wpmath0000004_4

Cylinder_(engine).html

  1. (Cylinder Volume) = π ( bore 2 ) 2 Stroke \,\text{(Cylinder Volume)}=\pi\cdot\left(\frac{\,\text{bore}}{2}\right)^{2}% \cdot\,\text{Stroke}
  2. (Engine Displacement) = (Cylinder Volume) (Number of Cylinders) \,\text{(Engine Displacement)}=\,\text{(Cylinder Volume)}\cdot\,\text{(Number % of Cylinders)}

Cylinder_(geometry).html

  1. r r
  2. h h
  3. 2 π r h 2πrh
  4. A = 2 π r h A=2πrh
  5. d d
  6. h = 2 r h=2r
  7. h = 2 r h=2r
  8. h h
  9. r r
  10. x x
  11. A ( x ) A(x)
  12. A ( x ) = π r 2 A(x)=\pi r^{2}
  13. A ( y ) = π r 2 A(y)=\pi r^{2}
  14. V V
  15. V o l u m e o f c y l i n d e r = lim | | Δ 0 | | i = 1 n A ( w i ) Δ i x {Volume\;of\;cylinder}=\lim_{||\Delta\to 0||}\sum_{i=1}^{n}A(w_{i})\Delta_{i}x
  16. = 0 h A ( y ) d y =\int_{0}^{h}A(y)\,dy
  17. = 0 h π r 2 d y =\int_{0}^{h}\pi r^{2}\,dy
  18. = π r 2 h =\pi\,r^{2}\,h\,
  19. = 0 h 0 2 π 0 r s d s d ϕ d z =\int_{0}^{h}\int_{0}^{2\pi}\int_{0}^{r}s\,\,ds\,d\phi\,dz
  20. = π r 2 h =\pi\,r^{2}\,h\,
  21. S = 2 π r h + 2 π r 2 S=2\pi rh+2\pi r^{2}
  22. S = L + 2 B S=L+2B
  23. e e
  24. a a
  25. r r
  26. α α
  27. e = cos α e=\cos\alpha\,
  28. a = r sin α a=\frac{r}{\sin\alpha}\,
  29. ( x a ) 2 + ( y b ) 2 = 1 \left(\frac{x}{a}\right)^{2}+\left(\frac{y}{b}\right)^{2}=1
  30. a = b a=b
  31. V = 0 h A ( x ) d x = 0 h π a b d x = π a b 0 h d x = π a b h V=\int_{0}^{h}A(x)dx=\int_{0}^{h}\pi abdx=\pi ab\int_{0}^{h}dx=\pi abh
  32. z z
  33. ( x a ) 2 + ( y b ) 2 = - 1 \left(\frac{x}{a}\right)^{2}+\left(\frac{y}{b}\right)^{2}=-1
  34. ( x a ) 2 - ( y b ) 2 = 1 \left(\frac{x}{a}\right)^{2}-\left(\frac{y}{b}\right)^{2}=1
  35. x 2 + 2 a y = 0 {x}^{2}+2a{y}=0\,
  36. v = ( α , β , γ ) \overrightarrow{v}=(\alpha,\beta,\gamma)\,
  37. ρ 2 = α 2 + β 2 + γ 2 \rho^{2}=\alpha^{2}+\beta^{2}+\gamma^{2}\,
  38. θ = arctan ( β α ) \theta=\arctan\left(\frac{\beta}{\alpha}\right)
  39. ϕ = arcsin ( γ ρ ) \phi=\arcsin\left(\frac{\gamma}{\rho}\right)
  40. A = - x sin ( θ ) + y cos ( θ ) c o s ( ϕ ) + z cos ( θ ) sin ( ϕ ) A=-x\sin(\theta)+y\cos(\theta)cos(\phi)+z\cos(\theta)\sin(\phi)
  41. B = - y sin ( ϕ ) + z cos ( ϕ ) B=-y\sin(\phi)+z\cos(\phi)
  42. A 2 + B 2 = R 2 A^{2}+B^{2}=R^{2}\,

Čech_cohomology.html

  1. 𝒰 \mathcal{U}
  2. N ( 𝒰 ) N(\mathcal{U})
  3. 𝒰 \mathcal{U}
  4. U 1 , U 2 𝒰 U_{1},U_{2}\in\mathcal{U}
  5. U 1 U 2 U_{1}\cap U_{2}\neq\emptyset
  6. { U 0 , , U k } \{U_{0},\ldots,U_{k}\}\,\!
  7. 𝒰 \mathcal{U}
  8. U 0 U k U_{0}\cap\cdots\cap U_{k}\neq\emptyset\,\!
  9. N ( 𝒰 ) N(\mathcal{U})
  10. 𝒰 \mathcal{U}
  11. 𝒰 \mathcal{U}
  12. N ( 𝒰 ) N(\mathcal{U})
  13. X X
  14. \mathcal{F}
  15. X X
  16. 𝒰 \mathcal{U}
  17. X X
  18. σ \sigma
  19. 𝒰 \mathcal{U}
  20. q + 1 q+1
  21. 𝒰 \mathcal{U}
  22. σ \sigma
  23. | σ | |\sigma|
  24. σ = ( U i ) i { 0 , , q } \sigma=(U_{i})_{i\in\{0,\ldots,q\}}
  25. σ \sigma
  26. σ \sigma
  27. j σ := ( U i ) i { 0 , , q } { j } . \partial_{j}\sigma:=(U_{i})_{i\in\{0,\ldots,q\}\setminus\{j\}}.
  28. σ \sigma
  29. σ := j = 0 q ( - 1 ) j + 1 j σ . \partial\sigma:=\sum_{j=0}^{q}(-1)^{j+1}\partial_{j}\sigma.
  30. 𝒰 \mathcal{U}
  31. \mathcal{F}
  32. ( | σ | ) \mathcal{F}(|\sigma|)
  33. 𝒰 \mathcal{U}
  34. \mathcal{F}
  35. C q ( 𝒰 , ) C^{q}(\mathcal{U},\mathcal{F})
  36. C q ( 𝒰 , ) C^{q}(\mathcal{U},\mathcal{F})
  37. ( C . ( 𝒰 , ) , δ ) (C^{\,\textbf{.}}(\mathcal{U},\mathcal{F}),\delta)
  38. δ q : C q ( 𝒰 , ) C q + 1 ( 𝒰 , ) \delta_{q}:C^{q}(\mathcal{U},\mathcal{F})\to C^{q+1}(\mathcal{U},\mathcal{F})
  39. ( δ q ω ) ( σ ) := j = 0 q + 1 ( - 1 ) j res | σ | | j σ | ω ( j σ ) \quad(\delta_{q}\omega)(\sigma):=\sum_{j=0}^{q+1}(-1)^{j}\mathrm{res}^{|% \partial_{j}\sigma|}_{|\sigma|}\omega(\partial_{j}\sigma)
  40. res | σ | | j σ | \mathrm{res}^{|\partial_{j}\sigma|}_{|\sigma|}
  41. ( | j σ | ) \mathcal{F}(|\partial_{j}\sigma|)
  42. ( | σ | ) . \mathcal{F}(|\sigma|).
  43. δ q + 1 δ q = 0 \delta_{q+1}\circ\delta_{q}=0
  44. Z q ( 𝒰 , ) := ker ( δ q : C q ( 𝒰 , ) C q + 1 ( 𝒰 , ) ) Z^{q}(\mathcal{U},\mathcal{F}):=\ker\left(\delta_{q}:C^{q}(\mathcal{U},% \mathcal{F})\to C^{q+1}(\mathcal{U},\mathcal{F})\right)
  45. j = 0 q - 1 ( - 1 ) j res | σ | | j σ | f ( j σ ) = 0 \sum_{j=0}^{q-1}(-1)^{j}\mathrm{res}^{|\partial_{j}\sigma|}_{|\sigma|}f(% \partial_{j}\sigma)=0
  46. { A , B , C } 𝒰 U := A B C , f ( B C ) | U - f ( A C ) | U + f ( A B ) | U = 0. \forall_{\{A,B,C\}\subset\mathcal{U}}\ U:=A\cap B\cap C,\ f(B\cap C)|_{U}-f(A% \cap C)|_{U}+f(A\cap B)|_{U}=0.
  47. B q ( 𝒰 , ) := im ( δ q - 1 : C q - 1 ( 𝒰 , ) C q ( 𝒰 , ) ) B^{q}(\mathcal{U},\mathcal{F}):=\mathrm{im}\left(\delta_{q-1}:C^{q-1}(\mathcal% {U},\mathcal{F})\to C^{q}(\mathcal{U},\mathcal{F})\right)
  48. { A , B } 𝒰 , U := A B , f ( U ) = ( δ h ) ( U ) = h ( A ) | U - h ( B ) | U . \forall_{\{A,B\}\subset\mathcal{U}},U:=A\cap B,f(U)=(\delta h)(U)=h(A)|_{U}-h(% B)|_{U}.
  49. 𝒰 \mathcal{U}
  50. \mathcal{F}
  51. ( C . ( 𝒰 , ) , δ ) (C^{\,\textbf{.}}(\mathcal{U},\mathcal{F}),\delta)
  52. H ˇ q ( 𝒰 , ) := H q ( ( C . ( 𝒰 , ) , δ ) ) = Z q ( 𝒰 , ) / B q ( 𝒰 , ) \check{H}^{q}(\mathcal{U},\mathcal{F}):=H^{q}((C^{\,\textbf{.}}(\mathcal{U},% \mathcal{F}),\delta))=Z^{q}(\mathcal{U},\mathcal{F})/B^{q}(\mathcal{U},% \mathcal{F})
  53. 𝒱 \mathcal{V}
  54. 𝒰 \mathcal{U}
  55. H ˇ * ( 𝒰 , ) H ˇ * ( 𝒱 , ) . \check{H}^{*}(\mathcal{U},\mathcal{F})\to\check{H}^{*}(\mathcal{V},\mathcal{F}).
  56. \mathcal{F}
  57. H ˇ ( X , ) := lim 𝒰 H ˇ ( 𝒰 , ) \check{H}(X,\mathcal{F}):=\underrightarrow{\lim}_{\mathcal{U}}\check{H}(% \mathcal{U},\mathcal{F})
  58. H ˇ ( X ; A ) \check{H}(X;A)
  59. H ˇ ( X , A ) \check{H}(X,\mathcal{F}_{A})
  60. A \mathcal{F}_{A}
  61. { x | ρ i ( x ) > 0 } \{x|\rho_{i}(x)>0\}
  62. H ˇ * ( X ; A ) \check{H}^{*}(X;A)
  63. H * ( X ; A ) H^{*}(X;A)\,
  64. H ˇ * ( X ; ) \check{H}^{*}(X;\mathbb{R})
  65. H ˇ 1 ( X ; ) = , \check{H}^{1}(X;\mathbb{Z})=\mathbb{Z},
  66. H 1 ( X ; ) = 0. H^{1}(X;\mathbb{Z})=0.
  67. 𝒰 \mathcal{U}
  68. 𝒰 \mathcal{U}
  69. H ˇ * ( 𝒰 ; ) \check{H}^{*}(\mathcal{U};\mathbb{R})
  70. H ˇ n ( X , F ) := lim 𝒰 H ˇ n ( 𝒰 , F ) . \check{H}^{n}(X,F):=\underrightarrow{\lim}_{\mathcal{U}}\check{H}^{n}(\mathcal% {U},F).
  71. H ˇ n ( 𝒰 , F ) \check{H}^{n}(\mathcal{U},F)
  72. 𝒰 × X r := 𝒰 × X × X 𝒰 . \mathcal{U}^{\times^{r}_{X}}:=\mathcal{U}\times_{X}\dots\times_{X}\mathcal{U}.
  73. H n ( X , F ) H ˇ n ( X , F ) H^{n}(X,F)\rightarrow\check{H}^{n}(X,F)
  74. N X 𝒰 : 𝒰 × X 𝒰 × X 𝒰 𝒰 × X 𝒰 𝒰 . N_{X}\mathcal{U}:\dots\rightarrow\mathcal{U}\times_{X}\mathcal{U}\times_{X}% \mathcal{U}\rightarrow\mathcal{U}\times_{X}\mathcal{U}\rightarrow\mathcal{U}.
  75. H ˇ n ( 𝒰 , F ) \check{H}^{n}(\mathcal{U},F)
  76. N X 𝒰 N_{X}\mathcal{U}
  77. H n ( X , F ) = lim K * H n ( F ( K * ) ) , H^{n}(X,F)=\underrightarrow{\lim}_{K_{*}}H^{n}(F(K_{*})),

Damping_factor.html

  1. Z load Z_{\mathrm{load}}
  2. Z source Z_{\mathrm{source}}
  3. D F DF
  4. D F = Z load Z source DF=\frac{Z_{\mathrm{load}}}{Z_{\mathrm{source}}}\,
  5. Z source Z_{\mathrm{source}}
  6. Z source = Z load D F Z_{\mathrm{source}}=\frac{Z_{\mathrm{load}}}{DF}\,
  7. Z source Z_{\mathrm{source}}
  8. Z load Z_{\mathrm{load}}
  9. Z load Z_{\mathrm{load}}
  10. Z source Z_{\mathrm{source}}

Dangling_pointer.html

  1. ( x 1 = = x n n o n e ) precondition [ k i l l ( x i ) ] s t a t e m e n t ( x 1 = = x n = n o n e ) postcondition \underbrace{(x_{1}=\dots=x_{n}\neq none)}_{\mathrm{precondition}}\Rightarrow% \underbrace{[kill(x_{i})]}_{\mathrm{}statement}\underbrace{(x_{1}=\dots=x_{n}=% none)}_{\mathrm{postcondition}}
  2. O ( 1 ) O(1)

Darboux_integral.html

  1. a = x 0 < x 1 < < x n = b . a=x_{0}<x_{1}<\cdots<x_{n}=b.\,\!
  2. P = ( x 0 , , x n ) P=(x_{0},\ldots,x_{n})\,\!
  3. M i = sup x [ x i - 1 , x i ] f ( x ) , m i = inf x [ x i - 1 , x i ] f ( x ) . \begin{aligned}\displaystyle M_{i}=\sup_{x\in[x_{i-1},x_{i}]}f(x),\\ \displaystyle m_{i}=\inf_{x\in[x_{i-1},x_{i}]}f(x).\end{aligned}
  4. U f , P = i = 1 n ( x i - x i - 1 ) M i . U_{f,P}=\sum_{i=1}^{n}(x_{i}-x_{i-1})M_{i}.\,\!
  5. L f , P = i = 1 n ( x i - x i - 1 ) m i . L_{f,P}=\sum_{i=1}^{n}(x_{i}-x_{i-1})m_{i}.\,\!
  6. U f = inf { U f , P : P is a partition of [ a , b ] } . U_{f}=\inf\{U_{f,P}\colon P\,\text{ is a partition of }[a,b]\}.\,\!
  7. L f = sup { L f , P : P is a partition of [ a , b ] } . L_{f}=\sup\{L_{f,P}\colon P\,\text{ is a partition of }[a,b]\}.\,\!
  8. L f a b ¯ f ( x ) d x U f a b ¯ f ( x ) d x \begin{aligned}\displaystyle L_{f}\equiv\underline{\int_{a}^{b}}f(x)\,dx&% \displaystyle\quad U_{f}\equiv\overline{\int_{a}^{b}}f(x)\,dx\end{aligned}
  9. a b f ( t ) d t = U f = L f , \int_{a}^{b}{f(t)\,dt}=U_{f}=L_{f},\,\!
  10. U f , P ϵ - L f , P ϵ < ϵ U_{f,P_{\epsilon}}-L_{f,P_{\epsilon}}<\epsilon
  11. ( b - a ) inf x [ a , b ] f ( x ) L f , P U f , P ( b - a ) sup x [ a , b ] f ( x ) (b-a)\inf_{x\in[a,b]}f(x)\leq L_{f,P}\leq U_{f,P}\leq(b-a)\sup_{x\in[a,b]}f(x)
  12. a b ¯ f ( x ) d x a b ¯ f ( x ) d x \underline{\int_{a}^{b}}f(x)\,dx\leq\overline{\int_{a}^{b}}f(x)\,dx
  13. a b ¯ f ( x ) d x = a c ¯ f ( x ) d x + c b ¯ f ( x ) d x a b ¯ f ( x ) d x = a c ¯ f ( x ) d x + c b ¯ f ( x ) d x \begin{aligned}\displaystyle\underline{\int_{a}^{b}}f(x)\,dx&\displaystyle=% \underline{\int_{a}^{c}}f(x)\,dx+\underline{\int_{c}^{b}}f(x)\,dx\\ \displaystyle\overline{\int_{a}^{b}}f(x)\,dx&\displaystyle=\overline{\int_{a}^% {c}}f(x)\,dx+\overline{\int_{c}^{b}}f(x)\,dx\end{aligned}
  14. a b ¯ f ( x ) d x + a b ¯ g ( x ) d x a b ¯ f ( x ) + g ( x ) d x a b ¯ f ( x ) d x + a b ¯ g ( x ) d x a b ¯ f ( x ) + g ( x ) d x \begin{aligned}\displaystyle\underline{\int_{a}^{b}}f(x)\,dx+\underline{\int_{% a}^{b}}g(x)\,dx&\displaystyle\leq\underline{\int_{a}^{b}}f(x)+g(x)\,dx\\ \displaystyle\overline{\int_{a}^{b}}f(x)\,dx+\overline{\int_{a}^{b}}g(x)\,dx&% \displaystyle\geq\overline{\int_{a}^{b}}f(x)+g(x)\,dx\end{aligned}
  15. a b ¯ c f ( x ) = c a b ¯ f ( x ) a b ¯ c f ( x ) = c a b ¯ f ( x ) \begin{aligned}\displaystyle\underline{\int_{a}^{b}}cf(x)&\displaystyle=c% \underline{\int_{a}^{b}}f(x)\\ \displaystyle\overline{\int_{a}^{b}}cf(x)&\displaystyle=c\overline{\int_{a}^{b% }}f(x)\end{aligned}
  16. a b ¯ c f ( x ) = c a b ¯ f ( x ) a b ¯ c f ( x ) = c a b ¯ f ( x ) \begin{aligned}\displaystyle\underline{\int_{a}^{b}}cf(x)&\displaystyle=c% \overline{\int_{a}^{b}}f(x)\\ \displaystyle\overline{\int_{a}^{b}}cf(x)&\displaystyle=c\underline{\int_{a}^{% b}}f(x)\end{aligned}
  17. F ( x ) = a x ¯ f ( t ) d t F(x)=\underline{\int_{a}^{x}}f(t)\,dt
  18. L f , P n = k = 1 n f ( x k - 1 ) ( x k - x k - 1 ) = k = 1 n k - 1 n 1 n = 1 n 2 k = 1 n [ k - 1 ] = 1 n 2 [ ( n - 1 ) n 2 ] \begin{aligned}\displaystyle L_{f,P_{n}}&\displaystyle=\sum_{k=1}^{n}f(x_{k-1}% )(x_{k}-x_{k-1})\\ &\displaystyle=\sum_{k=1}^{n}\frac{k-1}{n}\cdot\frac{1}{n}\\ &\displaystyle=\frac{1}{n^{2}}\sum_{k=1}^{n}[k-1]\\ &\displaystyle=\frac{1}{n^{2}}\left[\frac{(n-1)n}{2}\right]\end{aligned}
  19. U f , P n = k = 1 n f ( x k ) ( x k - x k - 1 ) = k = 1 n k n 1 n = 1 n 2 k = 1 n k = 1 n 2 [ ( n + 1 ) n 2 ] \begin{aligned}\displaystyle U_{f,P_{n}}&\displaystyle=\sum_{k=1}^{n}f(x_{k})(% x_{k}-x_{k-1})\\ &\displaystyle=\sum_{k=1}^{n}\frac{k}{n}\cdot\frac{1}{n}\\ &\displaystyle=\frac{1}{n^{2}}\sum_{k=1}^{n}k\\ &\displaystyle=\frac{1}{n^{2}}\left[\frac{(n+1)n}{2}\right]\end{aligned}
  20. U f , P n - L f , P n = 1 n \begin{aligned}\displaystyle U_{f,P_{n}}-L_{f,P_{n}}&\displaystyle=\frac{1}{n}% \end{aligned}
  21. U f , P n - L f , P n < ϵ \begin{aligned}\displaystyle U_{f,P_{n}}-L_{f,P_{n}}&\displaystyle<\epsilon% \end{aligned}
  22. 0 1 f ( x ) d x = lim n U f , P n = lim n L f , P n = 1 2 \begin{aligned}\displaystyle\int_{0}^{1}f(x)\,dx&\displaystyle=\lim_{n\to% \infty}U_{f,P_{n}}=\lim_{n\to\infty}L_{f,P_{n}}=\frac{1}{2}\end{aligned}
  23. f ( x ) = { 0 , if x is rational 1 , if x is irrational \begin{aligned}\displaystyle f(x)&\displaystyle=\begin{cases}0,&\,\text{if }x% \,\text{ is rational}\\ 1,&\,\text{if }x\,\text{ is irrational}\end{cases}\end{aligned}
  24. L f , P = k = 1 n ( x k - x k - 1 ) inf x [ x k - 1 , x k ] f = 0 U f , P = k = 1 n ( x k - x k - 1 ) sup x [ x k - 1 , x k ] f = 1 \begin{aligned}\displaystyle L_{f,P}&\displaystyle=\sum_{k=1}^{n}(x_{k}-x_{k-1% })\inf_{x\in[x_{k-1},x_{k}]}f=0\\ \displaystyle U_{f,P}&\displaystyle=\sum_{k=1}^{n}(x_{k}-x_{k-1})\sup_{x\in[x_% {k-1},x_{k}]}f=1\end{aligned}
  25. x 0 , , x n x_{0},\ldots,x_{n}\,\!
  26. y 0 , , y m y_{0},\ldots,y_{m}\,\!
  27. 0 i n 0\leq i\leq n\,\!
  28. x i = y r ( i ) . x_{i}=y_{r(i)}.\,\!
  29. P = ( y 0 , , y m ) P^{\prime}=(y_{0},\ldots,y_{m})\,\!
  30. P = ( x 0 , , x n ) , P=(x_{0},\ldots,x_{n}),\,\!
  31. U f , P U f , P U_{f,P}\geq U_{f,P^{\prime}}\,\!
  32. L f , P L f , P . L_{f,P}\leq L_{f,P^{\prime}}.\,\!
  33. L f , P 1 U f , P 2 . L_{f,P_{1}}\leq U_{f,P_{2}}.\,\!
  34. L f U f . L_{f}\leq U_{f}.\,\!
  35. P = ( x 0 , , x n ) P=(x_{0},\ldots,x_{n})\,\!
  36. T = ( t 1 , , t n ) T=(t_{1},\ldots,t_{n})\,\!
  37. x 0 t 1 x 1 x n - 1 t n x n x_{0}\leq t_{1}\leq x_{1}\leq\cdots\leq x_{n-1}\leq t_{n}\leq x_{n}\,\!
  38. L f , P R U f , P . L_{f,P}\leq R\leq U_{f,P}.\,\!

Darboux_vector.html

  1. s y m b o l ω = τ 𝐓 + κ 𝐁 ( 1 ) symbol{\omega}=\tau\mathbf{T}+\kappa\mathbf{B}\qquad\qquad(1)
  2. s y m b o l ω × 𝐓 = 𝐓 , symbol{\omega}\times\mathbf{T}=\mathbf{T^{\prime}},
  3. s y m b o l ω × 𝐍 = 𝐍 , symbol{\omega}\times\mathbf{N}=\mathbf{N^{\prime}},
  4. s y m b o l ω × 𝐁 = 𝐁 , symbol{\omega}\times\mathbf{B}=\mathbf{B^{\prime}},
  5. s y m b o l ω = s y m b o l ω 𝐓 + s y m b o l ω 𝐍 + s y m b o l ω 𝐁 . symbol{\omega}=symbol{\omega}_{\mathbf{T}}+symbol{\omega}_{\mathbf{N}}+symbol{% \omega}_{\mathbf{B}}.
  6. s y m b o l ω 𝐓 = lim Δ t 0 𝐓 ( t ) × 𝐓 ( t + Δ t ) 2 Δ t symbol{\omega}_{\mathbf{T}}=\lim_{\Delta t\rightarrow 0}{\mathbf{T}(t)\times% \mathbf{T}(t+\Delta t)\over 2\,\Delta t}
  7. = 𝐓 ( t ) × 𝐓 ( t ) 2 . ={\mathbf{T}(t)\times\mathbf{T^{\prime}}(t)\over 2}.
  8. s y m b o l ω 𝐍 = 1 2 𝐍 ( t ) × 𝐍 ( t ) , symbol{\omega}_{\mathbf{N}}={1\over 2}\ \mathbf{N}(t)\times\mathbf{N^{\prime}}% (t),
  9. s y m b o l ω 𝐁 = 1 2 𝐁 ( t ) × 𝐁 ( t ) . symbol{\omega}_{\mathbf{B}}={1\over 2}\ \mathbf{B}(t)\times\mathbf{B^{\prime}}% (t).
  10. s y m b o l ω 𝐓 = 1 2 𝐓 × 𝐓 = 1 2 κ 𝐓 × 𝐍 = 1 2 κ 𝐁 symbol{\omega}_{\mathbf{T}}={1\over 2}\mathbf{T}\times\mathbf{T^{\prime}}={1% \over 2}\kappa\mathbf{T}\times\mathbf{N}={1\over 2}\kappa\mathbf{B}
  11. s y m b o l ω 𝐍 = 1 2 𝐍 × 𝐍 = 1 2 ( - κ 𝐍 × 𝐓 + τ 𝐍 × 𝐁 ) = 1 2 ( κ 𝐁 + τ 𝐓 ) symbol{\omega}_{\mathbf{N}}={1\over 2}\mathbf{N}\times\mathbf{N^{\prime}}={1% \over 2}(-\kappa\mathbf{N}\times\mathbf{T}+\tau\mathbf{N}\times\mathbf{B})={1% \over 2}(\kappa\mathbf{B}+\tau\mathbf{T})
  12. s y m b o l ω 𝐁 = 1 2 𝐁 × 𝐁 = - 1 2 τ 𝐁 × 𝐍 = 1 2 τ 𝐓 symbol{\omega}_{\mathbf{B}}={1\over 2}\mathbf{B}\times\mathbf{B^{\prime}}=-{1% \over 2}\tau\mathbf{B}\times\mathbf{N}={1\over 2}\tau\mathbf{T}
  13. s y m b o l ω = 1 2 κ 𝐁 + 1 2 ( κ 𝐁 + τ 𝐓 ) + 1 2 τ 𝐓 = κ 𝐁 + τ 𝐓 , symbol{\omega}={1\over 2}\kappa\mathbf{B}+{1\over 2}(\kappa\mathbf{B}+\tau% \mathbf{T})+{1\over 2}\tau\mathbf{T}=\kappa\mathbf{B}+\tau\mathbf{T},

Darcy's_law.html

  1. Q = - κ A μ ( p b - p a ) L Q=\frac{-\kappa A}{\mu}\frac{(p_{b}-p_{a})}{L}
  2. κ \kappa
  3. q = - κ μ p q=\frac{-\kappa}{\mu}\nabla p
  4. p \nabla p
  5. ϕ \phi
  6. v = q ϕ v=\frac{q}{\phi}
  7. R e = ρ v d 30 μ Re=\frac{\rho vd_{30}}{\mu}
  8. D ( ρ u i ) / D t 0 D\left(\rho u_{i}\right)/Dt\approx 0
  9. μ 2 u i + ρ g i - i p = 0 \mu\nabla^{2}u_{i}+\rho g_{i}-\partial_{i}p=0
  10. μ \mu
  11. u i u_{i}
  12. i i
  13. g i g_{i}
  14. i i
  15. p p
  16. - ( κ i j ) - 1 μ ϕ u j + ρ g i - i p = 0 -\left(\kappa_{ij}\right)^{-1}\mu\phi u_{j}+\rho g_{i}-\partial_{i}p=0
  17. ϕ \phi
  18. κ i j \kappa_{ij}
  19. n n
  20. κ n i ( κ i j ) - 1 u j = δ n j u j = u n = - κ n i ϕ μ ( i p - ρ g i ) \kappa_{ni}\left(\kappa_{ij}\right)^{-1}u_{j}=\delta_{nj}u_{j}=u_{n}=-\frac{% \kappa_{ni}}{\phi\mu}\left(\partial_{i}p-\rho g_{i}\right)
  21. n n
  22. q n = - κ n i μ ( i p - ρ g i ) q_{n}=-\frac{\kappa_{ni}}{\mu}\left(\partial_{i}p-\rho g_{i}\right)
  23. κ i j = 0 \kappa_{ij}=0
  24. i j i\neq j
  25. κ = κ i i \kappa=\kappa_{ii}
  26. s y m b o l q = - κ μ ( s y m b o l p - ρ s y m b o l g ) symbol{q}=-\frac{\kappa}{\mu}\left(symbol{\nabla}p-\rho symbol{g}\right)
  27. τ q t + q = - κ h \tau\frac{\partial q}{\partial t}+q=-\kappa\nabla h
  28. β 2 q + q = - κ p \beta\nabla^{2}q+q=-\kappa\nabla p
  29. Q = κ A μ ( p x ) Q=\frac{\kappa A}{\mu}\left(\frac{\partial p}{\partial x}\right)
  30. p / x \partial p/\partial x
  31. p x = - μ κ q - ρ κ 1 q 2 \frac{\partial p}{\partial x}=-\frac{\mu}{\kappa}q-\frac{\rho}{\kappa_{1}}q^{2}
  32. κ 1 \kappa_{1}
  33. N = - ( κ μ p a + p b 2 + D k e f f ) 1 R g T p b - p a L N=-\left(\frac{\kappa}{\mu}\frac{p_{a}+p_{b}}{2}+D_{k}^{eff}\right)\frac{1}{R_% {g}T}\frac{p_{b}-p_{a}}{L}
  34. D k e f f D_{k}^{eff}
  35. p x = - R g T ( κ p μ + D K ) - 1 N \frac{\partial p}{\partial x}=-R_{g}T\left(\frac{\kappa p}{\mu}+D_{K}\right)^{% -1}N
  36. p x = - R g T ( κ p μ + D K ) - 1 p R g T q \frac{\partial p}{\partial x}=-R_{g}T\left(\frac{\kappa p}{\mu}+D_{K}\right)^{% -1}\dfrac{p}{R_{g}T}q
  37. q = - κ μ ( 1 + D k μ κ 1 p ) p x q=-\frac{\kappa}{\mu}\left(1+\frac{D_{k}\mu}{\kappa}\frac{1}{p}\right)\frac{% \partial p}{\partial x}
  38. q = - κ e f f μ p x q=-\frac{\kappa^{eff}}{\mu}\frac{\partial p}{\partial x}
  39. κ e f f = κ ( 1 + D k μ κ 1 p ) \kappa^{eff}=\kappa\left(1+\frac{D_{k}\mu}{\kappa}\frac{1}{p}\right)
  40. κ e f f = κ ( 1 + b p ) \kappa^{eff}=\kappa\left(1+\frac{b}{p}\right)

Dark_state.html

  1. ω \omega
  2. E 1 E_{1}
  3. E 2 = E 1 + ω E_{2}=E_{1}+\hbar\omega
  4. E 3 < E 2 E_{3}<E_{2}
  5. | 1 |1\rangle
  6. | 2 |2\rangle
  7. 1 2 S 1 / 2 1^{2}S_{1/2}
  8. 2 2 P 3 / 2 2^{2}P_{3/2}
  9. 2 2 P 3 / 2 2^{2}P_{3/2}
  10. | 1 | 3 |1\rangle\leftrightarrow|3\rangle
  11. | 2 | 3 |2\rangle\leftrightarrow|3\rangle
  12. | 1 | 2 |1\rangle\leftrightarrow|2\rangle
  13. H = H 0 + H 1 H=H_{0}+H_{1}
  14. H 0 = ω 1 | 1 1 | + ω 2 | 2 2 | + ω 3 | 3 3 | , H_{0}=\hbar\omega_{1}|1\rangle\langle 1|+\hbar\omega_{2}|2\rangle\langle 2|+% \hbar\omega_{3}|3\rangle\langle 3|,
  15. H 1 = - 2 ( Ω p e - i ω p t | 1 3 | + Ω c e - i ω c t | 2 3 | ) + H.c. , H_{1}=-\frac{\hbar}{2}\left(\Omega_{p}e^{-i\omega_{p}t}|1\rangle\langle 3|+% \Omega_{c}e^{-i\omega_{c}t}|2\rangle\langle 3|\right)+\mbox{H.c.}~{},
  16. Ω p \Omega_{p}
  17. Ω c \Omega_{c}
  18. ω p \omega_{p}
  19. ω c \omega_{c}
  20. ω 1 - ω 3 \omega_{1}-\omega_{3}
  21. ω 2 - ω 3 \omega_{2}-\omega_{3}
  22. | ψ ( t ) = c 1 ( t ) e - i ω 1 t | 1 + c 2 ( t ) e - i ω 2 t | 2 + c 3 ( t ) e - i ω 3 t | 3 . |\psi(t)\rangle=c_{1}(t)e^{-i\omega_{1}t}|1\rangle+c_{2}(t)e^{-i\omega_{2}t}|2% \rangle+c_{3}(t)e^{-i\omega_{3}t}|3\rangle.
  23. i | ψ ˙ = H | ψ i\hbar|\dot{\psi}\rangle=H|\psi\rangle
  24. c ˙ 1 = i 2 Ω p c 3 \dot{c}_{1}=\frac{i}{2}\Omega_{p}c_{3}
  25. c ˙ 2 = i 2 Ω c c 3 \dot{c}_{2}=\frac{i}{2}\Omega_{c}c_{3}
  26. c ˙ 3 = i 2 ( Ω p c 1 + Ω c c 2 ) . \dot{c}_{3}=\frac{i}{2}(\Omega_{p}c_{1}+\Omega_{c}c_{2}).
  27. | ψ ( 0 ) = c 1 ( 0 ) | 1 + c 2 ( 0 ) | 2 + c 3 ( 0 ) | 3 , |\psi(0)\rangle=c_{1}(0)|1\rangle+c_{2}(0)|2\rangle+c_{3}(0)|3\rangle,
  28. c 1 ( t ) = c 1 ( 0 ) [ Ω c 2 Ω 2 + Ω p 2 Ω 2 cos Ω t 2 ] + c 2 ( 0 ) [ - Ω p Ω c Ω 2 + Ω p Ω c Ω 2 cos Ω t 2 ] - i c 3 ( 0 ) Ω p Ω sin Ω t 2 c_{1}(t)=c_{1}(0)\left[\frac{\Omega_{c}^{2}}{\Omega^{2}}+\frac{\Omega_{p}^{2}}% {\Omega^{2}}\cos\frac{\Omega t}{2}\right]+c_{2}(0)\left[-\frac{\Omega_{p}% \Omega_{c}}{\Omega^{2}}+\frac{\Omega_{p}\Omega_{c}}{\Omega^{2}}\cos\frac{% \Omega t}{2}\right]\quad-ic_{3}(0)\frac{\Omega_{p}}{\Omega}\sin\frac{\Omega t}% {2}
  29. c 2 ( t ) = c 1 ( 0 ) [ - Ω p Ω c Ω 2 + Ω p Ω c Ω 2 cos Ω t 2 ] + c 2 ( 0 ) [ Ω p 2 Ω 2 + Ω c 2 Ω 2 cos Ω t 2 ] - i c 3 ( 0 ) Ω c Ω sin Ω t 2 c_{2}(t)=c_{1}(0)\left[-\frac{\Omega_{p}\Omega_{c}}{\Omega^{2}}+\frac{\Omega_{% p}\Omega_{c}}{\Omega^{2}}\cos\frac{\Omega t}{2}\right]+c_{2}(0)\left[\frac{% \Omega_{p}^{2}}{\Omega^{2}}+\frac{\Omega_{c}^{2}}{\Omega^{2}}\cos\frac{\Omega t% }{2}\right]\quad-ic_{3}(0)\frac{\Omega_{c}}{\Omega}\sin\frac{\Omega t}{2}
  30. c 3 ( t ) = - i c 1 ( 0 ) Ω p Ω sin Ω t 2 - i c 2 ( 0 ) Ω c Ω sin Ω t 2 + c 3 ( 0 ) cos Ω t 2 c_{3}(t)=-ic_{1}(0)\frac{\Omega_{p}}{\Omega}\sin\frac{\Omega t}{2}-ic_{2}(0)% \frac{\Omega_{c}}{\Omega}\sin\frac{\Omega t}{2}+c_{3}(0)\cos\frac{\Omega t}{2}
  31. Ω = Ω c 2 + Ω p 2 \Omega=\sqrt{\Omega_{c}^{2}+\Omega_{p}^{2}}
  32. c 1 ( 0 ) = Ω c Ω , c 2 ( 0 ) = - Ω p Ω , c 3 ( 0 ) = 0 , c_{1}(0)=\frac{\Omega_{c}}{\Omega},\qquad c_{2}(0)=-\frac{\Omega_{p}}{\Omega},% \qquad c_{3}(0)=0,
  33. | 3 |3\rangle
  34. θ \theta
  35. | D = cos θ | 1 - sin θ | 2 |D\rangle=\cos\theta|1\rangle-\sin\theta|2\rangle
  36. cos θ = Ω c Ω p 2 + Ω c 2 , sin θ = Ω p Ω p 2 + Ω c 2 . \cos\theta=\frac{\Omega_{c}}{\sqrt{\Omega_{p}^{2}+\Omega_{c}^{2}}},\qquad\sin% \theta=\frac{\Omega_{p}}{\sqrt{\Omega_{p}^{2}+\Omega_{c}^{2}}}.
  37. | 3 |3\rangle

Data-flow_analysis.html

  1. o u t b = t r a n s b ( i n b ) out_{b}=trans_{b}(in_{b})
  2. i n b = j o i n p p r e d b ( o u t p ) in_{b}=join_{p\in pred_{b}}(out_{p})
  3. t r a n s b trans_{b}
  4. b b
  5. i n b in_{b}
  6. o u t b out_{b}
  7. j o i n join
  8. p p r e d b p\in pred_{b}
  9. b b
  10. b b
  11. x 1 x_{1}
  12. o u t b = s s u c c b i n s out_{b}=\bigcup_{s\in succ_{b}}in_{s}
  13. i n b = ( o u t b - k i l l b ) g e n b in_{b}=(out_{b}-kill_{b})\cup gen_{b}

David_Hestenes.html

  1. i i\hbar
  2. i i
  3. / 2 \hbar/2

De_Casteljau's_algorithm.html

  1. β 0 , , β n \beta_{0},\ldots,\beta_{n}
  2. B ( t ) = i = 0 n β i b i , n ( t ) B(t)=\sum_{i=0}^{n}\beta_{i}b_{i,n}(t)
  3. b i , n ( t ) = ( n i ) ( 1 - t ) n - i t i b_{i,n}(t)={n\choose i}(1-t)^{n-i}t^{i}
  4. β i ( 0 ) := β i , i = 0 , , n \beta_{i}^{(0)}:=\beta_{i}\mbox{ , }~{}i=0,\ldots,n
  5. β i ( j ) := β i ( j - 1 ) ( 1 - t 0 ) + β i + 1 ( j - 1 ) t 0 , i = 0 , , n - j , j = 1 , , n \beta_{i}^{(j)}:=\beta_{i}^{(j-1)}(1-t_{0})+\beta_{i+1}^{(j-1)}t_{0}\mbox{ , }% ~{}i=0,\ldots,n-j\mbox{ , }~{}j=1,\ldots,n
  6. B B
  7. t 0 t_{0}
  8. n n
  9. B ( t 0 ) B(t_{0})
  10. B ( t 0 ) = β 0 ( n ) . B(t_{0})=\beta_{0}^{(n)}.
  11. B B
  12. t 0 t_{0}
  13. β 0 ( 0 ) , β 0 ( 1 ) , , β 0 ( n ) \beta_{0}^{(0)},\beta_{0}^{(1)},\ldots,\beta_{0}^{(n)}
  14. β 0 ( n ) , β 1 ( n - 1 ) , , β n ( 0 ) \beta_{0}^{(n)},\beta_{1}^{(n-1)},\ldots,\beta_{n}^{(0)}
  15. β 0 = β 0 ( 0 ) β 0 ( 1 ) β 1 = β 1 ( 0 ) β 0 ( n ) β n - 1 = β n - 1 ( 0 ) β n - 1 ( 1 ) β n = β n ( 0 ) \begin{matrix}\beta_{0}&=\beta_{0}^{(0)}&&&\\ &&\beta_{0}^{(1)}&&\\ \beta_{1}&=\beta_{1}^{(0)}&&&\\ &&&\ddots&\\ \vdots&&\vdots&&\beta_{0}^{(n)}\\ &&&&\\ \beta_{n-1}&=\beta_{n-1}^{(0)}&&&\\ &&\beta_{n-1}^{(1)}&&\\ \beta_{n}&=\beta_{n}^{(0)}&&&\\ \end{matrix}
  16. B ( t ) = i = 0 n β i ( 0 ) b i , n ( t ) , t [ 0 , 1 ] B(t)=\sum_{i=0}^{n}\beta_{i}^{(0)}b_{i,n}(t)\mbox{ , }~{}\qquad t\in[0,1]
  17. B 1 ( t ) = i = 0 n β 0 ( i ) b i , n ( t t 0 ) , t [ 0 , t 0 ] B_{1}(t)=\sum_{i=0}^{n}\beta_{0}^{(i)}b_{i,n}\left(\frac{t}{t_{0}}\right)\mbox% { , }~{}\qquad t\in[0,t_{0}]
  18. B 2 ( t ) = i = 0 n β i ( n - i ) b i , n ( t - t 0 1 - t 0 ) , t [ t 0 , 1 ] B_{2}(t)=\sum_{i=0}^{n}\beta_{i}^{(n-i)}b_{i,n}\left(\frac{t-t_{0}}{1-t_{0}}% \right)\mbox{ , }~{}\qquad t\in[t_{0},1]
  19. β 0 ( 0 ) = β 0 \beta_{0}^{(0)}=\beta_{0}
  20. β 1 ( 0 ) = β 1 \beta_{1}^{(0)}=\beta_{1}
  21. β 2 ( 0 ) = β 2 \beta_{2}^{(0)}=\beta_{2}
  22. β 0 ( 1 ) = β 0 ( 0 ) ( 1 - t 0 ) + β 1 ( 0 ) t 0 = β 0 ( 1 - t 0 ) + β 1 t 0 \beta_{0}^{(1)}=\beta_{0}^{(0)}(1-t_{0})+\beta_{1}^{(0)}t_{0}=\beta_{0}(1-t_{0% })+\beta_{1}t_{0}
  23. β 1 ( 1 ) = β 1 ( 0 ) ( 1 - t 0 ) + β 2 ( 0 ) t 0 = β 1 ( 1 - t 0 ) + β 2 t 0 \beta_{1}^{(1)}=\beta_{1}^{(0)}(1-t_{0})+\beta_{2}^{(0)}t_{0}=\beta_{1}(1-t_{0% })+\beta_{2}t_{0}
  24. β 0 ( 2 ) = β 0 ( 1 ) ( 1 - t 0 ) + β 1 ( 1 ) t 0 = β 0 ( 1 - t 0 ) ( 1 - t 0 ) + β 1 t 0 ( 1 - t 0 ) + β 1 ( 1 - t 0 ) t 0 + β 2 t 0 t 0 = β 0 ( 1 - t 0 ) 2 + β 1 2 t 0 ( 1 - t 0 ) + β 2 t 0 2 \begin{aligned}\displaystyle\beta_{0}^{(2)}&\displaystyle=\beta_{0}^{(1)}(1-t_% {0})+\beta_{1}^{(1)}t_{0}\\ &\displaystyle=\beta_{0}(1-t_{0})(1-t_{0})+\beta_{1}t_{0}(1-t_{0})+\beta_{1}(1% -t_{0})t_{0}+\beta_{2}t_{0}t_{0}\\ &\displaystyle=\beta_{0}(1-t_{0})^{2}+\beta_{1}2t_{0}(1-t_{0})+\beta_{2}t_{0}^% {2}\end{aligned}
  25. 𝐁 ( t ) = i = 0 n 𝐏 i b i , n ( t ) , t [ 0 , 1 ] \mathbf{B}(t)=\sum_{i=0}^{n}\mathbf{P}_{i}b_{i,n}(t)\mbox{ , }~{}t\in[0,1]
  26. 𝐏 i := ( x i y i z i ) \mathbf{P}_{i}:=\begin{pmatrix}x_{i}\\ y_{i}\\ z_{i}\end{pmatrix}
  27. B 1 ( t ) = i = 0 n x i b i , n ( t ) , t [ 0 , 1 ] B_{1}(t)=\sum_{i=0}^{n}x_{i}b_{i,n}(t)\mbox{ , }~{}t\in[0,1]
  28. B 2 ( t ) = i = 0 n y i b i , n ( t ) , t [ 0 , 1 ] B_{2}(t)=\sum_{i=0}^{n}y_{i}b_{i,n}(t)\mbox{ , }~{}t\in[0,1]
  29. B 3 ( t ) = i = 0 n z i b i , n ( t ) , t [ 0 , 1 ] B_{3}(t)=\sum_{i=0}^{n}z_{i}b_{i,n}(t)\mbox{ , }~{}t\in[0,1]
  30. P 0 , , P n \scriptstyle P_{0},...,P_{n}
  31. t : ( 1 - t ) \scriptstyle t:(1-t)
  32. t \scriptstyle t
  33. t \scriptstyle t
  34. t \scriptstyle t
  35. 𝐑 n \scriptstyle\mathbf{R}^{n}
  36. 𝐑 n + 1 \scriptstyle\mathbf{R}^{n+1}
  37. { ( x i , y i , z i ) } \scriptstyle\{(x_{i},y_{i},z_{i})\}
  38. { w i } \scriptstyle\{w_{i}\}
  39. { ( w i x i , w i y i , w i z i , w i ) } \scriptstyle\{(w_{i}x_{i},w_{i}y_{i},w_{i}z_{i},w_{i})\}
  40. 𝐑 4 \scriptstyle\mathbf{R}^{4}

