wpmath0000014_12

Stability_postulate.html

  1. X 1 , X 2 , , X n X_{1},X_{2},\dots,X_{n}\,
  2. p X j ( x ) = f ( x ) , p_{X_{j}}(x)=f(x),
  3. X n = max { X 1 , , X n } X^{\prime}_{n}=\max\{\,X_{1},\ldots,X_{n}\,\}\,
  4. F X n = [ F ( x ) ] n F_{X^{\prime}_{n}}={[F(x)]}^{n}\,
  5. ( a n X n + b n ) (a_{n}X^{\prime}_{n}+b_{n})\,
  6. a n , b n a_{n},b_{n}\,
  7. [ G ( x ) ] n = G ( a n x + b n ) {[G(x)]}^{n}=G{(a_{n}x+b_{n})}\,
  8. X i = Gumbel ( μ , β ) X_{i}=\textrm{Gumbel}(\mu,\beta)\,
  9. Y = min { X 1 , , X n } Y=\min\{\,X_{1},\ldots,X_{n}\,\}\,
  10. Y a n X + b n Y\sim a_{n}X+b_{n}\,
  11. a n = 1 a_{n}=1\,
  12. b n = β log ( n ) b_{n}=\beta\log(n)\,
  13. Y Gumbel ( μ - β log ( n ) , β ) Y\sim\textrm{Gumbel}(\mu-\beta\log(n),\beta)\,
  14. X i = EV ( μ , σ ) X_{i}=\textrm{EV}(\mu,\sigma)\,
  15. Y = max { X 1 , , X n } Y=\max\{\,X_{1},\ldots,X_{n}\,\}\,
  16. Y a n X + b n Y\sim a_{n}X+b_{n}\,
  17. a n = 1 a_{n}=1\,
  18. b n = σ log ( 1 n ) b_{n}=\sigma\log(\tfrac{1}{n})\,
  19. Y EV ( μ - σ log ( 1 n ) , σ ) Y\sim\textrm{EV}(\mu-\sigma\log(\tfrac{1}{n}),\sigma)\,
  20. X i = Frechet ( α , s , m ) X_{i}=\textrm{Frechet}(\alpha,s,m)\,
  21. Y = max { X 1 , , X n } Y=\max\{\,X_{1},\ldots,X_{n}\,\}\,
  22. Y a n X + b n Y\sim a_{n}X+b_{n}\,
  23. a n = n - 1 α a_{n}=n^{-\tfrac{1}{\alpha}}\,
  24. b n = m ( 1 - n - 1 α ) b_{n}=m\left(1-n^{-\tfrac{1}{\alpha}}\right)\,
  25. Y Frechet ( α , n 1 α s , m ) Y\sim\textrm{Frechet}(\alpha,n^{\tfrac{1}{\alpha}}s,m)\,

Standard_complex.html

  1. A A A A A A 0 , \cdots\rightarrow A\otimes A\otimes A\rightarrow A\otimes A\rightarrow A% \rightarrow 0\,,
  2. d ( a 0 a n + 1 ) = i = 0 n ( - 1 ) i a 0 a i a i + 1 a n + 1 . d(a_{0}\otimes\cdots\otimes a_{n+1})=\sum_{i=0}^{n}(-1)^{i}a_{0}\otimes\cdots% \otimes a_{i}a_{i+1}\otimes\cdots\otimes a_{n+1}\,.
  3. [ A A A A A ] [\cdots\rightarrow A\otimes A\otimes A\rightarrow A\otimes A]

Standard_Litres_Per_Minute.html

  1. 1 S L P M = 1 standard litre minute = 1.68875 Pa m 3 s 1\mathrm{SLPM}=1\frac{\mathrm{standard\,litre}}{\mathrm{minute}}=1.68875\frac{% \mathrm{Pa\cdot m^{3}}}{\mathrm{s}}

Standard_weight_in_fish.html

  1. W = a L b W=aL^{b}\!\,
  2. K = 100 ( W / L 3 ) K=100(W/L^{3})\!\,

Standardized_mean_of_a_contrast_variable.html

  1. d d
  2. δ . \delta.
  3. G 1 , G 2 , , G t G_{1},G_{2},\ldots,G_{t}
  4. μ 1 , μ 2 , , μ t \mu_{1},\mu_{2},\ldots,\mu_{t}
  5. σ 1 2 , σ 2 2 , , σ t 2 \sigma_{1}^{2},\sigma_{2}^{2},\ldots,\sigma_{t}^{2}
  6. V V
  7. V = i = 1 t c i G i , V=\sum_{i=1}^{t}c_{i}G_{i},
  8. c i c_{i}
  9. i = 1 t c i = 0 \sum_{i=1}^{t}c_{i}=0
  10. V V
  11. λ \lambda
  12. λ = E ( V ) stdev ( V ) = i = 1 t c i μ i Var ( i = 1 t c i G i ) = i = 1 t c i μ i i = 1 t c i 2 σ i 2 + 2 i = 1 t j = i c i c j σ i j \lambda=\frac{\operatorname{E}(V)}{\operatorname{stdev}(V)}=\frac{\sum_{i=1}^{% t}c_{i}\mu_{i}}{\sqrt{\,\text{Var}(\sum_{i=1}^{t}c_{i}G_{i})}}=\frac{\sum_{i=1% }^{t}c_{i}\mu_{i}}{\sqrt{\sum_{i=1}^{t}c_{i}^{2}\sigma_{i}^{2}+2\sum_{i=1}^{t}% \sum_{j=i}c_{i}c_{j}\sigma_{ij}}}
  13. σ i j \sigma_{ij}
  14. G i G_{i}
  15. G j G_{j}
  16. G 1 , G 2 , , G t G_{1},G_{2},\ldots,G_{t}
  17. λ = i = 1 t c i μ i i = 1 t c i 2 σ i 2 . \lambda=\frac{\sum_{i=1}^{t}c_{i}\mu_{i}}{\sqrt{\sum_{i=1}^{t}c_{i}^{2}\sigma_% {i}^{2}}}.
  18. λ \lambda
  19. λ - 5 \lambda\leq-5
  20. λ 5 \lambda\geq 5
  21. - 5 < λ - 3 -5<\lambda\leq-3
  22. 5 > λ 3 5>\lambda\geq 3
  23. - 3 < λ - 2 -3<\lambda\leq-2
  24. 3 > λ 2 3>\lambda\geq 2
  25. - 2 < λ - 1.645 -2<\lambda\leq-1.645
  26. 2 > λ 1.645 2>\lambda\geq 1.645
  27. - 1.645 < λ - 1.28 -1.645<\lambda\leq-1.28
  28. 1.645 > λ 1.28 1.645>\lambda\geq 1.28
  29. - 1.28 < λ - 1 -1.28<\lambda\leq-1
  30. 1.28 > λ 1 1.28>\lambda\geq 1
  31. - 1 < λ - 0.75 -1<\lambda\leq-0.75
  32. 1 > λ 0.75 1>\lambda\geq 0.75
  33. - 0.75 < λ < - 0.5 -0.75<\lambda<-0.5
  34. 0.75 > λ > 0.5 0.75>\lambda>0.5
  35. - 0.5 λ < - 0.25 -0.5\leq\lambda<-0.25
  36. 0.5 λ > 0.25 0.5\geq\lambda>0.25
  37. - 0.25 λ < 0 -0.25\leq\lambda<0
  38. 0.25 λ > 0 0.25\geq\lambda>0
  39. λ = 0 \lambda=0
  40. n i n_{i}
  41. Y i = ( Y i 1 , Y i 2 , , Y i n i ) Y_{i}=(Y_{i1},Y_{i2},\ldots,Y_{in_{i}})
  42. i th ( i = 1 , 2 , , t ) i\text{th}(i=1,2,\ldots,t)
  43. G i G_{i}
  44. Y i Y_{i}
  45. Y ¯ i = 1 n i j = 1 n i Y i j \bar{Y}_{i}=\frac{1}{n_{i}}\sum_{j=1}^{n_{i}}Y_{ij}
  46. s i 2 = 1 n i - 1 j = 1 n i ( Y i j - Y ¯ i ) 2 , s_{i}^{2}=\frac{1}{n_{i}-1}\sum_{j=1}^{n_{i}}(Y_{ij}-\bar{Y}_{i})^{2},
  47. N = i = 1 t n i N=\sum_{i=1}^{t}n_{i}
  48. MSE = 1 N - t i = 1 t ( n i - 1 ) s i 2 . \,\text{MSE }=\frac{1}{N-t}\sum_{i=1}^{t}(n_{i}-1)s_{i}^{2}.
  49. t t
  50. λ \lambda
  51. λ ^ MLE = i = 1 t c i Y ¯ i i = 1 t n i - 1 n i c i 2 s i 2 \hat{\lambda}\text{MLE }=\frac{\sum_{i=1}^{t}c_{i}\bar{Y}_{i}}{\sqrt{\sum_{i=1% }^{t}\frac{n_{i}-1}{n_{i}}c_{i}^{2}s_{i}^{2}}}
  52. λ ^ MM = i = 1 t c i Y ¯ i i = 1 t c i 2 s i 2 . \hat{\lambda}\text{MM}=\frac{\sum_{i=1}^{t}c_{i}\bar{Y}_{i}}{\sqrt{\sum_{i=1}^% {t}c_{i}^{2}s_{i}^{2}}}.
  53. t t
  54. λ \lambda
  55. λ ^ UMVUE = K N - t i = 1 t c i Y ¯ i i = 1 t MSE c i 2 \hat{\lambda}\text{UMVUE}=\sqrt{\frac{K}{N-t}}\frac{\sum_{i=1}^{t}c_{i}\bar{Y}% _{i}}{\sqrt{\sum_{i=1}^{t}\,\text{MSE }c_{i}^{2}}}
  56. K = 2 ( Γ ( N - t 2 ) ) 2 ( Γ ( N - t - 1 2 ) ) 2 K=\frac{2(\Gamma(\frac{N-t}{2}))^{2}}{(\Gamma(\frac{N-t-1}{2}))^{2}}
  57. T = i = 1 t c i Y ¯ i i = 1 t MSE c i 2 / n i noncentral t ( N - t , b λ ) T=\frac{\sum_{i=1}^{t}c_{i}\bar{Y}_{i}}{\sqrt{\sum_{i=1}^{t}\,\text{MSE }c_{i}% ^{2}/n_{i}}}\sim\,\text{noncentral }t(N-t,b\lambda)
  58. b = i = 1 t c i 2 i = 1 t c i 2 / n i . b=\sqrt{\frac{\sum_{i=1}^{t}c_{i}^{2}}{\sum_{i=1}^{t}c_{i}^{2}/n_{i}}}.
  59. n n
  60. ( Y 1 j , Y 2 j , , Y t j ) (Y_{1j},Y_{2j},\cdots,Y_{tj})
  61. t t
  62. G i G_{i}
  63. i = 1 , 2 , , t ; j = 1 , 2 , , n i=1,2,\cdots,t;j=1,2,\cdots,n
  64. j th j\text{th}
  65. V = i = 1 t c i G i V=\sum_{i=1}^{t}c_{i}G_{i}
  66. v j = i = 1 t c i Y i v_{j}=\sum_{i=1}^{t}c_{i}Y_{i}
  67. V ¯ \bar{V}
  68. s V 2 s_{V}^{2}
  69. V V
  70. λ ^ UMVUE = K n - 1 V ¯ s V \hat{\lambda}\text{UMVUE}=\sqrt{\frac{K}{n-1}}\frac{\bar{V}}{s_{V}}
  71. K = 2 ( Γ ( n - 1 2 ) ) 2 ( Γ ( n - 2 2 ) ) 2 . K=\frac{2(\Gamma(\frac{n-1}{2}))^{2}}{(\Gamma(\frac{n-2}{2}))^{2}}.
  72. T = V ¯ s V / n noncentral t ( n - 1 , n λ ) . T=\frac{\bar{V}}{s_{V}/\sqrt{n}}\sim\,\text{noncentral }t(n-1,\sqrt{n}\lambda).

Stanley_symmetric_function.html

  1. ( n 2 ) ! 1 n - 1 3 n - 2 5 n - 3 ( 2 n - 3 ) 1 \frac{{\left({{n}\atop{2}}\right)}!}{1^{n-1}\cdot 3^{n-2}\cdot 5^{n-3}\cdots(2% n-3)^{1}}
  2. ( n 2 ) {\left({{n}\atop{2}}\right)}

Star-mesh_transform.html

  1. z A B = z A z B 1 z z_{AB}=z_{A}z_{B}{\sum{1\over z}}
  2. z A z_{A}
  3. C 2 N {}_{N}C_{2}
  4. N > 3 N>3

State-merging.html

  1. H ( A | B ) = H ( A B ) - H ( B ) . H(A|B)\,=\,H(AB)-H(B)\,.
  2. H ( A ) H(A)
  3. H ( A ) := - T r ρ A log ρ A H(A):=-Tr\rho_{A}\log\rho_{A}
  4. I ( A : R ) := H ( A ) + H ( R ) - H ( A R ) I(A:R):=H(A)+H(R)-H(AR)

