wpmath0000014_11

Realized_variance.html

  1. 12 × R V \sqrt{12\times RV}
  2. P t = exp ( p t ) P_{t}=\exp{(p_{t})}
  3. p t = p 0 + 0 t σ s d B s , p_{t}=p_{0}+\int_{0}^{t}\sigma_{s}dB_{s},
  4. B s B_{s}
  5. σ s \sigma_{s}
  6. I V = 0 t σ s 2 d s , IV=\int_{0}^{t}\sigma_{s}^{2}ds,
  7. n n
  8. R V ( n ) = i = 1 n r i , n 2 , RV^{(n)}=\sum_{i=1}^{n}r_{i,n}^{2},
  9. r i , n = p i t n - p ( i - 1 ) t n , i = 1 , , n . r_{i,n}=p_{\frac{it}{n}}-p_{\frac{(i-1)t}{n}},\qquad i=1,\ldots,n.
  10. n n\rightarrow\infty
  11. R V ( n ) - I V 2 t 0 t σ s 4 d s , \frac{RV^{(n)}-IV}{\sqrt{2t\int_{0}^{t}\sigma_{s}^{4}ds}},
  12. n n

Recession_cone.html

  1. A A
  2. A A
  3. A X A\subset X
  4. recc ( A ) \operatorname{recc}(A)
  5. recc ( A ) = { y X : x A , λ 0 : x + λ y A } . \operatorname{recc}(A)=\{y\in X:\forall x\in A,\forall\lambda\geq 0:x+\lambda y% \in A\}.
  6. A A
  7. recc ( A ) = { y X : x A : x + y A } . \operatorname{recc}(A)=\{y\in X:\forall x\in A:x+y\in A\}.
  8. A A
  9. recc ( A ) = t > 0 t ( A - a ) \operatorname{recc}(A)=\bigcap_{t>0}t(A-a)
  10. a A . a\in A.
  11. A A
  12. 0 recc ( A ) 0\in\operatorname{recc}(A)
  13. A A
  14. recc ( A ) \operatorname{recc}(A)
  15. A X A\subset X
  16. X X
  17. d \mathbb{R}^{d}
  18. recc ( A ) = { 0 } \operatorname{recc}(A)=\{0\}
  19. A A
  20. A A
  21. A + recc ( A ) = A A+\operatorname{recc}(A)=A
  22. C X C\subseteq X
  23. C = { x X : ( t i ) i I ( 0 , ) , ( x i ) i I C : t i 0 , t i x i x } . C_{\infty}=\{x\in X:\exists(t_{i})_{i\in I}\subset(0,\infty),\exists(x_{i})_{i% \in I}\subset C:t_{i}\to 0,t_{i}x_{i}\to x\}.
  24. recc ( C ) C . \operatorname{recc}(C)\subseteq C_{\infty}.
  25. C = recc ( C ) C_{\infty}=\operatorname{recc}(C)
  26. C C
  27. A , B X A,B\subset X
  28. A A
  29. B B
  30. recc ( A ) recc ( B ) \operatorname{recc}(A)\cap\operatorname{recc}(B)
  31. A - B A-B
  32. A , B d A,B\subset\mathbb{R}^{d}
  33. y recc ( A ) \ { 0 } y\in\operatorname{recc}(A)\backslash\{0\}
  34. - y recc ( B ) -y\not\in\operatorname{recc}(B)
  35. A + B A+B

Rectified_10-cubes.html

  1. 2 \sqrt{2}
  2. 2 \sqrt{2}
  3. 2 \sqrt{2}
  4. 2 \sqrt{2}

Rectified_10-orthoplexes.html

  1. 2 \sqrt{2}
  2. 2 \sqrt{2}
  3. 2 \sqrt{2}
  4. 2 \sqrt{2}

Rectified_24-cell_honeycomb.html

  1. F ~ 4 {\tilde{F}}_{4}
  2. C ~ 4 {\tilde{C}}_{4}
  3. B ~ 4 {\tilde{B}}_{4}
  4. D ~ 4 {\tilde{D}}_{4}
  5. F ~ 4 {\tilde{F}}_{4}
  6. C ~ 4 {\tilde{C}}_{4}
  7. B ~ 4 {\tilde{B}}_{4}
  8. D ~ 4 {\tilde{D}}_{4}

Rectified_7-cubes.html

  1. 2 \sqrt{2}
  2. 2 \sqrt{2}
  3. 2 \sqrt{2}

Rectified_9-orthoplexes.html

  1. 2 \sqrt{2}

Rectified_Gaussian_distribution.html

  1. ( 0 , ) (0,\infty)
  2. X 𝒩 R ( μ , σ 2 ) X\sim\mathcal{N}^{\textrm{R}}(\mu,\sigma^{2})
  3. f ( x ; μ , σ 2 ) = Φ ( - μ σ ) δ ( x ) + 1 2 π σ 2 e - ( x - μ ) 2 2 σ 2 U ( x ) . f(x;\mu,\sigma^{2})=\Phi(-\frac{\mu}{\sigma})\delta(x)+\frac{1}{\sqrt{2\pi% \sigma^{2}}}\;e^{-\frac{(x-\mu)^{2}}{2\sigma^{2}}}\textrm{U}(x).
  4. Φ ( x ) \Phi(x)
  5. Φ ( x ) = 1 2 π - x e - t 2 / 2 d t x , \Phi(x)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{x}e^{-t^{2}/2}\,dt\quad x\in% \mathbb{R},
  6. δ ( x ) \delta(x)
  7. δ ( x ) = { + , x = 0 0 , x 0 \delta(x)=\begin{cases}+\infty,&x=0\\ 0,&x\neq 0\end{cases}
  8. U ( x ) \textrm{U}(x)
  9. U ( x ) = { 0 , x 0 , 1 , x > 0. \textrm{U}(x)=\begin{cases}0,&x\leq 0,\\ 1,&x>0.\end{cases}
  10. s 𝒩 ( μ , σ 2 ) , x = max ( 0 , s ) , s\sim\mathcal{N}(\mu,\sigma^{2}),x=\textrm{max}(0,s),
  11. x 𝒩 R ( μ , σ 2 ) x\sim\mathcal{N}^{\textrm{R}}(\mu,\sigma^{2})

Reduced_properties.html

  1. p p
  2. p c p_{c}
  3. p r = p p c p_{r}={p\over p_{c}}
  4. T r = T T c T_{r}={T\over T_{c}}
  5. v r = v p c R T c v_{r}=\frac{vp_{c}}{RT_{c}}\,

Reduced_viscosity.html

  1. η i \eta_{i}
  2. η i c \frac{\eta_{i}}{c}
  3. η i = η - η s η s \eta_{i}=\frac{\eta-\eta_{s}}{\eta_{s}}

Reeb_stability_theorem.html

  1. F F
  2. C 1 C^{1}
  3. k k
  4. M M
  5. L L
  6. U U
  7. L L
  8. F F
  9. π : U L \pi:U\to L
  10. L U L^{\prime}\subset U
  11. π | L : L L \pi|_{L^{\prime}}:L^{\prime}\to L
  12. y L y\in L
  13. π - 1 ( y ) \pi^{-1}(y)
  14. F F
  15. U U
  16. ( M n , F ) (M^{n},F)
  17. n 3 n\geq 3
  18. S i n g ( F ) Sing(F)
  19. F F
  20. C 1 C^{1}
  21. M M
  22. F F
  23. L L
  24. F F
  25. F F
  26. F F
  27. L L
  28. M M
  29. f : M S 1 f:M\to S^{1}
  30. S 1 S^{1}
  31. L L
  32. F F
  33. { f - 1 ( θ ) | θ S 1 } \{f^{-1}(\theta)|\theta\in S^{1}\}
  34. F F
  35. F F
  36. k 3 k\geq 3
  37. M M
  38. F F
  39. F F
  40. F F
  41. k k
  42. F F
  43. F F
  44. C 2 C^{2}

Reflectance_difference_spectroscopy.html

  1. R D S = 2 r α - r β r α + r β RDS=2\frac{r_{\alpha}-r_{\beta}}{r_{\alpha}+r_{\beta}}
  2. r α r_{\alpha}
  3. r β r_{\beta}

Reflected_Brownian_motion.html

  1. + d \scriptstyle\mathbb{R}^{d}_{+}
  2. Z ( t ) = X ( t ) + R Y ( t ) Z(t)=X(t)+RY(t)
  3. + d \scriptstyle\mathbb{R}^{d}_{+}
  4. { z + d : z j = 0 } \scriptstyle\{z\in\mathbb{R}^{d}_{+}:z_{j}=0\}
  5. ( Z ( t ) z ) = Φ ( z - μ t σ t 1 / 2 ) - e 2 μ z / σ 2 Φ ( - z - μ t σ t 1 / 2 ) \mathbb{P}(Z(t)\leq z)=\Phi\left(\frac{z-\mu t}{\sigma t^{1/2}}\right)-e^{2\mu z% /\sigma^{2}}\Phi\left(\frac{-z-\mu t}{\sigma t^{1/2}}\right)
  6. ( Z < z ) = 1 - e 2 μ z / σ 2 . \mathbb{P}(Z<z)=1-e^{2\mu z/\sigma^{2}}.
  7. Z ( t ) M ( t ) = sup s [ 0 , t ] X ( s ) . Z(t)\sim M(t)=\sup_{s\in[0,t]}X(s).
  8. p b p_{b}
  9. f ( x , p b ) = e - ( ( x - u ) / a ) 2 / 2 + e - ( ( x + u - 2 p b ) / a ) 2 / 2 a ( 2 π ) 1 / 2 f(x,p_{b})=\frac{e^{-((x-u)/a)^{2}/2}+e^{-((x+u-2p_{b})/a)^{2}/2}}{a(2\pi)^{1/% 2}}
  10. x p b x\geq p_{b}
  11. 2 Σ = R D + D R 2\Sigma=RD+DR^{\prime}
  12. p ( z 1 , z 2 , , z d ) = k = 1 d η k e - η k z k p(z_{1},z_{2},\ldots,z_{d})=\prod_{k=1}^{d}\eta_{k}e^{-\eta_{k}z_{k}}

Reflection_principle_(Wiener_process).html

  1. \approx
  2. ( W ( t ) : t 0 ) (W(t):t\geq 0)
  3. a > 0 a>0
  4. ( sup 0 s t W ( s ) a ) = 2 ( W ( t ) a ) \mathbb{P}\left(\sup_{0\leq s\leq t}W(s)\geq a\right)=2\mathbb{P}(W(t)\geq a)
  5. τ \tau
  6. τ \tau
  7. ( W τ ( t ) : t 0 ) (W^{\tau}(t):t\geq 0)
  8. W τ ( t ) = W ( t ) χ { t τ } + ( 2 W ( τ ) - W ( t ) ) χ { t > τ } W^{\tau}(t)=W(t)\chi_{\left\{}t\leq\tau\right\}+(2W(\tau)-W(t))\chi_{\left\{}t% >\tau\right\}
  9. τ = inf { t 0 : W ( t ) = a } \tau=\inf\left\{t\geq 0:W(t)=a\right\}
  10. τ a := inf { t : W ( t ) = a } \tau_{a}:=\inf\left\{t:W(t)=a\right\}
  11. τ a \tau_{a}
  12. X t := W ( t + τ a ) - a X_{t}:=W(t+\tau_{a})-a
  13. τ a W \mathcal{F}^{W}_{\tau_{a}}
  14. W ( s ) W(s)
  15. a a
  16. [ 0 , t ] [0,t]
  17. ( sup 0 s t W ( s ) a ) = ( sup 0 s t W ( s ) a , W ( t ) a ) + ( sup 0 s t W ( s ) a , W ( t ) < a ) = ( W ( t ) a ) + ( sup 0 s t W ( s ) a , X ( t - τ a ) < 0 ) \begin{aligned}\displaystyle\mathbb{P}(\sup_{0\leq s\leq t}W(s)\geq a)&% \displaystyle=\mathbb{P}(\sup_{0\leq s\leq t}W(s)\geq a,W(t)\geq a)+\mathbb{P}% (\sup_{0\leq s\leq t}W(s)\geq a,W(t)<a)\\ &\displaystyle=\mathbb{P}(W(t)\geq a)+\mathbb{P}(\sup_{0\leq s\leq t}W(s)\geq a% ,X(t-\tau_{a})<0)\\ \end{aligned}
  18. ( sup 0 s t W ( s ) a , X ( t - τ a ) < 0 ) \displaystyle\mathbb{P}(\sup_{0\leq s\leq t}W(s)\geq a,X(t-\tau_{a})<0)
  19. X ( t ) X(t)
  20. τ a W \mathcal{F}^{W}_{\tau_{a}}
  21. 1 / 2 1/2
  22. 0
  23. ( W ( t ) : t [ 0 , 1 ] ) (W(t):t\in[0,1])
  24. t max t\text{max}
  25. W ( t max ) = sup 0 s 1 W ( s ) W(t\text{max})=\sup_{0\leq s\leq 1}W(s)

Regenerative_process.html

  1. lim t 1 t 0 t X ( s ) d s = 𝔼 [ R ] 𝔼 [ τ ] . \lim_{t\to\infty}\frac{1}{t}\int_{0}^{t}X(s)ds=\frac{\mathbb{E}[R]}{\mathbb{E}% [\tau]}.
  2. τ \tau
  3. R = 0 τ X ( s ) d s R=\int_{0}^{\tau}X(s)ds

Region_(mathematics).html

  1. \R n \R^{n}
  2. \C n \C^{n}

Regular_semi-algebraic_system.html

  1. S S
  2. S 1 , , S e S_{1},\ldots,S_{e}
  3. S S
  4. S 1 , , S e S_{1},\ldots,S_{e}
  5. T T
  6. 𝐤 [ x 1 , , x n ] {{\mathbf{k}}}[x_{1},\ldots,x_{n}]
  7. 𝐱 = x 1 , , x n \mathbf{x}=x_{1},\ldots,x_{n}
  8. 𝐤 {{\mathbf{k}}}
  9. 𝐮 = u 1 , , u d \mathbf{u}=u_{1},\ldots,u_{d}
  10. 𝐲 = y 1 , , y n - d \mathbf{y}=y_{1},\ldots,y_{n-d}
  11. 𝐱 \mathbf{x}
  12. T T
  13. P 𝐤 [ 𝐱 ] P\subset{{\mathbf{k}}}[\mathbf{x}]
  14. P P
  15. T T
  16. P > := { p > 0 p P } P_{>}:=\{p>0\mid p\in P\}
  17. 𝒬 \mathcal{Q}
  18. 𝐤 [ 𝐱 ] {{\mathbf{k}}}[\mathbf{x}]
  19. 𝐮 \mathbf{u}
  20. R := [ 𝒬 , T , P > ] R:=[\mathcal{Q},T,P_{>}]
  21. 𝒬 \mathcal{Q}
  22. S S
  23. 𝐤 d {{\mathbf{k}}}^{d}
  24. [ T , P ] [T,P]
  25. u u
  26. S S
  27. u u
  28. S S
  29. [ T ( u ) , P ( u ) > ] [T(u),P(u)_{>}]
  30. R R
  31. Z 𝐤 ( R ) {Z}_{{{\mathbf{k}}}}(R)
  32. ( u , y ) 𝐤 d × 𝐤 n - d (u,y)\in{{{\mathbf{k}}}}^{d}\times{{{\mathbf{k}}}}^{n-d}
  33. 𝒬 ( u ) \mathcal{Q}(u)
  34. t ( u , y ) = 0 t(u,y)=0
  35. p ( u , y ) > 0 p(u,y)>0
  36. t T t\in T
  37. p P p\in P

RegularChains.html

  1. S S
  2. S 1 , , S n S_{1},\ldots,S_{n}
  3. S S
  4. S 1 , , S n S_{1},\ldots,S_{n}
  5. S 1 , , S n S_{1},\ldots,S_{n}
  6. S S
  7. F F
  8. F F
  9. N N
  10. N N

Regularized_canonical_correlation_analysis.html

  1. cov ( X , X ) \operatorname{cov}(X,X)
  2. cov ( Y , Y ) \operatorname{cov}(Y,Y)
  3. cov ( X , X ) + λ I X \operatorname{cov}(X,X)+\lambda I_{X}
  4. cov ( Y , Y ) + λ I Y \operatorname{cov}(Y,Y)+\lambda I_{Y}

Reinforced_concrete_column.html

  1. P n \displaystyle P_{\mathrm{n}}
  2. ϕ P n \mathrm{{\phi}P_{\mathrm{n}}}\,\!
  3. ϕ \mathrm{\phi}\,\!
  4. ϕ P n ( max ) \displaystyle{\phi}P_{\mathrm{n(max)}}
  5. ϕ = 0.75 \mathrm{\phi}=0.75\,\!
  6. ϕ P n ( max ) \displaystyle{\phi}P_{\mathrm{n(max)}}
  7. ϕ = 0.65 \mathrm{\phi}=0.65\,\!
  8. ρ s = 0.45 ( A g A c h - 1 ) f c f y t {\rho}_{s}=0.45(\frac{A_{g}}{A_{ch}}-1)\frac{f^{\prime}_{c}}{f_{yt}}

Reinforced_solid.html

  1. A r A_{r}
  2. A A
  3. ρ x \rho_{x}
  4. A r x A_{rx}
  5. A x A_{x}
  6. ρ y \rho_{y}
  7. A r y A_{ry}
  8. A y A_{y}
  9. ρ z \rho_{z}
  10. A r z A_{rz}
  11. A z A_{z}
  12. f y f_{y}
  13. [ σ x x - ρ x f y σ x y σ x z σ x y σ y y - ρ y f y σ y z σ x z σ y z σ z z - ρ z f y ] \left[{\begin{matrix}\sigma_{xx}-\rho_{x}f_{y}&\sigma_{xy}&\sigma_{xz}\\ \sigma_{xy}&\sigma_{yy}-\rho_{y}f_{y}&\sigma_{yz}\\ \sigma_{xz}&\sigma_{yz}&\sigma_{zz}-\rho_{z}f_{y}\\ \end{matrix}}\right]
  14. ρ x \rho_{x}
  15. ρ y \rho_{y}
  16. ρ z \rho_{z}
  17. ρ x \rho_{x}
  18. ρ y \rho_{y}
  19. ρ z \rho_{z}
  20. ρ x \rho_{x}
  21. f y f_{y}
  22. ρ y \rho_{y}
  23. f y f_{y}
  24. ρ z \rho_{z}
  25. f y f_{y}
  26. I 1 I_{1}
  27. I 2 I_{2}
  28. I 3 I_{3}
  29. σ y y σ z z - σ y z 2 \sigma_{yy}\sigma_{zz}-\sigma^{2}_{yz}
  30. I 1 ( σ y y σ z z - σ y z 2 ) - I 3 I_{1}(\sigma_{yy}\sigma_{zz}-\sigma^{2}_{yz})-I_{3}
  31. I 2 ( σ y y σ z z - σ y z 2 ) - I 3 ( σ y y + σ z z ) I_{2}(\sigma_{yy}\sigma_{zz}-\sigma^{2}_{yz})-I_{3}(\sigma_{yy}+\sigma_{zz})
  32. I 3 σ y y σ z z - σ y z 2 \frac{I_{3}}{\sigma_{yy}\sigma_{zz}-\sigma^{2}_{yz}}
  33. σ x x σ z z - σ x z 2 \sigma_{xx}\sigma_{zz}-\sigma^{2}_{xz}
  34. I 1 ( σ x x σ z z - σ x z 2 ) - I 3 I_{1}(\sigma_{xx}\sigma_{zz}-\sigma^{2}_{xz})-I_{3}
  35. I 2 ( σ x x σ z z - σ x z 2 ) - I 3 ( σ x x + σ z z ) I_{2}(\sigma_{xx}\sigma_{zz}-\sigma^{2}_{xz})-I_{3}(\sigma_{xx}+\sigma_{zz})
  36. I 3 σ x x σ z z - σ x z 2 \frac{I_{3}}{\sigma_{xx}\sigma_{zz}-\sigma^{2}_{xz}}
  37. σ x x σ y y - σ x y 2 \sigma_{xx}\sigma_{yy}-\sigma^{2}_{xy}
  38. I 1 ( σ x x σ y y - σ x y 2 ) - I 3 I_{1}(\sigma_{xx}\sigma_{yy}-\sigma^{2}_{xy})-I_{3}
  39. I 2 ( σ x x σ y y - σ x y 2 ) - I 3 ( σ x x + σ y y ) I_{2}(\sigma_{xx}\sigma_{yy}-\sigma^{2}_{xy})-I_{3}(\sigma_{xx}+\sigma_{yy})
  40. I 3 σ x x σ y y - σ x y 2 \frac{I_{3}}{\sigma_{xx}\sigma_{yy}-\sigma^{2}_{xy}}
  41. σ x x < 0 \sigma_{xx}<0
  42. σ y y - σ x y 2 σ x x + | σ y z - σ x z σ x y σ x x | \sigma_{yy}-\frac{\sigma^{2}_{xy}}{\sigma_{xx}}+|\sigma_{yz}-\frac{\sigma_{xz}% \sigma_{xy}}{\sigma_{xx}}|
  43. σ z z - σ x z 2 σ x x + | σ y z - σ x z σ x y σ x x | \sigma_{zz}-\frac{\sigma^{2}_{xz}}{\sigma_{xx}}+|\sigma_{yz}-\frac{\sigma_{xz}% \sigma_{xy}}{\sigma_{xx}}|
  44. σ y y < 0 \sigma_{yy}<0
  45. σ x x - σ x y 2 σ y y + | σ x z - σ y z σ x y σ y y | \sigma_{xx}-\frac{\sigma^{2}_{xy}}{\sigma_{yy}}+|\sigma_{xz}-\frac{\sigma_{yz}% \sigma_{xy}}{\sigma_{yy}}|
  46. σ z z - σ y z 2 σ y y + | σ x z - σ y z σ x y σ y y | \sigma_{zz}-\frac{\sigma^{2}_{yz}}{\sigma_{yy}}+|\sigma_{xz}-\frac{\sigma_{yz}% \sigma_{xy}}{\sigma_{yy}}|
  47. σ z z < 0 \sigma_{zz}<0
  48. σ x x - σ x z 2 σ z z + | σ x y - σ y z σ x z σ z z | \sigma_{xx}-\frac{\sigma^{2}_{xz}}{\sigma_{zz}}+|\sigma_{xy}-\frac{\sigma_{yz}% \sigma_{xz}}{\sigma_{zz}}|
  49. σ y y - σ y z 2 σ z z + | σ x y - σ x z σ y z σ z z | \sigma_{yy}-\frac{\sigma^{2}_{yz}}{\sigma_{zz}}+|\sigma_{xy}-\frac{\sigma_{xz}% \sigma_{yz}}{\sigma_{zz}}|
  50. σ y z + σ x z + σ x y \sigma_{yz}+\sigma_{xz}+\sigma_{xy}
  51. σ x z σ x y + σ y z σ x y + σ y z σ x z \sigma_{xz}\sigma_{xy}+\sigma_{yz}\sigma_{xy}+\sigma_{yz}\sigma_{xz}
  52. σ x x + σ x z + σ x y \sigma_{xx}+\sigma_{xz}+\sigma_{xy}
  53. σ y y + σ y z + σ x y \sigma_{yy}+\sigma_{yz}+\sigma_{xy}
  54. σ z z + σ y z + σ x z \sigma_{zz}+\sigma_{yz}+\sigma_{xz}
  55. - σ y z - σ x z + σ x y -\sigma_{yz}-\sigma_{xz}+\sigma_{xy}
  56. - σ x z σ x y - σ y z σ x y + σ y z σ x z -\sigma_{xz}\sigma_{xy}-\sigma_{yz}\sigma_{xy}+\sigma_{yz}\sigma_{xz}
  57. σ x x - σ x z + σ x y \sigma_{xx}-\sigma_{xz}+\sigma_{xy}
  58. σ y y - σ y z + σ x y \sigma_{yy}-\sigma_{yz}+\sigma_{xy}
  59. σ z z - σ y z - σ x z \sigma_{zz}-\sigma_{yz}-\sigma_{xz}
  60. σ y z - σ x z - σ x y \sigma_{yz}-\sigma_{xz}-\sigma_{xy}
  61. σ x z σ x y - σ y z σ x y - σ y z σ x z \sigma_{xz}\sigma_{xy}-\sigma_{yz}\sigma_{xy}-\sigma_{yz}\sigma_{xz}
  62. σ x x - σ x z - σ x y \sigma_{xx}-\sigma_{xz}-\sigma_{xy}
  63. σ y y + σ y z - σ x y \sigma_{yy}+\sigma_{yz}-\sigma_{xy}
  64. σ z z + σ y z - σ x z \sigma_{zz}+\sigma_{yz}-\sigma_{xz}
  65. - σ y z + σ x z - σ x y -\sigma_{yz}+\sigma_{xz}-\sigma_{xy}
  66. - σ x z σ x y + σ y z σ x y - σ y z σ x z -\sigma_{xz}\sigma_{xy}+\sigma_{yz}\sigma_{xy}-\sigma_{yz}\sigma_{xz}
  67. σ x x + σ x z - σ x y \sigma_{xx}+\sigma_{xz}-\sigma_{xy}
  68. σ y y - σ y z - σ x y \sigma_{yy}-\sigma_{yz}-\sigma_{xy}
  69. σ z z - σ y z + σ x z \sigma_{zz}-\sigma_{yz}+\sigma_{xz}
  70. σ x y σ x z σ y z < 0 \sigma_{xy}\sigma_{xz}\sigma_{yz}<0
  71. σ x x - σ x z σ x y σ y z \sigma_{xx}-\frac{\sigma_{xz}\sigma_{xy}}{\sigma_{yz}}
  72. σ y y - σ y z σ x y σ x z \sigma_{yy}-\frac{\sigma_{yz}\sigma_{xy}}{\sigma_{xz}}
  73. σ z z - σ y z σ x z σ x y \sigma_{zz}-\frac{\sigma_{yz}\sigma_{xz}}{\sigma_{xy}}
  74. I 1 I_{1}
  75. I 2 I_{2}
  76. I 3 I_{3}
  77. ρ x \rho_{x}
  78. ρ y \rho_{y}
  79. ρ z \rho_{z}
  80. f y f_{y}
  81. σ m \sigma_{m}
  82. m r m_{r}
  83. σ x x \sigma_{xx}
  84. σ y y \sigma_{yy}
  85. σ z z \sigma_{zz}
  86. σ y z \sigma_{yz}
  87. σ x z \sigma_{xz}
  88. σ x y \sigma_{xy}
  89. ρ x \rho_{x}
  90. ρ y \rho_{y}
  91. ρ z \rho_{z}
  92. σ m \sigma_{m}
  93. m r m_{r}
  94. ρ 1 \rho_{1}
  95. ρ 2 \rho_{2}
  96. ρ 3 \rho_{3}
  97. ρ x x \rho_{xx}
  98. ρ y y \rho_{yy}
  99. ρ z z \rho_{zz}
  100. ρ y z \rho_{yz}
  101. ρ x z \rho_{xz}
  102. ρ x y \rho_{xy}
  103. T i j T_{ij}
  104. ρ 1 \rho_{1}
  105. ρ 2 \rho_{2}
  106. ρ 3 \rho_{3}
  107. T i j T_{ij}
  108. T i j = [ σ x x σ x y σ x z σ x y σ y y σ y z σ x z σ y z σ z z ] i j - f y k [ ρ x x k ρ x y k ρ x z k ρ x y k ρ y y k ρ y z k ρ x z k ρ y z k ρ z z k ] - f y [ ρ x x ρ x y ρ x z ρ x y ρ y y ρ y z ρ x z ρ y z ρ z z ] T_{ij}=\left[{\begin{matrix}\sigma_{xx}&\sigma_{xy}&\sigma_{xz}\\ \sigma_{xy}&\sigma_{yy}&\sigma_{yz}\\ \sigma_{xz}&\sigma_{yz}&\sigma_{zz}\\ \end{matrix}}\right]_{ij}-f_{y}\sum_{k}\left[{\begin{matrix}\rho_{xxk}&\rho_{% xyk}&\rho_{xzk}\\ \rho_{xyk}&\rho_{yyk}&\rho_{yzk}\\ \rho_{xzk}&\rho_{yzk}&\rho_{zzk}\\ \end{matrix}}\right]-f_{y}\left[{\begin{matrix}\rho_{xx}&\rho_{xy}&\rho_{xz}\\ \rho_{xy}&\rho_{yy}&\rho_{yz}\\ \rho_{xz}&\rho_{yz}&\rho_{zz}\\ \end{matrix}}\right]
  109. i i
  110. j j
  111. k k

Reiss_relation.html

  1. f x x f y 2 - 2 f x y f x f y + f y y f x 2 f y 3 = 0 \sum\frac{f_{xx}f_{y}^{2}-2f_{xy}f_{x}f_{y}+f_{yy}f_{x}^{2}}{f_{y}^{3}}=0
  2. κ sin ( θ ) 3 = 0 \sum\frac{\kappa}{\sin(\theta)^{3}}=0

Relative_growth_rate.html

  1. P P
  2. d P d t \frac{dP}{dt}
  3. 1 P d P d t \frac{1}{P}\frac{dP}{dt}
  4. 1 P d P d t = k \frac{1}{P}\frac{dP}{dt}=k
  5. P ( t ) = exp ( k t ) P(t)=\exp(kt)
  6. P 0 P_{0}
  7. t t
  8. P ( t ) = P 0 2 t = P 0 exp ( ln ( 2 ) t ) P(t)=P_{0}2^{t}=P_{0}\exp(\ln(2)t)
  9. t t
  10. P ( 3 ) = P 0 2 3 P(3)=P_{0}2^{3}
  11. P ( T ) = P 0 8 T = P 0 exp ( ln ( 8 ) T ) P(T)=P_{0}8^{T}=P_{0}\exp(\ln(8)T)
  12. T T
  13. ln ( 2 ) \ln(2)
  14. ln ( 8 ) \ln(8)
  15. R G R = ( ln W 2 - ln W 1 ) / ( t 2 - t 1 ) RGR=(\ln W_{2}-\ln W_{1})/(t_{2}-t_{1})
  16. ln \ln
  17. t 1 t_{1}
  18. t 2 t_{2}
  19. W 1 W_{1}
  20. W 2 W_{2}

