wpmath0000002_3

Comparison_of_analog_and_digital_recording.html

  1. θ \scriptstyle\theta\,

Complete_graph.html

  1. n n
  2. K < s u b > n K<sub>n

Completeness_(statistics).html

  1. E ( g ( T ) ) = t = 0 n g ( t ) ( n t ) p t ( 1 - p ) n - t = ( 1 - p ) n t = 0 n g ( t ) ( n t ) ( p 1 - p ) t . \operatorname{E}(g(T))=\sum_{t=0}^{n}{g(t){n\choose t}p^{t}(1-p)^{n-t}}=(1-p)^% {n}\sum_{t=0}^{n}{g(t){n\choose t}\left(\frac{p}{1-p}\right)^{t}}.
  2. E ( g ( T ) ) = 0 E(g(T))=0
  3. t = 0 n g ( t ) ( n t ) ( p 1 - p ) t = 0. \sum_{t=0}^{n}g(t){n\choose t}\left(\frac{p}{1-p}\right)^{t}=0.
  4. t = 0 n g ( t ) ( n t ) r t = 0. \sum_{t=0}^{n}g(t){n\choose t}r^{t}=0.
  5. g ( t ) = 2 ( t - 0.5 ) , g(t)=2(t-0.5),\,
  6. s ( ( X 1 , X 2 ) ) = X 1 + X 2 s((X_{1},X_{2}))=X_{1}+X_{2}\,\!
  7. g g
  8. g ( s ( X 1 , X 2 ) ) = g ( X 1 + X 2 ) g(s(X_{1},X_{2}))=g(X_{1}+X_{2})\,\!
  9. x x
  10. exp ( - ( x - 2 θ ) 2 / 4 ) . \exp\left(-(x-2\theta)^{2}/4\right).
  11. - g ( x ) exp ( - ( x - 2 θ ) 2 / 4 ) d x . \int_{-\infty}^{\infty}g(x)\exp\left(-(x-2\theta)^{2}/4\right)\,dx.
  12. k ( θ ) - h ( x ) e x θ d x k(\theta)\int_{-\infty}^{\infty}h(x)e^{x\theta}\,dx\,\!
  13. h ( x ) = g ( x ) e - x 2 / 4 . h(x)=g(x)e^{-x^{2}/4}.\,\!

Completing_the_square.html

  1. a x 2 + b x + c ax^{2}+bx+c\,\!
  2. a ( ) 2 + constant . a(\cdots\cdots)^{2}+\mbox{constant}~{}.\,
  3. a x 2 + b x + c ax^{2}+bx+c\,\!
  4. a ( x + h ) 2 + k a(x+h)^{2}+k\,
  5. ( x + p ) 2 = x 2 + 2 p x + p 2 . (x+p)^{2}\,=\,x^{2}+2px+p^{2}.\,\!
  6. ( x + 3 ) 2 \displaystyle(x+3)^{2}
  7. x 2 + 10 x + 28. x^{2}+10x+28.\,\!
  8. ( x + 5 ) 2 = x 2 + 10 x + 25. (x+5)^{2}\,=\,x^{2}+10x+25.\,\!
  9. x 2 + 10 x + 28 = ( x + 5 ) 2 + 3. x^{2}+10x+28\,=\,(x+5)^{2}+3.
  10. x 2 + b x + c , x^{2}+bx+c,\,\!
  11. ( x + 1 2 b ) 2 = x 2 + b x + 1 4 b 2 . \left(x+\tfrac{1}{2}b\right)^{2}\,=\,x^{2}+bx+\tfrac{1}{4}b^{2}.
  12. x 2 + b x + c = ( x + 1 2 b ) 2 + k , x^{2}+bx+c\,=\,\left(x+\tfrac{1}{2}b\right)^{2}+k,
  13. x 2 + 6 x + 11 \displaystyle x^{2}+6x+11
  14. a x 2 + b x + c ax^{2}+bx+c\,\!
  15. 3 x 2 + 12 x + 27 \displaystyle 3x^{2}+12x+27
  16. a ( x - h ) 2 + k . a(x-h)^{2}+k.\,\!
  17. a x 2 + b x + c = a ( x - h ) 2 + k , where h = - b 2 a and k = c - a h 2 = c - b 2 4 a . ax^{2}+bx+c\;=\;a(x-h)^{2}+k,\quad\,\text{where}\quad h=-\frac{b}{2a}\quad\,% \text{and}\quad k=c-ah^{2}=c-\frac{b^{2}}{4a}.
  18. x 2 + b x + c = ( x - h ) 2 + k , where h = - b 2 and k = c - b 2 4 . x^{2}+bx+c\;=\;(x-h)^{2}+k,\quad\,\text{where}\quad h=-\frac{b}{2}\quad\,\text% {and}\quad k=c-\frac{b^{2}}{4}.
  19. x T A x + x T b + c = ( x - h ) T A ( x - h ) + k where h = - 1 2 A - 1 b and k = c - 1 4 b T A - 1 b x^{\mathrm{T}}Ax+x^{\mathrm{T}}b+c=(x-h)^{\mathrm{T}}A(x-h)+k\quad\,\text{% where}\quad h=-\frac{1}{2}A^{-1}b\quad\,\text{and}\quad k=c-\frac{1}{4}b^{% \mathrm{T}}A^{-1}b
  20. A A
  21. A A
  22. h h
  23. k k
  24. h = - ( A + A T ) - 1 b and k = c - h T A h = c - b T ( A + A T ) - 1 A ( A + A T ) - 1 b h=-(A+A^{\mathrm{T}})^{-1}b\quad\,\text{and}\quad k=c-h^{\mathrm{T}}Ah=c-b^{% \mathrm{T}}(A+A^{\mathrm{T}})^{-1}A(A+A^{\mathrm{T}})^{-1}b
  25. ( x - h ) 2 + k or a ( x - h ) 2 + k (x-h)^{2}+k\quad\,\text{or}\quad a(x-h)^{2}+k
  26. x 2 + 6 x + 5 = 0 , x^{2}+6x+5=0,\,\!
  27. ( x + 3 ) 2 - 4 = 0. (x+3)^{2}-4=0.\,\!
  28. ( x + 3 ) 2 = 4. (x+3)^{2}=4.\,\!
  29. x + 3 = - 2 or x + 3 = 2 , x+3=-2\quad\,\text{or}\quad x+3=2,
  30. x = - 5 or x = - 1. x=-5\quad\,\text{or}\quad x=-1.
  31. x 2 - 10 x + 18 = 0. x^{2}-10x+18=0.\,\!
  32. ( x - 5 ) 2 - 7 = 0 , (x-5)^{2}-7=0,\,\!
  33. ( x - 5 ) 2 = 7. (x-5)^{2}=7.\,\!
  34. x - 5 = - 7 or x - 5 = 7 , x-5=-\sqrt{7}\quad\,\text{or}\quad x-5=\sqrt{7},\,
  35. x = 5 - 7 or x = 5 + 7 . x=5-\sqrt{7}\quad\,\text{or}\quad x=5+\sqrt{7}.\,
  36. x = 5 ± 7 . x=5\pm\sqrt{7}.\,
  37. x 2 + 4 x + 5 = 0 ( x + 2 ) 2 + 1 = 0 ( x + 2 ) 2 = - 1 x + 2 = ± i x = - 2 ± i . \begin{array}[]{c}x^{2}+4x+5\,=\,0\\ (x+2)^{2}+1\,=\,0\\ (x+2)^{2}\,=\,-1\\ x+2\,=\,\pm i\\ x\,=\,-2\pm i.\end{array}
  38. 2 x 2 + 7 x + 6 = 0 x 2 + 7 2 x + 3 = 0 ( x + 7 4 ) 2 - 1 16 = 0 ( x + 7 4 ) 2 = 1 16 x + 7 4 = 1 4 or x + 7 4 = - 1 4 x = - 3 2 or x = - 2. \begin{array}[]{c}2x^{2}+7x+6\,=\,0\\ x^{2}+\tfrac{7}{2}x+3\,=\,0\\ \left(x+\tfrac{7}{4}\right)^{2}-\tfrac{1}{16}\,=\,0\\ \left(x+\tfrac{7}{4}\right)^{2}\,=\,\tfrac{1}{16}\\ x+\tfrac{7}{4}=\tfrac{1}{4}\quad\,\text{or}\quad x+\tfrac{7}{4}=-\tfrac{1}{4}% \\ x=-\tfrac{3}{2}\quad\,\text{or}\quad x=-2.\end{array}
  39. d x a x 2 + b x + c \int\frac{dx}{ax^{2}+bx+c}
  40. d x x 2 - a 2 = 1 2 a ln | x - a x + a | + C and d x x 2 + a 2 = 1 a arctan ( x a ) + C . \int\frac{dx}{x^{2}-a^{2}}=\frac{1}{2a}\ln\left|\frac{x-a}{x+a}\right|+C\quad% \,\text{and}\quad\int\frac{dx}{x^{2}+a^{2}}=\frac{1}{a}\arctan\left(\frac{x}{a% }\right)+C.
  41. d x x 2 + 6 x + 13 . \int\frac{dx}{x^{2}+6x+13}.
  42. d x ( x + 3 ) 2 + 4 = d x ( x + 3 ) 2 + 2 2 . \int\frac{dx}{(x+3)^{2}+4}\,=\,\int\frac{dx}{(x+3)^{2}+2^{2}}.
  43. d x ( x + 3 ) 2 + 4 = 1 2 arctan ( x + 3 2 ) + C . \int\frac{dx}{(x+3)^{2}+4}\,=\,\frac{1}{2}\arctan\left(\frac{x+3}{2}\right)+C.
  44. | z | 2 - b * z - b z * + c , |z|^{2}-b^{*}z-bz^{*}+c,\,
  45. | z - b | 2 - | b | 2 + c , |z-b|^{2}-|b|^{2}+c,\,\!
  46. | z - b | 2 = ( z - b ) ( z - b ) * = ( z - b ) ( z * - b * ) = z z * - z b * - b z * + b b * = | z | 2 - z b * - b z * + | b | 2 . \begin{aligned}\displaystyle|z-b|^{2}&\displaystyle{}=(z-b)(z-b)^{*}\\ &\displaystyle{}=(z-b)(z^{*}-b^{*})\\ &\displaystyle{}=zz^{*}-zb^{*}-bz^{*}+bb^{*}\\ &\displaystyle{}=|z|^{2}-zb^{*}-bz^{*}+|b|^{2}.\end{aligned}
  47. a x 2 + b y 2 + c , ax^{2}+by^{2}+c,\,\!
  48. z = a x + i b y . z=\sqrt{a}\,x+i\sqrt{b}\,y.
  49. | z | 2 = z z * = ( a x + i b y ) ( a x - i b y ) = a x 2 - i a b x y + i b a y x - i 2 b y 2 = a x 2 + b y 2 , \begin{aligned}\displaystyle|z|^{2}&\displaystyle{}=zz^{*}\\ &\displaystyle{}=(\sqrt{a}\,x+i\sqrt{b}\,y)(\sqrt{a}\,x-i\sqrt{b}\,y)\\ &\displaystyle{}=ax^{2}-i\sqrt{ab}\,xy+i\sqrt{ba}\,yx-i^{2}by^{2}\\ &\displaystyle{}=ax^{2}+by^{2},\end{aligned}
  50. a x 2 + b y 2 + c = | z | 2 + c . ax^{2}+by^{2}+c=|z|^{2}+c.\,\!
  51. a 2 + b 2 = a , a^{2}+b^{2}=a,
  52. ( a b b 1 - a ) \begin{pmatrix}a&b\\ b&1-a\end{pmatrix}
  53. a 2 + b 2 = a , a^{2}+b^{2}=a,
  54. ( a - 1 2 ) 2 + b 2 = 1 4 . (a-\tfrac{1}{2})^{2}+b^{2}=\tfrac{1}{4}.
  55. x 2 + b x = a . x^{2}+bx=a.\,
  56. u 2 + 2 u v u^{2}+2uv\,
  57. u 2 + v 2 u^{2}+v^{2}\,
  58. x + 1 x = ( x - 2 + 1 x ) + 2 = ( x - 1 x ) 2 + 2 \begin{aligned}\displaystyle x+{1\over x}&\displaystyle{}=\left(x-2+{1\over x}% \right)+2\\ &\displaystyle{}=\left(\sqrt{x}-{1\over\sqrt{x}}\right)^{2}+2\end{aligned}
  59. x 4 + 324. x^{4}+324.\,\!
  60. ( x 2 ) 2 + ( 18 ) 2 , (x^{2})^{2}+(18)^{2},\,\!
  61. x 4 + 324 = ( x 4 + 36 x 2 + 324 ) - 36 x 2 = ( x 2 + 18 ) 2 - ( 6 x ) 2 = a difference of two squares = ( x 2 + 18 + 6 x ) ( x 2 + 18 - 6 x ) = ( x 2 + 6 x + 18 ) ( x 2 - 6 x + 18 ) \begin{aligned}\displaystyle x^{4}+324&\displaystyle{}=(x^{4}+36x^{2}+324)-36x% ^{2}\\ &\displaystyle{}=(x^{2}+18)^{2}-(6x)^{2}=\,\text{a difference of two squares}% \\ &\displaystyle{}=(x^{2}+18+6x)(x^{2}+18-6x)\\ &\displaystyle{}=(x^{2}+6x+18)(x^{2}-6x+18)\end{aligned}

Complex_conjugate.html

  1. ρ e i ϕ \rho e^{i\phi}
  2. ρ e - i ϕ \rho e^{-i\phi}
  3. z z
  4. z ¯ \overline{z}
  5. z * z^{*}\!
  6. e i ϕ + c . c . e^{i\phi}+c.c.
  7. e i ϕ + e - i ϕ e^{i\phi}+e^{-i\phi}
  8. ( z + w ) ¯ = z ¯ + w ¯ \overline{(z+w)}=\overline{z}+\overline{w}\!
  9. z - w ¯ = z ¯ - w ¯ \overline{z-w}=\overline{z}-\overline{w}\!
  10. ( z w ) ¯ = z ¯ w ¯ \overline{(zw)}=\overline{z}\;\overline{w}\!
  11. ( z / w ) ¯ = z ¯ / w ¯ \overline{(z/w)}=\overline{z}/\overline{w}\!
  12. z ¯ = z \overline{z}=z\!
  13. z n ¯ = ( z ¯ ) n \overline{z^{n}}=(\overline{z})^{n}
  14. | z ¯ | = | z | \left|\overline{z}\right|=\left|z\right|
  15. | z | 2 = z z ¯ = z ¯ z {\left|z\right|}^{2}=z\overline{z}=\overline{z}z
  16. z ¯ ¯ = z \overline{\overline{z}}=z\!
  17. z - 1 = z ¯ | z | 2 z^{-1}=\frac{\overline{z}}{{\left|z\right|}^{2}}
  18. exp ( z ¯ ) = exp ( z ) ¯ \exp(\overline{z})=\overline{\exp(z)}\,\!
  19. log ( z ¯ ) = log ( z ) ¯ \log(\overline{z})=\overline{\log(z)}\,\!
  20. p p
  21. p ( z ) = 0 p(z)=0
  22. p ( z ¯ ) = 0 p(\overline{z})=0
  23. ϕ \phi\,
  24. ϕ ( z ) \phi(z)\,
  25. ϕ ( z ¯ ) = ϕ ( z ) ¯ . \phi(\overline{z})=\overline{\phi(z)}.\,\!
  26. σ ( z ) = z ¯ \sigma(z)=\overline{z}\,
  27. \mathbb{C}\,
  28. \mathbb{C}
  29. \mathbb{C}
  30. {\mathbb{C}}\,
  31. / \mathbb{C}/\mathbb{R}
  32. σ \sigma\,
  33. \mathbb{C}
  34. \mathbb{C}
  35. z = x + i y z=x+iy
  36. z = ρ e i θ z=\rho e^{i\theta}
  37. x = Re ( z ) = z + z ¯ 2 x=\operatorname{Re}\,(z)=\dfrac{z+\overline{z}}{2}
  38. y = Im ( z ) = z - z ¯ 2 i y=\operatorname{Im}\,(z)=\dfrac{z-\overline{z}}{2i}
  39. ρ = | z | = z z ¯ \rho=\left|z\right|=\sqrt{z\overline{z}}
  40. e i θ = e i arg z = z z ¯ e^{i\theta}=e^{i\arg z}=\sqrt{\dfrac{z}{\overline{z}}}
  41. θ = arg z = 1 i ln z z ¯ = ln z - ln z ¯ 2 i \theta=\arg z=\dfrac{1}{i}\ln\sqrt{\frac{z}{\overline{z}}}=\dfrac{\ln z-\ln% \overline{z}}{2i}
  42. z z\,
  43. z ¯ \overline{z}
  44. ρ \rho\,
  45. θ \theta
  46. z ¯ \overline{z}
  47. { z z r ¯ + z ¯ r = 0 } \{z\mid z\overline{r}+\overline{z}r=0\}
  48. r ¯ \overline{r}
  49. z r ¯ z\cdot\overline{r}
  50. z z\,
  51. r ¯ \overline{r}
  52. z - z 0 z ¯ - z 0 ¯ = u \frac{z-z_{0}}{\overline{z}-\overline{z_{0}}}=u
  53. z 0 z_{0}\,
  54. 𝐀𝐁 ¯ = ( 𝐀 ¯ ) ( 𝐁 ¯ ) \overline{\mathbf{AB}}=(\overline{\mathbf{A}})(\overline{\mathbf{B}})
  55. 𝐀 ¯ \overline{\mathbf{A}}
  56. 𝐀 \mathbf{A}
  57. ( 𝐀𝐁 ) * = 𝐁 * 𝐀 * (\mathbf{AB})^{*}=\mathbf{B}^{*}\mathbf{A}^{*}
  58. 𝐀 * \mathbf{A}^{*}
  59. 𝐀 \mathbf{A}
  60. a + b i + c j + d k a+bi+cj+dk
  61. a - b i - c j - d k a-bi-cj-dk
  62. ( z w ) * = w * z * . {\left(zw\right)}^{*}=w^{*}z^{*}.
  63. V V
  64. ϕ : V V \phi:V\rightarrow V\,
  65. ϕ 2 = id V \phi^{2}=\operatorname{id}_{V}\,
  66. ϕ 2 = ϕ ϕ \phi^{2}=\phi\circ\phi
  67. id V \operatorname{id}_{V}\,
  68. V V\,
  69. ϕ ( z v ) = z ¯ ϕ ( v ) \phi(zv)=\overline{z}\phi(v)
  70. v V v\in V\,
  71. z z\in{\mathbb{C}}\,
  72. ϕ ( v 1 + v 2 ) = ϕ ( v 1 ) + ϕ ( v 2 ) \phi(v_{1}+v_{2})=\phi(v_{1})+\phi(v_{2})\,
  73. v 1 V v_{1}\in V\,
  74. v 2 V v_{2}\in V\,
  75. ϕ \operatorname{\phi}
  76. V V
  77. ϕ \operatorname{\phi}
  78. \mathbb{R}
  79. V V
  80. V V

Complex_geometry.html

  1. 𝒪 \mathcal{O}
  2. 𝒪 * \mathcal{O}^{*}
  3. X \mathcal{M}_{X}
  4. U U\mapsto
  5. Γ ( U , 𝒪 X ) \Gamma(U,\mathcal{O}_{X})
  6. U i U_{i}
  7. X * / 𝒪 X * \mathcal{M}_{X}^{*}/\mathcal{O}_{X}^{*}
  8. Pic ( X ) \operatorname{Pic}(X)
  9. H 1 ( X , 𝒪 * ) H^{1}(X,\mathcal{O}^{*})
  10. 0 𝒪 𝒪 * 0 0\to\mathbb{Z}\to\mathcal{O}\to\mathcal{O}^{*}\to 0
  11. f exp ( 2 π i f ) f\mapsto\exp(2\pi if)
  12. Pic ( X ) H 2 ( X , ) . \operatorname{Pic}(X)\to H^{2}(X,\mathbb{Z}).
  13. \mathcal{L}
  14. c 1 ( ) c_{1}(\mathcal{L})
  15. \mathcal{L}
  16. D = a i V i , a i D=\sum a_{i}V_{i},\quad a_{i}\in\mathbb{Z}
  17. Div ( X ) \operatorname{Div}(X)
  18. H 0 ( X , * / 𝒪 * ) H^{0}(X,\mathcal{M}^{*}/\mathcal{O}^{*})
  19. * / 𝒪 * \mathcal{M}^{*}/\mathcal{O}^{*}
  20. Div ( X ) Pic ( X ) . \operatorname{Div}(X)\to\operatorname{Pic}(X).
  21. ( 1 , 1 ) (1,1)
  22. π : E X \pi:E\to X
  23. c 1 , c 2 , c_{1},c_{2},\dots
  24. c i ( E ) c_{i}(E)
  25. H 2 i ( X , ) H^{2i}(X,\mathbb{Z})
  26. c i ( f * ( E ) ) = f * ( c i ( E ) ) c_{i}(f^{*}(E))=f^{*}(c_{i}(E))
  27. f : Z X f:Z\to X
  28. c ( E F ) = c ( E ) c ( F ) c(E\oplus F)=c(E)\cup c(F)
  29. c = 1 + c 1 + c 2 + . c=1+c_{1}+c_{2}+\dots.
  30. c i ( E ) = 0 c_{i}(E)=0
  31. i > rk E i>\operatorname{rk}E
  32. - c 1 ( E 1 ) -c_{1}(E_{1})
  33. H 2 ( 𝐏 1 , ) H^{2}(\mathbb{C}\mathbf{P}^{1},\mathbb{Z})
  34. E 1 E_{1}
  35. 𝐏 1 \mathbb{C}\mathbf{P}^{1}
  36. ch ( L ) = e c 1 ( L ) \operatorname{ch}(L)=e^{c_{1}(L)}
  37. c i ( E ) t i = 1 r ( 1 + η i t ) \sum c_{i}(E)t^{i}=\prod_{1}^{r}(1+\eta_{i}t)
  38. ch ( E ) = e η i \operatorname{ch}(E)=\sum e^{\eta_{i}}

Complex_plane.html

  1. z = x + i y z=x+iy
  2. ( x , y ) = ( r cos θ , r sin θ ) ( r , θ ) = ( x 2 + y 2 , arctan y x ) . (x,y)=(r\cos\theta,r\sin\theta)\qquad(r,\theta)=\left(\sqrt{x^{2}+y^{2}},\quad% \arctan\frac{y}{x}\right).
  3. z = x + i y = | z | ( cos θ + i sin θ ) = | z | e i θ z=x+iy=|z|\left(\cos\theta+i\sin\theta\right)=|z|e^{i\theta}
  4. | z | = x 2 + y 2 ; θ = arg ( z ) = 1 i ln z | z | = - i ln z | z | . |z|=\sqrt{x^{2}+y^{2}};\quad\theta=\arg(z)=\frac{1}{i}\ln\frac{z}{|z|}=-i\ln% \frac{z}{|z|}.\,
  5. w w
  6. z z
  7. ( w z ¯ ) \Re(w\overline{z})
  8. z z
  9. z z
  10. a r g ( z ) arg(z)
  11. z z
  12. z = x + i y ; f ( z ) = w = u + i v z=x+iy;\qquad f(z)=w=u+iv
  13. w = f ( z ) = ± z = z 1 / 2 . w=f(z)=\pm\sqrt{z}=z^{1/2}.
  14. y = g ( x ) = x = x 1 / 2 y=g(x)=\sqrt{x}\ =x^{1/2}
  15. z = r e i θ and take w = z 1 / 2 = r e i θ / 2 ( 0 θ 2 π ) . z=re^{i\theta}\quad\mbox{and take}~{}\quad w=z^{1/2}=\sqrt{r}\,e^{i\theta/2}% \qquad(0\leq\theta\leq 2\pi).
  16. w = g ( z ) = ( z 2 - 1 ) 1 / 2 . w=g(z)=\left(z^{2}-1\right)^{1/2}.
  17. Γ ( z ) = e - γ z z n = 1 [ ( 1 + z n ) - 1 e z / n ] \Gamma(z)=\frac{e^{-\gamma z}}{z}\prod_{n=1}^{\infty}\left[\left(1+\frac{z}{n}% \right)^{-1}e^{z/n}\right]
  18. f ( z ) = n = 1 ( z 2 + n ) - 2 . f(z)=\sum_{n=1}^{\infty}\left(z^{2}+n\right)^{-2}.
  19. z 2 + n = 0 z = ± i n , z^{2}+n=0\quad\Leftrightarrow\quad z=\pm i\sqrt{n},
  20. f ( z ) = 1 + z 1 + z 1 + z 1 + z . f(z)=1+\cfrac{z}{1+\cfrac{z}{1+\cfrac{z}{1+\cfrac{z}{\ddots}}}}.
  21. w = f ( z ) = ± z = z 1 / 2 w=f(z)=\pm\sqrt{z}=z^{1/2}
  22. f ( z ) = z 1 / 2 f ( z ) = 1 2 z - 1 / 2 f(z)=z^{1/2}\quad\Rightarrow\quad f^{\prime}(z)={\textstyle\frac{1}{2}}z^{-1/2}
  23. w = g ( z ) = ( z 2 - 1 ) 1 / 2 , w=g(z)=\left(z^{2}-1\right)^{1/2},
  24. s = σ + j ω s=\sigma+j\omega

Component_(thermodynamics).html

  1. \rightleftharpoons
  2. \rightleftharpoons

Composite_Bézier_curve.html

  1. 𝐤 \mathbf{k}
  2. 𝐀 \mathbf{A}
  3. 𝐁 \mathbf{B}
  4. 𝐀 \mathbf{A^{\prime}}
  5. 𝐁 \mathbf{B^{\prime}}
  6. 𝐀 \displaystyle\mathbf{A}
  7. 𝐂 ( t ) = ( 1 - t ) 3 𝐀 + 3 ( 1 - t ) 2 t 𝐀 + 3 ( 1 - t ) t 2 𝐁 + t 3 𝐁 \mathbf{C}(t)=(1-t)^{3}\mathbf{A}+3(1-t)^{2}t\mathbf{A^{\prime}}+3(1-t)t^{2}% \mathbf{B^{\prime}}+t^{3}\mathbf{B}
  8. 𝐂 ( t = 0.5 ) \mathbf{C}(t=0.5)
  9. 𝐂 \displaystyle\mathbf{C}
  10. 0 8 + 3 8 𝐤 + 3 8 + 1 8 = 2 / 2 \frac{0}{8}\mathbf{+}\frac{3}{8}\mathbf{k}+\frac{3}{8}+\frac{1}{8}=\sqrt{2}/2
  11. 𝐤 = 4 3 ( 2 - 1 ) 0.5522847498 \mathbf{k}=\frac{4}{3}(\sqrt{2}-1)\approx 0.5522847498
  12. R R
  13. 𝐀 \mathbf{A}
  14. 𝐁 \mathbf{B}
  15. θ = 2 ϕ \theta=2\phi
  16. 𝐀 x \displaystyle\mathbf{A}_{x}
  17. 𝐀 x = 4 R - 𝐀 x 3 𝐀 y = ( R - 𝐀 x ) ( 3 R - 𝐀 x ) 3 𝐀 y 𝐁 x = 𝐀 x 𝐁 y = - 𝐀 y \begin{aligned}\displaystyle\mathbf{A^{\prime}}_{x}&\displaystyle=\frac{4R-% \mathbf{A}_{x}}{3}\\ \displaystyle\mathbf{A^{\prime}}_{y}&\displaystyle=\frac{(R-\mathbf{A}_{x})(3R% -\mathbf{A}_{x})}{3\mathbf{A}_{y}}\\ \displaystyle\mathbf{B^{\prime}}_{x}&\displaystyle=\mathbf{A^{\prime}}_{x}\\ \displaystyle\mathbf{B^{\prime}}_{y}&\displaystyle=-\mathbf{A^{\prime}}_{y}% \end{aligned}

Composite_number.html

  1. μ ( n ) = ( - 1 ) 2 x = 1 \mu(n)=(-1)^{2x}=1\,
  2. μ ( n ) = ( - 1 ) 2 x + 1 = - 1. \mu(n)=(-1)^{2x+1}=-1.\,
  3. μ ( 1 ) = 1 \mu(1)=1
  4. μ ( n ) = 0 \mu(n)=0
  5. { 1 , p , p 2 } \{1,p,p^{2}\}

Composition_series.html

  1. 1 = H 0 H 1 H n = G , 1=H_{0}\triangleleft H_{1}\triangleleft\cdots\triangleleft H_{n}=G,
  2. \mathbb{Z}
  3. C 1 C 2 C 6 C 12 , C 1 C 2 C 4 C 12 , C 1 C 3 C 6 C 12 \begin{aligned}&\displaystyle C_{1}\triangleleft C_{2}\triangleleft C_{6}% \triangleleft C_{12},\\ &\displaystyle C_{1}\triangleleft C_{2}\triangleleft C_{4}\triangleleft C_{12}% ,\\ &\displaystyle C_{1}\triangleleft C_{3}\triangleleft C_{6}\triangleleft C_{12}% \end{aligned}
  4. C 2 , C 3 , C 2 C_{2},C_{3},C_{2}
  5. C 2 , C 2 , C 3 C_{2},C_{2},C_{3}
  6. C 3 , C 2 , C 2 . C_{3},C_{2},C_{2}.
  7. { 0 } = J 0 J n = M \{0\}=J_{0}\subset\cdots\subset J_{n}=M
  8. A = X 0 X 1 X n = 0 A=X_{0}\supsetneq X_{1}\supsetneq\dots\supsetneq X_{n}=0

Compressibility.html

  1. β = - 1 V V p \beta=-\frac{1}{V}\frac{\partial V}{\partial p}
  2. β T = - 1 V ( V p ) T \beta_{T}=-\frac{1}{V}\left(\frac{\partial V}{\partial p}\right)_{T}
  3. β S = - 1 V ( V p ) S \beta_{S}=-\frac{1}{V}\left(\frac{\partial V}{\partial p}\right)_{S}
  4. c 2 = ( p ρ ) S c^{2}=\left(\frac{\partial p}{\partial\rho}\right)_{S}
  5. ρ \rho
  6. β S = 1 ρ c 2 \beta_{S}=\frac{1}{\rho c^{2}}
  7. Z = p V ¯ R T Z=\frac{p\underline{V}}{RT}
  8. V ¯ \underline{V}
  9. p = R T V ¯ p={RT\over{\underline{V}}}
  10. β S = β T - α 2 T ρ c p \beta_{S}=\beta_{T}-\frac{\alpha^{2}T}{\rho c_{p}}
  11. β T β S = γ \frac{\beta_{T}}{\beta_{S}}=\gamma
  12. γ \gamma\!
  13. × 10 6 \times 10^{–}6
  14. × 10 7 \times 10^{–}7
  15. × 10 7 \times 10^{–}7
  16. × 10 7 \times 10^{–}7
  17. × 10 7 \times 10^{–}7
  18. × 10 8 \times 10^{–}8
  19. × 10 7 \times 10^{–}7
  20. × 10 8 \times 10^{–}8
  21. × 10 8 \times 10^{–}8
  22. × 10 8 \times 10^{–}8
  23. × 10 8 \times 10^{–}8
  24. × 10 9 \times 10^{–}9
  25. × 10 10 \times 10^{–}10
  26. × 10 10 \times 10^{–}10
  27. × 10 10 \times 10^{–}10
  28. × 10 10 \times 10^{–}10
  29. × 10 11 \times 10^{–}11
  30. × 10 11 \times 10^{–}11
  31. × 10 11 \times 10^{–}11
  32. × 10 11 \times 10^{–}11

Compressible_flow.html

  1. μ = arcsin ( a / V ) = arcsin ( 1 / M ) \mu=\arcsin(a/V)=\arcsin(1/M)
  2. d P ( 1 - M 2 ) = ρ V 2 ( d A A ) dP(1-{M}^{2})=\rho{V}^{2}\left(\frac{dA}{A}\right)
  3. V m a x = 2 c p T t V_{max}=\sqrt{2c_{p}T_{t}}
  4. < m t p l > p r o p e r t y 1 p r o p e r t y 2 = f ( M , γ ) \frac{<}{m}tpl>{{property_{1}}}{{property_{2}}}=f(M,\gamma)
  5. A * = γ R T * {A}^{*}=\sqrt{\gamma R{T}^{*}}
  6. M 2 x * = < m t p l > V x a * {M}^{*}_{2x}=\frac{<}{m}tpl>{{V_{x}}}{{{a}^{*}}}
  7. M 2 y * = < m t p l > V y a * {M}^{*}_{2y}=\frac{<}{m}tpl>{{V_{y}}}{{{a}^{*}}}
  8. < m t p l > d M M = f d x D γ M 2 2 - 2 γ M 2 \frac{<}{m}tpl>{{dM}}{{M}}=\frac{{fdx}}{{D}}\frac{{\gamma M^{2}}}{{2-2\gamma M% ^{2}}}

Computability_theory.html

  1. Π 1 1 \Pi^{1}_{1}

Computational_fluid_dynamics.html

  1. t Q d V + F d 𝐀 = 0 , \frac{\partial}{\partial t}\iiint Q\,dV+\iint F\,d\mathbf{A}=0,
  2. Q Q
  3. F F
  4. V V
  5. 𝐀 \mathbf{A}
  6. R i = W i Q d V e R_{i}=\iiint W_{i}Q\,dV^{e}
  7. R i R_{i}
  8. i i
  9. Q Q
  10. W i W_{i}
  11. V e V^{e}
  12. Q t + F x + G y + H z = 0 \frac{\partial Q}{\partial t}+\frac{\partial F}{\partial x}+\frac{\partial G}{% \partial y}+\frac{\partial H}{\partial z}=0
  13. Q Q
  14. F F
  15. G G
  16. H H
  17. x x
  18. y y
  19. z z
  20. v ( x , y ) = a x + b y + c x y + d v(x,y)=ax+by+cxy+d
  21. k - ϵ k-\epsilon
  22. k k
  23. ϵ \epsilon
  24. R e 3 Re^{3}
  25. - 40 39 -\frac{40}{39}
  26. f V ( s y m b o l v ; s y m b o l x , t ) d s y m b o l v f_{V}(symbol{v};symbol{x},t)dsymbol{v}
  27. s y m b o l x symbol{x}
  28. s y m b o l v symbol{v}
  29. s y m b o l v + d s y m b o l v symbol{v}+dsymbol{v}

