wpmath0000001_3

Bose–Einstein_condensate.html

  1. T c = ( n ζ ( 3 / 2 ) ) 2 / 3 2 π 2 m k B 3.3125 2 n 2 / 3 m k B T_{c}=\left(\frac{n}{\zeta(3/2)}\right)^{2/3}\frac{2\pi\hbar^{2}}{mk_{B}}% \approx 3.3125\ \frac{\hbar^{2}n^{2/3}}{mk_{B}}
  2. T c \,T_{c}
  3. n \,n
  4. m \,m
  5. \hbar
  6. k B \,k_{B}
  7. ζ \,\zeta
  8. ζ ( 3 / 2 ) 2.6124. \,\zeta(3/2)\approx 2.6124.
  9. | 0 \scriptstyle|0\rangle
  10. | 1 \scriptstyle|1\rangle
  11. 2 N 2^{N}
  12. | 0 \scriptstyle|0\rangle
  13. | 1 \scriptstyle|1\rangle
  14. | 0 \scriptstyle|0\rangle
  15. | 1 \scriptstyle|1\rangle
  16. | 1 \scriptstyle|1\rangle
  17. | 0 \scriptstyle|0\rangle
  18. | 0 \scriptstyle|0\rangle
  19. | 1 \scriptstyle|1\rangle
  20. | 1 \scriptstyle|1\rangle
  21. | 0 \scriptstyle|0\rangle
  22. | 1 \scriptstyle|1\rangle
  23. e - E / k T e^{-E/kT}
  24. | 0 \scriptstyle|0\rangle
  25. | 0 \scriptstyle|0\rangle
  26. | 0 \scriptstyle|0\rangle
  27. P ( K ) = C e - K E / T = C p K . \,P(K)=Ce^{-KE/T}=Cp^{K}.
  28. N \scriptstyle N\rightarrow\infty
  29. n > 0 C n p n = p / ( 1 - p ) \scriptstyle\sum_{n>0}Cnp^{n}=p/(1-p)
  30. | k \scriptstyle|k\rangle
  31. N = V d 3 k ( 2 π ) 3 p ( k ) 1 - p ( k ) = V d 3 k ( 2 π ) 3 1 e k 2 2 m T - 1 \,N=V\int{d^{3}k\over(2\pi)^{3}}{p(k)\over 1-p(k)}=V\int{d^{3}k\over(2\pi)^{3}% }{1\over e^{k^{2}\over 2mT}-1}
  32. p ( k ) = e - k 2 2 m T . \,p(k)=e^{-k^{2}\over 2mT}.
  33. P = g / 2 n 2 P=g/2n^{2}
  34. ψ ( r ) \psi(\vec{r})
  35. | ψ ( r ) | 2 |\psi(\vec{r})|^{2}
  36. N = d r | ψ ( r ) | 2 N=\int d\vec{r}|\psi(\vec{r})|^{2}
  37. ψ ( r ) \psi(\vec{r})
  38. E = d r [ 2 2 m | ψ ( r ) | 2 + V ( r ) | ψ ( r ) | 2 + 1 2 U 0 | ψ ( r ) | 4 ] E=\int d\vec{r}\left[\frac{\hbar^{2}}{2m}|\nabla\psi(\vec{r})|^{2}+V(\vec{r})|% \psi(\vec{r})|^{2}+\frac{1}{2}U_{0}|\psi(\vec{r})|^{4}\right]
  39. ψ ( r ) \psi(\vec{r})
  40. i ψ ( r ) t = ( - 2 2 2 m + V ( r ) + U 0 | ψ ( r ) | 2 ) ψ ( r ) i\hbar\frac{\partial\psi(\vec{r})}{\partial t}=\left(-\frac{\hbar^{2}\nabla^{2% }}{2m}+V(\vec{r})+U_{0}|\psi(\vec{r})|^{2}\right)\psi(\vec{r})
  41. m \,m
  42. V ( r ) \,V(\vec{r})
  43. U 0 \,U_{0}
  44. T = 0 \ T=0
  45. ω p = < m t p l > p 2 2 m ( p 2 2 m + 2 U 0 n 0 ) {\omega_{p}}=\sqrt{\frac{<mtpl>{{p^{2}}}}{{2m}}\left({\frac{{{p^{2}}}}{{2m}}+2% {U_{0}}{n_{0}}}\right)}
  46. T = 0 \ T=0
  47. | ψ ( r ) | 2 |\psi(\vec{r})|^{2}
  48. ψ ( r ) = ϕ ( ρ , z ) e i θ \psi(\vec{r})=\phi(\rho,z)e^{i\ell\theta}
  49. ρ , z \rho,z
  50. θ \theta
  51. \ell
  52. ϕ ( ρ , z ) \phi(\rho,z)
  53. ψ ( r ) \psi(\vec{r})
  54. ψ ( r ) = ϕ ( ρ , z ) e i θ \psi(\vec{r})=\phi(\rho,z)e^{i\ell\theta}
  55. ϕ = n x 2 + x 2 \phi=\frac{nx}{\sqrt{2+x^{2}}}
  56. n 2 \,n^{2}
  57. x = ρ ξ , \,x=\frac{\rho}{\ell\xi},
  58. ξ \,\xi
  59. = 1 \ell=1
  60. ϵ v \epsilon_{v}
  61. ϵ v = π n 2 m ln ( 1.464 b ξ ) \epsilon_{v}=\pi n\frac{\hbar^{2}}{m}\ln\left(1.464\frac{b}{\xi}\right)
  62. b \,b
  63. b b
  64. > 1 \ell>1
  65. ϵ v 2 π n 2 m ln ( b ξ ) \epsilon_{v}\approx\ell^{2}\pi n\frac{\hbar^{2}}{m}\ln\left(\frac{b}{\xi}\right)
  66. \ell

Box–Muller_transform.html

  1. Z 0 = R cos ( Θ ) = - 2 ln U 1 cos ( 2 π U 2 ) Z_{0}=R\cos(\Theta)=\sqrt{-2\ln U_{1}}\cos(2\pi U_{2})\,
  2. Z 1 = R sin ( Θ ) = - 2 ln U 1 sin ( 2 π U 2 ) . Z_{1}=R\sin(\Theta)=\sqrt{-2\ln U_{1}}\sin(2\pi U_{2}).\,
  3. R 2 = - 2 ln U 1 R^{2}=-2\cdot\ln U_{1}\,
  4. Θ = 2 π U 2 . \Theta=2\pi U_{2}.\,
  5. s \scriptstyle\sqrt{s}
  6. π \scriptstyle\pi
  7. 2 π \scriptstyle 2\pi
  8. θ / ( 2 π ) \scriptstyle\theta/(2\pi)
  9. cos θ = cos 2 π U 2 \scriptstyle\cos\theta=\cos 2\pi U_{2}
  10. sin θ = sin 2 π U 2 \scriptstyle\sin\theta=\sin 2\pi U_{2}
  11. cos θ = u / R = u / s \scriptstyle\cos\theta=u/R=u/\sqrt{s}
  12. sin θ = v / R = v / s \scriptstyle\sin\theta=v/R=v/\sqrt{s}
  13. z 0 = - 2 ln U 1 cos ( 2 π U 2 ) = - 2 ln s ( u s ) = u - 2 ln s s z_{0}=\sqrt{-2\ln U_{1}}\cos(2\pi U_{2})=\sqrt{-2\ln s}\left(\frac{u}{\sqrt{s}% }\right)=u\cdot\sqrt{\frac{-2\ln s}{s}}
  14. z 1 = - 2 ln U 1 sin ( 2 π U 2 ) = - 2 ln s ( v s ) = v - 2 ln s s . z_{1}=\sqrt{-2\ln U_{1}}\sin(2\pi U_{2})=\sqrt{-2\ln s}\left(\frac{v}{\sqrt{s}% }\right)=v\cdot\sqrt{\frac{-2\ln s}{s}}.
  15. exp ( i z ) = e i z = cos ( z ) + i sin ( z ) , \mathrm{exp}(iz)=e^{iz}=\cos(z)+i\sin(z),\,
  16. 2 - 32 2^{-32}
  17. U 1 U_{1}
  18. U 2 U_{2}
  19. - 2 ln ( 2 - 32 ) cos ( 2 π 2 - 32 ) 6.66 \sqrt{-2\ln(2^{-32})}\cos(2\pi 2^{-32})\approx 6.66
  20. 2.74 × 10 - 11 2.74\times 10^{-11}
  21. μ \mu
  22. σ 2 \sigma^{2}
  23. Z Z
  24. X = Z σ + μ X=Z\sigma+\mu
  25. μ \mu
  26. σ \sigma

Boy's_surface.html

  1. z 1 \scriptstyle\|z\|\;\leq\;1
  2. g 1 = - 3 2 Im [ z ( 1 - z 4 ) z 6 + 5 z 3 - 1 ] g 2 = - 3 2 Re [ z ( 1 + z 4 ) z 6 + 5 z 3 - 1 ] g 3 = Im [ 1 + z 6 z 6 + 5 z 3 - 1 ] - 1 2 \begin{aligned}\displaystyle g_{1}&\displaystyle=-{3\over 2}\mathrm{Im}\left[{% z(1-z^{4})\over z^{6}+\sqrt{5}z^{3}-1}\right]\\ \displaystyle g_{2}&\displaystyle=-{3\over 2}\mathrm{Re}\left[{z(1+z^{4})\over z% ^{6}+\sqrt{5}z^{3}-1}\right]\\ \displaystyle g_{3}&\displaystyle=\mathrm{Im}\left[{1+z^{6}\over z^{6}+\sqrt{5% }z^{3}-1}\right]-{1\over 2}\\ \end{aligned}
  3. ( x y z ) = 1 g 1 2 + g 2 2 + g 3 2 ( g 1 g 2 g 3 ) \begin{pmatrix}x\\ y\\ z\end{pmatrix}=\frac{1}{g_{1}^{2}+g_{2}^{2}+g_{3}^{2}}\begin{pmatrix}g_{1}\\ g_{2}\\ g_{3}\end{pmatrix}
  4. - 1 z , \scriptstyle-{1\over z^{\star}},
  5. P ( z ) = ( x ( z ) , y ( z ) , z ( z ) ) \scriptstyle P(z)\;=\;(x(z),\,y(z),\,z(z))
  6. z 1 , \scriptstyle\|z\|\;\leq\;1,
  7. P ( z ) = P ( - 1 z ) P(z)=P\left(-{1\over z^{\star}}\right)
  8. z = z z = 1. \scriptstyle\|z\|\;=\;\sqrt{zz^{\star}}\;=\;1.
  9. z < 1 ? \scriptstyle\|z\|\;<\;1?
  10. - 1 z > 1 \left\|-{1\over z^{\star}}\right\|>1
  11. - 1 z = - z z z = - z z 2 -{1\over z^{\star}}={-z\over z^{\star}z}={-z\over\|z\|^{2}}
  12. z z 2 = 1 z , {\|z\|\over\|z\|^{2}}={1\over\|z\|},
  13. z < 1 , \scriptstyle\|z\|\;<\;1,
  14. 1 z > 1. {1\over\|z\|}>1.
  15. z 1 \scriptstyle\|z\|\;\leq\;1
  16. P ( - 1 z ) \scriptstyle P\left(-{1\over z^{\star}}\right)
  17. z = 1. \scriptstyle\|z\|\;=\;1.
  18. P ( z ) = P ( - z ) \scriptstyle P(z)\;=\;P(-z)
  19. z = 1 , \scriptstyle\|z\|\;=\;1,
  20. z 1 \scriptstyle\|z\|\;\leq\;1

Boyle's_law.html

  1. P 1 V P\propto\frac{1}{V}
  2. P V = k PV=k
  3. P 1 V 1 = P 2 V 2 . P_{1}V_{1}=P_{2}V_{2}.
  4. P V = k PV=k
  5. P 1 V 1 = P 2 V 2 . P_{1}V_{1}=P_{2}V_{2}.\,

BPP_(complexity).html

  1. B P P Σ 2 Π 2 \ BPP\subseteq\Sigma_{2}\cap\Pi_{2}
  2. 𝐢 . 𝐨 . - 𝐒𝐔𝐁𝐄𝐗𝐏 = ε > 0 𝐢 . 𝐨 . - 𝐃𝐓𝐈𝐌𝐄 ( 2 n ε ) . \mathbf{i.o.-SUBEXP}=\bigcap\nolimits_{\varepsilon>0}\mathbf{i.o.-DTIME}\left(% 2^{n^{\varepsilon}}\right).
  3. 𝐄 = 𝐃𝐓𝐈𝐌𝐄 ( 2 O ( n ) ) , \mathbf{E}=\mathbf{DTIME}\left(2^{O(n)}\right),

BQP.html

  1. { Q n : n } \{Q_{n}:n\in\mathbb{N}\}
  2. n n\in\mathbb{N}
  3. Pr ( Q | x | ( x ) = 1 ) 2 3 \mathrm{Pr}(Q_{|x|}(x)=1)\geq\tfrac{2}{3}
  4. Pr ( Q | x | ( x ) = 0 ) 2 3 \mathrm{Pr}(Q_{|x|}(x)=0)\geq\tfrac{2}{3}
  5. 𝐏 𝐁𝐏𝐏 𝐁𝐐𝐏 𝐀𝐖𝐏𝐏 𝐏𝐏 𝐏𝐒𝐏𝐀𝐂𝐄 \mathbf{P}\subseteq\mathbf{BPP}\subseteq\mathbf{BQP}\subseteq\mathbf{AWPP}% \subseteq\mathbf{PP}\subseteq\mathbf{PSPACE}

Bracket.html

  1. 2 + 3 × 4 2+3×4
  2. ( 2 + 3 ) × 4 (2+3)×4
  3. [ ( 2 + 3 ) × 4 ] 2 = 400 [(2+3)\times 4]^{2}=400
  4. f ( x ) f(x)
  5. f f
  6. x x
  7. ( 4 , 7 ) (4, 7)
  8. ( 0 , 5 ) (0,5)
  9. V ( t ) 2 = lim T 1 T - T / 2 T / 2 V ( t ) 2 d t . \left\langle V(t)^{2}\right\rangle=\lim_{T\to\infty}\frac{1}{T}\int_{-T/2}^{T/% 2}V(t)^{2}\,{\rm{d}}t.
  10. a , b \langle a,b\rangle
  11. a | b \langle a|b\rangle
  12. a | | b \langle a|\cdot|b\rangle
  13. a | O ^ | b \langle a|\hat{O}|b\rangle
  14. O ^ \hat{O}
  15. [ a , c ) [a,c)
  16. a a
  17. c c
  18. a a
  19. c c
  20. [ 5 , 12 ) [5, 12)
  21. 5 , 122 5, 122
  22. m v a r v a r g mvarvarg
  23. g < s u p > 1 h 1 g h g<sup> −1h^{ −1}gh

Brake.html

  1. K = m v 2 / 2 K=mv^{2}/2

Bra–ket_notation.html

  1. ϕ ψ \langle\phi\mid\psi\rangle
  2. ϕ | \langle\phi|
  3. | ψ |\psi\rangle
  4. [ ϕ ψ ] [\phi\mid\psi]
  5. x x
  6. p p
  7. ϕ ψ \langle\phi\mid\psi\rangle
  8. [ - F o r m u l a E r r o r - ] [-FormulaError-]
  9. [ - F o r m u l a E r r o r - ] [-FormulaError-]
  10. 𝐀 \mathbf{A}
  11. 𝐀 \mathbf{A}
  12. 𝐀 \mathbf{A}
  13. 𝐀 A x 𝐞 x + A y 𝐞 y + A z 𝐞 z = A x ( 1 0 0 ) + A y ( 0 1 0 ) + A z ( 0 0 1 ) \mathbf{A}\doteq\!\,A_{x}\mathbf{e}_{x}+A_{y}\mathbf{e}_{y}+A_{z}\mathbf{e}_{z% }=A_{x}\begin{pmatrix}1\\ 0\\ 0\end{pmatrix}+A_{y}\begin{pmatrix}0\\ 1\\ 0\end{pmatrix}+A_{z}\begin{pmatrix}0\\ 0\\ 1\end{pmatrix}
  14. = ( A x 0 0 ) + ( 0 A y 0 ) + ( 0 0 A z ) = ( A x A y A z ) =\begin{pmatrix}A_{x}\\ 0\\ 0\end{pmatrix}+\begin{pmatrix}0\\ A_{y}\\ 0\end{pmatrix}+\begin{pmatrix}0\\ 0\\ A_{z}\end{pmatrix}=\begin{pmatrix}A_{x}\\ A_{y}\\ A_{z}\\ \end{pmatrix}
  15. 𝐀 A 1 𝐞 1 + A 2 𝐞 2 + A 3 𝐞 3 = ( A 1 A 2 A 3 ) \mathbf{A}\doteq\!\,A_{1}\mathbf{e}_{1}+A_{2}\mathbf{e}_{2}+A_{3}\mathbf{e}_{3% }=\begin{pmatrix}A_{1}\\ A_{2}\\ A_{3}\\ \end{pmatrix}
  16. 𝐀 \mathbf{A}
  17. N N
  18. 𝐀 \mathbf{A}
  19. 𝐀 n = 1 N A n 𝐞 n = ( A 1 A 2 A N ) \mathbf{A}\doteq\!\,\sum_{n=1}^{N}A_{n}\mathbf{e}_{n}=\begin{pmatrix}A_{1}\\ A_{2}\\ \vdots\\ A_{N}\\ \end{pmatrix}
  20. 𝐀 \mathbf{A}
  21. 𝐀 \mathbf{A}
  22. 𝐀 , A , A ¯ \mathbf{A},\,\vec{A},\,\underline{A}
  23. | A |A\rangle
  24. | A = A x | e x + A y | e y + A z | e z ( A x A y A z ) , |A\rangle=A_{x}|e_{x}\rangle+A_{y}|e_{y}\rangle+A_{z}|e_{z}\rangle{\doteq\!\,}% \begin{pmatrix}A_{x}\\ A_{y}\\ A_{z}\end{pmatrix},
  25. | A = A 1 | e 1 + A 2 | e 2 + A 3 | e 3 ( A 1 A 2 A 3 ) , |A\rangle=A_{1}|e_{1}\rangle+A_{2}|e_{2}\rangle+A_{3}|e_{3}\rangle{\doteq\!\,}% \begin{pmatrix}A_{1}\\ A_{2}\\ A_{3}\end{pmatrix},
  26. | A = A 1 | 1 + A 2 | 2 + A 3 | 3 . |A\rangle=A_{1}|1\rangle+A_{2}|2\rangle+A_{3}|3\rangle~{}.
  27. | A |A\rangle
  28. A A
  29. A | B = the inner product of ket | A with ket | B \langle A|B\rangle=\,\text{the inner product of ket }|A\rangle\,\text{ with % ket }|B\rangle
  30. A | B A x * B x + A y * B y + A z * B z \langle A|B\rangle\doteq\!\,A_{x}^{*}B_{x}+A_{y}^{*}B_{y}+A_{z}^{*}B_{z}
  31. A i * A_{i}^{*}
  32. A | A | A x | 2 + | A y | 2 + | A z | 2 \langle A|A\rangle\doteq\!\,|A_{x}|^{2}+|A_{y}|^{2}+|A_{z}|^{2}
  33. A | B = ( A | ) ( | B ) \langle A|B\rangle=\left(\,\langle A|\,\right)\,\,\left(\,|B\rangle\,\right)
  34. A | \langle A|
  35. | B |B\rangle
  36. A | \langle A|
  37. | B |B\rangle
  38. A | B A 1 * B 1 + A 2 * B 2 + + A N * B N = ( A 1 * A 2 * A N * ) ( B 1 B 2 B N ) \langle A|B\rangle\doteq\!\,A_{1}^{*}B_{1}+A_{2}^{*}B_{2}+\cdots+A_{N}^{*}B_{N% }{=}\begin{pmatrix}A_{1}^{*}&A_{2}^{*}&\cdots&A_{N}^{*}\end{pmatrix}\begin{% pmatrix}B_{1}\\ B_{2}\\ \vdots\\ B_{N}\end{pmatrix}
  39. A | ( A 1 * A 2 * A N * ) \langle A|{\doteq\!\,}\begin{pmatrix}A_{1}^{*}&A_{2}^{*}&\cdots&A_{N}^{*}\end{pmatrix}
  40. | B ( B 1 B 2 B N ) |B\rangle{\doteq\!\,}\begin{pmatrix}B_{1}\\ B_{2}\\ \vdots\\ B_{N}\end{pmatrix}
  41. A | = | A , | A = A | \langle A|^{\dagger}=|A\rangle,\quad|A\rangle^{\dagger}=\langle A|
  42. ( A 1 * A 2 * A N * ) , \begin{pmatrix}A_{1}^{*}&A_{2}^{*}&\cdots&A_{N}^{*}\end{pmatrix},
  43. ( A 1 A 2 A N ) \begin{pmatrix}A_{1}\\ A_{2}\\ \vdots\\ A_{N}\end{pmatrix}
  44. B B
  45. | ψ |ψ\rangle
  46. c | ψ c|ψ\rangle
  47. c c
  48. | 1 + i | 2 |1\rangle+i|2\rangle
  49. | 1 |1\rangle
  50. | 2 |2\rangle
  51. U U
  52. | ψ |ψ\rangle
  53. U | ψ U|ψ\rangle
  54. U U
  55. | ψ |ψ\rangle
  56. 1 1
  57. 𝐫 \mathbf{r}
  58. | Ψ |Ψ\rangle
  59. 𝐫 \mathbf{r}
  60. Ψ ( 𝐫 ) = def 𝐫 | Ψ \Psi(\mathbf{r})\ \stackrel{\,\text{def}}{=}\ \langle\mathbf{r}|\Psi\rangle
  61. Ψ ( 𝐫 ) Ψ(\mathbf{r})
  62. A Ψ ( 𝐫 ) = def 𝐫 | A | Ψ . A\Psi(\mathbf{r})\ \stackrel{\,\text{def}}{=}\ \langle\mathbf{r}|A|\Psi\rangle.
  63. 𝐩 Ψ ( 𝐫 ) = def 𝐫 | 𝐩 | Ψ = - i Ψ ( 𝐫 ) . \mathbf{p}\Psi(\mathbf{r})\ \stackrel{\,\text{def}}{=}\ \langle\mathbf{r}|% \mathbf{p}|\Psi\rangle=-i\hbar\nabla\Psi(\mathbf{r}).
  64. | Ψ , \nabla|\Psi\rangle,
  65. 𝐫 | Ψ , \nabla\langle\mathbf{r}|\Psi\rangle,
  66. i ħ 𝐩 iħ\mathbf{p}
  67. φ ψ \langle φψ\rangle
  68. ψ ψ
  69. φ φ
  70. ψ ψ
  71. φ φ
  72. ψ ψ
  73. φ φ
  74. | z , | z |\uparrow_{z}\rangle,\;|\downarrow_{z}\rangle
  75. | z |\uparrow_{z}\rangle
  76. | z |\downarrow_{z}\rangle
  77. | ψ = a ψ | z + b ψ | z |\psi\rangle=a_{\psi}|\uparrow_{z}\rangle+b_{\psi}|\downarrow_{z}\rangle
  78. | x , | x |\uparrow_{x}\rangle,\;|\downarrow_{x}\rangle
  79. | ψ = c ψ | x + d ψ | x |\psi\rangle=c_{\psi}|\uparrow_{x}\rangle+d_{\psi}|\downarrow_{x}\rangle
  80. | ψ ( a ψ b ψ ) , OR | ψ ( c ψ d ψ ) |\psi\rangle{\doteq\!\,}\begin{pmatrix}a_{\psi}\\ b_{\psi}\end{pmatrix},\;\;\,\text{OR}\;\;|\psi\rangle{\doteq\!\,}\begin{% pmatrix}c_{\psi}\\ d_{\psi}\end{pmatrix}
  81. α̂ | α = α | α α̂|α\rangle=α|α\rangle
  82. α α
  83. α̂ α̂
  84. | α |α\rangle
  85. α α
  86. Ψ Ψ
  87. ψ ψ
  88. | α |α\rangle
  89. 2 \sqrt{2}
  90. | α / 2 |α/\sqrt{2}\rangle
  91. α α
  92. α α
  93. 𝐀 \mathbf{A}
  94. | ψ |ψ\rangle
  95. 𝐀 | ψ \mathbf{A}|ψ\rangle
  96. N N
  97. | ψ |ψ\rangle
  98. N × 1 N×1
  99. 𝐀 \mathbf{A}
  100. N × N N×N
  101. 𝐀 | ψ \mathbf{A}|ψ\rangle
  102. 𝐀 \mathbf{A}
  103. φ | \langle φ|
  104. φ | 𝐀 \langle φ|\mathbf{A}
  105. ( ϕ | A ) | ψ = ϕ | ( A | ψ ) , \bigg(\langle\phi|A\bigg)\;|\psi\rangle=\langle\phi|\;\bigg(A|\psi\rangle\bigg),
  106. ϕ | A | ψ \langle\phi|A|\psi\rangle
  107. N N
  108. φ | \langle φ|
  109. 1 × N 1×N
  110. 𝐀 \mathbf{A}
  111. N × N N×N
  112. φ | 𝐀 \langle φ|\mathbf{A}
  113. ψ | A | ψ , \langle\psi|A|\psi\rangle,
  114. 𝐀 \mathbf{A}
  115. | ψ |ψ\rangle
  116. H H
  117. φ | \langle φ|
  118. | ψ |ψ\rangle
  119. | ϕ ψ | |\phi\rangle\langle\psi|
  120. ( | ϕ ψ | ) ( x ) = ψ , x ϕ (|\phi\rangle\langle\psi|)(x)=\langle\psi,x\rangle\phi
  121. | ϕ ψ | ( ϕ 1 ϕ 2 ϕ N ) ( ψ 1 * ψ 2 * ψ N * ) = ( ϕ 1 ψ 1 * ϕ 1 ψ 2 * ϕ 1 ψ N * ϕ 2 ψ 1 * ϕ 2 ψ 2 * ϕ 2 ψ N * ϕ N ψ 1 * ϕ N ψ 2 * ϕ N ψ N * ) |\phi\rangle\,\langle\psi|{\doteq\!\,}\begin{pmatrix}\phi_{1}\\ \phi_{2}\\ \vdots\\ \phi_{N}\end{pmatrix}\begin{pmatrix}\psi_{1}^{*}&\psi_{2}^{*}&\cdots&\psi_{N}^% {*}\end{pmatrix}=\begin{pmatrix}\phi_{1}\psi_{1}^{*}&\phi_{1}\psi_{2}^{*}&% \cdots&\phi_{1}\psi_{N}^{*}\\ \phi_{2}\psi_{1}^{*}&\phi_{2}\psi_{2}^{*}&\cdots&\phi_{2}\psi_{N}^{*}\\ \vdots&\vdots&\ddots&\vdots\\ \phi_{N}\psi_{1}^{*}&\phi_{N}\psi_{2}^{*}&\cdots&\phi_{N}\psi_{N}^{*}\end{pmatrix}
  122. | ψ |ψ\rangle
  123. | ψ |ψ\rangle
  124. | ψ ψ | . |\psi\rangle\langle\psi|.
  125. | ψ |ψ\rangle
  126. ψ | \langle ψ|
  127. A | ψ A|ψ\rangle
  128. A A
  129. | ϕ = A | ψ |\phi\rangle=A|\psi\rangle
  130. ϕ | = ψ | A \qquad\langle\phi|=\langle\psi|A^{\dagger}
  131. A A
  132. N × N N×N
  133. A A
  134. ψ | A | ψ \langle ψ|A|ψ\rangle
  135. c c
  136. A A
  137. B B
  138. ϕ | ( c 1 | ψ 1 + c 2 | ψ 2 ) = c 1 ϕ | ψ 1 + c 2 ϕ | ψ 2 . \langle\phi|\;\bigg(c_{1}|\psi_{1}\rangle+c_{2}|\psi_{2}\rangle\bigg)=c_{1}% \langle\phi|\psi_{1}\rangle+c_{2}\langle\phi|\psi_{2}\rangle.
  139. ( c 1 ϕ 1 | + c 2 ϕ 2 | ) | ψ = c 1 ϕ 1 | ψ + c 2 ϕ 2 | ψ . \bigg(c_{1}\langle\phi_{1}|+c_{2}\langle\phi_{2}|\bigg)\;|\psi\rangle=c_{1}% \langle\phi_{1}|\psi\rangle+c_{2}\langle\phi_{2}|\psi\rangle.
  140. ψ | ( A | ϕ ) = ( ψ | A ) | ϕ = def ψ | A | ϕ \langle\psi|(A|\phi\rangle)=(\langle\psi|A)|\phi\rangle\,\stackrel{\,\text{def% }}{=}\,\langle\psi|A|\phi\rangle
  141. ( A | ψ ) ϕ | = A ( | ψ ϕ | ) = def A | ψ ϕ | (A|\psi\rangle)\langle\phi|=A(|\psi\rangle\langle\phi|)\,\stackrel{\,\text{def% }}{=}\,A|\psi\rangle\langle\phi|
  142. ( c 1 | ψ 1 + c 2 | ψ 2 ) = c 1 * ψ 1 | + c 2 * ψ 2 | . \left(c_{1}|\psi_{1}\rangle+c_{2}|\psi_{2}\rangle\right)^{\dagger}=c_{1}^{*}% \langle\psi_{1}|+c_{2}^{*}\langle\psi_{2}|~{}.
  143. ϕ | ψ * = ψ | ϕ . \langle\phi|\psi\rangle^{*}=\langle\psi|\phi\rangle~{}.
  144. ϕ | A | ψ * = ψ | A | ϕ \langle\phi|A|\psi\rangle^{*}=\langle\psi|A^{\dagger}|\phi\rangle
  145. ϕ | A B | ψ * = ψ | B A | ϕ . \langle\phi|A^{\dagger}B^{\dagger}|\psi\rangle^{*}=\langle\psi|BA|\phi\rangle~% {}.
  146. ( ( c 1 | ϕ 1 ψ 1 | ) + ( c 2 | ϕ 2 ψ 2 | ) ) = ( c 1 * | ψ 1 ϕ 1 | ) + ( c 2 * | ψ 2 ϕ 2 | ) . \left((c_{1}|\phi_{1}\rangle\langle\psi_{1}|)+(c_{2}|\phi_{2}\rangle\langle% \psi_{2}|)\right)^{\dagger}=(c_{1}^{*}|\psi_{1}\rangle\langle\phi_{1}|)+(c_{2}% ^{*}|\psi_{2}\rangle\langle\phi_{2}|)~{}.
  147. V V
  148. W W
  149. V W V⊗W
  150. V V
  151. W W
  152. | ψ |ψ\rangle
  153. V V
  154. | φ |φ\rangle
  155. W W
  156. V W V⊗W
  157. | ψ | ϕ , | ψ | ϕ , | ψ ϕ , | ψ , ϕ . |\psi\rangle|\phi\rangle\,,\quad|\psi\rangle\otimes|\phi\rangle\,,\quad|\psi% \phi\rangle\,,\quad|\psi,\phi\rangle\,.
  158. { e i | i } \{e_{i}\ |\ i\in\mathbb{N}\}
  159. H H
  160. , \langle\cdot,\cdot\rangle
  161. | ψ |ψ\rangle
  162. | ψ = i e i | ψ | e i , |\psi\rangle=\sum_{i\in\mathbb{N}}\langle e_{i}|\psi\rangle|e_{i}\rangle,
  163. | \langle\cdot|\cdot\rangle
  164. i | e i e i | = 1 ^ \sum_{i\in\mathbb{N}}|e_{i}\rangle\langle e_{i}|=\hat{1}
  165. v | w = v | i | e i e i | w = v | i | e i e i | j | e j e j | w = v | e i e i | e j e j | w \langle v|w\rangle=\langle v|\sum_{i\in\mathbb{N}}|e_{i}\rangle\langle e_{i}|w% \rangle=\langle v|\sum_{i\in\mathbb{N}}|e_{i}\rangle\langle e_{i}|\sum_{j\in% \mathbb{N}}|e_{j}\rangle\langle e_{j}|w\rangle=\langle v|e_{i}\rangle\langle e% _{i}|e_{j}\rangle\langle e_{j}|w\rangle
  166. ψ | ϕ \langle\psi|\phi\rangle
  167. ψ | e i = e i | ψ * \langle\psi|e_{i}\rangle=\langle e_{i}|\psi\rangle^{*}
  168. e i | ϕ \langle e_{i}|\phi\rangle
  169. 1 = d x | x x | = d p | p p | 1=∫dx|x\rangle\langle x|=∫dp|p\rangle\langle p|
  170. x x = δ ( x x ) \langle x′x\rangle=δ(x−x′)
  171. x p = e x p ( i x p / ħ ) / 2 π ħ ¯ \langle xp\rangle=exp(ixp/ħ)/√\overline{2πħ}
  172. \mathcal{H}
  173. h h\in\mathcal{H}
  174. \mathcal{H}
  175. | h |h\rangle
  176. | h |h\rangle\in\mathcal{H}
  177. * \mathcal{H}^{*}
  178. \mathcal{H}
  179. \mathcal{H}
  180. Φ : * \Phi:\mathcal{H}\to\mathcal{H}^{*}
  181. Φ ( h ) = ϕ h \Phi(h)=\phi_{h}
  182. g g\in\mathcal{H}
  183. ϕ h ( g ) = IP ( h , g ) = ( h , g ) = h , g = h | g \phi_{h}(g)=\mbox{IP}~{}(h,g)=(h,g)=\langle h,g\rangle=\langle h|g\rangle
  184. IP ( , ) , ( , ) , , \mbox{IP}~{}(\cdot,\cdot),(\cdot,\cdot),\langle\cdot,\cdot\rangle
  185. | \langle\cdot|\cdot\rangle
  186. ϕ h \phi_{h}
  187. g g
  188. h | \langle h|
  189. | g |g\rangle
  190. ϕ h = H = h | \phi_{h}=H=\langle h|
  191. g = G = | g g=G=|g\rangle
  192. ϕ h ( g ) = H ( g ) = H ( G ) = h | ( G ) = h | ( | g ) . \phi_{h}(g)=H(g)=H(G)=\langle h|(G)=\langle h|(|g\rangle).
  193. ( ϕ , ψ ) = ϕ ( x ) ψ ( x ) ¯ d x , (\phi,\psi)=\int\phi(x)\cdot\overline{\psi(x)}\,{\rm d}x\,,
  194. ψ | ϕ = d x ψ * ( x ) ϕ ( x ) . \langle\psi|\phi\rangle=\int{\rm d}x\,\psi^{*}(x)\cdot\phi(x)\,.