Debye_length.html

  1. N N
  2. j j
  3. q j q_{j}
  4. n j ( 𝐫 ) n_{j}(\mathbf{r})
  5. 𝐫 \mathbf{r}
  6. ε r \varepsilon_{r}
  7. Φ ( 𝐫 ) \Phi(\mathbf{r})
  8. ε 2 Φ ( 𝐫 ) = - j = 1 N q j n j ( 𝐫 ) - ρ E ( 𝐫 ) \varepsilon\nabla^{2}\Phi(\mathbf{r})=-\,\sum_{j=1}^{N}q_{j}\,n_{j}(\mathbf{r}% )-\rho_{E}(\mathbf{r})
  9. ε ε r ε 0 \varepsilon\equiv\varepsilon_{r}\varepsilon_{0}
  10. ε 0 \varepsilon_{0}
  11. ρ E \rho_{E}
  12. Φ ( 𝐫 ) \Phi(\mathbf{r})
  13. - q j Φ ( 𝐫 ) -q_{j}\,\nabla\Phi(\mathbf{r})
  14. T T
  15. n j ( 𝐫 ) n_{j}(\mathbf{r})
  16. j j
  17. n j ( 𝐫 ) = n j 0 exp ( - q j Φ ( 𝐫 ) k B T ) n_{j}(\mathbf{r})=n_{j}^{0}\,\exp\left(-\frac{q_{j}\,\Phi(\mathbf{r})}{k_{B}T}\right)
  18. k B k_{B}
  19. n j 0 n_{j}^{0}
  20. j j
  21. ε 2 Φ ( 𝐫 ) = - j = 1 N q j n j 0 exp ( - q j Φ ( 𝐫 ) k B T ) - ρ E ( 𝐫 ) \varepsilon\nabla^{2}\Phi(\mathbf{r})=-\,\sum_{j=1}^{N}q_{j}n_{j}^{0}\,\exp% \left(-\frac{q_{j}\,\Phi(\mathbf{r})}{k_{B}T}\right)-\rho_{E}(\mathbf{r})
  22. q j Φ ( 𝐫 ) k B T q_{j}\,\Phi(\mathbf{r})\ll k_{B}T
  23. exp ( - q j Φ ( 𝐫 ) k B T ) 1 - q j Φ ( 𝐫 ) k B T \exp\left(-\frac{q_{j}\,\Phi(\mathbf{r})}{k_{B}T}\right)\approx 1-\frac{q_{j}% \,\Phi(\mathbf{r})}{k_{B}T}
  24. ε 2 Φ ( 𝐫 ) = ( j = 1 N n j 0 q j 2 k B T ) Φ ( 𝐫 ) - j = 1 N n j 0 q j - ρ E ( 𝐫 ) \varepsilon\nabla^{2}\Phi(\mathbf{r})=\left(\sum_{j=1}^{N}\frac{n_{j}^{0}\,q_{% j}^{2}}{k_{B}T}\right)\,\Phi(\mathbf{r})-\,\sum_{j=1}^{N}n_{j}^{0}q_{j}-\rho_{% E}(\mathbf{r})
  25. ε \varepsilon
  26. λ D = ( ε k B T j = 1 N n j 0 q j 2 ) 1 / 2 \lambda_{D}=\left(\frac{\varepsilon\,k_{B}T}{\sum_{j=1}^{N}n_{j}^{0}\,q_{j}^{2% }}\right)^{1/2}
  27. λ D \lambda_{D}
  28. 2 Φ ( 𝐫 ) = λ D - 2 Φ ( 𝐫 ) - ρ E ( 𝐫 ) ε \nabla^{2}\Phi(\mathbf{r})=\lambda_{D}^{-2}\Phi(\mathbf{r})-\frac{\rho_{E}(% \mathbf{r})}{\varepsilon}
  29. ρ E = Q δ ( 𝐫 ) \rho_{E}=Q\delta(\mathbf{r})
  30. Φ ( 𝐫 ) = Q 4 π ε r e - r / λ D \Phi(\mathbf{r})=\frac{Q}{4\pi\varepsilon r}e^{-r/\lambda_{D}}
  31. λ B \lambda_{B}
  32. λ D = ( 4 π λ B j = 1 N n j 0 z j 2 ) - 1 / 2 \lambda_{D}=\left(4\pi\,\lambda_{B}\,\sum_{j=1}^{N}n_{j}^{0}\,z_{j}^{2}\right)% ^{-1/2}
  33. z j = q j / e z_{j}=q_{j}/e
  34. j j
  35. e e
  36. ε r = 1 \varepsilon_{r}=1
  37. λ D = ε 0 k B / q e 2 n e / T e + j z j 2 n j / T j \lambda_{D}=\sqrt{\frac{\varepsilon_{0}k_{B}/q_{e}^{2}}{n_{e}/T_{e}+\sum_{j}z_% {j}^{2}n_{j}/T_{j}}}
  38. λ D = ε 0 k B T e n e q e 2 \lambda_{D}=\sqrt{\frac{\varepsilon_{0}k_{B}T_{e}}{n_{e}q_{e}^{2}}}
  39. κ - 1 = ε r ε 0 k B T 2 N A e 2 I \kappa^{-1}=\sqrt{\frac{\varepsilon_{r}\varepsilon_{0}k_{B}T}{2N_{A}e^{2}I}}
  40. κ - 1 = ε r ε 0 R T 2 F 2 C 0 \kappa^{-1}=\sqrt{\frac{\varepsilon_{r}\varepsilon_{0}RT}{2F^{2}C_{0}}}
  41. κ - 1 = 1 8 π λ B N A I \kappa^{-1}=\frac{1}{\sqrt{8\pi\lambda_{B}N_{A}I}}
  42. λ B \lambda_{B}
  43. κ - 1 ( nm ) = 0.304 I ( M ) \kappa^{-1}(\mathrm{nm})=\frac{0.304}{\sqrt{I(\mathrm{M})}}
  44. L D = ε k B T q 2 N d \mathit{L}_{D}=\sqrt{\frac{\varepsilon k_{B}T}{q^{2}N_{d}}}

Decagonal_number.html

  1. D n = 4 n 2 - 3 n . D_{n}=4n^{2}-3n.
  2. D n = n 2 + 3 ( n 2 - n ) . D_{n}=n^{2}+3(n^{2}-n).

Decibel_watt.html

  1. Power in dBW = 10 log 10 Power in W 1 W \mbox{Power in dBW}~{}=10\log_{10}\frac{\mbox{Power in W}~{}}{1\mathrm{W}}

Decisional_Diffie–Hellman_assumption.html

  1. G G
  2. q q
  3. g g
  4. g a g^{a}
  5. g b g^{b}
  6. a , b q a,b\in\mathbb{Z}_{q}
  7. g a b g^{ab}
  8. G G
  9. n = log ( q ) n=\log(q)
  10. ( g a , g b , g a b ) (g^{a},g^{b},g^{ab})
  11. a a
  12. b b
  13. q \mathbb{Z}_{q}
  14. ( g a , g b , g c ) (g^{a},g^{b},g^{c})
  15. a , b , c a,b,c
  16. q \mathbb{Z}_{q}
  17. G G
  18. G G
  19. ( g a , g b , z ) (g^{a},g^{b},z)
  20. z = g a b z=g^{ab}
  21. l o g g log_{g}
  22. g a g^{a}
  23. z z
  24. ( g b ) a (g^{b})^{a}
  25. g a b g^{ab}
  26. ( g a , g b ) (g^{a},g^{b})
  27. k k
  28. p p
  29. ( p - 1 ) / k (p-1)/k
  30. k = 2 k=2
  31. E E
  32. G F ( p ) GF(p)
  33. p p
  34. E E
  35. G F ( p ) GF(p)
  36. p p
  37. p * \mathbb{Z}^{*}_{p}
  38. p p
  39. g a g^{a}
  40. g b g^{b}
  41. g a b g^{ab}
  42. g a b g^{ab}
  43. G F ( p ) GF(p)
  44. log 2 ( p ) \log^{2}(p)
  45. P , a P , b P , c P P,aP,bP,cP
  46. e ( P , c P ) e(P,cP)
  47. e ( a P , b P ) e(aP,bP)
  48. a b = c ab=c
  49. P P
  50. p p

Deck_(ship).html

  1. L O A + B e a m 16 {\sqrt{LOA}+Beam\over 16}
  2. [ L O A 3.28 + ( B e a m 3.28 ) 1.58 ] [\sqrt{LOA\cdot 3.28}+(Beam\cdot 3.28)\cdot 1.58]
  3. 0.07 + L W L 150 0.07+{LWL\over 150}
  4. 1.8 + L W L 1.8 1.8+{LWL\over 1.8}

Decomposition_of_spectrum_(functional_analysis).html

  1. T T
  2. X X
  3. λ \lambda
  4. T - λ T-\lambda
  5. X X
  6. T T
  7. T - λ T-\lambda
  8. σ ( T ) = σ p ( T ) σ c ( T ) σ r ( T ) . \sigma(T)=\sigma_{p}(T)\cup\sigma_{c}(T)\cup\sigma_{r}(T).
  9. ( T - λ ) x , φ = x , ( T * - λ ) φ = 0. \langle(T-\lambda)x,\varphi\rangle=\langle x,(T^{*}-{\lambda})\varphi\rangle=0.
  10. x X , x , ( T * - λ ) φ = ( T - λ ) x , φ = 0. \forall x\in X,\;\langle x,(T^{*}-\lambda)\varphi\rangle=\langle(T-\lambda)x,% \varphi\rangle=0.
  11. ( T h f ) ( s ) = h ( s ) f ( s ) . (T_{h}f)(s)=h(s)\cdot f(s).
  12. ( T h - λ ) f n p p = ( h - λ ) f n p p = S n | h - λ | p d μ 1 n p μ ( S n ) = 1 n p f n p p . \|(T_{h}-\lambda)f_{n}\|_{p}^{p}=\|(h-\lambda)f_{n}\|_{p}^{p}=\int_{S_{n}}|h-% \lambda\;|^{p}d\mu\leq\frac{1}{n^{p}}\;\mu(S_{n})=\frac{1}{n^{p}}\|f_{n}\|_{p}% ^{p}.
  13. s S , ( T h f ) ( s ) = λ f ( s ) , \forall s\in S,\;(T_{h}f)(s)=\lambda f(s),
  14. f n ( s ) = 1 h ( s ) - λ g n ( s ) f ( s ) . f_{n}(s)=\frac{1}{h(s)-\lambda}\cdot g_{n}(s)\cdot f(s).
  15. f n p n f p \|f_{n}\|_{p}\leq n\|f\|_{p}
  16. ( T h - λ ) f n f (T_{h}-\lambda)f_{n}\rightarrow f
  17. n 0 | x n | p < . \sum_{n\geq 0}|x_{n}|^{p}<\infty.
  18. T ( x 1 , x 2 , x 3 , ) = ( x 2 , x 3 , x 4 , ) . T(x_{1},x_{2},x_{3},\dots)=(x_{2},x_{3},x_{4},\dots).
  19. T * ( x 1 , x 2 , x 3 , ) = ( 0 , x 1 , x 2 , ) . T^{*}(x_{1},x_{2},x_{3},\dots)=(0,x_{1},x_{2},\dots).
  20. T x = λ x , i . e . ( x 2 , x 3 , x 4 , ) = λ ( x 1 , x 2 , x 3 , ) , Tx=\lambda x,\qquad i.e.\;(x_{2},x_{3},x_{4},\dots)=\lambda(x_{1},x_{2},x_{3},% \dots),
  21. x = x 1 ( 1 , λ , λ 2 , ) , x=x_{1}(1,\lambda,\lambda^{2},\dots),
  22. P ( T ) = sup λ σ ( T ) | P ( λ ) | . \|P(T)\|=\sup_{\lambda\in\sigma(T)}|P(\lambda)|.
  23. P P ( T ) P\rightarrow P(T)
  24. f h , f ( T ) h f\rightarrow\langle h,f(T)h\rangle
  25. σ ( T ) f d μ h = h , f ( T ) h . \int_{\sigma(T)}f\,d\mu_{h}=\langle h,f(T)h\rangle.
  26. σ ( T ) g d μ h = h , g ( T ) h . \int_{\sigma(T)}g\,d\mu_{h}=\langle h,g(T)h\rangle.
  27. k , g ( T ) h . \langle k,g(T)h\rangle.
  28. μ = μ ac + μ sc + μ pp \,\mu=\mu_{\mathrm{ac}}+\mu_{\mathrm{sc}}+\mu_{\mathrm{pp}}
  29. k , χ ( T ) k = σ ( T ) χ ( λ ) λ 2 d μ h ( λ ) = σ ( T ) χ ( λ ) d μ k ( λ ) . \langle k,\chi(T)k\rangle=\int_{\sigma(T)}\chi(\lambda)\cdot\lambda^{2}d\mu_{h% }(\lambda)=\int_{\sigma(T)}\chi(\lambda)\;d\mu_{k}(\lambda).
  30. λ 2 d μ h = d μ k \lambda^{2}d\mu_{h}=d\mu_{k}\,
  31. H = H ac H sc H pp . H=H_{\mathrm{ac}}\oplus H_{\mathrm{sc}}\oplus H_{\mathrm{pp}}.
  32. σ ( T ) = σ ac ( T ) σ sc ( T ) σ ¯ pp ( T ) . \sigma(T)=\sigma_{\mathrm{ac}}(T)\cup\sigma_{\mathrm{sc}}(T)\cup{\bar{\sigma}_% {\mathrm{pp}}(T)}.
  33. σ ( T ) = σ ¯ pp ( T ) σ ac ( T ) σ sc ( T ) . \sigma(T)={\bar{\sigma}_{\mathrm{pp}}(T)}\cup\sigma_{\mathrm{ac}}(T)\cup\sigma% _{\mathrm{sc}}(T).
  34. i = 1 m L 2 ( , μ i ) \oplus_{i=1}^{m}L^{2}(\mathbb{R},\mu_{i})
  35. μ i \mu_{i}
  36. L 2 ( , μ ) , L^{2}(\mathbb{R},\mu),
  37. f f
  38. L 2 ( ) L^{2}(\mathbb{R})
  39. x x\to\infty
  40. f ( x ) = { n if x [ n , n + 1 n 4 ] , 0 else. f(x)=\begin{cases}n&\,\text{if }x\in\left[n,n+\frac{1}{n^{4}}\right],\\ 0&\,\text{else.}\end{cases}
  41. x x\to\infty

Dedekind_sum.html

  1. ( ( ) ) : ((\,)):\mathbb{R}\rightarrow\mathbb{R}
  2. ( ( x ) ) = { x - x - 1 / 2 , if x ; 0 , if x . ((x))=\begin{cases}x-\lfloor x\rfloor-1/2,&\mbox{if }~{}x\in\mathbb{R}% \setminus\mathbb{Z};\\ 0,&\mbox{if }~{}x\in\mathbb{Z}.\end{cases}
  3. D ( a , b ; c ) = n mod c ( ( a n c ) ) ( ( b n c ) ) , D(a,b;c)=\sum_{n\bmod c}\left(\!\!\Bigg(\frac{an}{c}\Bigg)\!\!\right)\left(\!% \!\left(\frac{bn}{c}\right)\!\!\right),
  4. D ( a , b ; c ) = D ( b , a ; c ) , D(a,b;c)=D(b,a;c),
  5. n mod c ( ( n + x c ) ) = ( ( x ) ) , x . \sum_{n\bmod c}\left(\!\!\left(\frac{n+x}{c}\right)\!\!\right)=((x)),\qquad% \forall x\in\mathbb{R}.
  6. s ( b , c ) = - 1 c ω 1 ( 1 - ω b ) ( 1 - ω ) + 1 4 - 1 4 c , s(b,c)=\frac{-1}{c}\sum_{\omega}\frac{1}{(1-\omega^{b})(1-\omega)}+\frac{1}{4}% -\frac{1}{4c},
  7. ω \omega
  8. ω c = 1 \omega^{c}=1
  9. ω 1 \omega\not=1
  10. s ( b , c ) = 1 4 c n = 1 c - 1 cot ( π n c ) cot ( π n b c ) . s(b,c)=\frac{1}{4c}\sum_{n=1}^{c-1}\cot\left(\frac{\pi n}{c}\right)\cot\left(% \frac{\pi nb}{c}\right).
  11. s ( b , c ) + s ( c , b ) = 1 12 ( b c + 1 b c + c b ) - 1 4 . s(b,c)+s(c,b)=\frac{1}{12}\left(\frac{b}{c}+\frac{1}{bc}+\frac{c}{b}\right)-% \frac{1}{4}.
  12. 12 b c ( s ( b , c ) + s ( c , b ) ) = b 2 + c 2 - 3 b c + 1 , 12bc\left(s(b,c)+s(c,b)\right)=b^{2}+c^{2}-3bc+1,
  13. 12 b c s ( c , b ) = 0 mod k c 12bc\,s(c,b)=0\mod kc
  14. 12 b c s ( b , c ) = b 2 + 1 mod k c . 12bc\,s(b,c)=b^{2}+1\mod kc.
  15. δ = s ( a , c ) - a + d 12 c - s ( a , k ) + a + d 12 k \delta=s(a,c)-\frac{a+d}{12c}-s(a,k)+\frac{a+d}{12k}
  16. D ( a , b ; c ) + D ( b , c ; a ) + D ( c , a ; b ) = 1 12 a 2 + b 2 + c 2 a b c - 1 4 . D(a,b;c)+D(b,c;a)+D(c,a;b)=\frac{1}{12}\frac{a^{2}+b^{2}+c^{2}}{abc}-\frac{1}{% 4}.

Dedekind_zeta_function.html

  1. ζ K ( s ) = I 𝒪 K 1 ( N K / 𝐐 ( I ) ) s \zeta_{K}(s)=\sum_{I\subseteq\mathcal{O}_{K}}\frac{1}{(N_{K/\mathbf{Q}}(I))^{s}}
  2. ζ K ( s ) = P 𝒪 K 1 1 - ( N K / 𝐐 ( P ) ) - s , for Re ( s ) > 1. \zeta_{K}(s)=\prod_{P\subseteq\mathcal{O}_{K}}\frac{1}{1-(N_{K/\mathbf{Q}}(P))% ^{-s}},\,\text{ for Re}(s)>1.
  3. Γ 𝐑 ( s ) = π - s / 2 Γ ( s / 2 ) \Gamma_{\mathbf{R}}(s)=\pi^{-s/2}\Gamma(s/2)
  4. Γ 𝐂 ( s ) = 2 ( 2 π ) - s Γ ( s ) \Gamma_{\mathbf{C}}(s)=2(2\pi)^{-s}\Gamma(s)
  5. Λ K ( s ) = | Δ K | s / 2 Γ 𝐑 ( s ) r 1 Γ 𝐂 ( s ) r 2 ζ K ( s ) \Lambda_{K}(s)=\left|\Delta_{K}\right|^{s/2}\Gamma_{\mathbf{R}}(s)^{r_{1}}% \Gamma_{\mathbf{C}}(s)^{r_{2}}\zeta_{K}(s)
  6. Λ K ( s ) = Λ K ( 1 - s ) . \Lambda_{K}(s)=\Lambda_{K}(1-s).\;
  7. lim s 0 s - r ζ K ( s ) = - h ( K ) R ( K ) w ( K ) . \lim_{s\rightarrow 0}s^{-r}\zeta_{K}(s)=-\frac{h(K)R(K)}{w(K)}.
  8. ζ K ( s ) ζ 𝐐 ( s ) \frac{\zeta_{K}(s)}{\zeta_{\mathbf{Q}}(s)}
  9. ζ L ( s ) ζ K ( s ) \frac{\zeta_{L}(s)}{\zeta_{K}(s)}
  10. ζ K ( s ) \zeta_{K}(s)
  11. ζ L ( s ) \zeta_{L}(s)

Default_logic.html

  1. W , D \langle W,D\rangle
  2. W W
  3. D D
  4. Prerequisite : Justification 1 , , Justification n Conclusion \frac{\,\text{Prerequisite : Justification}_{1},\dots,\,\text{Justification}_{% n}}{\,\text{Conclusion}}
  5. P r e r e q u i s i t e Prerequisite
  6. J u s t i f i c a t i o n i Justification_{i}
  7. C o n c l u s i o n Conclusion
  8. W W
  9. D = { B i r d ( X ) : F l i e s ( X ) F l i e s ( X ) } D=\left\{\frac{Bird(X):Flies(X)}{Flies(X)}\right\}
  10. X X
  11. W = { B i r d ( C o n d o r ) , B i r d ( P e n g u i n ) , ¬ F l i e s ( P e n g u i n ) , F l i e s ( B e e ) } W=\{Bird(Condor),Bird(Penguin),\neg Flies(Penguin),Flies(Bee)\}
  12. B i r d ( C o n d o r ) Bird(Condor)
  13. F l i e s ( C o n d o r ) Flies(Condor)
  14. B i r d ( P e n g u i n ) Bird(Penguin)
  15. F l i e s ( P e n g u i n ) Flies(Penguin)
  16. B i r d ( P e n g u i n ) Bird(Penguin)
  17. F l i e s ( P e n g u i n ) Flies(Penguin)
  18. B i r d ( B e e ) Bird(Bee)
  19. F l i e s ( X ) Flies(X)
  20. B i r d ( X ) Bird(X)
  21. F F
  22. : ¬ F ¬ F \frac{:{\neg}F}{{\neg}F}
  23. ¬ F \neg F
  24. F F
  25. ¬ F \neg F
  26. ¬ F \neg F
  27. ¬ F \neg F
  28. { R e p u b l i c a n ( X ) : ¬ P a c i f i s t ( X ) ¬ P a c i f i s t ( X ) , Q u a k e r ( X ) : P a c i f i s t ( X ) P a c i f i s t ( X ) } , { R e p u b l i c a n ( N i x o n ) , Q u a k e r ( N i x o n ) } \left\langle\left\{\frac{Republican(X):\neg Pacifist(X)}{\neg Pacifist(X)},% \frac{Quaker(X):Pacifist(X)}{Pacifist(X)}\right\},\left\{Republican(Nixon),% Quaker(Nixon)\right\}\right\rangle
  29. P a c i f i s t ( N i x o n ) Pacifist(Nixon)
  30. P a c i f i s t ( N i x o n ) Pacifist(Nixon)
  31. α : β 1 , , β n γ \frac{\alpha:\beta_{1},\ldots,\beta_{n}}{\gamma}
  32. T T
  33. T α T\models\alpha
  34. T { β i } T\cup\{\beta_{i}\}
  35. T T
  36. T { γ } T\cup\{\gamma\}
  37. T T
  38. { : A ( b ) ¬ A ( b ) } , \left\langle\left\{\frac{:A(b)}{\neg A(b)}\right\},\emptyset\right\rangle
  39. A ( b ) A(b)
  40. A ( b ) A(b)
  41. ϕ : ψ ψ \frac{\phi:\psi}{\psi}
  42. P a c i f i s t ( N i x o n ) Pacifist(Nixon)
  43. ¬ P a c i f i s t ( N i x o n ) \neg Pacifist(Nixon)
  44. T T
  45. T T
  46. p : { r 1 , , r n } \langle p:\{r_{1},\ldots,r_{n}\}\rangle
  47. p p
  48. r 1 , , r n r_{1},\ldots,r_{n}
  49. p p
  50. x x \Box x\rightarrow x
  51. x x
  52. x \Box x
  53. x x \Box x\rightarrow x
  54. Σ 2 P \Sigma^{P}_{2}
  55. Π 2 P \Pi^{P}_{2}
  56. Σ 2 P \Sigma^{P}_{2}
  57. Δ 2 P [ l o g ] \Delta^{P[log]}_{2}
  58. Σ 2 P \Sigma^{P}_{2}

Deficit_round_robin.html

  1. Q i Q_{i}
  2. i i
  3. Q i Q_{i}
  4. i i
  5. Q i ( Q 1 + Q 2 + + Q N ) R \frac{Q_{i}}{(Q_{1}+Q_{2}+...+Q_{N})}R
  6. R R
  7. i i
  8. Q i Q_{i}

Degenerate_bilinear_form.html

  1. f ( x , y ) = 0 f(x,y)=0\,
  2. y V . y\in V.
  3. v ( x f ( x , v ) ) v\mapsto(x\mapsto f(x,v))
  4. f ( x , y ) = 0 f(x,y)=0\,
  5. y V y\in V
  6. V V * V\to V^{*}
  7. v ( x f ( x , v ) ) v\mapsto(x\mapsto f(x,v))
  8. f ( ϕ , ψ ) = ψ ( x ) ϕ ( x ) d x f(\phi,\psi)=\int\psi(x)\phi(x)dx
  9. f ( ϕ , ψ ) = 0 f(\phi,\psi)=0\,
  10. ϕ \,\phi
  11. ψ = 0. \psi=0.\,
  12. { x V f ( x , y ) = 0 for all y V } \{x\in V\mid f(x,y)=0\mbox{ for all }~{}y\in V\}
  13. x V x\in V
  14. f ( x , x ) = 0 f(x,x)=0
  15. f f

Dehn_twist.html

  1. S 1 × I , S^{1}\times I,
  2. e i θ e^{{\rm{i}}\theta}
  3. θ [ 0 , 2 π ] , \theta\in[0,2\pi],
  4. f ( s , t ) = ( s e i2 π t , t ) . \displaystyle f(s,t)=(se^{{\rm{i}}2\pi t},t).
  5. 𝕋 2 2 / 2 . \mathbb{T}^{2}\cong\mathbb{R}^{2}/\mathbb{Z}^{2}.
  6. γ a \gamma_{a}
  7. γ a \gamma_{a}
  8. a ( 0 ; 0 , 1 ) = { z : 0 < | z | < 1 } a(0;0,1)=\{z\in\mathbb{C}:0<|z|<1\}
  9. ( e i θ , t ) ( e i ( θ + 2 π t ) , t ) (e^{i\theta},t)\mapsto(e^{i(\theta+2\pi t)},t)
  10. γ a \gamma_{a}
  11. T a : 𝕋 2 𝕋 2 T_{a}:\mathbb{T}^{2}\to\mathbb{T}^{2}
  12. T a : π 1 ( 𝕋 2 ) π 1 ( 𝕋 2 ) : [ x ] [ T a ( x ) ] {T_{a}}_{\ast}:\pi_{1}(\mathbb{T}^{2})\to\pi_{1}(\mathbb{T}^{2}):[x]\mapsto[T_% {a}(x)]
  13. T a ( [ a ] ) = [ a ] {T_{a}}_{\ast}([a])=[a]
  14. T a ( [ b ] ) = [ b * a ] {T_{a}}_{\ast}([b])=[b*a]
  15. b * a b*a
  16. g g
  17. 3 g - 1 3g-1
  18. 2 g + 1 2g+1
  19. g > 1 g>1

