wpmath0000005_5

Følner_sequence.html

  1. G G
  2. X X
  3. F 1 , F 2 , F_{1},F_{2},\dots
  4. X X
  5. X X
  6. x X x\in X
  7. i i
  8. x F j x\in F_{j}
  9. j > i j>i
  10. lim i | g F i F i | | F i | = 0 \lim_{i\to\infty}\frac{|gF_{i}\,\triangle\,F_{i}|}{|F_{i}|}=0
  11. g g
  12. G G
  13. g F i gF_{i}
  14. F i F_{i}
  15. g g
  16. g f gf
  17. f f
  18. F i F_{i}
  19. \triangle
  20. A B A\triangle B
  21. A A
  22. B B
  23. | A | |A|
  24. A A
  25. g g
  26. F i F_{i}
  27. g g
  28. i i
  29. ( X , μ ) (X,\mu)
  30. lim i μ ( g F i F i ) μ ( F i ) = 0 \lim_{i\to\infty}\frac{\mu(gF_{i}\,\triangle\,F_{i})}{\mu(F_{i})}=0
  31. G G
  32. F i = G F_{i}=G
  33. i i
  34. F i F_{i}
  35. - i -i
  36. i i
  37. g F i gF_{i}
  38. g - i g-i
  39. g + i g+i
  40. i i
  41. 2 g 2g
  42. F i F_{i}
  43. 2 i + 1 2i+1
  44. 2 g / ( 2 i + 1 ) 2g/(2i+1)
  45. i i
  46. G G
  47. F i F_{i}
  48. μ \mu
  49. G G
  50. G G
  51. A A
  52. μ ( A ) = lim i | A F i | | F i | . \mu(A)=\lim_{i\to\infty}{|A\cap F_{i}|\over|F_{i}|}.
  53. U U
  54. [ n , ) [n,\infty)
  55. μ ( A ) = U - lim | A F i | | F i | . \mu(A)=U-\lim{|A\cap F_{i}|\over|F_{i}|}.
  56. μ \mu
  57. μ ( G ) = U - lim 1 = 1 \mu(G)=U-\lim 1=1
  58. μ \mu
  59. μ \mu
  60. | | g A F i | | F i | - | A F i | | F i | | = | | A g - 1 F i | | F i | - | A F i | | F i | | \left|{|gA\cap F_{i}|\over|F_{i}|}-{|A\cap F_{i}|\over|F_{i}|}\right|=\left|{|% A\cap g^{-1}F_{i}|\over|F_{i}|}-{|A\cap F_{i}|\over|F_{i}|}\right|
  61. | A ( g - 1 F i F i ) | | F i | 0 \leq{|A\cap(g^{-1}F_{i}\,\triangle\,F_{i})|\over|F_{i}|}\to 0

G-structure.html

  1. S p Sp
  2. A Q Ω 2 ( T M ) A^{Q}\to\Omega^{2}(TM)\,
  3. τ : Ω 1 ( Ad Q ) Ω 2 ( T M ) \tau:\Omega^{1}(\mathrm{Ad}_{Q})\to\Omega^{2}(TM)\,
  4. Ω 2 ( T M ) = Ω 2 , 0 ( T M ) im ( τ ) \Omega^{2}(TM)=\Omega^{2,0}(TM)\oplus\mathrm{im}(\tau)
  5. B ( J X , Y ) = B ( X , J Y ) = - J B ( X , Y ) . B(JX,Y)=B(X,JY)=-JB(X,Y).\,
  6. G G L ( n , 𝐑 ) G\to GL(n,\mathbf{R})
  7. G L ( n , 𝐑 ) GL(n,\mathbf{R})

Gabor_filter.html

  1. g ( x , y ; λ , θ , ψ , σ , γ ) = exp ( - x 2 + γ 2 y 2 2 σ 2 ) exp ( i ( 2 π x λ + ψ ) ) g(x,y;\lambda,\theta,\psi,\sigma,\gamma)=\exp\left(-\frac{x^{\prime 2}+\gamma^% {2}y^{\prime 2}}{2\sigma^{2}}\right)\exp\left(i\left(2\pi\frac{x^{\prime}}{% \lambda}+\psi\right)\right)
  2. g ( x , y ; λ , θ , ψ , σ , γ ) = exp ( - x 2 + γ 2 y 2 2 σ 2 ) cos ( 2 π x λ + ψ ) g(x,y;\lambda,\theta,\psi,\sigma,\gamma)=\exp\left(-\frac{x^{\prime 2}+\gamma^% {2}y^{\prime 2}}{2\sigma^{2}}\right)\cos\left(2\pi\frac{x^{\prime}}{\lambda}+% \psi\right)
  3. g ( x , y ; λ , θ , ψ , σ , γ ) = exp ( - x 2 + γ 2 y 2 2 σ 2 ) sin ( 2 π x λ + ψ ) g(x,y;\lambda,\theta,\psi,\sigma,\gamma)=\exp\left(-\frac{x^{\prime 2}+\gamma^% {2}y^{\prime 2}}{2\sigma^{2}}\right)\sin\left(2\pi\frac{x^{\prime}}{\lambda}+% \psi\right)
  4. x = x cos θ + y sin θ x^{\prime}=x\cos\theta+y\sin\theta\,
  5. y = - x sin θ + y cos θ y^{\prime}=-x\sin\theta+y\cos\theta\,
  6. λ \lambda
  7. θ \theta
  8. ψ \psi
  9. σ \sigma
  10. γ \gamma

Galactose—1-phosphate_uridylyltransferase.html

  1. \rightleftharpoons

Gallagher_Index.html

  1. LSq = 1 2 i = 1 n ( V i - S i ) 2 \mathrm{LSq}=\sqrt{\frac{1}{2}\sum_{i=1}^{n}(V_{i}-S_{i})^{2}}
  2. SLI = ( S - V ) 2 V \mathrm{SLI}=\sum{(S-V)^{2}\over V}

Gas_in_a_box.html

  1. p = h 2 L n x 2 + n y 2 + n z 2 n x , n y , n z = 1 , 2 , 3 , p=\frac{h}{2L}\sqrt{n_{x}^{2}+n_{y}^{2}+n_{z}^{2}}\qquad\qquad n_{x},n_{y},n_{% z}=1,2,3,\ldots
  2. n = n x 2 + n y 2 + n z 2 = 2 L p h n=\sqrt{n_{x}^{2}+n_{y}^{2}+n_{z}^{2}}=\frac{2Lp}{h}
  3. g = ( f 8 ) 4 3 π n 3 = 4 π f 3 ( L p h ) 3 g=\left(\frac{f}{8}\right)\frac{4}{3}\pi n^{3}=\frac{4\pi f}{3}\left(\frac{Lp}% {h}\right)^{3}
  4. d g = π 2 f n 2 d n = 4 π f V h 3 p 2 d p dg=\frac{\pi}{2}~{}fn^{2}\,dn=\frac{4\pi fV}{h^{3}}~{}p^{2}\,dp
  5. N i = g i Φ ( ϵ i ) N_{i}=\frac{g_{i}}{\Phi(\epsilon_{i})}
  6. g i \!g_{i}
  7. Φ ( ϵ i ) = { e β ( ϵ i - μ ) , for particles obeying Maxwell-Boltzmann statistics e β ( ϵ i - μ ) - 1 , for particles obeying Bose-Einstein statistics e β ( ϵ i - μ ) + 1 , for particles obeying Fermi-Dirac statistics \Phi(\epsilon_{i})=\begin{cases}e^{\beta(\epsilon_{i}-\mu)},&\mbox{for % particles obeying Maxwell-Boltzmann statistics }\\ e^{\beta(\epsilon_{i}-\mu)}-1,&\mbox{for particles obeying Bose-Einstein % statistics}\\ e^{\beta(\epsilon_{i}-\mu)}+1,&\mbox{for particles obeying Fermi-Dirac % statistics}\\ \end{cases}
  8. d N E = d g E Φ ( E ) dN_{E}=\frac{dg_{E}}{\Phi(E)}
  9. d g E \!dg_{E}
  10. P A P_{A}
  11. A A
  12. P A d A P_{A}dA
  13. A A
  14. A A
  15. A + d A A+dA
  16. P A d A = d N A N = d g A N Φ A P_{A}~{}dA=\frac{dN_{A}}{N}=\frac{dg_{A}}{N\Phi_{A}}
  17. d N A dN_{A}
  18. A A
  19. A A
  20. A + d A A+dA
  21. d g A dg_{A}
  22. A A
  23. A A
  24. A + d A A+dA
  25. Φ A - 1 \Phi_{A}^{-1}
  26. A A
  27. N N
  28. A P A d A = 1 \int_{A}P_{A}~{}dA=1
  29. P p P_{p}
  30. p p
  31. p + d p p+dp
  32. P p d p = V f N 4 π h 3 Φ p p 2 d p P_{p}~{}dp=\frac{Vf}{N}~{}\frac{4\pi}{h^{3}\Phi_{p}}~{}p^{2}dp
  33. P E P_{E}
  34. E E
  35. E + d E E+dE
  36. P E d E = P p d p d E d E P_{E}~{}dE=P_{p}\frac{dp}{dE}~{}dE
  37. E E
  38. p p
  39. E = p 2 2 m E=\frac{p^{2}}{2m}
  40. E = p c E=pc\,
  41. m m
  42. c c
  43. d g E = ( V f Λ 3 ) 2 π β 3 / 2 E 1 / 2 d E P E d E = 1 N ( V f Λ 3 ) 2 π β 3 / 2 E 1 / 2 Φ ( E ) d E \begin{aligned}\displaystyle dg_{E}&\displaystyle=\quad\ \left(\frac{Vf}{% \Lambda^{3}}\right)\frac{2}{\sqrt{\pi}}~{}\beta^{3/2}E^{1/2}~{}dE\\ \displaystyle P_{E}~{}dE&\displaystyle=\frac{1}{N}\left(\frac{Vf}{\Lambda^{3}}% \right)\frac{2}{\sqrt{\pi}}~{}\frac{\beta^{3/2}E^{1/2}}{\Phi(E)}~{}dE\\ \end{aligned}
  44. Λ = h 2 β 2 π m \Lambda=\sqrt{\frac{h^{2}\beta}{2\pi m}}
  45. ( V / N ) (V/N)
  46. d g E = ( V f Λ 3 ) 1 2 β 3 E 2 d E P E d E = 1 N ( V f Λ 3 ) 1 2 β 3 E 2 Φ ( E ) d E \begin{aligned}\displaystyle dg_{E}&\displaystyle=\quad\ \left(\frac{Vf}{% \Lambda^{3}}\right)\frac{1}{2}~{}\beta^{3}E^{2}~{}dE\\ \displaystyle P_{E}~{}dE&\displaystyle=\frac{1}{N}\left(\frac{Vf}{\Lambda^{3}}% \right)\frac{1}{2}~{}\frac{\beta^{3}E^{2}}{\Phi(E)}~{}dE\\ \end{aligned}
  47. Λ = c h β 2 π 1 / 3 \Lambda=\frac{ch\beta}{2\,\pi^{1/3}}
  48. Φ ( E ) = e β ( E - μ ) \Phi(E)=e^{\beta(E-\mu)}
  49. N = ( V f Λ 3 ) e β μ N=\left(\frac{Vf}{\Lambda^{3}}\right)\,\,e^{\beta\mu}
  50. P E d E = 2 β 3 E π e - β E d E P_{E}~{}dE=2\sqrt{\frac{\beta^{3}E}{\pi}}~{}e^{-\beta E}~{}dE
  51. Φ ( E ) = e β E z - 1 \Phi(E)=\frac{e^{\beta E}}{z}-1\,
  52. z = e β μ . z=e^{\beta\mu}.\,
  53. N = ( V f Λ 3 ) Li 3 / 2 ( z ) N=\left(\frac{Vf}{\Lambda^{3}}\right)\textrm{Li}_{3/2}(z)
  54. N = ( V f Λ c 3 ) ζ ( 3 / 2 ) . N=\left(\frac{Vf}{\Lambda_{c}^{3}}\right)\zeta(3/2).
  55. N = g 0 z 1 - z + ( V f Λ 3 ) Li 3 / 2 ( z ) N=\frac{g_{0}z}{1-z}+\left(\frac{Vf}{\Lambda^{3}}\right)\textrm{Li}_{3/2}(z)
  56. P ν d ν = h 3 N ( V f Λ 3 ) 1 2 β 3 ν 2 e ( h ν - μ ) / k T - 1 d ν P_{\nu}~{}d\nu=\frac{h^{3}}{N}\left(\frac{Vf}{\Lambda^{3}}\right)\frac{1}{2}~{% }\frac{\beta^{3}\nu^{2}}{e^{(h\nu-\mu)/kT}-1}~{}d\nu
  57. U ν d ν = ( N h ν V ) P ν d ν = 4 π f h ν 3 c 3 1 e ( h ν - μ ) / k T - 1 d ν . U_{\nu}~{}d\nu=\left(\frac{N\,h\nu}{V}\right)P_{\nu}~{}d\nu=\frac{4\pi fh\nu^{% 3}}{c^{3}}~{}\frac{1}{e^{(h\nu-\mu)/kT}-1}~{}d\nu.
  58. N = 16 π V c 3 h 3 β 3 Li 3 ( e μ / k T ) . N=\frac{16\,\pi V}{c^{3}h^{3}\beta^{3}}\,\mathrm{Li}_{3}\left(e^{\mu/kT}\right).
  59. U ν d ν = 8 π h ν 3 c 3 1 e h ν / k T - 1 d ν U_{\nu}~{}d\nu=\frac{8\pi h\nu^{3}}{c^{3}}~{}\frac{1}{e^{h\nu/kT}-1}~{}d\nu
  60. Φ ( E ) = e β ( E - μ ) + 1. \Phi(E)=e^{\beta(E-\mu)}+1.\,
  61. N = ( V f Λ 3 ) [ - Li 3 / 2 ( - z ) ] N=\left(\frac{Vf}{\Lambda^{3}}\right)\left[-\textrm{Li}_{3/2}(-z)\right]

Gas_in_a_harmonic_trap.html

  1. g i g_{i}
  2. [ n x , n y , n z ] [n_{x},n_{y},n_{z}]
  3. E = ω ( n x + n y + n z + 3 / 2 ) n i = 0 , 1 , 2 , E=\hbar\omega\left(n_{x}+n_{y}+n_{z}+3/2\right)~{}~{}~{}~{}~{}~{}~{}~{}n_{i}=0% ,1,2,\ldots
  4. f f
  5. f f
  6. f = 2 f=2
  7. n n
  8. E E
  9. g = f n 3 6 = f ( E / ω ) 3 6 g=f\,\frac{n^{3}}{6}=f\,\frac{(E/\hbar\omega)^{3}}{6}
  10. f f
  11. E E
  12. E + d E E+dE
  13. d g = 1 2 f n 2 d n = f ( ω β ) 3 1 2 β 3 E 2 d E dg=\frac{1}{2}\,fn^{2}\,dn=\frac{f}{(\hbar\omega\beta)^{3}}~{}\frac{1}{2}~{}% \beta^{3}E^{2}\,dE
  14. n i = 0 n_{i}=0
  15. ϵ i \epsilon_{i}
  16. N i = g i Φ N_{i}=\frac{g_{i}}{\Phi}
  17. Φ = e β ( ϵ i - μ ) \Phi=e^{\beta(\epsilon_{i}-\mu)}
  18. Φ = e β ( ϵ i - μ ) - 1 \Phi=e^{\beta(\epsilon_{i}-\mu)}-1
  19. Φ = e β ( ϵ i - μ ) + 1 \Phi=e^{\beta(\epsilon_{i}-\mu)}+1
  20. β = 1 / k T \beta=1/kT
  21. k k
  22. T T
  23. μ \mu
  24. d N dN
  25. E E
  26. E + d E E+dE
  27. d N = d g Φ dN=\frac{dg}{\Phi}
  28. A A
  29. P A d A P_{A}dA
  30. A A
  31. A A
  32. A + d A A+dA
  33. P A d A = d N N = d g N Φ P_{A}~{}dA=\frac{dN}{N}=\frac{dg}{N\Phi}
  34. A P A d A = 1 \int_{A}P_{A}~{}dA=1
  35. P E d E = 1 N ( f ( ω β ) 3 ) 1 2 β 3 E 2 Φ d E P_{E}~{}dE=\frac{1}{N}\,\left(\frac{f}{(\hbar\omega\beta)^{3}}\right)~{}\frac{% 1}{2}\frac{\beta^{3}E^{2}}{\Phi}\,dE
  36. Φ = e β ( E - μ ) \Phi=e^{\beta(E-\mu)}\,
  37. N N
  38. N = f ( ω β ) 3 e β μ N=\frac{f}{(\hbar\omega\beta)^{3}}~{}e^{\beta\mu}
  39. P E d E = β 3 E 2 e - β E 2 d E P_{E}~{}dE=\frac{\beta^{3}E^{2}e^{-\beta E}}{2}\,dE
  40. Φ = e β ϵ / z - 1 \Phi=e^{\beta\epsilon}/z-1\,
  41. z z
  42. z = e β μ z=e^{\beta\mu}\,
  43. N N
  44. N = f ( ω β ) 3 Li 3 ( z ) N=\frac{f}{(\hbar\omega\beta)^{3}}~{}\textrm{Li}_{3}(z)
  45. L i s ( z ) Li_{s}(z)
  46. ζ ( 3 ) \zeta(3)
  47. z z
  48. β \beta
  49. β \beta
  50. β c \beta_{c}
  51. z = 1 z=1
  52. N = f ( ω β c ) 3 ζ ( 3 ) N=\frac{f}{(\hbar\omega\beta_{c})^{3}}~{}\zeta(3)
  53. β = β c \beta=\beta_{c}
  54. N = g 0 z 1 - z + f ( ω β ) 3 Li 3 ( z ) N=\frac{g_{0}z}{1-z}+\frac{f}{(\hbar\omega\beta)^{3}}~{}\textrm{Li}_{3}(z)
  55. Φ = e β ( E - μ ) + 1 \Phi=e^{\beta(E-\mu)}+1\,
  56. 1 = f ( ω β ) 3 [ - Li 3 ( - z ) ] 1=\frac{f}{(\hbar\omega\beta)^{3}}~{}\left[-\textrm{Li}_{3}(-z)\right]
  57. L i s ( z ) Li_{s}(z)

Gauge_(bore_diameter).html

  1. 1 / 12 {1}/{12}
  2. d n = ( 6 × 453.59237 g n × π × 11.352 g / cm 3 ) 1 3 = 4.2416 cm × 1 n 3 d_{n}=\left(\frac{6\times 453.59237~{}\mathrm{g}}{n\times\pi\times 11.352~{}% \mathrm{g/cm}^{3}}\right)^{\frac{1}{3}}=4.2416~{}\mathrm{cm}\times\frac{1}{% \sqrt[3]{n}}

Gauge_anomaly.html

  1. n = 1 + D / 2 n=1+D/2
  2. D D
  3. δ ϵ \delta_{\epsilon}
  4. [ δ ϵ 1 , δ ϵ 2 ] = δ [ ϵ 1 , ϵ 2 ] \left[\delta_{\epsilon_{1}},\delta_{\epsilon_{2}}\right]\mathcal{F}=\delta_{% \left[\epsilon_{1},\epsilon_{2}\right]}\mathcal{F}
  5. \mathcal{F}
  6. δ ϵ S \delta_{\epsilon}S
  7. δ ϵ S = M d Ω ( d ) ( ϵ ) \delta_{\epsilon}S=\int_{M^{d}}\Omega^{(d)}(\epsilon)
  8. δ ϵ S = M d + 1 d Ω ( d ) ( ϵ ) . \delta_{\epsilon}S=\int_{M^{d+1}}d\Omega^{(d)}(\epsilon).
  9. [ δ ϵ 1 , δ ϵ 2 ] S = M d + 1 [ δ ϵ 1 d Ω ( d ) ( ϵ 2 ) - δ ϵ 2 d Ω ( d ) ( ϵ 1 ) ] = M d + 1 d Ω ( d ) ( [ ϵ 1 , ϵ 2 ] ) . \left[\delta_{\epsilon_{1}},\delta_{\epsilon_{2}}\right]S=\int_{M^{d+1}}\left[% \delta_{\epsilon_{1}}d\Omega^{(d)}(\epsilon_{2})-\delta_{\epsilon_{2}}d\Omega^% {(d)}(\epsilon_{1})\right]=\int_{M^{d+1}}d\Omega^{(d)}(\left[\epsilon_{1},% \epsilon_{2}\right]).
  10. δ ϵ 1 d Ω ( d ) ( ϵ 2 ) - δ ϵ 2 d Ω ( d ) ( ϵ 1 ) = d Ω ( d ) ( [ ϵ 1 , ϵ 2 ] ) . \delta_{\epsilon_{1}}d\Omega^{(d)}(\epsilon_{2})-\delta_{\epsilon_{2}}d\Omega^% {(d)}(\epsilon_{1})=d\Omega^{(d)}(\left[\epsilon_{1},\epsilon_{2}\right]).
  11. δ ϵ Ω ( d + 1 ) = d Ω ( d ) ( ϵ ) . \delta_{\epsilon}\Omega^{(d+1)}=d\Omega^{(d)}(\epsilon).
  12. Ω ( d + 2 ) = d Ω ( d + 1 ) \Omega^{(d+2)}=d\Omega^{(d+1)}
  13. δ ϵ Ω ( d + 2 ) = d δ ϵ Ω ( d + 1 ) = d 2 Ω ( d ) ( ϵ ) = 0 \delta_{\epsilon}\Omega^{(d+2)}=d\delta_{\epsilon}\Omega^{(d+1)}=d^{2}\Omega^{% (d)}(\epsilon)=0

Gauge_fixing.html

  1. φ \varphi
  2. 𝐄 = - φ - 𝐀 t , 𝐁 = × 𝐀 . {\mathbf{E}}=-\nabla\varphi-\frac{\partial{\mathbf{A}}}{\partial t}\,,\quad{% \mathbf{B}}=\nabla\times{\mathbf{A}}.
  3. 𝐀 𝐀 + ψ \mathbf{A}\rightarrow\mathbf{A}+\nabla\psi
  4. 𝐁 = × ( 𝐀 + ψ ) = × 𝐀 {\mathbf{B}}=\nabla\times({\mathbf{A}}+\nabla\psi)=\nabla\times{\mathbf{A}}
  5. 𝐄 = - φ - 𝐀 t - ψ t = - ( φ + ψ t ) - 𝐀 t {\mathbf{E}}=-\nabla\varphi-\frac{\partial{\mathbf{A}}}{\partial t}-\nabla% \frac{\partial{\psi}}{\partial t}=-\nabla\left(\varphi+\frac{\partial{\psi}}{% \partial t}\right)-\frac{\partial{\mathbf{A}}}{\partial t}
  6. φ φ - ψ t \varphi\rightarrow\varphi-\frac{\partial{\psi}}{\partial t}
  7. 𝐀 𝐀 + ψ . \mathbf{A}\rightarrow\mathbf{A}+\nabla\psi\,.
  8. 𝐀 ( 𝐫 , t ) = 0 . \nabla\cdot{\mathbf{A}}(\mathbf{r},t)=0\,.
  9. 𝐀 2 ( 𝐫 , t ) d 3 r = 𝐁 ( 𝐫 , t ) 𝐁 ( 𝐫 , t ) 4 π R d 3 r d 3 r \int\mathbf{A}^{2}(\mathbf{r},t)d^{3}r=\int\int\frac{\mathbf{B}(\mathbf{r},t)% \cdot\mathbf{B}(\mathbf{r^{\prime}},t)}{4\pi R}d^{3}rd^{3}r^{\prime}
  10. 𝐀 ( 𝐫 , t ) = - 𝐫 × 𝐁 / 2 {\mathbf{A}}(\mathbf{r},t)=-{\mathbf{r}}\times{\mathbf{B}}/2
  11. φ ( 𝐫 , t ) = 𝐄 ( 𝐫 , t ) 4 π R d 3 r - ψ ( 𝐫 , t ) t \varphi(\mathbf{r},t)=\int\frac{\nabla^{\prime}\cdot{\mathbf{E}}(\mathbf{r^{% \prime}},t)}{4\pi R}d^{3}r^{\prime}-\frac{\partial{\psi(\mathbf{r},t)}}{% \partial t}
  12. 𝐀 ( 𝐫 , t ) = × 𝐁 ( 𝐫 , t ) 4 π R d 3 r + ψ ( 𝐫 , t ) \mathbf{A}(\mathbf{r},t)=\nabla\times\int\frac{\mathbf{B}(\mathbf{r^{\prime}},% t)}{4\pi R}d^{3}r^{\prime}+\nabla\psi(\mathbf{r},t)
  13. 𝐀 + 1 c 2 φ t = 0 \nabla\cdot{\mathbf{A}}+\frac{1}{c^{2}}\frac{\partial\varphi}{\partial t}=0
  14. 𝐀 + 1 c φ t = 0. \nabla\cdot{\mathbf{A}}+\frac{1}{c}\frac{\partial\varphi}{\partial t}=0.
  15. μ A μ = 0. \partial^{\mu}A_{\mu}=0.
  16. 1 c 2 2 φ t 2 - 2 φ = ρ ε 0 \frac{1}{c^{2}}\frac{\partial^{2}\varphi}{\partial t^{2}}-\nabla^{2}{\varphi}=% \frac{\rho}{\varepsilon_{0}}
  17. 1 c 2 2 𝐀 t 2 - 2 𝐀 = μ 0 𝐉 \frac{1}{c^{2}}\frac{\partial^{2}\mathbf{A}}{\partial t^{2}}-\nabla^{2}{% \mathbf{A}}=\mu_{0}\mathbf{J}
  18. 2 ψ t 2 = c 2 2 ψ {\partial^{2}\psi\over\partial t^{2}}=c^{2}\nabla^{2}\psi
  19. μ μ A ν = μ 0 j ν \partial_{\mu}\partial^{\mu}A^{\nu}=\mu_{0}j^{\nu}
  20. μ μ A ν = 0 \partial_{\mu}\partial^{\mu}A^{\nu}=0
  21. μ j μ = 0 \partial_{\mu}j^{\mu}=0
  22. \mathcal{L}
  23. δ = - ( μ A μ ) 2 2 ξ \delta\mathcal{L}=-\frac{(\partial_{\mu}A^{\mu})^{2}}{2\xi}
  24. δ = B μ A μ + ξ 2 B 2 \delta\mathcal{L}=B\,\partial_{\mu}A^{\mu}+\frac{\xi}{2}B^{2}
  25. d D x [ ( A μ 1 ) 2 + ( A μ 2 ) 2 ] , \int d^{D}x\left[(A_{\mu}^{1})^{2}+(A_{\mu}^{2})^{2}\right]\,,
  26. 𝐀 μ = A μ a σ a . {\mathbf{A}}_{\mu}=A_{\mu}^{a}\sigma_{a}\,.
  27. d D x [ ( A μ 1 ) 2 + ( A μ 2 ) 2 + ( A μ 4 ) 2 + ( A μ 5 ) 2 + ( A μ 6 ) 2 + ( A μ 7 ) 2 ] , \int d^{D}x\left[(A_{\mu}^{1})^{2}+(A_{\mu}^{2})^{2}+(A_{\mu}^{4})^{2}+(A_{\mu% }^{5})^{2}+(A_{\mu}^{6})^{2}+(A_{\mu}^{7})^{2}\right]\,,
  28. 𝐀 μ = A μ a λ a {\mathbf{A}}_{\mu}=A_{\mu}^{a}\lambda_{a}
  29. φ = 0 \varphi=0
  30. 𝐫 𝐀 = 0 \mathbf{r}\cdot\mathbf{A}=0
  31. 𝐀 ( 𝐫 , t ) = - 𝐫 × 0 1 𝐁 ( u 𝐫 , t ) u d u \mathbf{A}(\mathbf{r},t)=-\mathbf{r}\times\int\limits_{0}^{1}\mathbf{B}(u% \mathbf{r},t)udu
  32. φ ( 𝐫 , t ) = - 𝐫 0 1 𝐄 ( u 𝐫 , t ) d u . \varphi(\mathbf{r},t)=-\mathbf{r}\cdot\int\limits_{0}^{1}\mathbf{E}(u\mathbf{r% },t)du.
  33. x μ A μ = 0 x^{\mu}A_{\mu}=0

Gaussian_binomial_coefficient.html

  1. ( m r ) q = { ( 1 - q m ) ( 1 - q m - 1 ) ( 1 - q m - r + 1 ) ( 1 - q ) ( 1 - q 2 ) ( 1 - q r ) r m 0 r > m {m\choose r}_{q}=\begin{cases}\frac{(1-q^{m})(1-q^{m-1})\cdots(1-q^{m-r+1})}{(% 1-q)(1-q^{2})\cdots(1-q^{r})}&r\leq m\\ 0&r>m\end{cases}
  2. [ k ] q = 1 - q k 1 - q = 0 i < k q i = 1 + q + q 2 + + q k - 1 ; [k]_{q}=\frac{1-q^{k}}{1-q}=\sum_{0\leq i<k}q^{i}=1+q+q^{2}+\cdots+q^{k-1};
  3. ( m r ) q = [ m ] q [ m - 1 ] q [ m - r + 1 ] q [ 1 ] q [ 2 ] q [ r ] q ( r m ) , {m\choose r}_{q}=\frac{[m]_{q}[m-1]_{q}\cdots[m-r+1]_{q}}{[1]_{q}[2]_{q}\cdots% [r]_{q}}\quad(r\leq m),
  4. ( m r ) q {\textstyle\left({{m}\atop{r}}\right)}_{q}
  5. ( m r ) . {\textstyle\left({{m}\atop{r}}\right)}.
  6. [ n ] q ! = [ 1 ] q [ 2 ] q [ n ] q [n]_{q}!=[1]_{q}[2]_{q}\cdots[n]_{q}
  7. ( m r ) q = [ m ] q ! [ r ] q ! [ m - r ] q ! ( r m ) , {m\choose r}_{q}=\frac{[m]_{q}!}{[r]_{q}!\,[m-r]_{q}!}\quad(r\leq m),
  8. ( m r ) q = ( m m - r ) q {\textstyle\left({{m}\atop{r}}\right)}_{q}={\textstyle\left({{m}\atop{m-r}}% \right)}_{q}
  9. ( m r ) {\textstyle\left({{m}\atop{r}}\right)}
  10. r r
  11. m m
  12. m m
  13. m m
  14. r r
  15. m m
  16. r r
  17. m r m−r
  18. ( m r ) q {\textstyle\left({{m}\atop{r}}\right)}_{q}
  19. d d
  20. r r
  21. m r m−r
  22. m m\rightarrow\infty
  23. ( r ) q = lim m ( m r ) q = 1 [ r ] q ! ( 1 - q ) r {\infty\choose r}_{q}=\lim_{m\rightarrow\infty}{m\choose r}_{q}=\frac{1}{[r]_{% q}!\,(1-q)^{r}}
  24. ( 0 0 ) q = ( 1 0 ) q = 1 {0\choose 0}_{q}={1\choose 0}_{q}=1
  25. ( 1 1 ) q = 1 - q 1 - q = 1 {1\choose 1}_{q}=\frac{1-q}{1-q}=1
  26. ( 2 1 ) q = 1 - q 2 1 - q = 1 + q {2\choose 1}_{q}=\frac{1-q^{2}}{1-q}=1+q
  27. ( 3 1 ) q = 1 - q 3 1 - q = 1 + q + q 2 {3\choose 1}_{q}=\frac{1-q^{3}}{1-q}=1+q+q^{2}
  28. ( 3 2 ) q = ( 1 - q 3 ) ( 1 - q 2 ) ( 1 - q ) ( 1 - q 2 ) = 1 + q + q 2 {3\choose 2}_{q}=\frac{(1-q^{3})(1-q^{2})}{(1-q)(1-q^{2})}=1+q+q^{2}
  29. ( 4 2 ) q = ( 1 - q 4 ) ( 1 - q 3 ) ( 1 - q ) ( 1 - q 2 ) = ( 1 + q 2 ) ( 1 + q + q 2 ) = 1 + q + 2 q 2 + q 3 + q 4 {4\choose 2}_{q}=\frac{(1-q^{4})(1-q^{3})}{(1-q)(1-q^{2})}=(1+q^{2})(1+q+q^{2}% )=1+q+2q^{2}+q^{3}+q^{4}
  30. r m - r r\rightarrow m-r
  31. ( m r ) q = ( m m - r ) q . {m\choose r}_{q}={m\choose m-r}_{q}.
  32. ( m 0 ) q = ( m m ) q = 1 , {m\choose 0}_{q}={m\choose m}_{q}=1\,,
  33. ( m 1 ) q = ( m m - 1 ) q = 1 - q m 1 - q = 1 + q + + q m - 1 m 1 . {m\choose 1}_{q}={m\choose m-1}_{q}=\frac{1-q^{m}}{1-q}=1+q+\cdots+q^{m-1}% \quad m\geq 1\,.
  34. ( m r ) 1 = ( m r ) {m\choose r}_{1}={m\choose r}
  35. ( m r ) q = q r ( m - 1 r ) q + ( m - 1 r - 1 ) q {m\choose r}_{q}=q^{r}{m-1\choose r}_{q}+{m-1\choose r-1}_{q}
  36. ( m r ) q = ( m - 1 r ) q + q m - r ( m - 1 r - 1 ) q . {m\choose r}_{q}={m-1\choose r}_{q}+q^{m-r}{m-1\choose r-1}_{q}.
  37. k = 0 n - 1 ( 1 + q k t ) = k = 0 n q k ( k - 1 ) / 2 ( n k ) q t k \prod_{k=0}^{n-1}(1+q^{k}t)=\sum_{k=0}^{n}q^{k(k-1)/2}{n\choose k}_{q}t^{k}
  38. k = 0 n - 1 1 ( 1 - q k t ) = k = 0 ( n + k - 1 k ) q t k . \prod_{k=0}^{n-1}\frac{1}{(1-q^{k}t)}=\sum_{k=0}^{\infty}{n+k-1\choose k}_{q}t% ^{k}.
  39. n n\rightarrow\infty
  40. k = 0 ( 1 + q k t ) = k = 0 q k ( k - 1 ) / 2 t k [ k ] q ! ( 1 - q ) k \prod_{k=0}^{\infty}(1+q^{k}t)=\sum_{k=0}^{\infty}\frac{q^{k(k-1)/2}t^{k}}{[k]% _{q}!\,(1-q)^{k}}
  41. k = 0 1 ( 1 - q k t ) = k = 0 t k [ k ] q ! ( 1 - q ) k . \prod_{k=0}^{\infty}\frac{1}{(1-q^{k}t)}=\sum_{k=0}^{\infty}\frac{t^{k}}{[k]_{% q}!\,(1-q)^{k}}.
  42. ( m m ) q = ( m 0 ) q = 1 {m\choose m}_{q}={m\choose 0}_{q}=1
  43. r m - r r\rightarrow m-r
  44. r m - r r\rightarrow m-r
  45. ( m r ) q = 1 - q m 1 - q m - r ( m - 1 r ) q {m\choose r}_{q}={{1-q^{m}}\over{1-q^{m-r}}}{m-1\choose r}_{q}
  46. ( n + m m ) q {n+m\choose m}_{q}
  47. ( n k ) q {n\choose k}_{q}
  48. ( n 1 ) q = 1 + q + q 2 + + q n - 1 {n\choose 1}_{q}=1+q+q^{2}+\cdots+q^{n-1}
  49. q k 2 - n k ( n k ) q 2 q^{k^{2}-nk}{n\choose k}_{q^{2}}
  50. q q
  51. q - 1 q^{-1}