Static_forces_and_virtual-particle_exchange.html

  1. m m
  2. M M
  3. 𝐅 = - G m M r 2 𝐫 ^ = m 𝐠 ( 𝐫 ) , \mathbf{F}=-G{mM\over{r}^{2}}\,\mathbf{\hat{r}}=m\mathbf{g}\left(\mathbf{r}% \right),
  4. 𝐫 ^ \mathbf{\hat{r}}
  5. M M
  6. m m
  7. 𝐅 = m 𝐠 ( 𝐫 ) , \mathbf{F}=m\mathbf{g}\left(\mathbf{r}\right),
  8. 𝐠 ( 𝐫 ) \mathbf{g}\left(\mathbf{r}\right)
  9. 𝐠 = - 4 π G ρ m , \nabla\cdot\mathbf{g}=-4\pi G\rho_{m},
  10. ρ m \rho_{m}
  11. q q
  12. Q Q
  13. 𝐅 = 1 4 π ε 0 q Q r 2 𝐫 ^ , \mathbf{F}={1\over 4\pi\varepsilon_{0}}{qQ\over r^{2}}\mathbf{\hat{r}},
  14. ε 0 \varepsilon_{0}
  15. r r
  16. 𝐫 ^ \mathbf{\hat{r}}
  17. Q Q
  18. q q
  19. 𝐅 = q 𝐄 ( 𝐫 ) , \mathbf{F}=q\mathbf{E}\left(\mathbf{r}\right),
  20. 𝐄 = ρ q ε 0 ; \nabla\cdot\mathbf{E}=\frac{\rho_{q}}{\varepsilon_{0}};
  21. ρ q \rho_{q}
  22. Z 0 | exp ( - i H ^ T ) | 0 = exp ( - i E T ) = D φ exp ( i 𝒮 [ φ ] ) = exp ( i W ) Z\equiv\langle 0|\exp\left(-i\hat{H}T\right)|0\rangle=\exp\left(-iET\right)=% \int D\varphi\;\exp\left(i\mathcal{S}[\varphi]\right)\;=\exp\left(iW\right)
  23. H ^ \hat{H}
  24. T T
  25. E E
  26. W = - E T W=-ET
  27. φ \varphi
  28. 𝒮 [ φ ] = d 4 x [ φ ( x ) ] \mathcal{S}[\varphi]=\int\mathrm{d}^{4}x\;{\mathcal{L}[\varphi(x)]\,}
  29. [ φ ( x ) ] \mathcal{L}[\varphi(x)]
  30. = c = 1 \hbar=c=1
  31. η μ ν = ( 1 0 0 0 0 - 1 0 0 0 0 - 1 0 0 0 0 - 1 ) . \eta_{\mu\nu}=\begin{pmatrix}1&0&0&0\\ 0&-1&0&0\\ 0&0&-1&0\\ 0&0&0&-1\end{pmatrix}.
  32. Z = exp [ i d 4 x ( 1 2 φ O ^ φ + J φ ) ] D φ Z=\int\exp\left[i\int d^{4}x\left(\frac{1}{2}\varphi\hat{O}\varphi+J\varphi% \right)\right]D\varphi
  33. O ^ \hat{O}
  34. φ \varphi
  35. J J
  36. Z exp ( i W ( J ) ) Z\propto\exp\left(iW\left(J\right)\right)
  37. W ( J ) = - 1 2 d 4 x d 4 y J ( x ) D ( x - y ) J ( y ) W\left(J\right)=-{1\over 2}\iint d^{4}x\;d^{4}y\;J\left(x\right)D\left(x-y% \right)J\left(y\right)
  38. D ( x - y ) D\left(x-y\right)
  39. O ^ D ( x - y ) = δ 4 ( x - y ) \hat{O}D\left(x-y\right)=\delta^{4}\left(x-y\right)
  40. J ( x ) = ( J 1 + J 2 , 0 , 0 , 0 ) J\left(x\right)=\left(J_{1}+J_{2},0,0,0\right)
  41. J 1 = a 1 δ 3 ( x - x 1 ) J_{1}=a_{1}\delta^{3}\left(\vec{x}-\vec{x}_{1}\right)
  42. J 2 = a 2 δ 3 ( x - x 2 ) J_{2}=a_{2}\delta^{3}\left(\vec{x}-\vec{x}_{2}\right)
  43. x 1 \vec{x}_{1}
  44. x 2 \vec{x}_{2}
  45. a 1 a_{1}
  46. a 2 a_{2}
  47. W ( J ) = - d 4 x d 4 y J 1 ( x ) 1 2 [ D ( x - y ) + D ( y - x ) ] J 2 ( y ) W\left(J\right)=-\iint d^{4}x\;d^{4}y\;J_{1}\left(x\right){1\over 2}\left[D% \left(x-y\right)+D\left(y-x\right)\right]J_{2}\left(y\right)
  48. W ( J ) = - T a 1 a 2 d 3 k ( 2 π ) 3 D ( k ) | k 0 = 0 exp ( i k ( x 1 - x 2 ) ) W\left(J\right)=-Ta_{1}a_{2}\int{d^{3}k\over(2\pi)^{3}}\;\;D\left(k\right)\mid% _{k_{0}=0}\;\exp\left(i\vec{k}\cdot\left(\vec{x}_{1}-\vec{x}_{2}\right)\right)
  49. D ( k ) D\left(k\right)
  50. 1 2 [ D ( x - y ) + D ( y - x ) ] {1\over 2}\left[D\left(x-y\right)+D\left(y-x\right)\right]
  51. E = - W T = a 1 a 2 d 3 k ( 2 π ) 3 D ( k ) | k 0 = 0 exp ( i k ( x 1 - x 2 ) ) E=-{W\over T}=a_{1}a_{2}\int{d^{3}k\over(2\pi)^{3}}\;\;D\left(k\right)\mid_{k_% {0}=0}\;\exp\left(i\vec{k}\cdot\left(\vec{x}_{1}-\vec{x}_{2}\right)\right)
  52. [ φ ( x ) ] = 1 2 [ ( φ ) 2 - m 2 φ 2 ] \mathcal{L}[\varphi(x)]={1\over 2}\left[\left(\partial\varphi\right)^{2}-m^{2}% \varphi^{2}\right]
  53. 2 φ + m 2 φ = 0 \partial^{2}\varphi+m^{2}\varphi=0
  54. Z = D φ exp { i d 4 x [ 1 2 ( ( φ ) 2 - m 2 φ 2 ) + J φ ] } Z=\int D\varphi\;\exp\left\{i\int d^{4}x\;\left[{1\over 2}\left(\left(\partial% \varphi\right)^{2}-m^{2}\varphi^{2}\right)+J\varphi\right]\right\}
  55. Z = D φ exp { i d 4 x [ - 1 2 φ ( 2 + m 2 ) φ + J φ ] } Z=\int D\varphi\;\exp\left\{i\int d^{4}x\;\left[-{1\over 2}\varphi\left(% \partial^{2}+m^{2}\right)\varphi+J\varphi\right]\right\}
  56. - ( 2 + m 2 ) D ( x - y ) = δ 4 ( x - y ) -\left(\partial^{2}+m^{2}\right)D\left(x-y\right)=\delta^{4}\left(x-y\right)
  57. D ( k ) | k 0 = 0 = - 1 k 2 + m 2 D\left(k\right)\mid_{k_{0}=0}\;=\;-{1\over\vec{k}^{2}+m^{2}}
  58. E = - a 1 a 2 4 π r exp ( - m r ) E=-{a_{1}a_{2}\over 4\pi r}\exp\left(-mr\right)
  59. r 2 = ( x 1 - x 2 ) 2 r^{2}=\left(\vec{x}_{1}-\vec{x}_{2}\right)^{2}
  60. 1 m {1\over m}
  61. [ φ ( x ) ] = - 1 4 F μ ν F μ ν + 1 2 m 2 A μ A μ + A μ J μ \mathcal{L}[\varphi(x)]=-{1\over 4}F_{\mu\nu}F^{\mu\nu}+{1\over 2}m^{2}A_{\mu}% A^{\mu}+A_{\mu}J^{\mu}
  62. F μ ν = μ A ν - ν A μ F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}
  63. μ J μ = 0 \partial_{\mu}J^{\mu}=0
  64. μ A μ = 0 \partial_{\mu}A^{\mu}=0
  65. J 0 J^{0}
  66. - 1 4 d 4 x F μ ν F μ ν = - 1 4 d 4 x ( μ A ν - ν A μ ) ( μ A ν - ν A μ ) -{1\over 4}\int d^{4}xF_{\mu\nu}F^{\mu\nu}=-{1\over 4}\int d^{4}x\left(% \partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}\right)\left(\partial^{\mu}A^{\nu}-% \partial^{\nu}A^{\mu}\right)
  67. = 1 2 d 4 x A ν ( 2 A ν - ν μ A μ ) = 1 2 d 4 x A μ ( η μ ν 2 ) A ν , ={1\over 2}\int d^{4}x\;A_{\nu}\left(\partial^{2}A^{\nu}-\partial^{\nu}% \partial_{\mu}A^{\mu}\right)={1\over 2}\int d^{4}x\;A^{\mu}\left(\eta_{\mu\nu}% \partial^{2}\right)A^{\nu},
  68. η μ α ( 2 + m 2 ) D α ν ( x - y ) = δ μ ν δ 4 ( x - y ) \eta_{\mu\alpha}\left(\partial^{2}+m^{2}\right)D^{\alpha\nu}\left(x-y\right)=% \delta_{\mu}^{\nu}\delta^{4}\left(x-y\right)
  69. D μ ν ( k ) | k 0 = 0 = η μ ν 1 - k 2 + m 2 . D_{\mu\nu}\left(k\right)\mid_{k_{0}=0}\;=\;\eta_{\mu\nu}{1\over-k^{2}+m^{2}}.
  70. D ( k ) | k 0 = 0 = 1 k 2 + m 2 D\left(k\right)\mid_{k_{0}=0}\;=\;{1\over\vec{k}^{2}+m^{2}}
  71. E = a 1 a 2 4 π r exp ( - m r ) E={a_{1}a_{2}\over 4\pi r}\exp\left(-mr\right)
  72. E = a 1 a 2 4 π r . E={a_{1}a_{2}\over 4\pi r}.
  73. a 1 a_{1}
  74. a 2 a_{2}
  75. ω 2 = ω p 2 + γ ( ω ) T e m k 2 . \omega^{2}=\omega_{p}^{2}+\gamma\left(\omega\right){T_{e}\over m}\vec{k}^{2}.
  76. ω \omega
  77. ω p 2 = 4 π n e 2 m \omega_{p}^{2}={4\pi ne^{2}\over m}
  78. e e
  79. m m
  80. T e T_{e}
  81. γ ( ω ) \gamma\left(\omega\right)
  82. γ ( ω ) \gamma\left(\omega\right)
  83. γ ( ω ) \gamma\left(\omega\right)
  84. k 2 + k D 2 = 0 \vec{k}^{2}+\vec{k}_{D}^{2}=0
  85. k D 2 = 4 π n e 2 T e k_{D}^{2}={4\pi ne^{2}\over T_{e}}
  86. D ( k ) | k 0 = 0 = 1 k 2 + k D 2 D\left(k\right)\mid_{k_{0}=0}\;=\;{1\over\vec{k}^{2}+k_{D}^{2}}
  87. - k 0 2 + k 2 + k D 2 - m T e k 0 2 = 0 , -k_{0}^{2}+\vec{k}^{2}+k_{D}^{2}-{m\over T_{e}}k_{0}^{2}=0,
  88. E = a 1 a 2 4 π r exp ( - k D r ) . E={a_{1}a_{2}\over 4\pi r}\exp\left(-k_{D}r\right).
  89. E = a 1 a 2 4 π r exp ( - k s r ) E={a_{1}a_{2}\over 4\pi r}\exp\left(-k_{s}r\right)
  90. k s 2 = 6 π n e 2 ϵ F k_{s}^{2}={6\pi ne^{2}\over\epsilon_{F}}
  91. ϵ F \epsilon_{F}
  92. ϵ F = 2 2 m ( 3 π 2 n ) 2 / 3 . \epsilon_{F}=\frac{\hbar^{2}}{2m}\left({3\pi^{2}n}\right)^{2/3}\,.
  93. μ = - e φ + ϵ F \mu=-e\varphi+\epsilon_{F}
  94. φ \varphi
  95. J 1 ( x ) = a 1 L B 1 2 π r δ 2 ( r ) J_{1}\left(x\right)={a_{1}\over L_{B}}{1\over 2\pi r}\delta^{2}\left(r\right)
  96. r r
  97. L B L_{B}
  98. D ( k ) | k 0 = 0 = 1 k 2 + k D s 2 D\left(k\right)\mid_{k_{0}=0}\;=\;{1\over\vec{k}^{2}+k_{Ds}^{2}}
  99. k D s k_{Ds}
  100. E = ( a 1 a 2 2 π L B ) 0 k d k k 2 + k D s 2 𝒥 0 ( k r 12 ) = ( a 1 a 2 2 π L B ) K 0 ( k D s r 12 ) E=\left({a_{1}\,a_{2}\over 2\pi L_{B}}\right)\int_{0}^{\infty}{{k\;dk\;}\over k% ^{2}+k_{Ds}^{2}}\mathcal{J}_{0}\left(kr_{12}\right)=\left({a_{1}\,a_{2}\over 2% \pi L_{B}}\right)K_{0}\left(k_{Ds}r_{12}\right)
  101. 𝒥 n ( x ) \mathcal{J}_{n}\left(x\right)
  102. K 0 ( x ) K_{0}\left(x\right)
  103. r 12 r_{12}
  104. 0 2 π d φ 2 π exp ( i p cos ( φ ) ) = 𝒥 0 ( p ) \int_{0}^{2\pi}{d\varphi\over 2\pi}\exp\left(ip\cos\left(\varphi\right)\right)% =\mathcal{J}_{0}\left(p\right)
  105. 0 k d k k 2 + m 2 𝒥 0 ( k r ) = K 0 ( m r ) . \int_{0}^{\infty}{{k\;dk\;}\over k^{2}+m^{2}}\mathcal{J}_{0}\left(kr\right)=K_% {0}\left(mr\right).
  106. k D s r 12 1 k_{Ds}r_{12}<<1
  107. K 0 ( k D s r 12 ) - ln ( k D s r 12 2 ) + 0.5772. K_{0}\left(k_{Ds}r_{12}\right)\rightarrow-\ln\left({k_{Ds}r_{12}\over 2}\right% )+0.5772.
  108. J 1 ( x ) = a 1 L b 1 2 π r δ 2 ( r - r B 1 ) J_{1}\left(x\right)={a_{1}\over L_{b}}{1\over 2\pi r}\delta^{2}\left(r-r_{B1}\right)
  109. r r
  110. L B L_{B}
  111. r B 1 = 4 π m 1 v 1 a 1 B = 2 m 1 ω c r_{B1}={\sqrt{4\pi}m_{1}v_{1}\over a_{1}B}=\sqrt{2\hbar\over m_{1}\omega_{c}}
  112. ω c = a 1 B 4 π m 1 c \omega_{c}={a_{1}B\over\sqrt{4\pi}m_{1}c}
  113. v 1 = 2 ω c m 1 v_{1}=\sqrt{2\hbar\omega_{c}\over m_{1}}
  114. E = ( a 1 a 2 2 π L B ) 0 k d k D ( k ) | k 0 = k B = 0 𝒥 0 ( k r B 1 ) 𝒥 0 ( k r B 2 ) 𝒥 0 ( k r 12 ) E=\left({a_{1}\,a_{2}\over 2\pi L_{B}}\right)\int_{0}^{\infty}{k\;dk\;}D\left(% k\right)\mid_{k_{0}=k_{B}=0}\mathcal{J}_{0}\left(kr_{B1}\right)\mathcal{J}_{0}% \left(kr_{B2}\right)\mathcal{J}_{0}\left(kr_{12}\right)
  115. r 12 r_{12}
  116. 𝒥 n ( x ) \mathcal{J}_{n}\left(x\right)
  117. 0 2 π d φ 2 π exp ( i p cos ( φ ) ) = 𝒥 0 ( p ) . \int_{0}^{2\pi}{d\varphi\over 2\pi}\exp\left(ip\cos\left(\varphi\right)\right)% =\mathcal{J}_{0}\left(p\right).
  118. μ = - e φ + N ω c = N 0 ω c \mu=-e\varphi+N\hbar\omega_{c}=N_{0}\hbar\omega_{c}
  119. - e φ -e\varphi
  120. N 0 N_{0}
  121. N N
  122. δ n = e φ ω c A M L B \delta n={e\varphi\over\hbar\omega_{c}A_{M}L_{B}}
  123. A M A_{M}
  124. ( k 2 + k B 2 ) φ = 0 \left(k^{2}+k_{B}^{2}\right)\varphi=0
  125. k B 2 = 4 π e 2 ω c A M L B . k_{B}^{2}={4\pi e^{2}\over\hbar\omega_{c}A_{M}L_{B}}.
  126. D ( k ) | k 0 = k B = 0 = 1 k 2 + k B 2 D\left(k\right)\mid_{k_{0}=k_{B}=0}={1\over k^{2}+k_{B}^{2}}
  127. E = ( a 1 a 2 2 π L B ) 0 k d k k 2 + k B 2 𝒥 0 ( k r B 1 ) 𝒥 0 ( k r B 2 ) 𝒥 0 ( k r 12 ) = ( 2 e 2 L B ) 0 k d k k 2 + k B 2 r B 2 𝒥 0 2 ( k ) 𝒥 0 ( k r 12 r B ) E=\left({a_{1}\,a_{2}\over 2\pi L_{B}}\right)\int_{0}^{\infty}{{k\;dk\;}\over k% ^{2}+k_{B}^{2}}\mathcal{J}_{0}\left(kr_{B1}\right)\mathcal{J}_{0}\left(kr_{B2}% \right)\mathcal{J}_{0}\left(kr_{12}\right)=\left({2e^{2}\over L_{B}}\right)% \int_{0}^{\infty}{{k\;dk\;}\over k^{2}+k_{B}^{2}r_{B}^{2}}\mathcal{J}_{0}^{2}% \left(k\right)\mathcal{J}_{0}\left(k{r_{12}\over r_{B}}\right)
  128. r l = l r B l = 0 , 1 , 2 , r_{\mathit{l}}=\sqrt{\mathit{l}}\;r_{B}\;\;\;\mathit{l}=0,1,2,\ldots
  129. l \mathit{l}
  130. l = 1 \mathit{l}=1
  131. l > 0 \mathit{l}>0
  132. l l \mathit{l}^{\prime}\geq\mathit{l}
  133. r l r_{\mathit{l}}
  134. E = ( 2 e 2 L B ) 0 k d k k 2 + k B 2 r l 2 𝒥 0 ( k ) 𝒥 0 ( l l k ) 𝒥 0 ( k r 12 r l ) . E=\left({2e^{2}\over L_{B}}\right)\int_{0}^{\infty}{{k\;dk\;}\over k^{2}+k_{B}% ^{2}r_{\mathit{l}}^{2}}\;\mathcal{J}_{0}\left(k\right)\;\mathcal{J}_{0}\left(% \sqrt{{\mathit{l}^{\prime}}\over{\mathit{l}}}\;k\right)\;\mathcal{J}_{0}\left(% k{r_{12}\over r_{\mathit{l}}}\right).
  135. l = l \mathit{l}=\mathit{l}^{\prime}
  136. k B r l k_{B}r_{\mathit{l}}
  137. r 12 = r 𝑙𝑙 = l + l r B . r_{12}=r_{\mathit{l}\mathit{l}^{\prime}}=\sqrt{\mathit{l}+\mathit{l}^{\prime}}% \;r_{B}.
  138. r 𝑙𝑙 r_{\mathit{l}\mathit{l}^{\prime}}
  139. l + l \mathit{l}+\mathit{l}^{\prime}
  140. r 𝑙𝑙 r_{\mathit{l}\mathit{l}^{\prime}}
  141. E = ( 2 e 2 L B ) 0 k d k k 2 + k B 2 r 𝑙𝑙 2 𝒥 0 ( cos θ k ) 𝒥 0 ( sin θ k ) 𝒥 0 ( k r 12 r 𝑙𝑙 ) E=\left({2e^{2}\over L_{B}}\right)\int_{0}^{\infty}{{k\;dk\;}\over k^{2}+k_{B}% ^{2}r_{\mathit{l}\mathit{l}^{\prime}}^{2}}\;\mathcal{J}_{0}\left(\cos\theta\;k% \right)\;\mathcal{J}_{0}\left(\sin\theta\;k\right)\;\mathcal{J}_{0}\left(k{r_{% 12}\over r_{\mathit{l}\mathit{l}^{\prime}}}\right)
  142. tan θ = l l . \tan\theta=\sqrt{\mathit{l}\over\mathit{l}^{\prime}}.
  143. r 12 r_{12}
  144. r 12 = r 𝑙𝑙 r_{12}=r_{\mathit{l}\mathit{l}^{\prime}}
  145. tan θ = l l \tan\theta=\sqrt{\mathit{l}\over\mathit{l}^{\prime}}
  146. l l = 1. {\mathit{l}\over\mathit{l}^{\prime}}=1.
  147. l = l = 1 \mathit{l}=\mathit{l}^{\prime}=1
  148. l l * = 1 2 {\mathit{l}\over\mathit{l}^{*}}={1\over 2}
  149. l * = l + l . {\mathit{l}^{*}}={\mathit{l}}+{\mathit{l}^{\prime}}.
  150. l = l . {\mathit{l}=\mathit{l}^{\prime}}.
  151. l l * = l * ± 1 2 l * {\mathit{l}\over\mathit{l}^{*}}=\;{\mathit{l}^{*}\pm 1\over 2\mathit{l}^{*}}
  152. l l * = 1 3 , 2 5 , 3 7 , etc., {\mathit{l}\over\mathit{l}^{*}}={1\over 3},{2\over 5},{3\over 7},\mbox{etc.,}~{}
  153. l l * = 2 3 , 3 5 , 4 7 , etc., {\mathit{l}\over\mathit{l}^{*}}={2\over 3},{3\over 5},{4\over 7},\mbox{etc.,}~{}
  154. 1 π r B 2 L B 1 n ! ( r r B ) 2 l exp ( - r 2 r B 2 ) . {1\over\pi r_{B}^{2}L_{B}}{1\over n!}\left({r\over r_{B}}\right)^{2\mathit{l}}% \exp\left(-{r^{2}\over r_{B}^{2}}\right).
  155. E = ( 2 e 2 L B ) 0 k d k k 2 + k B 2 r B 2 M ( l + 1 , 1 , - k 2 4 ) M ( l + 1 , 1 , - k 2 4 ) 𝒥 0 ( k r 12 r B ) E=\left({2e^{2}\over L_{B}}\right)\int_{0}^{\infty}{{k\;dk\;}\over k^{2}+k_{B}% ^{2}r_{B}^{2}}\;M\left(\mathit{l}+1,1,-{k^{2}\over 4}\right)\;M\left(\mathit{l% }^{\prime}+1,1,-{k^{2}\over 4}\right)\;\mathcal{J}_{0}\left(k{r_{12}\over r_{B% }}\right)
  156. M M
  157. 2 n ! 0 d r r 2 n + 1 exp ( - r 2 ) J 0 ( k r ) = M ( n + 1 , 1 , - k 2 4 ) . {2\over n!}\int_{0}^{\infty}{dr}\;r^{2n+1}\exp\left(-r^{2}\right)J_{0}\left(kr% \right)=M\left(n+1,1,-{k^{2}\over 4}\right).
  158. r 12 r_{12}
  159. l l * = 1 3 , 2 5 , 3 7 , etc., {\mathit{l}\over\mathit{l}^{*}}={1\over 3},{2\over 5},{3\over 7},\mbox{etc.,}~{}
  160. l l * = 2 3 , 3 5 , 4 7 , etc., {\mathit{l}\over\mathit{l}^{*}}={2\over 3},{3\over 5},{4\over 7},\mbox{etc.,}~{}
  161. J 1 ( x ) = a 1 v 1 δ 3 ( x - x 1 ) \vec{J}_{1}\left(\vec{x}\right)=a_{1}\vec{v}_{1}\delta^{3}\left(\vec{x}-\vec{x% }_{1}\right)
  162. J 2 \vec{J}_{2}
  163. J 1 ( k ) = a 1 v 1 exp ( i k x 1 ) . \vec{J}_{1}\left(\vec{k}\right)=a_{1}\vec{v}_{1}\exp\left(i\vec{k}\cdot\vec{x}% _{1}\right).
  164. J 1 ( k ) = a 1 [ 1 - k ^ k ^ ] v 1 exp ( i k x 1 ) + a 1 [ k ^ k ^ ] v 1 exp ( i k x 1 ) . \vec{J}_{1}\left(\vec{k}\right)=a_{1}\left[1-\hat{k}\hat{k}\right]\cdot\vec{v}% _{1}\exp\left(i\vec{k}\cdot\vec{x}_{1}\right)+a_{1}\left[\hat{k}\hat{k}\right]% \cdot\vec{v}_{1}\exp\left(i\vec{k}\cdot\vec{x}_{1}\right).
  165. k J = - k 0 J 0 0 , \vec{k}\cdot\vec{J}=-k_{0}J^{0}\rightarrow 0,
  166. k 0 k_{0}
  167. E = a 1 a 2 d 3 k ( 2 π ) 3 D ( k ) | k 0 = 0 v 1 [ 1 - k ^ k ^ ] v 2 exp ( i k ( x 1 - x 2 ) ) E=a_{1}a_{2}\int{d^{3}k\over(2\pi)^{3}}\;\;D\left(k\right)\mid_{k_{0}=0}\;\vec% {v}_{1}\cdot\left[1-\hat{k}\hat{k}\right]\cdot\vec{v}_{2}\;\exp\left(i\vec{k}% \cdot\left(x_{1}-x_{2}\right)\right)
  168. η μ α ( 2 + m 2 ) D α ν ( x - y ) = δ μ ν δ 4 ( x - y ) . \eta_{\mu\alpha}\left(\partial^{2}+m^{2}\right)D^{\alpha\nu}\left(x-y\right)=% \delta_{\mu}^{\nu}\delta^{4}\left(x-y\right).
  169. D ( k ) | k 0 = 0 = - 1 k 2 + m 2 , D\left(k\right)\mid_{k_{0}=0}\;=\;-{1\over\vec{k}^{2}+m^{2}},
  170. E = - a 1 a 2 d 3 k ( 2 π ) 3 v 1 [ 1 - k ^ k ^ ] v 2 k 2 + m 2 exp ( i k ( x 1 - x 2 ) ) E=-a_{1}a_{2}\int{d^{3}k\over(2\pi)^{3}}\;\;{\vec{v}_{1}\cdot\left[1-\hat{k}% \hat{k}\right]\cdot\vec{v}_{2}\over\vec{k}^{2}+m^{2}}\;\exp\left(i\vec{k}\cdot% \left(x_{1}-x_{2}\right)\right)
  171. E = - 1 2 a 1 a 2 4 π r e - m r { 2 ( m r ) 2 ( e m r - 1 ) - 2 m r } v 1 [ 1 + r ^ r ^ ] v 2 E=-{1\over 2}{a_{1}a_{2}\over 4\pi r}e^{-mr}\left\{{2\over\left(mr\right)^{2}}% \left(e^{mr}-1\right)-{2\over mr}\right\}\vec{v}_{1}\cdot\left[1+{\hat{r}}{% \hat{r}}\right]\cdot\vec{v}_{2}
  172. E = - 1 2 a 1 a 2 4 π r v 1 [ 1 + r ^ r ^ ] v 2 E=-{1\over 2}{a_{1}a_{2}\over 4\pi r}\vec{v}_{1}\cdot\left[1+{\hat{r}}{\hat{r}% }\right]\cdot\vec{v}_{2}
  173. c = 1 c=1
  174. k 0 2 = ω p 2 + k 2 , k_{0}^{2}=\omega_{p}^{2}+\vec{k}^{2},
  175. D ( k ) | k 0 = 0 = - 1 k 2 + ω p 2 . D\left(k\right)\mid_{k_{0}=0}\;=\;-{1\over\vec{k}^{2}+\omega_{p}^{2}}.
  176. ω p \omega_{p}
  177. E = - 1 2 a 1 a 2 4 π r v 1 [ 1 + r ^ r ^ ] v 2 e - ω p r { 2 ( ω p r ) 2 ( e ω p r - 1 ) - 2 ω p r } . E=-{1\over 2}{a_{1}a_{2}\over 4\pi r}\vec{v}_{1}\cdot\left[1+{\hat{r}}{\hat{r}% }\right]\cdot\vec{v}_{2}\;e^{-\omega_{p}r}\left\{{2\over\left(\omega_{p}r% \right)^{2}}\left(e^{\omega_{p}r}-1\right)-{2\over\omega_{p}r}\right\}.
  178. J 1 ( x ) = a 1 v 1 1 2 π r L B δ 2 ( r - r B 1 ) ( b ^ × r ^ ) \vec{J}_{1}\left(\vec{x}\right)=a_{1}v_{1}{1\over 2\pi rL_{B}}\;\delta^{2}% \left(r-r_{B1}\right)\;\left({\hat{b}\times\hat{r}}\right)
  179. r B 1 = 4 π m 1 v 1 a 1 B r_{B1}={\sqrt{4\pi}m_{1}v_{1}\over a_{1}B}
  180. b ^ \hat{b}
  181. L B L_{B}
  182. E = ( a 1 a 2 2 π L B ) v 1 v 2 0 k d k D ( k ) | k 0 = k B = 0 𝒥 1 ( k r B 1 ) 𝒥 1 ( k r B 2 ) 𝒥 0 ( k r 12 ) E=\left({a_{1}\,a_{2}\over 2\pi L_{B}}\right)v_{1}\,v_{2}\,\int_{0}^{\infty}{k% \;dk\;}D\left(k\right)\mid_{k_{0}=k_{B}=0}\mathcal{J}_{1}\left(kr_{B1}\right)% \mathcal{J}_{1}\left(kr_{B2}\right)\mathcal{J}_{0}\left(kr_{12}\right)
  183. r 12 r_{12}
  184. 𝒥 n ( x ) \mathcal{J}_{n}\left(x\right)
  185. 0 2 π d φ 2 π exp ( i p cos ( φ ) ) = 𝒥 0 ( p ) \int_{0}^{2\pi}{d\varphi\over 2\pi}\exp\left(ip\cos\left(\varphi\right)\right)% =\mathcal{J}_{0}\left(p\right)
  186. 0 2 π d φ 2 π cos ( φ ) exp ( i p cos ( φ ) ) = i 𝒥 1 ( p ) . \int_{0}^{2\pi}{d\varphi\over 2\pi}\cos\left(\varphi\right)\exp\left(ip\cos% \left(\varphi\right)\right)=i\mathcal{J}_{1}\left(p\right).
  187. - k 0 2 + k 2 + ω p 2 ( k 0 2 - ω p 2 ) ( k 0 2 - ω H 2 ) = 0 , -k_{0}^{2}+\vec{k}^{2}+\omega_{p}^{2}{\left(k_{0}^{2}-\omega_{p}^{2}\right)% \over\left(k_{0}^{2}-\omega_{H}^{2}\right)}=0,
  188. D ( k ) | k 0 = k B = 0 = - ( 1 k 2 + k X 2 ) D\left(k\right)\mid_{k_{0}=k_{B}=0}\;=\;-\left({1\over\vec{k}^{2}+k_{X}^{2}}\right)
  189. k X ω p 2 ω H k_{X}\equiv{\omega_{p}^{2}\over\omega_{H}}
  190. ω H 2 = ω p 2 + ω c 2 , \omega_{H}^{2}=\omega_{p}^{2}+\omega_{c}^{2},
  191. ω c = e B m c , \omega_{c}={eB\over mc},
  192. ω p 2 = 4 π n e 2 m . \omega_{p}^{2}={4\pi ne^{2}\over m}.
  193. E = - ( a 2 2 π L B ) v 2 0 k d k k 2 + k X 2 𝒥 1 2 ( k r B ) 𝒥 0 ( k r 12 ) E=-\left({a^{2}\over 2\pi L_{B}}\right)v^{2}\,\int_{0}^{\infty}{k\;dk\over\vec% {k}^{2}+k_{X}^{2}}\mathcal{J}_{1}^{2}\left(kr_{B}\right)\mathcal{J}_{0}\left(% kr_{12}\right)
  194. E = - E 0 I 1 ( μ ) K 1 ( μ ) E=-E_{0}\;I_{1}\left(\mu\right)K_{1}\left(\mu\right)
  195. E 0 = ( a 2 2 π L B ) v 2 E_{0}=\left({a^{2}\over 2\pi L_{B}}\right)v^{2}
  196. μ = ω p 2 r B ω H = k X r B \mu={\omega_{p}^{2}r_{B}\over\omega_{H}}=k_{X}\;r_{B}
  197. o k d k k 2 + m 2 𝒥 1 2 ( k r ) = I 1 ( m r ) K 1 ( m r ) . \int_{o}^{\infty}{k\;dk\over k^{2}+m^{2}}\mathcal{J}_{1}^{2}\left(kr\right)=I_% {1}\left(mr\right)K_{1}\left(mr\right).
  198. I 1 ( m r ) K 1 ( m r ) 1 2 [ 1 - 1 8 ( m r ) 2 ] . I_{1}\left(mr\right)K_{1}\left(mr\right)\rightarrow{1\over 2}\left[1-{1\over 8% }\left(mr\right)^{2}\right].
  199. I 1 ( m r ) K 1 ( m r ) 1 2 ( 1 m r ) . I_{1}\left(mr\right)K_{1}\left(mr\right)\rightarrow{1\over 2}\;\left({1\over mr% }\right).
  200. μ = ω p 2 r B ω H c = ( 2 e 2 r B L B c ) ν 1 + ω p 2 ω c 2 = 2 α ( r B L B ) ( 1 1 + ω p 2 ω c 2 ) ν \mu={\omega_{p}^{2}r_{B}\over\omega_{H}c}=\left({2e^{2}r_{B}\over L_{B}\hbar c% }\right){\nu\over\sqrt{1+{\omega_{p}^{2}\over\omega_{c}^{2}}}}=2\alpha\left({r% _{B}\over L_{B}}\right)\left({1\over\sqrt{1+{\omega_{p}^{2}\over\omega_{c}^{2}% }}}\right)\nu
  201. α \alpha
  202. ν = 2 π N c e B A \nu={2\pi N\hbar c\over eBA}
  203. μ \mu
  204. E = - E 0 2 [ 1 - 1 8 μ 2 ] E=-{E_{0}\over 2}\left[1-{1\over 8}\mu^{2}\right]
  205. E 0 = 4 π e 2 L B v 2 c 2 = 8 π e 2 L B ( ω c m c 2 ) E_{0}={4\pi}{e^{2}\over L_{B}}{v^{2}\over c^{2}}={8\pi}{e^{2}\over L_{B}}\left% ({\hbar\omega_{c}\over mc^{2}}\right)
  206. 1 2 m v 2 = ω c . {1\over 2}mv^{2}=\hbar\omega_{c}.
  207. T μ ν T^{\mu\nu}
  208. 00 00
  209. D ( k ) | k 0 = 0 = - 4 3 1 k 2 + m 2 D\left(k\right)\mid_{k_{0}=0}\;=\;-{4\over 3}{1\over\vec{k}^{2}+m^{2}}
  210. E = - 4 3 a 1 a 2 4 π r exp ( - m r ) E=-{4\over 3}{a_{1}a_{2}\over 4\pi r}\exp\left(-mr\right)

Stationary_subspace_analysis.html

  1. x ( t ) x(t)
  2. s 𝔰 ( t ) s^{\mathfrak{s}}(t)
  3. s 𝔫 ( t ) s^{\mathfrak{n}}(t)
  4. x ( t ) = A s ( t ) = [ A 𝔰 A 𝔫 ] [ s 𝔰 ( t ) s 𝔫 ( t ) ] , x(t)=As(t)=\begin{bmatrix}A^{\mathfrak{s}}&A^{\mathfrak{n}}\end{bmatrix}\begin% {bmatrix}s^{\mathfrak{s}}(t)\\ s^{\mathfrak{n}}(t)\\ \end{bmatrix},
  5. A A
  6. A 𝔰 A^{\mathfrak{s}}
  7. A 𝔫 A^{\mathfrak{n}}
  8. x ( t ) x(t)
  9. A - 1 A^{-1}
  10. x ( t ) x(t)
  11. s 𝔰 ( t ) s^{\mathfrak{s}}(t)
  12. A 𝔫 A^{\mathfrak{n}}
  13. s 𝔫 ( t ) s^{\mathfrak{n}}(t)
  14. A 𝔰 A^{\mathfrak{s}}

Steerable_filter.html

  1. x x
  2. y y

Stefan_adhesion.html

  1. F F
  2. R R
  3. η \eta
  4. h h
  5. t t
  6. F = 3 π R 4 2 h ( t ) 3 d h d t η . F=\frac{3\pi R^{4}}{2h(t)^{3}}\frac{dh}{dt}\eta\,.

Stella_octangula_number.html

  1. m 2 = n ( 2 n 2 - 1 ) m^{2}=n(2n^{2}-1)
  2. x 2 = y 3 - 2 y x^{2}=y^{3}-2y
  3. X = m / n X=m/\sqrt{n}
  4. Y = n Y=\sqrt{n}
  5. X 2 = 2 Y 4 - 1 X^{2}=2Y^{4}-1

Step_detection.html

  1. H [ m ] = i = 1 N j = 1 N Λ ( x i - m j , m i - m j , x i - x j , i - j ) H[m]=\sum_{i=1}^{N}\sum_{j=1}^{N}\Lambda(x_{i}-m_{j},m_{i}-m_{j},x_{i}-x_{j},i% -j)
  2. Λ \scriptstyle\Lambda
  3. Λ = 1 2 | x i - m j | 2 I ( i - j = 0 ) + γ | m i - m j | I ( i - j = 1 ) \Lambda=\frac{1}{2}\left|x_{i}-m_{j}\right|^{2}I(i-j=0)+\gamma\left|m_{i}-m_{j% }\right|I(i-j=1)
  4. γ \gamma
  5. Λ = min { 1 2 | m i - m j | 2 , W } \Lambda=\min\left\{\frac{1}{2}\left|m_{i}-m_{j}\right|^{2},W\right\}
  6. Λ = 1 - exp ( - β | m i - m j | 2 / 2 ) β I ( | i - j | W ) \Lambda=\frac{1-\exp(-\beta|m_{i}-m_{j}|^{2}/2)}{\beta}\cdot I(|i-j|\leq W)
  7. β > 0 \scriptstyle\beta>0
  8. Λ = 1 2 | x i - m j | 2 I ( i - j = 0 ) + γ | m i - m j | 0 I ( i - j = 1 ) \Lambda=\frac{1}{2}\left|x_{i}-m_{j}\right|^{2}I(i-j=0)+\gamma\left|m_{i}-m_{j% }\right|^{0}I(i-j=1)
  9. | x | 0 \scriptstyle\left|x\right|^{0}
  10. Λ \scriptstyle\Lambda
  11. u * = arg min u \R N γ u 0 + u - x p p u^{*}=\arg\min_{u\in\R^{N}}\gamma\|\nabla u\|_{0}+\|u-x\|_{p}^{p}
  12. u 0 = # { i : u i u i + 1 } \|\nabla u\|_{0}=\#\{i:u_{i}\neq u_{i+1}\}
  13. u - x p p = i = 1 N | u i - x i | p \|u-x\|_{p}^{p}=\sum_{i=1}^{N}|u_{i}-x_{i}|^{p}
  14. u * u^{*}
  15. u * \nabla u^{*}
  16. p = 2 p=2
  17. p = 1 p=1
  18. O ( N 2 ) O(N^{2})

Stereoscopic_acuity.html

  1. d γ = c a d z / z 2 d\gamma=cadz/z^{2}

Stericated_5-cubes.html

  1. ( ± 1 , ± 1 , ± 1 , ± 1 , ± ( 1 + 2 ) ) \left(\pm 1,\ \pm 1,\ \pm 1,\ \pm 1,\ \pm(1+\sqrt{2})\right)
  2. ( ± 1 , ± ( 1 + 2 ) , ± ( 1 + 2 ) , ± ( 1 + 2 ) , ± ( 1 + 2 2 ) ) \left(\pm 1,\ \pm(1+\sqrt{2}),\ \pm(1+\sqrt{2}),\ \pm(1+\sqrt{2}),\ \pm(1+2% \sqrt{2})\right)
  3. ( ± 1 , ± 1 , ± 1 , ± ( 1 + 2 ) , ± ( 1 + 2 2 ) ) \left(\pm 1,\ \pm 1,\ \pm 1,\ \pm(1+\sqrt{2}),\ \pm(1+2\sqrt{2})\right)
  4. ( 1 , 1 + 2 , 1 + 2 2 , 1 + 2 2 , 1 + 3 2 ) \left(1,\ 1+\sqrt{2},\ 1+2\sqrt{2},\ 1+2\sqrt{2},\ 1+3\sqrt{2}\right)
  5. ( 1 , 1 + 2 , 1 + 1 2 , 1 + 2 2 , 1 + 3 2 ) \left(1,\ 1+\sqrt{2},\ 1+1\sqrt{2},\ 1+2\sqrt{2},\ 1+3\sqrt{2}\right)
  6. ( ± 1 , ± 1 , ± 1 , ± 1 , ± ( 1 + 2 ) ) \left(\pm 1,\ \pm 1,\ \pm 1,\ \pm 1,\ \pm(1+\sqrt{2})\right)
  7. ( 1 , 1 , 1 + 2 , 1 + 2 2 , 1 + 3 2 ) \left(1,\ 1,\ 1+\sqrt{2},\ 1+2\sqrt{2},\ 1+3\sqrt{2}\right)
  8. ( 1 , 1 + 2 , 1 + 2 2 , 1 + 3 2 , 1 + 4 2 ) \left(1,\ 1+\sqrt{2},\ 1+2\sqrt{2},\ 1+3\sqrt{2},\ 1+4\sqrt{2}\right)

Stevo_Todorčević.html

  1. 2 0 = 2 2^{\aleph_{0}}=\aleph_{2}

Stochastic_discount_factor.html

  1. p 1 , , p n p_{1},...,p_{n}
  2. x ~ 1 , , x ~ n \tilde{x}_{1},...,\tilde{x}_{n}
  3. m ~ \tilde{m}
  4. E ( m ~ x ~ i ) = p i , i . E(\tilde{m}\tilde{x}_{i})=p_{i},\quad\forall i.
  5. x ~ i \tilde{x}_{i}
  6. m ~ \tilde{m}
  7. p i p_{i}
  8. R i = x ~ i / p i R_{i}=\tilde{x}_{i}/p_{i}
  9. E ( m ~ R ~ i ) = 1 , i , E(\tilde{m}\tilde{R}_{i})=1,\quad\forall i,
  10. E [ m ~ ( R ~ i - R ~ j ) ] = 0 , i , j . E[\tilde{m}(\tilde{R}_{i}-\tilde{R}_{j})]=0,\quad\forall i,j.
  11. E ( m ~ x ~ ) = p , E ( m ~ R ~ ) = 1. E(\tilde{m}\tilde{x})=p,E(\tilde{m}\tilde{R})=1.
  12. 1 = c o v ( m ~ , R ~ ) + E ( m ~ ) E ( R ~ ) . 1=cov(\tilde{m},\tilde{R})+E(\tilde{m})E(\tilde{R}).
  13. R ~ = R f \tilde{R}=R_{f}
  14. E ( m ~ ) = 1 / R f E(\tilde{m})=1/R_{f}
  15. R ~ \tilde{R}
  16. E ( R ~ ) - R f = - R f c o v ( m ~ , R ~ ) . E(\tilde{R})-R_{f}=-R_{f}cov(\tilde{m},\tilde{R}).
  17. E ( m ~ x ~ i ) E(\tilde{m}\,\tilde{x}_{i})
  18. m ~ \tilde{m}

Stochastic_equicontinuity.html

  1. { H n ( θ ) : n 1 } \{H_{n}(\theta):n\geq 1\}
  2. Θ \reals \Theta\rightarrow\reals
  3. Θ \Theta
  4. { H n ( θ ) } \{H_{n}(\theta)\}
  5. { H n ( θ ) } \{H_{n}(\theta)\}
  6. { H n } \{H_{n}\}
  7. ϵ > 0 \epsilon>0
  8. η > 0 \eta>0
  9. δ > 0 \delta>0
  10. lim sup n Pr ( sup θ Θ sup θ B ( θ , δ ) | H n ( θ ) - H n ( θ ) | > ϵ ) < η . \limsup_{n\rightarrow\infty}\Pr\left(\sup_{\theta\in\Theta}\sup_{\theta^{% \prime}\in B(\theta,\delta)}|H_{n}(\theta^{\prime})-H_{n}(\theta)|>\epsilon% \right)<\eta.

