ) r ( c + 1 - α ) r ( c + 1 - β ) r s r + b 0 s c r = 0 ( c - β ) ( c ) r ( c + 1 - γ ) r ( c + 1 - α ) r ( c + 1 - β ) r ( 1 c - β + k = 0 r - 1 ( 1 c + k + 1 c + 1 - γ + k - 1 c + 1 - α + k - 1 c + 1 - β + k ) ) s r \begin{aligned}\displaystyle\frac{\partial y}{\partial c}&\displaystyle=b_{0}s% ^{c}\ln(s)\sum_{r=0}^{\infty}\frac{(c-\beta)(c)_{r}(c+1-\gamma)_{r}}{(c+1-% \alpha)_{r}(c+1-\beta)_{r}}s^{r}\\ &\displaystyle\quad+b_{0}s^{c}\sum_{r=0}^{\infty}\frac{(c-\beta)(c)_{r}(c+1-% \gamma)_{r}}{(c+1-\alpha)_{r}(c+1-\beta)_{r}}\left(\frac{1}{c-\beta}+\sum_{k=0% }^{r-1}\left(\frac{1}{c+k}+\frac{1}{c+1-\gamma+k}-\frac{1}{c+1-\alpha+k}-\frac% {1}{c+1-\beta+k}\right)\right)s^{r}\end{aligned}
  110. y c = b 0 s c r = 0 ( c - β ) ( c ) r ( c + 1 - γ ) r ( c + 1 - α ) r ( c + 1 - β ) r ( ln ( s ) + 1 c - β + k = 0 r - 1 ( 1 c + k + 1 c + 1 - γ + k - 1 c + 1 - α + k - 1 c + 1 - β + k ) ) s r \frac{\partial y}{\partial c}=b_{0}s^{c}\sum_{r=0}^{\infty}\frac{(c-\beta)(c)_% {r}(c+1-\gamma)_{r}}{(c+1-\alpha)_{r}(c+1-\beta)_{r}}\left(\ln(s)+\frac{1}{c-% \beta}+\sum_{k=0}^{r-1}\left(\frac{1}{c+k}+\frac{1}{c+1-\gamma+k}-\frac{1}{c+1% -\alpha+k}-\frac{1}{c+1-\beta+k}\right)\right)s^{r}
  111. y = E { 1 ( β + 1 - α ) α - β - 1 r = α - β ( β ) r ( β + 1 - γ ) r ( 1 ) r ( 1 ) r + β - α x - r } + + F { x - α r = 0 ( α - β ) ( α ) r ( α + 1 - γ ) r ( 1 ) r ( α + 1 - β ) r ( ln ( x - 1 ) + 1 α - β + k = 0 r - 1 ( 1 α + k + 1 α + 1 + k - γ - 1 1 + k - 1 α + 1 + k - β ) ) x - r } \begin{aligned}\displaystyle y&\displaystyle=E\left\{\frac{1}{(\beta+1-\alpha)% _{\alpha-\beta-1}}\sum_{r=\alpha-\beta}^{\infty}\frac{(\beta)_{r}(\beta+1-% \gamma)_{r}}{(1)_{r}(1)_{r+\beta-\alpha}}x^{-r}\right\}+\\ &\displaystyle\quad+F\left\{x^{-\alpha}\sum_{r=0}^{\infty}\frac{(\alpha-\beta)% (\alpha)_{r}(\alpha+1-\gamma)_{r}}{(1)_{r}(\alpha+1-\beta)_{r}}\left(\ln\left(% x^{-1}\right)+\frac{1}{\alpha-\beta}+\sum_{k=0}^{r-1}\left(\frac{1}{\alpha+k}+% \frac{1}{\alpha+1+k-\gamma}-\frac{1}{1+k}-\frac{1}{\alpha+1+k-\beta}\right)% \right)x^{-r}\right\}\end{aligned}

Frostman_lemma.html

  1. μ ( B ( x , r ) ) r s \mu(B(x,r))\leq r^{s}
  2. C s ( A ) := sup { ( A × A d μ ( x ) d μ ( y ) | x - y | s ) - 1 : μ is a Borel measure and μ ( A ) = 1 } . C_{s}(A):=\sup\Bigl\{\Bigl(\int_{A\times A}\frac{d\mu(x)\,d\mu(y)}{|x-y|^{s}}% \Bigr)^{-1}:\mu\,\text{ is a Borel measure and }\mu(A)=1\Bigr\}.
  3. μ \mu
  4. dim H ( A ) = sup { s 0 : C s ( A ) > 0 } . \mathrm{dim}_{H}(A)=\sup\{s\geq 0:C_{s}(A)>0\}.

FSC15307+3253.html

  1. × 10 1 3 \times 10^{1}3
  2. z = 0.93 z=0.93

FTCS_scheme.html

  1. u t = F ( u , x , t , 2 u x 2 ) \frac{\partial u}{\partial t}=F\left(u,x,t,\frac{\partial^{2}u}{\partial x^{2}% }\right)
  2. u ( i Δ x , n Δ t ) = u i n u(i\,\Delta x,n\,\Delta t)=u_{i}^{n}\,
  3. u i n + 1 - u i n Δ t = F i n ( u , x , t , 2 u x 2 ) \frac{u_{i}^{n+1}-u_{i}^{n}}{\Delta t}=F_{i}^{n}\left(u,x,t,\frac{\partial^{2}% u}{\partial x^{2}}\right)
  4. F F
  5. u i n + 1 u_{i}^{n+1}
  6. u u
  7. ( n ) (n)
  8. u t = α 2 u x 2 \frac{\partial u}{\partial t}=\alpha\frac{\partial^{2}u}{\partial x^{2}}
  9. u i n + 1 - u i n Δ t = α Δ x 2 ( u i + 1 n - 2 u i n + u i - 1 n ) \frac{u_{i}^{n+1}-u_{i}^{n}}{\Delta t}=\frac{\alpha}{\Delta x^{2}}\left(u_{i+1% }^{n}-2u_{i}^{n}+u_{i-1}^{n}\right)
  10. r = α Δ t Δ x 2 r=\frac{\alpha\,\Delta t}{\Delta x^{2}}
  11. u i n + 1 = u i n + r ( u i + 1 n - 2 u i n + u i - 1 n ) u_{i}^{n+1}=u_{i}^{n}+r\left(u_{i+1}^{n}-2u_{i}^{n}+u_{i-1}^{n}\right)
  12. r = α Δ t Δ x 2 1 2 . r=\frac{\alpha\,\Delta t}{\Delta x^{2}}\leq\frac{1}{2}.
  13. Δ t \Delta t
  14. Δ t \Delta t

Fuchs'_theorem.html

  1. y ′′ + p ( x ) y + q ( x ) y = g ( x ) y^{\prime\prime}+p(x)y^{\prime}+q(x)y=g(x)\;
  2. p ( x ) p(x)
  3. q ( x ) q(x)
  4. g ( x ) g(x)
  5. x = a x=a
  6. a a
  7. y = n = 0 a n ( x - a ) n + s , a 0 0 y=\sum_{n=0}^{\infty}a_{n}(x-a)^{n+s},\quad a_{0}\neq 0
  8. y = y 0 ln ( x - a ) + n = 0 b n ( x - a ) n + r , b 0 0 y=y_{0}\ln(x-a)+\sum_{n=0}^{\infty}b_{n}(x-a)^{n+r},\quad b_{0}\neq 0
  9. p ( x ) p(x)
  10. q ( x ) q(x)
  11. g ( x ) g(x)

Fugacity_capacity.html

  1. C m = Z m f C_{m}=Z_{m}\cdot f
  2. f = M T / Σ m ( V m Z m ) f=M_{T}/\Sigma_{m}(V_{m}Z_{m})

Fugue_(hash_function).html

  1. SuperMix ( U ) = ROL ( M U + ( j 0 U j i 0 0 0 0 j 1 U j i 0 0 0 0 j 2 U j i 0 0 0 0 j 3 U j i ) M T ) \,\text{SuperMix}(U)=\,\text{ROL}\left(M\cdot U+\begin{pmatrix}\sum_{j\neq 0}U% _{j}^{i}&0&0&0\\ 0&\sum_{j\neq 1}U_{j}^{i}&0&0\\ 0&0&\sum_{j\neq 2}U_{j}^{i}&0\\ 0&0&0&\sum_{j\neq 3}U_{j}^{i}\end{pmatrix}\cdot M^{T}\right)
  2. M = ( 1 4 7 1 1 1 4 7 7 1 1 4 4 7 1 1 ) M=\begin{pmatrix}1&4&7&1\\ 1&1&4&7\\ 7&1&1&4\\ 4&7&1&1\end{pmatrix}
  3. U U
  4. M T M^{T}
  5. R O L ROL
  6. i i
  7. i i
  8. ROL ( W ) j i = W j - i ( mod 4 ) i \,\text{ROL}(W)_{j}^{i}=W_{j-i\;\;(\mathop{{\rm mod}}4)}^{i}

Function_application.html

  1. f ( x ) f(x)\,
  2. f x f\;x
  3. f : ( X × Y ) Z f:(X\times Y)\to Z
  4. f ( x , y ) f(x,y)\!
  5. f x , y f\;\langle x,y\rangle
  6. f : X ( Y Z ) f:X\to(Y\to Z)
  7. f x y f\;x\;y
  8. f ( x ) ( y ) f(x)(y)
  9. $ \$
  10. f $ x = f ( x ) f\mathop{\,\$\,}x=f(x)
  11. f ( g ( h ( j ( x ) ) ) ) f(g(h(j(x))))\,
  12. f $ g $ h $ j $ x f\mathop{\,\$\,}g\mathop{\,\$\,}h\mathop{\,\$\,}j\mathop{\,\$\,}x
  13. ( f g h j ) ( x ) (f\circ g\circ h\circ j)(x)

Fundamental_lemma_of_sieve_theory.html

  1. | A d | = w ( d ) d X + R d . \left|A_{d}\right|=\frac{w(d)}{d}X+R_{d}.
  2. η p ξ ( 1 - w ( p ) p ) - 1 < ( ln ξ ln η ) κ ( 1 + C ln η ) . \prod_{\eta\leq p\leq\xi}\left(1-\frac{w(p)}{p}\right)^{-1}<\left(\frac{\ln\xi% }{\ln\eta}\right)^{\kappa}\left(1+\frac{C}{\ln\eta}\right).
  3. S ( a , P , z ) = X p z , p P ( 1 - w ( p ) p ) { 1 + O ( u - u / 2 ) } + O ( d z u , d | P ( z ) | R d | ) . S(a,P,z)=X\prod_{p\leq z,p\in P}\left(1-\frac{w(p)}{p}\right)\{1+O(u^{-u/2})\}% +O\left(\sum_{d\leq z^{u},d|P(z)}|R_{d}|\right).
  4. η p ξ w ( p ) ln p p < κ ln ξ η + C . \sum_{\eta\leq p\leq\xi}\frac{w(p)\ln p}{p}<\kappa\ln\frac{\xi}{\eta}+C.
  5. S ( a , P , z ) = X p z , p P ( 1 - w ( p ) p ) { 1 + O ( e - u / 2 ) } . S(a,P,z)=X\prod_{p\leq z,\ p\in P}\left(1-\frac{w(p)}{p}\right)\{1+O(e^{-u/2})\}.

Fundamental_theorem_of_calculus.html

  1. A ( x + h ) - A ( x ) f ( x ) h A(x+h)-A(x)\approx f(x)h
  2. A ( x + h ) - A ( x ) = f ( x ) h + ( Red Excess ) A(x+h)-A(x)=f(x)h+(\rm Red\ Excess)
  3. f ( x ) = A ( x + h ) - A ( x ) h - ( Red Excess ) h f(x)=\frac{A(x+h)-A(x)}{h}-\frac{(\rm Red\ Excess)}{h}
  4. f ( x ) = lim h 0 A ( x + h ) - A ( x ) h f(x)=\lim_{h\to 0}\frac{A(x+h)-A(x)}{h}
  5. × \times
  6. \sum
  7. × \times
  8. v ( t ) × Δ t \sum v(t)\times\Delta t
  9. Δ t \Delta t
  10. x x
  11. t t
  12. v ( t ) v(t)
  13. d x = v ( t ) d t dx=v(t)dt
  14. x ( t ) x(t)
  15. v ( t ) v(t)
  16. t t
  17. F ( x ) = a x f ( t ) d t . F(x)=\int_{a}^{x}\!f(t)\,dt.
  18. F ( x ) = f ( x ) F^{\prime}(x)=f(x)\,
  19. a b f ( t ) d t = F ( b ) - F ( a ) . \int_{a}^{b}f(t)\,dt=F(b)-F(a).
  20. F ( x ) = f ( x ) . F^{\prime}(x)=f(x).
  21. a b f ( x ) d x = F ( b ) - F ( a ) \int_{a}^{b}f(x)\,dx=F(b)-F(a)
  22. F ( x ) = a x f ( t ) d t . F(x)=\int_{a}^{x}f(t)\,dt.
  23. F ( x 1 ) = a x 1 f ( t ) d t F(x_{1})=\int_{a}^{x_{1}}f(t)\,dt
  24. F ( x 1 + Δ x ) = a x 1 + Δ x f ( t ) d t . F(x_{1}+\Delta x)=\int_{a}^{x_{1}+\Delta x}f(t)\,dt.
  25. F ( x 1 + Δ x ) - F ( x 1 ) = a x 1 + Δ x f ( t ) d t - a x 1 f ( t ) d t . ( 1 ) F(x_{1}+\Delta x)-F(x_{1})=\int_{a}^{x_{1}+\Delta x}f(t)\,dt-\int_{a}^{x_{1}}f% (t)\,dt.\qquad(1)
  26. a x 1 f ( t ) d t + x 1 x 1 + Δ x f ( t ) d t = a x 1 + Δ x f ( t ) d t . \int_{a}^{x_{1}}f(t)\,dt+\int_{x_{1}}^{x_{1}+\Delta x}f(t)\,dt=\int_{a}^{x_{1}% +\Delta x}f(t)\,dt.
  27. a x 1 + Δ x f ( t ) d t - a x 1 f ( t ) d t = x 1 x 1 + Δ x f ( t ) d t . \int_{a}^{x_{1}+\Delta x}f(t)\,dt-\int_{a}^{x_{1}}f(t)\,dt=\int_{x_{1}}^{x_{1}% +\Delta x}f(t)\,dt.
  28. F ( x 1 + Δ x ) - F ( x 1 ) = x 1 x 1 + Δ x f ( t ) d t . ( 2 ) F(x_{1}+\Delta x)-F(x_{1})=\int_{x_{1}}^{x_{1}+\Delta x}f(t)\,dt.\qquad(2)
  29. c ( Δ x ) c(\Delta x)
  30. x 1 x 1 + Δ x f ( t ) d t = f ( c ( Δ x ) ) Δ x . \int_{x_{1}}^{x_{1}+\Delta x}f(t)\,dt=f\left(c(\Delta x)\right)\Delta x.
  31. c ( Δ x ) c(\Delta x)
  32. Δ x \Delta x
  33. F ( x 1 + Δ x ) - F ( x 1 ) = f ( c ) Δ x . F(x_{1}+\Delta x)-F(x_{1})=f(c)\Delta x.
  34. F ( x 1 + Δ x ) - F ( x 1 ) Δ x = f ( c ) . \frac{F(x_{1}+\Delta x)-F(x_{1})}{\Delta x}=f(c).
  35. lim Δ x 0 F ( x 1 + Δ x ) - F ( x 1 ) Δ x = lim Δ x 0 f ( c ) . \lim_{\Delta x\to 0}\frac{F(x_{1}+\Delta x)-F(x_{1})}{\Delta x}=\lim_{\Delta x% \to 0}f(c).
  36. F ( x 1 ) = lim Δ x 0 f ( c ) . ( 3 ) F^{\prime}(x_{1})=\lim_{\Delta x\to 0}f(c).\qquad(3)
  37. lim Δ x 0 x 1 = x 1 \lim_{\Delta x\to 0}x_{1}=x_{1}
  38. lim Δ x 0 x 1 + Δ x = x 1 . \lim_{\Delta x\to 0}x_{1}+\Delta x=x_{1}.\,
  39. lim Δ x 0 c = x 1 . \lim_{\Delta x\to 0}c=x_{1}.
  40. F ( x 1 ) = lim c x 1 f ( c ) . F^{\prime}(x_{1})=\lim_{c\to x_{1}}f(c).
  41. F ( x 1 ) = f ( x 1 ) . F^{\prime}(x_{1})=f(x_{1}).
  42. G ( x ) = a x f ( t ) d t G(x)=\int_{a}^{x}f(t)\,dt
  43. F ( a ) + c = G ( a ) = a a f ( t ) d t = 0 , F(a)+c=G(a)=\int_{a}^{a}f(t)\,dt=0,
  44. a b f ( x ) d x = G ( b ) = F ( b ) - F ( a ) . \int_{a}^{b}f(x)\,dx=G(b)=F(b)-F(a).
  45. a = x 0 < x 1 < x 2 < < x n - 1 < x n = b . a=x_{0}<x_{1}<x_{2}<\cdots<x_{n-1}<x_{n}=b.\,
  46. F ( b ) - F ( a ) = F ( x n ) - F ( x 0 ) . F(b)-F(a)=F(x_{n})-F(x_{0}).\,
  47. F ( b ) - F ( a ) \displaystyle F(b)-F(a)
  48. F ( b ) - F ( a ) = i = 1 n [ F ( x i ) - F ( x i - 1 ) ] . ( 1 ) F(b)-F(a)=\sum_{i=1}^{n}\,[F(x_{i})-F(x_{i-1})].\qquad(1)
  49. F ( c ) = F ( b ) - F ( a ) b - a . F^{\prime}(c)=\frac{F(b)-F(a)}{b-a}.
  50. F ( c ) ( b - a ) = F ( b ) - F ( a ) . F^{\prime}(c)(b-a)=F(b)-F(a).\,
  51. F ( x i ) - F ( x i - 1 ) = F ( c i ) ( x i - x i - 1 ) . F(x_{i})-F(x_{i-1})=F^{\prime}(c_{i})(x_{i}-x_{i-1}).
  52. F ( b ) - F ( a ) = i = 1 n [ F ( c i ) ( x i - x i - 1 ) ] . F(b)-F(a)=\sum_{i=1}^{n}\,[F^{\prime}(c_{i})(x_{i}-x_{i-1})].
  53. F ( c i ) = f ( c i ) . F^{\prime}(c_{i})=f(c_{i}).
  54. x i - x i - 1 x_{i}-x_{i-1}
  55. Δ x \Delta x
  56. i i
  57. F ( b ) - F ( a ) = i = 1 n [ f ( c i ) ( Δ x i ) ] . ( 2 ) F(b)-F(a)=\sum_{i=1}^{n}\,[f(c_{i})(\Delta x_{i})].\qquad(2)
  58. Δ x i \Delta x_{i}
  59. lim Δ x i 0 F ( b ) - F ( a ) = lim Δ x i 0 i = 1 n [ f ( c i ) ( Δ x i ) ] . \lim_{\|\Delta x_{i}\|\to 0}F(b)-F(a)=\lim_{\|\Delta x_{i}\|\to 0}\sum_{i=1}^{% n}\,[f(c_{i})(\Delta x_{i})].
  60. Δ x i \|\Delta x_{i}\|
  61. F ( b ) - F ( a ) = lim Δ x i 0 i = 1 n [ f ( c i ) ( Δ x i ) ] . F(b)-F(a)=\lim_{\|\Delta x_{i}\|\to 0}\sum_{i=1}^{n}\,[f(c_{i})(\Delta x_{i})].
  62. F ( b ) - F ( a ) = a b f ( x ) d x , F(b)-F(a)=\int_{a}^{b}f(x)\,dx,
  63. G ( x ) - G ( a ) = a x f ( t ) d t G(x)-G(a)=\int_{a}^{x}f(t)\,dt
  64. F ( x ) = a x f ( t ) d t = G ( x ) - G ( a ) F(x)=\int_{a}^{x}f(t)\,dt\ =G(x)-G(a)
  65. G ( x ) = 0 x f ( t ) d t G(x)=\int_{0}^{x}f(t)\,dt\,
  66. 2 5 x 2 d x . \int_{2}^{5}x^{2}\,dx.
  67. f ( x ) = x 2 f(x)=x^{2}\,
  68. F ( x ) = x 3 3 F(x)=\frac{x^{3}}{3}
  69. 2 5 x 2 d x = F ( 5 ) - F ( 2 ) = 5 3 3 - 2 3 3 = 125 3 - 8 3 = 117 3 = 39. \int_{2}^{5}x^{2}\,dx=F(5)-F(2)=\frac{5^{3}}{3}-\frac{2^{3}}{3}=\frac{125}{3}-% \frac{8}{3}=\frac{117}{3}=39.
  70. d d x 0 x t 3 d t \frac{d}{dx}\int_{0}^{x}t^{3}\,dt
  71. f ( t ) = t 3 f(t)=t^{3}\,
  72. F ( t ) = t 4 4 F(t)=\frac{t^{4}}{4}
  73. d d x 0 x t 3 d t = d d x F ( x ) - d d x F ( 0 ) = d d x x 4 4 = x 3 . \frac{d}{dx}\int_{0}^{x}t^{3}\,dt=\frac{d}{dx}F(x)-\frac{d}{dx}F(0)=\frac{d}{% dx}\frac{x^{4}}{4}=x^{3}.
  74. d d x 0 x t 3 d t = f ( x ) d x d x - f ( 0 ) d 0 d x = x 3 . \frac{d}{dx}\int_{0}^{x}t^{3}\,dt=f(x)\frac{dx}{dx}-f(0)\frac{d0}{dx}=x^{3}.
  75. F ( x ) = a x f ( t ) d t F(x)=\int_{a}^{x}f(t)\,dt
  76. F ( b ) - F ( a ) = a b f ( t ) d t . F(b)-F(a)=\int_{a}^{b}f(t)\,dt.
  77. γ f ( z ) d z = F ( γ ( b ) ) - F ( γ ( a ) ) . \int_{\gamma}f(z)\,dz=F(\gamma(b))-F(\gamma(a)).
  78. ω \omega
  79. M d ω = M ω . \int_{M}d\omega=\oint_{\partial M}\omega.
  80. ω \omega

