wpmath0000005_13

Split-quaternion.html

  1. ( u v v * u * ) \begin{pmatrix}u&v\\ v^{*}&u^{*}\end{pmatrix}
  2. ( 0 1 1 0 ) ( 1 0 0 - 1 ) = ( 0 - 1 1 0 ) \begin{pmatrix}0&1\\ 1&0\end{pmatrix}\begin{pmatrix}1&0\\ 0&-1\end{pmatrix}=\begin{pmatrix}0&-1\\ 1&0\end{pmatrix}
  3. ( a c b d ) q = ( a + d ) + ( c - b ) i + ( b + c ) j + ( a - d ) k 2 , \begin{pmatrix}a&c\\ b&d\end{pmatrix}\leftrightarrow q=\frac{(a+d)+(c-b)i+(b+c)j+(a-d)k}{2},
  4. m = p exp ( b p ) = sinh b + p cosh b = sinh b + i sinh a cosh b + r cosh a cosh b m~{}=p\exp{(bp)}=\sinh b+p\cosh b=\sinh b+i\sinh a~{}\cosh b+r\cosh a~{}\cosh b
  5. tan ϕ = x y = sinh b sinh a cosh b = tanh b sinh a \tan\phi=\frac{x}{y}=\frac{\sinh b}{\sinh a~{}\cosh b}=\frac{\tanh b}{\sinh a}
  6. lim b tan ϕ = 1 sinh a , \lim_{b\to\infty}\tan\phi=\frac{1}{\sinh a},
  7. q u - 1 q u q\mapsto u^{-1}qu
  8. u = e a v , v I or u = e a p , p J u=e^{av},\quad v\in I\quad\,\text{or}\quad u=e^{ap},\quad p\in J

Spray_(mathematics).html

  1. H ξ = ξ i x i | ( x , ξ ) - 2 G i ( x , ξ ) ξ i | ( x , ξ ) . H_{\xi}=\xi^{i}\frac{\partial}{\partial x^{i}}\Big|_{(x,\xi)}-2G^{i}(x,\xi)% \frac{\partial}{\partial\xi^{i}}\Big|_{(x,\xi)}.
  2. G i ( x , λ ξ ) = λ 2 G i ( x , ξ ) , λ > 0. G^{i}(x,\lambda\xi)=\lambda^{2}G^{i}(x,\xi),\quad\lambda>0.\,
  3. 𝒮 ( γ ) := a b L ( γ ( t ) , γ ˙ ( t ) ) d t \mathcal{S}(\gamma):=\int_{a}^{b}L(\gamma(t),\dot{\gamma}(t))dt
  4. d d s | s = 0 𝒮 ( γ s ) = | a b L ξ i X i - a b ( 2 L ξ j ξ i γ ¨ j + 2 L x j ξ i γ ˙ j - L x i ) X i d t , \frac{d}{ds}\Big|_{s=0}\mathcal{S}(\gamma_{s})=\Big|_{a}^{b}\frac{\partial L}{% \partial\xi^{i}}X^{i}-\int_{a}^{b}\Big(\frac{\partial^{2}L}{\partial\xi^{j}% \partial\xi^{i}}\ddot{\gamma}^{j}+\frac{\partial^{2}L}{\partial x^{j}\partial% \xi^{i}}\dot{\gamma}^{j}-\frac{\partial L}{\partial x^{i}}\Big)X^{i}dt,
  5. α ξ = α i ( x , ξ ) d x i | x T x * M \alpha_{\xi}=\alpha_{i}(x,\xi)dx^{i}|_{x}\in T_{x}^{*}M
  6. α i ( x , ξ ) = L ξ i ( x , ξ ) \alpha_{i}(x,\xi)=\tfrac{\partial L}{\partial\xi^{i}}(x,\xi)
  7. ξ T x M \xi\in T_{x}M
  8. α Ω 1 ( T M ) \alpha\in\Omega^{1}(TM)
  9. α ξ = α i ( x , ξ ) d x i | ( x , ξ ) T ξ * T M \alpha_{\xi}=\alpha_{i}(x,\xi)dx^{i}|_{(x,\xi)}\in T^{*}_{\xi}TM
  10. g ξ = g i j ( x , ξ ) ( d x i d x j ) | x g_{\xi}=g_{ij}(x,\xi)(dx^{i}\otimes dx^{j})|_{x}
  11. g i j ( x , ξ ) = 2 L ξ i ξ j ( x , ξ ) g_{ij}(x,\xi)=\tfrac{\partial^{2}L}{\partial\xi^{i}\partial\xi^{j}}(x,\xi)
  12. ξ T x M \xi\in T_{x}M
  13. g ξ \displaystyle g_{\xi}
  14. ξ T x M \xi\in T_{x}M
  15. g i j ( x , ξ ) \displaystyle g_{ij}(x,\xi)
  16. g i j ( x , ξ ) \displaystyle g^{ij}(x,\xi)
  17. E ( ξ ) = α ξ ( ξ ) - L ( ξ ) \displaystyle E(\xi)=\alpha_{\xi}(\xi)-L(\xi)
  18. d E = - ι H d α \displaystyle dE=-\iota_{H}d\alpha
  19. ι H d α = Y i 2 L ξ i x j d x j - X i 2 L ξ i x j d ξ j \iota_{H}d\alpha=Y^{i}\frac{\partial^{2}L}{\partial\xi^{i}\partial x^{j}}dx^{j% }-X^{i}\frac{\partial^{2}L}{\partial\xi^{i}\partial x^{j}}d\xi^{j}
  20. d E = ( 2 L x i ξ j ξ j - L x i ) d x i + ξ j 2 L ξ i x j d ξ i dE=\Big(\frac{\partial^{2}L}{\partial x^{i}\partial\xi^{j}}\xi^{j}-\frac{% \partial L}{\partial x^{i}}\Big)dx^{i}+\xi^{j}\frac{\partial^{2}L}{\partial\xi% ^{i}\partial x^{j}}d\xi^{i}
  21. G k ( x , ξ ) = g k i 2 ( 2 L ξ i x j ξ j - L x i ) . G^{k}(x,\xi)=\frac{g^{ki}}{2}\Big(\frac{\partial^{2}L}{\partial\xi^{i}\partial x% ^{j}}\xi^{j}-\frac{\partial L}{\partial x^{i}}\Big).
  22. d d s | s = 0 𝒮 ( γ s ) = | a b α i X i - a b g i k ( γ ¨ k + 2 G k ) X i d t , \frac{d}{ds}\Big|_{s=0}\mathcal{S}(\gamma_{s})=\Big|_{a}^{b}\alpha_{i}X^{i}-% \int_{a}^{b}g_{ik}(\ddot{\gamma}^{k}+2G^{k})X^{i}dt,
  23. L ( x , ξ ) = 1 2 F 2 ( x , ξ ) , L(x,\xi)=\tfrac{1}{2}F^{2}(x,\xi),
  24. F ( x , λ ξ ) = λ F ( x , ξ ) , λ > 0 F(x,\lambda\xi)=\lambda F(x,\xi),\quad\lambda>0
  25. α i = g i j ξ i , F 2 = g i j ξ i ξ j , E = α i ξ i - L = 1 2 F 2 . \alpha_{i}=g_{ij}\xi^{i},\quad F^{2}=g_{ij}\xi^{i}\xi^{j},\quad E=\alpha_{i}% \xi^{i}-L=\tfrac{1}{2}F^{2}.
  26. g i j ( λ ξ ) = g i j ( ξ ) , α i ( x , λ ξ ) = λ α i ( x , ξ ) , G i ( x , λ ξ ) = λ 2 G i ( x , ξ ) , g_{ij}(\lambda\xi)=g_{ij}(\xi),\quad\alpha_{i}(x,\lambda\xi)=\lambda\alpha_{i}% (x,\xi),\quad G^{i}(x,\lambda\xi)=\lambda^{2}G^{i}(x,\xi),
  27. F ( γ ( t ) , γ ˙ ( t ) ) = λ F(\gamma(t),\dot{\gamma}(t))=\lambda
  28. γ : [ a , b ] M \gamma:[a,b]\to M
  29. 𝒮 ( γ ) = ( b - a ) λ 2 2 = ( γ ) 2 2 ( b - a ) . \mathcal{S}(\gamma)=\frac{(b-a)\lambda^{2}}{2}=\frac{\ell(\gamma)^{2}}{2(b-a)}.
  30. γ : [ a , b ] M \gamma:[a,b]\to M
  31. h : T ( T M 0 ) T ( T M 0 ) ; h = 1 2 ( I - H J ) , h:T(TM\setminus 0)\to T(TM\setminus 0)\quad;\quad h=\tfrac{1}{2}\big(I-% \mathcal{L}_{H}J\big),
  32. v : T ( T M 0 ) T ( T M 0 ) ; v = 1 2 ( I + H J ) . v:T(TM\setminus 0)\to T(TM\setminus 0)\quad;\quad v=\tfrac{1}{2}\big(I+% \mathcal{L}_{H}J\big).
  33. T ( X , Y ) = J [ h X , h Y ] - v [ J X , h Y ) - v [ h X , J Y ] . \displaystyle T(X,Y)=J[hX,hY]-v[JX,hY)-v[hX,JY].
  34. H = Θ V + ϵ . \displaystyle H=\Theta V+\epsilon.
  35. τ = V v = 1 2 [ V , H ] - H J \tau=\mathcal{L}_{V}v=\tfrac{1}{2}\mathcal{L}_{[V,H]-H}J
  36. V ϵ + ϵ = τ Θ V . \mathcal{L}_{V}\epsilon+\epsilon=\tau\Theta V.
  37. ϵ | ξ = - 0 e - s ( Φ V - s ) * ( τ Θ V ) | Φ V s ( ξ ) d s . \epsilon|_{\xi}=\int\limits_{-\infty}^{0}e^{-s}(\Phi_{V}^{-s})_{*}(\tau\Theta V% )|_{\Phi_{V}^{s}(\xi)}ds.

Spurious_correlation.html

  1. x , y \displaystyle x,y
  2. x 1 / x 3 x_{1}/x_{3}
  3. x 2 / x 4 x_{2}/x_{4}
  4. x 1 , x 2 , x 3 , x 4 x_{1},x_{2},x_{3},x_{4}
  5. ρ = r 12 v 1 v 2 - r 14 v 1 v 4 - r 23 v 2 v 3 + r 34 v 3 v 4 v 1 2 + v 3 2 - 2 r 13 v 1 v 3 v 2 2 + v 4 2 - 2 r 24 v 2 v 4 \rho=\frac{r_{12}v_{1}v_{2}-r_{14}v_{1}v_{4}-r_{23}v_{2}v_{3}+r_{34}v_{3}v_{4}% }{\sqrt{v_{1}^{2}+v_{3}^{2}-2r_{13}v_{1}v_{3}}\sqrt{v_{2}^{2}+v_{4}^{2}-2r_{24% }v_{2}v_{4}}}
  6. v i v_{i}
  7. x i x_{i}
  8. r i j r_{ij}
  9. x i x_{i}
  10. x j x_{j}
  11. x 3 = x 4 x_{3}=x_{4}
  12. x 1 , x 2 , x 3 x_{1},x_{2},x_{3}
  13. ρ 0 = v 3 2 v 1 2 + v 3 2 v 2 2 + v 3 2 . \rho_{0}=\frac{v_{3}^{2}}{\sqrt{v_{1}^{2}+v_{3}^{2}}\sqrt{v_{2}^{2}+v_{3}^{2}}}.
  14. ρ 0 = 0.5 \rho_{0}=0.5

Square-cube_law.html

  1. A 2 = A 1 ( 2 1 ) 2 A_{2}=A_{1}\left(\frac{\ell_{2}}{\ell_{1}}\right)^{2}
  2. A 1 A_{1}
  3. A 2 A_{2}
  4. V 2 = V 1 ( 2 1 ) 3 V_{2}=V_{1}\left(\frac{\ell_{2}}{\ell_{1}}\right)^{3}
  5. V 1 V_{1}
  6. V 2 V_{2}
  7. 1 \ell_{1}
  8. 2 \ell_{2}
  9. F = M a F=Ma
  10. T = F A = M a A T=\frac{F}{A}=M\frac{a}{A}
  11. M = x 3 M M^{\prime}=x^{3}M
  12. A = x 2 A A^{\prime}=x^{2}A
  13. F = x 3 M a F^{\prime}=x^{3}Ma
  14. T = F A = x 3 x 2 × M a A = x × M a A = x × T \begin{aligned}\displaystyle T^{\prime}&\displaystyle=\frac{F^{\prime}}{A^{% \prime}}\\ &\displaystyle=\frac{x^{3}}{x^{2}}\times M\frac{a}{A}\\ &\displaystyle=x\times M\frac{a}{A}\\ &\displaystyle=x\times T\\ \end{aligned}

Stackelberg_competition.html

  1. P P
  2. P ( q 1 + q 2 ) P(q_{1}+q_{2})
  3. C i ( q i ) C_{i}(q_{i})
  4. Π 2 = P ( q 1 + q 2 ) q 2 - C 2 ( q 2 ) \Pi_{2}=P(q_{1}+q_{2})\cdot q_{2}-C_{2}(q_{2})
  5. q 2 q_{2}
  6. Π 2 \Pi_{2}
  7. q 1 q_{1}
  8. Π 2 \Pi_{2}
  9. q 2 q_{2}
  10. Π 2 \Pi_{2}
  11. q 2 q_{2}
  12. Π 2 q 2 = P ( q 1 + q 2 ) q 2 q 2 + P ( q 1 + q 2 ) - C 2 ( q 2 ) q 2 . \frac{\partial\Pi_{2}}{\partial q_{2}}=\frac{\partial P(q_{1}+q_{2})}{\partial q% _{2}}\cdot q_{2}+P(q_{1}+q_{2})-\frac{\partial C_{2}(q_{2})}{\partial q_{2}}.
  13. Π 2 q 2 = P ( q 1 + q 2 ) q 2 q 2 + P ( q 1 + q 2 ) - C 2 ( q 2 ) q 2 = 0. \frac{\partial\Pi_{2}}{\partial q_{2}}=\frac{\partial P(q_{1}+q_{2})}{\partial q% _{2}}\cdot q_{2}+P(q_{1}+q_{2})-\frac{\partial C_{2}(q_{2})}{\partial q_{2}}=0.
  14. q 2 q_{2}
  15. Π 1 = P ( q 1 + q 2 ( q 1 ) ) . q 1 - C 1 ( q 1 ) \Pi_{1}=P(q_{1}+q_{2}(q_{1})).q_{1}-C_{1}(q_{1})
  16. q 2 ( q 1 ) q_{2}(q_{1})
  17. q 1 q_{1}
  18. Π 1 \Pi_{1}
  19. q 2 ( q 1 ) q_{2}(q_{1})
  20. Π 1 \Pi_{1}
  21. q 1 q_{1}
  22. Π 1 \Pi_{1}
  23. q 1 q_{1}
  24. Π 1 q 1 = P ( q 1 + q 2 ) q 2 q 2 ( q 1 ) q 1 q 1 + P ( q 1 + q 2 ) q 1 q 1 + P ( q 1 + q 2 ( q 1 ) ) - C 1 ( q 1 ) q 1 . \frac{\partial\Pi_{1}}{\partial q_{1}}=\frac{\partial P(q_{1}+q_{2})}{\partial q% _{2}}\cdot\frac{\partial q_{2}(q_{1})}{\partial q_{1}}\cdot q_{1}+\frac{% \partial P(q_{1}+q_{2})}{\partial q_{1}}\cdot q_{1}+P(q_{1}+q_{2}(q_{1}))-% \frac{\partial C_{1}(q_{1})}{\partial q_{1}}.
  25. Π 1 q 1 = P ( q 1 + q 2 ) q 2 q 2 ( q 1 ) q 1 q 1 + P ( q 1 + q 2 ) q 1 q 1 + P ( q 1 + q 2 ( q 1 ) ) - C 1 ( q 1 ) q 1 = 0. \frac{\partial\Pi_{1}}{\partial q_{1}}=\frac{\partial P(q_{1}+q_{2})}{\partial q% _{2}}\cdot\frac{\partial q_{2}(q_{1})}{\partial q_{1}}\cdot q_{1}+\frac{% \partial P(q_{1}+q_{2})}{\partial q_{1}}\cdot q_{1}+P(q_{1}+q_{2}(q_{1}))-% \frac{\partial C_{1}(q_{1})}{\partial q_{1}}=0.
  26. P ( q 1 + q 2 ) = ( a - b ( q 1 + q 2 ) ) P(q_{1}+q_{2})=\bigg(a-b(q_{1}+q_{2})\bigg)
  27. 2 C i ( q i ) q i q j = 0 , j \frac{\partial^{2}C_{i}(q_{i})}{\partial q_{i}\cdot\partial q_{j}}=0,\forall j
  28. C i ( q i ) q j = 0 , j i \frac{\partial C_{i}(q_{i})}{\partial q_{j}}=0,j\neq\ i
  29. Π 2 = ( a - b ( q 1 + q 2 ) ) q 2 - C 2 ( q 2 ) . \Pi_{2}=\bigg(a-b(q_{1}+q_{2})\bigg)\cdot q_{2}-C_{2}(q_{2}).
  30. ( a - b ( q 1 + q 2 ) ) q 2 q 2 + a - b ( q 1 + q 2 ) - C 2 ( q 2 ) q 2 = 0 , \frac{\partial\bigg(a-b(q_{1}+q_{2})\bigg)}{\partial q_{2}}\cdot q_{2}+a-b(q_{% 1}+q_{2})-\frac{\partial C_{2}(q_{2})}{\partial q_{2}}=0,
  31. - b q 2 + a - b ( q 1 + q 2 ) - C 2 ( q 2 ) q 2 = 0 , \Rightarrow\ -bq_{2}+a-b(q_{1}+q_{2})-\frac{\partial C_{2}(q_{2})}{\partial q_% {2}}=0,
  32. q 2 = a - b q 1 - C 2 ( q 2 ) q 2 2 b . \Rightarrow\ q_{2}=\frac{a-bq_{1}-\frac{\partial C_{2}(q_{2})}{\partial q_{2}}% }{2b}.
  33. Π 1 = ( a - b ( q 1 + q 2 ( q 1 ) ) ) q 1 - C 1 ( q 1 ) . \Pi_{1}=\bigg(a-b(q_{1}+q_{2}(q_{1}))\bigg)\cdot q_{1}-C_{1}(q_{1}).
  34. q 2 ( q 1 ) q_{2}(q_{1})
  35. Π 1 = ( a - b ( q 1 + a - b q 1 - C 2 ( q 2 ) q 2 2 b ) ) q 1 - C 1 ( q 1 ) , \Pi_{1}=\bigg(a-b\bigg(q_{1}+\frac{a-bq_{1}-\frac{\partial C_{2}(q_{2})}{% \partial q_{2}}}{2b}\bigg)\bigg)\cdot q_{1}-C_{1}(q_{1}),
  36. Π 1 = ( a - b . q 1 + C 2 ( q 2 ) q 2 2 ) ) q 1 - C 1 ( q 1 ) . \Rightarrow\Pi_{1}=\bigg(\frac{a-b.q_{1}+\frac{\partial C_{2}(q_{2})}{\partial q% _{2}}}{2})\bigg)\cdot q_{1}-C_{1}(q_{1}).
  37. Π 1 q 1 = ( a - 2 b q 1 + C 2 ( q 2 ) q 2 2 ) - C 1 ( q 1 ) q 1 = 0. \frac{\partial\Pi_{1}}{\partial q_{1}}=\bigg(\frac{a-2bq_{1}+\frac{\partial C_% {2}(q_{2})}{\partial q_{2}}}{2}\bigg)-\frac{\partial C_{1}(q_{1})}{\partial q_% {1}}=0.
  38. q 1 q_{1}
  39. q 1 * q_{1}^{*}
  40. q 1 * = a + C 2 ( q 2 ) q 2 - 2 C 1 ( q 1 ) q 1 2 b . q_{1}^{*}=\frac{a+\frac{\partial C_{2}(q_{2})}{\partial q_{2}}-2\cdot\frac{% \partial C_{1}(q_{1})}{\partial q_{1}}}{2b}.
  41. q 2 * = a - b a + C 2 ( q 2 ) q 2 - 2 C 1 ( q 1 ) q 1 2 b - C 2 ( q 2 ) q 2 2 b , q_{2}^{*}=\frac{a-b\cdot\frac{a+\frac{\partial C_{2}(q_{2})}{\partial q_{2}}-2% \cdot\frac{\partial C_{1}(q_{1})}{\partial q_{1}}}{2b}-\frac{\partial C_{2}(q_% {2})}{\partial q_{2}}}{2b},
  42. q 2 * = a - 3 C 2 ( q 2 ) q 2 + 2 C 1 ( q 1 ) q 1 4 b . \Rightarrow q_{2}^{*}=\frac{a-3\cdot\frac{\partial C_{2}(q_{2})}{\partial q_{2% }}+2\cdot\frac{\partial C_{1}(q_{1})}{\partial q_{1}}}{4b}.
  43. ( q 1 * , q 2 * ) (q_{1}^{*},q_{2}^{*})
  44. q 2 q_{2}
  45. q 2 * ( 5000 - q 1 - q 2 - c 2 ) q_{2}*(5000-q_{1}-q_{2}-c_{2})
  46. q 2 = 5000 - q 1 - c 2 2 q_{2}=\frac{5000-q_{1}-c_{2}}{2}
  47. q 2 q_{2}
  48. q 1 q_{1}
  49. q 1 * ( 5000 - q 1 - q 2 - c 1 ) q_{1}*(5000-q_{1}-q_{2}-c_{1})
  50. q 2 q_{2}
  51. q 1 * ( 5000 - q 1 - 5000 - q 1 - c 2 2 - c 1 ) q_{1}*(5000-q_{1}-\frac{5000-q_{1}-c_{2}}{2}-c_{1})
  52. q 2 q_{2}
  53. q 1 = 5000 - 2 c 1 + c 2 2 q_{1}=\frac{5000-2c_{1}+c_{2}}{2}
  54. q 2 = 5000 + 2 c 1 - 3 c 2 4 q_{2}=\frac{5000+2c_{1}-3c_{2}}{4}
  55. c 1 = c 2 = 1000 c_{1}=c_{2}=1000
  56. ( 16 / 9 ) 10 6 (16/9)10^{6}

Stag_hunt.html

  1. a > b d > c a>b\geq d>c

Standard_Model_(mathematical_formulation).html

  1. S U ( 3 ) × S U ( 2 ) × SU(3) × SU(2) ×
  2. Q Q
  3. ψ ψ
  4. W 1 , W 2 , W 3 W_{1},W_{2},W_{3}
  5. B B
  6. φ φ
  7. \mathcal{L}
  8. S U ( 3 ) × S U ( 2 ) × U ( 1 ) SU(3) × SU(2) × U(1)
  9. B B
  10. φ φ
  11. S U ( 2 ) SU(2)
  12. W W
  13. φ φ
  14. G G
  15. ψ ψ
  16. ψ ψ
  17. ψ ψ
  18. P P
  19. P ψ = λ ψ Pψ=λψ
  20. λ λ
  21. P P
  22. ψ ψ
  23. ψ ψ
  24. P P
  25. φ φ
  26. ψ ψ
  27. μ μ
  28. B μ , W j μ B^{\mu},W_{j}^{\mu}
  29. G a μ G_{a}^{\mu}
  30. = γ μ B μ {\not}B=\gamma^{\mu}B_{\mu}
  31. ψ ψ
  32. ψ ψ
  33. ψ ν e , ψ ν μ \psi_{\nu_{\mathrm{e}}},\psi_{\nu_{\mu}}
  34. ψ ν τ \psi_{\nu_{\tau}}
  35. ψ ¯ \bar{\psi}
  36. ψ γ 0 \psi^{\dagger}\gamma^{0}
  37. \dagger
  38. ψ ψ
  39. n × 1 n× 1
  40. ψ ¯ \bar{\psi}
  41. 1 × n 1 ×n
  42. ψ ψ
  43. ψ L = 1 2 ( 1 - γ 5 ) ψ \psi^{L}=\frac{1}{2}(1-\gamma_{5})\psi
  44. ψ R = 1 2 ( 1 + γ 5 ) ψ \psi^{R}=\frac{1}{2}(1+\gamma_{5})\psi
  45. γ 5 \gamma_{5}
  46. U ( 1 ) U(1)
  47. ψ e L \psi^{L}_{\mathrm{e}}
  48. ψ e R \psi^{R}_{\mathrm{e}}
  49. W 1 , W 2 , W 3 W_{1},W_{2},W_{3}
  50. B B
  51. Z = cos θ W W 3 - sin θ W B Z=\cos\theta_{W}W_{3}-\sin\theta_{W}B
  52. A = sin θ W W 3 + cos θ W B A=\sin\theta_{W}W_{3}+\cos\theta_{W}B
  53. W ± = 1 2 ( W 1 i W 2 ) W^{\pm}=\frac{1}{\sqrt{2}}\left(W_{1}\mp iW_{2}\right)
  54. A A
  55. Z Z
  56. = 0 + I \mathcal{L}=\mathcal{L}_{0}+\mathcal{L}_{\mathrm{I}}
  57. ψ ψ
  58. ( i ∂̸ - m f c ) ψ f = 0 (i\hbar{\not}\partial-m_{f}c)\psi_{f}=0
  59. f f
  60. A A
  61. μ μ A ν = 0 \partial_{\mu}\partial^{\mu}A^{\nu}=0
  62. φ φ
  63. L L
  64. L L→∞
  65. F F
  66. F ( x ) = β 𝐩 r E 𝐩 - 1 2 ( a r ( 𝐩 ) u r ( 𝐩 ) e - i p x + b r ( 𝐩 ) v r ( 𝐩 ) e i p x ) F(x)=\beta\sum_{\mathbf{p}}\sum_{r}E_{\mathbf{p}}^{-\frac{1}{2}}\left(a_{r}(% \mathbf{p})u_{r}(\mathbf{p})e^{-\frac{ipx}{\hbar}}+b^{\dagger}_{r}(\mathbf{p})% v_{r}(\mathbf{p})e^{\frac{ipx}{\hbar}}\right)
  67. β β
  68. ψ f \psi_{f}
  69. m f c 2 / V \sqrt{m_{f}c^{2}/V}
  70. V = L 3 V=L^{3}
  71. c / 2 V \hbar c/\sqrt{2V}
  72. 𝐩 \mathbf{p}
  73. L L
  74. 2 π L ( n 1 , n 2 , n 3 ) \frac{2\pi\hbar}{L}(n_{1},n_{2},n_{3})
  75. n 1 , n 2 , n 3 n_{1},n_{2},n_{3}
  76. r r
  77. 1 1
  78. 2 2
  79. 1 1
  80. 3 3
  81. 𝐩 \mathbf{p}
  82. = m 2 c 4 + c 2 𝐩 2 =\sqrt{m^{2}c^{4}+c^{2}\mathbf{p}^{2}}
  83. m m
  84. b r ( 𝐩 ) b^{\dagger}_{r}(\mathbf{p})
  85. 𝐩 \mathbf{p}
  86. a a
  87. b b
  88. p = ( E 𝐩 / c , 𝐩 ) p=(E_{\mathbf{p}}/c,\mathbf{p})
  89. 𝐩 \mathbf{p}
  90. p x = p μ x μ px=p_{\mu}x^{\mu}
  91. L L→∞
  92. V V
  93. β β
  94. β β
  95. u r ( 𝐩 ) u_{r}(\mathbf{p})
  96. v r ( 𝐩 ) v_{r}(\mathbf{p})
  97. a r ( 𝐩 ) a^{\dagger}_{r}(\mathbf{p})
  98. a r ( 𝐩 ) a^{\dagger}_{r}(\mathbf{p})
  99. a r ( 𝐩 ) a^{\dagger}_{r}(\mathbf{p})
  100. 1 1
  101. 𝐩 \mathbf{p}
  102. \dagger
  103. a a
  104. b b
  105. u u
  106. v v
  107. u u
  108. v v
  109. a a
  110. b b
  111. S U ( 3 ) × S U ( 2 ) × U ( 1 ) SU(3) × SU(2) × U(1)
  112. i ψ ¯ γ μ μ ψ i\bar{\psi}\gamma^{\mu}\partial_{\mu}\psi
  113. ψ ψ
  114. F μ ν a = μ A ν a - ν A μ a + g f a b c A μ b A ν c F^{a}_{\mu\nu}=\partial_{\mu}A^{a}_{\nu}-\partial_{\nu}A^{a}_{\mu}+gf^{abc}A^{% b}_{\mu}A^{c}_{\nu}
  115. A A
  116. g g
  117. [ t a , t b ] = i f a b c t c , [t_{a},t_{b}]=if^{abc}t_{c},
  118. U ( 1 ) U(1)
  119. S U ( 2 ) SU(2)
  120. S U ( 3 ) SU(3)
  121. S U ( 3 ) × S U ( 2 ) × SU(3) × SU(2) ×
  122. G μ ν a G^{a}_{\mu\nu}
  123. a a
  124. 𝟖 \mathbf{8}
  125. g g
  126. W μ ν a W^{a}_{\mu\nu}
  127. S U ( 2 ) SU(2)
  128. a a
  129. 3 3
  130. g g
  131. W μ a W^{a}_{\mu}
  132. U ( 1 ) U(1)
  133. g g′
  134. kin = - 1 4 B μ ν B μ ν - 1 2 tr W μ ν W μ ν - 1 2 tr G μ ν G μ ν \mathcal{L}_{\rm{kin}}=-{1\over 4}B_{\mu\nu}B^{\mu\nu}-{1\over 2}\mathrm{tr}W_% {\mu\nu}W^{\mu\nu}-{1\over 2}\mathrm{tr}G_{\mu\nu}G^{\mu\nu}
  135. S U ( 2 ) SU(2)
  136. S U ( 3 ) SU(3)
  137. W W
  138. G G
  139. W W
  140. G G
  141. S U ( 2 ) SU(2)
  142. S U ( 3 ) SU(3)
  143. EW = ψ ψ ¯ γ μ ( i μ - g 1 2 Y W B μ - g 1 2 s y m b o l τ 𝐖 μ ) ψ \mathcal{L}_{\mathrm{EW}}=\sum_{\psi}\bar{\psi}\gamma^{\mu}\left(i\partial_{% \mu}-g^{\prime}{1\over 2}Y_{\mathrm{W}}B_{\mu}-g{1\over 2}symbol{\tau}\mathbf{% W}_{\mu}\right)\psi
  144. U ( 1 ) U(1)
  145. U ( 1 ) U(1)
  146. S U ( 2 ) SU(2)
  147. τ \mathbf{τ}
  148. S U ( 2 ) SU(2)
  149. [ - F o r m u l a E r r o r - ] [-FormulaError-]
  150. Q Q
  151. Q = T 3 + 1 2 Y W , Q=T_{3}+\tfrac{1}{2}Y_{W},
  152. 𝐣 μ = 1 2 ψ ¯ L γ \musymbol τ ψ L \mathbf{j}_{\mu}={1\over 2}\bar{\psi}_{L}\gamma_{\musymbol}{\tau}\psi_{L}
  153. j μ Y = 2 ( j μ e m - j μ 3 ) j_{\mu}^{Y}=2(j_{\mu}^{em}-j_{\mu}^{3})
  154. j μ e m j_{\mu}^{em}
  155. j μ 3 j_{\mu}^{3}
  156. ψ ψ
  157. - g 2 ( ν ¯ e e ¯ ) τ + γ μ ( W - ) μ ( ν e e ) = - g 2 ν ¯ e γ μ ( W - ) μ e -{g\over 2}(\bar{\nu}_{e}\;\bar{e})\tau^{+}\gamma_{\mu}(W^{-})^{\mu}\begin{% pmatrix}{\nu_{e}}\\ e\end{pmatrix}=-{g\over 2}\bar{\nu}_{e}\gamma_{\mu}(W^{-})^{\mu}e
  158. τ ± 1 2 ( τ 1 ± i τ 2 ) = ( 0 1 0 0 ) \tau^{\pm}\equiv{1\over 2}(\tau^{1}{\pm}i\tau^{2})=\begin{pmatrix}0&1\\ 0&0\end{pmatrix}
  159. U ( 1 ) U(1)
  160. S U ( 3 ) SU(3)
  161. QCD = i U ¯ ( μ - i g s G μ a T a ) γ μ U + i D ¯ ( μ - i g s G μ a T a ) γ μ D . \mathcal{L}_{\mathrm{QCD}}=i\overline{U}\left(\partial_{\mu}-ig_{s}G_{\mu}^{a}% T^{a}\right)\gamma^{\mu}U+i\overline{D}\left(\partial_{\mu}-ig_{s}G_{\mu}^{a}T% ^{a}\right)\gamma^{\mu}D.
  162. D D
  163. U U
  164. ψ ψ
  165. - m ψ ¯ ψ -m\bar{\psi}\psi
  166. ψ ψ
  167. - m ψ ¯ ψ = - m ( ψ ¯ L ψ R + ψ ¯ R ψ L ) -m\bar{\psi}\psi=-m(\bar{\psi}_{L}\psi_{R}+\bar{\psi}_{R}\psi_{L})
  168. ψ ¯ L ψ L \bar{\psi}_{L}\psi_{L}
  169. ψ ¯ R ψ R \bar{\psi}_{R}\psi_{R}
  170. S U ( 2 ) SU(2)
  171. ϕ = 1 2 ( ϕ + ϕ 0 ) , \phi=\frac{1}{\sqrt{2}}\begin{pmatrix}\phi^{+}\\ \phi^{0}\end{pmatrix},
  172. + +
  173. 0
  174. Q Q
  175. 1 1
  176. H = [ ( μ - i g W μ a t a - i g Y ϕ B μ ) ϕ ] 2 + μ 2 ϕ ϕ - λ ( ϕ ϕ ) 2 , \mathcal{L}_{H}=\left[\left(\partial_{\mu}-igW_{\mu}^{a}t^{a}-ig^{\prime}Y_{% \phi}B_{\mu}\right)\phi\right]^{2}+\mu^{2}\phi^{\dagger}\phi-\lambda(\phi^{% \dagger}\phi)^{2},
  177. λ > 0 λ>0
  178. ϕ 0 = v \langle\phi^{0}\rangle=v
  179. v v
  180. M W \displaystyle M_{W}
  181. Y U = U ¯ L G u U R ϕ 0 - D ¯ L G u U R ϕ - + U ¯ L G d D R ϕ + + D ¯ L G d D R ϕ 0 + h c \mathcal{L}_{YU}=\overline{U}_{L}G_{u}U_{R}\phi^{0}-\overline{D}_{L}G_{u}U_{R}% \phi^{-}+\overline{U}_{L}G_{d}D_{R}\phi^{+}+\overline{D}_{L}G_{d}D_{R}\phi^{0}% +hc
  182. 3 × 3 3 × 3
  183. i j ij
  184. i i
  185. j j
  186. Q = 0 Q=0
  187. - m 2 ( ν ¯ C ν + ν ¯ ν C ) -{m\over 2}\left(\overline{\nu}^{C}\nu+\overline{\nu}\nu^{C}\right)
  188. C C
  189. B B
  190. g g^{\prime}
  191. ( 𝟏 , 𝟏 , 0 ) (\mathbf{1},\mathbf{1},0)
  192. W W
  193. g w g_{w}
  194. ( 𝟏 , 𝟑 , 0 ) (\mathbf{1},\mathbf{3},0)
  195. G G
  196. g s g_{s}
  197. ( 𝟖 , 𝟏 , 0 ) (\mathbf{8},\mathbf{1},0)
  198. 1 / 2 {1}/{2}
  199. Q L Q_{L}
  200. 1 3 \textstyle\frac{1}{3}
  201. 0
  202. ( 𝟑 , 𝟐 , 1 3 ) (\mathbf{3},\mathbf{2},\textstyle\frac{1}{3})
  203. u L C u_{L}^{C}
  204. - 1 3 -\textstyle\frac{1}{3}
  205. 0
  206. ( 𝟑 ¯ , 𝟏 , - 4 3 ) (\bar{\mathbf{3}},\mathbf{1},-\textstyle\frac{4}{3})
  207. d L C d_{L}^{C}
  208. - 1 3 -\textstyle\frac{1}{3}
  209. 0
  210. ( 𝟑 ¯ , 𝟏 , 2 3 ) (\bar{\mathbf{3}},\mathbf{1},\textstyle\frac{2}{3})
  211. L L L_{L}
  212. 0
  213. 1 1
  214. ( 𝟏 , 𝟐 , - 1 ) (\mathbf{1},\mathbf{2},-1)
  215. e L C e_{L}^{C}
  216. 0
  217. - 1 -1
  218. ( 𝟏 , 𝟏 , 2 ) (\mathbf{1},\mathbf{1},2)
  219. H H
  220. ( 𝟏 , 𝟐 , 1 ) (\mathbf{1},\mathbf{2},1)
  221. - 1 -1
  222. - 1 / 2 -1/2
  223. - 1 -1
  224. 1 {1}
  225. + 1 +1
  226. 0
  227. + 2 +2
  228. 1 {1}
  229. 0
  230. + 1 / 2 +1/2
  231. - 1 -1
  232. 1 {1}
  233. 0
  234. 0
  235. 0
  236. 1 {1}
  237. + 2 / 3 +2/3
  238. + 1 / 2 +1/2
  239. + 1 / 3 +1/3
  240. 3 {3}
  241. - 2 / 3 -2/3
  242. 0
  243. - 4 / 3 -4/3
  244. 3 ¯ {\bar{3}}
  245. - 1 / 3 -1/3
  246. - 1 / 2 -1/2
  247. + 1 / 3 +1/3
  248. 3 {3}
  249. + 1 / 3 +1/3
  250. 0
  251. + 2 / 3 +2/3
  252. 3 ¯ {\bar{3}}
  253. - 1 -1
  254. - 1 / 2 -1/2
  255. - 1 -1
  256. 1 {1}
  257. + 1 +1
  258. 0
  259. + 2 +2
  260. 1 {1}
  261. 0
  262. + 1 / 2 +1/2
  263. - 1 -1
  264. 1 {1}
  265. 0
  266. 0
  267. 0
  268. 1 {1}
  269. + 2 / 3 +2/3
  270. + 1 / 2 +1/2
  271. + 1 / 3 +1/3
  272. 3 {3}
  273. - 2 / 3 -2/3
  274. 0
  275. - 4 / 3 -4/3
  276. 3 ¯ {\bar{3}}
  277. - 1 / 3 -1/3
  278. - 1 / 2 -1/2
  279. + 1 / 3 +1/3
  280. 3 {3}
  281. + 1 / 3 +1/3
  282. 0
  283. + 2 / 3 +2/3
  284. 3 ¯ {\bar{3}}
  285. - 1 -1
  286. - 1 / 2 -1/2
  287. - 1 -1
  288. 1 {1}
  289. + 1 +1
  290. 0
  291. + 2 +2
  292. 1 {1}
  293. 0
  294. + 1 / 2 +1/2
  295. - 1 -1
  296. 1 {1}
  297. 0
  298. 0
  299. 0
  300. 1 {1}
  301. + 2 / 3 +2/3
  302. + 1 / 2 +1/2
  303. + 1 / 3 +1/3
  304. 3 {3}
  305. - 2 / 3 -2/3
  306. 0
  307. - 4 / 3 -4/3
  308. 3 ¯ {\bar{3}}
  309. - 1 / 3 -1/3
  310. - 1 / 2 -1/2
  311. + 1 / 3 +1/3
  312. 3 {3}
  313. + 1 / 3 +1/3
  314. 0
  315. + 2 / 3 +2/3
  316. 3 ¯ {\bar{3}}
  317. M S ¯ \overline{MS}
  318. M S ¯ \overline{MS}
  319. M S ¯ \overline{MS}
  320. M S ¯ \overline{MS}
  321. M S ¯ \overline{MS}
  322. M S ¯ \overline{MS}
  323. M S ¯ \overline{MS}
  324. M S ¯ \overline{MS}
  325. ψ q ( x ) e i α / 3 ψ q \psi\text{q}(x)\to e^{i\alpha/3}\psi\text{q}
  326. E L e i β E L and ( e R ) c e i β ( e R ) c E_{L}\to e^{i\beta}E_{L}\,\text{ and }(e_{R})^{c}\to e^{i\beta}(e_{R})^{c}
  327. M L e i β M L and ( μ R ) c e i β ( μ R ) c M_{L}\to e^{i\beta}M_{L}\,\text{ and }(\mu_{R})^{c}\to e^{i\beta}(\mu_{R})^{c}
  328. T L e i β T L and ( τ R ) c e i β ( τ R ) c T_{L}\to e^{i\beta}T_{L}\,\text{ and }(\tau_{R})^{c}\to e^{i\beta}(\tau_{R})^{c}
  329. ( μ R ) c , ( τ R ) c (\mu_{R})^{c},(\tau_{R})^{c}
  330. ( e R ) c (e_{R})^{c}
  331. 1 3 {}_{\frac{1}{3}}
  332. - 1 3 {}_{-\frac{1}{3}}
  333. j ± = j 1 ± i j 2 j^{\pm}=j^{1}\pm ij^{2}
  334. j μ + = U ¯ i L γ μ D i L + ν ¯ i L γ μ l i L . j^{+}_{\mu}=\overline{U}_{iL}\gamma_{\mu}D_{iL}+\overline{\nu}_{iL}\gamma_{\mu% }l_{iL}.
  335. C C = g 2 ( j μ + W - μ + j μ - W + μ ) . \mathcal{L}_{CC}=\frac{g}{\sqrt{2}}(j_{\mu}^{+}W^{-\mu}+j_{\mu}^{-}W^{+\mu}).
  336. W 3 W^{3}
  337. j μ 3 = 1 2 ( U ¯ i L γ μ U i L - D ¯ i L γ μ D i L + ν ¯ i L γ μ ν i L - l ¯ i L γ μ l i L ) j_{\mu}^{3}=\frac{1}{2}(\overline{U}_{iL}\gamma_{\mu}U_{iL}-\overline{D}_{iL}% \gamma_{\mu}D_{iL}+\overline{\nu}_{iL}\gamma_{\mu}\nu_{iL}-\overline{l}_{iL}% \gamma_{\mu}l_{iL})
  338. j μ e m = 2 3 U ¯ i γ μ U i - 1 3 D ¯ i γ μ D i - l ¯ i γ μ l i . j_{\mu}^{em}=\frac{2}{3}\overline{U}_{i}\gamma_{\mu}U_{i}-\frac{1}{3}\overline% {D}_{i}\gamma_{\mu}D_{i}-\overline{l}_{i}\gamma_{\mu}l_{i}.
  339. N C = e j μ e m A μ + g cos θ W ( J μ 3 - sin 2 θ W J μ e m ) Z μ . \mathcal{L}_{NC}=ej_{\mu}^{em}A^{\mu}+\frac{g}{\cos\theta_{W}}(J_{\mu}^{3}-% \sin^{2}\theta_{W}J_{\mu}^{em})Z^{\mu}.

