wpmath0000002_1

Athlon_64.html

  1. CPU speed CPU multiplier DRAM divider = DRAM speed \frac{\mathrm{CPU~{}speed}}{\left\lceil\frac{\mathrm{CPU~{}multiplier}}{% \mathrm{DRAM~{}divider}}\right\rceil}=\mathrm{DRAM~{}speed}

Atomic_units.html

  1. m e m\text{e}
  2. e e
  3. = h / ( 2 π ) \hbar=h/(2\pi)
  4. k e = 1 / ( 4 π ϵ 0 ) k\text{e}=1/(4\pi\epsilon_{0})
  5. 137 137
  6. m = 3.4 m e m=3.4~{}m\text{e}
  7. m = 3.4 a . u . m=3.4~{}\mathrm{a.u.}
  8. L = 3.4 a.u. L=3.4~{}\,\text{a.u.}
  9. m = 3.4 m=3.4
  10. m e = 1 m\text{e}=1
  11. 3.4 m e = 3.4 3.4~{}m\text{e}=3.4
  12. m e \!m_{\mathrm{e}}
  13. e \!e
  14. = h / ( 2 π ) \hbar=h/(2\pi)
  15. k e = 1 / ( 4 π ϵ 0 ) k\text{e}=1/(4\pi\epsilon_{0})
  16. α = e 2 ( 4 π ϵ 0 ) c 1 / 137 \alpha=\frac{e^{2}}{(4\pi\epsilon_{0})\hbar c}\approx 1/137
  17. c \!c
  18. 1 / α 137 \!1/\alpha\approx 137
  19. r e = 1 4 π ϵ 0 e 2 m e c 2 r_{\mathrm{e}}=\frac{1}{4\pi\epsilon_{0}}\frac{e^{2}}{m_{\mathrm{e}}c^{2}}
  20. α 2 5.32 × 10 - 5 \!\alpha^{2}\approx 5.32\times 10^{-5}
  21. m p m_{\mathrm{p}}
  22. m p / m e 1836 m_{\mathrm{p}}/m_{\mathrm{e}}\approx 1836
  23. a 0 \!a_{0}
  24. 4 π ϵ 0 2 / ( m e e 2 ) = / ( m e c α ) 4\pi\epsilon_{0}\hbar^{2}/(m_{\mathrm{e}}e^{2})=\hbar/(m_{\mathrm{e}}c\alpha)
  25. E h \!E_{\mathrm{h}}
  26. m e e 4 / ( 4 π ϵ 0 ) 2 = α 2 m e c 2 m_{\mathrm{e}}e^{4}/(4\pi\epsilon_{0}\hbar)^{2}=\alpha^{2}m_{\mathrm{e}}c^{2}
  27. / E h \hbar/E_{\mathrm{h}}
  28. a 0 E h / = α c a_{0}E_{\mathrm{h}}/\hbar=\alpha c
  29. E h / a 0 \!E_{\mathrm{h}}/a_{0}
  30. E h / k B \!E_{\mathrm{h}}/k_{\mathrm{B}}
  31. E h / a 0 3 E_{\mathrm{h}}/{a_{0}}^{3}
  32. E h / ( e a 0 ) \!E_{\mathrm{h}}/(ea_{0})
  33. E h / e \!E_{\mathrm{h}}/e
  34. e a 0 ea_{0}
  35. e a 0 2 \frac{\hbar}{ea_{0}^{2}}
  36. e a 0 2 \frac{e}{a_{0}^{2}}
  37. μ B = e 2 m e = 1 / 2 \mu\text{B}=\frac{e\hbar}{2m\text{e}}=1/2
  38. μ B = e 2 m e c = α / 2 3.6 × 10 - 3 \mu\text{B}=\frac{e\hbar}{2m\text{e}c}=\alpha/2\approx 3.6\times 10^{-3}
  39. 1 / 2 {1}/{2}
  40. - 2 2 m e 2 ψ ( 𝐫 , t ) + V ( 𝐫 ) ψ ( 𝐫 , t ) = i ψ t ( 𝐫 , t ) -\frac{\hbar^{2}}{2m_{e}}\nabla^{2}\psi(\mathbf{r},t)+V(\mathbf{r})\psi(% \mathbf{r},t)=i\hbar\frac{\partial\psi}{\partial t}(\mathbf{r},t)
  41. - 1 2 2 ψ ( 𝐫 , t ) + V ( 𝐫 ) ψ ( 𝐫 , t ) = i ψ t ( 𝐫 , t ) -\frac{1}{2}\nabla^{2}\psi(\mathbf{r},t)+V(\mathbf{r})\psi(\mathbf{r},t)=i% \frac{\partial\psi}{\partial t}(\mathbf{r},t)
  42. H ^ = - 2 2 m e 2 - 1 4 π ϵ 0 e 2 r \hat{H}=-{{{\hbar^{2}}\over{2m_{e}}}\nabla^{2}}-{1\over{4\pi\epsilon_{0}}}{{e^% {2}}\over{r}}
  43. H ^ = - 1 2 2 - 1 r \hat{H}=-{{{1}\over{2}}\nabla^{2}}-{{1}\over{r}}
  44. 1 / α 137 1/\alpha\approx 137

Attractor.html

  1. f ( t , ( x , v ) ) = ( x + t v , v ) . f(t,(x,v))=(x+tv,v).
  2. x t + 1 = r x t ( 1 - x t ) , x_{t+1}=rx_{t}(1-x_{t}),
  3. x t = a x t - 1 x_{t}=ax_{t-1}
  4. d x / d t = a x dx/dt=ax
  5. f ( x ) = x 3 - 2 x 2 - 11 x + 12 f(x)=x^{3}-2x^{2}-11x+12

Audio_power.html

  1. P avg = 1 T 0 T v ( t ) i ( t ) d t P_{\mathrm{avg}}=\frac{1}{T}\int_{0}^{T}v(t)\cdot i(t)\,dt\,
  2. P avg = V rms I rms P_{\mathrm{avg}}=V_{\mathrm{rms}}\cdot I_{\mathrm{rms}}\,
  3. V rms I rms = V rms 2 R = V peak 2 2 R V_{\mathrm{rms}}\cdot I_{\mathrm{rms}}=\frac{V_{\mathrm{rms}}^{2}}{R}=\frac{V_% {\mathrm{peak}}^{2}}{2R}\,
  4. P avg = ( 6 V ) 2 2 ( 8 Ω ) = 2.25 W P_{\mathrm{avg}}={(6~{}\mathrm{V})^{2}\over 2(8~{}\Omega)}\,\ =2.25~{}\mathrm{W}