Gaussian_blur.html

  1. G ( x ) = 1 2 π σ 2 e - x 2 2 σ 2 G(x)=\frac{1}{\sqrt{2\pi\sigma^{2}}}e^{-\frac{x^{2}}{2\sigma^{2}}}
  2. G ( x , y ) = 1 2 π σ 2 e - x 2 + y 2 2 σ 2 G(x,y)=\frac{1}{2\pi\sigma^{2}}e^{-\frac{x^{2}+y^{2}}{2\sigma^{2}}}
  3. 6 σ \lceil 6\sigma\rceil
  4. 6 σ \lceil 6\sigma\rceil
  5. \lceil\cdot\rceil
  6. O ( w kernel w image h image ) + O ( h kernel w image h image ) O\left(w\text{kernel}w\text{image}h\text{image}\right)+O\left(h\text{kernel}w% \text{image}h\text{image}\right)
  7. O ( w kernel h kernel w image h image ) O\left(w\text{kernel}h\text{kernel}w\text{image}h\text{image}\right)
  8. 6 2 + 8 2 = 10 \sqrt{6^{2}+8^{2}}=10
  9. σ f \sigma_{f}
  10. σ X \sigma_{X}
  11. σ r \sigma_{r}
  12. σ r σ X σ f 2 π \sigma_{r}\approx\frac{\sigma_{X}}{\sigma_{f}2\sqrt{\pi}}

Gaussian_noise.html

  1. p p
  2. z z
  3. p G ( z ) = 1 σ 2 π e - ( z - μ ) 2 2 σ 2 p_{G}(z)=\frac{1}{\sigma\sqrt{2\pi}}e^{-\frac{(z-\mu)^{2}}{2\sigma^{2}}}
  4. z z
  5. μ \mu
  6. σ \sigma

Gauss–Kuzmin–Wirsing_operator.html

  1. h ( x ) = 1 / x - 1 / x . h(x)=1/x-\lfloor 1/x\rfloor.\,
  2. [ G f ] ( x ) = n = 1 1 ( x + n ) 2 f ( 1 x + n ) . [Gf](x)=\sum_{n=1}^{\infty}\frac{1}{(x+n)^{2}}f\left(\frac{1}{x+n}\right).
  3. 1 ln 2 1 1 + x \frac{1}{\ln 2}\ \frac{1}{1+x}
  4. x = [ 0 ; a 1 , a 2 , a 3 , ] x=[0;a_{1},a_{2},a_{3},\dots]\,
  5. 1 = | λ 1 | | λ 2 | | λ 3 | . 1=|\lambda_{1}|\geq|\lambda_{2}|\geq|\lambda_{3}|\geq\cdots.
  6. lim n λ n λ n + 1 = - ϕ 2 , where ϕ = 1 + 5 2 . \lim\limits_{n\rightarrow\infty}\frac{\lambda_{n}}{\lambda_{n+1}}=-\phi^{2},\,% \text{ where }\phi=\frac{1+\sqrt{5}}{2}.
  7. ( - 1 ) n + 1 λ n = ϕ - 2 n + C ϕ - 2 n n + d ( n ) ϕ - 2 n n , where C = 5 4 ζ ( 3 / 2 ) 2 π = 1.1019785625880999 + ; (-1)^{n+1}\lambda_{n}=\phi^{-2n}+C\cdot\frac{\phi^{-2n}}{\sqrt{n}}+d(n)\cdot% \frac{\phi^{-2n}}{n},\,\text{ where }C=\frac{\sqrt[4]{5}\cdot\zeta(3/2)}{2% \sqrt{\pi}}=1.1019785625880999_{+};
  8. d ( n ) d(n)
  9. ζ ( ) \zeta(\star)
  10. ζ ( s ) = 1 s - 1 - s 0 1 h ( x ) x s - 1 d x \zeta(s)=\frac{1}{s-1}-s\int_{0}^{1}h(x)x^{s-1}\;dx
  11. ζ ( s ) = s s - 1 - s 0 1 x [ G x s - 1 ] d x \zeta(s)=\frac{s}{s-1}-s\int_{0}^{1}x\left[Gx^{s-1}\right]\,dx
  12. g ( x ) = [ G f ] ( x ) g(x)=[Gf](x)
  13. f ( 1 - x ) = n = 0 ( - x ) n f ( n ) ( 1 ) n ! f(1-x)=\sum_{n=0}^{\infty}(-x)^{n}\frac{f^{(n)}(1)}{n!}
  14. ( - 1 ) m g ( m ) ( 1 ) m ! = n = 0 G m n ( - 1 ) n f ( n ) ( 1 ) n ! , (-1)^{m}\frac{g^{(m)}(1)}{m!}=\sum_{n=0}^{\infty}G_{mn}(-1)^{n}\frac{f^{(n)}(1% )}{n!},
  15. G m n = k = 0 n ( - 1 ) k ( n k ) ( k + m + 1 m ) [ ζ ( k + m + 2 ) - 1 ] . G_{mn}=\sum_{k=0}^{n}(-1)^{k}{n\choose k}{k+m+1\choose m}\left[\zeta(k+m+2)-1% \right].
  16. ζ ( s ) = s s - 1 - s n = 0 ( - 1 ) n ( s - 1 n ) t n \zeta(s)=\frac{s}{s-1}-s\sum_{n=0}^{\infty}(-1)^{n}{s-1\choose n}t_{n}
  17. t n t_{n}
  18. t n = m = 0 G m n ( m + 1 ) ( m + 2 ) . t_{n}=\sum_{m=0}^{\infty}\frac{G_{mn}}{(m+1)(m+2)}.
  19. t n = 1 - γ + k = 1 n ( - 1 ) k ( n k ) [ 1 k - ζ ( k + 1 ) k + 1 ] t_{n}=1-\gamma+\sum_{k=1}^{n}(-1)^{k}{n\choose k}\left[\frac{1}{k}-\frac{\zeta% (k+1)}{k+1}\right]
  20. γ \gamma
  21. t n t_{n}
  22. a n = t n - 1 2 ( n + 1 ) a_{n}=t_{n}-\frac{1}{2(n+1)}
  23. ( 2 n π ) 1 / 4 e - 4 π n cos ( 4 π n - 5 π 8 ) + 𝒪 ( e - 4 π n n 1 / 4 ) . \left(\frac{2n}{\pi}\right)^{1/4}e^{-\sqrt{4\pi n}}\cos\left(\sqrt{4\pi n}-% \frac{5\pi}{8}\right)+\mathcal{O}\left(\frac{e^{-\sqrt{4\pi n}}}{n^{1/4}}% \right).

Gear_inches.html

  1. gear inches = drive wheel diameter in inches × number of teeth in front chainring number of teeth in rear sprocket \,\text{gear inches}=\,\text{drive wheel diameter in inches}\times\frac{\,% \text{number of teeth in front chainring}}{\,\text{number of teeth in rear % sprocket}}
  2. development = drive wheel circumference in metres × number of teeth in front chainring number of teeth in rear sprocket \,\text{development}=\,\text{drive wheel circumference in metres}\times\frac{% \,\text{number of teeth in front chainring}}{\,\text{number of teeth in rear % sprocket}}

Gear_train.html

  1. v = r A ω A = r B ω B , v=r_{A}\omega_{A}=r_{B}\omega_{B},\!
  2. ω A ω B = r B r A = N B N A . \frac{\omega_{A}}{\omega_{B}}=\frac{r_{B}}{r_{A}}=\frac{N_{B}}{N_{A}}.
  3. M A = T B T A = N B N A . MA=\frac{T_{B}}{T_{A}}=\frac{N_{B}}{N_{A}}.
  4. ω A ω B = R . \frac{\omega_{A}}{\omega_{B}}=R.
  5. ω A = ω , ω B = ω / R . \omega_{A}=\omega,\quad\omega_{B}=\omega/R.\!
  6. F θ = T A ω A ω - T B ω B ω = T A - T B / R = 0. F_{\theta}=T_{A}\frac{\partial\omega_{A}}{\partial\omega}-T_{B}\frac{\partial% \omega_{B}}{\partial\omega}=T_{A}-T_{B}/R=0.
  7. M A = T B T A = R . MA=\frac{T_{B}}{T_{A}}=R.

Gelfond's_constant.html

  1. π \pi
  2. π \pi
  3. π \pi
  4. e π = ( e i π ) - i = ( - 1 ) - i , e^{\pi}=(e^{i\pi})^{-i}=(-1)^{-i},
  5. π \pi
  6. 2 2 2^{\sqrt{2}}
  7. π \pi
  8. π \pi
  9. e π 23.14069263277926900572908636794854738 . e^{\pi}\approx 23.14069263277926900572908636794854738\dots\,.
  10. k 0 = 1 2 \scriptstyle k_{0}\,=\,\tfrac{1}{\sqrt{2}}
  11. k n + 1 = 1 - 1 - k n 2 1 + 1 - k n 2 k_{n+1}=\frac{1-\sqrt{1-k_{n}^{2}}}{1+\sqrt{1-k_{n}^{2}}}
  12. n > 0 n>0
  13. ( 4 / k n + 1 ) 2 1 - n (4/k_{n+1})^{2^{1-n}}
  14. e π e^{\pi}
  15. V n = π n 2 R n Γ ( n 2 + 1 ) . V_{n}={\pi^{\frac{n}{2}}R^{n}\over\Gamma(\frac{n}{2}+1)}.
  16. R R
  17. Γ \Gamma
  18. V 2 n = π n n ! V_{2n}=\frac{\pi^{n}}{n!}
  19. n = 0 V 2 n = e π . \sum_{n=0}^{\infty}V_{2n}=e^{\pi}.\,

General_ledger.html

  1. Assets = Liabilities + (Shareholders or Owners equity) \mbox{Assets}~{}=\mbox{Liabilities}~{}+\mbox{(Shareholders or Owners equity)}~{}

Generalized_extreme_value_distribution.html

  1. ζ ( x ) \zeta(x)
  2. g k = Γ ( 1 - k ξ ) g_{k}=\Gamma(1-k\xi)
  3. { g 4 - 4 g 1 g 3 + 6 g 2 g 1 2 - 3 g 1 4 ( g 2 - g 1 2 ) 2 - 3 if ξ 0 , ξ < 1 4 , 12 5 if ξ = 0 , if ξ 1 4 . \begin{cases}\frac{g_{4}-4g_{1}g_{3}+6g_{2}g_{1}^{2}-3g_{1}^{4}}{(g_{2}-g_{1}^% {2})^{2}}-3&\,\text{if}\ \xi\neq 0,\xi<\frac{1}{4},\\ \frac{12}{5}&\,\text{if}\ \xi=0,\\ \infty&\,\text{if}\ \xi\geq\frac{1}{4}.\end{cases}
  4. log ( σ ) + γ ξ + ( γ + 1 ) \log(\sigma)\,+\,\gamma\xi\,+\,(\gamma+1)
  5. F ( x ; μ , σ , ξ ) = exp { - [ 1 + ξ ( x - μ σ ) ] - 1 / ξ } F(x;\mu,\sigma,\xi)=\exp\left\{-\left[1+\xi\left(\frac{x-\mu}{\sigma}\right)% \right]^{-1/\xi}\right\}
  6. 1 + ξ ( x - μ ) / σ > 0 1+\xi(x-\mu)/\sigma>0
  7. μ \mu\in\mathbb{R}
  8. σ > 0 \sigma>0
  9. ξ \xi\in\mathbb{R}
  10. ξ > 0 \xi>0
  11. x > μ - σ / ξ x>\mu-\sigma/\xi
  12. ξ < 0 \xi<0
  13. x < μ + σ / ( - ξ ) x<\mu+\sigma/(-\xi)
  14. ξ = 0 \xi=0
  15. ξ 0 \xi\to 0
  16. F ( x ; μ , σ , 0 ) = exp { - exp ( - x - μ σ ) } F(x;\mu,\sigma,0)=\exp\left\{-\exp\left(-\frac{x-\mu}{\sigma}\right)\right\}
  17. f ( x ; μ , σ , ξ ) = 1 σ [ 1 + ξ ( x - μ σ ) ] ( - 1 / ξ ) - 1 f(x;\mu,\sigma,\xi)=\frac{1}{\sigma}\left[1+\xi\left(\frac{x-\mu}{\sigma}% \right)\right]^{(-1/\xi)-1}
  18. exp { - [ 1 + ξ ( x - μ σ ) ] - 1 / ξ } \exp\left\{-\left[1+\xi\left(\frac{x-\mu}{\sigma}\right)\right]^{-1/\xi}\right\}
  19. x > μ - σ / ξ x>\mu-\sigma/\xi
  20. ξ > 0 \xi>0
  21. x < μ + σ / ( - ξ ) x<\mu+\sigma/(-\xi)
  22. ξ < 0 \xi<0
  23. ξ = 0 \xi=0
  24. f ( x ; μ , σ , ξ ) = 1 σ exp [ - ( x - μ σ ) ] f(x;\mu,\sigma,\xi)=\frac{1}{\sigma}\exp\left[-\left(\frac{x-\mu}{\sigma}% \right)\right]
  25. exp { - exp [ ( - x - μ σ ) ] } \exp\left\{-\exp\left[\left(-\frac{x-\mu}{\sigma}\right)\right]\right\}
  26. E ( X ) = μ - σ ξ + σ ξ g 1 , \operatorname{E}(X)=\mu-\frac{\sigma}{\xi}+\frac{\sigma}{\xi}g_{1},
  27. Var ( X ) = σ 2 ξ 2 ( g 2 - g 1 2 ) , \operatorname{Var}(X)=\frac{\sigma^{2}}{\xi^{2}}(g_{2}-g_{1}^{2}),
  28. Mode ( X ) = μ + σ ξ [ ( 1 + ξ ) - ξ - 1 ] . \operatorname{Mode}(X)=\mu+\frac{\sigma}{\xi}[(1+\xi)^{-\xi}-1].
  29. skewness ( X ) = g 3 - 3 g 1 g 2 + 2 g 1 3 ( g 2 - g 1 2 ) 3 / 2 \operatorname{skewness}(X)=\frac{g_{3}-3g_{1}g_{2}+2g_{1}^{3}}{(g_{2}-g_{1}^{2% })^{3/2}}
  30. g k = Γ ( 1 - k ξ ) g_{k}=\Gamma(1-k\xi)
  31. Γ ( t ) \Gamma(t)
  32. ξ \xi
  33. ξ = 0 \xi=0
  34. ξ > 0 \xi>0
  35. ξ < 0 \xi<0
  36. ξ = 0 \xi=0
  37. F ( x ; μ , σ , 0 ) = e - e - ( x - μ ) / σ for x . F(x;\mu,\sigma,0)=e^{-e^{-(x-\mu)/\sigma}}\;\;\;\,\text{for}\;\;x\in\mathbb{R}.
  38. ξ = α - 1 > 0 \xi=\alpha^{-1}>0
  39. F ( x ; μ , σ , ξ ) = { 0 x μ e - ( ( x - μ ) / σ ) - α x > μ . F(x;\mu,\sigma,\xi)=\begin{cases}0&x\leq\mu\\ e^{-((x-\mu)/\sigma)^{-\alpha}}&x>\mu.\end{cases}
  40. ξ = - α - 1 < 0 \xi=-\alpha^{-1}<0
  41. F ( x ; μ , σ , ξ ) = { e - ( - ( x - μ ) / σ ) α x < μ 1 x μ F(x;\mu,\sigma,\xi)=\begin{cases}e^{-(-(x-\mu)/\sigma)^{\alpha}}&x<\mu\\ 1&x\geq\mu\end{cases}
  42. σ > 0 \sigma>0
  43. t = μ - x t=\mu-x
  44. X X
  45. F ( x ; 0 , σ , α ) F(x;0,\sigma,\alpha)
  46. ln X \ln X
  47. F ( x ; ln σ , 1 / α , 0 ) F(x;\ln\sigma,1/\alpha,0)
  48. X X
  49. F ( x ; 0 , σ , - α ) F(x;0,\sigma,-\alpha)
  50. ln ( - X ) \ln(-X)
  51. F ( x ; - ln σ , 1 / α , 0 ) F(x;-\ln\sigma,1/\alpha,0)
  52. X GEV ( μ , σ , 0 ) X\sim\textrm{GEV}(\mu,\,\sigma,\,0)
  53. m X + b GEV ( m μ + b , m σ , 0 ) mX+b\sim\textrm{GEV}(m\mu+b,\,m\sigma,\,0)
  54. X Gumbel ( μ , σ ) X\sim\textrm{Gumbel}(\mu,\,\sigma)
  55. X GEV ( μ , σ , 0 ) X\sim\textrm{GEV}(\mu,\,\sigma,\,0)
  56. X Weibull ( σ , μ ) X\sim\textrm{Weibull}(\sigma,\,\mu)
  57. μ ( 1 - σ log X σ ) GEV ( μ , σ , 0 ) \mu\left(1-\sigma\mathrm{log}{\tfrac{X}{\sigma}}\right)\sim\textrm{GEV}(\mu,\,% \sigma,\,0)
  58. X GEV ( μ , σ , 0 ) X\sim\textrm{GEV}(\mu,\,\sigma,\,0)
  59. σ exp ( - X - μ μ σ ) Weibull ( σ , μ ) \sigma\exp(-\tfrac{X-\mu}{\mu\sigma})\sim\textrm{Weibull}(\sigma,\,\mu)
  60. X Exponential ( 1 ) X\sim\textrm{Exponential}(1)\,
  61. μ - σ log X GEV ( μ , σ , 0 ) \mu-\sigma\log{X}\sim\textrm{GEV}(\mu,\,\sigma,\,0)
  62. X GEV ( α , β , 0 ) X\sim\mathrm{GEV}(\alpha,\beta,0)\,
  63. Y GEV ( α , β , 0 ) Y\sim\mathrm{GEV}(\alpha,\beta,0)\,
  64. X - Y Logistic ( 0 , β ) X-Y\sim\mathrm{Logistic}(0,\beta)\,
  65. X GEV ( α , β , 0 ) X\sim\mathrm{GEV}(\alpha,\beta,0)\,
  66. Y GEV ( α , β , 0 ) Y\sim\mathrm{GEV}(\alpha,\beta,0)\,
  67. X + Y Logistic ( 2 α , β ) X+Y\sim\mathrm{Logistic}(2\alpha,\beta)\,

Generalized_function.html

  1. F = F ( x ) ~{}F=F(x)~{}
  2. F smooth F_{\rm smooth}
  3. F singular F_{\rm singular}
  4. F ~{}F~{}
  5. G ~{}G~{}
  6. ( 1 ) F G = F smooth G smooth + F smooth G singular + F singular G smooth . (1)~{}~{}~{}~{}~{}FG~{}=~{}F_{\rm smooth}~{}G_{\rm smooth}~{}+~{}F_{\rm smooth% }~{}G_{\rm singular}~{}+F_{\rm singular}~{}G_{\rm smooth}.
  7. δ ( x ) 2 = 0 ~{}\delta(x)^{2}=0~{}
  8. G = M / N G=M/N
  9. s = { a m : , n n m ; m } s=\{a_{m}:\mathbb{N}\to\mathbb{R},n\mapsto n^{m};~{}m\in\mathbb{Z}\}
  10. G s ( E , P ) = { f E p P , m : p ( f n ) = o ( n m ) } { f E p P , m : p ( f n ) = o ( n m ) } . G_{s}(E,P)=\frac{\{f\in E^{\mathbb{N}}\mid\forall p\in P,\exists m\in\mathbb{Z% }:p(f_{n})=o(n^{m})\}}{\{f\in E^{\mathbb{N}}\mid\forall p\in P,\forall m\in% \mathbb{Z}:p(f_{n})=o(n^{m})\}}.

Generalized_Kac–Moody_algebra.html

  1. c i j c_{ij}
  2. c i j = c j i c_{ij}=c_{ji}
  3. c i j 0 c_{ij}\leq 0
  4. i j i\neq j
  5. 2 c i j / c i i 2c_{ij}/c_{ii}
  6. c i i > 0. c_{ii}>0.
  7. e i e_{i}
  8. f i f_{i}
  9. h i h_{i}
  10. [ e i , f j ] = h i [e_{i},f_{j}]=h_{i}
  11. i = j i=j
  12. [ h i , e j ] = c i j e j [h_{i},e_{j}]=c_{ij}e_{j}
  13. [ h i , f j ] = - c i j f j [h_{i},f_{j}]=-c_{ij}f_{j}
  14. [ e i , [ e i , , [ e i , e j ] ] ] = [ f i , [ f i , , [ f i , f j ] ] ] = 0 [e_{i},[e_{i},\ldots,[e_{i},e_{j}]]]=[f_{i},[f_{i},\ldots,[f_{i},f_{j}]]]=0
  15. 1 - 2 c i j / c i i 1-2c_{ij}/c_{ii}
  16. e i e_{i}
  17. f i f_{i}
  18. c i i > 0 c_{ii}>0
  19. [ e i , e j ] = [ f i , f j ] = 0 [e_{i},e_{j}]=[f_{i},f_{j}]=0
  20. c i j = 0. c_{ij}=0.
  21. ( e i , f i ) = 1 (e_{i},f_{i})=1

Generalized_method_of_moments.html

  1. m ( θ 0 ) E [ g ( Y t , θ 0 ) ] = 0 , m(\theta_{0})\equiv\operatorname{E}[\,g(Y_{t},\theta_{0})\,]=0,
  2. m ^ ( θ ) 1 T t = 1 T g ( Y t , θ ) \hat{m}(\theta)\equiv\frac{1}{T}\sum_{t=1}^{T}g(Y_{t},\theta)
  3. m ^ ( θ ) E [ g ( Y t , θ ) ] = m ( θ ) \scriptstyle\hat{m}(\theta)\,\approx\;\operatorname{E}[g(Y_{t},\theta)]\,=\,m(\theta)
  4. m ^ ( θ 0 ) m ( θ 0 ) = 0 \scriptstyle\hat{m}(\theta_{0})\;\approx\;m(\theta_{0})\;=\;0
  5. θ ^ \scriptstyle\hat{\theta}
  6. m ^ ( θ ^ ) \scriptstyle\hat{m}(\;\!\hat{\theta}\;\!)
  7. m ^ ( θ ) \scriptstyle\hat{m}(\theta)
  8. m ^ ( θ ) W 2 = m ^ ( θ ) W m ^ ( θ ) , \|\hat{m}(\theta)\|^{2}_{W}=\hat{m}(\theta)^{\prime}\,W\hat{m}(\theta),
  9. W ^ \scriptstyle\hat{W}
  10. θ ^ = arg min θ Θ ( 1 T t = 1 T g ( Y t , θ ) ) W ^ ( 1 T t = 1 T g ( Y t , θ ) ) \hat{\theta}=\operatorname{arg}\min_{\theta\in\Theta}\bigg(\frac{1}{T}\sum_{t=% 1}^{T}g(Y_{t},\theta)\bigg)^{\prime}\hat{W}\bigg(\frac{1}{T}\sum_{t=1}^{T}g(Y_% {t},\theta)\bigg)
  11. W ^ \scriptstyle\hat{W}
  12. θ ^ 𝑝 θ 0 as T \hat{\theta}\xrightarrow{p}\theta_{0}\ \,\text{as}\ T\to\infty
  13. W ^ T 𝑝 W , \hat{W}_{T}\xrightarrow{p}W,
  14. W E [ g ( Y t , θ ) ] = 0 \,W\operatorname{E}[\,g(Y_{t},\theta)\,]=0
  15. θ = θ 0 , \,\theta=\theta_{0},
  16. Θ k \Theta\subset\mathbb{R}^{k}
  17. g ( Y , θ ) \,g(Y,\theta)
  18. E [ sup θ Θ g ( Y , θ ) ] < . \operatorname{E}[\,\textstyle\sup_{\theta\in\Theta}\lVert g(Y,\theta)\rVert\,]% <\infty.
  19. θ 0 \theta_{0}
  20. W E [ θ g ( Y t , θ 0 ) ] W\operatorname{E}[\nabla_{\theta}g(Y_{t},\theta_{0})]
  21. G = E [ θ g ( Y t , θ 0 ) ] , Ω = E [ g ( Y t , θ 0 ) g ( Y t , θ 0 ) ] G=\operatorname{E}[\,\nabla_{\!\theta}\,g(Y_{t},\theta_{0})\,],\qquad\Omega=% \operatorname{E}[\,g(Y_{t},\theta_{0})g(Y_{t},\theta_{0})^{\prime}\,]
  22. T ( θ ^ - θ 0 ) 𝑑 𝒩 [ 0 , ( G W G ) - 1 G W Ω W G ( G W G ) - 1 ] \sqrt{T}\big(\hat{\theta}-\theta_{0}\big)\ \xrightarrow{d}\ \mathcal{N}\big[0,% (G^{\prime}WG)^{-1}G^{\prime}W\Omega W^{\prime}G(G^{\prime}W^{\prime}G)^{-1}\big]
  23. θ ^ \hat{\theta}
  24. Θ k \Theta\subset\mathbb{R}^{k}
  25. g ( Y , θ ) \,g(Y,\theta)
  26. θ 0 \theta_{0}
  27. E [ g ( Y t , θ ) 2 ] < , \operatorname{E}[\,\lVert g(Y_{t},\theta)\rVert^{2}\,]<\infty,
  28. E [ sup θ N θ g ( Y t , θ ) ] < , \operatorname{E}[\,\textstyle\sup_{\theta\in N}\lVert\nabla_{\theta}g(Y_{t},% \theta)\rVert\,]<\infty,
  29. G W G G^{\prime}WG
  30. W Ω - 1 W\propto\ \Omega^{-1}
  31. T ( θ ^ - θ 0 ) 𝑑 𝒩 [ 0 , ( G Ω - 1 G ) - 1 ] \sqrt{T}\big(\hat{\theta}-\theta_{0}\big)\ \xrightarrow{d}\ \mathcal{N}\big[0,% (G^{\prime}\,\Omega^{-1}G)^{-1}\big]
  32. W = Ω - 1 W=\Omega^{-1}
  33. W = Ω - 1 W=\Omega^{-1}
  34. V ( W ) - V ( Ω - 1 ) \,V(W)-V(\Omega^{-1})
  35. = ( G W G ) - 1 G W Ω W G ( G W G ) - 1 - ( G Ω - 1 G ) - 1 \,=(G^{\prime}WG)^{-1}G^{\prime}W\Omega WG(G^{\prime}WG)^{-1}-(G^{\prime}% \Omega^{-1}G)^{-1}
  36. = ( G W G ) - 1 ( G W Ω W G - G W G ( G Ω - 1 G ) - 1 G W G ) ( G W G ) - 1 \,=(G^{\prime}WG)^{-1}\Big(G^{\prime}W\Omega WG-G^{\prime}WG(G^{\prime}\Omega^% {-1}G)^{-1}G^{\prime}WG\Big)(G^{\prime}WG)^{-1}
  37. = ( G W G ) - 1 G W Ω 1 / 2 ( I - Ω - 1 / 2 G ( G Ω - 1 G ) - 1 G Ω - 1 / 2 ) Ω 1 / 2 W G ( G W G ) - 1 \,=(G^{\prime}WG)^{-1}G^{\prime}W\Omega^{1/2}\Big(I-\Omega^{-1/2}G(G^{\prime}% \Omega^{-1}G)^{-1}G^{\prime}\Omega^{-1/2}\Big)\Omega^{1/2}WG(G^{\prime}WG)^{-1}
  38. = A ( I - B ) A , \,=A(I-B)A^{\prime},
  39. B 2 = B B^{2}=B
  40. I - B = ( I - B ) ( I - B ) I-B=(I-B)(I-B)^{\prime}
  41. = A ( I - B ) ( I - B ) A = ( A ( I - B ) ) ( A ( I - B ) ) 0 \,=A(I-B)(I-B)^{\prime}A^{\prime}=\Big(A(I-B)\Big)\Big(A(I-B)\Big)^{\prime}\geq 0
  42. m ^ ( θ ^ ) \hat{m}(\hat{\theta})
  43. m ^ ( θ ) = 0 \hat{m}(\theta)=0
  44. θ \theta
  45. θ 0 \theta_{0}
  46. m ( θ 0 ) = 0 m(\theta_{0})=0
  47. H 0 : m ( θ 0 ) = 0 H_{0}:\ m(\theta_{0})=0
  48. H 1 : m ( θ ) 0 , θ Θ H_{1}:\ m(\theta)\neq 0,\ \forall\theta\in\Theta
  49. H 0 H_{0}
  50. J T ( 1 T t = 1 T g ( Y t , θ ^ ) ) W ^ T ( 1 T t = 1 T g ( Y t , θ ^ ) ) 𝑑 χ k - 2 J\equiv T\cdot\bigg(\frac{1}{T}\sum_{t=1}^{T}g(Y_{t},\hat{\theta})\bigg)^{% \prime}\hat{W}_{T}\bigg(\frac{1}{T}\sum_{t=1}^{T}g(Y_{t},\hat{\theta})\bigg)\ % \xrightarrow{d}\ \chi^{2}_{k-\ell}
  51. H 0 , H_{0},
  52. θ ^ \hat{\theta}
  53. θ 0 \theta_{0}
  54. W ^ T \hat{W}_{T}
  55. Ω - 1 \Omega^{-1}
  56. Ω - 1 \Omega^{-1}
  57. Ω - 1 \Omega^{-1}
  58. H 1 H_{1}
  59. J 𝑝 J\ \xrightarrow{p}\ \infty
  60. H 1 H_{1}
  61. χ k - 2 \chi^{2}_{k-\ell}
  62. H 0 H_{0}
  63. J > q 0.95 χ k - 2 J>q_{0.95}^{\chi^{2}_{k-\ell}}
  64. H 0 H_{0}
  65. J < q 0.95 χ k - 2 J<q_{0.95}^{\chi^{2}_{k-\ell}}

Generic_polynomial.html

  1. x n + t 1 x n - 1 + + t n x^{n}+t_{1}x^{n-1}+\cdots+t_{n}
  2. H p 3 H_{p^{3}}
  3. x 2 - t x^{2}-t
  4. x 3 - t x 2 + ( t - 3 ) x + 1 x^{3}-tx^{2}+(t-3)x+1
  5. x 3 - t ( x + 1 ) x^{3}-t(x+1)
  6. ( x 2 - s ) ( x 2 - t ) (x^{2}-s)(x^{2}-t)
  7. x 4 - 2 s ( t 2 + 1 ) x 2 + s 2 t 2 ( t 2 + 1 ) x^{4}-2s(t^{2}+1)x^{2}+s^{2}t^{2}(t^{2}+1)
  8. x 4 - 2 s t x 2 + s 2 t ( t - 1 ) x^{4}-2stx^{2}+s^{2}t(t-1)
  9. x 4 + s x 2 - t ( x + 1 ) x^{4}+sx^{2}-t(x+1)
  10. x 5 + ( t - 3 ) x 4 + ( s - t + 3 ) x 3 + ( t 2 - t - 2 s - 1 ) x 2 + s x + t x^{5}+(t-3)x^{4}+(s-t+3)x^{3}+(t^{2}-t-2s-1)x^{2}+sx+t
  11. x 5 + s x 3 - t ( x + 1 ) x^{5}+sx^{3}-t(x+1)
  12. g d F G gd_{F}G
  13. \infty
  14. g d A 3 = 1 gd_{\mathbb{Q}}A_{3}=1
  15. g d S 3 = 1 gd_{\mathbb{Q}}S_{3}=1
  16. g d D 4 = 2 gd_{\mathbb{Q}}D_{4}=2
  17. g d S 4 = 2 gd_{\mathbb{Q}}S_{4}=2
  18. g d D 5 = 2 gd_{\mathbb{Q}}D_{5}=2
  19. g d S 5 = 2 gd_{\mathbb{Q}}S_{5}=2