Standard_rate_turn.html

  1. tan ( b a n k ) = T A S 2 / r g \tan(bank)=\frac{TAS^{2}/r}{g}
  2. r r
  3. g g
  4. tan ( b a n k ) = T A S ( k t ) 364 \tan(bank)=\frac{TAS(kt)}{364}
  5. A n g l e o f B a n k T A S ( k t ) 10 + 7 Angle\ of\ Bank\approx\frac{TAS(kt)}{10}+7
  6. R a d i u s o f t u r n i n N M = T A S ( k t ) r a t e o f t u r n ( i n d e g r e e s / s ) × 20 × π Radius\ of\ turn\ in\ NM=\frac{TAS(kt)}{rate\ of\ turn\ (in\ degrees/s)\times 2% 0\times\pi}
  7. R a d i u s o f t u r n i n f e e t = v e l o c i t y 2 11.29 × tan ( b a n k ) Radius\ of\ turn\ in\ feet=\frac{velocity^{2}}{11.29\times\tan(bank)}
  8. 11.29 = 9.8 × 3600 × 3600 1852 × 6076.12 11.29=\frac{9.8\times 3600\times 3600}{1852\times 6076.12}
  9. 9.8 = g r a v i t y i n m e t r e s p e r s e c o n d p e r s e c o n d 9.8=gravity\ in\ metres\ per\ second\ per\ second
  10. 3600 = s e c o n d s p e r h o u r 3600=seconds\ per\ hour
  11. 1852 = m e t r e s p e r n a u t i c a l m i l e 1852=metres\ per\ nautical\ mile
  12. 6076.12 = f e e t p e r n a u t i c a l m i l e 6076.12=feet\ per\ nautical\ mile

State_observer.html

  1. x ( k + 1 ) = A x ( k ) + B u ( k ) x(k+1)=Ax(k)+Bu(k)
  2. y ( k ) = C x ( k ) + D u ( k ) y(k)=Cx(k)+Du(k)
  3. k k
  4. x ( k ) x(k)
  5. u ( k ) u(k)
  6. y ( k ) y(k)
  7. y ( k ) y(k)
  8. L L
  9. x ^ ( k ) \hat{x}(k)
  10. y ^ ( k ) \hat{y}(k)
  11. x ^ ( k + 1 ) = A x ^ ( k ) + L [ y ( k ) - y ^ ( k ) ] + B u ( k ) \hat{x}(k+1)=A\hat{x}(k)+L\left[y(k)-\hat{y}(k)\right]+Bu(k)
  12. y ^ ( k ) = C x ^ ( k ) + D u ( k ) \hat{y}(k)=C\hat{x}(k)+Du(k)
  13. e ( k ) = x ^ ( k ) - x ( k ) e(k)=\hat{x}(k)-x(k)
  14. k k\rightarrow\infty
  15. e ( k + 1 ) = ( A - L C ) e ( k ) e(k+1)=(A-LC)e(k)
  16. A - L C A-LC
  17. K K
  18. u ( k ) = - K x ^ ( k ) u(k)=-K\hat{x}(k)
  19. x ^ ( k + 1 ) = A x ^ ( k ) + L ( y ( k ) - y ^ ( k ) ) - B K x ^ ( k ) \hat{x}(k+1)=A\hat{x}(k)+L\left(y(k)-\hat{y}(k)\right)-BK\hat{x}(k)
  20. y ^ ( k ) = C x ^ ( k ) - D K x ^ ( k ) \hat{y}(k)=C\hat{x}(k)-DK\hat{x}(k)
  21. x ^ ( k + 1 ) = ( A - B K ) x ^ ( k ) + L ( y ( k ) - y ^ ( k ) ) \hat{x}(k+1)=\left(A-BK\right)\hat{x}(k)+L\left(y(k)-\hat{y}(k)\right)
  22. y ^ ( k ) = ( C - D K ) x ^ ( k ) \hat{y}(k)=\left(C-DK\right)\hat{x}(k)
  23. K K
  24. L L
  25. A - L C A-LC
  26. A - B K A-BK
  27. L L
  28. A - L C A-LC
  29. x ˙ = A x + B u , \dot{x}=Ax+Bu,
  30. y = C x , y=Cx,
  31. x n , u m , y r x\in\mathbb{R}^{n},u\in\mathbb{R}^{m},y\in\mathbb{R}^{r}
  32. x ^ ˙ = A x ^ + B u + L ( y - C x ^ ) \dot{\hat{x}}=A\hat{x}+Bu+L\left(y-C\hat{x}\right)
  33. e = x - x ^ e=x-\hat{x}
  34. e ˙ = ( A - L C ) e \dot{e}=(A-LC)e
  35. A - L C A-LC
  36. L L
  37. [ A , C ] [A,C]
  38. e ( t ) 0 e(t)\rightarrow 0
  39. t t\rightarrow\infty
  40. L L
  41. x ˙ = f ( x ) \dot{x}=f(x)
  42. x n x\in\mathbb{R}^{n}
  43. y y\in\mathbb{R}
  44. y = h ( x ) . y=h(x).
  45. x ˙ = f ( x ) + B ( x ) u , \dot{x}=f(x)+B(x)u,
  46. y = h ( x ) , y=h(x),
  47. z = Φ ( x ) z=\Phi(x)
  48. z ˙ = A z + ϕ ( y ) , \dot{z}=Az+\phi(y),
  49. y = C z . y=Cz.
  50. z ^ ˙ = A z ^ + ϕ ( y ) - L ( C z ^ - y ) \dot{\hat{z}}=A\hat{z}+\phi(y)-L\left(C\hat{z}-y\right)
  51. e = z ^ - z e=\hat{z}-z
  52. e ˙ = ( A - L C ) e \dot{e}=(A-LC)e
  53. z = Φ ( x ) z=\Phi(x)
  54. z ˙ = A ( u ( t ) ) z + ϕ ( y , u ( t ) ) , \dot{z}=A(u(t))z+\phi(y,u(t)),
  55. y = C z , y=Cz,
  56. z ^ ˙ = A ( u ( t ) ) z ^ + ϕ ( y , u ( t ) ) - L ( t ) ( C z ^ - y ) \dot{\hat{z}}=A(u(t))\hat{z}+\phi(y,u(t))-L(t)\left(C\hat{z}-y\right)
  57. L ( t ) L(t)
  58. x ^ \hat{x}
  59. x ^ ˙ = [ H ( x ^ ) x ] - 1 M ( x ^ ) sgn ( V ( t ) - H ( x ^ ) ) \dot{\hat{x}}=\left[\frac{\partial H(\hat{x})}{\partial x}\right]^{-1}M(\hat{x% })\,\operatorname{sgn}(V(t)-H(\hat{x}))
  60. sgn ( ) \operatorname{sgn}(\mathord{\cdot})
  61. n n
  62. sgn ( z ) = [ sgn ( z 1 ) sgn ( z 2 ) sgn ( z i ) sgn ( z n ) ] \operatorname{sgn}(z)=\begin{bmatrix}\operatorname{sgn}(z_{1})\\ \operatorname{sgn}(z_{2})\\ \vdots\\ \operatorname{sgn}(z_{i})\\ \vdots\\ \operatorname{sgn}(z_{n})\end{bmatrix}
  63. z n z\in\mathbb{R}^{n}
  64. H ( x ) H(x)
  65. h ( x ) h(x)
  66. H ( x ) [ h 1 ( x ) h 2 ( x ) h 3 ( x ) h n ( x ) ] [ h ( x ) L f h ( x ) L f 2 h ( x ) L f n - 1 h ( x ) ] H(x)\triangleq\begin{bmatrix}h_{1}(x)\\ h_{2}(x)\\ h_{3}(x)\\ \vdots\\ h_{n}(x)\end{bmatrix}\triangleq\begin{bmatrix}h(x)\\ L_{f}h(x)\\ L_{f}^{2}h(x)\\ \vdots\\ L_{f}^{n-1}h(x)\end{bmatrix}
  67. L f i h L^{i}_{f}h
  68. h h
  69. f f
  70. x x
  71. H ( x ( t ) ) H(x(t))
  72. y ( t ) = h ( x ( t ) ) y(t)=h(x(t))
  73. n - 1 n-1
  74. H ( x ) H(x)
  75. H ( x ) H(x)
  76. M ( x ^ ) M(\hat{x})
  77. M ( x ^ ) diag ( m 1 ( x ^ ) , m 2 ( x ^ ) , , m n ( x ^ ) ) = [ m 1 ( x ^ ) m 2 ( x ^ ) m i ( x ^ ) m n ( x ^ ) ] M(\hat{x})\triangleq\operatorname{diag}(m_{1}(\hat{x}),m_{2}(\hat{x}),\ldots,m% _{n}(\hat{x}))=\begin{bmatrix}m_{1}(\hat{x})&&&&&\\ &m_{2}(\hat{x})&&&&\\ &&\ddots&&&\\ &&&m_{i}(\hat{x})&&\\ &&&&\ddots&\\ &&&&&m_{n}(\hat{x})\end{bmatrix}
  78. i { 1 , 2 , , n } i\in\{1,2,\dots,n\}
  79. m i ( x ^ ) > 0 m_{i}(\hat{x})>0
  80. V ( t ) V(t)
  81. V ( t ) [ v 1 ( t ) v 2 ( t ) v 3 ( t ) v i ( t ) v n ( t ) ] [ y ( t ) { m 1 ( x ^ ) sgn ( v 1 ( t ) - h 1 ( x ^ ( t ) ) ) } eq { m 2 ( x ^ ) sgn ( v 2 ( t ) - h 2 ( x ^ ( t ) ) ) } eq { m i - 1 ( x ^ ) sgn ( v i - 1 ( t ) - h i - 1 ( x ^ ( t ) ) ) } eq { m n - 1 ( x ^ ) sgn ( v n - 1 ( t ) - h n - 1 ( x ^ ( t ) ) ) } eq ] V(t)\triangleq\begin{bmatrix}v_{1}(t)\\ v_{2}(t)\\ v_{3}(t)\\ \vdots\\ v_{i}(t)\\ \vdots\\ v_{n}(t)\end{bmatrix}\triangleq\begin{bmatrix}y(t)\\ \{m_{1}(\hat{x})\operatorname{sgn}(v_{1}(t)-h_{1}(\hat{x}(t)))\}_{\,\text{eq}}% \\ \{m_{2}(\hat{x})\operatorname{sgn}(v_{2}(t)-h_{2}(\hat{x}(t)))\}_{\,\text{eq}}% \\ \vdots\\ \{m_{i-1}(\hat{x})\operatorname{sgn}(v_{i-1}(t)-h_{i-1}(\hat{x}(t)))\}_{\,% \text{eq}}\\ \vdots\\ \{m_{n-1}(\hat{x})\operatorname{sgn}(v_{n-1}(t)-h_{n-1}(\hat{x}(t)))\}_{\,% \text{eq}}\end{bmatrix}
  82. sgn ( ) \operatorname{sgn}(\mathord{\cdot})
  83. { } eq \{\ldots\}_{\,\text{eq}}
  84. sgn ( v i ( t ) - h i ( x ^ ( t ) ) ) \operatorname{sgn}(v_{i}(t)\!-\!h_{i}(\hat{x}(t)))
  85. e = H ( x ) - H ( x ^ ) e=H(x)-H(\hat{x})
  86. { e ˙ = d d t H ( x ) - d d t H ( x ^ ) = d d t H ( x ) - M ( x ^ ) sgn ( V ( t ) - H ( x ^ ( t ) ) ) , \begin{cases}\dot{e}=\frac{\operatorname{d}}{\operatorname{d}t}H(x)-\frac{% \operatorname{d}}{\operatorname{d}t}H(\hat{x})\\ =\frac{\operatorname{d}}{\operatorname{d}t}H(x)-M(\hat{x})\,\operatorname{sgn}% (V(t)-H(\hat{x}(t))),\end{cases}
  87. { [ e ˙ 1 e ˙ 2 e ˙ i e ˙ n - 1 e ˙ n ] = [ h ˙ 1 ( x ) h ˙ 2 ( x ) h ˙ i ( x ) h ˙ n - 1 ( x ) h ˙ n ( x ) ] d d t H ( x ) - M ( x ^ ) sgn ( V ( t ) - H ( x ^ ( t ) ) ) d d t H ( x ^ ) = [ h 2 ( x ) h 3 ( x ) h i + 1 ( x ) h n ( x ) L f n h ( x ) ] - [ m 1 sgn ( v 1 ( t ) - h 1 ( x ^ ( t ) ) ) m 2 sgn ( v 2 ( t ) - h 2 ( x ^ ( t ) ) ) m i sgn ( v i ( t ) - h i ( x ^ ( t ) ) ) m n - 1 sgn ( v n - 1 ( t ) - h n - 1 ( x ^ ( t ) ) ) m n sgn ( v n ( t ) - h n ( x ^ ( t ) ) ) ] = [ h 2 ( x ) - m 1 ( x ^ ) sgn ( v 1 ( t ) v 1 ( t ) = y ( t ) = h 1 ( x ) - h 1 ( x ^ ( t ) ) e 1 ) h 3 ( x ) - m 2 ( x ^ ) sgn ( v 2 ( t ) - h 2 ( x ^ ( t ) ) ) h i + 1 ( x ) - m i ( x ^ ) sgn ( v i ( t ) - h i ( x ^ ( t ) ) ) h n ( x ) - m n - 1 ( x ^ ) sgn ( v n - 1 ( t ) - h n - 1 ( x ^ ( t ) ) ) L f n h ( x ) - m n ( x ^ ) sgn ( v n ( t ) - h n ( x ^ ( t ) ) ) ] . \begin{cases}\begin{bmatrix}\dot{e}_{1}\\ \dot{e}_{2}\\ \vdots\\ \dot{e}_{i}\\ \vdots\\ \dot{e}_{n-1}\\ \dot{e}_{n}\end{bmatrix}=\mathord{\overbrace{\begin{bmatrix}\dot{h}_{1}(x)\\ \dot{h}_{2}(x)\\ \vdots\\ \dot{h}_{i}(x)\\ \vdots\\ \dot{h}_{n-1}(x)\\ \dot{h}_{n}(x)\end{bmatrix}}^{\tfrac{\operatorname{d}}{\operatorname{d}t}H(x)}% }-\mathord{\overbrace{M(\hat{x})\,\operatorname{sgn}(V(t)-H(\hat{x}(t)))}^{% \tfrac{\operatorname{d}}{\operatorname{d}t}H(\hat{x})}}=\begin{bmatrix}h_{2}(x% )\\ h_{3}(x)\\ \vdots\\ h_{i+1}(x)\\ \vdots\\ h_{n}(x)\\ L_{f}^{n}h(x)\end{bmatrix}-\begin{bmatrix}m_{1}\operatorname{sgn}(v_{1}(t)-h_{% 1}(\hat{x}(t)))\\ m_{2}\operatorname{sgn}(v_{2}(t)-h_{2}(\hat{x}(t)))\\ \vdots\\ m_{i}\operatorname{sgn}(v_{i}(t)-h_{i}(\hat{x}(t)))\\ \vdots\\ m_{n-1}\operatorname{sgn}(v_{n-1}(t)-h_{n-1}(\hat{x}(t)))\\ m_{n}\operatorname{sgn}(v_{n}(t)-h_{n}(\hat{x}(t)))\end{bmatrix}\\ =\begin{bmatrix}h_{2}(x)-m_{1}(\hat{x})\operatorname{sgn}(\mathord{\overbrace{% \mathord{\overbrace{v_{1}(t)}^{v_{1}(t)=y(t)=h_{1}(x)}}-h_{1}(\hat{x}(t))}^{e_% {1}}})\\ h_{3}(x)-m_{2}(\hat{x})\operatorname{sgn}(v_{2}(t)-h_{2}(\hat{x}(t)))\\ \vdots\\ h_{i+1}(x)-m_{i}(\hat{x})\operatorname{sgn}(v_{i}(t)-h_{i}(\hat{x}(t)))\\ \vdots\\ h_{n}(x)-m_{n-1}(\hat{x})\operatorname{sgn}(v_{n-1}(t)-h_{n-1}(\hat{x}(t)))\\ L_{f}^{n}h(x)-m_{n}(\hat{x})\operatorname{sgn}(v_{n}(t)-h_{n}(\hat{x}(t)))\end% {bmatrix}.\end{cases}
  88. m 1 ( x ^ ) | h 2 ( x ( t ) ) | m_{1}(\hat{x})\geq|h_{2}(x(t))|
  89. e ˙ 1 = h 2 ( x ^ ) - m 1 ( x ^ ) sgn ( e 1 ) \dot{e}_{1}=h_{2}(\hat{x})-m_{1}(\hat{x})\operatorname{sgn}(e_{1})
  90. e 1 = 0 e_{1}=0
  91. e 1 = 0 e_{1}=0
  92. v 2 ( t ) = { m 1 ( x ^ ) sgn ( e 1 ) } eq v_{2}(t)=\{m_{1}(\hat{x})\operatorname{sgn}(e_{1})\}_{\,\text{eq}}
  93. h 2 ( x ) h_{2}(x)
  94. v 2 ( t ) - h 2 ( x ^ ) = h 2 ( x ) - h 2 ( x ^ ) = e 2 v_{2}(t)-h_{2}(\hat{x})=h_{2}(x)-h_{2}(\hat{x})=e_{2}
  95. m 2 ( x ^ ) | h 3 ( x ( t ) ) | m_{2}(\hat{x})\geq|h_{3}(x(t))|
  96. e ˙ 2 = h 3 ( x ^ ) - m 2 ( x ^ ) sgn ( e 2 ) \dot{e}_{2}=h_{3}(\hat{x})-m_{2}(\hat{x})\operatorname{sgn}(e_{2})
  97. e 2 = 0 e_{2}=0
  98. e i = 0 e_{i}=0
  99. v i + 1 ( t ) = { } eq v_{i+1}(t)=\{\ldots\}_{\,\text{eq}}
  100. h i + 1 ( x ) h_{i+1}(x)
  101. m i + 1 ( x ^ ) | h i + 2 ( x ( t ) ) | m_{i+1}(\hat{x})\geq|h_{i+2}(x(t))|
  102. ( i + 1 ) (i+1)
  103. e ˙ i + 1 = h i + 2 ( x ^ ) - m i + 1 ( x ^ ) sgn ( e i + 1 ) \dot{e}_{i+1}=h_{i+2}(\hat{x})-m_{i+1}(\hat{x})\operatorname{sgn}(e_{i+1})
  104. e i + 1 = 0 e_{i+1}=0
  105. m i m_{i}
  106. m i m_{i}
  107. | h i ( x ( 0 ) ) | |h_{i}(x(0))|
  108. H : n n H:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}
  109. H ( x ) x B ( x ) \frac{\partial H(x)}{\partial x}B(x)
  110. x ^ ˙ = [ H ( x ^ ) x ] - 1 M ( x ^ ) sgn ( V ( t ) - H ( x ^ ) ) + B ( x ^ ) u . \dot{\hat{x}}=\left[\frac{\partial H(\hat{x})}{\partial x}\right]^{-1}M(\hat{x% })\operatorname{sgn}(V(t)-H(\hat{x}))+B(\hat{x})u.
  111. L L
  112. x ^ U ( k ) \hat{x}_{U}(k)
  113. e ( k ) = x ^ U ( k ) - x ( k ) e(k)=\hat{x}_{U}(k)-x(k)
  114. k k\rightarrow\infty
  115. x ^ L ( k ) \hat{x}_{L}(k)
  116. e ( k ) = x ^ L ( k ) - x ( k ) e(k)=\hat{x}_{L}(k)-x(k)
  117. x ^ U ( k ) x ( k ) x ^ L ( k ) \hat{x}_{U}(k)\geq x(k)\geq\hat{x}_{L}(k)

State_variable_filter.html

  1. R f 1 = R f 2 R_{f1}=R_{f2}
  2. C 1 = C 2 C_{1}=C_{2}
  3. R 1 = R 2 R_{1}=R_{2}
  4. F 0 = 1 2 π R f 1 C 1 F_{0}=\frac{1}{2\pi R_{f1}C_{1}}
  5. Q = ( 1 + R 4 R q ) ( 1 2 + R 1 R g ) Q=\left(1+\frac{R_{4}}{R_{q}}\right)\left(\frac{1}{2+\frac{R_{1}}{R_{g}}}\right)
  6. A H P = A L P = R 1 R G A_{HP}=A_{LP}=\frac{R_{1}}{R_{G}}

Static_spacetime.html

  1. K K
  2. × \times
  3. g [ ( t , x ) ] = - β ( x ) d t 2 + g S [ x ] g[(t,x)]=-\beta(x)dt^{2}+g_{S}[x]
  4. g S g_{S}
  5. β \beta
  6. K K
  7. t \partial_{t}
  8. K K
  9. λ \lambda
  10. λ = g ( K , K ) \lambda=g(K,K)
  11. λ \lambda
  12. g S g_{S}
  13. λ = - β ( x ) \lambda=-\beta(x)
  14. R μ ν = 0 R_{\mu\nu}=0

Stationary_phase_approximation.html

  1. I ( k ) = g ( x ) e i k f ( x ) d x I(k)=\int g(x)e^{ikf(x)}\,dx
  2. f ( x , t ) = 1 2 π F ( ω ) e i [ k ( ω ) x - ω t ] d ω f(x,t)=\frac{1}{2\pi}\int_{\mathbb{R}}F(\omega)e^{i[k(\omega)x-\omega t]}\,d\omega
  3. d d ω ( k ( ω ) x - ω t ) = 0 \frac{d}{d\omega}\mathopen{}\left(k(\omega)x-\omega t\right)\mathclose{}=0
  4. d k d ω = t x \frac{dk}{d\omega}=\frac{t}{x}
  5. ϕ = [ k ( ω 0 ) x - ω 0 t ] + 1 2 x k ′′ ( ω 0 ) ( ω - ω 0 ) 2 + \phi=\left[k(\omega_{0})x-\omega_{0}t\right]+\frac{1}{2}xk^{\prime\prime}(% \omega_{0})(\omega-\omega_{0})^{2}+\cdots
  6. f ( x , t ) 1 2 π 2 Re { e i [ k ( ω 0 ) x - ω 0 t ] | F ( ω 0 ) | e 1 2 i x k ′′ ( ω 0 ) ( ω - ω 0 ) 2 d ω } f(x,t)\approx\frac{1}{2\pi}\cdot 2\operatorname{Re}\left\{e^{i\left[k(\omega_{% 0})x-\omega_{0}t\right]}\left|F(\omega_{0})\right|\int_{\mathbb{R}}e^{\frac{1}% {2}ixk^{\prime\prime}(\omega_{0})(\omega-\omega_{0})^{2}}\,d\omega\right\}
  7. f ( x , t ) | F ( ω 0 ) | π 2 π x | k ′′ ( ω 0 ) | cos [ k ( ω 0 ) x - ω 0 t ± π 4 ] f(x,t)\approx\frac{\left|F(\omega_{0})\right|}{\pi}\sqrt{\frac{2\pi}{x\left|k^% {\prime\prime}(\omega_{0})\right|}}\cos\left[k(\omega_{0})x-\omega_{0}t\pm% \frac{\pi}{4}\right]
  8. ( x 1 2 + x 2 2 + + x j 2 ) - ( x j + 1 2 + x j + 2 2 + + x n 2 ) (x_{1}^{2}+x_{2}^{2}+\cdots+x_{j}^{2})-(x_{j+1}^{2}+x_{j+2}^{2}+\cdots+x_{n}^{% 2})
  9. g ( x ) = i h ( x i ) g(x)=\prod_{i}h(x_{i})
  10. J ( k ) = h ( x ) e i k f ( x ) d x J(k)=\int h(x)e^{ikf(x)}\,dx
  11. - 1 1 e i k x 2 d x = π k e i π / 4 + 𝒪 ( 1 k ) \int_{-1}^{1}e^{ikx^{2}}\,dx=\sqrt{\frac{\pi}{k}}e^{i\pi/4}+\mathcal{O}% \mathopen{}\left(\frac{1}{k}\right)\mathclose{}
  12. 2 π k f ′′ ( 0 ) \sqrt{\frac{2\pi}{kf^{\prime\prime}(0)}}

Stationary_set.html

  1. κ \kappa\,
  2. S κ , S\subseteq\kappa\,,
  3. S S\,
  4. κ , \kappa\,,
  5. S S\,
  6. S S\,
  7. C C\,
  8. S C S\cap C\,
  9. D D\,
  10. C D C\cap D\,
  11. ( S C ) D = S ( C D ) (S\cap C)\cap D=S\cap(C\cap D)\,
  12. ( S C ) (S\cap C)\,
  13. κ \kappa
  14. S κ S\subset\kappa
  15. κ \kappa
  16. κ S \kappa\setminus S
  17. κ \kappa
  18. κ \kappa
  19. ω = 0 \omega=\aleph_{0}
  20. κ \kappa
  21. κ \kappa
  22. κ \kappa
  23. S κ S\subset\kappa
  24. S S
  25. κ \kappa
  26. κ \kappa
  27. β \beta
  28. ω 1 \omega_{1}
  29. β \beta
  30. [ X ] λ [X]^{\lambda}
  31. λ \lambda
  32. X X
  33. | X | λ |X|\geq\lambda
  34. [ X ] λ [X]^{\lambda}
  35. X X
  36. λ \lambda
  37. [ X ] λ = { Y X : | Y | = λ } [X]^{\lambda}=\{Y\subset X:|Y|=\lambda\}
  38. S [ X ] λ S\subset[X]^{\lambda}
  39. [ X ] λ [X]^{\lambda}
  40. \subset
  41. λ \lambda
  42. X = ω 1 X=\omega_{1}
  43. λ = 0 \lambda=\aleph_{0}
  44. S [ ω 1 ] ω S\subset[\omega_{1}]^{\omega}
  45. S ω 1 S\cap\omega_{1}
  46. ω 1 \omega_{1}
  47. X X
  48. C 𝒫 ( X ) C\subset{\mathcal{P}}(X)
  49. F : [ X ] < ω X F:[X]^{<\omega}\to X
  50. C = { z : F [ [ z ] < ω ] z } C=\{z:F[[z]^{<\omega}]\subset z\}
  51. [ y ] < ω [y]^{<\omega}
  52. y y
  53. S 𝒫 ( X ) S\subset{\mathcal{P}}(X)
  54. 𝒫 ( X ) {\mathcal{P}}(X)
  55. 𝒫 ( X ) {\mathcal{P}}(X)
  56. M M
  57. X X
  58. F F
  59. M M
  60. S S
  61. M M
  62. S 𝒫 ( X ) S\subset{\mathcal{P}}(X)
  63. M M
  64. M M
  65. S S

Statistical_classification.html

  1. score ( 𝐗 i , k ) = s y m b o l β k 𝐗 i , \operatorname{score}(\mathbf{X}_{i},k)=symbol\beta_{k}\cdot\mathbf{X}_{i},

Stative_verb.html

  1. λ ( x ) : [ STATE x ] \lambda(x):\ [\operatorname{STATE}\ x]

Steady_state.html

  1. p t = 0 \frac{\partial p}{\partial t}=0

Steering_law.html

  1. T = a + b C d s W ( s ) T=a+b\int_{C}\frac{ds}{W(s)}
  2. T = a + b A W T=a+b\frac{A}{W}
  3. d s d T = W ( s ) b \frac{ds}{dT}=\frac{W(s)}{b}
  4. T goal = b log 2 ( A W + 1 ) T\text{goal}=b\log_{2}\left(\frac{A}{W}+1\right)
  5. = lim N i = 1 N b log 2 ( A / N W + 1 ) =\lim_{N\to\infty}\sum_{i=1}^{N}b\log_{2}\left(\frac{A/N}{W}+1\right)
  6. = lim N N b log 2 ( A N W + 1 ) =\lim_{N\to\infty}Nb\log_{2}\left(\frac{A}{NW}+1\right)
  7. = b lim N log 2 ( A N W + 1 ) 1 / N =b\lim_{N\to\infty}\frac{\log_{2}\left(\frac{A}{NW}+1\right)}{1/N}
  8. = b lim N 1 ( A N W + 1 ) A W ( - 1 / N 2 ) - 1 / N 2 =b\lim_{N\to\infty}\frac{\frac{1}{\left(\frac{A}{NW}+1\right)}\frac{A}{W}(-1/N% ^{2})}{-1/N^{2}}
  9. = b A W lim N 1 ( A N W + 1 ) =b\frac{A}{W}\lim_{N\to\infty}\frac{1}{\left(\frac{A}{NW}+1\right)}
  10. = b A W =b\frac{A}{W}
  11. = lim N i = 1 N b s i + 1 - s i W ( s i ) =\lim_{N\to\infty}\sum_{i=1}^{N}b\frac{s_{i+1}-s_{i}}{W(s_{i})}
  12. = b 0 A d s W ( s ) =b\int_{0}^{A}\frac{ds}{W(s)}
  13. T = a + b ( A / W ) 2 + ( A / t ) 2 . T=a+b\sqrt{(A/W)^{2}+(A/t)^{2}}.