Autocatalysis.html

  1. A + B k 2 B A+B\;\stackrel{k}{\rightarrow}\;2B
  2. v = k [ A ] [ B ] \ v=k[A][B]
  3. [ A ] = [ A ] 0 + [ B ] 0 1 + [ B ] 0 [ A ] 0 e ( [ A ] 0 + [ B ] 0 ) k t [A]=\frac{[A]_{0}+[B]_{0}}{1+\frac{[B]_{0}}{[A]_{0}}e^{([A]_{0}+[B]_{0})kt}}
  4. [ B ] = [ A ] 0 + [ B ] 0 1 + [ A ] 0 [ B ] 0 e - ( [ A ] 0 + [ B ] 0 ) k t [B]=\frac{[A]_{0}+[B]_{0}}{1+\frac{[A]_{0}}{[B]_{0}}e^{-([A]_{0}+[B]_{0})kt}}

Autoignition_temperature.html

  1. t i g t_{ig}\,
  2. T i g T_{ig}\,
  3. q ′′ q^{\prime\prime}\,
  4. t i g = ( π 4 ) ( k ρ c ) [ T i g - T o q ′′ ] 2 t_{ig}=\left(\frac{\pi}{4}\right)\left(k\rho c\right)\left[\frac{T_{ig}-T_{o}}% {q^{\prime\prime}}\right]^{2}
  5. T o T_{o}\,
  6. q ′′ q^{\prime\prime}\,

Automata_theory.html

  1. | | ||
  2. \cap
  3. \cap
  4. \cap
  5. \cap
  6. | | ||
  7. | | ||
  8. | | ||
  9. | | ||
  10. | | ||
  11. | | ||
  12. A i A i A_{i}\to A_{i}

Automorphic_number.html

  1. n > 1 n>1
  2. n n^{\prime}
  3. n = ( 3 n 2 - 2 n 3 ) mod 10 2 k . n^{\prime}=(3\cdot n^{2}-2\cdot n^{3})\ \bmod{10^{2k}}\,.
  4. n 0 ( mod 2 k ) , n 1 ( mod 5 k ) , n\equiv 0\;\;(\mathop{{\rm mod}}2^{k}),\quad n\equiv 1\;\;(\mathop{{\rm mod}}5% ^{k})\,,
  5. n 1 ( mod 2 k ) , n 0 ( mod 5 k ) . n\equiv 1\;\;(\mathop{{\rm mod}}2^{k}),\quad n\equiv 0\;\;(\mathop{{\rm mod}}5% ^{k})\,.
  6. k 1000 k\leq 1000
  7. 10 k + 1 10^{k}+1

Autonomous_system_(mathematics).html

  1. d d t x ( t ) = f ( x ( t ) ) \frac{d}{dt}x(t)=f(x(t))
  2. d d t x ( t ) = g ( x ( t ) , t ) \frac{d}{dt}x(t)=g(x(t),t)
  3. x 1 ( t ) x_{1}(t)
  4. d d t x ( t ) = f ( x ( t ) ) , x ( 0 ) = x 0 \frac{d}{dt}x(t)=f(x(t))\,\mathrm{,}\quad x(0)=x_{0}
  5. x 2 ( t ) = x 1 ( t - t 0 ) x_{2}(t)=x_{1}(t-t_{0})
  6. d d t x ( t ) = f ( x ( t ) ) , x ( t 0 ) = x 0 \frac{d}{dt}x(t)=f(x(t))\,\mathrm{,}\quad x(t_{0})=x_{0}
  7. s = t - t 0 s=t-t_{0}
  8. x 1 ( s ) = x 2 ( t ) x_{1}(s)=x_{2}(t)
  9. d s = d t ds=dt
  10. d d t x 2 ( t ) = d d t x 1 ( t - t 0 ) = d d s x 1 ( s ) = f ( x 1 ( s ) ) = f ( x 2 ( t ) ) \frac{d}{dt}x_{2}(t)=\frac{d}{dt}x_{1}(t-t_{0})=\frac{d}{ds}x_{1}(s)=f(x_{1}(s% ))=f(x_{2}(t))
  11. x 2 ( t 0 ) = x 1 ( t 0 - t 0 ) = x 1 ( 0 ) = x 0 x_{2}(t_{0})=x_{1}(t_{0}-t_{0})=x_{1}(0)=x_{0}
  12. y = ( 2 - y ) y y^{\prime}=(2-y)y
  13. x x
  14. ( 2 - y ) y (2-y)y
  15. x x
  16. y ( x ) = y ( x - x 0 ) y(x)=y(x-x_{0})
  17. x 0 x_{0}
  18. y = 0 y=0
  19. y = 2 y=2
  20. C 3 C_{3}
  21. n n
  22. n n
  23. d x d t = f ( x ) \frac{dx}{dt}=f(x)
  24. t + C = d x f ( x ) t+C=\int\frac{dx}{f(x)}
  25. d 2 x d t 2 = f ( x , x ) \frac{d^{2}x}{dt^{2}}=f(x,x^{\prime})
  26. v = d x d t v=\frac{dx}{dt}
  27. x x
  28. d 2 x d t 2 = d v d t = d x d t d v d x = v d v d x \frac{d^{2}x}{dt^{2}}=\frac{dv}{dt}=\frac{dx}{dt}\frac{dv}{dx}=v\frac{dv}{dx}
  29. v d v d x = f ( x , v ) v\frac{dv}{dx}=f(x,v)
  30. t t
  31. v v
  32. x x
  33. v v
  34. d x d t = v ( x ) t + C = d x v ( x ) \frac{dx}{dt}=v(x)\quad\Rightarrow\quad t+C=\int\frac{dx}{v(x)}
  35. f f
  36. x x^{\prime}
  37. d 2 x d t 2 = f ( x ) \frac{d^{2}x}{dt^{2}}=f(x)
  38. d x d t = ( d t d x ) - 1 \frac{dx}{dt}=\left(\frac{dt}{dx}\right)^{-1}
  39. x x
  40. d x d t = f ( x ) d t d x = 1 f ( x ) t + C = d x f ( x ) \frac{dx}{dt}=f(x)\quad\Rightarrow\quad\frac{dt}{dx}=\frac{1}{f(x)}\quad% \Rightarrow\quad t+C=\int\frac{dx}{f(x)}
  41. x x
  42. t t
  43. d 2 x d t 2 \displaystyle\frac{d^{2}x}{dt^{2}}
  44. t t
  45. x x
  46. d 2 x d t 2 = f ( x ) \frac{d^{2}x}{dt^{2}}=f(x)
  47. d d x ( 1 2 ( d t d x ) - 2 ) = f ( x ) \frac{d}{dx}\left(\frac{1}{2}\left(\frac{dt}{dx}\right)^{-2}\right)=f(x)
  48. ( d t d x ) - 2 = 2 f ( x ) d x + C 1 \left(\frac{dt}{dx}\right)^{-2}=2\int f(x)dx+C_{1}
  49. d t d x = ± 1 2 f ( x ) d x + C 1 \frac{dt}{dx}=\pm\frac{1}{\sqrt{2\int f(x)dx+C_{1}}}
  50. t + C 2 = ± d x 2 f ( x ) d x + C 1 t+C_{2}=\pm\int\frac{dx}{\sqrt{2\int f(x)dx+C_{1}}}
  51. d 2 x d t 2 = ( d x d t ) n f ( x ) \frac{d^{2}x}{dt^{2}}=\left(\frac{dx}{dt}\right)^{n}f(x)
  52. n n
  53. x x^{\prime}
  54. - ( d t d x ) - 3 d 2 t d x 2 = ( d t d x ) - n f ( x ) -\left(\frac{dt}{dx}\right)^{-3}\frac{d^{2}t}{dx^{2}}=\left(\frac{dt}{dx}% \right)^{-n}f(x)
  55. - ( d t d x ) n - 3 d 2 t d x 2 = f ( x ) -\left(\frac{dt}{dx}\right)^{n-3}\frac{d^{2}t}{dx^{2}}=f(x)
  56. d d x ( 1 2 - n ( d t d x ) n - 2 ) = f ( x ) \frac{d}{dx}\left(\frac{1}{2-n}\left(\frac{dt}{dx}\right)^{n-2}\right)=f(x)
  57. ( d t d x ) n - 2 = ( 2 - n ) f ( x ) d x + C 1 \left(\frac{dt}{dx}\right)^{n-2}=(2-n)\int f(x)dx+C_{1}
  58. t + C 2 = ( ( 2 - n ) f ( x ) d x + C 1 ) 1 n - 2 d x t+C_{2}=\int\left((2-n)\int f(x)dx+C_{1}\right)^{\frac{1}{n-2}}dx
  59. n n
  60. n = 2 n=2
  61. - ( d t d x ) - 1 d 2 t d x 2 = f ( x ) -\left(\frac{dt}{dx}\right)^{-1}\frac{d^{2}t}{dx^{2}}=f(x)
  62. - d d x ( ln ( d t d x ) ) = f ( x ) -\frac{d}{dx}\left(\ln\left(\frac{dt}{dx}\right)\right)=f(x)
  63. d t d x = C 1 e - f ( x ) d x \frac{dt}{dx}=C_{1}e^{-\int f(x)dx}
  64. t + C 2 = C 1 e - f ( x ) d x d x t+C_{2}=C_{1}\int e^{-\int f(x)dx}dx