Relative_scalar.html

  1. x ¯ j = x ¯ j ( x i ) \bar{x}^{j}=\bar{x}^{j}(x^{i})
  2. f ¯ ( x ¯ j ) = J w f ( x i ) \bar{f}(\bar{x}^{j})=J^{w}f(x^{i})
  3. J = | ( x 1 , , x n ) ( x ¯ 1 , , x ¯ n ) | , J=\begin{vmatrix}\displaystyle\frac{\partial(x_{1},\ldots,x_{n})}{\partial(% \bar{x}^{1},\ldots,\bar{x}^{n})}\end{vmatrix},
  4. w = 1 w=1
  5. w = 0 w=0
  6. x i x^{i}
  7. x ¯ j \bar{x}^{j}
  8. P P
  9. f ¯ ( x ¯ j ) = f ( x i ) \bar{f}(\bar{x}^{j})=f(x^{i})
  10. x ¯ j \bar{x}^{j}
  11. x i x^{i}
  12. f ¯ ( x ¯ j ) = f ( x i ( x ¯ j ) ) \bar{f}(\bar{x}^{j})=f(x^{i}(\bar{x}^{j}))
  13. f ( x i ) = f ¯ ( x ¯ j ( x i ) ) f(x^{i})=\bar{f}(\bar{x}^{j}(x^{i}))
  14. f ( x , y , z ) = 2 x + y + 5 f(x,y,z)=2x+y+5
  15. ( x , y , z ) (x,y,z)
  16. ( r , t , h ) (r,t,h)
  17. r = x 2 + y 2 r=\sqrt{x^{2}+y^{2}}\,
  18. t = arctan ( y / x ) t=\arctan(y/x)\,
  19. h = z h=z\,
  20. x = r cos ( t ) x=r\cos(t)\,
  21. y = r sin ( t ) y=r\sin(t)\,
  22. z = h . z=h.\,
  23. f ¯ ( x ¯ j ) = f ( x i ( x ¯ j ) ) \bar{f}(\bar{x}^{j})=f(x^{i}(\bar{x}^{j}))
  24. f ¯ ( r , t , h ) = 2 r cos ( t ) + r sin ( t ) + 5 \bar{f}(r,t,h)=2r\cos(t)+r\sin(t)+5
  25. P P
  26. ( x , y , z ) = ( 2 , 3 , 4 ) (x,y,z)=(2,3,4)
  27. ( r , t , h ) = ( 13 , arctan ( 3 / 2 ) , 4 ) (r,t,h)=(\sqrt{13},\arctan{(3/2)},4)
  28. f ( 2 , 3 , 4 ) = 12 f(2,3,4)=12
  29. f ¯ ( 13 , arctan ( 3 / 2 ) , 4 ) = 12 \bar{f}(\sqrt{13},\arctan{(3/2)},4)=12
  30. P P
  31. f ( x , y , z ) f(x,y,z)
  32. f ¯ ( r , t , h ) \bar{f}(r,t,h)
  33. f ¯ \bar{f}
  34. f f
  35. D D
  36. D D
  37. r r
  38. [ 0 , 2 ] [0,2]
  39. t t
  40. [ 0 , π / 2 ] [0,\pi/2]
  41. h h
  42. [ 0 , 2 ] [0,2]
  43. f f
  44. D D
  45. 0 2 0 2 2 - x 2 0 2 f ( x , y , z ) d z d y d x = 16 + 10 π \int_{0}^{2}\!\int_{0}^{\sqrt{2^{2}-x^{2}}}\!\int_{0}^{2}\!f(x,y,z)\,dz\,dy\,% dx=16+10\pi
  46. f ¯ \bar{f}
  47. 0 2 0 π / 2 0 2 f ¯ ( r , t , h ) d h d t d r = 12 + 10 π \int_{0}^{2}\!\int_{0}^{\pi/2}\!\int_{0}^{2}\!\bar{f}(r,t,h)\,dh\,dt\,dr=12+10\pi
  48. f ¯ \bar{f}
  49. r r
  50. 0 2 0 π / 2 0 2 f ¯ ( r , t , h ) r d h d t d r = 16 + 10 π \int_{0}^{2}\!\int_{0}^{\pi/2}\!\int_{0}^{2}\!\bar{f}(r,t,h)r\,dh\,dt\,dr=16+10\pi
  51. f ( x , y , z ) = 2 x + y + 5 f(x,y,z)=2x+y+5
  52. f ¯ ( r , t , h ) = ( 2 r cos ( t ) + r sin ( t ) + 5 ) r \bar{f}(r,t,h)=(2r\cos(t)+r\sin(t)+5)r
  53. f ( 2 , 3 , 4 ) = 12 f(2,3,4)=12
  54. f ¯ ( 13 , arctan ( 3 / 2 ) , 4 ) = 12 29 \bar{f}(\sqrt{13},\arctan{(3/2)},4)=12\sqrt{29}
  55. 0 2 0 2 2 - x 2 0 2 f ( x , y , z ) d z d y d x = 16 + 10 π \int_{0}^{2}\!\int_{0}^{\sqrt{2^{2}-x^{2}}}\!\int_{0}^{2}\!f(x,y,z)\,dz\,dy\,% dx=16+10\pi
  56. f ¯ \bar{f}
  57. 0 2 0 π / 2 0 2 f ¯ ( r , t , h ) d h d t d r = 16 + 10 π \int_{0}^{2}\!\int_{0}^{\pi/2}\!\int_{0}^{2}\!\bar{f}(r,t,h)\,dh\,dt\,dr=16+10\pi
  58. f ¯ \bar{f}
  59. 0 2 0 π / 2 0 2 f ¯ ( r , t , h ) r d h d t d r = 24 + 40 π / 3 \int_{0}^{2}\!\int_{0}^{\pi/2}\!\int_{0}^{2}\!\bar{f}(r,t,h)r\,dh\,dt\,dr=24+4% 0\pi/3

Relative_utilitarianism.html

  1. X X
  2. X X
  3. I I
  4. i I i\in I
  5. u i : X u_{i}:X\longrightarrow\mathbb{R}
  6. ( u i ) i I (u_{i})_{i\in I}
  7. X X
  8. x X x\in X
  9. U ( x ) := i I u i ( x ) . U(x):=\sum_{i\in I}u_{i}(x).
  10. ( u i ) i I (u_{i})_{i\in I}
  11. u i : X u_{i}:X\longrightarrow\mathbb{R}
  12. r i , s i r_{i},s_{i}\in\mathbb{R}
  13. s i > 0 s_{i}>0
  14. v i ( x ) := s i u i ( x ) + r i v_{i}(x):=s_{i}\,u_{i}(x)+r_{i}
  15. ( v i ) i I (v_{i})_{i\in I}
  16. r i r_{i}\in\mathbb{R}
  17. s i > 0 s_{i}>0
  18. i I i\in I
  19. V ( x ) := i I v i ( x ) , V(x):=\sum_{i\in I}v_{i}(x),
  20. V V
  21. U U
  22. i I i\in I
  23. m i := min x X u i ( x ) and M i := max x X u i ( x ) m_{i}\ :=\ \min_{x\in X}\,u_{i}(x)\quad\mbox{and}~{}\quad M_{i}\ :=\ \max_{x% \in X}\,u_{i}(x)
  24. X X
  25. X X
  26. u i u_{i}
  27. w i ( x ) := u i ( x ) - m i M i - m i w_{i}(x)\ :=\ \frac{u_{i}(x)-m_{i}}{M_{i}-m_{i}}
  28. x X x\in X
  29. w i : X w_{i}:X\longrightarrow\mathbb{R}
  30. X X
  31. W ( x ) := i I w i ( x ) . W(x):=\sum_{i\in I}w_{i}(x).

Relativistic_images.html

  1. α ^ > 3 π / 2 \hat{\alpha}>3\pi/2
  2. α ^ > 2 π \hat{\alpha}>2\pi
  3. α ^ e f f = α ^ - 2 n π \hat{\alpha}^{eff}=\hat{\alpha}-2n\pi
  4. α ^ \hat{\alpha}
  5. n n
  6. n = 0 n=0

Relativistic_similarity_parameter.html

  1. S = n e a 0 n c r S=\frac{n_{e}}{a_{0}n_{cr}}
  2. n e {n_{e}}
  3. n c r = m e ω 0 2 / 4 π e 2 {n_{cr}=m_{e}\omega_{0}^{2}/4\pi e^{2}}
  4. a 0 = e A / m e c 2 {a_{0}=eA/m_{e}c^{2}}
  5. m e {m_{e}}
  6. e {e}
  7. c {c}
  8. ω 0 {\omega_{0}}
  9. S {S}
  10. ( S 1 ) {(S\gg 1)}
  11. ( S 1 ) {(S\ll 1)}
  12. a 0 1 {a_{0}\gg 1}
  13. ω 0 τ \omega_{0}\tau
  14. R ω 0 / c {R\omega_{0}/c}
  15. S {S}
  16. τ {\tau}
  17. R {R}
  18. S {S}

Representation_up_to_homotopy.html

  1. : Γ ( A ) × Γ ( 𝔤 ( A ) ) Γ ( 𝔤 ( A ) ) : ϕ ψ := [ ϕ , ψ ] , \nabla\colon\Gamma(A)\times\Gamma(\mathfrak{g}(A))\to\Gamma(\mathfrak{g}(A)):% \nabla_{\!\phi\,}\psi:=[\phi,\psi],
  2. : Γ ( A ) × Γ ( ν ( A ) ) Γ ( ν ( A ) ) : ϕ X ¯ := [ ρ ( ϕ ) , X ] ¯ . \nabla\colon\Gamma(A)\times\Gamma(\nu(A))\to\Gamma(\nu(A)):\nabla_{\!\phi\,}% \overline{X}:=\overline{[\rho(\phi),X]}.
  3. H n ( A , 𝔤 ( A ) ) H d e f n ( A ) H n - 1 ( A , ν ( A ) ) H n - 1 ( A , 𝔤 ( A ) ) \dots\to H^{n}(A,\mathfrak{g}(A))\to H^{n}_{def}(A)\to H^{n-1}(A,\nu(A))\to H^% {n-1}(A,\mathfrak{g}(A))\to\dots
  4. D : Ω ( A , E ) Ω + 1 ( A , E ) , D\colon\Omega^{\bullet}(A,E)\to\Omega^{\bullet+1}(A,E),
  5. D ( α β ) = ( D α ) β + ( - 1 ) | α | α ( D β ) , D(\alpha\wedge\beta)=(D\alpha)\wedge\beta+(-1)^{|\alpha|}\alpha\wedge(D\beta),
  6. = \nabla\circ\partial=\partial\circ\nabla
  7. ω 2 + R = 0 , \partial\omega_{2}+R_{\nabla}=0,
  8. D = + + ω 2 + ω 3 + . D=\partial+\nabla+\omega_{2}+\omega_{3}+\cdots.\,
  9. D E Φ = Φ D E . D_{E}\circ\Phi=\Phi\circ D_{E}.\,
  10. Φ i Ω i ( A , Hom - i ( E , F ) ) \Phi_{i}\in\Omega^{i}(A,\mathrm{Hom}^{-i}(E,F))
  11. Φ n + d ( Φ n - 1 ) + [ ω 2 , Φ n - 2 ] + + [ ω n , Φ 0 ] = 0. \partial\Phi_{n}+d_{\nabla}(\Phi_{n-1})+[\omega_{2},\Phi_{n-2}]+\cdots+[\omega% _{n},\Phi_{0}]=0.
  12. ϕ b a s ψ := [ ϕ , ψ ] + ρ ( ψ ) ϕ , \nabla^{bas}_{\!\phi\,}\psi:=[\phi,\psi]+\nabla_{\!\rho(\psi)\,}\phi,
  13. ϕ b a s X := [ ρ ( ϕ ) , X ] + ρ ( X ϕ ) . \nabla^{bas}_{\!\phi\,}X:=[\rho(\phi),X]+\rho(\nabla_{\!X\,}\phi).
  14. R b a s ( ϕ , ψ ) ( X ) := X [ ϕ , ψ ] - [ X ϕ , ψ ] - [ ϕ , X ψ ] - ψ b a s X ϕ + ψ b a s X ϕ . R^{bas}(\phi,\psi)(X):=\nabla_{\!X\,}[\phi,\psi]-[\nabla_{\!X\,}\phi,\psi]-[% \phi,\nabla_{\!X\,}\psi]-\nabla_{\!\nabla^{bas}_{\!\psi\,}X\,}\phi+\nabla_{\!% \nabla^{bas}_{\!\psi\,}X\,}\phi.
  15. D = ρ + b a s + R b a s . D=\rho+\nabla^{bas}+R^{bas}.

Residual-resistance_ratio.html

  1. R R R = ρ 300 K ρ 0 K RRR={\rho_{300K}\over\rho_{0K}}

Residue_at_infinity.html

  1. \infty
  2. \mathbb{C}
  3. ^ \hat{\mathbb{C}}
  4. A ( 0 , R , ) A(0,R,\infty)
  5. R R
  6. Res ( f , ) = Res ( - 1 z 2 f ( 1 z ) , 0 ) \mathrm{Res}(f,\infty)=\mathrm{Res}\left({-1\over z^{2}}f\left({1\over z}% \right),0\right)
  7. f ( z ) f(z)
  8. f ( 1 / z ) f(1/z)
  9. r > R \forall r>R
  10. Res ( f , ) = - 1 2 π i C ( 0 , r ) f ( z ) d z \mathrm{Res}(f,\infty)={-1\over 2\pi i}\int_{C(0,r)}f(z)\,dz

Residue_curve.html

  1. d x d ξ = x - y \frac{dx}{d\xi}=x-y

Restricted_Boltzmann_machine.html

  1. W = ( w i , j ) W=(w_{i,j})
  2. h j h_{j}
  3. v i v_{i}
  4. a i a_{i}
  5. b j b_{j}
  6. ( v , h ) (v,h)
  7. E ( v , h ) = - i a i v i - j b j h j - i j v i w i , j h j E(v,h)=-\sum_{i}a_{i}v_{i}-\sum_{j}b_{j}h_{j}-\sum_{i}\sum_{j}v_{i}w_{i,j}h_{j}
  8. E ( v , h ) = - a T v - b T h - v T W h E(v,h)=-a^{\mathrm{T}}v-b^{\mathrm{T}}h-v^{\mathrm{T}}Wh
  9. P ( v , h ) = 1 Z e - E ( v , h ) P(v,h)=\frac{1}{Z}e^{-E(v,h)}
  10. Z Z
  11. e - E ( v , h ) e^{-E(v,h)}
  12. P ( v ) = 1 Z h e - E ( v , h ) P(v)=\frac{1}{Z}\sum_{h}e^{-E(v,h)}
  13. m m
  14. n n
  15. v v
  16. h h
  17. P ( v | h ) = i = 1 m P ( v i | h ) P(v|h)=\prod_{i=1}^{m}P(v_{i}|h)
  18. h h
  19. v v
  20. P ( h | v ) = j = 1 n P ( h j | v ) P(h|v)=\prod_{j=1}^{n}P(h_{j}|v)
  21. P ( h j = 1 | v ) = σ ( b j + i = 1 m w i , j v i ) P(h_{j}=1|v)=\sigma\left(b_{j}+\sum_{i=1}^{m}w_{i,j}v_{i}\right)\,
  22. P ( v i = 1 | h ) = σ ( a i + j = 1 n w i , j h j ) \,P(v_{i}=1|h)=\sigma\left(a_{i}+\sum_{j=1}^{n}w_{i,j}h_{j}\right)
  23. σ \sigma
  24. P ( v i k = 1 | h ) = exp ( a i k + Σ j W i j k h j ) Σ k = 1 K exp ( a i k + Σ j W i j k h j ) P(v_{i}^{k}=1|h)=\frac{\exp(a_{i}^{k}+\Sigma_{j}W_{ij}^{k}h_{j})}{\Sigma_{k=1}% ^{K}\exp(a_{i}^{k}+\Sigma_{j}W_{ij}^{k}h_{j})}
  25. V V
  26. v v
  27. arg max W v V P ( v ) \arg\max_{W}\prod_{v\in V}P(v)
  28. V V
  29. arg max W 𝔼 [ v V log P ( v ) ] \arg\max_{W}\mathbb{E}\left[\sum_{v\in V}\log P(v)\right]
  30. W W
  31. v v
  32. h h
  33. v v
  34. h h
  35. h h
  36. v v
  37. h h
  38. v v
  39. h h
  40. w i , j w_{i,j}
  41. Δ w i , j = ϵ ( v h 𝖳 - v h T ) \Delta w_{i,j}=\epsilon(vh^{\mathsf{T}}-v^{\prime}h^{\prime\mathsf{}}{T})
  42. a a
  43. b b

Restrictive_flow_orifice.html

  1. F l o w = Flow Rate of N2 at the same pressure * 1 Specific Gravity Flow=\,\text{Flow Rate of N2 at the same pressure}*\sqrt{\frac{1}{\,\text{% Specific Gravity}}}

Retina_Display.html

  1. 2 d r tan ( 0.5 ) 2dr\tan(0.5^{\circ})
  2. d d
  3. r r

Retkes_convergence_criterion.html

  1. { z k } k = 1 C \quad\{z_{k}\}_{k=1}^{\infty}\subset C
  2. z i z j z_{i}\neq z_{j}\quad
  3. i j \quad i\neq j\quad
  4. k = 1 z k = s lim n k = 1 n z k n Π k ( z 1 , , z n ) = s \sum_{k=1}^{\infty}z_{k}=s\quad\iff\quad\lim_{n\to\infty}\sum_{k=1}^{n}\frac{z% _{k}^{n}}{\Pi_{k}(z_{1},\ldots,z_{n})}=s
  5. Π k ( z 1 , , z n ) := ( z k - z 1 ) ( z k - z 2 ) ( z k - z k - 1 ) ( z k - z k + 1 ) ( z k - z n ) k = 1 , , n . \Pi_{k}(z_{1},\ldots,z_{n}):=(z_{k}-z_{1})(z_{k}-z_{2})\cdots(z_{k}-z_{k-1})(z% _{k}-z_{k+1})\cdots(z_{k}-z_{n})\quad k=1,\ldots,n.

Retkes_identities.html

  1. f ( u ) = u α f(u)=u^{\alpha}
  2. 0 u < 0\leq u<\infty
  3. 0 α 0\leq\alpha
  4. F ( n - 1 ) ( s ) = s α + n - 1 ( α + 1 ) ( α + 2 ) ( α + n - 1 ) . F^{(n-1)}(s)=\frac{s^{\alpha+n-1}}{(\alpha+1)(\alpha+2)\cdots(\alpha+n-1)}.
  5. f f
  6. α > 1 \alpha>1
  7. 0 < α < 1 0<\alpha<1
  8. α = 0 , 1 \alpha=0,1
  9. 1 < α 1 ( α + 1 ) ( α + 2 ) ( α + n - 1 ) i = 1 n x i α + n - 1 Π k ( x 1 , , x n ) < 1 n ! i = 1 n x i α 1<\alpha\quad\quad\quad\quad\frac{1}{(\alpha+1)(\alpha+2)\cdots(\alpha+n-1)}% \sum_{i=1}^{n}\frac{x_{i}^{\alpha+n-1}}{\Pi_{k}(x_{1},\ldots,x_{n})}<\frac{1}{% n!}\sum_{i=1}^{n}x_{i}^{\alpha}
  10. α = 1 i = 1 n x i n Π i ( x 1 , , x n ) = i = 1 n x i \alpha=1\quad\quad\quad\quad\sum_{i=1}^{n}\frac{x_{i}^{n}}{\Pi_{i}(x_{1},% \ldots,x_{n})}=\sum_{i=1}^{n}x_{i}
  11. 0 < α < 1 1 ( α + 1 ) ( α + 2 ) ( α + n - 1 ) i = 1 n x i α + n - 1 Π k ( x 1 , , x n ) > 1 n ! i = 1 n x i α 0<\alpha<1\quad\quad\frac{1}{(\alpha+1)(\alpha+2)\cdots(\alpha+n-1)}\sum_{i=1}% ^{n}\frac{x_{i}^{\alpha+n-1}}{\Pi_{k}(x_{1},\ldots,x_{n})}>\frac{1}{n!}\sum_{i% =1}^{n}x_{i}^{\alpha}
  12. α = 0 i = 1 n x i n - 1 Π i ( x 1 , , x n ) = 1. \alpha=0\quad\quad\quad\quad\sum_{i=1}^{n}\frac{x_{i}^{n-1}}{\Pi_{i}(x_{1},% \ldots,x_{n})}=1.
  13. α = 1 \quad\alpha=1
  14. i = 1 x i . \quad\sum_{i=1}^{\infty}x_{i}.
  15. x k 0 k = 1 , , n . x_{k}\neq 0\quad k=1,\ldots,n.
  16. 1 x k \quad\frac{1}{x_{k}}
  17. x k \quad x_{k}
  18. i = 1 n x i n Π i ( x 1 , , x n ) = i = 1 n x i \quad\sum_{i=1}^{n}\frac{x_{i}^{n}}{\Pi_{i}(x_{1},\ldots,x_{n})}=\sum_{i=1}^{n% }x_{i}
  19. i = 1 n x i n - 1 Π i ( x 1 , , x n ) = 1 \quad\sum_{i=1}^{n}\frac{x_{i}^{n-1}}{\Pi_{i}(x_{1},\ldots,x_{n})}=1
  20. i = 1 n 1 x i = ( - 1 ) n - 1 i = 1 n x i i = 1 n 1 x i 2 Π i ( x 1 , , x n ) \quad\sum_{i=1}^{n}\frac{1}{x_{i}}=(-1)^{n-1}\prod_{i=1}^{n}x_{i}\sum_{i=1}^{n% }\frac{1}{{x_{i}}^{2}\Pi_{i}(x_{1},\ldots,x_{n})}
  21. i = 1 n 1 x i = ( - 1 ) n - 1 i = 1 n 1 x i Π i ( x 1 , , x n ) \quad\prod_{i=1}^{n}\frac{1}{x_{i}}=(-1)^{n-1}\sum_{i=1}^{n}\frac{1}{x_{i}\Pi_% {i}(x_{1},\ldots,x_{n})}

Revitalizant.html

  1. nMe + mC Me n C m , \mathrm{nMe+mC\rightarrow Me_{n}C_{m}},

Reward-based_selection.html

  1. a ( g + 1 ) a^{\prime(g+1)}
  2. r ( g ) r^{(g)}
  3. a ( g + 1 ) a^{\prime(g+1)}
  4. Q ( g + 1 ) Q^{(g+1)}
  5. r ( g ) = 1 r^{(g)}=1
  6. a ( g + 1 ) a^{\prime(g+1)}
  7. Q ( g + 1 ) Q^{(g+1)}
  8. r ( g ) = 1 - r a n k ( a ( g + 1 ) ) μ if a ( g + 1 ) Q ( g + 1 ) r^{(g)}=1-\frac{rank(a^{\prime(g+1)})}{\mu}\mbox{ if }~{}a^{\prime(g+1)}\in Q^% {(g+1)}
  9. r a n k ( a ( g + 1 ) ) rank(a^{\prime(g+1)})
  10. μ \mu
  11. r ( g ) = a Q ( g + 1 ) Δ H ( a , Q ( g + 1 ) ) - a Q ( g ) Δ H ( a , Q ( g ) ) r^{(g)}=\sum_{a\in Q^{(g+1)}}\Delta{H}(a,Q^{(g+1)})-\sum_{a\in Q^{(g)}}\Delta{% H}(a,Q^{(g)})
  12. Δ H ( a , Q ( g ) \Delta{H}(a,Q^{(g)}
  13. a a
  14. Q ( g ) Q^{(g)}
  15. r ( g ) > 0 r^{(g)}>0
  16. k k
  17. r ( g ) = 1 2 k - 1 ( n d o m k ( Q ( g + 1 ) ) Δ H ( a , n d o m k ( Q ( g + 1 ) ) ) - n d o m k ( Q ( g ) ) Δ H ( a , n d o m k ( Q ( g ) ) ) ) r^{(g)}=\frac{1}{2^{k-1}}\left(\sum_{ndom_{k}(Q^{(g+1)})}\Delta{H}(a,ndom_{k}(% Q^{(g+1)}))-\sum_{ndom_{k}(Q^{(g)})}\Delta{H}(a,ndom_{k}(Q^{(g)}))\right)

Reynolds_operator.html

  1. φ ¯ \overline{φ}
  2. ϕ \phi
  3. ψ \psi
  4. a a
  5. , \langle\rangle,
  6. ϕ + ψ = ϕ + ψ , \langle\phi+\psi\rangle=\langle\phi\rangle+\langle\psi\rangle,\,
  7. a ϕ = a ϕ , \langle a\phi\rangle=a\langle\phi\rangle,\,
  8. ϕ ψ = ϕ ψ , \langle\langle\phi\rangle\psi\rangle=\langle\phi\rangle\langle\psi\rangle,\,
  9. ϕ = ϕ . \langle\langle\phi\rangle\rangle=\langle\phi\rangle.\,
  10. ϕ t = ϕ t , ϕ x = ϕ x , \left\langle\frac{\partial\phi}{\partial t}\right\rangle=\frac{\partial\langle% \phi\rangle}{\partial t},\qquad\left\langle\frac{\partial\phi}{\partial x}% \right\rangle=\frac{\partial\langle\phi\rangle}{\partial x},
  11. ϕ ( s y m b o l x , t ) d s y m b o l x d t = ϕ ( s y m b o l x , t ) d s y m b o l x d t . \left\langle\int\phi(symbol{x},t)\,dsymbol{x}\,dt\right\rangle=\int\langle\phi% (symbol{x},t)\rangle\,dsymbol{x}\,dt.

Rho_(disambiguation).html

  1. ρ + \rho^{+}
  2. ρ 0 \rho^{0}
  3. ρ - \rho^{-}

Ridit_scoring.html

  1. p j = P r o b ( x j ) . p_{j}=Prob({x_{j}}).
  2. w j = 0.5 p j + k < j p k . w_{j}=0.5p_{j}+\sum_{k<j}{p_{k}}.
  3. 1 j n 1\leq j\leq n
  4. k < j P r o b ( x k ) \sum_{k<j}{Prob(x_{k})}

Riemann_invariant.html

  1. l i ( A i j u j t + a i j u j x ) + l j b j = 0 l_{i}\left(A_{ij}\frac{\partial u_{j}}{\partial t}+a_{ij}\frac{\partial u_{j}}% {\partial x}\right)+l_{j}b_{j}=0
  2. A i j A_{ij}
  3. a i j a_{ij}
  4. 𝐀 \mathbf{A}
  5. 𝐚 \mathbf{a}
  6. l i l_{i}
  7. b i b_{i}
  8. m j ( β u j t + α u j x ) + l j b j = 0 m_{j}\left(\beta\frac{\partial u_{j}}{\partial t}+\alpha\frac{\partial u_{j}}{% \partial x}\right)+l_{j}b_{j}=0
  9. ( x , t ) (x,t)
  10. ( α , β ) (\alpha,\beta)
  11. x , t x,t
  12. x = X ( η ) , t = T ( η ) x=X(\eta),t=T(\eta)
  13. d u j d η = T u j t + X u j x \frac{du_{j}}{d\eta}=T^{\prime}\frac{\partial u_{j}}{\partial t}+X^{\prime}% \frac{\partial u_{j}}{\partial x}
  14. α = X ( η ) , β = T ( η ) \alpha=X^{\prime}(\eta),\beta=T^{\prime}(\eta)
  15. m j d u j d η + l j b j = 0 m_{j}\frac{du_{j}}{d\eta}+l_{j}b_{j}=0
  16. l i A i j = m j T l_{i}A_{ij}=m_{j}T^{\prime}
  17. l i a i j = m j X l_{i}a_{ij}=m_{j}X^{\prime}
  18. m j m_{j}
  19. l i ( A i j X - a i j T ) = 0 l_{i}(A_{ij}X^{\prime}-a_{ij}T^{\prime})=0
  20. | A i j X - a i j T | = 0 |A_{ij}X^{\prime}-a_{ij}T^{\prime}|=0
  21. 𝐀 \mathbf{A}
  22. u i t + a i j u j x = 0 \frac{\partial u_{i}}{\partial t}+a_{ij}\frac{\partial u_{j}}{\partial x}=0
  23. 𝐧 \mathbf{n}
  24. l i d u i d t = 0 l_{i}\frac{du_{i}}{dt}=0
  25. d x d t = λ \frac{dx}{dt}=\lambda
  26. l l
  27. 𝐀 \mathbf{A}
  28. λ s \lambda^{\prime}s
  29. 𝐀 \mathbf{A}
  30. | A - λ δ i j | = 0 |A-\lambda\delta_{ij}|=0
  31. d r i d t = l i d u i d t \frac{dr_{i}}{dt}=l_{i}\frac{du_{i}}{dt}
  32. μ l i d u i = d r i \mu l_{i}du_{i}=dr_{i}
  33. μ \mu
  34. d r i d t = 0 \frac{dr_{i}}{dt}=0
  35. d x d t = λ i \frac{dx}{dt}=\lambda_{i}
  36. r t k + λ k r x k = 0 , r_{t}^{k}+\lambda_{k}r_{x}^{k}=0,
  37. k = 1 , , N . k=1,...,N.
  38. ρ t + ρ u x + u ρ x = 0 \rho_{t}+\rho u_{x}+u\rho_{x}=0
  39. u t + u u x + ( c 2 / ρ ) ρ x = 0 u_{t}+uu_{x}+(c^{2}/\rho)\rho_{x}=0
  40. ( ρ u ) t + ( u ρ c 2 ρ u ) ( ρ u ) x = ( 0 0 ) \left(\begin{matrix}\rho\\ u\end{matrix}\right)_{t}+\left(\begin{matrix}u&\rho\\ \frac{c^{2}}{\rho}&u\end{matrix}\right)\left(\begin{matrix}\rho\\ u\end{matrix}\right)_{x}=\left(\begin{matrix}0\\ 0\end{matrix}\right)
  41. 𝐚 \mathbf{a}
  42. λ 2 - 2 u λ + u 2 - c 2 = 0 \lambda^{2}-2u\lambda+u^{2}-c^{2}=0
  43. λ = u ± c \lambda=u\pm c
  44. ( 1 c ρ ) , ( 1 - c ρ ) \left(\begin{matrix}1\\ \frac{c}{\rho}\end{matrix}\right),\left(\begin{matrix}1\\ -\frac{c}{\rho}\end{matrix}\right)
  45. r 1 = u + c ρ d ρ , r_{1}=u+\int\frac{c}{\rho}d\rho,
  46. r 2 = u - c ρ d ρ , r_{2}=u-\int\frac{c}{\rho}d\rho,
  47. c = ρ c=\sqrt{\rho}
  48. r 1 = u + 2 ρ , r_{1}=u+2\sqrt{\rho},
  49. r 2 = u - 2 ρ , r_{2}=u-2\sqrt{\rho},
  50. r 1 t + ( u + ρ ) r 1 x = 0 \frac{\partial r_{1}}{\partial t}+(u+\sqrt{\rho})\frac{\partial r_{1}}{% \partial x}=0
  51. r 2 t + ( u - ρ ) r 2 x = 0 \frac{\partial r_{2}}{\partial t}+(u-\sqrt{\rho})\frac{\partial r_{2}}{% \partial x}=0
  52. A v t + B v x = 0 A\frac{\partial v}{\partial t}+B\frac{\partial v}{\partial x}=0
  53. A - 1 A^{-1}
  54. 𝐀 \mathbf{A}

Riemann_sphere.html

  1. 𝐂 ^ , 𝐂 ¯ , or 𝐂 . \hat{\mathbf{C}},\quad\overline{\mathbf{C}},\quad\,\text{or}\quad\mathbf{C}_{% \infty}.
  2. z + = z+\infty=\infty
  3. z = z\cdot\infty=\infty
  4. z / 0 = and z / = 0 z/0=\infty\quad\,\text{and}\quad z/\infty=0
  5. z 0 z_{0}
  6. h ( z 0 ) h(z_{0})
  7. g ( z 0 ) g(z_{0})
  8. f ( z 0 ) f(z_{0})
  9. f ( z ) = 6 z 2 + 1 2 z 2 - 50 f(z)=\frac{6z^{2}+1}{2z^{2}-50}
  10. f ( z ) = 1 z f(z)=\frac{1}{z}\qquad
  11. ( α , β ) = ( λ α , λ β ) (\alpha,\beta)=(\lambda\alpha,\lambda\beta)
  12. ( α , β ) = ( ζ , 1 ) . (\alpha,\beta)=(\zeta,1).
  13. ( α , β ) = ( 1 , ξ ) . (\alpha,\beta)=(1,\xi).
  14. ( 1 , ξ ) = ( 1 / ξ , 1 ) = ( ζ , 1 ) = ( 1 , 1 / ζ ) (1,\xi)=(1/\xi,1)=(\zeta,1)=(1,1/\zeta)
  15. ζ = x + i y 1 - z = cot ( 1 2 ϕ ) e i θ . \zeta=\frac{x+iy}{1-z}=\cot(\tfrac{1}{2}\phi)\;e^{i\theta}.
  16. ξ = x - i y 1 + z = tan ( 1 2 ϕ ) e - i θ . \xi=\frac{x-iy}{1+z}=\tan(\tfrac{1}{2}\phi)\;e^{-i\theta}.
  17. 1 / K 1/\sqrt{K}
  18. d s 2 = ( 2 1 + | ζ | 2 ) 2 | d ζ | 2 = 4 ( 1 + ζ ζ ¯ ) 2 d ζ d ζ ¯ . ds^{2}=\left(\frac{2}{1+|\zeta|^{2}}\right)^{2}\,|d\zeta|^{2}=\frac{4}{\left(1% +\zeta\bar{\zeta}\right)^{2}}\,d\zeta\,d\bar{\zeta}.
  19. d s 2 = 4 ( 1 + u 2 + v 2 ) 2 ( d u 2 + d v 2 ) . ds^{2}=\frac{4}{\left(1+u^{2}+v^{2}\right)^{2}}\left(du^{2}+dv^{2}\right).
  20. f ( ζ ) = a ζ + b c ζ + d , f(\zeta)=\frac{a\zeta+b}{c\zeta+d},
  21. a d - b c 0 ad-bc\neq 0
  22. f ( α , β ) = ( a α + b β , c α + d β ) = ( α β ) ( a c b d ) . f(\alpha,\beta)=(a\alpha+b\beta,c\alpha+d\beta)=\begin{pmatrix}\alpha&\beta% \end{pmatrix}\begin{pmatrix}a&c\\ b&d\end{pmatrix}.