Future_of_an_expanding_universe.html

  1. 10 10 56 10^{10^{56}}
  2. 10 10 26 10^{10^{26}}
  3. 10 10 76 10^{10^{76}}
  4. 10 10 26 10^{10^{26}}
  5. 10 10 76 10^{10^{76}}
  6. 10 10 76 10^{10^{76}}

Fürer's_algorithm.html

  1. O ( n log n log log n ) O(n\log n\log\log n)
  2. n n
  3. n n
  4. n log n 2 O ( log * n ) n\log n\,2^{O(\log^{*}n)}
  5. log log n \log\log n
  6. 2 log * n 2^{\log^{*}n}
  7. n n
  8. O ( n log n 2 4 log * n ) O(n\log n2^{4\log^{*}n})
  9. O ( log * n ) O(\log^{*}n)
  10. O ( n log n 2 3 log * n ) O(n\log n2^{3\log^{*}n})

G2-structure.html

  1. G 2 G_{2}
  2. G 2 G_{2}
  3. G 2 G_{2}
  4. G 2 G_{2}

Gamma_ray.html

  1. I ( x ) = I 0 e - μ x I(x)=I_{0}\cdot e^{-\mu x}

Garside_element.html

  1. { r M for some x M , Δ = x r } , \{r\in M\mid\mbox{for some }~{}x\in M,\Delta=xr\},
  2. { M for some x M , Δ = x } , \{\ell\in M\mid\mbox{for some }~{}x\in M,\Delta=\ell x\},

Gas.html

  1. P V = n R T , PV=nRT,
  2. P = ρ R s T , P=\rho R_{s}T,
  3. R s R_{s}
  4. P V = k PV=k
  5. P 1 V 1 = P 2 V 2 . \qquad P_{1}V_{1}=P_{2}V_{2}.
  6. V 1 T 1 = V 2 T 2 \frac{V_{1}}{T_{1}}=\frac{V_{2}}{T_{2}}
  7. P 1 T 1 = P 2 T 2 \frac{P_{1}}{T_{1}}=\frac{P_{2}}{T_{2}}\,
  8. V 1 n 1 = V 2 n 2 \frac{V_{1}}{n_{1}}=\frac{V_{2}}{n_{2}}\,

Gas_diffusion_electrode.html

  1. p = 2 γ cos θ r p=\frac{2\ \gamma\cos\theta}{r}

Gas_pycnometer.html

  1. V s = V c + V r 1 - P 1 P 2 V_{s}=V_{c}+\frac{V_{r}}{1-\frac{P_{1}}{P_{2}}}

Gauge_gravitation_theory.html

  1. X X
  2. K = Γ + Θ K=\Gamma+\Theta
  3. Γ \Gamma
  4. Θ = Θ μ a d x μ ϑ a \Theta=\Theta_{\mu}^{a}dx^{\mu}\otimes\vartheta_{a}
  5. ϑ a = ϑ a λ λ \vartheta_{a}=\vartheta_{a}^{\lambda}\partial_{\lambda}
  6. K K
  7. Θ = θ = d x μ μ \Theta=\theta=dx^{\mu}\otimes\partial_{\mu}
  8. X X
  9. Θ \Theta
  10. Θ \Theta
  11. ϑ a \vartheta_{a}
  12. θ = ϑ a ϑ a \theta=\vartheta^{a}\otimes\vartheta_{a}
  13. ϑ a \vartheta^{a}
  14. P X P\to X
  15. X X
  16. F X FX
  17. X X
  18. T X T\to X
  19. X X
  20. T T
  21. X X
  22. F X FX
  23. G G
  24. P X P\to X
  25. H H
  26. P P
  27. H H
  28. P P
  29. H H
  30. P / H X P/H\to X
  31. G L ( 4 , ) GL(4,\mathbb{R})
  32. G L ( 4 , ) GL(4,\mathbb{R})
  33. F X FX
  34. X X
  35. F X / O ( 1 , 3 ) X FX/O(1,3)\to X
  36. S L ( 2 , ) SL(2,\mathbb{C})
  37. S O + ( 1 , 3 ) SO^{+}(1,3)
  38. T ( 3 ) T(3)

Gauss's_law_for_gravity.html

  1. V \oint_{\partial V}
  2. \nabla\cdot
  3. V 𝐠 d 𝐀 = V 𝐠 d V \oint_{\partial V}\mathbf{g}\cdot d\mathbf{A}=\int_{V}\nabla\cdot\mathbf{g}\ dV
  4. M = V ρ d V M=\int_{V}\rho\ dV
  5. V 𝐠 d V = - 4 π G V ρ d V \int_{V}\nabla\cdot\mathbf{g}\ dV=-4\pi G\int_{V}\rho\ dV
  6. V ( 𝐠 ) d V = V ( - 4 π G ρ ) d V . \int_{V}(\nabla\cdot\mathbf{g})\ dV=\int_{V}(-4\pi G\rho)\ dV.
  7. 𝐠 = - 4 π G ρ , \nabla\cdot\mathbf{g}=-4\pi G\rho,
  8. 𝐠 ( 𝐫 ) = - G M 𝐞 𝐫 r 2 \mathbf{g}(\mathbf{r})=-GM\frac{\mathbf{e_{r}}}{r^{2}}
  9. 𝐠 ( 𝐫 ) = - G ρ ( 𝐬 ) ( 𝐫 - 𝐬 ) | 𝐫 - 𝐬 | 3 d 3 𝐬 . \mathbf{g}(\mathbf{r})=-G\int\rho(\mathbf{s})\frac{(\mathbf{r}-\mathbf{s})}{|% \mathbf{r}-\mathbf{s}|^{3}}d^{3}\mathbf{s}.
  10. ( 𝐬 | 𝐬 | 3 ) = 4 π δ ( 𝐬 ) \nabla\cdot\left(\frac{\mathbf{s}}{|\mathbf{s}|^{3}}\right)=4\pi\delta(\mathbf% {s})
  11. 𝐠 ( 𝐫 ) = - 4 π G ρ ( 𝐬 ) δ ( 𝐫 - 𝐬 ) d 3 𝐬 . \nabla\cdot\mathbf{g}(\mathbf{r})=-4\pi G\int\rho(\mathbf{s})\ \delta(\mathbf{% r}-\mathbf{s})\ d^{3}\mathbf{s}.
  12. 𝐠 ( 𝐫 ) = - 4 π G ρ ( 𝐫 ) \nabla\cdot\mathbf{g}(\mathbf{r})=-4\pi G\rho(\mathbf{r})
  13. × 𝐠 = 0 \nabla\times\mathbf{g}=0
  14. V 𝐠 d 𝐀 = - 4 π G M . \oint_{\partial V}\mathbf{g}\cdot d\mathbf{A}=-4\pi GM.
  15. 𝐠 ( 𝐫 ) = g ( r ) 𝐞 𝐫 \mathbf{g}(\mathbf{r})=g(r)\mathbf{e_{r}}
  16. 4 π r 2 4\pi r^{2}
  17. g ( r ) V 𝐞 𝐫 d 𝐀 = - 4 π G M g(r)\oint_{\partial V}\mathbf{e_{r}}\cdot d\mathbf{A}=-4\pi GM
  18. g ( r ) ( 4 π r 2 ) = - 4 π G M g(r)(4\pi r^{2})=-4\pi GM
  19. g ( r ) = - G M / r 2 g(r)=-GM/r^{2}
  20. 𝐠 ( 𝐫 ) = - G M 𝐞 𝐫 r 2 \mathbf{g}(\mathbf{r})=-GM\frac{\mathbf{e_{r}}}{r^{2}}
  21. 𝐠 = - ϕ . \mathbf{g}=-\nabla\phi.
  22. 2 ϕ = 4 π G ρ . \nabla^{2}\phi=4\pi G\rho.
  23. r = | 𝐫 | r=|\mathbf{r}|
  24. 1 r 2 r ( r 2 ϕ r ) = 4 π G ρ ( r ) \frac{1}{r^{2}}\frac{\partial}{\partial r}\left(r^{2}\,\frac{\partial\phi}{% \partial r}\right)=4\pi G\rho(r)
  25. 𝐠 ( 𝐫 ) = - 𝐞 𝐫 ϕ r . \mathbf{g}(\mathbf{r})=-\mathbf{e_{r}}\frac{\partial\phi}{\partial r}.
  26. ( x , t ) = - ρ ( x , t ) ϕ ( x , t ) - 1 8 π G ( ϕ ( x , t ) ) 2 \mathcal{L}(\vec{x},t)=-\rho(\vec{x},t)\phi(\vec{x},t)-{1\over 8\pi G}(\nabla% \phi(\vec{x},t))^{2}
  27. 4 π G ρ ( x , t ) = 2 ϕ ( x , t ) . 4\pi G\rho(\vec{x},t)=\nabla^{2}\phi(\vec{x},t).

Gauss's_law_for_magnetism.html

  1. 𝐁 = × 𝐀 \mathbf{B}=\nabla\times\mathbf{A}
  2. × 𝐀 = × ( 𝐀 + ϕ ) \nabla\times\mathbf{A}=\nabla\times(\mathbf{A}+\nabla\phi)
  3. × ϕ = s y m b o l 0 \nabla\times\nabla\phi=symbol{0}
  4. 𝐁 = 4 π ρ m \nabla\cdot\mathbf{B}=4\pi\rho_{m}
  5. 𝐁 = ρ m \nabla\cdot\mathbf{B}=\rho_{m}
  6. 𝐁 = μ 0 ρ m \nabla\cdot\mathbf{B}=\mu_{0}\rho_{m}
  7. 𝐁 = × 𝐀 \mathbf{B}=\nabla\times\mathbf{A}

Gauss_circle_problem.html

  1. m 2 + n 2 r 2 . m^{2}+n^{2}\leq r^{2}.
  2. π r 2 \pi r^{2}
  3. N ( r ) N(r)
  4. N ( r ) = π r 2 + E ( r ) N(r)=\pi r^{2}+E(r)\,
  5. N ( 4 ) = 49 N(4)=49
  6. N ( 17 ) = 57 , N ( 18 ) = 61 , N ( 20 ) = 69 , N ( 5 ) = 81. N(\sqrt{17})=57,N(\sqrt{18})=61,N(\sqrt{20})=69,N(5)=81.
  7. E ( r ) E(r)
  8. 8 , 4 , 8 , 12 8,4,8,12
  9. 2 π r 2\pi r
  10. | E ( r ) | 2 2 π r . |E(r)|\leq 2\sqrt{2}\pi r.
  11. | E ( r ) | o ( r 1 / 2 ( log r ) 1 / 4 ) , |E(r)|\neq o\left(r^{1/2}(\log r)^{1/4}\right),
  12. | E ( r ) | = O ( r 1 / 2 + ε ) . |E(r)|=O\left(r^{1/2+\varepsilon}\right).
  13. 1 2 < t 131 208 = 0.6298 , \frac{1}{2}<t\leq\frac{131}{208}=0.6298\ldots,
  14. N ( r ) = 1 + 4 i = 0 ( r 2 4 i + 1 - r 2 4 i + 3 ) . N(r)=1+4\sum_{i=0}^{\infty}\left(\left\lfloor\frac{r^{2}}{4i+1}\right\rfloor-% \left\lfloor\frac{r^{2}}{4i+3}\right\rfloor\right).
  15. N ( r ) = n = 0 r 2 r 2 ( n ) . N(r)=\sum_{n=0}^{r^{2}}r_{2}(n).
  16. m 2 + n 2 r 2 . m^{2}+n^{2}\leq r^{2}.\,
  17. V ( r ) = 6 π r 2 + O ( r 1 + ε ) . V(r)=\frac{6}{\pi}r^{2}+O(r^{1+\varepsilon}).
  18. V ( r ) = 6 π r 2 + O ( r exp ( - c ( log r ) 3 / 5 ( log log r 2 ) - 1 / 5 ) ) V(r)=\frac{6}{\pi}r^{2}+O(r\exp(-c(\log r)^{3/5}(\log\log r^{2})^{-1/5}))

