wpmath0000015_3

G-expectation.html

  1. ( Ω , , ) (\Omega,\mathcal{F},\mathbb{P})
  2. ( W t ) t 0 (W_{t})_{t\geq 0}
  3. ( W t ) (W_{t})
  4. t = σ ( W s : s [ 0 , t ] ) \mathcal{F}_{t}=\sigma(W_{s}:s\in[0,t])
  5. X X
  6. T \mathcal{F}_{T}
  7. d Y t \displaystyle dY_{t}
  8. X X
  9. 𝔼 g [ X ] := Y 0 \mathbb{E}^{g}[X]:=Y_{0}
  10. X X
  11. Y t Y_{t}
  12. t t
  13. Z t Z_{t}
  14. m × d m\times d
  15. 𝔼 g [ X t ] := Y t \mathbb{E}^{g}[X\mid\mathcal{F}_{t}]:=Y_{t}
  16. 𝔼 g [ 1 A 𝔼 g [ X t ] ] = 𝔼 g [ 1 A X ] \mathbb{E}^{g}[1_{A}\mathbb{E}^{g}[X\mid\mathcal{F}_{t}]]=\mathbb{E}^{g}[1_{A}X]
  17. A t A\in\mathcal{F}_{t}
  18. 1 1
  19. g : [ 0 , T ] × m × m × d m g:[0,T]\times\mathbb{R}^{m}\times\mathbb{R}^{m\times d}\to\mathbb{R}^{m}
  20. g ( , y , z ) g(\cdot,y,z)
  21. t \mathcal{F}_{t}
  22. ( y , z ) m × m × d (y,z)\in\mathbb{R}^{m}\times\mathbb{R}^{m\times d}
  23. 0 T | g ( t , 0 , 0 ) | d t L 2 ( Ω , T , ) \int_{0}^{T}|g(t,0,0)|\,dt\in L^{2}(\Omega,\mathcal{F}_{T},\mathbb{P})
  24. | | |\cdot|
  25. m \mathbb{R}^{m}
  26. g g
  27. ( y , z ) (y,z)
  28. y 1 , y 2 m y_{1},y_{2}\in\mathbb{R}^{m}
  29. z 1 , z 2 m × d z_{1},z_{2}\in\mathbb{R}^{m\times d}
  30. | g ( t , y 1 , z 1 ) - g ( t , y 2 , z 2 ) | C ( | y 1 - y 2 | + | z 1 - z 2 | ) |g(t,y_{1},z_{1})-g(t,y_{2},z_{2})|\leq C(|y_{1}-y_{2}|+|z_{1}-z_{2}|)
  31. C C
  32. X L 2 ( Ω , t , ; m ) X\in L^{2}(\Omega,\mathcal{F}_{t},\mathbb{P};\mathbb{R}^{m})
  33. t \mathcal{F}_{t}
  34. ( Y , Z ) (Y,Z)
  35. g g
  36. g g
  37. t t
  38. g ( t , y , 0 ) 0 g(t,y,0)\equiv 0
  39. ( t , y ) [ 0 , T ] × m (t,y)\in[0,T]\times\mathbb{R}^{m}
  40. X L 2 ( Ω , t , ; m ) X\in L^{2}(\Omega,\mathcal{F}_{t},\mathbb{P};\mathbb{R}^{m})
  41. ( Y , Z ) (Y,Z)
  42. 𝔼 g [ X | t ] \mathbb{E}^{g}[X|\mathcal{F}_{t}]
  43. t t
  44. ρ g ( X ) := 𝔼 g [ - X ] \rho_{g}(X):=\mathbb{E}^{g}[-X]

Gabriel–Popescu_theorem.html

  1. 0 M 1 M 2 M 3 0 0\rightarrow M_{1}\rightarrow M_{2}\rightarrow M_{3}\rightarrow 0

Gagliardo–Nirenberg_interpolation_inequality.html

  1. 1 p = j n + ( 1 r - m n ) α + 1 - α q \frac{1}{p}=\frac{j}{n}+\left(\frac{1}{r}-\frac{m}{n}\right)\alpha+\frac{1-% \alpha}{q}
  2. j m α 1. \frac{j}{m}\leq\alpha\leq 1.
  3. D j u L p C D m u L r α u L q 1 - α . \|\mathrm{D}^{j}u\|_{L^{p}}\leq C\|\mathrm{D}^{m}u\|_{L^{r}}^{\alpha}\|u\|_{L^% {q}}^{1-\alpha}.
  4. D j u L p C 1 D m u L r α u L q 1 - α + C 2 u L s \|\mathrm{D}^{j}u\|_{L^{p}}\leq C_{1}\|\mathrm{D}^{m}u\|_{L^{r}}^{\alpha}\|u\|% _{L^{q}}^{1-\alpha}+C_{2}\|u\|_{L^{s}}

Gain-field_encoding.html

  1. r = f ( x ) g ( y ) r=f(x)g(y)
  2. r r
  3. f ( x ) f(x)
  4. g ( y ) g(y)
  5. x x
  6. y y

Galactofuranosylgalactofuranosylrhamnosyl-N-acetylglucosaminyl-diphospho-decaprenol_beta-1,5::1,6-galactofuranosyltransferase.html

  1. \rightleftharpoons

Galactonolactone_dehydrogenase.html

  1. \rightleftharpoons
  2. \rightleftharpoons

Galactosyl-N-acetylglucosaminylgalactosylglucosyl-ceramide_b-1,6-N-acetylglucosaminyltransferase.html

  1. \rightleftharpoons

Gall_stereographic_projection.html

  1. x = R λ / 2 x=R\lambda/\sqrt{2}
  2. y = R ( 1 + 2 / 2 ) tan φ 2 y=R(1+\sqrt{2}/2)\tan\frac{\varphi}{2}
  3. x = R λ x=R\lambda
  4. y = 2 R tan φ 2 y=2R\tan\frac{\varphi}{2}

Gallate_dioxygenase.html

  1. \rightleftharpoons

GalNAc-alpha-(1-4)-GalNAc-alpha-(1-3)-diNAcBac-PP-undecaprenol_alpha-1,4-N-acetyl-D-galactosaminyltransferase.html

  1. \rightleftharpoons

GalNAc5-diNAcBac-PP-undecaprenol_beta-1,3-glucosyltransferase.html

  1. \rightleftharpoons

Gamma::Gompertz_distribution.html

  1. f ( x ; b , s , β ) = b s e b x β s ( β - 1 + e b x ) s + 1 f(x;b,s,\beta)=\frac{bse^{bx}\beta^{s}}{\left(\beta-1+e^{bx}\right)^{s+1}}
  2. b > 0 b>0
  3. β , s > 0 \beta,s>0\,\!
  4. F ( x ; b , s , β ) \displaystyle F(x;b,s,\beta)
  5. E ( e - t x ) = { β s s b t + s b F 1 2 ( s + 1 , ( t / b ) + s ; ( t / b ) + s + 1 ; 1 - β ) , β 1 ; s b t + s b , β = 1. \displaystyle\,\text{E}(e^{-tx})=\begin{cases}\displaystyle\beta^{s}\frac{sb}{% t+sb}{\ }{{}_{2}\,\text{F}_{1}}(s+1,(t/b)+s;(t/b)+s+1;1-\beta),&\beta\neq 1;\\ \displaystyle\frac{sb}{t+sb},&\beta=1.\end{cases}
  6. F 1 2 ( a , b ; c ; z ) = k = 0 [ ( a ) k ( b ) k / ( c ) k ] z k / k ! {{}_{2}\,\text{F}_{1}}(a,b;c;z)=\sum_{k=0}^{\infty}[(a)_{k}(b)_{k}/(c)_{k}]z^{% k}/k!
  7. b . b\,\!.
  8. η \eta\,\!
  9. α \alpha\,\!
  10. β \beta\,\!
  11. α / β \alpha/\beta\,\!
  12. x x

Gamow–Teller_transition.html

  1. S = 1 S=1
  2. Δ J = 0 , ± 1 \Delta J=0,\pm 1
  3. S = 0 S=0
  4. Δ J = 0 \Delta J=0
  5. H ^ int = { G V 1 ^ τ ^ Fermi decay G A σ ^ τ ^ Gamow–Teller Decay \hat{H}\text{int}=\begin{cases}G_{V}\hat{1}\hat{\tau}&\,\text{Fermi decay}\\ G_{A}\hat{\sigma}\hat{\tau}&\,\text{Gamow–Teller Decay}\end{cases}
  6. τ ^ \hat{\tau}
  7. σ ^ \hat{\sigma}
  8. Δ J = 0 , ± 1 \Delta J=0,\pm 1
  9. 1 ^ \hat{1}
  10. J J
  11. G V G_{V}
  12. G A G_{A}
  13. B = n q - n q ¯ 3 B=\frac{n_{q}-n_{\bar{q}}}{3}
  14. n q n_{q}
  15. n q ¯ n_{\overline{q}}
  16. I = 1 2 I=\frac{1}{2}
  17. I z = { 1 2 up quark - 1 2 down quark I\text{z}=\begin{cases}\frac{1}{2}&\,\text{up quark}\\ -\frac{1}{2}&\,\text{down quark}\end{cases}
  18. I z = 1 2 ( n u - n d ) I\text{z}=\frac{1}{2}(n\text{u}-n\text{d})
  19. L n - n ¯ L\equiv n_{\ell}-n_{\bar{\ell}}
  20. n p + e - + ν ¯ e L : 0 = 0 + 1 - 1 \begin{matrix}&\,\text{n}&\rightarrow&\,\text{p}&+&\,\text{e}^{-}&+&\bar{\nu}% \text{e}\\ L:&0&=&0&+&1&-&1\end{matrix}
  21. S = 1 S=1
  22. S = 0 S=0
  23. Δ I = 0 \Delta I=0\Rightarrow
  24. O 6 8 14 N 7 * 7 14 + β + + ν e {}^{14}_{8}\,\text{O}_{6}\rightarrow{}^{14}_{7}\,\text{N}^{*}_{7}+\beta^{+}+% \nu\text{e}
  25. I i = 0 + I f = 0 + Δ I = 0 I_{i}=0^{+}\rightarrow I_{f}=0^{+}\Rightarrow\Delta I=0
  26. Δ π = 0 \Delta\pi=0\Rightarrow
  27. π ( Y m ) = ( - 1 ) \pi(Y_{\ell\,m})=(-1)^{\ell}
  28. N 7 * 7 14 {}^{14}_{7}\,\text{N}^{*}_{7}
  29. He 4 2 6 Li 3 3 6 + β - + ν ¯ e {}^{6}_{2}\,\text{He}_{4}\rightarrow{}^{6}_{3}\,\text{Li}_{3}+\beta^{-}+\bar{% \nu}\text{e}
  30. I i = 0 + I f = 1 + Δ I = 1 I_{i}=0^{+}\rightarrow I_{f}=1^{+}\Rightarrow\Delta I=1
  31. Δ π = 0 \Delta\pi=0\Rightarrow
  32. π ( Y m ) = ( - 1 ) \pi(Y_{\ell\,m})=(-1)^{\ell}\Rightarrow
  33. L = 1 L=1
  34. β + ν ¯ e \beta+\bar{\nu}\text{e}
  35. S = 1 S=1
  36. I i I f I_{i}\rightarrow I_{f}
  37. Δ I = I f - I i = { 0 I i = I f = 0 1 I i = 0 and I f = 1 \Delta I=I_{f}-I_{i}=\begin{cases}0&I_{i}=I_{f}=0\\ 1&I_{i}=0\,\text{ and }I_{f}=1\end{cases}
  38. Na 10 11 21 Ne 11 10 21 + β + + ν e {}^{21}_{11}\,\text{Na}_{10}\rightarrow{}^{21}_{10}\,\text{Ne}_{11}+\beta^{+}+% \nu\text{e}
  39. I i = 3 2 + I f = 3 2 + Δ I = 0 I_{i}=\frac{3}{2}^{+}\Rightarrow I_{f}=\frac{3}{2}^{+}\Rightarrow\Delta I=0
  40. Na 10 11 21 Ne 11 * 10 21 + β + + ν e {}^{21}_{11}\,\text{Na}_{10}\rightarrow{}^{21}_{10}\,\text{Ne}^{*}_{11}+\beta^% {+}+\nu\text{e}
  41. I i = 3 2 + I f = 5 2 + Δ I = 1 I_{i}=\frac{3}{2}^{+}\Rightarrow I_{f}=\frac{5}{2}^{+}\Rightarrow\Delta I=1
  42. y g F M F g GT M GT y\equiv\frac{g\text{F}M\text{F}}{g\text{GT}M\text{GT}}
  43. Δ I = 0 \Delta I=0
  44. Δ I = 1 \Delta I=1
  45. β + ν \beta+\nu
  46. S = 0 S=0
  47. S = 1 S=1
  48. I = L + S = 1 + 0 Δ I = 0 , 1 \vec{I}=\vec{L}+\vec{S}=\vec{1}+\vec{0}\Rightarrow\Delta I=0,1
  49. I = L + S = 1 + 1 Δ I = 0 , 1 , 2 \vec{I}=\vec{L}+\vec{S}=\vec{1}+\vec{1}\Rightarrow\Delta I=0,1,2
  50. Δ π = 1 \Delta\pi=1\Rightarrow
  51. Na 11 11 22 ( 3 + ) \displaystyle{}^{22}_{11}\,\text{Na}_{11}\left(3^{+}\right)
  52. W W
  53. M i , f M_{i,f}
  54. \hbar
  55. W = 2 π | M i , f | 2 × (Phase Space) = ln 2 t 1 / 2 W=\frac{2\pi}{\hbar}\left|M_{i,f}\right|^{2}\times\,\text{(Phase Space)}=\frac% {\ln 2}{t_{1/2}}
  56. | M i , f | 2 = ψ Daughter ϕ β ψ ν | H ^ int | ψ Parent \left|M_{i,f}\right|^{2}=\left\langle\psi\text{Daughter}\phi_{\beta}\psi_{\nu}% \right|\hat{H}\text{int}\left|\psi\text{Parent}\right\rangle
  57. H ^ int = { G V 1 ^ τ ^ Fermi decay G A σ ^ τ ^ Gamow–Teller Decay \hat{H}\text{int}=\begin{cases}G_{V}\hat{1}\hat{\tau}&\,\text{Fermi decay}\\ G_{A}\hat{\sigma}\hat{\tau}&\,\text{Gamow–Teller Decay}\end{cases}

Ganita_Kaumudi.html

  1. n x 2 + k 2 = y 2 nx^{2}+k^{2}=y^{2}
  2. 1 + 1 3 + 1 3 4 - 1 3 4 34 1+\tfrac{1}{3}+\tfrac{1}{3\cdot 4}-\tfrac{1}{3\cdot 4\cdot 34}
  3. 1 = 1 1 2 + 1 2 3 + 1 3 4 + + 1 ( n - 1 ) n + 1 n 1=\frac{1}{1\cdot 2}+\frac{1}{2\cdot 3}+\frac{1}{3\cdot 4}+\dots+\frac{1}{(n-1% )\cdot n}+\frac{1}{n}
  4. 1 = 1 2 + 1 3 + 1 3 2 + + 1 3 n - 2 + 1 2 3 n - 2 1=\frac{1}{2}+\frac{1}{3}+\frac{1}{3^{2}}+\dots+\frac{1}{3^{n-2}}+\frac{1}{2% \cdot 3^{n-2}}
  5. p / q p/q
  6. ( q + i ) / p (q+i)/p
  7. p q = 1 r + i q r \frac{p}{q}=\frac{1}{r}+\frac{i}{qr}
  8. n n
  9. k 1 , k 2 , , k n k_{1},k_{2},\dots,k_{n}
  10. 1 = ( k 2 - k 1 ) k 1 k 2 k 1 + ( k 3 - k 2 ) k 1 k 3 k 2 + + ( k n - k n - 1 ) k 1 k n k n - 1 + 1 k 1 k n 1=\frac{(k_{2}-k_{1})k_{1}}{k_{2}\cdot k_{1}}+\frac{(k_{3}-k_{2})k_{1}}{k_{3}% \cdot k_{2}}+\dots+\frac{(k_{n}-k_{n-1})k_{1}}{k_{n}\cdot k_{n-1}}+\frac{1% \cdot k_{1}}{k_{n}}
  11. a 1 , a 2 , , a n a_{1},a_{2},\dots,a_{n}
  12. i 1 , i 2 , , i n i_{1},i_{2},\dots,i_{n}
  13. i 1 = a 1 + 1 i_{1}=a_{1}+1
  14. i 2 = a 2 + i 1 i_{2}=a_{2}+i_{1}
  15. i 3 = a 3 + i 2 i_{3}=a_{3}+i_{2}
  16. 1 = a 1 1 i 1 + a 2 i 1 i 2 + a 3 i 2 i 3 + + a n i n - 1 i n + 1 i n 1=\frac{a_{1}}{1\cdot i_{1}}+\frac{a_{2}}{i_{1}\cdot i_{2}}+\frac{a_{3}}{i_{2}% \cdot i_{3}}+\dots+\frac{a_{n}}{i_{n-1}\cdot i_{n}}+\frac{1}{i_{n}}

Gassmann's_equation.html

  1. V P ( 1 ) V_{P}^{(1)}
  2. V S ( 1 ) V_{S}^{(1)}
  3. ρ ( 1 ) \rho^{(1)}
  4. V P ( 1 ) V_{\mathrm{P}}^{(1)}
  5. V S ( 1 ) V_{\mathrm{S}}^{(1)}
  6. ρ ( 1 ) \rho^{(1)}
  7. K sat ( 1 ) = ρ ( ( V P ( 1 ) ) 2 - 4 3 ( V S ( 1 ) ) 2 ) K_{\mathrm{sat}}^{(1)}=\rho\left((V_{\mathrm{P}}^{(1)})^{2}-\frac{4}{3}(V_{% \mathrm{S}}^{(1)})^{2}\right)
  8. μ sat ( 1 ) = ρ ( V S ( 1 ) ) 2 \mu_{\mathrm{sat}}^{(1)}=\rho(V_{\mathrm{S}}^{(1)})^{2}
  9. K sat ( 2 ) K mineral - K sat ( 2 ) - K fluid ( 2 ) ϕ ( K mineral - K fluid ( 2 ) ) = K sat ( 1 ) K mineral - K sat ( 1 ) - K fluid ( 1 ) ϕ ( K mineral - K fluid ( 1 ) ) \frac{K_{\mathrm{sat}}^{(2)}}{K_{\mathrm{mineral}}-K_{\mathrm{sat}}^{(2)}}-% \frac{K_{\mathrm{fluid}}^{(2)}}{\phi(K_{\mathrm{mineral}}-K_{\mathrm{fluid}}^{% (2)})}=\frac{K_{\mathrm{sat}}^{(1)}}{K_{\mathrm{mineral}}-K_{\mathrm{sat}}^{(1% )}}-\frac{K_{\mathrm{fluid}}^{(1)}}{\phi(K_{\mathrm{mineral}}-K_{\mathrm{fluid% }}^{(1)})}
  10. K sat ( 1 ) K_{\mathrm{sat}}^{(1)}
  11. K sat ( 2 ) K_{\mathrm{sat}}^{(2)}
  12. K fluid ( 1 ) K_{\mathrm{fluid}}^{(1)}
  13. K fluid ( 2 ) K_{\mathrm{fluid}}^{(2)}
  14. μ sat ( 2 ) = μ sat ( 1 ) \mu_{\mathrm{sat}}^{(2)}=\mu_{\mathrm{sat}}^{(1)}
  15. ρ ( 2 ) = ρ ( 1 ) + ϕ ( ρ fluid ( 2 ) - ρ fluid ( 1 ) ) \rho^{(2)}=\rho^{(1)}+\phi(\rho_{\mathrm{fluid}}^{(2)}-\rho_{\mathrm{fluid}}^{% (1)})
  16. V P ( 2 ) = K sat ( 2 ) + 4 3 μ sat ( 2 ) ρ ( 2 ) V_{\mathrm{P}}^{(2)}=\sqrt{\frac{K_{\mathrm{sat}}^{(2)}+\frac{4}{3}\mu_{% \mathrm{sat}}^{(2)}}{\rho^{(2)}}}
  17. V S ( 2 ) = μ sat ( 2 ) ρ ( 2 ) V_{\mathrm{S}}^{(2)}=\sqrt{\frac{\mu_{\mathrm{sat}}^{(2)}}{\rho^{(2)}}}
  18. K sat ( 2 ) K mineral - K sat ( 2 ) - K fluid ( 2 ) ϕ ( K mineral - K fluid ( 2 ) ) = K sat ( 1 ) K mineral - K sat ( 1 ) - K fluid ( 1 ) ϕ ( K mineral - K fluid ( 1 ) ) \frac{K_{\mathrm{sat}}^{(2)}}{K_{\mathrm{mineral}}-K_{\mathrm{sat}}^{(2)}}-% \frac{K_{\mathrm{fluid}}^{(2)}}{\phi(K_{\mathrm{mineral}}-K_{\mathrm{fluid}}^{% (2)})}=\frac{K_{\mathrm{sat}}^{(1)}}{K_{\mathrm{mineral}}-K_{\mathrm{sat}}^{(1% )}}-\frac{K_{\mathrm{fluid}}^{(1)}}{\phi(K_{\mathrm{mineral}}-K_{\mathrm{fluid% }}^{(1)})}
  19. S = K sat ( 1 ) K mineral - K sat ( 1 ) S=\frac{K_{\mathrm{sat}}^{(1)}}{K_{\mathrm{mineral}}-K_{\mathrm{sat}}^{(1)}}
  20. F 1 = K fluid ( 1 ) ϕ ( K mineral - K fluid ( 1 ) ) F 2 = K fluid ( 2 ) ϕ ( K mineral - K fluid ( 2 ) ) F_{1}=\frac{K_{\mathrm{fluid}}^{(1)}}{\phi(K_{\mathrm{mineral}}-K_{\mathrm{% fluid}}^{(1)})}\ \ \ \ F_{2}=\frac{K_{\mathrm{fluid}}^{(2)}}{\phi(K_{\mathrm{% mineral}}-K_{\mathrm{fluid}}^{(2)})}
  21. K sat ( 2 ) = K mineral 1 S - F 1 + F 2 + 1 K_{\mathrm{sat}}^{(2)}=\frac{K_{\mathrm{mineral}}}{\frac{1}{S-F_{1}+F_{2}}+1}
  22. K sat ( 2 ) = K mineral [ K sat ( 1 ) K mineral - K sat ( 1 ) - K fluid ( 1 ) ϕ ( K mineral - K fluid ( 1 ) ) + K fluid ( 2 ) ϕ ( K mineral - K fluid ( 2 ) ) ] - 1 + 1 K_{\mathrm{sat}}^{(2)}=\frac{K_{\mathrm{mineral}}}{\left[{\frac{K_{\mathrm{sat% }}^{(1)}}{K_{\mathrm{mineral}}-K_{\mathrm{sat}}^{(1)}}-\frac{K_{\mathrm{fluid}% }^{(1)}}{\phi(K_{\mathrm{mineral}}-K_{\mathrm{fluid}}^{(1)})}+\frac{K_{\mathrm% {fluid}}^{(2)}}{\phi(K_{\mathrm{mineral}}-K_{\mathrm{fluid}}^{(2)})}}\right]^{% -1}+1}
  23. μ sat = μ dry \mu_{\mathrm{sat}}=\mu_{\mathrm{dry}}

Gauge_group_(mathematics).html

  1. P X P\to X
  2. G G
  3. G ( X ) G(X)
  4. P ~ X \widetilde{P}\to X
  5. G G
  6. G ( X ) G(X)
  7. g ( x ) = 1 g(x)=1
  8. P ~ X \widetilde{P}\to X
  9. G 0 ( X ) G^{0}(X)
  10. G ( X ) G(X)
  11. G 0 ( X ) = { g ( x ) G ( X ) : g ( x 0 ) = 1 P ~ x 0 } G^{0}(X)=\{g(x)\in G(X)\quad:\quad g(x_{0})=1\in\widetilde{P}_{x_{0}}\}
  12. 1 P ~ x 0 1\in\widetilde{P}_{x_{0}}
  13. P ~ X \widetilde{P}\to X
  14. G ( X ) / G 0 ( X ) = G G(X)/G^{0}(X)=G
  15. G ¯ ( X ) = G ( X ) / Z \overline{G}(X)=G(X)/Z
  16. Z Z
  17. G ( X ) G(X)
  18. G ¯ ( X ) \overline{G}(X)
  19. G G
  20. G ¯ k ( X ) \overline{G}_{k}(X)
  21. G ( X ) G(X)
  22. G ¯ k ( X ) \overline{G}_{k}(X)
  23. A k A_{k}
  24. A k / G ¯ k ( X ) A_{k}/\overline{G}_{k}(X)

Gauge_theory_gravity.html

  1. x x = f ( x ) x\mapsto x^{\prime}=f(x)
  2. 𝗁 ¯ ( a , x ) 𝗁 ¯ ( a , x ) = 𝗁 ¯ ( f - 1 ( a ) , f ( x ) ) , \bar{\mathsf{h}}(a,x)\mapsto\bar{\mathsf{h}}^{\prime}(a,x)=\bar{\mathsf{h}}(f^% {-1}(a),f(x)),
  3. Ω ¯ ( a , x ) Ω ¯ ( a , x ) = R Ω ¯ ( a , x ) R - 2 a R R . \bar{\mathsf{\Omega}}(a,x)\mapsto\bar{\mathsf{\Omega}}^{\prime}(a,x)=R\bar{% \mathsf{\Omega}}(a,x)R^{\dagger}-2a\cdot\nabla RR^{\dagger}.
  4. a D = a 𝗁 ¯ ( ) + 1 2 Ω ( 𝗁 ( a ) ) a\cdot D=a\cdot\bar{\mathsf{h}}(\nabla)+\frac{1}{2}\mathsf{\Omega}(\mathsf{h}(% a))
  5. a 𝒟 = a 𝗁 ¯ ( ) + Ω ( 𝗁 ( a ) ) a\cdot\mathcal{D}=a\cdot\bar{\mathsf{h}}(\nabla)+\mathsf{\Omega}(\mathsf{h}(a))
  6. D μ = μ + 1 2 Ω μ D_{\mu}=\partial_{\mu}+\frac{1}{2}\Omega_{\mu}
  7. 𝒟 μ = μ + Ω μ × , \mathcal{D}_{\mu}=\partial_{\mu}+\Omega_{\mu}\times,
  8. [ D μ , D ν ] ψ = 1 2 𝖱 μ ν ψ [D_{\mu},D_{\nu}]\psi=\frac{1}{2}\mathsf{R}_{\mu\nu}\psi
  9. ( a b ) = 𝖱 ( 𝗁 ( a b ) ) \mathcal{R}(a\wedge b)=\mathsf{R}(\mathsf{h}(a\wedge b))
  10. S = [ 1 2 κ ( - 2 Λ ) + M ] ( det 𝗁 ) - 1 d 4 x . S=\int\left[{1\over 2\kappa}\left(\mathcal{R}-2\Lambda\right)+\mathcal{L}_{% \mathrm{M}}\right](\det\mathsf{h})^{-1}\,\mathrm{d}^{4}x.
  11. 𝒢 ( a ) - Λ a = κ 𝒯 ( a ) \mathcal{G}(a)-\Lambda a=\kappa\mathcal{T}(a)
  12. 𝒟 𝗁 ¯ ( a ) = κ 𝒮 𝗁 ¯ ( a ) , \mathcal{D}\wedge\bar{\mathsf{h}}(a)=\kappa\mathcal{S}\cdot\bar{\mathsf{h}}(a),
  13. 𝒯 \mathcal{T}
  14. 𝒮 \mathcal{S}
  15. { e ( a ) μ } \{{e_{(a)}}^{\mu}\}
  16. g μ = 𝗁 - 1 ( e μ ) g_{\mu}=\mathsf{h}^{-1}(e_{\mu})
  17. g μ = 𝗁 ¯ ( e μ ) g^{\mu}=\bar{\mathsf{h}}(e^{\mu})
  18. { e μ } \{e_{\mu}\}
  19. g μ ν = g μ g ν . g_{\mu\nu}=g_{\mu}\cdot g_{\nu}.

Gauss–Legendre_method.html

  1. 1 2 - 1 6 3 \tfrac{1}{2}-\tfrac{1}{6}\sqrt{3}
  2. 1 4 \tfrac{1}{4}
  3. 1 4 - 1 6 3 \tfrac{1}{4}-\tfrac{1}{6}\sqrt{3}
  4. 1 2 + 1 6 3 \tfrac{1}{2}+\tfrac{1}{6}\sqrt{3}
  5. 1 4 + 1 6 3 \tfrac{1}{4}+\tfrac{1}{6}\sqrt{3}
  6. 1 4 \tfrac{1}{4}
  7. 1 2 \tfrac{1}{2}
  8. 1 2 \tfrac{1}{2}
  9. 1 2 - 1 10 15 \tfrac{1}{2}-\tfrac{1}{10}\sqrt{15}
  10. 5 36 \tfrac{5}{36}
  11. 2 9 - 1 15 15 \tfrac{2}{9}-\tfrac{1}{15}\sqrt{15}
  12. 5 36 - 1 30 15 \tfrac{5}{36}-\tfrac{1}{30}\sqrt{15}
  13. 1 2 \tfrac{1}{2}
  14. 5 36 + 1 24 15 \tfrac{5}{36}+\tfrac{1}{24}\sqrt{15}
  15. 2 9 \tfrac{2}{9}
  16. 5 36 - 1 24 15 \tfrac{5}{36}-\tfrac{1}{24}\sqrt{15}
  17. 1 2 + 1 10 15 \tfrac{1}{2}+\tfrac{1}{10}\sqrt{15}
  18. 5 36 + 1 30 15 \tfrac{5}{36}+\tfrac{1}{30}\sqrt{15}
  19. 2 9 + 1 15 15 \tfrac{2}{9}+\tfrac{1}{15}\sqrt{15}
  20. 5 36 \tfrac{5}{36}
  21. 5 18 \tfrac{5}{18}
  22. 4 9 \tfrac{4}{9}
  23. 5 18 \tfrac{5}{18}

GDP-D-glucose_phosphorylase.html

  1. \rightleftharpoons

GDP-L-galactose_phosphorylase.html

  1. \rightleftharpoons

GDP-Man:Man2GlcNAc2-PP-dolichol_alpha-1,6-mannosyltransferase.html

  1. \rightleftharpoons

GDP-mannose:cellobiosyl-diphosphopolyprenol_alpha-mannosyltransferase.html

  1. \rightleftharpoons

Gelfand–Kirillov_dimension.html

  1. GKdim = sup V , M 0 lim sup n log n dim k M 0 V n \operatorname{GKdim}=\sup_{V,M_{0}}\limsup_{n\to\infty}\log_{n}\dim_{k}M_{0}V^% {n}
  2. V A V\subset A
  3. M 0 M M_{0}\subset M
  4. k [ x 1 , , x n ] k[x_{1},\dots,x_{n}]
  5. A n A_{n}

Gellan_tetrasaccharide_unsaturated_glucuronyl_hydrolase.html

  1. \rightleftharpoons

General_covariant_transformations.html

  1. X X
  2. X X
  3. π : Y X \pi:Y\to X
  4. ( x λ , y i ) (x^{\lambda},y^{i})\,
  5. Y Y
  6. X X
  7. X X
  8. Y Y
  9. Y Y
  10. u = u λ ( x μ ) λ + u i ( x μ , y j ) i u=u^{\lambda}(x^{\mu})\partial_{\lambda}+u^{i}(x^{\mu},y^{j})\partial_{i}
  11. Y Y
  12. τ = u λ λ \tau=u^{\lambda}\partial_{\lambda}
  13. X X
  14. X X
  15. τ = τ λ λ \tau=\tau^{\lambda}\partial_{\lambda}
  16. X X
  17. Y Y
  18. τ \tau
  19. Γ \Gamma
  20. Y Y
  21. τ \tau
  22. X X
  23. Γ τ = τ λ ( λ + Γ λ i i ) \Gamma\tau=\tau^{\lambda}(\partial_{\lambda}+\Gamma_{\lambda}^{i}\partial_{i})
  24. Y Y
  25. τ Γ τ \tau\to\Gamma\tau
  26. C ( X ) C^{\infty}(X)
  27. X X
  28. C ( Y ) C^{\infty}(Y)
  29. Y Y
  30. Γ \Gamma
  31. T X T\to X
  32. τ ~ \widetilde{\tau}
  33. T T
  34. τ \tau
  35. X X
  36. τ τ ~ \tau\to\widetilde{\tau}
  37. [ τ ~ , τ ~ ] = [ τ , τ ] ~ . [\widetilde{\tau},\widetilde{\tau}^{\prime}]=\widetilde{[\tau,\tau^{\prime}]}.
  38. τ ~ \widetilde{\tau}
  39. T T
  40. f f ~ f\to\widetilde{f}
  41. X X
  42. T X T\to X
  43. f ~ \widetilde{f}
  44. T T
  45. T T
  46. T X TX
  47. X X
  48. f f
  49. X X
  50. f ~ = T f \widetilde{f}=Tf
  51. T X TX
  52. T X TX
  53. ( x λ , x ˙ λ ) (x^{\lambda},\dot{x}^{\lambda})
  54. T X TX
  55. x ˙ μ = x μ x ν x ˙ ν . \dot{x}^{\prime\mu}=\frac{\partial x^{\prime\mu}}{\partial x^{\nu}}\dot{x}^{% \nu}.
  56. F X FX
  57. T X TX
  58. F X FX
  59. F X FX