Rigid_transformation.html

  1. d ( 𝐗 , 𝐘 ) 2 = ( X 1 - Y 1 ) 2 + ( X 2 - Y 2 ) 2 + + ( X n - Y n ) 2 = ( 𝐗 - 𝐘 ) ( 𝐗 - 𝐘 ) . d(\mathbf{X},\mathbf{Y})^{2}=(X_{1}-Y_{1})^{2}+(X_{2}-Y_{2})^{2}+\ldots+(X_{n}% -Y_{n})^{2}=(\mathbf{X}-\mathbf{Y})\cdot(\mathbf{X}-\mathbf{Y}).
  2. d ( g ( 𝐗 ) , g ( 𝐘 ) ) 2 = d ( 𝐗 , 𝐘 ) 2 . d(g(\mathbf{X}),g(\mathbf{Y}))^{2}=d(\mathbf{X},\mathbf{Y})^{2}.
  3. d ( 𝐯 + 𝐝 , 𝐰 + 𝐝 ) 2 = ( 𝐯 + 𝐝 - 𝐰 - 𝐝 ) ( 𝐯 + 𝐝 - 𝐰 - 𝐝 ) = ( 𝐯 - 𝐰 ) ( 𝐯 - 𝐰 ) = d ( 𝐯 , 𝐰 ) 2 . d(\mathbf{v}+\mathbf{d},\mathbf{w}+\mathbf{d})^{2}=(\mathbf{v}+\mathbf{d}-% \mathbf{w}-\mathbf{d})\cdot(\mathbf{v}+\mathbf{d}-\mathbf{w}-\mathbf{d})=(% \mathbf{v}-\mathbf{w})\cdot(\mathbf{v}-\mathbf{w})=d(\mathbf{v},\mathbf{w})^{2}.
  4. L ( 𝐕 ) = L ( a 𝐯 + b 𝐰 ) = a L ( 𝐯 ) + b L ( 𝐰 ) . L(\mathbf{V})=L(a\mathbf{v}+b\mathbf{w})=aL(\mathbf{v})+bL(\mathbf{w}).
  5. d ( [ L ] 𝐯 , [ L ] 𝐰 ) 2 = d ( 𝐯 , 𝐰 ) 2 , d([L]\mathbf{v},[L]\mathbf{w})^{2}=d(\mathbf{v},\mathbf{w})^{2},
  6. d ( [ L ] 𝐯 , [ L ] 𝐰 ) 2 = ( [ L ] 𝐯 - [ L ] 𝐰 ) ( [ L ] 𝐯 - [ L ] 𝐰 ) = ( [ L ] ( 𝐯 - 𝐰 ) ) ( [ L ] ( 𝐯 - 𝐰 ) ) . d([L]\mathbf{v},[L]\mathbf{w})^{2}=([L]\mathbf{v}-[L]\mathbf{w})\cdot([L]% \mathbf{v}-[L]\mathbf{w})=([L](\mathbf{v}-\mathbf{w}))\cdot([L](\mathbf{v}-% \mathbf{w})).
  7. d ( [ L ] 𝐯 , [ L ] 𝐰 ) 2 = ( 𝐯 - 𝐰 ) T [ L ] T [ L ] ( 𝐯 - 𝐰 ) . d([L]\mathbf{v},[L]\mathbf{w})^{2}=(\mathbf{v}-\mathbf{w})^{T}[L]^{T}[L](% \mathbf{v}-\mathbf{w}).
  8. [ L ] T [ L ] = [ I ] , [L]^{T}[L]=[I],
  9. det ( [ L ] T [ L ] ) = det [ L ] 2 = det [ I ] = 1 , \det([L]^{T}[L])=\det[L]^{2}=\det[I]=1,

Riley_slice.html

  1. ( 1 1 0 1 ) , ( 1 0 ρ 1 ) \begin{pmatrix}1&1\\ 0&1\\ \end{pmatrix},\begin{pmatrix}1&0\\ \rho&1\\ \end{pmatrix}

Ring_of_mixed_characteristic.html

  1. R R
  2. I I
  3. R / I R/I
  4. \mathbb{Z}
  5. p p
  6. 𝔽 p = / p \mathbb{F}_{p}=\mathbb{Z}/p\mathbb{Z}
  7. p p
  8. p p
  9. P P
  10. 𝒪 K \mathcal{O}_{K}
  11. K K
  12. 𝒪 K \mathcal{O}_{K}
  13. P P

Rips_machine.html

  1. \mathbb{R}
  2. \mathbb{R}
  3. \mathbb{R}
  4. \mathbb{R}
  5. \mathbb{R}

Risk_management_in_Indian_banks.html

  1. R R = p issue occurring p issue not occurring RR=\frac{p\text{issue occurring}}{p\text{issue not occurring}}
  2. T I R = I × R R TIR=I\times RR
  3. E = P ( n - X ) 2 E=\sqrt{\sum P(n-X)^{2}}

Risk_return_ratio.html

  1. R = ( P e n d - P s t a r t ) / P s t a r t R=(P_{end}-P_{start})/P_{start}
  2. P s t a r t P_{start}
  3. P e n d P_{end}
  4. M D D = M a x t ( s t a r t , e n d ) ( D D t ) where D D t = { 1 - P t P t - 1 / D D t - 1 if P t - P t - 1 < 0 0 otherwise MDD=Max_{t\in(start,end)}(DD_{t})\,\text{ where }DD_{t}=\begin{cases}% \displaystyle 1-\frac{P_{t}}{P_{t-1}/DD_{t-1}}&\,\text{if }P_{t}-P_{t-1}<0\\ 0&\,\text{otherwise}\end{cases}
  5. R R R = R / M D D RRR=R/MDD

Ritz's_Equation.html

  1. 𝐅 = q 1 q 2 4 π ϵ 0 r 2 [ [ 1 + 3 - k 4 ( v c ) 2 - 3 ( 1 - k ) 4 ( 𝐯 𝐫 c 2 ) 2 - r 2 c 2 ( 𝐚 𝐫 ) ] 𝐫 r - k + 1 2 c 2 ( 𝐯 𝐫 ) 𝐯 - r c 2 ( 𝐚 ) ] \mathbf{F}=\frac{q_{1}q_{2}}{4\pi\epsilon_{0}r^{2}}\left[\left[1+\frac{3-k}{4}% \left(\frac{v}{c}\right)^{2}-\frac{3(1-k)}{4}\left(\frac{\mathbf{v\cdot r}}{c^% {2}}\right)^{2}-\frac{r}{2c^{2}}(\mathbf{a\cdot r})\right]\frac{\mathbf{r}}{r}% -\frac{k+1}{2c^{2}}(\mathbf{v\cdot r})\mathbf{v}-\frac{r}{c^{2}}(\mathbf{a})\right]
  2. D D
  3. U x U_{x}
  4. U r U_{r}
  5. a x a_{x}
  6. F x = e D [ A 1 c o s ( ρ x ) + B 1 U x U r c 2 + C 1 ρ a x c 2 ] F_{x}=eD\left[A_{1}cos(\rho x)+B_{1}\frac{U_{x}U_{r}}{c^{2}}+C_{1}\frac{\rho a% _{x}}{c^{2}}\right]
  7. A 1 A_{1}
  8. B 1 B_{1}
  9. C 1 C_{1}
  10. u 2 / c 2 u^{2}/c^{2}
  11. u ρ 2 / c 2 u^{2}_{\rho}/c^{2}
  12. X + x ( t ) = X + x ( t ) - ( t - t ) v x X+x(t^{\prime})=X^{\prime}+x^{\prime}(t^{\prime})-(t-t^{\prime})v^{\prime}_{x}
  13. D α d t e d S ρ 2 = - e ρ c ρ 2 n d S d n D\alpha\frac{dt^{\prime}e^{\prime}dS}{\rho^{2}}=-\frac{e^{\prime}\partial\rho}% {c\rho^{2}\partial n}dSdn
  14. X X^{\prime}
  15. X X
  16. ρ n = ( X Y Z ) ( X Y Z ) = a e ρ 2 ( 1 + ρ a ρ c 2 ) \frac{\partial\rho}{\partial n}=\frac{\partial(XYZ)}{\partial(X^{\prime}Y^{% \prime}Z^{\prime})}=\frac{ae^{\prime}}{\rho^{2}}\left(1+\frac{\rho a^{\prime}_% {\rho}}{c^{2}}\right)
  17. ρ \rho
  18. U ρ < ρ Align g t ; U_{\rho}<\rho&gt;
  19. ρ = r ( 1 + r a r c 2 ) 1 / 2 \rho=r\left(1+\frac{ra^{\prime}_{r}}{c^{2}}\right)^{1/2}
  20. ρ x = r x + r 2 a x 2 c 2 \rho_{x}=r_{x}+\frac{r^{2}a^{\prime}_{x}}{2c^{2}}
  21. U ρ = v r - v r + r a r c U_{\rho}=v_{r}-v^{\prime}_{r}+\frac{ra^{\prime}_{r}}{c}
  22. F x = e e r 2 ( 1 + r a r c 2 ) [ A c o s ( r x ) ( 1 - 3 r a r 2 c 2 ) + A ( r a x 2 c 2 ) - B ( u x u r c 2 ) - C ( r a x c 2 ) ] F_{x}=\frac{ee^{\prime}}{r^{2}}\left(1+\frac{ra^{\prime}_{r}}{c^{2}}\right)% \left[Acos(rx)\left(1-\frac{3ra^{\prime}_{r}}{2c^{2}}\right)+A\left(\frac{ra^{% \prime}_{x}}{2c^{2}}\right)-B\left(\frac{u_{x}u_{r}}{c^{2}}\right)-C\left(% \frac{ra^{\prime}_{x}}{c^{2}}\right)\right]
  23. A = α 0 + α 1 u 2 c 2 + α 2 u r 2 c 2 + A=\alpha_{0}+\alpha_{1}\frac{u^{2}}{c^{2}}+\alpha_{2}\frac{u^{2}_{r}}{c^{2}}+...
  24. B = β 0 + β 1 u 2 c 2 + β 2 u r 2 c 2 + B=\beta_{0}+\beta_{1}\frac{u^{2}}{c^{2}}+\beta_{2}\frac{u^{2}_{r}}{c^{2}}+...
  25. C = γ 0 + γ 1 u 2 c 2 + γ 2 u r 2 c 2 + C=\gamma_{0}+\gamma_{1}\frac{u^{2}}{c^{2}}+\gamma_{2}\frac{u^{2}_{r}}{c^{2}}+...
  26. F x = e e r 2 [ ( α 0 + α 1 u x 2 c 2 + α 2 u r 2 c 2 ) c o s ( r x ) - β 0 u x u r c 2 - α 0 r a r 2 c 2 + ( r a x 2 c 2 ) ( α 0 - 2 γ 0 ) ] F_{x}=\frac{ee^{\prime}}{r^{2}}\left[\left(\alpha_{0}+\alpha_{1}\frac{u_{x}^{2% }}{c^{2}}+\alpha_{2}\frac{u^{2}_{r}}{c^{2}}\right)cos(rx)-\beta_{0}\frac{u_{x}% u_{r}}{c^{2}}-\alpha_{0}\frac{ra^{\prime}_{r}}{2c^{2}}+\left(\frac{ra^{\prime}% _{x}}{2c^{2}}\right)(\alpha_{0}-2\gamma_{0})\right]
  27. α 0 = 1 \alpha_{0}=1
  28. 2 γ 0 - 1 = 1 2\gamma_{0}-1=1
  29. γ 0 = 1 \gamma_{0}=1
  30. d 2 F x = i , j d e i d e j r 2 [ ( 1 + α 1 u x 2 c 2 + α 2 u r 2 c 2 ) c o s ( r x ) - β 0 u x u r c 2 - α 0 r a r 2 c 2 + r a x 2 c 2 ] d^{2}F_{x}=\sum_{i,j}\frac{de_{i}de_{j}^{\prime}}{r^{2}}\left[\left(1+\alpha_{% 1}\frac{u_{x}^{2}}{c^{2}}+\alpha_{2}\frac{u_{r}^{2}}{c^{2}}\right)cos(rx)-% \beta_{0}\frac{u_{x}u_{r}}{c^{2}}-\alpha_{0}\frac{ra^{\prime}_{r}}{2c^{2}}+% \frac{ra^{\prime}_{x}}{2c^{2}}\right]
  31. d q v = I d l dqv=Idl
  32. i , j d e i d e j = 0 \sum_{i,j}de_{i}de_{j}^{\prime}=0
  33. i , j d e i d e j u x 2 = - 2 d q d q w x w x \sum_{i,j}de_{i}de_{j}^{\prime}u^{2}_{x}=-2dqdq^{\prime}w_{x}w^{\prime}_{x}
  34. = - 2 I I d s d s c o s ϵ =-2II^{\prime}dsds^{\prime}cos\epsilon
  35. i , j d e i d e j u r 2 = - 2 d q d q w r w r \sum_{i,j}de_{i}de_{j}^{\prime}u^{2}_{r}=-2dqdq^{\prime}w_{r}w^{\prime}_{r}
  36. = - 2 I I d s d s c o s ( r d s ) c o s ( r d s ) =-2II^{\prime}dsds^{\prime}cos(rds)cos(rds)
  37. i , j d e i d e j u x u r = - d q d q ( w x w r + w x w r ) \sum_{i,j}de_{i}de_{j}^{\prime}u_{x}u_{r}=-dqdq^{\prime}(w_{x}w^{\prime}_{r}+w% ^{\prime}_{x}w_{r})
  38. = - I I d s d s [ c o s ( x d s ) c o s ( r d s ) + c o s ( r d s ) c o s ( x d s ) ] =-II^{\prime}dsds^{\prime}\left[cos(xds)cos(rds)+cos(rds)cos(xds^{\prime})\right]
  39. i , j d e i d e j a r = 0 \sum_{i,j}de_{i}de_{j}^{\prime}a^{\prime}_{r}=0
  40. i , j d e i d e j a x = 0 \sum_{i,j}de_{i}de_{j}^{\prime}a^{\prime}_{x}=0
  41. d 2 F x = I I d s d s r 2 [ [ 2 α 1 c o s ϵ + 2 α 2 c o s ( r d s ) c o s ( r d s ) ] c o s ( r x ) - β 0 c o s ( r d s ) c o s ( x d s ) - β 0 c o s ( r d s ) c o s ( x d s ) ] d^{2}F_{x}=\frac{II^{\prime}dsds^{\prime}}{r^{2}}\left[\left[2\alpha_{1}cos% \epsilon+2\alpha_{2}cos(rds)cos(rds^{\prime})\right]cos(rx)-\beta_{0}cos(rds^{% \prime})cos(xds)-\beta_{0}cos(rds)cos(xds^{\prime})\right]
  42. d 2 F x = - I I d s d s 2 r 2 [ [ ( 3 - k ) c o s ϵ - 3 ( 1 - k ) c o s ( r d s ) c o s ( r d s ) ] c o s ( r x ) - ( 1 + k ) c o s ( r d s ) c o s ( x d s ) - ( 1 + k ) c o s ( r d s ) c o s ( x d s ) ] d^{2}F_{x}=-\frac{II^{\prime}dsds^{\prime}}{2r^{2}}\left[\left[(3-k)cos% \epsilon-3(1-k)cos(rds)cos(rds^{\prime})\right]cos(rx)-(1+k)cos(rds^{\prime})% cos(xds)-(1+k)cos(rds)cos(xds^{\prime})\right]
  43. α 1 = 3 - k 4 \alpha_{1}=\frac{3-k}{4}
  44. α 2 = - 3 ( 1 - k ) 4 \alpha_{2}=-\frac{3(1-k)}{4}
  45. β 0 = 1 + k 2 \beta_{0}=\frac{1+k}{2}
  46. 𝐅 = q 1 q 2 4 π ϵ 0 r 2 [ [ 1 + 3 - k 4 ( v c ) 2 - 3 ( 1 - k ) 4 ( 𝐯 𝐫 c 2 ) 2 - r 2 c 2 ( 𝐚 𝐫 ) ] 𝐫 r - k + 1 2 c 2 ( 𝐯 𝐫 ) 𝐯 - r c 2 ( 𝐚 ) ] \mathbf{F}=\frac{q_{1}q_{2}}{4\pi\epsilon_{0}r^{2}}\left[\left[1+\frac{3-k}{4}% \left(\frac{v}{c}\right)^{2}-\frac{3(1-k)}{4}\left(\frac{\mathbf{v\cdot r}}{c^% {2}}\right)^{2}-\frac{r}{2c^{2}}(\mathbf{a\cdot r})\right]\frac{\mathbf{r}}{r}% -\frac{k+1}{2c^{2}}(\mathbf{v\cdot r})\mathbf{v}-\frac{r}{c^{2}}(\mathbf{a})\right]
  47. 𝐅 = q 1 q 2 4 π ϵ 0 r 2 [ [ 1 - ( v c ) 2 + 4.5 ( 𝐯 𝐫 c 2 ) 2 - r 2 c 2 ( 𝐚 𝐫 ) ] 𝐫 r - 4 c 2 ( 𝐯 𝐫 ) 𝐯 - r c 2 ( 𝐚 ) ] \mathbf{F}=\frac{q_{1}q_{2}}{4\pi\epsilon_{0}r^{2}}\left[\left[1-\left(\frac{v% }{c}\right)^{2}+4.5\left(\frac{\mathbf{v\cdot r}}{c^{2}}\right)^{2}-\frac{r}{2% c^{2}}(\mathbf{a\cdot r})\right]\frac{\mathbf{r}}{r}-\frac{4}{c^{2}}(\mathbf{v% \cdot r})\mathbf{v}-\frac{r}{c^{2}}(\mathbf{a})\right]

Rizza_manifold.html

  1. F ( x , c y ) = | c | F ( x , y ) c , x M , y T x M F(x,cy)=|c|F(x,y)\qquad\forall c\in\mathbb{C},\quad x\in M,\quad y\in T_{x}M

Robert_McNeill_Alexander.html

  1. v = 0.25. g - 0.5 . S L 1.67 . h - 1.17 {v}=0.25.{g^{-0.5}}.{SL^{1.67}}.{h^{-1.17}}

Robinson_Crusoe_economy.html

  1. Π \Pi
  2. Π = C - w L \Pi=C-wL\,
  3. C = Π + w L C=\Pi+wL\,
  4. F = 4 L f F=4L_{f}\,
  5. C = 8 L c C=8L_{c}\,
  6. L f + L c = 12 L_{f}+L_{c}=12\,
  7. F / 4 + C / 8 = 12 F/4+C/8=12\,
  8. = Δ C Δ F ={\Delta C\over{\Delta F}}\,
  9. = - 8 / 4 = - 2 =-8/4=-2\,
  10. F = 8 L f F=8L_{f}\,
  11. C = 4 L c C=4L_{c}\,
  12. L f + L c = 12 L_{f}+L_{c}=12\,
  13. = > F / 8 + C / 4 = 12 =>F/8+C/4=12\,
  14. = Δ C Δ F ={\Delta C\over{\Delta F}}\,
  15. = - 4 / 8 = - 1 / 2 =-4/8=-1/2\,

Rocket_propellant.html

  1. V f = V e ln ( M 0 / M f ) V_{f}=V_{e}\ln(M_{0}/M_{f})
  2. V f V_{f}
  3. V e V_{e}
  4. M 0 M_{0}
  5. M f M_{f}
  6. M M
  7. T c M \frac{\sqrt{T_{c}}}{M}
  8. T c \sqrt{T_{c}}
  9. M M

Rodion_Kuzmin.html

  1. x = 1 k 1 + 1 k 2 + x=\frac{1}{k_{1}+\frac{1}{k_{2}+\cdots}}
  2. Δ n ( s ) = { x n s } - log 2 ( 1 + s ) , \Delta_{n}(s)=\mathbb{P}\left\{x_{n}\leq s\right\}-\log_{2}(1+s),
  3. x n = 1 k n + 1 + 1 k n + 2 + . x_{n}=\frac{1}{k_{n+1}+\frac{1}{k_{n+2}+\cdots}}.
  4. | Δ n ( s ) | C e - α n , |\Delta_{n}(s)|\leq Ce^{-\alpha\sqrt{n}}~{},
  5. 2 2 = 2.6651441426902251886502972498731 2^{\sqrt{2}}=2.6651441426902251886502972498731\ldots
  6. f f
  7. f f^{\prime}
  8. f ( x ) λ > 0 \|f^{\prime}(x)\|\geq\lambda>0
  9. \|\cdot\|
  10. I I
  11. n I e 2 π i f ( n ) λ - 1 . \sum_{n\in I}e^{2\pi if(n)}\ll\lambda^{-1}.

Rogers_polynomials.html

  1. C n ( x ; β | q ) = ( β ; q ) n ( q ; q ) n e i n θ ϕ 1 2 ( q - n , β ; β - 1 q 1 - n ; q , q β - 1 e - 2 i θ ) C_{n}(x;\beta|q)=\frac{(\beta;q)_{n}}{(q;q)_{n}}e^{in\theta}{}_{2}\phi_{1}(q^{% -n},\beta;\beta^{-1}q^{1-n};q,q\beta^{-1}e^{-2i\theta})

Rogers–Szegő_polynomials.html

  1. h n ( x ; q ) = k = 0 n ( q ; q ) n ( q ; q ) k ( q ; q ) n - k x k h_{n}(x;q)=\sum_{k=0}^{n}\frac{(q;q)_{n}}{(q;q)_{k}(q;q)_{n-k}}x^{k}

Roll_forming.html

  1. F o r m i n g t i m e = [ L + n ( d ) ] / V Formingtime=[L+n(d)]/V

Rolling_cone_motion.html

  1. π \pi
  2. ω 2 ω 1 = sin α sin β \frac{\omega_{2}}{\omega_{1}}={\sin\alpha\over\sin\beta}
  3. α \alpha
  4. β \beta
  5. ω 1 \omega_{1}
  6. ω 2 \omega_{2}
  7. sin α \sin\alpha
  8. α \alpha

Root_of_unity_modulo_n.html

  1. x k 1 ( mod n ) x^{k}\equiv 1\;\;(\mathop{{\rm mod}}n)
  2. / n \mathbb{Z}/n\mathbb{Z}
  3. φ ( n ) \varphi(n)
  4. x k - 1 x^{k-1}
  5. x - 1 x-1
  6. j = 0 k - 1 x j 0 ( mod n ) \sum_{j=0}^{k-1}x^{j}\equiv 0\;\;(\mathop{{\rm mod}}n)
  7. ( x - 1 ) j = 0 k - 1 x j x k - 1 0 ( mod n ) . (x-1)\cdot\sum_{j=0}^{k-1}x^{j}\equiv x^{k}-1\equiv 0\;\;(\mathop{{\rm mod}}n).
  8. f ( n , k ) f(n,k)
  9. f ( n , 1 ) = 1 f(n,1)=1
  10. n 2 n\geq 2
  11. f ( n , λ ( n ) ) = φ ( n ) f(n,\lambda(n))=\varphi(n)
  12. φ \varphi
  13. n f ( n , k ) n\mapsto f(n,k)
  14. k l f ( n , k ) f ( n , l ) k\mid l\implies f(n,k)\mid f(n,l)
  15. f ( n , lcm ( a , b ) ) = lcm ( f ( n , a ) , f ( n , b ) ) f(n,\mathrm{lcm}(a,b))=\mathrm{lcm}(f(n,a),f(n,b))
  16. lcm \mathrm{lcm}
  17. p p
  18. i j f ( n , p i ) = p j \forall i\in\mathbb{N}\ \exists j\in\mathbb{N}\ f(n,p^{i})=p^{j}
  19. i i
  20. j j
  21. f f
  22. n n
  23. λ ( n ) \lambda(n)
  24. λ ( n ) \lambda(n)
  25. k k
  26. λ ( n ) \lambda(n)
  27. k k
  28. λ ( n ) \lambda(n)
  29. x x
  30. x λ ( n ) / k x^{\lambda(n)/k}
  31. gcd ( k , ) \mathrm{gcd}(k,\ell)
  32. k = gcd ( k , ) k=\mathrm{gcd}(k,\ell)
  33. gcd ( k , ) \mathrm{gcd}(k,\ell)
  34. g ( n , k ) g(n,k)
  35. g ( n , k ) = { > 0 if k λ ( n ) , 0 otherwise . g(n,k)=\begin{cases}>0&\,\text{if }k\mid\lambda(n),\\ 0&\,\text{otherwise}.\end{cases}
  36. k g ( n , k ) k\mapsto g(n,k)
  37. d ( λ ( n ) ) d(\lambda(n))
  38. d d
  39. g ( n , 1 ) = 1 g(n,1)=1
  40. g ( 4 , 2 ) = 1 g(4,2)=1
  41. g ( 2 n , 2 ) = 3 g(2^{n},2)=3
  42. n 3 n\geq 3
  43. g ( 2 n , 2 k ) = 2 k g(2^{n},2^{k})=2^{k}
  44. k [ 2 , n - 1 ) k\in[2,n-1)
  45. g ( n , 2 ) = 1 g(n,2)=1
  46. n 3 n\geq 3
  47. n n
  48. k g ( n , k ) = f ( n , λ ( n ) ) = φ ( n ) \sum_{k\in\mathbb{N}}g(n,k)=f(n,\lambda(n))=\varphi(n)
  49. φ \varphi
  50. f f
  51. g g
  52. f = 1 * g f={1}*g
  53. f ( n , k ) = d k g ( n , d ) f(n,k)=\sum_{d\mid k}g(n,d)
  54. g g
  55. f f
  56. x k 1 ( mod n ) x^{k}\equiv 1\;\;(\mathop{{\rm mod}}n)
  57. x 1 ( mod n ) x^{\ell}\equiv 1\;\;(\mathop{{\rm mod}}n)
  58. p prime dividing k , x k / p 1 ( mod n ) . \forall p\,\text{ prime dividing}\ k,\quad x^{k/p}\not\equiv 1\;\;(\mathop{{% \rm mod}}n).
  59. λ ( n ) \lambda(n)
  60. λ ( n ) \lambda(n)
  61. λ ( n ) \lambda(n)
  62. x λ ( n ) / k x^{\lambda(n)/k}
  63. k k
  64. λ ( n ) \lambda(n)
  65. x l x^{l}
  66. k k
  67. x l x^{l}
  68. k k
  69. k k
  70. l l
  71. x l x^{l}
  72. m m
  73. k k
  74. ( x l ) m 1 ( mod n ) (x^{l})^{m}\equiv 1\;\;(\mathop{{\rm mod}}n)
  75. k k
  76. l l
  77. l - 1 l^{-1}
  78. l l
  79. k k
  80. 1 ( ( x l ) m ) l - l x m ( mod n ) 1\equiv((x^{l})^{m})^{l^{-l}}\equiv x^{m}\;\;(\mathop{{\rm mod}}n)
  81. x x
  82. k k
  83. m m
  84. φ ( k ) \varphi(k)
  85. k k
  86. k k
  87. n n
  88. k + 1 , 2 k + 1 , 3 k + 1 , k+1,2k+1,3k+1,\dots
  89. p p
  90. λ ( p ) = p - 1 \lambda(p)=p-1
  91. m k + 1 mk+1
  92. λ ( m k + 1 ) = m k \lambda(mk+1)=mk
  93. n n
  94. k 1 k_{1}
  95. k 2 k_{2}
  96. k m k_{m}
  97. n n
  98. n n
  99. k 1 k_{1}
  100. k m k_{m}
  101. n n
  102. lcm ( k 1 , , k m ) \mathrm{lcm}(k_{1},...,k_{m})
  103. lcm ( k 1 , , k m ) \mathrm{lcm}(k_{1},...,k_{m})
  104. n n
  105. x x
  106. x lcm ( k 1 , , k m ) / k l x^{\mathrm{lcm}(k_{1},\dots,k_{m})/k_{l}}
  107. k l k_{l}
  108. n n
  109. k 1 k_{1}
  110. k m k_{m}
  111. n n
  112. k 1 , , k m k_{1},\dots,k_{m}
  113. λ ( n ) \lambda(n)
  114. lcm ( k 1 , , k m ) λ ( n ) \mathrm{lcm}(k_{1},\dots,k_{m})\mid\lambda(n)
  115. lcm ( k 1 , , k m ) \mathrm{lcm}(k_{1},...,k_{m})
  116. n n