Gauss–Hermite_quadrature.html

  1. - + e - x 2 f ( x ) d x . \int_{-\infty}^{+\infty}e^{-x^{2}}f(x)\,dx.
  2. - + e - x 2 f ( x ) d x i = 1 n w i f ( x i ) \int_{-\infty}^{+\infty}e^{-x^{2}}f(x)\,dx\approx\sum_{i=1}^{n}w_{i}f(x_{i})
  3. w i = 2 n - 1 n ! π n 2 [ H n - 1 ( x i ) ] 2 . w_{i}=\frac{2^{n-1}n!\sqrt{\pi}}{n^{2}[H_{n-1}(x_{i})]^{2}}.
  4. 𝒩 ( μ , σ 2 ) \mathcal{N}(\mu,\sigma^{2})
  5. E [ h ( y ) ] = - + 1 σ 2 π exp ( - ( y - μ ) 2 2 σ 2 ) h ( y ) d y E[h(y)]=\int_{-\infty}^{+\infty}\frac{1}{\sigma\sqrt{2\pi}}\exp\left(-\frac{(y% -\mu)^{2}}{2\sigma^{2}}\right)h(y)dy
  6. x = y - μ 2 σ y = 2 σ x + μ x=\frac{y-\mu}{\sqrt{2}\sigma}\Leftrightarrow y=\sqrt{2}\sigma x+\mu
  7. E [ h ( y ) ] = - + 1 π exp ( - x 2 ) h ( 2 σ x + μ ) d x E[h(y)]=\int_{-\infty}^{+\infty}\frac{1}{\sqrt{\pi}}\exp(-x^{2})h(\sqrt{2}% \sigma x+\mu)dx
  8. E [ h ( y ) ] 1 π i = 1 n w i h ( 2 σ x i + μ ) E[h(y)]\approx\frac{1}{\sqrt{\pi}}\sum_{i=1}^{n}w_{i}h(\sqrt{2}\sigma x_{i}+\mu)

Gauss–Jacobi_quadrature.html

  1. - 1 1 f ( x ) ( 1 - x ) α ( 1 + x ) β d x \int_{-1}^{1}f(x)(1-x)^{\alpha}(1+x)^{\beta}\,\mathrm{d}x
  2. - 1 1 f ( x ) ( 1 - x ) α ( 1 + x ) β d x λ 1 f ( x 1 ) + λ 2 f ( x 2 ) + + λ n f ( x n ) , \int_{-1}^{1}f(x)(1-x)^{\alpha}(1+x)^{\beta}\,\mathrm{d}x\approx\lambda_{1}f(x% _{1})+\lambda_{2}f(x_{2})+\cdots+\lambda_{n}f(x_{n}),
  3. λ i = - 2 n + α + β + 2 n + α + β + 1 Γ ( n + α + 1 ) Γ ( n + β + 1 ) Γ ( n + α + β + 1 ) ( n + 1 ) ! 2 α + β P n ( x i ) P n + 1 ( x i ) , \lambda_{i}=-\frac{2n+\alpha+\beta+2}{n+\alpha+\beta+1}\frac{\Gamma(n+\alpha+1% )\Gamma(n+\beta+1)}{\Gamma(n+\alpha+\beta+1)(n+1)!}\frac{2^{\alpha+\beta}}{P^{% \prime}_{n}(x_{i})P_{n+1}(x_{i})},

Gauss–Kronrod_quadrature_formula.html

  1. a b f ( x ) d x . \int_{a}^{b}f(x)\,dx.
  2. a b f ( x ) d x i = 1 n w i f ( x i ) . \int_{a}^{b}f(x)\,dx\approx\sum_{i=1}^{n}w_{i}f(x_{i}).
  3. n + 1 n+1
  4. n n
  5. 2 n + 1 2n+1
  6. ( 200 | G 7 - K 15 | ) 1.5 (200|G7-K15|)^{1.5}

Gauss–Laguerre_quadrature.html

  1. 0 + e - x f ( x ) d x . \int_{0}^{+\infty}e^{-x}f(x)\,dx.
  2. 0 + e - x f ( x ) d x i = 1 n w i f ( x i ) \int_{0}^{+\infty}e^{-x}f(x)\,dx\approx\sum_{i=1}^{n}w_{i}f(x_{i})
  3. w i = x i ( n + 1 ) 2 [ L n + 1 ( x i ) ] 2 . w_{i}=\frac{x_{i}}{(n+1)^{2}[L_{n+1}(x_{i})]^{2}}.
  4. f f
  5. 0 f ( x ) d x = 0 f ( x ) e x e - x d x = 0 g ( x ) e - x d x \int_{0}^{\infty}f\left(x\right)dx=\int_{0}^{\infty}f\left(x\right)e^{x}e^{-x}% dx=\int_{0}^{\infty}g\left(x\right)e^{-x}dx
  6. g ( x ) := e x f ( x ) g\left(x\right):=e^{x}f\left(x\right)
  7. x α x^{\alpha}
  8. α > - 1 \alpha>-1
  9. 0 + x α e - x f ( x ) d x . \int_{0}^{+\infty}x^{\alpha}e^{-x}f(x)\,dx.

General_mn-type_image_filter.html

  1. Z = Z 1 + Z 2 + Z 3 + + Z N Z=Z_{1}+Z_{2}+Z_{3}+\dots+Z_{N}
  2. Z m n = m 1 Z 1 + m 2 Z 2 + m 3 Z 3 + + m N Z N Z_{m_{n}}=m_{1}Z_{1}+m_{2}Z_{2}+m_{3}Z_{3}+\dots+m_{N}Z_{N}
  3. Y m n = Z m n k 2 + Z 2 - Z m n 2 Y_{m_{n}}=\frac{Z_{m_{n}}}{k^{2}+Z^{2}-Z_{m_{n}}^{2}}
  4. k = Z Y k=\sqrt{\frac{Z}{Y}}
  5. Y m n = m 1 Y 1 + m 2 Y 2 + m 3 Y 3 + + m N Y N Y_{m_{n}}=m_{1}Y_{1}+m_{2}Y_{2}+m_{3}Y_{3}+\dots+m_{N}Y_{N}
  6. Z m n = Y m n k 2 + Y 2 - Y m n 2 Z_{m_{n}}=\frac{Y_{m_{n}}}{k^{2}+Y^{2}-Y_{m_{n}}^{2}}

Generalized_estimating_equation.html

  1. μ i j \mu_{ij}
  2. V i V_{i}
  3. U ( β ) = i = 1 N μ i j β k V i - 1 { Y i - μ i ( β ) } U(\beta)=\sum_{i=1}^{N}\frac{\partial\mu_{ij}}{\partial\beta_{k}}V_{i}^{-1}\{Y% _{i}-\mu_{i}(\beta)\}\,\!

Generalized_Korteweg–de_Vries_equation.html

  1. t u + x 3 u + x f ( u ) = 0. \partial_{t}u+\partial_{x}^{3}u+\partial_{x}f(u)=0.\,

Generalized_logistic_distribution.html

  1. F ( x ; α ) = 1 ( 1 + exp ( - x ) ) α ( 1 + exp ( - x ) ) - α , α > 0. F(x;\alpha)=\frac{1}{(1+\exp(-x))^{\alpha}}\equiv(1+\exp(-x))^{-\alpha},\quad% \alpha>0.
  2. f ( x ; α ) = α exp ( - x ) ( 1 + exp ( - x ) ) α + 1 , α > 0. f(x;\alpha)=\frac{\alpha\exp(-x)}{\left(1+\exp(-x)\right)^{\alpha+1}},\quad% \alpha>0.
  3. F ( x ; α ) = 1 - exp ( - α x ) ( 1 + exp ( - x ) ) α , α > 0. F(x;\alpha)=1-\frac{\exp(-\alpha x)}{(1+\exp(-x))^{\alpha}},\quad\alpha>0.
  4. f ( x ; α ) = α ( e x + 2 ) e α ( - x ) ( e - x + 1 ) α e x + 1 , α > 0. f(x;\alpha)=\frac{\alpha\left(e^{x}+2\right)e^{\alpha(-x)}\left(e^{-x}+1\right% )^{\alpha}}{e^{x}+1},\quad\alpha>0.
  5. f ( x ; α ) = 1 B ( α , α ) exp ( - α x ) ( 1 + exp ( - x ) ) 2 α , α > 0. f(x;\alpha)=\frac{1}{B(\alpha,\alpha)}\frac{\exp(-\alpha x)}{(1+\exp(-x))^{2% \alpha}},\quad\alpha>0.
  6. M ( t ) = Γ ( α - t ) Γ ( α + t ) ( Γ ( α ) ) 2 , - α < t < α . M(t)=\frac{\Gamma(\alpha-t)\Gamma(\alpha+t)}{(\Gamma(\alpha))^{2}},\quad-% \alpha<t<\alpha.
  7. F ( x ; α ) = ( e x + 1 ) Γ ( α ) e α ( - x ) ( e - x + 1 ) 2 - 2 α F ~ 1 ( 1 , 1 - α ; α + 1 ; - e x ) B ( α , α ) , α > 0. F(x;\alpha)=\frac{\left(e^{x}+1\right)\Gamma(\alpha)e^{\alpha(-x)}\left(e^{-x}% +1\right)^{-2\alpha}\,_{2}\tilde{F}_{1}\left(1,1-\alpha;\alpha+1;-e^{x}\right)% }{B(\alpha,\alpha)},\quad\alpha>0.
  8. f ( x ; α , β ) = 1 B ( α , β ) exp ( - β x ) ( 1 + exp ( - x ) ) α + β , α , β > 0. f(x;\alpha,\beta)=\frac{1}{B(\alpha,\beta)}\frac{\exp(-\beta x)}{(1+\exp(-x))^% {\alpha+\beta}},\quad\alpha,\beta>0.
  9. M ( t ) = Γ ( β - t ) Γ ( α + t ) Γ ( α ) Γ ( β ) , - α < t < β . M(t)=\frac{\Gamma(\beta-t)\Gamma(\alpha+t)}{\Gamma(\alpha)\Gamma(\beta)},\quad% -\alpha<t<\beta.
  10. F ( x ; α , β ) = ( e x + 1 ) Γ ( α ) e β ( - x ) ( e - x + 1 ) 2 - α - β F ~ 1 ( 1 , 1 - β ; α + 1 ; - e x ) B ( α , β ) , α , β > 0. F(x;\alpha,\beta)=\frac{\left(e^{x}+1\right)\Gamma(\alpha)e^{\beta(-x)}\left(e% ^{-x}+1\right)^{-\alpha-\beta}\,_{2}\tilde{F}_{1}\left(1,1-\beta;\alpha+1;-e^{% x}\right)}{B(\alpha,\beta)},\quad\alpha,\beta>0.

Generalized_semi-infinite_programming.html

  1. min x X f ( x ) \min\limits_{x\in X}\;\;f(x)
  2. subject to: \mbox{subject to: }~{}
  3. g ( x , y ) 0 , y Y ( x ) g(x,y)\leq 0,\;\;\forall y\in Y(x)
  4. f : R n R f:R^{n}\to R
  5. g : R n × R m R g:R^{n}\times R^{m}\to R
  6. X R n X\subseteq R^{n}
  7. Y R m . Y\subseteq R^{m}.
  8. Y ( x ) Y(x)
  9. x X x\in X

Generalized_Verma_module.html

  1. 𝔤 \mathfrak{g}
  2. 𝔭 \mathfrak{p}
  3. 𝔤 \mathfrak{g}
  4. V V
  5. 𝔭 \mathfrak{p}
  6. M 𝔭 ( V ) := 𝒰 ( 𝔤 ) 𝒰 ( 𝔭 ) V M_{\mathfrak{p}}(V):=\mathcal{U}(\mathfrak{g})\otimes_{\mathcal{U}(\mathfrak{p% })}V
  7. 𝔤 \mathfrak{g}
  8. 𝒰 ( 𝔤 ) \mathcal{U}(\mathfrak{g})
  9. M 𝔭 ( λ ) M_{\mathfrak{p}}(\lambda)
  10. M 𝔭 ( λ ) M_{\mathfrak{p}}(\lambda)
  11. 𝔭 \mathfrak{p}
  12. 𝔭 \mathfrak{p}
  13. λ \lambda
  14. 𝔭 \mathfrak{p}
  15. 𝔤 \mathfrak{g}
  16. 𝔤 = j = - k k 𝔤 j \mathfrak{g}=\oplus_{j=-k}^{k}\mathfrak{g}_{j}
  17. 𝔭 = j 0 𝔤 j \mathfrak{p}=\oplus_{j\geq 0}\mathfrak{g}_{j}
  18. 𝔤 - := j < 0 𝔤 j \mathfrak{g}_{-}:=\oplus_{j<0}\mathfrak{g}_{j}
  19. 𝔤 - \mathfrak{g}_{-}
  20. 𝔤 0 \mathfrak{g}_{0}
  21. M 𝔭 ( V ) 𝒰 ( 𝔤 - ) V M_{\mathfrak{p}}(V)\simeq\mathcal{U}(\mathfrak{g}_{-})\otimes V
  22. v λ v_{\lambda}
  23. 1 v λ 1\otimes v_{\lambda}
  24. M 𝔭 ( λ ) M_{\mathfrak{p}}(\lambda)
  25. M λ M 𝔭 ( λ ) M_{\lambda}\to M_{\mathfrak{p}}(\lambda)
  26. ( 1 ) K λ := α S M s α λ M λ (1)\quad K_{\lambda}:=\sum_{\alpha\in S}M_{s_{\alpha}\cdot\lambda}\subset M_{\lambda}
  27. S Δ S\subset\Delta
  28. - α -\alpha
  29. 𝔭 \mathfrak{p}
  30. 𝔭 \mathfrak{p}
  31. s α s_{\alpha}
  32. s α λ s_{\alpha}\cdot\lambda
  33. s α s_{\alpha}
  34. M s α λ M_{s_{\alpha}\cdot\lambda}
  35. M λ M_{\lambda}
  36. M s α λ M λ M_{s_{\alpha}\cdot\lambda}\subset M_{\lambda}
  37. S = S=\emptyset
  38. 𝔭 \mathfrak{p}
  39. S = Δ S=\Delta
  40. 𝔭 = 𝔤 \mathfrak{p}=\mathfrak{g}
  41. M 𝔭 ( λ ) M_{\mathfrak{p}}(\lambda)
  42. λ ~ \tilde{\lambda}
  43. λ = w λ ~ \lambda=w\cdot\tilde{\lambda}
  44. \cdot
  45. M λ M_{\lambda}
  46. λ ~ \tilde{\lambda}
  47. λ ~ + δ \tilde{\lambda}+\delta
  48. 𝔤 \mathfrak{g}
  49. λ , μ \lambda,\mu
  50. M 𝔭 ( μ ) M 𝔭 ( λ ) M_{\mathfrak{p}}(\mu)\rightarrow M_{\mathfrak{p}}(\lambda)
  51. μ \mu
  52. λ \lambda
  53. W W
  54. 𝔤 \mathfrak{g}
  55. d i m ( H o m ( M 𝔭 ( μ ) , M 𝔭 ( λ ) ) ) dim(Hom(M_{\mathfrak{p}}(\mu),M_{\mathfrak{p}}(\lambda)))
  56. f : M μ M λ f:M_{\mu}\to M_{\lambda}
  57. K μ K_{\mu}
  58. K λ K_{\lambda}
  59. M μ M 𝔭 ( μ ) M_{\mu}\to M_{\mathfrak{p}}(\mu)
  60. M λ M 𝔭 ( λ ) M_{\lambda}\to M_{\mathfrak{p}}(\lambda)
  61. K μ K λ K_{\mu}\to K_{\lambda}
  62. M 𝔭 ( μ ) M 𝔭 ( λ ) M_{\mathfrak{p}}(\mu)\to M_{\mathfrak{p}}(\lambda)
  63. M μ M λ M_{\mu}\to M_{\lambda}
  64. S Δ S\subset\Delta
  65. - α -\alpha
  66. 𝔭 \mathfrak{p}
  67. M 𝔭 ( μ ) M 𝔭 ( λ ) M_{\mathfrak{p}}(\mu)\to M_{\mathfrak{p}}(\lambda)
  68. α S \alpha\in S
  69. M μ M_{\mu}
  70. M s α λ M_{s_{\alpha}\cdot\lambda}
  71. s α s_{\alpha}
  72. \cdot
  73. 𝔤 \mathfrak{g}
  74. 𝔤 \mathfrak{g}
  75. λ ~ \tilde{\lambda}
  76. 𝔤 \mathfrak{g}
  77. W 𝔭 W W^{\mathfrak{p}}\subset W
  78. w W 𝔭 w ( λ ~ ) w\in W^{\mathfrak{p}}\Leftrightarrow w(\tilde{\lambda})
  79. 𝔭 \mathfrak{p}
  80. W 𝔭 W 𝔭 \ W W^{\mathfrak{p}}\simeq W_{\mathfrak{p}}\backslash W
  81. W 𝔭 W_{\mathfrak{p}}
  82. 𝔭 \mathfrak{p}
  83. W 𝔭 W^{\mathfrak{p}}
  84. λ ~ \tilde{\lambda}
  85. w W 𝔭 M 𝔭 ( w λ ~ ) w\in W^{\mathfrak{p}}\mapsto M_{\mathfrak{p}}(w\cdot\tilde{\lambda})
  86. W 𝔭 W^{\mathfrak{p}}
  87. λ ~ \tilde{\lambda}
  88. μ = w λ ~ \mu=w^{\prime}\cdot\tilde{\lambda}
  89. λ = w λ ~ \lambda=w\cdot\tilde{\lambda}
  90. w w w\leq w^{\prime}
  91. M μ M λ M_{\mu}\to M_{\lambda}
  92. W 𝔭 W^{\mathfrak{p}}
  93. w = s γ w w^{\prime}=s_{\gamma}w
  94. γ \gamma
  95. M 𝔭 ( μ ) M 𝔭 ( λ ) M_{\mathfrak{p}}(\mu)\to M_{\mathfrak{p}}(\lambda)
  96. M 𝔭 ( μ ) M 𝔭 ( λ ) M_{\mathfrak{p}}(\mu)\to M_{\mathfrak{p}}(\lambda)
  97. w ′′ W w^{\prime\prime}\in W
  98. w w ′′ w w\leq w^{\prime\prime}\leq w^{\prime}
  99. w ′′ W 𝔭 w^{\prime\prime}\notin W^{\mathfrak{p}}
  100. λ ~ \tilde{\lambda}
  101. 𝔭 \mathfrak{p}
  102. 𝔭 \mathfrak{p}
  103. μ \mu
  104. λ \lambda
  105. λ ~ \tilde{\lambda}
  106. λ ~ + δ \tilde{\lambda}+\delta
  107. M 𝔭 ( μ ) M 𝔭 ( λ ) M_{\mathfrak{p}}(\mu)\to M_{\mathfrak{p}}(\lambda)

Genus_field.html

  1. p * = ± p 1 ( mod 4 ) if p is odd ; p^{*}=\pm p\equiv 1\;\;(\mathop{{\rm mod}}4)\,\text{ if }p\,\text{ is odd};
  2. 2 * = - 4 , 8 , - 8 according as m 3 ( mod 4 ) , 2 ( mod 8 ) , - 2 ( mod 8 ) . 2^{*}=-4,8,-8\,\text{ according as }m\equiv 3\;\;(\mathop{{\rm mod}}4),2\;\;(% \mathop{{\rm mod}}8),-2\;\;(\mathop{{\rm mod}}8).