Bremsstrahlung.html

  1. P = q 2 γ 4 6 π ε 0 c ( β ˙ 2 + ( β β ˙ ) 2 1 - β 2 ) , P=\frac{q^{2}\gamma^{4}}{6\pi\varepsilon_{0}c}\left(\dot{\beta}^{2}+\frac{(% \vec{\beta}\cdot\dot{\vec{\beta}})^{2}}{1-\beta^{2}}\right),
  2. β = v / c \vec{\beta}=\vec{v}/c
  3. γ \gamma
  4. β ˙ \dot{\vec{\beta}}
  5. β \vec{\beta}
  6. P = q 2 γ 6 6 π ε 0 c ( β ˙ 2 - ( β × β ˙ ) 2 ) . P=\frac{q^{2}\gamma^{6}}{6\pi\varepsilon_{0}c}\left(\dot{\beta}^{2}-(\vec{% \beta}\times\dot{\vec{\beta}})^{2}\right).
  7. P a v = q 2 a 2 γ 6 6 π ε 0 c 3 , P_{a\parallel v}=\frac{q^{2}a^{2}\gamma^{6}}{6\pi\varepsilon_{0}c^{3}},
  8. a v ˙ = β ˙ c a\equiv\dot{v}=\dot{\beta}c
  9. β β ˙ = 0 \vec{\beta}\cdot\dot{\vec{\beta}}=0
  10. P a v = q 2 a 2 γ 4 6 π ε 0 c 3 . P_{a\perp v}=\frac{q^{2}a^{2}\gamma^{4}}{6\pi\varepsilon_{0}c^{3}}.
  11. γ 4 \gamma^{4}
  12. a v a\perp v
  13. γ 6 \gamma^{6}
  14. a v a\parallel v
  15. E = γ m c 2 E=\gamma mc^{2}
  16. m - 4 m^{-4}
  17. m - 6 m^{-6}
  18. ( m p / m e ) 4 10 13 (m_{p}/m_{e})^{4}\approx 10^{13}
  19. d P d Ω = q 2 16 π 2 ε 0 c | n ^ × ( ( n ^ - β ) × β ˙ ) | 2 ( 1 - n ^ β ) 5 \frac{dP}{d\Omega}=\frac{q^{2}}{16\pi^{2}\varepsilon_{0}c}\frac{|\hat{n}\times% ((\hat{n}-\vec{\beta})\times\dot{\vec{\beta}})|^{2}}{(1-\hat{n}\cdot\vec{\beta% })^{5}}
  20. n ^ \hat{n}
  21. d Ω d\Omega
  22. d P a v d Ω = q 2 a 2 16 π 2 ε 0 c 3 sin 2 θ ( 1 - β cos θ ) 5 \frac{dP_{a\parallel v}}{d\Omega}=\frac{q^{2}a^{2}}{16\pi^{2}\varepsilon_{0}c^% {3}}\frac{\sin^{2}\theta}{(1-\beta\cos\theta)^{5}}
  23. θ \theta
  24. a \vec{a}
  25. ω \omega
  26. ω = ω p \omega=\omega_{p}
  27. T e > Z 2 E h T_{e}>Z^{2}E_{h}
  28. ω p / T e = 0.1 \hbar\omega_{p}/T_{e}=0.1
  29. k m k_{m}
  30. T e T_{e}
  31. 4 π 4\pi
  32. d P Br d ω = 8 2 3 π [ 1 - ω p 2 ω 2 ] 1 / 2 [ Z i 2 n i n e r e 3 ] [ ( m e c 2 ) 3 / 2 ( k B T e ) 1 / 2 ] E 1 ( y ) , {dP_{\mathrm{Br}}\over d\omega}={8\sqrt{2}\over 3\sqrt{\pi}}\left[1-{\omega_{p% }^{2}\over\omega^{2}}\right]^{1/2}\left[Z_{i}^{2}n_{i}n_{e}r_{e}^{3}\right]% \left[{\frac{(m_{e}c^{2})^{3/2}}{(k_{B}T_{e})^{1/2}}}\right]E_{1}(y),
  33. ω p \omega_{p}
  34. ω \omega
  35. n e , n i n_{e},n_{i}
  36. r e r_{e}
  37. m e m_{e}
  38. k B k_{B}
  39. c c
  40. ω < ω p \omega<\omega_{p}
  41. ω > ω p \omega>\omega_{p}
  42. E 1 E_{1}
  43. y y
  44. y = 1 2 ω 2 m e k m 2 k B T e y={1\over 2}{\omega^{2}m_{e}\over k_{m}^{2}k_{B}T_{e}}
  45. k m k_{m}
  46. k m = 1 / λ B k_{m}=1/\lambda_{B}
  47. k B T e > Z i 2 E h k_{B}T_{e}>Z_{i}^{2}E_{h}
  48. E h 27.2 E_{h}\approx 27.2
  49. λ B = / ( m e k B T e ) 1 / 2 \lambda_{B}=\hbar/(m_{e}k_{B}T_{e})^{1/2}
  50. k m 1 / l c k_{m}\propto 1/l_{c}
  51. l c l_{c}
  52. k m = 1 / λ B k_{m}=1/\lambda_{B}
  53. y = 1 2 [ ω k B T e ] 2 . y={1\over 2}\left[\frac{\hbar\omega}{k_{B}T_{e}}\right]^{2}.
  54. d P Br / d ω dP_{\mathrm{Br}}/d\omega
  55. ω \omega
  56. ω p \omega_{p}
  57. y 1 y\ll 1
  58. E 1 ( y ) - ln [ y e γ ] + O ( y ) E_{1}(y)\approx-\ln[ye^{\gamma}]+O(y)
  59. γ 0.577 \gamma\approx 0.577
  60. y > e - γ y>e^{-\gamma}
  61. P Br \displaystyle P_{\mathrm{Br}}
  62. G ( y p = 0 ) = 1 G(y_{p}=0)=1
  63. y p y_{p}
  64. k m = 1 / λ B k_{m}=1/\lambda_{B}
  65. P Br = 16 3 [ Z i 2 n i n e r e 3 ] [ c r e ( m e c 2 k B T e ) 1 / 2 ] α G ( y p ) P_{\mathrm{Br}}={16\over 3}\left[Z_{i}^{2}n_{i}n_{e}r_{e}^{3}\right]\left[{c% \over r_{e}}(m_{e}c^{2}k_{B}T_{e})^{1/2}\right]\alpha G(y_{p})
  66. α \alpha
  67. λ B \lambda_{B}
  68. G = 1 G=1
  69. P Br [ W / m 3 ] = Z i 2 n i n e [ 7.69 × 10 18 m - 3 ] 2 T e [ eV ] 1 / 2 . P_{\mathrm{Br}}[\textrm{W}/\textrm{m}^{3}]={Z_{i}^{2}n_{i}n_{e}\over\left[7.69% \times 10^{18}\textrm{m}^{-3}\right]^{2}}T_{e}[\textrm{eV}]^{1/2}.
  70. g B g_{B}
  71. ε ff = 1.4 × 10 - 27 T 1 / 2 n e n i Z 2 g B , \varepsilon_{\mathrm{ff}}=1.4\times 10^{-27}T^{1/2}n_{e}n_{i}Z^{2}g_{B},\,
  72. k B T e / m e c 2 . k_{B}T_{e}/m_{e}c^{2}\,.
  73. λ min = h c e V 1239.8 pm V in kV \lambda_{\min}=\frac{hc}{eV}\approx\frac{1239.8\,\text{ pm}}{V\,\text{ in kV}}\,
  74. d 4 σ \displaystyle d^{4}\sigma
  75. Z Z
  76. α fine 1 / 137 \alpha\text{fine}\approx 1/137
  77. \hbar
  78. c c
  79. E kin , i / f E_{\,\text{kin},i/f}
  80. E i , f E_{i,f}
  81. 𝐩 i , f \mathbf{p}_{i,f}
  82. E i , f = E kin , i / f + m e c 2 = m e 2 c 4 + 𝐩 i , f 2 c 2 , E_{i,f}=E_{\,\text{kin},i/f}+m_{e}c^{2}=\sqrt{m_{e}^{2}c^{4}+\mathbf{p}_{i,f}^% {2}c^{2}},
  83. m e m_{e}
  84. E f = E i - ω , E_{f}=E_{i}-\hbar\omega,
  85. ω \hbar\omega
  86. Θ i \displaystyle\Theta_{i}
  87. 𝐤 \mathbf{k}
  88. d Ω i \displaystyle d\Omega_{i}
  89. - 𝐪 2 \displaystyle-\mathbf{q}^{2}
  90. v Z c 137 v\gg\frac{Zc}{137}
  91. v v
  92. ω \omega
  93. Φ \Phi
  94. Θ f \Theta_{f}
  95. d 2 σ ( E i , ω , Θ i ) d ω d Ω i = j = 1 6 I j \frac{d^{2}\sigma(E_{i},\omega,\Theta_{i})}{d\omega\,d\Omega_{i}}=\sum\limits_% {j=1}^{6}I_{j}
  96. I 1 = 2 π A Δ 2 2 + 4 p i 2 p f 2 sin 2 Θ i ln ( Δ 2 2 + 4 p i 2 p f 2 sin 2 Θ i - Δ 2 2 + 4 p i 2 p f 2 sin 2 Θ i ( Δ 1 + Δ 2 ) + Δ 1 Δ 2 - Δ 2 2 - 4 p i 2 p f 2 sin 2 Θ i - Δ 2 2 + 4 p i 2 p f 2 sin 2 Θ i ( Δ 1 - Δ 2 ) + Δ 1 Δ 2 ) × [ 1 + c Δ 2 p f ( E i - c p i cos Θ i ) - p i 2 c 2 sin 2 Θ i ( E i - c p i cos Θ i ) 2 - 2 2 ω 2 p f Δ 2 c ( E i - c p i cos Θ i ) ( Δ 2 2 + 4 p i 2 p f 2 sin 2 Θ i ) ] , I 2 = - 2 π A c p f ( E i - c p i cos Θ i ) ln ( E f + p f c E f - p f c ) , I 3 = 2 π A ( Δ 2 E f + Δ 1 p f c ) 2 + 4 m 2 c 4 p i 2 p f 2 sin 2 Θ i × ln ( ( ( E f + p f c ) ( 4 p i 2 p f 2 sin 2 Θ i ( E f - p f c ) + ( Δ 1 + Δ 2 ) ( ( Δ 2 E f + Δ 1 p f c ) - ( Δ 2 E f + Δ 1 p f c ) 2 + 4 m 2 c 4 p i 2 p f 2 sin 2 Θ i ) ) ) ( ( E f - p f c ) ( 4 p i 2 p f 2 sin 2 Θ i ( - E f - p f c ) + ( Δ 1 - Δ 2 ) ( ( Δ 2 E f + Δ 1 p f c ) - ( Δ 2 E f + Δ 1 p f c ) 2 + 4 m 2 c 4 p i 2 p f 2 sin 2 Θ i ) ) ) - 1 ) × [ - ( Δ 2 2 + 4 p i 2 p f 2 sin 2 Θ i ) ( E f 3 + E f p f 2 c 2 ) + p f c ( 2 ( Δ 1 2 - 4 p i 2 p f 2 sin 2 Θ i ) E f p f c + Δ 1 Δ 2 ( 3 E f 2 + p f 2 c 2 ) ) ( Δ 2 E f + Δ 1 p f c ) 2 + 4 m 2 c 4 p i 2 p f 2 sin 2 Θ i - c ( Δ 2 E f + Δ 1 p f c ) p f ( E i - c p i cos Θ i ) - 4 E i 2 p f 2 ( 2 ( Δ 2 E f + Δ 1 p f c ) 2 - 4 m 2 c 4 p i 2 p f 2 sin 2 Θ i ) ( Δ 1 E f + Δ 2 p f c ) ( ( Δ 2 E f + Δ 1 p f c ) 2 + 4 m 2 c 4 p i 2 p f 2 sin 2 Θ i ) 2 + 8 p i 2 p f 2 m 2 c 4 sin 2 Θ i ( E i 2 + E f 2 ) - 2 2 ω 2 p i 2 sin 2 Θ i p f c ( Δ 2 E f + Δ 1 p f c ) + 2 2 ω 2 p f m 2 c 3 ( Δ 2 E f + Δ 1 p f c ) ( E i - c p i cos Θ i ) ( ( Δ 2 E f + Δ 1 p f c ) 2 + 4 m 2 c 4 p i 2 p f 2 sin 2 Θ i ) ] , I 4 = - 4 π A p f c ( Δ 2 E f + Δ 1 p f c ) ( Δ 2 E f + Δ 1 p f c ) 2 + 4 m 2 c 4 p i 2 p f 2 sin 2 Θ i - 16 π E i 2 p f 2 A ( Δ 2 E f + Δ 1 p f c ) 2 ( ( Δ 2 E f + Δ 1 p f c ) 2 + 4 m 2 c 4 p i 2 p f 2 sin 2 Θ i ) 2 , I 5 = 4 π A ( - Δ 2 2 + Δ 1 2 - 4 p i 2 p f 2 sin 2 Θ i ) ( ( Δ 2 E f + Δ 1 p f c ) 2 + 4 m 2 c 4 p i 2 p f 2 sin 2 Θ i ) × [ 2 ω 2 p f 2 E i - c p i cos Θ i × E f [ 2 Δ 2 2 ( Δ 2 2 - Δ 1 2 ) + 8 p i 2 p f 2 sin 2 Θ i ( Δ 2 2 + Δ 1 2 ) ] + p f c [ 2 Δ 1 Δ 2 ( Δ 2 2 - Δ 1 2 ) + 16 Δ 1 Δ 2 p i 2 p f 2 sin 2 Θ i ] Δ 2 2 + 4 p i 2 p f 2 sin 2 Θ i + 2 2 ω 2 p i 2 sin 2 Θ i ( 2 Δ 1 Δ 2 p f c + 2 Δ 2 2 E f + 8 p i 2 p f 2 sin 2 Θ i E f ) E i - c p i cos Θ i + 2 E i 2 p f 2 { 2 ( Δ 2 2 - Δ 1 2 ) ( Δ 2 E f + Δ 1 p f c ) 2 + 8 p i 2 p f 2 sin 2 Θ i [ ( Δ 1 2 + Δ 2 2 ) ( E f 2 + p f 2 c 2 ) + 4 Δ 1 Δ 2 E f p f c ] } ( ( Δ 2 E f + Δ 1 p f c ) 2 + 4 m 2 c 4 p i 2 p f 2 sin 2 Θ i ) + 8 p i 2 p f 2 sin 2 Θ i ( E i 2 + E f 2 ) ( Δ 2 p f c + Δ 1 E f ) E i - c p i cos Θ i ] , I 6 = 16 π E f 2 p i 2 sin 2 Θ i A ( E i - c p i cos Θ i ) 2 ( - Δ 2 2 + Δ 1 2 - 4 p i 2 p f 2 sin 2 Θ i ) , \begin{aligned}\displaystyle I_{1}&\displaystyle=\frac{2\pi A}{\sqrt{\Delta_{2% }^{2}+4p_{i}^{2}p_{f}^{2}\sin^{2}\Theta_{i}}}\ln\left(\frac{\Delta_{2}^{2}+4p_% {i}^{2}p_{f}^{2}\sin^{2}\Theta_{i}-\sqrt{\Delta_{2}^{2}+4p_{i}^{2}p_{f}^{2}% \sin^{2}\Theta_{i}}(\Delta_{1}+\Delta_{2})+\Delta_{1}\Delta_{2}}{-\Delta_{2}^{% 2}-4p_{i}^{2}p_{f}^{2}\sin^{2}\Theta_{i}-\sqrt{\Delta_{2}^{2}+4p_{i}^{2}p_{f}^% {2}\sin^{2}\Theta_{i}}(\Delta_{1}-\Delta_{2})+\Delta_{1}\Delta_{2}}\right)\\ &\displaystyle\times\left[1+\frac{c\Delta_{2}}{p_{f}(E_{i}-cp_{i}\cos\Theta_{i% })}-\frac{p_{i}^{2}c^{2}\sin^{2}\Theta_{i}}{(E_{i}-cp_{i}\cos\Theta_{i})^{2}}-% \frac{2\hbar^{2}\omega^{2}p_{f}\Delta_{2}}{c(E_{i}-cp_{i}\cos\Theta_{i})(% \Delta_{2}^{2}+4p_{i}^{2}p_{f}^{2}\sin^{2}\Theta_{i})}\right],\\ \displaystyle I_{2}&\displaystyle=-\frac{2\pi Ac}{p_{f}(E_{i}-cp_{i}\cos\Theta% _{i})}\ln\left(\frac{E_{f}+p_{f}c}{E_{f}-p_{f}c}\right),\\ \displaystyle I_{3}&\displaystyle=\frac{2\pi A}{\sqrt{(\Delta_{2}E_{f}+\Delta_% {1}p_{f}c)^{2}+4m^{2}c^{4}p_{i}^{2}p_{f}^{2}\sin^{2}\Theta_{i}}}\\ &\displaystyle\times\ln\Bigg(\Big((E_{f}+p_{f}c)(4p_{i}^{2}p_{f}^{2}\sin^{2}% \Theta_{i}(E_{f}-p_{f}c)+(\Delta_{1}+\Delta_{2})((\Delta_{2}E_{f}+\Delta_{1}p_% {f}c)\\ &\displaystyle-\sqrt{(\Delta_{2}E_{f}+\Delta_{1}p_{f}c)^{2}+4m^{2}c^{4}p_{i}^{% 2}p_{f}^{2}\sin^{2}\Theta_{i}}))\Big)\Big((E_{f}-p_{f}c)(4p_{i}^{2}p_{f}^{2}% \sin^{2}\Theta_{i}(-E_{f}-p_{f}c)\\ &\displaystyle+(\Delta_{1}-\Delta_{2})((\Delta_{2}E_{f}+\Delta_{1}p_{f}c)-% \sqrt{(\Delta_{2}E_{f}+\Delta_{1}p_{f}c)^{2}+4m^{2}c^{4}p_{i}^{2}p_{f}^{2}\sin% ^{2}\Theta_{i}}))\Big)^{-1}\Bigg)\\ &\displaystyle\times\left[-\frac{(\Delta_{2}^{2}+4p_{i}^{2}p_{f}^{2}\sin^{2}% \Theta_{i})(E_{f}^{3}+E_{f}p_{f}^{2}c^{2})+p_{f}c(2(\Delta_{1}^{2}-4p_{i}^{2}p% _{f}^{2}\sin^{2}\Theta_{i})E_{f}p_{f}c+\Delta_{1}\Delta_{2}(3E_{f}^{2}+p_{f}^{% 2}c^{2}))}{(\Delta_{2}E_{f}+\Delta_{1}p_{f}c)^{2}+4m^{2}c^{4}p_{i}^{2}p_{f}^{2% }\sin^{2}\Theta_{i}}\right.\\ &\displaystyle-\frac{c(\Delta_{2}E_{f}+\Delta_{1}p_{f}c)}{p_{f}(E_{i}-cp_{i}% \cos\Theta_{i})}\\ &\displaystyle-\frac{4E_{i}^{2}p_{f}^{2}(2(\Delta_{2}E_{f}+\Delta_{1}p_{f}c)^{% 2}-4m^{2}c^{4}p_{i}^{2}p_{f}^{2}\sin^{2}\Theta_{i})(\Delta_{1}E_{f}+\Delta_{2}% p_{f}c)}{((\Delta_{2}E_{f}+\Delta_{1}p_{f}c)^{2}+4m^{2}c^{4}p_{i}^{2}p_{f}^{2}% \sin^{2}\Theta_{i})^{2}}\\ &\displaystyle+\left.\frac{8p_{i}^{2}p_{f}^{2}m^{2}c^{4}\sin^{2}\Theta_{i}(E_{% i}^{2}+E_{f}^{2})-2\hbar^{2}\omega^{2}p_{i}^{2}\sin^{2}\Theta_{i}p_{f}c(\Delta% _{2}E_{f}+\Delta_{1}p_{f}c)+2\hbar^{2}\omega^{2}p_{f}m^{2}c^{3}(\Delta_{2}E_{f% }+\Delta_{1}p_{f}c)}{(E_{i}-cp_{i}\cos\Theta_{i})((\Delta_{2}E_{f}+\Delta_{1}p% _{f}c)^{2}+4m^{2}c^{4}p_{i}^{2}p_{f}^{2}\sin^{2}\Theta_{i})}\right],\\ \displaystyle I_{4}&\displaystyle=-\frac{4\pi Ap_{f}c(\Delta_{2}E_{f}+\Delta_{% 1}p_{f}c)}{(\Delta_{2}E_{f}+\Delta_{1}p_{f}c)^{2}+4m^{2}c^{4}p_{i}^{2}p_{f}^{2% }\sin^{2}\Theta_{i}}-\frac{16\pi E_{i}^{2}p_{f}^{2}A(\Delta_{2}E_{f}+\Delta_{1% }p_{f}c)^{2}}{((\Delta_{2}E_{f}+\Delta_{1}p_{f}c)^{2}+4m^{2}c^{4}p_{i}^{2}p_{f% }^{2}\sin^{2}\Theta_{i})^{2}},\\ \displaystyle I_{5}&\displaystyle=\frac{4\pi A}{(-\Delta_{2}^{2}+\Delta_{1}^{2% }-4p_{i}^{2}p_{f}^{2}\sin^{2}\Theta_{i})((\Delta_{2}E_{f}+\Delta_{1}p_{f}c)^{2% }+4m^{2}c^{4}p_{i}^{2}p_{f}^{2}\sin^{2}\Theta_{i})}\\ &\displaystyle\times\left[\frac{\hbar^{2}\omega^{2}p_{f}^{2}}{E_{i}-cp_{i}\cos% \Theta_{i}}\right.\\ &\displaystyle\times\frac{E_{f}[2\Delta_{2}^{2}(\Delta_{2}^{2}-\Delta_{1}^{2})% +8p_{i}^{2}p_{f}^{2}\sin^{2}\Theta_{i}(\Delta_{2}^{2}+\Delta_{1}^{2})]+p_{f}c[% 2\Delta_{1}\Delta_{2}(\Delta_{2}^{2}-\Delta_{1}^{2})+16\Delta_{1}\Delta_{2}p_{% i}^{2}p_{f}^{2}\sin^{2}\Theta_{i}]}{\Delta_{2}^{2}+4p_{i}^{2}p_{f}^{2}\sin^{2}% \Theta_{i}}\\ &\displaystyle+\frac{2\hbar^{2}\omega^{2}p_{i}^{2}\sin^{2}\Theta_{i}(2\Delta_{% 1}\Delta_{2}p_{f}c+2\Delta_{2}^{2}E_{f}+8p_{i}^{2}p_{f}^{2}\sin^{2}\Theta_{i}E% _{f})}{E_{i}-cp_{i}\cos\Theta_{i}}\\ &\displaystyle+\frac{2E_{i}^{2}p_{f}^{2}\{2(\Delta_{2}^{2}-\Delta_{1}^{2})(% \Delta_{2}E_{f}+\Delta_{1}p_{f}c)^{2}+8p_{i}^{2}p_{f}^{2}\sin^{2}\Theta_{i}[(% \Delta_{1}^{2}+\Delta_{2}^{2})(E_{f}^{2}+p_{f}^{2}c^{2})+4\Delta_{1}\Delta_{2}% E_{f}p_{f}c]\}}{((\Delta_{2}E_{f}+\Delta_{1}p_{f}c)^{2}+4m^{2}c^{4}p_{i}^{2}p_% {f}^{2}\sin^{2}\Theta_{i})}\\ &\displaystyle+\left.\frac{8p_{i}^{2}p_{f}^{2}\sin^{2}\Theta_{i}(E_{i}^{2}+E_{% f}^{2})(\Delta_{2}p_{f}c+\Delta_{1}E_{f})}{E_{i}-cp_{i}\cos\Theta_{i}}\right],% \\ \displaystyle I_{6}&\displaystyle=\frac{16\pi E_{f}^{2}p_{i}^{2}\sin^{2}\Theta% _{i}A}{(E_{i}-cp_{i}\cos\Theta_{i})^{2}(-\Delta_{2}^{2}+\Delta_{1}^{2}-4p_{i}^% {2}p_{f}^{2}\sin^{2}\Theta_{i})},\end{aligned}
  97. A \displaystyle A
  98. Z Z
  99. Z Z
  100. Z 2 Z^{2}

Bresenham's_line_algorithm.html

  1. ( x 0 , y 0 ) (x_{0},y_{0})
  2. ( x 1 , y 1 ) (x_{1},y_{1})
  3. x 0 x 1 x_{0}\leq x_{1}
  4. y 0 y 1 y_{0}\leq y_{1}
  5. x 1 - x 0 x_{1}-x_{0}
  6. y 1 - y 0 y_{1}-y_{0}
  7. x 0 x_{0}
  8. x 1 x_{1}
  9. y 0 y_{0}
  10. y 1 y_{1}
  11. y - y 0 y 1 - y 0 = x - x 0 x 1 - x 0 \frac{y-y_{0}}{y_{1}-y_{0}}=\frac{x-x_{0}}{x_{1}-x_{0}}
  12. y = y 1 - y 0 x 1 - x 0 ( x - x 0 ) + y 0 y=\frac{y_{1}-y_{0}}{x_{1}-x_{0}}(x-x_{0})+y_{0}
  13. ( y 1 - y 0 ) / ( x 1 - x 0 ) (y_{1}-y_{0})/(x_{1}-x_{0})
  14. y 0 y_{0}
  15. y = f ( x ) = m x + b y=f(x)=mx+b
  16. Δ y / Δ x \Delta y/\Delta x
  17. y = m x + b y = ( Δ y ) ( Δ x ) x + b ( Δ x ) y = ( Δ y ) x + ( Δ x ) b 0 = ( Δ y ) x - ( Δ x ) y + ( Δ x ) b \begin{aligned}\displaystyle y&\displaystyle=mx+b\\ \displaystyle y&\displaystyle=\frac{(\Delta y)}{(\Delta x)}x+b\\ \displaystyle(\Delta x)y&\displaystyle=(\Delta y)x+(\Delta x)b\\ \displaystyle 0&\displaystyle=(\Delta y)x-(\Delta x)y+(\Delta x)b\end{aligned}
  18. f ( x , y ) = 0 = A x + B y + C f(x,y)=0=Ax+By+C
  19. A = Δ y A=\Delta y
  20. B = - Δ x B=-\Delta x
  21. C = ( Δ x ) b C=(\Delta x)b
  22. f ( x , y ) = 0 f(x,y)=0
  23. ( x , y ) (x,y)
  24. f ( x , y ) 0 f(x,y)\neq 0
  25. y = 1 2 x + 1 y=\frac{1}{2}x+1
  26. f ( x , y ) = x - 2 y + 2 f(x,y)=x-2y+2
  27. f ( 2 , 2 ) = x - 2 y + 2 = ( 2 ) - 2 ( 2 ) + 2 = 2 - 4 + 2 = 0 f(2,2)=x-2y+2=(2)-2(2)+2=2-4+2=0
  28. f ( 2 , 3 ) = ( 2 ) - 2 ( 3 ) + 2 = 2 - 6 + 2 = - 2 f(2,3)=(2)-2(3)+2=2-6+2=-2
  29. f ( 2 , 1 ) = ( 2 ) - 2 ( 1 ) + 2 = 2 - 2 + 2 = 2 f(2,1)=(2)-2(1)+2=2-2+2=2
  30. f ( x 0 , y 0 ) = 0 f(x_{0},y_{0})=0
  31. ( x 0 + 1 , y 0 ) (x_{0}+1,y_{0})
  32. ( x 0 + 1 , y 0 + 1 ) (x_{0}+1,y_{0}+1)
  33. x 0 + 1 x_{0}+1
  34. f ( x 0 + 1 , y 0 + 1 / 2 ) f(x_{0}+1,y_{0}+1/2)
  35. ( x 0 + 1 , y 0 + 1 ) (x_{0}+1,y_{0}+1)
  36. ( x 0 + 1 , y 0 ) (x_{0}+1,y_{0})
  37. D = f ( x 0 + 1 , y 0 + 1 / 2 ) + f ( x 0 , y 0 ) D=f(x_{0}+1,y_{0}+1/2)+f(x_{0},y_{0})
  38. f ( x 0 , y 0 ) = 0 f(x_{0},y_{0})=0
  39. D = [ A ( x 0 + 1 ) + B ( y 0 + 1 2 ) + C ] - [ A x 0 + B y 0 + C ] D=\left[A(x_{0}+1)+B(y_{0}+\frac{1}{2})+C\right]-\left[Ax_{0}+By_{0}+C\right]
  40. = [ A x 0 + B y 0 + C + A + 1 2 B ] - [ A x 0 + B y 0 + C ] =\left[Ax_{0}+By_{0}+C+A+\frac{1}{2}B\right]-\left[Ax_{0}+By_{0}+C\right]
  41. = A + 1 2 B =A+\frac{1}{2}B
  42. ( x 0 + 1 , y 0 + 1 ) (x_{0}+1,y_{0}+1)
  43. ( x 0 + 1 , y 0 ) (x_{0}+1,y_{0})
  44. f ( x 0 + 2 , y 0 + 1 / 2 ) - f ( x 0 + 1 , y 0 + 1 / 2 ) = A = Δ y f(x_{0}+2,y_{0}+1/2)-f(x_{0}+1,y_{0}+1/2)=A=\Delta y
  45. f ( x 0 + 2 , y 0 + 3 / 2 ) - f ( x 0 + 1 , y 0 + 1 / 2 ) = A + B = Δ y - Δ x f(x_{0}+2,y_{0}+3/2)-f(x_{0}+1,y_{0}+1/2)=A+B=\Delta y-\Delta x
  46. ( x 0 + 2 , y 0 + 1 ) (x_{0}+2,y_{0}+1)
  47. ( x 0 + 2 , y 0 ) (x_{0}+2,y_{0})
  48. f ( x , y ) = x - 2 y + 2 f(x,y)=x-2y+2

Brewster's_angle.html

  1. θ B = arctan ( n 2 n 1 ) , \theta_{\mathrm{B}}=\arctan\!\left(\frac{n_{2}}{n_{1}}\right)\!,
  2. θ 1 + θ 2 = 90 , \theta_{1}+\theta_{2}=90^{\circ},
  3. n 1 sin θ 1 = n 2 sin θ 2 , n_{1}\sin\theta_{1}=n_{2}\sin\theta_{2},
  4. n 1 sin θ B = n 2 sin ( 90 - θ B ) = n 2 cos θ B . n_{1}\sin\theta_{\mathrm{B}}=n_{2}\sin(90^{\circ}-\theta_{\mathrm{B}})=n_{2}% \cos\theta_{\mathrm{B}}.
  5. θ B = arctan ( n 2 n 1 ) . \theta_{\mathrm{B}}=\arctan\!\left(\frac{n_{2}}{n_{1}}\right)\!.

Brightness.html

  1. μ = R + G + B 3 \mu={R+G+B\over 3}

Brightness_temperature.html

  1. ν \nu
  2. I ν = 2 h ν 3 c 2 1 e h ν k T - 1 I_{\nu}=\frac{2h\nu^{3}}{c^{2}}\frac{1}{e^{\frac{h\nu}{kT}}-1}
  3. I ν I_{\nu}
  4. ν \nu
  5. ν + d ν \nu+d\nu
  6. T T
  7. h h
  8. ν \nu
  9. c c
  10. k k
  11. ϵ \epsilon
  12. T b - 1 = k h ν ln [ 1 + e h ν k T - 1 ϵ ] T_{b}^{-1}=\frac{k}{h\nu}\,\,\text{ln}\left[1+\frac{e^{\frac{h\nu}{kT}}-1}{% \epsilon}\right]
  13. h ν k T h\nu\ll kT
  14. I ν = 2 ν 2 k T c 2 I_{\nu}=\frac{2\nu^{2}kT}{c^{2}}
  15. T b = ϵ T T_{b}=\epsilon T\,
  16. ν \nu
  17. T b = h ν k ln - 1 ( 1 + 2 h ν 3 I ν c 2 ) T_{b}=\frac{h\nu}{k}\ln^{-1}\left(1+\frac{2h\nu^{3}}{I_{\nu}c^{2}}\right)
  18. h ν k T h\nu\ll kT
  19. T b = I ν c 2 2 k ν 2 T_{b}=\frac{I_{\nu}c^{2}}{2k\nu^{2}}
  20. Δ ν ν \Delta\nu\ll\nu
  21. I I
  22. T b = I c 2 2 k ν 2 Δ ν T_{b}=\frac{Ic^{2}}{2k\nu^{2}\Delta\nu}
  23. I λ = 2 h c 2 λ 5 1 e h c k T λ - 1 I_{\lambda}=\frac{2hc^{2}}{\lambda^{5}}\frac{1}{e^{\frac{hc}{kT\lambda}}-1}
  24. T b = h c k λ ln - 1 ( 1 + 2 h c 2 I λ λ 5 ) T_{b}=\frac{hc}{k\lambda}\ln^{-1}\left(1+\frac{2hc^{2}}{I_{\lambda}\lambda^{5}% }\right)
  25. h c / λ k T hc/\lambda\ll kT
  26. T b = I λ λ 4 2 k c T_{b}=\frac{I_{\lambda}\lambda^{4}}{2kc}
  27. I I
  28. L c L_{c}
  29. T b = π I λ 2 L c 4 k c ln 2 T_{b}=\frac{\pi I\lambda^{2}L_{c}}{4kc\ln{2}}

Brouwer_fixed-point_theorem.html

  1. f ( x ) = x + 1 f(x)=x+1
  2. f ( x ) = x + 1 2 f(x)=\frac{x+1}{2}
  3. f ( r , θ ) = ( r , θ + π / 4 ) f(r,\theta)=(r,\theta+\pi/4)
  4. K = B ( 0 ) ¯ K=\overline{B(0)}
  5. n \mathbb{R}^{n}
  6. f : K K f:K\to K
  7. f f
  8. p B ( 0 ) p\in B(0)
  9. f f
  10. p p
  11. f f
  12. B ( 0 ) B(0)
  13. K K
  14. f f
  15. p B ( 0 ) p\in B(0)
  16. f f
  17. p p
  18. f f
  19. deg p ( f ) = x f - 1 ( p ) sign ( det ( D f ( x ) ) ) . \operatorname{deg}_{p}(f)=\sum_{x\in f^{-1}(p)}\operatorname{sign}\left(\det(% Df(x))\right).
  20. f f
  21. g g
  22. H t ( x ) = t f + ( 1 - t ) g H_{t}(x)=tf+(1-t)g
  23. 0 t 1 0\leq t\leq 1
  24. p p
  25. H t H_{t}
  26. deg p f = deg p g \deg_{p}f=\deg_{p}g
  27. K K
  28. H ( t , x ) = x - t f ( x ) H(t,x)=x-tf(x)
  29. g ( x ) = x - f ( x ) g(x)=x-f(x)
  30. g g
  31. g - 1 ( 0 ) g^{-1}(0)
  32. g - 1 ( 0 ) g^{-1}(0)
  33. 0 < B ω = B f * ( ω ) = B d f * ( ω ) = B f * ( d ω ) = B f * ( 0 ) = 0 0<\int_{\partial B}\omega=\int_{\partial B}f^{*}(\omega)=\int_{B}df^{*}(\omega% )=\int_{B}f^{*}(d\omega)=\int_{B}f^{*}(0)=0
  34. f : Δ n Δ n f:\Delta^{n}\to\Delta^{n}
  35. Δ n = { P n + 1 i = 0 n P i = 1 and P i 0 for all i } \Delta^{n}=\left\{P\in\mathbb{R}^{n+1}\mid\sum_{i=0}^{n}{P_{i}}=1\mbox{ and }~% {}P_{i}\geq 0\mbox{ for all }~{}i\right\}
  36. P Δ n P\in\Delta^{n}
  37. f ( P ) Δ n f(P)\in\Delta^{n}
  38. i = 0 n P i = 1 = i = 0 n f ( P ) i \sum_{i=0}^{n}{P_{i}}=1=\sum_{i=0}^{n}{f(P)_{i}}
  39. P Δ n P\in\Delta^{n}
  40. j { 0 , , n } j\in\{0,\cdots,n\}
  41. j j
  42. P P
  43. j j
  44. f ( P ) j P j f(P)_{j}\leq P_{j}
  45. P P
  46. Δ n \Delta^{n}
  47. j j
  48. Δ n \Delta^{n}
  49. P P
  50. j j
  51. f ( P ) j P j f(P)_{j}\leq P_{j}
  52. P P
  53. j : f ( P ) j P j \forall j:f(P)_{j}\leq P_{j}
  54. P P
  55. f ( P ) f(P)
  56. f ( P ) = P f(P)=P
  57. P P
  58. f f
  59. Δ n \Delta^{n}
  60. φ ( t ) := B det D g t ( x ) d x \varphi(t):=\int_{B}\operatorname{det}Dg^{t}(x)dx
  61. n ( - 1 ) n Tr ( f | H n ( B ) ) \displaystyle\sum_{n}(-1)^{n}\operatorname{Tr}(f|H_{n}(B))
  62. H 0 ( B ) H_{0}(B)
  63. y 0 = 1 - x 2 2 and y n = x n - 1 for n 1. y_{0}=\sqrt{1-\|x\|_{2}^{2}}\qquad\,\text{ and }\qquad y_{n}=x_{n-1}\quad\,% \text{ for }\quad n\geq 1.
  64. X X
  65. f : X X f:X\rightarrow X
  66. { U 1 , , U m } \{U_{1},\ldots,U_{m}\}
  67. U i U j U_{i}\cap U_{j}\neq\emptyset
  68. | i - j | 1 |i-j|\leq 1
  69. 1 / 2 {1}/{2}

Brown_dwarf.html

  1. 10 g / cm 3 ρ c 10 3 g / cm 3 10\,\mathrm{g/cm^{3}}\,\lesssim\,\rho_{c}\,\lesssim\,10^{3}\,\mathrm{{g}/{cm^{% 3}}}
  2. T c 3 × 10 6 K T_{c}\lesssim 3\times 10^{6}\,\mathrm{K}
  3. P c 10 5 Mbar . P_{c}\sim 10^{5}\,\mathrm{Mbar}.

Brownian_motion.html

  1. Δ \Delta
  2. ϕ ( Δ ) \phi(\Delta)
  3. ρ ( x , t + τ ) \displaystyle\rho(x,t+\tau)
  4. ρ t = 2 ρ x 2 - + Δ 2 2 τ ϕ ( Δ ) d Δ + higher order even moments \frac{\partial\rho}{\partial t}=\frac{\partial^{2}\rho}{\partial x^{2}}\cdot% \int_{-\infty}^{+\infty}\frac{\Delta^{2}}{2\,\tau}\cdot\phi(\Delta)\,\mathrm{d% }\Delta+\mathrm{higher\;order\;even\;moments}
  5. Δ \Delta
  6. D = - + Δ 2 2 τ ϕ ( Δ ) d Δ D=\int_{-\infty}^{+\infty}\frac{\Delta^{2}}{2\,\tau}\cdot\phi(\Delta)\,\mathrm% {d}\Delta
  7. ρ t = D 2 ρ x 2 , \frac{\partial\rho}{\partial t}=D\cdot\frac{\partial^{2}\rho}{\partial x^{2}},
  8. ρ ( x , t ) = N 4 π D t e - x 2 4 D t . \rho(x,t)=\frac{N}{\sqrt{4\pi Dt}}e^{-\frac{x^{2}}{4Dt}}.
  9. x 2 ¯ = 2 D t . \overline{x^{2}}=2\,D\,t.
  10. μ = 1 6 π η r \mu=\tfrac{1}{6\pi\eta r}
  11. ρ = ρ 0 e - m g h k B T , \rho=\rho_{0}e^{-\frac{mgh}{k_{B}T}},
  12. J = - D d ρ d h , J=-D\frac{d\rho}{dh},
  13. v = D m g k B T . v=\frac{Dmg}{k_{B}T}.
  14. x 2 ¯ 2 t = D = μ k B T = μ R T N = R T 6 π η r N . \frac{\overline{x^{2}}}{2t}=D=\mu k_{B}T=\frac{\mu RT}{N}=\frac{RT}{6\pi\eta rN}.
  15. k = p 0 / k k^{\prime}=p_{0}/k
  16. p 0 p_{0}
  17. ( Δ x ) 2 ¯ \overline{(\Delta x)^{2}}
  18. ( Δ x ) 2 ¯ = 2 D t = t 32 81 m u 2 π μ a = t 64 27 1 2 m u 2 3 π μ a , \overline{(\Delta x)^{2}}=2Dt=t\frac{32}{81}\frac{mu^{2}}{\pi\mu a}=t\frac{64}% {27}\frac{\frac{1}{2}mu^{2}}{3\pi\mu a},
  19. m u 2 / 2 mu^{2}/2
  20. [ m - ( n - m ) ] [ m + n - m ] = 2 m - n n , \frac{[m-(n-m)]}{[m+n-m]}=\frac{2m-n}{n},
  21. P m , n = ( n m ) 2 - n , P_{m,n}={\left({{n}\atop{m}}\right)}2^{-n},
  22. 2 m - n ¯ = m = n 2 n ( 2 m - n ) P m , n = n n ! 2 n [ ( n 2 ) ! ] 2 . \overline{2m-n}=\sum_{m=\frac{n}{2}}^{n}(2m-n)P_{m,n}=\frac{nn!}{2^{n}\left[% \left(\frac{n}{2}\right)!\right]^{2}}.
  23. n ! ( n e ) n 2 π n , n!\approx\left(\frac{n}{e}\right)^{n}\sqrt{2\pi n},
  24. 2 m - n ¯ 2 n π , \overline{2m-n}\approx\sqrt{\frac{2n}{\pi}},
  25. M U 2 / 2 MU^{2}/2
  26. m u 2 / 2 mu^{2}/2
  27. ( N N R ) = N ! N R ! ( N - N R ) ! {\left({{N}\atop{N_{R}}}\right)}=\frac{N!}{N_{R}!(N-N_{R})!}
  28. P N ( N R ) = N ! 2 N N R ! ( N - N R ) ! P_{N}(N_{R})=\frac{N!}{2^{N}N_{R}!(N-N_{R})!}
  29. W t - W s 𝒩 ( 0 , t - s ) W_{t}-W_{s}\sim\mathcal{N}(0,t-s)
  30. 0 s t 0\leq s\leq t
  31. 𝒩 ( μ , σ 2 ) \mathcal{N}(\mu,\sigma^{2})
  32. 0 s 1 t 1 s 2 t 2 0\leq s_{1}\leq t_{1}\leq s_{2}\leq t_{2}
  33. W t 1 - W s 1 W_{t_{1}}-W_{s_{1}}
  34. W t 2 - W s 2 W_{t_{2}}-W_{s_{2}}
  35. [ W t , W t ] = t [W_{t},W_{t}]=t
  36. 𝒩 ( 0 , 1 ) \mathcal{N}(0,1)
  37. 𝒜 \mathcal{A}
  38. Δ LB = 1 det ( g ) i = 1 m x i ( det ( g ) j = 1 m g i j x j ) , \Delta_{\mathrm{LB}}=\frac{1}{\sqrt{\det(g)}}\sum_{i=1}^{m}\frac{\partial}{% \partial x_{i}}\left(\sqrt{\det(g)}\sum_{j=1}^{m}g^{ij}\frac{\partial}{% \partial x_{j}}\right),
  39. v v_{\star}
  40. M V 2 m v 2 MV^{2}\approx mv_{\star}^{2}
  41. m M m\ll M
  42. v v_{\star}

Bucket_argument.html

  1. F Cfgl = m Ω 2 r , F_{\mathrm{Cfgl}}=m\mathit{\Omega}^{2}r\ ,
  2. F g = m g , F_{\mathrm{g}}=mg\ ,
  3. tan φ = F Cfgl F g = Ω 2 r g , \tan\varphi=\frac{F_{\mathrm{Cfgl}}}{F_{\mathrm{g}}}=\frac{\mathit{\Omega}^{2}% r}{g}\ ,
  4. tan φ = d h d r , \tan\varphi=\frac{\mathrm{d}h}{\mathrm{d}r}\ ,
  5. d h d r = Ω 2 r g , \frac{\mathrm{d}h}{\mathrm{d}r}=\frac{\mathit{\Omega}^{2}r}{g}\ ,
  6. h ( r ) = h ( 0 ) + 1 2 g ( Ω r ) 2 , h(r)=h(0)+\frac{1}{2g}\left(\mathit{\Omega}r\right)^{2}\ ,
  7. U Cfgl = - 1 2 m Ω 2 r 2 , {U}_{\mathrm{Cfgl}}=-\frac{1}{2}m\Omega^{2}r^{2}\ ,
  8. F Cfgl = - r U Cfgl F_{\mathrm{Cfgl}}=-\frac{\partial}{\partial r}{U}_{\mathrm{Cfgl}}
  9. = m Ω 2 r . =m\Omega^{2}r\ .
  10. h ( r ) h(r)\,
  11. g h ( r ) gh(r)
  12. U = U 0 + g h ( r ) - 1 2 Ω 2 r 2 {U}={U}_{0}+gh(r)-\frac{1}{2}\Omega^{2}r^{2}\,
  13. U 0 {U}_{0}
  14. h ( r ) = Ω 2 2 g r 2 + h ( 0 ) , h(r)=\frac{\Omega^{2}}{2g}r^{2}+h(0)\ ,

Buckingham_π_theorem.html

  1. f ( q 1 , q 2 , , q n ) = 0 f(q_{1},q_{2},\ldots,q_{n})=0\,
  2. F ( π 1 , π 2 , , π p ) = 0 F(\pi_{1},\pi_{2},\ldots,\pi_{p})=0\,
  3. π i = q 1 a 1 q 2 a 2 q n a n \pi_{i}=q_{1}^{a_{1}}\,q_{2}^{a_{2}}\cdots q_{n}^{a_{n}}\,
  4. / t 2 = 1 t - 2 \ell/t^{2}=\ell^{1}t^{-2}
  5. ( 1 , - 2 ) (1,-2)
  6. M [ a 1 a n ] M\begin{bmatrix}a_{1}\\ \vdots\\ a_{n}\end{bmatrix}
  7. q 1 a 1 q 2 a 2 q n a n . q_{1}^{a_{1}}\,q_{2}^{a_{2}}\cdots q_{n}^{a_{n}}.\,
  8. \ell
  9. D , T t , V / t . D\sim\ell,\ T\sim t,\ V\sim\ell/t.
  10. M = [ 1 0 1 0 1 - 1 ] M=\begin{bmatrix}1&0&1\\ 0&1&-1\end{bmatrix}
  11. \ell
  12. 1 t - 1 = / t \ell^{1}t^{-1}=\ell/t
  13. π = D a 1 T a 2 V a 3 \pi=D^{a_{1}}T^{a_{2}}V^{a_{3}}
  14. a = [ a 1 , a 2 , a 3 ] a=[a_{1},a_{2},a_{3}]
  15. a = [ - 1 1 1 ] . a=\begin{bmatrix}-1\\ 1\\ 1\end{bmatrix}.
  16. π = D - 1 T 1 V 1 = T V / D \pi=D^{-1}T^{1}V^{1}=TV/D
  17. π ( ) - 1 ( t ) 1 ( / t ) 1 1 \pi\sim(\ell)^{-1}(t)^{1}(\ell/t)^{1}\sim 1
  18. f ( π ) = 0 f(\pi)=0\,
  19. T = C D V T=\frac{CD}{V}
  20. C = f - 1 ( 0 ) C=f^{-1}(0)
  21. D = V T D=VT
  22. f ( π ) = 0 f(\pi)=0
  23. f ( π ) = π - 1 = V T / D - 1 = 0 f(\pi)=\pi-1=VT/D-1=0\,
  24. f ( T , M , L , g ) = 0. f(T,M,L,g)=0.\,
  25. f ( π ) = 0 f(\pi)=0\,
  26. π = T a 1 M a 2 L a 3 g a 4 \pi=T^{a_{1}}M^{a_{2}}L^{a_{3}}g^{a_{4}}\,
  27. T = t , M = m , L = , g = / t 2 . T=t,M=m,L=\ell,g=\ell/t^{2}.\,
  28. M = [ 1 0 0 - 2 0 1 0 0 0 0 1 1 ] M=\begin{bmatrix}1&0&0&-2\\ 0&1&0&0\\ 0&0&1&1\end{bmatrix}
  29. t - 2 m 0 1 t^{-2}m^{0}\ell^{1}
  30. a = [ 2 0 - 1 1 ] . a=\begin{bmatrix}2\\ 0\\ -1\\ 1\end{bmatrix}.
  31. π \displaystyle\pi
  32. π = ( t ) 2 ( m ) 0 ( ) - 1 ( / t 2 ) 1 = 1 \pi=(t)^{2}(m)^{0}(\ell)^{-1}(\ell/t^{2})^{1}=1\,
  33. g - 2 T - L \vec{g}-2\vec{T}-\vec{L}
  34. f ( g T 2 / L ) = 0. f(gT^{2}/L)=0.