Del_in_cylindrical_and_spherical_coordinates.html

  1. ( x , y , z ) (x,y,z)
  2. ( ρ , ϕ , z ) (ρ,ϕ,z)
  3. ( r , θ , ϕ ) (r,θ,ϕ)
  4. ( σ , τ , z ) (σ,τ,z)
  5. ρ = x 2 + y 2 ϕ = arctan ( y / x ) z = z \begin{aligned}\displaystyle\rho&\displaystyle=\sqrt{x^{2}+y^{2}}\\ \displaystyle\phi&\displaystyle=\arctan(y/x)\\ \displaystyle z&\displaystyle=z\end{aligned}
  6. x = ρ cos ϕ y = ρ sin ϕ z = z \begin{aligned}\displaystyle x&\displaystyle=\rho\cos\phi\\ \displaystyle y&\displaystyle=\rho\sin\phi\\ \displaystyle z&\displaystyle=z\end{aligned}
  7. x = r sin θ cos ϕ y = r sin θ sin ϕ z = r cos θ \begin{aligned}\displaystyle x&\displaystyle=r\sin\theta\cos\phi\\ \displaystyle y&\displaystyle=r\sin\theta\sin\phi\\ \displaystyle z&\displaystyle=r\cos\theta\end{aligned}
  8. x = σ τ y = 1 2 ( τ 2 - σ 2 ) z = z \begin{aligned}\displaystyle x&\displaystyle=\sigma\tau\\ \displaystyle y&\displaystyle=\tfrac{1}{2}\left(\tau^{2}-\sigma^{2}\right)\\ \displaystyle z&\displaystyle=z\end{aligned}
  9. r = x 2 + y 2 + z 2 θ = arccos ( z / r ) ϕ = arctan ( y / x ) \begin{aligned}\displaystyle r&\displaystyle=\sqrt{x^{2}+y^{2}+z^{2}}\\ \displaystyle\theta&\displaystyle=\arccos(z/r)\\ \displaystyle\phi&\displaystyle=\arctan(y/x)\end{aligned}
  10. r = ρ 2 + z 2 θ = arctan ( ρ / z ) ϕ = ϕ \begin{aligned}\displaystyle r&\displaystyle=\sqrt{\rho^{2}+z^{2}}\\ \displaystyle\theta&\displaystyle=\arctan{(\rho/z)}\\ \displaystyle\phi&\displaystyle=\phi\end{aligned}
  11. ρ = r sin θ ϕ = ϕ z = r cos θ \begin{aligned}\displaystyle\rho&\displaystyle=r\sin\theta\\ \displaystyle\phi&\displaystyle=\phi\\ \displaystyle z&\displaystyle=r\cos\theta\end{aligned}
  12. ρ cos ϕ = σ τ ρ sin ϕ = 1 2 ( τ 2 - σ 2 ) z = z \begin{aligned}\displaystyle\rho\cos\phi&\displaystyle=\sigma\tau\\ \displaystyle\rho\sin\phi&\displaystyle=\tfrac{1}{2}\left(\tau^{2}-\sigma^{2}% \right)\\ \displaystyle z&\displaystyle=z\end{aligned}
  13. s y m b o l ρ ^ = x 𝐱 ^ + y 𝐲 ^ x 2 + y 2 s y m b o l ϕ ^ = - y 𝐱 ^ + x 𝐲 ^ x 2 + y 2 𝐳 ^ = 𝐳 ^ \begin{aligned}\displaystyle symbol{\hat{\rho}}&\displaystyle=\frac{x\hat{% \mathbf{x}}+y\hat{\mathbf{y}}}{\sqrt{x^{2}+y^{2}}}\\ \displaystyle symbol{\hat{\phi}}&\displaystyle=\frac{-y\hat{\mathbf{x}}+x\hat{% \mathbf{y}}}{\sqrt{x^{2}+y^{2}}}\\ \displaystyle\mathbf{\hat{z}}&\displaystyle=\mathbf{\hat{z}}\end{aligned}
  14. 𝐱 ^ = cos \phisymbol ρ ^ - sin \phisymbol ϕ ^ 𝐲 ^ = sin \phisymbol ρ ^ + cos \phisymbol ϕ ^ 𝐳 ^ = 𝐳 ^ \begin{aligned}\displaystyle\hat{\mathbf{x}}&\displaystyle=\cos\phisymbol{\hat% {\rho}}-\sin\phisymbol{\hat{\phi}}\\ \displaystyle\hat{\mathbf{y}}&\displaystyle=\sin\phisymbol{\hat{\rho}}+\cos% \phisymbol{\hat{\phi}}\\ \displaystyle\mathbf{\hat{z}}&\displaystyle=\mathbf{\hat{z}}\end{aligned}
  15. 𝐱 ^ = sin θ cos \phisymbol r ^ + cos θ cos \phisymbol θ ^ - sin \phisymbol ϕ ^ 𝐲 ^ = sin θ sin \phisymbol r ^ + cos θ sin \phisymbol θ ^ + cos \phisymbol ϕ ^ 𝐳 ^ = cos θ s y m b o l r ^ - sin θ s y m b o l θ ^ \begin{aligned}\displaystyle\hat{\mathbf{x}}&\displaystyle=\sin\theta\cos% \phisymbol{\hat{r}}+\cos\theta\cos\phisymbol{\hat{\theta}}-\sin\phisymbol{\hat% {\phi}}\\ \displaystyle\hat{\mathbf{y}}&\displaystyle=\sin\theta\sin\phisymbol{\hat{r}}+% \cos\theta\sin\phisymbol{\hat{\theta}}+\cos\phisymbol{\hat{\phi}}\\ \displaystyle\mathbf{\hat{z}}&\displaystyle=\cos\theta symbol{\hat{r}}-\sin% \theta symbol{\hat{\theta}}\end{aligned}
  16. s y m b o l σ ^ = τ 𝐱 ^ - σ 𝐲 ^ τ 2 + σ 2 s y m b o l τ ^ = σ 𝐱 ^ + τ 𝐲 ^ τ 2 + σ 2 𝐳 ^ = 𝐳 ^ \begin{aligned}\displaystyle symbol{\hat{\sigma}}&\displaystyle=\frac{\tau\hat% {\mathbf{x}}-\sigma\hat{\mathbf{y}}}{\sqrt{\tau^{2}+\sigma^{2}}}\\ \displaystyle symbol{\hat{\tau}}&\displaystyle=\frac{\sigma\hat{\mathbf{x}}+% \tau\hat{\mathbf{y}}}{\sqrt{\tau^{2}+\sigma^{2}}}\\ \displaystyle\mathbf{\hat{z}}&\displaystyle=\mathbf{\hat{z}}\end{aligned}
  17. 𝐫 ^ = x 𝐱 ^ + y 𝐲 ^ + z 𝐳 ^ x 2 + y 2 + z 2 s y m b o l θ ^ = x z 𝐱 ^ + y z 𝐲 ^ - ( x 2 + y 2 ) 𝐳 ^ x 2 + y 2 x 2 + y 2 + z 2 s y m b o l ϕ ^ = - y 𝐱 ^ + x 𝐲 ^ x 2 + y 2 \begin{aligned}\displaystyle\mathbf{\hat{r}}&\displaystyle=\frac{x\hat{\mathbf% {x}}+y\hat{\mathbf{y}}+z\mathbf{\hat{z}}}{\sqrt{x^{2}+y^{2}+z^{2}}}\\ \displaystyle symbol{\hat{\theta}}&\displaystyle=\frac{xz\hat{\mathbf{x}}+yz% \hat{\mathbf{y}}-\left(x^{2}+y^{2}\right)\mathbf{\hat{z}}}{\sqrt{x^{2}+y^{2}}% \sqrt{x^{2}+y^{2}+z^{2}}}\\ \displaystyle symbol{\hat{\phi}}&\displaystyle=\frac{-y\hat{\mathbf{x}}+x\hat{% \mathbf{y}}}{\sqrt{x^{2}+y^{2}}}\end{aligned}
  18. 𝐫 ^ = ρ s y m b o l ρ ^ + z 𝐳 ^ ρ 2 + z 2 s y m b o l θ ^ = z s y m b o l ρ ^ - ρ 𝐳 ^ ρ 2 + z 2 s y m b o l ϕ ^ = s y m b o l ϕ ^ \begin{aligned}\displaystyle\mathbf{\hat{r}}&\displaystyle=\frac{\rho symbol{% \hat{\rho}}+z\mathbf{\hat{z}}}{\sqrt{\rho^{2}+z^{2}}}\\ \displaystyle symbol{\hat{\theta}}&\displaystyle=\frac{zsymbol{\hat{\rho}}-% \rho\mathbf{\hat{z}}}{\sqrt{\rho^{2}+z^{2}}}\\ \displaystyle symbol{\hat{\phi}}&\displaystyle=symbol{\hat{\phi}}\end{aligned}
  19. s y m b o l ρ ^ = sin θ 𝐫 ^ + cos θ s y m b o l θ ^ s y m b o l ϕ ^ = s y m b o l ϕ ^ 𝐳 ^ = cos θ 𝐫 ^ - sin θ s y m b o l θ ^ \begin{aligned}\displaystyle symbol{\hat{\rho}}&\displaystyle=\sin\theta% \mathbf{\hat{r}}+\cos\theta symbol{\hat{\theta}}\\ \displaystyle symbol{\hat{\phi}}&\displaystyle=symbol{\hat{\phi}}\\ \displaystyle\mathbf{\hat{z}}&\displaystyle=\cos\theta\mathbf{\hat{r}}-\sin% \theta symbol{\hat{\theta}}\end{aligned}
  20. \begin{matrix}\end{matrix}
  21. 𝐀 \mathbf{A}
  22. A x 𝐱 ^ + A y 𝐲 ^ + A z 𝐳 ^ A_{x}\hat{\mathbf{x}}+A_{y}\hat{\mathbf{y}}+A_{z}\mathbf{\hat{z}}
  23. A ρ s y m b o l ρ ^ + A ϕ s y m b o l ϕ ^ + A z 𝐳 ^ A_{\rho}symbol{\hat{\rho}}+A_{\phi}symbol{\hat{\phi}}+A_{z}\mathbf{\hat{z}}
  24. A r s y m b o l r ^ + A θ s y m b o l θ ^ + A ϕ s y m b o l ϕ ^ A_{r}symbol{\hat{r}}+A_{\theta}symbol{\hat{\theta}}+A_{\phi}symbol{\hat{\phi}}
  25. A σ s y m b o l σ ^ + A τ s y m b o l τ ^ + A ϕ 𝐳 ^ A_{\sigma}symbol{\hat{\sigma}}+A_{\tau}symbol{\hat{\tau}}+A_{\phi}\mathbf{\hat% {z}}
  26. f f
  27. f \nabla f
  28. f x 𝐱 ^ + f y 𝐲 ^ + f z 𝐳 ^ {\partial f\over\partial x}\hat{\mathbf{x}}+{\partial f\over\partial y}\hat{% \mathbf{y}}+{\partial f\over\partial z}\mathbf{\hat{z}}
  29. f ρ s y m b o l ρ ^ + 1 ρ f ϕ s y m b o l ϕ ^ + f z 𝐳 ^ {\partial f\over\partial\rho}symbol{\hat{\rho}}+{1\over\rho}{\partial f\over% \partial\phi}symbol{\hat{\phi}}+{\partial f\over\partial z}\mathbf{\hat{z}}
  30. f r s y m b o l r ^ + 1 r f θ s y m b o l θ ^ + 1 r sin θ f ϕ s y m b o l ϕ ^ {\partial f\over\partial r}symbol{\hat{r}}+{1\over r}{\partial f\over\partial% \theta}symbol{\hat{\theta}}+{1\over r\sin\theta}{\partial f\over\partial\phi}% symbol{\hat{\phi}}
  31. 1 σ 2 + τ 2 f σ s y m b o l σ ^ + 1 σ 2 + τ 2 f τ s y m b o l τ ^ + f z 𝐳 ^ \frac{1}{\sqrt{\sigma^{2}+\tau^{2}}}{\partial f\over\partial\sigma}symbol{\hat% {\sigma}}+\frac{1}{\sqrt{\sigma^{2}+\tau^{2}}}{\partial f\over\partial\tau}% symbol{\hat{\tau}}+{\partial f\over\partial z}\mathbf{\hat{z}}
  32. 𝐀 \nabla\cdot\mathbf{A}
  33. A x x + A y y + A z z {\partial A_{x}\over\partial x}+{\partial A_{y}\over\partial y}+{\partial A_{z% }\over\partial z}
  34. 1 ρ ( ρ A ρ ) ρ + 1 ρ A ϕ ϕ + A z z {1\over\rho}{\partial\left(\rho A_{\rho}\right)\over\partial\rho}+{1\over\rho}% {\partial A_{\phi}\over\partial\phi}+{\partial A_{z}\over\partial z}
  35. 1 r 2 ( r 2 A r ) r + 1 r sin θ θ ( A θ sin θ ) + 1 r sin θ A ϕ ϕ {1\over r^{2}}{\partial\left(r^{2}A_{r}\right)\over\partial r}+{1\over r\sin% \theta}{\partial\over\partial\theta}\left(A_{\theta}\sin\theta\right)+{1\over r% \sin\theta}{\partial A_{\phi}\over\partial\phi}
  36. 1 σ 2 + τ 2 ( ( σ 2 + τ 2 A σ ) σ + ( σ 2 + τ 2 A τ ) τ ) + A z z \frac{1}{\sqrt{\sigma^{2}+\tau^{2}}}\left({\partial(\sqrt{\sigma^{2}+\tau^{2}}% A_{\sigma})\over\partial\sigma}+{\partial(\sqrt{\sigma^{2}+\tau^{2}}A_{\tau})% \over\partial\tau}\right)+{\partial A_{z}\over\partial z}
  37. × 𝐀 \nabla\times\mathbf{A}
  38. ( A z y - A y z ) 𝐱 ^ + + ( A x z - A z x ) 𝐲 ^ + + ( A y x - A x y ) 𝐳 ^ \begin{aligned}\displaystyle\left(\frac{\partial A_{z}}{\partial y}-\frac{% \partial A_{y}}{\partial z}\right)&\displaystyle\hat{\mathbf{x}}+\\ \displaystyle+\left(\frac{\partial A_{x}}{\partial z}-\frac{\partial A_{z}}{% \partial x}\right)&\displaystyle\hat{\mathbf{y}}+\\ \displaystyle+\left(\frac{\partial A_{y}}{\partial x}-\frac{\partial A_{x}}{% \partial y}\right)&\displaystyle\mathbf{\hat{z}}\end{aligned}
  39. ( 1 ρ A z ϕ - A ϕ z ) s y m b o l ρ ^ + ( A ρ z - A z ρ ) s y m b o l ϕ ^ + 1 ρ ( ( ρ A ϕ ) ρ - A ρ ϕ ) 𝐳 ^ \begin{aligned}\displaystyle\left(\frac{1}{\rho}\frac{\partial A_{z}}{\partial% \phi}-\frac{\partial A_{\phi}}{\partial z}\right)&\displaystyle symbol{\hat{% \rho}}\\ \displaystyle+\left(\frac{\partial A_{\rho}}{\partial z}-\frac{\partial A_{z}}% {\partial\rho}\right)&\displaystyle symbol{\hat{\phi}}\\ \displaystyle+\frac{1}{\rho}\left(\frac{\partial\left(\rho A_{\phi}\right)}{% \partial\rho}-\frac{\partial A_{\rho}}{\partial\phi}\right)&\displaystyle% \mathbf{\hat{z}}\end{aligned}
  40. 1 r sin θ ( θ ( A ϕ sin θ ) - A θ ϕ ) s y m b o l r ^ + 1 r ( 1 sin θ A r ϕ - r ( r A ϕ ) ) s y m b o l θ ^ + 1 r ( r ( r A θ ) - A r θ ) s y m b o l ϕ ^ \begin{aligned}\displaystyle\frac{1}{r\sin\theta}\left(\frac{\partial}{% \partial\theta}\left(A_{\phi}\sin\theta\right)-\frac{\partial A_{\theta}}{% \partial\phi}\right)&\displaystyle symbol{\hat{r}}\\ \displaystyle+\frac{1}{r}\left(\frac{1}{\sin\theta}\frac{\partial A_{r}}{% \partial\phi}-\frac{\partial}{\partial r}\left(rA_{\phi}\right)\right)&% \displaystyle symbol{\hat{\theta}}\\ \displaystyle+\frac{1}{r}\left(\frac{\partial}{\partial r}\left(rA_{\theta}% \right)-\frac{\partial A_{r}}{\partial\theta}\right)&\displaystyle symbol{\hat% {\phi}}\end{aligned}
  41. ( 1 σ 2 + τ 2 A z τ - A τ z ) s y m b o l σ ^ - ( 1 σ 2 + τ 2 A z σ - A σ z ) s y m b o l τ ^ + 1 σ 2 + τ 2 ( ( σ 2 + τ 2 A σ ) τ - ( σ 2 + τ 2 A τ ) σ ) 𝐳 ^ \begin{aligned}\displaystyle\left(\frac{1}{\sqrt{\sigma^{2}+\tau^{2}}}\frac{% \partial A_{z}}{\partial\tau}-\frac{\partial A_{\tau}}{\partial z}\right)&% \displaystyle symbol{\hat{\sigma}}\\ \displaystyle-\left(\frac{1}{\sqrt{\sigma^{2}+\tau^{2}}}\frac{\partial A_{z}}{% \partial\sigma}-\frac{\partial A_{\sigma}}{\partial z}\right)&\displaystyle symbol% {\hat{\tau}}\\ \displaystyle+\frac{1}{\sqrt{\sigma^{2}+\tau^{2}}}\left(\frac{\partial\left(% \sqrt{\sigma^{2}+\tau^{2}}A_{\sigma}\right)}{\partial\tau}-\frac{\partial\left% (\sqrt{\sigma^{2}+\tau^{2}}A_{\tau}\right)}{\partial\sigma}\right)&% \displaystyle\mathbf{\hat{z}}\end{aligned}
  42. Δ f 2 f \Delta f\equiv\nabla^{2}f
  43. 2 f x 2 + 2 f y 2 + 2 f z 2 {\partial^{2}f\over\partial x^{2}}+{\partial^{2}f\over\partial y^{2}}+{% \partial^{2}f\over\partial z^{2}}
  44. 1 ρ ρ ( ρ f ρ ) + 1 ρ 2 2 f ϕ 2 + 2 f z 2 {1\over\rho}{\partial\over\partial\rho}\left(\rho{\partial f\over\partial\rho}% \right)+{1\over\rho^{2}}{\partial^{2}f\over\partial\phi^{2}}+{\partial^{2}f% \over\partial z^{2}}
  45. 1 r 2 r ( r 2 f r ) + 1 r 2 sin θ θ ( sin θ f θ ) + 1 r 2 sin 2 θ 2 f ϕ 2 {1\over r^{2}}{\partial\over\partial r}\!\left(r^{2}{\partial f\over\partial r% }\right)\!+\!{1\over r^{2}\!\sin\theta}{\partial\over\partial\theta}\!\left(% \sin\theta{\partial f\over\partial\theta}\right)\!+\!{1\over r^{2}\!\sin^{2}% \theta}{\partial^{2}f\over\partial\phi^{2}}
  46. 1 σ 2 + τ 2 ( 2 f σ 2 + 2 f τ 2 ) + 2 f z 2 \frac{1}{\sigma^{2}+\tau^{2}}\left(\frac{\partial^{2}f}{\partial\sigma^{2}}+% \frac{\partial^{2}f}{\partial\tau^{2}}\right)+\frac{\partial^{2}f}{\partial z^% {2}}
  47. Δ 𝐀 2 𝐀 \Delta\mathbf{A}\equiv\nabla^{2}\mathbf{A}
  48. Δ A x 𝐱 ^ + Δ A y 𝐲 ^ + Δ A z 𝐳 ^ \Delta A_{x}\hat{\mathbf{x}}+\Delta A_{y}\hat{\mathbf{y}}+\Delta A_{z}\mathbf{% \hat{z}}
  49. ( 𝐀 ) 𝐁 (\mathbf{A}\cdot\nabla)\mathbf{B}
  50. 𝐀 B x 𝐱 ^ + 𝐀 B y 𝐲 ^ + 𝐀 B z 𝐳 ^ \mathbf{A}\cdot\nabla B_{x}\hat{\mathbf{x}}+\mathbf{A}\cdot\nabla B_{y}\hat{% \mathbf{y}}+\mathbf{A}\cdot\nabla B_{z}\hat{\mathbf{z}}
  51. d 𝐥 = d x 𝐱 ^ + d y 𝐲 ^ + d z 𝐳 ^ d\mathbf{l}=dx\,\hat{\mathbf{x}}+dy\,\hat{\mathbf{y}}+dz\,\mathbf{\hat{z}}
  52. d 𝐥 = d ρ s y m b o l ρ ^ + ρ d ϕ s y m b o l ϕ ^ + d z 𝐳 ^ d\mathbf{l}=d\rho\,symbol{\hat{\rho}}+\rho\,d\phi\,symbol{\hat{\phi}}+dz\,% \mathbf{\hat{z}}
  53. d 𝐥 = d r 𝐫 ^ + r d θ s y m b o l θ ^ + r sin θ d ϕ s y m b o l ϕ ^ d\mathbf{l}=dr\,\mathbf{\hat{r}}+r\,d\theta\,symbol{\hat{\theta}}+r\,\sin% \theta\,d\phi\,symbol{\hat{\phi}}
  54. d 𝐥 = σ 2 + τ 2 d σ s y m b o l σ ^ + σ 2 + τ 2 d τ s y m b o l τ ^ + d z 𝐳 ^ d\mathbf{l}=\sqrt{\sigma^{2}+\tau^{2}}\,d\sigma\,symbol{\hat{\sigma}}+\sqrt{% \sigma^{2}+\tau^{2}}\,d\tau\,symbol{\hat{\tau}}+dz\,\mathbf{\hat{z}}
  55. d 𝐒 d\mathbf{S}
  56. d y d z 𝐱 ^ + d x d z 𝐲 ^ + d x d y 𝐳 ^ \begin{aligned}\displaystyle dy\,dz&\displaystyle\hat{\mathbf{x}}\\ \displaystyle+dx\,dz&\displaystyle\hat{\mathbf{y}}\\ \displaystyle+dx\,dy&\displaystyle\mathbf{\hat{z}}\end{aligned}
  57. ρ d ϕ d z s y m b o l ρ ^ + d ρ d z s y m b o l ϕ ^ + ρ d ρ d ϕ 𝐳 ^ \begin{aligned}\displaystyle\rho\,d\phi\,dz&\displaystyle symbol{\hat{\rho}}\\ \displaystyle+d\rho\,dz&\displaystyle symbol{\hat{\phi}}\\ \displaystyle+\rho\,d\rho\,d\phi&\displaystyle\mathbf{\hat{z}}\end{aligned}
  58. r 2 sin θ d θ d ϕ 𝐫 ^ + r sin θ d r d ϕ s y m b o l θ ^ + r d r d θ s y m b o l ϕ ^ \begin{aligned}\displaystyle r^{2}\sin\theta\,d\theta\,d\phi&\displaystyle% \mathbf{\hat{r}}\\ \displaystyle+r\sin\theta\,dr\,d\phi&\displaystyle symbol{\hat{\theta}}\\ \displaystyle+r\,dr\,d\theta&\displaystyle symbol{\hat{\phi}}\end{aligned}
  59. σ 2 + τ 2 d τ d z s y m b o l σ ^ + σ 2 + τ 2 d σ d z s y m b o l τ ^ + ( σ 2 + τ 2 ) d σ d τ 𝐳 ^ \begin{aligned}\displaystyle\sqrt{\sigma^{2}+\tau^{2}}\,d\tau\,dz&% \displaystyle symbol{\hat{\sigma}}\\ \displaystyle+\sqrt{\sigma^{2}+\tau^{2}}\,d\sigma\,dz&\displaystyle symbol{% \hat{\tau}}\\ \displaystyle+\left(\sigma^{2}+\tau^{2}\right)\,d\sigma\,d\tau&\displaystyle% \mathbf{\hat{z}}\end{aligned}
  60. d V dV
  61. d x d y d z dx\,dy\,dz
  62. ρ d ρ d ϕ d z \rho\,d\rho\,d\phi\,dz
  63. r 2 sin θ d r d θ d ϕ r^{2}\sin\theta\,dr\,d\theta\,d\phi
  64. ( σ 2 + τ 2 ) d σ d τ d z \left(\sigma^{2}+\tau^{2}\right)d\sigma\,d\tau\,dz
  65. div grad f f = 2 f Δ f \operatorname{div}\,\operatorname{grad}f\equiv\nabla\cdot\nabla f=\nabla^{2}f% \equiv\Delta f
  66. curl grad f × f = 𝟎 \operatorname{curl}\,\operatorname{grad}f\equiv\nabla\times\nabla f=\mathbf{0}
  67. div curl 𝐀 ( × 𝐀 ) = 0 \operatorname{div}\,\operatorname{curl}\mathbf{A}\equiv\nabla\cdot(\nabla% \times\mathbf{A})=0
  68. curl curl 𝐀 × ( × 𝐀 ) = ( 𝐀 ) - 2 𝐀 \operatorname{curl}\,\operatorname{curl}\mathbf{A}\equiv\nabla\times(\nabla% \times\mathbf{A})=\nabla(\nabla\cdot\mathbf{A})-\nabla^{2}\mathbf{A}
  69. Δ ( f g ) = f Δ g + 2 f g + g Δ f \Delta(fg)=f\Delta g+2\nabla f\cdot\nabla g+g\Delta f

Delta_Pavonis.html

  1. [ F e H ] = 0.33 \begin{smallmatrix}\left[\frac{Fe}{H}\right]\ =\ 0.33\end{smallmatrix}

Deltoid_curve.html

  1. x = 2 a cos ( t ) + a cos ( 2 t ) x=2a\cos(t)+a\cos(2t)\,
  2. y = 2 a sin ( t ) - a sin ( 2 t ) y=2a\sin(t)-a\sin(2t)\,
  3. z = 2 a e i t + a e - 2 i t z=2ae^{it}+ae^{-2it}
  4. ( x 2 + y 2 ) 2 + 18 a 2 ( x 2 + y 2 ) - 27 a 4 = 8 a ( x 3 - 3 x y 2 ) (x^{2}+y^{2})^{2}+18a^{2}(x^{2}+y^{2})-27a^{4}=8a(x^{3}-3xy^{2})\,
  5. r 4 + 18 a 2 r 2 - 27 a 4 = 8 a r 3 cos 3 θ . r^{4}+18a^{2}r^{2}-27a^{4}=8ar^{3}\cos 3\theta\,.
  6. t = 0 , ± 2 π 3 t=0,\,\pm\tfrac{2\pi}{3}
  7. x 3 - x 2 - ( 3 x + 1 ) y 2 = 0 , x^{3}-x^{2}-(3x+1)y^{2}=0,\,
  8. x 3 - x 2 + ( 3 x + 1 ) y 2 = 0 x^{3}-x^{2}+(3x+1)y^{2}=0\,
  9. 2 π a 2 2\pi a^{2}

Deltoidal_icositetrahedron.html

  1. arccos ( - 7 + 4 2 17 ) \arccos(-\frac{7+4\sqrt{2}}{17})
  2. 1 : ( 2 - 1 2 ) 1 : 1.292893 \scriptstyle 1:(2-\frac{1}{\sqrt{2}})\approx 1:1.292893\dots
  3. 6 29 - 2 2 \scriptstyle{6\sqrt{29-2\sqrt{2}}}
  4. 122 + 71 2 \scriptstyle{\sqrt{122+71\sqrt{2}}}

Denny's_paradox.html

  1. c m c_{m}
  2. c m = ( 4 g σ / ρ ) 1 / 2 c_{m}=\left(4g\sigma/\rho\right)^{1/2}
  3. g g
  4. σ \sigma
  5. ρ \rho
  6. c m = ( 4 g σ / ρ ) 1 / 2 0.23 m / s c_{m}=\left(4g\sigma/\rho\right)^{1/2}\simeq 0.23\mathrm{m/s}

Density_altitude.html

  1. DA = T SL γ [ 1 - ( P / P S L T / T S L ) Γ R g M - Γ R ] \mathrm{DA}=\frac{T\text{SL}}{\gamma}\left[1-\left(\frac{P/P_{SL}}{\mathrm{T}/% T_{SL}}\right)^{\frac{\Gamma R}{gM-\Gamma R}}\right]
  2. DA = \mathrm{DA}=
  3. P = P=
  4. P S L = P_{SL}=
  5. T = \mathrm{T}=
  6. T S L = T_{SL}=
  7. γ = \gamma=
  8. Γ = \Gamma=
  9. R = R=
  10. g = g=
  11. M = M=
  12. DA = 145442.16 [ 1 - ( 17.326 P 459.67 + T ) 0.235 ] \mathrm{DA}=145442.16\left[1-\left(\frac{17.326P}{459.67+T}\right)^{0.235}\right]
  13. DA = \mathrm{DA}=
  14. P = P=
  15. T = T=

Depth-limited_search.html

  1. | V | + | E | |V|+|E|
  2. | V | |V|
  3. | E | |E|

Derivation_(differential_algebra).html

  1. D ( a b ) = D ( a ) b + a D ( b ) . D(ab)=D(a)b+aD(b).
  2. D ( x 1 x 2 x n ) = i x 1 x i - 1 D ( x i ) x i + 1 x n = i D ( x i ) j i x j . D(x_{1}x_{2}\cdots x_{n})=\sum_{i}x_{1}\cdots x_{i-1}D(x_{i})x_{i+1}\cdots x_{% n}=\sum_{i}D(x_{i})\prod_{j\neq i}x_{j}.\,
  3. i , D ( x i ) i,\ D(x_{i})
  4. x 1 , x 2 , x i - 1 x_{1},x_{2},\cdots x_{i-1}
  5. Der K ( A , M ) Der k ( A , M ) , \operatorname{Der}_{K}(A,M)\subset\operatorname{Der}_{k}(A,M),\,
  6. [ D 1 , D 2 ] = D 1 D 2 - D 2 D 1 . [D_{1},D_{2}]=D_{1}\circ D_{2}-D_{2}\circ D_{1}.
  7. D ( a b ) = D ( a ) b + ϵ | a | | D | a D ( b ) , {D(ab)=D(a)b+\epsilon^{|a||D|}aD(b)},
  8. D ( a b ) = D ( a ) b + ( - 1 ) | a | a D ( b ) , {D(ab)=D(a)b+(-1)^{|a|}aD(b)},

Derived_category.html

  1. X - 1 d - 1 X 0 d 0 X 1 d 1 X 2 \cdots\to X^{-1}\xrightarrow{d^{-1}}X^{0}\xrightarrow{d^{0}}X^{1}\xrightarrow{% d^{1}}X^{2}\to\cdots
  2. X [ n ] i = X n + i , X[n]^{i}=X^{n+i},
  3. d X [ n ] = ( - 1 ) n d X . d_{X[n]}=(-1)^{n}d_{X}.
  4. 0 X Y Z 0 0\rightarrow X\rightarrow Y\rightarrow Z\rightarrow 0
  5. Hom D ( 𝒜 ) ( X , Y [ j ] ) = Ext 𝒜 j ( X , Y ) . \,\text{Hom}_{D(\mathcal{A})}(X,Y[j])=\,\text{Ext}^{j}_{\mathcal{A}}(X,Y).
  6. K ( 𝒜 ) D ( 𝒜 ) . K(\mathcal{A})\rightarrow D(\mathcal{A}).
  7. 0 X I 0 I 1 , 0\rightarrow X\rightarrow I^{0}\rightarrow I^{1}\rightarrow\cdots,\,
  8. D + ( 𝒜 ) K + ( Inj ( 𝒜 ) ) D^{+}(\mathcal{A})\rightarrow K^{+}(\mathrm{Inj}(\mathcal{A}))
  9. Hom D ( A ) ( X , Y ) = Hom K ( A ) ( X , Y ) . \mathrm{Hom}_{D(A)}(X,Y)=\mathrm{Hom}_{K(A)}(X,Y).
  10. 𝒜 F G 𝒞 , \mathcal{A}\stackrel{F}{\rightarrow}\mathcal{B}\stackrel{G}{\rightarrow}% \mathcal{C},\,

Design_matrix.html

  1. y = X β + ϵ , y=X\beta+\epsilon,
  2. β \beta
  3. ϵ \epsilon
  4. y i = β 0 + β 1 x i + ϵ i , y_{i}=\beta_{0}+\beta_{1}x_{i}+\epsilon_{i},\,
  5. β 0 \beta_{0}
  6. β 1 \beta_{1}
  7. [ y 1 y 2 y 3 y 4 y 5 y 6 y 7 ] = [ 1 x 1 1 x 2 1 x 3 1 x 4 1 x 5 1 x 6 1 x 7 ] [ β 0 β 1 ] + [ ϵ 1 ϵ 2 ϵ 3 ϵ 4 ϵ 5 ϵ 6 ϵ 7 ] \begin{bmatrix}y_{1}\\ y_{2}\\ y_{3}\\ y_{4}\\ y_{5}\\ y_{6}\\ y_{7}\end{bmatrix}=\begin{bmatrix}1&x_{1}\\ 1&x_{2}\\ 1&x_{3}\\ 1&x_{4}\\ 1&x_{5}\\ 1&x_{6}\\ 1&x_{7}\end{bmatrix}\begin{bmatrix}\beta_{0}\\ \beta_{1}\end{bmatrix}+\begin{bmatrix}\epsilon_{1}\\ \epsilon_{2}\\ \epsilon_{3}\\ \epsilon_{4}\\ \epsilon_{5}\\ \epsilon_{6}\\ \epsilon_{7}\end{bmatrix}
  8. y i y_{i}
  9. y i = β 0 + β 1 w i + β 2 x i + ϵ i y_{i}=\beta_{0}+\beta_{1}w_{i}+\beta_{2}x_{i}+\epsilon_{i}
  10. [ y 1 y 2 y 3 y 4 y 5 y 6 y 7 ] = [ 1 w 1 x 1 1 w 2 x 2 1 w 3 x 3 1 w 4 x 4 1 w 5 x 5 1 w 6 x 6 1 w 7 x 7 ] [ β 0 β 1 β 2 ] + [ ϵ 1 ϵ 2 ϵ 3 ϵ 4 ϵ 5 ϵ 6 ϵ 7 ] \begin{bmatrix}y_{1}\\ y_{2}\\ y_{3}\\ y_{4}\\ y_{5}\\ y_{6}\\ y_{7}\end{bmatrix}=\begin{bmatrix}1&w_{1}&x_{1}\\ 1&w_{2}&x_{2}\\ 1&w_{3}&x_{3}\\ 1&w_{4}&x_{4}\\ 1&w_{5}&x_{5}\\ 1&w_{6}&x_{6}\\ 1&w_{7}&x_{7}\end{bmatrix}\begin{bmatrix}\beta_{0}\\ \beta_{1}\\ \beta_{2}\end{bmatrix}+\begin{bmatrix}\epsilon_{1}\\ \epsilon_{2}\\ \epsilon_{3}\\ \epsilon_{4}\\ \epsilon_{5}\\ \epsilon_{6}\\ \epsilon_{7}\end{bmatrix}
  11. y i j = μ i + ϵ i j y_{ij}=\mu_{i}+\epsilon_{ij}
  12. [ y 1 y 2 y 3 y 4 y 5 y 6 y 7 ] = [ 1 0 0 1 0 0 1 0 0 0 1 0 0 1 0 0 0 1 0 0 1 ] [ μ 1 μ 2 μ 3 ] + [ ϵ 1 ϵ 2 ϵ 3 ϵ 4 ϵ 5 ϵ 6 ϵ 7 ] \begin{bmatrix}y_{1}\\ y_{2}\\ y_{3}\\ y_{4}\\ y_{5}\\ y_{6}\\ y_{7}\end{bmatrix}=\begin{bmatrix}1&0&0\\ 1&0&0\\ 1&0&0\\ 0&1&0\\ 0&1&0\\ 0&0&1\\ 0&0&1\end{bmatrix}\begin{bmatrix}\mu_{1}\\ \mu_{2}\\ \mu_{3}\end{bmatrix}+\begin{bmatrix}\epsilon_{1}\\ \epsilon_{2}\\ \epsilon_{3}\\ \epsilon_{4}\\ \epsilon_{5}\\ \epsilon_{6}\\ \epsilon_{7}\end{bmatrix}
  13. μ i \mu_{i}
  14. i i
  15. τ i \tau_{i}
  16. y i j = μ + τ i + ϵ i j y_{ij}=\mu+\tau_{i}+\epsilon_{ij}
  17. τ 1 \tau_{1}
  18. [ y 1 y 2 y 3 y 4 y 5 y 6 y 7 ] = [ 1 0 0 1 0 0 1 0 0 1 1 0 1 1 0 1 0 1 1 0 1 ] [ μ τ 2 τ 3 ] + [ ϵ 1 ϵ 2 ϵ 3 ϵ 4 ϵ 5 ϵ 6 ϵ 7 ] \begin{bmatrix}y_{1}\\ y_{2}\\ y_{3}\\ y_{4}\\ y_{5}\\ y_{6}\\ y_{7}\end{bmatrix}=\begin{bmatrix}1&0&0\\ 1&0&0\\ 1&0&0\\ 1&1&0\\ 1&1&0\\ 1&0&1\\ 1&0&1\end{bmatrix}\begin{bmatrix}\mu\\ \tau_{2}\\ \tau_{3}\end{bmatrix}+\begin{bmatrix}\epsilon_{1}\\ \epsilon_{2}\\ \epsilon_{3}\\ \epsilon_{4}\\ \epsilon_{5}\\ \epsilon_{6}\\ \epsilon_{7}\end{bmatrix}
  19. μ \mu
  20. τ i \tau_{i}
  21. i i
  22. τ 1 \tau_{1}

Detection_theory.html

  1. p ( H 1 ) = π 1 p(H1)=\pi_{1}
  2. p ( H 2 ) = π 2 p(H2)=\pi_{2}
  3. p ( H 1 | y ) = p ( y | H 1 ) π 1 p ( y ) p(H1|y)=\frac{p(y|H1)\cdot\pi_{1}}{p(y)}
  4. p ( H 2 | y ) = p ( y | H 2 ) π 2 p ( y ) p(H2|y)=\frac{p(y|H2)\cdot\pi_{2}}{p(y)}
  5. p ( y | H 1 ) π 1 + p ( y | H 2 ) π 2 p(y|H1)\cdot\pi_{1}+p(y|H2)\cdot\pi_{2}
  6. p ( y | H 2 ) π 2 p ( y | H 1 ) π 1 + p ( y | H 2 ) π 2 p ( y | H 1 ) π 1 p ( y | H 1 ) π 1 + p ( y | H 2 ) π 2 \frac{p(y|H2)\cdot\pi_{2}}{p(y|H1)\cdot\pi_{1}+p(y|H2)\cdot\pi_{2}}\geq\frac{p% (y|H1)\cdot\pi_{1}}{p(y|H1)\cdot\pi_{1}+p(y|H2)\cdot\pi_{2}}
  7. p ( y | H 2 ) p ( y | H 1 ) π 1 π 2 \Rightarrow\frac{p(y|H2)}{p(y|H1)}\geq\frac{\pi_{1}}{\pi_{2}}
  8. π 1 π 2 \frac{\pi_{1}}{\pi_{2}}
  9. τ M A P \tau_{MAP}
  10. p ( y | H 2 ) p ( y | H 1 ) \frac{p(y|H2)}{p(y|H1)}
  11. L ( y ) L(y)
  12. L ( y ) τ M A P L(y)\geq\tau_{MAP}
  13. U 11 U_{11}
  14. U 12 U_{12}
  15. U 21 U_{21}
  16. U 22 U_{22}
  17. U 11 - U 21 U_{11}-U_{21}
  18. U 22 - U 12 U_{22}-U_{12}
  19. P 11 P_{11}
  20. P 12 P_{12}
  21. U = P 11 U 11 + P 21 U 21 + P 12 U 12 + P 22 U 22 U=P_{11}\cdot U_{11}+P_{21}\cdot U_{21}+P_{12}\cdot U_{12}+P_{22}\cdot U_{22}
  22. U = P 11 U 11 + ( 1 - P 11 ) U 21 + P 12 U 12 + ( 1 - P 12 ) U 22 U=P_{11}\cdot U_{11}+(1-P_{11})\cdot U_{21}+P_{12}\cdot U_{12}+(1-P_{12})\cdot U% _{22}
  23. U = U 21 + U 22 + P 11 ( U 11 - U 21 ) - P 12 ( U 22 - U 12 ) U=U_{21}+U_{22}+P_{11}\cdot(U_{11}-U_{21})-P_{12}\cdot(U_{22}-U_{12})
  24. U = P 11 ( U 11 - U 21 ) - P 12 ( U 22 - U 12 ) U^{\prime}=P_{11}\cdot(U_{11}-U_{21})-P_{12}\cdot(U_{22}-U_{12})
  25. P 11 = π 1 R 1 p ( y | H 1 ) d y P_{11}=\pi_{1}\cdot\int_{R_{1}}p(y|H1)\,dy
  26. P 12 = π 2 R 1 p ( y | H 2 ) d y P_{12}=\pi_{2}\cdot\int_{R_{1}}p(y|H2)\,dy
  27. π 1 \pi_{1}
  28. π 2 \pi_{2}
  29. P ( H 1 ) P(H1)
  30. P ( H 2 ) P(H2)
  31. R 1 R_{1}
  32. U = R 1 { π 1 ( U 11 - U 21 ) p ( y | H 1 ) - π 2 ( U 22 - U 12 ) p ( y | H 2 ) } d y \Rightarrow U^{\prime}=\int_{R_{1}}\left\{\pi_{1}\cdot(U_{11}-U_{21})\cdot p(y% |H1)-\pi_{2}\cdot(U_{22}-U_{12})\cdot p(y|H2)\right\}\,dy
  33. U U^{\prime}
  34. U U
  35. R 1 R_{1}
  36. π 1 ( U 11 - U 21 ) p ( y | H 1 ) - π 2 ( U 22 - U 12 ) p ( y | H 2 ) > 0 \pi_{1}\cdot(U_{11}-U_{21})\cdot p(y|H1)-\pi_{2}\cdot(U_{22}-U_{12})\cdot p(y|% H2)>0
  37. π 2 ( U 22 - U 12 ) p ( y | H 2 ) π 1 ( U 11 - U 21 ) p ( y | H 1 ) \pi_{2}\cdot(U_{22}-U_{12})\cdot p(y|H2)\geq\pi_{1}\cdot(U_{11}-U_{21})\cdot p% (y|H1)
  38. L ( y ) p ( y | H 2 ) p ( y | H 1 ) π 1 ( U 11 - U 21 ) π 2 ( U 22 - U 12 ) τ B \Rightarrow L(y)\equiv\frac{p(y|H2)}{p(y|H1)}\geq\frac{\pi_{1}\cdot(U_{11}-U_{% 21})}{\pi_{2}\cdot(U_{22}-U_{12})}\equiv\tau_{B}

Deterministic_finite_automaton.html

  1. a Σ a\in\Sigma
  2. δ a : Q Q \delta_{a}:Q\rightarrow Q
  3. δ ( q , a ) = δ a ( q ) \delta(q,a)=\delta_{a}(q)
  4. q Q q\in Q
  5. δ a \delta_{a}
  6. δ b \delta_{b}
  7. δ ^ : Q × Σ Q \widehat{\delta}:Q\times\Sigma^{\star}\rightarrow Q
  8. a , b Σ a,b\in\Sigma
  9. δ ^ \widehat{\delta}
  10. δ ^ a b = δ a δ b \widehat{\delta}_{ab}=\delta_{a}\circ\delta_{b}
  11. \circ
  12. δ ^ ( q , ϵ ) = q . \widehat{\delta}(q,\epsilon)=q.
  13. ϵ \epsilon
  14. δ ^ ( q , w a ) = δ a ( δ ^ ( q , w ) ) . \widehat{\delta}(q,wa)=\delta_{a}(\widehat{\delta}(q,w)).
  15. w Σ * , a Σ w\in\Sigma^{*},a\in\Sigma
  16. q Q q\in Q
  17. δ ^ \widehat{\delta}
  18. w Σ * w\in\Sigma^{*}
  19. δ ^ \widehat{\delta}
  20. δ \delta
  21. n n
  22. k k
  23. n n
  24. k k
  25. 1 , , k 1,\ldots,k
  26. k k
  27. k 2 k\geq 2
  28. k k
  29. k k
  30. n n
  31. 1 1

DFT_matrix.html

  1. X = W x X=Wx
  2. x x
  3. W W
  4. X X
  5. W W
  6. W = ( ω j k N ) j , k = 0 , , N - 1 W=\left(\frac{\omega^{jk}}{\sqrt{N}}\right)_{j,k=0,\ldots,N-1}
  7. W = 1 N [ 1 1 1 1 1 1 ω ω 2 ω 3 ω N - 1 1 ω 2 ω 4 ω 6 ω 2 ( N - 1 ) 1 ω 3 ω 6 ω 9 ω 3 ( N - 1 ) 1 ω N - 1 ω 2 ( N - 1 ) ω 3 ( N - 1 ) ω ( N - 1 ) ( N - 1 ) ] , W=\frac{1}{\sqrt{N}}\begin{bmatrix}1&1&1&1&\cdots&1\\ 1&\omega&\omega^{2}&\omega^{3}&\cdots&\omega^{N-1}\\ 1&\omega^{2}&\omega^{4}&\omega^{6}&\cdots&\omega^{2(N-1)}\\ 1&\omega^{3}&\omega^{6}&\omega^{9}&\cdots&\omega^{3(N-1)}\\ \vdots&\vdots&\vdots&\vdots&\ddots&\vdots\\ 1&\omega^{N-1}&\omega^{2(N-1)}&\omega^{3(N-1)}&\cdots&\omega^{(N-1)(N-1)}\\ \end{bmatrix},
  8. ω = e - 2 π i N \omega=e^{-\frac{2\pi i}{N}}
  9. N N
  10. i = - 1 i=\sqrt{-1}
  11. 1 / N 1/\sqrt{N}
  12. 1 / N 1/\sqrt{N}
  13. O ( N 2 ) O(N^{2})
  14. 1 2 [ 1 1 1 - 1 ] \frac{1}{\sqrt{2}}\begin{bmatrix}1&1\\ 1&-1\end{bmatrix}
  15. 1 / 2 1/\sqrt{2}
  16. W = 1 4 [ 1 1 1 1 1 - i - 1 i 1 - 1 1 - 1 1 i - 1 - i ] W=\frac{1}{\sqrt{4}}\begin{bmatrix}1&1&1&1\\ 1&-i&-1&i\\ 1&-1&1&-1\\ 1&i&-1&-i\end{bmatrix}
  17. W = 1 8 [ ω 0 ω 0 ω 0 ω 0 ω 0 ω 1 ω 2 ω 7 ω 0 ω 2 ω 4 ω 14 ω 0 ω 3 ω 6 ω 21 ω 0 ω 4 ω 8 ω 28 ω 0 ω 5 ω 10 ω 35 ω 0 ω 7 ω 14 ω 49 ] W=\frac{1}{\sqrt{8}}\begin{bmatrix}\omega^{0}&\omega^{0}&\omega^{0}&\ldots&% \omega^{0}\\ \omega^{0}&\omega^{1}&\omega^{2}&\ldots&\omega^{7}\\ \omega^{0}&\omega^{2}&\omega^{4}&\ldots&\omega^{14}\\ \omega^{0}&\omega^{3}&\omega^{6}&\ldots&\omega^{21}\\ \omega^{0}&\omega^{4}&\omega^{8}&\ldots&\omega^{28}\\ \omega^{0}&\omega^{5}&\omega^{10}&\ldots&\omega^{35}\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ \omega^{0}&\omega^{7}&\omega^{14}&\ldots&\omega^{49}\\ \end{bmatrix}
  18. ω = e - 2 π i 8 = 1 2 - i 2 \omega=e^{-\frac{2\pi i}{8}}=\frac{1}{\sqrt{2}}-\frac{i}{\sqrt{2}}
  19. 1 / 8 1/\sqrt{8}
  20. 1 / N 1/\sqrt{N}

Diamond_anvil_cell.html

  1. p = F A p=\frac{F}{A}
  2. N 2 N_{2}

Diamond_principle.html

  1. κ \kappa
  2. S κ S\subseteq\kappa
  3. A α : α S \langle A_{\alpha}:\alpha\in S\rangle
  4. A α α A_{\alpha}\subseteq\alpha
  5. A κ , { α S : A α = A α } A\subseteq\kappa,\{\alpha\in S:A\cap\alpha=A_{\alpha}\}
  6. κ \kappa
  7. 2 κ = κ + 2^{\kappa}=\kappa^{+}

Diatomaceous_earth.html

  1. N = ( 1 - N f N F × A F A f ) × 100 N=\left(1-\frac{N_{f}}{N_{F}}\times\frac{A_{F}}{A_{f}}\right)\times 100

Difference_of_two_squares.html

  1. a 2 - b 2 = ( a + b ) ( a - b ) a^{2}-b^{2}=(a+b)(a-b)\,\!
  2. ( a + b ) ( a - b ) = a 2 + b a - a b - b 2 (a+b)(a-b)=a^{2}+ba-ab-b^{2}\,\!
  3. b a - a b = 0 ba-ab=0\,\!
  4. a 2 + b a - a b - b 2 a^{2}+ba-ab-b^{2}\,\!
  5. a 2 - b 2 a^{2}-b^{2}
  6. b a - a b = 0 ba-ab=0\,\!
  7. a 2 - b 2 a^{2}-b^{2}
  8. a ( a - b ) + b ( a - b ) a(a-b)+b(a-b)
  9. ( a + b ) ( a - b ) (a+b)(a-b)
  10. a 2 - b 2 = ( a + b ) ( a - b ) a^{2}-b^{2}=(a+b)(a-b)
  11. a 2 - b 2 a^{2}-b^{2}
  12. a + b a+b
  13. a - b a-b
  14. ( a + b ) ( a - b ) (a+b)(a-b)
  15. a 2 - b 2 = ( a + b ) ( a - b ) a^{2}-b^{2}=(a+b)(a-b)
  16. x 4 - 1 x^{4}-1
  17. x 4 - 1 = ( x 2 + 1 ) ( x 2 - 1 ) = ( x 2 + 1 ) ( x + 1 ) ( x - 1 ) x^{4}-1=(x^{2}+1)(x^{2}-1)=(x^{2}+1)(x+1)(x-1)
  18. x 2 - y 2 + x - y x^{2}-y^{2}+x-y
  19. ( x + y ) ( x - y ) (x+y)(x-y)
  20. x 2 - y 2 + x - y = ( x + y ) ( x - y ) + x - y = ( x - y ) ( x + y + 1 ) x^{2}-y^{2}+x-y=(x+y)(x-y)+x-y=(x-y)(x+y+1)
  21. z 2 + 5 z^{2}+5\,\!
  22. z 2 + 5 z^{2}+5\,\!
  23. = z 2 - i 2 5 =z^{2}-i^{2}\cdot 5
  24. = z 2 - ( i 5 ) 2 =z^{2}-(i\sqrt{5})^{2}
  25. = ( z + i 5 ) ( z - i 5 ) =(z+i\sqrt{5})(z-i\sqrt{5})
  26. ( z + i 5 ) (z+i\sqrt{5})
  27. ( z - i 5 ) (z-i\sqrt{5})
  28. 5 3 + 4 \dfrac{5}{\sqrt{3}+4}\,\!
  29. 5 3 + 4 \dfrac{5}{\sqrt{3}+4}\,\!
  30. = 5 3 + 4 × 3 - 4 3 - 4 =\dfrac{5}{\sqrt{3}+4}\times\dfrac{\sqrt{3}-4}{\sqrt{3}-4}\,\!
  31. = 5 ( 3 - 4 ) ( 3 + 4 ) ( 3 - 4 ) =\dfrac{5(\sqrt{3}-4)}{(\sqrt{3}+4)(\sqrt{3}-4)}\,\!
  32. = 5 ( 3 - 4 ) 3 2 - 4 2 =\dfrac{5(\sqrt{3}-4)}{\sqrt{3}^{2}-4^{2}}\,\!
  33. = 5 ( 3 - 4 ) 3 - 16 =\dfrac{5(\sqrt{3}-4)}{3-16}\,\!
  34. = - 5 ( 3 - 4 ) 13 . =-\dfrac{5(\sqrt{3}-4)}{13}.\,\!
  35. 3 + 4 \sqrt{3}+4\,\!
  36. 13 13\,\!
  37. 27 × 33 = ( 30 - 3 ) ( 30 + 3 ) 27\times 33=(30-3)(30+3)
  38. 27 × 33 27\times 33
  39. a 2 - b 2 a^{2}-b^{2}
  40. 30 2 - 3 2 = 891 30^{2}-3^{2}=891
  41. ( n + 1 ) 2 - n 2 = ( ( n + 1 ) + n ) ( ( n + 1 ) - n ) = 2 n + 1 \begin{array}[]{lcl}(n+1)^{2}-n^{2}&=&((n+1)+n)((n+1)-n)\\ &=&2n+1\end{array}
  42. ( n + k ) 2 - n 2 = ( ( n + k ) + n ) ( ( n + k ) - n ) = k ( 2 n + k ) \begin{array}[]{lcl}(n+k)^{2}-n^{2}&=&((n+k)+n)((n+k)-n)\\ &=&k(2n+k)\end{array}
  43. 𝐚 𝐚 - 𝐛 𝐛 = ( 𝐚 + 𝐛 ) ( 𝐚 - 𝐛 ) {\mathbf{a}}\cdot{\mathbf{a}}-{\mathbf{b}}\cdot{\mathbf{b}}=({\mathbf{a}}+{% \mathbf{b}})\cdot({\mathbf{a}}-{\mathbf{b}})\,\!
  44. 𝐚 \mathbf{a}
  45. 𝐛 \mathbf{b}
  46. a n - b n = ( a - b ) ( k = 0 n - 1 a n - 1 - k b k ) a^{n}-b^{n}=(a-b)(\sum_{k=0}^{n-1}a^{n-1-k}b^{k})

Differential_(mathematics).html

  1. ( C , d ) (C_{\bullet},d_{\bullet})