Geographical_distance.html

  1. D , D,\,\!
  2. P 1 P_{1}\,\!
  3. P 2 P_{2}\,\!
  4. ( ϕ 1 , λ 1 ) (\phi_{1},\lambda_{1})\,\!
  5. ( ϕ 2 , λ 2 ) , (\phi_{2},\lambda_{2}),\,\!
  6. P 1 P_{1}\,\!
  7. Δ ϕ = ϕ 2 - ϕ 1 ; Δ λ = λ 2 - λ 1 . \begin{aligned}\displaystyle\Delta\phi&\displaystyle=\phi_{2}-\phi_{1};\\ \displaystyle\Delta\lambda&\displaystyle=\lambda_{2}-\lambda_{1}.\end{aligned}\,\!
  8. ϕ m = ϕ 1 + ϕ 2 2 . \phi_{m}=\frac{\phi_{1}+\phi_{2}}{2}.\,\!
  9. θ = π 2 - ϕ ; \theta=\frac{\pi}{2}-\phi;\,\!
  10. θ = 90 - ϕ . \theta=90^{\circ}-\phi.\,\!
  11. R R\,\!
  12. D D\!
  13. Δ ϕ \Delta\phi\!
  14. Δ λ \Delta\lambda\!
  15. ϕ m \phi_{m}\!
  16. Δ λ \Delta\lambda\!
  17. λ 1 \lambda_{1}\!
  18. λ 2 \lambda_{2}\!
  19. ϕ m \phi_{m}\!
  20. ϕ 1 \phi_{1}\!
  21. λ 1 \lambda_{1}\!
  22. ϕ 2 \phi_{2}\!
  23. λ 2 \lambda_{2}\!
  24. D = R ( Δ ϕ ) 2 + ( cos ( ϕ m ) Δ λ ) 2 , D=R\sqrt{(\Delta\phi)^{2}+(\cos(\phi_{m})\Delta\lambda)^{2}},
  25. Δ ϕ \Delta\phi\,\!
  26. Δ λ \Delta\lambda\,\!
  27. ϕ m \phi_{m}\,\!
  28. cos ( ϕ m ) . \cos(\phi_{m}).\,\!
  29. 1 = ( π / 180 ) radians . 1^{\circ}=(\pi/180)\,\mathrm{radians}.
  30. D = ( K 1 Δ ϕ ) 2 + ( K 2 Δ λ ) 2 , D=\sqrt{(K_{1}\Delta\phi)^{2}+(K_{2}\Delta\lambda)^{2}},
  31. D D\,\!
  32. Δ ϕ \Delta\phi\,\!
  33. Δ λ \Delta\lambda\,\!
  34. ϕ m \phi_{m}\,\!
  35. cos ( ϕ m ) ; \cos(\phi_{m});\,\!
  36. K 1 = 111.13209 - 0.56605 cos ( 2 ϕ m ) + 0.00120 cos ( 4 ϕ m ) ; K 2 = 111.41513 cos ( ϕ m ) - 0.09455 cos ( 3 ϕ m ) + 0.00012 cos ( 5 ϕ m ) . \begin{aligned}\displaystyle K_{1}&\displaystyle=111.13209-0.56605\cos(2\phi_{% m})+0.00120\cos(4\phi_{m});\\ \displaystyle K_{2}&\displaystyle=111.41513\cos(\phi_{m})-0.09455\cos(3\phi_{m% })+0.00012\cos(5\phi_{m}).\end{aligned}\,\!
  37. K 1 = M π 180 K_{1}=M\frac{\pi}{180}\,\!
  38. K 2 = cos ( ϕ m ) N π 180 K_{2}=\cos(\phi_{m})N\frac{\pi}{180}\,\!
  39. M M\,\!
  40. N N\,\!
  41. M M\,\!
  42. N N\,\!
  43. D = R θ 1 2 s y m b o l + θ 2 2 - 2 θ 1 θ 2 cos ( Δ λ ) , D=R\sqrt{\theta^{2}_{1}\;symbol{+}\;\theta^{2}_{2}\;\mathbf{-}\;2\theta_{1}% \theta_{2}\cos(\Delta\lambda)},
  44. θ = π 180 ( 90 - ϕ ) . \theta=\frac{\pi}{180}(90^{\circ}-\phi).\,\!
  45. Δ X = cos ( ϕ 2 ) cos ( λ 2 ) - cos ( ϕ 1 ) cos ( λ 1 ) ; \displaystyle\Delta{X}=\cos(\phi_{2})\cos(\lambda_{2})-\cos(\phi_{1})\cos(% \lambda_{1});
  46. D = R C h D=RC_{h}
  47. D R D\ll R
  48. D ( D / R ) 2 / 24 D(D/R)^{2}/24
  49. ϕ 1 \scriptstyle\phi_{1}
  50. ϕ 2 \scriptstyle\phi_{2}
  51. β 1 \scriptstyle\beta_{1}
  52. β 2 \scriptstyle\beta_{2}
  53. tan β = ( 1 - f ) tan ϕ , \tan\beta=(1-f)\tan\phi,
  54. f f
  55. σ \sigma
  56. ( β 1 , λ 1 ) (\beta_{1},\;\lambda_{1})
  57. ( β 2 , λ 2 ) (\beta_{2},\;\lambda_{2})
  58. λ 1 \lambda_{1}\;
  59. λ 2 \lambda_{2}\;
  60. P = β 1 + β 2 2 Q = β 2 - β 1 2 P=\frac{\beta_{1}+\beta_{2}}{2}\qquad Q=\frac{\beta_{2}-\beta_{1}}{2}
  61. X = ( σ - sin σ ) sin 2 P cos 2 Q cos 2 σ 2 Y = ( σ + sin σ ) cos 2 P sin 2 Q sin 2 σ 2 X=(\sigma-\sin\sigma)\frac{\sin^{2}P\cos^{2}Q}{\cos^{2}\frac{\sigma}{2}}\qquad% \qquad Y=(\sigma+\sin\sigma)\frac{\cos^{2}P\sin^{2}Q}{\sin^{2}\frac{\sigma}{2}}
  62. distance = a ( σ - f 2 ( X + Y ) ) \mathrm{distance}=a\bigl(\sigma-\tfrac{f}{2}(X+Y)\bigr)
  63. a a
  64. A = 1 + e 2 cos 4 ϕ 1 , B = 1 + e 2 cos 2 ϕ 1 , A=\sqrt{1+e^{\prime 2}\cos^{4}\phi_{1}},\quad B=\sqrt{1+e^{\prime 2}\cos^{2}% \phi_{1}},
  65. e 2 = a 2 - b 2 b 2 = f ( 2 - f ) ( 1 - f ) 2 . e^{\prime 2}=\frac{a^{2}-b^{2}}{b^{2}}=\frac{f(2-f)}{(1-f)^{2}}.
  66. R = 1 + e 2 B 2 a . R^{\prime}=\frac{\sqrt{1+e^{\prime 2}}}{B^{2}}a.
  67. ϕ 1 = tan - 1 ( tan ϕ / B ) , Δ ϕ = Δ ϕ B [ 1 + 3 e 2 4 B 2 ( Δ ϕ ) sin ( 2 ϕ 1 + 2 3 Δ ϕ ) ] , Δ λ = A Δ λ , \begin{aligned}\displaystyle\phi_{1}^{\prime}&\displaystyle=\tan^{-1}(\tan\phi% /B),\\ \displaystyle\Delta\phi^{\prime}&\displaystyle=\frac{\Delta\phi}{B}\biggl[1+% \frac{3e^{\prime 2}}{4B^{2}}(\Delta\phi)\sin(2\phi_{1}+\tfrac{2}{3}\Delta\phi)% \biggr],\\ \displaystyle\Delta\lambda^{\prime}&\displaystyle=A\Delta\lambda,\end{aligned}
  68. Δ ϕ = ϕ 2 - ϕ 1 \Delta\phi=\phi_{2}-\phi_{1}
  69. Δ ϕ = ϕ 2 - ϕ 1 \Delta\phi^{\prime}=\phi_{2}^{\prime}-\phi_{1}^{\prime}
  70. Δ λ = λ 2 - λ 1 \Delta\lambda=\lambda_{2}-\lambda_{1}
  71. Δ λ = λ 2 - λ 1 \Delta\lambda^{\prime}=\lambda_{2}^{\prime}-\lambda_{1}^{\prime}

Geohash.html

  1. min round ( v a l u e ) max \min\leq\mathrm{round}(value)\leq\max

Geometric_and_material_buckling.html

  1. - D 2 Φ + Σ a Φ = 1 k ν Σ f Φ -D\nabla^{2}\Phi+\Sigma_{a}\Phi=\frac{1}{k}\nu\Sigma_{f}\Phi
  2. ν \nu
  3. Σ f \Sigma_{f}
  4. D = 1 3 Σ tr D=\frac{1}{3\Sigma_{\mathrm{tr}}}
  5. L = D Σ a L=\sqrt{\frac{D}{\Sigma_{a}}}
  6. - 2 Φ Φ = k k - 1 L 2 = B g 2 -\frac{\nabla^{2}\Phi}{\Phi}=\frac{\frac{k_{\infty}}{k}-1}{L^{2}}={B_{g}}^{2}
  7. ( π R ) 2 \left(\frac{\pi}{R}\right)^{2}
  8. ( π H ) 2 + ( 2.405 R ) 2 \left(\frac{\pi}{H}\right)^{2}+\left(\frac{2.405}{R}\right)^{2}
  9. ( π a ) 2 + ( π b ) 2 + ( π c ) 2 \left(\frac{\pi}{a}\right)^{2}+\left(\frac{\pi}{b}\right)^{2}+\left(\frac{\pi}% {c}\right)^{2}
  10. ( π R + δ ) 2 \left(\frac{\pi}{R+\delta}\right)^{2}
  11. ( π H + 2 δ ) 2 + ( 2.405 R + δ ) 2 \left(\frac{\pi}{H+2\delta}\right)^{2}+\left(\frac{2.405}{R+\delta}\right)^{2}
  12. ( π a + 2 δ ) 2 + ( π b + 2 δ ) 2 + ( π c + 2 δ ) 2 \left(\frac{\pi}{a+2\delta}\right)^{2}+\left(\frac{\pi}{b+2\delta}\right)^{2}+% \left(\frac{\pi}{c+2\delta}\right)^{2}
  13. k k_{\infty}
  14. k = ν Σ f Σ a k_{\infty}=\frac{\nu\Sigma_{f}}{\Sigma_{a}}
  15. B g 2 = k k - 1 L 2 = 1 k ν Σ f - Σ a D {B_{g}}^{2}=\frac{\frac{k_{\infty}}{k}-1}{L^{2}}=\frac{\frac{1}{k}\nu\Sigma_{f% }-\Sigma_{a}}{D}
  16. k = k eff = ν Σ f Σ a + D B g 2 k=k_{\mathrm{eff}}=\frac{\nu\Sigma_{f}}{\Sigma_{a}+D{B_{g}}^{2}}
  17. k = ν Σ f Σ a 1 + L 2 B g 2 k=\frac{\frac{\nu\Sigma_{f}}{\Sigma_{a}}}{1+L^{2}{B_{g}}^{2}}
  18. B g 2 = ν Σ f - Σ a D {B_{g}}^{2}=\frac{\nu\Sigma_{f}-\Sigma_{a}}{D}
  19. B m 2 = ν Σ f - Σ a D {B_{m}}^{2}=\frac{\nu\Sigma_{f}-\Sigma_{a}}{D}