Buffer_solution.html

  1. β = d n d ( p [ H + ] ) \beta=\frac{dn}{d(p[H^{+}])}
  2. d n d ( p H ) = 2.303 ( [ H + ] + C A K a [ H + ] ( K a + [ H + ] ) 2 + [ O H - ] ) \frac{dn}{d(pH)}=2.303\left([H^{+}]+\frac{C_{A}K_{a}[H^{+}]}{\left(K_{a}+[H^{+% }]\right)^{2}}+[OH^{-}]\right)
  3. K a = [ H + ] [ A - ] [ H A ] K_{a}=\frac{[H^{+}][A^{-}]}{[HA]}
  4. K a = x ( x + y ) C 0 - x K_{a}=\frac{x(x+y)}{C_{0}-x}
  5. x 2 + ( K a + y ) x - K a C 0 = 0 x^{2}+(K_{a}+y)x-K_{a}C_{0}=0
  6. C A = [ A 3 - ] + β 1 [ A 3 - ] [ H + ] + β 2 [ A 3 - ] [ H + ] 2 + β 3 [ A 3 - ] [ H + ] 3 C_{A}=[A^{3-}]+\beta_{1}[A^{3-}][H^{+}]+\beta_{2}[A^{3-}][H^{+}]^{2}+\beta_{3}% [A^{3-}][H^{+}]^{3}
  7. C H = [ H + ] + β 1 [ A 3 - ] [ H + ] + 2 β 2 [ A 3 - ] [ H + ] 2 + 3 β 3 [ A 3 - ] [ H + ] 3 - K w [ H ] - 1 C_{H}=[H^{+}]+\beta_{1}[A^{3-}][H^{+}]+2\beta_{2}[A^{3-}][H^{+}]^{2}+3\beta_{3% }[A^{3-}][H^{+}]^{3}-K_{w}[H]^{-1}
  8. log β 1 = p K a 3 , log β 2 = p K a 2 + p K a 3 , log β 3 = p K a 1 + p K a 2 + p K a 3 \log\beta_{1}=pK_{a3},\ \log\beta_{2}=pK_{a2}+pK_{a3},\ \log\beta_{3}=pK_{a1}+% pK_{a2}+pK_{a3}

Burali-Forti_paradox.html

  1. Ω \Omega
  2. Ω \Omega
  3. Ω + 1 \Omega+1
  4. Ω \Omega
  5. Ω \Omega
  6. Ω \Omega
  7. Ω < Ω + 1 \Omega<\Omega+1
  8. Ω + 1 < Ω \Omega+1<\Omega
  9. Ω \Omega
  10. α \alpha
  11. α \alpha
  12. Ω \Omega
  13. Ω \Omega
  14. Ω \Omega
  15. Ω \Omega
  16. α \alpha
  17. α \alpha
  18. Ω \Omega
  19. Ω \Omega
  20. P P

Busy_beaver.html

  1. S ( n ) = max { s ( M ) | M E n } = S(n)=\max\{s(M)|M\in E_{n}\}=\,\!
  2. S ( n ) Σ ( n ) S(n)\geq\Sigma(n)\,\!
  3. S ( n ) ( 2 n - 1 ) Σ ( 3 n + 3 ) S(n)\leq(2n-1)\Sigma(3n+3)\,\!
  4. S ( n ) < Σ ( 3 n + 6 ) ; S(n)<\Sigma(3n+6)\,\!;
  5. S ( n ) Σ ( n + 8 n / log 2 n + c ) . S(n)\leq\Sigma(n+\lceil 8n/\log_{2}n\rceil+c).\,
  6. Σ ( 2 k ) > 3 k - 2 3 > A ( k - 2 , k - 2 ) ( k 2 ) , \Sigma(2k)>3\uparrow^{k-2}3>A(k-2,k-2)\quad(k\geq 2),
  7. \uparrow
  8. Σ ( 10 ) > 3 3 = 3 3 3 3 = 3 3 3 . . . 3 \Sigma(10)>3\uparrow\uparrow\uparrow 3=3\uparrow\uparrow 3^{3^{3}}=3^{3^{3^{.^% {.^{.^{3}}}}}}
  9. Σ ( 12 ) > 3 3 = g 1 , \Sigma(12)>3\uparrow\uparrow\uparrow\uparrow 3=g_{1},
  10. 3 3 = 3 3 3 = 7 , 625 , 597 , 484 , 987 3\uparrow\uparrow 3=3^{3^{3}}=7,625,597,484,987
  11. Σ ( 8 ) \Sigma(8)
  12. S ( 10 ) > Σ ( 10 ) > 3 3 S(10)>\Sigma(10)>3\uparrow\uparrow\uparrow 3

Butterfly_effect.html

  1. f t f^{t}
  2. f t f^{t}
  3. 0 < d ( x , y ) < δ 0<d(x,y)<\delta
  4. d ( f τ ( x ) , f τ ( y ) ) > e a τ d ( x , y ) d(f^{\tau}(x),f^{\tau}(y))>\mathrm{e}^{a\tau}\,d(x,y)
  5. x n + 1 = 4 x n ( 1 - x n ) , 0 x 0 1 , x_{n+1}=4x_{n}(1-x_{n}),\quad 0\leq x_{0}\leq 1,
  6. x n = sin 2 ( 2 n θ π ) x_{n}=\sin^{2}(2^{n}\theta\pi)
  7. θ \theta
  8. θ = 1 π sin - 1 ( x 0 1 / 2 ) \theta=\tfrac{1}{\pi}\sin^{-1}(x_{0}^{1/2})
  9. θ \theta
  10. x n x_{n}
  11. θ \theta
  12. θ \theta
  13. x n x_{n}
  14. x n x_{n}

Bzip2.html

  1. i i
  2. i th i^{\mathrm{th}}

C*-algebra.html

  1. x * * = ( x * ) * = x x^{**}=(x^{*})^{*}=x
  2. ( x + y ) * = x * + y * (x+y)^{*}=x^{*}+y^{*}
  3. ( x y ) * = y * x * (xy)^{*}=y^{*}x^{*}
  4. ( λ x ) * = λ ¯ x * . (\lambda x)^{*}=\overline{\lambda}x^{*}.
  5. x * x = x x * . \|x^{*}x\|=\|x\|\|x^{*}\|.
  6. x x * = x 2 , \|xx^{*}\|=\|x\|^{2},
  7. x 2 = x * x = sup { | λ | : x * x - λ 1 is not invertible } . \|x\|^{2}=\|x^{*}x\|=\sup\{|\lambda|:x^{*}x-\lambda\,1\,\text{ is not % invertible}\}.
  8. π ( x y ) = π ( x ) π ( y ) \pi(xy)=\pi(x)\pi(y)\,
  9. π ( x * ) = π ( x ) * \pi(x^{*})=\pi(x)^{*}\,
  10. x x * = x 2 \lVert xx^{*}\rVert=\lVert x\rVert^{2}
  11. x e λ x xe_{\lambda}\rightarrow x
  12. 0 e λ e μ 1 whenever λ μ . 0\leq e_{\lambda}\leq e_{\mu}\leq 1\quad\mbox{ whenever }~{}\lambda\leq\mu.
  13. A = e min A A e A=\bigoplus_{e\in\min A}Ae
  14. A i I K ( H i ) , A\cong\bigoplus_{i\in I}K(H_{i}),

Caesar_cipher.html

  1. x x
  2. E n ( x ) = ( x + n ) mod 26. E_{n}(x)=(x+n)\mod{26}.
  3. D n ( x ) = ( x - n ) mod 26. D_{n}(x)=(x-n)\mod{26}.

Calcium_carbonate.html

  1. P CO 2 [ CO 2 ] = k H \frac{P_{\,\text{CO}_{2}}}{[\,\text{CO}_{2}]}\ =\ k\text{H}
  2. P CO 2 \scriptstyle P_{\,\text{CO}_{2}}
  3. P CO 2 \scriptstyle P_{\,\text{CO}_{2}}
  4. P CO 2 \scriptstyle P_{\,\text{CO}_{2}}
  5. P CO 2 \scriptstyle P_{\,\text{CO}_{2}}
  6. P CO 2 \scriptstyle P_{\,\text{CO}_{2}}
  7. P CO 2 \scriptstyle P_{\,\text{CO}_{2}}
  8. [ Ca 2 + ] [ A - ] 2 \scriptstyle[\mathrm{Ca}^{2+}]\simeq\frac{[\mathrm{A}^{-}]}{2}

Calculus.html

  1. d x dx
  2. f f
  3. f f′
  4. f ( x ) = 2 x f′(x)=2x
  5. f f
  6. f f
  7. y = m x + b y=mx+b
  8. x x
  9. y y
  10. b b
  11. m = rise run = change in y change in x = Δ y Δ x . m=\frac{\,\text{rise}}{\,\text{run}}=\frac{\,\text{change in }y}{\,\text{% change in }x}=\frac{\Delta y}{\Delta x}.
  12. y y
  13. x x
  14. f f
  15. a a
  16. f f
  17. ( a , f ( a ) ) (a,f(a))
  18. h h
  19. a + h a+h
  20. a a
  21. ( a + h , f ( a + h ) ) (a+h,f(a+h))
  22. ( a , f ( a ) ) (a,f(a))
  23. m = f ( a + h ) - f ( a ) ( a + h ) - a = f ( a + h ) - f ( a ) h . m=\frac{f(a+h)-f(a)}{(a+h)-a}=\frac{f(a+h)-f(a)}{h}.
  24. m m
  25. ( a , f ( a ) ) (a,f(a))
  26. ( a + h , f ( a + h ) ) (a+h,f(a+h))
  27. a a
  28. a a
  29. a + h a+h
  30. a a
  31. h h
  32. h h
  33. f f
  34. h h
  35. h h
  36. lim h 0 f ( a + h ) - f ( a ) h . \lim_{h\to 0}{f(a+h)-f(a)\over{h}}.
  37. f f
  38. a a
  39. f f
  40. f ( x ) f′(x)
  41. f ( 3 ) = lim h 0 ( 3 + h ) 2 - 3 2 h = lim h 0 9 + 6 h + h 2 - 9 h = lim h 0 6 h + h 2 h = lim h 0 ( 6 + h ) = 6. \begin{aligned}\displaystyle f^{\prime}(3)&\displaystyle=\lim_{h\to 0}{(3+h)^{% 2}-3^{2}\over{h}}\\ &\displaystyle=\lim_{h\to 0}{9+6h+h^{2}-9\over{h}}\\ &\displaystyle=\lim_{h\to 0}{6h+h^{2}\over{h}}\\ &\displaystyle=\lim_{h\to 0}(6+h)\\ &\displaystyle=6.\end{aligned}
  42. y = x 2 d y d x = 2 x . \begin{aligned}\displaystyle y&\displaystyle=x^{2}\\ \displaystyle\frac{dy}{dx}&\displaystyle=2x.\end{aligned}
  43. d y d x \frac{dy}{dx}
  44. d y dy
  45. y y
  46. d x dx
  47. x x
  48. d d x \frac{d}{dx}
  49. d d x ( x 2 ) = 2 x . \frac{d}{dx}(x^{2})=2x.
  50. d x dx
  51. x x
  52. d x dx
  53. d y dy
  54. F F
  55. f f
  56. f f
  57. F F
  58. Distance = Speed Time \mathrm{Distance}=\mathrm{Speed}\cdot\mathrm{Time}
  59. f ( x ) f(x)
  60. a a
  61. b b
  62. s s
  63. a a
  64. b b
  65. Δ x Δx
  66. f ( x ) f(x)
  67. h h
  68. Δ x Δx
  69. h h
  70. Δ x Δx
  71. h h
  72. f ( x ) = h f(x)=h
  73. Δ x Δx
  74. Δ x Δx
  75. \int\,
  76. a b f ( x ) d x . \int_{a}^{b}f(x)\,dx.
  77. d x dx
  78. Δ x Δx
  79. d x dx
  80. a b d x \int_{a}^{b}\cdots\,dx
  81. d x dx
  82. f ( x ) f(x)
  83. Δ x Δx
  84. f ( x ) d x . \int f(x)\,dx.
  85. C C
  86. y = 2 x y′=2x
  87. 2 x d x = x 2 + C . \int 2x\,dx=x^{2}+C.
  88. C C
  89. f f
  90. a a , b aa,b
  91. F F
  92. f f
  93. ( a , b ) (a,b)
  94. a b f ( x ) d x = F ( b ) - F ( a ) . \int_{a}^{b}f(x)\,dx=F(b)-F(a).
  95. x x
  96. ( a , b ) (a,b)
  97. d d x a x f ( t ) d t = f ( x ) . \frac{d}{dx}\int_{a}^{x}f(t)\,dt=f(x).

Calorimeter.html

  1. C p = W Δ H M Δ T C_{p}=\frac{W\Delta H}{M\Delta T}
  2. C p C_{p}
  3. Δ H \Delta H
  4. Δ T \Delta T
  5. W W
  6. M M
  7. Δ T = K d q d t = K C p β \Delta T=K{dq\over dt}=KC_{p}\,\beta
  8. d q d t = C p β + f ( t , T ) {dq\over dt}=C_{p}\beta+f(t,T)

Calorimetry.html

  1. p p
  2. p ( V , T ) p(V,T)
  3. V V
  4. T T
  5. δ V \delta V
  6. δ T \delta T
  7. δ Q \delta Q
  8. δ Q = C T ( V ) ( V , T ) δ V + C V ( T ) ( V , T ) δ T \delta Q\ =C^{(V)}_{T}(V,T)\,\delta V\,+\,C^{(T)}_{V}(V,T)\,\delta T
  9. C T ( V ) ( V , T ) C^{(V)}_{T}(V,T)
  10. T T
  11. V V
  12. C V ( T ) ( V , T ) C^{(T)}_{V}(V,T)
  13. V V
  14. T T
  15. C V ( T ) ( V , T ) C^{(T)}_{V}(V,T)
  16. C V ( V , T ) C_{V}(V,T)
  17. C V C_{V}
  18. p = p ( V , T ) p=p(V,T)
  19. δ Q \delta Q
  20. ( V , T ) (V,T)
  21. δ V \delta V
  22. δ V = 0 \delta V=0
  23. δ Q = C V δ T \delta Q=C_{V}\delta T
  24. δ T \delta T
  25. C V C_{V}
  26. δ p \delta p
  27. δ T \delta T
  28. δ Q \delta Q
  29. δ Q = C T ( p ) ( p , T ) δ p + C p ( T ) ( p , T ) δ T \delta Q\ =C^{(p)}_{T}(p,T)\,\delta p\,+\,C^{(T)}_{p}(p,T)\,\delta T
  30. C T ( p ) ( p , T ) C^{(p)}_{T}(p,T)
  31. p p
  32. T T
  33. C p ( T ) ( p , T ) C^{(T)}_{p}(p,T)
  34. p p
  35. T T
  36. C p ( T ) ( p , T ) C^{(T)}_{p}(p,T)
  37. C p ( p , T ) C_{p}(p,T)
  38. C p C_{p}
  39. C T ( p ) ( p , T ) = C T ( V ) ( V , T ) p V | ( V , T ) C^{(p)}_{T}(p,T)=\frac{C^{(V)}_{T}(V,T)}{\left.\cfrac{\partial p}{\partial V}% \right|_{(V,T)}}
  40. C p ( T ) ( p , T ) = C V ( T ) ( V , T ) - C T ( V ) ( V , T ) p T | ( V , T ) p V | ( V , T ) C^{(T)}_{p}(p,T)=C^{(T)}_{V}(V,T)-C^{(V)}_{T}(V,T)\frac{\left.\cfrac{\partial p% }{\partial T}\right|_{(V,T)}}{\left.\cfrac{\partial p}{\partial V}\right|_{(V,% T)}}
  41. p V | ( V , T ) \left.\frac{\partial p}{\partial V}\right|_{(V,T)}
  42. p ( V , T ) p(V,T)
  43. V V
  44. ( V , T ) (V,T)
  45. p T | ( V , T ) \left.\frac{\partial p}{\partial T}\right|_{(V,T)}
  46. p ( V , T ) p(V,T)
  47. T T
  48. ( V , T ) (V,T)
  49. C T ( V ) ( V , T ) C^{(V)}_{T}(V,T)
  50. C T ( p ) ( p , T ) C^{(p)}_{T}(p,T)
  51. γ ( V , T ) = C p ( T ) ( p , T ) C V ( T ) ( V , T ) \gamma(V,T)=\frac{C^{(T)}_{p}(p,T)}{C^{(T)}_{V}(V,T)}
  52. γ = C p C V \gamma=\frac{C_{p}}{C_{V}}
  53. P ( t 1 , t 2 ) P(t_{1},t_{2})
  54. V ( t ) V(t)
  55. T ( t ) T(t)
  56. t 1 t_{1}
  57. t 2 t_{2}
  58. Δ Q ( P ( t 1 , t 2 ) ) \Delta Q(P(t_{1},t_{2}))\,
  59. Δ \Delta
  60. Δ Q ( P ( t 1 , t 2 ) ) \Delta Q(P(t_{1},t_{2}))\,
  61. δ Q \delta Q
  62. Δ Q ( P ( t 1 , t 2 ) ) \Delta Q(P(t_{1},t_{2}))
  63. = P ( t 1 , t 2 ) Q ˙ ( t ) d t =\int_{P(t_{1},t_{2})}\dot{Q}(t)dt
  64. = P ( t 1 , t 2 ) C T ( V ) ( V , T ) V ˙ ( t ) d t + P ( t 1 , t 2 ) C V ( T ) ( V , T ) T ˙ ( t ) d t =\int_{P(t_{1},t_{2})}C^{(V)}_{T}(V,T)\,\dot{V}(t)\,dt\,+\,\int_{P(t_{1},t_{2}% )}C^{(T)}_{V}(V,T)\,\dot{T}(t)\,dt
  65. Q ˙ ( t ) \dot{Q}(t)
  66. δ Q \delta Q
  67. p ( V , T ) p(V,T)
  68. V V
  69. T T
  70. V ˙ ( t ) = d V d t | t \dot{V}(t)=\left.\frac{dV}{dt}\right|_{t}
  71. p ( V , T ) p(V,T)
  72. Q ˙ ( t ) = C T ( V ) ( V , T ) V ˙ ( t ) + C V ( T ) ( V , T ) T ˙ ( t ) \dot{Q}(t)\ =C^{(V)}_{T}(V,T)\,\dot{V}(t)\,+\,C^{(T)}_{V}(V,T)\,\dot{T}(t)
  73. t t
  74. Q ˙ ( t ) \dot{Q}(t)
  75. t t
  76. V ˙ ( t ) \dot{V}(t)
  77. t t
  78. T ˙ ( t ) \dot{T}(t)
  79. δ Q \delta Q
  80. Q ˙ ( t ) \dot{Q}(t)
  81. t t
  82. Q ( V , T ) Q(V,T)
  83. δ Q \delta Q
  84. q q
  85. δ Q \delta Q
  86. Δ Q ( P ( t 1 , t 2 ) ) \Delta Q(P(t_{1},t_{2}))
  87. P ( t 1 , t 2 ) P(t_{1},t_{2})
  88. V ( t ) V(t)
  89. T ( t ) T(t)
  90. Δ Q ( P ( t 1 , t 2 ) ) \Delta Q(P(t_{1},t_{2}))
  91. ( V , T ) (V,T)
  92. Q ˙ ( t ) \dot{Q}(t)
  93. t t
  94. Q Q
  95. Q ( V , T ) Q(V,T)
  96. α V ( V , T ) = p V | ( V , T ) p ( V , T ) \alpha_{V}(V,T)\ =\frac{\left.\cfrac{\partial p}{\partial V}\right|_{(V,T)}}{p% (V,T)}
  97. V V
  98. V ( T , p ) V(T,p)
  99. T T
  100. p p
  101. β p ( T , p ) = V T | ( T , p ) V ( T , p ) \beta_{p}(T,p)\ =\frac{\left.\cfrac{\partial V}{\partial T}\right|_{(T,p)}}{V(% T,p)}
  102. V V
  103. V ( T , p ) V(T,p)
  104. T T
  105. p p
  106. κ T ( T , p ) = - V p | ( T , p ) V ( T , p ) \kappa_{T}(T,p)\ =-\frac{\left.\cfrac{\partial V}{\partial p}\right|_{(T,p)}}{% V(T,p)}
  107. p = p ( V , T ) p=p(V,T)
  108. p T \frac{\partial p}{\partial T}
  109. β p ( T , p ) \beta_{p}(T,p)
  110. κ T ( T , p ) \kappa_{T}(T,p)
  111. p T = β p ( T , p ) κ T ( T , p ) \frac{\partial p}{\partial T}=\frac{\beta_{p}(T,p)}{\kappa_{T}(T,p)}
  112. U U
  113. U ( V , T ) U(V,T)
  114. ( V , T ) (V,T)
  115. U V \frac{\partial U}{\partial V}
  116. U T \frac{\partial U}{\partial T}
  117. δ Q = [ p ( V , T ) + U V | ( V , T ) ] δ V + U T | ( V , T ) δ T \delta Q\ =\left[p(V,T)\,+\,\left.\frac{\partial U}{\partial V}\right|_{(V,T)}% \right]\,\delta V\,+\,\left.\frac{\partial U}{\partial T}\right|_{(V,T)}\,\delta T
  118. C T ( V ) ( V , T ) = p ( V , T ) + U V | ( V , T ) C^{(V)}_{T}(V,T)=p(V,T)\,+\,\left.\frac{\partial U}{\partial V}\right|_{(V,T)}
  119. C V ( T ) ( V , T ) = U T | ( V , T ) C^{(T)}_{V}(V,T)=\left.\frac{\partial U}{\partial T}\right|_{(V,T)}
  120. U U
  121. U ( p , T ) U(p,T)
  122. ( p , T ) (p,T)
  123. U p \frac{\partial U}{\partial p}
  124. U T \frac{\partial U}{\partial T}
  125. V V
  126. V ( p , T ) V(p,T)
  127. ( p , T ) (p,T)
  128. V p \frac{\partial V}{\partial p}
  129. V T \frac{\partial V}{\partial T}
  130. δ Q = [ U p | ( p , T ) + p V p | ( p , T ) ] δ p + [ U T | ( p , T ) + p V T | ( p , T ) ] δ T \delta Q\ =\left[\left.\frac{\partial U}{\partial p}\right|_{(p,T)}\,+\,p\left% .\frac{\partial V}{\partial p}\right|_{(p,T)}\right]\delta p\,+\,\left[\left.% \frac{\partial U}{\partial T}\right|_{(p,T)}\,+\,p\left.\frac{\partial V}{% \partial T}\right|_{(p,T)}\right]\delta T
  131. C T ( p ) ( p , T ) = U p | ( p , T ) + p V p | ( p , T ) C^{(p)}_{T}(p,T)=\left.\frac{\partial U}{\partial p}\right|_{(p,T)}\,+\,p\left% .\frac{\partial V}{\partial p}\right|_{(p,T)}
  132. C p ( T ) ( p , T ) = U T | ( p , T ) + p V T | ( p , T ) C^{(T)}_{p}(p,T)=\left.\frac{\partial U}{\partial T}\right|_{(p,T)}\,+\,p\left% .\frac{\partial V}{\partial T}\right|_{(p,T)}
  133. C T ( V ) ( V , T ) C^{(V)}_{T}(V,T)
  134. C T ( p ) ( p , T ) C^{(p)}_{T}(p,T)
  135. C T ( V ) ( V , T ) p T | ( V , T ) 0 . C^{(V)}_{T}(V,T)\,\left.\frac{\partial p}{\partial T}\right|_{(V,T)}\geq 0\,.
  136. V a V_{a}
  137. V b V_{b}
  138. T + T^{+}
  139. V b V_{b}
  140. V c V_{c}
  141. V c V_{c}
  142. V d V_{d}
  143. T - T^{-}
  144. V d V_{d}
  145. V a V_{a}
  146. T + T^{+}
  147. Δ Q ( V a , V b ; T + ) = V a V b C T ( V ) ( V , T + ) d V \Delta Q(V_{a},V_{b};T^{+})\,=\,\,\,\,\,\,\,\,\int_{V_{a}}^{V_{b}}C^{(V)}_{T}(% V,T^{+})\,dV
  148. - Δ Q ( V c , V d ; T - ) = - V c V d C T ( V ) ( V , T - ) d V -\Delta Q(V_{c},V_{d};T^{-})\,=\,-\int_{V_{c}}^{V_{d}}C^{(V)}_{T}(V,T^{-})\,dV
  149. Δ Q ( V a , V b ; T + ; V c , V d ; T - ) = Δ Q ( V a , V b ; T + ) + Δ Q ( V c , V d ; T - ) = V a V b C T ( V ) ( V , T + ) d V + V c V d C T ( V ) ( V , T - ) d V \Delta Q(V_{a},V_{b};T^{+};V_{c},V_{d};T^{-})\,=\,\Delta Q(V_{a},V_{b};T^{+})% \,+\,\Delta Q(V_{c},V_{d};T^{-})\,=\,\int_{V_{a}}^{V_{b}}C^{(V)}_{T}(V,T^{+})% \,dV\,+\,\int_{V_{c}}^{V_{d}}C^{(V)}_{T}(V,T^{-})\,dV
  150. Δ U ( V a , V b ; T + ; V c , V d ; T - ) \Delta U(V_{a},V_{b};T^{+};V_{c},V_{d};T^{-})
  151. C T ( V ) ( V , T ) C^{(V)}_{T}(V,T)
  152. p = p ( V , T ) p=p(V,T)
  153. T T\,
  154. C T ( V ) ( V , T ) = T p T | ( V , T ) C^{(V)}_{T}(V,T)=T\left.\frac{\partial p}{\partial T}\right|_{(V,T)}
  155. C p ( p , T ) - C V ( V , T ) = [ p ( V , T ) + U V | ( V , T ) ] V T | ( p , T ) C_{p}(p,T)-C_{V}(V,T)=\left[p(V,T)\,+\,\left.\frac{\partial U}{\partial V}% \right|_{(V,T)}\right]\,\left.\frac{\partial V}{\partial T}\right|_{(p,T)}
  156. C p ( p , T ) - C V ( V , T ) = T V β p 2 ( T , p ) κ T ( T , p ) C_{p}(p,T)-C_{V}(V,T)=\frac{TV\,\beta_{p}^{2}(T,p)}{\kappa_{T}(T,p)}

Cam.html

  1. θ \theta
  2. L L
  3. ϕ \phi
  4. ϕ \phi
  5. π / 2 - θ / 2 \pi/2-\theta/2
  6. C C
  7. R R
  8. r r
  9. C = L / ( 1 - sin ϕ ) C=L/(1-\sin\phi)
  10. r = R - L sin ϕ / ( 1 - sin ϕ ) r=R-L\sin\phi/(1-\sin\phi)

Candela.html

  1. × 10 1 2 \times 10^{1}2
  2. 1 / 683 {1}/{683}
  3. I v ( λ ) = 683.002 y ¯ ( λ ) I e ( λ ) I_{\mathrm{v}}(\lambda)=683.002\cdot\overline{y}(\lambda)\cdot I_{\mathrm{e}}(\lambda)
  4. y ¯ ( λ ) \textstyle\overline{y}(\lambda)
  5. illuminance at point 𝐫 on d A , E v ( 𝐫 ) = i | 𝐚 ^ ( 𝐫 - 𝐫 i ) | | 𝐫 - 𝐫 i | 3 I i . \,\text{illuminance at point }\mathbf{r}\,\text{ on }dA\,\text{, }E_{v}(% \mathbf{r})=\sum_{i}{\frac{|\mathbf{\hat{a}}\cdot(\mathbf{r}-\mathbf{r}_{i})|}% {|\mathbf{r}-\mathbf{r}_{i}|^{3}}I_{i}}.
  6. E v ( r ) = I v r 2 . E_{v}(r)=\frac{I_{v}}{r^{2}}.

Cantor_set.html

  1. 1 / 3 {1}/{3}
  2. 2 / 3 {2}/{3}
  3. 1 / 3 {1}/{3}
  4. 2 / 3 {2}/{3}
  5. 1 / 9 {1}/{9}
  6. 2 / 9 {2}/{9}
  7. 1 / 3 {1}/{3}
  8. 2 / 3 {2}/{3}
  9. 7 / 9 {7}/{9}
  10. 8 / 9 {8}/{9}
  11. C n = C n - 1 3 ( 2 3 + C n - 1 3 ) C_{n}=\frac{C_{n-1}}{3}\cup\left(\frac{2}{3}+\frac{C_{n-1}}{3}\right)
  12. C 0 = [ 0 , 1 ] . C_{0}=[0,1].
  13. C = m = 1 k = 0 3 m - 1 - 1 ( [ 0 , 3 k + 1 3 m ] [ 3 k + 2 3 m , 1 ] ) C=\bigcap_{m=1}^{\infty}\bigcap_{k=0}^{3^{m-1}-1}\left(\left[0,\frac{3k+1}{3^{% m}}\right]\cup\left[\frac{3k+2}{3^{m}},1\right]\right)
  14. C = [ 0 , 1 ] m = 1 k = 0 3 m - 1 - 1 ( 3 k + 1 3 m , 3 k + 2 3 m ) . C=[0,1]\setminus\bigcup_{m=1}^{\infty}\bigcup_{k=0}^{3^{m-1}-1}\left(\frac{3k+% 1}{3^{m}},\frac{3k+2}{3^{m}}\right).
  15. n = 0 2 n 3 n + 1 = 1 3 + 2 9 + 4 27 + 8 81 + = 1 3 ( 1 1 - 2 3 ) = 1. \sum_{n=0}^{\infty}\frac{2^{n}}{3^{n+1}}=\frac{1}{3}+\frac{2}{9}+\frac{4}{27}+% \frac{8}{81}+\cdots=\frac{1}{3}\left(\frac{1}{1-\frac{2}{3}}\right)=1.
  16. f ( k = 1 a k 3 - k ) = k = 1 a k 2 2 - k . f\bigg(\sum_{k=1}^{\infty}a_{k}3^{-k}\bigg)=\sum_{k=1}^{\infty}\frac{a_{k}}{2}% 2^{-k}.
  17. f L ( x ) = x / 3 f_{L}(x)=x/3
  18. f R ( x ) = ( 2 + x ) / 3 f_{R}(x)=(2+x)/3
  19. f L ( C ) f R ( C ) C . f_{L}(C)\cong f_{R}(C)\cong C.
  20. f L f_{L}
  21. f R f_{R}
  22. { f L , f R } \{f_{L},f_{R}\}
  23. x x
  24. y y
  25. C C
  26. h : C C h:C\to C
  27. h ( x ) = y h(x)=y
  28. { 0 , 1 } \{0,1\}
  29. 2 = { ( x n ) | x n { 0 , 1 } for n } 2^{\mathbb{N}}=\{(x_{n})|x_{n}\in\{0,1\}\mbox{ for }~{}n\in\mathbb{N}\}
  30. 2 2^{\mathbb{N}}
  31. ( x n ) , ( y n ) 2 (x_{n}),(y_{n})\in 2^{\mathbb{N}}
  32. d ( ( x n ) , ( y n ) ) = 1 / k d((x_{n}),(y_{n}))=1/k
  33. k k
  34. x k y k x_{k}\neq y_{k}

Capillary.html

  1. J v = K f ( [ P c - P i ] - σ [ π c - π i ] ) \ J_{v}=K_{f}([P_{c}-P_{i}]-\sigma[\pi_{c}-\pi_{i}])
  2. ( [ P c - P i ] - σ [ π c - π i ] ) ([P_{c}-P_{i}]-\sigma[\pi_{c}-\pi_{i}])
  3. K f K_{f}
  4. J v J_{v}

Capsid.html

  1. T = h 2 + h k + k 2 T=h^{2}+h\cdot k+k^{2}
  2. ( h + k ) 2 - h k (h+k)^{2}-hk
  3. h k h\geq k

Carbon_dioxide.html

  1. K h = [ H 2 CO 3 ] [ CO 2 ( aq ) ] = 1.70 × 10 - 3 K_{\mathrm{h}}=\frac{\rm{[H_{2}CO_{3}]}}{\rm{[CO_{2}(aq)]}}=1.70\times 10^{-3}
  2. K a 1 = [ HCO 3 - ] [ H + ] [ H 2 CO 3 ] K_{a1}=\frac{\rm{[HCO_{3}^{-}][H^{+}]}}{\rm{[H_{2}CO_{3}]}}
  3. K a1 ( apparent ) = [ HCO 3 - ] [ H + ] [ H 2 CO 3 ] + [ CO 2 ( aq ) ] K_{\mathrm{a1}}{\rm{(apparent)}}=\frac{\rm{[HCO_{3}^{-}][H^{+}]}}{\rm{[H_{2}CO% _{3}]+[CO_{2}(aq)]}}

Carbon_nanotube.html

  1. d = a π ( n 2 + n m + m 2 ) = 78.3 ( ( n + m ) 2 - n m ) pm , d=\frac{a}{\pi}\sqrt{(n^{2}+nm+m^{2})}=78.3\sqrt{((n+m)^{2}-nm)}\rm pm,
  2. H ( t ) = β τ o ( 1 - e - t / τ o ) . H(t)={\beta}{\tau}_{o}({1-e^{-t/{\tau}_{o}}}).
  3. τ o {\tau}_{o}

Carbonic_acid.html

  1. K a 1 = [ H + ] [ H C O 3 - ] [ H 2 C O 3 ] K a ( a p p ) = [ H + ] [ H C O 3 - ] [ H 2 C O 3 ] + [ C O 2 ( a q ) ] K a 2 = [ H + ] [ C O 3 2 - ] [ H C O 3 - ] K_{a1}=\frac{[H^{+}][HCO_{3}^{-}]}{[H_{2}CO_{3}]}\qquad K_{a}{(app)}=\frac{[H^% {+}][HCO_{3}^{-}]}{[H_{2}CO_{3}]+[CO_{2}(aq)]}\qquad K_{a2}=\frac{[H^{+}][CO_{% 3}^{2-}]}{[HCO_{3}^{-}]}
  2. p C O 2 \scriptstyle p_{CO_{2}}
  3. K h = [ H 2 C O 3 ] [ C O 2 ] \scriptstyle K_{h}=\frac{[H_{2}CO_{3}]}{[CO_{2}]}
  4. [ C O 2 ] p C O 2 = 1 k H \scriptstyle\frac{[CO_{2}]}{p_{CO_{2}}}=\frac{1}{k_{\mathrm{H}}}
  5. [ H + ] [ O H - ] = 10 - 14 \scriptstyle[H^{+}][OH^{-}]=10^{-14}
  6. [ H + ] = [ O H - ] + [ H C O 3 - ] + 2 [ C O 3 2 - ] \scriptstyle[H^{+}]=[OH^{-}]+[HCO_{3}^{-}]+2[CO_{3}^{2-}]
  7. p C O 2 \scriptstyle p_{CO_{2}}
  8. p C O 2 \scriptstyle p_{CO_{2}}
  9. p C O 2 \scriptstyle p_{CO_{2}}
  10. p C O 2 = 3.5 × 10 - 4 \scriptstyle p_{CO_{2}}=3.5\times 10^{-4}
  11. p C O 2 \scriptstyle p_{CO_{2}}
  12. [ H + ] ( 10 - 14 + K h K a 1 k H p C O 2 ) 1 / 2 \scriptstyle[H^{+}]\simeq\left(10^{-14}+\frac{K_{h}K_{a1}}{k_{\mathrm{H}}}p_{% CO_{2}}\right)^{1/2}

Cardinal_number.html

  1. 0 , 1 , 2 , 3 , , n , ; 0 , 1 , 2 , , α , . 0,1,2,3,\ldots,n,\ldots;\aleph_{0},\aleph_{1},\aleph_{2},\ldots,\aleph_{\alpha% },\ldots.
  2. 0 \aleph_{0}
  3. 𝔠 \mathfrak{c}
  4. 0 \aleph_{0}
  5. ( 1 , 2 , 3 , ) . (\aleph_{1},\aleph_{2},\aleph_{3},\cdots).
  6. 𝔠 \mathfrak{c}
  7. 1 \aleph_{1}
  8. 0 \aleph_{0}
  9. \aleph
  10. 0 . \aleph_{0}.
  11. 1 \aleph_{1}
  12. α , \aleph_{\alpha},
  13. κ + κ . \kappa^{+}\nleq\kappa.
  14. | X | + | Y | = | X Y | . |X|+|Y|=|X\cup Y|.
  15. ( κ μ ) ( ( κ + ν μ + ν ) and ( ν + κ ν + μ ) ) . (\kappa\leq\mu)\rightarrow((\kappa+\nu\leq\mu+\nu)\mbox{ and }~{}(\nu+\kappa% \leq\nu+\mu)).
  16. κ + μ = max { κ , μ } . \kappa+\mu=\max\{\kappa,\mu\}\,.
  17. κ μ = max { κ , μ } . \kappa\cdot\mu=\max\{\kappa,\mu\}.
  18. ν μ = κ {\nu}^{\mu}={\kappa}
  19. μ λ = κ {\mu}^{\lambda}={\kappa}
  20. ν λ = κ {\nu}^{\lambda}={\kappa}
  21. 0 \aleph_{0}
  22. 2 0 . 2^{\aleph_{0}}.
  23. 𝔠 \mathfrak{c}
  24. 2 0 = 1 . 2^{\aleph_{0}}=\aleph_{1}.

Cardinality.html

  1. 0 < 1 < 2 < . \aleph_{0}<\aleph_{1}<\aleph_{2}<\ldots.
  2. α \alpha
  3. α + 1 \aleph_{\alpha+1}
  4. α \aleph_{\alpha}
  5. 0 \aleph_{0}
  6. 𝔠 \mathfrak{c}
  7. 𝔠 > 0 {\mathfrak{c}}>\aleph_{0}
  8. 𝔠 = 2 0 \mathfrak{c}=2^{\aleph_{0}}
  9. 1 = 2 0 \aleph_{1}=2^{\aleph_{0}}
  10. 2 0 2^{\aleph_{0}}
  11. 0 \aleph_{0}
  12. 0 \aleph_{0}
  13. 𝔠 \mathfrak{c}
  14. 0 \aleph_{0}
  15. 𝔠 \mathfrak{c}
  16. 0 \aleph_{0}
  17. 𝔠 = 2 0 > 0 \mathfrak{c}=2^{\aleph_{0}}>{\aleph_{0}}
  18. 𝔠 = 1 = 1 \mathfrak{c}=\aleph_{1}=\beth_{1}
  19. 𝔠 \mathfrak{c}
  20. 2 𝔠 = 2 > 𝔠 2^{\mathfrak{c}}=\beth_{2}>\mathfrak{c}
  21. 𝔠 2 = 𝔠 , \mathfrak{c}^{2}=\mathfrak{c},
  22. 𝔠 0 = 𝔠 , \mathfrak{c}^{\aleph_{0}}=\mathfrak{c},
  23. 𝔠 𝔠 = 2 𝔠 \mathfrak{c}^{\mathfrak{c}}=2^{\mathfrak{c}}
  24. 𝔠 2 = ( 2 0 ) 2 = 2 2 × 0 = 2 0 = 𝔠 , \mathfrak{c}^{2}=\left(2^{\aleph_{0}}\right)^{2}=2^{2\times{\aleph_{0}}}=2^{% \aleph_{0}}=\mathfrak{c},
  25. 𝔠 0 = ( 2 0 ) 0 = 2 0 × 0 = 2 0 = 𝔠 , \mathfrak{c}^{\aleph_{0}}=\left(2^{\aleph_{0}}\right)^{\aleph_{0}}=2^{{\aleph_% {0}}\times{\aleph_{0}}}=2^{\aleph_{0}}=\mathfrak{c},
  26. 𝔠 𝔠 = ( 2 0 ) 𝔠 = 2 𝔠 × 0 = 2 𝔠 . \mathfrak{c}^{\mathfrak{c}}=\left(2^{\aleph_{0}}\right)^{\mathfrak{c}}=2^{% \mathfrak{c}\times\aleph_{0}}=2^{\mathfrak{c}}.
  27. | A B | = | A | + | B | . \left|A\cup B\right|=\left|A\right|+\left|B\right|.
  28. | C D | + | C D | = | C | + | D | . \left|C\cup D\right|+\left|C\cap D\right|=\left|C\right|+\left|D\right|\,.