Computational_physics.html

  1. 10 23 10^{23}

Concave_function.html

  1. f ( ( 1 - t ) x + ( t ) y ) ( 1 - t ) f ( x ) + ( t ) f ( y ) . f((1-t)x+(t)y)\geq(1-t)f(x)+(t)f(y).
  2. f ( ( 1 - t ) x + ( t ) y ) > ( 1 - t ) f ( x ) + ( t ) f ( y ) f((1-t)x+(t)y)>(1-t)f(x)+(t)f(y)\,
  3. S ( a ) = { x : f ( x ) a } S(a)=\{x:f(x)\geq a\}
  4. f ( y ) f ( x ) + f ( x ) [ y - x ] f(y)\leq f(x)+f^{\prime}(x)[y-x]
  5. f ( x + y 2 ) f ( x ) + f ( y ) 2 f\left(\frac{x+y}{2}\right)\geq\frac{f(x)+f(y)}{2}
  6. f ( t x ) = f ( t x + ( 1 - t ) 0 ) t f ( x ) + ( 1 - t ) f ( 0 ) t f ( x ) f(tx)=f(tx+(1-t)\cdot 0)\geq tf(x)+(1-t)f(0)\geq tf(x)
  7. f ( a ) + f ( b ) = f ( ( a + b ) a a + b ) + f ( ( a + b ) b a + b ) a a + b f ( a + b ) + b a + b f ( a + b ) = f ( a + b ) f(a)+f(b)=f\left((a+b)\frac{a}{a+b}\right)+f\left((a+b)\frac{b}{a+b}\right)% \geq\frac{a}{a+b}f(a+b)+\frac{b}{a+b}f(a+b)=f(a+b)
  8. f ( x ) = - x 2 f(x)=-x^{2}
  9. g ( x ) = x g(x)=\sqrt{x}
  10. f ′′ ( x ) = - 2 f^{\prime\prime}(x)=-2
  11. g ′′ ( x ) = - 1 4 x 1.5 g^{\prime\prime}(x)=-\frac{1}{4x^{1.5}}
  12. f ( x ) = a x + b f(x)=ax+b
  13. [ 0 , π ] [0,\pi]
  14. f ( B ) = log | B | f(B)=\log|B|
  15. | B | |B|

Concrete_category.html

  1. / 2 × / 2 \mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/2\mathbb{Z}
  2. / 4 \mathbb{Z}/4\mathbb{Z}
  3. \ast
  4. D ( x ) = { a P : a x } D(x)=\{a\in P:a\leq x\}
  5. D ( x ) D ( y ) D(x)\hookrightarrow D(y)
  6. 2 X 2^{X}
  7. R X × Y R\subseteq X\times Y
  8. ρ : 2 X 2 Y \rho:2^{X}\rightarrow 2^{Y}
  9. ρ ( A ) = { y Y : x A . x R y } \rho(A)=\{y\in Y:\exists x{\in}A~{}.~{}xRy\}
  10. c ob C X ( c ) \coprod_{c\in\mathrm{ob}C}X(c)

Confidence_interval.html

  1. Pr θ , ϕ ( u ( X ) < θ < v ( X ) ) = γ for all ( θ , ϕ ) . {\Pr}_{\theta,\phi}(u(X)<\theta<v(X))=\gamma\,\text{ for all }(\theta,\phi).
  2. Pr θ , ϕ ( u ( X ) < θ < v ( X ) ) γ for all ( θ , ϕ ) {\Pr}_{\theta,\phi}(u(X)<\theta<v(X))\approx\gamma\,\text{ for all }(\theta,% \phi)\,
  3. Pr θ , ϕ ( u ( X ) < θ < v ( X ) ) γ for all ( θ , ϕ ) {\Pr}_{\theta,\phi}(u(X)<\theta<v(X))\geq\gamma\,\text{ for all }(\theta,\phi)\,
  4. μ ^ = X ¯ = 1 n i = 1 n X i . \hat{\mu}=\bar{X}=\frac{1}{n}\sum_{i=1}^{n}X_{i}.
  5. x ¯ = 1 25 i = 1 25 x i = 250.2 grams . \bar{x}=\frac{1}{25}\sum_{i=1}^{25}x_{i}=250.2\,\,\text{grams}.
  6. σ n = 2.5 g 25 = 0.5 grams \frac{\sigma}{\sqrt{n}}=\frac{2.5~{}\,\text{g}}{\sqrt{25}}=0.5\ \,\text{grams}
  7. Z = X ¯ - μ σ / n = X ¯ - μ 0.5 Z=\frac{\bar{X}-\mu}{\sigma/\sqrt{n}}=\frac{\bar{X}-\mu}{0.5}
  8. P ( - z Z z ) = 1 - α = 0.95. \!P(-z\leq Z\leq z)=1-\alpha=0.95.
  9. Φ ( z ) \displaystyle\Phi(z)
  10. 0.95 = 1 - α = P ( - z Z z ) = P ( - 1.96 X ¯ - μ σ / n 1.96 ) = P ( X ¯ - 1.96 σ n μ X ¯ + 1.96 σ n ) . \begin{aligned}\displaystyle 0.95&\displaystyle=1-\alpha=P(-z\leq Z\leq z)=P% \left(-1.96\leq\frac{\bar{X}-\mu}{\sigma/\sqrt{n}}\leq 1.96\right)\\ &\displaystyle=P\left(\bar{X}-1.96\frac{\sigma}{\sqrt{n}}\leq\mu\leq\bar{X}+1.% 96\frac{\sigma}{\sqrt{n}}\right)\end{aligned}.
  11. Lower endpoint = X ¯ - 1.96 σ n , \,\text{Lower endpoint}=\bar{X}-1.96\frac{\sigma}{\sqrt{n}},
  12. Upper endpoint = X ¯ + 1.96 σ n . \,\text{Upper endpoint}=\bar{X}+1.96\frac{\sigma}{\sqrt{n}}.
  13. 0.95 \displaystyle 0.95
  14. X ¯ \bar{X}
  15. μ \mu
  16. X ¯ - 0.98 \!\bar{X}-0{.}98
  17. X ¯ + 0.98. \!\bar{X}+0.98.
  18. ( x ¯ - 0.98 , x ¯ + 0.98 ) . (\bar{x}-0.98,\,\bar{x}+0.98).
  19. ( x ¯ - 0.98 ; x ¯ + 0.98 ) = ( 250.2 - 0.98 ; 250.2 + 0.98 ) = ( 249.22 ; 251.18 ) . (\bar{x}-0.98;\bar{x}+0.98)=(250.2-0.98;250.2+0.98)=(249.22;251.18).\,
  20. X ¯ = ( X 1 + + X n ) / n , \bar{X}=(X_{1}+\cdots+X_{n})/n\,,
  21. S 2 = 1 n - 1 i = 1 n ( X i - X ¯ ) 2 . S^{2}=\frac{1}{n-1}\sum_{i=1}^{n}\left(X_{i}-\bar{X}\,\right)^{2}.
  22. T = X ¯ - μ S / n T=\frac{\bar{X}-\mu}{S/\sqrt{n}}
  23. Pr ( - c T c ) = 0.95 \Pr\left(-c\leq T\leq c\right)=0.95\,
  24. Pr ( X ¯ - c S n μ X ¯ + c S n ) = 0.95 \Pr\left(\bar{X}-\frac{cS}{\sqrt{n}}\leq\mu\leq\bar{X}+\frac{cS}{\sqrt{n}}% \right)=0.95\,
  25. [ x ¯ - c s n , x ¯ + c s n ] , \left[\bar{x}-\frac{cs}{\sqrt{n}},\bar{x}+\frac{cs}{\sqrt{n}}\right],\,
  26. Pr θ , ϕ ( u ( X ) < Y < v ( X ) ) = γ for all ( θ , ϕ ) . {\Pr}_{\theta,\phi}(u(X)<Y<v(X))=\gamma\,\text{ for all }(\theta,\phi).\,
  27. Pr ( u ( x ) < Θ < v ( x ) | X = x ) = γ . \Pr(u(x)<\Theta<v(x)|X=x)=\gamma.\,

Conformal_geometry.html

  1. h = λ 2 g h=\lambda^{2}g\,
  2. C S O ( 1 , 1 ) = { ( e a 0 0 e b ) | a , b } CSO(1,1)=\left\{\left.\begin{pmatrix}e^{a}&0\\ 0&e^{b}\end{pmatrix}\right|a,b\in\mathbb{R}\right\}
  3. g = 2 d x d y . g=2\,dx\,dy~{}.
  4. ( Diff ( S 1 ) ) × ( Diff ( S 1 ) ) (\mathbb{Z}\rtimes\mathrm{Diff}(S^{1}))\times(\mathbb{Z}\rtimes\mathrm{Diff}(S% ^{1}))\,
  5. q ( z , z ¯ ) = z z ¯ q(z,\bar{z})=z\bar{z}\,
  6. g = d z d z ¯ . g=dz\,d\bar{z}.
  7. 𝐋 X d z = f ( z ) d z \mathbf{L}_{X}\,dz=f(z)\,dz
  8. 𝐋 X d z ¯ = f ( z ¯ ) d z ¯ \mathbf{L}_{X}\,d\bar{z}=f(\bar{z})\,d\bar{z}
  9. z a z + b c z + d z\mapsto\frac{az+b}{cz+d}
  10. q ( x 0 , x 1 , , x n + 1 ) = - 2 x 0 x n + 1 + x 1 2 + x 2 2 + + x n 2 . q(x_{0},x_{1},\ldots,x_{n+1})=-2x_{0}x_{n+1}+x_{1}^{2}+x_{2}^{2}+\cdots+x_{n}^% {2}.
  11. N = { ( x 0 , , x n + 1 ) | - 2 x 0 x n + 1 + x 1 2 + + x n 2 = 0 } . N=\left\{\left.(x_{0},\ldots,x_{n+1})\right|-2x_{0}x_{n+1}+x_{1}^{2}+\cdots+x_% {n}^{2}=0\right\}.
  12. z 2 + x 1 2 + x 2 2 + + x n 2 = 1. z^{2}+x_{1}^{2}+x_{2}^{2}+\cdots+x_{n}^{2}=1.
  13. x 0 = z + 1 2 , x 1 = x 1 , , x n = x n , x n + 1 = z - 1 2 . x_{0}=\frac{z+1}{\sqrt{2}},\,x_{1}=x_{1},\,\ldots,\,x_{n}=x_{n},\,x_{n+1}=% \frac{z-1}{\sqrt{2}}.
  14. x 0 = z + 1 κ ( x ) 2 , x 1 = x 1 , , x n = x n , x n + 1 = ( z - 1 ) κ ( x ) 2 x_{0}=\frac{z+1}{\kappa(x)\sqrt{2}},\,x_{1}=x_{1},\,\ldots,\,x_{n}=x_{n},\,x_{% n+1}=\frac{(z-1)\kappa(x)}{\sqrt{2}}
  15. g = d z 2 + d x 1 2 + d x 2 2 + + d x n 2 g=dz^{2}+dx_{1}^{2}+dx_{2}^{2}+\cdots+dx_{n}^{2}\,
  16. z 2 + x 1 2 + x 2 2 + + x n 2 . z^{2}+x_{1}^{2}+x_{2}^{2}+\cdots+x_{n}^{2}.\,
  17. [ g ] = { λ 2 g | λ > 0 } . [g]=\left\{\left.\lambda^{2}g\right|\lambda>0\right\}.\,
  18. 𝐲 𝐑 n ( 2 𝐲 | 𝐲 | 2 + 1 , | 𝐲 | 2 - 1 | 𝐲 | 2 + 1 ) S \sub 𝐑 n + 1 . \mathbf{y}\in\mathbf{R}^{n}\mapsto\left(\frac{2\mathbf{y}}{|\mathbf{y}|^{2}+1}% ,\frac{|\mathbf{y}|^{2}-1}{|\mathbf{y}|^{2}+1}\right)\in S\sub\mathbf{R}^{n+1}.
  19. x 0 = 2 | 𝐲 | 2 1 + | 𝐲 | 2 , x i = y i | 𝐲 | 2 + 1 , x n + 1 = 2 1 | 𝐲 | 2 + 1 . x_{0}=\sqrt{2}\frac{|\mathbf{y}|^{2}}{1+|\mathbf{y}|^{2}},x_{i}=\frac{y_{i}}{|% \mathbf{y}|^{2}+1},x_{n+1}=\sqrt{2}\frac{1}{|\mathbf{y}|^{2}+1}.
  20. x 0 = t 2 | 𝐲 | 2 1 + | 𝐲 | 2 , x i = t y i | 𝐲 | 2 + 1 , x n + 1 = t 2 1 | 𝐲 | 2 + 1 . x_{0}=t\sqrt{2}\frac{|\mathbf{y}|^{2}}{1+|\mathbf{y}|^{2}},x_{i}=t\frac{y_{i}}% {|\mathbf{y}|^{2}+1},x_{n+1}=t\sqrt{2}\frac{1}{|\mathbf{y}|^{2}+1}.
  21. ρ = - 2 x 0 x n + 1 + x 1 2 + x 2 2 + + x n 2 t 2 . \rho=\frac{-2x_{0}x_{n+1}+x_{1}^{2}+x_{2}^{2}+\cdots+x_{n}^{2}}{t^{2}}.
  22. t 2 g i j ( y ) d y i d y j + 2 ρ d t 2 + 2 t d t d ρ , t^{2}g_{ij}(y)\,dy^{i}\,dy^{j}+2\rho\,dt^{2}+2t\,dt\,d\rho,\,
  23. t ( y ) 2 g i j d y i d y j . t(y)^{2}g_{ij}\,dy^{i}\,dy^{j}.\,
  24. Q = ( 0 0 - 1 0 J 0 - 1 0 0 ) Q=\begin{pmatrix}0&0&-1\\ 0&J&0\\ -1&0&0\end{pmatrix}
  25. 𝐠 = 𝐠 - 1 𝐠 0 𝐠 1 \mathbf{g}=\mathbf{g}_{-1}\oplus\mathbf{g}_{0}\oplus\mathbf{g}_{1}
  26. 𝐠 - 1 = { ( 0 t p 0 0 0 J - 1 p 0 0 0 ) | p n } , 𝐠 - 1 = { ( 0 0 0 t q 0 0 0 q J - 1 0 ) | q ( n ) * } \mathbf{g}_{-1}=\left\{\left.\begin{pmatrix}0&^{t}p&0\\ 0&0&J^{-1}p\\ 0&0&0\end{pmatrix}\right|p\in\mathbb{R}^{n}\right\},\quad\mathbf{g}_{-1}=\left% \{\left.\begin{pmatrix}0&0&0\\ ^{t}q&0&0\\ 0&qJ^{-1}&0\end{pmatrix}\right|q\in(\mathbb{R}^{n})^{*}\right\}
  27. 𝐠 0 = { ( - a 0 0 0 A 0 0 0 a ) | A 𝔰 𝔬 ( p , q ) , a } . \mathbf{g}_{0}=\left\{\left.\begin{pmatrix}-a&0&0\\ 0&A&0\\ 0&0&a\end{pmatrix}\right|A\in\mathfrak{so}(p,q),a\in\mathbb{R}\right\}.

Conjugate_closure.html

  1. \varnothing

Conjugate_transpose.html

  1. A A
  2. A A
  3. A A
  4. ( s y m b o l A * ) i j = s y m b o l A j i ¯ (symbol{A}^{*})_{ij}=\overline{symbol{A}_{ji}}
  5. a + b i a+bi
  6. a - b i a-bi
  7. s y m b o l A * = ( s y m b o l A ¯ ) T = s y m b o l A T ¯ symbol{A}^{*}=(\overline{symbol{A}})^{\mathrm{T}}=\overline{symbol{A}^{\mathrm% {T}}}
  8. s y m b o l A T symbol{A}^{\mathrm{T}}\,\!
  9. s y m b o l A ¯ \overline{symbol{A}}\,\!
  10. A A
  11. s y m b o l A * symbol{A}^{*}\,\!
  12. s y m b o l A H symbol{A}^{\mathrm{H}}\,\!
  13. s y m b o l A symbol{A}^{\dagger}\,\!
  14. A A
  15. s y m b o l A + symbol{A}^{+}\,\!
  16. s y m b o l A * symbol{A}^{*}\,\!
  17. s y m b o l A * T symbol{A}^{*\mathrm{T}}\,\!
  18. s y m b o l A T * symbol{A}^{\mathrm{T}*}\,\!
  19. s y m b o l A = [ 1 - 2 - i 1 + i i ] symbol{A}=\begin{bmatrix}1&-2-i\\ 1+i&i\end{bmatrix}
  20. s y m b o l A * = [ 1 1 - i - 2 + i - i ] symbol{A}^{*}=\begin{bmatrix}1&1-i\\ -2+i&-i\end{bmatrix}
  21. A A
  22. a i j a_{ij}
  23. a i j = a j i ¯ a_{ij}=\overline{a_{ji}}
  24. a i j = - a j i ¯ a_{ij}=-\overline{a_{ji}}
  25. A A
  26. a d j ( A ) adj(A)
  27. A A
  28. A A
  29. a + i b ( a - b b a ) . a+ib\equiv\left(\begin{matrix}a&-b\\ b&a\end{matrix}\right).
  30. 2 \mathbb{R}^{2}
  31. \mathbb{C}
  32. A A
  33. B B
  34. r r
  35. A A
  36. r r
  37. A A
  38. B B
  39. A A
  40. A A
  41. A A
  42. A A
  43. A s y m b o l x , s y m b o l y = s y m b o l x , A * s y m b o l y \langle Asymbol{x},symbol{y}\rangle=\langle symbol{x},A^{*}symbol{y}\rangle
  44. A A
  45. 𝐱 \mathbf{x}
  46. n \mathbb{C}^{n}
  47. 𝐲 \mathbf{y}
  48. m \mathbb{C}^{m}
  49. , \langle\cdot,\cdot\rangle
  50. m \mathbb{C}^{m}
  51. n \mathbb{C}^{n}
  52. A A
  53. 𝐂 < s u p > n \mathbf{C}<sup>n

Conjugated_system.html

  1. E n + 1 - E n = ( 2 n + 1 ) 2 π 2 2 m L 2 E_{n+1}-E_{n}=\frac{(2n+1)\hbar^{2}\pi^{2}}{2mL^{2}}

Conjunctive_normal_form.html

  1. ¬ A ( B C ) \neg A\wedge(B\vee C)
  2. ( A B ) ( ¬ B C ¬ D ) ( D ¬ E ) (A\vee B)\wedge(\neg B\vee C\vee\neg D)\wedge(D\vee\neg E)
  3. A B A\lor B
  4. A B A\wedge B
  5. A A
  6. B B
  7. ¬ ( B C ) \neg(B\vee C)
  8. ( A B ) C (A\wedge B)\vee C
  9. A ( B ( D E ) ) . A\wedge(B\vee(D\wedge E)).
  10. ¬ B ¬ C \neg B\wedge\neg C
  11. ( A C ) ( B C ) (A\vee C)\wedge(B\vee C)
  12. A ( B D ) ( B E ) . A\wedge(B\vee D)\wedge(B\vee E).
  13. 2 n 2^{n}
  14. ( X 1 Y 1 ) ( X 2 Y 2 ) ( X n Y n ) . (X_{1}\wedge Y_{1})\vee(X_{2}\wedge Y_{2})\vee\dots\vee(X_{n}\wedge Y_{n}).
  15. ( X 1 X 2 X n ) ( Y 1 X 2 X n ) ( X 1 Y 2 X n ) ( Y 1 Y 2 X n ) ( Y 1 Y 2 Y n ) . (X_{1}\vee X_{2}\vee\cdots\vee X_{n})\wedge(Y_{1}\vee X_{2}\vee\cdots\vee X_{n% })\wedge(X_{1}\vee Y_{2}\vee\cdots\vee X_{n})\wedge(Y_{1}\vee Y_{2}\vee\cdots% \vee X_{n})\wedge\cdots\wedge(Y_{1}\vee Y_{2}\vee\cdots\vee Y_{n}).
  16. 2 n 2^{n}
  17. X i X_{i}
  18. Y i Y_{i}
  19. i i
  20. Z 1 , , Z n Z_{1},\ldots,Z_{n}
  21. ( Z 1 Z n ) ( ¬ Z 1 X 1 ) ( ¬ Z 1 Y 1 ) ( ¬ Z n X n ) ( ¬ Z n Y n ) . (Z_{1}\vee\cdots\vee Z_{n})\wedge(\neg Z_{1}\vee X_{1})\wedge(\neg Z_{1}\vee Y% _{1})\wedge\cdots\wedge(\neg Z_{n}\vee X_{n})\wedge(\neg Z_{n}\vee Y_{n}).
  22. Z i Z_{i}
  23. X i X_{i}
  24. Y i Y_{i}
  25. Z i Z_{i}
  26. Z i ¬ X i ¬ Y i Z_{i}\vee\neg X_{i}\vee\neg Y_{i}
  27. Z i X i Y i Z_{i}\equiv X_{i}\wedge Y_{i}
  28. Z i Z_{i}
  29. X i Y i X_{i}\wedge Y_{i}
  30. ( (
  31. l 11 l_{11}
  32. \lor
  33. \ldots
  34. \lor
  35. l 1 n 1 l_{1n_{1}}
  36. ) )
  37. \land
  38. \ldots
  39. \land
  40. ( (
  41. l m 1 l_{m1}
  42. \lor
  43. \ldots
  44. \lor
  45. l m n m l_{mn_{m}}
  46. ) )
  47. l i j l_{ij}
  48. { \{
  49. { \{
  50. l 11 l_{11}
  51. , ,
  52. \ldots
  53. , ,
  54. l 1 n 1 l_{1n_{1}}
  55. } \}
  56. , ,
  57. \ldots
  58. , ,
  59. { \{
  60. l m 1 l_{m1}
  61. , ,
  62. \ldots
  63. , ,
  64. l m n m l_{mn_{m}}
  65. } \}
  66. } \}
  67. X 1 X k X n X_{1}\vee\cdots\vee X_{k}\vee\cdots\vee X_{n}
  68. X 1 X k - 1 Z X_{1}\vee\cdots\vee X_{k-1}\vee Z
  69. ¬ Z X k X n \neg Z\vee X_{k}\cdots\vee X_{n}
  70. Z Z
  71. P Q P\rightarrow Q
  72. ¬ P Q \lnot P\lor Q
  73. P Q P\leftrightarrow Q
  74. ( P ¬ Q ) ( ¬ P Q ) (P\lor\lnot Q)\land(\lnot P\lor Q)
  75. \rightarrow
  76. \leftrightarrow
  77. ¬ ( P Q ) \lnot(P\lor Q)
  78. ( ¬ P ) ( ¬ Q ) (\lnot P)\land(\lnot Q)
  79. ¬ ( P Q ) \lnot(P\land Q)
  80. ( ¬ P ) ( ¬ Q ) (\lnot P)\lor(\lnot Q)
  81. ¬ ¬ P \lnot\lnot P
  82. P P
  83. ¬ ( x P ( x ) ) \lnot(\forall xP(x))
  84. x ¬ P ( x ) \exists x\lnot P(x)
  85. ¬ ( x P ( x ) ) \lnot(\exists xP(x))
  86. x ¬ P ( x ) \forall x\lnot P(x)
  87. ¬ \lnot
  88. ( x P ( x ) ) ( x Q ( x ) ) (\forall xP(x))\lor(\exists xQ(x))
  89. x [ y A n i m a l ( y ) ¬ L o v e s ( x , y ) ] [ y L o v e s ( y , x ) ] \forall x[\exists yAnimal(y)\land\lnot Loves(x,y)]\lor[\exists yLoves(y,x)]
  90. x [ y A n i m a l ( y ) ¬ L o v e s ( x , y ) ] [ z L o v e s ( z , x ) ] \forall x[\exists yAnimal(y)\land\lnot Loves(x,y)]\lor[\exists zLoves(z,x)]
  91. P ( x Q ( x ) ) P\land(\forall xQ(x))
  92. x ( P Q ( x ) ) \forall x(P\land Q(x))
  93. P ( x Q ( x ) ) P\lor(\forall xQ(x))
  94. x ( P Q ( x ) ) \forall x(P\lor Q(x))
  95. P ( x Q ( x ) ) P\land(\exists xQ(x))
  96. x ( P Q ( x ) ) \exists x(P\land Q(x))
  97. P ( x Q ( x ) ) P\lor(\exists xQ(x))
  98. x ( P Q ( x ) ) \exists x(P\lor Q(x))
  99. x x
  100. P P
  101. ¬ \lnot
  102. \land
  103. \lor
  104. x 1 x n y P ( y ) \forall x_{1}\ldots\forall x_{n}\;\exists y\;P(y)
  105. x 1 x n P ( f ( x 1 , , x n ) ) \forall x_{1}\ldots\forall x_{n}\;P(f(x_{1},\ldots,x_{n}))
  106. f f
  107. n n
  108. P ( Q R ) P\lor(Q\land R)
  109. ( P Q ) ( P R ) (P\lor Q)\land(P\lor R)
  110. \color r e d red {\color{red}{\,\text{red}}}
  111. x \forall x
  112. ( (
  113. y \forall y
  114. A n i m a l ( Animal(
  115. y y
  116. ) )
  117. \color r e d \color{red}\rightarrow
  118. L o v e s ( x , Loves(x,
  119. y y
  120. ) )
  121. ) )
  122. \rightarrow
  123. ( (
  124. \exists
  125. y y
  126. L o v e s ( Loves(
  127. y y
  128. , x ) ,x)
  129. ) )
  130. x \forall x
  131. ( (
  132. y \forall y
  133. ¬ \lnot
  134. A n i m a l ( Animal(
  135. y y
  136. ) )
  137. \lor
  138. L o v e s ( x , Loves(x,
  139. y y
  140. ) )
  141. ) )
  142. \color r e d \color{red}\rightarrow
  143. ( (
  144. \exists
  145. y y
  146. L o v e s ( Loves(
  147. y y
  148. , x ) ,x)
  149. ) )
  150. x \forall x
  151. \color r e d ¬ \color{red}\lnot
  152. ( (
  153. \color r e d y {\color{red}{\forall y}}
  154. ¬ \lnot
  155. A n i m a l ( Animal(
  156. y y
  157. ) )
  158. \lor
  159. L o v e s ( x , Loves(x,
  160. y y
  161. ) )
  162. ) )
  163. \lor
  164. ( (
  165. \exists
  166. y y
  167. L o v e s ( Loves(
  168. y y
  169. , x ) ,x)
  170. ) )
  171. x \forall x
  172. ( (
  173. y \exists y
  174. \color r e d ¬ \color{red}\lnot
  175. ( (
  176. ¬ \lnot
  177. A n i m a l ( Animal(
  178. y y
  179. ) )
  180. \color r e d \color{red}\lor
  181. L o v e s ( x , Loves(x,
  182. y y
  183. ) )
  184. ) )
  185. ) )
  186. \lor
  187. ( (
  188. \exists
  189. y y
  190. L o v e s ( Loves(
  191. y y
  192. , x ) ,x)
  193. ) )
  194. x \forall x
  195. ( (
  196. y \exists y
  197. \color r e d ¬ \color{red}\lnot
  198. \color r e d ¬ \color{red}\lnot
  199. A n i m a l ( Animal(
  200. y y
  201. ) )
  202. \land
  203. ¬ \lnot
  204. L o v e s ( x , Loves(x,
  205. y y
  206. ) )
  207. ) )
  208. \lor
  209. ( (
  210. \exists
  211. y y
  212. L o v e s ( Loves(
  213. y y
  214. , x ) ,x)
  215. ) )
  216. x \forall x
  217. ( (
  218. \color r e d y {\color{red}{\exists y}}
  219. A n i m a l ( Animal(
  220. y y
  221. ) )
  222. \land
  223. ¬ \lnot
  224. L o v e s ( x , Loves(x,
  225. y y
  226. ) )
  227. ) )
  228. \lor
  229. ( (
  230. \color r e d \color{red}\exists
  231. \color r e d y \color{red}y
  232. L o v e s ( Loves(
  233. y y
  234. , x ) ,x)
  235. ) )
  236. x \forall x
  237. ( (
  238. y \exists y
  239. A n i m a l ( Animal(
  240. y y
  241. ) )
  242. \land
  243. ¬ \lnot
  244. L o v e s ( x , Loves(x,
  245. y y
  246. ) )
  247. ) )
  248. \color r e d \color{red}\lor
  249. ( (
  250. \color r e d \color{red}\exists
  251. \color r e d z \color{red}z
  252. L o v e s ( Loves(
  253. z z
  254. , x ) ,x)
  255. ) )
  256. x \forall x
  257. z \exists z
  258. ( (
  259. \color r e d y {\color{red}{\exists y}}
  260. A n i m a l ( Animal(
  261. y y
  262. ) )
  263. \land
  264. ¬ \lnot
  265. L o v e s ( x , Loves(x,
  266. y y
  267. ) )
  268. ) )
  269. \color r e d \color{red}\lor
  270. L o v e s ( Loves(
  271. z z
  272. , x ) ,x)
  273. x \forall x
  274. \color r e d z {\color{red}{\exists z}}
  275. y \exists y
  276. ( (
  277. A n i m a l ( Animal(
  278. y y
  279. ) )
  280. \land
  281. ¬ \lnot
  282. L o v e s ( x , Loves(x,
  283. y y
  284. ) )
  285. ) )
  286. \lor
  287. L o v e s ( Loves(
  288. z z
  289. , x ) ,x)
  290. x \forall x
  291. \color r e d y {\color{red}{\exists y}}
  292. ( (
  293. A n i m a l ( Animal(
  294. y y
  295. ) )
  296. \land
  297. ¬ \lnot
  298. L o v e s ( x , Loves(x,
  299. y y
  300. ) )
  301. ) )
  302. \lor
  303. L o v e s ( Loves(
  304. g ( x ) g(x)
  305. , x ) ,x)
  306. ( (
  307. A n i m a l ( Animal(
  308. f ( x ) f(x)
  309. ) )
  310. \color r e d \color{red}\land
  311. ¬ \lnot
  312. L o v e s ( x , Loves(x,
  313. f ( x ) f(x)
  314. ) )
  315. ) )
  316. \color r e d \color{red}\lor
  317. L o v e s ( Loves(
  318. g ( x ) g(x)
  319. , x ) ,x)
  320. ( (
  321. A n i m a l ( Animal(
  322. f ( x ) f(x)
  323. ) )
  324. \color r e d \color{red}\lor
  325. L o v e s ( Loves(
  326. g ( x ) g(x)
  327. , x ) ,x)
  328. ) )
  329. \color r e d \color{red}\land
  330. ( (
  331. ¬ L o v e s ( x , f ( x ) ) \lnot Loves(x,f(x))
  332. \color r e d \color{red}\lor
  333. L o v e s ( g ( x ) , x ) Loves(g(x),x)
  334. ) )
  335. { \{
  336. { \{
  337. A n i m a l ( Animal(
  338. f ( x ) f(x)
  339. ) )
  340. , ,
  341. L o v e s ( Loves(
  342. g ( x ) g(x)
  343. , x ) ,x)
  344. } \}
  345. , ,
  346. { \{
  347. ¬ L o v e s ( x , f ( x ) ) \lnot Loves(x,f(x))
  348. , ,
  349. L o v e s ( g ( x ) , x ) Loves(g(x),x)
  350. } \}
  351. } \}
  352. g ( x ) g(x)
  353. x x
  354. f ( x ) f(x)
  355. x x
  356. x x
  357. f ( x ) f(x)
  358. x x
  359. g ( x ) g(x)
  360. ( A n i m a l ( f ( x ) ) L o v e s ( g ( x ) , x ) ) ( ¬ L o v e s ( x , f ( x ) ) L o v e s ( g ( x ) , x ) ) (Animal(f(x))\lor Loves(g(x),x))\land(\lnot Loves(x,f(x))\lor Loves(g(x),x))

Connection_(mathematics).html

  1. φ 0 ( x , y ) = ( 2 x 1 + x 2 + y 2 , 2 y 1 + x 2 + y 2 , 1 - x 2 - y 2 1 + x 2 + y 2 ) φ 1 ( x , y ) = ( 2 x 1 + x 2 + y 2 , 2 y 1 + x 2 + y 2 , x 2 + y 2 - 1 1 + x 2 + y 2 ) \begin{aligned}\displaystyle\varphi_{0}(x,y)&\displaystyle=\left(\frac{2x}{1+x% ^{2}+y^{2}},\frac{2y}{1+x^{2}+y^{2}},\frac{1-x^{2}-y^{2}}{1+x^{2}+y^{2}}\right% )\\ \displaystyle\varphi_{1}(x,y)&\displaystyle=\left(\frac{2x}{1+x^{2}+y^{2}},% \frac{2y}{1+x^{2}+y^{2}},\frac{x^{2}+y^{2}-1}{1+x^{2}+y^{2}}\right)\end{aligned}
  2. φ 0 - 1 ( X , Y , Z ) = ( X Z + 1 , Y Z + 1 ) , φ 1 - 1 ( X , Y , Z ) = ( - X Z - 1 , - Y Z - 1 ) , \begin{aligned}\displaystyle\varphi_{0}^{-1}(X,Y,Z)&\displaystyle=\left(\frac{% X}{Z+1},\frac{Y}{Z+1}\right),\\ \displaystyle\varphi_{1}^{-1}(X,Y,Z)&\displaystyle=\left(\frac{-X}{Z-1},\frac{% -Y}{Z-1}\right),\end{aligned}
  3. φ 01 ( x , y ) = φ 0 - 1 φ 1 ( x , y ) = ( x x 2 + y 2 , y x 2 + y 2 ) \varphi_{01}(x,y)=\varphi_{0}^{-1}\circ\varphi_{1}(x,y)=\left(\frac{x}{x^{2}+y% ^{2}},\frac{y}{x^{2}+y^{2}}\right)
  4. v ( P ) = J φ 0 ( φ 0 - 1 ( P ) ) v 0 ( φ 0 - 1 ( P ) ) ( 1 ) v(P)=J_{\varphi_{0}}(\varphi_{0}^{-1}(P))\cdot{v}_{0}(\varphi_{0}^{-1}(P))% \qquad(1)
  5. J φ 0 J_{\varphi_{0}}
  6. v ( P ) = J φ 1 ( φ 1 - 1 ( P ) ) v 1 ( φ 1 - 1 ( P ) ) . ( 2 ) v(P)=J_{\varphi_{1}}(\varphi_{1}^{-1}(P))\cdot{v}_{1}(\varphi_{1}^{-1}(P)).% \qquad(2)
  7. J φ 1 ( φ 1 - 1 ( P ) ) = J φ 0 ( φ 0 - 1 ( P ) ) J φ 01 ( φ 1 - 1 ( P ) ) . J_{\varphi_{1}}(\varphi_{1}^{-1}(P))=J_{\varphi_{0}}(\varphi_{0}^{-1}(P))\cdot J% _{\varphi_{01}}(\varphi_{1}^{-1}(P)).\,
  8. v 0 ( φ 0 - 1 ( P ) ) = J φ 01 ( φ 1 - 1 ( P ) ) v 1 ( φ 1 - 1 ( P ) ) . ( 3 ) {v}_{0}(\varphi_{0}^{-1}(P))=J_{\varphi_{01}}(\varphi_{1}^{-1}(P))\cdot{v}_{1}% (\varphi_{1}^{-1}(P)).\qquad(3)
  9. d d t v 0 ( φ 0 - 1 ( P ( t ) ) ) = ( d d t J φ 01 ( φ 1 - 1 ( P ( t ) ) ) ) v 1 ( φ 1 - 1 ( P ( t ) ) ) . \frac{d}{dt}{v}_{0}(\varphi_{0}^{-1}(P(t)))=\left(\frac{d}{dt}J_{\varphi_{01}}% (\varphi_{1}^{-1}(P(t)))\right)\cdot{v}_{1}(\varphi_{1}^{-1}(P(t))).
  10. ( d d t J φ 01 ( φ 1 - 1 ( P ( t ) ) ) ) \left(\frac{d}{dt}J_{\varphi_{01}}(\varphi_{1}^{-1}(P(t)))\right)
  11. u v = D u v + Γ ( φ ) { u , v } \nabla_{u}{v}=D_{u}{v}+\Gamma(\varphi)\{{u},{v}\}

Conservation_of_mass.html

  1. Δ m = Δ E / c 2 . \Delta m=\Delta E/c^{2}.