Geometric_calculus.html

  1. b F ( a ) = lim ϵ 0 F ( a + ϵ b ) - F ( a ) ϵ \nabla_{b}F(a)=\lim_{\epsilon\rightarrow 0}{\frac{F(a+\epsilon b)-F(a)}{% \epsilon}}
  2. { e i } \{e_{i}\}
  3. ( i ) (\partial_{i})
  4. ( e i ) (e_{i})
  5. i : F ( x e i F ( x ) ) \partial_{i}:F\mapsto(x\mapsto\nabla_{e_{i}}F(x))
  6. e i i e^{i}\partial_{i}
  7. F e i i F F\mapsto e^{i}\partial_{i}F
  8. F ( x e i e i F ( x ) ) F\mapsto(x\mapsto e^{i}\nabla_{e_{i}}F(x))
  9. = e i i \nabla=e^{i}\partial_{i}
  10. α a + β b = α a + β b \nabla_{\alpha a+\beta b}=\alpha\nabla_{a}+\beta\nabla_{b}
  11. a a
  12. a = ( a e i ) e i a=(a\cdot e^{i})e_{i}
  13. a = ( a e i ) e i = ( a e i ) e i = a ( e i e i ) = a \nabla_{a}=\nabla_{(a\cdot e^{i})e_{i}}=(a\cdot e^{i})\nabla_{e_{i}}=a\cdot(e^% {i}\nabla_{e^{i}})=a\cdot\nabla
  14. a F ( x ) \nabla_{a}F(x)
  15. a F ( x ) a\cdot\nabla F(x)
  16. F G = ( F ) G . \nabla FG=(\nabla F)G.
  17. ( F G ) \displaystyle\nabla(FG)
  18. e i F F e i e^{i}F\neq Fe^{i}
  19. ˙ F G ˙ = e i F ( i G ) , \dot{\nabla}F\dot{G}=e^{i}F(\partial_{i}G),
  20. ( F G ) = F G + ˙ F G ˙ \nabla(FG)=\nabla FG+\dot{\nabla}F\dot{G}
  21. F = F r - 1 = e i i F \nabla\cdot F=\langle\nabla F\rangle_{r-1}=e^{i}\cdot\partial_{i}F
  22. F = F r + 1 = e i i F . \nabla\wedge F=\langle\nabla F\rangle_{r+1}=e^{i}\wedge\partial_{i}F.
  23. F = F + F \nabla F=\nabla\cdot F+\nabla\wedge F
  24. F = div F \nabla\cdot F=\operatorname{div}F
  25. F = I curl F . \nabla\wedge F=I\,\operatorname{curl}F.
  26. { e 1 , e n } \{e_{1},\,...\,e_{n}\}
  27. e 1 e 2 e n e_{1}\wedge e_{2}\wedge\cdots\wedge e_{n}
  28. 1 k n 1\leq k\leq n
  29. { e i 1 , e i k } \{e_{i_{1}},\,...\,e_{i_{k}}\}
  30. e i 1 e i 2 e i k e_{i_{1}}\wedge e_{i_{2}}\wedge\cdots\wedge e_{i_{k}}
  31. { x i 1 e i 1 , x i k e i k } \{x^{i_{1}}e_{i_{1}},\,...\,x^{i_{k}}e_{i_{k}}\}
  32. { x i j } \{x^{i_{j}}\}
  33. d k X = ( d x i 1 e i 1 ) ( d x i 2 e i 2 ) ( d x i k e i k ) = ( e i 1 e i 2 e i k ) d x i 1 d x i 2 d x i k \begin{aligned}\displaystyle\mathrm{d}^{k}X&\displaystyle=\left(\mathrm{d}x^{i% _{1}}e_{i_{1}}\right)\wedge\left(\mathrm{d}x^{i_{2}}e_{i_{2}}\right)\wedge% \cdots\wedge\left(\mathrm{d}x^{i_{k}}e_{i_{k}}\right)\\ &\displaystyle=\left(e_{i_{1}}\wedge e_{i_{2}}\wedge\cdots\wedge e_{i_{k}}% \right)\mathrm{d}x^{i_{1}}\mathrm{d}x^{i_{2}}\cdots\mathrm{d}x^{i_{k}}\end{aligned}
  34. V F ( x ) d k X = V F ( x ) ( e i 1 e i 2 e i k ) d x i 1 d x i 2 d x i k \int_{V}F(x)\mathrm{d}^{k}X=\int_{V}F(x)\left(e_{i_{1}}\wedge e_{i_{2}}\wedge% \cdots\wedge e_{i_{k}}\right)\mathrm{d}x^{i_{1}}\mathrm{d}x^{i_{2}}\cdots% \mathrm{d}x^{i_{k}}
  35. { x i } \{x_{i}\}
  36. Δ U i ( x ) \Delta U_{i}(x)
  37. V F d U = lim n i = 1 n F ( x i ) Δ U i ( x ) . \int_{V}FdU=\lim_{n\rightarrow\infty}\sum_{i=1}^{n}F(x_{i})\Delta U_{i}(x).
  38. 𝖫 ( A ; x ) \mathsf{L}(A;x)
  39. 𝖫 ( A ; x ) = F ( x ) A I - 1 \mathsf{L}(A;x)=\langle F(x)AI^{-1}\rangle
  40. V 𝖫 ˙ ( ˙ d X ; x ) = V F ˙ ( x ) ˙ d X I - 1 = V F ˙ ( x ) ˙ | d X | = V F ( x ) | d X | . \begin{aligned}\displaystyle\int_{V}\dot{\mathsf{L}}\left(\dot{\nabla}dX;x% \right)&\displaystyle=\int_{V}\langle\dot{F}(x)\dot{\nabla}dXI^{-1}\rangle\\ &\displaystyle=\int_{V}\langle\dot{F}(x)\dot{\nabla}|dX|\rangle\\ &\displaystyle=\int_{V}\nabla\cdot F(x)|dX|.\end{aligned}
  41. V 𝖫 ( d S ; x ) = V F ( x ) d S I - 1 = V F ( x ) n ^ | d S | = V F ( x ) n ^ | d S | \begin{aligned}\displaystyle\oint_{\partial V}\mathsf{L}(dS;x)&\displaystyle=% \oint_{\partial V}\langle F(x)dSI^{-1}\rangle\\ &\displaystyle=\oint_{\partial V}\langle F(x)\hat{n}|dS|\rangle\\ &\displaystyle=\oint_{\partial V}F(x)\cdot\hat{n}|dS|\end{aligned}
  42. V F ( x ) | d X | = V F ( x ) n ^ | d S | . \int_{V}\nabla\cdot F(x)|dX|=\oint_{\partial V}F(x)\cdot\hat{n}|dS|.
  43. 𝒫 B ( A ) = ( A B - 1 ) B \mathcal{P}_{B}(A)=(A\cdot B^{-1})B
  44. \nabla
  45. \partial
  46. F = 𝒫 B ( ) F \partial F=\mathcal{P}_{B}(\nabla)F
  47. 𝒫 B ( F ) \mathcal{P}_{B}(\nabla F)
  48. a F = a F a\cdot\partial F=a\cdot\nabla F
  49. F \partial F
  50. a D F = 𝒫 B ( a F ) = 𝒫 B ( a 𝒫 B ( F ) ) a\cdot DF=\mathcal{P}_{B}(a\cdot\partial F)=\mathcal{P}_{B}(a\cdot\mathcal{P}_% {B}(\nabla F))
  51. a F = 𝒫 B ( a F ) + 𝒫 B ( a F ) , a\cdot\partial F=\mathcal{P}_{B}(a\cdot\partial F)+\mathcal{P}_{B}^{\perp}(a% \cdot\partial F),
  52. 𝖲 ( a ) \mathsf{S}(a)
  53. F × 𝖲 ( a ) = 𝒫 B ( a F ) , F\times\mathsf{S}(a)=\mathcal{P}_{B}^{\perp}(a\cdot\partial F),
  54. × \times
  55. { e i } \{e_{i}\}
  56. 𝖲 ( a ) = e i 𝒫 B ( a e i ) . \mathsf{S}(a)=e^{i}\wedge\mathcal{P}_{B}^{\perp}(a\cdot\partial e_{i}).
  57. [ a D , b D ] F = - ( 𝖲 ( a ) × 𝖲 ( b ) ) × F . [a\cdot D,\,b\cdot D]F=-(\mathsf{S}(a)\times\mathsf{S}(b))\times F.
  58. 𝖲 ( a ) × 𝖲 ( b ) \mathsf{S}(a)\times\mathsf{S}(b)
  59. 𝖱 ( a b ) = - 𝒫 B ( 𝖲 ( a ) × 𝖲 ( b ) ) . \mathsf{R}(a\wedge b)=-\mathcal{P}_{B}(\mathsf{S}(a)\times\mathsf{S}(b)).
  60. D F = D F r - 1 D\cdot F=\langle DF\rangle_{r-1}
  61. D F = D F r + 1 , D\wedge F=\langle DF\rangle_{r+1},
  62. { e i } \{e_{i}\}
  63. g i j = e i e j g_{ij}=e_{i}\cdot e_{j}
  64. Γ i j k = ( e i D e j ) e k \Gamma^{k}_{ij}=(e_{i}\cdot De_{j})\cdot e^{k}
  65. R i j k l = ( 𝖱 ( e i e j ) e k ) e l R_{ijkl}=(\mathsf{R}(e_{i}\wedge e_{j})\cdot e_{k})\cdot e_{l}
  66. ω = f I d x I = f i 1 , i 2 i k d x i 1 d x i 2 d x i k . \omega=f_{I}\mathrm{d}x^{I}=f_{i_{1},i_{2}\cdots i_{k}}\mathrm{d}x^{i_{1}}% \wedge\mathrm{d}x^{i_{2}}\wedge\cdots\wedge\mathrm{d}x^{i_{k}}.
  67. A = f i 1 , i 2 i k e i 1 e i 2 e i k A=f_{i_{1},i_{2}\cdots i_{k}}e^{i_{1}}\wedge e^{i_{2}}\wedge\cdots\wedge e^{i_% {k}}
  68. d k X = ( d x i 1 e i 1 ) ( d x i 2 e i 2 ) ( d x i k e i k ) = ( e i 1 e i 2 e i k ) d x i 1 d x i 2 d x i k . \begin{aligned}\displaystyle\mathrm{d}^{k}X&\displaystyle=\left(\mathrm{d}x^{i% _{1}}e_{i_{1}}\right)\wedge\left(\mathrm{d}x^{i_{2}}e_{i_{2}}\right)\wedge% \cdots\wedge\left(\mathrm{d}x^{i_{k}}e_{i_{k}}\right)\\ &\displaystyle=\left(e_{i_{1}}\wedge e_{i_{2}}\wedge\cdots\wedge e_{i_{k}}% \right)\mathrm{d}x^{i_{1}}\mathrm{d}x^{i_{2}}\cdots\mathrm{d}x^{i_{k}}\end{% aligned}.
  69. ω A d k X = A ( d k X ) , \omega\cong A^{\dagger}\cdot\mathrm{d}^{k}X=A\cdot\left(\mathrm{d}^{k}X\right)% ^{\dagger},
  70. d ω ( D A ) d k + 1 X = ( D A ) ( d k + 1 X ) , \mathrm{d}\omega\cong(D\wedge A)^{\dagger}\cdot\mathrm{d}^{k+1}X=(D\wedge A)% \cdot\left(\mathrm{d}^{k+1}X\right)^{\dagger},
  71. ω ( I - 1 A ) d k X , \star\omega\cong(I^{-1}A)^{\dagger}\cdot\mathrm{d}^{k}X,

Geometric_lattice.html

  1. x x
  2. y y
  3. x y x\vee y
  4. x y x\wedge y
  5. x x
  6. y y
  7. x : > y x:>y
  8. y < : x y<:x
  9. x > y x>y
  10. z z
  11. x x
  12. y y
  13. r ( x ) r(x)
  14. r ( x ) > r ( y ) r(x)>r(y)
  15. x > y x>y
  16. r ( x ) = r ( y ) + 1 r(x)=r(y)+1
  17. x : > y x:>y
  18. x x
  19. y y
  20. r ( x ) + r ( y ) r ( x y ) + r ( x y ) . r(x)+r(y)\geq r(x\wedge y)+r(x\vee y).\,
  21. r ( X ) + r ( Y ) r ( X Y ) + r ( X Y ) . r(X)+r(Y)\geq r(X\cup Y)+r(X\cap Y).\,
  22. L L
  23. L L
  24. L L
  25. L L
  26. L L
  27. L L

Geometry.html

  1. ( a , b , c ) (a,b,c)
  2. a 2 + b 2 = c 2 a^{2}+b^{2}=c^{2}
  3. 3 2 + 4 2 = 5 2 3^{2}+4^{2}=5^{2}
  4. 8 2 + 15 2 = 17 2 8^{2}+15^{2}=17^{2}
  5. 12 2 + 35 2 = 37 2 12^{2}+35^{2}=37^{2}

Geonets.html

  1. F S = allowable (test) value required (design) value FS=\frac{\,\text{allowable (test) value}}{\,\text{required (design) value}}
  2. F S = q a l l o w q r e q d FS=\frac{q_{allow}}{q_{reqd}}
  3. q = k i A q=kiA
  4. q = k i ( W × t ) q=ki(W\times t)
  5. q = ( k t ) i W q=(kt)iW
  6. k t = Θ = q i W kt=\mathit{\Theta}=\frac{q}{iW}