Stein's_example.html

  1. 𝐗 N ( s y m b o l θ , 1 ) . {\mathbf{X}}\sim N({symbol\theta},1).\,
  2. s y m b o l θ ^ = 𝐗 . \hat{symbol\theta}={\mathbf{X}}.\,
  3. E [ s y m b o l θ - s y m b o l θ ^ 2 ] . \operatorname{E}\left[\|{symbol\theta}-\hat{symbol\theta}\|^{2}\right].
  4. s y m b o l θ {symbol\theta}
  5. s y m b o l θ ^ 1 \hat{symbol\theta}_{1}
  6. s y m b o l θ ^ 2 \hat{symbol\theta}_{2}
  7. s y m b o l θ {symbol\theta}
  8. s y m b o l θ ^ 1 \hat{symbol\theta}_{1}
  9. s y m b o l θ ^ 2 \hat{symbol\theta}_{2}
  10. s y m b o l θ {symbol\theta}
  11. E [ ( θ i - θ ^ i ) 2 ] . \operatorname{E}\left[({\theta_{i}}-{\hat{\theta}_{i}})^{2}\right].
  12. θ 1 \theta_{1}
  13. X 1 X_{1}
  14. X 1 X_{1}
  15. θ 1 \theta_{1}
  16. X 1 X_{1}
  17. X i X_{i}
  18. θ ^ i \hat{\theta}_{i}

Stein_factorization.html

  1. f : X S f:X\to S
  2. f = g f f=g\circ f^{\prime}
  3. g : S S g:S^{\prime}\to S
  4. f : X S f^{\prime}:X\to S^{\prime}
  5. f * 𝒪 X = 𝒪 S f^{\prime}_{*}\mathcal{O}_{X}=\mathcal{O}_{S^{\prime}}
  6. f - 1 ( s ) f^{\prime-1}(s)
  7. s S s\in S
  8. s S s\in S
  9. f - 1 ( s ) f^{-1}(s)
  10. g - 1 ( s ) g^{-1}(s)
  11. S = S^{\prime}=
  12. f * 𝒪 X f_{*}\mathcal{O}_{X}
  13. g : S S g:S^{\prime}\to S
  14. 𝒪 X \mathcal{O}_{X}
  15. f : X S . f^{\prime}:X\to S^{\prime}.
  16. f * 𝒪 X = 𝒪 S f^{\prime}_{*}\mathcal{O}_{X}=\mathcal{O}_{S^{\prime}}
  17. f f^{\prime}

Step-growth_polymerization.html

  1. 1 1 - p n - 1 = 1 + ( n - 1 ) k t [ C O O H ] n - 1 \frac{1}{1-p^{n-1}}=1+(n-1)kt[COOH]^{n-1}
  2. r a t e = - d [ C O O H ] d t = k [ C O O H ] 2 [ O H ] rate=\frac{-d[COOH]}{dt}=k[COOH]^{2}[OH]
  3. r a t e = - d [ C O O H ] d t = k [ C O O H ] 3 rate=\frac{-d[COOH]}{dt}=k[COOH]^{3}
  4. 1 ( 1 - p ) 2 = 2 k t [ C O O H ] 2 + 1 = X n 2 \frac{1}{(1-p)^{2}}=2kt[COOH]^{2}+1=X^{2}_{n}
  5. ( t ) \sqrt{(}t)
  6. - d [ C O O H ] d t = k [ C O O H ] [ O H ] \frac{-d[COOH]}{dt}=k[COOH][OH]
  7. - d [ C O O H ] d t = k [ C O O H ] 2 \frac{-d[COOH]}{dt}=k[COOH]^{2}
  8. 1 1 - p = 1 + [ C O O H ] k t = X n \frac{1}{1-p}=1+[COOH]kt=X_{n}
  9. t t\,
  10. x A A + x B B A A - ( B B - A A ) x - 1 - B B xAA+xBB\rightarrow AA-(BB-AA)_{x-1}-BB
  11. x A B A - ( B - A ) x - 1 - B xAB\rightarrow A-(B-A)_{x-1}-B
  12. p x - 1 p^{x-1}\,
  13. ( 1 - p ) (1-p)\,
  14. P x = ( 1 - p ) p x - 1 P_{x}=(1-p)p^{x-1}\,
  15. N x N = ( 1 - p ) p x - 1 \frac{N_{x}}{N}=(1-p)p^{x-1}\,
  16. W x W o = x N x M o N o M o = x N x N o = x N x N N N o \frac{W_{x}}{W_{o}}=\frac{xN_{x}M_{o}}{N_{o}M_{o}}=\frac{xN_{x}}{N_{o}}=x\frac% {N_{x}}{N}\frac{N}{N_{o}}
  17. X n = 1 1 - p = N o N X_{n}=\frac{1}{1-p}=\frac{N_{o}}{N}
  18. W x W o = x ( 1 - p ) 2 p x - 1 \frac{W_{x}}{W_{o}}=x(1-p)^{2}p^{x-1}\,
  19. P D I = M w M n PDI=\frac{M_{w}}{M_{n}}
  20. P D I = 1 + p PDI=1+p\,
  21. X n = ( 1 + r ) ( 1 + r - 2 r p ) X_{n}=\frac{(1+r)}{(1+r-2rp)}
  22. r = N A A N B B r=\frac{N_{AA}}{N_{BB}}
  23. r = N A A ( N B B + 2 N B ) r=\frac{N_{AA}}{(N_{BB}+2N_{B})}
  24. f a v = N i \sdot f i N i f_{av}=\frac{\sum N_{i}\sdot f_{i}}{\sum N_{i}}
  25. x n = 2 2 - p f a v x_{n}=\frac{2}{2-pf_{av}}
  26. 2 ( N 0 - N ) N 0 \sdot f a v \frac{2(N_{0}-N)}{N_{0}\sdot f_{av}}

Steric_factor.html

  1. ρ \rho

Stiefel_manifold.html

  1. V k ( 𝔽 n ) = { A 𝔽 n × k : A A = 1 } . V_{k}(\mathbb{F}^{n})=\left\{A\in\mathbb{F}^{n\times k}:A^{\ast}A=1\right\}.
  2. dim V k ( n ) = n k - 1 2 k ( k + 1 ) \dim V_{k}(\mathbb{R}^{n})=nk-\frac{1}{2}k(k+1)
  3. dim V k ( n ) = 2 n k - k 2 \dim V_{k}(\mathbb{C}^{n})=2nk-k^{2}
  4. dim V k ( n ) = 4 n k - k ( 2 k - 1 ) . \dim V_{k}(\mathbb{H}^{n})=4nk-k(2k-1).
  5. V k ( n ) \displaystyle V_{k}(\mathbb{R}^{n})
  6. V k ( n ) SO ( n ) / SO ( n - k ) for k < n . V_{k}(\mathbb{R}^{n})\cong\mbox{SO}~{}(n)/\mbox{SO}~{}(n-k)\qquad\mbox{for }~{% }k<n.
  7. V k ( n ) SU ( n ) / SU ( n - k ) for k < n . V_{k}(\mathbb{C}^{n})\cong\mbox{SU}~{}(n)/\mbox{SU}~{}(n-k)\qquad\mbox{for }~{% }k<n.
  8. V 1 ( n ) = S n - 1 V 1 ( n ) = S 2 n - 1 V 1 ( n ) = S 4 n - 1 \begin{aligned}\displaystyle V_{1}(\mathbb{R}^{n})&\displaystyle=S^{n-1}\\ \displaystyle V_{1}(\mathbb{C}^{n})&\displaystyle=S^{2n-1}\\ \displaystyle V_{1}(\mathbb{H}^{n})&\displaystyle=S^{4n-1}\end{aligned}
  9. V n - 1 ( n ) SO ( n ) V n - 1 ( n ) SU ( n ) \begin{aligned}\displaystyle V_{n-1}(\mathbb{R}^{n})&\displaystyle\cong\mathrm% {SO}(n)\\ \displaystyle V_{n-1}(\mathbb{C}^{n})&\displaystyle\cong\mathrm{SU}(n)\end{aligned}
  10. V n ( n ) O ( n ) V n ( n ) U ( n ) V n ( n ) Sp ( n ) \begin{aligned}\displaystyle V_{n}(\mathbb{R}^{n})&\displaystyle\cong\mathrm{O% }(n)\\ \displaystyle V_{n}(\mathbb{C}^{n})&\displaystyle\cong\mathrm{U}(n)\\ \displaystyle V_{n}(\mathbb{H}^{n})&\displaystyle\cong\mathrm{Sp}(n)\end{aligned}
  11. X Y , X\hookrightarrow Y,
  12. V k ( X ) V k ( Y ) , V_{k}(X)\hookrightarrow V_{k}(Y),
  13. V n ( X ) V n ( X * ) . V_{n}(X)\stackrel{\sim}{\to}V_{n}(X^{*}).
  14. p : V k ( 𝔽 n ) G k ( 𝔽 n ) p:V_{k}(\mathbb{F}^{n})\to G_{k}(\mathbb{F}^{n})
  15. O ( k ) \displaystyle\mathrm{O}(k)
  16. V k - 1 ( n - 1 ) V k ( n ) S n - 1 V_{k-1}(\mathbb{R}^{n-1})\to V_{k}(\mathbb{R}^{n})\to S^{n-1}
  17. π n - k V k ( n ) \pi_{n-k}V_{k}(\mathbb{R}^{n})\simeq\mathbb{Z}
  18. π n - k V k ( n ) 2 \pi_{n-k}V_{k}(\mathbb{R}^{n})\simeq\mathbb{Z}_{2}

Stieltjes_constants.html

  1. γ k \gamma_{k}
  2. ζ ( s ) = 1 s - 1 + n = 0 ( - 1 ) n n ! γ n ( s - 1 ) n . \zeta(s)=\frac{1}{s-1}+\sum_{n=0}^{\infty}\frac{(-1)^{n}}{n!}\gamma_{n}\;(s-1)% ^{n}.
  3. γ 0 = γ = 0.577 \gamma_{0}=\gamma=0.577\dots
  4. γ n = lim m { k = 1 m ln n k k - ln n + 1 m n + 1 } . \gamma_{n}=\lim_{m\rightarrow\infty}{\left\{\sum_{k=1}^{m}\frac{\ln^{n}k}{k}-% \frac{\ln^{n+1}\!m}{n+1}\right\}}.
  5. γ n = ( - 1 ) n n ! 2 π 0 2 π e - n i x ζ ( e i x + 1 ) d x . \gamma_{n}=\frac{(-1)^{n}n!}{2\pi}\int_{0}^{2\pi}e^{-nix}\zeta\left(e^{ix}+1% \right)dx.
  6. γ n = 1 2 δ n , 0 + 1 i 0 d x e 2 π x - 1 { ln n ( 1 - i x ) 1 - i x - ln n ( 1 + i x ) 1 + i x } , n = 0 , 1 , 2 , \gamma_{n}\,=\,\frac{1}{2}\delta_{n,0}+\,\frac{1}{i}\!\int\limits_{0}^{\infty}% \!\frac{dx}{e^{2\pi x}-1}\left\{\frac{\ln^{n}(1-ix)}{1-ix}-\frac{\ln^{n}(1+ix)% }{1+ix}\right\}\,,\qquad\quad n=0,1,2,\ldots
  7. γ n = - π 2 ( n + 1 ) - + ln n + 1 ( 1 2 ± i x ) cosh 2 π x d x n = 0 , 1 , 2 , \gamma_{n}\,=\,-\frac{\pi}{2(n+1)}\!\int\limits_{-\infty}^{+\infty}\frac{\ln^{% n+1}\!\big(\frac{1}{2}\pm ix\big)}{\cosh^{2}\!\pi x}\,dx\qquad\qquad\qquad% \qquad\qquad\qquad n=0,1,2,\ldots
  8. γ 1 = - [ γ - ln 2 2 ] ln 2 + i 0 d x e π x + 1 { ln ( 1 - i x ) 1 - i x - ln ( 1 + i x ) 1 + i x } γ 1 = - γ 2 - 0 [ 1 1 - e - x - 1 x ] e - x ln x d x \begin{array}[]{l}\displaystyle\gamma_{1}=-\left[\gamma-\frac{\ln 2}{2}\right]% \ln 2+\,i\!\int\limits_{0}^{\infty}\!\frac{dx}{e^{\pi x}+1}\left\{\frac{\ln(1-% ix)}{1-ix}-\frac{\ln(1+ix)}{1+ix}\right\}\\ \displaystyle\gamma_{1}=-\gamma^{2}-\int\limits_{0}^{\infty}\left[\frac{1}{1-e% ^{-x}}-\frac{1}{x}\right]e^{-x}\ln x\,dx\end{array}
  9. γ 1 = ln 2 2 k = 2 ( - 1 ) k k log 2 k ( 2 log 2 k - log 2 2 k ) \gamma_{1}\,=\,\frac{\ln 2}{2}\sum_{k=2}^{\infty}\frac{(-1)^{k}}{k}\,\lfloor% \log_{2}{k}\rfloor\cdot\big(2\log_{2}{k}-\lfloor\log_{2}{2k}\rfloor\big)
  10. B 2 k B_{2k}
  11. γ m = k = 1 n ln m k k - ln m + 1 n m + 1 - ln m n 2 n - k = 1 N - 1 B 2 k ( 2 k ) ! [ ln m x x ] x = n ( 2 k - 1 ) + θ B 2 N ( 2 N ) ! [ ln m x x ] x = n ( 2 N - 1 ) , 0 < θ < 1 \gamma_{m}\,=\,\sum_{k=1}^{n}\frac{\,\ln^{m}\!k\,}{k}-\frac{\,\ln^{m+1}\!n\,}{% m+1}-\frac{\,\ln^{m}\!n\,}{2n}-\sum_{k=1}^{N-1}\frac{\,B_{2k}\,}{(2k)!}\left[% \frac{\ln^{m}\!x}{x}\right]^{(2k-1)}_{x=n}+\theta\cdot\frac{\,B_{2N}\,}{(2N)!}% \left[\frac{\ln^{m}\!x}{x}\right]^{(2N-1)}_{x=n}\,,\qquad 0<\theta<1
  12. n = 1 H n - ( γ + ln n ) n = - γ 1 - 1 2 γ 2 + 1 12 π 2 n = 1 H n ( 2 ) - ( γ + ln n ) 2 n = - γ 2 - 2 γ γ 1 - 2 3 γ 3 + 5 3 ζ ( 3 ) \begin{array}[]{l}\displaystyle\sum_{n=1}^{\infty}\frac{\,H_{n}-(\gamma+\ln n)% \,}{n}\,=\,\,-\gamma_{1}-\frac{1}{2}\gamma^{2}+\frac{1}{12}\pi^{2}\\ \displaystyle\sum_{n=1}^{\infty}\frac{\,H^{(2)}_{n}-(\gamma+\ln n)^{2}\,}{n}\,% =\,\,-\gamma_{2}-2\gamma\gamma_{1}-\frac{2}{3}\gamma^{3}+\frac{5}{3}\zeta(3)% \end{array}
  13. [ ] \left[{\cdot\atop\cdot}\right]
  14. γ m = 1 2 δ m , 0 + ( - 1 ) m m ! π n = 1 1 n n ! k = 0 1 2 n ( - 1 ) k [ 2 k + 2 m + 1 ] [ n 2 k + 1 ] ( 2 π ) 2 k + 1 , m = 0 , 1 , 2 , , \gamma_{m}\,=\,\frac{1}{2}\delta_{m,0}+\frac{\,(-1)^{m}m!\,}{\pi}\sum_{n=1}^{% \infty}\frac{1}{\,n\cdot n!\,}\sum_{k=0}^{\lfloor\!\frac{1}{2}n\!\rfloor}\frac% {\,(-1)^{k}\cdot\left[{2k+2\atop m+1}\right]\cdot\left[{n\atop 2k+1}\right]\,}% {\,(2\pi)^{2k+1}\,}\,,\qquad m=0,1,2,...,
  15. γ m = 1 2 δ m , 0 + ( - 1 ) m m ! k = 1 N [ 2 k m + 1 ] B 2 k ( 2 k ) ! + θ ( - 1 ) m m ! [ 2 N + 2 m + 1 ] B 2 N + 2 ( 2 N + 2 ) ! , 0 < θ < 1 \gamma_{m}\,=\,\frac{1}{2}\delta_{m,0}+(-1)^{m}m!\cdot\!\sum_{k=1}^{N}\frac{\,% \left[{2k\atop m+1}\right]\cdot B_{2k}\,}{(2k)!}\,+\,\theta\cdot\frac{\,(-1)^{% m}m!\!\cdot\left[{2N+2\atop m+1}\right]\cdot B_{2N+2}\,}{(2N+2)!}\,,\qquad 0<% \theta<1
  16. | γ n | { 2 ( n - 1 ) ! π n , n = 1 , 3 , 5 , 4 ( n - 1 ) ! π n , n = 2 , 4 , 6 , \big|\gamma_{n}\big|\,\leqslant\,\begin{cases}\displaystyle\frac{2\,(n-1)!}{% \pi^{n}}\,,&n=1,3,5,\ldots\\ \displaystyle\frac{4\,(n-1)!}{\pi^{n}}\,,&n=2,4,6,\ldots\end{cases}
  17. | γ n | < 10 - 4 e n ln ln n , n 5 |\gamma_{n}|<10^{-4}e^{n\ln\ln n}\,,\qquad n\geqslant 5
  18. 2 π exp ( v tan v ) = n cos ( v ) v 2\pi\exp(v\tan v)=n\frac{\cos(v)}{v}
  19. 0 < v < π / 2 0<v<\pi/2
  20. u = v tan v u=v\tan v
  21. γ n B n e n A cos ( a n + b ) \gamma_{n}\sim\frac{B}{\sqrt{n}}e^{nA}\cos(an+b)
  22. A = 1 2 ln ( u 2 + v 2 ) - u u 2 + v 2 A=\frac{1}{2}\ln(u^{2}+v^{2})-\frac{u}{u^{2}+v^{2}}
  23. B = 2 2 π u 2 + v 2 [ ( u + 1 ) 2 + v 2 ] 1 / 4 B=\frac{2\sqrt{2\pi}\sqrt{u^{2}+v^{2}}}{[(u+1)^{2}+v^{2}]^{1/4}}
  24. a = tan - 1 ( v u ) + v u 2 + v 2 a=\tan^{-1}\left(\frac{v}{u}\right)+\frac{v}{u^{2}+v^{2}}
  25. b = tan - 1 ( v u ) - 1 2 ( v u + 1 ) . b=\tan^{-1}\left(\frac{v}{u}\right)-\frac{1}{2}\left(\frac{v}{u+1}\right).
  26. ζ ( s , a ) = 1 s - 1 + n = 0 ( - 1 ) n n ! γ n ( a ) ( s - 1 ) n . \zeta(s,a)=\frac{1}{s-1}+\sum_{n=0}^{\infty}\frac{(-1)^{n}}{n!}\gamma_{n}(a)\;% (s-1)^{n}.
  27. γ n ( a ) = lim m { k = 0 m ln n ( k + a ) k + a - ln n + 1 ( m + a ) n + 1 } , n = 0 , 1 , 2 , a 0 , - 1 , - 2 , \gamma_{n}(a)\,=\,\lim_{m\to\infty}\left\{\sum_{k=0}^{m}\frac{\ln^{n}(k+a)}{k+% a}-\frac{\ln^{n+1}(m+a)}{n+1}\right\}\,,\qquad\;\begin{array}[]{l}n=0,1,2,% \ldots\\ a\neq 0,-1,-2,\ldots\end{array}
  28. γ n ( a ) = [ 1 2 a - ln a n + 1 ] ln n a - i 0 d x e 2 π x - 1 { ln n ( a - i x ) a - i x - ln n ( a + i x ) a + i x } , n = 0 , 1 , 2 , a 0 , - 1 , - 2 , \gamma_{n}(a)\,=\,\left[\frac{1}{2a}-\frac{\ln{a}}{n+1}\right]\ln^{n}\!{a}-i\!% \int\limits_{0}^{\infty}\!\frac{dx}{e^{2\pi x}-1}\left\{\frac{\ln^{n}(a-ix)}{a% -ix}-\frac{\ln^{n}(a+ix)}{a+ix}\right\}\,,\qquad\;\begin{array}[]{l}n=0,1,2,% \ldots\\ a\neq 0,-1,-2,\ldots\end{array}
  29. γ n ( a + 1 ) = γ n ( a ) - ln n a a , n = 0 , 1 , 2 , a 0 , - 1 , - 2 , \gamma_{n}(a+1)\,=\,\gamma_{n}(a)-\frac{\,\ln^{n}\!a\,}{a}\,,\qquad\;\begin{% array}[]{l}n=0,1,2,\ldots\\ a\neq 0,-1,-2,\ldots\end{array}
  30. l = 0 n - 1 γ p ( a + l n ) = ( - 1 ) p n [ ln n p + 1 - Ψ ( a n ) ] ln p n + n r = 0 p - 1 ( - 1 ) r ( p r ) γ p - r ( a n ) ln r n , n = 2 , 3 , 4 , \sum_{l=0}^{n-1}\gamma_{p}\!\left(\!a+\frac{l}{\,n\,}\right)=\,(-1)^{p}n\!% \left[\frac{\ln n}{\,p+1\,}-\Psi(an)\right]\!\ln^{p}\!n\,+\,n\sum_{r=0}^{p-1}(% -1)^{r}{\left({{p}\atop{r}}\right)}\gamma_{p-r}(an)\cdot\ln^{r}\!{n}\,,\qquad% \qquad n=2,3,4,\ldots
  31. ( p r ) {\left({{p}\atop{r}}\right)}
  32. γ 1 ( m n ) - γ 1 ( 1 - m n ) = 2 π l = 1 n - 1 sin 2 π m l n ln Γ ( l n ) - π ( γ + ln 2 π n ) cot m π n \gamma_{1}\biggl(\frac{m}{n}\biggr)-\gamma_{1}\biggl(1-\frac{m}{n}\biggr)=2\pi% \sum_{l=1}^{n-1}\sin\frac{2\pi ml}{n}\cdot\ln\Gamma\biggl(\frac{l}{n}\biggr)-% \pi(\gamma+\ln 2\pi n)\cot\frac{m\pi}{n}
  33. γ 1 ( r m ) = γ 1 + γ 2 + γ ln 2 π m + ln 2 π ln m + 1 2 ln 2 m + ( γ + ln 2 π m ) Ψ ( r m ) + π l = 1 m - 1 sin 2 π r l m ln Γ ( l m ) + l = 1 m - 1 cos 2 π r l m ζ ′′ ( 0 , l m ) , r = 1 , 2 , 3 , , m - 1 . \begin{array}[]{ll}\displaystyle\gamma_{1}\biggl(\frac{r}{m}\biggr)=&% \displaystyle\gamma_{1}+\gamma^{2}+\gamma\ln 2\pi m+\ln 2\pi\cdot\ln{m}+\frac{% 1}{2}\ln^{2}\!{m}+(\gamma+\ln 2\pi m)\cdot\Psi\!\left(\!\frac{r}{m}\!\right)\\ &\displaystyle\qquad+\pi\sum_{l=1}^{m-1}\sin\frac{2\pi rl}{m}\cdot\ln\Gamma% \biggl(\frac{l}{m}\biggr)+\sum_{l=1}^{m-1}\cos\frac{2\pi rl}{m}\cdot\zeta^{% \prime\prime}\!\left(\!0,\,\frac{l}{m}\!\right)\end{array}\,,\qquad\quad r=1,2% ,3,\ldots,m-1\,.
  34. r = 0 m - 1 γ 1 ( a + r m ) = m ln m Ψ ( a m ) - m 2 ln 2 m + m γ 1 ( a m ) , a r = 1 m - 1 γ 1 ( r m ) = ( m - 1 ) γ 1 - m γ ln m - m 2 ln 2 m r = 1 2 m - 1 ( - 1 ) r γ 1 ( r 2 m ) = - γ 1 + m ( 2 γ + ln 2 + 2 ln m ) ln 2 r = 0 2 m - 1 ( - 1 ) r γ 1 ( 2 r + 1 4 m ) = m { 4 π ln Γ ( 1 4 ) - π ( 4 ln 2 + 3 ln π + ln m + γ ) } r = 1 m - 1 γ 1 ( r m ) cos 2 π r k m = - γ 1 + m ( γ + ln 2 π m ) ln ( 2 sin k π m ) + m 2 { ζ ′′ ( 0 , k m ) + ζ ′′ ( 0 , 1 - k m ) } , k = 1 , 2 , , m - 1 r = 1 m - 1 γ 1 ( r m ) sin 2 π r k m = π 2 ( γ + ln 2 π m ) ( 2 k - m ) - π m 2 { ln π - ln sin k π m } + m π ln Γ ( k m ) , k = 1 , 2 , , m - 1 r = 1 m - 1 γ 1 ( r m ) cot π r m = π 6 { ( 1 - m ) ( m - 2 ) γ + 2 ( m 2 - 1 ) ln 2 π - ( m 2 + 2 ) ln m } - 2 π l = 1 m - 1 l ln Γ ( l m ) r = 1 m - 1 r m γ 1 ( r m ) = 1 2 { ( m - 1 ) γ 1 - m γ ln m - m 2 ln 2 m } - π 2 m ( γ + ln 2 π m ) l = 1 m - 1 l cot π l m - π 2 l = 1 m - 1 cot π l m ln Γ ( l m ) \begin{array}[]{ll}\displaystyle\sum_{r=0}^{m-1}\gamma_{1}\!\left(\!a+\frac{r}% {\,m\,}\right)=\,m\ln{m}\cdot\Psi(am)-\frac{m}{2}\ln^{2}\!m+m\gamma_{1}(am)\,,% \qquad a\in\mathbb{C}\\ \displaystyle\sum_{r=1}^{m-1}\gamma_{1}\!\left(\!\frac{r}{\,m\,}\right)=\,(m-1% )\gamma_{1}-m\gamma\ln{m}-\frac{m}{2}\ln^{2}\!m\\ \displaystyle\sum_{r=1}^{2m-1}(-1)^{r}\gamma_{1}\biggl(\!\frac{r}{2m}\!\biggr)% \,=\,-\gamma_{1}+m(2\gamma+\ln 2+2\ln m)\ln 2\\ \displaystyle\sum_{r=0}^{2m-1}(-1)^{r}\gamma_{1}\biggl(\!\frac{2r+1}{4m}\!% \biggr)\,=\,m\left\{4\pi\ln\Gamma\biggl(\frac{1}{4}\biggr)-\pi\big(4\ln 2+3\ln% \pi+\ln m+\gamma\big)\!\right\}\\ \displaystyle\sum_{r=1}^{m-1}\gamma_{1}\biggl(\!\frac{r}{m}\!\biggr)\!\cdot% \cos\dfrac{2\pi rk}{m}\,=\,-\gamma_{1}+m(\gamma+\ln 2\pi m)\ln\!\left(\!2\sin% \frac{\,k\pi\,}{m}\!\right)+\frac{m}{2}\left\{\zeta^{\prime\prime}\!\left(\!0,% \,\frac{k}{m}\!\right)+\,\zeta^{\prime\prime}\!\left(\!0,\,1-\frac{k}{m}\!% \right)\!\right\}\,,\qquad k=1,2,\ldots,m-1\\ \displaystyle\sum_{r=1}^{m-1}\gamma_{1}\biggl(\!\frac{r}{m}\!\biggr)\!\cdot% \sin\dfrac{2\pi rk}{m}\,=\,\frac{\pi}{2}(\gamma+\ln 2\pi m)(2k-m)-\frac{\pi m}% {2}\left\{\ln\pi-\ln\sin\frac{k\pi}{m}\right\}+m\pi\ln\Gamma\biggl(\frac{k}{m}% \biggr)\,,\qquad k=1,2,\ldots,m-1\\ \displaystyle\sum_{r=1}^{m-1}\gamma_{1}\biggl(\!\frac{r}{m}\!\biggr)\cdot\cot% \frac{\pi r}{m}=\,\displaystyle\frac{\pi}{6}\Big\{\!(1-m)(m-2)\gamma+2(m^{2}-1% )\ln 2\pi-(m^{2}+2)\ln{m}\Big\}-2\pi\!\sum_{l=1}^{m-1}l\!\cdot\!\ln\Gamma\!% \left(\!\frac{l}{m}\!\right)\\ \displaystyle\sum_{r=1}^{m-1}\frac{r}{m}\cdot\gamma_{1}\biggl(\!\frac{r}{m}\!% \biggr)=\,\frac{1}{2}\left\{\!(m-1)\gamma_{1}-m\gamma\ln{m}-\frac{m}{2}\ln^{2}% \!{m}\!\right\}-\frac{\pi}{2m}(\gamma+\ln 2\pi m)\!\sum_{l=1}^{m-1}l\!\cdot\!% \cot\frac{\pi l}{m}-\frac{\pi}{2}\!\sum_{l=1}^{m-1}\cot\frac{\pi l}{m}\cdot\ln% \Gamma\biggl(\!\frac{l}{m}\!\biggr)\end{array}
  35. γ 1 ( 1 2 ) = - 2 γ ln 2 - ln 2 2 + γ 1 = - 1.353459680 \gamma_{1}\!\left(\!\frac{1}{\,2\,}\!\right)=-2\gamma\ln 2-\ln^{2}\!2+\gamma_{% 1}\,=\,-1.353459680\ldots
  36. γ 1 ( 1 4 ) = 2 π ln Γ ( 1 4 ) - 3 π 2 ln π - 7 2 ln 2 2 - ( 3 γ + 2 π ) ln 2 - γ π 2 + γ 1 = - 5.518076350 γ 1 ( 3 4 ) = - 2 π ln Γ ( 1 4 ) + 3 π 2 ln π - 7 2 ln 2 2 - ( 3 γ - 2 π ) ln 2 + γ π 2 + γ 1 = - 0.3912989024 γ 1 ( 1 3 ) = - 3 γ 2 ln 3 - 3 4 ln 2 3 + π 4 3 { ln 3 - 8 ln 2 π - 2 γ + 12 ln Γ ( 1 3 ) } + γ 1 = - 3.259557515 \begin{array}[]{l}\displaystyle\gamma_{1}\!\left(\!\frac{1}{\,4\,}\!\right)=\,% 2\pi\ln\Gamma\!\left(\!\frac{1}{\,4\,}\!\right)-\frac{3\pi}{2}\ln\pi-\frac{7}{% 2}\ln^{2}\!2-(3\gamma+2\pi)\ln 2-\frac{\gamma\pi}{2}+\gamma_{1}\,=\,-5.5180763% 50\ldots\\ \displaystyle\gamma_{1}\!\left(\!\frac{3}{\,4\,}\!\right)=\,-2\pi\ln\Gamma\!% \left(\!\frac{1}{\,4\,}\!\right)+\frac{3\pi}{2}\ln\pi-\frac{7}{2}\ln^{2}\!2-(3% \gamma-2\pi)\ln 2+\frac{\gamma\pi}{2}+\gamma_{1}\,=\,-0.3912989024\ldots\\ \displaystyle\gamma_{1}\!\left(\!\frac{1}{\,3\,}\!\right)=\,-\frac{3\gamma}{2}% \ln 3-\frac{3}{4}\ln^{2}\!3+\frac{\pi}{4\sqrt{3\,}}\left\{\ln 3-8\ln 2\pi-2% \gamma+12\ln\Gamma\!\left(\!\frac{1}{\,3\,}\!\right)\!\right\}+\,\gamma_{1}\,=% \,-3.259557515\ldots\end{array}
  37. γ 1 ( 2 3 ) = - 3 γ 2 ln 3 - 3 4 ln 2 3 - π 4 3 { ln 3 - 8 ln 2 π - 2 γ + 12 ln Γ ( 1 3 ) } + γ 1 = - 0.5989062842 γ 1 ( 1 6 ) = - 3 γ 2 ln 3 - 3 4 ln 2 3 - ln 2 2 - ( 3 ln 3 + 2 γ ) ln 2 + 3 π 3 2 ln Γ ( 1 6 ) - π 2 3 { 3 ln 3 + 11 ln 2 + 15 2 ln π + 3 γ } + γ 1 = - 10.74258252 γ 1 ( 5 6 ) = - 3 γ 2 ln 3 - 3 4 ln 2 3 - ln 2 2 - ( 3 ln 3 + 2 γ ) ln 2 - 3 π 3 2 ln Γ ( 1 6 ) + π 2 3 { 3 ln 3 + 11 ln 2 + 15 2 ln π + 3 γ } + γ 1 = - 0.2461690038 \begin{array}[]{l}\displaystyle\gamma_{1}\!\left(\!\frac{2}{\,3\,}\!\right)=\,% -\frac{3\gamma}{2}\ln 3-\frac{3}{4}\ln^{2}\!3-\frac{\pi}{4\sqrt{3\,}}\left\{% \ln 3-8\ln 2\pi-2\gamma+12\ln\Gamma\!\left(\!\frac{1}{\,3\,}\!\right)\!\right% \}+\,\gamma_{1}\,=\,-0.5989062842\ldots\\ \displaystyle\gamma_{1}\!\left(\!\frac{1}{\,6\,}\!\right)=\,-\frac{3\gamma}{2}% \ln 3-\frac{3}{4}\ln^{2}\!3-\ln^{2}\!2-(3\ln 3+2\gamma)\ln 2+\frac{3\pi\sqrt{3% \,}}{2}\ln\Gamma\!\left(\!\frac{1}{\,6\,}\!\right)\\ \displaystyle\qquad\qquad\quad-\frac{\pi}{2\sqrt{3\,}}\left\{3\ln 3+11\ln 2+% \frac{15}{2}\ln\pi+3\gamma\right\}+\,\gamma_{1}\,=\,-10.74258252\ldots\\ \displaystyle\gamma_{1}\!\left(\!\frac{5}{\,6\,}\!\right)=\,-\frac{3\gamma}{2}% \ln 3-\frac{3}{4}\ln^{2}\!3-\ln^{2}\!2-(3\ln 3+2\gamma)\ln 2-\frac{3\pi\sqrt{3% \,}}{2}\ln\Gamma\!\left(\!\frac{1}{\,6\,}\!\right)\\ \displaystyle\qquad\qquad\quad+\frac{\pi}{2\sqrt{3\,}}\left\{3\ln 3+11\ln 2+% \frac{15}{2}\ln\pi+3\gamma\right\}+\,\gamma_{1}\,=\,-0.2461690038\ldots\end{array}
  38. γ 1 ( 1 5 ) = γ 1 + 5 2 { ζ ′′ ( 0 , 1 5 ) + ζ ′′ ( 0 , 4 5 ) } + π 10 + 2 5 2 ln Γ ( 1 5 ) + π 10 - 2 5 2 ln Γ ( 2 5 ) + { 5 2 ln 2 - 5 2 ln ( 1 + 5 ) - 5 4 ln 5 - π 25 + 10 5 10 } γ - 5 2 { ln 2 + ln 5 + ln π + π 25 - 10 5 10 } ln ( 1 + 5 ) + 5 2 ln 2 2 + 5 ( 1 - 5 ) 8 ln 2 5 + 3 5 4 ln 2 ln 5 + 5 2 ln 2 ln π + 5 4 ln 5 ln π - π ( 2 25 + 10 5 + 5 25 + 2 5 ) 20 ln 2 - π ( 4 25 + 10 5 - 5 5 + 2 5 ) 40 ln 5 - π ( 5 5 + 2 5 + 25 + 10 5 ) 10 ln π = - 8.030205511 γ 1 ( 1 8 ) = γ 1 + 2 { ζ ′′ ( 0 , 1 8 ) + ζ ′′ ( 0 , 7 8 ) } + 2 π 2 ln Γ ( 1 8 ) - π 2 ( 1 - 2 ) ln Γ ( 1 4 ) - { 1 + 2 2 π + 4 ln 2 + 2 ln ( 1 + 2 ) } γ - 1 2 ( π + 8 ln 2 + 2 ln π ) ln ( 1 + 2 ) - 7 ( 4 - 2 ) 4 ln 2 2 + 1 2 ln 2 ln π - π ( 10 + 11 2 ) 4 ln 2 - π ( 3 + 2 2 ) 2 ln π = - 16.64171976 γ 1 ( 1 12 ) = γ 1 + 3 { ζ ′′ ( 0 , 1 12 ) + ζ ′′ ( 0 , 11 12 ) } + 4 π ln Γ ( 1 4 ) + 3 π 3 ln Γ ( 1 3 ) - { 2 + 3 2 π + 3 2 ln 3 - 3 ( 1 - 3 ) ln 2 + 2 3 ln ( 1 + 3 ) } γ - 2 3 ( 3 ln 2 + ln 3 + ln π ) ln ( 1 + 3 ) - 7 - 6 3 2 ln 2 2 - 3 4 ln 2 3 + 3 3 ( 1 - 3 ) 2 ln 3 ln 2 + 3 ln 2 ln π - π ( 17 + 8 3 ) 2 3 ln 2 + π ( 1 - 3 ) 3 4 ln 3 - π 3 ( 2 + 3 ) ln π = - 29.84287823 \begin{array}[]{ll}\displaystyle\gamma_{1}\biggl(\!\frac{1}{5}\!\biggr)=&% \displaystyle\!\!\!\gamma_{1}+\frac{\sqrt{5}}{2}\!\left\{\zeta^{\prime\prime}% \!\left(\!0,\,\frac{1}{5}\!\right)+\zeta^{\prime\prime}\!\left(\!0,\,\frac{4}{% 5}\!\right)\!\right\}+\frac{\pi\sqrt{10+2\sqrt{5}}}{2}\ln\Gamma\biggl(\!\frac{% 1}{5}\!\biggr)\\ &\displaystyle+\frac{\pi\sqrt{10-2\sqrt{5}}}{2}\ln\Gamma\biggl(\!\frac{2}{5}\!% \biggr)+\left\{\!\frac{\sqrt{5}}{2}\ln{2}-\frac{\sqrt{5}}{2}\ln\!\big(1+\sqrt{% 5}\big)-\frac{5}{4}\ln 5-\frac{\pi\sqrt{25+10\sqrt{5}}}{10}\right\}\!\cdot% \gamma\\ &\displaystyle-\frac{\sqrt{5}}{2}\left\{\ln 2+\ln 5+\ln\pi+\frac{\pi\sqrt{25-1% 0\sqrt{5}}}{10}\right\}\!\cdot\ln\!\big(1+\sqrt{5})+\frac{\sqrt{5}}{2}\ln^{2}% \!2+\frac{\sqrt{5}\big(1-\sqrt{5}\big)}{8}\ln^{2}\!5\\ &\displaystyle+\frac{3\sqrt{5}}{4}\ln 2\cdot\ln 5+\frac{\sqrt{5}}{2}\ln 2\cdot% \ln\pi+\frac{\sqrt{5}}{4}\ln 5\cdot\ln\pi-\frac{\pi\big(2\sqrt{25+10\sqrt{5}}+% 5\sqrt{25+2\sqrt{5}}\big)}{20}\ln 2\\ &\displaystyle-\frac{\pi\big(4\sqrt{25+10\sqrt{5}}-5\sqrt{5+2\sqrt{5}}\big)}{4% 0}\ln 5-\frac{\pi\big(5\sqrt{5+2\sqrt{5}}+\sqrt{25+10\sqrt{5}}\big)}{10}\ln\pi% \\ &\displaystyle=-8.030205511\ldots\\ \displaystyle\gamma_{1}\biggl(\!\frac{1}{8}\!\biggr)=&\displaystyle\!\!\!% \gamma_{1}+\sqrt{2}\left\{\zeta^{\prime\prime}\!\left(\!0,\,\frac{1}{8}\!% \right)+\zeta^{\prime\prime}\!\left(\!0,\,\frac{7}{8}\right)\!\right\}+2\pi% \sqrt{2}\ln\Gamma\biggl(\!\frac{1}{8}\!\biggr)-\pi\sqrt{2}\big(1-\sqrt{2}\big)% \ln\Gamma\biggl(\!\frac{1}{4}\!\biggr)\\ &\displaystyle-\left\{\!\frac{1+\sqrt{2}}{2}\pi+4\ln{2}+\sqrt{2}\ln\!\big(1+% \sqrt{2}\big)\!\right\}\!\cdot\gamma-\frac{1}{\sqrt{2}}\big(\pi+8\ln 2+2\ln\pi% \big)\!\cdot\ln\!\big(1+\sqrt{2})\\ &\displaystyle-\frac{7\big(4-\sqrt{2}\big)}{4}\ln^{2}\!2+\frac{1}{\sqrt{2}}\ln 2% \cdot\ln\pi-\frac{\pi\big(10+11\sqrt{2}\big)}{4}\ln 2-\frac{\pi\big(3+2\sqrt{2% }\big)}{2}\ln\pi\\ &\displaystyle=-16.64171976\ldots\\ \displaystyle\gamma_{1}\biggl(\!\frac{1}{12}\!\biggr)=&\displaystyle\!\!\!% \gamma_{1}+\sqrt{3}\left\{\zeta^{\prime\prime}\!\left(\!0,\,\frac{1}{12}\!% \right)+\zeta^{\prime\prime}\!\left(\!0,\,\frac{11}{12}\right)\!\right\}+4\pi% \ln\Gamma\biggl(\!\frac{1}{4}\!\biggr)+3\pi\sqrt{3}\ln\Gamma\biggl(\!\frac{1}{% 3}\!\biggr)\\ &\displaystyle-\left\{\!\frac{2+\sqrt{3}}{2}\pi+\frac{3}{2}\ln 3-\sqrt{3}(1-% \sqrt{3})\ln{2}+2\sqrt{3}\ln\!\big(1+\sqrt{3}\big)\!\right\}\!\cdot\gamma\\ &\displaystyle-2\sqrt{3}\big(3\ln 2+\ln 3+\ln\pi\big)\!\cdot\ln\!\big(1+\sqrt{% 3})-\frac{7-6\sqrt{3}}{2}\ln^{2}\!2-\frac{3}{4}\ln^{2}\!3\\ &\displaystyle+\frac{3\sqrt{3}(1-\sqrt{3})}{2}\ln 3\cdot\ln 2+\sqrt{3}\ln 2% \cdot\ln\pi-\frac{\pi\big(17+8\sqrt{3}\big)}{2\sqrt{3}}\ln 2\\ &\displaystyle+\frac{\pi\big(1-\sqrt{3}\big)\sqrt{3}}{4}\ln 3-\pi\sqrt{3}(2+% \sqrt{3})\ln\pi=-29.84287823\ldots\end{array}
  39. γ 2 ( r m ) = γ 2 + 2 3 l = 1 m - 1 cos 2 π r l m ζ ′′′ ( 0 , l m ) - 2 ( γ + ln 2 π m ) l = 1 m - 1 cos 2 π r l m ζ ′′ ( 0 , l m ) + π l = 1 m - 1 sin 2 π r l m ζ ′′ ( 0 , l m ) - 2 π ( γ + ln 2 π m ) l = 1 m - 1 sin 2 π r l m ln Γ ( l m ) - 2 γ 1 ln m - γ 3 - [ ( γ + ln 2 π m ) 2 - π 2 12 ] Ψ ( r m ) + π 3 12 cot π r m - γ 2 ln ( 4 π 2 m 3 ) + π 2 12 ( γ + ln m ) - γ ( ln 2 2 π + 4 ln m ln 2 π + 2 ln 2 m ) - { ln 2 2 π + 2 ln 2 π ln m + 2 3 ln 2 m } ln m , r = 1 , 2 , 3 , , m - 1 . \begin{array}[]{rl}\displaystyle\gamma_{2}\biggl(\frac{r}{m}\biggr)=\,\gamma_{% 2}+\frac{2}{3}\!\sum_{l=1}^{m-1}\cos\frac{2\pi rl}{m}\cdot\zeta^{\prime\prime% \prime}\!\left(\!0,\,\frac{l}{m}\!\right)-2(\gamma+\ln 2\pi m)\!\sum_{l=1}^{m-% 1}\cos\frac{2\pi rl}{m}\cdot\zeta^{\prime\prime}\!\left(\!0,\,\frac{l}{m}\!% \right)\\ \displaystyle\quad+\pi\!\sum_{l=1}^{m-1}\sin\frac{2\pi rl}{m}\cdot\zeta^{% \prime\prime}\!\left(\!0,\,\frac{l}{m}\!\right)-2\pi(\gamma+\ln 2\pi m)\!\sum_% {l=1}^{m-1}\sin\frac{2\pi rl}{m}\cdot\ln\Gamma\biggl(\frac{l}{m}\biggr)-2% \gamma_{1}\ln{m}\\ \displaystyle\quad-\gamma^{3}-\left[(\gamma+\ln 2\pi m)^{2}-\frac{\pi^{2}}{12}% \right]\!\cdot\!\Psi\!\biggl(\frac{r}{m}\biggr)+\frac{\pi^{3}}{12}\cot\frac{% \pi r}{m}-\gamma^{2}\ln\big(4\pi^{2}m^{3}\big)+\frac{\pi^{2}}{12}(\gamma+\ln{m% })\\ \displaystyle\quad-\gamma\big(\ln^{2}\!{2\pi}+4\ln{m}\cdot\ln{2\pi}+2\ln^{2}\!% {m}\big)-\left\{\!\ln^{2}\!{2\pi}+2\ln{2\pi}\cdot\ln{m}+\frac{2}{3}\ln^{2}\!{m% }\!\right\}\!\ln{m}\end{array}\,,\qquad\quad r=1,2,3,\ldots,m-1\,.