Differential_diagnosis.html

  1. Pr ( Presentation is caused by condition in individual ) Pr ( Presentation has occurred in individual ) = Pr ( Presentation WHOIFPI by condition ) Pr ( Presentation WHOIFPI ) \displaystyle\frac{\Pr(\,\text{Presentation is caused by condition in % individual})}{\Pr(\,\text{Presentation has occurred in individual})}=\frac{\Pr% (\,\text{Presentation WHOIFPI by condition})}{\Pr(\,\text{Presentation WHOIFPI% })}
  2. Pr ( Presentation is caused by condition in individual ) = Pr ( Presentation WHOIFPI by condition ) Pr ( Presentation WHOIFPI ) \Pr(\,\text{Presentation is caused by condition in individual})=\frac{\Pr(\,% \text{Presentation WHOIFPI by condition})}{\Pr(\,\text{Presentation WHOIFPI})}
  3. Pr ( Presentation WHOIFPI ) \displaystyle\Pr(\,\text{Presentation WHOIFPI})
  4. Pr ( Presentation WHOIFPI by condition ) = Pr ( Condition WHOIFPI ) r condition presentation , \Pr(\,\text{Presentation WHOIFPI by condition})=\Pr(\,\text{Condition WHOIFPI}% )\cdot r_{\,\text{condition}\rightarrow\,\text{presentation}},
  5. Pr ( Condition WHOIFPI ) R R condition Pr ( Condition in population ) , \Pr(\,\text{Condition WHOIFPI})\approx RR\text{condition}\cdot\Pr(\,\text{% Condition in population}),
  6. Pr ( PH in population ) = 0.5 years 1 4000 per year = 1 8000 \Pr(\,\text{PH in population})=0.5\,\text{ years}\cdot\frac{1}{\,\text{4000 % per year}}=\frac{1}{8000}
  7. Pr ( PH WHOIFPI ) R R P H Pr ( PH in population ) = 10 1 8000 = 1 800 = 0.00125 \Pr(\,\text{PH WHOIFPI})\approx RR_{PH}\cdot\Pr(\,\text{PH in population})=10% \cdot\frac{1}{8000}=\frac{1}{800}=0.00125
  8. Pr ( Hypercalcemia WHOIFPI by PH ) \displaystyle\Pr(\,\text{Hypercalcemia WHOIFPI by PH})
  9. Pr ( cancer in population ) = 0.5 years 1 250 per year = 1 500 \Pr(\,\text{cancer in population})=0.5\,\text{ years}\cdot\frac{1}{\,\text{250% per year}}=\frac{1}{500}
  10. Pr ( cancer WHOIFPI ) R R cancer Pr ( cancer in population ) = 1 1 500 = 1 500 = 0.002. \Pr(\,\text{cancer WHOIFPI})\approx RR\text{cancer}\cdot\Pr(\,\text{cancer in % population})=1\cdot\frac{1}{500}=\frac{1}{500}=0.002.
  11. Pr ( Hypercalcemia WHOIFPI by cancer ) \displaystyle\Pr(\,\text{Hypercalcemia WHOIFPI by cancer})
  12. Pr ( no disease in population ) \displaystyle\Pr(\,\text{no disease in population})
  13. Pr ( no disease WHOIFPI ) = 0.997. \Pr(\,\text{no disease WHOIFPI})=0.997.\,
  14. r no disease hypercalcemia = 0.0014 r_{\,\text{no disease}\rightarrow\,\text{hypercalcemia}}=0.0014
  15. Pr ( Hypercalcemia WHOIFPI by no disease ) \displaystyle\Pr(\,\text{Hypercalcemia WHOIFPI by no disease})
  16. Pr ( hypercalcemia WHOIFPI ) \displaystyle\Pr(\,\text{hypercalcemia WHOIFPI})
  17. Pr ( hypercalcemia is caused by PH in individual ) \displaystyle\Pr(\,\text{hypercalcemia is caused by PH in individual})
  18. Pr ( hypercalcemia is caused by cancer in individual ) \displaystyle\Pr(\,\text{hypercalcemia is caused by cancer in individual})
  19. Pr ( hypercalcemia is caused by other conditions in individual ) \displaystyle\Pr(\,\text{hypercalcemia is caused by other conditions in % individual})
  20. Pr ( hypercalcemia is present despite no disease in individual ) \displaystyle\Pr(\,\text{hypercalcemia is present despite no disease in % individual})
  21. odds = probability 1 - probability \,\text{odds}=\frac{\,\text{probability}}{1-\,\text{probability}}
  22. probability = odds odds + 1 \,\text{probability}=\frac{\,\text{odds}}{\,\text{odds}+1}
  23. Odds ( PostBT P H ) = Odds ( PreBT P H ) L H ( B T ) = 0.595 7 = 4.16 , \operatorname{Odds}(\,\text{PostBT}_{PH})=\operatorname{Odds}(\,\text{PreBT}_{% PH})\cdot LH(BT)=0.595\cdot 7=4.16,
  24. Pr ( PostBT P H ) = Odds ( PostBT P H ) Odds ( PostBT P H ) + 1 = 4.16 4.16 + 1 = 0.806 = 80.6 % \Pr(\,\text{PostBT}_{PH})=\frac{\operatorname{Odds}(\,\text{PostBT}_{PH})}{% \operatorname{Odds}(\,\text{PostBT}_{PH})+1}=\frac{4.16}{4.16+1}=0.806=80.6\%
  25. Pr ( PostBT r e s t ) = 100 % - 80.6 % = 19.4 % \Pr(\,\text{PostBT}_{rest})=100\%-80.6\%=19.4\%
  26. Pr ( PreBTrest ) = 6.0 % + 14.9 % + 41.8 % = 62.7 % \Pr(\,\text{PreBT}\text{rest})=6.0\%+14.9\%+41.8\%=62.7\%
  27. Correcting factor = Pr ( PostBTrest ) Pr ( PreBTrest ) = 19.4 62.7 = 0.309 \,\text{Correcting factor}=\frac{\Pr(\,\text{PostBT}\text{rest})}{\Pr(\,\text{% PreBT}\text{rest})}=\frac{19.4}{62.7}=0.309
  28. Pr ( PostBTcancer ) = Pr ( PreBTcancer ) Correcting factor = 6.0 % 0.309 = 1.9 % \Pr(\,\text{PostBT}\text{cancer})=\Pr(\,\text{PreBT}\text{cancer})\cdot\,\text% {Correcting factor}=6.0\%\cdot 0.309=1.9\%

Differential_of_the_first_kind.html

  1. x k d x Q ( x ) \int\frac{x^{k}\,dx}{\sqrt{Q(x)}}

Differential_structure.html

  1. n \mathbb{R}^{n}
  2. φ i : M W i U i n \varphi_{i}:M\supset W_{i}\rightarrow U_{i}\subset\mathbb{R}^{n}
  3. n \mathbb{R}^{n}
  4. φ i : W i U i , \varphi_{i}:W_{i}\rightarrow U_{i},\,
  5. φ j : W j U j . \varphi_{j}:W_{j}\rightarrow U_{j}.\,
  6. W i j = W i W j W_{ij}=W_{i}\cap W_{j}\;
  7. U i j = φ i ( W i j ) , U_{ij}=\varphi_{i}\left(W_{ij}\right),\,
  8. U j i = φ j ( W i j ) U_{ji}=\varphi_{j}\left(W_{ij}\right)
  9. φ i j : U i j U j i \varphi_{ij}:U_{ij}\rightarrow U_{ji}
  10. φ i j ( x ) = φ j ( φ i - 1 ( x ) ) . \varphi_{ij}(x)=\varphi_{j}\left(\varphi_{i}^{-1}\left(x\right)\right).
  11. φ i , φ j \varphi_{i},\,\varphi_{j}
  12. U i j , U j i U_{ij},\,U_{ji}
  13. φ i j , φ j i \varphi_{ij},\,\varphi_{ji}
  14. b 2 b_{2}
  15. b 2 > 18 b_{2}>18
  16. S 4 , P 2 , S^{4},{\mathbb{C}}P^{2},...
  17. 4 , S 3 × , M 4 { * } , {\mathbb{R}}^{4},S^{3}\times{\mathbb{R}},M^{4}\setminus\{*\},...

Differentiation_under_the_integral_sign.html

  1. d d x ( a ( x ) b ( x ) f ( x , t ) d t ) = f ( x , b ( x ) ) b ( x ) - f ( x , a ( x ) ) a ( x ) + a ( x ) b ( x ) x f ( x , t ) d t . \frac{\mathrm{d}}{\mathrm{d}x}\left(\int_{a(x)}^{b(x)}f(x,t)\,\mathrm{d}t% \right)=f(x,b(x))\cdot b^{\prime}(x)-f(x,a(x))\cdot a^{\prime}(x)+\int_{a(x)}^% {b(x)}\frac{\partial}{\partial x}f(x,t)\;\mathrm{d}t.
  2. d d t D ( t ) F ( 𝐱 , t ) d V = D ( t ) t F ( 𝐱 , t ) d V + D ( t ) F ( 𝐱 , t ) 𝐯 b d 𝚺 \frac{\mathrm{d}}{\mathrm{d}t}\int_{D(t)}F(\vec{\textbf{x}},t)\,\mathrm{d}V=% \int_{D(t)}\frac{\partial}{\partial t}\,F(\vec{\textbf{x}},t)\,\mathrm{d}V+% \int_{\partial D(t)}\,F(\vec{\textbf{x}},t)\,\vec{\textbf{v}}_{b}\cdot\mathrm{% d}\mathbf{\Sigma}
  3. F ( 𝐱 , t ) F(\vec{\textbf{x}},t)\,
  4. 𝐯 b \vec{\textbf{v}}_{b}\,
  5. d d t Ω ( t ) ω = Ω ( t ) i 𝐯 ( d x ω ) + Ω ( t ) i 𝐯 ω + Ω ( t ) ω ˙ , \frac{\mathrm{d}}{\mathrm{d}t}\int_{\Omega(t)}\omega=\int_{\Omega(t)}i_{\vec{% \textbf{v}}}(\mathrm{d}_{x}\omega)+\int_{\partial\Omega(t)}i_{\vec{\textbf{v}}% }\omega+\int_{\Omega(t)}\dot{\omega},\,
  6. 𝐯 \vec{\textbf{v}}\,
  7. 𝐯 = 𝐱 t \vec{\textbf{v}}=\frac{\partial\vec{\textbf{x}}}{\partial t}\,
  8. ω ˙ \dot{\omega}\,
  9. b ( a b f ( x ) d x ) = f ( b ) , a ( a b f ( x ) d x ) = - f ( a ) . \frac{\partial}{\partial b}\left(\int_{a}^{b}f(x)\;\mathrm{d}x\right)=f(b),% \qquad\frac{\partial}{\partial a}\left(\int_{a}^{b}f(x)\;\mathrm{d}x\right)=-f% (a).
  10. b ( a b f ( x ) d x ) = lim Δ b 0 1 Δ b [ a b + Δ b f ( x ) d x - a b f ( x ) d x ] = lim Δ b 0 1 Δ b b b + Δ b f ( x ) d x = lim Δ b 0 1 Δ b [ f ( b ) Δ b + 𝒪 ( Δ b 2 ) ] = f ( b ) a ( a b f ( x ) d x ) = lim Δ a 0 1 Δ a [ a + Δ a b f ( x ) d x - a b f ( x ) d x ] = lim Δ a 0 1 Δ a a + Δ a a f ( x ) d x = lim Δ a 0 1 Δ a [ - f ( a ) Δ a + 𝒪 ( Δ a 2 ) ] = - f ( a ) . \begin{aligned}\displaystyle\frac{\partial}{\partial b}\left(\int_{a}^{b}f(x)% \;\mathrm{d}x\right)&\displaystyle=\lim_{\Delta b\to 0}\frac{1}{\Delta b}\left% [\int_{a}^{b+\Delta b}f(x)\,\mathrm{d}x-\int_{a}^{b}f(x)\,\mathrm{d}x\right]\\ &\displaystyle=\lim_{\Delta b\to 0}\frac{1}{\Delta b}\int_{b}^{b+\Delta b}f(x)% \,\mathrm{d}x\\ &\displaystyle=\lim_{\Delta b\to 0}\frac{1}{\Delta b}\left[f(b)\Delta b+% \mathcal{O}\left(\Delta b^{2}\right)\right]\\ &\displaystyle=f(b)\\ \displaystyle\frac{\partial}{\partial a}\left(\int_{a}^{b}f(x)\;\mathrm{d}x% \right)&\displaystyle=\lim_{\Delta a\to 0}\frac{1}{\Delta a}\left[\int_{a+% \Delta a}^{b}f(x)\,\mathrm{d}x-\int_{a}^{b}f(x)\,\mathrm{d}x\right]\\ &\displaystyle=\lim_{\Delta a\to 0}\frac{1}{\Delta a}\int_{a+\Delta a}^{a}f(x)% \,\mathrm{d}x\\ &\displaystyle=\lim_{\Delta a\to 0}\frac{1}{\Delta a}\left[-f(a)\,\Delta a+% \mathcal{O}\left(\Delta a^{2}\right)\right]\\ &\displaystyle=-f(a).\end{aligned}
  11. φ ( α ) = a b f ( x , α ) d x . \varphi(\alpha)=\int_{a}^{b}f(x,\alpha)\;\mathrm{d}x.
  12. φ \varphi
  13. d φ d α = a b α f ( x , α ) d x . \frac{\mathrm{d}\varphi}{\mathrm{d}\alpha}=\int_{a}^{b}\frac{\partial}{% \partial\alpha}\,f(x,\alpha)\,\mathrm{d}x.\,
  14. | f ( x , α + Δ α ) - f ( x , α ) | < ε . |f(x,\alpha+\Delta\alpha)-f(x,\alpha)|<\varepsilon.
  15. Δ φ \displaystyle\Delta\varphi
  16. α f ( x , α ) \frac{\partial}{\partial\alpha}\,f(x,\alpha)
  17. x [ a , b ] | f ( x , α + Δ α ) - f ( x , α ) Δ α - f α | < ε . \forall x\in[a,b]\quad\left|\frac{f(x,\alpha+\Delta\alpha)-f(x,\alpha)}{\Delta% \alpha}-\frac{\partial f}{\partial\alpha}\right|<\varepsilon.
  18. Δ φ Δ α = a b f ( x , α + Δ α ) - f ( x , α ) Δ α d x = a b f ( x , α ) α d x + R \frac{\Delta\varphi}{\Delta\alpha}=\int_{a}^{b}\frac{f(x,\alpha+\Delta\alpha)-% f(x,\alpha)}{\Delta\alpha}\;\mathrm{d}x=\int_{a}^{b}\frac{\partial\,f(x,\alpha% )}{\partial\alpha}\,\mathrm{d}x+R
  19. | R | < a b ε d x = ε ( b - a ) . |R|<\int_{a}^{b}\varepsilon\;\mathrm{d}x=\varepsilon(b-a).
  20. lim Δ α 0 Δ φ Δ α = d φ d α = a b α f ( x , α ) d x . \lim_{{\Delta\alpha}\rightarrow 0}\frac{\Delta\varphi}{\Delta\alpha}=\frac{% \mathrm{d}\varphi}{\mathrm{d}\alpha}=\int_{a}^{b}\frac{\partial}{\partial% \alpha}\,f(x,\alpha)\,\mathrm{d}x.\,
  21. a b f ( x , α ) d x = φ ( α ) , \int_{a}^{b}f(x,\alpha)\;\mathrm{d}x=\varphi(\alpha),
  22. Δ φ = φ ( α + Δ α ) - φ ( α ) = a + Δ a b + Δ b f ( x , α + Δ α ) d x - a b f ( x , α ) d x = a + Δ a a f ( x , α + Δ α ) d x + a b f ( x , α + Δ α ) d x + b b + Δ b f ( x , α + Δ α ) d x - a b f ( x , α ) d x = - a a + Δ a f ( x , α + Δ α ) d x + a b [ f ( x , α + Δ α ) - f ( x , α ) ] d x + b b + Δ b f ( x , α + Δ α ) d x . \begin{aligned}\displaystyle\Delta\varphi&\displaystyle=\varphi(\alpha+\Delta% \alpha)-\varphi(\alpha)\\ &\displaystyle=\int_{a+\Delta a}^{b+\Delta b}f(x,\alpha+\Delta\alpha)\;\mathrm% {d}x\,-\int_{a}^{b}f(x,\alpha)\;\mathrm{d}x\\ &\displaystyle=\int_{a+\Delta a}^{a}f(x,\alpha+\Delta\alpha)\;\mathrm{d}x+\int% _{a}^{b}f(x,\alpha+\Delta\alpha)\;\mathrm{d}x+\int_{b}^{b+\Delta b}f(x,\alpha+% \Delta\alpha)\;\mathrm{d}x-\int_{a}^{b}f(x,\alpha)\;\mathrm{d}x\\ &\displaystyle=-\int_{a}^{a+\Delta a}\,f(x,\alpha+\Delta\alpha)\;\mathrm{d}x+% \int_{a}^{b}[f(x,\alpha+\Delta\alpha)-f(x,\alpha)]\;\mathrm{d}x+\int_{b}^{b+% \Delta b}\,f(x,\alpha+\Delta\alpha)\;\mathrm{d}x.\end{aligned}
  23. a b f ( x ) d x = ( b - a ) f ( ξ ) , \int_{a}^{b}f(x)\;\mathrm{d}x=(b-a)f(\xi),
  24. d φ d α = a b α f ( x , α ) d x \frac{\mathrm{d}\varphi}{\mathrm{d}\alpha}=\int_{a}^{b}\frac{\partial}{% \partial\alpha}\,f(x,\alpha)\,\mathrm{d}x
  25. d φ d α = a b α f ( x , α ) d x + f ( b , α ) b α - f ( a , α ) a α . \frac{\mathrm{d}\varphi}{\mathrm{d}\alpha}=\int_{a}^{b}\frac{\partial}{% \partial\alpha}\,f(x,\alpha)\,\mathrm{d}x+f(b,\alpha)\frac{\partial b}{% \partial\alpha}-f(a,\alpha)\frac{\partial a}{\partial\alpha}.
  26. φ ( α ) = 0 1 α x 2 + α 2 d x = { 0 α = 0 arctan ( 1 α ) α 0 \varphi(\alpha)=\int_{0}^{1}\frac{\alpha}{x^{2}+\alpha^{2}}\;\mathrm{d}x=% \begin{cases}0&\alpha=0\\ \arctan\left(\tfrac{1}{\alpha}\right)&\alpha\neq 0\end{cases}
  27. d d α φ ( α ) = 0 1 α ( α x 2 + α 2 ) d x = 0 1 x 2 - α 2 ( x 2 + α 2 ) 2 d x = - x x 2 + α 2 | 0 1 = - 1 1 + α 2 , \frac{\mathrm{d}}{\mathrm{d}\alpha}\varphi(\alpha)=\int_{0}^{1}\frac{\partial}% {\partial\alpha}\left(\frac{\alpha}{x^{2}+\alpha^{2}}\right)\;\mathrm{d}x=\int% _{0}^{1}\frac{x^{2}-\alpha^{2}}{(x^{2}+\alpha^{2})^{2}}\mathrm{d}x=-\frac{x}{x% ^{2}+\alpha^{2}}\bigg|_{0}^{1}=-\frac{1}{1+\alpha^{2}},
  28. d d x sin x cos x cosh t 2 d t \displaystyle\frac{\mathrm{d}}{\mathrm{d}x}\int_{\sin x}^{\cos x}\cosh t^{2}\;% \mathrm{d}t
  29. φ ( α ) = 0 π ln ( 1 - 2 α cos ( x ) + α 2 ) d x | α | > 1. \varphi(\alpha)=\int_{0}^{\pi}\,\ln(1-2\alpha\cos(x)+\alpha^{2})\;\mathrm{d}x% \qquad|\alpha|>1.
  30. d d α φ ( α ) = 0 π - 2 cos ( x ) + 2 α 1 - 2 α cos ( x ) + α 2 d x = 1 α 0 π ( 1 - 1 - α 2 1 - 2 α cos ( x ) + α 2 ) d x = π α - 2 α { arctan ( 1 + α 1 - α tan ( x 2 ) ) } | 0 π . \begin{aligned}\displaystyle\frac{\mathrm{d}}{\mathrm{d}\alpha}\,\varphi(% \alpha)&\displaystyle=\int_{0}^{\pi}\frac{-2\cos(x)+2\alpha}{1-2\alpha\cos(x)+% \alpha^{2}}\;\mathrm{d}x\\ &\displaystyle=\frac{1}{\alpha}\int_{0}^{\pi}\,\left(1-\frac{1-\alpha^{2}}{1-2% \alpha\cos(x)+\alpha^{2}}\,\right)\,\mathrm{d}x\\ &\displaystyle=\frac{\pi}{\alpha}-\frac{2}{\alpha}\left\{\,\arctan\left(\frac{% 1+\alpha}{1-\alpha}\cdot\tan\left(\frac{x}{2}\right)\right)\,\right\}\,\bigg|_% {0}^{\pi}.\end{aligned}
  31. { 1 + α 1 - α tan ( x 2 ) 0 | α | < 1 1 + α 1 - α tan ( x 2 ) 0 | α | > 1 \begin{cases}\frac{1+\alpha}{1-\alpha}\tan\left(\frac{x}{2}\right)\geq 0&|% \alpha|<1\\ \frac{1+\alpha}{1-\alpha}\tan\left(\frac{x}{2}\right)\leq 0&|\alpha|>1\end{cases}
  32. arctan ( 1 + α 1 - α tan ( x 2 ) ) | 0 π = { π 2 | α | < 1 - π 2 | α | > 1 \arctan\left(\frac{1+\alpha}{1-\alpha}\cdot\tan\left(\frac{x}{2}\right)\right)% \,\bigg|_{0}^{\pi}=\begin{cases}\frac{\pi}{2}&|\alpha|<1\\ -\frac{\pi}{2}&|\alpha|>1\end{cases}
  33. d d α φ ( α ) = { 0 | α | < 1 2 π α | α | > 1 \frac{\mathrm{d}}{\mathrm{d}\alpha}\,\varphi(\alpha)=\begin{cases}0&|\alpha|<1% \\ \frac{2\pi}{\alpha}&|\alpha|>1\end{cases}
  34. α α
  35. φ ( α ) = { C 1 | α | < 1 2 π ln | α | + C 2 | α | > 1 \varphi(\alpha)=\begin{cases}C_{1}&|\alpha|<1\\ 2\pi\ln|\alpha|+C_{2}&|\alpha|>1\end{cases}
  36. φ ( 0 ) φ(0)
  37. φ ( 0 ) = 0 π ln ( 1 ) d x = 0 π 0 d x = 0 \varphi(0)=\int_{0}^{\pi}\ln(1)\;\mathrm{d}x=\int_{0}^{\pi}0\;\mathrm{d}x=0
  38. φ ( α ) φ(α)
  39. φ ( α ) = { 0 | α | < 1 2 π ln | α | | α | > 1 \varphi(\alpha)=\begin{cases}0&|\alpha|<1\\ 2\pi\ln|\alpha|&|\alpha|>1\end{cases}
  40. α = ± 1 α=±1
  41. 𝐈 = 0 π 2 1 ( a cos 2 ( x ) + b sin 2 ( x ) ) 2 d x , a , b > 0. \textbf{I}=\int_{0}^{\frac{\pi}{2}}\frac{1}{\left(a\cos^{2}(x)+b\sin^{2}(x)% \right)^{2}}\;\mathrm{d}x,\qquad a,b>0.
  42. 𝐉 \displaystyle\,\textbf{J}
  43. 𝐉 a = - 0 π 2 cos 2 x d x ( a cos 2 x + b sin 2 x ) 2 \frac{\partial\textbf{J}}{\partial a}=-\int_{0}^{\frac{\pi}{2}}\frac{\cos^{2}x% \;\mathrm{d}x}{\left(a\cos^{2}x+b\sin^{2}x\right)^{2}}
  44. 𝐉 a = a ( π 2 a b ) = - π 4 a 3 b . \frac{\partial\textbf{J}}{\partial a}=\frac{\partial}{\partial a}\left(\frac{% \pi}{2\sqrt{ab}}\right)=-\frac{\pi}{4\sqrt{a^{3}b}}.
  45. 0 π 2 cos 2 ( x ) d x ( a cos 2 ( x ) + b sin 2 ( x ) ) 2 = π 4 a 3 b . \int_{0}^{\frac{\pi}{2}}\frac{\cos^{2}(x)\;\mathrm{d}x}{\left(a\cos^{2}(x)+b% \sin^{2}(x)\right)^{2}}=\frac{\pi}{4\sqrt{a^{3}b}}.
  46. 𝐉 b \frac{\partial\textbf{J}}{\partial b}
  47. 0 π 2 sin 2 x d x ( a cos 2 x + b sin 2 x ) 2 = π 4 a b 3 . \int_{0}^{\frac{\pi}{2}}\frac{\sin^{2}x\mathrm{d}x}{\left(a\cos^{2}x+b\sin^{2}% x\right)^{2}}=\frac{\pi}{4\sqrt{ab^{3}}}.
  48. 𝐈 = 0 π 2 1 ( a cos 2 x + b sin 2 x ) 2 d x = π 4 a b ( 1 a + 1 b ) , \textbf{I}=\int_{0}^{\frac{\pi}{2}}\frac{1}{\left(a\cos^{2}x+b\sin^{2}x\right)% ^{2}}\;\mathrm{d}x=\frac{\pi}{4\sqrt{ab}}\left(\frac{1}{a}+\frac{1}{b}\right),
  49. 𝐈 n = 0 π 2 1 ( a cos 2 x + b sin 2 x ) n d x , \textbf{I}_{n}=\int_{0}^{\frac{\pi}{2}}\frac{1}{\left(a\cos^{2}x+b\sin^{2}x% \right)^{n}}\;\mathrm{d}x,
  50. 𝐈 n - 1 a + 𝐈 n - 1 b + ( n - 1 ) 𝐈 n = 0. \frac{\partial\textbf{I}_{n-1}}{\partial a}+\frac{\partial\textbf{I}_{n-1}}{% \partial b}+(n-1)\textbf{I}_{n}=0.
  51. 𝐈 ( α ) = 0 π 2 ln ( 1 + cos α cos x ) cos x d x , 0 < α < π . \textbf{I}(\alpha)=\int_{0}^{\frac{\pi}{2}}\frac{\ln\,(1+\cos\alpha\,\cos\,x)}% {\cos x}\;\mathrm{d}x,\qquad 0<\alpha<\pi.
  52. d d α 𝐈 ( α ) = 0 π 2 α ( ln ( 1 + cos α cos x ) cos x ) d x [ 6 p t ] = - 0 π 2 sin α 1 + cos α cos x d x [ 6 p t ] = - 0 π 2 sin α ( cos 2 x 2 + sin 2 x 2 ) + cos α ( cos 2 x 2 - sin 2 x 2 ) d x [ 6 p t ] = - sin α 1 - cos α 0 π 2 1 cos 2 x 2 1 1 + cos α 1 - cos α + tan 2 x 2 d x [ 6 p t ] = - 2 sin α 1 - cos α 0 π 2 1 2 sec 2 x 2 2 cos 2 α 2 2 sin 2 α 2 + tan 2 x 2 d x [ 6 p t ] = - 2 ( 2 sin α 2 cos α 2 ) 2 sin 2 α 2 0 π 2 1 cot 2 α 2 + tan 2 x 2 d ( tan x 2 ) [ 6 p t ] = - 2 cot α 2 0 π 2 1 cot 2 α 2 + tan 2 x 2 d ( tan x 2 ) [ 6 p t ] = - 2 arctan ( tan α 2 tan x 2 ) | 0 π 2 = - α \begin{aligned}\displaystyle\frac{\mathrm{d}}{\mathrm{d}\alpha}\,\textbf{I}(% \alpha)&\displaystyle=\int_{0}^{\frac{\pi}{2}}\frac{\partial}{\partial\alpha}% \left(\frac{\ln(1+\cos\alpha\cos x)}{\cos x}\right)\,\mathrm{d}x\\ \displaystyle[6pt]&\displaystyle=-\int_{0}^{\frac{\pi}{2}}\frac{\sin\alpha}{1+% \cos\alpha\cos x}\,\mathrm{d}x\\ \displaystyle[6pt]&\displaystyle=-\int_{0}^{\frac{\pi}{2}}\frac{\sin\alpha}{% \left(\cos^{2}\frac{x}{2}+\sin^{2}\frac{x}{2}\right)+\cos\alpha\left(\cos^{2}% \frac{x}{2}-\sin^{2}\frac{x}{2}\right)}\,\mathrm{d}x\\ \displaystyle[6pt]&\displaystyle=-\frac{\sin\alpha}{1-\cos\alpha}\int_{0}^{% \frac{\pi}{2}}\frac{1}{\cos^{2}\frac{x}{2}}\frac{1}{\frac{1+\cos\alpha}{1-\cos% \alpha}+\tan^{2}\frac{x}{2}}\,\mathrm{d}x\\ \displaystyle[6pt]&\displaystyle=-\frac{2\sin\alpha}{1-\cos\alpha}\int_{0}^{% \frac{\pi}{2}}\frac{\frac{1}{2}\sec^{2}\frac{x}{2}}{\frac{2\cos^{2}\frac{% \alpha}{2}}{2\sin^{2}\frac{\alpha}{2}}+\tan^{2}\frac{x}{2}}\,\mathrm{d}x\\ \displaystyle[6pt]&\displaystyle=-\frac{2\left(2\sin\frac{\alpha}{2}\cos\frac{% \alpha}{2}\right)}{2\sin^{2}\frac{\alpha}{2}}\int_{0}^{\frac{\pi}{2}}\,\frac{1% }{\cot^{2}\frac{\alpha}{2}+\tan^{2}\frac{x}{2}}\mathrm{d}\left(\tan\frac{x}{2}% \right)\\ \displaystyle[6pt]&\displaystyle=-2\cot\frac{\alpha}{2}\int_{0}^{\frac{\pi}{2}% }\frac{1}{\cot^{2}\frac{\alpha}{2}+\tan^{2}\frac{x}{2}}\,\mathrm{d}\left(\tan% \frac{x}{2}\right)\\ \displaystyle[6pt]&\displaystyle=-2\arctan\left(\tan\frac{\alpha}{2}\tan\frac{% x}{2}\right)\bigg|_{0}^{\frac{\pi}{2}}\\ &\displaystyle=-\alpha\end{aligned}
  53. 𝐈 ( α ) = C - α 2 2 \,\textbf{I}(\alpha)=C-\frac{\alpha^{2}}{2}
  54. 𝐈 ( α ) = π 2 8 - α 2 2 . \textbf{I}(\alpha)=\frac{\pi^{2}}{8}-\frac{\alpha^{2}}{2}.
  55. 0 2 π e cos θ cos ( sin θ ) d θ . \int_{0}^{2\pi}e^{\cos\theta}\cos(\sin\theta)\;\mathrm{d}\theta.
  56. f ( φ ) = 0 2 π e φ cos θ cos ( φ sin θ ) d θ . f(\varphi)=\int_{0}^{2\pi}e^{\varphi\cos\theta}\cos(\varphi\sin\theta)\;% \mathrm{d}\theta.
  57. d f d φ \displaystyle\frac{\mathrm{d}f}{\mathrm{d}\varphi}
  58. d f d φ = 0 \frac{\mathrm{d}f}{\mathrm{d}\varphi}=0
  59. f ( 1 ) - f ( 0 ) = f ( 0 ) f ( 1 ) d f = 0 1 0 d φ = 0 f(1)-f(0)=\int_{f(0)}^{f(1)}\;\mathrm{d}f=\int_{0}^{1}0\;\mathrm{d}\varphi=0
  60. 0 sin x x d x \displaystyle\int_{0}^{\infty}\;\frac{\sin\,x}{x}\;\mathrm{d}x

Diffusion_equation.html

  1. ϕ ( 𝐫 , t ) t = D 2 ϕ ( 𝐫 , t ) , \frac{\partial\phi(\mathbf{r},t)}{\partial t}=D\nabla^{2}\phi(\mathbf{r},t),
  2. ϕ t + 𝐣 = 0 , \frac{\partial\phi}{\partial t}+\nabla\cdot\mathbf{j}=0,
  3. 𝐣 = - D ( ϕ , 𝐫 ) ϕ ( 𝐫 , t ) . \mathbf{j}=-D(\phi,\mathbf{r})\,\nabla\phi(\mathbf{r},t).
  4. ϕ ( 𝐫 , t ) t = [ D ( ϕ , 𝐫 ) ] ϕ ( 𝐫 , t ) + tr [ D ( ϕ , 𝐫 ) ( T ϕ ( 𝐫 , t ) ) ] \frac{\partial\phi(\mathbf{r},t)}{\partial t}=\nabla\cdot\left[D(\phi,\mathbf{% r})\right]\nabla\phi(\mathbf{r},t)+{\rm tr}\Big[D(\phi,\mathbf{r})\big(\nabla% \nabla^{T}\phi(\mathbf{r},t)\big)\Big]

Digital_Signal_1.html

  1. ( 8 bits channel × 24 channels frame + 1 framing bit frame ) × 8 000 frames second \displaystyle\left(8\,\frac{\mathrm{bits}}{\mathrm{channel}}\times 24\,\frac{% \mathrm{channels}}{\mathrm{frame}}+1\,\frac{\mathrm{framing\ bit}}{\mathrm{% frame}}\right)\times 8\,000\,\frac{\mathrm{frames}}{\mathrm{second}}

Digital_watermarking.html

  1. ( m = m 1 m n , n 𝒩 \left(m=m_{1}\ldots m_{n},\;n\in\mathcal{N}\right.
  2. n = | m | ) \left.n=|m|\right)
  3. M = { 0 , 1 } n M=\{0,1\}^{n}

DIIS.html

  1. m m
  2. 𝐞 m + 1 = i = 1 m c i 𝐞 i . \mathbf{e}_{m+1}=\sum_{i=1}^{m}\ c_{i}\mathbf{e}_{i}.
  3. 𝐩 \displaystyle\mathbf{p}
  4. λ λ
  5. L = 𝐞 m + 1 2 - λ ( i c i - 1 ) , = i j c j B j i c i - λ ( i c i - 1 ) , where B i j = 𝐞 j , 𝐞 i . \begin{aligned}\displaystyle L&\displaystyle=\left\|\mathbf{e}_{m+1}\right\|^{% 2}-\lambda\left(\sum_{i}\ c_{i}-1\right),\\ &\displaystyle=\sum_{ij}c_{j}B_{ji}c_{i}-\lambda\left(\sum_{i}\ c_{i}-1\right)% ,\,\text{ where }B_{ij}=\langle\mathbf{e}_{j},\mathbf{e}_{i}\rangle.\end{aligned}
  6. L L
  7. ( m + 1 ) (m+1)
  8. m m
  9. 𝐩 m + 1 = i = 1 m c i 𝐩 i . \mathbf{p}_{m+1}=\sum_{i=1}^{m}c_{i}\mathbf{p}_{i}.

Dilatant.html

  1. η = K γ ˙ n - 1 \eta=K\dot{\gamma}^{n-1}
  2. V = π R ( - H 12 π h 2 + 64 C k T Γ 2 e κ h κ 2 ) V=\pi R\left(\frac{-H}{12\pi h^{2}}+\frac{64CkT\Gamma^{2}e^{\kappa}h}{\kappa^{% 2}}\right)
  3. Γ \Gamma
  4. κ \kappa

Dilation_(morphology).html

  1. A B = b B A b . A\oplus B=\bigcup_{b\in B}A_{b}.
  2. A B = B A = a A B a A\oplus B=B\oplus A=\bigcup_{a\in A}B_{a}
  3. A B = { z E ( B s ) z A } A\oplus B=\{z\in E\mid(B^{s})_{z}\cap A\neq\varnothing\}
  4. B s = { x E - x B } B^{s}=\{x\in E\mid-x\in B\}
  5. A C A\subseteq C
  6. A B C B A\oplus B\subseteq C\oplus B
  7. A A B A\subseteq A\oplus B
  8. ( A B ) C = A ( B C ) (A\oplus B)\oplus C=A\oplus(B\oplus C)
  9. { , - } \mathbb{R}\cup\{\infty,-\infty\}
  10. \mathbb{R}
  11. \infty
  12. - -\infty
  13. ( f b ) ( x ) = sup y E [ f ( y ) + b ( x - y ) ] , (f\oplus b)(x)=\sup_{y\in E}[f(y)+b(x-y)],
  14. b ( x ) = { 0 , x B , - , otherwise , b(x)=\left\{\begin{array}[]{ll}0,&x\in B,\\ -\infty,&\,\text{otherwise},\end{array}\right.
  15. B E B\subseteq E
  16. ( f b ) ( x ) = sup y E [ f ( y ) + b ( x - y ) ] = sup z E [ f ( x - z ) + b ( z ) ] = sup z B [ f ( x - z ) ] . (f\oplus b)(x)=\sup_{y\in E}[f(y)+b(x-y)]=\sup_{z\in E}[f(x-z)+b(z)]=\sup_{z% \in B}[f(x-z)].
  17. ( L , ) (L,\leq)
  18. \wedge
  19. \vee
  20. \varnothing
  21. { X i } \{X_{i}\}
  22. δ : L L \delta:L\rightarrow L
  23. i δ ( X i ) = δ ( i X i ) , \bigvee_{i}\delta(X_{i})=\delta\left(\bigvee_{i}X_{i}\right),
  24. δ ( ) = . \delta(\varnothing)=\varnothing.

Dilution_of_precision_(GPS).html

  1. G D O P = Δ ( Output Location ) Δ ( Measured Data ) GDOP=\frac{\Delta({\rm Output\ Location})}{\Delta({\rm Measured\ Data})}
  2. Δ ( Measured Data ) \Delta({\rm Measured\ Data})
  3. x x
  4. y y
  5. z z
  6. t t
  7. ( ( x i - x ) R i , ( y i - y ) R i , ( z i - z ) R i ) \scriptstyle\left(\frac{(x_{i}-x)}{R_{i}},\frac{(y_{i}-y)}{R_{i}},\frac{(z_{i}% -z)}{R_{i}}\right)
  8. R i = ( x i - x ) 2 + ( y i - y ) 2 + ( z i - z ) 2 \scriptstyle R_{i}=\sqrt{(x_{i}-x)^{2}+(y_{i}-y)^{2}+(z_{i}-z)^{2}}
  9. x , y \scriptstyle\ x,\ y
  10. z \scriptstyle\ z
  11. x i , y i \scriptstyle\ x_{i},y_{i}
  12. z i \scriptstyle\ z_{i}
  13. A = [ ( x 1 - x ) R 1 ( y 1 - y ) R 1 ( z 1 - z ) R 1 - 1 ( x 2 - x ) R 2 ( y 2 - y ) R 2 ( z 2 - z ) R 2 - 1 ( x 3 - x ) R 3 ( y 3 - y ) R 3 ( z 3 - z ) R 3 - 1 ( x 4 - x ) R 4 ( y 4 - y ) R 4 ( z 4 - z ) R 4 - 1 ] A=\begin{bmatrix}\frac{(x_{1}-x)}{R_{1}}&\frac{(y_{1}-y)}{R_{1}}&\frac{(z_{1}-% z)}{R_{1}}&-1\\ \frac{(x_{2}-x)}{R_{2}}&\frac{(y_{2}-y)}{R_{2}}&\frac{(z_{2}-z)}{R_{2}}&-1\\ \frac{(x_{3}-x)}{R_{3}}&\frac{(y_{3}-y)}{R_{3}}&\frac{(z_{3}-z)}{R_{3}}&-1\\ \frac{(x_{4}-x)}{R_{4}}&\frac{(y_{4}-y)}{R_{4}}&\frac{(z_{4}-z)}{R_{4}}&-1\end% {bmatrix}
  14. σ t \sigma_{t}
  15. σ t \sigma_{t}
  16. Q = ( A T A ) - 1 Q=\left(A^{T}A\right)^{-1}
  17. Q = [ σ x 2 σ x y σ x z σ x t σ x y σ y 2 σ y z σ y t σ x z σ y z σ z 2 σ z t σ x t σ y t σ z t σ t 2 ] Q=\begin{bmatrix}\sigma_{x}^{2}&\sigma_{xy}&\sigma_{xz}&\sigma_{xt}\\ \sigma_{xy}&\sigma_{y}^{2}&\sigma_{yz}&\sigma_{yt}\\ \sigma_{xz}&\sigma_{yz}&\sigma_{z}^{2}&\sigma_{zt}\\ \sigma_{xt}&\sigma_{yt}&\sigma_{zt}&\sigma_{t}^{2}\end{bmatrix}
  18. P D O P \displaystyle PDOP
  19. H D O P = σ x 2 + σ y 2 \scriptstyle HDOP=\sqrt{\sigma_{x}^{2}+\sigma_{y}^{2}}
  20. V D O P = σ z 2 \scriptstyle\ VDOP=\sqrt{\sigma_{z}^{2}}

Dilworth's_theorem.html

  1. P P
  2. P P
  3. P P
  4. P P
  5. a a
  6. P P
  7. k k
  8. P := P { a } P^{\prime}:=P\setminus\{a\}
  9. k k
  10. C 1 , , C k C_{1},\dots,C_{k}
  11. A 0 A_{0}
  12. k k
  13. A 0 C i A_{0}\cap C_{i}\neq\emptyset
  14. i = 1 , 2 , , k i=1,2,\dots,k
  15. i = 1 , 2 , , k i=1,2,\dots,k
  16. x i x_{i}
  17. C i C_{i}
  18. k k
  19. P P^{\prime}
  20. A := { x 1 , x 2 , , x k } A:=\{x_{1},x_{2},\dots,x_{k}\}
  21. A A
  22. A i A_{i}
  23. k k
  24. x i x_{i}
  25. i i
  26. j j
  27. A i C j A_{i}\cap C_{j}\neq\emptyset
  28. y A i C j y\in A_{i}\cap C_{j}
  29. y x j y\leq x_{j}
  30. x j x_{j}
  31. x i x j x_{i}\not\geq x_{j}
  32. x i y x_{i}\not\geq y
  33. i i
  34. j j
  35. x j x i x_{j}\not\geq x_{i}
  36. A A
  37. P P
  38. a x i a\geq x_{i}
  39. i { 1 , 2 , , k } i\in\{1,2,\dots,k\}
  40. K K
  41. { a } { z C i : z x i } \{a\}\cup\{z\in C_{i}:z\leq x_{i}\}
  42. x i x_{i}
  43. P K P\setminus K
  44. k k
  45. P K P\setminus K
  46. k - 1 k-1
  47. A { x i } A\setminus\{x_{i}\}
  48. k - 1 k-1
  49. P K P\setminus K
  50. P P
  51. k k
  52. a x i a\not\geq x_{i}
  53. i { 1 , 2 , , k } i\in\{1,2,\dots,k\}
  54. A { a } A\cup\{a\}
  55. k + 1 k+1
  56. P P
  57. a a
  58. P P
  59. P P
  60. k + 1 k+1
  61. { a } , C 1 , C 2 , , C k \{a\},C_{1},C_{2},\dots,C_{k}
  62. width ( B n ) = ( n n / 2 ) . \mbox{width}~{}(B_{n})={n\choose\lfloor{n/2}\rfloor}.