Generalized_beta_distribution.html

  1. G B ( y ; a , b , c , p , q ) = | a | y a p - 1 ( 1 - ( 1 - c ) ( y / b ) a ) q - 1 b a p B ( p , q ) ( 1 + c ( y / b ) a ) p + q for 0 < y a < b a 1 - c , GB(y;a,b,c,p,q)=\frac{|a|y^{ap-1}(1-(1-c)(y/b)^{a})^{q-1}}{b^{ap}B(p,q)(1+c(y/% b)^{a})^{p+q}}\quad\quad\,\text{ for }0<y^{a}<\frac{b^{a}}{1-c},
  2. 0 c 1 0\leq c\leq 1
  3. b b
  4. p p
  5. q q
  6. E G B ( Y h ) = b h B ( p + h / a , q ) B ( p , q ) F 1 2 [ p + h / a , h / a ; c p + q + h / a ; ] , \operatorname{E}_{GB}(Y^{h})=\frac{b^{h}B(p+h/a,q)}{B(p,q)}{}_{2}F_{1}\begin{% bmatrix}p+h/a,h/a;c\\ p+q+h/a;\end{bmatrix},
  7. F 1 2 {}_{2}F_{1}
  8. 0 < y a < b a 0<y^{a}<b^{a}
  9. b b
  10. p p
  11. q q
  12. G B 1 ( y ; a , b , p , q ) = G B ( y ; a , b , c = 0 , p , q ) . GB1(y;a,b,p,q)=GB(y;a,b,c=0,p,q).
  13. E G B 1 ( Y h ) = b h B ( p + h / a , q ) B ( p , q ) . \operatorname{E}_{GB1}(Y^{h})=\frac{b^{h}B(p+h/a,q)}{B(p,q)}.
  14. B 1 ( y ; b , p , q ) = G B 1 ( y ; a = 1 , b , p , q ) , B1(y;b,p,q)=GB1(y;a=1,b,p,q),
  15. G G ( y ; a , β , p ) = lim q G B 1 ( y ; a , b = q 1 / a β , p , q ) , GG(y;a,\beta,p)=\lim_{q\to\infty}GB1(y;a,b=q^{1/a}\beta,p,q),
  16. P A R E T O ( y ; b , p ) = G B 1 ( y ; a = - 1 , b , p , q = 1 ) . PARETO(y;b,p)=GB1(y;a=-1,b,p,q=1).
  17. G B 2 ( y ; a , b , p , q ) = | a | y a p - 1 b a p B ( p , q ) ( 1 + ( y / b ) a ) p + q GB2(y;a,b,p,q)=\frac{|a|y^{ap-1}}{b^{ap}B(p,q)(1+(y/b)^{a})^{p+q}}
  18. 0 < y < 0<y<\infty
  19. G B 2 ( y ; a , b , p , q ) = G B ( y ; a , b , c = 1 , p , q ) . GB2(y;a,b,p,q)=GB(y;a,b,c=1,p,q).
  20. E G B 2 ( Y h ) = b h B ( p + h / a , q - h / a ) B ( p , q ) . \operatorname{E}_{GB2}(Y^{h})=\frac{b^{h}B(p+h/a,q-h/a)}{B(p,q)}.
  21. B ( y ; b , c , p , q ) = y p - 1 ( 1 - ( 1 - c ) ( y / b ) ) q - 1 b p B ( p , q ) ( 1 + c ( y / b ) ) p + q B(y;b,c,p,q)=\frac{y^{p-1}(1-(1-c)(y/b))^{q-1}}{b^{p}B(p,q)(1+c(y/b))^{p+q}}
  22. 0 < y < b / ( 1 - c ) 0<y<b/(1-c)
  23. B ( y ; b , c , p , q ) = G B ( y ; a = 1 , b , c , p , q ) . B(y;b,c,p,q)=GB(y;a=1,b,c,p,q).
  24. G G ( y ; a , β , p ) = lim q G B 2 ( y , a , b = q 1 / a β , p , q ) = | a | y a p - 1 e - ( y / β ) a β a Γ ( p ) GG(y;a,\beta,p)=\lim_{q\rightarrow\infty}GB2(y,a,b=q^{1/a}\beta,p,q)=\frac{|a|% y^{ap-1}e^{-(y/\beta)^{a}}}{\beta^{a}\Gamma(p)}
  25. h h
  26. E ( Y G G h ) = β h Γ ( p + h / a ) Γ ( p ) . \operatorname{E}(Y_{GG}^{h})=\frac{\beta^{h}\Gamma(p+h/a)}{\Gamma(p)}.
  27. Y G B ( y ; a , b , c , p , q ) Y\sim GB(y;a,b,c,p,q)
  28. Z = ln ( Y ) Z=\ln(Y)
  29. E G B ( z ; δ , σ , c , p , q ) = e p ( z - δ ) / σ ( 1 - ( 1 - c ) e ( z - δ ) / σ ) q - 1 | σ | B ( p , q ) ( 1 + c e ( z - δ ) / σ ) p + q EGB(z;\delta,\sigma,c,p,q)=\frac{e^{p(z-\delta)/\sigma}(1-(1-c)e^{(z-\delta)/% \sigma})^{q-1}}{|\sigma|B(p,q)(1+ce^{(z-\delta)/\sigma})^{p+q}}
  30. - < z - δ σ < ln ( 1 1 - c ) -\infty<\frac{z-\delta}{\sigma}<\ln(\frac{1}{1-c})
  31. M E G B ( Z ) = e δ t B ( p + t σ , q ) B ( p , q ) F 1 2 [ p + t σ , t σ ; c p + q + t σ ; ] . M_{EGB}(Z)=\frac{e^{\delta t}B(p+t\sigma,q)}{B(p,q)}{}_{2}F_{1}\begin{bmatrix}% p+t\sigma,t\sigma;c\\ p+q+t\sigma;\end{bmatrix}.
  32. G = ( 1 2 μ ) E ( | Y - X | ) = ( P 1 2 μ ) 0 0 | x - y | f ( x ) f ( y ) d x d y = 1 - 0 ( 1 - F ( y ) ) 2 d y 0 ( 1 - F ( y ) ) d y P = ( 1 2 μ ) E ( | Y - μ | ) = ( 1 2 μ ) 0 | y - μ | f ( y ) d y T = E ( ln ( Y / μ ) Y / μ ) = 0 ( y / μ ) ln ( y / μ ) f ( y ) d y \begin{aligned}\displaystyle G=\left({\frac{1}{2\mu}}\right)\operatorname{E}(|% Y-X|)=\left(P{\frac{1}{2\mu}}\right)\int_{0}^{\infty}\int_{0}^{\infty}|x-y|f(x% )f(y)\,dxdy\\ \displaystyle=1-\frac{\int_{0}^{\infty}(1-F(y))^{2}\,dy}{\int_{0}^{\infty}(1-F% (y))\,dy}\\ \displaystyle P=\left(\frac{1}{2\mu}\right)\operatorname{E}(|Y-\mu|)=\left(% \frac{1}{2\mu}\right)\int_{0}^{\infty}|y-\mu|f(y)\,dy\\ \displaystyle T=\operatorname{E}(\ln(Y/\mu)^{Y/\mu})=\int_{0}^{\infty}(y/\mu)% \ln(y/\mu)f(y)\,dy\end{aligned}
  33. h ( s ) = f ( s ) 1 - F ( s ) h(s)=\frac{f(s)}{1-F(s)}

Generalized_distributive_law.html

  1. a * ( b + c ) = a * b + a * c a*(b+c)=a*b+a*c
  2. a a
  3. b + c b+c
  4. a * b + a * c a*b+a*c
  5. a * b + a * c a*b+a*c
  6. a * ( b + c ) a*(b+c)
  7. α ( a , b ) = def c , d , e A f ( a , c , b ) g ( a , d , e ) \alpha(a,\,b)\stackrel{\mathrm{def}}{=}\displaystyle\sum\limits_{c,d,e\in A}f(% a,\,c,\,b)\,g(a,\,d,\,e)
  8. f ( ) f(\cdot)
  9. g ( ) g(\cdot)
  10. a , b , c , d , e A a,b,c,d,e\in A
  11. | A | = q |A|=q
  12. c c
  13. d d
  14. e e
  15. q 2 q^{2}
  16. ( a , b ) (a,b)
  17. q 3 q^{3}
  18. ( c , d , e ) (c,d,e)
  19. α ( a , b ) \alpha(a,\,b)
  20. 2 q 2 q 3 = 2 q 5 2\cdot q^{2}\cdot q^{3}=2q^{5}
  21. O ( n 5 ) O(n^{5})
  22. α ( a , b ) = def c A f ( a , c , b ) d , e A g ( a , d , e ) \alpha(a,\,b)\stackrel{\mathrm{def}}{=}\displaystyle\sum\limits_{c\in A}f(a,\,% c,\,b)\cdot\sum_{d,\,e\in A}g(a,\,d,\,e)
  23. α ( a , b ) \alpha(a,\,b)
  24. α 1 ( a , b ) α 2 ( a ) \alpha_{1}(a,\,b)\cdot\alpha_{2}(a)
  25. α 1 ( a , b ) = def c A f ( a , c , b ) \alpha_{1}(a,b)\stackrel{\mathrm{def}}{=}\displaystyle\sum\limits_{c\in A}f(a,% \,c,\,b)
  26. α 2 ( a ) = def d , e A g ( a , d , e ) \alpha_{2}(a)\stackrel{\mathrm{def}}{=}\displaystyle\sum\limits_{d,\,e\in A}g(% a,\,d,\,e)
  27. q 3 q^{3}
  28. α 1 ( a , b ) \alpha_{1}(a,\,b)
  29. α 2 ( a ) \alpha_{2}(a)
  30. q 2 q^{2}
  31. α 1 ( a , b ) α 2 ( a ) \alpha_{1}(a,\,b)\cdot\alpha_{2}(a)
  32. α ( a , b ) \alpha(a,\,b)
  33. q 3 + q 3 + q 2 = 2 q 3 + q 2 q^{3}+q^{3}+q^{2}=2q^{3}+q^{2}
  34. α ( a , b ) \alpha(a,b)
  35. O ( n 3 ) O(n^{3})
  36. O ( n 5 ) O(n^{5})
  37. K K
  38. + +
  39. . .
  40. ( K , + ) (K,\,+)
  41. ( K , . ) (K,\,.)
  42. p 1 , , p n p_{1},\ldots,p_{n}
  43. p 1 A 1 , , p n A n p_{1}\in A_{1},\ldots,p_{n}\in A_{n}
  44. A A
  45. | A i | = q i |A_{i}|=q_{i}
  46. i = 1 , , n i=1,\ldots,n
  47. S = { i 1 , , i r } S=\{i_{1},\ldots,i_{r}\}
  48. S { 1 , , n } S\,\subset\{1,\ldots,n\}
  49. A S = A i 1 × × A i r A_{S}=A_{i_{1}}\times\cdots\times A_{i_{r}}
  50. p S = ( p i 1 , , p i r ) p_{S}=(p_{i_{1}},\ldots,p_{i_{r}})
  51. q S = | A S | q_{S}=|A_{S}|
  52. 𝐀 = A 1 × × A n \mathbf{A}=A_{1}\times\cdots\times A_{n}
  53. 𝐩 = { p 1 , , p n } \mathbf{p}=\{p_{1},\ldots,p_{n}\}
  54. S = { S j } j = 1 M S=\{S_{j}\}_{j=1}^{M}
  55. S j { 1 , , n } S_{j}\subset\{1,...\,,n\}
  56. α i : A S i R \alpha_{i}:A_{S_{i}}\rightarrow R
  57. R R
  58. p S i p_{S_{i}}
  59. α i \alpha_{i}
  60. β : 𝐀 R \beta:\mathbf{A}\rightarrow R
  61. β ( p 1 , , p n ) = i = 1 M α ( p S i ) \beta(p_{1},...\,,p_{n})=\prod_{i=1}^{M}\alpha(p_{S_{i}})
  62. i = 1 , , M i=1,...\,,M
  63. S i S_{i}
  64. β \beta
  65. β i : A S i R \beta_{i}:A_{S_{i}}\rightarrow R
  66. β i ( p S i ) = p S i c A S i c β ( p ) \beta_{i}(p_{S_{i}})\,=\displaystyle\sum\limits_{p_{S_{i}^{c}}\in A_{S_{i}^{c}% }}\beta(p)
  67. S i c S_{i}^{c}
  68. S i S_{i}
  69. { 1 , , n } \mathbf{\{}1,...\,,n\}
  70. β i ( p S i ) \beta_{i}(p_{S_{i}})
  71. i t h i^{th}
  72. S i S_{i}
  73. i t h i^{th}
  74. M q 1 q 2 q 3 q n Mq_{1}q_{2}q_{3}\cdots q_{n}
  75. q 1 q 2 q n q_{1}q_{2}\cdots q_{n}
  76. ( M - 1 ) q 1 q 2 q n (M-1)q_{1}q_{2}...q_{n}
  77. i th i\text{th}
  78. p 1 , p 2 , p 3 , p 4 , p_{1},\,p_{2},\,p_{3},\,p_{4},
  79. p 5 p_{5}
  80. p 1 A 1 , p 2 A 2 , p 3 A 3 , p 4 A 4 , p_{1}\in A_{1},p_{2}\in A_{2},p_{3}\in A_{3},p_{4}\in A_{4},
  81. p 5 A 5 p_{5}\in A_{5}
  82. M = 4 M=4
  83. S = { { 1 , 2 , 5 } , { 2 , 4 } , { 1 , 4 } , { 2 } } S=\{\{1,2,5\},\{2,4\},\{1,4\},\{2\}\}
  84. f ( p 1 , p 2 , p 5 ) f(p_{1},p_{2},p_{5})
  85. g ( p 3 , p 4 ) g(p_{3},p_{4})
  86. α ( p 1 , p 4 ) \alpha(p_{1},\,p_{4})
  87. β ( p 2 ) \beta(p_{2})
  88. α ( p 1 , p 4 ) = p 2 A 2 , p 3 A 3 , p 5 A 5 f ( p 1 , p 2 , p 5 ) g ( p 2 , p 4 ) \alpha(p_{1},\,p_{4})=\displaystyle\sum\limits_{p_{2}\in A_{2},\,p_{3}\in A_{3% },\,p_{5}\in A_{5}}f(p_{1},\,p_{2},\,p_{5})\cdot g(p_{2},\,p_{4})
  89. β ( p 2 ) = p 1 A 1 , p 3 A 3 , p 4 A 4 , p 5 A 5 f ( p 1 , p 2 , p 5 ) g ( p 2 , p 4 ) \beta(p_{2})=\sum\limits_{p_{1}\in A_{1},\,p_{3}\in A_{3},\,p_{4}\in A_{4},\,p% _{5}\in A_{5}}f(p_{1},\,p_{2},\,p_{5})\cdot g(p_{2},\,p_{4})
  90. { p 1 , p 2 , p 5 } \{p_{1},p_{2},p_{5}\}
  91. ( f ( p 1 , p 2 , p 5 ) (f(p_{1},p_{2},p_{5})
  92. { p 2 , p 4 } \{p_{2},p_{4}\}
  93. g ( p 2 , p 4 ) g(p_{2},p_{4})
  94. { p 1 , p 4 } \{p_{1},p_{4}\}
  95. 1 1
  96. { p 2 } \{p_{2}\}
  97. 1 1
  98. α ( p 1 , p 4 ) \alpha(p_{1},p_{4})
  99. 3 r d 3^{rd}
  100. β ( p 2 ) \beta(p_{2})
  101. 4 t h 4^{th}
  102. p 1 , p 2 , p 3 , p 4 , r 1 , r 2 , r 3 , r 4 { 0 , 1 } p_{1},p_{2},p_{3},p_{4},r_{1},r_{2},r_{3},r_{4}\in\{0,1\}
  103. f ( r 1 , r 2 , r 3 , r 4 ) f(r_{1},r_{2},r_{3},r_{4})
  104. { r 1 , r 2 , r 3 , r 4 } \{r_{1},r_{2},r_{3},r_{4}\}
  105. f ( r 1 , r 2 , r 3 , r 4 ) f(r_{1},r_{2},r_{3},r_{4})
  106. { p 1 , r 1 } \{p_{1},r_{1}\}
  107. ( - 1 ) p 1 r 1 (-1)^{p_{1}r_{1}}
  108. { p 2 , r 2 } \{p_{2},r_{2}\}
  109. ( - 1 ) p 2 r 2 (-1)^{p_{2}r_{2}}
  110. { p 3 , r 3 } \{p_{3},r_{3}\}
  111. ( - 1 ) p 3 r 3 (-1)^{p_{3}r_{3}}
  112. { p 4 , r 4 } \{p_{4},r_{4}\}
  113. ( - 1 ) p 4 r 4 (-1)^{p_{4}r_{4}}
  114. { p 1 , p 2 , p 3 , p 4 } \{p_{1},p_{2},p_{3},p_{4}\}
  115. 1 1
  116. F ( p 1 , p 2 , p 3 , p 4 , r 1 , r 2 , r 3 , r 4 ) = f ( p 1 , p 2 , p 3 , p 4 ) ( - 1 ) p 1 r 1 + p 2 r 2 + p 3 r 3 + p 4 r 4 F(p_{1},p_{2},p_{3},p_{4},r_{1},r_{2},r_{3},r_{4})=f(p_{1},p_{2},p_{3},p_{4})% \cdot(-1)^{p_{1}r_{1}+p_{2}r_{2}+p_{3}r_{3}+p_{4}r_{4}}
  117. p 1 , p 2 , p 3 , p 4 p_{1},p_{2},p_{3},p_{4}
  118. F ( p 1 , p 2 , p 3 , p 4 ) = r 1 , r 2 , r 3 , r 4 f ( r 1 , r 2 , r 3 , r 4 ) ( - 1 ) p 1 r 1 + p 2 r 2 + p 3 r 3 + p 4 r 4 . F(p_{1},p_{2},p_{3},p_{4})=\displaystyle\sum\limits_{r_{1},r_{2},r_{3},r_{4}}f% (r_{1},r_{2},r_{3},r_{4})\cdot(-1)^{p_{1}r_{1}+p_{2}r_{2}+p_{3}r_{3}+p_{4}r_{4% }}.
  119. f ( ) f(\cdot)
  120. S S
  121. S S
  122. T T
  123. v i v_{i}
  124. v j v_{j}
  125. i j i\neq j
  126. S i S j S_{i}\cap S_{j}
  127. v i v_{i}
  128. v j v_{j}
  129. { p 2 } \{p_{2}\}
  130. { p 3 , p 2 } \{p_{3},p_{2}\}
  131. { p 2 , p 1 } \{p_{2},p_{1}\}
  132. { p 3 , p 4 } \{p_{3},p_{4}\}
  133. { p 3 } \{p_{3}\}
  134. { p 1 , p 4 } \{p_{1},p_{4}\}
  135. { p 1 } \{p_{1}\}
  136. { p 4 } \{p_{4}\}
  137. { p 2 , p 4 } \{p_{2},p_{4}\}
  138. { p 1 , p 2 } \{p_{1},p_{2}\}
  139. { p 2 , p 3 } \{p_{2},p_{3}\}
  140. { p 3 , p 4 } \{p_{3},p_{4}\}
  141. { p 1 , p 4 } \{p_{1},p_{4}\}
  142. { p 1 , p 2 \{p_{1},p_{2}
  143. p 4 } p_{4}\}
  144. { p 2 , p 3 \{p_{2},p_{3}
  145. p 4 } p_{4}\}
  146. v i v_{i}
  147. v j v_{j}
  148. v i v_{i}
  149. v j v_{j}
  150. μ i , j \mu_{i,j}
  151. A S i S j R A_{S_{i}\cap S_{j}}\rightarrow R
  152. i i
  153. j j
  154. μ i , j \mu_{i,j}
  155. 1 1
  156. μ i , j ( p S i S j ) \mu_{i,j}(p_{S_{i}\cap S_{j}})
  157. p S i S j A S i S j α i ( p S i ) v k adj v i , k j μ k , j ( p S k S i ) ( 1 ) \sum_{p_{S_{i}\setminus S_{j}}\in A_{S_{i}\setminus S_{j}}}\alpha_{i}(p_{S_{i}% })\prod_{{v_{k}\operatorname{adj}v_{i}},{k\neq j}}\mu_{k,j}(p_{S_{k}\cap S_{i}% })(1)
  158. v k adj v i v_{k}\operatorname{adj}v_{i}
  159. v k v_{k}
  160. v i v_{i}
  161. σ i : A S i R \sigma_{i}:A_{S_{i}}\rightarrow R
  162. v i v_{i}
  163. α ( p S i ) \alpha(p_{S_{i}})
  164. σ i \sigma_{i}
  165. σ ( p S i ) = α i ( p S i ) v k adj v i μ k , j ( p S k S i ) ( 2 ) . \sigma(p_{S_{i}})=\alpha_{i}(p_{S_{i}})\prod_{v_{k}\operatorname{adj}v_{i}}\mu% _{k,j}(p_{S_{k}\cap S_{i}})(2).
  166. v 0 v_{0}
  167. v 0 v_{0}
  168. v 0 v_{0}
  169. v 0 v_{0}
  170. v 0 v_{0}
  171. p 2 p_{2}
  172. Round Message or State Computation \,\text{Round Message or State Computation}
  173. 1. μ 8 , 4 ( p 4 ) = α 8 ( p 4 ) 1.\mu_{8,4}(p_{4})=\alpha_{8}(p_{4})
  174. 2. μ 8 , 4 ( p 4 ) = Σ p 2 α 9 ( p 2 , p 4 ) 2.\mu_{8,4}(p_{4})=\Sigma_{p_{2}}\alpha_{9}(p_{2},p_{4})
  175. 3. μ 5 , 2 ( p 3 ) = α 5 ( p 3 ) 3.\mu_{5,2}(p_{3})=\alpha_{5}(p_{3})
  176. 4. μ 6 , 3 ( p 1 ) = Σ p 4 α 6 ( p 1 , p 4 ) 4.\mu_{6,3}(p_{1})=\Sigma_{p_{4}}\alpha_{6}(p_{1},p_{4})
  177. 5. μ 7 , 3 ( p 1 ) = α 7 ( p 1 ) 5.\mu_{7,3}(p_{1})=\alpha_{7}(p_{1})
  178. 6. μ 4 , 2 ( p 3 ) = Σ p 4 α 4 ( p 3 , p 4 ) . μ 8 , 4 ( p 4 ) . μ 9 , 4 ( p 4 ) 6.\mu_{4,2}(p_{3})=\Sigma_{p_{4}}\alpha_{4}(p_{3},p_{4}).\mu_{8,4}(p_{4}).\mu_% {9,4}(p_{4})
  179. 7. μ 3 , 1 ( p 2 ) = Σ p 1 α 3 ( p 2 , p 1 ) . μ 6 , 3 ( p 1 ) . μ 7 , 3 ( p 1 ) 7.\mu_{3,1}(p_{2})=\Sigma_{p_{1}}\alpha_{3}(p_{2},p_{1}).\mu_{6,3}(p_{1}).\mu_% {7,3}(p_{1})
  180. 8. μ 2 , 1 ( p 2 ) = Σ p 3 α 2 ( p 3 , p 2 ) . μ 4 , 2 ( p 3 ) . μ 5 , 2 ( p 3 ) 8.\mu_{2,1}(p_{2})=\Sigma_{p_{3}}\alpha_{2}(p_{3},p_{2}).\mu_{4,2}(p_{3}).\mu_% {5,2}(p_{3})
  181. 9. σ 1 ( p 2 ) = α 1 ( p 2 ) . μ 2 , 1 ( p 2 ) . μ 3 , 1 ( p 2 ) 9.\sigma_{1}(p_{2})=\alpha_{1}(p_{2}).\mu_{2,1}(p_{2}).\mu_{3,1}(p_{2})
  182. Σ v d ( v ) | A S ( v ) | \Sigma_{v}d(v)|A_{S_{(v)}}|
  183. S ( v ) S(v)
  184. v v
  185. d ( v ) d(v)
  186. v v
  187. Σ v V d ( v ) | A S ( v ) | \Sigma_{v\in V}d(v)|A_{S_{(v)}}|
  188. G L D G_{LD}
  189. M M
  190. v 1 , v 2 , v 3 , , v M v_{1},v_{2},v_{3},\ldots,v_{M}
  191. e i , j : v i v j e_{i,j}:v_{i}\leftrightarrow v_{j}
  192. ω i , j = | S i S j | \omega_{i,j}=|S_{i}\cap S_{j}|
  193. x k S i S j x_{k}\in S_{i}\cap S_{j}
  194. x k x_{k}
  195. e i , j e_{i,j}
  196. ω m a x \omega_{max}
  197. G L D G_{LD}
  198. ω * = Σ i = 1 M | S i | - n \omega^{*}=\Sigma^{M}_{i=1}|S_{i}|-n
  199. T {}^{\prime}T^{\prime}
  200. V {}^{\prime}V^{\prime}
  201. E {}^{\prime}E^{\prime}
  202. E {}^{\prime}E^{\prime}
  203. E = { ( 1 , 2 ) , ( 2 , 1 ) , ( 1 , 3 ) , ( 3 , 1 ) , ( 4 , 2 ) , ( 2 , 4 ) , ( 5 , 2 ) , ( 2 , 5 ) , ( 6 , 3 ) , ( 3 , 6 ) , ( 7 , 3 ) , ( 3 , 7 ) , ( 8 , 4 ) , ( 4 , 8 ) , ( 9 , 4 ) , ( 4 , 9 ) } E=\{(1,2),(2,1),(1,3),(3,1),(4,2),(2,4),(5,2),(2,5),(6,3),(3,6),(7,3),(3,7),(8% ,4),(4,8),(9,4),(4,9)\}
  204. E E
  205. E E
  206. = \mathcal{E}=
  207. E 1 , E 2 , E 3 , , E N E_{1},E_{2},E_{3},\ldots,E_{N}
  208. E N E_{N}
  209. N t h N^{th}
  210. = { E 1 , E 2 , E 3 , , E N } \mathcal{E}=\{E_{1},E_{2},E_{3},\ldots,E_{N}\}
  211. V × { 0 , 1 , 2 , 3 , , N } V\times\{0,1,2,3,\ldots,N\}
  212. v i ( t ) v_{i}(t)
  213. t { 0 , 1 , 2 , 3 , , N } t\in\{0,1,2,3,\ldots,N\}
  214. v j v_{j}
  215. j th j\text{th}
  216. σ ( p S i ) = α i ( p S i ) v k adj v i μ k , j ( p S k S i ) \sigma(p_{S_{i}})=\alpha_{i}(p_{S_{i}})\prod_{v_{k}\operatorname{adj}v_{i}}\mu% _{k,j}(p_{S_{k}\cap S_{i}})
  217. v i ( 0 ) v_{i}(0)
  218. v j ( N ) v_{j}(N)
  219. a b + a c ab+ac
  220. a ( b + c ) a(b+c)
  221. μ v , w ( p v w ) = p v w A S ( v ) S ( w ) α v ( p v ) u a d j v u v μ u , v ( p u v ) \mu_{v,w}(p_{v\cap w})=\sum_{p_{v\setminus w}\in A_{S(v)\setminus S(w)}}\alpha% _{v}(p_{v})\prod_{uadjv_{u\neq v}}\mu_{u,v}(p_{u\cap v})
  222. σ v ( p v ) = α v ( p v ) u adj v μ v , w ( p v w ) \sigma_{v}(p_{v})=\alpha_{v}(p_{v})\prod_{u\operatorname{adj}v}\mu_{v,w}(p_{v% \cap w})
  223. v 0 v_{0}
  224. v v
  225. v 0 v_{0}
  226. ( v , w ) (v,w)
  227. p u v p_{u\cap v}
  228. q v w - 1 q_{v\setminus w}-1
  229. q v w ( d ( v ) - 1 ) q_{v\setminus w}(d(v)-1)
  230. | A S ( v ) S ( w ) | |A_{S(v)\ S(w)}|
  231. q v w q_{v\setminus w}
  232. x v w x_{v\cap w}
  233. q v w = def | A S ( v ) S ( w ) | q_{v\cap w}\stackrel{\mathrm{def}}{=}|A_{S(v)\cap S(w)}|
  234. p v w p_{v\cap w}
  235. ( q v w ) ( q v w - 1 ) = q v - q v w (q_{v\cap w})(q_{v\setminus w}-1)=q_{v}-q_{v\cap w}
  236. ( q v w ) q v w . ( d ( v ) - 1 ) = ( d ( v ) - 1 ) q v (q_{v\cap w})q_{v\setminus w}.(d(v)-1)=(d(v)-1)q_{v}
  237. v 0 v_{0}
  238. v v 0 ( q v - q v w ) \sum_{v\neq v0}(q_{v}-q_{v\cap w})
  239. v v 0 ( d ( v ) - 1 ) q v \sum_{v\neq v0}(d(v)-1)q_{v}
  240. v 0 v_{0}
  241. d ( v 0 ) q 0 d(v_{0})q_{0}
  242. v v 0 ( q v - q v w ) \sum_{v\neq v_{0}}(q_{v}-q_{v\cap w})
  243. v v 0 ( d ( v ) - 1 ) q v + d ( v 0 ) q v 0 \sum_{v\neq v_{0}}(d(v)-1)q_{v}+d(v_{0})q_{v_{0}}
  244. χ ( T ) = v V d ( v ) q v - e E q e \chi(T)=\sum_{v\in V}d(v)q_{v}-\sum_{e\in E}q_{e}
  245. ( 1 ) (1)
  246. e = ( v , w ) e=(v,w)
  247. q v w q_{v\cap w}
  248. e = ( v , w ) e=(v,w)
  249. χ ( e ) = q v + q w - q v w \chi(e)=q_{v}+q_{w}-q_{v\cap w}
  250. ( 1 ) (1)
  251. χ ( T ) = e E χ ( e ) \chi(T)=\sum_{e\in E}\chi(e)
  252. χ ( 1 , 2 ) = q 2 + q 2 q 3 - q 2 \chi(1,2)=q_{2}+q_{2}q_{3}-q_{2}
  253. χ ( 2 , 4 ) = q 3 q 4 + q 2 q 3 - q 3 \chi(2,4)=q_{3}q_{4}+q_{2}q_{3}-q_{3}
  254. χ ( 2 , 5 ) = q 3 + q 2 q 3 - q 3 \chi(2,5)=q_{3}+q_{2}q_{3}-q_{3}
  255. χ ( 4 , 8 ) = q 4 + q 3 q 4 - q 4 \chi(4,8)=q_{4}+q_{3}q_{4}-q_{4}
  256. χ ( 4 , 9 ) = q 2 q 4 + q 3 q 4 - q 4 \chi(4,9)=q_{2}q_{4}+q_{3}q_{4}-q_{4}
  257. χ ( 1 , 3 ) = q 2 + q 2 q 1 - q 2 \chi(1,3)=q_{2}+q_{2}q_{1}-q_{2}
  258. χ ( 3 , 7 ) = q 1 + q 1 q 2 - q 1 \chi(3,7)=q_{1}+q_{1}q_{2}-q_{1}
  259. χ ( 3 , 6 ) = q 1 q 4 + q 1 q 2 - q 1 \chi(3,6)=q_{1}q_{4}+q_{1}q_{2}-q_{1}
  260. 3 q 2 q 3 + 3 q 3 q 4 + 3 q 1 q 2 + q 2 q 4 + q 1 q 4 - q 1 - q 3 - q 4 3q_{2}q_{3}+3q_{3}q_{4}+3q_{1}q_{2}+q_{2}q_{4}+q_{1}q_{4}-q_{1}-q_{3}-q_{4}
  261. O ( v d ( v ) d ( v ) q v ) O(\sum_{v}d(v)d(v)q_{v})
  262. 3 ( d - 2 ) 3(d-2)
  263. d d
  264. ( a 1 , , a d ) (a_{1},\ldots,a_{d})
  265. d d
  266. d - 1 d-1
  267. a i a_{i}
  268. 3 ( d - 2 ) 3(d-2)
  269. d ( d - 2 ) d(d-2)
  270. b 1 = a 1 , b 2 = b 1 a 2 = a 1 a 2 , b d - 1 = b d - 2 a d - 1 = a 1 a 2 a d - 1 b_{1}=a_{1},b_{2}=b_{1}\cdot a_{2}=a_{1}\cdot a_{2},b_{d-1}=b_{d-2}\cdot a_{d-% 1}=a_{1}a_{2}\cdots a_{d-1}
  271. c d = a d , c d - 1 = a d - 1 c d = a d - 1 a d , , c 2 = a 2 c 3 = a 2 a 3 a d c_{d}=a_{d},c_{d-1}=a_{d-1}c_{d}=a_{d-1}\cdot a_{d},\ldots,c_{2}=a_{2}\cdot c_% {3}=a_{2}a_{3}\cdots a_{d}
  272. 2 ( d - 2 ) 2(d-2)
  273. m j m_{j}
  274. a i a_{i}
  275. a j a_{j}
  276. m 1 = c 2 , m 2 = b 1 c 3 m_{1}=c_{2},m_{2}=b_{1}\cdot c_{3}
  277. d - 2 d-2
  278. 3 ( d - 2 ) 3(d-2)
  279. χ ( T ) \chi(T)

Generalized_multivariate_log-gamma_distribution.html

  1. s y m b o l Y G - MVLG ( δ , ν , s y m b o l λ , s y m b o l μ ) symbol{Y}\sim\mathrm{G}\,\text{-}\mathrm{MVLG}(\delta,\nu,symbol{\lambda},% symbol{\mu})
  2. s y m b o l Y = ( Y 1 , , Y k ) symbol{Y}=(Y_{1},\dots,Y_{k})
  3. f ( y 1 , , y k ) = δ ν n = 0 ( 1 - δ ) n i = 1 k μ i λ i - ν - n [ Γ ( ν + n ) ] k - 1 Γ ( ν ) n ! exp { ( ν + n ) i = 1 k μ i y i - i = 1 k 1 λ i exp { μ i y i } } , f(y_{1},\dots,y_{k})=\delta^{\nu}\sum_{n=0}^{\infty}\frac{(1-\delta)^{n}\prod_% {i=1}^{k}\mu_{i}\lambda_{i}^{-\nu-n}}{[\Gamma(\nu+n)]^{k-1}\Gamma(\nu)n!}\exp% \bigg\{(\nu+n)\sum_{i=1}^{k}\mu_{i}y_{i}-\sum_{i=1}^{k}\frac{1}{\lambda_{i}}% \exp\{\mu_{i}y_{i}\}\bigg\},
  4. s y m b o l y k , ν > 0 , λ j > 0 , μ j > 0 symbol{y}\in\mathbb{R}^{k},\nu>0,\lambda_{j}>0,\mu_{j}>0
  5. j = 1 , , k , δ = det ( s y m b o l Ω ) 1 k - 1 , j=1,\dots,k,\delta=\det(symbol{\Omega})^{\frac{1}{k-1}},
  6. s y m b o l Ω = ( 1 abs ( ρ 12 ) abs ( ρ 1 k ) abs ( ρ 12 ) 1 abs ( ρ 2 k ) abs ( ρ 1 k ) abs ( ρ 2 k ) 1 ) , symbol{\Omega}=\left(\begin{array}[]{cccc}1&\sqrt{\mathrm{abs}(\rho_{12})}&% \cdots&\sqrt{\mathrm{abs}(\rho_{1k})}\\ \sqrt{\mathrm{abs}(\rho_{12})}&1&\cdots&\sqrt{\mathrm{abs}(\rho_{2k})}\\ \vdots&\vdots&\ddots&\vdots\\ \sqrt{\mathrm{abs}(\rho_{1k})}&\sqrt{\mathrm{abs}(\rho_{2k})}&\cdots&1\end{% array}\right),
  7. ρ i j \rho_{ij}
  8. Y i Y_{i}
  9. Y j Y_{j}
  10. det ( ) \det(\cdot)
  11. abs ( ) \mathrm{abs}(\cdot)
  12. s y m b o l g = ( δ , ν , s y m b o l λ T , s y m b o l μ T ) symbol{g}=(\delta,\nu,symbol{\lambda}^{T},symbol{\mu}^{T})
  13. M s y m b o l Y ( s y m b o l t ) = δ ν ( i = 1 k λ i t i / μ i ) n = 0 Γ ( ν + n ) Γ ( ν ) n ! ( 1 - δ ) n i = 1 k Γ ( ν + n + t i / μ i ) Γ ( ν + n ) . M_{symbol{Y}}(symbol{t})=\delta^{\nu}\bigg(\prod_{i=1}^{k}\lambda_{i}^{t_{i}/% \mu_{i}}\bigg)\sum_{n=0}^{\infty}\frac{\Gamma(\nu+n)}{\Gamma(\nu)n!}(1-\delta)% ^{n}\prod_{i=1}^{k}\frac{\Gamma(\nu+n+t_{i}/\mu_{i})}{\Gamma(\nu+n)}.
  14. r th r\text{th}
  15. Y i Y_{i}
  16. μ i r = [ ( λ i / δ ) t i / μ i Γ ( ν ) k = 0 r ( r k ) [ ln ( λ i / δ ) μ i ] r - k k Γ ( ν + t i / μ i ) t i k ] t i = 0 . {\mu_{i}}^{\prime}_{r}=\left[\frac{(\lambda_{i}/\delta)^{t_{i}/\mu_{i}}}{% \Gamma(\nu)}\sum_{k=0}^{r}{\left({{r}\atop{k}}\right)}\left[\frac{\ln(\lambda_% {i}/\delta)}{\mu_{i}}\right]^{r-k}\frac{\partial^{k}\Gamma(\nu+t_{i}/\mu_{i})}% {\partial t_{i}^{k}}\right]_{t_{i}=0}.
  17. Y i Y_{i}
  18. E ( Y i ) = 1 μ i [ ln ( λ i / δ ) + ϝ ( ν ) ] , \operatorname{E}(Y_{i})=\frac{1}{\mu_{i}}\big[\ln(\lambda_{i}/\delta)+\digamma% (\nu)\big],
  19. var ( Z i ) = ϝ [ 1 ] ( ν ) / ( μ i ) 2 \operatorname{var}(Z_{i})=\digamma^{[1]}(\nu)/(\mu_{i})^{2}
  20. ϝ ( ν ) \digamma(\nu)
  21. ϝ [ 1 ] ( ν ) \digamma^{[1]}(\nu)
  22. ν \nu
  23. s y m b o l T G - MVGB ( δ , ν , s y m b o l λ , s y m b o l μ ) symbol{T}\sim\mathrm{G}\,\text{-}\mathrm{MVGB}(\delta,\nu,symbol{\lambda},% symbol{\mu})
  24. f ( t 1 , , t k ; δ , ν , s y m b o l λ , s y m b o l μ ) ) = δ ν n = 0 ( 1 - δ ) n i = 1 k μ i λ i - ν - n [ Γ ( ν + n ) ] k - 1 Γ ( ν ) n ! exp { - ( ν + n ) i = 1 k μ i t i - i = 1 k 1 λ i exp { - μ i t i } } , t i . f(t_{1},\dots,t_{k};\delta,\nu,symbol{\lambda},symbol{\mu}))=\delta^{\nu}\sum_% {n=0}^{\infty}\frac{(1-\delta)^{n}\prod_{i=1}^{k}\mu_{i}\lambda_{i}^{-\nu-n}}{% [\Gamma(\nu+n)]^{k-1}\Gamma(\nu)n!}\exp\bigg\{-(\nu+n)\sum_{i=1}^{k}\mu_{i}t_{% i}-\sum_{i=1}^{k}\frac{1}{\lambda_{i}}\exp\{-\mu_{i}t_{i}\}\bigg\},\quad t_{i}% \in\mathbb{R}.

