wpmath0000008_13

Vectorization_(mathematics).html

  1. vec ( A ) = [ a 1 , 1 , , a m , 1 , a 1 , 2 , , a m , 2 , , a 1 , n , , a m , n ] T \mathrm{vec}(A)=[a_{1,1},\ldots,a_{m,1},a_{1,2},\ldots,a_{m,2},\ldots,a_{1,n},% \ldots,a_{m,n}]^{T}
  2. a i , j a_{i,j}
  3. ( i , j ) (i,j)
  4. A A
  5. T {}^{T}
  6. 𝐑 m × n := 𝐑 m 𝐑 n 𝐑 m n \mathbf{R}^{m\times n}:=\mathbf{R}^{m}\otimes\mathbf{R}^{n}\cong\mathbf{R}^{mn}
  7. A A
  8. [ a b c d ] \begin{bmatrix}a&b\\ c&d\end{bmatrix}
  9. vec ( A ) = [ a c b d ] \mathrm{vec}(A)=\begin{bmatrix}a\\ c\\ b\\ d\end{bmatrix}
  10. vec ( A B C ) = ( C T A ) vec ( B ) \mbox{vec}~{}(ABC)=(C^{T}\otimes A)\mbox{vec}~{}(B)
  11. ad ( X ) A = A X - X A \mbox{ad}~{}_{A}(X)=AX-XA
  12. vec ( ad ( X ) A ) = ( I n A - A T I n ) vec ( X ) \mbox{vec}~{}(\mbox{ad}~{}_{A}(X))=(I_{n}\otimes A-A^{T}\otimes I_{n})\mbox{% vec}~{}(X)
  13. I n I_{n}
  14. vec ( A B C ) = ( I n A B ) vec ( C ) = ( C T B T I k ) vec ( A ) \mbox{vec}~{}(ABC)=(I_{n}\otimes AB)\mbox{vec}~{}(C)=(C^{T}B^{T}\otimes I_{k})% \mbox{vec}~{}(A)
  15. vec ( A B ) = ( I m A ) vec ( B ) = ( B T I k ) vec ( A ) \mbox{vec}~{}(AB)=(I_{m}\otimes A)\mbox{vec}~{}(B)=(B^{T}\otimes I_{k})\mbox{% vec}~{}(A)
  16. \circ
  17. \circ
  18. [ a b b d ] \begin{bmatrix}a&b\\ b&d\end{bmatrix}
  19. [ a b d ] \begin{bmatrix}a\\ b\\ d\end{bmatrix}

Vehicle_routing_problem.html

  1. C i j C_{ij}
  2. t i j t_{ij}
  3. min i V j V c i j x i j \,\text{min}\sum_{i\in V}\sum_{j\in V}c_{ij}x_{ij}
  4. i V x i j = 1 j V \ { 0 } \sum_{i\in V}x_{ij}=1~{}~{}~{}~{}\forall j\in V\backslash\left\{0\right\}
  5. j V x i j = 1 i V \ { 0 } \sum_{j\in V}x_{ij}=1~{}~{}~{}~{}\forall i\in V\backslash\left\{0\right\}
  6. i V x i 0 = K \sum_{i\in V}x_{i0}=K
  7. j V x 0 j = K \sum_{j\in V}x_{0j}=K
  8. i V j S x i j r ( s ) S V \ { 0 } , S \sum_{i\notin V}\sum_{j\in S}x_{ij}\geq r(s)~{}~{}~{}~{}\forall S\subseteq V% \backslash\left\{0\right\},S\neq\emptyset
  9. x i j 0 , 1 i , j V x_{ij}\in{0,1}~{}~{}~{}~{}\forall i,j\in V
  10. i S j S x i j | S | - r ( s ) \sum_{i\in S}\sum_{j\in S}x_{ij}\leq|S|-r(s)
  11. u i - u j + C x i j C - d j i , j V \ { 0 } , i j s.t. d i + d j C u_{i}-u_{j}+Cx_{ij}\leq C-d_{j}~{}~{}~{}~{}~{}~{}\forall i,j\in V\backslash\{0% \},i\neq j~{}~{}~{}~{}\,\text{s.t. }d_{i}+d_{j}\leq C
  12. d i u i C i V \ { 0 } d_{i}\leq u_{i}\leq C~{}~{}~{}~{}~{}~{}\forall i\in V\backslash\{0\}
  13. u i , i V \ { 0 } u_{i},~{}i\in V\backslash\{0\}
  14. x i j = 0 x_{ij}=0
  15. u i C u_{i}\leq C
  16. u j d j u_{j}\geq d_{j}
  17. x i j = 1 x_{ij}=1
  18. u j u i + d j u_{j}\geq u_{i}+d_{j}

Verdier_duality.html

  1. f : X Y f:X\to Y
  2. \mathcal{F}
  3. 𝒢 \mathcal{G}
  4. [ R f ! , 𝒢 ] [ , f ! 𝒢 ] . [Rf_{!}\mathcal{F},\mathcal{G}]\cong[\mathcal{F},f^{!}\mathcal{G}].\,\!
  5. R o m ( R f ! , 𝒢 ) R f R o m ( , f ! 𝒢 ) R\,\mathcal{H}om(Rf_{!}\mathcal{F},\mathcal{G})\cong Rf_{\ast}R\,\mathcal{H}om% (\mathcal{F},f^{!}\mathcal{G})
  6. ω X = p ! ( k ) , \omega_{X}=p^{!}(k),\,\!
  7. D : D b ( X ) D b ( X ) D\colon D^{b}(X)\to D^{b}(X)\,\!
  8. D ( ) = R o m ( , ω X ) . D(\mathcal{F})=R\,\mathcal{H}om(\mathcal{F},\omega_{X}).\,\!
  9. [ R p ! k X , k ] [ k X , p ! k ] . [Rp_{!}k_{X},k]\cong[k_{X},p^{!}k].\,\!
  10. k X I X = I X 0 I X 1 k_{X}\to I^{\bullet}_{X}=I^{0}_{X}\to I^{1}_{X}\to\cdots
  11. R p ! k X = p ! I X = Γ c ( X ; I X ) Rp_{!}k_{X}=p_{!}I^{\bullet}_{X}=\Gamma_{c}(X;I^{\bullet}_{X})
  12. Hom ( Γ c ( X ; I X ) , k ) = Γ c ( X ; I X 2 ) Γ c ( X ; I X 1 ) Γ c ( X ; I X 0 ) 0 \mathrm{Hom}^{\bullet}(\Gamma_{c}(X;I^{\bullet}_{X}),k)=\cdots\to\Gamma_{c}(X;% I^{2}_{X})^{\vee}\to\Gamma_{c}(X;I^{1}_{X})^{\vee}\to\Gamma_{c}(X;I^{0}_{X})^{% \vee}\to 0
  13. [ R p ! k X , k ] H 0 ( Hom ( Γ c ( X ; I X ) , k ) ) = H c 0 ( X ; k X ) . [Rp_{!}k_{X},k]\cong H^{0}(\mathrm{Hom}^{\bullet}(\Gamma_{c}(X;I^{\bullet}_{X}% ),k))=H^{0}_{c}(X;k_{X})^{\vee}.
  14. p ! k = k X [ n ] , p^{!}k=k_{X}[n],
  15. [ k X , k X [ n ] ] H n ( Hom ( k X , k X ) ) = H n ( X ; k X ) . [k_{X},k_{X}[n]]\cong H^{n}(\mathrm{Hom}^{\bullet}(k_{X},k_{X}))=H^{n}(X;k_{X}).
  16. H c 0 ( X ; k X ) H n ( X ; k X ) . H^{0}_{c}(X;k_{X})^{\vee}\cong H^{n}(X;k_{X}).
  17. H c i ( X ; k X ) H n - i ( X ; k X ) . H^{i}_{c}(X;k_{X})^{\vee}\cong H^{n-i}(X;k_{X}).

Version_space_learning.html

  1. H 1 H 2 H n H_{1}\lor H_{2}\lor...\lor H_{n}
  2. n n
  3. x x
  4. x x

Vertical_tangent.html

  1. lim h 0 f ( a + h ) - f ( a ) h = + or lim h 0 f ( a + h ) - f ( a ) h = - . \lim_{h\to 0}\frac{f(a+h)-f(a)}{h}={+\infty}\quad\,\text{or}\quad\lim_{h\to 0}% \frac{f(a+h)-f(a)}{h}={-\infty}.
  2. lim x a f ( x ) = + , \lim_{x\to a}f^{\prime}(x)={+\infty}\,\text{,}
  3. lim x a f ( x ) = - , \lim_{x\to a}f^{\prime}(x)={-\infty}\,\text{,}
  4. lim h 0 - f ( a + h ) - f ( a ) h = + and lim h 0 + f ( a + h ) - f ( a ) h = - , \lim_{h\to 0^{-}}\frac{f(a+h)-f(a)}{h}={+\infty}\quad\,\text{and}\quad\lim_{h% \to 0^{+}}\frac{f(a+h)-f(a)}{h}={-\infty}\,\text{,}
  5. lim x a - f ( x ) = - and lim x a + f ( x ) = + , \lim_{x\to a^{-}}f^{\prime}(x)={-\infty}\quad\,\text{and}\quad\lim_{x\to a^{+}% }f^{\prime}(x)={+\infty}\,\text{,}
  6. \infty
  7. - -\infty
  8. f ( x ) = x 3 f(x)=\sqrt[3]{x}
  9. lim x 0 f ( x ) = lim x 0 1 x 2 3 = . \lim_{x\to 0}f^{\prime}(x)\;=\;\lim_{x\to 0}\frac{1}{\sqrt[3]{x^{2}}}\;=\;\infty.
  10. g ( x ) = x 2 3 g(x)=\sqrt[3]{x^{2}}
  11. lim x 0 - g ( x ) = lim x 0 - 1 x 3 = - , \lim_{x\to 0^{-}}g^{\prime}(x)\;=\;\lim_{x\to 0^{-}}\frac{1}{\sqrt[3]{x}}\;=\;% {-\infty}\,\text{,}
  12. lim x 0 + g ( x ) = lim x 0 + 1 x 3 = + . \lim_{x\to 0^{+}}g^{\prime}(x)\;=\;\lim_{x\to 0^{+}}\frac{1}{\sqrt[3]{x}}\;=\;% {+\infty}\,\text{.}

Virial_stress.html

  1. τ i j = 1 Ω k Ω ( - m ( k ) ( u i ( k ) - u ¯ i ) ( u j ( k ) - u ¯ j ) + 1 2 Ω ( x i ( ) - x i ( k ) ) f j ( k ) ) \tau_{ij}=\frac{1}{\Omega}\sum_{k\in\Omega}\left(-m^{(k)}(u_{i}^{(k)}-\bar{u}_% {i})(u_{j}^{(k)}-\bar{u}_{j})+\frac{1}{2}\sum_{\ell\in\Omega}(x_{i}^{(\ell)}-x% _{i}^{(k)})f_{j}^{(k\ell)}\right)
  2. k k
  3. \ell
  4. Ω \Omega
  5. m ( k ) m^{(k)}
  6. u i ( k ) u_{i}^{(k)}
  7. u ¯ j \bar{u}_{j}
  8. x i ( k ) x_{i}^{(k)}
  9. f i ( k ) f_{i}^{(k\ell)}
  10. k k
  11. \ell
  12. τ i j = 1 2 Ω k , Ω ( x i ( ) - x i ( k ) ) f j ( k ) \tau_{ij}=\frac{1}{2\Omega}\sum_{k,\ell\in\Omega}(x_{i}^{(\ell)}-x_{i}^{(k)})f% _{j}^{(k\ell)}

Virtual_screening.html

  1. O ( N 2 ) O(N^{2})

Visibility_Scorecard.html

  1. b k b_{k}
  2. v v
  3. α = k = 1 N b k v \alpha\,=\sum_{k=1}^{N}b_{k}v

Vizing's_conjecture.html

  1. γ ( G H ) γ ( G ) γ ( H ) . \gamma(G\,\Box\,H)\geq\gamma(G)\gamma(H).\,
  2. C 4 C 4 C_{4}\,\Box\,C_{4}
  3. G = K 1 , n K 1 , n G=K_{1,n}\,\Box\,K_{1,n}
  4. γ ( K 1 , n K 1 , n ) = n + 1 \gamma(K_{1,n}\,\Box\,K_{1,n})=n+1
  5. γ ( G H ) = γ ( G ) γ ( H ) \gamma(G\,\Box\,H)=\gamma(G)\gamma(H)
  6. γ ( G H ) min { γ ( G ) | V ( H ) | , γ ( H ) | V ( G ) | } . \gamma(G\,\Box\,H)\leq\min\{\gamma(G)|V(H)|,\gamma(H)|V(G)|\}.

Vizing's_theorem.html

  1. Δ Δ
  2. Δ Δ
  3. Δ Δ
  4. Δ + 1 Δ+1
  5. Δ = 1 Δ=1
  6. G G
  7. Δ ( G ) = 1 Δ(G)=1
  8. Δ = 2 Δ=2
  9. G G
  10. Δ = 2 Δ=2
  11. Δ Δ
  12. G G
  13. G G
  14. n n
  15. p ( n ) p(n)
  16. n n
  17. p ( n ) p(n)
  18. n n
  19. p ( n ) p(n)
  20. Δ + 1 Δ+1
  21. Δ Δ
  22. u v uv
  23. P P
  24. u u
  25. p p
  26. u u
  27. c c
  28. p p
  29. q q
  30. u q uq
  31. c c
  32. p q pq
  33. P P
  34. P P
  35. v v
  36. w w
  37. P P
  38. c c
  39. u u
  40. w w
  41. u w uw
  42. c c
  43. p p
  44. u p up
  45. p p
  46. u v uv
  47. v v
  48. P P
  49. w w
  50. u u
  51. c c
  52. u w uw
  53. d d
  54. u u
  55. c c
  56. d d
  57. P P
  58. O ( n ) O(n)
  59. n n
  60. m m
  61. O ( m n ) O(mn)
  62. O ( m n log n ) O(m\sqrt{n}\log n)
  63. Δ + 1 Δ+1
  64. ( 3 / 2 ) Δ (3/2)Δ

Voice_inversion.html

  1. p p
  2. s - p s-p
  3. s s

Void_ratio.html

  1. e = V V V S = V V V T - V V = ϕ 1 - ϕ e=\frac{V_{V}}{V_{S}}=\frac{V_{V}}{V_{T}-V_{V}}=\frac{\phi}{1-\phi}
  2. ϕ = V V V T = V V V S + V V = e 1 + e \phi=\frac{V_{V}}{V_{T}}=\frac{V_{V}}{V_{S}+V_{V}}=\frac{e}{1+e}
  3. e e
  4. ϕ \phi
  5. ϕ \phi
  6. e = V V V S = V V V T - V V = n 1 - n e=\frac{V_{V}}{V_{S}}=\frac{V_{V}}{V_{T}-V_{V}}=\frac{n}{1-n}
  7. n = V V V T = V V V S + V V = e 1 + e n=\frac{V_{V}}{V_{T}}=\frac{V_{V}}{V_{S}+V_{V}}=\frac{e}{1+e}
  8. e e
  9. n n

Voigt_effect.html

  1. n n_{\parallel}
  2. ( n (n_{\perp}
  3. E i = ( cos β sin β 0 ) e - i ω ( t - n 1 z / c ) \vec{E_{i}}=\begin{pmatrix}\cos\beta\\ \sin\beta\\ 0\end{pmatrix}e^{-i\omega(t-n_{1}z/c)}
  4. m = ( cos ϕ sin ϕ 0 ) \vec{m}=\begin{pmatrix}\cos\phi\\ \sin\phi\\ 0\end{pmatrix}
  5. δ β \delta\beta
  6. δ β = R e [ B 1 + n 0 2 Q 2 2 n 0 ( n 0 2 - 1 ) ] sin [ 2 ( ϕ - β ) ] \delta\beta=Re\Big[\frac{B_{1}+n_{0}^{2}Q^{2}}{2n_{0}(n_{0}^{2}-1)}\Big]\sin[2% (\phi-\beta)]
  7. Q Q
  8. n 0 n_{0}
  9. B 1 B_{1}
  10. E i \vec{E}_{i}
  11. E r \vec{E}_{r}
  12. E t \vec{E}_{t}
  13. E i = ( cos β sin β 0 ) e - i ω ( t - n 1 z / c ) \vec{E_{i}}=\begin{pmatrix}\cos\beta\\ \sin\beta\\ 0\end{pmatrix}e^{-i\omega(t-n_{1}z/c)}
  14. m = ( cos ϕ sin ϕ 0 ) \vec{m}=\begin{pmatrix}\cos\phi\\ \sin\phi\\ 0\end{pmatrix}
  15. ϕ \phi
  16. E r = ( cos β + δ β sin β + δ β 0 ) e - i ω ( t + n 1 z / c ) \vec{E_{r}}=\begin{pmatrix}\cos\beta+\delta\beta\\ \sin\beta+\delta\beta\\ 0\end{pmatrix}e^{-i\omega(t+n_{1}z/c)}
  17. δ β \delta\beta
  18. δ β \delta\beta
  19. m \vec{m}
  20. m = M / M s \vec{m}=\vec{M}/M_{s}
  21. M s M_{s}
  22. ϵ r \epsilon_{r}
  23. ( 1 ) ϵ r = ϵ [ 1 0 i Q m y 0 1 - i Q m x - i Q m y i Q m x 1 ] + [ B 1 m x 2 B 2 m x m y 0 B 2 m x m y B 1 m y 2 0 0 0 B 1 m z 2 ] (1)\qquad\epsilon_{r}=\epsilon\begin{bmatrix}1&0&iQm_{y}\\ 0&1&-iQm_{x}\\ -iQm_{y}&iQm_{x}&1\end{bmatrix}+\begin{bmatrix}B_{1}m_{x}^{2}&B_{2}m_{x}m_{y}&% 0\\ B_{2}m_{x}m_{y}&B_{1}m_{y}^{2}&0\\ 0&0&B_{1}m_{z}^{2}\end{bmatrix}
  24. ϵ \epsilon
  25. Q Q
  26. B 1 B_{1}
  27. B 2 B_{2}
  28. m i 2 m_{i}^{2}
  29. m i m_{i}
  30. m i = M i / M s m_{i}=M_{i}/M_{s}
  31. B 1 = B 2 B_{1}=B_{2}
  32. n = k c / ω \vec{n}=\vec{k}c/\omega
  33. ( 2 ) n 2 E - n ( n E ) = ϵ E (2)\qquad n^{2}\vec{E}-\vec{n}(\vec{n}\cdot\vec{E})=\epsilon\vec{E}
  34. E \vec{E}
  35. D = ϵ E \vec{D}=\epsilon\vec{E}
  36. D = 0 \vec{\nabla}\cdot\vec{D}=0
  37. k z \vec{k}\parallel\vec{z}
  38. D = ( D x D y 0 ) \vec{D}=\begin{pmatrix}D_{x}\\ D_{y}\\ 0\end{pmatrix}
  39. D \vec{D}
  40. ( 3 ) { ( ϵ x x - 1 - 1 n 2 ) D x + ϵ x y - 1 D y = 0 ϵ y x - 1 D x + ( ϵ y y - 1 - 1 n 2 ) D y = 0 (3)\quad\left\{\begin{matrix}(\epsilon_{xx}^{-1}-\frac{1}{n^{2}})D_{x}+% \epsilon_{xy}^{-1}D_{y}=0\\ \\ \epsilon_{yx}^{-1}D_{x}+(\epsilon_{yy}^{-1}-\frac{1}{n^{2}})D_{y}=0\end{matrix% }\right.
  41. ϵ i j - 1 \epsilon_{ij}^{-1}
  42. i j ij
  43. ϵ r \epsilon_{r}
  44. n 2 = ϵ n^{2}=\epsilon
  45. Q Q
  46. B 1 B_{1}
  47. n 2 = ϵ + B 1 n_{\parallel}^{2}=\epsilon+B_{1}
  48. n 2 = ϵ ( 1 - Q 2 ) n_{\perp}^{2}=\epsilon(1-Q^{2})
  49. D \vec{D}
  50. E \vec{E}
  51. ( 4 ) D = ( cos ( ϕ ) sin ( ϕ ) 0 ) D = ( - sin ( ϕ ) cos ( ϕ ) 0 ) E = ϵ - 1 D = ( cos ( ϕ ) B 1 + ϵ sin ( ϕ ) B 1 + ϵ 0 ) E = ϵ - 1 D = ( sin ( ϕ ) ( Q 2 - 1 ) ϵ cos ( ϕ ) ( 1 - Q 2 ) ϵ - i Q ( 1 - Q 2 ) ϵ ) (4)\qquad\vec{D}_{\parallel}=\begin{pmatrix}\cos(\phi)\\ \sin(\phi)\\ 0\end{pmatrix}\qquad\vec{D}_{\perp}=\begin{pmatrix}-\sin(\phi)\\ \cos(\phi)\\ 0\end{pmatrix}\qquad\vec{E}_{\parallel}=\epsilon^{-1}\vec{D}_{\parallel}=% \begin{pmatrix}\frac{\cos(\phi)}{B1+\epsilon}\\ \frac{\sin(\phi)}{B1+\epsilon}\\ 0\end{pmatrix}\qquad\vec{E}_{\perp}=\epsilon^{-1}\vec{D}_{\perp}=\begin{% pmatrix}\frac{\sin(\phi)}{(Q^{2}-1)\epsilon}\\ \frac{\cos(\phi)}{(1-Q^{2})\epsilon}\\ \frac{-iQ}{(1-Q^{2})\epsilon}\end{pmatrix}
  52. E r = ( E r x E r y 0 ) \vec{E_{r}}=\begin{pmatrix}E_{rx}\\ E_{ry}\\ 0\end{pmatrix}
  53. E \vec{E}
  54. H \vec{H}
  55. H \vec{H}
  56. × H = 1 c D t \vec{\nabla}\times\vec{H}=\frac{1}{c}\frac{\partial\vec{D}}{\partial t}
  57. E t = α E + β E \vec{E}_{t}=\alpha\vec{E}_{\parallel}+\beta\vec{E}_{\perp}
  58. ( 5 ) { α ( D y n ) + β ( D y n ) + E r y = E 0 y α ( D x n ) + β ( D x n ) + E r x = E 0 x α E x + β E x - E r x = E 0 x α E y + β E y - E r y = E 0 y (5)\quad\left\{\begin{matrix}\alpha\Big(\frac{D_{y\parallel}}{n_{\parallel}}% \Big)+\beta\Big(\frac{D_{y\perp}}{n_{\perp}}\Big)+E_{ry}=E_{0y}\\ \\ \alpha\Big(\frac{D_{x\parallel}}{n_{\parallel}}\Big)+\beta\Big(\frac{D_{x\perp% }}{n_{\perp}}\Big)+E_{rx}=E_{0x}\\ \\ \alpha E_{x\parallel}+\beta E_{x\perp}-E_{rx}=E_{0x}\\ \\ \alpha E_{y\parallel}+\beta E_{y\perp}-E_{ry}=E_{0y}\end{matrix}\right.
  59. E r x = E 0 ( 1 - n n ) cos ( β ) + ( n - n ) cos ( β - 2 ϕ ) ( 1 + n ) ( 1 + n ) E_{rx}=E_{0}\frac{(1-n_{\perp}n_{\parallel})\cos(\beta)+(n_{\perp}-n_{% \parallel})\cos(\beta-2\phi)}{(1+n_{\parallel})(1+n_{\perp})}
  60. E r y = E 0 ( 1 - n n ) sin ( β ) - ( n - n ) sin ( β - 2 ϕ ) ( 1 + n ) ( 1 + n ) E_{ry}=E_{0}\frac{(1-n_{\perp}n_{\parallel})\sin(\beta)-(n_{\perp}-n_{% \parallel})\sin(\beta-2\phi)}{(1+n_{\parallel})(1+n_{\perp})}
  61. δ β \delta\beta
  62. ψ \psi
  63. χ = E r y / E r x \chi=E_{ry}/E_{rx}
  64. tan 2 δ β = 2 R e ( χ ) 1 - | χ | 2 sin ( 2 ψ K ) = 2 Im ( χ ) 1 - | χ | 2 \tan 2\delta\beta=\frac{2Re(\chi)}{1-|\chi|^{2}}\qquad\sin(2\psi_{K})=\frac{2% \,\text{Im}(\chi)}{1-|\chi|^{2}}
  65. R e ( χ ) Re(\chi)
  66. I m ( χ ) Im(\chi)
  67. χ \chi
  68. ( 6 ) χ = ( B 1 + n 0 2 Q 2 ) 2 n 0 ( n 0 2 - 1 ) sin [ 2 ( ϕ - β ) ] cos ( β ) 2 + tan ( β ) (6)\qquad\chi=\frac{(B_{1}+n_{0}^{2}Q^{2})}{2n_{0}(n_{0}^{2}-1)}\frac{\sin[2(% \phi-\beta)]}{\cos(\beta)^{2}}+\tan(\beta)
  69. ( 7 ) δ β = R e [ B 1 + n 0 2 Q 2 2 n 0 ( n 0 2 - 1 ) ] sin [ 2 ( ϕ - β ) ] (7)\qquad\delta\beta=Re\Big[\frac{B_{1}+n_{0}^{2}Q^{2}}{2n_{0}(n_{0}^{2}-1)}% \Big]\sin[2(\phi-\beta)]
  70. T < T c 2 T<\frac{T_{c}}{2}
  71. H 1 H_{1}
  72. H 2 H_{2}
  73. R e [ B 1 + n 0 2 Q 2 2 n 0 ( n 0 2 - 1 ) ] P V o i g t = 0.5 m r a d Re\Big[\frac{B_{1}+n_{0}^{2}Q^{2}}{2n_{0}(n_{0}^{2}-1)}\Big]P_{Voigt}=0.5mrad
  74. H 1 H_{1}
  75. H 2 H_{2}
  76. P V o i g t P_{Voigt}