Diminishing_returns.html

  1. i {i}
  2. 1 2 i - 1 \frac{1}{2^{i-1}}
  3. D = 1 X i = 1 X 1 2 i - 1 \,\text{D}=\frac{1}{X}\sum_{i=1}^{X}\frac{1}{2^{i-1}}
  4. X \,\text{X}
  5. 1 2 i - 1 \frac{1}{2^{i-1}}
  6. i {i}
  7. X \,\text{X}
  8. D = 1 3 i = 1 3 1 2 i - 1 \,\text{D}=\frac{1}{3}\sum_{i=1}^{3}\frac{1}{2^{i-1}}
  9. = 1 3 ( 1 2 1 - 1 + 1 2 2 - 1 + 1 2 3 - 1 ) =\frac{1}{3}\cdot\left(\frac{1}{2^{1-1}}+\frac{1}{2^{2-1}}+\frac{1}{2^{3-1}}\right)
  10. = 1 3 ( 1 2 0 + 1 2 1 + 1 2 2 ) =\frac{1}{3}\cdot\left(\frac{1}{2^{0}}+\frac{1}{2^{1}}+\frac{1}{2^{2}}\right)
  11. = 1 3 ( 1 1 + 1 2 + 1 4 ) = 1.75 3 =\frac{1}{3}\cdot\left(\frac{1}{1}+\frac{1}{2}+\frac{1}{4}\right)=\frac{1.75}{3}
  12. = 0.58 3 ¯ t/kg =0.58\overline{3}\,\text{ t/kg}

Dini_test.html

  1. ω f ( δ ; t ) = max | ε | δ | f ( t ) - f ( t + ε ) | \left.\right.\omega_{f}(\delta;t)=\max_{|\varepsilon|\leq\delta}|f(t)-f(t+% \varepsilon)|
  2. ω f ( δ ) = max t ω f ( δ ; t ) \left.\right.\omega_{f}(\delta)=\max_{t}\omega_{f}(\delta;t)
  3. 0 π 1 δ ω f ( δ ; t ) d δ < . \int_{0}^{\pi}\frac{1}{\delta}\omega_{f}(\delta;t)\,d\delta<\infty.
  4. ω f = log - 2 ( δ - 1 ) \omega_{f}=\log^{-2}(\delta^{-1})
  5. log - 1 ( δ - 1 ) \log^{-1}(\delta^{-1})
  6. ω f ( δ ) = o ( log 1 δ ) - 1 . \omega_{f}(\delta)=o\left(\log\frac{1}{\delta}\right)^{-1}.
  7. ω f ( δ ) = O ( log 1 δ ) - 1 . \omega_{f}(\delta)=O\left(\log\frac{1}{\delta}\right)^{-1}.
  8. 0 π 1 δ Ω ( δ ) d δ = \int_{0}^{\pi}\frac{1}{\delta}\Omega(\delta)\,d\delta=\infty
  9. ω f ( δ ; 0 ) < Ω ( δ ) \left.\right.\omega_{f}(\delta;0)<\Omega(\delta)

Dinitz_conjecture.html

  1. χ l ( K n , n ) = n \chi^{\prime}_{l}(K_{n,n})=n
  2. n n
  3. K n , n K_{n,n}
  4. n n

Dirac_adjoint.html

  1. ψ ¯ ψ γ 0 \bar{\psi}\equiv\psi^{\dagger}\gamma^{0}
  2. λ λ - 1 \lambda^{\dagger}\neq\lambda^{-1}
  3. ψ λ ψ \psi\mapsto\lambda\psi
  4. ψ ψ λ \psi^{\dagger}\mapsto\psi^{\dagger}\lambda^{\dagger}
  5. ψ ¯ ( λ ψ ) γ 0 \bar{\psi}\mapsto\left(\lambda\psi\right)^{\dagger}\gamma^{0}
  6. ψ ¯ ψ ¯ λ - 1 \bar{\psi}\mapsto\bar{\psi}\lambda^{-1}
  7. J μ = c ψ ¯ γ μ ψ J^{\mu}=c\bar{\psi}\gamma^{\mu}\psi
  8. s y m b o l J = ( c ρ , s y m b o l j ) symbolJ=(c\rho,symbolj)
  9. ( γ 0 ) 2 = I \left(\gamma^{0}\right)^{2}=I
  10. ρ = ψ ψ \rho=\psi^{\dagger}\psi

Dirac_measure.html

  1. { x , y , z } \{x,y,z\}
  2. δ x \delta_{x}
  3. δ x ( A ) = 1 A ( x ) = { 0 , x A ; 1 , x A . \delta_{x}(A)=1_{A}(x)=\begin{cases}0,&x\not\in A;\\ 1,&x\in A.\end{cases}
  4. 1 A 1_{A}
  5. A A
  6. X f ( y ) d δ x ( y ) = f ( x ) , \int_{X}f(y)\,\mathrm{d}\delta_{x}(y)=f(x),
  7. X f ( y ) δ x ( y ) d y = f ( x ) , \int_{X}f(y)\delta_{x}(y)\,\mathrm{d}y=f(x),

Dirac_operator.html

  1. D 2 = Δ , D^{2}=\Delta,\,
  2. ψ ( x , y ) = [ χ ( x , y ) η ( x , y ) ] \psi(x,y)=\begin{bmatrix}\chi(x,y)\\ \eta(x,y)\end{bmatrix}
  3. D = - i σ x x - i σ y y , D=-i\sigma_{x}\partial_{x}-i\sigma_{y}\partial_{y},\,
  4. D = γ μ μ / , D=\gamma^{\mu}\partial_{\mu}\ \equiv\partial\!\!\!/,
  5. D = j = 1 n e j x j D=\sum_{j=1}^{n}e_{j}\frac{\partial}{\partial x_{j}}
  6. j = 1 n e j ( x ) Γ ~ e j ( x ) \sum_{j=1}^{n}e_{j}(x)\tilde{\Gamma}_{e_{j}(x)}
  7. Γ ~ \tilde{\Gamma}
  8. f ( x 1 , , x k ) ( x 1 ¯ f x 2 ¯ f x k ¯ f ) f(x_{1},\ldots,x_{k})\mapsto\begin{pmatrix}\partial_{\underline{x_{1}}}f\\ \partial_{\underline{x_{2}}}f\\ \ldots\\ \partial_{\underline{x_{k}}}f\\ \end{pmatrix}
  9. x i = ( x i 1 , x i 2 , , x i n ) x_{i}=(x_{i1},x_{i2},\ldots,x_{in})
  10. x i ¯ = j e j x i j \partial_{\underline{x_{i}}}=\sum_{j}e_{j}\cdot\partial_{x_{ij}}

Dirac_string.html

  1. 2 π 2\pi
  2. H 1 ( M ) H_{1}(M)
  3. F = d A F=dA
  4. d F = 0 dF=0
  5. d F = 0 dF=0

Direct_digital_synthesizer.html

  1. f c l k / f o f_{clk}/f_{o}
  2. f c l k / 2 f_{clk}/2

Direct_methanol_fuel_cell.html

  1. CH 3 OH + H 2 O 6 H + + 6 e - + CO 2 \mathrm{CH_{3}OH+H_{2}O\to 6\ H^{+}+6\ e^{-}+CO_{2}}
  2. 3 2 O 2 + 6 H + + 6 e - 3 H 2 O \mathrm{\frac{3}{2}O_{2}+6\ H^{+}+6\ e^{-}\to 3\ H_{2}O}
  3. CH 3 OH + 3 2 O 2 2 H 2 O + CO 2 \mathrm{CH_{3}OH+\frac{3}{2}O_{2}\to 2\ H_{2}O+CO_{2}}
  4. CH 3 OH + 6 OH - 5 H 2 O + 6 e - + CO 2 \mathrm{CH_{3}OH+6\ OH^{-}\to 5\ H_{2}O+6\ e^{-}+CO_{2}}
  5. 3 2 O 2 + 3 H 2 O + 6 e - 6 OH - \mathrm{\frac{3}{2}O_{2}+3\ H_{2}O+6\ e^{-}\to 6\ OH^{-}}
  6. CH 3 OH + 3 2 O 2 2 H 2 O + CO 2 \mathrm{CH_{3}OH+\frac{3}{2}O_{2}\to 2\ H_{2}O+CO_{2}}

Dirichlet_distribution.html

  1. Dir ( s y m b o l α ) \operatorname{Dir}(symbol\alpha)
  2. s y m b o l α symbol\alpha
  3. x i x_{i}
  4. α i - 1 \alpha_{i}-1
  5. α i \alpha_{i}
  6. f ( x 1 , , x K ; α 1 , , α K ) = 1 B ( α ) i = 1 K x i α i - 1 , f\left(x_{1},\cdots,x_{K};\alpha_{1},\cdots,\alpha_{K}\right)=\frac{1}{\mathrm% {B}(\alpha)}\prod_{i=1}^{K}x_{i}^{\alpha_{i}-1},
  7. x 1 , , x K - 1 > 0 \displaystyle x_{1},\cdots,x_{K-1}>0
  8. B ( s y m b o l α ) = i = 1 K Γ ( α i ) Γ ( i = 1 K α i ) , \qquadsymbol α = ( α 1 , , α K ) . \mathrm{B}(symbol\alpha)=\frac{\prod_{i=1}^{K}\Gamma(\alpha_{i})}{\Gamma\left(% \sum_{i=1}^{K}\alpha_{i}\right)},\qquadsymbol{\alpha}=(\alpha_{1},\cdots,% \alpha_{K}).
  9. s y m b o l x symbolx
  10. s y m b o l x 1 = 1 \|symbolx\|_{1}=1
  11. s y m b o l α symbol\alpha
  12. f ( x 1 , , x K - 1 ; α ) = Γ ( α K ) Γ ( α ) K i = 1 K x i α - 1 . f(x_{1},\dots,x_{K-1};\alpha)=\frac{\Gamma(\alpha K)}{\Gamma(\alpha)^{K}}\prod% _{i=1}^{K}x_{i}^{\alpha-1}.
  13. α s y m b o l n \alpha symboln
  14. s y m b o l n = ( n 1 , , n K ) symboln=(n_{1},\dots,n_{K})
  15. s y m b o l n symboln
  16. n i n_{i}
  17. X = ( X 1 , , X K ) Dir ( α ) X=(X_{1},\ldots,X_{K})\sim\operatorname{Dir}(\alpha)
  18. X K = 1 - i = 1 K - 1 X i X_{K}=1-\sum_{i=1}^{K-1}X_{i}
  19. α 0 = i = 1 K α i . \alpha_{0}=\sum_{i=1}^{K}\alpha_{i}.
  20. E [ X i ] = α i α 0 , \mathrm{E}[X_{i}]=\frac{\alpha_{i}}{\alpha_{0}},
  21. Var [ X i ] = α i ( α 0 - α i ) α 0 2 ( α 0 + 1 ) . \mathrm{Var}[X_{i}]=\frac{\alpha_{i}(\alpha_{0}-\alpha_{i})}{\alpha_{0}^{2}(% \alpha_{0}+1)}.
  22. i j i\neq j
  23. Cov [ X i , X j ] = - α i α j α 0 2 ( α 0 + 1 ) . \mathrm{Cov}[X_{i},X_{j}]=\frac{-\alpha_{i}\alpha_{j}}{\alpha_{0}^{2}(\alpha_{% 0}+1)}.
  24. E [ i = 1 K x i β i ] = B ( s y m b o l α + s y m b o l β ) B ( s y m b o l α ) = Γ ( i = 1 K α i ) Γ ( i = 1 K α i + β i ) × i = 1 K Γ ( α i + β i ) Γ ( α i ) . E\left[\prod_{i=1}^{K}x_{i}^{\beta_{i}}\right]=\frac{B\left(symbol{\alpha}+% symbol{\beta}\right)}{B\left(symbol{\alpha}\right)}=\frac{\Gamma\left(\sum_{i=% 1}^{K}\alpha_{i}\right)}{\Gamma\left(\sum_{i=1}^{K}\alpha_{i}+\beta_{i}\right)% }\times\prod_{i=1}^{K}\frac{\Gamma\left(\alpha_{i}+\beta_{i}\right)}{\Gamma% \left(\alpha_{i}\right)}.
  25. x i = α i - 1 α 0 - K , α i > 1. x_{i}=\frac{\alpha_{i}-1}{\alpha_{0}-K},\qquad\alpha_{i}>1.
  26. X i Beta ( α i , ( j = 1 K α j ) - α i ) . X_{i}\sim\operatorname{Beta}\left(\alpha_{i},\left(\sum\nolimits_{j=1}^{K}{% \alpha_{j}}\right)-\alpha_{i}\right).
  27. s y m b o l α = ( α 1 , , α K ) = concentration hyperparameter 𝐩 \midsymbol α = ( p 1 , , p K ) Dir ( K , s y m b o l α ) 𝕏 𝐩 = ( 𝐱 1 , , 𝐱 K ) Cat ( K , 𝐩 ) \begin{array}[]{rcccl}symbol\alpha&=&\left(\alpha_{1},\cdots,\alpha_{K}\right)% &=&\,\text{concentration hyperparameter}\\ \mathbf{p}\midsymbol\alpha&=&\left(p_{1},\cdots,p_{K}\right)&\sim&% \operatorname{Dir}(K,symbol\alpha)\\ \mathbb{X}\mid\mathbf{p}&=&\left(\mathbf{x}_{1},\cdots,\mathbf{x}_{K}\right)&% \sim&\operatorname{Cat}(K,\mathbf{p})\end{array}
  28. 𝐜 = ( c 1 , , c K ) = number of occurrences of category i 𝐩 𝕏 , s y m b o l α Dir ( K , 𝐜 + s y m b o l α ) = Dir ( K , c 1 + α 1 , , c K + α K ) \begin{array}[]{rcccl}\mathbf{c}&=&\left(c_{1},\cdots,c_{K}\right)&=&\,\text{% number of occurrences of category }i\\ \mathbf{p}\mid\mathbb{X},symbol\alpha&\sim&\operatorname{Dir}(K,\mathbf{c}+% symbol\alpha)&=&\operatorname{Dir}\left(K,c_{1}+\alpha_{1},\cdots,c_{K}+\alpha% _{K}\right)\end{array}
  29. log ( X i ) \log(X_{i})
  30. E [ log ( X i ) ] = ψ ( α i ) - ψ ( α 0 ) \operatorname{E}[\log(X_{i})]=\psi(\alpha_{i})-\psi(\alpha_{0})
  31. Cov [ log ( X i ) , log ( X j ) ] = ψ ( α i ) δ i j - ψ ( α 0 ) \operatorname{Cov}[\log(X_{i}),\log(X_{j})]=\psi^{\prime}(\alpha_{i})\delta_{% ij}-\psi^{\prime}(\alpha_{0})
  32. ψ \psi
  33. ψ \psi^{\prime}
  34. δ i j \delta_{ij}
  35. E [ log ( X i ) ] \operatorname{E}[\log(X_{i})]
  36. H ( X ) = log B ( s y m b o l α ) + ( α 0 - K ) ψ ( α 0 ) - j = 1 K ( α j - 1 ) ψ ( α j ) H(X)=\log\mathrm{B}(symbol\alpha)+(\alpha_{0}-K)\psi(\alpha_{0})-\sum_{j=1}^{K% }(\alpha_{j}-1)\psi(\alpha_{j})
  37. λ = 1 \lambda=1
  38. F R ( λ ) = ( 1 - λ ) - 1 ( - log B ( α ) + j = 1 K log Γ ( λ ( α i - 1 ) + 1 ) - log Γ ( λ ( α 0 - d ) + d ) ) F_{R}(\lambda)=(1-\lambda)^{-1}\left(-\log\mathrm{B}(\alpha)+\sum_{j=1}^{K}% \log\Gamma(\lambda(\alpha_{i}-1)+1)-\log\Gamma(\lambda(\alpha_{0}-d)+d)\right)
  39. λ \lambda
  40. X = ( X 1 , , X K ) Dir ( α 1 , , α K ) X=(X_{1},\cdots,X_{K})\sim\operatorname{Dir}(\alpha_{1},\cdots,\alpha_{K})
  41. X = ( X 1 , , X i + X j , , X K ) Dir ( α 1 , , α i + α j , , α K ) . X^{\prime}=(X_{1},\cdots,X_{i}+X_{j},\cdots,X_{K})\sim\operatorname{Dir}\left(% \alpha_{1},\cdots,\alpha_{i}+\alpha_{j},\cdots,\alpha_{K}\right).
  42. X i X_{i}
  43. X = ( X 1 , , X K ) Dir ( α ) X=(X_{1},\ldots,X_{K})\sim\operatorname{Dir}(\alpha)
  44. X ( - K ) X^{(-K)}
  45. X ( - K ) = ( X 1 1 - X K , X 2 1 - X K , , X K - 1 1 - X K ) , X^{(-K)}=\left(\frac{X_{1}}{1-X_{K}},\frac{X_{2}}{1-X_{K}},\cdots,\frac{X_{K-1% }}{1-X_{K}}\right),
  46. X 2 , , X K - 1 X_{2},\ldots,X_{K-1}
  47. C F ( s 1 , , s k - 1 ) = 𝔼 ( e i ( s 1 x 1 + + s k - 1 x k - 1 ) ) = Ψ [ k - 1 ] ( α 1 , , α k ; α ; i s 1 , i s k - 1 ) CF\left(s_{1},\ldots,s_{k-1}\right)=\mathbb{E}\left(e^{i\left(s_{1}x_{1}+% \cdots+s_{k-1}x_{k-1}\right)}\right)=\Psi^{\left[k-1\right]}\left(\alpha_{1},% \ldots,\alpha_{k};\alpha;is_{1},\ldots is_{k-1}\right)
  48. Ψ [ m ] ( a 1 , , a m ; c ; z 1 , z m ) = ( a ) k 1 ( a m ) k m z 1 a 1 z m a m ( c ) k k 1 ! k m ! . \Psi^{\left[m\right]}\left(a_{1},\ldots,a_{m};c;z_{1},\ldots z_{m}\right)=\sum% \frac{\left(a\right)_{k_{1}}\cdots\left(a_{m}\right)_{k_{m}}\,z_{1}^{a_{1}}% \cdots z_{m}^{a^{m}}}{\left(c\right)_{k}\,k_{1}!\cdots k_{m}!}.
  49. k 1 , , k m k_{1},\ldots,k_{m}
  50. k = k 1 + + k m k=k_{1}+\cdots+k_{m}
  51. Ψ [ m ] = Γ ( c ) 2 π i L e t t a 1 + + a m - c j = 1 m ( t - z j ) - a j d t \Psi^{\left[m\right]}=\frac{\Gamma(c)}{2\pi i}\int_{L}e^{t}\,t^{a_{1}+\cdots+a% _{m}-c}\,\prod_{j=1}^{m}\left(t-z_{j}\right)^{-a_{j}}\,dt
  52. - -\infty
  53. - -\infty
  54. Y 1 Gamma ( α 1 , θ ) , , Y K Gamma ( α K , θ ) Y_{1}\sim\operatorname{Gamma}(\alpha_{1},\theta),\cdots,Y_{K}\sim\operatorname% {Gamma}(\alpha_{K},\theta)
  55. V = i = 1 K Y i Gamma ( i = 1 K α i , θ ) , V=\sum_{i=1}^{K}Y_{i}\sim\operatorname{Gamma}\left(\sum_{i=1}^{K}\alpha_{i},% \theta\right),
  56. X = ( X 1 , , X K ) = ( Y 1 V , , Y K V ) Dir ( α 1 , , α K ) . X=(X_{1},\cdots,X_{K})=\left(\frac{Y_{1}}{V},\cdots,\frac{Y_{K}}{V}\right)\sim% \operatorname{Dir}\left(\alpha_{1},\cdots,\alpha_{K}\right).
  57. x = ( x 1 , , x K ) x=(x_{1},\cdots,x_{K})
  58. ( α 1 , , α K ) (\alpha_{1},\cdots,\alpha_{K})
  59. y 1 , , y K y_{1},\ldots,y_{K}
  60. Gamma ( α i , 1 ) = y i α i - 1 e - y i Γ ( α i ) , \textrm{Gamma}(\alpha_{i},1)=\frac{y_{i}^{\alpha_{i}-1}\;e^{-y_{i}}}{\Gamma(% \alpha_{i})},\!
  61. x i = y i j = 1 K y j . x_{i}=\frac{y_{i}}{\sum_{j=1}^{K}y_{j}}.
  62. x 1 x_{1}
  63. Beta ( α 1 , i = 2 K α i ) \textrm{Beta}\left(\alpha_{1},\sum_{i=2}^{K}\alpha_{i}\right)
  64. x 2 , , x K - 1 x_{2},\ldots,x_{K-1}
  65. j = 2 , , K - 1 j=2,\ldots,K-1
  66. ϕ j \phi_{j}
  67. Beta ( α j , i = j + 1 K α i ) , \textrm{Beta}\left(\alpha_{j},\sum_{i=j+1}^{K}\alpha_{i}\right),
  68. x j = ( 1 - i = 1 j - 1 x i ) ϕ j . x_{j}=\left(1-\sum_{i=1}^{j-1}x_{i}\right)\phi_{j}.
  69. x K = 1 - i = 1 K - 1 x i . x_{K}=1-\sum_{i=1}^{K-1}x_{i}.

Discharge_(hydrology).html

  1. u ¯ \bar{u}
  2. Q = A u ¯ Q=A\,\bar{u}
  3. Q Q
  4. A A
  5. u ¯ \bar{u}

Discrete_valuation.html

  1. ν : K { } \nu:K\to\mathbb{Z}\cup\{\infty\}
  2. ν ( x y ) = ν ( x ) + ν ( y ) \nu(x\cdot y)=\nu(x)+\nu(y)
  3. ν ( x + y ) min { ν ( x ) , ν ( y ) } \nu(x+y)\geq\min\big\{\nu(x),\nu(y)\big\}
  4. ν ( x ) = x = 0 \nu(x)=\infty\iff x=0
  5. x , y K x,y\in K
  6. 0 , 0,\infty
  7. ν \nu
  8. 𝒪 K := { x K ν ( x ) 0 } \mathcal{O}_{K}:=\left\{x\in K\mid\nu(x)\geq 0\right\}
  9. K K
  10. ν : A \Z { } \nu:A\rightarrow\Z\cup\{\infty\}
  11. A A
  12. K = Quot ( A ) K=\,\text{Quot}(A)
  13. 𝒪 K \mathcal{O}_{K}
  14. A A
  15. p p
  16. x x\in\mathbb{Q}
  17. x = p j a b x=p^{j}\frac{a}{b}
  18. j , a , b \Z j,a,b\in\Z
  19. p p
  20. a , b a,b
  21. ν ( x ) = j \nu(x)=j
  22. \Q \Q
  23. X X
  24. K = M ( X ) K=M(X)
  25. X \C { } X\to\C\cup\{\infty\}
  26. p X p\in X
  27. K K
  28. ν ( f ) = j \nu(f)=j
  29. j j
  30. f ( z ) / ( z - p ) j f(z)/(z-p)^{j}
  31. p p
  32. ν ( f ) = j > 0 \nu(f)=j>0
  33. f f
  34. j j
  35. p p
  36. ν ( f ) = j < 0 \nu(f)=j<0
  37. f f
  38. - j -j
  39. p p
  40. p p

Discrete_valuation_ring.html

  1. | x - y | = 2 - ν ( x - y ) |x-y|=2^{-\nu(x-y)}

Discrete_wavelet_transform.html

  1. 2 n 2^{n}
  2. 2 n - 1 2^{n}-1
  3. 2 d 2^{d}
  4. O ( n ) O(n)
  5. [ 1 1 1 1 1 - i - 1 i 1 - 1 1 - 1 1 i - 1 - i ] \begin{bmatrix}1&1&1&1\\ 1&-i&-1&i\\ 1&-1&1&-1\\ 1&i&-1&-i\end{bmatrix}
  6. [ 1 1 1 1 1 1 - 1 - 1 1 - 1 0 0 0 0 1 - 1 ] \begin{bmatrix}1&1&1&1\\ 1&1&-1&-1\\ 1&-1&0&0\\ 0&0&1&-1\end{bmatrix}
  7. ( 1 , 0 , 0 , 0 ) \displaystyle(1,0,0,0)
  8. ( 1 4 , 1 4 , 1 4 , 1 4 ) \displaystyle\left(\frac{1}{4},\frac{1}{4},\frac{1}{4},\frac{1}{4}\right)
  9. ( 1 4 , 1 4 , 1 4 , 1 4 ) \displaystyle\left(\frac{1}{4},\frac{1}{4},\frac{1}{4},\frac{1}{4}\right)
  10. 1 / 4 1/4
  11. 1 / 2 1/2
  12. x x
  13. g g
  14. y [ n ] = ( x * g ) [ n ] = k = - x [ k ] g [ n - k ] y[n]=(x*g)[n]=\sum\limits_{k=-\infty}^{\infty}{x[k]g[n-k]}
  15. h h
  16. y low [ n ] = k = - x [ k ] h [ 2 n - k ] y_{\mathrm{low}}[n]=\sum\limits_{k=-\infty}^{\infty}{x[k]h[2n-k]}
  17. y high [ n ] = k = - x [ k ] g [ 2 n - k ] y_{\mathrm{high}}[n]=\sum\limits_{k=-\infty}^{\infty}{x[k]g[2n-k]}
  18. \downarrow
  19. ( y k ) [ n ] = y [ k n ] (y\downarrow k)[n]=y[kn]
  20. y low = ( x * g ) 2 y_{\mathrm{low}}=(x*g)\downarrow 2
  21. y high = ( x * h ) 2 y_{\mathrm{high}}=(x*h)\downarrow 2
  22. x * g x*g
  23. 2 n 2^{n}
  24. n n
  25. f n f_{n}
  26. 0
  27. < m t p l > f n / 8 <mtpl>{{f_{n}}}/8
  28. < m t p l > f n / 8 <mtpl>{{f_{n}}}/8
  29. < m t p l > f n / 4 <mtpl>{{f_{n}}}/4
  30. < m t p l > f n / 4 <mtpl>{{f_{n}}}/4
  31. < m t p l > f n / 2 <mtpl>{{f_{n}}}/2
  32. < m t p l > f n / 2 <mtpl>{{f_{n}}}/2
  33. f n f_{n}
  34. ψ ( t ) \psi(t)
  35. ψ j , k ( t ) = 1 2 j ψ ( t - k 2 j 2 j ) \psi_{j,k}(t)=\frac{1}{\sqrt{2^{j}}}\psi\left(\frac{t-k2^{j}}{2^{j}}\right)
  36. j j
  37. k k
  38. γ \gamma
  39. x ( t ) x(t)
  40. x ( t ) x(t)
  41. x ( t ) x(t)
  42. 2 N 2^{N}
  43. γ j k = - x ( t ) 1 2 j ψ ( t - k 2 j 2 j ) d t \gamma_{jk}=\int_{-\infty}^{\infty}x(t)\frac{1}{\sqrt{2^{j}}}\psi\left(\frac{t% -k2^{j}}{2^{j}}\right)dt
  44. j j
  45. γ j k \gamma_{jk}
  46. k k
  47. γ j k \gamma_{jk}
  48. x ( t ) x(t)
  49. h ( t ) = 1 2 j ψ ( - t 2 j ) h(t)=\frac{1}{\sqrt{2^{j}}}\psi\left(\frac{-t}{2^{j}}\right)
  50. 1 , 2 j , 2 2 j , , 2 N 1,2^{j},2^{2j},...,2^{N}
  51. j j
  52. h [ n ] h[n]
  53. g [ n ] g[n]
  54. ψ ( t ) \psi(t)
  55. ψ = [ 1 , - 1 ] \psi=[1,-1]
  56. h [ n ] = 1 2 [ - 1 , 1 ] h[n]=\frac{1}{\sqrt{2}}[-1,1]
  57. g [ n ] g[n]
  58. h [ n ] h[n]
  59. x * h x*h
  60. x * g x*g
  61. g [ n ] g[n]
  62. T ( N ) = 2 N + T ( N / 2 ) T(N)=2N+T(N/2)
  63. h [ n ] h[n]
  64. g [ n ] g[n]
  65. h [ n ] = [ - 2 2 , 2 2 ] g [ n ] = [ 2 2 , 2 2 ] h[n]=\left[\frac{-\sqrt{2}}{2},\frac{\sqrt{2}}{2}\right]g[n]=\left[\frac{\sqrt% {2}}{2},\frac{\sqrt{2}}{2}\right]

Disdyakis_dodecahedron.html

  1. arccos ( - 71 + 12 2 97 ) \arccos(-\frac{71+12\sqrt{2}}{97})
  2. 6 7 783 + 436 2 \tfrac{6}{7}\scriptstyle{\sqrt{783+436\sqrt{2}}}
  3. 1 7 3 ( 2194 + 1513 2 ) \tfrac{1}{7}\scriptstyle{\sqrt{3(2194+1513\sqrt{2})}}
  4. n 7. n\geq 7.

Disjoint-set_data_structure.html

  1. O ( n log ( n ) ) O(n\log(n))
  2. n n
  3. n n
  4. A A
  5. B B
  6. A A
  7. B B
  8. B B
  9. A A
  10. B B
  11. L L
  12. x x
  13. x x
  14. x x
  15. x x
  16. n n
  17. x x
  18. n n
  19. log 2 ( n ) \log_{2}(n)
  20. log 2 ( n ) \log_{2}(n)
  21. n n
  22. O ( n log ( n ) ) O(n\log(n))
  23. O ( 1 ) O(1)
  24. O ( log n ) O(\log n)
  25. O ( α ( n ) ) O(\alpha(n))
  26. α ( n ) \alpha(n)
  27. n = f ( x ) = A ( x , x ) n=f(x)=A(x,x)
  28. A A
  29. α ( n ) \alpha(n)
  30. α ( n ) \alpha(n)
  31. n n
  32. Ω ( α ( n ) ) \Omega(\alpha(n))

Disjunction_and_existence_properties.html

  1. ( x ) φ ( x ) (\exists x\in\mathbb{N})\varphi(x)
  2. φ ( n ¯ ) \varphi(\bar{n})
  3. n n\in\mathbb{N}
  4. n ¯ \bar{n}
  5. T T
  6. ( x ) ( y ) φ ( x , y ) (\forall x\in\mathbb{N})(\exists y\in\mathbb{N})\varphi(x,y)
  7. f e f_{e}
  8. ( x ) φ ( x , f e ( x ) ) (\forall x)\varphi(x,f_{e}(x))
  9. ( f : ) ψ ( f ) (\exists f\colon\mathbb{N}\to\mathbb{N})\psi(f)
  10. f e f_{e}
  11. ψ ( f e ) \psi(f_{e})
  12. \mathbb{N}
  13. \mathbb{N}
  14. 𝟏 \mathbf{1}
  15. 𝟏 \mathbf{1}
  16. A B ( n ) [ ( n = 0 A ) ( n 0 B ) ] A\vee B\equiv(\exists n)[(n=0\to A)\wedge(n\neq 0\to B)]
  17. A B A\vee B
  18. T T
  19. n : ( n = 0 A ) ( n 0 B ) \exists n\colon(n=0\to A)\wedge(n\neq 0\to B)
  20. s s
  21. ( s ¯ = 0 A ) ( s ¯ 0 B ) (\bar{s}=0\to A)\wedge(\bar{s}\neq 0\to B)
  22. s ¯ \bar{s}
  23. s s
  24. s = 0 s=0
  25. A A
  26. s 0 s\neq 0
  27. B B

Disk_algebra.html

  1. A ( 𝐃 ) = H ( 𝐃 ) C ( 𝐃 ¯ ) , A(\mathbf{D})=H^{\infty}(\mathbf{D})\cap C(\overline{\mathbf{D}}),
  2. H ( 𝐃 ) H^{\infty}(\mathbf{D})
  3. f = sup { | f ( z ) | z 𝐃 } = max { | f ( z ) | z 𝐃 ¯ } , \|f\|=\sup\{|f(z)|\mid z\in\mathbf{D}\}=\max\{|f(z)|\mid z\in\overline{\mathbf% {D}}\},

Dislocation.html

  1. σ x x = - μ b 2 π ( 1 - ν ) y ( 3 x 2 + y 2 ) ( x 2 + y 2 ) 2 \sigma_{xx}=\frac{-\mu b}{2\pi(1-\nu)}\frac{y(3x^{2}+y^{2})}{(x^{2}+y^{2})^{2}}
  2. σ y y = μ b 2 π ( 1 - ν ) y ( x 2 - y 2 ) ( x 2 + y 2 ) 2 \sigma_{yy}=\frac{\mu b}{2\pi(1-\nu)}\frac{y(x^{2}-y^{2})}{(x^{2}+y^{2})^{2}}
  3. τ x y = μ b 2 π ( 1 - ν ) x ( x 2 - y 2 ) ( x 2 + y 2 ) 2 \tau_{xy}=\frac{\mu b}{2\pi(1-\nu)}\frac{x(x^{2}-y^{2})}{(x^{2}+y^{2})^{2}}
  4. τ r = - μ b 2 π r \tau_{r}=\frac{-\mu b}{2\pi r}
  5. τ ρ 1 / 2 \tau\propto\rho^{1/2}
  6. τ h o m G = 7.4 × 10 - 2 \frac{\tau_{hom}}{G}=7.4\times 10^{-2}
  7. τ h o m \tau_{hom}\,\!
  8. τ m = G 2 π . \tau_{m}=\frac{G}{2\pi\ }.\,

Dispersion_relation.html

  1. λ \lambda
  2. v = v ( λ ) . v=v(\lambda).\,
  3. v ( λ ) = λ f ( λ ) . v(\lambda)=\lambda\ f(\lambda).\,
  4. ω = 2 π f \omega=2\pi f
  5. k = 2 π / λ k=2\pi/\lambda
  6. ω ( k ) = v ( k ) k . \omega(k)=v(k)\ k.\,
  7. A ( x , t ) = A 0 e 2 π i x - v t λ = A 0 e i ( k x - ω t ) , A(x,t)=A_{0}e^{2\pi i\frac{x-vt}{\lambda}}=A_{0}e^{i(kx-\omega t)},
  8. ω = c k . \omega=ck.\,
  9. v = ω k = d ω d k = c ; v=\frac{\omega}{k}=\frac{d\omega}{dk}=c;
  10. E 2 = ( m c 2 ) 2 + ( p c ) 2 E^{2}=(mc^{2})^{2}+(pc)^{2}\,
  11. E = p c E=pc\,
  12. E = m c 2 + p 2 2 m E=mc^{2}+\frac{p^{2}}{2m}
  13. m m
  14. m c 2 mc^{2}
  15. p 2 / 2 m p^{2}/2m
  16. p = m v p=mv
  17. E = ω , p = k . E=\hbar\omega,\quad p=\hbar k.
  18. ω = k 2 2 m . \omega=\frac{\hbar k^{2}}{2m}.
  19. ω k \frac{\partial\omega}{\partial k}
  20. ω = g k , \omega=\sqrt{gk},
  21. v p = ω k = g k v_{p}=\frac{\omega}{k}=\sqrt{\frac{g}{k}}
  22. v g = d ω d k = 1 2 v p . v_{g}=\frac{d{\omega}}{dk}=\frac{1}{2}v_{p}.
  23. ω = k T μ \omega=k\sqrt{\frac{T}{\mu}}
  24. ω 2 = T μ k 2 + α k 4 \omega^{2}=\frac{T}{\mu}k^{2}+\alpha k^{4}
  25. α \alpha

Displacement_(vector).html

  1. s y m b o l s = s y m b o l R f - R i = \Deltasymbol R symbol{s}=symbol{R_{f}-R_{i}}=\Deltasymbol{R}
  2. s y m b o l v = d s y m b o l s d t symbol{v}=\frac{\,\text{d}symbol{s}}{\,\text{d}t}
  3. s y m b o l a = d s y m b o l v d t = d 2 s y m b o l s d t 2 symbol{a}=\frac{\,\text{d}symbol{v}}{\,\text{d}t}=\frac{\,\text{d}^{2}symbol{s% }}{\,\text{d}t^{2}}
  4. s y m b o l j = d s y m b o l a d t = d 2 s y m b o l v d t 2 = d 3 s y m b o l s d t 3 symbol{j}=\frac{\,\text{d}symbol{a}}{\,\text{d}t}=\frac{\,\text{d}^{2}symbol{v% }}{\,\text{d}t^{2}}=\frac{\,\text{d}^{3}symbol{s}}{\,\text{d}t^{3}}

Dissipation_factor.html

  1. ESR = σ ε ω 2 C \,\text{ESR}=\frac{\sigma}{\varepsilon\omega^{2}C}
  2. σ \sigma
  3. ω \omega
  4. ε \varepsilon
  5. C C
  6. DF = i 2 ESR i 2 | X c | = ω C ESR = σ ε ω = 1 Q \,\text{DF}=\frac{i^{2}\,\text{ESR}}{i^{2}|X_{c}|}=\omega C\cdot\,\text{ESR}=% \frac{\sigma}{\varepsilon\omega}=\frac{1}{Q}
  7. tan δ = ESR | X c | = DF \tan\delta=\frac{\,\text{ESR}}{\left|X_{c}\right|}=\,\text{DF}
  8. ESR \,\text{ESR}
  9. X c X_{c}

Distance_(graph_theory).html

  1. d ( u , v ) d(u,v)
  2. u u
  3. v v
  4. u u
  5. v v
  6. d ( u , v ) d(u,v)
  7. d ( v , u ) d(v,u)
  8. ϵ ( v ) \epsilon(v)
  9. v v
  10. v v
  11. r r
  12. r = min v V ϵ ( v ) r=\min_{v\in V}\epsilon(v)
  13. d d
  14. d d
  15. d = max v V ϵ ( v ) d=\max_{v\in V}\epsilon(v)
  16. r r
  17. r r
  18. v v
  19. ϵ ( v ) = r \epsilon(v)=r
  20. d d
  21. d d
  22. v v
  23. ϵ ( v ) = d \epsilon(v)=d
  24. v v
  25. u u
  26. v v
  27. u u
  28. u u
  29. v v
  30. d ( u , v ) = ϵ ( u ) d(u,v)=\epsilon(u)
  31. ϵ ( u ) = ϵ ( v ) \epsilon(u)=\epsilon(v)
  32. u u
  33. u u
  34. v v
  35. ϵ ( v ) > ϵ ( u ) \epsilon(v)>\epsilon(u)
  36. u = v u=v
  37. v v

Distance_geometry.html

  1. det [ 0 d ( A B ) 2 d ( A C ) 2 1 d ( A B ) 2 0 d ( B C ) 2 1 d ( A C ) 2 d ( B C ) 2 0 1 1 1 1 0 ] = 0 , \det\begin{bmatrix}0&d(AB)^{2}&d(AC)^{2}&1\\ d(AB)^{2}&0&d(BC)^{2}&1\\ d(AC)^{2}&d(BC)^{2}&0&1\\ 1&1&1&0\end{bmatrix}=0,
  2. det [ 0 d ( A B ) 2 d ( A C ) 2 d ( A D ) 2 1 d ( A B ) 2 0 d ( B C ) 2 d ( B D ) 2 1 d ( A C ) 2 d ( B C ) 2 0 d ( C D ) 2 1 d ( A D ) 2 d ( B D ) 2 d ( C D ) 2 0 1 1 1 1 1 0 ] = 0 , \det\begin{bmatrix}0&d(AB)^{2}&d(AC)^{2}&d(AD)^{2}&1\\ d(AB)^{2}&0&d(BC)^{2}&d(BD)^{2}&1\\ d(AC)^{2}&d(BC)^{2}&0&d(CD)^{2}&1\\ d(AD)^{2}&d(BD)^{2}&d(CD)^{2}&0&1\\ 1&1&1&1&0\end{bmatrix}=0,
  3. det [ 0 d ( A B ) 2 d ( A C ) 2 d ( A D ) 2 d ( A E ) 2 1 d ( A B ) 2 0 d ( B C ) 2 d ( B D ) 2 d ( B E ) 2 1 d ( A C ) 2 d ( B C ) 2 0 d ( C D ) 2 d ( C E ) 2 1 d ( A D ) 2 d ( B D ) 2 d ( C D ) 2 0 d ( D E ) 2 1 d ( A E ) 2 d ( B E ) 2 d ( C E ) 2 d ( D E ) 2 0 1 1 1 1 1 1 0 ] = 0 , \det\begin{bmatrix}0&d(AB)^{2}&d(AC)^{2}&d(AD)^{2}&d(AE)^{2}&1\\ d(AB)^{2}&0&d(BC)^{2}&d(BD)^{2}&d(BE)^{2}&1\\ d(AC)^{2}&d(BC)^{2}&0&d(CD)^{2}&d(CE)^{2}&1\\ d(AD)^{2}&d(BD)^{2}&d(CD)^{2}&0&d(DE)^{2}&1\\ d(AE)^{2}&d(BE)^{2}&d(CE)^{2}&d(DE)^{2}&0&1\\ 1&1&1&1&1&0\end{bmatrix}=0,

Distributed_knowledge.html

  1. φ \varphi
  2. φ ψ \varphi\Rightarrow\psi
  3. ( K b φ K a ( φ ψ ) ) D a , b ψ (K_{b}\varphi\land K_{a}(\varphi\Rightarrow\psi))\Rightarrow D_{a,b}\psi