Genetic_load.html

  1. 𝐀 1 𝐀 n \mathbf{A}_{1}\dots\mathbf{A}_{n}
  2. w 1 w n w_{1}\dots w_{n}
  3. p 1 p n p_{1}\dots p_{n}
  4. L L
  5. L = w max - w ¯ w max ( 1 ) L={{w_{\max}-\bar{w}}\over w_{\max}}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}(1)
  6. w max w_{\max}
  7. w 1 w n w_{1}\dots w_{n}
  8. w ¯ \bar{w}
  9. w ¯ = i = 1 n p i w i ( 2 ) \bar{w}={\sum_{i=1}^{n}{p_{i}w_{i}}}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}(2)
  10. i th i^{\mathrm{th}}
  11. 𝐀 i \mathbf{A}_{i}
  12. w i w_{i}
  13. p i p_{i}
  14. w max = 1 w_{\max}=1
  15. L = 1 - w ¯ . ( 3 ) L=1-\bar{w}.~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}(3)

Genotype_frequency.html

  1. f ( a ) \displaystyle f({a})
  2. f ( a a ) \displaystyle f({aa})
  3. f ( 𝐀𝐀 ) = p 2 f(\mathbf{AA})=p^{2}
  4. f ( 𝐀𝐚 ) = 2 p q f(\mathbf{Aa})=2pq
  5. f ( 𝐚𝐚 ) = q 2 f(\mathbf{aa})=q^{2}
  6. f ( 𝐀𝐀 ) = p 2 f(\mathbf{AA})=p^{2}
  7. f ( 𝐀𝐚 ) = 2 p q f(\mathbf{Aa})=2pq
  8. f ( 𝐚𝐚 ) = q 2 f(\mathbf{aa})=q^{2}

Geometric_invariant_theory.html

  1. g f ( v ) = f ( g - 1 v ) , g G , v V . g\cdot f(v)=f(g^{-1}v),\quad g\in G,v\in V.
  2. V / / G = Spec A = Spec R ( V ) G . V/\!\!/G=\operatorname{Spec}A=\operatorname{Spec}R(V)^{G}.
  3. deg ( V ) rank ( V ) < deg ( W ) rank ( W ) \displaystyle\frac{\deg(V)}{\hbox{rank}(V)}<\frac{\deg(W)}{\hbox{rank}(W)}

Geometrical_frustration.html

  1. = G - I k ν , k μ S k ν S k μ , \mathcal{H}=\sum_{G}\,-I_{k_{\nu},k_{\mu}}\,\,S_{k_{\nu}}\cdot S_{k_{\mu}}\,,
  2. I k ν , k μ , I_{k_{\nu},k_{\mu}}\,,
  3. ± 1 \pm 1
  4. S k ν S k μ S_{k_{\nu}}\cdot S_{k_{\mu}}
  5. P W P_{W}
  6. P W = I 1 , 2 I 2 , 3 I 3 , 4 I 4 , 1 P_{W}=I_{1,2}\,I_{2,3}\,I_{3,4}\,I_{4,1}
  7. P W = I 1 , 2 I 2 , 3 I 3 , 1 , P_{W}=I_{1,2}\,I_{2,3}\,I_{3,1}\,,
  8. \mathcal{H}
  9. I i , k ϵ i I i , k ϵ k , S i ϵ i S i , S k ϵ k S k . I_{i,k}\to\epsilon_{i}I_{i,k}\epsilon_{k},\quad S_{i}\to\epsilon_{i}S_{i},% \quad S_{k}\to\epsilon_{k}S_{k}\,.
  10. ϵ i \epsilon_{i}
  11. ϵ k \epsilon_{k}
  12. 2 π 2\pi
  13. 2 π 2\pi
  14. 2 π 2\pi
  15. { 3 , 3 , 5 } \{3,3,5\}
  16. l l
  17. r r
  18. l 1.05 r l\simeq 1.05r
  19. { 3 , 3 , 5 } \{3,3,5\}
  20. S 3 S^{3}
  21. ( τ = ( 1 + 5 ) / 2 ) (\tau=(1+\surd 5)/2)
  22. { 3 , 3 , 5 } \{3,3,5\}

Geostationary_ring.html

  1. J 2 J_{2}

Gilbert–Varshamov_bound.html

  1. A q ( n , d ) A_{q}(n,d)
  2. C C
  3. 𝔽 q \mathbb{F}_{q}
  4. A q ( n , d ) q n j = 0 d - 1 ( n j ) ( q - 1 ) j . A_{q}(n,d)\geq\frac{q^{n}}{\sum_{j=0}^{d-1}{\left({{n}\atop{j}}\right)}(q-1)^{% j}}.
  5. C C
  6. n n
  7. d d
  8. | C | = A q ( n , d ) . |C|=A_{q}(n,d).\,
  9. x 𝔽 q n x\in\mathbb{F}_{q}^{n}
  10. c x C c_{x}\in C
  11. d ( x , c x ) d(x,c_{x})
  12. x x
  13. c x c_{x}
  14. d ( x , c x ) d - 1 d(x,c_{x})\leq d-1
  15. | C | |C|
  16. 𝔽 q n \mathbb{F}_{q}^{n}
  17. c C c\in C
  18. 𝔽 q n = c C B ( c , d - 1 ) . \mathbb{F}_{q}^{n}=\cup_{c\in C}B(c,d-1).\,
  19. j = 0 d - 1 ( n j ) ( q - 1 ) j \sum_{j=0}^{d-1}{\left({{n}\atop{j}}\right)}(q-1)^{j}
  20. d - 1 d-1
  21. n n
  22. ( q - 1 ) (q-1)
  23. 𝔽 q n \mathbb{F}_{q}^{n}
  24. | 𝔽 q n | = | c C B ( c , d - 1 ) | c C | B ( c , d - 1 ) | = | C | j = 0 d - 1 ( n j ) ( q - 1 ) j \begin{aligned}\displaystyle|\mathbb{F}_{q}^{n}|&\displaystyle=|\cup_{c\in C}B% (c,d-1)|\\ \\ &\displaystyle\leq\sum_{c\in C}|B(c,d-1)|\\ \\ &\displaystyle=|C|\sum_{j=0}^{d-1}{\left({{n}\atop{j}}\right)}(q-1)^{j}\\ \\ \end{aligned}
  25. A q ( n , d ) q n j = 0 d - 1 ( n j ) ( q - 1 ) j A_{q}(n,d)\geq\frac{q^{n}}{\sum_{j=0}^{d-1}{\left({{n}\atop{j}}\right)}(q-1)^{% j}}
  26. | 𝔽 q n | = q n |\mathbb{F}_{q}^{n}|=q^{n}
  27. A q ( n , d ) q k A_{q}(n,d)\geq q^{k}
  28. q k < q n j = 0 d - 2 ( n - 1 j ) ( q - 1 ) j . q^{k}<\frac{q^{n}}{\sum_{j=0}^{d-2}{\left({{n-1}\atop{j}}\right)}(q-1)^{j}}.

Gimel.html

  1. \gimel

Gires–Tournois_etalon.html

  1. r = - r 1 - e - i δ 1 - r 1 e - i δ r=-\frac{r_{1}-e^{-i\delta}}{1-r_{1}e^{-i\delta}}
  2. δ = 4 π λ n t cos θ t \delta=\frac{4\pi}{\lambda}nt\cos\theta_{t}
  3. r 1 r_{1}
  4. | r | = 1 |r|=1
  5. δ \delta
  6. Φ \Phi
  7. r 1 r_{1}
  8. r 1 = R r_{1}=\sqrt{R}
  9. R R
  10. Φ \Phi
  11. r = e i Φ . r=e^{i\Phi}.
  12. tan ( Φ 2 ) = - 1 + R 1 - R tan ( δ 2 ) \tan\left(\frac{\Phi}{2}\right)=-\frac{1+\sqrt{R}}{1-\sqrt{R}}\tan\left(\frac{% \delta}{2}\right)
  13. Φ = δ \Phi=\delta
  14. Φ \Phi
  15. δ \delta

Glass_electrode.html

  1. E = E 0 + R T z i F ln [ a i + j ( k i j a j z i / z j ) ] E=E^{0}+\frac{RT}{z_{i}F}\ln\left[a_{i}+\sum_{j}\left(k_{ij}a_{j}^{z_{i}/z_{j}% }\right)\right]
  2. E = E 0 + R T F ln ( a H + + k H + , Na + a Na + ) E=E^{0}+\frac{RT}{F}\ln\left(a_{\,\text{H}^{+}}+k_{\,\text{H}^{+},\,\text{Na}^% {+}}a_{\,\text{Na}^{+}}\right)
  3. E = E 0 - 2.303 R T F pH E=E^{0}-\frac{2.303RT}{F}\,\text{pH}

Gliese_229.html

  1. L V L V = 10 0.4 ( M V - M V ) \scriptstyle\frac{L_{V_{\ast}}}{L_{V_{\odot}}}=10^{0.4\left(M_{V_{\odot}}-M_{V% _{\ast}}\right)}
  2. M b o l = 4.75 \scriptstyle M_{bol_{\odot}}=4.75
  3. L b o l L b o l = 10 0.4 ( M b o l - M b o l ) \scriptstyle\frac{L_{bol_{\ast}}}{L_{bol_{\odot}}}=10^{0.4\left(M_{bol_{\odot}% }-M_{bol_{\ast}}\right)}

Gliese_710.html

  1. B C = - 0.814 \scriptstyle BC=-0.814
  2. M b o l = 7.416 \scriptstyle M_{bol_{\ast}}=7.416
  3. M b o l = 4.73 \scriptstyle M_{bol_{\odot}}=4.73
  4. L b o l L b o l = 10 0.4 ( M b o l - M b o l ) \scriptstyle\frac{L_{bol_{\ast}}}{L_{bol_{\odot}}}=10^{0.4\left(M_{bol_{\odot}% }-M_{bol_{\ast}}\right)}
  5. M V = 8.23 \scriptstyle M_{V_{\ast}}=8.23
  6. M V = 4.83 \scriptstyle M_{V_{\odot}}=4.83
  7. L V L V = 10 0.4 ( M V - M V ) \scriptstyle\frac{L_{V_{\ast}}}{L_{V_{\odot}}}=10^{0.4\left(M_{V_{\odot}}-M_{V% _{\ast}}\right)}

GLIMMER.html

  1. P ( S / M ) = x = 1 n IMM 8 ( S x ) \operatorname{P(S/M)=\sum_{x=1}^{n}{IMM_{8}(S_{x})}}
  2. S x S_{x}
  3. I M M 8 ( S x ) IMM_{8}(S_{x})
  4. 8 t h 8^{th}
  5. IMM k ( S x ) = Y k ( S x - 1 ) * P k ( S x ) + [ 1 - Y k ( S ( x - 1 ) ] * IMM k - 1 ( S x ) \operatorname{IMM_{k}(S_{x})=Y_{k}(S_{x-1})*P_{k}(S_{x})+[1-{Y_{k}(S_{(}x-1)]*% IMM_{k-1}(S_{x})}}
  6. Y k ( S x - 1 ) Y_{k}(S_{x-1})
  7. P k ( S x ) P_{k}(S_{x})
  8. k t h k^{th}
  9. S x S_{x}
  10. P i ( S x ) = P ( s x / S x , j ) = f ( S x , j ) / be [ acgt ] f ( S x , i , b ) \operatorname{P_{i}(S_{x})=P(s_{x}/S_{x,j})=f(S_{x,j})/\sum_{be{[acgt]}}% \operatorname{f(S_{x,i},b)}}
  11. Y i ( S x ) Y_{i}(S_{x})
  12. P i ( S x ) P_{i}(S_{x})
  13. Y i ( S x ) Y_{i}(S_{x})
  14. S x , i S_{x,i}
  15. Y i ( S x ) Y_{i}(S_{x})
  16. Y Y
  17. S x , i S_{x,i}
  18. f ( S x , i , a ) f(S_{x,i},a)
  19. f ( S x , i , c ) f(S_{x,i},c)
  20. f ( S x , i , g ) f(S_{x,i},g)
  21. f ( S x , i , t ) f(S_{x,i},t)
  22. I M M i - 1 ( S x , i - 1 , a ) IMM_{i-1}(S_{x,{i-1}},a)
  23. I M M i - 1 ( S x , i - 1 , c ) IMM_{i-1}(S_{x,{i-1}},c)
  24. I M M i - 1 ( S x , i - 1 , g ) IMM_{i-1}(S_{x,{i-1}},g)
  25. I M M i - 1 ( S x , i - 1 , t ) IMM_{i-1}(S_{x,{i-1}},t)
  26. X 2 X^{2}

Glossary_of_game_theory.html

  1. \mathbb{R}
  2. N \mathrm{N}
  3. Σ = i N Σ i \Sigma\ =\prod_{i\in\mathrm{N}}\Sigma\ ^{i}
  4. Σ i \Sigma\ ^{i}
  5. σ i \sigma\ _{i}
  6. Σ i \Sigma\ ^{i}
  7. σ - i \sigma\ _{-i}
  8. Σ - i = j N , j i Σ j \Sigma\ ^{-i}=\prod_{j\in\mathrm{N},j\neq i}\Sigma\ ^{j}
  9. Γ \Gamma
  10. N \mathbb{R}^{\mathrm{N}}
  11. π : i N Σ i N \pi\ :\prod_{i\in\mathrm{N}}\Sigma\ ^{i}\to\mathbb{R}^{\mathrm{N}}
  12. π : i N Σ i Γ \pi\ :\prod_{i\in\mathrm{N}}\Sigma\ ^{i}\to\Gamma
  13. ν : Γ N \nu\ :\Gamma\ \to\mathbb{R}^{\mathrm{N}}
  14. ν : 2 ( N ) \nu\ :2^{\mathbb{P}(N)}\to\mathbb{R}
  15. N N
  16. ν ( N ) \nu(N)
  17. ν : Γ N \nu\ :\Gamma\ \to\mathbb{R}^{\mathrm{N}}
  18. σ - i \sigma\ _{-i}
  19. τ i \tau\ _{i}
  20. σ i Σ i π ( σ i , σ - i ) π ( τ i , σ - i ) \forall\sigma\ _{i}\in\ \Sigma\ ^{i}\quad\quad\pi\ (\sigma\ _{i},\sigma\ _{-i}% )\leq\pi\ (\tau\ _{i},\sigma\ _{-i})
  21. S N \mathrm{S}\subseteq\mathrm{N}
  22. m m\in\mathbb{N}
  23. a A , \exist σ n Σ n s . t . σ - n Σ - n : Γ ( σ - n , σ n ) = a \forall a\in\mathrm{A},\;\exist\sigma\ _{n}\in\Sigma\ ^{n}\;s.t.\;\forall% \sigma\ _{-n}\in\Sigma\ ^{-n}:\;\Gamma\ (\sigma\ _{-n},\sigma\ _{n})=a
  24. a A , σ - n Σ - n \exist σ n Σ n s . t . Γ ( σ - n , σ n ) = a \forall a\in\mathrm{A},\;\forall\sigma\ _{-n}\in\Sigma\ ^{-n}\;\exist\sigma\ _% {n}\in\Sigma\ ^{n}\;s.t.\;\Gamma\ (\sigma\ _{-n},\sigma\ _{n})=a
  25. α \alpha
  26. β \beta
  27. j N ν j ( a ) ν j ( b ) \forall j\in\mathrm{N}\;\quad\nu\ _{j}(a)\leq\ \nu\ _{j}(b)
  28. i N s . t . ν i ( a ) f o r s t r i c t d o m i n a t i o n . < b r / > A n o u t c o m e ′′′ a ′′′ i s ( s t r i c t l y ) ′′′ d o m i n a t e d ′′′ i f i t i s ( s t r i c t l y ) ′′′ d o m i n a t e d ′′′ b y s o m e o t h e r ′′′ o u t c o m e ′′′ . < b r / > A n o u t c o m e ′′′ a ′′′ i s d o m i n a t e d f o r a ′′′ c o a l i t i o n ′′′ S ′′′ ′′′ i f a l l p l a y e r s i n ′′′ S ′′′ p r e f e r s o m e o t h e r o u t c o m e t o ′′′ a ′′′ . S e e a l s o ′′′ C o n d o r c e t w i n n e r ′′′ . ; D o m i n a t e d s t r a t e g y : w e s a y t h a t s t r a t e g y i s ( s t r o n g l y ) d o m i n a t e d b y s t r a t e g y < m a t h > τ i \exists i\in\mathrm{N}\;s.t.\;\nu\ _{i}(a)forstrictdomination.<br/>Anoutcome^{% \prime\prime\prime}a^{\prime\prime\prime}is(strictly)^{\prime\prime\prime}% dominated^{\prime\prime\prime}ifitis(strictly)^{\prime\prime\prime}dominated^{% \prime\prime\prime}bysomeother^{\prime\prime\prime}outcome^{\prime\prime\prime% }.<br/>Anoutcome^{\prime\prime\prime}a^{\prime\prime\prime}isdominatedfora^{% \prime\prime\prime}coalition^{\prime\prime\prime}{}^{\prime\prime\prime}S^{% \prime\prime\prime}ifallplayersin^{\prime\prime\prime}S^{\prime\prime\prime}% prefersomeotheroutcometo^{\prime\prime\prime}a^{\prime\prime\prime}.Seealso^{% \prime\prime\prime}Condorcetwinner^{\prime\prime\prime}.\par ;% Dominatedstrategy:wesaythatstrategyis(strongly)dominatedbystrategy<math>\tau\ % _{i}
  29. σ - i \sigma\ _{-i}
  30. τ i \tau\ _{i}
  31. σ - i Σ - i π ( σ i , σ - i ) π ( τ i , σ - i ) \forall\sigma\ _{-i}\in\ \Sigma\ ^{-i}\quad\quad\pi\ (\sigma\ _{i},\sigma\ _{-% i})\leq\pi\ (\tau\ _{i},\sigma\ _{-i})
  32. σ - i Σ - i s . t . π ( σ i , σ - i ) . < b r / > A s t r a t e g y ′′′ σ ′′′ i s ( s t r i c t l y ) ′′′ d o m i n a t e d ′′′ i f i t i s ( s t r i c t l y ) ′′′ d o m i n a t e d ′′′ b y s o m e o t h e r ′′′ s t r a t e g y ′′′ . ; D u m m y : A p l a y e r ′′′ i ′′′ i s a d u m m y i f h e h a s n o e f f e c t o n t h e o u t c o m e o f t h e g a m e . I . e . i f t h e o u t c o m e o f t h e g a m e i s i n s e n s i t i v e t o p l a y e r ′′′ i ′′′′ s s t r a t e g y . A n t o n y m s : ′′ s a y ′′ , ′′ v e t o ′′ , ′′ d i c t a t o r ′′ . ; E f f e c t i v e n e s s : A c o a l i t i o n ( o r a s i n g l e p l a y e r ) ′′′ S ′′′ i s ′′ e f f e c t i v e f o r ′′ a ′′′ ′′′ i f i t c a n f o r c e ′′′ a ′′′ t o b e t h e o u t c o m e o f t h e g a m e . ′′′ S ′′′ i s α - e f f e c t i v e i f t h e m e m b e r s o f ′′′ S ′′′ h a v e s t r a t e g i e s s . t . n o m a t t e r w h a t t h e c o m p l e m e n t o f ′′′ S ′′′ d o e s , t h e o u t c o m e w i l l b e ′′′ a ′′′ . S ′′′ ′′′ i s β - e f f e c t i v e i f f o r a n y s t r a t e g i e s o f t h e c o m p l e m e n t o f ′′′ S ′′′ , t h e m e m b e r s o f ′′′ S ′′′ c a n a n s w e r w i t h s t r a t e g i e s t h a t e n s u r e o u t c o m e ′′′ a ′′′ . ; F i n i t e g a m e : i s a g a m e w i t h f i n i t e l y m a n y p l a y e r s , e a c h o f w h i c h h a s a f i n i t e s e t o f ′′′ s t r a t e g i e s ′′′ . ; G r a n d c o a l i t i o n : r e f e r s t o t h e c o a l i t i o n c o n t a i n i n g a l l p l a y e r s . I n c o o p e r a t i v e g a m e s i t i s o f t e n a s s u m e d t h a t t h e g r a n d c o a l i t i o n f o r m s a n d t h e p u r p o s e o f t h e g a m e i s t o f i n d s t a b l e i m p u t a t i o n s . ; M i x e d s t r a t e g y : f o r p l a y e r ′′′ i ′′′ i s a p r o b a b i l i t y d i s t r i b u t i o n ′′′ P ′′′ o n < m a t h > Σ i \exists\sigma\ _{-i}\in\ \Sigma\ ^{-i}\quad s.t.\quad\pi\ (\sigma\ _{i},\sigma% \ _{-i}).<br/>Astrategy^{\prime\prime\prime}σ^{\prime\prime\prime}is(strictly)% ^{\prime\prime\prime}dominated^{\prime\prime\prime}ifitis(strictly)^{\prime% \prime\prime}dominated^{\prime\prime\prime}bysomeother^{\prime\prime\prime}% strategy^{\prime\prime\prime}.\par ;Dummy:Aplayer^{\prime\prime\prime}i^{% \prime\prime\prime}isadummyifhehasnoeffectontheoutcomeofthegame.I.e.% iftheoutcomeofthegameisinsensitivetoplayer^{\prime\prime\prime}i^{\prime\prime% \prime\prime}sstrategy.\par \par Antonyms:^{\prime\prime}say^{\prime\prime},^% {\prime\prime}veto^{\prime\prime},^{\prime\prime}dictator^{\prime\prime}.\par % ;Effectiveness:Acoalition(orasingleplayer)^{\prime\prime\prime}S^{\prime\prime% \prime}is^{\prime\prime}effectivefor^{\prime\prime}{}^{\prime\prime\prime}a^{% \prime\prime\prime}ifitcanforce^{\prime\prime\prime}a^{\prime\prime\prime}% tobetheoutcomeofthegame.^{\prime\prime\prime}S^{\prime\prime\prime}isα-% effectiveifthemembersof^{\prime\prime\prime}S^{\prime\prime\prime}% havestrategiess.t.nomatterwhatthecomplementof^{\prime\prime\prime}S^{\prime% \prime\prime}does,theoutcomewillbe^{\prime\prime\prime}a^{\prime\prime\prime}.% \par \par {}^{\prime\prime\prime}S^{\prime\prime\prime}isβ-% effectiveifforanystrategiesofthecomplementof^{\prime\prime\prime}S^{\prime% \prime\prime},themembersof^{\prime\prime\prime}S^{\prime\prime\prime}% cananswerwithstrategiesthatensureoutcome^{\prime\prime\prime}a^{\prime\prime% \prime}.\par ;Finitegame:isagamewithfinitelymanyplayers,% eachofwhichhasafinitesetof^{\prime\prime\prime}strategies^{\prime\prime\prime}% .\par \par ;Grandcoalition:referstothecoalitioncontainingallplayers.% Incooperativegamesitisoftenassumedthatthegrandcoalitionformsandthepurposeofthegameistofindstableimputations% .\par \par ;Mixedstrategy:forplayer^{\prime\prime\prime}i^{\prime\prime\prime}% isaprobabilitydistribution^{\prime\prime\prime}P^{\prime\prime\prime}on<math>% \Sigma\ ^{i}
  33. ν : Γ N \nu\ :\Gamma\ \to\mathbb{R}^{\mathrm{N}}
  34. σ = ( σ i ) i N \sigma\ =(\sigma\ _{i})_{i\in\mathrm{N}}
  35. σ i \sigma\ _{i}
  36. σ \sigma
  37. i N τ i Σ i π ( τ , σ - i ) π ( σ ) \forall i\in\mathrm{N}\quad\forall\tau\ _{i}\in\ \Sigma\ ^{i}\quad\pi\ (\tau\ % ,\sigma\ _{-i})\leq\pi\ (\sigma\ )
  38. γ Γ i N ν i ( γ ) = c o n s t . \forall\gamma\ \in\Gamma\ \sum_{i\in\mathrm{N}}\nu\ _{i}(\gamma\ )=const.

Glove_problem.html

  1. min ( M / 2 + N , M + N / 2 ) \min\left(\lceil M/2\rceil+N,M+\lceil N/2\rceil\right)

Gluing_axiom.html

  1. ( U ) i ( U i ) i , j ( U i U j ) {\mathcal{F}}(U)\rightarrow\prod_{i}{\mathcal{F}}(U_{i}){{{}\atop% \longrightarrow}\atop{\longrightarrow\atop{}}}\prod_{i,j}{\mathcal{F}}(U_{i}% \cap U_{j})
  2. i , j U i U j i U i U \coprod_{i,j}U_{i}\cap U_{j}{{{}\atop\longrightarrow}\atop{\longrightarrow% \atop{}}}\coprod_{i}U_{i}\rightarrow U

Gluon_condensate.html

  1. G μ ν G μ ν \langle G_{\mu\nu}G^{\mu\nu}\rangle

Gnomonic_projection.html

  1. r ( d ) = R tan ( d / R ) r(d)=R\,\tan(d/R)
  2. r ( d ) = 1 cos 2 ( d / R ) r^{\prime}(d)=\frac{1}{\cos^{2}(d/R)}
  3. 1 cos ( d / R ) \frac{1}{\cos(d/R)}
  4. R = 1 R=1

Golden_Rule_savings_rate.html

  1. y = f ( k ) y=f(k)
  2. s f ( k ) = ( n + d ) k sf(k)=(n+d)k
  3. f ( k ) f(k)
  4. c = ( 1 - s ) f ( k ) c=(1-s)f(k)
  5. c = f ( k ) - ( n + d ) k c=f(k)-(n+d)k
  6. k G k^{G}
  7. k = k G k=k^{G}
  8. d c / d k = 0 dc/dk=0
  9. d f / d k - ( n + d ) = 0 df/dk-(n+d)=0
  10. Golden rule for capital/labour ratio: d f d k = ( n + d ) \mbox{Golden rule for capital/labour ratio: }~{}\frac{df}{dk}=(n+d)
  11. k = k G k=k^{G}
  12. y G = f ( k G ) y^{G}=f(k^{G})
  13. i G = ( n + d ) k G i^{G}=(n+d)k^{G}
  14. Golden rule savings rate: s G = ( n + d ) k G f ( k G ) \mbox{Golden rule savings rate: }~{}s^{G}=\frac{(n+d)k^{G}}{f(k^{G})}
  15. Golden rule savings rate: s G = m p k G a p k G \mbox{Golden rule savings rate: }~{}s^{G}=\frac{mpk^{G}}{apk^{G}}
  16. m p k G mpk^{G}
  17. d f ( k ) / d k df(k)/dk
  18. a p k G apk^{G}
  19. f ( k ) / k f(k)/k
  20. k G k^{G}
  21. y G y^{G}
  22. a p k G apk^{G}
  23. s G s^{G}
  24. f ( k ) f(k)
  25. y = f ( k ) = k a y=f(k)=k^{a}
  26. a p k = k ( a - 1 ) apk=k^{(a-1)}
  27. m p k = a k ( a - 1 ) mpk=ak^{(a-1)}
  28. s G = a s^{G}=a
  29. k G = ( a / ( n + d ) ) 1 / ( 1 - a ) k^{G}=(a/(n+d))^{1/(1-a)}
  30. y G = ( a / ( n + d ) ) a / ( 1 - a ) y^{G}=(a/(n+d))^{a/(1-a)}

Golden_section_search.html

  1. f ( x ) f(x)
  2. f ( x ) f(x)
  3. x 1 x_{1}
  4. x 2 x_{2}
  5. x 3 x_{3}
  6. f 2 f_{2}
  7. f 1 f_{1}
  8. f 3 f_{3}
  9. x 1 x_{1}
  10. x 3 x_{3}
  11. x 4 x_{4}
  12. x 4 x_{4}
  13. x 2 x_{2}
  14. x 3 x_{3}
  15. f 4 a f_{4a}
  16. x 1 x_{1}
  17. x 4 x_{4}
  18. x 1 x_{1}
  19. x 2 x_{2}
  20. x 4 x_{4}
  21. f 4 b f_{4b}
  22. x 2 x_{2}
  23. x 3 x_{3}
  24. x 2 x_{2}
  25. x 4 x_{4}
  26. x 3 x_{3}
  27. x 1 x_{1}
  28. x 4 x_{4}
  29. x 2 x_{2}
  30. x 3 x_{3}
  31. x 4 = x 1 + ( x 3 - x 2 ) x_{4}=x_{1}+(x_{3}-x_{2})
  32. x 2 x_{2}
  33. x 1 x_{1}
  34. x 3 x_{3}
  35. x 1 , x 2 , x 4 x_{1},x_{2},x_{4}
  36. x 2 , x 4 , x 3 x_{2},x_{4},x_{3}
  37. x 2 x_{2}
  38. x 1 x_{1}
  39. x 3 x_{3}
  40. f ( x 4 ) f(x_{4})
  41. f ( x 4 ) f(x_{4})
  42. f 4 a f_{4a}
  43. x 1 x_{1}
  44. x 2 x_{2}
  45. x 4 x_{4}
  46. c a = a b . \frac{c}{a}=\frac{a}{b}.
  47. f ( x 4 ) f(x_{4})
  48. f 4 b f_{4b}
  49. x 2 x_{2}
  50. x 4 x_{4}
  51. x 3 x_{3}
  52. c ( b - c ) = a b . \frac{c}{(b-c)}=\frac{a}{b}.
  53. ( b a ) 2 = b a + 1 \left(\frac{b}{a}\right)^{2}=\frac{b}{a}+1
  54. b a = φ \frac{b}{a}=\varphi
  55. φ = 1 + 5 2 = 1.618033988 \varphi=\frac{1+\sqrt{5}}{2}=1.618033988\ldots
  56. x 1 x_{1}
  57. x 2 x_{2}
  58. x 3 x_{3}
  59. x 4 x_{4}
  60. | x 3 - x 1 | < τ ( | x 2 | + | x 4 | ) |x_{3}-x_{1}|<\tau(|x_{2}|+|x_{4}|)\,
  61. τ \tau
  62. | x | |x|
  63. x x
  64. x x
  65. f ( x ) f(x)
  66. τ = ϵ \tau=\sqrt{\epsilon}
  67. ϵ \epsilon
  68. f ( x ) f(x)
  69. φ = ( - 1 + 5 ) / 2 \varphi=(-1+\sqrt{5})/2
  70. f ( x ) f(x)
  71. f ( x 2 ) f(x_{2})
  72. f ( x ) f(x)