Georges_Reeb.html

  1. \mathbb{N}

German_tank_problem.html

  1. N m + m k - 1 = 16.5 N\approx m+\frac{m}{k}-1=16.5
  2. Pr ( N = n ) = { 0 if n < m k - 1 k ( m - 1 k - 1 ) ( n k ) if n m \Pr(N=n)=\begin{cases}0&\,\text{if }n<m\\ \frac{k-1}{k}\frac{{\left({{m-1}\atop{k-1}}\right)}}{{\left({{n}\atop{k}}% \right)}}&\,\text{if }n\geq m\end{cases}
  3. N μ ± σ = 19.5 ± 10 μ = ( m - 1 ) k - 1 k - 2 σ = ( k - 1 ) ( m - 1 ) ( m - k + 1 ) ( k - 3 ) ( k - 2 ) 2 \begin{aligned}\displaystyle N&\displaystyle\approx\mu\pm\sigma=19.5\pm 10\\ \displaystyle\mu&\displaystyle=(m-1)\frac{k-1}{k-2}\\ \displaystyle\sigma&\displaystyle=\sqrt{\frac{(k-1)(m-1)(m-k+1)}{(k-3)(k-2)^{2% }}}\end{aligned}
  4. N ^ \hat{N}
  5. N ^ = m ( 1 + k - 1 ) - 1 \hat{N}=m\left(1+k^{-1}\right)-1
  6. var ( N ^ ) = 1 k ( N - k ) ( N + 1 ) ( k + 2 ) N 2 k 2 for small samples k N \operatorname{var}(\hat{N})=\frac{1}{k}\frac{(N-k)(N+1)}{(k+2)}\approx\frac{N^% {2}}{k^{2}}\,\text{ for small samples }k\ll N
  7. N ^ = m + m - k k = m + m k - 1 - 1 = m ( 1 + k - 1 ) - 1 \hat{N}=m+\frac{m-k}{k}=m+mk^{-1}-1=m\left(1+k^{-1}\right)-1
  8. ( m - k ) / k (m-k)/k
  9. - k -k
  10. ( m - 1 k - 1 ) / ( N k ) {\textstyle\left({{m-1}\atop{k-1}}\right)}\big/{\textstyle\left({{N}\atop{k}}% \right)}
  11. ( ) {\textstyle\left({{\cdot}\atop{\cdot}}\right)}
  12. μ = m = k N m ( m - 1 k - 1 ) ( N k ) = k ( N + 1 ) k + 1 \begin{aligned}\displaystyle\mu&\displaystyle=\sum_{m=k}^{N}m\frac{{\textstyle% \left({{m-1}\atop{k-1}}\right)}}{{\textstyle\left({{N}\atop{k}}\right)}}=\frac% {k(N+1)}{k+1}\end{aligned}
  13. N = μ ( 1 + k - 1 ) - 1 \begin{aligned}\displaystyle N&\displaystyle=\mu\left(1+k^{-1}\right)-1\end{aligned}
  14. μ ( 1 + k - 1 ) - 1 = E [ m ( 1 + k - 1 ) - 1 ] \begin{aligned}\displaystyle\mu\left(1+k^{-1}\right)-1&\displaystyle=E\left[m% \left(1+k^{-1}\right)-1\right]\end{aligned}
  15. N ^ = m ( 1 + k - 1 ) - 1. \begin{aligned}\displaystyle\hat{N}&\displaystyle=m\left(1+k^{-1}\right)-1.% \end{aligned}
  16. 0.025 1 / 5 0.48 , \scriptstyle 0.025^{1/5}\;\approx\;0.48,\,
  17. 0.975 1 / 5 0.995 \scriptstyle 0.975^{1/5}\;\approx\;0.995
  18. [ 1.005 m , 2.08 m ] \scriptstyle\left[1.005m,\,2.08m\right]
  19. 0.05 1 / 5 0.55 , \scriptstyle 0.05^{1/5}\;\approx\;0.55,
  20. [ m , m / 0.05 1 / k ] = [ m , m 20 1 / k ] \scriptstyle\left[m,\,m/0.05^{1/k}\right]\;=\;\left[m,\,m\cdot 20^{1/k}\right]
  21. 2 m \scriptstyle 2m
  22. [ m , 20 m ] \scriptstyle[m,\,20m]
  23. 1.5 m \scriptstyle 1.5m
  24. [ m , 4.5 m ] \scriptstyle[m,\,4.5m]
  25. 1.2 m \scriptstyle 1.2m
  26. [ m , 1.82 m ] \scriptstyle[m,\,1.82m]
  27. 1.1 m \scriptstyle 1.1m
  28. [ m , 1.35 m ] \scriptstyle[m,\,1.35m]
  29. 1.05 m \scriptstyle 1.05m
  30. [ m , 1.16 m ] \scriptstyle[m,\,1.16m]
  31. ( N = n M = m , K = k ) \scriptstyle(N=n\mid M=m,K=k)
  32. N \scriptstyle N
  33. n \scriptstyle n
  34. K \scriptstyle K
  35. k \scriptstyle k
  36. M \scriptstyle M
  37. m \scriptstyle m
  38. ( N = n M = m , K = k ) \scriptstyle(N=n\mid M=m,K=k)
  39. ( n m , k ) \scriptstyle(n\mid m,k)
  40. ( n m , k ) = ( m n , k ) ( n k ) ( m k ) (n\mid m,k)=(m\mid n,k)\frac{(n\mid k)}{(m\mid k)}
  41. ( m n , k ) = ( M = m N = n , K = k ) \scriptstyle(m\mid n,k)=(M=m\mid N=n,K=k)
  42. m \scriptstyle m
  43. n \scriptstyle n
  44. k \scriptstyle k
  45. ( m n , k ) = { ( m - 1 k - 1 ) ( n k ) if k m n 0 otherwise (m\mid n,k)=\begin{cases}\frac{{\left({{m-1}\atop{k-1}}\right)}}{{\left({{n}% \atop{k}}\right)}}&\,\text{if }k\leq m\leq n\\ 0&\,\text{otherwise}\end{cases}
  46. ( n k ) \scriptstyle{\left({{n}\atop{k}}\right)}
  47. k \scriptstyle k
  48. n \scriptstyle n
  49. ( m k ) = ( M = m K = k ) \scriptstyle(m\mid k)=(M=m\mid K=k)
  50. ( m k ) \scriptstyle(m\mid k)
  51. n \scriptstyle n
  52. ( m k ) = ( m k ) 1 = ( m k ) n = 0 ( n m , k ) = ( m k ) n = 0 ( m n , k ) ( n k ) ( m k ) = n = 0 ( m n , k ) ( n k ) \begin{aligned}\displaystyle(m\mid k)&\displaystyle=(m\mid k)\cdot 1\\ &\displaystyle=(m\mid k){\sum_{n=0}^{\infty}(n\mid m,k)}\\ &\displaystyle=(m\mid k){\sum_{n=0}^{\infty}(m\mid n,k)\frac{(n\mid k)}{(m\mid k% )}}\\ &\displaystyle=\sum_{n=0}^{\infty}(m\mid n,k)(n\mid k)\end{aligned}
  53. ( n k ) = ( N = n K = k ) \scriptstyle(n\mid k)=(N=n\mid K=k)
  54. ( n k ) = { 1 Ω - k if k n < Ω 0 otherwise (n\mid k)=\begin{cases}\frac{1}{\Omega-k}&\,\text{if }k\leq n<\Omega\\ 0&\,\text{otherwise}\end{cases}
  55. Ω \Omega
  56. f ( n ) = lim Ω { 1 Ω - k if k n < Ω 0 otherwise f(n)=\lim_{\Omega\rightarrow\infty}\begin{cases}\frac{1}{\Omega-k}&\,\text{if % }k\leq n<\Omega\\ 0&\,\text{otherwise}\end{cases}
  57. f ( n ) = 0 \scriptstyle f(n)=0
  58. ( n m , k ) = { ( m n , k ) n = m Ω - 1 ( m n , k ) if m n < Ω 0 otherwise (n\mid m,k)=\begin{cases}\frac{(m\mid n,k)}{\sum_{n=m}^{\Omega-1}(m\mid n,k)}&% \,\text{if }m\leq n<\Omega\\ 0&\,\text{otherwise}\end{cases}
  59. n = m ( m n , k ) < \scriptstyle\sum_{n=m}^{\infty}(m\mid n,k)<\infty
  60. Ω \scriptstyle\Omega
  61. ( n m , k ) = { 0 if n < m ( m n , k ) n = m ( m n , k ) if n m (n\mid m,k)=\begin{cases}0&\,\text{if }n<m\\ \frac{(m\mid n,k)}{\sum_{n=m}^{\infty}(m\mid n,k)}&\,\text{if }n\geq m\end{cases}
  62. n n
  63. ( N = n M = m k , K = k 2 ) = { 0 if n < m k - 1 k ( m - 1 k - 1 ) ( n k ) if n m (N=n\mid M=m\geq k,K=k\geq 2)=\begin{cases}0&\,\text{if }n<m\\ \frac{k-1}{k}\frac{{\left({{m-1}\atop{k-1}}\right)}}{{\left({{n}\atop{k}}% \right)}}&\,\text{if }n\geq m\end{cases}
  64. N \scriptstyle N
  65. n \scriptstyle n
  66. ( N > n M = m k , K = k 2 ) = { 1 if n < m ( m - 1 k - 1 ) ( n k - 1 ) if n m (N>n\mid M=m\geq k,K=k\geq 2)=\begin{cases}1&\,\text{if }n<m\\ \frac{{\left({{m-1}\atop{k-1}}\right)}}{{\left({{n}\atop{k-1}}\right)}}&\,% \text{if }n\geq m\end{cases}
  67. N N
  68. ( m - 1 ) ( k - 1 ) k - 2 \frac{(m-1)(k-1)}{k-2}
  69. N \scriptstyle N
  70. ( m - 1 ) ( k - 1 ) ( m + 1 - k ) ( k - 2 ) 2 ( k - 3 ) \sqrt{\frac{(m-1)(k-1)(m+1-k)}{(k-2)^{2}(k-3)}}
  71. n = m 1 ( n k ) = k k - 1 1 ( m - 1 k - 1 ) \sum_{n=m}^{\infty}\frac{1}{{\left({{n}\atop{k}}\right)}}=\frac{k}{k-1}\frac{1% }{{\left({{m-1}\atop{k-1}}\right)}}
  72. n = m d n n k = 1 k - 1 1 m k - 1 \int_{n=m}^{\infty}\frac{dn}{n^{k}}=\frac{1}{k-1}\frac{1}{m^{k-1}}
  73. ( M = m N = n , K = 1 ) = ( m n ) = [ m n ] n (M=m\mid N=n,K=1)=(m\mid n)=\frac{[m\leq n]}{n}
  74. m \scriptstyle m
  75. ( n ) = [ n m ] n \mathcal{L}(n)=\frac{[n\geq m]}{n}
  76. n ( n ) = n = m 1 n = \sum_{n}\mathcal{L}(n)=\sum_{n=m}^{\infty}\frac{1}{n}=\infty
  77. n ( n ) [ n < Ω ] = n = m Ω - 1 1 n = H Ω - 1 - H m - 1 \begin{aligned}\displaystyle\sum_{n}\mathcal{L}(n)[n<\Omega]&\displaystyle=% \sum_{n=m}^{\Omega-1}\frac{1}{n}\\ &\displaystyle=H_{\Omega-1}-H_{m-1}\end{aligned}
  78. H n H_{n}
  79. Ω \scriptstyle\Omega
  80. ( N = n M = m , K = 1 ) \displaystyle(N=n\mid M=m,K=1)
  81. N \scriptstyle N
  82. n n ( n m ) = n = m Ω - 1 1 H Ω - 1 - H m - 1 = Ω - m H Ω - 1 - H m - 1 Ω - m log ( Ω - 1 m - 1 ) \begin{aligned}\displaystyle\sum_{n}n\cdot(n\mid m)&\displaystyle=\sum_{n=m}^{% \Omega-1}\frac{1}{H_{\Omega-1}-H_{m-1}}\\ &\displaystyle=\frac{\Omega-m}{H_{\Omega-1}-H_{m-1}}\\ &\displaystyle\approx\frac{\Omega-m}{\log\left(\frac{\Omega-1}{m-1}\right)}% \end{aligned}
  83. ( M = m N = n , K = 2 ) = ( m n ) = [ m n ] m - 1 ( n 2 ) (M=m\mid N=n,K=2)=(m\mid n)=[m\leq n]\frac{m-1}{{\left({{n}\atop{2}}\right)}}
  84. ( n ) = [ n m ] m - 1 ( n 2 ) \mathcal{L}(n)=[n\geq m]\frac{m-1}{{\left({{n}\atop{2}}\right)}}
  85. n ( n ) = m - 1 1 n = m 1 ( n 2 ) = m - 1 1 2 2 - 1 1 ( m - 1 2 - 1 ) = 2 \begin{aligned}\displaystyle\sum_{n}\mathcal{L}(n)&\displaystyle=\frac{m-1}{1}% \sum_{n=m}^{\infty}\frac{1}{{\left({{n}\atop{2}}\right)}}\\ &\displaystyle=\frac{m-1}{1}\cdot\frac{2}{2-1}\cdot\frac{1}{{\left({{m-1}\atop% {2-1}}\right)}}\\ &\displaystyle=2\end{aligned}
  86. ( N = n M = m , K = 2 ) = ( n m ) = ( n ) n ( n ) = [ n m ] m - 1 n ( n - 1 ) \begin{aligned}&\displaystyle(N=n\mid M=m,K=2)\\ \displaystyle=&\displaystyle(n\mid m)\\ \displaystyle=&\displaystyle\frac{\mathcal{L}(n)}{\sum_{n}\mathcal{L}(n)}\\ \displaystyle=&\displaystyle[n\geq m]\frac{m-1}{n(n-1)}\end{aligned}
  87. N ~ \scriptstyle\tilde{N}
  88. n [ n N ~ ] ( n m ) = 1 2 \sum_{n}[n\geq\tilde{N}](n\mid m)=\frac{1}{2}
  89. m - 1 N ~ - 1 = 1 2 \frac{m-1}{\tilde{N}-1}=\frac{1}{2}
  90. N ~ = 2 m - 1 \tilde{N}=2m-1
  91. μ = n n ( n m ) = m - 1 1 n = m 1 n - 1 = \mu=\sum_{n}n\cdot(n\mid m)=\frac{m-1}{1}\sum_{n=m}^{\infty}\frac{1}{n-1}=\infty
  92. ( M = m N = n , K = k 2 ) = ( m n , k ) = [ m n ] ( m - 1 k - 1 ) ( n k ) \begin{aligned}&\displaystyle(M=m\mid N=n,K=k\geq 2)\\ \displaystyle=&\displaystyle(m\mid n,k)\\ \displaystyle=&\displaystyle[m\leq n]\frac{{\left({{m-1}\atop{k-1}}\right)}}{{% \left({{n}\atop{k}}\right)}}\end{aligned}
  93. ( n ) = [ n m ] ( m - 1 k - 1 ) ( n k ) \mathcal{L}(n)=[n\geq m]\frac{{\left({{m-1}\atop{k-1}}\right)}}{{\left({{n}% \atop{k}}\right)}}
  94. n ( n ) \displaystyle\sum_{n}\mathcal{L}(n)
  95. ( N = n M = m , K = k 2 ) = ( n m , k ) = ( n ) n ( n ) = [ n m ] k - 1 k ( m - 1 k - 1 ) ( n k ) = [ n m ] m - 1 n ( m - 2 k - 2 ) ( n - 1 k - 1 ) = [ n m ] m - 1 n m - 2 n - 1 k - 1 k - 2 ( m - 3 k - 3 ) ( n - 2 k - 2 ) \begin{aligned}&\displaystyle(N=n\mid M=m,K=k\geq 2)=(n\mid m,k)\\ \displaystyle=&\displaystyle\frac{\mathcal{L}(n)}{\sum_{n}\mathcal{L}(n)}\\ \displaystyle=&\displaystyle[n\geq m]\frac{k-1}{k}\frac{{\left({{m-1}\atop{k-1% }}\right)}}{{\left({{n}\atop{k}}\right)}}\\ \displaystyle=&\displaystyle[n\geq m]\frac{m-1}{n}\frac{{\left({{m-2}\atop{k-2% }}\right)}}{{\left({{n-1}\atop{k-1}}\right)}}\\ \displaystyle=&\displaystyle[n\geq m]\frac{m-1}{n}\frac{m-2}{n-1}\frac{k-1}{k-% 2}\frac{{\left({{m-3}\atop{k-3}}\right)}}{{\left({{n-2}\atop{k-2}}\right)}}% \end{aligned}
  96. ( N > x M = m , K = k ) = { 1 if x < m n = x + 1 ( n m , k ) if x m = [ x < m ] + [ x m ] n = x + 1 k - 1 k ( m - 1 k - 1 ) ( N k ) = [ x < m ] + [ x m ] k - 1 k ( m - 1 k - 1 ) 1 n = x + 1 1 ( n k ) = [ x < m ] + [ x m ] k - 1 k ( m - 1 k - 1 ) 1 k k - 1 1 ( x k - 1 ) = [ x < m ] + [ x m ] ( m - 1 k - 1 ) ( x k - 1 ) \begin{aligned}&\displaystyle(N>x\mid M=m,K=k)\\ \displaystyle=&\displaystyle\begin{cases}1&\,\text{if }x<m\\ \sum_{n=x+1}^{\infty}(n\mid m,k)&\,\text{if }x\geq m\end{cases}\\ \displaystyle=&\displaystyle[x<m]+[x\geq m]\sum_{n=x+1}^{\infty}\frac{k-1}{k}% \frac{{\left({{m-1}\atop{k-1}}\right)}}{{\left({{N}\atop{k}}\right)}}\\ \displaystyle=&\displaystyle[x<m]+[x\geq m]\frac{k-1}{k}\frac{{\left({{m-1}% \atop{k-1}}\right)}}{1}\sum_{n=x+1}^{\infty}\frac{1}{{\left({{n}\atop{k}}% \right)}}\\ \displaystyle=&\displaystyle[x<m]+[x\geq m]\frac{k-1}{k}\frac{{\left({{m-1}% \atop{k-1}}\right)}}{1}\cdot\frac{k}{k-1}\frac{1}{{\left({{x}\atop{k-1}}\right% )}}\\ \displaystyle=&\displaystyle[x<m]+[x\geq m]\frac{{\left({{m-1}\atop{k-1}}% \right)}}{{\left({{x}\atop{k-1}}\right)}}\end{aligned}
  97. ( N x M = m , K = k ) = 1 - ( N > x M = m , K = k ) = [ x m ] ( 1 - ( m - 1 k - 1 ) ( x k - 1 ) ) \begin{aligned}&\displaystyle(N\leq x\mid M=m,K=k)\\ \displaystyle=&\displaystyle 1-(N>x\mid M=m,K=k)\\ \displaystyle=&\displaystyle[x\geq m]\left(1-\frac{{\left({{m-1}\atop{k-1}}% \right)}}{{\left({{x}\atop{k-1}}\right)}}\right)\end{aligned}
  98. μ = n n ( N = n M = m , K = k ) = n n [ n m ] m - 1 n ( m - 2 k - 2 ) ( n - 1 k - 1 ) = m - 1 1 ( m - 2 k - 2 ) 1 n = m 1 ( n - 1 k - 1 ) = m - 1 1 ( m - 2 k - 2 ) 1 k - 1 k - 2 1 ( m - 2 k - 2 ) = m - 1 1 k - 1 k - 2 \begin{aligned}\displaystyle\mu&\displaystyle=\sum_{n}n\cdot(N=n\mid M=m,K=k)% \\ &\displaystyle=\sum_{n}n[n\geq m]\frac{m-1}{n}\frac{{\left({{m-2}\atop{k-2}}% \right)}}{{\left({{n-1}\atop{k-1}}\right)}}\\ &\displaystyle=\frac{m-1}{1}\frac{{\left({{m-2}\atop{k-2}}\right)}}{1}\sum_{n=% m}^{\infty}\frac{1}{{\left({{n-1}\atop{k-1}}\right)}}\\ &\displaystyle=\frac{m-1}{1}\frac{{\left({{m-2}\atop{k-2}}\right)}}{1}\cdot% \frac{k-1}{k-2}\frac{1}{{\left({{m-2}\atop{k-2}}\right)}}\\ &\displaystyle=\frac{m-1}{1}\frac{k-1}{k-2}\end{aligned}
  99. σ 2 + μ 2 = n n 2 ( N = n M = m , K = k ) \sigma^{2}+\mu^{2}=\sum_{n}n^{2}\cdot(N=n\mid M=m,K=k)
  100. σ 2 + μ 2 - μ = n n ( n - 1 ) ( N = n M = m , K = k ) = n = m n ( n - 1 ) m - 1 n m - 2 n - 1 k - 1 k - 2 ( m - 3 k - 3 ) ( n - 2 k - 2 ) = m - 1 1 m - 2 1 k - 1 k - 2 ( m - 3 k - 3 ) 1 n = m 1 ( n - 2 k - 2 ) = m - 1 1 m - 2 1 k - 1 k - 2 ( m - 3 k - 3 ) 1 k - 2 k - 3 1 ( m - 3 k - 3 ) = m - 1 1 m - 2 1 k - 1 k - 3 \begin{aligned}\displaystyle\sigma^{2}+\mu^{2}-\mu&\displaystyle=\sum_{n}n(n-1% )\cdot(N=n\mid M=m,K=k)\\ &\displaystyle=\sum_{n=m}^{\infty}n(n-1)\frac{m-1}{n}\frac{m-2}{n-1}\frac{k-1}% {k-2}\frac{{\left({{m-3}\atop{k-3}}\right)}}{{\left({{n-2}\atop{k-2}}\right)}}% \\ &\displaystyle=\frac{m-1}{1}\frac{m-2}{1}\frac{k-1}{k-2}\cdot\frac{{\left({{m-% 3}\atop{k-3}}\right)}}{1}\sum_{n=m}^{\infty}\frac{1}{{\left({{n-2}\atop{k-2}}% \right)}}\\ &\displaystyle=\frac{m-1}{1}\frac{m-2}{1}\frac{k-1}{k-2}\frac{{\left({{m-3}% \atop{k-3}}\right)}}{1}\frac{k-2}{k-3}\frac{1}{{\left({{m-3}\atop{k-3}}\right)% }}\\ &\displaystyle=\frac{m-1}{1}\frac{m-2}{1}\frac{k-1}{k-3}\\ &\end{aligned}
  101. σ = m - 1 1 m - 2 1 k - 1 k - 3 + μ - μ 2 = ( k - 1 ) ( m - 1 ) ( m - k + 1 ) ( k - 3 ) ( k - 2 ) 2 \begin{aligned}\displaystyle\sigma&\displaystyle=\sqrt{\frac{m-1}{1}\frac{m-2}% {1}\frac{k-1}{k-3}+\mu-\mu^{2}}\\ &\displaystyle=\sqrt{\frac{(k-1)(m-1)(m-k+1)}{(k-3)(k-2)^{2}}}\\ &\end{aligned}
  102. σ 2 μ = m - k + 1 ( k - 3 ) ( k - 2 ) \frac{\sigma^{2}}{\mu}=\frac{m-k+1}{(k-3)(k-2)}

GGH_encryption_scheme.html

  1. B B
  2. L L
  3. U U
  4. L L
  5. B = U B B^{\prime}=UB
  6. ( λ 1 , , λ n ) (\lambda_{1},...,\lambda_{n})
  7. - M < λ i < M -M<\lambda_{i}<M
  8. m = ( λ 1 , , λ n ) m=(\lambda_{1},...,\lambda_{n})
  9. e e
  10. B B^{\prime}
  11. v = λ i b i v=\sum\lambda_{i}b_{i}^{\prime}
  12. v = m B v=m\cdot B^{\prime}
  13. m m
  14. b b^{\prime}
  15. c = v + e = m B + e c=v+e=m\cdot B^{\prime}+e
  16. c B - 1 = ( m B + e ) B - 1 = m U B B - 1 + e B - 1 = m U + e B - 1 c\cdot B^{-1}=(m\cdot B^{\prime}+e)B^{-1}=m\cdot U\cdot B\cdot B^{-1}+e\cdot B% ^{-1}=m\cdot U+e\cdot B^{-1}
  17. e B - 1 e\cdot B^{-1}
  18. m = m U U - 1 m=m\cdot U\cdot U^{-1}
  19. L 2 L\subset\mathbb{R}^{2}
  20. B B
  21. B - 1 B^{-1}
  22. B = ( 7 0 0 3 ) B=\begin{pmatrix}7&0\\ 0&3\\ \end{pmatrix}
  23. B - 1 = ( < c o d e > 1 7 0 0 1 3 < / c o d e > < b r / > < c o d e > ) < / c o d e > B^{-1}=\begin{pmatrix}\par <code>\frac{1}{7}& 0\\  0& \frac{1}{3}\\       </code><br/><code>\end{pmatrix}</code>
  24. U = ( 2 3 3 5 ) U=\begin{pmatrix}2&3\\ 3&5\\ \end{pmatrix}
  25. U - 1 = ( 5 - 3 - 3 2 ) U^{-1}=\begin{pmatrix}5&-3\\ -3&2\\ \end{pmatrix}
  26. B = U B = ( 14 9 21 15 ) B^{\prime}=UB=\begin{pmatrix}14&9\\ 21&15\\ \end{pmatrix}
  27. m = ( 3 , - 7 ) m=(3,-7)
  28. e = ( 1 , - 1 ) e=(1,-1)
  29. c = m B + e = ( - 104 , - 79 ) . c=mB^{\prime}+e=(-104,-79).\,
  30. c B - 1 = ( - 104 7 , - 79 3 ) . cB^{-1}=\left(\frac{-104}{7},\frac{-79}{3}\right).
  31. ( - 15 , - 26 ) (-15,-26)
  32. m = ( - 15 , - 26 ) U - 1 = ( 3 , - 7 ) . m=(-15,-26)U^{-1}=(3,-7).\,