Divergent_series.html

  1. 1 + 1 2 + 1 3 + 1 4 + 1 5 + = n = 1 1 n . 1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\cdots=\sum_{n=1}^{\infty}% \frac{1}{n}.
  2. 1 - 1 + 1 - 1 + 1-1+1-1+\cdots
  3. f : f:\mathbb{N}\rightarrow\mathbb{N}
  4. N N\in\mathbb{N}
  5. G ( r , c ) = k = 0 c r k = c + k = 0 c r k + 1 (stability) = c + r k = 0 c r k (linearity) = c + r G ( r , c ) , hence G ( r , c ) = c 1 - r , unless it is infinite \begin{aligned}\displaystyle G(r,c)&\displaystyle=\sum_{k=0}^{\infty}cr^{k}&&% \\ &\displaystyle=c+\sum_{k=0}^{\infty}cr^{k+1}&&\displaystyle\mbox{ (stability) % }\\ &\displaystyle=c+r\sum_{k=0}^{\infty}cr^{k}&&\displaystyle\mbox{ (linearity) }% \\ &\displaystyle=c+r\,G(r,c),&&\displaystyle\mbox{ hence }\\ \displaystyle G(r,c)&\displaystyle=\frac{c}{1-r},\mbox{unless it is infinite}&% &\\ \end{aligned}
  6. p n p 0 + p 1 + + p n 0. \frac{p_{n}}{p_{0}+p_{1}+\cdots+p_{n}}\rightarrow 0.
  7. t m = p m s 0 + p m - 1 s 1 + + p 0 s m p 0 + p 1 + + p m t_{m}=\frac{p_{m}s_{0}+p_{m-1}s_{1}+\cdots+p_{0}s_{m}}{p_{0}+p_{1}+\cdots+p_{m}}
  8. p n k = ( n + k - 1 k - 1 ) p_{n}^{k}={n+k-1\choose k-1}
  9. f ( x ) = n = 0 a n exp ( - λ n x ) f(x)=\sum_{n=0}^{\infty}a_{n}\exp(-\lambda_{n}x)
  10. A λ ( s ) = lim x 0 + f ( x ) . A_{\lambda}(s)=\lim_{x\rightarrow 0^{+}}f(x).
  11. f ( x ) = n = 0 a n e - n x = n = 0 a n z n , f(x)=\sum_{n=0}^{\infty}a_{n}e^{-nx}=\sum_{n=0}^{\infty}a_{n}z^{n},
  12. A ( s ) = lim z 1 - n = 0 a n z n . A(s)=\lim_{z\rightarrow 1^{-}}\sum_{n=0}^{\infty}a_{n}z^{n}.
  13. f ( x ) = a 1 + a 2 2 - 2 x + a 3 3 - 3 x + . f(x)=a_{1}+a_{2}2^{-2x}+a_{3}3^{-3x}+\cdots.
  14. f ( s ) = a 1 1 s + a 2 2 s + a 3 3 s + f(s)=\frac{a_{1}}{1^{s}}+\frac{a_{2}}{2^{s}}+\frac{a_{3}}{3^{s}}+\cdots
  15. f ( s ) = 1 a 1 s + 1 a 2 s + 1 a 3 s + f(s)=\frac{1}{a_{1}^{s}}+\frac{1}{a_{2}^{s}}+\frac{1}{a_{3}^{s}}+\cdots
  16. ζ ( - s ) = n = 1 n s = 1 s + 2 s + 3 s + = - B s + 1 s + 1 \zeta(-s)=\sum_{n=1}^{\infty}n^{s}=1^{s}+2^{s}+3^{s}+\ldots=-\frac{B_{s+1}}{s+1}
  17. lim x n p n ( a 0 + + a n ) x n n p n x n , \lim_{x\rightarrow\infty}\frac{\sum_{n}p_{n}(a_{0}+\cdots+a_{n})x^{n}}{\sum_{n% }p_{n}x^{n}},
  18. lim n + H ( n ) 2 π h Z e - h 2 H ( n ) / 2 ( a 0 + + a h ) \lim_{n\rightarrow+\infty}\sqrt{\frac{H(n)}{2\pi}}\sum_{h\in Z}e^{-h^{2}H(n)/2% }(a_{0}+\cdots+a_{h})
  19. μ n = x n d μ \mu_{n}=\int x^{n}d\mu
  20. a ( x ) = a 0 x 0 μ 0 + a 1 x 1 μ 1 + a(x)=\frac{a_{0}x^{0}}{\mu_{0}}+\frac{a_{1}x^{1}}{\mu_{1}}+\cdots
  21. a ( x ) d μ \int a(x)d\mu
  22. 0 e - t a n t n n ! d t . \int_{0}^{\infty}e^{-t}\sum\frac{a_{n}t^{n}}{n!}dt.
  23. 0 e - t a n t n α Γ ( n α + 1 ) d t \int_{0}^{\infty}e^{-t}\sum\frac{a_{n}t^{n\alpha}}{\Gamma(n\alpha+1)}dt
  24. lim x 1 n x a n n x [ x n ] = s \lim_{x\rightarrow\infty}\sum_{1\leq n\leq x}a_{n}\frac{n}{x}\left[\frac{x}{n}% \right]=s
  25. lim y 0 + n 1 a n n y e - n y 1 - e - n y = s \lim_{y\rightarrow 0^{+}}\sum_{n\geq 1}a_{n}\frac{nye^{-ny}}{1-e^{-ny}}=s
  26. lim ζ 1 - n Γ ( 1 + ζ n ) Γ ( 1 + n ) a n = s \lim_{\zeta\rightarrow 1^{-}}\sum_{n}\frac{\Gamma(1+\zeta n)}{\Gamma(1+n)}a_{n% }=s
  27. lim δ 0 n a n Γ ( 1 + δ n ) = s \lim_{\delta\rightarrow 0}\sum_{n}\frac{a_{n}}{\Gamma(1+\delta n)}=s
  28. lim h 0 n a n ( sin n h n h ) k = s \lim_{h\rightarrow 0}\sum_{n}a_{n}\left(\frac{\sin nh}{nh}\right)^{k}=s
  29. lim h 0 2 π n sin 2 n h n 2 h ( a 1 + a n ) = s \lim_{h\rightarrow 0}\frac{2}{\pi}\sum_{n}\frac{\sin^{2}nh}{n^{2}h}(a_{1}+% \cdots a_{n})=s
  30. A λ ( x ) = a 0 + + a n for λ n < x λ n + 1 A_{\lambda}(x)=a_{0}+\cdots+a_{n}\,\text{ for }\lambda_{n}<x\leq\lambda_{n+1}
  31. lim ω κ ω κ 0 ω A λ ( x ) ( ω - x ) κ - 1 d x \lim_{\omega\rightarrow\infty}\frac{\kappa}{\omega^{\kappa}}\int_{0}^{\omega}A% _{\lambda}(x)(\omega-x)^{\kappa-1}dx
  32. lim m a 0 + a 1 m m + 1 + a 2 m ( m - 1 ) ( m + 1 ) ( m + 2 ) + = s \lim_{m\rightarrow\infty}a_{0}+a_{1}\frac{m}{m+1}+a_{2}\frac{m(m-1)}{(m+1)(m+2% )}+\cdots=s

Divide-and-conquer_eigenvalue_algorithm.html

  1. m × m m\times m
  2. 4 3 m 3 \frac{4}{3}m^{3}
  3. 8 3 m 3 \frac{8}{3}m^{3}
  4. T = [ T 1 0 0 T 2 ] . T=\begin{bmatrix}T_{1}&0\\ 0&T_{2}\end{bmatrix}.
  5. T T
  6. T 1 T_{1}
  7. T 2 T_{2}
  8. m × m m\times m
  9. T T
  10. T 1 T_{1}
  11. n × n n\times n
  12. T 2 T_{2}
  13. ( m - n ) × ( m - n ) (m-n)\times(m-n)
  14. T T
  15. n n
  16. n m / 2 n\approx m/2
  17. T T
  18. T 1 T_{1}
  19. T ^ 1 \hat{T}_{1}
  20. t n n t_{nn}
  21. T ^ 1 \hat{T}_{1}
  22. t n n - β t_{nn}-\beta
  23. T ^ 2 \hat{T}_{2}
  24. t n + 1 , n + 1 t_{n+1,n+1}
  25. t n + 1 , n + 1 - β t_{n+1,n+1}-\beta
  26. T ^ 1 \hat{T}_{1}
  27. T ^ 2 \hat{T}_{2}
  28. T ^ 1 = Q 1 D 1 Q 1 T \hat{T}_{1}=Q_{1}D_{1}Q_{1}^{T}
  29. T ^ 2 = Q 2 D 2 Q 2 T \hat{T}_{2}=Q_{2}D_{2}Q_{2}^{T}
  30. z T = ( q 1 T , q 2 T ) z^{T}=(q_{1}^{T},q_{2}^{T})
  31. q 1 T q_{1}^{T}
  32. Q 1 Q_{1}
  33. q 2 T q_{2}^{T}
  34. Q 2 Q_{2}
  35. T = [ Q 1 Q 2 ] ( [ D 1 D 2 ] + β z z T ) [ Q 1 T Q 2 T ] T=\begin{bmatrix}Q_{1}&\\ &Q_{2}\end{bmatrix}\left(\begin{bmatrix}D_{1}&\\ &D_{2}\end{bmatrix}+\beta zz^{T}\right)\begin{bmatrix}Q_{1}^{T}&\\ &Q_{2}^{T}\end{bmatrix}
  36. D + w w T D+ww^{T}
  37. D D
  38. w w
  39. e i e_{i}
  40. D + w w T D+ww^{T}
  41. ( D + w w T ) e i = D e i = d i e i (D+ww^{T})e_{i}=De_{i}=d_{i}e_{i}
  42. λ \lambda
  43. ( D + w w T ) q = λ q (D+ww^{T})q=\lambda q
  44. q q
  45. ( D - λ I ) q + w ( w T q ) = 0 (D-\lambda I)q+w(w^{T}q)=0
  46. q + ( D - λ I ) - 1 w ( w T q ) = 0 q+(D-\lambda I)^{-1}w(w^{T}q)=0
  47. w T q + w T ( D - λ I ) - 1 w ( w T q ) = 0 w^{T}q+w^{T}(D-\lambda I)^{-1}w(w^{T}q)=0
  48. w T q w^{T}q
  49. w w
  50. q q
  51. w T q w^{T}q
  52. q q
  53. D D
  54. ( D + w w T ) q = λ q (D+ww^{T})q=\lambda q
  55. q q
  56. D D
  57. w T q w^{T}q
  58. 1 + w T ( D - λ I ) - 1 w = 0 1+w^{T}(D-\lambda I)^{-1}w=0
  59. 1 + j = 1 m w j 2 d j - λ = 0. 1+\sum_{j=1}^{m}\frac{w_{j}^{2}}{d_{j}-\lambda}=0.
  60. Θ ( m ) \Theta(m)
  61. m m
  62. Θ ( m 2 ) \Theta(m^{2})
  63. n m / 2 n\approx m/2
  64. T ( m ) = 2 × T ( m 2 ) + Θ ( m 2 ) T(m)=2\times T\left(\frac{m}{2}\right)+\Theta(m^{2})
  65. a = b = 2 a=b=2
  66. log b a = 1 \log_{b}a=1
  67. Θ ( m 2 ) = Ω ( m 1 ) \Theta(m^{2})=\Omega(m^{1})
  68. T ( m ) = Θ ( m 2 ) T(m)=\Theta(m^{2})
  69. 4 3 m 3 \frac{4}{3}m^{3}
  70. Θ ( m 2 ) \Theta(m^{2})
  71. 8 3 m 3 \frac{8}{3}m^{3}
  72. Θ ( m 3 ) \Theta(m^{3})
  73. 6 m 3 \approx 6m^{3}
  74. 4 3 m 3 \approx\frac{4}{3}m^{3}
  75. Θ ( m 3 ) \Theta(m^{3})
  76. Q Q
  77. 8 3 m 3 \frac{8}{3}m^{3}
  78. 9 m 3 \approx 9m^{3}
  79. 4 m 3 \approx 4m^{3}
  80. Q Q
  81. z z

Divided_differences.html

  1. ( x 0 , y 0 ) , , ( x k , y k ) (x_{0},y_{0}),\ldots,(x_{k},y_{k})
  2. [ y ν ] := y ν , ν { 0 , , k } [y_{\nu}]:=y_{\nu},\qquad\nu\in\{0,\ldots,k\}
  3. [ y ν , , y ν + j ] := [ y ν + 1 , , y ν + j ] - [ y ν , , y ν + j - 1 ] x ν + j - x ν , ν { 0 , , k - j } , j { 1 , , k } . [y_{\nu},\ldots,y_{\nu+j}]:=\frac{[y_{\nu+1},\ldots,y_{\nu+j}]-[y_{\nu},\ldots% ,y_{\nu+j-1}]}{x_{\nu+j}-x_{\nu}},\qquad\nu\in\{0,\ldots,k-j\},\ j\in\{1,% \ldots,k\}.
  4. [ y ν ] := y ν , ν { 0 , , k } [y_{\nu}]:=y_{\nu},\qquad\nu\in\{0,\ldots,k\}
  5. [ y ν , , y ν - j ] := [ y ν , , y ν - j + 1 ] - [ y ν - 1 , , y ν - j ] x ν - x ν - j , ν { j , , k } , j { 1 , , k } . [y_{\nu},\ldots,y_{\nu-j}]:=\frac{[y_{\nu},\ldots,y_{\nu-j+1}]-[y_{\nu-1},% \ldots,y_{\nu-j}]}{x_{\nu}-x_{\nu-j}},\qquad\nu\in\{j,\ldots,k\},\ j\in\{1,% \ldots,k\}.
  6. ( x 0 , f ( x 0 ) ) , , ( x k , f ( x k ) ) (x_{0},f(x_{0})),\ldots,(x_{k},f(x_{k}))
  7. f [ x ν ] := f ( x ν ) , ν { 0 , , k } f[x_{\nu}]:=f(x_{\nu}),\qquad\nu\in\{0,\ldots,k\}
  8. f [ x ν , , x ν + j ] := f [ x ν + 1 , , x ν + j ] - f [ x ν , , x ν + j - 1 ] x ν + j - x ν , ν { 0 , , k - j } , j { 1 , , k } . f[x_{\nu},\ldots,x_{\nu+j}]:=\frac{f[x_{\nu+1},\ldots,x_{\nu+j}]-f[x_{\nu},% \ldots,x_{\nu+j-1}]}{x_{\nu+j}-x_{\nu}},\qquad\nu\in\{0,\ldots,k-j\},\ j\in\{1% ,\ldots,k\}.
  9. [ x 0 , , x n ] f , [x_{0},\ldots,x_{n}]f,
  10. [ x 0 , , x n ; f ] , [x_{0},\ldots,x_{n};f],
  11. D [ x 0 , , x n ] f D[x_{0},\ldots,x_{n}]f
  12. ν \nu
  13. [ y 0 ] \displaystyle\mathopen{[}y_{0}]
  14. x 0 y 0 = [ y 0 ] [ y 0 , y 1 ] x 1 y 1 = [ y 1 ] [ y 0 , y 1 , y 2 ] [ y 1 , y 2 ] [ y 0 , y 1 , y 2 , y 3 ] x 2 y 2 = [ y 2 ] [ y 1 , y 2 , y 3 ] [ y 2 , y 3 ] x 3 y 3 = [ y 3 ] \begin{matrix}x_{0}&y_{0}=[y_{0}]&&&\\ &&[y_{0},y_{1}]&&\\ x_{1}&y_{1}=[y_{1}]&&[y_{0},y_{1},y_{2}]&\\ &&[y_{1},y_{2}]&&[y_{0},y_{1},y_{2},y_{3}]\\ x_{2}&y_{2}=[y_{2}]&&[y_{1},y_{2},y_{3}]&\\ &&[y_{2},y_{3}]&&\\ x_{3}&y_{3}=[y_{3}]&&&\\ \end{matrix}
  15. ( f + g ) [ x 0 , , x n ] = f [ x 0 , , x n ] + g [ x 0 , , x n ] (f+g)[x_{0},\dots,x_{n}]=f[x_{0},\dots,x_{n}]+g[x_{0},\dots,x_{n}]
  16. ( λ f ) [ x 0 , , x n ] = λ f [ x 0 , , x n ] (\lambda\cdot f)[x_{0},\dots,x_{n}]=\lambda\cdot f[x_{0},\dots,x_{n}]
  17. ( f g ) [ x 0 , , x n ] = f [ x 0 ] g [ x 0 , , x n ] + f [ x 0 , x 1 ] g [ x 1 , , x n ] + + f [ x 0 , , x n ] g [ x n ] (f\cdot g)[x_{0},\dots,x_{n}]=f[x_{0}]\cdot g[x_{0},\dots,x_{n}]+f[x_{0},x_{1}% ]\cdot g[x_{1},\dots,x_{n}]+\dots+f[x_{0},\dots,x_{n}]\cdot g[x_{n}]
  18. f [ x 0 , , x n ] = f ( n ) ( ξ ) n ! f[x_{0},\dots,x_{n}]=\frac{f^{(n)}(\xi)}{n!}
  19. T f ( x 0 , , x n ) = ( f [ x 0 ] f [ x 0 , x 1 ] f [ x 0 , x 1 , x 2 ] f [ x 0 , , x n ] 0 f [ x 1 ] f [ x 1 , x 2 ] f [ x 1 , , x n ] 0 0 0 f [ x n ] ) T_{f}(x_{0},\dots,x_{n})=\begin{pmatrix}f[x_{0}]&f[x_{0},x_{1}]&f[x_{0},x_{1},% x_{2}]&\ldots&f[x_{0},\dots,x_{n}]\\ 0&f[x_{1}]&f[x_{1},x_{2}]&\ldots&f[x_{1},\dots,x_{n}]\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ 0&0&0&\ldots&f[x_{n}]\end{pmatrix}
  20. T f + g x = T f x + T g x T_{f+g}x=T_{f}x+T_{g}x
  21. T f g x = T f x T g x T_{f\cdot g}x=T_{f}x\cdot T_{g}x
  22. T f x T_{f}x
  23. f ( x 0 ) , , f ( x n ) f(x_{0}),\dots,f(x_{n})
  24. δ ξ \delta_{\xi}
  25. δ ξ ( t ) = { 1 : t = ξ , 0 : else . \delta_{\xi}(t)=\begin{cases}1&:t=\xi,\\ 0&:\mbox{else}~{}.\end{cases}
  26. f δ ξ = f ( ξ ) δ ξ f\cdot\delta_{\xi}=f(\xi)\cdot\delta_{\xi}
  27. δ ξ \delta_{\xi}
  28. T δ x i x T_{\delta_{x_{i}}}x
  29. T f x T_{f}x
  30. T f x T δ x i x = f ( x i ) T δ x i x T_{f}x\cdot T_{\delta_{x_{i}}}x=f(x_{i})\cdot T_{\delta_{x_{i}}}x
  31. T δ x i x T_{\delta_{x_{i}}}x
  32. T δ x i x T_{\delta_{x_{i}}}x
  33. i i
  34. T δ x i x T_{\delta_{x_{i}}}x
  35. U x Ux
  36. U ( x 0 , x 1 , x 2 , x 3 ) = ( 1 1 ( x 1 - x 0 ) 1 ( x 2 - x 0 ) ( x 2 - x 1 ) 1 ( x 3 - x 0 ) ( x 3 - x 1 ) ( x 3 - x 2 ) 0 1 1 ( x 2 - x 1 ) 1 ( x 3 - x 1 ) ( x 3 - x 2 ) 0 0 1 1 ( x 3 - x 2 ) 0 0 0 1 ) U(x_{0},x_{1},x_{2},x_{3})=\begin{pmatrix}1&\frac{1}{(x_{1}-x_{0})}&\frac{1}{(% x_{2}-x_{0})\cdot(x_{2}-x_{1})}&\frac{1}{(x_{3}-x_{0})\cdot(x_{3}-x_{1})\cdot(% x_{3}-x_{2})}\\ 0&1&\frac{1}{(x_{2}-x_{1})}&\frac{1}{(x_{3}-x_{1})\cdot(x_{3}-x_{2})}\\ 0&0&1&\frac{1}{(x_{3}-x_{2})}\\ 0&0&0&1\end{pmatrix}
  37. T f x T_{f}x
  38. U x diag ( f ( x 0 ) , , f ( x n ) ) = T f x U x Ux\cdot\operatorname{diag}(f(x_{0}),\dots,f(x_{n}))=T_{f}x\cdot Ux
  39. f [ x 0 ] = f ( x 0 ) f [ x 0 , x 1 ] = f ( x 0 ) ( x 0 - x 1 ) + f ( x 1 ) ( x 1 - x 0 ) f [ x 0 , x 1 , x 2 ] = f ( x 0 ) ( x 0 - x 1 ) ( x 0 - x 2 ) + f ( x 1 ) ( x 1 - x 0 ) ( x 1 - x 2 ) + f ( x 2 ) ( x 2 - x 0 ) ( x 2 - x 1 ) f [ x 0 , x 1 , x 2 , x 3 ] = f ( x 0 ) ( x 0 - x 1 ) ( x 0 - x 2 ) ( x 0 - x 3 ) + f ( x 1 ) ( x 1 - x 0 ) ( x 1 - x 2 ) ( x 1 - x 3 ) + f ( x 2 ) ( x 2 - x 0 ) ( x 2 - x 1 ) ( x 2 - x 3 ) + f ( x 3 ) ( x 3 - x 0 ) ( x 3 - x 1 ) ( x 3 - x 2 ) f [ x 0 , , x n ] = j = 0 n f ( x j ) k { 0 , , n } { j } ( x j - x k ) \begin{aligned}\displaystyle f[x_{0}]&\displaystyle=f(x_{0})\\ \displaystyle f[x_{0},x_{1}]&\displaystyle=\frac{f(x_{0})}{(x_{0}-x_{1})}+% \frac{f(x_{1})}{(x_{1}-x_{0})}\\ \displaystyle f[x_{0},x_{1},x_{2}]&\displaystyle=\frac{f(x_{0})}{(x_{0}-x_{1})% \cdot(x_{0}-x_{2})}+\frac{f(x_{1})}{(x_{1}-x_{0})\cdot(x_{1}-x_{2})}+\frac{f(x% _{2})}{(x_{2}-x_{0})\cdot(x_{2}-x_{1})}\\ \displaystyle f[x_{0},x_{1},x_{2},x_{3}]&\displaystyle=\frac{f(x_{0})}{(x_{0}-% x_{1})\cdot(x_{0}-x_{2})\cdot(x_{0}-x_{3})}+\frac{f(x_{1})}{(x_{1}-x_{0})\cdot% (x_{1}-x_{2})\cdot(x_{1}-x_{3})}+\frac{f(x_{2})}{(x_{2}-x_{0})\cdot(x_{2}-x_{1% })\cdot(x_{2}-x_{3})}+\\ &\displaystyle\quad\quad\frac{f(x_{3})}{(x_{3}-x_{0})\cdot(x_{3}-x_{1})\cdot(x% _{3}-x_{2})}\\ \displaystyle f[x_{0},\dots,x_{n}]&\displaystyle=\sum_{j=0}^{n}\frac{f(x_{j})}% {\prod_{k\in\{0,\dots,n\}\setminus\{j\}}(x_{j}-x_{k})}\end{aligned}
  40. q q
  41. q ( ξ ) = ( ξ - x 0 ) ( ξ - x n ) q(\xi)=(\xi-x_{0})\cdots(\xi-x_{n})
  42. f [ x 0 , , x n ] = j = 0 n f ( x j ) q ( x j ) . f[x_{0},\dots,x_{n}]=\sum_{j=0}^{n}\frac{f(x_{j})}{q^{\prime}(x_{j})}.
  43. x k = x k + n + 1 = x k - ( n + 1 ) x_{k}=x_{k+n+1}=x_{k-(n+1)}
  44. k < 0 k<0
  45. n < k n<k
  46. x - 1 x_{-1}
  47. x n x_{n}
  48. x - 2 x_{-2}
  49. x n - 1 x_{n-1}
  50. x - n x_{-n}
  51. x 0 x_{0}
  52. f [ x 0 , , x n ] = j = 0 n f ( x j ) k = j - n j - 1 ( x j - x k ) = j = 0 n f ( x j ) k = j + 1 j + n ( x j - x k ) f[x_{0},\dots,x_{n}]=\sum_{j=0}^{n}\frac{f(x_{j})}{\prod\limits_{k=j-n}^{j-1}(% x_{j}-x_{k})}=\sum_{j=0}^{n}\frac{f(x_{j})}{\prod\limits_{k=j+1}^{j+n}(x_{j}-x% _{k})}
  53. f [ x 0 , , x n ] = j = 0 n lim x x j [ f ( x j ) ( x - x j ) k = 0 n ( x - x k ) ] f[x_{0},\dots,x_{n}]=\sum_{j=0}^{n}\lim_{x\rightarrow x_{j}}\left[\frac{f(x_{j% })(x-x_{j})}{\prod\limits_{k=0}^{n}(x-x_{k})}\right]
  54. p p
  55. q q
  56. deg p < deg q \mathrm{deg}\ p<\mathrm{deg}\ q
  57. q q
  58. q ( ξ ) = ( ξ - x 1 ) ( ξ - x n ) q(\xi)=(\xi-x_{1})\cdot\dots\cdot(\xi-x_{n})
  59. p ( ξ ) q ( ξ ) = ( t p ( t ) ξ - t ) [ x 1 , , x n ] . \frac{p(\xi)}{q(\xi)}=\left(t\to\frac{p(t)}{\xi-t}\right)[x_{1},\dots,x_{n}].
  60. x j x_{j}
  61. f f
  62. f ( x ) = p ( x ) + q ( x ) d ( x ) f(x)=p(x)+q(x)\cdot d(x)
  63. f f
  64. q q
  65. p ( ξ ) q ( ξ ) = ( t f ( t ) ξ - t ) [ x 1 , , x n ] . \frac{p(\xi)}{q(\xi)}=\left(t\to\frac{f(t)}{\xi-t}\right)[x_{1},\dots,x_{n}].
  66. f [ x 0 , , x n ] = 1 n ! x 0 x n f ( n ) ( t ) B n - 1 ( t ) d t f[x_{0},\ldots,x_{n}]=\frac{1}{n!}\int_{x_{0}}^{x_{n}}f^{(n)}(t)B_{n-1}(t)\,dt
  67. B n - 1 B_{n-1}
  68. n - 1 n-1
  69. x 0 , , x n x_{0},\dots,x_{n}
  70. f ( n ) f^{(n)}
  71. n n
  72. f f
  73. B n - 1 B_{n-1}
  74. f ( y ) - f ( x ) y - x f ( x ) \frac{f(y)-f(x)}{y-x}\approx f^{\prime}(x)
  75. x y x\approx y
  76. f ( y ) = f ( x ) + f ( x ) ( y - x ) + f ′′ ( x ) ( y - x ) 2 2 ! + f ′′′ ( x ) ( y - x ) 3 3 ! + f(y)=f(x)+f^{\prime}(x)\cdot(y-x)+f^{\prime\prime}(x)\cdot\frac{(y-x)^{2}}{2!}% +f^{\prime\prime\prime}(x)\cdot\frac{(y-x)^{3}}{3!}+\dots
  77. f ( y ) - f ( x ) y - x = f ( x ) + f ′′ ( x ) y - x 2 ! + f ′′′ ( x ) ( y - x ) 2 3 ! + \Rightarrow\frac{f(y)-f(x)}{y-x}=f^{\prime}(x)+f^{\prime\prime}(x)\cdot\frac{y% -x}{2!}+f^{\prime\prime\prime}(x)\cdot\frac{(y-x)^{2}}{3!}+\dots
  78. y - x y-x
  79. x x
  80. y y
  81. x = m - h , y = m + h x=m-h,y=m+h
  82. m = x + y 2 , h = y - x 2 m=\frac{x+y}{2},h=\frac{y-x}{2}
  83. f ( m + h ) = f ( m ) + f ( m ) h + f ′′ ( m ) h 2 2 ! + f ′′′ ( m ) h 3 3 ! + f(m+h)=f(m)+f^{\prime}(m)\cdot h+f^{\prime\prime}(m)\cdot\frac{h^{2}}{2!}+f^{% \prime\prime\prime}(m)\cdot\frac{h^{3}}{3!}+\dots
  84. f ( m - h ) = f ( m ) - f ( m ) h + f ′′ ( m ) h 2 2 ! - f ′′′ ( m ) h 3 3 ! + f(m-h)=f(m)-f^{\prime}(m)\cdot h+f^{\prime\prime}(m)\cdot\frac{h^{2}}{2!}-f^{% \prime\prime\prime}(m)\cdot\frac{h^{3}}{3!}+\dots
  85. f ( y ) - f ( x ) y - x = f ( m + h ) - f ( m - h ) 2 h = f ( m ) + f ′′′ ( m ) h 2 3 ! + \frac{f(y)-f(x)}{y-x}=\frac{f(m+h)-f(m-h)}{2\cdot h}=f^{\prime}(m)+f^{\prime% \prime\prime}(m)\cdot\frac{h^{2}}{3!}+\dots
  86. f f
  87. f [ x 0 , , x n ] f[x_{0},\dots,x_{n}]
  88. p n ( x ) = x n . p_{n}(x)=x^{n}.
  89. f = f ( 0 ) p 0 + f ( 0 ) p 1 + f ′′ ( 0 ) 2 ! p 2 + f ′′′ ( 0 ) 3 ! p 3 + f=f(0)\cdot p_{0}+f^{\prime}(0)\cdot p_{1}+\frac{f^{\prime\prime}(0)}{2!}\cdot p% _{2}+\frac{f^{\prime\prime\prime}(0)}{3!}\cdot p_{3}+\dots
  90. f [ x 0 , , x n ] = f ( 0 ) p 0 [ x 0 , , x n ] + f ( 0 ) p 1 [ x 0 , , x n ] + f ′′ ( 0 ) 2 ! p 2 [ x 0 , , x n ] + f ′′′ ( 0 ) 3 ! p 3 [ x 0 , , x n ] + f[x_{0},\dots,x_{n}]=f(0)\cdot p_{0}[x_{0},\dots,x_{n}]+f^{\prime}(0)\cdot p_{% 1}[x_{0},\dots,x_{n}]+\frac{f^{\prime\prime}(0)}{2!}\cdot p_{2}[x_{0},\dots,x_% {n}]+\frac{f^{\prime\prime\prime}(0)}{3!}\cdot p_{3}[x_{0},\dots,x_{n}]+\dots
  91. n n
  92. j < n p j [ x 0 , , x n ] = 0 p n [ x 0 , , x n ] = 1 p n + 1 [ x 0 , , x n ] = x 0 + + x n p n + m [ x 0 , , x n ] = a { 0 , , n } m with a 1 a 2 a m j a x j . \begin{array}[]{llcl}\forall j<n&p_{j}[x_{0},\dots,x_{n}]&=&0\\ &p_{n}[x_{0},\dots,x_{n}]&=&1\\ &p_{n+1}[x_{0},\dots,x_{n}]&=&x_{0}+\dots+x_{n}\\ &p_{n+m}[x_{0},\dots,x_{n}]&=&\sum_{a\in\{0,\dots,n\}^{m}\,\text{ with }a_{1}% \leq a_{2}\leq\dots\leq a_{m}}\prod_{j\in a}x_{j}.\\ \end{array}
  93. f ( n ) ( 0 ) n ! \frac{f^{(n)}(0)}{n!}
  94. f f
  95. t n = ( 1 - x 0 t ) ( 1 - x n t ) ( p 0 [ x 0 , , x n ] + p 1 [ x 0 , , x n ] t + p 2 [ x 0 , , x n ] t 2 + ) . t^{n}=(1-x_{0}\cdot t)\dots\cdot(1-x_{n}\cdot t)\cdot(p_{0}[x_{0},\dots,x_{n}]% +p_{1}[x_{0},\dots,x_{n}]\cdot t+p_{2}[x_{0},\dots,x_{n}]\cdot t^{2}+\dots).
  96. p n p_{n}
  97. p n [ h ] p_{n}[h]
  98. n n
  99. J J
  100. J = ( x 0 1 0 0 0 0 x 1 1 0 0 0 0 x 2 1 0 0 0 0 0 x n ) J=\begin{pmatrix}x_{0}&1&0&0&\cdots&0\\ 0&x_{1}&1&0&\cdots&0\\ 0&0&x_{2}&1&&0\\ \vdots&\vdots&&\ddots&\ddots&\\ 0&0&0&0&&x_{n}\end{pmatrix}
  101. x 0 , , x n x_{0},\dots,x_{n}
  102. J n J^{n}
  103. n n
  104. φ ( p ) \varphi(p)
  105. p p
  106. p p
  107. φ M ( p ) \varphi_{\mathrm{M}}(p)
  108. J J
  109. φ ( p ) ( ξ ) = a 0 + a 1 ξ + + a n ξ n \varphi(p)(\xi)=a_{0}+a_{1}\cdot\xi+\dots+a_{n}\cdot\xi^{n}
  110. φ M ( p ) ( J ) = a 0 + a 1 J + + a n J n \varphi_{\mathrm{M}}(p)(J)=a_{0}+a_{1}\cdot J+\dots+a_{n}\cdot J^{n}
  111. = ( φ ( p ) [ x 0 ] φ ( p ) [ x 0 , x 1 ] φ ( p ) [ x 0 , x 1 , x 2 ] φ ( p ) [ x 0 , , x n ] 0 φ ( p ) [ x 1 ] φ ( p ) [ x 1 , x 2 ] φ ( p ) [ x 1 , , x n ] 0 0 0 φ ( p ) [ x n ] ) =\begin{pmatrix}\varphi(p)[x_{0}]&\varphi(p)[x_{0},x_{1}]&\varphi(p)[x_{0},x_{% 1},x_{2}]&\ldots&\varphi(p)[x_{0},\dots,x_{n}]\\ 0&\varphi(p)[x_{1}]&\varphi(p)[x_{1},x_{2}]&\ldots&\varphi(p)[x_{1},\dots,x_{n% }]\\ \vdots&\ddots&\ddots&\ddots&\vdots\\ 0&\ldots&0&0&\varphi(p)[x_{n}]\end{pmatrix}
  112. p p
  113. f f
  114. J J
  115. x 0 , , x n x_{0},\dots,x_{n}
  116. J J
  117. ( x 0 , y 0 ) , , ( x n - 1 , y n - 1 ) (x_{0},y_{0}),\ldots,(x_{n-1},y_{n-1})
  118. x ν = x 0 + ν h , h > 0 , ν = 0 , , n - 1 x_{\nu}=x_{0}+\nu h\mbox{ , }~{}h>0\mbox{ , }~{}\nu=0,\ldots,n-1
  119. ( 0 ) y i := y i \triangle^{(0)}y_{i}:=y_{i}
  120. ( k ) y i := ( k - 1 ) y i + 1 - ( k - 1 ) y i , k 1. \triangle^{(k)}y_{i}:=\triangle^{(k-1)}y_{i+1}-\triangle^{(k-1)}y_{i}\mbox{ , % }~{}k\geq 1.
  121. y 0 y 0 y 1 2 y 0 y 1 3 y 0 y 2 2 y 1 y 2 y 3 \begin{matrix}y_{0}&&&\\ &\triangle y_{0}&&\\ y_{1}&&\triangle^{2}y_{0}&\\ &\triangle y_{1}&&\triangle^{3}y_{0}\\ y_{2}&&\triangle^{2}y_{1}&\\ &\triangle y_{2}&&\\ y_{3}&&&\\ \end{matrix}

Divisibility_rule.html

  1. p 1 n p 2 m p 3 q p_{1}^{n}p_{2}^{m}p_{3}^{q}
  2. 10 1 ( mod 3 ) 10\equiv 1\;\;(\mathop{{\rm mod}}3)
  3. 10 n 1 n 1 ( mod 3 ) 10^{n}\equiv 1^{n}\equiv 1\;\;(\mathop{{\rm mod}}3)
  4. 100 a + 10 b + 1 c ( 1 ) a + ( 1 ) b + ( 1 ) c ( mod 3 ) 100\cdot a+10\cdot b+1\cdot c\equiv(1)a+(1)b+(1)c\;\;(\mathop{{\rm mod}}3)
  5. 10 - 1 ( mod 11 ) 10\equiv-1\;\;(\mathop{{\rm mod}}11)
  6. 10 n ( - 1 ) n { 1 , if n is even - 1 , if n is odd ( mod 11 ) . 10^{n}\equiv(-1)^{n}\equiv\begin{cases}1,&\mbox{if }~{}n\mbox{ is even}\\ -1,&\mbox{if }~{}n\mbox{ is odd}\end{cases}\;\;(\mathop{{\rm mod}}11).
  7. 1000 a + 100 b + 10 c + 1 d ( - 1 ) a + ( 1 ) b + ( - 1 ) c + ( 1 ) d ( mod 11 ) 1000\cdot a+100\cdot b+10\cdot c+1\cdot d\equiv(-1)a+(1)b+(-1)c+(1)d\;\;(% \mathop{{\rm mod}}11)
  8. 100 a + b 100\cdot a+b
  9. ( 98 + 2 ) a + b (98+2)\cdot a+b
  10. 98 a + 2 a + b , 98\cdot a+2\cdot a+b,
  11. 2 a + b , 2\cdot a+b,
  12. 10 a + b , 10\cdot a+b,
  13. 20 a + 2 b , 20\cdot a+2\cdot b,
  14. ( 21 - 1 ) a + 2 b . (21-1)\cdot a+2\cdot b.
  15. - 1 a + 2 b , -1\cdot a+2\cdot b,
  16. a - 2 b . a-2\cdot b.
  17. 10 n = 2 n 5 n 0 ( mod 2 n or 5 n ) 10^{n}=2^{n}\cdot 5^{n}\equiv 0\;\;(\mathop{{\rm mod}}2^{n}\mathrm{\ or\ }5^{n})
  18. 10 n y + z , 10^{n}\cdot y+z,
  19. x = 10 n y + z z ( mod 2 n or 5 n ) x=10^{n}\cdot y+z\equiv z\;\;(\mathop{{\rm mod}}2^{n}\mathrm{\ or\ }5^{n})
  20. 10 y + z , 10\cdot y+z,
  21. - 2 x y - 2 z ( mod 7 ) , -2x\equiv y-2z\;\;(\mathop{{\rm mod}}7),

Dixon's_factorization_method.html

  1. x 2 y 2 ( mod N ) , x ± y ( mod N ) . x^{2}\equiv y^{2}\quad(\hbox{mod }N),\qquad x\not\equiv\pm y\quad(\hbox{mod }N).
  2. N \sqrt{N}
  3. N \sqrt{N}
  4. a 1 a n a_{1}\ldots a_{n}
  5. a i 2 mod N = j = 1 m b j e i j a_{i}^{2}\mod N=\prod_{j=1}^{m}b_{j}^{e_{ij}}
  6. b 1 b m b_{1}\ldots b_{m}
  7. e i j e_{ij}
  8. a i a_{i}
  9. a i a_{i}
  10. z 2 p i P p i a i ( mod N ) z^{2}\equiv\prod_{p_{i}\in P}p_{i}^{a_{i}}\;\;(\mathop{{\rm mod}}N)
  11. z 1 2 z 2 2 z k 2 p i P p i a i , 1 + a i , 2 + + a i , k ( mod N ) ( where a i , 1 + a i , 2 + + a i , k 0 ( mod 2 ) ) {z_{1}^{2}z_{2}^{2}\cdots z_{k}^{2}\equiv\prod_{p_{i}\in P}p_{i}^{a_{i,1}+a_{i% ,2}+\cdots+a_{i,k}}\ \;\;(\mathop{{\rm mod}}N)\quad(\,\text{where }a_{i,1}+a_{% i,2}+\cdots+a_{i,k}\equiv 0\;\;(\mathop{{\rm mod}}2))}
  12. 84923 = 292 \left\lceil\sqrt{84923}\right\rceil=292
  13. 513 2 mod 84923 = 8400 = 2 4 3 5 2 7 513^{2}\mod 84923=8400=2^{4}\cdot 3\cdot 5^{2}\cdot 7
  14. 537 2 mod 84923 = 33600 = 2 6 3 5 2 7 537^{2}\mod 84923=33600=2^{6}\cdot 3\cdot 5^{2}\cdot 7
  15. ( 513 537 ) 2 mod 84923 = 2 10 3 2 5 4 7 2 (513\cdot 537)^{2}\mod 84923=2^{10}\cdot 3^{2}\cdot 5^{4}\cdot 7^{2}
  16. ( 513 537 ) 2 mod 84923 = ( 275481 ) 2 mod 84923 \displaystyle{}\qquad(513\cdot 537)^{2}\mod 84923=(275481)^{2}\mod 84923
  17. 20712 2 mod 84923 = ( 2 5 3 5 2 7 ) 2 mod 84923 = 16800 2 mod 84923. 20712^{2}\mod 84923=(2^{5}\cdot 3\cdot 5^{2}\cdot 7)^{2}\mod 84923=16800^{2}% \mod 84923.
  18. log 2 z \log_{2}z
  19. N 1 / a N^{1/a}
  20. a - a a^{-a}
  21. exp ( log N log log N ) \exp\left(\sqrt{\log N\log\log N}\right)
  22. O ( exp ( 2 2 log n log log n ) ) O\left(\exp\left(2\sqrt{2}\sqrt{\log n\log\log n}\right)\right)
  23. L n [ 1 / 2 , 2 2 ] L_{n}[1/2,2\sqrt{2}]

Domineering.html

  1. { 2 | - 1 2 } = 3 4 ± 5 4 \textstyle\left\{2\left|-\frac{1}{2}\right.\right\}=\frac{3}{4}\pm\frac{5}{4}

Donald_C._Spencer.html

  1. ¯ \bar{\partial}

Donald_McKay.html

  1. 1.34 × LWL 1.34\times\sqrt{\mbox{LWL}~{}}

Doping_(semiconductor).html

  1. n = p = n i . n=p=n_{i}.
  2. n 0 p 0 = n i 2 n_{0}\cdot p_{0}=n_{i}^{2}
  3. n e = N C ( T ) exp ( ( E F - E C ) / k T ) , n h = N V ( T ) exp ( ( E V - E F ) / k T ) , n_{e}=N_{\rm C}(T)\exp((E_{\rm F}-E_{\rm C})/kT),\quad n_{h}=N_{\rm V}(T)\exp(% (E_{\rm V}-E_{\rm F})/kT),
  4. n i 2 = n h n e = N V ( T ) N C ( T ) exp ( ( E V - E C ) / k T ) , n_{i}^{2}=n_{h}n_{e}=N_{\rm V}(T)N_{\rm C}(T)\exp((E_{\rm V}-E_{\rm C})/kT),
  5. N C ( T ) = 2 ( 2 π m e * k T / h 2 ) 3 / 2 N V ( T ) = 2 ( 2 π m v * k T / h 2 ) 3 / 2 . N_{\rm C}(T)=2(2\pi m_{e}^{*}kT/h^{2})^{3/2}\quad N_{\rm V}(T)=2(2\pi m_{v}^{*% }kT/h^{2})^{3/2}.
  6. Si 30 ( n , γ ) 31 Si 31 P + β - ( T 1 / 2 = 2.62 h ) . {}^{30}\mathrm{Si}\,(n,\gamma)\,^{31}\mathrm{Si}\rightarrow\,^{31}\mathrm{P}+% \beta^{-}\;(\mathrm{T}_{1/2}=2.62h).

Double-elimination_tournament.html

  1. N N
  2. N N
  3. N N
  4. N N
  5. 2 N 2N
  6. N N
  7. n n
  8. 2 n - 2 2n-2
  9. 2 n - 1 2n-1

Double_coset.html

  1. H \ G / K . H\backslash G/K.
  2. H g 1 K Hg_{1}K
  3. K g 2 L Kg_{2}L
  4. H g 1 K = i H a i \textstyle Hg_{1}K=\coprod_{i}Ha_{i}
  5. K g 2 L = j b j L \textstyle Kg_{2}L=\coprod_{j}b_{j}L
  6. c g = | { ( i , j ) : a i b j H g } | c_{g}=\left|\{(i,j):a_{i}b_{j}\in Hg\}\right|
  7. H g 1 K Hg_{1}K
  8. K g 2 L Kg_{2}L
  9. H g 1 K K g 2 L = g H \ G c g H g L \textstyle Hg_{1}K\cdot Kg_{2}L=\sum_{g\in H\backslash G}c_{g}HgL
  10. Γ \ GL 2 + ( ) / Γ \Gamma\backslash\mathrm{GL}_{2}^{+}(\mathbb{Q})/\Gamma
  11. T m T_{m}
  12. Γ 0 ( N ) g Γ 0 ( N ) \Gamma_{0}(N)g\Gamma_{0}(N)
  13. Γ 1 ( N ) g Γ 1 ( N ) \Gamma_{1}(N)g\Gamma_{1}(N)
  14. g = ( 1 0 0 m ) g=\left(\begin{smallmatrix}1&0\\ 0&m\end{smallmatrix}\right)
  15. d \langle d\rangle
  16. Γ 1 ( N ) ( a b c d ) Γ 1 ( N ) \Gamma_{1}(N)\left(\begin{smallmatrix}a&b\\ c&d\end{smallmatrix}\right)\Gamma_{1}(N)
  17. d ( / N ) × d\in(\mathbb{Z}/N\mathbb{Z})^{\times}
  18. ( a b c d ) Γ 0 ( N ) \left(\begin{smallmatrix}a&b\\ c&d\end{smallmatrix}\right)\in\Gamma_{0}(N)

Downhill_creep.html

  1. q s = k d S q_{s}=k_{d}S\,\!
  2. k d k_{d}\,\!
  3. S S\,\!
  4. q s = k d S 1 - ( S / S c ) 2 q_{s}=\frac{k_{d}S}{1-(S/S_{c})^{2}}\,\!
  5. S c S_{c}\,\!