Consistency.html

  1. Φ \Phi
  2. Φ \Phi
  3. ϕ \phi
  4. Φ ϕ \Phi\vdash\phi
  5. Φ ¬ ϕ \Phi\vdash\lnot\phi
  6. Φ \Phi
  7. Φ \Phi
  8. Φ \Phi
  9. ϕ \phi
  10. Φ \Phi
  11. ϕ \phi
  12. ϕ \phi
  13. Φ \Phi
  14. Φ \Phi
  15. Φ \Phi
  16. Φ \Phi
  17. Φ \Phi
  18. ϕ \phi
  19. Φ ϕ \Phi\cup\phi
  20. ϕ Φ \phi\in\Phi
  21. Φ \Phi
  22. x ϕ \exists x\phi
  23. t t
  24. ( x ϕ ϕ t x ) Φ (\exists x\phi\to\phi{t\over x})\in\Phi
  25. Φ \Phi
  26. ϕ , Φ ϕ . \phi,\;\Phi\vdash\phi.
  27. Φ \Phi
  28. \mathfrak{I}
  29. Φ \mathfrak{I}\vDash\Phi
  30. Φ \Phi
  31. ϕ \phi
  32. Φ ϕ \Phi\vdash\phi
  33. ( Φ { ¬ ϕ } ) \left(\Phi\cup\{\lnot\phi\}\right)
  34. Φ \Phi
  35. Φ ϕ \Phi\vdash\phi
  36. ( Φ { ϕ } ) \left(\Phi\cup\{\phi\}\right)
  37. Φ \Phi
  38. ( Φ { ϕ } ) \left(\Phi\cup\{\phi\}\right)
  39. ( Φ { ¬ ϕ } ) \left(\Phi\cup\{\lnot\phi\}\right)
  40. Φ \Phi
  41. ϕ \phi
  42. ψ \psi
  43. Φ ϕ \Phi\vdash\phi
  44. ϕ Φ \phi\in\Phi
  45. ϕ Φ \phi\in\Phi
  46. ¬ ϕ Φ \lnot\phi\in\Phi
  47. ( ϕ ψ ) Φ (\phi\psi)\in\Phi
  48. ϕ Φ \phi\in\Phi
  49. ψ Φ \psi\in\Phi
  50. ( ϕ ψ ) Φ (\phi\to\psi)\in\Phi
  51. ϕ Φ \phi\in\Phi
  52. ψ Φ \psi\in\Phi
  53. x ϕ Φ \exists x\phi\in\Phi
  54. t t
  55. ϕ t x Φ \phi{t\over x}\in\Phi
  56. Φ \Phi
  57. S S
  58. \sim
  59. S S
  60. t 0 t 1 t_{0}\sim t_{1}
  61. t 0 t 1 Φ \;t_{0}\equiv t_{1}\in\Phi
  62. t ¯ \overline{t}\!
  63. t t\!
  64. T Φ := { t ¯ | t T S } T_{\Phi}:=\{\;\overline{t}\;|\;t\in T^{S}\}
  65. T S T^{S}\!
  66. S S\!
  67. S S
  68. 𝔗 Φ \mathfrak{T}_{\Phi}
  69. T Φ T_{\Phi}\!
  70. Φ \Phi
  71. n n
  72. R S R\in S
  73. R 𝔗 Φ t 0 ¯ t n - 1 ¯ R^{\mathfrak{T}_{\Phi}}\overline{t_{0}}\ldots\overline{t_{n-1}}
  74. R t 0 t n - 1 Φ \;Rt_{0}\ldots t_{n-1}\in\Phi
  75. n n
  76. f S f\in S
  77. f 𝔗 Φ ( t 0 ¯ t n - 1 ¯ ) := f t 0 t n - 1 ¯ f^{\mathfrak{T}_{\Phi}}(\overline{t_{0}}\ldots\overline{t_{n-1}}):=\overline{% ft_{0}\ldots t_{n-1}}
  78. c S c\in S
  79. c 𝔗 Φ := c ¯ c^{\mathfrak{T}_{\Phi}}:=\overline{c}
  80. Φ := ( 𝔗 Φ , β Φ ) \mathfrak{I}_{\Phi}:=(\mathfrak{T}_{\Phi},\beta_{\Phi})
  81. Φ \Phi
  82. β Φ ( x ) := x ¯ \beta_{\Phi}(x):=\bar{x}
  83. ϕ \phi
  84. Φ ϕ \;\mathfrak{I}_{\Phi}\vDash\phi
  85. ϕ Φ \;\phi\in\Phi
  86. \sim
  87. \sim
  88. t 0 , , t n - 1 t_{0},\ldots,t_{n-1}
  89. Φ Φ \mathfrak{I}_{\Phi}\vDash\Phi

Constant_factor_rule_in_differentiation.html

  1. g ( x ) = k f ( x ) . g(x)=k\cdot f(x).
  2. g ( x ) = lim h 0 g ( x + h ) - g ( x ) h g^{\prime}(x)=\lim_{h\to 0}\frac{g(x+h)-g(x)}{h}
  3. g ( x ) = lim h 0 k f ( x + h ) - k f ( x ) h g^{\prime}(x)=\lim_{h\to 0}\frac{k\cdot f(x+h)-k\cdot f(x)}{h}
  4. g ( x ) = lim h 0 k ( f ( x + h ) - f ( x ) ) h g^{\prime}(x)=\lim_{h\to 0}\frac{k(f(x+h)-f(x))}{h}
  5. g ( x ) = k lim h 0 f ( x + h ) - f ( x ) h g^{\prime}(x)=k\lim_{h\to 0}\frac{f(x+h)-f(x)}{h}
  6. g ( x ) = k f ( x ) . g^{\prime}(x)=k\cdot f^{\prime}(x).
  7. d ( k f ( x ) ) d x = k d ( f ( x ) ) d x . \frac{d(k\cdot f(x))}{dx}=k\cdot\frac{d(f(x))}{dx}.
  8. d ( - y ) d x = - d y d x . \frac{d(-y)}{dx}=-\frac{dy}{dx}.

Constant_factor_rule_in_integration.html

  1. k d y d x d x = k d y d x d x . \int k\frac{dy}{dx}dx=k\int\frac{dy}{dx}dx.\quad
  2. y = d y d x d x . y=\int\frac{dy}{dx}dx.
  3. k y = k d y d x d x . (1) ky=k\int\frac{dy}{dx}dx.\quad\mbox{(1)}~{}
  4. d ( k y ) d x = k d y d x . \frac{d\left(ky\right)}{dx}=k\frac{dy}{dx}.
  5. k y = k d y d x d x . (2) ky=\int k\frac{dy}{dx}dx.\quad\mbox{(2)}~{}
  6. k y = k d y d x d x ky=k\int\frac{dy}{dx}dx
  7. k y = k d y d x d x . ky=\int k\frac{dy}{dx}dx.
  8. k d y d x d x = k d y d x d x . (3) \int k\frac{dy}{dx}dx=k\int\frac{dy}{dx}dx.\quad\mbox{(3)}~{}
  9. u = d y d x . u=\frac{dy}{dx}.
  10. k u d x = k u d x . \int kudx=k\int udx.
  11. y = u . y=u.\,
  12. k y d x = k y d x . \int kydx=k\int ydx.
  13. - y d x = - y d x . \int-ydx=-\int ydx.

Constant_of_integration.html

  1. f ( x ) f(x)
  2. F ( x ) F(x)
  3. f ( x ) f(x)
  4. f ( x ) f(x)
  5. F ( x ) + C F(x)+C
  6. F ( x ) F(x)
  7. f ( x ) f(x)
  8. ( F ( x ) + C ) = F ( x ) + C = F ( x ) (F(x)+C)^{\prime}=F\,^{\prime}(x)+C\,^{\prime}=F\,^{\prime}(x)
  9. cos ( x ) \cos(x)
  10. sin ( x ) \sin(x)
  11. sin ( x ) + 1 \sin(x)+1
  12. sin ( x ) - π \sin(x)-\pi
  13. cos ( x ) \cos(x)
  14. cos ( x ) \cos(x)
  15. cos ( x ) d x = sin ( x ) + C . \int\cos(x)\,dx=\sin(x)+C.
  16. cos ( x ) \cos(x)
  17. d d x [ sin ( x ) + C ] \displaystyle\frac{d}{dx}[\sin(x)+C]
  18. 2 sin ( x ) cos ( x ) 2\sin(x)\cos(x)
  19. 2 sin ( x ) cos ( x ) d x \displaystyle\int 2\sin(x)\cos(x)\,dx
  20. cos ( x ) \cos(x)
  21. f ( x ) f(x)
  22. d y d x = f ( x ) \frac{dy}{dx}=f(x)
  23. d d x \frac{d}{dx}
  24. d d x \frac{d}{dx}
  25. d d x \frac{d}{dx}
  26. F : F:\mathbb{R}\rightarrow\mathbb{R}
  27. G : G:\mathbb{R}\rightarrow\mathbb{R}
  28. F ( x ) = G ( x ) F\,^{\prime}(x)=G\,^{\prime}(x)
  29. F ( x ) - G ( x ) = C F(x)-G(x)=C
  30. [ F ( x ) - G ( x ) ] = 0 [F(x)-G(x)]^{\prime}=0
  31. C = F ( a ) C=F(a)
  32. a x 0 d t = F ( x ) - F ( a ) = F ( x ) - C , \begin{aligned}\displaystyle\int_{a}^{x}0\,dt&\displaystyle=F(x)-F(a)\\ &\displaystyle=F(x)-C,\end{aligned}
  33. F ( x ) = C F(x)=C
  34. d x / x \textstyle\int dx/x
  35. tan x d x , \textstyle\int\tan x\,dx,
  36. 1 x d x = { ln | x | + C - x < 0 ln | x | + C + x > 0 \int{1\over x}\,dx=\begin{cases}\ln\left|x\right|+C^{-}&x<0\\ \ln\left|x\right|+C^{+}&x>0\end{cases}
  37. F ( x ) F(x)
  38. G ( x ) = 0 G(x)=0

Constant_term.html

  1. x 2 + 2 x + 3 , x^{2}+2x+3,
  2. a x 2 + b x + c , ax^{2}+bx+c,
  3. ( x + 1 ) ( x - 2 ) (x+1)(x-2)
  4. x 2 + 2 x y + y 2 - 2 x + 2 y - 4 x^{2}+2xy+y^{2}-2x+2y-4
  5. a 0 + a 1 x + a 2 x 2 + a 3 x 3 + , a_{0}+a_{1}x+a_{2}x^{2}+a_{3}x^{3}+\cdots,
  6. ( x - 3 ) 2 + 4 (x-3)^{2}+4

Constraint_satisfaction_problem.html

  1. X , D , C \langle X,D,C\rangle
  2. X = { X 1 , , X n } X=\{X_{1},\ldots,X_{n}\}
  3. D = { D 1 , , D n } D=\{D_{1},\ldots,D_{n}\}
  4. C = { C 1 , , C m } C=\{C_{1},\ldots,C_{m}\}
  5. X i X_{i}
  6. D i D_{i}
  7. C j C C_{j}\in C
  8. t j , R j \langle t_{j},R_{j}\rangle
  9. t j X t_{j}\subset X
  10. k k
  11. R j R_{j}
  12. k k
  13. D j D_{j}
  14. v v
  15. t j , R j \langle t_{j},R_{j}\rangle
  16. t j t_{j}
  17. R j R_{j}

Construction_of_the_real_numbers.html

  1. A A\,
  2. ( A , B ) (A,B)\,
  3. A A
  4. B B
  5. r r
  6. 𝐐 \,\textbf{Q}
  7. r r
  8. r 𝐐 r\neq\,\textbf{Q}
  9. r r
  10. x , y 𝐐 x,y\in\,\textbf{Q}
  11. x < y x<y
  12. y r y\in r
  13. x r x\in r
  14. r r
  15. x r x\in r
  16. y r y\in r
  17. y x y\leq x
  18. 𝐑 \,\textbf{R}
  19. A A
  20. 𝐐 \,\textbf{Q}
  21. x y x y x\leq y\Leftrightarrow x\subseteq y
  22. q q
  23. { x 𝐐 : x < q } \{x\in\,\textbf{Q}:x<q\}
  24. A + B := { a + b : a A and b B } A+B:=\{a+b:a\in A\and b\in B\}
  25. A - B := { a - b : a A and b ( 𝐐 B ) } A-B:=\{a-b:a\in A\and b\in(\,\textbf{Q}\setminus B)\}
  26. 𝐐 B \,\textbf{Q}\setminus B
  27. B B
  28. 𝐐 \,\textbf{Q}
  29. { x : x 𝐐 and x B } \{x:x\in\,\textbf{Q}\and x\notin B\}
  30. - B := { a - b : a < 0 and b ( 𝐐 B ) } -B:=\{a-b:a<0\and b\in(\,\textbf{Q}\setminus B)\}
  31. A , B 0 A,B\geq 0
  32. A × B := { a × b : a 0 and a A and b 0 and b B } { x Q : x < 0 } A\times B:=\{a\times b:a\geq 0\and a\in A\and b\geq 0\and b\in B\}\cup\{x\in% \mathrm{Q}:x<0\}
  33. A A\,
  34. B B\,
  35. A × B = - ( A × - B ) = - ( - A × B ) = ( - A × - B ) A\times B=-(A\times-B)=-(-A\times B)=(-A\times-B)\,
  36. A A\,
  37. B B\,
  38. A 0 and B > 0 A\geq 0\mbox{ and }~{}B>0
  39. A / B := { a / b : a A and b ( 𝐐 B ) } A/B:=\{a/b:a\in A\and b\in(\,\textbf{Q}\setminus B)\}
  40. A A\,
  41. B B\,
  42. A / B = - ( A / - B ) = - ( - A / B ) = - A / - B A/B=-(A/{-B})=-(-A/B)=-A/{-B}\,
  43. A A\,
  44. B B\,
  45. S S
  46. 𝐑 \,\textbf{R}
  47. 𝐑 \,\textbf{R}
  48. S \bigcup S
  49. A = { x 𝐐 : x < 0 x × x < 2 } A=\{x\in\,\textbf{Q}:x<0x\times x<2\}
  50. A A
  51. A × A = 2 A\times A=2\,
  52. A A\,
  53. x x\,
  54. x × x < 2 x\times x<2\,
  55. y y\,
  56. x < y x<y\,
  57. y × y < 2 . y\times y<2\,.
  58. y = 2 x + 2 x + 2 y=\frac{2x+2}{x+2}\,
  59. A × A 2 A\times A\leq 2
  60. r r\,
  61. x x\,
  62. A A
  63. r < x × x r<x\times x\,
  64. \mathbb{Z}
  65. f : f:\mathbb{Z}\to\mathbb{Z}
  66. { f ( n + m ) - f ( m ) - f ( n ) : n , m } \{f(n+m)-f(m)-f(n):n,m\in\mathbb{Z}\}
  67. f , g f,g
  68. { f ( n ) - g ( n ) : n } \{f(n)-g(n):n\in\mathbb{Z}\}
  69. [ f ] [f]
  70. f f
  71. 0 [ f ] 0\leq[f]
  72. f f
  73. f f
  74. + \mathbb{Z}^{+}

Consumer_choice.html

  1. R + 2 R^{2}_{+}
  2. ( x , y ) (x,y)
  3. x 0 x\geq 0
  4. y 0 y\geq 0
  5. u ( x , y ) = x α y β u(x,y)=x^{\alpha}\cdot y^{\beta}
  6. ( x , y ) (x,y)
  7. x p X + y p Y xp_{X}+yp_{Y}
  8. x p X + y p Y income xp_{X}+yp_{Y}\leq\mathrm{income}
  9. P ( L ) P(L)
  10. Δ y 1 n \Delta y_{1}^{n}
  11. m m^{\prime}
  12. m m
  13. p 1 p_{1}^{\prime}
  14. Δ y 1 n = y 1 ( p 1 , m ) - y 1 ( p 1 , m ) . \Delta y_{1}^{n}=y_{1}(p_{1}^{\prime},m)-y_{1}(p_{1}^{\prime},m^{\prime}).
  15. Δ y 1 s \Delta y_{1}^{s}
  16. Y \ Y
  17. Y \ Y
  18. p 1 \ p_{1}
  19. p 1 \ p_{1}^{\prime}

Consumer_price_index.html

  1. C P I 2 C P I 1 = P r i c e 2 P r i c e 1 \frac{CPI_{2}}{CPI_{1}}=\frac{Price_{2}}{Price_{1}}
  2. C P I = updated cost base period cost × 100 CPI=\frac{\,\text{updated cost}}{\,\text{base period cost}}\times 100
  3. C P I = i = 1 n C P I i * w e i g h t i CPI=\sum_{i=1}^{n}CPI_{i}*weight_{i}
  4. w e i g h t i weight_{i}

Continuous_wave.html

  1. B n = B K B_{n}=BK
  2. B n B_{n}
  3. B B
  4. K K

Contract_theory.html

  1. max w ( ) E [ y ( e ^ ) - w ( y ( e ^ ) ) ] \max_{w(\cdot)}E\left[y(\hat{e})-w(y(\hat{e}))\right]
  2. E [ u ( w ( y ( e ) ) ) - c ( e ) ] u ¯ E\left[u(w(y(e)))-c(e)\right]\geq\bar{u}
  3. e ^ = arg max e E [ u ( w ( y ( e ) ) ) - c ( e ) ] u ¯ \hat{e}=\arg\max_{e}E\left[u(w(y(e)))-c(e)\right]\geq\bar{u}
  4. w ( ) w(\cdot)
  5. y y
  6. e e
  7. c ( e ) c(e)
  8. u ¯ \bar{u}
  9. u ( ) u(\cdot)

Contradiction.html

  1. \bot
  2. \top
  3. \vdash
  4. φ \varphi
  5. φ \varphi\vdash\bot
  6. φ \varphi
  7. φ ψ \vdash\varphi\rightarrow\psi
  8. ψ \psi
  9. ψ \bot\rightarrow\psi
  10. φ \varphi
  11. φ \vdash\varphi
  12. φ \varphi
  13. ¬ φ \neg\varphi\vdash\bot
  14. φ \varphi
  15. ¬ φ \neg\varphi\vdash\bot
  16. φ \varphi
  17. A ¬ A A\vee\neg A
  18. \Rightarrow\Leftarrow

Controllability.html

  1. 𝐱 ˙ ( t ) = A ( t ) 𝐱 ( t ) + B ( t ) 𝐮 ( t ) \dot{\mathbf{x}}(t)=A(t)\mathbf{x}(t)+B(t)\mathbf{u}(t)
  2. 𝐲 ( t ) = C ( t ) 𝐱 ( t ) + D ( t ) 𝐮 ( t ) . \mathbf{y}(t)=C(t)\mathbf{x}(t)+D(t)\mathbf{u}(t).
  3. u u
  4. x 0 x_{0}
  5. t 0 t_{0}
  6. x 1 x_{1}
  7. t 1 > t 0 t_{1}>t_{0}
  8. x 1 - ϕ ( t 0 , t 1 ) x 0 x_{1}-\phi(t_{0},t_{1})x_{0}
  9. W ( t 0 , t 1 ) = t 0 t 1 ϕ ( t 0 , t ) B ( t ) B ( t ) T ϕ ( t 0 , t ) T d t W(t_{0},t_{1})=\int_{t_{0}}^{t_{1}}\phi(t_{0},t)B(t)B(t)^{T}\phi(t_{0},t)^{T}dt
  10. ϕ \phi
  11. W ( t 0 , t 1 ) W(t_{0},t_{1})
  12. η 0 \eta_{0}
  13. W ( t 0 , t 1 ) η = x 1 - ϕ ( t 0 , t 1 ) x 0 W(t_{0},t_{1})\eta=x_{1}-\phi(t_{0},t_{1})x_{0}
  14. u ( t ) = - B ( t ) T ϕ ( t 0 , t ) T η 0 u(t)=-B(t)^{T}\phi(t_{0},t)^{T}\eta_{0}
  15. W W
  16. W ( t 0 , t 1 ) W(t_{0},t_{1})
  17. W ( t 0 , t 1 ) W(t_{0},t_{1})
  18. t 1 t 0 t_{1}\geq t_{0}
  19. W ( t 0 , t 1 ) W(t_{0},t_{1})
  20. d d t W ( t , t 1 ) = A ( t ) W ( t , t 1 ) + W ( t , t 1 ) A ( t ) T - B ( t ) B ( t ) T , W ( t 1 , t 1 ) = 0 \frac{d}{dt}W(t,t_{1})=A(t)W(t,t_{1})+W(t,t_{1})A(t)^{T}-B(t)B(t)^{T},\;W(t_{1% },t_{1})=0
  21. W ( t 0 , t 1 ) W(t_{0},t_{1})
  22. W ( t 0 , t 1 ) = W ( t 0 , t ) + ϕ ( t 0 , t ) W ( t , t 1 ) ϕ ( t 0 , t ) T W(t_{0},t_{1})=W(t_{0},t)+\phi(t_{0},t)W(t,t_{1})\phi(t_{0},t)^{T}
  23. 𝐱 ˙ ( t ) = A 𝐱 ( t ) + B 𝐮 ( t ) \dot{\mathbf{x}}(t)=A\mathbf{x}(t)+B\mathbf{u}(t)
  24. 𝐲 ( t ) = C 𝐱 ( t ) + D 𝐮 ( t ) \mathbf{y}(t)=C\mathbf{x}(t)+D\mathbf{u}(t)
  25. 𝐱 \mathbf{x}
  26. n × 1 n\times 1
  27. 𝐲 \mathbf{y}
  28. m × 1 m\times 1
  29. 𝐮 \mathbf{u}
  30. r × 1 r\times 1
  31. A A
  32. n × n n\times n
  33. B B
  34. n × r n\times r
  35. C C
  36. m × n m\times n
  37. D D
  38. m × r m\times r
  39. n × n r n\times nr
  40. R = [ B A B A 2 B A n - 1 B ] R=\begin{bmatrix}B&AB&A^{2}B&...&A^{n-1}B\end{bmatrix}
  41. rank ( R ) = n \operatorname{rank}(R)=n
  42. k k\in\mathbb{Z}
  43. 𝐱 ( k + 1 ) = A 𝐱 ( k ) + B 𝐮 ( k ) \,\textbf{x}(k+1)=A\,\textbf{x}(k)+B\,\textbf{u}(k)
  44. A A
  45. n × n n\times n
  46. B B
  47. n × r n\times r
  48. 𝐮 \mathbf{u}
  49. r r
  50. r × 1 r\times 1
  51. n × n r n\times nr
  52. 𝒞 = [ B A B A 2 B A n - 1 B ] \mathcal{C}=\begin{bmatrix}B&AB&A^{2}B&\cdots&A^{n-1}B\end{bmatrix}
  53. rank ( 𝒞 ) = n \operatorname{rank}(\mathcal{C})=n
  54. 𝒞 \mathcal{C}
  55. n n
  56. n n
  57. 𝒞 \mathcal{C}
  58. n n
  59. u ( k ) u(k)
  60. 𝐱 ( 0 ) \,\textbf{x}(0)
  61. 𝐱 ( 1 ) = A 𝐱 ( 0 ) + B 𝐮 ( 0 ) , \,\textbf{x}(1)=A\,\textbf{x}(0)+B\,\textbf{u}(0),
  62. 𝐱 ( 2 ) = A 𝐱 ( 1 ) + B 𝐮 ( 1 ) = A 2 𝐱 ( 0 ) + A B 𝐮 ( 0 ) + B 𝐮 ( 1 ) , \,\textbf{x}(2)=A\,\textbf{x}(1)+B\,\textbf{u}(1)=A^{2}\,\textbf{x}(0)+AB\,% \textbf{u}(0)+B\,\textbf{u}(1),
  63. 𝐱 ( n ) = B 𝐮 ( n - 1 ) + A B 𝐮 ( n - 2 ) + + A n - 1 B 𝐮 ( 0 ) + A n 𝐱 ( 0 ) \,\textbf{x}(n)=B\,\textbf{u}(n-1)+AB\,\textbf{u}(n-2)+\cdots+A^{n-1}B\,% \textbf{u}(0)+A^{n}\,\textbf{x}(0)
  64. 𝐱 ( n ) - A n 𝐱 ( 0 ) = [ B A B A n - 1 B ] [ 𝐮 T ( n - 1 ) 𝐮 T ( n - 2 ) 𝐮 T ( 0 ) ] T . \,\textbf{x}(n)-A^{n}\,\textbf{x}(0)=[B\,\,AB\,\,\cdots\,\,A^{n-1}B][\,\textbf% {u}^{T}(n-1)\,\,\,\textbf{u}^{T}(n-2)\,\,\cdots\,\,\,\textbf{u}^{T}(0)]^{T}.
  65. 𝐱 ( n ) \,\textbf{x}(n)
  66. n = 2 n=2
  67. r = 1 r=1
  68. B B
  69. A B AB
  70. 2 × 1 2\times 1
  71. [ B A B ] \begin{bmatrix}B&AB\end{bmatrix}
  72. B B
  73. A B AB
  74. B B
  75. A B AB
  76. k = 0 k=0
  77. x ( 1 ) = A 𝐱 ( 0 ) + B 𝐮 ( 0 ) = B 𝐮 ( 0 ) x(1)=A\,\textbf{x}(0)+B\,\textbf{u}(0)=B\,\textbf{u}(0)
  78. k = 1 k=1
  79. x ( 2 ) = A 𝐱 ( 1 ) + B 𝐮 ( 1 ) = A B 𝐮 ( 0 ) + B 𝐮 ( 1 ) x(2)=A\,\textbf{x}(1)+B\,\textbf{u}(1)=AB\,\textbf{u}(0)+B\,\textbf{u}(1)
  80. k = 0 k=0
  81. B B
  82. k = 1 k=1
  83. A B AB
  84. B B
  85. k = 2 k=2
  86. n n
  87. n = 2 n=2
  88. C C
  89. C C
  90. n = 3 n=3
  91. 𝐱 ˙ = 𝐟 ( 𝐱 ) + i = 1 m 𝐠 i ( 𝐱 ) u i \dot{\mathbf{x}}=\mathbf{f(x)}+\sum_{i=1}^{m}\mathbf{g}_{i}(\mathbf{x})u_{i}
  92. x 0 x_{0}
  93. R R
  94. n n
  95. n n
  96. x x
  97. R = [ 𝐠 1 𝐠 m [ ad 𝐠 i k 𝐠 𝐣 ] [ ad 𝐟 k 𝐠 𝐢 ] ] . R=\begin{bmatrix}\mathbf{g}_{1}&\cdots&\mathbf{g}_{m}&[\mathrm{ad}^{k}_{% \mathbf{g}_{i}}\mathbf{\mathbf{g}_{j}}]&\cdots&[\mathrm{ad}^{k}_{\mathbf{f}}% \mathbf{\mathbf{g}_{i}}]\end{bmatrix}.
  98. [ ad 𝐟 k 𝐠 ] [\mathrm{ad}^{k}_{\mathbf{f}}\mathbf{\mathbf{g}}]
  99. [ ad 𝐟 k 𝐠 ] = [ 𝐟 j [ 𝐟 , 𝐠 ] ] . [\mathrm{ad}^{k}_{\mathbf{f}}\mathbf{\mathbf{g}}]=\begin{bmatrix}\mathbf{f}&% \cdots&j&\cdots&\mathbf{[\mathbf{f},\mathbf{g}]}\end{bmatrix}.
  100. A A
  101. B B
  102. C C
  103. D D
  104. m × ( n + 1 ) r m\times(n+1)r
  105. [ C B C A B C A 2 B C A n - 1 B D ] \begin{bmatrix}CB&CAB&CA^{2}B&\cdots&CA^{n-1}B&D\end{bmatrix}
  106. m m

Converse_(logic).html

  1. x . S ( x ) P ( x ) \forall x.S(x)\to P(x)

Convex_function.html

  1. f ( x ) f(x)
  2. f ( x ) = x 2 f(x)=x^{2}
  3. f ( x ) = e x f(x)=e^{x}
  4. f : X 𝐑 f:X→\mathbf{R}
  5. f f
  6. x 1 , x 2 X , t [ 0 , 1 ] : f ( t x 1 + ( 1 - t ) x 2 ) t f ( x 1 ) + ( 1 - t ) f ( x 2 ) . \forall x_{1},x_{2}\in X,\forall t\in[0,1]:\qquad f(tx_{1}+(1-t)x_{2})\leq tf(% x_{1})+(1-t)f(x_{2}).
  7. f f
  8. x 1 x 2 X , t ( 0 , 1 ) : f ( t x 1 + ( 1 - t ) x 2 ) < t f ( x 1 ) + ( 1 - t ) f ( x 2 ) . \forall x_{1}\neq x_{2}\in X,\forall t\in(0,1):\qquad f(tx_{1}+(1-t)x_{2})<tf(% x_{1})+(1-t)f(x_{2}).
  9. f f
  10. f −f
  11. f f
  12. R ( x 1 , x 2 ) = f ( x 1 ) - f ( x 2 ) x 1 - x 2 R(x_{1},x_{2})=\frac{f(x_{1})-f(x_{2})}{x_{1}-x_{2}}
  13. f f
  14. f f
  15. C C
  16. f f
  17. f f
  18. C C
  19. f f
  20. C C
  21. C C
  22. x 1 , x 2 C : f ( x 1 + x 2 2 ) f ( x 1 ) + f ( x 2 ) 2 . \forall x_{1},x_{2}\in C:\qquad f\left(\frac{x_{1}+x_{2}}{2}\right)\leq\frac{f% (x_{1})+f(x_{2})}{2}.
  23. f ( x ) f ( y ) + f ( y ) ( x - y ) f(x)\geq f(y)+f^{\prime}(y)(x-y)
  24. f ( c ) = 0 f′(c)=0
  25. c c
  26. f ( x ) f(x)
  27. f f
  28. f f
  29. f f
  30. E ( f ( X ) ) f ( E ( X ) ) E(f(X))≥f(E(X))
  31. E E
  32. f f
  33. f ( 0 ) 0 f(0)≤0
  34. f f
  35. f f
  36. f ( t x ) = f ( t x + ( 1 - t ) 0 ) t f ( x ) + ( 1 - t ) f ( 0 ) t f ( x ) , t [ 0 , 1 ] . f(tx)=f(tx+(1-t)\cdot 0)\leq tf(x)+(1-t)f(0)\leq tf(x),\quad\forall t\in[0,1].
  37. f ( a ) + f ( b ) = f ( ( a + b ) a a + b ) + f ( ( a + b ) b a + b ) a a + b f ( a + b ) + b a + b f ( a + b ) = f ( a + b ) f(a)+f(b)=f\left((a+b)\frac{a}{a+b}\right)+f\left((a+b)\frac{b}{a+b}\right)% \leq\frac{a}{a+b}f(a+b)+\frac{b}{a+b}f(a+b)=f(a+b)
  38. f f
  39. g g
  40. m ( x ) = max { f ( x ) , g ( x ) } m(x)=\max\{f(x),g(x)\}
  41. h ( x ) = f ( x ) + g ( x ) . h(x)=f(x)+g(x).
  42. f f
  43. g g
  44. g g
  45. h ( x ) = g ( f ( x ) ) h(x)=g(f(x))
  46. f ( x ) f(x)
  47. e f ( x ) e^{f(x)}
  48. e x e^{x}
  49. f f
  50. g g
  51. h ( x ) = g ( f ( x ) ) h(x)=g(f(x))
  52. f ( x ) f(x)
  53. x 𝐑 n x\in\mathbf{R}^{n}
  54. g ( x ) = f ( A x + b ) g(x)=f(Ax+b)
  55. A 𝐑 m × n , b 𝐑 m . A\in\mathbf{R}^{m\times n},b\in\mathbf{R}^{m}.
  56. f ( x , y ) f(x,y)
  57. x x
  58. g ( x ) = sup y C f ( x , y ) g(x)=\sup_{y\in C}f(x,y)
  59. g ( x ) > - g(x)>-\infty
  60. x x
  61. f ( x , y ) f(x,y)
  62. ( x , y ) (x,y)
  63. g ( x ) = inf y C f ( x , y ) g(x)=\inf_{y\in C}f(x,y)
  64. x x
  65. g ( x ) > - g(x)>-\infty
  66. f ( x ) f(x)
  67. g ( x , t ) = t f ( x / t ) g(x,t)=tf(x/t)
  68. { ( x , t ) | x t Dom ( f ) , t > 0 } \left\{(x,t)|\tfrac{x}{t}\in\,\text{Dom}(f),t>0\right\}
  69. f ( x ) f(x)
  70. f ( x ) = sup n ( a n x + b n ) f(x)=\sup_{n}(a_{n}x+b_{n})
  71. ( a n , b n ) (a_{n},b_{n})
  72. f f
  73. m > 0 m>0
  74. x , y x,y
  75. ( f ( x ) - f ( y ) ) T ( x - y ) m x - y 2 2 (\nabla f(x)-\nabla f(y))^{T}(x-y)\geq m\|x-y\|_{2}^{2}
  76. f ( x ) - f ( y ) , ( x - y ) m x - y 2 \langle\nabla f(x)-\nabla f(y),(x-y)\rangle\geq m\|x-y\|^{2}
  77. \|\cdot\|
  78. f ( y ) f ( x ) + f ( x ) T ( y - x ) + m 2 y - x 2 2 f(y)\geq f(x)+\nabla f(x)^{T}(y-x)+\frac{m}{2}\|y-x\|_{2}^{2}
  79. t [ 0 , 1 ] t\in[0,1]
  80. f ( t x + ( 1 - t ) y ) t f ( x ) + ( 1 - t ) f ( y ) - 1 2 m t ( 1 - t ) x - y 2 2 f(tx+(1-t)y)\leq tf(x)+(1-t)f(y)-\frac{1}{2}mt(1-t)\|x-y\|_{2}^{2}
  81. f f
  82. f f
  83. 2 f ( x ) m I \nabla^{2}f(x)\succeq mI
  84. 2 f \nabla^{2}f
  85. \succeq
  86. 2 f ( x ) - m I \nabla^{2}f(x)-mI
  87. 2 f ( x ) \nabla^{2}f(x)
  88. 2 f ( x ) \nabla^{2}f(x)
  89. f ′′ ( x ) f^{\prime\prime}(x)
  90. f ′′ ( x ) m f^{\prime\prime}(x)\geq m
  91. f ′′ ( x ) 0 f^{\prime\prime}(x)\geq 0
  92. 2 f ( x ) \nabla^{2}f(x)
  93. f ( y ) = f ( x ) + f ( x ) T ( y - x ) + 1 2 ( y - x ) T 2 f ( z ) ( y - x ) f(y)=f(x)+\nabla f(x)^{T}(y-x)+\frac{1}{2}(y-x)^{T}\nabla^{2}f(z)(y-x)
  94. z { t x + ( 1 - t ) y : t [ 0 , 1 ] } z\in\{tx+(1-t)y:t\in[0,1]\}
  95. ( y - x ) T 2 f ( z ) ( y - x ) m ( y - x ) T ( y - x ) (y-x)^{T}\nabla^{2}f(z)(y-x)\geq m(y-x)^{T}(y-x)
  96. f f
  97. x f ( x ) - m 2 x 2 x\mapsto f(x)-\frac{m}{2}\|x\|^{2}
  98. f f
  99. f f
  100. f ′′ ( x ) 0 f^{\prime\prime}(x)\geq 0
  101. x x
  102. f f
  103. f ′′ ( x ) > 0 f^{\prime\prime}(x)>0
  104. x x
  105. f f
  106. f ′′ ( x ) m > 0 f^{\prime\prime}(x)\geq m>0
  107. x x
  108. f f
  109. ( x n ) (x_{n})
  110. f ′′ ( x n ) = 1 n f^{\prime\prime}(x_{n})=\frac{1}{n}
  111. f ′′ ( x n ) > 0 , f^{\prime\prime}(x_{n})>0,
  112. f ′′ ( x ) f^{\prime\prime}(x)
  113. f f
  114. X X
  115. f ′′ ( x ) > 0 f^{\prime\prime}(x)>0
  116. x X x\in X
  117. ϕ \phi
  118. f f
  119. t 0 , 11 t∈0,11
  120. f ( t x + ( 1 - t ) y ) t f ( x ) + ( 1 - t ) f ( y ) - t ( 1 - t ) ϕ ( x - y ) f(tx+(1-t)y)\leq tf(x)+(1-t)f(y)-t(1-t)\phi(\|x-y\|)
  121. ϕ \phi
  122. ϕ ( α ) = m 2 α 2 \phi(\alpha)=\frac{m}{2}\alpha^{2}
  123. f ( x ) = x 2 f(x)=x^{2}
  124. f ′′ ( x ) = 2 > 0 f^{\prime\prime}(x)=2>0
  125. f f
  126. f ( x ) = x 4 f(x)=x^{4}
  127. f ′′ ( x ) = 12 x 2 0 f^{\prime\prime}(x)=12x^{2}\geq 0
  128. f f
  129. f ( x ) = | x | f(x)=|x|
  130. f ( x ) = | x | p f(x)=|x|^{p}
  131. 1 p 1≤p
  132. f ( x ) = e x f(x)=e^{x}
  133. f ′′ ( x ) = e x > 0 f^{\prime\prime}(x)=e^{x}>0
  134. g ( x ) = e f ( x ) g(x)=e^{f(x)}
  135. f f
  136. f f
  137. - log det ( X ) -\log\,\text{det}(X)
  138. f f
  139. f ( a + b ) = f ( a ) + f ( b ) . f(a+b)=f(a)+f(b).
  140. f ( x ) = a T x + b f(x)=a^{T}x+b
  141. f ( x ) = x f(x)=\sqrt{x}
  142. h ( x ) = x 2 h(x)=x^{2}
  143. k ( x ) = - x k(x)=-x
  144. f ′′ ( x ) = 2 x 3 f^{\prime\prime}(x)=\frac{2}{x^{3}}
  145. f ( x ) f(x)
  146. ( 0 , + ) (0,+∞)
  147. ( , 0 ) (−∞,0)
  148. f ( x ) = x < s u p > 2 f(x)=x<sup>−2