Gibbs_lemma.html

  1. ϕ = i = 1 n f i ( x i ) \phi=\sum_{i=1}^{n}f_{i}(x_{i})
  2. ϕ \phi
  3. x i = X \sum x_{i}=X
  4. x i 0 x_{i}\geq 0
  5. x 0 = ( x 1 0 , , x n 0 ) x^{0}=(x_{1}^{0},\ldots,x_{n}^{0})
  6. f i f_{i}
  7. λ \lambda
  8. f i ( x i 0 ) = λ if x i 0 > 0 λ if x i 0 = 0. \begin{aligned}\displaystyle f^{\prime}_{i}(x_{i}^{0})&\displaystyle=\lambda% \mbox{ if }~{}x_{i}^{0}>0\\ &\displaystyle\leq\lambda\mbox{ if }x_{i}^{0}=0.\end{aligned}

Girolami_method.html

  1. V S = i V i V_{S}\,=\,\sum_{i}V_{i}
  2. d = M 5 V S d\,=\,\frac{M}{5\cdot V_{S}}

Glossary_of_fuel_cell_terms.html

  1. 5 / 9 {5}/{9}
  2. Δ H v \Delta{}_{v}H
  3. Re \mathrm{Re}
  4. V ρ {\mathrm{V}}\rho
  5. μ / L \mu/L
  6. k k
  7. η t h \eta_{th}\,

Godement_resolution.html

  1. x X x\in X
  2. F x F_{x}
  3. U X U\subset X
  4. Gode ( F ) ( U ) := x U F x . \operatorname{Gode}(F)(U):=\prod_{x\in U}F_{x}.
  5. U V U\subset V
  6. Gode ( F ) ( V ) Gode ( F ) ( U ) \operatorname{Gode}(F)(V)\rightarrow\operatorname{Gode}(F)(U)
  7. F Gode ( F ) F\to\operatorname{Gode}(F)
  8. Y = x X { x } \textstyle Y=\coprod_{x\in X}\{x\}
  9. G 0 ( F ) = Gode ( F ) G_{0}(F)=\operatorname{Gode}(F)
  10. d 0 : F G 0 ( F ) d_{0}\colon F\rightarrow G_{0}(F)
  11. i > 0 i>0
  12. G i ( F ) G_{i}(F)
  13. Gode ( coker ( d i - 1 ) ) \operatorname{Gode}(\operatorname{coker}(d_{i-1}))
  14. d i : G i - 1 G i d_{i}\colon G_{i-1}\rightarrow G_{i}

Goldbeter–Koshland_kinetics.html

  1. z = [ Z ] [ Z ] 0 = G ( v 1 , v 2 , J 1 , J 2 ) \displaystyle z=\frac{[Z]}{[Z]_{0}}=G(v_{1},v_{2},J_{1},J_{2})
  2. v 1 = k 1 [ X ] ; v 2 \displaystyle v_{1}=k_{1}[X];\ v_{2}
  3. d [ Z ] d t = ! 0 \displaystyle\frac{d[Z]}{dt}\ \stackrel{!}{=}\ 0
  4. r 1 = k 1 [ X ] [ Z P ] K M 1 + [ Z P ] r_{1}=\frac{k_{1}[X][Z_{P}]}{K_{M1}+[Z_{P}]}
  5. r 2 = k 2 [ Y ] [ Z ] K M 2 + [ Z ] r_{2}=\frac{k_{2}[Y][Z]}{K_{M2}+[Z]}
  6. d [ Z ] d t = r 1 - r 2 = k 1 [ X ] ( [ Z ] 0 - [ Z ] ) K M 1 + ( [ Z ] 0 - [ Z ] ) \displaystyle\frac{d[Z]}{dt}=r_{1}-r_{2}=\frac{k_{1}[X]([Z]_{0}-[Z])}{K_{M1}+(% [Z]_{0}-[Z])}
  7. z = [ Z ] [ Z ] 0 ; v 1 = k 1 [ X ] ; v 2 \displaystyle z=\frac{[Z]}{[Z]_{0}};\ v_{1}=k_{1}[X];\ v_{2}
  8. v 1 ( 1 - z ) J 1 + ( 1 - z ) \displaystyle\frac{v_{1}(1-z)}{J_{1}+(1-z)}
  9. z = [ Z ] [ Z ] 0 = G ( v 1 , v 2 , J 1 , J 2 ) \displaystyle z=\frac{[Z]}{[Z]_{0}}=G(v_{1},v_{2},J_{1},J_{2})

Golden_rhombus.html

  1. p q = φ \frac{p}{q}=\varphi\!
  2. φ \varphi\!
  3. 2 arctan 1 φ = arctan 2 63.43495 2\arctan\frac{1}{\varphi}=\arctan{2}\approx 63.43495
  4. 2 arctan φ = arctan 1 + arctan 3 116.56505 2\arctan\varphi=\arctan{1}+\arctan{3}\approx 116.56505
  5. q = 1 q=1
  6. e = 1 2 p 2 + q 2 = 1 2 1 + φ 2 = 1 4 10 + 2 5 0.95106 \begin{array}[]{rcl}e&=&\tfrac{1}{2}\sqrt{p^{2}+q^{2}}\\ &=&\tfrac{1}{2}\sqrt{1+\varphi^{2}}\\ &=&\tfrac{1}{4}\sqrt{10+2\sqrt{5}}\\ &\approx&0.95106\end{array}
  7. p = φ e = 2 1 + 5 10 + 2 5 1.70130 \begin{array}[]{rcl}p&=&\frac{\varphi}{e}\\ &=&2\frac{1+\sqrt{5}}{\sqrt{10+2\sqrt{5}}}\\ &\approx&1.70130\end{array}
  8. q = 1 e = 4 1 10 + 2 5 1.05146 \begin{array}[]{rcl}q&=&\frac{1}{e}\\ &=&4\frac{1}{\sqrt{10+2\sqrt{5}}}\\ &\approx&1.05146\end{array}

Gompertz_distribution.html

  1. f ( x ; η , b ) = b η e b x e η exp ( - η e b x ) for x 0 , f\left(x;\eta,b\right)=b\eta e^{bx}e^{\eta}\exp\left(-\eta e^{bx}\right)\,% \text{for }x\geq 0,\,
  2. b > 0 b>0\,\!
  3. η > 0 \eta>0\,\!
  4. F ( x ; η , b ) = 1 - exp ( - η ( e b x - 1 ) ) , F\left(x;\eta,b\right)=1-\exp\left(-\eta\left(e^{bx}-1\right)\right),
  5. η , b > 0 , \eta,b>0,
  6. x 0 . x\geq 0\,.
  7. E ( e - t X ) = η e η E t / b ( η ) \,\text{E}\left(e^{-tX}\right)=\eta e^{\eta}\,\text{E}_{t/b}\left(\eta\right)
  8. E t / b ( η ) = 1 e - η v v - t / b d v , t > 0. \,\text{E}_{t/b}\left(\eta\right)=\int_{1}^{\infty}e^{-\eta v}v^{-t/b}dv,\ t>0.
  9. η \eta\,\!
  10. η 1 , \eta\geq 1,\,
  11. 0 < η < 1 , 0<\eta<1,\,
  12. x * = ( 1 / b ) ln ( 1 / η ) with 0 < F ( x * ) < 1 - e - 1 = 0.632121 x^{*}=\left(1/b\right)\ln\left(1/\eta\right)\text{with }0<F\left(x^{*}\right)<% 1-e^{-1}=0.632121
  13. f 1 f_{1}
  14. f 2 f_{2}
  15. D K L ( f 1 f 2 ) = 0 f 1 ( x ; b 1 , η 1 ) ln f 1 ( x ; b 1 , η 1 ) f 2 ( x ; b 2 , η 2 ) d x = ln e η 1 b 1 η 1 e η 2 b 2 η 2 + e η 1 [ ( b 2 b 1 - 1 ) Ei ( - η 1 ) + η 2 η 1 b 2 b 1 Γ ( b 2 b 1 + 1 , η 1 ) ] - ( η 1 + 1 ) \begin{aligned}\displaystyle D_{KL}(f_{1}\parallel f_{2})&\displaystyle=\int_{% 0}^{\infty}f_{1}(x;b_{1},\eta_{1})\,\ln\frac{f_{1}(x;b_{1},\eta_{1})}{f_{2}(x;% b_{2},\eta_{2})}dx\\ &\displaystyle=\ln\frac{e^{\eta_{1}}\,b_{1}\,\eta_{1}}{e^{\eta_{2}}\,b_{2}\,% \eta_{2}}+e^{\eta_{1}}\left[\left(\frac{b_{2}}{b_{1}}-1\right)\,\operatorname{% Ei}(-\eta_{1})+\frac{\eta_{2}}{\eta_{1}^{\frac{b_{2}}{b_{1}}}}\,\Gamma\left(% \frac{b_{2}}{b_{1}}+1,\eta_{1}\right)\right]-(\eta_{1}+1)\end{aligned}
  16. Ei ( ) \operatorname{Ei}(\cdot)
  17. Γ ( , ) \Gamma(\cdot,\cdot)
  18. b . b\,\!.
  19. η \eta\,\!
  20. α \alpha\,\!
  21. β \beta\,\!
  22. α / β \alpha/\beta\,\!
  23. x x

Good_prime.html

  1. p n 2 > p ( n - i ) p ( n + i ) p_{n}^{2}>p_{(n-i)}\cdot p_{(n+i)}
  2. 5 2 > 3 7 5^{2}>3\cdot 7
  3. 5 2 > 2 11 5^{2}>2\cdot 11

Goodman_and_Kruskal's_lambda.html

  1. λ \lambda
  2. λ \lambda
  3. λ = ε 1 - ε 2 ε 1 . \lambda=\frac{\varepsilon_{1}-\varepsilon_{2}}{\varepsilon_{1}}.
  4. ε 1 \varepsilon_{1}
  5. ε 2 \varepsilon_{2}
  6. λ = 128 - ( 30 + 98 ) 128 = 0 \lambda=\frac{128-(30+98)}{128}=0

Goodman_relation.html

  1. σ a = σ fat × ( 1 - σ m σ ts ) . \sigma\text{a}=\sigma\text{fat}\times\left(1-\frac{\sigma\text{m}}{\sigma\text% {ts}}\right).
  2. σ a \sigma\text{a}
  3. σ m \sigma\text{m}
  4. σ fat \sigma\text{fat}
  5. σ ts \sigma\text{ts}

Goodwin_model_(economics).html

  1. q = min ( a , k σ ) q=\min\left(a\ell,\frac{k}{\sigma}\right)
  2. a = k σ = q a\ell=\frac{k}{\sigma}=q
  3. v = n v=\frac{\ell}{n}
  4. d v / d t v = g v = g - β . \frac{dv/dt}{v}=g_{v}=g_{\ell}-\beta.
  5. d / d t = g = g q - α \frac{d\ell/dt}{\ell}=g_{\ell}=g_{q}-\alpha
  6. d w / d t w = g w = ρ v - γ . \frac{dw/dt}{w}=g_{w}=\rho v-\gamma.
  7. u = w q = w a . u=\frac{w\ell}{q}=\frac{w}{a}.
  8. d u / d t u = g u = g w - α \frac{du/dt}{u}=g_{u}=g_{w}-\alpha
  9. d k / d t k = g k = g q = s ( 1 - u ) ( q / k ) - δ . \frac{dk/dt}{k}=g_{k}=g_{q}=s(1-u)(q/k)-\delta.
  10. d v / d t v = g v = s ( 1 - u ) σ - ( δ + α + β ) . \frac{dv/dt}{v}=g_{v}=\frac{s(1-u)}{\sigma}-(\delta+\alpha+\beta).
  11. d v / d t v = g v = s ( 1 - u ) σ - ( δ + α + β ) \frac{dv/dt}{v}=g_{v}=\frac{s(1-u)}{\sigma}-(\delta+\alpha+\beta)
  12. d u / d t u = g u = ρ v - γ - α \frac{du/dt}{u}=g_{u}=\rho v-\gamma-\alpha
  13. u * = 1 - ( δ + α + β ) σ s u*=1-\frac{(\delta+\alpha+\beta)\sigma}{s}
  14. v * = γ + α ρ v*=\frac{\gamma+\alpha}{\rho}

Google_matrix.html

  1. A i , j A_{i,j}
  2. j j
  3. i i
  4. k j k_{j}
  5. k j k_{j}
  6. a a
  7. G i j = α S i j + ( 1 - α ) 1 N ( 1 ) G_{ij}=\alpha S_{ij}+(1-\alpha)\frac{1}{N}\;\;\;\;\;\;\;\;\;\;\;(1)
  8. α \alpha
  9. S S
  10. α \alpha
  11. ( 1 - α ) (1-\alpha)
  12. G G
  13. α = 1 \alpha=1
  14. 0 < α < 1 0<\alpha<1
  15. λ = 1 \lambda=1
  16. P i P_{i}
  17. P i P_{i}
  18. K i K_{i}
  19. P 1 / K β P\propto 1/K^{\beta}
  20. β 0.9 \beta\approx 0.9
  21. N P 1 / P ν N_{P}\propto 1/P^{\nu}
  22. ν = 1 + 1 / β 2.1 \nu=1+1/\beta\approx 2.1
  23. λ = 1 \lambda=1
  24. 0 < α 0<\alpha
  25. λ i α λ i \lambda_{i}\rightarrow\alpha\lambda_{i}
  26. λ = 1 \lambda=1
  27. α \alpha
  28. λ i < 1 \lambda_{i}<1
  29. λ = 1 \lambda=1
  30. λ = 1 \lambda=1
  31. 1 - α 0.15 1-\alpha\approx 0.15
  32. G G
  33. λ i \lambda_{i}
  34. α = 1 \alpha=1
  35. α = 1 \alpha=1
  36. G G
  37. λ = 1 \lambda=1
  38. λ \lambda
  39. G G
  40. N = 285509 N=285509
  41. α = 0.85 \alpha=0.85
  42. | λ i | |\lambda_{i}|
  43. λ \lambda
  44. d 1.3 d\approx 1.3
  45. G G
  46. G G

Googol.html

  1. 2 ( 100 / log 10 2 ) 2^{(100/\mathrm{log}_{10}2)}

Gordon–Newell_theorem.html

  1. P i j P_{ij}
  2. P i j P_{ij}
  3. j = 1 m P i j = 1 \scriptstyle{\sum_{j=1}^{m}P_{ij}=1}
  4. ( k 1 , k 2 , , k m ) \scriptstyle{(k_{1},k_{2},\ldots,k_{m})}
  5. S ( K , m ) = { 𝐤 m such that i = 1 m k i = K and k i 0 i } . S(K,m)=\left\{\mathbf{k}\in\mathbb{Z}^{m}\,\text{ such that }\sum_{i=1}^{m}k_{% i}=K\,\text{ and }k_{i}\geq 0\quad\forall i\right\}.
  6. π ( k 1 , k 2 , , k m ) = 1 G ( K ) i = 1 m ( e i μ i ) k i \pi(k_{1},k_{2},\ldots,k_{m})=\frac{1}{G(K)}\prod_{i=1}^{m}\left(\frac{e_{i}}{% \mu_{i}}\right)^{k_{i}}
  7. G ( K ) = 𝐤 S ( K , m ) i = 1 m ( e i μ i ) k i , G(K)=\sum_{\mathbf{k}\in S(K,m)}\prod_{i=1}^{m}\left(\frac{e_{i}}{\mu_{i}}% \right)^{k_{i}},
  8. e i = j = 1 m e j p j i for 1 i m . e_{i}=\sum_{j=1}^{m}e_{j}p_{ji}\,\text{ for }1\leq i\leq m.\,

Gossen's_laws.html

  1. U / x i p i = U / x j p j ( i , j ) \frac{\partial U/\partial x_{i}}{p_{i}}=\frac{\partial U/\partial x_{j}}{p_{j}% }\,\forall\left(i,j\right)
  2. U U
  3. x i x_{i}
  4. i i
  5. p i p_{i}
  6. i i