Volterra_series.html

  1. y ( t ) = h 0 + n = 1 N a b a b h n ( τ 1 , . . , τ n ) j = 1 n x ( t - τ j ) d τ j y(t)=h_{0}+\sum_{n=1}^{N}{\int_{a}^{b}\cdots\int_{a}^{b}{h_{n}(\tau_{1},.\,.\,% ,\tau_{n})\prod^{n}_{j=1}{x(t-\tau_{j})d\tau_{j}}}}
  2. h 0 h_{0}
  3. y y
  4. h n ( τ 1 , . . , τ n ) h_{n}(\tau_{1},.\,.\,,\tau_{n})
  5. h n ( t 1 , t 2 , , t n ) h_{n}(t_{1},t_{2},\ldots,t_{n})
  6. t 1 , t 2 , , t n t_{1},t_{2},\ldots,t_{n}
  7. a 0 a\geq 0
  8. x ( t ) x(t)
  9. y ( n ) = h 0 + p = 1 P τ 1 = a b τ p = a b h p ( τ 1 , . . , τ p ) j = 1 p x ( n - τ j ) , y(n)=h_{0}+\sum_{p=1}^{P}{\sum_{\tau_{1}=a}^{b}\cdots\sum_{\tau_{p}=a}^{b}{h_{% p}(\tau_{1},.\,.\,,\tau_{p})\prod^{p}_{j=1}{x(n-\tau_{j})}}},
  10. h p ( τ 1 , . . , τ p ) h_{p}(\tau_{1},.\,.\,,\tau_{p})
  11. a 0 a\geq 0
  12. h p ( τ 1 , . . , τ p ) h_{p}(\tau_{1},.\,.\,,\tau_{p})
  13. τ 1 , . . , τ p \tau_{1},.\,.\,,\tau_{p}
  14. τ 1 = 0 M τ 2 = τ 1 M τ p = τ p - 1 M h p ( τ 1 , . . , τ p ) j = 1 p x ( n - τ j ) . \sum_{\tau_{1}=0}^{M}\sum_{\tau_{2}=\tau_{1}}^{M}\cdots\sum_{\tau_{p}=\tau_{p-% 1}}^{M}{h_{p}(\tau_{1},.\,.\,,\tau_{p})\prod^{p}_{j=1}{x(n-\tau_{j})}}.
  15. y ( n ) = p H p x ( n ) p G p x ( n ) y(n)=\sum_{p}{H_{p}x(n)}\equiv\sum_{p}{G_{p}x(n)}
  16. E { H i x ( n ) G j x ( n ) } = 0 ; i < j E\{H_{i}x(n)G_{j}x(n)\}=0;\qquad i<j
  17. E { G i x ( n ) G j x ( n ) } = 0 ; i j E\{G_{i}x(n)G_{j}x(n)\}=0;\qquad i\neq j
  18. H i x ( n ) H_{i}x(n)
  19. H p ¯ * x ( n ) = j = 1 p ¯ x ( n - τ j ) H^{*}_{\overline{p}}x(n)=\prod^{\overline{p}}_{j=1}{x(n-\tau_{j})}
  20. E { y ( n ) H p ¯ * x ( n ) } = E { p = 0 G p x ( n ) H p ¯ * x ( n ) } E\left\{y(n)H^{*}_{\overline{p}}x(n)\right\}=E\left\{\sum_{p=0}^{\infty}{G_{p}% x(n)H^{*}_{\overline{p}}x(n)}\right\}
  21. τ 1 τ 2 τ P \tau_{1}\neq\tau_{2}\neq\ldots\neq\tau_{P}
  22. A = σ x 2 A=\sigma^{2}_{x}
  23. E { y ( n ) j = 1 p ¯ x ( n - τ j ) } = E { G p ¯ x ( n ) j = 1 p ¯ x ( n - τ j ) } = p ¯ ! A p ¯ k p ¯ ( τ 1 , . . , τ p ¯ ) E\left\{y(n)\prod^{\overline{p}}_{j=1}{x(n-\tau_{j})}\right\}=E\left\{G_{% \overline{p}}x(n)\prod^{\overline{p}}_{j=1}{x(n-\tau_{j})}\right\}=\overline{p% }!A^{\overline{p}}k_{\overline{p}}(\tau_{1},.\,.\,,\tau_{\overline{p}})
  24. τ i τ j , i , j {\tau_{i}\neq\tau_{j},\,\forall i,j}
  25. k p ( τ 1 , . . , τ p ) = E { y ( n ) x ( n - τ 1 ) x ( n - τ p ) } < m t p l > p ! A p . k_{p}(\tau_{1},.\,.\,,\tau_{p})=\frac{E\left\{{y(n)x(n-\tau_{1})\cdots x(n-% \tau_{p})}\right\}}{<}mtpl>{{p!A^{p}}}.
  26. k p ( τ 1 , . . , τ p ) = E { ( y ( n ) - m = 0 p - 1 G m x ( n ) ) x ( n - τ 1 ) x ( n - τ p ) } p ! A p k_{p}(\tau_{1},.\,.\,,\tau_{p})\!=\!\frac{E\left\{{\left({y(n)\!-\!\!\!\sum% \limits_{m=0}^{p-1}{\!G_{m}x(n)}}\!\!\right)\!x(n-\tau_{1})\cdots x(n-\tau_{p}% )}\right\}}{p!A^{p}}
  27. σ x \sigma_{x}
  28. σ x \sigma_{x}
  29. σ x \sigma_{x}
  30. σ x \sigma_{x}
  31. k 0 ( 0 ) = E { y ( 0 ) ( n ) } k_{0}^{(0)}=E\{y^{(0)}(n)\}
  32. k 1 ( 1 ) ( τ 1 ) = 1 A 1 E { y ( 1 ) ( n ) x ( 1 ) ( n - τ 1 ) } k_{1}^{(1)}(\tau_{1})=\frac{1}{A_{1}}E\left\{y^{(1)}(n)\,x^{(1)}(n-\tau_{1})\right\}
  33. k 2 ( 2 ) ( τ 1 , τ 2 ) = 1 2 ! A 2 2 { E { y ( 2 ) ( n ) i = 1 2 x ( 2 ) ( n - τ i ) } - A 2 k 0 ( 2 ) δ τ 1 τ 2 } k_{2}^{(2)}(\tau_{1},\tau_{2})=\frac{1}{2!A_{2}^{2}}\,\left\{E\left\{y^{(2)}(n% )\,\prod_{i=1}^{2}{x^{(2)}(n-\tau_{i})}\right\}-A_{2}k_{0}^{(2)}\delta_{\tau_{% 1}\tau_{2}}\right\}
  34. k 3 ( 3 ) ( τ 1 , τ 2 , τ 3 ) = 1 3 ! A 3 3 { E { y ( 3 ) ( n ) i = 1 3 x ( 3 ) ( n - τ i ) } - A 3 2 [ k 1 ( 3 ) ( τ 1 ) δ τ 2 τ 3 + k 1 ( 3 ) ( τ 2 ) δ τ 1 τ 3 + k 1 ( 3 ) ( τ 3 ) δ τ 1 τ 2 ] } k_{3}^{(3)}(\tau_{1},\tau_{2},\tau_{3})=\frac{1}{3!A_{3}^{3}}\left\{E\left\{y^% {(3)}(n)\,\prod_{i=1}^{3}{x^{(3)}(n-\tau_{i})}\right\}-A_{3}^{2}\left[k_{1}^{(% 3)}(\tau_{1})\delta_{\tau_{2}\tau_{3}}+k_{1}^{(3)}(\tau_{2})\delta_{\tau_{1}% \tau_{3}}+k_{1}^{(3)}(\tau_{3})\delta_{\tau_{1}\tau_{2}}\right]\right\}
  35. h 5 = k 5 ( 5 ) h_{5}=\,k_{5}^{(5)}
  36. h 4 = k 4 ( 4 ) h_{4}=\,k_{4}^{(4)}
  37. h 3 = k 3 ( 3 ) - 10 A 3 τ 4 k 5 ( 5 ) ( τ 1 , τ 2 , τ 3 , τ 4 , τ 4 ) h_{3}=\,k_{3}^{(3)}-10A_{3}\sum_{\tau_{4}}{k_{5}^{(5)}(\tau_{1},\tau_{2},\tau_% {3},\tau_{4},\tau_{4})}
  38. h 2 = k 2 ( 2 ) - 6 A 2 τ 3 k 4 ( 4 ) ( τ 1 , τ 2 , τ 3 , τ 3 ) h_{2}=\,k_{2}^{(2)}-6A_{2}\sum_{\tau_{3}}{k_{4}^{(4)}(\tau_{1},\tau_{2},\tau_{% 3},\tau_{3})}
  39. h 1 = k 1 ( 1 ) - 3 A 1 τ 2 k 3 ( 3 ) ( τ 1 , τ 2 , τ 2 ) + 15 A 1 2 τ 2 τ 3 k 5 ( 5 ) ( τ 1 , τ 2 , τ 2 , τ 3 , τ 3 ) h_{1}=\,k_{1}^{(1)}-3A_{1}\sum_{\tau_{2}}{k_{3}^{(3)}(\tau_{1},\tau_{2},\tau_{% 2})}+15A_{1}^{2}\sum_{\tau 2}{\sum_{\tau_{3}}{k_{5}^{(5)}(\tau_{1},\tau_{2},% \tau_{2},\tau_{3},\tau_{3})}}
  40. h 0 = k 0 ( 0 ) - A 0 τ 1 k 2 ( 2 ) ( τ 1 , τ 1 ) + 3 A 0 2 τ 1 τ 2 k 4 ( 4 ) ( τ 1 , τ 1 , τ 2 , τ 2 ) h_{0}=\,k_{0}^{(0)}-A_{0}\sum_{\tau_{1}}{k_{2}^{(2)}(\tau_{1},\tau_{1})}+3A_{0% }^{2}\sum_{\tau_{1}}{\sum_{\tau_{2}}{k_{4}^{(4)}(\tau_{1},\tau_{1},\tau_{2},% \tau_{2})}}

Von_Mises–Fisher_distribution.html

  1. κ = 1 \kappa=1
  2. κ = 10 \kappa=10
  3. κ = 100 \kappa=100
  4. μ \mu
  5. ( p - 1 ) (p-1)
  6. p \mathbb{R}^{p}
  7. p = 2 p=2
  8. 𝐱 \mathbf{x}\,
  9. f p ( 𝐱 ; μ , κ ) = C p ( κ ) exp ( κ μ T 𝐱 ) f_{p}(\mathbf{x};\mu,\kappa)=C_{p}(\kappa)\exp\left({\kappa\mu^{T}\mathbf{x}}\right)
  10. κ 0 , μ = 1 \kappa\geq 0,\left\|\mu\right\|=1\,
  11. C p ( κ ) C_{p}(\kappa)\,
  12. C p ( κ ) = κ p / 2 - 1 ( 2 π ) p / 2 I p / 2 - 1 ( κ ) . C_{p}(\kappa)=\frac{\kappa^{p/2-1}}{(2\pi)^{p/2}I_{p/2-1}(\kappa)}.\,
  13. I v I_{v}
  14. v v
  15. p = 3 p=3
  16. C 3 ( κ ) = κ 4 π sinh κ = κ 2 π ( e κ - e - κ ) . C_{3}(\kappa)=\frac{\kappa}{4\pi\sinh\kappa}=\frac{\kappa}{2\pi(e^{\kappa}-e^{% -\kappa})}.\,
  17. μ \mu\,
  18. κ \kappa\,
  19. κ \kappa\,
  20. μ \mu\,
  21. κ > 0 \kappa>0\,
  22. κ = 0 \kappa=0\,
  23. p = 3 p=3
  24. x i x_{i}
  25. A p ( κ ) = I p / 2 ( κ ) I p / 2 - 1 ( κ ) . A_{p}(\kappa)=\frac{I_{p/2}(\kappa)}{I_{p/2-1}(\kappa)}.\,
  26. μ \mu\,
  27. κ \kappa\,
  28. μ = i N x i i N x i , \mu=\frac{\sum_{i}^{N}x_{i}}{\|\sum_{i}^{N}x_{i}\|},
  29. κ = A p - 1 ( R ¯ ) . \kappa=A_{p}^{-1}(\bar{R}).
  30. κ \kappa\,
  31. A p ( κ ) = i N x i N = R ¯ . A_{p}(\kappa)=\frac{\|\sum_{i}^{N}x_{i}\|}{N}=\bar{R}.
  32. κ \kappa
  33. κ ^ = R ¯ ( p - R ¯ 2 ) 1 - R ¯ 2 , \hat{\kappa}=\frac{\bar{R}(p-\bar{R}^{2})}{1-\bar{R}^{2}},
  34. κ ^ 1 = κ ^ - A p ( κ ^ ) - R ¯ 1 - A p ( κ ^ ) 2 - p - 1 κ ^ A p ( κ ^ ) , \hat{\kappa}_{1}=\hat{\kappa}-\frac{A_{p}(\hat{\kappa})-\bar{R}}{1-A_{p}(\hat{% \kappa})^{2}-\frac{p-1}{\hat{\kappa}}A_{p}(\hat{\kappa})},
  35. κ ^ 2 = κ ^ 1 - A p ( κ ^ 1 ) - R ¯ 1 - A p ( κ ^ 1 ) 2 - p - 1 κ ^ 1 A p ( κ ^ 1 ) . \hat{\kappa}_{2}=\hat{\kappa}_{1}-\frac{A_{p}(\hat{\kappa}_{1})-\bar{R}}{1-A_{% p}(\hat{\kappa}_{1})^{2}-\frac{p-1}{\hat{\kappa}_{1}}A_{p}(\hat{\kappa}_{1})}.
  36. σ ^ = ( d N R ¯ 2 ) 1 / 2 \hat{\sigma}=\left(\frac{d}{N\bar{R}^{2}}\right)^{1/2}
  37. d = 1 - 1 N i N ( μ T x i ) 2 d=1-\frac{1}{N}\sum_{i}^{N}(\mu^{T}x_{i})^{2}
  38. 100 ( 1 - α ) % 100(1-\alpha)\%
  39. μ \mu
  40. q = arcsin ( e α 1 / 2 σ ^ ) , q=\arcsin(e_{\alpha}^{1/2}\hat{\sigma}),
  41. e α = - ln ( α ) . e_{\alpha}=-\ln(\alpha).
  42. α = 0.05 , e α = - ln ( 0.05 ) = 2.996 , \alpha=0.05,e_{\alpha}=-\ln(0.05)=2.996,
  43. q = arcsin ( 1.731 σ ^ ) . q=\arcsin(1.731\hat{\sigma}).

Von_Zeipel_theorem.html

  1. F F
  2. g eff g_{\textrm{eff}}
  3. F = - L ( P ) 4 π G M * ( P ) g eff F=-\frac{L(P)}{4\pi GM_{*}(P)}g_{\textrm{eff}}
  4. L L
  5. M * M_{*}
  6. P P
  7. T eff T_{\textrm{eff}}
  8. θ \theta
  9. T eff ( θ ) g eff 1 / 4 ( θ ) T_{\textrm{eff}}(\theta)\sim g_{\textrm{eff}}^{1/4}(\theta)

W-shingling.html

  1. r ( A , B ) = | S ( A ) S ( B ) | | S ( A ) S ( B ) | r(A,B)={{|S(A)\cap S(B)|}\over{|S(A)\cup S(B)|}}

Wasserstein_metric.html

  1. M d ( x , x 0 ) p d μ ( x ) < + . \int_{M}d(x,x_{0})^{p}\,\mathrm{d}\mu(x)<+\infty.
  2. W p ( μ , ν ) := ( inf γ Γ ( μ , ν ) M × M d ( x , y ) p d γ ( x , y ) ) 1 / p , W_{p}(\mu,\nu):=\left(\inf_{\gamma\in\Gamma(\mu,\nu)}\int_{M\times M}d(x,y)^{p% }\,\mathrm{d}\gamma(x,y)\right)^{1/p},
  3. W p ( μ , ν ) p = inf 𝐄 [ d ( X , Y ) p ] , W_{p}(\mu,\nu)^{p}=\inf\mathbf{E}\big[d(X,Y)^{p}\big],
  4. W 1 ( μ , ν ) = sup { M f ( x ) d ( μ - ν ) ( x ) | continuous f : M , Lip ( f ) 1 } , W_{1}(\mu,\nu)=\sup\left\{\left.\int_{M}f(x)\,\mathrm{d}(\mu-\nu)(x)\right|% \mbox{continuous }~{}f:M\to\mathbb{R},\mathrm{Lip}(f)\leq 1\right\},
  5. ρ ( μ , ν ) := sup { M f ( x ) d ( μ - ν ) ( x ) | continuous f : M [ - 1 , 1 ] } . \rho(\mu,\nu):=\sup\left\{\left.\int_{M}f(x)\,\mathrm{d}(\mu-\nu)(x)\right|% \mbox{continuous }~{}f:M\to[-1,1]\right\}.
  6. 2 W 1 ( μ , ν ) C ρ ( μ , ν ) , 2W_{1}(\mu,\nu)\leq C\rho(\mu,\nu),

Watermarking_attack.html

  1. P \scriptstyle P
  2. I V \scriptstyle IV
  3. P 1 I V 1 = P 2 I V 2 \scriptstyle P_{1}\,\oplus\,IV_{1}\;=\;P_{2}\,\oplus\,IV_{2}
  4. P 1 , P 2 \scriptstyle P_{1},\,P_{2}
  5. P 1 P 2 = I V 1 I V 2 \scriptstyle P_{1}\,\oplus\,P_{2}\;=\;IV_{1}\,\oplus\,IV_{2}

Wave-making_resistance.html

  1. c c
  2. c = g 2 π l c=\sqrt{\frac{g}{2\pi}l}
  3. l l
  4. g g
  5. g g
  6. c in knots 1.341 × length in ft 4 3 × length in ft \mbox{c in knots}~{}\approx 1.341\times\sqrt{\mbox{length in ft}~{}}\approx% \frac{4}{3}\times\sqrt{\mbox{length in ft}~{}}
  7. c in knots 2.429 × length in m 6 × length in m 2.5 × length in m \mbox{c in knots}~{}\approx 2.429\times\sqrt{\mbox{length in m}~{}}\approx% \sqrt{6\times\mbox{length in m}~{}}\approx 2.5\times\sqrt{\mbox{length in m}~{}}

Waveshaper.html

  1. y = f ( a ( t ) x ( t ) ) y=f(a(t)x(t))
  2. f ( x ) = a n x n + a n - 1 x n - 1 + + a 2 x 2 + a 1 x + a 0 = n = 0 N a n x n f(x)=a_{n}x^{n}+a_{n-1}x^{n-1}+\cdots+a_{2}x^{2}+a_{1}x+a_{0}=\sum_{n=0}^{N}a_% {n}x^{n}
  3. n = 0 N a n ( α c o s ( ω t + ϕ ) ) n \sum_{n=0}^{N}a_{n}(\alpha cos(\omega t+\phi))^{n}
  4. n = 0 N a n ( α e j ( ω t + ϕ ) + e - j ( ω t + ϕ ) 2 ) n = a 0 + n = 1 N a n α n 2 n - 1 ( e j ( ω t + ϕ ) + e - j ( ω t + ϕ ) ) n 2 \sum_{n=0}^{N}a_{n}\Bigg(\alpha\frac{e^{j(\omega t+\phi)}+e^{-j(\omega t+\phi)% }}{2}\Bigg)^{n}=a_{0}+\sum_{n=1}^{N}\frac{a_{n}\alpha^{n}}{2^{n-1}}\frac{(e^{j% (\omega t+\phi)}+e^{-j(\omega t+\phi)})^{n}}{2}
  5. a 0 + n = 1 N [ a n α n 2 n - 1 k = 0 n ( n k ) e j ( n - k ) ( ω t + ϕ ) e - j k ( ω t + ϕ ) 2 ] = a 0 + n = 1 N [ a n α n 2 n - 1 k = 0 n ( n k ) e j ( n - 2 k ) ( ω t + ϕ ) 2 ] a_{0}+\sum_{n=1}^{N}\Bigg[{\frac{a_{n}\alpha^{n}}{2^{n-1}}\sum_{k=0}^{n}{{n% \choose k}\frac{e^{j(n-k)(\omega t+\phi)}e^{-jk(\omega t+\phi)}}{2}}\Bigg]}=a_% {0}+\sum_{n=1}^{N}\Bigg[{\frac{a_{n}\alpha^{n}}{2^{n-1}}\sum_{k=0}^{n}{{n% \choose k}\frac{e^{j(n-2k)(\omega t+\phi)}}{2}}\Bigg]}
  6. = a 0 + n = 1 N [ a n α n 2 n - 1 k = 0 n / 2 ( n k ) cos ( ( n - 2 k ) ( ω t + ϕ ) ) ] =a_{0}+\sum_{n=1}^{N}\Bigg[{\frac{a_{n}\alpha^{n}}{2^{n-1}}\sum_{k=0}^{\lfloor n% /2\rfloor}{{n\choose k}\cos{((n-2k)(\omega t+\phi))}}\Bigg]}
  7. N ω N\omega
  8. x n x^{n}
  9. a n a_{n}
  10. a n α n 2 n - 1 \frac{a_{n}\alpha^{n}}{2^{n-1}}

Weak_localization.html

  1. σ ( B ) - σ ( 0 ) = + e 2 2 π 2 [ l n ( B ϕ B ) - ψ ( 1 2 + B ϕ B ) ] \sigma(B)-\sigma(0)=+{e^{2}\over 2\pi^{2}\hbar}\left[ln\left({B_{\phi}\over B}% \right)-\psi\left({1\over 2}+{B_{\phi}\over B}\right)\right]
  2. + e 2 π 2 [ l n ( B S O + B e B ) - ψ ( 1 2 + B S O + B e B ) ] +{e^{2}\over\pi^{2}\hbar}\left[ln\left({B_{SO}+B_{e}\over B}\right)-\psi\left(% {1\over 2}+{B_{SO}+B_{e}\over B}\right)\right]
  3. - 3 e 2 2 π 2 [ l n ( ( 4 / 3 ) B S O + B ϕ B ) - ψ ( 1 2 + ( 4 / 3 ) B S O + B ϕ B ) ] -{3e^{2}\over 2\pi^{2}\hbar}\left[ln\left({(4/3)B_{SO}+B_{\phi}\over B}\right)% -\psi\left({1\over 2}+{(4/3)B_{SO}+B_{\phi}\over B}\right)\right]
  4. ψ \psi
  5. B ϕ B_{\phi}
  6. B S O B_{SO}
  7. B e B_{e}
  8. B i = / 4 e l i 2 {B_{i}=\hbar/4el_{i}^{2}}
  9. l ϕ l_{\phi}
  10. l S O l_{SO}
  11. l e l_{e}
  12. B S O B ϕ B_{SO}>>B_{\phi}
  13. σ ( B ) - σ ( 0 ) = α e 2 2 π 2 ( l n ( B ϕ B ) - ψ ( 1 2 + B ϕ B ) ) \sigma(B)-\sigma(0)=\alpha{e^{2}\over 2\pi^{2}\hbar}\left(ln\left({B_{\phi}% \over B}\right)-\psi\left({1\over 2}+{B_{\phi}\over B}\right)\right)
  14. α \alpha

Weakly_o-minimal_structure.html

  1. ( M , < , ) (M,<,...)
  2. M M
  3. I = { r M : a < r < b } I=\{r\in M\,:\,a<r<b\}
  4. M { ± } M\cup\{\pm\infty\}
  5. ( M , < , ) (M,<,...)
  6. C C
  7. C C
  8. α < π ( 0 , α ) . \bigcup_{\alpha<\pi}(0,\alpha).