Goldman_equation.html

  1. M M
  2. A A
  3. E m = R T F ln ( i N P M i + [ M i + ] out + j M P A j - [ A j - ] in i N P M i + [ M i + ] in + j M P A j - [ A j - ] out ) E_{m}=\frac{RT}{F}\ln{\left(\frac{\sum_{i}^{N}P_{M^{+}_{i}}[M^{+}_{i}]_{% \mathrm{out}}+\sum_{j}^{M}P_{A^{-}_{j}}[A^{-}_{j}]_{\mathrm{in}}}{\sum_{i}^{N}% P_{M^{+}_{i}}[M^{+}_{i}]_{\mathrm{in}}+\sum_{j}^{M}P_{A^{-}_{j}}[A^{-}_{j}]_{% \mathrm{out}}}\right)}
  4. K x Na 1 - x Cl \mathrm{K}_{x}\mathrm{Na}_{1-x}\mathrm{Cl}
  5. E m , K x Na 1 - x Cl = R T F ln ( P Na [ Na + ] out + P K [ K + ] out + P Cl [ Cl - ] in P Na [ Na + ] in + P K [ K + ] in + P Cl [ Cl - ] out ) E_{m,\mathrm{K}_{x}\mathrm{\,\text{Na}}_{1-x}\mathrm{Cl}}=\frac{RT}{F}\ln{% \left(\frac{P_{\,\text{Na}}[\,\text{Na}^{+}]_{\mathrm{out}}+P_{\,\text{K}}[\,% \text{K}^{+}]_{\mathrm{out}}+P_{\,\text{Cl}}[\,\text{Cl}^{-}]_{\mathrm{in}}}{P% _{\,\text{Na}}[\,\text{Na}^{+}]_{\mathrm{in}}+P_{\,\text{K}}[\,\text{K}^{+}]_{% \mathrm{in}}+P_{\,\text{Cl}}[\,\text{Cl}^{-}]_{\mathrm{out}}}\right)}
  6. E m , Na = R T F ln ( P Na [ Na + ] out P Na [ Na + ] in ) = R T F ln ( [ Na + ] out [ Na + ] in ) E_{m,\,\text{Na}}=\frac{RT}{F}\ln{\left(\frac{P_{\,\text{Na}}[\,\text{Na}^{+}]% _{\mathrm{out}}}{P_{\,\text{Na}}[\,\text{Na}^{+}]_{\mathrm{in}}}\right)}=\frac% {RT}{F}\ln{\left(\frac{[\,\text{Na}^{+}]_{\mathrm{out}}}{[\,\text{Na}^{+}]_{% \mathrm{in}}}\right)}
  7. E m E_{m}
  8. P ion P_{\mathrm{ion}}
  9. [ i o n ] out [ion]_{\mathrm{out}}
  10. [ i o n ] in [ion]_{\mathrm{in}}
  11. R R
  12. T T
  13. F F
  14. E X = 61.5 mV log ( [ X + ] out [ X + ] in ) = - 61.5 mV log ( [ X - ] out [ X - ] in ) E_{X}=61.5\ \mathrm{mV}\log{\left(\frac{[X^{+}]_{\mathrm{out}}}{[X^{+}]_{% \mathrm{in}}}\right)}=-61.5\ \mathrm{mV}\log{\left(\frac{[X^{-}]_{\mathrm{out}% }}{[X^{-}]_{\mathrm{in}}}\right)}
  15. R 8.3 J K mol R\approx\frac{8.3\ \mathrm{J}}{\mathrm{K}\cdot\mathrm{mol}}
  16. F 9.6 × 10 4 J mol V F\approx\frac{9.6\times 10^{4}\ \mathrm{J}}{\mathrm{mol}\cdot\mathrm{V}}
  17. T = 37 C = 310 K T=37\ ^{\circ}\mathrm{C}=310\ \mathrm{K}
  18. E X = R T z F ln X o X i E_{X}=\frac{RT}{zF}\ln\frac{X_{o}}{X_{i}}
  19. E X .0267 V z ln X o X i = 26.7 mV z ln X o X i 61.5 mV z log X o X i since ln 10 2.30 \begin{aligned}\displaystyle E_{X}&\displaystyle\approx\frac{.0267\ \mathrm{V}% }{z}\ln\frac{X_{o}}{X_{i}}\\ &\displaystyle=\frac{26.7\ \mathrm{mV}}{z}\ln\frac{X_{o}}{X_{i}}\\ &\displaystyle\approx\frac{61.5\ \mathrm{mV}}{z}\log\frac{X_{o}}{X_{i}}&% \displaystyle\,\text{ since }\ln 10\approx 2.30\end{aligned}
  20. j A = - D A ( d [ A ] d z - n A F R T E m L [ A ] ) j_{\mathrm{A}}=-D_{\mathrm{A}}\left(\frac{d\left[\mathrm{A}\right]}{dz}-\frac{% n_{\mathrm{A}}F}{RT}\frac{E_{m}}{L}\left[\mathrm{A}\right]\right)
  21. d [ A ] - j A D A + n A F E m R T L [ A ] = d z \frac{d\left[\mathrm{A}\right]}{-\frac{j_{\mathrm{A}}}{D_{\mathrm{A}}}+\frac{n% _{\mathrm{A}}FE_{m}}{RTL}\left[\mathrm{A}\right]}=dz
  22. j A = μ n A P A [ A ] out - [ A ] in e n μ 1 - e n μ j_{\mathrm{A}}=\mu n_{\mathrm{A}}P_{\mathrm{A}}\frac{\left[\mathrm{A}\right]_{% \mathrm{out}}-\left[\mathrm{A}\right]_{\mathrm{in}}e^{n\mu}}{1-e^{n\mu}}
  23. μ = F E m R T \mu=\frac{FE_{m}}{RT}
  24. P A = D A L P_{\mathrm{A}}=\frac{D_{\mathrm{A}}}{L}
  25. J A = q A j A J_{A}=q_{\mathrm{A}}j_{\mathrm{A}}
  26. J t o t = A J A = 0 J_{tot}=\sum_{A}J_{A}=0
  27. w - v e μ = 0 w-ve^{\mu}=0
  28. F E m R T = μ = ln w v \frac{FE_{m}}{RT}=\mu=\ln\frac{w}{v}
  29. w = cations C P C [ C + ] out + anions A P A [ A - ] in w=\sum_{\mathrm{cations\ C}}P_{\mathrm{C}}\left[\mathrm{C}^{+}\right]_{\mathrm% {out}}+\sum_{\mathrm{anions\ A}}P_{\mathrm{A}}\left[\mathrm{A}^{-}\right]_{% \mathrm{in}}
  30. v = cations C P C [ C + ] in + anions A P A [ A - ] out v=\sum_{\mathrm{cations\ C}}P_{\mathrm{C}}\left[\mathrm{C}^{+}\right]_{\mathrm% {in}}+\sum_{\mathrm{anions\ A}}P_{\mathrm{A}}\left[\mathrm{A}^{-}\right]_{% \mathrm{out}}

Goppa_code.html

  1. X X
  2. 𝔽 q \mathbb{F}_{q}
  3. 𝔽 q \mathbb{F}_{q}
  4. 𝔽 q \mathbb{F}_{q}
  5. 𝒫 \mathcal{P}
  6. 𝔽 q \mathbb{F}_{q}
  7. P i P_{i}
  8. 𝒫 \mathcal{P}
  9. L ( G ) L(G)
  10. 𝒫 \mathcal{P}
  11. D = P 1 + P 2 + + P n D=P_{1}+P_{2}+\cdots+P_{n}
  12. P i P_{i}
  13. \in
  14. 𝔽 q n \mathbb{F}_{q}^{n}
  15. 𝔽 q \mathbb{F}_{q}
  16. 𝔽 q n \mathbb{F}_{q}^{n}
  17. 𝔽 q \mathbb{F}_{q}
  18. [ f 1 ( P 1 ) f 1 ( P n ) f k ( P 1 ) f k ( P n ) ] \begin{bmatrix}f_{1}(P_{1})&...&f_{1}(P_{n})\\ ...&...&...\\ f_{k}(P_{1})&...&f_{k}(P_{n})\end{bmatrix}
  19. α : L ( G ) 𝔽 n \alpha:L(G)\longrightarrow\mathbb{F}^{n}
  20. f ( f ( P 1 ) , , f ( P n ) ) f\longmapsto(f(P_{1}),\dots,f(P_{n}))
  21. k = l ( G ) - l ( G - D ) k=l(G)-l(G-D)
  22. d n - deg ( G ) d\geq n-\deg(G)
  23. C ( D , G ) L ( G ) / ker ( α ) , C(D,G)\cong L(G)/\ker(\alpha),
  24. ker ( α ) = L ( G - D ) \ker(\alpha)=L(G-D)
  25. f ker ( α ) f\in\ker(\alpha)
  26. f ( P i ) = 0 , i = 1 , , n f(P_{i})=0,i=1,\dots,n
  27. div ( f ) > D \mathrm{div}(f)>D
  28. f L ( G - D ) f\in L(G-D)
  29. f L ( G - D ) f\in L(G-D)
  30. div ( f ) > D \mathrm{div}(f)>D
  31. P i < G , i = 1 , , n P_{i}<G,i=1,\dots,n
  32. - D -D
  33. f ( P i ) = 0 , i = 1 , , n f(P_{i})=0,i=1,\dots,n
  34. d n - deg ( G ) d\geq n-\deg(G)
  35. α ( f ) \alpha(f)
  36. f ( P i ) = 0 f(P_{i})=0
  37. n - d n-d
  38. P i P_{i}
  39. P i 1 , , P i n - d P_{i_{1}},\dots,P_{i_{n-d}}
  40. f L ( G - P i 1 - - P i n - d ) f\in L(G-P_{i_{1}}-\dots-P_{i_{n-d}})
  41. div ( f ) + G - P i 1 - - P i n - d > 0 \mathrm{div}(f)+G-P_{i_{1}}-\dots-P_{i_{n-d}}>0
  42. deg ( div ( f ) ) = 0 \deg(\mathrm{div}(f))=0
  43. deg ( G ) - ( n - d ) 0 \deg(G)-(n-d)\geq 0
  44. d n - deg ( G ) d\geq n-\deg(G)
  45. P i P_{i}

Gospel_of_the_Nazarenes.html

  1. 𝔓 \mathfrak{P}

GPS_meteorology.html

  1. z z
  2. 1 cos z \frac{1}{\cos z}

Graetz_number.html

  1. Gz = D H L Re Pr \mathrm{Gz}={D_{H}\over L}\mathrm{Re}\,\mathrm{Pr}
  2. Gz = D H L Re Sc \mathrm{Gz}={D_{H}\over L}\mathrm{Re}\,\mathrm{Sc}

Grain_boundary.html

  1. R = [ a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 ] R=\begin{bmatrix}a_{11}&a_{12}&a_{13}\\ a_{21}&a_{22}&a_{23}\\ a_{31}&a_{32}&a_{33}\end{bmatrix}
  2. 2 cos θ + 1 = a 11 + a 22 + a 33 2\cos\;\theta\;+1=a_{11}+a_{22}+a_{33}\,\!
  3. [ ( a 32 - a 23 ) , ( a 13 - a 31 ) , ( a 21 - a 12 ) ] [(a_{32}-a_{23}),(a_{13}-a_{31}),(a_{21}-a_{12})]\,\!
  4. γ s = γ 0 θ ( A - ln θ ) \gamma_{s}=\gamma_{0}\theta(A-\ln\theta)\,\!
  5. θ = b / h \theta=b/h\,\!
  6. γ 0 = G b / 4 π ( 1 - ν ) \gamma_{0}=Gb/4\pi(1-\nu)\,\!
  7. A = 1 + l n ( b / 2 π r 0 ) A=1+ln(b/2\pi r_{0})\,\!
  8. G G
  9. ν \nu
  10. r 0 r_{0}
  11. Σ \Sigma
  12. M = M 0 exp ( - Q R T ) M=M_{0}\exp\left(-\frac{Q}{RT}\right)\,\!

Granger_causality.html

  1. X X
  2. Y Y
  3. [ Y ( t + 1 ) A | ( t ) ] [ Y ( t + 1 ) A | - X ( t ) ] \mathbb{P}[Y(t+1)\in A|\mathcal{I}(t)]\neq\mathbb{P}[Y(t+1)\in A|\mathcal{I}_{% -X}(t)]
  4. \mathbb{P}
  5. A A
  6. ( t ) \mathcal{I}(t)
  7. - X ( t ) \mathcal{I}_{-X}(t)
  8. t t
  9. X X
  10. X X
  11. Y Y
  12. y t = a 0 + a 1 y t - 1 + a 2 y t - 2 + + a m y t - m + residual t . y_{t}=a_{0}+a_{1}y_{t-1}+a_{2}y_{t-2}+\cdots+a_{m}y_{t-m}+\mathrm{residual}_{t}.
  13. y t = a 0 + a 1 y t - 1 + a 2 y t - 2 + + a m y t - m + b p x t - p + + b q x t - q + residual t . y_{t}=a_{0}+a_{1}y_{t-1}+a_{2}y_{t-2}+\cdots+a_{m}y_{t-m}+b_{p}x_{t-p}+\cdots+% b_{q}x_{t-q}+\mathrm{residual}_{t}.
  14. X ( t ) d × 1 X(t)\in\mathbb{R}^{d\times 1}
  15. t = 1 , , T t=1,\ldots,T
  16. d d
  17. L L
  18. X ( t ) = τ = 1 L A τ X ( t - τ ) + ε ( t ) , X(t)=\sum_{\tau=1}^{L}A_{\tau}X(t-\tau)+\varepsilon(t),
  19. ε ( t ) \varepsilon(t)
  20. X i X_{i}
  21. X j X_{j}
  22. A τ ( j , i ) A_{\tau}(j,i)
  23. τ = 1 , , L \tau=1,\ldots,L

Graph_isomorphism_problem.html

  1. X X
  2. X X

Graph_labeling.html

  1. E \|E\|
  2. E \|E\|
  3. k - j \|k-j\|
  4. G := ( V , E ) G:=(V,E)
  5. E \|E\|

Graph_property.html

  1. k k
  2. k k

Graph_rewriting.html

  1. L R L\rightarrow R
  2. L L
  3. R R
  4. r = ( L K R ) r=(L\leftarrow K\rightarrow R)
  5. L K R L\supseteq K\subseteq R
  6. K L K\rightarrow L
  7. k : K G k\colon K\rightarrow G
  8. m : L G m\colon L\rightarrow G
  9. L L
  10. G G
  11. L L
  12. R R
  13. G G
  14. K K
  15. K K
  16. G G
  17. r : L R r\colon L\rightarrow R

Graphical_timeline_of_the_Big_Bang.html

  1. 10 log 10 10\cdot\log_{10}
  2. 10 log 10 0.000001 = 10 ( - 6 ) = - 60 10\cdot\log_{10}0.000001=10\cdot(-6)=-60
  3. 10 - 30 10 = 10 - 3 = 0.001 10^{-\frac{30}{10}}=10^{-3}=0.001

Graveyard_orbit.html

  1. Δ H \Delta{H}\,
  2. Δ H = 235 km + ( 1000 C R A m ) km \Delta{H}=235\mbox{ km}~{}+\left(1000C_{R}\frac{A}{m}\right)\mbox{ km}~{}
  3. C R C_{R}\,
  4. A m \frac{A}{m}\,
  5. Δ H \Delta{H}

Gravitational_anomaly.html

  1. n = 1 + D / 2 n=1+D/2
  2. D D

Gravitational_microlensing.html

  1. θ E = 4 G M c 2 d S - d L d S d L \theta_{E}=\sqrt{\frac{4GM}{c^{2}}\frac{d_{S}-d_{L}}{d_{S}d_{L}}}
  2. θ E \theta_{E}
  3. θ E \theta_{E}
  4. θ E \theta_{E}
  5. θ E \theta_{E}
  6. A ( u ) = u 2 + 2 u u 2 + 4 A(u)=\frac{u^{2}+2}{u\sqrt{u^{2}+4}}
  7. t E t_{E}
  8. θ E \theta_{E}
  9. t E t_{E}
  10. u ( t ) = u m i n 2 + ( t - t 0 t E ) 2 u(t)=\sqrt{u_{min}^{2}+\left(\frac{t-t_{0}}{t_{E}}\right)^{2}}
  11. t E t_{E}
  12. θ E \theta_{E}
  13. r ~ E \tilde{r}_{E}
  14. r ~ E = 4 G M c 2 d S d L d S - d L \tilde{r}_{E}=\sqrt{\frac{4GM}{c^{2}}\frac{d_{S}d_{L}}{d_{S}-d_{L}}}
  15. μ E = t E - 1 \vec{\mu}_{E}={t_{E}}^{-1}
  16. π E = r ~ E - 1 \vec{\pi}_{E}={\tilde{r}_{E}}^{-1}
  17. M = c 2 4 G θ E r ~ E M=\frac{c^{2}}{4G}\theta_{E}\tilde{r}_{E}
  18. π L = π E θ E + π S \pi_{L}=\pi_{E}\theta_{E}+\pi_{S}
  19. μ L = μ E θ E + μ S \mu_{L}=\mu_{E}\theta_{E}+\mu_{S}
  20. t E t_{E}
  21. θ E \theta_{E}
  22. π E \pi_{E}
  23. t S t_{S}
  24. θ S \theta_{S}
  25. θ E = θ S t E t S \theta_{E}=\theta_{S}\frac{t_{E}}{t_{S}}
  26. θ S / θ E \theta_{S}/\theta_{E}
  27. t S t_{S}
  28. π E \vec{\pi}_{E}
  29. π E \pi_{E}

Gravitational_plane_wave.html

  1. d s 2 = [ a ( u ) ( x 2 - y 2 ) + 2 b ( u ) x y ] d u 2 + 2 d u d v + d x 2 + d y 2 ds^{2}=[a(u)(x^{2}-y^{2})+2b(u)xy]du^{2}+2dudv+dx^{2}+dy^{2}
  2. a ( u ) , b ( u ) a(u),b(u)
  3. a = 2 G M ( 2 r N S ) 2 9 × 10 11 m / s 2 10 11 g a=\frac{2GM\odot}{(2rNS)^{2}}\simeq 9\times 10^{11}m/s^{2}\simeq 10^{11}g

Gravity_darkening.html

  1. F centrifugal = m Ω 2 ρ F_{\,\text{centrifugal}}=m\Omega^{2}\rho
  2. m m
  3. Ω \Omega
  4. ρ \rho
  5. ρ \rho

Greek_letters_used_in_mathematics,_science,_and_engineering.html

  1. \Alpha α \Alpha\,\alpha\,
  2. ϝ \digamma\,
  3. K κ ϰ K\,\kappa\,\varkappa\,
  4. \Omicron \omicron \Omicron\,\omicron\,
  5. Υ υ \Upsilon\,\upsilon\,
  6. B β B\,\beta\,
  7. \Zeta ζ \Zeta\,\zeta\,
  8. Λ λ \Lambda\,\lambda\,
  9. Π π ϖ \Pi\,\pi\,\varpi\,
  10. Φ ϕ φ \Phi\,\phi\,\varphi\,
  11. Γ γ \Gamma\,\gamma\,
  12. \Eta η \Eta\,\eta\,
  13. \Mu μ \Mu\,\mu\,
  14. \Rho ρ ϱ \Rho\,\rho\,\varrho\,
  15. \Chi χ \Chi\,\chi\,
  16. Δ δ \Delta\,\delta\,
  17. Θ θ ϑ \Theta\,\theta\,\vartheta\,
  18. \Nu ν \Nu\,\nu\,
  19. Σ σ ς \Sigma\,\sigma\,\varsigma\,
  20. Ψ ψ \Psi\,\psi\,
  21. \Epsilon ϵ ε \Epsilon\,\epsilon\,\varepsilon\,
  22. \Iota ι \Iota\,\iota\,
  23. Ξ ξ \Xi\,\xi\,
  24. \Tau τ \Tau\,\tau\,
  25. Ω ω \Omega\,\omega\,
  26. ω , ω ω , ω ω ω , \omega,\omega^{\omega},\omega^{\omega^{\omega}},\dots
  27. ( ε / ε 0 ) (\varepsilon/\varepsilon_{0})
  28. ω \omega
  29. \varnothing
  30. χ 2 \chi^{2}
  31. C ω C^{\omega}
  32. \mathbb{N}

Green–Kubo_relations.html

  1. γ \gamma
  2. γ = 0 A ˙ ( t ) A ˙ ( 0 ) d t \gamma=\int_{0}^{\infty}\langle\dot{A}(t)\dot{A}(0)\rangle dt
  3. I = σ V . I=\sigma V.\,
  4. S x y S_{xy}
  5. γ \gamma
  6. γ = def u x / y \gamma\ \stackrel{\mathrm{def}}{=}\ \partial u_{x}/\partial y
  7. S x y = η γ . S_{xy}=\eta\gamma.\,
  8. S x y = η ( γ ) γ . S_{xy}=\eta(\gamma)\gamma.\,
  9. J = L ( F e = 0 ) F e . J=L(F_{e}=0)F_{e}.\,
  10. L ( F e = 0 ) = β V 0 d s J ( 0 ) J ( s ) F e = 0 , L(F_{e}=0)=\beta V\;\int_{0}^{\infty}{ds}\left\langle{J(0)J(s)}\right\rangle_{% F_{e}=0},\,
  11. β = 1 k T \beta=\frac{1}{kT}
  12. L ( F e ) = β V 0 d s J ( 0 ) J ( s ) F e , L(F_{e})=\beta V\;\int_{0}^{\infty}{ds}\left\langle{J(0)J(s)}\right\rangle_{F_% {e}},\,
  13. F e = 0 F_{e}=0
  14. J ( 0 ) F e = 0 \left\langle{J(0)}\right\rangle_{F_{e}}=0
  15. J ( t ) F e 0 \left\langle{J(t)}\right\rangle_{F_{e}}\neq 0
  16. J ( t ; F e ) = J ( 0 ) exp [ - β V 0 t J ( - s ) F e d s ] F e . \left\langle{J(t;F_{e})}\right\rangle=\left\langle{J(0)\exp[-\beta V\int_{0}^{% t}{J(-s)F_{e}\;ds]}}\right\rangle_{F_{e}}.\,
  17. Ω ¯ t = - β J ¯ t V F e . \bar{\Omega}_{t}=-\beta\overline{J}_{t}VF_{e}.\,
  18. F e 2 t F_{e}^{2}t
  19. lim t , F e 0 1 t ln ( < m t p l > p ( β J ¯ t = A ) p ( β J ¯ t = - A ) ) = - lim t , F e 0 A V F e , F e 2 t = c . \lim_{t\to\infty,F_{e}\to 0}\frac{1}{t}\ln\left({\frac{<}{m}tpl>{{p(\beta% \overline{J}_{t}=A)}}{{p(\beta\overline{J}_{t}=-A)}}}\right)=-\lim_{t\to\infty% ,F_{e}\to 0}AVF_{e},\quad F_{e}^{2}t=c.\,
  20. lim t , F e 0 1 t ln ( < m t p l > p ( J ¯ t ) = A p ( J ¯ t ) = - A ) = lim t , F e 0 2 A J F e t σ J ¯ ( t ) 2 . \lim_{t\to\infty,F_{e}\to 0}\frac{1}{t}\ln\left({\frac{<}{m}tpl>{{p(\overline{% J}_{t})=A}}{{p(\overline{J}_{t})=-A}}}\right)=\lim_{t\to\infty,F_{e}\to 0}% \frac{{2A\left\langle J\right\rangle_{F_{e}}}}{{t\sigma_{\overline{J}(t)}^{2}}}.
  21. L ( 0 ) = β V 0 d t J ( 0 ) J ( t ) F e = 0 . L(0)=\beta V\;\int_{0}^{\infty}{dt}\left\langle{J(0)J(t)}\right\rangle_{F_{e}=% 0}.\,

Gross_margin.html

  1. Gross Margin Percentage = Revenue - COGS Revenue * 100 % \,\text{Gross Margin Percentage}=\frac{\,\text{Revenue - COGS}}{\,\text{% Revenue}}*100\%
  2. $ 200 - $ 100 $ 200 * 100 % = 50 % \frac{\$200-\$100}{\$200}*100\%=50\%
  3. gross margin = markup 1 + markup \,\text{gross margin}=\frac{\,\text{markup}}{1+\,\text{markup}}
  4. gross margin = 1 1 + 1 = 0.5 = 50 % \,\text{gross margin}=\frac{1}{1+1}=0.5=50\%
  5. gross margin = 0.667 1 + 0.667 = 0.4 = 40 % \,\text{gross margin}=\frac{0.667}{1+0.667}=0.4=40\%
  6. markup = gross margin 1 - gross margin \,\text{markup}=\frac{\,\text{gross margin}}{1-\,\text{gross margin}}
  7. markup = 0.5 1 - 0.5 = 1 = 100 % \,\text{markup}=\frac{0.5}{1-0.5}=1=100\%
  8. markup = 0.4 1 - 0.4 = 0.667 = 66.7 % \,\text{markup}=\frac{0.4}{1-0.4}=0.667=66.7\%

Grothendieck_group.html

  1. i : M K i\colon M\rightarrow K
  2. f : M A f\colon M\rightarrow A
  3. g : K A g\colon K\rightarrow A
  4. f = g i . f=g\circ i.
  5. { ( x + y ) - ( x + y ) x , y M } \{(x+^{\prime}y)-^{\prime}(x+y)\mid x,y\in M\}
  6. n - m n - m n + m = n + m n-m\sim n^{\prime}-m^{\prime}\Leftrightarrow n+m^{\prime}=n^{\prime}+m
  7. n := [ n - 0 ] n:=[n-0]
  8. - n := [ 0 - n ] -n:=[0-n]
  9. C ( M ) C^{\infty}(M)
  10. { [ X ] | X R - Mod } \{[X]|X\in R\mathrm{-Mod}\}
  11. 0 A B C 0 0\to A\to B\to C\to 0
  12. [ A ] - [ B ] + [ C ] = 0 [A]-[B]+[C]=0
  13. χ ( A ) - χ ( B ) + χ ( C ) = 0 \chi(A)-\chi(B)+\chi(C)=0
  14. 𝒜 \mathcal{A}
  15. χ : G 0 ( R ) Hom K ( R , K ) \chi:G_{0}(R)\to\mathrm{Hom}_{K}(R,K)
  16. 𝐅 ¯ p \overline{\mathbf{F}}_{p}
  17. G 0 ( 𝐅 ¯ p [ G ] ) BCh ( G ) G_{0}(\overline{\mathbf{F}}_{p}[G])\to\mathrm{BCh}(G)
  18. 0 0 A n A n + 1 A m - 1 A m 0 0 \cdots\to 0\to 0\to A^{n}\to A^{n+1}\to\cdots\to A^{m-1}\to A^{m}\to 0\to 0\to\cdots
  19. [ A ] = i ( - 1 ) i [ A i ] = i ( - 1 ) i [ H i ( A ) ] G 0 ( R ) . [A^{\ast}]=\sum_{i}(-1)^{i}[A^{i}]=\sum_{i}(-1)^{i}[H^{i}(A^{\ast})]\in G_{0}(% R).
  20. 𝒜 \mathcal{A}
  21. 𝒜 \mathcal{A}
  22. [ A ] - [ B ] + [ C ] = 0 [A]-[B]+[C]=0
  23. A B C A\hookrightarrow B\twoheadrightarrow C
  24. ϕ : Ob ( 𝒜 ) G \phi:\mathrm{Ob}(\mathcal{A})\to G
  25. 𝒜 \mathcal{A}
  26. χ : Ob ( 𝒜 ) X \chi\colon\mathrm{Ob}(\mathcal{A})\to X
  27. 𝒜 \mathcal{A}
  28. A B C A\hookrightarrow B\twoheadrightarrow C
  29. χ ( A ) - χ ( B ) + χ ( C ) = 0 \chi(A)-\chi(B)+\chi(C)=0
  30. 𝒜 := R \mathcal{A}:=R
  31. 𝒜 \mathcal{A}
  32. A A B B A\hookrightarrow A\oplus B\twoheadrightarrow B
  33. ( Iso ( 𝒜 ) , ) (\mathrm{Iso}(\mathcal{A}),\oplus)
  34. Iso ( 𝒜 ) \mathrm{Iso}(\mathcal{A})
  35. 𝒜 \mathcal{A}
  36. [ V ] = [ k dim ( V ) ] [V]=[k^{\mbox{dim}~{}(V)}]
  37. K 0 ( Vect fin ) K_{0}(\mathrm{Vect}_{\mathrm{fin}})
  38. 0 k l k m k n 0 0\to k^{l}\to k^{m}\to k^{n}\to 0
  39. [ k l + n ] = [ k l ] + [ k n ] = ( l + n ) [ k ] . [k^{l+n}]=[k^{l}]+[k^{n}]=(l+n)[k].
  40. [ V ] = dim ( V ) [ k ] [V]=\operatorname{dim}(V)[k]
  41. K 0 ( Vect fin ) K_{0}(\mathrm{Vect}_{\mathrm{fin}})
  42. [ V * ] = χ ( V * ) [ k ] [V^{*}]=\chi(V^{*})[k]
  43. χ \chi
  44. χ ( V * ) = i ( - 1 ) i dim V = i ( - 1 ) i dim H i ( V * ) . \chi(V^{*})=\sum_{i}(-1)^{i}\operatorname{dim}V=\sum_{i}(-1)^{i}\operatorname{% dim}H^{i}(V^{*}).
  45. ( X , 𝒪 X ) (X,\mathcal{O}_{X})
  46. 𝒜 \mathcal{A}
  47. ( X , 𝒪 X ) (X,\mathcal{O}_{X})
  48. 𝒜 \mathcal{A}
  49. 𝒜 \mathcal{A}
  50. 𝒜 \mathcal{A}

Grothendieck–Riemann–Roch_theorem.html

  1. K 0 ( X ) K_{0}(X)\,
  2. ch : K 0 ( X ) A ( X , ) , \mbox{ch}~{}\colon K_{0}(X)\to A(X,{\mathbb{Q}}),
  3. A d ( X , ) A_{d}(X,{\mathbb{Q}})\,
  4. H 2 d i m ( X ) - 2 d ( X , ) . H^{2\mathrm{dim}(X)-2d}(X,{\mathbb{Q}}).
  5. f : X Y f\colon X\to Y\,
  6. \bull {\mathcal{F}^{\bull}}
  7. X . X.
  8. f ! = ( - 1 ) i R i f * : K 0 ( X ) K 0 ( Y ) f_{\mbox{!}~{}}=\sum(-1)^{i}R^{i}f_{*}\colon K_{0}(X)\to K_{0}(Y)
  9. f * : A ( X ) A ( Y ) , f_{*}\colon A(X)\to A(Y),\,
  10. ch ( f ! \bull ) td ( Y ) = f * ( ch ( \bull ) td ( X ) ) . \mbox{ch}~{}(f_{\mbox{!}~{}}{\mathcal{F}}^{\bull})\mbox{td}~{}(Y)=f_{*}(\mbox{% ch}~{}({\mathcal{F}}^{\bull})\mbox{td}~{}(X)).
  11. ch ( f ! \bull ) = f * ( ch ( \bull ) td ( T f ) ) , \mbox{ch}~{}(f_{\mbox{!}~{}}{\mathcal{F}}^{\bull})=f_{*}(\mbox{ch}~{}({% \mathcal{F}}^{\bull})\mbox{td}~{}(T_{f})),

Ground_expression.html

  1. C C
  2. V V
  3. F F
  4. P P

Groundwater_flow_equation.html

  1. Δ M s t o r Δ t = M i n Δ t - M o u t Δ t - M g e n Δ t \frac{\Delta M_{stor}}{\Delta t}=\frac{M_{in}}{\Delta t}-\frac{M_{out}}{\Delta t% }-\frac{M_{gen}}{\Delta t}
  2. S s h t = - q - G . S_{s}\frac{\partial h}{\partial t}=-\nabla\cdot q-G.
  3. S s h t = - ( - K h ) - G . S_{s}\frac{\partial h}{\partial t}=-\nabla\cdot(-K\nabla h)-G.
  4. S s h t = K 2 h - G . S_{s}\frac{\partial h}{\partial t}=K\nabla^{2}h-G.
  5. h t = α 2 h - G . \frac{\partial h}{\partial t}=\alpha\nabla^{2}h-G.
  6. h t = α [ 2 h x 2 + 2 h y 2 + 2 h z 2 ] - G . \frac{\partial h}{\partial t}=\alpha\left[\frac{\partial^{2}h}{\partial x^{2}}% +\frac{\partial^{2}h}{\partial y^{2}}+\frac{\partial^{2}h}{\partial z^{2}}% \right]-G.
  7. h t = α [ 2 h r 2 + 1 r h r + 1 r 2 2 h θ 2 + 2 h z 2 ] - G . \frac{\partial h}{\partial t}=\alpha\left[\frac{\partial^{2}h}{\partial r^{2}}% +\frac{1}{r}\frac{\partial h}{\partial r}+\frac{1}{r^{2}}\frac{\partial^{2}h}{% \partial\theta^{2}}+\frac{\partial^{2}h}{\partial z^{2}}\right]-G.
  8. 0 = α 2 h 0=\alpha\nabla^{2}h
  9. h / z = 0 \partial h/\partial z=0
  10. δ x δ y \delta x\delta y
  11. h / z = 0 \partial h/\partial z=0
  12. q x / z = 0 \partial q_{x}/\partial z=0
  13. K / z = 0 \partial K/\partial z=0
  14. Q x = 0 b q x d z = - K b h x Q_{x}=\int_{0}^{b}q_{x}dz=-Kb\frac{\partial h}{\partial x}
  15. Q y = 0 b q y d z = - K b h y Q_{y}=\int_{0}^{b}q_{y}dz=-Kb\frac{\partial h}{\partial y}
  16. n b t = ( K b h ) + N . \frac{\partial nb}{\partial t}=\nabla\cdot(Kb\nabla h)+N.
  17. S h t = ( K H h ) + N . S\frac{\partial h}{\partial t}=\nabla\cdot(KH\nabla h)+N.
  18. S y h t = ( K h h ) + N . S_{y}\frac{\partial h}{\partial t}=\nabla\cdot(Kh\nabla h)+N.
  19. ( K h 2 ) = - 2 N . \nabla\cdot(K\nabla h^{2})=-2N.
  20. 2 h 2 = - 2 N K . \nabla^{2}h^{2}=-\frac{2N}{K}.