Avalanche.html

  1. Pref = 1 2 ρ v 2 \textrm{Pref}=\frac{1}{2}\,{\rho}\,{v^{2}}\,\!

Avalanche_diode.html

  1. i 2 t i^{2}t

Avalanche_photodiode.html

  1. M = 1 1 - 0 L α ( x ) d x M=\frac{1}{1-\int_{0}^{L}\alpha(x)\,dx}
  2. α \alpha
  3. ENF = κ M + ( 2 - 1 M ) ( 1 - κ ) \,\text{ENF}=\kappa M+\left(2-\frac{1}{M}\right)\left(1-\kappa\right)
  4. κ \kappa\,
  5. κ \kappa\,

Avogadro's_law.html

  1. V n V\propto n\,
  2. V n = k \frac{V}{n}=k
  3. V 1 n 1 = V 2 n 2 \frac{V_{1}}{n_{1}}=\frac{V_{2}}{n_{2}}
  4. V n = k \frac{V}{n}=k
  5. p 1 V 1 T 1 n 1 = p 2 V 2 T 2 n 2 = c o n s t a n t \frac{p_{1}\cdot V_{1}}{T_{1}\cdot n_{1}}=\frac{p_{2}\cdot V_{2}}{T_{2}\cdot n% _{2}}=constant
  6. p V = n R T pV=nRT
  7. V m = V n = R T p = ( 8.314 Jmol - 1 K - 1 ) ( 273.15 K ) 101.325 kPa = 22.41 dm 3 mol - 1 = 22.41 liters / mol V_{\rm m}=\frac{V}{n}=\frac{RT}{p}=\frac{(8.314\mathrm{J}\mathrm{mol}^{-1}% \mathrm{K}^{-1})(273.15\mathrm{K})}{101.325\mathrm{kPa}}=22.41\mathrm{dm}^{3}% \mathrm{mol}^{-1}=22.41\mathrm{liters}/\mathrm{mol}

Axial_precession.html

  1. T = 3 G m r 3 ( C - A ) sin δ cos δ ( sin α - cos α 0 ) \overrightarrow{T}=\frac{3Gm}{r^{3}}(C-A)\sin\delta\cos\delta\begin{pmatrix}% \sin\alpha\\ -\cos\alpha\\ 0\end{pmatrix}
  2. T x = 3 2 G m a 3 ( 1 - e 2 ) 3 / 2 ( C - A ) sin ϵ cos ϵ T_{x}=\frac{3}{2}\frac{Gm}{a^{3}(1-e^{2})^{3/2}}(C-A)\sin\epsilon\cos\epsilon
  3. a a
  4. a 3 ( 1 - e 2 ) 3 / 2 a^{3}(1-e^{2})^{3/2}
  5. ϵ \epsilon\,\!
  6. d ψ d t = T x C ω sin ϵ \frac{d\psi}{dt}=\frac{T_{x}}{C\omega\sin\epsilon}
  7. d ψ S d t = 3 2 [ G m a 3 ( 1 - e 2 ) 3 / 2 ] S [ ( C - A ) C cos ϵ ω ] E \frac{d\psi_{S}}{dt}=\frac{3}{2}\left[\frac{Gm}{a^{3}(1-e^{2})^{3/2}}\right]_{% S}\left[\frac{(C-A)}{C}\frac{\cos\epsilon}{\omega}\right]_{E}
  8. d ψ L d t = 3 2 [ G m ( 1 - 1.5 sin 2 i ) a 3 ( 1 - e 2 ) 3 / 2 ] L [ ( C - A ) C cos ϵ ω ] E \frac{d\psi_{L}}{dt}=\frac{3}{2}\left[\frac{Gm(1-1.5\sin^{2}i)}{a^{3}(1-e^{2})% ^{3/2}}\right]_{L}\left[\frac{(C-A)}{C}\frac{\cos\epsilon}{\omega}\right]_{E}
  9. ( 1 - 1.5 sin 2 i ) (1-1.5\sin^{2}i)
  10. e ′′ 2 = a 2 - c 2 a 2 + c 2 e^{\prime\prime 2}=\frac{\mathrm{a}^{2}-\mathrm{c}^{2}}{\mathrm{a}^{2}+\mathrm% {c}^{2}}
  11. ϵ \epsilon\,\!
  12. {}^{−}
  13. + {}^{+}
  14. + {}^{+}
  15. + {}^{+}