Web_interoperability.html

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

Wedderburn's_little_theorem.html

  1. Br ( k ) = H 2 ( k al / k ) \operatorname{Br}(k)=H^{2}(k^{\,\text{al}}/k)
  2. H 1 ( k al / k ) H^{1}(k^{\,\text{al}}/k)
  3. a a x , a x a : A A a\mapsto ax,a\mapsto xa:A\to A
  4. q n - 1 = q - 1 + q n - 1 q d - 1 q^{n}-1=q-1+\sum{q^{n}-1\over q^{d}-1}
  5. Φ f ( q ) \Phi_{f}(q)
  6. x n - 1 = m | n Φ m ( x ) x^{n}-1=\prod_{m|n}\Phi_{m}(x)
  7. x d - 1 = m | d Φ m ( x ) x^{d}-1=\prod_{m|d}\Phi_{m}(x)
  8. Φ n ( q ) \Phi_{n}(q)
  9. q n - 1 q d - 1 {q^{n}-1\over q^{d}-1}
  10. Φ n ( q ) \Phi_{n}(q)
  11. | Φ n ( q ) | q - 1 |\Phi_{n}(q)|\leq q-1
  12. | Φ n ( q ) | > q - 1 |\Phi_{n}(q)|>q-1
  13. Φ n ( x ) = ( x - ζ ) \Phi_{n}(x)=\prod(x-\zeta)
  14. | Φ n ( q ) | = | q - ζ | |\Phi_{n}(q)|=\prod|q-\zeta|
  15. | q - ζ | > | q - 1 | |q-\zeta|>|q-1|
  16. | Φ n ( q ) | > q - 1 |\Phi_{n}(q)|>q-1

Weierstrass_product_inequality.html

  1. ( 1 - a ) ( 1 - b ) ( 1 - c ) ( 1 - d ) + a + b + c + d 1. (1-a)(1-b)(1-c)(1-d)+a+b+c+d\geq 1.\,

Weight_(strings).html

  1. a a
  2. a a
  3. A A
  4. a A a\in A
  5. A A
  6. c A * c\in A^{*}
  7. A * A^{*}
  8. A A
  9. A A
  10. a a
  11. c c
  12. wt a ( c ) \mathrm{wt}_{a}(c)
  13. a a
  14. c c
  15. A A
  16. A A
  17. wt ( c ) \mathrm{wt}(c)
  18. c c
  19. c c
  20. A = { x , y , z } A=\{x,y,z\}
  21. c = y x x z y y z x y z z y x c=yxxzyyzxyzzyx
  22. y y
  23. y y
  24. c c
  25. wt y ( c ) = 5 \mathrm{wt}_{y}(c)=5
  26. A = 𝐙 3 = { 0 , 1 , 2 } A=\mathbf{Z}_{3}=\{0,1,2\}
  27. c = 002001200 c=002001200
  28. wt 0 ( c ) = 6 \mathrm{wt}_{0}(c)=6
  29. wt 1 ( c ) = 1 \mathrm{wt}_{1}(c)=1
  30. wt 2 ( c ) = 2 \mathrm{wt}_{2}(c)=2
  31. wt ( c ) = wt 1 ( c ) + wt 2 ( c ) = 3 \mathrm{wt}(c)=\mathrm{wt}_{1}(c)+\mathrm{wt}_{2}(c)=3

Weighted_fair_queueing.html

  1. N N
  2. w i w_{i}
  3. i i
  4. w i ( w 1 + w 2 + + w N ) R \frac{w_{i}}{(w_{1}+w_{2}+...+w_{N})}R
  5. w i = 1 / c i w_{i}=1/c_{i}
  6. c i c_{i}
  7. i i
  8. R i := w i ( w 1 + w 2 + + w N ) R Ri:=\frac{w_{i}}{(w_{1}+w_{2}+...+w_{N})}R

Weighted_Majority_Algorithm.html

  1. 𝐎 ( 𝐥𝐨𝐠 | 𝐀 | + 𝐦 ) \mathbf{O(log|A|+m)}
  2. 𝐱 i \mathbf{x}_{i}
  3. 𝐦 \mathbf{m}

Weil_group.html

  1. W E c {W}_{E}^{c}
  2. w x w - 1 = || w || x \displaystyle wxw^{-1}=||w||x

Weissberger's_model.html

  1. L = { 1.33 f 0.284 d 0.588 , if 14 < d 400 0.45 f 0.284 d , if 0 < d 14 L=\begin{cases}1.33\,f^{0.284}\,d^{0.588}\,\mbox{, if }~{}14<d\leq 400\\ 0.45\,f^{0.284}\,d\,\,\,\,\,\,\,\,\,\,\mbox{, if }~{}0<d\leq 14\end{cases}

Weitzenböck_identity.html

  1. M α , δ β := M d α , β \int_{M}\langle\alpha,\delta\beta\rangle:=\int_{M}\langle d\alpha,\beta\rangle
  2. - , - \langle-,-\rangle
  3. Δ = d δ + δ d . \Delta=d\delta+\delta d.
  4. : Ω p M T * M Ω p M \nabla:\Omega^{p}M\rightarrow T^{*}M\otimes\Omega^{p}M
  5. Δ = * \Delta^{\prime}=\nabla^{*}\nabla
  6. * \nabla^{*}
  7. \nabla
  8. Δ - Δ = A \Delta^{\prime}-\Delta=A
  9. A = 1 2 R ( θ , θ ) # , # + Ric ( θ , # ) A=\frac{1}{2}\langle R(\theta,\theta)\#,\#\rangle+\operatorname{Ric}(\theta,\#)\,
  10. θ : T * M Ω p M Ω p + 1 M \theta:T^{*}M\otimes\Omega^{p}M\rightarrow\Omega^{p+1}M
  11. # : Ω p + 1 M T * M Ω p M \#:\Omega^{p+1}M\rightarrow T^{*}M\otimes\Omega^{p}M
  12. : S M T * M S M . \nabla:SM\rightarrow T^{*}M\otimes SM.
  13. Δ = * \Delta^{\prime}=\nabla^{*}\nabla
  14. Δ - Δ = - 1 4 S c \Delta^{\prime}-\Delta=-\frac{1}{4}Sc
  15. ¯ \bar{\partial}
  16. Δ = ¯ * ¯ + ¯ ¯ * \Delta=\bar{\partial}^{*}\bar{\partial}+\bar{\partial}\bar{\partial}^{*}
  17. Δ = - k k k ¯ \Delta^{\prime}=-\sum_{k}\nabla_{k}\nabla_{\bar{k}}
  18. α = α i 1 i 2 i p j ¯ 1 j ¯ 2 j ¯ q \alpha=\alpha_{i_{1}i_{2}\dots i_{p}\bar{j}_{1}\bar{j}_{2}\dots\bar{j}_{q}}
  19. A ( α ) = - k , j s Ric j ¯ α k ¯ α i 1 i 2 i p j ¯ 1 j ¯ 2 k ¯ j ¯ q A(\alpha)=-\sum_{k,j_{s}}\operatorname{Ric}_{\bar{j}_{\alpha}}^{\bar{k}}\alpha% _{i_{1}i_{2}\dots i_{p}\bar{j}_{1}\bar{j}_{2}\dots\bar{k}\dots\bar{j}_{q}}

Welch's_t_test.html

  1. t = X ¯ 1 - X ¯ 2 s 1 2 N 1 + s 2 2 N 2 t\quad=\quad{\;\overline{X}_{1}-\overline{X}_{2}\;\over\sqrt{\;{s_{1}^{2}\over N% _{1}}\;+\;{s_{2}^{2}\over N_{2}}\quad}}\,
  2. X ¯ 1 \overline{X}_{1}
  3. s 1 2 s_{1}^{2}
  4. N 1 N_{1}
  5. 1 1
  6. ν \nu
  7. ν ( s 1 2 N 1 + s 2 2 N 2 ) 2 s 1 4 N 1 2 ν 1 + s 2 4 N 2 2 ν 2 \nu\quad\approx\quad{{\left(\;{s_{1}^{2}\over N_{1}}\;+\;{s_{2}^{2}\over N_{2}% }\;\right)^{2}}\over{\quad{s_{1}^{4}\over N_{1}^{2}\nu_{1}}\;+\;{s_{2}^{4}% \over N_{2}^{2}\nu_{2}}\quad}}
  8. ν 1 \nu_{1}
  9. N 1 - 1 N_{1}-1
  10. 1 1
  11. ν 2 \nu_{2}
  12. N 2 - 1 N_{2}-1
  13. 2 2
  14. ν \nu
  15. μ 1 \mu_{1}
  16. μ 2 \mu_{2}
  17. σ 1 2 \sigma_{1}^{2}
  18. σ 2 2 \sigma_{2}^{2}
  19. N 1 N_{1}
  20. N 2 N_{2}
  21. A 1 = 27.5 , 21.0 , 19.0 , 23.6 , 17.0 , 17.9 , 16.9 , 20.1 , 21.9 , 22.6 , 23.1 , 19.6 , 19.0 , 21.7 , 21.4 A1={27.5,21.0,19.0,23.6,17.0,17.9,16.9,20.1,21.9,22.6,23.1,19.6,19.0,21.7,21.4}
  22. A 2 = 27.1 , 22.0 , 20.8 , 23.4 , 23.4 , 23.5 , 25.8 , 22.0 , 24.8 , 20.2 , 21.9 , 22.1 , 22.9 , 20.5 , 24.4 A2={27.1,22.0,20.8,23.4,23.4,23.5,25.8,22.0,24.8,20.2,21.9,22.1,22.9,20.5,24.4}
  23. σ 1 2 \sigma_{1}^{2}
  24. σ 2 2 \sigma_{2}^{2}
  25. N 1 N_{1}
  26. N 2 N_{2}
  27. A 1 = 17.2 , 20.9 , 22.6 , 18.1 , 21.7 , 21.4 , 23.5 , 24.2 , 14.7 , 21.8 A1={17.2,20.9,22.6,18.1,21.7,21.4,23.5,24.2,14.7,21.8}
  28. A 2 = 21.5 , 22.8 , 21.0 , 23.0 , 21.6 , 23.6 , 22.5 , 20.7 , 23.4 , 21.8 , 20.7 , 21.7 , 21.5 , 22.5 , 23.6 , 21.5 , 22.5 , 23.5 , 21.5 , 21.8 A2={21.5,22.8,21.0,23.0,21.6,23.6,22.5,20.7,23.4,21.8,20.7,21.7,21.5,22.5,23.6% ,21.5,22.5,23.5,21.5,21.8}
  29. σ 1 2 \sigma_{1}^{2}
  30. σ 2 2 \sigma_{2}^{2}
  31. N 1 N_{1}
  32. N 2 N_{2}
  33. A 1 = 19.8 , 20.4 , 19.6 , 17.8 , 18.5 , 18.9 , 18.3 , 18.9 , 19.5 , 22.0 A1={19.8,20.4,19.6,17.8,18.5,18.9,18.3,18.9,19.5,22.0}
  34. A 2 = 28.2 , 26.6 , 20.1 , 23.3 , 25.2 , 22.1 , 17.7 , 27.6 , 20.6 , 13.7 , 23.2 , 17.5 , 20.6 , 18.0 , 23.9 , 21.6 , 24.3 , 20.4 , 24.0 , 13.2 A2={28.2,26.6,20.1,23.3,25.2,22.1,17.7,27.6,20.6,13.7,23.2,17.5,20.6,18.0,23.9% ,21.6,24.3,20.4,24.0,13.2}
  35. μ 1 - μ 2 \mu_{1}-\mu_{2}
  36. N 1 N_{1}
  37. X ¯ 1 \overline{X}_{1}
  38. s 1 2 s_{1}^{2}
  39. N 2 N_{2}
  40. X ¯ 2 \overline{X}_{2}
  41. s 2 2 s_{2}^{2}
  42. t t
  43. ν \nu
  44. P P
  45. P s i m P_{sim}
  46. t t
  47. ν \nu
  48. P P
  49. P s i m P_{sim}

Welch–Satterthwaite_equation.html

  1. n n
  2. χ = i = 1 n k i s i 2 . \chi^{\prime}=\sum_{i=1}^{n}k_{i}s_{i}^{2}.
  3. k i k_{i}
  4. 1 n i = 1 v i + 1 \frac{1}{n_{i}}=\frac{1}{v_{i}+1}
  5. $\mathbf{ }$
  6. ν χ ( i = 1 n k i s i 2 ) 2 i = 1 n ( k i s i 2 ) 2 ν i \nu_{\chi^{\prime}}\approx\frac{\displaystyle\left(\sum_{i=1}^{n}k_{i}s_{i}^{2% }\right)^{2}}{\displaystyle\sum_{i=1}^{n}\frac{(k_{i}s_{i}^{2})^{2}}{\nu_{i}}}
  7. σ < s u b > i 2 σ<sub>i^{2}

Werckmeister_temperament.html

  1. 1 1 \frac{1}{1}
  2. 256 243 \frac{256}{243}
  3. 64 81 2 \frac{64}{81}\sqrt{2}
  4. 32 27 \frac{32}{27}
  5. 256 243 2 4 \frac{256}{243}\sqrt[4]{2}
  6. 4 3 \frac{4}{3}
  7. 1024 729 \frac{1024}{729}
  8. 8 9 8 4 \frac{8}{9}\sqrt[4]{8}
  9. 128 81 \frac{128}{81}
  10. 1024 729 2 4 \frac{1024}{729}\sqrt[4]{2}
  11. 16 9 \frac{16}{9}
  12. 128 81 2 4 \frac{128}{81}\sqrt[4]{2}
  13. 1 1 \frac{1}{1}
  14. 120 120
  15. 16384 19683 2 3 \frac{16384}{19683}\sqrt[3]{2}
  16. 114 1 5 114\frac{1}{5}
  17. 114 1 2 114\frac{1}{2}
  18. 8 9 2 3 \frac{8}{9}\sqrt[3]{2}
  19. 107 1 5 107\frac{1}{5}
  20. 32 27 \frac{32}{27}
  21. 101 1 5 101\frac{1}{5}
  22. 64 81 4 3 \frac{64}{81}\sqrt[3]{4}
  23. 95 3 5 95\frac{3}{5}
  24. 4 3 \frac{4}{3}
  25. 90 90
  26. 1024 729 \frac{1024}{729}
  27. 85 1 3 85\frac{1}{3}
  28. 32 27 2 3 \frac{32}{27}\sqrt[3]{2}
  29. 80 1 5 80\frac{1}{5}
  30. 8192 6561 2 3 \frac{8192}{6561}\sqrt[3]{2}
  31. 76 2 15 76\frac{2}{15}
  32. 256 243 4 3 \frac{256}{243}\sqrt[3]{4}
  33. 71 7 10 71\frac{7}{10}
  34. 9 4 2 3 \frac{9}{4\sqrt[3]{2}}
  35. 67 1 5 67\frac{1}{5}
  36. 4096 2187 \frac{4096}{2187}
  37. 64 64
  38. 1 1 \frac{1}{1}
  39. 8 9 2 4 \frac{8}{9}\sqrt[4]{2}
  40. 9 8 \frac{9}{8}
  41. 2 4 \sqrt[4]{2}
  42. 8 9 2 \frac{8}{9}\sqrt{2}
  43. 9 8 2 4 \frac{9}{8}\sqrt[4]{2}
  44. 2 \sqrt{2}
  45. 3 2 \frac{3}{2}
  46. 128 81 \frac{128}{81}
  47. 8 4 \sqrt[4]{8}
  48. 3 8 4 \frac{3}{\sqrt[4]{8}}
  49. 4 3 2 \frac{4}{3}\sqrt{2}
  50. 196 = 7 × 7 × 4 196=7\times 7\times 4

West_number.html

  1. W n = W o P V f ( T H + T K ) ( T H - T K ) = B n ( T H + T K ) ( T H - T K ) W_{n}=\frac{Wo}{PVf}\frac{(T_{H}+T_{K})}{(T_{H}-T_{K})}=B_{n}\frac{(T_{H}+T_{K% })}{(T_{H}-T_{K})}
  2. W o = W n P V f ( T H - T K ) ( T H + T K ) W_{o}=W_{n}PVf\frac{(T_{H}-T_{K})}{(T_{H}+T_{K})}

Weyl's_lemma_(Laplace_equation).html

  1. Ω \Omega
  2. n n
  3. n \mathbb{R}^{n}
  4. Δ \Delta
  5. u L loc 1 ( Ω ) u\in L_{\mathrm{loc}}^{1}(\Omega)
  6. Ω u ( x ) Δ ϕ ( x ) d x = 0 \int_{\Omega}u(x)\Delta\phi(x)\,dx=0
  7. ϕ C c ( Ω ) \phi\in C_{c}^{\infty}(\Omega)
  8. u C ( Ω ) u\in C^{\infty}(\Omega)
  9. Δ u = 0 \Delta u=0
  10. Ω \Omega
  11. Ω \Omega
  12. Ω \partial\Omega
  13. u u
  14. ϕ ϵ \phi_{\epsilon}
  15. u ϵ = ϕ ϵ u u_{\epsilon}=\phi_{\epsilon}\ast u
  16. u ϵ u_{\epsilon}
  17. ϵ 0 \epsilon\to 0
  18. u u
  19. T D ( Ω ) T\in D^{\prime}(\Omega)
  20. T , Δ ϕ = 0 \langle T,\Delta\phi\rangle=0
  21. ϕ C c ( Ω ) \phi\in C_{c}^{\infty}(\Omega)
  22. T = T u T=T_{u}
  23. u C ( Ω ) u\in C^{\infty}(\Omega)
  24. P P
  25. P u Pu
  26. u u
  27. u u
  28. Δ u = 0 \Delta u=0
  29. u u
  30. 0
  31. u C ( Ω ) u\in C^{\infty}(\Omega)
  32. Δ u = 0 \Delta u=0

Weyl_scalar.html

  1. { Ψ 0 , Ψ 1 , Ψ 2 , Ψ 3 , Ψ 4 } \{\Psi_{0},\Psi_{1},\Psi_{2},\Psi_{3},\Psi_{4}\}
  2. { l a , n a , m a , m ¯ a } \{l^{a},n^{a},m^{a},\bar{m}^{a}\}
  3. { ( - , + , + , + ) ; l a n a = - 1 , m a m ¯ a = 1 } \{(-,+,+,+);l^{a}n_{a}=-1\,,m^{a}\bar{m}_{a}=1\}
  4. Ψ 0 := C α β γ δ l α m β l γ m δ , \Psi_{0}:=C_{\alpha\beta\gamma\delta}l^{\alpha}m^{\beta}l^{\gamma}m^{\delta}\ ,
  5. Ψ 1 := C α β γ δ l α n β l γ m δ , \Psi_{1}:=C_{\alpha\beta\gamma\delta}l^{\alpha}n^{\beta}l^{\gamma}m^{\delta}\ ,
  6. Ψ 2 := C α β γ δ l α m β m ¯ γ n δ , \Psi_{2}:=C_{\alpha\beta\gamma\delta}l^{\alpha}m^{\beta}\bar{m}^{\gamma}n^{% \delta}\ ,
  7. Ψ 3 := C α β γ δ l α n β m ¯ γ n δ , \Psi_{3}:=C_{\alpha\beta\gamma\delta}l^{\alpha}n^{\beta}\bar{m}^{\gamma}n^{% \delta}\ ,
  8. Ψ 4 := C α β γ δ n α m ¯ β n γ m ¯ δ . \Psi_{4}:=C_{\alpha\beta\gamma\delta}n^{\alpha}\bar{m}^{\beta}n^{\gamma}\bar{m% }^{\delta}\ .
  9. { ( + , - , - , - ) ; l a n a = 1 , m a m ¯ a = - 1 } \{(+,-,-,-);l^{a}n_{a}=1\,,m^{a}\bar{m}_{a}=-1\}
  10. Ψ i \Psi_{i}
  11. Ψ i - Ψ i \Psi_{i}\mapsto-\Psi_{i}
  12. Ψ 0 = D σ - δ κ - ( ρ + ρ ¯ ) σ - ( 3 ε - ε ¯ ) σ + ( τ - π ¯ + α ¯ + 3 β ) κ , \Psi_{0}=D\sigma-\delta\kappa-(\rho+\bar{\rho})\sigma-(3\varepsilon-\bar{% \varepsilon})\sigma+(\tau-\bar{\pi}+\bar{\alpha}+3\beta)\kappa\,,
  13. Ψ 1 = D β - δ ε - ( α + π ) σ - ( ρ ¯ - ε ¯ ) β + ( μ + γ ) κ + ( α ¯ - π ¯ ) ε , \Psi_{1}=D\beta-\delta\varepsilon-(\alpha+\pi)\sigma-(\bar{\rho}-\bar{% \varepsilon})\beta+(\mu+\gamma)\kappa+(\bar{\alpha}-\bar{\pi})\varepsilon\,,
  14. Ψ 2 = δ ¯ τ - Δ ρ - ( ρ μ ¯ + σ λ ) + ( β ¯ - α - τ ¯ ) τ + ( γ + γ ¯ ) ρ + ν κ - 2 Λ , \Psi_{2}=\bar{\delta}\tau-\Delta\rho-(\rho\bar{\mu}+\sigma\lambda)+(\bar{\beta% }-\alpha-\bar{\tau})\tau+(\gamma+\bar{\gamma})\rho+\nu\kappa-2\Lambda\,,
  15. Ψ 3 = δ ¯ γ - Δ α + ( ρ + ε ) ν - ( τ + β ) λ + ( γ ¯ - μ ¯ ) α + ( β ¯ - τ ¯ ) γ . \Psi_{3}=\bar{\delta}\gamma-\Delta\alpha+(\rho+\varepsilon)\nu-(\tau+\beta)% \lambda+(\bar{\gamma}-\bar{\mu})\alpha+(\bar{\beta}-\bar{\tau})\gamma\,.
  16. Ψ 4 = δ ν - Δ λ - ( μ + μ ¯ ) λ - ( 3 γ - γ ¯ ) λ + ( 3 α + β ¯ + π - τ ¯ ) ν . \Psi_{4}=\delta\nu-\Delta\lambda-(\mu+\bar{\mu})\lambda-(3\gamma-\bar{\gamma})% \lambda+(3\alpha+\bar{\beta}+\pi-\bar{\tau})\nu\,.
  17. Λ \Lambda
  18. Ψ 2 \Psi_{2}
  19. Λ := R 24 \Lambda:=\frac{R}{24}
  20. g a b g_{ab}
  21. Ψ 2 \Psi_{2}
  22. Ψ 1 \Psi_{1}
  23. Ψ 3 \Psi_{3}
  24. Ψ 0 \Psi_{0}
  25. Ψ 4 \Psi_{4}
  26. Ψ 1 \Psi_{1}
  27. Ψ 3 \Psi_{3}
  28. Ψ 4 \Psi_{4}
  29. Ψ 4 = 1 2 ( h ¨ θ ^ θ ^ - h ¨ ϕ ^ ϕ ^ ) + i h ¨ θ ^ ϕ ^ = - h ¨ + + i h ¨ × . \Psi_{4}=\frac{1}{2}\left(\ddot{h}_{\hat{\theta}\hat{\theta}}-\ddot{h}_{\hat{% \phi}\hat{\phi}}\right)+i\ddot{h}_{\hat{\theta}\hat{\phi}}=-\ddot{h}_{+}+i% \ddot{h}_{\times}\ .
  30. h + h_{+}
  31. h × h_{\times}
  32. Ψ 2 \Psi_{2}
  33. Ψ 0 \Psi_{0}
  34. Ψ 4 \Psi_{4}
  35. Ψ 0 \Psi_{0}

Wheat_and_chessboard_problem.html

  1. T 64 = 1 + 2 + 4 + + 9 , 223 , 372 , 036 , 854 , 775 , 808 = 18 , 446 , 744 , 073 , 709 , 551 , 615 T_{64}=1+2+4+\cdots+9,223,372,036,854,775,808=18,446,744,073,709,551,615
  2. T 64 T_{64}
  3. T 64 = 2 0 + 2 1 + 2 2 + + 2 63 T_{64}=2^{0}+2^{1}+2^{2}+\cdots+2^{63}
  4. i = 0 63 2 i . \sum_{i=0}^{63}2^{i}.\,
  5. T 64 = 2 64 - 1. T_{64}=2^{64}-1.\,
  6. s = 2 0 + 2 1 + 2 2 + + 2 63 . s=2^{0}+2^{1}+2^{2}+\cdots+2^{63}.
  7. 2 s = 2 1 + 2 2 + 2 3 + + 2 63 + 2 64 . 2s=2^{1}+2^{2}+2^{3}+\cdots+2^{63}+2^{64}.
  8. 2 s - s = - 2 0 + 2 64 2s-s=-2^{0}+2^{64}
  9. s = 2 64 - 1. \therefore s=2^{64}-1.\,

Whitehead's_point-free_geometry.html

  1. x x . x\leq x.
  2. ( x z and z y ) x y . (x\leq z\and z\leq y)\rightarrow x\leq y.
  3. ( x y and y x ) x = y . (x\leq y\and y\leq x)\rightarrow x=y.
  4. z [ x z and y z ] . \exists z[x\leq z\and y\leq z].
  5. x < y z [ x < z < y ] . x<y\rightarrow\exists z[x<z<y].
  6. y z [ y < x and x < z ] . \exists yz[y<x\and x<z].
  7. z [ z < x z < y ] x y . \forall z[z<x\rightarrow z<y]\rightarrow x\leq y.
  8. C x x . \ Cxx.
  9. C x y C y x . Cxy\rightarrow Cyx.
  10. z [ C z x C z y ] x = y . \forall z[Czx\leftrightarrow Czy]\rightarrow x=y.
  11. y [ y < x ] . \exists y[y<x].
  12. z [ C z x and C z y ] . \exists z[Czx\and Czy].
  13. y z [ ( y x ) and ( z x ) and ¬ C y z ] . \exists yz[(y\leq x)\and(z\leq x)\and\neg Cyz].