Stochastic_measurement_procedure.html

  1. 𝔇 \mathfrak{D}
  2. C D ( β ) C_{D}^{(\beta)}
  3. { x } \{x\}
  4. C D ( β ) ( { x } ) C_{D}^{(\beta)}(\{x\})
  5. β \beta
  6. β \beta
  7. C D ( β ) ( { x } ) C_{D}^{(\beta)}(\{x\})
  8. β \beta
  9. β \beta
  10. β = 1.0 \beta=1.0
  11. β \beta
  12. β \beta
  13. C D ( β ) ( { x } ) C_{D}^{(\beta)}(\{x\})
  14. { x } \{x\}
  15. A X ( β ) A_{X}^{(\beta)}
  16. β \beta
  17. C D ( β ) C_{D}^{(\beta)}
  18. A X ( β ) A_{X}^{(\beta)}
  19. C D ( { x } ) = { d | x ϵ A X ( β ) ( { d } ) } C_{D}(\{x\})=\{d|x\epsilon A_{X}^{(\beta)}(\{d\})\}
  20. C D ( { x } ) C_{D}(\{x\})
  21. { x } \{x\}
  22. 𝔅 X , D \mathfrak{B}_{X,D}
  23. A X ( β ) A_{X}^{(\beta)}
  24. β \beta
  25. A X ( β ) ( { d } ) A_{X}^{(\beta)}(\{d\})
  26. d ϵ 𝔇 d\epsilon\mathfrak{D}
  27. { x } \{x\}
  28. A X ( β ) ( { d } ) A_{X}^{(\beta)}(\{d\})

Stochastic_prediction_procedure.html

  1. A X ( β ) A^{(\beta)}_{X}
  2. A X ( β ) A^{(\beta)}_{X}
  3. A X ( β ) A^{(\beta)}_{X}
  4. β \beta
  5. β \beta
  6. A X ( β ) A^{(\beta)}_{X}
  7. A X ( β ) A^{(\beta)}_{X}
  8. A X ( β ) A_{X}^{(\beta)}
  9. A X ( β ) A_{X}^{(\beta)}
  10. 𝔛 \mathfrak{X}
  11. 𝔛 \mathfrak{X}
  12. β \beta

Stock_correlation_network.html

  1. C i j Cij
  2. i i
  3. j j
  4. d i j dij
  5. d i j dij
  6. ( 2 ( 1 - C i j ) ) 0.5 (2(1-Cij))^{0.5}
  7. d i j dij
  8. i i
  9. j j
  10. C i j Cij

Stomatal_conductance.html

  1. C v L - C v 1 R v s + R 1 = C v 1 - C v 2 R 2 \frac{C_{vL}-C_{v1}}{R_{vs}+R_{1}}=\frac{C_{v1}-C_{v2}}{R_{2}}
  2. C v L C_{vL}
  3. C v 1 C_{v1}
  4. C v 2 C_{v2}
  5. R v s R_{vs}
  6. R 1 R_{1}
  7. R 2 R_{2}
  8. R v s = 1 - h 1 h 2 - h 1 R 2 - R 1 R_{vs}=\frac{1-h_{1}}{h_{2}-h_{1}}R_{2}-R_{1}
  9. g v s = 1 R v s g_{vs}=\frac{1}{R_{vs}}
  10. R R
  11. Δ t = ( R + A ) l Δ h 1 - h \Delta t=\frac{\left(R+A\right)l\Delta h}{1-h}
  12. t t
  13. h h
  14. h h
  15. l l
  16. A A
  17. R v s = A f ( 1 h - 1 ) - R v a R_{vs}=\frac{A}{f}\left(\frac{1}{h}-1\right)-R_{va}
  18. R v s R_{vs}
  19. R v a R_{va}
  20. A A
  21. f f
  22. h h
  23. g g
  24. g = g 0 + a 1 A n ( c s - Γ ) ( 1 + D s D 0 ) g=g_{0}+\frac{a_{1}A_{n}}{(c_{s}-\Gamma)(1+\frac{D_{s}}{D_{0}})}
  25. g g
  26. g 0 g_{0}
  27. g g
  28. A n A_{n}
  29. D s D_{s}
  30. c s c_{s}
  31. Γ \Gamma
  32. a 1 a1
  33. D 0 D0

Stoner–Wohlfarth_model.html

  1. 𝐌 \mathbf{M}
  2. 𝐇 \mathbf{H}
  3. h h
  4. φ φ
  5. θ θ
  6. V V
  7. η = E 2 K u V = 1 4 - 1 4 cos ( 2 ( ϕ - θ ) ) - h cos ϕ , \eta=\frac{E}{2K_{u}V}=\frac{1}{4}-\frac{1}{4}\cos\left(2\left(\phi-\theta% \right)\right)-h\cos\phi,\,
  8. η ϕ = 1 2 sin ( 2 ( ϕ - θ ) ) + h sin ϕ = 0. \frac{\partial\eta}{\partial\phi}=\frac{1}{2}\sin\left(2\left(\phi-\theta% \right)\right)+h\sin\phi=0.\,
  9. 2 η ϕ 2 = cos ( 2 ( ϕ - θ ) ) + h cos ϕ > 0. \frac{\partial^{2}\eta}{\partial\phi^{2}}=\cos\left(2\left(\phi-\theta\right)% \right)+h\cos\phi>0.\,
  10. θ θ
  11. φ φ
  12. h h
  13. φ φ
  14. 0
  15. π π
  16. φ φ
  17. π π
  18. 2 π
  19. φ = 0 φ=0
  20. π π
  21. h = ± h=±∞
  22. h h
  23. 1 / 2 h 1 / 2 −1/2≤h≤1/2
  24. h h
  25. h = 0.5 h=0.5
  26. h > 0 h>0
  27. h = 0.5 h=−0.5
  28. h = 0.5 h=0.5
  29. θ = 0 θ=0
  30. θ θ
  31. h s = ( 1 - t 2 + t 4 ) 1 / 2 1 + t 2 , h_{s}=\frac{\left(1-t^{2}+t^{4}\right)^{1/2}}{1+t^{2}},\,
  32. t = tan 1 / 3 θ . t=\tan^{1/3}\theta.\,
  33. 𝐡 \mathbf{h}
  34. h 2 / 3 + h 2 / 3 = 1. h_{\parallel}^{2/3}+h_{\perp}^{2/3}=1.\,
  35. 0.5 Align l t ; | h | Align l t ; 1 0.5&lt;|h|&lt;1
  36. h = 0.5 h=0.5
  37. 0.5 0.5
  38. h = 1 h=1
  39. 1 1
  40. 0.5 0.5
  41. M r s / M s M_{rs}/M_{s}
  42. 0.5 0.5
  43. H c / 2 K u H_{c}/2K_{u}
  44. 0.479 0.479
  45. H c r / 2 K u H_{cr}/2K_{u}
  46. 0.524 0.524
  47. χ 0 / 2 K u \chi_{0}/2K_{u}
  48. 2 / 3 2/3
  49. H c r / H c H_{cr}/H_{c}
  50. 1.09 1.09
  51. M a f ( H ) = M r s - M i r ( H ) M d f ( H ) = M r s - 2 M i r ( H ) . \begin{aligned}\displaystyle M_{af}(H)&\displaystyle=M_{rs}-M_{ir}(H)\\ \displaystyle M_{df}(H)&\displaystyle=M_{rs}-2M_{ir}(H)\end{aligned}.\,

Straightening_theorem_for_vector_fields.html

  1. X X
  2. y 1 , , y n y_{1},\dots,y_{n}
  3. X = / y 1 X=\partial/\partial y_{1}
  4. X X
  5. n \mathbb{R}^{n}
  6. X = j f j ( x ) x j X=\sum_{j}f_{j}(x){\partial\over\partial x_{j}}
  7. x x
  8. 0
  9. f = ( f 1 , , f n ) f=(f_{1},\dots,f_{n})
  10. f ( 0 ) = ( 1 , 0 , , 0 ) . f(0)=(1,0,\dots,0).
  11. Φ ( t , p ) \Phi(t,p)
  12. x ˙ = f ( x ) , x ( 0 ) = p \dot{x}=f(x),x(0)=p
  13. ψ ( x 1 , , x n ) = Φ ( x 1 , ( 0 , x 2 , , x n ) ) . \psi(x_{1},\dots,x_{n})=\Phi(x_{1},(0,x_{2},\dots,x_{n})).
  14. Φ \Phi
  15. ψ \psi
  16. x 1 ψ ( x ) = f ( ψ ( x ) ) {\partial\over\partial x_{1}}\psi(x)=f(\psi(x))
  17. ψ ( 0 , x 2 , , x n ) = Φ ( 0 , ( 0 , x 2 , , x n ) ) = ( 0 , x 2 , , x n ) \psi(0,x_{2},\dots,x_{n})=\Phi(0,(0,x_{2},\dots,x_{n}))=(0,x_{2},\dots,x_{n})
  18. d ψ d\psi
  19. 0
  20. y = ψ - 1 ( x ) y=\psi^{-1}(x)
  21. 0
  22. x = ψ ( y ) x=\psi(y)
  23. x j y 1 = f j ( ψ ( y ) ) = f j ( x ) {\partial x_{j}\over\partial y_{1}}=f_{j}(\psi(y))=f_{j}(x)
  24. y 1 = X {\partial\over\partial y_{1}}=X

Strange–Rahman–Smith_equation.html

  1. Δ v \Delta v
  2. x x
  3. x + Δ x x+\Delta\,x
  4. T T
  5. T + Δ T T+\Delta T
  6. d v d x = d v k G T d T x 2 \frac{dv}{dx}=\frac{dv\,k_{GT}}{d\,T\,x^{2}}
  7. k G T k_{GT}

Stratifold.html

  1. X X\to\mathbb{R}
  2. g ( f 1 , , f n ) : X g\circ(f_{1},\dots,f_{n}):X\to\mathbb{R}
  3. g : n g:\mathbb{R}^{n}\to\mathbb{R}
  4. f i C f_{i}\in C
  5. T x X T_{x}X
  6. X i = { x X : T x X X_{i}=\{x\in X\colon T_{x}X
  7. } \}
  8. M n = M M_{n}=M
  9. ( S i , C | S i ) (S_{i},C|_{S_{i}})
  10. C x C ( S i ) x C_{x}\to C^{\infty}(S_{i})_{x}
  11. ρ : U \R \rho\colon U\to\R
  12. ρ ( x ) 0 \rho(x)\neq 0
  13. supp ( ρ ) U \,\text{supp}(\rho)\subset U
  14. ( T , T ) (T,\partial T)
  15. T - T T-\partial T
  16. T \partial T
  17. S , S X S,S^{\prime}\to X
  18. S X S\to X
  19. S H k X SH_{k}X
  20. S H k ( point ) = 0 SH_{k}(\,\text{point})=0
  21. S H 0 ( point ) SH_{0}(\,\text{point})\cong\mathbb{Z}
  22. S H k ( X ) H k ( X ) SH_{k}(X)\cong H_{k}(X)
  23. S H k G ( X ) SH_{k}^{G}(X)
  24. S H k G ( X ) H k - dim ( G ) G ( X ) SH_{k}^{G}(X)\cong H_{k-\dim(G)}^{G}(X)
  25. Ω O ( point ) / 2 [ t ] \Omega^{O}(\,\text{point})\to\mathbb{Z}/2[t]
  26. Ω S O ( point ) [ t ] \Omega^{SO}(\,\text{point})\to\mathbb{Z}[t]
  27. i : N M i:N\hookrightarrow M
  28. H k ( M ) H k + dim ( N ) - dim ( M ) ( N ) H_{k}(M)\to H_{k+\dim(N)-\dim(M)}(N)
  29. x H k ( M ) x\in H_{k}(M)
  30. f : S M f:S\to M
  31. H k + dim ( N ) - dim ( M ) ( N ) H_{k+\dim(N)-\dim(M)}(N)

Stream_power_law.html

  1. E = K A m S n E=KA^{m}S^{n}

Streeter–Phelps_equation.html

  1. D t = k 1 L t - k 2 D \frac{\partial D}{\partial t}=k_{1}L_{t}-k_{2}D
  2. D = k 1 L a k 2 - k 1 ( e - k 1 t - e - k 2 t ) + D a e - k 2 t D=\frac{k_{1}L_{a}}{k_{2}-k_{1}}(e^{-k_{1}t}-e^{-k_{2}t})+D_{a}e^{-k_{2}t}
  3. D D
  4. D = D O s a t - D O D=DO_{sat}-DO
  5. D D
  6. g m 3 \tfrac{\mathrm{g}}{\mathrm{m}^{3}}
  7. k 1 k_{1}
  8. d - 1 d^{-1}
  9. k 2 k_{2}
  10. d - 1 d^{-1}
  11. L a L_{a}
  12. L a L_{a}
  13. g m 3 \tfrac{\mathrm{g}}{\mathrm{m}^{3}}
  14. L t L_{t}
  15. L t = L a e - k 1 t L_{t}=L_{a}e^{-k_{1}t}
  16. D a D_{a}
  17. [ g m 3 ] [\tfrac{\mathrm{g}}{\mathrm{m}^{3}}]
  18. t t
  19. [ d ] [d]
  20. k 1 k_{1}
  21. d - 1 d^{-1}
  22. k 2 k_{2}
  23. d - 1 d^{-1}
  24. t c r i t = 1 k 2 - k 1 ln [ k 2 k 1 ( 1 - D a ( k 2 - k 1 ) L a k 1 ) ] t_{crit}=\frac{1}{k_{2}-k_{1}}\ln{\left[\frac{k_{2}}{k_{1}}\left({1-\frac{D_{a% }(k_{2}-k_{1})}{L_{a}k_{1}}}\right)\right]}
  25. D c r i t D_{crit}
  26. t c r i t t_{crit}
  27. D O c r i t = D O s a t - D c r i t DO_{crit}=DO_{sat}-D_{crit}
  28. D O c r i t DO_{crit}
  29. D O c r i t DO_{crit}
  30. x c r i t = v t c r i t x_{crit}=vt_{crit}
  31. v v
  32. k 2 = K v a H - b k_{2}=Kv^{a}H^{-b}
  33. K K
  34. v v
  35. H H
  36. a a
  37. b b
  38. k 2 = 2.148 v 0.878 H - 1.48 k_{2}=2.148v^{0.878}H^{-1.48}
  39. k 2 k_{2}
  40. [ d - 1 ] [d^{-1}]
  41. v v
  42. H H
  43. k 1 k_{1}
  44. k 2 k_{2}
  45. k = k 20 θ ( T - 20 ) k=k_{20}\theta^{(T-20)}
  46. k 20 k_{20}
  47. T T
  48. k 1 k_{1}
  49. k 2 k_{2}
  50. D c r i t D_{crit}
  51. x c r i t x_{crit}
  52. D O s a t DO_{sat}
  53. L a = L s Q s + L b Q b Q s + Q b L_{a}=\frac{L_{s}Q_{s}+L_{b}Q_{b}}{Q_{s}+Q_{b}}
  54. D O 0 = D O s Q s + D O b Q b Q s + Q b DO_{0}=\frac{DO_{s}Q_{s}+DO_{b}Q_{b}}{Q_{s}+Q_{b}}
  55. L a L_{a}
  56. L a L_{a}
  57. g m 3 \tfrac{\mathrm{g}}{\mathrm{m}^{3}}
  58. L b L_{b}
  59. [ g m 3 ] [\tfrac{\mathrm{g}}{\mathrm{m}^{3}}]
  60. L s L_{s}
  61. [ g m 3 ] [\tfrac{\mathrm{g}}{\mathrm{m}^{3}}]
  62. D O 0 DO_{0}
  63. [ g m 3 ] [\tfrac{\mathrm{g}}{\mathrm{m}^{3}}]
  64. D O b DO_{b}
  65. [ g m 3 ] [\tfrac{\mathrm{g}}{\mathrm{m}^{3}}]
  66. D O s DO_{s}
  67. [ g m 3 ] [\tfrac{\mathrm{g}}{\mathrm{m}^{3}}]
  68. Q b Q_{b}
  69. [ m 3 s ] [\tfrac{m^{3}}{s}]
  70. Q s Q_{s}
  71. [ m 3 s ] [\tfrac{m^{3}}{s}]
  72. d L d t = - k 1 L + L b \frac{dL}{dt}=-k_{1}L+L_{b}
  73. k 1 k_{1}
  74. d - 1 d^{-1}
  75. L L
  76. [ g m 3 ] [\tfrac{\mathrm{g}}{\mathrm{m}^{3}}]
  77. L b L_{b}
  78. [ g m 3 ] [\tfrac{\mathrm{g}}{\mathrm{m}^{3}}]
  79. k r = k 1 + k 3 k_{r}=k_{1}+k_{3}
  80. k 1 k_{1}
  81. d - 1 d^{-1}
  82. k 3 k_{3}
  83. d - 1 d^{-1}
  84. d L d t = - k r L \frac{dL}{dt}=-k_{r}L
  85. L L
  86. [ g m 3 ] [\tfrac{\mathrm{g}}{\mathrm{m}^{3}}]
  87. k r k_{r}
  88. d - 1 d^{-1}
  89. d D d t = - k 2 D + S H \frac{dD}{dt}=-k_{2}D+\frac{S}{H}
  90. H H
  91. S S
  92. [ g m 2 d ] [\tfrac{g}{m^{2}d}]
  93. [ g m 3 ] [\tfrac{\mathrm{g}}{\mathrm{m}^{3}}]
  94. k 2 k_{2}
  95. d - 1 d^{-1}
  96. g m 2 d \tfrac{g}{m^{2}d}
  97. g m 2 d \tfrac{g}{m^{2}d}
  98. d D d t = k N N - k 2 D \frac{dD}{dt}=k_{N}N-k_{2}D
  99. k N k_{N}
  100. [ d - 1 ] [d^{-1}]
  101. N N
  102. k N k_{N}
  103. d - 1 d^{-1}
  104. d D d t = - k 2 D + ( R + P ) a v g \frac{dD}{dt}=-k_{2}D+(R+P)_{avg}
  105. R R
  106. [ m g L d ] [\tfrac{mg}{Ld}]
  107. P P
  108. [ m g L d ] [\tfrac{mg}{Ld}]
  109. P ( t ) = P m a x s i n ( π f + ( t - t s ) ) P(t)=P_{max}sin\left(\frac{\pi}{f}+(t-t_{s})\right)
  110. P ( t ) P(t)
  111. [ m g L d ] [\tfrac{mg}{Ld}]
  112. P m a x P_{max}
  113. [ m g L d ] [\tfrac{mg}{Ld}]
  114. f f
  115. 1 2 \tfrac{1}{2}
  116. t s t_{s}
  117. [ d ] [d]
  118. ( P - R ) (P-R)
  119. m g L d \tfrac{mg}{Ld}

Strength_(mathematical_logic).html

  1. α \alpha
  2. β \beta
  3. β \beta
  4. α \alpha

Stretched_grid_method.html

  1. n \ n
  2. Π = D j = 1 n R j 2 \Pi=D\sum_{j=1}^{n}{R_{j}}^{2}
  3. n \ n
  4. R j \ R_{j}
  5. j \ j
  6. D \ D
  7. j \ j
  8. R = ( X 12 - X 11 ) 2 + ( X 22 - X 21 ) 2 \ R=\sqrt{(X_{12}-X_{11})^{2}+(X_{22}-X_{21})^{2}}
  9. { X } \{\ X\}
  10. { X / } \{\ X^{/}\}
  11. { X } \{\ X\}
  12. { X } = { X / } + { Δ X } \{\ X\}=\{\ X^{/}\}+\{\Delta\ X\}
  13. { X } \{\ X\}
  14. Π \ \Pi
  15. { Δ X } \{\Delta\ X\}
  16. Π Δ X k l = 0 \frac{\partial\Pi}{\partial\Delta X_{kl}}=0
  17. l \ l
  18. k \ k
  19. [ A ] { Δ X 1 } = { B 1 } [\ A]\{\Delta X_{1}\}=\{\ B_{1}\}
  20. [ A ] { Δ X 2 } = { B 2 } [\ A]\{\Delta X_{2}\}=\{\ B_{2}\}
  21. [ A ] [\ A]
  22. { Δ X 1 } \{\Delta\ X_{1}\}
  23. { Δ X 2 } \{\Delta\ X_{2}\}
  24. { B 1 } \{\ B_{1}\}
  25. { B 2 } \{\ B_{2}\}
  26. [ A ] { Δ X 3 } = { B 3 } [\ A]\{\Delta X_{3}\}=\{\ B_{3}\}
  27. Π \ \Pi
  28. j = 1 , 2 , 3 \ j=1,2,3
  29. Π = j = 1 n D j R j 2 + i = 1 3 ( k = 1 m C i k Δ X i k 2 - k = 1 m P i k Δ X i k ) \Pi=\sum_{j=1}^{n}D_{j}R_{j}^{2}+\sum_{i=1}^{3}\left(\sum_{k=1}^{m}C_{ik}% \Delta X_{ik}^{2}-\sum_{k=1}^{m}P_{ik}\Delta X_{ik}\right)
  30. n \ n
  31. m \ m
  32. R j \ R_{j}
  33. j \ j
  34. D j \ D_{j}
  35. j \ j
  36. Δ X i k \ \Delta X_{ik}
  37. k \ k
  38. i \ i
  39. C i k \ C_{ik}
  40. k \ k
  41. i \ i
  42. P i k \ P_{ik}
  43. k \ k
  44. i \ i
  45. I 1 = E 1 ( u , v ) d u 2 + 2 F 1 ( u , v ) d u d v + G 1 ( u , v ) d v 2 I_{1}=E_{1}(u,v)\operatorname{d}u^{2}+2F_{1}(u,v)\operatorname{d}u% \operatorname{d}v+G_{1}(u,v)\operatorname{d}v^{2}
  46. I 2 = E 2 ( u , v ) d u 2 + 2 F 2 ( u , v ) d u d v + G 2 ( u , v ) d v 2 I_{2}=E_{2}(u,v)\operatorname{d}u^{2}+2F_{2}(u,v)\operatorname{d}u% \operatorname{d}v+G_{2}(u,v)\operatorname{d}v^{2}
  47. I 2 = λ I 1 \sqrt{I_{2}}=\lambda\sqrt{I_{1}}
  48. λ \ \lambda
  49. ( u , v ) \ (u,v)
  50. ( u + d u , v + d v ) \ (u+\operatorname{d}u,v+\operatorname{d}v)
  51. λ \ \lambda
  52. Π = D j = 1 n S j w j ( λ I 1 - I 2 ) 2 d s \Pi=D\sum_{j=1}^{n}\oint_{S_{j}}w_{j}\left(\lambda\sqrt{I_{1}}-\sqrt{I_{2}}% \right)^{2}\operatorname{d}s
  53. n \ n
  54. w j \ w_{j}
  55. Π \ \Pi
  56. D \ D
  57. w j = 1 \ w_{j}=1
  58. Π = D j = 1 n S j w j ( λ R j - L j ) 2 d s \Pi=D\sum_{j=1}^{n}\oint_{S_{j}}w_{j}\left(\lambda R_{j}-L_{j}\right)^{2}% \operatorname{d}s
  59. R j \ R_{j}
  60. j \ j
  61. L j \ L_{j}
  62. j \ j
  63. λ \ \lambda
  64. j \ j
  65. R = ( X 12 - X 11 ) 2 + ( X 22 - X 21 ) 2 + ( X 32 - X 31 ) 2 R=\sqrt{(X_{12}-X_{11})^{2}+(X_{22}-X_{21})^{2}+(X_{32}-X_{31})^{2}}
  66. L = ( x 12 - x 11 ) 2 + ( x 22 - x 21 ) 2 L=\sqrt{(x_{12}-x_{11})^{2}+(x_{22}-x_{21})^{2}}
  67. X i k \ X_{ik}
  68. x i k \ x_{ik}
  69. x 32 = x 31 = 0 \ x_{32}=x_{31}=0
  70. { x } \{\ x\}
  71. { X } \{\ X\}
  72. { x } = { X } + { Δ X } \{\ x\}=\{\ X\}+\{\Delta\ X\}
  73. { Δ X } \{\Delta\ X\}
  74. Π Δ X k l = 0 \frac{\partial\Pi}{\partial\Delta X_{kl}}=0
  75. [ A ] { Δ X 1 } = { B 1 } + { Δ P 1 } [\ A]\{\Delta X_{1}\}=\{\ B_{1}\}+\{\Delta P_{1}\}
  76. [ A ] { Δ X 2 } = { B 2 } + { Δ P 2 } [\ A]\{\Delta X_{2}\}=\{\ B_{2}\}+\{\Delta P_{2}\}
  77. { Δ P 1 } \{\ \Delta P_{1}\}
  78. { Δ P 2 } \{\ \Delta P_{2}\}
  79. { Δ P l t } = - { j = 1 N λ R m L m ( x l m - x l t ) } \{\Delta P_{lt}\}=-\left\{\sum_{j=1}^{N}\lambda\frac{R_{m}}{L_{m}}(x_{lm}-x_{% lt})\right\}
  80. N \ N
  81. t \ t
  82. l \ l

Strictly_singular_operator.html

  1. T : l p l q T:l_{p}\to l_{q}
  2. 1 q , p < 1\leq q,p<\infty
  3. p q p\neq q
  4. l p l_{p}
  5. l q l_{q}
  6. T : c 0 l p T:c_{0}\to l_{p}
  7. T : l p c 0 T:l_{p}\to c_{0}
  8. 1 p < 1\leq p<\infty
  9. c 0 c_{0}

Strictly_standardized_mean_difference.html

  1. β \beta
  2. μ 1 \mu_{1}
  3. σ 1 2 \sigma_{1}^{2}
  4. μ 2 \mu_{2}
  5. σ 2 2 \sigma_{2}^{2}
  6. σ 12 . \sigma_{12}.
  7. β = μ 1 - μ 2 σ 1 2 + σ 2 2 - 2 σ 12 . \beta=\frac{\mu_{1}-\mu_{2}}{\sqrt{\sigma_{1}^{2}+\sigma_{2}^{2}-2\sigma_{12}}}.
  8. β = μ 1 - μ 2 σ 1 2 + σ 2 2 . \beta=\frac{\mu_{1}-\mu_{2}}{\sqrt{\sigma_{1}^{2}+\sigma_{2}^{2}}}.
  9. σ 2 \sigma^{2}
  10. β = μ 1 - μ 2 2 σ . \beta=\frac{\mu_{1}-\mu_{2}}{\sqrt{2}\sigma}.
  11. σ 12 \sigma_{12}
  12. D D
  13. μ D \mu_{D}
  14. σ D 2 \sigma_{D}^{2}
  15. β = μ D σ D . \beta=\frac{\mu_{D}}{\sigma_{D}}.
  16. X ¯ 1 , X ¯ 2 \bar{X}_{1},\bar{X}_{2}
  17. s 1 2 , s 2 2 s_{1}^{2},s_{2}^{2}
  18. β ^ = X ¯ 1 - X ¯ 2 s 1 2 + s 2 2 . \hat{\beta}=\frac{\bar{X}_{1}-\bar{X}_{2}}{\sqrt{s_{1}^{2}+s_{2}^{2}}}.
  19. β ^ = X ¯ 1 - X ¯ 2 2 K ( ( n 1 - 1 ) s 1 2 + ( n 2 - 1 ) s 2 2 ) , \hat{\beta}=\frac{\bar{X}_{1}-\bar{X}_{2}}{\sqrt{\frac{2}{K}((n_{1}-1)s_{1}^{2% }+(n_{2}-1)s_{2}^{2})}},
  20. n 1 , n 2 n_{1},n_{2}
  21. K n 1 + n 2 - 3.48 K\approx n_{1}+n_{2}-3.48
  22. n n
  23. D ¯ \bar{D}
  24. s D 2 s_{D}^{2}
  25. β ^ = D ¯ s D . \hat{\beta}=\frac{\bar{D}}{s_{D}}.
  26. β ^ = Γ ( n - 1 2 ) Γ ( n - 2 2 ) 2 n - 1 D ¯ s D . \hat{\beta}=\frac{\Gamma(\frac{n-1}{2})}{\Gamma(\frac{n-2}{2})}\sqrt{\frac{2}{% n-1}}\frac{\bar{D}}{s_{D}}.
  27. X ¯ P , X ¯ N \bar{X}_{P},\bar{X}_{N}
  28. s P 2 , s N 2 s_{P}^{2},s_{N}^{2}
  29. n P , n N n_{P},n_{N}
  30. β ^ = X ¯ P - X ¯ N 2 K ( ( n P - 1 ) s P 2 + ( n N - 1 ) s N 2 ) , \hat{\beta}=\frac{\bar{X}_{P}-\bar{X}_{N}}{\sqrt{\frac{2}{K}((n_{P}-1)s_{P}^{2% }+(n_{N}-1)s_{N}^{2})}},
  31. K n P + n N - 3.48 K\approx n_{P}+n_{N}-3.48
  32. β ^ = X ¯ P - X ¯ N s P 2 + s N 2 . \hat{\beta}=\frac{\bar{X}_{P}-\bar{X}_{N}}{\sqrt{s_{P}^{2}+s_{N}^{2}}}.
  33. β ^ = X ~ P - X ~ N 1.4826 s ~ P 2 + s ~ N 2 , \hat{\beta}=\frac{\tilde{X}_{P}-\tilde{X}_{N}}{1.4826\sqrt{\tilde{s}_{P}^{2}+% \tilde{s}_{N}^{2}}},
  34. X ~ P , X ~ N , s ~ P , s ~ N \tilde{X}_{P},\tilde{X}_{N},\tilde{s}_{P},\tilde{s}_{N}
  35. β - 2 \beta\leq-2
  36. β - 3 \beta\leq-3
  37. β - 5 \beta\leq-5
  38. β - 7 \beta\leq-7
  39. - 2 < β - 1 -2<\beta\leq-1
  40. - 3 < β - 2 -3<\beta\leq-2
  41. - 5 < β - 3 -5<\beta\leq-3
  42. - 7 < β - 5 -7<\beta\leq-5
  43. - 1 < β - 0.5 -1<\beta\leq-0.5
  44. - 2 < β - 1 -2<\beta\leq-1
  45. - 3 < β - 2 -3<\beta\leq-2
  46. - 5 < β - 3 -5<\beta\leq-3
  47. β > - 0.5 \beta>-0.5
  48. β > - 1 \beta>-1
  49. β > - 2 \beta>-2
  50. β > - 3 \beta>-3
  51. β \beta
  52. β - 5 \beta\leq-5
  53. β 5 \beta\geq 5
  54. - 5 < β - 3 -5<\beta\leq-3
  55. 5 > β 3 5>\beta\geq 3
  56. - 3 < β - 2 -3<\beta\leq-2
  57. 3 > β 2 3>\beta\geq 2
  58. - 2 < β - 1.645 -2<\beta\leq-1.645
  59. 2 > β 1.645 2>\beta\geq 1.645
  60. - 1.645 < β - 1.28 -1.645<\beta\leq-1.28
  61. 1.645 > β 1.28 1.645>\beta\geq 1.28
  62. - 1.28 < β - 1 -1.28<\beta\leq-1
  63. 1.28 > β 1 1.28>\beta\geq 1
  64. - 1 < β - 0.75 -1<\beta\leq-0.75
  65. 1 > β 0.75 1>\beta\geq 0.75
  66. - 0.75 < β < - 0.5 -0.75<\beta<-0.5
  67. 0.75 > β > 0.5 0.75>\beta>0.5
  68. - 0.5 β < - 0.25 -0.5\leq\beta<-0.25
  69. 0.5 β > 0.25 0.5\geq\beta>0.25
  70. - 0.25 β < 0 -0.25\leq\beta<0
  71. 0.25 β > 0 0.25\geq\beta>0
  72. β = 0 \beta=0
  73. X i X_{i}
  74. n N n_{N}
  75. X ¯ N \bar{X}_{N}
  76. X ~ N \tilde{X}_{N}
  77. s N s_{N}
  78. s ~ N \tilde{s}_{N}
  79. SSMD = X i - X ¯ N s N 2 ( n N - 1 ) / K , \,\text{SSMD}=\frac{X_{i}-\bar{X}_{N}}{s_{N}\sqrt{2(n_{N}-1)/K}},
  80. K n N - 2.48 K\approx n_{N}-2.48
  81. SSMD* = X i - X ~ N 1.4826 s ~ N 2 ( n N - 1 ) / K \,\text{SSMD*}=\frac{X_{i}-\tilde{X}_{N}}{1.4826\tilde{s}_{N}\sqrt{2(n_{N}-1)/% K}}
  82. n n
  83. d ¯ i \bar{d}_{i}
  84. s i 2 s_{i}^{2}
  85. SSMD = Γ ( n - 1 2 ) Γ ( n - 2 2 ) 2 n - 1 d ¯ i s i \,\text{SSMD}=\frac{\Gamma(\frac{n-1}{2})}{\Gamma(\frac{n-2}{2})}\sqrt{\frac{2% }{n-1}}\frac{\bar{d}_{i}}{s_{i}}