Grönwall's_inequality.html

  1. I I
  2. [ a , ) [a, ∞)
  3. [ a , b ] [a,b]
  4. [ a , b ) [a,b)
  5. β β
  6. u u
  7. I I
  8. u u
  9. I I
  10. I I
  11. a a
  12. b b
  13. u ( t ) β ( t ) u ( t ) , t I , u^{\prime}(t)\leq\beta(t)\,u(t),\qquad t\in I^{\circ},
  14. u u
  15. y ( t ) = β ( t ) y ( t ) y′(t)=β(t)y(t)
  16. u ( t ) u ( a ) exp ( a t β ( s ) d s ) u(t)\leq u(a)\exp\biggl(\int_{a}^{t}\beta(s)\,\mathrm{d}s\biggr)
  17. t I t∈I
  18. β β
  19. u u
  20. v ( t ) = exp ( a t β ( s ) d s ) , t I . v(t)=\exp\biggl(\int_{a}^{t}\beta(s)\,\mathrm{d}s\biggr),\qquad t\in I.
  21. v v
  22. v ( t ) = β ( t ) v ( t ) , t I , v^{\prime}(t)=\beta(t)\,v(t),\qquad t\in I^{\circ},
  23. v ( a ) = 1 v(a)=1
  24. v ( t ) > 0 v(t)>0
  25. t I t∈I
  26. d d t u ( t ) v ( t ) = u ( t ) v ( t ) - v ( t ) u ( t ) v 2 ( t ) β ( t ) u ( t ) v ( t ) - β ( t ) v ( t ) u ( t ) v 2 ( t ) = 0 , t I , \frac{d}{dt}\frac{u(t)}{v(t)}=\frac{u^{\prime}(t)\,v(t)-v^{\prime}(t)\,u(t)}{v% ^{2}(t)}\leq\frac{\beta(t)\,u(t)\,v(t)-\beta(t)\,v(t)\,u(t)}{v^{2}(t)}=0,% \qquad t\in I^{\circ},
  27. u ( t ) / v ( t ) u(t)/v(t)
  28. a a
  29. I I
  30. u ( t ) v ( t ) u ( a ) v ( a ) = u ( a ) , t I , \frac{u(t)}{v(t)}\leq\frac{u(a)}{v(a)}=u(a),\qquad t\in I,
  31. I I
  32. [ a , ) [a,∞)
  33. [ a , b ] [a,b]
  34. [ a , b ) [a,b)
  35. α α
  36. β β
  37. u u
  38. I I
  39. β β
  40. u u
  41. α α
  42. I I
  43. β β
  44. u u
  45. u ( t ) α ( t ) + a t β ( s ) u ( s ) d s , t I , u(t)\leq\alpha(t)+\int_{a}^{t}\beta(s)u(s)\,\mathrm{d}s,\qquad\forall t\in I,
  46. u ( t ) α ( t ) + a t α ( s ) β ( s ) exp ( s t β ( r ) d r ) d s , t I . u(t)\leq\alpha(t)+\int_{a}^{t}\alpha(s)\beta(s)\exp\biggl(\int_{s}^{t}\beta(r)% \,\mathrm{d}r\biggr)\mathrm{d}s,\qquad t\in I.
  47. α α
  48. u ( t ) α ( t ) exp ( a t β ( s ) d s ) , t I . u(t)\leq\alpha(t)\exp\biggl(\int_{a}^{t}\beta(s)\,\mathrm{d}s\biggr),\qquad t% \in I.
  49. α α
  50. u u
  51. u u
  52. β β
  53. u u
  54. v ( s ) = exp ( - a s β ( r ) d r ) a s β ( r ) u ( r ) d r , s I . v(s)=\exp\biggl({-}\int_{a}^{s}\beta(r)\,\mathrm{d}r\biggr)\int_{a}^{s}\beta(r% )u(r)\,\mathrm{d}r,\qquad s\in I.
  55. v ( s ) = ( u ( s ) - a s β ( r ) u ( r ) d r α ( s ) ) β ( s ) exp ( - a s β ( r ) d r ) , s I , v^{\prime}(s)=\biggl(\underbrace{u(s)-\int_{a}^{s}\beta(r)u(r)\,\mathrm{d}r}_{% \leq\,\alpha(s)}\biggr)\beta(s)\exp\biggl({-}\int_{a}^{s}\beta(r)\mathrm{d}r% \biggr),\qquad s\in I,
  56. β β
  57. v v
  58. v ( a ) = 0 v(a)=0
  59. a a
  60. t t
  61. v ( t ) a t α ( s ) β ( s ) exp ( - a s β ( r ) d r ) d s . v(t)\leq\int_{a}^{t}\alpha(s)\beta(s)\exp\biggl({-}\int_{a}^{s}\beta(r)\,% \mathrm{d}r\biggr)\mathrm{d}s.
  62. v ( t ) v(t)
  63. a t β ( s ) u ( s ) d s = exp ( a t β ( r ) d r ) v ( t ) a t α ( s ) β ( s ) exp ( a t β ( r ) d r - a s β ( r ) d r = s t β ( r ) d r ) d s . \begin{aligned}\displaystyle\int_{a}^{t}\beta(s)u(s)\,\mathrm{d}s&% \displaystyle=\exp\biggl(\int_{a}^{t}\beta(r)\,\mathrm{d}r\biggr)v(t)\\ &\displaystyle\leq\int_{a}^{t}\alpha(s)\beta(s)\exp\biggl(\underbrace{\int_{a}% ^{t}\beta(r)\,\mathrm{d}r-\int_{a}^{s}\beta(r)\,\mathrm{d}r}_{=\,\int_{s}^{t}% \beta(r)\,\mathrm{d}r}\biggr)\mathrm{d}s.\end{aligned}
  64. α α
  65. α ( s ) α ( t ) α(s)≤α(t)
  66. u ( t ) α ( t ) + ( - α ( t ) exp ( s t β ( r ) d r ) ) | s = a s = t = α ( t ) exp ( a t β ( r ) d r ) , t I . \begin{aligned}\displaystyle u(t)&\displaystyle\leq\alpha(t)+\biggl({-}\alpha(% t)\exp\biggl(\int_{s}^{t}\beta(r)\,\mathrm{d}r\biggr)\biggr)\biggr|^{s=t}_{s=a% }\\ &\displaystyle=\alpha(t)\exp\biggl(\int_{a}^{t}\beta(r)\,\mathrm{d}r\biggr),% \qquad t\in I.\end{aligned}
  67. I I
  68. [ a , ) [a,∞)
  69. [ a , b ] [a,b]
  70. [ a , b ) [a,b)
  71. α α
  72. u u
  73. I I
  74. μ μ
  75. I I
  76. t I t∈I
  77. μ μ
  78. u u
  79. μ μ
  80. [ a , t ) | u ( s ) | μ ( d s ) < , t I , \int_{[a,t)}|u(s)|\,\mu(\mathrm{d}s)<\infty,\qquad t\in I,
  81. u u
  82. u ( t ) α ( t ) + [ a , t ) u ( s ) μ ( d s ) , t I . u(t)\leq\alpha(t)+\int_{[a,t)}u(s)\,\mu(\mathrm{d}s),\qquad t\in I.
  83. α α
  84. t I t∈I
  85. α α
  86. μ μ
  87. [ a , t ) | α ( s ) | μ ( d s ) < , t I , \int_{[a,t)}|\alpha(s)|\,\mu(\mathrm{d}s)<\infty,\qquad t\in I,
  88. u u
  89. u ( t ) α ( t ) + [ a , t ) α ( s ) exp ( μ ( I s , t ) ) μ ( d s ) u(t)\leq\alpha(t)+\int_{[a,t)}\alpha(s)\exp\bigl(\mu(I_{s,t})\bigr)\,\mu(% \mathrm{d}s)
  90. t I t∈I
  91. ( s , t ) (s,t)
  92. α α
  93. u u
  94. α α
  95. u u
  96. u u
  97. u u
  98. μ μ
  99. μ μ
  100. [ 0 , 1 ] [0, 1]
  101. u ( 0 ) = 0 u(0)=0
  102. u ( t ) = 1 / t u(t)=1/t
  103. t t∈
  104. ( 0 , 1 ] (0,1]
  105. α α
  106. α α
  107. u u
  108. μ μ
  109. μ μ
  110. β β
  111. u ( t ) α ( t ) + a t α ( s ) β ( s ) exp ( s t β ( r ) d r ) d s , t I . u(t)\leq\alpha(t)+\int_{a}^{t}\alpha(s)\beta(s)\exp\biggl(\int_{s}^{t}\beta(r)% \,\mathrm{d}r\biggr)\,\mathrm{d}s,\qquad t\in I.
  112. α α
  113. β β
  114. μ μ
  115. c c
  116. u ( t ) α ( t ) + c a t α ( s ) exp ( c ( t - s ) ) d s , t I . u(t)\leq\alpha(t)+c\int_{a}^{t}\alpha(s)\exp\bigl(c(t-s)\bigr)\,\mathrm{d}s,% \qquad t\in I.
  117. α α
  118. u ( t ) α ( t ) + c α ( t ) a t exp ( c ( t - s ) ) d s = α ( t ) exp ( c ( t - a ) ) , t I . u(t)\leq\alpha(t)+c\alpha(t)\int_{a}^{t}\exp\bigl(c(t-s)\bigr)\,\mathrm{d}s=% \alpha(t)\exp(c(t-a)),\qquad t\in I.
  119. n n
  120. n n
  121. n n
  122. u ( t ) α ( t ) + [ a , t ) α ( s ) k = 0 n - 1 μ k ( A k ( s , t ) ) μ ( d s ) + R n ( t ) u(t)\leq\alpha(t)+\int_{[a,t)}\alpha(s)\sum_{k=0}^{n-1}\mu^{\otimes k}(A_{k}(s% ,t))\,\mu(\mathrm{d}s)+R_{n}(t)
  123. R n ( t ) := [ a , t ) u ( s ) μ n ( A n ( s , t ) ) μ ( d s ) , t I , R_{n}(t):=\int_{[a,t)}u(s)\mu^{\otimes n}(A_{n}(s,t))\,\mu(\mathrm{d}s),\qquad t% \in I,
  124. A n ( s , t ) = { ( s 1 , , s n ) I s , t n s 1 < s 2 < < s n } , n 1 , A_{n}(s,t)=\{(s_{1},\ldots,s_{n})\in I_{s,t}^{n}\mid s_{1}<s_{2}<\cdots<s_{n}% \},\qquad n\geq 1,
  125. n n
  126. μ 0 ( A 0 ( s , t ) ) := 1. \mu^{\otimes 0}(A_{0}(s,t)):=1.
  127. n = 0 n=0
  128. n n
  129. n + 1 n+1
  130. u u
  131. R n ( t ) [ a , t ) α ( s ) μ n ( A n ( s , t ) ) μ ( d s ) + R ~ n ( t ) R_{n}(t)\leq\int_{[a,t)}\alpha(s)\mu^{\otimes n}(A_{n}(s,t))\,\mu(\mathrm{d}s)% +\tilde{R}_{n}(t)
  132. R ~ n ( t ) := [ a , t ) ( [ a , q ) u ( s ) μ ( d s ) ) μ n ( A n ( q , t ) ) μ ( d q ) , t I . \tilde{R}_{n}(t):=\int_{[a,t)}\biggl(\int_{[a,q)}u(s)\,\mu(\mathrm{d}s)\biggr)% \mu^{\otimes n}(A_{n}(q,t))\,\mu(\mathrm{d}q),\qquad t\in I.
  133. R ~ n ( t ) = [ a , t ) u ( s ) ( s , t ) μ n ( A n ( q , t ) ) μ ( d q ) = μ n + 1 ( A n + 1 ( s , t ) ) μ ( d s ) = R n + 1 ( t ) , t I . \tilde{R}_{n}(t)=\int_{[a,t)}u(s)\underbrace{\int_{(s,t)}\mu^{\otimes n}(A_{n}% (q,t))\,\mu(\mathrm{d}q)}_{=\,\mu^{\otimes n+1}(A_{n+1}(s,t))}\,\mu(\mathrm{d}% s)=R_{n+1}(t),\qquad t\in I.
  134. n + 1 n+1
  135. n n
  136. I I
  137. μ n ( A n ( s , t ) ) ( μ ( I s , t ) ) n n ! \mu^{\otimes n}(A_{n}(s,t))\leq\frac{\bigl(\mu(I_{s,t})\bigr)^{n}}{n!}
  138. t I t∈I
  139. n = 0 n=0
  140. n 1 n≥1
  141. A n , σ ( s , t ) = { ( s 1 , , s n ) I s , t n s σ ( 1 ) < s σ ( 2 ) < < s σ ( n ) } . A_{n,\sigma}(s,t)=\{(s_{1},\ldots,s_{n})\in I_{s,t}^{n}\mid s_{\sigma(1)}<s_{% \sigma(2)}<\cdots<s_{\sigma(n)}\}.
  142. σ S n A n , σ ( s , t ) I s , t n . \bigcup_{\sigma\in S_{n}}A_{n,\sigma}(s,t)\subset I_{s,t}^{n}.
  143. σ S n μ n ( A n , σ ( s , t ) ) μ n ( I s , t n ) = ( μ ( I s , t ) ) n . \sum_{\sigma\in S_{n}}\mu^{\otimes n}(A_{n,\sigma}(s,t))\leq\mu^{\otimes n}% \bigl(I_{s,t}^{n}\bigr)=\bigl(\mu(I_{s,t})\bigr)^{n}.
  144. n n
  145. μ μ
  146. n ! n!
  147. t I t∈I
  148. { ( s 1 , , s n ) I s , t n s i = s j } \{(s_{1},\ldots,s_{n})\in I_{s,t}^{n}\mid s_{i}=s_{j}\}
  149. n n
  150. μ μ
  151. I s , t n σ S n A n , σ ( s , t ) 1 i < j n { ( s 1 , , s n ) I s , t n s i = s j } , I_{s,t}^{n}\subset\bigcup_{\sigma\in S_{n}}A_{n,\sigma}(s,t)\cup\bigcup_{1\leq i% <j\leq n}\{(s_{1},\ldots,s_{n})\in I_{s,t}^{n}\mid s_{i}=s_{j}\},
  152. n n
  153. | R n ( t ) | ( μ ( I a , t ) ) n n ! [ a , t ) | u ( s ) | μ ( d s ) , t I . |R_{n}(t)|\leq\frac{\bigl(\mu(I_{a,t})\bigr)^{n}}{n!}\int_{[a,t)}|u(s)|\,\mu(% \mathrm{d}s),\qquad t\in I.
  154. u u
  155. lim n R n ( t ) = 0 , t I . \lim_{n\to\infty}R_{n}(t)=0,\qquad t\in I.
  156. k = 0 n - 1 μ k ( A k ( s , t ) ) k = 0 n - 1 ( μ ( I s , t ) ) k k ! exp ( μ ( I s , t ) ) \sum_{k=0}^{n-1}\mu^{\otimes k}(A_{k}(s,t))\leq\sum_{k=0}^{n-1}\frac{\bigl(\mu% (I_{s,t})\bigr)^{k}}{k!}\leq\exp\bigl(\mu(I_{s,t})\bigr)
  157. I I
  158. α α
  159. u u
  160. t I t∈I
  161. k = 0 n - 1 μ k ( A k ( s , t ) ) = k = 0 n - 1 ( μ ( I s , t ) ) k k ! exp ( μ ( I s , t ) ) as n \sum_{k=0}^{n-1}\mu^{\otimes k}(A_{k}(s,t))=\sum_{k=0}^{n-1}\frac{\bigl(\mu(I_% {s,t})\bigr)^{k}}{k!}\to\exp\bigl(\mu(I_{s,t})\bigr)\qquad\,\text{as }n\to\infty
  162. α α

Guanidine_nitrate.html

  1. 1.7 k N x s k g 1.7kN\ x\ \tfrac{s}{kg}

Guiding_center.html

  1. m m
  2. q q
  3. B B
  4. ω c = | q | B / m . \omega_{c}=|q|B/m.\,\!
  5. v v_{\perp}
  6. ρ L = v / ω c . \rho_{\operatorname{L}}=v_{\perp}/\omega_{c}.\,\!
  7. F \vec{F}
  8. v f = 1 q F × B B 2 . \vec{v}_{f}=\frac{1}{q}\frac{\vec{F}\times\vec{B}}{B^{2}}.
  9. v g = m q g × B B 2 \vec{v}_{g}=\frac{m}{q}\frac{\vec{g}\times\vec{B}}{B^{2}}
  10. E × B \vec{E}\times\vec{B}
  11. v E = E × B B 2 \vec{v}_{E}=\frac{\vec{E}\times\vec{B}}{B^{2}}
  12. v E = ( 1 + 1 4 ρ L 2 2 ) E × B B 2 \vec{v}_{E}=\left(1+\frac{1}{4}\rho_{L}^{2}\nabla^{2}\right)\frac{\vec{E}% \times\vec{B}}{B^{2}}
  13. K = 1 2 m v 2 K_{\|}=\frac{1}{2}mv_{\|}^{2}
  14. K = 1 2 m v 2 K_{\perp}=\frac{1}{2}mv_{\perp}^{2}
  15. v B = K q B B × B B 2 \vec{v}_{\nabla B}=\frac{K_{\perp}}{qB}\frac{\vec{B}\times\nabla B}{B^{2}}
  16. v R = 2 K q B R c × B R c 2 B \vec{v}_{R}=\frac{2K_{\|}}{qB}\frac{\vec{R}_{c}\times\vec{B}}{R_{c}^{2}B}
  17. R c \vec{R}_{c}
  18. v inertial = v ω c b × d b d t , \vec{v}_{\rm inertial}=\frac{v_{\|}}{\omega_{c}}\,\vec{b}\times\frac{d\vec{b}}% {dt},
  19. b = B / B \vec{b}=\vec{B}/B
  20. v ω c b × [ b t + ( v E b ) ] . \frac{v_{\|}}{\omega_{c}}\,\vec{b}\times\left[\frac{\partial\vec{b}}{\partial t% }+(\vec{v}_{E}\cdot\nabla\vec{b})\right].
  21. v R + v B = 2 K + K q B R c × B R c 2 B \vec{v}_{R}+\vec{v}_{\nabla B}=\frac{2K_{\|}+K_{\perp}}{qB}\frac{\vec{R}_{c}% \times\vec{B}}{R_{c}^{2}B}
  22. 2 K + K 2K_{\|}+K_{\perp}
  23. 2 k B T 2k_{B}T
  24. k B T / 2 k_{B}T/2
  25. K K_{\|}
  26. k B T k_{B}T
  27. K K_{\perp}
  28. B \nabla B
  29. × B = 0 \nabla\times\vec{B}=0
  30. × B = 1 r r ( r B θ ) z ^ = 0 \nabla\times\vec{B}=\frac{1}{r}\frac{\partial}{\partial r}\left(rB_{\theta}% \right)\hat{z}=0
  31. r B θ rB_{\theta}
  32. B = - B R c R c 2 \nabla B=-B\frac{\vec{R}_{c}}{R_{c}^{2}}
  33. v B = - K q B × R c R c 2 B 2 \vec{v}_{\nabla B}=-\frac{K_{\perp}}{q}\frac{\vec{B}\times\vec{R}_{c}}{R_{c}^{% 2}B^{2}}
  34. v p = m q B 2 d E d t \vec{v}_{p}=\frac{m}{qB^{2}}\frac{d\vec{E}}{dt}
  35. v D = - p × B q n B 2 \vec{v}_{D}=-\frac{\nabla p\times\vec{B}}{qnB^{2}}

Gupta–Bleuler_formalism.html

  1. k \vec{k}
  2. k a ; ϵ μ | k b ; ϵ ν = ( - η μ ν ) 1 2 | k a | δ ( k a - k b ) \langle\vec{k}_{a};\epsilon_{\mu}|\vec{k}_{b};\epsilon_{\nu}\rangle=(-\eta_{% \mu\nu}){1\over 2|\vec{k}_{a}|}\delta(\vec{k}_{a}-\vec{k}_{b})
  3. 1 2 | k a | {1\over 2|\vec{k}_{a}|}
  4. k ϵ = 0 k\cdot\epsilon=0
  5. A A
  6. μ μ A = 0 \partial^{\mu}\partial_{\mu}A=0
  7. χ | μ A μ | ψ = 0 \langle\chi|\partial^{\mu}A_{\mu}|\psi\rangle=0
  8. μ A μ = 0 \partial^{\mu}A_{\mu}=0
  9. χ | O | ψ \langle\chi|O|\psi\rangle
  10. A A

Gyromagnetic_ratio.html

  1. f = γ 2 π B . f=\frac{\gamma}{2\pi}B.
  2. γ = q 2 m \gamma=\frac{q}{2m}
  3. μ = I A = q v 2 π r × π r 2 = q 2 m × m v r = q 2 m L . \mu=IA=\frac{qv}{2\pi r}\times\pi r^{2}=\frac{q}{2m}\times mvr=\frac{q}{2m}L.
  4. | γ e | = | - e | 2 m e g e = g e μ B / , |\gamma_{\mathrm{e}}|=\frac{|-e|}{2m_{\mathrm{e}}}g_{\mathrm{e}}=g_{\mathrm{e}% }\mu_{\mathrm{B}}/\hbar,
  5. g e = 2 ( 1 + α 2 π + ) , g_{e}=2(1+\frac{\alpha}{2\pi}+\cdots),
  6. α \alpha
  7. g e = 2.0023193043617 ( 15 ) . ~{}g_{\mathrm{e}}=2.0023193043617(15).
  8. | γ e | = 1.760 859 708 ( 39 ) × 10 11 rad s T \left|\gamma_{\mathrm{e}}\right|=1.760\,859\,708(39)\times 10^{11}\,\mathrm{\ % \frac{rad}{s\cdot T}}
  9. | γ e 2 π | = 28 024.952 66 ( 62 ) MHz T . \left|\frac{\gamma_{\mathrm{e}}}{2\pi}\right|=28\,024.952\,66(62)\mathrm{\ % \frac{MHz}{T}}.
  10. γ n = e 2 m p g n = g n μ N / , \gamma_{n}=\frac{e}{2m_{p}}g_{n}=g_{n}\mu_{\mathrm{N}}/\hbar,
  11. μ N \mu_{\mathrm{N}}
  12. g n g_{n}
  13. γ n \gamma_{n}
  14. γ n / ( 2 π ) \gamma_{n}/(2\pi)

H2.html

  1. H 2 H^{2}
  2. h 2 h^{2}

Hadamard_three-circle_theorem.html

  1. f ( z ) f(z)
  2. r 1 | z | r 3 . r_{1}\leq\left|z\right|\leq r_{3}.
  3. M ( r ) M(r)
  4. | f ( z ) | |f(z)|
  5. | z | = r . |z|=r.
  6. log M ( r ) \log M(r)
  7. log ( r ) . \log(r).
  8. f ( z ) f(z)
  9. c z λ cz^{\lambda}
  10. λ \lambda
  11. c c
  12. log M ( r ) \log M(r)
  13. log ( r ) . \log(r).
  14. log ( r 3 r 1 ) log M ( r 2 ) log ( r 3 r 2 ) log M ( r 1 ) + log ( r 2 r 1 ) log M ( r 3 ) \log\left(\frac{r_{3}}{r_{1}}\right)\log M(r_{2})\leq\log\left(\frac{r_{3}}{r_% {2}}\right)\log M(r_{1})+\log\left(\frac{r_{2}}{r_{1}}\right)\log M(r_{3})
  15. r 1 < r 2 < r 3 . r_{1}<r_{2}<r_{3}.

Hagen_number.html

  1. Hg = - 1 ρ d p d x L 3 ν 2 \mathrm{Hg}=-\frac{1}{\rho}\frac{\mathrm{d}p}{\mathrm{d}x}\frac{L^{3}}{\nu^{2}}
  2. d p d x \frac{\mathrm{d}p}{\mathrm{d}x}
  3. d p d x = ρ g β Δ T , \frac{\mathrm{d}p}{\mathrm{d}x}=\rho g\beta\Delta T,
  4. Hg = - 1 ρ d p d x L 3 δ 2 \mathrm{Hg}=-\frac{1}{\rho}\frac{\mathrm{d}p}{\mathrm{d}x}\frac{L^{3}}{\delta^% {2}}
  5. δ \delta

Hagenbach-Bischoff_quota.html

  1. total votes total seats + 1 \frac{\mbox{total}~{}\;\mbox{votes}~{}}{\mbox{total}~{}\;\mbox{seats}~{}+1}
  2. 100 2 + 1 = 33 + 1 3 \frac{100}{2+1}=33+\frac{1}{3}

Half-integer.html

  1. n + 1 2 n+{1\over 2}
  2. n n
  3. + 1 2 . \mathbb{Z}+{1\over 2}.
  4. 1 2 \frac{1}{2}\mathbb{Z}
  5. V n ( R ) = π n / 2 Γ ( n 2 + 1 ) R n . V_{n}(R)=\frac{\pi^{n/2}}{\Gamma(\frac{n}{2}+1)}R^{n}.
  6. Γ ( 1 2 + n ) = ( 2 n - 1 ) ! ! 2 n π = ( 2 n ) ! 4 n n ! π \Gamma\left(\frac{1}{2}+n\right)=\frac{(2n-1)!!}{2^{n}}\,\sqrt{\pi}={(2n)!% \over 4^{n}n!}\sqrt{\pi}

Halo_nucleus.html

  1. r = r A 1 3 , r=r_{\circ}A^{\frac{1}{3}},
  2. r r_{\circ}

Hamiltonian_vector_field.html

  1. ω : T M T * M , \omega:TM\to T^{*}M,
  2. Ω : T * M T M , Ω = ω - 1 . \Omega:T^{*}M\to TM,\quad\Omega=\omega^{-1}.
  3. d H ( Y ) = ω ( X H , Y ) \mathrm{d}H(Y)=\omega(X_{H},Y)\,
  4. ω = i d q i d p i , \omega=\sum_{i}\mathrm{d}q^{i}\wedge\mathrm{d}p_{i},
  5. \Chi H = ( H p i , - H q i ) = Ω d H , \Chi_{H}=\left(\frac{\partial H}{\partial p_{i}},-\frac{\partial H}{\partial q% ^{i}}\right)=\Omega\,\mathrm{d}H,
  6. Ω = [ 0 I n - I n 0 ] , \Omega=\begin{bmatrix}0&I_{n}\\ -I_{n}&0\\ \end{bmatrix},
  7. d H = [ H q i H p i ] . \mathrm{d}H=\begin{bmatrix}\frac{\partial H}{\partial q^{i}}\\ \frac{\partial H}{\partial p_{i}}\end{bmatrix}.
  8. X H = / q i ; X_{H}=\partial/\partial q^{i};
  9. X H = - / p i ; X_{H}=-\partial/\partial p^{i};
  10. H = 1 / 2 ( p i ) 2 H=1/2\sum(p_{i})^{2}
  11. X H = p i / q i ; X_{H}=\sum p_{i}\partial/\partial q^{i};
  12. H = 1 / 2 a i j q i q j , a i j = a j i H=1/2\sum a_{ij}q^{i}q^{j},a_{ij}=a_{ji}
  13. X H = - a i j q i / p j . X_{H}=-\sum a_{ij}q_{i}\partial/\partial p^{j}.
  14. q ˙ i = H p i \dot{q}^{i}=\frac{\partial H}{\partial p_{i}}
  15. p ˙ i = - H q i . \dot{p}_{i}=-\frac{\partial H}{\partial q^{i}}.
  16. d H , γ ˙ = ω ( X H ( γ ) , X H ( γ ) ) = 0 \langle dH,\dot{\gamma}\rangle=\omega(X_{H}(\gamma),X_{H}(\gamma))=0
  17. X H ω = 0 \mathcal{L}_{X_{H}}\omega=0
  18. { f , g } = ω ( X g , X f ) = d g ( X f ) = X f g \{f,g\}=\omega(X_{g},X_{f})=dg(X_{f})=\mathcal{L}_{X_{f}}g
  19. X \mathcal{L}_{X}
  20. X { f , g } = [ X f , X g ] , X_{\{f,g\}}=[X_{f},X_{g}],
  21. { { f , g } , h } + { { g , h } , f } + { { h , f } , g } = 0 , \{\{f,g\},h\}+\{\{g,h\},f\}+\{\{h,f\},g\}=0,

Hamming_bound.html

  1. A q ( n , d ) \ A_{q}(n,d)
  2. q q
  3. C \ C
  4. n n
  5. d d
  6. q q
  7. n n
  8. 𝒜 q n , \mathcal{A}_{q}^{n}\,\text{,}
  9. 𝒜 q \mathcal{A}_{q}
  10. q q
  11. A q ( n , d ) q n k = 0 t ( n k ) ( q - 1 ) k \ A_{q}(n,d)\leq\frac{q^{n}}{\sum_{k=0}^{t}{\left({{n}\atop{k}}\right)}(q-1)^{% k}}
  12. t = d - 1 2 . t=\left\lfloor\frac{d-1}{2}\right\rfloor.
  13. d d
  14. t = 1 2 ( d - 1 ) t=\left\lfloor\frac{1}{2}(d-1)\right\rfloor
  15. t t
  16. c C c\in C
  17. t t
  18. c c
  19. t t
  20. n n
  21. ( q - 1 ) (q-1)
  22. 𝒜 q n \mathcal{A}_{q}^{n}
  23. m = k = 0 t ( n k ) ( q - 1 ) k m=\begin{matrix}\sum_{k=0}^{t}{\left({{n}\atop{k}}\right)}(q-1)^{k}\end{matrix}
  24. A q ( n , d ) A_{q}(n,d)
  25. C C
  26. 𝒜 q n \mathcal{A}_{q}^{n}
  27. | 𝒜 q n | = q n |\mathcal{A}_{q}^{n}|=q^{n}
  28. A q ( n , d ) × m = A q ( n , d ) × k = 0 t ( n k ) ( q - 1 ) k q n . A_{q}(n,d)\times m=A_{q}(n,d)\times\begin{matrix}\sum_{k=0}^{t}{\left({{n}% \atop{k}}\right)}(q-1)^{k}\end{matrix}\leq q^{n}.
  29. A q ( n , d ) q n k = 0 t ( n k ) ( q - 1 ) k . A_{q}(n,d)\leq\frac{q^{n}}{\begin{matrix}\sum_{k=0}^{t}{\left({{n}\atop{k}}% \right)}(q-1)^{k}\end{matrix}}.
  30. A q ( n , d ) A_{q}(n,d)
  31. 𝒜 q n \mathcal{A}_{q}^{n}
  32. 𝒜 q n \mathcal{A}_{q}^{n}
  33. t = 1 2 ( d - 1 ) t\,=\,\left\lfloor\frac{1}{2}(d-1)\right\rfloor
  34. 𝒜 q n \scriptstyle\mathcal{A}_{q}^{n}

Hanbury_Brown_and_Twiss_effect.html

  1. ρ 0 π - π + \rho^{0}\rightarrow\pi^{-}\pi^{+}
  2. ω \omega
  3. ϕ \phi
  4. i 1 = E 2 sin 2 ( ω t ) i_{1}=E^{2}\sin^{2}(\omega t)\,
  5. i 2 = E 2 sin 2 ( ω t + ϕ ) = E 2 ( sin ( ω t ) cos ( ϕ ) + sin ( ϕ ) cos ( ω t ) ) 2 i_{2}=E^{2}\sin^{2}(\omega t+\phi)=E^{2}(\sin(\omega t)\cos(\phi)+\sin(\phi)% \cos(\omega t))^{2}\,
  6. i 1 i 2 = lim T E 4 T 0 T sin 2 ( ω t ) ( sin ( ω t ) cos ( ϕ ) + sin ( ϕ ) cos ( ω t ) ) 2 d t \langle i_{1}i_{2}\rangle=\lim_{T\rightarrow\infty}\frac{E^{4}}{T}\int^{T}_{0}% \sin^{2}(\omega t)(\sin(\omega t)\cos(\phi)+\sin(\phi)\cos(\omega t))^{2}\,dt
  7. = E 4 4 + E 4 8 cos ( 2 ϕ ) , =\frac{E^{4}}{4}+\frac{E^{4}}{8}\cos(2\phi),
  8. Δ i = i - i \Delta i=i-\langle i\rangle
  9. i \langle i\rangle
  10. Δ i 1 Δ i 2 = ( i 1 - i 1 ) ( i 2 - i 2 ) = i 1 i 2 - i 1 i 2 - i 2 i 1 + i 1 i 2 \langle\Delta i_{1}\Delta i_{2}\rangle=\langle(i_{1}-\langle i_{1}\rangle)(i_{% 2}-\langle i_{2}\rangle)\rangle=\langle i_{1}i_{2}\rangle-\langle i_{1}\langle i% _{2}\rangle\rangle-\langle i_{2}\langle i_{1}\rangle\rangle+\langle i_{1}% \rangle\langle i_{2}\rangle
  11. = i 1 i 2 - i 1 i 2 , =\langle i_{1}i_{2}\rangle-\langle i_{1}\rangle\langle i_{2}\rangle,
  12. E 2 / 2 E^{2}/2
  13. Δ i 1 Δ i 2 = E 4 8 cos ( 2 ϕ ) , \langle\Delta i_{1}\Delta i_{2}\rangle=\frac{E^{4}}{8}\cos(2\phi),
  14. E 2 / 2 E^{2}/2
  15. sin 2 ( ω t ) \sin^{2}(\omega t)
  16. i 1 = i r i_{1}=i_{r}
  17. i 2 = i t = i - i r i_{2}=i_{t}=i-i_{r}\,
  18. Δ i 1 Δ i 2 = - E 4 8 cos ( 2 ϕ ) , \langle\Delta i_{1}\Delta i_{2}\rangle=-\frac{E^{4}}{8}\cos(2\phi),
  19. g ( 2 ) ( τ ) g^{(2)}(\tau)
  20. a a
  21. b b
  22. A A
  23. B B
  24. a a
  25. A A
  26. b b
  27. B B
  28. a a
  29. B B
  30. b b
  31. A A
  32. A | a B | b \langle A|a\rangle\langle B|b\rangle
  33. B | a A | b \langle B|a\rangle\langle A|b\rangle
  34. a a
  35. b b
  36. A B AB