Carl_Friedrich_Gauss.html

  1. 1 + α β γ .1 + etc. 1+\frac{\alpha\beta}{\gamma.1}+\mbox{etc.}~{}

Carmichael_number.html

  1. n n
  2. b n - 1 1 ( mod n ) b^{n-1}\equiv 1\;\;(\mathop{{\rm mod}}n)
  3. 1 < b < n 1<b<n
  4. n n
  5. n n
  6. n n
  7. p p
  8. n n
  9. p - 1 n - 1 p-1\mid n-1
  10. p - 1 n - 1 p-1\mid n-1
  11. - 1 -1
  12. 561 = 3 11 17 561=3\cdot 11\cdot 17
  13. 2 560 2\mid 560
  14. 10 560 10\mid 560
  15. 16 560 16\mid 560
  16. 1105 = 5 13 17 ( 4 1104 ; 12 1104 ; 16 1104 ) 1105=5\cdot 13\cdot 17\qquad(4\mid 1104;\quad 12\mid 1104;\quad 16\mid 1104)
  17. 1729 = 7 13 19 ( 6 1728 ; 12 1728 ; 18 1728 ) 1729=7\cdot 13\cdot 19\qquad(6\mid 1728;\quad 12\mid 1728;\quad 18\mid 1728)
  18. 2465 = 5 17 29 ( 4 2464 ; 16 2464 ; 28 2464 ) 2465=5\cdot 17\cdot 29\qquad(4\mid 2464;\quad 16\mid 2464;\quad 28\mid 2464)
  19. 2821 = 7 13 31 ( 6 2820 ; 12 2820 ; 30 2820 ) 2821=7\cdot 13\cdot 31\qquad(6\mid 2820;\quad 12\mid 2820;\quad 30\mid 2820)
  20. 6601 = 7 23 41 ( 6 6600 ; 22 6600 ; 40 6600 ) 6601=7\cdot 23\cdot 41\qquad(6\mid 6600;\quad 22\mid 6600;\quad 40\mid 6600)
  21. 8911 = 7 19 67 ( 6 8910 ; 18 8910 ; 66 8910 ) . 8911=7\cdot 19\cdot 67\qquad(6\mid 8910;\quad 18\mid 8910;\quad 66\mid 8910).
  22. ( 6 k + 1 ) ( 12 k + 1 ) ( 18 k + 1 ) (6k+1)(12k+1)(18k+1)
  23. n n
  24. n 2 / 7 n^{2/7}
  25. n n
  26. k = 3 , 4 , 5 , k=3,4,5,\ldots
  27. 561 = 3 11 17 561=3\cdot 11\cdot 17\,
  28. 41041 = 7 11 13 41 41041=7\cdot 11\cdot 13\cdot 41\,
  29. 825265 = 5 7 17 19 73 825265=5\cdot 7\cdot 17\cdot 19\cdot 73\,
  30. 321197185 = 5 19 23 29 37 137 321197185=5\cdot 19\cdot 23\cdot 29\cdot 37\cdot 137\,
  31. 5394826801 = 7 13 17 23 31 67 73 5394826801=7\cdot 13\cdot 17\cdot 23\cdot 31\cdot 67\cdot 73\,
  32. 232250619601 = 7 11 13 17 31 37 73 163 232250619601=7\cdot 11\cdot 13\cdot 17\cdot 31\cdot 37\cdot 73\cdot 163\,
  33. 9746347772161 = 7 11 13 17 19 31 37 41 641 9746347772161=7\cdot 11\cdot 13\cdot 17\cdot 19\cdot 31\cdot 37\cdot 41\cdot 6% 41\,
  34. 41041 = 7 11 13 41 41041=7\cdot 11\cdot 13\cdot 41\,
  35. 62745 = 3 5 47 89 62745=3\cdot 5\cdot 47\cdot 89\,
  36. 63973 = 7 13 19 37 63973=7\cdot 13\cdot 19\cdot 37\,
  37. 75361 = 11 13 17 31 75361=11\cdot 13\cdot 17\cdot 31\,
  38. 101101 = 7 11 13 101 101101=7\cdot 11\cdot 13\cdot 101\,
  39. 126217 = 7 13 19 73 126217=7\cdot 13\cdot 19\cdot 73\,
  40. 172081 = 7 13 31 61 172081=7\cdot 13\cdot 31\cdot 61\,
  41. 188461 = 7 13 19 109 188461=7\cdot 13\cdot 19\cdot 109\,
  42. 278545 = 5 17 29 113 278545=5\cdot 17\cdot 29\cdot 113\,
  43. 340561 = 13 17 23 67 340561=13\cdot 17\cdot 23\cdot 67\,
  44. C ( X ) C(X)
  45. X X
  46. n n
  47. C ( 10 n ) C(10^{n})
  48. C ( X ) < X exp ( - k 1 ( log X log log X ) 1 2 ) C(X)<X\exp\left({-k_{1}\left(\log X\log\log X\right)^{\frac{1}{2}}}\right)
  49. k 1 k_{1}
  50. C ( X ) < X exp ( - k 2 log X log log log X log log X ) C(X)<X\exp\left(\frac{-k_{2}\log X\log\log\log X}{\log\log X}\right)
  51. k 2 k_{2}
  52. C ( X ) C(X)
  53. X = 10 n X=10^{n}
  54. n n
  55. C ( X ) > X 2 7 . C(X)>X^{\frac{2}{7}}.
  56. C ( X ) > X 0.332 C(X)>X^{0.332}
  57. 1 / 3 1/3
  58. X 1 - o ( 1 ) X^{1-o(1)}
  59. X 1 - { 1 + o ( 1 ) } log log log X log log X X^{1-{\frac{\{1+o(1)\}\log\log\log X}{\log\log X}}}
  60. 𝔭 \mathfrak{p}
  61. 𝒪 K {\mathcal{O}}_{K}
  62. α N ( 𝔭 ) α mod 𝔭 \alpha^{{\rm N}(\mathfrak{p})}\equiv\alpha\bmod{\mathfrak{p}}
  63. α \alpha
  64. 𝒪 K {\mathcal{O}}_{K}
  65. N ( 𝔭 ) {\rm N}(\mathfrak{p})
  66. 𝔭 \mathfrak{p}
  67. m p m mod p m^{p}\equiv m\bmod p
  68. 𝔞 \mathfrak{a}
  69. 𝒪 K {\mathcal{O}}_{K}
  70. α N ( 𝔞 ) α mod 𝔞 \alpha^{{\rm N}(\mathfrak{a})}\equiv\alpha\bmod{\mathfrak{a}}
  71. α 𝒪 K \alpha\in{\mathcal{O}}_{K}
  72. N ( 𝔞 ) {\rm N}(\mathfrak{a})
  73. 𝔞 \mathfrak{a}
  74. 𝐐 \mathbf{Q}
  75. 𝔞 \mathfrak{a}
  76. 𝔞 = ( a ) \mathfrak{a}=(a)
  77. 𝒪 K {\mathcal{O}}_{K}
  78. p 𝒪 K p{\mathcal{O}}_{K}
  79. 𝒪 K {\mathcal{O}}_{K}
  80. gcd ( x = 1 n - 1 x n - 1 , n ) = 1 \gcd\left(\sum_{x=1}^{n-1}x^{n-1},n\right)=1
  81. 10 12 10^{12}
  82. 10 18 10^{18}
  83. n n
  84. p i p j - 1 p_{i}\mid p_{j}-1
  85. p i p_{i}
  86. p j p_{j}
  87. n n
  88. n n
  89. p j - 1 n - 1 p_{j}-1\mid n-1
  90. p i n - 1 p_{i}\mid n-1
  91. p i p_{i}
  92. n n

Carnot_heat_engine.html

  1. Δ S H = Q H / T H \Delta S_{H}=Q_{H}/T_{H}
  2. Δ S C = Q C / T C \Delta S_{C}=Q_{C}/T_{C}
  3. η I \eta_{I}
  4. η I = W Q H = 1 - T C T H ( 1 ) \eta_{I}=\frac{W}{Q_{H}}=1-\frac{T_{C}}{T_{H}}\quad\quad\quad\quad\quad\quad% \quad\quad\quad(1)
  5. W W
  6. Q H Q_{H}
  7. T C T_{C}
  8. T H T_{H}
  9. η \eta
  10. W W
  11. Q H Q_{H}
  12. Q C Q_{C}
  13. W = Q H - Q C ( 2 ) W=Q_{H}-Q_{C}\quad\quad\quad\quad\quad(2)
  14. T H T_{H}
  15. T H T_{H}
  16. Δ S H \Delta S_{H}
  17. Δ S H = Q i n d Q H T ( 3 ) \Delta S_{H}=\int_{Q_{in}}\frac{dQ_{H}}{T}\quad\quad\quad\quad(3)
  18. T T
  19. T H T_{H}
  20. Q H T H = d Q H T H Δ S H ( 4 ) \frac{Q_{H}}{T_{H}}=\frac{\int dQ_{H}}{T_{H}}\leq\Delta S_{H}\quad\quad\quad% \quad\quad(4)
  21. Δ S C \Delta S_{C}
  22. Δ S C = Q o u t d Q C T d Q C T C = Q C T C ( 5 ) \Delta S_{C}=\int_{Q_{out}}\frac{dQ_{C}}{T}\leq\frac{\int dQ_{C}}{T_{C}}=\frac% {Q_{C}}{T_{C}}\quad\quad\quad\quad(5)
  23. T T
  24. T C T_{C}
  25. Δ S H = Δ S C ( 6 ) \Delta S_{H}=\Delta S_{C}\quad\quad\quad\quad(6)
  26. Q C T C Q H T H ( 7 ) \frac{Q_{C}}{T_{C}}\geq\frac{Q_{H}}{T_{H}}\quad\quad\quad\quad(7)
  27. W Q H ( 1 - T C T H ) ( 8 ) \frac{W}{Q_{H}}\leq(1-\frac{T_{C}}{T_{H}})\quad\quad\quad\quad\quad(8)
  28. η η I ( 9 ) \eta\leq\eta_{I}\quad\quad\quad\quad\quad\quad(9)
  29. η = W Q H \eta=\frac{W}{Q_{H}}
  30. η I \eta_{I}
  31. T H T_{H}
  32. T C T_{C}
  33. Q C Q_{C}
  34. T C T_{C}
  35. Q H Q_{H}
  36. T H T_{H}

Carrier-to-receiver_noise_density.html

  1. C k T , \frac{C}{kT},

Carson_bandwidth_rule.html

  1. C B R = 2 ( Δ f + f m ) CBR=2(\Delta f+f_{m})
  2. Δ f \Delta f
  3. f m f_{m}
  4. 10 log ( 0.02 0.98 ) 10\log\left(\frac{0.02}{0.98}\right)

Cartesian_coordinate_system.html

  1. \R 2 = \R × \R \R^{2}=\R\times\R
  2. \R \R
  3. \R n \R^{n}
  4. ( x 1 , y 1 ) (x_{1},y_{1})
  5. ( x 2 , y 2 ) (x_{2},y_{2})
  6. d = ( x 2 - x 1 ) 2 + ( y 2 - y 1 ) 2 . d=\sqrt{(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}}.
  7. ( x 1 , y 1 , z 1 ) (x_{1},y_{1},z_{1})
  8. ( x 2 , y 2 , z 2 ) (x_{2},y_{2},z_{2})
  9. d = ( x 2 - x 1 ) 2 + ( y 2 - y 1 ) 2 + ( z 2 - z 1 ) 2 , d=\sqrt{(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}+(z_{2}-z_{1})^{2}},
  10. ( x , y ) = ( x + a , y + b ) . (x^{\prime},y^{\prime})=(x+a,y+b).
  11. θ \theta
  12. x = x cos θ - y sin θ x^{\prime}=x\cos\theta-y\sin\theta
  13. y = x sin θ + y cos θ . y^{\prime}=x\sin\theta+y\cos\theta.
  14. ( x , y ) = ( ( x cos θ - y sin θ ) , ( x sin θ + y cos θ ) ) . (x^{\prime},y^{\prime})=((x\cos\theta-y\sin\theta\,),(x\sin\theta+y\cos\theta% \,)).
  15. θ \theta
  16. x = x cos 2 θ + y sin 2 θ x^{\prime}=x\cos 2\theta+y\sin 2\theta
  17. y = x sin 2 θ - y cos 2 θ . y^{\prime}=x\sin 2\theta-y\cos 2\theta.
  18. ( x , y ) = ( ( x cos 2 θ + y sin 2 θ ) , ( x sin 2 θ - y cos 2 θ ) ) . (x^{\prime},y^{\prime})=((x\cos 2\theta+y\sin 2\theta\,),(x\sin 2\theta-y\cos 2% \theta\,)).
  19. ( x , y ) (x^{\prime},y^{\prime})
  20. ( x , y ) (x,y)
  21. ( x , y ) = ( x , y ) A + b (x^{\prime},y^{\prime})=(x,y)A+b\,
  22. x = x A 11 + y A 21 + b 1 x^{\prime}=xA_{11}+yA_{21}+b_{1}\,
  23. y = x A 12 + y A 22 + b 2 , y^{\prime}=xA_{12}+yA_{22}+b_{2},\,
  24. A = ( A 11 A 12 A 21 A 22 ) . A=\begin{pmatrix}A_{11}&A_{12}\\ A_{21}&A_{22}\end{pmatrix}.
  25. A 11 A 21 + A 12 A 22 = 0 A_{11}A_{21}+A_{12}A_{22}=0
  26. A 11 2 + A 12 2 = A 21 2 + A 22 2 = 1. A_{11}^{2}+A_{12}^{2}=A_{21}^{2}+A_{22}^{2}=1.
  27. A 11 A 22 - A 21 A 12 = 1. A_{11}A_{22}-A_{21}A_{12}=1.
  28. A 11 A 22 - A 21 A 12 = - 1. A_{11}A_{22}-A_{21}A_{12}=-1.
  29. ( A 11 A 21 b 1 A 12 A 22 b 2 0 0 1 ) ( x y 1 ) = ( x y 1 ) . \begin{pmatrix}A_{11}&A_{21}&b_{1}\\ A_{12}&A_{22}&b_{2}\\ 0&0&1\end{pmatrix}\begin{pmatrix}x\\ y\\ 1\end{pmatrix}=\begin{pmatrix}x^{\prime}\\ y^{\prime}\\ 1\end{pmatrix}.
  30. ( x , y ) = ( m x , m y ) . (x^{\prime},y^{\prime})=(mx,my).
  31. ( x , y ) = ( x + y s , y ) (x^{\prime},y^{\prime})=(x+ys,y)\,
  32. ( x , y ) = ( x , x s + y ) (x^{\prime},y^{\prime})=(x,xs+y)\,
  33. 𝐫 \mathbf{r}
  34. 𝐫 = x 𝐢 + y 𝐣 \mathbf{r}=x\mathbf{i}+y\mathbf{j}
  35. 𝐢 = ( 1 0 ) \mathbf{i}=\begin{pmatrix}1\\ 0\end{pmatrix}
  36. 𝐣 = ( 0 1 ) \mathbf{j}=\begin{pmatrix}0\\ 1\end{pmatrix}
  37. ( x , y , z ) (x,y,z)
  38. 𝐫 = x 𝐢 + y 𝐣 + z 𝐤 \mathbf{r}=x\mathbf{i}+y\mathbf{j}+z\mathbf{k}
  39. 𝐤 = ( 0 0 1 ) \mathbf{k}=\begin{pmatrix}0\\ 0\\ 1\end{pmatrix}

Casimir_effect.html

  1. E = 1 2 ω . {E}=\begin{matrix}\frac{1}{2}\end{matrix}\hbar\omega\ .
  2. E n E_{n}
  3. E = 1 2 n E n \langle E\rangle=\frac{1}{2}\sum_{n}E_{n}
  4. E = ω / 2 E=\hbar\omega/2
  5. E n E_{n}
  6. E n ( s ) E_{n}(s)
  7. E ( s ) \langle E(s)\rangle
  8. δ s \delta s
  9. F ( p ) = - δ E ( s ) δ s | p . F(p)=-\left.\frac{\delta\langle E(s)\rangle}{\delta s}\right|_{p}.\,
  10. ψ n ( x , y , z ; t ) = e - i ω n t e i k x x + i k y y sin ( k n z ) \psi_{n}(x,y,z;t)=e^{-i\omega_{n}t}e^{ik_{x}x+ik_{y}y}\sin\left(k_{n}z\right)
  11. ψ \psi
  12. k x k_{x}
  13. k y k_{y}
  14. k n = n π a k_{n}=\frac{n\pi}{a}
  15. ω n = c k x 2 + k y 2 + n 2 π 2 a 2 \omega_{n}=c\sqrt{{k_{x}}^{2}+{k_{y}}^{2}+\frac{n^{2}\pi^{2}}{a^{2}}}
  16. E = 2 2 A d k x d k y ( 2 π ) 2 n = 1 ω n \langle E\rangle=\frac{\hbar}{2}\cdot 2\int\frac{Adk_{x}dk_{y}}{(2\pi)^{2}}% \sum_{n=1}^{\infty}\omega_{n}
  17. E ( s ) A = d k x d k y ( 2 π ) 2 n = 1 ω n | ω n | - s . \frac{\langle E(s)\rangle}{A}=\hbar\int\frac{dk_{x}dk_{y}}{(2\pi)^{2}}\sum_{n=% 1}^{\infty}\omega_{n}|\omega_{n}|^{-s}.
  18. s 0 s\to 0
  19. E ( s ) A = c 1 - s 4 π 2 n 0 2 π q d q | q 2 + π 2 n 2 a 2 | ( 1 - s ) / 2 , \frac{\langle E(s)\rangle}{A}=\frac{\hbar c^{1-s}}{4\pi^{2}}\sum_{n}\int_{0}^{% \infty}2\pi qdq\left|q^{2}+\frac{\pi^{2}n^{2}}{a^{2}}\right|^{(1-s)/2},
  20. q 2 = k x 2 + k y 2 q^{2}=k_{x}^{2}+k_{y}^{2}
  21. q q
  22. 2 π 2\pi
  23. E ( s ) A = - c 1 - s π 2 - s 2 a 3 - s 1 3 - s n | n | 3 - s . \frac{\langle E(s)\rangle}{A}=-\frac{\hbar c^{1-s}\pi^{2-s}}{2a^{3-s}}\frac{1}% {3-s}\sum_{n}|n|^{3-s}.
  24. E A = lim s 0 E ( s ) A = - c π 2 6 a 3 ζ ( - 3 ) . \frac{\langle E\rangle}{A}=\lim_{s\to 0}\frac{\langle E(s)\rangle}{A}=-\frac{% \hbar c\pi^{2}}{6a^{3}}\zeta(-3).
  25. ζ ( - 3 ) = 1 / 120 \zeta(-3)=1/120
  26. E A = - c π 2 3 240 a 3 . \frac{\langle E\rangle}{A}=\frac{-\hbar c\pi^{2}}{3\cdot 240a^{3}}.
  27. F c / A F_{c}/A
  28. F c A = - d d a E A = - c π 2 240 a 4 {F_{c}\over A}=-\frac{d}{da}\frac{\langle E\rangle}{A}=-\frac{\hbar c\pi^{2}}{% 240a^{4}}
  29. \hbar
  30. c c
  31. a a
  32. \hbar
  33. F c / A F_{c}/A
  34. | ω n | - s |\omega_{n}|^{-s}
  35. E ( t ) = 1 2 n | ω n | exp ( - t | ω n | ) \langle E(t)\rangle=\frac{1}{2}\sum_{n}\hbar|\omega_{n}|\exp(-t|\omega_{n}|)
  36. t 0 + t\to 0^{+}
  37. E ( t ) = C t 3 + finite \langle E(t)\rangle=\frac{C}{t^{3}}+\textrm{finite}\,
  38. E ( t ) = 1 2 n | ω n | exp ( - t 2 | ω n | 2 ) \langle E(t)\rangle=\frac{1}{2}\sum_{n}\hbar|\omega_{n}|\exp(-t^{2}|\omega_{n}% |^{2})
  39. E ( s ) = 1 2 n | ω n | | ω n | - s \langle E(s)\rangle=\frac{1}{2}\sum_{n}\hbar|\omega_{n}||\omega_{n}|^{-s}

Casorati–Weierstrass_theorem.html

  1. z 0 z_{0}
  2. U \ { z 0 } U\ \backslash\ \{z_{0}\}
  3. z 0 z_{0}
  4. z 0 z_{0}
  5. f ( V \ { z 0 } ) f(V\ \backslash\ \{z_{0}\})
  6. z 0 z_{0}
  7. z 0 z_{0}
  8. f ( z ) = e 1 / z . f(z)=e^{1/z}.\,
  9. f ( z ) = n = 0 1 n ! z - n . f(z)=\displaystyle\sum_{n=0}^{\infty}\frac{1}{n!}z^{-n}.
  10. f ( z ) = - e 1 z z 2 f^{\prime}(z)=\frac{-e^{\frac{1}{z}}}{z^{2}}
  11. z = r e i θ z=re^{i\theta}
  12. f ( z ) = e 1 r e - i θ = e 1 r cos ( θ ) e - 1 r i sin ( θ ) . f(z)=e^{\frac{1}{r}e^{-i\theta}}=e^{\frac{1}{r}\cos(\theta)}e^{-\frac{1}{r}i% \sin(\theta)}.
  13. | f ( z ) | = | e 1 r cos θ | | e - 1 r i sin ( θ ) | = e 1 r cos θ . \left|f(z)\right|=\left|e^{\frac{1}{r}\cos\theta}\right|\left|e^{-\frac{1}{r}i% \sin(\theta)}\right|=e^{\frac{1}{r}\cos\theta}.
  14. f ( z ) f(z)\rightarrow\infty
  15. r 0 r\rightarrow 0
  16. cos θ < 0 \cos\theta<0
  17. f ( z ) 0 f(z)\rightarrow 0
  18. r 0 r\rightarrow 0
  19. f ( z ) = e R [ cos ( R tan θ ) - i sin ( R tan θ ) ] f(z)=e^{R}\left[\cos\left(R\tan\theta\right)-i\sin\left(R\tan\theta\right)\right]
  20. | f ( z ) | = e R . \left|f(z)\right|=e^{R}.\,
  21. | f ( z ) | \left|f(z)\right|
  22. z 0 z\rightarrow 0
  23. θ π 2 \theta\rightarrow\frac{\pi}{2}
  24. [ cos ( R tan θ ) - i sin ( R tan θ ) ] \left[\cos\left(R\tan\theta\right)-i\sin\left(R\tan\theta\right)\right]\,
  25. g ( z ) = 1 f ( z ) - b g(z)=\frac{1}{f(z)-b}
  26. f ( z ) = 1 g ( z ) + b f(z)=\frac{1}{g(z)}+b
  27. lim z z 0 g ( z ) . \lim_{z\to z_{0}}g(z).

Casting_(metalworking).html

  1. t = B ( V A ) n t=B\left(\frac{V}{A}\right)^{n}

Catalan's_conjecture.html

  1. A x n - B y m = C Ax^{n}-By^{m}=C
  2. | A x n - B y m | x λ n |Ax^{n}-By^{m}|\gg x^{\lambda n}

Catalan's_constant.html

  1. G = β ( 2 ) = n = 0 ( - 1 ) n ( 2 n + 1 ) 2 = 1 1 2 - 1 3 2 + 1 5 2 - 1 7 2 + G=\beta(2)=\sum_{n=0}^{\infty}\frac{(-1)^{n}}{(2n+1)^{2}}=\frac{1}{1^{2}}-% \frac{1}{3^{2}}+\frac{1}{5^{2}}-\frac{1}{7^{2}}+\cdots\!
  2. G = 0 1 0 1 1 1 + x 2 y 2 d x d y G=\int_{0}^{1}\int_{0}^{1}\frac{1}{1+x^{2}y^{2}}\,dx\,dy\!
  3. G = - 0 1 ln t 1 + t 2 d t G=-\int_{0}^{1}\frac{\ln t}{1+t^{2}}\,dt\!
  4. G = 0 π / 4 t sin t cos t d t G=\int_{0}^{\pi/4}\frac{t}{\sin t\cos t}\;dt\!
  5. G = 1 4 - π / 2 π / 2 t sin t d t G=\frac{1}{4}\int_{-\pi/2}^{\pi/2}\frac{t}{\sin t}\;dt\!
  6. G = 0 π / 4 ln ( cot ( t ) ) d t G=\int_{0}^{\pi/4}\ln(\cot(t))\,dt\!
  7. G = 0 arctan ( e - t ) d t G=\int_{0}^{\infty}\arctan(e^{-t})\,dt\!
  8. G = 1 ln t 1 + t 2 d t G=\int_{1}^{\infty}\frac{\ln t}{1+t^{2}}\,dt\!
  9. G = 1 2 0 1 K ( t ) d t G=\frac{1}{2}\int_{0}^{1}\mathrm{K}(t)\,dt\!
  10. Γ ( x + 1 ) = x ! \Gamma(x+1)=x!
  11. G = π 4 0 1 Γ ( 1 + x 2 ) Γ ( 1 - x 2 ) d x = π 2 0 1 2 Γ ( 1 + y ) Γ ( 1 - y ) d y G=\frac{\pi}{4}\int_{0}^{1}\Gamma(1+\tfrac{x}{2})\Gamma(1-\tfrac{x}{2})\,dx=% \frac{\pi}{2}\int_{0}^{\tfrac{1}{2}}\Gamma(1+y)\Gamma(1-y)\,dy
  12. G = Ti 2 ( 1 ) = 0 1 arctan t t d t . G=\,\text{Ti}_{2}(1)=\int_{0}^{1}\frac{\arctan t}{t}\,dt.\!
  13. ψ 1 ( 1 4 ) = π 2 + 8 G \psi_{1}\left(\tfrac{1}{4}\right)=\pi^{2}+8G
  14. ψ 1 ( 3 4 ) = π 2 - 8 G . \psi_{1}\left(\tfrac{3}{4}\right)=\pi^{2}-8G.
  15. G = 4 π log ( G ( 3 8 ) G ( 7 8 ) G ( 1 8 ) G ( 5 8 ) ) + 4 π log ( Γ ( 3 8 ) Γ ( 1 8 ) ) + π 2 log ( 1 + 2 2 ( 2 - 2 ) ) G=4\pi\log\left(\frac{G(\tfrac{3}{8})G(\tfrac{7}{8})}{G(\tfrac{1}{8})G(\tfrac{% 5}{8})}\right)+4\pi\log\left(\frac{\Gamma(\tfrac{3}{8})}{\Gamma(\tfrac{1}{8})}% \right)+\frac{\pi}{2}\log\left(\frac{1+\sqrt{2}}{2\,(2-\sqrt{2})}\right)
  16. Φ ( z , s , α ) \Phi(z,s,\alpha)
  17. Φ ( z , s , α ) = n = 0 z n ( n + α ) s . \Phi(z,s,\alpha)=\sum_{n=0}^{\infty}\frac{z^{n}}{(n+\alpha)^{s}}.
  18. G = 1 4 Φ ( - 1 , 2 , 1 2 ) . G=\tfrac{1}{4}\,\Phi(-1,2,\tfrac{1}{2}).
  19. G \displaystyle G
  20. G = 1 8 π log ( 2 + 3 ) + 3 8 n = 0 ( n ! ) 2 ( 2 n ) ! ( 2 n + 1 ) 2 . G=\tfrac{1}{8}\pi\log(2+\sqrt{3})+\tfrac{3}{8}\sum_{n=0}^{\infty}\frac{(n!)^{2% }}{(2n)!(2n+1)^{2}}.

Category_of_sets.html

  1. V ω V_{\omega}

Catenary.html

  1. y = A c o s h ( B x ) y=Acosh(Bx)
  2. A B = 1 AB=1
  3. y = a cosh ( x a ) = a 2 ( e x / a + e - x / a ) y=a\,\cosh\left({x\over a}\right)={a\over 2}\,\left(e^{x/a}+e^{-x/a}\right)\,
  4. tan φ = s a . \tan\varphi=\frac{s}{a}.\,
  5. d φ d s = cos 2 φ a \frac{d\varphi}{ds}=\frac{\cos^{2}\varphi}{a}\,
  6. φ \varphi
  7. κ = a s 2 + a 2 . \kappa=\frac{a}{s^{2}+a^{2}}.\,
  8. ρ = a sec 2 φ \rho=a\sec^{2}\varphi\,
  9. d 𝐫 d s = 𝐮 \frac{d\mathbf{r}}{ds}=\mathbf{u}\,
  10. T cos φ = T 0 T\cos\varphi=T_{0}\,
  11. T sin φ = λ g s , T\sin\varphi=\lambda gs,\,
  12. d y d x = tan φ = λ g s T 0 . \frac{dy}{dx}=\tan\varphi=\frac{\lambda gs}{T_{0}}.\,
  13. a = T 0 λ g a=\frac{T_{0}}{\lambda g}\,
  14. d y d x = s a \frac{dy}{dx}=\frac{s}{a}\,
  15. d y d x = s a , \frac{dy}{dx}=\frac{s}{a},\,
  16. d s d x = 1 + ( d y d x ) 2 = a 2 + s 2 a . \frac{ds}{dx}=\sqrt{1+\left(\dfrac{dy}{dx}\right)^{2}}=\frac{\sqrt{a^{2}+s^{2}% }}{a}.\,
  17. d x d s = 1 d s d x = a a 2 + s 2 \frac{dx}{ds}=\frac{1}{\frac{ds}{dx}}=\frac{a}{\sqrt{a^{2}+s^{2}}}\,
  18. d y d s = d y d x d s d x = s a 2 + s 2 . \frac{dy}{ds}=\frac{\frac{dy}{dx}}{\frac{ds}{dx}}=\frac{s}{\sqrt{a^{2}+s^{2}}}.\,
  19. y = a 2 + s 2 + β y=\sqrt{a^{2}+s^{2}}+\beta\,
  20. y = a 2 + s 2 , y 2 = a 2 + s 2 . y=\sqrt{a^{2}+s^{2}},\ y^{2}=a^{2}+s^{2}.\,
  21. x = a arcsinh ( s / a ) + α . x=a\ \operatorname{arcsinh}(s/a)+\alpha.\,
  22. x = a arcsinh ( s / a ) , s = a sinh x a . x=a\ \operatorname{arcsinh}(s/a),\ s=a\sinh{x\over a}.\,
  23. y = a cosh x a . y=a\cosh\frac{x}{a}.\,
  24. s = a tan φ s=a\tan\varphi\,
  25. d x d φ = d x d s d s d φ = cos φ a sec 2 φ = a sec φ \frac{dx}{d\varphi}=\frac{dx}{ds}\frac{ds}{d\varphi}=\cos\varphi\cdot a\sec^{2% }\varphi=a\sec\varphi\,
  26. d y d φ = d y d s d s d φ = sin φ a sec 2 φ = a tan φ sec φ . \frac{dy}{d\varphi}=\frac{dy}{ds}\frac{ds}{d\varphi}=\sin\varphi\cdot a\sec^{2% }\varphi=a\tan\varphi\sec\varphi.\,
  27. x = a ln ( sec φ + tan φ ) + α , x=a\ln(\sec\varphi+\tan\varphi)+\alpha,\,
  28. y = a sec φ + β . y=a\sec\varphi+\beta.\,
  29. sec φ + tan φ = e x / a , \sec\varphi+\tan\varphi=e^{x/a},\,
  30. sec φ - tan φ = e - x / a . \sec\varphi-\tan\varphi=e^{-x/a}.\,
  31. y = a sec φ = a cosh x a , y=a\sec\varphi=a\cosh\tfrac{x}{a},\,
  32. s = a tan φ = a sinh x a . s=a\tan\varphi=a\sinh\tfrac{x}{a}.\,
  33. y = a cosh x a y=a\cosh\tfrac{x}{a}\,
  34. v = a cosh x 2 a - a cosh x 1 a . v=a\cosh\tfrac{x_{2}}{a}-a\cosh\tfrac{x_{1}}{a}.\,
  35. s = a sinh x 2 a - a sinh x 1 a . s=a\sinh\tfrac{x_{2}}{a}-a\sinh\tfrac{x_{1}}{a}.\,
  36. s 2 - v 2 = a 2 ( - 2 + 2 cosh x 2 - x 1 a ) = 4 a 2 sinh 2 h 2 a , s^{2}-v^{2}=a^{2}(-2+2\cosh\tfrac{x_{2}-x_{1}}{a})=4a^{2}\sinh^{2}\tfrac{h}{2a% },\,
  37. s 2 - v 2 = 2 a sinh h 2 a . \sqrt{s^{2}-v^{2}}=2a\sinh\tfrac{h}{2a}.\,
  38. w d s , \int w\ ds,\,
  39. T cos φ = T 0 T\cos\varphi=T_{0}\,
  40. T sin φ = w d s , T\sin\varphi=\int w\ ds,\,
  41. d y d x = tan φ = 1 T 0 w d s . \frac{dy}{dx}=\tan\varphi=\frac{1}{T_{0}}\int w\ ds.\,
  42. w = T 0 d d s d y d x = T 0 d 2 y d x 2 1 + ( d y d x ) 2 . w=T_{0}\frac{d}{ds}\frac{dy}{dx}=\frac{T_{0}\frac{d^{2}y}{dx^{2}}}{\sqrt{1+% \left(\frac{dy}{dx}\right)^{2}}}.\,
  43. w = T 0 ρ cos 2 φ . w=\frac{T_{0}}{\rho\cos^{2}\varphi}.\,
  44. d y d x = tan φ = w T 0 x . \frac{dy}{dx}=\tan\varphi=\frac{w}{T_{0}}x.\,
  45. y = w 2 T 0 x 2 + β y=\frac{w}{2T_{0}}x^{2}+\beta\,
  46. T cos φ = T 0 , T\cos\varphi=T_{0},\,
  47. T sin φ = 1 c T d s . T\sin\varphi=\frac{1}{c}\int T\ ds.\,
  48. c tan φ = sec φ d s c\tan\varphi=\int\sec\varphi\ ds\,
  49. c = ρ cos φ c=\rho\cos\varphi\,
  50. y = c ln sec x c . y=c\ln\sec\frac{x}{c}.\,
  51. x = c φ , s = ln tan 1 4 ( π + 2 φ ) . x=c\varphi,\ s=\ln\tan\tfrac{1}{4}(\pi+2\varphi).\,
  52. s = ( 1 + T E ) p , s=\left(1+\frac{T}{E}\right)p,\,
  53. d s d p = 1 + T E . \frac{ds}{dp}=1+\frac{T}{E}.
  54. T cos φ = T 0 , T\cos\varphi=T_{0},\,
  55. T sin φ = λ 0 g p , T\sin\varphi=\lambda_{0}gp,\,
  56. d y d x = tan φ = λ 0 g p T 0 , T = T 0 2 + λ 0 2 g 2 p 2 , \frac{dy}{dx}=\tan\varphi=\frac{\lambda_{0}gp}{T_{0}},\ T=\sqrt{T_{0}^{2}+% \lambda_{0}^{2}g^{2}p^{2}},\,
  57. a = T 0 λ 0 g a=\frac{T_{0}}{\lambda_{0}g}\,
  58. d y d x = tan φ = p a , T = T 0 a a 2 + p 2 . \frac{dy}{dx}=\tan\varphi=\frac{p}{a},\ T=\frac{T_{0}}{a}\sqrt{a^{2}+p^{2}}.
  59. d x d s = cos φ = T 0 T \frac{dx}{ds}=\cos\varphi=\frac{T_{0}}{T}
  60. d y d s = sin φ = λ 0 g p T , \frac{dy}{ds}=\sin\varphi=\frac{\lambda_{0}gp}{T},
  61. d x d p = T 0 T d s d p = T 0 ( 1 T + 1 E ) = a a 2 + p 2 + T 0 E \frac{dx}{dp}=\frac{T_{0}}{T}\frac{ds}{dp}=T_{0}(\frac{1}{T}+\frac{1}{E})=% \frac{a}{\sqrt{a^{2}+p^{2}}}+\frac{T_{0}}{E}
  62. d y d p = λ 0 g p T d s d p = T 0 p a ( 1 T + 1 E ) = p a 2 + p 2 + T 0 p E a . \frac{dy}{dp}=\frac{\lambda_{0}gp}{T}\frac{ds}{dp}=\frac{T_{0}p}{a}(\frac{1}{T% }+\frac{1}{E})=\frac{p}{\sqrt{a^{2}+p^{2}}}+\frac{T_{0}p}{Ea}.
  63. x = a arcsinh ( p / a ) + T 0 E p + α , x=a\operatorname{arcsinh}(p/a)+\frac{T_{0}}{E}p+\alpha,
  64. y = a 2 + p 2 + T 0 2 E a p 2 + β . y=\sqrt{a^{2}+p^{2}}+\frac{T_{0}}{2Ea}p^{2}+\beta.
  65. x = a arcsinh ( p / a ) + T 0 E p , x=a\ \operatorname{arcsinh}(p/a)+\frac{T_{0}}{E}p,\,
  66. y = a 2 + p 2 + T 0 2 E a p 2 y=\sqrt{a^{2}+p^{2}}+\frac{T_{0}}{2Ea}p^{2}\,
  67. 𝐓 = T 𝐮 , \mathbf{T}=T\mathbf{u},\,
  68. 𝐓 ( s + Δ s ) - 𝐓 ( s ) + 𝐆 Δ s 0. \mathbf{T}(s+\Delta s)-\mathbf{T}(s)+\mathbf{G}\Delta s\approx\mathbf{0}.\,
  69. d 𝐓 d s + 𝐆 = 0. \frac{d\mathbf{T}}{ds}+\mathbf{G}=\mathbf{0}.\,

Cauchy's_integral_theorem.html

  1. γ \!\,\gamma
  2. γ f ( z ) d z = 0. \oint_{\gamma}f(z)\,dz=0.
  3. π \pi
  4. U = { z : | z - z 0 | < r } U=\{z:|z-z_{0}|<r\}
  5. γ ( t ) = e i t t [ 0 , 2 π ] \gamma(t)=e^{it}\quad t\in\left[0,2\pi\right]
  6. γ 1 z d z = 0 2 π i e i t e i t d t = 0 2 π i d t = 2 π i \oint_{\gamma}\frac{1}{z}\,dz=\int_{0}^{2\pi}{ie^{it}\over e^{it}}\,dt=\int_{0% }^{2\pi}i\,dt=2\pi i
  7. f ( z ) = 1 / z f(z)=1/z
  8. z = 0 z=0
  9. γ f ( z ) d z = F ( b ) - F ( a ) . \int_{\gamma}f(z)\,dz=F(b)-F(a).
  10. U ¯ \textstyle\overline{U}
  11. γ \gamma
  12. U ¯ \textstyle\overline{U}
  13. γ k \gamma_{k}
  14. γ k \gamma_{k}
  15. γ f ( z ) d z = 0. \oint_{\gamma}f(z)\,dz=0.
  16. f = u + i v f=u+iv
  17. γ \gamma
  18. f f
  19. d z dz
  20. f = u + i v \displaystyle f=u+iv
  21. d z = d x + i d y \displaystyle dz=dx+i\,dy
  22. γ f ( z ) d z = γ ( u + i v ) ( d x + i d y ) = γ ( u d x - v d y ) + i γ ( v d x + u d y ) \oint_{\gamma}f(z)\,dz=\oint_{\gamma}(u+iv)(dx+i\,dy)=\oint_{\gamma}(u\,dx-v\,% dy)+i\oint_{\gamma}(v\,dx+u\,dy)
  23. γ \gamma
  24. D D
  25. γ \gamma
  26. γ ( u d x - v d y ) = D ( - v x - u y ) d x d y \oint_{\gamma}(u\,dx-v\,dy)=\iint_{D}\left(-\frac{\partial v}{\partial x}-% \frac{\partial u}{\partial y}\right)\,dx\,dy
  27. γ ( v d x + u d y ) = D ( u x - v y ) d x d y \oint_{\gamma}(v\,dx+u\,dy)=\iint_{D}\left(\frac{\partial u}{\partial x}-\frac% {\partial v}{\partial y}\right)\,dx\,dy
  28. D D
  29. u u
  30. v v
  31. u x = v y {\partial u\over\partial x}={\partial v\over\partial y}
  32. u y = - v x {\partial u\over\partial y}=-{\partial v\over\partial x}
  33. D ( - v x - u y ) d x d y = D ( u y - u y ) d x d y = 0 \iint_{D}\left(-\frac{\partial v}{\partial x}-\frac{\partial u}{\partial y}% \right)\,dx\,dy=\iint_{D}\left(\frac{\partial u}{\partial y}-\frac{\partial u}% {\partial y}\right)\,dx\,dy=0
  34. D ( u x - v y ) d x d y = D ( u x - u x ) d x d y = 0 \iint_{D}\left(\frac{\partial u}{\partial x}-\frac{\partial v}{\partial y}% \right)\,dx\,dy=\iint_{D}\left(\frac{\partial u}{\partial x}-\frac{\partial u}% {\partial x}\right)\,dx\,dy=0
  35. γ f ( z ) d z = 0 \oint_{\gamma}f(z)\,dz=0