Axiom_of_infinity.html

  1. 𝐈 ( 𝐈 and x 𝐈 ( ( x { x } ) 𝐈 ) ) . \exists\mathbf{I}\,(\in\mathbf{I}\,\and\,\forall x\in\mathbf{I}\,(\,(x\cup\{x% \})\in\mathbf{I})).
  2. n ( n 𝐍 ( [ n = k ( n = k { k } ) ] and m n [ m = k n ( m = k { k } ) ] ) ) . \forall n(n\in\mathbf{N}\iff([n=\,\,\,\,\exists k(n=k\cup\{k\})]\,\,\and\,\,% \forall m\in n[m=\,\,\,\,\exists k\in n(m=k\cup\{k\})])).
  3. n ( n 𝐍 ( [ k ( ¬ k n ) k j ( j n ( j k j = k ) ) ] and \forall n(n\in\mathbf{N}\iff([\forall k(\lnot k\in n)\exists k\forall j(j\in n% \iff(j\in k\lor j=k))]\and
  4. m ( m n [ k ( ¬ k m ) k ( k n and j ( j m ( j k j = k ) ) ) ] ) ) ) . \forall m(m\in n\Rightarrow[\forall k(\lnot k\in m)\exists k(k\in n\and\forall j% (j\in m\iff(j\in k\lor j=k)))]))).
  5. Φ ( x ) \Phi(x)
  6. Φ ( x ) = ( x y ( y x ( y { y } x ) ) ) \Phi(x)=(\varnothing\in x\wedge\forall y(y\in x\to(y\cup\{y\}\in x)))
  7. W W
  8. x ( x W I ( Φ ( I ) x I ) ) . \forall x(x\in W\leftrightarrow\forall I(\Phi(I)\to x\in I)).
  9. I I
  10. W = { x I : J ( Φ ( J ) x J ) } W=\{x\in I:\forall J(\Phi(J)\to x\in J)\}
  11. W W
  12. I I
  13. x W x\in W
  14. x x
  15. x x
  16. I I
  17. W W
  18. x x
  19. W W^{\prime}
  20. W W W^{\prime}\subseteq W
  21. W W
  22. W W W\subseteq W^{\prime}
  23. W W^{\prime}
  24. W = W W=W^{\prime}
  25. ω \omega
  26. I ω I\subseteq\omega
  27. ω I \omega\subseteq I
  28. I = ω I=\omega
  29. ω \omega
  30. V ω V_{\omega}\!
  31. 0 \aleph_{0}

Babinet's_principle.html

  1. Z metal Z slot = η 2 4 , Z\text{metal}\,Z\text{slot}=\frac{\eta^{2}}{4},
  2. η \eta
  3. J \vec{J}
  4. A \vec{A}
  5. η = μ ϵ \eta=\sqrt{\frac{\mu}{\epsilon}}

Backtracking.html

  1. \wedge
  2. \wedge
  3. \cdots
  4. \wedge

Baire_space.html

  1. X A X\setminus A
  2. \mathbb{R}
  3. \mathbb{R}
  4. [ 0 , 1 ] [0,1]
  5. \mathbb{R}
  6. m = 1 n = 1 ( r n - 1 2 n + m , r n + 1 2 n + m ) \bigcap_{m=1}^{\infty}\bigcup_{n=1}^{\infty}\left(r_{n}-{1\over 2^{n+m}},r_{n}% +{1\over 2^{n+m}}\right)
  7. { r n } n = 1 \left\{r_{n}\right\}_{n=1}^{\infty}
  8. \mathbb{R}