Hankel_transform.html

  1. F ν ( k ) = 0 f ( r ) J ν ( k r ) r d r F_{\nu}(k)=\int_{0}^{\infty}f(r)J_{\nu}(kr)\,r\operatorname{d}\!r
  2. J ν J_{\nu}
  3. ν \nu
  4. ν - 1 2 \nu\geq-\frac{1}{2}
  5. f ( r ) = 0 F ν ( k ) J ν ( k r ) k d k f(r)=\int_{0}^{\infty}F_{\nu}(k)J_{\nu}(kr)k\operatorname{d}\!k
  6. 0 | f ( r ) | r 1 2 d r < . \int_{0}^{\infty}|f(r)|\,r^{\frac{1}{2}}\operatorname{d}\!r<\infty.
  7. f ( r ) = ( 1 + r ) - 3 / 2 f(r)=(1+r)^{-3/2}
  8. h ν ( k ) = 0 g ( r ) J ν ( k r ) k r d r h_{\nu}(k)=\int_{0}^{\infty}g(r)J_{\nu}(kr)\,\sqrt{kr}\operatorname{d}\!r
  9. If g ( r ) = f ( r ) r then h ν ( k ) = F ν ( k ) k . \,\text{If }g(r)=f(r)\sqrt{r}\,\text{ then }h_{\nu}(k)=F_{\nu}(k)\sqrt{k}.
  10. g ( r ) = 0 h ν ( k ) J ν ( k r ) k r d k g(r)=\int_{0}^{\infty}h_{\nu}(k)J_{\nu}(kr)\,\sqrt{kr}\operatorname{d}\!k
  11. 0 | g ( r ) | d r < \int_{0}^{\infty}|g(r)|\operatorname{d}\!r<\infty
  12. 0 J ν ( k r ) J ν ( k r ) r d r = δ ( k - k ) k , k , k > 0. \int_{0}^{\infty}J_{\nu}(kr)J_{\nu}(k^{\prime}r)r\operatorname{d}\!r=\frac{% \delta(k-k^{\prime})}{k},\qquad k,k^{\prime}>0.
  13. 0 f ( r ) g ( r ) r d r = 0 F ν ( k ) G ν ( k ) k d k . \int_{0}^{\infty}f(r)g(r)r\operatorname{d}\!r=\int_{0}^{\infty}F_{\nu}(k)G_{% \nu}(k)k\operatorname{d}\!k.
  14. 0 | f ( r ) | 2 r d r = 0 | F ν ( k ) | 2 k d k , \int_{0}^{\infty}|f(r)|^{2}r\operatorname{d}\!r=\int_{0}^{\infty}|F_{\nu}(k)|^% {2}k\operatorname{d}\!k,
  15. 2 2
  16. 2 2
  17. f ( 𝐫 ) f(\mathbf{r})
  18. 𝐫 \mathbf{r}
  19. F ( 𝐤 ) = f ( 𝐫 ) e i 𝐤 𝐫 d 𝐫 . F(\mathbf{k})=\iint f(\mathbf{r})e^{i\mathbf{k}\cdot\mathbf{r}}\operatorname{d% }\!\mathbf{r}.
  20. ( r , θ ) (r,θ)
  21. 𝐤 \mathbf{k}
  22. θ = 0 θ=0
  23. F ( 𝐤 ) = r = 0 θ = 0 2 π f ( r , θ ) e i k r cos ( θ ) r d θ d r F(\mathbf{k})=\int_{r=0}^{\infty}\int_{\theta=0}^{2\pi}f(r,\theta)e^{ikr\cos(% \theta)}\,r\operatorname{d}\!\theta\operatorname{d}\!r
  24. θ θ
  25. 𝐤 \mathbf{k}
  26. 𝐫 \mathbf{r}
  27. f f
  28. θ θ
  29. f ( r ) f(r)
  30. θ θ
  31. F ( 𝐤 ) = F ( k ) = 2 π 0 f ( r ) J 0 ( k r ) r d r F(\mathbf{k})=F(k)=2\pi\int_{0}^{\infty}f(r)J_{0}(kr)r\operatorname{d}\!r
  32. 2 π
  33. f ( r ) f(r)
  34. f ( 𝐫 ) = 1 ( 2 π ) 2 F ( 𝐤 ) e - i 𝐤 𝐫 d 𝐤 = 1 2 π 0 F ( k ) J 0 ( k r ) k d k f(\mathbf{r})=\frac{1}{(2\pi)^{2}}\iint F(\mathbf{k})e^{-i\mathbf{k}\cdot% \mathbf{r}}\operatorname{d}\!\mathbf{k}=\frac{1}{2\pi}\int_{0}^{\infty}F(k)J_{% 0}(kr)k\operatorname{d}\!k
  35. f ( r ) f(r)
  36. 1 2 π 1\frac{2}{π}
  37. F ( k ) F(k)
  38. n n
  39. n n
  40. F ( 𝐤 ) = f ( 𝐫 ) e - i 𝐤 𝐫 d n 𝐫 F(\mathbf{k})=\int f(\mathbf{r})e^{-i\mathbf{k}\cdot\mathbf{r}}d^{n}\mathbf{r}
  41. f f
  42. k n / 2 - 1 F ( k ) = ( 2 π ) n / 2 0 r n / 2 - 1 f ( r ) J n / 2 - 1 ( k r ) r d r k^{n/2-1}F(k)=(2\pi)^{n/2}\int_{0}^{\infty}r^{n/2-1}f(r)J_{n/2-1}(kr)r\,dr
  43. f f
  44. f ( r , θ ) = m = - f m ( r ) e i m θ , f(r,\theta)=\sum_{m=-\infty}^{\infty}f_{m}(r)e^{im\theta},
  45. 𝐤 \mathbf{k}
  46. θ = 0 θ=0
  47. F ( 𝐤 ) = 0 r d r 0 2 π d θ f ( r , θ ) e i k r cos ( θ - θ k ) = m 0 r d r 0 2 π d θ f m ( r ) e i m θ e i k r cos ( θ - θ k ) = m e i m θ k 0 r d r f m ( r ) 0 2 π d φ e i m φ e i k r cos φ φ = θ - θ k = m e i m θ k 0 r d r f m ( r ) 2 π i m J m ( k r ) = 2 π m i m e i m θ k 0 f m ( r ) J m ( k r ) r d r . = 2 π m i m e i m θ k F m ( k ) \begin{aligned}\displaystyle F(\mathbf{k})&\displaystyle=\int_{0}^{\infty}r% \operatorname{d}\!r\,\int_{0}^{2\pi}\operatorname{d}\!\theta\,f(r,\theta)e^{% ikr\cos(\theta-\theta_{k})}\\ &\displaystyle=\sum_{m}\int_{0}^{\infty}r\operatorname{d}\!r\,\int_{0}^{2\pi}% \operatorname{d}\!\theta\,f_{m}(r)e^{im\theta}e^{ikr\cos(\theta-\theta_{k})}\\ &\displaystyle=\sum_{m}e^{im\theta_{k}}\int_{0}^{\infty}r\operatorname{d}\!r\,% f_{m}(r)\int_{0}^{2\pi}\operatorname{d}\!\varphi\,e^{im\varphi}e^{ikr\cos% \varphi}&&\displaystyle\varphi=\theta-\theta_{k}\\ &\displaystyle=\sum_{m}e^{im\theta_{k}}\int_{0}^{\infty}r\operatorname{d}\!r\,% f_{m}(r)2\pi i^{m}J_{m}(kr)\\ &\displaystyle=2\pi\sum_{m}i^{m}e^{im\theta_{k}}\int_{0}^{\infty}f_{m}(r)J_{m}% (kr)r\operatorname{d}\!r.\\ &\displaystyle=2\pi\sum_{m}i^{m}e^{im\theta_{k}}F_{m}(k)\end{aligned}
  48. m m
  49. R R
  50. f m ( r ) = r m t 0 f m t ( 1 - ( r R ) 2 ) t , 0 r R . f_{m}(r)=r^{m}\sum_{t\geq 0}f_{mt}\left(1-\left(\tfrac{r}{R}\right)^{2}\right)% ^{t},\qquad 0\leq r\leq R.
  51. F ( 𝐤 ) = 2 π m i m e i m θ k t f m t 0 R r m ( 1 - ( r R ) 2 ) t J m ( k r ) r d r ( * ) = 2 π m i m e i m θ k R m + 2 t f m t 0 1 x m ( 1 - x 2 ) t J m ( k x R ) x d x x = r R = 2 π m i m e i m θ k R m + 2 t f m t t ! 2 t ( k R ) 1 + t J m + t + 1 ( k R ) . \begin{aligned}\displaystyle F(\mathbf{k})&\displaystyle=2\pi\sum_{m}i^{m}e^{% im\theta_{k}}\sum_{t}f_{mt}\int_{0}^{R}r^{m}\left(1-\left(\tfrac{r}{R}\right)^% {2}\right)^{t}J_{m}(kr)r\operatorname{d}\!r&&\displaystyle(*)\\ &\displaystyle=2\pi\sum_{m}i^{m}e^{im\theta_{k}}R^{m+2}\sum_{t}f_{mt}\int_{0}^% {1}x^{m}(1-x^{2})^{t}J_{m}(kxR)x\operatorname{d}\!x&&\displaystyle x=\tfrac{r}% {R}\\ &\displaystyle=2\pi\sum_{m}i^{m}e^{im\theta_{k}}R^{m+2}\sum_{t}f_{mt}\frac{t!2% ^{t}}{(kR)^{1+t}}J_{m+t+1}(kR).\end{aligned}
  52. A A
  53. F F
  54. H H
  55. F A = H . FA=H.
  56. 1 1
  57. f ( r ) f(r)\,
  58. F 0 ( k ) F_{0}(k)\,
  59. 1 1\,
  60. δ ( k ) k \tfrac{\delta(k)}{k}
  61. 1 r \tfrac{1}{r}
  62. 1 k \tfrac{1}{k}
  63. r r\,
  64. - 1 k 3 -\tfrac{1}{k^{3}}
  65. r 3 r^{3}\,
  66. 9 k 5 \tfrac{9}{k^{5}}
  67. r m r^{m}\,
  68. 2 m + 1 Γ ( m 2 + 1 ) k m + 2 Γ ( - m 2 ) , - 2 < ( m ) < - 1 2 \frac{2^{m+1}\Gamma\left(\tfrac{m}{2}+1\right)}{k^{m+2}\Gamma\left(-\tfrac{m}{% 2}\right)},\qquad-2<\Re(m)<-\tfrac{1}{2}
  69. 1 r 2 + z 2 \frac{1}{\sqrt{r^{2}+z^{2}}}\,
  70. e - k | z | k \frac{e^{-k|z|}}{k}
  71. 1 r 2 + z 2 \frac{1}{r^{2}+z^{2}}
  72. K 0 ( k z ) z 𝐂 K_{0}(kz)\qquad z\in\mathbf{C}
  73. e i a r r \tfrac{e^{iar}}{r}
  74. i a 2 - k 2 a > 0 , k < a . \frac{i}{\sqrt{a^{2}-k^{2}}}\qquad a>0,k<a.
  75. \,
  76. 1 k 2 - a 2 a > 0 , k > a . \frac{1}{\sqrt{k^{2}-a^{2}}}\qquad a>0,k>a.
  77. e - 1 2 a 2 r 2 e^{-\frac{1}{2}a^{2}r^{2}}
  78. 1 a 2 e - k 2 2 a 2 \frac{1}{a^{2}}e^{-\tfrac{k^{2}}{2a^{2}}}
  79. 1 r J 0 ( l r ) e - s r \frac{1}{r}J_{0}(lr)e^{-sr}\,
  80. 2 π ( k + l ) 2 + s 2 K ( 4 k l ( k + l ) 2 + s 2 ) \frac{2}{\pi\sqrt{(k+l)^{2}+s^{2}}}K\bigg(\sqrt{\frac{4kl}{(k+l)^{2}+s^{2}}}\bigg)
  81. - r 2 f ( r ) -r^{2}f(r)\,
  82. d 2 F 0 d k 2 + 1 k d F 0 d k \frac{\operatorname{d}^{2}\!F_{0}}{\operatorname{d}\!k^{2}}+\frac{1}{k}\frac{% \operatorname{d}\!F_{0}}{\operatorname{d}\!k}
  83. f ( r ) f(r)\,
  84. F ν ( k ) F_{\nu}(k)\,
  85. r s r^{s}\,
  86. 2 s + 1 k s + 2 Γ ( 1 2 ( 2 + ν + s ) ) Γ ( 1 2 ( ν - s ) ) \frac{2^{s+1}}{k^{s+2}}\frac{\Gamma\left(\tfrac{1}{2}(2+\nu+s)\right)}{\Gamma(% \tfrac{1}{2}(\nu-s))}
  87. r ν - 2 s Γ ( s , r 2 h ) r^{\nu-2s}\Gamma\left(s,r^{2}h\right)\,
  88. 1 2 ( k 2 ) 2 s - ν - 2 γ ( 1 - s + ν , k 2 4 h ) \tfrac{1}{2}\left(\tfrac{k}{2}\right)^{2s-\nu-2}\gamma\left(1-s+\nu,\tfrac{k^{% 2}}{4h}\right)\,
  89. e - r 2 r ν U ( a , b , r 2 ) e^{-r^{2}}r^{\nu}U\left(a,b,r^{2}\right)\,
  90. Γ ( 2 + ν - b ) 2 Γ ( 2 + ν - b + a ) ( k 2 ) ν e 1 - k 2 4 F 1 ( a , 2 + a - b + ν , k 2 4 ) \frac{\Gamma(2+\nu-b)}{2\Gamma(2+\nu-b+a)}\left(\tfrac{k}{2}\right)^{\nu}e^{-% \frac{k^{2}}{4}}\,_{1}F_{1}\left(a,2+a-b+\nu,\tfrac{k^{2}}{4}\right)
  91. r n J μ ( l r ) e - s r r^{n}J_{\mu}(lr)e^{-sr}\,
  92. - r 2 f ( r ) -r^{2}f(r)\,
  93. d 2 F ν d k 2 + 1 k d F ν d k - ν 2 k 2 F ν \frac{\operatorname{d}^{2}\!F_{\nu}}{\operatorname{d}\!k^{2}}+\frac{1}{k}\frac% {\operatorname{d}\!F_{\nu}}{\operatorname{d}\!k}-\frac{\nu^{2}}{k^{2}}F_{\nu}
  94. K ( z ) K(z)
  95. d 2 F 0 d k 2 + 1 k d F 0 d k \frac{\operatorname{d}^{2}\!F_{0}}{\operatorname{d}\!k^{2}}+\frac{1}{k}\frac{% \operatorname{d}\!F_{0}}{\operatorname{d}\!k}
  96. ( k , θ ) (k,θ)
  97. R n m ( r ) = ( - 1 ) n - m 2 0 J n + 1 ( k ) J m ( k r ) d k R_{n}^{m}(r)=(-1)^{\frac{n-m}{2}}\int_{0}^{\infty}J_{n+1}(k)J_{m}(kr)% \operatorname{d}\!k
  98. n m 0 n−m≥0

Hardy's_theorem.html

  1. f f
  2. R R
  3. f f
  4. I ( r ) = 1 2 π 0 2 π | f ( r e i θ ) | d θ I(r)=\frac{1}{2\pi}\int_{0}^{2\pi}\!\left|f(re^{i\theta})\right|\,d\theta
  5. 0 < r < R , 0<r<R,

Hardy–Littlewood_circle_method.html

  1. f ( z ) = a n z n f(z)=\sum a_{n}z^{n}
  2. I n = f ( z ) z - ( n + 1 ) d z = 2 π i a n I_{n}=\int f(z)z^{-(n+1)}\,dz=2\pi ia_{n}
  3. ζ = exp ( 2 π i r s ) . \zeta\ =\exp\left(\frac{2\pi ir}{s}\right).

Harnack's_principle.html

  1. u 1 ( z ) u_{1}(z)
  2. u 2 ( z ) u_{2}(z)
  3. G G
  4. u 1 ( z ) u 2 ( z ) u_{1}(z)\leq u_{2}(z)\leq...
  5. G G
  6. lim n u n ( z ) \lim_{n\to\infty}u_{n}(z)
  7. G G
  8. G G
  9. u ( z ) = lim n u n ( z ) u(z)=\lim_{n\to\infty}u_{n}(z)
  10. G G

Harrod–Domar_model.html

  1. Y = f ( K ) \ Y=f(K)
  2. d Y d K = c d Y d K = Y K \ \frac{dY}{dK}=c\Rightarrow\frac{dY}{dK}=\frac{Y}{K}
  3. f ( 0 ) = 0 \ f(0)=0
  4. s Y = S = I \ sY=S=I
  5. Δ K = I - δ K \ \Delta\ K=I-\delta\ K
  6. c = d Y d K = Y ( t + 1 ) - Y ( t ) K ( t ) + s Y ( t ) - δ K ( t ) - K ( t ) \displaystyle c=\frac{dY}{dK}=\frac{Y(t+1)-Y(t)}{K(t)+sY(t)-\delta\ K(t)-K(t)}
  7. Y ˙ \ \dot{Y}
  8. Y = c K l o g ( Y ) = l o g ( c ) + l o g ( K ) . \ Y=cK\Rightarrow log(Y)=log(c)+log(K).
  9. d log ( Y ) d t = d log ( K ) d t Y ˙ Y = K ˙ K . \ \frac{d\log(Y)}{dt}=\frac{d\log(K)}{dt}\Rightarrow\frac{\dot{Y}}{Y}=\frac{% \dot{K}}{K}.
  10. K ˙ K = I K - δ = s Y K - δ \ \frac{\dot{K}}{K}=\frac{I}{K}-\delta\ =s\frac{Y}{K}-\delta
  11. Y ˙ Y = s c - δ \ \Rightarrow\frac{\dot{Y}}{Y}=sc-\delta

Hash_join.html

  1. r r
  2. R R
  3. r r
  4. S S
  5. S S
  6. S S
  7. S S
  8. R R
  9. S S
  10. k k

Haugh_unit.html

  1. H U = 100 * l o g ( h - 1.7 w 0.37 + 7.6 ) HU=100*log(h-1.7w^{0.37}+7.6)

Havriliak–Negami_relaxation.html

  1. ε ^ ( ω ) = ε + Δ ε ( 1 + ( i ω τ ) α ) β , \hat{\varepsilon}(\omega)=\varepsilon_{\infty}+\frac{\Delta\varepsilon}{(1+(i% \omega\tau)^{\alpha})^{\beta}},
  2. ε \varepsilon_{\infty}
  3. Δ ε = ε s - ε \Delta\varepsilon=\varepsilon_{s}-\varepsilon_{\infty}
  4. ε s \varepsilon_{s}
  5. τ \tau
  6. α \alpha
  7. β \beta
  8. β = 1 \beta=1
  9. α = 1 \alpha=1
  10. ε \varepsilon^{\prime}
  11. ε ′′ \varepsilon^{\prime\prime}
  12. ε ^ ( ω ) = ε ( ω ) - i ε ′′ ( ω ) \hat{\varepsilon}(\omega)=\varepsilon^{\prime}(\omega)-i\varepsilon^{\prime% \prime}(\omega)
  13. ε ( ω ) = Δ ε ( 1 + 2 ( ω τ ) α cos ( π α / 2 ) + ( ω τ ) 2 α ) - β / 2 cos ( β ϕ ) \varepsilon^{\prime}(\omega)=\Delta\varepsilon\left(1+2(\omega\tau)^{\alpha}% \cos(\pi\alpha/2)+(\omega\tau)^{2\alpha}\right)^{-\beta/2}\cos(\beta\phi)
  14. ε ′′ ( ω ) = Δ ε ( 1 + 2 ( ω τ ) α cos ( π α / 2 ) + ( ω τ ) 2 α ) - β / 2 sin ( β ϕ ) \varepsilon^{\prime\prime}(\omega)=\Delta\varepsilon\left(1+2(\omega\tau)^{% \alpha}\cos(\pi\alpha/2)+(\omega\tau)^{2\alpha}\right)^{-\beta/2}\sin(\beta\phi)
  15. ϕ = arctan ( ( ω τ ) α sin ( π α / 2 ) 1 + ( ω τ ) α cos ( π α / 2 ) ) \phi=\arctan\left({(\omega\tau)^{\alpha}\sin(\pi\alpha/2)\over 1+(\omega\tau)^% {\alpha}\cos(\pi\alpha/2)}\right)
  16. ω max = ( sin ( π α 2 ( β + 1 ) ) sin ( π α β 2 ( β + 1 ) ) ) 1 / α τ - 1 \omega_{\rm max}=\left({\sin\left({\pi\alpha\over 2(\beta+1)}\right)\over\sin% \left({\pi\alpha\beta\over 2(\beta+1)}\right)}\right)^{1/\alpha}\tau^{-1}
  17. ε ^ ( ω ) - ϵ Δ ε = τ D = 0 1 1 + i ω τ D g ( ln τ D ) d ln τ D {\hat{\varepsilon}(\omega)-\epsilon_{\infty}\over\Delta\varepsilon}=\int_{\tau% _{D}=0}^{\infty}{1\over 1+i\omega\tau_{D}}g(\ln\tau_{D})d\ln\tau_{D}
  18. g ( ln τ D ) = 1 π ( τ D / τ ) α β sin ( β θ ) ( ( τ D / τ ) 2 α + 2 ( τ D / τ ) α cos ( π α ) + 1 ) β / 2 g(\ln\tau_{D})={1\over\pi}{(\tau_{D}/\tau)^{\alpha\beta}\sin(\beta\theta)\over% ((\tau_{D}/\tau)^{2\alpha}+2(\tau_{D}/\tau)^{\alpha}\cos(\pi\alpha)+1)^{\beta/% 2}}
  19. θ = arctan ( sin ( π α ) ( τ D / τ ) α + cos ( π α ) ) \theta=\arctan\left({\sin(\pi\alpha)\over(\tau_{D}/\tau)^{\alpha}+\cos(\pi% \alpha)}\right)
  20. θ = arctan ( sin ( π α ) ( τ D / τ ) α + cos ( π α ) ) + π \theta=\arctan\left({\sin(\pi\alpha)\over(\tau_{D}/\tau)^{\alpha}+\cos(\pi% \alpha)}\right)+\pi
  21. ln τ D = ln τ + Ψ ( β ) + Eu α \langle\ln\tau_{D}\rangle=\ln\tau+{\Psi(\beta)+{\rm Eu}\over\alpha}
  22. Ψ \Psi
  23. Eu {\rm Eu}
  24. ε ^ ( ω ) \hat{\varepsilon}(\omega)
  25. X ( t ) = ε δ ( t ) + Δ ε τ ( t τ ) α β - 1 E α , α β β ( - ( t / τ ) α ) , X(t)=\varepsilon_{\infty}\delta(t)+\frac{\Delta\varepsilon}{\tau}\left(\frac{t% }{\tau}\right)^{\alpha\beta-1}E_{\alpha,\alpha\beta}^{\beta}(-(t/\tau)^{\alpha% }),
  26. δ ( t ) \delta(t)
  27. E α , β γ ( z ) = 1 Γ ( γ ) k = 0 Γ ( γ + k ) z k k ! Γ ( α k + β ) E_{\alpha,\beta}^{\gamma}(z)=\frac{1}{\Gamma(\gamma)}\sum_{k=0}^{\infty}\frac{% \Gamma(\gamma+k)z^{k}}{k!\Gamma(\alpha k+\beta)}
  28. E α , β γ ( z ) E_{\alpha,\beta}^{\gamma}(z)

Heat-affected_zone.html

  1. Q = ( V × I × 60 S × 1000 ) × Efficiency Q=\left(\frac{V\times I\times 60}{S\times 1000}\right)\times\mathrm{Efficiency}

Heat_capacity_ratio.html

  1. C P C_{P}
  2. C V C_{V}
  3. γ \gamma
  4. κ \kappa
  5. k k
  6. γ = C P C V = c P c V \gamma=\frac{C_{P}}{C_{V}}=\frac{c_{P}}{c_{V}}
  7. C C
  8. c c
  9. P P
  10. V V
  11. C V Δ T C_{V}\Delta T
  12. Δ T \Delta T
  13. C V C_{V}
  14. C P C_{P}
  15. C P C_{P}
  16. C V C_{V}
  17. C P C_{P}
  18. C V C_{V}
  19. P d V PdV
  20. C V C_{V}
  21. H = C P T H=C_{P}T
  22. U = C V T U=C_{V}T
  23. γ = H U \gamma=\frac{H}{U}
  24. γ \gamma
  25. R R
  26. C P = γ n R γ - 1 and C V = n R γ - 1 C_{P}=\frac{\gamma nR}{\gamma-1}\qquad\mbox{and}~{}\qquad C_{V}=\frac{nR}{% \gamma-1}
  27. n n
  28. C V C_{V}
  29. C P C_{P}
  30. C V C_{V}
  31. C V = C P - n R C_{V}=C_{P}-nR
  32. γ \gamma
  33. f f
  34. γ = 1 + 2 f or f = 2 γ - 1 \gamma\ =1+\frac{2}{f}\qquad\mbox{or}~{}\qquad f=\frac{2}{\gamma-1}
  35. γ = 5 3 1.67 \gamma\ =\frac{5}{3}\approx 1.67
  36. γ = 7 5 = 1.4 \gamma=\frac{7}{5}=1.4
  37. γ \gamma
  38. C P C_{P}
  39. C V C_{V}
  40. C P C_{P}
  41. C V + n R C_{V}+nR
  42. P V PV
  43. γ \gamma
  44. C P - C V = n R C_{P}-C_{V}=nR
  45. C P C V \frac{C_{P}}{C_{V}}
  46. C V C_{V}
  47. C P - C V = - T ( V T ) P 2 ( V P ) T = - T ( P T ) V 2 ( P V ) T C_{P}-C_{V}\ =\ -T\frac{{\left({\frac{\partial V}{\partial T}}\right)_{P}^{2}}% }{\left(\frac{\partial V}{\partial P}\right)_{T}}\ =\ -T\frac{{\left({\frac{% \partial P}{\partial T}}\right)}_{V}^{2}}{\left(\frac{\partial P}{\partial V}% \right)_{T}}
  48. C P C_{P}
  49. C V C_{V}
  50. C V C_{V}
  51. P V γ = constant PV^{\gamma}=\,\text{constant}
  52. P P
  53. V V

Heat_current.html

  1. d Q d t \frac{dQ}{dt}
  2. Q Q
  3. t t
  4. Q t = - k S T d S \frac{\partial Q}{\partial t}=-k\oint_{S}{\overrightarrow{\nabla}T\cdot\,% \overrightarrow{dS}}
  5. . Q t . \big.\frac{\partial Q}{\partial t}\big.
  6. d S \overrightarrow{dS}
  7. . Δ Q Δ t = - k A Δ T Δ x \big.\frac{\Delta Q}{\Delta t}=-kA\frac{\Delta T}{\Delta x}
  8. Δ T \Delta T
  9. Δ x \Delta x
  10. W = σ A T 4 W=\sigma\cdot A\cdot T^{4}
  11. σ \sigma
  12. A A
  13. T T

Heat_transfer_coefficient.html

  1. h = q Δ T h{=\frac{q}{\Delta T}}
  2. Q ˙ \dot{Q}
  3. < 0.2 W / c m 2 <0.2W/cm^{2}
  4. 1 / ( U A ) = 1 / ( h 1 A 1 ) + d x w / ( k A ) + 1 / ( h 2 A 2 ) 1/(U\cdot A)=1/(h_{1}\cdot A_{1})+dx_{w}/(k\cdot A)+1/(h_{2}\cdot A_{2})
  5. U U
  6. A A
  7. A 1 A_{1}
  8. A 2 A_{2}
  9. k k
  10. h h
  11. d x w dx_{w}
  12. 1 / U = 1 / h 1 + d x w / k + 1 / h 2 1/U=1/h_{1}+dx_{w}/k+1/h_{2}
  13. U = 1 / ( 1 / h 1 + d x w / k + 1 / h 2 ) U=1/(1/h_{1}+dx_{w}/k+1/h_{2})
  14. d x w dx_{w}
  15. T f T_{f}
  16. T s T_{s}
  17. T {{T}_{\infty}}
  18. T f = T s + T 2 {{T}_{f}}=\frac{{{T}_{s}}+{{T}_{\infty}}}{2}
  19. h = k L ( 0.825 + 0.387 Ra L 1 / 6 ( 1 + ( 0.492 / Pr ) 9 / 16 ) 8 / 27 ) 2 {\mathrm{h}}\ =\frac{k}{L}\left({0.825+\frac{0.387\mathrm{Ra}_{L}^{1/6}}{\left% (1+(0.492/\mathrm{Pr})^{9/16}\right)^{8/27}}}\right)^{2}
  20. R a L < 10 9 Ra_{L}<10^{9}
  21. h = k L ( 0.68 + 0.67 Ra L 1 / 4 ( 1 + ( 0.492 / Pr ) 9 / 16 ) 4 / 9 ) Ra L 10 9 {\mathrm{h}}\ =\frac{k}{L}\left(0.68+\frac{0.67\mathrm{Ra}_{L}^{1/4}}{\left(1+% (0.492/\mathrm{Pr})^{9/16}\right)^{4/9}}\right)\,\quad\mathrm{Ra}_{L}\leq 10^{9}
  22. D D
  23. D L 35 G r L 1 4 \frac{D}{L}\geq\frac{35}{Gr_{L}^{\frac{1}{4}}}
  24. G r L Gr_{L}
  25. h = k 0.54 Ra L 1 / 4 L 10 5 Ra L 2 × 10 7 {\mathrm{h}}\ =\frac{k0.54\mathrm{Ra}_{L}^{1/4}}{L}\,\quad 10^{5}\leq\mathrm{% Ra}_{L}\leq 2\times 10^{7}
  26. h = k 0.14 Ra L 1 / 3 L 2 × 10 7 Ra L 3 × 10 10 {\mathrm{h}}\ =\frac{k0.14\mathrm{Ra}_{L}^{1/3}}{L}\,\quad 2\times 10^{7}\leq% \mathrm{Ra}_{L}\leq 3\times 10^{10}
  27. h = k 0.27 Ra L 1 / 4 L 3 × 10 5 Ra L 10 10 {\mathrm{h}}\ =\frac{k0.27\mathrm{Ra}_{L}^{1/4}}{L}\,\quad 3\times 10^{5}\leq% \mathrm{Ra}_{L}\leq 10^{10}
  28. 60 o 60^{o}
  29. g g
  30. 10 - 5 < Ra D < 10 12 10^{-5}<\mathrm{Ra}_{D}<10^{12}
  31. h = k D ( 0.6 + 0.387 Ra D 1 / 6 ( 1 + ( 0.559 / Pr ) 9 / 16 ) 8 / 27 ) 2 {\mathrm{h}}\ =\frac{k}{D}\left({0.6+\frac{0.387\mathrm{Ra}_{D}^{1/6}}{\left(1% +(0.559/\mathrm{Pr})^{9/16}\,\right)^{8/27}\,}}\right)^{2}
  32. 1 Ra D 10 5 1\leq\mathrm{Ra}_{D}\leq 10^{5}
  33. Nu D = 2 + 0.43 Ra D 1 / 4 {\mathrm{Nu}}_{D}\ =2+0.43\mathrm{Ra}_{D}^{1/4}
  34. D D
  35. μ b {\mu}_{b}
  36. μ w {\mu}_{w}
  37. 𝑁𝑢 D = 1.86 ( 𝑅𝑒 𝑃𝑟 ) 1 3 ( D L ) 1 3 ( μ b μ w ) 0.14 \mathit{Nu}_{D}={1.86}\cdot{{{\left(\mathit{Re}\cdot\mathit{Pr}\right)}^{{}^{1% }\!\!\diagup\!\!{}_{3}\;}}}{{\left(\frac{D}{L}\right)}^{{}^{1}\!\!\diagup\!\!{% }_{3}\;}}{{\left(\frac{{{\mu}_{b}}}{{{\mu}_{w}}}\right)}^{0.14}}
  38. h d k = 0.023 ( j d μ ) 0.8 ( μ c p k ) n {hd\over k}={0.023}\,\left({jd\over\mu}\right)^{0.8}\,\left({\mu c_{p}\over k}% \right)^{n}
  39. d d
  40. k k
  41. μ \mu
  42. j j
  43. c p c_{p}
  44. x x
  45. L L
  46. Δ T s a t = 22.5 q 0.5 exp ( - P / 8.7 ) \Delta T_{sat}=22.5\cdot{q}^{0.5}\exp(-P/8.7)
  47. Δ T s a t \Delta T_{sat}
  48. h w a l l = k x h_{wall}={k\over x}
  49. h w a l l = 2 k d i ln ( d o / d i ) h_{wall}={2k\over{d_{i}\ln(d_{o}/d_{i})}}
  50. h = h 1 + h 2 + h=h_{1}+h_{2}+\cdots
  51. 1 h = 1 h 1 + 1 h 2 + {1\over h}={1\over h_{1}}+{1\over h_{2}}+\dots
  52. q = ( 1 1 h + t k ) A Δ T q=\left({1\over{{1\over h}+{t\over k}}}\right)\cdot A\cdot\Delta T
  53. Δ T \Delta T
  54. U U
  55. U U
  56. q = U A Δ T L M q=UA\Delta T_{LM}
  57. q q
  58. U U
  59. A A
  60. Δ T L M \Delta T_{LM}
  61. 1 U A = 1 h A + R \frac{1}{UA}=\sum\frac{1}{hA}+\sum R
  62. R = x k A R=\frac{x}{k\cdot A}
  63. 1 U f P \frac{1}{U_{f}P}
  64. 1 U P + R f H P H + R f C P C \frac{1}{UP}+\frac{R_{fH}}{P_{H}}+\frac{R_{fC}}{P_{C}}
  65. U f U_{f}
  66. W m 2 K \frac{W}{m^{2}K}
  67. P P
  68. m m
  69. U U
  70. W m 2 K \frac{W}{m^{2}K}
  71. R f C R_{fC}
  72. m 2 K W \frac{m^{2}K}{W}
  73. R f H R_{fH}
  74. m 2 K W \frac{m^{2}K}{W}
  75. P C P_{C}
  76. m m
  77. P H P_{H}
  78. m m
  79. P P
  80. U P UP
  81. R f R_{f}
  82. d f k f \frac{d_{f}}{k_{f}}
  83. d f d_{f}
  84. m m
  85. k f k_{f}
  86. W m K \frac{W}{mK}

Heaviside_condition.html

  1. R = G = 0 \scriptstyle R=G=0
  2. G C = R L . \frac{G}{C}=\frac{R}{L}.
  3. V in V out = e γ x \frac{V_{\mathrm{in}}}{V_{\mathrm{out}}}=e^{\gamma x}
  4. x x
  5. γ = α + j β \gamma=\alpha+j\beta\,
  6. v = ω β v=\frac{\omega}{\beta}
  7. γ 2 = ( α + j β ) 2 = ( R + j ω L ) ( G + j ω C ) \gamma^{2}=(\alpha+j\beta)^{2}=(R+j\omega L)(G+j\omega C)\,
  8. ( A + j ω B ) 2 \scriptstyle(A+j\omega B)^{2}
  9. ( R + j ω L ) \scriptstyle(R+j\omega L)
  10. ( G + j ω C ) \scriptstyle(G+j\omega C)
  11. R G = j ω L j ω C \frac{R}{G}=\frac{j\omega L}{j\omega C}
  12. α = R G \alpha=\sqrt{RG}
  13. β = ω L C \beta=\omega\sqrt{LC}
  14. v = 1 L C v=\frac{1}{\sqrt{LC}}
  15. Z 0 = R + j ω L G + j ω C Z_{0}=\sqrt{\frac{R+j\omega L}{G+j\omega C}}
  16. Z 0 = L C , Z_{0}=\sqrt{\frac{L}{C}},
  17. Z 0 = L / C \scriptstyle Z_{0}=\sqrt{L/C}
  18. R = 0 , G = 0 \scriptstyle R=0,\ G=0
  19. G C R L . \frac{G}{C}\ll\frac{R}{L}.