Whitening_transformation.html

  1. M M
  2. Y Y
  3. X X
  4. X X
  5. M M
  6. 0
  7. M M
  8. X X
  9. M - 1 / 2 M^{-1/2}
  10. M M
  11. X X
  12. M = E [ X X T ] M=\operatorname{E}[XX^{T}]
  13. M M
  14. M 1 / 2 M^{1/2}
  15. M 1 / 2 M 1 / 2 = M M^{1/2}M^{1/2}=M
  16. M M
  17. M 1 / 2 M^{1/2}
  18. Y = M - 1 / 2 X Y=M^{-1/2}X
  19. Cov ( Y ) = E [ Y Y T ] = M - 1 / 2 E [ X X T ] ( M - 1 / 2 ) T = M - 1 / 2 M M - 1 / 2 = I \operatorname{Cov}(Y)=\operatorname{E}[YY^{T}]=M^{-1/2}\operatorname{E}[XX^{T}% ](M^{-1/2})^{T}=M^{-1/2}MM^{-1/2}=I
  20. M M
  21. M 1 / 2 M^{1/2}
  22. X X
  23. X X
  24. Y Y
  25. m m
  26. m m
  27. M M

Whitney_extension_theorem.html

  1. f ( x ) = | α | m D α f ( y ) α ! ( x - y ) α + | α | = m R α ( x , y ) ( x - y ) α α ! f({x})=\sum_{|\alpha|\leq m}\frac{D^{\alpha}f({y})}{\alpha!}\cdot({x}-{y})^{% \alpha}+\sum_{|\alpha|=m}R_{\alpha}({x},{y})\frac{({x}-{y})^{\alpha}}{\alpha!}
  2. f α ( x ) = | β | m - | α | f α + β ( y ) β ! ( x - y ) β + R α ( x , y ) f_{\alpha}({x})=\sum_{|\beta|\leq m-|\alpha|}\frac{f_{\alpha+\beta}({y})}{% \beta!}({x}-{y})^{\beta}+R_{\alpha}({x},{y})
  3. | α | m |\alpha|\leq m
  4. E : C ( 𝐑 + ) C ( 𝐑 ) , \displaystyle{E:C^{\infty}(\mathbf{R}^{+})\rightarrow C^{\infty}(\mathbf{R}),}
  5. E ( f ) ( x ) = m = 1 a m f ( - b m x ) φ ( - b m x ) ( x < 0 ) , \displaystyle{E(f)(x)=\sum_{m=1}^{\infty}a_{m}f(-b_{m}x)\varphi(-b_{m}x)\,\,\,% (x<0),}
  6. g ( z ) = m = 1 a m z m g(z)=\sum_{m=1}^{\infty}a_{m}z^{m}
  7. W ( z ) = j 1 ( 1 - z / 2 j ) , W(z)=\prod_{j\geq 1}(1-z/2^{j}),
  8. M ( z ) = j 1 ( - 1 ) j W ( 2 j ) ( z - 2 j ) M(z)=\sum_{j\geq 1}(-1)^{j}\over W^{\prime}(2^{j})(z-2^{j})
  9. g ( z ) = W ( z ) M ( z ) \displaystyle{g(z)=W(z)M(z)}
  10. C ( Ω ¯ ) C ( 𝐑 n ) \displaystyle{C^{\infty}(\overline{\Omega})\rightarrow C^{\infty}(\mathbf{R}^{% n})}

Wick's_theorem.html

  1. A ^ \hat{A}
  2. B ^ \hat{B}
  3. A ^ B ^ A ^ B ^ - : A ^ B ^ : \hat{A}^{\bullet}\,\hat{B}^{\bullet}\equiv\hat{A}\,\hat{B}\,-\mathopen{:}\hat{% A}\,\hat{B}\mathclose{:}
  4. : O ^ : \mathopen{:}\hat{O}\mathclose{:}
  5. O ^ \hat{O}
  6. A ^ \hat{A}
  7. B ^ \hat{B}
  8. A ^ \hat{A}
  9. B ^ \hat{B}
  10. N N
  11. a ^ i \hat{a}_{i}^{\dagger}
  12. a ^ i \hat{a}_{i}
  13. i = 1 , 2 , 3 , N i=1,2,3\ldots,N
  14. [ a ^ i , a ^ j ] = δ i j [\hat{a}_{i},\hat{a}_{j}^{\dagger}]=\delta_{ij}
  15. δ i j \delta_{ij}
  16. a ^ i a ^ j = a ^ i a ^ j - : a ^ i a ^ j : = 0 \hat{a}_{i}^{\bullet}\,\hat{a}_{j}^{\bullet}=\hat{a}_{i}\,\hat{a}_{j}\,-% \mathopen{:}\,\hat{a}_{i}\,\hat{a}_{j}\,\mathclose{:}\,=0
  17. a ^ i a ^ j = a ^ i a ^ j - : a ^ i a ^ j : = 0 \hat{a}_{i}^{\dagger\bullet}\,\hat{a}_{j}^{\dagger\bullet}=\hat{a}_{i}^{% \dagger}\,\hat{a}_{j}^{\dagger}\,-\,\mathopen{:}\hat{a}_{i}^{\dagger}\,\hat{a}% _{j}^{\dagger}\,\mathclose{:}\,=0
  18. a ^ i a ^ j = a ^ i a ^ j - : a ^ i a ^ j : = 0 \hat{a}_{i}^{\dagger\bullet}\,\hat{a}_{j}^{\bullet}=\hat{a}_{i}^{\dagger}\,% \hat{a}_{j}\,-\mathopen{:}\,\hat{a}_{i}^{\dagger}\,\hat{a}_{j}\,\mathclose{:}% \,=0
  19. a ^ i a ^ j = a ^ i a ^ j - : a ^ i a ^ j : = δ i j \hat{a}_{i}^{\bullet}\,\hat{a}_{j}^{\dagger\bullet}=\hat{a}_{i}\,\hat{a}_{j}^{% \dagger}\,-\mathopen{:}\,\hat{a}_{i}\,\hat{a}_{j}^{\dagger}\,\mathclose{:}\,=% \delta_{ij}
  20. i , j = 1 , , N i,j=1,\ldots,N
  21. a ^ i \hat{a}_{i}
  22. a ^ i \hat{a}_{i}^{\dagger}
  23. [ a ^ i , a ^ j ] = 0 \left[\hat{a}_{i}^{\dagger},\hat{a}_{j}^{\dagger}\right]=0
  24. [ a ^ i , a ^ j ] = 0 \left[\hat{a}_{i},\hat{a}_{j}\right]=0
  25. [ a ^ i , a ^ j ] = δ i j \left[\hat{a}_{i},\hat{a}_{j}^{\dagger}\right]=\delta_{ij}
  26. i , j = 1 , , N i,j=1,\ldots,N
  27. [ A ^ , B ^ ] A ^ B ^ - B ^ A ^ \left[\hat{A},\hat{B}\right]\equiv\hat{A}\hat{B}-\hat{B}\hat{A}
  28. δ i j \delta_{ij}
  29. a ^ i \hat{a}_{i}
  30. a ^ i \hat{a}_{i}^{\dagger}
  31. a ^ i a ^ j = a ^ j a ^ i + δ i j = a ^ j a ^ i + a ^ i a ^ j = : a ^ i a ^ j : + a ^ i a ^ j \hat{a}_{i}\,\hat{a}_{j}^{\dagger}=\hat{a}_{j}^{\dagger}\,\hat{a}_{i}+\delta_{% ij}=\hat{a}_{j}^{\dagger}\,\hat{a}_{i}+\hat{a}_{i}^{\bullet}\,\hat{a}_{j}^{% \dagger\bullet}=\,\mathopen{:}\,\hat{a}_{i}\,\hat{a}_{j}^{\dagger}\,\mathclose% {:}+\hat{a}_{i}^{\bullet}\,\hat{a}_{j}^{\dagger\bullet}
  32. a ^ i a ^ j \hat{a}_{i}\,\hat{a}_{j}^{\dagger}
  33. : a ^ i a ^ j : + a ^ i a ^ j \,\mathopen{:}\,\hat{a}_{i}\,\hat{a}_{j}^{\dagger}\,\mathclose{:}+\hat{a}_{i}^% {\bullet}\,\hat{a}_{j}^{\dagger\bullet}
  34. a ^ i a ^ j a ^ k = ( a ^ j a ^ i + δ i j ) a ^ k = a ^ j a ^ i a ^ k + δ i j a ^ k = a ^ j a ^ i a ^ k + a ^ i a ^ j a ^ k = : a ^ i a ^ j a ^ k : + : a ^ i a ^ j a ^ k : \hat{a}_{i}\,\hat{a}_{j}^{\dagger}\,\hat{a}_{k}=(\hat{a}_{j}^{\dagger}\,\hat{a% }_{i}+\delta_{ij})\hat{a}_{k}=\hat{a}_{j}^{\dagger}\,\hat{a}_{i}\,\hat{a}_{k}+% \delta_{ij}\hat{a}_{k}=\hat{a}_{j}^{\dagger}\,\hat{a}_{i}\,\hat{a}_{k}+\hat{a}% _{i}^{\bullet}\,\hat{a}_{j}^{\dagger\bullet}\hat{a}_{k}=\,\mathopen{:}\,\hat{a% }_{i}\,\hat{a}_{j}^{\dagger}\hat{a}_{k}\,\mathclose{:}+\mathopen{:}\,\hat{a}_{% i}^{\bullet}\,\hat{a}_{j}^{\dagger\bullet}\,\hat{a}_{k}\mathclose{:}
  35. a ^ i a ^ j a ^ k a ^ l = ( a ^ j a ^ i + δ i j ) ( a ^ l a ^ k + δ k l ) \hat{a}_{i}\,\hat{a}_{j}^{\dagger}\,\hat{a}_{k}\,\hat{a}_{l}^{\dagger}=(\hat{a% }_{j}^{\dagger}\,\hat{a}_{i}+\delta_{ij})(\hat{a}_{l}^{\dagger}\,\hat{a}_{k}+% \delta_{kl})
  36. = a ^ j a ^ i a ^ l a ^ k + δ k l a ^ j a ^ i + δ i j a ^ l a ^ k + δ i j δ k l =\hat{a}_{j}^{\dagger}\,\hat{a}_{i}\,\hat{a}_{l}^{\dagger}\,\hat{a}_{k}+\delta% _{kl}\hat{a}_{j}^{\dagger}\,\hat{a}_{i}+\delta_{ij}\hat{a}_{l}^{\dagger}\hat{a% }_{k}+\delta_{ij}\delta_{kl}
  37. = a ^ j ( a ^ l a ^ i + δ i l ) a ^ k + δ k l a ^ j a ^ i + δ i j a ^ l a ^ k + δ i j δ k l =\hat{a}_{j}^{\dagger}(\hat{a}_{l}^{\dagger}\,\hat{a}_{i}+\delta_{il})\hat{a}_% {k}+\delta_{kl}\hat{a}_{j}^{\dagger}\,\hat{a}_{i}+\delta_{ij}\hat{a}_{l}^{% \dagger}\hat{a}_{k}+\delta_{ij}\delta_{kl}
  38. = a ^ j a ^ l a ^ i a ^ k + δ i l a ^ j a ^ k + δ k l a ^ j a ^ i + δ i j a ^ l a ^ k + δ i j δ k l =\hat{a}_{j}^{\dagger}\hat{a}_{l}^{\dagger}\,\hat{a}_{i}\hat{a}_{k}+\delta_{il% }\hat{a}_{j}^{\dagger}\,\hat{a}_{k}+\delta_{kl}\hat{a}_{j}^{\dagger}\,\hat{a}_% {i}+\delta_{ij}\hat{a}_{l}^{\dagger}\hat{a}_{k}+\delta_{ij}\delta_{kl}
  39. = : a ^ i a ^ j a ^ k a ^ l : + : a ^ i a ^ j a ^ k a ^ l : + : a ^ i a ^ j a ^ k a ^ l : + : a ^ i a ^ j a ^ k a ^ l : + : a ^ i a ^ j a ^ k a ^ l : =\,\mathopen{:}\hat{a}_{i}\,\hat{a}_{j}^{\dagger}\,\hat{a}_{k}\,\hat{a}_{l}^{% \dagger}\,\mathclose{:}+\mathopen{:}\,\hat{a}_{i}^{\bullet}\,\hat{a}_{j}^{% \dagger}\,\hat{a}_{k}\,\hat{a}_{l}^{\dagger\bullet}\,\mathclose{:}+\mathopen{:% }\,\hat{a}_{i}\,\hat{a}_{j}^{\dagger}\,\hat{a}_{k}^{\bullet}\,\hat{a}_{l}^{% \dagger\bullet}\,\mathclose{:}+\mathopen{:}\,\hat{a}_{i}^{\bullet}\,\hat{a}_{j% }^{\dagger\bullet}\,\hat{a}_{k}\,\hat{a}_{l}^{\dagger}\,\mathclose{:}+\,% \mathopen{:}\hat{a}_{i}^{\bullet}\,\hat{a}_{j}^{\dagger\bullet}\,\hat{a}_{k}^{% \bullet\bullet}\,\hat{a}_{l}^{\dagger\bullet\bullet}\mathclose{:}
  40. {}^{\bullet}
  41. a ^ i a ^ j a ^ k a ^ l \hat{a}_{i}\,\hat{a}_{j}^{\dagger}\,\hat{a}_{k}\,\hat{a}_{l}^{\dagger}
  42. A ^ B ^ C ^ D ^ E ^ F ^ \hat{A}\hat{B}\hat{C}\hat{D}\hat{E}\hat{F}\ldots
  43. A ^ B ^ C ^ D ^ E ^ F ^ = : A ^ B ^ C ^ D ^ E ^ F ^ : + singles : A ^ B ^ C ^ D ^ E ^ F ^ : + doubles : A ^ B ^ C ^ D ^ E ^ F ^ : + \begin{aligned}\displaystyle\hat{A}\hat{B}\hat{C}\hat{D}\hat{E}\hat{F}\ldots&% \displaystyle=\mathopen{:}\hat{A}\hat{B}\hat{C}\hat{D}\hat{E}\hat{F}\ldots% \mathclose{:}\\ &\displaystyle\quad+\sum\text{singles}\mathopen{:}\hat{A}^{\bullet}\hat{B}^{% \bullet}\hat{C}\hat{D}\hat{E}\hat{F}\ldots\mathclose{:}\\ &\displaystyle\quad+\sum\text{doubles}\mathopen{:}\hat{A}^{\bullet}\hat{B}^{% \bullet\bullet}\hat{C}^{\bullet\bullet}\hat{D}^{\bullet}\hat{E}\hat{F}\ldots% \mathclose{:}\\ &\displaystyle\quad+\ldots\end{aligned}
  44. N = 2 N=2
  45. f ^ i \hat{f}_{i}^{\dagger}
  46. f ^ i \hat{f}_{i}
  47. i = 1 , 2 i=1,2
  48. f ^ 1 f ^ 2 f ^ 1 f ^ 2 = : f ^ 1 f ^ 2 f ^ 1 f ^ 2 : - : f ^ 1 f ^ 2 f ^ 1 f ^ 2 : + : f ^ 1 f ^ 2 f ^ 1 f ^ 2 : + : f ^ 1 f ^ 2 f ^ 1 f ^ 2 : - : f ^ 1 f ^ 2 f ^ 1 f ^ 2 : - : f ^ 1 f ^ 2 f ^ 1 f ^ 2 : + : f ^ 1 f ^ 2 f ^ 1 f ^ 2 : \begin{array}[]{ll}\hat{f}_{1}\,\hat{f}_{2}\,\hat{f}_{1}^{\dagger}\,\hat{f}_{2% }^{\dagger}&=\,\mathopen{:}\hat{f}_{1}\,\hat{f}_{2}\,\hat{f}_{1}^{\dagger}\,% \hat{f}_{2}^{\dagger}\,\mathclose{:}\\ &-\,\mathopen{:}\hat{f}_{1}^{\bullet}\,\hat{f}_{2}\,\hat{f}_{1}^{\dagger% \bullet}\,\hat{f}_{2}^{\dagger}\,\mathclose{:}+\,\mathopen{:}\hat{f}_{1}^{% \bullet}\,\hat{f}_{2}\,\hat{f}_{1}^{\dagger}\,\hat{f}_{2}^{\dagger\bullet}\,% \mathclose{:}+\,\mathopen{:}\hat{f}_{1}\,\hat{f}_{2}^{\bullet}\,\hat{f}_{1}^{% \dagger\bullet}\,\hat{f}_{2}^{\dagger}\,\mathclose{:}-\mathopen{:}\hat{f}_{1}% \,\hat{f}_{2}^{\bullet}\,\hat{f}_{1}^{\dagger}\,\hat{f}_{2}^{\dagger\bullet}\,% \mathclose{:}\\ &-\mathopen{:}\hat{f}_{1}^{\bullet\bullet}\,\hat{f}_{2}^{\bullet}\,\hat{f}_{1}% ^{\dagger\bullet\bullet}\,\hat{f}_{2}^{\dagger\bullet}\,\mathclose{:}+% \mathopen{:}\hat{f}_{1}^{\bullet\bullet}\,\hat{f}_{2}^{\bullet}\,\hat{f}_{1}^{% \dagger\bullet}\,\hat{f}_{2}^{\dagger\bullet\bullet}\mathclose{:}\end{array}
  49. 𝒞 ( x 1 , x 2 ) = 0 | 𝒯 ϕ i ( x 1 ) ϕ i ( x 2 ) | 0 = ϕ i ( x 1 ) ϕ i ( x 2 ) ¯ = i Δ F ( x 1 - x 2 ) = i d 4 k ( 2 π ) 4 e - i k ( x 1 - x 2 ) ( k 2 - m 2 ) + i ϵ . \mathcal{C}(x_{1},x_{2})=\left\langle 0|\mathcal{T}\phi_{i}(x_{1})\phi_{i}(x_{% 2})|0\right\rangle=\overline{\phi_{i}(x_{1})\phi_{i}(x_{2})}=i\Delta_{F}(x_{1}% -x_{2})=i\int{\frac{d^{4}k}{(2\pi)^{4}}\frac{e^{-ik(x_{1}-x_{2})}}{(k^{2}-m^{2% })+i\epsilon}}.
  50. A B ¯ = 𝒯 A B - : A B : \overline{AB}=\mathcal{T}AB-\mathopen{:}AB\mathclose{:}
  51. 𝒯 Π k = 1 m ϕ ( x k ) = : Π ϕ i ( x k ) : + α , β ϕ ( x α ) ϕ ( x β ) ¯ : Π k α , β ϕ i ( x k ) : + \mathcal{T}\Pi_{k=1}^{m}\phi(x_{k})=\mathopen{:}\Pi\phi_{i}(x_{k})\mathclose{:% }+\sum_{\alpha,\beta}\overline{\phi(x_{\alpha})\phi(x_{\beta})}\mathopen{:}\Pi% _{k\not=\alpha,\beta}\phi_{i}(x_{k})\mathclose{:}+
  52. + ( α , β ) , ( γ , δ ) ϕ ( x α ) ϕ ( x β ) ¯ ϕ ( x γ ) ϕ ( x δ ) ¯ : Π k α , β , γ , δ ϕ i ( x k ) : + . \mathcal{+}\sum_{(\alpha,\beta),(\gamma,\delta)}\overline{\phi(x_{\alpha})\phi% (x_{\beta})}\;\overline{\phi(x_{\gamma})\phi(x_{\delta})}\mathopen{:}\Pi_{k% \not=\alpha,\beta,\gamma,\delta}\phi_{i}(x_{k})\mathclose{:}+\cdots.
  53. F m i ( x ) = 0 | 𝒯 ϕ i ( x 1 ) ϕ i ( x 2 ) | 0 = pairs ϕ ( x 1 ) ϕ ( x 2 ) ¯ ϕ ( x m - 1 ) ϕ ( x m ¯ ) F_{m}^{i}(x)=\left\langle 0|\mathcal{T}\phi_{i}(x_{1})\phi_{i}(x_{2})|0\right% \rangle=\sum_{\mathrm{pairs}}\overline{\phi(x_{1})\phi(x_{2})}\cdots\overline{% \phi(x_{m-1})\phi(x_{m}})
  54. G p ( n ) = 0 | 𝒯 : v i ( y 1 ) : : v i ( y n ) : ϕ i ( x 1 ) ϕ i ( x p ) | 0 G_{p}^{(n)}=\left\langle 0|\mathcal{T}\mathopen{:}v_{i}(y_{1})\mathclose{:}% \dots\mathopen{:}v_{i}(y_{n})\mathclose{:}\phi_{i}(x_{1})\cdots\phi_{i}(x_{p})% |0\right\rangle
  55. v = g y 4 : v i ( y 1 ) : = : ϕ i ( y 1 ) ϕ i ( y 1 ) ϕ i ( y 1 ) ϕ i ( y 1 ) : v=gy^{4}\Rightarrow\mathopen{:}v_{i}(y_{1})\mathclose{:}=\mathopen{:}\phi_{i}(% y_{1})\phi_{i}(y_{1})\phi_{i}(y_{1})\phi_{i}(y_{1})\mathclose{:}