Baja_California.html

  1. M w M_{\mathrm{w}}

Baker–Campbell–Hausdorff_formula.html

  1. X X
  2. Y Y
  3. e x p exp
  4. l o g log
  5. 𝔤 , \mathfrak{g},
  6. Z = l o g ( e x p ( X ) e x p ( Y ) ) Z=log(exp(X)exp(Y))
  7. X X
  8. Y Y
  9. Z Z
  10. 𝔤 \mathfrak{g}
  11. S S S⊗S
  12. Δ ( X ) = X 1 + 1 X Δ(X)=X⊗1+1⊗X
  13. Δ ( X Y ) = Δ ( X ) Δ ( Y ) Δ(XY)=Δ(X)Δ(Y)
  14. r = e x p ( s ) r=exp(s)
  15. Δ ( r ) = r r Δ(r)=r⊗r
  16. Δ ( s ) = s 1 + 1 s Δ(s)=s⊗1 + 1⊗s
  17. e x p ( X ) e x p ( Y ) exp(X)exp(Y)
  18. l o g ( e x p ( X ) e x p ( Y ) ) log(exp(X)exp(Y))
  19. Z Z
  20. 𝔤 \mathfrak{g}
  21. exp : 𝔤 G \exp:\mathfrak{g}\rightarrow G
  22. log ( exp X exp Y ) = n > 0 ( - 1 ) n - 1 n r i + s i > 0 1 i n ( i = 1 n ( r i + s i ) ) - 1 r 1 ! s 1 ! r n ! s n ! [ X r 1 Y s 1 X r 2 Y s 2 X r n Y s n ] , \log(\exp X\exp Y)=\sum_{n>0}\frac{(-1)^{n-1}}{n}\sum_{\begin{smallmatrix}{r_{% i}+s_{i}>0}\\ {1\leq i\leq n}\end{smallmatrix}}\frac{(\sum_{i=1}^{n}(r_{i}+s_{i}))^{-1}}{r_{% 1}!s_{1}!\cdots r_{n}!s_{n}!}[X^{r_{1}}Y^{s_{1}}X^{r_{2}}Y^{s_{2}}\ldots X^{r_% {n}}Y^{s_{n}}],
  23. s n s_{n}
  24. r n r_{n}
  25. [ X r 1 Y s 1 X r n Y s n ] = [ X , [ X , [ X r 1 , [ Y , [ Y , [ Y s 1 , [ X , [ X , [ X r n , [ Y , [ Y , Y s n ] ] ] ] . [X^{r_{1}}Y^{s_{1}}\ldots X^{r_{n}}Y^{s_{n}}]=[\underbrace{X,[X,\ldots[X}_{r_{% 1}},[\underbrace{Y,[Y,\ldots[Y}_{s_{1}},\,\ldots\,[\underbrace{X,[X,\ldots[X}_% {r_{n}},[\underbrace{Y,[Y,\ldots Y}_{s_{n}}]]\ldots]].
  26. s n > 1 s_{n}>1
  27. s n = 0 s_{n}=0
  28. r n > 1 r_{n}>1
  29. Z ( Y , X ) = Z ( X , Y ) Z(Y,X)=−Z(−X,−Y)
  30. X X , Y XX,Y
  31. X + Y X+Y
  32. X X , Y XX,Y
  33. Y Y
  34. log ( exp X exp Y ) = X + ad X e ad X e ad X - 1 Y + O ( Y 2 ) , \log(\exp X\exp Y)=X+\frac{\,\text{ad}_{X}~{}e^{\operatorname{ad}_{X}}}{e^{% \operatorname{ad}_{X}}-1}~{}Y+O(Y^{2}),
  35. Y Y
  36. X X , Y = = s Y XX,Y==sY
  37. Z = X + s Y / ( 1 e x p ( s ) ) Z=X+sY/ (1 − exp(−s))
  38. e X e Y = e exp ( s ) Y e X , e^{X}e^{Y}=e^{\exp(s)~{}Y}e^{X}~{},
  39. e X e Y e - X = e exp ( s ) Y . e^{X}e^{Y}e^{-X}=e^{\exp(s)~{}Y}~{}.
  40. log ( exp X exp Y ) = X + ( 0 1 ψ ( e ad X e t ad Y ) d t ) Y , \log(\exp X\exp Y)=X+\left(\int^{1}_{0}\psi\left(e^{\operatorname{ad}_{X}}~{}e% ^{t\,\,\text{ad}_{Y}}\right)\,dt\right)\,Y,
  41. ψ ( x ) = def x log x x - 1 = 1 - n = 1 ( 1 - x ) n n ( n + 1 ) , \psi(x)~{}\stackrel{\,\text{def}}{=}~{}\frac{x\log x}{x-1}=1-\sum^{\infty}_{n=% 1}{(1-x)^{n}\over n(n+1)}~{},
  42. ψ ( e y ) = n = 0 B n y n / n ! \psi(e^{y})=\sum_{n=0}^{\infty}B_{n}~{}y^{n}/n!
  43. G \sub GL ( n , ) G\sub\mbox{GL}~{}(n,\mathbb{R})
  44. exp X = e X = n = 0 X n n ! . \exp X=e^{X}=\sum_{n=0}^{\infty}{\frac{X^{n}}{n!}}.
  45. e Z = e X e Y , e^{Z}=e^{X}e^{Y},\,\!
  46. e x p exp
  47. l o g log
  48. Z = n > 0 ( - 1 ) n - 1 n r i + s i > 0 1 i n X r 1 Y s 1 X r n Y s n r 1 ! s 1 ! r n ! s n ! , || X || + || Y || < log 2 , || Z || < log 2. Z=\sum_{n>0}\frac{(-1)^{n-1}}{n}\sum_{\begin{smallmatrix}r_{i}+s_{i}>0\\ 1\leq i\leq n\end{smallmatrix}}\frac{X^{r_{1}}Y^{s_{1}}\cdots X^{r_{n}}Y^{s_{n% }}}{r_{1}!s_{1}!\cdots r_{n}!s_{n}!},\quad||X||+||Y||<\log 2,||Z||<\log 2.
  49. z 1 = X + Y z_{1}=X+Y\,\!
  50. z 2 = 1 2 ( X Y - Y X ) z_{2}=\frac{1}{2}(XY-YX)
  51. z 3 = 1 12 ( X 2 Y + X Y 2 - 2 X Y X + Y 2 X + Y X 2 - 2 Y X Y ) z_{3}=\frac{1}{12}(X^{2}Y+XY^{2}-2XYX+Y^{2}X+YX^{2}-2YXY)
  52. z 4 = 1 24 ( X 2 Y 2 - 2 X Y X Y - Y 2 X 2 + 2 Y X Y X ) . z_{4}=\frac{1}{24}(X^{2}Y^{2}-2XYXY-Y^{2}X^{2}+2YXYX).
  53. e t ( X + Y ) = e t X e t Y e - t 2 2 [ X , Y ] e t 3 6 ( 2 [ Y , [ X , Y ] ] + [ X , [ X , Y ] ] ) e - t 4 24 ( [ [ [ X , Y ] , X ] , X ] + 3 [ [ [ X , Y ] , X ] , Y ] + 3 [ [ [ X , Y ] , Y ] , Y ] ) e^{t(X+Y)}=e^{tX}~{}e^{tY}~{}e^{-\frac{t^{2}}{2}[X,Y]}~{}e^{\frac{t^{3}}{6}(2[% Y,[X,Y]]+[X,[X,Y]])}~{}e^{\frac{-t^{4}}{24}([[[X,Y],X],X]+3[[[X,Y],X],Y]+3[[[X% ,Y],Y],Y])}\cdots
  54. G G
  55. 𝐠 \mathbf{g}
  56. 𝐠 \mathbf{g}
  57. X 𝐠 X∈\mathbf{g}
  58. A G A∈G
  59. 𝐠 \mathbf{g}
  60. Ad e X = e ad X , \operatorname{Ad}_{e^{X}}=e^{\operatorname{ad}_{X}},
  61. Ad e X Y = e X Y e - X = e ad X Y = Y + [ X , Y ] + 1 2 ! [ X , [ X , Y ] ] + 1 3 ! [ X , [ X , [ X , Y ] ] ] + . \operatorname{Ad}_{e^{X}}Y=e^{X}Ye^{-X}=e^{\operatorname{ad}_{X}}Y=Y+\left[X,Y% \right]+\frac{1}{2!}[X,[X,Y]]+\frac{1}{3!}[X,[X,[X,Y]]]+\cdots.
  62. s s
  63. d d s f ( s ) Y = d d s ( e s X Y e - s X ) = X e s X Y e - s X - e s X Y e - s X X = ad X ( e s X Y e - s X ) \frac{d}{ds}f(s)Y=\frac{d}{ds}\left(e^{sX}Ye^{-sX}\right)=Xe^{sX}Ye^{-sX}-e^{% sX}Ye^{-sX}X=\operatorname{ad}_{X}(e^{sX}Ye^{-sX})
  64. f ( s ) = ad X f ( s ) , f ( 0 ) = 1 f ( s ) = e s ad X . f^{\prime}(s)=\operatorname{ad}_{X}f(s),\qquad f(0)=1\qquad\Longrightarrow% \qquad f(s)=e^{s\operatorname{ad}_{X}}.
  65. X , Y X , Y X,YX,Y
  66. X X
  67. Y Y
  68. e s X Y e - s X = Y + s [ X , Y ] . e^{sX}Ye^{-sX}=Y+s[X,Y]~{}.
  69. d g d s = ( X + e s X Y e - s X ) g ( s ) = ( X + Y + s [ X , Y ] ) g ( s ) , \frac{dg}{ds}=\Bigl(X+e^{sX}Ye^{-sX}\Bigr)g(s)=(X+Y+s[X,Y])~{}g(s)~{},
  70. g ( s ) = e s ( X + Y ) + s 2 2 [ X , Y ] , g(s)=e^{s(X+Y)+\frac{s^{2}}{2}[X,Y]}~{},
  71. e X e Y = e X + Y + 1 2 [ X , Y ] . e^{X}e^{Y}=e^{X+Y+\frac{1}{2}[X,Y]}~{}.
  72. X , Y X , Y X,YX,Y
  73. e X e Y = e ( Y + [ X , Y ] + 1 2 ! [ X , [ X , Y ] ] + 1 3 ! [ X , [ X , [ X , Y ] ] ] + ) e X . e^{X}e^{Y}=e^{(Y+\left[X,Y\right]+\frac{1}{2!}[X,[X,Y]]+\frac{1}{3!}[X,[X,[X,Y% ]]]+\cdots)}~{}e^{X}.
  74. â â
  75. â â
  76. e v a ^ - v * a ^ = e v a ^ e - v * a ^ e - | v | 2 / 2 , e^{v\hat{a}^{\dagger}-v^{*}\hat{a}}=e^{v\hat{a}^{\dagger}}e^{-v^{*}\hat{a}}e^{% -|v|^{2}/2},
  77. v v
  78. e v a ^ - v * a ^ e u a ^ - u * a ^ = e ( v + u ) a ^ - ( v * + u * ) a ^ e ( v u * - u v * ) / 2 , e^{v\hat{a}^{\dagger}-v^{*}\hat{a}}e^{u\hat{a}^{\dagger}-u^{*}\hat{a}}=e^{(v+u% )\hat{a}^{\dagger}-(v^{*}+u^{*})\hat{a}}e^{(vu^{*}-uv^{*})/2},
  79. 𝐑 \mathbf{R}
  80. 𝐂 \mathbf{C}
  81. e < s u p > Z = e X e Y e<sup>Z=e^{X}e^{Y}