Heawood_conjecture.html

  1. γ ( g ) = 7 + 1 + 48 g 2 , \gamma(g)=\left\lfloor\frac{7+\sqrt{1+48g}}{2}\right\rfloor,
  2. x \left\lfloor x\right\rfloor
  3. γ ( χ ) = 7 + 49 - 24 χ 2 . \gamma(\chi)=\left\lfloor\frac{7+\sqrt{49-24\chi}}{2}\right\rfloor.

Heawood_graph.html

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

Heckscher–Ohlin_model.html

  1. A = K 1 / 3 L 2 / 3 A={{K}^{1/3}}{{L}^{2/3}}
  2. F = K 1 / 2 L 1 / 2 F={{K}^{1/2}}{{L}^{1/2}}
  3. 𝐅 𝐂 = 𝐕 𝐂 - s C 𝐕 \mathbf{F_{C}}=\mathbf{V_{C}}-s_{C}\mathbf{V}
  4. 𝐅 𝐂 \mathbf{F_{C}}
  5. c c
  6. 𝐕 𝐂 \mathbf{V_{C}}
  7. c c
  8. s C s_{C}
  9. c c
  10. 𝐕 \mathbf{V}

Hellmann–Feynman_theorem.html

  1. d E d λ = ψ λ * d H ^ λ d λ ψ λ d V , \frac{\mathrm{d}E}{\mathrm{d}{\lambda}}=\int{\psi^{*}_{\lambda}\frac{\mathrm{d% }{\hat{H}_{\lambda}}}{\mathrm{d}{\lambda}}\psi_{\lambda}\ \mathrm{d}V},
  2. H ^ λ \hat{H}_{\lambda}
  3. λ \lambda\,
  4. ψ λ \psi_{\lambda}\,
  5. λ \lambda\,
  6. E E\,
  7. d V \mathrm{d}V\,
  8. H ^ λ | ψ λ = E λ | ψ λ , \hat{H}_{\lambda}|\psi_{\lambda}\rangle=E_{\lambda}|\psi_{\lambda}\rangle,
  9. ψ λ | ψ λ = 1 d d λ ψ λ | ψ λ = 0. \langle\psi_{\lambda}|\psi_{\lambda}\rangle=1\Rightarrow\frac{\mathrm{d}}{% \mathrm{d}\lambda}\langle\psi_{\lambda}|\psi_{\lambda}\rangle=0.
  10. d E λ d λ \displaystyle\frac{\mathrm{d}E_{\lambda}}{\mathrm{d}\lambda}
  11. E [ ψ , λ ] = ψ | H ^ λ | ψ ψ | ψ . E[\psi,\lambda]=\frac{\langle\psi|\hat{H}_{\lambda}|\psi\rangle}{\langle\psi|% \psi\rangle}.
  12. E λ = E [ ψ λ , λ ] , E_{\lambda}=E[\psi_{\lambda},\lambda],
  13. ψ λ \psi_{\lambda}
  14. δ E [ ψ , λ ] δ ψ ( x ) | ψ = ψ λ = 0. \left.\frac{\delta E[\psi,\lambda]}{\delta\psi(x)}\right|_{\psi=\psi_{\lambda}% }=0.
  15. d E λ d λ = E [ ψ λ , λ ] λ + δ E [ ψ , λ ] δ ψ ( x ) d ψ λ ( x ) d λ d x . \frac{dE_{\lambda}}{d\lambda}=\frac{\partial E[\psi_{\lambda},\lambda]}{% \partial\lambda}+\int\frac{\delta E[\psi,\lambda]}{\delta\psi(x)}\frac{d\psi_{% \lambda}(x)}{d\lambda}dx.
  16. H ^ = T ^ + U ^ - i = 1 N α = 1 M Z α | 𝐫 i - 𝐑 α | + α M β > α M Z α Z β | 𝐑 α - 𝐑 β | . \hat{H}=\hat{T}+\hat{U}-\sum_{i=1}^{N}\sum_{\alpha=1}^{M}\frac{Z_{\alpha}}{|% \mathbf{r}_{i}-\mathbf{R}_{\alpha}|}+\sum_{\alpha}^{M}\sum_{\beta>\alpha}^{M}% \frac{Z_{\alpha}Z_{\beta}}{|\mathbf{R}_{\alpha}-\mathbf{R}_{\beta}|}.
  17. F X γ = - E X γ = - ψ | H ^ X γ | ψ . F_{X_{\gamma}}=-\frac{\partial E}{\partial X_{\gamma}}=-\bigg\langle\psi\bigg|% \frac{\partial\hat{H}}{\partial X_{\gamma}}\bigg|\psi\bigg\rangle.
  18. H ^ X γ = X γ ( - i = 1 N α = 1 M Z α | 𝐫 i - 𝐑 α | + α M β > α M Z α Z β | 𝐑 α - 𝐑 β | ) , = Z γ i = 1 N x i - X γ | 𝐫 i - 𝐑 γ | 3 - Z γ α γ M Z α X α - X γ | 𝐑 α - 𝐑 γ | 3 . \begin{aligned}\displaystyle\frac{\partial\hat{H}}{\partial X_{\gamma}}&% \displaystyle=\frac{\partial}{\partial X_{\gamma}}\left(-\sum_{i=1}^{N}\sum_{% \alpha=1}^{M}\frac{Z_{\alpha}}{|\mathbf{r}_{i}-\mathbf{R}_{\alpha}|}+\sum_{% \alpha}^{M}\sum_{\beta>\alpha}^{M}\frac{Z_{\alpha}Z_{\beta}}{|\mathbf{R}_{% \alpha}-\mathbf{R}_{\beta}|}\right),\\ &\displaystyle=Z_{\gamma}\sum_{i=1}^{N}\frac{x_{i}-X_{\gamma}}{|\mathbf{r}_{i}% -\mathbf{R}_{\gamma}|^{3}}-Z_{\gamma}\sum_{\alpha\neq\gamma}^{M}Z_{\alpha}% \frac{X_{\alpha}-X_{\gamma}}{|\mathbf{R}_{\alpha}-\mathbf{R}_{\gamma}|^{3}}.% \end{aligned}
  19. F X γ = - Z γ ( d 𝐫 ρ ( 𝐫 ) x - X γ | 𝐫 - 𝐑 γ | 3 - α γ M Z α X α - X γ | 𝐑 α - 𝐑 γ | 3 ) . F_{X_{\gamma}}=-Z_{\gamma}\left(\int\mathrm{d}\mathbf{r}\ \rho(\mathbf{r})% \frac{x-X_{\gamma}}{|\mathbf{r}-\mathbf{R}_{\gamma}|^{3}}-\sum_{\alpha\neq% \gamma}^{M}Z_{\alpha}\frac{X_{\alpha}-X_{\gamma}}{|\mathbf{R}_{\alpha}-\mathbf% {R}_{\gamma}|^{3}}\right).
  20. H ^ l = - 2 2 μ r 2 ( d d r ( r 2 d d r ) - l ( l + 1 ) ) - Z e 2 r , \hat{H}_{l}=-\frac{\hbar^{2}}{2\mu r^{2}}\left(\frac{\mathrm{d}}{\mathrm{d}r}% \left(r^{2}\frac{\mathrm{d}}{\mathrm{d}r}\right)-l(l+1)\right)-\frac{Ze^{2}}{r},
  21. H ^ l l = 2 2 μ r 2 ( 2 l + 1 ) . \frac{\partial\hat{H}_{l}}{\partial l}=\frac{\hbar^{2}}{2\mu r^{2}}(2l+1).
  22. 1 r 2 \frac{1}{r^{2}}
  23. ψ n l | 1 r 2 | ψ n l = 2 μ 2 1 2 l + 1 ψ n l | H ^ l l | ψ n l = 2 μ 2 1 2 l + 1 E n l = 2 μ 2 1 2 l + 1 E n n n l = 2 μ 2 1 2 l + 1 Z 2 μ e 4 2 n 3 = Z 2 μ 2 e 4 4 n 3 ( l + 1 / 2 ) . \begin{aligned}\displaystyle\bigg\langle\psi_{nl}\bigg|\frac{1}{r^{2}}\bigg|% \psi_{nl}\bigg\rangle&\displaystyle=\frac{2\mu}{\hbar^{2}}\frac{1}{2l+1}\bigg% \langle\psi_{nl}\bigg|\frac{\partial\hat{H}_{l}}{\partial l}\bigg|\psi_{nl}% \bigg\rangle\\ &\displaystyle=\frac{2\mu}{\hbar^{2}}\frac{1}{2l+1}\frac{\partial E_{n}}{% \partial l}\\ &\displaystyle=\frac{2\mu}{\hbar^{2}}\frac{1}{2l+1}\frac{\partial E_{n}}{% \partial n}\frac{\partial n}{\partial l}\\ &\displaystyle=\frac{2\mu}{\hbar^{2}}\frac{1}{2l+1}\frac{Z^{2}\mu e^{4}}{\hbar% ^{2}n^{3}}\\ &\displaystyle=\frac{Z^{2}\mu^{2}e^{4}}{\hbar^{4}n^{3}(l+1/2)}.\end{aligned}
  24. Ψ λ ( t ) | H λ λ | Ψ λ ( t ) = i t Ψ λ ( t ) | Ψ λ ( t ) λ \bigg\langle\Psi_{\lambda}(t)\bigg|\frac{\partial H_{\lambda}}{\partial\lambda% }\bigg|\Psi_{\lambda}(t)\bigg\rangle=i\hbar\frac{\partial}{\partial t}\bigg% \langle\Psi_{\lambda}(t)\bigg|\frac{\partial\Psi_{\lambda}(t)}{\partial\lambda% }\bigg\rangle
  25. i Ψ λ ( t ) t = H λ Ψ λ ( t ) i\hbar\frac{\partial\Psi_{\lambda}(t)}{\partial t}=H_{\lambda}\Psi_{\lambda}(t)
  26. Ψ λ ( t ) | H λ λ | Ψ λ ( t ) = λ Ψ λ ( t ) | H λ | Ψ λ ( t ) - Ψ λ ( t ) λ | H λ | Ψ λ ( t ) - Ψ λ ( t ) | H λ | Ψ λ ( t ) λ = i λ Ψ λ ( t ) | Ψ λ ( t ) t - i Ψ λ ( t ) λ | Ψ λ ( t ) t + i Ψ λ ( t ) t | Ψ λ ( t ) λ = i Ψ λ ( t ) | 2 Ψ λ ( t ) λ t + i Ψ λ ( t ) t | Ψ λ ( t ) λ = i t Ψ λ ( t ) | Ψ λ ( t ) λ \begin{aligned}\displaystyle\bigg\langle\Psi_{\lambda}(t)\bigg|\frac{\partial H% _{\lambda}}{\partial\lambda}\bigg|\Psi_{\lambda}(t)\bigg\rangle&\displaystyle=% \frac{\partial}{\partial\lambda}\langle\Psi_{\lambda}(t)|H_{\lambda}|\Psi_{% \lambda}(t)\rangle-\bigg\langle\frac{\partial\Psi_{\lambda}(t)}{\partial% \lambda}\bigg|H_{\lambda}\bigg|\Psi_{\lambda}(t)\bigg\rangle-\bigg\langle\Psi_% {\lambda}(t)\bigg|H_{\lambda}\bigg|\frac{\partial\Psi_{\lambda}(t)}{\partial% \lambda}\bigg\rangle\\ &\displaystyle=i\hbar\frac{\partial}{\partial\lambda}\bigg\langle\Psi_{\lambda% }(t)\bigg|\frac{\partial\Psi_{\lambda}(t)}{\partial t}\bigg\rangle-i\hbar\bigg% \langle\frac{\partial\Psi_{\lambda}(t)}{\partial\lambda}\bigg|\frac{\partial% \Psi_{\lambda}(t)}{\partial t}\bigg\rangle+i\hbar\bigg\langle\frac{\partial% \Psi_{\lambda}(t)}{\partial t}\bigg|\frac{\partial\Psi_{\lambda}(t)}{\partial% \lambda}\bigg\rangle\\ &\displaystyle=i\hbar\bigg\langle\Psi_{\lambda}(t)\bigg|\frac{\partial^{2}\Psi% _{\lambda}(t)}{\partial\lambda\partial t}\bigg\rangle+i\hbar\bigg\langle\frac{% \partial\Psi_{\lambda}(t)}{\partial t}\bigg|\frac{\partial\Psi_{\lambda}(t)}{% \partial\lambda}\bigg\rangle\\ &\displaystyle=i\hbar\frac{\partial}{\partial t}\bigg\langle\Psi_{\lambda}(t)% \bigg|\frac{\partial\Psi_{\lambda}(t)}{\partial\lambda}\bigg\rangle\end{aligned}

Helly's_theorem.html

  1. n > d n>d
  2. d + 1 d+1
  3. j = 1 n X j . \bigcap_{j=1}^{n}X_{j}\neq\varnothing.
  4. d + 1 d+1
  5. n = d + 2 n=d+2
  6. j = 1 , , n j=1,...,n
  7. A A
  8. p p
  9. p j = 1 n X j . p\in\bigcap_{j=1}^{n}X_{j}.
  10. p X < s u b > j . p∈X<sub>j.

Helly–Bray_theorem.html

  1. g ( x ) d F n ( x ) n g ( x ) d F ( x ) \int_{\mathbb{R}}g(x)\,dF_{n}(x)\quad\xrightarrow[n\to\infty]{}\quad\int_{% \mathbb{R}}g(x)\,dF(x)
  2. S g d P n n S g d P , \int_{S}g\,dP_{n}\quad\xrightarrow[n\to\infty]{}\quad\int_{S}g\,dP,

Helmholtz_coil.html

  1. h h
  2. R R
  3. h = R h=R
  4. 2 B / x 2 = 0 \partial^{2}B/\partial x^{2}=0
  5. 4 B / x 4 \partial^{4}B/\partial x^{4}
  6. h h
  7. 2 B / x 2 \partial^{2}B/\partial x^{2}
  8. x x
  9. x = 0 x=0
  10. x 2 x^{2}
  11. x 4 x^{4}
  12. R / 2 R/2
  13. x = ± R / 2 x=\pm R/2
  14. B = ( 4 5 ) 3 / 2 μ 0 n I R , B={\left(\frac{4}{5}\right)}^{3/2}\frac{\mu_{0}nI}{R},
  15. μ 0 \mu_{0}
  16. 4 π × 10 - 7 T m/A 4\pi\times 10^{-7}\,\text{ T}\cdot\,\text{m/A}
  17. B 1 ( x ) = μ 0 I R 2 2 ( R 2 + x 2 ) 3 / 2 . B_{1}(x)=\frac{\mu_{0}IR^{2}}{2(R^{2}+x^{2})^{3/2}}.
  18. μ 0 \mu_{0}\;
  19. 4 π × 10 - 7 T m/A = 1.257 × 10 - 6 T m/A , 4\pi\times 10^{-7}\,\text{ T}\cdot\,\text{m/A}=1.257\times 10^{-6}\,\text{ T}% \cdot\,\text{m/A},
  20. I I\;
  21. R R\;
  22. x x\;
  23. B 1 ( x ) = μ 0 n I R 2 2 ( R 2 + x 2 ) 3 / 2 . B_{1}(x)=\frac{\mu_{0}nIR^{2}}{2(R^{2}+x^{2})^{3/2}}.
  24. B 1 ( R 2 ) = μ 0 n I R 2 2 ( R 2 + ( R / 2 ) 2 ) 3 / 2 . B_{1}\left(\frac{R}{2}\right)=\frac{\mu_{0}nIR^{2}}{2(R^{2}+(R/2)^{2})^{3/2}}.
  25. B ( R 2 ) \displaystyle B\left(\frac{R}{2}\right)

Hematopoietic_stem_cell.html

  1. ρ = L / M \rho=L/M

Hensel's_lemma.html

  1. p p
  2. p p
  3. p p
  4. f ( x ) f(x)
  5. f ( r ) 0 ( mod p k ) f(r)\equiv 0\;\;(\mathop{{\rm mod}}p^{k})
  6. f ( r ) 0 ( mod p ) f^{\prime}(r)\not\equiv 0\;\;(\mathop{{\rm mod}}p)
  7. f ( s ) 0 ( mod p k + m ) f(s)\equiv 0\;\;(\mathop{{\rm mod}}p^{k+m})
  8. r s ( mod p k ) . r\equiv s\;\;(\mathop{{\rm mod}}p^{k}).
  9. s = r + t p k s=r+tp^{k}
  10. t = - f ( r ) p k ( f ( r ) - 1 ) . t=-\frac{f(r)}{p^{k}}\cdot(f^{\prime}(r)^{-1}).
  11. f ( r ) - 1 f^{\prime}(r)^{-1}
  12. / p m \mathbb{Z}/p^{m}\mathbb{Z}
  13. f ( r ) 0 ( mod p ) f^{\prime}(r)\equiv 0\;\;(\mathop{{\rm mod}}p)
  14. r s ( mod p k ) r\equiv s\;\;(\mathop{{\rm mod}}p^{k})
  15. f ( r + t p k ) f(r+tp^{k})
  16. f ( r + t p k ) = f ( r ) + t p k f ( r ) + O ( p 2 k ) . f(r+tp^{k})=f(r)+tp^{k}\cdot f^{\prime}(r)+O(p^{2k}).
  17. f ( s ) 0 ( mod p k + m ) f(s)\equiv 0\;\;(\mathop{{\rm mod}}p^{k+m})
  18. 0 f ( r + t p k ) f ( r ) + t p k f ( r ) ( mod p k + m ) 0\equiv f(r+tp^{k})\equiv f(r)+tp^{k}\cdot f^{\prime}(r)\;\;(\mathop{{\rm mod}% }p^{k+m})
  19. f ( r ) = z p k f(r)=zp^{k}
  20. 0 ( z + t f ( r ) ) p k ( mod p k + m ) 0\equiv(z+tf^{\prime}(r))p^{k}\;\;(\mathop{{\rm mod}}p^{k+m})
  21. 0 z + t f ( r ) ( mod p m ) . 0\equiv z+tf^{\prime}(r)\;\;(\mathop{{\rm mod}}p^{m}).
  22. / p m \mathbb{Z}/p^{m}\mathbb{Z}
  23. f ( r ) f^{\prime}(r)
  24. f ( r ) f^{\prime}(r)
  25. p m p^{m}
  26. p m p^{m}
  27. p k + m p^{k+m}
  28. f ( s ) f ( r ) 0 ( mod p ) f^{\prime}(s)\equiv f^{\prime}(r)\not\equiv 0\;\;(\mathop{{\rm mod}}p)
  29. f ( x ) 0 ( mod p k ) f(x)\equiv 0\;\;(\mathop{{\rm mod}}p^{k})
  30. f ( r k ) 0 ( mod p ) f^{\prime}(r_{k})\not\equiv 0\;\;(\mathop{{\rm mod}}p)
  31. f ( r ) 0 mod p k f(r)\equiv 0\,\bmod{p^{k}}
  32. f ( r ) 0 mod p , f^{\prime}(r)\equiv 0\,\bmod{p},
  33. s r mod p k f ( s ) f ( r ) mod p k + 1 s\equiv r\,\bmod p^{k}\Rightarrow f(s)\equiv f(r)\,\bmod p^{k+1}
  34. f ( r + t p k ) 0 mod p k + 1 f(r+tp^{k})\equiv 0\,\bmod{p^{k+1}}\,
  35. f ( r ) 0 mod p k + 1 , f(r)\not\equiv 0\,\bmod{p^{k+1}},
  36. f ( r ) 0 mod p k + 1 , f(r)\equiv 0\,\bmod{p^{k+1}},
  37. t f ( r k ) - ( f ( r k ) / p k ) mod p m tf^{\prime}(r_{k})\equiv-(f(r_{k})/p^{k})\,\bmod{p^{m}}\,
  38. - ( f ( r k ) / p k ) / f ( r k ) \ -(f(r_{k})/p^{k})/f^{\prime}(r_{k})
  39. r k + 1 = r k + t p k = r k - f ( r k ) f ( r k ) . r_{k+1}=r_{k}+tp^{k}=r_{k}-\frac{f(r_{k})}{f^{\prime}(r_{k})}.
  40. m = 1 m=1
  41. f ( r ) = r 2 - a 0 mod p f(r)=r^{2}-a\equiv 0\,\bmod{p}
  42. f ( r ) = 2 r 0 mod p f^{\prime}(r)=2r\not\equiv 0\,\bmod{p}
  43. r k + 1 r k mod p k , r k 2 a mod p k . r_{k+1}\equiv r_{k}\,\bmod{p^{k}},\quad r_{k}^{2}\equiv a\,\bmod{p^{k}}.
  44. x 2 - 2 = 0 x^{2}-2=0
  45. r 1 = 3 r_{1}=3
  46. r 2 r_{2}
  47. f ( r 1 ) = 3 2 - 2 = 7 f(r_{1})=3^{2}-2=7
  48. f ( r 1 ) / p 1 = 7 / 7 = 1 f(r_{1})/p^{1}=7/7=1
  49. f ( r 1 ) = 2 r 1 = 6 f^{\prime}(r_{1})=2r_{1}=6
  50. t f ( r 1 ) - ( f ( r 1 ) / p k - 1 ) mod p , tf^{\prime}(r_{1})\equiv-(f(r_{1})/p^{k-1})\,\bmod{p},
  51. t 6 - 1 mod 7 t\cdot 6\equiv-1\,\bmod{7}
  52. t = 1 \Rightarrow t=1
  53. r 2 = r 1 + t p 1 = 3 + 1 7 = 10 = 13 7 . r_{2}=r_{1}+tp^{1}=3+1\cdot 7=10=13_{7}.
  54. 10 2 2 mod 7 2 10^{2}\equiv 2\,\bmod{7^{2}}
  55. r 3 = 108 = 3 + 7 + 2 7 2 = 213 7 r_{3}=108=3+7+2\cdot 7^{2}=213_{7}
  56. 3 + 7 + 2 7 2 + 6 7 3 + 7 4 + 2 7 5 + 7 6 + 2 7 7 + 4 7 8 + . 3+7+2\cdot 7^{2}+6\cdot 7^{3}+7^{4}+2\cdot 7^{5}+7^{6}+2\cdot 7^{7}+4\cdot 7^{% 8}+\cdots.
  57. r 1 = 4 r_{1}=4
  58. 𝔪 A \mathfrak{m}_{A}
  59. f ( x ) A [ x ] f(x)\in A[x]
  60. f ( a ) 0 mod f ( a ) 2 𝔪 f(a)\equiv 0\,\bmod{f^{\prime}(a)^{2}\mathfrak{m}}
  61. f ( b ) = 0 f(b)=0
  62. b a mod f ( a ) 𝔪 . b\equiv a\,\bmod{f^{\prime}(a)\mathfrak{m}}.
  63. f ( a ) 0 mod 𝔪 f(a)\equiv 0\,\bmod{\mathfrak{m}}
  64. b a mod 𝔪 . b\equiv a\,\bmod{\mathfrak{m}}.

Heptatonic_scale.html

  1. 2 ( 4 ! 2 ! 2 ! ) 2 = 2 6 2 = 2 36 = 72 2\cdot\left(\frac{4!}{2!\cdot 2!}\right)^{2}=2\cdot 6^{2}=2\cdot 36=72

Herbert_Scarf.html

  1. ( S , s ) (S,s)
  2. K K
  3. K 0 K\geq 0
  4. f ( x ) f(x)
  5. K K
  6. f ( x ) + a [ f ( x ) - f ( x - b ) b ] f ( x + a ) + K f(x)+a[\frac{f(x)-f(x-b)}{b}]\leq f(x+a)+K
  7. a , b a,\,b
  8. x . x.
  9. 0
  10. K K
  11. ( S , s ) (S,s)
  12. s s
  13. S S
  14. K K
  15. ( S , s ) (S,s)
  16. ( S , s ) (S,s)
  17. N N
  18. S N S\subseteq N
  19. S \mathbb{R}^{S}
  20. S S
  21. S S
  22. S S
  23. V ( S ) V(S)
  24. V ( S ) V(S)
  25. N \mathbb{R}^{N}
  26. x i x_{i}
  27. x V ( S ) x\in V(S)
  28. i S . i\in S.
  29. V ( S ) V(S)
  30. S S
  31. V ( S ) V(S)
  32. x V ( S ) x\in V(S)
  33. y N y\in\mathbb{R}^{N}
  34. y i x i y_{i}\leq x_{i}
  35. i S i\in S
  36. y v ( S ) y\in v(S)
  37. x V ( N ) x\in V(N)
  38. S S
  39. y V ( S ) y\in V(S)
  40. y i > x i y_{i}>x_{i}
  41. i S i\in S
  42. x x
  43. V ( N ) V(N)
  44. Ω \Omega
  45. δ ( S ) \delta(S)
  46. S Ω S\in\Omega
  47. i S Ω δ ( S ) = 1 \sum_{i\in S\in\Omega}\delta(S)=1
  48. i N . i\in N.
  49. N N
  50. Ω \Omega
  51. u u
  52. V ( N ) V(N)
  53. u u
  54. V ( S ) V(S)
  55. S Ω S\in\Omega
  56. A A
  57. C C
  58. n × m n\times m
  59. 𝐀 = [ 1 0 0 a ( 1 , n + 1 ) a ( 1 , m ) 0 1 0 a ( 2 , n + 2 ) a ( 2 , m ) 0 0 1 a ( n , n + 1 ) a ( n , m ) ] \mathbf{A}=\begin{bmatrix}1&0&\cdots&0&a(1,n+1)&\cdots&a(1,m)\\ 0&1&\cdots&0&a(2,n+2)&\cdots&a(2,m)\\ &&\cdots&&&\cdots&\\ 0&0&\cdots&1&a(n,n+1)&\cdots&a(n,m)\end{bmatrix}
  60. 𝐂 = [ c ( 1 , 1 ) c ( 1 , n ) c ( 1 , n + 1 ) c ( 1 , m ) c ( 2 , 1 ) c ( 2 , n ) c ( 2 , n + 2 ) c ( 2 , m ) c ( n , 1 ) c ( n , n ) c ( n , n + 1 ) c ( n , m ) ] \mathbf{C}=\begin{bmatrix}c(1,1)&\cdots&c(1,n)&c(1,n+1)&\cdots&c(1,m)\\ c(2,1)&\cdots&c(2,n)&c(2,n+2)&\cdots&c(2,m)\\ &\cdots&&&\cdots&\\ c(n,1)&\cdots&c(n,n)&c(n,n+1)&\cdots&c(n,m)\end{bmatrix}
  61. A A
  62. C C
  63. i i
  64. c ( i , i ) c(i,i)
  65. c ( i , j ) c(i,j)
  66. C C
  67. n n
  68. k k
  69. n < k m n<k\leq m
  70. c ( i , j ) c ( i , k ) c(i,j)\geq c(i,k)
  71. A A
  72. C C
  73. n × m n\times m
  74. b b
  75. { x A x = b and x 0 } \{x\mid Ax=b\,\mbox{and}~{}\,x\geq 0\}
  76. A x = b Ax=b
  77. x 0 x\geq 0
  78. u i = min c ( i , j ) u_{i}=\min c(i,j)
  79. j j
  80. k k
  81. u i c ( i , k ) u_{i}\geq c(i,k)
  82. i i
  83. f : S n S n f:S^{n}\to S^{n}
  84. S n S^{n}
  85. f ( p * ) = p * f(p^{*})=p^{*}
  86. S n = { x + n i = 1 n x i = 1 } S^{n}=\{x\in\mathbb{R}^{n}_{+}\mid\sum_{i=1}^{n}x_{i}=1\}
  87. S n S^{n}
  88. { 1 , 2 , , n } \{1,2,\cdots,n\}
  89. x x
  90. l ( x ) = min { j f j ( x ) x j > 0 } l(x)=\min\{j\mid f_{j}(x)\leq x_{j}>0\}
  91. x j = 0 x_{j}=0
  92. l ( x ) j . l(x)\neq j.
  93. n n
  94. n = 3 n=3
  95. n . n.
  96. l ( x 1 ) = 1 l(x^{1})=1
  97. l ( x 2 ) = 2 l(x^{2})=2
  98. l ( x 3 ) = 3 l(x^{3})=3
  99. x 1 = ( 1 , 0 , 0 ) x^{1}=(1,0,0)
  100. x 2 = ( 0 , 1 , 0 ) x^{2}=(0,1,0)
  101. x 3 = ( 0 , 0 , 1 ) x^{3}=(0,0,1)
  102. S 3 . S^{3}.
  103. max ( a ( 0 , 1 ) h 1 + a ( 0 , 2 ) h 2 + + a ( 0 , n ) h n ) \displaystyle\max(a(0,1)h_{1}+a(0,2)h_{2}+\cdots+a(0,n)h_{n})
  104. h 1 , h 2 , , h n h_{1},h_{2},\cdots,h_{n}
  105. b 1 , b 2 , , b m b_{1},b_{2},\cdots,b_{m}
  106. h = ( h 1 , h 2 , , h n ) h=(h_{1},h_{2},\cdots,h_{n})
  107. h h
  108. N ( h ) N(h)
  109. N ( h ) = { h } + N ( 0 ) N(h)=\{h\}+N(0)
  110. k N ( h ) k\in N(h)
  111. h N ( k ) . h\in N(k).
  112. N ( h ) N(h)
  113. h . h.
  114. h h
  115. h + k h+k
  116. k N ( h ) k\in N(h^{\prime})
  117. h h
  118. A = ( a ( i , j ) ) A=(a(i,j))

Heterodyne_detection.html

  1. E sig cos ( ω sig t + φ ) E_{\mathrm{sig}}\cos(\omega_{\mathrm{sig}}t+\varphi)\,
  2. E LO cos ( ω LO t ) . E_{\mathrm{LO}}\cos(\omega_{\mathrm{LO}}t).\,
  3. I ( E sig cos ( ω sig t + φ ) + E LO cos ( ω LO t ) ) 2 I\propto\left(E_{\mathrm{sig}}\cos(\omega_{\mathrm{sig}}t+\varphi)+E_{\mathrm{% LO}}\cos(\omega_{\mathrm{LO}}t)\right)^{2}
  4. = E sig 2 2 ( 1 + cos ( 2 ω sig t + 2 φ ) ) =\frac{E_{\mathrm{sig}}^{2}}{2}\left(1+\cos(2\omega_{\mathrm{sig}}t+2\varphi)\right)
  5. + E LO 2 2 ( 1 + cos ( 2 ω LO t ) ) +\frac{E_{\mathrm{LO}}^{2}}{2}(1+\cos(2\omega_{\mathrm{LO}}t))
  6. + E sig E LO [ cos ( ( ω sig + ω LO ) t + φ ) + cos ( ( ω sig - ω LO ) t + φ ) ] +E_{\mathrm{sig}}E_{\mathrm{LO}}\left[\cos((\omega_{\mathrm{sig}}+\omega_{% \mathrm{LO}})t+\varphi)+\cos((\omega_{\mathrm{sig}}-\omega_{\mathrm{LO}})t+% \varphi)\right]
  7. = E sig 2 + E LO 2 2 c o n s t a n t c o m p o n e n t + E sig 2 2 cos ( 2 ω sig t + 2 φ ) + E LO 2 2 cos ( 2 ω LO t ) + E sig E LO cos ( ( ω sig + ω LO ) t + φ ) h i g h f r e q u e n c y c o m p o n e n t =\underbrace{\frac{E_{\mathrm{sig}}^{2}+E_{\mathrm{LO}}^{2}}{2}}_{constant\;% component}+\underbrace{\frac{E_{\mathrm{sig}}^{2}}{2}\cos(2\omega_{\mathrm{sig% }}t+2\varphi)+\frac{E_{\mathrm{LO}}^{2}}{2}\cos(2\omega_{\mathrm{LO}}t)+E_{% \mathrm{sig}}E_{\mathrm{LO}}\cos((\omega_{\mathrm{sig}}+\omega_{\mathrm{LO}})t% +\varphi)}_{high\;frequency\;component}
  8. + E sig E LO cos ( ( ω sig - ω LO ) t + φ ) b e a t c o m p o n e n t . +\underbrace{E_{\mathrm{sig}}E_{\mathrm{LO}}\cos((\omega_{\mathrm{sig}}-\omega% _{\mathrm{LO}})t+\varphi)}_{beat\;component}.
  9. 2 ω sig 2\omega_{\mathrm{sig}}
  10. 2 ω LO 2\omega_{\mathrm{LO}}
  11. ω sig + ω LO \omega_{\mathrm{sig}}+\omega_{\mathrm{LO}}
  12. ω sig - ω LO \omega_{\mathrm{sig}}-\omega_{\mathrm{LO}}
  13. ω LO \omega_{\mathrm{LO}}
  14. ω sig \omega_{\mathrm{sig}}
  15. E sig E_{\mathrm{sig}}
  16. E LO E_{\mathrm{LO}}

Hicksian_demand_function.html

  1. h ( p , u ¯ ) = arg min x i p i x i h(p,\bar{u})=\arg\min_{x}\sum_{i}p_{i}x_{i}
  2. subject to u ( x ) u ¯ {\rm subject~{}to}\ \ u(x)\geq\bar{u}
  3. u ¯ \bar{u}
  4. x i x_{i}
  5. x ( p , w ) x(p,w)
  6. w w
  7. h ( p , u ) = x ( p , e ( p , u ) ) , h(p,u)=x(p,e(p,u)),
  8. e ( p , u ) e(p,u)
  9. h ( p , v ( p , w ) ) = x ( p , w ) , h(p,v(p,w))=x(p,w),
  10. v ( p , w ) v(p,w)
  11. u ( x ) u(x)
  12. h ( p , u ) = p e ( p , u ) . h(p,u)=\nabla_{p}e(p,u).
  13. u ( x ) u(x)
  14. h ( p , u ) h(p,u)
  15. a > 0 a>0
  16. h ( a p , u ) = h ( p , u ) h(ap,u)=h(p,u)
  17. i p i x i \sum_{i}p_{i}x_{i}
  18. i a p i x i \sum_{i}ap_{i}x_{i}
  19. u ( x ) u ¯ u(x)\geq\bar{u}
  20. u ( x ) = u ¯ u(x)=\bar{u}
  21. u ¯ \bar{u}

Hidden_transformation.html

  1. c c
  2. C ( x , y ) C(x,y)
  3. ( 1 , 2 ) (1,2)
  4. ( 2 , 0 ) (2,0)
  5. x x
  6. 1 1
  7. c = ( 1 , 2 ) c=(1,2)
  8. 2 2
  9. c = ( 2 , 0 ) c=(2,0)
  10. y y

High_resolution_electron_energy_loss_spectroscopy.html

  1. Δ E / E \Delta E/E
  2. E i = E s + Δ E E_{i}=E_{s}+\Delta E
  3. k i , | | = k s , | | + q | | + G k_{i,||}=k_{s,||}+q_{||}+G
  4. P + p e I ω t P+pe^{I\omega t}
  5. ϵ \epsilon\rightarrow\infty
  6. ϵ \epsilon
  7. i | p z | f \left\langle i\right|p_{z}\left|f\right\rangle
  8. Δ E M E i \frac{\Delta E_{M}}{E_{i}}
  9. 2 / 5 \sqrt{2/5}

Highly_abundant_number.html

  1. σ ( n ) > σ ( m ) \sigma(n)>\sigma(m)

Hilbert's_fourteenth_problem.html

  1. R := K k [ x 1 , , x n ] . R:=K\cap k[x_{1},\dots,x_{n}]\ .

Hilbert's_irreducibility_theorem.html

  1. f 1 ( X 1 , , X r , Y 1 , , Y s ) , , f n ( X 1 , , X r , Y 1 , , Y s ) f_{1}(X_{1},\ldots,X_{r},Y_{1},\ldots,Y_{s}),\ldots,f_{n}(X_{1},\ldots,X_{r},Y% _{1},\ldots,Y_{s})\,
  2. [ X 1 , , X r , Y 1 , , Y s ] . \mathbb{Q}[X_{1},\ldots,X_{r},Y_{1},\ldots,Y_{s}].\,
  3. f 1 ( a 1 , , a r , Y 1 , , Y s ) , , f n ( a 1 , , a r , Y 1 , , Y s ) f_{1}(a_{1},\ldots,a_{r},Y_{1},\ldots,Y_{s}),\ldots,f_{n}(a_{1},\ldots,a_{r},Y% _{1},\ldots,Y_{s})\,
  4. [ Y 1 , , Y s ] . \mathbb{Q}[Y_{1},\ldots,Y_{s}].\,
  5. r \mathbb{Q}^{r}
  6. n = r = s = 1 n=r=s=1
  7. n = r = s = 1 n=r=s=1
  8. f = f 1 f=f_{1}
  9. E = ( X 1 , , X r ) , E=\mathbb{Q}(X_{1},\ldots,X_{r}),
  10. g ( x ) [ x ] g(x)\in\mathbb{Z}[x]
  11. [ x ] \mathbb{Z}[x]
  12. n = r = s = 1 n=r=s=1
  13. f 1 ( X , Y ) = Y 2 - g ( X ) f_{1}(X,Y)\,=Y^{2}-g(X)

Hilbert's_syzygy_theorem.html

  1. k [ x 1 , , x n ] . k[x_{1},\ldots,x_{n}].