Conway_chained_arrow_notation.html

  1. n + 1 n+1
  2. p p
  3. q q
  4. X X
  5. p p
  6. p p
  7. p q p\to q
  8. p q p^{q}
  9. 1 \to 1
  10. X 1 X\to 1
  11. X X
  12. X p ( q + 1 ) X\to p\to(q+1)
  13. X ( X ( ( X ( X ) q ) ) q ) q X\to(X\to(\cdots(X\to(X)\to q)\cdots)\to q)\to q
  14. X 1 ( q + 1 ) = X X\to 1\to(q+1)=X
  15. X ( p + 1 ) ( q + 1 ) = X ( X p ( q + 1 ) ) q X\to(p+1)\to(q+1)=X\to(X\to p\to(q+1))\to q
  16. p q r = p [ r + 2 ] q = p q = p r q . r arrows \begin{matrix}p\to q\to r=p[r+2]q=p&\underbrace{\uparrow\uparrow\uparrow\dots% \uparrow\uparrow\uparrow}&\!\!\!q=p\uparrow^{r}q.\\ &\!\!\!r\,\text{ arrows}\end{matrix}
  17. 2 3 2 = 2 3 = 2 2 2 = 16 2\rightarrow 3\rightarrow 2={}^{3}2=2^{2^{2}}=16
  18. 2 ( 3 2 ) = 2 ( 3 2 ) = 2 3 2 = 512 2\rightarrow\left(3\rightarrow 2\right)=2^{(3^{2})}=2^{3^{2}}=512
  19. ( 2 3 ) 2 = ( 2 3 ) 2 = 64 \left(2\rightarrow 3\right)\rightarrow 2=\left(2^{3}\right)^{2}=64
  20. 2 2 2 2^{2^{\dots^{2}}}
  21. a b 2 2 a\to b\to 2\to 2
  22. = a b 2 ( 1 + 1 ) =a\to b\to 2\to(1+1)
  23. = a b ( a b ) 1 =a\to b\to(a\to b)\to 1
  24. = a b a b =a\to b\to a^{b}
  25. = a [ a b + 2 ] b =a[a^{b}+2]b
  26. a b 3 2 a\to b\to 3\to 2
  27. = a b 3 ( 1 + 1 ) =a\to b\to 3\to(1+1)
  28. = a b ( a b ( a b ) 1 ) 1 =a\to b\to(a\to b\to(a\to b)\to 1)\to 1
  29. = a b ( a b a b ) =a\to b\to(a\to b\to a^{b})
  30. = a [ a b 2 2 + 2 ] b =a[a\to b\to 2\to 2+2]b
  31. a b 4 2 a\to b\to 4\to 2
  32. = a b ( a b ( a b a b ) ) =a\to b\to(a\to b\to(a\to b\to a^{b}))
  33. = a [ a b 3 2 + 2 ] b =a[a\to b\to 3\to 2+2]b
  34. f ( p ) = X p f(p)=X\to p
  35. X p 2 = f p ( 1 ) X\to p\to 2=f^{p}(1)
  36. X = a b X=a\to b
  37. f ( p ) = a [ p + 2 ] b f(p)=a[p+2]b
  38. a b p 2 = a [ a b ( p - 1 ) 2 + 2 ] b = f p ( 1 ) a\to b\to p\to 2=a[a\to b\to(p-1)\to 2+2]b=f^{p}(1)
  39. 10 6 3 2 = 10 [ 10 [ 1000002 ] 6 + 2 ] 6 10\to 6\to 3\to 2=10[10[1000002]6+2]6\!
  40. a b 2 3 a\to b\to 2\to 3
  41. = a b 2 ( 2 + 1 ) =a\to b\to 2\to(2+1)
  42. = a b ( a b ) 2 =a\to b\to(a\to b)\to 2
  43. = a b a b 2 =a\to b\to a^{b}\to 2
  44. = f a b ( 1 ) =f^{a^{b}}(1)
  45. g q ( p ) = X p q g_{q}(p)=X\to p\to q
  46. X p q + 1 = g q p ( 1 ) X\to p\to q+1=g_{q}^{p}(1)
  47. g q + 1 ( p ) = g q p ( 1 ) g_{q+1}(p)=g_{q}^{p}(1)
  48. g 2 ( p ) = a b p 2 = f p ( 1 ) g_{2}(p)=a\to b\to p\to 2=f^{p}(1)
  49. g 3 ( p ) = g 2 p ( 1 ) g_{3}(p)=g_{2}^{p}(1)
  50. a b 2 3 = g 3 ( 2 ) = g 2 2 ( 1 ) = g 2 ( g 2 ( 1 ) ) = f f ( 1 ) ( 1 ) = f a b ( 1 ) a\to b\to 2\to 3=g_{3}(2)=g_{2}^{2}(1)=g_{2}(g_{2}(1))=f^{f(1)}(1)=f^{a^{b}}(1)
  51. G G\!
  52. f ( n ) = 3 3 n f(n)=3\rightarrow 3\rightarrow n\!
  53. G = f 64 ( 4 ) G=f^{64}(4)\,
  54. 3 3 64 2 < G < 3 3 65 2 3\rightarrow 3\rightarrow 64\rightarrow 2<G<3\rightarrow 3\rightarrow 65% \rightarrow 2\,
  55. f 64 ( 1 ) f^{64}(1)\,
  56. = 3 3 ( 3 3 ( ( 3 3 ( 3 3 1 ) ) ) ) =3\rightarrow 3\rightarrow(3\rightarrow 3\rightarrow(\cdots(3\rightarrow 3% \rightarrow(3\rightarrow 3\rightarrow 1))\cdots))\,
  57. 3 3 3\rightarrow 3
  58. = 3 3 ( 3 3 ( ( 3 3 ( 3 3 ) 1 ) ) 1 ) 1 =3\rightarrow 3\rightarrow(3\rightarrow 3\rightarrow(\cdots(3\rightarrow 3% \rightarrow(3\rightarrow 3)\rightarrow 1)\cdots)\rightarrow 1)\rightarrow 1\,
  59. = 3 3 64 2 ; =3\rightarrow 3\rightarrow 64\rightarrow 2;\,
  60. f 64 ( 4 ) = G ; f^{64}(4)=G;\,
  61. = 3 3 ( 3 3 ( ( 3 3 ( 3 3 4 ) ) ) ) =3\rightarrow 3\rightarrow(3\rightarrow 3\rightarrow(\cdots(3\rightarrow 3% \rightarrow(3\rightarrow 3\rightarrow 4))\cdots))\,
  62. 3 3 3\rightarrow 3
  63. f 64 ( 27 ) f^{64}(27)\,
  64. = 3 3 ( 3 3 ( ( 3 3 ( 3 3 27 ) ) ) ) =3\rightarrow 3\rightarrow(3\rightarrow 3\rightarrow(\cdots(3\rightarrow 3% \rightarrow(3\rightarrow 3\rightarrow 27))\cdots))\,
  65. 3 3 3\rightarrow 3
  66. = 3 3 ( 3 3 ( ( 3 3 ( 3 3 ( 3 3 ) ) ) ) ) =3\rightarrow 3\rightarrow(3\rightarrow 3\rightarrow(\cdots(3\rightarrow 3% \rightarrow(3\rightarrow 3\rightarrow(3\rightarrow 3)))\cdots))\,
  67. 3 3 3\rightarrow 3
  68. = 3 3 65 2 =3\rightarrow 3\rightarrow 65\rightarrow 2\,
  69. f 64 ( 1 ) < f 64 ( 4 ) < f 64 ( 27 ) f^{64}(1)<f^{64}(4)<f^{64}(27)\,
  70. 3 3 3 3 = 3 3 ( 3 3 27 2 ) 2 = f 3 3 27 2 ( 1 ) = f f 27 ( 1 ) ( 1 ) 3\rightarrow 3\rightarrow 3\rightarrow 3~{}~{}=~{}~{}3\rightarrow 3\rightarrow% (3\rightarrow 3\rightarrow 27\rightarrow 2)\rightarrow 2\,~{}~{}=~{}~{}f^{3% \rightarrow 3\rightarrow 27\rightarrow 2}(1)~{}~{}=~{}~{}f^{f^{27}(1)}(1)
  71. 3 3 27 2 3\rightarrow 3\rightarrow 27\rightarrow 2

Cooperative_binding.html

  1. Y ¯ \bar{Y}
  2. Y ¯ = [ bound sites ] [ bound sites ] + [ unbound sites ] = [ bound sites ] [ total sites ] \bar{Y}=\frac{[\,\text{bound sites}]}{[\,\text{bound sites}]+[\,\text{unbound % sites}]}=\frac{[\,\text{bound sites}]}{[\,\text{total sites}]}
  3. Y ¯ = 0 \bar{Y}=0
  4. Y ¯ = 1 \bar{Y}=1
  5. Y ¯ \bar{Y}
  6. Y ¯ = K [ X ] n 1 + K [ X ] n = [ X ] n K * + [ X ] n = [ X ] n K d n + [ X ] n \bar{Y}=\frac{K\cdot{}[X]^{n}}{1+K\cdot{}[X]^{n}}=\frac{[X]^{n}}{K^{*}+[X]^{n}% }=\frac{[X]^{n}}{K_{d}^{n}+[X]^{n}}
  7. n n
  8. [ X ] [X]
  9. K K
  10. K * K^{*}
  11. EC 50 \mathrm{EC}_{50}
  12. K d K_{d}
  13. n < 1 n<1
  14. n > 1 n>1
  15. n n
  16. log Y ¯ 1 - Y ¯ = n log [ X ] - n log K d \log\frac{\bar{Y}}{1-\bar{Y}}=n\cdot{}\log[X]-n\cdot{}\log K_{d}
  17. log Y ¯ 1 - Y ¯ \log\frac{\bar{Y}}{1-\bar{Y}}
  18. log [ X ] \log[X]
  19. n H n_{H}
  20. log ( K d ) \log(K_{d})
  21. K i K_{i}
  22. Y ¯ = 1 4 K I [ X ] + 2 K I I [ X ] 2 + 3 K I I I [ X ] 3 + 4 K I V [ X ] 4 1 + K I [ X ] + K I I [ X ] 2 + K I I I [ X ] 3 + K I V [ X ] 4 \bar{Y}=\frac{1}{4}\cdot{}\frac{K_{I}[X]+2K_{II}[X]^{2}+3K_{III}[X]^{3}+4K_{IV% }[X]^{4}}{1+K_{I}[X]+K_{II}[X]^{2}+K_{III}[X]^{3}+K_{IV}[X]^{4}}
  23. Y ¯ = 1 n K I [ X ] + 2 K I I [ X ] 2 + + n K n [ X ] n 1 + K I [ X ] + K I I [ X ] 2 + + K n [ X ] n \bar{Y}=\frac{1}{n}\frac{K_{I}[X]+2K_{II}[X]^{2}+\ldots+nK_{n}[X]^{n}}{1+K_{I}% [X]+K_{II}[X]^{2}+\ldots+K_{n}[X]^{n}}
  24. K i K_{i}
  25. K 1 K_{1}
  26. K 2 K_{2}
  27. Y ¯ \bar{Y}
  28. Y ¯ = K 1 [ X ] + 2 K 1 K 2 [ X ] 2 + + n ( K 1 K 2 K n ) [ X ] n 1 + K 1 [ X ] + K 1 K 2 [ X ] 2 + + ( K 1 K 2 K n ) [ X ] n \bar{Y}=\frac{K_{1}[X]+2K_{1}K_{2}[X]^{2}+\ldots+n\left(K_{1}K_{2}\ldots K_{n}% \right)[X]^{n}}{1+K_{1}[X]+K_{1}K_{2}[X]^{2}+\ldots+\left(K_{1}K_{2}\ldots K_{% n}\right)[X]^{n}}
  29. K 1 K_{1}
  30. K 2 K_{2}
  31. K K
  32. K 1 = n K , K 2 = n - 1 2 K , K n = 1 n K K_{1}=nK,K_{2}=\frac{n-1}{2}K,\ldots K_{n}=\frac{1}{n}K
  33. K i = n - i + 1 i K K_{i}=\frac{n-i+1}{i}K
  34. K i K_{i}
  35. i > 1 i>1
  36. K K
  37. α \alpha
  38. Y ¯ = K [ X ] + 3 α K 2 [ X ] 2 + 3 α K 3 3 [ X ] 3 + α K 4 6 [ X ] 4 1 + 4 K [ X ] + 6 α K 2 [ X ] 2 + 4 α K 3 3 [ X ] 3 + α K 4 6 [ X ] 4 \bar{Y}=\frac{K[X]+3\alpha{}K^{2}[X]^{2}+3\alpha{}^{3}K^{3}[X]^{3}+\alpha{}^{6% }K^{4}[X]^{4}}{1+4K[X]+6\alpha{}K^{2}[X]^{2}+4\alpha{}^{3}K^{3}[X]^{3}+\alpha{% }^{6}K^{4}[X]^{4}}
  39. Y ¯ = K A B 3 ( K X K t [ X ] ) + 3 K A B 4 K B B ( K X K t [ X ] ) 2 + 3 K A B 3 K B B 3 ( K X K t [ X ] ) 3 + K B B 6 ( K X K t [ X ] ) 4 1 + 4 K A B 3 ( K X K t [ X ] ) + 6 K A B 4 K B B ( K X K t [ X ] ) 2 + 4 K A B 3 K B B 3 ( K X K t [ X ] ) 3 + K B B 6 ( K X K t [ X ] ) 4 \bar{Y}=\frac{K_{AB}^{3}(K_{X}K_{t}[X])+3K_{AB}^{4}K_{BB}(K_{X}K_{t}[X])^{2}+3% K_{AB}^{3}K_{BB}^{3}(K_{X}K_{t}[X])^{3}+K_{BB}^{6}(K_{X}K_{t}[X])^{4}}{1+4K_{% AB}^{3}(K_{X}K_{t}[X])+6K_{AB}^{4}K_{BB}(K_{X}K_{t}[X])^{2}+4K_{AB}^{3}K_{BB}^% {3}(K_{X}K_{t}[X])^{3}+K_{BB}^{6}(K_{X}K_{t}[X])^{4}}
  40. K X K_{X}
  41. K t K_{t}
  42. K A B K_{AB}
  43. K B B K_{BB}
  44. N s N_{s}
  45. Y ¯ \bar{Y}
  46. L = [ T 0 ] [ R 0 ] L=\frac{\left[T_{0}\right]}{\left[R_{0}\right]}
  47. c = K d R K d T c=\frac{K_{d}^{R}}{K_{d}^{T}}
  48. c = 1 c=1
  49. α = [ X ] K d R \alpha=\frac{[X]}{K_{d}^{R}}
  50. Y ¯ = α ( 1 + α ) n - 1 + L c α ( 1 + c α ) n - 1 ( 1 + α ) n + L ( 1 + c α ) n \bar{Y}=\frac{\alpha(1+\alpha)^{n-1}+Lc\alpha(1+c\alpha)^{n-1}}{(1+\alpha)^{n}% +L(1+c\alpha)^{n}}
  51. R ¯ \bar{R}
  52. R R
  53. R ¯ \bar{R}
  54. R ¯ \bar{R}
  55. R ¯ = ( 1 + α ) n ( 1 + α ) n + L ( 1 + c α ) n \bar{R}=\frac{(1+\alpha)^{n}}{(1+\alpha)^{n}+L(1+c\alpha)^{n}}
  56. Y ¯ \bar{Y}
  57. R ¯ \bar{R}

Coproduct.html

  1. X = j J X j X=\coprod_{j\in J}X_{j}
  2. X = j J X j . X=\bigoplus_{j\in J}X_{j}.
  3. f = j J f j : j J X j Y f=\coprod_{j\in J}f_{j}:\coprod_{j\in J}X_{j}\to Y
  4. Hom C ( j J X j , Y ) j J Hom C ( X j , Y ) \operatorname{Hom}_{C}\left(\coprod_{j\in J}X_{j},Y\right)\cong\prod_{j\in J}% \operatorname{Hom}_{C}(X_{j},Y)
  5. ( f j ) j J j J Hom ( X j , Y ) (f_{j})_{j\in J}\in\prod_{j\in J}\operatorname{Hom}(X_{j},Y)
  6. j J f j Hom ( j J X j , Y ) . \coprod_{j\in J}f_{j}\in\operatorname{Hom}\left(\coprod_{j\in J}X_{j},Y\right).
  7. ( f i j ) j J . (f\circ i_{j})_{j\in J}.
  8. X ( Y Z ) ( X Y ) Z X Y Z X\oplus(Y\oplus Z)\cong(X\oplus Y)\oplus Z\cong X\oplus Y\oplus Z
  9. X 0 0 X X X\oplus 0\cong 0\oplus X\cong X
  10. X Y Y X . X\oplus Y\cong Y\oplus X.

Correlation_and_dependence.html

  1. ρ X , Y = corr ( X , Y ) = cov ( X , Y ) σ X σ Y = E [ ( X - μ X ) ( Y - μ Y ) ] σ X σ Y , \rho_{X,Y}=\mathrm{corr}(X,Y)={\mathrm{cov}(X,Y)\over\sigma_{X}\sigma_{Y}}={E[% (X-\mu_{X})(Y-\mu_{Y})]\over\sigma_{X}\sigma_{Y}},
  2. r x y = i = 1 n ( x i - x ¯ ) ( y i - y ¯ ) ( n - 1 ) s x s y = i = 1 n ( x i - x ¯ ) ( y i - y ¯ ) i = 1 n ( x i - x ¯ ) 2 i = 1 n ( y i - y ¯ ) 2 , r_{xy}=\frac{\sum\limits_{i=1}^{n}(x_{i}-\bar{x})(y_{i}-\bar{y})}{(n-1)s_{x}s_% {y}}=\frac{\sum\limits_{i=1}^{n}(x_{i}-\bar{x})(y_{i}-\bar{y})}{\sqrt{\sum% \limits_{i=1}^{n}(x_{i}-\bar{x})^{2}\sum\limits_{i=1}^{n}(y_{i}-\bar{y})^{2}}},
  3. r x y = x i y i - n x ¯ y ¯ ( n - 1 ) s x s y = n x i y i - x i y i n x i 2 - ( x i ) 2 n y i 2 - ( y i ) 2 . r_{xy}=\frac{\sum x_{i}y_{i}-n\bar{x}\bar{y}}{(n-1)s_{x}s_{y}}=\frac{n\sum x_{% i}y_{i}-\sum x_{i}\sum y_{i}}{\sqrt{n\sum x_{i}^{2}-(\sum x_{i})^{2}}~{}\sqrt{% n\sum y_{i}^{2}-(\sum y_{i})^{2}}}.
  4. E ( Y X ) = E ( Y ) + r σ y X - E ( X ) σ x , E(Y\mid X)=E(Y)+r\sigma_{y}\frac{X-E(X)}{\sigma_{x}},

Correlation_function.html

  1. C ( s , t ) = corr ( X ( s ) , X ( t ) ) , C(s,t)=\operatorname{corr}(X(s),X(t)),
  2. corr \operatorname{corr}
  3. C i j ( s , t ) = corr ( X i ( s ) , X j ( t ) ) C_{ij}(s,t)=\operatorname{corr}(X_{i}(s),X_{j}(t))
  4. C i 1 i 2 i n ( s 1 , s 2 , , s n ) = X i 1 ( s 1 ) X i 2 ( s 2 ) X i n ( s n ) . C_{i_{1}i_{2}\cdots i_{n}}(s_{1},s_{2},\cdots,s_{n})=\langle X_{i_{1}}(s_{1})X% _{i_{2}}(s_{2})\cdots X_{i_{n}}(s_{n})\rangle.
  5. i j i_{j}

Correlation_function_(astronomy).html

  1. ξ ( r ) \xi(r)
  2. P ( k ) P(k)
  3. ξ ( r ) = 1 2 π 2 d k k 2 P ( k ) sin ( k r ) k r \xi(r)=\frac{1}{2\pi^{2}}\int dk\,k^{2}P(k)\,\frac{\sin(kr)}{kr}

Corrosion.html

  1. N R i = M i S f i t NR_{i}=\frac{M_{i}}{Sf_{i}t}
  2. N R x i = 2 ρ D i π t NRx_{i}=2\rho\sqrt{\frac{D_{i}}{\pi t}}
  3. N R h = ρ r h NRh=\rho r_{h}

Coset.html

  1. { x V : x = a + n , n W } \{x\in V\colon x=a+n,n\in W\}
  2. G / H G/H
  3. { g H : g G } \{gH:g\in G\}
  4. H \ G H\backslash G
  5. { H g : g G } \{Hg:g\in G\}
  6. K \ G / H K\backslash G/H
  7. { K g H : g G } \{KgH:g\in G\}

Cotangent_bundle.html

  1. \mathcal{I}
  2. / 2 \mathcal{I}/\mathcal{I}^{2}
  3. Γ T * M = Δ * ( / 2 ) . \Gamma T^{*}M=\Delta^{*}(\mathcal{I}/\mathcal{I}^{2}).
  4. ϕ : M N \phi\colon M\to N
  5. ϕ * T * N \phi^{*}T^{*}N
  6. ϕ * ( T * N ) T * M \phi^{*}(T^{*}N)\to T^{*}M
  7. θ ( x , p ) = 𝔦 = 1 n p i d x i . \theta_{(x,p)}=\sum_{{\mathfrak{i}}=1}^{n}p_{i}dx^{i}.
  8. θ ( x , ω ) = π * ω . \theta_{(x,\omega)}=\pi^{*}\omega.
  9. n × n \mathbb{R}^{n}\times\mathbb{R}^{n}
  10. y i d x i y_{i}dx_{i}
  11. d y i and d x i dy_{i}{\and}dx_{i}
  12. M M
  13. T * M \!\,T^{*}\!M

Counting_measure.html

  1. Σ \Sigma
  2. X X
  3. μ \mu
  4. ( X , Σ ) (X,\Sigma)
  5. Σ [ 0 , + ] \Sigma\rightarrow[0,+\infty]
  6. μ ( A ) = { | A | if A is finite + if A is infinite \mu(A)=\begin{cases}|A|&\,\text{if }A\,\text{ is finite}\\ +\infty&\,\text{if }A\,\text{ is infinite}\end{cases}
  7. A Σ A\in\Sigma
  8. | A | |A|
  9. A A
  10. ( X , Σ ) (X,\Sigma)
  11. X X
  12. f : X [ 0 , ) f\colon X\to[0,\infty)
  13. μ \mu
  14. ( X , Σ ) (X,\Sigma)
  15. μ ( A X ) := a A f ( a ) , \mu(A\subseteq X):=\sum_{a\in A}f(a),
  16. y Y y := sup F Y , | F | < { y F y } . \sum_{y\in Y\subseteq\mathbb{R}}y:=\sup_{F\subseteq Y,|F|<\infty}\left\{\sum_{% y\in F}y\right\}.

Covariance.html

  1. σ ( X , Y ) = E [ ( X - E [ X ] ) ( Y - E [ Y ] ) ] , \sigma(X,Y)=\operatorname{E}{\big[(X-\operatorname{E}[X])(Y-\operatorname{E}[Y% ])\big]},
  2. σ ( X , Y ) \displaystyle\sigma(X,Y)
  3. E [ X Y ] E [ X ] E [ Y ] \operatorname{E}[XY]\approx\operatorname{E}[X]\operatorname{E}[Y]
  4. 𝐗 m \mathbf{X}\in\mathbb{R}^{m}
  5. 𝐘 n \mathbf{Y}\in\mathbb{R}^{n}
  6. σ ( 𝐗 , 𝐘 ) \displaystyle\sigma(\mathbf{X},\mathbf{Y})
  7. 𝐗 = [ X 1 X 2 X m ] T \mathbf{X}=\begin{bmatrix}X_{1}&X_{2}&\dots&X_{m}\end{bmatrix}^{\mathrm{T}}
  8. Σ ( 𝐗 ) = σ ( 𝐗 , 𝐗 ) . \Sigma(\mathbf{X})=\sigma(\mathbf{X},\mathbf{X}).
  9. cov ( X , Y ) = 1 n 2 i = 1 n j = 1 n 1 2 ( x i - x j ) ( y i - y j ) T = 1 n 2 i j > i ( x i - x j ) ( y i - y j ) T . \operatorname{cov}(X,Y)=\frac{1}{n^{2}}\sum_{i=1}^{n}\sum_{j=1}^{n}\frac{1}{2}% (x_{i}-x_{j})\cdot(y_{i}-y_{j})^{\mathrm{T}}=\frac{1}{n^{2}}\sum_{i}\sum_{j>i}% (x_{i}-x_{j})\cdot(y_{i}-y_{j})^{\mathrm{T}}.
  10. σ ( X , X ) = σ 2 ( X ) . \sigma(X,X)=\sigma^{2}(X).
  11. σ ( X , a ) \displaystyle\sigma(X,a)
  12. σ 2 ( i = 1 n a i X i ) = i = 1 n a i 2 σ 2 ( X i ) + 2 i , j : i < j a i a j σ ( X i , X j ) = i , j a i a j σ ( X i , X j ) \sigma^{2}\left(\sum_{i=1}^{n}a_{i}X_{i}\right)=\sum_{i=1}^{n}a_{i}^{2}\sigma^% {2}(X_{i})+2\sum_{i,j\,:\,i<j}a_{i}a_{j}\sigma(X_{i},X_{j})=\sum_{i,j}{a_{i}a_% {j}\sigma(X_{i},X_{j})}
  13. 𝐗 \mathbf{X}
  14. Σ ( 𝐗 ) \Sigma(\mathbf{X})
  15. A A
  16. 𝐗 \mathbf{X}
  17. A 𝐗 A\mathbf{X}
  18. Σ ( A 𝐗 ) = A Σ ( 𝐗 ) A T \Sigma(A\mathbf{X})=A\,\Sigma(\mathbf{X})\,A^{\mathrm{T}}
  19. E [ X Y ] = E [ X ] E [ Y ] . \operatorname{E}\left[XY\right]=\operatorname{E}[X]\cdot\operatorname{E}[Y].
  20. σ ( X , Y ) \displaystyle\sigma(X,Y)
  21. | σ ( X , Y ) | σ 2 ( X ) σ 2 ( Y ) |\sigma(X,Y)|\leq\sqrt{\sigma^{2}(X)\sigma^{2}(Y)}
  22. Z = X - σ ( X , Y ) σ 2 ( Y ) Y . Z=X-\frac{\sigma(X,Y)}{\sigma^{2}(Y)}Y.
  23. 0 σ 2 ( Z ) \displaystyle 0\leq\sigma^{2}(Z)
  24. q ¯ ¯ = [ [ q j k ] ] \textstyle\overline{\overline{q}}=\left[[q_{jk}]\right]
  25. q j k = 1 N - 1 i = 1 N ( X i j - X ¯ j ) ( X i k - X ¯ k ) q_{jk}=\frac{1}{N-1}\sum_{i=1}^{N}\left(X_{ij}-\bar{X}_{j}\right)\left(X_{ik}-% \bar{X}_{k}\right)
  26. j j
  27. k k
  28. 𝐗 \textstyle\mathbf{X}
  29. N - 1 \textstyle N-1
  30. N \textstyle N
  31. E ( X ) E(X)
  32. 𝐗 ¯ \mathbf{\bar{X}}
  33. E ( X ) E(X)
  34. q j k = 1 N i = 1 N ( X i j - E ( X j ) ) ( X i k - E ( X k ) ) q_{jk}=\frac{1}{N}\sum_{i=1}^{N}\left(X_{ij}-E(X_{j})\right)\left(X_{ik}-E(X_{% k})\right)