Stirling_transform.html

  1. b n = k = 1 n { n k } a k , b_{n}=\sum_{k=1}^{n}\left\{\begin{matrix}n\\ k\end{matrix}\right\}a_{k},
  2. { n k } \left\{\begin{matrix}n\\ k\end{matrix}\right\}
  3. a n = k = 1 n s ( n , k ) b k , a_{n}=\sum_{k=1}^{n}s(n,k)b_{k},
  4. f ( x ) = n = 1 a n n ! x n f(x)=\sum_{n=1}^{\infty}{a_{n}\over n!}x^{n}
  5. g ( x ) = n = 1 b n n ! x n g(x)=\sum_{n=1}^{\infty}{b_{n}\over n!}x^{n}
  6. g ( x ) = f ( e x - 1 ) . g(x)=f(e^{x}-1).\,

Stochastic_differential_equation.html

  1. x ˙ i = d x i d t = f i ( 𝐱 ) + m = 1 n g i m ( 𝐱 ) η m ( t ) , \dot{x}_{i}=\frac{dx_{i}}{dt}=f_{i}(\mathbf{x})+\sum_{m=1}^{n}g_{i}^{m}(% \mathbf{x})\eta_{m}(t),\,
  2. 𝐱 = { x i | 1 i k } \mathbf{x}=\{x_{i}|1\leq i\leq k\}
  3. f i f_{i}
  4. g i g_{i}
  5. η m \eta_{m}
  6. g i g_{i}
  7. g ( x ) x g(x)\propto x
  8. η m \eta_{m}
  9. η m \eta_{m}
  10. d X t = μ ( X t , t ) d t + σ ( X t , t ) d B t , \mathrm{d}X_{t}=\mu(X_{t},t)\,\mathrm{d}t+\sigma(X_{t},t)\,\mathrm{d}B_{t},
  11. B B
  12. X t + s - X t = t t + s μ ( X u , u ) d u + t t + s σ ( X u , u ) d B u . X_{t+s}-X_{t}=\int_{t}^{t+s}\mu(X_{u},u)\mathrm{d}u+\int_{t}^{t+s}\sigma(X_{u}% ,u)\,\mathrm{d}B_{u}.
  13. d X t = μ X t d t + σ X t d B t . \mathrm{d}X_{t}=\mu X_{t}\,\mathrm{d}t+\sigma X_{t}\,\mathrm{d}B_{t}.
  14. μ : n × [ 0 , T ] n ; \mu:\mathbb{R}^{n}\times[0,T]\to\mathbb{R}^{n};
  15. σ : n × [ 0 , T ] n × m ; \sigma:\mathbb{R}^{n}\times[0,T]\to\mathbb{R}^{n\times m};
  16. | μ ( x , t ) | + | σ ( x , t ) | C ( 1 + | x | ) ; \big|\mu(x,t)\big|+\big|\sigma(x,t)\big|\leq C\big(1+|x|\big);
  17. | μ ( x , t ) - μ ( y , t ) | + | σ ( x , t ) - σ ( y , t ) | D | x - y | ; \big|\mu(x,t)-\mu(y,t)\big|+\big|\sigma(x,t)-\sigma(y,t)\big|\leq D|x-y|;
  18. | σ | 2 = i , j = 1 n | σ i j | 2 . |\sigma|^{2}=\sum_{i,j=1}^{n}|\sigma_{ij}|^{2}.
  19. 𝔼 [ | Z | 2 ] < + . \mathbb{E}\big[|Z|^{2}\big]<+\infty.
  20. d X t = μ ( X t , t ) d t + σ ( X t , t ) d B t for t [ 0 , T ] ; \mathrm{d}X_{t}=\mu(X_{t},t)\,\mathrm{d}t+\sigma(X_{t},t)\,\mathrm{d}B_{t}% \mbox{ for }~{}t\in[0,T];
  21. X 0 = Z ; X_{0}=Z;
  22. 𝔼 [ 0 T | X t | 2 d t ] < + . \mathbb{E}\left[\int_{0}^{T}|X_{t}|^{2}\,\mathrm{d}t\right]<+\infty.
  23. d X t = ( a ( t ) X t + c ( t ) ) d t + ( b ( t ) X t + d ( t ) ) d W t dX_{t}=(a(t)X_{t}+c(t))dt+(b(t)X_{t}+d(t))dW_{t}
  24. X t = Φ t , t 0 ( X t 0 + t 0 t Φ s , t 0 - 1 ( c ( s ) - b ( s ) d ( s ) ) d s + t 0 t Φ s , t 0 - 1 d ( s ) d W s ) X_{t}=\Phi_{t,t_{0}}\left(X_{t_{0}}+\int_{t_{0}}^{t}\Phi^{-1}_{s,t_{0}}(c(s)-b% (s)d(s))ds+\int_{t_{0}}^{t}\Phi^{-1}_{s,t_{0}}d(s)dW_{s}\right)
  25. Φ t , t 0 = exp ( t 0 t ( a ( s ) - b 2 ( s ) 2 ) d s + t 0 t b ( s ) d W s ) \Phi_{t,t_{0}}=\exp\left(\int_{t_{0}}^{t}\left(a(s)-\frac{b^{2}(s)}{2}\right)% ds+\int_{t_{0}}^{t}b(s)dW_{s}\right)
  26. d X t = 1 2 f ( X t ) f ( X t ) d t + f ( X t ) W t dX_{t}=\frac{1}{2}f(X_{t})f^{\prime}(X_{t})dt+f(X_{t})W_{t}
  27. f f
  28. d X t = f ( X t ) W t dX_{t}=f(X_{t})\circ W_{t}
  29. X t = h - 1 ( W t + h ( X 0 ) ) X_{t}=h^{-1}(W_{t}+h(X_{0}))
  30. h ( x ) = x d s f ( s ) h(x)=\int^{x}\frac{ds}{f(s)}
  31. d X t = ( α f ( X t ) + 1 2 f ( X t ) f ( X t ) ) d t + f ( X t ) W t dX_{t}=\left(\alpha f(X_{t})+\frac{1}{2}f(X_{t})f^{\prime}(X_{t})\right)dt+f(X% _{t})W_{t}
  32. f f
  33. d X t = α f ( X t ) d t + f ( X t ) W t dX_{t}=\alpha f(X_{t})dt+f(X_{t})\circ W_{t}
  34. d Y t = α d t + d W t dY_{t}=\alpha dt+dW_{t}
  35. Y t = h ( X t ) Y_{t}=h(X_{t})
  36. h h
  37. X t = h - 1 ( α t + W t + h ( X 0 ) ) X_{t}=h^{-1}(\alpha t+W_{t}+h(X_{0}))

Stock_duration.html

  1. D d d m = 1 + g K e - g D_{ddm}=\frac{1+g}{K_{e}-g}
  2. D d d m D_{ddm}
  3. K e K_{e}
  4. g g

Stokes_parameters.html

  1. S 0 = I S 1 = I p cos 2 ψ cos 2 χ S 2 = I p sin 2 ψ cos 2 χ S 3 = I p sin 2 χ \begin{aligned}\displaystyle S_{0}&\displaystyle=I\\ \displaystyle S_{1}&\displaystyle=Ip\cos 2\psi\cos 2\chi\\ \displaystyle S_{2}&\displaystyle=Ip\sin 2\psi\cos 2\chi\\ \displaystyle S_{3}&\displaystyle=Ip\sin 2\chi\end{aligned}
  2. I I
  3. p p
  4. 2 ψ 2\psi
  5. 2 χ 2\chi
  6. ( S 1 , S 2 , S 3 ) (S_{1},S_{2},S_{3})
  7. I I
  8. p p
  9. ψ \psi
  10. χ \chi
  11. I \displaystyle I
  12. S = ( S 0 S 1 S 2 S 3 ) = ( I Q U V ) \vec{S}\ =\begin{pmatrix}S_{0}\\ S_{1}\\ S_{2}\\ S_{3}\end{pmatrix}=\begin{pmatrix}I\\ Q\\ U\\ V\end{pmatrix}
  13. ( 1 1 0 0 ) \begin{pmatrix}1\\ 1\\ 0\\ 0\end{pmatrix}
  14. ( 1 - 1 0 0 ) \begin{pmatrix}1\\ -1\\ 0\\ 0\end{pmatrix}
  15. ( 1 0 1 0 ) \begin{pmatrix}1\\ 0\\ 1\\ 0\end{pmatrix}
  16. ( 1 0 - 1 0 ) \begin{pmatrix}1\\ 0\\ -1\\ 0\end{pmatrix}
  17. ( 1 0 0 1 ) \begin{pmatrix}1\\ 0\\ 0\\ 1\end{pmatrix}
  18. ( 1 0 0 - 1 ) \begin{pmatrix}1\\ 0\\ 0\\ -1\end{pmatrix}
  19. ( 1 0 0 0 ) \begin{pmatrix}1\\ 0\\ 0\\ 0\end{pmatrix}
  20. k \vec{k}
  21. E 1 E_{1}
  22. E 2 E_{2}
  23. ( ϵ ^ 1 , ϵ ^ 2 ) (\hat{\epsilon}_{1},\hat{\epsilon}_{2})
  24. ( E 1 , E 2 ) (E_{1},E_{2})
  25. ϕ \phi
  26. Ψ \Psi
  27. Ψ \Psi
  28. I I
  29. Q Q
  30. U U
  31. V V
  32. I E x 2 + E y 2 = E a 2 + E b 2 = E l 2 + E r 2 , Q E x 2 - E y 2 , U E a 2 - E b 2 , V E l 2 - E r 2 . \begin{matrix}I&\equiv&\langle E_{x}^{2}\rangle+\langle E_{y}^{2}\rangle\\ &=&\langle E_{a}^{2}\rangle+\langle E_{b}^{2}\rangle\\ &=&\langle E_{l}^{2}\rangle+\langle E_{r}^{2}\rangle,\\ Q&\equiv&\langle E_{x}^{2}\rangle-\langle E_{y}^{2}\rangle,\\ U&\equiv&\langle E_{a}^{2}\rangle-\langle E_{b}^{2}\rangle,\\ V&\equiv&\langle E_{l}^{2}\rangle-\langle E_{r}^{2}\rangle.\end{matrix}
  33. x ^ , y ^ \hat{x},\hat{y}
  34. a ^ , b ^ \hat{a},\hat{b}
  35. l ^ , r ^ \hat{l},\hat{r}
  36. l ^ = ( x ^ + i y ^ ) / 2 \hat{l}=(\hat{x}+i\hat{y})/\sqrt{2}
  37. I | E x | 2 + | E y | 2 = | E a | 2 + | E b | 2 = | E l | 2 + | E r | 2 Q | E x | 2 - | E y | 2 , U | E a | 2 - | E b | 2 , V | E l | 2 - | E r | 2 . \begin{matrix}I\equiv|E_{x}|^{2}+|E_{y}|^{2}=|E_{a}|^{2}+|E_{b}|^{2}=|E_{l}|^{% 2}+|E_{r}|^{2}\\ Q\equiv|E_{x}|^{2}-|E_{y}|^{2},\\ U\equiv|E_{a}|^{2}-|E_{b}|^{2},\\ V\equiv|E_{l}|^{2}-|E_{r}|^{2}.\end{matrix}
  38. x ^ , y ^ \hat{x},\hat{y}
  39. I = | E x | 2 + | E y | 2 , Q = | E x | 2 - | E y | 2 , U = 2 Re ( E x E y * ) , V = - 2 Im ( E x E y * ) , \begin{matrix}I&=&|E_{x}|^{2}+|E_{y}|^{2},\\ Q&=&|E_{x}|^{2}-|E_{y}|^{2},\\ U&=&2\mbox{Re}~{}(E_{x}E_{y}^{*}),\\ V&=&-2\mbox{Im}~{}(E_{x}E_{y}^{*}),\\ \end{matrix}
  40. ( a ^ , b ^ ) (\hat{a},\hat{b})
  41. I = | E a | 2 + | E b | 2 , Q = - 2 Re ( E a * E b ) , U = | E a | 2 - | E b | 2 , V = 2 Im ( E a * E b ) . \begin{matrix}I&=&|E_{a}|^{2}+|E_{b}|^{2},\\ Q&=&-2\mbox{Re}~{}(E_{a}^{*}E_{b}),\\ U&=&|E_{a}|^{2}-|E_{b}|^{2},\\ V&=&2\mbox{Im}~{}(E_{a}^{*}E_{b}).\\ \end{matrix}
  42. ( l ^ , r ^ ) (\hat{l},\hat{r})
  43. I = | E l | 2 + | E r | 2 , Q = 2 Re ( E l * E r ) , U = - 2 Im ( E l * E r ) , V = | E r | 2 - | E l | 2 . \begin{matrix}I&=&|E_{l}|^{2}+|E_{r}|^{2},\\ Q&=&2\mbox{Re}~{}(E_{l}^{*}E_{r}),\\ U&=&-2\mbox{Im}~{}(E_{l}^{*}E_{r}),\\ V&=&|E_{r}|^{2}-|E_{l}|^{2}.\\ \end{matrix}
  44. Q 2 + U 2 + V 2 = I 2 , \begin{matrix}Q^{2}+U^{2}+V^{2}=I^{2},\end{matrix}
  45. Q 2 + U 2 + V 2 I 2 . \begin{matrix}Q^{2}+U^{2}+V^{2}\leq I^{2}.\end{matrix}
  46. I p I_{p}
  47. Q 2 + U 2 + V 2 = I p 2 , \begin{matrix}Q^{2}+U^{2}+V^{2}=I_{p}^{2},\end{matrix}
  48. I p / I I_{p}/I
  49. L | L | e i 2 θ Q + i U . \begin{matrix}L&\equiv&|L|e^{i2\theta}\\ &\equiv&Q+iU.\\ \end{matrix}
  50. θ θ + θ \theta\rightarrow\theta+\theta^{\prime}
  51. I I
  52. V V
  53. L e i 2 θ L , Q Re ( e i 2 θ L ) , U Im ( e i 2 θ L ) . \begin{matrix}L&\rightarrow&e^{i2\theta^{\prime}}L,\\ Q&\rightarrow&\mbox{Re}~{}\left(e^{i2\theta^{\prime}}L\right),\\ U&\rightarrow&\mbox{Im}~{}\left(e^{i2\theta^{\prime}}L\right).\\ \end{matrix}
  54. I 0 , V , L , \begin{matrix}I&\geq&0,\\ V&\in&\mathbb{R},\\ L&\in&\mathbb{C},\\ \end{matrix}
  55. I I
  56. | V | |V|
  57. | L | |L|
  58. I p = | L | 2 + | V | 2 I_{p}=\sqrt{|L|^{2}+|V|^{2}}
  59. θ = 1 2 arg ( L ) , h = sgn ( V ) . \begin{matrix}\theta&=&\frac{1}{2}\arg(L),\\ h&=&\operatorname{sgn}(V).\\ \end{matrix}
  60. Q = Re ( L ) Q=\mbox{Re}~{}(L)
  61. U = Im ( L ) U=\mbox{Im}~{}(L)
  62. | L | = Q 2 + U 2 , θ = 1 2 tan - 1 ( U / Q ) . \begin{matrix}|L|&=&\sqrt{Q^{2}+U^{2}},\\ \theta&=&\frac{1}{2}\tan^{-1}(U/Q).\\ \end{matrix}
  63. I p = A 2 + B 2 , Q = ( A 2 - B 2 ) cos ( 2 θ ) , U = ( A 2 - B 2 ) sin ( 2 θ ) , V = 2 A B h . \begin{matrix}I_{p}&=&A^{2}+B^{2},\\ Q&=&(A^{2}-B^{2})\cos(2\theta),\\ U&=&(A^{2}-B^{2})\sin(2\theta),\\ V&=&2ABh.\\ \end{matrix}
  64. A = 1 2 ( I p + | L | ) B = 1 2 ( I p - | L | ) θ = 1 2 arg ( L ) h = sgn ( V ) . \begin{matrix}A&=&\sqrt{\frac{1}{2}(I_{p}+|L|)}\\ B&=&\sqrt{\frac{1}{2}(I_{p}-|L|)}\\ \theta&=&\frac{1}{2}\arg(L)\\ h&=&\operatorname{sgn}(V).\\ \end{matrix}

Stopping_power.html

  1. d E k / d x \mathrm{d}E_{k}/\mathrm{d}x

Stopping_time.html

  1. τ \tau
  2. { τ t } t \{\tau\leq t\}\in\mathcal{F}_{t}
  3. t \mathcal{F}_{t}
  4. τ \tau
  5. { τ t } \{\tau\leq t\}
  6. t \mathcal{F}_{t}
  7. { τ t } \{\tau\leq t\}
  8. t \mathcal{F}_{t}
  9. ( B t ) t 0 (B_{t})_{t\geq 0}
  10. B t B_{t}
  11. ( Ω , , ) (\Omega,\mathcal{F},\mathbb{P})
  12. t \mathcal{F}_{t}
  13. ( B s ) - 1 ( A ) (B_{s})^{-1}(A)
  14. 0 s t 0\leq s\leq t
  15. A A\subseteq\mathbb{R}
  16. t \mathcal{F}_{t}
  17. τ := t 0 \tau:=t_{0}
  18. t 0 t_{0}
  19. a . a\in\mathbb{R}.
  20. τ := inf { t 0 | B t > a } \tau:=\inf\{t\geq 0\,|\,B_{t}>a\}
  21. τ := inf { t 1 | B s > 0 for all s [ t - 1 , t ] } \tau:=\inf\{t\geq 1\,|\,B_{s}>0\,\text{ for all }s\in[t-1,t]\}
  22. ( Ω , , { t } t 0 , ) \left(\Omega,\mathcal{F},\left\{\mathcal{F}_{t}\right\}_{t\geq 0},\mathbb{P}\right)
  23. τ 1 τ 2 \tau_{1}\wedge\tau_{2}
  24. τ 1 τ 2 \tau_{1}\vee\tau_{2}
  25. X t τ = X min ( t , τ ) X^{\tau}_{t}=X_{\min(t,\tau)}
  26. 𝟏 { τ n > 0 } X τ n \mathbf{1}_{\{\tau_{n}>0\}}X^{\tau_{n}}
  27. 𝟏 { τ n > 0 } X τ n \mathbf{1}_{\{\tau_{n}>0\}}X^{\tau_{n}}
  28. 𝔼 [ 𝟏 { τ n > 0 } X τ n ] < \mathbb{E}\left[\mathbf{1}_{\{\tau_{n}>0\}}X^{\tau_{n}}\right]<\infty

Strain_(chemistry).html

  1. l n K e q = - Δ G o R T lnK_{eq}=\frac{-\Delta{G^{o}}}{RT}\,
  2. Δ G o = Δ H o - T Δ S o . \Delta{G^{o}}=\Delta{H^{o}}-T\Delta{S^{o}}\,.

Strain_energy.html

  1. U = 1 2 V σ ϵ = 1 2 V E ϵ 2 = 1 2 V E σ 2 U=\frac{1}{2}V\sigma\epsilon=\frac{1}{2}VE\epsilon^{2}=\frac{1}{2}\frac{V}{E}% \sigma^{2}
  2. E = σ ϵ E=\frac{\sigma}{\epsilon}

STRIPS.html

  1. P , O , I , G \langle P,O,I,G\rangle
  2. P P
  3. O O
  4. α , β , γ , δ \langle\alpha,\beta,\gamma,\delta\rangle
  5. I I
  6. G G
  7. N , M \langle N,M\rangle
  8. P , O , I , G \langle P,O,I,G\rangle
  9. succ : 2 P × O 2 P , \operatorname{succ}:2^{P}\times O\rightarrow 2^{P},
  10. 2 P 2^{P}
  11. P P
  12. succ \operatorname{succ}
  13. C P C\subseteq P
  14. succ ( C , α , β , γ , δ ) \operatorname{succ}(C,\langle\alpha,\beta,\gamma,\delta\rangle)
  15. C \ δ γ C\backslash\delta\cup\gamma
  16. α C \alpha\subseteq C
  17. β C = \beta\cap C=\varnothing
  18. C C
  19. succ \operatorname{succ}
  20. succ ( C , [ ] ) = C \operatorname{succ}(C,[\ ])=C
  21. succ ( C , [ a 1 , a 2 , , a n ] ) = succ ( succ ( C , a 1 ) , [ a 2 , , a n ] ) \operatorname{succ}(C,[a_{1},a_{2},\ldots,a_{n}])=\operatorname{succ}(% \operatorname{succ}(C,a_{1}),[a_{2},\ldots,a_{n}])
  22. [ a 1 , a 2 , , a n ] [a_{1},a_{2},\ldots,a_{n}]
  23. G = N , M G=\langle N,M\rangle
  24. F = succ ( I , [ a 1 , a 2 , , a n ] ) F=\operatorname{succ}(I,[a_{1},a_{2},\ldots,a_{n}])
  25. N F N\subseteq F
  26. M F = M\cap F=\varnothing
  27. A t At
  28. A t ( r o o m 1 ) At(room1)
  29. I I

Strong_antichain.html

  1. x , y A [ x y ¬ z X [ z x z y ] ] . \forall x,y\in A\;[x\neq y\rightarrow\neg\exists z\in X\;[z\leq x\land z\leq y% ]].

Strong_topology_(polar_topology).html

  1. ( X , Y , , ) (X,Y,\langle,\rangle)
  2. 𝔽 {\mathbb{F}}
  3. {\mathbb{R}}
  4. {\mathbb{C}}
  5. {\mathcal{B}}
  6. B X B\subseteq X
  7. Y Y
  8. y Y sup x B | x , y | < . \forall y\in Y\qquad\sup_{x\in B}|\langle x,y\rangle|<\infty.
  9. β ( Y , X ) \beta(Y,X)
  10. Y Y
  11. Y Y
  12. || y || B = sup x B | x , y | , y Y , B . ||y||_{B}=\sup_{x\in B}|\langle x,y\rangle|,\qquad y\in Y,\qquad B\in{\mathcal% {B}}.
  13. X X
  14. X X^{\prime}
  15. f : X 𝔽 f:X\to{\mathbb{F}}
  16. β ( X , X ) \beta(X^{\prime},X)
  17. X X
  18. X X^{\prime}
  19. || f || B = sup x B | f ( x ) | , f X , ||f||_{B}=\sup_{x\in B}|f(x)|,\qquad f\in X^{\prime},
  20. B B
  21. X X
  22. X X^{\prime}
  23. X X
  24. X β X^{\prime}_{\beta}
  25. X X
  26. X X^{\prime}
  27. X X^{\prime}
  28. X X^{\prime}
  29. β ( X , X ) \beta(X,X^{\prime})
  30. X X
  31. X X
  32. X X
  33. β ( X , X ) \beta(X,X^{\prime})
  34. X X
  35. X X
  36. ( X , X ) (X,X^{\prime})

Strongly_regular_graph.html

  1. ( v - k - 1 ) μ = k ( k - λ - 1 ) (v-k-1)\mu=k(k-\lambda-1)
  2. k - λ - 1 k-\lambda-1
  3. k × ( k - λ - 1 ) k\times(k-\lambda-1)
  4. ( v - k - 1 ) (v-k-1)
  5. ( v - k - 1 ) × μ (v-k-1)\times\mu
  6. A J = J A = k J , AJ=JA=kJ,
  7. A 2 + ( μ - λ ) A + ( μ - k ) I = μ J , {A}^{2}+(\mu-\lambda){A}+(\mu-k){I}=\mu{J},
  8. 1 2 [ ( λ - μ ) + ( λ - μ ) 2 + 4 ( k - μ ) ] \frac{1}{2}\left[(\lambda-\mu)+\sqrt{(\lambda-\mu)^{2}+4(k-\mu)}\right]
  9. 1 2 [ ( v - 1 ) - 2 k + ( v - 1 ) ( λ - μ ) ( λ - μ ) 2 + 4 ( k - μ ) ] \frac{1}{2}\left[(v-1)-\frac{2k+(v-1)(\lambda-\mu)}{\sqrt{(\lambda-\mu)^{2}+4(% k-\mu)}}\right]
  10. 1 2 [ ( λ - μ ) - ( λ - μ ) 2 + 4 ( k - μ ) ] \frac{1}{2}\left[(\lambda-\mu)-\sqrt{(\lambda-\mu)^{2}+4(k-\mu)}\right]
  11. 1 2 [ ( v - 1 ) + 2 k + ( v - 1 ) ( λ - μ ) ( λ - μ ) 2 + 4 ( k - μ ) ] \frac{1}{2}\left[(v-1)+\frac{2k+(v-1)(\lambda-\mu)}{\sqrt{(\lambda-\mu)^{2}+4(% k-\mu)}}\right]
  12. 2 k + ( v - 1 ) ( λ - μ ) = 0 2k+(v-1)(\lambda-\mu)=0
  13. srg ( v , 1 2 ( v - 1 ) , 1 4 ( v - 5 ) , 1 4 ( v - 1 ) ) . \,\text{srg}\left(v,\tfrac{1}{2}(v-1),\tfrac{1}{4}(v-5),\tfrac{1}{4}(v-1)% \right).
  14. 2 k + ( v - 1 ) ( λ - μ ) 0 2k+(v-1)(\lambda-\mu)\neq 0

Structural_equation_modeling.html

  1. 𝐴𝐼𝐶 = 2 k - 2 ln ( L ) \mathit{AIC}=2k-2\ln(L)\,

Structure_formation.html

  1. H H
  2. R = c / H R=c/H
  3. c c
  4. ρ ( 𝐱 , t ) \rho(\mathbf{x},t)
  5. ρ \rho
  6. ρ ^ ( 𝐤 , t ) \hat{\rho}(\mathbf{k},t)
  7. ρ ^ ( 𝐤 , t ) ρ ^ ( 𝐤 , t ) = f ( k ) δ ( 3 ) ( 𝐤 - 𝐤 ) \langle\hat{\rho}(\mathbf{k},t)\hat{\rho}(\mathbf{k}^{\prime},t)\rangle=f(k)% \delta^{(3)}(\mathbf{k}-\mathbf{k^{\prime}})
  8. δ ( 3 ) \delta^{(3)}
  9. k = | 𝐤 | k=|\mathbf{k}|
  10. 𝐤 \mathbf{k}
  11. ρ ^ ( 𝐤 , t ) ρ ^ ( 𝐤 , t ) = k n s - 1 δ ( 3 ) ( 𝐤 - 𝐤 ) \langle\hat{\rho}(\mathbf{k},t)\hat{\rho}(\mathbf{k}^{\prime},t)\rangle=k^{n_{% s}-1}\delta^{(3)}(\mathbf{k}-\mathbf{k^{\prime}})
  12. n s - 1 n_{s}-1

Stueckelberg_action.html

  1. = - 1 4 ( μ A ν - ν A μ ) ( μ A ν - ν A μ ) + 1 2 ( μ ϕ + m A μ ) ( μ ϕ + m A μ ) \mathcal{L}=-\frac{1}{4}(\partial^{\mu}A^{\nu}-\partial^{\nu}A^{\mu})(\partial% _{\mu}A_{\nu}-\partial_{\nu}A_{\mu})+\frac{1}{2}(\partial^{\mu}\phi+mA^{\mu})(% \partial_{\mu}\phi+mA_{\mu})
  2. S t = - 1 4 C μ ν C μ ν + g X C μ 𝒥 X μ - 1 2 ( μ σ + M 1 C μ + M 2 B μ ) 2 . \mathcal{L}_{St}=-\frac{1}{4}C_{\mu\nu}C^{\mu\nu}+g_{X}C_{\mu}\mathcal{J}_{X}^% {\mu}-\frac{1}{2}\left(\partial_{\mu}\sigma+M_{1}C_{\mu}+M_{2}B_{\mu}\right)^{% 2}.
  3. M 1 M_{1}
  4. M 2 M_{2}
  5. σ \sigma
  6. Z S t Z^{\prime}_{St}
  7. M 2 / M 1 0 M_{2}/M_{1}\to 0

Sturm's_theorem.html

  1. p p
  2. p p
  3. p p
  4. p p
  5. p 0 , p 1 , , p m p_{0},p_{1},\dots,p_{m}
  6. p ( ξ ) = 0 p(ξ)=0
  7. p p
  8. p 0 ( x ) \displaystyle p_{0}(x)
  9. m m
  10. d e g ( p ) deg(p)
  11. p p
  12. p p
  13. p 0 , p 1 , p 2 , , p m . p_{0},p_{1},p_{2},\ldots,p_{m}.
  14. p p
  15. p p
  16. σ ( ξ ) σ(ξ)
  17. p 0 ( ξ ) , p 1 ( ξ ) , p 2 ( ξ ) , , p m ( ξ ) . p_{0}(\xi),p_{1}(\xi),p_{2}(\xi),\ldots,p_{m}(\xi).
  18. p p
  19. ( a , b (a,b
  20. σ ( a ) σ ( b ) σ(a)−σ(b)
  21. p p
  22. σ ( a ) σ ( b ) σ(a)−σ(b)
  23. p p
  24. ( a , b (a,b
  25. a a
  26. b b
  27. p p
  28. p ( x ) = x 4 + x 3 - x - 1 p(x)=x^{4}+x^{3}-x-1
  29. p 0 ( x ) = p ( x ) = x 4 + x 3 - x - 1 p 1 ( x ) = p ( x ) = 4 x 3 + 3 x 2 - 1 \begin{aligned}\displaystyle p_{0}(x)&\displaystyle=p(x)=x^{4}+x^{3}-x-1\\ \displaystyle p_{1}(x)&\displaystyle=p^{\prime}(x)=4x^{3}+3x^{2}-1\end{aligned}
  30. - 3 16 x 2 - 3 4 x - 15 16 -\tfrac{3}{16}x^{2}-\tfrac{3}{4}x-\tfrac{15}{16}
  31. 1 −1
  32. p 2 ( x ) = 3 16 x 2 + 3 4 x + 15 16 p_{2}(x)=\tfrac{3}{16}x^{2}+\tfrac{3}{4}x+\tfrac{15}{16}
  33. 1 −1
  34. p 3 ( x ) = - 32 x - 64 p_{3}(x)=-32x-64
  35. 1 −1
  36. p 4 ( x ) = - 3 16 p_{4}(x)=-\tfrac{3}{16}
  37. p 0 ( x ) = x 4 + x 3 - x - 1 p_{0}(x)=x^{4}+x^{3}-x-1
  38. p 1 ( x ) = 4 x 3 + 3 x 2 - 1 p_{1}(x)=4x^{3}+3x^{2}-1
  39. p 2 ( x ) = 3 16 x 2 + 3 4 x + 15 16 p_{2}(x)=\tfrac{3}{16}x^{2}+\tfrac{3}{4}x+\tfrac{15}{16}
  40. p 3 ( x ) = - 32 x - 64 p_{3}(x)=-32x-64
  41. p 4 ( x ) = - 3 16 p_{4}(x)=-\tfrac{3}{16}
  42. −∞
  43. −∞
  44. + + + +−++−
  45. + +
  46. + +
  47. + +
  48. + + + +++−−
  49. −∞
  50. 3 1 = 2. 3−1=2.
  51. 1 −1
  52. 1 1
  53. p p
  54. ξ ξ
  55. x x
  56. ξ ξ
  57. ξ ξ
  58. x x
  59. ξ ξ
  60. x x
  61. ξ ξ
  62. ξ ξ
  63. p ( ξ ) p(ξ)
  64. p p
  65. ξ ξ
  66. p p
  67. ξ ξ
  68. ξ ξ
  69. p p
  70. q q
  71. i i
  72. σ ( x ) σ(x)
  73. x x
  74. d d
  75. ξ ξ
  76. ξ ξ
  77. q q
  78. a = b = M −a=b=M
  79. M M
  80. M , M −−M,M
  81. M = 1 + max i = 0 n - 1 | a i | | a n | . M=1+\frac{\max_{i=0}^{n-1}|a_{i}|}{|a_{n}|}.
  82. x x
  83. x −x
  84. x x
  85. p ( x ) = a n x n + p(x)=a_{n}x^{n}+\cdots
  86. s i g n ( p ( x ) ) sign(p(−x))
  87. ξ ξ
  88. ξ ξ
  89. k k
  90. ξ ξ
  91. k 1 k−1
  92. p p
  93. ξ ξ
  94. p ( x ) = k = 0 i p k x k , p ~ ( x ) = k = 0 i - 1 p ~ k x k , p ~ i - 1 0 , p(x)=\sum_{k=0}^{i}p_{k}x^{k},\qquad\widetilde{p}(x)=\sum_{k=0}^{i-1}% \widetilde{p}_{k}x^{k},\quad\widetilde{p}_{i-1}\neq 0,
  95. rem ( p , p ~ ) = p ~ i - 1 p ( x ) - p i p ~ ( x ) ( x + p i - 1 p i - p ~ i - 2 p ~ i - 1 ) , \,\text{rem}\left(p,\widetilde{p}\right)=\widetilde{p}_{i-1}\cdot p(x)-p_{i}% \cdot\widetilde{p}(x)\left(x+\frac{p_{i-1}}{p_{i}}-\frac{\widetilde{p}_{i-2}}{% \widetilde{p}_{i-1}}\right),
  96. ξ ξ
  97. a a , b aa,b
  98. a a , b aa,b
  99. ( X < s u b > 0 , X 1 , , X r ) (X<sub>0,X_{1},...,X_{r})

