wpmath0000009_12

Stream_thrust_averaging.html

  1. F = ( ρ 𝐕 d 𝐀 ) 𝐕 𝐟 + p d 𝐀 𝐟 . F=\int\left(\rho\mathbf{V}\cdot d\mathbf{A}\right)\mathbf{V}\cdot\mathbf{f}+% \int pd\mathbf{A}\cdot\mathbf{f}.
  2. m ˙ = ρ 𝐕 d 𝐀 . \dot{m}=\int\rho\mathbf{V}\cdot d\mathbf{A}.
  3. H = 1 m ˙ ( ρ 𝐕 d 𝐀 ) ( h + | 𝐕 | 2 2 ) , H={1\over\dot{m}}\int\left({\rho\mathbf{V}\cdot d\mathbf{A}}\right)\left(h+{|% \mathbf{V}|^{2}\over 2}\right),
  4. U ¯ 2 ( 1 - R 2 C p ) - U ¯ F m ˙ + H R C p = 0. \overline{U}^{2}\left({1-{R\over 2C_{p}}}\right)-\overline{U}{F\over\dot{m}}+{% HR\over C_{p}}=0.
  5. U ¯ \overline{U}
  6. ρ ¯ = m ˙ U ¯ A , \overline{\rho}={\dot{m}\over\overline{U}A},
  7. p ¯ = F A - ρ ¯ U ¯ 2 , \overline{p}={F\over A}-{\overline{\rho}\overline{U}^{2}},
  8. h ¯ = p ¯ C p ρ ¯ R . \overline{h}={\overline{p}C_{p}\over\overline{\rho}R}.
  9. s = C p ln ( T ¯ T 1 ) + R ln ( p ¯ p 1 ) . \nabla s=C_{p}\ln({\overline{T}\over T_{1}})+R\ln({\overline{p}\over p_{1}}).
  10. T 1 T_{1}
  11. p 1 p_{1}
  12. s = C p ln ( T ¯ ) + R ln ( p ¯ ) . \nabla s=C_{p}\ln(\overline{T})+R\ln(\overline{p}).

Streaming_current.html

  1. I s t r = - ϵ r s ϵ 0 a 2 π η Δ P L ζ I_{str}=-\frac{\epsilon_{rs}\epsilon_{0}a^{2}\pi}{\eta}\frac{\Delta P}{L}\zeta
  2. I c = K L a 2 π U s t r L I_{c}=K_{L}a^{2}\pi\frac{U_{str}}{L}
  3. U s t r = ϵ r s ϵ 0 ζ η K L Δ P U_{str}=\frac{\epsilon_{rs}\epsilon_{0}\zeta}{\eta K_{L}}\Delta P
  4. κ a 1 \kappa a\gg 1

Streamline_(swimming).html

  1. R = 1 / 2 D p A v 2 R=1/2DpAv^{2}

Stress_majorization.html

  1. σ \sigma
  2. m m
  3. σ ( X ) = i < j n w i j ( d i j ( X ) - δ i j ) 2 \sigma(X)=\sum_{i<j\leq n}w_{ij}(d_{ij}(X)-\delta_{ij})^{2}
  4. w i j 0 w_{ij}\geq 0
  5. ( i , j ) (i,j)
  6. d i j ( X ) d_{ij}(X)
  7. i i
  8. j j
  9. δ i j \delta_{ij}
  10. m m
  11. w i j w_{ij}
  12. X X
  13. σ ( X ) \sigma(X)
  14. m m
  15. σ ( X ) \sigma(X)
  16. σ \sigma
  17. σ \sigma
  18. Z Z
  19. σ \sigma
  20. σ ( X ) = i < j n w i j ( d i j ( X ) - δ i j ) 2 = i < j w i j δ i j 2 + i < j w i j d i j 2 ( X ) - 2 i < j w i j δ i j d i j ( X ) \sigma(X)=\sum_{i<j\leq n}w_{ij}(d_{ij}(X)-\delta_{ij})^{2}=\sum_{i<j}w_{ij}% \delta_{ij}^{2}+\sum_{i<j}w_{ij}d_{ij}^{2}(X)-2\sum_{i<j}w_{ij}\delta_{ij}d_{% ij}(X)
  21. C C
  22. X V X X^{\prime}VX
  23. i < j w i j δ i j d i j ( X ) = tr X B ( X ) X tr X B ( Z ) Z \sum_{i<j}w_{ij}\delta_{ij}d_{ij}(X)=\,\operatorname{tr}\,X^{\prime}B(X)X\geq% \,\operatorname{tr}\,X^{\prime}B(Z)Z
  24. B ( Z ) B(Z)
  25. b i j = - w i j δ i j d i j ( Z ) b_{ij}=-\frac{w_{ij}\delta_{ij}}{d_{ij}(Z)}
  26. d i j ( Z ) 0 , i j d_{ij}(Z)\neq 0,i\neq j
  27. b i j = 0 b_{ij}=0
  28. d i j ( Z ) = 0 , i j d_{ij}(Z)=0,i\neq j
  29. b i i = - j = 1 , j i n b i j b_{ii}=-\sum_{j=1,j\neq i}^{n}b_{ij}
  30. τ ( X , Z ) \tau(X,Z)
  31. σ ( X ) = C + tr X V X - 2 tr X B ( X ) X \sigma(X)=C+\,\operatorname{tr}\,X^{\prime}VX-2\,\operatorname{tr}\,X^{\prime}% B(X)X
  32. C + tr X V X - 2 tr X B ( Z ) Z = τ ( X , Z ) \leq C+\,\operatorname{tr}\,X^{\prime}VX-2\,\operatorname{tr}\,X^{\prime}B(Z)Z% =\tau(X,Z)
  33. Z X k - 1 Z\leftarrow X^{k-1}
  34. X k min X τ ( X , Z ) X^{k}\leftarrow\min_{X}\tau(X,Z)
  35. σ ( X k - 1 ) - σ ( X k ) < ϵ \sigma(X^{k-1})-\sigma(X^{k})<\epsilon
  36. δ i j \delta_{ij}
  37. w i j w_{ij}
  38. δ i j - α \delta_{ij}^{-\alpha}
  39. α \alpha
  40. α = 2 \alpha=2

String_duality.html

  1. w R / L s t 2 wR/L_{st}^{2}
  2. L s t L_{st}
  3. L s t 2 / R L_{st}^{2}/R

String_operations.html

  1. ε \varepsilon
  2. s s
  3. t t
  4. s t s\cdot t
  5. s t st
  6. s ε = s = ε s s\cdot\varepsilon=s=\varepsilon\cdot s
  7. s ( t u ) = ( s t ) u s\cdot(t\cdot u)=(s\cdot t)\cdot u
  8. ( b l ) ( ε a h ) = b l a h = b l a h (\langle b\rangle\cdot\langle l\rangle)\cdot(\varepsilon\cdot\langle ah\rangle% )=\langle bl\rangle\cdot\langle ah\rangle=\langle blah\rangle
  9. S S
  10. T T
  11. S T S\cdot T
  12. S S
  13. T T
  14. S T = { s t s S t T } S\cdot T=\{s\cdot t\mid s\in S\land t\in T\}
  15. \cdot
  16. { ε } \{\varepsilon\}
  17. { } \{\}
  18. S { ε } = S = { ε } S S\cdot\{\varepsilon\}=S=\{\varepsilon\}\cdot S
  19. S { } = { } = { } S S\cdot\{\}=\{\}=\{\}\cdot S
  20. S ( T U ) = ( S T ) U S\cdot(T\cdot U)=(S\cdot T)\cdot U
  21. D = { 0 , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 } D=\{\langle 0\rangle,\langle 1\rangle,\langle 2\rangle,\langle 3\rangle,% \langle 4\rangle,\langle 5\rangle,\langle 6\rangle,\langle 7\rangle,\langle 8% \rangle,\langle 9\rangle\}
  22. D D D D\cdot D\cdot D
  23. Alph ( s ) \operatorname{Alph}(s)
  24. S S
  25. S S
  26. Alph ( S ) = s S Alph ( s ) \operatorname{Alph}(S)=\bigcup_{s\in S}\operatorname{Alph}(s)
  27. { a , c , o } \{\langle a\rangle,\langle c\rangle,\langle o\rangle\}
  28. c a c a o \langle cacao\rangle
  29. D D
  30. D D D D\cdot D\cdot D
  31. f ( L ) = s L f ( s ) f(L)=\bigcup_{s\in L}f(s)
  32. Σ \Sigma
  33. Σ \Sigma
  34. π Σ ( s ) \pi_{\Sigma}(s)\,
  35. π Σ ( s ) = { ε if s = ε the empty string π Σ ( t ) if s = t a and a Σ π Σ ( t ) a if s = t a and a Σ \pi_{\Sigma}(s)=\begin{cases}\varepsilon&\mbox{if }~{}s=\varepsilon\mbox{ the % empty string}\\ \pi_{\Sigma}(t)&\mbox{if }~{}s=ta\mbox{ and }~{}a\notin\Sigma\\ \pi_{\Sigma}(t)a&\mbox{if }~{}s=ta\mbox{ and }~{}a\in\Sigma\end{cases}
  36. ε \varepsilon
  37. π Σ ( L ) = { π Σ ( s ) | s L } \pi_{\Sigma}(L)=\{\pi_{\Sigma}(s)\ |\ s\in L\}
  38. s / a s/a
  39. ( s a ) / b = { s if a = b ε if a b (sa)/b=\begin{cases}s&\mbox{if }~{}a=b\\ \varepsilon&\mbox{if }~{}a\neq b\end{cases}
  40. ε / a = ε \varepsilon/a=\varepsilon
  41. S M S\subset M
  42. M M
  43. S / a = { s M | s a S } S/a=\{s\in M\ |\ sa\in S\}
  44. S M S\subset M
  45. M M
  46. S = { ( s , t ) M × M | S / s = S / t } \sim_{S}\;\,=\,\{(s,t)\in M\times M\ |\ S/s=S/t\}
  47. { S / m | m M } \{S/m\ |\ m\in M\}
  48. s ÷ a s\div a
  49. ( s a ) ÷ b = { s if a = b ( s ÷ b ) a if a b (sa)\div b=\begin{cases}s&\mbox{if }~{}a=b\\ (s\div b)a&\mbox{if }~{}a\neq b\end{cases}
  50. ε ÷ a = ε \varepsilon\div a=\varepsilon
  51. π Σ ( s ) ÷ a = π Σ ( s ÷ a ) \pi_{\Sigma}(s)\div a=\pi_{\Sigma}(s\div a)
  52. Pref L ( s ) = { t | s = t u for t , u Alph ( L ) * } \operatorname{Pref}_{L}(s)=\{t\ |\ s=tu\mbox{ for }t,u\in\operatorname{Alph}(L% )^{*}\}
  53. s L s\in L
  54. Pref ( L ) = s L Pref L ( s ) = { t | s = t u ; s L ; t , u Alph ( L ) * } \operatorname{Pref}(L)=\bigcup_{s\in L}\operatorname{Pref}_{L}(s)=\left\{t\ |% \ s=tu;s\in L;t,u\in\operatorname{Alph}(L)^{*}\right\}
  55. L = { a b c } then Pref ( L ) = { ε , a , a b , a b c } L=\left\{abc\right\}\mbox{ then }~{}\operatorname{Pref}(L)=\left\{\varepsilon,% a,ab,abc\right\}
  56. Pref ( L ) = L \operatorname{Pref}(L)=L
  57. Pref ( Pref ( L ) ) = Pref ( L ) \operatorname{Pref}(\operatorname{Pref}(L))=\operatorname{Pref}(L)
  58. \sqsubseteq
  59. s t s\sqsubseteq t
  60. s Pref L ( t ) s\in\operatorname{Pref}_{L}(t)

Strip_algebra.html

  1. e - f - v = 2 ( g - 1 ) , e-f-v=2(g-1),\,
  2. ( n , m ) [ T + , T - ] (n,m)[T_{+},T_{-}]

Strong_partition_cardinal.html

  1. k k
  2. [ k ] k [k]^{k}
  3. k k
  4. k k
  5. k k
  6. k k

Strong_product_of_graphs.html

  1. u u
  2. v v
  3. u u
  4. v v
  5. u u
  6. v v
  7. u u
  8. v v
  9. u u
  10. v v
  11. u u
  12. v v
  13. Θ ( G ) = sup k α ( G k ) k = lim k α ( G k ) k , \Theta(G)=\sup_{k}\sqrt[k]{\alpha(G^{k})}=\lim_{k\rightarrow\infty}\sqrt[k]{% \alpha(G^{k})},
  14. ϑ ( G H ) = ϑ ( G ) ϑ ( H ) . \vartheta(G\boxtimes H)=\vartheta(G)\vartheta(H).

Strongly_monotone.html

  1. A : X X * A:X\to X^{*}
  2. c > 0 s.t. A u - A v , u - v c u - v 2 u , v X . \exists\,c>0\mbox{ s.t. }~{}\langle Au-Av,u-v\rangle\geq c\|u-v\|^{2}\quad% \forall u,v\in X.

Structure_theorem_for_finitely_generated_modules_over_a_principal_ideal_domain.html

  1. M M
  2. R R
  3. ( d 1 ) ( d 2 ) ( d n ) (d_{1})\supseteq(d_{2})\supseteq\cdots\supseteq(d_{n})
  4. M M
  5. M i R / ( d i ) = R / ( d 1 ) R / ( d 2 ) R / ( d n ) . M\cong\bigoplus_{i}R/(d_{i})=R/(d_{1})\oplus R/(d_{2})\oplus\cdots\oplus R/(d_% {n}).
  6. d i d_{i}
  7. d 1 d 2 d n d_{1}\mid d_{2}\mid\cdots\mid d_{n}
  8. d i = 0 d_{i}=0
  9. M M
  10. R R
  11. d i m M > 1 dimM>1
  12. d i d_{i}
  13. d i d_{i}
  14. R f i R / ( d i ) = R f R / ( d 1 ) R / ( d 2 ) R / ( d n - f ) R^{f}\oplus\bigoplus_{i}R/(d_{i})=R^{f}\oplus R/(d_{1})\oplus R/(d_{2})\oplus% \cdots\oplus R/(d_{n-f})
  15. d i d_{i}
  16. d i d_{i}
  17. i R / ( q i ) \bigoplus_{i}R/(q_{i})
  18. ( q i ) R (q_{i})\neq R
  19. ( q i ) (q_{i})
  20. q i q_{i}
  21. q i q_{i}
  22. ( q i ) = ( p i r i ) = ( p i ) r i (q_{i})=(p_{i}^{r_{i}})=(p_{i})^{r_{i}}
  23. q i = 0 q_{i}=0
  24. R R
  25. R / ( q i ) R/(q_{i})
  26. q i = 0 q_{i}=0
  27. R f ( i R / ( q i ) ) R^{f}\oplus(\bigoplus_{i}R/(q_{i}))
  28. q i q_{i}
  29. R r R g R^{r}\to R^{g}
  30. R n R^{n}
  31. M = t M F M=tM\oplus F
  32. N p = { m t M i , m p i = 0 } N_{p}=\{m\in tM\mid\exists i,mp^{i}=0\}
  33. R = K R=K
  34. R = R=\mathbb{Z}
  35. R = K [ T ] R=K[T]
  36. K [ T ] K[T]
  37. K [ T ] / p ( T ) K[T]/p(T)
  38. 0 < < T < M . 0<\cdots<T<M.
  39. M 𝐙 𝐙 / 2 M\approx\mathbf{Z}\oplus\mathbf{Z}/2
  40. ( 1 , 0 ) , ( 0 , 1 ) (1,0),(0,1)
  41. ( 1 , 1 ) , ( 0 , 1 ) (1,1),(0,1)
  42. [ 1 1 0 1 ] \begin{bmatrix}1&1\\ 0&1\end{bmatrix}
  43. 𝐙 \mathbf{Z}
  44. 𝐙 / 2 \mathbf{Z}/2

Sturm_separation_theorem.html

  1. u \displaystyle u
  2. v \displaystyle v
  3. W [ u , v ] \displaystyle W[u,v]
  4. W [ u , v ] ( x ) W ( x ) 0 W[u,v](x)\equiv W(x)\neq 0
  5. x \displaystyle x
  6. I \displaystyle I
  7. W ( x ) < 0 x I W(x)<0\mbox{ }~{}\forall\mbox{ }~{}x\in I
  8. u ( x ) v ( x ) - u ( x ) v ( x ) 0. u(x)v^{\prime}(x)-u^{\prime}(x)v(x)\neq 0.
  9. x = x 0 \displaystyle x=x_{0}
  10. W ( x 0 ) = - u ( x 0 ) v ( x 0 ) W(x_{0})=-u^{\prime}\left(x_{0}\right)v\left(x_{0}\right)
  11. u ( x 0 ) u^{\prime}\left(x_{0}\right)
  12. v ( x 0 ) v\left(x_{0}\right)
  13. x = x 1 \displaystyle x=x_{1}
  14. W ( x 1 ) = - u ( x 1 ) v ( x 1 ) W(x_{1})=-u^{\prime}\left(x_{1}\right)v\left(x_{1}\right)
  15. x = x 0 \displaystyle x=x_{0}
  16. x = x 1 \displaystyle x=x_{1}
  17. u ( x ) \displaystyle u(x)
  18. u ( x 1 ) < 0 u^{\prime}\left(x_{1}\right)<0
  19. W ( x ) < 0 \displaystyle W(x)<0
  20. v ( x 1 ) < 0 v\left(x_{1}\right)<0
  21. u ( x ) > 0 x ( x 0 , x 1 ] \displaystyle u^{\prime}(x)>0\mbox{ }~{}\forall\mbox{ }~{}x\in\left(x_{0},x_{1% }\right]
  22. u ( x ) \displaystyle u(x)
  23. x \displaystyle x
  24. x = x 1 \displaystyle x=x_{1}
  25. x = x 1 \displaystyle x=x_{1}
  26. u ( x 1 ) = 0 u^{\prime}\left(x_{1}\right)=0
  27. u ( x 1 ) 0 u^{\prime}\left(x_{1}\right)\leq 0
  28. u ( x 1 ) 0 u^{\prime}\left(x_{1}\right)\leq 0
  29. ( x 0 , x 1 ) \left(x_{0},x_{1}\right)
  30. v ( x ) \displaystyle v(x)
  31. x * ( x 0 , x 1 ) x^{*}\in\left(x_{0},x_{1}\right)
  32. v ( x * ) = 0 v\left(x^{*}\right)=0
  33. ( x 0 , x 1 ) \left(x_{0},x_{1}\right)

Sturm–Picone_comparison_theorem.html

  1. p i , q i , p_{i},\,q_{i},\,
  2. ( p 1 ( x ) y ) + q 1 ( x ) y = 0 (p_{1}(x)y^{\prime})^{\prime}+q_{1}(x)y=0\,
  3. ( p 2 ( x ) y ) + q 2 ( x ) y = 0 (p_{2}(x)y^{\prime})^{\prime}+q_{2}(x)y=0\,
  4. 0 < p 2 ( x ) p 1 ( x ) 0<p_{2}(x)\leq p_{1}(x)\,
  5. q 1 ( x ) q 2 ( x ) . q_{1}(x)\leq q_{2}(x).\,

Subdivided_interval_categories.html

  1. [ n ] [n]
  2. n n\in\mathbb{N}
  3. [ n ] [n]
  4. 0 , 1 , 2 , , n 0,1,2,\ldots,n
  5. H o m ( i , j ) Hom(i,j)
  6. i , j [ n ] i,j\in[n]
  7. j < i j<i
  8. i j i\leq j
  9. Δ \Delta
  10. X : Δ o p S e t s X:\Delta^{op}\rightarrow Sets
  11. 𝒞 \mathcal{C}
  12. 𝒞 [ 1 ] \mathcal{C}^{[1]}
  13. 𝒞 . \mathcal{C}.

Subgradient_method.html

  1. f : n f:\mathbb{R}^{n}\to\mathbb{R}
  2. n \mathbb{R}^{n}
  3. x ( k + 1 ) = x ( k ) - α k g ( k ) x^{(k+1)}=x^{(k)}-\alpha_{k}g^{(k)}
  4. g ( k ) g^{(k)}
  5. f f
  6. x ( k ) x^{(k)}
  7. f f
  8. f \nabla f
  9. - g ( k ) -g^{(k)}
  10. f f
  11. x ( k ) x^{(k)}
  12. f best f_{\rm{best}}
  13. f best ( k ) = min { f best ( k - 1 ) , f ( x ( k ) ) } . f_{\rm{best}}^{(k)}=\min\{f_{\rm{best}}^{(k-1)},f(x^{(k)})\}.
  14. α k = α . \alpha_{k}=\alpha.
  15. α k = γ / g ( k ) 2 \alpha_{k}=\gamma/\lVert g^{(k)}\rVert_{2}
  16. x ( k + 1 ) - x ( k ) 2 = γ . \lVert x^{(k+1)}-x^{(k)}\rVert_{2}=\gamma.
  17. α k 0 , k = 1 α k 2 < , k = 1 α k = . \alpha_{k}\geq 0,\qquad\sum_{k=1}^{\infty}\alpha_{k}^{2}<\infty,\qquad\sum_{k=% 1}^{\infty}\alpha_{k}=\infty.
  18. α k 0 , lim k α k = 0 , k = 1 α k = . \alpha_{k}\geq 0,\qquad\lim_{k\to\infty}\alpha_{k}=0,\qquad\sum_{k=1}^{\infty}% \alpha_{k}=\infty.
  19. α k = γ k / g ( k ) 2 \alpha_{k}=\gamma_{k}/\lVert g^{(k)}\rVert_{2}
  20. γ k 0 , lim k γ k = 0 , k = 1 γ k = . \gamma_{k}\geq 0,\qquad\lim_{k\to\infty}\gamma_{k}=0,\qquad\sum_{k=1}^{\infty}% \gamma_{k}=\infty.
  21. lim k f best ( k ) - f * < ϵ \lim_{k\to\infty}f_{\rm{best}}^{(k)}-f^{*}<\epsilon
  22. f ( x ) f(x)
  23. x 𝒞 x\in\mathcal{C}
  24. 𝒞 \mathcal{C}
  25. x ( k + 1 ) = P ( x ( k ) - α k g ( k ) ) x^{(k+1)}=P\left(x^{(k)}-\alpha_{k}g^{(k)}\right)
  26. P P
  27. 𝒞 \mathcal{C}
  28. g ( k ) g^{(k)}
  29. f f
  30. x ( k ) . x^{(k)}.
  31. f 0 ( x ) f_{0}(x)
  32. f i ( x ) 0 , i = 1 , , m f_{i}(x)\leq 0,\quad i=1,\dots,m
  33. f i f_{i}
  34. x ( k + 1 ) = x ( k ) - α k g ( k ) x^{(k+1)}=x^{(k)}-\alpha_{k}g^{(k)}
  35. α k > 0 \alpha_{k}>0
  36. g ( k ) g^{(k)}
  37. x . x.
  38. g ( k ) = { f 0 ( x ) if f i ( x ) 0 i = 1 m f j ( x ) for some j such that f j ( x ) > 0 g^{(k)}=\begin{cases}\partial f_{0}(x)&\,\text{ if }f_{i}(x)\leq 0\;\forall i=% 1\dots m\\ \partial f_{j}(x)&\,\text{ for some }j\,\text{ such that }f_{j}(x)>0\end{cases}
  39. f \partial f
  40. f f

Subgroup_test.html

  1. G G
  2. H H
  3. G G
  4. a a
  5. b b
  6. H H
  7. a b - 1 ab^{-1}
  8. H H
  9. H H
  10. G G

Subjective_logic.html

  1. ω x A \omega^{A}_{x}
  2. A A\,\!
  3. x x\,\!
  4. ω ( A : x ) \omega(A:x)\,\!
  5. x x\,\!
  6. X X\,\!
  7. x x\,\!
  8. x x\,\!
  9. ω x = ( b , d , u , a ) \omega_{x}=(b,d,u,a)\,\!
  10. b b\,\!
  11. d d\,\!
  12. u u\,\!
  13. a a\,\!
  14. b + d + u = 1 b+d+u=1\,\!
  15. b , d , u , a [ 0 , 1 ] b,d,u,a\in[0,1]\,\!
  16. b = 1 b=1\,\!
  17. d = 1 d=1\,\!
  18. b + d = 1 b+d=1\,\!
  19. b + d < 1 b+d<1\,\!
  20. b + d = 0 b+d=0\,\!
  21. E = b + a u E=b+au\,\!
  22. ( b , d , u ) (b,d,u)\,\!
  23. E E\,\!
  24. Beta ( α , β ) \mathrm{Beta}(\alpha,\beta)\,\!
  25. α \alpha\,\!
  26. β \beta\,\!
  27. ω = ( b , d , u , a ) \omega=(b,d,u,a)\,\!
  28. Beta ( α , β ) where { α = 2 b / u + 2 a β = 2 d / u + 2 ( 1 - a ) \mathrm{Beta}(\alpha,\beta)\mbox{ where }~{}\begin{cases}\alpha&=2b/u+2a\\ \beta&=2d/u+2(1-a)\end{cases}\,\!
  29. X X\,\!
  30. x i x_{i}\,\!
  31. X X\,\!
  32. ω X = ( b , u , a ) \omega_{X}=(\vec{b},u,\vec{a})\,\!
  33. b \vec{b}\,\!
  34. X X\,\!
  35. u u\,\!
  36. a \vec{a}\,\!
  37. X X\,\!
  38. u + b ( x i ) = 1 u+\sum\vec{b}(x_{i})=1\,\!
  39. a ( x i ) = 1 \sum\vec{a}(x_{i})=1\,\!
  40. b ( x i ) , u , a ( x i ) [ 0 , 1 ] \vec{b}(x_{i}),u,\vec{a}(x_{i})\in[0,1]\,\!
  41. Dir ( α ) \mathrm{Dir}(\vec{\alpha})\,\!
  42. α \vec{\alpha}\,\!
  43. ω X = ( b , u , a ) \omega_{X}=(\vec{b},u,\vec{a})\,\!
  44. Dir ( α ) \mathrm{Dir}(\vec{\alpha})
  45. α ( x i ) = 2 b ( x i ) / u + 2 a ( x i ) \vec{\alpha}(x_{i})=2\vec{b}(x_{i})/u+2\vec{a}(x_{i})\,\!
  46. ω x y A = ω x A + ω y A \omega^{A}_{x\cup y}=\omega^{A}_{x}+\omega^{A}_{y}\,\!
  47. ω x \ y A = ω x A - ω y A \omega^{A}_{x\backslash y}=\omega^{A}_{x}-\omega^{A}_{y}\,\!
  48. ω x y A = ω x A ω y A \omega^{A}_{x\land y}=\omega^{A}_{x}\cdot\omega^{A}_{y}\,\!
  49. ω x ¯ y A = ω x A / ω y A \omega^{A}_{x\overline{\land}y}=\omega^{A}_{x}/\omega^{A}_{y}\,\!
  50. ω x y A = ω x A ω y A \omega^{A}_{x\lor y}=\omega^{A}_{x}\sqcup\omega^{A}_{y}\,\!
  51. ω x ¯ y A = ω x A ¯ ω y A \omega^{A}_{x\overline{\lor}y}=\omega^{A}_{x}\;\overline{\sqcup}\;\omega^{A}_{% y}\,\!
  52. ω x ¯ A = ¬ ω x A \omega^{A}_{\overline{x}}\;\;=\lnot\omega^{A}_{x}\,\!
  53. ω y x A = ω x A ( ω y | x A , ω y | x ¯ A ) \omega^{A}_{y\|x}=\omega^{A}_{x}\circledcirc(\omega^{A}_{y|x},\omega^{A}_{y|% \overline{x}})\,\!
  54. ω y ¯ x A = ω x A ¯ ( ω x | y A , ω x | y ¯ A , a y ) \omega^{A}_{y\overline{\|}x}=\omega^{A}_{x}\;\overline{\circledcirc}\;(\omega^% {A}_{x|y},\omega^{A}_{x|\overline{y}},a_{y})\,\!
  55. ω x A : B = ω B A ω x B \omega^{A:B}_{x}=\omega^{A}_{B}\otimes\omega^{B}_{x}\,\!
  56. ω x A B = ω x A ω x B \omega^{A\diamond B}_{x}=\omega^{A}_{x}\oplus\omega^{B}_{x}\,\!
  57. ω x A ¯ B = ω x A ¯ ω x B \omega^{A\underline{\diamond}B}_{x}=\omega^{A}_{x}\;\underline{\oplus}\;\omega% ^{B}_{x}\,\!
  58. A A\,\!
  59. B B\,\!
  60. C C\,\!
  61. A : B : C A:B:C\,\!
  62. [ A , B ] : [ B , C ] [A,B]:[B,C]\,\!
  63. ω x ( y z ) ω ( x y ) ( x z ) \omega_{x\land(y\lor z)}\neq\omega_{(x\land y)\lor(x\land z)}\,\!
  64. x ( y z ) ( x y ) ( x z ) x\land(y\lor z)\Leftrightarrow(x\land y)\lor(x\land z)\,\!
  65. ω x ( y z ) = ω ( x y ) ( x z ) \omega_{x\land(y\cup z)}=\omega_{(x\land y)\cup(x\land z)}\,\!
  66. ω x y ¯ = ω x ¯ y ¯ \omega_{\overline{x\land y}}=\omega_{\overline{x}\lor\overline{y}}\,\!
  67. [ A , B ] [A,B]\,\!
  68. A A\,\!
  69. B B\,\!
  70. ( [ A , B ] : [ B , D ] ) ( [ A , C ] : [ C , D ] ) ([A,B]:[B,D])\diamond([A,C]:[C,D])\,\!
  71. A A\,\!
  72. B B\,\!
  73. C C\,\!
  74. D D\,\!
  75. A A\,\!
  76. D D\,\!
  77. ω D A = ( ω B A ω D B ) ( ω C A ω D C ) \omega^{A}_{D}=(\omega^{A}_{B}\otimes\omega^{B}_{D})\oplus(\omega^{A}_{C}% \otimes\omega^{C}_{D})\,\!
  78. X X\,\!
  79. Y Y\,\!
  80. Z Z\,\!
  81. X X\,\!
  82. Y Y\,\!
  83. Z Z\,\!