Rosati_involution.html

  1. A A
  2. A ^ = Pic 0 ( A ) \hat{A}=\mathrm{Pic}^{0}(A)
  3. a A a\in A
  4. T a : A A T_{a}:A\to A
  5. a a
  6. T a ( x ) = x + a T_{a}(x)=x+a
  7. D D
  8. A A
  9. ϕ D : A A ^ \phi_{D}:A\to\hat{A}
  10. ϕ D ( a ) = [ T a * D - D ] \phi_{D}(a)=[T_{a}^{*}D-D]
  11. ϕ D \phi_{D}
  12. D D
  13. End ( A ) \mathrm{End}(A)\otimes\mathbb{Q}
  14. ϕ D \phi_{D}
  15. ψ End ( A ) \psi\in\mathrm{End}(A)\otimes\mathbb{Q}
  16. ψ = ϕ D - 1 ψ ^ ϕ D \psi^{\prime}=\phi_{D}^{-1}\circ\hat{\psi}\circ\phi_{D}
  17. ψ ^ : A ^ A ^ \hat{\psi}:\hat{A}\to\hat{A}
  18. ψ * \psi^{*}
  19. Pic ( A ) \mathrm{Pic}(A)
  20. NS ( A ) \mathrm{NS}(A)
  21. A A
  22. ϕ D \phi_{D}
  23. Φ : NS ( A ) End ( A ) \Phi:\mathrm{NS}(A)\otimes\mathbb{Q}\to\mathrm{End}(A)\otimes\mathbb{Q}
  24. Φ E = ϕ D - 1 ϕ E \Phi_{E}=\phi_{D}^{-1}\circ\phi_{E}
  25. Φ \Phi
  26. { ψ End ( A ) : ψ = ψ } \{\psi\in\mathrm{End}(A)\otimes\mathbb{Q}:\psi^{\prime}=\psi\}
  27. E F = 1 2 Φ - 1 ( Φ E Φ F + Φ F Φ E ) E\star F=\frac{1}{2}\Phi^{-1}(\Phi_{E}\circ\Phi_{F}+\Phi_{F}\circ\Phi_{E})
  28. NS ( A ) \mathrm{NS}(A)\otimes\mathbb{Q}

Rotating_unbalance.html

  1. e e
  2. m m
  3. U = m * r U=m*r
  4. U = m * r * d U=m*r*d

Rotation_map.html

  1. v v
  2. i i
  3. i i
  4. v v
  5. v v
  6. Rot G : [ N ] × [ D ] [ N ] × [ D ] \mathrm{Rot}_{G}:[N]\times[D]\rightarrow[N]\times[D]
  7. Rot G ( v , i ) = ( w , j ) \mathrm{Rot}_{G}(v,i)=(w,j)
  8. Rot G \mathrm{Rot}_{G}
  9. Rot G Rot G \mathrm{Rot}_{G}\circ\mathrm{Rot}_{G}
  10. Rot G \mathrm{Rot}_{G}
  11. π \pi
  12. v Rot G ( v , i ) = ( v [ i ] , π ( i ) ) \forall v\ \mathrm{Rot}_{G}(v,i)=(v[i],\pi(i))
  13. π \pi

Rotational_partition_function.html

  1. Z Z
  2. ζ \zeta
  3. E j E_{j}
  4. E j = i E j i = E j t r a n s + E j r o t + E j v i b + E j e E_{j}=\sum_{i}E_{j}^{i}=E_{j}^{trans}+E_{j}^{rot}+E_{j}^{vib}+E_{j}^{e}
  5. g j = i g j i = g j t r a n s g j r o t g j v i b g j e , g_{j}=\prod_{i}g_{j}^{i}=g_{j}^{trans}g_{j}^{rot}g_{j}^{vib}g_{j}^{e},
  6. ζ = j g j e - E j / k B T \zeta=\sum_{j}g_{j}e^{-E_{j}/k_{B}T}
  7. ζ = i ζ i = ζ t r a n s ζ r o t ζ v i b ζ e . \zeta=\prod_{i}\zeta^{i}=\zeta^{trans}\zeta^{rot}\zeta^{vib}\zeta^{e}.
  8. E j r o t = J 2 2 I = j ( j + 1 ) 2 2 I = j ( j + 1 ) ϵ . E_{j}^{rot}=\frac{{J}^{2}}{2I}=\frac{j(j+1)\hbar^{2}}{2I}=j(j+1)\epsilon.
  9. g j g_{j}
  10. j j
  11. 2 j + 1 2j+1
  12. ζ r o t = j = 0 g j e - E j / k T = j = 0 ( 2 j + 1 ) e - j ( j + 1 ) ϵ / k T . \zeta^{rot}=\sum_{j=0}^{\infty}g_{j}e^{-E_{j}/kT}=\sum_{j=0}^{\infty}(2j+1)e^{% -j(j+1)\epsilon/kT}.
  13. E j r o t E_{j}^{rot}
  14. k B T k_{B}T
  15. T = 300 K T=300K
  16. ζ r o t \zeta^{rot}
  17. ζ \zeta
  18. 10 2 10^{2}
  19. ζ r o t \zeta^{rot}
  20. T T

Rouché–Capelli_theorem.html

  1. n \mathbb{R}^{n}
  2. A = [ 1 1 2 1 1 1 2 2 2 ] , A=\begin{bmatrix}1&1&2\\ 1&1&1\\ 2&2&2\\ \end{bmatrix},
  3. ( A | B ) = [ 1 1 2 3 1 1 1 1 2 2 2 2 ] . (A|B)=\left[\begin{array}[]{ccc|c}1&1&2&3\\ 1&1&1&1\\ 2&2&2&2\end{array}\right].
  4. A = [ 1 1 2 1 1 1 2 2 2 ] , A=\begin{bmatrix}1&1&2\\ 1&1&1\\ 2&2&2\\ \end{bmatrix},
  5. ( A | B ) = [ 1 1 2 3 1 1 1 1 2 2 2 5 ] . (A|B)=\left[\begin{array}[]{ccc|c}1&1&2&3\\ 1&1&1&1\\ 2&2&2&5\end{array}\right].

Rouse_model.html

  1. ν \nu

RST_model.html

  1. S CGHS = 1 2 π d 2 x - g { e - 2 ϕ [ R + 4 ( ϕ ) 2 + 4 λ 2 ] - i = 1 N 1 2 ( f i ) 2 } S_{\,\text{CGHS}}=\frac{1}{2\pi}\int d^{2}x\,\sqrt{-g}\left\{e^{-2\phi}\left[R% +4\left(\nabla\phi\right)^{2}+4\lambda^{2}\right]-\sum^{N}_{i=1}\frac{1}{2}% \left(\nabla f_{i}\right)^{2}\right\}
  2. S RST = - κ 8 π d 2 x - g [ R 1 2 R - 2 ϕ R ] S_{\,\text{RST}}=-\frac{\kappa}{8\pi}\int d^{2}x\,\sqrt{-g}\left[R\frac{1}{% \nabla^{2}}R-2\phi R\right]
  3. ( N - 24 ) / 12 (N-24)/12
  4. N / 12 N/12
  5. S RST = - κ π d x + d x - [ + ρ - ρ + ϕ + - ρ ] S_{\,\text{RST}}=-\frac{\kappa}{\pi}\int dx^{+}\,dx^{-}\left[\partial_{+}\rho% \partial_{-}\rho+\phi\partial_{+}\partial_{-}\rho\right]

Rule_of_marteloio.html

  1. 70 miles sin D = ritorno sin 45 = avanzo sin 90 \frac{70\,\text{ miles}}{\sin D}=\frac{\,\text{ritorno}}{\sin 45}=\frac{\,% \text{avanzo}}{\sin 90}
  2. 1 / 2 {1}/{2}
  3. 1 / 5 {1}/{5}
  4. 1 / 5 {1}/{5}
  5. 26 100 = x 65 \frac{26}{100}=\frac{x}{65}
  6. < v a r > x = 65 × 26 ÷ 100 <var>x=65×26÷100
  7. 1 / 10 {1}/{10}
  8. 1 / 5 {1}/{5}
  9. 1 / 10 {1}/{10}
  10. 1 / 5 {1}/{5}
  11. 9 / 10 {9}/{10}

Runcinated_5-cubes.html

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

Runcinated_5-orthoplexes.html

  1. ( 0 , 1 , 2 , 3 , 4 ) \left(0,1,2,3,4\right)

S-adenosylmethionine_synthetase_enzyme.html

  1. \rightleftharpoons

S2P_(complexity).html

  1. L L
  2. S 2 P S_{2}^{P}
  3. x L x\in L
  4. P ( x , y , z ) = 1 P(x,y,z)=1
  5. x L x\notin L
  6. P ( x , y , z ) = 0 P(x,y,z)=0
  7. Σ 2 P \Sigma_{2}^{P}
  8. Π 2 P \Pi_{2}^{P}
  9. Σ 2 P Π 2 P \Sigma_{2}^{P}\cap\Pi_{2}^{P}
  10. Δ 2 P \Delta_{2}^{P}
  11. P S 2 P = S 2 P P^{S_{2}^{P}}=S_{2}^{P}
  12. S 2 S_{2}
  13. S 2 P = S 2 P S_{2}P=S_{2}^{P}
  14. S 2 S_{2}

S_(set_theory).html

  1. \exist s [ s < r F x s ] . \exist s[s<r\land Fxs].
  2. s t \exist r [ t < r s < r ] . \forall s\forall t\exist r[t<r\land s<r]\,.
  3. \exist r \exist u [ u < r t [ t < r \exist s [ t < s s < r ] ] ] . \exist r\exist u[u<r\land\forall t[t<r\rightarrow\exist s[t<s\land s<r]]]\,.
  4. r x [ F x r [ y ( y x B y r ) and ¬ B x r ] ] . \forall r\forall x[Fxr\leftrightarrow[\forall y(y\in x\rightarrow Byr)\and% \lnot Bxr]]\,.
  5. \exist r y [ A ( y ) B y r ] \exist x y [ y x A ( y ) ] . \exist r\forall y[A(y)\rightarrow Byr]\rightarrow\exist x\forall y[y\in x% \leftrightarrow A(y)]\,.

Safety_factor_(plasma_physics).html

  1. θ \theta
  2. ζ \zeta
  3. ϕ \phi
  4. i = 2 π R B p r B t i=2\pi\cdot\frac{R\cdot B_{p}}{r\cdot B_{t}}
  5. r r
  6. B p B_{p}
  7. B t B_{t}
  8. i i
  9. q = 2 π i = r B t R B p q=\frac{2\pi}{i}=\frac{r\cdot B_{t}}{R\cdot B_{p}}
  10. q = d ϕ d θ q=\frac{d\phi}{d\theta}

Saffman–Delbrück_model.html

  1. D 3 D = k B T 6 π η a D_{3D}=\frac{k_{B}T}{6\pi\eta a}
  2. a a
  3. h h
  4. μ m \mu_{m}
  5. μ f \mu_{f}
  6. D s d = k B T 4 π μ m h [ ln ( 2 L s d / a ) - γ ] D_{sd}=\frac{k_{B}T}{4\pi\mu_{m}h}\left[\ln(2L_{sd}/a)-\gamma\right]
  7. L s d = h μ m 2 μ f L_{sd}=\frac{h\mu_{m}}{2\mu_{f}}
  8. γ 0.577 \gamma\approx 0.577
  9. L s d L_{sd}
  10. a L s d a\ll L_{sd}
  11. a a\approx
  12. D s d D_{sd}
  13. L s d = 1 μ m L_{sd}=1\mu m
  14. a a
  15. D s d D_{sd}
  16. a a
  17. a L s d a\gg L_{sd}
  18. D k B T 8 η m h a L s d a D\to\frac{k_{B}T}{8\eta_{m}ha}\frac{L_{sd}}{a}
  19. D = k B T 4 π η m h [ ln ( 2 / ϵ ) - γ + 4 ϵ / π - ( ϵ 2 / 2 ) ln ( 2 / ϵ ) ] [ 1 - ( ϵ 3 / π ) ln ( 2 / ϵ ) + c 1 ϵ b 1 / ( 1 + c 2 ϵ b 2 ) ] - 1 D=\frac{k_{B}T}{4\pi\eta_{m}h}\left[\ln(2/\epsilon)-\gamma+4\epsilon/\pi-(% \epsilon^{2}/2)\ln(2/\epsilon)\right]\left[1-(\epsilon^{3}/\pi)\ln(2/\epsilon)% +c_{1}\epsilon^{b_{1}}/(1+c_{2}\epsilon^{b_{2}})\right]^{-1}
  20. ϵ = a / L s d \epsilon=a/L_{sd}
  21. b 1 = 2.74819 b_{1}=2.74819
  22. b 2 = 0.51465 b_{2}=0.51465
  23. c 1 = 0.73761 c_{1}=0.73761
  24. c 2 = 0.52119 c_{2}=0.52119
  25. a a
  26. a - 1 a^{-1}
  27. ln ( a ) \ln(a)

Sanov's_theorem.html

  1. x n = x 1 , x 2 , , x n x^{n}=x_{1},x_{2},\ldots,x_{n}
  2. p ^ x n \hat{p}_{x^{n}}
  3. { x n : p ^ x n A } \{x^{n}:\hat{p}_{x^{n}}\in A\}
  4. q n ( x n ) ( n + 1 ) | X | 2 - n D KL ( p * | | q ) q^{n}(x^{n})\leq(n+1)^{|X|}2^{-nD_{\mathrm{KL}}(p^{*}||q)}
  5. q n ( x n ) q^{n}(x^{n})
  6. q ( x 1 ) q ( x 2 ) q ( x n ) q(x_{1})q(x_{2})\cdots q(x_{n})
  7. p * p^{*}
  8. lim n 1 n log q n ( x n ) = - D KL ( p * | | q ) . \lim_{n\to\infty}\frac{1}{n}\log q^{n}(x^{n})=-D_{\mathrm{KL}}(p^{*}||q).

Satake_isomorphism.html

  1. c [ G ( K ) / G ( O ) ] G ( O ) K 0 ( G L - R e p ) . \mathbb{C}_{c}[G(K)/G(O)]^{G(O)}\cong K_{0}(G^{L}-Rep).

Saturated_absorption_spectroscopy.html

  1. n ( v ) d v = N m 2 π k B T e - m v 2 2 k B T d v , n(v)dv=N\sqrt{\frac{m}{2\pi k_{B}T}}e^{-\frac{mv^{2}}{2k_{B}T}}dv,
  2. N N
  3. k B k_{B}
  4. m m
  5. ω l a b = ω 0 ( 1 ± v c ) , \omega_{lab}=\omega_{0}\left(1\pm\frac{v}{c}\right),
  6. ω 0 \omega_{0}
  7. v v
  8. ω 0 \omega_{0}
  9. ω l a b \omega_{lab}
  10. Δ ω l a b = ω 0 8 k B T ln 2 m c 2 \Delta\omega_{lab}=\omega_{0}\sqrt{\frac{8k_{B}T\ln 2}{mc^{2}}}
  11. Δ ω l a b 2 π 500 MHz Γ / 2 2 π 3 MHz \Delta\omega_{lab}\approx 2\pi\cdot 500\mbox{ MHz}~{}\gg\Gamma/2\approx 2\pi% \cdot 3\mbox{ MHz}~{}

Sauer–Shelah_lemma.html

  1. = { S 1 , S 2 , } \mathcal{F}=\{S_{1},S_{2},\dots\}
  2. T T
  3. T T
  4. \mathcal{F}
  5. T T
  6. T T
  7. T S i T\cap S_{i}
  8. T T
  9. \mathcal{F}
  10. \mathcal{F}
  11. \mathcal{F}
  12. n n
  13. | | > i = 0 k - 1 ( n i ) |\mathcal{F}|>\sum_{i=0}^{k-1}{{\left({{n}\atop{i}}\right)}}
  14. \mathcal{F}
  15. k k
  16. \mathcal{F}
  17. k k
  18. \mathcal{F}
  19. i = 0 k ( n i ) = O ( n k ) \sum_{i=0}^{k}{{\left({{n}\atop{i}}\right)}}=O(n^{k})
  20. \mathcal{F}
  21. | | = i = 0 k - 1 ( n i ) |\mathcal{F}|=\sum_{i=0}^{k-1}{{\left({{n}\atop{i}}\right)}}
  22. k k
  23. \mathcal{F}
  24. { 1 , 2 , n } \{1,2,\dots n\}
  25. k k
  26. \mathcal{F}
  27. | | |\mathcal{F}|
  28. i = 0 k - 1 ( n i ) \sum_{i=0}^{k-1}{{\textstyle\left({{n}\atop{i}}\right)}}
  29. n n
  30. k k
  31. | | > i = 0 k - 1 ( n i ) |\mathcal{F}|>\sum_{i=0}^{k-1}{{\textstyle\left({{n}\atop{i}}\right)}}
  32. k k
  33. \mathcal{F}
  34. | | |\mathcal{F}|
  35. x x
  36. \mathcal{F}
  37. \mathcal{F}
  38. x x
  39. x x
  40. | | |\mathcal{F}|
  41. x x
  42. S S
  43. \mathcal{F}
  44. S S
  45. S S
  46. S { x } S\cup\{x\}
  47. \mathcal{F}
  48. S S
  49. \mathcal{F}
  50. \mathcal{F}
  51. \mathcal{F}
  52. | | |\mathcal{F}|
  53. O ( d ϵ 2 log d ϵ ) O(\tfrac{d}{\epsilon^{2}}\log\tfrac{d}{\epsilon})
  54. O ( d ϵ log 1 ϵ ) O(\tfrac{d}{\epsilon}\log\tfrac{1}{\epsilon})
  55. d ϵ ln 1 ϵ + 2 d ϵ ln ln 1 ϵ + 6 d ϵ \tfrac{d}{\epsilon}\ln\tfrac{1}{\epsilon}+\tfrac{2d}{\epsilon}\ln\ln\tfrac{1}{% \epsilon}+\tfrac{6d}{\epsilon}
  56. O ( d ϵ log 1 ϵ ) O(\tfrac{d}{\epsilon}\log\tfrac{1}{\epsilon})
  57. O ( d ϵ log 2 1 ϵ ) O(\tfrac{d}{\epsilon}\log^{2}\tfrac{1}{\epsilon})
  58. x y x\cup y

Scale_of_temperature.html

  1. p V pV
  2. p V pV
  3. T = 1 n R lim p 0 p V . T={1\over nR}\lim_{p\to 0}{pV}.
  4. η = w c y q H = q H - q C q H = 1 - q C q H ( 1 ) \eta=\frac{w_{cy}}{q_{H}}=\frac{q_{H}-q_{C}}{q_{H}}=1-\frac{q_{C}}{q_{H}}% \qquad(1)
  5. q C q H = f ( T H , T C ) ( 2 ) . \frac{q_{C}}{q_{H}}=f(T_{H},T_{C})\qquad(2).
  6. f ( T 1 , T 3 ) = q 3 q 1 = q 2 q 3 q 1 q 2 = f ( T 1 , T 2 ) f ( T 2 , T 3 ) . f(T_{1},T_{3})=\frac{q_{3}}{q_{1}}=\frac{q_{2}q_{3}}{q_{1}q_{2}}=f(T_{1},T_{2}% )f(T_{2},T_{3}).
  7. T 1 T_{1}
  8. f ( T 2 , T 3 ) = f ( T 1 , T 3 ) f ( T 1 , T 2 ) = 273.16 f ( T 1 , T 3 ) 273.16 f ( T 1 , T 2 ) . f(T_{2},T_{3})=\frac{f(T_{1},T_{3})}{f(T_{1},T_{2})}=\frac{273.16\cdot f(T_{1}% ,T_{3})}{273.16\cdot f(T_{1},T_{2})}.
  9. T = 273.16 f ( T 1 , T ) T=273.16\cdot f(T_{1},T)\,
  10. f ( T 2 , T 3 ) = T 3 T 2 , f(T_{2},T_{3})=\frac{T_{3}}{T_{2}},
  11. q C q H = f ( T H , T C ) = T C T H . ( 3 ) . \frac{q_{C}}{q_{H}}=f(T_{H},T_{C})=\frac{T_{C}}{T_{H}}.\qquad(3).
  12. η = 1 - q C q H = 1 - T C T H ( 4 ) . \eta=1-\frac{q_{C}}{q_{H}}=1-\frac{T_{C}}{T_{H}}\qquad(4).

Scanning_electrochemical_microscopy.html

  1. i T , = 4 n F C D a i_{T,\infty}=4nFCDa

Scanning_joule_expansion_microscopy.html

  1. F < m t p l 1 2 d C d z V 2 F<mtpl>{{=}}\frac{1}{2}{\operatorname{d}C\over\operatorname{d}z}V^{2}
  2. V 2 V^{2}
  3. Δ L < m t p l α C E T L Δ T \Delta\,L<mtpl>{{=}}\alpha\,\!_{CET}L\Delta\,T
  4. α C E T \alpha\,\!_{CET}
  5. ρ C p d T d t < m t p l ( k T ) + Q \rho\,\!C_{p}{\operatorname{d}T\over\operatorname{d}t}<mtpl>{{=}}\nabla\left(k% \nabla\ T\right)+Q
  6. ρ C p T n t - T n t - 1 Δ t < m t p l k n T n - 1 t - T n t Δ x 2 + k n + 1 T n + 1 t - T n t Δ x 2 + Q n \rho\,\!C_{p}\frac{T_{n}^{t}-T_{n}^{t-1}}{\Delta\ t}<mtpl>{{=}}k_{n}\frac{T_{n% -1}^{t}-T_{n}^{t}}{\Delta\ x^{2}}+k_{n+1}\frac{T_{n+1}^{t}-T_{n}^{t}}{\Delta\ % x^{2}}+Q_{n}
  7. T n t T_{n}^{t}
  8. Δ L < m t p l α C E T L Δ T \Delta\,L<mtpl>{{=}}\alpha\,\!_{CET}L\Delta\,T
  9. α C E T \alpha\,\!_{CET}

Schläfli_graph.html

  1. \square

Schmidt-Samoa_cryptosystem.html

  1. N = p 2 q N=p^{2}q
  2. d = N - 1 mod lcm ( p - 1 , q - 1 ) d=N^{-1}\mod\,\text{lcm}(p-1,q-1)
  3. c = m N mod N . c=m^{N}\mod N.
  4. m = c d mod p q , m=c^{d}\mod pq,
  5. p = 7 , q = 11 , N = p 2 q = 539 , d = N - 1 mod lcm ( p - 1 , q - 1 ) = 29 p=7,q=11,N=p^{2}q=539,d=N^{-1}\mod\,\text{lcm}(p-1,q-1)=29
  6. m = 32 , c = m N mod N = 373 m=32,c=m^{N}\mod N=373
  7. m = c d mod p q = 373 29 mod p q = 373 29 mod 77 = 32 m=c^{d}\mod pq=373^{29}\mod pq=373^{29}\mod 77=32

Schneider–Lang_theorem.html

  1. m ( ρ 1 + ρ 2 ) [ K : ] . m\leq(\rho_{1}+\rho_{2})[K:\mathbb{Q}].\,
  2. [ ( z ) ] 2 = 4 [ ( z ) ] 3 - g 2 ( z ) - g 3 , [\wp^{\prime}(z)]^{2}=4[\wp(z)]^{3}-g_{2}\wp(z)-g_{3},\,

Scholar_Indices_and_Impact.html

  1. g = ( α - 1 α - 2 ) α - 1 α 1 T α g=\left(\frac{\alpha-1}{\alpha-2}\right)^{\frac{\alpha-1}{\alpha}}\frac{1}{T^{% \alpha}}
  2. h = T 1 / α h=T^{1/\alpha}
  3. g = ( α - 1 α - 2 ) α - 1 α h > h g=\left(\frac{\alpha-1}{\alpha-2}\right)^{\frac{\alpha-1}{\alpha}}h>h
  4. h ( x ) x y a ( y ) h(x)\leftarrow\sum_{x\rightarrow y}a(y)
  5. a ( x ) y x h ( y ) a(x)\leftarrow\sum_{y\rightarrow x}h(y)
  6. h = ( 1 / λ h ) A A T h \vec{h}=(1/\lambda_{h})AA^{T}\vec{h}
  7. a = ( 1 / λ a ) A T A a \vec{a}=(1/\lambda_{a})A^{T}A\vec{a}
  8. h i p r i n c i p a l e i g e n v e c t o r ( A A T ) h_{i}\leftarrow principal~{}eigen~{}vector(AA^{T})
  9. a i p r i n c i p a l e i g e n v e c t o r ( A T A ) a_{i}\leftarrow principal~{}eigen~{}vector(A^{T}A)
  10. h = A * a i h=A*a_{i}
  11. a = A T * h i a=A^{T}*h_{i}

Schottky's_theorem.html

  1. log | f ( z ) | 1 + | z | 1 - | z | ( 7 + max ( 0 , log | f ( 0 ) | ) ) \log|f(z)|\leq\frac{1+|z|}{1-|z|}(7+\max(0,\log|f(0)|))

Schröder–Hipparchus_number.html

  1. x n = i = 1 n N ( n , i ) k i - 1 = i = 1 n 1 n ( n i ) ( n i - 1 ) k i - 1 . x_{n}=\sum_{i=1}^{n}N(n,i)\,k^{i-1}=\sum_{i=1}^{n}\frac{1}{n}{n\choose i}{n% \choose i-1}k^{i-1}.
  2. S ( n ) = 1 n ( ( 6 n - 9 ) S ( n - 1 ) - ( n - 3 ) S ( n - 2 ) ) . S(n)=\frac{1}{n}\left((6n-9)S(n-1)-(n-3)S(n-2)\right).

Schrödinger_group.html

  1. [ J i , J j ] = i ϵ i j k J k , [J_{i},J_{j}]=i\epsilon_{ijk}J_{k},\,\!
  2. [ J i , P j ] = i ϵ i j k P k , [J_{i},P_{j}]=i\epsilon_{ijk}P_{k},\,\!
  3. [ J i , K j ] = i ϵ i j k K k , [J_{i},K_{j}]=i\epsilon_{ijk}K_{k},\,\!
  4. [ P i , P j ] = 0 , [ K i , K j ] = 0 , [ K i , P j ] = i δ i j M , [P_{i},P_{j}]=0,[K_{i},K_{j}]=0,[K_{i},P_{j}]=i\delta_{ij}M,\,\!
  5. [ H , J i ] = 0 , [ H , P i ] = 0 , [ H , K i ] = i P i . [H,J_{i}]=0,[H,P_{i}]=0,[H,K_{i}]=iP_{i}.\,\!
  6. [ H , C ] = i D , [ C , D ] = - 2 i C , [ H , D ] = 2 i H , [H,C]=iD,[C,D]=-2iC,[H,D]=2iH,\,\!
  7. [ P i , D ] = i P i , [ K i , D ] = - i K i , [P_{i},D]=iP_{i},[K_{i},D]=-iK_{i},\,\!
  8. [ P i , C ] = - i K i , [ K i , C ] = 0 , [P_{i},C]=-iK_{i},[K_{i},C]=0,\,\!
  9. [ J i , C ] = [ J i , D ] = 0. [J_{i},C]=[J_{i},D]=0.\,\!