Ball_(mathematics).html

  1. n n
  2. n n
  3. n n
  4. n n
  5. n n
  6. r r
  7. x x
  8. x x
  9. n n
  10. r r
  11. r r
  12. x x
  13. n n
  14. n n
  15. n n
  16. n n
  17. n n
  18. R R
  19. n n
  20. V n ( R ) = π n / 2 Γ ( n 2 + 1 ) R n , V_{n}(R)=\frac{\pi^{n/2}}{\Gamma(\frac{n}{2}+1)}R^{n},
  21. V 2 k ( R ) = π k k ! R 2 k , V_{2k}(R)=\frac{\pi^{k}}{k!}R^{2k},
  22. V 2 k + 1 ( R ) = 2 k + 1 π k ( 2 k + 1 ) ! ! R 2 k + 1 = 2 ( k ! ) ( 4 π ) k ( 2 k + 1 ) ! R 2 k + 1 . V_{2k+1}(R)=\frac{2^{k+1}\pi^{k}}{(2k+1)!!}R^{2k+1}=\frac{2(k!)(4\pi)^{k}}{(2k% +1)!}R^{2k+1}.
  23. M M
  24. d d
  25. M M
  26. d d
  27. r r
  28. p p
  29. M M
  30. B B
  31. r r
  32. p p
  33. B B
  34. p p
  35. r r
  36. B r ( p ) = { x M d ( x , p ) < r } , B_{r}(p)=\{x\in M\mid d(x,p)<r\},
  37. B B
  38. t t
  39. p p
  40. B B
  41. p p
  42. r r
  43. B r [ p ] = { x M d ( x , p ) r } . B_{r}[p]=\{x\in M\mid d(x,p)\leq r\}.
  44. p p
  45. r r
  46. B B
  47. r r
  48. p p
  49. B r ( p ) ¯ \overline{B_{r}(p)}
  50. B r ( p ) B r ( p ) ¯ B_{r}(p)\subseteq\overline{B_{r}(p)}
  51. B r ( p ) ¯ B r [ p ] \overline{B_{r}(p)}\subseteq B_{r}[p]
  52. B r ( p ) ¯ = B r [ p ] \overline{B_{r}(p)}=B_{r}[p]
  53. X X
  54. B 1 ( p ) ¯ = { p } \overline{B_{1}(p)}=\{p\}
  55. B 1 [ p ] = X B_{1}[p]=X
  56. p X p\in X
  57. d d
  58. V V
  59. d d
  60. x x
  61. y y
  62. x x
  63. y y
  64. B B
  65. r r
  66. p p
  67. B B
  68. r r
  69. p p
  70. p p
  71. \R n \R^{n}
  72. p p
  73. L L
  74. p p
  75. B ( r ) = { x \R n : i = 1 n | x i | p < r p } . B(r)=\left\{x\in\R^{n}\,:\,\sum_{i=1}^{n}\left|x_{i}\right|^{p}<r^{p}\right\}.
  76. n n
  77. L L
  78. L L
  79. p p
  80. n n
  81. L L
  82. L L
  83. L L
  84. p p
  85. p p
  86. X X
  87. R R
  88. n n
  89. n n
  90. X X
  91. n n
  92. X X
  93. n n
  94. X X
  95. X X
  96. n n
  97. n n
  98. n n
  99. n n
  100. n n
  101. ( 0 , 1 ) n \R n (0,1)^{n}\subseteq\R^{n}
  102. n n
  103. n n
  104. n n
  105. n n
  106. m m
  107. n n
  108. m m
  109. n n
  110. B B
  111. n n
  112. B B
  113. n n
  114. n n
  115. 1 \ell_{1}

Ball_bearing.html

  1. ν \nu

Band_gap.html

  1. E g ( T ) = E g ( 0 ) - α T 2 T + β E_{g}(T)=E_{g}(0)-\frac{\alpha T^{2}}{T+\beta}
  2. e ( - Δ E k T ) e^{\left(\frac{-\Delta E}{kT}\right)}

Barcan_formula.html

  1. x F x x F x \forall x\Box Fx\rightarrow\Box\forall xFx
  2. x F x x F x \Diamond\exists xFx\to\exists x\Diamond Fx
  3. x F x x F x \Box\forall xFx\rightarrow\forall x\Box Fx

Bark_scale.html

  1. Bark = 13 arctan ( 0.00076 f ) + 3.5 arctan ( ( f / 7500 ) 2 ) \,\text{Bark}=13\arctan(0.00076f)+3.5\arctan((f/7500)^{2})\,
  2. Critical band rate (bark) = [ ( 26.81 f ) / ( 1960 + f ) ] - 0.53 \,\text{Critical band rate (bark)}=[(26.81f)/(1960+f)]-0.53\,
  3. Critical bandwidth (Hz) = 52548 / ( z 2 - 52.56 z + 690.39 ) \,\text{Critical bandwidth (Hz)}=52548/(z^{2}-52.56z+690.39)\,

Baroclinity.html

  1. p × ρ \nabla p\times\nabla\rho
  2. D ω D t ω t + ( V ) ω = ( ω ) V - ω ( V ) + 1 ρ 2 ρ × p baroclinic contribution \frac{D\vec{\omega}}{Dt}\equiv\frac{\partial\vec{\omega}}{\partial t}+(\vec{V}% \cdot\vec{\nabla})\vec{\omega}=(\vec{\omega}\cdot\vec{\nabla})\vec{V}-\vec{% \omega}(\vec{\nabla}\cdot\vec{V})+\underbrace{\frac{1}{\rho^{2}}\vec{\nabla}% \rho\times\vec{\nabla}p}_{\,\text{baroclinic contribution}}
  3. V \vec{V}
  4. ω \vec{\omega}
  5. p p
  6. ρ \rho
  7. 1 ρ 2 ρ × p \frac{1}{\rho^{2}}\nabla\rho\times\nabla p