Covariance_and_contravariance_of_vectors.html

  1. 𝐯 = v i 𝐞 i . \mathbf{v}=v^{i}\mathbf{e}_{i}.
  2. 𝐯 = v i 𝐞 i . \mathbf{v}=v_{i}\mathbf{e}^{i}.
  3. ( v 1 , v 2 , v 3 ) . (v_{1},v_{2},v_{3}).\,
  4. f = f x 1 x ^ 1 + f x 2 x ^ 2 + f x 3 x ^ 3 \nabla f=\frac{\partial f}{\partial x^{1}}\widehat{x}^{1}+\frac{\partial f}{% \partial x^{2}}\widehat{x}^{2}+\frac{\partial f}{\partial x^{3}}\widehat{x}^{3}
  5. 𝐟 𝐟 = ( i a 1 i X i , , i a n i X i ) = 𝐟 A \mathbf{f}\mapsto\mathbf{f}^{\prime}=\left(\sum_{i}a^{i}_{1}X_{i},\dots,\sum_{% i}a^{i}_{n}X_{i}\right)=\mathbf{f}A
  6. a j i a^{i}_{j}
  7. Y j = i a j i X i . Y_{j}=\sum_{i}a^{i}_{j}X_{i}.
  8. v = i v i [ 𝐟 ] X i , v=\sum_{i}v^{i}[\mathbf{f}]X_{i},
  9. 𝐯 [ 𝐟 ] = [ v 1 [ 𝐟 ] v 2 [ 𝐟 ] v n [ 𝐟 ] ] \mathbf{v}[\mathbf{f}]=\begin{bmatrix}v^{1}[\mathbf{f}]\\ v^{2}[\mathbf{f}]\\ \vdots\\ v^{n}[\mathbf{f}]\end{bmatrix}
  10. v = 𝐟 𝐯 [ 𝐟 ] . v=\mathbf{f}\,\mathbf{v}[\mathbf{f}].
  11. v = 𝐟 𝐯 [ 𝐟 ] . v=\mathbf{f^{\prime}}\,\mathbf{v}[\mathbf{f^{\prime}}].
  12. 𝐟 𝐯 [ 𝐟 ] = v = 𝐟 𝐯 [ 𝐟 ] . \mathbf{f}\,\mathbf{v}[\mathbf{f}]=v=\mathbf{f^{\prime}}\,\mathbf{v}[\mathbf{f% ^{\prime}}].
  13. 𝐟 𝐯 [ 𝐟 ] = 𝐟 A 𝐯 [ 𝐟 A ] , \mathbf{f}\,\mathbf{v}[\mathbf{f}]=\mathbf{f}A\,\mathbf{v}[\mathbf{f}A],
  14. 𝐯 [ 𝐟 A ] = A - 1 𝐯 [ 𝐟 ] . \mathbf{v}[\mathbf{f}A]=A^{-1}\mathbf{v}[\mathbf{f}].
  15. v i [ 𝐟 A ] = j a ~ j i v j [ 𝐟 ] v^{i}[\mathbf{f}A]=\sum_{j}\tilde{a}^{i}_{j}v^{j}[\mathbf{f}]
  16. a ~ j i \tilde{a}^{i}_{j}
  17. 𝐟 𝐟 \mathbf{f}\longrightarrow\mathbf{f^{\prime}}
  18. v [ 𝐟 ] v [ 𝐟 ] v[\mathbf{f}]\longleftarrow v[\mathbf{f^{\prime}}]
  19. α ( X i ) = α i [ 𝐟 ] , i = 1 , 2 , , n . \alpha(X_{i})=\alpha_{i}[\mathbf{f}],\quad i=1,2,\dots,n.
  20. α i [ 𝐟 A ] = α ( Y i ) = α ( j a i j X j ) = j a i j α ( X j ) = j a i j α j [ 𝐟 ] . \begin{array}[]{rcl}\alpha_{i}[\mathbf{f}A]&=&\alpha(Y_{i})\\ &=&\alpha\left(\sum_{j}a^{j}_{i}X_{j}\right)\\ &=&\sum_{j}a^{j}_{i}\alpha(X_{j})\\ &=&\sum_{j}a^{j}_{i}\alpha_{j}[\mathbf{f}]\end{array}.
  21. α [ 𝐟 ] = [ α 1 [ 𝐟 ] , α 2 [ 𝐟 ] , , α n [ 𝐟 ] ] \mathbf{\alpha}[\mathbf{f}]=\begin{bmatrix}\alpha_{1}[\mathbf{f}],\alpha_{2}[% \mathbf{f}],\dots,\alpha_{n}[\mathbf{f}]\end{bmatrix}
  22. α [ 𝐟 A ] = α [ 𝐟 ] A . \alpha[\mathbf{f}A]=\alpha[\mathbf{f}]A.
  23. 𝐟 𝐟 \mathbf{f}\longrightarrow\mathbf{f^{\prime}}
  24. α [ 𝐟 ] α [ 𝐟 ] \alpha[\mathbf{f}]\longrightarrow\alpha[\mathbf{f^{\prime}}]
  25. α T [ 𝐟 A ] = A T α T [ 𝐟 ] . \alpha^{\mathrm{T}}[\mathbf{f}A]=A^{\mathrm{T}}\alpha^{\mathrm{T}}[\mathbf{f}].
  26. x i [ 𝐟 ] ( v ) = v i [ 𝐟 ] . x^{i}[\mathbf{f}](v)=v^{i}[\mathbf{f}].
  27. x i [ 𝐟 A ] = k = 1 n a ~ k i x k [ 𝐟 ] . x^{i}[\mathbf{f}A]=\sum_{k=1}^{n}\tilde{a}^{i}_{k}x^{k}[\mathbf{f}].
  28. X 1 = x 1 , , X n = x n . X_{1}=\frac{\partial}{\partial x^{1}},\dots,X_{n}=\frac{\partial}{\partial x^{% n}}.
  29. Y 1 = y 1 , , Y n = y n , Y_{1}=\frac{\partial}{\partial y^{1}},\dots,Y_{n}=\frac{\partial}{\partial y^{% n}},
  30. 𝐟 = 𝐟 J - 1 , J = ( y i x j ) i , j = 1 n . \mathbf{f}^{\prime}=\mathbf{f}J^{-1},\quad J=\left(\frac{\partial y^{i}}{% \partial x^{j}}\right)_{i,j=1}^{n}.
  31. y i = j = 1 n x j y i x j . \frac{\partial}{\partial y^{i}}=\sum_{j=1}^{n}\frac{\partial x^{j}}{\partial y% ^{i}}\frac{\partial}{\partial x^{j}}.
  32. / x i \partial/\partial x^{i}
  33. v = i = 1 n v i [ 𝐟 ] X i = 𝐟 𝐯 [ 𝐟 ] . v=\sum_{i=1}^{n}v^{i}[\mathbf{f}]X_{i}=\mathbf{f}\ \mathbf{v}[\mathbf{f}].
  34. 𝐯 [ 𝐟 ] = 𝐯 [ 𝐟 J - 1 ] = J 𝐯 [ 𝐟 ] . \mathbf{v}[\mathbf{f}^{\prime}]=\mathbf{v}[\mathbf{f}J^{-1}]=J\,\mathbf{v}[% \mathbf{f}].
  35. v i [ 𝐟 ] = j = 1 n y i x j v j [ 𝐟 ] . v^{i}[\mathbf{f}^{\prime}]=\sum_{j=1}^{n}\frac{\partial y^{i}}{\partial x^{j}}% v^{j}[\mathbf{f}].
  36. α ( w ) = g ( v , w ) \alpha(w)=g(v,w)
  37. Y i ( X j ) = δ j i , Y^{i}(X_{j})=\delta^{i}_{j},
  38. v \displaystyle v
  39. 𝐯 [ 𝐟 A ] = A - 1 𝐯 [ 𝐟 ] , 𝐯 [ 𝐟 A ] = A T 𝐯 [ 𝐟 ] . \mathbf{v}[\mathbf{f}A]=A^{-1}\mathbf{v}[\mathbf{f}],\quad\mathbf{v}^{\sharp}[% \mathbf{f}A]=A^{T}\mathbf{v}^{\sharp}[\mathbf{f}].
  40. 𝐞 1 , 𝐞 2 \mathbf{e}_{1},\mathbf{e}_{2}
  41. 𝐞 1 , 𝐞 2 \mathbf{e}^{1},\mathbf{e}^{2}
  42. 𝐞 1 𝐞 1 = 1 , 𝐞 1 𝐞 2 = 0 𝐞 2 𝐞 1 = 0 , 𝐞 2 𝐞 2 = 1. \begin{aligned}\displaystyle\mathbf{e}^{1}\cdot\mathbf{e}_{1}=1,&\displaystyle% \quad\mathbf{e}^{1}\cdot\mathbf{e}_{2}=0\\ \displaystyle\mathbf{e}^{2}\cdot\mathbf{e}_{1}=0,&\displaystyle\quad\mathbf{e}% ^{2}\cdot\mathbf{e}_{2}=1.\end{aligned}
  43. 𝐞 1 = 1 2 𝐞 1 - 1 2 𝐞 2 \mathbf{e}^{1}=\frac{1}{2}\mathbf{e}_{1}-\frac{1}{\sqrt{2}}\mathbf{e}_{2}
  44. 𝐞 2 = - 1 2 𝐞 1 + 2 𝐞 2 . \mathbf{e}^{2}=-\frac{1}{\sqrt{2}}\mathbf{e}_{1}+2\mathbf{e}_{2}.
  45. R = [ 1 / 2 - 1 / 2 - 1 / 2 2 ] , R=\begin{bmatrix}1/2&-1/\sqrt{2}\\ -1/\sqrt{2}&2\end{bmatrix},
  46. [ 𝐞 1 𝐞 2 ] = [ 𝐞 1 𝐞 2 ] [ 1 / 2 - 1 / 2 - 1 / 2 2 ] . [\mathbf{e}^{1}\ \mathbf{e}^{2}]=[\mathbf{e}_{1}\ \mathbf{e}_{2}]\begin{% bmatrix}1/2&-1/\sqrt{2}\\ -1/\sqrt{2}&2\end{bmatrix}.
  47. v = 3 2 𝐞 1 + 2 𝐞 2 v=\frac{3}{2}\mathbf{e}_{1}+2\mathbf{e}_{2}
  48. v 1 = 3 2 , v 2 = 2. v^{1}=\frac{3}{2},\quad v^{2}=2.
  49. v = v 1 𝐞 1 + v 2 𝐞 2 = v 1 𝐞 1 + v 2 𝐞 2 v=v_{1}\mathbf{e}^{1}+v_{2}\mathbf{e}^{2}=v^{1}\mathbf{e}_{1}+v^{2}\mathbf{e}_% {2}
  50. [ v 1 v 2 ] = R - 1 [ v 1 v 2 ] = [ 4 2 2 1 ] [ v 1 v 2 ] = [ 6 + 2 2 2 + 3 / 2 ] . \begin{aligned}\displaystyle\begin{bmatrix}v_{1}\\ v_{2}\end{bmatrix}&\displaystyle=R^{-1}\begin{bmatrix}v^{1}\\ v^{2}\end{bmatrix}\\ &\displaystyle=\begin{bmatrix}4&\sqrt{2}\\ \sqrt{2}&1\end{bmatrix}\begin{bmatrix}v^{1}\\ v^{2}\end{bmatrix}=\begin{bmatrix}6+2\sqrt{2}\\ 2+3/\sqrt{2}\end{bmatrix}\end{aligned}.
  51. 𝐞 1 = 𝐞 2 × 𝐞 3 𝐞 1 ( 𝐞 2 × 𝐞 3 ) ; 𝐞 2 = 𝐞 3 × 𝐞 1 𝐞 2 ( 𝐞 3 × 𝐞 1 ) ; 𝐞 3 = 𝐞 1 × 𝐞 2 𝐞 3 ( 𝐞 1 × 𝐞 2 ) . \mathbf{e}^{1}=\frac{\mathbf{e}_{2}\times\mathbf{e}_{3}}{\mathbf{e}_{1}\cdot(% \mathbf{e}_{2}\times\mathbf{e}_{3})};\qquad\mathbf{e}^{2}=\frac{\mathbf{e}_{3}% \times\mathbf{e}_{1}}{\mathbf{e}_{2}\cdot(\mathbf{e}_{3}\times\mathbf{e}_{1})}% ;\qquad\mathbf{e}^{3}=\frac{\mathbf{e}_{1}\times\mathbf{e}_{2}}{\mathbf{e}_{3}% \cdot(\mathbf{e}_{1}\times\mathbf{e}_{2})}.
  52. 𝐞 i 𝐞 j = δ j i , \mathbf{e}^{i}\cdot\mathbf{e}_{j}=\delta^{i}_{j},
  53. q 1 = 𝐯 𝐞 1 ; q 2 = 𝐯 𝐞 2 ; q 3 = 𝐯 𝐞 3 . q^{1}=\mathbf{v}\cdot\mathbf{e}^{1};\qquad q^{2}=\mathbf{v}\cdot\mathbf{e}^{2}% ;\qquad q^{3}=\mathbf{v}\cdot\mathbf{e}^{3}.\,
  54. q 1 = 𝐯 𝐞 1 ; q 2 = 𝐯 𝐞 2 ; q 3 = 𝐯 𝐞 3 . q_{1}=\mathbf{v}\cdot\mathbf{e}_{1};\qquad q_{2}=\mathbf{v}\cdot\mathbf{e}_{2}% ;\qquad q_{3}=\mathbf{v}\cdot\mathbf{e}_{3}.\,
  55. 𝐯 = q i 𝐞 i = q 1 𝐞 1 + q 2 𝐞 2 + q 3 𝐞 3 \mathbf{v}=q_{i}\mathbf{e}^{i}=q_{1}\mathbf{e}^{1}+q_{2}\mathbf{e}^{2}+q_{3}% \mathbf{e}^{3}\,
  56. 𝐯 = q i 𝐞 i = q 1 𝐞 1 + q 2 𝐞 2 + q 3 𝐞 3 . \mathbf{v}=q^{i}\mathbf{e}_{i}=q^{1}\mathbf{e}_{1}+q^{2}\mathbf{e}_{2}+q^{3}% \mathbf{e}_{3}.\,
  57. 𝐯 = ( 𝐯 𝐞 i ) 𝐞 i = ( 𝐯 𝐞 i ) 𝐞 i \mathbf{v}=(\mathbf{v}\cdot\mathbf{e}_{i})\mathbf{e}^{i}=(\mathbf{v}\cdot% \mathbf{e}^{i})\mathbf{e}_{i}\,
  58. q i = 𝐯 𝐞 i = ( q j 𝐞 j ) 𝐞 i = ( 𝐞 j 𝐞 i ) q j q_{i}=\mathbf{v}\cdot\mathbf{e}_{i}=(q^{j}\mathbf{e}_{j})\cdot\mathbf{e}_{i}=(% \mathbf{e}_{j}\cdot\mathbf{e}_{i})q^{j}\,
  59. q i = 𝐯 𝐞 i = ( q j 𝐞 j ) 𝐞 i = ( 𝐞 j 𝐞 i ) q j . q^{i}=\mathbf{v}\cdot\mathbf{e}^{i}=(q_{j}\mathbf{e}^{j})\cdot\mathbf{e}^{i}=(% \mathbf{e}^{j}\cdot\mathbf{e}^{i})q_{j}.\,
  60. 𝐞 1 , , 𝐞 n \mathbf{e}_{1},\dots,\mathbf{e}_{n}
  61. 𝐞 i = e i j 𝐞 j \mathbf{e}^{i}=e^{ij}\mathbf{e}_{j}
  62. e i j = 𝐞 i 𝐞 j . e_{ij}=\mathbf{e}_{i}\cdot\mathbf{e}_{j}.
  63. 𝐞 i 𝐞 k = e i j 𝐞 j 𝐞 k = e i j e j k = δ k i . \mathbf{e}^{i}\cdot\mathbf{e}_{k}=e^{ij}\mathbf{e}_{j}\cdot\mathbf{e}_{k}=e^{% ij}e_{jk}=\delta^{i}_{k}.
  64. 𝐯 = q i 𝐞 i = q i 𝐞 i \mathbf{v}=q_{i}\mathbf{e}^{i}=q^{i}\mathbf{e}_{i}\,
  65. q i = 𝐯 𝐞 i = ( q j 𝐞 j ) 𝐞 i = q j e j i q_{i}=\mathbf{v}\cdot\mathbf{e}_{i}=(q^{j}\mathbf{e}_{j})\cdot\mathbf{e}_{i}=q% ^{j}e_{ji}
  66. q i = 𝐯 𝐞 i = ( q j 𝐞 j ) 𝐞 i = q j e j i . q^{i}=\mathbf{v}\cdot\mathbf{e}^{i}=(q_{j}\mathbf{e}^{j})\cdot\mathbf{e}^{i}=q% _{j}e^{ji}.\,
  67. d x μ d τ \frac{dx^{\mu}}{d\tau}
  68. x μ x^{\mu}\!
  69. τ \tau\!
  70. ϕ x μ \frac{\partial\phi}{\partial x^{\mu}}
  71. ϕ \phi\!
  72. 𝐱 𝐟𝐱 . \mathbf{x}\mapsto\mathbf{f}\mathbf{x}.

Covariance_matrix.html

  1. 𝐗 = [ X 1 X n ] \mathbf{X}=\begin{bmatrix}X_{1}\\ \vdots\\ X_{n}\end{bmatrix}
  2. Σ i j = cov ( X i , X j ) = E [ ( X i - μ i ) ( X j - μ j ) ] \Sigma_{ij}=\mathrm{cov}(X_{i},X_{j})=\mathrm{E}\begin{bmatrix}(X_{i}-\mu_{i})% (X_{j}-\mu_{j})\end{bmatrix}
  3. μ i = E ( X i ) \mu_{i}=\mathrm{E}(X_{i})\,
  4. Σ = [ E [ ( X 1 - μ 1 ) ( X 1 - μ 1 ) ] E [ ( X 1 - μ 1 ) ( X 2 - μ 2 ) ] E [ ( X 1 - μ 1 ) ( X n - μ n ) ] E [ ( X 2 - μ 2 ) ( X 1 - μ 1 ) ] E [ ( X 2 - μ 2 ) ( X 2 - μ 2 ) ] E [ ( X 2 - μ 2 ) ( X n - μ n ) ] E [ ( X n - μ n ) ( X 1 - μ 1 ) ] E [ ( X n - μ n ) ( X 2 - μ 2 ) ] E [ ( X n - μ n ) ( X n - μ n ) ] ] . \Sigma=\begin{bmatrix}\mathrm{E}[(X_{1}-\mu_{1})(X_{1}-\mu_{1})]&\mathrm{E}[(X% _{1}-\mu_{1})(X_{2}-\mu_{2})]&\cdots&\mathrm{E}[(X_{1}-\mu_{1})(X_{n}-\mu_{n})% ]\\ \\ \mathrm{E}[(X_{2}-\mu_{2})(X_{1}-\mu_{1})]&\mathrm{E}[(X_{2}-\mu_{2})(X_{2}-% \mu_{2})]&\cdots&\mathrm{E}[(X_{2}-\mu_{2})(X_{n}-\mu_{n})]\\ \\ \vdots&\vdots&\ddots&\vdots\\ \\ \mathrm{E}[(X_{n}-\mu_{n})(X_{1}-\mu_{1})]&\mathrm{E}[(X_{n}-\mu_{n})(X_{2}-% \mu_{2})]&\cdots&\mathrm{E}[(X_{n}-\mu_{n})(X_{n}-\mu_{n})]\end{bmatrix}.
  5. Σ - 1 \Sigma^{-1}
  6. Σ = E [ ( 𝐗 - E [ 𝐗 ] ) ( 𝐗 - E [ 𝐗 ] ) T ] \Sigma=\mathrm{E}\left[\left(\mathbf{X}-\mathrm{E}[\mathbf{X}]\right)\left(% \mathbf{X}-\mathrm{E}[\mathbf{X}]\right)^{\rm T}\right]
  7. σ 2 = var ( X ) = E [ ( X - E ( X ) ) 2 ] = E [ ( X - E ( X ) ) ( X - E ( X ) ) ] . \sigma^{2}=\mathrm{var}(X)=\mathrm{E}[(X-\mathrm{E}(X))^{2}]=\mathrm{E}[(X-% \mathrm{E}(X))\cdot(X-\mathrm{E}(X))].\,
  8. Σ \Sigma
  9. 𝐗 \mathbf{X}
  10. 𝐗 \mathbf{X}
  11. corr ( 𝐗 ) = ( diag ( Σ ) ) - 1 2 Σ ( diag ( Σ ) ) - 1 2 \,\text{corr}(\mathbf{X})=\left(\,\text{diag}(\Sigma)\right)^{-\frac{1}{2}}\,% \Sigma\,\left(\,\text{diag}(\Sigma)\right)^{-\frac{1}{2}}
  12. diag ( Σ ) \,\text{diag}(\Sigma)
  13. Σ \Sigma
  14. X i X_{i}
  15. i = 1 , , n i=1,\dots,n
  16. X i / σ ( X i ) X_{i}/\sigma(X_{i})
  17. i = 1 , , n i=1,\dots,n
  18. Σ \Sigma
  19. X X
  20. X X
  21. var ( 𝐗 ) = cov ( 𝐗 ) = E [ ( 𝐗 - E [ 𝐗 ] ) ( 𝐗 - E [ 𝐗 ] ) T ] . \operatorname{var}(\mathbf{X})=\operatorname{cov}(\mathbf{X})=\mathrm{E}\left[% (\mathbf{X}-\mathrm{E}[\mathbf{X}])(\mathbf{X}-\mathrm{E}[\mathbf{X}])^{\rm T}% \right].
  22. cov ( 𝐗 , 𝐘 ) = E [ ( 𝐗 - E [ 𝐗 ] ) ( 𝐘 - E [ 𝐘 ] ) T ] . \operatorname{cov}(\mathbf{X},\mathbf{Y})=\mathrm{E}\left[(\mathbf{X}-\mathrm{% E}[\mathbf{X}])(\mathbf{Y}-\mathrm{E}[\mathbf{Y}])^{\rm T}\right].
  23. Σ \Sigma
  24. Σ = E [ ( 𝐗 - E [ 𝐗 ] ) ( 𝐗 - E [ 𝐗 ] ) T ] \Sigma=\mathrm{E}\left[\left(\mathbf{X}-\mathrm{E}[\mathbf{X}]\right)\left(% \mathbf{X}-\mathrm{E}[\mathbf{X}]\right)^{\rm T}\right]
  25. s y m b o l μ = E ( 𝐗 ) symbol{\mu}=\mathrm{E}(\,\textbf{X})
  26. Σ = E ( 𝐗𝐗 T ) - s y m b o l μ s y m b o l μ T \Sigma=\mathrm{E}(\mathbf{XX^{\rm T}})-symbol{\mu}symbol{\mu}^{\rm T}
  27. Σ \Sigma\,
  28. cov ( 𝐀𝐗 + 𝐚 ) = 𝐀 cov ( 𝐗 ) 𝐀 T \operatorname{cov}(\mathbf{AX}+\mathbf{a})=\mathbf{A}\,\operatorname{cov}(% \mathbf{X})\,\mathbf{A^{\rm T}}
  29. cov ( 𝐗 , 𝐘 ) = cov ( 𝐘 , 𝐗 ) T \operatorname{cov}(\mathbf{X},\mathbf{Y})=\operatorname{cov}(\mathbf{Y},% \mathbf{X})^{\rm T}
  30. cov ( 𝐗 1 + 𝐗 2 , 𝐘 ) = cov ( 𝐗 1 , 𝐘 ) + cov ( 𝐗 2 , 𝐘 ) \operatorname{cov}(\mathbf{X}_{1}+\mathbf{X}_{2},\mathbf{Y})=\operatorname{cov% }(\mathbf{X}_{1},\mathbf{Y})+\operatorname{cov}(\mathbf{X}_{2},\mathbf{Y})
  31. var ( 𝐗 + 𝐘 ) = var ( 𝐗 ) + cov ( 𝐗 , 𝐘 ) + cov ( 𝐘 , 𝐗 ) + var ( 𝐘 ) \operatorname{var}(\mathbf{X}+\mathbf{Y})=\operatorname{var}(\mathbf{X})+% \operatorname{cov}(\mathbf{X},\mathbf{Y})+\operatorname{cov}(\mathbf{Y},% \mathbf{X})+\operatorname{var}(\mathbf{Y})
  32. cov ( 𝐀𝐗 + 𝐚 , 𝐁 T 𝐘 + 𝐛 ) = 𝐀 cov ( 𝐗 , 𝐘 ) 𝐁 \operatorname{cov}(\mathbf{AX}+\mathbf{a},\mathbf{B}^{\rm T}\mathbf{Y}+\mathbf% {b})=\mathbf{A}\,\operatorname{cov}(\mathbf{X},\mathbf{Y})\,\mathbf{B}
  33. 𝐗 \mathbf{X}
  34. 𝐘 \mathbf{Y}
  35. cov ( 𝐗 , 𝐘 ) = 𝟎 \operatorname{cov}(\mathbf{X},\mathbf{Y})=\mathbf{0}
  36. 𝐗 , 𝐗 1 \mathbf{X},\mathbf{X}_{1}
  37. 𝐗 2 \mathbf{X}_{2}
  38. 𝐘 \mathbf{Y}
  39. 𝐚 \mathbf{a}
  40. 𝐛 \mathbf{b}
  41. 𝐀 \mathbf{A}
  42. 𝐁 \mathbf{B}
  43. s y m b o l μ X , Y symbol\mu_{X,Y}
  44. s y m b o l Σ X , Y symbol\Sigma_{X,Y}
  45. s y m b o l X symbol{X}
  46. s y m b o l Y symbol{Y}
  47. s y m b o l μ X , Y = [ s y m b o l μ X s y m b o l μ Y ] , s y m b o l Σ X , Y = [ s y m b o l Σ 𝑋𝑋 s y m b o l Σ 𝑋𝑌 s y m b o l Σ 𝑌𝑋 s y m b o l Σ 𝑌𝑌 ] symbol\mu_{X,Y}=\begin{bmatrix}symbol\mu_{X}\\ symbol\mu_{Y}\end{bmatrix},\qquad symbol\Sigma_{X,Y}=\begin{bmatrix}symbol% \Sigma_{\mathit{XX}}&symbol\Sigma_{\mathit{XY}}\\ symbol\Sigma_{\mathit{YX}}&symbol\Sigma_{\mathit{YY}}\end{bmatrix}
  48. s y m b o l Σ X X = var ( s y m b o l X ) , s y m b o l Σ Y Y = var ( s y m b o l Y ) , symbol\Sigma_{XX}=\mbox{var}~{}(symbol{X}),symbol\Sigma_{YY}=\mbox{var}~{}(% symbol{Y}),
  49. s y m b o l Σ X Y = s y m b o l Σ 𝑌𝑋 T = cov ( s y m b o l X , s y m b o l Y ) symbol\Sigma_{XY}=symbol\Sigma^{T}_{\mathit{YX}}=\mbox{cov}~{}(symbol{X},% symbol{Y})
  50. s y m b o l Σ X X symbol\Sigma_{XX}
  51. s y m b o l Σ Y Y symbol\Sigma_{YY}
  52. s y m b o l X symbol{X}
  53. s y m b o l Y symbol{Y}
  54. s y m b o l X symbol{X}
  55. s y m b o l Y symbol{Y}
  56. s y m b o l x , s y m b o l y 𝒩 ( s y m b o l μ X , Y , s y m b o l Σ X , Y ) symbol{x},symbol{y}\sim\ \mathcal{N}(symbol\mu_{X,Y},symbol\Sigma_{X,Y})
  57. s y m b o l Y symbol{Y}
  58. s y m b o l X symbol{X}
  59. s y m b o l y | s y m b o l x 𝒩 ( s y m b o l μ Y | X , s y m b o l Σ Y | X ) symbol{y}|symbol{x}\sim\ \mathcal{N}(symbol\mu_{Y|X},symbol\Sigma_{Y|X})
  60. s y m b o l μ Y | X = s y m b o l μ Y + s y m b o l Σ Y X s y m b o l Σ X X - 1 ( 𝐱 - s y m b o l μ X ) symbol\mu_{Y|X}=symbol\mu_{Y}+symbol\Sigma_{YX}symbol\Sigma_{XX}^{-1}\left(% \mathbf{x}-symbol\mu_{X}\right)
  61. s y m b o l Σ Y | X = s y m b o l Σ Y Y - s y m b o l Σ 𝑌𝑋 s y m b o l Σ 𝑋𝑋 - 1 s y m b o l Σ 𝑋𝑌 . symbol\Sigma_{Y|X}=symbol\Sigma_{YY}-symbol\Sigma_{\mathit{YX}}symbol\Sigma_{% \mathit{XX}}^{-1}symbol\Sigma_{\mathit{XY}}.
  62. 𝐜 T Σ = cov ( 𝐜 T 𝐗 , 𝐗 ) \mathbf{c}^{\rm T}\Sigma=\operatorname{cov}(\mathbf{c}^{\rm T}\mathbf{X},% \mathbf{X})
  63. 𝐝 T Σ 𝐜 = cov ( 𝐝 T 𝐗 , 𝐜 T 𝐗 ) \mathbf{d}^{\rm T}\Sigma\mathbf{c}=\operatorname{cov}(\mathbf{d}^{\rm T}% \mathbf{X},\mathbf{c}^{\rm T}\mathbf{X})
  64. 𝐜 T Σ 𝐜 \mathbf{c}^{\rm T}\Sigma\mathbf{c}
  65. c - μ | Σ + | c - μ \langle c-\mu|\Sigma^{+}|c-\mu\rangle
  66. 𝐛 \mathbf{b}
  67. ( p × 1 ) (p\times 1)
  68. var ( 𝐛 T 𝐗 ) = 𝐛 T var ( 𝐗 ) 𝐛 , \operatorname{var}(\mathbf{b}^{\rm T}\mathbf{X})=\mathbf{b}^{\rm T}% \operatorname{var}(\mathbf{X})\mathbf{b},\,
  69. 𝐗 \mathbf{X}
  70. var ( 𝐌 1 / 2 𝐗 ) = 𝐌 1 / 2 ( var ( 𝐗 ) ) 𝐌 1 / 2 = 𝐌 . \operatorname{var}(\mathbf{M}^{1/2}\mathbf{X})=\mathbf{M}^{1/2}(\operatorname{% var}(\mathbf{X}))\mathbf{M}^{1/2}=\mathbf{M}.\,
  71. var ( z ) = E [ ( z - μ ) ( z - μ ) * ] \operatorname{var}(z)=\operatorname{E}\left[(z-\mu)(z-\mu)^{*}\right]
  72. z z
  73. z * z^{*}
  74. Z Z
  75. E [ ( Z - μ ) ( Z - μ ) ] , \operatorname{E}\left[(Z-\mu)(Z-\mu)^{\dagger}\right],
  76. Z Z^{\dagger}
  77. 𝐌 𝐗 \mathbf{M}_{\mathbf{X}}
  78. 𝐌 𝐘 \mathbf{M}_{\mathbf{Y}}
  79. 𝐐 𝐗 \mathbf{Q}_{\mathbf{X}}
  80. 𝐐 𝐗𝐘 \mathbf{Q}_{\mathbf{XY}}
  81. 𝐐 𝐗 = 1 n - 1 𝐌 𝐗 T 𝐌 𝐗 , 𝐐 𝐗𝐘 = 1 n - 1 𝐌 𝐗 T 𝐌 𝐘 \mathbf{Q}_{\mathbf{X}}=\frac{1}{n-1}\mathbf{M}_{\mathbf{X}}^{T}\mathbf{M}_{% \mathbf{X}},\qquad\mathbf{Q}_{\mathbf{XY}}=\frac{1}{n-1}\mathbf{M}_{\mathbf{X}% }^{T}\mathbf{M}_{\mathbf{Y}}
  82. 𝐐 𝐗 = 1 n 𝐌 𝐗 T 𝐌 𝐗 , 𝐐 𝐗𝐘 = 1 n 𝐌 𝐗 T 𝐌 𝐘 \mathbf{Q}_{\mathbf{X}}=\frac{1}{n}\mathbf{M}_{\mathbf{X}}^{T}\mathbf{M}_{% \mathbf{X}},\qquad\mathbf{Q}_{\mathbf{XY}}=\frac{1}{n}\mathbf{M}_{\mathbf{X}}^% {T}\mathbf{M}_{\mathbf{Y}}

Covering_space.html

  1. p : C X p\colon C\to X\,
  2. \mathbb{R}
  3. U ( n ) \mathrm{U}(n)
  4. SU ( n ) × \mathrm{SU}(n)\times\mathbb{R}
  5. f ( c γ ) = ( f c ) γ f\cdot(c\cdot\gamma)=(f\cdot c)\cdot\gamma
  6. Γ p ( c ) = { [ γ ] : γ C is a closed curve in C passing through c C } \Gamma_{p}(c)=\{[\gamma]:\gamma_{C}\mbox{ is a closed curve in }~{}C\mbox{ % passing through }c\in C\}
  7. π 1 : TopCov ( X ) GpdCov ( π 1 X ) \pi_{1}:\mathrm{TopCov}(X)\to\mathrm{GpdCov}(\pi_{1}X)
  8. C n ( T ) C n - 1 ( T ) C 0 ( T ) 𝜀 𝐙 , \cdots\overset{\partial}{\to}C_{n}(T)\overset{\partial}{\to}C_{n-1}(T)\overset% {\partial}{\to}\cdots\overset{\partial}{\to}C_{0}(T)\overset{\varepsilon}{\to}% \mathbf{Z},

Coxeter_group.html

  1. r 1 , r 2 , , r n ( r i r j ) m i j = 1 \left\langle r_{1},r_{2},\ldots,r_{n}\mid(r_{i}r_{j})^{m_{ij}}=1\right\rangle
  2. m i i = 1 m_{ii}=1
  3. m i j 2 m_{ij}\geq 2
  4. i j i\neq j
  5. m i j = m_{ij}=\infty
  6. ( r i r j ) m (r_{i}r_{j})^{m}
  7. r i = r i - 1 r_{i}=r_{i}^{-1}
  8. ( r i r j ) 2 = r i r j r i r j = r i r j r i - 1 r j - 1 (r_{i}r_{j})^{2}=r_{i}r_{j}r_{i}r_{j}=r_{i}r_{j}r_{i}^{-1}r_{j}^{-1}
  9. ( x y ) k (xy)^{k}
  10. ( y x ) k (yx)^{k}
  11. y ( x y ) k y - 1 = ( y x ) k y y - 1 = ( y x ) k y(xy)^{k}y^{-1}=(yx)^{k}yy^{-1}=(yx)^{k}
  12. I ~ 1 {\tilde{I}}_{1}
  13. A ~ 3 {\tilde{A}}_{3}
  14. [ 1 2 2 1 ] \left[\begin{smallmatrix}1&2\\ 2&1\\ \end{smallmatrix}\right]
  15. [ 1 3 3 1 ] \left[\begin{smallmatrix}1&3\\ 3&1\\ \end{smallmatrix}\right]
  16. [ 1 1 ] \left[\begin{smallmatrix}1&\infty\\ \infty&1\\ \end{smallmatrix}\right]
  17. [ 1 3 2 3 1 3 2 3 1 ] \left[\begin{smallmatrix}1&3&2\\ 3&1&3\\ 2&3&1\end{smallmatrix}\right]
  18. [ 1 4 2 4 1 3 2 3 1 ] \left[\begin{smallmatrix}1&4&2\\ 4&1&3\\ 2&3&1\end{smallmatrix}\right]
  19. [ 1 3 2 2 3 1 3 3 2 3 1 2 2 3 2 1 ] \left[\begin{smallmatrix}1&3&2&2\\ 3&1&3&3\\ 2&3&1&2\\ 2&3&2&1\end{smallmatrix}\right]
  20. [ 1 3 2 3 3 1 3 2 2 3 1 3 3 2 3 1 ] \left[\begin{smallmatrix}1&3&2&3\\ 3&1&3&2\\ 2&3&1&3\\ 3&2&3&1\end{smallmatrix}\right]
  21. [ 2 0 0 2 ] \left[\begin{smallmatrix}2&0\\ 0&2\end{smallmatrix}\right]
  22. [ 2 - 1 - 1 2 ] \left[\begin{smallmatrix}2&-1\\ -1&2\end{smallmatrix}\right]
  23. [ 2 - 2 - 2 2 ] \left[\begin{smallmatrix}2&-2\\ -2&2\end{smallmatrix}\right]
  24. [ 2 - 1 0 - 1 2 - 1 0 - 1 2 ] \left[\begin{smallmatrix}2&-1&0\\ -1&2&-1\\ 0&-1&2\end{smallmatrix}\right]
  25. [ 2 - 2 0 - 2 2 - 1 0 - 1 2 ] \left[\begin{smallmatrix}2&-\sqrt{2}&0\\ -\sqrt{2}&2&-1\\ 0&-1&2\end{smallmatrix}\right]
  26. [ 2 - 1 0 0 - 1 2 - 1 - 1 0 - 1 2 0 0 - 1 0 2 ] \left[\begin{smallmatrix}2&-1&0&0\\ -1&2&-1&-1\\ 0&-1&2&0\\ 0&-1&0&2\end{smallmatrix}\right]
  27. [ 2 - 1 0 - 1 - 1 2 - 1 0 0 - 1 2 - 1 - 1 0 - 1 2 ] \left[\begin{smallmatrix}2&-1&0&-1\\ -1&2&-1&0\\ 0&-1&2&-1\\ -1&0&-1&2\end{smallmatrix}\right]
  28. ( r i r j ) k (r_{i}r_{j})^{k}
  29. π / k \pi/k
  30. r i r j r_{i}r_{j}
  31. 2 π / k 2\pi/k
  32. r i 2 r_{i}^{2}
  33. ( r i r j ) k (r_{i}r_{j})^{k}
  34. A n , B C n , D n , A_{n},BC_{n},D_{n},
  35. I 2 ( p ) , I_{2}(p),
  36. E 6 , E 7 , E 8 , F 4 , H 3 , E_{6},E_{7},E_{8},F_{4},H_{3},
  37. H 4 . H_{4}.
  38. A n , B C n , A_{n},BC_{n},
  39. D n , D_{n},
  40. E 6 , E 7 , E 8 , F 4 , E_{6},E_{7},E_{8},F_{4},
  41. I 2 ( 6 ) , I_{2}(6),
  42. G 2 . G_{2}.
  43. H 3 H_{3}
  44. H 4 , H_{4},
  45. I 2 ( p ) I_{2}(p)
  46. I 2 ( 3 ) A 2 , I 2 ( 4 ) B C 2 , I_{2}(3)\cong A_{2},I_{2}(4)\cong BC_{2},
  47. I 2 ( 6 ) G 2 I_{2}(6)\cong G_{2}
  48. H 3 , H_{3},
  49. H 4 , H_{4},
  50. I 2 ( p ) I_{2}(p)
  51. p = 3 , 4 , p=3,4,
  52. 6 6
  53. A ~ n {\tilde{A}}_{n}
  54. B ~ n {\tilde{B}}_{n}
  55. C ~ n {\tilde{C}}_{n}
  56. D ~ n {\tilde{D}}_{n}
  57. E ~ 6 {\tilde{E}}_{6}
  58. E ~ 7 {\tilde{E}}_{7}
  59. E ~ 8 {\tilde{E}}_{8}
  60. F ~ 4 {\tilde{F}}_{4}
  61. G ~ 2 {\tilde{G}}_{2}
  62. I ~ 1 {\tilde{I}}_{1}
  63. v ( - 1 ) l ( v ) v\to(-1)^{l(v)}
  64. G { ± 1 } , G\to\{\pm 1\},