Wick_product.html

  1. X 1 , , X k \langle X_{1},\dots,X_{k}\rangle\,
  2. = 1 \langle\rangle=1\,
  3. X 1 , , X k X i = X 1 , , X i - 1 , X ^ i , X i + 1 , , X k , {\partial\langle X_{1},\dots,X_{k}\rangle\over\partial X_{i}}=\langle X_{1},% \dots,X_{i-1},\widehat{X}_{i},X_{i+1},\dots,X_{k}\rangle,
  4. X ^ i \widehat{X}_{i}
  5. E X 1 , , X k = 0 for k 1. \operatorname{E}\langle X_{1},\dots,X_{k}\rangle=0\mbox{ for }~{}k\geq 1.\,
  6. X = X - E X , \langle X\rangle=X-\operatorname{E}X,\,
  7. X , Y = X Y - E Y X - E X Y + 2 ( E X ) ( E Y ) - E ( X Y ) . \langle X,Y\rangle=XY-\operatorname{E}Y\cdot X-\operatorname{E}X\cdot Y+2(% \operatorname{E}X)(\operatorname{E}Y)-\operatorname{E}(XY).\,
  8. X , Y , Z = X Y Z - E Y X Z - E Z X Y - E X Y Z + 2 ( E Y ) ( E Z ) X + 2 ( E X ) ( E Z ) Y + 2 ( E X ) ( E Y ) Z - E ( X Z ) Y - E ( X Y ) Z - E ( Y Z ) X \begin{aligned}\displaystyle\langle X,Y,Z\rangle=&\displaystyle XYZ\\ &\displaystyle-\operatorname{E}Y\cdot XZ\\ &\displaystyle-\operatorname{E}Z\cdot XY\\ &\displaystyle-\operatorname{E}X\cdot YZ\\ &\displaystyle+2(\operatorname{E}Y)(\operatorname{E}Z)\cdot X\\ &\displaystyle+2(\operatorname{E}X)(\operatorname{E}Z)\cdot Y\\ &\displaystyle+2(\operatorname{E}X)(\operatorname{E}Y)\cdot Z\\ &\displaystyle-\operatorname{E}(XZ)\cdot Y\\ &\displaystyle-\operatorname{E}(XY)\cdot Z\\ &\displaystyle-\operatorname{E}(YZ)\cdot X\\ \end{aligned}
  9. : X 1 , , X k : :X_{1},\dots,X_{k}:\,
  10. X \langle X\rangle\,
  11. X n = X , , X X^{\prime n}=\langle X,\dots,X\rangle\,
  12. P n ( X ) = X , , X = X n P_{n}(X)=\langle X,\dots,X\rangle=X^{\prime n}\,
  13. P n ( x ) = n P n - 1 ( x ) , P_{n}^{\prime}(x)=nP_{n-1}(x),\,
  14. X n = B n ( X ) X^{\prime n}=B_{n}(X)\,
  15. X n = H n ( X ) X^{\prime n}=H_{n}(X)\,
  16. ( a X + b Y ) n = i = 0 n ( n i ) a i b n - i X i Y n - i (aX+bY)^{{}^{\prime}n}=\sum_{i=0}^{n}{n\choose i}a^{i}b^{n-i}X^{{}^{\prime}i}Y% ^{{}^{\prime}{n-i}}
  17. exp ( a X ) = def i = 0 a i i ! X i \langle\operatorname{exp}(aX)\rangle\ \stackrel{\mathrm{def}}{=}\ \sum_{i=0}^{% \infty}\frac{a^{i}}{i!}X^{{}^{\prime}i}

Widom_scaling.html

  1. α , α , β , γ , γ \alpha,\alpha^{\prime},\beta,\gamma,\gamma^{\prime}
  2. δ \delta
  3. M ( t , 0 ) ( - t ) β M(t,0)\simeq(-t)^{\beta}
  4. t 0 t\uparrow 0
  5. M ( 0 , H ) | H | 1 / δ sign ( H ) M(0,H)\simeq|H|^{1/\delta}\mathrm{sign}(H)
  6. H 0 H\rightarrow 0
  7. χ T ( t , 0 ) { ( t ) - γ , for t 0 ( - t ) - γ , for t 0 \chi_{T}(t,0)\simeq\begin{cases}(t)^{-\gamma},&\textrm{for}\ t\downarrow 0\\ (-t)^{-\gamma^{\prime}},&\textrm{for}\ t\uparrow 0\end{cases}
  8. c H ( t , 0 ) { ( t ) - α for t 0 ( - t ) - α for t 0 c_{H}(t,0)\simeq\begin{cases}(t)^{-\alpha}&\textrm{for}\ t\downarrow 0\\ (-t)^{-\alpha^{\prime}}&\textrm{for}\ t\uparrow 0\end{cases}
  9. t T - T c T c t\equiv\frac{T-T_{c}}{T_{c}}
  10. H ( t ) M | M | δ - 1 f ( t / | M | 1 / β ) H(t)\simeq M|M|^{\delta-1}f(t/|M|^{1/\beta})
  11. f f
  12. f ( t / | M | 1 / β ) 1 + const × ( t / | M | 1 / β ) ω + f(t/|M|^{1/\beta})\approx 1+{\rm const}\times(t/|M|^{1/\beta})^{\omega}+\dots
  13. ω \omega
  14. f ( t , H ) f(t,H)
  15. d d
  16. f r f_{r}
  17. f s f_{s}
  18. f s ( λ p t , λ q H ) = λ d f s ( t , H ) f_{s}(\lambda^{p}t,\lambda^{q}H)=\lambda^{d}f_{s}(t,H)\,
  19. λ q M ( λ p t , λ q H ) = λ d M ( t , H ) \lambda^{q}M(\lambda^{p}t,\lambda^{q}H)=\lambda^{d}M(t,H)\,
  20. H = 0 H=0
  21. λ = ( - t ) - 1 / p \lambda=(-t)^{-1/p}
  22. M ( t , 0 ) = ( - t ) d - q p M ( - 1 , 0 ) , M(t,0)=(-t)^{\frac{d-q}{p}}M(-1,0),
  23. t 0 t\uparrow 0
  24. β \beta
  25. β = d - q p ν 2 ( d - 2 + η ) . \beta=\frac{d-q}{p}\equiv\frac{\nu}{2}(d-2+\eta).
  26. t = 0 t=0
  27. λ = H - 1 / q \lambda=H^{-1/q}
  28. δ = q d - q d + 2 - η d - 2 + η . \delta=\frac{q}{d-q}\equiv\frac{d+2-\eta}{d-2+\eta}.
  29. q p = ν 2 ( d + 2 - η ) , 1 p = ν . \frac{q}{p}=\frac{\nu}{2}(d+2-\eta),~{}\frac{1}{p}=\nu.
  30. χ T \chi_{T}
  31. λ 2 q χ T ( λ p t , λ q H ) = λ d χ T ( t , H ) \lambda^{2q}\chi_{T}(\lambda^{p}t,\lambda^{q}H)=\lambda^{d}\chi_{T}(t,H)\,
  32. λ = ( t ) - 1 / p \lambda=(t)^{-1/p}
  33. t 0 t\downarrow 0
  34. λ = ( - t ) - 1 / p \lambda=(-t)^{-1/p}
  35. t 0 t\uparrow 0
  36. γ = γ = 2 q - d p \gamma=\gamma^{\prime}=\frac{2q-d}{p}\,
  37. c H c_{H}
  38. λ 2 p c H ( λ p t , λ q H ) = λ d c H ( t , H ) \lambda^{2p}c_{H}(\lambda^{p}t,\lambda^{q}H)=\lambda^{d}c_{H}(t,H)\,
  39. λ = ( t ) - 1 / p \lambda=(t)^{-1/p}
  40. t 0 t\downarrow 0
  41. λ = ( - t ) - 1 / p \lambda=(-t)^{-1/p}
  42. t 0 ) t\uparrow 0)
  43. α = α = 2 - d p = 2 - ν d \alpha=\alpha^{\prime}=2-\frac{d}{p}=2-\nu d
  44. p , q p,q\in\mathbb{R}
  45. α = α = 2 - ν d , \alpha=\alpha^{\prime}=2-\nu d,
  46. γ = γ = β ( δ - 1 ) = ν ( 2 - η ) . \gamma=\gamma^{\prime}=\beta(\delta-1)=\nu(2-\eta).

Wiener_deconvolution.html

  1. y ( t ) = ( h * x ) ( t ) + n ( t ) \ y(t)=(h*x)(t)+n(t)
  2. * *
  3. x ( t ) \ x(t)
  4. t \ t
  5. h ( t ) \ h(t)
  6. n ( t ) \ n(t)
  7. x ( t ) \ x(t)
  8. y ( t ) \ y(t)
  9. g ( t ) \ g(t)
  10. x ( t ) \ x(t)
  11. x ^ ( t ) = ( g * y ) ( t ) \ \hat{x}(t)=(g*y)(t)
  12. x ^ ( t ) \ \hat{x}(t)
  13. x ( t ) \ x(t)
  14. g ( t ) \ g(t)
  15. G ( f ) = H * ( f ) S ( f ) | H ( f ) | 2 S ( f ) + N ( f ) \ G(f)=\frac{H^{*}(f)S(f)}{|H(f)|^{2}S(f)+N(f)}
  16. G ( f ) \ G(f)
  17. H ( f ) \ H(f)
  18. g \ g
  19. h \ h
  20. f \ f
  21. S ( f ) \ S(f)
  22. x ( t ) \ x(t)
  23. N ( f ) \ N(f)
  24. n ( t ) \ n(t)
  25. * {}^{*}
  26. X ^ ( f ) = G ( f ) Y ( f ) \ \hat{X}(f)=G(f)Y(f)
  27. X ^ ( f ) \ \hat{X}(f)
  28. Y ( f ) \ Y(f)
  29. x ^ ( t ) \hat{x}(t)
  30. y ( t ) y(t)
  31. X ^ ( f ) \ \hat{X}(f)
  32. x ^ ( t ) \ \hat{x}(t)
  33. t \ t
  34. f \ f
  35. G ( f ) \displaystyle G(f)
  36. 1 / H ( f ) \ 1/H(f)
  37. SNR ( f ) = S ( f ) / N ( f ) \ \mathrm{SNR}(f)=S(f)/N(f)
  38. ϵ ( f ) = 𝔼 | X ( f ) - X ^ ( f ) | 2 \ \epsilon(f)=\mathbb{E}\left|X(f)-\hat{X}(f)\right|^{2}
  39. 𝔼 \ \mathbb{E}
  40. X ^ ( f ) \ \hat{X}(f)
  41. ϵ ( f ) = 𝔼 | X ( f ) - G ( f ) Y ( f ) | 2 = 𝔼 | X ( f ) - G ( f ) [ H ( f ) X ( f ) + V ( f ) ] | 2 = 𝔼 | [ 1 - G ( f ) H ( f ) ] X ( f ) - G ( f ) V ( f ) | 2 \begin{aligned}\displaystyle\epsilon(f)&\displaystyle=\mathbb{E}\left|X(f)-G(f% )Y(f)\right|^{2}\\ &\displaystyle=\mathbb{E}\left|X(f)-G(f)\left[H(f)X(f)+V(f)\right]\right|^{2}% \\ &\displaystyle=\mathbb{E}\big|\left[1-G(f)H(f)\right]X(f)-G(f)V(f)\big|^{2}% \end{aligned}
  42. ϵ ( f ) \displaystyle\epsilon(f)
  43. 𝔼 { X ( f ) V * ( f ) } = 𝔼 { V ( f ) X * ( f ) } = 0 \ \mathbb{E}\Big\{X(f)V^{*}(f)\Big\}=\mathbb{E}\Big\{V(f)X^{*}(f)\Big\}=0
  44. S ( f ) = 𝔼 | X ( f ) | 2 \ S(f)=\mathbb{E}|X(f)|^{2}
  45. N ( f ) = 𝔼 | V ( f ) | 2 \ N(f)=\mathbb{E}|V(f)|^{2}
  46. ϵ ( f ) = [ 1 - G ( f ) H ( f ) ] [ 1 - G ( f ) H ( f ) ] * S ( f ) + G ( f ) G * ( f ) N ( f ) \epsilon(f)=\Big[1-G(f)H(f)\Big]\Big[1-G(f)H(f)\Big]^{*}S(f)+G(f)G^{*}(f)N(f)
  47. G ( f ) \ G(f)
  48. d ϵ ( f ) d G ( f ) = G * ( f ) N ( f ) - H ( f ) [ 1 - G ( f ) H ( f ) ] * S ( f ) = 0 \ \frac{d\epsilon(f)}{dG(f)}=G^{*}(f)N(f)-H(f)\Big[1-G(f)H(f)\Big]^{*}S(f)=0

Wiener–Ikehara_theorem.html

  1. f ( s ) - 1 s - 1 f(s)-\frac{1}{s-1}
  2. n = 1 a ( n ) n - s \sum_{n=1}^{\infty}a(n)n^{-s}
  3. ( s ) b \Re(s)\geq b\,
  4. n X a ( n ) c b X b . \sum_{n\leq X}a(n)\sim\frac{c}{b}X^{b}.
  5. ( s ) = 1. \Re(s)=1.\,

Wigner_D-matrix.html

  1. [ J x , J y ] = i J z , [ J z , J x ] = i J y , [ J y , J z ] = i J x , [J_{x},J_{y}]=iJ_{z},\quad[J_{z},J_{x}]=iJ_{y},\quad[J_{y},J_{z}]=iJ_{x},
  2. \hbar
  3. J 2 = J x 2 + J y 2 + J z 2 J^{2}=J_{x}^{2}+J_{y}^{2}+J_{z}^{2}
  4. J z J_{z}
  5. J 2 J^{2}
  6. J 2 | j m = j ( j + 1 ) | j m , J z | j m = m | j m , J^{2}|jm\rangle=j(j+1)|jm\rangle,\quad J_{z}|jm\rangle=m|jm\rangle,
  7. ( α , β , γ ) = e - i α J z e - i β J y e - i γ J z , \mathcal{R}(\alpha,\beta,\gamma)=e^{-i\alpha J_{z}}e^{-i\beta J_{y}}e^{-i% \gamma J_{z}},
  8. D m m j ( α , β , γ ) j m | ( α , β , γ ) | j m = e - i m α d m m j ( β ) e - i m γ . D^{j}_{m^{\prime}m}(\alpha,\beta,\gamma)\equiv\langle jm^{\prime}|\mathcal{R}(% \alpha,\beta,\gamma)|jm\rangle=e^{-im^{\prime}\alpha}d^{j}_{m^{\prime}m}(\beta% )e^{-im\gamma}.
  9. d m m j ( β ) = j m | e - i β J y | j m d^{j}_{m^{\prime}m}(\beta)=\langle jm^{\prime}|e^{-i\beta J_{y}}|jm\rangle
  10. d m m j ( β ) = [ ( j + m ) ! ( j - m ) ! ( j + m ) ! ( j - m ) ! ] 1 / 2 s [ ( - 1 ) m - m + s ( j + m - s ) ! s ! ( m - m + s ) ! ( j - m - s ) ! ( cos β 2 ) 2 j + m - m - 2 s ( sin β 2 ) m - m + 2 s ] . \begin{array}[]{lcl}d^{j}_{m^{\prime}m}(\beta)&=&[(j+m^{\prime})!(j-m^{\prime}% )!(j+m)!(j-m)!]^{1/2}\sum\limits_{s}\left[\frac{(-1)^{m^{\prime}-m+s}}{(j+m-s)% !s!(m^{\prime}-m+s)!(j-m^{\prime}-s)!}\right.\\ &&\left.\cdot\left(\cos\frac{\beta}{2}\right)^{2j+m-m^{\prime}-2s}\left(\sin% \frac{\beta}{2}\right)^{m^{\prime}-m+2s}\right].\end{array}
  11. ( - 1 ) m - m + s (-1)^{m^{\prime}-m+s}
  12. ( - 1 ) s i m - m (-1)^{s}\,i^{m-m^{\prime}}
  13. P k ( a , b ) ( cos β ) P^{(a,b)}_{k}(\cos\beta)
  14. a a\,
  15. b b\,
  16. k = min ( j + m , j - m , j + m , j - m ) . k=\min(j+m,\,j-m,\,j+m^{\prime},\,j-m^{\prime}).
  17. If k = { j + m : a = m - m ; λ = m - m j - m : a = m - m ; λ = 0 j + m : a = m - m ; λ = 0 j - m : a = m - m ; λ = m - m \hbox{If}\quad k=\begin{cases}j+m:&\quad a=m^{\prime}-m;\quad\lambda=m^{\prime% }-m\\ j-m:&\quad a=m-m^{\prime};\quad\lambda=0\\ j+m^{\prime}:&\quad a=m-m^{\prime};\quad\lambda=0\\ j-m^{\prime}:&\quad a=m^{\prime}-m;\quad\lambda=m^{\prime}-m\\ \end{cases}
  18. b = 2 j - 2 k - a b=2j-2k-a\,
  19. d m m j ( β ) = ( - 1 ) λ ( 2 j - k k + a ) 1 / 2 ( k + b b ) - 1 / 2 ( sin β 2 ) a ( cos β 2 ) b P k ( a , b ) ( cos β ) , d^{j}_{m^{\prime}m}(\beta)=(-1)^{\lambda}{\left({{2j-k}\atop{k+a}}\right)}^{1/% 2}{\left({{k+b}\atop{b}}\right)}^{-1/2}\left(\sin\frac{\beta}{2}\right)^{a}% \left(\cos\frac{\beta}{2}\right)^{b}P^{(a,b)}_{k}(\cos\beta),
  20. a , b 0. a,b\geq 0.\,
  21. ( x , y , z ) = ( 1 , 2 , 3 ) (x,\,y,\,z)=(1,\,2,\,3)
  22. 𝒥 ^ 1 = i ( cos α cot β α + sin α β - cos α sin β γ ) 𝒥 ^ 2 = i ( sin α cot β α - cos α β - sin α sin β γ ) 𝒥 ^ 3 = - i α , \begin{array}[]{lcl}\hat{\mathcal{J}}_{1}&=&i\left(\cos\alpha\cot\beta\,{% \partial\over\partial\alpha}\,+\sin\alpha\,{\partial\over\partial\beta}\,-{% \cos\alpha\over\sin\beta}\,{\partial\over\partial\gamma}\,\right)\\ \hat{\mathcal{J}}_{2}&=&i\left(\sin\alpha\cot\beta\,{\partial\over\partial% \alpha}\,-\cos\alpha\;{\partial\over\partial\beta}\,-{\sin\alpha\over\sin\beta% }\,{\partial\over\partial\gamma}\,\right)\\ \hat{\mathcal{J}}_{3}&=&-i\;{\partial\over\partial\alpha},\end{array}
  23. 𝒫 ^ 1 = i ( cos γ sin β α - sin γ β - cot β cos γ γ ) 𝒫 ^ 2 = i ( - sin γ sin β α - cos γ β + cot β sin γ γ ) 𝒫 ^ 3 = - i γ , \begin{array}[]{lcl}\hat{\mathcal{P}}_{1}&=&\,i\left({\cos\gamma\over\sin\beta% }{\partial\over\partial\alpha}-\sin\gamma{\partial\over\partial\beta}-\cot% \beta\cos\gamma{\partial\over\partial\gamma}\right)\\ \hat{\mathcal{P}}_{2}&=&\,i\left(-{\sin\gamma\over\sin\beta}{\partial\over% \partial\alpha}-\cos\gamma{\partial\over\partial\beta}+\cot\beta\sin\gamma{% \partial\over\partial\gamma}\right)\\ \hat{\mathcal{P}}_{3}&=&-i{\partial\over\partial\gamma},\\ \end{array}
  24. [ 𝒥 1 , 𝒥 2 ] = i 𝒥 3 , and [ 𝒫 1 , 𝒫 2 ] = - i 𝒫 3 \left[\mathcal{J}_{1},\,\mathcal{J}_{2}\right]=i\mathcal{J}_{3},\qquad\hbox{% and}\qquad\left[\mathcal{P}_{1},\,\mathcal{P}_{2}\right]=-i\mathcal{P}_{3}
  25. 𝒫 i \mathcal{P}_{i}
  26. [ 𝒫 i , 𝒥 j ] = 0 , i , j = 1 , 2 , 3 , \left[\mathcal{P}_{i},\,\mathcal{J}_{j}\right]=0,\quad i,\,j=1,\,2,\,3,
  27. 𝒥 2 𝒥 1 2 + 𝒥 2 2 + 𝒥 3 2 = 𝒫 2 𝒫 1 2 + 𝒫 2 2 + 𝒫 3 2 . \mathcal{J}^{2}\equiv\mathcal{J}_{1}^{2}+\mathcal{J}_{2}^{2}+\mathcal{J}_{3}^{% 2}=\mathcal{P}^{2}\equiv\mathcal{P}_{1}^{2}+\mathcal{P}_{2}^{2}+\mathcal{P}_{3% }^{2}.
  28. 𝒥 2 = 𝒫 2 = - 1 sin 2 β ( 2 α 2 + 2 γ 2 - 2 cos β 2 α γ ) - 2 β 2 - cot β β . \mathcal{J}^{2}=\mathcal{P}^{2}=-\frac{1}{\sin^{2}\beta}\left(\frac{\partial^{% 2}}{\partial\alpha^{2}}+\frac{\partial^{2}}{\partial\gamma^{2}}-2\cos\beta% \frac{\partial^{2}}{\partial\alpha\partial\gamma}\right)-\frac{\partial^{2}}{% \partial\beta^{2}}-\cot\beta\frac{\partial}{\partial\beta}.
  29. 𝒥 i \mathcal{J}_{i}
  30. 𝒥 3 D m m j ( α , β , γ ) * = m D m m j ( α , β , γ ) * , \mathcal{J}_{3}\,D^{j}_{m^{\prime}m}(\alpha,\beta,\gamma)^{*}=m^{\prime}\,D^{j% }_{m^{\prime}m}(\alpha,\beta,\gamma)^{*},
  31. ( 𝒥 1 ± i 𝒥 2 ) D m m j ( α , β , γ ) * = j ( j + 1 ) - m ( m ± 1 ) D m ± 1 , m j ( α , β , γ ) * . (\mathcal{J}_{1}\pm i\mathcal{J}_{2})\,D^{j}_{m^{\prime}m}(\alpha,\beta,\gamma% )^{*}=\sqrt{j(j+1)-m^{\prime}(m^{\prime}\pm 1)}\,D^{j}_{m^{\prime}\pm 1,m}(% \alpha,\beta,\gamma)^{*}.
  32. 𝒫 i \mathcal{P}_{i}
  33. 𝒫 3 D m m j ( α , β , γ ) * = m D m m j ( α , β , γ ) * , \mathcal{P}_{3}\,D^{j}_{m^{\prime}m}(\alpha,\beta,\gamma)^{*}=m\,D^{j}_{m^{% \prime}m}(\alpha,\beta,\gamma)^{*},
  34. ( 𝒫 1 i 𝒫 2 ) D m m j ( α , β , γ ) * = j ( j + 1 ) - m ( m ± 1 ) D m , m ± 1 j ( α , β , γ ) * . (\mathcal{P}_{1}\mp i\mathcal{P}_{2})\,D^{j}_{m^{\prime}m}(\alpha,\beta,\gamma% )^{*}=\sqrt{j(j+1)-m(m\pm 1)}\,D^{j}_{m^{\prime},m\pm 1}(\alpha,\beta,\gamma)^% {*}.
  35. 𝒥 2 D m m j ( α , β , γ ) * = 𝒫 2 D m m j ( α , β , γ ) * = j ( j + 1 ) D m m j ( α , β , γ ) * . \mathcal{J}^{2}\,D^{j}_{m^{\prime}m}(\alpha,\beta,\gamma)^{*}=\mathcal{P}^{2}% \,D^{j}_{m^{\prime}m}(\alpha,\beta,\gamma)^{*}=j(j+1)D^{j}_{m^{\prime}m}(% \alpha,\beta,\gamma)^{*}.
  36. { 𝒥 i } \{\mathcal{J}_{i}\}
  37. { - 𝒫 i } \{-\mathcal{P}_{i}\}
  38. ( α , β , γ ) \mathcal{R}(\alpha,\beta,\gamma)
  39. T T\,
  40. j m | ( α , β , γ ) | j m = j m | T ( α , β , γ ) T | j m = ( - 1 ) m - m j , - m | ( α , β , γ ) | j , - m * , \langle jm^{\prime}|\mathcal{R}(\alpha,\beta,\gamma)|jm\rangle=\langle jm^{% \prime}|T^{\,\dagger}\mathcal{R}(\alpha,\beta,\gamma)T|jm\rangle=(-1)^{m^{% \prime}-m}\langle j,-m^{\prime}|\mathcal{R}(\alpha,\beta,\gamma)|j,-m\rangle^{% *},
  41. D m m j ( α , β , γ ) = ( - 1 ) m - m D - m , - m j ( α , β , γ ) * . D^{j}_{m^{\prime}m}(\alpha,\beta,\gamma)=(-1)^{m^{\prime}-m}D^{j}_{-m^{\prime}% ,-m}(\alpha,\beta,\gamma)^{*}.
  42. T T\,
  43. T T^{\dagger}\,
  44. T | j m = ( - 1 ) j - m | j , - m T|jm\rangle=(-1)^{j-m}|j,-m\rangle
  45. ( - 1 ) 2 j - m - m = ( - 1 ) m - m (-1)^{2j-m^{\prime}-m}=(-1)^{m^{\prime}-m}
  46. D m k j ( α , β , γ ) D^{j}_{mk}(\alpha,\beta,\gamma)
  47. α \alpha
  48. β , \beta,
  49. γ \gamma
  50. 0 2 π d α 0 π sin β d β 0 2 π d γ D m k j ( α , β , γ ) D m k j ( α , β , γ ) = 8 π 2 2 j + 1 δ m m δ k k δ j j . \int_{0}^{2\pi}d\alpha\int_{0}^{\pi}\sin\beta d\beta\int_{0}^{2\pi}d\gamma\,\,% D^{j^{\prime}}_{m^{\prime}k^{\prime}}(\alpha,\beta,\gamma)^{\ast}D^{j}_{mk}(% \alpha,\beta,\gamma)=\frac{8\pi^{2}}{2j+1}\delta_{m^{\prime}m}\delta_{k^{% \prime}k}\delta_{j^{\prime}j}.
  51. 𝐃 j ( α , β , γ ) 𝐃 j ( α , β , γ ) \mathbf{D}^{j}(\alpha,\beta,\gamma)\otimes\mathbf{D}^{j^{\prime}}(\alpha,\beta% ,\gamma)
  52. D m k j ( α , β , γ ) D m k j ( α , β , γ ) = J = | j - j | j + j M = - J J K = - J J j m j m | J M j k j k | J K D M K J ( α , β , γ ) D^{j}_{mk}(\alpha,\beta,\gamma)D^{j^{\prime}}_{m^{\prime}k^{\prime}}(\alpha,% \beta,\gamma)=\sum_{J=|j-j^{\prime}|}^{j+j^{\prime}}\sum_{M=-J}^{J}\sum_{K=-J}% ^{J}\langle jmj^{\prime}m^{\prime}|JM\rangle\langle jkj^{\prime}k^{\prime}|JK% \rangle D^{J}_{MK}(\alpha,\beta,\gamma)
  53. j m j m | J M \langle jmj^{\prime}m^{\prime}|JM\rangle
  54. l l
  55. D m 0 ( α , β , 0 ) = 4 π 2 + 1 Y m * ( β , α ) = ( - m ) ! ( + m ) ! P m ( cos β ) e - i m α D^{\ell}_{m0}(\alpha,\beta,0)=\sqrt{\frac{4\pi}{2\ell+1}}Y_{\ell}^{m*}(\beta,% \alpha)=\sqrt{\frac{(\ell-m)!}{(\ell+m)!}}\,P_{\ell}^{m}(\cos{\beta})\,e^{-im\alpha}
  56. d m 0 ( β ) = ( - m ) ! ( + m ) ! P m ( cos β ) d^{\ell}_{m0}(\beta)=\sqrt{\frac{(\ell-m)!}{(\ell+m)!}}\,P_{\ell}^{m}(\cos{% \beta})
  57. D 0 , 0 ( α , β , γ ) = d 0 , 0 ( β ) = P ( cos β ) . D^{\ell}_{0,0}(\alpha,\beta,\gamma)=d^{\ell}_{0,0}(\beta)=P_{\ell}(\cos\beta).
  58. α \alpha
  59. β \beta
  60. ( Y m ) * = ( - 1 ) m Y - m . \left(Y_{\ell}^{m}\right)^{*}=(-1)^{m}Y_{\ell}^{-m}.
  61. D - m s ( α , β , - γ ) = ( - 1 ) m 4 π 2 + 1 Y m s ( β , α ) e i s γ . D^{\ell}_{-ms}(\alpha,\beta,-\gamma)=(-1)^{m}\sqrt{\frac{4\pi}{2{\ell}+1}}{}_{% s}Y_{{\ell}m}(\beta,\alpha)e^{is\gamma}.
  62. m , m \ell\gg m,m^{\prime}
  63. D m m ( α , β , γ ) e - i m α - i m γ J m - m ( β ) D^{\ell}_{mm^{\prime}}(\alpha,\beta,\gamma)\approx e^{-im\alpha-im^{\prime}% \gamma}J_{m-m^{\prime}}(\ell\beta)
  64. J m - m ( β ) J_{m-m^{\prime}}(\ell\beta)
  65. β \ell\beta
  66. d 1 / 2 , 1 / 2 1 / 2 = cos ( θ / 2 ) d_{1/2,1/2}^{1/2}=\cos(\theta/2)
  67. d 1 / 2 , - 1 / 2 1 / 2 = - sin ( θ / 2 ) d_{1/2,-1/2}^{1/2}=-\sin(\theta/2)
  68. d 1 , 1 1 = 1 + cos θ 2 d_{1,1}^{1}=\frac{1+\cos\theta}{2}
  69. d 1 , 0 1 = - sin θ 2 d_{1,0}^{1}=\frac{-\sin\theta}{\sqrt{2}}
  70. d 1 , - 1 1 = 1 - cos θ 2 d_{1,-1}^{1}=\frac{1-\cos\theta}{2}
  71. d 0 , 0 1 = cos θ d_{0,0}^{1}=\cos\theta
  72. d 3 / 2 , 3 / 2 3 / 2 = 1 + cos θ 2 cos θ 2 d_{3/2,3/2}^{3/2}=\frac{1+\cos\theta}{2}\cos\frac{\theta}{2}
  73. d 3 / 2 , 1 / 2 3 / 2 = - 3 1 + cos θ 2 sin θ 2 d_{3/2,1/2}^{3/2}=-\sqrt{3}\frac{1+\cos\theta}{2}\sin\frac{\theta}{2}
  74. d 3 / 2 , - 1 / 2 3 / 2 = 3 1 - cos θ 2 cos θ 2 d_{3/2,-1/2}^{3/2}=\sqrt{3}\frac{1-\cos\theta}{2}\cos\frac{\theta}{2}
  75. d 3 / 2 , - 3 / 2 3 / 2 = - 1 - cos θ 2 sin θ 2 d_{3/2,-3/2}^{3/2}=-\frac{1-\cos\theta}{2}\sin\frac{\theta}{2}
  76. d 1 / 2 , 1 / 2 3 / 2 = 3 cos θ - 1 2 cos θ 2 d_{1/2,1/2}^{3/2}=\frac{3\cos\theta-1}{2}\cos\frac{\theta}{2}
  77. d 1 / 2 , - 1 / 2 3 / 2 = - 3 cos θ + 1 2 sin θ 2 d_{1/2,-1/2}^{3/2}=-\frac{3\cos\theta+1}{2}\sin\frac{\theta}{2}
  78. d 2 , 2 2 = 1 4 ( 1 + cos θ ) 2 d_{2,2}^{2}=\frac{1}{4}\left(1+\cos\theta\right)^{2}
  79. d 2 , 1 2 = - 1 2 sin θ ( 1 + cos θ ) d_{2,1}^{2}=-\frac{1}{2}\sin\theta\left(1+\cos\theta\right)
  80. d 2 , 0 2 = 3 8 sin 2 θ d_{2,0}^{2}=\sqrt{\frac{3}{8}}\sin^{2}\theta
  81. d 2 , - 1 2 = - 1 2 sin θ ( 1 - cos θ ) d_{2,-1}^{2}=-\frac{1}{2}\sin\theta\left(1-\cos\theta\right)
  82. d 2 , - 2 2 = 1 4 ( 1 - cos θ ) 2 d_{2,-2}^{2}=\frac{1}{4}\left(1-\cos\theta\right)^{2}
  83. d 1 , 1 2 = 1 2 ( 2 cos 2 θ + cos θ - 1 ) d_{1,1}^{2}=\frac{1}{2}\left(2\cos^{2}\theta+\cos\theta-1\right)
  84. d 1 , 0 2 = - 3 8 sin 2 θ d_{1,0}^{2}=-\sqrt{\frac{3}{8}}\sin 2\theta
  85. d 1 , - 1 2 = 1 2 ( - 2 cos 2 θ + cos θ + 1 ) d_{1,-1}^{2}=\frac{1}{2}\left(-2\cos^{2}\theta+\cos\theta+1\right)
  86. d 0 , 0 2 = 1 2 ( 3 cos 2 θ - 1 ) d_{0,0}^{2}=\frac{1}{2}\left(3\cos^{2}\theta-1\right)
  87. d m , m j = ( - 1 ) m - m d m , m j = d - m , - m j d_{m^{\prime},m}^{j}=(-1)^{m-m^{\prime}}d_{m,m^{\prime}}^{j}=d_{-m,-m^{\prime}% }^{j}