Sturmian_word.html

  1. s X s\in X
  2. n ( w ) \mathcal{L}_{n}(w)
  3. n ( w ) \mathcal{L}_{n}(w)
  4. x [ 0 , 1 ) x\in[0,1)
  5. θ ( 0 , ) \theta\in(0,\infty)
  6. f ( t ) = θ t + x f(t)=\theta t+x
  7. x [ 0 , 1 ) x\in[0,1)
  8. θ ( 0 , 1 ) \theta\in(0,1)
  9. w n = n θ + x - ( n - 1 ) θ + x w_{n}=\lfloor n\theta+x\rfloor-\lfloor(n-1)\theta+x\rfloor
  10. n n
  11. w n = n θ + x - ( n - 1 ) θ + x w_{n}=\lceil n\theta+x\rceil-\lceil(n-1)\theta+x\rceil
  12. n n
  13. θ [ 0 , 1 ) \theta\in[0,1)
  14. T θ : [ 0 , 1 ) [ 0 , 1 ) T_{\theta}:[0,1)\to[0,1)
  15. t t + θ mod 1 t\mapsto t+\theta\mod 1
  16. x [ 0 , 1 ) x\in[0,1)
  17. x n = { 1 if T θ n ( x ) [ 0 , θ ) 0 else x_{n}=\left\{\begin{array}[]{cl}1&\,\text{ if }T_{\theta}^{n}(x)\in[0,\theta)% \\ 0&\,\text{ else}\end{array}\right.
  18. x [ 0 , 1 ) x\in[0,1)
  19. θ ( 0 , ) \theta\in(0,\infty)
  20. 1 / ϕ 1/\phi
  21. ϕ \phi
  22. ( a n ) n (a_{n})_{n\in\mathbb{N}}
  23. α \alpha
  24. ρ \rho
  25. α \alpha
  26. a n = α ( n + 1 ) + ρ - α n + ρ - α a_{n}=\lfloor\alpha(n+1)+\rho\rfloor-\lfloor\alpha n+\rho\rfloor-\lfloor\alpha\rfloor
  27. n n
  28. α \alpha
  29. 0 < α < 1 0<\alpha<1
  30. ( α + k ) ( n + 1 ) + ρ - ( α + k ) n + ρ - α + k = a n . \lfloor(\alpha+k)(n+1)+\rho\rfloor-\lfloor(\alpha+k)n+\rho\rfloor-\lfloor% \alpha+k\rfloor=a_{n}.
  31. α \alpha
  32. c α c_{\alpha}
  33. ρ = 0 \rho=0
  34. α \alpha
  35. 0 < α < 1 0<\alpha<1
  36. c α c_{\alpha}
  37. α \alpha
  38. c α c_{\alpha}
  39. ( s n ) n 0 (s_{n})_{n\geq 0}
  40. [ 0 ; d 1 + 1 , d 2 , , d n , ] [0;d_{1}+1,d_{2},\ldots,d_{n},\ldots]
  41. α \alpha
  42. s 0 = 1 s_{0}=1
  43. s 1 = 0 s_{1}=0
  44. s n + 1 = s n d n s n - 1 for n > 0 s_{n+1}=s_{n}^{d_{n}}s_{n-1}\,\text{ for }n>0
  45. ( s n ) n > 0 (s_{n})_{n>0}
  46. c α c_{\alpha}
  47. ( s n ) n 0 (s_{n})_{n\geq 0}
  48. c α c_{\alpha}

Subgroup_series.html

  1. 1 = A 0 A 1 A n = G . 1=A_{0}\leq A_{1}\leq\cdots\leq A_{n}=G.
  2. 1 = A 0 A 1 A n = G . 1=A_{0}\triangleleft A_{1}\triangleleft\cdots\triangleleft A_{n}=G.
  3. 1 G 1\triangleleft G
  4. 1 = A 0 A 1 A n = G 1=A_{0}\leq A_{1}\leq\cdots\leq A_{n}=G
  5. G = B 0 B 1 B n = 1. G=B_{0}\geq B_{1}\geq\cdots\geq B_{n}=1.
  6. 1 = A 0 A 1 G 1=A_{0}\leq A_{1}\leq\cdots\leq G
  7. G = B 0 B 1 1 G=B_{0}\geq B_{1}\geq\cdots\geq 1
  8. 1 = A 0 A 1 G 1=A_{0}\leq A_{1}\leq\cdots\leq G
  9. A i A_{i}
  10. 1 = A 0 A 1 A ω A ω + 1 = G 1=A_{0}\leq A_{1}\leq\cdots\leq A_{\omega}\leq A_{\omega+1}=G
  11. A λ := α < λ A α A_{\lambda}:=\bigcup_{\alpha<\lambda}A_{\alpha}
  12. A λ := α < λ A α A_{\lambda}:=\bigcap_{\alpha<\lambda}A_{\alpha}
  13. 1 A - 1 A 0 A 1 G 1\leq\cdots\leq A_{-1}\leq A_{0}\leq A_{1}\leq\cdots\leq G
  14. A i + 1 / A i Z ( G / A i ) A_{i+1}/A_{i}\subseteq Z(G/A_{i})
  15. i = 0 , 1 , , n - 2 i=0,1,\ldots,n-2

Sublinear_function.html

  1. f : V 𝐅 f:V\rightarrow\mathbf{F}
  2. \mathbb{R}
  3. f ( γ x ) = γ f ( x ) f(\gamma x)=\gamma f\left(x\right)
  4. γ 𝐅 \gamma\in\mathbf{F}
  5. f ( x + y ) f ( x ) + f ( y ) f(x+y)\leq f(x)+f(y)
  6. f : + f:\mathbb{Z^{+}}\rightarrow\mathbb{R}
  7. f ( n ) o ( n ) f(n)\in o(n)
  8. o \,o
  9. f ( n ) o ( n ) f(n)\in o(n)
  10. c > 0 \,c>0
  11. n 0 \,n_{0}
  12. n n 0 0 f ( n ) < c n n\geq n_{0}\Rightarrow 0\leq f(n)<c\cdot n
  13. g g
  14. f f
  15. g g
  16. \mathbb{R}

Subshift_of_finite_type.html

  1. V V
  2. n n
  3. n × n n\times n
  4. Σ A + = { ( x 0 , x 1 , ) : x j V , A x j x j + 1 = 1 , j } . \Sigma_{A}^{+}=\left\{(x_{0},x_{1},\ldots):x_{j}\in V,A_{x_{j}x_{j+1}}=1,j\in% \mathbb{N}\right\}.
  5. Σ A = { ( , x - 1 , x 0 , x 1 , ) : x j V , A x j x j + 1 = 1 , j } . \Sigma_{A}=\left\{(\ldots,x_{-1},x_{0},x_{1},\ldots):x_{j}\in V,A_{x_{j}x_{j+1% }}=1,j\in\mathbb{Z}\right\}.
  6. ( T ( x ) ) j = x j + 1 . \displaystyle(T(x))_{j}=x_{j+1}.
  7. V V^{\mathbb{Z}}
  8. V = Π n V = { x = ( , x - 1 , x 0 , x 1 , ) : x k V k } V^{\mathbb{Z}}=\Pi_{n\in\mathbb{Z}}V=\{x=(\ldots,x_{-1},x_{0},x_{1},\ldots):x_% {k}\in V\;\forall k\in\mathbb{Z}\}
  9. C t [ a 0 , , a s ] = { x V : x t = a 0 , , x t + s = a s } C_{t}[a_{0},\ldots,a_{s}]=\{x\in V^{\mathbb{Z}}:x_{t}=a_{0},\ldots,x_{t+s}=a_{% s}\}
  10. n × n n\times n
  11. P = ( p i j ) P=(p_{ij})
  12. p i j 0 p_{ij}\geq 0
  13. j = 1 n p i j = 1 \sum_{j=1}^{n}p_{ij}=1
  14. π = ( π i ) \pi=(\pi_{i})
  15. π i 0 , π i = 1 \pi_{i}\geq 0,\sum\pi_{i}=1
  16. i = 1 n π i p i j = π j \sum_{i=1}^{n}\pi_{i}p_{ij}=\pi_{j}
  17. p i j = 0 p_{ij}=0
  18. A i j = 0 A_{ij}=0
  19. μ ( C t [ a 0 , , a s ] ) = π a 0 p a 0 , a 1 p a s - 1 , a s \mu(C_{t}[a_{0},\ldots,a_{s}])=\pi_{a_{0}}p_{a_{0},a_{1}}\cdots p_{a_{s-1},a_{% s}}
  20. s μ = - i = 1 n π i j = 1 n p i j log p i j s_{\mu}=-\sum_{i=1}^{n}\pi_{i}\sum_{j=1}^{n}p_{ij}\log p_{ij}
  21. ζ ( z ) = exp ( n = 1 | Fix ( T n ) | z n n ) , \zeta(z)=\exp(\sum_{n=1}^{\infty}\left|\textrm{Fix}(T^{n})\right|\frac{z^{n}}{% n}),
  22. ζ ( z ) = γ ( 1 - z | γ | ) - 1 \zeta(z)=\prod_{\gamma}(1-z^{|\gamma|})^{-1}
  23. ζ ( z ) = ( det ( I - z A ) ) - 1 . \zeta(z)=(\det(I-zA))^{-1}\ .

Substitution_model.html

  1. π i Q i j = π j Q j i \pi_{i}Q_{ij}=\pi_{j}Q_{ji}
  2. π i P ( t ) i j = π j P ( t ) j i \pi_{i}P(t)_{ij}=\pi_{j}P(t)_{ji}
  3. Q i j Q_{ij}
  4. Q i i = - { j j i } Q i j , Q_{ii}=-{\sum_{\{j\mid j\neq i\}}Q_{ij}}\,,
  5. π Q = 0 . \pi\,Q=0\,.
  6. P ( t ) P(t)
  7. P i j ( t ) P_{ij}(t)
  8. lim t P i j ( t ) = π j , \lim_{t\rightarrow\infty}P_{ij}(t)=\pi_{j}\,,
  9. π P ( t ) = π \pi P(t)=\pi
  10. P ( t ) = e Q t = n = 0 Q n t n n ! , P(t)=e^{Qt}=\sum_{n=0}^{\infty}Q^{n}\frac{t^{n}}{n!}\,,
  11. Λ = ( λ 1 0 0 λ 4 ) , \Lambda=\begin{pmatrix}\lambda_{1}&\ldots&0\\ \vdots&\ddots&\vdots\\ 0&\ldots&\lambda_{4}\end{pmatrix}\,,
  12. { λ i } \{\lambda_{i}\}
  13. P ( t ) = e Q t = e U - 1 ( Λ t ) U = U - 1 e Λ t U , P(t)=e^{Qt}=e^{U^{-1}(\Lambda t)U}=U^{-1}e^{\Lambda t}\,U\,,
  14. e Λ t = ( e λ 1 t 0 0 e λ 4 t ) . e^{\Lambda t}=\begin{pmatrix}e^{\lambda_{1}t}&\ldots&0\\ \vdots&\ddots&\vdots\\ 0&\ldots&e^{\lambda_{4}t}\end{pmatrix}\,.
  15. π = ( π 1 , π 2 , π 3 , π 4 ) \vec{\pi}=(\pi_{1},\pi_{2},\pi_{3},\pi_{4})
  16. Q = ( - ( x 1 + x 2 + x 3 ) x 1 x 2 x 3 π 1 x 1 π 2 - ( π 1 x 1 π 2 + x 4 + x 5 ) x 4 x 5 π 1 x 2 π 3 π 2 x 4 π 3 - ( π 1 x 2 π 3 + π 2 x 4 π 3 + x 6 ) x 6 π 1 x 3 π 4 π 2 x 5 π 4 π 3 x 6 π 4 - ( π 1 x 3 π 4 + π 2 x 5 π 4 + π 3 x 6 π 4 ) ) Q=\begin{pmatrix}{-(x_{1}+x_{2}+x_{3})}&x_{1}&x_{2}&x_{3}\\ {\pi_{1}x_{1}\over\pi_{2}}&{-({\pi_{1}x_{1}\over\pi_{2}}+x_{4}+x_{5})}&x_{4}&x% _{5}\\ {\pi_{1}x_{2}\over\pi_{3}}&{\pi_{2}x_{4}\over\pi_{3}}&{-({\pi_{1}x_{2}\over\pi% _{3}}+{\pi_{2}x_{4}\over\pi_{3}}+x_{6})}&x_{6}\\ {\pi_{1}x_{3}\over\pi_{4}}&{\pi_{2}x_{5}\over\pi_{4}}&{\pi_{3}x_{6}\over\pi_{4% }}&{-({\pi_{1}x_{3}\over\pi_{4}}+{\pi_{2}x_{5}\over\pi_{4}}+{\pi_{3}x_{6}\over% \pi_{4}})}\end{pmatrix}
  17. μ \mu
  18. μ \mu
  19. n 2 - n 2 {{n^{2}-n}\over 2}
  20. μ \mu
  21. n 2 - n 2 + ( n - 1 ) - 1 = 1 2 n 2 + 1 2 n - 2. {{n^{2}-n}\over 2}+(n-1)-1={1\over 2}n^{2}+{1\over 2}n-2.
  22. 4 3 = 64 4^{3}=64
  23. 20 × 19 × 3 2 + 63 - 1 = 632 {{20\times 19\times 3}\over 2}+63-1=632

Successor.html

  1. S ( n ) = n + 1 S(n)=n+1

Suffix_array.html

  1. | |
  2. 𝒪 ( n ) \mathcal{O}(n)
  3. 𝒪 ( n ) \mathcal{O}(n)
  4. 𝒪 ( n ) \mathcal{O}(n)
  5. 𝒪 ( n ) \mathcal{O}(n)
  6. S = S [ 1 ] S [ 2 ] S [ n ] S=S[1]S[2]...S[n]
  7. S [ i , j ] S[i,j]
  8. S S
  9. i i
  10. j j
  11. A A
  12. S S
  13. S S
  14. A [ i ] A[i]
  15. i i
  16. S S
  17. 1 < i n 1<i\leq n
  18. S [ A [ i - 1 ] , n ] < S [ A [ i ] , n ] S[A[i-1],n]<S[A[i],n]
  19. S S
  20. S [ i ] S[i]
  21. A A
  22. A [ i ] A[i]
  23. A [ i ] A[i]
  24. A [ 3 ] A[3]
  25. S S
  26. n n
  27. 4 4
  28. 4 n 4n
  29. 20 n 20n
  30. 𝒪 ( n log n ) \mathcal{O}(n\log n)
  31. σ \sigma
  32. 𝒪 ( n log σ ) \mathcal{O}(n\log\sigma)
  33. σ = 4 \sigma=4
  34. n = 3.4 × 10 9 n=3.4\times 10^{9}
  35. 𝒪 ( n ) \mathcal{O}(n)
  36. 𝒪 ( n ) \mathcal{O}(n)
  37. 𝒪 ( n ) \mathcal{O}(n)
  38. 𝒪 ( n log n ) \mathcal{O}(n\log n)
  39. 𝒪 ( n ) \mathcal{O}(n)
  40. 𝒪 ( n 2 log n ) \mathcal{O}(n^{2}\log n)
  41. Θ ( n ) \Theta(n)
  42. n n
  43. 𝒪 ( n log n ) \mathcal{O}(n\log n)
  44. P P
  45. S S

Sulfamic_acid.html

  1. \rightleftarrows

Sum-free_sequence.html

  1. { n k } k \{n_{k}\}_{k\in\mathbb{N}}
  2. k k
  3. n k n_{k}
  4. R R
  5. 2.065 < R < 3.0752 2.065<R<3.0752
  6. A ( x ) A(x)
  7. x x
  8. A ( x ) = o ( x ) A(x)=o(x)
  9. x i x_{i}
  10. A ( x i ) = O ( x φ - 1 ) A(x_{i})=O(x^{\varphi-1})
  11. φ \varphi
  12. x x
  13. A ( x ) = Ω ( x 2 / 7 ) A(x)=\Omega(x^{2/7})
  14. A ( x ) = Ω ( x 1 / 3 ) A(x)=\Omega(x^{1/3})
  15. A ( x ) = Ω ( x 1 / 2 - ε ) A(x)=\Omega(x^{1/2-\varepsilon})

Superabundant_number.html

  1. σ ( m ) m < σ ( n ) n \frac{\sigma(m)}{m}<\frac{\sigma(n)}{n}
  2. n = i = 1 k ( p i ) a i n=\prod_{i=1}^{k}(p_{i})^{a_{i}}
  3. a 1 a 2 a k 1. a_{1}\geq a_{2}\geq\cdots\geq a_{k}\geq 1.
  4. p k p_{k}
  5. p k # . p_{k}\#.
  6. σ ( n ) e γ n log log n < 1 \frac{\sigma(n)}{e^{\gamma}n\log\log n}<1

Superadditivity.html

  1. a n + m a n + a m a_{n+m}\geq a_{n}+a_{m}\,
  2. f ( x + y ) f ( x ) + f ( y ) f(x+y)\geq f(x)+f(y)\,
  3. f ( x ) = x 2 f(x)=x^{2}
  4. ( x + y ) (x+y)
  5. x x
  6. y y
  7. x x
  8. y y
  9. f ( x ) f ( x + y ) - f ( y ) f(x)\leq f(x+y)-f(y)
  10. f ( 0 ) f ( 0 + y ) - f ( y ) = 0 f(0)\leq f(0+y)-f(y)=0

Superatom.html

  1. n = 1 , 2 , 3 , n=1, 2, 3, …
  2. x = 1 13 x=1–13
  3. x = 1 14 x=1–14
  4. n e = N ν A - M - z n_{e}=N\nu_{A}-M-z
  5. N N
  6. v v
  7. M M
  8. z z

Superconducting_quantum_computing.html

  1. L L
  2. C C
  3. Φ \Phi
  4. Q Q
  5. [ Φ , Q ] = i [\Phi,Q]=i\hbar
  6. H = Φ 2 2 L + Q 2 2 C H=\frac{\Phi^{2}}{2L}+\frac{Q^{2}}{2C}
  7. E = ω 0 ( n + 1 2 ) E=\hbar\omega_{0}(n+\frac{1}{2})
  8. ω 0 = 1 L C \omega_{0}=\frac{1}{\sqrt{LC}}

Superdiamagnetism.html

  1. χ v \chi_{v}

Superformula.html

  1. r r
  2. φ \varphi
  3. r ( φ ) = [ | cos ( m φ 4 ) a | n 2 + | sin ( m φ 4 ) b | n 3 ] - 1 n 1 . r\left(\varphi\right)=\left[\left|\frac{\cos\left(\frac{m\varphi}{4}\right)}{a% }\right|^{n_{2}}+\left|\frac{\sin\left(\frac{m\varphi}{4}\right)}{b}\right|^{n% _{3}}\right]^{-\frac{1}{n_{1}}}.
  4. x = r 1 ( θ ) cos θ r 2 ( ϕ ) cos ϕ , x=r_{1}(\theta)\cos\theta\cdot r_{2}(\phi)\cos\phi,
  5. y = r 1 ( θ ) sin θ r 2 ( ϕ ) cos ϕ , y=r_{1}(\theta)\sin\theta\cdot r_{2}(\phi)\cos\phi,
  6. z = r 2 ( ϕ ) sin ϕ , z=r_{2}(\phi)\sin\phi,
  7. ϕ \phi

Superior_highly_composite_number.html

  1. 2 2
  2. 2 2
  3. 2 3 2⋅3
  4. 6 6
  5. 2 < s u p > 2 3 2<sup>2⋅3
  6. d ( n ) n ε d ( k ) k ε \frac{d(n)}{n^{\varepsilon}}\geq\frac{d(k)}{k^{\varepsilon}}
  7. d ( n ) n ε > d ( k ) k ε \frac{d(n)}{n^{\varepsilon}}>\frac{d(k)}{k^{\varepsilon}}
  8. e p ( x ) = 1 p x - 1 e_{p}(x)=\left\lfloor\frac{1}{\sqrt[x]{p}-1}\right\rfloor\quad
  9. s ( x ) = p p e p ( x ) \quad s(x)=\prod_{p\in\mathbb{P}}p^{e_{p}(x)}\quad
  10. p > 2 x p>2^{x}
  11. e p ( x ) = 0 e_{p}(x)=0
  12. s ( x ) s(x)
  13. p 2 x p\geq 2^{x}
  14. e p ( x ) e_{p}(x)
  15. 1 / x 1/x
  16. ε \varepsilon
  17. s s^{\prime}
  18. I \R + I\subset\R^{+}
  19. x I : s ( x ) = s \forall x\in I:s(x)=s^{\prime}
  20. π 1 , π 2 , \pi_{1},\pi_{2},\ldots\in\mathbb{P}
  21. s n s_{n}
  22. s n = i = 1 n π i s_{n}=\prod_{i=1}^{n}\pi_{i}
  23. π i \pi_{i}

Superlattice.html

  1. d = a + b d=a+b
  2. ψ \psi
  3. E z ( k z ) E_{z}(k_{z})
  4. 2 π / d 2\pi/d
  5. d = a + b d=a+b
  6. E z ( k z ) = Δ 2 ( 1 - cos ( k z d ) ) \ E_{z}(k_{z})=\frac{\Delta}{2}(1-\cos(k_{z}d))
  7. 2 π / d 2\pi/d
  8. m * = 2 2 E / k 2 | k = 0 \ {m^{*}=\frac{\hbar^{2}}{\partial^{2}E/\partial k^{2}}}|_{k=0}
  9. 2 π / d 2\pi/d
  10. E 2 - E 1 E_{2}-E_{1}
  11. e i 𝐤 𝐫 / 2 π e^{i\mathbf{k}\cdot\mathbf{r}}/2\pi
  12. f k ( z ) f_{k}(z)
  13. ( E c ( z ) - z 2 2 m c ( z ) z + 2 𝐤 2 2 m c ( z ) ) f k ( z ) = E f k ( z ) \left(E_{c}(z)-\frac{\partial}{\partial z}\frac{\hbar^{2}}{2m_{c}(z)}\frac{% \partial}{\partial z}+\frac{\hbar^{2}\mathbf{k}^{2}}{2m_{c}(z)}\right)f_{k}(z)% =Ef_{k}(z)
  14. E c ( z ) E_{c}(z)
  15. m c ( z ) m_{c}(z)
  16. f k ( z ) = ϕ q , 𝐤 ( z ) f_{k}(z)=\phi_{q,\mathbf{k}}(z)
  17. E ν ( q , 𝐤 ) E^{\nu}(q,\mathbf{k})
  18. E ν ( q , 𝐤 ) E ν ( q , 𝟎 ) + ϕ q , 𝐤 2 𝐤 2 2 m c ( z ) ϕ q , 𝐤 E^{\nu}(q,\mathbf{k})\approx E^{\nu}(q,\mathbf{0})+\langle\phi_{q,\mathbf{k}}% \mid\frac{\hbar^{2}\mathbf{k}^{2}}{2m_{c}(z)}\mid\phi_{q,\mathbf{k}}\rangle
  19. ϕ q , 𝟎 ( z ) \phi_{q,\mathbf{0}}(z)
  20. E k = 2 𝐤 2 2 m w E_{k}=\frac{\hbar^{2}\mathbf{k}^{2}}{2m_{w}}
  21. m w m_{w}
  22. ϕ 0 ( z ) \phi_{0}(z)
  23. E 0 E_{0}
  24. Φ j ( z ) = Φ 0 ( z - j d ) \Phi_{j}(z)=\Phi_{0}(z-jd)
  25. Φ 0 ( z ) \Phi_{0}(z)
  26. V ( x , y ) = - V 0 ( cos 2 π x / a + cos 2 π y / a ) , V 0 > 0 V(x,y)=-V_{0}(\cos 2\pi x/a+\cos 2\pi y/a),V_{0}>0
  27. | V 0 | E f |V_{0}|\gg E_{f}
  28. E 1 / a 2 E\propto 1/a^{2}
  29. | V 0 | E f |V_{0}|\ll E_{f}

Superposition_principle.html

  1. F ( x 1 + x 2 ) = F ( x 1 ) + F ( x 2 ) F(x_{1}+x_{2})=F(x_{1})+F(x_{2})\,
  2. F ( a x ) = a F ( x ) F(ax)=aF(x)\,
  3. a a
  4. F ( y ) = 0 F(y)=0
  5. G ( y ) = z G(y)=z
  6. F ( y 1 ) = F ( y 2 ) = = 0 F ( y 1 + y 2 + ) = 0 F(y_{1})=F(y_{2})=\cdots=0\ \Rightarrow\ F(y_{1}+y_{2}+\cdots)=0
  7. G ( y 1 ) + G ( y 2 ) = G ( y 1 + y 2 ) G(y_{1})+G(y_{2})=G(y_{1}+y_{2})
  8. x ˙ = A x + B ( u 1 + u 2 ) , x ( 0 ) = x 0 . \dot{x}=Ax+B(u_{1}+u_{2}),x(0)=x_{0}.
  9. x ˙ 1 = A x 1 + B u 1 , x 1 ( 0 ) = x 0 . \dot{x}_{1}=Ax_{1}+Bu_{1},x_{1}(0)=x_{0}.
  10. x ˙ 2 = A x 2 + B u 2 , x 2 ( 0 ) = 0. \dot{x}_{2}=Ax_{2}+Bu_{2},x_{2}(0)=0.
  11. x = x 1 + x 2 . x=x_{1}+x_{2}.
  12. x ˙ = A x + B ( u 1 + u 2 ) + ϕ ( c T x ) , x ( 0 ) = x 0 . \dot{x}=Ax+B(u_{1}+u_{2})+\phi(c^{T}x),x(0)=x_{0}.
  13. ϕ \phi
  14. x ˙ 1 = A x 1 + B u 1 + ϕ ( y d ) , x 1 ( 0 ) = x 0 . \dot{x}_{1}=Ax_{1}+Bu_{1}+\phi(y_{d}),x_{1}(0)=x_{0}.
  15. x ˙ 2 = A x 2 + B u 2 + ϕ ( c T x 1 + c T x 2 ) - ϕ ( y d ) , x 2 ( 0 ) = 0. \dot{x}_{2}=Ax_{2}+Bu_{2}+\phi(c^{T}x_{1}+c^{T}x_{2})-\phi(y_{d}),x_{2}(0)=0.
  16. x = x 1 + x 2 . x=x_{1}+x_{2}.

Superquadrics.html

  1. | x | r + | y | s + | z | t = 1 \left|x\right|^{r}+\left|y\right|^{s}+\left|z\right|^{t}=1
  2. | x A | r + | y B | s + | z C | t 1 \left|\frac{x}{A}\right|^{r}+\left|\frac{y}{B}\right|^{s}+\left|\frac{z}{C}% \right|^{t}\leq 1
  3. x ( u , v ) = A c ( v , 2 r ) c ( u , 2 r ) y ( u , v ) = B c ( v , 2 s ) s ( u , 2 s ) z ( u , v ) = C s ( v , 2 t ) - π 2 v π 2 , - π u < π , \begin{aligned}\displaystyle x(u,v)&\displaystyle{}=Ac\left(v,\frac{2}{r}% \right)c\left(u,\frac{2}{r}\right)\\ \displaystyle y(u,v)&\displaystyle{}=Bc\left(v,\frac{2}{s}\right)s\left(u,% \frac{2}{s}\right)\\ \displaystyle z(u,v)&\displaystyle{}=Cs\left(v,\frac{2}{t}\right)\\ &\displaystyle-\frac{\pi}{2}\leq v\leq\frac{\pi}{2},\quad-\pi\leq u<\pi,\end{aligned}
  4. c ( ω , m ) = sgn ( cos ω ) | cos ω | m s ( ω , m ) = sgn ( sin ω ) | sin ω | m \begin{aligned}\displaystyle c(\omega,m)&\displaystyle{}=\operatorname{sgn}(% \cos\omega)|\cos\omega|^{m}\\ \displaystyle s(\omega,m)&\displaystyle{}=\operatorname{sgn}(\sin\omega)|\sin% \omega|^{m}\end{aligned}
  5. sgn ( x ) = { - 1 , x < 0 0 , x = 0 + 1 , x > 0. \operatorname{sgn}(x)=\begin{cases}-1,&x<0\\ 0,&x=0\\ +1,&x>0.\end{cases}

Supertrace.html

  1. T = ( T 00 T 01 T 10 T 11 ) T=\begin{pmatrix}T_{00}&T_{01}\\ T_{10}&T_{11}\end{pmatrix}
  2. T ( 𝐞 j ) = 𝐞 i T j i . T(\mathbf{e}_{j})=\mathbf{e}_{i}T^{i}_{j}.\,
  3. 𝐞 i = 𝐞 i A i i \mathbf{e}_{i^{\prime}}=\mathbf{e}_{i}A^{i}_{i^{\prime}}
  4. 𝐞 i = 𝐞 i ( A - 1 ) i i , \mathbf{e}_{i}=\mathbf{e}_{i^{\prime}}(A^{-1})^{i^{\prime}}_{i},\,
  5. str ( A - 1 T A ) = ( - 1 ) | i | ( A - 1 ) j i T k j A i k = ( - 1 ) | i | ( - 1 ) ( | i | + | j | ) ( | i | + | j | ) T k j A i k ( A - 1 ) j i = ( - 1 ) | j | T j j = str ( T ) . \operatorname{str}(A^{-1}TA)=(-1)^{|i^{\prime}|}(A^{-1})^{i^{\prime}}_{j}T^{j}% _{k}A^{k}_{i^{\prime}}=(-1)^{|i^{\prime}|}(-1)^{(|i^{\prime}|+|j|)(|i^{\prime}% |+|j|)}T^{j}_{k}A^{k}_{i^{\prime}}(A^{-1})^{i^{\prime}}_{j}=(-1)^{|j|}T^{j}_{j% }=\operatorname{str}(T).
  6. str ( A - 1 T A ) = ( - 1 ) | i | ( A - 1 ) j i T k j A i k = ( - 1 ) | i | ( - 1 ) ( 1 + | j | + | k | ) ( | i | + | j | ) T k j ( A - 1 ) j i A i k = ( - 1 ) | j | T j j = str ( T ) . \operatorname{str}(A^{-1}TA)=(-1)^{|i^{\prime}|}(A^{-1})^{i^{\prime}}_{j}T^{j}% _{k}A^{k}_{i^{\prime}}=(-1)^{|i^{\prime}|}(-1)^{(1+|j|+|k|)(|i^{\prime}|+|j|)}% T^{j}_{k}(A^{-1})^{i^{\prime}}_{j}A^{k}_{i^{\prime}}=(-1)^{|j|}T^{j}_{j}=% \operatorname{str}(T).
  7. str ( T 1 T 2 ) = ( - 1 ) | T 1 | | T 2 | str ( T 2 T 1 ) \operatorname{str}(T_{1}T_{2})=(-1)^{|T_{1}||T_{2}|}\operatorname{str}(T_{2}T_% {1})
  8. str [ M 2 ] = s ( - 1 ) 2 s ( 2 s + 1 ) tr [ m s 2 ] . \operatorname{str}[M^{2}]=\sum_{s}(-1)^{2s}(2s+1)\operatorname{tr}[m_{s}^{2}].
  9. M M
  10. V e f f 1 - l o o p = 1 64 π 2 str [ M 4 ln ( M 2 Λ 2 ) ] = 1 64 π 2 tr [ m B 4 ln ( m B 2 Λ 2 ) - m F 4 ln ( m F 2 Λ 2 ) ] V_{eff}^{1-loop}=\dfrac{1}{64\pi^{2}}\operatorname{str}\bigg[M^{4}\ln\Big(% \dfrac{M^{2}}{\Lambda^{2}}\Big)\bigg]=\dfrac{1}{64\pi^{2}}\operatorname{tr}% \bigg[m_{B}^{4}\ln\Big(\dfrac{m_{B}^{2}}{\Lambda^{2}}\Big)-m_{F}^{4}\ln\Big(% \dfrac{m_{F}^{2}}{\Lambda^{2}}\Big)\bigg]
  11. m B m_{B}
  12. m F m_{F}
  13. Λ \Lambda

Supplee's_paradox.html

  1. v v
  2. γ 2 = 1 1 - v 2 / c 2 \gamma^{2}=\frac{1}{1-v^{2}/c^{2}}
  3. g ( γ 2 - 1 ) g(\gamma^{2}-1)
  4. γ 2 \gamma^{2}

Surface_brightness.html

  1. S = m + 2.5 log 10 A . S=m+2.5\cdot\log_{10}A.
  2. S ( m a g / a r c s e c 2 ) = M + 21.572 - 2.5 log 10 S ( L / p c 2 ) , S(mag/arcsec^{2})=M_{\odot}+21.572-2.5\log_{10}S(L_{\odot}/pc^{2}),
  3. M M_{\odot}
  4. L L_{\odot}

Surface_gravity.html

  1. × 10 1 2 \times 10^{1}2
  2. × 10 1 2 \times 10^{1}2
  3. 0.107 0.532 2 = 0.38 \frac{0.107}{0.532^{2}}=0.38
  4. g = G M r 2 g={\frac{GM}{r^{2}}}
  5. g = 4 π 3 G ρ r g={\frac{4\pi}{3}G\rho r}
  6. κ \kappa
  7. k a k^{a}
  8. k a a k b = κ k b k^{a}\nabla_{a}k^{b}=\kappa k^{b}
  9. k a k a - 1 k^{a}k_{a}\rightarrow-1
  10. r r\rightarrow\infty
  11. κ 0 \kappa\geq 0
  12. k a k^{a}
  13. k a a = t k^{a}\partial_{a}=\frac{\partial}{\partial t}
  14. k a a = t + Ω ϕ k^{a}\partial_{a}=\frac{\partial}{\partial t}+\Omega\frac{\partial}{\partial\phi}
  15. Ω \Omega
  16. k a k^{a}
  17. k a a k b = κ k b k^{a}\nabla_{a}k^{b}=\kappa k^{b}
  18. - k a b k a = κ k b -k^{a}\nabla^{b}k_{a}=\kappa k^{b}
  19. ( t , r , θ , ϕ ) (t,r,\theta,\phi)
  20. k a = ( 1 , 0 , 0 , 0 ) k^{a}=(1,0,0,0)
  21. v = t + r + 2 M ln | r - 2 M | v=t+r+2M\ln|r-2M|
  22. d s 2 = - ( 1 - 2 M r ) d v 2 + 2 d v d r + r 2 ( d θ 2 + sin 2 θ d ϕ 2 ) . ds^{2}=-\left(1-\frac{2M}{r}\right)dv^{2}+2dvdr+r^{2}(d\theta^{2}+\sin^{2}% \theta d\phi^{2}).
  23. k v = A t v k t k^{v}=A_{t}^{v}k^{t}
  24. k a = ( 1 , 0 , 0 , 0 ) k^{a^{\prime}}=(1,0,0,0)
  25. k a = ( - 1 + 2 M r , 1 , 0 , 0 ) . k_{a^{\prime}}=\left(-1+\frac{2M}{r},1,0,0\right).
  26. k a a k b = κ k b k^{a}\nabla_{a}k^{b}=\kappa k^{b}
  27. - 1 2 r ( - 1 + 2 M r ) = κ . -\frac{1}{2}\frac{\partial}{\partial r}\left(-1+\frac{2M}{r}\right)=\kappa.
  28. M M
  29. κ = 1 4 M . \kappa=\frac{1}{4M}.
  30. κ = r + - r - 2 ( r + 2 + a 2 ) = M 2 - Q 2 - J 2 / M 2 2 M 2 - Q 2 + 2 M M 2 - Q 2 - J 2 / M 2 \kappa=\frac{r_{+}-r_{-}}{2(r_{+}^{2}+a^{2})}=\frac{\sqrt{M^{2}-Q^{2}-J^{2}/M^% {2}}}{2M^{2}-Q^{2}+2M\sqrt{M^{2}-Q^{2}-J^{2}/M^{2}}}
  31. Q Q
  32. J J
  33. r ± := M ± M 2 - Q 2 - J 2 / M 2 r_{\pm}:=M\pm\sqrt{M^{2}-Q^{2}-J^{2}/M^{2}}
  34. a := J / M a:=J/M

Surface_plasmon_resonance.html

  1. K ( ω ) = ω c ε 1 ε 2 μ 1 μ 2 ε 1 μ 1 + ε 2 μ 2 K(\omega)=\frac{\omega}{c}\sqrt{\frac{\varepsilon_{1}\varepsilon_{2}\mu_{1}\mu% _{2}}{\varepsilon_{1}\mu_{1}+\varepsilon_{2}\mu_{2}}}
  2. ϵ \epsilon
  3. μ \mu
  4. K D = k d k a K_{D}=\frac{k_{\,\text{d}}}{k_{\,\text{a}}}

Suspension_(topology).html

  1. S X = ( X × I ) / { ( x 1 , 0 ) ( x 2 , 0 ) and ( x 1 , 1 ) ( x 2 , 1 ) for all x 1 , x 2 X } SX=(X\times I)/\{(x_{1},0)\sim(x_{2},0)\mbox{ and }~{}(x_{1},1)\sim(x_{2},1)% \mbox{ for all }~{}x_{1},x_{2}\in X\}
  2. f : X Y , f:X\rightarrow Y,
  3. S f : S X S Y Sf:SX\rightarrow SY
  4. S f ( [ x , t ] ) := [ f ( x ) , t ] . Sf([x,t]):=[f(x),t].
  5. S S
  6. S X SX
  7. X S 0 , X\star S^{0},
  8. S 0 S^{0}
  9. S X SX
  10. X X
  11. Σ X = ( X × I ) / ( X × { 0 } X × { 1 } { x 0 } × I ) \Sigma X=(X\times I)/(X\times\{0\}\cup X\times\{1\}\cup\{x_{0}\}\times I)
  12. Σ X S 1 X \Sigma X\cong S^{1}\wedge X
  13. Ω \Omega
  14. X X
  15. Ω X \Omega X
  16. Maps * ( Σ X , Y ) Maps * ( X , Ω Y ) \operatorname{Maps}_{*}\left(\Sigma X,Y\right)\cong\operatorname{Maps}_{*}% \left(X,\Omega Y\right)
  17. Maps * ( X , Y ) \operatorname{Maps}_{*}\left(X,Y\right)

Symbolic_integration.html

  1. d F d x = f ( x ) . \frac{dF}{dx}=f(x).
  2. F ( x ) = f ( x ) d x . F(x)=\int f(x)\,dx.
  3. x 2 d x = x 3 3 + C \int x^{2}\,dx=\frac{x^{3}}{3}+C
  4. - 1 1 x 2 d x = [ x 3 3 ] - 1 1 = 1 3 3 - ( - 1 ) 3 3 = 2 3 \int_{-1}^{1}x^{2}\,dx=\left[\frac{x^{3}}{3}\right]_{-1}^{1}=\frac{1^{3}}{3}-% \frac{(-1)^{3}}{3}=\frac{2}{3}
  5. - 1 1 x 2 d x 0.6667 \int_{-1}^{1}x^{2}\,dx\approx 0.6667

Symmetric_bilinear_form.html

  1. B B
  2. ( u , v ) (u,v)
  3. V V
  4. B ( u , v ) = B ( v , u ) B(u,v)=B(v,u)
  5. u u
  6. v v
  7. V V
  8. B : V × V K B:V\times V\rightarrow K
  9. B ( u , v ) = B ( v , u ) u , v V B(u,v)=B(v,u)\ \quad\forall u,v\in V
  10. B ( u + v , w ) = B ( u , w ) + B ( v , w ) u , v , w V B(u+v,w)=B(u,w)+B(v,w)\ \quad\forall u,v,w\in V
  11. B ( λ v , w ) = λ B ( v , w ) λ K , v , w V B(\lambda v,w)=\lambda B(v,w)\ \quad\forall\lambda\in K,\forall v,w\in V
  12. f , g V f,g\in V
  13. B ( f , g ) = 0 1 f ( t ) g ( t ) d t B(f,g)=\int_{0}^{1}f(t)g(t)dt
  14. C = { e 1 , , e n } C=\{e_{1},\ldots,e_{n}\}
  15. A i j = B ( e i , e j ) A_{ij}=B(e_{i},e_{j})
  16. B ( v , w ) B(v,w)
  17. x 𝖳 A y = y 𝖳 A x . x^{\mathsf{T}}Ay=y^{\mathsf{T}}Ax.
  18. [ e 1 e n ] = [ e 1 e n ] S \begin{bmatrix}e^{\prime}_{1}&\cdots&e^{\prime}_{n}\end{bmatrix}=\begin{% bmatrix}e_{1}&\cdots&e_{n}\end{bmatrix}S
  19. A = S 𝖳 A S . A^{\prime}=S^{\mathsf{T}}AS.
  20. A x = 0 x 𝖳 A = 0. Ax=0\Longleftrightarrow x^{\mathsf{T}}A=0.
  21. C = { e 1 , , e n } C=\{e_{1},\ldots,e_{n}\}
  22. B ( e i , e j ) = 0 i j . B(e_{i},e_{j})=0\ \forall i\neq j.
  23. C = { e 1 , , e n } C=\{e_{1},\ldots,e_{n}\}
  24. C = { e 1 , , e n } C^{\prime}=\{e^{\prime}_{1},\ldots,e^{\prime}_{n}\}
  25. e i = { e i if B ( e i , e i ) = 0 e i B ( e i , e i ) if B ( e i , e i ) > 0 e i - B ( e i , e i ) if B ( e i , e i ) < 0 e^{\prime}_{i}=\begin{cases}e_{i}&\,\text{if }B(e_{i},e_{i})=0\\ \frac{e_{i}}{\sqrt{B(e_{i},e_{i})}}&\,\text{if }B(e_{i},e_{i})>0\\ \frac{e_{i}}{\sqrt{-B(e_{i},e_{i})}}&\,\text{if }B(e_{i},e_{i})<0\end{cases}
  26. C = { e 1 , , e n } C=\{e_{1},\ldots,e_{n}\}
  27. C = { e 1 , , e n } C^{\prime}=\{e^{\prime}_{1},\ldots,e^{\prime}_{n}\}
  28. e i = { e i if B ( e i , e i ) = 0 e i / B ( e i , e i ) if B ( e i , e i ) 0 e^{\prime}_{i}=\begin{cases}e_{i}&\,\text{if }\;B(e_{i},e_{i})=0\\ e_{i}/\sqrt{B(e_{i},e_{i})}&\,\text{if }\;B(e_{i},e_{i})\neq 0\\ \end{cases}
  29. α : D ( V ) D ( V ) : W W . \alpha:D(V)\rightarrow D(V):W\mapsto W^{\perp}.