Schubert_polynomial.html

  1. 𝔖 w \mathfrak{S}_{w}
  2. x 1 , x 2 , \ x_{1},x_{2},\ldots
  3. w w
  4. S S_{\infty}
  5. 1 , 2 , 3 , 1,2,3,\ldots
  6. [ x 1 , x 2 , ] \mathbb{Z}[x_{1},x_{2},\ldots]
  7. Fl ( m ) \,\text{Fl}(m)
  8. ( x 1 , x 2 , , x m ] / I \mathbb{Z}(x_{1},x_{2},\ldots,x_{m}]/I
  9. I I
  10. 𝔖 w \mathfrak{S}_{w}
  11. ( w ) \ell(w)
  12. w w
  13. Fl ( m ) \,\text{Fl}(m)
  14. m m
  15. w 0 w_{0}
  16. S n S_{n}
  17. 𝔖 w 0 = x 1 n - 1 x 2 n - 2 x n - 1 1 \mathfrak{S}_{w_{0}}=x_{1}^{n-1}x_{2}^{n-2}\cdots x_{n-1}^{1}
  18. i 𝔖 w = 𝔖 w s i \partial_{i}\mathfrak{S}_{w}=\mathfrak{S}_{ws_{i}}
  19. w ( i ) > w ( i + 1 ) w(i)>w(i+1)
  20. s i s_{i}
  21. ( i , i + 1 ) (i,i+1)
  22. i \partial_{i}
  23. P P
  24. ( P - s i P ) / ( x i - x i + 1 ) (P-s_{i}P)/(x_{i}-x_{i+1})
  25. 𝔖 w = w - 1 w 0 x 1 n - 1 x 2 n - 2 x n - 1 1 \mathfrak{S}_{w}=\partial_{w^{-1}w_{0}}x_{1}^{n-1}x_{2}^{n-2}\cdots x_{n-1}^{1}
  26. 𝔖 i d = 1 \mathfrak{S}_{id}=1
  27. s n s_{n}
  28. ( n , n + 1 ) (n,n+1)
  29. 𝔖 s n = x 1 + + x n \mathfrak{S}_{s_{n}}=x_{1}+\ldots+x_{n}
  30. w ( i ) < w ( i + 1 ) w(i)<w(i+1)
  31. i r i\neq r
  32. 𝔖 w \mathfrak{S}_{w}
  33. s λ ( x 1 , , x r ) s_{\lambda}(x_{1},\ldots,x_{r})
  34. λ \lambda
  35. ( w ( r ) - r , , w ( 2 ) - 2 , w ( 1 ) - 1 ) (w(r)-r,\cdots,w(2)-2,w(1)-1)
  36. 𝔖 24531 ( x ) = x 1 x 3 2 x 4 x 2 2 + x 1 2 x 3 x 4 x 2 2 + x 1 2 x 3 2 x 4 x 2 \mathfrak{S}_{24531}(x)=x_{1}x_{3}^{2}x_{4}x_{2}^{2}+x_{1}^{2}x_{3}x_{4}x_{2}^% {2}+x_{1}^{2}x_{3}^{2}x_{4}x_{2}
  37. c β γ α c^{\alpha}_{\beta\gamma}
  38. 𝔖 β 𝔖 γ = α c β γ α 𝔖 α \mathfrak{S}_{\beta}\mathfrak{S}_{\gamma}=\sum_{\alpha}c^{\alpha}_{\beta\gamma% }\mathfrak{S}_{\alpha}
  39. 𝔖 w ( x 1 , x 2 , , y 1 , y 2 , ) \mathfrak{S}_{w}(x_{1},x_{2},\ldots,y_{1},y_{2},\ldots)
  40. y i y_{i}
  41. 0
  42. 𝔖 w ( x 1 , x 2 , , y 1 , y 2 , ) \mathfrak{S}_{w}(x_{1},x_{2},\ldots,y_{1},y_{2},\ldots)
  43. 𝔖 w ( x 1 , x 2 , , y 1 , y 2 , ) = i + j n ( x i - y j ) \mathfrak{S}_{w}(x_{1},x_{2},\ldots,y_{1},y_{2},\ldots)=\prod\limits_{i+j\leq n% }(x_{i}-y_{j})
  44. w w
  45. 1 , , n 1,\ldots,n
  46. i 𝔖 w = 𝔖 w s i \partial_{i}\mathfrak{S}_{w}=\mathfrak{S}_{ws_{i}}
  47. w ( i ) > w ( i + 1 ) w(i)>w(i+1)

Schwarzschild_coordinates.html

  1. d s 2 = - f ( r ) 2 d t 2 + g ( r ) 2 d r 2 + r 2 ( d θ 2 + sin 2 θ d ϕ 2 ) , ds^{2}=-f(r)^{2}\,dt^{2}+g(r)^{2}\,dr^{2}+r^{2}\left(d\theta^{2}+\sin^{2}% \theta\,d\phi^{2}\right),
  2. - < t < , r 0 < r < r 1 , 0 < θ < π , - π < ϕ < π -\infty<t<\infty,\,r_{0}<r<r_{1},\,0<\theta<\pi,\,-\pi<\phi<\pi
  3. t \partial_{t}
  4. ϕ , sin ϕ θ + cot θ cos ϕ ϕ , cos ϕ θ - cot θ sin ϕ ϕ \partial_{\phi},\;\;\sin\phi\,\partial_{\theta}+\cot\theta\,\cos\phi\,\partial% _{\phi},\;\;\cos\phi\,\partial_{\theta}-\cot\theta\,\sin\phi\,\partial_{\phi}
  5. X = t \vec{X}=\partial_{t}
  6. t = t 0 t=t_{0}
  7. t = t 0 , r = r 0 t=t_{0},\,r=r_{0}
  8. d σ 2 = r 0 2 ( d θ 2 + sin 2 θ d ϕ 2 ) , 0 < θ < π , - π < ϕ < π d\sigma^{2}=r_{0}^{2}\left(d\theta^{2}+\sin^{2}\theta\,d\phi^{2}\right),\;0<% \theta<\pi,\;-\pi<\phi<\pi
  9. A = 4 π r 0 2 A=4\pi r_{0}^{2}
  10. K = 1 / r 0 2 K=1/r_{0}^{2}
  11. θ , ϕ \theta,\phi
  12. θ \theta
  13. ϕ \phi
  14. r \partial_{r}
  15. g ( r ) d r g(r)dr
  16. Δ ρ = r 1 r 2 g ( r ) d r \Delta\rho=\int_{r_{1}}^{r_{2}}g(r)dr
  17. r = r 0 , θ = θ 0 , ϕ = ϕ 0 r=r_{0},\theta=\theta_{0},\phi=\phi_{0}
  18. f ( r ) d t f(r)dt
  19. Δ τ = t 1 t 2 f ( r ) d t \Delta\tau=\int_{t_{1}}^{t_{2}}f(r)dt
  20. t = t 0 , r = r 0 , θ = 0 t=t_{0},\,r=r_{0},\,\theta=0
  21. t = t 0 , r = r 0 , θ = π t=t_{0},\,r=r_{0},\,\theta=\pi
  22. ϕ = 0 \phi=0
  23. t = t 0 t=t_{0}
  24. r = 0 r=0
  25. ϕ \partial_{\phi}
  26. ϕ \phi
  27. r 1 > 0 r_{1}>0
  28. r 2 < r_{2}<\infty
  29. t = t 0 t=t_{0}
  30. t = 0 , θ = π / 2 t=0,\theta=\pi/2
  31. d ρ 2 = g ( r ) 2 d r 2 + r 2 d ϕ 2 , r 1 < r < r 2 , - π < ϕ < π d\rho^{2}=g(r)^{2}dr^{2}+r^{2}d\phi^{2},\;\;r_{1}<r<r_{2},\,-\pi<\phi<\pi
  32. ( z , r , ϕ ) ( h ( r ) , r cos ϕ , r sin ϕ ) (z,r,\phi)\rightarrow(h(r),\,r\cos\phi,\,r\sin\phi)
  33. r = ( h ( r ) , cos ϕ , sin ϕ ) , ϕ = ( 0 , - r sin ϕ , r cos ϕ ) \partial_{r}=(h^{\prime}(r),\,\cos\phi,\,\sin\phi),\;\;\partial_{\phi}=(0,-r% \sin\phi,r\cos\phi)
  34. d ρ 2 = ( 1 + h ( r ) 2 ) d r 2 + r 2 d ϕ 2 , r 1 < r < r 2 , - π < ϕ < π d\rho^{2}=\left(1+h^{\prime}(r)^{2}\right)\,dr^{2}+r^{2}\,d\phi^{2},\;r_{1}<r<% r_{2},\,-\pi<\phi<\pi
  35. h ( r ) = 1 - g ( r ) 2 h^{\prime}(r)=\sqrt{1-g(r)^{2}}
  36. g ( r ) = h ( r ) = sin ( r ) g(r)=h(r)=\sin(r)
  37. g ( r ) = h ( r ) = sinh ( r ) g(r)=h(r)=\sinh(r)
  38. σ 0 = - f ( r ) d t \sigma^{0}=-f(r)\,dt
  39. σ 1 = g ( r ) d r \sigma^{1}=g(r)\,dr
  40. σ 2 = r d θ \sigma^{2}=rd\theta\,
  41. σ 3 = r sin θ d ϕ \sigma^{3}=r\sin\theta\,d\phi
  42. d σ 0 = - f ( r ) d r d t = f ( r ) g ( r ) d t σ 1 d\sigma^{0}=-f^{\prime}(r)\,dr\wedge dt=\frac{f^{\prime}(r)}{g(r)}\,dt\wedge% \sigma^{1}
  43. d σ 1 = 0 d\sigma^{1}=0\,
  44. d σ 2 = d r d θ d\sigma^{2}=dr\wedge d\theta
  45. d σ 3 = sin θ d r d ϕ + r cos θ d θ d ϕ = - ( sin θ d ϕ g ( r ) σ 1 + cos θ d ϕ σ 2 ) d\sigma^{3}=\sin\theta\,dr\wedge d\phi+r\,\cos\theta\,d\theta\wedge d\phi=-% \left(\frac{\sin\theta\,d\phi}{g(r)}\wedge\sigma^{1}+\cos\theta\,d\phi\wedge% \sigma^{2}\right)
  46. d σ m ^ = - ω m ^ n ^ σ n ^ d\sigma^{\hat{m}}=-{\omega^{\hat{m}}}_{\hat{n}}\,\wedge\sigma^{\hat{n}}
  47. d t , d r , d θ , d ϕ dt,\,dr,\,d\theta,d\phi
  48. ω m ^ n ^ {\omega^{\hat{m}}}_{\hat{n}}
  49. ω 0 1 = f d t g {\omega^{0}}_{1}=\frac{f^{\prime}\,dt}{g}
  50. ω 0 2 = 0 {\omega^{0}}_{2}=0
  51. ω 0 3 = 0 {\omega^{0}}_{3}=0
  52. ω 1 2 = - d θ g {\omega^{1}}_{2}=-\frac{d\theta}{g}
  53. ω 1 3 = - sin θ d ϕ g {\omega^{1}}_{3}=-\frac{\sin\theta\,d\phi}{g}
  54. ω 2 3 = - cos θ d ϕ {\omega^{2}}_{3}=-\cos\theta\,d\phi
  55. Ω m ^ n ^ = d ω m ^ n ^ - ω m ^ ^ ω ^ n ^ {\Omega^{\hat{m}}}_{\hat{n}}=d{\omega^{\hat{m}}}_{\hat{n}}-{\omega^{\hat{m}}}_% {\hat{\ell}}\wedge{\omega^{\hat{\ell}}}_{\hat{n}}
  56. Ω m ^ n ^ = R m ^ n ^ | i ^ j ^ | σ i ^ σ j ^ {\Omega^{\hat{m}}}_{\hat{n}}={R^{\hat{m}}}_{\hat{n}|\hat{i}\hat{j}|}\,\sigma^{% \hat{i}}\wedge\sigma^{\hat{j}}
  57. R 0 101 = - f ′′ g + f g f g 3 {R^{0}}_{101}=\frac{-f^{\prime\prime}\,g+f^{\prime}\,g^{\prime}}{f\,g^{3}}
  58. R 0 202 = - f r f g 2 = R 0 303 {R^{0}}_{202}=\frac{-f^{\prime}}{r\,f\,g^{2}}={R^{0}}_{303}
  59. R 1 212 = g r g 3 = R 1 313 {R^{1}}_{212}=\frac{g^{\prime}}{r\,g^{3}}={R^{1}}_{313}
  60. R 2 323 = - 1 + g 2 r 2 g 2 {R^{2}}_{323}=\frac{-1+g^{2}}{r^{2}\,g^{2}}
  61. R m ^ n ^ i ^ j ^ R_{\hat{m}\hat{n}\hat{i}\hat{j}}
  62. [ R 0101 R 0102 R 0103 R 0123 R 0131 R 0112 R 0201 R 0202 R 0203 R 0223 R 0231 R 0212 R 0301 R 0302 R 0303 R 0323 R 0331 R 0312 R 2301 R 2302 R 2303 R 2323 R 2331 R 2312 R 3101 R 3102 R 3103 R 3123 R 3131 R 3112 R 1201 R 1202 R 1203 R 1223 R 1231 R 1212 ] = [ E B B T L ] \left[\begin{matrix}R_{0101}&R_{0102}&R_{0103}&R_{0123}&R_{0131}&R_{0112}\\ R_{0201}&R_{0202}&R_{0203}&R_{0223}&R_{0231}&R_{0212}\\ R_{0301}&R_{0302}&R_{0303}&R_{0323}&R_{0331}&R_{0312}\\ R_{2301}&R_{2302}&R_{2303}&R_{2323}&R_{2331}&R_{2312}\\ R_{3101}&R_{3102}&R_{3103}&R_{3123}&R_{3131}&R_{3112}\\ R_{1201}&R_{1202}&R_{1203}&R_{1223}&R_{1231}&R_{1212}\end{matrix}\right]=\left% [\begin{matrix}E&B\\ B^{T}&L\end{matrix}\right]
  63. X = e 0 = 1 f ( r ) t \vec{X}=\vec{e}_{0}=\frac{1}{f(r)}\,\partial_{t}
  64. E [ X ] 11 = f ′′ g - f g f g 3 , E [ X ] 22 = E [ X ] 33 = f r f g 2 E[\vec{X}]_{11}=\frac{f^{\prime\prime}\,g-f^{\prime}\,g^{\prime}}{f\,g^{3}},\;% E[\vec{X}]_{22}=E[\vec{X}]_{33}=\frac{f^{\prime}}{r\,f\,g^{2}}
  65. X \vec{X}
  66. L [ X ] 11 = 1 - g 2 r 2 g 2 , L [ X ] 22 = L [ X ] 33 = - g r g 3 L[\vec{X}]_{11}=\frac{1-g^{2}}{r^{2}\,g^{2}},\;L[\vec{X}]_{22}=L[\vec{X}]_{33}% =\frac{-g^{\prime}}{r\,g^{3}}
  67. e 0 = 1 f ( r ) t \vec{e}_{0}=\frac{1}{f(r)}\,\partial_{t}
  68. e 1 = 1 g ( r ) r \vec{e}_{1}=\frac{1}{g(r)}\,\partial_{r}
  69. e 2 = 1 r θ \vec{e}_{2}=\frac{1}{r}\,\partial_{\theta}
  70. e 3 = 1 r sin θ ϕ \vec{e}_{3}=\frac{1}{r\sin\theta}\,\partial_{\phi}
  71. 1 g ( r ) \frac{1}{g(r)}
  72. d s 2 = - f ( t , r ) 2 d t 2 + g ( t , r ) 2 d r 2 + r 2 ( d θ 2 + sin 2 θ d ϕ 2 ) , ds^{2}=-f(t,r)^{2}\,dt^{2}+g(t,r)^{2}\,dr^{2}+r^{2}\left(d\theta^{2}+\sin^{2}% \theta\,d\phi^{2}\right),
  73. - < t < , r 0 < r < r 1 , 0 < θ < π , - π < ϕ < π -\infty<t<\infty,\,r_{0}<r<r_{1},\,0<\theta<\pi,\,-\pi<\phi<\pi
  74. d s 2 = - f ( r ) 2 d t 2 + g ( r ) 2 d r 2 + d x 2 + d y 2 ( 1 + x 2 + y 2 ) 2 , - < t , x , y < , r 1 < r < r 2 ds^{2}=-f(r)^{2}\,dt^{2}+g(r)^{2}\,dr^{2}+\frac{dx^{2}+dy^{2}}{(1+x^{2}+y^{2})% ^{2}},\;-\infty<t,x,y<\infty,r_{1}<r<r_{2}
  75. X \vec{X}
  76. t \partial_{t}
  77. x \partial_{x}
  78. / x \partial/\partial x

SCOP_formalism.html

  1. ( Σ , \Mu , L , μ , ν ) (\Sigma,\Mu,\mathit{L},\mu,\nu)
  2. Σ \Sigma
  3. \Mu \Mu
  4. L \mathit{L}
  5. μ : Σ × \Mu × Σ [ 0 , 1 ] , ( p , e , q ) μ ( p , e , q ) \mu:\Sigma\times\Mu\times\Sigma\to[0,1],~{}(p,e,q)\mapsto\mu(p,e,q)
  6. p p
  7. q q
  8. e e
  9. ν : Σ × L [ 0 , 1 ] , ( p , a ) [ 0 , 1 ] \nu:\Sigma\times\mathit{L}\to[0,1],~{}(p,a)\mapsto[0,1]
  10. a a
  11. p p
  12. 𝟏 \mathbf{1}
  13. p ^ \hat{p}
  14. μ ( p ^ , 𝟏 , p ^ ) = 1 \mu(\hat{p},\mathbf{1},\hat{p})=1
  15. μ ( p , 𝟏 , p ^ ) = 1 \mu(p,\mathbf{1},\hat{p})=1
  16. p Σ , p p ^ p\in\Sigma,p\neq\hat{p}
  17. e e
  18. p p
  19. μ ( p , e , p ) = 1 \mu(p,e,p)=1
  20. p p
  21. e e
  22. q q

Scoring_systems_for_Sailing_at_the_Summer_Olympics.html

  1. P o i n t s = 101 + 1000 l o g A - 1000 l o g N Points=101+1000logA-1000logN

Sea_ice_concentration.html

  1. T b = T b 0 + i = 1 n ( T b i - T b 0 ) C i \vec{T}_{b}=\vec{T}_{b0}+\sum_{i=1}^{n}(\vec{T}_{bi}-\vec{T}_{b0})C_{i}
  2. T b \vec{T}_{b}
  3. T b 0 \vec{T}_{b0}
  4. T b i \vec{T}_{bi}

Sea_ice_emissivity_modelling.html

  1. ϵ * = a V b + b \epsilon^{*}=aV_{b}+b
  2. ϵ * \epsilon^{*}
  3. ϵ e f f = ϵ 1 + V b ϵ 1 ( ϵ 2 - ϵ 1 ) / ( ϵ 1 + P ( ϵ 2 - ϵ 1 ) 1 - P V b ( ϵ 2 - ϵ 1 ) / [ ϵ 1 + P ( ϵ 2 - ϵ 1 ) ] \epsilon_{eff}=\epsilon_{1}+\frac{V_{b}\epsilon_{1}(\epsilon_{2}-\epsilon_{1})% /(\epsilon_{1}+P(\epsilon_{2}-\epsilon 1)}{1-PV_{b}(\epsilon_{2}-\epsilon_{1})% /\left[\epsilon_{1}+P(\epsilon_{2}-\epsilon_{1})\right]}
  4. ϵ 1 \epsilon_{1}
  5. ϵ 2 \epsilon_{2}
  6. ϵ i \epsilon_{i}
  7. Δ z i \Delta z_{i}
  8. T i T_{i}\uparrow
  9. T i T_{i}\downarrow
  10. T i - τ i ( 1 - R i ) T i + 1 - τ i R i T i = ( 1 - τ i ) T i T_{i}\uparrow-\tau_{i}(1-R_{i})T_{i+1}\uparrow-\tau_{i}R_{i}T_{i}\downarrow=(1% -\tau_{i})T_{i}
  11. T i - τ i ( 1 - R i - 1 ) T i - 1 - τ i R i - 1 T i = ( 1 - τ i ) T i T_{i}\downarrow-\tau_{i}(1-R_{i-1})T_{i-1}\downarrow-\tau_{i}R_{i-1}T_{i}% \uparrow=(1-\tau_{i})T_{i}
  12. τ i \tau_{i}
  13. τ i = exp ( - α i Δ z i cos θ i ) \tau_{i}=\exp\left(-\frac{\alpha_{i}\,\Delta z_{i}}{\cos\theta_{i}}\right)
  14. θ i \theta_{i}
  15. Δ z i \Delta z_{i}
  16. α i \alpha_{i}
  17. α i = 4 π ν c Imag n i \alpha_{i}=\frac{4\pi\nu}{c}\mathrm{Imag}\,n_{i}
  18. ν \nu
  19. n i = ϵ i n_{i}=\sqrt{\epsilon_{i}}

Sea_ice_growth_processes.html

  1. h Q * = k ( T s - T w ) hQ^{*}=k(T_{s}-T_{w})
  2. Q * = Q E [ e ( T s ) ] + Q H ( T s ) + Q L W ( T s 4 ) + Q S W Q^{*}=Q_{E}\left[e(T_{s})\right]+Q_{H}(T_{s})+Q_{LW}(T_{s}^{4})+Q_{SW}
  3. g = Q * L ρ g=\frac{Q^{*}}{L\rho}
  4. ρ \rho
  5. S = S 0 f ( g ) S=S_{0}f(g)
  6. f ( g ) = 0.12 0.12 + 0.88 exp ( - 4.2 × 10 4 g ) f(g)=\frac{0.12}{0.12+0.88\exp(-4.2\times 10^{4}g)}
  7. V b = S ρ i S b ρ b - S ρ b + S ρ i V_{b}=\frac{S\rho_{i}}{S_{b}\rho_{b}-S\rho_{b}+S\rho_{i}}
  8. ρ i \rho_{i}
  9. ρ b \rho_{b}
  10. V b = 10 - 3 S ( - 49.185 T + 0.532 ) V_{b}=10^{-3}S\left(-\frac{49.185}{T}+0.532\right)
  11. S ( T 2 ) S ( T 1 ) = S b ( T 2 ) ( 1 - 1 / ρ i ) S b ( T 1 ) ρ b ( T 2 ) ρ b ( T 1 ) exp { c ρ i [ S b ( T 1 ) - S b ( T 2 ) ] } \frac{S(T_{2})}{S(T_{1})}=\frac{S_{b}(T_{2})\left(1-1/\rho_{i}\right)}{S_{b}(T% _{1})}\frac{\rho_{b}(T_{2})}{\rho_{b}(T_{1})}\exp\left\{\frac{c}{\rho_{i}\left% [S_{b}(T_{1})-S_{b}(T_{2})\right]}\right\}
  12. S = exp ( a h + b ) S=\exp(ah+b)

Sea_level_equation.html

  1. S = N - U , S=N-U,
  2. S S
  3. N N
  4. U U
  5. S ( θ , λ , t ) = ρ i γ G s i I + ρ w γ G s o S + S E - ρ i γ G s i I ¯ - ρ w γ G o o S ¯ , S(\theta,\lambda,t)=\frac{\rho_{i}}{\gamma}G_{s}\otimes_{i}I+\frac{\rho_{w}}{% \gamma}G_{s}\otimes_{o}S+S^{E}-\frac{\rho_{i}}{\gamma}\overline{G_{s}\otimes_{% i}I}-\frac{\rho_{w}}{\gamma}\overline{G_{o}\otimes_{o}S},
  6. θ \theta
  7. λ \lambda
  8. t t
  9. ρ i \rho_{i}
  10. ρ w \rho_{w}
  11. γ \gamma
  12. G s = G s ( h , k ) G_{s}=G_{s}(h,k)
  13. h h
  14. k k
  15. I = I ( θ , λ , t ) I=I(\theta,\lambda,t)
  16. S E = S E ( t ) S^{E}=S^{E}(t)
  17. S S
  18. i \otimes_{i}
  19. o \otimes_{o}

Second-order_co-occurrence_pointwise_mutual_information.html

  1. f b ( t i , w ) > 0 f^{b}(t_{i},w)>0
  2. f pmi ( t i , w ) = log 2 f b ( t i , w ) × m f t ( t i ) f t ( w ) , f\text{pmi}(t_{i},w)=\log_{2}\frac{f^{b}(t_{i},w)\times m}{f^{t}(t_{i})f^{t}(w% )},
  3. f t ( t i ) f^{t}(t_{i})
  4. t i t_{i}
  5. f b ( t i , w ) f^{b}(t_{i},w)
  6. t i t_{i}
  7. w w
  8. m m
  9. w w
  10. X w X^{w}
  11. w w
  12. β \beta
  13. f pmi ( t i , w ) > 0 f\text{pmi}(t_{i},w)>0
  14. X w X^{w}
  15. X i w X_{i}^{w}
  16. X w = { X i w } X^{w}=\{X_{i}^{w}\}
  17. i = 1 , 2 , , β i=1,2,\ldots,\beta
  18. f pmi ( X 1 w , w ) f pmi ( X 2 w , w ) f pmi ( X β - 1 w , w ) f pmi ( X β w , w ) f\text{pmi}(X_{1}^{w},w)\geq f\text{pmi}(X_{2}^{w},w)\geq\cdots f\text{pmi}(X_% {\beta-1}^{w},w)\geq f\text{pmi}(X_{\beta}^{w},w)
  19. β \beta
  20. β \beta
  21. w 1 w_{1}
  22. w 2 w_{2}
  23. f ( w 1 , w 2 , β ) = i = 1 β ( f pmi ( X i w 1 , w 2 ) ) γ f(w_{1},w_{2},\beta)=\sum_{i=1}^{\beta}(f\text{pmi}(X_{i}^{w_{1}},w_{2}))^{\gamma}
  24. f pmi ( X i w 1 , w 2 ) > 0 f\text{pmi}(X_{i}^{w_{1}},w_{2})>0
  25. X w 2 X^{w_{2}}
  26. X w 1 X^{w_{1}}
  27. w 2 w_{2}
  28. w 1 w_{1}
  29. γ \gamma
  30. β \beta
  31. w 1 w_{1}
  32. w 2 w_{2}
  33. β = β 1 \beta=\beta_{1}
  34. β \beta
  35. w 2 w_{2}
  36. w 1 w_{1}
  37. β = β 2 \beta=\beta_{2}
  38. f ( w 1 , w 2 , β 1 ) = i = 1 β 1 ( f pmi ( X i w 1 , w 2 ) ) γ f(w_{1},w_{2},\beta_{1})=\sum_{i=1}^{\beta_{1}}(f\text{pmi}(X_{i}^{w_{1}},w_{2% }))^{\gamma}
  39. f ( w 2 , w 1 , β 2 ) = i = 1 β 2 ( f pmi ( X i w 2 , w 1 ) ) γ f(w_{2},w_{1},\beta_{2})=\sum_{i=1}^{\beta_{2}}(f\text{pmi}(X_{i}^{w_{2}},w_{1% }))^{\gamma}
  40. w 1 w_{1}
  41. w 2 w_{2}
  42. Sim ( w 1 , w 2 ) = f ( w 1 , w 2 , β 1 ) β 1 + f ( w 2 , w 1 , β 2 ) β 2 . \mathrm{Sim}(w_{1},w_{2})=\frac{f(w_{1},w_{2},\beta_{1})}{\beta_{1}}+\frac{f(w% _{2},w_{1},\beta_{2})}{\beta_{2}}.
  43. 0
  44. 1 1
  45. r i r_{i}
  46. s j s_{j}
  47. λ \lambda
  48. λ = 20 \lambda=20

Secondary_leading-tone_chord.html

  1. v i i o x y \frac{vii^{ox}}{y}

Secondary_supertonic_chord.html

  1. i i V \frac{ii}{V}
  2. V V \frac{V}{V}
  3. V ( I ) \frac{V}{(I)}

Security_characteristic_line.html

  1. SCL : R i , t - R f = α i + β i ( R M , t - R f ) + ϵ i , t \mathrm{SCL}:R_{i,t}-R_{f}=\alpha_{i}+\beta_{i}\,(R_{M,t}-R_{f})+\epsilon_{i,t}

Selberg's_zeta_function_conjecture.html

  1. ε > 0 \varepsilon>0
  2. T 0 = T 0 ( ε ) > 0 T_{0}=T_{0}(\varepsilon)>0
  3. c = c ( ε ) > 0 , c=c(\varepsilon)>0,
  4. T T 0 T\geq T_{0}
  5. H = T 0.5 + ε H=T^{0.5+\varepsilon}
  6. N ( T + H ) - N ( T ) c H log T N(T+H)-N(T)\geq cH\log T
  7. H = T 0.5 + ε . H=T^{0.5+\varepsilon}.
  8. ε \varepsilon
  9. 0 < ε < 0.001 , 0<\varepsilon<0.001,
  10. H = T a + ε , H=T^{a+\varepsilon},
  11. a = 27 82 = 1 3 - 1 246 , a=\tfrac{27}{82}=\tfrac{1}{3}-\tfrac{1}{246},
  12. ζ ( 1 2 + i t ) ; \zeta\Bigl(\tfrac{1}{2}+it\Bigr);

Self-sampling_assumption.html

  1. \approx

Self-sustainability.html

  1. E E
  2. p ( v ) p(v)
  3. v v
  4. p ( v ) = P ( E = v ) p(v)=P(E=v)
  5. S S
  6. S = p ( 0 ) S=p(0)
  7. e t e_{t}
  8. t t
  9. S = lim t e t / t S=\lim_{t\rightarrow\infty}{e_{t}}/{t}

Semantic_compression.html

  1. c u m f ( k i ) = f ( k i ) + j c u m f ( k j ) cumf(k_{i})=f(k_{i})+\sum_{j}cumf(k_{j})
  2. k i k_{i}
  3. k j k_{j}

Semi-infinite.html

  1. ( c , ) (c,\infty)
  2. ( - , c ) (-\infty,c)
  3. \R \R

Semi-inner-product.html

  1. V V
  2. \mathbb{C}
  3. V × V V\times V
  4. \mathbb{C}
  5. [ , ] [\cdot,\cdot]
  6. [ f + g , h ] = [ f , h ] + [ g , h ] f , g , h V [f+g,h]=[f,h]+[g,h]\quad\forall f,g,h\in V
  7. [ α f , g ] = α [ f , g ] α , f , g V , [\alpha f,g]=\alpha[f,g]\quad\forall\alpha\in\mathbb{C},\ \forall f,g\in V,
  8. [ f , α g ] = α ¯ [ f , g ] α , f , g V , [f,\alpha g]=\overline{\alpha}[f,g]\quad\forall\alpha\in\mathbb{C},\ \forall f% ,g\in V,
  9. [ f , f ] 0 and [ f , f ] = 0 if and only if f = 0 , [f,f]\geq 0\,\text{ and }[f,f]=0\,\text{ if and only if }f=0,
  10. | [ f , g ] | [ f , f ] 1 / 2 [ g , g ] 1 / 2 f , g V . \left|[f,g]\right|\leq[f,f]^{1/2}[g,g]^{1/2}\quad\forall f,g\in V.
  11. [ f , g ] [ g , f ] ¯ [f,g]\neq\overline{[g,f]}
  12. [ f , g + h ] [ f , g ] + [ f , h ] . [f,g+h]\neq[f,g]+[f,h].\,
  13. [ , ] [\cdot,\cdot]
  14. V V
  15. f := [ f , f ] 1 / 2 , f V \|f\|:=[f,f]^{1/2},\quad f\in V
  16. V V
  17. V V
  18. \|\cdot\|
  19. V V
  20. V V
  21. f = [ f , f ] 1 / 2 , f V . \|f\|=[f,f]^{1/2},\ \ \forall f\in V.
  22. n \mathbb{C}^{n}
  23. p \ell^{p}
  24. 1 p < + 1\leq p<+\infty
  25. x p := ( j = 1 p | x j | p ) 1 / p \|x\|_{p}:=\biggl(\sum_{j=1}^{p}|x_{j}|^{p}\biggr)^{1/p}
  26. [ x , y ] := j = 1 n x j y j ¯ | y j | p - 2 y p p - 2 , x , y n { 0 } , 1 < p < + , [x,y]:=\frac{\sum_{j=1}^{n}x_{j}\overline{y_{j}}|y_{j}|^{p-2}}{\|y\|_{p}^{p-2}% },\quad x,y\in\mathbb{C}^{n}\setminus\{0\},\ \ 1<p<+\infty,
  27. [ x , y ] := j = 1 n x j sgn ( y j ¯ ) , x , y n , p = 1 , [x,y]:=\sum_{j=1}^{n}x_{j}\operatorname{sgn}(\overline{y_{j}}),\quad x,y\in% \mathbb{C}^{n},\ \ p=1,
  28. sgn ( t ) := { t | t | , t { 0 } , 0 , t = 0. \operatorname{sgn}(t):=\left\{\begin{array}[]{ll}\frac{t}{|t|},&t\in\mathbb{C}% \setminus\{0\},\\ 0,&t=0.\end{array}\right.
  29. L p ( Ω , d μ ) L^{p}(\Omega,d\mu)
  30. p p
  31. ( Ω , μ ) (\Omega,\mu)
  32. 1 p < + 1\leq p<+\infty
  33. f p := ( Ω | f ( t ) | p d μ ( t ) ) 1 / p \|f\|_{p}:=\left(\int_{\Omega}|f(t)|^{p}d\mu(t)\right)^{1/p}
  34. [ f , g ] := Ω f ( t ) g ( t ) ¯ | g ( t ) | p - 2 d μ ( t ) g p p - 2 , f , g L p ( Ω , d μ ) { 0 } , 1 < p < + , [f,g]:=\frac{\int_{\Omega}f(t)\overline{g(t)}|g(t)|^{p-2}d\mu(t)}{\|g\|_{p}^{p% -2}},\ \ f,g\in L^{p}(\Omega,d\mu)\setminus\{0\},\ \ 1<p<+\infty,
  35. [ f , g ] := Ω f ( t ) sgn ( g ( t ) ¯ ) d μ ( t ) , f , g L 1 ( Ω , d μ ) . [f,g]:=\int_{\Omega}f(t)\operatorname{sgn}(\overline{g(t)})d\mu(t),\ \ f,g\in L% ^{1}(\Omega,d\mu).