Submarine_earthquake.html

  1. M w M_{w}

Subspace_theorem.html

  1. | L 1 ( x ) L n ( x ) | < | x | - ϵ |L_{1}(x)\cdots L_{n}(x)|<|x|^{-\epsilon}
  2. | a i - x i / y | < y - ( 1 + 1 / n + ϵ ) , i = 1 , , n . |a_{i}-x_{i}/y|<y^{-(1+1/n+\epsilon)},\quad i=1,\ldots,n.

Substrate_(chemistry).html

  1. S P S\rightarrow P
  2. S + C P + C S+C\rightarrow P+C
  3. E + S E S E P E + P E+S\rightleftharpoons ES\rightarrow EP\rightleftharpoons E+P

Subtractor.html

  1. X i X_{i}
  2. Y i Y_{i}
  3. B i B_{i}
  4. D i D_{i}
  5. B i + 1 B_{i+1}
  6. X i - Y i - B i X_{i}-Y_{i}-B_{i}
  7. - 2 B i + 1 + D i -2B_{i+1}+D_{i}
  8. D i = X i Y i B i D_{i}=X_{i}\oplus Y_{i}\oplus B_{i}
  9. B i + 1 = X i < ( Y i + B i ) B_{i+1}=X_{i}<(Y_{i}+B_{i})
  10. - B = B ¯ + 1 -B=\bar{B}+1
  11. A - B = A + ( - B ) = A + B ¯ + 1 \begin{aligned}\displaystyle A-B&\displaystyle=A+(-B)\\ &\displaystyle=A+\bar{B}+1\\ \end{aligned}
  12. B = b ¯ a B=\overline{b}\cdot a
  13. D = y x D=y\oplus x
  14. B = x ¯ y B=\overline{x}\cdot y
  15. D = ( X Y ) Z D=(X\oplus Y)\oplus Z
  16. B = X ¯ ( Y Z ) + Y Z B=\overline{X}\cdot(Y\oplus Z)+Y\cdot Z

Succinct_data_structure.html

  1. Z Z
  2. Z + O ( 1 ) Z+O(1)
  3. Z + o ( Z ) Z+o(Z)
  4. O ( Z ) O(Z)
  5. 2 Z 2Z
  6. Z + Z Z+\sqrt{Z}
  7. Z + lg Z Z+\lg Z
  8. Z + 3 Z+3
  9. k k
  10. S S
  11. U = [ 0 n ) = { 0 , 1 , , n - 1 } U=[0\dots n)=\{0,1,\dots,n-1\}
  12. B [ 0 n ) B[0\dots n)
  13. B [ i ] = 1 B[i]=1
  14. i S i\in S
  15. 𝐫𝐚𝐧𝐤 q ( x ) = | { k [ 0 x ] : B [ k ] = q } | \mathbf{rank}_{q}(x)=|\{k\in[0\dots x]:B[k]=q\}|
  16. 𝐬𝐞𝐥𝐞𝐜𝐭 q ( x ) = min { k [ 0 n ) : 𝐫𝐚𝐧𝐤 q ( k ) = x } \mathbf{select}_{q}(x)=\min\{k\in[0\dots n):\mathbf{rank}_{q}(k)=x\}
  17. q { 0 , 1 } q\in\{0,1\}
  18. 𝐫𝐚𝐧𝐤 q ( x ) \mathbf{rank}_{q}(x)
  19. q q
  20. x x
  21. 𝐬𝐞𝐥𝐞𝐜𝐭 q ( x ) \mathbf{select}_{q}(x)
  22. x x
  23. q q
  24. n + o ( n ) n+o(n)
  25. o ( n ) o(n)
  26. B B
  27. l = lg 2 n l=\lg^{2}n
  28. s = lg n / 2 s=\lg n/2
  29. R l [ 0 n / l ) R_{l}[0\dots n/l)
  30. lg n \lg n
  31. ( n / l ) lg n = n / lg n (n/l)\lg n=n/\lg n
  32. R s [ 0 l / s ) R_{s}[0\dots l/s)
  33. l / s = 2 lg n l/s=2\lg n
  34. lg l = lg lg 2 n = 2 lg lg n \lg l=\lg\lg^{2}n=2\lg\lg n
  35. ( n / s ) lg l = 4 n lg lg n / lg n (n/s)\lg l=4n\lg\lg n/\lg n
  36. R p R_{p}
  37. s s
  38. i [ 0 , s ) i\in[0,s)
  39. 2 s s lg s = O ( n lg n lg lg n ) 2^{s}s\lg s=O(\sqrt{n}\lg n\lg\lg n)
  40. o ( n ) o(n)
  41. O ( 1 ) O(1)
  42. n + o ( n ) n+o(n)
  43. 𝐫𝐚𝐧𝐤 1 ( x ) \mathbf{rank}_{1}(x)
  44. 𝐫𝐚𝐧𝐤 1 ( x ) = R l [ x / l ] + R s [ x / s ] + R p [ x x / s , x mod s ] \mathbf{rank}_{1}(x)=R_{l}[\lfloor x/l\rfloor]+R_{s}[\lfloor x/s\rfloor]+R_{p}% [x\lfloor x/s\rfloor,x\,\text{ mod }s]
  45. R p R_{p}
  46. O ( lg n ) O(\lg n)
  47. 3 n / lg lg n + O ( n lg n lg lg n ) = o ( n ) 3n/\lg\lg n+O(\sqrt{n}\lg n\lg\lg n)=o(n)
  48. O ( ) O(\cdot)
  49. n + o ( n ) n+o(n)
  50. ( n m ) \textstyle{\left({{n}\atop{m}}\right)}
  51. m m
  52. [ n ) [n)
  53. n n
  54. m m
  55. ( m , n ) = lg ( n m ) \textstyle\mathcal{B}(m,n)=\lceil\lg{\left({{n}\atop{m}}\right)}\rceil
  56. B B
  57. ( m , n ) + o ( ( m , n ) ) \mathcal{B}(m,n)+o(\mathcal{B}(m,n))
  58. ( m , n ) + O ( m + n lg lg n / lg n ) \mathcal{B}(m,n)+O(m+n\lg\lg n/\lg n)
  59. ( m , n ) + O ( n t t / lg t n + n 3 / 4 ) \mathcal{B}(m,n)+O(nt^{t}/\lg^{t}n+n^{3/4})
  60. O ( t ) O(t)
  61. s e l e c t select
  62. n n
  63. 2 n + o ( n ) 2n+o(n)
  64. n n
  65. ( 2 n n ) {{\textstyle\left({{2n}\atop{n}}\right)}}
  66. / ( n + 1 ) /(n+1)
  67. n n
  68. 4 n 4^{n}
  69. log 2 ( 4 n ) = 2 n \log_{2}(4^{n})=2n
  70. 2 2

Sum-free_set.html

  1. a + b = c a+b=c
  2. a , b , c A a,b,c\in A
  3. O ( 2 N / 2 ) O(2^{N/2})

Summation_of_Grandi's_series.html

  1. σ n = 1 2 \sigma_{n}=\frac{1}{2}
  2. σ n = 1 2 + 1 2 n \sigma_{n}=\frac{1}{2}+\frac{1}{2n}
  3. 2 3 ( 2 2 m - 1 2 2 m + 2 ) to 1 3 ( 1 - 2 - 2 m ) , \frac{2}{3}\left(\frac{2^{2m}-1}{2^{2m}+2}\right)\;\mathrm{to}\;\frac{1}{3}(1-% 2^{-2m}),
  4. F ( x ) = x - x 2 + x 4 - x 8 + = x - [ ( x 2 ) - ( x 2 ) 2 + ( x 2 ) 4 - ] = x - F ( x 2 ) . \begin{array}[]{rcl}F(x)&=&\displaystyle x-x^{2}+x^{4}-x^{8}+\cdots\\ &=&\displaystyle x-\left[(x^{2})-(x^{2})^{2}+(x^{2})^{4}-\cdots\right]\\ &=&\displaystyle x-F(x^{2}).\end{array}
  5. F ( x ) = Ψ ( x ) + Φ ( x ) F(x)=\Psi(x)+\Phi(x)
  6. Φ ( x ) = n = 0 ( - 1 ) n n ! ( 1 + 2 n ) ( log 1 x ) n \Phi(x)=\sum_{n=0}^{\infty}\frac{(-1)^{n}}{n!(1+2^{n})}\left(\log\frac{1}{x}% \right)^{n}
  7. S φ = lim δ 0 m = 0 ( - 1 ) m φ ( δ m ) = 1 2 . S_{\varphi}=\lim_{\delta\downarrow 0}\sum_{m=0}^{\infty}(-1)^{m}\varphi(\delta m% )=\frac{1}{2}.
  8. S φ = lim δ 0 m = 0 [ φ ( 2 k δ ) - φ ( 2 k δ - δ ) ] = lim δ 0 m = 0 φ ( 2 k δ + c k ) ( - δ ) = - 1 2 0 φ ( x ) d x = - 1 2 φ ( x ) | 0 = 1 2 . \begin{array}[]{rcl}S_{\varphi}&=&\displaystyle\lim_{\delta\downarrow 0}\sum_{% m=0}^{\infty}\left[\varphi(2k\delta)-\varphi(2k\delta-\delta)\right]\\ &=&\displaystyle\lim_{\delta\downarrow 0}\sum_{m=0}^{\infty}\varphi^{\prime}(2% k\delta+c_{k})(-\delta)\\ &=&\displaystyle-\frac{1}{2}\int_{0}^{\infty}\varphi^{\prime}(x)\,dx=-\frac{1}% {2}\varphi(x)|_{0}^{\infty}=\frac{1}{2}.\end{array}
  9. 1 - x + x 2 2 ! - x 3 3 ! + x 4 4 ! - = e - x 1-x+\frac{x^{2}}{2!}-\frac{x^{3}}{3!}+\frac{x^{4}}{4!}-\cdots=e^{-x}
  10. 0 e - x e - x d x = 0 e - 2 x d x = 1 2 . \int_{0}^{\infty}e^{-x}e^{-x}\,dx=\int_{0}^{\infty}e^{-2x}\,dx=\frac{1}{2}.
  11. { ω n } \{\omega_{n}\}
  12. n sgn ( ω n ) \sum_{n}\operatorname{sgn}(\omega_{n})\;
  13. sgn ( ω n ) = ± 1 \operatorname{sgn}(\omega_{n})=\pm 1
  14. lim t 0 n sgn ( ω n ) e - t | ω n | \lim_{t\to 0}\sum_{n}\operatorname{sgn}(\omega_{n})e^{-t|\omega_{n}|}
  15. 1 / x = 1 - ( x - 1 ) + ( x - 1 ) 2 - ( x - 1 ) 3 + ( x - 1 ) 4 - 1/x=1-(x-1)+(x-1)^{2}-(x-1)^{3}+(x-1)^{4}-...
  16. 1 = ( x - 1 ) - ( x - 1 ) 2 + ( x - 1 ) 3 - ( x - 1 ) 4 + + 1 - ( x - 1 ) + ( x - 1 ) 2 - ( x - 1 ) 3 + 1=(x-1)-(x-1)^{2}+(x-1)^{3}-(x-1)^{4}+...+1-(x-1)+(x-1)^{2}-(x-1)^{3}+...
  17. 1 = 1 1=1
  18. 1 / 2 = 1 - 1 + 1 - 1 + 1 - 1/2=1-1+1-1+1-...
  19. d χ = e - k ( log x ) 2 x - 1 d x d\chi=e^{-k(\log x)^{2}}x^{-1}dx

Sums_of_powers.html

  1. 1 k + 2 k + 3 k + + n k 1^{k}+2^{k}+3^{k}+\cdots+n^{k}
  2. x k + y k = z k x^{k}+y^{k}=z^{k}
  3. | x / a | k + | y / b | k = 1 |x/a|^{k}+|y/b|^{k}=1
  4. k = 4 , a = b k=4,a=b
  5. a 4 + b 4 + c 4 + d 4 = ( a + b + c + d ) 4 a^{4}+b^{4}+c^{4}+d^{4}=(a+b+c+d)^{4}
  6. i = 1 n a i k = j = 1 m b j k . \sum_{i=1}^{n}a_{i}^{k}=\sum_{j=1}^{m}b_{j}^{k}.
  7. φ n + 1 = φ n + φ n - 1 . \varphi^{n+1}=\varphi^{n}+\varphi^{n-1}.

Supercontinuum.html

  1. 2 {}_{2}
  2. L fiss L_{\mathrm{fiss}}
  3. L fiss = L D N = τ 0 2 | β 2 | γ P 0 L_{\mathrm{fiss}}=\frac{L_{D}}{N}=\sqrt{\frac{\tau^{2}_{0}}{|\beta_{2}|\gamma P% _{0}}}
  4. L D L_{D}
  5. N N
  6. L fiss L_{\mathrm{fiss}}
  7. L MI L_{\mathrm{MI}}
  8. L MI L_{\mathrm{MI}}
  9. L MI = n dB 20 γ P 0 lg 10 4 γ P 0 L_{\mathrm{MI}}=\frac{n_{\mathrm{dB}}}{20\gamma P_{0}\lg 10}\sim\frac{4}{% \gamma P_{0}}
  10. n dB n_{\mathrm{dB}}
  11. L MI L fiss L_{\mathrm{MI}}\ll L_{\mathrm{fiss}}
  12. 4 2 γ P 0 τ 0 2 | β 2 | = N 2 4^{2}\ll\frac{\gamma P_{0}\tau_{0}^{2}}{|\beta_{2}|}=N^{2}
  13. N = 16 N=16
  14. 2 {}_{2}
  15. 5 {}_{5}
  16. 2 {}_{2}

Superegg.html

  1. | x 2 + y 2 r | p + | z h | p 1 \left|\frac{\sqrt{x^{2}+y^{2}}}{r}\right|^{p}+\left|\frac{z}{h}\right|^{p}\leq 1

Superfluid_film.html

  1. 2 π m / k 2\pi\sqrt{m/k}

Superprocess.html

  1. ( α , d , β ) (\alpha,d,\beta)
  2. X ( t , d x ) X(t,dx)
  3. × d \mathbb{R}\times\mathbb{R}^{d}
  4. Φ ( s ) = 1 1 + β ( 1 - s ) 1 + β + s \Phi(s)=\frac{1}{1+\beta}(1-s)^{1+\beta}+s
  5. α \alpha
  6. Δ α \Delta_{\alpha}
  7. α = 2 \alpha=2
  8. ( 2 , d , 1 ) (2,d,1)
  9. Δ u - u 2 = 0 o n d . \Delta u-u^{2}=0\ on\ \mathbb{R}^{d}.

Supersymmetry_as_a_quantum_group.html

  1. ( - 1 ) F 2 = 1 {(-1)^{F}}^{2}=1
  2. ϵ ( ( - 1 ) F ) = 1 \epsilon((-1)^{F})=1
  3. Δ ( - 1 ) F = ( - 1 ) F ( - 1 ) F \Delta(-1)^{F}=(-1)^{F}\otimes(-1)^{F}
  4. S ( - 1 ) F = ( - 1 ) F S(-1)^{F}=(-1)^{F}
  5. 2 \mathbb{Z}_{2}
  6. = 1 2 [ 1 1 + ( - 1 ) F 1 + 1 ( - 1 ) F - ( - 1 ) F ( - 1 ) F ] \mathcal{R}=\frac{1}{2}\left[1\otimes 1+(-1)^{F}\otimes 1+1\otimes(-1)^{F}-(-1% )^{F}\otimes(-1)^{F}\right]
  7. 2 \mathbb{Z}_{2}
  8. Δ x = x 1 + 1 x \Delta x=x\otimes 1+1\otimes x
  9. Δ y = y 1 + ( - 1 ) F y \Delta y=y\otimes 1+(-1)^{F}\otimes y
  10. \mathcal{R}

Supervaluationism.html

  1. p ¬ p p\vee\neg p
  2. p p
  3. p p

Supnick_matrix.html

  1. 1 i < k n 1\leq i<k\leq n
  2. 1 j < l n 1\leq j<l\leq n
  3. a i j + a k l a i l + a k j a_{ij}+a_{kl}\leq a_{il}+a_{kj}\,
  4. a i j = a j i . a_{ij}=a_{ji}.\,
  5. S = [ s i j ] = [ α i + α j ] ; S=[s_{ij}]=[\alpha_{i}+\alpha_{j}];\,

Support_function.html

  1. n \mathbb{R}^{n}
  2. n \mathbb{R}^{n}
  3. h A : n h_{A}\colon\mathbb{R}^{n}\to\mathbb{R}
  4. n \mathbb{R}^{n}
  5. h A ( x ) = sup { x a : a A } , h_{A}(x)=\sup\{x\cdot a:a\in A\},
  6. x n x\in\mathbb{R}^{n}
  7. { y n : y x h A ( x ) } \{y\in\mathbb{R}^{n}:y\cdot x\leqslant h_{A}(x)\}
  8. H ( x ) = { y n : y x = h A ( x ) } H(x)=\{y\in\mathbb{R}^{n}:y\cdot x=h_{A}(x)\}
  9. h A ( x ) = x a h_{A}(x)=x\cdot a
  10. h B 1 ( x ) = | x | h_{B_{1}}(x)=|x|
  11. h A ( x ) = | x a | h_{A}(x)=|x\cdot a|
  12. \infty
  13. h A ( u ; x ) = h A H ( u ) ( x ) x n . h_{A}^{\prime}(u;x)=h_{A\cap H(u)}(x)\qquad x\in\mathbb{R}^{n}.
  14. h A ( α x ) = α h A ( x ) , α 0 , x n , h_{A}(\alpha x)=\alpha h_{A}(x),\qquad\alpha\geq 0,x\in\mathbb{R}^{n},
  15. h A ( x + y ) h A ( x ) + h A ( y ) , x , y n . h_{A}(x+y)\leq h_{A}(x)+h_{A}(y),\qquad x,y\in\mathbb{R}^{n}.
  16. n \mathbb{R}^{n}
  17. n \mathbb{R}^{n}
  18. h α A ( x ) = α h A ( x ) , α 0 , x n h_{\alpha A}(x)=\alpha h_{A}(x),\qquad\alpha\geq 0,x\in\mathbb{R}^{n}
  19. h A + b ( x ) = h A ( x ) + x b , x , b n . h_{A+b}(x)=h_{A}(x)+x\cdot b,\qquad x,b\in\mathbb{R}^{n}.
  20. h A + B ( x ) = h A ( x ) + h B ( x ) , x n , h_{A+B}(x)=h_{A}(x)+h_{B}(x),\qquad x\in\mathbb{R}^{n},
  21. A + B := { a + b n a A , b B } . A+B:=\{\,a+b\in\mathbb{R}^{n}\mid a\in A,\ b\in B\,\}.
  22. d H ( A , B ) = h A - h B d_{\mathrm{H}}(A,B)=\|h_{A}-h_{B}\|_{\infty}
  23. τ \tau
  24. \mapsto
  25. τ \tau
  26. τ \tau
  27. x x N ( x ) {x}\mapsto{x}\cdot N({x})
  28. x M {x}\in M

Support_polygon.html

  1. C 1 , , C N C_{1},\ldots,C_{N}
  2. C k C_{k}
  3. F C k FC_{k}
  4. F C k FC_{k}
  5. f 1 , , f N f_{1},\ldots,f_{N}
  6. f 1 , , f N f_{1},\ldots,f_{N}
  7. k = 1 N f k + G = 0 \sum_{k=1}^{N}f_{k}+G=0
  8. k = 1 N f k × C k + G × C M = 0 \sum_{k=1}^{N}f_{k}\times C_{k}+G\times CM=0
  9. f k F C k f_{k}\in FC_{k}
  10. k k
  11. G G
  12. C M CM
  13. f 1 , , f N f_{1},\ldots,f_{N}
  14. C M CM
  15. C M + α G CM+\alpha G
  16. C M CM
  17. - G -G

Suspension_polymerization.html

  1. d ¯ = k D v R ν m ϵ D s N ν l C s \bar{d}=k{D_{v}\cdot R\cdot\nu_{m}\cdot\epsilon\over D_{s}\cdot N\cdot\nu_{l}% \cdot C_{s}}

Swift–Hohenberg_equation.html

  1. u t = r u - ( 1 + 2 ) 2 u + N ( u ) \frac{\partial u}{\partial t}=ru-(1+\nabla^{2})^{2}u+N(u)

Swizzling_(computer_graphics).html

  1. A = ( 1 , 2 , 3 , 4 ) T A=(1,2,3,4)^{T}
  2. A A
  3. A . w w x y = [ 0 0 0 1 0 0 0 1 1 0 0 0 0 1 0 0 ] [ 1 2 3 4 ] = [ 4 4 1 2 ] A.wwxy=\begin{bmatrix}0&0&0&1\\ 0&0&0&1\\ 1&0&0&0\\ 0&1&0&0\end{bmatrix}\begin{bmatrix}1\\ 2\\ 3\\ 4\end{bmatrix}=\begin{bmatrix}4\\ 4\\ 1\\ 2\end{bmatrix}

Sylvester's_criterion.html

  1. {}\quad\vdots
  2. \mathbb{R}
  3. λ = x T A x x T x = x T B T B x x T x = B x 2 x 2 0. \lambda={\frac{x^{T}Ax}{x^{T}x}}={\frac{x^{T}B^{T}Bx}{x^{T}x}}={\frac{\|Bx\|^{% 2}}{\|x\|^{2}}}\geq 0.
  4. 𝐀 \displaystyle\mathbf{A}
  5. u 11 , u 22 , , u 11 \scriptstyle\sqrt{u_{11}},\scriptstyle\sqrt{u_{22}},\ldots,\scriptstyle\sqrt{u% _{11}}
  6. 𝐑 = L D = [ 1 0 0 r 12 / r 11 1 0 r 1 n / r 11 r 2 n / r 22 1 ] [ r 11 0 0 0 r 22 0 0 0 r n n ] . \mathbf{R}=LD=\begin{bmatrix}1&0&\cdots&0\\ r_{12}/r_{11}&1&\cdots&0\\ \vdots&\vdots&&\vdots\\ r_{1n}/r_{11}&r_{2n}/r_{22}&\cdots&1\end{bmatrix}\begin{bmatrix}r_{11}&0&% \cdots&0\\ 0&r_{22}&\cdots&0\\ \vdots&\vdots&&\vdots\\ 0&0&\cdots&r_{nn}\end{bmatrix}.
  7. A = B T B A=B^{T}B
  8. A = B T B = R T Q T Q R = R T R A=B^{T}B=R^{T}Q^{T}QR=R^{T}R

Sylvester's_determinant_theorem.html

  1. det ( I p + A B ) = det ( I n + B A ) , \det(I_{p}+AB)=\det(I_{n}+BA),
  2. M M
  3. - A -A
  4. B B
  5. I n I_{n}
  6. I p I_{p}
  7. M = ( I p - A B I n ) M=\begin{pmatrix}I_{p}&-A\\ B&I_{n}\end{pmatrix}
  8. M M
  9. M = ( I p 0 B I n ) ( I p - A 0 I n + B A ) M=\begin{pmatrix}I_{p}&0\\ B&I_{n}\end{pmatrix}\begin{pmatrix}I_{p}&-A\\ 0&I_{n}+BA\end{pmatrix}
  10. det ( M ) = det ( I n + B A ) \det(M)=\det(I_{n}+BA)
  11. M M
  12. M = ( I p + A B - A 0 I n ) ( I p 0 B I n ) M=\begin{pmatrix}I_{p}+AB&-A\\ 0&I_{n}\end{pmatrix}\begin{pmatrix}I_{p}&0\\ B&I_{n}\end{pmatrix}
  13. det ( M ) = det ( I p + A B ) \det(M)=\det(I_{p}+AB)
  14. det ( I n + B A ) = det ( I p + A B ) \det(I_{n}+BA)=\det(I_{p}+AB)

Sylvester's_formula.html

  1. f ( A ) f(A)
  2. A A
  3. A A
  4. A A
  5. f ( A ) = i = 1 k f ( λ i ) A i , f(A)=\sum_{i=1}^{k}f(\lambda_{i})~{}A_{i}~{},
  6. A A
  7. A A
  8. A A
  9. A A
  10. A A
  11. k k
  12. λ λ
  13. f f
  14. f ( A ) f(A)
  15. f f
  16. m m
  17. f f
  18. A = [ 1 3 4 2 ] . A=\begin{bmatrix}1&3\\ 4&2\end{bmatrix}.
  19. A 1 = c 1 r 1 = [ 3 4 ] [ 1 / 7 1 / 7 ] = [ 3 / 7 3 / 7 4 / 7 4 / 7 ] = A + 2 I 5 - ( - 2 ) A 2 = c 2 r 2 = [ 1 / 7 - 1 / 7 ] [ 4 - 3 ] = [ 4 / 7 - 3 / 7 - 4 / 7 3 / 7 ] = A - 5 I - 2 - 5 . \begin{aligned}\displaystyle A_{1}&\displaystyle=c_{1}r_{1}=\begin{bmatrix}3\\ 4\end{bmatrix}\begin{bmatrix}1/7&1/7\end{bmatrix}=\begin{bmatrix}3/7&3/7\\ 4/7&4/7\end{bmatrix}=\frac{A+2I}{5-(-2)}\\ \displaystyle A_{2}&\displaystyle=c_{2}r_{2}=\begin{bmatrix}1/7\\ -1/7\end{bmatrix}\begin{bmatrix}4&-3\end{bmatrix}=\begin{bmatrix}4/7&-3/7\\ -4/7&3/7\end{bmatrix}=\frac{A-5I}{-2-5}.\end{aligned}
  20. f ( A ) = f ( 5 ) A 1 + f ( - 2 ) A 2 . f(A)=f(5)A_{1}+f(-2)A_{2}.\,
  21. f f
  22. 1 5 [ 3 / 7 3 / 7 4 / 7 4 / 7 ] - 1 2 [ 4 / 7 - 3 / 7 - 4 / 7 3 / 7 ] = [ - 0.2 0.3 0.4 - 0.1 ] . \frac{1}{5}\begin{bmatrix}3/7&3/7\\ 4/7&4/7\end{bmatrix}-\frac{1}{2}\begin{bmatrix}4/7&-3/7\\ -4/7&3/7\end{bmatrix}=\begin{bmatrix}-0.2&0.3\\ 0.4&-0.1\end{bmatrix}.