Geodesics_on_an_ellipsoid.html

  1. \color w h i t e . cos α d s = ρ d ϕ = - d R / sin ϕ , sin α d s = R d λ , {\color{white}.}\qquad\cos\alpha\,ds=\rho\,d\phi=-dR/\sin\phi,\quad\sin\alpha% \,ds=R\,d\lambda,
  2. d s 2 = ρ 2 d ϕ 2 + R 2 d λ 2 ds^{2}=\rho^{2}\,d\phi^{2}+R^{2}\,d\lambda^{2}
  3. d s = ρ 2 ϕ 2 + R 2 d λ L ( ϕ , ϕ ) d λ , \begin{aligned}\displaystyle ds&\displaystyle=\sqrt{\rho^{2}\phi^{\prime 2}+R^% {2}}\,d\lambda\\ &\displaystyle\equiv L(\phi,\phi^{\prime})\,d\lambda,\end{aligned}
  4. s 12 = λ 1 λ 2 L ( ϕ , ϕ ) d λ , s_{12}=\int_{\lambda_{1}}^{\lambda_{2}}L(\phi,\phi^{\prime})\,d\lambda,
  5. L - ϕ L ϕ = const. L-\phi^{\prime}\frac{\partial L}{\partial\phi^{\prime}}=\,\text{const.}
  6. R sin α = const. R\sin\alpha=\,\text{const.}
  7. d α = sin ϕ d λ . d\alpha=\sin\phi\,d\lambda.
  8. \color w h i t e . d ϕ d s = cos α ρ ; d λ d s = sin α ν cos ϕ ; d α d s = tan ϕ sin α ν . {\color{white}.}\qquad\displaystyle\frac{d\phi}{ds}=\frac{\cos\alpha}{\rho};% \quad\frac{d\lambda}{ds}=\frac{\sin\alpha}{\nu\cos\phi};\quad\frac{d\alpha}{ds% }=\frac{\tan\phi\sin\alpha}{\nu}.
  9. R = a cos β R=a\cos\beta
  10. sin α 1 cos β 1 = sin α 2 cos β 2 . \sin\alpha_{1}\cos\beta_{1}=\sin\alpha_{2}\cos\beta_{2}.
  11. \color w h i t e . cos α d σ = d β , sin α d σ = cos β d ω . {\color{white}.}\qquad\cos\alpha\,d\sigma=d\beta,\quad\sin\alpha\,d\sigma=\cos% \beta\,d\omega.
  12. 1 a d s d σ = d λ d ω = sin β sin ϕ . \frac{1}{a}\frac{ds}{d\sigma}=\frac{d\lambda}{d\omega}=\frac{\sin\beta}{\sin% \phi}.
  13. R 2 a 2 + Z 2 b 2 = 1 , \frac{R^{2}}{a^{2}}+\frac{Z^{2}}{b^{2}}=1,
  14. R sin ϕ a 2 - Z cos ϕ b 2 = 0 ; \frac{R\sin\phi}{a^{2}}-\frac{Z\cos\phi}{b^{2}}=0;
  15. R a = cos β = cos ϕ 1 - e 2 sin 2 ϕ . \frac{R}{a}=\cos\beta=\frac{\cos\phi}{\sqrt{1-e^{2}\sin^{2}\phi}}.
  16. tan β = 1 - e 2 tan ϕ = ( 1 - f ) tan ϕ , \tan\beta=\sqrt{1-e^{2}}\tan\phi=(1-f)\tan\phi,
  17. sin β sin ϕ = 1 - e 2 cos 2 β , \frac{\sin\beta}{\sin\phi}=\sqrt{1-e^{2}\cos^{2}\beta},
  18. 1 a d s d σ = d λ d ω = 1 - e 2 cos 2 β . \frac{1}{a}\frac{ds}{d\sigma}=\frac{d\lambda}{d\omega}=\sqrt{1-e^{2}\cos^{2}% \beta}.
  19. sin β = sin β ( σ ; α 0 ) = cos α 0 sin σ , \sin\beta=\sin\beta(\sigma;\alpha_{0})=\cos\alpha_{0}\sin\sigma,
  20. \color w h i t e . s b = 0 σ 1 - e 2 cos 2 β ( σ ; α 0 ) 1 - f d σ = 0 σ 1 + k 2 sin 2 σ d σ , {\color{white}.}\qquad\begin{aligned}\displaystyle\frac{s}{b}&\displaystyle=% \int_{0}^{\sigma}\frac{\sqrt{1-e^{2}\cos^{2}\beta(\sigma^{\prime};\alpha_{0})}% }{1-f}\,d\sigma^{\prime}\\ &\displaystyle=\int_{0}^{\sigma}\sqrt{1+k^{2}\sin^{2}\sigma^{\prime}}\,d\sigma% ^{\prime},\end{aligned}
  21. k = e cos α 0 , k=e^{\prime}\cos\alpha_{0},
  22. d ω = sin α 0 cos 2 β d σ , d\omega=\frac{\sin\alpha_{0}}{\cos^{2}\beta}\,d\sigma,
  23. \color w h i t e . λ - λ 0 = ( 1 - f ) sin α 0 0 σ 1 + k 2 sin 2 σ 1 - cos 2 α 0 sin 2 σ d σ = ω - sin α 0 0 σ e 2 1 + 1 - e 2 cos 2 β ( σ ; α 0 ) d σ = ω - f sin α 0 0 σ 2 - f 1 + ( 1 - f ) 1 + k 2 sin 2 σ d σ , {\color{white}.}\qquad\begin{aligned}\displaystyle\lambda-\lambda_{0}&% \displaystyle=(1-f)\sin\alpha_{0}\int_{0}^{\sigma}\frac{\sqrt{1+k^{2}\sin^{2}% \sigma^{\prime}}}{1-\cos^{2}\alpha_{0}\sin^{2}\sigma^{\prime}}\,d\sigma^{% \prime}\\ &\displaystyle=\omega-\sin\alpha_{0}\int_{0}^{\sigma}\frac{e^{2}}{1+\sqrt{1-e^% {2}\cos^{2}\beta(\sigma^{\prime};\alpha_{0})}}\,d\sigma^{\prime}\\ &\displaystyle=\omega-f\sin\alpha_{0}\int_{0}^{\sigma}\frac{2-f}{1+(1-f)\sqrt{% 1+k^{2}\sin^{2}\sigma^{\prime}}}\,d\sigma^{\prime},\end{aligned}
  24. I = B 0 σ + j = 1 B j sin 2 j σ I=B_{0}\sigma+\sum_{j=1}^{\infty}B_{j}\sin 2j\sigma
  25. \color w h i t e . s b = E ( σ , i k ) , {\color{white}.}\qquad\displaystyle\frac{s}{b}=E(\sigma,ik),
  26. \color w h i t e . λ = ( 1 - f ) sin α 0 G ( σ , cos 2 α 0 , i k ) = χ - e 2 1 + e 2 sin α 0 H ( σ , - e 2 , i k ) , {\color{white}.}\qquad\begin{aligned}\displaystyle\lambda&\displaystyle=(1-f)% \sin\alpha_{0}G(\sigma,\cos^{2}\alpha_{0},ik)\\ &\displaystyle=\chi-\frac{e^{\prime 2}}{\sqrt{1+e^{\prime 2}}}\sin\alpha_{0}H(% \sigma,-e^{\prime 2},ik),\\ \end{aligned}
  27. tan χ = 1 + e 2 1 + k 2 sin 2 σ tan ω , \tan\chi=\sqrt{\frac{1+e^{\prime 2}}{1+k^{2}\sin^{2}\sigma}}\tan\omega,
  28. G ( ϕ , α 2 , k ) = 0 ϕ 1 - k 2 sin 2 θ 1 - α 2 sin 2 θ d θ = k 2 α 2 F ( ϕ , k ) + ( 1 - k 2 α 2 ) Π ( ϕ , α 2 , k ) , H ( ϕ , α 2 , k ) = 0 ϕ cos 2 θ ( 1 - α 2 sin 2 θ ) 1 - k 2 sin 2 θ d θ = 1 α 2 F ( ϕ , k ) + ( 1 - 1 α 2 ) Π ( ϕ , α 2 , k ) , \begin{aligned}\displaystyle G(\phi,\alpha^{2},k)&\displaystyle=\int_{0}^{\phi% }\frac{\sqrt{1-k^{2}\sin^{2}\theta}}{1-\alpha^{2}\sin^{2}\theta}\,d\theta\\ &\displaystyle=\frac{k^{2}}{\alpha^{2}}F(\phi,k)+\biggl(1-\frac{k^{2}}{\alpha^% {2}}\biggr)\Pi(\phi,\alpha^{2},k),\\ \displaystyle H(\phi,\alpha^{2},k)&\displaystyle=\int_{0}^{\phi}\frac{\cos^{2}% \theta}{(1-\alpha^{2}\sin^{2}\theta)\sqrt{1-k^{2}\sin^{2}\theta}}\,d\theta\\ &\displaystyle=\frac{1}{\alpha^{2}}F(\phi,k)+\biggl(1-\frac{1}{\alpha^{2}}% \biggr)\Pi(\phi,\alpha^{2},k),\end{aligned}
  29. sin α 0 = sin α cos β = tan ω cot σ , cos σ = cos β cos ω = tan α 0 cot α , cos α = cos ω cos α 0 = cot σ tan β , sin β = cos α 0 sin σ = cot α tan ω , sin ω = sin σ sin α = tan β tan α 0 . \begin{aligned}\displaystyle\sin\alpha_{0}&\displaystyle=\sin\alpha\cos\beta=% \tan\omega\cot\sigma,\\ \displaystyle\cos\sigma&\displaystyle=\cos\beta\cos\omega=\tan\alpha_{0}\cot% \alpha,\\ \displaystyle\cos\alpha&\displaystyle=\cos\omega\cos\alpha_{0}=\cot\sigma\tan% \beta,\\ \displaystyle\sin\beta&\displaystyle=\cos\alpha_{0}\sin\sigma=\cot\alpha\tan% \omega,\\ \displaystyle\sin\omega&\displaystyle=\sin\sigma\sin\alpha=\tan\beta\tan\alpha% _{0}.\end{aligned}
  30. \color w h i t e . ϕ 1 0 , | ϕ 2 | | ϕ 1 | , 0 λ 12 π . {\color{white}.}\qquad\phi_{1}\leq 0,\quad\left|\phi_{2}\right|\leq\left|\phi_% {1}\right|,\quad 0\leq\lambda_{12}\leq\pi.
  31. tan α 1 = cos β 2 sin ω 12 cos β 1 sin β 2 - sin β 1 cos β 2 cos ω 12 , \tan\alpha_{1}=\frac{\cos\beta_{2}\sin\omega_{12}}{\cos\beta_{1}\sin\beta_{2}-% \sin\beta_{1}\cos\beta_{2}\cos\omega_{12}},
  32. g ( α 1 ) λ 12 ( α 1 ; ϕ 1 , ϕ 2 ) - λ 12 = 0 , g(\alpha_{1})\equiv\lambda_{12}(\alpha_{1};\phi_{1},\phi_{2})-\lambda_{12}=0,
  33. λ 12 = ω 12 - f sin α 0 σ 1 σ 2 2 - f 1 + ( 1 - f ) 1 + k 2 sin 2 σ d σ = ω 12 - f sin α 0 I ( σ 1 , σ 2 ; α 0 ) . \begin{aligned}\displaystyle\lambda_{12}&\displaystyle=\omega_{12}-f\sin\alpha% _{0}\int_{\sigma_{1}}^{\sigma_{2}}\frac{2-f}{1+(1-f)\sqrt{1+k^{2}\sin^{2}% \sigma^{\prime}}}\,d\sigma^{\prime}\\ &\displaystyle=\omega_{12}-f\sin\alpha_{0}I(\sigma_{1},\sigma_{2};\alpha_{0}).% \end{aligned}
  34. ω 12 = λ 12 + f sin α 0 I ( σ 1 , σ 2 ; α 0 ) . \omega_{12}=\lambda_{12}+f\sin\alpha_{0}I(\sigma_{1},\sigma_{2};\alpha_{0}).
  35. \color w h i t e . d 2 t ( s ) d s 2 = K ( s ) t ( s ) , {\color{white}.}\qquad\displaystyle\frac{d^{2}t(s)}{ds^{2}}=K(s)t(s),
  36. t ( s 2 ) = C m ( s 1 , s 2 ) + D M ( s 1 , s 2 ) t(s_{2})=Cm(s_{1},s_{2})+DM(s_{1},s_{2})
  37. m ( s 1 , s 1 ) = 0 , d m ( s 1 , s 2 ) d s 2 | s 2 = s 1 = 1 , M ( s 1 , s 1 ) = 1 , d M ( s 1 , s 2 ) d s 2 | s 2 = s 1 = 0. \begin{aligned}\displaystyle m(s_{1},s_{1})&\displaystyle=0,\quad\left.\frac{% dm(s_{1},s_{2})}{ds_{2}}\right|_{s_{2}=s_{1}}=1,\\ \displaystyle M(s_{1},s_{1})&\displaystyle=1,\quad\left.\frac{dM(s_{1},s_{2})}% {ds_{2}}\right|_{s_{2}=s_{1}}=0.\end{aligned}
  38. m 12 + m 21 = 0. m_{12}+m_{21}=0.
  39. d m 12 d s 2 = M 21 , d M 12 d s 2 = - 1 - M 12 M 21 m 12 . \begin{aligned}\displaystyle\frac{dm_{12}}{ds_{2}}&\displaystyle=M_{21},\\ \displaystyle\frac{dM_{12}}{ds_{2}}&\displaystyle=-\frac{1-M_{12}M_{21}}{m_{12% }}.\end{aligned}
  40. m 13 = m 12 M 23 + m 23 M 21 , M 13 = M 12 M 23 - ( 1 - M 12 M 21 ) m 23 m 12 , M 31 = M 32 M 21 - ( 1 - M 23 M 32 ) m 12 m 23 . \begin{aligned}\displaystyle m_{13}&\displaystyle=m_{12}M_{23}+m_{23}M_{21},\\ \displaystyle M_{13}&\displaystyle=M_{12}M_{23}-(1-M_{12}M_{21})\frac{m_{23}}{% m_{12}},\\ \displaystyle M_{31}&\displaystyle=M_{32}M_{21}-(1-M_{23}M_{32})\frac{m_{12}}{% m_{23}}.\end{aligned}
  41. K = 1 ρ ν = ( 1 - e 2 sin 2 ϕ ) 2 b 2 = b 2 a 4 ( 1 - e 2 cos 2 β ) 2 . K=\frac{1}{\rho\nu}=\frac{(1-e^{2}\sin^{2}\phi)^{2}}{b^{2}}=\frac{b^{2}}{a^{4}% (1-e^{2}\cos^{2}\beta)^{2}}.
  42. m 12 / b = 1 + k 2 sin 2 σ 2 cos σ 1 sin σ 2 - 1 + k 2 sin 2 σ 1 sin σ 1 cos σ 2 - cos σ 1 cos σ 2 ( J ( σ 2 ) - J ( σ 1 ) ) , M 12 = cos σ 1 cos σ 2 + 1 + k 2 sin 2 σ 2 1 + k 2 sin 2 σ 1 sin σ 1 sin σ 2 - sin σ 1 cos σ 2 ( J ( σ 2 ) - J ( σ 1 ) ) 1 + k 2 sin 2 σ 1 , \begin{aligned}\displaystyle m_{12}/b&\displaystyle=\sqrt{1+k^{2}\sin^{2}% \sigma_{2}}\,\cos\sigma_{1}\sin\sigma_{2}-\sqrt{1+k^{2}\sin^{2}\sigma_{1}}\,% \sin\sigma_{1}\cos\sigma_{2}\\ &\displaystyle\quad-\cos\sigma_{1}\cos\sigma_{2}\bigl(J(\sigma_{2})-J(\sigma_{% 1})\bigr),\\ \displaystyle M_{12}&\displaystyle=\cos\sigma_{1}\cos\sigma_{2}+\frac{\sqrt{1+% k^{2}\sin^{2}\sigma_{2}}}{\sqrt{1+k^{2}\sin^{2}\sigma_{1}}}\sin\sigma_{1}\sin% \sigma_{2}\\ &\displaystyle\quad-\frac{\sin\sigma_{1}\cos\sigma_{2}\bigl(J(\sigma_{2})-J(% \sigma_{1})\bigr)}{\sqrt{1+k^{2}\sin^{2}\sigma_{1}}},\end{aligned}
  43. J ( σ ) = 0 σ k 2 sin 2 σ 1 + k 2 sin 2 σ d σ = E ( σ , i k ) - F ( σ , i k ) . \begin{aligned}\displaystyle J(\sigma)&\displaystyle=\int_{0}^{\sigma}\frac{k^% {2}\sin^{2}\sigma^{\prime}}{\sqrt{1+k^{2}\sin^{2}\sigma^{\prime}}}\,d\sigma^{% \prime}\\ &\displaystyle=E(\sigma,ik)-F(\sigma,ik).\end{aligned}
  44. m 12 = sin ( K s 12 ) / K , M 12 = cos ( K s 12 ) . m_{12}=\sin(\sqrt{K}s_{12})/\sqrt{K},\quad M_{12}=\cos(\sqrt{K}s_{12}).
  45. x 2 / 3 + y 2 / 3 = 1 x^{2/3}+y^{2/3}=1
  46. x = cos 3 θ , y = sin 3 θ . x=\cos^{3}\theta,\quad y=\sin^{3}\theta.
  47. x cos γ + y sin γ = 1 , \frac{x}{\cos\gamma}+\frac{y}{\sin\gamma}=1,
  48. T = d T = 1 K cos ϕ d ϕ d λ , T=\int dT=\int\frac{1}{K}\cos\phi\,d\phi\,d\lambda,
  49. Γ = K d T = cos ϕ d ϕ d λ , \Gamma=\int K\,dT=\int\cos\phi\,d\phi\,d\lambda,
  50. Γ = 2 π - j θ j \Gamma=2\pi-\sum_{j}\theta_{j}
  51. T = R 2 2 Γ + ( 1 K - R 2 2 ) cos ϕ d ϕ d λ = R 2 2 Γ + ( b 2 ( 1 - e 2 sin 2 ϕ ) 2 - R 2 2 ) cos ϕ d ϕ d λ , \begin{aligned}\displaystyle T&\displaystyle=R_{2}^{2}\,\Gamma+\int\biggl(% \frac{1}{K}-R_{2}^{2}\biggr)\cos\phi\,d\phi\,d\lambda\\ &\displaystyle=R_{2}^{2}\,\Gamma+\int\biggl(\frac{b^{2}}{(1-e^{2}\sin^{2}\phi)% ^{2}}-R_{2}^{2}\biggr)\cos\phi\,d\phi\,d\lambda,\end{aligned}
  52. S 12 = R 2 2 ( α 2 - α 1 ) + b 2 λ 1 λ 2 ( 1 2 ( 1 - e 2 sin 2 ϕ ) + tanh - 1 ( e sin ϕ ) 2 e sin ϕ - R 2 2 b 2 ) sin ϕ d λ , S_{12}=R_{2}^{2}(\alpha_{2}-\alpha_{1})+b^{2}\int_{\lambda_{1}}^{\lambda_{2}}% \biggl(\frac{1}{2(1-e^{2}\sin^{2}\phi)}+\frac{\tanh^{-1}(e\sin\phi)}{2e\sin% \phi}-\frac{R_{2}^{2}}{b^{2}}\biggr)\sin\phi\,d\lambda,
  53. S 12 = R 2 2 E 12 - e 2 a 2 cos α 0 sin α 0 σ 1 σ 2 t ( e 2 ) - t ( k 2 sin 2 σ ) e 2 - k 2 sin 2 σ sin σ 2 d σ , S_{12}=R_{2}^{2}E_{12}-e^{2}a^{2}\cos\alpha_{0}\sin\alpha_{0}\int_{\sigma_{1}}% ^{\sigma_{2}}\frac{t(e^{\prime 2})-t(k^{2}\sin^{2}\sigma)}{e^{\prime 2}-k^{2}% \sin^{2}\sigma}\frac{\sin\sigma}{2}\,d\sigma,
  54. t ( x ) = 1 + x + 1 + x sinh - 1 x x , t(x)=1+x+\sqrt{1+x}\frac{\sinh^{-1}\!\sqrt{x}}{\sqrt{x}},
  55. tan E 12 2 = sin 1 2 ( β 2 + β 1 ) cos 1 2 ( β 2 - β 1 ) tan ω 12 2 . \tan\frac{E_{12}}{2}=\frac{\sin\tfrac{1}{2}(\beta_{2}+\beta_{1})}{\cos\tfrac{1% }{2}(\beta_{2}-\beta_{1})}\tan\frac{\omega_{12}}{2}.
  56. h = X 2 a 2 + Y 2 b 2 + Z 2 c 2 = 1 , h=\frac{X^{2}}{a^{2}}+\frac{Y^{2}}{b^{2}}+\frac{Z^{2}}{c^{2}}=1,
  57. h | h | = ( cos ϕ cos λ cos ϕ sin λ sin ϕ ) . \frac{\nabla h}{\left|\nabla h\right|}=\left(\begin{array}[]{c}\cos\phi\cos% \lambda\\ \cos\phi\sin\lambda\\ \sin\phi\end{array}\right).
  58. X = a cos ϕ cos λ , Y = b cos ϕ sin λ , Z = c sin ϕ . \begin{aligned}\displaystyle X&\displaystyle=a\cos\phi^{\prime}\cos\lambda^{% \prime},\\ \displaystyle Y&\displaystyle=b\cos\phi^{\prime}\sin\lambda^{\prime},\\ \displaystyle Z&\displaystyle=c\sin\phi^{\prime}.\end{aligned}
  59. X = a cos ω a 2 - b 2 sin 2 β - c 2 cos 2 β a 2 - c 2 , Y = b cos β sin ω , Z = c sin β a 2 sin 2 ω + b 2 cos 2 ω - c 2 a 2 - c 2 . \begin{aligned}\displaystyle X&\displaystyle=a\cos\omega\frac{\sqrt{a^{2}-b^{2% }\sin^{2}\beta-c^{2}\cos^{2}\beta}}{\sqrt{a^{2}-c^{2}}},\\ \displaystyle Y&\displaystyle=b\cos\beta\sin\omega,\\ \displaystyle Z&\displaystyle=c\sin\beta\frac{\sqrt{a^{2}\sin^{2}\omega+b^{2}% \cos^{2}\omega-c^{2}}}{\sqrt{a^{2}-c^{2}}}.\end{aligned}
  60. d s 2 ( a 2 - b 2 ) sin 2 ω + ( b 2 - c 2 ) cos 2 β = b 2 sin 2 β + c 2 cos 2 β a 2 - b 2 sin 2 β - c 2 cos 2 β d β 2 + a 2 sin 2 ω + b 2 cos 2 ω a 2 sin 2 ω + b 2 cos 2 ω - c 2 d ω 2 \begin{aligned}\displaystyle\frac{ds^{2}}{(a^{2}-b^{2})\sin^{2}\omega+(b^{2}-c% ^{2})\cos^{2}\beta}&\displaystyle=\frac{b^{2}\sin^{2}\beta+c^{2}\cos^{2}\beta}% {a^{2}-b^{2}\sin^{2}\beta-c^{2}\cos^{2}\beta}d\beta^{2}\\ &\displaystyle\qquad+\frac{a^{2}\sin^{2}\omega+b^{2}\cos^{2}\omega}{a^{2}\sin^% {2}\omega+b^{2}\cos^{2}\omega-c^{2}}d\omega^{2}\end{aligned}
  61. d β d s = 1 ( a 2 - b 2 ) sin 2 ω + ( b 2 - c 2 ) cos 2 β a 2 - b 2 sin 2 β - c 2 cos 2 β b 2 sin 2 β + c 2 cos 2 β cos α , d ω d s = 1 ( a 2 - b 2 ) sin 2 ω + ( b 2 - c 2 ) cos 2 β a 2 sin 2 ω + b 2 cos 2 ω - c 2 a 2 sin 2 ω + b 2 cos 2 ω sin α , d α d s = 1 ( ( a 2 - b 2 ) sin 2 ω + ( b 2 - c 2 ) cos 2 β ) 3 / 2 × ( ( a 2 - b 2 ) cos ω sin ω a 2 sin 2 ω + b 2 cos 2 ω - c 2 a 2 sin 2 ω + b 2 cos 2 ω cos α + ( b 2 - c 2 ) cos β sin β a 2 - b 2 sin 2 β - c 2 cos 2 β b 2 sin 2 β + c 2 cos 2 β sin α ) . \begin{aligned}\displaystyle\frac{d\beta}{ds}&\displaystyle=\frac{1}{\sqrt{(a^% {2}-b^{2})\sin^{2}\omega+(b^{2}-c^{2})\cos^{2}\beta}}\frac{\sqrt{a^{2}-b^{2}% \sin^{2}\beta-c^{2}\cos^{2}\beta}}{\sqrt{b^{2}\sin^{2}\beta+c^{2}\cos^{2}\beta% }}\cos\alpha,\\ \displaystyle\frac{d\omega}{ds}&\displaystyle=\frac{1}{\sqrt{(a^{2}-b^{2})\sin% ^{2}\omega+(b^{2}-c^{2})\cos^{2}\beta}}\frac{\sqrt{a^{2}\sin^{2}\omega+b^{2}% \cos^{2}\omega-c^{2}}}{\sqrt{a^{2}\sin^{2}\omega+b^{2}\cos^{2}\omega}}\sin% \alpha,\\ \displaystyle\frac{d\alpha}{ds}&\displaystyle=\frac{1}{((a^{2}-b^{2})\sin^{2}% \omega+(b^{2}-c^{2})\cos^{2}\beta)^{3/2}}\times\\ &\displaystyle\quad\biggl(\frac{(a^{2}-b^{2})\cos\omega\sin\omega\sqrt{a^{2}% \sin^{2}\omega+b^{2}\cos^{2}\omega-c^{2}}}{\sqrt{a^{2}\sin^{2}\omega+b^{2}\cos% ^{2}\omega}}\cos\alpha\\ &\displaystyle\qquad+\frac{(b^{2}-c^{2})\cos\beta\sin\beta\sqrt{a^{2}-b^{2}% \sin^{2}\beta-c^{2}\cos^{2}\beta}}{\sqrt{b^{2}\sin^{2}\beta+c^{2}\cos^{2}\beta% }}\sin\alpha\biggr).\end{aligned}
  62. δ = b 2 sin 2 β + c 2 cos 2 β d β a 2 - b 2 sin 2 β - c 2 cos 2 β ( b 2 - c 2 ) cos 2 β - γ - a 2 sin 2 ω + b 2 cos 2 ω d ω a 2 sin 2 ω + b 2 cos 2 ω - c 2 ( a 2 - b 2 ) sin 2 ω + γ . \begin{aligned}\displaystyle\delta&\displaystyle=\int\frac{\sqrt{b^{2}\sin^{2}% \beta+c^{2}\cos^{2}\beta}\,d\beta}{\sqrt{a^{2}-b^{2}\sin^{2}\beta-c^{2}\cos^{2% }\beta}\sqrt{(b^{2}-c^{2})\cos^{2}\beta-\gamma}}\\ &\displaystyle\quad-\int\frac{\sqrt{a^{2}\sin^{2}\omega+b^{2}\cos^{2}\omega}\,% d\omega}{\sqrt{a^{2}\sin^{2}\omega+b^{2}\cos^{2}\omega-c^{2}}\sqrt{(a^{2}-b^{2% })\sin^{2}\omega+\gamma}}.\end{aligned}
  63. γ = ( b 2 - c 2 ) cos 2 β sin 2 α - ( a 2 - b 2 ) sin 2 ω cos 2 α , \gamma=(b^{2}-c^{2})\cos^{2}\beta\sin^{2}\alpha-(a^{2}-b^{2})\sin^{2}\omega% \cos^{2}\alpha,
  64. d s ( a 2 - b 2 ) sin 2 ω + ( b 2 - c 2 ) cos 2 β = b 2 sin 2 β + c 2 cos 2 β d β a 2 - b 2 sin 2 β - c 2 cos 2 β ( b 2 - c 2 ) cos 2 β - γ = a 2 sin 2 ω + b 2 cos 2 ω d ω a 2 sin 2 ω + b 2 cos 2 ω - c 2 ( a 2 - b 2 ) sin 2 ω + γ . \begin{aligned}\displaystyle\frac{ds}{(a^{2}-b^{2})\sin^{2}\omega+(b^{2}-c^{2}% )\cos^{2}\beta}&\displaystyle=\frac{\sqrt{b^{2}\sin^{2}\beta+c^{2}\cos^{2}% \beta}\,d\beta}{\sqrt{a^{2}-b^{2}\sin^{2}\beta-c^{2}\cos^{2}\beta}\sqrt{(b^{2}% -c^{2})\cos^{2}\beta-\gamma}}\\ &\displaystyle=\frac{\sqrt{a^{2}\sin^{2}\omega+b^{2}\cos^{2}\omega}\,d\omega}{% \sqrt{a^{2}\sin^{2}\omega+b^{2}\cos^{2}\omega-c^{2}}\sqrt{(a^{2}-b^{2})\sin^{2% }\omega+\gamma}}.\end{aligned}
  65. d s = b 2 sin 2 β + c 2 cos 2 β ( b 2 - c 2 ) cos 2 β - γ d β a 2 - b 2 sin 2 β - c 2 cos 2 β + a 2 sin 2 ω + b 2 cos 2 ω ( a 2 - b 2 ) sin 2 ω + γ d ω a 2 sin 2 ω + b 2 cos 2 ω - c 2 . \begin{aligned}\displaystyle ds&\displaystyle=\frac{\sqrt{b^{2}\sin^{2}\beta+c% ^{2}\cos^{2}\beta}\sqrt{(b^{2}-c^{2})\cos^{2}\beta-\gamma}\,d\beta}{\sqrt{a^{2% }-b^{2}\sin^{2}\beta-c^{2}\cos^{2}\beta}}\\ &\displaystyle\quad{}+\frac{\sqrt{a^{2}\sin^{2}\omega+b^{2}\cos^{2}\omega}% \sqrt{(a^{2}-b^{2})\sin^{2}\omega+\gamma}\,d\omega}{\sqrt{a^{2}\sin^{2}\omega+% b^{2}\cos^{2}\omega-c^{2}}}.\end{aligned}

Geometric_mean_theorem.html

  1. h 2 = p q h = p q h^{2}=pq\Leftrightarrow h=\sqrt{pq}
  2. h = p q h=\sqrt{pq}
  3. h 2 = p q . h^{2}=pq.
  4. A D C \triangle ADC
  5. B D C \triangle BDC
  6. A D C = C D B \angle ADC=\angle CDB
  7. C A D = 90 - D B C = B C D \angle CAD=90^{\circ}-\angle DBC=\angle BCD
  8. h p = q h h 2 = p q h = p q ( h , p , q > 0 ) \frac{h}{p}=\frac{q}{h}\,\Leftrightarrow\,h^{2}=pq\,\Leftrightarrow\,h=\sqrt{% pq}\qquad(h,p,q>0)
  9. A B C \triangle ABC
  10. h 2 = p q h^{2}=pq
  11. h 2 = p q h^{2}=pq
  12. h p = q h \tfrac{h}{p}=\tfrac{q}{h}
  13. A D C = C D B \angle ADC=\angle CDB
  14. A D C \triangle ADC
  15. B D C \triangle BDC
  16. A C B = A C D + D C B = A C D + ( 90 - A C D ) = 90 \angle ACB=\angle ACD+\angle DCB=\angle ACD+(90^{\circ}-\angle ACD)=90^{\circ}

Geometrothermodynamics.html

  1. 𝒯 \mathcal{T}
  2. Z A = { Φ , E a , I a } Z^{A}=\{\Phi,E^{a},I^{a}\}
  3. Φ \Phi
  4. E a E^{a}
  5. a = 1 , 2 , , n a=1,2,\ldots,n
  6. I a I^{a}
  7. Θ = d Φ - δ a b I a d E b \Theta=d\Phi-\delta_{ab}I^{a}dE^{b}
  8. δ a b = diag ( + 1 , , + 1 ) \delta_{ab}={\rm diag}(+1,\ldots,+1)
  9. Θ ( d Θ ) n 0 \Theta\wedge(d\Theta)^{n}\neq 0
  10. n n
  11. { Z A } { Z ~ A } = { Φ ~ , E ~ a , I ~ a } , Φ = Φ ~ - δ k l E ~ k I ~ l , E i = - I ~ i , E j = E ~ j , I i = E ~ i , I j = I ~ j , \{Z^{A}\}\longrightarrow\{\widetilde{Z}^{A}\}=\{\tilde{\Phi},\tilde{E}^{a},% \tilde{I}^{a}\}\ ,\quad\Phi=\tilde{\Phi}-\delta_{kl}\tilde{E}^{k}\tilde{I}^{l}% ,\quad E^{i}=-\tilde{I}^{i},\quad E^{j}=\tilde{E}^{j},\quad I^{i}=\tilde{E}^{i% },\quad I^{j}=\tilde{I}^{j}\ ,
  12. i j i\cup j
  13. { 1 , , n } \{1,\ldots,n\}
  14. k , l = 1 , , i k,l=1,\ldots,i
  15. i = { 1 , , n } i=\{1,\ldots,n\}
  16. i = i=\emptyset
  17. 𝒯 \mathcal{T}
  18. G G
  19. ( 𝒯 , Θ , G ) (\mathcal{T},\Theta,G)
  20. 𝒯 \mathcal{E}\subset\mathcal{T}
  21. φ : 𝒯 \varphi:\mathcal{E}\rightarrow\mathcal{T}
  22. φ : { E a } { Φ , E a , I a } \varphi:\{E^{a}\}\mapsto\{\Phi,E^{a},I^{a}\}
  23. Φ = Φ ( E a ) \Phi=\Phi(E^{a})
  24. I a = I a ( E a ) I^{a}=I^{a}(E^{a})
  25. φ * ( Θ ) = φ * ( d Φ - δ a b I a d E b ) = 0 \varphi^{*}(\Theta)=\varphi^{*}(d\Phi-\delta_{ab}I^{a}dE^{b})=0
  26. φ * \varphi^{*}
  27. φ \varphi
  28. \mathcal{E}
  29. g = φ * ( G ) g=\varphi^{*}(G)
  30. \mathcal{E}
  31. Φ = Φ ( E a ) \Phi=\Phi(E^{a})
  32. G I = ( d Φ - δ a b I a d E b ) 2 + Λ ( ξ a b E a I b ) ( δ c d d E c d I d ) , δ a b = diag ( 1 , , 1 ) G^{I}=(d\Phi-\delta_{ab}I^{a}dE^{b})^{2}+\Lambda\,(\xi_{ab}E^{a}I^{b})\left(% \delta_{cd}dE^{c}dI^{d}\right)\ ,\quad\delta_{ab}={\rm diag}(1,\ldots,1)
  33. G I I = ( d Φ - δ a b I a d E b ) 2 + Λ ( ξ a b E a I b ) ( η c d d E c d I d ) , η a b = diag ( - 1 , 1 , , 1 ) G^{II}=(d\Phi-\delta_{ab}I^{a}dE^{b})^{2}+\Lambda\,(\xi_{ab}E^{a}I^{b})\left(% \eta_{cd}dE^{c}dI^{d}\right)\ ,\quad\eta_{ab}={\rm diag}(-1,1,\ldots,1)
  34. ξ a b \xi_{ab}
  35. δ a b \delta_{ab}
  36. η a b \eta_{ab}
  37. Λ \Lambda
  38. Z A Z^{A}
  39. G I G^{I}
  40. G I I G^{II}
  41. G I I I = ( d Φ - δ a b I a d E b ) 2 + Λ ( E a I a ) 2 k + 1 ( d E a d I a ) , E a = δ a b E b , I a = δ a b I b . G^{III}=(d\Phi-\delta_{ab}I^{a}dE^{b})^{2}+\Lambda\,(E_{a}I_{a})^{2k+1}\left(% dE^{a}dI^{a}\right)\ ,\quad E_{a}=\delta_{ab}E^{b}\ ,\quad I_{a}=\delta_{ab}I^% {b}\ .
  42. {\mathcal{E}}
  43. g a b = Z A E a Z B E b G A B . g_{ab}=\frac{\partial Z^{A}}{\partial E^{a}}\frac{\partial Z^{B}}{\partial E^{% b}}G_{AB}\ .