DSPACE.html

  1. \mathbb{N}
  2. 𝒞 \mathcal{C}
  3. | 𝒞 | 2 c . s ( n ) |\mathcal{C}|\leq 2^{c.s(n)}
  4. | 𝒞 | |\mathcal{C}|
  5. | 𝒞 | |\mathcal{C}|
  6. | S | | 𝒞 | | 𝒞 | ( 2 c . s ( n ) ) 2 c . s ( n ) = 2 c . s ( n ) .2 c . s ( n ) < 2 2 2 c . s ( n ) = 2 2 o ( log log n ) = o ( n ) |S|\leq|\mathcal{C}|^{|\mathcal{C}|}\leq(2^{c.s(n)})^{2^{c.s(n)}}=2^{c.s(n).2^% {c.s(n)}}<2^{2^{2c.s(n)}}=2^{2^{o(\log\log n)}}=o(n)
  7. 𝒞 i ( x ) \mathcal{C}_{i}(x)
  8. 𝒞 j ( x ) \mathcal{C}_{j}(x)
  9. k DSPACE ( 2 n k ) \bigcup_{k\in\mathbb{N}}\mbox{DSPACE}~{}(2^{n^{k}})
  10. f : f:\mathbb{N}\to\mathbb{N}
  11. O ( f ( n ) ) O(f(n))
  12. o ( f ( n ) ) o(f(n))
  13. DSPACE [ s ( n ) ] NSPACE [ s ( n ) ] DSPACE [ ( s ( n ) ) 2 ] . \mbox{DSPACE}~{}[s(n)]\subseteq\mbox{NSPACE}~{}[s(n)]\subseteq\mbox{DSPACE}~{}% [(s(n))^{2}].
  14. NTIME ( t ( n ) ) DSPACE ( t ( n ) ) \mbox{NTIME}~{}(t(n))\subseteq\mbox{DSPACE}~{}(t(n))

DTIME.html

  1. P = k DTIME ( n k ) \mbox{P}~{}=\bigcup_{k\in\mathbb{N}}\mbox{DTIME}~{}(n^{k})
  2. DTIME ( n ) \mbox{DTIME}~{}\left(n\right)
  3. EXPTIME = k DTIME ( 2 n k ) . \mbox{EXPTIME}~{}=\bigcup_{k\in\mathbb{N}}\mbox{DTIME}~{}\left(2^{n^{k}}\right).
  4. P EXPTIME \mbox{P}~{}\subsetneq\mbox{EXPTIME}~{}
  5. \in
  6. ϵ \epsilon
  7. \in
  8. ϵ \epsilon
  9. 𝖣𝖳𝖨𝖬𝖤 ( O ( n ) ) 𝖭𝖳𝖨𝖬𝖤 ( O ( n ) ) \mathsf{DTIME}(O(n))\neq\mathsf{NTIME}(O(n))

Dual_basis.html

  1. B = { v i } i I B=\{v_{i}\}_{i\in I}
  2. B * = { v i } i I B^{*}=\{v^{i}\}_{i\in I}
  3. v i ( v j ) = δ j i = { 1 if i = j 0 if i j , v^{i}(v_{j})=\delta^{i}_{j}=\begin{cases}1&\,\text{if }i=j\\ 0&\,\text{if }i\neq j\,\text{,}\end{cases}
  4. δ j i \delta^{i}_{j}
  5. A A
  6. R R
  7. A A
  8. R - 𝐌𝐨𝐝 R\,\text{-}\mathbf{Mod}
  9. A A
  10. A A^{\ast}
  11. Hom R ( A , R ) \,\text{Hom}_{R}(A,R)
  12. R R
  13. A A
  14. R R
  15. A A
  16. A A^{\ast\ast}
  17. Hom R ( A , R ) \,\text{Hom}_{R}(A^{\ast},R)
  18. F F
  19. R R
  20. R R
  21. X X
  22. F F
  23. δ x y \delta_{xy}
  24. X X
  25. δ x y = 1 \delta_{xy}=1
  26. x = y x=y
  27. δ x y = 0 \delta_{xy}=0
  28. x y x\neq y
  29. S = { f x : F R | f x ( y ) = δ x y } S=\{f_{x}:F\to R\;|\;f_{x}(y)=\delta_{xy}\}
  30. f x Hom R ( F , R ) f_{x}\in\,\text{Hom}_{R}(F,R)
  31. F F
  32. X X
  33. S S
  34. F F^{\ast}
  35. F F^{\ast}
  36. R R
  37. w = i K α i v i w=\sum_{i\in K}\alpha_{i}v^{i}
  38. w ( v j ) = ( i K α i v i ) ( v j ) = 0 w(v_{j})=(\sum_{i\in K}\alpha_{i}v^{i})(v_{j})=0
  39. e i , e j = δ j i . \left\langle e^{i},e_{j}\right\rangle=\delta^{i}_{j}.
  40. 𝐱 = x 1 𝐢 1 + x 2 𝐢 2 + x 3 𝐢 3 \mathbf{x}=x^{1}\mathbf{i}_{1}+x^{2}\mathbf{i}_{2}+x^{3}\mathbf{i}_{3}
  41. 𝐢 k \mathbf{i}_{k}
  42. 𝐱 \mathbf{x}
  43. x k = 𝐱 𝐢 k x^{k}=\mathbf{x}\cdot\mathbf{i}_{k}
  44. x i = 𝐱 𝐞 i ( i = 1 , 2 , 3 ) x^{i}=\mathbf{x}\cdot\mathbf{e}^{i}\qquad(i=1,2,3)
  45. 𝐞 k = 𝐞 k = 𝐢 k \mathbf{e}^{k}=\mathbf{e}_{k}=\mathbf{i}_{k}
  46. { 𝐞 1 , 𝐞 2 } = { ( 1 0 ) , ( 0 1 ) } \{\mathbf{e}_{1},\mathbf{e}_{2}\}=\left\{\begin{pmatrix}1\\ 0\end{pmatrix},\begin{pmatrix}0\\ 1\end{pmatrix}\right\}
  47. { 𝐞 1 , 𝐞 2 } = { ( 1 0 ) , ( 0 1 ) } . \{\mathbf{e}^{1},\mathbf{e}^{2}\}=\left\{\begin{pmatrix}1&0\end{pmatrix},% \begin{pmatrix}0&1\end{pmatrix}\right\}\,\text{.}
  48. 𝐞 1 = ( 𝐞 2 × 𝐞 3 V ) T , 𝐞 2 = ( 𝐞 3 × 𝐞 1 V ) T , 𝐞 3 = ( 𝐞 1 × 𝐞 2 V ) T . \mathbf{e}^{1}=\left(\frac{\mathbf{e}_{2}\times\mathbf{e}_{3}}{V}\right)\text{% T},\ \mathbf{e}^{2}=\left(\frac{\mathbf{e}_{3}\times\mathbf{e}_{1}}{V}\right)% \text{T},\ \mathbf{e}^{3}=\left(\frac{\mathbf{e}_{1}\times\mathbf{e}_{2}}{V}% \right)\text{T}.
  49. T {}^{T}
  50. V = ( 𝐞 1 ; 𝐞 2 ; 𝐞 3 ) = 𝐞 1 ( 𝐞 2 × 𝐞 3 ) = 𝐞 2 ( 𝐞 3 × 𝐞 1 ) = 𝐞 3 ( 𝐞 1 × 𝐞 2 ) V\,=\,\left(\mathbf{e}_{1};\mathbf{e}_{2};\mathbf{e}_{3}\right)\,=\,\mathbf{e}% _{1}\cdot(\mathbf{e}_{2}\times\mathbf{e}_{3})\,=\,\mathbf{e}_{2}\cdot(\mathbf{% e}_{3}\times\mathbf{e}_{1})\,=\,\mathbf{e}_{3}\cdot(\mathbf{e}_{1}\times% \mathbf{e}_{2})
  51. 𝐞 1 , 𝐞 2 \mathbf{e}_{1},\,\mathbf{e}_{2}
  52. 𝐞 3 . \mathbf{e}_{3}.

Dual_basis_in_a_field_extension.html

  1. B 1 = α 0 , α 1 , , α m - 1 B_{1}={\alpha_{0},\alpha_{1},\ldots,\alpha_{m-1}}
  2. B 2 = γ 0 , γ 1 , , γ m - 1 B_{2}={\gamma_{0},\gamma_{1},\ldots,\gamma_{m-1}}
  3. Tr ( α i γ j ) = { 0 , if i j 1 , otherwise \operatorname{Tr}(\alpha_{i}\cdot\gamma_{j})=\left\{\begin{matrix}0,&% \operatorname{if}\ i\neq j\\ 1,&\operatorname{otherwise}\end{matrix}\right.
  4. Tr ( β ) = i = 0 m - 1 β p i \operatorname{Tr}(\beta)=\sum_{i=0}^{m-1}\beta^{p^{i}}

Duckworth–Lewis_method.html

  1. Team 2’s par score = Team 1’s score × Team 2’s resources Team 1’s resources \,\text{Team 2's par score }=\,\text{ Team 1's score}\times\frac{\,\text{Team % 2's resources}}{\,\text{Team 1's resources}}
  2. Team 2’s par score = Team 1’s score × Team 2’s resources \,\text{Team 2's par score }=\,\text{ Team 1's score}\times\,\text{Team 2's resources}
  3. Total resources available = Resources at start of innings - Resources lost by first interruption - Resources lost by second interruption - Resources lost by third interruption - etc \begin{matrix}\,\text{Total}\\ \,\text{resources}\\ \,\text{available}\end{matrix}\ \ \ =\ \ \ \begin{matrix}\,\text{Resources}\\ \,\text{at start}\\ \,\text{of innings}\end{matrix}\ \ \ \ -\ \ \ \ \ \ \ \ \ \ \ \ \ \begin{% matrix}\,\text{Resources lost by}\\ \,\text{first interruption}\end{matrix}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ -\ % \ \ \ \ \ \ \ \ \ \ \begin{matrix}\,\text{Resources lost by}\\ \,\text{second interruption}\end{matrix}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ -\ \ \ % \ \ \ \ \ \ \ \ \ \begin{matrix}\,\text{Resources lost by}\\ \,\text{third interruption}\end{matrix}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ -\ \ \ \ % \ \,\text{etc}...
  4. = Resources at start of innings - ( Resources remaining at first interruption - Resources remaining at restart ) - ( Resources remaining at second interruption - Resources remaining at restart ) - ( Resources remaining at third interruption - Resources remaining at restart ) - =\ \ \ \begin{matrix}\,\text{Resources}\\ \,\text{at start}\\ \,\text{of innings}\end{matrix}\ \ \ -\ \ \ \left(\begin{matrix}\,\text{% Resources}\\ \,\text{remaining}\\ \,\text{at first}\\ \,\text{interruption}\end{matrix}\ -\ \begin{matrix}\,\text{Resources}\\ \,\text{remaining}\\ \,\text{at restart}\end{matrix}\right)\ \ \ -\ \ \ \left(\begin{matrix}\,\text% {Resources}\\ \,\text{remaining}\\ \,\text{at second}\\ \,\text{interruption}\end{matrix}\ -\ \begin{matrix}\,\text{Resources}\\ \,\text{remaining}\\ \,\text{at restart}\end{matrix}\right)\ \ \ -\ \ \ \left(\begin{matrix}\,\text% {Resources}\\ \,\text{remaining}\\ \,\text{at third}\\ \,\text{interruption}\end{matrix}\ -\ \begin{matrix}\,\text{Resources}\\ \,\text{remaining}\\ \,\text{at restart}\end{matrix}\right)\ \ \ -\ \ \ ...

Dunkl_operator.html

  1. T i f ( x ) = x i f ( x ) + v R + k v f ( x ) - f ( x σ v ) x , v v i T_{i}f(x)=\frac{\partial}{\partial x_{i}}f(x)+\sum_{v\in R_{+}}k_{v}\frac{f(x)% -f(x\sigma_{v})}{\left\langle x,v\right\rangle}v_{i}
  2. v i v_{i}
  3. T i ( T j f ( x ) ) = T j ( T i f ( x ) ) T_{i}(T_{j}f(x))=T_{j}(T_{i}f(x))

Dupin_cyclide.html

  1. \R 3 \R^{3}
  2. \C 3 \C^{3}
  3. A r 4 + i = 1 3 P i x i r 2 + i , j = 1 3 Q i j x i x j + i = 1 3 R i x i + B = 0 Ar^{4}+\sum_{i=1}^{3}P_{i}x_{i}r^{2}+\sum_{i,j=1}^{3}Q_{ij}x_{i}x_{j}+\sum_{i=% 1}^{3}R_{i}x_{i}+B=0

Dutch_book.html

  1. 1 1 + 1 = 0.5 \frac{1}{1+1}=0.5
  2. 1 3 + 1 = 0.25 \frac{1}{3+1}=0.25
  3. 1 4 + 1 = 0.2 \frac{1}{4+1}=0.2
  4. 1 9 + 1 = 0.1 \frac{1}{9+1}=0.1

Dyadic.html

  1. a A a\in A
  2. b B b\in B

Dynamical_friction.html

  1. 𝐟 d y n = M d 𝐯 M d t = - 4 π Ln ( Λ ) G 2 M 2 ρ v M 3 [ erf ( X ) - 2 X π e - X 2 ] 𝐯 M \,\textbf{f}_{dyn}=M\frac{d\,\textbf{v}_{M}}{dt}=-\frac{4\pi\mbox{Ln}~{}(% \Lambda)G^{2}M^{2}\rho}{v_{M}^{3}}\left[\mbox{erf}~{}(X)-\frac{2X}{\sqrt{\pi}}% e^{-X^{2}}\right]\,\textbf{v}_{M}
  2. v M {v_{M}}
  3. X = v M / ( 2 σ ) X=v_{M}/(\sqrt{2}\sigma)
  4. σ \sigma
  5. erf \mbox{erf}~{}
  6. ρ \rho
  7. Ln ( Λ ) \mbox{Ln}~{}(\Lambda)
  8. 𝐟 d y n \,\textbf{f}_{dyn}
  9. f d y n C G 2 M 2 ρ v M 2 f_{dyn}\approx C\frac{G^{2}M^{2}\rho}{v^{2}_{M}}
  10. C C
  11. v M v_{M}

Dynamical_simulation.html

  1. F = m a \vec{F}=m\vec{a}
  2. d ( 𝐈 s y m b o l ω ) d t = j = 1 N τ j \frac{\mathrm{d}(\mathbf{I}symbol{\omega})}{\mathrm{d}t}=\sum_{j=1}^{N}\tau_{j}
  3. 𝐈 \mathbf{I}
  4. ω \vec{\omega}
  5. τ j \tau_{j}
  6. I 1 ω ˙ 1 + ( I 3 - I 2 ) ω 2 ω 3 = N 1 I 2 ω ˙ 2 + ( I 1 - I 3 ) ω 3 ω 1 = N 2 I 3 ω ˙ 3 + ( I 2 - I 1 ) ω 1 ω 2 = N 3 \begin{matrix}I_{1}\dot{\omega}_{1}+(I_{3}-I_{2})\omega_{2}\omega_{3}&=&N_{1}% \\ I_{2}\dot{\omega}_{2}+(I_{1}-I_{3})\omega_{3}\omega_{1}&=&N_{2}\\ I_{3}\dot{\omega}_{3}+(I_{2}-I_{1})\omega_{1}\omega_{2}&=&N_{3}\end{matrix}
  7. ω {\omega}
  8. I ω ˙ 1 = N 1 I ω ˙ 2 = N 2 I ω ˙ 3 = N 3 \begin{matrix}I\dot{\omega}_{1}&=&N_{1}\\ I\dot{\omega}_{2}&=&N_{2}\\ I\dot{\omega}_{3}&=&N_{3}\end{matrix}
  9. ω {\omega}
  10. I = def V l 2 ( m ) d m = V l 2 ( v ) ρ ( v ) d v = V l 2 ( x , y , z ) ρ ( x , y , z ) d x d y d z I\ \stackrel{\mathrm{def}}{=}\ \int_{V}l^{2}(m)\,dm=\iiint_{V}l^{2}(v)\,\rho(v% )\,dv=\iiint_{V}l^{2}(x,y,z)\,\rho(x,y,z)\,dx\,dy\,dz\!

Dynamometer.html

  1. P = τ ω P=\tau\cdot\omega
  2. P = F v P=F\cdot v
  3. P hp = τ lb ft ω RPM 5252 P_{\mathrm{hp}}={\tau_{\mathrm{lb\cdot ft}}\cdot\omega_{\mathrm{RPM}}\over 5252}
  4. P kW = τ N m ω RPM 9549 P_{\mathrm{kW}}={\tau_{\mathrm{N\cdot m}}\cdot\omega_{\mathrm{RPM}}\over 9549}

E._T._Whittaker.html

  1. E = m c 2 E=mc^{2}

E7_(mathematics).html

  1. ( 8 4 ) \begin{pmatrix}8\\ 4\end{pmatrix}
  2. 4 × ( 6 2 ) 4\times\begin{pmatrix}6\\ 2\end{pmatrix}
  3. ( ± 1 2 , ± 1 2 , ± 1 2 , ± 1 2 , ± 1 2 , ± 1 2 , ± 1 2 ) \left(\pm{1\over 2},\pm{1\over 2},\pm{1\over 2},\pm{1\over 2},\pm{1\over 2},% \pm{1\over 2},\pm{1\over\sqrt{2}}\right)
  4. ( 0 , 0 , 0 , 0 , 0 , 0 , ± 2 ) . \left(0,0,0,0,0,0,\pm\sqrt{2}\right).
  5. ± 2 \pm\sqrt{2}
  6. [ 1 - 1 0 0 0 0 0 0 1 - 1 0 0 0 0 0 0 1 - 1 0 0 0 0 0 0 1 - 1 0 0 0 0 0 0 1 1 0 - 1 2 - 1 2 - 1 2 - 1 2 - 1 2 - 1 2 2 2 0 0 0 0 1 - 1 0 ] . \begin{bmatrix}1&-1&0&0&0&0&0\\ 0&1&-1&0&0&0&0\\ 0&0&1&-1&0&0&0\\ 0&0&0&1&-1&0&0\\ 0&0&0&0&1&1&0\\ -\frac{1}{2}&-\frac{1}{2}&-\frac{1}{2}&-\frac{1}{2}&-\frac{1}{2}&-\frac{1}{2}&% \frac{\sqrt{2}}{2}\\ 0&0&0&0&1&-1&0\\ \end{bmatrix}.
  7. [ 2 - 1 0 0 0 0 0 - 1 2 - 1 0 0 0 0 0 - 1 2 - 1 0 0 0 0 0 - 1 2 - 1 0 - 1 0 0 0 - 1 2 - 1 0 0 0 0 0 - 1 2 0 0 0 0 - 1 0 0 2 ] . \begin{bmatrix}2&-1&0&0&0&0&0\\ -1&2&-1&0&0&0&0\\ 0&-1&2&-1&0&0&0\\ 0&0&-1&2&-1&0&-1\\ 0&0&0&-1&2&-1&0\\ 0&0&0&0&-1&2&0\\ 0&0&0&-1&0&0&2\end{bmatrix}.
  8. C 1 = p q - q p + T r [ P Q ] - T r [ Q P ] C_{1}=pq-qp+Tr[PQ]-Tr[QP]
  9. C 2 = ( p q + T r [ P Q ] ) 2 + p T r [ Q Q ~ ] + q T r [ P P ~ ] + T r [ P ~ Q ~ ] C_{2}=(pq+Tr[P\circ Q])^{2}+pTr[Q\circ\tilde{Q}]+qTr[P\circ\tilde{P}]+Tr[% \tilde{P}\circ\tilde{Q}]
  10. P ~ det ( P ) P - 1 \tilde{P}\equiv\det(P)P^{-1}
  11. A B = ( A B + B A ) / 2 A\circ B=(AB+BA)/2
  12. C 2 = T r [ ( X Y ) 2 ] - 1 4 T r [ X Y ] 2 + 1 96 ϵ i j k l m n o p ( X i j X k l X m n X o p + Y i j Y k l Y m n Y o p ) C_{2}=Tr[(XY)^{2}]-\dfrac{1}{4}Tr[XY]^{2}+\frac{1}{96}\epsilon_{ijklmnop}\left% (X^{ij}X^{kl}X^{mn}X^{op}+Y^{ij}Y^{kl}Y^{mn}Y^{op}\right)
  13. 1 gcd ( 2 , q - 1 ) q 63 ( q 18 - 1 ) ( q 14 - 1 ) ( q 12 - 1 ) ( q 10 - 1 ) ( q 8 - 1 ) ( q 6 - 1 ) ( q 2 - 1 ) \frac{1}{\mathrm{gcd}(2,q-1)}q^{63}(q^{18}-1)(q^{14}-1)(q^{12}-1)(q^{10}-1)(q^% {8}-1)(q^{6}-1)(q^{2}-1)

E8_(mathematics).html

  1. [ J i j , J k ] = δ j k J i - δ j J i k - δ i k J j + δ i J j k [J_{ij},J_{k\ell}]=\delta_{jk}J_{i\ell}-\delta_{j\ell}J_{ik}-\delta_{ik}J_{j% \ell}+\delta_{i\ell}J_{jk}
  2. [ J i j , Q a ] = 1 4 ( γ i γ j - γ j γ i ) a b Q b , [J_{ij},Q_{a}]=\frac{1}{4}(\gamma_{i}\gamma_{j}-\gamma_{j}\gamma_{i})_{ab}Q_{b},
  3. [ Q a , Q b ] = γ a c [ i γ c b j ] J i j . [Q_{a},Q_{b}]=\gamma^{[i}_{ac}\gamma^{j]}_{cb}J_{ij}.
  4. ( ± 1 , ± 1 , 0 , 0 , 0 , 0 , 0 , 0 ) (\pm 1,\pm 1,0,0,0,0,0,0)\,
  5. ( ± 1 2 , ± 1 2 , ± 1 2 , ± 1 2 , ± 1 2 , ± 1 2 , ± 1 2 , ± 1 2 ) \left(\pm\tfrac{1}{2},\pm\tfrac{1}{2},\pm\tfrac{1}{2},\pm\tfrac{1}{2},\pm% \tfrac{1}{2},\pm\tfrac{1}{2},\pm\tfrac{1}{2},\pm\tfrac{1}{2}\right)\,
  6. A i j = 2 ( α i , α j ) ( α i , α i ) A_{ij}=2\frac{(\alpha_{i},\alpha_{j})}{(\alpha_{i},\alpha_{i})}
  7. [ 2 - 1 0 0 0 0 0 0 - 1 2 - 1 0 0 0 0 0 0 - 1 2 - 1 0 0 0 0 0 0 - 1 2 - 1 0 0 0 0 0 0 - 1 2 - 1 0 - 1 0 0 0 0 - 1 2 - 1 0 0 0 0 0 0 - 1 2 0 0 0 0 0 - 1 0 0 2 ] . \left[\begin{smallmatrix}2&-1&0&0&0&0&0&0\\ -1&2&-1&0&0&0&0&0\\ 0&-1&2&-1&0&0&0&0\\ 0&0&-1&2&-1&0&0&0\\ 0&0&0&-1&2&-1&0&-1\\ 0&0&0&0&-1&2&-1&0\\ 0&0&0&0&0&-1&2&0\\ 0&0&0&0&-1&0&0&2\end{smallmatrix}\right].
  8. [ 1 - 1 0 0 0 0 0 0 0 1 - 1 0 0 0 0 0 0 0 1 - 1 0 0 0 0 0 0 0 1 - 1 0 0 0 0 0 0 0 1 - 1 0 0 0 0 0 0 0 1 1 0 - 1 2 - 1 2 - 1 2 - 1 2 - 1 2 - 1 2 - 1 2 - 1 2 0 0 0 0 0 1 - 1 0 ] . \left[\begin{smallmatrix}1&-1&0&0&0&0&0&0\\ 0&1&-1&0&0&0&0&0\\ 0&0&1&-1&0&0&0&0\\ 0&0&0&1&-1&0&0&0\\ 0&0&0&0&1&-1&0&0\\ 0&0&0&0&0&1&1&0\\ -\frac{1}{2}&-\frac{1}{2}&-\frac{1}{2}&-\frac{1}{2}&-\frac{1}{2}&-\frac{1}{2}&% -\frac{1}{2}&-\frac{1}{2}\\ 0&0&0&0&0&1&-1&0\\ \end{smallmatrix}\right].
  9. q 120 ( q 30 - 1 ) ( q 24 - 1 ) ( q 20 - 1 ) ( q 18 - 1 ) ( q 14 - 1 ) ( q 12 - 1 ) ( q 8 - 1 ) ( q 2 - 1 ) q^{120}(q^{30}-1)(q^{24}-1)(q^{20}-1)(q^{18}-1)(q^{14}-1)(q^{12}-1)(q^{8}-1)(q% ^{2}-1)

Earth's_orbit.html

  1. R H = a ( m 3 M ) 1 3 \begin{smallmatrix}R_{H}=a\left(\frac{m}{3M}\right)^{\frac{1}{3}}\end{smallmatrix}
  2. ( 1 3 332 , 946 ) 1 3 = 0.01 \begin{smallmatrix}\left(\frac{1}{3\cdot 332,946}\right)^{\frac{1}{3}}=0.01% \end{smallmatrix}

Earth–Moon–Earth_communication.html

  1. ( 4 π d λ ) 2 (\frac{4\pi d}{\lambda})^{2}
  2. 3 * 10 8 3*10^{8}
  3. 300 F \frac{300}{F}
  4. 4 * π * 10 3 * F * d 300 4*\pi*10^{3}*F*\frac{d}{300}
  5. 32.45 + 20 log F + 20 log d 32.45+20\log{F}+20\log{d}
  6. P r = P t * G t * G r * L o s s P_{r}=P_{t}*G_{t}*G_{r}*Loss
  7. ρ * λ 2 / ( 4 * p i * ) 3 * d 4 \rho*\lambda^{2}/(4*pi*)^{3}*d^{4}
  8. L o s s E M E ( dB ) = 100.4 + 20 log ( F ) + 40 log ( d ) - 10 log ( ρ ) Loss_{EME}\mathrm{(dB)}=100.4+20\log(F)+40\log(d)-10\log(\rho)
  9. ρ = 0.065 * D 2 * π / 4 \rho=0.065*\mathrm{D}^{2}*\pi/4
  10. D \mathrm{D}
  11. ρ = 6.25 * 10 11 m 2 \rho=6.25*10^{11}m^{2}

Eaton_Hodgkinson.html

  1. W = 26 A d L W=\frac{26Ad}{L}

Eccentric_anomaly.html

  1. a a\,\!
  2. x 2 a 2 + y 2 b 2 = 1 , \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\ ,
  3. cos E = x a and sin E = y b . \cos E=\frac{x}{a}\quad\mathrm{and}\quad\sin E=\frac{y}{b}\ .
  4. sin E = y b . \sin E^{\prime}=\frac{y}{b}\ .
  5. y b = 1 - ( x a ) 2 = 1 - cos 2 E = sin E , \frac{y}{b}=\sqrt{1-\left(\frac{x}{a}\right)^{2}}=\sqrt{1-\cos^{2}E}=\sin E\ ,
  6. e = 1 - ( b a ) 2 . e=\sqrt{1-\left(\frac{b}{a}\right)^{2}}\ .
  7. r 2 \displaystyle r^{2}
  8. r = a ( 1 - e cos E ) . r=a\left(1-e\cdot\cos{E}\right)\ .
  9. cos E = x a = a e + r cos θ a = e + ( 1 - e cos E ) cos θ cos E = e + cos θ 1 + e cos θ \cos E=\frac{x}{a}=\frac{ae+r\cos\theta}{a}=e+(1-e\cos E)\cos\theta\ \to\cos E% =\frac{e+\cos\theta}{1+e\cos\theta}
  10. sin E = 1 - cos 2 E = 1 - e 2 sin θ 1 + e cos θ . \sin E=\sqrt{1-\cos^{2}E}=\frac{\sqrt{1-e^{2}}\,\sin\theta}{1+e\cos\theta}\ .
  11. tan E = sin E cos E = 1 - e 2 sin θ e + cos θ . \tan E=\frac{\sin E}{\cos E}=\frac{\sqrt{1-e^{2}}\sin\theta}{e+\cos\theta}\ .
  12. tan θ 2 = 1 + e 1 - e tan E 2 \tan\frac{\theta}{2}=\sqrt{\frac{1+e}{1-e}}\cdot\tan\frac{E}{2}
  13. r = a ( 1 - e 2 ) 1 + e cos θ . r=\frac{a\left(1-e^{2}\right)}{1+e\cos\theta}\ .
  14. E E
  15. M M
  16. M = E - e sin E M=E-e\cdot\sin E
  17. E E
  18. M M

Eccentricity_vector.html

  1. 𝐞 \mathbf{e}\,
  2. 𝐞 = 𝐯 × 𝐡 μ - 𝐫 | 𝐫 | \mathbf{e}={\mathbf{v}\times\mathbf{h}\over{\mu}}-{\mathbf{r}\over{\left|% \mathbf{r}\right|}}
  3. 𝐯 \mathbf{v}\,\!
  4. 𝐡 \mathbf{h}\,\!
  5. 𝐫 × 𝐯 \mathbf{r}\times\mathbf{v}
  6. 𝐫 \mathbf{r}\,\!
  7. μ \mu\,\!
  8. 𝐞 = | 𝐯 | 2 𝐫 μ - ( 𝐫 𝐯 ) 𝐯 μ - 𝐫 | 𝐫 | \mathbf{e}={\mathbf{\left|v\right|}^{2}\mathbf{r}\over{\mu}}-{(\mathbf{r}\cdot% \mathbf{v})\mathbf{v}\over{\mu}}-{\mathbf{r}\over{\left|\mathbf{r}\right|}}

Economic_base_analysis.html

  1. L Q = e i / e E i / E LQ=\frac{e_{i}/e}{E_{i}/E}
  2. e i = e_{i}=
  3. e = e=
  4. E i = E_{i}=
  5. E = E=

Economic_model.html

  1. π ( x , t ) = x p ( x ) - C ( x ) - t x \pi(x,t)=xp(x)-C(x)-tx\quad
  2. p ( x ) p(x)
  3. x x
  4. x p ( x ) xp(x)
  5. C ( x ) C(x)
  6. x x
  7. t t
  8. π ( x , t ) x = ( x p ( x ) - C ( x ) ) x - t = 0 \frac{\partial\pi(x,t)}{\partial x}=\frac{\partial(xp(x)-C(x))}{\partial x}-t=0
  9. 2 ( x p ( x ) - C ( x ) ) 2 x = 2 π ( x , t ) x 2 , \frac{\partial^{2}(xp(x)-C(x))}{\partial^{2}x}={\partial^{2}\pi(x,t)\over% \partial x^{2}},

Economic_order_quantity.html

  1. P P
  2. Q Q
  3. Q * Q^{*}
  4. D D
  5. K K
  6. h h
  7. T C = P D + D K Q + h Q 2 TC=PD+{\frac{DK}{Q}}+{\frac{hQ}{2}}
  8. 0 = - D K Q 2 + h 2 {0}=-{\frac{DK}{Q^{2}}}+{\frac{h}{2}}
  9. Q * 2 = 2 D K h Q^{*2}={\frac{2DK}{h}}
  10. T C = D K Q + h Q 2 + P D = h 2 Q ( Q - 2 D K / h ) 2 + 2 h D K + P D , TC={\frac{DK}{Q}}+{\frac{hQ}{2}}+PD={\frac{h}{2Q}}(Q-\sqrt{2DK/h})^{2}+\sqrt{2% hDK}+PD,
  11. Q = 2 D K / h , Q=\sqrt{2DK/h},
  12. T C m i n = 2 h D K + P D . TC_{min}=\sqrt{2hDK}+PD.
  13. 2 D * K h \sqrt{\frac{2D*K}{h}}
  14. = 2 * 10000 * 2 8 * 0.02 =\sqrt{\frac{2*10000*2}{8*0.02}}
  15. = 10000 500 = 20 ={\frac{10000}{500}}=20
  16. = P * D + K ( D / E O Q ) + h ( E O Q / 2 ) =P*D+K(D/EOQ)+h(EOQ/2)
  17. = 8 * 10000 + 2 ( 10000 / 500 ) + 0.16 ( 500 / 2 ) = $ 80080 =8*10000+2(10000/500)+0.16(500/2)=\$80080
  18. = 8 * 10000 + 2 ( 10000 / 300 ) + 0.16 ( 300 / 2 ) = $ 80091 =8*10000+2(10000/300)+0.16(300/2)=\$80091

Econophysics.html

  1. P x - 4 , P\propto x^{-4}\,,

Edge_coloring.html

  1. k k
  2. k k
  3. Δ Δ
  4. Δ + 1 Δ+1
  5. Δ Δ
  6. 3 Δ / 2 3Δ/2
  7. Δ + 1 Δ+1
  8. n n
  9. n 1 n−1
  10. n n
  11. n n
  12. ( n 1 ) (n−1)
  13. n n
  14. n n
  15. ( n 1 ) / 2 (n−1)/2
  16. 1 / n 1/n
  17. n n
  18. n 1 n−1
  19. 2 2
  20. n = 3 n=3
  21. n = 6 n=6
  22. n n
  23. n + 1 n+1
  24. n n
  25. G G
  26. L ( G ) L(G)
  27. G G
  28. G G
  29. k k
  30. k k
  31. k k
  32. k k
  33. G G
  34. χ ( G ) χ′(G)
  35. k k
  36. k k
  37. χ ( G ) χ(G)
  38. G G
  39. G G
  40. m m
  41. β β
  42. m / β m/β
  43. m = 24 m=24
  44. β = 7 β=7
  45. k k
  46. k + 1 k+1
  47. k k
  48. χ m / β χ′≥m/β
  49. m = 15 m=15
  50. β = 5 β=5
  51. G G
  52. Δ ( G ) Δ(G)
  53. G G
  54. χ ( G ) Δ ( G ) χ′(G)≥Δ(G)
  55. Δ Δ
  56. v v
  57. Δ Δ
  58. Δ ( G ) Δ(G)
  59. Δ ( G ) + 1 Δ(G)+1
  60. χ ( G ) = Δ ( G ) χ′(G)=Δ(G)
  61. χ ( G ) χ′(G)
  62. Δ ( G ) Δ(G)
  63. μ ( G ) μ(G)
  64. μ ( G ) μ(G)
  65. Δ ( G ) = 2 μ ( G ) Δ(G)=2μ(G)
  66. μ ( G ) μ(G)
  67. 3 μ ( G ) 3μ(G)
  68. 3 μ ( G ) 3μ(G)
  69. χ ( G ) ( 3 / 2 ) Δ ( G ) χ′(G)≤(3/2)Δ(G)
  70. G G
  71. G G
  72. χ ( G ) Δ ( G ) + μ ( G ) χ′(G)≤Δ(G)+μ(G)
  73. μ ( G ) = 1 μ(G)=1
  74. Δ Δ
  75. Δ Δ
  76. O ( m l o g Δ ) O(mlogΔ)
  77. m m
  78. O ( m l o g m ) O(mlogm)
  79. Δ 7 Δ≥7
  80. Δ Δ
  81. Δ 9 Δ≥9
  82. Δ + 1 Δ+1
  83. G G
  84. H H
  85. G G
  86. u u
  87. v v
  88. u u
  89. v v
  90. G G
  91. H H
  92. H H
  93. G G
  94. H H
  95. G G
  96. O ( m l o g Δ ) O(mlogΔ)
  97. 3 Δ 2 3\left\lceil\frac{\Delta}{2}\right\rceil
  98. 3 Δ 2 \left\lfloor\frac{3\Delta}{2}\right\rfloor
  99. 2 Δ 1 2Δ−1
  100. Δ + Δ / 2 \Delta+\sqrt{\Delta/2}
  101. n n
  102. n / 2 n/2
  103. m m
  104. w w
  105. w w
  106. n n
  107. k k
  108. k k
  109. k ! k!
  110. k 3 k≠3
  111. k k
  112. k = 3 k=3
  113. k k
  114. G ( 6 n + 3 , 2 ) G(6n+3,2)
  115. n 2 n≥2
  116. G ( 9 , 2 ) G(9,2)
  117. G G
  118. P P
  119. Q Q
  120. Q Q
  121. P > Q P>Q
  122. P P
  123. Q Q
  124. P P
  125. i i
  126. j j
  127. l a ( G ) la(G)
  128. G G
  129. G G
  130. l a ( G ) χ ( G ) 2 l a ( G ) la(G)≤χ′(G)≤2la(G)
  131. la ( G ) Δ + 1 2 \mathop{\mathrm{la}}(G)\leq\left\lceil\frac{\Delta+1}{2}\right\rceil
  132. 2 l a ( G ) 2 χ ( G ) 2 l a ( G ) 2la(G)−2≤χ′(G)≤2la(G)
  133. l a ( G ) la(G)
  134. χ ( G ) 2 l a ( G ) χ′(G)≤2la(G)
  135. G G
  136. k k
  137. k k
  138. n n
  139. G G
  140. a ( G ) a^{\prime}(G)
  141. G G
  142. a ( G ) Δ + 2 a^{\prime}(G)\leq\Delta+2
  143. Δ \Delta
  144. G G
  145. a ( G ) 3.74 ( Δ - 1 ) a^{\prime}(G)\leq\lceil 3.74(\Delta-1)\rceil
  146. G G
  147. c c
  148. G G
  149. c Δ log Δ c\Delta\log\Delta
  150. a ( G ) Δ + 2 a^{\prime}(G)\leq\Delta+2
  151. ϵ > 0 \epsilon>0
  152. g g
  153. G G
  154. g g
  155. a ( G ) ( 1 + ϵ ) Δ a^{\prime}(G)\leq(1+\epsilon)\Delta
  156. d d
  157. d d
  158. k k
  159. K < s u b > n K<sub>n
  160. n n
  161. ( n 1 ) (n−1)

Edgeworth_series.html

  1. F F
  2. F F
  3. f f
  4. F F
  5. κ r \kappa_{r}
  6. Ψ Ψ
  7. ψ ψ
  8. γ r \gamma_{r}
  9. Ψ Ψ
  10. f ( t ) = exp [ r = 1 κ r ( i t ) r r ! ] f(t)=\exp\left[\sum_{r=1}^{\infty}\kappa_{r}\frac{(it)^{r}}{r!}\right]
  11. ψ ( t ) = exp [ r = 1 γ r ( i t ) r r ! ] , \psi(t)=\exp\left[\sum_{r=1}^{\infty}\gamma_{r}\frac{(it)^{r}}{r!}\right],
  12. f ( t ) = exp [ r = 1 ( κ r - γ r ) ( i t ) r r ! ] ψ ( t ) . f(t)=\exp\left[\sum_{r=1}^{\infty}(\kappa_{r}-\gamma_{r})\frac{(it)^{r}}{r!}% \right]\psi(t)\,.
  13. ( i t ) r ψ ( t ) (it)^{r}\psi(t)
  14. ( - 1 ) r [ D r Ψ ] ( - x ) (-1)^{r}[D^{r}\Psi](-x)
  15. D D
  16. x x
  17. x x
  18. - x -x
  19. F F
  20. F ( x ) = exp [ r = 1 ( κ r - γ r ) ( - D ) r r ! ] Ψ ( x ) . F(x)=\exp\left[\sum_{r=1}^{\infty}(\kappa_{r}-\gamma_{r})\frac{(-D)^{r}}{r!}% \right]\Psi(x)\,.
  21. Ψ Ψ
  22. F F
  23. μ = κ 1 \mu=\kappa_{1}
  24. σ 2 = κ 2 \sigma^{2}=\kappa_{2}
  25. F ( x ) = exp [ r = 3 κ r ( - D ) r r ! ] 1 2 π σ exp [ - ( x - μ ) 2 2 σ 2 ] . F(x)=\exp\left[\sum_{r=3}^{\infty}\kappa_{r}\frac{(-D)^{r}}{r!}\right]\frac{1}% {\sqrt{2\pi}\sigma}\exp\left[-\frac{(x-\mu)^{2}}{2\sigma^{2}}\right].
  26. γ r = 0 \gamma_{r}=0
  27. r r
  28. F ( x ) 1 2 π σ exp [ - ( x - μ ) 2 2 σ 2 ] [ 1 + κ 3 3 ! σ 3 H 3 ( x - μ σ ) + κ 4 4 ! σ 4 H 4 ( x - μ σ ) ] , F(x)\approx\frac{1}{\sqrt{2\pi}\sigma}\exp\left[-\frac{(x-\mu)^{2}}{2\sigma^{2% }}\right]\left[1+\frac{\kappa_{3}}{3!\sigma^{3}}H_{3}\left(\frac{x-\mu}{\sigma% }\right)+\frac{\kappa_{4}}{4!\sigma^{4}}H_{4}\left(\frac{x-\mu}{\sigma}\right)% \right]\,,
  29. H 3 ( x ) = x 3 - 3 x H_{3}(x)=x^{3}-3x
  30. H 4 ( x ) = x 4 - 6 x 2 + 3 H_{4}(x)=x^{4}-6x^{2}+3
  31. F ( x ) F(x)
  32. exp ( - ( x 2 ) / 4 ) \exp(-(x^{2})/4)
  33. Y n = 1 n i = 1 n X i - μ σ . Y_{n}=\frac{1}{\sqrt{n}}\sum_{i=1}^{n}\frac{X_{i}-\mu}{\sigma}.
  34. lim n F n ( x ) = Φ ( x ) - x 1 2 π e - 1 2 q 2 d q \lim_{n\to\infty}F_{n}(x)=\Phi(x)\equiv\int_{-\infty}^{x}\tfrac{1}{\sqrt{2\pi}% }e^{-\frac{1}{2}q^{2}}dq
  35. Ψ ( x ) = 1 2 π exp ( - 1 2 x 2 ) \Psi(x)=\frac{1}{\sqrt{2\pi}}\exp(-\tfrac{1}{2}x^{2})
  36. κ 1 F ( n ) - γ 1 = 0 , \kappa^{F(n)}_{1}-\gamma_{1}=0,
  37. κ 2 F ( n ) - γ 2 = 0 , \kappa^{F(n)}_{2}-\gamma_{2}=0,
  38. κ r F ( n ) - γ r = κ r σ r n r / 2 - 1 = λ r n r / 2 - 1 ; r 3. \kappa^{F(n)}_{r}-\gamma_{r}=\frac{\kappa_{r}}{\sigma^{r}n^{r/2-1}}=\frac{% \lambda_{r}}{n^{r/2-1}};\qquad r\geq 3.
  39. f n ( t ) = [ 1 + j = 1 P j ( i t ) n j / 2 ] exp ( - t 2 / 2 ) , f_{n}(t)=\left[1+\sum_{j=1}^{\infty}\frac{P_{j}(it)}{n^{j/2}}\right]\exp(-t^{2% }/2)\,,
  40. F n ( x ) = Φ ( x ) + j = 1 P j ( - D ) n j / 2 Φ ( x ) . F_{n}(x)=\Phi(x)+\sum_{j=1}^{\infty}\frac{P_{j}(-D)}{n^{j/2}}\Phi(x)\,.
  41. F n ( x ) = Φ ( x ) - 1 n 1 2 ( 1 6 λ 3 Φ ( 3 ) ( x ) ) + 1 n ( 1 24 λ 4 Φ ( 4 ) ( x ) + 1 72 λ 3 2 Φ ( 6 ) ( x ) ) - 1 n 3 2 ( 1 120 λ 5 Φ ( 5 ) ( x ) + 1 144 λ 3 λ 4 Φ ( 7 ) ( x ) + 1 1296 λ 3 3 Φ ( 9 ) ( x ) ) + 1 n 2 ( 1 720 λ 6 Φ ( 6 ) ( x ) + ( 1 1152 λ 4 2 + 1 720 λ 3 λ 5 ) Φ ( 8 ) ( x ) + 1 1728 λ 3 2 λ 4 Φ ( 10 ) ( x ) + 1 31104 λ 3 4 Φ ( 12 ) ( x ) ) + O ( n - 5 2 ) . \begin{aligned}\displaystyle F_{n}(x)&\displaystyle=\Phi(x)\\ &\displaystyle\quad-\frac{1}{n^{\frac{1}{2}}}\left(\tfrac{1}{6}\lambda_{3}\,% \Phi^{(3)}(x)\right)\\ &\displaystyle\quad+\frac{1}{n}\left(\tfrac{1}{24}\lambda_{4}\,\Phi^{(4)}(x)+% \tfrac{1}{72}\lambda_{3}^{2}\,\Phi^{(6)}(x)\right)\\ &\displaystyle\quad-\frac{1}{n^{\frac{3}{2}}}\left(\tfrac{1}{120}\lambda_{5}\,% \Phi^{(5)}(x)+\tfrac{1}{144}\lambda_{3}\lambda_{4}\,\Phi^{(7)}(x)+\tfrac{1}{12% 96}\lambda_{3}^{3}\,\Phi^{(9)}(x)\right)\\ &\displaystyle\quad+\frac{1}{n^{2}}\left(\tfrac{1}{720}\lambda_{6}\,\Phi^{(6)}% (x)+\left(\tfrac{1}{1152}\lambda_{4}^{2}+\tfrac{1}{720}\lambda_{3}\lambda_{5}% \right)\Phi^{(8)}(x)+\tfrac{1}{1728}\lambda_{3}^{2}\lambda_{4}\,\Phi^{(10)}(x)% +\tfrac{1}{31104}\lambda_{3}^{4}\,\Phi^{(12)}(x)\right)\\ &\displaystyle\quad+O\left(n^{-\frac{5}{2}}\right).\end{aligned}
  42. Φ ( · ) Φ(·)
  43. X i χ 2 ( k = 2 ) i = 1 , 2 , 3 X_{i}\sim\chi^{2}(k=2)\qquad i=1,2,3
  44. X ¯ = 1 3 i = 1 3 X i \bar{X}=\frac{1}{3}\sum_{i=1}^{3}X_{i}
  45. X ¯ \bar{X}
  46. X ¯ Gamma ( α = n k / 2 , θ = 2 / n ) \bar{X}\sim\mathrm{Gamma}\left(\alpha=n\cdot k/2,\theta=2/n\right)
  47. Gamma ( α = 3 , θ = 2 / 3 ) \mathrm{Gamma}\left(\alpha=3,\theta=2/3\right)
  48. X ¯ n N ( k , 2 k / n ) = N ( 2 , 4 / 3 ) \bar{X}\xrightarrow{n\to\infty}N(k,2\cdot k/n)=N(2,4/3)

Eduard_Study.html

  1. q = a + b i + c j + d k q=a+bi+cj+dk\!
  2. e 0 = 1 , e 1 = i , e 2 = j , e 3 = k , e_{0}=1,\ e_{1}=i,\ e_{2}=j,\ e_{3}=k,\!
  3. ε 0 = ε , ε 1 = ε i , ε 2 = ε j , ε 3 = ε k . \varepsilon_{0}=\varepsilon,\ \varepsilon_{1}=\varepsilon i,\ \varepsilon_{2}=% \varepsilon j,\ \varepsilon_{3}=\varepsilon k.\!
  4. i 2 = - 1 , ε 2 = 0 , i ε = - ε i = η . i^{2}=-1,\ \varepsilon^{2}=0,\ i\varepsilon=-\varepsilon i=\eta.\!