String_girdling_Earth.html

  1. 1 2 π \tfrac{1}{2\pi}
  2. 2 π {2\pi}

String_group.html

  1. 0 K ( Z , 2 ) String ( n ) Spin ( n ) 0 0\rightarrow K(Z,2)\rightarrow\,\text{String}(n)\rightarrow\,\text{Spin}(n)\rightarrow 0

String_kernel.html

  1. D D
  2. K : D × D K:D\times D\rightarrow\mathbb{R}
  3. K K
  4. K ( x , y ) = φ ( x ) φ ( y ) K(x,y)=\varphi(x)\cdot\varphi(y)
  5. φ \varphi
  6. Σ \Sigma
  7. φ u : { Σ n Σ n s 𝐢 : u = s 𝐢 λ l ( 𝐢 ) \varphi_{u}:\left\{\begin{array}[]{l}\Sigma^{n}\rightarrow\mathbb{R}^{\Sigma^{% n}}\\ s\mapsto\sum_{\mathbf{i}:u=s_{\mathbf{i}}}\lambda^{l(\mathbf{i})}\end{array}\right.
  8. 𝐢 \mathbf{i}
  9. u u
  10. n n
  11. λ \lambda
  12. 0
  13. 1 1
  14. ϕ ( x ) \phi(x)

String_topology.html

  1. x H p ( M ) x\in H_{p}(M)
  2. y H q ( M ) y\in H_{q}(M)
  3. x × y H p + q ( M × M ) x\times y\in H_{p+q}(M\times M)
  4. M M × M M\hookrightarrow M\times M
  5. H p + q - d ( M ) H_{p+q-d}(M)
  6. Ω X \Omega X
  7. m : Ω X × Ω X Ω X m:\Omega X\times\Omega X\to\Omega X
  8. S 1 S^{1}
  9. γ : Map ( 8 , M ) L M \gamma:{\rm Map}(8,M)\to LM
  10. L M × L M LM\times LM
  11. γ \gamma
  12. x H p ( L M ) x\in H_{p}(LM)
  13. y H q ( L M ) y\in H_{q}(LM)
  14. x × y x\times y
  15. H p + q ( L M × L M ) H_{p+q}(LM\times LM)
  16. i ! : H p + q ( L M × L M ) H p + q - d ( Map ( 8 , M ) ) . i^{!}:H_{p+q}(LM\times LM)\to H_{p+q-d}({\rm Map}(8,M)).
  17. M a p ( 8 , M ) L M × L M Map(8,M)\subset LM\times LM
  18. γ \gamma
  19. H p + q - d ( L M ) H_{p+q-d}(LM)
  20. H H
  21. L n M = Map ( S n , M ) L^{n}M={\rm Map}(S^{n},M)
  22. h * ( Map ( N , M ) ) \mathcal{}h_{*}({\rm Map}(N,M))
  23. h * L n M \mathcal{}h_{*}L^{n}M
  24. e v : L M M ev:LM\to M
  25. Ω M \Omega M
  26. L E L B LE\to LB
  27. E B E\to B
  28. S 1 × L M L M S^{1}\times LM\to LM
  29. H * ( S 1 ) H * ( L M ) H * ( L M ) H_{*}(S^{1})\otimes H_{*}(LM)\to H_{*}(LM)
  30. [ S 1 ] H 1 ( S 1 ) [S^{1}]\in H_{1}(S^{1})
  31. Δ : H * ( L M ) H * + 1 ( L M ) \Delta:H_{*}(LM)\to H_{*+1}(LM)
  32. H * ( L M ) \mathcal{}H_{*}(LM)
  33. n 1 n\geq 1
  34. H * ( L M ) p H * ( L M ) q H_{*}(LM)^{\otimes p}\to H_{*}(LM)^{\otimes q}
  35. ( H * ( L M ) , H * ( L M ) ) \mathcal{}(H_{*}(LM),H_{*}(LM))
  36. n 1 n\geq 1
  37. H * ( L M ) p H * ( L M ) q H_{*}(LM)^{\otimes p}\to H_{*}(LM)^{\otimes q}

Strong_duality.html

  1. F = F * * F=F^{**}
  2. F F
  3. F * * F^{**}
  4. F F

Strongly_measurable_functions.html

  1. ( X , Y ) \mathcal{L}(X,Y)
  2. x X x\in X

Strömgren_integral.html

  1. 15 4 π 4 0 x t 7 e 2 t ( e t - 1 ) 3 d t . \frac{15}{4\pi^{4}}\int_{0}^{x}\frac{t^{7}e^{2t}}{(e^{t}-1)^{3}}\,dt.

Structural_acoustics.html

  1. 2 w x 2 = 1 c L 2 2 w t 2 {\partial^{2}w\over\partial x^{2}}={1\over c_{L}^{2}}{\partial^{2}w\over% \partial t^{2}}
  2. w w
  3. c L c_{L}
  4. c L c_{L}
  5. B B
  6. ρ \rho
  7. c L = B ρ {c_{L}}={\sqrt{B\over\rho}}
  8. E E
  9. B B
  10. 2 w x 2 = 1 c s 2 2 w t 2 {\partial^{2}w\over\partial x^{2}}={1\over c_{s}^{2}}{\partial^{2}w\over% \partial t^{2}}
  11. G G
  12. E E
  13. B B
  14. [ - ω 2 M + j ω B + ( 1 + j η ) K ] d = F {[-\omega^{2}{M}+j\omega{B}+(1+j\eta){K}]}{{d}={F}}
  15. p ( r , θ ) = j ω ρ 0 a 2 v n J 1 ( k a sin θ ) k a sin θ e j k r r {p(r,\theta)}={j\omega\rho_{0}a^{2}v_{n}{J_{1}(ka\sin\theta)\over ka\sin\theta% }{e^{jkr}\over r}}
  16. v n v_{n}
  17. J 1 J_{1}
  18. P r a d = R 0 < | v | > 2 {P_{rad}}={R_{0}<|v|>^{2}}
  19. < | v | Align g t ; <|v|&gt;

Structure_of_liquids_and_glasses.html

  1. S ( q ) - 1 = 4 π ρ q 0 [ g ( r ) - 1 ] sin ( q r ) d r S(q)-1=\frac{4\pi\rho}{q}\int_{0}^{\infty}[g(r)-1]\sin{(qr)}{d}r
  2. g ( r ) - 1 = 1 2 π ρ r 0 [ S ( q ) - 1 ] sin ( q r ) d q g(r)-1=\frac{1}{2\pi\rho r}\int_{0}^{\infty}[S(q)-1]\sin{(qr)}{d}q
  3. g ( r ) = n ( r ) ρ 4 π r 2 Δ r g(r)=\frac{n(r)}{\rho 4\pi r^{2}\Delta r}
  4. g a b ( r ) = 1 N a N b i = 1 N a j = 1 N b δ ( | 𝐫 i j | - r ) g_{ab}(r)=\frac{1}{N_{a}N_{b}}\sum\limits_{i=1}^{N_{a}}\sum\limits_{j=1}^{N_{b% }}\langle\delta(|\mathbf{r}_{ij}|-r)\rangle

Structured_derivations.html

  1. \bullet
  2. 3 x + 6 = 16 - x 3x+6=16-x
  3. \bullet
  4. 3 x + 6 = 16 - x 3x+6=16-x
  5. \Leftrightarrow
  6. 3 x + 6 - 6 = 16 - x - 6 3x+6-6=16-x-6
  7. \Leftrightarrow
  8. 3 x + 6 - 6 + x = 16 - x - 6 + x 3x+6-6+x=16-x-6+x
  9. \Leftrightarrow
  10. 4 x = 10 4x=10
  11. \Leftrightarrow
  12. x = 2.5 x=2.5
  13. \square
  14. \Vdash
  15. \bullet
  16. 0.04 m 0.04m
  17. 0.72 m 0.72m
  18. \Vdash
  19. = =
  20. 0.04 m 0.72 m \dfrac{0.04m}{0.72m}
  21. = =
  22. 1 18 \dfrac{1}{18}
  23. = =
  24. \approx
  25. 5.6 % 5.6\%
  26. \square
  27. \ldots
  28. \bullet
  29. \Vdash
  30. = =
  31. salt mass total mass \dfrac{\,\text{salt mass}}{\,\text{total mass}}
  32. = =
  33. \bullet
  34. = =
  35. 0.04 m 0.04m
  36. \ldots
  37. 0.04 m total mass \dfrac{0.04m}{\,\text{total mass}}
  38. = =
  39. \bullet
  40. = =
  41. ( 100 - 28 % ) m (100-28\%)\cdot m
  42. = =
  43. 0.72 m 0.72m
  44. \ldots
  45. 0.04 m 0.72 m \dfrac{0.04m}{0.72m}
  46. = =
  47. 1 18 \dfrac{1}{18}
  48. \approx
  49. 5.6 % 5.6\%
  50. \square

Stuart_number.html

  1. N = B 2 L c σ ρ U = Ha 2 Re \mathrm{N}=\frac{B^{2}L_{c}\sigma}{\rho U}=\frac{\mathrm{Ha}^{2}}{\mathrm{Re}}

Studie_II.html

  1. 5 25 \sqrt[25]{5}
  2. 2 12 : 1 \sqrt[12]{2}:1

Stuttering_equivalence.html

  1. π \pi
  2. π \pi^{\prime}
  3. π s t π \pi\sim_{st}\pi^{\prime}
  4. π \pi
  5. π \pi^{\prime}
  6. k th k^{\mathrm{th}}
  7. L ( \sdot ) L(\sdot)
  8. k th k^{\mathrm{th}}
  9. π = s 0 , s 1 , \pi=s_{0},s_{1},\ldots
  10. π = r 0 , r 1 , \pi^{\prime}=r_{0},r_{1},\ldots
  11. π s t π \pi\sim_{st}\pi^{\prime}
  12. 0 = i 0 < i 1 < i 2 < 0=i_{0}<i_{1}<i_{2}<\ldots
  13. 0 = j 0 < j 1 < j 2 < 0=j_{0}<j_{1}<j_{2}<\ldots
  14. k 0 k\geq 0
  15. L ( s i k ) = L ( s i k + 1 ) = = L ( s i k + 1 - 1 ) = L ( r j k ) = L ( r j k + 1 ) = = L ( r j k + 1 - 1 ) L(s_{i_{k}})=L(s_{i_{k}+1})=\ldots=L(s_{i_{k+1}-1})=L(r_{j_{k}})=L(r_{j_{k}+1}% )=\ldots=L(r_{j_{k+1}-1})

Submodular_set_function.html

  1. Ω \Omega
  2. f : 2 Ω f:2^{\Omega}\rightarrow\mathbb{R}
  3. 2 Ω 2^{\Omega}
  4. Ω \Omega
  5. X , Y Ω X,Y\subseteq\Omega
  6. X Y X\subseteq Y
  7. x Ω \ Y x\in\Omega\backslash Y
  8. f ( X { x } ) - f ( X ) f ( Y { x } ) - f ( Y ) f(X\cup\{x\})-f(X)\geq f(Y\cup\{x\})-f(Y)
  9. S , T Ω S,T\subseteq\Omega
  10. f ( S ) + f ( T ) f ( S T ) + f ( S T ) f(S)+f(T)\geq f(S\cup T)+f(S\cap T)
  11. X Ω X\subseteq\Omega
  12. x 1 , x 2 Ω \ X x_{1},x_{2}\in\Omega\backslash X
  13. f ( X { x 1 } ) + f ( X { x 2 } ) f ( X { x 1 , x 2 } ) + f ( X ) f(X\cup\{x_{1}\})+f(X\cup\{x_{2}\})\geq f(X\cup\{x_{1},x_{2}\})+f(X)
  14. f f
  15. T S T\subseteq S
  16. f ( T ) f ( S ) f(T)\leq f(S)
  17. f ( S ) = i S w i f(S)=\sum_{i\in S}w_{i}
  18. i , w i 0 \forall i,w_{i}\geq 0
  19. f ( S ) = min ( B , i S w i ) f(S)=\min(B,\sum_{i\in S}w_{i})
  20. w i 0 w_{i}\geq 0
  21. B 0 B\geq 0
  22. Ω = { E 1 , E 2 , , E n } \Omega=\{E_{1},E_{2},\ldots,E_{n}\}
  23. Ω \Omega^{\prime}
  24. f ( S ) = | E i S E i | f(S)=|\cup_{E_{i}\in S}E_{i}|
  25. S Ω S\subseteq\Omega
  26. Ω = { X 1 , X 2 , , X n } \Omega=\{X_{1},X_{2},\ldots,X_{n}\}
  27. S Ω S\subseteq\Omega
  28. H ( S ) H(S)
  29. H ( S ) H(S)
  30. S S
  31. Ω = { e 1 , e 2 , , e n } \Omega=\{e_{1},e_{2},\dots,e_{n}\}
  32. f f
  33. S Ω S\subseteq\Omega
  34. f ( S ) = f ( Ω - S ) f(S)=f(\Omega-S)
  35. Ω = { v 1 , v 2 , , v n } \Omega=\{v_{1},v_{2},\dots,v_{n}\}
  36. S Ω S\subseteq\Omega
  37. f ( S ) f(S)
  38. e = ( u , v ) e=(u,v)
  39. u S u\in S
  40. v Ω - S v\in\Omega-S
  41. Ω = { X 1 , X 2 , , X n } \Omega=\{X_{1},X_{2},\ldots,X_{n}\}
  42. S Ω S\subseteq\Omega
  43. f ( S ) = I ( S ; Ω - S ) f(S)=I(S;\Omega-S)
  44. I ( S ; Ω - S ) I(S;\Omega-S)
  45. Ω = { v 1 , v 2 , , v n } \Omega=\{v_{1},v_{2},\dots,v_{n}\}
  46. S Ω S\subseteq\Omega
  47. f ( S ) f(S)
  48. e = ( u , v ) e=(u,v)
  49. u S u\in S
  50. v Ω - S v\in\Omega-S
  51. x = { x 1 , x 2 , , x n } {x}=\{x_{1},x_{2},\dots,x_{n}\}
  52. 0 x i 1 0\leq x_{i}\leq 1
  53. f L ( x ) = 𝔼 ( f ( { i | x i λ } ) ) f^{L}({x})=\mathbb{E}(f(\{i|x_{i}\geq\lambda\}))
  54. λ \lambda
  55. [ 0 , 1 ] [0,1]
  56. x = { x 1 , x 2 , , x n } {x}=\{x_{1},x_{2},\ldots,x_{n}\}
  57. 0 x i 1 0\leq x_{i}\leq 1
  58. F ( x ) = S Ω f ( S ) i S x i i S ( 1 - x i ) F({x})=\sum_{S\subseteq\Omega}f(S)\prod_{i\in S}x_{i}\prod_{i\notin S}(1-x_{i})
  59. x = { x 1 , x 2 , , x n } {x}=\{x_{1},x_{2},\dots,x_{n}\}
  60. 0 x i 1 0\leq x_{i}\leq 1
  61. f - ( x ) = min ( S α S f ( S ) : S α S 1 S = x , S α S = 1 , α S 0 ) f^{-}({x})=\min(\sum_{S}\alpha_{S}f(S):\sum_{S}\alpha_{S}1_{S}={x},\sum_{S}% \alpha_{S}=1,\alpha_{S}\geq 0)
  62. f L ( x ) = f - ( x ) f^{L}({x})=f^{-}({x})
  63. x = { x 1 , x 2 , , x n } {x}=\{x_{1},x_{2},\dots,x_{n}\}
  64. 0 x i 1 0\leq x_{i}\leq 1
  65. f + ( x ) = max ( S α S f ( S ) : S α S 1 S = x , S α S = 1 , α S 0 ) f^{+}({x})=\max(\sum_{S}\alpha_{S}f(S):\sum_{S}\alpha_{S}1_{S}={x},\sum_{S}% \alpha_{S}=1,\alpha_{S}\geq 0)
  66. f 1 , f 2 , , f k f_{1},f_{2},\ldots,f_{k}
  67. α 1 , α 2 , , α k \alpha_{1},\alpha_{2},\ldots,\alpha_{k}
  68. g g
  69. g ( S ) = i = 1 k α i f i ( S ) g(S)=\sum_{i=1}^{k}\alpha_{i}f_{i}(S)
  70. f f
  71. g ( S ) = f ( Ω S ) g(S)=f(\Omega\setminus S)
  72. g ( S ) = min ( f ( S ) , c ) g(S)=\min(f(S),c)
  73. c c
  74. f f
  75. f : 2 Ω + f:2^{\Omega}\rightarrow\mathbb{R}_{+}
  76. g : 2 Ω + g:2^{\Omega}\rightarrow\mathbb{R}_{+}
  77. g ( S ) = ϕ ( f ( S ) ) g(S)=\phi(f(S))
  78. ϕ \phi
  79. T T
  80. Ω \Omega
  81. T T
  82. p p
  83. 𝔼 [ f ( T ) ] p f ( Ω ) + ( 1 - p ) f ( ) \mathbb{E}[f(T)]\geq pf(\Omega)+(1-p)f(\varnothing)
  84. \varnothing
  85. S S
  86. 1 i l , A i Ω 1\leq i\leq l,A_{i}\subseteq\Omega
  87. S i S_{i}
  88. A i A_{i}
  89. S i S_{i}
  90. p i p_{i}
  91. S = i = 1 l S i S=\cup_{i=1}^{l}S_{i}
  92. 𝔼 [ f ( S ) ] R [ l ] Π i R p i Π i R ( 1 - p i ) f ( i R A i ) \mathbb{E}[f(S)]\geq\sum_{R\subseteq[l]}\Pi_{i\in R}p_{i}\Pi_{i\notin R}(1-p_{% i})f(\cup_{i\in R}A_{i})
  93. S Ω S\subseteq\Omega
  94. 1 - 1 / e 1-1/e
  95. 1 - 1 / e 1-1/e

Sudhansu_Datta_Majumdar.html

  1. d s 2 = - U ( x , y , z ) - 2 d t 2 + U ( x , y , z ) 2 ( d x 2 + d y 2 + d z 2 ) , ds^{2}=-U(x,y,z)^{-2}dt^{2}+U(x,y,z)^{2}(dx^{2}+dy^{2}+dz^{2}),
  2. A μ A_{\mu}
  3. Φ ( x ) \Phi(x)
  4. Φ ( x ) = A t ( x ) = U - 1 ( x ) , \Phi(x)=A_{t}(x)=U^{-1}(x),
  5. 2 U ( x , y , z ) = 2 U x 2 + 2 U y 2 + 2 U z 2 = 0 , \nabla^{2}U(x,y,z)=\frac{\partial^{2}U}{\partial x^{2}}+\frac{\partial^{2}U}{% \partial y^{2}}+\frac{\partial^{2}U}{\partial z^{2}}=0,

Sufficient_dimension_reduction.html

  1. 𝐱 \,\textbf{x}
  2. y | 𝐱 y|\,\textbf{x}
  3. y y
  4. 𝐱 \,\textbf{x}
  5. R ( 𝐱 ) R(\,\textbf{x})
  6. 𝐱 \,\textbf{x}
  7. k \mathbb{R}^{k}
  8. R ( 𝐱 ) R(\,\textbf{x})
  9. 𝐱 \,\textbf{x}
  10. R ( 𝐱 ) R(\,\textbf{x})
  11. y | R ( 𝐱 ) y|R(\,\textbf{x})
  12. y | 𝐱 y|\,\textbf{x}
  13. 𝐱 \,\textbf{x}
  14. y | 𝐱 y|\,\textbf{x}
  15. y y
  16. 𝐱 \,\textbf{x}
  17. p 3 p\geq 3
  18. R ( 𝐱 ) R(\,\textbf{x})
  19. y y
  20. R ( 𝐱 ) R(\,\textbf{x})
  21. y | 𝐱 y|\,\textbf{x}
  22. 𝐱 \,\textbf{x}
  23. R ( 𝐱 ) = A T 𝐱 R(\,\textbf{x})=A^{T}\,\textbf{x}
  24. A A
  25. p × k p\times k
  26. k p k\leq p
  27. y | 𝐱 y|\,\textbf{x}
  28. y | A T 𝐱 y|A^{T}\,\textbf{x}
  29. y y
  30. A T 𝐱 A^{T}\,\textbf{x}
  31. A A
  32. η \eta
  33. A A
  34. η \eta
  35. 𝒮 ( η ) \mathcal{S}(\eta)
  36. F y | x = F y | η T x , F_{y|x}=F_{y|\eta^{T}x},
  37. F F
  38. y 𝐱 | η T 𝐱 , y\perp\!\!\!\perp\,\textbf{x}\,|\,\eta^{T}\,\textbf{x},
  39. y y
  40. 𝐱 \,\textbf{x}
  41. η T 𝐱 \eta^{T}\,\textbf{x}
  42. 𝒮 ( η ) \mathcal{S}(\eta)
  43. y | 𝐱 y|\,\textbf{x}
  44. d d
  45. 𝐱 \,\textbf{x}
  46. y | 𝐱 y|\,\textbf{x}
  47. 𝐱 \,\textbf{x}
  48. d \mathbb{R}^{d}
  49. 𝒮 \mathcal{S}
  50. y | 𝐱 y|\,\textbf{x}
  51. y | 𝐱 y|\,\textbf{x}
  52. 𝒮 \mathcal{S}
  53. d d
  54. y | 𝐱 y|\,\textbf{x}
  55. 𝒮 \mathcal{S}
  56. η \eta
  57. η T 𝐱 \eta^{T}\,\textbf{x}
  58. 𝒮 \mathcal{S}
  59. y | 𝐱 y|\,\textbf{x}
  60. 𝒮 𝒮 d r s \mathcal{S}\subset\mathcal{S}_{drs}
  61. 𝒮 d r s \mathcal{S}_{drs}
  62. 𝒮 y | x \mathcal{S}_{y|x}
  63. y | 𝐱 y|\,\textbf{x}
  64. 𝒮 d r s \cap\mathcal{S}_{drs}
  65. 𝒮 y | x \mathcal{S}_{y|x}
  66. 𝒮 y | x \mathcal{S}_{y|x}
  67. 𝒮 d r s \cap\mathcal{S}_{drs}
  68. 𝒮 y | x \mathcal{S}_{y|x}
  69. 𝒮 y | x \mathcal{S}_{y|x}
  70. 𝒮 1 \mathcal{S}_{1}
  71. 𝒮 2 \mathcal{S}_{2}
  72. y | 𝐱 y|\,\textbf{x}
  73. 𝐱 \,\textbf{x}
  74. f ( a ) > 0 f(a)>0
  75. a Ω x a\in\Omega_{x}
  76. f ( a ) = 0 f(a)=0
  77. Ω x \Omega_{x}
  78. 𝒮 1 𝒮 2 \mathcal{S}_{1}\cap\mathcal{S}_{2}
  79. 𝒮 y | x \mathcal{S}_{y|x}
  80. 𝐱 \,\textbf{x}
  81. y = α + β T 𝐱 + ε , where ε 𝐱 . y=\alpha+\beta^{T}\,\textbf{x}+\varepsilon,\,\text{ where }\varepsilon\perp\!% \!\!\perp\,\textbf{x}.
  82. y | 𝐱 y|\,\textbf{x}
  83. y | β T 𝐱 y|\beta^{T}\,\textbf{x}
  84. β \beta
  85. β \beta
  86. β = 𝟎 \beta=\,\textbf{0}
  87. d = 1 d=1
  88. β ^ \hat{\beta}
  89. β \beta
  90. β ^ \hat{\beta}
  91. 𝒮 y | x \mathcal{S}_{y|x}
  92. y y
  93. β ^ T 𝐱 \hat{\beta}^{T}\,\textbf{x}

Sulfur_tetrachloride.html

  1. SCl 2 + Cl 2 193 K SCl 4 \rm SCl_{2}+Cl_{2}\xrightarrow{193~{}K}SCl_{4}
  2. SCl 4 - 15 C SCl 2 + Cl 2 \mathrm{SCl_{4}\ \xrightarrow{-15~{}^{\circ}C}\ SCl_{2}+Cl_{2}}
  3. SCl 4 + H 2 O SOCl 2 + 2 H C l \mathrm{SCl_{4}+H_{2}O\ \xrightarrow{}\ SOCl_{2}+2HCl}
  4. SCl 4 + 2 H 2 O SO 2 + 4 H C l \mathrm{SCl_{4}+2H_{2}O\ \xrightarrow{}\ SO_{2}+4HCl}
  5. SCl 4 + 2 H N O 3 + 2 H 2 O H 2 SO 4 + 2 N O 2 + 4 H C l \mathrm{SCl_{4}+2HNO_{3}+2H_{2}O\ \xrightarrow{}\ H_{2}SO_{4}+2NO_{2}\uparrow+% 4HCl}
  6. SCl 4 + 6 N a O H Na 2 SO 3 + 4 N a C l + 3 H 2 O \mathrm{SCl_{4}+6NaOH\ \xrightarrow{}\ Na_{2}SO_{3}+4NaCl+3H_{2}O}

Sum-of-squares_optimization.html

  1. min u \R n c T u \min_{u\in\R^{n}}c^{T}u
  2. a k , 0 ( x ) + a k , 1 ( x ) u 1 + + a k , n ( x ) u n SOS ( k = 1 , , N s ) . a_{k,0}(x)+a_{k,1}(x)u_{1}+\cdots+a_{k,n}(x)u_{n}\in\,\text{SOS}\quad(k=1,% \ldots,N_{s}).
  3. c \R n c\in\R^{n}
  4. { a k , j } \{a_{k,j}\}
  5. u \R n u\in\R^{n}
  6. p p
  7. { f i } i = 1 m \{f_{i}\}_{i=1}^{m}
  8. p = i = 1 m f i 2 p=\sum_{i=1}^{m}f_{i}^{2}
  9. p = x 2 - 4 x y + 7 y 2 p=x^{2}-4xy+7y^{2}
  10. p = f 1 2 + f 2 2 p=f_{1}^{2}+f_{2}^{2}
  11. f 1 = ( x - 2 y ) and f 2 = 3 y . f_{1}=(x-2y)\,\text{ and }f_{2}=\sqrt{3}y.
  12. p p
  13. p ( x ) 0 p(x)\geq 0
  14. x \R n x\in\R^{n}
  15. p ( x ) = x T Q x p(x)=x^{T}Qx
  16. Q Q
  17. p ( x ) = z ( x ) T Q z ( x ) , p(x)=z(x)^{T}Qz(x),
  18. z z
  19. d \leq d
  20. p p
  21. Q Q
  22. p ( x ) = z ( x ) T Q z ( x ) p(x)=z(x)^{T}Qz(x)

Sum_activity_of_peripheral_deiodinases.html

  1. G ^ D = β 31 ( K M 1 + [ F T 4 ] ) ( 1 + K 30 [ T B G ] ) [ F T 3 ] α 31 [ F T 4 ] \hat{G}_{D}={{\beta_{31}(K_{M1}+[FT_{4}])(1+K_{30}[TBG])[FT_{3}]}\over{\alpha_% {31}[FT_{4}]}}
  2. G ^ D = β 31 ( K M 1 + [ F T 4 ] ) [ T T 3 ] α 31 [ F T 4 ] \hat{G}_{D}={{\beta_{31}(K_{M1}+[FT_{4}])[TT_{3}]}\over{\alpha_{31}[FT_{4}]}}
  3. α 31 \alpha_{31}
  4. β 31 \beta_{31}

Sum_of_squares.html

  1. π \pi

Sumner's_conjecture.html

  1. n n
  2. ( 2 n - 2 ) (2n-2)
  3. P P
  4. K 1 , n - 1 K_{1,n-1}
  5. P P
  6. 2 n - 3 2n-3
  7. n - 2 n-2
  8. P P
  9. n - 1 n-1
  10. 2 n - 2 2n-2
  11. n - 3 2 n-\frac{3}{2}
  12. n - 3 2 = n - 1 \left\lceil n-\frac{3}{2}\right\rceil=n-1
  13. P P
  14. n n
  15. f ( n ) f(n)
  16. f ( n ) = 2 n + o ( n ) f(n)=2n+o(n)
  17. n n
  18. f ( n ) f(n)
  19. f ( n ) 3 n - 3 f(n)\leq 3n-3
  20. g ( k ) g(k)
  21. n + g ( k ) n+g(k)
  22. k k
  23. h ( n , Δ ) h(n,\Delta)
  24. n n
  25. Δ \Delta
  26. h ( n , Δ ) h(n,\Delta)
  27. Δ \Delta
  28. h ( n , Δ ) h(n,\Delta)
  29. n + o ( n ) n+o(n)
  30. 2 n - 2 2n-2
  31. n n
  32. n n
  33. ( 2 n - 2 ) (2n-2)
  34. ( 2 n - 2 ) (2n-2)
  35. n n
  36. n n
  37. n 8 n\geq 8
  38. n n
  39. n + k - 1 n+k-1
  40. k k
  41. G G
  42. 2 n - 2 2n-2
  43. G G
  44. G G
  45. n n
  46. n n
  47. f ( n ) f(n)

Supercapacitor.html

  1. C total = C 1 C 2 C 1 + C 2 C\text{total}=\frac{C_{1}\cdot C_{2}}{C_{1}+C_{2}}
  2. C = ε A d C=\varepsilon\frac{A}{d}
  3. W = 1 2 C DC V DC 2 W=\frac{1}{2}\cdot C\text{DC}\cdot V\text{DC}^{2}
  4. C total = I discharge t 2 - t 1 V 1 - V 2 C\text{total}=I\text{discharge}\cdot\frac{t_{2}-t_{1}}{V_{1}-V_{2}}
  5. R i = Δ V 2 I discharge R\text{i}=\frac{\Delta V_{2}}{I\text{discharge}}
  6. τ \tau
  7. τ = R i C \tau=R\text{i}\cdot C
  8. P loss = R i I 2 P\text{loss}=R\text{i}\cdot I^{2}
  9. W max = 1 2 C total V loaded 2 W\text{max}=\frac{1}{2}\cdot C\text{total}\cdot V\text{loaded}^{2}
  10. W eff = 1 2 C ( V max 2 - V min 2 ) W\text{eff}=\frac{1}{2}\ C\cdot\ (V\text{max}^{2}-V\text{min}^{2})
  11. P max = 1 4 V 2 R i P\text{max}=\frac{1}{4}\cdot\frac{V^{2}}{R_{i}}
  12. P eff = 1 8 V 2 R i P\text{eff}=\frac{1}{8}\cdot\frac{V^{2}}{R_{i}}
  13. L x = L 0 2 T 0 - T x 10 L_{x}=L_{0}\cdot 2^{\frac{T_{0}-T_{x}}{10}}
  14. t = C ( U charge - U min ) I t=\frac{C\cdot(U\text{charge}-U\text{min})}{I}
  15. t = 1 2 P C ( U charge 2 - U min 2 ) . t=\frac{1}{2P}\cdot C\cdot(U\text{charge}^{2}-U\text{min}^{2}).