Cauchy_distribution.html

  1. f ( x ; 0 , 1 ) = 1 π ( 1 + x 2 ) . f(x;0,1)=\frac{1}{\pi(1+x^{2})}.\!
  2. F ( x ; 0 , 1 ) = 1 π arctan ( x ) + 1 2 F(x;0,1)=\frac{1}{\pi}\arctan\left(x\right)+\frac{1}{2}
  3. f ( x ; x 0 , γ ) f(x;x_{0},\gamma)
  4. ( x 0 , γ ) (x_{0},\gamma)
  5. E [ ln ( 1 + X 2 ) ] = ln ( 4 ) \operatorname{E}\!\left[\ln(1+X^{2})\right]=\ln(4)
  6. f ( x ; x 0 , γ ) = 1 π γ [ 1 + ( x - x 0 γ ) 2 ] = 1 π γ [ γ 2 ( x - x 0 ) 2 + γ 2 ] , f(x;x_{0},\gamma)=\frac{1}{\pi\gamma\left[1+\left(\frac{x-x_{0}}{\gamma}\right% )^{2}\right]}={1\over\pi\gamma}\left[{\gamma^{2}\over(x-x_{0})^{2}+\gamma^{2}}% \right],
  7. Amplitude (or height) = 1 π γ . \,\text{Amplitude (or height)}=\frac{1}{\pi\gamma}.
  8. f ( x ; 0 , 1 ) = 1 π ( 1 + x 2 ) . f(x;0,1)=\frac{1}{\pi(1+x^{2})}.\!
  9. f ( x ; x 0 , γ , I ) = I [ 1 + ( x - x 0 γ ) 2 ] = I [ γ 2 ( x - x 0 ) 2 + γ 2 ] , f(x;x_{0},\gamma,I)=\frac{I}{\left[1+\left(\frac{x-x_{0}}{\gamma}\right)^{2}% \right]}=I\left[{\gamma^{2}\over(x-x_{0})^{2}+\gamma^{2}}\right],
  10. I = 1 π γ . I=\frac{1}{\pi\gamma}.\!
  11. F ( x ; x 0 , γ ) = 1 π arctan ( x - x 0 γ ) + 1 2 F(x;x_{0},\gamma)=\frac{1}{\pi}\arctan\left(\frac{x-x_{0}}{\gamma}\right)+% \frac{1}{2}
  12. Q ( p ; x 0 , γ ) = x 0 + γ tan [ π ( p - 1 2 ) ] . Q(p;x_{0},\gamma)=x_{0}+\gamma\,\tan\left[\pi\left(p-\tfrac{1}{2}\right)\right].
  13. Q ( p ; γ ) = γ π sec 2 [ π ( p - 1 2 ) ] . Q^{\prime}(p;\gamma)=\gamma\,\pi\,{\sec}^{2}\left[\pi\left(p-\tfrac{1}{2}% \right)\right].\!
  14. h ( γ ) = 0 1 log ( Q ( p ; γ ) ) d p = log ( γ ) + log ( 4 π ) . h(\gamma)=\int_{0}^{1}\log\,(Q^{\prime}(p;\gamma))\,\mathrm{d}p=\log(\gamma)\,% +\,\log(4\,\pi).\!
  15. ϕ X ¯ ( t ) = E [ e i X ¯ t ] \phi_{\overline{X}}(t)=\mathrm{E}\left[e^{i\overline{X}t}\right]
  16. X ¯ \overline{X}
  17. ϕ X ( t ; x 0 , γ ) = E [ e i X t ] = - f ( x ; x 0 , γ ) e i x t d x = e i x 0 t - γ | t | . \phi_{X}(t;x_{0},\gamma)=\mathrm{E}\left[e^{iXt}\right]=\int_{-\infty}^{\infty% }f(x;x_{0},\gamma)e^{ixt}\,dx=e^{ix_{0}t-\gamma|t|}.
  18. f ( x ; x 0 , γ ) = 1 2 π - ϕ X ( t ; x 0 , γ ) e - i x t d t f(x;x_{0},\gamma)=\frac{1}{2\pi}\int_{-\infty}^{\infty}\phi_{X}(t;x_{0},\gamma% )e^{-ixt}\,dt\!
  19. - x f ( x ) d x . ( 1 ) \int_{-\infty}^{\infty}xf(x)\,dx.\qquad\qquad(1)\!
  20. a x f ( x ) d x + - a x f ( x ) d x . ( 2 ) \int_{a}^{\infty}xf(x)\,dx+\int_{-\infty}^{a}xf(x)\,dx.\qquad\qquad(2)\!
  21. lim a - a a x f ( x ) d x , \lim_{a\to\infty}\int_{-a}^{a}xf(x)\,dx,\!
  22. lim a - 2 a a x f ( x ) d x , \lim_{a\to\infty}\int_{-2a}^{a}xf(x)\,dx,\!
  23. E [ X 2 ] \displaystyle\mathrm{E}[X^{2}]
  24. - \infty-\infty
  25. x ¯ = 1 n i = 1 n x i \bar{x}=\frac{1}{n}\sum_{i=1}^{n}x_{i}
  26. ^ ( x 0 , γ x 1 , , x n ) = - n log ( γ π ) - i = 1 n log ( 1 + ( x i - x 0 γ ) 2 ) \hat{\ell}(\!x_{0},\gamma\mid x_{1},\ldots,x_{n})=-n\log(\gamma\pi)-\sum_{i=1}% ^{n}\log\left(1+\left(\frac{x_{i}-x_{0}}{\gamma}\right)^{2}\right)
  27. i = 1 n x i - x 0 γ 2 + [ x i - x 0 ] 2 = 0 \sum_{i=1}^{n}\frac{x_{i}-x_{0}}{\gamma^{2}+[x_{i}-\!x_{0}]^{2}}=0
  28. i = 1 n γ 2 γ 2 + [ x i - x 0 ] 2 - n 2 = 0 \sum_{i=1}^{n}\frac{\gamma^{2}}{\gamma^{2}+[x_{i}-x_{0}]^{2}}-\frac{n}{2}=0
  29. i = 1 n γ 2 γ 2 + [ x i - x 0 ] 2 \sum_{i=1}^{n}\frac{\gamma^{2}}{\gamma^{2}+[x_{i}-x_{0}]^{2}}
  30. min | x i - x 0 | γ max | x i - x 0 | . \min|x_{i}-x_{0}|\leq\gamma\leq\max|x_{i}-x_{0}|.
  31. Z = X - i X + i Z=\frac{X-i}{X+i}
  32. P c c ( θ ; ζ ) = 1 2 π 1 - | ζ | 2 | e i θ - ζ | 2 P_{cc}(\theta;\zeta)=\frac{1}{2\pi}\frac{1-|\zeta|^{2}}{|e^{i\theta}-\zeta|^{2}}
  33. ζ = ψ - i ψ + i \zeta=\frac{\psi-i}{\psi+i}
  34. ψ = μ + i γ \psi=\mu+i\gamma\,
  35. P c c ( θ ; ζ ) P_{cc}(\theta;\zeta)
  36. P w c ( θ ; ψ ) P_{wc}(\theta;\psi)
  37. P w c ( θ ; ψ ) = P c c ( θ , e i ψ ) P_{wc}(\theta;\psi)=P_{cc}(\theta,e^{i\psi})\,
  38. E [ Z r ] = ζ r , E [ Z ¯ r ] = ζ ¯ r \operatorname{E}[Z^{r}]=\zeta^{r},\quad\operatorname{E}[\bar{Z}^{r}]=\bar{% \zeta}^{r}
  39. n - 1 U ( z , ζ ^ ) = n - 1 U ( z j , ζ ^ ) = 0 n^{-1}U\left(z,\hat{\zeta}\right)=n^{-1}\sum U\left(z_{j},\hat{\zeta}\right)=0
  40. ζ ( r + 1 ) = U ( n - 1 U ( z , ζ ( r ) ) , - ζ ( r ) ) \zeta^{(r+1)}=U\left(n^{-1}U(z,\zeta^{(r)}),\,-\zeta^{(r)}\right)\,
  41. μ ^ \hat{\mu}
  42. γ ^ \hat{\gamma}
  43. μ ^ ± i γ ^ = i 1 + ζ ^ 1 - ζ ^ . \hat{\mu}\pm i\hat{\gamma}=i\frac{1+\hat{\zeta}}{1-\hat{\zeta}}.
  44. ζ ^ \hat{\zeta}
  45. 1 4 π p n ( χ ( t , ζ ) ) ( 1 - | t | 2 ) 2 , \frac{1}{4\pi}\frac{p_{n}(\chi(t,\zeta))}{(1-|t|^{2})^{2}},
  46. χ ( t , ζ ) = | t - ζ | 2 4 ( 1 - | t | 2 ) ( 1 - | ζ | 2 ) \chi(t,\zeta)=\frac{|t-\zeta|^{2}}{4(1-|t|^{2})(1-|\zeta|^{2})}
  47. ϕ X ( t ) = e i x 0 ( t ) - γ ( t ) , \phi_{X}(t)=e^{ix_{0}(t)-\gamma(t)},\!
  48. x 0 ( a t ) = a x 0 ( t ) , x_{0}(at)=ax_{0}(t),
  49. γ ( a t ) = | a | γ ( t ) , \gamma(at)=|a|\gamma(t),
  50. f ( x , y ; x 0 , y 0 , γ ) = 1 2 π [ γ ( ( x - x 0 ) 2 + ( y - y 0 ) 2 + γ 2 ) 1.5 ] . f(x,y;x_{0},y_{0},\gamma)={1\over 2\pi}\left[{\gamma\over((x-x_{0})^{2}+(y-y_{% 0})^{2}+\gamma^{2})^{1.5}}\right].
  51. f ( 𝐱 ; μ , 𝚺 , k ) = Γ ( 1 + k 2 ) Γ ( 1 2 ) π k 2 | 𝚺 | 1 2 [ 1 + ( 𝐱 - μ ) T 𝚺 - 1 ( 𝐱 - μ ) ] 1 + k 2 . f({\mathbf{x}};{\mathbf{\mu}},{\mathbf{\Sigma}},k)=\frac{\Gamma\left(\frac{1+k% }{2}\right)}{\Gamma(\frac{1}{2})\pi^{\frac{k}{2}}\left|{\mathbf{\Sigma}}\right% |^{\frac{1}{2}}\left[1+({\mathbf{x}}-{\mathbf{\mu}})^{T}{\mathbf{\Sigma}}^{-1}% ({\mathbf{x}}-{\mathbf{\mu}})\right]^{\frac{1+k}{2}}}.
  52. X Cauchy ( x 0 , γ ) X\sim\textrm{Cauchy}(x_{0},\gamma)\,
  53. k X + l Cauchy ( x 0 k + l , γ | k | ) kX+l\sim\textrm{Cauchy}(x_{0}{k}+l,\gamma|k|)\,
  54. X Cauchy ( x 0 , γ 0 ) X\sim\textrm{Cauchy}(x_{0},\gamma_{0})\,
  55. Y Cauchy ( x 1 , γ 1 ) Y\sim\textrm{Cauchy}(x_{1},\gamma_{1})\,
  56. X + Y Cauchy ( x 0 + x 1 , γ 0 + γ 1 ) X+Y\sim\textrm{Cauchy}(x_{0}+x_{1},\gamma_{0}+\gamma_{1})\,
  57. X Cauchy ( 0 , γ ) X\sim\textrm{Cauchy}(0,\gamma)\,
  58. 1 X Cauchy ( 0 , 1 γ ) \tfrac{1}{X}\sim\textrm{Cauchy}(0,\tfrac{1}{\gamma})\,
  59. ψ = x 0 + i γ \psi=x_{0}+i\gamma
  60. a X + b c X + d \frac{aX+b}{cX+d}
  61. ( a ψ + b c ψ + d ) \left(\frac{a\psi+b}{c\psi+d}\right)
  62. X - i X + i \frac{X-i}{X+i}
  63. ( ψ - i ψ + i ) \left(\frac{\psi-i}{\psi+i}\right)
  64. Cauchy ( 0 , 1 ) t ( d f = 1 ) \textrm{Cauchy}(0,1)\sim\textrm{t}(df=1)\,
  65. Cauchy ( μ , σ ) t ( d f = 1 ) ( μ , σ ) \textrm{Cauchy}(\mu,\sigma)\sim\textrm{t}_{(df=1)}(\mu,\sigma)\,
  66. X , Y N ( 0 , 1 ) X , Y X,Y\sim\textrm{N}(0,1)\,X,Y
  67. X Y Cauchy ( 0 , 1 ) \tfrac{X}{Y}\sim\textrm{Cauchy}(0,1)\,
  68. X U ( 0 , 1 ) X\sim\textrm{U}(0,1)\,
  69. tan ( π ( X - 1 2 ) ) Cauchy ( 0 , 1 ) \tan\left({\pi\left(X-\tfrac{1}{2}\right)}\right)\sim\textrm{Cauchy}(0,1)\,

Cauchy_sequence.html

  1. x 1 , x 2 , x 3 , x_{1},x_{2},x_{3},\ldots
  2. | x m - x n | < ε , |x_{m}-x_{n}|<\varepsilon,
  3. x m - x n x_{m}-x_{n}
  4. | x m - x n | |x_{m}-x_{n}|
  5. d ( x m , x n ) d(x_{m},x_{n})
  6. x m x_{m}
  7. x n x_{n}
  8. x 1 , x 2 , x 3 , x_{1},x_{2},x_{3},\ldots
  9. d ( x m , x n ) < ε . d(x_{m},x_{n})<\varepsilon.
  10. x 0 = 1 , x n + 1 = x n + 2 x n 2 x_{0}=1,x_{n+1}=\frac{x_{n}+\frac{2}{x_{n}}}{2}
  11. x n = F n / F n - 1 x_{n}=F_{n}/F_{n-1}\,
  12. ϕ \phi
  13. ϕ 2 = ϕ + 1 \phi^{2}=\phi+1
  14. φ = ( 1 + 5 ) / 2 \varphi=(1+\sqrt{5})/2
  15. X = ( 0 , 2 ) X=(0,2)
  16. x n = 1 / n x_{n}=1/n
  17. d > 0 d>0
  18. x n x_{n}
  19. n > 1 / d n>1/d
  20. ( 0 , d ) (0,d)
  21. n = 1 x n \sum_{n=1}^{\infty}x_{n}
  22. ( s m ) (s_{m})
  23. s m = n = 1 m x n s_{m}=\sum_{n=1}^{m}x_{n}
  24. s p - s q = n = q + 1 p x n . s_{p}-s_{q}=\sum_{n=q+1}^{p}x_{n}.
  25. f : M N f\colon M\rightarrow N
  26. ( f ( x n ) ) (f(x_{n}))
  27. ( x n ) (x_{n})
  28. ( y n ) (y_{n})
  29. ( x n + y n ) (x_{n}+y_{n})
  30. ( x n y n ) (x_{n}y_{n})
  31. X X
  32. B B
  33. X X
  34. x k x_{k}
  35. V B V\in B
  36. N N
  37. n , m > N , x n - x m n,m>N,x_{n}-x_{m}
  38. V V
  39. X X
  40. d d
  41. ( x k ) (x_{k})
  42. G G
  43. U U
  44. G G
  45. N N
  46. m , n > N m,n>N
  47. x n x m - 1 U x_{n}x_{m}^{-1}\in U
  48. G G
  49. G G
  50. ( x k ) (x_{k})
  51. ( y k ) (y_{k})
  52. U U
  53. G G
  54. N N
  55. m , n > N m,n>N
  56. x n y m - 1 U x_{n}y_{m}^{-1}\in U
  57. y n x m - 1 = ( x m y n - 1 ) - 1 U - 1 y_{n}x_{m}^{-1}=(x_{m}y_{n}^{-1})^{-1}\in U^{-1}
  58. x n z l - 1 = x n y m - 1 y m z l - 1 U U ′′ x_{n}z_{l}^{-1}=x_{n}y_{m}^{-1}y_{m}z_{l}^{-1}\in U^{\prime}U^{\prime\prime}
  59. U U^{\prime}
  60. U ′′ U^{\prime\prime}
  61. U U ′′ U U^{\prime}U^{\prime\prime}\subseteq U
  62. G G
  63. H = ( H r ) H=(H_{r})
  64. G G
  65. ( x n ) (x_{n})
  66. G G
  67. H H
  68. r r
  69. N N
  70. m , n > N , x n x m - 1 H r \forall m,n>N,x_{n}x_{m}^{-1}\in H_{r}
  71. G G
  72. H H
  73. C C
  74. C 0 C_{0}
  75. r , N , n > N , x n H r \forall r,\exists N,\forall n>N,x_{n}\in H_{r}
  76. C C
  77. C / C 0 C/C_{0}
  78. G G
  79. H H
  80. ( G / H r ) (G/H_{r})
  81. H H
  82. H r H_{r}
  83. ( G / H ) H (G/H)_{H}
  84. H H
  85. ( x 1 , x 2 , x 3 , ) (x_{1},x_{2},x_{3},...)
  86. X X
  87. α \alpha
  88. k m , n > α ( k ) , | x m - x n | < 1 / k \forall k\forall m,n>\alpha(k),|x_{m}-x_{n}|<1/k
  89. α ( k ) \alpha(k)
  90. N N
  91. r r
  92. 1 / k 1/k
  93. α ( k ) = k \alpha(k)=k
  94. α ( k ) = 2 k \alpha(k)=2^{k}
  95. u n : n \langle u_{n}:n\in\mathbb{N}\rangle
  96. u H u_{H}
  97. u K u_{K}
  98. st ( u H - u K ) = 0 \,\mathrm{st}(u_{H}-u_{K})=0

Cauchy–Riemann_equations.html

  1. u x = v y \dfrac{\partial u}{\partial x}=\dfrac{\partial v}{\partial y}
  2. u y = - v x \dfrac{\partial u}{\partial y}=-\dfrac{\partial v}{\partial x}
  3. i f x = f y . {i\dfrac{\partial f}{\partial x}}=\dfrac{\partial f}{\partial y}.
  4. ( a - b b a ) , \begin{pmatrix}a&-b\\ b&\;\;a\end{pmatrix},
  5. a = u / x = v / y \scriptstyle a=\partial u/\partial x=\partial v/\partial y
  6. b = v / x = - u / y \scriptstyle b=\partial v/\partial x=-\partial u/\partial y
  7. f ( z ) = u ( z ) + i v ( z ) f(z)=u(z)+i\cdot v(z)
  8. lim h 0 h 𝐂 f ( z 0 + h ) - f ( z 0 ) h = f ( z 0 ) \lim_{\underset{h\in\mathbf{C}}{h\to 0}}\frac{f(z_{0}+h)-f(z_{0})}{h}=f^{% \prime}(z_{0})
  9. lim h 0 h 𝐑 f ( z 0 + h ) - f ( z 0 ) h = f x ( z 0 ) . \lim_{\underset{h\in\mathbf{R}}{h\to 0}}\frac{f(z_{0}+h)-f(z_{0})}{h}=\frac{% \partial f}{\partial x}(z_{0}).
  10. lim h 0 h 𝐑 f ( z 0 + i h ) - f ( z 0 ) i h = 1 i f y ( z 0 ) . \lim_{\underset{h\in\mathbf{R}}{h\to 0}}\frac{f(z_{0}+ih)-f(z_{0})}{ih}=\frac{% 1}{i}\frac{\partial f}{\partial y}(z_{0}).
  11. i f x ( z 0 ) = f y ( z 0 ) , i\frac{\partial f}{\partial x}(z_{0})=\frac{\partial f}{\partial y}(z_{0}),
  12. f ( z 0 + Δ z ) - f ( z 0 ) = f x Δ x + f y Δ y + η ( Δ z ) Δ z f(z_{0}+\Delta z)-f(z_{0})=f_{x}\Delta x+f_{y}\Delta y+\eta(\Delta z)\Delta z\,
  13. Δ z + Δ z ¯ = 2 Δ x \Delta z+\Delta\bar{z}=2\Delta x
  14. Δ z - Δ z ¯ = 2 i Δ y \Delta z-\Delta\bar{z}=2i\Delta y
  15. Δ f ( z 0 ) = f x - i f y 2 Δ z + f x + i f y 2 Δ z ¯ + η ( Δ z ) Δ z \Delta f(z_{0})=\frac{f_{x}-if_{y}}{2}\Delta z+\frac{f_{x}+if_{y}}{2}\Delta% \bar{z}+\eta(\Delta z)\Delta z\,
  16. z = 1 2 ( x - i y ) , z ¯ = 1 2 ( x + i y ) , \frac{\partial}{\partial z}=\frac{1}{2}\Bigl(\frac{\partial}{\partial x}-i% \frac{\partial}{\partial y}\Bigr),\;\;\;\frac{\partial}{\partial\bar{z}}=\frac% {1}{2}\Bigl(\frac{\partial}{\partial x}+i\frac{\partial}{\partial y}\Bigr),
  17. Δ z 0 , Δ z ¯ 0 \Delta z\rightarrow 0,\Delta\bar{z}\rightarrow 0
  18. d f d z | z = z 0 = f z | z = z 0 + f z ¯ | z = z 0 d z ¯ d z + η ( Δ z ) , ( Δ z 0 ) . \left.\frac{df}{dz}\right|_{z=z_{0}}=\left.\frac{\partial f}{\partial z}\right% |_{z=z_{0}}+\left.\frac{\partial f}{\partial\bar{z}}\right|_{z=z_{0}}\cdot% \frac{\bar{dz}}{dz}+\eta(\Delta z),\;\;\;\;(\Delta z\neq 0).
  19. d z ¯ / d z = 1 \bar{dz}/dz=1
  20. d z ¯ / d z = - 1 \bar{dz}/dz=-1
  21. d z ¯ / d z \bar{dz}/dz
  22. ( f / z ¯ ) = 0 (\partial f/\partial\bar{z})=0
  23. z = z 0 z=z_{0}
  24. z ¯ \bar{z}
  25. x + i y ¯ := x - i y \overline{x+iy}:=x-iy
  26. f z ¯ = 0 \dfrac{\partial f}{\partial\bar{z}}=0
  27. z ¯ \bar{z}
  28. u , v u,v
  29. u = u x 𝐢 + u y 𝐣 \nabla u=\frac{\partial u}{\partial x}\mathbf{i}+\frac{\partial u}{\partial y}% \mathbf{j}
  30. 2 u x 2 + 2 u y 2 = 0. \frac{\partial^{2}u}{\partial x^{2}}+\frac{\partial^{2}u}{\partial y^{2}}=0.
  31. u v = 0 \nabla u\cdot\nabla v=0
  32. v = c o n s t v=const
  33. u = c o n s t u=const
  34. u = c o n s t u=const
  35. v = c o n s t v=const
  36. u = 0 \nabla u=0
  37. u = c o n s t u=const
  38. f ¯ = [ u - v ] \bar{f}=\begin{bmatrix}u\\ -v\end{bmatrix}
  39. f ¯ \bar{f}
  40. ( - v ) x - u y = 0. \frac{\partial(-v)}{\partial x}-\frac{\partial u}{\partial y}=0.
  41. u x + ( - v ) y = 0. \frac{\partial u}{\partial x}+\frac{\partial(-v)}{\partial y}=0.
  42. v d x + u d y v\,dx+u\,dy
  43. J = [ 0 1 - 1 0 ] . J=\begin{bmatrix}0&1\\ -1&0\end{bmatrix}.
  44. J 2 = - I J^{2}=-I
  45. f ( x , y ) = [ u ( x , y ) v ( x , y ) ] . f(x,y)=\begin{bmatrix}u(x,y)\\ v(x,y)\end{bmatrix}.
  46. D f ( x , y ) = [ u x u y v x v y ] Df(x,y)=\begin{bmatrix}\frac{\partial u}{\partial x}&\frac{\partial u}{% \partial y}\\ \frac{\partial v}{\partial x}&\frac{\partial v}{\partial y}\end{bmatrix}
  47. u n = v s , v n = - u s \frac{\partial u}{\partial n}=\frac{\partial v}{\partial s},\quad\frac{% \partial v}{\partial n}=-\frac{\partial u}{\partial s}
  48. u r = 1 r v θ , v r = - 1 r u θ . {\partial u\over\partial r}={1\over r}{\partial v\over\partial\theta},\quad{% \partial v\over\partial r}=-{1\over r}{\partial u\over\partial\theta}.
  49. f r = 1 i r f θ . {\partial f\over\partial r}={1\over ir}{\partial f\over\partial\theta}.
  50. u x - v y = α ( x , y ) \frac{\partial u}{\partial x}-\frac{\partial v}{\partial y}=\alpha(x,y)
  51. u y + v x = β ( x , y ) \frac{\partial u}{\partial y}+\frac{\partial v}{\partial x}=\beta(x,y)
  52. f z ¯ = φ ( z , z ¯ ) \frac{\partial f}{\partial\bar{z}}=\varphi(z,\bar{z})
  53. f ( ζ , ζ ¯ ) = 1 2 π i D φ ( z , z ¯ ) d z d z ¯ z - ζ f(\zeta,\bar{\zeta})=\frac{1}{2\pi i}\iint_{D}\varphi(z,\bar{z})\frac{dz\wedge d% \bar{z}}{z-\zeta}
  54. f ( z ) = { exp ( - z - 4 ) if z 0 0 if z = 0 f(z)=\begin{cases}\exp(-z^{-4})&\mathrm{if\ }z\not=0\\ 0&\mathrm{if\ }z=0\end{cases}
  55. ¯ \bar{\partial}
  56. f z ¯ = 0 , {\partial f\over\partial\bar{z}}=0,
  57. f z ¯ = 1 2 ( f x + i f y ) . {\partial f\over\partial\bar{z}}={1\over 2}\left({\partial f\over\partial x}+i% {\partial f\over\partial y}\right).
  58. z = x + i y z=x+iy
  59. z x + I y z\equiv x+Iy
  60. I σ 1 σ 2 I\equiv\sigma_{1}\sigma_{2}
  61. σ 1 x + σ 2 y \nabla\equiv\sigma_{1}\partial_{x}+\sigma_{2}\partial_{y}
  62. f = u + I v f=u+Iv
  63. f = 0 \nabla f=0
  64. 0 = f = ( σ 1 x + σ 2 y ) ( u + σ 1 σ 2 v ) = σ 1 x u + σ 1 σ 1 σ 2 = σ 2 x v + σ 2 y u + σ 2 σ 1 σ 2 = - σ 1 y v = 0 0=\nabla f=(\sigma_{1}\partial_{x}+\sigma_{2}\partial_{y})(u+\sigma_{1}\sigma_% {2}v)=\sigma_{1}\partial_{x}u+\underbrace{\sigma_{1}\sigma_{1}\sigma_{2}}_{=% \sigma_{2}}\partial_{x}v+\sigma_{2}\partial_{y}u+\underbrace{\sigma_{2}\sigma_% {1}\sigma_{2}}_{=-\sigma_{1}}\partial_{y}v=0
  65. σ 1 \sigma_{1}
  66. σ 2 \sigma_{2}
  67. f = σ 1 ( x u - y v ) + σ 2 ( x v + y u ) = 0 \nabla f=\sigma_{1}(\partial_{x}u-\partial_{y}v)+\sigma_{2}(\partial_{x}v+% \partial_{y}u)=0
  68. { x u - y v = 0 x v + y u = 0 \begin{cases}\partial_{x}u-\partial_{y}v=0\\ \partial_{x}v+\partial_{y}u=0\end{cases}
  69. { u x = v y u y = - v x \begin{cases}\dfrac{\partial u}{\partial x}=\dfrac{\partial v}{\partial y}\\ \dfrac{\partial u}{\partial y}=-\dfrac{\partial v}{\partial x}\end{cases}
  70. f : Ω 𝐑 n f:\Omega\to\mathbf{R}^{n}
  71. D f T D f = ( det ( D f ) ) 2 / n I Df^{T}Df=(\det(Df))^{2/n}I
  72. D f T Df^{T}
  73. n = 2 n=2
  74. n > 2 n>2

Cauchy–Schwarz_inequality.html

  1. | x , y | 2 x , x y , y , |\langle x,y\rangle|^{2}\leq\langle x,x\rangle\cdot\langle y,y\rangle,
  2. , \langle\cdot,\cdot\rangle
  3. | x , y | x y . |\langle x,y\rangle|\leq\|x\|\cdot\|y\|.\,
  4. x 1 , , x n x_{1},\ldots,x_{n}\in\mathbb{C}
  5. y 1 , , y n y_{1},\ldots,y_{n}\in\mathbb{C}
  6. | x 1 y ¯ 1 + + x n y ¯ n | 2 ( | x 1 | 2 + + | x n | 2 ) ( | y 1 | 2 + + | y n | 2 ) . |x_{1}\bar{y}_{1}+\cdots+x_{n}\bar{y}_{n}|^{2}\leq(|x_{1}|^{2}+\cdots+|x_{n}|^% {2})(|y_{1}|^{2}+\cdots+|y_{n}|^{2}).
  7. | i = 1 n x i y ¯ i | 2 j = 1 n | x j | 2 k = 1 n | y k | 2 . \left|\sum_{i=1}^{n}x_{i}\bar{y}_{i}\right|^{2}\leq\sum_{j=1}^{n}|x_{j}|^{2}% \sum_{k=1}^{n}|y_{k}|^{2}.
  8. | u , v | u v , \big|\langle u,v\rangle\big|\leq\left\|u\right\|\left\|v\right\|,
  9. u , v 0 \langle u,v\rangle\neq 0
  10. u \left\|u\right\|
  11. v \left\|v\right\|
  12. z = u - u , v v , v v . z=u-\frac{\langle u,v\rangle}{\langle v,v\rangle}v.
  13. z , v = u - u , v v , v v , v = u , v - u , v v , v v , v = 0 , \langle z,v\rangle=\left\langle u-\frac{\langle u,v\rangle}{\langle v,v\rangle% }v,v\right\rangle=\langle u,v\rangle-\frac{\langle u,v\rangle}{\langle v,v% \rangle}\langle v,v\rangle=0,
  14. u = u , v v , v v + z , u=\frac{\langle u,v\rangle}{\langle v,v\rangle}v+z,
  15. u 2 = | u , v v , v | 2 v 2 + z 2 = | u , v | 2 v 2 + z 2 | u , v | 2 v 2 , \left\|u\right\|^{2}=\left|\frac{\langle u,v\rangle}{\langle v,v\rangle}\right% |^{2}\left\|v\right\|^{2}+\left\|z\right\|^{2}=\frac{|\langle u,v\rangle|^{2}}% {\left\|v\right\|^{2}}+\left\|z\right\|^{2}\geq\frac{|\langle u,v\rangle|^{2}}% {\left\|v\right\|^{2}},
  16. v 2 \left\|v\right\|^{2}
  17. z 2 = 0 \left\|z\right\|^{2}=0
  18. z = 0 z=0
  19. u , v = 0 \langle u,v\rangle=0
  20. u 0 u\neq 0
  21. v 0 v\neq 0
  22. λ = | u , v | u , v \lambda=\frac{|\langle u,v\rangle|}{\langle u,v\rangle}
  23. | λ | = 1 |\lambda|=1
  24. 0 λ u u - v v 2 = | λ | 2 u 2 u 2 - 2 Re ( λ u u , v v ) + v 2 v 2 = 2 - 2 λ u , v u v . 0\leq\left\|\frac{\lambda u}{\|u\|}-\frac{v}{\|v\|}\right\|^{2}=|\lambda|^{2}% \frac{\|u\|^{2}}{\|u\|^{2}}-2\,\text{Re}\left(\left\langle\frac{\lambda u}{\|u% \|},\frac{v}{\|v\|}\right\rangle\right)+\frac{\|v\|^{2}}{\|v\|^{2}}=2-2\frac{% \lambda\langle u,v\rangle}{\|u\|\|v\|}.
  25. | u , v | = λ u , v u v . |\langle u,v\rangle|=\lambda\langle u,v\rangle\leq\|u\|\|v\|.
  26. n \mathbb{R}^{n}
  27. ( i = 1 n x i y i ) 2 ( i = 1 n x i 2 ) ( i = 1 n y i 2 ) . \left(\sum_{i=1}^{n}x_{i}y_{i}\right)^{2}\leq\left(\sum_{i=1}^{n}x_{i}^{2}% \right)\left(\sum_{i=1}^{n}y_{i}^{2}\right).
  28. ( x 1 z + y 1 ) 2 + + ( x n z + y n ) 2 = ( ( x i 2 ) ) z 2 + 2 ( ( x i y i ) ) z + ( y i 2 ) (x_{1}z+y_{1})^{2}+\cdots+(x_{n}z+y_{n})^{2}=\left(\sum(x_{i}^{2})\right)\cdot z% ^{2}+2\cdot\left(\sum(x_{i}\cdot y_{i})\right)\cdot z+\sum(y_{i}^{2})
  29. ( ( x i y i ) ) 2 - x i 2 y i 2 0 , \left(\sum(x_{i}\cdot y_{i})\right)^{2}-\sum{x_{i}^{2}}\cdot\sum{y_{i}^{2}}% \leq 0,
  30. n \mathbb{R}^{n}
  31. i = 1 n j = 1 n ( x i y j - x j y i ) 2 = i = 1 n x i 2 j = 1 n y j 2 + j = 1 n x j 2 i = 1 n y i 2 - 2 i = 1 n x i y i j = 1 n x j y j , \sum_{i=1}^{n}\sum_{j=1}^{n}\left(x_{i}y_{j}-x_{j}y_{i}\right)^{2}=\sum_{i=1}^% {n}x_{i}^{2}\sum_{j=1}^{n}y_{j}^{2}+\sum_{j=1}^{n}x_{j}^{2}\sum_{i=1}^{n}y_{i}% ^{2}-2\sum_{i=1}^{n}x_{i}y_{i}\sum_{j=1}^{n}x_{j}y_{j},
  32. 1 2 i = 1 n j = 1 n ( x i y j - x j y i ) 2 = i = 1 n x i 2 i = 1 n y i 2 - ( i = 1 n x i y i ) 2 . \frac{1}{2}\sum_{i=1}^{n}\sum_{j=1}^{n}\left(x_{i}y_{j}-x_{j}y_{i}\right)^{2}=% \sum_{i=1}^{n}x_{i}^{2}\sum_{i=1}^{n}y_{i}^{2}-\left(\sum_{i=1}^{n}x_{i}y_{i}% \right)^{2}.
  33. i = 1 n x i 2 i = 1 n y i 2 - ( i = 1 n x i y i ) 2 0. \sum_{i=1}^{n}x_{i}^{2}\sum_{i=1}^{n}y_{i}^{2}-\left(\sum_{i=1}^{n}x_{i}y_{i}% \right)^{2}\geq 0.
  34. x 1 , x 2 , y 1 x_{1},~{}x_{2},~{}y_{1}
  35. y 2 y_{2}~{}
  36. | x y | = x y | cos θ | x y . |x\cdot y|=\|x\|\|y\||\cos\theta|\leq\|x\|\|y\|.
  37. x , x y , y = | x , y | 2 + | x × y | 2 \langle x,x\rangle\cdot\langle y,y\rangle=|\langle x,y\rangle|^{2}+|x\times y|% ^{2}
  38. | n f ( x ) g ( x ) ¯ d x | 2 n | f ( x ) | 2 d x n | g ( x ) | 2 d x . \left|\int_{\mathbb{R}^{n}}f(x)\overline{g(x)}\,dx\right|^{2}\leq\int_{\mathbb% {R}^{n}}\left|f(x)\right|^{2}\,dx\cdot\int_{\mathbb{R}^{n}}\left|g(x)\right|^{% 2}\,dx.
  39. x + y 2 \displaystyle\|x+y\|^{2}
  40. cos θ x y = x , y x y . \cos\theta_{xy}=\frac{\langle x,y\rangle}{\|x\|\|y\|}.
  41. Var ( Y ) Cov ( Y , X ) Cov ( Y , X ) Var ( X ) . \,\text{Var}\left(Y\right)\geq\frac{\,\text{Cov}\left(Y,X\right)\,\text{Cov}% \left(Y,X\right)}{\,\text{Var}\left(X\right)}.
  42. X , Y E ( X Y ) , \langle X,Y\rangle\triangleq\operatorname{E}(XY),
  43. | E ( X Y ) | 2 E ( X 2 ) E ( Y 2 ) . |\operatorname{E}(XY)|^{2}\leq\operatorname{E}(X^{2})\operatorname{E}(Y^{2}).
  44. | Cov ( X , Y ) | 2 = | E ( ( X - μ ) ( Y - ν ) ) | 2 = | X - μ , Y - ν | 2 X - μ , X - μ Y - ν , Y - ν = E ( ( X - μ ) 2 ) E ( ( Y - ν ) 2 ) = Var ( X ) Var ( Y ) , \begin{aligned}\displaystyle|\operatorname{Cov}(X,Y)|^{2}&\displaystyle=|% \operatorname{E}((X-\mu)(Y-\nu))|^{2}\\ &\displaystyle=|\langle X-\mu,Y-\nu\rangle|^{2}\\ &\displaystyle\leq\langle X-\mu,X-\mu\rangle\langle Y-\nu,Y-\nu\rangle\\ &\displaystyle=\operatorname{E}((X-\mu)^{2})\operatorname{E}((Y-\nu)^{2})\\ &\displaystyle=\operatorname{Var}(X)\operatorname{Var}(Y),\end{aligned}
  45. ϕ ( g ) = g , 1 . \phi(g)=\langle g,1\rangle.
  46. | ϕ ( g * f ) | 2 ϕ ( f * f ) ϕ ( g * g ) , |\phi(g^{*}f)|^{2}\leq\phi(f^{*}f)\phi(g^{*}g),
  47. M = [ f * g * ] [ f g ] = [ f * f f * g g * f g * g ] . M=\begin{bmatrix}f^{*}\\ g^{*}\end{bmatrix}\begin{bmatrix}f&g\end{bmatrix}=\begin{bmatrix}f^{*}f&f^{*}g% \\ g^{*}f&g^{*}g\end{bmatrix}.
  48. M = ( I 2 ϕ ) ( M ) = [ ϕ ( f * f ) ϕ ( f * g ) ϕ ( g * f ) ϕ ( g * g ) ] M^{\prime}=(I_{2}\otimes\phi)(M)=\begin{bmatrix}\phi(f^{*}f)&\phi(f^{*}g)\\ \phi(g^{*}f)&\phi(g^{*}g)\end{bmatrix}
  49. ϕ ( f * f ) ϕ ( g * g ) - | ϕ ( g * f ) | 2 0 i.e. ϕ ( f * f ) ( g * g ) | ϕ ( g * f ) | 2 . \phi\left(f^{*}f\right)\phi\left(g^{*}g\right)-\left|\phi\left(g^{*}f\right)% \right|^{2}\geq 0\quad\,\text{i.e.}\quad\phi\left(f^{*}f\right)\left(g^{*}g% \right)\geq\left|\phi\left(g^{*}f\right)\right|^{2}.
  50. Φ \Phi
  51. a a
  52. Φ ( a * a ) Φ ( a * ) Φ ( a ) \Phi(a^{*}a)\geq\Phi(a^{*})\Phi(a)
  53. Φ ( a * a ) Φ ( a ) Φ ( a * ) \Phi(a^{*}a)\geq\Phi(a)\Phi(a^{*})
  54. φ ( a * a ) 1 φ ( a ) * φ ( a ) = | φ ( a ) | 2 \varphi(a^{*}a)\cdot 1\geq\varphi(a)^{*}\varphi(a)=|\varphi(a)|^{2}
  55. φ \varphi
  56. a a
  57. a = a * a=a^{*}
  58. Φ ( a ) * Φ ( a ) Φ ( 1 ) Φ ( a * a ) \Phi(a)^{*}\Phi(a)\leq\|\Phi(1)\|\Phi(a^{*}a)
  59. Φ ( a * b ) 2 Φ ( a * a ) Φ ( b * b ) . \|\Phi(a^{*}b)\|^{2}\leq\|\Phi(a^{*}a)\|\cdot\|\Phi(b^{*}b)\|.
  60. M = [ a * 0 b * 0 ] [ a b 0 0 ] = [ a * a a * b b * a b * b ] . M=\begin{bmatrix}a^{*}&0\\ b^{*}&0\end{bmatrix}\begin{bmatrix}a&b\\ 0&0\end{bmatrix}=\begin{bmatrix}a^{*}a&a^{*}b\\ b^{*}a&b^{*}b\end{bmatrix}.
  61. ( I 2 Φ ) M = [ Φ ( a * a ) Φ ( a * b ) Φ ( b * a ) Φ ( b * b ) ] (I_{2}\otimes\Phi)M=\begin{bmatrix}\Phi(a^{*}a)&\Phi(a^{*}b)\\ \Phi(b^{*}a)&\Phi(b^{*}b)\end{bmatrix}
  62. [ a b 0 0 ] \begin{bmatrix}a&b\\ 0&0\end{bmatrix}
  63. [ 1 a 0 0 ] . \begin{bmatrix}1&a\\ 0&0\end{bmatrix}.