Sylvester_equation.html

  1. A X + X B = C . AX+XB=C.
  2. vec \operatorname{vec}
  3. ( I n A + B T I n ) vec X = vec C , (I_{n}\otimes A+B^{T}\otimes I_{n})\operatorname{vec}X=\operatorname{vec}C,
  4. I n I_{n}
  5. n × n n\times n
  6. n 2 × n 2 n^{2}\times n^{2}
  7. n × n n\times n
  8. A A
  9. B B
  10. X X
  11. C C
  12. A A
  13. - B -B
  14. S : M n M n S:M_{n}\rightarrow M_{n}
  15. X A X + X B X\mapsto AX+XB
  16. A A
  17. - B -B
  18. f ( z ) f(z)
  19. g ( z ) g(z)
  20. 1 1
  21. p ( z ) p(z)
  22. q ( z ) q(z)
  23. p ( z ) f ( z ) + q ( z ) g ( z ) = 1 p(z)f(z)+q(z)g(z)=1
  24. f ( A ) = 0 = g ( - B ) f(A)=0=g(-B)
  25. g ( A ) q ( A ) = I g(A)q(A)=I
  26. X X
  27. S ( X ) = 0 S(X)=0
  28. A X = - X B AX=-XB
  29. X = q ( A ) g ( A ) X = q ( A ) X g ( - B ) = 0 X=q(A)g(A)X=q(A)Xg(-B)=0
  30. S S
  31. C C
  32. X X
  33. s s
  34. A A
  35. - B -B
  36. s s
  37. A T A^{T}
  38. v v
  39. w w
  40. A T w = s w A^{T}w=sw
  41. B v = - s v Bv=-sv
  42. C C
  43. C v = w ¯ Cv=\overline{w}
  44. w w
  45. A X + X B = C AX+XB=C
  46. X X
  47. < ( A X + X B ) v , w < C v , w < w ¯ , w Align g t ; <(AX+XB)v,w>=<Cv,w>=<\overline{w},w&gt;
  48. [ A C 0 B ] \begin{bmatrix}A&C\\ 0&B\end{bmatrix}
  49. [ A 0 0 B ] \begin{bmatrix}A&0\\ 0&B\end{bmatrix}
  50. [ I n X 0 I m ] [ A C 0 B ] [ I n - X 0 I m ] = [ A 0 0 B ] . \begin{bmatrix}I_{n}&X\\ 0&I_{m}\end{bmatrix}\begin{bmatrix}A&C\\ 0&B\end{bmatrix}\begin{bmatrix}I_{n}&-X\\ 0&I_{m}\end{bmatrix}=\begin{bmatrix}A&0\\ 0&B\end{bmatrix}.
  51. A A
  52. B B
  53. ( n 3 ) (n^{3})

Symbol_of_a_differential_operator.html

  1. P = p ( x , D ) = | α | k a α ( x ) D α . P=p(x,D)=\sum_{|\alpha|\leq k}a_{\alpha}(x)D^{\alpha}.
  2. σ P ( ξ ) = p ( x , ξ ) = | α | k a α ( x ) ξ α . \sigma_{P}(\xi)=p(x,\xi)=\sum_{|\alpha|\leq k}a_{\alpha}(x)\xi^{\alpha}.
  3. σ P ( ξ ) = | α | = k a α ξ α \sigma_{P}(\xi)=\sum_{|\alpha|=k}a_{\alpha}\xi^{\alpha}
  4. P f ( x ) = 1 2 π 𝐑 d e i x ξ p ( x , i ξ ) f ^ ( ξ ) d ξ . Pf(x)=\frac{1}{2\pi}\int_{\mathbf{R}^{d}}e^{ix\cdot\xi}p(x,i\xi)\hat{f}(\xi)\,% d\xi.
  5. P : C ( E ) C ( F ) P:C^{\infty}(E)\to C^{\infty}(F)
  6. k k
  7. P u ( x ) = | α | = k P α ( x ) α u x α + lower order terms Pu(x)=\sum_{|\alpha|=k}P^{\alpha}(x)\frac{\partial^{\alpha}u}{\partial x^{% \alpha}}+\,\text{lower order terms}
  8. P α ( x ) : E F P^{\alpha}(x):E\to F
  9. σ P : S k ( T * X ) E F \sigma_{P}:S^{k}(T^{*}X)\otimes E\to F
  10. ( P u ) ν = μ P ν μ u μ (Pu)_{\nu}=\sum_{\mu}P_{\nu\mu}u_{\mu}
  11. P ν μ = α P ν μ α x α . P_{\nu\mu}=\sum_{\alpha}P_{\nu\mu}^{\alpha}\frac{\partial}{\partial x^{\alpha}}.
  12. ( σ P ( ξ ) u ) ν = | α | = k μ P ν μ α ( x ) ξ α u μ . (\sigma_{P}(\xi)u)_{\nu}=\sum_{|\alpha|=k}\sum_{\mu}P_{\nu\mu}^{\alpha}(x)\xi_% {\alpha}u^{\mu}.
  13. σ P \sigma_{P}
  14. T x * X T^{*}_{x}X
  15. Hom ( E x , F x ) \operatorname{Hom}(E_{x},F_{x})
  16. P P
  17. θ T * X \theta\in T^{*}X
  18. σ P ( θ , , θ ) \sigma_{P}(\theta,\dots,\theta)

Symmetric_rank-one.html

  1. x f ( x ) x\mapsto f(x)
  2. f \nabla f
  3. B B
  4. f f
  5. x 0 x_{0}
  6. f ( x 0 + Δ x ) = f ( x 0 ) + f ( x 0 ) T Δ x + 1 2 Δ x T B Δ x f(x_{0}+\Delta x)=f(x_{0})+\nabla f(x_{0})^{T}\Delta x+\frac{1}{2}\Delta x^{T}% {B}\Delta x
  7. f ( x 0 + Δ x ) = f ( x 0 ) + B Δ x \nabla f(x_{0}+\Delta x)=\nabla f(x_{0})+B\Delta x
  8. B B
  9. B B
  10. B k B_{k}
  11. B k + 1 = B k + ( y k - B k Δ x k ) ( y k - B k Δ x k ) T ( y k - B k Δ x k ) T Δ x k B_{k+1}=B_{k}+\frac{(y_{k}-B_{k}\Delta x_{k})(y_{k}-B_{k}\Delta x_{k})^{T}}{(y% _{k}-B_{k}\Delta x_{k})^{T}\Delta x_{k}}
  12. y k = f ( x k + Δ x k ) - f ( x k ) y_{k}=\nabla f(x_{k}+\Delta x_{k})-\nabla f(x_{k})
  13. H k = B k - 1 H_{k}=B_{k}^{-1}
  14. H k + 1 = H k + ( Δ x k - H k y k ) ( Δ x k - H k y k ) T ( Δ x k - H k y k ) T y k H_{k+1}=H_{k}+\frac{(\Delta x_{k}-H_{k}y_{k})(\Delta x_{k}-H_{k}y_{k})^{T}}{(% \Delta x_{k}-H_{k}y_{k})^{T}y_{k}}
  15. | Δ x k T ( y k - B k Δ x k ) | r Δ x k y k - B k Δ x k |\Delta x_{k}^{T}(y_{k}-B_{k}\Delta x_{k})|\geq r\|\Delta x_{k}\|\cdot\|y_{k}-% B_{k}\Delta x_{k}\|
  16. r ( 0 , 1 ) r\in(0,1)
  17. 10 - 8 10^{-8}

Symmetric_successive_overrelaxation.html

  1. A = D + L + L T A=D+L+L^{T}
  2. M = ( D + L ) D - 1 ( D + L ) T M=(D+L)D^{-1}(D+L)^{T}
  3. ω \omega
  4. M ( ω ) = ω 2 - ω ( 1 ω D + L ) ( D ) - 1 ( 1 ω D + L ) T M(\omega)={\omega\over{2-\omega}}\left({1\over\omega}D+L\right)\left(D\right)^% {-1}\left({1\over\omega}D+L\right)^{T}

Synchronization_(alternating_current).html

  1. f = p n s 120 f=\frac{pn_{s}}{120\ }

Synchronization_of_chaos.html

  1. x = F ( x ) + α ( y - x ) x^{\prime}=F(x)+\alpha(y-x)
  2. y = F ( y ) + α ( x - y ) y^{\prime}=F(y)+\alpha(x-y)
  3. F F
  4. α \alpha
  5. x ( t ) = y ( t ) x(t)=y(t)
  6. x ( t ) = y ( t ) x(t)=y(t)
  7. v = x - y v=x-y
  8. v v
  9. v = D F ( x ( t ) ) v - 2 α v v^{\prime}=DF(x(t))v-2\alpha v
  10. D F ( x ( t ) ) DF(x(t))
  11. α = 0 \alpha=0
  12. u = D F ( x ( t ) ) u , u^{\prime}=DF(x(t))u,
  13. u ( t ) u ( 0 ) e λ t \|u(t)\|\leq\|u(0)\|e^{\lambda t}
  14. λ \lambda
  15. v = u e - 2 α t v=ue^{-2\alpha t}
  16. v v
  17. u u
  18. v ( t ) u ( 0 ) e ( - 2 α + λ ) t \|v(t)\|\leq\|u(0)\|e^{(-2\alpha+\lambda)t}
  19. α c = λ / 2 \alpha_{c}=\lambda/2
  20. α > α c \alpha>\alpha_{c}

Systematic_code.html

  1. G G
  2. G = [ I k | P ] G=[I_{k}|P]
  3. I k I_{k}
  4. k k

Sz.-Nagy's_dilation_theorem.html

  1. T n = P H U n | H , n 0. T^{n}=P_{H}U^{n}|_{H},\quad n\geq 0.
  2. T 1 \|T\|\leq 1
  3. U = [ S D S * 0 - S * ] . U=\begin{bmatrix}S&D_{S^{*}}\\ 0&-S^{*}\end{bmatrix}.
  4. n 0 H \oplus_{n\geq 0}H
  5. S = [ T 0 0 D T 0 0 0 I 0 0 0 I ] . S=\begin{bmatrix}T&0&0&\cdots&\\ D_{T}&0&0&&\\ 0&I&0&\ddots\\ 0&0&I&\ddots\\ \vdots&&\ddots&\ddots\end{bmatrix}.
  6. T n = P H S n | H = P H ( Q H U | H ) n | H = P H U n | H . T^{n}=P_{H}S^{n}|_{H}=P_{H}(Q_{H^{\prime}}U|_{H^{\prime}})^{n}|_{H}=P_{H}U^{n}% |_{H}.
  7. ( X ) \mathcal{R}(X)

Sørensen–Dice_coefficient.html

  1. Q S = 2 C A + B = 2 | A B | | A | + | B | QS=\frac{2C}{A+B}=\frac{2|A\cap B|}{|A|+|B|}
  2. s = 2 | X Y | | X | + | Y | s=\frac{2|X\cap Y|}{|X|+|Y|}
  3. s v = 2 | A B | | A | 2 + | B | 2 s_{v}=\frac{2|A\cdot B|}{|A|^{2}+|B|^{2}}
  4. s = 2 n t n x + n y s=\frac{2n_{t}}{n_{x}+n_{y}}
  5. d = 1 - 2 | X Y | | X | + | Y | d=1-\frac{2|X\cap Y|}{|X|+|Y|}

T-criterion.html

  1. σ ¯ \bar{\sigma}
  2. ϵ ¯ \bar{\epsilon}
  3. Θ \Theta
  4. σ ¯ - ϵ ¯ \bar{\sigma}-\bar{\epsilon}
  5. Θ \Theta
  6. d ϵ i , j d\epsilon_{i,j}
  7. d ϵ i , j e d\epsilon_{i,j}^{e}
  8. d ϵ i , j p d\epsilon_{i,j}^{p}
  9. d ϵ i , j = d ϵ i , j e + d ϵ i , j p d\epsilon_{i,j}=d\epsilon_{i,j}^{e}+d\epsilon_{i,j}^{p}
  10. d ϵ i , j e d\epsilon_{i,j}^{e}
  11. d ϵ i , j e = 1 2 G ( d σ i , j - 3 ν 1 + ν δ i , j d p ) d\epsilon_{i,j}^{e}=\cfrac{1}{2G}(d{\sigma}_{i,j}-\cfrac{3\nu}{1+{\nu}}{\delta% }_{i,j}dp)
  12. G = E 2 ( 1 + ν ) G=\cfrac{E}{2(1+\nu)}
  13. ν \nu
  14. δ i , j {\delta}_{i,j}
  15. d ϵ i , j p d\epsilon_{i,j}^{p}
  16. d ϵ i , j p = s i , j d λ d\epsilon_{i,j}^{p}=s_{i,j}d{\lambda}
  17. s i , j = σ i , j - δ i , j p s_{i,j}={\sigma}_{i,j}-{\delta}_{i,j}p
  18. d λ d{\lambda}
  19. d w p = σ i , j d ϵ i , j p = σ i , j s i , j d λ dw_{p}={\sigma}_{i,j}d{\epsilon}_{i,j}^{p}={\sigma}_{i,j}s_{i,j}d{\lambda}
  20. d T dT
  21. Π {\Pi}
  22. d T = d Π = p d Θ + σ d ϵ = d T V + d T D * dT=d{\Pi}=pd{\Theta}+{\sigma}d{\epsilon}=dT_{V}+dT_{D}^{*}
  23. Θ = ϵ 11 + ϵ 22 + ϵ 33 {\Theta}={\epsilon}_{11}+{\epsilon}_{22}+{\epsilon}_{33}
  24. p = 1 3 ( σ 11 + σ 22 + σ 33 ) p=\cfrac{1}{3}({\sigma}_{11}+{\sigma}_{22}+{\sigma}_{33})
  25. σ ¯ = 1 2 2 [ ( σ 11 - σ 22 ) 2 + ( σ 22 - σ 33 ) 2 + ( σ 33 - σ 11 ) 2 ] 1 / 2 \bar{\sigma}=\cfrac{1}{2}\sqrt{2}[({\sigma}_{11}-{\sigma}_{22})^{2}+({\sigma}_% {22}-{\sigma}_{33})^{2}+({\sigma}_{33}-{\sigma}_{11})^{2}]^{1/2}
  26. ϵ ¯ = 1 ′′′ 2 2 [ ( ϵ 11 - ϵ 22 ) 2 + ( ϵ 22 - ϵ 33 ) 2 + ( ϵ 33 - ϵ 11 ) 2 ] 1 / 2 \bar{\epsilon}=\cfrac{1^{\prime\prime\prime}}{2}\sqrt{2}[({\epsilon}_{11}-{% \epsilon}_{22})^{2}+({\epsilon}_{22}-{\epsilon}_{33})^{2}+({\epsilon}_{33}-{% \epsilon}_{11})^{2}]^{1/2}
  27. d T V = p d Θ dT_{V}=pd\Theta
  28. d T D * = σ ¯ d ϵ ¯ dT_{D}^{*}=\bar{\sigma}d\bar{\epsilon}
  29. d λ d{\lambda}
  30. d λ d{\lambda}
  31. d λ d{\lambda}
  32. σ ¯ = σ ¯ ( ϵ ¯ ) \bar{\sigma}=\bar{\sigma}(\bar{\epsilon})
  33. H ( σ ¯ , ϵ ¯ ) = d σ ¯ d ϵ ¯ H(\bar{\sigma},\bar{\epsilon})=\cfrac{d\bar{\sigma}}{d\bar{\epsilon}}
  34. d λ d{\lambda}
  35. d λ = 3 2 σ ¯ 2 d w p ( H ) d{\lambda}=\cfrac{3}{2\bar{\sigma}^{2}}dw_{p}(H)
  36. d w p ( H ) dw_{p}(H)
  37. d T D = σ ¯ d ϵ ¯ e dT_{D}=\bar{\sigma}d\bar{\epsilon}^{e}
  38. ϵ ¯ e \bar{\epsilon}^{e}
  39. T D = σ ¯ d ϵ ¯ e = 1 6 G σ ¯ 2 T_{D}=\int\bar{\sigma}d\bar{\epsilon}^{e}=\cfrac{1}{6G}\bar{\sigma}^{2}
  40. d T V = p d Θ dT_{V}=pd\Theta
  41. T V = p d Θ = 1 2 K p 2 = 1 2 K Θ 2 T_{V}=\int{pd\Theta}=\cfrac{1}{2K}p^{2}=\cfrac{1}{2}K{\Theta}^{2}
  42. Θ {\Theta}
  43. Θ = 1 2 K p {\Theta}=\cfrac{1}{2K}p
  44. d Θ = 1 K d p d{\Theta}=\cfrac{1}{K}dp
  45. Θ = ϵ 11 + ϵ 22 + ϵ 33 {\Theta}={\epsilon}_{11}+{\epsilon}_{22}+{\epsilon}_{33}
  46. p = 1 3 ( σ 11 + σ 22 + σ 33 ) p=\cfrac{1}{3}({\sigma}_{11}+{\sigma}_{22}+{\sigma}_{33})
  47. K = E 3 ( 1 - 2 ν ) K=\cfrac{E}{3(1-2\nu)}
  48. T V , 0 T_{V,0}
  49. T D , 0 T_{D,0}

T-function.html

  1. x i = x i + f ( x 0 , , x i - 1 ) x_{i}^{\prime}=x_{i}+f(x_{0},\cdots,x_{i-1})

T-J_model.html

  1. H ^ = - t < i j > σ ( a ^ i σ a ^ j σ + a ^ j σ a ^ i σ ) + J < i j > ( S i S j - n i n j / 4 ) \hat{H}=-t\sum_{<ij>\sigma}\left(\hat{a}^{\dagger}_{i\sigma}\hat{a}_{j\sigma}+% \hat{a}^{\dagger}_{j\sigma}\hat{a}_{i\sigma}\right)+J\sum_{<ij>}(\vec{S}_{i}% \cdot\vec{S}_{j}-n_{i}n_{j}/4)
  2. < i j > \sum_{<ij>}
  3. a ^ i σ , a ^ j σ \hat{a}^{\dagger}_{i\sigma},\hat{a}_{j\sigma}
  4. σ \sigma
  5. t t
  6. J J
  7. J = 4 t 2 / U J=4t^{2}/U
  8. U U
  9. n i = σ a ^ i σ a ^ i σ n_{i}=\sum_{\sigma}\hat{a}^{\dagger}_{i\sigma}\hat{a}_{i\sigma}
  10. S i , S j \vec{S}_{i},\vec{S}_{j}

Table_of_thermodynamic_equations.html

  1. β = 1 / k B T \beta=1/k_{B}T\,\!
  2. τ = k B T \tau=k_{B}T\,\!
  3. τ = k B ( U / S ) N \tau=k_{B}\left(\partial U/\partial S\right)_{N}\,\!
  4. 1 / τ = 1 / k B ( S / U ) N 1/\tau=1/k_{B}\left(\partial S/\partial U\right)_{N}\,\!
  5. S = - k B i p i ln p i S=-k_{B}\sum_{i}p_{i}\ln p_{i}
  6. S = ( F / T ) V S=\left(\partial F/\partial T\right)_{V}\,\!
  7. S = ( G / T ) N , P S=\left(\partial G/\partial T\right)_{N,P}\,\!
  8. P = - ( F / V ) T , N P=-\left(\partial F/\partial V\right)_{T,N}\,\!
  9. P = - ( U / V ) S , N P=-\left(\partial U/\partial V\right)_{S,N}\,\!
  10. U = i E i U=\sum_{i}E_{i}\!
  11. H = U + p V H=U+pV\,\!
  12. G = H - T S G=H-TS\,\!
  13. μ i = ( U / N i ) N j i , S , V \mu_{i}=\left(\partial U/\partial N_{i}\right)_{N_{j\neq i},S,V}\,\!
  14. μ i = ( F / N i ) T , V \mu_{i}=\left(\partial F/\partial N_{i}\right)_{T,V}\,\!
  15. μ i = ( G / N i ) T , P \mu_{i}=\left(\partial G/\partial N_{i}\right)_{T,P}\,\!
  16. μ i / τ = - 1 / k B ( S / N i ) U , V \mu_{i}/\tau=-1/k_{B}\left(\partial S/\partial N_{i}\right)_{U,V}\,\!
  17. F = U - T S F=U-TS\,\!
  18. Ω = U - T S - μ N \Omega=U-TS-\mu N\,\!
  19. Φ = S - U / T \Phi=S-U/T\,\!
  20. Ξ = Φ - p V / T \Xi=\Phi-pV/T\,\!
  21. C = Q / T C=\partial Q/\partial T\,\!
  22. C p = H / T C_{p}=\partial H/\partial T\,\!
  23. C m p = 2 Q / m T C_{mp}=\partial^{2}Q/\partial m\partial T\,\!
  24. C n p = 2 Q / n T C_{np}=\partial^{2}Q/\partial n\partial T\,\!
  25. C V = Q / T C_{V}=\partial Q/\partial T\,\!
  26. C m V = 2 Q / m T C_{mV}=\partial^{2}Q/\partial m\partial T\,\!
  27. C n V = 2 Q / n T C_{nV}=\partial^{2}Q/\partial n\partial T\,\!
  28. L = Q / m L=\partial Q/\partial m\,\!
  29. γ = C p / C V = c p / c V = C m p / C m V \gamma=C_{p}/C_{V}=c_{p}/c_{V}=C_{mp}/C_{mV}\,\!
  30. T \nabla T\,\!
  31. P = d Q / d t P=\mathrm{d}Q/\mathrm{d}t\,\!
  32. I = d P / d A I=\mathrm{d}P/\mathrm{d}A
  33. Q = 𝐪 d 𝐒 d t Q=\iint\mathbf{q}\cdot\mathrm{d}\mathbf{S}\mathrm{d}t\,\!
  34. Δ Q = 0 , Δ U = - Δ W \Delta Q=0,\quad\Delta U=-\Delta W\,\!
  35. Δ U = 0 , Δ W = Δ H \Delta U=0,\quad\Delta W=\Delta H\,\!
  36. W = k T N ln ( V 2 / V 1 ) W=kTN\ln(V_{2}/V_{1})\,\!
  37. Δ W = p Δ V , Δ Q = Δ U + p δ V \Delta W=p\Delta V,\quad\Delta Q=\Delta U+p\delta V\,\!
  38. Δ W = 0 , Δ Q = Δ U \Delta W=0,\quad\Delta Q=\Delta U\,\!
  39. p 1 V 1 γ = p 2 V 2 γ p_{1}V_{1}^{\gamma}=p_{2}V_{2}^{\gamma}\,\!
  40. T 1 V 1 γ - 1 = T 2 V 2 γ - 1 T_{1}V_{1}^{\gamma-1}=T_{2}V_{2}^{\gamma-1}\,\!
  41. Δ U = 0 \Delta U=0\,\!
  42. Δ W = V 1 V 2 p d V \Delta W=\int_{V_{1}}^{V_{2}}p\mathrm{d}V\,\!
  43. Δ W = cycle p d V \Delta W=\oint_{\mathrm{cycle}}p\mathrm{d}V\,\!
  44. p V = n R T = k T N pV=nRT=kTN\,\!
  45. p 1 V 1 p 2 V 2 = n 1 T 1 n 2 T 2 = N 1 T 1 N 2 T 2 \frac{p_{1}V_{1}}{p_{2}V_{2}}=\frac{n_{1}T_{1}}{n_{2}T_{2}}=\frac{N_{1}T_{1}}{% N_{2}T_{2}}\,\!
  46. p = N m v 2 3 V = n M m v 2 3 V = 1 3 ρ v 2 p=\frac{Nm\langle v^{2}\rangle}{3V}=\frac{nM_{m}\langle v^{2}\rangle}{3V}=% \frac{1}{3}\rho\langle v^{2}\rangle\,\!
  47. Q = 0 Q=0
  48. δ W = - p d V \delta W=-pdV\;
  49. - p Δ V -p\Delta V\;
  50. 0 0\;
  51. - n R T ln V 2 V 1 -nRT\ln\frac{V_{2}}{V_{1}}\;
  52. n R T ln P 2 P 1 nRT\ln\frac{P_{2}}{P_{1}}\;
  53. P V γ ( V f 1 - γ - V i 1 - γ ) 1 - γ = C V ( T 2 - T 1 ) \frac{PV^{\gamma}(V_{f}^{1-\gamma}-V_{i}^{1-\gamma})}{1-\gamma}=C_{V}\left(T_{% 2}-T_{1}\right)
  54. C p = 5 2 n R C_{p}=\frac{5}{2}nR\;
  55. C p = 7 2 n R C_{p}=\frac{7}{2}nR\;
  56. C V = 3 2 n R C_{V}=\frac{3}{2}nR\;
  57. C V = 5 2 n R C_{V}=\frac{5}{2}nR\;
  58. Δ U = C v Δ T \Delta U=C_{v}\Delta T\;
  59. Q + W Q+W\;
  60. Q p - p Δ V Q_{p}-p\Delta V\;
  61. Q Q\;
  62. C V ( T 2 - T 1 ) C_{V}\left(T_{2}-T_{1}\right)\;
  63. 0 0\;
  64. Q = - W Q=-W\;
  65. W W\;
  66. C V ( T 2 - T 1 ) C_{V}\left(T_{2}-T_{1}\right)\;
  67. H = U + p V H=U+pV\;
  68. C p ( T 2 - T 1 ) C_{p}\left(T_{2}-T_{1}\right)\;
  69. Q V + V Δ p Q_{V}+V\Delta p\;
  70. 0 0\;
  71. C p ( T 2 - T 1 ) C_{p}\left(T_{2}-T_{1}\right)\;
  72. Δ S = C v ln T 2 T 1 + n R ln V 2 V 1 \Delta S=C_{v}\ln{T_{2}\over T_{1}}+nR\ln{V_{2}\over V_{1}}
  73. Δ S = C p ln T 2 T 1 - n R ln p 2 p 1 \Delta S=C_{p}\ln{T_{2}\over T_{1}}-nR\ln{p_{2}\over p_{1}}
  74. C p ln T 2 T 1 C_{p}\ln\frac{T_{2}}{T_{1}}\;
  75. C V ln T 2 T 1 C_{V}\ln\frac{T_{2}}{T_{1}}\;
  76. n R ln V 2 V 1 nR\ln\frac{V_{2}}{V_{1}}\;
  77. Q T \frac{Q}{T}\;
  78. C p ln V 2 V 1 + C V ln p 2 p 1 = 0 C_{p}\ln\frac{V_{2}}{V_{1}}+C_{V}\ln\frac{p_{2}}{p_{1}}=0\;
  79. \;
  80. V T \frac{V}{T}\;
  81. p T \frac{p}{T}\;
  82. p V pV\;
  83. p V γ pV^{\gamma}\;
  84. S = k B ( ln Ω ) S=k_{B}(\ln\Omega)
  85. d S = δ Q T dS=\frac{\delta Q}{T}
  86. θ = k B T / m c 2 \theta=k_{B}T/mc^{2}\,\!
  87. P ( v ) = 4 π ( m 2 π k B T ) 3 / 2 v 2 e - m v 2 / 2 k B T P\left(v\right)=4\pi\left(\frac{m}{2\pi k_{B}T}\right)^{3/2}v^{2}e^{-mv^{2}/2k% _{B}T}\,\!
  88. f ( p ) = 1 4 π m 3 c 3 θ K 2 ( 1 / θ ) e - γ ( p ) / θ f(p)=\frac{1}{4\pi m^{3}c^{3}\theta K_{2}(1/\theta)}e^{-\gamma(p)/\theta}
  89. S = - k B i P i ln P i = k B ln Ω S=-k_{B}\sum_{i}P_{i}\ln P_{i}=k_{\mathrm{B}}\ln\Omega\,\!
  90. P i = 1 / Ω P_{i}=1/\Omega\,\!
  91. Δ S = Q 1 Q 2 d Q T \Delta S=\int_{Q_{1}}^{Q_{2}}\frac{\mathrm{d}Q}{T}\,\!
  92. Δ S = k B N ln V 2 V 1 + N C V ln T 2 T 1 \Delta S=k_{B}N\ln\frac{V_{2}}{V_{1}}+NC_{V}\ln\frac{T_{2}}{T_{1}}\,\!
  93. 𝐅 S = - T S \mathbf{F}_{\mathrm{S}}=-T\nabla S\,\!
  94. E k = 1 2 k T \langle E_{\mathrm{k}}\rangle=\frac{1}{2}kT\,\!
  95. U = d f E k = d f 2 k T U=d_{f}\langle E_{\mathrm{k}}\rangle=\frac{d_{f}}{2}kT\,\!
  96. v = 8 k B T π m \langle v\rangle=\sqrt{\frac{8k_{B}T}{\pi m}}\,\!
  97. v rms = v 2 = 3 k B T m v_{\mathrm{rms}}=\sqrt{\langle v^{2}\rangle}=\sqrt{\frac{3k_{B}T}{m}}\,\!
  98. v mode = 2 k B T m v_{\mathrm{mode}}=\sqrt{\frac{2k_{B}T}{m}}\,\!
  99. = 1 / 2 n σ \ell=1/\sqrt{2}n\sigma\,\!
  100. d U = δ Q - δ W dU=\delta Q-\delta W
  101. 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}
  102. 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}
  103. 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}
  104. 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}
  105. U ( S , V ) U(S,V)\,
  106. H ( S , P ) H(S,P)\,
  107. F ( T , V ) F(T,V)\,
  108. G ( T , P ) G(T,P)\,
  109. ( T V ) S = - ( P S ) V = 2 U S V \left(\frac{\partial T}{\partial V}\right)_{S}=-\left(\frac{\partial P}{% \partial S}\right)_{V}=\frac{\partial^{2}U}{\partial S\partial V}
  110. ( T P ) S = + ( V S ) P = 2 H S P \left(\frac{\partial T}{\partial P}\right)_{S}=+\left(\frac{\partial V}{% \partial S}\right)_{P}=\frac{\partial^{2}H}{\partial S\partial P}
  111. + ( S V ) T = ( P T ) V = - 2 F T V +\left(\frac{\partial S}{\partial V}\right)_{T}=\left(\frac{\partial P}{% \partial T}\right)_{V}=-\frac{\partial^{2}F}{\partial T\partial V}
  112. - ( S P ) T = ( V T ) P = 2 G T P -\left(\frac{\partial S}{\partial P}\right)_{T}=\left(\frac{\partial V}{% \partial T}\right)_{P}=\frac{\partial^{2}G}{\partial T\partial P}
  113. ( S U ) V , N = 1 T \left({\partial S\over\partial U}\right)_{V,N}={1\over T}
  114. ( S V ) N , U = p T \left({\partial S\over\partial V}\right)_{N,U}={p\over T}
  115. ( S N ) V , U = - μ T \left({\partial S\over\partial N}\right)_{V,U}=-{\mu\over T}
  116. ( T S ) V = T C V \left({\partial T\over\partial S}\right)_{V}={T\over C_{V}}
  117. ( T S ) P = T C P \left({\partial T\over\partial S}\right)_{P}={T\over C_{P}}
  118. - ( p V ) T = 1 V K T -\left({\partial p\over\partial V}\right)_{T}={1\over{VK_{T}}}
  119. H = - T 2 ( ( G / T ) T ) p H=-T^{2}\left(\frac{\partial\left(G/T\right)}{\partial T}\right)_{p}
  120. U = - T 2 ( ( F / T ) T ) V U=-T^{2}\left(\frac{\partial\left(F/T\right)}{\partial T}\right)_{V}
  121. G = - V 2 ( ( F / V ) V ) T G=-V^{2}\left(\frac{\partial\left(F/V\right)}{\partial V}\right)_{T}
  122. ( H p ) T = V - T ( V T ) P \left(\frac{\partial H}{\partial p}\right)_{T}=V-T\left(\frac{\partial V}{% \partial T}\right)_{P}
  123. ( U V ) T = T ( P T ) V - P \left(\frac{\partial U}{\partial V}\right)_{T}=T\left(\frac{\partial P}{% \partial T}\right)_{V}-P
  124. U = N k B T 2 ( ln Z T ) V U=Nk_{B}T^{2}\left(\frac{\partial\ln Z}{\partial T}\right)_{V}~{}
  125. S = U T + N * S = U T + N k B ln Z - N k ln N + N k S=\frac{U}{T}+N*~{}S=\frac{U}{T}+Nk_{B}\ln Z-Nk\ln N+Nk~{}
  126. Z t = ( 2 π m k B T ) 3 2 V h 3 Z_{t}=\frac{(2\pi mk_{B}T)^{\frac{3}{2}}V}{h^{3}}
  127. Z v = 1 1 - e - h ω 2 π k B T Z_{v}=\frac{1}{1-e^{\frac{-h\omega}{2\pi k_{B}T}}}
  128. Z r = 2 I k B T σ ( h 2 π ) 2 Z_{r}=\frac{2Ik_{B}T}{\sigma(\frac{h}{2\pi})^{2}}
  129. μ J T = ( T p ) H \mu_{JT}=\left(\frac{\partial T}{\partial p}\right)_{H}
  130. K T = - 1 V ( V p ) T , N K_{T}=-{1\over V}\left({\partial V\over\partial p}\right)_{T,N}
  131. α p = 1 V ( V T ) p \alpha_{p}=\frac{1}{V}\left(\frac{\partial V}{\partial T}\right)_{p}
  132. C p = ( Q r e v T ) p = ( U T ) p + p ( V T ) p = ( H T ) p = T ( S T ) p C_{p}=\left({\partial Q_{rev}\over\partial T}\right)_{p}=\left({\partial U% \over\partial T}\right)_{p}+p\left({\partial V\over\partial T}\right)_{p}=% \left({\partial H\over\partial T}\right)_{p}=T\left({\partial S\over\partial T% }\right)_{p}
  133. C V = ( Q r e v T ) V = ( U T ) V = T ( S T ) V C_{V}=\left({\partial Q_{rev}\over\partial T}\right)_{V}=\left({\partial U% \over\partial T}\right)_{V}=T\left({\partial S\over\partial T}\right)_{V}
  134. ( T p ) H ( p H ) T ( H T ) p = - 1 \left(\frac{\partial T}{\partial p}\right)_{H}\left(\frac{\partial p}{\partial H% }\right)_{T}\left(\frac{\partial H}{\partial T}\right)_{p}=-1
  135. ( T p ) H = - ( H p ) T ( T H ) p = - 1 ( H T ) p ( H p ) T \begin{aligned}\displaystyle\left(\frac{\partial T}{\partial p}\right)_{H}&% \displaystyle=-\left(\frac{\partial H}{\partial p}\right)_{T}\left(\frac{% \partial T}{\partial H}\right)_{p}\\ &\displaystyle=\frac{-1}{\left(\frac{\partial H}{\partial T}\right)_{p}}\left(% \frac{\partial H}{\partial p}\right)_{T}\end{aligned}
  136. C p = ( H T ) p C_{p}=\left(\frac{\partial H}{\partial T}\right)_{p}
  137. ( T p ) H = - 1 C p ( H p ) T \Rightarrow\left(\frac{\partial T}{\partial p}\right)_{H}=-\frac{1}{C_{p}}% \left(\frac{\partial H}{\partial p}\right)_{T}
  138. d U = δ Q r e v - δ W r e v , dU=\delta Q_{rev}-\delta W_{rev},
  139. δ S = δ Q r e v T , δ W r e v = p δ V \delta S=\frac{\delta Q_{rev}}{T},\delta W_{rev}=p\delta V
  140. d U = T δ S - p δ V dU=T\delta S-p\delta V
  141. ( U T ) V = T ( S T ) V - p ( V T ) V ; C V = ( U T ) V \left(\frac{\partial U}{\partial T}\right)_{V}=T\left(\frac{\partial S}{% \partial T}\right)_{V}-p\left(\frac{\partial V}{\partial T}\right)_{V};C_{V}=% \left(\frac{\partial U}{\partial T}\right)_{V}
  142. C V = T ( S T ) V \Rightarrow C_{V}=T\left(\frac{\partial S}{\partial T}\right)_{V}
  143. I = σ ϵ ( T external 4 - T system 4 ) I=\sigma\epsilon\left(T_{\mathrm{external}}^{4}-T_{\mathrm{system}}^{4}\right)\,\!
  144. Δ U = N C V Δ T \Delta U=NC_{V}\Delta T\,\!
  145. C p - C V = n R C_{p}-C_{V}=nR\,\!
  146. λ net = j λ j \lambda_{\mathrm{net}}=\sum_{j}\lambda_{j}\,\!
  147. 1 λ net = j ( 1 λ j ) \frac{1}{\lambda}_{\mathrm{net}}=\sum_{j}\left(\frac{1}{\lambda}_{j}\right)\,\!
  148. η = | W Q H | \eta=\left|\frac{W}{Q_{H}}\right|\,\!
  149. η c = 1 - | Q L Q H | = 1 - T L T H \eta_{c}=1-\left|\frac{Q_{L}}{Q_{H}}\right|=1-\frac{T_{L}}{T_{H}}\,\!
  150. K = | Q L W | K=\left|\frac{Q_{L}}{W}\right|\,\!
  151. K C = | Q L | | Q H | - | Q L | = T L T H - T L K_{C}=\frac{|Q_{L}|}{|Q_{H}|-|Q_{L}|}=\frac{T_{L}}{T_{H}-T_{L}}\,\!