Effective_field_theory.html

  1. n \displaystyle n

Egerton_Gospel.html

  1. 𝔓 \mathfrak{P}
  2. 𝔓 \mathfrak{P}
  3. 𝔓 \mathfrak{P}
  4. 𝔓 \mathfrak{P}
  5. 𝔓 \mathfrak{P}

Egyptian_numerals.html

  1. 2 / 3 {2}/{3}
  2. 3 / 4 {3}/{4}
  3. 1 / 3 {1}/{3}
  4. = 1 3 =\frac{1}{3}
  5. 1 / 2 {1}/{2}
  6. 2 / 3 {2}/{3}
  7. 3 / 4 {3}/{4}
  8. = 1 2 =\frac{1}{2}
  9. = 2 3 =\frac{2}{3}
  10. = 3 4 =\frac{3}{4}
  11. = 1 331 =\frac{1}{331}

Ehrenfeucht–Fraïssé_game.html

  1. γ \gamma
  2. ω \omega
  3. 𝔄 \mathfrak{A}
  4. 𝔅 \mathfrak{B}
  5. G n ( 𝔄 , 𝔅 ) G_{n}(\mathfrak{A},\mathfrak{B})
  6. a 1 a_{1}
  7. 𝔄 \mathfrak{A}
  8. b 1 b_{1}
  9. 𝔅 \mathfrak{B}
  10. 𝔄 \mathfrak{A}
  11. b 1 b_{1}
  12. 𝔅 \mathfrak{B}
  13. a 1 a_{1}
  14. 𝔄 \mathfrak{A}
  15. a 2 a_{2}
  16. 𝔄 \mathfrak{A}
  17. b 2 b_{2}
  18. 𝔅 \mathfrak{B}
  19. a 2 a_{2}
  20. b 2 b_{2}
  21. 𝔄 \mathfrak{A}
  22. 𝔅 \mathfrak{B}
  23. n - 2 n-2
  24. a 1 , , a n a_{1},\dots,a_{n}
  25. 𝔄 \mathfrak{A}
  26. b 1 , , b n b_{1},\dots,b_{n}
  27. 𝔅 \mathfrak{B}
  28. { 1 , , n } \{1,\dots,n\}
  29. 𝔄 \mathfrak{A}
  30. i i
  31. a i a_{i}
  32. 𝔅 \mathfrak{B}
  33. i i
  34. b i b_{i}
  35. 𝔄 n 𝔅 \mathfrak{A}\stackrel{n}{\sim}\mathfrak{B}
  36. G n ( 𝔄 , 𝔅 ) G_{n}(\mathfrak{A},\mathfrak{B})
  37. 𝔄 𝔅 \mathfrak{A}\sim\mathfrak{B}
  38. 𝔄 𝔅 \mathfrak{A}\sim\mathfrak{B}
  39. 𝔄 \mathfrak{A}
  40. 𝔅 \mathfrak{B}
  41. \exists
  42. \forall

Eigenplane.html

  1. M [ 𝐬 𝐭 ] = [ 𝐬 𝐭 ] Λ θ M\;[\mathbf{s}\;\mathbf{t}]\;=\;[\mathbf{s}\;\mathbf{t}]\Lambda_{\theta}

Eightfold_Way_(physics).html

  1. 1 2 \frac{1}{2}
  2. 3 2 \frac{3}{2}
  3. up quark ( 1 0 0 ) , down quark ( 0 1 0 ) , strange quark ( 0 0 1 ) \,\text{up quark}\rightarrow\begin{pmatrix}1\\ 0\\ 0\end{pmatrix},\qquad\,\text{down quark}\rightarrow\begin{pmatrix}0\\ 1\\ 0\end{pmatrix},\qquad\,\text{strange quark}\rightarrow\begin{pmatrix}0\\ 0\\ 1\end{pmatrix}
  4. ( x y z ) A ( x y z ) , where A is in S U ( 3 ) \begin{pmatrix}x\\ y\\ z\end{pmatrix}\mapsto A\begin{pmatrix}x\\ y\\ z\end{pmatrix},\quad\,\text{where }A\,\text{ is in }SU(3)
  5. A = ( 0 1 0 - 1 0 0 0 0 1 ) A=\begin{pmatrix}0&1&0\\ -1&0&0\\ 0&0&1\end{pmatrix}
  6. | ψ |\psi\rangle
  7. A | ψ A|\psi\rangle

Eilenberg–MacLane_space.html

  1. i = 1 k 𝐒 1 \textstyle\bigvee_{i=1}^{k}\mathbf{S}^{1}
  2. H n ( K ( G , n ) ; G ) = Hom ( H n ( K ( G , n ) ; 𝐙 ) , G ) = Hom ( π n ( K ( G , n ) ) , G ) = Hom ( G , G ) , H^{n}(K(G,n);G)=\mathrm{Hom}(H_{n}(K(G,n);\mathbf{Z}),G)=\mathrm{Hom}(\pi_{n}(% K(G,n)),G)=\mathrm{Hom}(G,G),
  3. u H n ( K ( G , n ) ; G ) u\in H^{n}(K(G,n);G)
  4. f f * u f\mapsto f^{*}u
  5. n n
  6. a : G G a:G\to G^{\prime}
  7. K ( a , n ) = { [ f ] : f : K ( G , n ) K ( G , n ) , H n ( f ) = a } , K(a,n)=\{[f]:f:K(G,n)\to K(G^{\prime},n),H_{n}(f)=a\},
  8. K ( a b , n ) K ( a , n ) K ( b , n ) and 1 K ( 1 , n ) , K(a\circ b,n)\supset K(a,n)\circ K(b,n)\mbox{ and }~{}1\in K(1,n),
  9. [ f ] [f]
  10. f f
  11. S T := { s t : s S , t T } . S\circ T:=\{s\circ t:s\in S,t\in T\}.

Einselection.html

  1. | ϵ i |\epsilon_{i}\rangle
  2. ϵ i | ϵ j = δ i j \langle\epsilon_{i}|\epsilon_{j}\rangle=\delta_{ij}
  3. | ψ |\psi\rangle
  4. Ψ ( t ) = - Tr ( ρ Ψ ( t ) log ρ Ψ ( t ) ) \mathcal{H}_{\Psi}(t)=-\operatorname{Tr}\left(\rho_{\Psi}(t)\log\rho_{\Psi}(t)\right)
  5. ρ Ψ ( t ) \rho_{\Psi}\left(t\right)
  6. ρ Ψ ( 0 ) = | Ψ Ψ | \rho_{\Psi}(0)=|\Psi\rangle\langle\Psi|
  7. | Ψ \left|\Psi\right\rangle
  8. Ψ \mathcal{H}_{\Psi}\,
  9. | Ψ \left|\Psi\right\rangle
  10. t t

Einstein_manifold.html

  1. Ric = k g , \mathrm{Ric}=k\,g,
  2. R a b = k g a b . R_{ab}=k\,g_{ab}.
  3. R = n k R=nk\,
  4. R a b - 1 2 g a b R + g a b Λ = 8 π T a b , R_{ab}-\frac{1}{2}g_{ab}R+g_{ab}\Lambda=8\pi T_{ab},
  5. R a b = 2 Λ n - 2 g a b . R_{ab}=\frac{2\Lambda}{n-2}\,g_{ab}.

Einstein_tensor.html

  1. 𝐆 \mathbf{G}
  2. 𝐆 = 𝐑 - 1 2 𝐠 R , \mathbf{G}=\mathbf{R}-\frac{1}{2}\mathbf{g}R,
  3. 𝐑 \mathbf{R}
  4. 𝐠 \mathbf{g}
  5. R R
  6. G μ ν = R μ ν - 1 2 g μ ν R . G_{\mu\nu}=R_{\mu\nu}-{1\over 2}g_{\mu\nu}R.
  7. G μ ν = G ν μ G_{\mu\nu}=G_{\nu\mu}
  8. μ G μ ν = 0 . \nabla_{\mu}G^{\mu\nu}=0\,.
  9. G α β \displaystyle G_{\alpha\beta}
  10. δ β α \delta^{\alpha}_{\beta}
  11. Γ β γ α \Gamma^{\alpha}_{\beta\gamma}
  12. Γ β γ α = 1 2 g α ϵ ( g β ϵ , γ + g γ ϵ , β - g β γ , ϵ ) . \Gamma^{\alpha}_{\beta\gamma}=\frac{1}{2}g^{\alpha\epsilon}(g_{\beta\epsilon,% \gamma}+g_{\gamma\epsilon,\beta}-g_{\beta\gamma,\epsilon}).
  13. 2 × ( 6 + 6 + 9 + 9 ) = 60 2\times(6+6+9+9)=60
  14. G α β \displaystyle G_{\alpha\beta}
  15. g α [ β , γ ] ϵ = 1 2 ( g α β , γ ϵ - g α γ , β ϵ ) . g_{\alpha[\beta,\gamma]\epsilon}\,=\frac{1}{2}(g_{\alpha\beta,\gamma\epsilon}-% g_{\alpha\gamma,\beta\epsilon}).
  16. g μ ν g^{\mu\nu}
  17. n n
  18. g μ ν G μ ν \displaystyle g^{\mu\nu}G_{\mu\nu}
  19. G G\,
  20. R R\,
  21. G μ ν = 8 π G c 4 T μ ν . G_{\mu\nu}=\frac{8\pi G}{c^{4}}T_{\mu\nu}.
  22. G μ ν = 8 π T μ ν . G_{\mu\nu}=8\pi\,T_{\mu\nu}.
  23. μ G μ ν = 0. \nabla_{\mu}G^{\mu\nu}=0.
  24. μ T μ ν = 0. \nabla_{\mu}T^{\mu\nu}=0.
  25. ξ μ \xi^{\mu}
  26. μ ( - g T μ ξ ν ν ) = 0 \partial_{\mu}(\sqrt{-g}T^{\mu}{}_{\nu}\xi^{\nu})=0
  27. g μ ν g_{\mu\nu}

Einstein–Hilbert_action.html

  1. S = 1 2 κ R - g d 4 x , S={1\over 2\kappa}\int R\sqrt{-g}\,\mathrm{d}^{4}x\;,
  2. g = det ( g μ ν ) g=\det(g_{\mu\nu})
  3. R R
  4. κ = 8 π G c - 4 \kappa=8\pi Gc^{-4}
  5. G G
  6. c c
  7. S S
  8. M \mathcal{L}_{\mathrm{M}}
  9. S = [ 1 2 κ R + M ] - g d 4 x S=\int\left[{1\over 2\kappa}\,R+\mathcal{L}_{\mathrm{M}}\right]\sqrt{-g}\,% \mathrm{d}^{4}x
  10. 0 = δ S = [ 1 2 κ δ ( - g R ) δ g μ ν + δ ( - g M ) δ g μ ν ] δ g μ ν d 4 x = [ 1 2 κ ( δ R δ g μ ν + R - g δ - g δ g μ ν ) + 1 - g δ ( - g M ) δ g μ ν ] δ g μ ν - g d 4 x . \begin{aligned}\displaystyle 0&\displaystyle=\delta S\\ &\displaystyle=\int\left[{1\over 2\kappa}\frac{\delta(\sqrt{-g}R)}{\delta g^{% \mu\nu}}+\frac{\delta(\sqrt{-g}\mathcal{L}_{\mathrm{M}})}{\delta g^{\mu\nu}}% \right]\delta g^{\mu\nu}\mathrm{d}^{4}x\\ &\displaystyle=\int\left[{1\over 2\kappa}\left(\frac{\delta R}{\delta g^{\mu% \nu}}+\frac{R}{\sqrt{-g}}\frac{\delta\sqrt{-g}}{\delta g^{\mu\nu}}\right)+% \frac{1}{\sqrt{-g}}\frac{\delta(\sqrt{-g}\mathcal{L}_{\mathrm{M}})}{\delta g^{% \mu\nu}}\right]\delta g^{\mu\nu}\sqrt{-g}\,\mathrm{d}^{4}x.\end{aligned}
  11. δ g μ ν \delta g^{\mu\nu}
  12. δ R δ g μ ν + R - g δ - g δ g μ ν = - 2 κ 1 - g δ ( - g M ) δ g μ ν , \frac{\delta R}{\delta g^{\mu\nu}}+\frac{R}{\sqrt{-g}}\frac{\delta\sqrt{-g}}{% \delta g^{\mu\nu}}=-2\kappa\frac{1}{\sqrt{-g}}\frac{\delta(\sqrt{-g}\mathcal{L% }_{\mathrm{M}})}{\delta g^{\mu\nu}},
  13. T μ ν := - 2 - g δ ( - g M ) δ g μ ν = - 2 δ M δ g μ ν + g μ ν M . T_{\mu\nu}:=\frac{-2}{\sqrt{-g}}\frac{\delta(\sqrt{-g}\mathcal{L}_{\mathrm{M}}% )}{\delta g^{\mu\nu}}=-2\frac{\delta\mathcal{L}_{\mathrm{M}}}{\delta g^{\mu\nu% }}+g_{\mu\nu}\mathcal{L}_{\mathrm{M}}.
  14. R ρ σ μ ν = μ Γ ν σ ρ - ν Γ μ σ ρ + Γ μ λ ρ Γ ν σ λ - Γ ν λ ρ Γ μ σ λ . {R^{\rho}}_{\sigma\mu\nu}=\partial_{\mu}\Gamma^{\rho}_{\nu\sigma}-\partial_{% \nu}\Gamma^{\rho}_{\mu\sigma}+\Gamma^{\rho}_{\mu\lambda}\Gamma^{\lambda}_{\nu% \sigma}-\Gamma^{\rho}_{\nu\lambda}\Gamma^{\lambda}_{\mu\sigma}.
  15. Γ μ ν λ \Gamma^{\lambda}_{\mu\nu}
  16. δ R ρ σ μ ν = μ δ Γ ν σ ρ - ν δ Γ μ σ ρ + δ Γ μ λ ρ Γ ν σ λ + Γ μ λ ρ δ Γ ν σ λ - δ Γ ν λ ρ Γ μ σ λ - Γ ν λ ρ δ Γ μ σ λ . \delta{R^{\rho}}_{\sigma\mu\nu}=\partial_{\mu}\delta\Gamma^{\rho}_{\nu\sigma}-% \partial_{\nu}\delta\Gamma^{\rho}_{\mu\sigma}+\delta\Gamma^{\rho}_{\mu\lambda}% \Gamma^{\lambda}_{\nu\sigma}+\Gamma^{\rho}_{\mu\lambda}\delta\Gamma^{\lambda}_% {\nu\sigma}-\delta\Gamma^{\rho}_{\nu\lambda}\Gamma^{\lambda}_{\mu\sigma}-% \Gamma^{\rho}_{\nu\lambda}\delta\Gamma^{\lambda}_{\mu\sigma}.
  17. δ Γ ν μ ρ \delta\Gamma^{\rho}_{\nu\mu}
  18. λ ( δ Γ ν μ ρ ) = λ ( δ Γ ν μ ρ ) + Γ σ λ ρ δ Γ ν μ σ - Γ ν λ σ δ Γ σ μ ρ - Γ μ λ σ δ Γ ν σ ρ . \nabla_{\lambda}(\delta\Gamma^{\rho}_{\nu\mu})=\partial_{\lambda}(\delta\Gamma% ^{\rho}_{\nu\mu})+\Gamma^{\rho}_{\sigma\lambda}\delta\Gamma^{\sigma}_{\nu\mu}-% \Gamma^{\sigma}_{\nu\lambda}\delta\Gamma^{\rho}_{\sigma\mu}-\Gamma^{\sigma}_{% \mu\lambda}\delta\Gamma^{\rho}_{\nu\sigma}.
  19. δ R ρ = σ μ ν μ ( δ Γ ν σ ρ ) - ν ( δ Γ μ σ ρ ) . \delta R^{\rho}{}_{\sigma\mu\nu}=\nabla_{\mu}(\delta\Gamma^{\rho}_{\nu\sigma})% -\nabla_{\nu}(\delta\Gamma^{\rho}_{\mu\sigma}).
  20. δ R μ ν δ R ρ = μ ρ ν ρ ( δ Γ ν μ ρ ) - ν ( δ Γ ρ μ ρ ) . \delta R_{\mu\nu}\equiv\delta R^{\rho}{}_{\mu\rho\nu}=\nabla_{\rho}(\delta% \Gamma^{\rho}_{\nu\mu})-\nabla_{\nu}(\delta\Gamma^{\rho}_{\rho\mu}).
  21. R = g μ ν R μ ν . R=g^{\mu\nu}R_{\mu\nu}.\!
  22. g μ ν g^{\mu\nu}
  23. δ R = R μ ν δ g μ ν + g μ ν δ R μ ν = R μ ν δ g μ ν + σ ( g μ ν δ Γ ν μ σ - g μ σ δ Γ ρ μ ρ ) . \begin{aligned}\displaystyle\delta R&\displaystyle=R_{\mu\nu}\delta g^{\mu\nu}% +g^{\mu\nu}\delta R_{\mu\nu}\\ &\displaystyle=R_{\mu\nu}\delta g^{\mu\nu}+\nabla_{\sigma}\left(g^{\mu\nu}% \delta\Gamma^{\sigma}_{\nu\mu}-g^{\mu\sigma}\delta\Gamma^{\rho}_{\rho\mu}% \right).\end{aligned}
  24. σ g μ ν = 0 \nabla_{\sigma}g^{\mu\nu}=0
  25. σ ( g μ ν δ Γ ν μ σ - g μ σ δ Γ ρ μ ρ ) \nabla_{\sigma}(g^{\mu\nu}\delta\Gamma^{\sigma}_{\nu\mu}-g^{\mu\sigma}\delta% \Gamma^{\rho}_{\rho\mu})
  26. - g \sqrt{-g}
  27. - g A ; a a = ( - g A a ) , a or - g μ A μ = μ ( - g A μ ) \sqrt{-g}A^{a}_{;a}=(\sqrt{-g}A^{a})_{,a}\;\mathrm{or}\;\sqrt{-g}\nabla_{\mu}A% ^{\mu}=\partial_{\mu}\left(\sqrt{-g}A^{\mu}\right)
  28. δ g μ ν \delta g^{\mu\nu}
  29. δ R δ g μ ν = R μ ν . \frac{\delta R}{\delta g^{\mu\nu}}=R_{\mu\nu}.
  30. δ g = δ det ( g μ ν ) = g g μ ν δ g μ ν \,\!\delta g=\delta\det(g_{\mu\nu})=g\,g^{\mu\nu}\delta g_{\mu\nu}
  31. g μ ν g_{\mu\nu}\!
  32. δ - g = - 1 2 - g δ g = 1 2 - g ( g μ ν δ g μ ν ) = - 1 2 - g ( g μ ν δ g μ ν ) . \begin{aligned}\displaystyle\delta\sqrt{-g}&\displaystyle=-\frac{1}{2\sqrt{-g}% }\delta g&\displaystyle=\frac{1}{2}\sqrt{-g}(g^{\mu\nu}\delta g_{\mu\nu})&% \displaystyle=-\frac{1}{2}\sqrt{-g}(g_{\mu\nu}\delta g^{\mu\nu})\,.\end{aligned}
  33. g μ ν δ g μ ν = - g μ ν δ g μ ν g_{\mu\nu}\delta g^{\mu\nu}=-g^{\mu\nu}\delta g_{\mu\nu}
  34. δ g μ ν = - g μ α ( δ g α β ) g β ν . \delta g^{\mu\nu}=-g^{\mu\alpha}(\delta g_{\alpha\beta})g^{\beta\nu}\,.
  35. 1 - g δ - g δ g μ ν = - 1 2 g μ ν . \frac{1}{\sqrt{-g}}\frac{\delta\sqrt{-g}}{\delta g^{\mu\nu}}=-\frac{1}{2}g_{% \mu\nu}.
  36. R μ ν - 1 2 g μ ν R = 8 π G c 4 T μ ν , R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R=\frac{8\pi G}{c^{4}}T_{\mu\nu},
  37. κ = 8 π G c 4 \kappa=\frac{8\pi G}{c^{4}}
  38. S = [ 1 2 κ ( R - 2 Λ ) + M ] - g d 4 x S=\int\left[{1\over 2\kappa}\left(R-2\Lambda\right)+\mathcal{L}_{\mathrm{M}}% \right]\sqrt{-g}\,\mathrm{d}^{4}x
  39. R μ ν - 1 2 g μ ν R + Λ g μ ν = 8 π G c 4 T μ ν . R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R+\Lambda g_{\mu\nu}=\frac{8\pi G}{c^{4}}T_{% \mu\nu}\,.

Electric_displacement_field.html

  1. 𝐃 ε 0 𝐄 + 𝐏 , \mathbf{D}\equiv\varepsilon_{0}\mathbf{E}+\mathbf{P},
  2. ε 0 \varepsilon_{0}
  3. ρ = ρ f + ρ b \rho=\rho\text{f}+\rho\text{b}
  4. ρ = ρ f - 𝐏 . \rho=\rho\text{f}-\nabla\cdot\mathbf{P}.
  5. 𝐄 = 1 ε 0 ρ = 1 ε 0 ( ρ f - 𝐏 ) \nabla\cdot\mathbf{E}=\frac{1}{\varepsilon_{0}}\rho=\frac{1}{\varepsilon_{0}}(% \rho\text{f}-\nabla\cdot\mathbf{P})
  6. ( ε 0 𝐄 + 𝐏 ) = ρ f \nabla\cdot(\varepsilon_{0}\mathbf{E}+\mathbf{P})=\rho\text{f}
  7. 𝐃 = ρ - ρ b = ρ f \nabla\cdot\mathbf{D}=\rho-\rho\text{b}=\rho\text{f}
  8. × 𝐃 = ε 0 × 𝐄 + × 𝐏 , \nabla\times\mathbf{D}=\varepsilon_{0}\nabla\times\mathbf{E}+\nabla\times% \mathbf{P},
  9. × 𝐃 = × 𝐏 \nabla\times\mathbf{D}=\nabla\times\mathbf{P}
  10. 𝐏 = ε 0 χ 𝐄 , \mathbf{P}=\varepsilon_{0}\chi\mathbf{E},
  11. χ \chi
  12. 𝐃 = ε 0 ( 1 + χ ) 𝐄 = ε 𝐄 \mathbf{D}=\varepsilon_{0}(1+\chi)\mathbf{E}=\varepsilon\mathbf{E}
  13. 𝐃 ( ω ) = ε ( ω ) 𝐄 ( ω ) , \mathbf{D(\omega)}=\varepsilon(\omega)\mathbf{E}(\omega),
  14. ω \omega
  15. ( 𝐃 𝟏 - 𝐃 𝟐 ) 𝐧 ^ = D 1 , - D 2 , = σ f (\mathbf{D_{1}}-\mathbf{D_{2}})\cdot\hat{\mathbf{n}}=D_{1,\perp}-D_{2,\perp}=% \sigma\text{f}
  16. 𝐧 ^ \mathbf{\hat{n}}
  17. A 𝐃 d 𝐀 = Q free \oint_{A}\mathbf{D}\cdot\mathrm{d}\mathbf{A}=Q\text{free}
  18. | 𝐃 | = Q free A |\mathbf{D}|=\frac{Q\text{free}}{A}
  19. C = Q free V Q free | 𝐄 | d = A d ε , C=\frac{Q\text{free}}{V}\approx\frac{Q\text{free}}{|\mathbf{E}|d}=\frac{A}{d}\varepsilon,

Electric_susceptibility.html

  1. χ e \chi_{\,\text{e}}
  2. 𝐏 = ε 0 χ e 𝐄 , {\mathbf{P}}=\varepsilon_{0}\chi_{\,\text{e}}{\mathbf{E}},
  3. 𝐏 \mathbf{P}
  4. ε 0 \varepsilon_{0}
  5. χ e \chi_{\,\text{e}}
  6. 𝐄 \mathbf{E}
  7. ε r \varepsilon_{\textrm{r}}
  8. χ e = ε r - 1 \chi_{\,\text{e}}\ =\varepsilon_{\,\text{r}}-1
  9. χ e = 0 \chi_{\,\text{e}}\ =0
  10. 𝐃 = ε 0 𝐄 + 𝐏 = ε 0 ( 1 + χ e ) 𝐄 = ε r ε 0 𝐄 . \mathbf{D}\ =\ \varepsilon_{0}\mathbf{E}+\mathbf{P}\ =\ \varepsilon_{0}(1+\chi% _{\,\text{e}})\mathbf{E}\ =\ \varepsilon_{\,\text{r}}\varepsilon_{0}\mathbf{E}.
  11. 𝐩 = ε 0 α 𝐄 local \mathbf{p}=\varepsilon_{0}\alpha\mathbf{E_{\,\text{local}}}
  12. 𝐏 = N 𝐩 = N ε 0 α 𝐄 local , \mathbf{P}=N\mathbf{p}=N\varepsilon_{0}\alpha\mathbf{E}\text{local},
  13. χ e 𝐄 = N α 𝐄 local \chi_{\,\text{e}}\mathbf{E}=N\alpha\mathbf{E}_{\,\text{local}}
  14. χ e = N α \chi_{\,\text{e}}\ =N\alpha
  15. P = P 0 + ε 0 χ ( 1 ) E + ε 0 χ ( 2 ) E 2 + ε 0 χ ( 3 ) E 3 + . P=P_{0}+\varepsilon_{0}\chi^{(1)}E+\varepsilon_{0}\chi^{(2)}E^{2}+\varepsilon_% {0}\chi^{(3)}E^{3}+\cdots.
  16. P 0 = 0 P_{0}=0
  17. χ ( 1 ) \chi^{(1)}
  18. χ ( n ) \chi^{(n)}
  19. χ ( n ) \chi^{(n)}
  20. 𝐏 ( t ) = ε 0 - t χ e ( t - t ) 𝐄 ( t ) d t . \mathbf{P}(t)=\varepsilon_{0}\int_{-\infty}^{t}\chi_{\,\text{e}}(t-t^{\prime})% \mathbf{E}(t^{\prime})\,dt^{\prime}.
  21. χ e ( Δ t ) \chi_{\,\text{e}}(\Delta t)
  22. χ e ( Δ t ) = 0 \chi_{\,\text{e}}(\Delta t)=0
  23. Δ t < 0 \Delta t<0
  24. χ e ( Δ t ) = χ e δ ( Δ t ) \chi_{\,\text{e}}(\Delta t)=\chi_{\,\text{e}}\delta(\Delta t)
  25. 𝐏 ( ω ) = ε 0 χ e ( ω ) 𝐄 ( ω ) . \mathbf{P}(\omega)=\varepsilon_{0}\chi_{\,\text{e}}(\omega)\mathbf{E}(\omega).
  26. χ e ( Δ t ) = 0 \chi_{\,\text{e}}(\Delta t)=0
  27. Δ t < 0 \Delta t<0
  28. χ e ( 0 ) \chi_{\,\text{e}}(0)

Electricity_meter.html

  1. P = 3600 K h t P={{3600\cdot Kh}\over t}

Electromagnetic_cavity.html

  1. [ - 2 2 m 2 + V ( 𝐫 ) ] ψ ( 𝐫 ) = E ψ ( 𝐫 ) , \left[-\frac{\hbar^{2}}{2m}\nabla^{2}+V(\mathbf{r})\right]\psi(\mathbf{r})=E% \psi(\mathbf{r}),

Electromagnetic_four-potential.html

  1. A α = ( ϕ / c , 𝐀 ) A^{\alpha}=\left(\phi/c,\mathbf{A}\right)\,\!
  2. A α = ( ϕ , 𝐀 ) A^{\alpha}=(\phi,\mathbf{A})
  3. 𝐄 = - ϕ - 𝐀 t \mathbf{E}=-\mathbf{\nabla}\phi-\frac{\partial\mathbf{A}}{\partial t}
  4. 𝐄 = - ϕ - 1 c 𝐀 t \mathbf{E}=-\mathbf{\nabla}\phi-\frac{1}{c}\frac{\partial\mathbf{A}}{\partial t}
  5. 𝐁 = × 𝐀 . \mathbf{B}=\mathbf{\nabla}\times\mathbf{A}.
  6. 𝐁 = × 𝐀 . \mathbf{B}=\mathbf{\nabla}\times\mathbf{A}.
  7. F μ ν = μ A ν - ν A μ . F^{\mu\nu}=\partial^{\mu}A^{\nu}-\partial^{\nu}A^{\mu}.
  8. α A α = 0 \partial_{\alpha}A^{\alpha}=0
  9. A α = μ 0 J α \Box A^{\alpha}=\mu_{0}J^{\alpha}
  10. A α = 4 π c J α \Box A^{\alpha}=\frac{4\pi}{c}J^{\alpha}
  11. = 1 c 2 2 t 2 - 2 \Box=\frac{1}{c^{2}}\frac{\partial^{2}}{\partial t^{2}}-\nabla^{2}
  12. ϕ = - ρ ϵ 0 \Box\phi=-\frac{\rho}{\epsilon_{0}}
  13. ϕ = 4 π ρ \Box\phi=4\pi\rho
  14. 𝐀 = - μ 0 𝐣 \Box\mathbf{A}=-\mu_{0}\mathbf{j}
  15. 𝐀 = 4 π c 𝐣 \Box\mathbf{A}=\frac{4\pi}{c}\mathbf{j}
  16. ϕ ( 𝐫 , t ) = 1 4 π ϵ 0 d 3 x ρ ( 𝐫 , t r ) | 𝐫 - 𝐫 | \phi(\mathbf{r},t)=\frac{1}{4\pi\epsilon_{0}}\int\mathrm{d}^{3}x^{\prime}\frac% {\rho(\mathbf{r}^{\prime},t_{r})}{\left|\mathbf{r}-\mathbf{r}^{\prime}\right|}
  17. 𝐀 ( 𝐫 , t ) = μ 0 4 π d 3 x 𝐣 ( 𝐫 , t r ) | 𝐫 - 𝐫 | , \mathbf{A}(\mathbf{r},t)=\frac{\mu_{0}}{4\pi}\int\mathrm{d}^{3}x^{\prime}\frac% {\mathbf{j}(\mathbf{r}^{\prime},t_{r})}{\left|\mathbf{r}-\mathbf{r}^{\prime}% \right|},
  18. t r = t - | 𝐫 - 𝐫 | c t_{r}=t-\frac{\left|\mathbf{r}-\mathbf{r}^{\prime}\right|}{c}
  19. ρ ( 𝐫 , t r ) = [ ρ ( 𝐫 , t ) ] , \rho(\mathbf{r}^{\prime},t_{r})=[\rho(\mathbf{r}^{\prime},t)],

Electron-beam_lithography.html

  1. D A = T I D\cdot A=T\cdot I\,
  2. T T
  3. I I
  4. D D
  5. A A
  6. d p = 2 e 2 / b v dp=2e^{2}/bv
  7. T = ( d p ) 2 / 2 m = e 4 / E b 2 T=(dp)^{2}/2m=e^{4}/Eb^{2}
  8. E = ( 1 / 2 ) m v 2 E=(1/2)mv^{2}
  9. E E

Electron_mobility.html

  1. v d \,v_{d}
  2. v d = μ E \,v_{d}=\mu E
  3. v d = μ E \,v_{d}=\mu E
  4. - μ e 𝐄 -\mu_{e}\mathbf{E}
  5. n e μ e 𝐄 ne\mu_{e}\mathbf{E}
  6. σ = n e μ e \sigma=ne\mu_{e}
  7. σ = p e μ h \sigma=pe\mu_{h}
  8. σ = e ( n μ e + p μ h ) . \sigma=e(n\mu_{e}+p\mu_{h}).
  9. m * v e m i t 2 2 ω p h o n o n ( o p t . ) \frac{m^{*}v_{emit}^{2}}{2}\approx\hbar\omega_{phonon(opt.)}
  10. μ = q m * τ ¯ \mu=\frac{q}{m^{*}}\overline{\tau}
  11. τ ¯ \overline{τ}
  12. 1 μ = 1 μ impurities + 1 μ lattice \frac{1}{\mu}=\frac{1}{\mu_{\rm impurities}}+\frac{1}{\mu_{\rm lattice}}
  13. μ impurities \mu_{\rm impurities}
  14. μ lattice \mu_{\rm lattice}
  15. 1 μ = 1 μ impurities + 1 μ lattice + 1 μ defects + \frac{1}{\mu}=\frac{1}{\mu_{\rm impurities}}+\frac{1}{\mu_{\rm lattice}}+\frac% {1}{\mu_{\rm defects}}+\cdots
  16. 1 τ = 1 τ impurities + 1 τ lattice + 1 τ defects + \frac{1}{\tau}=\frac{1}{\tau_{\rm impurities}}+\frac{1}{\tau_{\rm lattice}}+% \frac{1}{\tau_{\rm defects}}+\cdots
  17. 1 τ v Σ \frac{1}{\tau}\propto\left\langle v\right\rangle\Sigma
  18. Σ \Sigma
  19. v \left\langle v\right\rangle
  20. v T \left\langle v\right\rangle\sim\sqrt{T}
  21. Σ d e f v - 4 {\Sigma}_{def}\propto{\left\langle v\right\rangle}^{-4}
  22. v \left\langle v\right\rangle
  23. μ p h T - 3 / 2 {\mu}_{ph}\sim T^{-3/2}
  24. μ d e f T 3 / 2 {\mu}_{def}\sim T^{3/2}
  25. F H n = - q ( v n × B z ) \overrightarrow{F}_{Hn}=-q(\overrightarrow{v}_{n}\times\overrightarrow{B}_{z})
  26. F H p = + q ( v p × B z ) \overrightarrow{F}_{Hp}=+q(\overrightarrow{v}_{p}\times\overrightarrow{B}_{z})
  27. F y = ( - q ) ξ y + ( - q ) [ v n × B z ] = 0 \overrightarrow{F}_{y}=(-q)\overrightarrow{\xi}_{y}+(-q)[\overrightarrow{v}_{n% }\times\overrightarrow{B}_{z}]=0
  28. - q ξ y + q v x B z = 0 \Rightarrow-q\xi_{y}+qv_{x}B_{z}=0
  29. ξ y = v x B z \xi_{y}=v_{x}B_{z}
  30. I = - q n v x t W I=-qnv_{x}tW
  31. ξ y = - I B n q t W = + R H n I B t W \xi_{y}=-\frac{IB}{nqtW}=+\frac{R_{Hn}IB}{tW}
  32. R H n = - 1 n q R_{Hn}=-\frac{1}{nq}
  33. ξ y = V H W \xi_{y}=\frac{V_{H}}{W}
  34. R H n = - 1 n q = V H n t I B R_{Hn}=-\frac{1}{nq}=\frac{V_{Hn}t}{IB}
  35. R H p = 1 p q = V H p t I B R_{Hp}=\frac{1}{pq}=\frac{V_{Hp}t}{IB}
  36. μ n = ( - n q ) μ n ( - 1 n q ) = - σ n R H n \mu_{n}=(-nq)\mu_{n}(-\frac{1}{nq})=-\sigma_{n}R_{Hn}
  37. = - σ n V H n t I B =-\frac{\sigma_{n}V_{Hn}t}{IB}
  38. μ p = σ p V H p t I B \mu_{p}=\frac{\sigma_{p}V_{Hp}t}{IB}
  39. μ = m s a t 2 2 L W 1 C i \mu=m_{sat}^{2}\frac{2L}{W}\frac{1}{C_{i}}
  40. I D = μ C i 2 W L ( V G S - V t h ) 2 . I_{D}=\frac{\mu C_{i}}{2}\frac{W}{L}(V_{GS}-V_{th})^{2}.
  41. I D V G S I_{D}\propto V_{GS}
  42. μ = m l i n L W 1 V D S 1 C i \mu=m_{lin}\frac{L}{W}\frac{1}{V_{DS}}\frac{1}{C_{i}}
  43. I D = μ C i W L ( ( V G S - V t h ) V D S - V D S 2 2 ) I_{D}=\mu C_{i}\frac{W}{L}\left((V_{GS}-V_{th})V_{DS}-\frac{V_{DS}^{2}}{2}\right)
  44. 10 18 cm - 3 10^{18}\mathrm{cm}^{-3}
  45. μ = μ o + μ 1 1 + ( N N ref ) α \mu=\mu_{o}+\frac{\mu_{1}}{1+(\frac{N}{N\text{ref}})^{\alpha}}
  46. μ n ( N D ) = 65 + 1265 1 + ( N D 8.5 × 10 16 ) 0.72 \mu_{n}(N_{D})=65+\frac{1265}{1+(\frac{N_{D}}{8.5\times 10^{16}})^{0.72}}
  47. μ p ( N A ) = 48 + 447 1 + ( N A 6.3 × 10 16 ) 0.76 \mu_{p}(N_{A})=48+\frac{447}{1+(\frac{N_{A}}{6.3\times 10^{16}})^{0.76}}
  48. μ n ( N A ) = 232 + 1180 1 + ( N A 8 × 10 16 ) 0.9 \mu_{n}(N_{A})=232+\frac{1180}{1+(\frac{N_{A}}{8\times 10^{16}})^{0.9}}
  49. μ p ( N D ) = 130 + 370 1 + ( N D 8 × 10 17 ) 1.25 \mu_{p}(N_{D})=130+\frac{370}{1+(\frac{N_{D}}{8\times 10^{17}})^{1.25}}