Geometry_of_binary_search_trees.html

  1. ρ A = sup X cost A ( X ) cost opt ( X ) . \rho_{A}=\sup_{X}\frac{\mathrm{cost}_{A}(X)}{\mathrm{cost}_{\mathrm{opt}}(X)}.

Geoneutrino.html

  1. U 92 238 Pb 82 206 + 8 α + 6 e - + 6 ν ¯ e + 51.698 MeV U 92 235 Pb 82 207 + 7 α + 4 e - + 4 ν ¯ e + 46.402 MeV Th 90 232 Pb 82 208 + 6 α + 4 e - + 4 ν ¯ e + 42.652 MeV K 19 40 89.3 % Ca 20 40 + e - + ν ¯ e + 1.311 MeV K 19 40 + e - 10.7 % Ar 18 40 + ν e + 1.505 MeV \begin{array}[]{rcl}{}_{~{}92}^{238}\,\text{U}&\longrightarrow&{}_{~{}82}^{206% }\,\text{Pb}+8\alpha+6e^{-}+6\bar{\nu}_{e}+51.698\,\,\text{MeV}\\ {}_{~{}92}^{235}\,\text{U}&\longrightarrow&{}_{~{}82}^{207}\,\text{Pb}+7\alpha% +4e^{-}+4\bar{\nu}_{e}+46.402\,\,\text{MeV}\\ {}_{~{}90}^{232}\,\text{Th}&\longrightarrow&{}_{~{}82}^{208}\,\text{Pb}+6% \alpha+4e^{-}+4\bar{\nu}_{e}+42.652\,\,\text{MeV}\\ {}_{19}^{40}\,\text{K}&\stackrel{89.3\,\%}{\longrightarrow}&{}_{20}^{40}\,% \text{Ca}+e^{-}+\bar{\nu}_{e}+1.311\,\,\text{MeV}\\ {}_{19}^{40}\,\text{K}+e^{-}&\stackrel{10.7\,\%}{\longrightarrow}&{}_{18}^{40}% \,\text{Ar}+\nu_{e}+1.505\,\,\text{MeV}\end{array}
  2. d ϕ ( E ν ¯ e , r ) d E ν ¯ e = 10 λ X N A M d n ( E ν ¯ e ) d E ν ¯ e V d 3 r A ( r ) ρ ( r ) P e e ( E ν ¯ e , | r - r | ) 4 π | r - r | 2 \frac{\mathrm{d}\phi(E_{\bar{\nu}_{e}},\vec{r})}{\mathrm{d}E_{\bar{\nu}_{e}}}=% 10\frac{\lambda XN_{A}}{M}\frac{\mathrm{d}n(E_{\bar{\nu}_{e}})}{\mathrm{d}E_{% \bar{\nu}_{e}}}\int\limits_{V}\mathrm{d}^{3}\vec{r}^{\prime}\frac{A(\vec{r}^{% \prime})\rho(\vec{r}^{\prime})P_{ee}(E_{\bar{\nu}_{e}},|\vec{r}-\vec{r}^{% \prime}|)}{4\pi|\vec{r}-\vec{r}^{\prime}|^{2}}
  3. ϕ ( r ) = 10 n ν ¯ e P e e λ X N A M V d 3 r A ( r ) ρ ( r ) 4 π | r - r | 2 \phi(\vec{r})=10\frac{n_{\bar{\nu}_{e}}\langle P_{ee}\rangle\lambda XN_{A}}{M}% \int\limits_{V}\mathrm{d}^{3}\vec{r}^{\prime}\frac{A(\vec{r}^{\prime})\rho(% \vec{r}^{\prime})}{4\pi|\vec{r}-\vec{r}^{\prime}|^{2}}
  4. ν ¯ e + p e + + n \bar{\nu}_{e}+p\rightarrow e^{+}+n
  5. e + + e - γ + γ e^{+}+e^{-}\rightarrow\gamma+\gamma
  6. n + p d + γ n+p\rightarrow d+\gamma

GEORGE_(programming_language).html

  1. a + b a+b
  2. a x 2 + b x + c ax^{2}+bx+c
  3. y = a x 2 + b x + c y=ax^{2}+bx+c

Geranial_dehydrogenase.html

  1. \rightleftharpoons

Geraniol_8-hydroxylase.html

  1. \rightleftharpoons

Geranyl-pyrophosphate—olivetolic_acid_geranyltransferase.html

  1. \rightleftharpoons

Geranyl_diphosphate_2-C-methyltransferase.html

  1. \rightleftharpoons

Geranyl_diphosphate_diphosphatase.html

  1. \rightleftharpoons

Geranylfarnesyl_diphosphate_synthase.html

  1. \rightleftharpoons

Geranylgeraniol_18-hydroxylase.html

  1. \rightleftharpoons

Geranylgeranyl_diphosphate_diphosphatase.html

  1. \rightleftharpoons

Geranylgeranyl_diphosphate_reductase.html

  1. \rightleftharpoons

Germacrene_A_alcohol_dehydrogenase.html

  1. \rightleftharpoons
  2. \rightleftharpoons
  3. \rightleftharpoons

Germacrene_A_hydroxylase.html

  1. \rightleftharpoons

Gilbreath_shuffle.html

  1. x x 2 + c x\mapsto x^{2}+c
  2. x = 0 x=0

GlcA-beta-(1-2)-D-Man-alpha-(1-3)-D-Glc-beta-(1-4)-D-Glc-alpha-1-diphospho-ditrans,octacis-undecaprenol_4-beta-mannosyltransferase.html

  1. \rightleftharpoons

Glennie's_identity.html

  1. A A

Globotriosylceramide_b-1,6-N-acetylgalactosaminyl-transferase.html

  1. \rightleftharpoons

Glossary_of_algebraic_geometry.html

  1. k ( y ¯ ) k(\bar{y})

Glossary_of_commutative_algebra.html

  1. 𝔭 \mathfrak{p}
  2. 1 \leq 1

Glucose-6-phosphate_dehydrogenase_(coenzyme-F420).html

  1. \rightleftharpoons

Glucosyl-3-phosphoglycerate_phosphatase.html

  1. \rightleftharpoons

Glucosyl-3-phosphoglycerate_synthase.html

  1. \rightleftharpoons

Glucosylceramide_beta-1,4-galactosyltransferase.html

  1. \rightleftharpoons

Glucosylglycerate_synthase.html

  1. \rightleftharpoons

Glucuronosyl-N-acetylgalactosaminyl-proteoglycan_4-b-N-acetylgalactosaminyltransferase.html

  1. \rightleftharpoons

Glucuronosyl-N-acetylglucosaminyl-proteoglycan_4-alpha-N-acetylglucosaminyltransferase.html

  1. \rightleftharpoons

Glucuronyl-galactosyl-proteoglycan_4-a-N-acetylglucosaminyltransferase.html

  1. \rightleftharpoons

Glucuronylgalactosylproteoglycan_4-b-N-acetylgalactosaminyltransferase.html

  1. \rightleftharpoons

Gluon_field_strength_tensor.html

  1. a , b , c , n a,b,c,n
  2. α , β , μ , ν α,β,μ,ν
  3. G G
  4. F F
  5. F ¯ \overline{F}
  6. G α β = ± 1 g s [ D α , D β ] , G_{\alpha\beta}=\pm\frac{1}{g_{s}}[D_{\alpha},D_{\beta}]\,,
  7. D μ = μ ± i g s t a 𝒜 μ a , D_{\mu}=\partial_{\mu}\pm ig_{s}t_{a}\mathcal{A}^{a}_{\mu}\,,
  8. i i
  9. a a
  10. μ μ
  11. 𝒜 μ = t a 𝒜 μ a \mathcal{A}_{\mu}=t_{a}\mathcal{A}^{a}_{\mu}
  12. 𝒜 μ \mathcal{A}_{\mu}
  13. 𝒜 μ a \mathcal{A}^{a}_{\mu}
  14. G α β = α 𝒜 β - β 𝒜 α ± i g s [ 𝒜 α , 𝒜 β ] G_{\alpha\beta}=\partial_{\alpha}\mathcal{A}_{\beta}-\partial_{\beta}\mathcal{% A}_{\alpha}\pm ig_{s}[\mathcal{A}_{\alpha},\mathcal{A}_{\beta}]
  15. t a 𝒜 α a = 𝒜 α t_{a}\mathcal{A}^{a}_{\alpha}=\mathcal{A}_{\alpha}
  16. [ t a , t b ] = i f a b c t c [t_{a},t_{b}]=if^{abc}t_{c}
  17. G α β = α t a 𝒜 β a - β t a 𝒜 α a ± i g s [ t b , t c ] 𝒜 α b 𝒜 β c = t a ( α 𝒜 β a - β 𝒜 α a ± i 2 g s 𝒜 α b 𝒜 β c ) = t a G α β a , \begin{aligned}\displaystyle G_{\alpha\beta}&\displaystyle=\partial_{\alpha}t_% {a}\mathcal{A}^{a}_{\beta}-\partial_{\beta}t_{a}\mathcal{A}^{a}_{\alpha}\pm ig% _{s}\left[t_{b},t_{c}\right]\mathcal{A}^{b}_{\alpha}\mathcal{A}^{c}_{\beta}\\ &\displaystyle=t_{a}\left(\partial_{\alpha}\mathcal{A}^{a}_{\beta}-\partial_{% \beta}\mathcal{A}^{a}_{\alpha}\pm i^{2}g_{s}\mathcal{A}^{b}_{\alpha}\mathcal{A% }^{c}_{\beta}\right)\\ &\displaystyle=t_{a}G^{a}_{\alpha\beta}\\ \end{aligned}\,,
  18. G α β a = α 𝒜 β a - β 𝒜 α a g s f a b c 𝒜 α b 𝒜 β c , G^{a}_{\alpha\beta}=\partial_{\alpha}\mathcal{A}^{a}_{\beta}-\partial_{\beta}% \mathcal{A}^{a}_{\alpha}\mp g_{s}f^{abc}\mathcal{A}^{b}_{\alpha}\mathcal{A}^{c% }_{\beta}\,,
  19. a , b , c = 1 , 2 , , 8 a,b,c=1,2,...,8
  20. 𝐆 = d s y m b o l 𝒜 g s s y m b o l 𝒜 s y m b o l 𝒜 , \mathbf{G}=\mathrm{d}symbol{\mathcal{A}}\mp g_{s}\,symbol{\mathcal{A}}\wedge symbol% {\mathcal{A}}\,,
  21. s y m b o l 𝒜 symbol{\mathcal{A}}
  22. 𝐆 \mathbf{G}
  23. s y m b o l 𝒜 symbol{\mathcal{A}}
  24. F F
  25. A A
  26. F α β = α A β - β A α , F_{\alpha\beta}=\partial_{\alpha}A_{\beta}-\partial_{\beta}A_{\alpha}\,,
  27. 𝐅 = d 𝐀 . \mathbf{F}=\mathrm{d}\mathbf{A}\,.
  28. = - 1 2 tr ( G α β G α β ) + ψ ¯ ( i D μ ) γ μ ψ \mathcal{L}=-\frac{1}{2}\mathrm{tr}\left(G_{\alpha\beta}G^{\alpha\beta}\right)% +\bar{\psi}\left(iD_{\mu}\right)\gamma^{\mu}\psi
  29. ( i γ μ D μ - m c ) ψ = 0 (i\hbar\gamma^{\mu}D_{\mu}-mc)\psi=0
  30. [ D μ , G μ ν ] = g s j ν \left[D_{\mu},G^{\mu\nu}\right]=g_{s}j^{\nu}
  31. j ν = t b j b ν , j b ν = ψ ¯ γ ν t b ψ , j^{\nu}=t^{b}j_{b}^{\nu}\,,\quad j_{b}^{\nu}=\bar{\psi}\gamma^{\nu}t^{b}\psi\,,
  32. ν j ν = 0 . \partial_{\nu}j^{\nu}=0\,.

Glutaryl-CoA_dehydrogenase_(non-decarboxylating).html

  1. \rightleftharpoons

Glutathione_amide-dependent_peroxidase.html

  1. \rightleftharpoons

Glutathione_amide_reductase.html

  1. \rightleftharpoons

Glutathione_hydrolase.html

  1. \rightleftharpoons

Glycerate_2-kinase.html

  1. \rightleftharpoons

Glycerol-3-phosphate_2-O-acyltransferase.html

  1. \rightleftharpoons

Glycerol-3-phosphate_dehydrogenase_(quinone).html

  1. \rightleftharpoons

Glycine::sarcosine::dimethylglycine_N-methyltransferase.html

  1. \rightleftharpoons
  2. \rightleftharpoons
  3. \rightleftharpoons
  4. \rightleftharpoons

Glycine::sarcosine_N-methyltransferase.html

  1. \rightleftharpoons
  2. \rightleftharpoons
  3. \rightleftharpoons

Glycine_N-phenylacetyltransferase.html

  1. \rightleftharpoons

Glycine_oxidase.html

  1. \rightleftharpoons
  2. \rightleftharpoons
  3. \rightleftharpoons

Glycopeptide_alpha-N-acetylgalactosaminidase.html

  1. \rightleftharpoons

Glycosyltransferase_DesVII.html

  1. \rightleftharpoons

GNU_Archimedes.html

  1. f t + 1 k E ( k ) r f + q F ( r ) k f = [ f t ] collision \frac{\partial f}{\partial t}+\frac{1}{\hbar}\nabla_{k}E(k)\nabla_{r}f+\frac{% qF(r)}{\hbar}\nabla_{k}f=\left[\frac{\partial f}{\partial t}\right]_{\mathrm{% collision}}
  2. v = 1 k E ( k ) v=\frac{1}{\hbar}\nabla_{k}E(k)
  3. d r d t = 1 k E ( k ) \frac{dr}{dt}=\frac{1}{\hbar}\nabla_{k}E(k)
  4. d k d t = q F ( r ) \frac{dk}{dt}=\frac{qF(r)}{\hbar}

Goldschmidt_tolerance_factor.html

  1. t = r A + r 0 2 ( r B + r 0 ) t={r_{A}+r_{0}\over\sqrt{2}(r_{B}+r_{0})}
  2. a = 2 ( r A + r 0 ) = 2 ( r B + r 0 ) a=\sqrt{2}(r_{A}+r_{0})=2(r_{B}+r_{0})

Google_Flu_Trends.html

  1. logit ( P ) = β 0 + β 1 × logit ( Q ) + ϵ \operatorname{logit}(P)=\beta_{0}+\beta_{1}\times\operatorname{logit}(Q)+\epsilon

Gowers_norm.html

  1. f U d ( G ) 2 d = 𝐄 x , h 1 , , h d G ω 1 , , ω d { 0 , 1 } J ω 1 + + ω d f ( x + h 1 ω 1 + + h d ω d ) . \|f\|_{U^{d}(G)}^{2^{d}}=\mathbf{E}_{x,h_{1},\ldots,h_{d}\in G}\prod_{\omega_{% 1},\ldots,\omega_{d}\in\{0,1\}}J^{\omega_{1}+\cdots+\omega_{d}}f\left({x+h_{1}% \omega_{1}+\cdots+h_{d}\omega_{d}}\right)\ .
  2. f U d [ N ] = f ~ U d ( / N ~ ) / 1 [ N ] U d ( / N ~ ) \|f\|_{U^{d}[N]}=\|\tilde{f}\|_{U^{d}(\mathbb{Z}/\tilde{N}\mathbb{Z})}/\|1_{[N% ]}\|_{U^{d}(\mathbb{Z}/\tilde{N}\mathbb{Z})}
  3. N ~ \tilde{N}
  4. 1 [ N ] 1_{[N]}
  5. f ~ ( x ) \tilde{f}(x)
  6. f ( x ) f(x)
  7. x [ N ] x\in[N]
  8. 0
  9. x x
  10. N ~ \tilde{N}
  11. N ~ > 2 d N \tilde{N}>2^{d}N
  12. 𝔽 \mathbb{F}
  13. δ > 0 \delta>0
  14. c > 0 c>0
  15. 𝔽 \mathbb{F}
  16. f f
  17. V V
  18. f U d [ V ] δ \|f\|_{U^{d}[V]}\geq\delta
  19. P : V / P\colon V\to\mathbb{R}/\mathbb{Z}
  20. | 1 | V | x V f ( x ) e ( - P ( x ) ) | c , \left|\frac{1}{|V|}\sum_{x\in V}f(x)e(-P(x))\right|\geq c,
  21. e ( x ) := e 2 π i x e(x):=e^{2\pi ix}
  22. U d [ N ] U^{d}[N]
  23. δ > 0 \delta>0
  24. δ \mathcal{M}_{\delta}
  25. c , C c,C
  26. N N
  27. f : [ N ] f\colon[N]\to\mathbb{C}
  28. f U d [ N ] δ \|f\|_{U^{d}[N]}\geq\delta
  29. G / Γ δ G/\Gamma\in\mathcal{M}_{\delta}
  30. F ( g n x ) F(g^{n}x)
  31. g G , x G / Γ g\in G,\ x\in G/\Gamma
  32. F : G / Γ F\colon G/\Gamma\to\mathbb{C}
  33. C C
  34. | 1 N n = 0 N - 1 f ( n ) F ( g n x ¯ ) | c . \left|\frac{1}{N}\sum_{n=0}^{N-1}f(n)\overline{F(g^{n}x})\right|\geq c.

Grain_128a.html

  1. 0 < w 32 0<w\leq 32
  2. I V 0 IV_{0}
  3. I V 0 = 1 IV_{0}=1
  4. I V 0 = 0 IV_{0}=0
  5. b b
  6. s s
  7. f f
  8. g g
  9. h h
  10. f ( x ) = 1 + x 32 + x 47 + x 58 + x 90 + x 121 + x 128 f(x)=1+x^{32}+x^{47}+x^{58}+x^{90}+x^{121}+x^{128}
  11. g ( x ) = 1 + x 32 + x 37 + x 72 + x 102 + x 128 + x 44 x 60 + x 61 x 125 + x 63 x 67 x 69 x 101 + x 80 x 88 + x 110 x 111 + x 115 x 117 + x 46 x 50 x 58 + x 103 x 104 x 106 + x 33 x 35 x 36 x 40 g(x)=1+x^{32}+x^{37}+x^{72}+x^{102}+x^{128}+x^{44}x^{60}+x^{61}x^{125}+x^{63}x% ^{67}x^{69}x^{101}+x^{80}x^{88}+x^{110}x^{111}+x^{115}x^{117}+x^{46}x^{50}x^{5% 8}+x^{103}x^{104}x^{106}+x^{33}x^{35}x^{36}x^{40}
  12. h ( x ) = b i + 12 s i + 8 + s i + 13 s i + 20 + b i + 95 s i + 42 + s i + 60 s i + 79 + b i + 12 b i + 95 s i + 94 h(x)=b_{i+12}s_{i+8}+s_{i+13}s_{i+20}+b_{i+95}s_{i+42}+s_{i+60}s_{i+79}+b_{i+1% 2}b_{i+95}s_{i+94}
  13. b i + 128 = s i + b i + b i + 26 + b i + 56 + b i + 91 + b i + 96 + b i + 3 b i + 67 + b i + 11 b i + 13 + b i + 17 b i + 18 + b i + 27 b i + 59 + b i + 40 b i + 48 + b i + 61 b i + 65 + b i + 68 b i + 84 + b i + 88 b i + 92 b i + 93 b i + 95 + b i + 22 b i + 24 b i + 25 + b i + 70 b i + 78 b i + 82 b_{i+128}=s_{i}+b_{i}+b_{i+26}+b_{i+56}+b_{i+91}+b_{i+96}+b_{i+3}b_{i+67}+b_{i% +11}b_{i+13}+b_{i+17}b_{i+18}+b_{i+27}b_{i+59}+b_{i+40}b_{i+48}+b_{i+61}b_{i+6% 5}+b_{i+68}b_{i+84}+b_{i+88}b_{i+92}b_{i+93}b_{i+95}+b_{i+22}b_{i+24}b_{i+25}+% b_{i+70}b_{i+78}b_{i+82}
  14. s i + 128 = s i + s i + 7 + s i + 38 + s i + 70 + s i + 81 + s i + 96 s_{i+128}=s_{i}+s_{i+7}+s_{i+38}+s_{i+70}+s_{i+81}+s_{i+96}
  15. y y
  16. y i = h ( x ) + s i + 93 + b i + 2 + b i + 12 + b i + 36 + b i + 45 + b i + 64 + b i + 73 + b i + 89 y_{i}=h(x)+s_{i+93}+b_{i+2}+b_{i+12}+b_{i+36}+b_{i+45}+b_{i+64}+b_{i+73}+b_{i+% 89}
  17. I V IV
  18. I V 0 IV_{0}
  19. s i = I V i s_{i}=IV_{i}
  20. 0 i 95 0\leq i\leq 95
  21. s i = 1 s_{i}=1
  22. 96 i 126 96\leq i\leq 126
  23. s 127 = 0 s_{127}=0
  24. k k
  25. b i = k i b_{i}=k_{i}
  26. 0 i 127 0\leq i\leq 127
  27. g g
  28. f f
  29. z z
  30. y y
  31. I V 0 IV_{0}
  32. 2 w 2w
  33. w w
  34. z i = y 2 w + 2 i z_{i}=y_{2w+2i}
  35. z i = y i z_{i}=y_{i}
  36. w w
  37. w w
  38. r r
  39. a a
  40. m m
  41. L L
  42. m + 1 m+1
  43. m L = 1 m_{L}=1
  44. m 1 = 1 m1=1
  45. m 2 = 10 m2=10
  46. 0 j 31 0\leq j\leq 31
  47. 0 i L 0\leq i\leq L
  48. a i j a_{i}^{j}
  49. 2 w 2w
  50. y y
  51. r i = y i + 31 r_{i}=y_{i+31}
  52. 0 i 31 0\leq i\leq 31
  53. a 0 j = y j a_{0}^{j}=y_{j}
  54. 0 j 31 0\leq j\leq 31
  55. y y
  56. r i + 31 = y 64 + 2 i + 1 r_{i+31}=y_{64+2i+1}
  57. a i + 1 j = a i j + m i r i + j a_{i+1}^{j}=a_{i}^{j}+m_{i}r_{i+j}
  58. 0 i L 0\leq i\leq L
  59. t t
  60. t i = a L + 1 i t_{i}=a_{L+1}^{i}
  61. 0 i 31 0\leq i\leq 31

Grain_yield_monitor.html

  1. m ˙ \dot{m}
  2. v v
  3. k i l o m e t e r h r \tfrac{kilometer}{hr}
  4. m i l e h r \tfrac{mile}{hr}
  5. w w
  6. t o n n e h a \tfrac{tonne}{ha}
  7. b u a c r e \tfrac{bu}{acre}
  8. Y i e l d = m ˙ * b u 56 l b * 3600 s h r w * v * 5280 f t m i * a c r e 43560 f t 2 Yield=\dfrac{\dot{m}*\dfrac{bu}{56lb}*\dfrac{3600s}{hr}}{w*v*\dfrac{5280ft}{mi% }*\dfrac{acre}{43560ft^{2}}}
  9. Y i e l d = m ˙ * t o n n e 1 , 000 k g * 3600 s h r w * v * 1000 m k m * h a 10 , 000 m 2 Yield=\dfrac{\dot{m}*\dfrac{tonne}{1,000kg}*\dfrac{3600s}{hr}}{w*v*\dfrac{1000% m}{km}*\dfrac{ha}{10,000m^{2}}}

Graph_power.html

  1. O ( Δ k / 2 ) O(\Delta^{\lfloor k/2\rfloor})
  2. max ( d + 5 , 3 d 2 + 1 ) \max\left(d+5,\frac{3d}{2}+1\right)
  3. 5 d 3 + O ( 1 ) \frac{5d}{3}+O(1)

Gray_level_size_zone_matrix.html

  1. G S f GS_{f}
  2. G S f ( s n , g m ) GS_{f}(s_{n},g_{m})
  3. s n s_{n}
  4. g m g_{m}

Green's_function_number.html

  1. G = 0 G=0
  2. G n = 0 \frac{\partial G}{\partial n}=0
  3. k G n + h G = 0 k\frac{\partial G}{\partial n}+hG=0
  4. ϕ \phi
  5. ϕ \phi
  6. θ \theta
  7. α \alpha
  8. δ \delta
  9. 0 < x < 0<x<\infty
  10. x = x=\infty
  11. 2 G x 2 + 1 α δ ( t - τ ) δ ( x - x ) = 1 α G t ; 0 < x < ; t > 0 G n | x = 0 = 0 ; G | x is bounded ; G | t < τ = 0 ; \begin{aligned}\displaystyle\dfrac{\partial^{2}G}{\partial x^{2}}+\dfrac{1}{% \alpha}\delta(t-\tau)\delta(x-x^{\prime})=\dfrac{1}{\alpha}\dfrac{\partial G}{% \partial t};&\displaystyle\;\;0<x<\infty;\;\;\;t>0\\ \displaystyle\dfrac{\partial G}{\partial n}|_{x=0}=0;\;\;G|_{x\rightarrow% \infty}\mbox{ is bounded}~{};\;\;G|_{t<\tau}=0;&\end{aligned}
  12. G | r = 0 is bounded ; k G n | r = a + h G | r = a = 0 ; G | t < τ = 0 ; G|_{r=0}\mbox{ is bounded}~{};\;\;k\dfrac{\partial G}{\partial n}|_{r=a}+hG|_{% r=a}=0;\;\;G|_{t<\tau}=0;
  13. k k
  14. h h
  15. 1 r r ( r G r ) + 1 α δ ( t - τ ) δ ( r - r ) = 1 α G t ; a < r < ; t > 0 G | r = a = 0 ; G | r is bounded ; G | t < τ = 0 ; \begin{aligned}\displaystyle\dfrac{1}{r}\dfrac{\partial}{\partial r}\left(r% \dfrac{\partial G}{\partial r}\right)+\dfrac{1}{\alpha}\delta(t-\tau)\delta(r-% r^{\prime})=\dfrac{1}{\alpha}\dfrac{\partial G}{\partial t};&\displaystyle\;\;% a<r<\infty;\;\;\;t>0\\ \displaystyle G|_{r=a}=0;\;\;G|_{r\rightarrow\infty}\mbox{ is bounded}~{};\;\;% G|_{t<\tau}=0;&\end{aligned}
  16. G | r = 0 is bounded ; G n | r = b = 0 ; G | t < τ = 0 ; G|_{r=0}\mbox{ is bounded}~{};\;\;\dfrac{\partial G}{\partial n}|_{r=b}=0;\;\;% G|_{t<\tau}=0;

Greenberg–Hastings_cellular_automaton.html

  1. C i , j C_{i,j}
  2. t = 0 , 1 , 2 , t=0,1,2,...
  3. C i , j C_{i,j}
  4. S i , j t { 0 , 1 , 2 } . S_{i,j}^{t}\in\{0,1,2\}.
  5. S i , j t = 1 S_{i,j}^{t}=1
  6. S i , j t + 1 = 2 S_{i,j}^{t+1}=2
  7. S i , j t = 2 S_{i,j}^{t}=2
  8. S i , j t + 1 = 0. S_{i,j}^{t+1}=0.
  9. S i , j t = 0 S_{i,j}^{t}=0
  10. S i , j t = 1 S_{i^{\prime},j^{\prime}}^{t}=1
  11. C i , j , C_{i^{\prime},j^{\prime}},
  12. S i , j t + 1 = 1 S_{i,j}^{t+1}=1
  13. S i , j t + 1 = 0. S_{i,j}^{t+1}=0.

Grigoriy_Yablonsky.html

  1. A 2 + 2 Z 2 A Z A_{2}+2Z\rightleftharpoons 2AZ
  2. B + Z B Z B+Z\rightleftharpoons BZ
  3. A Z + B Z A B + 2 Z AZ+BZ\to AB+2Z
  4. B + Z ( B Z ) B+Z\rightleftharpoons(BZ)
  5. A 2 , B A_{2},B
  6. O 2 O_{2}
  7. C O 2 CO_{2}
  8. x ˙ = 2 q 1 z 2 - 2 q 5 x 2 - q 3 x y \dot{x}=2q_{1}z^{2}-2q_{5}x^{2}-q_{3}xy
  9. y ˙ = q 2 z - q 6 y - q 3 x y \dot{y}=q_{2}z-q_{6}y-q_{3}xy
  10. s ˙ = q 4 z - k q 4 s \dot{s}=q_{4}z-kq_{4}s
  11. α r A i r j β r j A j , \alpha_{r}A_{i_{r}}\to\sum_{j}\beta_{rj}A_{j}\,,
  12. A i A_{i}
  13. α r , β r j 0 \alpha_{r},\beta_{rj}\geq 0
  14. 2 P t ( + O 2 ) 2 P t ; P t O ( + C O ) P t ( + C O 2 ) . 2Pt(+O_{2})\rightleftharpoons 2Pt;\;\;PtO(+CO)\rightleftharpoons Pt(+CO_{2}% \uparrow).
  15. α r m i r = j β r j m j for some m j > 0 and all r , \alpha_{r}m_{i_{r}}=\sum_{j}\beta_{rj}m_{j}\,\text{ for some }m_{j}>0\,\text{ % and all }r,
  16. W r = k r c i r α r , W_{r}=k_{r}c_{i_{r}}^{\alpha_{r}},
  17. c i c_{i}
  18. A i A_{i}
  19. 𝐜 ( t ) = ( c i ( t ) ) \mathbf{c}(t)=(c_{i}(t))
  20. m ( 𝐜 ) = m i c i m(\mathbf{c})=\sum m_{i}c_{i}
  21. l 1 l_{1}
  22. 𝐜 ( 1 ) ( t ) , 𝐜 ( 2 ) ( t ) \mathbf{c}^{(1)}(t),\mathbf{c}^{(2)}(t)
  23. 𝐜 ( 1 ) - 𝐜 ( 2 ) = i m i | c i ( 1 ) - c i ( 2 ) | , \|\mathbf{c}^{(1)}-\mathbf{c}^{(2)}\|=\sum_{i}m_{i}|c^{(1)}_{i}-{c}^{(2)}_{i}|,
  24. i α i A i j β j A j \sum_{i}\alpha_{i}A_{i}\to\sum_{j}\beta_{j}A_{j}
  25. γ i = β i - α i \gamma_{i}=\beta_{i}-\alpha_{i}

Grossberg_network.html

  1. ϵ d n ( t ) / d t = - n ( t ) + ( b + - n ( t ) ) p + - ( n ( t ) + b - ) p - \epsilon dn(t)/dt=-n(t)+(b^{+}-n(t))p^{+}-(n(t)+b^{-})p^{-}\,

Grothendieck_local_duality.html

  1. Ω ¯ \overline{Ω}
  2. Ext R i ( M , Ω ¯ ) Hom R ( H m d - i ( M ) , E ( k ) ) \operatorname{Ext}_{R}^{i}(M,\overline{\Omega})\cong\operatorname{Hom}_{R}(H_{% m}^{d-i}(M),E(k))