Semi-orthogonal_matrix.html

  1. A T A = I or A A T = I . A^{T}A=I\,\text{ or }AA^{T}=I.\,
  2. A T A = I n , A^{T}A=I_{n},\,
  3. A x 2 = x 2 \|Ax\|_{2}=\|x\|_{2}\,
  4. [ 1 0 ] \begin{bmatrix}1\\ 0\end{bmatrix}

Semilinear_response.html

  1. f ( t ) f(t)
  2. S ~ ( ω ) \tilde{S}(\omega)
  3. f ( t ) f ( 0 ) \langle f(t)f(0)\rangle
  4. G G
  5. G = G [ S ~ ( ω ) ] = - η ( ω ) S ~ ( ω ) d ω G=G[\tilde{S}(\omega)]=\int_{-\infty}^{\infty}\eta(\omega)\tilde{S}(\omega)\,d\omega
  6. G G
  7. η ( ω ) \eta(\omega)
  8. [ A ] S ~ ( ω ) λ S ~ ( ω ) implies G λ G [A]\ \ \ \ \tilde{S}(\omega)\mapsto\lambda\tilde{S}(\omega)\,\text{ implies }G% \mapsto\lambda G
  9. [ B ] S ~ ( ω ) S ~ 1 ( ω ) + S ~ 2 ( ω ) implies G G 1 + G 2 [B]\ \ \ \ \tilde{S}(\omega)\mapsto\tilde{S}_{1}(\omega)+\tilde{S}_{2}(\omega)% \,\text{ implies }G\mapsto G_{1}+G_{2}
  10. G = [ [ G n m ] ] G=[[G_{nm}]]
  11. [ [ G n m ] ] [[G_{nm}]]
  12. G = n m G n m G=\sum_{nm}G_{nm}
  13. G = [ n m G n m - 1 ] - 1 G=\left[\sum_{nm}G_{nm}^{-1}\right]^{-1}
  14. [ [ λ A ] ] = λ [ [ A ] ] [[\lambda A]]=\lambda[[A]]
  15. [ [ A + B ] ] [ [ A ] ] + [ [ B ] ] [[A+B]]\neq[[A]]+[[B]]
  16. G n m G_{nm}
  17. G = G [ S ~ ( ω ) ] = [ - μ ( ω ) S ~ ( ω ) - 1 d ω ] - 1 G=G[\tilde{S}(\omega)]=\left[\int_{-\infty}^{\infty}\mu(\omega)\tilde{S}(% \omega)^{-1}\,d\omega\right]^{-1}

Semiorder.html

  1. 1 n + 1 ( 2 n n ) , \frac{1}{n+1}{\left({{2n}\atop{n}}\right)},

Sendov's_conjecture.html

  1. f ( z ) = ( z - r 1 ) ( z - r n ) , ( n 2 ) f(z)=(z-r_{1})\cdots(z-r_{n}),\qquad(n\geq 2)

Separation_relation.html

  1. A 0 A n / / A 1 A n + 1 . A_{0}A_{n}//A_{1}A_{n+1}.
  2. A 1 A n / / A 2 M A_{1}A_{n}//A_{2}M
  3. A 1 P / / A 2 M A_{1}P//A_{2}M
  4. A 1 A n / / P M A_{1}A_{n}//PM

Serial_module.html

  1. N 1 N 2 N_{1}\subseteq N_{2}
  2. N 2 N 1 N_{2}\subseteq N_{1}
  3. / n \mathbb{Z}/n\mathbb{Z}
  4. n > 1 n>1
  5. R = i = 1 n e i R R=\oplus_{i=1}^{n}e_{i}R
  6. 𝔽 [ G ] \mathbb{F}[G]
  7. R End B ( P ) R\cong\mathrm{End}_{B}(P)
  8. U V U\rightarrow V
  9. V U V\rightarrow U
  10. [ U ] e = [ V ] e [U]_{e}=[V]_{e}
  11. U V U\rightarrow V
  12. V U V\rightarrow U
  13. U 1 U n U_{1}\oplus\dots\oplus U_{n}
  14. V 1 V t V_{1}\oplus\dots\oplus V_{t}
  15. σ \sigma
  16. τ \tau
  17. [ U i ] m = [ V σ ( i ) ] m [U_{i}]_{m}=[V_{\sigma(i)}]_{m}
  18. [ U i ] e = [ V τ ( i ) ] e [U_{i}]_{e}=[V_{\tau(i)}]_{e}
  19. A / J ( A ) B / J ( B ) A/J(A)\cong B/J(B)

Series-parallel_partial_order.html

  1. L ( P | | Q ) = ( | P | + | Q | ) ! | P | ! | Q | ! L ( P ) L ( Q ) , L(P||Q)=\frac{(|P|+|Q|)!}{|P|!|Q|!}L(P)L(Q),

Series_multisection.html

  1. n = - a n x n \sum_{n=-\infty}^{\infty}a_{n}\cdot x^{n}
  2. m = - a c m + d x c m + d \sum_{m=-\infty}^{\infty}a_{cm+d}\cdot x^{cm+d}
  3. F ( x ) F(x)
  4. m = - a c m + d x c m + d = 1 c k = 0 c - 1 w - k d F ( w k x ) , \sum_{m=-\infty}^{\infty}a_{cm+d}\cdot x^{cm+d}=\tfrac{1}{c}\cdot\sum_{k=0}^{c% -1}w^{-kd}\cdot F(w^{k}\cdot x),
  5. w = e 2 π i c w=e^{\frac{2\pi i}{c}}
  6. ( 1 + x ) q = ( q 0 ) x 0 + ( q 1 ) x + ( q 2 ) x 2 + (1+x)^{q}={q\choose 0}x^{0}+{q\choose 1}x+{q\choose 2}x^{2}+\cdots
  7. ( q d ) + ( q d + c ) + ( q d + 2 c ) + = 1 c k = 0 c - 1 ( 2 cos π k c ) q cos π ( q - 2 d ) k c . {q\choose d}+{q\choose d+c}+{q\choose d+2c}+\cdots=\frac{1}{c}\cdot\sum_{k=0}^% {c-1}\left(2\cos\frac{\pi k}{c}\right)^{q}\cdot\cos\frac{\pi(q-2d)k}{c}.

Servo_bandwidth.html

  1. 1 2 \tfrac{1}{\sqrt{2}}
  2. 1 2 \tfrac{1}{\sqrt{2}}
  3. 1 2 ) \tfrac{1}{\sqrt{2}})

Seven_states_of_randomness.html

  1. q w \sqrt[w]{q}

Shadowing_lemma.html

  1. ( x n ) (x_{n})
  2. x n + 1 x_{n+1}
  3. f ( x n ) f(x_{n})
  4. ( x n ) , x n U , d ( x n + 1 , f ( x n ) ) < ε ( y n ) , y n + 1 = f ( y n ) , such that n x n U δ ( y n ) . \forall(x_{n}),\,x_{n}\in U,\,d(x_{n+1},f(x_{n}))<\varepsilon\quad\exists(y_{n% }),\,\,y_{n+1}=f(y_{n}),\quad\,\text{such that}\,\,\forall n\,\,x_{n}\in U_{% \delta}(y_{n}).

Shapley–Folkman_lemma.html

  1. \bullet
  2. \circ
  3. \oslash
  4. 1 / N {1}/{N}
  5. \bullet
  6. \circ
  7. \oslash
  8. x = 1 d D q d + D + 1 n N q n x=\sum_{1\leq{d}\leq{D}}{q_{d}}+\sum_{D+1\leq{n}\leq{N}}{q_{n}}
  9. x 1 d D Conv ( Q d ) + D + 1 n N Q n x\in{\sum_{1\leq{d}\leq{D}}{\operatorname{Conv}{(Q_{d})}}+\sum_{D+1\leq{n}\leq% {N}}{Q_{n}}}

Shearer's_inequality.html

  1. H [ ( X 1 , , X d ) ] 1 r i = 1 n H [ ( X j ) j S i ] H[(X_{1},\dots,X_{d})]\leq\frac{1}{r}\sum_{i=1}^{n}H[(X_{j})_{j\in S_{i}}]
  2. ( X j ) j S i (X_{j})_{j\in S_{i}}
  3. X j X_{j}
  4. S i S_{i}
  5. S i S_{i}

Shelling_(topology).html

  1. Δ \Delta
  2. C 1 , C 2 , C_{1},C_{2},\ldots
  3. Δ \Delta
  4. B k := ( i = 1 k - 1 C i ) C k B_{k}:=\left(\bigcup_{i=1}^{k-1}C_{i}\right)\cap C_{k}
  5. ( dim C k - 1 ) (\dim C_{k}-1)
  6. k = 2 , 3 , k=2,3,\ldots
  7. C k C_{k}
  8. B k B_{k}
  9. C k C_{k}
  10. B k B_{k}
  11. C k C_{k}
  12. C k C_{k}
  13. Δ \Delta
  14. Δ \Delta

Shimizu_L-function.html

  1. L ( M , V , s ) = μ { M - 0 } / V sign N ( μ ) | N ( μ ) | s L(M,V,s)=\sum_{\mu\in\{M-0\}/V}\frac{\operatorname{sign}N(\mu)}{|N(\mu)|^{s}}

Shimura_correspondence.html

  1. f f
  2. ( 2 k + 1 ) / 2 (2k+1)/2
  3. χ \chi
  4. n = 1 Λ ( n ) n - s = p ( 1 - ω p p - s + ( χ p ) 2 p 2 k - 1 - 2 s ) - 1 , \sum^{\infty}_{n=1}\Lambda(n)n^{-s}=\prod_{p}(1-\omega_{p}p^{-s}+(\chi_{p})^{2% }p^{2k-1-2s})^{-1}\ ,
  5. ω p \omega_{p}
  6. T ( p 2 ) T(p^{2})
  7. F ( z ) = n = 1 Λ ( n ) q n F(z)=\sum^{\infty}_{n=1}\Lambda(n)q^{n}
  8. χ 2 \chi^{2}

Shinnar–Le_Roux_algorithm.html

  1. [ B 1 ( t ) , ϕ ( t ) ] S L R [ A N ( z ) , B N ( z ) ] [B_{1}(t),\phi(t)]\Longleftarrow SLR\Longrightarrow[A_{N}(z),B_{N}(z)]
  2. [ A N ( z ) , B N ( z ) ] [A_{N}(z),B_{N}(z)]
  3. [ A N ( z ) , B N ( z ) ] [A_{N}(z),B_{N}(z)]
  4. N - 1 N-1
  5. A N ( z ) A_{N}(z)

Shockley–Ramo_theorem.html

  1. i = E v q v i=E_{v}qv

Short_supermultiplet.html

  1. 2 N / 2 2^{N/2}
  2. N N

Shunt_impedance.html

  1. V \scriptstyle V_{\parallel}
  2. P \scriptstyle P
  3. R \scriptstyle R
  4. R = | V | 2 P R=\frac{|V_{\parallel}|^{2}}{P}
  5. | V | \scriptstyle|V_{\parallel}|
  6. R 0 \scriptstyle R_{0}
  7. V 0 \scriptstyle V_{0}
  8. R 0 = V 0 2 P . R_{0}=\frac{V_{0}^{2}}{P}.
  9. Q \scriptstyle Q
  10. P \scriptstyle P
  11. R = Q | V | 2 ω W , R=Q\frac{|V_{\parallel}|^{2}}{\omega W},
  12. R Q \scriptstyle\frac{R}{Q}
  13. R = | V | 2 P 0 = Q | V | 2 ω W R_{\perp}=\frac{|V_{\perp}|^{2}}{P_{0}}=Q\frac{|V_{\perp}|^{2}}{\omega W}

Siacci's_theorem.html

  1. 𝐫 = r 𝐞 r . \mathbf{r}=r\mathbf{e}_{r}.
  2. 𝐯 = d 𝐫 d t = s ˙ 𝐞 t = v 𝐞 t , \mathbf{v}=\frac{d\mathbf{r}}{dt}=\dot{s}\mathbf{e}_{t}=v\mathbf{e}_{t},
  3. 𝐡 = 𝐫 × m 𝐯 = h 𝐤 , \mathbf{h}=\mathbf{r}\times m\mathbf{v}=h\mathbf{k},
  4. 𝐫 = q 𝐞 t - p 𝐞 n \mathbf{r}=q\mathbf{e}_{t}-p\mathbf{e}_{n}
  5. 𝐚 = - κ v 2 r p 𝐞 r + ( h 2 ) 2 p 2 𝐞 t = S r 𝐞 r + S t 𝐞 t . \mathbf{a}=-\frac{\kappa v^{2}r}{p}\mathbf{e}_{r}+\frac{(h^{2})^{\prime}}{2p^{% 2}}\mathbf{e}_{t}=S_{r}\mathbf{e}_{r}+S_{t}\mathbf{e}_{t}.
  6. S r = - f ( r ) = - κ r h 2 p 3 , S t = 0. S_{r}=-f(r)=-\frac{\kappa rh^{2}}{p^{3}},\qquad S_{t}=0.
  7. κ = 1 r d p d r , \kappa=\frac{1}{r}\frac{dp}{dr},
  8. f ( r ) = h 2 p 3 d p d r . f(r)=\frac{h^{2}}{p^{3}}\frac{dp}{dr}.
  9. 1 2 h 2 p 2 + f ( r ) = c o n s t a n t \frac{1}{2}\frac{h^{2}}{p^{2}}+\int f(r)=constant
  10. 𝐚 = - κ v 2 r p 𝐞 r + ( v d v d s + κ v 2 q p ) 𝐞 t . \mathbf{a}=-\frac{\kappa v^{2}r}{p}\mathbf{e}_{r}+\left(v\frac{dv}{ds}+\frac{% \kappa v^{2}q}{p}\right)\mathbf{e}_{t}.

Side_looking_airborne_radar.html

  1. u = ( u , v , w ) t \vec{u}=(u,v,w)^{t}
  2. δ g = τ c 0 2 cos γ \delta_{g}=\frac{\tau c_{0}}{2\cos\gamma}
  3. τ \tau
  4. c 0 c_{0}
  5. γ \gamma
  6. τ \tau
  7. δ g \delta_{g}
  8. τ \tau
  9. δ g \delta_{g}
  10. δ r = c 0 / 2 B \delta_{r}=c_{0}/2B
  11. δ A z = ρ λ 2 L = H λ L sin γ \delta_{Az}=\frac{\rho\lambda}{2L}=\frac{H\lambda}{L\sin\gamma}
  12. λ \lambda
  13. L L
  14. ρ \rho
  15. H H

Siegel_domain.html

  1. ( z ) V \Im(z)\in V\,
  2. ( z ) - F ( w , w ) V \Im(z)-F(w,w)\in V\,
  3. ( z ) - L t ( w , w ) V \Im(z)-\Re L_{t}(w,w)\in V\,

Siemens_(unit).html

  1. G = 1 R = I V G=\frac{1}{R}=\frac{I}{V}
  2. S = Ω - 1 = A V \mathrm{S}=\Omega^{-1}=\dfrac{\mathrm{A}}{\mathrm{V}}
  3. \mho

Sieved_ultraspherical_polynomials.html

  1. 2 x c n λ ( x ; k ) = c n + 1 λ ( x ; k ) + c n - 1 λ ( x ; k ) 2xc_{n}^{\lambda}(x;k)=c_{n+1}^{\lambda}(x;k)+c_{n-1}^{\lambda}(x;k)
  2. 2 x ( m + λ ) c m k λ ( x ; k ) = ( m + 2 λ ) c m k + 1 λ ( x ; k ) + m c m k - 1 λ ( x ; k ) 2x(m+\lambda)c_{mk}^{\lambda}(x;k)=(m+2\lambda)c_{mk+1}^{\lambda}(x;k)+mc_{mk-% 1}^{\lambda}(x;k)
  3. 2 x B n - 1 λ ( x ; k ) = B n λ ( x ; k ) + B n - 2 λ ( x ; k ) 2xB_{n-1}^{\lambda}(x;k)=B_{n}^{\lambda}(x;k)+B_{n-2}^{\lambda}(x;k)
  4. 2 x ( m + λ ) B m k - 1 λ ( x ; k ) = m B m k λ ( x ; k ) + ( m + 2 λ ) B m k - 2 λ ( x ; k ) 2x(m+\lambda)B_{mk-1}^{\lambda}(x;k)=mB_{mk}^{\lambda}(x;k)+(m+2\lambda)B_{mk-% 2}^{\lambda}(x;k)

Sieverts'_law.html

  1. K 2 = C N 2 p N 2 K^{2}={C_{N}^{2}\over{p_{N_{2}}}}
  2. C N = K p N 2 C_{N}={K\sqrt{p_{N_{2}}}}

Sigma-martingale.html

  1. d \mathbb{R}^{d}
  2. X = ( X t ) t = 0 T X=(X_{t})_{t=0}^{T}
  3. d \mathbb{R}^{d}
  4. ϕ \phi
  5. + \mathbb{R}_{+}
  6. X = ϕ M . X=\phi\cdot M.\,

Sigma_(disambiguation).html

  1. Σ n 0 \Sigma^{0}_{n}
  2. Σ n 1 \Sigma^{1}_{n}
  3. Σ i P \Sigma^{P}_{i}

Sigma_coordinate_system.html

  1. σ \sigma
  2. p p
  3. p 0 p_{0}
  4. p T p_{T}
  5. σ 0 = 1 \sigma_{0}=1

Signalizer_functor.html

  1. p p^{\prime}
  2. G G
  3. G G
  4. p p^{\prime}
  5. p p
  6. G . G.
  7. a A a\in A
  8. θ ( a ) \theta(a)
  9. C G ( a ) . C_{G}(a).
  10. a , b A a,b\in A
  11. θ ( a ) C G ( b ) θ ( b ) . \theta(a)\cap C_{G}(b)\subseteq\theta(b).
  12. θ ( a ) \theta(a)
  13. θ \theta
  14. θ , \theta,
  15. W = θ ( a ) a A , a 1 W=\langle\theta(a)\mid a\in A,a\neq 1\rangle
  16. G G
  17. θ ( a ) \theta(a)
  18. p p^{\prime}
  19. θ \theta
  20. A A
  21. W W
  22. A A
  23. A A
  24. W W
  25. p p^{\prime}
  26. θ \theta
  27. θ \theta
  28. A A
  29. p p^{\prime}
  30. H H
  31. G G
  32. H C G ( a ) θ ( a ) H\cap C_{G}(a)\subseteq\theta(a)
  33. a A . a\in A.
  34. θ ( a ) \theta(a)
  35. θ \theta
  36. W W
  37. W W
  38. θ \theta
  39. θ \theta
  40. W W
  41. θ \theta
  42. A A
  43. A A
  44. G G
  45. A A
  46. p p^{\prime}
  47. M M
  48. G , G,
  49. θ ( a ) = M C G ( a ) \theta(a)=M\cap C_{G}(a)
  50. a A . a\in A.
  51. θ \theta
  52. A A
  53. p p^{\prime}
  54. θ ( a ) = O p ( C G ( a ) ) . \theta(a)=O_{p^{\prime}}(C_{G}(a)).
  55. θ ( a ) \theta(a)
  56. A A
  57. p p^{\prime}
  58. G G
  59. A A
  60. θ \theta
  61. a A , a\in A,
  62. C G ( a ) C_{G}(a)
  63. p p
  64. p p
  65. θ \theta
  66. P × Q P\times Q
  67. A × B A\times B
  68. A A
  69. A A
  70. E E
  71. X . X.
  72. E E
  73. X X
  74. X = C X ( E 0 ) E 0 E , and E / E 0 cyclic X=\langle C_{X}(E_{0})\mid E_{0}\subseteq E,\,\text{ and }E/E_{0}\,\text{ % cyclic }\rangle
  75. q q
  76. X , X,
  77. X X
  78. E E
  79. q q
  80. X X
  81. q q
  82. X X
  83. X X
  84. E E
  85. E / C E ( X ) E/C_{E}(X)
  86. θ \theta
  87. W W
  88. G , G,
  89. θ \theta
  90. A A
  91. G G
  92. B B
  93. A . A.
  94. W = θ ( a ) a A , a 1 = θ ( b ) b B , b 1 W=\langle\theta(a)\mid a\in A,a\neq 1\rangle=\langle\theta(b)\mid b\in B,b\neq 1\rangle
  95. θ ( a ) \theta(a)
  96. θ ( a ) = θ ( a ) C G ( b ) b B , b 1 θ ( b ) b B , b 1 . \theta(a)=\langle\theta(a)\cap C_{G}(b)\mid b\in B,b\neq 1\rangle\subseteq% \langle\theta(b)\mid b\in B,b\neq 1\rangle.
  97. θ \theta
  98. g G g\in G
  99. a A a\in A
  100. θ ( a g ) = θ ( a ) g . \theta(a^{g})=\theta(a)^{g}.\,
  101. g . g.
  102. a O p ( C G ( a ) ) a\mapsto O_{p^{\prime}}(C_{G}(a))
  103. θ \theta
  104. B B
  105. W . W.
  106. G G
  107. A , A,
  108. θ \theta
  109. G . G.

Simple_Dietz_method.html

  1. R = B - A - C A + C / 2 R=\frac{B-A-C}{A+C/2}
  2. R R
  3. A A
  4. B B
  5. C C
  6. R 1 R_{1}
  7. R 2 R_{2}
  8. R = w 1 × R 1 + w 2 × R 2 R=w_{1}\times R_{1}+w_{2}\times R_{2}
  9. w 1 w_{1}
  10. w 2 w_{2}
  11. w i = A i + C i 2 A 1 + A 2 + C 1 + C 2 2 w_{i}=\frac{A_{i}+\frac{C_{i}}{2}}{A_{1}+A_{2}+\frac{C_{1}+C_{2}}{2}}
  12. R = I ÷ 1 / 2 ( A + B - I ) R=I\div{1/2}(A+B-I)
  13. M 1 M_{1}
  14. M 2 M_{2}
  15. M 2 = M 1 + C + I M_{2}={M_{1}}+C+I
  16. R = I ÷ 1 / 2 ( M 1 + M 2 - I ) R=I\div{1/2}({M_{1}}+{M_{2}}-I)
  17. R = M 2 - M 1 - C 1 / 2 ( M 1 + M 2 - M 2 + M 1 + C ) R=\frac{{M_{2}}-{M_{1}}-C}{{1/2}({M_{1}}+{M_{2}}-{M_{2}}+{M_{1}}+C)}
  18. R = M 2 - M 1 - C M 1 + C / 2 R=\frac{{M_{2}}-{M_{1}}-C}{{M_{1}}+C/2}
  19. M 2 = M 1 + C + R M 1 + R C / 2 {M_{2}}={M_{1}}+C+{RM_{1}}+RC/2

Simplectic_honeycomb.html

  1. A ~ 2 {\tilde{A}}_{2}
  2. A ~ 3 {\tilde{A}}_{3}
  3. A ~ n {\tilde{A}}_{n}
  4. x + y + x+y+...\in\mathbb{Z}
  5. A ~ 2 + {\tilde{A}}_{2+}
  6. A ~ 1 {\tilde{A}}_{1}
  7. A ~ 2 {\tilde{A}}_{2}
  8. A ~ 3 {\tilde{A}}_{3}
  9. A ~ 4 {\tilde{A}}_{4}
  10. A ~ 5 {\tilde{A}}_{5}
  11. A ~ 6 {\tilde{A}}_{6}
  12. A ~ 7 {\tilde{A}}_{7}
  13. A ~ 8 {\tilde{A}}_{8}
  14. A ~ 9 {\tilde{A}}_{9}
  15. A ~ 10 {\tilde{A}}_{10}
  16. A ~ 11 {\tilde{A}}_{11}
  17. A ~ 2 {\tilde{A}}_{2}
  18. A ~ 4 {\tilde{A}}_{4}
  19. A ~ 6 {\tilde{A}}_{6}
  20. A ~ 8 {\tilde{A}}_{8}
  21. A ~ 10 {\tilde{A}}_{10}
  22. A ~ 3 {\tilde{A}}_{3}
  23. A ~ 3 {\tilde{A}}_{3}
  24. A ~ 5 {\tilde{A}}_{5}
  25. A ~ 7 {\tilde{A}}_{7}
  26. A ~ 9 {\tilde{A}}_{9}
  27. C ~ 1 {\tilde{C}}_{1}
  28. C ~ 2 {\tilde{C}}_{2}
  29. C ~ 3 {\tilde{C}}_{3}
  30. C ~ 4 {\tilde{C}}_{4}
  31. C ~ 5 {\tilde{C}}_{5}

Single_domain_(magnetic).html

  1. μ = V M s \mu=VM_{s}
  2. V V
  3. M s M_{s}
  4. r r\,\!
  5. r c r_{c}\,\!
  6. r < r c r<r_{c}\,\!
  7. r c r_{c}\,\!
  8. 10 10
  9. 100 100

Singular_submodule.html

  1. 𝒵 ( M ) = { m M ann ( m ) e R } \mathcal{Z}(M)=\{m\in M\mid\mathrm{ann}(m)\subseteq_{e}R\}\,
  2. 𝒵 ( M ) \mathcal{Z}(M)
  3. t o r s ( M ) = 𝒵 ( M ) tors(M)=\mathcal{Z}(M)
  4. 𝒵 ( R R ) \mathcal{Z}(R_{R})
  5. 𝒵 ( R R ) \mathcal{Z}(R_{R})
  6. 𝒵 ( R R ) \mathcal{Z}(_{R}R)
  7. 𝒵 ( R R ) 𝒵 ( R R ) \mathcal{Z}(R_{R})\neq\mathcal{Z}(_{R}R)
  8. 𝒵 ( M ) = M \mathcal{Z}(M)=M\,
  9. 𝒵 ( M ) = { 0 } \mathcal{Z}(M)=\{0\}\,
  10. 𝒵 ( R R ) = { 0 } \mathcal{Z}(R_{R})=\{0\}\,
  11. 𝒵 ( R R ) R \mathcal{Z}(R_{R})\subsetneq R\,
  12. 𝒵 ( M ) soc ( M ) = { 0 } \mathcal{Z}(M)\cdot\mathrm{soc}(M)=\{0\}\,
  13. soc ( M ) \mathrm{soc}(M)\,
  14. f ( 𝒵 ( M ) ) 𝒵 ( N ) f(\mathcal{Z}(M))\subseteq\mathcal{Z}(N)\,
  15. 𝒵 ( N ) = N 𝒵 ( M ) \mathcal{Z}(N)=N\cap\mathcal{Z}(M)\,
  16. t ( M / t ( M ) ) = { 0 } t(M/t(M))=\{0\}\,
  17. 𝒵 ( M / 𝒵 ( M ) ) = { 0 } \mathcal{Z}(M/\mathcal{Z}(M))=\{0\}\,
  18. 𝒵 ( R R ) = J ( R ) \mathcal{Z}(R_{R})=J(R)\,
  19. S = End ( E ( R R ) ) S=\mathrm{End}(E(R_{R}))\,
  20. J ( S ) = { 0 } J(S)=\{0\}\,
  21. Q m a x r ( R ) Q_{max}^{r}(R)
  22. Q m a x r ( R ) Q_{max}^{r}(R)
  23. Q m a x r ( R ) Q_{max}^{r}(R)

Singularity_spectrum.html

  1. D ( α ) D(\alpha)
  2. f ( x ) f(x)
  3. D ( α ) = D F { x , α ( x ) = α } D(\alpha)=D_{F}\{x,\alpha(x)=\alpha\}
  4. α ( x ) \alpha(x)
  5. α ( x ) \alpha(x)
  6. f ( x ) f(x)
  7. x x
  8. D F { } D_{F}\{\cdot\}

Size_(statistics).html

  1. α = P ( test rejects H 0 | H 0 ) . \alpha=P(\,\text{test rejects }H_{0}|H_{0}).\,
  2. α = sup h H 0 P ( test rejects H 0 | h ) . \alpha=\sup_{h\in H_{0}}P(\,\text{test rejects }H_{0}|h).\,
  3. α \alpha
  4. α \alpha