CpG_site.html

  1. ( number of C p G s * length of sequence ) (\,\text{number of }CpGs*\,\text{length of sequence})
  2. ( number of C * number of G ) (\,\text{number of }C*\,\text{number of }G)
  3. ( ( number of C + number of G ) / 2 ) 2 ((\,\text{number of }C+\,\text{number of }G)/2)^{2}

CPU_power_dissipation.html

  1. P c p u = P d y n + P s c + P l e a k P_{cpu}=P_{dyn}+P_{sc}+P_{leak}
  2. P = C V 2 f P=CV^{2}f
  3. C C
  4. f f
  5. V V

Cramer's_rule.html

  1. n n
  2. n n
  3. A x = b Ax=b
  4. n × n n×n
  5. A A
  6. x = ( x 1 , , x n ) T x=(x_{1},\ldots,x_{n})^{\mathrm{T}}
  7. x i = det ( A i ) det ( A ) i = 1 , , n x_{i}=\frac{\det(A_{i})}{\det(A)}\qquad i=1,\ldots,n
  8. A i A_{i}
  9. i i
  10. A A
  11. b b
  12. A X = B AX=B
  13. n × n n×n
  14. A A
  15. X X
  16. B B
  17. n × m n×m
  18. 1 i 1 < i 2 < < i k n 1\leq i_{1}<i_{2}<\ldots<i_{k}\leq n
  19. 1 j 1 < j 2 < < j k n 1\leq j_{1}<j_{2}<\ldots<j_{k}\leq n
  20. k × k k×k
  21. X X
  22. I := ( i 1 , , i k ) I:=(i_{1},\ldots,i_{k})
  23. J := ( j 1 , , j k ) J:=(j_{1},\ldots,j_{k})
  24. A B ( I , J ) A_{B}(I,J)
  25. n × n n×n
  26. i s i_{s}
  27. A A
  28. j s j_{s}
  29. s = 1 , , k s=1,\ldots,k
  30. det X I , J = det ( A B ( I , J ) ) det ( A ) . \det X_{I,J}=\frac{\det(A_{B}(I,J))}{\det(A)}.
  31. k = 1 k=1
  32. 𝐑 \mathbf{R}
  33. A A
  34. d e t ( A ) det(A)
  35. A A
  36. A A
  37. A A
  38. n n
  39. n n
  40. x 1 , , x n x_{1},\ldots,x_{n}
  41. A A
  42. a 11 x 1 + a 12 x 2 + + a 1 n x n = b 1 a 21 x 1 + a 22 x 2 + + a 2 n x n = b 2 a n 1 x 1 + a n 2 x 2 + + a n n x n = b n . \begin{matrix}a_{11}x_{1}+a_{12}x_{2}+\cdots+a_{1n}x_{n}&=&b_{1}\\ a_{21}x_{1}+a_{22}x_{2}+\cdots+a_{2n}x_{n}&=&b_{2}\\ \vdots&\vdots&\vdots\\ a_{n1}x_{1}+a_{n2}x_{2}+\cdots+a_{nn}x_{n}&=&b_{n}.\end{matrix}
  43. A 𝐱 = 𝐛 A\mathbf{x}=\mathbf{b}
  44. x j = L ( j ) 𝐛 det ( A ) . x_{j}=\frac{L_{(j)}\cdot\mathbf{b}}{\det(A)}.
  45. A A
  46. A A
  47. A A
  48. n × n n×n
  49. A A
  50. d e t ( A ) det(A)
  51. 1 det ( A ) M = A - 1 , \frac{1}{\det(A)}M=A^{-1},
  52. A A
  53. n × n n×n
  54. Adj ( A ) A = det ( A ) I \mathrm{Adj}(A)A=\mathrm{det}(A)I
  55. A A
  56. d e t ( A ) det(A)
  57. A A
  58. A - 1 = 1 det ( A ) Adj ( A ) . A^{-1}=\frac{1}{\operatorname{det}(A)}\operatorname{Adj}(A).
  59. A A
  60. d e t ( A ) 0 det(A)≠0
  61. A A
  62. { a 1 x + b 1 y = \color r e d c 1 a 2 x + b 2 y = \color r e d c 2 \left\{\begin{matrix}a_{1}x+b_{1}y&={\color{red}c_{1}}\\ a_{2}x+b_{2}y&={\color{red}c_{2}}\end{matrix}\right.
  63. [ a 1 b 1 a 2 b 2 ] [ x y ] = [ \color r e d c 1 \color r e d c 2 ] . \begin{bmatrix}a_{1}&b_{1}\\ a_{2}&b_{2}\end{bmatrix}\begin{bmatrix}x\\ y\end{bmatrix}=\begin{bmatrix}{\color{red}c_{1}}\\ {\color{red}c_{2}}\end{bmatrix}.
  64. x x
  65. y y
  66. x = | \color r e d c 1 b 1 \color r e d c 2 b 2 | | a 1 b 1 a 2 b 2 | = \color r e d c 1 b 2 - b 1 \color r e d c 2 a 1 b 2 - b 1 a 2 y = | a 1 \color r e d c 1 a 2 \color r e d c 2 | | a 1 b 1 a 2 b 2 | = a 1 \color r e d c 2 - \color r e d c 1 a 2 a 1 b 2 - b 1 a 2 \begin{aligned}\displaystyle x&\displaystyle=\frac{\begin{vmatrix}{\color{red}% {c_{1}}}&b_{1}\\ {\color{red}{c_{2}}}&b_{2}\end{vmatrix}}{\begin{vmatrix}a_{1}&b_{1}\\ a_{2}&b_{2}\end{vmatrix}}={{\color{red}c_{1}}b_{2}-b_{1}{\color{red}c_{2}}% \over a_{1}b_{2}-b_{1}a_{2}}\\ \displaystyle y&\displaystyle=\frac{\begin{vmatrix}a_{1}&{\color{red}{c_{1}}}% \\ a_{2}&{\color{red}{c_{2}}}\end{vmatrix}}{\begin{vmatrix}a_{1}&b_{1}\\ a_{2}&b_{2}\end{vmatrix}}={a_{1}{\color{red}c_{2}}-{\color{red}c_{1}}a_{2}% \over a_{1}b_{2}-b_{1}a_{2}}\end{aligned}
  67. 3 × 3 3×3
  68. { a 1 x + b 1 y + c 1 z = \color r e d d 1 a 2 x + b 2 y + c 2 z = \color r e d d 2 a 3 x + b 3 y + c 3 z = \color r e d d 3 \left\{\begin{matrix}a_{1}x+b_{1}y+c_{1}z&={\color{red}d_{1}}\\ a_{2}x+b_{2}y+c_{2}z&={\color{red}d_{2}}\\ a_{3}x+b_{3}y+c_{3}z&={\color{red}d_{3}}\end{matrix}\right.
  69. [ a 1 b 1 c 1 a 2 b 2 c 2 a 3 b 3 c 3 ] [ x y z ] = [ \color r e d d 1 \color r e d d 2 \color r e d d 3 ] . \begin{bmatrix}a_{1}&b_{1}&c_{1}\\ a_{2}&b_{2}&c_{2}\\ a_{3}&b_{3}&c_{3}\end{bmatrix}\begin{bmatrix}x\\ y\\ z\end{bmatrix}=\begin{bmatrix}{\color{red}d_{1}}\\ {\color{red}d_{2}}\\ {\color{red}d_{3}}\end{bmatrix}.
  70. x , y x,y
  71. z z
  72. x = | \color r e d d 1 b 1 c 1 \color r e d d 2 b 2 c 2 \color r e d d 3 b 3 c 3 | | a 1 b 1 c 1 a 2 b 2 c 2 a 3 b 3 c 3 | , y = | a 1 \color r e d d 1 c 1 a 2 \color r e d d 2 c 2 a 3 \color r e d d 3 c 3 | | a 1 b 1 c 1 a 2 b 2 c 2 a 3 b 3 c 3 | , and z = | a 1 b 1 \color r e d d 1 a 2 b 2 \color r e d d 2 a 3 b 3 \color r e d d 3 | | a 1 b 1 c 1 a 2 b 2 c 2 a 3 b 3 c 3 | . x=\frac{\begin{vmatrix}{\color{red}d_{1}}&b_{1}&c_{1}\\ {\color{red}d_{2}}&b_{2}&c_{2}\\ {\color{red}d_{3}}&b_{3}&c_{3}\end{vmatrix}}{\begin{vmatrix}a_{1}&b_{1}&c_{1}% \\ a_{2}&b_{2}&c_{2}\\ a_{3}&b_{3}&c_{3}\end{vmatrix}},\quad y=\frac{\begin{vmatrix}a_{1}&{\color{red% }d_{1}}&c_{1}\\ a_{2}&{\color{red}d_{2}}&c_{2}\\ a_{3}&{\color{red}d_{3}}&c_{3}\end{vmatrix}}{\begin{vmatrix}a_{1}&b_{1}&c_{1}% \\ a_{2}&b_{2}&c_{2}\\ a_{3}&b_{3}&c_{3}\end{vmatrix}},\,\text{ and }z=\frac{\begin{vmatrix}a_{1}&b_{% 1}&{\color{red}d_{1}}\\ a_{2}&b_{2}&{\color{red}d_{2}}\\ a_{3}&b_{3}&{\color{red}d_{3}}\end{vmatrix}}{\begin{vmatrix}a_{1}&b_{1}&c_{1}% \\ a_{2}&b_{2}&c_{2}\\ a_{3}&b_{3}&c_{3}\end{vmatrix}}.
  73. F ( x , y , u , v ) = 0 F(x,y,u,v)=0
  74. G ( x , y , u , v ) = 0 G(x,y,u,v)=0
  75. x = X ( u , v ) x=X(u,v)
  76. y = Y ( u , v ) . y=Y(u,v).
  77. x u \dfrac{\partial x}{\partial u}
  78. d F \displaystyle dF
  79. d F \displaystyle dF
  80. F x x u + F y y u \displaystyle\frac{\partial F}{\partial x}\frac{\partial x}{\partial u}+\frac{% \partial F}{\partial y}\frac{\partial y}{\partial u}
  81. x u = | - F u F y - G u G y | | F x F y G x G y | . \frac{\partial x}{\partial u}=\frac{\begin{vmatrix}-\frac{\partial F}{\partial u% }&\frac{\partial F}{\partial y}\\ -\frac{\partial G}{\partial u}&\frac{\partial G}{\partial y}\end{vmatrix}}{% \begin{vmatrix}\frac{\partial F}{\partial x}&\frac{\partial F}{\partial y}\\ \frac{\partial G}{\partial x}&\frac{\partial G}{\partial y}\end{vmatrix}}.
  82. x u = - ( ( F , G ) ( u , y ) ) ( ( F , G ) ( x , y ) ) . \frac{\partial x}{\partial u}=-\frac{\left(\frac{\partial(F,G)}{\partial(u,y)}% \right)}{\left(\frac{\partial(F,G)}{\partial(x,y)}\right)}.
  83. x v , y u , y v . \frac{\partial x}{\partial v},\frac{\partial y}{\partial u},\frac{\partial y}{% \partial v}.
  84. a 11 x 1 + a 12 x 2 = b 1 a 21 x 1 + a 22 x 2 = b 2 \begin{matrix}a_{11}x_{1}+a_{12}x_{2}&=b_{1}\\ a_{21}x_{1}+a_{22}x_{2}&=b_{2}\end{matrix}
  85. x 1 ( a 11 a 21 ) + x 2 ( a 12 a 22 ) = ( b 1 b 2 ) . x_{1}{\left({{a_{11}}\atop{a_{21}}}\right)}+x_{2}{\left({{a_{12}}\atop{a_{22}}% }\right)}={\left({{b_{1}}\atop{b_{2}}}\right)}.
  86. ( a 11 a 21 ) {\left({{a_{11}}\atop{a_{21}}}\right)}
  87. ( a 12 a 22 ) {\left({{a_{12}}\atop{a_{22}}}\right)}
  88. | a 11 a 12 a 21 a 22 | . \begin{vmatrix}a_{11}&a_{12}\\ a_{21}&a_{22}\end{vmatrix}.
  89. n n
  90. n n
  91. n n
  92. x 1 ( a 11 a 21 ) x_{1}{\left({{a_{11}}\atop{a_{21}}}\right)}
  93. ( a 12 a 22 ) {\left({{a_{12}}\atop{a_{22}}}\right)}
  94. x 1 x_{1}
  95. ( b 1 b 2 ) = x 1 ( a 11 a 21 ) + x 2 ( a 12 a 22 ) {\left({{b_{1}}\atop{b_{2}}}\right)}=x_{1}{\left({{a_{11}}\atop{a_{21}}}\right% )}+x_{2}{\left({{a_{12}}\atop{a_{22}}}\right)}
  96. ( a 12 a 22 ) {\left({{a_{12}}\atop{a_{22}}}\right)}
  97. | b 1 a 12 b 2 a 22 | = | a 11 x 1 a 12 a 21 x 1 a 22 | = x 1 | a 11 a 12 a 21 a 22 | \begin{vmatrix}b_{1}&a_{12}\\ b_{2}&a_{22}\end{vmatrix}=\begin{vmatrix}a_{11}x_{1}&a_{12}\\ a_{21}x_{1}&a_{22}\end{vmatrix}=x_{1}\begin{vmatrix}a_{11}&a_{12}\\ a_{21}&a_{22}\end{vmatrix}
  98. x 1 x_{1}
  99. X 1 = [ x 1 0 0 0 x 2 1 0 0 x 3 0 1 0 x n 0 0 1 ] X_{1}=\begin{bmatrix}x_{1}&0&0&\dots&0\\ x_{2}&1&0&\dots&0\\ x_{3}&0&1&\dots&0\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ x_{n}&0&0&\dots&1\end{bmatrix}
  100. A A
  101. X 1 X_{1}
  102. A - 1 b , A - 1 v 2 , , A - 1 v n A^{-1}b,A^{-1}v_{2},\ldots,A^{-1}v_{n}
  103. v k v_{k}
  104. k k
  105. A A
  106. A 1 A_{1}
  107. b , v 2 , , v n b,v_{2},\ldots,v_{n}
  108. x 1 = det ( X 1 ) = det ( A - 1 ) det ( A 1 ) = det ( A 1 ) det ( A ) . x_{1}=\det(X_{1})=\det(A^{-1})\det(A_{1})=\frac{\det(A_{1})}{\det(A)}.
  109. x j x_{j}
  110. x 1 , x 2 , x 3 x_{1},x_{2},x_{3}
  111. a 11 x 1 + a 12 x 2 + a 13 x 3 = c 1 a 21 x 1 + a 22 x 2 + a 23 x 3 = c 2 a 31 x 1 + a 32 x 2 + a 33 x 3 = c 3 \begin{aligned}\displaystyle a_{11}x_{1}+a_{12}x_{2}+a_{13}x_{3}&\displaystyle% =c_{1}\\ \displaystyle a_{21}x_{1}+a_{22}x_{2}+a_{23}x_{3}&\displaystyle=c_{2}\\ \displaystyle a_{31}x_{1}+a_{32}x_{2}+a_{33}x_{3}&\displaystyle=c_{3}\end{aligned}
  112. 𝐞 1 , 𝐞 2 , 𝐞 3 \mathbf{e}_{1},\mathbf{e}_{2},\mathbf{e}_{3}
  113. 𝒢 3 \mathcal{G}_{3}
  114. a 11 𝐞 1 x 1 + a 12 𝐞 1 x 2 + a 13 𝐞 1 x 3 = c 1 𝐞 1 a 21 𝐞 2 x 1 + a 22 𝐞 2 x 2 + a 23 𝐞 2 x 3 = c 2 𝐞 2 a 31 𝐞 3 x 1 + a 32 𝐞 3 x 2 + a 33 𝐞 3 x 3 = c 3 𝐞 3 \begin{aligned}\displaystyle a_{11}\mathbf{e}_{1}x_{1}+a_{12}\mathbf{e}_{1}x_{% 2}+a_{13}\mathbf{e}_{1}x_{3}&\displaystyle=c_{1}\mathbf{e}_{1}\\ \displaystyle a_{21}\mathbf{e}_{2}x_{1}+a_{22}\mathbf{e}_{2}x_{2}+a_{23}% \mathbf{e}_{2}x_{3}&\displaystyle=c_{2}\mathbf{e}_{2}\\ \displaystyle a_{31}\mathbf{e}_{3}x_{1}+a_{32}\mathbf{e}_{3}x_{2}+a_{33}% \mathbf{e}_{3}x_{3}&\displaystyle=c_{3}\mathbf{e}_{3}\end{aligned}
  115. 𝐚 1 = a 11 𝐞 1 + a 21 𝐞 2 + a 31 𝐞 3 𝐚 2 = a 12 𝐞 1 + a 22 𝐞 2 + a 32 𝐞 3 𝐚 3 = a 13 𝐞 1 + a 23 𝐞 2 + a 33 𝐞 3 \begin{aligned}\displaystyle\mathbf{a}_{1}&\displaystyle=a_{11}\mathbf{e}_{1}+% a_{21}\mathbf{e}_{2}+a_{31}\mathbf{e}_{3}\\ \displaystyle\mathbf{a}_{2}&\displaystyle=a_{12}\mathbf{e}_{1}+a_{22}\mathbf{e% }_{2}+a_{32}\mathbf{e}_{3}\\ \displaystyle\mathbf{a}_{3}&\displaystyle=a_{13}\mathbf{e}_{1}+a_{23}\mathbf{e% }_{2}+a_{33}\mathbf{e}_{3}\end{aligned}
  116. 𝐜 = c 1 𝐞 1 + c 2 𝐞 2 + c 3 𝐞 3 = x 1 𝐚 1 + x 2 𝐚 2 + x 3 𝐚 3 \begin{aligned}\displaystyle\mathbf{c}&\displaystyle=c_{1}\mathbf{e}_{1}+c_{2}% \mathbf{e}_{2}+c_{3}\mathbf{e}_{3}\\ &\displaystyle=x_{1}\mathbf{a}_{1}+x_{2}\mathbf{a}_{2}+x_{3}\mathbf{a}_{3}\end% {aligned}
  117. x k x_{k}
  118. 𝐜 𝐚 2 𝐚 3 = x 1 𝐚 1 𝐚 2 𝐚 3 𝐜 𝐚 1 𝐚 3 = x 2 𝐚 2 𝐚 1 𝐚 3 𝐜 𝐚 1 𝐚 2 = x 3 𝐚 3 𝐚 1 𝐚 2 x 1 = 𝐜 𝐚 2 𝐚 3 𝐚 1 𝐚 2 𝐚 3 x 2 = 𝐜 𝐚 1 𝐚 3 𝐚 2 𝐚 1 𝐚 3 = 𝐚 1 𝐜 𝐚 3 𝐚 1 𝐚 2 𝐚 3 x 3 = 𝐜 𝐚 1 𝐚 2 𝐚 3 𝐚 1 𝐚 2 = 𝐚 1 𝐚 2 𝐜 𝐚 1 𝐚 2 𝐚 3 \begin{aligned}\displaystyle\mathbf{c}\wedge\mathbf{a}_{2}\wedge\mathbf{a}_{3}% &\displaystyle=x_{1}\mathbf{a}_{1}\wedge\mathbf{a}_{2}\wedge\mathbf{a}_{3}\\ \displaystyle\mathbf{c}\wedge\mathbf{a}_{1}\wedge\mathbf{a}_{3}&\displaystyle=% x_{2}\mathbf{a}_{2}\wedge\mathbf{a}_{1}\wedge\mathbf{a}_{3}\\ \displaystyle\mathbf{c}\wedge\mathbf{a}_{1}\wedge\mathbf{a}_{2}&\displaystyle=% x_{3}\mathbf{a}_{3}\wedge\mathbf{a}_{1}\wedge\mathbf{a}_{2}\\ \displaystyle x_{1}&\displaystyle=\frac{\mathbf{c}\wedge\mathbf{a}_{2}\wedge% \mathbf{a}_{3}}{\mathbf{a}_{1}\wedge\mathbf{a}_{2}\wedge\mathbf{a}_{3}}\\ \displaystyle x_{2}&\displaystyle=\frac{\mathbf{c}\wedge\mathbf{a}_{1}\wedge% \mathbf{a}_{3}}{\mathbf{a}_{2}\wedge\mathbf{a}_{1}\wedge\mathbf{a}_{3}}=\frac{% \mathbf{a}_{1}\wedge\mathbf{c}\wedge\mathbf{a}_{3}}{\mathbf{a}_{1}\wedge% \mathbf{a}_{2}\wedge\mathbf{a}_{3}}\\ \displaystyle x_{3}&\displaystyle=\frac{\mathbf{c}\wedge\mathbf{a}_{1}\wedge% \mathbf{a}_{2}}{\mathbf{a}_{3}\wedge\mathbf{a}_{1}\wedge\mathbf{a}_{2}}=\frac{% \mathbf{a}_{1}\wedge\mathbf{a}_{2}\wedge\mathbf{c}}{\mathbf{a}_{1}\wedge% \mathbf{a}_{2}\wedge\mathbf{a}_{3}}\end{aligned}
  119. n n
  120. n n
  121. k k
  122. x k x_{k}
  123. x k = 𝐚 1 ( 𝐜 ) k 𝐚 n 𝐚 1 𝐚 k 𝐚 n = ( 𝐚 1 ( 𝐜 ) k 𝐚 n ) ( 𝐚 1 𝐚 k 𝐚 n ) - 1 = ( 𝐚 1 ( 𝐜 ) k 𝐚 n ) ( 𝐚 1 𝐚 k 𝐚 n ) ( 𝐚 1 𝐚 k 𝐚 n ) ( 𝐚 1 𝐚 k 𝐚 n ) = ( 𝐚 1 ( 𝐜 ) k 𝐚 n ) ( 𝐚 1 𝐚 k 𝐚 n ) ( - 1 ) n ( n - 1 ) 2 ( 𝐚 n 𝐚 k 𝐚 1 ) ( 𝐚 1 𝐚 k 𝐚 n ) = ( 𝐚 n ( 𝐜 ) k 𝐚 1 ) ( 𝐚 1 𝐚 k 𝐚 n ) ( 𝐚 n 𝐚 k 𝐚 1 ) ( 𝐚 1 𝐚 k 𝐚 n ) \begin{aligned}\displaystyle x_{k}&\displaystyle=\frac{\mathbf{a}_{1}\wedge% \cdots\wedge(\mathbf{c})_{k}\wedge\cdots\wedge\mathbf{a}_{n}}{\mathbf{a}_{1}% \wedge\cdots\wedge\mathbf{a}_{k}\wedge\cdots\wedge\mathbf{a}_{n}}\\ &\displaystyle=(\mathbf{a}_{1}\wedge\cdots\wedge(\mathbf{c})_{k}\wedge\cdots% \wedge\mathbf{a}_{n})(\mathbf{a}_{1}\wedge\cdots\wedge\mathbf{a}_{k}\wedge% \cdots\wedge\mathbf{a}_{n})^{-1}\\ &\displaystyle=\frac{(\mathbf{a}_{1}\wedge\cdots\wedge(\mathbf{c})_{k}\wedge% \cdots\wedge\mathbf{a}_{n})(\mathbf{a}_{1}\wedge\cdots\wedge\mathbf{a}_{k}% \wedge\cdots\wedge\mathbf{a}_{n})}{(\mathbf{a}_{1}\wedge\cdots\wedge\mathbf{a}% _{k}\wedge\cdots\wedge\mathbf{a}_{n})(\mathbf{a}_{1}\wedge\cdots\wedge\mathbf{% a}_{k}\wedge\cdots\wedge\mathbf{a}_{n})}\\ &\displaystyle=\frac{(\mathbf{a}_{1}\wedge\cdots\wedge(\mathbf{c})_{k}\wedge% \cdots\wedge\mathbf{a}_{n})\cdot(\mathbf{a}_{1}\wedge\cdots\wedge\mathbf{a}_{k% }\wedge\cdots\wedge\mathbf{a}_{n})}{(-1)^{\frac{n(n-1)}{2}}(\mathbf{a}_{n}% \wedge\cdots\wedge\mathbf{a}_{k}\wedge\cdots\wedge\mathbf{a}_{1})\cdot(\mathbf% {a}_{1}\wedge\cdots\wedge\mathbf{a}_{k}\wedge\cdots\wedge\mathbf{a}_{n})}\\ &\displaystyle=\frac{(\mathbf{a}_{n}\wedge\cdots\wedge(\mathbf{c})_{k}\wedge% \cdots\wedge\mathbf{a}_{1})\cdot(\mathbf{a}_{1}\wedge\cdots\wedge\mathbf{a}_{k% }\wedge\cdots\wedge\mathbf{a}_{n})}{(\mathbf{a}_{n}\wedge\cdots\wedge\mathbf{a% }_{k}\wedge\cdots\wedge\mathbf{a}_{1})\cdot(\mathbf{a}_{1}\wedge\cdots\wedge% \mathbf{a}_{k}\wedge\cdots\wedge\mathbf{a}_{n})}\end{aligned}
  124. x k x_{k}
  125. x k = ( 𝐚 n ( 𝐜 ) k 𝐚 1 ) ( 𝐚 1 𝐚 k 𝐚 n ) ( 𝐚 n 𝐚 k 𝐚 1 ) ( 𝐚 1 𝐚 k 𝐚 n ) [ 8 p t ] = | 𝐚 1 𝐚 1 𝐚 1 ( 𝐜 ) k 𝐚 1 𝐚 n 𝐚 k 𝐚 1 𝐚 k ( 𝐜 ) k 𝐚 k 𝐚 n 𝐚 n 𝐚 1 𝐚 n ( 𝐜 ) k 𝐚 n 𝐚 n | | 𝐚 1 𝐚 1 𝐚 1 𝐚 k 𝐚 1 𝐚 n 𝐚 k 𝐚 1 𝐚 k 𝐚 k 𝐚 k 𝐚 n 𝐚 n 𝐚 1 𝐚 n 𝐚 k 𝐚 n 𝐚 n | - 1 [ 8 p t ] = | 𝐚 1 𝐚 k 𝐚 n | | 𝐚 1 ( 𝐜 ) k 𝐚 n | | 𝐚 1 𝐚 k 𝐚 n | - 1 | 𝐚 1 𝐚 k 𝐚 n | - 1 [ 8 p t ] = | 𝐚 1 ( 𝐜 ) k 𝐚 n | | 𝐚 1 𝐚 k 𝐚 n | - 1 [ 8 p t ] = | a 11 c 1 a 1 n a k 1 c k a k n a n 1 c n a n n | | a 11 a 1 k a 1 n a k 1 a k k a k n a n 1 a n k a n n | - 1 \begin{aligned}\displaystyle x_{k}&\displaystyle=\frac{(\mathbf{a}_{n}\wedge% \cdots\wedge(\mathbf{c})_{k}\wedge\cdots\wedge\mathbf{a}_{1})\cdot(\mathbf{a}_% {1}\wedge\cdots\wedge\mathbf{a}_{k}\wedge\cdots\wedge\mathbf{a}_{n})}{(\mathbf% {a}_{n}\wedge\cdots\wedge\mathbf{a}_{k}\wedge\cdots\wedge\mathbf{a}_{1})\cdot(% \mathbf{a}_{1}\wedge\cdots\wedge\mathbf{a}_{k}\wedge\cdots\wedge\mathbf{a}_{n}% )}\\ \displaystyle[8pt]&\displaystyle=\begin{vmatrix}\mathbf{a}_{1}\cdot\mathbf{a}_% {1}&\cdots&\mathbf{a}_{1}\cdot(\mathbf{c})_{k}&\cdots&\mathbf{a}_{1}\cdot% \mathbf{a}_{n}\\ \vdots&\ddots&\vdots&\ddots&\vdots\\ \mathbf{a}_{k}\cdot\mathbf{a}_{1}&\cdots&\mathbf{a}_{k}\cdot(\mathbf{c})_{k}&% \cdots&\mathbf{a}_{k}\cdot\mathbf{a}_{n}\\ \vdots&\ddots&\vdots&\ddots&\vdots\\ \mathbf{a}_{n}\cdot\mathbf{a}_{1}&\cdots&\mathbf{a}_{n}\cdot(\mathbf{c})_{k}&% \cdots&\mathbf{a}_{n}\cdot\mathbf{a}_{n}\end{vmatrix}\begin{vmatrix}\mathbf{a}% _{1}\cdot\mathbf{a}_{1}&\cdots&\mathbf{a}_{1}\cdot\mathbf{a}_{k}&\cdots&% \mathbf{a}_{1}\cdot\mathbf{a}_{n}\\ \vdots&\ddots&\vdots&\ddots&\vdots\\ \mathbf{a}_{k}\cdot\mathbf{a}_{1}&\cdots&\mathbf{a}_{k}\cdot\mathbf{a}_{k}&% \cdots&\mathbf{a}_{k}\cdot\mathbf{a}_{n}\\ \vdots&\ddots&\vdots&\ddots&\vdots\\ \mathbf{a}_{n}\cdot\mathbf{a}_{1}&\cdots&\mathbf{a}_{n}\cdot\mathbf{a}_{k}&% \cdots&\mathbf{a}_{n}\cdot\mathbf{a}_{n}\end{vmatrix}^{-1}\\ \displaystyle[8pt]&\displaystyle=\begin{vmatrix}\mathbf{a}_{1}\\ \vdots\\ \mathbf{a}_{k}\\ \vdots\\ \mathbf{a}_{n}\end{vmatrix}\begin{vmatrix}\mathbf{a}_{1}&\cdots&(\mathbf{c})_{% k}&\cdots&\mathbf{a}_{n}\end{vmatrix}\begin{vmatrix}\mathbf{a}_{1}\\ \vdots\\ \mathbf{a}_{k}\\ \vdots\\ \mathbf{a}_{n}\end{vmatrix}^{-1}\begin{vmatrix}\mathbf{a}_{1}&\cdots&\mathbf{a% }_{k}&\cdots&\mathbf{a}_{n}\end{vmatrix}^{-1}\\ \displaystyle[8pt]&\displaystyle=\begin{vmatrix}\mathbf{a}_{1}&\cdots&(\mathbf% {c})_{k}&\cdots&\mathbf{a}_{n}\end{vmatrix}\begin{vmatrix}\mathbf{a}_{1}&% \cdots&\mathbf{a}_{k}&\cdots&\mathbf{a}_{n}\end{vmatrix}^{-1}\\ \displaystyle[8pt]&\displaystyle=\begin{vmatrix}a_{11}&\ldots&c_{1}&\cdots&a_{% 1n}\\ \vdots&\ddots&\vdots&\ddots&\vdots\\ a_{k1}&\cdots&c_{k}&\cdots&a_{kn}\\ \vdots&\ddots&\vdots&\ddots&\vdots\\ a_{n1}&\cdots&c_{n}&\cdots&a_{nn}\end{vmatrix}\begin{vmatrix}a_{11}&\ldots&a_{% 1k}&\cdots&a_{1n}\\ \vdots&\ddots&\vdots&\ddots&\vdots\\ a_{k1}&\cdots&a_{kk}&\cdots&a_{kn}\\ \vdots&\ddots&\vdots&\ddots&\vdots\\ a_{n1}&\cdots&a_{nk}&\cdots&a_{nn}\end{vmatrix}^{-1}\end{aligned}
  126. ( 𝐜 ) < s u b > k (\mathbf{c})<sub>k