Guruswami–Sudan_list_decoding_algorithm.html

  1. δ / 2 \delta/2
  2. δ / 2 \delta/2
  3. δ / 2 \delta/2
  4. 1 - 2 R 1-\sqrt{2R}
  5. 1 - R 1-\sqrt{R}
  6. δ \delta
  7. 𝔽 \mathbb{F}
  8. ( x i , y i ) i = 1 n {(x_{i},y_{i})_{i=1}^{n}}
  9. F × F F\times F
  10. d d
  11. t t
  12. f : F F f:F\mapsto F
  13. f ( x ) f(x)
  14. x x
  15. i | f ( x i ) = y i i|f(x_{i})=y_{i}
  16. t \geq t
  17. t t
  18. t ( n + d + 1 ) / 2 t\geq(n+d+1)/2
  19. ( 1 , k ) (1,k)
  20. t = ( 2 n d ) t=(\sqrt{2nd})
  21. 1 - ( R 2 ) 1-\left(\frac{R}{2}\right)
  22. R < 0.07 R<0.07
  23. w x , w y , ϵ Z + w_{x},w_{y},\epsilon Z^{+}
  24. ( w x , w y ) (w_{x},w_{y})
  25. q i j x i y j q_{ij}x^{i}y^{j}
  26. i w x + j w y iw_{x}+jw_{y}
  27. ( w x , w y ) (w_{x},w_{y})
  28. Q ( x , y ) = i j q i j x i y j Q(x,y)=\sum_{ij}q_{ij}x^{i}y^{j}
  29. ( w x , w y ) (w_{x},w_{y})
  30. 3 x y 3xy
  31. x , y x,y
  32. n , d , t n,d,t
  33. ( x 1 , y 1 ) ( x n , y n ) (x_{1},y_{1})\cdots(x_{n},y_{n})
  34. Q : F 2 F Q:F^{2}\mapsto F
  35. Q ( x , y ) Q(x,y)
  36. m + l d m+ld
  37. i i
  38. \in
  39. Q ( x i , y i ) = 0 , Q(x_{i},y_{i})=0,
  40. Q Q
  41. f f
  42. ( y - f ( x ) ) (y-f(x))
  43. f ( x i ) = y i f(x_{i})=y_{i}
  44. i i
  45. \in
  46. Q : F 2 F Q:F^{2}\mapsto F
  47. Q ( x , y ) Q(x,y)
  48. ( 1 , k ) (1,k)
  49. D D
  50. Q ( x , y ) = j = 0 l k = 0 m + ( l - j ) d q k j x k y j Q(x,y)=\sum_{j=0}^{l}\sum_{k=0}^{m+(l-j)d}q_{kj}x^{k}y^{j}
  51. q k j q_{kj}
  52. j = 0 l k = 0 m + ( l - j ) d q k j x k y j = 0 \sum_{j=0}^{l}\sum_{k=0}^{m+(l-j)d}q_{kj}x^{k}y^{j}=0
  53. i ϵ [ n ] i\epsilon[n]
  54. q k j q_{kj}
  55. ( m + 1 ) ( l + 1 ) + d ( l + 1 2 ) > n (m+1)(l+1)+d\begin{pmatrix}l+1\\ 2\end{pmatrix}>n
  56. Q ( x , y ) Q(x,y)
  57. Q ( x , y ) Q(x,y)
  58. d e g X ( Q ) deg_{X}(Q)
  59. X X
  60. Q ( x , y ) Q(x,y)
  61. d e g Y ( Q ) deg_{Y}(Q)
  62. Y Y
  63. Q ( x , y ) Q(x,y)
  64. Q ( x , y ) Q(x,y)
  65. m + l d m+ld
  66. q j k = 0 q_{jk}=0
  67. ( m + 1 ) ( l + 1 ) + d ( l + 1 2 ) > n (m+1)(l+1)+d\begin{pmatrix}l+1\\ 2\end{pmatrix}>n
  68. Q ( x , y ) Q(x,y)
  69. Q ( x , y ) Q(x,y)
  70. f ( x ) f(x)
  71. t > m + l d t>m+ld
  72. ( y - f ( x ) ) (y-f(x))
  73. Q ( x , y ) Q(x,y)
  74. p ( x ) = Q ( x , f ( x ) ) p(x)=Q(x,f(x))
  75. x x
  76. m + l d m+ld
  77. q j k x k y j q_{jk}x^{k}y^{j}
  78. Q ( x ) Q(x)
  79. Q Q
  80. ( 1 , d ) (1,d)
  81. m + l d m+ld
  82. k + j d m + l d k+jd\leq m+ld
  83. q k j x k f ( x ) j q_{kj}x^{k}f(x)^{j}
  84. x x
  85. k + j d m + l d k+jd\leq m+ld
  86. p ( x ) p(x)
  87. m + l d m+ld
  88. p ( x ) p(x)
  89. Q ( x i , f ( x i ) ) Q(x_{i},f(x_{i}))
  90. y i = f ( x i ) y_{i}=f(x_{i})
  91. p ( x i ) p(x_{i})
  92. m + l d m+ld
  93. p p
  94. Q ( x , f ( x ) ) 0 Q(x,f(x))\equiv 0
  95. m m
  96. l l
  97. m + l d < t m+ld<t
  98. ( m + 1 ) ( l + 1 ) + d ( l + 1 2 ) > n (m+1)(l+1)+d\begin{pmatrix}l+1\\ 2\end{pmatrix}>n
  99. l l
  100. m m
  101. m m
  102. ( n + 1 - d ( l + 1 2 ) ) / 2 - 1 (n+1-d\begin{pmatrix}l+1\\ 2\end{pmatrix})/2-1
  103. t t
  104. n + 1 l + 1 + d l 2 \frac{n+1}{l+1}+\frac{dl}{2}
  105. l l
  106. l = 2 ( n + 1 ) d - 1 l=\sqrt{\frac{2(n+1)}{d}}-1
  107. l l
  108. m m
  109. t t
  110. m ( n + 1 ) d 2 - ( n + 1 ) d 2 + d 2 - 1 = d 2 - 1 m\geq\sqrt{\frac{(n+1)d}{2}}-\sqrt{\frac{(n+1)d}{2}}+\frac{d}{2}-1=\frac{d}{2}-1
  111. t > 2 ( n + 1 ) d 2 d - d 2 - 1 t>\sqrt{\frac{2(n+1)d^{2}}{d}}-\frac{d}{2}-1
  112. t > 2 ( n + 1 ) d - d 2 - 1 t>\sqrt{2(n+1)d}-\frac{d}{2}-1
  113. ( n , k ) (n,k)
  114. 𝔽 = G F ( q ) \mathbb{F}=GF(q)
  115. ( α 1 , α 2 , , α n ) (\alpha_{1},\alpha_{2},\ldots,\alpha_{n})
  116. r r
  117. β = ( β 1 , β 2 , , β n ) \beta=(\beta_{1},\beta_{2},\ldots,\beta_{n})
  118. \in
  119. 𝔽 n \mathbb{F}^{n}
  120. k \leq k
  121. Q ( x , y ) Q(x,y)
  122. Q ( x , y ) Q(x,y)
  123. r r
  124. ( 0 , 0 ) (0,0)
  125. Q ( x , y ) Q(x,y)
  126. r \leq r
  127. f ( x ) f(x)
  128. f ( x ) f(x)
  129. \qquad
  130. d e g x f ( x ) deg_{x}f(x)
  131. = =
  132. max i I { i } \max_{i\in I}\{i\}
  133. Q ( x , y ) = y - 4 x 2 Q(x,y)=y-4x^{2}
  134. Q ( x , y ) Q(x,y)
  135. Q ( x , y ) = y + 6 x 2 Q(x,y)=y+6x^{2}
  136. Q ( x , y ) Q(x,y)
  137. Q ( x , y ) = ( y - 4 x 2 ) ( y + 6 x 2 ) Q(x,y)=(y-4x^{2})(y+6x^{2})
  138. Q ( x , y ) = y 2 + 6 x 2 y - 4 x 2 y - 24 x 4 Q(x,y)=y^{2}+6x^{2}y-4x^{2}y-24x^{4}
  139. Q ( x , y ) Q(x,y)
  140. Q ( x , y ) = [ ( y - β ) - 4 ( x - α ) 2 ) ] [ ( y - β ) + 6 ( x - α ) 2 ) ] Q(x,y)=[(y-\beta)-4(x-\alpha)^{2})][(y-\beta)+6(x-\alpha)^{2})]
  141. Q ( x , y ) Q(x,y)
  142. ( α , β ) (\alpha,\beta)
  143. Q ( x , y ) Q(x,y)
  144. r r
  145. ( α , β ) (\alpha,\beta)
  146. Q ( x , y ) Q(x,y)
  147. r r
  148. ( α , β ) (\alpha,\beta)
  149. ( α , β ) ( 0 , 0 ) (\alpha,\beta)\neq(0,0)
  150. ( f ( α 1 ) , f ( α 2 ) , , f ( α n ) ) (f(\alpha_{1}),f(\alpha_{2}),\ldots,f(\alpha_{n}))
  151. ( α 1 , α 2 , , α n ) (\alpha_{1},\alpha_{2},\ldots,\alpha_{n})
  152. ( β 1 , β 2 , , β n ) (\beta_{1},\beta_{2},\ldots,\beta_{n})
  153. ( β 1 , β 2 , , β n ) (\beta_{1},\beta_{2},\ldots,\beta_{n})
  154. Q ( x , y ) Q(x,y)
  155. ( 1 , k ) - (1,k)-
  156. d d
  157. Q Q
  158. r r
  159. ( α i , β i ) (\alpha_{i},\beta_{i})
  160. 1 i n 1\leq i\leq n
  161. Q ( α i , β i ) = 0 Q(\alpha_{i},\beta_{i})=0\,
  162. Q ( x , y ) Q(x,y)
  163. y - p ( x ) y-p(x)
  164. p ( α i ) = β i p(\alpha_{i})=\beta_{i}
  165. t t
  166. i i
  167. 0 i n 0\leq i\leq n
  168. p ( x ) p(x)
  169. k \leq k
  170. k \leq k
  171. ( r + 1 2 ) \begin{pmatrix}r+1\\ 2\end{pmatrix}
  172. a i a_{i}
  173. Q ( x , y ) = i = 0 , j = 0 i = m , j = p a i , j x i y j Q(x,y)=\sum_{i=0,j=0}^{i=m,j=p}a_{i,j}x^{i}y^{j}
  174. deg x Q ( x , y ) = m \deg_{x}Q(x,y)=m
  175. deg y Q ( x , y ) = p \deg_{y}Q(x,y)=p
  176. Q ( x + α , y + β ) Q(x+\alpha,y+\beta)
  177. = =
  178. u = 0 , v = 0 r \sum_{u=0,v=0}^{r}
  179. Q u , v Q_{u,v}
  180. ( α , β ) (\alpha,\beta)
  181. x u x^{u}
  182. y v y^{v}
  183. Q u , v Q_{u,v}
  184. ( x , y ) (x,y)
  185. = =
  186. i = 0 , j = 0 i = m , j = p \sum_{i=0,j=0}^{i=m,j=p}
  187. ( i u ) \begin{pmatrix}i\\ u\end{pmatrix}
  188. ( j v ) \begin{pmatrix}j\\ v\end{pmatrix}
  189. a i , j a_{i,j}
  190. x i - u x^{i-u}
  191. y j - v y^{j-v}
  192. Q ( x + α , y + β ) = i , j a i , j ( x + α ) i ( y + β ) j Q(x+\alpha,y+\beta)=\sum_{i,j}a_{i,j}(x+\alpha)^{i}(y+\beta)^{j}
  193. Q ( x + α , y + β ) = i , j a i , j ( u ( i u ) x u α i - u ) ( v ( i v ) y v β j - v ) Q(x+\alpha,y+\beta)=\sum_{i,j}a_{i,j}\Bigg(\sum_{u}\begin{pmatrix}i\\ u\end{pmatrix}x^{u}\alpha^{i-u}\Bigg)\Bigg(\sum_{v}\begin{pmatrix}i\\ v\end{pmatrix}y^{v}\beta^{j-v}\Bigg)
  194. Q ( x + α , y + β ) = u , v x u y v ( i , j ( i u ) ( i v ) a i , j α i - u β j - v ) Q(x+\alpha,y+\beta)=\sum_{u,v}x^{u}y^{v}\Bigg(\sum_{i,j}\begin{pmatrix}i\\ u\end{pmatrix}\begin{pmatrix}i\\ v\end{pmatrix}a_{i,j}\alpha^{i-u}\beta^{j-v}\Bigg)
  195. Q ( x + α , y + β ) = u , v Q(x+\alpha,y+\beta)=\sum_{u,v}
  196. Q u , v ( α , β ) x u y v Q_{u,v}(\alpha,\beta)x^{u}y^{v}
  197. Q ( x , y ) Q(x,y)
  198. r r
  199. ( α , β ) (\alpha,\beta)
  200. Q u , v Q_{u,v}
  201. ( α , β ) (\alpha,\beta)
  202. \equiv
  203. 0
  204. 0 u + v r - 1 0\leq u+v\leq r-1
  205. u u
  206. r - v r-v
  207. 0 v r - 1 0\leq v\leq r-1
  208. v = 0 r - 1 r - v \sum_{v=0}^{r-1}{r-v}
  209. = =
  210. ( r + 1 2 ) \begin{pmatrix}r+1\\ 2\end{pmatrix}
  211. ( r + 1 2 ) \begin{pmatrix}r+1\\ 2\end{pmatrix}
  212. ( u , v ) (u,v)
  213. a i a_{i}
  214. Q ( x , p ( x ) ) 0 Q(x,p(x))\equiv 0
  215. y - p ( x ) y-p(x)
  216. Q ( x , y ) Q(x,y)
  217. y - p ( x ) y-p(x)
  218. Q ( x , y ) Q(x,y)
  219. Q ( x , y ) Q(x,y)
  220. Q ( x , y ) = L ( x , y ) ( y - p ( x ) ) Q(x,y)=L(x,y)(y-p(x))
  221. + +
  222. R ( x ) R(x)
  223. L ( x , y ) L(x,y)
  224. Q ( x , y ) Q(x,y)
  225. y - p ( x ) y-p(x)
  226. R ( x ) R(x)
  227. y y
  228. p ( x ) p(x)
  229. Q ( x , p ( x ) ) Q(x,p(x))
  230. \equiv
  231. 0
  232. R ( x ) R(x)
  233. \equiv
  234. 0
  235. p ( α ) = β p(\alpha)=\beta
  236. ( x - α ) r (x-\alpha)^{r}
  237. Q ( x , p ( x ) ) Q(x,p(x))
  238. Q ( x , y ) Q(x,y)
  239. = =
  240. u , v \sum_{u,v}
  241. Q u , v Q_{u,v}
  242. ( α , β ) (\alpha,\beta)
  243. ( x - α ) u (x-\alpha)^{u}
  244. ( y - β ) v (y-\beta)^{v}
  245. Q ( x , p ( x ) ) Q(x,p(x))
  246. = =
  247. u , v \sum_{u,v}
  248. Q u , v Q_{u,v}
  249. ( α , β ) (\alpha,\beta)
  250. ( x - α ) u (x-\alpha)^{u}
  251. ( p ( x ) - β ) v (p(x)-\beta)^{v}
  252. p ( α ) p(\alpha)
  253. = =
  254. β \beta
  255. ( p ( x ) - β ) (p(x)-\beta)
  256. ( x - α ) (x-\alpha)
  257. = =
  258. 0
  259. ( x - α ) u (x-\alpha)^{u}
  260. ( p ( x ) - β ) v (p(x)-\beta)^{v}
  261. ( x - α ) u + v (x-\alpha)^{u+v}
  262. = =
  263. 0
  264. ( x - α ) r (x-\alpha)^{r}
  265. Q ( x , p ( x ) ) Q(x,p(x))
  266. t r > D t\cdot r>D
  267. t > D r t>\frac{D}{r}
  268. D ( D + 2 ) 2 ( k - 1 ) > n ( r + 1 2 ) \frac{D(D+2)}{2(k-1)}>n\begin{pmatrix}r+1\\ 2\end{pmatrix}
  269. Q ( x , y ) Q(x,y)
  270. D = k n r ( r - 1 ) D=\sqrt{knr(r-1)}\,
  271. t = k n ( 1 - 1 r ) t=\left\lceil{\sqrt{kn(1-\frac{1}{r})}}\right\rceil
  272. r = 2 k n r=2kn
  273. t > k n - 1 2 > k n t>\left\lceil{\sqrt{kn-\frac{1}{2}}}\right\rceil>\left\lceil{\sqrt{kn}}\right\rceil
  274. 1 - R 1-\sqrt{R}

Hagen–Rubens_relation.html

  1. R 1 - 2 2 ϵ 0 ω σ R\approx 1-2\sqrt{\frac{2\epsilon_{0}\omega}{\sigma}}
  2. ω \omega
  3. σ \sigma
  4. ϵ 0 \epsilon_{0}

Halcyon_RB80.html

  1. F O 2 l o o p F_{O_{2}loop}
  2. F O 2 l o o p = ( P a m b * K b e l l o w s * K E + 1 ) F O 2 f e e d - 1 P a m b * K b e l l o w s * K E F_{O_{2}loop}=\frac{(P_{amb}*K_{bellows}*K_{E}+1)F_{O_{2}feed}-1}{P_{amb}*K_{% bellows}*K_{E}}
  3. P a m b P_{amb}
  4. K b e l l o w s K_{bellows}
  5. K E K_{E}
  6. F O 2 f e e d F_{O_{2}feed}

Hall_plane.html

  1. F = G F ( p n ) F=GF(p^{n})
  2. f ( x ) = x 2 - r x - s f(x)=x^{2}-rx-s
  3. ( a , b ) ( c , d ) = ( a c - b d - 1 f ( c ) , a d - b c + b r ) (a,b)\circ(c,d)=(ac-bd^{-1}f(c),ad-bc+br)
  4. d 0 d\neq 0
  5. ( a , b ) ( c , 0 ) = ( a c , b c ) (a,b)\circ(c,0)=(ac,bc)
  6. \ell
  7. π \pi\setminus\ell
  8. \ell
  9. \ell
  10. \ell
  11. f ( x ) = x 2 + 1 f(x)=x^{2}+1
  12. g ( x ) = x 2 - x - 1 g(x)=x^{2}-x-1
  13. h ( x ) = x 2 + x - 1 h(x)=x^{2}+x-1

Halved_cube_graph.html

  1. 1 2 Q n = Q n - 1 2 \frac{1}{2}Q_{n}=Q_{n-1}^{2}
  2. 1 2 Q 3 \tfrac{1}{2}Q_{3}
  3. 1 2 Q 4 \tfrac{1}{2}Q_{4}
  4. 1 2 Q 5 \tfrac{1}{2}Q_{5}

Hamiltonian_constraint_of_LQG.html

  1. q a b ( x ) q_{ab}(x)
  2. K a b ( x ) K^{ab}(x)
  3. q a b q_{ab}
  4. π a b = q ( K a b - q a b K c c ) \pi^{ab}=\sqrt{q}(K^{ab}-q^{ab}K_{c}^{c})
  5. H = det ( q ) ( K a b K a b - ( K a a ) 2 - R ) H=\sqrt{\det(q)}(K_{ab}K^{ab}-(K_{a}^{a})^{2}-\;^{3}R)
  6. R 3 \;{}^{3}R
  7. q a b ( x ) q_{ab}(x)
  8. S U ( 2 ) SU(2)
  9. A a i A_{a}^{i}
  10. E ~ i a \tilde{E}_{i}^{a}
  11. E ~ i a = det ( q ) E i a \tilde{E}_{i}^{a}=\sqrt{\det(q)}E_{i}^{a}
  12. det ( q ) q a b = E ~ i a E ~ j b δ i j \det(q)q^{ab}=\tilde{E}_{i}^{a}\tilde{E}_{j}^{b}\delta^{ij}
  13. i i
  14. S U ( 2 ) SU(2)
  15. A a i = Γ a i - i K a i A_{a}^{i}=\Gamma_{a}^{i}-iK_{a}^{i}
  16. Γ a i \Gamma_{a}^{i}
  17. Γ a i j \Gamma_{a\;\;i}^{\;\;j}
  18. Γ a i = Γ a j k ϵ j k i \Gamma_{a}^{i}=\Gamma_{ajk}\epsilon^{jki}
  19. K a i = K a b E ~ a i / det ( q ) K_{a}^{i}=K_{ab}\tilde{E}^{ai}/\sqrt{\det(q)}
  20. H = ϵ i j k F a b k E ~ i a E ~ j b det ( q ) H={\epsilon_{ijk}F_{ab}^{k}\tilde{E}_{i}^{a}\tilde{E}_{j}^{b}\over\sqrt{\det(q% )}}
  21. F a b k F_{ab}^{k}
  22. A a i A_{a}^{i}
  23. 1 / det ( q ) 1/\sqrt{\det(q)}
  24. H = 0 H=0
  25. H ~ = det ( q ) H = ϵ i j k F a b k E ~ i a E ~ j b = 0 \tilde{H}=\sqrt{\det(q)}H=\epsilon_{ijk}F_{ab}^{k}\tilde{E}_{i}^{a}\tilde{E}_{% j}^{b}=0
  26. ( + , + , + , + ) (+,+,+,+)
  27. t t
  28. A a i = Γ a i + β K a i A_{a}^{i}=\Gamma_{a}^{i}+\beta K_{a}^{i}
  29. H = - ζ ϵ i j k F a b k E ~ i a E ~ j b det ( q ) + 2 ζ β 2 - 1 β 2 ( E ~ i a E ~ j b - E ~ j a E ~ i b ) det ( q ) ( A a i - Γ a i ) ( A b j - Γ b j ) = H E + H H=-\zeta\frac{\epsilon_{ijk}F_{ab}^{k}\tilde{E}_{i}^{a}\tilde{E}_{j}^{b}}{% \sqrt{\det(q)}}+2{\zeta\beta^{2}-1\over\beta^{2}}\frac{(\tilde{E}_{i}^{a}% \tilde{E}_{j}^{b}-\tilde{E}_{j}^{a}\tilde{E}_{i}^{b})}{\sqrt{\det(q)}}(A_{a}^{% i}-\Gamma_{a}^{i})(A_{b}^{j}-\Gamma_{b}^{j})=H_{E}+H^{\prime}
  30. β \beta
  31. ζ \zeta
  32. Γ a i \Gamma_{a}^{i}
  33. β = i \beta=i
  34. H E H_{E}
  35. β = ± 1 \beta=\pm 1
  36. 1 / det ( q ) 1/\sqrt{\det(q)}
  37. β \beta
  38. 1 / det ( q ) 1/\sqrt{\det(q)}
  39. { A c k , V } = ϵ a b c ϵ i j k E ~ i a E ~ j b det ( q ) \{A_{c}^{k},V\}={\epsilon_{abc}\epsilon^{ijk}\tilde{E}_{i}^{a}\tilde{E}_{j}^{b% }\over\sqrt{\det(q)}}
  40. V V
  41. V = d 3 x det ( q ) = 1 6 d 3 x | E ~ i a E ~ j b E ~ k c ϵ i j k ϵ a b c | V=\int d^{3}x\sqrt{\det(q)}={1\over 6}\int d^{3}x\sqrt{|\tilde{E}_{i}^{a}% \tilde{E}_{j}^{b}\tilde{E}_{k}^{c}\epsilon^{ijk}\epsilon_{abc}|}
  42. H E = { A c k , V } F a b k ϵ ~ a b c H_{E}=\{A_{c}^{k},V\}F_{ab}^{k}\tilde{\epsilon}^{abc}
  43. Γ a i \Gamma_{a}^{i}
  44. E ~ i a \tilde{E}^{a}_{i}
  45. D a E b i = 0 D_{a}E_{b}^{i}=0
  46. c g a b = 0 \nabla_{c}g_{ab}=0
  47. det ( E ~ ) = | det ( E ) | 2 \det(\tilde{E})=|\det(E)|^{2}
  48. E ~ i a \tilde{E}_{i}^{a}
  49. Γ a i = 1 2 ϵ i j k E ~ k b [ E ~ a , b j - E ~ b , a j + E ~ j c E ~ a l E ~ c , b l ] + 1 4 ϵ i j k E ~ k b [ 2 E ~ a j ( det ( E ~ ) ) , b det ( E ~ ) - E ~ b j ( det ( E ~ ) ) , a det ( E ~ ) ] \Gamma_{a}^{i}={1\over 2}\epsilon^{ijk}\tilde{E}_{k}^{b}[\tilde{E}^{j}_{a,b}-% \tilde{E}^{j}_{b,a}+\tilde{E}_{j}^{c}\tilde{E}_{a}^{l}\tilde{E}^{l}_{c,b}]+{1% \over 4}\epsilon^{ijk}\tilde{E}_{k}^{b}\Big[2\tilde{E}_{a}^{j}{(\det(\tilde{E}% ))_{,b}\over\det(\tilde{E})}-\tilde{E}_{b}^{j}{(\det(\tilde{E}))_{,a}\over\det% (\tilde{E})}\Big]
  50. K = d 3 x K a i E ~ i a K=\int d^{3}xK_{a}^{i}\tilde{E}_{i}^{a}
  51. K a i = K a b E ~ a i / det ( q ) K_{a}^{i}=K_{ab}\tilde{E}^{ai}/\sqrt{\det(q)}
  52. K a i = { A a i , K } K_{a}^{i}=\{A_{a}^{i},K\}
  53. { Γ a i , K } = 0 \{\Gamma_{a}^{i},K\}=0
  54. β K \beta K
  55. E ~ i a E ~ i a / β \tilde{E}_{i}^{a}\mapsto\tilde{E}_{i}^{a}/\beta
  56. Γ a i \Gamma_{a}^{i}
  57. A a i - Γ a i = β K a i = β { A a i , K } A_{a}^{i}-\Gamma_{a}^{i}=\beta K_{a}^{i}=\beta\{A_{a}^{i},K\}
  58. A a i A_{a}^{i}
  59. K K
  60. H = ϵ a b c ϵ i j k { A a i , K } { A b j , K } { A c k , V } H^{\prime}=\epsilon^{abc}\epsilon_{ijk}\{A_{a}^{i},K\}\{A_{b}^{j},K\}\{A_{c}^{% k},V\}
  61. K K
  62. K K
  63. K = - { V , d 3 x H E } K=-\{V,\int d^{3}xH_{E}\}
  64. L = - d 4 x - det ( g ) ( - g μ ν μ φ ν φ - V ( φ ) ) L=-\int d^{4}x\sqrt{-\det(g)}(-g^{\mu\nu}\partial_{\mu}\varphi\partial_{\nu}% \varphi-V(\varphi))
  65. μ , ν \mu,\nu
  66. π ~ = δ L / δ φ ˙ \tilde{\pi}=\delta L/\delta\dot{\varphi}
  67. H = d 3 x N ( π ~ 2 det ( q ) + det ( q ) ( q a b a φ b φ + V ( φ ) ) ) + N a π ~ a φ H=\int d^{3}xN\left({\tilde{\pi}^{2}\over\sqrt{\det(q)}}+\sqrt{\det(q)}(q^{ab}% \partial_{a}\varphi\partial_{b}\varphi+V(\varphi))\right)+N^{a}\tilde{\pi}% \partial_{a}\varphi
  68. N N
  69. N a N^{a}
  70. H = d 3 x N det ( q ) ( π ~ 2 + E ~ i a E ~ b i a φ b φ + det ( q ) V ( φ ) ) + N a π ~ a φ H=\int d^{3}x{N\over\sqrt{\det(q)}}\left(\tilde{\pi}^{2}+\tilde{E}_{i}^{a}% \tilde{E}^{bi}\partial_{a}\varphi\partial_{b}\varphi+\det(q)V(\varphi)\right)+% N^{a}\tilde{\pi}\partial_{a}\varphi
  71. N a N^{a}
  72. N N
  73. C ( N ) φ = d 3 x N a π ~ a φ C(\vec{N})_{\varphi}=\int d^{3}xN^{a}\tilde{\pi}\partial_{a}\varphi
  74. H ( N ) φ = d 3 x N det ( q ) ( π ~ 2 + E ~ i a E ~ b i a φ b φ + det ( q ) V ( φ ) ) H(N)_{\varphi}=\int d^{3}x{N\over\sqrt{\det(q)}}\left(\tilde{\pi}^{2}+\tilde{E% }_{i}^{a}\tilde{E}^{bi}\partial_{a}\varphi\partial_{b}\varphi+\det(q)V(\varphi% )\right)
  75. 8 π G β 8\pi G\beta
  76. γ I \gamma^{I}
  77. γ I e I a ( x ) = γ a ( x ) \gamma^{I}e_{I}^{a}(x)=\gamma^{a}(x)
  78. ψ e i ϵ I J ( x ) σ I J ψ \psi\mapsto e^{i\epsilon^{IJ}(x)\sigma_{IJ}}\psi
  79. ϵ I J \epsilon_{IJ}
  80. ω μ I J \omega_{\mu}^{IJ}
  81. a ψ = ( a - i 4 ω a I J σ I J ) ψ \nabla_{a}\psi=(\partial_{a}-{i\over 4}\omega_{a}^{IJ}\sigma_{IJ})\psi
  82. ( i γ a a - m ) ψ = 0 (i\gamma^{a}\nabla_{a}-m)\psi=0
  83. S D i r a c = 1 2 d 4 x - d e t ( g ) [ Ψ ¯ γ I E I a a Ψ - a Ψ ¯ γ I E I a Ψ ] S_{Dirac}={1\over 2}\int_{\mathcal{M}}d^{4}x\sqrt{-det(g)}[\overline{\Psi}% \gamma^{I}E_{I}^{a}\nabla_{a}\Psi-\overline{\nabla_{a}\Psi}\gamma^{I}E_{I}^{a}\Psi]
  84. Ψ = ( ψ , η ) \Psi=(\psi,\eta)
  85. Ψ ¯ = ( Ψ * ) T γ 0 \overline{\Psi}=(\Psi^{*})^{T}\gamma^{0}
  86. a \nabla_{a}
  87. E a I E_{a}^{I}
  88. L = - d 4 x - det ( g ) ( g μ α g ν β μ ν α β ) L=-\int d^{4}x\sqrt{-\det(g)}(g^{\mu\alpha}g^{\nu\beta}\mathcal{F}_{\mu\nu}% \mathcal{F}_{\alpha\beta})
  89. μ ν = μ 𝒜 ν - ν 𝒜 μ \mathcal{F}^{\mu\nu}=\nabla^{\mu}\mathcal{A}^{\nu}-\nabla^{\nu}\mathcal{A}^{\mu}
  90. 0 a = a \mathcal{F}^{0a}=\mathcal{E}^{a}
  91. a b = ϵ a b c B c \mathcal{F}^{ab}=\epsilon^{abc}B_{c}
  92. a = - a 𝒜 0 - 𝒜 ˙ a \mathcal{E}^{a}=-\nabla_{a}\mathcal{A}_{0}-\dot{\mathcal{A}}_{a}
  93. B a = ϵ a b c b 𝒜 c B^{a}=\epsilon^{abc}\nabla_{b}\mathcal{A}_{c}
  94. H ( N , N a , Λ ) = 1 2 Σ d 3 x N q a b d e t ( q ) [ ~ a ~ b + B a B b ] + N a a b ~ a + Λ a ~ a H(N,N^{a},\Lambda)={1\over 2}\int_{\Sigma}d^{3}xN{q_{ab}\over\sqrt{det(q)}}[% \tilde{\mathcal{E}}^{a}\tilde{\mathcal{E}}^{b}+B^{a}B^{b}]+N^{a}\mathcal{F}_{% ab}\tilde{\mathcal{E}}^{a}+\Lambda\nabla_{a}\tilde{\mathcal{E}}^{a}
  95. B a = ϵ a b c B c ~ a = - q N 0 a B^{a}=\epsilon^{abc}B_{c}\qquad\qquad\tilde{\mathcal{E}}^{a}=-\sqrt{q}N% \mathcal{F}^{0a}
  96. 𝒜 a \mathcal{A}_{a}
  97. ~ a \tilde{\mathcal{E}}^{a}
  98. H = 1 2 Σ d 3 x q a b d e t ( q ) [ ~ I a ~ I b + B I a B I b ] H={1\over 2}\int_{\Sigma}d^{3}x{q_{ab}\over\sqrt{det(q)}}[\tilde{\mathcal{E}}^% {a}_{I}\tilde{\mathcal{E}}^{b}_{I}+B^{a}_{I}B^{b}_{I}]
  99. H = H E i n s t e i n + H M a x w e l + H Y a n g - M i l l s + H D i r a c + H H i g g s H=H_{Einstein}+H_{Maxwel}+H_{Yang-Mills}+H_{Dirac}+H_{Higgs}
  100. H ( N ) = d 3 x N { A c k , V } F a b k ϵ a b c H(N)=\int d^{3}xN\{A_{c}^{k},V\}F_{ab}^{k}\epsilon^{abc}
  101. h γ [ A ] = 𝒫 exp { - s 0 s 1 d s γ ˙ a A a i ( γ ( s ) ) T i } h_{\gamma}[A]=\mathcal{P}\exp\left\{-\int_{s_{0}}^{s_{1}}ds\dot{\gamma}^{a}A_{% a}^{i}(\gamma(s))T_{i}\right\}
  102. 𝒫 \mathcal{P}
  103. s s
  104. T i T_{i}
  105. s u ( 2 ) su(2)
  106. [ T i , T j ] = 2 i ϵ i j k T k [T^{i},T^{j}]=2i\epsilon^{ijk}T^{k}
  107. T r ( T i T j ) - T r ( T j T i ) = 2 i ϵ i j k T r ( T k ) Tr(T^{i}T^{j})-Tr(T^{j}T^{i})=2i\epsilon^{ijk}Tr(T^{k})
  108. T r ( T i ) = 0 Tr(T^{i})=0
  109. Ψ [ A ] = γ Ψ [ γ ] s γ [ A ] \Psi[A]=\sum_{\gamma}\Psi[\gamma]s_{\gamma}[A]
  110. Ψ [ γ ] = [ d A ] Ψ [ A ] s γ [ A ] \Psi[\gamma]=\int[dA]\Psi[A]s_{\gamma}[A]
  111. ψ [ x ] = d k ψ ( k ) exp ( i k x ) \psi[x]=\int dk\psi(k)\exp(ikx)
  112. O ^ \hat{O}
  113. Φ [ A ] = O ^ Ψ [ A ] \Phi[A]=\hat{O}\Psi[A]
  114. Φ [ γ ] \Phi[\gamma]
  115. Φ [ γ ] = [ d A ] Φ [ A ] s γ [ A ] \Phi[\gamma]=\int[dA]\Phi[A]s_{\gamma}[A]
  116. O ^ \hat{O}^{\prime}
  117. Ψ [ γ ] \Psi[\gamma]
  118. O ^ Ψ [ γ ] = [ d A ] s γ [ A ] O ^ Ψ [ A ] \hat{O}^{\prime}\Psi[\gamma]=\int[dA]s_{\gamma}[A]\hat{O}\Psi[A]
  119. O ^ Ψ [ γ ] = [ d A ] ( O ^ s γ [ A ] ) Ψ [ A ] \hat{O}^{\prime}\Psi[\gamma]=\int[dA](\hat{O}^{\dagger}s_{\gamma}[A])\Psi[A]
  120. O ^ \hat{O}^{\dagger}
  121. O ^ \hat{O}
  122. Ψ [ A ] \Psi[A]
  123. O ^ \hat{O}^{\prime}
  124. O ^ \hat{O}^{\dagger}
  125. A A
  126. s γ [ A ] s_{\gamma}[A]
  127. γ \gamma
  128. O ^ \hat{O}^{\prime}
  129. γ \gamma
  130. Ψ [ γ ] \Psi[\gamma]
  131. h ^ γ Ψ [ η ] = h γ Ψ [ η ] \hat{h}_{\gamma}\Psi[\eta]=h_{\gamma}\Psi[\eta]
  132. Δ \Delta
  133. v ( Δ ) v(\Delta)
  134. s i ( Δ ) s_{i}(\Delta)
  135. i = 1 , 2 , 3 i=1,2,3
  136. v ( Δ ) v(\Delta)
  137. α i j = s i ( Δ ) s i j ( Δ ) s j ( Δ ) - 1 \alpha_{ij}=s_{i}(\Delta)\cdot s_{ij}(\Delta)\cdot s_{j}(\Delta)^{-1}
  138. s i ( Δ ) s_{i}(\Delta)
  139. s i s_{i}
  140. s j s_{j}
  141. v ( Δ ) v(\Delta)
  142. s i j s_{ij}
  143. v ( Δ ) v(\Delta)
  144. s j s_{j}
  145. h γ [ A ] = 𝒫 exp { - s 0 s 1 d s γ ˙ a A a i ( γ ( s ) ) T i } I - ( s k a ) A a i T i h_{\gamma}[A]=\mathcal{P}\exp\left\{-\int_{s_{0}}^{s_{1}}ds\dot{\gamma}^{a}A_{% a}^{i}(\gamma(s))T_{i}\right\}\approx I-(s_{k}^{a})A_{a}^{i}T_{i}
  146. lim Δ v ( Δ ) h s k = I - A c s k c \lim_{\Delta\rightarrow v(\Delta)}h_{s_{k}}=I-A_{c}s_{k}^{c}
  147. s k c s_{k}^{c}
  148. s k s_{k}
  149. lim Δ v ( Δ ) h α i j = I + 1 2 F a b s i a s j b \lim_{\Delta v\rightarrow(\Delta)}h_{\alpha_{ij}}=I+{1\over 2}F_{ab}s_{i}^{a}s% _{j}^{b}
  150. H Δ ( N ) = Δ N ( v ( Δ ) ) ϵ i j k T r ( h α i j h s k { h s k - 1 , V } ) H_{\Delta}(N)=\sum_{\Delta}N(v(\Delta))\epsilon^{ijk}Tr\big(h_{\alpha_{ij}}h_{% s_{k}}\{h_{s_{k}}^{-1},V\}\big)
  151. Δ \Delta
  152. H Δ ( N ) = Δ N ( v ( Δ ) ) ϵ i j k T r ( ( I + 1 2 F a b s i a s j b ) ( I - A c s k c ) { ( I + A d s k d ) , V } ) H_{\Delta}(N)=\sum_{\Delta}N(v(\Delta))\epsilon^{ijk}Tr\big((I+{1\over 2}F_{ab% }s_{i}^{a}s_{j}^{b})(I-A_{c}s_{k}^{c})\{(I+A_{d}s_{k}^{d}),V\}\big)
  153. s k c s_{k}^{c}
  154. h s k h_{s_{k}}
  155. α i j \alpha_{ij}
  156. A c = A c i T i A_{c}=A_{c}^{i}T_{i}
  157. T i T_{i}
  158. h α i j h_{\alpha_{ij}}
  159. F a b F_{ab}
  160. s s
  161. Ψ | \langle\Psi|
  162. Ψ , s = Ψ , s \langle\Psi,s\rangle=\langle\Psi,s^{\prime}\rangle
  163. s s
  164. s s^{\prime}
  165. H ^ ( N ) \hat{H}(N)
  166. H ^ ( N ) Ψ , s = lim Δ v Δ Ψ , H ^ Δ ( N ) s \langle\hat{H}(N)\Psi,s\rangle=\lim_{\Delta\rightarrow v}\sum_{\Delta}\langle% \Psi,\hat{H}_{\Delta}(N)s\rangle
  167. Ψ \Psi
  168. C a ( x ) = 0 C^{a}(x)=0
  169. x Σ x\in\Sigma
  170. K i n \mathcal{H}_{Kin}
  171. H ( x ) H(x)
  172. D i f f \mathcal{H}_{Diff}
  173. D i f f \mathcal{H}_{Diff}
  174. P h y s \mathcal{H}_{Phys}
  175. D i f f \mathcal{H}_{Diff}
  176. { H ( N ) , H ( M ) } = C ( K ) \{H(N),H(M)\}=C(\vec{K})
  177. K a = E ~ i a E ~ b i ( N b M - M b N ) / ( det ( q ) ) K^{a}=\tilde{E}_{i}^{a}\tilde{E}^{bi}(N\partial_{b}M-M\partial_{b}N)/(\det(q))
  178. { H ( N ) , H ( M ) } \{H(N),H(M)\}
  179. C \vec{C}
  180. H ^ ( x ) \hat{H}(x)
  181. K i n \mathcal{H}_{Kin}
  182. ~ a , B a \tilde{\mathcal{E}}^{a},B^{a}
  183. q a b / q q_{ab}/\sqrt{q}
  184. q a b q ( x ) δ i j { A a i ( x ) , V } { A b j ( x ) , V } {q_{ab}\over\sqrt{q}}(x)\propto\delta_{ij}\{A_{a}^{i}(x),\sqrt{V}\}\{A_{b}^{j}% (x),\sqrt{V}\}
  185. h ^ ( x , λ ) Ψ = e i λ φ ( x ) Ψ \hat{h}(x,\lambda)\Psi=e^{i\lambda\varphi(x)}\Psi
  186. P ( f ) = d 3 x π φ ( x ) f ( x ) P(f)=\int d^{3}x\pi_{\varphi}(x)f(x)
  187. π φ \pi_{\varphi}
  188. f ( x ) f(x)
  189. { h ( x , λ ) , P ( f ) } = i λ f ( x ) h ( x , λ ) \{h(x,\lambda),P(f)\}=i\lambda f(x)h(x,\lambda)
  190. [ h ^ ( x , λ ) , P ^ ( f ) ] = i λ f ( x ) h ^ ( x , λ ) [\hat{h}(x,\lambda),\hat{P}(f)]=i\lambda f(x)\hat{h}(x,\lambda)
  191. E a i E^{i}_{a}
  192. E E
  193. E E
  194. H ( x ) = 0 H(x)=0
  195. M = d 3 x [ H ( x ) ] 2 det q ( x ) M=\int d^{3}x{[H(x)]^{2}\over\sqrt{\det q(x)}}
  196. H ( x ) H(x)
  197. M M
  198. H ( x ) H(x)
  199. H ( x ) H(x)
  200. M M
  201. M M
  202. C ( N ) C(\vec{N})
  203. { M , C ( N ) } = 0 \{M,C(\vec{N})\}=0
  204. s u ( 2 ) su(2)
  205. { M , M } = 0 \{M,M\}=0
  206. M = d 3 x H ( x ) 2 det ( q ) ( x ) = d 3 x ( H [ det ( q ) ] 1 / 4 ) ( x ) d 3 y δ ( x , y ) ( H [ det ( q ) ] 1 / 4 ) ( y ) M=\int d^{3}x{H(x)^{2}\over\sqrt{\det(q)}(x)}=\int d^{3}x({H\over[\det(q)]^{1/% 4}})(x)\int d^{3}y\delta(x,y)({H\over[\det(q)]^{1/4}})(y)
  207. χ ϵ ( x , y ) \chi_{\epsilon}(x,y)
  208. lim ϵ 0 χ ϵ ( x , y ) / ϵ 3 = δ ( x , y ) \lim_{\epsilon\rightarrow 0}\chi_{\epsilon}(x,y)/\epsilon^{3}=\delta(x,y)
  209. χ ϵ ( x , x ) = 1 \chi_{\epsilon}(x,x)=1
  210. V ϵ , x = d 3 y χ ϵ ( x , y ) det ( q ) ( y ) V_{\epsilon,x}=\int d^{3}y\chi_{\epsilon}(x,y)\sqrt{\det(q)}(y)
  211. { A , V ϵ } \{A,\sqrt{V_{\epsilon}}\}
  212. { A , V } \{A,V\}
  213. [ det ( q ) ] 1 / 4 [\det(q)]^{1/4}
  214. M = d 3 x ϵ a b c { A c k , V ϵ } F a b k ( x ) d 3 y χ ϵ ( x , y ) ϵ a b c { A c k , V ϵ } F a b k ( y ) M=\int d^{3}x\epsilon^{abc}\{A_{c}^{k},\sqrt{V}_{\epsilon}\}F_{ab}^{k}(x)\int d% ^{3}y\chi_{\epsilon}(x,y)\epsilon^{a^{\prime}b^{\prime}c^{\prime}}\{A_{c^{% \prime}}^{k^{\prime}},\sqrt{V}_{\epsilon}\}F_{a^{\prime}b^{\prime}}^{k^{\prime% }}(y)
  215. M = lim ϵ 0 Δ , Δ χ ( v ( Δ ) , v ( Δ ) ) C ϵ ( Δ ) ¯ C ϵ ( Δ ) M=\lim_{\epsilon\rightarrow 0}\sum_{\Delta,\Delta^{\prime}}\chi(v(\Delta),v(% \Delta^{\prime}))\overline{C_{\epsilon}(\Delta)}C_{\epsilon}(\Delta^{\prime})
  216. C ϵ ( Δ ) C_{\epsilon}(\Delta)
  217. H Δ H_{\Delta}
  218. C ϵ ( Δ ) C_{\epsilon}(\Delta)
  219. H Δ H_{\Delta}
  220. K i n \mathcal{H}_{Kin}
  221. M ^ \hat{M}
  222. K i n \mathcal{H}_{Kin}
  223. D i f f \mathcal{H}_{Diff}
  224. M ^ \hat{M}
  225. Q M Q_{M}
  226. M ^ \hat{M}
  227. Q M Q_{M}
  228. M M
  229. D i f f \mathcal{H}_{Diff}
  230. D i f f \mathcal{H}_{Diff}
  231. K i n \mathcal{H}_{Kin}