Causality.html

  1. α τ ί α αἰτία
  2. P ( c a n c e r | s m o k i n g ) P(cancer|smoking)
  3. P ( c a n c e r | d o ( s m o k i n g ) ) P(cancer|do(smoking))
  4. X X
  5. Y Y
  6. X X
  7. d d
  8. X X
  9. Y Y
  10. X X
  11. X Y Z X\rightarrow Y\rightarrow Z
  12. X Y Z X\leftarrow Y\rightarrow Z
  13. X Y Z X\rightarrow Y\leftarrow Z
  14. X X
  15. Z Z
  16. Y Y
  17. X X
  18. Z Z
  19. X X
  20. Z Z
  21. y i = a 0 + a 1 x 1 , i + a 2 x 2 , i + + a k x k , i + e i y_{i}=a_{0}+a_{1}x_{1,i}+a_{2}x_{2,i}+...+a_{k}x_{k,i}+e_{i}
  22. y i y_{i}
  23. x j , i x_{j,i}
  24. e i e_{i}
  25. x j x_{j}
  26. a j a_{j}
  27. a j = 0 a_{j}=0
  28. a j 0 a_{j}\neq 0
  29. x j x_{j}
  30. a j = 0 a_{j}=0
  31. x j x_{j}
  32. a j 0 a_{j}\neq 0
  33. x j x_{j}
  34. x j x_{j}
  35. x j x_{j}
  36. x j x_{j}
  37. x j x_{j}
  38. x j x_{j}

Celestial_coordinate_system.html

  1. x , y x,y
  2. x x
  3. a a
  4. A A
  5. δ δ
  6. α α
  7. h h
  8. β β
  9. λ λ
  10. b b
  11. l l
  12. S G B SGB
  13. S G L SGL
  14. A A
  15. a a
  16. α \alpha
  17. δ \delta
  18. h h
  19. λ \lambda
  20. β \beta
  21. l l
  22. b b
  23. λ 0 \lambda_{0}
  24. ϕ 0 \phi_{0}
  25. ε \varepsilon
  26. θ L \theta_{\rm L}
  27. θ G \theta_{\rm G}
  28. h = θ L - α h=\theta_{L}-\alpha
  29. h = θ G - λ o - α h=\theta_{G}-\lambda_{o}-\alpha
  30. α = θ L - h \alpha=\theta_{L}-h
  31. α = θ G - λ o - h \alpha=\theta_{G}-\lambda_{o}-h
  32. tan λ = sin α cos ε + tan δ sin ε cos α ; { cos β sin λ = cos δ sin α cos ε + sin δ sin ε ; cos β cos λ = cos δ cos α . \tan\lambda={\sin\alpha\cos\varepsilon+\tan\delta\sin\varepsilon\over\cos% \alpha};\qquad\qquad\begin{cases}\cos\beta\sin\lambda=\cos\delta\sin\alpha\cos% \varepsilon+\sin\delta\sin\varepsilon;\\ \cos\beta\cos\lambda=\cos\delta\cos\alpha.\end{cases}
  33. sin β = sin δ cos ε - cos δ sin ε sin α \sin\beta=\sin\delta\cos\varepsilon-\cos\delta\sin\varepsilon\sin\alpha
  34. [ cos β cos λ cos β sin λ sin β ] = [ 1 0 0 0 cos ε sin ε 0 - sin ε cos ε ] [ cos δ cos α cos δ sin α sin δ ] \begin{bmatrix}\cos\beta\cos\lambda\\ \cos\beta\sin\lambda\\ \sin\beta\end{bmatrix}=\begin{bmatrix}1&0&0\\ 0&\cos\varepsilon&\sin\varepsilon\\ 0&-\sin\varepsilon&\cos\varepsilon\end{bmatrix}\begin{bmatrix}\cos\delta\cos% \alpha\\ \cos\delta\sin\alpha\\ \sin\delta\end{bmatrix}
  35. tan α = sin λ cos ε - tan β sin ε cos λ ; { cos δ sin α = cos β sin λ cos ε - sin β sin ε ; cos δ cos α = cos β cos λ . \tan\alpha={\sin\lambda\cos\varepsilon-\tan\beta\sin\varepsilon\over\cos% \lambda};\qquad\qquad\begin{cases}\cos\delta\sin\alpha=\cos\beta\sin\lambda% \cos\varepsilon-\sin\beta\sin\varepsilon;\\ \cos\delta\cos\alpha=\cos\beta\cos\lambda.\end{cases}
  36. sin δ = sin β cos ε + cos β sin ε sin λ \sin\delta=\sin\beta\cos\varepsilon+\cos\beta\sin\varepsilon\sin\lambda
  37. [ cos δ cos α cos δ sin α sin δ ] = [ 1 0 0 0 cos ε - sin ε 0 sin ε cos ε ] [ cos β cos λ cos β sin λ sin β ] \begin{bmatrix}\cos\delta\cos\alpha\\ \cos\delta\sin\alpha\\ \sin\delta\end{bmatrix}=\begin{bmatrix}1&0&0\\ 0&\cos\varepsilon&-\sin\varepsilon\\ 0&\sin\varepsilon&\cos\varepsilon\end{bmatrix}\begin{bmatrix}\cos\beta\cos% \lambda\\ \cos\beta\sin\lambda\\ \sin\beta\end{bmatrix}
  38. a a
  39. tan A = sin h cos h sin ϕ o - tan δ cos ϕ o { cos a sin A = cos δ sin h cos a cos A = cos δ cos h sin ϕ o - sin δ cos ϕ o \tan A={\sin h\over\cos h\sin\phi_{o}-\tan\delta\cos\phi_{o}}\qquad\qquad% \begin{cases}\cos a\sin A=\cos\delta\sin h\\ \cos a\cos A=\cos\delta\cos h\sin\phi_{o}-\sin\delta\cos\phi_{o}\end{cases}
  40. sin a = sin ϕ o sin δ + cos ϕ o cos δ cos h \sin a=\sin\phi_{o}\sin\delta+\cos\phi_{o}\cos\delta\cos h
  41. [ cos a cos A cos a sin A sin a ] = [ sin ϕ o 0 - cos ϕ o 0 1 0 cos ϕ o 0 sin ϕ o ] [ cos δ cos h cos δ sin h sin δ ] \begin{bmatrix}\cos a\cos A\\ \cos a\sin A\\ \sin a\end{bmatrix}=\begin{bmatrix}\sin\phi_{o}&0&-\cos\phi_{o}\\ 0&1&0\\ \cos\phi_{o}&0&\sin\phi_{o}\end{bmatrix}\begin{bmatrix}\cos\delta\cos h\\ \cos\delta\sin h\\ \sin\delta\end{bmatrix}
  42. tan h = sin A cos A sin ϕ o + tan a cos ϕ o { cos δ sin h = cos a sin A cos δ cos h = sin a cos ϕ o + cos a cos A sin ϕ o \tan h={\sin A\over\cos A\sin\phi_{o}+\tan a\cos\phi_{o}}\qquad\qquad\begin{% cases}\cos\delta\sin h=\cos a\sin A\\ \cos\delta\cos h=\sin a\cos\phi_{o}+\cos a\cos A\sin\phi_{o}\end{cases}
  43. sin δ = sin ϕ o sin a - cos ϕ o cos a cos A \sin\delta=\sin\phi_{o}\sin a-\cos\phi_{o}\cos a\cos A
  44. [ cos δ cos h cos δ sin h sin δ ] = [ sin ϕ o 0 cos ϕ o 0 1 0 - cos ϕ o 0 sin ϕ o ] [ cos a cos A cos a sin A sin a ] \begin{bmatrix}\cos\delta\cos h\\ \cos\delta\sin h\\ \sin\delta\end{bmatrix}=\begin{bmatrix}\sin\phi_{o}&0&\cos\phi_{o}\\ 0&1&0\\ -\cos\phi_{o}&0&\sin\phi_{o}\end{bmatrix}\begin{bmatrix}\cos a\cos A\\ \cos a\sin A\\ \sin a\end{bmatrix}
  45. l = 303 - arctan ( sin ( 192 .25 - α ) cos ( 192 .25 - α ) sin 27 .4 - tan δ cos 27 .4 ) l=303^{\circ}-\arctan\left({\sin(192^{\circ}.25-\alpha)\over\cos(192^{\circ}.2% 5-\alpha)\sin 27^{\circ}.4-\tan\delta\cos 27^{\circ}.4}\right)
  46. sin b = sin δ sin 27 .4 + cos δ cos 27 .4 cos ( 192 .25 - α ) \sin b=\sin\delta\sin 27^{\circ}.4+\cos\delta\cos 27^{\circ}.4\cos(192^{\circ}% .25-\alpha)
  47. α = arctan ( sin ( l - 123 ) cos ( l - 123 ) sin 27 .4 - tan b cos 27 .4 ) + 12 .25 \alpha=\arctan\left({\sin(l-123^{\circ})\over\cos(l-123^{\circ})\sin 27^{\circ% }.4-\tan b\cos 27^{\circ}.4}\right)+12^{\circ}.25
  48. sin δ = sin b sin 27 .4 + cos b cos 27 .4 cos ( l - 123 ) \sin\delta=\sin b\sin 27^{\circ}.4+\cos b\cos 27^{\circ}.4\cos(l-123^{\circ})
  49. 2 π
  50. 2 π
  51. A A
  52. A A
  53. h h
  54. a a
  55. λ < s u b > o λ<sub>o
  56. A A
  57. A A
  58. A A
  59. δ δ
  60. φ φ
  61. a a
  62. φ φ
  63. a a
  64. A A
  65. δ δ
  66. a a
  67. φ φ
  68. a a
  69. A A
  70. φ φ

Center_(group_theory).html

  1. Z ( G ) = { z G g G , z g = g z } . Z(G)=\{z\in G\mid\forall g\in G,zg=gz\}.
  2. ϕ g ( h ) = g h g - 1 . \phi_{g}(h)=ghg^{-1}.\,
  3. G / Z ( G ) Inn ( G ) . G/Z(G)\cong\rm{Inn}(G).
  4. Out ( G ) \operatorname{Out}(G)
  5. 1 Z ( G ) G Aut ( G ) Out ( G ) 1. 1\to Z(G)\to G\to\operatorname{Aut}(G)\to\operatorname{Out}(G)\to 1.
  6. ( 1 0 z 0 1 0 0 0 1 ) \begin{pmatrix}1&0&z\\ 0&1&0\\ 0&0&1\end{pmatrix}
  7. Q 8 = { 1 , - 1 , i , - i , j , - j , k , - k } Q_{8}=\{1,-1,i,-i,j,-j,k,-k\}
  8. { 1 , - 1 } \{1,-1\}
  9. GL n ( F ) \mathrm{GL}_{n}(F)
  10. { s I n | s F { 0 } } \{sI_{n}|s\in F\setminus\{0\}\}
  11. O ( n , F ) O(n,F)
  12. { I n , - I n } \{I_{n},-I_{n}\}
  13. G / Z ( G ) G/Z(G)
  14. G / Z ( G ) G/Z(G)
  15. G / Z ( G ) G/Z(G)
  16. Q 8 Q_{8}
  17. G 0 = G G 1 = G 0 / Z ( G 0 ) G 2 = G 1 / Z ( G 1 ) G_{0}=G\to G_{1}=G_{0}/Z(G_{0})\to G_{2}=G_{1}/Z(G_{1})\to\cdots
  18. G G i G\to G_{i}
  19. Z i ( G ) . Z^{i}(G).
  20. ( i + 1 ) (i+1)
  21. 1 Z ( G ) Z 2 ( G ) 1\leq Z(G)\leq Z^{2}(G)\leq\cdots
  22. Z i ( G ) = Z i + 1 ( G ) Z^{i}(G)=Z^{i+1}(G)
  23. G i G_{i}
  24. Z 0 ( G ) = Z 1 ( G ) Z^{0}(G)=Z^{1}(G)
  25. Z 1 ( G ) = Z 2 ( G ) Z^{1}(G)=Z^{2}(G)

Centimetre–gram–second_system_of_units.html

  1. v = d x d t v=\frac{dx}{dt}
  2. F = m d 2 x d t 2 F=m\frac{d^{2}x}{dt^{2}}
  3. E = F d x E=\int\vec{F}\cdot\vec{dx}
  4. p = F L 2 p=\frac{F}{L^{2}}
  5. η = τ / d v d x \eta=\tau/\frac{dv}{dx}
  6. q = I t q=I\cdot t
  7. F = k C q q d 2 F=k_{\rm C}\frac{q\cdot q^{\prime}}{d^{2}}
  8. q q
  9. q q^{\prime}
  10. k C k_{\rm C}
  11. d F d L = 2 k A I I d \frac{dF}{dL}=2k_{\rm A}\frac{I\,I^{\prime}}{d}
  12. I = q / t I=q/t\,
  13. I = q / t I^{\prime}=q^{\prime}/t
  14. k A k_{\rm A}
  15. k C k_{\rm C}
  16. k A k_{\rm A}
  17. k C / k A = c 2 k_{\rm C}/k_{\rm A}=c^{2}
  18. k C = 1 k_{\rm C}=1
  19. 2 / c 2 2/c^{2}
  20. k A = 1 k_{\rm A}=1
  21. k A = 1 / 2 k_{\rm A}=1/2
  22. 𝐅 = α L q 𝐯 × 𝐁 . \mathbf{F}=\alpha_{\rm L}q\;\mathbf{v}\times\mathbf{B}\;.
  23. d 𝐁 = α B I d 𝐥 × 𝐫 ^ r 2 , d\mathbf{B}=\alpha_{\rm B}\frac{Id\mathbf{l}\times\mathbf{\hat{r}}}{r^{2}}\;,
  24. 𝐫 ^ \mathbf{\hat{r}}
  25. k A = α L α B k_{\rm A}=\alpha_{\rm L}\cdot\alpha_{\rm B}\;
  26. k A = 1 k_{\rm A}=1
  27. α L = α B = 1 \alpha_{\rm L}=\alpha_{\rm B}=1\;
  28. 𝐃 = ϵ 0 𝐄 + λ 𝐏 \mathbf{D}=\epsilon_{0}\mathbf{E}+\lambda\mathbf{P}
  29. 𝐇 = 𝐁 / μ 0 - λ 𝐌 \mathbf{H}=\mathbf{B}/\mu_{0}-\lambda^{\prime}\mathbf{M}
  30. 4 π k C ϵ 0 4\pi k_{\rm C}\epsilon_{0}
  31. k C ϵ 0 = 1 k_{\rm C}\epsilon_{0}=1
  32. k C k_{\rm C}
  33. α B \alpha_{\rm B}
  34. ϵ 0 \epsilon_{0}
  35. μ 0 \mu_{0}
  36. k A = k C c 2 k_{\rm A}=\frac{k_{\rm C}}{c^{2}}
  37. α L = k C α B c 2 \alpha_{\rm L}=\frac{k_{\rm C}}{\alpha_{\rm B}c^{2}}
  38. λ = 4 π k C ϵ 0 \lambda=4\pi k_{\rm C}\cdot\epsilon_{0}
  39. λ \lambda^{\prime}
  40. 1 4 π \frac{1}{4\pi}
  41. 1 4 π c \frac{1}{4\pi c}
  42. 1 4 π c 2 \frac{1}{4\pi c^{2}}
  43. c 2 b \frac{c^{2}}{b}
  44. 1 b \frac{1}{b}
  45. b 4 π c 2 \frac{b}{4\pi c^{2}}
  46. 4 π b \frac{4\pi}{b}
  47. 1 b \frac{1}{b}
  48. b = 10 7 A 2 / N = 10 7 m / H = 4 π / μ 0 = 4 π ϵ 0 c 2 b=10^{7}\,\mathrm{A}^{2}/\mathrm{N}=10^{7}\,\mathrm{m/H}=4\pi/\mu_{0}=4\pi% \epsilon_{0}c^{2}\;
  49. k C = k 1 = k E k_{\rm C}=k_{1}=k_{\rm E}
  50. α B = α k 2 = k B \alpha_{\rm B}=\alpha\cdot k_{2}=k_{\rm B}
  51. k A = k 2 = k E / c 2 k_{\rm A}=k_{2}=k_{\rm E}/c^{2}
  52. α L = k 3 = k F \alpha_{\rm L}=k_{3}=k_{\rm F}
  53. E = 4 π k C ρ B = 0 × E = - α L B t × B = 4 π α B J + α B k C E t \begin{array}[]{ccl}\vec{\nabla}\cdot\vec{E}&=&4\pi k_{\rm C}\rho\\ \vec{\nabla}\cdot\vec{B}&=&0\\ \vec{\nabla}\times\vec{E}&=&\displaystyle{-\alpha_{\rm L}\frac{\partial\vec{B}% }{\partial t}}\\ \vec{\nabla}\times\vec{B}&=&\displaystyle{4\pi\alpha_{\rm B}\vec{J}+\frac{% \alpha_{\rm B}}{k_{\rm C}}\frac{\partial\vec{E}}{\partial t}}\end{array}
  54. α L \alpha_{\rm L}
  55. c - 1 c^{-1}
  56. E \vec{E}
  57. B \vec{B}
  58. k C = 1 k_{\rm C}=1
  59. 1 Fr = 1 statcoulomb = 1 esu charge = 1 cm dyne = 1 g 1 / 2 cm 3 / 2 s - 1 \mathrm{1\,Fr=1\,statcoulomb=1\,esu\;charge=1\,cm\sqrt{dyne}=1\,g^{1/2}\cdot cm% ^{3/2}\cdot s^{-1}}
  60. 1 Fr / s = 1 statampere = 1 esu current = 1 ( cm / s ) dyne = 1 g 1 / 2 cm 3 / 2 s - 2 \mathrm{1\,Fr/s=1\,statampere=1\,esu\;current=1\,(cm/s)\sqrt{dyne}=1\,g^{1/2}% \cdot cm^{3/2}\cdot s^{-2}}
  61. k A = 1 k_{\rm A}=1
  62. 1 Bi = 1 abampere = 1 emu current = 1 dyne = 1 g 1 / 2 cm 1 / 2 s - 1 \mathrm{1\,Bi=1\,abampere=1\,emu\;current=1\,\sqrt{dyne}=1\,g^{1/2}\cdot cm^{1% /2}\cdot s^{-1}}
  63. 1 Bi s = 1 abcoulomb = 1 emu charge = 1 s dyne = 1 g 1 / 2 cm 1 / 2 \mathrm{1\,Bi\cdot s=1\,abcoulomb=1\,emu\,charge=1\,s\cdot\sqrt{dyne}=1\,g^{1/% 2}\cdot cm^{1/2}}
  64. k C / k A = c 2 k_{\rm C}/k_{\rm A}=c^{2}
  65. 1 statcoulomb 1 abcoulomb = 1 statampere 1 abampere = c - 1 \mathrm{\frac{1\,statcoulomb}{1\,abcoulomb}}=\mathrm{\frac{1\,statampere}{1\,% abampere}}=c^{-1}
  66. 1 statvolt 1 abvolt = 1 stattesla 1 gauss = c \mathrm{\frac{1\,statvolt}{1\,abvolt}}=\mathrm{\frac{1\,stattesla}{1\,gauss}}=c
  67. 1 statohm 1 abohm = 1 statvolt 1 abvolt × 1 abampere 1 statampere = c 2 \mathrm{\frac{1\,statohm}{1\,abohm}}=\mathrm{\frac{1\,statvolt}{1\,abvolt}}% \times\mathrm{\frac{1\,abampere}{1\,statampere}}=c^{2}
  68. k C = 1 4 π ϵ 0 = μ 0 ( c / 100 ) 2 4 π = 10 - 7 10 - 4 c 2 = 10 - 11 c 2 . k_{\rm C}=\frac{1}{4\pi\epsilon_{0}}=\frac{\mu_{0}(c/100)^{2}}{4\pi}=10^{-7}% \cdot 10^{-4}\cdot c^{2}=10^{-11}\cdot c^{2}.
  69. 1 1 r - 1 R \frac{1}{\frac{1}{r}-\frac{1}{R}}
  70. \hbar
  71. 4 π ϵ 0 4\pi\epsilon_{0}
  72. 1 1
  73. c c
  74. ϵ 0 \epsilon_{0}
  75. 4 π 4\pi
  76. \hbar

Central_limit_theorem.html

  1. S n := X 1 + + X n n S_{n}:=\frac{X_{1}+\cdots+X_{n}}{n}
  2. n \sqrt{n}
  3. n \sqrt{n}
  4. n \sqrt{n}
  5. n ( ( 1 n i = 1 n X i ) - μ ) 𝑑 N ( 0 , σ 2 ) . \sqrt{n}\bigg(\bigg(\frac{1}{n}\sum_{i=1}^{n}X_{i}\bigg)-\mu\bigg)\ % \xrightarrow{d}\ N(0,\;\sigma^{2}).
  6. n \sqrt{n}
  7. lim n Pr [ n ( S n - μ ) z ] = Φ ( z / σ ) , \lim_{n\to\infty}\Pr[\sqrt{n}(S_{n}-\mu)\leq z]=\Phi(z/\sigma),
  8. lim n sup z 𝐑 | Pr [ n ( S n - μ ) z ] - Φ ( z / σ ) | = 0 , \lim_{n\to\infty}\sup_{z\in{\mathbf{R}}}\bigl|\Pr[\sqrt{n}(S_{n}-\mu)\leq z]-% \Phi(z/\sigma)\bigr|=0,
  9. σ i 2 {σ}_{i}^{2}
  10. s n 2 = i = 1 n σ i 2 s_{n}^{2}=\sum_{i=1}^{n}\sigma_{i}^{2}
  11. lim n 1 s n 2 + δ i = 1 n E [ | X i - μ i | 2 + δ ] = 0 \lim_{n\to\infty}\frac{1}{s_{n}^{2+\delta}}\sum_{i=1}^{n}\operatorname{E}\big[% \,|X_{i}-\mu_{i}|^{2+\delta}\,\big]=0
  12. 1 s n i = 1 n ( X i - μ i ) 𝑑 𝒩 ( 0 , 1 ) . \frac{1}{s_{n}}\sum_{i=1}^{n}(X_{i}-\mu_{i})\ \xrightarrow{d}\ \mathcal{N}(0,% \;1).
  13. lim n 1 s n 2 i = 1 n E [ ( X i - μ i ) 2 𝟏 { | X i - μ i | > ε s n } ] = 0 \lim_{n\to\infty}\frac{1}{s_{n}^{2}}\sum_{i=1}^{n}\operatorname{E}\big[(X_{i}-% \mu_{i})^{2}\cdot\mathbf{1}_{\{|X_{i}-\mu_{i}|>\varepsilon s_{n}\}}\big]=0
  14. 1 s n i = 1 n ( X i - μ i ) \frac{1}{s_{n}}\sum_{i=1}^{n}\left(X_{i}-\mu_{i}\right)
  15. 𝐗 𝐢 = [ X i ( 1 ) X i ( k ) ] \mathbf{X_{i}}=\begin{bmatrix}X_{i(1)}\\ \vdots\\ X_{i(k)}\end{bmatrix}
  16. [ X 1 ( 1 ) X 1 ( k ) ] + [ X 2 ( 1 ) X 2 ( k ) ] + + [ X n ( 1 ) X n ( k ) ] = [ i = 1 n [ X i ( 1 ) ] i = 1 n [ X i ( k ) ] ] = i = 1 n 𝐗 𝐢 \begin{bmatrix}X_{1(1)}\\ \vdots\\ X_{1(k)}\end{bmatrix}+\begin{bmatrix}X_{2(1)}\\ \vdots\\ X_{2(k)}\end{bmatrix}+\cdots+\begin{bmatrix}X_{n(1)}\\ \vdots\\ X_{n(k)}\end{bmatrix}=\begin{bmatrix}\sum_{i=1}^{n}\left[X_{i(1)}\right]\\ \vdots\\ \sum_{i=1}^{n}\left[X_{i(k)}\right]\end{bmatrix}=\sum_{i=1}^{n}\mathbf{X_{i}}
  17. 1 n i = 1 n 𝐗 𝐢 = 1 n [ i = 1 n X i ( 1 ) i = 1 n X i ( k ) ] = [ X ¯ i ( 1 ) X ¯ i ( k ) ] = 𝐗 ¯ 𝐧 \frac{1}{n}\sum_{i=1}^{n}\mathbf{X_{i}}=\frac{1}{n}\begin{bmatrix}\sum_{i=1}^{% n}X_{i(1)}\\ \vdots\\ \sum_{i=1}^{n}X_{i(k)}\end{bmatrix}=\begin{bmatrix}\bar{X}_{i(1)}\\ \vdots\\ \bar{X}_{i(k)}\end{bmatrix}=\mathbf{\bar{X}_{n}}
  18. 1 n i = 1 n [ 𝐗 𝐢 - E ( X i ) ] = 1 n i = 1 n ( 𝐗 𝐢 - μ ) = n ( 𝐗 ¯ n - μ ) \frac{1}{\sqrt{n}}\sum_{i=1}^{n}\left[\mathbf{X_{i}}-E\left(X_{i}\right)\right% ]=\frac{1}{\sqrt{n}}\sum_{i=1}^{n}(\mathbf{X_{i}}-\mu)=\sqrt{n}\left(\mathbf{% \overline{X}}_{n}-\mu\right)
  19. n ( 𝐗 ¯ n - μ ) D 𝒩 k ( 0 , Σ ) \sqrt{n}\left(\mathbf{\overline{X}}_{n}-\mu\right)\ \stackrel{D}{\rightarrow}% \ \mathcal{N}_{k}(0,\Sigma)
  20. Σ = [ Var ( X 1 ( 1 ) ) Cov ( X 1 ( 1 ) , X 1 ( 2 ) ) Cov ( X 1 ( 1 ) , X 1 ( 3 ) ) Cov ( X 1 ( 1 ) , X 1 ( k ) ) Cov ( X 1 ( 2 ) , X 1 ( 1 ) ) Var ( X 1 ( 2 ) ) Cov ( X 1 ( 2 ) , X 1 ( 3 ) ) Cov ( X 1 ( 2 ) , X 1 ( k ) ) Cov ( X 1 ( 3 ) , X 1 ( 1 ) ) Cov ( X 1 ( 3 ) , X 1 ( 2 ) ) Var ( X 1 ( 3 ) ) Cov ( X 1 ( 3 ) , X 1 ( k ) ) Cov ( X 1 ( k ) , X 1 ( 1 ) ) Cov ( X 1 ( k ) , X 1 ( 2 ) ) Cov ( X 1 ( k ) , X 1 ( 3 ) ) Var ( X 1 ( k ) ) ] . \Sigma=\begin{bmatrix}{\operatorname{Var}\left(X_{1(1)}\right)}&\operatorname{% Cov}\left(X_{1(1)},X_{1(2)}\right)&\operatorname{Cov}\left(X_{1(1)},X_{1(3)}% \right)&\cdots&\operatorname{Cov}\left(X_{1(1)},X_{1(k)}\right)\\ \operatorname{Cov}\left(X_{1(2)},X_{1(1)}\right)&\operatorname{Var}\left(X_{1(% 2)}\right)&\operatorname{Cov}\left(X_{1(2)},X_{1(3)}\right)&\cdots&% \operatorname{Cov}\left(X_{1(2)},X_{1(k)}\right)\\ \operatorname{Cov}\left(X_{1(3)},X_{1(1)}\right)&\operatorname{Cov}\left(X_{1(% 3)},X_{1(2)}\right)&\operatorname{Var}\left(X_{1(3)}\right)&\cdots&% \operatorname{Cov}\left(X_{1(3)},X_{1(k)}\right)\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ \operatorname{Cov}\left(X_{1(k)},X_{1(1)}\right)&\operatorname{Cov}\left(X_{1(% k)},X_{1(2)}\right)&\operatorname{Cov}\left(X_{1(k)},X_{1(3)}\right)&\cdots&% \operatorname{Var}\left(X_{1(k)}\right)\\ \end{bmatrix}.
  21. σ 2 = lim n E ( S n 2 ) n \sigma^{2}=\lim_{n}\frac{E(S_{n}^{2})}{n}
  22. S n / ( σ n ) S_{n}/(\sigma\sqrt{n})
  23. σ 2 = E ( X 1 2 ) + 2 k = 1 E ( X 1 X 1 + k ) , \sigma^{2}=E(X_{1}^{2})+2\sum_{k=1}^{\infty}E(X_{1}X_{1+k}),
  24. n α n δ 2 ( 2 + δ ) < . \sum_{n}\alpha_{n}^{\frac{\delta}{2(2+\delta)}}<\infty.
  25. 1 n k = 1 n E ( ( M k - M k - 1 ) 2 | M 1 , , M k - 1 ) 1 \frac{1}{n}\sum_{k=1}^{n}\mathrm{E}((M_{k}-M_{k-1})^{2}|M_{1},\dots,M_{k-1})\to 1
  26. 1 n k = 1 n E ( ( M k - M k - 1 ) 2 ; | M k - M k - 1 | > ε n ) 0 \frac{1}{n}\sum_{k=1}^{n}\mathrm{E}\Big((M_{k}-M_{k-1})^{2};|M_{k}-M_{k-1}|>% \varepsilon\sqrt{n}\Big)\to 0
  27. M n / n M_{n}/\sqrt{n}
  28. φ Y ( t ) = 1 - t 2 2 + o ( t 2 ) , t 0 \varphi_{Y}(t)=1-{t^{2}\over 2}+o(t^{2}),\quad t\rightarrow 0
  29. Z n = n X ¯ n - n μ σ n = i = 1 n Y i n Z_{n}=\frac{n\overline{X}_{n}-n\mu}{\sigma\sqrt{n}}=\sum_{i=1}^{n}{Y_{i}\over% \sqrt{n}}
  30. φ Z n \displaystyle\varphi_{Z_{n}}
  31. [ φ Y ( t n ) ] n = [ 1 - t 2 2 n + o ( t 2 n ) ] n e - t 2 / 2 , n . \left[\varphi_{Y}\left({t\over\sqrt{n}}\right)\right]^{n}=\left[1-{t^{2}\over 2% n}+o\left({t^{2}\over n}\right)\right]^{n}\,\rightarrow\,e^{-t^{2}/2},\quad n% \rightarrow\infty.
  32. f ( n ) = a 1 φ 1 ( n ) + a 2 φ 2 ( n ) + O ( φ 3 ( n ) ) ( n ) . f(n)=a_{1}\varphi_{1}(n)+a_{2}\varphi_{2}(n)+O(\varphi_{3}(n))\qquad(n% \rightarrow\infty).
  33. lim n f ( n ) φ 1 ( n ) = a 1 . \lim_{n\to\infty}\frac{f(n)}{\varphi_{1}(n)}=a_{1}.
  34. lim n f ( n ) - a 1 φ 1 ( n ) φ 2 ( n ) = a 2 . \lim_{n\to\infty}\frac{f(n)-a_{1}\varphi_{1}(n)}{\varphi_{2}(n)}=a_{2}.
  35. S n - n μ n ξ , \frac{S_{n}-n\mu}{\sqrt{n}}\rightarrow\xi,
  36. S n μ n + ξ n . S_{n}\approx\mu n+\xi\sqrt{n}.\,
  37. S n - a n b n Ξ , \frac{S_{n}-a_{n}}{b_{n}}\rightarrow\Xi,
  38. S n a n + Ξ b n . S_{n}\approx a_{n}+\Xi b_{n}.\,
  39. n log log n \sqrt{n\log\log n}
  40. X 1 + + X n n \frac{X_{1}+\cdots+X_{n}}{\sqrt{n}}
  41. | ( a X 1 + + X n n b ) - 1 2 π a b e - t 2 / 2 d t | C n \bigg|\mathbb{P}\Big(a\leq\frac{X_{1}+\cdots+X_{n}}{\sqrt{n}}\leq b\Big)-\frac% {1}{\sqrt{2\pi}}\int_{a}^{b}\mathrm{e}^{-t^{2}/2}\,\mathrm{d}t\bigg|\leq\frac{% C}{n}
  42. | ( a c 1 X 1 + + c n X n b ) - 1 2 π a b e - t 2 / 2 d t | C ( c 1 4 + + c n 4 ) . \bigg|\mathbb{P}(a\leq c_{1}X_{1}+\cdots+c_{n}X_{n}\leq b)-\frac{1}{\sqrt{2\pi% }}\int_{a}^{b}\mathrm{e}^{-t^{2}/2}\,\mathrm{d}t\bigg|\leq C(c_{1}^{4}+\dots+c% _{n}^{4}).
  43. ( X 1 + + X n ) / n (X_{1}+\cdots+X_{n})/\sqrt{n}
  44. r 1 2 + r 2 2 + = and r k 2 r 1 2 + + r k 2 0 , r_{1}^{2}+r_{2}^{2}+\cdots=\infty\,\text{ and }\frac{r_{k}^{2}}{r_{1}^{2}+% \cdots+r_{k}^{2}}\to 0,
  45. X 1 + + X k r 1 2 + + r k 2 \frac{X_{1}+\cdots+X_{k}}{\sqrt{r_{1}^{2}+\cdots+r_{k}^{2}}}
  46. X n - E ( X n ) Var ( X n ) \frac{X_{n}-\mathrm{E}(X_{n})}{\sqrt{\operatorname{Var}(X_{n})}}
  47. 3 \sqrt{3}
  48. n \sqrt{n}

Central_moment.html

  1. μ n = E [ ( X - E [ X ] ) n ] = - + ( x - μ ) n f ( x ) d x . \mu_{n}=\operatorname{E}\left[(X-\operatorname{E}[X])^{n}\right]=\int_{-\infty% }^{+\infty}(x-\mu)^{n}f(x)\,dx.
  2. μ n ( X + c ) = μ n ( X ) . \mu_{n}(X+c)=\mu_{n}(X).\,
  3. μ n ( c X ) = c n μ n ( X ) . \mu_{n}(cX)=c^{n}\mu_{n}(X).\,
  4. μ n ( X + Y ) = μ n ( X ) + μ n ( Y ) provided 1 n 3. \mu_{n}(X+Y)=\mu_{n}(X)+\mu_{n}(Y)\,\text{ provided }1\leq n\leq 3.\,
  5. μ n = E [ ( X - E [ X ] ) n ] = j = 0 n ( n j ) ( - 1 ) n - j μ j μ n - j , \mu_{n}=\mathrm{E}\left[\left(X-\mathrm{E}\left[X\right]\right)^{n}\right]=% \sum_{j=0}^{n}{n\choose j}(-1)^{n-j}\mu^{\prime}_{j}\mu^{n-j},
  6. μ j = - + x j f ( x ) d x = E [ X j ] \mu^{\prime}_{j}=\int_{-\infty}^{+\infty}x^{j}f(x)\,dx=\mathrm{E}\left[X^{j}\right]
  7. μ = μ 1 \mu=\mu^{\prime}_{1}
  8. μ 0 = 1 \mu^{\prime}_{0}=1
  9. μ 2 = μ 2 - μ 2 \mu_{2}=\mu^{\prime}_{2}-\mu^{2}\,
  10. Var ( X ) = E [ X 2 ] - ( E [ X ] ) 2 \mathrm{Var}\left(X\right)=\mathrm{E}\left[X^{2}\right]-\left(\mathrm{E}\left[% X\right]\right)^{2}
  11. μ 3 = μ 3 - 3 μ μ 2 + 2 μ 3 \mu_{3}=\mu^{\prime}_{3}-3\mu\mu^{\prime}_{2}+2\mu^{3}\,
  12. μ 4 = μ 4 - 4 μ μ 3 + 6 μ 2 μ 2 - 3 μ 4 . \mu_{4}=\mu^{\prime}_{4}-4\mu\mu^{\prime}_{3}+6\mu^{2}\mu^{\prime}_{2}-3\mu^{4% }.\,
  13. μ 5 = μ 5 - 5 μ μ 4 + 10 μ 2 μ 3 - 10 μ 3 μ 2 + 4 μ 5 . \mu_{5}=\mu^{\prime}_{5}-5\mu\mu^{\prime}_{4}+10\mu^{2}\mu^{\prime}_{3}-10\mu^% {3}\mu^{\prime}_{2}+4\mu^{5}.\,
  14. 5 μ 4 μ 1 - μ 5 μ 0 = 5 μ 4 μ - μ 5 = 5 μ 5 - μ 5 = 4 μ 5 5\mu^{4}\mu^{\prime}_{1}-\mu^{5}\mu^{\prime}_{0}=5\mu^{4}\mu-\mu^{5}=5\mu^{5}-% \mu^{5}=4\mu^{5}
  15. μ j , k = E [ ( X - E [ X ] ) j ( Y - E [ Y ] ) k ] = - + - + ( x - μ X ) j ( y - μ Y ) k f ( x , y ) d x d y . \mu_{j,k}=\operatorname{E}\left[(X-\operatorname{E}[X])^{j}(Y-\operatorname{E}% [Y])^{k}\right]=\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}(x-\mu_{X})^{j% }(y-\mu_{Y})^{k}f(x,y)\,dx\,dy.

Central_tendency.html

  1. f 2 ( c ) = x - c 2 f_{2}(c)=\|x-c\|_{2}
  2. f 1 ( c ) = x - c 1 f_{1}(c)=\|x-c\|_{1}
  3. | θ - μ | σ 3 , \frac{|\theta-\mu|}{\sigma}\leq\sqrt{3},
  4. | ν - μ | σ 0.6 , \frac{|\nu-\mu|}{\sigma}\leq\sqrt{0.6},
  5. | θ - ν | σ 3 , \frac{|\theta-\nu|}{\sigma}\leq\sqrt{3},
  6. | ν - μ | σ 1. \frac{|\nu-\mu|}{\sigma}\leq 1.