Superficial_velocity.html

  1. u s = Q A u_{s}=\frac{Q}{A}
  2. u s = ϕ u u_{s}=\phi u
  3. ϕ \phi

Superfluid_vacuum_theory.html

  1. E 2 | p | 2 E^{2}\propto|\vec{p}|^{2}
  2. E | p | 2 E\propto|\vec{p}|^{2}

Superhedging_price.html

  1. ρ ( - X ) \rho(-X)
  2. ρ \rho
  3. ρ ( X ) = sup Q EMM 𝔼 Q [ - X ] \rho(X)=\sup_{Q\in\mathrm{EMM}}\mathbb{E}^{Q}[-X]
  4. ρ \rho
  5. A = { - V T : ( V t ) t = 0 T is the price of a self-financing portfolio at each time } A=\{-V_{T}:(V_{t})_{t=0}^{T}\,\text{ is the price of a self-financing % portfolio at each time}\}
  6. inf Q EMM 𝔼 Q [ X ] \inf_{Q\in\mathrm{EMM}}\mathbb{E}^{Q}[X]
  7. - ρ ( X ) -\rho(X)
  8. ρ t ( X ) = ess sup Q E M M 𝔼 Q [ - X | t ] . \rho_{t}(X)=\operatorname*{ess\sup}_{Q\in EMM}\mathbb{E}^{Q}[-X|\mathcal{F}_{t% }].

Superimposed_code.html

  1. n = N ( 1 - 2 - 1 r ) n=N(1-2^{-\frac{1}{r}})
  2. F = ( v N ) n F=(\frac{v}{N})^{n}

Superpartient_ratio.html

  1. n + a n , \frac{n+a}{n}\,,
  2. n + 1 n \tfrac{n+1}{n}

Supporting_functional.html

  1. C X C\subset X
  2. ϕ : X \phi:X\to\mathbb{R}
  3. x 0 x_{0}
  4. ϕ ( x ) ϕ ( x 0 ) \phi(x)\leq\phi(x_{0})
  5. x C x\in C
  6. h C : X * h_{C}:X^{*}\to\mathbb{R}
  7. X * X^{*}
  8. X X
  9. h C ( x * ) = x * ( x 0 ) h_{C}\left(x^{*}\right)=x^{*}\left(x_{0}\right)
  10. h C h_{C}
  11. ϕ : X \phi:X\to\mathbb{R}
  12. x 0 x_{0}
  13. ϕ ( x ) = x * ( x ) \phi(x)=x^{*}(x)
  14. x X x\in X
  15. ϕ \phi
  16. x 0 C x_{0}\in C
  17. ϕ ( x 0 ) = σ = sup x C ϕ ( x ) > inf x C ϕ ( x ) \phi\left(x_{0}\right)=\sigma=\sup_{x\in C}\phi(x)>\inf_{x\in C}\phi(x)
  18. H = ϕ - 1 ( σ ) H=\phi^{-1}(\sigma)
  19. x 0 x_{0}

Surface_acoustic_wave_sensor.html

  1. f 0 = v p p f_{0}=\frac{v_{p}}{p}
  2. ρ \scriptstyle\rho
  3. v E / ρ v\propto\sqrt{E/\rho}

Surface_modification_of_biomaterials_with_proteins.html

  1. C D = k 1 λ N A CD=k_{1}\cdot\frac{\lambda}{NA}
  2. C D \,CD
  3. k 1 \,k_{1}
  4. λ \,\lambda
  5. N A \,NA

Surface_plasmon_polariton.html

  1. E = E 0 exp [ i ( k x x + k z z - ω t ) ] E=E_{0}\exp[i(k_{x}x+k_{z}z-\omega t)]\,
  2. k z 1 ε 1 + k z 2 ε 2 = 0 \frac{k_{z1}}{\varepsilon_{1}}+\frac{k_{z2}}{\varepsilon_{2}}=0
  3. k x 2 + k z i 2 = ε i ( ω c ) 2 i = 1 , 2 k_{x}^{2}+k_{zi}^{2}=\varepsilon_{i}\left(\frac{\omega}{c}\right)^{2}\qquad i=% 1,2
  4. k x = ω c ( ε 1 ε 2 ε 1 + ε 2 ) 1 / 2 . k_{x}=\frac{\omega}{c}\left(\frac{\varepsilon_{1}\varepsilon_{2}}{\varepsilon_% {1}+\varepsilon_{2}}\right)^{1/2}.
  5. ε ( ω ) = 1 - ω P 2 ω 2 , \varepsilon(\omega)=1-\frac{\omega_{P}^{2}}{\omega^{2}},
  6. ω P = n e 2 ε 0 m * \omega_{P}=\sqrt{\frac{ne^{2}}{{\varepsilon_{0}}m^{*}}}
  7. ε 0 {\varepsilon_{0}}
  8. ω S P = ω P / 1 + ε 2 . \omega_{SP}=\omega_{P}/\sqrt{1+\varepsilon_{2}}.
  9. ω S P = ω P / 2 . \omega_{SP}=\omega_{P}/\sqrt{2}.
  10. k x = k x + i k x ′′ = [ ω c ( ε 1 ε 2 ε 1 + ε 2 ) 1 / 2 ] + i [ ω c ( ε 1 ε 2 ε 1 + ε 2 ) 3 / 2 ε 1 ′′ 2 ( ε 1 ) 2 ] . k_{x}=k_{x}^{\prime}+ik_{x}^{\prime\prime}=\left[\frac{\omega}{c}\left(\frac{% \varepsilon_{1}^{\prime}\varepsilon_{2}}{\varepsilon_{1}^{\prime}+\varepsilon_% {2}}\right)^{1/2}\right]+i\left[\frac{\omega}{c}\left(\frac{\varepsilon_{1}^{% \prime}\varepsilon_{2}}{\varepsilon_{1}^{\prime}+\varepsilon_{2}}\right)^{3/2}% \frac{\varepsilon_{1}^{\prime\prime}}{2(\varepsilon_{1}^{\prime})^{2}}\right].
  11. L = 1 2 k x ′′ . L=\frac{1}{2k_{x}^{\prime\prime}}.
  12. z i = λ 2 π ( | ε 1 | + ε 2 ε i 2 ) 1 / 2 z_{i}=\frac{\lambda}{2\pi}\left(\frac{|\varepsilon_{1}^{\prime}|+\varepsilon_{% 2}}{\varepsilon_{i}^{2}}\right)^{1/2}
  13. Δ k = k S P - k x , photon \Delta k=k_{SP}-k_{x,\,\text{photon}}
  14. k S P P = k x , photon ± n k grating = ω c sin θ 0 ± n 2 π a k_{SPP}=k_{x,\,\text{photon}}\pm n\ k\text{grating}=\frac{\omega}{c}\sin{% \theta_{0}}\pm n\frac{2\pi}{a}
  15. k grating k\text{grating}
  16. θ 0 \theta_{0}
  17. G ( x , y ) = 1 A A z ( x , y ) z ( x - x , y - y ) d x d y G(x,y)=\frac{1}{A}\int_{A}z(x^{\prime},y^{\prime})\ z(x^{\prime}-x,y^{\prime}-% y)\,dx^{\prime}\,dy^{\prime}
  18. z ( x , y ) z(x,y)
  19. ( x , y ) (x,y)
  20. A A
  21. G ( x , y ) = δ 2 exp ( - r 2 σ 2 ) G(x,y)=\delta^{2}\exp\left(-\frac{r^{2}}{\sigma^{2}}\right)
  22. δ \delta
  23. r r
  24. ( x , y ) (x,y)
  25. σ \sigma
  26. | s ( k surf ) | 2 = 1 4 π σ 2 δ 2 exp ( - σ 2 k surf 2 4 ) |s(k\text{surf})|^{2}=\frac{1}{4\pi}\sigma^{2}\delta^{2}\exp\left(-\frac{% \sigma^{2}k\text{surf}^{2}}{4}\right)
  27. s s
  28. k surf k\text{surf}
  29. s s
  30. k = 2 π a k=\frac{2\pi}{a}
  31. s s
  32. d I dI
  33. d Ω d\Omega
  34. I 0 I_{0}
  35. d I d Ω I 0 = 4 ε 0 cos θ 0 π 4 λ 4 | t 012 p | 2 | W | 2 | s ( k surf ) | 2 \frac{dI}{d\Omega\ I_{0}}=\frac{4\sqrt{\varepsilon_{0}}}{\cos{\theta_{0}}}% \frac{\pi^{4}}{\lambda^{4}}|t_{012}^{p}|^{2}\ |W|^{2}|s(k\text{surf})|^{2}
  36. | W | 2 |W|^{2}
  37. | W | 2 = A ( θ , | ε 1 | ) sin 2 ψ [ ( 1 + sin 2 θ / | ε 1 | ) 1 / 2 - sin θ ] 2 |W|^{2}=A(\theta,|\varepsilon_{1}|)\ \sin^{2}{\psi}\ [(1+\sin^{2}\theta/|% \varepsilon_{1}|)^{1/2}-\sin{\theta}]^{2}
  38. A ( θ , | ε 1 | ) = | ε 1 | + 1 | ε 1 | - 1 4 1 + tan θ / | ε 1 | A(\theta,|\varepsilon_{1}|)=\frac{|\varepsilon_{1}|+1}{|\varepsilon_{1}|-1}% \frac{4}{1+\tan{\theta}/|\varepsilon_{1}|}
  39. ψ \psi
  40. θ \theta
  41. ψ = 0 \psi=0
  42. | W | 2 = 0 |W|^{2}=0
  43. d I d Ω I 0 = 0 \frac{dI}{d\Omega\ I_{0}}=0

Surface_properties_of_transition_metal_oxides.html

  1. H o = p K B H + - l o g [ B H + ] [ B ] H_{o}=pK_{BH^{+}}-log\frac{[BH^{+}]}{[B]}
  2. n n
  3. p p
  4. n = N c e - ( E c - E f ) k T n=N_{c}e^{\frac{-(E_{c}-E_{f})}{kT}}
  5. p = N v e - ( E f - E v ) k T p=N_{v}e^{\frac{-(E_{f}-E_{v})}{kT}}
  6. P E o 2 | μ i f | 2 P\propto{E_{o}}^{2}{|\mu_{if}|}^{2}
  7. Φ k C T ( k C T + k R ) \Phi\propto{\frac{k_{CT}}{(k_{CT}+k_{R})}}
  8. 1 / 2 1/2

Surfactants_in_paint.html

  1. Δ G em = γ 3 V R \Delta G_{\,\text{em}}=\gamma{3V\ \over R}
  2. R = 3 L s ϕ d ϕ s R={3L_{\,\text{s}}\phi_{\,\text{d}}\over\phi_{\,\text{s}}}

Surrogate_data_testing.html

  1. H 0 H_{0}
  2. H 0 H_{0}

Survival_function.html

  1. S ( t ) = P ( { T > t } ) = t f ( u ) d u = 1 - F ( t ) . S(t)=P(\{T>t\})=\int_{t}^{\infty}f(u)\,du=1-F(t).
  2. S ( u ) S ( t ) S(u)\leq S(t)
  3. u > t u>t
  4. S ( t ) = 1 - F ( t ) S(t)=1-F(t)

Suspension_(dynamical_systems).html

  1. X X
  2. f : X X f:X\to X
  3. r : X + r:X\to\mathbb{R}^{+}
  4. X r = { ( x , t ) : 0 t r ( x ) , x X } / ( x , r ( x ) ) ( f x , 0 ) . X_{r}=\{(x,t):0\leq t\leq r(x),x\in X\}/(x,r(x))\sim(fx,0).
  5. ( X , f ) (X,f)
  6. r r
  7. f t : X r X r f_{t}:X_{r}\to X_{r}
  8. T t : X × X × , ( x , s ) ( x , s + t ) T_{t}:X\times\mathbb{R}\to X\times\mathbb{R},(x,s)\mapsto(x,s+t)
  9. r ( x ) 1 r(x)\equiv 1
  10. ( X , f ) (X,f)

SVJ.html

  1. d S = μ S d t + ν S d Z 1 + ( e α + δ ϵ - 1 ) S d q dS=\mu S\,dt+\sqrt{\nu}S\,dZ_{1}+(e^{\alpha+\delta\epsilon}-1)Sdq
  2. d ν = - λ ( ν - ν ¯ ) d t + η ν d Z 2 d\nu=-\lambda(\nu-\overline{\nu})dt+\eta\sqrt{\nu}dZ_{2}
  3. corr ( d Z 1 , d Z 2 ) = ρ \operatorname{corr}(dZ_{1},dZ_{2})=\rho

Switching_lemma.html

  1. exp ( Ω ( n 1 k - 1 ) ) \exp\left(\Omega\left(n^{\frac{1}{k-1}}\right)\right)
  2. d d

Symbols_for_zero.html

  1. \emptyset
  2. 𝒪 \mathcal{O}

Symplectic_frame_bundle.html

  1. ( M , ω ) (M,\omega)\,
  2. Sp ( n , ) {\mathrm{Sp}}(n,{\mathbb{R}})
  3. π 𝐑 : 𝐑 M \pi_{\mathbf{R}}\colon{\mathbf{R}}\to M\,
  4. F M \mathrm{F}M\,
  5. ω \omega\,
  6. u F p ( M ) u\in\mathrm{F}_{p}(M)\,
  7. p M , p\in M\,,
  8. ( 𝐞 1 , , 𝐞 n , 𝐟 1 , , 𝐟 n ) ({\mathbf{e}}_{1},\dots,{\mathbf{e}}_{n},{\mathbf{f}}_{1},\dots,{\mathbf{f}}_{% n})\,
  9. p p\,
  10. T p ( M ) T_{p}(M)\,
  11. ω p ( 𝐞 j , 𝐞 k ) = ω p ( 𝐟 j , 𝐟 k ) = 0 \omega_{p}({\mathbf{e}}_{j},{\mathbf{e}}_{k})=\omega_{p}({\mathbf{f}}_{j},{% \mathbf{f}}_{k})=0\,
  12. ω p ( 𝐞 j , 𝐟 k ) = δ j k \omega_{p}({\mathbf{e}}_{j},{\mathbf{f}}_{k})=\delta_{jk}\,
  13. j , k = 1 , , n j,k=1,\dots,n\,
  14. p M p\in M\,
  15. 𝐑 p {\mathbf{R}}_{p}\,
  16. Sp ( n , ) {\mathrm{Sp}}(n,{\mathbb{R}})
  17. π 𝐑 : 𝐑 M \pi_{\mathbf{R}}\colon{\mathbf{R}}\to M\,
  18. T p ( M ) T_{p}(M)\,
  19. π 𝐑 : 𝐑 M \pi_{\mathbf{R}}\colon{\mathbf{R}}\to M\,
  20. F M \mathrm{F}M\,
  21. M M\,

Symplectic_spinor_bundle.html

  1. π 𝐏 : 𝐏 M \pi_{\mathbf{P}}\colon{\mathbf{P}}\to M\,
  2. 2 n 2n
  3. ( M , ω ) , (M,\omega),\,
  4. π 𝐐 : 𝐐 M \pi_{\mathbf{Q}}\colon{\mathbf{Q}}\to M\,
  5. 𝐐 {\mathbf{Q}}\,
  6. ( 𝐏 , F 𝐏 ) ({\mathbf{P}},F_{\mathbf{P}})
  7. ( M , ω ) , (M,\omega),\,
  8. π 𝐑 : 𝐑 M \pi_{\mathbf{R}}\colon{\mathbf{R}}\to M\,
  9. ρ : Mp ( n , ) Sp ( n , ) . \rho\colon{\mathrm{Mp}}(n,{\mathbb{R}})\to{\mathrm{Sp}}(n,{\mathbb{R}}).\,
  10. 𝐐 {\mathbf{Q}}\,
  11. 𝐐 = 𝐏 × 𝔪 L 2 ( n ) {\mathbf{Q}}={\mathbf{P}}\times_{\mathfrak{m}}L^{2}({\mathbb{R}}^{n})\,
  12. 𝐏 {\mathbf{P}}
  13. 𝔪 : Mp ( n , ) U ( L 2 ( n ) ) , {\mathfrak{m}}\colon{\mathrm{Mp}}(n,{\mathbb{R}})\to{\mathrm{U}}(L^{2}({% \mathbb{R}}^{n})),\,
  14. Mp ( n , ) . {\mathrm{Mp}}(n,{\mathbb{R}}).\,
  15. U ( 𝐖 ) {\mathrm{U}}({\mathbf{W}})\,
  16. 𝐖 . {\mathbf{W}}.\,
  17. Mp ( n , ) {\mathrm{Mp}}(n,{\mathbb{R}})
  18. L 2 ( n ) . L^{2}({\mathbb{R}}^{n}).\,

Synchronization_networks.html

  1. θ i ( t ) \theta_{i}(t)
  2. i i
  3. ω i \omega_{i}
  4. g ( ω ) = γ π [ γ 2 + ( ω - ω 0 ) 2 ) g(\omega)=\frac{\gamma}{\pi[\gamma^{2}+(\omega-\omega_{0})^{2})}
  5. γ \gamma
  6. ω 0 \omega_{0}
  7. d θ i d t = ω i + 1 N j = 1 N K i j sin ( θ j - θ i ) , i = 1 , , N \frac{d\theta_{i}}{dt}=\omega_{i}+\frac{1}{N}\sum^{N}_{j=1}K_{ij}\sin(\theta_{% j}-\theta_{i}),i=1,...,N
  8. N N
  9. K K
  10. i i
  11. j j
  12. r ( t ) = | 1 N j = 1 N e i θ j ( t ) | r(t)=\bigg|\frac{1}{N}\sum^{N}_{j=1}e^{i\theta_{j}(t)}\bigg|
  13. K c K_{c}
  14. N N\to\infty
  15. t t\to\infty
  16. r = { 0 , K < K c 1 - ( K c / K ) , K K c r=\begin{cases}0,&K<K_{c}\\ \sqrt{1-(K_{c}/K)},&K\geq K_{c}\end{cases}
  17. K c = 2 γ K_{c}=2\gamma
  18. r = 0 r=0\to
  19. r = e i θ r=e^{i\theta}\to
  20. K c K_{c}

Synchronous_context-free_grammar.html

  1. X X\to
  2. X 1 X_{1}
  3. X 2 X_{2}
  4. X 2 X_{2}
  5. X 1 X_{1}

Synge's_world_function.html

  1. x , x x,x^{\prime}
  2. x x
  3. x x
  4. γ ( λ ) \gamma(\lambda)
  5. x x
  6. x x^{\prime}
  7. λ \lambda
  8. γ ( λ 0 ) = x \gamma(\lambda_{0})=x^{\prime}
  9. γ ( λ 1 ) = x \gamma(\lambda_{1})=x
  10. σ ( x , x ) = 1 2 ( λ 1 - λ 0 ) λ 0 λ 1 g μ ν ( z ) t μ t ν d λ \sigma(x,x^{\prime})=\frac{1}{2}(\lambda_{1}-\lambda_{0})\int_{\lambda_{0}}^{% \lambda_{1}}g_{\mu\nu}(z)t^{\mu}t^{\nu}d\lambda
  11. σ ( x , x ) \sigma(x,x^{\prime})
  12. x x
  13. x x^{\prime}
  14. σ ( x , x ) = 1 2 η α β ( x - x ) α ( x - x ) β \sigma(x,x^{\prime})=\frac{1}{2}\eta_{\alpha\beta}(x-x^{\prime})^{\alpha}(x-x^% {\prime})^{\beta}

System_size_expansion.html

  1. P ( X , t ) P(X,t)
  2. X X
  3. t t
  4. X X
  5. Ω \Omega
  6. 𝐗 \mathbf{X}
  7. 𝐱 = 𝐗 / Ω \mathbf{x}=\mathbf{X}/\Omega
  8. ϕ \mathbf{\phi}
  9. 𝐱 \mathbf{x}
  10. 𝐗 \mathbf{X}
  11. N N
  12. R R
  13. d P ( 𝐗 , t ) d t = Ω j = 1 R ( i = 1 N 𝔼 - S i j - 1 ) f j ( 𝐱 , Ω ) P ( 𝐗 , t ) . \frac{dP(\mathbf{X},t)}{dt}=\Omega\sum_{j=1}^{R}\left(\prod_{i=1}^{N}\mathbb{E% }^{-S_{ij}}-1\right)f_{j}(\mathbf{x},\Omega)P(\mathbf{X},t).
  14. Ω \Omega
  15. 𝔼 \mathbb{E}
  16. S i j S_{ij}
  17. S i j S_{ij}
  18. i i
  19. j j
  20. f j f_{j}
  21. j j
  22. 𝐱 \mathbf{x}
  23. Ω \Omega
  24. 𝔼 - S i j \mathbb{E}^{-S_{ij}}
  25. S i j S_{ij}
  26. i i
  27. 𝔼 - S 23 f ( x 1 , x 2 , x 3 ) = f ( x 1 , x 2 - S 23 , x 3 ) \mathbb{E}^{-S_{23}}f(x_{1},x_{2},x_{3})=f(x_{1},x_{2}-S_{23},x_{3})
  28. j j
  29. 𝐗 \mathbf{X}
  30. j j
  31. j j
  32. 𝐗 \mathbf{X^{\prime}}
  33. 𝐗 \mathbf{X}
  34. 𝐗 \mathbf{X^{\prime}}
  35. X 1 X_{1}
  36. X 2 X_{2}
  37. X 1 X 2 X_{1}\rightarrow X_{2}
  38. N = 2 N=2
  39. R = 1 R=1
  40. 𝐗 = { n 1 , n 2 } \mathbf{X}=\{n_{1},n_{2}\}
  41. n 1 , n 2 n_{1},n_{2}
  42. X 1 X_{1}
  43. X 2 X_{2}
  44. f 1 ( 𝐱 , Ω ) = n 1 Ω = x 1 f_{1}(\mathbf{x},\Omega)=\frac{n_{1}}{\Omega}=x_{1}
  45. X 1 X_{1}
  46. ( - 1 , 1 ) T (-1,1)^{T}
  47. d P ( 𝐗 , t ) d t \displaystyle\frac{dP(\mathbf{X},t)}{dt}
  48. 𝚫 𝐗 = { 1 , - 1 } \mathbf{\Delta X}=\{1,-1\}
  49. 𝐗 \mathbf{X}
  50. 𝐗 \mathbf{X}^{\prime}
  51. X i X_{i}
  52. i i
  53. ξ \xi
  54. Ω 1 / 2 \Omega^{1/2}
  55. X i = Ω ϕ i + Ω 1 / 2 ξ i . X_{i}=\Omega\phi_{i}+\Omega^{1/2}\xi_{i}.
  56. 𝐗 \mathbf{X}
  57. ξ \xi
  58. P ( 𝐗 , t ) = P ( Ω ϕ + Ω 1 / 2 ξ ) = Π ( ξ , t ) . P(\mathbf{X},t)=P(\Omega\mathbf{\phi}+\Omega^{1/2}\mathbf{\xi})=\Pi(\mathbf{% \xi},t).
  59. f f
  60. 𝔼 \mathbb{E}
  61. f j ( 𝐱 ) = f j ( ϕ + Ω - 1 / 2 ξ ) = f j ( ϕ ) + Ω - 1 / 2 i = 1 N f j ( ϕ ) ϕ i ξ i + O ( Ω - 1 ) . f_{j}(\mathbf{x})=f_{j}(\mathbf{\phi}+\Omega^{-1/2}\mathbf{\xi})=f_{j}(\mathbf% {\phi})+\Omega^{-1/2}\sum_{i=1}^{N}\frac{\partial f^{\prime}_{j}(\mathbf{\phi}% )}{\partial\phi_{i}}\xi_{i}+O(\Omega^{-1}).
  62. 𝔼 f ( n ) f ( n + 1 ) \mathbb{E}f(n)\rightarrow f(n+1)
  63. 𝔼 f ( ξ ) f ( ξ + Ω - 1 / 2 ) \mathbb{E}f(\xi)\rightarrow f(\xi+\Omega^{-1/2})
  64. i = 1 N 𝔼 - S i j 1 - Ω - 1 / 2 i S i j ξ i + Ω - 1 2 i k S i j S k j 2 ξ i ξ k + O ( Ω - 3 / 2 ) . \prod_{i=1}^{N}\mathbb{E}^{-S_{ij}}\simeq 1-\Omega^{-1/2}\sum_{i}S_{ij}\frac{% \partial}{\partial\xi_{i}}+\frac{\Omega^{-1}}{2}\sum_{i}\sum_{k}S_{ij}S_{kj}% \frac{\partial^{2}}{\partial\xi_{i}\,\partial\xi_{k}}+O(\Omega^{-3/2}).
  65. Π ( ξ , t ) t - Ω 1 / 2 i = 1 N ϕ i t Π ( ξ , t ) ξ i = Ω j = 1 R ( - Ω - 1 / 2 i S i j ξ i + Ω - 1 2 i k S i j S k j 2 ξ i ξ k + O ( Ω - 3 / 2 ) ) × ( f j ( ϕ ) + Ω - 1 / 2 i f j ( ϕ ) ϕ i ξ i + O ( Ω - 1 ) ) Π ( ξ , t ) . \begin{aligned}&\displaystyle{}\quad\frac{\partial\Pi(\mathbf{\xi},t)}{% \partial t}-\Omega^{1/2}\sum_{i=1}^{N}\frac{\partial\phi_{i}}{\partial t}\frac% {\partial\Pi(\mathbf{\xi},t)}{\partial\xi_{i}}\\ &\displaystyle=\Omega\sum_{j=1}^{R}\left(-\Omega^{-1/2}\sum_{i}S_{ij}\frac{% \partial}{\partial\xi_{i}}+\frac{\Omega^{-1}}{2}\sum_{i}\sum_{k}S_{ij}S_{kj}% \frac{\partial^{2}}{\partial\xi_{i}\,\partial\xi_{k}}+O(\Omega^{-3/2})\right)% \\ &\displaystyle{}\qquad\times\left(f_{j}(\mathbf{\phi})+\Omega^{-1/2}\sum_{i}% \frac{\partial f^{\prime}_{j}(\mathbf{\phi})}{\partial\phi_{i}}\xi_{i}+O(% \Omega^{-1})\right)\Pi(\mathbf{\xi},t).\end{aligned}
  66. Ω \Omega
  67. Ω 1 / 2 \Omega^{1/2}
  68. i = 1 N ϕ i t Π ( ξ , t ) ξ i = i = 1 N j = 1 R S i j f j ( ϕ ) Π ( ξ , t ) ξ j . \sum_{i=1}^{N}\frac{\partial\phi_{i}}{\partial t}\frac{\partial\Pi(\mathbf{\xi% },t)}{\partial\xi_{i}}=\sum_{i=1}^{N}\sum_{j=1}^{R}S_{ij}f^{\prime}_{j}(% \mathbf{\phi})\frac{\partial\Pi(\mathbf{\xi},t)}{\partial\xi_{j}}.
  69. ϕ i t = j = 1 R S i j f j ( ϕ ) . \frac{\partial\phi_{i}}{\partial t}=\sum_{j=1}^{R}S_{ij}f^{\prime}_{j}(\mathbf% {\phi}).
  70. Ω 0 \Omega^{0}
  71. Π ( ξ , t ) t = j ( i k - S i j f j ϕ k ( ξ k Π ( ξ , t ) ) ξ i + 1 2 f j i k S i j S k j 2 Π ( ξ , t ) ξ i ξ k ) , \frac{\partial\Pi(\mathbf{\xi},t)}{\partial t}=\sum_{j}\left(\sum_{ik}-S_{ij}% \frac{\partial f^{\prime}_{j}}{\partial\phi_{k}}\frac{\partial(\xi_{k}\Pi(% \mathbf{\xi},t))}{\partial\xi_{i}}+\frac{1}{2}f^{\prime}_{j}\sum_{ik}S_{ij}S_{% kj}\frac{\partial^{2}\Pi(\mathbf{\xi},t)}{\partial\xi_{i}\,\partial\xi_{k}}% \right),
  72. Π ( ξ , t ) t = i k A i k ( ξ k Π ) ξ i + 1 2 i k [ 𝐁𝐁 T ] i k 2 Π ξ i ξ k , \frac{\partial\Pi(\mathbf{\xi},t)}{\partial t}=\sum_{ik}A_{ik}\frac{\partial(% \xi_{k}\Pi)}{\partial\xi_{i}}+\frac{1}{2}\sum_{ik}[\mathbf{BB}^{T}]_{ik}\frac{% \partial^{2}\Pi}{\partial\xi_{i}\,\partial\xi_{k}},
  73. A i k = j = 1 R S i j f j ϕ k = ( 𝐒 i 𝐟 ) ϕ k , A_{ik}=\sum_{j=1}^{R}S_{ij}\frac{\partial f^{\prime}_{j}}{\partial\phi_{k}}=% \frac{\partial(\mathbf{S}_{i}\cdot\mathbf{f})}{\partial\phi_{k}},
  74. [ 𝐁𝐁 T ] i k = j = 1 R S i j S k j f j ( ϕ ) = [ 𝐒 diag ( f ( ϕ ) ) 𝐒 T ] i k . [\mathbf{BB}^{T}]_{ik}=\sum_{j=1}^{R}S_{ij}S_{kj}f^{\prime}_{j}(\mathbf{\phi})% =[\mathbf{S}\,\mbox{diag}~{}(f(\mathbf{\phi}))\,\mathbf{S}^{T}]_{ik}.
  75. Π \Pi
  76. 𝐀 \mathbf{A}
  77. 𝐁𝐁 T \mathbf{BB}^{T}
  78. Ω \Omega
  79. O ( Ω - 1 / 2 ) O(\Omega^{-1/2})
  80. 𝐟 \mathbf{f}
  81. S S
  82. Π \Pi

Szegő_limit_theorems.html

  1. T n ( ϕ ) k , l = ϕ ^ ( k - l ) , 0 k , l n - 1 , T_{n}(\phi)_{k,l}=\widehat{\phi}(k-l),\quad 0\leq k,l\leq n-1,
  2. ϕ ^ ( k ) = 1 2 π 0 2 π ϕ ( e i θ ) e - i k θ d θ \widehat{\phi}(k)=\frac{1}{2\pi}\int_{0}^{2\pi}\phi(e^{i\theta})e^{-ik\theta}% \,d\theta
  3. lim n det T n ( ϕ ) det T n - 1 ( ϕ ) = exp { 1 2 π 0 2 π log ϕ ( e i θ ) d θ } . \lim_{n\to\infty}\frac{\det T_{n}(\phi)}{\det T_{n-1}(\phi)}=\exp\left\{\frac{% 1}{2\pi}\int_{0}^{2\pi}\log\phi(e^{i\theta})\,d\theta\right\}.
  4. lim n det T n ( ϕ ) G n ( ϕ ) = exp { k = 1 k | ( log ϕ ) ^ ( k ) | 2 } . \lim_{n\to\infty}\frac{\det T_{n}(\phi)}{G^{n}(\phi)}=\exp\left\{\sum_{k=1}^{% \infty}k\left|\widehat{(\log\phi)}(k)\right|^{2}\right\}.