Symmetric_polynomial.html

  1. n n
  2. P P
  3. σ σ
  4. 1 , 2 , , n 1,2,...,n
  5. X 1 3 + X 2 3 - 7 X_{1}^{3}+X_{2}^{3}-7
  6. 4 X 1 2 X 2 2 + X 1 3 X 2 + X 1 X 2 3 + ( X 1 + X 2 ) 4 4X_{1}^{2}X_{2}^{2}+X_{1}^{3}X_{2}+X_{1}X_{2}^{3}+(X_{1}+X_{2})^{4}
  7. X 1 X 2 X 3 - 2 X 1 X 2 - 2 X 1 X 3 - 2 X 2 X 3 X_{1}X_{2}X_{3}-2X_{1}X_{2}-2X_{1}X_{3}-2X_{2}X_{3}\,
  8. 1 i < j n ( X i - X j ) 2 \prod_{1\leq i<j\leq n}(X_{i}-X_{j})^{2}
  9. X 1 - X 2 X_{1}-X_{2}\,
  10. X 1 X_{1}
  11. X 2 X_{2}
  12. X 2 - X 1 X_{2}-X_{1}
  13. X 1 4 X 2 2 X 3 + X 1 X 2 4 X 3 2 + X 1 2 X 2 X 3 4 X_{1}^{4}X_{2}^{2}X_{3}+X_{1}X_{2}^{4}X_{3}^{2}+X_{1}^{2}X_{2}X_{3}^{4}
  14. X 1 4 X 2 2 X 3 + X 1 X 2 4 X 3 2 + X 1 2 X 2 X 3 4 + X 1 4 X 2 X 3 2 + X 1 X 2 2 X 3 4 + X 1 2 X 2 4 X 3 X_{1}^{4}X_{2}^{2}X_{3}+X_{1}X_{2}^{4}X_{3}^{2}+X_{1}^{2}X_{2}X_{3}^{4}+X_{1}^% {4}X_{2}X_{3}^{2}+X_{1}X_{2}^{2}X_{3}^{4}+X_{1}^{2}X_{2}^{4}X_{3}
  15. P = t n + a n - 1 t n - 1 + + a 2 t 2 + a 1 t + a 0 P=t^{n}+a_{n-1}t^{n-1}+\cdots+a_{2}t^{2}+a_{1}t+a_{0}
  16. P = t n + a n - 1 t n - 1 + + a 2 t 2 + a 1 t + a 0 = ( t - x 1 ) ( t - x 2 ) ( t - x n ) . P=t^{n}+a_{n-1}t^{n-1}+\cdots+a_{2}t^{2}+a_{1}t+a_{0}=(t-x_{1})(t-x_{2})\cdots% (t-x_{n}).
  17. a n - 1 = - x 1 - x 2 - - x n a n - 2 = x 1 x 2 + x 1 x 3 + + x 2 x 3 + + x n - 1 x n = 1 i < j n x i x j a 1 = ( - 1 ) n - 1 ( x 2 x 3 x n + x 1 x 3 x 4 x n + + x 1 x 2 x n - 2 x n + x 1 x 2 x n - 1 ) = ( - 1 ) n - 1 i = 1 n j i x j a 0 = ( - 1 ) n x 1 x 2 x n . \begin{aligned}\displaystyle a_{n-1}&\displaystyle=-x_{1}-x_{2}-\cdots-x_{n}\\ \displaystyle a_{n-2}&\displaystyle=x_{1}x_{2}+x_{1}x_{3}+\cdots+x_{2}x_{3}+% \cdots+x_{n-1}x_{n}=\textstyle\sum_{1\leq i<j\leq n}x_{i}x_{j}\\ &\displaystyle{}\ \,\vdots\\ \displaystyle a_{1}&\displaystyle=(-1)^{n-1}(x_{2}x_{3}\cdots x_{n}+x_{1}x_{3}% x_{4}\cdots x_{n}+\cdots+x_{1}x_{2}\cdots x_{n-2}x_{n}+x_{1}x_{2}\cdots x_{n-1% })=\textstyle(-1)^{n-1}\sum_{i=1}^{n}\prod_{j\neq i}x_{j}\\ \displaystyle a_{0}&\displaystyle=(-1)^{n}x_{1}x_{2}\cdots x_{n}.\\ \end{aligned}
  18. ( - 1 ) n - i (-1)^{n-i}
  19. X 1 3 + X 2 3 - 7 = e 1 ( X 1 , X 2 ) 3 - 3 e 2 ( X 1 , X 2 ) e 1 ( X 1 , X 2 ) - 7 X_{1}^{3}+X_{2}^{3}-7=e_{1}(X_{1},X_{2})^{3}-3e_{2}(X_{1},X_{2})e_{1}(X_{1},X_% {2})-7
  20. m ( 3 , 1 , 1 ) ( X 1 , X 2 , X 3 ) = X 1 3 X 2 X 3 + X 1 X 2 3 X 3 + X 1 X 2 X 3 3 m_{(3,1,1)}(X_{1},X_{2},X_{3})=X_{1}^{3}X_{2}X_{3}+X_{1}X_{2}^{3}X_{3}+X_{1}X_% {2}X_{3}^{3}
  21. m ( 3 , 2 , 1 ) ( X 1 , X 2 , X 3 ) = X 1 3 X 2 2 X 3 + X 1 3 X 2 X 3 2 + X 1 2 X 2 3 X 3 + X 1 2 X 2 X 3 3 + X 1 X 2 3 X 3 2 + X 1 X 2 2 X 3 3 . m_{(3,2,1)}(X_{1},X_{2},X_{3})=X_{1}^{3}X_{2}^{2}X_{3}+X_{1}^{3}X_{2}X_{3}^{2}% +X_{1}^{2}X_{2}^{3}X_{3}+X_{1}^{2}X_{2}X_{3}^{3}+X_{1}X_{2}^{3}X_{3}^{2}+X_{1}% X_{2}^{2}X_{3}^{3}.
  22. e k ( X 1 , , X n ) = m α ( X 1 , , X n ) e_{k}(X_{1},\ldots,X_{n})=m_{\alpha}(X_{1},\ldots,X_{n})
  23. p k ( X 1 , , X n ) = X 1 k + X 2 k + + X n k . p_{k}(X_{1},\ldots,X_{n})=X_{1}^{k}+X_{2}^{k}+\cdots+X_{n}^{k}.
  24. p 3 ( X 1 , X 2 ) = 3 2 p 2 ( X 1 , X 2 ) p 1 ( X 1 , X 2 ) - 1 2 p 1 ( X 1 , X 2 ) 3 . p_{3}(X_{1},X_{2})=\textstyle\frac{3}{2}p_{2}(X_{1},X_{2})p_{1}(X_{1},X_{2})-% \frac{1}{2}p_{1}(X_{1},X_{2})^{3}.
  25. m ( 2 , 1 ) ( X 1 , X 2 ) = X 1 2 X 2 + X 1 X 2 2 m_{(2,1)}(X_{1},X_{2})=X_{1}^{2}X_{2}+X_{1}X_{2}^{2}
  26. m ( 2 , 1 ) ( X 1 , X 2 ) = 1 2 p 1 ( X 1 , X 2 ) 3 - 1 2 p 2 ( X 1 , X 2 ) p 1 ( X 1 , X 2 ) . m_{(2,1)}(X_{1},X_{2})=\textstyle\frac{1}{2}p_{1}(X_{1},X_{2})^{3}-\frac{1}{2}% p_{2}(X_{1},X_{2})p_{1}(X_{1},X_{2}).
  27. m ( 2 , 1 ) ( X 1 , X 2 , X 3 ) = X 1 2 X 2 + X 1 X 2 2 + X 1 2 X 3 + X 1 X 3 2 + X 2 2 X 3 + X 2 X 3 2 = p 1 ( X 1 , X 2 , X 3 ) p 2 ( X 1 , X 2 , X 3 ) - p 3 ( X 1 , X 2 , X 3 ) . \begin{aligned}\displaystyle m_{(2,1)}(X_{1},X_{2},X_{3})&\displaystyle=X_{1}^% {2}X_{2}+X_{1}X_{2}^{2}+X_{1}^{2}X_{3}+X_{1}X_{3}^{2}+X_{2}^{2}X_{3}+X_{2}X_{3% }^{2}\\ &\displaystyle=p_{1}(X_{1},X_{2},X_{3})p_{2}(X_{1},X_{2},X_{3})-p_{3}(X_{1},X_% {2},X_{3}).\end{aligned}
  28. h 3 ( X 1 , X 2 , X 3 ) = X 1 3 + X 1 2 X 2 + X 1 2 X 3 + X 1 X 2 2 + X 1 X 2 X 3 + X 1 X 3 2 + X 2 3 + X 2 2 X 3 + X 2 X 3 2 + X 3 3 . h_{3}(X_{1},X_{2},X_{3})=X_{1}^{3}+X_{1}^{2}X_{2}+X_{1}^{2}X_{3}+X_{1}X_{2}^{2% }+X_{1}X_{2}X_{3}+X_{1}X_{3}^{2}+X_{2}^{3}+X_{2}^{2}X_{3}+X_{2}X_{3}^{2}+X_{3}% ^{3}.
  29. h 3 ( X 1 , X 2 , X 3 ) = m ( 3 ) ( X 1 , X 2 , X 3 ) + m ( 2 , 1 ) ( X 1 , X 2 , X 3 ) + m ( 1 , 1 , 1 ) ( X 1 , X 2 , X 3 ) = ( X 1 3 + X 2 3 + X 3 3 ) + ( X 1 2 X 2 + X 1 2 X 3 + X 1 X 2 2 + X 1 X 3 2 + X 2 2 X 3 + X 2 X 3 2 ) + ( X 1 X 2 X 3 ) . \begin{aligned}\displaystyle h_{3}(X_{1},X_{2},X_{3})&\displaystyle=m_{(3)}(X_% {1},X_{2},X_{3})+m_{(2,1)}(X_{1},X_{2},X_{3})+m_{(1,1,1)}(X_{1},X_{2},X_{3})\\ &\displaystyle=(X_{1}^{3}+X_{2}^{3}+X_{3}^{3})+(X_{1}^{2}X_{2}+X_{1}^{2}X_{3}+% X_{1}X_{2}^{2}+X_{1}X_{3}^{2}+X_{2}^{2}X_{3}+X_{2}X_{3}^{2})+(X_{1}X_{2}X_{3})% .\\ \end{aligned}
  30. n = 2 n=2
  31. X 1 3 + X 2 3 - 7 = - 2 h 1 ( X 1 , X 2 ) 3 + 3 h 1 ( X 1 , X 2 ) h 2 ( X 1 , X 2 ) - 7. X_{1}^{3}+X_{2}^{3}-7=-2h_{1}(X_{1},X_{2})^{3}+3h_{1}(X_{1},X_{2})h_{2}(X_{1},% X_{2})-7.
  32. i = 0 k ( - 1 ) i e i ( X 1 , , X n ) h k - i ( X 1 , , X n ) = 0 \sum_{i=0}^{k}(-1)^{i}e_{i}(X_{1},\ldots,X_{n})h_{k-i}(X_{1},\ldots,X_{n})=0

Symmetric_space.html

  1. γ ( 0 ) = p \gamma(0)=p
  2. f ( γ ( t ) ) = γ ( - t ) . f(\gamma(t))=\gamma(-t).
  3. G σ = { g G : σ ( g ) = g } . G^{\sigma}=\{g\in G:\sigma(g)=g\}.
  4. 𝔤 \mathfrak{g}
  5. 𝔥 \mathfrak{h}
  6. 𝔪 \mathfrak{m}
  7. 𝔤 \mathfrak{g}
  8. 𝔤 = 𝔥 𝔪 \mathfrak{g}=\mathfrak{h}\oplus\mathfrak{m}
  9. [ 𝔥 , 𝔥 ] 𝔥 , [ 𝔥 , 𝔪 ] 𝔪 , [ 𝔪 , 𝔪 ] 𝔥 . [\mathfrak{h},\mathfrak{h}]\subset\mathfrak{h},\;[\mathfrak{h},\mathfrak{m}]% \subset\mathfrak{m},\;[\mathfrak{m},\mathfrak{m}]\subset\mathfrak{h}.
  10. 𝔥 \mathfrak{h}
  11. 𝔤 \mathfrak{g}
  12. 𝔪 \mathfrak{m}
  13. 𝔥 \mathfrak{h}
  14. 𝔥 \mathfrak{h}
  15. 𝔤 \mathfrak{g}
  16. 𝔪 \mathfrak{m}
  17. 𝔥 \mathfrak{h}
  18. 𝔤 \mathfrak{g}
  19. 𝔥 \mathfrak{h}
  20. 𝔪 \mathfrak{m}
  21. σ : G G , h s p h s p \sigma:G\to G,h\mapsto s_{p}\circ h\circ s_{p}
  22. s p : M M , h K h σ ( h - 1 h ) K s_{p}:M\to M,h^{\prime}K\mapsto h\sigma(h^{-1}h^{\prime})K
  23. SO ( n ) \mathrm{SO}(n)
  24. SU ( n ) \mathrm{SU}(n)
  25. Sp ( n ) \mathrm{Sp}(n)
  26. SU ( n ) \mathrm{SU}(n)\,
  27. SO ( n ) \mathrm{SO}(n)\,
  28. ( n - 1 ) ( n + 2 ) / 2 (n-1)(n+2)/2
  29. n \mathbb{C}^{n}
  30. SU ( 2 n ) \mathrm{SU}(2n)\,
  31. Sp ( n ) \mathrm{Sp}(n)\,
  32. ( n - 1 ) ( 2 n + 1 ) (n-1)(2n+1)
  33. 2 n \mathbb{C}^{2n}
  34. SU ( p + q ) \mathrm{SU}(p+q)\,
  35. S ( U ( p ) × U ( q ) ) \mathrm{S}(\mathrm{U}(p)\times\mathrm{U}(q))\,
  36. 2 p q 2pq
  37. p + q \mathbb{C}^{p+q}
  38. SO ( p + q ) \mathrm{SO}(p+q)\,
  39. SO ( p ) × SO ( q ) \mathrm{SO}(p)\times\mathrm{SO}(q)\,
  40. p q pq
  41. p + q \mathbb{R}^{p+q}
  42. SO ( 2 n ) \mathrm{SO}(2n)\,
  43. U ( n ) \mathrm{U}(n)\,
  44. n ( n - 1 ) n(n-1)
  45. 2 n \mathbb{R}^{2n}
  46. Sp ( n ) \mathrm{Sp}(n)\,
  47. U ( n ) \mathrm{U}(n)\,
  48. n ( n + 1 ) n(n+1)
  49. n \mathbb{H}^{n}
  50. Sp ( p + q ) \mathrm{Sp}(p+q)\,
  51. Sp ( p ) × Sp ( q ) \mathrm{Sp}(p)\times\mathrm{Sp}(q)\,
  52. 4 p q 4pq
  53. p + q \mathbb{H}^{p+q}
  54. E 6 E_{6}\,
  55. Sp ( 4 ) / { ± I } \mathrm{Sp}(4)/\{\pm I\}\,
  56. E 6 E_{6}\,
  57. SU ( 6 ) SU ( 2 ) \mathrm{SU}(6)\cdot\mathrm{SU}(2)\,
  58. ( 𝕆 ) P 2 (\mathbb{C}\otimes\mathbb{O})P^{2}
  59. ( ) P 2 (\mathbb{C}\otimes\mathbb{H})P^{2}
  60. E 6 E_{6}\,
  61. SO ( 10 ) SO ( 2 ) \mathrm{SO}(10)\cdot\mathrm{SO}(2)\,
  62. ( 𝕆 ) P 2 (\mathbb{C}\otimes\mathbb{O})P^{2}
  63. E 6 E_{6}\,
  64. F 4 F_{4}\,
  65. ( 𝕆 ) P 2 (\mathbb{C}\otimes\mathbb{O})P^{2}
  66. 𝕆 2 \mathbb{OP}^{2}
  67. E 7 E_{7}\,
  68. SU ( 8 ) / { ± I } \mathrm{SU}(8)/\{\pm I\}\,
  69. E 7 E_{7}\,
  70. SO ( 12 ) SU ( 2 ) \mathrm{SO}(12)\cdot\mathrm{SU}(2)\,
  71. ( 𝕆 ) P 2 (\mathbb{H}\otimes\mathbb{O})P^{2}
  72. 𝕆 \mathbb{H}\otimes\mathbb{O}
  73. E 7 E_{7}\,
  74. E 6 SO ( 2 ) E_{6}\cdot\mathrm{SO}(2)\,
  75. ( 𝕆 ) P 2 (\mathbb{H}\otimes\mathbb{O})P^{2}
  76. ( 𝕆 ) P 2 (\mathbb{C}\otimes\mathbb{O})P^{2}
  77. E 8 E_{8}\,
  78. Spin ( 16 ) / { ± v o l } \mathrm{Spin}(16)/\{\pm vol\}\,
  79. ( 𝕆 𝕆 ) P 2 (\mathbb{O}\otimes\mathbb{O})P^{2}
  80. E 8 E_{8}\,
  81. E 7 SU ( 2 ) E_{7}\cdot\mathrm{SU}(2)\,
  82. ( 𝕆 𝕆 ) P 2 (\mathbb{O}\otimes\mathbb{O})P^{2}
  83. ( 𝕆 ) P 2 (\mathbb{H}\otimes\mathbb{O})P^{2}
  84. F 4 F_{4}\,
  85. Sp ( 3 ) SU ( 2 ) \mathrm{Sp}(3)\cdot\mathrm{SU}(2)\,
  86. 𝕆 P 2 \mathbb{O}P^{2}
  87. P 2 \mathbb{H}P^{2}
  88. F 4 F_{4}\,
  89. Spin ( 9 ) \mathrm{Spin}(9)\,
  90. 𝕆 P 2 \mathbb{O}P^{2}
  91. G 2 G_{2}\,
  92. SO ( 4 ) \mathrm{SO}(4)\,
  93. 𝕆 \mathbb{O}
  94. \mathbb{H}
  95. ( 𝐀 𝐁 ) n , (\mathbf{A}\otimes\mathbf{B})^{n},
  96. 𝔤 = 𝔥 𝔪 \mathfrak{g}=\mathfrak{h}\oplus\mathfrak{m}
  97. 𝔪 \mathfrak{m}
  98. 𝔥 \mathfrak{h}
  99. 𝔥 \mathfrak{h}
  100. 𝔤 \mathfrak{g}
  101. 𝔤 \mathfrak{g}
  102. 𝔤 \mathfrak{g}
  103. 𝔤 \mathfrak{g}
  104. 𝔤 \mathfrak{g}
  105. 𝔤 c \mathfrak{g}^{c}
  106. 𝔤 \mathfrak{g}
  107. 𝔤 c \mathfrak{g}^{c}
  108. 𝔤 \mathfrak{g}
  109. 𝔤 c \mathfrak{g}^{c}
  110. 𝔤 c \mathfrak{g}^{c}
  111. 𝔤 c \mathfrak{g}^{c}

Symmetrical_components.html

  1. V a b c = [ V a V b V c ] = [ V a , 0 V b , 0 V c , 0 ] + [ V a , 1 V b , 1 V c , 1 ] + [ V a , 2 V b , 2 V c , 2 ] V_{abc}=\begin{bmatrix}V_{a}\\ V_{b}\\ V_{c}\end{bmatrix}=\begin{bmatrix}V_{a,0}\\ V_{b,0}\\ V_{c,0}\end{bmatrix}+\begin{bmatrix}V_{a,1}\\ V_{b,1}\\ V_{c,1}\end{bmatrix}+\begin{bmatrix}V_{a,2}\\ V_{b,2}\\ V_{c,2}\end{bmatrix}
  2. 2 3 π \scriptstyle\frac{2}{3}\pi
  3. α \scriptstyle\alpha
  4. α e 2 3 π i \alpha\equiv e^{\frac{2}{3}\pi i}
  5. V 0 V a , 0 = V b , 0 = V c , 0 V_{0}\equiv V_{a,0}=V_{b,0}=V_{c,0}
  6. V 1 \displaystyle V_{1}
  7. V a b c = [ V 0 V 0 V 0 ] + [ V 1 α 2 V 1 α V 1 ] + [ V 2 α V 2 α 2 V 2 ] = [ 1 1 1 1 α 2 α 1 α α 2 ] [ V 0 V 1 V 2 ] = 𝐀 V 012 \begin{aligned}\displaystyle V_{abc}&\displaystyle=\begin{bmatrix}V_{0}\\ V_{0}\\ V_{0}\end{bmatrix}+\begin{bmatrix}V_{1}\\ \alpha^{2}V_{1}\\ \alpha V_{1}\end{bmatrix}+\begin{bmatrix}V_{2}\\ \alpha V_{2}\\ \alpha^{2}V_{2}\end{bmatrix}\\ &\displaystyle=\begin{bmatrix}1&1&1\\ 1&\alpha^{2}&\alpha\\ 1&\alpha&\alpha^{2}\end{bmatrix}\begin{bmatrix}V_{0}\\ V_{1}\\ V_{2}\end{bmatrix}\\ &\displaystyle=\,\textbf{A}V_{012}\end{aligned}
  8. V 012 = [ V 0 V 1 V 2 ] , 𝐀 = [ 1 1 1 1 α 2 α 1 α α 2 ] V_{012}=\begin{bmatrix}V_{0}\\ V_{1}\\ V_{2}\end{bmatrix},\,\textbf{A}=\begin{bmatrix}1&1&1\\ 1&\alpha^{2}&\alpha\\ 1&\alpha&\alpha^{2}\end{bmatrix}
  9. V 012 = 𝐀 - 1 V a b c V_{012}=\,\textbf{A}^{-1}V_{abc}
  10. 𝐀 - 1 = 1 3 [ 1 1 1 1 α α 2 1 α 2 α ] \,\textbf{A}^{-1}=\frac{1}{3}\begin{bmatrix}1&1&1\\ 1&\alpha&\alpha^{2}\\ 1&\alpha^{2}&\alpha\end{bmatrix}
  11. V ( a b ) = V ( a ) - V ( b ) ; V ( b c ) = V ( b ) - V ( c ) ; V ( c a ) = V ( c ) - V ( a ) \scriptstyle V_{(ab)}=V_{(a)}-V_{(b)};\;V_{(bc)}=V_{(b)}-V_{(c)};\;V_{(ca)}=V_% {(c)}-V_{(a)}

Symplectic_vector_field.html

  1. ( M , ω ) (M,\omega)
  2. M M
  3. ω \omega
  4. X 𝔛 ( M ) X\in\mathfrak{X}(M)
  5. 𝔛 ( M ) \mathfrak{X}(M)
  6. X ω = 0 \mathcal{L}_{X}\omega=0
  7. H 1 ( M ) H^{1}(M)
  8. H 1 ( M ) H^{1}(M)

Syriac_Sinaiticus.html

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

System_of_imprimitivity.html

  1. U g W = { U g w : w W } . U_{g}W=\{U_{g}w:w\in W\}.
  2. H = W X W . H=\bigoplus_{W\in X}W.
  3. W X c W v W = 0 , v W W { 0 } \sum_{W\in X}c_{W}v_{W}=0,\quad v_{W}\in W\setminus\{0\}
  4. G 0 = { g G : U g W 0 W 0 } . G_{0}=\{g\in G:U_{g}W_{0}\subseteq W_{0}\}.
  5. χ U ( g ) = tr ( U g ) . \chi_{U}(g)=\operatorname{tr}(U_{g}).
  6. χ U ( g ) = 1 | G 0 | { x G : x - 1 g x G 0 } χ V ( x - 1 g x ) , g G . \chi_{U}(g)=\frac{1}{|G_{0}|}\sum_{\{x\in G:{x}^{-1}\,g\,x\in G_{0}\}}\chi_{V}% ({x}^{-1}\ g\ x),\quad\forall g\in G.
  7. [ L g ψ ] ( h ) = ψ ( g - 1 h ) . [\operatorname{L}_{g}\psi](h)=\psi(g^{-1}h).
  8. W x = { ψ H : ψ ( g ) = 0 , g x } . W_{x}=\{\psi\in H:\psi(g)=0,\quad\forall g\neq x\}.
  9. G × X X , ( g , x ) g x . G\times X\rightarrow X,\quad(g,x)\mapsto g\cdot x.
  10. U g π ( A ) U g - 1 = π ( g A ) . U_{g}\pi(A)U_{g^{-1}}=\pi(g\cdot A).
  11. μ ( g - 1 A ) = μ ( A ) \mu(g^{-1}A)=\mu(A)\quad
  12. [ U g ψ ] ( x ) = ψ ( g - 1 x ) . [U_{g}\psi](x)=\psi(g^{-1}x).\quad
  13. s ( g , x ) = [ d μ d g - 1 μ ] ( x ) [ 0 , ) s(g,x)=\bigg[\frac{d\mu}{dg^{-1}\mu}\bigg](x)\in[0,\infty)
  14. g - 1 μ ( A ) = μ ( g A ) . g^{-1}\mu(A)=\mu(gA).\quad
  15. s : G × X [ 0 , ) s:G\times X\rightarrow[0,\infty)
  16. s ( g , x ) = [ d μ d g - 1 μ ] ( x ) [ 0 , ) s(g,x)=\bigg[\frac{d\mu}{dg^{-1}\mu}\bigg](x)\in[0,\infty)
  17. Φ : G × X U ( H ) \Phi:G\times X\rightarrow\operatorname{U}(H)
  18. Φ ( e , x ) = I \Phi(e,x)=I\quad
  19. Φ ( g h , x ) = Φ ( g , h x ) Φ ( h , x ) \Phi(gh,x)=\Phi(g,h\cdot x)\Phi(h,x)
  20. [ U g ψ ] ( x ) = s ( g , g - 1 x ) Φ ( g , g - 1 x ) ψ ( g - 1 x ) . [U_{g}\psi](x)=\sqrt{s(g,g^{-1}x)}\ \Phi(g,g^{-1}x)\ \psi(g^{-1}x).
  21. X H d μ ( x ) . \int_{X}^{\oplus}Hd\mu(x).
  22. π ( A ) ψ = 1 A ψ , X H d μ ( x ) X H d μ ( x ) , \pi(A)\psi=1_{A}\psi,\quad\int_{X}^{\oplus}Hd\mu(x)\rightarrow\int_{X}^{\oplus% }Hd\mu(x),
  23. G x = { g G : g x = x } G_{x}=\{g\in G:g\cdot x=x\}
  24. Φ : G × X U ( H ) \Phi:G\times X\rightarrow\operatorname{U}(H)
  25. [ h χ ] ( n ) = χ ( h - 1 n h ) . [h\cdot\chi](n)=\chi(h^{-1}nh).
  26. g g x 0 . g\mapsto g\cdot x_{0}.
  27. [ 1 x z 0 1 y 0 0 1 ] . \begin{bmatrix}1&x&z\\ 0&1&y\\ 0&0&1\end{bmatrix}.
  28. H = { [ 1 w 0 0 1 0 0 0 1 ] : w } H=\bigg\{\begin{bmatrix}1&w&0\\ 0&1&0\\ 0&0&1\end{bmatrix}:w\in\mathbb{R}\bigg\}
  29. N = { [ 1 0 t 0 1 s 0 0 1 ] : s , t } . N=\bigg\{\begin{bmatrix}1&0&t\\ 0&1&s\\ 0&0&1\end{bmatrix}:s,t\in\mathbb{R}\bigg\}.
  30. [ w ] - 1 [ s t ] [ w ] = [ s - w s + t ] = [ 1 0 - w 1 ] [ s t ] [w]^{-1}\begin{bmatrix}s\\ t\end{bmatrix}[w]=\begin{bmatrix}s\\ -ws+t\end{bmatrix}=\begin{bmatrix}1&0\\ -w&1\end{bmatrix}\begin{bmatrix}s\\ t\end{bmatrix}
  31. [ 1 - w 0 1 ] . \begin{bmatrix}1&-w\\ 0&1\end{bmatrix}.
  32. ( π [ s , t ] ψ ) ( x ) = e i t y 0 e i s x ψ ( x ) . (\pi[s,t]\psi)(x)=e^{ity_{0}}e^{isx}\psi(x).\quad
  33. ( π [ w ] ψ ) ( x ) = ψ ( x + w y 0 ) . (\pi[w]\psi)(x)=\psi(x+wy_{0}).\quad
  34. π [ s , t ] = e i s x 0 . \pi[s,t]=e^{isx_{0}}.\quad
  35. π [ w ] = e i λ w . \pi[w]=e^{i\lambda w}.\quad

Szemerédi_regularity_lemma.html

  1. G G
  2. V V
  3. X , Y X,Y
  4. V V
  5. ( X , Y ) (X,Y)
  6. d ( X , Y ) := | E ( X , Y ) | | X | | Y | d(X,Y):=\frac{\left|E(X,Y)\right|}{|X||Y|}
  7. E ( X , Y ) E(X,Y)
  8. X X
  9. Y Y
  10. ε > 0 ε > 0
  11. X X
  12. Y Y
  13. ε ε
  14. A X A⊆X
  15. B Y B⊆Y
  16. | d ( X , Y ) - d ( A , B ) | ε . \left|d(X,Y)-d(A,B)\right|\leq\varepsilon.
  17. V V
  18. k k
  19. V < s u b > 1 , , V k V<sub>1, ...,V_{k}
  20. ε > 0 ε > 0
  21. m m
  22. M M
  23. G G
  24. M M
  25. k k
  26. m k M m≤k≤M
  27. ε ε
  28. G G
  29. k k
  30. ε ε
  31. V < s u b > 0 V<sub>0

T-square_(fractal).html

  1. log 3 log 2 = 1.5849... \textstyle{\frac{\log{3}}{\log{2}}=1.5849...}

T-structure.html

  1. D n D^{\geq n}
  2. D n D^{\leq n}
  3. H m ( X ) = 0 H^{m}(X)=0
  4. m < n m<n
  5. m > n m>n
  6. D 0 D 1 D^{\leq 0}\subset D^{\leq 1}
  7. D 1 D 0 ; D^{\geq 1}\subset D^{\geq 0};
  8. Hom ( D 0 , D 1 ) = 0 ; \operatorname{Hom}(D^{\leq 0},D^{\geq 1})=0;
  9. X 0 X X 1 X^{\leq 0}\to X\to X^{\geq 1}\to
  10. X 0 D 0 X^{\leq 0}\in D^{\leq 0}
  11. X 1 D 1 . X^{\geq 1}\in D^{\geq 1}.
  12. D 0 D^{\leq 0}
  13. D 1 D^{\geq 1}
  14. S 0 S_{0}
  15. D 1 := { X D , H o m ( S 0 [ - n ] , X ) = 0 , n 0 } D^{\geq 1}:=\{X\in D,Hom(S_{0}[-n],X)=0,n\geq 0\}
  16. D 0 := { Y D , H o m ( Y , D 1 ) = 0 } D^{\leq 0}:=\{Y\in D,Hom(Y,D^{\geq 1})=0\}
  17. S 0 S_{0}
  18. D 0 D 0 D^{\leq 0}\cap D^{\geq 0}
  19. D 0 D^{\leq 0}
  20. D 1 D^{\geq 1}
  21. D 0 D^{\leq 0}
  22. H o m ( X , D 1 ) = 0 Hom(X,D^{\geq 1})=0
  23. X 0 X^{\leq 0}
  24. X 0 X^{\geq 0}
  25. X τ 0 := X 0 X\mapsto\tau^{\leq 0}:=X^{\leq 0}
  26. D 0 D D^{\leq 0}\to D
  27. 1 \geq 1
  28. τ 0 X X τ 1 X . \tau^{\leq 0}X\to X\to\tau^{\geq 1}X\to.
  29. H n H^{n}
  30. H n := τ n τ n H^{n}:=\tau^{\leq n}\tau^{\geq n}
  31. X Y Z X\to Y\to Z\to
  32. H i ( X ) H i ( Y ) H i ( Z ) H i + 1 ( X ) . \cdots\to H^{i}(X)\to H^{i}(Y)\to H^{i}(Z)\to H^{i+1}(X)\to\cdots.
  33. i * F Z , j * F U [ 1 ] i_{*}F_{Z},j_{*}F_{U}[1]
  34. i : Z X i:Z\to X
  35. F Z F_{Z}
  36. U U
  37. F U F_{U}
  38. D b ( P e r v ( X ) ) D^{b}(Perv(X))
  39. P n P n + 1 \dots\to P^{n}\to P^{n+1}\to\dots
  40. P n P^{n}
  41. S p 0 Sp^{\leq 0}
  42. \leq
  43. \geq
  44. S p 0 Sp^{\geq 0}
  45. D 0 D 1 D^{\leq 0}\subset D^{\leq 1}
  46. D 1 D 0 ; D^{\geq 1}\subset D^{\geq 0};
  47. D 0 D 1 D^{\leq 0}\supset D^{\leq 1}
  48. D 1 D 0 ; D^{\geq 1}\supset D^{\geq 0};

Table_of_Lie_groups.html

  1. π 0 \pi_{0}
  2. π 1 \pi_{1}
  3. π 0 \pi_{0}
  4. π 1 \pi_{1}
  5. { [ a b 0 0 ] : a , b } \left\{\left[\begin{smallmatrix}a&b\\ 0&0\end{smallmatrix}\right]:a,b\in\mathbb{R}\right\}
  6. π 0 \pi_{0}
  7. π 1 \pi_{1}
  8. \otimes

Tachyphylaxis.html

  1. A p B A\xrightarrow{\ \ p\ \ }B
  2. R A p ( S ) B q R\xrightarrow{}A\xrightarrow{\ \ p(S)\ \ }B\xrightarrow{\ \ q\ \ }

Tag_cloud.html

  1. t i t_{i}
  2. t m i n t_{min}
  3. t m a x t_{max}
  4. s i = f max ( t i - t min ) t max - t min s_{i}=\left\lceil\frac{f_{\mathrm{max}}\cdot(t_{i}-t_{\mathrm{min}})}{t_{% \mathrm{max}}-t_{\mathrm{min}}}\right\rceil
  5. t i > t min t_{i}>t_{\mathrm{min}}
  6. s i = 1 s_{i}=1
  7. s i s_{i}
  8. f max f_{\mathrm{max}}
  9. t i t_{i}
  10. t min t_{\mathrm{min}}
  11. t max t_{\mathrm{max}}

Tailwind.html

  1. A = Angle of the wind from the direction of travel A=\,\text{Angle of the wind from the direction of travel}
  2. W S = The measured total wind speed WS=\,\text{The measured total wind speed}
  3. C W = Crosswind CW=\,\text{Crosswind}
  4. H W = Headwind HW=\,\text{Headwind}
  5. C W = sin ( A ) W S CW=\sin(A)\cdot WS
  6. H W = cos ( A ) W S HW=\cos(A)\cdot WS
  7. A = 60 A=60^{\circ}
  8. Crosswind = sin [ 60 ] 15 𝗄𝗇𝗈𝗍𝗌 13 𝗄𝗇𝗈𝗍𝗌 \,\text{Crosswind}=\sin[60^{\circ}]\cdot 15\mathsf{knots}\approx 13\mathsf{knots}
  9. Headwind = cos [ 60 ] 15 𝗄𝗇𝗈𝗍𝗌 7.5 𝗄𝗇𝗈𝗍𝗌 \,\text{Headwind}=\cos[60^{\circ}]\cdot 15\mathsf{knots}\approx 7.5\mathsf{knots}
  10. A = Angle of the wind from the direction of travel A=\,\text{Angle of the wind from the direction of travel}
  11. W S = The measured total wind speed WS=\,\text{The measured total wind speed}
  12. C W = Crosswind CW=\,\text{Crosswind}
  13. H W = Headwind HW=\,\text{Headwind}
  14. C W = sin ( A ) W S CW=\sin(A)\cdot WS
  15. H W = cos ( A ) W S HW=\cos(A)\cdot WS
  16. A = 30 A=30^{\circ}
  17. Crosswind = sin [ 30 ] 15 𝗄𝗇𝗈𝗍𝗌 7.5 𝗄𝗇𝗈𝗍𝗌 \,\text{Crosswind}=\sin[30^{\circ}]\cdot 15\mathsf{knots}\approx 7.5\mathsf{knots}
  18. Tailwind = cos [ 30 ] 15 𝗄𝗇𝗈𝗍𝗌 13 𝗄𝗇𝗈𝗍𝗌 \,\text{Tailwind}=\cos[30^{\circ}]\cdot 15\mathsf{knots}\approx 13\mathsf{knots}

Tangent_cone.html

  1. C : y 2 = x 3 + x 2 C:y^{2}=x^{3}+x^{2}
  2. x = y , x = - y . x=y,\quad x=-y.
  3. gr m O X , x = i 0 m i / m i + 1 . \operatorname{gr}_{m}O_{X,x}=\bigoplus_{i\geq 0}m^{i}/m^{i+1}.