Barotropic_vorticity_equation.html

  1. D η D t = 0 , \frac{D\eta}{Dt}=0,
  2. D D t \frac{D}{Dt}
  3. η = ζ + f \eta=\zeta+f
  4. ζ \zeta
  5. f = 2 Ω sin ϕ , f=2\Omega\sin\phi,
  6. Ω \Omega
  7. Ω \Omega
  8. ϕ \phi
  9. D ζ D t = - v β , \frac{D\zeta}{Dt}=-v\beta,
  10. β = f / y \beta=\partial f/\partial y
  11. y y
  12. v v

Barrel_shifter.html

  1. 128 × log 2 ( 128 ) = 128 × 7 = 896 \scriptstyle 128\times\log_{2}(128)=128\times 7=896
  2. 64 × log 2 ( 64 ) = 64 × 6 = 384 \scriptstyle 64\times\log_{2}(64)=64\times 6=384
  3. 32 × log 2 ( 32 ) = 32 × 5 = 160 \scriptstyle 32\times\log_{2}(32)=32\times 5=160
  4. 16 × log 2 ( 16 ) = 16 × 4 = 64 \scriptstyle 16\times\log_{2}(16)=16\times 4=64
  5. 8 × log 2 ( 8 ) = 8 × 3 = 24 \scriptstyle 8\times\log_{2}(8)=8\times 3=24

Barycenter.html

  1. r 1 = a m 2 m 1 + m 2 = a 1 + m 1 / m 2 r_{1}=a\cdot{m_{2}\over m_{1}+m_{2}}={a\over 1+m_{1}/m_{2}}
  2. a R 1 m 2 m 1 {a\over R_{1}}\cdot{m_{2}\over m_{1}}
  3. a R m p l a n e t m > 1 a m p l a n e t > R m 2.3 × 10 11 m E a r t h km 1530 m E a r t h AU {a\over R_{\bigodot}}\cdot{m_{planet}\over m_{\bigodot}}>1\;\Rightarrow\;{a% \cdot m_{planet}}>{R_{\bigodot}\cdot m_{\bigodot}}\approx 2.3\times 10^{11}\;m% _{Earth}\;\mbox{km}~{}\approx 1530\;m_{Earth}\;\mbox{AU}~{}
  4. 1 1 - e > r 1 R 1 > 1 1 + e {1\over{1-e}}>{r_{1}\over R_{1}}>{1\over{1+e}}

Baryonic_dark_matter.html

  1. \sim

Base64.html

  1. 4 n / 3 4\lceil n/3\rceil

Bayesian_network.html

  1. m m
  2. m m
  3. 2 m 2^{m}
  4. 2 m 2^{m}
  5. P ( G , S , R ) = P ( G S , R ) P ( S R ) P ( R ) \mathrm{P}(G,S,R)=\mathrm{P}(G\mid S,R)\mathrm{P}(S\mid R)\mathrm{P}(R)
  6. P ( R = T G = T ) = P ( G = T , R = T ) P ( G = T ) = S { T , F } P ( G = T , S , R = T ) S , R { T , F } P ( G = T , S , R ) \mathrm{P}(\mathit{R}=T\mid\mathit{G}=T)=\frac{\mathrm{P}(\mathit{G}=T,\mathit% {R}=T)}{\mathrm{P}(\mathit{G}=T)}=\frac{\sum_{\mathit{S}\in\{T,F\}}\mathrm{P}(% \mathit{G}=T,\mathit{S},\mathit{R}=T)}{\sum_{\mathit{S},\mathit{R}\in\{T,F\}}% \mathrm{P}(\mathit{G}=T,\mathit{S},\mathit{R})}
  7. P ( G , S , R ) \mathrm{P}(G,S,R)
  8. P ( G = T , S = T , R = T ) = P ( G = T S = T , R = T ) P ( S = T R = T ) P ( R = T ) = 0.99 × 0.01 × 0.2 = 0.00198. \begin{aligned}\displaystyle\mathrm{P}(G=T,&\displaystyle\,S=T,R=T)\\ &\displaystyle=\mathrm{P}(G=T\mid S=T,R=T)\mathrm{P}(S=T\mid R=T)\mathrm{P}(R=% T)\\ &\displaystyle=0.99\times 0.01\times 0.2\\ &\displaystyle=0.00198.\end{aligned}
  9. P ( R = T G = T ) = 0.00198 T T T + 0.1584 T F T 0.00198 T T T + 0.288 T T F + 0.1584 T F T + 0.0 T F F = 891 2491 35.77 % . \begin{aligned}\displaystyle\mathrm{P}(R=T\mid G=T)&\displaystyle=\frac{0.0019% 8_{TTT}+0.1584_{TFT}}{0.00198_{TTT}+0.288_{TTF}+0.1584_{TFT}+0.0_{TFF}}\\ &\displaystyle=\frac{891}{2491}\approx 35.77\%.\end{aligned}
  10. P ( S , R do ( G = T ) ) = P ( S R ) P ( R ) \mathrm{P}(S,R\mid\,\text{do}(G=T))=P(S\mid R)P(R)
  11. P ( G S , R ) \mathrm{P}(G\mid S,R)
  12. P ( R do ( G = T ) ) = P ( R ) \mathrm{P}(R\mid\,\text{do}(G=T))=P(R)
  13. P ( R , G do ( S = T ) ) = P ( R ) P ( G R , S = T ) P(R,G\mid\,\text{do}(S=T))=P(R)P(G\mid R,S=T)
  14. P ( S = T R ) P(S=T\mid R)
  15. do ( x ) \,\text{do}(x)
  16. P ( Y , Z do ( x ) ) = P ( Y , Z , X = x ) / P ( X = x Z ) P(Y,Z\mid\,\text{do}(x))=P(Y,Z,X=x)/P(X=x\mid Z)
  17. 2 10 = 1024 2^{10}=1024
  18. 10 2 3 = 80 10\cdot 2^{3}=80
  19. X Y Z X\rightarrow Y\rightarrow Z
  20. X Y Z X\leftarrow Y\rightarrow Z
  21. X Y Z X\rightarrow Y\leftarrow Z
  22. X X
  23. Z Z
  24. Y Y
  25. X X
  26. Z Z
  27. X X
  28. Z Z
  29. x x\,\!
  30. θ \theta
  31. p ( θ ) p(\theta)
  32. p ( x θ ) p(x\mid\theta)
  33. p ( θ x ) p ( x θ ) p ( θ ) p(\theta\mid x)\propto p(x\mid\theta)p(\theta)
  34. θ \theta
  35. φ \varphi
  36. p ( θ ) p(\theta)
  37. p ( θ φ ) p(\theta\mid\varphi)
  38. p ( φ ) p(\varphi)
  39. φ \varphi
  40. p ( θ , φ | x ) p ( x | θ ) p ( θ | φ ) p ( φ ) . p(\theta,\varphi|x)\propto p(x|\theta)p(\theta|\varphi)p(\varphi).
  41. φ \varphi
  42. ψ \psi\,\!
  43. x 1 , , x n x_{1},\dots,x_{n}\,\!
  44. σ \sigma\,\!
  45. x i N ( θ i , σ 2 ) x_{i}\sim N(\theta_{i},\sigma^{2})
  46. θ i \theta_{i}
  47. θ i \theta_{i}
  48. θ i = x i \theta_{i}=x_{i}
  49. θ i \theta_{i}
  50. x i N ( θ i , σ 2 ) , x_{i}\sim N(\theta_{i},\sigma^{2}),
  51. θ i N ( φ , τ 2 ) \theta_{i}\sim N(\varphi,\tau^{2})
  52. φ \varphi\sim
  53. τ \tau\sim
  54. ( 0 , ) \in(0,\infty)
  55. n 3 n\geq 3
  56. θ i \theta_{i}
  57. τ \tau\,\!
  58. p ( x ) = v V p ( x v | x pa ( v ) ) p(x)=\prod_{v\in V}p\left(x_{v}\,\big|\,x_{\operatorname{pa}(v)}\right)
  59. P ( X 1 = x 1 , , X n = x n ) = v = 1 n P ( X v = x v X v + 1 = x v + 1 , , X n = x n ) \mathrm{P}(X_{1}=x_{1},\ldots,X_{n}=x_{n})=\prod_{v=1}^{n}\mathrm{P}\left(X_{v% }=x_{v}\mid X_{v+1}=x_{v+1},\ldots,X_{n}=x_{n}\right)
  60. P ( X 1 = x 1 , , X n = x n ) = v = 1 n P ( X v = x v X j = x j \mathrm{P}(X_{1}=x_{1},\ldots,X_{n}=x_{n})=\prod_{v=1}^{n}\mathrm{P}(X_{v}=x_{% v}\mid X_{j}=x_{j}
  61. X j X_{j}\,
  62. X v ) X_{v}\,)
  63. X v X V de ( v ) X pa ( v ) for all v V X_{v}\perp\!\!\!\perp X_{V\setminus\operatorname{de}(v)}\mid X_{\operatorname{% pa}(v)}\quad\,\text{for all }v\in V
  64. P ( X v = x v X i = x i \mathrm{P}(X_{v}=x_{v}\mid X_{i}=x_{i}
  65. X i X_{i}\,
  66. X v ) = P ( X v = x v X j = x j X_{v}\,)=P(X_{v}=x_{v}\mid X_{j}=x_{j}
  67. X j X_{j}\,
  68. X v ) X_{v}\,)
  69. X u X v X Z X_{u}\perp\!\!\!\perp X_{v}\mid X_{Z}
  70. a b c and a b c a\longrightarrow b\longrightarrow c\qquad\,\text{and}\qquad a\longleftarrow b\longleftarrow c