Tachyonic_antitelephone.html

  1. A A
  2. B B
  3. Δ t = t 1 - t 0 = B - A a . \Delta t=t_{1}-t_{0}=\frac{B-A}{a}.
  4. Δ t \displaystyle\Delta t^{\prime}
  5. v v
  6. a a
  7. c c
  8. a > 1 a>1
  9. S S
  10. ( t , x ) = ( t , a t ) (t,x)=(t,at)
  11. S S^{\prime}
  12. v v
  13. x = L x^{\prime}=L
  14. L L
  15. t = γ ( 1 - a v ) t t^{\prime}=\gamma\left(1-av\right)t
  16. x = γ ( a - v ) t x^{\prime}=\gamma\left(a-v\right)t
  17. x = L x^{\prime}=L
  18. t = L γ ( a - v ) t=\tfrac{L}{\gamma(a-v)}
  19. t = 1 - a v a - v L t^{\prime}=\frac{1-av}{a-v}L
  20. L a \tfrac{L}{a}
  21. T = L a + t = ( 1 a + 1 - a v a - v ) L T=\frac{L}{a}+t^{\prime}=\left(\frac{1}{a}+\frac{1-av}{a-v}\right)L
  22. v > 2 a 1 + a 2 v>\tfrac{2a}{1+a^{2}}
  23. T < 0 T<0
  24. 1 γ = 1 - ( v / c ) 2 \frac{1}{\gamma}=\sqrt{1-{(v/c)^{2}}}
  25. t \displaystyle t^{\prime}
  26. γ = 1 1 - ( v / c ) 2 \gamma=\frac{1}{\sqrt{1-{(v/c)^{2}}}}
  27. γ = 1 0.6 \gamma=\frac{1}{0.6}

Tail_sequence.html

  1. s s α | α < γ s\equiv\langle s_{\alpha}|\alpha<\gamma\rangle

Tajima's_D.html

  1. S S
  2. π \pi
  3. E [ π ] = θ = E [ S i = 1 n - 1 1 i ] = 4 N μ E[\pi]=\theta=E\left[\frac{S}{\sum_{i=1}^{n-1}\frac{1}{i}}\right]=4N\mu
  4. E [ π ] = θ = E [ S i = 1 n - 1 1 i ] = 2 N μ E[\pi]=\theta=E\left[\frac{S}{\sum_{i=1}^{n-1}\frac{1}{i}}\right]=2N\mu
  5. S S
  6. π \pi
  7. D D\,
  8. θ \theta\,
  9. d d\,
  10. d d\,
  11. V ^ ( d ) \sqrt{\hat{V}(d)}
  12. D = d V ^ ( d ) D=\frac{d}{\sqrt{\hat{V}(d)}}
  13. D D\,
  14. D D\,
  15. D = d V ^ ( d ) = k ^ - S a 1 [ e 1 S + e 2 S ( S - 1 ) ] D=\frac{d}{\sqrt{\hat{V}(d)}}=\frac{\hat{k}-\frac{S}{a_{1}}}{\sqrt{[e_{1}S+e_{% 2}S(S-1)]}}
  16. e 1 = c 1 a 1 e_{1}=\frac{c_{1}}{a_{1}}
  17. e 2 = c 2 a 1 2 + a 2 e_{2}=\frac{c_{2}}{a_{1}^{2}+a_{2}}
  18. c 1 = b 1 - 1 a 1 c_{1}=b_{1}-\frac{1}{a_{1}}
  19. c 2 = b 2 - n + 2 a 1 n + a 2 a 1 2 c_{2}=b_{2}-\frac{n+2}{a_{1}n}+\frac{a_{2}}{a_{1}^{2}}
  20. b 1 = n + 1 3 ( n - 1 ) b_{1}=\frac{n+1}{3(n-1)}
  21. b 2 = 2 ( n 2 + n + 3 ) 9 n ( n - 1 ) b_{2}=\frac{2(n^{2}+n+3)}{9n(n-1)}
  22. a 1 = i = 1 n - 1 1 i a_{1}=\sum_{i=1}^{n-1}\frac{1}{i}
  23. a 2 = i = 1 n - 1 1 i 2 a_{2}=\sum_{i=1}^{n-1}\frac{1}{i^{2}}
  24. k ^ \hat{k}\,
  25. S a 1 \frac{S}{a_{1}}
  26. n n\,
  27. N N\,
  28. ( i , j ) (i,j)
  29. k ^ = i < j k i j ( n 2 ) \hat{k}=\frac{\sum\sum_{i<j}k_{ij}}{{\left({{n}\atop{2}}\right)}}
  30. S S\,
  31. E ( S ) = a 1 M E(S)=a_{1}M\,
  32. M = 4 N μ M=4N\mu\,
  33. θ = 4 N μ \theta=4N\mu\,
  34. 3 + 2 + 2 + 3 + 1 + 3 + 2 + 2 + 1 + 1 10 = 2 {3+2+2+3+1+3+2+2+1+1\over 10}=2
  35. d = 2 - 1.92 = .08 d=2-1.92=.08

Tanaka's_formula.html

  1. | B t | = 0 t sgn ( B s ) d B s + L t |B_{t}|=\int_{0}^{t}\operatorname{sgn}(B_{s})\,dB_{s}+L_{t}
  2. 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}
  3. L t = lim ε 0 1 2 ε | { s [ 0 , t ] | B s ( - ε , + ε ) } | . L_{t}=\lim_{\varepsilon\downarrow 0}\frac{1}{2\varepsilon}|\{s\in[0,t]|B_{s}% \in(-\varepsilon,+\varepsilon)\}|.
  4. f ( x ) = | x | f(x)=|x|
  5. f ( x ) = sgn ( x ) f^{\prime}(x)=\operatorname{sgn}(x)
  6. f ′′ ( x ) = 2 δ ( x ) f^{\prime\prime}(x)=2\delta(x)
  7. x 2 2 | ε | + | ε | 2 . \frac{x^{2}}{2|\varepsilon|}+\frac{|\varepsilon|}{2}.

Tanaka_equation.html

  1. d X t = sgn ( X t ) d B t , \mathrm{d}X_{t}=\operatorname{sgn}(X_{t})\,\mathrm{d}B_{t},
  2. sgn ( x ) = { + 1 , x 0 ; - 1 , x < 0. \operatorname{sgn}(x)=\begin{cases}+1,&x\geq 0;\\ -1,&x<0.\end{cases}
  3. B ^ \hat{B}
  4. B ~ \tilde{B}
  5. B ~ t = 0 t sgn ( B ^ s ) d B ^ s = 0 t sgn ( X s ) d X s , \tilde{B}_{t}=\int_{0}^{t}\operatorname{sgn}\big(\hat{B}_{s}\big)\,\mathrm{d}% \hat{B}_{s}=\int_{0}^{t}\operatorname{sgn}\big(X_{s}\big)\,\mathrm{d}X_{s},
  6. d B ~ t = sgn ( X t ) d X t . \mathrm{d}\tilde{B}_{t}=\operatorname{sgn}(X_{t})\,\mathrm{d}X_{t}.
  7. d X t = sgn ( X t ) d B ~ t , \mathrm{d}X_{t}=\operatorname{sgn}(X_{t})\,\mathrm{d}\tilde{B}_{t},

Tangent_developable.html

  1. γ ( t ) \gamma(t)
  2. γ ( t ) \gamma(t)
  3. γ \gamma
  4. t t
  5. γ \gamma
  6. γ \gamma
  7. ( s , t ) γ ( t ) + s γ ( t ) . (s,t)\mapsto\gamma(t)+s\gamma^{\prime}(t).

Tangent_measure.html

  1. T a , r ( x ) = x - a r , T_{a,r}(x)=\frac{x-a}{r},
  2. T a , r # μ ( A ) = μ ( a + r A ) T_{a,r\#}\mu(A)=\mu(a+rA)
  3. a + r A = { a + r x : x A } . a+rA=\{a+rx:x\in A\}.
  4. lim i c i T a , r i # μ = ν \lim_{i\rightarrow\infty}c_{i}T_{a,r_{i}\#}\mu=\nu
  5. lim i Ω φ d ( c i T a , r i # μ ) = Ω φ d ν . \lim_{i\rightarrow\infty}\int_{\Omega}\varphi\,\mathrm{d}(c_{i}T_{a,r_{i}\#}% \mu)=\int_{\Omega}\varphi\,\mathrm{d}\nu.
  6. lim sup r 0 μ ( B ( a , 2 r ) ) μ ( B ( a , r ) ) < \limsup_{r\downarrow 0}\frac{\mu(B(a,2r))}{\mu(B(a,r))}<\infty
  7. 0 < lim sup r 0 μ ( B ( a , r ) ) r s < 0<\limsup_{r\downarrow 0}\frac{\mu(B(a,r))}{r^{s}}<\infty
  8. 0 < s < 0<s<\infty
  9. ν Tan ( μ , a ) \nu\in\mathrm{Tan}(\mu,a)
  10. c > 0 c>0
  11. c ν Tan ( μ , a ) c\nu\in\mathrm{Tan}(\mu,a)
  12. ν Tan ( μ , a ) \nu\in\mathrm{Tan}(\mu,a)
  13. r > 0 r>0
  14. T 0 , r # ν Tan ( μ , a ) T_{0,r\#}\nu\in\mathrm{Tan}(\mu,a)
  15. ν Tan ( μ , a ) \nu\in\mathrm{Tan}(\mu,a)
  16. T x , 1 # ν Tan ( μ , a ) T_{x,1\#}\nu\in\mathrm{Tan}(\mu,a)
  17. μ a , r r 0 * θ H k P , \mu_{a,r}\xrightarrow[r\to 0]{*}\theta H^{k}\lfloor_{P},
  18. μ a , r ( A ) = 1 r n - 1 μ ( a + r A ) . \mu_{a,r}(A)=\frac{1}{r^{n-1}}\mu(a+rA).

Tangential_and_normal_components.html

  1. S S
  2. x x
  3. 𝐯 \mathbf{v}
  4. x . x.
  5. 𝐯 \mathbf{v}
  6. 𝐯 = 𝐯 + 𝐯 \mathbf{v}=\mathbf{v}_{\parallel}+\mathbf{v}_{\perp}
  7. n ^ \hat{n}
  8. S S
  9. x . x.
  10. 𝐯 = ( 𝐯 n ^ ) n ^ \mathbf{v}_{\perp}=(\mathbf{v}\cdot\hat{n})\hat{n}
  11. 𝐯 = 𝐯 - 𝐯 \mathbf{v}_{\parallel}=\mathbf{v}-\mathbf{v}_{\perp}
  12. \cdot
  13. 𝐯 = - n ^ × ( n ^ × 𝐯 ) , \mathbf{v}_{\parallel}=-\hat{n}\times(\hat{n}\times\mathbf{v}),
  14. × \times
  15. n ^ \hat{n}
  16. p N p\in N
  17. T p N T p M T p M / T p N T_{p}N\to T_{p}M\to T_{p}M/T_{p}N
  18. T p M / T p N T_{p}M/T_{p}N
  19. T p M = T p N N p N := ( T p N ) T_{p}M=T_{p}N\oplus N_{p}N:=(T_{p}N)^{\perp}
  20. v T p M v\in T_{p}M
  21. v = v + v v=v_{\parallel}+v_{\perp}
  22. v T p N v_{\parallel}\in T_{p}N
  23. v N p N := ( T p N ) v_{\perp}\in N_{p}N:=(T_{p}N)^{\perp}
  24. g i g_{i}
  25. g i g_{i}

Tangential_quadrilateral.html

  1. a + c = b + d = a + b + c + d 2 = s . a+c=b+d=\frac{a+b+c+d}{2}=s.
  2. B E + B F = D E + D F \displaystyle BE+BF=DE+DF
  3. A E - E C = A F - F C . \displaystyle AE-EC=AF-FC.
  4. tan A B D 2 tan B D C 2 = tan A D B 2 tan D B C 2 . \tan{\frac{\angle ABD}{2}}\cdot\tan{\frac{\angle BDC}{2}}=\tan{\frac{\angle ADB% }{2}}\cdot\tan{\frac{\angle DBC}{2}}.
  5. R a R c = R b R d R_{a}R_{c}=R_{b}R_{d}
  6. K = r s , \displaystyle K=r\cdot s,
  7. K = 1 2 p 2 q 2 - ( a c - b d ) 2 \displaystyle K=\tfrac{1}{2}\sqrt{p^{2}q^{2}-(ac-bd)^{2}}
  8. K = ( e + f + g + h ) ( e f g + f g h + g h e + h e f ) . \displaystyle K=\sqrt{(e+f+g+h)(efg+fgh+ghe+hef)}.
  9. K = a b c d - ( e g - f h ) 2 . K=\sqrt{abcd-(eg-fh)^{2}}.
  10. a b c d \sqrt{abcd}
  11. K = a b c d sin A + C 2 = a b c d sin B + D 2 . \displaystyle K=\sqrt{abcd}\sin\frac{A+C}{2}=\sqrt{abcd}\sin\frac{B+D}{2}.
  12. K = a b c d K=\sqrt{abcd}
  13. K = ( I A I C + I B I D ) sin A + C 2 K=\left(IA\cdot IC+IB\cdot ID\right)\sin\frac{A+C}{2}
  14. K = a b sin B 2 csc D 2 sin B + D 2 . K=ab\sin{\frac{B}{2}}\csc{\frac{D}{2}}\sin\frac{B+D}{2}.
  15. K = 1 2 | ( a c - b d ) tan θ | , K=\tfrac{1}{2}|(ac-bd)\tan{\theta}|,
  16. K a b c d K\leq\sqrt{abcd}
  17. s 4 r s\geq 4r
  18. K 4 r 2 K\geq 4r^{2}
  19. r = K s = K a + c = K b + d r=\frac{K}{s}=\frac{K}{a+c}=\frac{K}{b+d}
  20. r = e f g + f g h + g h e + h e f e + f + g + h . \displaystyle r=\sqrt{\frac{efg+fgh+ghe+hef}{e+f+g+h}}.
  21. r = 2 ( σ - u v x ) ( σ - v x y ) ( σ - x y u ) ( σ - y u v ) u v x y ( u v + x y ) ( u x + v y ) ( u y + v x ) r=2\sqrt{\frac{(\sigma-uvx)(\sigma-vxy)(\sigma-xyu)(\sigma-yuv)}{uvxy(uv+xy)(% ux+vy)(uy+vx)}}
  22. σ = 1 2 ( u v x + v x y + x y u + y u v ) \sigma=\tfrac{1}{2}(uvx+vxy+xyu+yuv)
  23. sin A 2 = e f g + f g h + g h e + h e f ( e + f ) ( e + g ) ( e + h ) , \sin{\frac{A}{2}}=\sqrt{\frac{efg+fgh+ghe+hef}{(e+f)(e+g)(e+h)}},
  24. sin B 2 = e f g + f g h + g h e + h e f ( f + e ) ( f + g ) ( f + h ) , \sin{\frac{B}{2}}=\sqrt{\frac{efg+fgh+ghe+hef}{(f+e)(f+g)(f+h)}},
  25. sin C 2 = e f g + f g h + g h e + h e f ( g + e ) ( g + f ) ( g + h ) , \sin{\frac{C}{2}}=\sqrt{\frac{efg+fgh+ghe+hef}{(g+e)(g+f)(g+h)}},
  26. sin D 2 = e f g + f g h + g h e + h e f ( h + e ) ( h + f ) ( h + g ) . \sin{\frac{D}{2}}=\sqrt{\frac{efg+fgh+ghe+hef}{(h+e)(h+f)(h+g)}}.
  27. sin φ = ( e + f + g + h ) ( e f g + f g h + g h e + h e f ) ( e + f ) ( f + g ) ( g + h ) ( h + e ) . \sin{\varphi}=\sqrt{\frac{(e+f+g+h)(efg+fgh+ghe+hef)}{(e+f)(f+g)(g+h)(h+e)}}.
  28. p = e + g f + h ( ( e + g ) ( f + h ) + 4 f h ) , \displaystyle p=\sqrt{\frac{e+g}{f+h}\Big((e+g)(f+h)+4fh\Big)},
  29. q = f + h e + g ( ( e + g ) ( f + h ) + 4 e g ) . \displaystyle q=\sqrt{\frac{f+h}{e+g}\Big((e+g)(f+h)+4eg\Big)}.
  30. k = 2 ( e f g + f g h + g h e + h e f ) ( e + f ) ( g + h ) ( e + g ) ( f + h ) , \displaystyle k=\frac{2(efg+fgh+ghe+hef)}{\sqrt{(e+f)(g+h)(e+g)(f+h)}},
  31. l = 2 ( e f g + f g h + g h e + h e f ) ( e + h ) ( f + g ) ( e + g ) ( f + h ) \displaystyle l=\frac{2(efg+fgh+ghe+hef)}{\sqrt{(e+h)(f+g)(e+g)(f+h)}}
  32. k 2 l 2 = b d a c . \frac{k^{2}}{l^{2}}=\frac{bd}{ac}.
  33. B W D Y \tfrac{BW}{DY}
  34. B M D M \tfrac{BM}{DM}
  35. A B C D = I A I B I C I D , B C D A = I B I C I D I A . \frac{AB}{CD}=\frac{IA\cdot IB}{IC\cdot ID},\quad\quad\frac{BC}{DA}=\frac{IB% \cdot IC}{ID\cdot IA}.
  36. A B B C = I B 2 + I A I B I C I D . AB\cdot BC=IB^{2}+\frac{IA\cdot IB\cdot IC}{ID}.
  37. I A I C + I B I D = A B B C C D D A . IA\cdot IC+IB\cdot ID=\sqrt{AB\cdot BC\cdot CD\cdot DA}.
  38. I A I C = I B I D . IA\cdot IC=IB\cdot ID.
  39. I M p I M q = I A I C I B I D = e + g f + h \frac{IM_{p}}{IM_{q}}=\frac{IA\cdot IC}{IB\cdot ID}=\frac{e+g}{f+h}
  40. a b c d / s \sqrt{abcd}/s
  41. 1 r 1 + 1 r 3 = 1 r 2 + 1 r 4 . \frac{1}{r_{1}}+\frac{1}{r_{3}}=\frac{1}{r_{2}}+\frac{1}{r_{4}}.
  42. 1 h 1 + 1 h 3 = 1 h 2 + 1 h 4 . \frac{1}{h_{1}}+\frac{1}{h_{3}}=\frac{1}{h_{2}}+\frac{1}{h_{4}}.
  43. 1 r a + 1 r c = 1 r b + 1 r d . \frac{1}{r_{a}}+\frac{1}{r_{c}}=\frac{1}{r_{b}}+\frac{1}{r_{d}}.
  44. R 1 + R 3 = R 2 + R 4 . R_{1}+R_{3}=R_{2}+R_{4}.
  45. 1 R a + 1 R c = 1 R b + 1 R d . \frac{1}{R_{a}}+\frac{1}{R_{c}}=\frac{1}{R_{b}}+\frac{1}{R_{d}}.
  46. a ( A P B ) + c ( C P D ) = b ( B P C ) + d ( D P A ) \frac{a}{\triangle(APB)}+\frac{c}{\triangle(CPD)}=\frac{b}{\triangle(BPC)}+% \frac{d}{\triangle(DPA)}
  47. a p 2 q 2 + c p 1 q 1 = b p 1 q 2 + d p 2 q 1 ap_{2}q_{2}+cp_{1}q_{1}=bp_{1}q_{2}+dp_{2}q_{1}
  48. ( p 1 + q 1 - a ) ( p 2 + q 2 - c ) ( p 1 + q 1 + a ) ( p 2 + q 2 + c ) = ( p 2 + q 1 - b ) ( p 1 + q 2 - d ) ( p 2 + q 1 + b ) ( p 1 + q 2 + d ) \frac{(p_{1}+q_{1}-a)(p_{2}+q_{2}-c)}{(p_{1}+q_{1}+a)(p_{2}+q_{2}+c)}=\frac{(p% _{2}+q_{1}-b)(p_{1}+q_{2}-d)}{(p_{2}+q_{1}+b)(p_{1}+q_{2}+d)}
  49. ( a + p 1 - q 1 ) ( c + p 2 - q 2 ) ( a - p 1 + q 1 ) ( c - p 2 + q 2 ) = ( b + p 2 - q 1 ) ( d + p 1 - q 2 ) ( b - p 2 + q 1 ) ( d - p 1 + q 2 ) . \frac{(a+p_{1}-q_{1})(c+p_{2}-q_{2})}{(a-p_{1}+q_{1})(c-p_{2}+q_{2})}=\frac{(b% +p_{2}-q_{1})(d+p_{1}-q_{2})}{(b-p_{2}+q_{1})(d-p_{1}+q_{2})}.
  50. A W W B = D Y Y C \frac{AW}{WB}=\frac{DY}{YC}
  51. A C B D = A W + C Y B X + D Z \frac{AC}{BD}=\frac{AW+CY}{BX+DZ}

Tango_tree.html

  1. O ( log log n ) O(\log\log n)
  2. O ( log log n ) O(\log\log n)
  3. O ( log n ) O(\log n)
  4. ( k + 1 ) O ( log log n ) (k+1)O(\log\log n)
  5. k + 1 k+1
  6. O ( log log n ) O(\log\log n)
  7. log n \log n
  8. ( k + 1 ) O ( log log n ) (k+1)O(\log\log n)
  9. O ( log log n ) O(\log\log n)
  10. ( k + 1 ) O ( log log n ) (k+1)O(\log\log n)

Taylor_dispersion.html

  1. s y m b o l u = w s y m b o l z ^ = w 0 ( 1 - r 2 / a 2 ) s y m b o l z ^ symbol{u}=w\hat{symbol{z}}=w_{0}(1-r^{2}/a^{2})\hat{symbol{z}}
  2. c t + s y m b o l w s y m b o l c = D 2 c \frac{\partial c}{\partial t}+symbol{w}\cdot symbol{\nabla}c=D\nabla^{2}c
  3. w ( r ) = w ¯ + w ( r ) w(r)=\bar{w}+w^{\prime}(r)
  4. c ( r , z ) = c ¯ ( z ) + c ( r , z ) c(r,z)=\bar{c}(z)+c^{\prime}(r,z)
  5. c ¯ t + w ¯ c ¯ z = D ( 1 + a 2 w ¯ 2 48 D 2 ) 2 c ¯ z 2 \frac{\partial\bar{c}}{\partial t}+\bar{w}\frac{\partial\bar{c}}{\partial z}=D% \left(1+\frac{a^{2}\bar{w}^{2}}{48D^{2}}\right)\frac{\partial^{2}\bar{c}}{% \partial z^{2}}
  6. D eff = D ( 1 + 1 192 𝑃𝑒 d 2 ) , D_{\mathrm{eff}}=D\left(1+\frac{1}{192}\mathit{Pe}_{d}^{2}\right)\,,
  7. 𝑃𝑒 d = d w ¯ / D \mathit{Pe}_{d}=d\bar{w}/D
  8. d = 2 a d=2a
  9. c c ¯ c^{\prime}\ll\bar{c}
  10. z z
  11. z z
  12. r r
  13. L L
  14. z z
  15. L a 2 D w ¯ = 𝑃𝑒 d d 4 L\gg\frac{a^{2}}{D}\bar{w}=\frac{\mathit{Pe}_{d}\,d}{4}

TCP-Illinois.html

  1. α / W \alpha/W
  2. W W
  3. β W \beta W
  4. α \alpha
  5. β \beta
  6. d a d_{a}
  7. α = f 1 ( d a ) , β = f 2 ( d a ) \alpha=f_{1}(d_{a}),\beta=f_{2}(d_{a})
  8. f 1 ( ) f_{1}(\cdot)
  9. f 2 ( ) f_{2}(\cdot)
  10. f 1 ( ) f_{1}(\cdot)
  11. f 2 ( ) f_{2}(\cdot)
  12. α = f 1 ( d a ) = { α m a x if d a d 1 κ 1 κ 2 + d a otherwise. \alpha=f_{1}(d_{a})=\left\{\begin{array}[]{ll}\alpha_{max}&\mbox{if }~{}d_{a}% \leq d_{1}\\ \frac{\kappa_{1}}{\kappa_{2}+d_{a}}&\mbox{otherwise.}\end{array}\right.
  13. β = f 2 ( d a ) = { β m i n if d a d 2 κ 3 + κ 4 d a if d 2 < d a < d 3 β m a x otherwise. \beta=f_{2}(d_{a})=\left\{\begin{array}[]{ll}\beta_{min}&\mbox{if }~{}d_{a}% \leq d_{2}\\ \kappa_{3}+\kappa_{4}d_{a}&\mbox{if }~{}d_{2}<d_{a}<d_{3}\\ \beta_{max}&\mbox{otherwise.}\end{array}\right.
  14. f 1 ( ) f_{1}(\cdot)
  15. f 2 ( ) f_{2}(\cdot)
  16. κ 1 κ 2 + d 1 = α m a x \frac{\kappa_{1}}{\kappa_{2}+d_{1}}=\alpha_{max}
  17. β m i n = κ 3 + κ 4 d 2 \beta_{min}=\kappa_{3}+\kappa_{4}d_{2}
  18. β m a x = κ 3 + κ 4 d 3 \beta_{max}=\kappa_{3}+\kappa_{4}d_{3}
  19. d m d_{m}
  20. α m i n = f 1 ( d m ) \alpha_{min}=f_{1}(d_{m})
  21. κ 1 κ 2 + d m = α m i n \frac{\kappa_{1}}{\kappa_{2}+d_{m}}=\alpha_{min}
  22. κ 1 = ( d m - d 1 ) α m i n α m a x α m a x - α m i n and κ 2 = ( d m - d 1 ) α m i n α m a x - α m i n - d 1 , κ 3 = β m i n d 3 - β m a x d 2 d 3 - d 2 and κ 4 = β m a x - β m i n d 3 - d 2 . \begin{array}[]{lcl}\kappa_{1}=\frac{(d_{m}-d_{1})\alpha_{min}\alpha_{max}}{% \alpha_{max}-\alpha_{min}}&\mbox{and}&\kappa_{2}=\frac{(d_{m}-d_{1})\alpha_{% min}}{\alpha_{max}-\alpha_{min}}-d_{1}\,,\\ \kappa_{3}=\frac{\beta_{min}d_{3}-\beta_{max}d_{2}}{d_{3}-d_{2}}&\mbox{and}&% \kappa_{4}=\frac{\beta_{max}-\beta_{min}}{d_{3}-d_{2}}\,.\end{array}

Technetium-99m.html

  1. Tc 43 99 m γ 141 keV 6 h Tc 43 99 β - 249 keV 211 000 y Ru 44 99 \mathrm{{}^{99m}_{\ \ 43}Tc\ \xrightarrow[6\ h]{\gamma\ 141\ keV}\ {}^{99}_{43% }Tc\ \xrightarrow[211\ 000\ y]{\beta^{-}\ 249\ keV}\ {}^{99}_{44}Ru}

Telegraph_process.html

  1. t P ( a , t | x , t 0 ) = - λ P ( a , t | x , t 0 ) + μ P ( b , t | x , t 0 ) \partial_{t}P(a,t|x,t_{0})=-\lambda P(a,t|x,t_{0})+\mu P(b,t|x,t_{0})
  2. t P ( b , t | x , t 0 ) = λ P ( a , t | x , t 0 ) - μ P ( b , t | x , t 0 ) . \partial_{t}P(b,t|x,t_{0})=\lambda P(a,t|x,t_{0})-\mu P(b,t|x,t_{0}).
  3. X s = a μ + b λ μ + λ . \langle X\rangle_{s}=\frac{a\mu+b\lambda}{\mu+\lambda}.
  4. var { X } s = ( a - b ) 2 μ λ ( μ + λ ) 2 . \operatorname{var}\{X\}_{s}=\frac{(a-b)^{2}\mu\lambda}{(\mu+\lambda)^{2}}.
  5. X ( t ) , X ( s ) s = exp ( - ( λ + μ ) | t - s | ) var { X } s . \langle X(t),X(s)\rangle_{s}=\exp(-(\lambda+\mu)|t-s|)\operatorname{var}\{X\}_% {s}.