Hilbert's_twenty-first_problem.html

  1. n n
  2. p + 1 p+1
  3. 2 ( n - 1 ) p 2(n-1)p

Hill's_muscle_model.html

  1. ( v + b ) ( F + a ) = b ( F 0 + a ) \left(v+b\right)(F+a)=b(F_{0}+a)
  2. F F
  3. v v
  4. F 0 F_{0}
  5. a a
  6. b = a v 0 / F 0 b=a\cdot v_{0}/F_{0}
  7. v 0 v_{0}
  8. F = 0 F=0
  9. F C E F_{CE}
  10. F S E F_{SE}
  11. F P E F_{PE}
  12. F = F P E + F S E and F C E = F S E . F=F_{PE}+F_{SE}\quad\mathrm{and}\quad F_{CE}=F_{SE}\;.
  13. L L
  14. L C E L_{CE}
  15. L S E L_{SE}
  16. L P E L_{PE}
  17. L = L P E and L = L C E + L S E . L=L_{PE}\quad\mathrm{and}\quad L=L_{CE}+L_{SE}\;.
  18. F D = k ( L ˙ D ) a F_{D}=k(\dot{L}_{D})^{a}
  19. k k
  20. a a

Hille–Yosida_theorem.html

  1. T ( 0 ) = I T(0)=I\quad
  2. T ( s + t ) = T ( s ) T ( t ) , t , s 0. T(s+t)=T(s)\circ T(t),\quad\forall t,s\geq 0.
  3. t T ( t ) x t\mapsto T(t)x
  4. h - 1 ( T ( h ) x - x ) h^{-1}\bigg(T(h)x-x\bigg)
  5. t T ( t ) x . t\mapsto T(t)x.
  6. T ( t ) M e ω t \|T(t)\|\leq M{\rm e}^{\omega t}
  7. ( λ I - A ) - n M ( λ - ω ) n . \|(\lambda I-A)^{-n}\|\leq\frac{M}{(\lambda-\omega)^{n}}.
  8. ( λ I - A ) - 1 1 λ . \|(\lambda I-A)^{-1}\|\leq\frac{1}{\lambda}.

History_of_general_relativity.html

  1. R μ ν = T μ ν R_{\mu\nu}=T_{\mu\nu}\,
  2. R μ ν R_{\mu\nu}
  3. T μ ν T_{\mu\nu}
  4. R μ ν - 1 2 R g μ ν = T μ ν R_{\mu\nu}-{1\over 2}Rg_{\mu\nu}=T_{\mu\nu}
  5. R R
  6. g μ ν g_{\mu\nu}
  7. R μ ν - 1 2 R g μ ν + Λ g μ ν = T μ ν R_{\mu\nu}-{1\over 2}Rg_{\mu\nu}+\Lambda g_{\mu\nu}=T_{\mu\nu}

History_of_science_and_technology_in_the_Indian_subcontinent.html

  1. 10 12 10^{12}
  2. ( 3 , 4 , 5 ) (3,4,5)
  3. ( 5 , 12 , 13 ) (5,12,13)
  4. ( 8 , 15 , 17 ) (8,15,17)
  5. ( 7 , 24 , 25 ) (7,24,25)
  6. ( 12 , 35 , 37 ) (12,35,37)
  7. arctan x \arctan x
  8. π \pi
  9. π \pi
  10. 104348 / 33215 104348/33215
  11. π \pi
  12. 3.141592653 3.141592653
  13. π \pi

History_of_special_relativity.html

  1. 1 / 1 - v 2 / c 2 \scriptstyle{1/\sqrt{1-{v^{2}}/{c^{2}}}}
  2. t = t - v x / c 2 \scriptstyle{t^{\prime}=t-vx/c^{2}}
  3. v / c \scriptstyle{v/c}
  4. t = t - v x / c 2 \scriptstyle{t^{\prime}=t-vx/c^{2}}
  5. v / c \scriptstyle{v/c}
  6. v / c \scriptstyle{v/c}
  7. m = ( 4 / 3 ) E / c 2 \scriptstyle{m=(4/3)E/c^{2}}
  8. m \scriptstyle{m}
  9. E \scriptstyle{E}
  10. m = E / c 2 \scriptstyle{m=E/c^{2}}
  11. E = m c 2 \scriptstyle{E=mc^{2}}
  12. e / m \scriptstyle{e/m}
  13. e \scriptstyle{e}
  14. m \scriptstyle{m}
  15. e / m \scriptstyle{e/m}
  16. E / c 2 \scriptstyle{E/c^{2}}
  17. m = ( 8 / 3 ) E / c 2 \scriptstyle{m=(8/3)E/c^{2}}
  18. m = ( 4 / 3 ) E / c 2 \scriptstyle{m=(4/3)E/c^{2}}
  19. i t \scriptstyle{it}
  20. i = - 1 \scriptstyle{i=\sqrt{-1}}
  21. t = t - v x / c 2 \scriptstyle{t^{\prime}=t-{vx}/{c^{2}}}
  22. 1 - v 2 / c 2 \scriptstyle{\sqrt{1-{v^{2}}/{c^{2}}}}
  23. x 2 + y 2 + z 2 - c 2 t 2 \scriptstyle{x^{2}+y^{2}+z^{2}-c^{2}t^{2}}
  24. c t - 1 \scriptstyle{ct\sqrt{-1}}
  25. E k i n = m c 2 ( 1 1 - v 2 c 2 - 1 ) E_{kin}=mc^{2}\left(\frac{1}{\sqrt{1-\frac{v^{2}}{c^{2}}}}-1\right)
  26. E = m c 2 E=mc^{2}

History_of_superconductivity.html

  1. 2 e V = ϕ t 2eV=\hbar\frac{\partial\phi}{\partial t}

HITS_algorithm.html

  1. p \forall p
  2. auth ( p ) = 1 \mathrm{auth}(p)=1
  3. hub ( p ) = 1 \mathrm{hub}(p)=1
  4. p \forall p
  5. auth ( p ) \mathrm{auth}(p)
  6. auth ( p ) = i = 1 n hub ( i ) \mathrm{auth}(p)=\displaystyle\sum_{i=1}^{n}\mathrm{hub}(i)
  7. p \forall p
  8. hub ( p ) \mathrm{hub}(p)
  9. hub ( p ) = i = 1 n auth ( i ) \mathrm{hub}(p)=\displaystyle\sum_{i=1}^{n}\mathrm{auth}(i)

Hodge_index_theorem.html

  1. H H = d H\cdot H=d

Hodograph.html

  1. V 0 \vec{V}_{0}
  2. V 4 \vec{V}_{4}
  3. V 3 \vec{V}_{3}

Holdrian_comma.html

  1. 2 53 \sqrt[53]{2}
  2. 2 55 \sqrt[55]{2}
  3. 2 53 \sqrt[53]{2}

Hollaback_Girl.html

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

Holonomic.html

  1. e k = x k e_{k}={\partial\over\partial x^{k}}
  2. x j x_{j}\,\!
  3. t t\,\!

Homeomorphism_group.html

  1. MCG ( X ) = Homeo ( X ) / Homeo 0 ( X ) {\rm MCG}(X)={\rm Homeo}(X)/{\rm Homeo}_{0}(X)
  2. MCG ( X ) = π 0 ( Homeo ( X ) ) {\rm MCG}(X)=\pi_{0}({\rm Homeo}(X))
  3. 1 Homeo 0 ( X ) Homeo ( X ) MCG ( X ) 1. 1\rightarrow{\rm Homeo}_{0}(X)\rightarrow{\rm Homeo}(X)\rightarrow{\rm MCG}(X)% \rightarrow 1.

Homogeneous_catalysis.html

  1. \overrightarrow{\leftarrow}

Homogeneous_polynomial.html

  1. x 5 + 2 x 3 y 2 + 9 x y 4 x^{5}+2x^{3}y^{2}+9xy^{4}
  2. x 3 + 3 x 2 y + z 7 x^{3}+3x^{2}y+z^{7}
  3. P ( λ x 1 , , λ x n ) = λ d P ( x 1 , , x n ) , P(\lambda x_{1},\ldots,\lambda x_{n})=\lambda^{d}\,P(x_{1},\ldots,x_{n})\,,
  4. λ \lambda
  5. P ( x 1 , , x n ) = 0 P ( λ x 1 , , λ x n ) = 0 , P(x_{1},\ldots,x_{n})=0\quad\Rightarrow\quad P(\lambda x_{1},\ldots,\lambda x_% {n})=0,
  6. λ . \lambda.
  7. R = K [ x 1 , , x n ] R=K[x_{1},\ldots,x_{n}]
  8. R d . R_{d}.
  9. R R
  10. R d R_{d}
  11. R d R_{d}
  12. ( d + n - 1 n - 1 ) = ( d + n - 1 d ) = ( d + n - 1 ) ! d ! ( n - 1 ) ! . {\left({{d+n-1}\atop{n-1}}\right)}={\left({{d+n-1}\atop{d}}\right)}=\frac{(d+n% -1)!}{d!(n-1)!}.
  13. P h ( x 0 , x 1 , , x n ) = x 0 d P ( x 1 x 0 , , x n x 0 ) , {{}^{h}\!P}(x_{0},x_{1},\dots,x_{n})=x_{0}^{d}P\left(\frac{x_{1}}{x_{0}},\dots% ,\frac{x_{n}}{x_{0}}\right),
  14. P = x 3 3 + x 1 x 2 + 7 , P=x_{3}^{3}+x_{1}x_{2}+7,
  15. P h = x 3 3 + x 0 x 1 x 2 + 7 x 0 3 . {}^{h}\!P=x_{3}^{3}+x_{0}x_{1}x_{2}+7x_{0}^{3}.
  16. P ( x 1 , , x n ) = P h ( 1 , x 1 , , x n ) . P(x_{1},\dots,x_{n})={{}^{h}\!P}(1,x_{1},\dots,x_{n}).

Honey_flow.html

  1. 7000 forager bees × 10 trips in good flying weather per day × 70 mg of nectar during honey flow per trip and bee × 1 kg 1 , 000 , 000 mg 5 kg/day 7000\,\text{ forager bees}\times\frac{10\,\text{ trips in good flying weather}% }{\,\text{ per day}}\times\frac{70\,\text{ mg of nectar during honey flow}}{\,% \text{per trip and bee}}\times\frac{1\,\text{ kg}}{1,000,000\,\text{ mg}}% \approx 5\,\text{ kg/day}

Hopf_link.html

  1. σ 1 2 . \sigma_{1}^{2}.\,

Horn-satisfiability.html

  1. l l
  2. l l
  3. ¬ l \neg l
  4. l l
  5. ¬ l \neg l

Horn_antenna.html

  1. a E = 2 λ L E a H = 3 λ L H a_{E}=\sqrt{2\lambda L_{E}}\qquad a_{H}=\sqrt{3\lambda L_{H}}
  2. d = 3 λ L d=\sqrt{3\lambda L}
  3. G = 4 π A λ 2 e A G=\frac{4\pi A}{\lambda^{2}}e_{A}
  4. G = ( π d λ ) 2 e A G=\left(\frac{\pi d}{\lambda}\right)^{2}e_{A}

Host–guest_chemistry.html

  1. H + G H G H+G\rightleftharpoons\ HG
  2. K a = [ H G ] e q [ H ] e q [ G ] e q K_{a}=\frac{[HG]_{eq}}{[H]_{eq}[G]_{eq}}
  3. K a K_{a}
  4. K d K_{d}
  5. K d = [ H ] e q [ G ] e q [ H G ] e q = 1 K a K_{d}=\frac{[H]_{eq}[G]_{eq}}{[HG]_{eq}}=\frac{1}{K_{a}}
  6. K a K_{a}
  7. K d K_{d}
  8. K a K_{a}
  9. Δ G = R T ln K a \Delta G=RT\ln{K_{a}}
  10. Δ G = Δ H - T Δ S \Delta G=\Delta H-T\Delta S\,
  11. H G HG
  12. [ H G ] [HG]
  13. [ G ] o [G]_{o}
  14. [ H ] e q [H]_{eq}
  15. [ G ] e q [G]_{eq}
  16. [ H G ] [HG]
  17. [ G ] o [G]_{o}
  18. [ H ] o [H]_{o}
  19. [ G ] o [G]_{o}
  20. [ H ] o [H]_{o}
  21. [ H G ] e q [HG]_{eq}
  22. [ H ] e q [H]_{eq}
  23. [ H ] [H]
  24. [ H G ] e q [HG]_{eq}
  25. [ H ] e q [H]_{eq}
  26. [ H ] o [H]_{o}
  27. [ H G ] e q [HG]_{eq}
  28. [ H ] e q [H]_{eq}
  29. [ H ] o = [ H G ] e q + [ H ] e q [H]_{o}=[HG]_{eq}+[H]_{eq}
  30. [ H ] [H]
  31. [ H ] e q = [ H ] o - [ H G ] e q [H]_{eq}=[H]_{o}-[HG]_{eq}
  32. K a = [ H G ] e q [ H e q ] [ G e q ] K_{a}=\frac{[HG]_{eq}}{[H_{eq}][G_{eq}]}
  33. K a = [ H G ] e q [ G ] e q ( [ H ] o - [ H G ] e q ) K_{a}=\frac{[HG]_{eq}}{[G]_{eq}([H]_{o}-[HG]_{eq})}
  34. [ H G ] e q [HG]_{eq}
  35. [ H G ] e q = K a [ G ] e q [ H ] o 1 + K a [ G ] e q [HG]_{eq}=\frac{K_{a}[G]_{eq}[H]_{o}}{1+K_{a}[G]_{eq}}
  36. [ H G ] e q [HG]_{eq}
  37. [ H ] o [H]_{o}
  38. [ H G ] [HG]
  39. [ H ] o [H]_{o}
  40. [ G ] o [G]_{o}
  41. [ G ] o [G]_{o}
  42. [ H ] o [H]_{o}
  43. [ G ] o [ H ] o [G]_{o}>>>[H]_{o}
  44. [ G ] o = [ G ] e q [G]_{o}=[G]_{eq}
  45. [ H G ] [HG]
  46. [ H G ] e q = K a [ G ] o [ H ] o 1 + K a [ G ] o [HG]_{eq}=\frac{K_{a}[G]_{o}[H]_{o}}{1+K_{a}[G]_{o}}
  47. [ G ] [G]
  48. [ G ] o [G]_{o}
  49. [ G ] o [G]_{o}
  50. [ H G ] e q = K a [ G ] o [ H ] o 1 + K a [ G ] o [HG]_{eq}=\frac{K_{a}[G]_{o}[H]_{o}}{1+K_{a}[G]_{o}}
  51. [ H G ] e q = K a [ G ] o [ H ] o 1 [HG]_{eq}=\frac{K_{a}[G]_{o}[H]_{o}}{1}
  52. [ G o ] [G_{o}]
  53. ( [ G ] o 1 ) ([G]_{o}>>>1)
  54. [ H G e q ] [HG_{eq}]
  55. [ G ] o [G]_{o}
  56. [ G ] o [G]_{o}
  57. [ H G ] e q = K a [ G ] o [ H ] o 1 + K a [ G ] o [HG]_{eq}=\frac{K_{a}[G]_{o}[H]_{o}}{1+K_{a}[G]_{o}}
  58. [ H G ] e q = K a [ H ] o 1 + K a [HG]_{eq}=\frac{K_{a}[H]_{o}}{1+K_{a}}
  59. [ G ] [G]
  60. Δ G \Delta G
  61. K a K_{a}
  62. Δ G \Delta G
  63. Δ H \Delta H
  64. Δ S \Delta S
  65. K a K_{a}
  66. Δ H \Delta H
  67. Δ G = Δ H - T Δ S \Delta G=\Delta H-T\Delta S
  68. K a K_{a}
  69. Δ H 0 \Delta H_{0}
  70. Q = V Δ H 0 [ H . G ] Q={V\Delta H_{0}[H.G]}
  71. Q = V Δ H 0 K a [ H 0 ] [ G ] 1 + K a [ G ] Q=\frac{V\Delta H_{0}K_{a}[H_{0}][G]}{1+K_{a}[G]}
  72. [ H 0 ] [H_{0}]
  73. [ G ] [G]
  74. V V
  75. K a K_{a}
  76. Δ G \Delta G
  77. Δ H \Delta H
  78. Δ S \Delta S
  79. K a K_{a}
  80. Δ G \Delta G
  81. Δ H \Delta H
  82. Δ S \Delta S
  83. K a K_{a}
  84. Δ G \Delta G
  85. Δ H \Delta H
  86. Δ S \Delta S
  87. 1.2 * 10 5 M - 1 1.2*10^{5}M^{-1}
  88. B a Z n 2 L ( C l O 4 ) 2 BaZn_{2}L(ClO_{4})_{2}
  89. Δ G = Δ H - T Δ S \Delta G=\Delta H-T\Delta S
  90. Δ G S = Δ G A + Δ G B - Δ G A B \Delta G_{S}^{\circ}=\Delta G_{A}^{\circ}+\Delta G_{B}^{\circ}-\Delta G_{AB}^{\circ}
  91. Δ H S = Δ H A + Δ H B - Δ H A B \Delta H_{S}^{\circ}=\Delta H_{A}^{\circ}+\Delta H_{B}^{\circ}-\Delta H_{AB}^{\circ}
  92. T Δ G S = T Δ H A + T Δ H B - T Δ S A B \ T\Delta G_{S}^{\circ}=T\Delta H_{A}^{\circ}+T\Delta H_{B}^{\circ}-T\Delta S_% {AB}^{\circ}
  93. Δ G A \Delta G_{A}^{\circ}
  94. Δ G B \Delta G_{B}^{\circ}
  95. Δ G S \Delta G_{S}^{\circ}
  96. Δ G A B \Delta G_{AB}^{\circ}
  97. Δ G S \Delta G_{S}^{\circ}
  98. Δ G A \Delta G_{A}^{\circ}
  99. Δ G B \Delta G_{B}^{\circ}
  100. Δ G S \Delta G_{S}^{\circ}

Hotelling's_rule.html

  1. P ( t ) P ( t ) = δ , \frac{P^{\prime}(t)}{P(t)}=\delta,
  2. P ( t ) P(t)
  3. δ \delta
  4. p ( t ) p(t)
  5. c ( t ) c(t)
  6. t t
  7. Π ( t ) = p ( t ) - c ( t ) , \Pi(t)=p(t)-c(t),
  8. Π ( t ) \Pi(t)
  9. Π ( t ) Π ( t ) = r \frac{\Pi^{\prime}(t)}{\Pi(t)}=r
  10. P ( t ) P ( t ) = r ( 1 - c p ( t ) ) , \frac{P^{\prime}(t)}{P(t)}=r\left(1-\frac{c}{p(t)}\right),

Hubbard_model.html

  1. \uparrow
  2. \downarrow
  3. H = - t i , j , σ ( c i , σ c j , σ + c j , σ c i , σ ) + U i = 1 N n i n i , H=-t\sum_{\langle i,j\rangle,\sigma}(c^{\dagger}_{i,\sigma}c_{j,\sigma}+c^{% \dagger}_{j,\sigma}c_{i,\sigma})+U\sum_{i=1}^{N}n_{i\uparrow}n_{i\downarrow},
  4. i , j \langle i,j\rangle
  5. ( Ni 2 + O 2 - ) 2 Ni 3 + O 2 - + Ni 1 + O 2 - (\mathrm{Ni}^{2+}\mathrm{O}^{2-})_{2}\longrightarrow\mathrm{Ni}^{3+}\mathrm{O}% ^{2-}+\mathrm{Ni}^{1+}\mathrm{O}^{2-}
  6. Ni 2 + O 2 - Ni 1 + O 1 - \mathrm{Ni}^{2+}\mathrm{O}^{2-}\longrightarrow\mathrm{Ni}^{1+}\mathrm{O}^{1-}
  7. U U

Hull_speed.html

  1. v h u l l 1.34 × L W L v_{hull}\approx 1.34\times\sqrt{L_{WL}}
  2. L W L L_{WL}
  3. v h u l l v_{hull}
  4. L W L \sqrt{L_{WL}}

Hull–White_model.html

  1. d r ( t ) = [ θ ( t ) - α ( t ) r ( t ) ] d t + σ ( t ) d W ( t ) dr(t)=\left[\theta(t)-\alpha(t)r(t)\right]\,dt+\sigma(t)\,dW(t)\,\!
  2. d f ( r ( t ) ) = [ θ ( t ) + u - α ( t ) f ( r ( t ) ) ] d t + σ 1 ( t ) d W 1 ( t ) d\,f(r(t))=\left[\theta(t)+u-\alpha(t)\,f(r(t))\right]dt+\sigma_{1}(t)\,dW_{1}% (t)\!
  3. u \displaystyle u
  4. d u = - b u d t + σ 2 d W 2 ( t ) du=-bu\,dt+\sigma_{2}\,dW_{2}(t)
  5. θ \theta
  6. α \alpha
  7. θ \theta
  8. σ \sigma
  9. r ( t ) = e - α t r ( 0 ) + θ α ( 1 - e - α t ) + σ e - α t 0 t e α u d W ( u ) r(t)=e^{-\alpha t}r(0)+\frac{\theta}{\alpha}\left(1-e^{-\alpha t}\right)+% \sigma e^{-\alpha t}\int_{0}^{t}e^{\alpha u}\,dW(u)\,\!
  10. r ( t ) 𝒩 ( e - α t r ( 0 ) + θ α ( 1 - e - α t ) , σ 2 2 α ( 1 - e - 2 α t ) ) . r(t)\sim\mathcal{N}\left(e^{-\alpha t}r(0)+\frac{\theta}{\alpha}\left(1-e^{-% \alpha t}\right),\frac{\sigma^{2}}{2\alpha}\left(1-e^{-2\alpha t}\right)\right).
  11. 𝒩 ( μ , σ 2 ) \mathcal{N}(\mu,\sigma^{2})
  12. μ \mu
  13. σ 2 \sigma^{2}
  14. θ ( t ) \theta(t)
  15. r ( t ) = e - α t r ( 0 ) + 0 t e α ( s - t ) θ ( s ) d s + σ e - α t 0 t e α u d W ( u ) r(t)=e^{-\alpha t}r(0)+\int_{0}^{t}e^{\alpha(s-t)}\theta(s)ds+\sigma e^{-% \alpha t}\int_{0}^{t}e^{\alpha u}\,dW(u)\,\!
  16. r ( t ) 𝒩 ( e - α t r ( 0 ) + 0 t e α ( s - t ) θ ( s ) d s , σ 2 2 α ( 1 - e - 2 α t ) ) . r(t)\sim\mathcal{N}\left(e^{-\alpha t}r(0)+\int_{0}^{t}e^{\alpha(s-t)}\theta(s% )ds,\frac{\sigma^{2}}{2\alpha}\left(1-e^{-2\alpha t}\right)\right).
  17. P ( S , T ) = A ( S , T ) exp ( - B ( S , T ) r ( S ) ) P(S,T)=A(S,T)\exp(-B(S,T)r(S))\!
  18. B ( S , T ) = 1 - exp ( - α ( T - S ) ) α B(S,T)=\frac{1-\exp(-\alpha(T-S))}{\alpha}\,
  19. A ( S , T ) = P ( 0 , T ) P ( 0 , S ) exp ( - B ( S , T ) log ( P ( 0 , S ) ) S - σ 2 ( exp ( - α T ) - exp ( - α S ) ) 2 ( exp ( 2 α S ) - 1 ) 4 α 3 ) A(S,T)=\frac{P(0,T)}{P(0,S)}\exp\left(\,-B(S,T)\frac{\partial\log(P(0,S))}{% \partial S}-\frac{\sigma^{2}(\exp(-\alpha T)-\exp(-\alpha S))^{2}(\exp(2\alpha S% )-1)}{4\alpha^{3}}\right)\,
  20. V ( t ) = P ( t , S ) 𝔼 S [ V ( S ) | ( t ) ] . V(t)=P(t,S)\mathbb{E}_{S}[V(S)|\mathcal{F}(t)].\,
  21. 𝔼 S \mathbb{E}_{S}
  22. F V ( t , T ) F_{V}(t,T)
  23. F V ( t , T ) = V ( t ) / P ( t , T ) F_{V}(t,T)=V(t)/P(t,T)
  24. F V ( t , T ) = 𝔼 T [ V ( T ) | ( t ) ] . F_{V}(t,T)=\mathbb{E}_{T}[V(T)|\mathcal{F}(t)].\,
  25. V ( S ) = ( K - P ( S , T ) ) + . V(S)=(K-P(S,T))^{+}.\,
  26. E S [ ( K - P ( S , T ) ) + ] = K N ( - d 2 ) - F ( t , S , T ) N ( d 1 ) {E}_{S}[(K-P(S,T))^{+}]=KN(-d_{2})-F(t,S,T)N(d_{1})\,
  27. d 1 = log ( F / K ) + σ P 2 S / 2 σ P S d_{1}=\frac{\log(F/K)+\sigma_{P}^{2}S/2}{\sigma_{P}\sqrt{S}}\,
  28. d 2 = d 1 - σ P S . d_{2}=d_{1}-\sigma_{P}\sqrt{S}.\,
  29. P ( 0 , S ) K N ( - d 2 ) - P ( 0 , T ) N ( - d 1 ) P(0,S)KN(-d_{2})-P(0,T)N(-d_{1})\,
  30. S σ P = σ α ( 1 - exp ( - α ( T - S ) ) ) 1 - exp ( - 2 α S ) 2 α \sqrt{S}\sigma_{P}=\frac{\sigma}{\alpha}(1-\exp(-\alpha(T-S)))\sqrt{\frac{1-% \exp(-2\alpha S)}{2\alpha}}\,

Hund's_rules.html

  1. 2 S + 1 2S+1
  2. S S
  3. S S\,
  4. L L\,
  5. J J\,
  6. s y m b o l J = s y m b o l L + s y m b o l S symbol{J}=symbol{L}+symbol{S}
  7. J J\,
  8. S S
  9. L L
  10. ( S = 1 ) (S=1)
  11. Z Z
  12. L = 3 L=3
  13. L = 1 L=1
  14. ( M L = 4 , M S = 1 ) (M_{L}=4,M_{S}=1)
  15. ( M L = 2 , M S = + 1 / 2 ) (M_{L}=2,M_{S}=+1/2)
  16. L L\,
  17. S S\,
  18. Δ E \displaystyle\Delta E
  19. ζ ( L , S ) \zeta(L,S)\,
  20. J J\,
  21. P 3 {}^{3}\!P\,
  22. J = 2 , 1 , 0 J=2,1,0\,
  23. P 0 3 {}^{3}\!P_{0}\,
  24. P 3 {}^{3}\!P\,
  25. J = 2 , 1 , 0 J=2,1,0\,
  26. P 2 3 {}^{3}\!P_{2}\,
  27. L = 0 L=0\,
  28. J J\,
  29. S S\,
  30. S = 3 / 2 , L = 0 S=3/2,\ L=0
  31. J = S = 3 / 2 J=S=3/2

Hurwitz_matrix.html

  1. p ( z ) = a 0 z n + a 1 z n - 1 + + a n - 1 z + a n p(z)=a_{0}z^{n}+a_{1}z^{n-1}+\cdots+a_{n-1}z+a_{n}
  2. n × n n\times n
  3. H = ( a 1 a 3 a 5 0 0 0 a 0 a 2 a 4 0 a 1 a 3 a 0 a 2 0 0 a 1 a n a 0 a n - 1 0 0 a n - 2 a n a n - 3 a n - 1 0 0 0 0 a n - 4 a n - 2 a n ) . H=\begin{pmatrix}a_{1}&a_{3}&a_{5}&\dots&\dots&\dots&0&0&0\\ a_{0}&a_{2}&a_{4}&&&&\vdots&\vdots&\vdots\\ 0&a_{1}&a_{3}&&&&\vdots&\vdots&\vdots\\ \vdots&a_{0}&a_{2}&\ddots&&&0&\vdots&\vdots\\ \vdots&0&a_{1}&&\ddots&&a_{n}&\vdots&\vdots\\ \vdots&\vdots&a_{0}&&&\ddots&a_{n-1}&0&\vdots\\ \vdots&\vdots&0&&&&a_{n-2}&a_{n}&\vdots\\ \vdots&\vdots&\vdots&&&&a_{n-3}&a_{n-1}&0\\ 0&0&0&\dots&\dots&\dots&a_{n-4}&a_{n-2}&a_{n}\end{pmatrix}.
  4. p p
  5. H ( p ) H(p)
  6. Δ 1 ( p ) \displaystyle\Delta_{1}(p)
  7. Δ k ( p ) \Delta_{k}(p)
  8. A A
  9. A A
  10. Re [ λ i ] < 0 \mathop{\mathrm{Re}}[\lambda_{i}]<0\,
  11. λ i \lambda_{i}
  12. A A
  13. x ˙ = A x \dot{x}=Ax
  14. x ( t ) 0 x(t)\to 0
  15. t . t\to\infty.
  16. G ( s ) G(s)
  17. G G
  18. G G
  19. G ( s ) , G(s),
  20. s , s,
  21. A A
  22. x ˙ ( t ) = A x ( t ) + B u ( t ) \dot{x}(t)=Ax(t)+Bu(t)
  23. y ( t ) = C x ( t ) + D u ( t ) y(t)=Cx(t)+Du(t)\,