Śleszyński–Pringsheim_theorem.html

  1. a 1 b 1 + a 2 b 2 + a 3 b 3 + \cfrac{a_{1}}{b_{1}+\cfrac{a_{2}}{b_{2}+\cfrac{a_{3}}{b_{3}+\ddots}}}
  2. f = n { A n B n - A n - 1 B n - 1 } , f=\sum_{n}\left\{\frac{A_{n}}{B_{n}}-\frac{A_{n-1}}{B_{n-1}}\right\},

T-matrix_method.html

  1. 𝐌 1 , 𝐍 1 \mathbf{M}^{1},\mathbf{N}^{1}
  2. 𝐌 3 , 𝐍 3 \mathbf{M}^{3},\mathbf{N}^{3}
  3. 𝐄 i n c = n = 1 m = - n n a m n 𝐌 m n 1 + b m n 𝐍 m n 1 . \mathbf{E}_{inc}=\sum_{n=1}^{\infty}\sum_{m=-n}^{n}a_{mn}\mathbf{M}^{1}_{mn}+b% _{mn}\mathbf{N}^{1}_{mn}.
  4. 𝐄 s c a t = n = 1 m = - n n f m n 𝐌 m n 3 + g m n 𝐍 m n 3 . \mathbf{E}_{scat}=\sum_{n=1}^{\infty}\sum_{m=-n}^{n}f_{mn}\mathbf{M}^{3}_{mn}+% g_{mn}\mathbf{N}^{3}_{mn}.
  5. ( a m n b m n ) = T ( f m n g m n ) \begin{pmatrix}a_{mn}\\ b_{mn}\end{pmatrix}=T\begin{pmatrix}f_{mn}\\ g_{mn}\end{pmatrix}
  6. 𝐧 × ( 𝐄 s c a t + 𝐄 i n c ) = 𝐄 i n t \mathbf{n}\times(\mathbf{E}_{scat}+\mathbf{E}_{inc})=\mathbf{E}_{int}
  7. 𝐧 × ( 𝐇 s c a t + 𝐇 i n c ) = 𝐇 i n t \mathbf{n}\times(\mathbf{H}_{scat}+\mathbf{H}_{inc})=\mathbf{H}_{int}
  8. 𝐧 \mathbf{n}

T_pad.html

  1. Z 1 Z_{1}\,\!
  2. Z 2 Z_{2}\,\!
  3. A = V out V in A=\frac{V_{\mathrm{out}}}{V_{\mathrm{in}}}
  4. L = V in V out L=\frac{V_{\mathrm{in}}}{V_{\mathrm{out}}}
  5. L dB = 20 log L L_{\mathrm{dB}}=20\log L\,\!
  6. L = e γ L=e^{\gamma}\,
  7. γ \gamma\,
  8. Z iT = Z 2 + Z Y Z_{\mathrm{iT}}=\sqrt{Z^{2}+\frac{Z}{Y}}
  9. Y i Π = Y 2 + Y Z Y_{\mathrm{i\Pi}}=\sqrt{Y^{2}+\frac{Y}{Z}}
  10. L L1 = Z iT Y i Π e γ L L_{\mathrm{L1}}=\sqrt{Z_{\mathrm{iT}}Y_{\mathrm{i\Pi}}}\ e^{\gamma_{\mathrm{L}}}
  11. γ L = sinh - 1 Z Y \gamma_{\mathrm{L}}=\sinh^{-1}{\sqrt{ZY}}
  12. L L2 = Z i Π Y iT e γ L L_{\mathrm{L2}}=\sqrt{Z_{\mathrm{i\Pi}}Y_{\mathrm{iT}}}\ e^{\gamma_{\mathrm{L}}}
  13. L T = L L1 L L2 = e 2 γ L = e γ T L_{\mathrm{T}}=L_{\mathrm{L1}}L_{\mathrm{L2}}=e^{2\gamma_{\mathrm{L}}}=e^{% \gamma_{\mathrm{T}}}\,
  14. γ T = 2 γ L = 2 sinh - 1 R 1 2 R 2 \gamma_{\mathrm{T}}=2\gamma_{\mathrm{L}}=2\sinh^{-1}{\sqrt{\frac{R_{1}}{2R_{2}% }}}\,
  15. Z 0 = R 1 2 + 2 R 1 R 2 Z_{0}=\sqrt{{R_{1}}^{2}+2R_{1}R_{2}}
  16. R 1 = Z 0 tanh ( γ T 2 ) R_{1}=Z_{0}\tanh\left(\frac{\gamma_{\mathrm{T}}}{2}\right)
  17. R 2 = Z 0 2 - R 1 2 2 R 1 R_{2}=\frac{{Z_{0}}^{2}-{R_{1}}^{2}}{2R_{1}}

Table_of_simple_cubic_graphs.html

  1. v v
  2. p p
  3. j = i + d i mod p , 2 d i p - 2. j=i+d_{i}\quad(\bmod\,p),\quad 2\leq d_{i}\leq p-2.
  4. p p
  5. d p - 1 - i = - d i mod p , i = 0 , 1 , p / 2 - 1 d_{p-1-i}=-d_{i}\quad(\bmod\,p),\quad i=0,1,\ldots p/2-1
  6. 5 , - 9 , 7 , - 7 , 9 , - 55 < s u p > 44 5,-9,7,-7,9,-55<sup>44

Tabulation_hashing.html

  1. k k

Tait_equation.html

  1. β 0 ( P ) = - 1 V 0 ( P ) ( V P ) T = 0.4343 C V 0 ( P ) ( B + P ) \beta_{0}^{(P)}=\frac{-1}{V_{0}^{(P)}}\left(\frac{\partial V}{\partial P}% \right)_{T}=\frac{0.4343C}{V_{0}^{(P)}(B+P)}
  2. V 0 ( P ) = V 0 ( 1 ) - C log 10 B + P B + P ( 1 ) V_{0}^{(P)}=V_{0}^{(1)}-C\log_{10}\frac{B+P}{B+P^{(1)}}
  3. β 0 ( P ) \beta_{0}^{(P)}
  4. V 0 V_{0}
  5. B B
  6. C C

Tangent_half-angle_substitution.html

  1. sin x \displaystyle\sin x
  2. t = tan x 2 . t=\tan\frac{x}{2}.
  3. sin x = 2 sin x 2 cos x 2 = 2 t cos 2 x 2 = 2 t sec 2 x 2 = 2 t 1 + t 2 . \begin{aligned}\displaystyle\sin x&\displaystyle=2\sin\frac{x}{2}\cos\frac{x}{% 2}\\ &\displaystyle=2t\cos^{2}\frac{x}{2}\\ &\displaystyle=\frac{2t}{\sec^{2}\frac{x}{2}}\\ &\displaystyle=\frac{2t}{1+t^{2}}.\end{aligned}
  4. cos x = 1 - 2 sin 2 x 2 = 1 - 2 t 2 cos 2 x 2 = 1 - 2 t 2 sec 2 x 2 = 1 - 2 t 2 1 + t 2 = 1 - t 2 1 + t 2 . \begin{aligned}\displaystyle\cos x&\displaystyle=1-2\sin^{2}\frac{x}{2}\\ &\displaystyle=1-2t^{2}\cos^{2}\frac{x}{2}\\ &\displaystyle=1-\frac{2t^{2}}{\sec^{2}\frac{x}{2}}\\ &\displaystyle=1-\frac{2t^{2}}{1+t^{2}}\\ &\displaystyle=\frac{1-t^{2}}{1+t^{2}}.\end{aligned}
  5. d t d x \displaystyle\frac{\mathrm{d}t}{\mathrm{d}x}
  6. csc x d x = d x sin x = d t t t = tan x 2 = ln t + C = ln tan x 2 + C . \begin{aligned}\displaystyle\int\csc x\,\mathrm{d}x&\displaystyle=\int\frac{% \mathrm{d}x}{\sin x}&\\ &\displaystyle=\int\frac{\mathrm{d}t}{t}&\displaystyle t=\tan\frac{x}{2}\\ &\displaystyle=\ln t+C\\ &\displaystyle=\ln\,\tan\frac{x}{2}+C.\end{aligned}
  7. 0 2 π d x 2 + cos x = x = 0 x = π d x 2 + cos x + x = π x = 2 π d x 2 + cos x = t = 0 t = d x 2 + cos x + t = - t = 0 d x 2 + cos x t = tan x 2 = t = - t = d x 2 + cos x = - 2 d t 3 + t 2 = 2 3 - d u 1 + u 2 t = u 3 = 2 π 3 . \begin{aligned}\displaystyle\int_{0}^{2\pi}\frac{\mathrm{d}x}{2+\cos x}&% \displaystyle=\int_{x=0}^{x=\pi}\frac{\mathrm{d}x}{2+\cos x}+\int_{x=\pi}^{x=2% \pi}\frac{\mathrm{d}x}{2+\cos x}&&\\ &\displaystyle=\int_{t=0}^{t=\infty}\frac{\mathrm{d}x}{2+\cos x}+\int_{t=-% \infty}^{t=0}\frac{\mathrm{d}x}{2+\cos x}&\displaystyle t&\displaystyle=\tan% \frac{x}{2}\\ &\displaystyle=\int_{t=-\infty}^{t=\infty}\frac{\mathrm{d}x}{2+\cos x}&&\\ &\displaystyle=\int_{-\infty}^{\infty}\frac{2\,\mathrm{d}t}{3+t^{2}}&&\\ &\displaystyle=\frac{2}{\sqrt{3}}\int_{-\infty}^{\infty}\frac{\mathrm{d}u}{1+u% ^{2}}&\displaystyle t&\displaystyle=u\sqrt{3}\\ &\displaystyle=\frac{2\pi}{\sqrt{3}}.&&\end{aligned}
  8. t = 0 t=0
  9. t = tan x 2 t=\tan\frac{x}{2}
  10. x = π x=\pi
  11. ( 1 - t 2 1 + t 2 , 2 t 1 + t 2 ) \left(\frac{1-t^{2}}{1+t^{2}},\frac{2t}{1+t^{2}}\right)
  12. sinh x = 2 t 1 - t 2 , \sinh x=\frac{2t}{1-t^{2}},
  13. cosh x = 1 + t 2 1 - t 2 , \cosh x=\frac{1+t^{2}}{1-t^{2}},
  14. tanh x = 2 t 1 + t 2 , \tanh x=\frac{2t}{1+t^{2}},
  15. d x = 2 1 - t 2 d t . dx=\frac{2}{1-t^{2}}dt.

Tangential_trapezoid.html

  1. A B + C D = B C + D A . AB+CD=BC+DA.
  2. A + D = B + C = π . A+D=B+C=\pi.
  3. e h = f g . eh=fg.
  4. K = a + b | b - a | a b ( a - c ) ( c - b ) . K=\frac{a+b}{|b-a|}\sqrt{ab(a-c)(c-b)}.
  5. K = e f g h 4 ( e + f + g + h ) . K=\sqrt[4]{efgh}(e+f+g+h).
  6. r = K a + b = a b ( a - c ) ( c - b ) | b - a | . r=\frac{K}{a+b}=\frac{\sqrt{ab(a-c)(c-b)}}{|b-a|}.
  7. r = e f g h 4 . r=\sqrt[4]{efgh}.
  8. r = e h = f g . r=\sqrt{eh}=\sqrt{fg}.
  9. r = a b a + b . r=\frac{ab}{a+b}.
  10. K = a b \displaystyle K=ab
  11. P = 2 ( a + b ) . \displaystyle P=2(a+b).
  12. r = 1 2 a b . r=\tfrac{1}{2}\sqrt{ab}.

Tangle_(mathematics).html

  1. 𝐑 2 × [ 0 , 1 ] \mathbf{R}^{2}\times[0,1]
  2. ( a 0 , a 1 , a 2 , ) (a_{0},a_{1},a_{2},\dots)
  3. [ a n , a n - 1 , a n - 2 , ] [a_{n},a_{n-1},a_{n-2},\dots]
  4. \infty

Tate_duality.html

  1. H r ( k , M ) × H 2 - r ( k , M ) H 2 ( k , G m ) = Q / Z \displaystyle H^{r}(k,M)\times H^{2-r}(k,M^{\prime})\rightarrow H^{2}(k,G_{m})% =Q/Z

Tautological_consequence.html

  1. Q Q
  2. P 1 P_{1}
  3. P 2 P_{2}
  4. P n P_{n}
  5. P 1 P_{1}
  6. P 2 P_{2}
  7. P n P_{n}
  8. Q Q
  9. Q Q
  10. P 1 P_{1}
  11. P 2 P_{2}
  12. P n P_{n}
  13. P 1 P_{1}
  14. P 2 P_{2}
  15. P n P_{n}
  16. Q Q
  17. a a
  18. b b
  19. c c
  20. a and b c \frac{a\and b}{\therefore c}

Tautology_(rule_of_inference).html

  1. P P P PP\Leftrightarrow P
  2. P and P P P\and P\Leftrightarrow P
  3. \Leftrightarrow
  4. ϕ \phi
  5. ϕ \vdash\phi
  6. ϕ \models\phi
  7. P P P PP\vdash P\,
  8. P and P P P\and P\vdash P\,
  9. \vdash
  10. P P
  11. P P PP
  12. P and P P\and P
  13. P P P \frac{PP}{\therefore P}
  14. P and P P \frac{P\and P}{\therefore P}
  15. P P PP
  16. P and P P\and P
  17. P P
  18. ( P P ) P (PP)\to P\,
  19. ( P and P ) P (P\and P)\to P\,
  20. P P

Taxation_in_Norway.html

  1. M T R = ( 7 , 8 + 28 + 12 ) + 14 , 1 100 + 14 , 1 = 0 , 5425 = 54 , 3 % MTR=\frac{(7,8+28+12)+14,1}{100+14,1}=0,5425=54,3\%

Taylor's_law.html

  1. var ( Y ) = a μ b \,\text{var}\,(Y)=a\mu^{b}
  2. s 2 = a m + b m 2 s^{2}=am+bm^{2}
  3. log V x = log V 1 + b log x \log V_{x}=\log V_{1}+b\log x\,
  4. K b * ( s ; θ , λ ) = λ κ b ( θ ) [ ( 1 + s θ ) α - 1 ] K^{*}_{b}(s;\theta,\lambda)=\lambda\kappa_{b}(\theta)\left[\left(1+{s\over% \theta}\right)^{\alpha}-1\right]
  5. κ b ( θ ) = ( α - 1 ) α ( θ ( α - 1 ) ) α \kappa_{b}(\theta)={(\alpha-1)\over\alpha}\left({\theta\over(\alpha-1)}\right)% ^{\alpha}
  6. α = ( b - 2 ) / ( b - 1 ) \alpha=(b-2)/(b-1)
  7. a = λ 1 / ( α - 1 ) a=\lambda^{1/(\alpha-1)}
  8. s i 2 = a m i b s_{i}^{2}=am_{i}^{b}
  9. log s i 2 = log a + b log m i \log s_{i}^{2}=\log a+b\log m_{i}
  10. μ 1 = E ( x ) \mu_{1}=E(x)
  11. μ 2 = E ( y ) = E ( c x ) = c E ( x ) = c μ 1 \mu_{2}=E(y)=E(cx)=cE(x)=c\mu_{1}
  12. σ 1 2 = Σ ( x - μ 1 ) 2 \sigma^{2}_{1}=\Sigma(x-\mu_{1})^{2}
  13. σ 2 2 = Σ ( y - μ 2 ) = Σ ( c x - c μ 1 ) 2 = c 2 Σ ( x - μ 1 ) 2 = c 2 σ 1 2 \sigma^{2}_{2}=\Sigma(y-\mu_{2})=\Sigma(cx-c\mu_{1})^{2}=c^{2}\Sigma(x-\mu_{1}% )^{2}=c^{2}\sigma^{2}_{1}
  14. σ 1 2 = a μ 1 b \sigma_{1}^{2}=a\mu_{1}^{b}
  15. σ 2 2 = a μ 2 b = c 2 σ 1 2 = c b a μ 1 b \sigma_{2}^{2}=a\mu_{2}^{b}=c^{2}\sigma_{1}^{2}=c^{b}a\mu_{1}^{b}
  16. b = f - φ + ( f - φ ) 2 - 4 r 2 f φ 2 r f b=\frac{f-\varphi+\sqrt{(f-\varphi)^{2}-4r^{2}f\varphi}}{2r\sqrt{f}}
  17. s 2 = ( c n d ) ( m b ) s^{2}=(cn^{d})(m^{b})
  18. S = V + P S=V+P
  19. V = a m b V=am^{b}
  20. P = m P=m
  21. S = V + P = a m b + m S=V+P=am^{b}+m
  22. s 2 = n p ( 1 - p ) s^{2}=np(1-p)
  23. v a r o b s = log ( a ) + b log ( v a r b i n ) var_{obs}=\log(a)+b\log(var_{bin})
  24. p = 1 - e - m p=1-e^{-m}
  25. p = 1 - e - m log ( s 2 / m ) ( s 2 / m - 1 ) - 1 p=1-e^{-m\log(s^{2}/m)(s^{2}/m-1)^{-1}}
  26. p = 1 - e - m log ( a m b - 1 ) ( a m b - 1 - 1 ) - 1 p=1-e^{-m\log(am^{b-1})(am^{b-1}-1)^{-1}}
  27. k = m 2 / ( s 2 - m ) k=m^{2}/(s^{2}-m)
  28. k = m a m b - 1 - 1 k=\frac{m}{am^{b-1}-1}
  29. p = 1 - e - m log ( a m b - 1 ) ( a m b - 1 - 1 ) - 1 p=1-e^{-m\log(am^{b-1})(am^{b-1}-1)^{-1}}
  30. P ( x ) = P ( x - 1 ) k + x - 1 x m k - 1 m k - 1 - 1 P(x)=P(x-1)\frac{k+x-1}{x}\frac{mk^{-1}}{mk^{-1}-1}
  31. P ( 0 ) = ( 1 + m / k ) - k P(0)=(1+m/k)^{-k}
  32. CI = t ( P ( x ) ( 1 - P ( x ) ) N ) 1 / 2 \mathrm{CI}=t(\frac{P(x)(1-P(x))}{N})^{1/2}
  33. m = w 1 1 - w 2 m=\frac{w_{1}}{1-w_{2}}
  34. s 2 = w 1 ( 1 - w 2 ) 2 s^{2}=\frac{w_{1}}{(1-w_{2})^{2}}
  35. w 1 ( 1 - w 2 ) = m \frac{w_{1}}{(1-w_{2})}=m
  36. w 2 ( 1 - w 2 ) = s 2 - m m \frac{w_{2}}{(1-w_{2})}=\frac{s^{2}-m}{m}
  37. p n = ( a + b n ) p n - 1 p_{n}=(a+\frac{b}{n})p_{n-1}
  38. a = - log ( 1 - w 2 ) a=-\log(1-w_{2})
  39. b = 1 b=1
  40. J n = n 2 s 2 - m m J_{n}=\sqrt{\frac{n}{2}}\frac{s^{2}-m}{m}
  41. N S = n - 1 2 ( s 2 m - 1 ) NS=\sqrt{\frac{n-1}{2}}(\frac{s^{2}}{m}-1)
  42. T = ( n - 1 ) s 2 m T=\frac{(n-1)s^{2}}{m}
  43. J n = n 2 ( a m b - 1 - 1 ) J_{n}=\sqrt{\frac{n}{2}}(am^{b-1}-1)
  44. N t + 1 = r N t N_{t+1}=rN_{t}
  45. v a r ( r ) = s 2 ( log ( r ) ) var(r)=s^{2}(\log(r))
  46. T E = 2 log ( N ) V a r ( r ) ( log ( K ) - log ( N ) 2 ) T_{E}=\frac{2\log(N)}{Var(r)}(\log(K)-\frac{\log(N)}{2})
  47. P ( t ) = 1 - e t T E P(t)=1-e^{\frac{t}{T_{E}}}
  48. H = m - a m ( b - 1 ) H=m-am^{(b-1)}
  49. m > a 1 2 - b m>a^{\frac{1}{2-b}}
  50. n = ( t / D ) 2 / m n=(t/D)^{2}/m
  51. n = ( t / D ) 2 ( m + k ) / ( m k ) n=(t/D)^{2}(m+k)/(mk)
  52. n = ( t / D ) 2 a m ( b - 2 ) n=(t/D)^{2}am^{(b-2)}
  53. n = a m b / D 2 n=am^{b}/D^{2}\,
  54. n = ( t / d m ) 2 a m ( b - 2 ) n=(t/d_{m})^{2}am^{(b-2)}
  55. d m = C I 2 m d_{m}=\frac{CI}{2m}
  56. n = ( t / d p ) 2 p - 1 q n=(t/d_{p})^{2}p^{-1}q
  57. d p = C I 2 p d_{p}=\frac{CI}{2p}
  58. N = ( t D p i ) 2 1 - p p N=(\frac{t}{D_{pi}})^{2}\frac{1-p}{p}
  59. log T = log ( D 2 ) - a b - 2 + ( log n ) b - 1 b - 2 \log T=\frac{\log(D^{2})-a}{b-2}+(\log n)\frac{b-1}{b-2}
  60. n = t | m - T | - 2 a m b n=t|m-T|^{-2}am^{b}
  61. n = t | m - T | - 2 p q n=t|m-T|^{-2}pq
  62. D = ( a n 1 - b T b - 2 ) 1 / 2 D=(an^{1-b}T^{b-2})^{1/2}
  63. T n ( a n 1 - b D 2 ) 1 2 - b T_{n}\geq(\frac{an^{1-b}}{D^{2}})^{\frac{1}{2-b}}
  64. T n α - 1 D 2 - β - 1 n T_{n}\geq\frac{\alpha-1}{D^{2}-\frac{\beta-1}{n}}
  65. s i 2 = a m i + b m i 2 s_{i}^{2}=am_{i}+bm_{i}^{2}\,
  66. p 0 = exp ( - a m b ) p_{0}=\exp(-am^{b})
  67. log m = c + d log p 0 \log m=c+d\log p_{0}
  68. P 1 = 1 - e x p ( - e x p ( l o g e ( A 2 a ) b - 2 + l o g e ( n ) ( b - 1 b - 2 - 1 ) - c d ) ) P_{1}=1-exp(-exp(\frac{\frac{log_{e}(\frac{A^{2}}{a})}{b-2}+log_{e}(n)(\frac{b% -1}{b-2}-1)-c}{d}))
  69. N = n P 1 N=nP_{1}
  70. A 2 = D 2 z α 2 2 A^{2}=\frac{D^{2}}{z^{2}_{\frac{\alpha}{2}}}
  71. log ( m ) = log ( a ) + b log ( - log ( p 0 ) ) \log(m)=\log(a)+b\log(-\log(p_{0}))
  72. p 0 = e x p ( - A m B ) p_{0}=exp(-Am^{B})
  73. P = 1 - a e b m P=1-ae^{bm}
  74. m a = m ( 1 - v a r ( log ( m i ) ) 2 ) m_{a}=m(1-\frac{var(\log(m_{i}))}{2})
  75. m a = m e ( M S E / 2 ) m_{a}=me^{(MSE/2)}
  76. V a r ( m ) = m 2 ( c 1 + c 2 - c 3 + M S E ) Var(m)=m^{2}(c_{1}+c_{2}-c_{3}+MSE)
  77. c 1 = β 2 ( 1 - p 0 ) n p 0 log e ( p 0 ) 2 c_{1}=\frac{\beta^{2}(1-p_{0})}{np_{0}\log_{e}(p_{0})^{2}}
  78. c 2 = M S E N + s β 2 ( log e ( log e ( p 0 ) ) - p 2 ) c_{2}=\frac{MSE}{N}+s_{\beta}^{2}(\log_{e}(\log_{e}(p_{0}))-p^{2})
  79. c 3 = exp ( a + ( b - 2 ) [ α - β log e ( p 0 ) ] ) n c_{3}=\frac{\exp(a+(b-2)[\alpha-\beta\log_{e}(p_{0})])}{n}
  80. s 2 = a + b log e ( m ) s^{2}=a+b\log_{e}(m)
  81. s 2 = a p b ( 1 - p ) c s^{2}=ap^{b}(1-p)^{c}
  82. log ( s 2 ) = log ( a ) + b log ( p ) + c log ( 1 - p ) . \log(s^{2})=\log(a)+b\log(p)+c\log(1-p).
  83. log ( s 2 / n 2 ) = a + b log ( p ( 1 - p ) / n ) \log(s^{2}/n^{2})=a+b\log(p(1-p)/n)
  84. 1 / k = a m b - 2 - 1 / m 1/k=am^{b-2}-1/m
  85. A x k + x = N log ( 1 + m / k ) \sum\frac{A_{x}}{k+x}=N\log(1+m/k)
  86. U = s 2 - m + m 2 / k U=s^{2}-m+m^{2}/k
  87. V a r ( U ) = 2 m p 2 q ( 1 - R 2 - log ( 1 - R ) - R ) + p 4 ( 1 - R ) - k - 1 - k R N ( - log ( 1 - R ) - R ) 2 Var(U)=2mp^{2}q(\frac{1-R^{2}}{-\log(1-R)-R})+p^{4}\frac{(1-R)^{-k}-1-kR}{N(-% \log(1-R)-R)^{2}}
  88. σ 2 = μ + a μ p \sigma^{2}=\mu+a\mu^{p}
  89. k = m 2 / ( s 2 - m ) k=m^{2}/(s^{2}-m)
  90. k i = a + b m i k_{i}=a+bm_{i}
  91. C = 100 ( s 2 - m ) 0.5 m C=\frac{100(s^{2}-m)^{0.5}}{m}
  92. I c = Σ x 2 ( Σ x ) 2 I_{c}=\frac{\Sigma x^{2}}{(\Sigma x)^{2}}
  93. I c = s 2 + ( n m ) 2 ( n m ) 2 = 1 n 2 s 2 m 2 + 1 I_{c}=\frac{s^{2}+(nm)^{2}}{(nm)^{2}}=\frac{1}{n^{2}}\frac{s^{2}}{m^{2}}+1
  94. I c = a m b - 2 n 2 + 1 I_{c}=\frac{am^{b-2}}{n^{2}}+1
  95. IMC = m + s 2 / m - 1 \mathrm{IMC}=m+s^{2}/m-1
  96. IP = I M C / m \mathrm{IP}=IMC/m
  97. s 2 = m + m 2 k s^{2}=m+\frac{m^{2}}{k}
  98. S E ( I P ) = 1 k 2 [ v a r ( k ) + k ( k + 1 ) ( k + m ) m q ] SE(IP)=\frac{1}{k^{2}}[var(k)+\frac{k(k+1)(k+m)}{mq}]
  99. IMC = m + a - 1 m 1 - b - 1 \mathrm{IMC}=m+a^{-1}m^{1-b}-1
  100. IP = 1 + a - 1 m - b - 1 m \mathrm{IP}=1+a^{-1}m^{-b}-\frac{1}{m}
  101. y i = m i + s 2 / m i - 1 y_{i}=m_{i}+s^{2}/m_{i}-1
  102. n = ( t D ) 2 ( a + 1 m + b - 1 ) n=(\frac{t}{D})^{2}(\frac{a+1}{m}+b-1)
  103. N u = i m c + t ( i ( a + 1 ) m c + ( b - 1 ) m c 2 ) 1 / 2 N_{u}=im_{c}+t(i(a+1)m_{c}+(b-1)m_{c}^{2})^{1/2}
  104. N l = i m c - t ( i ( a + 1 ) m c + ( b - 1 ) m c 2 ) 1 / 2 N_{l}=im_{c}-t(i(a+1)m_{c}+(b-1)m_{c}^{2})^{1/2}
  105. T n = a + 1 D 2 - b - 1 n T_{n}=\frac{a+1}{D^{2}-\frac{b-1}{n}}
  106. T n = ( 1 - n N ) a + 1 D 2 - ( 1 - n N ) b - 1 n T_{n}=(1-\frac{n}{N})\frac{a+1}{D^{2}-(1-\frac{n}{N})\frac{b-1}{n}}
  107. I m = x ( x - 1 ) n m ( m - 1 ) I_{m}=\frac{\sum x(x-1)}{nm(m-1)}
  108. I m = n x 2 - x ( x ) 2 - x I_{m}=n\frac{\sum x^{2}-\sum x}{(\sum x)^{2}-\sum x}
  109. I m = n I M C ( n m - 1 ) I_{m}=\frac{nIMC}{(nm-1)}
  110. I m ( x - 1 ) + n - x I_{m}(\sum x-1)+n-\sum x
  111. z = I m - 1 ( 2 / n m 2 ) z=\frac{I_{m}-1}{(2/nm^{2})}
  112. M u = χ 0.975 2 - k + x x - 1 M_{u}=\frac{\chi^{2}_{0.975}-k+\sum x}{\sum x-1}
  113. M c = χ 0.025 2 - k + x x - 1 M_{c}=\frac{\chi^{2}_{0.025}-k+\sum x}{\sum x-1}
  114. I p = 0.5 + 0.5 ( I d - M c k - M c ) I_{p}=0.5+0.5(\frac{I_{d}-M_{c}}{k-M_{c}})
  115. I p = 0.5 ( I d - 1 M u - 1 ) I_{p}=0.5(\frac{I_{d}-1}{M_{u}-1})
  116. I p = - 0.5 ( I d - 1 M u - 1 ) I_{p}=-0.5(\frac{I_{d}-1}{M_{u}-1})
  117. I p = - 0.5 + 0.5 ( I d - M u M u ) I_{p}=-0.5+0.5(\frac{I_{d}-M_{u}}{M_{u}})
  118. 1 k = m * m - 1 \frac{1}{k}=\frac{m^{*}}{m}-1
  119. ID = ( n - 1 ) s 2 m \mathrm{ID}=\frac{(n-1)s^{2}}{m}
  120. x n > 3 \frac{\sum x}{n}>3
  121. ID = ( n - 1 ) a m b - 1 \mathrm{ID}=(n-1)am^{b-1}
  122. ICS = s 2 / m - 1 \mathrm{ICS}=s^{2}/m-1
  123. ICS = a m b - 1 - 1 \mathrm{ICS}=am^{b-1}-1
  124. C x = s 2 / m - 1 n m - 1 C_{x}=\frac{s^{2}/m-1}{nm-1}
  125. C x = a m b - 1 - 1 ( n m - 1 ) C_{x}=\frac{am^{b-1}-1}{(nm-1)}
  126. s 2 = n p ( 1 - p ) s^{2}=np(1-p)
  127. D = v a r o b s v a r b i n = s 2 n p ( 1 - p ) D=\frac{var_{obs}}{var_{bin}}=\frac{s^{2}}{np(1-p)}
  128. C = D ( n N - 1 ) - n N ( 2 N ( n 2 - n ) ) 1 / 2 C=\frac{D(nN-1)-nN}{(2N(n^{2}-n))^{1/2}}
  129. ρ = 1 - x i ( T - x i ) p ( 1 - p ) N T ( T - 1 ) \rho=1-\frac{\sum x_{i}(T-x_{i})}{p(1-p)NT(T-1)}
  130. ρ = D - 1 n - 1 \rho=\frac{D-1}{n-1}
  131. D = 1 + ( n - 1 ) θ 1 + θ D=1+\frac{(n-1)\theta}{1+\theta}
  132. m 0 = e x p ( l o g ( a ) 1 - b ) m_{0}=exp(\frac{log(a)}{1-b})
  133. v a r ( s 2 - m ) = 2 t 2 n - 1 var(s^{2}-m)=\frac{2t^{2}}{n-1}
  134. O T = n - 1 2 s 2 - m m O_{T}=\sqrt{\frac{n-1}{2}}\frac{s^{2}-m}{m}
  135. θ = s 2 m \theta=\frac{s^{2}}{m}
  136. V θ = 2 n ( n - 1 ) 2 V_{\theta}=\frac{2n}{(n-1)^{2}}
  137. V θ = 2 n - 1 V_{\theta}=\frac{2}{n-1}
  138. θ = a m b - 1 \theta=am^{b-1}
  139. θ = s 2 m = 1 n ( x i - n N ) 2 \theta=\frac{s^{2}}{m}=\frac{1}{n}\sum{(x_{i}-\frac{n}{N})^{2}}
  140. E ( θ ) = N N - 1 E(\theta)=\frac{N}{N-1}
  141. V a r ( θ ) = ( N - 1 ) 2 N 3 - 2 N - 3 n N 2 Var(\theta)=\frac{(N-1)^{2}}{N^{3}}-\frac{2N-3}{nN^{2}}
  142. V a r ( θ ) 2 N ( 1 - 1 n ) Var(\theta)\sim\ \frac{2}{N}(1-\frac{1}{n})