Tempered_representation.html

  1. ( 1 + σ ) r g / Ξ (1+\sigma)^{r}g/\Xi

Template:Elastic_moduli.html

  1. K = K=\,
  2. E = E=\,
  3. λ = \lambda=\,
  4. G = G=\,
  5. ν = \nu=\,
  6. M = M=\,
  7. ( K , E ) (K,\,E)
  8. K K
  9. E E
  10. 3 K ( 3 K - E ) 9 K - E \tfrac{3K(3K-E)}{9K-E}
  11. 3 K E 9 K - E \tfrac{3KE}{9K-E}
  12. 3 K - E 6 K \tfrac{3K-E}{6K}
  13. 3 K ( 3 K + E ) 9 K - E \tfrac{3K(3K+E)}{9K-E}
  14. ( K , λ ) (K,\,\lambda)
  15. K K
  16. 9 K ( K - λ ) 3 K - λ \tfrac{9K(K-\lambda)}{3K-\lambda}
  17. λ \lambda
  18. 3 ( K - λ ) 2 \tfrac{3(K-\lambda)}{2}
  19. λ 3 K - λ \tfrac{\lambda}{3K-\lambda}
  20. 3 K - 2 λ 3K-2\lambda\,
  21. ( K , G ) (K,\,G)
  22. K K
  23. 9 K G 3 K + G \tfrac{9KG}{3K+G}
  24. K - 2 G 3 K-\tfrac{2G}{3}
  25. G G
  26. 3 K - 2 G 2 ( 3 K + G ) \tfrac{3K-2G}{2(3K+G)}
  27. K + 4 G 3 K+\tfrac{4G}{3}
  28. ( K , ν ) (K,\,\nu)
  29. K K
  30. 3 K ( 1 - 2 ν ) 3K(1-2\nu)\,
  31. 3 K ν 1 + ν \tfrac{3K\nu}{1+\nu}
  32. 3 K ( 1 - 2 ν ) 2 ( 1 + ν ) \tfrac{3K(1-2\nu)}{2(1+\nu)}
  33. ν \nu
  34. 3 K ( 1 - ν ) 1 + ν \tfrac{3K(1-\nu)}{1+\nu}
  35. ( K , M ) (K,\,M)
  36. K K
  37. 9 K ( M - K ) 3 K + M \tfrac{9K(M-K)}{3K+M}
  38. 3 K - M 2 \tfrac{3K-M}{2}
  39. 3 ( M - K ) 4 \tfrac{3(M-K)}{4}
  40. 3 K - M 3 K + M \tfrac{3K-M}{3K+M}
  41. M M
  42. ( E , λ ) (E,\,\lambda)
  43. E + 3 λ + R 6 \tfrac{E+3\lambda+R}{6}
  44. E E
  45. λ \lambda
  46. E - 3 λ + R 4 \tfrac{E-3\lambda+R}{4}
  47. 2 λ E + λ + R \tfrac{2\lambda}{E+\lambda+R}
  48. E - λ + R 2 \tfrac{E-\lambda+R}{2}
  49. R = E 2 + 9 λ 2 + 2 E λ R=\sqrt{E^{2}+9\lambda^{2}+2E\lambda}
  50. ( E , G ) (E,\,G)
  51. E G 3 ( 3 G - E ) \tfrac{EG}{3(3G-E)}
  52. E E
  53. G ( E - 2 G ) 3 G - E \tfrac{G(E-2G)}{3G-E}
  54. G G
  55. E 2 G - 1 \tfrac{E}{2G}-1
  56. G ( 4 G - E ) 3 G - E \tfrac{G(4G-E)}{3G-E}
  57. ( E , ν ) (E,\,\nu)
  58. E 3 ( 1 - 2 ν ) \tfrac{E}{3(1-2\nu)}
  59. E E
  60. E ν ( 1 + ν ) ( 1 - 2 ν ) \tfrac{E\nu}{(1+\nu)(1-2\nu)}
  61. E 2 ( 1 + ν ) \tfrac{E}{2(1+\nu)}
  62. ν \nu
  63. E ( 1 - ν ) ( 1 + ν ) ( 1 - 2 ν ) \tfrac{E(1-\nu)}{(1+\nu)(1-2\nu)}
  64. ( E , M ) (E,\,M)
  65. 3 M - E + S 6 \tfrac{3M-E+S}{6}
  66. E E
  67. M - E + S 4 \tfrac{M-E+S}{4}
  68. 3 M + E - S 8 \tfrac{3M+E-S}{8}
  69. E - M + S 4 M \tfrac{E-M+S}{4M}
  70. M M
  71. S = ± E 2 + 9 M 2 - 10 E M S=\pm\sqrt{E^{2}+9M^{2}-10EM}
  72. ν 0 \nu\geq 0
  73. ν 0 \nu\leq 0
  74. ( λ , G ) (\lambda,\,G)
  75. λ + 2 G 3 \lambda+\tfrac{2G}{3}
  76. G ( 3 λ + 2 G ) λ + G \tfrac{G(3\lambda+2G)}{\lambda+G}
  77. λ \lambda
  78. G G
  79. λ 2 ( λ + G ) \tfrac{\lambda}{2(\lambda+G)}
  80. λ + 2 G \lambda+2G\,
  81. ( λ , ν ) (\lambda,\,\nu)
  82. λ ( 1 + ν ) 3 ν \tfrac{\lambda(1+\nu)}{3\nu}
  83. λ ( 1 + ν ) ( 1 - 2 ν ) ν \tfrac{\lambda(1+\nu)(1-2\nu)}{\nu}
  84. λ \lambda
  85. λ ( 1 - 2 ν ) 2 ν \tfrac{\lambda(1-2\nu)}{2\nu}
  86. ν \nu
  87. λ ( 1 - ν ) ν \tfrac{\lambda(1-\nu)}{\nu}
  88. ν = 0 λ = 0 \nu=0\Leftrightarrow\lambda=0
  89. ( λ , M ) (\lambda,\,M)
  90. M + 2 λ 3 \tfrac{M+2\lambda}{3}
  91. ( M - λ ) ( M + 2 λ ) M + λ \tfrac{(M-\lambda)(M+2\lambda)}{M+\lambda}
  92. λ \lambda
  93. M - λ 2 \tfrac{M-\lambda}{2}
  94. λ M + λ \tfrac{\lambda}{M+\lambda}
  95. M M
  96. ( G , ν ) (G,\,\nu)
  97. 2 G ( 1 + ν ) 3 ( 1 - 2 ν ) \tfrac{2G(1+\nu)}{3(1-2\nu)}
  98. 2 G ( 1 + ν ) 2G(1+\nu)\,
  99. 2 G ν 1 - 2 ν \tfrac{2G\nu}{1-2\nu}
  100. G G
  101. ν \nu
  102. 2 G ( 1 - ν ) 1 - 2 ν \tfrac{2G(1-\nu)}{1-2\nu}
  103. ( G , M ) (G,\,M)
  104. M - 4 G 3 M-\tfrac{4G}{3}
  105. G ( 3 M - 4 G ) M - G \tfrac{G(3M-4G)}{M-G}
  106. M - 2 G M-2G\,
  107. G G
  108. M - 2 G 2 M - 2 G \tfrac{M-2G}{2M-2G}
  109. M M
  110. ( ν , M ) (\nu,\,M)
  111. M ( 1 + ν ) 3 ( 1 - ν ) \tfrac{M(1+\nu)}{3(1-\nu)}
  112. M ( 1 + ν ) ( 1 - 2 ν ) 1 - ν \tfrac{M(1+\nu)(1-2\nu)}{1-\nu}
  113. M ν 1 - ν \tfrac{M\nu}{1-\nu}
  114. M ( 1 - 2 ν ) 2 ( 1 - ν ) \tfrac{M(1-2\nu)}{2(1-\nu)}
  115. ν \nu
  116. M M

Template:Infobox_planet::doc.html

  1. v m a x = 2 π a 2 1 - e 2 T a ( 1 - e ) v_{max}=2\pi a^{2}\frac{\sqrt{1-e^{2}}}{Ta(1-e)}
  2. v m i n = 2 π a 2 1 - e 2 T a ( 1 + e ) v_{min}=2\pi a^{2}\frac{\sqrt{1-e^{2}}}{Ta(1+e)}
  3. 4 a E ( e ) 4aE(e)
  4. v o = 4 a E ( e ) T v_{o}=\frac{4aE(e)}{T}
  5. π / 2 \pi/2
  6. v o = 2 π a T [ 1 - e 2 4 - 3 e 4 64 - ] v_{o}=\frac{2\pi a}{T}\left[1-\frac{e^{2}}{4}-\frac{3e^{4}}{64}-\dots\right]
  7. v o π T [ 3 ( a + b ) - ( 3 a + b ) ( a + 3 b ) ] v_{o}\approx\frac{\pi}{T}\left[3(a+b)-\sqrt{(3a+b)(a+3b)}\right]
  8. b = a 1 - e 2 b=a\sqrt{1-e^{2}}\,\!
  9. g spherical = G M r 2 g_{\rm spherical}=\frac{GM}{r^{2}}\,\!
  10. × 10 11 \times 10^{−}11
  11. r max r_{\rm max}
  12. g outer = G M r max 2 . g_{\rm outer}=\frac{GM}{r_{\rm max}^{2}}\,.\!
  13. g centrifugal = - ( 2 π T ) 2 r eq g_{\rm centrifugal}=-\left(\frac{2\pi}{T}\right)^{2}r_{\rm eq}
  14. r eq r_{\rm eq}
  15. g effective g gravitational + g centrifugal = g gravitational - | g centrifugal | . g_{\rm effective}\approx g_{\rm gravitational}+g_{\rm centrifugal}=g_{\rm gravitational% }-|g_{\rm centrifugal}|\ .
  16. v e = 2 g r v_{e}=\sqrt{2gr}
  17. L 0 L_{0}
  18. T = ( ( 1 - α ) L 0 ϵ σ 16 π a 2 ) 1 4 T=\left(\frac{(1-\alpha)L_{0}}{\epsilon\sigma 16\pi a^{2}}\right)^{\frac{1}{4}}

Tensor-hom_adjunction.html

  1. - X -\otimes X
  2. Hom ( X , - ) \operatorname{Hom}(X,-)
  3. Hom ( Y X , Z ) Hom ( Y , Hom ( X , Z ) ) . \operatorname{Hom}(Y\otimes X,Z)\cong\operatorname{Hom}(Y,\operatorname{Hom}(X% ,Z)).
  4. 𝒞 = Mod R and 𝒟 = Mod S . \mathcal{C}=\mathrm{Mod}_{R}\quad\,\text{and}\quad\mathcal{D}=\mathrm{Mod}_{S}.
  5. F ( Y ) = Y R X for Y 𝒞 F(Y)=Y\otimes_{R}X\quad\,\text{for }Y\in\mathcal{C}
  6. G ( Z ) = Hom S ( X , Z ) for Z 𝒟 G(Z)=\operatorname{Hom}_{S}(X,Z)\quad\,\text{for }Z\in\mathcal{D}
  7. Hom S ( Y R X , Z ) Hom R ( Y , Hom S ( X , Z ) ) . \operatorname{Hom}_{S}(Y\otimes_{R}X,Z)\cong\operatorname{Hom}_{R}(Y,% \operatorname{Hom}_{S}(X,Z)).
  8. ε : F G 1 𝒞 \varepsilon:FG\to 1_{\mathcal{C}}
  9. ε Z : Hom S ( X , Z ) R X Z \varepsilon_{Z}:\operatorname{Hom}_{S}(X,Z)\otimes_{R}X\to Z
  10. ϕ Hom R ( X , Z ) and x X , \phi\in\operatorname{Hom}_{R}(X,Z)\quad\,\text{and}\quad x\in X,
  11. ε ( ϕ x ) = ϕ ( x ) . \varepsilon(\phi\otimes x)=\phi(x).
  12. η : 1 𝒟 G F \eta:1_{\mathcal{D}}\to GF
  13. η Y : Y Hom S ( X , Y R X ) \eta_{Y}:Y\to\operatorname{Hom}_{S}(X,Y\otimes_{R}X)
  14. η Y ( y ) Hom S ( X , Y R X ) \eta_{Y}(y)\in\operatorname{Hom}_{S}(X,Y\otimes_{R}X)
  15. η Y ( y ) ( t ) = y t for t X . \eta_{Y}(y)(t)=y\otimes t\quad\,\text{for }t\in X.
  16. ε F Y F ( η Y ) : Y R X Hom S ( X , Y ) R X Y R X \varepsilon_{FY}\circ F(\eta_{Y}):Y\otimes_{R}X\to\operatorname{Hom}_{S}(X,Y)% \otimes_{R}X\to Y\otimes_{R}X
  17. ε F Y F ( η Y ) ( y x ) = η Y ( y ) ( x ) = y x . \varepsilon_{FY}\circ F(\eta_{Y})(y\otimes x)=\eta_{Y}(y)(x)=y\otimes x.
  18. G ( ε Z ) η G Z : Hom S ( X , Z ) Hom S ( X , Hom S ( X , Z ) R X ) Hom S ( X , Z ) . G(\varepsilon_{Z})\circ\eta_{GZ}:\operatorname{Hom}_{S}(X,Z)\to\operatorname{% Hom}_{S}(X,\operatorname{Hom}_{S}(X,Z)\otimes_{R}X)\to\operatorname{Hom}_{S}(X% ,Z).
  19. G ( ε Z ) η G Z ( ϕ ) G(\varepsilon_{Z})\circ\eta_{GZ}(\phi)
  20. G ( ε Z ) η G Z ( ϕ ) ( x ) = ε Z ( ϕ x ) = ϕ ( x ) G(\varepsilon_{Z})\circ\eta_{GZ}(\phi)(x)=\varepsilon_{Z}(\phi\otimes x)=\phi(x)
  21. G ( ε Z ) η G Z ( ϕ ) = ϕ . G(\varepsilon_{Z})\circ\eta_{GZ}(\phi)=\phi.

Tensor_product_of_quadratic_forms.html

  1. v w V W v\otimes w\in V\otimes W
  2. q ( v w ) = q 1 ( v ) q 2 ( w ) q(v\otimes w)=q_{1}(v)q_{2}(w)
  3. q 1 a 1 , , a n q_{1}\cong\langle a_{1},...,a_{n}\rangle
  4. q 2 b 1 , , b m q_{2}\cong\langle b_{1},...,b_{m}\rangle
  5. q 1 q 2 = q a 1 b 1 , a 1 b 2 , a 1 b m , a 2 b 1 , , a 2 b m , , a n b 1 , a n b m . q_{1}\otimes q_{2}=q\cong\langle a_{1}b_{1},a_{1}b_{2},...a_{1}b_{m},a_{2}b_{1% },...,a_{2}b_{m},...,a_{n}b_{1},...a_{n}b_{m}\rangle.

Term_test.html

  1. lim n a n 0 \lim_{n\to\infty}a_{n}\neq 0
  2. n = 1 a n \sum_{n=1}^{\infty}a_{n}
  3. lim n a n = 0 , \lim_{n\to\infty}a_{n}=0,
  4. n = 1 a n \sum_{n=1}^{\infty}a_{n}
  5. lim n a n = 0 , \lim_{n\to\infty}a_{n}=0,
  6. n = 1 1 n p , \sum_{n=1}^{\infty}\frac{1}{n^{p}},
  7. n = 1 a n \sum_{n=1}^{\infty}a_{n}
  8. lim n a n = 0. \lim_{n\to\infty}a_{n}=0.
  9. lim n s n = s \lim_{n\to\infty}s_{n}=s
  10. lim n a n = lim n ( s n - s n - 1 ) = lim n s n - lim n s n - 1 = s - s = 0. \lim_{n\to\infty}a_{n}=\lim_{n\to\infty}(s_{n}-s_{n-1})=\lim_{n\to\infty}s_{n}% -\lim_{n\to\infty}s_{n-1}=s-s=0.
  11. ε > 0 \varepsilon>0
  12. | a n + 1 + a n + 2 + + a n + p | < ε |a_{n+1}+a_{n+2}+\ldots+a_{n+p}|<\varepsilon
  13. lim n a n = 0. \lim_{n\to\infty}a_{n}=0.

Tesseractic_honeycomb.html

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

Tetracontagon.html

  1. A = 10 t 2 cot π 40 A=10t^{2}\cot\frac{\pi}{40}
  2. r = 1 2 t cot π 40 r=\frac{1}{2}t\cot\frac{\pi}{40}
  3. cot π 40 \cot\frac{\pi}{40}
  4. x 8 - 8 x 7 - 60 x 6 - 8 x 5 + 134 x 4 + 8 x 3 - 60 x 2 + 8 x + 1 x^{8}-8x^{7}-60x^{6}-8x^{5}+134x^{4}+8x^{3}-60x^{2}+8x+1
  5. R = 1 2 t csc π 40 R=\frac{1}{2}t\csc\frac{\pi}{40}
  6. sin π 40 \sin\frac{\pi}{40}
  7. cos π 40 \cos\frac{\pi}{40}
  8. sin π 40 = 1 4 ( 2 - 1 ) 1 2 ( 2 + 2 ) ( 5 + 5 ) - 1 8 2 - 2 ( 1 + 2 ) ( 5 - 1 ) \sin\frac{\pi}{40}=\frac{1}{4}(\sqrt{2}-1)\sqrt{\frac{1}{2}(2+\sqrt{2})(5+% \sqrt{5})}-\frac{1}{8}\sqrt{2-\sqrt{2}}(1+\sqrt{2})(\sqrt{5}-1)
  9. cos π 40 = 1 8 ( 2 - 1 ) 2 + 2 ( 5 - 1 ) + 1 4 ( 1 + 2 ) 1 2 ( 2 - 2 ) ( 5 + 5 ) \cos\frac{\pi}{40}=\frac{1}{8}(\sqrt{2}-1)\sqrt{2+\sqrt{2}}(\sqrt{5}-1)+\frac{% 1}{4}(1+\sqrt{2})\sqrt{\frac{1}{2}(2-\sqrt{2})(5+\sqrt{5})}

Tetrad_formalism.html

  1. { e ( a ) = e ( a ) μ μ } a = 1 4 \scriptstyle\{e_{(a)}=e_{(a)}^{\mu}\partial_{\mu}\}_{a=1\dots 4}
  2. { e ( a ) = e μ ( a ) d x μ } a = 1 4 \scriptstyle\{e^{(a)}=e^{(a)}_{\mu}dx^{\mu}\}_{a=1\dots 4}
  3. e ( a ) ( e ( b ) ) = e μ ( a ) e ( b ) μ = δ ( b ) ( a ) , e^{(a)}(e_{(b)})=e^{(a)}_{\mu}e^{\mu}_{(b)}=\delta^{(a)}_{(b)},
  4. δ ( b ) ( a ) \delta^{(a)}_{(b)}
  5. e ( a ) μ e_{(a)}^{\mu}
  6. x μ x^{\mu}
  7. { e ( a ) } a = 1 4 \scriptstyle\{e_{(a)}\}_{a=1\dots 4}
  8. M \scriptstyle M
  9. T M M × 4 \scriptstyle TM\cong M\times{\mathbb{R}^{4}}
  10. { μ } \scriptstyle\{\partial_{\mu}\}
  11. { d x μ } \{dx^{\mu}\}
  12. φ = ( φ 1 , , φ n ) \scriptstyle\varphi=(\varphi^{1},\ldots,\varphi^{n})
  13. n \scriptstyle\mathbb{R}^{n}
  14. f \scriptstyle f
  15. μ [ f ] f φ - 1 x μ . \partial_{\mu}[f]\equiv\frac{\partial f\circ\varphi^{-1}}{\partial x^{\mu}}.
  16. d x μ = d φ μ dx^{\mu}=d\varphi^{\mu}
  17. M M
  18. \scriptstyle\otimes
  19. g a b \scriptstyle g_{ab}
  20. g g
  21. g = g μ ν d x μ d x ν where g μ ν = g ( μ , ν ) g=g_{\mu\nu}dx^{\mu}dx^{\nu}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\,\text{where}~{}% g_{\mu\nu}=g(\partial_{\mu},\partial_{\nu})
  22. g = g a b e ( a ) e ( b ) where g a b = g ( e ( a ) , e ( b ) ) g=g_{ab}e^{(a)}e^{(b)}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\,\text{where}~{}g_{ab}% =g(e_{(a)},e_{(b)})
  23. e ( a ) = e μ ( a ) d x μ e^{(a)}=e^{(a)}_{\mu}dx^{\mu}
  24. g = g a b e ( a ) e ( b ) = g a b e μ ( a ) e ν ( b ) d x μ d x ν = g μ ν d x μ d x ν g=g_{ab}e^{(a)}e^{(b)}=g_{ab}e^{(a)}_{\mu}e^{(b)}_{\nu}dx^{\mu}dx^{\nu}=g_{\mu% \nu}dx^{\mu}dx^{\nu}
  25. g μ ν = g a b e μ ( a ) e ν ( b ) g_{\mu\nu}=g_{ab}e^{(a)}_{\mu}e^{(b)}_{\nu}
  26. d x μ = e ( a ) μ e ( a ) dx^{\mu}=e^{\mu}_{(a)}e^{(a)}
  27. g = g μ ν d x μ d x ν = g μ ν e ( a ) μ e ( b ) ν e ( a ) e ( b ) = g a b e ( a ) e ( b ) g=g_{\mu\nu}dx^{\mu}dx^{\nu}=g_{\mu\nu}e_{(a)}^{\mu}e_{(b)}^{\nu}e^{(a)}e^{(b)% }=g_{ab}e^{(a)}e^{(b)}
  28. g a b = g μ ν e ( a ) μ e ( b ) ν g_{ab}=g_{\mu\nu}e_{(a)}^{\mu}e_{(b)}^{\nu}
  29. μ ν = ν μ \partial_{\mu}\partial_{\nu}=\partial_{\nu}\partial_{\mu}
  30. [ e a , e b ] = e a e b - e b e a 0 [e_{a},e_{b}]=e_{a}e_{b}-e_{b}e_{a}\neq 0
  31. X , Y X,Y
  32. R ( X , Y ) = ( X Y - Y X - [ X , Y ] ) R(X,Y)=(\nabla_{X}\nabla_{Y}-\nabla_{Y}\nabla_{X}-\nabla_{[X,Y]})
  33. R ν σ τ μ = d x μ ( ( σ τ - τ σ ) ν ) . R^{\mu}_{\ \nu\sigma\tau}=dx^{\mu}((\nabla_{\sigma}\nabla_{\tau}-\nabla_{\tau}% \nabla_{\sigma})\partial_{\nu}).
  34. R b c d a = e a ( ( c d - d c ) e b ) (wrong!) R^{a}_{\ bcd}=e^{a}((\nabla_{c}\nabla_{d}-\nabla_{d}\nabla_{c})e_{b})~{}~{}~{}% ~{}~{}~{}~{}~{}~{}\,\text{(wrong!)}
  35. ( c d - d c ) (\nabla_{c}\nabla_{d}-\nabla_{d}\nabla_{c})
  36. R b c d a = e a ( ( c d - d c - f c d e e ) e b ) R^{a}_{\ bcd}=e^{a}((\nabla_{c}\nabla_{d}-\nabla_{d}\nabla_{c}-f^{e}_{cd}% \nabla_{e})e_{b})
  37. [ e a , e b ] = f a b c e c [e_{a},e_{b}]=f^{c}_{ab}e_{c}
  38. ( c d - d c - f c d e e ) (\nabla_{c}\nabla_{d}-\nabla_{d}\nabla_{c}-f^{e}_{cd}\nabla_{e})

The_Dark_Energy_Survey.html

  1. m - M = 5 log 10 d L + 25 m-M=5\log_{10}d_{L}+25
  2. d L d_{L}
  3. d L d_{L}
  4. d L = c H 0 ( 1 + z ) 0 z d z Ω M ( 1 + z ) 3 + Ω D E ( 1 + z ) 3 ( 1 + w ) + Ω k ( 1 + z ) 2 d_{L}=\frac{c}{H_{0}}(1+z)\int_{0}^{z}\frac{dz}{\sqrt{\Omega_{M}(1+z)^{3}+% \Omega_{DE}(1+z)^{3(1+w)}+\Omega_{k}(1+z)^{2}}}
  5. d L d_{L}
  6. H 0 H_{0}
  7. Ω M \Omega_{M}
  8. Ω D E \Omega_{DE}
  9. Ω k \Omega_{k}
  10. w w
  11. P ( k ) P(k)
  12. H ( z ) H(z)
  13. d A ( z ) d_{A}(z)
  14. H ( z ) H(z)
  15. d A d_{A}
  16. d A = Δ χ Δ θ d_{A}=\frac{\Delta\chi}{\Delta\theta}
  17. Δ χ \Delta\chi
  18. Δ θ \Delta\theta
  19. d A d_{A}
  20. H - 1 ( z ) H^{-1}(z)
  21. d A = 1 1 + z 0 z d z H ( z ) d_{A}=\frac{1}{1+z}\int_{0}^{z}\frac{dz}{H(z)}
  22. Δ χ \Delta\chi
  23. Ω M \Omega_{M}
  24. Ω B \Omega_{B}
  25. H - 1 ( z ) H^{-1}(z)
  26. Δ z \Delta z
  27. Δ z = H ( z ) Δ χ \Delta z=H(z)\Delta\chi
  28. Δ z \Delta z
  29. d A ( z ) d_{A}(z)
  30. 1 / H ( z ) 1/H(z)
  31. w 0 - w a w_{0}-w_{a}
  32. Δ z \Delta z
  33. a p a_{p}
  34. w ( a ) w{(a)}
  35. w 0 - w a w_{0}-w_{a}
  36. [ σ ( w p ) σ ( w a ) ] - 1 {[\sigma{(w_{p})}\sigma{(w_{a})}]}^{-1}
  37. Δ σ z \Delta\sigma_{z}
  38. Δ z bias \Delta z_{\rm{bias}}
  39. z bias z_{\rm{bias}}
  40. w 0 w_{0}
  41. w a w_{a}
  42. M M
  43. n ( > M , z ) n(>M,z)
  44. d 2 V d^{2}V
  45. d A dA
  46. z z
  47. r r
  48. d Ω d\Omega
  49. d l dl
  50. ( z , z + d z ) (z,z+dz)
  51. d 2 V = d A d l d^{2}V=dA\ dl
  52. d A dA
  53. d A = a ( t e ) r d θ a ( t e ) r sin ( θ ) d ϕ = a e 2 r 2 d Ω = a 0 2 r 2 d Ω ( 1 + z ) 2 dA=a(t_{e})r\ d\theta a(t_{e})r\sin(\theta)d\phi=a_{e}^{2}r^{2}d\Omega=\frac{a% _{0}^{2}r^{2}d\Omega}{(1+z)^{2}}
  54. a ( t ) a(t)
  55. d l dl
  56. ( z , z + d z ) (z,z+dz)
  57. d t dt
  58. d l = c d t dl=c\ dt
  59. z z
  60. d l = d a a ˙ = d z ( 1 + z ) a a ˙ = d z H ( z ) ( 1 + z ) dl=\frac{da}{\dot{a}}=\frac{dz}{(1+z)}\frac{a}{\dot{a}}=\frac{dz}{H(z)(1+z)}
  61. H ( z ) H(z)
  62. H ( z ) = H 0 i Ω i ( 1 + z ) 3 ( 1 - w i ) + Ω k ( 1 + z ) 2 H(z)=H_{0}\sqrt{\sum_{i}\Omega_{i}(1+z)^{3(1-w_{i})}+\Omega_{k}(1+z)^{2}}
  63. w i w_{i}
  64. d A dA
  65. d l dl
  66. d 2 V = d A d l = a 0 2 r 2 ( z ) H ( z ) ( 1 + z ) 3 d l d z d^{2}V=dA\ dl=\frac{a_{0}^{2}r^{2}(z)}{H(z)(1+z)^{3}}dldz
  67. n ( > M , z ) n(>M,z)
  68. d 2 N ( > M , z ) d z d Ω = n ( > M , z ) d 2 V d z d Ω = n ( > M , z ) a 0 2 r 2 ( z ) H ( z ) ( 1 + z ) 3 \frac{d^{2}N(>M,z)}{dzd\Omega}=n(>M,z)\frac{d^{2}V}{dzd\Omega}=\frac{n(>M,z)a_% {0}^{2}r^{2}(z)}{H(z)(1+z)^{3}}
  69. d 2 N ( > λ ) / d z d Ω d^{2}N(>\lambda)/dzd\Omega
  70. z z
  71. λ \lambda
  72. λ \lambda
  73. d 2 N ( > λ ) / d z d Ω d^{2}N(>\lambda)/dzd\Omega
  74. d 2 N ( > λ , z ) d z d Ω = r 2 ( z ) H ( z ) 0 f ( > λ , z ) d λ 0 p ( λ | M , z ) d n ( M , z ) d M d M \frac{d^{2}N(>\lambda,z)}{dzd\Omega}=\frac{r^{2}(z)}{H(z)}\int_{0}^{\infty}{f(% >\lambda,z)d\lambda}\int_{0}^{\infty}{p(\lambda|M,z)\frac{dn(M,z)}{dM}dM}
  75. f ( > λ , z ) f(>\lambda,z)
  76. λ \lambda
  77. p ( λ | M , z ) p(\lambda|M,z)
  78. M M
  79. z z
  80. λ \lambda
  81. d n ( M , z ) / d M dn(M,z)/dM
  82. 10 \sim 10
  83. z 1 z\sim 1
  84. Λ \Lambda