Wilks's_lambda_distribution.html

  1. A W p ( Σ , m ) B W p ( Σ , n ) A\sim W_{p}(\Sigma,m)\qquad B\sim W_{p}(\Sigma,n)
  2. m p m\geq p
  3. λ = det ( A ) det ( A + B ) = 1 det ( I + A - 1 B ) Λ ( p , m , n ) \lambda=\frac{\det(A)}{\det(A+B)}=\frac{1}{\det(I+A^{-1}B)}\sim\Lambda(p,m,n)
  4. n + m n+m
  5. u i B ( m + i - p 2 , p 2 ) u_{i}\sim B\left(\frac{m+i-p}{2},\frac{p}{2}\right)
  6. i = 1 n u i Λ ( p , m , n ) . \prod_{i=1}^{n}u_{i}\sim\Lambda(p,m,n).
  7. ( p + n + 1 2 - m ) log Λ ( p , m , n ) χ n p 2 . \left(\frac{p+n+1}{2}-m\right)\log\Lambda(p,m,n)\sim\chi^{2}_{np}.

Winkel_tripel_projection.html

  1. x = 1 2 [ λ cos φ 1 + 2 cos φ sin λ 2 sinc α ] x=\frac{1}{2}\left[\lambda\cos\varphi_{1}+\frac{2\cos\varphi\sin\frac{\lambda}% {2}}{\mathrm{sinc}\,\alpha}\right]
  2. y = 1 2 [ φ + sin φ sinc α ] y=\frac{1}{2}\left[\varphi+\frac{\sin\varphi}{\mathrm{sinc}\,\alpha}\right]
  3. λ \lambda
  4. φ \varphi
  5. φ 1 \varphi_{1}
  6. α = arccos [ cos φ cos λ 2 ] \alpha=\arccos\left[\cos\varphi\cos\frac{\lambda}{2}\right]
  7. sinc α \mathrm{sinc}\,\alpha
  8. φ 1 = arccos 2 π \varphi_{1}=\arccos\frac{2}{\pi}\,

Winsorized_mean.html

  1. x 2 + x 2 + x 3 + x 4 + x 5 + x 6 + x 7 + x 8 + x 9 + x 9 10 . \frac{\overbrace{x_{2}+x_{2}}+x_{3}+x_{4}+x_{5}+x_{6}+x_{7}+x_{8}+\overbrace{x% _{9}+x_{9}}}{10}.\,

Witt_group.html

  1. 𝐙 8 [ s , t ] / 2 s , 2 t , s 2 , t 2 , s t - 4 . \mathbf{Z}_{8}[s,t]/\langle 2s,2t,s^{2},t^{2},st-4\rangle\ .
  2. ( d 1 , e 1 ) + ( d 2 , e 2 ) = ( ( - 1 ) e 1 e 2 d 1 d 2 , e 1 + e 2 ) (d_{1},e_{1})+(d_{2},e_{2})=((-1)^{e_{1}e_{2}}d_{1}d_{2},e_{1}+e_{2})
  3. ( d 1 , e 1 ) ( d 2 , e 2 ) = ( d 1 e 2 d 2 e 1 , e 1 e 2 ) . (d_{1},e_{1})\cdot(d_{2},e_{2})=(d_{1}^{e_{2}}d_{2}^{e_{1}},e_{1}e_{2})\ .
  4. W ( K ) = W ( k ) π W ( k ) W(K)=W(k)\oplus\langle\pi\rangle\cdot W(k)
  5. ( d 1 , e 1 , f 1 ) + ( d 2 , e 2 , f 2 ) = ( ( - 1 ) e 1 e 2 d 1 d 2 , e 1 + e 2 , [ d 1 , d 2 ] [ - d 1 d 2 , ( - 1 ) e 1 e 2 ] f 1 f 2 ) (d_{1},e_{1},f_{1})+(d_{2},e_{2},f_{2})=((-1)^{e_{1}e_{2}}d_{1}d_{2},e_{1}+e_{% 2},[d_{1},d_{2}][-d_{1}d_{2},(-1)^{e_{1}e_{2}}]f_{1}f_{2})
  6. ( d 1 , e 1 , f 1 ) ( d 2 , e 2 , f 2 ) = ( d 1 e 2 d 2 e 1 , e 1 e 2 , [ d 1 , d 2 ] 1 + e 1 e 2 f 1 e 2 f 2 e 1 ) . (d_{1},e_{1},f_{1})\cdot(d_{2},e_{2},f_{2})=(d_{1}^{e_{2}}d_{2}^{e_{1}},e_{1}e% _{2},[d_{1},d_{2}]^{1+e_{1}e_{2}}f_{1}^{e_{2}}f_{2}^{e_{1}})\ .
  7. W ( 𝐐 ) W ( 𝐑 ) p W ( 𝐐 p ) . W(\mathbf{Q})\rightarrow W(\mathbf{R})\oplus\prod_{p}W(\mathbf{Q}_{p})\ .
  8. 𝐙 W ( 𝐐 ) 𝐙 / 2 p 2 W ( 𝐅 p ) 0 \mathbf{Z}\rightarrow W(\mathbf{Q})\rightarrow\mathbf{Z}/2\oplus\prod_{p\neq 2% }W(\mathbf{F}_{p})\rightarrow 0
  9. W ( 𝐐 ) 𝐙 𝐙 / 2 p 2 W ( 𝐅 p ) W(\mathbf{Q})\cong\mathbf{Z}\oplus\mathbf{Z}/2\oplus\prod_{p\neq 2}W(\mathbf{F% }_{p})

Witt_vector.html

  1. p p
  2. p \mathbb{Z}_{p}
  3. a 0 + a 1 p 1 + a 2 p 2 + a_{0}+a_{1}p^{1}+a_{2}p^{2}+\cdots
  4. a a
  5. { 0 , 1 , 2 , , p - 1 } \{0,1,2,...,p-1\}
  6. p - 1 p-1
  7. 1 1
  8. p p
  9. x p - x = 0 x^{p}-x=0
  10. p \mathbb{Z}_{p}
  11. 𝔽 p \mathbb{F}_{p}
  12. p p
  13. p p
  14. 𝔽 p × \mathbb{F}_{p}^{\times}
  15. ω : 𝔽 p × p × \omega:\mathbb{F}_{p}^{\times}\rightarrow\mathbb{Z}_{p}^{\times}
  16. p p
  17. ω ( 𝔽 p × ) { 0 } \omega(\mathbb{F}_{p}^{\times})\cup\{0\}
  18. ω ( 𝔽 p × ) { 0 } \omega(\mathbb{F}_{p}^{\times})\cup\{0\}
  19. p p
  20. p p
  21. p \mathbb{Z}_{p}
  22. p p
  23. 𝔽 p \mathbb{F}_{p}
  24. p p
  25. ( n 0 , n 1 , ) (n_{0},n_{1},...)
  26. n i / p ( i + 1 ) n_{i}\in\mathbb{Z}/p^{(i+1)}\mathbb{Z}
  27. n i n j mod p i n_{i}\equiv n_{j}\mod p^{i}
  28. i < j i<j
  29. p p
  30. a 0 + a 1 p 1 + a 2 p 2 + a_{0}+a_{1}p^{1}+a_{2}p^{2}+\cdots
  31. a a
  32. { 0 , 1 , 2 , , p - 1 } \{0,1,2,...,p-1\}
  33. p \mathbb{Z}_{p}
  34. a i a_{i}
  35. p j p^{j}
  36. \mathbb{Z}
  37. p \mathbb{Z}_{p}
  38. 𝔽 p \mathbb{F}_{p}
  39. p \mathbb{Z}_{p}
  40. 𝔽 p \prod_{\mathbb{N}}\mathbb{F}_{p}
  41. a + b = c a+b=c
  42. c 0 a 0 + b 0 mod p c_{0}\equiv a_{0}+b_{0}\mod p
  43. c 0 + c 1 p a 0 + a 1 p + b 0 + b 1 p mod p 2 c_{0}+c_{1}p\equiv a_{0}+a_{1}p+b_{0}+b_{1}p\mod p^{2}
  44. c 0 + c 1 p + c 2 p 2 a 0 + a 1 p + a 2 p 2 + b 0 + b 1 p + b 2 p 2 mod p 3 c_{0}+c_{1}p+c_{2}p^{2}\equiv a_{0}+a_{1}p+a_{2}p^{2}+b_{0}+b_{1}p+b_{2}p^{2}% \mod p^{3}
  45. p \mathbb{Z}_{p}
  46. 𝔽 p \mathbb{F}_{p}
  47. 0
  48. p * \mathbb{Z}_{p}^{*}
  49. 𝔽 p * \mathbb{F}_{p}^{*}
  50. ω : 𝔽 p * p * \omega:\mathbb{F}_{p}^{*}\rightarrow\mathbb{Z}_{p}^{*}
  51. ω \omega
  52. ω ( k ) = ω ( i ) + ω ( j ) mod p \omega(k)=\omega(i)+\omega(j)\mod p
  53. p \mathbb{Z}_{p}
  54. i + j = k i+j=k
  55. 𝔽 p \mathbb{F}_{p}
  56. m ω = id 𝔽 p m\circ\omega=\mathrm{id}_{\mathbb{F}_{p}}
  57. m : p p / p p 𝔽 p m:\mathbb{Z}_{p}\rightarrow\mathbb{Z}_{p}/p\mathbb{Z}_{p}\cong\mathbb{F}_{p}
  58. x p - 1 - 1 = 0 x^{p-1}-1=0
  59. 3 \mathbb{Z}_{3}
  60. 2 2
  61. x 2 - 1 = 0 x^{2}-1=0
  62. / 9 \mathbb{Z}/9\mathbb{Z}
  63. x 2 mod 3 x\equiv 2\mod 3
  64. 8 8
  65. / 27 \mathbb{Z}/27\mathbb{Z}
  66. x 2 - 1 = 0 x^{2}-1=0
  67. x 2 mod 9 x\equiv 2\mod 9
  68. 26 26
  69. ( x p - 1 - 1 , ( p - 1 ) x p - 2 ) = 1 (x^{p-1}-1,(p-1)x^{p-2})=1
  70. / p n \mathbb{Z}/p^{n}\mathbb{Z}
  71. a 0 + a 1 p 1 + a 2 p 2 + a_{0}+a_{1}p^{1}+a_{2}p^{2}+...
  72. j { 0 , 1 , 2 , , p - 1 } j\in\{0,1,2,...,p-1\}
  73. ω ( j ) \omega(j)
  74. a 0 = j a_{0}=j
  75. p p
  76. p p
  77. b = a 0 + a 1 p 1 + a 2 p 2 + b=a_{0}+a_{1}p^{1}+a_{2}p^{2}+...
  78. b - ω ( a 0 ) = a 1 p 1 + a 2 p 2 + b-\omega(a_{0})=a^{\prime}_{1}p^{1}+a^{\prime}_{2}p^{2}+...
  79. ω ( a 1 ) p \omega(a^{\prime}_{1})p
  80. a i a_{i}
  81. p p
  82. a i p = a i a_{i}^{p}=a_{i}
  83. c 0 = a 0 + b 0 c_{0}=a_{0}+b_{0}
  84. p \mathbb{Z}_{p}
  85. 𝔽 p \mathbb{F}_{p}
  86. c 0 p ( a 0 + b 0 ) p mod p 2 c_{0}^{p}\equiv(a_{0}+b_{0})^{p}\mod p^{2}
  87. c 0 - a 0 - b 0 ( a 0 + b 0 ) p - a 0 - b 0 ( p 1 ) a 0 p - 1 b 0 + + ( p 1 ) a 0 b 0 p - 1 mod p 2 c_{0}-a_{0}-b_{0}\equiv(a_{0}+b_{0})^{p}-a_{0}-b_{0}\equiv{\left({{p}\atop{1}}% \right)}a_{0}^{p-1}b_{0}+...+{\left({{p}\atop{1}}\right)}a_{0}b_{0}^{p-1}\mod p% ^{2}
  88. ( p i ) {\left({{p}\atop{i}}\right)}
  89. p p
  90. p p
  91. c 1 c_{1}
  92. c 1 a 1 + b 1 - a 0 p - 1 b 0 - p - 1 2 a 0 p - 2 b 0 2 - - a 0 b 0 p - 1 mod p c_{1}\equiv a_{1}+b_{1}-a_{0}^{p-1}b_{0}-\frac{p-1}{2}a_{0}^{p-2}b_{0}^{2}-...% -a_{0}b_{0}^{p-1}\mod p
  93. c 1 c_{1}
  94. mod p \mod p
  95. 𝔽 p \mathbb{F}_{p}
  96. c 2 c_{2}
  97. c 1 = c 1 p ( a 1 + b 1 - a 0 p - 1 b 0 - p - 1 2 a 0 p - 2 b 0 2 - - a 0 b 0 p - 1 ) p mod p c_{1}=c_{1}^{p}\equiv(a_{1}+b_{1}-a_{0}^{p-1}b_{0}-\frac{p-1}{2}a_{0}^{p-2}b_{% 0}^{2}-...-a_{0}b_{0}^{p-1})^{p}\mod p
  98. c 0 c_{0}
  99. p p
  100. c 0 = c 0 p 2 ( a 0 + b 0 ) p 2 c_{0}=c_{0}^{p^{2}}\equiv(a_{0}+b_{0})^{p^{2}}
  101. ( p 2 i ) {\left({{p^{2}}\atop{i}}\right)}
  102. p 2 p^{2}
  103. i = p d i=pd
  104. a i b p 2 - i = a d b p - d a^{i}b^{p^{2}-i}=a^{d}b^{p-d}
  105. c 1 p c_{1}^{p}
  106. p 2 p^{2}
  107. c 0 a 0 + b 0 mod p c_{0}\equiv a_{0}+b_{0}\mod p
  108. c 0 p + c 1 p a 0 p + a 1 p + b 0 p + b 1 p mod p 2 c_{0}^{p}+c_{1}p\equiv a_{0}^{p}+a_{1}p+b_{0}^{p}+b_{1}p\mod p^{2}
  109. c 0 p 2 + c 1 p p + c 2 p 2 a 0 p 2 + a 1 p p + a 2 p 2 + b 0 p 2 + b 1 p p + b 2 p 2 mod p 3 c_{0}^{p^{2}}+c_{1}^{p}p+c_{2}p^{2}\equiv a_{0}^{p^{2}}+a_{1}^{p}p+a_{2}p^{2}+% b_{0}^{p^{2}}+b_{1}^{p}p+b_{2}p^{2}\mod p^{3}
  110. ( X 0 , X 1 , X 2 , ) (X_{0},X_{1},X_{2},...)
  111. W i W_{i}
  112. W 0 = X 0 W_{0}=X_{0}\,
  113. W 1 = X 0 p + p X 1 W_{1}=X_{0}^{p}+pX_{1}
  114. W 2 = X 0 p 2 + p X 1 p + p 2 X 2 W_{2}=X_{0}^{p^{2}}+pX_{1}^{p}+p^{2}X_{2}
  115. W n = i p i X i p n - i . W_{n}=\sum_{i}p^{i}X_{i}^{p^{n-i}}.
  116. ( W 0 , W 1 , W 2 , ) (W_{0},W_{1},W_{2},...)
  117. ( X 0 , X 1 , X 2 , ) (X_{0},X_{1},X_{2},...)
  118. ( X ( 0 ) , X ( 1 ) , X ( 2 ) , ) (X^{(0)},X^{(1)},X^{(2)},...)
  119. ( X + Y ) i (X+Y)_{i}
  120. ( X Y ) i (XY)_{i}
  121. X ( i ) + Y ( i ) = ( X + Y ) ( i ) X^{(i)}+Y^{(i)}=(X+Y)^{(i)}
  122. X ( i ) Y ( i ) = ( X Y ) ( i ) X^{(i)}Y^{(i)}=(XY)^{(i)}
  123. ( X 0 , X 1 , ) + ( Y 0 , Y 1 , ) = ( X 0 + Y 0 , X 1 + Y 1 + ( X 0 p + Y 0 p - ( X 0 + Y 0 ) p ) / p , ) (X_{0},X_{1},...)+(Y_{0},Y_{1},...)=(X_{0}+Y_{0},X_{1}+Y_{1}+(X_{0}^{p}+Y_{0}^% {p}-(X_{0}+Y_{0})^{p})/p,...)
  124. ( X 0 , X 1 , ) × ( Y 0 , Y 1 , ) = ( X 0 Y 0 , X 0 p Y 1 + X 1 Y 0 p + p X 1 Y 1 , ) (X_{0},X_{1},...)\times(Y_{0},Y_{1},...)=(X_{0}Y_{0},X_{0}^{p}Y_{1}+X_{1}Y_{0}% ^{p}+pX_{1}Y_{1},...)
  125. W 1 = X 1 W_{1}=X_{1}\,
  126. W 2 = X 1 2 + 2 X 2 W_{2}=X_{1}^{2}+2X_{2}
  127. W 3 = X 1 3 + 3 X 3 W_{3}=X_{1}^{3}+3X_{3}
  128. W 4 = X 1 4 + 2 X 2 2 + 4 X 4 W_{4}=X_{1}^{4}+2X_{2}^{2}+4X_{4}
  129. W n = d | n d X d n / d . W_{n}=\sum_{d|n}dX_{d}^{n/d}.
  130. ( W 1 , W 2 , W 3 , ) (W_{1},W_{2},W_{3},...)
  131. ( X 1 , X 2 , X 3 , ) (X_{1},X_{2},X_{3},...)
  132. ( X ( 1 ) , X ( 2 ) , X ( 3 ) , ) (X^{(1)},X^{(2)},X^{(3)},...)
  133. X X
  134. f X ( t ) = n 1 ( 1 - X n t n ) = n 0 A n t n f_{X}(t)=\prod_{n\geq 1}(1-X_{n}t^{n})=\sum_{n\geq 0}A_{n}t^{n}
  135. n 1 n\geq 1
  136. 𝒮 n \mathcal{S}_{n}
  137. { 1 , 2 , , n } \{1,2,...,n\}
  138. n n
  139. A n = S 𝒮 ( - 1 ) | S | i S X i A_{n}=\sum_{S\in\mathcal{S}}(-1)^{|S|}\sum_{i\in S}{X_{i}}
  140. d d t log f X ( t ) = n 1 d d t ( 1 - X n t n ) = - n 1 d 1 X n d t n d d = - m 1 d | m m d X m d d m t m = - m 1 X ( m ) t m m \frac{d}{dt}\log f_{X}(t)=\sum_{n\geq 1}\frac{d}{dt}(1-X_{n}t^{n})=-\sum_{n% \geq 1}\sum_{d\geq 1}\frac{X_{n}^{d}t^{nd}}{d}=-\sum_{m\geq 1}\frac{\sum_{d|m}% \frac{m}{d}X_{\frac{m}{d}}^{d}}{m}t^{m}=-\sum_{m\geq 1}\frac{X^{(m)}t^{m}}{m}
  141. f Z ( t ) = f X ( t ) f Y ( t ) f_{Z}(t)=f_{X}(t)f_{Y}(t)
  142. Z = X + Y Z=X+Y
  143. C n = 0 i n A n B n - i C_{n}=\sum_{0\leq i\leq n}A_{n}B_{n-i}
  144. A n , B n , C n A_{n},B_{n},C_{n}
  145. f X ( t ) , f Y ( t ) , f Z ( t ) f_{X}(t),f_{Y}(t),f_{Z}(t)
  146. Z n = 0 i n A n B n - i - S 𝒮 , S { n } ( - 1 ) | S | i S Z i Z_{n}=\sum_{0\leq i\leq n}A_{n}B_{n-i}-\sum_{S\in\mathcal{S},S\neq\{n\}}(-1)^{% |S|}\sum_{i\in S}{Z_{i}}
  147. A n A_{n}
  148. X 1 , , X n X_{1},...,X_{n}
  149. B n B_{n}
  150. Z n Z_{n}
  151. X 1 , , X n , Y 1 , , Y n X_{1},...,X_{n},Y_{1},...,Y_{n}
  152. W = X Y W=XY
  153. d d t log f W ( t ) = - m 1 X ( m ) Y ( m ) t m m \frac{d}{dt}\log f_{W}(t)=-\sum_{m\geq 1}\frac{X^{(m)}Y^{(m)}t^{m}}{m}
  154. m 1 X ( m ) Y ( m ) m t m = m 1 d | m d X d m / d e | m e Y e m / e m t m \sum_{m\geq 1}\frac{X^{(m)}Y^{(m)}}{m}t^{m}=\sum_{m\geq 1}\frac{\sum_{d|m}dX_{% d}^{m/d}\sum_{e|m}eY_{e}^{m/e}}{m}t^{m}
  155. m , d , e {m,d,e}
  156. m + , d | m , e | m m\in\mathbb{Z}^{+},d|m,e|m
  157. d , e , n {d,e,n}
  158. d , e , n + d,e,n\in\mathbb{Z}^{+}
  159. n = m / [ d , e ] n=m/[d,e]
  160. [ d , e ] [d,e]
  161. d , e 1 d e [ d , e ] n 1 ( X d [ d , e ] / d Y e [ d , e ] / e t [ d , e ] ) n n \sum_{d,e\geq 1}\frac{\frac{de}{[d,e]}\sum_{n\geq 1}(X_{d}^{[d,e]/d}Y_{e}^{[d,% e]/e}t^{[d,e]})^{n}}{n}
  162. f W ( t ) = d , e 1 ( 1 - X d [ d , e ] / d Y e [ d , e ] / e t [ d , e ] ) d e / [ d , e ] = n 0 D n t n f_{W}(t)=\prod_{d,e\geq 1}(1-X_{d}^{[d,e]/d}Y_{e}^{[d,e]/e}t^{[d,e]})^{de/[d,e% ]}=\sum_{n\geq 0}D_{n}t^{n}
  163. D n D_{n}
  164. X 1 , , X n , Y 1 , , Y n X_{1},...,X_{n},Y_{1},...,Y_{n}
  165. f W ( t ) = n 1 ( 1 - W n t n ) f_{W}(t)=\prod_{n\geq 1}(1-W_{n}t^{n})
  166. W n W_{n}
  167. X 1 , , X n , Y 1 , , Y n X_{1},...,X_{n},Y_{1},...,Y_{n}
  168. R R
  169. R n R^{n}
  170. 𝔸 n \mathbb{A}_{\mathbb{Z}}^{n}
  171. 𝔸 n \mathbb{A}_{\mathbb{Z}}^{n}
  172. 𝒪 ¯ n \underline{\mathcal{O}}^{n}
  173. 𝕎 n \mathbb{W}_{n}
  174. 𝔸 n \mathbb{A}_{\mathbb{Z}}^{n}
  175. 𝕎 n 𝒪 ¯ n \mathbb{W}_{n}\rightarrow\underline{\mathcal{O}}^{n}
  176. G a G_{a}