Taylor–Goldstein_equation.html

  1. g g
  2. L ρ L_{\rho}
  3. 𝐮 = [ U ( z ) + u ( x , z , t ) , 0 , w ( x , z , t ) ] , \mathbf{u}=\left[U(z)+u^{\prime}(x,z,t),0,w^{\prime}(x,z,t)\right],\,
  4. ( U ( z ) , 0 , 0 ) (U(z),0,0)
  5. 𝐮 exp ( i α ( x - c t ) ) \mathbf{u}^{\prime}\propto\exp(i\alpha(x-ct))
  6. u x = d ϕ ~ / d z , u z = - i α ϕ ~ u_{x}^{\prime}=d\tilde{\phi}/dz,u_{z}^{\prime}=-i\alpha\tilde{\phi}
  7. ( U - c ) 2 ( d 2 ϕ ~ d z 2 - α 2 ϕ ~ ) + [ N 2 - ( U - c ) d 2 U d z 2 ] ϕ ~ = 0 , (U-c)^{2}\left({d^{2}\tilde{\phi}\over dz^{2}}-\alpha^{2}\tilde{\phi}\right)+% \left[N^{2}-(U-c){d^{2}U\over dz^{2}}\right]\tilde{\phi}=0,
  8. N = g L ρ N=\sqrt{g\over L_{\rho}}
  9. c c
  10. c c
  11. N N
  12. z = z 1 z=z_{1}
  13. z = z 2 : z=z_{2}:
  14. α ϕ ~ = d ϕ ~ d z = 0 at z = z 1 and z = z 2 . \alpha\tilde{\phi}={d\tilde{\phi}\over dz}=0\quad\,\text{ at }z=z_{1}\,\text{ % and }z=z_{2}.

Technological_theory_of_social_production.html

  1. L L
  2. K K
  3. Y Y
  4. L L
  5. K K
  6. Y = Y ( K , L ) . Y=Y(K,L).
  7. K K^{\prime}
  8. L L^{\prime}
  9. Y = Y ( K , L ) . Y=Y(K^{\prime},L^{\prime}).
  10. K = A K ( t ) K K^{\prime}=A_{K}(t)K
  11. L = A L ( t ) L L^{\prime}=A_{L}(t)L
  12. K K
  13. L L
  14. P = A K ( t ) K P=A_{K}(t)K
  15. L = L L^{\prime}=L
  16. K K
  17. Y Y
  18. Y = Y ( K , L , P ) Y=Y(K,L,P)
  19. K K
  20. L L
  21. P P
  22. P P
  23. K K
  24. L L
  25. P P
  26. d K d t = I - μ K , d L d t = ( λ ¯ I K - μ ) L , d P d t = ( ε ¯ I K - μ ) P . \frac{dK}{dt}=I-\mu K,\quad\frac{dL}{dt}=\left(\bar{\lambda}\frac{I}{K}-\mu% \right)L,\quad\frac{dP}{dt}=\left(\bar{\varepsilon}\frac{I}{K}-\mu\right)P.
  27. I I
  28. λ \lambda
  29. ε \varepsilon
  30. λ ¯ = λ K / L \bar{\lambda}=\lambda K/L
  31. ε ¯ = ε K / P . \bar{\varepsilon}=\varepsilon K/P.
  32. μ \mu
  33. L L
  34. K K
  35. P P
  36. P P
  37. L L
  38. K K
  39. L L
  40. P P
  41. Y = { ξ K , ξ > 0 Y 0 L L 0 ( L 0 L P P 0 ) α , 0 < α < 1 Y=\begin{cases}\xi K,&\xi>0\\ Y_{0}\frac{L}{L_{0}}\left(\frac{L_{0}}{L}\frac{P}{P_{0}}\right)^{\alpha},&0<% \alpha<1\end{cases}
  42. Y 0 Y_{0}
  43. L 0 L_{0}
  44. P 0 P_{0}
  45. ξ \xi
  46. α \alpha
  47. 1 ξ d ξ d t = ln ( L 0 L P P 0 ) d α d t , α = 1 - λ ¯ ε ¯ - λ ¯ \frac{1}{\xi}\frac{d\xi}{dt}=\ln\left(\frac{L_{0}}{L}\frac{P}{P_{0}}\right)% \frac{d\alpha}{dt},\quad\alpha=\frac{1-\bar{\lambda}}{\bar{\varepsilon}-\bar{% \lambda}}
  48. δ ~ ( t ) δ = 1 K d K d t , ν ~ ( t ) ν = 1 L d L d t , η ~ ( t ) η = 1 P d P d t , \tilde{\delta}(t)\geq\delta=\frac{1}{K}\,\frac{dK}{dt},\quad\tilde{\nu}(t)\geq% \nu=\frac{1}{L}\,\frac{dL}{dt},\quad\tilde{\eta}(t)\geq\eta=\frac{1}{P}\,\frac% {dP}{dt},
  49. I = ( δ + μ ) K = min { ( δ ~ + μ ) K ( ν ~ + μ ) K / λ ¯ ( η ~ + μ ) K / ε ¯ . I=(\delta+\mu)K=\min\begin{cases}(\tilde{\delta}+\mu)K\\ (\tilde{\nu}+\mu)K/\bar{\lambda}\\ (\tilde{\eta}+\mu)K/\bar{\varepsilon}\end{cases}.
  50. d λ ¯ d t = - 1 τ ( λ ¯ - ν ~ + μ δ ~ + μ ) , d ε ¯ d t = - 1 τ ( ε ¯ - η ~ + μ δ ~ + μ ) , \frac{d\bar{\lambda}}{dt}=-\frac{1}{\tau}\left(\bar{\lambda}-\frac{\tilde{\nu}% +\mu}{\tilde{\delta}+\mu}\right),\quad\frac{d\bar{\varepsilon}}{dt}=-\frac{1}{% \tau}\left(\bar{\varepsilon}-\frac{\tilde{\eta}+\mu}{\tilde{\delta}+\mu}\right),
  51. τ \tau
  52. ν ~ \tilde{\nu}
  53. η ~ \tilde{\eta}
  54. ν ~ \tilde{\nu}
  55. η ~ \tilde{\eta}
  56. P / L P/L
  57. B = Y L = Y 0 L 0 ( L 0 L P P 0 ) α . B=\frac{Y}{L}=\frac{Y_{0}}{L_{0}}\left(\frac{L_{0}}{L}\frac{P}{P_{0}}\right)^{% \alpha}.
  58. 1 B d B d t = ( 1 - λ ¯ ) ( ν + μ ) λ ¯ + 1 ξ d ξ d t \frac{1}{B}\frac{dB}{dt}=\frac{(1-\bar{\lambda})(\nu+\mu)}{\bar{\lambda}}+% \frac{1}{\xi}\frac{d\xi}{dt}
  59. λ ¯ \bar{\lambda}
  60. λ ¯ = 1 \bar{\lambda}=1
  61. λ ¯ < 1 \bar{\lambda}<1
  62. K = K 0 e δ t , L = L 0 e ν t , P = P 0 e η t . K=K_{0}e^{\delta t},\quad L=L_{0}e^{\nu t},\quad P=P_{0}e^{\eta t}.
  63. Y = Y 0 e [ ν + α ( η - ν ) ] t = Y 0 e δ t . Y=Y_{0}e^{[\nu+\alpha(\eta-\nu)]t}=Y_{0}e^{\delta\,t}.
  64. ( 1 - α ) ν (1-\alpha)\nu
  65. α η \alpha\eta
  66. ν 0 \nu\rightarrow 0
  67. α \alpha
  68. δ \delta

Teichmüller_character.html

  1. ω ( x ) = lim n x p n \omega(x)=\lim_{n\rightarrow\infty}x^{p^{n}}

Template:Automatic_taxobox::doc::gory_technical_details.html

  1. O ( n 2 ) O(n^{2})\,
  2. n n
  3. n 2 + n 2 + k 2 + k 2 \frac{n^{2}+n}{2}+\frac{k^{2}+k}{2}
  4. k k

Template:Did_you_know_nominations::Cherry_blossom_front.html

  1. D T S = exp ( ( 9.5 × 10 3 ) . ( T - 288.2 288.2 T ) ) DTS=\exp((9.5\times 10^{3}).(\frac{T-288.2}{288.2T}))

Template:Did_you_know_nominations::Log-Cauchy_distribution.html

  1. γ \gamma
  2. μ \mu
  3. σ \sigma
  4. γ \gamma

Template:Did_you_know_nominations::Sparse_Distributed_Memory.html

  1. 2 n 2^{n}
  2. 2 n 2^{n}

Template:Material_properties_equations_(thermodynamics).html

  1. c = c=
  2. T T
  3. S \partial S
  4. N N
  5. T \partial T
  6. β = - \beta=-
  7. 1 1
  8. V \partial V
  9. V V
  10. p \partial p
  11. α = \alpha=
  12. 1 1
  13. V \partial V
  14. V V
  15. T \partial T

Template:Mvar::doc.html

  1. x < s u b > 1 x<sub>1

Template:Oiiint::doc.html

  1. Ω {\scriptstyle\Omega}
  2. F s d Σ \frac{\partial F}{\partial s}{\rm d}\Sigma
  3. Ω {\scriptstyle\Omega}
  4. F s d Σ \frac{\partial F}{\partial s}{\rm d}\Sigma
  5. Ω {\scriptstyle\Omega}
  6. F s d Σ \frac{\partial F}{\partial s}{\rm d}\Sigma

Template:Oiint::sandbox.html

  1. C E d s y m b o l = t \oint_{C}{E}\cdot{\rm d}symbol{\ell}=\frac{\partial}{\partial t}
  2. B d S {B}\cdot{\rm d}{S}
  3. S F d S = Σ \frac{}{}_{S}{F}\cdot{\rm d}{S}=\Sigma

Template:Simple_recursion::doc.html

  1. φ - 1 = φ - 1 \varphi^{-1}=\varphi-1

Template:Smallmath::doc.html

  1. 9 1 2 = 3 9^{\frac{1}{2}}=3

Template:Sqrt::doc.html

  1. 2 \sqrt{2}

Template:Table_of_thermodynamic_potentials.html

  1. U U
  2. ( T d S - p d V + i μ i d N i ) \int(TdS-pdV+\sum_{i}\mu_{i}dN_{i})
  3. S , V , { N i } S,V,\{N_{i}\}
  4. F F
  5. U - T S U-TS
  6. T , V , { N i } T,V,\{N_{i}\}
  7. H H
  8. U + p V U+pV
  9. S , p , { N i } S,p,\{N_{i}\}
  10. G G
  11. U + p V - T S U+pV-TS
  12. T , p , { N i } T,p,\{N_{i}\}
  13. Ω \Omega
  14. Φ G \Phi_{G}
  15. U - T S - U-TS-
  16. i \sum_{i}\,
  17. μ i N i \mu_{i}N_{i}
  18. T , V , { μ i } T,V,\{\mu_{i}\}

Tennis_(paper_game).html

  1. S i , t S_{i,t}
  2. B t B_{t}
  3. B 0 = 0 B_{0}=0
  4. S i , 0 = 50 S_{i,0}=50
  5. 0 Z i , t S i , t - 1 0\leq Z_{i,t}\leq S_{i,t-1}
  6. Z i , t = 0 Z_{i,t}=0
  7. S i , t = 0 S_{i,t}=0
  8. S i , t = S i , t - 1 - Z i , t S_{i,t}=S_{i,t-1}-Z_{i,t}
  9. Z 1 , t > Z 2 , t Z_{1,t}>Z_{2,t}
  10. B t = 1 B_{t}=1
  11. B t = B t - 1 + 1 B_{t}=B_{t-1}+1
  12. Z 1 , t < Z 2 , t Z_{1,t}<Z_{2,t}
  13. Z 1 , t = Z 2 , t Z_{1,t}=Z_{2,t}
  14. Z 1 , t Z_{1,t}
  15. Z 2 , t Z_{2,t}
  16. S 1 , t S_{1,t}
  17. S 2 , t S_{2,t}
  18. B t B_{t}
  19. Z 1 , t Z_{1,t}
  20. Z 2 , t Z_{2,t}
  21. S 1 , t S_{1,t}
  22. S 2 , t S_{2,t}
  23. B t B_{t}

Tennis_racket_theorem.html

  1. I 1 ω ˙ 1 \displaystyle I_{1}\dot{\omega}_{1}
  2. I 1 , I 2 , I 3 I_{1},I_{2},I_{3}
  3. I 1 > I 2 > I 3 I_{1}>I_{2}>I_{3}
  4. ω 1 , ω 2 , ω 3 \omega_{1},\omega_{2},\omega_{3}
  5. I 1 I_{1}
  6. ω ˙ 1 ~{}\dot{\omega}_{1}
  7. ω 1 ~{}\omega_{1}
  8. ω ˙ 3 \dot{\omega}_{3}
  9. I 2 I 3 ω ¨ 2 = ( I 3 - I 1 ) ( I 1 - I 2 ) ( ω 1 ) 2 ω 2 i.e. ω ¨ 2 = (negative quantity) × ω 2 \begin{aligned}\displaystyle I_{2}I_{3}\ddot{\omega}_{2}&\displaystyle=(I_{3}-% I_{1})(I_{1}-I_{2})(\omega_{1})^{2}\omega_{2}\\ \displaystyle\,\text{i.e.}~{}~{}~{}~{}\ddot{\omega}_{2}&\displaystyle=\,\text{% (negative quantity)}\times\omega_{2}\end{aligned}
  10. ω 2 \omega_{2}
  11. I 3 I_{3}
  12. I 2 I_{2}
  13. ω ˙ 2 \dot{\omega}_{2}
  14. ω 2 ~{}\omega_{2}
  15. ω ˙ 3 \dot{\omega}_{3}
  16. I 1 I 3 ω ¨ 1 = ( I 2 - I 3 ) ( I 1 - I 2 ) ( ω 2 ) 2 ω 1 i.e. ω ¨ 1 = (positive quantity) × ω 1 \begin{aligned}\displaystyle I_{1}I_{3}\ddot{\omega}_{1}&\displaystyle=(I_{2}-% I_{3})(I_{1}-I_{2})(\omega_{2})^{2}\omega_{1}\\ \displaystyle\,\text{i.e.}~{}~{}~{}~{}\ddot{\omega}_{1}&\displaystyle=\,\text{% (positive quantity)}\times\omega_{1}\end{aligned}
  17. ω 1 \omega_{1}

Tensile_testing.html

  1. 3 / 8 {3}/{8}
  2. 3 / 8 {3}/{8}
  3. 5 / 16 {5}/{16}
  4. 5 / 32 {5}/{32}
  5. 3 / 32 {3}/{32}
  6. 5 / 8 {5}/{8}
  7. ε = Δ L L 0 = L - L 0 L 0 \varepsilon=\frac{\Delta L}{L_{0}}=\frac{L-L_{0}}{L_{0}}
  8. σ = F n A \sigma=\frac{F_{n}}{A}

Tensiometer_(soil_science).html

  1. Ψ m \Psi_{m}

Tensor_glyph.html

  1. 3 × 3 3\times 3

Terasecond_and_longer.html

  1. 10 10 10 76.66 10^{10^{10^{76.66}}}
  2. 10 10 10 76.66 10^{10^{10^{76.66}}}
  3. 10 10 10 120 10^{10^{10^{120}}}
  4. 10 10 10 120 10^{10^{10^{120}}}
  5. 10 10 10 10 13 10^{10^{10^{10^{13}}}}
  6. 10 10 10 10 13 10^{10^{10^{10^{13}}}}

Teresa_Cohen.html

  1. n = 1 n - p , \sum_{n=1}^{\infty}n^{-p},

Test_(assessment).html

  1. 6.14 * 7.95 = 48.813 6.14*7.95=48.813
  2. 6 * 8 = 48 6*8=48

Test_and_evaluation_master_plan.html

  1. - 40 o -40^{o}

Testing_and_performance_of_IC_engines.html

  1. b p = 2 π N τ 60 bp={2\pi N\tau\over 60}
  2. τ \tau\,
  3. N N\,
  4. b p bp
  5. i p = p A R k 60 ip={pARk\over 60}
  6. p p\,
  7. A A\,
  8. k k\,
  9. E = b p i p E={bp\over ip}
  10. E = b p b p + f p E={bp\over{bp+fp}}
  11. f p = i p - b p {fp={ip-bp}}
  12. ( F / A ) = A c t u a l f u e l - a i r r a t i o / S t o i c h i o m e t r i c f u e l - a i r r a t i o (F/A)={Actualfuel-airratio/Stoichiometricfuel-airratio}
  13. ( A / F ) = A c t u a l a i r - f u e l r a t i o / S t o i c h i o m e t r i c a i r - f u e l r a t i o (A/F)={Actualair-fuelratio/Stoichiometricair-fuelratio}
  14. W = 2 π r F W={2\pi rF}
  15. p = i p 60 L A R K p={ip60\over LARK}
  16. p p\,
  17. i p ip\,
  18. A A\,
  19. R R\,
  20. k k\,
  21. f m e p = i m e p - b m e p fmep={imep-bmep}
  22. τ = b m e p A R K 2 π \tau={bmepARK\over 2\pi}

Tests_of_relativistic_energy_and_momentum.html

  1. E k = 1 2 m v 2 , p = m v . E_{k}=\tfrac{1}{2}mv^{2},\quad p=mv.\,
  2. E 2 - ( p c ) 2 = ( m c 2 ) 2 E^{2}-(pc)^{2}=(mc^{2})^{2}\,
  3. E 0 E_{0}
  4. E E
  5. E k E_{k}
  6. p p
  7. E 0 = m c 2 , E = γ m c 2 , E k = ( γ - 1 ) m c 2 , p = γ m v E_{0}=mc^{2},\quad E=\gamma mc^{2},\quad E_{k}=(\gamma-1)mc^{2},\quad p=\gamma mv
  8. γ = 1 / 1 - ( v / c ) 2 \gamma=1/\sqrt{1-(v/c)^{2}}
  9. M = γ m M=\gamma m\,
  10. m T = m γ m_{T}=m\gamma
  11. M M
  12. M m = p m v = E m c 2 = γ \frac{M}{m}=\frac{p}{mv}=\frac{E}{mc^{2}}=\gamma
  13. p 2 / ( m γ ) p^{2}/(m\gamma)
  14. Δ v / c = ( - 1.3 ± 2.7 ) × 10 - 6 \Delta v/c=(-1.3\pm 2.7)\times 10^{-6}
  15. Δ v / c = 2 × 10 - 7 \Delta v/c=2\times 10^{-7}
  16. p = m v p=mv
  17. Δ v / c = 10 - 5 \Delta v/c=10^{-5}
  18. E 0 = m c 2 E_{0}=mc^{2}
  19. γ m c 2 \gamma mc^{2}

Tetranitrogen.html

  1. N 4 + + C H 4 - C H 4 + N 4 N_{4}^{+}\xrightarrow[-CH_{4}^{+}]{+CH_{4}}N_{4}
  2. N 4 + O 2 - e - N 4 + N_{4}\xrightarrow[-e^{-}]{+O_{2}}N_{4}^{+}

Textual_variants_in_the_Gospel_of_Matthew.html

  1. 𝔓 \mathfrak{P}
  2. 𝔓 \mathfrak{P}
  3. 𝔓 \mathfrak{P}
  4. 𝔓 \mathfrak{P}
  5. 𝔓 \mathfrak{P}
  6. 𝔓 \mathfrak{P}
  7. 𝔓 \mathfrak{P}
  8. 𝔓 \mathfrak{P}
  9. 𝔓 \mathfrak{P}
  10. 𝔓 \mathfrak{P}
  11. 𝔓 \mathfrak{P}
  12. 𝔓 \mathfrak{P}
  13. 𝔓 \mathfrak{P}
  14. 𝔓 \mathfrak{P}

The_Chinkees_Are_Coming.html

  1. [ - F o r m u l a E r r o r - ] [-FormulaError-]

The_Entropy_Influence_Conjecture.html

  1. f : { - 1 , 1 } n { - 1 , 1 } f:\{-1,1\}^{n}\to\{-1,1\}\!
  2. f = S [ n ] f ^ ( S ) x S f=\sum_{S\subset[n]}\hat{f}(S)x_{S}\!
  3. x S = i S x i x_{S}=\prod_{i\in S}x_{i}\!
  4. I ( f ) = S | S | f ^ 2 ( S ) I(f)=\sum_{S}|S|\hat{f}^{2}(S)\!
  5. H ( f ) = - S f ^ 2 ( S ) log f ^ 2 ( S ) H(f)=-\sum_{S}\hat{f}^{2}(S)\log\hat{f}^{2}(S)\!
  6. x log x = 0 x\log x=0\!
  7. x = 0 x=0\!
  8. f : { - 1 , 1 } n { - 1 , 1 } f:\{-1,1\}^{n}\to\{-1,1\}\!
  9. H ( f ) C I ( f ) H(f)\leq CI(f)\!