Size_effect_on_structural_strength.html

  1. D D
  2. P f P_{f}
  3. σ k \sigma_{k}
  4. P 1 ( σ k ) P_{1}(\sigma_{k})
  5. N N
  6. 1 - P f = k = 1 N [ 1 - P 1 ( σ k ) ] 1-P_{f}=\prod_{k=1}^{N}[1-P_{1}(\sigma_{k})]
  7. P 1 ( σ k ) P_{1}(\sigma_{k})
  8. m m
  9. σ N \sigma_{N}
  10. P f = 1 - e - ( σ N / S 0 ) m where S 0 = s 0 ( l 0 / D ) n d / m Ψ - 1 / m P_{f}\;=\;1\,-\,e^{-(\sigma_{N}/S_{0})^{m}}\ \,\text{where}\ S_{0}=s_{0}(l_{0}% /D)^{n_{d}/m}\Psi^{-1/m}
  11. S 0 S_{0}
  12. m m
  13. Ψ = V [ σ ^ ( ξ ) ] m d V ( ξ ) \Psi=\int_{V}\left[{\hat{\sigma}}(\xi)\right]^{m}\,\mbox{d}~{}V(\xi)
  14. V V
  15. ξ \xi
  16. σ ^ ( ξ ) \hat{\sigma}(\xi)
  17. n d n_{d}
  18. n d n_{d}
  19. l 0 l_{0}
  20. N e q N_{eq}
  21. 10 4 10^{4}
  22. N e q = ( D / l 0 ) n d Ψ N_{eq}=(D/l_{0})^{n_{d}}\Psi
  23. P f P_{f}
  24. N e q < N N_{eq}<N
  25. N e q N N_{eq}\ll N
  26. σ ¯ N = C s ( l 0 / D ) n / m , C s = [ Γ ( 1 + m - 1 ) Ψ - 1 / m s 0 \bar{\sigma}_{N}\;=\;C_{s}(l_{0}/D)^{n/m},\;C_{s}\;=\;[\Gamma(1+m^{-1})\Psi^{-% 1/m}s_{0}
  27. w = [ Γ ( 1 + 2 m - 1 ) Γ - 2 ( 1 + m - 1 ) - 1 ] 1 / 2 w\;=\;[\Gamma(1+2m^{-1})\Gamma^{-2}(1+m^{-1})-1]^{1/2}
  28. Γ \Gamma
  29. D D
  30. m m
  31. σ N \sigma_{N}
  32. m m
  33. σ ¯ N \bar{\sigma}_{N}
  34. D D
  35. log σ ¯ N \log\bar{\sigma}_{N}
  36. log D \log D
  37. - 1 / m -1/m
  38. N e q N_{eq}
  39. D 0 D_{0}
  40. l 0 l_{0}
  41. D D
  42. D 0 D_{0}
  43. P P
  44. P P
  45. σ 1 = M c / I = 3 P L / 2 b D 2 \sigma_{1}=Mc/I=3PL/2bD^{2}
  46. M = P L / 4 M=PL/4
  47. D D
  48. c = D / 2 , I = b h 3 / 12 c=D/2,\;I=bh^{3}/12
  49. b b
  50. σ ¯ \bar{\sigma}
  51. l 0 / 2 l_{0}/2
  52. σ ¯ \bar{\sigma}
  53. σ 1 - σ n l 0 / 2 \sigma_{1}-\sigma^{\prime}_{n}l_{0}/2
  54. σ n \sigma^{\prime}_{n}
  55. 2 σ 1 / D 2\sigma_{1}/D
  56. σ ¯ = f t \bar{\sigma}=f^{\prime}_{t}
  57. σ ¯ \bar{\sigma}
  58. f t f^{\prime}_{t}
  59. P / b D = σ N P/bD=\sigma_{N}
  60. σ 0 / ( 1 - D b / D ) \sigma_{0}/(1-D_{b}/D)
  61. σ 0 = ( 2 D / 3 L ) f t \sigma_{0}=(2D/3L)f^{\prime}_{t}
  62. L / D L/D
  63. l 0 / D l_{0}/D
  64. σ N = σ 0 ( 1 + r l 0 D ) 1 / r \sigma_{N}=\sigma_{0}\left(1+\frac{rl_{0}}{D}\right)^{1/r}
  65. σ N \sigma_{N}
  66. D D
  67. D / l 0 D/l_{0}\to\infty
  68. r r
  69. r = 1 r=1
  70. r 1.45 r\approx 1.45
  71. ϵ n \epsilon^{\prime}_{n}
  72. D D\to\infty
  73. σ N \sigma_{N}
  74. σ ¯ N = σ 0 [ ( l 0 D ) r n d / m + r l 0 D ] 1 / r \bar{\sigma}_{N}=\sigma_{0}\left[\left(\frac{l_{0}}{D}\right)^{rn_{d}/m}+\ % \frac{rl_{0}}{D}\ \right]^{1/r}
  75. r , m r,m
  76. r n d / m < 1 rn_{d}/m<1
  77. m m\rightarrow\infty
  78. σ N / f t \sigma_{N}/f^{\prime}_{t}
  79. D / D 0 D/D_{0}
  80. s s
  81. s 2 s^{2}
  82. m m
  83. N e q < 10 4 N_{eq}<10^{4}
  84. P f P_{f}
  85. N N
  86. N e q N_{eq}
  87. N e q = 10 4 N_{eq}=10^{4}
  88. m m
  89. σ N \sigma_{N}
  90. a a
  91. h h
  92. k k
  93. k k
  94. a / D a/D
  95. U ¯ D 2 \bar{U}D^{2}
  96. G f D G_{f}D
  97. G f G_{f}
  98. U ¯ = σ N 2 / 2 E \bar{U}=\sigma_{N}^{2}/2E
  99. E E
  100. D D
  101. D 2 D^{2}
  102. D D
  103. σ N \sigma_{N}
  104. D D
  105. h h
  106. σ N = B f t ( 1 + D D 0 ) - 1 / 2 \sigma_{N}=Bf^{\prime}_{t}\left(1+\frac{D}{D_{0}}\right)^{-1/2}
  107. B , f t , D 0 B,f^{\prime}_{t},D_{0}
  108. f t f^{\prime}_{t}
  109. B B
  110. B f t = E G f g ( α 0 ) c f , D 0 = c f g ( α 0 ) g ( α 0 ) Bf^{\prime}_{t}=\sqrt{\frac{EG_{f}}{g^{\prime}(\alpha_{0})c_{f}}},\;\;\;\;D_{0% }=c_{f}\frac{g^{\prime}(\alpha_{0})}{g(\alpha_{0})}
  111. c f c_{f}\approx
  112. α 0 = a / D \alpha_{0}=a/D
  113. g ( α 0 ) = k 2 ( α 0 ) g(\alpha_{0})=k^{2}(\alpha_{0})
  114. k ( α 0 ) = K ( α 0 ) b D / P k(\alpha_{0})=K(\alpha_{0})b\sqrt{D}/P
  115. K K
  116. σ N \sigma_{N}
  117. G f G_{f}
  118. c f c_{f}
  119. σ \sigma
  120. w w
  121. σ = f ( w ) \sigma=f(w)
  122. G f G_{f}
  123. l c h = E G f / f t 2 l_{ch}=EG_{f}/{f^{\prime}_{t}}^{2}
  124. h h
  125. σ = f ^ ( ϵ ) \sigma=\hat{f}(\epsilon)
  126. ϵ = w / h \epsilon=w/h
  127. h h
  128. G f G_{f}
  129. h h
  130. < 10 - 6 <10^{-6}
  131. 10 - 3 10^{-3}
  132. G f G_{f}

Skew_binary_number_system.html

  1. 2 n + 1 - 1 2^{n+1}-1
  2. 2 ( 2 n + 1 - 1 ) + 1 = 2 n + 2 - 1 2(2^{n+1}-1)+1=2^{n+2}-1

Skew_gradient.html

  1. f ( z ( x , y ) ) = u ( x , y ) + i v ( x , y ) f(z(x,y))=u(x,y)+iv(x,y)
  2. u ( x , y ) = v ( x , y ) \nabla^{\perp}u(x,y)=\nabla v(x,y)
  3. u ( x , y ) = ( - u y , u x ) \nabla^{\perp}u(x,y)=(-\frac{\partial u}{\partial y},\frac{\partial u}{% \partial x})
  4. u ( x , y ) u ( x , y ) = 0 , u = u \nabla u(x,y)\cdot\nabla^{\perp}u(x,y)=0,\rVert\nabla u\rVert=\rVert\nabla^{% \perp}u\rVert

Skinny_triangle.html

  1. b r θ b\simeq r\theta\,
  2. a r e a 1 2 θ r 2 area\simeq\frac{1}{2}\theta r^{2}\,
  3. sin θ θ , θ 1 \sin\theta\simeq\theta,\quad\theta\ll 1\,
  4. cos θ = sin ( π 2 - θ ) 1 , θ 1 \cos\theta=\sin\left(\frac{\pi}{2}-\theta\right)\simeq 1,\quad\theta\ll 1
  5. θ \scriptstyle\theta
  6. b sin θ = r sin ( π - θ 2 ) \frac{b}{\sin\theta}=\frac{r}{\sin\left(\frac{\pi-\theta}{2}\right)}
  7. π - θ 2 \scriptstyle\frac{\pi-\theta}{2}
  8. b θ r 1 \frac{b}{\theta}\simeq\frac{r}{1}
  9. a r e a = sin θ 2 r 2 area=\frac{\sin\theta}{2}r^{2}
  10. a r e a 1 2 θ r 2 area\simeq\frac{1}{2}\theta r^{2}\,
  11. b h θ b\simeq h\theta
  12. tan θ θ , θ 1 \tan\theta\simeq\theta,\quad\theta\ll 1
  13. b = h tan θ b=h\tan\theta

Skip_graph.html

  1. 1 1 - p \frac{1}{1-p}
  2. l o g n ( 1 - p ) ( l o g ( 1 - p ) ) \frac{logn}{(1-p)(log(1-p))}
  3. l o g n ( 1 - p ) ( l o g ( 1 - p ) ) \frac{logn}{(1-p)(log(1-p))}
  4. 1 l o g n \frac{1}{logn}
  5. 1 l o g n \frac{1}{logn}

Skolem–Mahler–Lech_theorem.html

  1. F ( i ) = F ( i - 2 ) + F ( i - 4 ) F(i)=F(i-2)+F(i-4)

Skot_(unit).html

  1. 1 10 3 π \frac{1}{10^{3}\pi}
  2. 1 10 7 π \frac{1}{10^{7}\pi}

Slater's_condition.html

  1. Minimize f 0 ( x ) \,\text{Minimize }\;f_{0}(x)
  2. subject to: \,\text{subject to: }
  3. f i ( x ) 0 , i = 1 , , m f_{i}(x)\leq 0,i=1,\ldots,m
  4. A x = b Ax=b
  5. f 0 , , f m f_{0},\ldots,f_{m}
  6. x relint ( D ) x\in\operatorname{relint}(D)
  7. D = i = 0 m dom ( f i ) D=\cap_{i=0}^{m}\operatorname{dom}(f_{i})
  8. f i ( x ) < 0 , i = 1 , , m f_{i}(x)<0,i=1,\ldots,m
  9. A x = b . Ax=b.\,
  10. k k
  11. f 1 , , f k f_{1},\ldots,f_{k}
  12. x relint ( D ) x\in\operatorname{relint}(D)
  13. f i ( x ) 0 , i = 1 , , k , f_{i}(x)\leq 0,i=1,\ldots,k,
  14. f i ( x ) < 0 , i = k + 1 , , m , f_{i}(x)<0,i=k+1,\ldots,m,
  15. A x = b . Ax=b.\,
  16. Minimize f 0 ( x ) \,\text{Minimize }\;f_{0}(x)
  17. subject to: \,\text{subject to: }
  18. f i ( x ) K i 0 , i = 1 , , m f_{i}(x)\leq_{K_{i}}0,i=1,\ldots,m
  19. A x = b Ax=b
  20. f 0 f_{0}
  21. f i f_{i}
  22. K i K_{i}
  23. i i
  24. x relint ( D ) x\in\operatorname{relint}(D)
  25. f i ( x ) < K i 0 , i = 1 , , m f_{i}(x)<_{K_{i}}0,i=1,\ldots,m
  26. A x = b Ax=b

Slepian–Wolf_coding.html

  1. R X H ( X | Y ) , R_{X}\geq H(X|Y),\,
  2. R Y H ( Y | X ) , R_{Y}\geq H(Y|X),\,
  3. R X + R Y H ( X , Y ) . R_{X}+R_{Y}\geq H(X,Y).\,
  4. H ( X ) H(X)
  5. H ( Y ) H(Y)
  6. X X
  7. Y Y
  8. H ( X ) H(X)
  9. H ( Y ) H(Y)
  10. X X
  11. Y Y
  12. X X
  13. Y Y
  14. H ( X , Y ) H(X,Y)
  15. Y Y
  16. R Y = H ( Y ) R_{Y}=H(Y)
  17. Y Y
  18. H ( X | Y ) H(X|Y)
  19. X X

Sliding_criterion_(geotechnical_engineering).html

  1. s ′′ l i d i n g - a n g l e ′′ {}^{\prime\prime}sliding-angle^{\prime\prime}
  2. = R l * R s * I m * K a 0.0113 =\frac{Rl*Rs*Im*Ka}{0.0113}

Slip_ratio_(gas–liquid_flow).html

  1. S = u G u L = U G ( 1 - ϵ G ) U L ϵ G = ρ L x ( 1 - ϵ G ) ρ G ( 1 - x ) ϵ G S=\frac{u_{G}}{u_{L}}=\frac{U_{G}(1-\epsilon_{G})}{U_{L}\epsilon_{G}}=\frac{% \rho_{L}x(1-\epsilon_{G})}{\rho_{G}(1-x)\epsilon_{G}}
  2. S = 1 - x ( 1 - ρ L ρ G ) S=\sqrt{1-x(1-\frac{\rho_{L}}{\rho_{G}}})

Smale_conjecture.html

  1. Diff ( S 3 ) O ( 4 ) \scriptstyle{\mathrm{Diff}}(S^{3})\simeq{\mathrm{O}}(4)

Smear_(optics).html

  1. M T F s m e a r ( u ) = s i n ( π α u ) π α u MTF_{smear}(u)=\frac{sin(\pi\alpha u)}{\pi\alpha u}
  2. α \alpha

Smooth_completion.html

  1. y 2 = P ( x ) y^{2}=P(x)
  2. ( x , y ) 2 (x,y)\in\mathbb{C}^{2}
  3. P P
  4. x x
  5. 2 \mathbb{C}\mathbb{P}^{2}
  6. 2 \mathbb{C}\mathbb{P}^{2}
  7. P ( x ) P(x)
  8. 2 g - 2 + r > 0 2g-2+r>0
  9. 2 g + r - 1 2g+r-1

SND_Experiment.html

  1. ϕ a 0 ( 980 ) γ , f 0 ( 975 ) γ \phi\to a_{0}(980)\gamma,~{}~{}f_{0}(975)\gamma
  2. e + e - ρ , ω , ϕ π 0 γ , η γ e^{+}e^{-}\to\rho,\omega,\phi\to\pi^{0}\gamma,\eta\gamma
  3. ϕ η γ \phi\to\eta^{\prime}\gamma
  4. ϕ a 0 ( 980 ) γ , f 0 ( 975 ) γ , π π γ , η π γ \phi\to a_{0}(980)\gamma,~{}~{}f_{0}(975)\gamma,~{}~{}\pi\pi\gamma,~{}~{}\eta\pi\gamma
  5. ρ , ω π π γ \rho,\omega\to\pi\pi\gamma
  6. ϕ ω π 0 , π π , η π π \phi\to\omega\pi^{0},~{}~{}\pi\pi,~{}~{}\eta\pi\pi
  7. ρ π + π - π 0 \rho\to\pi^{+}\pi^{-}\pi^{0}
  8. ω π + π - \omega\to\pi^{+}\pi^{-}
  9. ρ , ω , ϕ η e + e - , π 0 e + e - \rho,\omega,\phi\to\eta e^{+}e^{-},~{}~{}\pi^{0}e^{+}e^{-}
  10. e + e - 2 π , 3 π , 4 π , 5 π e^{+}e^{-}\to 2\pi,~{}~{}3\pi,~{}~{}4\pi,~{}~{}5\pi
  11. e + e - ω π , η π π , ϕ π e^{+}e^{-}\to\omega\pi,~{}~{}\eta\pi\pi,~{}~{}\phi\pi
  12. e + e - K + K - , K S 0 K L 0 , K K π e^{+}e^{-}\to K^{+}K^{-},~{}~{}K^{0}_{S}K^{0}_{L},~{}~{}KK\pi
  13. e + e - 3 γ , e + e - γ e^{+}e^{-}\to 3\gamma,~{}~{}e^{+}e^{-}\gamma
  14. e + e - 4 γ , e + e - γ γ , e + e - e + e - e^{+}e^{-}\to 4\gamma,~{}~{}e^{+}e^{-}\gamma\gamma,~{}~{}e^{+}e^{-}e^{+}e^{-}
  15. e + e - 5 γ , 3 γ e + e - , 4 e γ e^{+}e^{-}\to 5\gamma,~{}~{}3\gamma e^{+}e^{-},~{}~{}4e\gamma
  16. K S 0 γ γ , 3 π 0 , 2 π 0 γ , π 0 γ γ , π 0 e + e - K^{0}_{S}\to\gamma\gamma,~{}~{}3\pi^{0},~{}~{}2\pi^{0}\gamma,~{}~{}\pi^{0}% \gamma\gamma,~{}~{}\pi^{0}e^{+}e^{-}
  17. η 3 γ , e + e - , 4 e \eta\to 3\gamma,~{}~{}e^{+}e^{-},~{}~{}4e
  18. e + e - η , a 0 , f 0 , a 2 , f 2 e^{+}e^{-}\to\eta^{\prime},~{}~{}a_{0},~{}~{}f_{0},~{}~{}a_{2},~{}~{}f_{2}

SO_(complexity).html

  1. k k
  2. k k
  3. k k
  4. Σ 1 \Sigma^{1}
  5. Π 1 \Pi^{1}
  6. n O ( 1 ) n^{O(1)}
  7. 2 n O ( 1 ) 2^{n^{O(1)}}

Software_reliability_testing.html

  1. l n [ n ( T ) T ] = - α l n ( T ) + b ; . . E q : 1 \begin{aligned}\displaystyle ln\left[\frac{n\left(T\right)}{T}\right]=-\alpha ln% \left(T\right)+b;\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ .....Eq:1\end{aligned}
  2. n ( T ) = K T 1 - α ; E q : 2 \begin{aligned}\displaystyle n\left(T\right)=KT^{1-\alpha};\ \ \ \ \ \ \ \ \ % \ \ \ \ \ \ \ \ \ \ ......Eq:2\end{aligned}
  3. N e w T e s t c a s e s ( c u r r e n t r e l e a s e ) = ( N R ) * T \begin{aligned}\displaystyle NewTestcases_{(currentrelease)}=\left(\frac{N}{R}% \right)*T\end{aligned}

Soil_resistivity.html

  1. ρ E = 4 π a R W 1 + 2 a a 2 + 4 b 2 - a a 2 + b 2 \rho_{E}=\frac{4\cdot{\pi}\cdot a\cdot R_{W}}{1+\frac{2\cdot a}{\sqrt{a^{2}+4% \cdot b^{2}}}-\frac{a}{\sqrt{a^{2}+b^{2}}}}\,
  2. ρ E = 2 π a R W \rho_{E}=2\cdot\pi\cdot a\cdot R_{W}\,
  3. ρ E = π c ( c + a ) a R S \rho_{E}={\pi}\frac{c\cdot(c+a)}{a}R_{S}\,
  4. R W = ρ E 2 π a W R_{W}=\frac{\rho_{E}}{2\cdot\pi\cdot a_{W}}\,
  5. a W = a S + 2 c 3 a_{W}=\frac{a_{S}+2c}{3}\,
  6. ρ E = π c ( c + a S ) a S R S \rho_{E}={\pi}\frac{c\cdot(c+a_{S})}{a_{S}}R_{S}\,

Solar_cell_efficiency.html

  1. η = P m G × A c \eta=\frac{P_{m}}{G\times A_{c}}
  2. F F = P m V O C × I S C = η × A c × E V O C × I S C . FF=\frac{P_{m}}{V_{OC}\times I_{SC}}=\frac{\eta\times A_{c}\times E}{V_{OC}% \times I_{SC}}.

Solvency_cone.html

  1. Π \Pi
  2. d d
  3. Π = ( π i j ) 1 i , j d \Pi=\left(\pi^{ij}\right)_{1\leq i,j\leq d}
  4. m d m\leq d
  5. m = d m=d
  6. K ( Π ) d K(\Pi)\subset\mathbb{R}^{d}
  7. e i , 1 i m e^{i},1\leq i\leq m
  8. π i j e i - e j , 1 i , j d \pi^{ij}e^{i}-e^{j},1\leq i,j\leq d
  9. K K
  10. K d K\subseteq\mathbb{R}^{d}
  11. K + d K\supseteq\mathbb{R}^{d}_{+}
  12. { K t ( ω ) } t = 0 T \left\{K_{t}(\omega)\right\}_{t=0}^{T}
  13. K + = { w d : v K : 0 w T v } K^{+}=\left\{w\in\mathbb{R}^{d}:\forall v\in K:0\leq w^{T}v\right\}
  14. K = { x 2 : ( 1 , 1 ) x 0 } K=\{x\in\mathbb{R}^{2}:(1,1)x\geq 0\}
  15. K = { x 2 : ( 2 , 1 ) x 0 , ( 1 , 2 ) x 0 } K=\{x\in\mathbb{R}^{2}:(2,1)x\geq 0,(1,2)x\geq 0\}
  16. ( 0 , t ) ( 1 , 0 ) ( 0 , 1 2 ) (0,t)\rightarrow(1,0)\rightarrow(0,\frac{1}{2})
  17. t < 1 2 t<\frac{1}{2}
  18. ( 1 , 0 ) ( 0 , t ) ( t 2 , 0 ) (1,0)\rightarrow(0,t)\rightarrow(\frac{t}{2},0)
  19. t > 2 t>2
  20. K K
  21. K = + d K=\mathbb{R}^{d}_{+}

Somatotype_and_constitutional_psychology.html

  1. - 0.7182 + ( 0.145 * X ) - ( 0.00068 * X 2 ) + ( 0.0000014 * X 3 ) -0.7182+(0.145*X)-(0.00068*X^{2})+(0.0000014*X^{3})
  2. X = ( t r i c e p S k i n f o l d M M + s u b s c a p u l a r S k i n f o l d M M + s u p r a s p i n a l S k i n f o l d M M ) * ( 170.18 h e i g h t C M ) X=(tricepSkinfoldMM+subscapularSkinfoldMM+supraspinalSkinfoldMM)*(\frac{170.18% }{heightCM})
  3. ( 0.858 * H u m e r u s B r e a d t h C M ) + ( 0.601 * F e m u r B r e a d t h C M ) + ( 0.188 * u p p e r A r m G i r t h C M ) + ( 0.161 * m a x C a l f G i r t h C M ) - ( 0.131 * h e i g h t C M ) + 4.5 (0.858*HumerusBreadthCM)+(0.601*FemurBreadthCM)+(0.188*upperArmGirthCM)+(0.161% *maxCalfGirthCM)-(0.131*heightCM)+4.5
  4. PI = m a s s h e i g h t 3 \,\text{PI}=\frac{mass}{height^{3}}
  5. P I > 40.74 PI>40.74
  6. ( 0.732 * P I ) - 28.58 (0.732*PI)-28.58
  7. 39.65 < P I < 40.74 39.65<PI<40.74
  8. ( 0.463 * P I ) - 17.615 (0.463*PI)-17.615
  9. P I < 39.65 PI<39.65
  10. 0.5 0.5

Sonine_formula.html

  1. 0 J z ( a t ) J z ( b t ) J z ( c t ) t 1 - z d t = 2 z - 1 Δ ( a , b , c ) 2 z - 1 π 1 / 2 Γ ( z + 1 2 ) ( a b c ) z \int_{0}^{\infty}J_{z}(at)J_{z}(bt)J_{z}(ct)t^{1-z}\,dt=\frac{2^{z-1}\Delta(a,% b,c)^{2z-1}}{\pi^{1/2}\Gamma(z+\tfrac{1}{2})(abc)^{z}}

Sound_intensity_probe.html

  1. I = p v I=pv
  2. p p
  3. v v

Space-time_Fourier_transform.html

  1. f = f ( r , t ) f=f(r,t)
  2. k x = l π L k_{x}=\frac{l\pi}{L}
  3. k y = m π W k_{y}=\frac{m\pi}{W}
  4. k z = n π D k_{z}=\frac{n\pi}{D}
  5. f ( k , ω ) = Ω T sin ( k x x ) sin ( k y y ) sin ( k z z ) exp ( - i ω t ) d t d x d y d z f\left(k,\omega\right)=\int_{\Omega}\int_{T}\sin(k_{x}x)\sin(k_{y}y)\sin(k_{z}% z)\exp(-i\omega t)\,dt\,dx\,dy\,dz
  6. Ω \Omega

Space_cardioid.html

  1. X ( t ) = ( ( k + cos ( t ) ) cos ( t ) , ( j + cos ( t ) ) sin ( t ) , sin ( t ) ) X(t)=((k+\cos(t))\cos(t),(j+\cos(t))\sin(t),\sin(t))\,

Space_tether.html

  1. L c = σ ρ g L_{c}=\frac{\sigma}{\rho g}
  2. V = σ ρ V=\sqrt{\frac{\sigma}{\rho}}
  3. V c = 2 σ ρ V_{c}=\sqrt{\frac{2\sigma}{\rho}}
  4. A ( r ) = M v 2 T R e δ T v 2 2 ( 1 - r 2 R 2 ) A(r)=\frac{Mv^{2}}{TR}\mathrm{e}^{\frac{\delta}{T}\frac{v^{2}}{2}\left(1-\frac% {r^{2}}{R^{2}}\right)}
  5. δ \delta
  6. M m = π δ T V 2 2 e ( δ T V 2 2 ) erf ( δ T V 2 2 ) \frac{M}{m}=\sqrt{\pi\frac{\delta}{T}\frac{V^{2}}{2}}\mathrm{e}^{\left(\frac{% \delta}{T}\frac{V^{2}}{2}\right)}\mathrm{erf}\left(\sqrt{\frac{\delta}{T}\frac% {V^{2}}{2}}\right)
  7. V r = V / V c V_{r}=V/V_{c}\,
  8. V c = 2 T δ V_{c}=\sqrt{\frac{2T}{\delta}}
  9. M m = π V r e V r 2 erf ( V r ) \frac{M}{m}=\sqrt{\pi}V_{r}\mathrm{e}^{{V_{r}}^{2}}\mathrm{erf}({V_{r}})

Sparse_distributed_memory.html

  1. F : { 0 , 1 } n - > { 0 , 1 } F:\{0,1\}^{n}->\{0,1\}
  2. i = 0 , , n - 1 i=0,...,n-1
  3. F t F_{t}
  4. F t { 0 , 1 } F_{t}\in\{0,1\}
  5. x i , t x_{i,t}
  6. x i , t { 0 , 1 } x_{i,t}\in\{0,1\}
  7. w i w_{i}
  8. c c
  9. S t = i = 0 n - 1 w i x i , t S_{t}=\sum_{i=0}^{n-1}w_{i}x_{i,t}
  10. 𝐒 t = { 1 if S t c , 0 otherwise . \mathbf{S}_{t}=\begin{cases}1&\,\text{if }S_{t}>=c,\\ 0&\,\text{otherwise }.\end{cases}
  11. w 0 , . . , w n 1 w_{0},..,w_{n_{1}}
  12. 𝐚 i = { 1 if w i > 0 , 0 if w i < 0. \mathbf{a}_{i}=\begin{cases}1&\,\text{if }w_{i}>0,\\ 0&\,\text{if }w_{i}<0.\end{cases}
  13. S ( w ) S(w)
  14. S ( w ) = w i > 0 w i S(w)=\sum_{w_{i}>0}w_{i}
  15. s ( w ) s(w)
  16. s ( w ) = w i < 0 w i s(w)=\sum_{w_{i}<0}w_{i}
  17. s ( w ) < c < S ( w ) s(w)<c<S(w)
  18. 2 1000 2^{1000}
  19. d N d\in N
  20. d n d\leqslant n
  21. f ~ ( d ) = { 1 2 [ 1 - N ( z < w s h a r e d ( d ) θ ) + N ( z < - w s h a r e d ( d ) θ ) ] - d n } 2 \tilde{f}(d)=\left\{\frac{1}{2}\cdot\left[1-N\left(z<\frac{w\cdot shared(d)}{% \sqrt{\theta}}\right)+N\left(z<\frac{-w\cdot shared(d)}{\sqrt{\theta}}\right)% \right]-\frac{d}{n}\right\}^{2}
  22. d d
  23. h h
  24. s s
  25. H H
  26. w w
  27. θ \theta
  28. h h
  29. s h a r e d ( d ) shared(d)
  30. d d
  31. N = 2 n N=2^{n}
  32. | x | |x|
  33. x - | d | x-|d|
  34. d = 0 d=0
  35. d = n d=n
  36. d = 0 d=0
  37. d = 0 d=0
  38. n / 2 n/2
  39. n / 2 n/2
  40. f ( x ) = 1 + e - a x \!f(x)=1+e^{-ax}
  41. f ( x ) = 1 - [ 1 1 + e - a ( x - c ) ] f(x)=1-[\frac{1}{1+e^{-a(x-c)}}]

Specific-information.html

  1. I V I_{V}
  2. I S I_{S}
  3. I s s i I_{ssi}
  4. X X
  5. Y = y Y=y
  6. I ( X ; Y = y ) I(X;Y=y)

Specific_fan_power.html

  1. S F P = P q v SFP={{\sum P}\over q_{v}}
  2. P {\sum P}
  3. q v {q_{v}}
  4. [ S F P ] k W m 3 / s W l / s k J m 3 k P a [SFP]\equiv{kW\over m^{3}/s}\equiv{W\over l/s}\equiv{kJ\over m^{3}}\equiv{kPa}
  5. η t o t S F P = Δ p t o t \eta_{tot}\cdot SFP=\Delta p_{tot}
  6. η t o t \eta_{tot}
  7. Δ p t o t \Delta p_{tot}
  8. η t o t = 1 \eta_{tot}=1
  9. η t o t < 0.6 \eta_{tot}<0.6

Spectral_abscissa.html

  1. η ( A ) \eta(A)
  2. η ( A ) = max i { Re ( λ i ) } \eta(A)=\max_{i}\{{\rm Re}(\lambda_{i})\}\,
  3. η ( A ) < 0 \eta(A)<0

Spectral_shape_analysis.html

  1. Δ f := div grad f . \Delta f:=\operatorname{div}\;\operatorname{grad}f.
  2. Δ ϕ i + λ i ϕ i = 0. \Delta\phi_{i}+\lambda_{i}\phi_{i}=0.\,
  3. ϕ i \phi_{i}
  4. λ i \lambda_{i}
  5. x x
  6. x x
  7. h t ( x , y ) = i = 0 exp ( - λ i t ) ϕ i ( x ) ϕ i ( y ) . h_{t}(x,y)=\sum_{i=0}^{\infty}\exp(-\lambda_{i}t)\phi_{i}(x)\phi_{i}(y).
  8. h t ( x , x ) h_{t}(x,x)
  9. t j t_{j}
  10. s ( x ) = ( ϕ 1 ( x ) , ϕ 2 ( x ) , , ϕ N ( x ) ) s(x)=\left(\phi_{1}(x),\phi_{2}(x),...,\phi_{N}(x)\right)
  11. x x

Sphere–cylinder_intersection.html

  1. r r
  2. x 2 + y 2 = r 2 . x^{2}+y^{2}=r^{2}.
  3. R R
  4. ( a , 0 , 0 ) (a,0,0)
  5. ( x - a ) 2 + y 2 + z 2 = R 2 . (x-a)^{2}+y^{2}+z^{2}=R^{2}.
  6. a + R < r a+R<r
  7. R < r R<r
  8. a + R = r a+R=r
  9. ( r , 0 , 0 ) (r,0,0)
  10. a = 0 a=0
  11. r r
  12. z = ± R 2 - r 2 ; z=\pm\sqrt{R^{2}-r^{2}};
  13. r = R r=R
  14. z = 0 z=0
  15. z 2 + ( r 2 - R 2 + a 2 ) = 2 a x . z^{2}+(r^{2}-R^{2}+a^{2})=2ax.
  16. x x
  17. z z
  18. - r < x < r -r<x<r
  19. ( - b , 0 , 0 ) (-b,0,0)
  20. b = R 2 - r 2 - a 2 2 a . b=\frac{R^{2}-r^{2}-a^{2}}{2a}.
  21. R > r + a R>r+a
  22. x < r x<r
  23. r r
  24. ϕ \phi
  25. ( x , y , z ) = ( r cos ϕ , r sin ϕ , ± 2 a ( b + r cos ϕ ) ) . (x,y,z)=\left(r\cos\phi,r\sin\phi,\pm\sqrt{2a(b+r\cos\phi)}\right).
  26. ( - r , 0 , ± R 2 - ( r + a ) 2 ) ; ( 0 , ± r , ± R 2 - ( r - a ) ( r + a ) ) ; ( + r , 0 , ± R 2 - ( r - a ) 2 ) . \left(-r,0,\pm\sqrt{R^{2}-(r+a)^{2}}\right);\quad\left(0,\pm r,\pm\sqrt{R^{2}-% (r-a)(r+a)}\right);\quad\left(+r,0,\pm\sqrt{R^{2}-(r-a)^{2}}\right).
  27. R < r + a R<r+a
  28. ϕ \phi
  29. - ϕ 0 < ϕ < + ϕ 0 -\phi_{0}<\phi<+\phi_{0}
  30. cos ϕ 0 = - b / r \cos\phi_{0}=-b/r
  31. ( - b , ± r 2 - b 2 , 0 ) ; ( 0 , ± r , ± R 2 - ( r - a ) ( r + a ) ) ; ( + r , 0 , ± R 2 - ( r - a ) 2 ) . \left(-b,\pm\sqrt{r^{2}-b^{2}},0\right);\quad\left(0,\pm r,\pm\sqrt{R^{2}-(r-a% )(r+a)}\right);\quad\left(+r,0,\pm\sqrt{R^{2}-(r-a)^{2}}\right).
  32. R = r + a R=r+a
  33. ( r , 0 , 0 ) (r,0,0)
  34. ( x , y , z ) = ( r cos ϕ , r sin ϕ , 2 a r cos ϕ 2 ) , (x,y,z)=\left(r\cos\phi,r\sin\phi,2\sqrt{ar}\cos\frac{\phi}{2}\right),
  35. ϕ \phi
  36. a = r , R = 2 r a=r,R=2r
  37. ( x , y , z ) = ( r cos ϕ , r sin ϕ , R cos ϕ 2 ) . (x,y,z)=\left(r\cos\phi,r\sin\phi,R\cos\frac{\phi}{2}\right).