Womersley_number.html

  1. α \alpha
  2. α 2 = transient inertial force viscous force = ρ ω U μ U L - 2 = ω L 2 μ ρ - 1 = ω L 2 ν , \alpha^{2}=\frac{\,\text{transient inertial force}}{\,\text{viscous force}}=% \frac{\rho\omega U}{\mu UL^{-2}}=\frac{\omega L^{2}}{\mu\rho^{-1}}=\frac{% \omega L^{2}}{\nu}\,,
  3. α = L ( ω ρ μ ) 1 2 . \alpha=L\left(\frac{\omega\rho}{\mu}\right)^{\frac{1}{2}}\,.
  4. α = ( 2 π Re Sr ) 1 / 2 . \alpha=\left(2\pi\,\mathrm{Re}\,\mathrm{Sr}\right)^{1/2}\,.
  5. α \alpha
  6. α \alpha
  7. δ \delta
  8. δ = ( L / α ) = ( L Stk ) , \delta=\left(L/\alpha\right)=\left(\frac{L}{\sqrt{\mathrm{Stk}}}\right),

Woodward_effect.html

  1. δ m 0 = 1 4 π G [ 1 ρ 0 c 2 P t - ( 1 ρ 0 c 2 ) 2 P 2 V ] \delta m_{0}=\frac{1}{4\pi G}\left[\frac{1}{\rho_{0}c^{2}}\frac{\partial P}{% \partial t}-\left(\frac{1}{\rho_{0}c^{2}}\right)^{2}\frac{P^{2}}{V}\right]
  2. m 0 m_{0}
  3. G G
  4. c c
  5. ρ 0 \rho_{0}
  6. V V
  7. P P
  8. - ( t ϕ c 2 ) 2 -\left(\frac{\partial}{\partial t}\frac{\phi}{c^{2}}\right)^{2}
  9. ϕ c 2 1 \frac{\phi}{c^{2}}\approx 1
  10. 1 ρ 0 c 2 P t \tfrac{1}{\rho_{0}c^{2}}\tfrac{\partial P}{\partial t}
  11. - ( 1 ρ 0 c 2 ) 2 P 2 V -\left(\tfrac{1}{\rho_{0}c^{2}}\right)^{2}\tfrac{P^{2}}{V}
  12. = =
  13. = =
  14. = =

Word_error_rate.html

  1. 𝑊𝐸𝑅 = S + D + I N \mathit{WER}=\frac{S+D+I}{N}
  2. 𝑊𝐸𝑅 = S + D + I S + D + C \mathit{WER}=\frac{S+D+I}{S+D+C}
  3. 𝑊𝐴𝑐𝑐 = 1 - 𝑊𝐸𝑅 = N - S - D - I N = H - I N \mathit{WAcc}=1-\mathit{WER}=\frac{N-S-D-I}{N}=\frac{H-I}{N}
  4. 𝑊𝐸𝑅 = S + 0.5 D + 0.5 I N \mathit{WER}=\frac{S+0.5D+0.5I}{N}

Wrapping_(graphics).html

  1. x x ( mod w ) x^{\prime}\equiv x\;\;(\mathop{{\rm mod}}w)
  2. x = x - ( x - x m i n ) / ( x m a x - x m i n ) * ( x m a x - x m i n ) x^{\prime}=x-\lfloor(x-x_{min})/(x_{max}-x_{min})\rfloor*(x_{max}-x_{min})
  3. x m a x x_{max}
  4. x m i n x_{min}

X-ray_standing_waves.html

  1. Y p Y_{p}
  2. Y p ( Ω ) = 1 + R + 2 C R f H cos ( ν - 2 π P H ) Y_{p}(\Omega)=1+R+2C\sqrt{R}f_{H}\cos(\nu-2\pi P_{H})
  3. R R
  4. ν \nu
  5. Y p Y_{p}
  6. f H f_{H}
  7. P H P_{H}

Xenon_oxytetrafluoride.html

  1. 2 X e O F 4 + 4 H 2 O 2 X e + 8 H F + 3 O 2 \rm 2XeOF_{4}+4H_{2}O\longrightarrow 2Xe+8HF+3O_{2}
  2. XeOF 4 + H 2 O XeO 2 F 2 + 2 H F \rm XeOF_{4}+H_{2}O\longrightarrow XeO_{2}F_{2}+2HF
  3. XeO 2 F 2 + H 2 O XeO 3 + 2 H F \rm XeO_{2}F_{2}+H_{2}O\longrightarrow XeO_{3}+2HF
  4. 2 X e O 3 2 X e + 3 O 2 \rm 2XeO_{3}\longrightarrow 2Xe+3O_{2}

Xxencoding.html

  1. n n

Yamartino_method.html

  1. s a = 1 n i = 1 n sin θ i , s_{a}=\frac{1}{n}\sum_{i=1}^{n}\sin\theta_{i},
  2. c a = 1 n i = 1 n cos θ i . c_{a}=\frac{1}{n}\sum_{i=1}^{n}\cos\theta_{i}.
  3. θ a = arctan ( c a , s a ) . \theta_{a}=\arctan\ (c_{a},s_{a}).
  4. σ θ = arcsin ( ε ) [ 1 + ( 2 3 - 1 ) ε 3 ] , \sigma_{\theta}=\arcsin(\varepsilon)\left[1+\left(\tfrac{2}{\sqrt{3}}-1\right)% \varepsilon^{3}\right],
  5. ε = 1 - ( s a 2 + c a 2 ) . \varepsilon=\sqrt{1-(s^{2}_{a}+c^{2}_{a})}.
  6. ε \varepsilon
  7. ε \varepsilon
  8. π 2 \tfrac{\pi}{2}
  9. π 3 \tfrac{\pi}{\sqrt{3}}

Yield_surface.html

  1. σ 1 , σ 2 , σ 3 \sigma_{1},\sigma_{2},\sigma_{3}
  2. I 1 , J 2 , J 3 I_{1},J_{2},J_{3}
  3. f ( σ 1 , σ 2 , σ 3 ) = 0 f(\sigma_{1},\sigma_{2},\sigma_{3})=0\,
  4. σ i \sigma_{i}
  5. f ( I 1 , J 2 , J 3 ) = 0 f(I_{1},J_{2},J_{3})=0\,
  6. I 1 I_{1}
  7. J 2 , J 3 J_{2},J_{3}
  8. f ( p , q , r ) = 0 f(p,q,r)=0\,
  9. p , q p,q
  10. I 1 I_{1}
  11. J 2 J_{2}
  12. r r
  13. J 2 , J 3 J_{2},J_{3}
  14. f ( ξ , ρ , θ ) = 0 f(\xi,\rho,\theta)=0\,
  15. ξ , ρ \xi,\rho
  16. I 1 I_{1}
  17. J 2 J_{2}
  18. θ \theta
  19. I 1 I_{1}
  20. s y m b o l σ symbol{\sigma}
  21. J 2 , J 3 J_{2},J_{3}
  22. s y m b o l s symbol{s}
  23. I 1 = Tr ( s y m b o l σ ) = σ 1 + σ 2 + σ 3 J 2 = 1 2 s y m b o l s : s y m b o l s = 1 6 [ ( σ 1 - σ 2 ) 2 + ( σ 2 - σ 3 ) 2 + ( σ 3 - σ 1 ) 2 ] J 3 = det ( s y m b o l s ) = 1 3 ( s y m b o l s \cdotsymbol s ) : s y m b o l s = s 1 s 2 s 3 \begin{aligned}\displaystyle I_{1}&\displaystyle=\,\text{Tr}(symbol{\sigma})=% \sigma_{1}+\sigma_{2}+\sigma_{3}\\ \displaystyle J_{2}&\displaystyle=\tfrac{1}{2}symbol{s}:symbol{s}=\tfrac{1}{6}% \left[(\sigma_{1}-\sigma_{2})^{2}+(\sigma_{2}-\sigma_{3})^{2}+(\sigma_{3}-% \sigma_{1})^{2}\right]\\ \displaystyle J_{3}&\displaystyle=\det(symbol{s})=\tfrac{1}{3}(symbol{s}% \cdotsymbol{s}):symbol{s}=s_{1}s_{2}s_{3}\end{aligned}
  24. σ 1 , σ 2 , σ 3 \sigma_{1},\sigma_{2},\sigma_{3}
  25. s y m b o l σ symbol{\sigma}
  26. s 1 , s 2 , s 3 s_{1},s_{2},s_{3}
  27. s y m b o l s symbol{s}
  28. s y m b o l s = s y m b o l σ - I 1 3 s y m b o l I symbol{s}=symbol{\sigma}-\tfrac{I_{1}}{3}\,symbol{I}
  29. s y m b o l I symbol{I}
  30. p , q , r p,q,r\,
  31. p = 1 3 I 1 : q = 3 J 2 = σ eq ; r = 3 ( 1 2 J 3 ) 1 / 3 p=\tfrac{1}{3}~{}I_{1}~{}:~{}~{}q=\sqrt{3~{}J_{2}}=\sigma_{\mathrm{eq}}~{};~{}% ~{}r=3\left(\tfrac{1}{2}\,J_{3}\right)^{1/3}
  32. σ eq \sigma_{\mathrm{eq}}
  33. J 3 J_{3}
  34. r r
  35. ξ , ρ , θ \xi,\rho,\theta\,
  36. ξ = 1 3 I 1 = 3 p ; ρ = 2 J 2 = 2 3 q ; cos ( 3 θ ) = ( r q ) 3 = 3 3 2 J 3 J 2 3 / 2 \xi=\tfrac{1}{\sqrt{3}}~{}I_{1}=\sqrt{3}~{}p~{};~{}~{}\rho=\sqrt{2J_{2}}=\sqrt% {\tfrac{2}{3}}~{}q~{};~{}~{}\cos(3\theta)=\left(\tfrac{r}{q}\right)^{3}=\tfrac% {3\sqrt{3}}{2}~{}\cfrac{J_{3}}{J_{2}^{3/2}}
  37. ξ - ρ \xi-\rho\,
  38. θ \theta
  39. θ \theta
  40. J 2 , J 3 J_{2},J_{3}
  41. [ σ 1 σ 2 σ 3 ] = 1 3 [ ξ ξ ξ ] + 2 3 ρ [ cos θ cos ( θ - 2 π 3 ) cos ( θ + 2 π 3 ) ] = 1 3 [ ξ ξ ξ ] + 2 3 ρ [ cos θ - sin ( π 6 - θ ) - sin ( π 6 + θ ) ] . \begin{bmatrix}\sigma_{1}\\ \sigma_{2}\\ \sigma_{3}\end{bmatrix}=\tfrac{1}{\sqrt{3}}\begin{bmatrix}\xi\\ \xi\\ \xi\end{bmatrix}+\sqrt{\tfrac{2}{3}}~{}\rho~{}\begin{bmatrix}\cos\theta\\ \cos\left(\theta-\tfrac{2\pi}{3}\right)\\ \cos\left(\theta+\tfrac{2\pi}{3}\right)\end{bmatrix}=\tfrac{1}{\sqrt{3}}\begin% {bmatrix}\xi\\ \xi\\ \xi\end{bmatrix}+\sqrt{\tfrac{2}{3}}~{}\rho~{}\begin{bmatrix}\cos\theta\\ -\sin\left(\tfrac{\pi}{6}-\theta\right)\\ -\sin\left(\tfrac{\pi}{6}+\theta\right)\end{bmatrix}\,.
  42. sin ( 3 θ ) = 3 3 2 J 3 J 2 3 / 2 \sin(3\theta)=~{}\tfrac{3\sqrt{3}}{2}~{}\cfrac{J_{3}}{J_{2}^{3/2}}
  43. σ 1 σ 2 σ 3 \sigma_{1}\geq\sigma_{2}\geq\sigma_{3}
  44. [ σ 1 σ 2 σ 3 ] = 1 3 [ ξ ξ ξ ] + ρ 2 [ cos θ + sin θ 3 2 sin θ 3 sin θ 3 - cos θ ] . \begin{bmatrix}\sigma_{1}\\ \sigma_{2}\\ \sigma_{3}\end{bmatrix}=\tfrac{1}{\sqrt{3}}\begin{bmatrix}\xi\\ \xi\\ \xi\end{bmatrix}+\tfrac{\rho}{\sqrt{2}}~{}\begin{bmatrix}\cos\theta+\tfrac{% \sin\theta}{\sqrt{3}}\\ \tfrac{2\sin\theta}{\sqrt{3}}\\ \tfrac{\sin\theta}{\sqrt{3}}-\cos\theta\end{bmatrix}\,.
  45. 1 2 max ( | σ 1 - σ 2 | , | σ 2 - σ 3 | , | σ 3 - σ 1 | ) = S s y = 1 2 S y \tfrac{1}{2}{\max(|\sigma_{1}-\sigma_{2}|,|\sigma_{2}-\sigma_{3}|,|\sigma_{3}-% \sigma_{1}|)=S_{sy}=\tfrac{1}{2}S_{y}}\!
  46. S s y S_{sy}
  47. S y S_{y}
  48. σ 1 , σ 2 \sigma_{1},\sigma_{2}
  49. ( σ 1 - σ 2 ) 2 + ( σ 2 - σ 3 ) 2 + ( σ 3 - σ 1 ) 2 = 2 S y 2 {(\sigma_{1}-\sigma_{2})^{2}+(\sigma_{2}-\sigma_{3})^{2}+(\sigma_{3}-\sigma_{1% })^{2}=2{S_{y}}^{2}}\!
  50. S y S_{y}
  51. σ 1 , σ 2 \sigma_{1},\sigma_{2}
  52. 3 I 2 = σ eq - γ 1 I 1 1 - γ 1 σ eq - γ 2 I 1 1 - γ 2 3I_{2}^{\prime}=\frac{\sigma_{\mathrm{eq}}-\gamma_{1}I_{1}}{1-\gamma_{1}}\frac% {\sigma_{\mathrm{eq}}-\gamma_{2}I_{1}}{1-\gamma_{2}}
  53. γ 1 = γ 2 = 0 \gamma_{1}=\gamma_{2}=0
  54. γ 1 = γ 2 ] 0 , 1 [ \gamma_{1}=\gamma_{2}\in]0,1[
  55. γ 1 ] 0 , 1 [ , γ 2 = 0 \gamma_{1}\in]0,1[,\gamma_{2}=0
  56. I 1 = 0 I_{1}=0
  57. γ 1 = - γ 2 ] 0 , 1 [ \gamma_{1}=-\gamma_{2}\in]0,1[
  58. I 1 = 1 2 ( 1 γ 1 + 1 γ 2 ) I_{1}=\frac{1}{2}\,\bigg(\frac{1}{\gamma_{1}}+\frac{1}{\gamma_{2}}\bigg)
  59. γ 1 ] 0 , 1 [ , γ 2 < 0 \gamma_{1}\in]0,1[,\gamma_{2}<0
  60. γ 1 ] 0 , 1 [ , γ 2 ] 0 , γ 1 [ \gamma_{1}\in]0,1[,\gamma_{2}\in]0,\gamma_{1}[
  61. I 1 = 0 I_{1}=0
  62. γ 1 = - γ 2 = a i \gamma_{1}=-\gamma_{2}=a\,i
  63. i = - 1 i=\sqrt{-1}
  64. γ 1 , 2 = b ± a i \gamma_{1,2}=b\pm a\,i
  65. i = - 1 i=\sqrt{-1}
  66. σ - σ + = 1 1 - γ 1 - γ 2 , ( 3 τ * σ + ) 2 = 1 ( 1 - γ 1 ) ( 1 - γ 2 ) \frac{\sigma_{-}}{\sigma_{+}}=\frac{1}{1-\gamma_{1}-\gamma_{2}},\qquad\bigg(% \sqrt{3}\,\frac{\tau_{*}}{\sigma_{+}}\bigg)^{2}=\frac{1}{(1-\gamma_{1})(1-% \gamma_{2})}
  67. ν + in = - 1 + 2 ( γ 1 + γ 2 ) - 3 γ 1 γ 2 - 2 + γ 1 + γ 2 \nu_{+}^{\mathrm{in}}=\frac{-1+2(\gamma_{1}+\gamma_{2})-3\gamma_{1}\gamma_{2}}% {-2+\gamma_{1}+\gamma_{2}}
  68. ν - in = - - 1 + γ 1 2 + γ 2 2 - γ 1 γ 2 ( - 2 + γ 1 + γ 2 ) ( - 1 + γ 1 + γ 2 ) \nu_{-}^{\mathrm{in}}=-\frac{-1+\gamma_{1}^{2}+\gamma_{2}^{2}-\gamma_{1}\,% \gamma_{2}}{(-2+\gamma_{1}+\gamma_{2})\,(-1+\gamma_{1}+\gamma_{2})}
  69. ν + in [ 0.48 , 1 2 ] \nu_{+}^{\mathrm{in}}\in\bigg[\,0.48,\,\frac{1}{2}\,\bigg]
  70. ν + in ] - 1 , ν + el ] \nu_{+}^{\mathrm{in}}\in]-1,~{}\nu_{+}^{\mathrm{el}}\,]
  71. 3 I 2 1 + c 3 cos 3 θ + c 6 cos 2 3 θ 1 + c 3 + c 6 = σ eq - γ 1 I 1 1 - γ 1 σ eq - γ 2 I 1 1 - γ 2 3I_{2}^{\prime}\frac{1+c_{3}\cos 3\theta+c_{6}\cos^{2}3\theta}{1+c_{3}+c_{6}}=% \frac{\sigma_{\mathrm{eq}}-\gamma_{1}I_{1}}{1-\gamma_{1}}\frac{\sigma_{\mathrm% {eq}}-\gamma_{2}I_{1}}{1-\gamma_{2}}
  72. m + 1 2 max ( | σ 1 - σ 2 | + K ( σ 1 + σ 2 ) , | σ 1 - σ 3 | + K ( σ 1 + σ 3 ) , | σ 2 - σ 3 | + K ( σ 2 + σ 3 ) ) = S y c \frac{m+1}{2}\max\Big(|\sigma_{1}-\sigma_{2}|+K(\sigma_{1}+\sigma_{2})~{},~{}~% {}|\sigma_{1}-\sigma_{3}|+K(\sigma_{1}+\sigma_{3})~{},~{}~{}|\sigma_{2}-\sigma% _{3}|+K(\sigma_{2}+\sigma_{3})\Big)=S_{yc}
  73. m = S y c S y t ; K = m - 1 m + 1 m=\frac{S_{yc}}{S_{yt}};K=\frac{m-1}{m+1}
  74. S y c S_{yc}
  75. S y t S_{yt}
  76. S y c = S y t S_{yc}=S_{yt}
  77. K K
  78. σ 1 , σ 2 \sigma_{1},\sigma_{2}
  79. ( m - 1 2 ) ( σ 1 + σ 2 + σ 3 ) + ( m + 1 2 ) ( σ 1 - σ 2 ) 2 + ( σ 2 - σ 3 ) 2 + ( σ 3 - σ 1 ) 2 2 = S y c \bigg(\frac{m-1}{2}\bigg)(\sigma_{1}+\sigma_{2}+\sigma_{3})+\bigg(\frac{m+1}{2% }\bigg)\sqrt{\frac{(\sigma_{1}-\sigma_{2})^{2}+(\sigma_{2}-\sigma_{3})^{2}+(% \sigma_{3}-\sigma_{1})^{2}}{2}}=S_{yc}
  80. m = S y c S y t m=\frac{S_{yc}}{S_{yt}}
  81. S y c S_{yc}
  82. S y t S_{yt}
  83. S y c = S y t S_{yc}=S_{yt}
  84. σ 1 , σ 2 \sigma_{1},\sigma_{2}
  85. σ 1 = - σ 2 \sigma_{1}=-\sigma_{2}
  86. σ 1 = σ 2 \sigma_{1}=\sigma_{2}
  87. S y c = 1 2 [ ( σ 1 - σ 2 ) 2 + ( σ 2 - σ 3 ) 2 + ( σ 3 - σ 1 ) 2 ] 1 / 2 - c 0 - c 1 ( σ 1 + σ 2 + σ 3 ) - c 2 ( σ 1 + σ 2 + σ 3 ) 2 S_{yc}=\tfrac{1}{\sqrt{2}}\left[(\sigma_{1}-\sigma_{2})^{2}+(\sigma_{2}-\sigma% _{3})^{2}+(\sigma_{3}-\sigma_{1})^{2}\right]^{1/2}-c_{0}-c_{1}~{}(\sigma_{1}+% \sigma_{2}+\sigma_{3})-c_{2}~{}(\sigma_{1}+\sigma_{2}+\sigma_{3})^{2}
  88. c 0 , c 1 , c 2 c_{0},c_{1},c_{2}
  89. c 2 c_{2}
  90. σ c \sigma_{c}
  91. σ t \sigma_{t}
  92. σ b \sigma_{b}
  93. c 1 = ( σ t - σ c ( σ t + σ c ) ) ( 4 σ b 2 - σ b ( σ c + σ t ) + σ c σ t 4 σ b 2 + 2 σ b ( σ t - σ c ) - σ c σ t ) c 2 = ( 1 ( σ t + σ c ) ) ( σ b ( 3 σ t - σ c ) - 2 σ c σ t 4 σ b 2 + 2 σ b ( σ t - σ c ) - σ c σ t ) c 0 = σ c + 3 ( c 1 σ c - c 2 σ c 2 ) \begin{aligned}\displaystyle c_{1}=&\displaystyle\left(\cfrac{\sigma_{t}-% \sigma_{c}}{(\sigma_{t}+\sigma_{c})}\right)\left(\cfrac{4\sigma_{b}^{2}-\sigma% _{b}(\sigma_{c}+\sigma_{t})+\sigma_{c}\sigma_{t}}{4\sigma_{b}^{2}+2\sigma_{b}(% \sigma_{t}-\sigma_{c})-\sigma_{c}\sigma_{t}}\right)\\ \displaystyle c_{2}=&\displaystyle\left(\cfrac{1}{(\sigma_{t}+\sigma_{c})}% \right)\left(\cfrac{\sigma_{b}(3\sigma_{t}-\sigma_{c})-2\sigma_{c}\sigma_{t}}{% 4\sigma_{b}^{2}+2\sigma_{b}(\sigma_{t}-\sigma_{c})-\sigma_{c}\sigma_{t}}\right% )\\ \displaystyle c_{0}=&\displaystyle\sigma_{c}+\sqrt{3}(c_{1}\sigma_{c}-c_{2}% \sigma_{c}^{2})\end{aligned}
  94. f ( I 1 , J 2 , J 3 ) = 0 . f(I_{1},J_{2},J_{3})=0~{}.
  95. f ( ξ , ρ , θ ) = 0 . f(\xi,\rho,\theta)=0~{}.
  96. f ( p , q , θ ) = F ( p ) + q g ( θ ) = 0 , f(p,q,\theta)=F(p)+\frac{q}{g(\theta)}=0,
  97. F ( p ) F(p)
  98. F ( p ) = { - M p c ( ϕ - ϕ m ) [ 2 ( 1 - α ) ϕ + α ] , ϕ [ 0 , 1 ] , + , ϕ [ 0 , 1 ] , F(p)=\left\{\begin{array}[]{ll}-Mp_{c}\sqrt{(\phi-\phi^{m})[2(1-\alpha)\phi+% \alpha]},&\phi\in[0,1],\\ +\infty,&\phi\notin[0,1],\end{array}\right.
  99. ϕ = p + c p c + c , \phi=\frac{p+c}{p_{c}+c},
  100. g ( θ ) g(\theta)
  101. g ( θ ) = 1 cos [ β π 6 - 1 3 cos - 1 ( γ cos 3 θ ) ] , g(\theta)=\frac{1}{\cos[\beta\frac{\pi}{6}-\frac{1}{3}\cos^{-1}(\gamma\cos 3% \theta)]},
  102. M > 0 , p c > 0 , c 0 , 0 < α < 2 , m > 1 defining F ( p ) , 0 β 2 , 0 γ < 1 defining g ( θ ) , \underbrace{M>0,~{}p_{c}>0,~{}c\geq 0,~{}0<\alpha<2,~{}m>1}_{\mbox{defining}~{% }~{}\displaystyle{F(p)}},~{}~{}~{}\underbrace{0\leq\beta\leq 2,~{}0\leq\gamma<% 1}_{\mbox{defining}~{}~{}\displaystyle{g(\theta)}},
  103. cos 3 θ = 3 3 2 I 3 I 2 3 2 \cos 3\theta=\frac{3\sqrt{3}}{2}\frac{I_{3}^{\prime}}{I_{2}^{\prime\frac{3}{2}}}
  104. ( 3 I 2 ) 3 1 + c 3 cos 3 θ + c 6 cos 2 3 θ 1 + c 3 + c 6 = ( σ eq - γ 1 I 1 1 - γ 1 ) 6 - l - m ( σ eq - γ 2 I 1 1 - γ 2 ) l σ eq m (3I_{2}^{\prime})^{3}\frac{1+c_{3}\cos 3\theta+c_{6}\cos^{2}3\theta}{1+c_{3}+c% _{6}}=\displaystyle\left(\frac{\sigma_{\mathrm{eq}}-\gamma_{1}\,I_{1}}{1-% \gamma_{1}}\right)^{6-l-m}\,\left(\frac{\sigma_{\mathrm{eq}}-\gamma_{2}\,I_{1}% }{1-\gamma_{2}}\right)^{l}\,\sigma_{\mathrm{eq}}^{m}
  105. c 3 c_{3}
  106. c 6 c_{6}
  107. π \pi
  108. c 6 = 1 4 ( 2 + c 3 ) , c 6 = 1 4 ( 2 - c 3 ) , c 6 5 12 c 3 2 - 1 3 , c_{6}=\frac{1}{4}(2+c_{3}),\qquad c_{6}=\frac{1}{4}(2-c_{3}),\qquad c_{6}\geq% \frac{5}{12}\,c_{3}^{2}-\frac{1}{3},
  109. γ 1 [ 0 , 1 [ \gamma_{1}\in[0,\,1[
  110. γ 2 \gamma_{2}
  111. γ 2 [ 0 , γ 1 [ \gamma_{2}\in[0,\,\gamma_{1}[
  112. γ 2 < 0 \gamma_{2}<0
  113. l 0 l\geq 0
  114. m 0 m\geq 0
  115. l + m < 6 l+m<6
  116. l = m = 0 l=m=0
  117. l = 0 l=0