Gossen's_second_law.html

  1. U / x i p i = U / x j p j ( i , j ) \frac{\partial U/\partial x_{i}}{p_{i}}=\frac{\partial U/\partial x_{j}}{p_{j}% }~{}\forall\left(i,j\right)
  2. U U
  3. x i x_{i}
  4. i i
  5. p i p_{i}
  6. i i
  7. U / x i \partial U/\partial x_{i}
  8. U ( x 1 , x 2 , , x n ) U\left(x_{1},x_{2},\dots,x_{n}\right)
  9. W i = 1 n ( p i x i ) W\geq\sum_{i=1}^{n}\left(p_{i}\cdot x_{i}\right)
  10. W W
  11. ( x 1 , x 2 , , x n , λ ) = U ( x 1 , x 2 , , x n ) + λ [ W - i = 1 n ( p i x i ) ] \mathcal{L}\left(x_{1},x_{2},\dots,x_{n},\lambda\right)=U\left(x_{1},x_{2},% \dots,x_{n}\right)+\lambda\cdot\left[W-\sum_{i=1}^{n}\left(p_{i}\cdot x_{i}% \right)\right]
  12. λ = 0 \frac{\partial\mathcal{L}}{\partial\lambda}=0
  13. W W
  14. x i = 0 i \frac{\partial\mathcal{L}}{\partial x_{i}}=0~{}~{}\forall i
  15. U x i - λ p i = 0 i \frac{\partial U}{\partial x_{i}}-\lambda\cdot p_{i}=0~{}~{}\forall i
  16. U / x i p i = λ i \frac{\partial U/\partial x_{i}}{p_{i}}=\lambda~{}~{}\forall i
  17. λ \lambda
  18. U / x i p i = U / x j p j ( i , j ) \frac{\partial U/\partial x_{i}}{p_{i}}=\frac{\partial U/\partial x_{j}}{p_{j}% }~{}\forall\left(i,j\right)

GPRS_roaming_exchange.html

  1. N ( N - 1 ) 2 \tfrac{N(N-1)}{2}
  2. N N

Grammar_systems_theory.html

  1. 𝔸 \mathbb{A}
  2. 𝔸 \mathbb{A}
  3. 𝕃 𝔸 = { ( f m t n f r ) + : 1 m k ; 1 n ; 1 r k } , \mathbb{L_{A}}=\{(f^{m}t^{n}f^{r})^{+}:1\leq m\leq k;1\leq n\leq\ell;1\leq r% \leq k\},
  4. 𝔾 𝔸 \mathbb{G_{A}}
  5. 𝕃 𝔸 \mathbb{L_{A}}
  6. 𝔸 \mathbb{A}
  7. 𝔸 \mathbb{A}

Gran_plot.html

  1. E = E 0 + s log { H + } E=E^{0}+s\log\{H^{+}\}
  2. [ H + ] = 10 E - E 0 s o r [ H + ] = 10 - p H [H^{+}]=10^{\frac{E-E^{0}}{s}}\ or\ [H^{+}]=10^{-pH}
  3. [ H + ] = C i v i - c O H v v i + v [H^{+}]=\frac{C_{i}v_{i}-c_{OH}v}{v_{i}+v}
  4. C i v i - c O H v = ( v i + v ) 10 E - E 0 s o r = ( v i + v ) 10 - p H C_{i}v_{i}-c_{OH}v=(v_{i}+v)10^{\frac{E-E^{0}}{s}}\ or\ =(v_{i}+v)10^{-pH}
  5. ( v i + v ) 10 E - E 0 s o r ( v i + v ) 10 - p H (v_{i}+v)10^{\frac{E-E^{0}}{s}}\ or\ (v_{i}+v)10^{-pH}
  6. C i v i = c O H v C_{i}v_{i}=c_{OH}v
  7. K w = [ H + ] i [ O H - ] i K_{w}=[H^{+}]_{i}[OH^{-}]_{i}
  8. v 0 v_{0}
  9. [ H + ] 0 [H^{+}]_{0}
  10. [ O H - ] 0 [OH^{-}]_{0}
  11. v i v_{i}
  12. v 0 [ H + ] 0 - v i [ O H - ] 0 v 0 + v i { [ H + ] i or 10 - p H i when v 0 [ H + ] 0 > v i [ O H - ] 0 (acidic region) = 0 when v 0 [ H + ] 0 = v i [ O H - ] 0 (equivalence point) - [ O H - ] i or - K w 10 p H i when v 0 [ H + ] 0 < v i [ O H - ] 0 (alkaline region) \frac{v_{0}[H^{+}]_{0}-v_{i}[OH^{-}]_{0}}{v_{0}+v_{i}}\begin{cases}\approx[H^{% +}]_{i}\,\text{ or }10^{-pH_{i}}&\,\text{ when }v_{0}[H^{+}]_{0}>v_{i}[OH^{-}]% _{0}\,\text{ (acidic region)}\\ =0&\,\text{ when }v_{0}[H^{+}]_{0}=v_{i}[OH^{-}]_{0}\,\text{ (equivalence % point)}\\ \approx-[OH^{-}]_{i}\,\text{ or }-K_{w}10^{pH_{i}}&\,\text{ when }v_{0}[H^{+}]% _{0}<v_{i}[OH^{-}]_{0}\,\text{ (alkaline region)}\end{cases}
  13. v e = v i v_{e}=v_{i}
  14. ( v 0 + v i ) 10 - p H i vs. v i ({v_{0}+v_{i}})10^{-pH_{i}}\,\text{ vs. }v_{i}
  15. - [ O H - ] 0 -[OH^{-}]_{0}
  16. ( v 0 + v i ) 10 p H i vs. v i ({v_{0}+v_{i}})10^{pH_{i}}\,\text{ vs. }v_{i}
  17. [ O H - ] 0 / K w [OH^{-}]_{0}/K_{w}
  18. v e = v 0 [ H + ] 0 / [ O H - ] 0 v_{e}=v_{0}[H^{+}]_{0}/[OH^{-}]_{0}
  19. [ H + ] 0 [H^{+}]_{0}
  20. [ O H - ] 0 [OH^{-}]_{0}
  21. v 0 [ O H - ] 0 - v i [ H + ] 0 v 0 + v i { [ O H - ] i or K w 10 p H i when v 0 [ O H - ] 0 > v i [ H + ] 0 (alkaline region) = 0 when v 0 [ O H - ] 0 = v i [ H + ] 0 (equivalence point) - [ H + ] i or - 10 - p H i when v 0 [ O H - ] 0 < v i [ H + ] 0 (acidic region) \frac{v_{0}[OH^{-}]_{0}-v_{i}[H^{+}]_{0}}{v_{0}+v_{i}}\begin{cases}\approx[OH^% {-}]_{i}\,\text{ or }K_{w}10^{pH_{i}}&\,\text{ when }v_{0}[OH^{-}]_{0}>v_{i}[H% ^{+}]_{0}\,\text{ (alkaline region)}\\ =0&\,\text{ when }v_{0}[OH^{-}]_{0}=v_{i}[H^{+}]_{0}\,\text{ (equivalence % point)}\\ \approx-[H^{+}]_{i}\,\text{ or }-10^{-pH_{i}}&\,\text{ when }v_{0}[OH^{-}]_{0}% <v_{i}[H^{+}]_{0}\,\text{ (acidic region)}\end{cases}
  22. ( v 0 + v i ) 10 p H i vs. v i ({v_{0}+v_{i}})10^{pH_{i}}\,\text{ vs. }v_{i}
  23. - [ H + ] 0 / K w -[H^{+}]_{0}/K_{w}
  24. ( v 0 + v i ) 10 - p H i vs. v i ({v_{0}+v_{i}})10^{-pH_{i}}\,\text{ vs. }v_{i}
  25. [ H + ] 0 [H^{+}]_{0}
  26. v e = v 0 [ O H - ] 0 / [ H + ] 0 v_{e}=v_{0}[OH^{-}]_{0}/[H^{+}]_{0}
  27. K a = [ H + ] i [ A - ] i [ H A ] i K_{a}=\frac{[H^{+}]_{i}[A^{-}]_{i}}{[HA]_{i}}
  28. v 0 v_{0}
  29. [ H A ] 0 [HA]_{0}
  30. [ O H - ] 0 [OH^{-}]_{0}
  31. [ H A ] i v 0 [ H A ] 0 - v i [ O H - ] 0 v 0 + v i [HA]_{i}\approx\frac{v_{0}[HA]_{0}-v_{i}[OH^{-}]_{0}}{v_{0}+v_{i}}
  32. [ A - ] i v i [ O H - ] 0 v 0 + v i [A^{-}]_{i}\approx\frac{v_{i}[OH^{-}]_{0}}{v_{0}+v_{i}}
  33. K a 10 - p H i v i [ O H - ] 0 v 0 [ H A ] 0 - v i [ O H - ] 0 K_{a}\approx\frac{10^{-pH_{i}}v_{i}[OH^{-}]_{0}}{v_{0}[HA]_{0}-v_{i}[OH^{-}]_{% 0}}
  34. K a ( v 0 [ H A ] 0 [ O H - ] 0 - v i ) 10 - p H i v i K_{a}(v_{0}\frac{[HA]_{0}}{[OH^{-}]_{0}}-v_{i})\approx 10^{-pH_{i}}v_{i}
  35. v e = v 0 [ H A ] 0 [ O H - ] 0 v_{e}=v_{0}\frac{[HA]_{0}}{[OH^{-}]_{0}}
  36. K a ( v e - v i ) 10 - p H i v i K_{a}(v_{e}-v_{i})\approx 10^{-pH_{i}}v_{i}
  37. 10 - p H i v i 10^{-pH_{i}}v_{i}
  38. v i v_{i}
  39. - K a -K_{a}
  40. v e v_{e}
  41. [ H A ] 0 [HA]_{0}
  42. [ O H - ] 0 [OH^{-}]_{0}
  43. v 0 v_{0}
  44. ( v e - v e ) [ H + ] 0 = 2 v 0 [ C O 2 ] 0 (v_{e}-v_{e}^{\prime})[H^{+}]_{0}=2v_{0}[CO_{2}]_{0}
  45. ( v e - v e ) [ O H - ] 0 = 2 v e [ C O 2 ] 0 (v_{e}^{\prime}-v_{e})[OH^{-}]_{0}=2v_{e}^{\prime}[CO_{2}]_{0}
  46. v e v_{e}^{\prime}
  47. E i E_{i}
  48. H + H^{+}
  49. p H i pH_{i}
  50. - l o g 10 [ H + ] i = b 0 - b 1 E i -log_{10}[H^{+}]_{i}=b_{0}-b_{1}E_{i}
  51. b 0 b_{0}
  52. n F E 0 / R T nFE_{0}/RT
  53. b 1 b_{1}
  54. n F / R T nF/RT
  55. - b 1 E i -b_{1}E_{i}
  56. p H i pH_{i}
  57. ( v 0 + v i ) 10 b 1 E i ({v_{0}+v_{i}})10^{b_{1}E_{i}}
  58. v i v_{i}
  59. - 10 b 0 [ O H - ] 0 -10^{b_{0}}[OH^{-}]_{0}
  60. ( v 0 + v i ) 10 - b 1 E i ({v_{0}+v_{i}})10^{-b_{1}E_{i}}
  61. v i v_{i}
  62. 10 - b 0 [ O H - ] 0 / K w 10^{-b_{0}}[OH^{-}]_{0}/K_{w}
  63. v e = v 0 [ H + ] 0 / [ O H - ] 0 v_{e}=v_{0}[H^{+}]_{0}/[OH^{-}]_{0}
  64. b 0 b_{0}
  65. b 1 b_{1}
  66. - [ O H - ] 0 10 b 0 -[OH^{-}]_{0}10^{b_{0}}
  67. b 0 b_{0}
  68. [ O H - ] 0 [OH^{-}]_{0}
  69. K w K_{w}
  70. [ H + ] 0 10 b 0 [H^{+}]_{0}10^{b_{0}}
  71. b 0 b_{0}
  72. - [ H + ] 0 10 - b 0 / K w -[H^{+}]_{0}10^{-b_{0}}/K_{w}
  73. K w K_{w}
  74. [ H + ] 0 [H^{+}]_{0}
  75. b 1 b_{1}
  76. [ H + ] i [H^{+}]_{i}
  77. [ H + ] 0 [H^{+}]_{0}
  78. [ O H - ] 0 [OH^{-}]_{0}
  79. b 0 b_{0}
  80. b 1 b_{1}
  81. - l o g 10 [ H + ] i = b 0 - b 1 E i -log_{10}[H^{+}]_{i}=b_{0}-b_{1}E_{i}
  82. b 1 b_{1}
  83. ( v 0 + v i ) 10 b 1 E i - b 0 ({v_{0}+v_{i}})10^{b_{1}E_{i}-b_{0}}
  84. v i v_{i}
  85. ( v 0 + v i ) K w 10 - b 1 E i + b 0 ({v_{0}+v_{i}})K_{w}10^{-b_{1}E_{i}+b_{0}}
  86. v i v_{i}
  87. v e v_{e}
  88. v e v_{e}^{\prime}
  89. v e [ O H - ] 0 / v e v_{e}[OH^{-}]_{0}/v_{e}^{\prime}
  90. [ H + ] i [H^{+}]_{i}
  91. ( v 0 [ H + ] 0 - v i [ O H - ] 0 ) / ( v 0 + v i ) (v_{0}[H^{+}]_{0}-v_{i}[OH^{-}]_{0})/(v_{0}+v_{i})
  92. ( v e [ H + ] 0 - v i v e [ O H - ] 0 / v e ) / ( v 0 + v i ) (v_{e}[H^{+}]_{0}-v_{i}v_{e}[OH^{-}]_{0}/v_{e}^{\prime})/(v_{0}+v_{i})
  93. - l o g 10 [ H + ] i = b 0 - b 1 E i -log_{10}[H^{+}]_{i}=b_{0}-b_{1}E_{i}
  94. E i = b 0 / b 1 + ( 1 / b 1 ) l o g 10 [ H + ] i E_{i}=b_{0}/b_{1}+(1/b_{1})log_{10}[H^{+}]_{i}
  95. E i E_{i}
  96. l o g 10 [ H + ] i log_{10}[H^{+}]_{i}
  97. [ H + ] i [H^{+}]_{i}
  98. [ O H - ] i [OH^{-}]_{i}
  99. 1 / b 1 1/b_{1}
  100. b 0 / b 1 b_{0}/b_{1}
  101. b 0 b_{0}
  102. b 1 b_{1}
  103. b 1 b_{1}
  104. b 0 b_{0}
  105. v e v_{e}
  106. [ O H - ] 0 [OH^{-}]_{0}
  107. ( v 0 [ H + ] 0 - v i [ O H - ] 0 ) / ( v 0 + v i ) = [ H + ] i - K w / [ H + ] i (v_{0{}}[H^{+}]_{0}-v_{i}[OH^{-}]_{0})/(v_{0}+v_{i})=[H^{+}]_{i}-K_{w}/[H^{+}]% _{i}
  108. [ H + ] i [H^{+}]_{i}
  109. H + H^{+}
  110. S 1 S^{1}
  111. - l o g 10 [ S 1 ] i = b 0 - b 1 E i -log_{10}[S^{1}]_{i}=b_{0}-b_{1}E_{i}
  112. S 2 S^{2}
  113. S 1 S^{1}
  114. ( v 0 + v i ) 10 b 1 E i ({v_{0}+v_{i}})10^{b_{1}E_{i}}
  115. ( v 0 + v i ) 10 - b 1 E i ({v_{0}+v_{i}})10^{-b_{1}E_{i}}
  116. v i v_{i}
  117. v 0 [ S 0 ] 0 / [ S 1 ] 0 v_{0}[S^{0}]_{0}/[S^{1}]_{0}
  118. S 1 S^{1}
  119. S 2 S^{2}
  120. v 0 [ S 1 ] 0 / [ S 0 ] 0 v_{0}[S^{1}]_{0}/[S^{0}]_{0}
  121. K s p K_{sp}
  122. K w K_{w}
  123. v 0 [ C l - ] 0 - v i [ A g + ] 0 v 0 + v i { [ C l - ] i or K s p 10 - b 1 E i + b 0 when v 0 [ C l - ] 0 > v i [ A g + ] 0 (before equivalence) = 0 when v 0 [ C l - ] 0 = v i [ A g + ] 0 (equivalence point) - [ A g + ] i or - 10 b 1 E i - b 0 when v 0 [ C l - ] 0 < v i [ A g + ] 0 (after equivalence) \frac{v_{0}[Cl^{-}]_{0}-v_{i}[Ag^{+}]_{0}}{v_{0}+v_{i}}\begin{cases}\approx[Cl% ^{-}]_{i}\,\text{ or }K_{sp}10^{-b_{1}E_{i}+b_{0}}&\,\text{ when }v_{0}[Cl^{-}% ]_{0}>v_{i}[Ag^{+}]_{0}\,\text{ (before equivalence)}\\ =0&\,\text{ when }v_{0}[Cl^{-}]_{0}=v_{i}[Ag^{+}]_{0}\,\text{ (equivalence % point)}\\ \approx-[Ag^{+}]_{i}\,\text{ or }-10^{b_{1}E_{i}-b_{0}}&\,\text{ when }v_{0}[% Cl^{-}]_{0}<v_{i}[Ag^{+}]_{0}\,\text{ (after equivalence)}\end{cases}
  124. ( v 0 + v i ) 10 - b 1 E i vs. v i ({v_{0}+v_{i}})10^{-b_{1}E_{i}}\,\text{ vs. }v_{i}
  125. - [ A g + ] 0 10 - b 0 / K s p -[Ag^{+}]_{0}10^{-b_{0}}/K_{sp}
  126. ( v 0 + v i ) 10 b 1 E i vs. v i ({v_{0}+v_{i}})10^{b_{1}E_{i}}\,\text{ vs. }v_{i}
  127. [ A g + ] 0 10 b 0 [Ag^{+}]_{0}10^{b_{0}}
  128. v e = v 0 [ C l - ] 0 / [ A g + ] 0 v_{e}=v_{0}[Cl^{-}]_{0}/[Ag^{+}]_{0}
  129. 10 p H i 10^{pH_{i}}
  130. 10 - p H i 10^{-pH_{i}}