Cramér's_conjecture.html

  1. p n + 1 - p n = O ( ( log p n ) 2 ) , p_{n+1}-p_{n}=O((\log p_{n})^{2}),
  2. lim sup n p n + 1 - p n ( log p n ) 2 = 1 , \limsup_{n\rightarrow\infty}\frac{p_{n+1}-p_{n}}{(\log p_{n})^{2}}=1,
  3. p n + 1 - p n = O ( p n log p n ) p_{n+1}-p_{n}=O(\sqrt{p_{n}}\,\log p_{n})
  4. lim sup n p n + 1 - p n log p n = . \limsup_{n\to\infty}\frac{p_{n+1}-p_{n}}{\log p_{n}}=\infty.
  5. lim sup n p n + 1 - p n ( log p n ) 2 = c , \limsup_{n\rightarrow\infty}\frac{p_{n+1}-p_{n}}{(\log p_{n})^{2}}=c,
  6. c = 1. c=1.
  7. c 2 e - γ 1.1229 c\geq 2e^{-\gamma}\approx 1.1229\ldots
  8. γ \gamma
  9. G ( x ) ln ( x ) ( ln ( x ) - ln ln ( ( x ) ) ) , G(x)\sim\ln(x)(\ln(x)-\ln\ln((x))),
  10. x x
  11. G ( x ) ln 2 ( x ) G(x)\sim\ln^{2}(x)
  12. R = log ( p n ) p n + 1 - p n . R=\frac{\log(p_{n})}{\sqrt{p_{n+1}-p_{n}}}.
  13. R R
  14. 1 / R 2 1/R^{2}

Credit_default_swap.html

  1. t 0 t_{0}
  2. t 1 t_{1}
  3. t 2 t_{2}
  4. t 3 t_{3}
  5. t 4 t_{4}
  6. N N
  7. c c
  8. N c / 4 Nc/4
  9. N ( 1 - R ) N(1-R)
  10. R R
  11. N c / 4 Nc/4
  12. t i - 1 t_{i-1}
  13. t i t_{i}
  14. p i p_{i}
  15. 1 - p i 1-p_{i}
  16. δ 1 \delta_{1}
  17. δ 4 \delta_{4}
  18. t 1 t_{1}
  19. 0 0\,
  20. N ( 1 - R ) δ 1 N(1-R)\delta_{1}\,
  21. 1 - p 1 1-p_{1}\,
  22. t 2 t_{2}
  23. - N c 4 δ 1 -\frac{Nc}{4}\delta_{1}
  24. N ( 1 - R ) δ 2 N(1-R)\delta_{2}\,
  25. p 1 ( 1 - p 2 ) p_{1}(1-p_{2})\,
  26. t 3 t_{3}
  27. - N c 4 ( δ 1 + δ 2 ) -\frac{Nc}{4}(\delta_{1}+\delta_{2})
  28. N ( 1 - R ) δ 3 N(1-R)\delta_{3}\,
  29. p 1 p 2 ( 1 - p 3 ) p_{1}p_{2}(1-p_{3})\,
  30. t 4 t_{4}
  31. - N c 4 ( δ 1 + δ 2 + δ 3 ) -\frac{Nc}{4}(\delta_{1}+\delta_{2}+\delta_{3})
  32. N ( 1 - R ) δ 4 N(1-R)\delta_{4}\,
  33. p 1 p 2 p 3 ( 1 - p 4 ) p_{1}p_{2}p_{3}(1-p_{4})\,
  34. - N c 4 ( δ 1 + δ 2 + δ 3 + δ 4 ) -\frac{Nc}{4}(\delta_{1}+\delta_{2}+\delta_{3}+\delta_{4})
  35. 0 0\,
  36. p 1 × p 2 × p 3 × p 4 p_{1}\times p_{2}\times p_{3}\times p_{4}
  37. p 1 p_{1}
  38. p 2 p_{2}
  39. p 3 p_{3}
  40. p 4 p_{4}
  41. t t
  42. t + Δ t t+\Delta t
  43. p = exp ( - s ( t ) Δ t / ( 1 - R ) ) p=\exp(-s(t)\Delta t/(1-R))
  44. s ( t ) s(t)
  45. t t
  46. P V PV\,
  47. = =\,
  48. ( 1 - p 1 ) N ( 1 - R ) δ 1 (1-p_{1})N(1-R)\delta_{1}\,
  49. + p 1 ( 1 - p 2 ) [ N ( 1 - R ) δ 2 - N c 4 δ 1 ] +p_{1}(1-p_{2})[N(1-R)\delta_{2}-\frac{Nc}{4}\delta_{1}]
  50. + p 1 p 2 ( 1 - p 3 ) [ N ( 1 - R ) δ 3 - N c 4 ( δ 1 + δ 2 ) ] +p_{1}p_{2}(1-p_{3})[N(1-R)\delta_{3}-\frac{Nc}{4}(\delta_{1}+\delta_{2})]
  51. + p 1 p 2 p 3 ( 1 - p 4 ) [ N ( 1 - R ) δ 4 - N c 4 ( δ 1 + δ 2 + δ 3 ) ] +p_{1}p_{2}p_{3}(1-p_{4})[N(1-R)\delta_{4}-\frac{Nc}{4}(\delta_{1}+\delta_{2}+% \delta_{3})]
  52. - p 1 p 2 p 3 p 4 ( δ 1 + δ 2 + δ 3 + δ 4 ) N c 4 -p_{1}p_{2}p_{3}p_{4}(\delta_{1}+\delta_{2}+\delta_{3}+\delta_{4})\frac{Nc}{4}

Critical_mass.html

  1. k = 1 k=1
  2. k < 1 k<1
  3. k > 1 k>1
  4. - 1 = n σ \ell^{-1}=n\sigma
  5. \ell
  6. R c s s n σ R_{c}\simeq\ell\sqrt{s}\simeq\frac{\sqrt{s}}{n\sigma}
  7. 1 = f σ m s ρ 2 / 3 M 1 / 3 1=\frac{f\sigma}{m\sqrt{s}}\rho^{2/3}M^{1/3}
  8. 1 = f σ m s Σ 1=\frac{f^{\prime}\sigma}{m\sqrt{s}}\Sigma
  9. f f^{\prime}
  10. L L
  11. L L

Cross_elasticity_of_demand.html

  1. - 20 % 10 % = - 2 \frac{-20\%}{10\%}=-2

Cross_product.html

  1. 𝐚 × 𝐛 = 𝐚 𝐛 sin θ 𝐧 \mathbf{a}\times\mathbf{b}=\left\|\mathbf{a}\right\|\left\|\mathbf{b}\right\|% \sin\theta\ \mathbf{n}
  2. 𝐢 \displaystyle\mathbf{i}
  3. 𝐤 × 𝐣 = - 𝐢 𝐢 × 𝐤 = - 𝐣 𝐣 × 𝐢 = - 𝐤 \begin{aligned}\displaystyle\mathbf{k\times j}&\displaystyle=-\mathbf{i}\\ \displaystyle\mathbf{i\times k}&\displaystyle=-\mathbf{j}\\ \displaystyle\mathbf{j\times i}&\displaystyle=-\mathbf{k}\end{aligned}
  4. 𝐢 × 𝐢 = 𝐣 × 𝐣 = 𝐤 × 𝐤 = 𝟎 \mathbf{i}\times\mathbf{i}=\mathbf{j}\times\mathbf{j}=\mathbf{k}\times\mathbf{% k}=\mathbf{0}
  5. 𝐮 \displaystyle\mathbf{u}
  6. 𝐮 × 𝐯 = \displaystyle\mathbf{u}\times\mathbf{v}=
  7. 𝐮 × 𝐯 = \displaystyle\mathbf{u}\times\mathbf{v}=
  8. s 1 = u 2 v 3 - u 3 v 2 s 2 = u 3 v 1 - u 1 v 3 s 3 = u 1 v 2 - u 2 v 1 \begin{aligned}\displaystyle s_{1}&\displaystyle=u_{2}v_{3}-u_{3}v_{2}\\ \displaystyle s_{2}&\displaystyle=u_{3}v_{1}-u_{1}v_{3}\\ \displaystyle s_{3}&\displaystyle=u_{1}v_{2}-u_{2}v_{1}\end{aligned}
  9. ( s 1 s 2 s 3 ) = ( u 2 v 3 - u 3 v 2 u 3 v 1 - u 1 v 3 u 1 v 2 - u 2 v 1 ) \begin{pmatrix}s_{1}\\ s_{2}\\ s_{3}\end{pmatrix}=\begin{pmatrix}u_{2}v_{3}-u_{3}v_{2}\\ u_{3}v_{1}-u_{1}v_{3}\\ u_{1}v_{2}-u_{2}v_{1}\end{pmatrix}
  10. 𝐮 × 𝐯 = | 𝐢 𝐣 𝐤 u 1 u 2 u 3 v 1 v 2 v 3 | \mathbf{u\times v}=\begin{vmatrix}\mathbf{i}&\mathbf{j}&\mathbf{k}\\ u_{1}&u_{2}&u_{3}\\ v_{1}&v_{2}&v_{3}\\ \end{vmatrix}
  11. 𝐮 × 𝐯 = ( u 2 v 3 𝐢 + u 3 v 1 𝐣 + u 1 v 2 𝐤 ) - ( u 3 v 2 𝐢 + u 1 v 3 𝐣 + u 2 v 1 𝐤 ) . \mathbf{u\times v}=(u_{2}v_{3}\mathbf{i}+u_{3}v_{1}\mathbf{j}+u_{1}v_{2}% \mathbf{k})-(u_{3}v_{2}\mathbf{i}+u_{1}v_{3}\mathbf{j}+u_{2}v_{1}\mathbf{k}).
  12. 𝐮 × 𝐯 = | u 2 u 3 v 2 v 3 | 𝐢 - | u 1 u 3 v 1 v 3 | 𝐣 + | u 1 u 2 v 1 v 2 | 𝐤 \mathbf{u\times v}=\begin{vmatrix}u_{2}&u_{3}\\ v_{2}&v_{3}\end{vmatrix}\mathbf{i}-\begin{vmatrix}u_{1}&u_{3}\\ v_{1}&v_{3}\end{vmatrix}\mathbf{j}+\begin{vmatrix}u_{1}&u_{2}\\ v_{1}&v_{2}\end{vmatrix}\mathbf{k}
  13. A = 𝐚 × 𝐛 = 𝐚 𝐛 sin θ . A=\left\|\mathbf{a}\times\mathbf{b}\right\|=\left\|\mathbf{a}\right\|\left\|% \mathbf{b}\right\|\sin\theta.\,\!
  14. 𝐚 ( 𝐛 × 𝐜 ) = 𝐛 ( 𝐜 × 𝐚 ) = 𝐜 ( 𝐚 × 𝐛 ) . \mathbf{a}\cdot(\mathbf{b}\times\mathbf{c})=\mathbf{b}\cdot(\mathbf{c}\times% \mathbf{a})=\mathbf{c}\cdot(\mathbf{a}\times\mathbf{b}).
  15. V = | 𝐚 ( 𝐛 × 𝐜 ) | . V=|\mathbf{a}\cdot(\mathbf{b}\times\mathbf{c})|.
  16. 𝐚 × 𝐛 = - 𝐛 × 𝐚 , \mathbf{a}\times\mathbf{b}=-\mathbf{b}\times\mathbf{a},
  17. 𝐚 × ( 𝐛 + 𝐜 ) = ( 𝐚 × 𝐛 ) + ( 𝐚 × 𝐜 ) , \mathbf{a}\times(\mathbf{b}+\mathbf{c})=(\mathbf{a}\times\mathbf{b})+(\mathbf{% a}\times\mathbf{c}),
  18. ( r 𝐚 ) × 𝐛 = 𝐚 × ( r 𝐛 ) = r ( 𝐚 × 𝐛 ) . (r\mathbf{a})\times\mathbf{b}=\mathbf{a}\times(r\mathbf{b})=r(\mathbf{a}\times% \mathbf{b}).
  19. 𝐚 × ( 𝐛 × 𝐜 ) + 𝐛 × ( 𝐜 × 𝐚 ) + 𝐜 × ( 𝐚 × 𝐛 ) = 0. \mathbf{a}\times(\mathbf{b}\times\mathbf{c})+\mathbf{b}\times(\mathbf{c}\times% \mathbf{a})+\mathbf{c}\times(\mathbf{a}\times\mathbf{b})=\mathbf{0}.
  20. 𝟎 \displaystyle\mathbf{0}
  21. 𝐜 = 𝐛 + t 𝐚 , \mathbf{c}=\mathbf{b}+t\mathbf{a},
  22. 𝐚 × ( 𝐛 - 𝐜 ) = 𝟎 \mathbf{a}\times(\mathbf{b}-\mathbf{c})=\mathbf{0}
  23. 𝐚 ( 𝐛 - 𝐜 ) = 0 , \mathbf{a}\cdot(\mathbf{b}-\mathbf{c})=0,
  24. ( R 𝐚 ) × ( R 𝐛 ) = R ( 𝐚 × 𝐛 ) (R\mathbf{a})\times(R\mathbf{b})=R(\mathbf{a}\times\mathbf{b})
  25. R R
  26. det ( R ) = 1 \det(R)=1
  27. ( M 𝐚 ) × ( M 𝐛 ) = ( det M ) M - T ( 𝐚 × 𝐛 ) (M\mathbf{a})\times(M\mathbf{b})=(\det M)M^{-T}(\mathbf{a}\times\mathbf{b})
  28. M \scriptstyle M
  29. M - T \scriptstyle M^{-T}
  30. M \scriptstyle M
  31. 𝐚 × 𝐛 N S ( [ 𝐚 𝐛 ] ) . \mathbf{a}\times\mathbf{b}\in NS\left(\begin{bmatrix}\mathbf{a}\\ \mathbf{b}\end{bmatrix}\right).
  32. 𝐚 × 𝐛 + 𝐜 × 𝐝 = ( 𝐚 - 𝐜 ) × ( 𝐛 - 𝐝 ) + 𝐚 × 𝐝 + 𝐜 × 𝐛 . \mathbf{a}\times\mathbf{b}+\mathbf{c}\times\mathbf{d}=(\mathbf{a}-\mathbf{c})% \times(\mathbf{b}-\mathbf{d})+\mathbf{a}\times\mathbf{d}+\mathbf{c}\times% \mathbf{b}.
  33. d d t ( 𝐚 × 𝐛 ) = d 𝐚 d t × 𝐛 + 𝐚 × d 𝐛 d t \frac{d}{dt}(\mathbf{a}\times\mathbf{b})=\frac{d\mathbf{a}}{dt}\times\mathbf{b% }+\mathbf{a}\times\frac{d\mathbf{b}}{dt}
  34. 𝐚 ( 𝐛 × 𝐜 ) , \mathbf{a}\cdot(\mathbf{b}\times\mathbf{c}),
  35. 𝐚 ( 𝐛 × 𝐜 ) = 𝐛 ( 𝐜 × 𝐚 ) = 𝐜 ( 𝐚 × 𝐛 ) , \mathbf{a}\cdot(\mathbf{b}\times\mathbf{c})=\mathbf{b}\cdot(\mathbf{c}\times% \mathbf{a})=\mathbf{c}\cdot(\mathbf{a}\times\mathbf{b}),
  36. 𝐚 × ( 𝐛 × 𝐜 ) = 𝐛 ( 𝐚 𝐜 ) - 𝐜 ( 𝐚 𝐛 ) . \mathbf{a}\times(\mathbf{b}\times\mathbf{c})=\mathbf{b}(\mathbf{a}\cdot\mathbf% {c})-\mathbf{c}(\mathbf{a}\cdot\mathbf{b}).
  37. × ( × 𝐟 ) = ( 𝐟 ) - ( ) 𝐟 = ( 𝐟 ) - 2 𝐟 , \begin{aligned}\displaystyle\nabla\times(\nabla\times\mathbf{f})&\displaystyle% =\nabla(\nabla\cdot\mathbf{f})-(\nabla\cdot\nabla)\mathbf{f}\\ &\displaystyle=\nabla(\nabla\cdot\mathbf{f})-\nabla^{2}\mathbf{f},\\ \end{aligned}
  38. ( 𝐚 × 𝐛 ) × ( 𝐚 × 𝐜 ) = ( 𝐚 ( 𝐛 × 𝐜 ) ) 𝐚 (\mathbf{a}\times\mathbf{b})\times(\mathbf{a}\times\mathbf{c})=(\mathbf{a}% \cdot(\mathbf{b}\times\mathbf{c}))\mathbf{a}
  39. ( 𝐚 × 𝐛 ) ( 𝐜 × 𝐝 ) = 𝐛 T ( ( 𝐜 T 𝐚 ) I - 𝐜𝐚 T ) 𝐝 (\mathbf{a}\times\mathbf{b})\cdot(\mathbf{c}\times\mathbf{d})=\mathbf{b}^{T}((% \mathbf{c}^{T}\mathbf{a})I-\mathbf{c}\mathbf{a}^{T})\mathbf{d}
  40. 𝐚 × 𝐛 2 = 𝐚 2 𝐛 2 - ( 𝐚 𝐛 ) 2 . \left\|\mathbf{a}\times\mathbf{b}\right\|^{2}=\left\|\mathbf{a}\right\|^{2}% \left\|\mathbf{b}\right\|^{2}-(\mathbf{a}\cdot\mathbf{b})^{2}.
  41. 𝐚 𝐛 = 𝐚 𝐛 cos θ , \mathbf{a\cdot b}=\left\|\mathbf{a}\right\|\left\|\mathbf{b}\right\|\cos\theta,
  42. 𝐚 × 𝐛 2 = 𝐚 2 𝐛 2 ( 1 - cos 2 θ ) . \left\|\mathbf{a\times b}\right\|^{2}=\left\|\mathbf{a}\right\|^{2}\left\|% \mathbf{b}\right\|^{2}\left(1-\cos^{2}\theta\right).
  43. 𝐚 × 𝐛 = 𝐚 𝐛 | sin θ | , \left\|\mathbf{a}\times\mathbf{b}\right\|=\left\|\mathbf{a}\right\|\left\|% \mathbf{b}\right\|\left|\sin\theta\right|,
  44. 𝐚 × 𝐛 2 = det [ 𝐚 𝐚 𝐚 𝐛 𝐚 𝐛 𝐛 𝐛 ] = 𝐚 2 𝐛 2 - ( 𝐚 𝐛 ) 2 . \left\|\mathbf{a}\times\mathbf{b}\right\|^{2}=\det\begin{bmatrix}\mathbf{a}% \cdot\mathbf{a}&\mathbf{a}\cdot\mathbf{b}\\ \mathbf{a}\cdot\mathbf{b}&\mathbf{b}\cdot\mathbf{b}\\ \end{bmatrix}=\left\|\mathbf{a}\right\|^{2}\left\|\mathbf{b}\right\|^{2}-(% \mathbf{a}\cdot\mathbf{b})^{2}.
  45. 1 i < j n ( a i b j - a j b i ) 2 = 𝐚 2 𝐛 2 - ( 𝐚 𝐛 ) 2 , \sum_{1\leq i<j\leq n}\left(a_{i}b_{j}-a_{j}b_{i}\right)^{2}=\left\|\mathbf{a}% \right\|^{2}\left\|\mathbf{b}\right\|^{2}-(\mathbf{a\cdot b})^{2}\ ,
  46. | 𝐚 × 𝐛 | 2 = 1 i < j 3 ( a i b j - a j b i ) 2 = ( a 1 b 2 - b 1 a 2 ) 2 + ( a 2 b 3 - a 3 b 2 ) 2 + ( a 3 b 1 - a 1 b 3 ) 2 . |\mathbf{a}\times\mathbf{b}|^{2}=\sum_{1\leq i<j\leq 3}\left(a_{i}b_{j}-a_{j}b% _{i}\right)^{2}=(a_{1}b_{2}-b_{1}a_{2})^{2}+(a_{2}b_{3}-a_{3}b_{2})^{2}+(a_{3}% b_{1}-a_{1}b_{3})^{2}\ .
  47. 𝐚 × 𝐛 = det [ 𝐢 𝐣 𝐤 a 1 a 2 a 3 b 1 b 2 b 3 ] . \mathbf{a}\times\mathbf{b}=\det\begin{bmatrix}\mathbf{i}&\mathbf{j}&\mathbf{k}% \\ a_{1}&a_{2}&a_{3}\\ b_{1}&b_{2}&b_{3}\\ \end{bmatrix}.
  48. ( 𝐚 × 𝐛 ) ( 𝐜 × 𝐝 ) = ( 𝐚 𝐜 ) ( 𝐛 𝐝 ) - ( 𝐚 𝐝 ) ( 𝐛 𝐜 ) . (\mathbf{a}\times\mathbf{b})\cdot(\mathbf{c}\times\mathbf{d})=(\mathbf{a}\cdot% \mathbf{c})(\mathbf{b}\cdot\mathbf{d})-(\mathbf{a}\cdot\mathbf{d})(\mathbf{b}% \cdot\mathbf{c}).
  49. d d ϕ | ϕ = 0 R ( ϕ , s y m b o l n ) s y m b o l x = s y m b o l n × s y m b o l x \left.{d\over d\phi}\right|_{\phi=0}R(\phi,symbol{n})symbol{x}=symbol{n}\times symbol% {x}
  50. 𝐚 × 𝐛 = [ 𝐚 ] × 𝐛 = [ 0 - a 3 a 2 a 3 0 - a 1 - a 2 a 1 0 ] [ b 1 b 2 b 3 ] \mathbf{a}\times\mathbf{b}=[\mathbf{a}]_{\times}\mathbf{b}=\begin{bmatrix}\,0&% \!-a_{3}&\,\,a_{2}\\ \,\,a_{3}&0&\!-a_{1}\\ -a_{2}&\,\,a_{1}&\,0\end{bmatrix}\begin{bmatrix}b_{1}\\ b_{2}\\ b_{3}\end{bmatrix}
  51. 𝐚 × 𝐛 = [ 𝐛 ] × T 𝐚 = [ 0 b 3 - b 2 - b 3 0 b 1 b 2 - b 1 0 ] [ a 1 a 2 a 3 ] \mathbf{a}\times\mathbf{b}=[\mathbf{b}]_{\times}^{\mathrm{T}}\mathbf{a}=\begin% {bmatrix}\,0&\,\,b_{3}&\!-b_{2}\\ -b_{3}&0&\,\,b_{1}\\ \,\,b_{2}&\!-b_{1}&\,0\end{bmatrix}\begin{bmatrix}a_{1}\\ a_{2}\\ a_{3}\end{bmatrix}
  52. [ 𝐚 ] × = def [ 0 - a 3 a 2 a 3 0 - a 1 - a 2 a 1 0 ] . [\mathbf{a}]_{\times}\stackrel{\rm def}{=}\begin{bmatrix}\,\,0&\!-a_{3}&\,\,\,% a_{2}\\ \,\,\,a_{3}&0&\!-a_{1}\\ \!-a_{2}&\,\,a_{1}&\,\,0\end{bmatrix}.
  53. 𝐚 = 𝐜 × 𝐝 \mathbf{a}=\mathbf{c}\times\mathbf{d}
  54. [ 𝐚 ] × = 𝐝𝐜 T - 𝐜𝐝 T . [\mathbf{a}]_{\times}=\mathbf{d}\mathbf{c}^{\mathrm{T}}-\mathbf{c}\mathbf{d}^{% \mathrm{T}}.
  55. 𝐚 = 𝐜 × 𝐝 = ( c 2 d 3 - c 3 d 2 c 3 d 1 - c 1 d 3 c 1 d 2 - c 2 d 1 ) \mathbf{a}=\mathbf{c}\times\mathbf{d}=\begin{pmatrix}c_{2}d_{3}-c_{3}d_{2}\\ c_{3}d_{1}-c_{1}d_{3}\\ c_{1}d_{2}-c_{2}d_{1}\end{pmatrix}
  56. [ 𝐚 ] × = [ 0 c 2 d 1 - c 1 d 2 c 3 d 1 - c 1 d 3 c 1 d 2 - c 2 d 1 0 c 3 d 2 - c 2 d 3 c 1 d 3 - c 3 d 1 c 2 d 3 - c 3 d 2 0 ] [\mathbf{a}]_{\times}=\begin{bmatrix}0&c_{2}d_{1}-c_{1}d_{2}&c_{3}d_{1}-c_{1}d% _{3}\\ c_{1}d_{2}-c_{2}d_{1}&0&c_{3}d_{2}-c_{2}d_{3}\\ c_{1}d_{3}-c_{3}d_{1}&c_{2}d_{3}-c_{3}d_{2}&0\end{bmatrix}
  57. 𝐜𝐝 T = [ c 1 d 1 c 1 d 2 c 1 d 3 c 2 d 1 c 2 d 2 c 2 d 3 c 3 d 1 c 3 d 2 c 3 d 3 ] \mathbf{c}\mathbf{d}^{\mathrm{T}}=\begin{bmatrix}c_{1}d_{1}&c_{1}d_{2}&c_{1}d_% {3}\\ c_{2}d_{1}&c_{2}d_{2}&c_{2}d_{3}\\ c_{3}d_{1}&c_{3}d_{2}&c_{3}d_{3}\end{bmatrix}
  58. 𝐝𝐜 T = [ c 1 d 1 c 2 d 1 c 3 d 1 c 1 d 2 c 2 d 2 c 3 d 2 c 1 d 3 c 2 d 3 c 3 d 3 ] \mathbf{d}\mathbf{c}^{\mathrm{T}}=\begin{bmatrix}c_{1}d_{1}&c_{2}d_{1}&c_{3}d_% {1}\\ c_{1}d_{2}&c_{2}d_{2}&c_{3}d_{2}\\ c_{1}d_{3}&c_{2}d_{3}&c_{3}d_{3}\end{bmatrix}
  59. 𝐝𝐜 T - 𝐜𝐝 T = [ 0 c 2 d 1 - c 1 d 2 c 3 d 1 - c 1 d 3 c 1 d 2 - c 2 d 1 0 c 3 d 2 - c 2 d 3 c 1 d 3 - c 3 d 1 c 2 d 3 - c 3 d 2 0 ] \mathbf{d}\mathbf{c}^{\mathrm{T}}-\mathbf{c}\mathbf{d}^{\mathrm{T}}=\begin{% bmatrix}0&c_{2}d_{1}-c_{1}d_{2}&c_{3}d_{1}-c_{1}d_{3}\\ c_{1}d_{2}-c_{2}d_{1}&0&c_{3}d_{2}-c_{2}d_{3}\\ c_{1}d_{3}-c_{3}d_{1}&c_{2}d_{3}-c_{3}d_{2}&0\end{bmatrix}
  60. [ 𝐚 ] × 𝐚 = 𝟎 [\mathbf{a}]_{\times}\,\mathbf{a}=\mathbf{0}
  61. 𝐚 T [ 𝐚 ] × = 𝟎 \mathbf{a}^{\mathrm{T}}\,[\mathbf{a}]_{\times}=\mathbf{0}
  62. 𝐛 T [ 𝐚 ] × 𝐛 = 0. \mathbf{b}^{\mathrm{T}}\,[\mathbf{a}]_{\times}\,\mathbf{b}=0.
  63. 𝐚 × 𝐛 = 𝐜 c m = i = 1 3 j = 1 3 k = 1 3 η m i ε i j k a j b k \mathbf{a\times b}=\mathbf{c}\Leftrightarrow\ c^{m}=\sum_{i=1}^{3}\sum_{j=1}^{% 3}\sum_{k=1}^{3}\eta^{mi}\varepsilon_{ijk}a^{j}b^{k}
  64. i , j , k \scriptstyle i,j,k
  65. 𝐚 × 𝐛 = 𝐜 c m = η m i ε i j k a j b k \mathbf{a\times b}=\mathbf{c}\Leftrightarrow\ c^{m}=\eta^{mi}\varepsilon_{ijk}% a^{j}b^{k}
  66. η m i ε i j k a j = [ 𝐚 ] × . \eta^{mi}\varepsilon_{ijk}a^{j}=[\mathbf{a}]_{\times}.
  67. 𝐚 = 𝐛 × 𝐜 \mathbf{a}=\mathbf{b}\times\mathbf{c}
  68. 𝐚 = [ a x a y a z ] , 𝐛 = [ b x b y b z ] , 𝐜 = [ c x c y c z ] \mathbf{a}=\begin{bmatrix}a_{x}\\ a_{y}\\ a_{z}\end{bmatrix},\mathbf{b}=\begin{bmatrix}b_{x}\\ b_{y}\\ b_{z}\end{bmatrix},\mathbf{c}=\begin{bmatrix}c_{x}\\ c_{y}\\ c_{z}\end{bmatrix}
  69. a x = b y c z - b z c y a_{x}=b_{y}c_{z}-b_{z}c_{y}\,
  70. a y = b z c x - b x c z a_{y}=b_{z}c_{x}-b_{x}c_{z}\,
  71. a z = b x c y - b y c x . a_{z}=b_{x}c_{y}-b_{y}c_{x}.\,
  72. 𝐚 = 𝐛 × 𝐜 \mathbf{a}=\mathbf{b}\times\mathbf{c}
  73. 𝐚 = [ b x b y b z ] × [ c x c y c z ] . \mathbf{a}=\begin{bmatrix}b_{x}\\ b_{y}\\ b_{z}\end{bmatrix}\times\begin{bmatrix}c_{x}\\ c_{y}\\ c_{z}\end{bmatrix}.
  74. a x a_{x}
  75. b x b_{x}
  76. c x c_{x}
  77. a x = [ b y b z ] × [ c y c z ] . a_{x}=\begin{bmatrix}b_{y}\\ b_{z}\end{bmatrix}\times\begin{bmatrix}c_{y}\\ c_{z}\end{bmatrix}.
  78. a y a_{y}
  79. a y a_{y}
  80. a z a_{z}
  81. a y = [ b z b x ] × [ c z c x ] , a z = [ b x b y ] × [ c x c y ] a_{y}=\begin{bmatrix}b_{z}\\ b_{x}\end{bmatrix}\times\begin{bmatrix}c_{z}\\ c_{x}\end{bmatrix},a_{z}=\begin{bmatrix}b_{x}\\ b_{y}\end{bmatrix}\times\begin{bmatrix}c_{x}\\ c_{y}\end{bmatrix}
  82. a x a_{x}
  83. a x a_{x}
  84. a x = b y c z - b z c y . a_{x}=b_{y}c_{z}-b_{z}c_{y}.\,
  85. a y a_{y}
  86. a z a_{z}
  87. p 1 = ( x 1 , y 1 ) \scriptstyle p_{1}=(x_{1},y_{1})
  88. p 2 = ( x 2 , y 2 ) \scriptstyle p_{2}=(x_{2},y_{2})
  89. p 3 = ( x 3 , y 3 ) \scriptstyle p_{3}=(x_{3},y_{3})
  90. p 1 , p 2 \scriptstyle p_{1},p_{2}
  91. p 1 , p 3 \scriptstyle p_{1},p_{3}
  92. P = ( x 2 - x 1 ) ( y 3 - y 1 ) - ( y 2 - y 1 ) ( x 3 - x 1 ) \scriptstyle P=(x_{2}-x_{1})(y_{3}-y_{1})-(y_{2}-y_{1})(x_{3}-x_{1})
  93. p 1 \scriptstyle p_{1}
  94. p 2 \scriptstyle p_{2}
  95. p 3 \scriptstyle p_{3}
  96. P \scriptstyle P
  97. p 3 \scriptstyle p_{3}
  98. p 1 , p 2 \scriptstyle p_{1},p_{2}
  99. 𝐋 \scriptstyle\mathbf{L}
  100. 𝐋 = 𝐫 × 𝐩 \mathbf{L}=\mathbf{r}\times\mathbf{p}\,
  101. 𝐫 \scriptstyle\mathbf{r}
  102. 𝐩 \scriptstyle\mathbf{p}
  103. 𝐌 \scriptstyle\mathbf{M}
  104. 𝐅 B \scriptstyle\mathbf{F}_{\mathrm{B}}
  105. 𝐌 A = 𝐫 AB × 𝐅 B \mathbf{M}_{\mathrm{A}}=\mathbf{r}_{\mathrm{AB}}\times\mathbf{F}_{\mathrm{B}}\,
  106. τ \scriptstyle\mathbf{\tau}
  107. 𝐫 \scriptstyle\mathbf{r}
  108. 𝐩 \scriptstyle\mathbf{p}
  109. 𝐅 \scriptstyle\mathbf{F}
  110. 𝐋 \scriptstyle\mathbf{L}
  111. 𝐌 \scriptstyle\mathbf{M}
  112. 𝐯 P - 𝐯 Q = ω × ( 𝐫 P - 𝐫 Q ) \mathbf{v}_{P}-\mathbf{v}_{Q}=\mathbf{\omega}\times\left(\mathbf{r}_{P}-% \mathbf{r}_{Q}\right)\,
  113. 𝐫 \scriptstyle\mathbf{r}
  114. 𝐯 \scriptstyle\mathbf{v}
  115. ω \scriptstyle\mathbf{\omega}
  116. 𝐫 \scriptstyle\mathbf{r}
  117. 𝐯 \scriptstyle\mathbf{v}
  118. ω \scriptstyle\mathbf{\omega}
  119. q e q_{e}
  120. 𝐅 = q e ( 𝐄 + 𝐯 × 𝐁 ) \mathbf{F}=q_{e}\,\left(\mathbf{E}+\mathbf{v}\times\mathbf{B}\right)
  121. 𝐯 \scriptstyle\mathbf{v}
  122. 𝐅 \scriptstyle\mathbf{F}
  123. 𝐄 \scriptstyle\mathbf{E}
  124. 𝐁 \scriptstyle\mathbf{B}
  125. a × b = * ( a b ) . a\times b=*(a\wedge b)\,.
  126. 𝐑 3 . \scriptstyle\mathbf{R}^{3}.
  127. { x , y , z } , \scriptstyle\{x,y,z\},
  128. [ x , y ] = z , [ x , z ] = [ y , z ] = 0. \scriptstyle[x,y]=z,[x,z]=[y,z]=0.
  129. V × V × V 𝐑 , \scriptstyle V\times V\times V\to\mathbf{R},
  130. V × V V * , \scriptstyle V\times V\to V^{*},
  131. V 𝐑 \scriptstyle V\to\mathbf{R}
  132. V V * , \scriptstyle V\to V^{*},
  133. V × V V , \scriptstyle V\times V\to V,
  134. ( a , b , - ) \scriptstyle(a,b,-)
  135. V 𝐑 \scriptstyle V\to\mathbf{R}
  136. a × b . \scriptstyle a\times b.
  137. Vol ( a , b , c ) = ( a × b ) c . \scriptstyle\mathrm{Vol}(a,b,c)=(a\times b)\cdot c.
  138. ( 0 , n ) \scriptstyle(0,n)
  139. ( 1 , n - 1 ) \scriptstyle(1,n-1)
  140. n - 1 \scriptstyle n-1
  141. ( n - 1 ) \scriptstyle(n-1)
  142. ( n - 2 , 2 ) \scriptstyle(n-2,2)
  143. ( k , n - k ) \scriptstyle(k,n-k)
  144. ( n - 1 ) \scriptstyle(n-1)
  145. n - 1 \scriptstyle n-1
  146. v 1 , , v n - 1 \scriptstyle v_{1},\dots,v_{n-1}
  147. 𝐑 n , \scriptstyle\mathbf{R}^{n},
  148. v n = v 1 × × v n - 1 \scriptstyle v_{n}=v_{1}\times\cdots\times v_{n-1}
  149. v i , \scriptstyle v_{i},
  150. v i , \scriptstyle v_{i},
  151. v i , \scriptstyle v_{i},
  152. v 1 , , v n \scriptstyle v_{1},\dots,v_{n}
  153. e 1 × × e n - 1 = e n \scriptstyle e_{1}\times\cdots\times e_{n-1}=e_{n}
  154. e 2 × × e n = e 1 , \scriptstyle e_{2}\times\cdots\times e_{n}=e_{1},
  155. ( n - 1 ) \scriptstyle(n-1)
  156. ( 𝐯 1 , , 𝐯 n - 1 ) = | v 1 1 v 1 n v n - 1 1 v n - 1 n 𝐞 1 𝐞 n | . \bigwedge(\mathbf{v}_{1},\dots,\mathbf{v}_{n-1})=\begin{vmatrix}v_{1}{}^{1}&% \cdots&v_{1}{}^{n}\\ \vdots&\ddots&\vdots\\ v_{n-1}{}^{1}&\cdots&v_{n-1}{}^{n}\\ \mathbf{e}_{1}&\cdots&\mathbf{e}_{n}\end{vmatrix}.
  157. ( n - 1 ) \scriptstyle(n-1)
  158. ω \omega
  159. Ω \Omega
  160. 𝐯 P - 𝐯 Q = Ω ( 𝐫 P - 𝐫 Q ) \mathbf{v}_{P}-\mathbf{v}_{Q}={\Omega}\cdot\left(\mathbf{r}_{P}-\mathbf{r}_{Q}% \right)\,
  161. R N × N R^{N\times N}
  162. Ω d R d t R T \Omega\triangleq\frac{dR}{dt}R^{\mathrm{T}}
  163. Ω = [ ω ] × = [ 0 - ω 3 ω 2 ω 3 0 - ω 1 - ω 2 ω 1 0 ] \Omega=[\omega]_{\times}=\begin{bmatrix}\,\,0&\!-\omega_{3}&\,\,\,\omega_{2}\\ \,\,\,\omega_{3}&0&\!-\omega_{1}\\ \!-\omega_{2}&\,\,\omega_{1}&\,\,0\end{bmatrix}
  164. L L
  165. 𝐱 \mathbf{x}
  166. 𝐩 \mathbf{p}
  167. L i j = x i p j - p i x j L_{ij}=x_{i}p_{j}-p_{i}x_{j}
  168. 𝐱 \mathbf{x}
  169. 𝐩 \mathbf{p}
  170. N N
  171. V × × V 𝐑 . \scriptstyle V\times\cdots\times V\to\mathbf{R}.
  172. 𝐑 n , \scriptstyle\mathbf{R}^{n},
  173. S L \scriptstyle SL