The_Foundations_of_Arithmetic.html

  1. N x : F x Nx:Fx

Theorems_and_definitions_in_linear_algebra.html

  1. V V
  2. u \vec{u}
  3. v \vec{v}
  4. w \vec{w}
  5. V V
  6. c c
  7. d d
  8. V V
  9. u + v is in V . \vec{u}+\vec{v}\,\text{ is in }V\,\text{.}
  10. u + v = v + u \vec{u}+\vec{v}=\vec{v}+\vec{u}
  11. u + ( v + w ) = ( u + v ) + w \vec{u}+(\vec{v}+\vec{w})=(\vec{u}+\vec{v})+\vec{w}
  12. V has a 𝐳𝐞𝐫𝐨 𝐯𝐞𝐜𝐭𝐨𝐫 0 such that for every u in V , u + 0 = u V\,\text{ has a }\mathbf{zero}\,\text{ }\mathbf{vector}\,\text{ }\vec{0}\,% \text{ such that for every }\vec{u}\,\text{ in }V\,\text{, }\vec{u}+\vec{0}=% \vec{u}
  13. For every u in V , there is a vector in V denoted by - u such that u + ( - u ) = 0 . \,\text{For every }\vec{u}\,\text{ in }V\,\text{, there is a vector in }V\,% \text{ denoted by }-\vec{u}\,\text{ such that }\vec{u}+(-\vec{u})=\vec{0}\,% \text{.}
  14. c u is in V . c\vec{u}\,\text{ is in }V\,\text{.}
  15. c ( u + v ) = c u + c v c(\vec{u}+\vec{v})=c\vec{u}+c\vec{v}
  16. ( c + d ) u = c u + d u (c+d)\vec{u}=c\vec{u}+d\vec{u}
  17. c ( d u ) = ( c d ) u c(d\vec{u})=(cd)\vec{u}
  18. 1 ( u ) = u 1(\vec{u})=\vec{u}
  19. W W
  20. V V
  21. W W
  22. V V
  23. If u and v are in W , then u + v is in W . \,\text{If }\vec{u}\,\text{ and }\vec{v}\,\text{ are in }W\,\text{, then }\vec% {u}+\vec{v}\,\text{ is in }W\,\text{.}
  24. If u is in W and c is any scalar, then c u is in W . \,\text{If }\vec{u}\,\text{ is in }W\,\text{ and }c\,\text{ is any scalar, % then }c\vec{u}\,\text{ is in }W\,\text{.}
  25. v \vec{v}
  26. V V
  27. u 1 \vec{u}_{1}
  28. u 2 \vec{u}_{2}
  29. \dots
  30. u k \vec{u}_{k}
  31. V V
  32. v \vec{v}
  33. v = c 1 u 1 + c 2 u 2 + + c k u k \vec{v}=c_{1}\vec{u}_{1}+c_{2}\vec{u}_{2}+\dots+c_{k}\vec{u}_{k}
  34. c 1 c_{1}
  35. c 2 c_{2}
  36. \dots
  37. c k c_{k}
  38. n n
  39. n n
  40. | A | \left|A\right|
  41. x 1 = det ( A 1 ) det ( A ) , x 2 = det ( A 2 ) det ( A ) , , x n = det ( A n ) det ( A ) x_{1}=\frac{\det(A_{1})}{\det(A)},\qquad x_{2}=\frac{\det(A_{2})}{\det(A)},% \qquad\dots,\qquad x_{n}=\frac{\det(A_{n})}{\det(A)}
  42. i i
  43. A i A_{i}
  44. S = { v 1 S=\{\vec{v}_{1}
  45. v 2 \vec{v}_{2}
  46. \dots
  47. v n } \vec{v}_{n}\}
  48. V V
  49. S S
  50. V V
  51. S S
  52. F n F_{n}
  53. V V
  54. W W
  55. T : V W T:V\to W
  56. V V
  57. W W
  58. u \vec{u}
  59. v \vec{v}
  60. V V
  61. c c
  62. T ( u + v ) = T ( u ) + T ( v ) T(\vec{u}+\vec{v})=T(\vec{u})+T(\vec{v})
  63. T ( c u ) = c T ( u ) T(c\vec{u})=cT(\vec{u})
  64. \color B l u e 2.1 {\color{Blue}~{}2.1}
  65. \color B l u e 2.2 {\color{Blue}~{}2.2}
  66. β = v 1 , v 2 , , v n \beta={v_{1},v_{2},\ldots,v_{n}}
  67. R ( T ) = span ( T ( β ) ) = span ( T ( v 1 ) , T ( v 2 ) , , T ( v n ) ) \mathrm{R(T)}=\mathrm{span}(T(\beta\mathrm{))}=\mathrm{span}({T(v_{1}),T(v_{2}% ),\ldots,T(v_{n})})
  68. \color B l u e 2.3 {\color{Blue}~{}2.3}
  69. nullity ( T ) + rank ( T ) = dim ( V ) . \mathrm{nullity}(T)+\mathrm{rank}(T)=\dim(V).
  70. \color B l u e 2.4 {\color{Blue}~{}2.4}
  71. T : V W T:V\to W
  72. T T
  73. ker ( T ) = { 0 } \operatorname{ker}(T)=\{\vec{0}\}
  74. \color B l u e 2.5 {\color{Blue}~{}2.5}
  75. \color B l u e 2.6 {\color{Blue}~{}2.6}
  76. w 1 , w 2 , , w n = {w_{1},w_{2},\ldots,w_{n}}=
  77. v 1 , v 2 , , v n {v_{1},v_{2},\ldots,v_{n}}
  78. w 1 , w 2 , , w n w_{1},w_{2},\ldots,w_{n}
  79. T ( v i ) = w i \mathrm{T}(v_{i})=w_{i}
  80. i = 1 , 2 , , n . i=1,2,\ldots,n.
  81. v 1 , v 2 , , v n {v_{1},v_{2},\ldots,v_{n}}
  82. U ( v i ) = T ( v i ) U(v_{i})=T(v_{i})
  83. i = 1 , 2 , , n , i=1,2,\ldots,n,
  84. \color B l u e 2.7 {\color{Blue}~{}2.7}
  85. a a
  86. a T + U a\mathrm{T}+\mathrm{U}
  87. \color B l u e 2.8 {\color{Blue}~{}2.8}
  88. [ T + U ] β γ = [ T ] β γ + [ U ] β γ [T+U]_{\beta}^{\gamma}=[T]_{\beta}^{\gamma}+[U]_{\beta}^{\gamma}
  89. [ a T ] β γ = a [ T ] β γ [aT]_{\beta}^{\gamma}=a[T]_{\beta}^{\gamma}
  90. a a
  91. \color B l u e 2.9 {\color{Blue}~{}2.9}
  92. \color B l u e 2.10 {\color{Blue}~{}2.10}
  93. \mathcal{L}
  94. a a
  95. a a
  96. a a
  97. a a
  98. \color B l u e 2.11 {\color{Blue}~{}2.11}
  99. [ U T ] α γ = [ U ] β γ [ T ] α β [UT]_{\alpha}^{\gamma}=[U]_{\beta}^{\gamma}[T]_{\alpha}^{\beta}
  100. \mathcal{L}
  101. \color B l u e 2.12 {\color{Blue}~{}2.12}
  102. a a
  103. a a
  104. a a
  105. a a
  106. a 1 , a 2 , , a k a_{1},a_{2},\ldots,a_{k}
  107. A ( i = 1 k a i B i ) = i = 1 k a i A B i A\Bigg(\sum_{i=1}^{k}a_{i}B_{i}\Bigg)=\sum_{i=1}^{k}a_{i}AB_{i}
  108. ( i = 1 k a i C i ) A = i = 1 k a i C i A \Bigg(\sum_{i=1}^{k}a_{i}C_{i}\Bigg)A=\sum_{i=1}^{k}a_{i}C_{i}A
  109. \color B l u e 2.13 {\color{Blue}~{}2.13}
  110. j ( 1 j p ) j(1\leq j\leq p)
  111. u j u_{j}
  112. v j v_{j}
  113. u j = A v j u_{j}=Av_{j}
  114. v j = B e j v_{j}=Be_{j}
  115. e j e_{j}
  116. \color B l u e 2.14 {\color{Blue}~{}2.14}
  117. [ T ( u ) ] γ = [ T ] β γ [ u ] β [T(u)]_{\gamma}=[T]_{\beta}^{\gamma}[u]_{\beta}
  118. \color B l u e 2.15 {\color{Blue}~{}2.15}
  119. [ L A ] β γ = A [L_{A}]_{\beta}^{\gamma}=A
  120. a a
  121. a a
  122. a a
  123. C = [ L A ] β γ \mathrm{C}=[L_{A}]_{\beta}^{\gamma}
  124. L I n = I F n L_{I_{n}}=I_{F^{n}}
  125. \color B l u e 2.16 {\color{Blue}~{}2.16}
  126. \color B l u e 2.17 {\color{Blue}~{}2.17}
  127. \color B l u e 2.18 {\color{Blue}~{}2.18}
  128. [ T ] β γ [T]_{\beta}^{\gamma}
  129. [ T - 1 ] γ β = ( [ T ] β γ ) - 1 [T^{-1}]_{\gamma}^{\beta}=([T]_{\beta}^{\gamma})^{-1}
  130. \color B l u e 2.19 {\color{Blue}~{}2.19}
  131. \color B l u e 2.20 {\color{Blue}~{}2.20}
  132. Φ ~{}\Phi
  133. \mathcal{L}
  134. Φ ( T ) = [ T ] β γ ~{}\Phi(T)=[T]_{\beta}^{\gamma}
  135. \mathcal{L}
  136. \mathcal{L}
  137. \color B l u e 2.21 {\color{Blue}~{}2.21}
  138. \color B l u e 2.22 {\color{Blue}~{}2.22}
  139. Q = [ I V ] β β Q=[I_{V}]_{\beta^{\prime}}^{\beta}
  140. Q Q
  141. v v\in
  142. [ v ] β = Q [ v ] β ~{}[v]_{\beta}=Q[v]_{\beta^{\prime}}
  143. \color B l u e 2.23 {\color{Blue}~{}2.23}
  144. [ T ] β = Q - 1 [ T ] β Q ~{}[T]_{\beta^{\prime}}=Q^{-1}[T]_{\beta}Q
  145. a x 2 + b x y + c y 2 + d x + e y + f = 0 ax^{2}+bxy+cy^{2}+dx+ey+f=0
  146. X = P X X=PX^{\prime}
  147. x y xy
  148. P P
  149. | P | = 1 \left|P\right|=1
  150. A A
  151. P T A P = [ λ 1 0 0 λ 2 ] P^{T}AP=\begin{bmatrix}\lambda_{1}&0\\ 0&\lambda_{2}\end{bmatrix}
  152. λ 1 \lambda_{1}
  153. λ 2 \lambda_{2}
  154. A A
  155. λ 1 ( x ) 2 + λ 2 ( y ) 2 + [ d e ] P X + f = 0 \lambda_{1}(x^{\prime})^{2}+\lambda_{2}(y^{\prime})^{2}+\begin{bmatrix}d&e\\ \end{bmatrix}PX^{\prime}+f=0
  156. \color B l u e 2.26 {\color{Blue}~{}2.26}
  157. \color B l u e 2.26 {\color{Blue}~{}2.26}
  158. \color B l u e 2.27 {\color{Blue}~{}2.27}
  159. x x
  160. x ( k ) x^{(k)}
  161. \color B l u e 2.28 {\color{Blue}~{}2.28}
  162. C \mathrm{C}^{\infty}
  163. \color B l u e 2.29 {\color{Blue}~{}2.29}
  164. f ( t ) = e c t , f ( t ) = c e c t f(t)=e^{ct},f^{\prime}(t)=ce^{ct}
  165. \color B l u e 2.30 {\color{Blue}~{}2.30}
  166. y + a 0 y = 0 y^{\prime}+a_{0}y=0
  167. { e - a 0 t } \{e^{-a_{0}t}\}
  168. e c t e^{ct}
  169. \color B l u e 2.31 {\color{Blue}~{}2.31}
  170. e c t e^{ct}
  171. \color B l u e 2.32 {\color{Blue}~{}2.32}
  172. \color B l u e 2.33 {\color{Blue}~{}2.33}
  173. c 1 , c 2 , , c n c_{1},c_{2},\ldots,c_{n}
  174. { e c 1 t , e c 2 t , , e c n t } \{e^{c_{1}t},e^{c_{2}t},\ldots,e^{c_{n}t}\}
  175. c 1 , c 2 , , c n c_{1},c_{2},\ldots,c_{n}
  176. { e c 1 t , e c 2 t , , e c n t } \{e^{c_{1}t},e^{c_{2}t},\ldots,e^{c_{n}t}\}
  177. β = { e c 1 t , e c 2 t , , e c n t } \beta=\{e^{c_{1}t},e^{c_{2}t},\ldots,e^{c_{n}t}\}
  178. \color B l u e 2.34 {\color{Blue}~{}2.34}
  179. ( t - c 1 ) 1 n ( t - c 2 ) 2 n ( t - c k ) k n , (t-c_{1})^{n}_{1}(t-c_{2})^{n}_{2}\cdots(t-c_{k})^{n}_{k},
  180. n 1 , n 2 , , n k n_{1},n_{2},\ldots,n_{k}
  181. c 1 , c 2 , , c n c_{1},c_{2},\ldots,c_{n}
  182. { e c 1 t , t e c 1 t , , t n 1 - 1 e c 1 t , , e c k t , t e c k t , , t n k - 1 e c k t } \{e^{c_{1}t},te^{c_{1}t},\ldots,t^{n_{1}-1}e^{c_{1}t},\ldots,e{c_{k}t},te^{c_{% k}t},\ldots,t^{n_{k}-1}e^{c_{k}t}\}
  183. P P
  184. P - 1 = P T P^{-1}=P^{T}
  185. A A
  186. n × n n\times n
  187. A A
  188. A A
  189. λ \lambda
  190. A A
  191. k k
  192. λ \lambda
  193. k k
  194. λ \lambda
  195. k k
  196. A A
  197. A A
  198. n × n n\times n
  199. I n I_{n}
  200. If A is n × n , then the following statements are equivalent: \,\text{If }A\,\text{ is }n\,\text{ × }n\,\text{, then the following % statements are equivalent:}
  201. A is invertible. A\,\text{ is invertible.}
  202. A x = b has a unique solution for every n × 1 column matrix b . A\vec{x}=\vec{b}\,\text{ has a unique solution for every }n\times 1\,\text{ % column matrix }\vec{b}\,\text{.}
  203. A x = 0 has only the trivial solution. A\vec{x}=\vec{0}\,\text{ has only the trivial solution.}
  204. A is row-equivalent to I n . A\,\text{ is row-equivalent to }I_{n}\,\text{.}
  205. A can be written as the product of elementary matrices. A\,\text{ can be written as the product of elementary matrices.}
  206. det ( A ) 0 \det(A)\neq 0
  207. rk ( A ) = n number of columns. \operatorname{rk}(A)=n\,\text{ number of columns.}
  208. nul ( A ) = 0 \operatorname{nul}(A)=0
  209. All of the n -row vectors of A are linearly independent. \,\text{All of the }n\,\text{-row vectors of }A\,\text{ are linearly % independent.}
  210. All of the n -column vectors of A are linearly independent. \,\text{All of the }n\,\text{-column vectors of }A\,\text{ are linearly % independent.}
  211. A = ( a b c d ) A=\begin{pmatrix}a&b\\ c&d\\ \end{pmatrix}
  212. a d - b c ad-bc
  213. u , v , u,v,
  214. w w
  215. k k
  216. det ( u + k v w ) = det ( u w ) + k det ( v w ) \det\begin{pmatrix}u+kv\\ w\\ \end{pmatrix}=\det\begin{pmatrix}u\\ w\\ \end{pmatrix}+k\det\begin{pmatrix}v\\ w\\ \end{pmatrix}
  217. det ( w u + k v ) = det ( w u ) + k det ( w v ) \det\begin{pmatrix}w\\ u+kv\\ \end{pmatrix}=\det\begin{pmatrix}w\\ u\\ \end{pmatrix}+k\det\begin{pmatrix}w\\ v\\ \end{pmatrix}
  218. \in
  219. A - 1 = 1 det ( A ) ( A 22 - A 12 - A 21 A 11 ) A^{-1}=\frac{1}{\det(A)}\begin{pmatrix}A_{22}&-A_{12}\\ -A_{21}&A_{11}\\ \end{pmatrix}
  220. \color B l u e 5.1 {\color{Blue}~{}5.1}
  221. β = v 1 , v 2 , , v n \beta={v_{1},v_{2},\ldots,v_{n}}
  222. D j j D_{jj}
  223. v j v_{j}
  224. 1 j n 1\leq j\leq n
  225. \color B l u e 5.2 {\color{Blue}~{}5.2}
  226. \color B l u e 5.3 {\color{Blue}~{}5.3}
  227. \color B l u e 5.4 {\color{Blue}~{}5.4}
  228. \color B l u e 5.5 {\color{Blue}~{}5.5}
  229. λ 1 , λ 2 , , λ k , \lambda_{1},\lambda_{2},\ldots,\lambda_{k},
  230. v 1 , v 2 , , v k v_{1},v_{2},\ldots,v_{k}
  231. λ i \lambda_{i}
  232. v i v_{i}
  233. 1 i k 1\leq i\leq k
  234. v 1 , v 2 , , v k v_{1},v_{2},\ldots,v_{k}
  235. \color B l u e 5.6 {\color{Blue}~{}5.6}
  236. \color B l u e 5.7 {\color{Blue}~{}5.7}
  237. m m
  238. 1 dim ( E λ ) m 1\leq\dim(E_{\lambda})\leq m
  239. \color B l u e 5.8 {\color{Blue}~{}5.8}
  240. λ 1 , λ 2 , , λ k , \lambda_{1},\lambda_{2},\ldots,\lambda_{k},
  241. i = 1 , 2 , , k , i=1,2,\ldots,k,
  242. S i S_{i}
  243. E λ i E_{\lambda_{i}}
  244. S = S 1 S 2 S k S=S_{1}\cup S_{2}\cup\cdots\cup S_{k}
  245. \color B l u e 5.9 {\color{Blue}~{}5.9}
  246. λ 1 , λ 2 , , λ k \lambda_{1},\lambda_{2},\ldots,\lambda_{k}
  247. λ i \lambda_{i}
  248. dim ( E λ i ) \dim(E_{\lambda_{i}})
  249. i i
  250. β i \beta_{i}
  251. E λ i E_{\lambda_{i}}
  252. i i
  253. β = β 1 β 2 β k \beta=\beta_{1}\cup\beta_{2}\cup\cup\beta_{k}
  254. b a s i s 2 basis^{2}
  255. \color B l u e 6.1 {\color{Blue}~{}6.1}
  256. x , y + z = x , y + x , z . \langle x,y+z\rangle=\langle x,y\rangle+\langle x,z\rangle.
  257. x , c y = c ¯ x , y . \langle x,cy\rangle=\bar{c}\langle x,y\rangle.
  258. x , 0 = 0 , x = 0. \langle x,\mathit{0}\rangle=\langle\mathit{0},x\rangle=0.
  259. x , x = 0 \langle x,x\rangle=0
  260. x = 0. x=\mathit{0}.
  261. x , y = x , z \langle x,y\rangle=\langle x,z\rangle
  262. x x\in
  263. y = z y=z
  264. \color B l u e 6.2 {\color{Blue}~{}6.2}
  265. c x = | c | x \|cx\|=|c|\cdot\|x\|
  266. x = 0 \|x\|=0
  267. x = 0 x=0
  268. x 0 \|x\|\geq 0
  269. | x , y | x y |\langle x,y\rangle|\leq\|x\|\cdot\|y\|
  270. x + y x + y \|x+y\|\leq\|x\|+\|y\|
  271. \color B l u e 6.3 {\color{Blue}~{}6.3}
  272. S = { v 1 , v 2 , , v k } S=\{v_{1},v_{2},\ldots,v_{k}\}
  273. y y
  274. y = i = 1 n y , v i v i 2 v i y=\sum_{i=1}^{n}{\langle y,v_{i}\rangle\over\|v_{i}\|^{2}}v_{i}
  275. \color B l u e 6.4 {\color{Blue}~{}6.4}
  276. { w 1 , w 2 , , w n } \{w_{1},w_{2},\ldots,w_{n}\}
  277. { v 1 , v 2 , , v n } \{v_{1},v_{2},\ldots,v_{n}\}
  278. v 1 = w 1 v_{1}=w_{1}
  279. v k = w k - j = 1 k - 1 w k , v j v j 2 v j v_{k}=w_{k}-\sum_{j=1}^{k-1}{\langle w_{k},v_{j}\rangle\over\|v_{j}\|^{2}}v_{j}
  280. \color B l u e 6.5 {\color{Blue}~{}6.5}
  281. { v 1 , v 2 , , v n } \{v_{1},v_{2},\ldots,v_{n}\}
  282. x = i = 1 n x , v i v i x=\sum_{i=1}^{n}\langle x,v_{i}\rangle v_{i}
  283. { v 1 , v 2 , , v n } \{v_{1},v_{2},\ldots,v_{n}\}
  284. i i
  285. j j
  286. A i j = T ( v j ) , v i A_{ij}=\langle T(v_{j}),v_{i}\rangle
  287. \color B l u e 6.6 {\color{Blue}~{}6.6}
  288. y y
  289. u u
  290. z z
  291. y = u + z y=u+z
  292. { v 1 , v 2 , , v k } \{v_{1},v_{2},\ldots,v_{k}\}
  293. u = i = 1 k y , v i v i u=\sum_{i=1}^{k}\langle y,v_{i}\rangle v_{i}
  294. u u
  295. y y
  296. x x
  297. y - x y - u \|y-x\|\geq\|y-u\|
  298. x = u x=u
  299. \color B l u e 6.7 {\color{Blue}~{}6.7}
  300. S = { v 1 , v 2 , , v k } S=\{v_{1},v_{2},\ldots,v_{k}\}
  301. n n
  302. { v 1 , v 2 , , v k , v k + 1 , , v n } \{v_{1},v_{2},\ldots,v_{k},v_{k+1},\ldots,v_{n}\}
  303. S 1 = { v k + 1 , v k + 2 , , v n } S_{1}=\{v_{k+1},v_{k+2},\ldots,v_{n}\}
  304. \color B l u e 6.8 {\color{Blue}~{}6.8}
  305. g g
  306. y y
  307. g ( x ) = x , y \rm{g}(x)=\langle x,y\rangle
  308. x x
  309. \color B l u e 6.9 {\color{Blue}~{}6.9}
  310. T ( x ) , y = x , T * ( y ) \langle\rm{T}(x),y\rangle=\langle x,\rm{T}^{*}(y)\rangle
  311. x , y x,y
  312. \color B l u e 6.10 {\color{Blue}~{}6.10}
  313. [ T * ] β = [ T ] β * [T^{*}]_{\beta}=[T]^{*}_{\beta}
  314. \color B l u e 6.11 {\color{Blue}~{}6.11}
  315. c c
  316. c ¯ \bar{c}
  317. c c
  318. c ¯ \bar{c}
  319. c c
  320. \color B l u e 6.12 {\color{Blue}~{}6.12}
  321. y y
  322. x 0 x_{0}
  323. ( A * A ) x 0 = A * y (A*A)x_{0}=A*y
  324. A x 0 - Y A x - y \|Ax_{0}-Y\|\leq\|Ax-y\|
  325. x x
  326. y y
  327. A x , y m = x , A * y n \langle Ax,y\rangle_{m}=\langle x,A*y\rangle_{n}
  328. \color B l u e 6.13 {\color{Blue}~{}6.13}
  329. A x = b Ax=b
  330. s s
  331. A x = b Ax=b
  332. s s
  333. s s
  334. A x = b Ax=b
  335. u u
  336. ( A A * ) u = b (AA*)u=b
  337. s = A * u s=A*u

Theory_of_tides.html

  1. ζ t + 1 a cos ( φ ) [ λ ( u D ) + φ ( v D cos ( φ ) ) ] = 0 , u t - v ( 2 Ω sin ( φ ) ) + 1 a cos ( φ ) λ ( g ζ + U ) = 0 and v t + u ( 2 Ω sin ( φ ) ) + 1 a φ ( g ζ + U ) = 0 , \begin{aligned}\displaystyle\frac{\partial\zeta}{\partial t}&\displaystyle+% \frac{1}{a\cos(\varphi)}\left[\frac{\partial}{\partial\lambda}(uD)+\frac{% \partial}{\partial\varphi}\left(vD\cos(\varphi)\right)\right]=0,\\ \displaystyle\frac{\partial u}{\partial t}&\displaystyle-v\left(2\Omega\sin(% \varphi)\right)+\frac{1}{a\cos(\varphi)}\frac{\partial}{\partial\lambda}\left(% g\zeta+U\right)=0\qquad\,\text{and}\\ \displaystyle\frac{\partial v}{\partial t}&\displaystyle+u\left(2\Omega\sin(% \varphi)\right)+\frac{1}{a}\frac{\partial}{\partial\varphi}\left(g\zeta+U% \right)=0,\end{aligned}

Therapeutic_inertia.html

  1. h v - c v \frac{h}{v}-\frac{c}{v}
  2. 4 5 - 2 5 = 0.4 = 40 % \frac{4}{5}-\frac{2}{5}=0.4=40\%
  3. 1 - c h 1-\frac{c}{h}
  4. 1 - 2 4 = 0.5 = 50 % 1-\frac{2}{4}=0.5=50\%

Thermal_effusivity.html

  1. e = ( k ρ c p ) 1 / 2 e={(k\rho c_{p})}^{1/2}
  2. ρ \rho
  3. c p c_{p}
  4. ρ \rho
  5. c p c_{p}
  6. T m = T 1 + ( T 2 - T 1 ) e 2 ( e 2 + e 1 ) T_{m}=T_{1}+(T_{2}-T_{1}){e_{2}\over(e_{2}+e_{1})}

Thermal_oxidation.html

  1. Si + 2 H 2 O SiO 2 + 2 H 2 ( g ) \rm Si+2H_{2}O\rightarrow SiO_{2}+2H_{2\ (g)}
  2. Si + O 2 SiO 2 \rm Si+O_{2}\rightarrow SiO_{2}\,
  3. τ = X o 2 B + X o ( B A ) \tau=\frac{X_{o}^{2}}{B}+\frac{X_{o}}{(\frac{B}{A})}
  4. X o ( t ) = A / 2 [ 1 + 4 B A 2 ( t + τ ) - 1 ] X_{o}(t)=A/2\cdot\left[\sqrt{1+\frac{4B}{A^{2}}(t+\tau)}-1\right]

Thermodynamics_of_the_universe.html

  1. 0 = d Q = d U + P d V 0=dQ=dU+PdV
  2. Q Q
  3. U U
  4. P P
  5. V V
  6. u U / V u\equiv U/V
  7. d u = d ( U V ) = d U V - U d V V 2 = - ( p + u ) d V V = - 3 ( p + u ) d a a du=d\left({U\over V}\right)={dU\over V}-U{dV\over V^{2}}=-(p+u){dV\over V}=-3(% p+u){da\over a}
  8. a 3 a^{3}
  9. a a
  10. d t dt
  11. ρ = u \rho=u
  12. u u
  13. ρ \rho
  14. p = u / 3 p=u/3
  15. p u p<<u
  16. d u = - 4 u d a a du=-4u{da\over a}
  17. u u
  18. a - 4 a^{-4}
  19. d u = - 3 u d a a du=-3u{da\over a}
  20. u u
  21. a - 3 a^{-3}
  22. a - 3 a^{-3}
  23. T T
  24. T a - 3 Ta^{-3}
  25. T T
  26. a a
  27. a a
  28. a - 4 a^{-4}
  29. a a
  30. Λ \Lambda
  31. k k
  32. a 1 a\ll 1
  33. a ˙ 2 a 2 ρ {{\dot{a}}^{2}}\propto{a^{2}}\rho
  34. ρ = u \rho=u
  35. a t 1 / 2 a\propto t^{1/2}
  36. a t 2 / 3 a\propto t^{2/3}
  37. u ˙ = - 3 ( p + u ) a ˙ a {\dot{u}}=-3(p+u)\frac{\dot{a}}{a}
  38. p = - u p=-u

Third_derivative.html

  1. d 3 y d x 3 , f ′′′ ( x ) , or d 3 d x 3 [ f ( x ) ] . \frac{d^{3}y}{dx^{3}},\quad f^{\prime\prime\prime}(x),\quad\,\text{or }\frac{d% ^{3}}{dx^{3}}[f(x)].
  2. f ( x ) = x 4 f(x)=x^{4}
  3. f ( x ) = 4 x 3 f^{\prime}(x)=4x^{3}
  4. f ′′ ( x ) = 12 x 2 f^{\prime\prime}(x)=12x^{2}
  5. f ′′′ ( x ) = 24 x f^{\prime\prime\prime}(x)=24x
  6. d 3 d x 3 [ x 4 ] = 24 x \frac{d^{3}}{dx^{3}}[x^{4}]=24x
  7. f ( x ) f(x)
  8. f ( x ) f(x)
  9. d 3 d x 3 [ f ( x ) ] = d d x [ f ′′ ( x ) ] \frac{d^{3}}{dx^{3}}[f(x)]=\frac{d}{dx}[f^{\prime\prime}(x)]
  10. j ( t ) = d 3 r d t 3 {j}(t)=\frac{d^{3}{r}}{dt^{3}}