Centripetal_force.html

  1. F = m a c = m v 2 r F=ma_{c}=\frac{mv^{2}}{r}
  2. a c a_{c}
  3. F = m r ω 2 . F=mr\omega^{2}.\,
  4. F = m r 4 π 2 T 2 . F=mr\frac{4\pi^{2}}{T^{2}}.
  5. F = γ m v 2 r F=\frac{\gamma mv^{2}}{r}
  6. γ = 1 1 - v 2 / c 2 \gamma=\frac{1}{\sqrt{1-v^{2}/c^{2}}}
  7. 𝐫 \,\textbf{r}
  8. r r
  9. θ \theta
  10. x ^ \hat{x}
  11. y ^ \hat{y}
  12. 𝐫 = r cos ( θ ) x ^ + r sin ( θ ) y ^ . \,\textbf{r}=r\cos(\theta)\hat{x}+r\sin(\theta)\hat{y}.
  13. r r
  14. ω \omega
  15. θ = ω t \theta=\omega t
  16. t t
  17. 𝐯 \,\textbf{v}
  18. 𝐚 \,\textbf{a}
  19. 𝐫 = r cos ( ω t ) x ^ + r sin ( ω t ) y ^ \,\textbf{r}=r\cos(\omega t)\hat{x}+r\sin(\omega t)\hat{y}
  20. 𝐫 ˙ = 𝐯 = - r ω sin ( ω t ) x ^ + r ω cos ( ω t ) y ^ \dot{\,\textbf{r}}=\,\textbf{v}=-r\omega\sin(\omega t)\hat{x}+r\omega\cos(% \omega t)\hat{y}
  21. 𝐫 ¨ = 𝐚 = - r ω 2 cos ( ω t ) x ^ - r ω 2 sin ( ω t ) y ^ \ddot{\,\textbf{r}}=\,\textbf{a}=-r\omega^{2}\cos(\omega t)\hat{x}-r\omega^{2}% \sin(\omega t)\hat{y}
  22. 𝐚 = - ω 2 ( r cos ( ω t ) x ^ + r sin ( ω t ) y ^ ) \,\textbf{a}=-\omega^{2}(r\cos(\omega t)\hat{x}+r\sin(\omega t)\hat{y})
  23. 𝐫 \,\textbf{r}
  24. 𝐚 = - ω 2 𝐫 . \,\textbf{a}=-\omega^{2}\,\textbf{r}.
  25. | 𝛀 | = d θ d t = ω , |\mathbf{\Omega}|=\frac{\mathrm{d}\theta}{\mathrm{d}t}=\omega\ ,
  26. d s y m b o l = 𝛀 × 𝐫 ( t ) d t , \mathrm{d}symbol{\ell}=\mathbf{\Omega}\times\mathbf{r}(t)\mathrm{d}t\ ,
  27. d 𝐫 d t = lim Δ t 0 𝐫 ( t + Δ t ) - 𝐫 ( t ) Δ t = d s y m b o l d t . \frac{\mathrm{d}\mathbf{r}}{\mathrm{d}t}=\lim_{{\Delta}t\to 0}\frac{\mathbf{r}% (t+{\Delta}t)-\mathbf{r}(t)}{{\Delta}t}=\frac{\mathrm{d}symbol{\ell}}{\mathrm{% d}t}\ .
  28. 𝐯 = def d 𝐫 d t = d 𝐬𝐲𝐦𝐛𝐨𝐥 d t = 𝛀 × 𝐫 ( t ) . \mathbf{v}\ \stackrel{\mathrm{def}}{=}\ \frac{\mathrm{d}\mathbf{r}}{\mathrm{d}% t}=\frac{\mathrm{d}\mathbf{symbol{\ell}}}{\mathrm{d}t}=\mathbf{\Omega}\times% \mathbf{r}(t)\ .
  29. 𝐚 = def d 𝐯 d t = 𝛀 × d 𝐫 ( t ) d t = 𝛀 × [ 𝛀 × 𝐫 ( t ) ] . \mathbf{a}\ \stackrel{\mathrm{def}}{=}\ \frac{\mathrm{d}\mathbf{v}}{d\mathrm{t% }}=\mathbf{\Omega}\times\frac{\mathrm{d}\mathbf{r}(t)}{\mathrm{d}t}=\mathbf{% \Omega}\times\left[\mathbf{\Omega}\times\mathbf{r}(t)\right]\ .
  30. 𝐚 × ( 𝐛 × 𝐜 ) = 𝐛 ( 𝐚 𝐜 ) - 𝐜 ( 𝐚 𝐛 ) . \mathbf{a}\times\left(\mathbf{b}\times\mathbf{c}\right)=\mathbf{b}\left(% \mathbf{a}\cdot\mathbf{c}\right)-\mathbf{c}\left(\mathbf{a}\cdot\mathbf{b}% \right)\ .
  31. 𝐚 = - | 𝛀 | 2 𝐫 ( t ) . \mathbf{a}=-{|\mathbf{\Omega|}}^{2}\mathbf{r}(t)\ .
  32. | 𝐚 | = | 𝐫 ( t ) | ( d θ d t ) 2 = r ω 2 |\mathbf{a}|=|\mathbf{r}(t)|\left(\frac{\mathrm{d}\theta}{\mathrm{d}t}\right)^% {2}=r{\omega}^{2}
  33. | 𝐅 h | = m | 𝐠 | sin θ cos θ = m | 𝐠 | tan θ . |\mathbf{F}_{\mathrm{h}}|=m|\mathbf{g}|\frac{\mathrm{sin}\ \theta}{\mathrm{cos% }\ \theta}=m|\mathbf{g}|\mathrm{tan}\ \theta\ .
  34. | 𝐅 c | = m | 𝐚 c | = m | 𝐯 | 2 r . |\mathbf{F}_{\mathrm{c}}|=m|\mathbf{a}_{\mathrm{c}}|=\frac{m|\mathbf{v}|^{2}}{% r}\ .
  35. m | 𝐠 | tan θ = m | 𝐯 | 2 r , m|\mathbf{g}|\mathrm{tan}\ \theta=\frac{m|\mathbf{v}|^{2}}{r}\ ,
  36. tan θ = | 𝐯 | 2 | 𝐠 | r . \mathrm{tan}\ \theta=\frac{|\mathbf{v}|^{2}}{|\mathbf{g}|r}\ .
  37. 𝐯 = r d 𝐮 r d t = r d d t ( cos θ 𝐢 + sin θ 𝐣 ) \mathbf{v}=r\frac{\mathrm{d}\mathbf{u}_{\mathrm{r}}}{\mathrm{d}t}=r\frac{% \mathrm{d}}{\mathrm{d}t}\left(\mathrm{cos}\ \theta\ \mathbf{i}+\mathrm{sin}\ % \theta\ \mathbf{j}\right)
  38. = r d θ d t ( - sin θ 𝐢 + cos θ 𝐣 ) =r\frac{d\theta}{dt}\left(-\mathrm{sin}\ \theta\ \mathbf{i}+\mathrm{cos}\ % \theta\ \mathbf{j}\right)\,
  39. = r d θ d t 𝐮 θ =r\frac{\mathrm{d}\theta}{\mathrm{d}t}\mathbf{u}_{\mathrm{\theta}}\,
  40. = ω r 𝐮 θ =\omega r\mathbf{u}_{\mathrm{\theta}}\,
  41. d 𝐮 θ d t = - d θ d t 𝐮 r = - ω 𝐮 r , {\frac{\mathrm{d}\mathbf{u}_{\mathrm{\theta}}}{\mathrm{d}t}=-\frac{\mathrm{d}% \theta}{\mathrm{d}t}\mathbf{u}_{\mathrm{r}}=-\omega\mathbf{u}_{\mathrm{r}}}\ ,
  42. 𝐚 = r ( d ω d t 𝐮 θ - ω 2 𝐮 r ) . \mathbf{a}=r\left(\frac{\mathrm{d}\omega}{\mathrm{d}t}\mathbf{u}_{\mathrm{% \theta}}-\omega^{2}\mathbf{u}_{\mathrm{r}}\right)\ .
  43. 𝐚 r = - ω 2 r 𝐮 r = - | 𝐯 | 2 r 𝐮 r \mathbf{a}_{\mathrm{r}}=-\omega^{2}r\ \mathbf{u}_{\mathrm{r}}=-\frac{|\mathbf{% v}|^{2}}{r}\ \mathbf{u}_{\mathrm{r}}
  44. 𝐚 θ = r d ω d t 𝐮 θ = d | 𝐯 | d t 𝐮 θ , \ \mathbf{a}_{\mathrm{\theta}}=r\ \frac{\mathrm{d}\omega}{\mathrm{d}t}\ % \mathbf{u}_{\mathrm{\theta}}=\frac{\mathrm{d}|\mathbf{v}|}{\mathrm{d}t}\ % \mathbf{u}_{\mathrm{\theta}}\ ,
  45. 𝐫 = ρ 𝐮 ρ , \mathbf{r}=\rho\mathbf{u}_{\rho}\ ,
  46. 𝐯 = d ρ d t 𝐮 ρ + ρ d 𝐮 ρ d t . \mathbf{v}=\frac{\mathrm{d}\rho}{\mathrm{d}t}\mathbf{u}_{\rho}+\rho\frac{% \mathrm{d}\mathbf{u}_{\rho}}{\mathrm{d}t}\ .
  47. d 𝐮 ρ = 𝐮 θ d θ , \mathrm{d}\mathbf{u}_{\rho}=\mathbf{u}_{\theta}\mathrm{d}\theta\ ,
  48. d 𝐮 ρ d t = 𝐮 θ d θ d t . \frac{\mathrm{d}\mathbf{u}_{\rho}}{\mathrm{d}t}=\mathbf{u}_{\theta}\frac{% \mathrm{d}\theta}{\mathrm{d}t}\ .
  49. d 𝐮 θ d t = - d θ d t 𝐮 ρ . \frac{\mathrm{d}\mathbf{u}_{\theta}}{\mathrm{d}t}=-\frac{\mathrm{d}\theta}{% \mathrm{d}t}\mathbf{u}_{\rho}\ .
  50. 𝐯 = d ρ d t 𝐮 ρ + ρ 𝐮 θ d θ d t = v ρ 𝐮 ρ + v θ 𝐮 θ = 𝐯 ρ + 𝐯 θ . \mathbf{v}=\frac{\mathrm{d}\rho}{\mathrm{d}t}\mathbf{u}_{\rho}+\rho\mathbf{u}_% {\theta}\frac{\mathrm{d}\theta}{\mathrm{d}t}=v_{\rho}\mathbf{u}_{\rho}+v_{% \theta}\mathbf{u}_{\theta}=\mathbf{v}_{\rho}+\mathbf{v}_{\theta}\ .
  51. 𝐚 = d 2 ρ d t 2 𝐮 ρ + d ρ d t d 𝐮 ρ d t + d ρ d t 𝐮 θ d θ d t + ρ d 𝐮 θ d t d θ d t + ρ 𝐮 θ d 2 θ d t 2 . \mathbf{a}=\frac{\mathrm{d}^{2}\rho}{\mathrm{d}t^{2}}\mathbf{u}_{\rho}+\frac{% \mathrm{d}\rho}{\mathrm{d}t}\frac{\mathrm{d}\mathbf{u}_{\rho}}{\mathrm{d}t}+% \frac{\mathrm{d}\rho}{\mathrm{d}t}\mathbf{u}_{\theta}\frac{\mathrm{d}\theta}{% \mathrm{d}t}+\rho\frac{\mathrm{d}\mathbf{u}_{\theta}}{\mathrm{d}t}\frac{% \mathrm{d}\theta}{\mathrm{d}t}+\rho\mathbf{u}_{\theta}\frac{\mathrm{d}^{2}% \theta}{\mathrm{d}t^{2}}\ .
  52. 𝐚 = d 2 ρ d t 2 𝐮 ρ + 2 d ρ d t 𝐮 θ d θ d t - ρ 𝐮 ρ ( d θ d t ) 2 + ρ 𝐮 θ d 2 θ d t 2 , \mathbf{a}=\frac{\mathrm{d}^{2}\rho}{\mathrm{d}t^{2}}\mathbf{u}_{\rho}+2\frac{% \mathrm{d}\rho}{\mathrm{d}t}\mathbf{u}_{\theta}\frac{\mathrm{d}\theta}{\mathrm% {d}t}-\rho\mathbf{u}_{\rho}\left(\frac{\mathrm{d}\theta}{\mathrm{d}t}\right)^{% 2}+\rho\mathbf{u}_{\theta}\frac{\mathrm{d}^{2}\theta}{\mathrm{d}t^{2}}\ ,
  53. = 𝐮 ρ [ d 2 ρ d t 2 - ρ ( d θ d t ) 2 ] + 𝐮 θ [ 2 d ρ d t d θ d t + ρ d 2 θ d t 2 ] =\mathbf{u}_{\rho}\left[\frac{\mathrm{d}^{2}\rho}{\mathrm{d}t^{2}}-\rho\left(% \frac{\mathrm{d}\theta}{\mathrm{d}t}\right)^{2}\right]+\mathbf{u}_{\theta}% \left[2\frac{\mathrm{d}\rho}{\mathrm{d}t}\frac{\mathrm{d}\theta}{\mathrm{d}t}+% \rho\frac{\mathrm{d}^{2}\theta}{\mathrm{d}t^{2}}\right]
  54. = 𝐮 ρ [ d v ρ d t - v θ 2 ρ ] + 𝐮 θ [ 2 ρ v ρ v θ + ρ d d t v θ ρ ] . =\mathbf{u}_{\rho}\left[\frac{\mathrm{d}v_{\rho}}{\mathrm{d}t}-\frac{v_{\theta% }^{2}}{\rho}\right]+\mathbf{u}_{\theta}\left[\frac{2}{\rho}v_{\rho}v_{\theta}+% \rho\frac{\mathrm{d}}{\mathrm{d}t}\frac{v_{\theta}}{\rho}\right]\ .
  55. 𝐚 = 𝐮 ρ [ - ρ ( d θ d t ) 2 ] + 𝐮 θ [ ρ d 2 θ d t 2 ] \mathbf{a}=\mathbf{u}_{\rho}\left[-\rho\left(\frac{\mathrm{d}\theta}{\mathrm{d% }t}\right)^{2}\right]+\mathbf{u}_{\theta}\left[\rho\frac{\mathrm{d}^{2}\theta}% {\mathrm{d}t^{2}}\right]
  56. = 𝐮 ρ [ - v 2 r ] + 𝐮 θ [ d v d t ] =\mathbf{u}_{\rho}\left[-\frac{v^{2}}{r}\right]+\mathbf{u}_{\theta}\left[\frac% {\mathrm{d}v}{\mathrm{d}t}\right]
  57. v = v θ . v=v_{\theta}.
  58. s = s ( t ) . s=s(t)\ .
  59. 1 ρ ( s ) = κ ( s ) = d θ d s . \frac{1}{\rho(s)}=\kappa(s)=\frac{\mathrm{d}\theta}{\mathrm{d}s}\ .
  60. ω ( s ) = d θ d t = d θ d s d s d t = 1 ρ ( s ) d s d t = v ( s ) ρ ( s ) , \omega(s)=\frac{\mathrm{d}\theta}{\mathrm{d}t}=\frac{\mathrm{d}\theta}{\mathrm% {d}s}\frac{\mathrm{d}s}{\mathrm{d}t}=\frac{1}{\rho(s)}\ \frac{\mathrm{d}s}{% \mathrm{d}t}=\frac{v(s)}{\rho(s)}\ ,
  61. v ( s ) = d s d t . v(s)=\frac{\mathrm{d}s}{\mathrm{d}t}\ .
  62. d 𝐮 n ( s ) d s = 𝐮 t ( s ) d θ d s = 𝐮 t ( s ) 1 ρ ; \frac{d\mathbf{u}_{\mathrm{n}}(s)}{ds}=\mathbf{u}_{\mathrm{t}}(s)\frac{d\theta% }{ds}=\mathbf{u}_{\mathrm{t}}(s)\frac{1}{\rho}\ ;
  63. d 𝐮 t ( s ) d s = - 𝐮 n ( s ) d θ d s = - 𝐮 n ( s ) 1 ρ . \frac{d\mathbf{u}_{\mathrm{t}}(s)}{\mathrm{d}s}=-\mathbf{u}_{\mathrm{n}}(s)% \frac{\mathrm{d}\theta}{\mathrm{d}s}=-\mathbf{u}_{\mathrm{n}}(s)\frac{1}{\rho}\ .
  64. 𝐯 ( t ) = v 𝐮 t ( s ) ; \mathbf{v}(t)=v\mathbf{u}_{\mathrm{t}}(s)\ ;
  65. 𝐚 ( t ) = d v d t 𝐮 t ( s ) - v 2 ρ 𝐮 n ( s ) ; \mathbf{a}(t)=\frac{\mathrm{d}v}{\mathrm{d}t}\mathbf{u}_{\mathrm{t}}(s)-\frac{% v^{2}}{\rho}\mathbf{u}_{\mathrm{n}}(s)\ ;
  66. d v d t = d v d s d s d t = d v d s v . \frac{\mathrm{\mathrm{d}}v}{\mathrm{\mathrm{d}}t}=\frac{\mathrm{d}v}{\mathrm{d% }s}\ \frac{\mathrm{d}s}{\mathrm{d}t}=\frac{\mathrm{d}v}{\mathrm{d}s}\ v\ .
  67. 𝐫 ( s ) = [ x ( s ) , y ( s ) ] . \mathbf{r}(s)=\left[x(s),\ y(s)\right]\ .
  68. d 𝐫 ( s ) = [ d x ( s ) , d y ( s ) ] = [ x ( s ) , y ( s ) ] d s , \mathrm{d}\mathbf{r}(s)=\left[\mathrm{d}x(s),\ \mathrm{d}y(s)\right]=\left[x^{% \prime}(s),\ y^{\prime}(s)\right]\mathrm{d}s\ ,
  69. [ x ( s ) 2 + y ( s ) 2 ] = 1 . \left[x^{\prime}(s)^{2}+y^{\prime}(s)^{2}\right]=1\ .
  70. 𝐮 t ( s ) = [ x ( s ) , y ( s ) ] , \mathbf{u}_{\mathrm{t}}(s)=\left[x^{\prime}(s),\ y^{\prime}(s)\right]\ ,
  71. 𝐮 n ( s ) = [ y ( s ) , - x ( s ) ] , \mathbf{u}_{\mathrm{n}}(s)=\left[y^{\prime}(s),\ -x^{\prime}(s)\right]\ ,
  72. sin θ = y ( s ) x ( s ) 2 + y ( s ) 2 = y ( s ) ; \sin\theta=\frac{y^{\prime}(s)}{\sqrt{x^{\prime}(s)^{2}+y^{\prime}(s)^{2}}}=y^% {\prime}(s)\ ;
  73. cos θ = x ( s ) x ( s ) 2 + y ( s ) 2 = x ( s ) . \cos\theta=\frac{x^{\prime}(s)}{\sqrt{x^{\prime}(s)^{2}+y^{\prime}(s)^{2}}}=x^% {\prime}(s)\ .
  74. 1 ρ = d θ d s . \frac{1}{\rho}=\frac{\mathrm{d}\theta}{\mathrm{d}s}\ .
  75. d sin θ d s = cos θ d θ d s = 1 ρ cos θ = 1 ρ x ( s ) . \frac{\mathrm{d}\sin\theta}{\mathrm{d}s}=\cos\theta\frac{\mathrm{d}\theta}{% \mathrm{d}s}=\frac{1}{\rho}\cos\theta\ =\frac{1}{\rho}x^{\prime}(s)\ .
  76. d sin θ d s = d d s y ( s ) x ( s ) 2 + y ( s ) 2 \frac{\mathrm{d}\sin\theta}{\mathrm{d}s}=\frac{\mathrm{d}}{\mathrm{d}s}\frac{y% ^{\prime}(s)}{\sqrt{x^{\prime}(s)^{2}+y^{\prime}(s)^{2}}}
  77. = y ′′ ( s ) x ( s ) 2 - y ( s ) x ( s ) x ′′ ( s ) ( x ( s ) 2 + y ( s ) 2 ) 3 / 2 , =\frac{y^{\prime\prime}(s)x^{\prime}(s)^{2}-y^{\prime}(s)x^{\prime}(s)x^{% \prime\prime}(s)}{\left(x^{\prime}(s)^{2}+y^{\prime}(s)^{2}\right)^{3/2}}\ ,
  78. d θ d s = 1 ρ = y ′′ ( s ) x ( s ) - y ( s ) x ′′ ( s ) \frac{\mathrm{d}\theta}{\mathrm{d}s}=\frac{1}{\rho}=y^{\prime\prime}(s)x^{% \prime}(s)-y^{\prime}(s)x^{\prime\prime}(s)
  79. = y ′′ ( s ) x ( s ) = - x ′′ ( s ) y ( s ) , =\frac{y^{\prime\prime}(s)}{x^{\prime}(s)}=-\frac{x^{\prime\prime}(s)}{y^{% \prime}(s)}\ ,
  80. x ( s ) x ′′ ( s ) + y ( s ) y ′′ ( s ) = 0 . x^{\prime}(s)x^{\prime\prime}(s)+y^{\prime}(s)y^{\prime\prime}(s)=0\ .
  81. 𝐚 ( s ) = d d t 𝐯 ( s ) \mathbf{a}(s)=\frac{\mathrm{d}}{\mathrm{d}t}\mathbf{v}(s)
  82. = d d t [ d s d t ( x ( s ) , y ( s ) ) ] =\frac{\mathrm{d}}{\mathrm{d}t}\left[\frac{\mathrm{d}s}{\mathrm{d}t}\left(x^{% \prime}(s),\ y^{\prime}(s)\right)\right]
  83. = ( d 2 s d t 2 ) 𝐮 t ( s ) + ( d s d t ) 2 ( x ′′ ( s ) , y ′′ ( s ) ) =\left(\frac{\mathrm{d}^{2}s}{\mathrm{d}t^{2}}\right)\mathbf{u}_{\mathrm{t}}(s% )+\left(\frac{\mathrm{d}s}{\mathrm{d}t}\right)^{2}\left(x^{\prime\prime}(s),\ % y^{\prime\prime}(s)\right)
  84. = ( d 2 s d t 2 ) 𝐮 t ( s ) - ( d s d t ) 2 1 ρ 𝐮 n ( s ) , =\left(\frac{\mathrm{d}^{2}s}{\mathrm{d}t^{2}}\right)\mathbf{u}_{\mathrm{t}}(s% )-\left(\frac{\mathrm{d}s}{\mathrm{d}t}\right)^{2}\frac{1}{\rho}\mathbf{u}_{% \mathrm{n}}(s)\ ,
  85. x = α cos s α ; y = α sin s α . x=\alpha\cos\frac{s}{\alpha}\ ;\ y=\alpha\sin\frac{s}{\alpha}\ .
  86. x 2 + y 2 = α 2 , x^{2}+y^{2}=\alpha^{2}\ ,
  87. y ( s ) = cos s α ; x ( s ) = - sin s α , y^{\prime}(s)=\cos\frac{s}{\alpha}\ ;\ x^{\prime}(s)=-\sin\frac{s}{\alpha}\ ,
  88. y ′′ ( s ) = - 1 α sin s α ; x ′′ ( s ) = - 1 α cos s α . y^{\prime\prime}(s)=-\frac{1}{\alpha}\sin\frac{s}{\alpha}\ ;\ x^{\prime\prime}% (s)=-\frac{1}{\alpha}\cos\frac{s}{\alpha}\ .
  89. x ( s ) 2 + y ( s ) 2 = 1 ; 1 ρ = y ′′ ( s ) x ( s ) - y ( s ) x ′′ ( s ) = 1 α . x^{\prime}(s)^{2}+y^{\prime}(s)^{2}=1\ ;\ \frac{1}{\rho}=y^{\prime\prime}(s)x^% {\prime}(s)-y^{\prime}(s)x^{\prime\prime}(s)=\frac{1}{\alpha}\ .
  90. 𝐮 t ( s ) = [ - sin s α , cos s α ] ; 𝐮 n ( s ) = [ cos s α , sin s α ] , \mathbf{u}_{\mathrm{t}}(s)=\left[-\sin\frac{s}{\alpha}\ ,\ \cos\frac{s}{\alpha% }\right]\ ;\ \mathbf{u}_{\mathrm{n}}(s)=\left[\cos\frac{s}{\alpha}\ ,\ \sin% \frac{s}{\alpha}\right]\ ,
  91. d d s 𝐮 t ( s ) = - 1 α [ cos s α , sin s α ] = - 1 α 𝐮 n ( s ) ; \frac{\mathrm{d}}{\mathrm{d}s}\mathbf{u}_{\mathrm{t}}(s)=-\frac{1}{\alpha}% \left[\cos\frac{s}{\alpha}\ ,\ \sin\frac{s}{\alpha}\right]=-\frac{1}{\alpha}% \mathbf{u}_{\mathrm{n}}(s)\ ;
  92. d d s 𝐮 n ( s ) = 1 α [ - sin s α , cos s α ] = 1 α 𝐮 t ( s ) . \ \frac{\mathrm{d}}{\mathrm{d}s}\mathbf{u}_{\mathrm{n}}(s)=\frac{1}{\alpha}% \left[-\sin\frac{s}{\alpha}\ ,\ \cos\frac{s}{\alpha}\right]=\frac{1}{\alpha}% \mathbf{u}_{\mathrm{t}}(s)\ .
  93. s ( t ) = 0 t d t v ( t ) , s(t)=\int_{0}^{t}\ dt^{\prime}\ v(t^{\prime})\ ,
  94. 𝐯 = v ( t ) 𝐮 t ( s ) , \mathbf{v}=v(t)\mathbf{u}_{\mathrm{t}}(s)\ ,
  95. 𝐚 = d v d t 𝐮 t ( s ) + v d d t 𝐮 t ( s ) = d v d t 𝐮 t ( s ) - v 1 α 𝐮 n ( s ) d s d t \mathbf{a}=\frac{\mathrm{d}v}{\mathrm{d}t}\mathbf{u}_{\mathrm{t}}(s)+v\frac{% \mathrm{d}}{\mathrm{d}t}\mathbf{u}_{\mathrm{t}}(s)=\frac{\mathrm{d}v}{\mathrm{% d}t}\mathbf{u}_{\mathrm{t}}(s)-v\frac{1}{\alpha}\mathbf{u}_{\mathrm{n}}(s)% \frac{\mathrm{d}s}{\mathrm{d}t}
  96. 𝐚 = d v d t 𝐮 t ( s ) - v 2 α 𝐮 n ( s ) , \mathbf{a}=\frac{\mathrm{d}v}{\mathrm{d}t}\mathbf{u}_{\mathrm{t}}(s)-\frac{v^{% 2}}{\alpha}\mathbf{u}_{\mathrm{n}}(s)\ ,

Cepstrum.html

  1. = | - 1 { log ( | { f ( t ) } | 2 ) } | 2 =\left|\mathcal{F}^{-1}\left\{\mbox{log}~{}(\left|\mathcal{F}\left\{f(t)\right% \}\right|^{2})\right\}\right|^{2}
  2. = I F T ( l o g ( F T ( t h e s i g n a l ) ) + j 2 π m =IFT(log(FT(thesignal))+j2πm
  3. m m
  4. x 1 * x 2 x 1 + x 2 x_{1}*x_{2}\rightarrow x^{\prime}_{1}+x^{\prime}_{2}

Chain_rule.html

  1. ( f g ) = ( f g ) g . (f\circ g)^{\prime}=(f^{\prime}\circ g)\cdot g^{\prime}.
  2. F ( x ) = f ( g ( x ) ) g ( x ) . F^{\prime}(x)=f^{\prime}(g(x))g^{\prime}(x).
  3. d z d x = d z d y d y d x . \frac{dz}{dx}=\frac{dz}{dy}\cdot\frac{dy}{dx}.
  4. a + b z + c z 2 \sqrt{a+bz+cz^{2}}
  5. a + b z + c z 2 a+bz+cz^{2}
  6. ( f g ) ( t ) = f ( g ( t ) ) g ( t ) . (f\circ g)^{\prime}(t)=f^{\prime}(g(t))\cdot g^{\prime}(t).
  7. ( f g ) ( t ) = ( - 10.1325 e - 0.0001 ( 4000 - 4.9 t 2 ) ) ( - 9.8 t ) . (f\circ g)^{\prime}(t)=\big(\mathord{-}10.1325e^{-0.0001(4000-4.9t^{2})}\big)% \cdot\big(\mathord{-}9.8t\big).
  8. ( f g ) ( c ) = f ( g ( c ) ) g ( c ) . (f\circ g)^{\prime}(c)=f^{\prime}(g(c))\cdot g^{\prime}(c).
  9. ( f g ) = ( f g ) g . (f\circ g)^{\prime}=(f^{\prime}\circ g)\cdot g^{\prime}.\,
  10. d y d x = d y d u d u d x . \frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}.
  11. d y d x | x = c = d y d u | u = g ( c ) d u d x | x = c . \left.\frac{dy}{dx}\right|_{x=c}=\left.\frac{dy}{du}\right|_{u=g(c)}\cdot\left% .\frac{du}{dx}\right|_{x=c}.\,
  12. g ( t ) g(t)
  13. t t
  14. f ( h ) f(h)
  15. h h
  16. f f
  17. g g
  18. f f′
  19. g g′
  20. f f
  21. g g
  22. [ u v a l , u 2126.5 , u u = \xb 0 C / k m ] [ u v a l , u 2.5 , u u = k m / h ] = [ u v a l , u 21216.25 , u u = \xb 0 C / h ] [u^{\prime}val^{\prime},u^{\prime}\u{2}2126.5^{\prime},u^{\prime}u=\xb 0C/km^{% \prime}]⋅[u^{\prime}val^{\prime},u^{\prime}2.5^{\prime},u^{\prime}u=km/h^{% \prime}]=[u^{\prime}val^{\prime},u^{\prime}\u{2}21216.25^{\prime},u^{\prime}u=% \xb 0C/h^{\prime}]
  23. f f′
  24. g g
  25. g g′
  26. g g
  27. f f′
  28. y = e sin x 2 . y=e^{\sin{x^{2}}}.
  29. y \displaystyle y
  30. d y d u \displaystyle\frac{dy}{du}
  31. ( f g h ) ( a ) = f ( ( g h ) ( a ) ) ( g h ) ( a ) = f ( ( g h ) ( a ) ) g ( h ( a ) ) h ( a ) . (f\circ g\circ h)^{\prime}(a)=f^{\prime}((g\circ h)(a))\cdot(g\circ h)^{\prime% }(a)=f^{\prime}((g\circ h)(a))\cdot g^{\prime}(h(a))\cdot h^{\prime}(a).
  32. d y d x = d y d u | u = g ( h ( a ) ) d u d v | v = h ( a ) d v d x | x = a , \frac{dy}{dx}=\left.\frac{dy}{du}\right|_{u=g(h(a))}\cdot\left.\frac{du}{dv}% \right|_{v=h(a)}\cdot\left.\frac{dv}{dx}\right|_{x=a},
  33. d y d x = d y d u d u d v d v d x . \frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dv}\cdot\frac{dv}{dx}.
  34. d y d x = e sin x 2 cos x 2 2 x . \frac{dy}{dx}=e^{\sin{x^{2}}}\cdot\cos{x^{2}}\cdot 2x.
  35. ( f g h ) ( a ) = ( f g ) ( h ( a ) ) h ( a ) = f ( g ( h ( a ) ) ) g ( h ( a ) ) h ( a ) . (f\circ g\circ h)^{\prime}(a)=(f\circ g)^{\prime}(h(a))\cdot h^{\prime}(a)=f^{% \prime}(g(h(a)))\cdot g^{\prime}(h(a))\cdot h^{\prime}(a).
  36. f 1 f 2 f n - 1 f n f_{1}\circ f_{2}\circ\ldots\circ f_{n-1}\circ f_{n}
  37. f a . . b = f a f a + 1 f b - 1 f b f_{a..b}=f_{a}\circ f_{a+1}\circ\ldots\circ f_{b-1}\circ f_{b}
  38. f a . . a = f a f_{a..a}=f_{a}
  39. f a . . b ( x ) = x f_{a..b}(x)=x
  40. b < a b<a
  41. D f 1.. n = ( D f 1 f 2.. n ) ( D f 2 f 3.. n ) ( D f n - 1 f n . . n ) D f n = k = 1 n [ D f k f ( k + 1 ) . . n ] Df_{1..n}=(Df_{1}\circ f_{2..n})(Df_{2}\circ f_{3..n})\ldots(Df_{n-1}\circ f_{% n..n})Df_{n}=\prod_{k=1}^{n}\left[Df_{k}\circ f_{(k+1)..n}\right]
  42. f 1.. n ( x ) = f 1 ( f 2.. n ( x ) ) f 2 ( f 3.. n ( x ) ) f n - 1 ( f n . . n ( x ) ) f n ( x ) = k = 1 n f k ( f ( k + 1.. n ) ( x ) ) f_{1..n}^{\prime}(x)=f_{1}^{\prime}\left(f_{2..n}(x)\right)\;f_{2}^{\prime}% \left(f_{3..n}(x)\right)\;\ldots\;f_{n-1}^{\prime}\left(f_{n..n}(x)\right)\;f_% {n}^{\prime}(x)=\prod_{k=1}^{n}f_{k}^{\prime}\left(f_{(k+1..n)}(x)\right)
  43. d d x ( f ( x ) g ( x ) ) \displaystyle\frac{d}{dx}\left(\frac{f(x)}{g(x)}\right)
  44. f ( x ) 1 g ( x ) + f ( x ) ( - 1 g ( x ) 2 g ( x ) ) = f ( x ) g ( x ) - f ( x ) g ( x ) g ( x ) 2 , f^{\prime}(x)\cdot\frac{1}{g(x)}+f(x)\cdot\left(-\frac{1}{g(x)^{2}}\cdot g^{% \prime}(x)\right)=\frac{f^{\prime}(x)g(x)-f(x)g^{\prime}(x)}{g(x)^{2}},
  45. f ( g ( x ) ) = x . f(g(x))=x.
  46. f ( g ( x ) ) g ( x ) = 1. f^{\prime}(g(x))g^{\prime}(x)=1.
  47. f ( g ( f ( y ) ) ) g ( f ( y ) ) = 1 f ( y ) g ( f ( y ) ) = 1 f ( y ) = 1 g ( f ( y ) ) . \begin{aligned}\displaystyle f^{\prime}(g(f(y)))g^{\prime}(f(y))&\displaystyle% =1\\ \displaystyle f^{\prime}(y)g^{\prime}(f(y))&\displaystyle=1\\ \displaystyle f^{\prime}(y)=\frac{1}{g^{\prime}(f(y))}.\end{aligned}
  48. d d y ln y = 1 e ln y = 1 y . \frac{d}{dy}\ln y=\frac{1}{e^{\ln y}}=\frac{1}{y}.
  49. d y d x = d y d u d u d x \frac{dy}{dx}=\frac{dy}{du}\frac{du}{dx}
  50. d 2 y d x 2 = d 2 y d u 2 ( d u d x ) 2 + d y d u d 2 u d x 2 \frac{d^{2}y}{dx^{2}}=\frac{d^{2}y}{du^{2}}\left(\frac{du}{dx}\right)^{2}+% \frac{dy}{du}\frac{d^{2}u}{dx^{2}}
  51. d 3 y d x 3 = d 3 y d u 3 ( d u d x ) 3 + 3 d 2 y d u 2 d u d x d 2 u d x 2 + d y d u d 3 u d x 3 \frac{d^{3}y}{dx^{3}}=\frac{d^{3}y}{du^{3}}\left(\frac{du}{dx}\right)^{3}+3\,% \frac{d^{2}y}{du^{2}}\frac{du}{dx}\frac{d^{2}u}{dx^{2}}+\frac{dy}{du}\frac{d^{% 3}u}{dx^{3}}
  52. d 4 y d x 4 = d 4 y d u 4 ( d u d x ) 4 + 6 d 3 y d u 3 ( d u d x ) 2 d 2 u d x 2 + d 2 y d u 2 ( 4 d u d x d 3 u d x 3 + 3 ( d 2 u d x 2 ) 2 ) + d y d u d 4 u d x 4 . \frac{d^{4}y}{dx^{4}}=\frac{d^{4}y}{du^{4}}\left(\frac{du}{dx}\right)^{4}+6\,% \frac{d^{3}y}{du^{3}}\left(\frac{du}{dx}\right)^{2}\frac{d^{2}u}{dx^{2}}+\frac% {d^{2}y}{du^{2}}\left(4\,\frac{du}{dx}\frac{d^{3}u}{dx^{3}}+3\,\left(\frac{d^{% 2}u}{dx^{2}}\right)^{2}\right)+\frac{dy}{du}\frac{d^{4}u}{dx^{4}}.
  53. ( f g ) ( a ) = lim x a f ( g ( x ) ) - f ( g ( a ) ) x - a . (f\circ g)^{\prime}(a)=\lim_{x\to a}\frac{f(g(x))-f(g(a))}{x-a}.
  54. lim x a f ( g ( x ) ) - f ( g ( a ) ) g ( x ) - g ( a ) g ( x ) - g ( a ) x - a . \lim_{x\to a}\frac{f(g(x))-f(g(a))}{g(x)-g(a)}\cdot\frac{g(x)-g(a)}{x-a}.
  55. Q ( y ) = { f ( y ) - f ( g ( a ) ) y - g ( a ) , y g ( a ) , f ( g ( a ) ) , y = g ( a ) . Q(y)=\begin{cases}\frac{f(y)-f(g(a))}{y-g(a)},&y\neq g(a),\\ f^{\prime}(g(a)),&y=g(a).\end{cases}
  56. Q ( g ( x ) ) g ( x ) - g ( a ) x - a . Q(g(x))\cdot\frac{g(x)-g(a)}{x-a}.
  57. g ( a + h ) - g ( a ) = g ( a ) h + ε ( h ) h . g(a+h)-g(a)=g^{\prime}(a)h+\varepsilon(h)h.\,
  58. f ( g ( a ) + k ) - f ( g ( a ) ) = f ( g ( a ) ) k + η ( k ) k . f(g(a)+k)-f(g(a))=f^{\prime}(g(a))k+\eta(k)k.\,
  59. f ( g ( a + h ) ) - f ( g ( a ) ) = f ( g ( a ) + g ( a ) h + ε ( h ) h ) - f ( g ( a ) ) . f(g(a+h))-f(g(a))=f(g(a)+g^{\prime}(a)h+\varepsilon(h)h)-f(g(a)).
  60. f ( g ( a ) + k h ) - f ( g ( a ) ) = f ( g ( a ) ) k h + η ( k h ) k h . f(g(a)+k_{h})-f(g(a))=f^{\prime}(g(a))k_{h}+\eta(k_{h})k_{h}.\,
  61. f ( g ( a ) ) g ( a ) h + [ f ( g ( a ) ) ε ( h ) + η ( k h ) g ( a ) + η ( k h ) ε ( h ) ] h . f^{\prime}(g(a))g^{\prime}(a)h+[f^{\prime}(g(a))\varepsilon(h)+\eta(k_{h})g^{% \prime}(a)+\eta(k_{h})\varepsilon(h)]h.\,
  62. Q ( y ) = f ( g ( a ) ) + η ( y - g ( a ) ) . Q(y)=f^{\prime}(g(a))+\eta(y-g(a)).\,
  63. y = f ( x ) y=f(x)
  64. x = g ( t ) x=g(t)
  65. Δ t 0 \Delta t\not=0
  66. Δ x = g ( t + Δ t ) - g ( t ) \Delta x=g(t+\Delta t)-g(t)
  67. Δ y = f ( x + Δ x ) - f ( x ) \Delta y=f(x+\Delta x)-f(x)
  68. Δ y Δ t = Δ y Δ x Δ x Δ t \frac{\Delta y}{\Delta t}=\frac{\Delta y}{\Delta x}\frac{\Delta x}{\Delta t}
  69. d y d t = d y d x d x d t \frac{dy}{dt}=\frac{dy}{dx}\frac{dx}{dt}
  70. D 𝐚 ( f g ) = D g ( 𝐚 ) f D 𝐚 g , D_{\mathbf{a}}(f\circ g)=D_{g(\mathbf{a})}f\circ D_{\mathbf{a}}g,
  71. D ( f g ) = D f D g . D(f\circ g)=Df\circ Dg.
  72. J f g ( 𝐚 ) = J f ( g ( 𝐚 ) ) J g ( 𝐚 ) , J_{f\circ g}(\mathbf{a})=J_{f}(g(\mathbf{a}))J_{g}(\mathbf{a}),
  73. J f g = ( J f g ) J g . J_{f\circ g}=(J_{f}\circ g)J_{g}.
  74. J g ( a ) \displaystyle J_{g}(a)
  75. ( y 1 , , y k ) ( x 1 , , x n ) = ( y 1 , , y k ) ( u 1 , , u m ) ( u 1 , , u m ) ( x 1 , , x n ) . \frac{\partial(y_{1},\ldots,y_{k})}{\partial(x_{1},\ldots,x_{n})}=\frac{% \partial(y_{1},\ldots,y_{k})}{\partial(u_{1},\ldots,u_{m})}\frac{\partial(u_{1% },\ldots,u_{m})}{\partial(x_{1},\ldots,x_{n})}.
  76. ( y 1 , , y k ) x i = ( y 1 , , y k ) ( u 1 , , u m ) ( u 1 , , u m ) x i . \frac{\partial(y_{1},\ldots,y_{k})}{\partial x_{i}}=\frac{\partial(y_{1},% \ldots,y_{k})}{\partial(u_{1},\ldots,u_{m})}\frac{\partial(u_{1},\ldots,u_{m})% }{\partial x_{i}}.
  77. ( y 1 , , y k ) x i = = 1 m ( y 1 , , y k ) u u x i . \frac{\partial(y_{1},\ldots,y_{k})}{\partial x_{i}}=\sum_{\ell=1}^{m}\frac{% \partial(y_{1},\ldots,y_{k})}{\partial u_{\ell}}\frac{\partial u_{\ell}}{% \partial x_{i}}.
  78. y x i = = 1 m y u u x i . \frac{\partial y}{\partial x_{i}}=\sum_{\ell=1}^{m}\frac{\partial y}{\partial u% _{\ell}}\frac{\partial u_{\ell}}{\partial x_{i}}.
  79. y x i = f 𝐮 x i . \frac{\partial y}{\partial x_{i}}=\nabla f\cdot\frac{\partial\mathbf{u}}{% \partial x_{i}}.
  80. u r = u x x r + u y y r = ( 2 x ) ( sin ( t ) ) + ( 2 ) ( 0 ) = 2 r sin 2 ( t ) , \frac{\partial u}{\partial r}=\frac{\partial u}{\partial x}\frac{\partial x}{% \partial r}+\frac{\partial u}{\partial y}\frac{\partial y}{\partial r}=(2x)(% \sin(t))+(2)(0)=2r\sin^{2}(t),
  81. u t = u x x t + u y y t = ( 2 x ) ( r cos ( t ) ) + ( 2 ) ( 2 sin ( t ) cos ( t ) ) = ( 2 r sin ( t ) ) ( r cos ( t ) ) + 4 sin ( t ) cos ( t ) = 2 ( r 2 + 2 ) sin ( t ) cos ( t ) . \begin{aligned}\displaystyle\frac{\partial u}{\partial t}&\displaystyle=\frac{% \partial u}{\partial x}\frac{\partial x}{\partial t}+\frac{\partial u}{% \partial y}\frac{\partial y}{\partial t}\\ &\displaystyle=(2x)(r\cos(t))+(2)(2\sin(t)\cos(t))\\ &\displaystyle=(2r\sin(t))(r\cos(t))+4\sin(t)\cos(t)\\ &\displaystyle=2(r^{2}+2)\sin(t)\cos(t).\end{aligned}
  82. 2 y x i x j = k ( y u k 2 u k x i x j ) + k , ( 2 y u k u u k x i u x j ) . \frac{\partial^{2}y}{\partial x_{i}\partial x_{j}}=\sum_{k}\left(\frac{% \partial y}{\partial u_{k}}\frac{\partial^{2}u_{k}}{\partial x_{i}\partial x_{% j}}\right)+\sum_{k,\ell}\left(\frac{\partial^{2}y}{\partial u_{k}\partial u_{% \ell}}\frac{\partial u_{k}}{\partial x_{i}}\frac{\partial u_{\ell}}{\partial x% _{j}}\right).