Crystal_optics.html

  1. 𝐃 = ε 0 𝐄 + 𝐏 \mathbf{D}=\varepsilon_{0}\mathbf{E}+\mathbf{P}
  2. 𝐏 = χ ε 0 𝐄 \mathbf{P}=\chi\varepsilon_{0}\mathbf{E}
  3. 𝐃 = ε 0 𝐄 + χ ε 0 𝐄 = ε 0 ( 1 + χ ) 𝐄 = ε 𝐄 \mathbf{D}=\varepsilon_{0}\mathbf{E}+\chi\varepsilon_{0}\mathbf{E}=\varepsilon% _{0}(1+\chi)\mathbf{E}=\varepsilon\mathbf{E}
  4. ε = ε 0 ( 1 + χ ) \varepsilon=\varepsilon_{0}(1+\chi)
  5. n = 1 + χ n=\sqrt{1+\chi}
  6. 𝐏 = ε 0 s y m b o l χ 𝐄 . \mathbf{P}=\varepsilon_{0}symbol{\chi}\mathbf{E}.
  7. ( P x P y P z ) = ε 0 ( χ x x χ x y χ x z χ y x χ y y χ y z χ z x χ z y χ z z ) ( E x E y E z ) \begin{pmatrix}P_{x}\\ P_{y}\\ P_{z}\end{pmatrix}=\varepsilon_{0}\begin{pmatrix}\chi_{xx}&\chi_{xy}&\chi_{xz}% \\ \chi_{yx}&\chi_{yy}&\chi_{yz}\\ \chi_{zx}&\chi_{zy}&\chi_{zz}\end{pmatrix}\begin{pmatrix}E_{x}\\ E_{y}\\ E_{z}\end{pmatrix}
  8. P i = ε 0 j { x , y , z } χ i j E j . P_{i}=\varepsilon_{0}\sum_{j\in\{x,y,z\}}\chi_{ij}E_{j}\quad.
  9. P x = ε 0 χ x x E x P_{x}=\varepsilon_{0}\chi_{xx}E_{x}
  10. P y = ε 0 χ y y E y P_{y}=\varepsilon_{0}\chi_{yy}E_{y}
  11. P z = ε 0 χ z z E z P_{z}=\varepsilon_{0}\chi_{zz}E_{z}
  12. 𝐃 = ε 0 𝐄 + 𝐏 = ε 0 𝐄 + ε 0 s y m b o l χ 𝐄 = ε 0 ( I + s y m b o l χ ) 𝐄 = ε 0 s y m b o l ε 𝐄 . \mathbf{D}=\varepsilon_{0}\mathbf{E}+\mathbf{P}=\varepsilon_{0}\mathbf{E}+% \varepsilon_{0}symbol{\chi}\mathbf{E}=\varepsilon_{0}(I+symbol{\chi})\mathbf{E% }=\varepsilon_{0}symbol{\varepsilon}\mathbf{E}.
  13. n x x = ( 1 + χ x x ) 1 / 2 = ( ε x x ) 1 / 2 . n_{xx}=(1+\chi_{xx})^{1/2}=(\varepsilon_{xx})^{1/2}.
  14. n y y = ( 1 + χ y y ) 1 / 2 = ( ε y y ) 1 / 2 . n_{yy}=(1+\chi_{yy})^{1/2}=(\varepsilon_{yy})^{1/2}.

Crystal_radio.html

  1. f = 1 2 π L C f=\frac{1}{2\pi\sqrt{LC}}\,
  2. P = U 2 / R P=U^{2}/R

Cube_root.html

  1. 8 3 , \sqrt[3]{8},
  2. - 1 + 3 i -1+\sqrt{3}i
  3. - 1 - 3 i . -1-\sqrt{3}i.
  4. 3 i , 3 3 2 - 3 2 i , and - 3 3 2 - 3 2 i . 3i,\quad\frac{3\sqrt{3}}{2}-\frac{3}{2}i,\quad\,\text{and}\quad-\frac{3\sqrt{3% }}{2}-\frac{3}{2}i.
  5. 3 . \sqrt[3]{}.
  6. 8 3 8^{3}
  7. 8 , - 4 + 4 i 3 , and - 4 - 4 i 3 . 8,\,-4+4i\sqrt{3},\,\,\text{and}-4-4i\sqrt{3}.
  8. y 3 = x . y^{3}=x.
  9. 1 3 = { 1 - 1 2 + 3 2 i - 1 2 - 3 2 i . \sqrt[3]{1}=\begin{cases}\ \ 1\\ -\frac{1}{2}+\frac{\sqrt{3}}{2}i\\ -\frac{1}{2}-\frac{\sqrt{3}}{2}i.\end{cases}
  10. x 1 / 3 = exp ( 1 3 ln x ) x^{1/3}=\exp(\tfrac{1}{3}\ln{x})
  11. x = r exp ( i θ ) x=r\exp(i\theta)\,
  12. - π < θ π -\pi<\theta\leq\pi
  13. x 3 = r 3 exp ( 1 3 i θ ) . \sqrt[3]{x}=\sqrt[3]{r}\exp(\tfrac{1}{3}i\theta).
  14. - 8 3 \sqrt[3]{-8}
  15. - 2 -2
  16. 1 + i 3 . 1+i\sqrt{3}.
  17. x = { r exp ( i ( θ ) ) , r exp ( i ( θ + 2 π ) ) , r exp ( i ( θ - 2 π ) ) . x=\begin{cases}r\exp\bigl(i(\theta)\bigr),\\ r\exp\bigl(i(\theta+2\pi)\bigr),\\ r\exp\bigl(i(\theta-2\pi)\bigr).\end{cases}
  18. x 3 = { r 3 exp ( i ( 1 3 θ ) ) , r 3 exp ( i ( 1 3 θ + 2 3 π ) ) , r 3 exp ( i ( 1 3 θ - 2 3 π ) ) . \sqrt[3]{x}=\begin{cases}\sqrt[3]{r}\exp\bigl(i(\tfrac{1}{3}\theta)\bigr),\\ \sqrt[3]{r}\exp\bigl(i(\tfrac{1}{3}\theta+\tfrac{2}{3}\pi)\bigr),\\ \sqrt[3]{r}\exp\bigl(i(\tfrac{1}{3}\theta-\tfrac{2}{3}\pi)\bigr).\end{cases}
  19. - 8 3 \sqrt[3]{-8}
  20. a a
  21. x n + 1 = 1 3 ( a x n 2 + 2 x n ) . x_{n+1}=\frac{1}{3}\left(\frac{a}{x_{n}^{2}}+2x_{n}\right).
  22. x n × x n × a x n 2 = a x_{n}\times x_{n}\times\frac{a}{x_{n}^{2}}=a
  23. x n + 1 = x n ( x n 3 + 2 a 2 x n 3 + a ) . x_{n+1}=x_{n}\left(\frac{x_{n}^{3}+2a}{2x_{n}^{3}+a}\right).
  24. x 0 x_{0}
  25. z 3 = x 3 + y 3 = x + y 3 x 2 + 2 y 2 x + 4 y 9 x 2 + 5 y 2 x + 7 y 15 x 2 + 8 y 2 x + \sqrt[3]{z}=\sqrt[3]{x^{3}+y}=x+\cfrac{y}{3x^{2}+\cfrac{2y}{2x+\cfrac{4y}{9x^{% 2}+\cfrac{5y}{2x+\cfrac{7y}{15x^{2}+\cfrac{8y}{2x+\ddots}}}}}}
  26. = x + 2 x y 3 ( 2 z - y ) - y - 2 4 y 2 9 ( 2 z - y ) - 5 7 y 2 15 ( 2 z - y ) - 8 10 y 2 21 ( 2 z - y ) - . =x+\cfrac{2x\cdot y}{3(2z-y)-y-\cfrac{2\cdot 4y^{2}}{9(2z-y)-\cfrac{5\cdot 7y^% {2}}{15(2z-y)-\cfrac{8\cdot 10y^{2}}{21(2z-y)-\ddots}}}}.

Cubic_centimetre.html

  1. 1 1 , 000 , 000 \frac{1}{1,000,000}
  2. 1 1 , 000 \frac{1}{1,000}
  3. d = π 4 × b 2 × s × n d={\pi\over 4}\times b^{2}\times s\times n

Cubic_function.html

  1. f ( x ) = a x 3 + b x 2 + c x + d , f(x)=ax^{3}+bx^{2}+cx+d,\,
  2. a x 3 + b x 2 + c x + d = 0. ax^{3}+bx^{2}+cx+d=0.\,
  3. x 3 + p x 2 + q x = N x^{3}+px^{2}+qx=N
  4. p , q 0 p,q\neq 0
  5. q = 0 q=0
  6. x 3 + 12 x = 6 x 2 + 35 x^{3}+12x=6x^{2}+35\,
  7. x = - b ± b 2 - 3 a c 3 a . x=\frac{-b\pm\sqrt{b^{2}-3ac}}{3a}.
  8. a x 3 + b x 2 + c x + d = 0 ( 1 ) ax^{3}+bx^{2}+cx+d=0\qquad(1)
  9. a 0 . a\neq 0\,.
  10. Δ = 18 a b c d - 4 b 3 d + b 2 c 2 - 4 a c 3 - 27 a 2 d 2 . \Delta=18abcd-4b^{3}d+b^{2}c^{2}-4ac^{3}-27a^{2}d^{2}.\,
  11. a x 3 + b x 2 + c x + d = 0 ax^{3}+bx^{2}+cx+d=0
  12. x k = - 1 3 a ( b + u k C + Δ 0 u k C ) , k { 1 , 2 , 3 } x_{k}=-\frac{1}{3a}\left(b\ +\ u_{k}C\ +\ \frac{\Delta_{0}}{u_{k}C}\right)\ ,% \qquad k\in\{1,2,3\}
  13. u 1 = 1 , u 2 = - 1 + i 3 2 , u 3 = - 1 - i 3 2 u_{1}=1\ ,\qquad u_{2}={-1+i\sqrt{3}\over 2}\ ,\qquad u_{3}={-1-i\sqrt{3}\over 2}
  14. C = Δ 1 + Δ 1 2 - 4 Δ 0 3 2 3 \color w h i t e . C=\sqrt[3]{\frac{\Delta_{1}+\sqrt{\Delta_{1}^{2}-4\Delta_{0}^{3}}}{2}}\qquad% \qquad{\color{white}.}
  15. Δ 0 = b 2 - 3 a c Δ 1 = 2 b 3 - 9 a b c + 27 a 2 d \begin{aligned}\displaystyle\Delta_{0}&\displaystyle=b^{2}-3ac\\ \displaystyle\Delta_{1}&\displaystyle=2b^{3}-9abc+27a^{2}d\end{aligned}
  16. Δ 1 2 - 4 Δ 0 3 = - 27 a 2 Δ , \Delta_{1}^{2}-4\Delta_{0}^{3}=-27\,a^{2}\,\Delta\ ,
  17. Δ \Delta
  18. \sqrt{~{}~{}}
  19. 3 \sqrt[3]{~{}~{}}
  20. x 2 x_{2}
  21. x 3 x_{3}
  22. x k = - 1 3 a ( b + u k C + u ¯ k C ¯ ) x_{k}=-\frac{1}{3a}\left(b\ +\ u_{k}C\ +\ \bar{u}_{k}\bar{C}\right)
  23. C ¯ = Δ 1 - Δ 1 2 - 4 Δ 0 3 2 3 \bar{C}=\sqrt[3]{\frac{\Delta_{1}-\sqrt{\Delta_{1}^{2}-4\Delta_{0}^{3}}}{2}}
  24. u ¯ k \bar{u}_{k}
  25. u k u_{k}
  26. C C ¯ = Δ 0 C\bar{C}=\Delta_{0}
  27. Δ 1 2 - 4 Δ 0 3 \Delta_{1}^{2}-4\Delta_{0}^{3}
  28. Δ 0 \Delta\neq 0
  29. Δ 0 = 0 , \Delta_{0}=0,
  30. Δ 1 2 - 4 Δ 0 3 = Δ 1 2 \sqrt{\Delta_{1}^{2}-4\Delta_{0}^{3}}=\sqrt{\Delta_{1}^{2}}
  31. C 0 , C\neq 0,
  32. Δ 1 2 = Δ 1 , \sqrt{\Delta_{1}^{2}}=\Delta_{1},
  33. Δ 1 . \Delta_{1}.
  34. Δ = 0 \Delta=0
  35. Δ 0 = 0 , \ \Delta_{0}=0,
  36. x 1 = x 2 = x 3 = - b 3 a . x_{1}=x_{2}=x_{3}=-\frac{b}{3a}.
  37. Δ = 0 \Delta=0
  38. Δ 0 0 , \Delta_{0}\neq 0,
  39. x 1 = x 2 = 9 a d - b c 2 Δ 0 , x_{1}=x_{2}=\frac{9ad-bc}{2\Delta_{0}},
  40. x 3 = 4 a b c - 9 a 2 d - b 3 a Δ 0 . x_{3}=\frac{4abc-9a^{2}d-b^{3}}{a\Delta_{0}}.
  41. a a
  42. t - b 3 a t-\frac{b}{3a}
  43. x x
  44. t 3 + p t + q = 0 ( 2 ) t^{3}+pt+q=0\qquad(2)
  45. p = 3 a c - b 2 3 a 2 q = 2 b 3 - 9 a b c + 27 a 2 d 27 a 3 . \begin{aligned}\displaystyle p=&\displaystyle\frac{3ac-b^{2}}{3a^{2}}\\ \displaystyle q=&\displaystyle\frac{2b^{3}-9abc+27a^{2}d}{27a^{3}}.\end{aligned}
  46. p p
  47. q q
  48. x = t - b 3 a x=t-\frac{b}{3a}
  49. t 3 + p t + q = 0 . ( 2 ) t^{3}+pt+q=0\,.\qquad(2)
  50. u + v = t u+v=t\,
  51. u 3 + v 3 + ( 3 u v + p ) ( u + v ) + q = 0 ( 3 ) u^{3}+v^{3}+(3uv+p)(u+v)+q=0\qquad(3)\,
  52. 3 u v + p = 0 3uv+p=0\,
  53. u 3 + v 3 = - q u^{3}+v^{3}=-q
  54. u 3 v 3 = - p 3 / 27 u^{3}v^{3}=-p^{3}/27
  55. u 3 u^{3}
  56. v 3 v^{3}
  57. z 2 + q z - p 3 27 = 0 . z^{2}+qz-{p^{3}\over 27}=0\,.
  58. q 2 4 + p 3 27 > 0 . \frac{q^{2}}{4}+\frac{p^{3}}{27}>0\,.
  59. u u
  60. v v
  61. u 3 = - q 2 + q 2 4 + p 3 27 u^{3}=-{q\over 2}+\sqrt{{q^{2}\over 4}+{p^{3}\over 27}}
  62. v 3 = - q 2 - q 2 4 + p 3 27 v^{3}=-{q\over 2}-\sqrt{{q^{2}\over 4}+{p^{3}\over 27}}
  63. t 1 = u + v = - q 2 + q 2 4 + p 3 27 3 + - q 2 - q 2 4 + p 3 27 . 3 t_{1}=u+v=\sqrt[3]{-{q\over 2}+\sqrt{{q^{2}\over 4}+{p^{3}\over 27}}}+\sqrt[3]% {-{q\over 2}-\sqrt{{q^{2}\over 4}+{p^{3}\over 27}}.}
  64. q 2 4 + p 3 27 > 0 , \frac{q^{2}}{4}+\frac{p^{3}}{27}>0\,,
  65. u v uv
  66. - 1 2 + i 3 2 \,\tfrac{-1}{2}+i\tfrac{\sqrt{3}}{2}\,
  67. - 1 2 - i 3 2 \,\tfrac{-1}{2}-i\tfrac{\sqrt{3}}{2}\,
  68. q 2 4 + p 3 27 \frac{q^{2}}{4}+\frac{p^{3}}{27}\,
  69. u 3 u^{3}
  70. v 3 v^{3}
  71. v = - p 3 u v=-\frac{p}{3u}
  72. u = - q 2 - q 2 4 + p 3 27 3 ( 4 ) u=\sqrt[3]{-{q\over 2}-\sqrt{{q^{2}\over 4}+{p^{3}\over 27}}}\qquad(4)
  73. t = u - p 3 u . t=u-\frac{p}{3u}\,.
  74. t t
  75. u u
  76. v v
  77. u 0 u\neq 0
  78. p = 0 p=0
  79. q 0 q\neq 0
  80. t t
  81. p = q = 0 p=q=0
  82. t = 0 t=0
  83. p = q = 0 p=q=0
  84. t = 0. t=0.\,
  85. p = 0 p=0
  86. q 0 q\neq 0
  87. u = - q 3 and v = 0 u=-\sqrt[3]{q}\,\text{ and }v=0
  88. - q -q
  89. p 0 p\neq 0
  90. q = 0 q=0
  91. u = < m t p l > p 3 and v = - p 3 , u=\sqrt{<}mtpl>{{p\over 3}}\qquad\,\text{and}\qquad v=-\sqrt{{p\over 3}},
  92. t = u + v = 0 , t = ω 1 u - p 3 ω 1 u = - p , t = u ω 1 - ω 1 p 3 u = - - p , t=u+v=0,\qquad t=\omega_{1}u-{p\over 3\omega_{1}u}=\sqrt{-p},\qquad t={u\over% \omega_{1}}-{\omega_{1}p\over 3u}=-\sqrt{-p},
  93. ω 1 = e i 2 π 3 = - 1 2 + 3 2 i . \omega_{1}=e^{i\frac{2\pi}{3}}=-\tfrac{1}{2}+\tfrac{\sqrt{3}}{2}i.
  94. 4 p 3 + 27 q 2 = 0 and p 0 4p^{3}+27q^{2}=0\,\text{ and }p\neq 0
  95. p and q p\,\text{ and }q
  96. t 1 = t 2 = - 3 q 2 p and t 3 = 3 q p . t_{1}=t_{2}=-\frac{3q}{2p}\quad\,\text{and}\quad t_{3}=\frac{3q}{p}\,.
  97. t t
  98. x x
  99. b 3 a \frac{b}{3a}
  100. p p
  101. q q
  102. a , b , c , d a,b,c,d
  103. t 3 + p t + q = 0 , t^{3}+pt+q=0,
  104. t = w - p 3 w t=w-\frac{p}{3w}
  105. w 3 + q - p 3 27 w 3 = 0. w^{3}+q-\frac{p^{3}}{27w^{3}}=0.
  106. w 6 + q w 3 - p 3 27 = 0 w^{6}+qw^{3}-\frac{p^{3}}{27}=0
  107. t 1 = w 1 - p 3 w 1 , t 2 = w 2 - p 3 w 2 and t 3 = w 3 - p 3 w 3 . t_{1}=w_{1}-\frac{p}{3w_{1}},\quad t_{2}=w_{2}-\frac{p}{3w_{2}}\quad\,\text{% and}\quad t_{3}=w_{3}-\frac{p}{3w_{3}}.
  108. ζ = - 1 2 + 3 2 i \zeta=-\tfrac{1}{2}+\tfrac{\sqrt{3}}{2}i
  109. ζ 2 + ζ + 1 = 0 \zeta^{2}+\zeta+1=0
  110. s 0 = x 0 + x 1 + x 2 , s_{0}=x_{0}+x_{1}+x_{2},\,
  111. s 1 = x 0 + ζ x 1 + ζ 2 x 2 , s_{1}=x_{0}+\zeta x_{1}+\zeta^{2}x_{2},\,
  112. s 2 = x 0 + ζ 2 x 1 + ζ x 2 . s_{2}=x_{0}+\zeta^{2}x_{1}+\zeta x_{2}.\,
  113. x 0 = 1 3 ( s 0 + s 1 + s 2 ) , x_{0}=\tfrac{1}{3}(s_{0}+s_{1}+s_{2}),\,
  114. x 1 = 1 3 ( s 0 + ζ 2 s 1 + ζ s 2 ) , x_{1}=\tfrac{1}{3}(s_{0}+\zeta^{2}s_{1}+\zeta s_{2}),\,
  115. x 2 = 1 3 ( s 0 + ζ s 1 + ζ 2 s 2 ) . x_{2}=\tfrac{1}{3}(s_{0}+\zeta s_{1}+\zeta^{2}s_{2}).\,
  116. s 0 s_{0}
  117. - b / a -b/a
  118. s 1 s_{1}
  119. s 2 s_{2}
  120. s 0 s_{0}
  121. s 1 s_{1}
  122. ζ s 1 \zeta s_{1}
  123. s 2 s_{2}
  124. ζ 2 s 2 \zeta^{2}s_{2}
  125. s 1 s_{1}
  126. ζ 2 s 1 \zeta^{2}s_{1}
  127. s 2 s_{2}
  128. ζ s 2 \zeta s_{2}
  129. x 1 x_{1}
  130. x 2 x_{2}
  131. s 1 s_{1}
  132. s 2 s_{2}
  133. ζ . \zeta.
  134. s 1 3 s_{1}^{3}
  135. s 2 3 s_{2}^{3}
  136. s 1 s 2 s_{1}s_{2}
  137. ζ 3 = 1 \zeta^{3}=1
  138. s 1 s 2 s_{1}s_{2}
  139. s 1 3 + s 2 3 s_{1}^{3}+s_{2}^{3}
  140. x 1 x_{1}
  141. x 2 x_{2}
  142. s 1 s_{1}
  143. s 2 s_{2}
  144. S 3 S_{3}
  145. s 1 3 + s 2 3 s_{1}^{3}+s_{2}^{3}
  146. s 1 s 2 s_{1}s_{2}
  147. s 1 3 + s 2 3 = A s_{1}^{3}+s_{2}^{3}=A
  148. s 1 s 2 = B s_{1}s_{2}=B
  149. s 1 3 s_{1}^{3}
  150. s 2 3 s_{2}^{3}
  151. z 2 - A z + B 3 = 0 . z^{2}-Az+B^{3}=0\,.
  152. s 1 s_{1}
  153. s 2 s_{2}
  154. u u
  155. v v
  156. E 1 = x 0 + x 1 + x 2 E_{1}=x_{0}+x_{1}+x_{2}
  157. E 2 = x 0 x 1 + x 1 x 2 + x 2 x 0 E_{2}=x_{0}x_{1}+x_{1}x_{2}+x_{2}x_{0}
  158. E 3 = x 0 x 1 x 2 E_{3}=x_{0}x_{1}x_{2}
  159. ζ 3 = 1 \zeta^{3}=1
  160. s 1 3 = x 0 3 + x 1 3 + x 2 3 + 3 ζ ( x 0 2 x 1 + x 1 2 x 2 + x 2 2 x 0 ) + 3 ζ 2 ( x 0 x 1 2 + x 1 x 2 2 + x 2 x 0 2 ) + 6 x 0 x 1 x 2 . s_{1}^{3}=x_{0}^{3}+x_{1}^{3}+x_{2}^{3}+3\zeta(x_{0}^{2}x_{1}+x_{1}^{2}x_{2}+x% _{2}^{2}x_{0})+3\zeta^{2}(x_{0}x_{1}^{2}+x_{1}x_{2}^{2}+x_{2}x_{0}^{2})+6x_{0}% x_{1}x_{2}\,.
  161. s 2 3 s_{2}^{3}
  162. ζ \zeta
  163. ζ 2 \zeta^{2}
  164. ζ 2 + ζ = - 1 \zeta^{2}+\zeta=-1
  165. A = s 1 3 + s 2 3 = 2 ( x 0 3 + x 1 3 + x 2 3 ) - 3 ( x 0 2 x 1 + x 1 2 x 2 + x 2 2 x 0 + x 0 x 1 2 + x 1 x 2 2 + x 2 x 0 2 ) + 12 x 0 x 1 x 2 , A=s_{1}^{3}+s_{2}^{3}=2(x_{0}^{3}+x_{1}^{3}+x_{2}^{3})-3(x_{0}^{2}x_{1}+x_{1}^% {2}x_{2}+x_{2}^{2}x_{0}+x_{0}x_{1}^{2}+x_{1}x_{2}^{2}+x_{2}x_{0}^{2})+12x_{0}x% _{1}x_{2}\,,
  166. A = s 1 3 + s 2 3 = 2 E 1 3 - 9 E 1 E 2 + 27 E 3 . A=s_{1}^{3}+s_{2}^{3}=2E_{1}^{3}-9E_{1}E_{2}+27E_{3}\,.
  167. B = s 1 s 2 = x 0 2 + x 1 2 + x 2 2 + ( ζ + ζ 2 ) ( x 0 x 1 + x 1 x 2 + x 2 x 0 ) = E 1 2 - 3 E 2 . B=s_{1}s_{2}=x_{0}^{2}+x_{1}^{2}+x_{2}^{2}+(\zeta+\zeta^{2})(x_{0}x_{1}+x_{1}x% _{2}+x_{2}x_{0})=E_{1}^{2}-3E_{2}\,.
  168. E 1 = - b / a E_{1}=-b/a
  169. E 2 = c / a E_{2}=c/a
  170. E 3 = - d / a E_{3}=-d/a
  171. E 1 = 0 E_{1}=0
  172. E 2 = p E_{2}=p
  173. E 3 = - q E_{3}=-q
  174. A = - 27 q A=-27q
  175. B = - 3 p B=-3p
  176. x 0 = 1 3 ( s 1 + s 2 ) x_{0}=\tfrac{1}{3}(s_{1}+s_{2})
  177. s 1 s 2 = - 3 p s_{1}s_{2}=-3p
  178. x 0 = u + v x_{0}=u+v
  179. u v = - 1 3 p . uv=-\frac{1}{3}p\,.
  180. u u
  181. v v
  182. s 1 = 3 u s_{1}=3u
  183. s 2 = 3 v s_{2}=3v
  184. t 3 + p t + q = 0 t^{3}+pt+q=0
  185. t = u cos θ . t=u\cos\theta\,.
  186. u u
  187. 4 cos 3 θ - 3 cos θ - cos ( 3 θ ) = 0 . 4\cos^{3}\theta-3\cos\theta-\cos(3\theta)=0\,.
  188. u = 2 - p 3 u=2\sqrt{-\frac{p}{3}}
  189. u 3 4 \frac{u^{3}}{4}
  190. 4 cos 3 θ - 3 cos θ - 3 q 2 p - 3 p = 0 . 4\cos^{3}\theta-3\cos\theta-\frac{3q}{2p}\sqrt{\frac{-3}{p}}=0\,.
  191. cos ( 3 θ ) = 3 q 2 p - 3 p \cos(3\theta)=\frac{3q}{2p}\sqrt{\frac{-3}{p}}
  192. t k = 2 - p 3 cos ( 1 3 arccos ( 3 q 2 p - 3 p ) - 2 π k 3 ) for k = 0 , 1 , 2 . t_{k}=2\sqrt{-\frac{p}{3}}\cos\left(\frac{1}{3}\arccos\left(\frac{3q}{2p}\sqrt% {\frac{-3}{p}}\right)-\frac{2\pi k}{3}\right)\quad\,\text{for}\quad k=0,1,2\,.
  193. p < 0 p<0
  194. 4 p 3 + 27 q 2 0 , 4p^{3}+27q^{2}\leq 0\,,
  195. p < 0 p<0
  196. C ( p , q ) C(p,q)
  197. - π arccos ( u ) π -\pi\leq\arccos(u)\leq\pi
  198. - 1 u 1 , -1\leq u\leq 1\,,
  199. t 0 = C ( p , q ) , t 2 = - C ( p , - q ) , t 1 = - t 0 - t 2 . t_{0}=C(p,q),\qquad t_{2}=-C(p,-q),\qquad t_{1}=-t_{0}-t_{2}\,.
  200. t 0 t 1 t 2 . t_{0}\geq t_{1}\geq t_{2}\,.
  201. t 0 = - 2 | q | q - p 3 cosh ( 1 3 arcosh ( - 3 | q | 2 p - 3 p ) ) if 4 p 3 + 27 q 2 > 0 and p < 0 , t_{0}=-2\frac{|q|}{q}\sqrt{-\frac{p}{3}}\cosh\left(\frac{1}{3}\operatorname{% arcosh}\left(\frac{-3|q|}{2p}\sqrt{\frac{-3}{p}}\right)\right)\quad\,\text{if % }\quad 4p^{3}+27q^{2}>0\,\text{ and }p<0\,,
  202. t 0 = - 2 p 3 sinh ( 1 3 arsinh ( 3 q 2 p 3 p ) ) if p > 0 . t_{0}=-2\sqrt{\frac{p}{3}}\sinh\left(\frac{1}{3}\operatorname{arsinh}\left(% \frac{3q}{2p}\sqrt{\frac{3}{p}}\right)\right)\quad\,\text{if }\quad p>0\,.
  203. p = ± 3 p=\pm 3
  204. t 0 t_{0}
  205. p = - 3 p=-3
  206. C 1 3 ( q ) C_{\frac{1}{3}}(q)
  207. S 1 3 ( q ) , S_{\frac{1}{3}}(q),
  208. p = 3 p=3
  209. a x 3 + b x 2 + c x + d = 0 ax^{3}+bx^{2}+cx+d=0
  210. ( x - r ) ( a x 2 + ( b + a r ) x + c + b r + a r 2 ) = a x 3 + b x 2 + c x + d . \left(x-r\right)\left(ax^{2}+(b+ar)x+c+br+ar^{2}\right)=ax^{3}+bx^{2}+cx+d\,.
  211. a x 2 + ( b + a r ) x + c + b r + a r 2 ax^{2}+(b+ar)x+c+br+ar^{2}
  212. - b - r a ± b 2 - 4 a c - 2 a b r - 3 a 2 r 2 2 a \frac{-b-ra\pm\sqrt{b^{2}-4ac-2abr-3a^{2}r^{2}}}{2a}
  213. x 3 + b x 2 + c x + d = 0 x^{3}+bx^{2}+cx+d=0
  214. t 3 + p t + q = 0 t^{3}+pt+q=0
  215. t k = 2 - p 3 cos ( 1 3 arccos ( 3 q 2 p - 3 p ) - k 2 π 3 ) for k = 0 , 1 , 2 . t_{k}=2\sqrt{-\frac{p}{3}}\cos\left(\frac{1}{3}\arccos\left(\frac{3q}{2p}\sqrt% {\frac{-3}{p}}\right)-k\frac{2\pi}{3}\right)\quad\,\text{for}\quad k=0,1,2\,.
  216. arccos ( 3 q 2 p - 3 p ) \arccos\left(\frac{3q}{2p}\sqrt{\frac{-3}{p}}\right)
  217. 1 3 \tfrac{1}{3}
  218. - k 2 π 3 -k\frac{2\pi}{3}
  219. 2 - p 3 2\sqrt{-\frac{p}{3}}
  220. x 3 + b x 2 + c x + d = 0 x^{3}+bx^{2}+cx+d=0
  221. x = t - b 3 x=t-\tfrac{b}{3}
  222. t = x + b 3 t=x+\tfrac{b}{3}
  223. t t
  224. O M ¯ \scriptstyle\overline{OM}
  225. tan O R H \scriptstyle\sqrt{\tan ORH}
  226. slope of line RH \scriptstyle\sqrt{\,\text{slope of line RH}}
  227. B E ¯ \scriptstyle\overline{BE}
  228. D A ¯ \scriptstyle\overline{DA}
  229. g ± h i , g\pm hi,
  230. π 3 \tfrac{\pi}{3}
  231. π 3 \tfrac{\pi}{3}
  232. π 3 \tfrac{\pi}{3}
  233. x 3 + a 2 x = b x^{3}+a^{2}x=b
  234. b > 0 , b>0,
  235. y = x 2 / a , y=x^{2}/a,
  236. [ 0 , b / a 2 ] [0,b/a^{2}]
  237. x x
  238. x 4 a 2 = x ( b a 2 - x ) . \frac{x^{4}}{a^{2}}=x\,(\frac{b}{a^{2}}-x)\,.
  239. y 2 + x ( x - b a 2 ) = 0 , y^{2}+x\,(x-\frac{b}{a^{2}})=0,
  240. y < s u p > 2 y<sup>2