Tata_Institute_of_Fundamental_Research.html

  1. 3 \mathbb{C}^{3}

Taut_foliation.html

  1. 3 \mathbb{R}^{3}
  2. ( M , ) \left(M,{\mathcal{F}}\right)
  3. \mathcal{F}
  4. \mathcal{F}
  5. \mathcal{F}

Tautological_one-form.html

  1. θ = i p i d q i \theta=\sum_{i}p_{i}dq^{i}
  2. ω = - d θ = i d q i d p i \omega=-d\theta=\sum_{i}dq^{i}\wedge dp_{i}
  3. Q Q
  4. M = T * Q M=T^{*}Q
  5. π : M Q \pi:M\to Q
  6. T π : T M T Q T_{\pi}:TM\to TQ
  7. q = π ( m ) q=\pi(m)
  8. m : T q Q m:T_{q}Q\to\mathbb{R}
  9. θ m \theta_{m}
  10. θ m = m T π \theta_{m}=m\circ T_{\pi}
  11. θ m : T m M \theta_{m}:T_{m}M\to\mathbb{R}
  12. θ : M T * M \theta:M\to T^{*}M
  13. β : Q T * Q \beta:Q\to T^{*}Q
  14. β * \beta^{*}
  15. β * θ = β \beta^{*}\theta=\beta
  16. β * θ = β * ( i p i d q i ) = i β * p i d q i = i β i d q i = β . \beta^{*}\theta=\beta^{*}(\sum_{i}p_{i}\,dq^{i})=\sum_{i}\beta^{*}p_{i}\,dq^{i% }=\sum_{i}\beta_{i}\,dq^{i}=\beta.
  17. β * ω = - β * d θ = - d ( β * θ ) = - d β \beta^{*}\omega=-\beta^{*}d\theta=-d(\beta^{*}\theta)=-d\beta
  18. X H X_{H}
  19. S = θ ( X H ) S=\theta(X_{H})
  20. S ( E ) = i p i d q i S(E)=\sum_{i}\oint p_{i}\,dq^{i}
  21. E E
  22. H = E = c o n s t . H=E=const.
  23. g : T Q T * Q g:TQ\to T^{*}Q
  24. Θ = g * θ \Theta=g^{*}\theta
  25. Ω = - d Θ = g * ω \Omega=-d\Theta=g^{*}\omega
  26. ( q 1 , , q n , q ˙ 1 , , q ˙ n ) (q^{1},\ldots,q^{n},\dot{q}^{1},\ldots,\dot{q}^{n})
  27. Θ = i j g i j q ˙ i d q j \Theta=\sum_{ij}g_{ij}\dot{q}^{i}dq^{j}
  28. Ω = i j g i j d q i d q ˙ j + i j k g i j q k q ˙ i d q j d q k \Omega=\sum_{ij}g_{ij}\;dq^{i}\wedge d\dot{q}^{j}+\sum_{ijk}\frac{\partial g_{% ij}}{\partial q^{k}}\;\dot{q}^{i}\,dq^{j}\wedge dq^{k}
  29. T * Q T^{*}Q

Tax_wedge.html

  1. P c Pc
  2. P s Ps

Teichmüller_space.html

  1. ( x , y ) x + y - 1 (x,y)\mapsto x+y\sqrt{-1}
  2. { φ : U 𝐑 2 } \{\varphi:U\rightarrow{\mathbf{R}}^{2}\}
  3. { ψ : V 𝐑 2 } \{\psi:V\rightarrow{\mathbf{R}}^{2}\}
  4. f : X X f:X\rightarrow X
  5. { φ f } = { ψ } \{\varphi\circ f\}=\{\psi\}
  6. f f
  7. f : X X f:X\rightarrow X
  8. f : X X f:X\rightarrow X
  9. Homeo ( X ) {\rm Homeo}(X)
  10. ( φ , f ) φ f (\varphi,f)\mapsto\varphi\circ f
  11. Homeo 0 ( X ) {\rm Homeo}_{0}(X)
  12. Homeo ( X ) {\rm Homeo}(X)
  13. Homeo 0 ( X ) {\rm Homeo}_{0}(X)
  14. Homeo ( X ) {\rm Homeo}(X)

Teltron_tube.html

  1. F L F_{L}
  2. F Z F_{Z}
  3. F L \displaystyle F_{L}
  4. e m = v B r \frac{e}{m}=\frac{v}{B\cdot r}
  5. e U = 1 2 m v 2 e\cdot U=\frac{1}{2}\cdot m\cdot v^{2}
  6. e m = 2 U r 2 B 2 \frac{e}{m}=\frac{2\,U}{r^{2}\cdot B^{2}}
  7. - e m - 1.7588202 10 11 C kg \frac{-e}{m}\approx-1{.}7588202\cdot 10^{11}\,\mathrm{\frac{C}{kg}}
  8. m = e r 2 B 2 2 U 9.1094 10 - 31 kg \ m=\frac{e\cdot r^{2}\cdot B^{2}}{2\,U}\approx 9{.}1094\cdot 10^{-31}\,% \mathrm{kg}

Temperature_gradient.html

  1. T = T ( x , y , z ) T=T(x,y,z)
  2. T = ( T x , T y , T z ) \nabla T=\begin{pmatrix}{\frac{\partial T}{\partial x}},{\frac{\partial T}{% \partial y}},{\frac{\partial T}{\partial z}}\end{pmatrix}

Temporal_difference_learning.html

  1. r t r_{t}
  2. V ¯ t \bar{V}_{t}
  3. γ \gamma
  4. V ¯ t = i = 0 γ i r t + i \bar{V}_{t}=\sum_{i=0}^{\infty}\gamma^{i}r_{t+i}
  5. 0 γ < 1 0\leq\gamma<1
  6. V ¯ t = r t + i = 1 γ i r t + i \bar{V}_{t}=r_{t}+\sum_{i=1}^{\infty}\gamma^{i}r_{t+i}
  7. V ¯ t = r t + i = 0 γ i + 1 r t + i + 1 \bar{V}_{t}=r_{t}+\sum_{i=0}^{\infty}\gamma^{i+1}r_{t+i+1}
  8. V ¯ t = r t + γ i = 0 γ i r t + i + 1 \bar{V}_{t}=r_{t}+\gamma\sum_{i=0}^{\infty}\gamma^{i}r_{t+i+1}
  9. V ¯ t = r t + γ V ¯ t + 1 \bar{V}_{t}=r_{t}+\gamma\bar{V}_{t+1}
  10. r t = V ¯ t - γ V ¯ t + 1 r_{t}=\bar{V}_{t}-\gamma\bar{V}_{t+1}
  11. λ \lambda
  12. 0 λ 1 0\leq\lambda\leq 1
  13. λ \lambda
  14. λ = 1 \lambda=1

Tensor_density.html

  1. 𝔗 β α = ( det [ x ¯ ι x γ ] ) W x α x ¯ δ x ¯ ϵ x β 𝔗 ¯ ϵ δ , {\mathfrak{T}}^{\alpha}_{\beta}=\left(\det{\left[\frac{\partial\bar{x}^{\iota}% }{\partial{x}^{\gamma}}\right]}\right)^{W}\,\frac{\partial{x}^{\alpha}}{% \partial\bar{x}^{\delta}}\,\frac{\partial\bar{x}^{\epsilon}}{\partial{x}^{% \beta}}\,\bar{\mathfrak{T}}^{\delta}_{\epsilon}\,,
  2. 𝔗 ¯ \bar{\mathfrak{T}}
  3. x ¯ \bar{x}
  4. 𝔗 {\mathfrak{T}}
  5. x {x}
  6. 𝔗 β α = sgn ( det [ x ¯ ι x γ ] ) ( det [ x ¯ ι x γ ] ) W x α x ¯ δ x ¯ ϵ x β 𝔗 ¯ ϵ δ , {\mathfrak{T}}^{\alpha}_{\beta}=\operatorname{sgn}\left(\det{\left[\frac{% \partial\bar{x}^{\iota}}{\partial{x}^{\gamma}}\right]}\right)\left(\det{\left[% \frac{\partial\bar{x}^{\iota}}{\partial{x}^{\gamma}}\right]}\right)^{W}\,\frac% {\partial{x}^{\alpha}}{\partial\bar{x}^{\delta}}\,\frac{\partial\bar{x}^{% \epsilon}}{\partial{x}^{\beta}}\,\bar{\mathfrak{T}}^{\delta}_{\epsilon}\,,
  7. 𝔗 β α = | det [ x ¯ ι x γ ] | W x α x ¯ δ x ¯ ϵ x β 𝔗 ¯ ϵ δ . {\mathfrak{T}}^{\alpha}_{\beta}=\left|\det{\left[\frac{\partial\bar{x}^{\iota}% }{\partial{x}^{\gamma}}\right]}\right|^{W}\,\frac{\partial{x}^{\alpha}}{% \partial\bar{x}^{\delta}}\,\frac{\partial\bar{x}^{\epsilon}}{\partial{x}^{% \beta}}\,\bar{\mathfrak{T}}^{\delta}_{\epsilon}\,.
  8. 𝔗 β α = sgn ( det [ x ¯ ι x γ ] ) | det [ x ¯ ι x γ ] | W x α x ¯ δ x ¯ ϵ x β 𝔗 ¯ ϵ δ . {\mathfrak{T}}^{\alpha}_{\beta}=\operatorname{sgn}\left(\det{\left[\frac{% \partial\bar{x}^{\iota}}{\partial{x}^{\gamma}}\right]}\right)\left|\det{\left[% \frac{\partial\bar{x}^{\iota}}{\partial{x}^{\gamma}}\right]}\right|^{W}\,\frac% {\partial{x}^{\alpha}}{\partial\bar{x}^{\delta}}\,\frac{\partial\bar{x}^{% \epsilon}}{\partial{x}^{\beta}}\,\bar{\mathfrak{T}}^{\delta}_{\epsilon}\,.
  9. W W
  10. W W
  11. W W
  12. W W
  13. 𝔗 α β {\mathfrak{T}}_{\alpha\beta}
  14. 𝔗 α β {\mathfrak{T}}_{\alpha\beta}
  15. det 𝔗 α β \det{\mathfrak{T}}_{\alpha\beta}
  16. N W + 2 NW+2
  17. 𝔗 α β {\mathfrak{T}}^{\alpha\beta}
  18. det 𝔗 α β \det{\mathfrak{T}}^{\alpha\beta}
  19. N W 2 NW−2
  20. det 𝔗 β α \det{\mathfrak{T}}^{\alpha}_{~{}\beta}
  21. T μ ν T_{\mu\nu}
  22. T μ ν = x ¯ κ x μ T ¯ κ λ x ¯ λ x ν , T_{\mu\nu}=\frac{\partial\bar{x}^{\kappa}}{\partial{x}^{\mu}}\bar{T}_{\kappa% \lambda}\frac{\partial\bar{x}^{\lambda}}{\partial{x}^{\nu}}\,,
  23. det ( T ¯ κ λ ) \det(\bar{T}_{\kappa\lambda})
  24. | det [ x ¯ ι x γ ] | = det ( T μ ν ) det ( T ¯ κ λ ) . \left|\det{\left[\frac{\partial\bar{x}^{\iota}}{\partial{x}^{\gamma}}\right]}% \right|=\sqrt{\frac{\det({T}_{\mu\nu})}{\det(\bar{T}_{\kappa\lambda})}}\,.
  25. g κ λ {g}_{\kappa\lambda}
  26. x ¯ ι \bar{x}^{\iota}
  27. g ¯ κ λ = η κ λ = \bar{g}_{\kappa\lambda}=\eta_{\kappa\lambda}=
  28. det ( g ¯ κ λ ) = det ( η κ λ ) = \det(\bar{g}_{\kappa\lambda})=\det(\eta_{\kappa\lambda})=
  29. | det [ x ¯ ι x γ ] | = - g , \left|\det{\left[\frac{\partial\bar{x}^{\iota}}{\partial{x}^{\gamma}}\right]}% \right|=\sqrt{-{g}}\,,
  30. g = det ( g μ ν ) {g}=\det({g}_{\mu\nu})
  31. g μ ν {g}_{\mu\nu}
  32. 𝔗 ν μ \mathfrak{T}^{\mu\dots}_{\nu\dots}
  33. 𝔗 ν μ = - g W T ν μ , \mathfrak{T}^{\mu\dots}_{\nu\dots}=\sqrt{-g}\;^{W}T^{\mu\dots}_{\nu\dots}\,,
  34. T ν μ T^{\mu\dots}_{\nu\dots}\,
  35. g κ λ = η κ λ g_{\kappa\lambda}=\eta_{\kappa\lambda}
  36. 𝔗 ν μ \mathfrak{T}^{\mu\dots}_{\nu\dots}
  37. T ν μ T^{\mu\dots}_{\nu\dots}\,
  38. 𝔗 ν ; α μ = - g W T ν ; α μ = - g W ( - g - W 𝔗 ν μ ) ; α . \mathfrak{T}^{\mu\dots}_{\nu\dots;\alpha}=\sqrt{-g}\;^{W}T^{\mu\dots}_{\nu% \dots;\alpha}=\sqrt{-g}\;^{W}(\sqrt{-g}\;^{-W}\mathfrak{T}^{\mu\dots}_{\nu% \dots})_{;\alpha}\,.
  39. - W Γ δ α δ 𝔗 ν μ -W\,\Gamma^{\delta}_{\delta\alpha}\,\mathfrak{T}^{\mu\dots}_{\nu\dots}\,
  40. ( 𝔗 ν μ 𝔖 τ σ ) ; α = ( 𝔗 ν ; α μ ) 𝔖 τ σ + 𝔗 ν μ ( 𝔖 τ ; α σ ) , (\mathfrak{T}^{\mu\dots}_{\nu\dots}\mathfrak{S}^{\sigma\dots}_{\tau\dots})_{;% \alpha}=(\mathfrak{T}^{\mu\dots}_{\nu\dots;\alpha})\mathfrak{S}^{\sigma\dots}_% {\tau\dots}+\mathfrak{T}^{\mu\dots}_{\nu\dots}(\mathfrak{S}^{\sigma\dots}_{% \tau\dots;\alpha})\,,
  41. g κ λ g_{\kappa\lambda}
  42. g κ λ ; α = 0 ( - g W ) ; α = ( - g W ) , α - W Γ δ α δ - g W = W 2 g κ λ g κ λ , α - g W - W Γ δ α δ - g W = 0 . \begin{aligned}\displaystyle g_{\kappa\lambda;\alpha}&\displaystyle=0\\ \displaystyle(\sqrt{-g}\;^{W})_{;\alpha}&\displaystyle=(\sqrt{-g}\;^{W})_{,% \alpha}-W\Gamma^{\delta}_{\delta\alpha}\sqrt{-g}\;^{W}=\frac{W}{2}g^{\kappa% \lambda}g_{\kappa\lambda,\alpha}\sqrt{-g}\;^{W}-W\Gamma^{\delta}_{\delta\alpha% }\sqrt{-g}\;^{W}=0\,.\end{aligned}
  43. - g \sqrt{-g}
  44. 𝔍 μ \mathfrak{J}^{\mu}
  45. 𝔍 2 \mathfrak{J}^{2}
  46. d x 3 d x 4 d x 1 dx^{3}\,dx^{4}\,dx^{1}
  47. 𝔍 μ = J μ - g \mathfrak{J}^{\mu}=J^{\mu}\sqrt{-g}
  48. J μ J^{\mu}\,
  49. 𝔣 μ \mathfrak{f}_{\mu}
  50. d x 1 d x 2 d x 3 d x 4 dx^{1}\,dx^{2}\,dx^{3}\,dx^{4}
  51. ε α β γ δ g α κ g β λ g γ μ g δ ν = ε κ λ μ ν g , \varepsilon^{\alpha\beta\gamma\delta}\,g_{\alpha\kappa}\,g_{\beta\lambda}\,g_{% \gamma\mu}g_{\delta\nu}\,=\,\varepsilon_{\kappa\lambda\mu\nu}\,g\,,
  52. g g
  53. ε κ λ μ ν \varepsilon_{\kappa\lambda\mu\nu}
  54. g = det ( g ρ σ ) = 1 4 ! ε α β γ δ ε κ λ μ ν g α κ g β λ g γ μ g δ ν , g=\det\left(g_{\rho\sigma}\right)=\frac{1}{4!}\varepsilon^{\alpha\beta\gamma% \delta}\varepsilon^{\kappa\lambda\mu\nu}g_{\alpha\kappa}g_{\beta\lambda}g_{% \gamma\mu}g_{\delta\nu}\,,
  55. x [ u o v e r b a r , u x ] x→[u^{\prime}overbar^{\prime},u^{\prime}x^{\prime}]
  56. [ u o v e r b a r , u x ] x [u^{\prime}overbar^{\prime},u^{\prime}x^{\prime}]→x

Term_symbol.html

  1. P = ( - 1 ) i l i , P=(-1)^{\sum_{i}l_{i}}\ ,\!
  2. N = ( t e ) = t ! e ! ( t - e ) ! . N={t\choose e}={t!\over{e!\,(t-e)!}}.
  3. N = 6 ! 2 ! 4 ! = 15 N={6!\over{2!\,4!}}=15
  4. M = i = 1 e m i M=\sum_{i=1}^{e}m_{i}
  5. X ( L , S , l ) = { L - L 3 , if S = 1 / 2 and 0 L < l l - L 3 , if S = 1 / 2 and l L 3 l - 1 L 3 - L - l 2 + L - l + 1 2 , if S = 3 / 2 and 0 L < l L 3 - L - l 2 , if S = 3 / 2 and l L 3 l - 3 0 , other cases X(L,S,l)=\begin{cases}L-\lfloor\frac{L}{3}\rfloor,&\,\text{if }S=1/2\,\text{ % and }0\leq L<l\\ l-\lfloor\frac{L}{3}\rfloor,&\,\text{if }S=1/2\,\text{ and }l\leq L\leq 3l-1\\ \lfloor\frac{L}{3}\rfloor-\lfloor\frac{L-l}{2}\rfloor+\lfloor\frac{L-l+1}{2}% \rfloor,&\,\text{if }S=3/2\,\text{ and }0\leq L<l\\ \lfloor\frac{L}{3}\rfloor-\lfloor\frac{L-l}{2}\rfloor,&\,\text{if }S=3/2\,% \text{ and }l\leq L\leq 3l-3\\ 0,&\,\text{ other cases}\end{cases}
  6. x \lfloor x\rfloor
  7. α , β , γ , \alpha,\beta,\gamma,

Ternary_plot.html

  1. ( 𝐬 - 𝐩 ) - ( ( 𝐬 - 𝐩 ) 𝐧 ^ ) 𝐧 ^ \|(\mathbf{s}-\mathbf{p})-((\mathbf{s}-\mathbf{p})\cdot\mathbf{\hat{n}})% \mathbf{\hat{n}}\|
  2. 𝐩 = ( a b c ) \mathbf{p}=\begin{pmatrix}a\\ b\\ c\end{pmatrix}
  3. 𝐬 = ( 0 K 0 ) \mathbf{s}=\begin{pmatrix}0\\ K\\ 0\end{pmatrix}
  4. 𝐧 ^ = ( 0 K 0 ) - ( 0 0 K ) || ( 0 K 0 ) - ( 0 0 K ) || = ( 0 K - K ) 0 2 + K 2 + ( - K ) 2 = ( 0 1 / 2 - 1 / 2 ) \mathbf{\hat{n}}=\frac{\Big(\begin{smallmatrix}0\\ K\\ 0\end{smallmatrix}\Big)-\Big(\begin{smallmatrix}0\\ 0\\ K\end{smallmatrix}\Big)}{\Big|\Big|\Big(\begin{smallmatrix}0\\ K\\ 0\end{smallmatrix}\Big)-\Big(\begin{smallmatrix}0\\ 0\\ K\end{smallmatrix}\Big)\Big|\Big|}=\frac{\Big(\begin{smallmatrix}0\\ K\\ -K\end{smallmatrix}\Big)}{\sqrt{0^{2}+K^{2}+(-K)^{2}}}=\begin{pmatrix}0\\ \;\;1/\sqrt{2}\\ -1/\sqrt{2}\end{pmatrix}
  5. a = || ( - a K - b - c ) - ( ( - a K - b - c ) ( 0 1 / 2 - 1 / 2 ) ) ( 0 1 / 2 - 1 / 2 ) || = || ( - a K - b - c ) - ( 0 + K - b 2 + c 2 ) ( 0 1 / 2 - 1 / 2 ) || = || ( - a K - b - K - b + c 2 - c + K - b + c 2 ) || = || ( - a K - b - c 2 K - b - c 2 ) || = ( - a ) 2 + ( K - b - c 2 ) 2 + ( K - b - c 2 ) 2 = a 2 + ( K - b - c ) 2 2 \begin{aligned}\displaystyle a^{\prime}&\displaystyle=\bigg|\bigg|\Big(\begin{% smallmatrix}-a\\ K-b\\ -c\end{smallmatrix}\Big)-\bigg(\Big(\begin{smallmatrix}-a\\ K-b\\ -c\end{smallmatrix}\Big)\cdot\Big(\begin{smallmatrix}0\\ \;\;1/\sqrt{2}\\ -1/\sqrt{2}\end{smallmatrix}\Big)\bigg)\Big(\begin{smallmatrix}0\\ \;\;1/\sqrt{2}\\ -1/\sqrt{2}\end{smallmatrix}\Big)\bigg|\bigg|\\ &\displaystyle=\bigg|\bigg|\Big(\begin{smallmatrix}-a\\ K-b\\ -c\end{smallmatrix}\Big)-\Big(0+\tfrac{K-b}{\sqrt{2}}+\tfrac{c}{\sqrt{2}}\Big)% \Big(\begin{smallmatrix}0\\ \;\;1/\sqrt{2}\\ -1/\sqrt{2}\end{smallmatrix}\Big)\bigg|\bigg|\\ &\displaystyle=\bigg|\bigg|\bigg(\begin{smallmatrix}-a\\ K-b-\tfrac{K-b+c}{2}\\ -c+\tfrac{K-b+c}{2}\end{smallmatrix}\bigg)\bigg|\bigg|=\bigg|\bigg|\bigg(% \begin{smallmatrix}-a\\ \tfrac{K-b-c}{2}\\ \tfrac{K-b-c}{2}\end{smallmatrix}\bigg)\bigg|\bigg|\\ &\displaystyle=\sqrt{(-a)^{2}+\big(\tfrac{K-b-c}{2}\big)^{2}+\big(\tfrac{K-b-c% }{2}\big)^{2}}=\sqrt{a^{2}+\tfrac{(K-b-c)^{2}}{2}}\\ \end{aligned}
  6. a = a 2 + ( a + b + c - b - c ) 2 2 = a 2 + a 2 2 = a 3 2 a^{\prime}=\sqrt{a^{2}+\tfrac{(a+b+c-b-c)^{2}}{2}}=\sqrt{a^{2}+\tfrac{a^{2}}{2% }}=a\sqrt{\tfrac{3}{2}}
  7. b = b 3 2 b^{\prime}=b\sqrt{\tfrac{3}{2}}
  8. c = c 3 2 c^{\prime}=c\sqrt{\tfrac{3}{2}}
  9. a = 100 % a=100\%
  10. ( x , y ) = ( 0 , 0 ) (x,y)=(0,0)
  11. b = 100 % b=100\%
  12. ( 1 , 0 ) (1,0)
  13. c = 100 % c=100\%
  14. ( 1 2 , 3 2 ) \left(\tfrac{1}{2},\tfrac{\sqrt{3}}{2}\right)
  15. ( a , b , c ) (a,b,c)
  16. ( 1 2 2 b + c a + b + c , 3 2 c a + b + c ) . \left(\tfrac{1}{2}\tfrac{2b+c}{a+b+c},\tfrac{\sqrt{3}}{2}\tfrac{c}{a+b+c}% \right).

Tests_of_general_relativity.html

  1. γ = β = 0 \gamma=\beta=0
  2. v / c v/c
  3. E = h f E=hf
  4. E = m c 2 E=mc^{2}

Tests_of_special_relativity.html

  1. γ = 1 / 1 - v 2 / c 2 \gamma=1/\sqrt{1-v^{2}/c^{2}}
  2. x = γ ( x - v t ) , y = y , z = z , t = γ ( t - v x c 2 ) x^{\prime}=\gamma(x-vt),\ y^{\prime}=y,\ z^{\prime}=z,\ t^{\prime}=\gamma\left% (t-\frac{vx}{c^{2}}\right)

Teth.html

  1. \otimes

Tetrad_(index_notation).html

  1. x i , i = 1 , , n x_{i}\;\,\text{,}\qquad i=1,\dots,n
  2. e 1 = R d α e_{1}=R\,d\alpha
  3. e 2 = R sin ( α ) d θ e_{2}=R\,\sin{(\alpha)}d\theta
  4. e 3 = R sin ( α ) sin ( θ ) d ϕ e_{3}=R\,\sin{(\alpha)}\sin{(\theta)}d\phi
  5. d e 1 = 0 de_{1}=0
  6. d e 2 = R cos ( α ) d α d θ de_{2}=R\cos{(\alpha)}d\alpha\wedge d\theta
  7. d e 3 = R ( cos ( α ) sin ( θ ) d α d ϕ + sin ( α ) cos ( θ ) d θ d ϕ ) de_{3}=R(\cos{(\alpha)}\sin{(\theta)}d\alpha\wedge d\phi+\sin{(\alpha)}\cos{(% \theta)}d\theta\wedge d\phi)
  8. d 𝐀 e = d e + A e = 0 d_{\mathbf{A}}e=de+A\wedge e=0
  9. A 12 = - cos ( α ) d θ A_{12}=-\cos{(\alpha)}\,d\theta
  10. A 13 = - cos ( α ) sin ( θ ) d ϕ A_{13}=-\cos{(\alpha)}\,\sin{(\theta)}d\phi
  11. A 23 = - cos ( θ ) d ϕ A_{23}=-\cos{(\theta)}\,d\phi
  12. 𝐅 = d 𝐀 + 𝐀 𝐀 \mathbf{F}=d\mathbf{A}+\mathbf{A}\wedge\mathbf{A}
  13. F 12 = sin ( α ) d α d θ F_{12}=\sin{(\alpha)}d\alpha\wedge d\theta
  14. F 13 = sin ( α ) sin ( θ ) d α d ϕ F_{13}=\sin{(\alpha)}\sin{(\theta)}d\alpha\wedge d\phi
  15. F 23 = sin 2 ( α ) sin ( θ ) d θ d ϕ F_{23}=\sin^{2}{(\alpha)}\sin{(\theta)}d\theta\wedge d\phi

TetraVex.html

  1. n × n n\times{}n
  2. n ( n - 1 ) n(n-1)
  3. 4 n 4n
  4. 2 n ( n - 1 ) + 4 n = 2 n ( n + 1 ) 2n(n-1)+4n=2n(n+1)
  5. 10 2 n ( n + 1 ) 10^{2n(n+1)}

Texture_(crystalline).html

  1. O D F ODF
  2. O D F ODF
  3. O D F ODF
  4. s y m b o l g symbol{g}
  5. O D F ( s y m b o l g ) = 1 V d V ( s y m b o l g ) d g . ODF(symbol{g})=\frac{1}{V}\frac{dV(symbol{g})}{dg}.
  6. s y m b o l g symbol{g}
  7. O D F ODF
  8. O D F ODF
  9. O D F ODF
  10. O D F ODF
  11. O D F ODF
  12. O D F ODF
  13. O D F ODF

Thabit_number.html

  1. 3 2 n - 1 3\cdot 2^{n}-1
  2. 9 2 2 n - 1 - 1 9\cdot 2^{2n-1}-1
  3. 2 n ( 3 2 n - 1 - 1 ) ( 3 2 n - 1 ) 2^{n}(3\cdot 2^{n-1}-1)(3\cdot 2^{n}-1)
  4. 2 n ( 9 2 2 n - 1 - 1 ) . 2^{n}(9\cdot 2^{2n-1}-1).

The_Swallow's_Tail.html

  1. \int
  2. V = x 5 + a x 3 + b x 2 + c x , V=x^{5}+ax^{3}+bx^{2}+cx,

Theodorus_of_Cyrene.html

  1. x 2 = n y 2 x^{2}=ny^{2}
  2. n n
  3. n n
  4. x x
  5. y y

Theory_of_planned_behavior.html

  1. B I < m t p l ( W 1 ) A B [ ( b ) + ( e ) ] + ( W 2 ) S N [ ( n ) + ( m ) ] + ( W 3 ) P B C [ ( c ) + ( p ) ] BI<mtpl>{{=}}(W_{1})AB[(b)+(e)]+(W_{2})SN[(n)+(m)]+(W_{3})PBC[(c)+(p)]\,\!

Thermal_de_Broglie_wavelength.html

  1. Λ \Lambda
  2. V V
  3. N N
  4. V N Λ 3 1 , or ( V N ) 1 / 3 Λ \displaystyle\frac{V}{N\Lambda^{3}}\leq 1\ ,{\rm or}\ \left(\frac{V}{N}\right)% ^{1/3}\leq\Lambda
  5. V N Λ 3 1 , or ( V N ) 1 / 3 Λ \displaystyle\frac{V}{N\Lambda^{3}}\gg 1\ ,{\rm or}\ \left(\frac{V}{N}\right)^% {1/3}\gg\Lambda
  6. Λ = h p \Lambda=\frac{h}{p}
  7. p p
  8. E K = p 2 2 m E_{K}=\frac{p^{2}}{2m}
  9. Λ = h 2 m E K \Lambda=\frac{h}{\sqrt{2mE_{K}}}
  10. E K = π k B T E_{K}=\pi k_{B}T
  11. Λ = h 2 2 π m k B T = h 2 π m k B T , \Lambda=\sqrt{\frac{h^{2}}{2\pi mk_{B}T}}=\frac{h}{\sqrt{2\pi mk_{B}T}},
  12. h h
  13. m m
  14. k B k_{B}
  15. T T
  16. Λ = c h 2 π 1 / 3 k B T \Lambda=\frac{ch}{2\pi^{1/3}k_{B}T}
  17. n n
  18. E E
  19. p p
  20. E = a p s E=ap^{s}\,
  21. a a
  22. s s
  23. Λ = h π ( a k B T ) 1 / s [ Γ ( n / 2 + 1 ) Γ ( n / s + 1 ) ] 1 / n \Lambda=\frac{h}{\sqrt{\pi}}\left(\frac{a}{k_{B}T}\right)^{1/s}\left[\frac{% \Gamma(n/2+1)}{\Gamma(n/s+1)}\right]^{1/n}
  24. n = 3 n=3
  25. n = 3 n=3
  26. E = p c E=p c
  27. m m
  28. Λ \Lambda

Thermal_desorption_spectroscopy.html

  1. r ( σ ) = - d σ d t = v ( σ ) σ n * e - E act ( σ ) / R T r(\sigma)=-\frac{\mathrm{d}\sigma}{\mathrm{d}t}=v(\sigma)\sigma^{n}*e^{-E_{% \mathrm{act}}(\sigma)/RT}
  2. r ( σ ) r(\sigma)
  3. σ \sigma
  4. n n
  5. σ \sigma
  6. v ( σ ) v(\sigma)
  7. σ \sigma
  8. E act ( σ ) E_{\mathrm{act}}(\sigma)
  9. σ \sigma
  10. R R
  11. T T
  12. T ( t ) = T 0 + ( β * t ) \begin{aligned}\displaystyle T(t)=T_{0}+(\beta*t)\end{aligned}
  13. β \beta
  14. T 0 T_{0}
  15. t t
  16. d P d t + P / α = d ( a * r ( t ) ) d t \frac{\mathrm{d}P}{\mathrm{d}t}+P/\alpha=\frac{\mathrm{d}(a*r(t))}{\mathrm{d}t}
  17. P P
  18. t t
  19. a = A / K V a=A/KV
  20. A A
  21. K K
  22. V V
  23. r ( t ) r(t)
  24. α = V / S \alpha=V/S
  25. S S
  26. V V
  27. S S
  28. P / α P/\alpha
  29. d P d t \frac{\mathrm{d}P}{\mathrm{d}t}
  30. a * r ( t ) = d P d t a*r(t)=\frac{\mathrm{d}P}{\mathrm{d}t}
  31. r ( t ) = - d σ d t = v n σ n * e - E act / R T r(t)=-\frac{\mathrm{d}\sigma}{\mathrm{d}t}=v_{n}\sigma^{n}*e^{-E_{\mathrm{act}% }/RT}
  32. r ( t ) r(t)
  33. n n
  34. σ \sigma
  35. v n v_{n}
  36. E act E_{\mathrm{act}}
  37. R R
  38. T T
  39. T m T_{m}
  40. E act / R T m 2 = v 1 / β * e - E act / R T E_{\mathrm{act}}/{RT_{m}}^{2}=v_{1}/\beta*e^{-E_{\mathrm{act}}/RT}
  41. E act / R T m 2 = σ 0 v 2 / β * e - E act / R T E_{\mathrm{act}}/{RT_{m}}^{2}=\sigma_{0}v_{2}/\beta*e^{-E_{\mathrm{act}}/RT}
  42. ln ( σ 0 T m 2 ) = - E act / R T + ln ( β * - E act / v 2 R ) \ln({\sigma_{0}{T_{m}}^{2}})={-E_{\mathrm{act}}/RT}+\ln({\beta*-E_{\mathrm{act% }}/v_{2}R})
  43. σ 0 \sigma_{0}
  44. ln ( σ 0 T m ) \ln({\sigma_{0}T_{m}})
  45. 1 / T m 1/T_{m}
  46. - E act / R -E_{\mathrm{act}}/R
  47. T m T_{m}
  48. n n
  49. β \beta
  50. E act E_{\mathrm{act}}
  51. T m T_{m}