Spherical_sector.html

  1. V = 2 π r 2 h 3 . V=\frac{2\pi r^{2}h}{3}\;.
  2. V = 2 π r 3 3 ( 1 - cos φ ) , V=\frac{2\pi r^{3}}{3}(1-\cos\varphi),
  3. φ φ
  4. A = 2 π r h . A=2\pi rh\;.
  5. V = 0 2 π 0 ϕ 0 r r 2 sin ϕ d r d ϕ d θ = 2 π r 3 3 0 ϕ sin ϕ d ϕ = 2 π r 3 3 ( 1 - cos ϕ ) V=\int_{0}^{2\pi}\!\int_{0}^{\phi}\!\int_{0}^{r}\!r^{2}\sin{\phi}\,\mathrm{d}r% \,\mathrm{d}\phi\,\mathrm{d}\theta=\frac{2\pi r^{3}}{3}\int_{0}^{\phi}\!\sin{% \phi}\,\mathrm{d}\phi=\frac{2\pi r^{3}}{3}(1-\cos\phi)
  6. ϕ \phi
  7. θ \theta

Spherical_tokamak.html

  1. P net = η capture ( P fusion - P conduction - P radiation ) P\text{net}=\eta\text{capture}\left(P\text{fusion}-P\text{conduction}-P\text{% radiation}\right)
  2. β = p p m a g = n k B T ( B 2 / 2 μ 0 ) \beta=\frac{p}{p_{mag}}=\frac{nk_{B}T}{(B^{2}/2\mu_{0})}
  3. β = μ 0 p B 2 . \beta=\frac{\mu_{0}p}{\langle B^{2}\rangle}.
  4. B 2 \langle B^{2}\rangle
  5. B 2 = B θ 2 + B ρ 2 \scriptstyle\langle B^{2}\rangle=\langle B_{\theta}^{2}+B_{\rho}^{2}\rangle
  6. B 0 \scriptstyle B_{0}
  7. q = 2 π B 0 a 2 μ 0 R 0 I ( 1 + κ 2 2 ) . q_{\star}=\frac{2\pi B_{0}a^{2}}{\mu_{0}R_{0}I}\left(\frac{1+\kappa^{2}}{2}% \right).
  8. B 0 \scriptstyle B_{0}
  9. R \scriptstyle R
  10. I \scriptstyle I
  11. κ \scriptstyle\kappa
  12. a / R , \scriptstyle a/R,
  13. β crit = 5 B N ( 1 + κ 2 2 ) ϵ q . \beta\text{crit}=5\langle B_{N}\rangle\left(\frac{1+\kappa^{2}}{2}\right)\frac% {\epsilon}{q_{\star}}.
  14. ϵ \epsilon
  15. 1 / A 1/A
  16. B N \langle B_{N}\rangle
  17. q q_{\star}
  18. q q_{\star}
  19. q q_{\star}
  20. q = 1 + ( 3 4 ) 4 / 5 1.8. q_{\star}=1+\left(\frac{3}{4}\right)^{4/5}\approx 1.8.
  21. β max = 0.072 ( 1 + κ 2 2 ) ϵ . \beta\text{max}=0.072\left(\frac{1+\kappa^{2}}{2}\right)\epsilon.
  22. κ \kappa
  23. β max = 0.072 ( 1 + 2 2 2 ) 1 1.25 = 0.14. \beta\text{max}=0.072\left(\frac{1+2^{2}}{2}\right)\frac{1}{1.25}=0.14.
  24. β max = 0.072 ( 1 + 2 2 2 ) 1 5 / 2 = 0.072. \beta\text{max}=0.072\left(\frac{1+2^{2}}{2}\right)\frac{1}{5/2}=0.072.
  25. β max \beta\text{max}\,
  26. B 0 = ( 1 - ϵ B - ϵ ) B max . B_{0}=({1-\epsilon_{B}-\epsilon}){B\text{max}}.\,
  27. p = β max ( 1 + κ 2 ) ϵ ( 1 - ϵ B - ϵ ) 2 G ( ϵ ) ( B max ) 2 . {\langle p\rangle}=\beta\text{max}\left(1+\kappa^{2}\right)\epsilon\left({1-% \epsilon_{B}-\epsilon}\right)^{2}G(\epsilon)\left(B\text{max}\right)^{2}.
  28. B 0 B_{0}
  29. ϵ 0 \epsilon_{0}
  30. B max B\text{max}
  31. p = 0.14 ( 1 + 2 2 ) ( 1 1.25 ) ( 1 - 1 1.25 ) 2 2.5 7.5 2 = 2.6 atmospheres . {\langle p\rangle}=0.14\left(1+2^{2}\right)\left(\frac{1}{1.25}\right)\left(1-% \frac{1}{1.25}\right)^{2}2.5\cdot 7.5^{2}=2.6\,\text{ atmospheres}.
  32. B max B\text{max}
  33. ϵ \epsilon
  34. ϵ b \epsilon_{b}
  35. p = 0.072 ( 1 + 2 2 ) ( 1 0.4 ) ( 1 - 1 0.24 - 1 0.4 ) 2 1.2 15 2 = 7.7 atmospheres . {\langle p\rangle}=0.072\left(1+2^{2}\right)\left(\frac{1}{0.4}\right)\left(1-% \frac{1}{0.24}-\frac{1}{0.4}\right)^{2}1.2\cdot 15^{2}=7.7\,\text{ atmospheres}.
  36. B max . B\text{max}.

Spherically_complete_field.html

  1. B 1 B 2 n 𝐍 B n . B_{1}\supseteq B_{2}\supseteq\cdots\Rightarrow\bigcap_{n\in{\mathbf{N}}}B_{n}\neq.

Spin_angular_momentum_of_light.html

  1. L L
  2. R R
  3. ± \pm\hbar
  4. \hbar
  5. ± \pm
  6. L L
  7. R R
  8. z z
  9. 𝐒 = ϵ 0 ( 𝐄 × 𝐀 ) d 3 𝐫 , \mathbf{S}=\epsilon_{0}\int\left(\mathbf{E}\times\mathbf{A}\right)d^{3}\mathbf% {r},
  10. 𝐄 \mathbf{E}
  11. 𝐀 \mathbf{A}
  12. ϵ 0 \epsilon_{0}
  13. 𝐒 = ϵ 0 2 i ω ( 𝐄 × 𝐄 ) d 3 𝐫 . \mathbf{S}=\frac{\epsilon_{0}}{2i\omega}\int\left(\mathbf{E}^{\ast}\times% \mathbf{E}\right)d^{3}\mathbf{r}.
  14. 𝐒 = 𝐤 𝐮 𝐤 ( a ^ 𝐤 , L a ^ 𝐤 , L - a ^ 𝐤 , R a ^ 𝐤 , R ) , \mathbf{S}=\sum_{\mathbf{k}}\hbar\mathbf{u}_{\mathbf{k}}\left(\hat{a}^{\dagger% }_{\mathbf{k},L}\hat{a}_{\mathbf{k},L}-\hat{a}^{\dagger}_{\mathbf{k},R}\hat{a}% _{\mathbf{k},R}\right),
  15. 𝐮 𝐤 \mathbf{u}_{\mathbf{k}}
  16. a ^ 𝐤 , π \hat{a}^{\dagger}_{\mathbf{k},\pi}
  17. a ^ 𝐤 , π \hat{a}_{\mathbf{k},\pi}
  18. 𝐤 \mathbf{k}
  19. π \pi
  20. 𝐒 z = ± . \mathbf{S}_{z}=\pm\hbar.
  21. | ± = 1 2 ( 1 ± i ) . |\pm\rangle=\frac{1}{\sqrt{2}}\left(\begin{array}[l]{c}1\\ \pm i\end{array}\right).

Spin–lattice_relaxation_in_the_rotating_frame.html

  1. M x y ( 0 ) M_{xy}(0)
  2. M x y ( t S L ) = M x y ( 0 ) e - t S L / T 1 r h o M_{xy}(t_{SL})=M_{xy}(0)e^{-t_{SL}/T_{1rho}}\,

Spiral_antenna.html

  1. s s
  2. w w
  3. r 1 r_{1}
  4. r 2 r_{2}
  5. f low = c / 2 π r 2 ) f_{\,\text{low}}=c/2\pi r_{2})
  6. ( f high = c / 2 π r 1 ) (f_{\,\text{high}}=c/2\pi r_{1})
  7. c c
  8. ( r , θ ) (r,\theta)
  9. r r
  10. θ \theta
  11. r = a + b θ r=a+b\theta
  12. a a
  13. b b

Split_normal_distribution.html

  1. f ( x ; μ , σ 1 , σ 2 ) = A exp ( - ( x - μ ) 2 2 σ 1 2 ) if x < μ f(x;\mu,\sigma_{1},\sigma_{2})=A\exp\left(-\frac{(x-\mu)^{2}}{2\sigma_{1}^{2}}% \right)\quad\,\text{if }x<\mu
  2. f ( x ; μ , σ 1 , σ 2 ) = A exp ( - ( x - μ ) 2 2 σ 2 2 ) otherwise f(x;\mu,\sigma_{1},\sigma_{2})=A\exp\left(-\frac{(x-\mu)^{2}}{2\sigma_{2}^{2}}% \right)\quad\,\text{otherwise}
  3. A = 2 / π ( σ 1 + σ 2 ) - 1 . \quad A=\sqrt{2/\pi}(\sigma_{1}+\sigma_{2})^{-1}.
  4. σ 1 2 = σ 2 2 = σ * 2 \sigma_{1}^{2}=\sigma_{2}^{2}=\sigma_{*}^{2}
  5. σ * 2 \sigma_{*}^{2}
  6. σ 1 2 = σ 2 2 = σ * 2 \sigma_{1}^{2}=\sigma_{2}^{2}=\sigma_{*}^{2}
  7. 𝒮 𝒩 ( μ , σ 2 , γ ) \mathcal{SN}(\mu,\,\sigma^{2},\gamma)
  8. 𝒮 𝒩 ( μ , σ 2 , ξ ) \mathcal{SN}(\mu,\,\sigma^{2},\xi)
  9. σ 2 = σ 1 2 ( 1 + γ ) = σ 2 2 ( 1 - γ ) γ = σ 2 - σ 1 σ 2 + σ 1 ξ = 2 / π ( σ 2 - σ 1 ) γ = sgn ( ξ ) 1 - ( 1 + 2 β - 1 β ) 2 , where β = π ξ 2 2 σ 2 . \begin{aligned}\displaystyle\sigma^{2}&\displaystyle=\sigma_{1}^{2}(1+\gamma)=% \sigma_{2}^{2}(1-\gamma)\\ \displaystyle\gamma&\displaystyle=\frac{\sigma_{2}-\sigma_{1}}{\sigma_{2}+% \sigma_{1}}\\ \displaystyle\xi&\displaystyle=\sqrt{2/\pi}(\sigma_{2}-\sigma_{1})\\ \displaystyle\gamma&\displaystyle=\operatorname{sgn}(\xi)\sqrt{1-\left(\frac{% \sqrt{1+2\beta}-1}{\beta}\right)^{2}},\quad\,\text{where}\quad\beta=\frac{\pi% \xi^{2}}{2\sigma^{2}}.\end{aligned}
  10. L ( μ ) = - [ x i : x i < μ ( x i - μ ) 2 ] 1 / 3 - [ x i : x i > μ ( x i - μ ) 2 ] 1 / 3 L(\mu)=-\left[\sum_{x_{i}:x_{i}<\mu}(x_{i}-\mu)^{2}\right]^{1/3}-\left[\sum_{x% _{i}:x_{i}>\mu}(x_{i}-\mu)^{2}\right]^{1/3}
  11. μ ^ \hat{\mu}
  12. σ ^ 1 2 = - L ( μ ) N [ x i : x i < μ ( x i - μ ) 2 ] 2 / 3 , \hat{\sigma}_{1}^{2}=\frac{-L(\mu)}{N}\left[\sum_{x_{i}:x_{i}<\mu}(x_{i}-\mu)^% {2}\right]^{2/3},
  13. σ ^ 2 2 = - L ( μ ) N [ x i : x i > μ ( x i - μ ) 2 ] 2 / 3 , \hat{\sigma}_{2}^{2}=\frac{-L(\mu)}{N}\left[\sum_{x_{i}:x_{i}>\mu}(x_{i}-\mu)^% {2}\right]^{2/3},

Spray_characteristics.html

  1. Q f = Q w / S p g Q_{f}={Q_{w}}/\sqrt{S{p_{g}}}
  2. Q 2 = Q 1 P 2 / P 1 {Q_{2}}={Q_{1}}\sqrt{P_{2}/P_{1}}
  3. F l F_{l}
  4. F l = C Q Δ P {F_{l}}=CQ\sqrt{\Delta P}
  5. C = 2 D tan ( θ / 2 ) {C}=2D\tan(\theta/2)
  6. R S F = D V 0.9 - D V 0.1 D V 0.5 {RSF}=\frac{D_{V0.9}-D_{V0.1}}{D_{V0.5}}

Spring_system.html

  1. Δ 𝐋 = [ 1 - 1 0 0 1 - 1 ] 𝐱 - L \Delta\mathbf{L}=\begin{bmatrix}1&-1&0\\ 0&1&-1\end{bmatrix}\mathbf{x}-L
  2. F springs = - K Δ L = - K ( A 𝐱 - L ) = - K A 𝐱 - K L F\text{springs}=-K\Delta L=-K(A\mathbf{x}-L)=-KA\mathbf{x}-KL
  3. A A^{\top}
  4. F nodes = - A K A 𝐱 - A K L = 0 F\text{nodes}=-A^{\top}KA\mathbf{x}-A^{\top}KL=0
  5. A K A 𝐱 = - A K L A^{\top}KA\mathbf{x}=-A^{\top}KL
  6. A K A A^{\top}KA
  7. x 1 = 2 x_{1}=2
  8. A K A = [ 1 - 1 0 - 1 2 - 1 0 - 1 1 ] A^{\top}KA=\begin{bmatrix}1&-1&0\\ -1&2&-1\\ 0&-1&1\end{bmatrix}
  9. x 1 = 2 x_{1}=2
  10. A K A 𝐱 = [ 1 - 1 0 - 1 2 - 1 0 - 1 1 ] [ 2 x 2 x 3 ] = - A K L = [ - 1 - 1 2 ] A^{\top}KA\mathbf{x}=\begin{bmatrix}1&-1&0\\ -1&2&-1\\ 0&-1&1\end{bmatrix}\begin{bmatrix}2\\ x_{2}\\ x_{3}\end{bmatrix}=-A^{\top}KL=\begin{bmatrix}-1\\ -1\\ 2\end{bmatrix}
  11. [ 2 - 2 0 ] + [ - 1 0 2 - 1 - 1 1 ] [ x 2 x 3 ] = [ - 1 - 1 2 ] \begin{bmatrix}2\\ -2\\ 0\end{bmatrix}+\begin{bmatrix}-1&0\\ 2&-1\\ -1&1\end{bmatrix}\begin{bmatrix}x_{2}\\ x_{3}\end{bmatrix}=\begin{bmatrix}-1\\ -1\\ 2\end{bmatrix}
  12. [ - 2 0 ] + [ 2 - 1 - 1 1 ] [ x 2 x 3 ] = [ - 1 2 ] \begin{bmatrix}-2\\ 0\end{bmatrix}+\begin{bmatrix}2&-1\\ -1&1\end{bmatrix}\begin{bmatrix}x_{2}\\ x_{3}\end{bmatrix}=\begin{bmatrix}-1\\ 2\end{bmatrix}
  13. [ 2 - 1 - 1 1 ] [ x 2 x 3 ] = [ 1 2 ] \begin{bmatrix}2&-1\\ -1&1\end{bmatrix}\begin{bmatrix}x_{2}\\ x_{3}\end{bmatrix}=\begin{bmatrix}1\\ 2\end{bmatrix}
  14. [ x 2 x 3 ] = [ 2 - 1 - 1 1 ] - 1 [ 1 2 ] = [ 3 5 ] \begin{bmatrix}x_{2}\\ x_{3}\end{bmatrix}=\begin{bmatrix}2&-1\\ -1&1\end{bmatrix}^{-1}\begin{bmatrix}1\\ 2\end{bmatrix}=\begin{bmatrix}3\\ 5\end{bmatrix}
  15. x 1 = 2 x_{1}=2
  16. x 2 = 3 x_{2}=3
  17. x 3 = 5 x_{3}=5

Spt_function.html

  1. S ( q ) = 1 ( q ) n = 1 q n m = 1 n - 1 ( 1 - q m ) 1 - q n S(q)=\frac{1}{(q)_{\infty}}\sum_{n=1}^{\infty}\frac{q^{n}\prod_{m=1}^{n-1}(1-q% ^{m})}{1-q^{n}}
  2. ( q ) = n = 1 ( 1 - q n ) (q)_{\infty}=\prod_{n=1}^{\infty}(1-q^{n})
  3. S ( q ) S(q)
  4. E 2 ( z ) E_{2}(z)
  5. η ( z ) \eta(z)
  6. q = e 2 π i z q=e^{2\pi iz}
  7. S ~ ( z ) := q - 1 / 24 S ( q ) - 1 12 E 2 ( z ) η ( z ) \tilde{S}(z):=q^{-1/24}S(q)-\frac{1}{12}\frac{E_{2}(z)}{\eta(z)}
  8. S L 2 ( ) SL_{2}(\mathbb{Z})
  9. χ η - 1 \chi_{\eta}^{-1}
  10. χ η \chi_{\eta}
  11. η ( z ) \eta(z)
  12. spt ( 5 n + 4 ) 0 mod ( 5 ) \mathrm{spt}(5n+4)\equiv 0\mod(5)
  13. spt ( 7 n + 5 ) 0 mod ( 7 ) \mathrm{spt}(7n+5)\equiv 0\mod(7)
  14. spt ( 13 n + 6 ) 0 mod ( 13 ) \mathrm{spt}(13n+6)\equiv 0\mod(13)

Square-integrable_function.html

  1. - | f ( x ) | 2 d x < , \int_{-\infty}^{\infty}|f(x)|^{2}\,dx<\infty,
  2. ( - , + ) (-\infty,+\infty)
  3. f , g = A f ( x ) ¯ g ( x ) d x \langle f,g\rangle=\int_{A}\overline{f(x)}g(x)\,dx
  4. ( - , + ) (-\infty,+\infty)
  5. f , f < . \langle f,f\rangle<\infty.\,
  6. ( L 2 , , 2 ) \left(L_{2},\langle\cdot,\cdot\rangle_{2}\right)
  7. L 2 L_{2}
  8. L 2 L_{2}
  9. , 2 \langle\cdot,\cdot\rangle_{2}

Square_packing_in_a_square.html

  1. 2 + 2 4 / 5 2+2\sqrt{4/5}

Square_root_of_a_2_by_2_matrix.html

  1. M = ( A B C D ) M=\left(\begin{array}[]{cc}A&B\\ C&D\end{array}\right)
  2. s = ± δ , t = ± τ + 2 s . s=\pm\sqrt{\delta},\quad\quad t=\pm\sqrt{\tau+2s}.
  3. R = 1 t ( A + s B C D + s ) . R=\frac{1}{t}\left(\begin{array}[]{cc}A+s&B\\ C&D+s\end{array}\right).
  4. R 2 = 1 t 2 ( ( A + s ) 2 + B C ( A + s ) B + B ( D + s ) C ( A + s ) + ( D + s ) C ( D + s ) 2 + B C ) = 1 A + D + 2 s ( A ( A + D + 2 s ) ( A + D + 2 s ) B C ( A + D + 2 s ) D ( A + D + 2 s ) ) = M . \begin{array}[]{rcl}R^{2}&=&\displaystyle\frac{1}{t^{2}}\left(\begin{array}[]{% cc}(A+s)^{2}+BC&(A+s)B+B(D+s)\\ C(A+s)+(D+s)C&(D+s)^{2}+BC\end{array}\right)\\ &=&\displaystyle\frac{1}{A+D+2s}\left(\begin{array}[]{cc}A(A+D+2s)&(A+D+2s)B\\ C(A+D+2s)&D(A+D+2s)\end{array}\right)\;=\;M.\end{array}
  5. τ \tau
  6. R = ( 0 0 c 0 ) and R = ( 0 b 0 0 ) R=\left(\begin{array}[]{cc}0&0\\ c&0\end{array}\right)\quad\,\text{and}\quad R=\left(\begin{array}[]{cc}0&b\\ 0&0\end{array}\right)
  7. R = ( a 0 0 d ) R=\left(\begin{array}[]{cc}a&0\\ 0&d\end{array}\right)
  8. ( 1 0 0 1 ) \bigl(\begin{smallmatrix}\\ 1&0\\ 0&1\end{smallmatrix}\bigr)
  9. 1 t ( s r r - s ) , \tfrac{1}{t}\bigl(\begin{smallmatrix}\\ s&r\\ r&-s\end{smallmatrix}\bigr),
  10. 1 t ( s - r - r - s ) , \tfrac{1}{t}\bigl(\begin{smallmatrix}\\ s&-r\\ -r&-s\end{smallmatrix}\bigr),
  11. 1 t ( - s r r s ) , \tfrac{1}{t}\bigl(\begin{smallmatrix}\\ -s&r\\ r&s\end{smallmatrix}\bigr),
  12. 1 t ( - s - r - r s ) , \tfrac{1}{t}\bigl(\begin{smallmatrix}\\ -s&-r\\ -r&s\end{smallmatrix}\bigr),
  13. ( 1 0 0 ± 1 ) , \bigl(\begin{smallmatrix}\\ 1&0\\ 0&\pm 1\end{smallmatrix}\bigr),
  14. ( - 1 0 0 ± 1 ) , \bigl(\begin{smallmatrix}\\ -1&0\\ 0&\pm 1\end{smallmatrix}\bigr),
  15. ( r , s , t ) (r,s,t)
  16. r 2 + s 2 = t 2 . r^{2}+s^{2}=t^{2}.
  17. r 2 + s 2 = t 2 r^{2}+s^{2}=t^{2}
  18. R = ( a 0 C / ( a + d ) d ) . R=\left(\begin{array}[]{cc}a&0\\ C/(a+d)&d\end{array}\right).

Stability_(learning_theory).html

  1. H H
  2. S S
  3. H H
  4. d d
  5. n n
  6. O ( d n ) O\left(\sqrt{\frac{d}{n}}\right)
  7. H H
  8. H H
  9. L L
  10. L L
  11. ( x , y ) (x,y)
  12. f f
  13. X X
  14. Y Y
  15. X X
  16. Y Y
  17. f f
  18. H H
  19. S = { z 1 = ( x 1 , y 1 ) , . . , z m = ( x m , y m ) } S=\{z_{1}=(x_{1},\ y_{1})\ ,..,\ z_{m}=(x_{m},\ y_{m})\}
  20. m m
  21. Z = X × Y Z=X\times Y
  22. L L
  23. Z m Z_{m}
  24. H H
  25. S S
  26. f S f_{S}
  27. X X
  28. Y Y
  29. L L
  30. S S
  31. V V
  32. f f
  33. z = ( x , y ) z=(x,y)
  34. V ( f , z ) = V ( f ( x ) , y ) V(f,z)=V(f(x),y)
  35. f f
  36. I S [ f ] = 1 n V ( f , z i ) I_{S}[f]=\frac{1}{n}\sum V(f,z_{i})
  37. f f
  38. I [ f ] = 𝔼 z V ( f , z ) I[f]=\mathbb{E}_{z}V(f,z)
  39. S | i = { z 1 , , z i - 1 , z i + 1 , , z m } S^{|i}=\{z_{1},...,\ z_{i-1},\ z_{i+1},...,\ z_{m}\}
  40. S i = { z 1 , , z i - 1 , z i , z i + 1 , , z m } S^{i}=\{z_{1},...,\ z_{i-1},\ z_{i}^{^{\prime}},\ z_{i+1},...,\ z_{m}\}
  41. L L
  42. i { 1 , , m } , 𝔼 S , z [ | V ( f S , z ) - V ( f S | i , z ) | ] β . \forall i\in\{1,...,m\},\mathbb{E}_{S,z}[|V(f_{S},z)-V(f_{S^{|i}},z)|]\leq\beta.
  43. L L
  44. i { 1 , , m } , 𝔼 S [ | V ( f S , z i ) - V ( f S | i , z i ) | ] β . \forall i\in\ \{1,...,m\},\mathbb{E}_{S}[|V(f_{S},z_{i})-V(f_{S^{|i}},z_{i})|]% \leq\beta.
  45. L L
  46. S Z m , i { 1 , , m } , | 𝔼 z [ V ( f S , z ) ] - 𝔼 z [ V ( f S | i , z ) ] | β \forall S\in Z^{m},\forall i\in\{1,...,m\},|\mathbb{E}_{z}[V(f_{S},z)]-\mathbb% {E}_{z}[V(f_{S^{|i}},z)]|\leq\beta
  47. L L
  48. S Z m , i { 1 , , m } , sup z Z | V ( f S , z ) - V ( f S i , z ) | β \forall S\in Z^{m},\forall i\in\{1,...,m\},\sup_{z\in Z}|V(f_{S},z)-V(f_{S^{i}% },z)|\leq\beta
  49. S Z m , i { 1 , , m } , S { sup z Z | V ( f S , z ) - V ( f S i , z ) | β } 1 - δ \forall S\in Z^{m},\forall i\in\{1,...,m\},\mathbb{P}_{S}\{\sup_{z\in Z}|V(f_{% S},z)-V(f_{S^{i}},z)|\leq\beta\}\geq 1-\delta
  50. L L
  51. i { 1 , , m } , S { sup z Z | V ( f S , z i ) - V ( f S | i , z i ) | β C V } 1 - δ C V \forall i\in\{1,...,m\},\mathbb{P}_{S}\{\sup_{z\in Z}|V(f_{S},z_{i})-V(f_{S^{|% i}},z_{i})|\leq\beta_{CV}\}\geq 1-\delta_{CV}
  52. E l o o e r r Eloo_{err}
  53. L L
  54. E l o o e r r Eloo_{err}
  55. β E L m \beta_{EL}^{m}
  56. δ E L m \delta_{EL}^{m}
  57. i { 1 , , m } , S { | I [ f S ] - 1 m i = 1 m V ( f S | i , z i ) | β E L m } 1 - δ E L m \forall i\in\{1,...,m\},\mathbb{P}_{S}\{|I[f_{S}]-\frac{1}{m}\sum_{i=1}^{m}V(f% _{S^{|i}},z_{i})|\leq\beta_{EL}^{m}\}\geq 1-\delta_{EL}^{m}
  58. β E L m \beta_{EL}^{m}
  59. δ E L m \delta_{EL}^{m}
  60. n inf n\rightarrow\inf
  61. E l o o e r r Eloo_{err}
  62. C C
  63. k k
  64. n n