Hamilton–Jacobi–Einstein_equation.html

  1. S S
  2. Ψ Ψ
  3. | Ψ |Ψ\rangle
  4. Ψ = ρ e i S / \Psi=\sqrt{\rho}e^{iS/\hbar}
  5. Ψ Ψ
  6. ρ = Ψ * Ψ = | Ψ | \sqrt{ρ}=\sqrt{Ψ*Ψ}=|Ψ|
  7. ħ ħ
  8. i Ψ t = H ^ Ψ , i\hbar\frac{\partial\Psi}{\partial t}=\hat{H}\Psi\,,
  9. ħ 0 ħ→0
  10. - S t = H , -\frac{\partial S}{\partial t}=H\,,
  11. x x
  12. p p
  13. E E
  14. t t
  15. Δ x Δ p 2 , Δ E Δ t 2 , \Delta x\Delta p\geq\frac{\hbar}{2},\quad\Delta E\Delta t\geq\frac{\hbar}{2}\,,
  16. 𝐫 \mathbf{r}
  17. 𝐫 = ( x , y , z ) \mathbf{r}=(x,y,z)
  18. t t
  19. g i j = g i j ( 𝐫 ) . g_{ij}=g_{ij}(\mathbf{r})\,.
  20. g g
  21. R R
  22. δ δ
  23. d d
  24. π i j ( 𝐫 ) = π i j = δ S δ g i j , \pi^{ij}(\mathbf{r})=\pi^{ij}=\frac{\delta S}{\delta g_{ij}}\,,
  25. g g
  26. π π
  27. q q
  28. p = S / q p=∂S/∂q
  29. g i j g i j + δ g i j , g_{ij}\rightarrow g_{ij}+\delta g_{ij}\,,
  30. δ S = δ S δ g i j ( 𝐫 ) δ g i j ( 𝐫 ) d 3 𝐫 = 0 , \delta S=\int\frac{\delta S}{\delta g_{ij}(\mathbf{r})}\delta g_{ij}(\mathbf{r% })\mathrm{d}^{3}\mathbf{r}=0\,,
  31. Ψ = n c n ψ n , \Psi=\sum_{n}c_{n}\psi_{n}\,,
  32. δ S = S n + 1 - S n = 0 , \delta S=S_{n+1}-S_{n}=0\,,
  33. S 1 = S 2 = = S n = . S_{1}=S_{2}=\cdots=S_{n}=\cdots\,.
  34. Ψ Ψ

Hammer_retroazimuthal_projection.html

  1. x = R K cos ϕ 1 sin ( λ - λ 0 ) x=RK\cos\phi_{1}\sin(\lambda-\lambda_{0})
  2. y = - R K [ sin ϕ 1 cos ϕ - cos ϕ 1 sin ϕ cos ( λ - λ 0 ) ] y=-RK[\sin\phi_{1}\cos\phi-\cos\phi_{1}\sin\phi\cos(\lambda-\lambda_{0})]
  3. K = z / sin z K=z/\sin z
  4. cos z = sin ϕ 1 sin ϕ + cos ϕ 1 cos ϕ cos ( λ - λ 0 ) \cos z=\sin\phi_{1}\sin\phi+\cos\phi_{1}\cos\phi\cos(\lambda-\lambda_{0})

Hanner_polytope.html

  1. 1 \ell_{1}
  2. \ell_{\infty}

Haplotype_estimation.html

  1. N N
  2. 2 N - 1 2^{N-1}
  3. O ( K 2 ) O(K^{2})
  4. O ( K ) O(K)

Harcourt's_theorem.html

  1. a a + b b + c c = 2 K . aa^{\prime}+bb^{\prime}+cc^{\prime}=2K.
  2. - a a + b b + c c = 2 K . -aa^{\prime}+bb^{\prime}+cc^{\prime}=2K.
  3. a a + b b + c c = 2 K . aa^{\prime}+bb^{\prime}+cc^{\prime}=2K.

Hardy–Littlewood_inequality.html

  1. n f ( x ) g ( x ) d x n f * ( x ) g * ( x ) d x \int_{\mathbb{R}^{n}}f(x)g(x)\,dx\leq\int_{\mathbb{R}^{n}}f^{*}(x)g^{*}(x)\,dx
  2. f ( x ) = 0 χ f ( x ) > r d r f(x)=\int_{0}^{\infty}\chi_{f(x)>r}\,dr
  3. g ( x ) = 0 χ g ( x ) > s d s g(x)=\int_{0}^{\infty}\chi_{g(x)>s}\,ds
  4. χ f ( x ) > r \chi_{f(x)>r}
  5. E f = { x X : f ( x ) > r } E_{f}=\left\{x\in X:f(x)>r\right\}\,
  6. χ g ( x ) > s \chi_{g(x)>s}
  7. E g = { x X : g ( x ) > s } E_{g}=\left\{x\in X:g(x)>s\right\}\,
  8. n f ( x ) g ( x ) d x = n 0 0 χ f ( x ) > r χ g ( x ) > s d r d s d x \int_{\mathbb{R}^{n}}f(x)g(x)\,dx=\displaystyle\int_{\mathbb{R}^{n}}\int_{0}^{% \infty}\int_{0}^{\infty}\chi_{f(x)>r}\chi_{g(x)>s}\,dr\,ds\,dx
  9. = 0 0 n χ f ( x ) > r g ( x ) > s d x d r d s =\int_{0}^{\infty}\int_{0}^{\infty}\int_{\mathbb{R}^{n}}\chi_{f(x)>r\cap g(x)>% s}\,dx\,dr\,ds
  10. = 0 0 μ ( { f ( x ) > r } { g ( x ) > s } ) d r d s =\int_{0}^{\infty}\int_{0}^{\infty}\mu\left(\left\{f(x)>r\right\}\cap\left\{g(% x)>s\right\}\right)\,dr\,ds
  11. 0 0 min ( μ ( f ( x ) > r ) ; μ ( g ( x ) > s ) ) d r d s \leq\int_{0}^{\infty}\int_{0}^{\infty}\min\left(\mu\left(f(x)>r\right);\mu% \left(g(x)>s\right)\right)\,dr\,ds
  12. = 0 0 min ( μ ( f * ( x ) > r ) ; μ ( g * ( x ) > s ) ) d r d s =\int_{0}^{\infty}\int_{0}^{\infty}\min\left(\mu\left(f^{*}(x)>r\right);\mu% \left(g^{*}(x)>s\right)\right)\,dr\,ds
  13. = 0 0 μ ( { f ( x ) > r } { g ( x ) > s } ) d r d s =\int_{0}^{\infty}\int_{0}^{\infty}\mu\left(\left\{f^{\ast}(x)>r\right\}\cap% \left\{g^{\ast}(x)>s\right\}\right)\,dr\,ds
  14. = n f * ( x ) g * ( x ) d x =\int_{\mathbb{R}^{n}}f^{*}(x)g^{*}(x)\,dx

HARP_(algorithm).html

  1. I k ( 𝐲 , t ) I_{k}(\mathbf{y},t)
  2. 𝐲 = [ y 1 , y 2 ] T \mathbf{y}=[y_{1},y_{2}]^{T}
  3. t t
  4. I k ( 𝐲 , t ) = D k ( 𝐲 , t ) e j ϕ k ( 𝐲 , t ) I_{k}(\mathbf{y},t)=D_{k}(\mathbf{y},t)e^{j\phi_{k}(\mathbf{y},t)}
  5. D k D_{k}
  6. ϕ k \phi_{k}
  7. k k
  8. I k ( 𝐲 , t ) I_{k}(\mathbf{y},t)
  9. [ - π , + π ) [-\pi,+\pi)
  10. a k ( 𝐲 , t ) a_{k}(\mathbf{y},t)
  11. a k ( 𝐲 , t ) = m o d ( ϕ k ( 𝐲 , t ) + π , 2 π ) - π a_{k}(\mathbf{y},t)=mod(\phi_{k}(\mathbf{y},t)+\pi,2\pi)-\pi
  12. ϕ k \phi_{k}
  13. a k a_{k}
  14. a k a_{k}
  15. 𝐲 m \mathbf{y}_{m}
  16. t m t_{m}
  17. 𝐲 m + 1 \mathbf{y}_{m+1}
  18. t m + 1 t_{m+1}
  19. ϕ k ( 𝐲 m + 1 , t m + 1 ) = ϕ k ( 𝐲 m , t m ) \phi_{k}(\mathbf{y}_{m+1},t_{m+1})=\phi_{k}(\mathbf{y}_{m},t_{m})
  20. y ( n + 1 ) = y ( n ) - [ ϕ k ( 𝐲 ( n ) , t m ) ] - 1 [ ϕ k ( 𝐲 ( n ) , t m ) - ϕ k ( 𝐲 m , t m ) ] y^{(n+1)}=y^{(n)}-[\nabla\phi_{k}(\mathbf{y}^{(n)},t_{m})]^{-1}[\phi_{k}(% \mathbf{y}^{(n)},t_{m})-\phi_{k}(\mathbf{y}_{m},t_{m})]
  21. ϕ k \phi_{k}
  22. a k a_{k}
  23. ϕ k \phi_{k}
  24. a k a_{k}
  25. ± 1 \pm 1

Hasse_invariant_of_an_algebra.html

  1. inv L / K : Br ( L / K ) / . \underset{L/K}{\operatorname{inv}}:\operatorname{Br}(L/K)\rightarrow\mathbb{Q}% /\mathbb{Z}.
  2. 0 Br ( K ) v S Br ( K v ) 𝐐 / 𝐙 0 , 0\rightarrow\textrm{Br}(K)\rightarrow\bigoplus_{v\in S}\textrm{Br}(K_{v})% \rightarrow\mathbf{Q}/\mathbf{Z}\rightarrow 0,

Hausdorff_completion.html

  1. G ^ \widehat{G}
  2. G n G_{n}
  3. lim G / G n \underleftarrow{\lim}G/G_{n}
  4. G / G n G/G_{n}
  5. G G ^ G\to\widehat{G}
  6. G n G_{n}
  7. gr ( G ) gr ( G ^ ) \operatorname{gr}(G)\to\operatorname{gr}(\widehat{G})
  8. gr ( G ) \operatorname{gr}(G)

HBJ_model.html

  1. p p
  2. n n
  3. m m
  4. τ \tau
  5. σ \sigma
  6. m m
  7. T c o m p T_{comp}
  8. T c o m m T_{comm}
  9. q q
  10. q q
  11. ( τ + σ m ) (\tau+\sigma m)
  12. m m
  13. O ( τ + σ m ) O(\tau+\sigma m)
  14. O ( l o g ( p ) ( τ + σ m ) ) O(log(p)(\tau+\sigma m))
  15. O ( l o g ( p ) ( τ + σ m ) ) O(log(p)(\tau+\sigma m))
  16. O ( l o g ( p ) n p ( τ + σ m ) ) O(log(p){n\over p}(\tau+\sigma m))
  17. O ( p ( τ + σ m ) ) ) O(p(\tau+\sigma m)))

HD_108063.html

  1. ± \pm

HD_114613.html

  1. M M\!\,
  2. m m\!\,

Heat_transfer_physics.html

  1. 𝐪 = - ρ c p T t + i , j s ˙ i - j , \nabla\cdot\mathbf{q}=-\rho c_{p}\frac{\partial T}{\partial t}+\sum_{i,j}\dot{% s}_{i-j},
  2. s ˙ \textstyle\dot{s}
  3. 2 π 0 0 π 2\pi\textstyle\int_{0}^{\infty}\int_{0}^{\pi}
  4. s ˙ i - j \dot{s}_{i\mbox{-}~{}j}
  5. s ˙ i - j \dot{s}_{i\mbox{-}~{}j}
  6. φ = φ o + i α φ d i α | o d i α + 1 2 i , j α , β 2 φ d i α d j β | o d i α d j β + 1 6 i , j , k α , β , γ 3 φ d i α d j β d k γ | o d i α d j β d k γ + + \qquad\qquad\langle\varphi\rangle=\langle\varphi\rangle_{\mathrm{o}}+\sum_{i}% \sum_{\alpha}\frac{\partial\langle\varphi\rangle}{\partial d_{i\alpha}}|_{% \mathrm{o}}d_{i\alpha}+\frac{1}{2}\sum_{i,j}\sum_{\alpha,\beta}\frac{\partial^% {2}\langle\varphi\rangle}{\partial d_{i\alpha}\partial d_{j\beta}}|_{\mathrm{o% }}d_{i\alpha}d_{j\beta}+\frac{1}{6}\sum_{i,j,k}\sum_{\alpha,\beta,\gamma}\frac% {\partial^{3}\langle\varphi\rangle}{\partial d_{i\alpha}\partial d_{j\beta}% \partial d_{k\gamma}}|_{\mathrm{o}}d_{i\alpha}d_{j\beta}d_{k\gamma}+...+
  7. φ o + 1 2 i , j α , β Γ α β d i α d j β , \qquad\qquad\ \ \ \ \ \ \ \ \ \approx\langle\varphi\rangle_{\mathrm{o}}+\frac{% 1}{2}\sum_{i,j}\sum_{\alpha,\beta}\Gamma_{\alpha\beta}d_{i\alpha}d_{j\beta},
  8. m j d 2 𝐝 ( j l , t ) d t 2 = - j l s y m b o l Γ ( j j l l ) 𝐝 ( j l , T ) , \qquad\qquad m_{j}\frac{d^{2}\mathbf{d}(jl,t)}{dt^{2}}=-\sum_{j^{\prime}l^{% \prime}}symbol{\Gamma}{\left({{j\ j^{\prime}}\atop{l\ l^{\prime}}}\right)}% \cdot\mathbf{d}(j^{\prime}l^{\prime},T),
  9. 𝐌 ω p 2 ( s y m b o l κ p , α ) 𝐬 α ( s y m b o l κ p ) = 𝐃 ( s y m b o l κ p ) 𝐬 α ( s y m b o l κ p ) , \qquad\qquad\mathbf{M}\omega_{p}^{2}(symbol{\kappa}_{p},\alpha)\mathbf{s}_{% \alpha}(symbol{\kappa}_{p})=\mathbf{D}(symbol{\kappa}_{p})\mathbf{s}_{\alpha}(% symbol{\kappa}_{p}),
  10. H p = x 1 2 m 𝐩 2 ( 𝐱 ) + 1 2 𝐱 , 𝐱 𝐝 i ( 𝐱 ) D i j ( 𝐱 - 𝐱 ) 𝐝 j ( 𝐱 ) , \qquad\qquad\mathrm{H}_{p}=\sum_{x}\frac{1}{2m}\mathbf{p}^{2}(\mathbf{x})+% \frac{1}{2}\sum_{\mathbf{x},\mathbf{x}^{\prime}}\mathbf{d}_{i}(\mathbf{x})D_{% ij}(\mathbf{x}-\mathbf{x}^{\prime})\mathbf{d}_{j}(\mathbf{x}^{\prime}),
  11. b κ , α = 1 N 1 / 2 κ p , α e - i ( s y m b o l κ p 𝐱 ) 𝐬 α ( s y m b o l κ p ) [ ( m ω p , α 2 ) 1 / 2 𝐝 ( 𝐱 ) + i ( 1 2 m ω p , α ) 1 / 2 𝐩 ( 𝐱 ) ] , \qquad\qquad b_{\kappa,\alpha}=\frac{1}{N^{1/2}}\sum_{\kappa_{p},\alpha}e^{-i(% symbol{\kappa}_{p}\cdot\mathbf{x})}\mathbf{s}_{\alpha}(symbol{\kappa}_{p})% \cdot[(\frac{m\omega_{p,\alpha}}{2\hbar})^{1/2}\mathbf{d}(\mathbf{x})+i(\frac{% 1}{2\hbar m\omega_{p,\alpha}})^{1/2}\mathbf{p}(\mathbf{x})],
  12. b κ , α = 1 N 1 / 2 κ p , α e i ( s y m b o l κ p 𝐱 ) 𝐬 α ( s y m b o l κ p ) [ ( m ω p , α 2 ) 1 / 2 𝐝 ( 𝐱 ) - i ( 1 2 m ω p , α ) 1 / 2 𝐩 ( 𝐱 ) ] . \qquad\qquad b_{\kappa,\alpha}^{\dagger}=\frac{1}{N^{1/2}}\sum_{\kappa_{p},% \alpha}e^{i(symbol{\kappa}_{p}\cdot\mathbf{x})}\mathbf{s}_{\alpha}(symbol{% \kappa}_{p})\cdot[(\frac{m\omega_{p,\alpha}}{2\hbar})^{1/2}\mathbf{d}(\mathbf{% x})-i(\frac{1}{2\hbar m\omega_{p,\alpha}})^{1/2}\mathbf{p}(\mathbf{x})].
  13. c v , p = d E p d T | v = 9 k B m ( T T D ) 3 n 0 T D / T x 4 e x ( e x - 1 ) 2 d x ( x = ω k B T ) , \qquad\qquad c_{v,p}=\frac{dE_{p}}{dT}|_{v}=\frac{9k_{\mathrm{B}}}{m}(\frac{T}% {T_{D}})^{3}n\int_{0}^{T_{D}/T}\frac{x^{4}e^{x}}{(e^{x}-1)^{2}}dx\qquad(x=% \frac{\hbar\omega}{k_{\mathrm{B}}T}),
  14. k i = 1 3 n i c v , i u i λ i , \qquad\qquad k_{i}=\frac{1}{3}n_{i}c_{v,i}u_{i}\lambda_{i},
  15. k p , 𝐬 = 1 8 π 3 α c v , p τ p ( 𝐮 p , g 𝐬 ) 2 d κ for component along 𝐬 , \qquad\qquad k_{p,\mathbf{s}}=\frac{1}{8\pi^{3}}\sum_{\alpha}\int c_{v,p}\tau_% {p}(\mathbf{u}_{p,g}\cdot\mathbf{s})^{2}d\kappa\ \ \ \ \ \mathrm{for\ % component\ along\ }\mathbf{s},
  16. k p = 1 6 π 3 α c v , p τ p u p , g 2 κ 2 d κ for isotropic conductivity . \qquad\qquad k_{p}=\frac{1}{6\pi^{3}}\sum_{\alpha}\int c_{v,p}\tau_{p}{u}_{p,g% }^{2}\kappa^{2}d\kappa\ \ \ \ \ \ \ \ \mathrm{for\ isotropic\ conductivity}.
  17. k p = ( 48 π 2 ) 1 / 3 k B 3 T 3 a h P 2 T D 0 T / T D τ p x 4 e x ( e x - 1 ) 2 d x , \qquad\qquad k_{p}=(48\pi^{2})^{1/3}\frac{k_{\mathrm{B}}^{3}T^{3}}{ah_{\mathrm% {P}}^{2}T_{\mathrm{D}}}\int_{0}^{T/T_{\mathrm{D}}}\tau_{p}\frac{x^{4}e^{x}}{(e% ^{x}-1)^{2}}dx,
  18. k p = k p , S = 3.1 × 10 12 M V a 1 / 3 T D , 3 T γ G 2 N o 2 / 3 high temperatures ( T > 0.2 T D , phonon - phonon scattering only ) , \qquad\qquad k_{p}=k_{p,S}=\frac{3.1\times 10^{12}\langle M\rangle V_{a}^{1/3}% T_{D,\infty}^{3}}{T\langle\gamma_{G}^{2}\rangle N_{o}^{2/3}}\qquad\mathrm{\ % high\ temperatures}\ (T>0.2T_{D},\mathrm{\ phonon\mbox{-}~{}phonon\ scattering% \ only)},
  19. H e = - 2 2 m e 2 + φ c ( 𝐱 ) , \qquad\qquad\mathrm{H}_{e}=-\frac{\hbar^{2}}{2m_{e}}\nabla^{2}+\varphi_{c}(% \mathbf{x}),
  20. H e ψ e , 𝐱 ( 𝐱 ) = E e ( s y m b o l κ e ) ψ e , 𝐱 ( 𝐱 ) , \qquad\qquad\mathrm{H}_{e}\psi_{e,\mathbf{x}}(\mathbf{x})=E_{e}(symbol{\kappa}% _{e})\psi_{e,\mathbf{x}}(\mathbf{x}),
  21. 𝐣 e = 𝐀 e e E F e c + 𝐀 e t 1 T , and \qquad\qquad\mathbf{j}_{e}=\mathbf{A}_{ee}\cdot\nabla\frac{E_{\mathrm{F}}}{e_{% c}}+\mathbf{A}_{et}\cdot\nabla\frac{1}{T},\ \ \mathrm{and}
  22. 𝐪 = 𝐀 t e E F e c + 𝐀 t t 1 T . \qquad\qquad\mathbf{q}=\mathbf{A}_{te}\cdot\nabla\frac{E_{\mathrm{F}}}{e_{c}}+% \mathbf{A}_{tt}\cdot\nabla\frac{1}{T}.
  23. 𝐣 e = α e e 𝐞 e - α e t T 2 T ( 𝐞 e = α e e - 1 𝐣 e + α e e - 1 α e t T 2 T ) , \qquad\qquad\mathbf{j}_{e}=\alpha_{ee}\mathbf{e}_{e}-\frac{\alpha_{et}}{T^{2}}% \nabla T\qquad(\mathbf{e}_{e}=\alpha_{ee}^{-1}\mathbf{j}_{e}+\frac{\alpha_{ee}% ^{-1}\alpha_{et}}{T^{2}}\nabla T),
  24. 𝐪 = α t e α e e - 1 𝐣 e - α t t - α t e α e e - 1 α e t T 2 T . \qquad\qquad\mathbf{q}=\alpha_{te}\alpha_{ee}^{-1}\mathbf{j}_{e}-\frac{\alpha_% {tt}-\alpha_{te}\alpha_{ee}^{-1}\alpha_{et}}{T^{2}}\nabla T.
  25. σ e = 1 ρ e = α e e , k e = α t t - α t e α e e - 1 α e t T 2 , and α S = α e t α e e - 1 T 2 ( α S = α P T ) . \qquad\qquad\sigma_{e}=\frac{1}{\rho_{e}}=\alpha_{ee},\ \ k_{e}=\frac{\alpha_{% tt}-\alpha_{te}\alpha_{ee}^{-1}\alpha_{et}}{T^{2}},\mathrm{and}\ \alpha_{% \mathrm{S}}=\frac{\alpha_{et}\alpha_{ee}^{-1}}{T^{2}}\ \ (\alpha_{\mathrm{S}}=% \alpha_{\mathrm{P}}T).
  26. α S , mix = 1 q S mix N = k B q ln ( 1 - f e o f e o ) , \qquad\qquad\alpha_{\mathrm{S,mix}}=\frac{1}{q}\frac{\partial S_{\mathrm{mix}}% }{\partial N}=\frac{k_{\mathrm{B}}}{q}\mathrm{ln}(\frac{1-f_{e}^{\mathrm{o}}}{% f_{e}^{\mathrm{o}}}),
  27. S vib = - F mix T = 3 N k B T 0 ω { ω 2 k B T coth ( ω 2 k B T ) - ln [ 2 s i n h ( ω 2 k B T ) ] } D p ( ω ) d ω , \qquad\qquad S_{\mathrm{vib}}=-\frac{\partial F_{\mathrm{mix}}}{\partial T}=3% Nk_{\mathrm{B}}T\int_{0}^{\omega}\{\frac{\hbar\omega}{2k_{\mathrm{B}}T}\mathrm% {coth}(\frac{\hbar\omega}{2k_{\mathrm{B}}T})-\mathrm{ln}[2\mathrm{sinh}(\frac{% \hbar\omega}{2k_{\mathrm{B}}T})]\}D_{p}(\omega)d\omega,
  28. 𝐣 e = - e c 3 p 𝐮 e f e = - e c 3 k B T p 𝐮 e τ e ( - f e o E e ) ( 𝐮 e 𝐅 t e ) , \qquad\qquad\mathbf{j}_{e}=-\frac{e_{c}}{\hbar^{3}}\sum_{p}\mathbf{u}_{e}f_{e}% ^{\prime}=-\frac{e_{c}}{\hbar^{3}k_{\mathrm{B}}T}\sum_{p}\mathbf{u}_{e}\tau_{e% }(-\frac{\partial f_{e}^{\mathrm{o}}}{\partial E_{e}})(\mathbf{u}_{e}\cdot% \mathbf{F}_{te}),
  29. 𝐪 = 1 3 p ( E e - E F ) 𝐮 e f e = 1 3 k B T p 𝐮 e τ e ( - f e o E e ) ( E e - E F ) ( 𝐮 e 𝐅 t e ) , \qquad\qquad\mathbf{q}=\frac{1}{\hbar^{3}}\sum_{p}(E_{e}-E_{\mathrm{F}})% \mathbf{u}_{e}f_{e}^{\prime}=\frac{1}{\hbar^{3}k_{\mathrm{B}}T}\sum_{p}\mathbf% {u}_{e}\tau_{e}(-\frac{\partial f_{e}^{\mathrm{o}}}{\partial E_{e}})(E_{e}-E_{% \mathrm{F}})(\mathbf{u}_{e}\cdot\mathbf{F}_{te}),
  30. γ ˙ e \dot{\gamma}_{e}
  31. τ e = 1 / γ ˙ e \tau_{e}=1/\dot{\gamma}_{e}
  32. translational E f , t , n = π 2 2 2 m ( n x 2 L 2 + n y 2 L 2 + n z 2 L 2 ) and Z f , t i = 0 g f , t , i exp ( - E f , t , i k B T ) = V ( m k B T 2 π 2 ) 3 / 2 , \qquad\qquad\mathrm{translational}\ \ \ \ \ \ \ \ \ \ \ \ E_{f,t,n}=\frac{\pi^% {2}\hbar^{2}}{2m}(\frac{n_{x}^{2}}{L^{2}}+\frac{n_{y}^{2}}{L^{2}}+\frac{n_{z}^% {2}}{L^{2}})\ \ \ \mathrm{and}\ \ \ Z_{f,t}\sum_{i=0}^{\infty}g_{f,t,i}\mathrm% {exp}(-\frac{E_{f,t,i}}{k_{\mathrm{B}}T})=V(\frac{mk_{\mathrm{B}}T}{2\pi\hbar^% {2}})^{3/2},
  33. vibrational E f , v , l = ω f , v ( 1 + 1 2 ) and Z f , v j = 0 exp [ - ( l + 1 2 ) ω f , v k B T ] = exp ( - T f , v / 2 T ) 1 - exp ( - T f , v / T ) , \qquad\qquad\mathrm{vibrational}\ \ \ \ \ \ \ \ \ \ \ \ \ \ E_{f,v,l}=\hbar% \omega_{f,v}(1+\frac{1}{2})\ \ \mathrm{and}\ \ Z_{f,v}\sum_{j=0}^{\infty}% \mathrm{exp}[-(l+\frac{1}{2})\frac{\hbar\omega_{f,v}}{k_{\mathrm{B}}T}]=\frac{% \mathrm{exp}(-T_{f,v}/2T)}{1-\mathrm{exp}(-T_{f,v}/T)},
  34. rotational E f , r , j = 2 2 I f and Z f , r j = 0 ( 2 j + 1 ) exp [ - - 2 j ( j + 1 ) 2 I f k B T ] T T f , r ( 1 + T f , r 3 T + T f , r 2 15 T 2 + ) , \qquad\qquad\mathrm{rotational}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ E_{f,r,j}=\frac% {\hbar^{2}}{2I_{f}}\ \ \mathrm{and}\ \ Z_{f,r}\sum_{j=0}^{\infty}(2j+1)\mathrm% {exp}[-\frac{-\hbar^{2}j(j+1)}{2I_{f}k_{\mathrm{B}}T}]\approx\frac{T}{T_{f,r}}% (1+\frac{T_{f,r}}{3T}+\frac{T_{f,r}^{2}}{15T^{2}}+...),
  35. total E f = i E f , i = E f , t + E f , v + E f , r + and Z f = i Z f , i = Z f , t Z f , v Z f , r . \qquad\qquad\mathrm{total}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ % E_{f}=\sum_{i}E_{f,i}=E_{f,t}+E_{f,v}+E_{f,r}+...\ \ \mathrm{and}\ \ Z_{f}=% \prod_{i}Z_{f,i}=Z_{f,t}Z_{f,v}Z_{f,r}....
  36. e f = ( k B T 2 / m ) ( ln Z f / T ) | N , V . e_{f}=(k_{\mathrm{B}}T^{2}/m)(\partial\mathrm{ln}Z_{f}/\partial T)|_{N,V}.
  37. monatomic ideal gas c v , f = e f T | V = 3 R g 2 M , \qquad\qquad\mathrm{monatomic\ ideal\ gas}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ % \ \ \ c_{v,f}=\frac{\partial e_{f}}{\partial T}|_{V}=\frac{3R_{g}}{2M},
  38. diatomic ideal gas c v , f = R g M { 3 2 + ( T f , v T ) 2 exp ( T f , v , i / T ) [ exp ( T f , v , i / T ) - 1 ] 2 + 1 + 2 15 ( T f , v T ) 2 } , \qquad\qquad\mathrm{diatomic\ ideal\ gas}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ % \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ c_{v,f}=\frac{R_{g}}{M}\{\frac{3}{2}+(\frac{% T_{f,v}}{T})^{2}\frac{\mathrm{exp}(T_{f,v,i}/T)}{[\mathrm{exp}(T_{f,v,i}/T)-1]% ^{2}}+1+\frac{2}{15}(\frac{T_{f,v}}{T})^{2}\},
  39. nonlinear , polyatomic ideal gas c v , f = R g M { 3 + j = 1 3 N o - 6 ( T f , v T ) 2 exp ( T f , v , i / T ) [ exp ( T f , v , i / T ) - 1 ] 2 } . \qquad\qquad\mathrm{nonlinear,\ polyatomic\ ideal\ gas}\ \ \ \ \ \ \ \ \ c_{v,% f}=\frac{R_{g}}{M}\{3+\textstyle\sum_{j=1}^{3N_{o}-6}\displaystyle(\frac{T_{f,% v}}{T})^{2}\frac{\mathrm{exp}(T_{f,v,i}/T)}{[\mathrm{exp}(T_{f,v,i}/T)-1]^{2}}\}.
  40. k f = 1 3 n f c p , f u f 2 τ f - f , \qquad\qquad\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ k_{f}=\frac{1}{3}n_{f}c_% {p,f}\langle u_{f}^{2}\rangle\tau_{f\mbox{-}~{}f},
  41. H p h = 1 2 ( ϵ o 𝐞 e 2 + μ o - 1 𝐛 e 2 ) d V = α ω p h , α ( c α c α + 1 2 ) , \qquad\qquad\mathrm{H}_{ph}=\frac{1}{2}\int(\epsilon_{\mathrm{o}}\mathbf{e}_{e% }^{2}+\mu_{\mathrm{o}}^{-1}\mathbf{b}_{e}^{2})dV=\sum_{\alpha}\hbar\omega_{ph,% \alpha}(c_{\alpha}^{\dagger}c_{\alpha}+\frac{1}{2}),
  42. 𝐚 e ( 𝐱 , t ) = α ( 2 ϵ o ω p h , α V ) 1 / 2 𝐬 p h , α ( c α e i s y m b o l κ α 𝐱 + c α e - i s y m b o l κ α 𝐱 ) , \qquad\qquad\mathbf{a}_{e}(\mathbf{x},t)=\sum_{\alpha}(\frac{\hbar}{2\epsilon_% {\mathrm{o}}\omega_{ph,\alpha}V})^{1/2}\mathbf{s}_{ph,\alpha}(c_{\alpha}e^{% isymbol{\kappa}_{\alpha}\cdot\mathbf{x}}+c_{\alpha}^{\dagger}e^{-isymbol{% \kappa}_{\alpha}\cdot\mathbf{x}}),
  43. d I b , ω = D p h , b , ω f p h u p h d ω p h 4 π \qquad\qquad dI_{b,\omega}=\frac{D_{ph,b,\omega}f_{ph}u_{ph}d\omega_{ph}}{4\pi}
  44. = ω p h 3 4 π 3 u p h 2 =\frac{\hbar\omega_{ph}^{3}}{4\pi^{3}u_{ph}^{2}}
  45. 1 e ω p h / k B T - 1 d ω p h or d I b , λ = 4 π u p h 2 d λ p h λ p h 5 ( e 2 π u p h / λ p h k B T - 1 ) \frac{1}{e^{\hbar\omega_{ph}/k_{\mathrm{B}}T}-1}d\omega_{ph}\ \mathrm{or}\ dI_% {b,\lambda}=\frac{4\pi\hbar u_{ph}^{2}d\lambda_{ph}}{\lambda_{ph}^{5}(e^{2\pi% \hbar u_{ph}/\lambda_{ph}k_{\mathrm{B}}T}-1)}\
  46. , ,
  47. E b = 0 d E b , λ = σ SB T 4 , where σ SB = π 2 k B 4 60 3 u p h 2 \qquad\qquad E_{b}=\int_{0}^{\infty}dE_{b,\lambda}=\sigma_{\mathrm{SB}}T^{4}\ % \mathrm{,\ where}\ \sigma_{\mathrm{SB}}=\frac{\pi^{2}k_{\mathrm{B}}^{4}}{60% \hbar^{3}u_{ph}^{2}}\ \ \
  48. . .
  49. s ˙ f , p h - e \textstyle\dot{s}_{f,ph-e}
  50. s ˙ f , p h , i \textstyle\dot{s}_{f,ph,i}
  51. f p h t + u p h 𝐬 f p h = f p h t | s + u p h σ p h , ω [ f p h ( ω p h , T ) - f p h ( 𝐬 ) ] + s ˙ f , p h , i . \qquad\qquad\frac{\partial f_{ph}}{\partial t}+u_{ph}\mathbf{s}\cdot\nabla f_{% ph}=\frac{\partial f_{ph}}{\partial t}|_{s}+u_{ph}\sigma_{ph,\omega}[f_{ph}(% \omega_{ph},T)-f_{ph}(\mathbf{s})]+\dot{s}_{f,ph,i}.
  52. I p h , ω ( ω p h , 𝐬 ) u p h t + 𝐬 I p h , ω ( ω p h , 𝐬 ) = I p h , ω ( ω p h , 𝐬 ) u p h t | s + \qquad\qquad\frac{\partial I_{ph,\omega}(\omega_{ph},\mathbf{s})}{u_{ph}% \partial t}+\mathbf{s}\cdot\nabla I_{ph,\omega}(\omega_{ph},\mathbf{s})=\frac{% \partial I_{ph,\omega}(\omega_{ph},\mathbf{s})}{u_{ph}\partial t}|_{s}+
  53. σ p h , ω [ I p h , ω ( ω p h , T ) - I p h ( ω p h , 𝐬 ) ] + s ˙ p h , i . \sigma_{ph,\omega}[I_{ph,\omega}(\omega_{ph},T)-I_{ph}(\omega_{ph},\mathbf{s})% ]+\dot{s}_{ph,i}.
  54. 𝐪 r = 𝐪 p h = 0 4 π 𝐬 I p h , ω d Ω d ω . \textstyle\mathbf{q}_{r}=\mathbf{q}_{ph}=\int_{0}^{\infty}\int_{4\pi}\mathbf{s% }I_{ph,\omega}d\Omega d\omega.
  55. σ p h , ω = ω γ ˙ p h , a n e u p h , \sigma_{ph,\omega}=\frac{\hbar\omega\dot{\gamma}_{ph,a}n_{e}}{u_{ph}},
  56. γ ˙ p h , a \dot{\gamma}_{ph,a}
  57. H p h - e = - e c m e ( a + a ) 𝐚 e 𝐩 e = - ( ω p h , α 2 ϵ o V ) 1 / 2 ( 𝐬 p h , α e c 𝐱 e ) ( a + a ) ( c e i κ x + c e - i κ x ) , \qquad\qquad\mathrm{H}_{ph-e}=-\frac{e_{c}}{m_{e}}(a+a^{\dagger})\mathbf{a}_{e% }\cdot\mathbf{p}_{e}=-(\frac{\hbar\omega_{ph,\alpha}}{2\epsilon_{o}V})^{1/2}(% \mathbf{s}_{ph,\alpha}\cdot e_{c}\mathbf{x}_{e})(a+a^{\dagger})(ce^{i\mathrm{% \kappa}\cdot\mathrm{x}}+c^{\dagger}e^{-i\mathrm{\kappa}\cdot\mathrm{x}}),\ \ \
  58. ϵ e , c , ω = 4 π 2 ω 2 V i \isin VB , j \isin CB κ w κ | p i j | 2 δ ( E κ , j - E κ , i - ω ) , \qquad\qquad\ \ \ \ \ \ \ \ \ \ \ \epsilon_{e,c,\omega}=\frac{4\pi^{2}}{\omega% ^{2}V}\sum_{i\isin\mathrm{VB},j\isin\mathrm{CB}}\sum_{\kappa}w_{\kappa}|p_{ij}% |^{2}\delta(E_{\kappa,j}-E_{\kappa,i}-\hbar\omega),\ \ \ \qquad\
  59. ϵ e , r , ω = 1 + 4 π 0 d ω ω ϵ e , c , ω ω 2 - ω 2 . \qquad\qquad\ \ \ \ \ \ \ \ \ \ \ \epsilon_{e,r,\omega}=1+\frac{4}{\pi}\mathbb% {P}\int_{0}^{\infty}\mathrm{d}\omega^{\prime}\frac{\omega^{\prime}\epsilon_{e,% c,\omega^{\prime}}}{\omega^{\prime 2}-\omega^{2}}.
  60. \mathbb{P}
  61. ϵ e , ω ϵ e , = 1 + j ω LO , j 2 - ω TO , j 2 ω TO , j 2 - ω 2 - i γ ω , \frac{\epsilon_{e,\omega}}{\epsilon_{e,\infty}}=1+\sum_{j}\frac{\omega_{% \mathrm{LO},j}^{2}-\omega_{\mathrm{TO},j}^{2}}{\omega_{\mathrm{TO},j}^{2}-% \omega^{2}-i\gamma\omega},
  62. n e , A π 2 u p h 2 γ ˙ p h , e , s p ω e , g 2 ω d ω = n e , A π ω e , g | s y m b o l μ e | 2 3 ϵ o u p h ω d ω \frac{n_{e,A}\pi^{2}u_{ph}^{2}\dot{\gamma}_{ph,e,sp}}{\omega_{e,g}^{2}\int_{% \omega}\mathrm{d}\omega}=\frac{n_{e,A}\pi\omega_{e,g}|symbol{\mu}_{e}|^{2}}{3% \epsilon_{\mathrm{o}}\hbar u_{ph}\int_{\omega}\mathrm{d}\omega}
  63. γ ˙ p h , e , s p \dot{\gamma}_{ph,e,sp}
  64. ω d ω \textstyle\int_{\omega}\mathrm{d}\omega
  65. 2 ω κ ω u p h = ( 2 n e , c e c 2 τ e ω ϵ o ϵ e m e u p h 2 ) 1 / 2 \frac{2\omega\kappa_{\omega}}{u_{ph}}=(\frac{2n_{e,c}e_{c}^{2}\langle\langle% \tau_{e}\rangle\rangle\omega}{\epsilon_{\mathrm{o}}\epsilon_{e}m_{e}u_{ph}^{2}% })^{1/2}
  66. τ e \langle\langle\tau_{e}\rangle\rangle
  67. e c 2 | φ v | e c 𝐱 | φ c | 2 D p h - e [ f e o ( E e , v ) - f e o ( E e , c ) ] ϵ o 2 m e , e 2 u p h n ω ω \frac{e_{c}^{2}|\langle\varphi_{v}|e_{c}\mathbf{x}|\varphi_{c}\rangle|^{2}D_{% ph-e}[f_{e}^{\mathrm{o}}(E_{e,v})-f_{e}^{\mathrm{o}}(E_{e,c})]}{\epsilon_{% \mathrm{o}}^{2}m_{e,e}^{2}u_{ph}n_{\omega}\omega}
  68. a p h - e - p , a ( ω - Δ E e , g + ω p ) 2 exp ( ω p / k B T ) - 1 \frac{a_{ph\mathrm{-}e\mathrm{-}p,a}(\hbar\omega-\Delta E_{e,g}+\hbar\omega_{p% })^{2}}{\mathrm{exp}(\hbar\omega_{p}/k_{\mathrm{B}}T)-1}
  69. a p h - e - p , e ( ω - Δ E e , g - ω p ) 2 1 - exp ( - ω p / k B T ) \frac{a_{ph-e-p,e}(\hbar\omega-\Delta E_{e,g}-\hbar\omega_{p})^{2}}{1-\mathrm{% exp}(-\hbar\omega_{p}/k_{\mathrm{B}}T)}