Thermal_expansion.html

  1. α V = 1 V ( V T ) p \alpha_{V}=\frac{1}{V}\,\left(\frac{\partial V}{\partial T}\right)_{p}
  2. α L = 1 L d L d T \alpha_{L}=\frac{1}{L}\,\frac{dL}{dT}
  3. L L
  4. d L / d T dL/dT
  5. Δ L L = α L Δ T \frac{\Delta L}{L}=\alpha_{L}\Delta T
  6. Δ T \Delta T
  7. ϵ thermal \epsilon_{\mathrm{thermal}}
  8. ϵ thermal = ( L final - L initial ) L initial \epsilon_{\mathrm{thermal}}=\frac{(L_{\mathrm{final}}-L_{\mathrm{initial}})}{L% _{\mathrm{initial}}}
  9. L initial L_{\mathrm{initial}}
  10. L final L_{\mathrm{final}}
  11. ϵ thermal Δ T \epsilon_{\mathrm{thermal}}\propto\Delta T
  12. ϵ thermal = α L Δ T \epsilon_{\mathrm{thermal}}=\alpha_{L}\Delta T
  13. Δ T = ( T final - T initial ) \Delta T=(T_{\mathrm{final}}-T_{\mathrm{initial}})
  14. α L \alpha_{L}
  15. α A = 1 A d A d T \alpha_{A}=\frac{1}{A}\,\frac{dA}{dT}
  16. A A
  17. d A / d T dA/dT
  18. Δ A A = α A Δ T \frac{\Delta A}{A}=\alpha_{A}\Delta T
  19. δ T \delta T
  20. α V = 1 V d V d T \alpha_{V}=\frac{1}{V}\,\frac{dV}{dT}
  21. V V
  22. d V / d T dV/dT
  23. Δ V V = α V Δ T \frac{\Delta V}{V}=\alpha_{V}\Delta T
  24. Δ V / V \Delta V/V
  25. Δ T \Delta T
  26. Δ V V = T i T f α V ( T ) d T \frac{\Delta V}{V}=\int_{T_{i}}^{T_{f}}\alpha_{V}(T)\,dT
  27. α V ( T ) \alpha_{V}(T)
  28. T i T_{i}
  29. T f T_{f}
  30. α V = 3 α L \alpha_{V}=3\alpha_{L}
  31. V = L 3 V=L^{3}
  32. V + Δ V = ( L + Δ L ) 3 = L 3 + 3 L 2 Δ L + 3 L Δ L 2 + Δ L 3 L 3 + 3 L 2 Δ L = V + 3 V Δ L L V+\Delta V=(L+\Delta L)^{3}=L^{3}+3L^{2}\Delta L+3L\Delta L^{2}+\Delta L^{3}% \approx L^{3}+3L^{2}\Delta L=V+3V{\Delta L\over L}
  33. Δ V = α V L 3 Δ T \Delta V=\alpha_{V}L^{3}\Delta T
  34. Δ L = α L L Δ T \Delta L=\alpha_{L}L\Delta T
  35. V + Δ V = ( L + L α V Δ T ) 3 = L 3 + 3 L 3 α L Δ T + 3 L 3 α L 2 Δ T 2 + L 3 α L 3 Δ T 3 L 3 + 3 L 3 α L Δ T V+\Delta V=(L+L\alpha_{V}\Delta T)^{3}=L^{3}+3L^{3}\alpha_{L}\Delta T+3L^{3}% \alpha_{L}^{2}\Delta T^{2}+L^{3}\alpha_{L}^{3}\Delta T^{3}\approx L^{3}+3L^{3}% \alpha_{L}\Delta T
  36. Δ T \Delta T
  37. Δ L \Delta L
  38. Δ T \Delta T
  39. α A = 2 α L \alpha_{A}=2\alpha_{L}
  40. L 2 L^{2}
  41. Δ T \Delta T
  42. α L \frac{}{}\alpha_{L}
  43. p v = T pv=T\,
  44. ln ( v ) + ln ( p ) = ln ( T ) \ln\left(v\right)+\ln\left(p\right)=\ln\left(T\right)
  45. γ p 1 v ( v T ) p = ( d ( ln v ) d T ) p = d ( ln T ) d T = 1 T . \gamma_{p}\equiv\frac{1}{v}\left(\frac{\partial v}{\partial T}\right)_{p}=% \left(\frac{d(\ln v)}{dT}\right)_{p}=\frac{d(\ln T)}{dT}=\frac{1}{T}.
  46. p p
  47. V T = i V i T + i V i E T \frac{\partial V}{\partial T}=\sum_{i}\frac{\partial V_{i}}{\partial T}+\sum_{% i}\frac{\partial V_{i}^{E}}{\partial T}
  48. α = i α i V i + i α i E V i E \alpha=\sum_{i}\alpha_{i}V_{i}+\sum_{i}\alpha_{i}^{E}V_{i}^{E}
  49. V E ¯ i T = R ( l n ( γ i ) ) P + R T 2 T P l n ( γ i ) \frac{\partial\bar{V^{E}}_{i}}{\partial T}=R\frac{\partial(ln(\gamma_{i}))}{% \partial P}+RT{\partial^{2}\over\partial T\partial P}ln(\gamma_{i})
  50. × 10 - 6 \times 10^{-}6
  51. α 0.020 M P \alpha\approx\frac{0.020}{M_{P}}
  52. α 0.038 M P - 7.0 10 - 6 K - 1 \alpha\approx\frac{0.038}{M_{P}}-7.0\cdot 10^{-6}\,\mathrm{K}^{-1}

Thermal_shock.html

  1. R T = k σ T ( 1 - ν ) α E R_{\mathrm{T}}=\frac{k\sigma_{\mathrm{T}}(1-\nu)}{\alpha E}\,
  2. k k
  3. σ T \sigma_{\mathrm{T}}
  4. α \alpha
  5. E E
  6. ν \nu
  7. P h P_{\mathrm{h}}
  8. P s P_{\mathrm{s}}
  9. q q
  10. P h = q P s . P_{\mathrm{h}}=qP_{\mathrm{s}}.
  11. q = 1 - ω s / ω p q=1-\omega_{\mathrm{s}}/\omega_{\mathrm{p}}
  12. ω p \omega_{\mathrm{p}}
  13. ω s \omega_{\mathrm{s}}
  14. P s , max = 3 R T q L 2 h , P_{\mathrm{s,max}}=3\frac{R_{\mathrm{T}}}{q}\frac{L^{2}}{h},\,
  15. h h
  16. L L
  17. Δ T \Delta T
  18. P s , max = 2 k Δ T q L 2 h P_{\mathrm{s,max}}=2\frac{k\Delta T}{q}\frac{L^{2}}{h}\,
  19. P s , max = R L 2 h P_{\mathrm{s,max}}=R\frac{L^{2}}{h}\,
  20. R = min { 3 R T / q 2 k Δ T / q R=\textrm{min}\left\{\begin{array}[]{c}3R_{\mathrm{T}}/q\\ 2k\Delta T/q\end{array}\right.
  21. Q Q
  22. β \beta
  23. L L
  24. h h
  25. P = R 2 Q β 3 P=\frac{R^{2}}{Q\beta^{3}}
  26. β \beta
  27. β ~{}\beta~{}
  28. P P
  29. β max = ( R 2 P Q ) 1 3 . ~{}\beta_{\mathrm{max}}=\left(\frac{R^{2}}{PQ}\right)^{\frac{1}{3}}.

Thermal_wind.html

  1. Φ 2 - Φ 1 = R T ¯ ln [ p 1 p 2 ] \Phi_{2}-\Phi_{1}=\ R\bar{T}\ln\left[\frac{p_{1}}{p_{2}}\right]
  2. R \,R\,
  3. Φ n \,\Phi_{n}\,
  4. p n \,p_{n}\,
  5. T ¯ \bar{T}
  6. 𝐯 g = 1 f 𝐤 × p Φ \mathbf{v}_{g}=\frac{1}{f}\mathbf{k}\times\nabla_{p}\Phi
  7. f \;f\;
  8. 𝐤 \mathbf{k}
  9. p 0 \,p_{0}\,
  10. p 1 \,p_{1}\,
  11. 𝐯 T = 1 f 𝐤 × p ( Φ 1 - Φ 0 ) \mathbf{v}_{T}=\frac{1}{f}\mathbf{k}\times\nabla_{p}(\Phi_{1}-\Phi_{0})
  12. 𝐯 T = R f ln [ p 0 p 1 ] 𝐤 × p T ¯ \mathbf{v}_{T}=\frac{R}{f}\ln\left[\frac{p_{0}}{p_{1}}\right]\mathbf{k}\times% \nabla_{p}\bar{T}
  13. f \;f\;

Thermodynamic_equations.html

  1. p = W t = ( m g ) h t p=\frac{W}{t}=\frac{(mg)h}{t}
  2. β T \beta_{T}
  3. β S \beta_{S}
  4. α \alpha
  5. d U = δ Q - δ W dU=\delta Q-\delta W\,
  6. d U dU\,
  7. δ Q \delta Q\,
  8. δ W \delta W\,
  9. δ \delta
  10. d U = δ Q + δ W dU=\delta Q+\delta W
  11. d S 0 dS\geq 0
  12. S = 0 \,S=0
  13. T = 0 \,T=0
  14. 𝐉 u = L u u ( 1 / T ) - L u r ( m / T ) \mathbf{J}_{u}=L_{uu}\,\nabla(1/T)-L_{ur}\,\nabla(m/T)\!
  15. 𝐉 r = L r u ( 1 / T ) - L r r ( m / T ) \mathbf{J}_{r}=L_{ru}\,\nabla(1/T)-L_{rr}\,\nabla(m/T)\!
  16. d U = T d S - p d V + i = 1 k μ i d N i dU=TdS-pdV+\sum_{i=1}^{k}\mu_{i}dN_{i}
  17. d U = ( U S ) V , { N i } d S + ( U V ) S , { N i } d V + i ( U N i ) S , V , { N j i } d N i dU=\left(\frac{\partial U}{\partial S}\right)_{V,\{N_{i}\}}dS+\left(\frac{% \partial U}{\partial V}\right)_{S,\{N_{i}\}}dV+\sum_{i}\left(\frac{\partial U}% {\partial N_{i}}\right)_{S,V,\{N_{j\neq i}\}}dN_{i}
  18. ( U S ) V , { N i } = T \left(\frac{\partial U}{\partial S}\right)_{V,\{N_{i}\}}=T
  19. ( U V ) S , { N i } = - p \left(\frac{\partial U}{\partial V}\right)_{S,\{N_{i}\}}=-p
  20. ( U N i ) S , V , { N j i } = μ i \left(\frac{\partial U}{\partial N_{i}}\right)_{S,V,\{N_{j\neq i}\}}=\mu_{i}
  21. d S dS
  22. ( S V ) U , { N i } = p T \left(\frac{\partial S}{\partial V}\right)_{U,\{N_{i}\}}=\frac{p}{T}
  23. d U ( S , V , N i ) = T d S - p d V + i μ i d N i dU\left(S,V,{N_{i}}\right)=TdS-pdV+\sum_{i}\mu_{i}dN_{i}
  24. d H ( S , p , N i ) = T d S + V d p + i μ i d N i dH\left(S,p,N_{i}\right)=TdS+Vdp+\sum_{i}\mu_{i}dN_{i}
  25. d F ( T , V , N i ) = - S d T - p d V + i μ i d N i dF\left(T,V,N_{i}\right)=-SdT-pdV+\sum_{i}\mu_{i}dN_{i}
  26. d G ( T , p , N i ) = - S d T + V d p + i μ i d N i dG\left(T,p,N_{i}\right)=-SdT+Vdp+\sum_{i}\mu_{i}dN_{i}
  27. d Φ = i Φ X i d X i d\Phi=\sum_{i}\frac{\partial\Phi}{\partial X_{i}}dX_{i}
  28. X i X_{i}
  29. γ i \gamma_{i}
  30. X i X_{i}
  31. γ i = Φ X i \gamma_{i}=\frac{\partial\Phi}{\partial X_{i}}
  32. ( F V ) T , { N i } = - p \left(\frac{\partial F}{\partial V}\right)_{T,\{N_{i}\}}=-p
  33. U = T S - p V + i μ i N i U=TS-pV+\sum_{i}\mu_{i}N_{i}\,
  34. F = - p V + i μ i N i F=-pV+\sum_{i}\mu_{i}N_{i}\,
  35. H = T S + i μ i N i H=TS+\sum_{i}\mu_{i}N_{i}\,
  36. G = i μ i N i G=\sum_{i}\mu_{i}N_{i}\,
  37. 0 = S d T - V d p + i N i d μ i 0=SdT-Vdp+\sum_{i}N_{i}d\mu_{i}\,
  38. ( T V ) S , N = - ( p S ) V , N ~{}\left({\partial T\over\partial V}\right)_{S,N}=-\left({\partial p\over% \partial S}\right)_{V,N}~{}
  39. ( T p ) S , N = ( V S ) p , N ~{}\left({\partial T\over\partial p}\right)_{S,N}=\left({\partial V\over% \partial S}\right)_{p,N}~{}
  40. ( T V ) p , N = - ( p S ) T , N ~{}\left({\partial T\over\partial V}\right)_{p,N}=-\left({\partial p\over% \partial S}\right)_{T,N}~{}
  41. ( T p ) V , N = ( V S ) T , N ~{}\left({\partial T\over\partial p}\right)_{V,N}=\left({\partial V\over% \partial S}\right)_{T,N}~{}
  42. β T or S = - 1 V ( V p ) T , N or S , N ~{}\beta_{T\,\text{ or }S}=-{1\over V}\left({\partial V\over\partial p}\right)% _{T,N\,\text{ or }S,N}
  43. C p or V = T N ( S T ) p or V ~{}C_{p\,\text{ or }V}=\frac{T}{N}\left({\partial S\over\partial T}\right)_{p% \,\text{ or }V}~{}
  44. α p = 1 V ( V T ) p \alpha_{p}=\frac{1}{V}\left(\frac{\partial V}{\partial T}\right)_{p}

Thermodynamic_versus_kinetic_reaction_control.html

  1. ln ( [ A ] t [ B ] t ) = ln ( k A k B ) = - Δ E a R T \ln\left(\frac{[A]_{t}}{[B]_{t}}\right)=\ln\left(\frac{k_{A}}{k_{B}}\right)=-% \frac{\Delta E_{a}}{RT}
  2. ln ( [ A ] [ B ] ) = ln K e q = - Δ G R T \ln\left(\frac{[A]_{\infty}}{[B]_{\infty}}\right)=\ln\ K_{eq}=-\frac{\Delta G^% {\circ}}{RT}
  3. [ A ] t / [ B ] t {[A]_{t}}/{[B]_{t}}
  4. Δ E a \Delta E_{a}
  5. Δ G \Delta G^{\circ}
  6. Δ E a \Delta E_{a}
  7. Δ S \Delta S^{\ddagger}
  8. Δ G \Delta G^{\circ}
  9. Δ S \Delta S^{\circ}

Thermogravimetric_analysis.html

  1. d w d t \frac{dw}{dt}
  2. 40.078 + ( 2 × 26.982 ) + ( 4 × 15.999 ) = 158.038 g/mol 40.078+(2\times 26.982)+(4\times 15.999)=158.038~{}\,\text{g/mol}
  3. 40.078 + ( 2 × 26.982 ) + ( 18 × 12.011 ) + ( 37 × 1.008 ) + ( 9 × 15.999 ) + ( 3 × 14.007 ) = 533.548 g/mol 40.078+(2\times 26.982)+(18\times 12.011)+(37\times 1.008)+(9\times 15.999)+(3% \times 14.007)=533.548~{}\,\text{g/mol}
  4. molecular weight of CaAl 2 O 4 molecular weight of CaAl 2 C 18 H 37 O 9 N 3 × 100 = 158.038 g/mol 533.548 g/mol × 100 = 29.6 % \frac{\,\text{molecular weight of CaAl}_{2}\,\text{O}_{4}}{\,\text{molecular % weight of CaAl}_{2}\,\text{C}_{18}H_{37}\,\text{O}_{9}\,\text{N}_{3}}\times 10% 0=\frac{158.038~{}\,\text{g/mol}}{533.548~{}\,\text{g/mol}}\times 100=29.6\%

Thiele::Small.html

  1. F s = 1 2 π C ms M ms F_{\rm s}=\frac{1}{2\pi\cdot\sqrt{C_{\rm ms}\cdot M_{\rm ms}}}
  2. Q es = 2 π F s M ms R e ( B l ) 2 Q_{\rm es}=\frac{2\pi\cdot F_{\rm s}\cdot M_{\rm ms}\cdot R_{\rm e}}{(Bl)^{2}}
  3. Q ms = 2 π F s M ms R ms Q_{\rm ms}=\frac{2\pi\cdot F_{\rm s}\cdot M_{\rm ms}}{R_{\rm ms}}
  4. Q ts = Q ms Q es Q ms + Q es Q_{\rm ts}=\frac{Q_{\rm ms}\cdot Q_{\rm es}}{Q_{\rm ms}+Q_{\rm es}}
  5. V as = ρ c 2 S d 2 C ms V_{\rm as}=\rho\cdot c^{2}\cdot S_{\rm d}^{2}\cdot C_{\rm ms}
  6. Z m a x = R e ( 1 + Q m s Q e s ) Z_{max}=R_{e}(1+\frac{Q_{ms}}{Q_{es}})
  7. E B P = F s Q e s EBP=\frac{F_{s}}{Q_{es}}
  8. η 0 = ( ρ B l 2 S d 2 2 π c M m s 2 R e ) × 100 % \eta_{0}=\left(\frac{\rho\cdot Bl^{2}\cdot S_{d}^{2}}{2\cdot\pi\cdot c\cdot M_% {ms}^{2}\cdot R_{e}}\right)\times 100\%
  9. η 0 = ( 4 π 2 F s 3 V a s c 3 Q e s ) × 100 % \eta_{0}=\left(\frac{4\cdot\pi^{2}\cdot F_{s}^{3}\cdot V_{as}}{c^{3}\cdot Q_{% es}}\right)\times 100\%

Thom_space.html

  1. \infty
  2. Φ : H k ( B ; 𝐙 2 ) H ~ k + n ( T ( E ) ; 𝐙 2 ) , \Phi\colon H^{k}(B;\mathbf{Z}_{2})\to\tilde{H}^{k+n}(T(E);\mathbf{Z}_{2}),
  3. H * ( E , E - B ; Λ ) H^{*}(E,E-B;\Lambda)
  4. H * ( E ; Λ ) H^{*}(E;\Lambda)
  5. p * : H * ( B ; Λ ) H * ( E ; Λ ) p^{*}:H^{*}(B;\Lambda)\to H^{*}(E;\Lambda)
  6. Φ ( b ) = p * ( b ) u . \Phi(b)=p^{*}(b)\cup u.
  7. H ~ n ( T ( E ) ) = H n ( S p h ( E ) , B ) H n ( E , E - B ) . \tilde{H}^{n}(T(E))=H^{n}(Sph(E),B)\simeq H^{n}(E,E-B).
  8. S q i : H m ( - ; 𝐙 2 ) H m + i ( - ; 𝐙 2 ) , Sq^{i}\colon H^{m}(-;\mathbf{Z}_{2})\to H^{m+i}(-;\mathbf{Z}_{2}),
  9. w i ( p ) = Φ - 1 ( S q i ( Φ ( 1 ) ) ) = Φ - 1 ( S q i ( U ) ) . w_{i}(p)=\Phi^{-1}(Sq^{i}(\Phi(1)))=\Phi^{-1}(Sq^{i}(U)).\,
  10. M O ( n ) = T ( γ n ) MO(n)=T(\gamma^{n})
  11. π * M O \pi_{*}MO
  12. S p h ( E ) Sph(E)
  13. ( S p h ( E ) , S p h ( E ) - B , B ) (Sph(E),Sph(E)-B,B)
  14. S p h ( E ) - B Sph(E)-B
  15. H n ( S p h ( E ) , B ) H n ( S p h ( E ) , S p h ( E ) - B ) H^{n}(Sph(E),B)\simeq H^{n}(Sph(E),Sph(E)-B)
  16. H n ( E , E - B ) H^{n}(E,E-B)

Thomas_precession.html

  1. v v
  2. v −v
  3. O O
  4. O O
  5. 𝐮 \mathbf{u}
  6. 𝐯 \mathbf{v}
  7. θ θ
  8. Σ Σ′
  9. Σ Σ
  10. Σ Σ′′
  11. Σ Σ′
  12. Σ Σ
  13. Σ Σ′′
  14. ε ε
  15. 𝐮 × 𝐯 \mathbf{u}×\mathbf{v}
  16. Σ Σ
  17. Σ Σ′′
  18. Σ Σ
  19. Σ Σ′′
  20. Σ Σ
  21. Σ , Σ Σ Σ,Σ′Σ′′
  22. Σ Σ′
  23. 𝐮 \mathbf{u}
  24. Σ Σ
  25. Σ Σ′′
  26. 𝐯 \mathbf{v}
  27. Σ Σ′
  28. Σ Σ′
  29. Σ Σ′
  30. Σ Σ
  31. Σ Σ
  32. Σ Σ′
  33. Σ Σ′
  34. Σ Σ′′
  35. Σ Σ′′
  36. B ( 𝐰 ) B(\mathbf{w})
  37. 𝐰 \mathbf{w}
  38. γ γ
  39. Σ Σ′′
  40. Σ Σ
  41. 𝐮 −\mathbf{u}
  42. 𝐯 −\mathbf{v}
  43. θ θ
  44. Σ Σ′
  45. Σ Σ
  46. Σ Σ′′
  47. Σ Σ′
  48. Σ Σ′′
  49. Σ Σ
  50. ε ε
  51. ( 𝐮 × 𝐯 ) −(\mathbf{u}×\mathbf{v})
  52. Σ Σ′′
  53. Σ Σ
  54. Σ Σ′′
  55. Σ Σ
  56. ε ε
  57. 𝐮 × 𝐯 \mathbf{u}×\mathbf{v}
  58. Σ Σ
  59. 𝐮 −\mathbf{u}
  60. Σ Σ′
  61. Σ Σ′
  62. 𝐯 −\mathbf{v}
  63. Σ Σ′′
  64. 𝐮 𝐮 \mathbf{u}→−\mathbf{u}
  65. 𝐯 𝐯 \mathbf{v}→−\mathbf{v}
  66. Σ Σ
  67. Σ Σ′′
  68. ( 𝐯 ) ( 𝐮 ) 𝐯 𝐮 (−\mathbf{v})⊕(−\mathbf{u})≡−\mathbf{v}⊕\mathbf{u}
  69. Σ Σ′′
  70. Σ Σ
  71. Σ Σ′′
  72. Σ Σ
  73. Σ Σ
  74. Σ Σ
  75. Σ Σ′′
  76. ( Σ , Σ ) (Σ,Σ′)
  77. ( Σ , Σ ) (Σ′,Σ′′)
  78. B ( 𝐯 ) B ( 𝐮 ) = B ( 𝐮 𝐯 ) R ( 𝐮 , 𝐯 ) , B(\mathbf{v})B(\mathbf{u})=B(\mathbf{u}\oplus\mathbf{v})R(\mathbf{u},\mathbf{v% })\,,
  79. B ( 𝐯 ) B ( 𝐮 ) = R ( 𝐮 , 𝐯 ) B ( 𝐯 𝐮 ) , B(\mathbf{v})B(\mathbf{u})=R(\mathbf{u},\mathbf{v})B(\mathbf{v}\oplus\mathbf{u% })\,,
  80. B ( 𝐮 ) B(\mathbf{u})
  81. 𝐮 \mathbf{u}
  82. R ( 𝐮 , 𝐯 ) R(\mathbf{u},\mathbf{v})
  83. 𝐮 \mathbf{u}
  84. 𝐯 \mathbf{v}
  85. 𝐮 \mathbf{u}
  86. 𝐯 \mathbf{v}
  87. B ( 𝐮 ) B(\mathbf{u})
  88. B ( 𝐮 ) B ( 𝐮 ) = 1 B(\mathbf{u})B(−\mathbf{u})=1
  89. R ( 𝐮 , 𝐯 ) R(\mathbf{u},\mathbf{v})
  90. R ( 𝐮 , 𝐯 ) R ( 𝐯 , 𝐮 ) = 1 R(\mathbf{u},\mathbf{v})R(\mathbf{v},\mathbf{u})=1
  91. R ( 𝐮 , 𝐯 ) R ( 𝐮 , 𝐯 ) R(−\mathbf{u},−\mathbf{v})≡R(\mathbf{u},\mathbf{v})
  92. Σ Σ′′
  93. Σ Σ
  94. B ( 𝐯 ) B ( 𝐮 ) = B ( 𝐰 d ) R ( 𝐮 , 𝐯 ) , B(\mathbf{v})B(\mathbf{u})=B(\mathbf{w}_{d})R(\mathbf{u},\mathbf{v})\,,
  95. Σ Σ
  96. Σ Σ′′
  97. B ( - 𝐮 ) B ( - 𝐯 ) = R ( 𝐯 , 𝐮 ) B ( - 𝐰 d ) . B(-\mathbf{u})B(-\mathbf{v})=R(\mathbf{v},\mathbf{u})B(-\mathbf{w}_{d})\,.
  98. Σ Σ
  99. Σ Σ′′
  100. B ( - 𝐮 ) B ( - 𝐯 ) = B ( - 𝐰 i ) R ( - 𝐯 , - 𝐮 ) , B(-\mathbf{u})B(-\mathbf{v})=B(-\mathbf{w}_{i})R(-\mathbf{v},-\mathbf{u})\,,
  101. Σ Σ′′
  102. Σ Σ
  103. B ( 𝐯 ) B ( 𝐮 ) = R ( 𝐮 , 𝐯 ) B ( 𝐰 i ) . B(\mathbf{v})B(\mathbf{u})=R(\mathbf{u},\mathbf{v})B(\mathbf{w}_{i})\,.
  104. B ( 𝐰 i ) = R - 1 ( 𝐮 , 𝐯 ) B ( 𝐰 d ) R ( 𝐮 , 𝐯 ) . B(\mathbf{w}_{i})=R^{-1}(\mathbf{u},\mathbf{v})B(\mathbf{w}_{d})R(\mathbf{u},% \mathbf{v})\,.
  105. Σ Σ
  106. Σ Σ′′
  107. Σ Σ
  108. Σ Σ′′
  109. Σ Σ′′
  110. t , t + Δ t t,t+Δt
  111. Δ t 0 Δt→0
  112. 𝐮 𝐯 = 𝐮 + 𝐯 1 + 𝐮 𝐯 c 2 + 1 c 2 γ u 1 + γ u 𝐮 × ( 𝐮 × 𝐯 ) 1 + 𝐮 𝐯 c 2 , \mathbf{u}\oplus\mathbf{v}=\frac{\mathbf{u}+\mathbf{v}}{1+\frac{\mathbf{u}% \cdot\mathbf{v}}{c^{2}}}+\frac{1}{c^{2}}\frac{\gamma_{u}}{1+\gamma_{u}}\frac{% \mathbf{u}\times(\mathbf{u}\times\mathbf{v})}{1+\frac{\mathbf{u}\cdot\mathbf{v% }}{c^{2}}},
  113. 𝐮 𝐯 = { 1 1 + 𝐮 𝐯 c 2 + 1 c 2 γ u 1 + γ u 𝐮 𝐯 1 + 𝐮 𝐯 c 2 } 𝐮 + α u 1 1 + 𝐮 𝐯 c 2 𝐯 , \mathbf{u}\oplus\mathbf{v}=\left\{\frac{1}{1+\frac{\mathbf{u}\cdot\mathbf{v}}{% c^{2}}}+\frac{1}{c^{2}}\frac{\gamma_{u}}{1+\gamma_{u}}\frac{\mathbf{u}\cdot% \mathbf{v}}{1+\frac{\mathbf{u}\cdot\mathbf{v}}{c^{2}}}\right\}\mathbf{u}+% \alpha_{u}\frac{1}{1+\frac{\mathbf{u}\cdot\mathbf{v}}{c^{2}}}\mathbf{v},
  114. 𝐮 \mathbf{u}
  115. Σ Σ′
  116. Σ Σ
  117. 𝐯 \mathbf{v}
  118. Σ Σ′′
  119. Σ Σ′
  120. 𝐰 = 𝐮 𝐯 \mathbf{w}=\mathbf{u}⊕\mathbf{v}
  121. Σ Σ′
  122. Σ Σ
  123. | 𝐮 𝐯 | = | 𝐯 𝐮 | |\mathbf{u}\oplus\mathbf{v}|=|\mathbf{v}\oplus\mathbf{u}|
  124. 𝐮 𝐯 𝐯 𝐮 \mathbf{u}\oplus\mathbf{v}\neq\mathbf{v}\oplus\mathbf{u}
  125. 𝐮 ( 𝐯 𝐰 ) ( 𝐮 𝐯 ) 𝐰 \mathbf{u}\oplus(\mathbf{v}\oplus\mathbf{w})\neq(\mathbf{u}\oplus\mathbf{v})% \oplus\mathbf{w}
  126. ( - 𝐮 ) ( - 𝐯 ) = - ( 𝐮 𝐯 ) (-\mathbf{u})\oplus(-\mathbf{v})=-(\mathbf{u}\oplus\mathbf{v})
  127. 𝐮 \mathbf{u}
  128. 𝐮 B ( 𝐮 ) . \mathbf{u}\leftrightarrow B(\mathbf{u}).
  129. 𝐮 𝐯 B ( 𝐮 𝐯 ) . \mathbf{u}\oplus\mathbf{v}\leftrightarrow B(\mathbf{u}\oplus\mathbf{v}).
  130. B ( 𝐯 ) B ( 𝐮 ) , B(\mathbf{v})B(\mathbf{u}),
  131. B ( 𝐮 𝐯 ) B ( 𝐯 ) B ( 𝐮 ) . B(\mathbf{u}\oplus\mathbf{v})\neq B(\mathbf{v})B(\mathbf{u}).
  132. B ( 𝐯 ) B ( 𝐮 ) = B ( 𝐮 𝐯 ) R b ( 𝐮 , 𝐯 ) , B ( 𝐯 ) B ( 𝐮 ) = R a ( 𝐮 , 𝐯 ) B ( 𝐮 𝐯 ) . \begin{aligned}\displaystyle B(\mathbf{v})B(\mathbf{u})&\displaystyle=B(% \mathbf{u}\oplus\mathbf{v})R_{b}(\mathbf{u},\mathbf{v}),\\ \displaystyle B(\mathbf{v})B(\mathbf{u})&\displaystyle=R_{a}(\mathbf{u},% \mathbf{v})B(\mathbf{u}\oplus\mathbf{v}).\end{aligned}
  133. B ( s y m b o l ζ ) = e - s y m b o l ζ 𝐊 , B(symbol\zeta)=e^{-symbol\zeta\cdot\mathbf{K}},
  134. ς \mathbf{ς}
  135. 𝐬𝐨 ( 3 , 1 ) \mathbf{so}(3,1)
  136. ( K 1 , K 2 , K 3 ) = ( [ 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 ] , [ 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 ] , [ 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 ] ) . (K_{1},K_{2},K_{3})=\left(\left[\begin{smallmatrix}0&1&0&0\\ 1&0&0&0\\ 0&0&0&0\\ 0&0&0&0\end{smallmatrix}\right],\left[\begin{smallmatrix}0&0&1&0\\ 0&0&0&0\\ 1&0&0&0\\ 0&0&0&0\end{smallmatrix}\right],\left[\begin{smallmatrix}0&0&0&1\\ 0&0&0&0\\ 0&0&0&0\\ 1&0&0&0\end{smallmatrix}\right]\right).
  137. ς \mathbf{ς}
  138. s y m b o l ζ = s y m b o l β ^ tanh - 1 β symbol\zeta=symbol\hat{\beta}\tanh^{-1}\beta
  139. β β
  140. ς ς
  141. β β
  142. β \mathbf{β}
  143. 𝐮 \mathbf{u}
  144. s y m b o l β = 𝐮 c , symbol\beta={\mathbf{u}\over c},
  145. B ( s y m b o l ζ ) = e - tanh - 1 β s y m b o l β ^ 𝐊 = e - tanh - 1 β c β 𝐮 𝐊 B ( 𝐮 ) . B(symbol\zeta)=e^{-\tanh^{-1}\beta symbol\hat{\beta}\cdot\mathbf{K}}=e^{-{% \tanh^{-1}\beta\over c\beta}\mathbf{u}\cdot\mathbf{K}}\equiv B(\mathbf{u}).
  146. Λ = B ( 𝐰 ) R . \Lambda=B(\mathbf{w})R.
  147. 𝐰 \mathbf{w}
  148. R R
  149. Λ Λ
  150. x μ = Λ μ ν x ν . x^{\prime\mu}={\Lambda^{\mu}}_{\nu}x^{\nu}.
  151. ( Λ - 1 ) ν μ Λ μ ρ x ρ = ( Λ - 1 ) ν μ x μ , {(\Lambda^{-1})^{\nu}}_{\mu}{\Lambda^{\mu}}_{\rho}x^{\rho}={(\Lambda^{-1})^{% \nu}}_{\mu}x^{\prime\mu},
  152. x ν = Λ μ ν x μ . x^{\nu}={\Lambda_{\mu}}^{\nu}x^{\prime\mu}.
  153. x = ( c t , 0 , 0 , 0 ) . x′=(ct′,0,0,0).
  154. x ν = Λ 0 ν x 0 , x^{\nu}={\Lambda_{0}}^{\nu}x^{\prime 0},
  155. x = ( c t x 1 x 2 x 3 ) = ( Λ 0 0 c t Λ 0 1 c t Λ 0 2 c t Λ 0 3 c t ) x=\begin{pmatrix}ct\\ x_{1}\\ x_{2}\\ x_{3}\end{pmatrix}=\begin{pmatrix}{\Lambda_{0}}^{0}ct^{\prime}\\ {\Lambda_{0}}^{1}ct^{\prime}\\ {\Lambda_{0}}^{2}ct^{\prime}\\ {\Lambda_{0}}^{3}ct^{\prime}\end{pmatrix}
  156. Λ - 1 = ( B ( 𝐰 ) R ) - 1 = R - 1 B ( - w ) , \Lambda^{-1}=(B(\mathbf{w})R)^{-1}=R^{-1}B(\mathbf{-}w),
  157. B ( - 𝐰 ) = [ γ γ β x γ β y γ β z γ β x 1 + ( γ - 1 ) β x 2 β 2 ( γ - 1 ) β x β y β 2 ( γ - 1 ) β x β z β 2 γ β y ( γ - 1 ) β y β x β 2 1 + ( γ - 1 ) β y 2 β 2 ( γ - 1 ) β y β z β 2 γ β z ( γ - 1 ) β z β x β 2 ( γ - 1 ) β z β y β 2 1 + ( γ - 1 ) β z 2 β 2 ] B(-\mathbf{w})=\begin{bmatrix}\gamma&\gamma\beta_{x}&\gamma\beta_{y}&\gamma% \beta_{z}\\ \gamma\beta_{x}&1+(\gamma-1)\dfrac{\beta_{x}^{2}}{\beta^{2}}&(\gamma-1)\dfrac{% \beta_{x}\beta_{y}}{\beta^{2}}&(\gamma-1)\dfrac{\beta_{x}\beta_{z}}{\beta^{2}}% \\ \gamma\beta_{y}&(\gamma-1)\dfrac{\beta_{y}\beta_{x}}{\beta^{2}}&1+(\gamma-1)% \dfrac{\beta_{y}^{2}}{\beta^{2}}&(\gamma-1)\dfrac{\beta_{y}\beta_{z}}{\beta^{2% }}\\ \gamma\beta_{z}&(\gamma-1)\dfrac{\beta_{z}\beta_{x}}{\beta^{2}}&(\gamma-1)% \dfrac{\beta_{z}\beta_{y}}{\beta^{2}}&1+(\gamma-1)\dfrac{\beta_{z}^{2}}{\beta^% {2}}\\ \end{bmatrix}
  158. x = ( c t x 1 x 2 x 3 ) = ( γ c t γ β x c t γ β y c t γ β z c t ) = ( γ c t γ w x t γ w y t γ w z t ) = γ ( c t w x t w y t w z t ) , x=\begin{pmatrix}ct\\ x_{1}\\ x_{2}\\ x_{3}\end{pmatrix}=\begin{pmatrix}\gamma ct^{\prime}\\ \gamma\beta_{x}ct^{\prime}\\ \gamma\beta_{y}ct^{\prime}\\ \gamma\beta_{z}ct^{\prime}\end{pmatrix}=\begin{pmatrix}\gamma ct^{\prime}\\ \gamma w_{x}t^{\prime}\\ \gamma w_{y}t^{\prime}\\ \gamma w_{z}t^{\prime}\end{pmatrix}=\gamma\begin{pmatrix}ct^{\prime}\\ w_{x}t^{\prime}\\ w_{y}t^{\prime}\\ w_{z}t^{\prime}\end{pmatrix},
  159. x x
  160. 1 c t 𝐱 = 𝐰 c = s y m b o l β = ( x 1 c t x 2 c t x 3 c t ) = ( β x β y β z ) = ( Λ 0 1 / Λ 0 0 Λ 0 2 / Λ 0 0 Λ 0 3 / Λ 0 0 ) . \frac{1}{ct}\mathbf{x}=\frac{\mathbf{w}}{c}=symbol\beta=\begin{pmatrix}\frac{x% _{1}}{ct}\\ \frac{x_{2}}{ct}\\ \frac{x_{3}}{ct}\end{pmatrix}=\begin{pmatrix}\beta_{x}\\ \beta_{y}\\ \beta_{z}\end{pmatrix}=\begin{pmatrix}{\Lambda_{0}}^{1}/{\Lambda_{0}}^{0}\\ {\Lambda_{0}}^{2}/{\Lambda_{0}}^{0}\\ {\Lambda_{0}}^{3}/{\Lambda_{0}}^{0}\end{pmatrix}.
  161. Λ Λ
  162. β \mathbf{β}
  163. 𝐰 \mathbf{w}
  164. 𝐰 \mathbf{w}
  165. 𝐰 \mathbf{w}
  166. B ( 𝐰 ) B(−\mathbf{w})
  167. R R
  168. R = B ( - 𝐰 ) Λ . R=B(-\mathbf{w})\Lambda.
  169. Λ = R B ( 𝐰 ) , \Lambda=RB(\mathbf{w}),
  170. R = Λ B ( - 𝐰 ) . R=\Lambda B(-\mathbf{w}).
  171. 𝐮 \mathbf{u}
  172. 𝐯 \mathbf{v}
  173. B ( 𝐯 ) B ( 𝐮 ) B(\mathbf{v})B(\mathbf{u})
  174. B ( 𝐮 ) B(\mathbf{u})
  175. B ( 𝐯 ) B(\mathbf{v})
  176. B ( 𝐰 ) B(−\mathbf{w})
  177. B ( 𝐰 ) B ( 𝐯 ) B ( 𝐮 ) B(−\mathbf{w})B(\mathbf{v})B(\mathbf{u})
  178. ω 9.5 10 - 7 arcseconds / day . \omega\approx 9.5\cdot 10^{-7}\,\mathrm{arcseconds}/\mathrm{day}.