Thomas_Kilgore_Sherwood.html

  1. S h = K c L 𝒟 Sh=\frac{K_{c}L}{\mathcal{D}}
  2. K c K_{c}
  3. L L
  4. 𝒟 \mathcal{D}

Threshold_graph.html

  1. S S
  2. v v
  3. w ( v ) w(v)
  4. v , u v,u
  5. u v uv
  6. w ( u ) + w ( v ) S w(u)+w(v)\geq S
  7. T T
  8. v v
  9. a ( v ) a(v)
  10. X V X\subseteq V
  11. X X
  12. v X a ( v ) T . \sum_{v\in X}a(v)\geq T.
  13. ϵ \epsilon
  14. u u
  15. j j
  16. ϵ u u j \epsilon uuj
  17. ϵ u j \epsilon uj
  18. ϵ u u u j u u j \epsilon uuujuuj

Thymidylate_synthase.html

  1. \rightleftharpoons

Time_slicing_(digital_broadcasting).html

  1. T B T_{B}
  2. B B B_{B}
  3. R B R_{B}
  4. T B = B B R B 0.96 T_{B}=\frac{B_{B}}{R_{B}\cdot 0.96}
  5. T O N T_{ON}
  6. T O N = T B + T S y n c T_{ON}=T_{B}+T_{Sync}
  7. R C R_{C}
  8. R C = R B T O N + T O F F R_{C}=\frac{R_{B}}{T_{ON}+T_{OFF}}
  9. T O F F = R B R C - T O N T_{OFF}=\frac{R_{B}}{R_{C}}-T_{ON}
  10. P P
  11. P = ( 1 - R C ( 1 R B 0.96 + T S y n c B B ) ) 100 % P=(1-R_{C}\cdot(\frac{1}{R_{B}\cdot 0.96}+\frac{T_{Sync}}{B_{B}}))\cdot 100\%

Timeline_of_algebra.html

  1. x x
  2. x 2 x^{2}
  3. x 3 x^{3}
  4. 1 / x 1/x
  5. 1 / x 2 1/x^{2}
  6. 1 / x 3 1/x^{3}
  7. x P - N = 0 x^{P}-N=0

Tip-speed_ratio.html

  1. v v
  2. λ = Tip speed of blade Wind speed \lambda=\frac{\mbox{Tip speed of blade}~{}}{\mbox{Wind speed}~{}}
  3. ω \omega
  4. ω \omega
  5. λ = ω R v \lambda=\frac{\omega\,\text{R}}{v}
  6. C p C_{p}
  7. C p C_{p}
  8. C p = C p m a x C_{p}=C_{p~{}max}
  9. C p m a x C_{p~{}max}
  10. N = 120 f P N=\frac{120f}{P}
  11. N N
  12. f f
  13. P P

TITAN2D.html

  1. h t Change in mass over time + h u ¯ x + h v ¯ y Total spatial variation of x,y mass fluxes = 0 {\underbrace{\partial h\over\partial t}}_{\begin{smallmatrix}\,\text{Change}\\ \,\text{in mass}\\ \,\text{over time}\end{smallmatrix}}+\underbrace{{\partial\overline{hu}\over% \partial x}+{\partial\overline{hv}\over\partial y}}_{\begin{smallmatrix}\,% \text{Total spatial}\\ \,\text{variation of}\\ \,\text{x,y mass fluxes}\end{smallmatrix}}=0
  2. h u ¯ t Change in x mass flux over time + x ( h u 2 ¯ + 1 2 k a p g z h 2 ) + h u v ¯ y Total spatial variation of x,y momentum fluxes in x-direction = - h k a p sgn ( u y ) h g z y sin ϕ i n t Dissipative internal friction force in x-direction - u u 2 + v 2 [ g z h ( 1 + u r x g x ) ] tan ϕ b e d Dissipative basal friction force in x-direction + g x h Driving gravitational force in x-direction {\underbrace{\partial\overline{hu}\over\partial t}}_{\begin{smallmatrix}\,% \text{Change in}\\ \,\text{x mass flux}\\ \,\text{over time}\end{smallmatrix}}+\underbrace{{\partial\over\partial x}% \left(\overline{hu^{2}}+{1\over 2}{k_{ap}g_{z}h^{2}}\right)+{\partial\overline% {huv}\over\partial y}}_{\begin{smallmatrix}\,\text{Total spatial variation}\\ \,\text{of x,y momentum fluxes}\\ \,\text{in x-direction}\end{smallmatrix}}=\underbrace{-hk_{ap}\operatorname{% sgn}\left({\partial u\over\partial y}\right){\partial hg_{z}\over\partial y}% \sin\phi_{int}}_{\begin{smallmatrix}\,\text{Dissipative internal}\\ \,\text{friction force}\\ \,\text{in x-direction}\end{smallmatrix}}-\underbrace{{u\over\sqrt{u^{2}+v^{2}% }}\left[g_{z}h\left(1+{u\over r_{x}g_{x}}\right)\right]\tan\phi_{bed}}_{\begin% {smallmatrix}\,\text{Dissipative basal}\\ \,\text{friction force}\\ \,\text{in x-direction}\end{smallmatrix}}+\underbrace{g_{x}h}_{\begin{% smallmatrix}\,\text{Driving}\\ \,\text{gravitational}\\ \,\text{force in}\\ \,\text{x-direction}\end{smallmatrix}}
  3. h v ¯ t Change in y mass flux over time + h u v ¯ x + y ( h v 2 ¯ + 1 2 k a p g z h 2 ) Total spatial variation of x,y momentum fluxes in y-direction = - h k a p sgn ( v x ) h g z x sin ϕ i n t Dissipative internal friction force in y-direction - v u 2 + v 2 [ g z h ( 1 + v r y g y ) ] tan ϕ b e d Dissipative basal friction force in y-direction + g y h Driving gravitational force in y-direction {\underbrace{\partial\overline{hv}\over\partial t}}_{\begin{smallmatrix}\,% \text{Change in}\\ \,\text{y mass flux}\\ \,\text{over time}\end{smallmatrix}}+\underbrace{{\partial\overline{huv}\over% \partial x}+{\partial\over\partial y}\left(\overline{hv^{2}}+{1\over 2}{k_{ap}% g_{z}h^{2}}\right)}_{\begin{smallmatrix}\,\text{Total spatial variation}\\ \,\text{of x,y momentum fluxes}\\ \,\text{in y-direction}\end{smallmatrix}}=\underbrace{-hk_{ap}\operatorname{% sgn}\left({\partial v\over\partial x}\right){\partial hg_{z}\over\partial x}% \sin\phi_{int}}_{\begin{smallmatrix}\,\text{Dissipative internal}\\ \,\text{friction force}\\ \,\text{in y-direction}\end{smallmatrix}}-\underbrace{{v\over\sqrt{u^{2}+v^{2}% }}\left[g_{z}h\left(1+{v\over r_{y}g_{y}}\right)\right]\tan\phi_{bed}}_{\begin% {smallmatrix}\,\text{Dissipative basal}\\ \,\text{friction force}\\ \,\text{in y-direction}\end{smallmatrix}}+\underbrace{g_{y}h}_{\begin{% smallmatrix}\,\text{Driving}\\ \,\text{gravitational}\\ \,\text{force in}\\ \,\text{y-direction}\end{smallmatrix}}

Toda_oscillator.html

  1. x ~{}x~{}
  2. z ~{}z~{}
  3. z ~{}z~{}
  4. d 2 x d z 2 + D ( x ) d x d z + Φ ( x ) = 0 , \frac{{\rm d^{2}}x}{{\rm d}z^{2}}+D(x)\frac{{\rm d}x}{{\rm d}z}+\Phi^{\prime}(% x)=0,
  5. D ( x ) = u e x + v ~{}D(x)=ue^{x}+v~{}
  6. Φ ( x ) = e x - x - 1 ~{}\Phi(x)=e^{x}-x-1~{}
  7. z ~{}z~{}
  8. t ~{}t~{}
  9. z = t / t 0 ~{}z=t/t_{0}~{}
  10. t 0 ~{}t_{0}~{}
  11. x ˙ = d x d z ~{}\dot{x}=\frac{{\rm d}x}{{\rm d}z}
  12. x ~{}x~{}
  13. x ¨ = d 2 x d z 2 ~{}\ddot{x}=\frac{{\rm d}^{2}x}{{\rm d}z^{2}}~{}
  14. D ~{}D~{}
  15. u ~{}u~{}
  16. v ~{}v~{}
  17. x ~{}x~{}
  18. Φ ( x ) = e x - x - 1 ~{}\Phi(x)=e^{x}-x-1~{}
  19. x ~{}x~{}
  20. x ~{}x~{}
  21. exp ( x ) ~{}\exp(x)~{}
  22. x ~{}x~{}
  23. u = v = 0 ~{}u=v=0~{}
  24. 10 - 4 ~{}10^{-4}~{}
  25. x = x ( t ) ~{}x=x(t)~{}
  26. u = v = 0 ~{}u=v=0~{}
  27. E = 1 2 ( d x d z ) 2 + Φ ( x ) ~{}E=\frac{1}{2}\left(\frac{{\rm d}x}{{\rm d}z}\right)^{2}+\Phi(x)~{}
  28. z ~{}z~{}
  29. x ~{}x~{}
  30. z ~{}z~{}
  31. z = ± x x max d a 2 E - Φ ( a ) z=\pm\int_{x}^{x_{\max}}\!\!\frac{{\rm d}a}{\sqrt{2}\sqrt{E-\Phi(a)}}
  32. x min ~{}x_{\min}~{}
  33. x max ~{}x_{\max}~{}
  34. x ~{}x~{}
  35. x ˙ ( 0 ) = 0 \dot{x}(0)=0
  36. x max / x max = 2 γ ~{}x_{\max}/x_{\max}=2\gamma~{}
  37. δ = x max - x min 1 \delta=\frac{x_{\max}-x_{\min}}{1}
  38. δ = ln sin ( γ ) γ \delta=\ln\frac{\sin(\gamma)}{\gamma}
  39. E = E ( γ ) = γ tanh ( γ ) + ln sinh γ γ - 1 E=E(\gamma)=\frac{\gamma}{\tanh(\gamma)}+\ln\frac{\sinh\gamma}{\gamma}-1
  40. γ ~{}\gamma~{}
  41. γ = 5 ~{}\gamma=5~{}
  42. E E
  43. γ ~{}\gamma~{}
  44. γ 1 ~{}\gamma\ll 1~{}
  45. T ( γ ) = 2 π ( 1 + γ 2 24 + O ( γ 4 ) ) ~{}T(\gamma)=2\pi\left(1+\frac{\gamma^{2}}{24}+O(\gamma^{4})\right)~{}
  46. γ 1 ~{}\gamma\gg 1~{}
  47. T ( γ ) = 4 γ 1 / 2 ( 1 + O ( 1 / γ ) ) ~{}T(\gamma)=4\gamma^{1/2}\left(1+O(1/\gamma)\right)~{}
  48. γ > 0 ~{}\gamma>0~{}
  49. T = T ( γ ) ~{}T=T(\gamma)~{}
  50. k ( γ ) = 2 π T ( γ ) ~{}k(\gamma)=\frac{2\pi}{T(\gamma)}~{}
  51. k fit ( γ ) = 2 π T fit ( γ ) = k\text{fit}(\gamma)=\frac{2\pi}{T\text{fit}(\gamma)}=
  52. ( 10630 + 674 γ + 695.2419 γ 2 + 191.4489 γ 3 + 16.86221 γ 4 + 4.082607 γ 5 + γ 6 10630 + 674 γ + 2467 γ 2 + 303.2428 γ 3 + 164.6842 γ 4 + 36.6434 γ 5 + 3.9596 γ 6 + 0.8983 γ 7 + 16 π 4 γ 8 ) 1 / 4 \left(\frac{10630+674\gamma+695.2419\gamma^{2}+191.4489\gamma^{3}+16.86221% \gamma^{4}+4.082607\gamma^{5}+\gamma^{6}}{10630+674\gamma+2467\gamma^{2}+303.2% 428\gamma^{3}+164.6842\gamma^{4}+36.6434\gamma^{5}+3.9596\gamma^{6}+0.8983% \gamma^{7}+\frac{16}{\pi^{4}}\gamma^{8}}\right)^{1/4}
  53. 22 × 10 - 9 22\times 10^{-9}
  54. u ~{}u~{}
  55. v ~{}v~{}
  56. u ~{}u~{}
  57. v ~{}v~{}

Tolman–Oppenheimer–Volkoff_equation.html

  1. d P ( r ) d r = - G r 2 [ ρ ( r ) + P ( r ) c 2 ] [ M ( r ) + 4 π r 3 P ( r ) c 2 ] [ 1 - 2 G M ( r ) c 2 r ] - 1 \frac{dP(r)}{dr}=-\frac{G}{r^{2}}\left[\rho(r)+\frac{P(r)}{c^{2}}\right]\left[% M(r)+4\pi r^{3}\frac{P(r)}{c^{2}}\right]\left[1-\frac{2GM(r)}{c^{2}r}\right]^{% -1}\;
  2. d s 2 = e ν ( r ) c 2 d t 2 - ( 1 - 2 G M ( r ) / r c 2 ) - 1 d r 2 - r 2 ( d θ 2 + sin 2 θ d ϕ 2 ) ds^{2}=e^{\nu(r)}c^{2}dt^{2}-(1-2GM(r)/rc^{2})^{-1}dr^{2}-r^{2}(d\theta^{2}+% \sin^{2}\theta d\phi^{2})\;
  3. d ν ( r ) d r = - ( 2 P ( r ) + ρ ( r ) c 2 ) d P ( r ) d r \frac{d\nu(r)}{dr}=-\left(\frac{2}{P(r)+\rho(r)c^{2}}\right)\frac{dP(r)}{dr}\;
  4. d s 2 = ( 1 - 2 G M 0 / r c 2 ) c 2 d t 2 - ( 1 - 2 G M 0 / r c 2 ) - 1 d r 2 - r 2 ( d θ 2 + sin 2 θ d ϕ 2 ) ds^{2}=(1-2GM_{0}/rc^{2})c^{2}dt^{2}-(1-2GM_{0}/rc^{2})^{-1}dr^{2}-r^{2}(d% \theta^{2}+\sin^{2}\theta d\phi^{2})\;
  5. d M ( r ) d r = 4 π ρ ( r ) r 2 \frac{dM(r)}{dr}=4\pi\rho(r)r^{2}\;
  6. M 0 = M ( r B ) = 0 r B 4 π ρ ( r ) r 2 d r M_{0}=M(r_{B})=\int_{0}^{r_{B}}4\pi\rho(r)r^{2}\;dr\;
  7. M 1 = 0 r B 4 π ρ ( r ) r 2 1 - 2 G M ( r ) / r c 2 d r M_{1}=\int_{0}^{r_{B}}\frac{4\pi\rho(r)r^{2}}{\sqrt{1-2GM(r)/rc^{2}}}\;dr\;
  8. δ M = 0 r B 4 π ρ ( r ) r 2 ( ( 1 - 2 G M ( r ) / r c 2 ) - 1 / 2 - 1 ) d r \delta M=\int_{0}^{r_{B}}4\pi\rho(r)r^{2}((1-2GM(r)/rc^{2})^{-1/2}-1)\;dr\;
  9. ( J = 0 ) (J=0)
  10. c 2 d τ 2 = g 00 c 2 d t 2 + g 11 d r 2 + g 22 d θ 2 + g 33 d ϕ 2 c^{2}d\tau^{2}=g_{00}c^{2}dt^{2}+g_{11}dr^{2}+g_{22}d\theta^{2}+g_{33}d\phi^{2}\;
  11. c 2 d τ 2 = e ν ( r ) c 2 d t 2 - e λ ( r ) d r 2 - r 2 d θ 2 - r 2 sin 2 θ d ϕ 2 c^{2}d\tau^{2}=e^{\nu(r)}c^{2}dt^{2}-e^{\lambda(r)}dr^{2}-r^{2}d\theta^{2}-r^{% 2}\sin^{2}\theta d\phi^{2}\;
  12. g 00 = e ν ( r ) g_{00}=e^{\nu(r)}\;
  13. g 11 = - e λ ( r ) g_{11}=-e^{\lambda(r)}\;
  14. T 00 = ρ ( r ) c 2 g 00 = ρ ( r ) e ν ( r ) c 2 T_{00}=\rho(r)c^{2}g_{00}=\rho(r)e^{\nu(r)}c^{2}\;
  15. T 11 = P ( r ) g 11 = - P ( r ) e λ ( r ) T_{11}=P(r)g_{11}=-P(r)e^{\lambda(r)}\;
  16. ρ ( r ) \rho(r)
  17. P ( r ) P(r)
  18. G 11 = - r ν ( r ) + e λ ( r ) - 1 r 2 G_{11}=\frac{-r\nu^{\prime}(r)+e^{\lambda(r)}-1}{r^{2}}\;
  19. 8 π G c 4 T 11 = G 11 \frac{8\pi G}{c^{4}}T_{11}=G_{11}\;
  20. - 8 π G c 4 P ( r ) e λ ( r ) = - r ν ( r ) + e λ ( r ) - 1 r 2 -\frac{8\pi G}{c^{4}}P(r)e^{\lambda(r)}=\frac{-r\nu^{\prime}(r)+e^{\lambda(r)}% -1}{r^{2}}\;
  21. ν ( r ) \nu^{\prime}(r)
  22. d ν ( r ) d r = 1 r ( 8 π G r 2 e λ ( r ) P ( r ) c 4 + e λ ( r ) - 1 ) \frac{d\nu(r)}{dr}=\frac{1}{r}\left(\frac{8\pi Gr^{2}e^{\lambda(r)}P(r)}{c^{4}% }+e^{\lambda(r)}-1\right)\;
  23. d P ( r ) d r = - ( T 00 g 00 + T 11 g 11 2 ) d ν ( r ) d r = - ( ρ ( r ) c 2 + P ( r ) 2 ) d ν ( r ) d r \frac{dP(r)}{dr}=-\left(\frac{T_{00}g^{00}+T_{11}g^{11}}{2}\right)\frac{d\nu(r% )}{dr}=-\left(\frac{\rho(r)c^{2}+P(r)}{2}\right)\frac{d\nu(r)}{dr}\;
  24. T 00 T_{00}
  25. T 11 T_{11}
  26. g 00 g^{00}
  27. g 11 g^{11}
  28. g 00 = ( g 00 ) - 1 = ( e ν ( r ) ) - 1 = e - ν ( r ) g^{00}=\left(g_{00}\right)^{-1}=\left(e^{\nu(r)}\right)^{-1}=e^{-\nu(r)}\;
  29. g 11 = ( g 11 ) - 1 = ( - e λ ( r ) ) - 1 = - e - λ ( r ) g^{11}=\left(g_{11}\right)^{-1}=\left(-e^{\lambda(r)}\right)^{-1}=-e^{-\lambda% (r)}\;
  30. 8 π G c 4 T β α = G β α \frac{8\pi G}{c^{4}}T^{\alpha}_{\beta}=G^{\alpha}_{\beta}
  31. T r r = T θ θ = T ϕ ϕ = ± P T^{r}_{r}=T^{\theta}_{\theta}=T^{\phi}_{\phi}={\pm}P
  32. T r r = T θ θ T^{r}_{r}=T^{\theta}_{\theta}
  33. ν , λ , λ \nu^{\prime},\lambda^{\prime},\lambda
  34. r r
  35. d P ( r ) d r \frac{dP(r)}{dr}
  36. ν , λ , λ \nu^{\prime},\lambda^{\prime},\lambda
  37. r r
  38. d ν ( r ) d r = - d P ( r ) d r ( 2 ρ ( r ) c 2 + P ( r ) ) = 1 r ( 8 π G r 2 e λ ( r ) P ( r ) c 4 + e λ ( r ) - 1 ) \frac{d\nu(r)}{dr}=-\frac{dP(r)}{dr}\left(\frac{2}{\rho(r)c^{2}+P(r)}\right)=% \frac{1}{r}\left(\frac{8\pi Gr^{2}e^{\lambda(r)}P(r)}{c^{4}}+e^{\lambda(r)}-1% \right)\;
  39. P ( r ) P^{\prime}(r)
  40. d P ( r ) d r = - ( ρ ( r ) c 2 + P ( r ) ) ( ( e λ ( r ) - 1 ) c 4 + 8 π G r 2 e λ ( r ) P ( r ) ) 2 c 4 r \frac{dP(r)}{dr}=-\frac{\left(\rho(r)c^{2}+P(r)\right)\left((e^{\lambda(r)}-1)% c^{4}+8\pi Gr^{2}e^{\lambda(r)}P(r)\right)}{2c^{4}r}\;
  41. r s = 2 G M ( r ) c 2 r_{s}=\frac{2GM(r)}{c^{2}}\;
  42. e λ ( r ) - 1 = r s r - r s e^{\lambda(r)}-1=\frac{r_{s}}{r-r_{s}}\;
  43. e λ ( r ) = ( 1 - r s r ) - 1 e^{\lambda(r)}=\left(1-\frac{r_{s}}{r}\right)^{-1}\;
  44. d P ( r ) d r = - ( ρ ( r ) c 2 + P ( r ) ) ( c 4 r s + 8 π G r 3 P ( r ) ) 2 c 4 r ( r - r s ) \frac{dP(r)}{dr}=-\frac{\left(\rho(r)c^{2}+P(r)\right)\left(c^{4}r_{s}+8\pi Gr% ^{3}P(r)\right)}{2c^{4}r\left(r-r_{s}\right)}\;
  45. 2 G c 2 2Gc^{2}
  46. c 2 c^{2}
  47. d P ( r ) d r = - G ( ρ ( r ) c 2 + P ( r ) ) ( M ( r ) + 4 π r 3 P ( r ) c 2 ) r ( c 2 r - 2 G M ( r ) ) \frac{dP(r)}{dr}=-\frac{G\left(\rho(r)c^{2}+P(r)\right)\left(M(r)+4\pi r^{3}% \frac{P(r)}{c^{2}}\right)}{r\left(c^{2}r-2GM(r)\right)}\;
  48. r ( r - r s ) = r 2 ( 1 - r s r ) r(r-r_{s})=r^{2}\left(1-\frac{r_{s}}{r}\right)\;
  49. c 2 c^{2}
  50. d P ( r ) d r = - G r 2 ( ρ ( r ) + P ( r ) c 2 ) ( M ( r ) + 4 π r 3 P ( r ) c 2 ) ( 1 - 2 G M ( r ) c 2 r ) - 1 \frac{dP(r)}{dr}=-\frac{G}{r^{2}}\left(\rho(r)+\frac{P(r)}{c^{2}}\right)\left(% M(r)+4\pi r^{3}\frac{P(r)}{c^{2}}\right)\left(1-\frac{2GM(r)}{c^{2}r}\right)^{% -1}\;

Tonelli's_theorem_(functional_analysis).html

  1. F [ u ] = Ω f ( u ( x ) ) d x . F[u]=\int_{\Omega}f(u(x))\,\mathrm{d}x.
  2. m u f ( u ) { ± } \mathbb{R}^{m}\ni u\mapsto f(u)\in\mathbb{R}\cup\{\pm\infty\}

Topological_derivative.html

  1. Ω \Omega
  2. d \mathbb{R}^{d}
  3. d 2 d\geq 2
  4. ω ε ( x ~ ) = x ~ + ε ω \omega_{\varepsilon}(\tilde{x})=\tilde{x}+\varepsilon\omega
  5. ε \varepsilon
  6. x ~ \tilde{x}
  7. Ω \Omega
  8. ω \omega
  9. d \mathbb{R}^{d}
  10. Ψ \Psi
  11. Ψ ε \Psi_{\varepsilon}
  12. Ω ε = Ω \ ω ε ¯ \Omega_{\varepsilon}=\Omega\backslash\overline{\omega_{\varepsilon}}
  13. Φ ( Ψ ε ( x ~ ) ) \Phi(\Psi_{\varepsilon}(\tilde{x}))
  14. Φ ( Ψ ε ( x ~ ) ) = Φ ( Ψ ) + f ( ε ) g ( x ~ ) + o ( f ( ε ) ) \Phi(\Psi_{\varepsilon}(\tilde{x}))=\Phi(\Psi)+f(\varepsilon)g(\tilde{x})+o(f(% \varepsilon))
  15. Φ ( Ψ ) \Phi(\Psi)
  16. f ( ε ) f(\varepsilon)
  17. Φ ( Ψ ) \Phi(\Psi)
  18. o ( f ( ε ) ) o(f(\varepsilon))
  19. g ( x ~ ) g(\tilde{x})
  20. Φ \Phi
  21. x ~ \tilde{x}
  22. Ω \Omega
  23. J ( Ω ) J(\Omega)
  24. O ( N l o g ( N ) ) O(Nlog(N))
  25. N N
  26. u L 2 ( Ω ) u\in L^{2}(\Omega)
  27. L u + n Lu+n
  28. E E
  29. Ω \Omega
  30. u u
  31. E E
  32. L : L 2 ( Ω ) E L:L^{2}(\Omega)\rightarrow E
  33. E E
  34. . E \|.\|_{E}
  35. u H 1 ( Ω ) u\in H^{1}(\Omega)
  36. C 1 / 2 u L 2 ( Ω ) 2 + L u - v E 2 \|C^{1/2}\nabla u\|_{L^{2}(\Omega)}^{2}+\|Lu-v\|_{E}^{2}
  37. C C
  38. u u
  39. E = L 2 ( Ω ) E=L^{2}(\Omega)
  40. L u = u Lu=u
  41. E = L 2 ( Ω ) E=L^{2}(\Omega)
  42. L u = ϕ u Lu=\phi\ast u
  43. ϕ \phi
  44. E = L 2 ( Ω \ ω ) E=L^{2}(\Omega\backslash\omega)
  45. L u = u | Ω \ ω Lu=u|_{\Omega\backslash\omega}
  46. ω Ω \omega\subset\Omega
  47. J Ω ( u Ω ) = 1 2 Ω u Ω 2 J_{\Omega}(u_{\Omega})=\frac{1}{2}\int_{\Omega}u_{\Omega}^{2}
  48. g ( x , n ) = - π c ( u 0 . n ) ( p 0 . n ) - π ( u 0 . n ) 2 g(x,n)=-\pi c(\nabla u_{0}.n)(\nabla p_{0}.n)-\pi(\nabla u_{0}.n)^{2}
  49. n n
  50. c c
  51. u 0 u_{0}
  52. p 0 p_{0}
  53. - ( c u 0 ) + L * L u 0 = L * v -\nabla(c\nabla u_{0})+L^{*}Lu_{0}=L^{*}v
  54. Ω \Omega
  55. n u 0 = 0 \partial_{n}u_{0}=0
  56. Ω \partial\Omega
  57. - ( c p 0 ) + L * L p 0 = Δ u 0 -\nabla(c\nabla p_{0})+L^{*}Lp_{0}=\Delta u_{0}
  58. Ω \Omega
  59. n p 0 = 0 \partial_{n}p_{0}=0
  60. Ω \partial\Omega
  61. C C

Topological_entropy_in_physics.html

  1. S L α L - γ + 𝒪 ( L - ν ) , ν > 0 S_{L}\;\longrightarrow\;\alpha L-\gamma+\mathcal{O}(L^{-\nu})\;,\qquad\nu>0\,\!

Topologically_stratified_space.html

  1. = X - 1 X 0 X 1 X n = X \emptyset=X_{-1}\subset X_{0}\subset X_{1}\ldots\subset X_{n}=X
  2. X i X i - 1 X_{i}\smallsetminus X_{i-1}
  3. U X U\subset X
  4. U i × C L U\cong\mathbb{R}^{i}\times CL
  5. C L CL
  6. X i X i - 1 X_{i}\smallsetminus X_{i-1}

Torsion-free_abelian_group.html

  1. G , * \langle G,*\rangle
  2. a + a + + a a+a+\cdots+a

Total_dynamic_head.html

  1. h total = P 2 - P 1 ρ g {\rm h_{total}=\frac{P_{2}-P_{1}}{\rho g}}

Total_electron_content.html

  1. τ p iono = - κ TEC f 2 \tau_{p}^{\mathrm{iono}}=-\kappa\frac{\mathrm{TEC}}{f^{2}}
  2. τ g iono = - τ p iono \tau_{g}^{\mathrm{iono}}=-\tau_{p}^{\mathrm{iono}}
  3. κ = q 2 / ( 8 π 2 m e ϵ 0 ) = c 2 r e / ( 2 π ) \kappa=q^{2}/(8\pi^{2}m_{e}\epsilon_{0})=c^{2}r_{e}/(2\pi)

Total_functional_programming.html

  1. x . x ÷ 0 = 0 \forall x\in\mathbb{N}.x\div 0=0

Total_pressure.html

  1. p 0 p_{0}
  2. p 0 = p + q + ρ g z p_{0}=p+q+\rho gz\,
  3. p 0 = p + q p_{0}=p+q\,

Tower_Mounted_Amplifier.html

  1. F 1 F_{1}
  2. G 1 G_{1}
  3. S y s t e m N o i s e F i g u r e = F 1 + F 2 - 1 G 1 + F 3 - 1 G 1 × G 2 + + F n - 1 G 1 × G 2 × G 3 × × G n - 1 SystemNoiseFigure=F_{1}+\frac{F_{2}-1}{G_{1}}+\frac{F_{3}-1}{G_{1}\times G_{2}% }+\cdots+\frac{F_{n}-1}{G_{1}\times G_{2}\times G_{3}\times\cdots\times G_{n-1}}
  4. F 1 F_{1}
  5. F 2 F_{2}
  6. G 1 G_{1}
  7. G 1 G_{1}
  8. G 2 G_{2}
  9. S y s t e m N o i s e F i g u r e = F 1 + F 2 - 1 G 1 SystemNoiseFigure=F_{1}+\frac{F_{2}-1}{G_{1}}
  10. F 1 F_{1}
  11. G 1 G_{1}
  12. G 1 G_{1}
  13. F 2 - 1 F_{2}-1
  14. G 1 G_{1}
  15. F 2 F_{2}
  16. F 1 F_{1}
  17. G 1 G_{1}
  18. F 2 F_{2}
  19. G 2 G_{2}
  20. F 3 F_{3}
  21. G 3 G_{3}
  22. F 3 F_{3}
  23. G 2 G_{2}
  24. G 1 1 G_{1}\approx 1
  25. G 2 G_{2}
  26. G 3 G_{3}
  27. G 2 G_{2}
  28. G 3 G_{3}
  29. S y s t e m N o i s e F i g u r e = F 1 + F 2 - 1 1 + F 3 - 1 1 × G 2 + F 4 - 1 1 × G 2 × G 3 SystemNoiseFigure=F_{1}+\frac{F_{2}-1}{1}+\frac{F_{3}-1}{1\times G_{2}}+\frac{% F_{4}-1}{1\times G_{2}\times G_{3}}
  30. N o i s e F i g u r e ( i n d B ) = 10 × log 10 ( F s ) NoiseFigure(indB)=10\times\log_{10}(Fs)