Chaitin's_constant.html

  1. Ω F = p P F 2 - | p | \Omega_{F}=\sum_{p\in P_{F}}2^{-|p|}
  2. | p | \left|p\right|
  3. N N
  4. Ω \Omega
  5. N N
  6. p p
  7. p p
  8. p p
  9. P = { p 1 , p 2 , } P=\{p_{1},p_{2},\ldots\}
  10. p P 2 - | p | \sum_{p\in P}2^{-|p|}
  11. i S i \bigcup_{i\in\mathbb{N}}S_{i}
  12. Δ 2 0 \Delta^{0}_{2}

Chandrasekhar_limit.html

  1. M \begin{smallmatrix}M_{\odot}\end{smallmatrix}
  2. P = K 1 ρ 5 3 P=K_{1}\rho^{5\over 3}
  3. ρ \rho
  4. K 1 K_{1}
  5. P = K 2 ρ 4 3 P=K_{2}\rho^{4\over 3}
  6. P = K 1 ρ 5 3 P=K_{1}\rho^{5\over 3}
  7. P = K 2 ρ 4 3 P=K_{2}\rho^{4\over 3}
  8. M limit = ω 3 0 3 π 2 ( c G ) 3 / 2 1 ( μ e m H ) 2 , M_{\rm limit}=\frac{\omega_{3}^{0}\sqrt{3\pi}}{2}\left(\frac{\hbar c}{G}\right% )^{3/2}\frac{1}{(\mu_{e}m_{H})^{2}},
  9. \hbar
  10. ω 3 0 2.018236 \omega_{3}^{0}\approx 2.018236
  11. c / G \sqrt{\hbar c/G}
  12. M P l 3 m H 2 . \frac{M_{Pl}^{3}}{m_{H}^{2}}.
  13. × 10 3 0 \times 10^{3}0

Channel_reliability.html

  1. C h R = 100 ( 1 - T o T s ) = 100 T a T s ChR=100(1-\frac{T_{o}}{T_{s}})=100\frac{T_{a}}{T_{s}}

Chaos_theory.html

  1. δ 𝐙 0 \delta\mathbf{Z}_{0}
  2. | δ 𝐙 ( t ) | e λ t | δ 𝐙 0 | |\delta\mathbf{Z}(t)|\approx e^{\lambda t}|\delta\mathbf{Z}_{0}|
  3. 5 - 5 8 \tfrac{5-\sqrt{5}}{8}
  4. 5 + 5 8 \tfrac{5+\sqrt{5}}{8}
  5. 5 - 5 8 \tfrac{5-\sqrt{5}}{8}
  6. d x d t \displaystyle\frac{\mathrm{d}x}{\mathrm{d}t}
  7. x x
  8. y y
  9. z z
  10. t t
  11. σ \sigma
  12. ρ \rho
  13. β \beta
  14. J ( x , x ¨ , x ˙ , x ) = 0 J\left(\overset{...}{x},\ddot{x},\dot{x},x\right)=0
  15. x x
  16. d 3 x d t 3 + A d 2 x d t 2 + d x d t - | x | + 1 = 0. \frac{\mathrm{d}^{3}x}{\mathrm{d}t^{3}}+A\frac{\mathrm{d}^{2}x}{\mathrm{d}t^{2% }}+\frac{\mathrm{d}x}{\mathrm{d}t}-|x|+1=0.
  17. R A = R / A = 5 R / 3 R_{A}=R/A=5R/3
  18. 1 / 2 π R C 1/2\pi RC

Characteristic_impedance.html

  1. Z 0 = R + j ω L G + j ω C Z_{0}=\sqrt{\frac{R+j\omega L}{G+j\omega C}}
  2. R R
  3. L L
  4. G G
  5. C C
  6. j j
  7. ω \omega
  8. V + I + = Z 0 = - V - I - \frac{V^{+}}{I^{+}}=Z_{0}=-\frac{V^{-}}{I^{-}}
  9. + +
  10. - -
  11. Z 0 = L C Z_{0}=\sqrt{\frac{L}{C}}
  12. 𝑆𝐼𝐿 = V LL 2 Z 0 \mathit{SIL}=\frac{{V_{\mathrm{LL}}}^{2}}{Z_{0}}
  13. V LL V_{\mathrm{LL}}

Characteristic_subgroup.html

  1. φ ( H ) = H \varphi(H)=H
  2. H char G . H\mathrel{{}\operatorname{char}{}}G.
  3. x g x g - 1 x\mapsto gxg^{-1}
  4. V = { e , a , b , a b } V=\left\{e,a,b,ab\right\}
  5. T ( e ) = e , T ( a ) = b , T ( b ) = a , T ( a b ) = a b T(e)=e,T(a)=b,T(b)=a,T(ab)=ab
  6. H char G H\,\mathrm{char}\,G
  7. Aut G Aut G / H \mbox{Aut}~{}\,G\to\mbox{Aut}~{}\,G/H
  8. End G End G / H \mbox{End}~{}\,G\to\mbox{End}~{}\,G/H

Charge-coupled_device.html

  1. g = ( 1 + P ) N g=(1+P)^{N}
  2. P ( n ) = ( n - m + 1 ) m - 1 ( m - 1 ) ! ( g - 1 + 1 m ) m exp ( - n - m + 1 g - 1 + 1 m ) P\left(n\right)=\frac{\left(n-m+1\right)^{m-1}}{\left(m-1\right)!\left(g-1+% \frac{1}{m}\right)^{m}}\exp\left(-\frac{n-m+1}{g-1+\frac{1}{m}}\right)
  3. n m n\geq m

Charles_Sanders_Peirce.html

  1. \therefore
  2. \therefore
  3. \therefore

Chemical_affinity.html

  1. A = - ( G ξ ) P , T . A=-\left(\frac{\partial G}{\partial\xi}\right)_{P,T}.
  2. A = - Δ r G . A=-\Delta_{r}G.\,
  3. A = d Q d ξ . A=\frac{{\mathrm{d}}Q^{\prime}}{{\mathrm{d}}\xi}.\,

Chemical_equilibrium.html

  1. α A + β B σ S + τ T \alpha A+\beta B\rightleftharpoons\sigma S+\tau T
  2. forward reaction rate = k + A α B β \mbox{forward reaction rate}~{}=k_{+}{A}^{\alpha}{B}^{\beta}\,\!
  3. backward reaction rate = k - S σ T τ \mbox{backward reaction rate}~{}=k_{-}{S}^{\sigma}{T}^{\tau}\,\!
  4. k + { A } α { B } β = k - { S } σ { T } τ k_{+}\left\{A\right\}^{\alpha}\left\{B\right\}^{\beta}=k_{-}\left\{S\right\}^{% \sigma}\left\{T\right\}^{\tau}\,
  5. K c = k + k - = { S } σ { T } τ { A } α { B } β K_{c}=\frac{k_{+}}{k_{-}}=\frac{\{S\}^{\sigma}\{T\}^{\tau}}{\{A\}^{\alpha}\{B% \}^{\beta}}
  6. K = { C H 3 C O 2 - } { H 3 O + } { C H 3 C O 2 H } K=\frac{\{CH_{3}CO_{2}^{-}\}\{H_{3}O^{+}\}}{\{CH_{3}CO_{2}H\}}
  7. Δ r G = - R T ln K e q \Delta_{r}G^{\ominus}=-RT\ln K_{eq}
  8. K c = [ S ] σ [ T ] τ [ A ] α [ B ] β K_{c}=\frac{[S]^{\sigma}[T]^{\tau}}{[A]^{\alpha}[B]^{\beta}}
  9. ( d G d ξ ) T , p = 0 \left(\frac{dG}{d\xi}\right)_{T,p}=0~{}
  10. α A + β B σ S + τ T \alpha A+\beta B\rightleftharpoons\sigma S+\tau T
  11. α μ A + β μ B = σ μ S + τ μ T \alpha\mu_{A}+\beta\mu_{B}=\sigma\mu_{S}+\tau\mu_{T}\,
  12. μ A = μ A + R T ln { A } \mu_{A}=\mu_{A}^{\ominus}+RT\ln\{A\}\,
  13. μ A \mu_{A}^{\ominus}~{}
  14. d G = V d p - S d T + i = 1 k μ i d N i dG=Vdp-SdT+\sum_{i=1}^{k}\mu_{i}dN_{i}
  15. d N i = ν i d ξ dN_{i}=\nu_{i}d\xi\,
  16. ν i \nu_{i}~{}
  17. d ξ d\xi~{}
  18. ( d G d ξ ) T , p = i = 1 k μ i ν i = Δ r G T , p \left(\frac{dG}{d\xi}\right)_{T,p}=\sum_{i=1}^{k}\mu_{i}\nu_{i}=\Delta_{r}G_{T% ,p}
  19. Δ r G T , p = σ μ S + τ μ T - α μ A - β μ B \Delta_{r}G_{T,p}=\sigma\mu_{S}+\tau\mu_{T}-\alpha\mu_{A}-\beta\mu_{B}\,
  20. Δ r G T , p = ( σ μ S + τ μ T ) - ( α μ A + β μ B ) + ( σ R T ln { S } + τ R T ln { T } ) - ( α R T ln { A } + β R T ln { B } ) \Delta_{r}G_{T,p}=(\sigma\mu_{S}^{\ominus}+\tau\mu_{T}^{\ominus})-(\alpha\mu_{% A}^{\ominus}+\beta\mu_{B}^{\ominus})+(\sigma RT\ln\{S\}+\tau RT\ln\{T\})-(% \alpha RT\ln\{A\}+\beta RT\ln\{B\})
  21. Δ r G T , p = i = 1 k μ i ν i + R T ln { S } σ { T } τ { A } α { B } β \Delta_{r}G_{T,p}=\sum_{i=1}^{k}\mu_{i}^{\ominus}\nu_{i}+RT\ln\frac{\{S\}^{% \sigma}\{T\}^{\tau}}{\{A\}^{\alpha}\{B\}^{\beta}}
  22. i = 1 k μ i ν i = Δ r G \sum_{i=1}^{k}\mu_{i}^{\ominus}\nu_{i}=\Delta_{r}G^{\ominus}
  23. Q r = { S } σ { T } τ { A } α { B } β Q_{r}=\frac{\{S\}^{\sigma}\{T\}^{\tau}}{\{A\}^{\alpha}\{B\}^{\beta}}
  24. ( d G d ξ ) T , p = Δ r G T , p = Δ r G + R T ln Q r \left(\frac{dG}{d\xi}\right)_{T,p}=\Delta_{r}G_{T,p}=\Delta_{r}G^{\ominus}+RT% \ln Q_{r}
  25. ( d G d ξ ) T , p = Δ r G T , p = 0 \left(\frac{dG}{d\xi}\right)_{T,p}=\Delta_{r}G_{T,p}=0
  26. 0 = Δ r G + R T ln K e q 0=\Delta_{r}G^{\ominus}+RT\ln K_{eq}
  27. Δ r G = - R T ln K e q \Delta_{r}G^{\ominus}=-RT\ln K_{eq}
  28. Q r = K e q Q_{r}=K_{eq}~{}
  29. ξ = ξ e q \xi=\xi_{eq}~{}
  30. Q r K e q Q_{r}\neq K_{eq}~{}
  31. ( d G d ξ ) T , p = Δ r G + R T ln Q r \left(\frac{dG}{d\xi}\right)_{T,p}=\Delta_{r}G^{\ominus}+RT\ln Q_{r}~{}
  32. Δ r G = - R T ln K e q \Delta_{r}G^{\ominus}=-RT\ln K_{eq}~{}
  33. ( d G d ξ ) T , p = R T ln ( Q r K e q ) \left(\frac{dG}{d\xi}\right)_{T,p}=RT\ln\left(\frac{Q_{r}}{K_{eq}}\right)~{}
  34. i i~{}
  35. Q r = ( a j ) ν j ( a i ) ν i Q_{r}=\frac{\prod(a_{j})^{\nu_{j}}}{\prod(a_{i})^{\nu_{i}}}~{}
  36. Q r < K e q Q_{r}<K_{eq}~{}
  37. ( d G d ξ ) T , p < 0 \left(\frac{dG}{d\xi}\right)_{T,p}<0~{}
  38. j j~{}
  39. Q r > K e q Q_{r}>K_{eq}~{}
  40. ( d G d ξ ) T , p > 0 \left(\frac{dG}{d\xi}\right)_{T,p}>0~{}
  41. K = [ S ] σ [ T ] τ [ A ] α [ B ] β × γ S σ γ T τ γ A α γ B β = K c Γ K=\frac{{[S]}^{\sigma}{[T]}^{\tau}...}{{[A]}^{\alpha}{[B]}^{\beta}...}\times% \frac{{\gamma_{S}}^{\sigma}{\gamma_{T}}^{\tau}...}{{\gamma_{A}}^{\alpha}{% \gamma_{B}}^{\beta}...}=K_{c}\Gamma
  42. μ = μ Θ + R T ln ( f b a r ) = μ Θ + R T ln ( p b a r ) + R T ln γ \mu=\mu^{\Theta}+RT\ln\left(\frac{f}{bar}\right)=\mu^{\Theta}+RT\ln\left(\frac% {p}{bar}\right)+RT\ln\gamma
  43. I = 1 2 i = 1 N c i z i 2 I=\frac{1}{2}\sum_{i=1}^{N}c_{i}z_{i}^{2}
  44. K c = K Γ K_{c}=\frac{K}{\Gamma}
  45. \rightleftharpoons
  46. \rightleftharpoons
  47. K c = [ C H 3 C O 2 - ] [ H 3 O + ] [ C H 3 C O 2 H ] [ H 2 O ] K_{c}=\frac{[{CH_{3}CO_{2}}^{-}][{H_{3}O}^{+}]}{[{CH_{3}CO_{2}H}][{H_{2}O}]}
  48. K = [ C H 3 C O 2 - ] [ H 3 O + ] [ C H 3 C O 2 H ] = K c K=\frac{[{CH_{3}CO_{2}}^{-}][{H_{3}O}^{+}]}{[{CH_{3}CO_{2}H}]}=K_{c}
  49. H 2 O + H 2 O H 3 O + + O H - H_{2}O+H_{2}O\rightleftharpoons H_{3}O^{+}+OH^{-}
  50. K w = [ H + ] [ O H - ] K_{w}=[H^{+}][OH^{-}]\,
  51. 2 C O C O 2 + C 2CO\rightleftharpoons CO_{2}+C
  52. K c = [ C O 2 ] [ C O ] 2 K_{c}=\frac{[CO_{2}]}{[CO]^{2}}
  53. H 2 A H A - + H + : K 1 = [ H A - ] [ H + ] [ H 2 A ] H_{2}A\rightleftharpoons HA^{-}+H^{+}:K_{1}=\frac{[HA^{-}][H^{+}]}{[H_{2}A]}
  54. H A - A 2 - + H + : K 2 = [ A 2 - ] [ H + ] [ H A - ] HA^{-}\rightleftharpoons A^{2-}+H^{+}:K_{2}=\frac{[A^{2-}][H^{+}]}{[HA^{-}]}
  55. β D \beta_{D}
  56. H 2 A A 2 - + 2 H + : β D = [ A 2 - ] [ H + ] 2 [ H 2 A ] = K 1 K 2 H_{2}A\rightleftharpoons A^{2-}+2H^{+}:\beta_{D}=\frac{[A^{2-}][H^{+}]^{2}}{[H% _{2}A]}=K_{1}K_{2}
  57. A 2 - + H + H A - : β 1 = [ H A - ] [ A 2 - ] [ H + ] A^{2-}+H^{+}\rightleftharpoons HA^{-}:\beta_{1}=\frac{[HA^{-}]}{[A^{2-}][H^{+}]}
  58. A 2 - + 2 H + H 2 A : β 2 = [ H 2 A ] [ A 2 - ] [ H + ] 2 A^{2-}+2H^{+}\rightleftharpoons H_{2}A:\beta_{2}=\frac{[H_{2}A]}{[A^{2-}][H^{+% }]^{2}}
  59. d ln K d T = Δ H m Θ R T 2 \frac{d\ln K}{dT}=\frac{{\Delta H_{m}}^{\Theta}}{RT^{2}}
  60. d ln K d ( 1 / T ) = - Δ H m Θ R \frac{d\ln K}{d(1/T)}=-\frac{{\Delta H_{m}}^{\Theta}}{R}
  61. T A = [ A ] + [ H A ] + [ H 2 A ] T_{A}=[A]+[HA]+[H_{2}A]\,
  62. T H = [ H ] + [ H A ] + 2 [ H 2 A ] - [ O H ] T_{H}=[H]+[HA]+2[H_{2}A]-[OH]\,
  63. T A = [ A ] + β 1 [ A ] [ H ] + β 2 [ A ] [ H ] 2 T_{A}=[A]+\beta_{1}[A][H]+\beta_{2}[A][H]^{2}\,
  64. T H = [ H ] + β 1 [ A ] [ H ] + 2 β 2 [ A ] [ H ] 2 - K w [ H ] - 1 T_{H}=[H]+\beta_{1}[A][H]+2\beta_{2}[A][H]^{2}-K_{w}[H]^{-1}\,
  65. T A = [ A ] + i p i β i [ A ] p i [ B ] q i T_{A}=[A]+\sum_{i}{p_{i}\beta_{i}[A]^{p_{i}}[B]^{q_{i}}}
  66. T B = [ B ] + i q i β i [ A ] p i [ B ] q i T_{B}=[B]+\sum_{i}{q_{i}\beta_{i}[A]^{p_{i}}[B]^{q_{i}}}
  67. d G = j = 1 m μ j d N j = 0 dG=\sum_{j=1}^{m}\mu_{j}\,dN_{j}=0
  68. j = 1 m a i j N j = b i 0 \sum_{j=1}^{m}a_{ij}N_{j}=b_{i}^{0}
  69. a i j a_{ij}
  70. 𝒢 = G + i = 1 k λ i ( j = 1 m a i j N j - b i 0 ) = 0 \mathcal{G}=G+\sum_{i=1}^{k}\lambda_{i}\left(\sum_{j=1}^{m}a_{ij}N_{j}-b_{i}^{% 0}\right)=0
  71. λ i \lambda_{i}
  72. N j N_{j}
  73. 𝒢 N j = 0 \frac{\partial\mathcal{G}}{\partial N_{j}}=0
  74. 𝒢 λ i = 0 \frac{\partial\mathcal{G}}{\partial\lambda_{i}}=0
  75. N j N_{j}
  76. λ i \lambda_{i}
  77. N j N_{j}

Chemical_reaction.html

  1. a A + b B c C + d D \mathrm{a\ A+b\ B\longrightarrow c\ C+d\ D}
  2. AB A + B \mathrm{AB\longrightarrow A+B}
  3. A + B AB \mathrm{A+B\longrightarrow AB}
  4. HA + B A + HB \mathrm{HA+B\longrightarrow A+HB}
  5. N a C l ( a q ) + A g N O 3 ( a q ) N a N O 3 ( a q ) + A g C l ( s ) NaCl_{(aq)}+AgNO_{3(aq)}\longrightarrow NaNO_{3(aq)}+AgCl_{(s)}
  6. Δ G = Δ H - T Δ S . \mathrm{\Delta G=\Delta H-T\cdot\Delta S}.
  7. 2 C O ( g ) + M o O 2 ( s ) 2 C O 2 ( g ) + M o ( s ) ; Δ H o = + 21.86 kJ at 298 K 2CO_{(g)}+MoO_{2(s)}\longrightarrow 2CO_{2(g)}+Mo_{(s)};\ \mathrm{\Delta H^{o}% =+21.86\ kJ\ at\ 298\ K}
  8. C O ( g ) + H 2 O ( v ) C O 2 ( g ) + H 2 ( g ) CO_{(g)}+H_{2}O_{(v)}\rightleftharpoons CO_{2(g)}+H_{2(g)}
  9. d U = T d S - p d V + μ d n \mathrm{d}U=T\,\mathrm{d}S-p\,\mathrm{d}V+\mu\,\mathrm{d}n\!
  10. v = - d [ A ] d t = k [ A ] . v=-\frac{d[\mathrm{A}]}{dt}=k\cdot[\mathrm{A}].
  11. [ A ] ( t ) = [ A ] 0 e - k t . \mathrm{[A]}(t)=\mathrm{[A]}_{0}\cdot e^{-k\cdot t}.
  12. k = k 0 e - E a / k B T k=k_{0}e^{{-E_{a}}/{k_{B}T}}
  13. A + B A B A+B\longrightarrow AB
  14. 8 F e + S 8 8 F e S 8Fe+S_{8}\longrightarrow 8FeS
  15. A B A + B AB\longrightarrow A+B
  16. 2 H 2 O 2 H 2 + O 2 2H_{2}O\longrightarrow 2H_{2}+O_{2}
  17. A + B C A C + B A+BC\longrightarrow AC+B
  18. M g + 2 H 2 O M g ( O H ) 2 + H 2 Mg+2H_{2}O\longrightarrow Mg(OH)_{2}+H_{2}
  19. A B + C D A D + C B AB+CD\longrightarrow AD+CB
  20. P b ( N O 3 ) 2 + 2 K I P b I 2 + 2 K N O 3 Pb(NO_{3})_{2}+2KI\longrightarrow PbI_{2}+2KNO_{3}
  21. 2 N a ( s ) + C l 2 ( g ) 2 N a C l ( s ) 2Na_{(s)}+Cl_{2(g)}\longrightarrow 2NaCl_{(s)}
  22. H A + B A - + H B + HA+B\rightleftharpoons A^{-}+HB^{+}
  23. X + R - H X - H + R \mathrm{X{\cdot}+R{-}H\longrightarrow X{-}H+R{\cdot}}
  24. R + X 2 R - X + X \mathrm{R{\cdot}+X_{2}\longrightarrow R{-}X+X{\cdot}}

Chemical_thermodynamics.html

  1. Δ U f reactants \Delta{U_{f}^{\circ}}_{\mathrm{reactants}}
  2. Δ U f products \Delta{U_{f}^{\circ}}_{\mathrm{products}}
  3. G = G ( T , P , { N i } ) . G=G(T,P,\{N_{i}\})\,.
  4. d G = - S d T + V d P + i μ i d N i dG=-SdT+VdP+\sum_{i}\mu_{i}dN_{i}\,
  5. μ i = ( G N i ) T , P , N j i , e t c . . \mu_{i}=\left(\frac{\partial G}{\partial N_{i}}\right)_{T,P,N_{j\neq i},etc.}\,.
  6. ( d G ) T , P = i μ i d N i . (dG)_{T,P}=\sum_{i}\mu_{i}dN_{i}\,.
  7. ( d G ) T , P = ( G ξ ) T , P d ξ . (dG)_{T,P}=\left(\frac{\partial G}{\partial\xi}\right)_{T,P}d\xi.\,
  8. ν i = N i / ξ \nu_{i}=\partial N_{i}/\partial\xi\,
  9. ( G ξ ) T , P = i μ i ν i = - 𝔸 \left(\frac{\partial G}{\partial\xi}\right)_{T,P}=\sum_{i}\mu_{i}\nu_{i}=-% \mathbb{A}\,
  10. ( d G ) T , P = - 𝔸 d ξ . (dG)_{T,P}=-\mathbb{A}\,d\xi\,.
  11. ( d G ) T , P = - k 𝔸 k d ξ k . (dG)_{T,P}=-\sum_{k}\mathbb{A}_{k}\,d\xi_{k}\,.
  12. 𝔸 ξ ˙ 0 . \mathbb{A}\ \dot{\xi}\leq 0\,.
  13. d G = - S d T + V d P - k 𝔸 k d ξ k + W dG=-SdT+VdP-\sum_{k}\mathbb{A}_{k}\,d\xi_{k}+W^{\prime}\,
  14. d G T , P = - k 𝔸 k d ξ k + W . dG_{T,P}=-\sum_{k}\mathbb{A}_{k}\,d\xi_{k}+W^{\prime}.\,

Chemisorption.html

  1. Δ G = Δ H - T Δ S \Delta G=\Delta H-T\Delta S
  2. E ( { R i } ) = E e l ( { R i } ) + V ion-ion ( { R i } ) E(\{R_{i}\})=E_{el}(\{R_{i}\})+V_{\,\text{ion-ion}}(\{R_{i}\})
  3. E e l E_{el}
  4. V i o n - i o n V_{ion-ion}

Chemistry.html

  1. × 10 2 3 \times 10^{2}3
  2. e - E / k T e^{-E/kT}
  3. Δ G 0 \Delta G\leq 0\,

Chemotaxis.html

  1. φ \varphi
  2. φ \nabla\varphi
  3. J J
  4. J = χ C φ J=\chi C\nabla\varphi
  5. C C
  6. χ \chi
  7. χ \chi
  8. φ \varphi
  9. χ ( φ ) \chi(\varphi)

Chinese_remainder_theorem.html

  1. x x
  2. { x a 1 ( mod n 1 ) x a k ( mod n k ) \begin{cases}x\equiv a_{1}&\;\;(\mathop{{\rm mod}}n_{1})\\ \quad\cdots\\ x\equiv a_{k}&\;\;(\mathop{{\rm mod}}n_{k})\end{cases}
  3. x x
  4. x y ( mod n i ) , 1 i k x y ( mod N ) . x\equiv y\;\;(\mathop{{\rm mod}}n_{i}),\quad 1\leq i\leq k\qquad% \Longleftrightarrow\qquad x\equiv y\;\;(\mathop{{\rm mod}}N).
  5. x x
  6. a i a j ( mod gcd ( n i , n j ) ) for all i and j a_{i}\equiv a_{j}\;\;(\mathop{{\rm mod}}\gcd(n_{i},n_{j}))\qquad\,\text{for % all }i\,\text{ and }j
  7. x x
  8. n = p 1 r 1 p k r k n=p_{1}^{r_{1}}\cdots p_{k}^{r_{k}}
  9. 𝐙 / n 𝐙 𝐙 / p 1 r 1 𝐙 × × 𝐙 / p k r k 𝐙 \mathbf{Z}/n\mathbf{Z}\cong\mathbf{Z}/p_{1}^{r_{1}}\mathbf{Z}\times\cdots% \times\mathbf{Z}/p_{k}^{r_{k}}\mathbf{Z}
  10. k k
  11. R R
  12. R R
  13. R R
  14. a b a−b
  15. a b a−b
  16. N N
  17. x x
  18. a ( m o d b ) a(modb)
  19. a a
  20. b b
  21. k = 2 k=2
  22. { x a 1 ( mod n 1 ) x a 2 ( mod n 2 ) \begin{cases}x\equiv a_{1}&\;\;(\mathop{{\rm mod}}n_{1})\\ x\equiv a_{2}&\;\;(\mathop{{\rm mod}}n_{2})\end{cases}
  23. n 2 [ n 2 - 1 ] n 1 + n 1 [ n 1 - 1 ] n 2 = 1 n_{2}\left[n_{2}^{-1}\right]_{n_{1}}+n_{1}\left[n_{1}^{-1}\right]_{n_{2}}=1
  24. x x
  25. x = x n 2 [ n 2 - 1 ] n 1 + x n 1 [ n 1 - 1 ] n 2 x=xn_{2}\left[n_{2}^{-1}\right]_{n_{1}}+xn_{1}\left[n_{1}^{-1}\right]_{n_{2}}
  26. x n 2 [ n 2 - 1 ] n 1 1 + x n 1 0 [ n 1 - 1 ] n 2 x × 1 + x × 0 × [ n 1 - 1 ] n 2 x ( mod n 1 ) x\underbrace{n_{2}\left[n_{2}^{-1}\right]_{n_{1}}}_{1}+x\underbrace{n_{1}}_{0}% \left[n_{1}^{-1}\right]_{n_{2}}\equiv x\times 1+x\times 0\times\left[n_{1}^{-1% }\right]_{n_{2}}\equiv x\;\;(\mathop{{\rm mod}}n_{1})
  27. x a 1 n 2 [ n 2 - 1 ] n 1 + a 2 n 1 [ n 1 - 1 ] n 2 x\equiv a_{1}n_{2}\left[n_{2}^{-1}\right]_{n_{1}}+a_{2}n_{1}\left[n_{1}^{-1}% \right]_{n_{2}}
  28. a 1 n 2 [ n 2 - 1 ] n 1 + a 2 n 1 [ n 1 - 1 ] n 2 a 1 × 1 + a 2 × 0 × [ n 1 - 1 ] n 2 a 1 ( mod n 1 ) a_{1}n_{2}\left[n_{2}^{-1}\right]_{n_{1}}+a_{2}n_{1}\left[n_{1}^{-1}\right]_{n% _{2}}\equiv a_{1}\times 1+a_{2}\times 0\times\left[n_{1}^{-1}\right]_{n_{2}}% \equiv a_{1}\;\;(\mathop{{\rm mod}}n_{1})
  29. k k
  30. x := [ i a i N n i [ ( N n i ) - 1 ] n i ] N x:=\left[\sum_{i}a_{i}\frac{N}{n_{i}}\left[\left(\frac{N}{n_{i}}\right)^{-1}% \right]_{n_{i}}\right]_{N}
  31. x x
  32. { x 2 ( mod 3 ) x 3 ( mod 4 ) x 1 ( mod 5 ) \begin{cases}x\equiv 2\;\;(\mathop{{\rm mod}}3)\\ x\equiv 3\;\;(\mathop{{\rm mod}}4)\\ x\equiv 1\;\;(\mathop{{\rm mod}}5)\end{cases}
  33. 3 × 4 × 5 = 60 3×4×5=60
  34. x 11 ( mod 60 ) x\equiv 11\;\;(\mathop{{\rm mod}}60)
  35. t , s t,s
  36. u u
  37. { x = 2 + 3 t x = 3 + 4 s x = 1 + 5 u \begin{cases}x=2+3t\\ x=3+4s\\ x=1+5u\end{cases}
  38. x x
  39. 2 + 3 t \displaystyle 2+3t
  40. t = 3 + 4 s t=3+4s
  41. s s
  42. t t
  43. x = 2 + 3 t = 2 + 3 ( 3 + 4 s ) = 11 + 12 s x=2+3t=2+3(3+4s)=11+12s
  44. x x
  45. 11 + 12 s \displaystyle 11+12s
  46. s = 0 + 5 u s=0+5u
  47. u u
  48. x = 11 + 12 s = 11 + 12 ( 5 u ) = 11 + 60 u x=11+12s=11+12(5u)=11+60u
  49. x a i ( mod n i ) , i = 1 , , k . x\equiv a_{i}\;\;(\mathop{{\rm mod}}n_{i}),\qquad i=1,\cdots,k.
  50. i i
  51. 1 1
  52. j i j≠i
  53. e i { 1 ( mod n i ) 0 ( mod n j ) j i e_{i}\equiv\begin{cases}1\;\;(\mathop{{\rm mod}}n_{i})\\ 0\;\;(\mathop{{\rm mod}}n_{j})&j\neq i\end{cases}
  54. x = i = 1 k a i e i x=\sum_{i=1}^{k}a_{i}e_{i}
  55. x x
  56. { x 2 ( mod 3 ) x 3 ( mod 4 ) x 1 ( mod 5 ) \begin{cases}x\equiv 2&\;\;(\mathop{{\rm mod}}3)\\ x\equiv 3&\;\;(\mathop{{\rm mod}}4)\\ x\equiv 1&\;\;(\mathop{{\rm mod}}5)\end{cases}
  57. x x
  58. ( 13 ) × 3 + 2 × 20 = 1 (−13)×3+2×20=1
  59. x x
  60. ( 11 ) × 4 + 3 × 15 = 1 (−11)×4+3×15=1
  61. x x
  62. 5 × 5 + ( 2 ) × 12 = 1 5×5+(−2)×12=1
  63. x x
  64. 2 × 40 + 3 × 45 + 1 × ( 24 ) = 191 2×40+3×45+1×(−24)=191
  65. e < s u b > 1 = 20 , e 2 = 15 e<sub>1=−20,e_{2}=−15
  66. ( 70 ) 2 + ( 21 ) 3 + ( 15 ) 2 = 233 23 ( m o d 105 ) (70)2+(21)3+(15)2=233≡23(mod105)
  67. R R
  68. R R
  69. R / u R R/uR
  70. f : R / u R R / u 1 R × × R / u k R f ( x + u R ) = ( x + u 1 R , , x + u k R ) \begin{aligned}\displaystyle f:R/uR&\displaystyle\to R/u_{1}R\times\cdots% \times R/u_{k}R\\ \displaystyle f(x+uR)&\displaystyle=(x+u_{1}R,\ldots,x+u_{k}R)\end{aligned}
  71. 𝐙 \mathbf{Z}
  72. x a i ( mod u i ) 1 i k x\equiv a_{i}\;\;(\mathop{{\rm mod}}u_{i})\qquad 1\leq i\leq k
  73. x x
  74. x x
  75. u u
  76. i i
  77. r r
  78. s s
  79. R R
  80. r u i + s u / u i = 1 ru_{i}+su/u_{i}=1
  81. e i δ i j ( mod u j R ) . e_{i}\equiv\delta_{ij}\;\;(\mathop{{\rm mod}}u_{j}R).
  82. g : R / u 1 R × × R / u k R R / u R g ( a 1 + u 1 R , , a k + u k R ) = i = 1 k a i e i + u R \begin{aligned}\displaystyle g:R/u_{1}R\times\cdots\times R/u_{k}R&% \displaystyle\to R/uR\\ \displaystyle g(a_{1}+u_{1}R,\ldots,a_{k}+u_{k}R)&\displaystyle=\sum_{i=1}^{k}% a_{i}e_{i}+uR\end{aligned}
  83. R R
  84. i j i≠j
  85. I I
  86. R / I R/I
  87. f : R / I \displaystyle f:R/I
  88. R R
  89. 2 2
  90. R R
  91. R / ( I 1 I k ) R / I 1 × × R / I k \scriptstyle R/(I_{1}\,\cap\,\cdots\,\cap\,I_{k})\,\simeq\,R/I_{1}\,\times\,% \cdots\,\times\,R/I_{k}
  92. R R
  93. r r
  94. P ( x ) 𝐂 x x P(x)∈\mathbf{C}xx
  95. P ( k ) ( λ j ) = a j , k 1 j r , 0 k < ν j . P^{(k)}(\lambda_{j})=a_{j,k}\qquad 1\leq j\leq r,\quad 0\leq k<\nu_{j}.
  96. A j ( x ) := k = 0 ν j - 1 a j , k k ! ( x - λ j ) k A_{j}(x):=\sum_{k=0}^{\nu_{j}-1}\frac{a_{j,k}}{k!}(x-\lambda_{j})^{k}
  97. r r
  98. P ( x ) A j ( x ) ( mod ( x - λ j ) ν j ) , 1 j r P(x)\equiv A_{j}(x)\;\;(\mathop{{\rm mod}}(x-\lambda_{j})^{\nu_{j}}),\qquad 1% \leq j\leq r
  99. 𝐂 x x \mathbf{C}xx
  100. P ( x ) P(x)
  101. deg ( P ) < n := j ν j . \deg(P)<n:=\sum_{j}\nu_{j}.
  102. Q = i = 1 r ( x - λ i ) ν i Q j = Q ( x - λ j ) ν j \begin{aligned}\displaystyle Q&\displaystyle=\prod_{i=1}^{r}(x-\lambda_{i})^{% \nu_{i}}\\ \displaystyle Q_{j}&\displaystyle=\frac{Q}{(x-\lambda_{j})^{\nu_{j}}}\end{aligned}
  103. 1 Q \frac{1}{Q}
  104. r r
  105. 1 Q = i = 1 r S i ( x - λ i ) ν i \frac{1}{Q}=\sum_{i=1}^{r}\frac{S_{i}}{(x-\lambda_{i})^{\nu_{i}}}
  106. 1 = i = 1 r S i Q i . 1=\sum_{i=1}^{r}S_{i}Q_{i}.
  107. i = 1 r A i S i Q i = A j + i = 1 r ( A i - A j ) S i Q i A j ( mod ( x - λ j ) ν j ) 1 j r \sum_{i=1}^{r}A_{i}S_{i}Q_{i}=A_{j}+\sum_{i=1}^{r}(A_{i}-A_{j})S_{i}Q_{i}% \equiv A_{j}\;\;(\mathop{{\rm mod}}(x-\lambda_{j})^{\nu_{j}})\qquad 1\leq j\leq r
  108. Q Q
  109. n n
  110. M M
  111. k k
  112. k k
  113. i I α i f i = 0 \sum_{i\in I}\alpha_{i}f_{i}=0
  114. k k
  115. k k
  116. k k
  117. k M M kMM
  118. M M
  119. k k
  120. i I α i f i = 0 , \sum_{i\in I}\alpha_{i}f_{i}=0,
  121. i I α i F i = 0. \sum_{i\in I}\alpha_{i}F_{i}=0.
  122. i , j I ; i j i,j∈I;i≠j
  123. k k
  124. k M M kMM
  125. i I i∈I
  126. i j i≠j
  127. ϕ : k [ M ] / K \displaystyle\phi:k[M]/K
  128. K = i I Ker F i = i I Ker F i . K=\prod_{i\in I}\mathrm{Ker}F_{i}=\bigcap_{i\in I}\mathrm{Ker}F_{i}.
  129. Φ : k [ M ] i I k [ M ] / Ker F i Φ ( x ) = ( x + Ker F i ) i I \begin{aligned}\displaystyle\Phi:k[M]&\displaystyle\to\prod_{i\in I}k[M]/% \mathrm{Ker}F_{i}\\ \displaystyle\Phi(x)&\displaystyle=\left(x+\mathrm{Ker}F_{i}\right)_{i\in I}% \end{aligned}
  130. Φ Φ
  131. ψ : k [ M ] \displaystyle\psi:k[M]
  132. i I α i F i = 0 \sum_{i\in I}\alpha_{i}F_{i}=0
  133. i I α i u i = 0 \sum_{i\in I}\alpha_{i}u_{i}=0
  134. ψ ψ
  135. ψ ψ
  136. i I α i u i = 0 \sum_{i\in I}\alpha_{i}u_{i}=0
  137. ( u i ) i I i I k . \left(u_{i}\right)_{i\in I}\in\prod_{i\in I}k.
  138. R R
  139. x x
  140. y y
  141. I I
  142. x x
  143. J J
  144. x y + 1 xy+1
  145. I + J = R I+J=R
  146. I J I J I∩J≠IJ
  147. I I
  148. x x
  149. J J
  150. y = 1 x y=−\frac{1}{x}
  151. p = ( x y + 1 ) x I J p=(xy+1)x∈I∩J
  152. R R
  153. R R
  154. x x
  155. y y
  156. y = x y=x
  157. q J q∈J
  158. q q
  159. y y
  160. q q
  161. y = 1 x y=−\frac{1}{x}
  162. q I J q∈IJ
  163. y y
  164. I J IJ
  165. x x
  166. x x
  167. y y
  168. x x
  169. p = ( x y + 1 ) x I J p=(xy+1)x∉IJ
  170. y y
  171. p p
  172. x y x xyx
  173. x x
  174. I J I J I∩J≠IJ
  175. I + J = R I+J=R
  176. I J = I J + J I I∩J=IJ+JI
  177. I J = ( I J ) ( I + J ) I J + J I I∩J=(I∩J)(I+J)⊂IJ+JI
  178. R R
  179. R / ( I 1 I m ) R / I 1 R / I m R/(I_{1}\cap\cdots\cap I_{m})\to R/I_{1}\oplus\cdots\oplus R/I_{m}
  180. I < s u b > 1 I m I<sub>1∩...∩I_{m}