The_Musical_Moment.html

  1. \flat

The_Prisoner_of_Benda.html

  1. π π
  2. k k
  3. n n
  4. 1... n 1...n
  5. π = ( 1 2 k k + 1 n 2 3 1 k + 1 n ) . \pi=\begin{pmatrix}1&2&...&k&k+1&...&n\\ 2&3&...&1&k+1&...&n\end{pmatrix}.
  6. x x
  7. y y
  8. π * = ( 1 2 k k + 1 n x y 2 3 1 k + 1 n x y ) . \pi^{*}=\begin{pmatrix}1&2&...&k&k+1&...&n&x&y\\ 2&3&...&1&k+1&...&n&x&y\end{pmatrix}.
  9. a b ab
  10. a a
  11. b b
  12. σ σ
  13. σ = ( x 1 ) ( x 2 ) ( x i ) ( y i + 1 ) ( y i + 2 ) ( y k ) ( x i + 1 ) ( y 1 ) . \sigma=(x~{}~{}1)(x~{}~{}2)\cdots(x~{}~{}i)(y~{}~{}i+1)(y~{}~{}i+2)\cdots(y~{}% ~{}k)(x~{}~{}i+1)(y~{}~{}1).
  14. n n
  15. x , y x,y
  16. π * σ = ( 1 2 n x y 1 2 n y x ) . \pi^{*}\sigma=\begin{pmatrix}1&2&...&n&x&y\\ 1&2&...&n&y&x\end{pmatrix}.
  17. σ σ
  18. k k
  19. x x
  20. y y
  21. ( x y ) (xy)
  22. π π
  23. n n
  24. x x
  25. y y
  26. ( x y ) (xy)

The_Product_Space.html

  1. RCA c , i = x ( c , i ) / i x ( c , i ) c x ( c , i ) / c , i x ( c , i ) \,\text{RCA}_{c,i}=\frac{{x(c,i)}/{\sum_{i}x(c,i)}}{{\sum_{c}x(c,i)}/{\sum_{c,% i}x(c,i)}}
  2. ϕ i , j = min { Pr ( RCA x i 1 RCA x j 1 ) , Pr ( RCA x j 1 RCA x i ) 1 } \phi_{i,j}=\min\{\Pr(\,\text{RCA}x_{i}\geq 1\mid\,\text{RCA}x_{j}\geq 1),\Pr(% \,\text{RCA}x_{j}\geq 1\mid\,\text{RCA}x_{i})\geq 1\}
  3. Pr ( RCA i 1 RCA j 1 ) \Pr(\,\text{RCA}_{i}\geq 1\mid\,\text{RCA}_{j}\geq 1)
  4. ω j k = i x i ϕ i j i ϕ i j \omega_{j}^{k}=\frac{{\displaystyle\sum_{i}x_{i}\phi_{ij}}}{{\displaystyle\sum% _{i}\phi_{ij}}}
  5. H j = k = 1 T ω j k / T k = T + 1 N ω j k / ( N - T ) H_{j}=\frac{{\displaystyle\sum_{k=1}^{T}\omega_{j}^{k}/T}}{{\displaystyle\sum_% {k=T+1}^{N}\omega_{j}^{k}/(N-T)}}
  6. H j H_{j}

Theil–Sen_estimator.html

  1. m m
  2. x x
  3. x x
  4. m m
  5. y y
  6. b b
  7. x x
  8. x x
  9. 1 - 1 2 29.3 % 1-\frac{1}{\sqrt{2}}\approx 29.3\%
  10. n n
  11. x x
  12. O ( n l o g n ) O(nlogn)
  13. O ( n log n ) O(n\sqrt{\log n})

Theoretical_gravity.html

  1. g ϕ = 9.780327 ( 1 + 0.00516323 sin 2 ( ϕ ) + 0.00002269 sin 4 ( ϕ ) ) m s 2 \ g_{\phi}=9.780327\left(1+0.00516323\sin^{2}(\phi)+0.00002269\sin^{4}(\phi)% \right)\,\frac{\mathrm{m}}{\mathrm{s}^{2}}
  2. g ϕ = ( 9.8061999 - 0.0259296 cos ( 2 ϕ ) + 0.0000567 cos 2 ( 2 ϕ ) ) m s 2 \ g_{\phi}=\left(9.8061999-0.0259296\cos(2\phi)+0.0000567\cos^{2}(2\phi)\right% )\,\frac{\mathrm{m}}{\mathrm{s}^{2}}
  3. g ϕ = ( 9.7803267714 1 + 0.00193185138639 sin 2 ϕ 1 - 0.00669437999013 sin 2 ϕ ) m s 2 \ g_{\phi}=\left(9.7803267714~{}\frac{1+0.00193185138639\sin^{2}\phi}{\sqrt{1-% 0.00669437999013\sin^{2}\phi}}\right)\,\frac{\mathrm{m}}{\mathrm{s}^{2}}

Theory_of_indispensable_attributes.html

  1. \mathcal{E}
  2. \mathcal{M}
  3. \mathcal{E}
  4. \mathcal{F}
  5. \mathcal{E}
  6. \mathcal{M}
  7. \mathcal{F}

Theory_of_solar_cells.html

  1. I = I L - I D - I S H I=I_{L}-I_{D}-I_{SH}
  2. V j = V + I R S V_{j}=V+IR_{S}
  3. I D = I 0 { exp [ q V j n k T ] - 1 } I_{D}=I_{0}\left\{\exp\left[\frac{qV_{j}}{nkT}\right]-1\right\}
  4. k T / q 0.0259 kT/q\approx 0.0259
  5. I S H = V j R S H I_{SH}=\frac{V_{j}}{R_{SH}}
  6. I = I L - I 0 { exp [ q ( V + I R S ) n k T ] - 1 } - V + I R S R S H . I=I_{L}-I_{0}\left\{\exp\left[\frac{q(V+IR_{S})}{nkT}\right]-1\right\}-\frac{V% +IR_{S}}{R_{SH}}.
  7. V O C n k T q ln ( I L I 0 + 1 ) . V_{OC}\approx\frac{nkT}{q}\ln\left(\frac{I_{L}}{I_{0}}+1\right).
  8. I S C I L . I_{SC}\approx I_{L}.
  9. J = J L - J 0 { exp [ q ( V + J r S ) n k T ] - 1 } - V + J r S r S H J=J_{L}-J_{0}\left\{\exp\left[\frac{q(V+Jr_{S})}{nkT}\right]-1\right\}-\frac{V% +Jr_{S}}{r_{SH}}
  10. V V
  11. V O C = k T q ln ( I S C I 0 + 1 ) . V_{OC}=\frac{kT}{q}\ln\left(\frac{I_{SC}}{I_{0}}+1\right).

Thermal_ellipsoid.html

  1. U 1 U_{1}
  2. U 2 U_{2}
  3. U 3 U_{3}

Thermal_fluctuations.html

  1. 𝒱 \mathcal{V}
  2. 2 m 2m
  3. V V
  4. E \sqrt{E}
  5. E 2 m \sqrt{E}^{2m}
  6. 𝒱 = ( C E ) m Γ ( m + 1 ) , \mathcal{V}=\frac{(C\cdot E)^{m}}{\Gamma(m+1)},
  7. C C
  8. Γ \Gamma
  9. 2 m 2m
  10. Ω ( E ) = 𝒱 E = C m E m - 1 Γ ( m ) , \Omega(E)=\frac{\partial\mathcal{V}}{\partial E}=\frac{C^{m}\cdot E^{m-1}}{% \Gamma(m)},
  11. m Γ ( m ) = Γ ( m + 1 ) m\Gamma(m)=\Gamma(m+1)
  12. Ω ( E ) \Omega(E)
  13. 𝒵 ( β ) = 0 e - β E Ω ( E ) d E , \mathcal{Z}(\beta)=\int_{0}^{\infty}e^{-\beta E}\Omega(E)\,dE,
  14. β \beta
  15. E E^{\star}
  16. f ( E ; β ) = e - β E 𝒵 ( β ) Ω ( E ) , f(E;\beta)=\frac{e^{-\beta E}}{\mathcal{Z}(\beta)}\Omega(E),
  17. 𝒵 ( β ) \mathcal{Z}(\beta)
  18. E = - ln 𝒵 β , ( E - E ) 2 = ( Δ E ) 2 = 2 ln 𝒵 β 2 , \langle E\rangle=-\frac{\partial\ln\mathcal{Z}}{\partial\beta},\qquad\ \langle% (E-\langle E\rangle)^{2}\rangle=\langle(\Delta E)^{2}\rangle=\frac{\partial^{2% }\ln\mathcal{Z}}{\partial\beta^{2}},
  19. Ω ( E ) \Omega(E)
  20. e - β E Ω ( E ) e^{-\beta E}\Omega(E)
  21. E \langle E\rangle
  22. E E^{\star}
  23. f ( E ; β ) = e - β E 𝒵 ( β ) Ω ( E ) exp { - ( E - E ) 2 / 2 ( Δ E ) 2 } 2 π ( Δ E ) 2 . f(E;\beta)=\frac{e^{-\beta E}}{\mathcal{Z}(\beta)}\Omega(E)\approx\frac{\exp\{% -(E-\langle E\rangle)^{2}/2\langle(\Delta E)^{2}\rangle\}}{\sqrt{2\pi\langle(% \Delta E)^{2}\rangle}}.
  24. f ( E ; β ) f(E;\beta)
  25. Ω \Omega
  26. E m E^{m}
  27. β - m \beta^{-m}
  28. E = E E=\langle E\rangle
  29. Ω ( E ) = e β ( E ) E 𝒵 ( β ( E ) ) 2 π ( Δ E ) 2 . \Omega(\langle E\rangle)=\frac{e^{\beta(\langle E\rangle)\langle E\rangle}% \mathcal{Z}(\beta(\langle E\rangle))}{\sqrt{2\pi\langle(\Delta E)^{2}\rangle}}.
  30. β ( E ) = m / E \beta(\langle E\rangle)=m/\langle E\rangle
  31. ( Δ E ) 2 = E 2 / m \langle(\Delta E)^{2}\rangle=\langle E\rangle^{2}/m
  32. Ω ( E ) = E m - 1 m 2 π m m m e - m \Omega(\langle E\rangle)=\frac{\langle E\rangle^{m-1}m}{\sqrt{2\pi m}m^{m}e^{-% m}}
  33. m ! = Γ ( m + 1 ) m!=\Gamma(m+1)
  34. f ( E ; β ) = β ( β E ) m - 1 Γ ( m ) e - β E f(E;\beta)=\beta\frac{(\beta E)^{m-1}}{\Gamma(m)}e^{-\beta E}
  35. x x
  36. w ( x ) d x w(x)dx
  37. x x
  38. S S
  39. w ( x ) exp ( S ( x ) ) . w(x)\propto\exp\left(S(x)\right).
  40. w ( x ) = 1 2 π x 2 exp ( - x 2 2 x 2 ) . w(x)=\frac{1}{\sqrt{2\pi\langle x^{2}\rangle}}\exp\left(-\frac{x^{2}}{2\langle x% ^{2}\rangle}\right).
  41. x 2 \langle x^{2}\rangle
  42. w ( x 1 , x 2 , , x n ) d x 1 d x 2 d x n w(x_{1},x_{2},\ldots,x_{n})dx_{1}dx_{2}\ldots dx_{n}
  43. w = i , j = 1 n 1 ( 2 π ) n / 2 x i x j exp ( - x i x j 2 x i x j ) , w=\prod_{i,j=1\ldots n}\frac{1}{\left(2\pi\right)^{n/2}\sqrt{\langle x_{i}x_{j% }\rangle}}\exp\left(-\frac{x_{i}x_{j}}{2\langle x_{i}x_{j}\rangle}\right),
  44. x i x j \langle x_{i}x_{j}\rangle
  45. x i x j x_{i}x_{j}
  46. T , V , P T,V,P
  47. S S
  48. x i x j \langle x_{i}x_{j}\rangle
  49. k B k_{B}
  50. C P C_{P}
  51. C V C_{V}
  52. Δ T \Delta T
  53. Δ V \Delta V
  54. Δ S \Delta S
  55. Δ P \Delta P
  56. Δ T \Delta T
  57. T 2 C V \frac{T^{2}}{C_{V}}
  58. 0
  59. T T
  60. T 2 C V ( P T ) V \frac{T^{2}}{C_{V}}\left(\frac{\partial P}{\partial T}\right)_{V}
  61. Δ V \Delta V
  62. 0
  63. - T ( V P ) T -T\left(\frac{\partial V}{\partial P}\right)_{T}
  64. T ( V T ) P T\left(\frac{\partial V}{\partial T}\right)_{P}
  65. - T -T
  66. Δ S \Delta S
  67. T T
  68. T ( V T ) P T\left(\frac{\partial V}{\partial T}\right)_{P}
  69. C P C_{P}
  70. 0
  71. Δ P \Delta P
  72. T 2 C v ( P T ) V \frac{T^{2}}{C_{v}}\left(\frac{\partial P}{\partial T}\right)_{V}
  73. - T -T
  74. 0
  75. - T ( P V ) S -T\left(\frac{\partial P}{\partial V}\right)_{S}

Thermal_ionization.html

  1. µ µ
  2. ϕ ϕ
  3. n + n 0 = g + g 0 exp ( W - Δ E I k T ) \frac{n_{+}}{n_{0}}=\frac{g_{+}}{g_{0}}\exp\Bigg(\frac{W-\Delta E_{I}}{kT}\Bigg)
  4. n + n 0 \frac{n_{+}}{n_{0}}
  5. g + g 0 \frac{g_{+}}{g_{0}}
  6. e e
  7. W W
  8. Δ E I \Delta E_{I}
  9. k k
  10. T T
  11. Δ E A \Delta E_{A}

Thermal_lag.html

  1. T h e r m a l l a g ( s ) = 1 2 * α * Ω * L Thermal\ lag(s)={\sqrt{1\over{2*\alpha*\Omega}}*L}

Thermal_rocket.html

  1. I s p I_{sp}
  2. I s p = V e / g o = 1 g o 3 k b T m I_{sp}=V_{e}/g_{o}=\frac{1}{g_{o}}\sqrt{\frac{3k_{b}T}{m}}
  3. g o g_{o}
  4. k b k_{b}

Thermochemical_cycle.html

  1. \rightleftharpoons
  2. Δ H = Δ G + T Δ S \Delta H=\Delta G+T\Delta S
  3. Δ G = Δ G 0 - ( T - T 0 ) Δ S 0 \Delta G=\Delta G^{0}-(T-T^{0})\Delta S^{0}
  4. i Δ H i 0 = Δ H 0 \sum_{i}{\Delta H^{0}_{i}}=\Delta H^{0}
  5. i Δ S i 0 = Δ S 0 \sum_{i}{\Delta S^{0}_{i}}=\Delta S^{0}
  6. Δ G = i Δ G i \Delta G=\sum_{i}{\Delta G_{i}}
  7. Δ G = p ( Δ G i 0 - ( T i - T 0 ) Δ S i 0 ) + n ( Δ G i 0 - ( T i - T 0 ) Δ S i 0 ) \Delta G=\sum_{p}{(\Delta G^{0}_{i}-(T_{i}-T^{0})\Delta S^{0}_{i})}+\sum_{n}{(% \Delta G^{0}_{i}-(T_{i}-T^{0})\Delta S^{0}_{i})}
  8. Δ G = Δ G 0 + p ( T i - T 0 ) ( - Δ S i 0 ) + n ( T i - T 0 ) ( - Δ S i 0 ) \Delta G=\Delta G^{0}+\sum_{p}{(T_{i}-T^{0})(-\Delta S^{0}_{i})}+\sum_{n}{(T_{% i}-T^{0})(-\Delta S^{0}_{i})}
  9. p ( T i - T 0 ) ( - Δ S i 0 ) \sum_{p}{(T_{i}-T^{0})(-\Delta S^{0}_{i})}
  10. n ( T i - T 0 ) ( - Δ S i 0 ) \sum_{n}{(T_{i}-T^{0})(-\Delta S^{0}_{i})}
  11. Δ G = Δ G 0 - ( T H - T 0 ) p Δ S i 0 \Delta G=\Delta G^{0}-(T_{H}-T^{0})\sum_{p}{\Delta S^{0}_{i}}
  12. p Δ S i 0 Δ G 0 ( T H - T 0 ) \sum_{p}{\Delta S^{0}_{i}}\geq\frac{\Delta G^{0}}{(T_{H}-T^{0})}
  13. W Q T H - T 0 T H \frac{W}{Q}\leq\frac{T_{H}-T^{0}}{T_{H}}
  14. Q = i q i Q=\sum_{i}{q_{i}}
  15. q i = T H Δ S i q_{i}=T_{H}\Delta S_{i}
  16. Q = T H p Δ S i Q=T_{H}\sum_{p}{\Delta S_{i}}
  17. p Δ S i 0 Δ G 0 - W a d d ( T H - T 0 ) \sum_{p}{\Delta S^{0}_{i}}\geq\frac{\Delta G^{0}-W_{add}}{(T_{H}-T^{0})}
  18. p Δ S i 0 Δ G 0 ( ( 1 + f ) T H - T 0 ) \sum_{p}{\Delta S^{0}_{i}}\geq\frac{\Delta G^{0}}{((1+f)T_{H}-T^{0})}
  19. Δ H 1 0 Δ S 1 0 < Δ H 1 0 + Δ H 2 0 Δ S 1 0 + Δ S 2 0 \frac{\Delta H^{0}_{1}}{\Delta S^{0}_{1}}<\frac{\Delta H^{0}_{1}+\Delta H^{0}_% {2}}{\Delta S^{0}_{1}+\Delta S^{0}_{2}}

Thermoporometry_and_cryoporometry.html

  1. T 2 T_{2}

Thermoremanent_magnetization.html

  1. T C \scriptstyle T\text{C}
  2. T 1 \scriptstyle T_{1}
  3. T 1 , T 2 , \scriptstyle T_{1},T_{2},\ldots
  4. T 1 \scriptstyle T_{1}
  5. T 2 \scriptstyle T_{2}
  6. M A \scriptstyle M\text{A}
  7. H \scriptstyle H
  8. M B \scriptstyle M\text{B}
  9. M A ( H ) \scriptstyle M\text{A}(H)
  10. M B ( H ) \scriptstyle M\text{B}(H)
  11. H \scriptstyle H
  12. H \scriptstyle H
  13. M A \scriptstyle M\text{A}
  14. A \scriptstyle A
  15. M B \scriptstyle M\text{B}
  16. B \scriptstyle B
  17. M A \scriptstyle M\text{A}
  18. M B \scriptstyle M\text{B}
  19. M A B \scriptstyle M_{\,\text{A}\cup\,\text{B}}
  20. M A B = M A + M B \scriptstyle M_{\,\text{A}\cup\,\text{B}}=M\text{A}+M\text{B}
  21. A \scriptstyle A
  22. B \scriptstyle B
  23. T B \scriptstyle T\text{B}
  24. T B \scriptstyle T\text{B}
  25. T UB \scriptstyle T\text{UB}
  26. T B \scriptstyle T\text{B}

Theta_model.html

  1. f f
  2. g g
  3. h h
  4. I I
  5. x x
  6. y y
  7. θ \theta
  8. ε \varepsilon
  9. p p
  10. q q
  11. f f
  12. h h
  13. y y
  14. g g
  15. y ˙ \dot{y}
  16. x ˙ \dot{x}
  17. y ˙ \dot{y}
  18. x ˙ \dot{x}
  19. x x
  20. y y
  21. x ˙ = f ( x ) + ε 2 g ( x , y , ε ) \dot{x}=f(x)+\varepsilon^{2}g(x,y,\varepsilon)
  22. y ˙ = ε h ( x , y , ε ) \dot{y}=\varepsilon h(x,y,\varepsilon)
  23. x x
  24. p p
  25. x p x\in\mathbb{R}^{p}
  26. y y
  27. q q
  28. y q y\in\mathbb{R}^{q}
  29. ε \varepsilon
  30. f f
  31. g g
  32. h h
  33. x ˙ = f ( x ) \dot{x}=f(x)
  34. 2 \mathbb{R}^{2}
  35. x = 0 x=0
  36. y ˙ = h ( 0 , y , 0 ) \dot{y}=h(0,y,0)
  37. ε = 0 \varepsilon=0
  38. x ˙ = f ( x ) \dot{x}=f(x)
  39. x = 0 x=0
  40. x = 0 x=0\;
  41. y ˙ = h ( 0 , y , 0 ) \dot{y}=h(0,y,0)
  42. d θ d t = 1 - cos θ + ( 1 + cos θ ) I ( t ) , θ S 1 \frac{d\theta}{dt}=1-\cos\theta+(1+\cos\theta)I(t),\;\;\;\theta\in S^{1}
  43. θ \theta
  44. I ( t ) I(t)
  45. θ \theta
  46. θ = π \theta=\pi
  47. ( I < 0 ) (I<0)
  48. 2 \mathbb{R}^{2}
  49. S 1 S^{1}
  50. 2 \mathbb{R}^{2}
  51. ( I > 0 ) (I>0)
  52. θ ˙ \dot{\theta}
  53. I ( t ) = 0 I(t)=0
  54. d x d t = x 2 + I \frac{dx}{dt}=x^{2}+I
  55. x ( t ) x(t)\;
  56. - -\infty
  57. + +\infty
  58. T = π I T=\frac{\pi}{\sqrt{I}}
  59. I 0 + I\rightarrow 0^{+}
  60. I ( t ) I(t)
  61. I ( t ) := sin ( α t ) I(t):=\sin(\alpha t)
  62. α \alpha
  63. α t ( 0 , π ) \alpha t\in(0,\pi)
  64. I ( t ) I(t)
  65. θ \theta
  66. π \pi
  67. α t \alpha t
  68. π \pi
  69. α t \alpha t
  70. α t = π / 2 \alpha t=\pi/2
  71. α t = π \alpha t=\pi
  72. θ \theta
  73. θ = π \theta=\pi
  74. α t ( π , 2 π ) \alpha t\in(\pi,2\pi)
  75. S 1 × 2 S^{1}\times\mathbb{R}^{2}
  76. x 1 ˙ = f ¯ ( x 1 ) + ε 2 g ¯ ( x 1 , y , ε ) x 1 S 1 \dot{x_{1}}=\overline{f}(x_{1})+\varepsilon^{2}\overline{g}(x_{1},y,% \varepsilon)\;\;\;\;\;\;x_{1}\in S^{1}
  77. y ˙ = ε h ¯ ( x 1 , y , ε ) y q \dot{y}=\varepsilon\overline{h}(x_{1},y,\varepsilon)\;\;\;\;\;\;y\in\mathbb{R}% ^{q}
  78. ε 0 \varepsilon\rightarrow 0
  79. θ ˙ = ( 1 - cos θ ) + ( 1 + cos θ ) g ¯ ( 0 , y , 0 ) \dot{\theta}=(1-\cos\theta)+(1+\cos\theta)\overline{g}(0,y,0)
  80. y ˙ = 1 c h ¯ ( 0 , y , 0 ) \dot{y}=\frac{1}{c}\overline{h}(0,y,0)
  81. θ π \theta\neq\pi
  82. θ = π \theta=\pi
  83. I ( t ) := g ¯ ( 0 , y , 0 ) I(t):=\overline{g}(0,y,0)
  84. θ ˙ = f ( θ ) + g ( θ ) S ( t ) \dot{\theta}=f(\theta)+g(\theta)S(t)
  85. S ( t ) S(t)
  86. f f
  87. g g
  88. f ( θ ) = ( 1 - cos θ ) + I ( 1 + cos θ ) f(\theta)=(1-\cos\theta)+I(1+\cos\theta)\;
  89. g ( θ ) = ( 1 + cos θ ) g(\theta)=(1+\cos\theta)\;
  90. Z ( θ ) = K ( 1 + cos θ ) Z(\theta)=K(1+\cos\theta)\;
  91. τ u ˙ = a 0 ( u - u r e s t ) ( u - u c ) + R m I \tau\dot{u}=a_{0}(u-u_{rest})(u-u_{c})+R_{m}I
  92. a 0 a_{0}
  93. u u
  94. u r e s t u_{rest}
  95. u c u_{c}
  96. R m R_{m}
  97. τ \tau
  98. u c > u r e s t u_{c}>u_{rest}
  99. I = 0 I=0
  100. I I
  101. u u
  102. u r u_{r}
  103. + 20 +20
  104. - 80 -80
  105. Δ I \Delta I
  106. u ˙ = u 2 + Δ I \dot{u}=u^{2}+\Delta I
  107. θ ˙ = 1 - cos θ + ( 1 + cos θ ) Δ I \dot{\theta}=1-\cos\theta+(1+\cos\theta)\Delta I
  108. u ( t ) = tan ( θ / 2 ) u(t)=\tan(\theta/2)\;
  109. d tan ( x ) / d x = 1 / cos 2 ( x ) d\tan(x)/dx=1/\cos^{2}(x)
  110. u ˙ = 1 cos 2 ( θ / 2 ) 1 2 θ ˙ = u 2 + Δ I \dot{u}=\frac{1}{\cos^{2}(\theta/2)}\frac{1}{2}\dot{\theta}=u^{2}+\Delta I
  111. θ \theta
  112. θ ˙ = 2 [ cos 2 ( θ / 2 ) tan 2 ( θ / 2 ) + cos 2 ( θ / 2 ) Δ I ] = 2 [ sin 2 ( θ / 2 ) + cos 2 ( θ / 2 ) Δ I ] \dot{\theta}=2[\cos^{2}(\theta/2)\tan^{2}(\theta/2)+\cos^{2}(\theta/2)\Delta I% ]=2[\sin^{2}(\theta/2)+\cos^{2}(\theta/2)\Delta I]
  113. cos 2 ( x / 2 ) = 1 + cos ( x ) 2 \cos^{2}(x/2)=\frac{1+\cos(x)}{2}
  114. sin 2 ( x / 2 ) = 1 - cos ( x ) 2 \sin^{2}(x/2)=\frac{1-\cos(x)}{2}
  115. θ ˙ \dot{\theta}
  116. θ ˙ = 2 [ 1 - cos θ 2 + ( 1 + cos θ 2 ) Δ I ] = 1 - cos θ + ( 1 + cos θ ) Δ I \dot{\theta}=2[\frac{1-\cos\theta}{2}+(\frac{1+\cos\theta}{2})\Delta I]=1-\cos% \theta+(1+\cos\theta)\Delta I
  117. u ( t ) = tan ( θ / 2 ) u(t)=\tan(\theta/2)\;

Thin-film_thickness_monitor.html

  1. F m = F i T m T i F_{m}=F_{i}\,\frac{T_{m}}{T_{i}}

Thirring–Wess_model.html

  1. A μ A^{\mu}
  2. ( F μ ν ) 2 4 + μ 2 2 ( A μ ) 2 {(F^{\mu\nu})^{2}\over 4}+{\mu^{2}\over 2}(A^{\mu})^{2}
  3. F μ ν = μ A ν - ν A μ F^{\mu\nu}=\partial^{\mu}A^{\nu}-\partial^{\nu}A^{\mu}
  4. μ \mu
  5. ψ \psi
  6. ψ ¯ ( i / - m ) ψ \overline{\psi}(i\partial\!\!\!/-m)\psi
  7. m m
  8. q A μ ( ψ ¯ γ μ ψ ) qA^{\mu}(\bar{\psi}\gamma^{\mu}\psi)
  9. α 2 ( μ A μ ) 2 {\alpha\over 2}(\partial^{\mu}A^{\mu})^{2}
  10. α 0 \alpha\geq 0
  11. α > 0 \alpha>0
  12. α = 0 \alpha=0
  13. m = 0 m=0
  14. α = 1 \alpha=1
  15. α = 0 \alpha=0
  16. α = 0 \alpha=0
  17. α 0 \alpha\geq 0

Thomas_Spencer_(mathematical_physicist).html

  1. J x , y | x - y | - 2 J_{x,y}\sim|x-y|^{-2}

Thompson_order_formula.html

  1. | G | = C G ( t ) C G ( z ) x a ( x ) C G ( x ) |G|=C_{G}(t)C_{G}(z)\sum_{x}\frac{a(x)}{C_{G}(x)}
  2. | G | C G ( z ) | G | C G ( t ) = x a ( x ) | G | C G ( x ) \frac{|G|}{C_{G}(z)}\frac{|G|}{C_{G}(t)}=\sum_{x}a(x)\frac{|G|}{C_{G}(x)}

Thompson_sampling.html

  1. 𝒳 \mathcal{X}
  2. 𝒜 \mathcal{A}
  3. \mathbb{R}
  4. x 𝒳 x\in\mathcal{X}
  5. a 𝒜 a\in\mathcal{A}
  6. r r\in\mathbb{R}
  7. Θ \Theta
  8. θ \theta
  9. P ( θ ) P(\theta)
  10. 𝒟 = { ( x ; a ; r ) } \mathcal{D}=\{(x;a;r)\}
  11. P ( r | θ , a , x ) P(r|\theta,a,x)
  12. P ( θ | 𝒟 ) P ( 𝒟 | θ ) P ( θ ) P(\theta|\mathcal{D})\propto P(\mathcal{D}|\theta)P(\theta)
  13. P ( 𝒟 | θ ) P(\mathcal{D}|\theta)
  14. a 𝒜 a^{\ast}\in\mathcal{A}
  15. 𝕀 [ 𝔼 ( r | a , x , θ ) = max a 𝔼 ( r | a , x , θ ) ] P ( θ | 𝒟 ) d θ , \int\mathbb{I}[\mathbb{E}(r|a,x,\theta)=\max_{a^{\prime}}\mathbb{E}(r|a^{% \prime},x,\theta)]P(\theta|\mathcal{D})\,d\theta,
  16. 𝕀 \mathbb{I}
  17. θ \theta^{\ast}
  18. P ( θ | 𝒟 ) P(\theta|\mathcal{D})
  19. a a^{\ast}
  20. 𝔼 [ r | θ , a , x ] \mathbb{E}[r|\theta^{\ast},a^{\ast},x]
  21. a 1 , a 2 , , a T a_{1},a_{2},\ldots,a_{T}
  22. T T
  23. o 1 , o 2 , , o T o_{1},o_{2},\ldots,o_{T}
  24. T T
  25. a T + 1 a_{T+1}
  26. P ( a T + 1 | a ^ 1 : T , o 1 : T ) , P(a_{T+1}|\hat{a}_{1:T},o_{1:T}),
  27. a ^ t \hat{a}_{t}
  28. a t a_{t}
  29. θ Θ \theta\in\Theta
  30. P ( a T + 1 | a ^ 1 : T , o 1 : T ) = Θ P ( a T + 1 | θ , a ^ 1 : T , o 1 : T ) P ( θ | a ^ 1 : T , o 1 : T ) d θ P(a_{T+1}|\hat{a}_{1:T},o_{1:T})=\int_{\Theta}P(a_{T+1}|\theta,\hat{a}_{1:T},o% _{1:T})P(\theta|\hat{a}_{1:T},o_{1:T})\,d\theta
  31. P ( θ | a ^ 1 : T , o 1 : T ) P(\theta|\hat{a}_{1:T},o_{1:T})
  32. θ \theta
  33. a 1 : T a_{1:T}
  34. o 1 : T o_{1:T}
  35. θ \theta^{\ast}
  36. P ( θ | a ^ 1 : T , o 1 : T ) P(\theta|\hat{a}_{1:T},o_{1:T})
  37. o 1 , o 2 , , o T o_{1},o_{2},\ldots,o_{T}
  38. a 1 , a 2 , , a T a_{1},a_{2},\ldots,a_{T}
  39. a T + 1 a^{\ast}_{T+1}
  40. P ( a T + 1 | θ , a ^ 1 : T , o 1 : T ) P(a_{T+1}|\theta^{\ast},\hat{a}_{1:T},o_{1:T})