Townsend_(unit).html

  1. E E
  2. N N
  3. 1 Td = 10 - 21 V m 2 = 10 - 17 V cm 2 . 1\,{\rm Td}=10^{-21}\,{\rm V\cdot m^{2}}=10^{-17}\,{\rm V\cdot cm^{2}}.
  4. E = 2.5 10 4 V / m E=2.5\cdot 10^{4}\,{\rm V/m}
  5. N = 2.5 10 25 m - 3 N=2.5\cdot 10^{25}\,{\rm m^{-3}}
  6. E / N = 10 - 21 V m 2 E/N=10^{-21}\,{\rm V\cdot m^{2}}
  7. 1 Td 1\,{\rm Td}
  8. E / N E/N
  9. E E
  10. N N
  11. N N
  12. E E

Törnqvist_index.html

  1. p i , t - 1 p_{i,t-1}
  2. q i , t q_{i,t}
  3. P t P_{t}
  4. P t P t - 1 = i = 1 n ( p i t p i , t - 1 ) 1 2 [ p i , t - 1 q i , t - 1 j = 1 n ( p j , t - 1 q j , t - 1 ) + p i , t q i , t j = 1 n ( p j , t q j , t ) ] \frac{P_{t}}{P_{t-1}}=\prod_{i=1}^{n}\left(\frac{p_{it}}{p_{i,t-1}}\right)^{% \frac{1}{2}\left[\frac{p_{i,t-1}q_{i,t-1}}{\sum_{j=1}^{n}\left(p_{j,t-1}q_{j,t% -1}\right)}+\frac{p_{i,t}q_{i,t}}{\sum_{j=1}^{n}\left(p_{j,t}q_{j,t}\right)}% \right]}
  5. p t - 1 p_{t-1}
  6. q t - 1 q_{t-1}
  7. p t p_{t}
  8. q t q_{t}
  9. P t P t - 1 = i = 1 n ( p i t p i , t - 1 ) 1 2 [ p i , t - 1 q i , t - 1 p t - 1 q t - 1 + p i , t q i , t p t q t ] \frac{P_{t}}{P_{t-1}}=\prod_{i=1}^{n}\left(\frac{p_{it}}{p_{i,t-1}}\right)^{% \frac{1}{2}\left[\frac{p_{i,t-1}q_{i,t-1}}{p_{t-1}\cdot q_{t-1}}+\frac{p_{i,t}% q_{i,t}}{p_{t}\cdot q_{t}}\right]}
  10. P t P_{t}
  11. l n P t P t - 1 = 1 2 i = 1 n ( p i , t - 1 q i , t - 1 p t - 1 q t - 1 + p i , t q i , t p t q t ) l n ( p i , t p i , t - 1 ) ln\frac{P_{t}}{P_{t-1}}=\frac{1}{2}\sum_{i=1}^{n}\left(\frac{p_{i,t-1}q_{i,t-1% }}{p_{t-1}q_{t-1}}+\frac{p_{i,t}q_{i,t}}{p_{t}q_{t}}\right)ln\left(\frac{p_{i,% t}}{p_{i,t-1}}\right)
  12. Q t Q t - 1 = i = 1 n ( q i , t q i , t - 1 ) 1 2 [ p i , t - 1 q i , t - 1 j = 1 n ( p j , t - 1 q j , t - 1 ) + p i , t q i , t j = 1 n ( p j , t q j , t ) ] \frac{Q_{t}}{Q_{t-1}}=\prod_{i=1}^{n}\left(\frac{q_{i,t}}{q_{i,t-1}}\right)^{% \frac{1}{2}\left[\frac{p_{i,t-1}q_{i,t-1}}{\sum_{j=1}^{n}\left(p_{j,t-1}q_{j,t% -1}\right)}+\frac{p_{i,t}q_{i,t}}{\sum_{j=1}^{n}\left(p_{j,t}q_{j,t}\right)}% \right]}

Trace_monoid.html

  1. Σ * \Sigma^{*}
  2. Σ \Sigma
  3. I I
  4. Σ \Sigma
  5. \sim
  6. Σ * \Sigma^{*}
  7. u v u\sim v\,
  8. x , y Σ * x,y\in\Sigma^{*}
  9. ( a , b ) I (a,b)\in I
  10. u = x a b y u=xaby
  11. v = x b a y v=xbay
  12. u , v , x u,v,x
  13. y y
  14. Σ * \Sigma^{*}
  15. a a
  16. b b
  17. Σ \Sigma
  18. \sim
  19. Σ * \Sigma^{*}
  20. D \equiv_{D}
  21. u v u\equiv v
  22. ( w 0 , w 1 , , w n ) (w_{0},w_{1},\cdots,w_{n})
  23. u w 0 u\sim w_{0}\,
  24. v w n v\sim w_{n}\,
  25. w i w i + 1 w_{i}\sim w_{i+1}\,
  26. 0 i < n 0\leq i<n
  27. 𝕄 ( D ) \mathbb{M}(D)
  28. 𝕄 ( D ) = Σ * / D . \mathbb{M}(D)=\Sigma^{*}/\equiv_{D}.
  29. ϕ D : Σ * 𝕄 ( D ) \phi_{D}:\Sigma^{*}\to\mathbb{M}(D)
  30. Σ = { a , b , c } \Sigma=\{a,b,c\}
  31. D = { a , b } × { a , b } { a , c } × { a , c } = { a , b } 2 { a , c } 2 = { ( a , b ) , ( b , a ) , ( a , c ) , ( c , a ) , ( a , a ) , ( b , b ) , ( c , c ) } \begin{matrix}D&=&\{a,b\}\times\{a,b\}\quad\cup\quad\{a,c\}\times\{a,c\}\\ &=&\{a,b\}^{2}\cup\{a,c\}^{2}\\ &=&\{(a,b),(b,a),(a,c),(c,a),(a,a),(b,b),(c,c)\}\end{matrix}
  32. I D = { ( b , c ) , ( c , b ) } I_{D}=\{(b,c)\,,\,(c,b)\}
  33. b , c b,c
  34. a b a b a b b c a abababbca
  35. [ a b a b a b b c a ] D = { a b a b a b b c a , a b a b a b c b a , a b a b a c b b a } [abababbca]_{D}=\{abababbca\,,\;abababcba\,,\;ababacbba\}
  36. [ a b a b a b b c a ] D [abababbca]_{D}
  37. w v w\equiv v
  38. ( w ÷ a ) ( v ÷ a ) (w\div a)\equiv(v\div a)
  39. w ÷ a w\div a
  40. w v w\equiv v
  41. x w y x v y xwy\equiv xvy
  42. u a v b ua\equiv vb
  43. a b a\neq b
  44. ( a , b ) I D (a,b)\in I_{D}
  45. u = w b u=wb
  46. v = w a v=wa
  47. w v w\equiv v
  48. π Σ ( w ) π Σ ( v ) \pi_{\Sigma}(w)\equiv\pi_{\Sigma}(v)
  49. u v x y uv\equiv xy
  50. z 1 , z 2 , z 3 z_{1},z_{2},z_{3}
  51. z 4 z_{4}
  52. ( w 2 , w 3 ) I D (w_{2},w_{3})\in I_{D}
  53. w 2 Σ w_{2}\in\Sigma
  54. w 3 Σ w_{3}\in\Sigma
  55. w 2 w_{2}
  56. z 2 z_{2}
  57. w 3 w_{3}
  58. z 3 z_{3}
  59. u z 1 z 2 , v z 3 z 4 , u\equiv z_{1}z_{2},\qquad v\equiv z_{3}z_{4},
  60. x z 1 z 3 , y z 2 z 4 . x\equiv z_{1}z_{3},\qquad y\equiv z_{2}z_{4}.
  61. ψ : Σ * M \psi:\Sigma^{*}\to M\,
  62. ψ ( w ) = ψ ( ε ) \psi(w)=\psi(\varepsilon)
  63. w = ε w=\varepsilon
  64. ( a , b ) I D (a,b)\in I_{D}
  65. ψ ( a b ) = ψ ( b a ) \psi(ab)=\psi(ba)\,
  66. ψ ( u a ) = ψ ( v ) \psi(ua)=\psi(v)\,
  67. ψ ( u ) = ψ ( v ÷ a ) \psi(u)=\psi(v\div a)
  68. ψ ( u a ) = ψ ( v b ) \psi(ua)=\psi(vb)\,
  69. a b a\neq b
  70. ( a , b ) I D (a,b)\in I_{D}
  71. ψ : Σ * M \psi:\Sigma^{*}\to M\,
  72. 𝕄 ( D ) \mathbb{M}(D)
  73. Σ * \Sigma^{*}
  74. 𝕄 ( D ) \mathbb{M}(D)
  75. L Σ * L\subset\Sigma^{*}
  76. L = [ L ] D L=\bigcup[L]_{D}
  77. [ L ] D = { [ w ] D | w L } [L]_{D}=\{[w]_{D}|w\in L\}
  78. T = { w | [ w ] D T } \bigcup T=\{w|[w]_{D}\in T\}

Trace_operator.html

  1. Ω \Omega
  2. n \mathbb{R}^{n}
  3. Ω . \partial\Omega.
  4. u u
  5. C 1 C^{1}
  6. Ω ¯ \bar{\Omega}
  7. Ω , \Omega,
  8. Ω . \partial\Omega.
  9. u u
  10. Ω \partial\Omega
  11. Ω . \Omega.
  12. u u
  13. u u
  14. ( u n ) (u_{n})
  15. C 1 C^{1}
  16. Ω . \Omega.
  17. u | Ω u_{|\partial\Omega}
  18. u u
  19. Ω \partial\Omega
  20. ( u n | Ω ) (u_{n|\partial\Omega})
  21. p 1 p\geq 1
  22. T : C 1 ( Ω ¯ ) L p ( Ω ) T:C^{1}(\bar{\Omega})\to L^{p}(\partial\Omega)
  23. C 1 C^{1}
  24. Ω \Omega
  25. L p ( Ω ) , L^{p}(\partial\Omega),
  26. T u = u | Ω . Tu=u_{|\partial\Omega}.\,
  27. T T
  28. W 1 , p ( Ω ) . W^{1,p}(\Omega).
  29. C C
  30. Ω \Omega
  31. p , p,
  32. T u L p ( Ω ) C u W 1 , p ( Ω ) \|Tu\|_{L^{p}(\partial\Omega)}\leq C\|u\|_{W^{1,p}(\Omega)}
  33. u u
  34. C 1 ( Ω ¯ ) . C^{1}(\bar{\Omega}).
  35. C 1 C^{1}
  36. Ω ¯ \bar{\Omega}
  37. W 1 , p ( Ω ) W^{1,p}(\Omega)
  38. T T
  39. T : W 1 , p ( Ω ) L p ( Ω ) T:W^{1,p}(\Omega)\to L^{p}(\partial\Omega)\,
  40. W 1 , p ( Ω ) . W^{1,p}(\Omega).
  41. T T
  42. u | Ω u_{|\partial\Omega}
  43. u u
  44. W 1 , p ( Ω ) W^{1,p}(\Omega)
  45. T u . Tu.
  46. u u
  47. W 1 , p ( Ω ) , W^{1,p}(\Omega),
  48. ( u n ) (u_{n})
  49. C 1 C^{1}
  50. Ω ¯ , \bar{\Omega},
  51. u n u_{n}
  52. u u
  53. W 1 , p ( Ω ) . W^{1,p}(\Omega).
  54. u n | Ω u_{n|\partial\Omega}
  55. L p ( Ω ) . L^{p}(\partial\Omega).
  56. u | Ω = lim n u n | Ω . u_{|\partial\Omega}=\lim_{n\to\infty}u_{n\,|\partial\Omega}.\,
  57. ( u n ) (u_{n})
  58. u . u.
  59. { - Δ u = f in Ω u | Ω = 0. \begin{cases}-\Delta u=f\,\text{ in }\Omega\\ u_{|\partial\Omega}=0.\end{cases}
  60. f f
  61. Ω ¯ . \bar{\Omega}.
  62. H 0 1 ( Ω ) H^{1}_{0}(\Omega)
  63. W 1 , 2 ( Ω ) W^{1,2}(\Omega)
  64. H 1 ( Ω ) H^{1}(\Omega)
  65. u u
  66. H 0 1 ( Ω ) H^{1}_{0}(\Omega)
  67. Ω u ( x ) v ( x ) d x = Ω f ( x ) v ( x ) d x \int_{\Omega}\!\nabla u(x)\cdot\nabla v(x)\,dx=\int_{\Omega}\!f(x)v(x)\,dx
  68. v v
  69. H 0 1 ( Ω ) . H^{1}_{0}(\Omega).

Trade-weighted_US_dollar_index.html

  1. t t
  2. I t = I t - 1 × j = 1 N ( t ) ( e j , t e j , t - 1 ) w j , t I_{t}=I_{t-1}\times\prod_{j=1}^{N(t)}\left(\frac{e_{j,t}}{e_{j,t-1}}\right)^{w% _{j,t}}
  3. I t I_{t}
  4. I t - 1 I_{t-1}
  5. t t
  6. t - 1 t-1
  7. N ( t ) N(t)
  8. t t
  9. e j , t e_{j,t}
  10. e j , t - 1 e_{j,t-1}
  11. j j
  12. t t
  13. t - 1 t-1
  14. w j , t w_{j,t}
  15. j j
  16. t t
  17. j = 1 N ( t ) w j , t = 1 \sum_{j=1}^{N(t)}w_{j,t}=1
  18. I t = I t - 1 × j = 1 N ( t ) ( e j , t p t p j , t e j , t - 1 p t - 1 p j , t - 1 ) w j , t I_{t}=I_{t-1}\times\prod_{j=1}^{N(t)}\left(\frac{e_{j,t}\cdot\frac{p_{t}}{p_{j% ,t}}}{e_{j,t-1}\cdot\frac{p_{t-1}}{p_{j,t-1}}}\right)^{w_{j,t}}
  19. p t p_{t}
  20. p t - 1 p_{t-1}
  21. t t
  22. t - 1 t-1
  23. p j , t p_{j,t}
  24. p j , t - 1 p_{j,t-1}
  25. j j
  26. t t
  27. t - 1 t-1

Transaldolase.html

  1. \rightleftharpoons

Transfer-matrix_method.html

  1. 𝒵 = 𝐯 0 { k = 1 N 𝐖 k } 𝐯 N + 1 \mathcal{Z}=\mathbf{v}_{0}\cdot\left\{\prod_{k=1}^{N}\mathbf{W}_{k}\right\}% \cdot\mathbf{v}_{N+1}
  2. 𝒵 = tr { k = 1 N 𝐖 k } \mathcal{Z}=\mathrm{tr}\left\{\prod_{k=1}^{N}\mathbf{W}_{k}\right\}
  3. m m
  4. Pr m ( x ) = tr [ k = 1 x 𝐖 k 𝐏𝐣 k = x + 1 N 𝐖 k ] tr [ k = 1 N 𝐖 k ] \mathrm{Pr_{m}(x)}=\frac{\mathrm{tr}\left[\prod_{k=1}^{x}\mathbf{W}_{k}\mathbf% {Pj}\prod_{k^{\prime}=x+1}^{N}\mathbf{W}_{k^{\prime}}\right]}{\mathrm{tr}\left% [\prod_{k=1}^{N}\mathbf{W}_{k}\right]}
  5. P j Pj
  6. m m
  7. P j μ ν = δ μ ν δ μ m Pj_{\mu\nu}=\delta_{\mu\nu}\delta_{\mu m}

Transition_dipole_moment.html

  1. 𝐝 n m \scriptstyle{\mathbf{d}_{nm}}
  2. m \scriptstyle{m}
  3. n \scriptstyle{n}
  4. | ψ a |\psi_{a}\rangle
  5. | ψ b |\psi_{b}\rangle
  6. (t.d.m.) \,\text{(t.d.m.)}
  7. ( t.d.m. a b ) = ψ a | ( q 𝐫 ) | ψ b = q ψ a * ( 𝐫 ) 𝐫 ψ b ( 𝐫 ) d 3 𝐫 (\,\text{t.d.m. }a\rightarrow b)=\langle\psi_{a}|(q\mathbf{r})|\psi_{b}\rangle% =q\int\psi_{a}^{*}(\mathbf{r})\,\mathbf{r}\,\psi_{b}(\mathbf{r})\,d^{3}\mathbf% {r}
  8. d 3 𝐫 \int d^{3}\mathbf{r}
  9. d x d y d z \iiint dx\,dy\,dz
  10. ( x-component of t.d.m. a b ) = ψ a | ( q x ) | ψ b = q ψ a * ( 𝐫 ) x ψ b ( 𝐫 ) d 3 𝐫 (\,\text{x-component of t.d.m. }a\rightarrow b)=\langle\psi_{a}|(qx)|\psi_{b}% \rangle=q\int\psi_{a}^{*}(\mathbf{r})\,x\,\psi_{b}(\mathbf{r})\,d^{3}\mathbf{r}
  11. ( t.d.m. a b ) = ψ a | ( q 1 𝐫 1 + q 2 𝐫 2 + ) | ψ b = (\,\text{t.d.m. }a\rightarrow b)=\langle\psi_{a}|(q_{1}\mathbf{r}_{1}+q_{2}% \mathbf{r}_{2}+\cdots)|\psi_{b}\rangle=
  12. = ψ a * ( 𝐫 1 , 𝐫 2 , ) ( q 1 𝐫 1 + q 2 𝐫 2 + ) ψ b ( 𝐫 1 , 𝐫 2 , ) d 3 𝐫 1 d 3 𝐫 2 =\int\psi_{a}^{*}(\mathbf{r}_{1},\mathbf{r}_{2},\ldots)\,(q_{1}\mathbf{r}_{1}+% q_{2}\mathbf{r}_{2}+\cdots)\,\psi_{b}(\mathbf{r}_{1},\mathbf{r}_{2},\ldots)\,d% ^{3}\mathbf{r}_{1}\,d^{3}\mathbf{r}_{2}\cdots
  13. ψ a | 𝐫 | ψ b = i ( E b - E a ) m ψ a | 𝐩 | ψ b \langle\psi_{a}|\mathbf{r}|\psi_{b}\rangle=\frac{i\hbar}{(E_{b}-E_{a})m}% \langle\psi_{a}|\mathbf{p}|\psi_{b}\rangle
  14. [ H , x ] = [ p 2 2 m + V ( x , y , z ) , x ] = [ p 2 2 m , x ] = 1 2 m ( p x [ p x , x ] + [ p x , x ] p x ) = - i p x / m [H,x]=[\frac{p^{2}}{2m}+V(x,y,z),x]=[\frac{p^{2}}{2m},x]=\frac{1}{2m}(p_{x}[p_% {x},x]+[p_{x},x]p_{x})=-i\hbar p_{x}/m
  15. ψ a | ( H x - x H ) | ψ b = - i m ψ a | p x | ψ b \langle\psi_{a}|(Hx-xH)|\psi_{b}\rangle=\frac{-i\hbar}{m}\langle\psi_{a}|p_{x}% |\psi_{b}\rangle
  16. ψ a | ( H x - x H ) | ψ b = ( ψ a | H ) x | ψ b - ψ a | x ( H | ψ b ) = ( E a - E b ) ψ a | x | ψ b \langle\psi_{a}|(Hx-xH)|\psi_{b}\rangle=(\langle\psi_{a}|H)x|\psi_{b}\rangle-% \langle\psi_{a}|x(H|\psi_{b}\rangle)=(E_{a}-E_{b})\langle\psi_{a}|x|\psi_{b}\rangle
  17. + q \scriptstyle{+q}
  18. - q \scriptstyle{-q}
  19. 𝐫 \scriptstyle{\mathbf{r}}
  20. 𝐩 = q 𝐫 \mathbf{p}=q\mathbf{r}
  21. | τ | = | q 𝐫 | | 𝐄 | sin θ |\mathbf{\tau}|=|q\mathbf{r}||\mathbf{E}|\sin\theta
  22. 𝐝 n m \scriptstyle{\mathbf{d}_{nm}}
  23. ω \scriptstyle{\omega}
  24. ω \scriptstyle{\hbar\omega}
  25. e \scriptstyle{e}
  26. 𝐑 α \scriptstyle{\mathbf{R}_{\alpha}}
  27. π \scriptstyle{\pi}
  28. π * \scriptstyle{\pi^{*}}
  29. 𝐫 \scriptstyle{\mathbf{r}}
  30. ψ 1 * μ ψ 2 d τ \int\psi_{1}^{*}\mu\psi_{2}d\tau

Transition_state_theory.html

  1. d ln K d T = Δ U R T 2 \frac{d\ln K}{dT}=\frac{\Delta U}{RT^{2}}
  2. d ln k d T = Δ E R T 2 \frac{d\ln k}{dT}=\frac{\Delta E}{RT^{2}}
  3. k = A e - E a / R T k=Ae^{-E_{a}/RT}
  4. k exp ( - Δ G R T ) k\propto\exp\left(\frac{-\Delta^{\ddagger}G^{\ominus}}{RT}\right)
  5. k exp ( Δ S R ) exp ( - Δ H R T ) k\propto\exp\left(\frac{\Delta^{\ddagger}S^{\ominus}}{R}\right)\exp\left(\frac% {-\Delta^{\ddagger}H^{\ominus}}{RT}\right)
  6. d ln k d T = a - b T R T 2 \frac{d\ln k}{dT}=\frac{a-bT}{RT^{2}}
  7. AB k 1 k - 1 A + B \mathrm{AB}\overset{k_{1}}{\underset{}{}}{k_{-1}}{\begin{smallmatrix}% \displaystyle\longrightarrow\\ \displaystyle\longleftarrow\end{smallmatrix}}\mathrm{A}+\mathrm{B}
  8. k 1 = k B T h ( 1 1 - exp ( - h ν k B T ) ) exp ( - E R T ) k_{1}=\frac{k_{\mathrm{B}}T}{h}\left(\frac{1}{1-\exp\left(\frac{-h\nu}{k_{B}T}% \right)}\right)\exp\left(\frac{-E^{\ominus}}{RT}\right)
  9. E \textstyle E^{\ominus}
  10. A + B [ AB ] P \mathrm{A}+\mathrm{B}\rightleftharpoons[\mathrm{AB}]^{\ddagger}\to\mathrm{P}
  11. [ A B r ] = [ A B l ] = 1 2 [ A B ] [AB_{\mathrm{r}}]^{\ddagger}=[AB_{\mathrm{l}}]^{\ddagger}=\frac{1}{2}[AB]^{\ddagger}
  12. K = [ A B ] [ A ] [ B ] K^{\ddagger\ominus}=\frac{[AB]^{\ddagger}}{[A][B]}
  13. [ AB ] = K [ A ] [ B ] [\mathrm{AB}]^{\ddagger}=K^{\ddagger\ominus}[\mathrm{A}][\mathrm{B}]
  14. d [ P ] d t = k [ AB ] \Dagger = k K \Dagger [ A ] [ B ] = k [ A ] [ B ] \frac{d[P]}{dt}=k^{\ddagger\ominus}[\mathrm{AB}]^{\Dagger}=k^{\ddagger}K^{% \Dagger}[A][B]=k[A][B]
  15. k = k \Dagger K \Dagger k=k^{\Dagger}K^{\Dagger}
  16. ν \nu
  17. κ \kappa
  18. k \Dagger = κ ν k^{\Dagger}=\kappa\nu
  19. K \Dagger = k B T h ν K \Dagger K^{\Dagger}=\frac{k_{B}T}{h\nu}K^{\Dagger^{\prime}}
  20. K \Dagger = e - Δ G \Dagger R T K^{\Dagger^{\prime}}=e^{\frac{-\Delta G^{\Dagger}}{RT}}
  21. k = k \Dagger K \Dagger = κ k B T h e - Δ G \Dagger R T k=k^{\Dagger}K^{\Dagger}=\kappa\frac{k_{B}T}{h}e^{\frac{-\Delta G^{\Dagger}}{% RT}}
  22. k = κ k B T h e Δ S \Dagger R e - Δ H \Dagger R T k=\kappa\frac{k_{B}T}{h}e^{\frac{\Delta S^{\Dagger}}{R}}e^{\frac{-\Delta H^{% \Dagger}}{RT}}
  23. E + I k 1 k 2 E I k 3 k 4 E I * E+I\overset{k_{1}}{\underset{}{}}{k_{2}}{\begin{smallmatrix}\displaystyle% \longrightarrow\\ \displaystyle\longleftarrow\end{smallmatrix}}EI\overset{k_{3}}{\underset{}{}}{% k_{4}}{\begin{smallmatrix}\displaystyle\longrightarrow\\ \displaystyle\longleftarrow\end{smallmatrix}}EI^{*}
  24. K i = k 2 k 1 K_{i}=\frac{k_{2}}{k_{1}}
  25. K i * = K i k 4 k 3 + k 4 K_{i}^{*}=\frac{K_{i}k_{4}}{k_{3}+k_{4}}

Transition_system.html

  1. p 𝛼 q . p\overset{\alpha}{\rightarrow}q.\,

Transverse_measure.html

  1. v + S = { v + s V | s S } v+S=\{v+s\in V|s\in S\}
  2. μ ( E ) = λ 1 ( { x 𝐑 | ( x , 0 ) E 𝐑 2 } ) . \mu(E)=\lambda^{1}\big(\{x\in\mathbf{R}|(x,0)\in E\subseteq\mathbf{R}^{2}\}% \big).

Traveler's_dilemma.html

  1. S = { 2 , , 100 } S=\{2,...,100\}
  2. F : S × S F:S\times S\rightarrow\mathbb{R}
  3. F ( x , y ) = min ( x , y ) + 2 sgn ( y - x ) F(x,y)=\min(x,y)+2\cdot\operatorname{sgn}(y-x)
  4. F ( y , x ) F(y,x)

Tree_homomorphism.html

  1. T 1 T_{1}
  2. T 2 T_{2}
  3. ϕ \phi
  4. T 1 T_{1}
  5. T 2 T_{2}
  6. ϕ \phi
  7. T 1 T_{1}
  8. T 2 T_{2}
  9. n 2 n_{2}
  10. n 1 n_{1}
  11. T 1 T_{1}
  12. ϕ ( n 2 ) \phi(n_{2})
  13. ϕ ( n 1 ) \phi(n_{1})
  14. T 2 T_{2}
  15. n T 1 n\in T_{1}
  16. n n
  17. T 1 T_{1}
  18. ϕ ( n ) \phi(n)
  19. T 2 T_{2}

Triangle_center.html

  1. { \begin{cases}\\ \\ \end{cases}
  2. { \begin{cases}\\ \end{cases}
  3. { \begin{cases}\\ \\ \end{cases}
  4. { \begin{cases}\\ \\ \end{cases}

Tribimaximal_mixing.html

  1. [ | U e 1 | 2 | U e 2 | 2 | U e 3 | 2 | U μ 1 | 2 | U μ 2 | 2 | U μ 3 | 2 | U τ 1 | 2 | U τ 2 | 2 | U τ 3 | 2 ] = [ 2 3 1 3 0 1 6 1 3 1 2 1 6 1 3 1 2 ] . \begin{bmatrix}|U_{e1}|^{2}&|U_{e2}|^{2}&|U_{e3}|^{2}\\ |U_{\mu 1}|^{2}&|U_{\mu 2}|^{2}&|U_{\mu 3}|^{2}\\ |U_{\tau 1}|^{2}&|U_{\tau 2}|^{2}&|U_{\tau 3}|^{2}\end{bmatrix}=\begin{bmatrix% }\frac{2}{3}&\frac{1}{3}&0\\ \frac{1}{6}&\frac{1}{3}&\frac{1}{2}\\ \frac{1}{6}&\frac{1}{3}&\frac{1}{2}\end{bmatrix}.
  2. θ 12 = sin - 1 ( 1 3 ) 35.3 θ 23 = 45 θ 13 = 0 δ = 0. \begin{matrix}\theta_{12}=\sin^{-1}\left({\frac{1}{\sqrt{3}}}\right)\simeq 35.% 3^{\circ}&\theta_{23}=45^{\circ}\\ \theta_{13}=0&\delta=0.\end{matrix}
  3. ν 2 \nu_{2}
  4. ν e \nu_{e}
  5. ν μ \nu_{\mu}
  6. ν τ \nu_{\tau}
  7. ν 3 \nu_{3}
  8. ν μ \nu_{\mu}
  9. ν τ \nu_{\tau}
  10. ν e \nu_{e}
  11. ν 3 \nu_{3}
  12. | U e 3 | 2 = 0 |U_{e3}|^{2}=0
  13. P e e 1 / 3 \langle P_{ee}\rangle\simeq 1/3
  14. P e e 5 / 9 \langle P_{ee}\rangle\simeq 5/9
  15. ν 3 \nu_{3}
  16. P μ μ 1 / 2 \langle P_{\mu\mu}\rangle\simeq 1/2
  17. ν e \nu_{e}
  18. ν μ \nu_{\mu}
  19. | U e 3 | 2 = 0 |U_{e3}|^{2}=0
  20. ν μ \nu_{\mu}
  21. ν τ \nu_{\tau}
  22. ( P μ μ = P τ τ 7 / 18 ) (P_{\mu\mu}=P_{\tau\tau}\simeq 7/18)
  23. U = [ cos θ sin θ 0 - sin θ / 2 cos θ / 2 1 2 sin θ / 2 - cos θ / 2 1 2 ] . U=\begin{bmatrix}\cos\theta&\sin\theta&0\\ -\sin\theta/\sqrt{2}&\cos\theta/\sqrt{2}&\frac{1}{\sqrt{2}}\\ \sin\theta/\sqrt{2}&-\cos\theta/\sqrt{2}&\frac{1}{\sqrt{2}}\end{bmatrix}.
  24. θ = sin - 1 ( 1 3 ) \theta=\sin^{-1}\left({\frac{1}{\sqrt{3}}}\right)