Heavy_traffic_approximation.html

  1. X ^ n ( t ) = X ( n t ) - 𝔼 ( X ( n t ) ) n \hat{X}_{n}(t)=\frac{X(nt)-\mathbb{E}(X(nt))}{\sqrt{n}}
  2. β = ( 1 - ρ ) s \beta=(1-\rho)\sqrt{s}
  3. j j
  4. j j
  5. T j T_{j}
  6. S j S_{j}
  7. ρ j = λ j μ j \rho_{j}=\frac{\lambda_{j}}{\mu_{j}}
  8. 1 λ j = E ( T j ) \frac{1}{\lambda_{j}}=E(T_{j})
  9. 1 μ j = E ( S j ) \frac{1}{\mu_{j}}=E(S_{j})
  10. W q , j W_{q,j}
  11. α j = - E [ S j - T j ] \alpha_{j}=-E[S_{j}-T_{j}]
  12. β j 2 = var [ S j - T j ] ; \beta_{j}^{2}=\operatorname{var}[S_{j}-T_{j}];
  13. T j 𝑑 T T_{j}\xrightarrow{d}T
  14. S j 𝑑 S S_{j}\xrightarrow{d}S
  15. ρ j 1 \rho_{j}\rightarrow 1
  16. 2 α j β j 2 W q , j 𝑑 exp ( 1 ) \frac{2\alpha_{j}}{\beta_{j}^{2}}W_{q,j}\xrightarrow{d}\exp(1)
  17. Var [ S - T ] > 0 \operatorname{Var}[S-T]>0
  18. δ > 0 \delta>0
  19. E [ S j 2 + δ ] E[S_{j}^{2+\delta}]
  20. E [ T j 2 + δ ] E[T_{j}^{2+\delta}]
  21. C C
  22. j j
  23. U ( n ) = S ( n ) - T ( n ) U^{(n)}=S^{(n)}-T^{(n)}
  24. W q ( n ) W_{q}^{(n)}
  25. W q ( n ) = max ( W q ( n - 1 ) + U ( n - 1 ) , 0 ) W_{q}^{(n)}=\max(W_{q}^{(n-1)}+U^{(n-1)},0)
  26. W q ( n ) = max ( U ( 1 ) + + U ( n - 1 ) , U ( 2 ) + + U ( n - 1 ) , , U ( n - 1 ) , 0 ) W_{q}^{(n)}=\max(U^{(1)}+\cdots+U^{(n-1)},U^{(2)}+\cdots+U^{(n-1)},\ldots,U^{(% n-1)},0)
  27. P ( k ) = i = 1 k U ( n - i ) P^{(k)}=\sum_{i=1}^{k}U^{(n-i)}
  28. U ( i ) U^{(i)}
  29. α = - E [ U ( i ) ] \alpha=-E[U^{(i)}]
  30. β 2 = var [ U ( i ) ] \beta^{2}=\operatorname{var}[U^{(i)}]
  31. E [ P ( k ) ] = - k α E[P^{(k)}]=-k\alpha
  32. var [ P ( k ) ] = k β 2 \operatorname{var}[P^{(k)}]=k\beta^{2}
  33. W q ( n ) = max n - 1 k 0 P ( k ) ; W_{q}^{(n)}=\max_{n-1\geq k\geq 0}P^{(k)};
  34. W q ( ) = sup k 0 P ( k ) W_{q}^{(\infty)}=\sup_{k\geq 0}P^{(k)}
  35. n n
  36. W q ( n ) W_{q}^{(n)}
  37. P ( 0 ) = 0 P^{(0)}=0
  38. P ( k ) P^{(k)}
  39. ρ \rho
  40. \infty
  41. P ( t ) 𝒩 ( - α t , β 2 t ) P^{(t)}\ \sim\ \mathcal{N}(-\alpha t,\beta^{2}t)
  42. X X
  43. μ \mu
  44. σ \sigma
  45. M t sup 0 s t X ( s ) M_{t}\sup_{0\leq s\leq t}X(s)
  46. μ 0 \mu\leq 0
  47. lim t P ( M t > x ) = exp ( 2 μ x / σ 2 ) , x 0 ; \lim_{t\rightarrow\infty}P(M_{t}>x)=\exp(2\mu x/\sigma^{2}),x\geq 0;
  48. lim t P ( M t x ) = 1 , x 0. \lim_{t\rightarrow\infty}P(M_{t}\geq x)=1,x\geq 0.
  49. W q ( ) exp ( 2 α β 2 ) W_{q}^{(\infty)}\thicksim\exp\left(\frac{2\alpha}{\beta^{2}}\right)
  50. λ \lambda
  51. E [ S ] = 1 μ E[S]=\frac{1}{\mu}
  52. var [ S ] = σ B 2 \operatorname{var}[S]=\sigma_{B}^{2}
  53. W q = ρ 2 + λ 2 σ B 2 2 λ ( 1 - ρ ) W_{q}=\frac{\rho^{2}+\lambda^{2}\sigma_{B}^{2}}{2\lambda(1-\rho)}
  54. W q ( H ) = λ ( 1 λ 2 + σ B 2 ) 2 ( 1 - ρ ) . W_{q}^{(H)}=\frac{\lambda(\frac{1}{\lambda^{2}}+\sigma_{B}^{2})}{2(1-\rho)}.
  55. W q ( H ) - W q W q = 1 - ρ 2 ρ 2 + λ 2 σ B 2 \frac{W_{q}^{(H)}-W_{q}}{W_{q}}=\frac{1-\rho^{2}}{\rho^{2}+\lambda^{2}\sigma_{% B}^{2}}
  56. ρ 1 \rho\rightarrow 1
  57. W q ( H ) - W q W q 0. \frac{W_{q}^{(H)}-W_{q}}{W_{q}}\rightarrow 0.

Heegner's_lemma.html

  1. y 2 = a 4 x 4 + a 3 x 3 + a 2 x 2 + a 1 x + a 0 y^{2}=a_{4}x^{4}+a_{3}x^{3}+a_{2}x^{2}+a_{1}x+a_{0}\,

Height_zeta_function.html

  1. N ( S , H , B ) = { x S : H ( x ) B } . N(S,H,B)=\sharp\{x\in S:H(x)\leq B\}.
  2. Z ( S , H ; s ) = x S H ( x ) - s . Z(S,H;s)=\sum_{x\in S}H(x)^{-s}.
  3. N c B a ( log B ) t - 1 N\sim cB^{a}(\log B)^{t-1}

Heinrich_August_Rothe.html

  1. T ( n ) = T ( n - 1 ) + ( n - 1 ) T ( n - 2 ) T(n)=T(n-1)+(n-1)T(n-2)

Henneberg_surface.html

  1. x ( u , v ) = 2 cos ( v ) sinh ( u ) - ( 2 / 3 ) cos ( 3 v ) sinh ( 3 u ) y ( u , v ) = 2 sin ( v ) sinh ( u ) + ( 2 / 3 ) sin ( 3 v ) sinh ( 3 u ) z ( u , v ) = 2 cos ( 2 v ) cosh ( 2 u ) \begin{aligned}\displaystyle x(u,v)&\displaystyle=2\cos(v)\sinh(u)-(2/3)\cos(3% v)\sinh(3u)\\ \displaystyle y(u,v)&\displaystyle=2\sin(v)\sinh(u)+(2/3)\sin(3v)\sinh(3u)\\ \displaystyle z(u,v)&\displaystyle=2\cos(2v)\cosh(2u)\end{aligned}

Henry_adsorption_constant.html

  1. X = K H P X=K_{H}P
  2. K H = lim ϱ 0 ϱ s ϱ ( z ) , K_{H}=\lim_{\varrho\rightarrow 0}\frac{\varrho_{s}}{\varrho(z)},
  3. ϱ ( z ) \varrho(z)
  4. ϱ s \varrho_{s}
  5. K H = - x [ exp ( - β u ) - exp ( - β u 0 ) ] d x - x [ 1 - exp ( - β u ) ] d x . K_{H}=\int\limits_{-\infty}^{x^{\prime}}\big[\exp(-\beta u)-\exp(-\beta u_{0})% \big]dx-\int\limits_{x^{\prime}}^{\infty}\big[1-\exp(-\beta u)\big]dx.
  6. x x^{\prime}
  7. u = u ( x ) u=u(x)
  8. u 0 u_{0}
  9. β = 1 / k B T \beta=1/k_{B}T
  10. k B k_{B}
  11. T T
  12. x 0 = - - 0 θ ~ ( x ) d x + 0 φ ~ ( x ) d x , x_{0}=-\int\limits_{-\infty}^{0}\widetilde{\theta}(x)dx+\int\limits_{0}^{% \infty}\widetilde{\varphi}(x)dx,
  13. θ ~ = exp ( - β u ) - exp ( - β u 0 ) 1 - exp ( - β u 0 ) \widetilde{\theta}=\frac{\exp{(-\beta u)}-\exp{(-\beta u_{0})}}{1-\exp{(-\beta u% _{0})}}
  14. φ ~ = 1 - exp ( - β u ) 1 - exp ( - β u 0 ) , \widetilde{\varphi}=\frac{1-\exp{(-\beta u)}}{1-\exp{(-\beta u_{0})}},
  15. K H ( x ) = [ x - x 0 ( T ) ] [ 1 - exp ( - β u 0 ) ] K_{H}(x^{\prime})=[x^{\prime}-x_{0}(T)][1-\exp(-\beta u_{0})]
  16. K H K_{H}
  17. x 0 x_{0}
  18. k H = lim ϱ 0 ϱ ( z ) ϱ ( z ) = exp ( - β u 0 ) , k_{H}=\lim_{\varrho\rightarrow 0}\frac{\varrho(z^{\prime})}{\varrho(z)}=\exp(-% \beta u_{0}),
  19. ϱ ( z ) \varrho(z^{\prime})
  20. K H = - x [ k H u ~ ( x ) - k H ] d x - x [ 1 - k H u ~ ( x ) ] d x K_{H}=\int\limits_{-\infty}^{x^{\prime}}\big[k_{H}^{\widetilde{u}(x)}-k_{H}% \big]dx-\int\limits_{x^{\prime}}^{\infty}\big[1-k_{H}^{\widetilde{u}(x)}\big]dx
  21. u ~ ( x ) = u ( x ) u 0 . \widetilde{u}(x)=\frac{u(x)}{u_{0}}.
  22. x = ± x=\pm\infty
  23. K H = - [ exp ( - β u ) - 1 ] d x . K_{H}=\int\limits_{-\infty}^{\infty}\big[\exp(-\beta u)-1\big]dx.
  24. K H K_{H}
  25. u u
  26. K H = - x exp ( - β u ) d x - x [ 1 - exp ( - β u ) ] d x , K_{H}=\int\limits_{-\infty}^{x^{\prime}}\exp(-\beta u)dx-\int\limits_{x^{% \prime}}^{\infty}\big[1-\exp(-\beta u)\big]dx,
  27. K H ( x ) = x - x 0 ( T ) , K_{H}(x^{\prime})=x^{\prime}-x_{0}(T),
  28. x 0 = - - 0 θ ( x ) d x + 0 φ ( x ) d x . x_{0}=-\int\limits_{-\infty}^{0}\theta(x)dx+\int\limits_{0}^{\infty}\varphi(x)dx.
  29. θ = exp ( - β u ) \theta=\exp{(-\beta u)}
  30. φ = 1 - exp ( - β u ) . \varphi=1-\exp{(-\beta u)}.
  31. x 0 = x s t e p , x_{0}=x_{step},
  32. x s t e p x_{step}
  33. K H ( x ) = x - x s t e p . K_{H}(x^{\prime})=x^{\prime}-x_{step}.
  34. u ( x ) u(x)
  35. x x^{\prime}
  36. x x^{\prime}
  37. x 0 x_{0}
  38. x 0 ± R x_{0}\pm R
  39. R R
  40. x x^{\prime}
  41. K H K_{H}
  42. x = x 0 x^{\prime}=x_{0}
  43. x x^{\prime}
  44. u 0 < 0 u_{0}<0
  45. x x^{\prime}
  46. K H K_{H}

Hermann_Kinkelin.html

  1. n F ( n x ) = f ( x ) + f ( x + 1 n ) + f ( x + 2 n ) + f ( x + n - 1 n ) . \scriptstyle nF(nx)=f(x)+f(x+\frac{1}{n})+f(x+\frac{2}{n})+\ldots f(x+\frac{n-% 1}{n}).

Hesperidin_6-O-alpha-L-rhamnosyl-beta-D-glucosidase.html

  1. \rightleftharpoons

Hesse_pencil.html

  1. λ ( x 3 + y 3 + z 3 ) + μ x y z = 0. \lambda(x^{3}+y^{3}+z^{3})+\mu xyz=0.
  2. λ , μ \lambda,\mu
  3. ( x , y , z ) (x,y,z)
  4. λ \lambda
  5. μ \mu
  6. λ \lambda
  7. μ \mu

Hessian_automatic_differentiation.html

  1. n n
  2. f : n f:\mathbb{R}^{n}\rightarrow\mathbb{R}
  3. x n x\in\mathbb{R}^{n}
  4. H ( x ) n × n H(x)\in\mathbb{R}^{n\times n}
  5. u n u\in\mathbb{R}^{n}
  6. H ( x ) u H(x)u
  7. H ( x ) e i H(x)e_{i}
  8. i = 1 , , n i=1,\ldots,n
  9. f ( x ) u T f ( x ) f(x)\rightarrow u^{T}\nabla f(x)
  10. u T f ( x ) u^{T}\nabla f(x)
  11. ( u f ( x ) ) = u T H ( x ) = ( H ( x ) u ) T \nabla\left(u\cdot\nabla f(x)\right)=u^{T}H(x)=(H(x)u)^{T}
  12. H H
  13. x n x\in\mathbb{R}^{n}

Hexaprenyl-diphosphate_synthase_((2E,6E)-farnesyl-diphosphate_specific).html

  1. \rightleftharpoons

Hexaprenyl_diphosphate_synthase_(geranylgeranyl-diphosphate_specific).html

  1. \rightleftharpoons

Hierarchical_generalized_linear_model.html

  1. y y
  2. u u
  3. g g
  4. η = X β \eta=X\beta
  5. v = v ( u ) v=v(u)
  6. u u
  7. y | u y|u
  8. u u
  9. y u f ( θ , ϕ ) y\mid u\sim\ f(\theta,\,\phi)
  10. u f u ( α ) . u\sim\ f_{u}(\alpha).
  11. g ( E ( y ) ) = g ( μ ) = η = X β + v g(E(y))=g(\mu)=\eta=X\beta+v\,
  12. g g
  13. μ = E ( y ) \mu=E(y)
  14. η = X β + v \eta=X\beta+v
  15. v = v ( u ) v=v(u)
  16. u u
  17. β \beta
  18. u u
  19. u u
  20. X β + v = ( X β + a ) + ( v - a ) X\beta+v=(X\beta+a)+(v-a)\,
  21. a a
  22. E ( v ) = 0 E(v)=0
  23. y u y\mid u
  24. u u
  25. g g
  26. v v
  27. u u
  28. u u
  29. v v
  30. y u y\mid u
  31. u u
  32. y u y\mid u
  33. u u
  34. y u y\mid u
  35. u u
  36. η = 𝐱 s y m b o l β \eta=\mathbf{x}symbol\beta
  37. 𝐯 ( 𝐮 ) \mathbf{v(u)}

Hierarchical_network_model.html

  1. P ( k ) c k - γ P\left(k\right)\sim ck^{-\gamma}\,
  2. γ = 1 + l n M l n ( M - 1 ) \gamma=1+\frac{lnM}{ln(M-1)}
  3. C ( k ) k - β C\left(k\right)\sim k^{-\beta}\,

Higgs_field_(classical).html

  1. G G
  2. P X P\to X
  3. H H
  4. h h
  5. P / G X P/G\to X
  6. P P / G X P\to P/G\to X
  7. P P / G P\to P/G
  8. H H
  9. H H
  10. E P / G X E\to P/G\to X
  11. E P / G E\to P/G
  12. P P / G P\to P/G
  13. P P / G P\to P/G
  14. P X P\to X
  15. X X
  16. F X / O ( 1 , 3 ) X FX/O(1,3)\to X
  17. F X FX
  18. X X
  19. G L ( 4 , ) GL(4,\mathbb{R})

High-field_domain.html

  1. C d S CdS
  2. C d S CdS
  3. G a A s GaAs
  4. C d S CdS
  5. d n d x = e k T ( j e μ - n F ) \frac{dn}{dx}=\frac{e}{kT}\left(\frac{j}{e\mu}-nF\right)
  6. d F d x = e ϵ ϵ 0 ( n - n a ) \frac{dF}{dx}=\frac{e}{\epsilon\epsilon_{0}}\left(n-n_{a}\right)
  7. < v a r > n F <var>nF

High-velocity_cloud.html

  1. M \begin{smallmatrix}M_{\odot}\end{smallmatrix}
  2. M \begin{smallmatrix}M_{\odot}\end{smallmatrix}
  3. M \begin{smallmatrix}M_{\odot}\end{smallmatrix}
  4. M \begin{smallmatrix}M_{\odot}\end{smallmatrix}
  5. M \begin{smallmatrix}M_{\odot}\end{smallmatrix}
  6. M \begin{smallmatrix}M_{\odot}\end{smallmatrix}
  7. M \begin{smallmatrix}M_{\odot}\end{smallmatrix}

History_of_experiments.html

  1. 2 h / g \sqrt{2h/g}
  2. d = 1 2 g t 2 \ d=\frac{1}{2}gt^{2}

History_of_Maxwell's_equations.html

  1. c = 1 μ 0 ε 0 . c=\frac{1}{\sqrt{\mu_{0}\varepsilon_{0}}}\ .
  2. 1 / μ 0 ε 0 1/\sqrt{\mu_{0}\varepsilon_{0}}
  3. 𝐉 tot = 𝐉 + 𝐃 t \mathbf{J}_{\mathrm{tot}}=\mathbf{J}+\frac{\partial\mathbf{D}}{\partial t}
  4. μ 𝐇 = × 𝐀 \mu\mathbf{H}=\nabla\times\mathbf{A}
  5. × 𝐇 = 𝐉 tot \nabla\times\mathbf{H}=\mathbf{J}_{\mathrm{tot}}
  6. 𝐄 = μ 𝐯 × 𝐇 - 𝐀 t - ϕ \mathbf{E}=\mu\mathbf{v}\times\mathbf{H}-\frac{\partial\mathbf{A}}{\partial t}% -\nabla\phi
  7. 𝐄 = 1 ε 𝐃 \mathbf{E}=\frac{1}{\varepsilon}\mathbf{D}
  8. 𝐄 = 1 σ 𝐉 \mathbf{E}=\frac{1}{\sigma}\mathbf{J}
  9. 𝐃 = ρ \nabla\cdot\mathbf{D}=\rho
  10. 𝐉 = - ρ t \nabla\cdot\mathbf{J}=-\frac{\partial\rho}{\partial t}
  11. 𝐉 tot = 0 \nabla\cdot\mathbf{J}_{\mathrm{tot}}=0

Hochster–Roberts_theorem.html

  1. f 1 , , f d f_{1},\cdots,f_{d}
  2. k [ V ] G k[V]^{G}
  3. k [ f 1 , , f d ] k[f_{1},\cdots,f_{d}]

Holographic_interference_microscopy.html

  1. I i m + I_{im+}
  2. I b + I_{b+}
  3. I i m + ( x , y ) = 2 I 0 [ 1 + c o s f ( x , y ) ] I_{im+}(x^{\prime},y^{\prime})=2I_{0}[1+cosf(x,y)]
  4. I b + = 4 I 0 I_{b+}=4I_{0}
  5. I i m - I_{im-}
  6. I b - I_{b-}
  7. I i m - ( x , y ) = 2 I 0 [ 1 - c o s f ( x , y ) ] I_{im-}(x^{\prime},y^{\prime})=2I_{0}[1-cosf(x,y)]
  8. I b - = 0 I_{b-}=0
  9. f ( x , y ) f(x,y)
  10. I 0 I_{0}
  11. h ( x , y ) h(x^{\prime},y^{\prime})
  12. f ( x , y ) f(x,y)
  13. h ( x , y ) = f ( x , y ) T / 2 π h(x^{\prime},y^{\prime})=f(x,y)T/2\pi
  14. T T

Holonomic_basis.html

  1. e k = x k . e_{k}={\partial\over\partial x^{k}}.
  2. [ e i , e j ] = 0. [e_{i},e_{j}]=0.

Homospermidine_synthase.html

  1. \rightleftharpoons
  2. \rightleftharpoons

Horst_Schubert.html

  1. S 3 S^{3}

HOSVD-based_canonical_form_of_TP_functions_and_qLPV_models.html

  1. f ( 𝐱 ) = 𝒮 n = 1 N 𝐰 n ( x n ) , f(\mathbf{x})=\mathcal{S}\boxtimes_{n=1}^{N}\mathbf{w}_{n}(x_{n}),
  2. 𝐱 Ω R N \mathbf{x}\in\Omega\subset R^{N}
  3. 𝐰 n ( x n ) \mathbf{w}_{n}(x_{n})
  4. n = 1 N n=1\ldots N
  5. 𝒮 \mathcal{S}
  6. 𝒮 = 𝒜 n = 1 N 𝐔 n \mathcal{S}=\mathcal{A}\boxtimes_{n=1}^{N}\mathbf{U}_{n}
  7. f ( 𝐱 ) = 𝒮 n = 1 N 𝐰 n ( x n ) = ( 𝒜 n = 1 N 𝐔 n ) n = 1 N 𝐰 n ( x n ) , f(\mathbf{x})=\mathcal{S}\boxtimes_{n=1}^{N}\mathbf{w}_{n}(x_{n})=\left(% \mathcal{A}\boxtimes_{n=1}^{N}\mathbf{U}_{n}\right)\boxtimes_{n=1}^{N}\mathbf{% w}_{n}(x_{n}),
  8. f ( 𝐱 ) = 𝒜 n = 1 N ( 𝐰 n ( x n ) 𝐔 n ) = 𝒜 n = 1 N 𝐰 n ( x n ) , f(\mathbf{x})=\mathcal{A}\boxtimes_{n=1}^{N}\left(\mathbf{w}_{n}(x_{n})\mathbf% {U}_{n}\right)=\mathcal{A}\boxtimes_{n=1}^{N}\mathbf{w^{\prime}}_{n}(x_{n}),
  9. 𝐰 n ( x n ) , \mathbf{w^{\prime}}_{n}(x_{n}),
  10. 𝐰 n ( x n ) \mathbf{w}_{n}(x_{n})
  11. 𝐔 n \mathbf{U}_{n}
  12. 𝒜 \mathcal{A}
  13. f ( 𝐱 ) = 𝒜 n = 1 N 𝐰 n ( x n ) , f(\mathbf{x})=\mathcal{A}\boxtimes_{n=1}^{N}\mathbf{w}_{n}(x_{n}),
  14. f ( 𝐱 ) f(\mathbf{x})
  15. w n , i n ( x n ) w_{n,i_{n}}(x_{n})
  16. i n = 1 , , r n i_{n}=1,\dots,r_{n}
  17. i n i_{n}
  18. n n
  19. n = 1 , , N n=1,\dots,N
  20. 𝐰 n ( x n ) \mathbf{w}_{n}(x_{n})
  21. n : a n b n w ~ n , i ( p n ) w ~ n , j ( p n ) d p n = δ i , j , 1 i , j I n , \forall n:\int_{a_{n}}^{b_{n}}\tilde{w}_{n,i}(p_{n})\tilde{w}_{n,j}(p_{n})\,dp% _{n}=\delta_{i,j},\quad 1\leq i,j\leq I_{n},
  22. δ i , j \delta_{i,j}
  23. δ i j = 1 \delta_{ij}=1
  24. i = j i=j
  25. δ i j = 0 \delta_{ij}=0
  26. i j i\neq j
  27. 𝒜 i n = i {\mathcal{A}}_{i_{n}=i}
  28. 𝒜 i n = i {\mathcal{A}}_{i_{n}=i}
  29. 𝒜 i n = j {\mathcal{A}}_{i_{n}=j}
  30. n , i n,i
  31. j : 𝒜 i n = i , 𝒜 i n = j = 0 j:\left\langle{\mathcal{A}}_{i_{n}=i},{\mathcal{A}}_{i_{n}=j}\right\rangle=0
  32. i j i\neq j
  33. 𝒜 i n = 1 𝒜 i n = 2 𝒜 i n = r n > 0 \left\|{\mathcal{A}}_{i_{n}=1}\right\|\geq\left\|{\mathcal{A}}_{i_{n}=2}\right% \|\geq\ldots\geq\left\|{\mathcal{A}}_{i_{n}=r_{n}}\right\|>0
  34. n = 1 , , N + 2 n=1,\ldots,N+2
  35. n n
  36. f ( 𝐱 ) f(\mathbf{x})
  37. 𝒜 i n = i \left\|{\mathcal{A}}_{i_{n}=i}\right\|
  38. σ i ( n ) \sigma_{i}^{(n)}
  39. n n
  40. 𝒜 \mathcal{A}
  41. 𝒜 {\mathcal{A}}
  42. n n
  43. f ( 𝐱 ) f(\mathbf{x})
  44. n n
  45. r a n k n ( f ( 𝐱 ) ) rank_{n}(f(\mathbf{x}))
  46. n n

Hotel_rating.html

  1. \bigstar
  2. 𝐒 \bigstar\mathbf{S}
  3. \bigstar\bigstar
  4. 𝐒 \bigstar\bigstar\mathbf{S}
  5. \bigstar\bigstar\bigstar
  6. 𝐒 \bigstar\bigstar\bigstar\mathbf{S}
  7. \bigstar\bigstar\bigstar\bigstar
  8. 𝐒 \bigstar\bigstar\bigstar\bigstar\mathbf{S}
  9. \bigstar\bigstar\bigstar\bigstar\bigstar
  10. 𝐒 \bigstar\bigstar\bigstar\bigstar\bigstar\mathbf{S}

Hua's_identity.html

  1. a - ( a - 1 + ( b - 1 - a ) - 1 ) - 1 = a b a a-(a^{-1}+(b^{-1}-a)^{-1})^{-1}=aba
  2. a b 0 , 1 ab\neq 0,1
  3. b b
  4. - b - 1 -b^{-1}
  5. ( a + a b - 1 a ) - 1 + ( a + b ) - 1 = a - 1 . (a+ab^{-1}a)^{-1}+(a+b)^{-1}=a^{-1}.
  6. σ : K L \sigma:K\to L
  7. σ \sigma
  8. σ ( a + b ) = σ ( a ) + σ ( b ) , σ ( 1 ) = 1 , σ ( a - 1 ) = σ ( a ) - 1 , \sigma(a+b)=\sigma(a)+\sigma(b),\quad\sigma(1)=1,\quad\sigma(a^{-1})=\sigma(a)% ^{-1},
  9. σ \sigma
  10. ( a - a b a ) ( a - 1 + ( b - 1 - a ) - 1 ) = a b ( b - 1 - a ) ( a - 1 + ( b - 1 - a ) - 1 ) = 1. (a-aba)(a^{-1}+(b^{-1}-a)^{-1})=ab(b^{-1}-a)(a^{-1}+(b^{-1}-a)^{-1})=1.