Young_model.html

  1. L = G B G M ( h B h M d 2 ) 2 β L\;=\;G_{B}\;G_{M}\;\left(\frac{h_{B}\;h_{M}}{d^{2}}\right)^{2}\beta
  2. β \beta

Young_stellar_object.html

  1. α \alpha\,
  2. α = d log ( λ F λ ) d log ( λ ) \alpha=\frac{d\log(\lambda F_{\lambda})}{d\log(\lambda)}
  3. λ \lambda\,
  4. F λ F_{\lambda}
  5. α \alpha\,
  6. μ m {\mu}m
  7. λ < 10 μ m {\lambda}<10{\mu}m
  8. λ < 20 μ m {\lambda}<20{\mu}m
  9. α > 0.3 {\alpha}>0.3
  10. 0.3 > α > - 0.3 0.3>{\alpha}>-0.3
  11. - 0.3 > α > - 1.6 -0.3>{\alpha}>-1.6
  12. α < - 1.6 {\alpha}<-1.6

Yuktibhāṣā.html

  1. r θ = r sin θ cos θ - ( 1 / 3 ) r ( sin θ ) 3 ( cos θ ) 3 + ( 1 / 5 ) r ( sin θ ) 5 ( cos θ ) 5 - ( 1 / 7 ) r ( sin θ ) 7 ( cos θ ) 7 + r\theta={\frac{r\sin\theta}{\cos\theta}}-(1/3)\,r\,{\frac{\left(\sin\theta% \right)^{3}}{\left(\cos\theta\right)^{3}}}+(1/5)\,r\,{\frac{\left(\sin\theta% \right)^{5}}{\left(\cos\theta\right)^{5}}}-(1/7)\,r\,{\frac{\left(\sin\theta% \right)^{7}}{\left(\cos\theta\right)^{7}}}+\cdots
  2. θ = tan θ - ( 1 / 3 ) tan 3 θ + ( 1 / 5 ) tan 5 θ - \theta=\tan\theta-(1/3)\tan^{3}\theta+(1/5)\tan^{5}\theta-\cdots
  3. π 4 = 1 - 1 3 + 1 5 - 1 7 + + ( - 1 ) n 2 n + 1 + \frac{\pi}{4}=1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\cdots+\frac{(-1)^{n}}{2n+% 1}+\cdots
  4. π = 12 ( 1 - 1 3 3 + 1 5 3 2 - 1 7 3 3 + ) \pi=\sqrt{12}\left(1-{1\over 3\cdot 3}+{1\over 5\cdot 3^{2}}-{1\over 7\cdot 3^% {3}}+\cdots\right)
  5. n 2 + 1 4 n 3 + 5 n \frac{n^{2}+1}{4n^{3}+5n}
  6. π 4 \frac{\pi}{4}

Z-matrix_(mathematics).html

  1. Z = ( z i j ) ; z i j 0 , i j . Z=(z_{ij});\quad z_{ij}\leq 0,\quad i\neq j.

Zahorski_theorem.html

  1. G δ σ {G_{\delta}}_{\sigma}

Zero-order_hold.html

  1. x ZOH ( t ) = n = - x [ n ] rect ( t - T / 2 - n T T ) x_{\mathrm{ZOH}}(t)\,=\sum_{n=-\infty}^{\infty}x[n]\cdot\mathrm{rect}\left(% \frac{t-T/2-nT}{T}\right)
  2. rect ( ) \mathrm{rect}()
  3. rect ( t - T / 2 T ) \mathrm{rect}\left(\frac{t-T/2}{T}\right)
  4. x ZOH ( t ) x_{\mathrm{ZOH}}(t)\,
  5. x s ( t ) \displaystyle x_{s}(t)
  6. x ZOH ( t ) = n = - x [ n ] rect ( t - n T T - 1 2 ) x_{\mathrm{ZOH}}(t)\,=\sum_{n=-\infty}^{\infty}x[n]\cdot\mathrm{rect}\left(% \frac{t-nT}{T}-\frac{1}{2}\right)
  7. h ZOH ( t ) = 1 T rect ( t T - 1 2 ) = { 1 T if 0 t < T 0 otherwise h_{\mathrm{ZOH}}(t)\,=\frac{1}{T}\mathrm{rect}\left(\frac{t}{T}-\frac{1}{2}% \right)=\begin{cases}\frac{1}{T}&\mbox{if }~{}0\leq t<T\\ 0&\mbox{otherwise}\end{cases}
  8. H ZOH ( f ) = { h ZOH ( t ) } = 1 - e - i 2 π f T i 2 π f T = e - i π f T sinc ( f T ) H_{\mathrm{ZOH}}(f)\,=\mathcal{F}\{h_{\mathrm{ZOH}}(t)\}\,=\frac{1-e^{-i2\pi fT% }}{i2\pi fT}=e^{-i\pi fT}\mathrm{sinc}(fT)
  9. sinc ( x ) \mathrm{sinc}(x)
  10. sin ( π x ) π x \frac{\sin(\pi x)}{\pi x}
  11. H ZOH ( s ) = { h ZOH ( t ) } = 1 - e - s T s H_{\mathrm{ZOH}}(s)\,=\mathcal{L}\{h_{\mathrm{ZOH}}(t)\}\,=\frac{1-e^{-sT}}{s}

Zero-stage.html

  1. w 2 = ( w 2 T 3 / P 3 ) * ( P 3 / P 2 ) * ( T 2 / T 3 ) * ( P 2 / T 2 ) w_{2}=(w_{2}\sqrt{T_{3}}/P_{3})*(P_{3}/P_{2})*(\sqrt{T_{2}/T_{3}})*(P_{2}/% \sqrt{T_{2}})\,
  2. w 2 w_{2}\,
  3. ( w 2 T 3 / P 3 ) (w_{2}\sqrt{T_{3}}/P_{3})\,
  4. ( P 3 / P 2 ) (P_{3}/P_{2})\,
  5. T 2 / T 3 T_{2}/T_{3}\,
  6. P 3 / P 2 P_{3}/P_{2}\,
  7. P 2 P_{2}\,
  8. T 2 T_{2}\,
  9. ( P 3 / P 2 ) (P_{3}/P_{2})\,
  10. w 2 w_{2}\,

Zero_object_(algebra).html

  1. R R
  2. R R
  3. κ 0 = 0 κ0=0
  4. κ R κ∈R
  5. 0 × 0 = 0 0×0=0
  6. R R
  7. r r
  8. R R
  9. r = r × 1 = r × 0 = 0. r=r\times 1=r\times 0=0.\,
  10. [ 0 0 ] \begin{bmatrix}0\\ 0\end{bmatrix}
  11. [ ] \begin{bmatrix}\\ \end{bmatrix}
  12. R R
  13. R R
  14. A A
  15. A A
  16. 0
  17. A A
  18. 0
  19. 0 A 0∈A
  20. A A
  21. 1 1
  22. 1 0 1≠0
  23. 1 0 1≠0
  24. 1 0 1≠0
  25. 0

Zerosumfree_monoid.html

  1. ( M , 0 , + ) (M,0,+)
  2. ( a , b M ) a + b = 0 a = b = 0 (\forall a,b\in M)\ a+b=0\implies a=b=0\!
  3. 0 + 0 0+0

Zipper_(data_structure).html

  1. T ( A , R ) = 1 + A R 2 T(A,R)=1+A\cdot R^{2}
  2. A A
  3. R R
  4. d T ( A , R ) d R = A 2 R . \frac{dT(A,R)}{dR}=A\cdot 2\cdot R.
  5. R = T ( A , R ) , R=T(A,R),
  6. T T
  7. A A
  8. R R
  9. d T ( A , R ) d R | R = T ( A , R ) \frac{dT(A,R)}{dR}|_{R=T(A,R)}
  10. T ( A , R ) | R = T ( A , R ) . T(A,R)|_{R=T(A,R)}.

Zsigmondy's_theorem.html

  1. ( a n ) n 1 (a_{n})_{n\geq 1}
  2. 𝒵 ( a n ) = { n 1 : a n has no primitive prime divisors } . \mathcal{Z}(a_{n})=\{n\geq 1:a_{n}\,\text{ has no primitive prime divisors}\}.
  3. n n
  4. a n a_{n}
  5. a m a_{m}
  6. m < n m<n
  7. 𝒵 ( a n - b n ) { 1 , 2 , 6 } \mathcal{Z}(a^{n}-b^{n})\subset\{1,2,6\}
  8. { 1 , 2 , 6 , 12 } \{1,2,6,12\}
  9. { 1 } \{1\}
  10. ( a n ) n 1 (a_{n})_{n\geq 1}
  11. 𝒵 ( a n ) { 1 n 30 } \mathcal{Z}(a_{n})\subseteq\{1\leq n\leq 30\}
  12. ( W n ) n 1 (W_{n})_{n\geq 1}
  13. 𝒵 ( W n ) \mathcal{Z}(W_{n})
  14. 𝒵 ( W n ) \mathcal{Z}(W_{n})
  15. 𝒵 ( W n ) \mathcal{Z}(W_{n})

Δ18O.html

  1. δ 18 O = ( ( O 18 O 16 ) s a m p l e ( O 18 O 16 ) s t a n d a r d - 1 ) * 1000 o / o o \delta^{18}O=\Biggl(\frac{\bigl(\frac{{}^{18}O}{{}^{16}O}\bigr)_{sample}}{% \bigl(\frac{{}^{18}O}{{}^{16}O}\bigr)_{standard}}-1\Biggr)*1000\ ^{o}\!/\!_{oo}
  2. T ( deg C ) = 16.9 - 4.0 × δ 18 O calcite - δ 18 O seawater T(\,\text{deg C})=16.9-4.0\times\mathrm{\delta^{18}O_{calcite}}-\mathrm{\delta% ^{18}O_{seawater}}

Multifractal_system.html

  1. s s
  2. s ( x + a ) - s ( x ) a h ( x ) . s(\vec{x}+\vec{a})-s(\vec{x})\sim a^{h(\vec{x})}.
  3. h ( x ) h(\vec{x})
  4. x \vec{x}
  5. s s
  6. X X
  7. ζ ( q ) , q \zeta(q),\ q\in{\mathbb{R}}
  8. a a
  9. T X ( a ) T_{X}(a)
  10. a a
  11. a a
  12. a a
  13. T X ( a ) q a ζ ( q ) \langle T_{X}(a)^{q}\rangle\sim a^{\zeta(q)}
  14. q q
  15. D ( h ) D(h)
  16. ζ ( q ) \zeta(q)
  17. D ( h ) D(h)
  18. ζ ( q ) \zeta(q)
  19. ζ ( q ) \zeta(q)
  20. D ( h ) D(h)
  21. ζ ( q ) \zeta(q)
  22. T X ( a ) T_{X}(a)
  23. a a
  24. f ( α ) f(\alpha)
  25. α \alpha
  26. f ( α ) f(\alpha)
  27. α \alpha
  28. P P
  29. m m
  30. i i
  31. ϵ \epsilon
  32. α \alpha
  33. P P
  34. P [ i , ϵ ] ϵ - α i α i log P [ i , ϵ ] log ϵ - 1 P_{[i,\epsilon]}\varpropto\epsilon^{-\alpha_{i}}\therefore\alpha_{i}\varpropto% \frac{\log{P_{[i,\epsilon]}}}{\log{\epsilon^{-1}}}
  35. ϵ \epsilon
  36. i i
  37. ϵ \epsilon
  38. m [ i , ϵ ] m_{[i,\epsilon]}
  39. i i
  40. ϵ \epsilon
  41. N ϵ N_{\epsilon}
  42. ϵ \epsilon
  43. M ϵ = i = 1 N ϵ m [ i , ϵ ] = M_{\epsilon}=\sum_{i=1}^{N_{\epsilon}}m_{[i,\epsilon]}=
  44. ϵ \epsilon
  45. i i
  46. P P
  47. Q Q
  48. Q = 1 Q=1
  49. Q = 0 Q=0
  50. N ϵ N_{\epsilon}
  51. ϵ \epsilon
  52. τ \tau
  53. I ( Q ) [ ϵ ] ϵ τ ( Q ) I_{{(Q)}_{[\epsilon]}}\varpropto\epsilon^{\tau_{(Q)}}
  54. τ ( Q ) \tau_{(Q)}
  55. ϵ \epsilon
  56. Q Q
  57. τ ( Q ) = lim ϵ 0 [ log I ( Q ) [ ϵ ] log ϵ ] \tau_{(Q)}={\lim_{\epsilon\to 0}{\left[\frac{\log{I_{{(Q)}_{[\epsilon]}}}}{% \log{\epsilon}}\right]}}
  58. D ( Q ) = lim ϵ 0 [ log I ( Q ) [ ϵ ] log ϵ - 1 ] ( 1 - Q ) - 1 D_{(Q)}={\lim_{\epsilon\to 0}{\left[\frac{\log{I_{{(Q)}_{[\epsilon]}}}}{\log{% \epsilon^{-1}}}\right]}}{(1-Q)^{-1}}
  59. D ( Q ) = τ ( Q ) Q - 1 D_{(Q)}=\frac{\tau_{(Q)}}{Q-1}
  60. τ ( Q ) = D ( Q ) ( Q - 1 ) \tau_{{(Q)}}=D_{(Q)}\left(Q-1\right)
  61. τ ( Q ) = α ( Q ) Q - f ( α ( Q ) ) \tau_{(Q)}=\alpha_{(Q)}Q-f_{\left(\alpha_{(Q)}\right)}
  62. α ( Q ) \alpha_{(Q)}
  63. A ϵ , Q = i = 1 N ϵ μ i , ϵ Q P i , ϵ Q A_{\epsilon,Q}=\sum_{i=1}^{N_{\epsilon}}{\mu_{{i,\epsilon}_{Q}}{P_{{i,\epsilon% }_{Q}}}}
  64. f ( α < m t p l > ( Q ) 1.65 64 τ ( Q ) 1.66 65 τ ( Q ) [ ϵ ] 1.67 66 ϵ 1.68 67 τ ( Q ) [ ϵ ] = i = 1 N ϵ P [ i , ϵ ] Q - 1 N ϵ f_{\left(\alpha_{<}mtpl>{{(Q)}}$\par \par \@@section{subsection}{S1.SS65}{1.65% }{1.65}{{\@tag[][]{1.65}64}}{{\@tag[][]{1.65}64}}\par $\tau_{(Q)}$\par \par % \@@section{subsection}{S1.SS66}{1.66}{1.66}{{\@tag[][]{1.66}65}}{{\@tag[][]{1.% 66}65}}\par $\tau_{{(Q)}_{[\epsilon]}}$\par \par \@@section{subsection}{S1.SS6% 7}{1.67}{1.67}{{\@tag[][]{1.67}66}}{{\@tag[][]{1.67}66}}\par $\epsilon$\par % \par \@@section{subsection}{S1.SS68}{1.68}{1.68}{{\@tag[][]{1.68}67}}{{\@tag[]% []{1.68}67}}\par $$\tau_{(Q)_{[\epsilon]}}=\frac{\sum_{i=1}^{N_{\epsilon}}{P_{% [i,\epsilon]}^{Q-1}}}{N_{\epsilon}}$$\end{document}}