Hurwitz_problem.html

  1. ( x 2 + y 2 ) ( u 2 + v 2 ) = ( x u - y v ) 2 + ( x v + y u ) 2 , (x^{2}+y^{2})(u^{2}+v^{2})=(xu-yv)^{2}+(xv+yu)^{2}\ ,
  2. ( x 1 2 + + x r 2 ) ( y 1 2 + + y s 2 ) = ( z 1 2 + + z n 2 ) , (x_{1}^{2}+\cdots+x_{r}^{2})\cdot(y_{1}^{2}+\cdots+y_{s}^{2})=(z_{1}^{2}+% \cdots+z_{n}^{2})\ ,

HVDC_converter.html

  1. V dc = V av = 3 V LLpeak π cos ( α ) - 6 f L c I d {V_{\mathrm{dc}}=V_{\mathrm{av}}=\frac{3V_{\mathrm{LLpeak}}}{\pi}\cos(\alpha)}% -{6fL_{\mathrm{c}}I_{\mathrm{d}}}
  2. γ = 180 - α - μ \gamma=180-\alpha-\mu
  3. I v = I d 3 + I ac 2 {I_{\mathrm{v}}=\frac{I_{\mathrm{d}}}{3}+\frac{I_{\mathrm{ac}}}{2}}
  4. I v = I d 3 - I ac 2 {I_{\mathrm{v}}=\frac{I_{\mathrm{d}}}{3}-\frac{I_{\mathrm{ac}}}{2}}

Hybrid_difference_scheme.html

  1. t ( ρ ϕ ) + ( ρ 𝐮 ϕ ) = ( Γ grad ϕ ) + S ϕ \frac{\partial}{\partial t}(\rho\phi)+\nabla(\rho\mathbf{u}\phi)\,=\nabla(% \Gamma\cdot\operatorname{grad}\phi)+S_{\phi}
  2. \;
  3. ρ \rho
  4. 𝐮 \mathbf{u}
  5. Γ \Gamma
  6. S ϕ S_{\phi}
  7. ϕ \phi
  8. 𝐮 \mathbf{u}
  9. x ( ρ u ϕ ) = x ( Γ ϕ x ) , 0 < x < L \frac{\partial}{\partial x}(\rho u\phi)\,=\frac{\partial}{\partial x}\left(% \Gamma\frac{\partial\phi}{\partial x}\right),\quad 0<x<L\;
  10. \;
  11. ϕ ( 0 ) = ϕ 0 \phi(0)\,=\phi_{0}
  12. ϕ ( L ) = ϕ L \phi(L)\,=\phi_{L}
  13. ϕ 0 \phi_{0}
  14. ϕ L \phi_{L}
  15. C V ( ρ 𝐮 ϕ ) d V = A 𝐧 ( ρ 𝐮 ϕ ) d A \int_{CV}\nabla(\rho\mathbf{u}\phi)dV\,=\int_{A}\mathbf{n}\cdot(\rho\mathbf{u}% \phi)dA
  16. \;
  17. ( ρ u A ϕ ) r - ( ρ u A ϕ ) l \left(\rho uA\phi\right)_{r}-\left(\rho uA\phi\right)_{l}
  18. ( Γ A ϕ x ) r - ( Γ A ϕ x ) l \left(\Gamma A\frac{\partial\phi}{\partial x}\right)_{r}-\left(\Gamma A\frac{% \partial\phi}{\partial x}\right)_{l}
  19. \;
  20. ( ρ u A ) r - ( ρ u A ) l \left(\rho uA\right)_{r}-\left(\rho uA\right)_{l}
  21. \;
  22. F = ρ u A F\,=\rho uA
  23. \;
  24. \;
  25. D = Γ A δ x D\,=\frac{\Gamma A}{\delta x}
  26. \;
  27. F r ϕ r - F l ϕ l = D r ( ϕ R - ϕ N ) - D l ( ϕ N - ϕ L ) F_{r}\phi_{r}-F_{l}\phi_{l}\,=D_{r}(\phi_{R}-\phi_{N})-D_{l}(\phi_{N}-\phi_{L})
  28. \;
  29. F r - F l = 0 F_{r}-F_{l}\,=0
  30. \;
  31. P e = F D = ρ u Γ / δ x Pe\,=\frac{F}{D}\,=\frac{\rho u}{\Gamma/\delta x}
  32. \;
  33. ϕ r = ϕ R + ϕ N 2 \phi_{r}\,=\frac{\phi_{R}+\phi_{N}}{2}
  34. \;
  35. \;
  36. ϕ l = ϕ N + ϕ L 2 \phi_{l}\,=\frac{\phi_{N}+\phi_{L}}{2}
  37. \;
  38. a N ϕ N = a R ϕ R + a L ϕ L a_{N}\phi_{N}\,=a_{R}\phi_{R}+a_{L}\phi_{L}
  39. \;
  40. a L a_{L}
  41. a R a_{R}
  42. a N a_{N}
  43. D l + F l 2 D_{l}+\frac{F_{l}}{2}
  44. D r + F r 2 D_{r}+\frac{F_{r}}{2}
  45. a r + a l + ( F r - F l ) a_{r}+a_{l}+(F_{r}-F_{l})
  46. ϕ l = ϕ L \phi_{l}\,=\phi_{L}
  47. \;
  48. \;
  49. ϕ r = ϕ N \phi_{r}\,=\phi_{N}
  50. \;
  51. \;
  52. \;
  53. ϕ l = ϕ N \phi_{l}\,=\phi_{N}
  54. \;
  55. a L a_{L}
  56. a R a_{R}
  57. a N a_{N}
  58. D l + max ( F l , 0 ) D_{l}+\max(F_{l},0)
  59. D r + max ( F r , 0 ) D_{r}+\max(F_{r},0)
  60. a r + a l + ( F r - F l ) a_{r}+a_{l}+(F_{r}-F_{l})
  61. ϕ l = [ ( 1 + 2 P e l ) ϕ L 2 + ( 1 - 2 P e l ) ϕ R 2 ] \phi_{l}\,=\left[\left(1+\frac{2}{Pe_{l}}\right)\frac{\phi_{L}}{2}+\left(1-% \frac{2}{Pe_{l}}\right)\frac{\phi_{R}}{2}\right]\,
  62. \;
  63. \;
  64. - 2 < P e l < 2 -2<Pe_{l}<2
  65. \;
  66. ϕ l = ϕ L \phi_{l}\,=\phi_{L}
  67. \;
  68. \;
  69. P e l 2 Pe_{l}\,\text{≥}2
  70. \;
  71. ϕ l = ϕ N \phi_{l}\,=\phi_{N}
  72. \;
  73. \;
  74. P e l - 2 Pe_{l}\,\text{≤}-2
  75. \;
  76. a L a_{L}
  77. a R a_{R}
  78. a N a_{N}
  79. max [ F l , ( D l + F l 2 ) , 0 ] \max\left[F_{l},\left(D_{l}+\frac{F_{l}}{2}\right),0\right]
  80. max [ F r , ( D r + F r 2 ) , 0 ] \max\left[F_{r},\left(D_{r}+\frac{F_{r}}{2}\right),0\right]
  81. a r + a l + ( F r - F l ) a_{r}+a_{l}+(F_{r}-F_{l})

Hydraucone.html

  1. r z = C , C constant rz=C,C\rightarrow\mbox{constant}~{}
  2. z = C ln ( r ) z=C\mathrm{ln}(r)
  3. z = C r z=Cr

Hydrogenase_(NAD+,_ferredoxin).html

  1. \rightleftharpoons

Hydroquinone_1,2-dioxygenase.html

  1. \rightleftharpoons

Hydroxylamine_dehydrogenase.html

  1. \rightleftharpoons
  2. \rightleftharpoons

Hydroxylamine_oxidoreductase.html

  1. \rightleftharpoons

Hygromycin_B_4-O-kinase.html

  1. \rightleftharpoons

Hypertopology.html

  1. i : x { x } ¯ , i:x\mapsto\overline{\{x\}},

Hypoxia-inducible_factor-asparagine_dioxygenase.html

  1. \rightleftharpoons

Hypoxia-inducible_factor-proline_dioxygenase.html

  1. \rightleftharpoons

Iacono's_working_set_structure.html

  1. n n
  2. x x
  3. x x
  4. O ( log n ) O(\log n)
  5. x x
  6. O ( log w ( x ) ) O(\log w(x))
  7. w ( x ) w(x)
  8. x x
  9. n n
  10. T 1 , T 2 , , T k T_{1},T_{2},\ldots,T_{k}
  11. Q 1 , Q 2 , Q k Q_{1},Q_{2},\ldots Q_{k}
  12. k = log log n k=\lceil\log\log n\rceil
  13. 1 i k 1\leq i\leq k
  14. T i T_{i}
  15. Q i Q_{i}
  16. i < k i<k
  17. T i T_{i}
  18. Q i Q_{i}
  19. 2 2 i 2^{2^{i}}
  20. T k T_{k}
  21. Q k Q_{k}
  22. n - i = 1 k - 1 2 2 i n-\sum_{i=1}^{k-1}2^{2^{i}}
  23. n n
  24. x x
  25. y y
  26. Q i Q_{i}
  27. w ( x ) < w ( y ) w(x)<w(y)
  28. 1 i < k 1\leq i<k
  29. Q i Q_{i}
  30. Q i + 1 Q_{i+1}
  31. h h
  32. j j
  33. h h
  34. j j
  35. h h
  36. j j
  37. h < j h<j
  38. h i < j h\leq i<j
  39. Q i Q_{i}
  40. Q i + 1 Q_{i+1}
  41. T i T_{i}
  42. T i + 1 T_{i+1}
  43. O ( i = h j log | T i | ) = O ( i = h j log 2 2 i ) = O ( 2 j ) O(\sum_{i=h}^{j}\log|T_{i}|)=O(\sum_{i=h}^{j}\log 2^{2^{i}})=O(2^{j})
  44. j < h j<h
  45. j < i h j<i\leq h
  46. Q i Q_{i}
  47. Q i - 1 Q_{i-1}
  48. T i T_{i}
  49. T i - 1 T_{i-1}
  50. O ( i = j h log | T i | ) = O ( i = j h log 2 2 i ) = O ( 2 h ) O(\sum_{i=j}^{h}\log|T_{i}|)=O(\sum_{i=j}^{h}\log 2^{2^{i}})=O(2^{h})
  51. T h T_{h}
  52. T j T_{j}
  53. x x
  54. x x
  55. T 1 , T 2 , T k T_{1},T_{2},\ldots T_{k}
  56. T j T_{j}
  57. x x
  58. x x
  59. T j T_{j}
  60. T 1 T_{1}
  61. 1 1
  62. j j
  63. O ( i = 1 j log 2 2 i ) = O ( 2 j ) O(\sum_{i=1}^{j}\log 2^{2^{i}})=O(2^{j})
  64. O ( i = j k log 2 2 i ) = O ( 2 k ) = O ( log n ) O(\sum_{i=j}^{k}\log 2^{2^{i}})=O(2^{k})=O(\log n)
  65. T 1 , T 2 , , T j - 1 T_{1},T_{2},\ldots,T_{j-1}
  66. x x
  67. T j - 1 T_{j-1}
  68. x x
  69. w ( x ) > | T j - 1 | = 2 2 j - 1 w(x)>|T_{j-1}|=2^{2^{j-1}}
  70. O ( 2 j ) = O ( log 2 2 j - 1 ) = O ( log w ( x ) ) O(2^{j})=O(\log 2^{2^{j-1}})=O(\log w(x))
  71. x x
  72. x x
  73. T 1 T_{1}
  74. Q 1 Q_{1}
  75. 1 1
  76. k k
  77. | T k | = 2 2 k |T_{k}|=2^{2^{k}}
  78. k k
  79. T k T_{k}
  80. Q k Q_{k}
  81. 1 1
  82. k k
  83. O ( 2 k ) = O ( 2 log log n ) = O ( log n ) O(2^{k})=O(2^{\log\log n})=O(\log n)
  84. x x
  85. Q 1 Q_{1}
  86. x x
  87. x x
  88. T j T_{j}
  89. x x
  90. T j T_{j}
  91. Q j Q_{j}
  92. k k
  93. j j
  94. T j T_{j}
  95. 2 2 j 2^{2^{j}}
  96. O ( 2 k ) = O ( log n ) O(2^{k})=O(\log n)
  97. O ( log n ) O(\log n)
  98. O ( log w ( x ) ) O(\log w(x))

IGR_J17091-3624.html

  1. M \begin{smallmatrix}M_{\odot}\end{smallmatrix}

Incoherent_broad-band_cavity-enhanced_absorption_spectroscopy.html

  1. α ( λ ) = ( I 0 ( λ ) I ( λ ) - 1 ) 1 - R eff ( λ ) d \alpha(\lambda)=\left(\frac{I_{0}(\lambda)}{I(\lambda)}-1\right)\frac{1-R\text% {eff}(\lambda)}{d}
  2. I 0 = I i n 1 - R 1 + R I_{0}=I_{in}\frac{1-R}{1+R}
  3. I = I i n ( 1 - R ) 2 ( 1 - L ) n = 0 R 2 n ( 1 - L ) 2 n I=I_{in}(1-R)^{2}(1-L)\sum_{n=0}^{\infty}R^{2n}(1-L)^{2n}
  4. α = 1 d | ln ( 1 2 R 2 ( 4 R 2 + ( I 0 I ( R 2 - 1 ) ) 2 + I 0 I ( R 2 - 1 ) ) ) | \alpha=\frac{1}{d}\left|\ln\left(\frac{1}{2R^{2}}\left(\sqrt{4R^{2}+\left(% \frac{I_{0}}{I}(R^{2}-1)\right)^{2}}+\frac{I_{0}}{I}(R^{2}-1)\right)\right)\right|
  5. α 1 d ( I 0 I - 1 ) ( 1 - R ) \alpha\approx\frac{1}{d}\left(\frac{I_{0}}{I}-1\right)(1-R)
  6. α m i n = 1 d ( Δ I m i n I 0 ) ( 1 - R ) \alpha_{min}=\frac{1}{d}\left(\frac{\Delta I_{min}}{I_{0}}\right)(1-R)
  7. R eff ( λ ) = 1 - σ ( λ ) n d ( I 0 I - 1 ) - 1 R\text{eff}(\lambda)=1-\sigma(\lambda)nd\left(\frac{I_{0}}{I}-1\right)^{-1}
  8. b 1 Σ g + ( v = 2 ) X 3 Σ g - ( v ′′ = 0 ) b^{1}\Sigma_{g}^{+}(v^{^{\prime}}=2)\leftarrow X^{3}\Sigma_{g}^{-}(v^{{}^{% \prime\prime}}=0)

Indole-2-monooxygenase.html

  1. \rightleftharpoons

Indole-3-carboxylate_decarboxylase.html

  1. \rightleftharpoons

Indole-3-pyruvate_monooxygenase.html

  1. \rightleftharpoons

Indolin-2-one_monooxygenase.html

  1. \rightleftharpoons

Induction-recursion_(type_theory).html

  1. D D
  2. f f
  3. a : A a:A
  4. d : D d:D
  5. g : A T y p e g:A\to Type
  6. h : A D h:A\to D
  7. i : A B D i:A\to B\to D
  8. j : g a j:g\ a
  9. D D
  10. k : ( f d ) D k:(f\ d)\to D
  11. d d
  12. D D
  13. f f
  14. k : D A k:D\to A
  15. D D
  16. l : ( A D ) A l:(A\to D)\to A
  17. D D
  18. A A
  19. m : z d m:z\ d
  20. d d
  21. D D
  22. f f
  23. U U
  24. T T
  25. U U
  26. U U
  27. T T
  28. U U
  29. U U
  30. c o n s t r u c t o r Π ( u : U ) ( u : ( x : T ( u ) ) U ) : U constructor_{\Pi}(u:U)(u^{\prime}:(x:T(u))U):U
  31. u u
  32. U U
  33. u u^{\prime}
  34. : U :U
  35. U U
  36. T ( c o n s t r u c t o r Π ( u , u ) ) T(constructor_{\Pi}(u,u^{\prime}))
  37. Π ( T ( u ) , ( x ) T ( u ( x ) ) ) \Pi(T(u),(x)T(u^{\prime}(x)))
  38. T T
  39. T T
  40. T T

Inertial_manifold.html

  1. p ( t ) p(t)
  2. q ( t ) q(t)
  3. a a
  4. d p d t = a p - p q , d q d t = - q + p 2 - 2 q 2 . \frac{dp}{dt}=ap-pq\,,\qquad\frac{dq}{dt}=-q+p^{2}-2q^{2}.
  5. \mathcal{M}
  6. q = p 2 / ( 1 + 2 a ) q=p^{2}/(1+2a)
  7. \mathcal{M}
  8. d q d t = d d t p 2 1 + 2 a = 2 p d p d t 1 + 2 a = 2 a p 2 1 + 2 a - 2 p 4 ( 1 + 2 a ) 2 \frac{dq}{dt}=\frac{d}{dt}\frac{p^{2}}{1+2a}=\frac{2p\frac{dp}{dt}}{1+2a}=% \frac{2ap^{2}}{1+2a}-\frac{2p^{4}}{(1+2a)^{2}}
  9. - q + p 2 - 2 q 2 = - p 2 1 + 2 a + p 2 - 2 ( p 2 1 + 2 a ) 2 = 2 a p 2 1 + 2 a - 2 p 4 ( 1 + 2 a ) 2 . -q+p^{2}-2q^{2}=-\frac{p^{2}}{1+2a}+p^{2}-2\left(\frac{p^{2}}{1+2a}\right)^{2}% =\frac{2ap^{2}}{1+2a}-\frac{2p^{4}}{(1+2a)^{2}}.
  10. \mathcal{M}
  11. d q d t - q \frac{dq}{dt}\approx-q
  12. \mathcal{M}
  13. d p d t = a p - 1 1 + 2 a p 3 \frac{dp}{dt}=ap-\frac{1}{1+2a}p^{3}
  14. u ( t ) u(t)
  15. u ( t ) u(t)
  16. H = n H=\mathbb{R}^{n}
  17. H H
  18. u ( t ) u(t)
  19. H H
  20. d u / d t = F ( u ( t ) ) {du}/{dt}=F(u(t))
  21. u ( 0 ) = u 0 u(0)=u_{0}
  22. S : H H S:H\to H
  23. u ( t ) = S ( t ) u 0 u(t)=S(t)u_{0}
  24. t 0 t\geq 0
  25. u 0 u_{0}
  26. S ( t ) S(t)
  27. \mathcal{M}
  28. \mathcal{M}
  29. S ( t ) S(t)\mathcal{M}\subset\mathcal{M}
  30. t 0 t\geq 0
  31. \mathcal{M}
  32. u 0 H u_{0}\in H
  33. c j > 0 c_{j}>0
  34. dist ( S ( t ) u 0 , ) c 1 exp ( - c 2 t ) \,\text{dist}(S(t)u_{0},\mathcal{M})\leq c_{1}\exp(-c_{2}t)
  35. d u / d t = F ( u ) du/dt=F(u)
  36. \mathcal{M}
  37. \mathcal{M}
  38. u 0 u_{0}
  39. v 0 v_{0}\in\mathcal{M}
  40. τ 0 \tau\geq 0
  41. dist ( S ( t ) u 0 , S ( t + τ ) v 0 ) 0 \,\text{dist}(S(t)u_{0},S(t+\tau)v_{0})\to 0
  42. t t\to\infty
  43. d u / d t + A u + f ( u ) = 0 du/dt+Au+f(u)=0
  44. A A
  45. D ( A ) H D(A)\subset H
  46. f : D ( A ) H f:D(A)\to H
  47. H H
  48. v j v_{j}
  49. A v j = λ j v j Av_{j}=\lambda_{j}v_{j}
  50. j = 1 , 2 , j=1,2,\ldots
  51. 0 < λ 1 λ 2 0<\lambda_{1}\leq\lambda_{2}\leq\cdots
  52. m m
  53. P P
  54. H H
  55. v 1 , , v m v_{1},\ldots,v_{m}
  56. Q = I - P Q=I-P
  57. v m + 1 , v m + 2 , v_{m+1},v_{m+2},\ldots
  58. Φ : P H Q H \Phi:PH\to QH
  59. λ m + 1 - λ m c ( λ m + 1 + λ m ) \lambda_{m+1}-\lambda_{m}\geq c(\sqrt{\lambda_{m+1}}+\sqrt{\lambda_{m}})
  60. c c
  61. A A
  62. m m
  63. p ( t ) = P u ( t ) p(t)=Pu(t)
  64. q ( t ) = Q u ( t ) q(t)=Qu(t)
  65. d u / d t + A u + f ( u ) = 0 du/dt+Au+f(u)=0
  66. P H PH
  67. Q H QH
  68. d p / d t + A p + P f ( p + q ) = 0 dp/dt+Ap+Pf(p+q)=0
  69. d q / d t + A q + Q f ( p + q ) = 0 dq/dt+Aq+Qf(p+q)=0
  70. M M
  71. q ( t ) = Φ ( p ( t ) ) q(t)=\Phi(p(t))
  72. - d Φ d p [ A p + P f ( p + Φ ( p ) ) ] + A Φ ( p ) + Q f ( p + Φ ( p ) ) = 0. -\frac{d\Phi}{dp}\left[Ap+Pf(p+\Phi(p))\right]+A\Phi(p)+Qf(p+\Phi(p))=0.
  73. p p

Infrastructure-based_development.html

  1. Y = F ( K , G , N , Z ) = Z K α G β N 1 - α - β Y=F(K,G,N,Z)=ZK^{\alpha}G^{\beta}N^{1-\alpha-\beta}

Inherent_viscosity.html

  1. η i n h = ln η r e l c \eta_{inh}=\frac{\ln\eta_{rel}}{c}
  2. η r e l \eta_{rel}
  3. η r e l = η η 0 \eta_{rel}=\frac{\eta}{\eta_{0}}
  4. η \eta
  5. η 0 \eta_{0}

Initial_and_final_state_radiation.html

  1. | i |i\rangle
  2. | f |f\rangle
  3. S f i = f | S | i , S_{fi}=\langle f|S|i\rangle\;,
  4. S S
  5. e + e - 2 γ e^{+}e^{-}\to 2\gamma

Inositol-1,3,4-trisphosphate_5::6-kinase.html

  1. \rightleftharpoons
  2. \rightleftharpoons

Inositol-hexakisphosphate_kinase.html

  1. \rightleftharpoons
  2. \rightleftharpoons

Inositol-polyphosphate_5-phosphatase.html

  1. \rightleftharpoons
  2. \rightleftharpoons

Inositol-polyphosphate_multikinase.html

  1. \rightleftharpoons
  2. \rightleftharpoons
  3. \rightleftharpoons

Interacting_particle_system.html

  1. ( X ( t ) ) t + (X(t))_{t\in\mathbb{R}^{+}}
  2. Ω = S G \Omega=S^{G}
  3. G G
  4. S S
  5. c Λ ( η , ξ ) > 0 c_{\Lambda}(\eta,\xi)>0
  6. Λ G \Lambda\subset G
  7. η , ξ Ω \eta,\xi\in\Omega
  8. η i = ξ i \eta_{i}=\xi_{i}
  9. i Λ i\notin\Lambda
  10. η \eta
  11. ξ \xi
  12. c Λ ( η , d ξ ) c_{\Lambda}(\eta,d\xi)
  13. S Λ S^{\Lambda}
  14. L L
  15. f f
  16. L L
  17. L f ( η ) = Λ ξ : ξ Λ c = η Λ c c Λ ( η , d ξ ) [ f ( ξ ) - f ( η ) ] Lf(\eta)=\sum_{\Lambda}\int_{\xi:\xi_{\Lambda^{c}}=\eta_{\Lambda^{c}}}c_{% \Lambda}(\eta,d\xi)[f(\xi)-f(\eta)]
  18. G = d G=\mathbb{Z}^{d}
  19. S = { - 1 , + 1 } S=\{-1,+1\}
  20. c Λ = 0 c_{\Lambda}=0
  21. Λ { i } \Lambda\neq\{i\}
  22. i G i\in G
  23. c i ( η , η i ) = exp [ - β j : | j - i | = 1 η i η j ] c_{i}(\eta,\eta^{i})=\exp[-\beta\sum_{j:|j-i|=1}\eta_{i}\eta_{j}]
  24. η i \eta^{i}
  25. η \eta
  26. i i
  27. β \beta

Interband_cascade_laser.html

  1. g t h = α w g + α m i r r Γ g_{th}=\frac{\alpha_{wg}+\alpha_{mirr}}{\Gamma}

Interdependent_networks.html

  1. 1 - p 1-p
  2. d P d p \frac{dP^{\infty}}{dp}
  3. p c p_{c}
  4. p > p c p>p_{c}
  5. A A
  6. B B
  7. N N
  8. A i A_{i}
  9. A A
  10. B i B_{i}
  11. B B
  12. A i A_{i}
  13. B i B_{i}
  14. 1 - p 1-p
  15. A A
  16. A A
  17. B B
  18. A A
  19. 1 - p 1-p
  20. B B
  21. A A
  22. B B
  23. A A
  24. B B
  25. p c p_{c}
  26. q q
  27. q c q_{c}

Interpolation_inequality.html

  1. u 0 0 C u 1 1 α 1 u 2 2 α 2 u n n α n , n 2 , \|u_{0}\|_{0}\leq C\|u_{1}\|_{1}^{\alpha_{1}}\|u_{2}\|_{2}^{\alpha_{2}}\dots\|% u_{n}\|_{n}^{\alpha_{n}},\quad n\geq 2,
  2. 2 | u ( x ) | 4 d x 2 2 | u ( x ) | 2 d x 2 | u ( x ) | 2 d x , \int_{\mathbb{R}^{2}}|u(x)|^{4}\,\mathrm{d}x\leq 2\int_{\mathbb{R}^{2}}|u(x)|^% {2}\,\mathrm{d}x\int_{\mathbb{R}^{2}}|\nabla u(x)|^{2}\,\mathrm{d}x,
  3. u L 4 2 4 u L 2 1 / 2 u L 2 1 / 2 . \|u\|_{L^{4}}\leq\sqrt[4]{2}\,\|u\|_{L^{2}}^{1/2}\,\|\nabla u\|_{L^{2}}^{1/2}.
  4. f g L s f L r g L p , \|f\star g\|_{L^{s}}\leq\|f\|_{L^{r}}\|g\|_{L^{p}},
  5. 1 r + 1 p = 1 + 1 s . \frac{1}{r}+\frac{1}{p}=1+\frac{1}{s}.

Interspiro_DCSC.html

  1. V O 2 V_{O_{2}}
  2. V R M V_{RM}
  3. K E K_{E}
  4. K E = V R M V O 2 K_{E}=\frac{V_{RM}}{V_{O_{2}}}
  5. V R M = K E * V O 2 V_{RM}=K_{E}*V_{O_{2}}
  6. d F O 2 l o o p dF_{O_{2}loop}
  7. V l o o p * d F O 2 l o o p = ( Q f e e d * F O 2 f e e d - V O 2 - ( Q f e e d - V O 2 ) * F O 2 l o o p ) d t V_{loop}*dF_{O_{2}loop}=(Q_{feed}*F_{O_{2}feed}-V_{O_{2}}-(Q_{feed}-V_{O_{2}})% *F_{O_{2}loop})dt
  8. V l o o p V_{loop}
  9. Q f e e d Q_{feed}
  10. F O 2 f e e d F_{O_{2}feed}
  11. V O 2 V_{O_{2}}
  12. d F O 2 l o o p d t = ( Q f e e d * F O 2 f e e d - V O 2 ( t ) - ( Q f e e d - V O 2 ) * F O 2 l o o p ( t ) ) V l o o p \frac{dF_{O_{2}loop}}{dt}=\frac{(Q_{feed}*F_{O_{2}feed}-V_{O_{2}}(t)-(Q_{feed}% -V_{O_{2}})*F_{O_{2}loop}(t))}{V_{loop}}
  13. F O 2 l o o p ( t ) = Q f e e d * F O 2 f e e d - V O 2 Q f e e d - V O 2 + ( F O 2 l o o p s t a r t - Q f e e d * F O 2 f e e d - V O 2 Q f e e d - V O 2 ) * e - Q f e e d - V O 2 V l o o p t F_{O_{2}loop}(t)=\frac{Q_{feed}*F_{O_{2}feed}-V_{O_{2}}}{Q_{feed}-V_{O_{2}}}+(% F_{O_{2}loop}^{start}-\frac{Q_{feed}*F_{O_{2}feed}-V_{O_{2}}}{Q_{feed}-V_{O_{2% }}})*e^{-\frac{Q_{feed}-V_{O_{2}}}{V_{loop}}t}
  14. F O 2 l o o p F_{O_{2}loop}
  15. F O 2 l o o p = ( Q f e e d * F O 2 f e e d - V O 2 ) ( Q f e e d - V O 2 ) F_{O_{2}loop}=\frac{(Q_{feed}*F_{O_{2}feed}-V_{O_{2}})}{(Q_{feed}-V_{O_{2}})}
  16. Q f e e d Q_{feed}
  17. V O 2 V_{O_{2}}
  18. F O 2 f e e d F_{O_{2}feed}
  19. Q f e e d = K d o s a g e * V R M Q_{feed}=K_{dosage}*V_{RM}
  20. Q f e e d = K d o s a g e * K E * V O 2 Q_{feed}=K_{dosage}*K_{E}*V_{O_{2}}
  21. F O 2 l o o p = ( K d o s a g e * K E * V O 2 * F O 2 f e e d - V O 2 ) ( K d o s a g e * K E * V O 2 - V O 2 ) F_{O_{2}loop}=\frac{(K_{dosage}*K_{E}*V_{O_{2}}*F_{O_{2}feed}-V_{O_{2}})}{(K_{% dosage}*K_{E}*V_{O_{2}}-V_{O_{2}})}
  22. F O 2 l o o p = ( K d o s a g e * K E * F O 2 f e e d - 1 ) ( K d o s a g e * K E * - 1 ) F_{O_{2}loop}=\frac{(K_{dosage}*K_{E}*F_{O_{2}feed}-1)}{(K_{dosage}*K_{E}*-1)}

Interval_contractor.html

  1. C ( [ x ] ) [ x ] C([x])\subset[x]
  2. C ( [ x ] ) X = [ x ] X C([x])\cap X=[x]\cap X
  3. [ x ] [ y ] C ( [ x ] ) C ( [ y ] ) [x]\subset[y]\Rightarrow C([x])\subset C([y])
  4. C ( [ x ] ) = [ [ x ] X ] C([x])=[[x]\cap X]
  5. C ( { x } ) = { x } X C(\{x\})=\{x\}\cap X
  6. C C ( [ x ] ) = C ( [ x ] ) . C\circ C([x])=C([x]).
  7. [ x ] ( k ) x C ( [ x ] ( k ) ) { x } X . [x](k)\rightarrow x\Rightarrow C([x](k))\rightarrow\{x\}\cap X.
  8. ( C 1 C 2 ) ( [ x ] ) = C 1 ( [ x ] ) C 2 ( [ x ] ) (C_{1}\cap C_{2})([x])=C_{1}([x])\cap C_{2}([x])
  9. ( C 1 C 2 ) ( [ x ] ) = [ C 1 ( [ x ] ) C 2 ( [ x ] ) ] (C_{1}\cup C_{2})([x])=[C_{1}([x])\cup C_{2}([x])]
  10. ( C 1 C 2 ) ( [ x ] ) = C 1 ( C 2 ( [ x ] ) ) (C_{1}\circ C_{2})([x])=C_{1}(C_{2}([x]))
  11. C ( [ x ] ) = C C C C ( [ x ] ) C^{\infty}([x])=C\circ C\circ C\circ\cdots\circ C([x])
  12. ( x 1 + x 2 ) x 3 [ 1 , 2 ] . (x_{1}+x_{2})\cdot x_{3}\in[1,2].
  13. a = x 1 + x 2 a=x_{1}+x_{2}
  14. b = a x 3 b=a\cdot x_{3}
  15. x 3 = b a x_{3}=\frac{b}{a}
  16. a = b x 3 a=\frac{b}{x_{3}}
  17. x 1 = a - x 2 x_{1}=a-x_{2}
  18. x 2 = a - x 1 x_{2}=a-x_{1}
  19. C ( [ x 1 ] , [ x 2 ] , [ x 3 ] ) C([x_{1}],[x_{2}],[x_{3}])
  20. [ a ] = [ x 1 ] + [ x 2 ] [a]=[x_{1}]+[x_{2}]
  21. [ b ] = [ a ] [ x 3 ] [b]=[a]\cdot[x_{3}]
  22. [ b ] = [ b ] [ 1 , 2 ] [b]=[b]\cap[1,2]
  23. [ x 3 ] = [ x 3 ] [ b ] [ a ] [x_{3}]=[x_{3}]\ \cap\ \frac{[b]}{[a]}
  24. [ a ] = [ a ] [ b ] [ x 3 ] [a]=[a]\cap\frac{[b]}{[x_{3}]}
  25. [ x 1 ] = [ x 1 ] [ a ] - [ x 2 ] [x_{1}]=[x_{1}]\ \cap\ \ [a]-[x_{2}]
  26. [ x 2 ] = [ x 2 ] [ a ] - [ x 1 ] [x_{2}]=[x_{2}]\ \cap\ \ [a]-[x_{1}]