Bean_machine.html

  1. ( n k ) {n\choose k}
  2. ( n k ) p k ( 1 - p ) n - k {n\choose k}p^{k}(1-p)^{n-k}

Becquerel.html

  1. m m
  2. m a m_{a}
  3. t 1 / 2 t_{1/2}
  4. A B q = m m a N A ln ( 2 ) t 1 / 2 A_{Bq}=\frac{m}{m_{a}}N_{A}\frac{\ln(2)}{t_{1/2}}
  5. N A N_{A}
  6. × 10 2 3 \times 10^{2}3
  7. A A
  8. A B q = n N A ln ( 2 ) t 1 / 2 A_{Bq}=nN_{A}\frac{\ln(2)}{t_{1/2}}
  9. t 1 / 2 t_{1/2}

Belief.html

  1. S S
  2. P P
  3. P P
  4. S S
  5. P P
  6. S S
  7. P P

Bell_number.html

  1. 30 × 1 = 2 × 15 = 3 × 10 = 5 × 6 = 2 × 3 × 5 30\times 1=2\times 15=3\times 10=5\times 6=2\times 3\times 5
  2. B n + 1 = k = 0 n ( n k ) B k . B_{n+1}=\sum_{k=0}^{n}{\left({{n}\atop{k}}\right)}B_{k}.
  3. ( n k ) {\textstyle\left({{n}\atop{k}}\right)}
  4. B n = k = 0 n { n k } . B_{n}=\sum_{k=0}^{n}\left\{{n\atop k}\right\}.
  5. { n k } \left\{{n\atop k}\right\}
  6. B n + m = k = 0 n j = 0 m { m j } ( n k ) j n - k B k . B_{n+m}=\sum_{k=0}^{n}\sum_{j=0}^{m}\left\{{m\atop j}\right\}{n\choose k}j^{n-% k}B_{k}.
  7. B ( x ) = n = 0 B n n ! x n = e e x - 1 . B(x)=\sum_{n=0}^{\infty}\frac{B_{n}}{n!}x^{n}=e^{e^{x}-1}.
  8. SET ( SET 1 ( 𝒵 ) ) . \mathrm{S\scriptstyle ET}(\mathrm{S\scriptstyle ET}_{\geq 1}(\mathcal{Z})).
  9. 𝒵 \mathcal{Z}
  10. SET 1 \mathrm{S\scriptstyle ET}_{\geq 1}
  11. SET \mathrm{S\scriptstyle ET}
  12. SET \mathrm{S\scriptstyle ET}
  13. B ( x ) = e x B ( x ) B^{\prime}(x)=e^{x}B(x)
  14. B n = 1 e k = 0 k n k ! . B_{n}=\frac{1}{e}\sum_{k=0}^{\infty}\frac{k^{n}}{k!}.
  15. B p + n B n + B n + 1 ( mod p ) B_{p+n}\equiv B_{n}+B_{n+1}\;\;(\mathop{{\rm mod}}p)
  16. B p m + n m B n + B n + 1 ( mod p ) . B_{p^{m}+n}\equiv mB_{n}+B_{n+1}\;\;(\mathop{{\rm mod}}p).
  17. p p - 1 p - 1 \frac{p^{p}-1}{p-1}
  18. B n = n ! 2 π i e γ e e z z n + 1 d z . B_{n}=\frac{n!}{2\pi ie}\int_{\gamma}\frac{e^{e^{z}}}{z^{n+1}}\,dz.
  19. B n < ( 0.792 n ln ( n + 1 ) ) n ; B_{n}<\left(\frac{0.792n}{\ln(n+1)}\right)^{n}~{};
  20. ε > 0 \varepsilon>0
  21. n > n 0 ( ε ) n>n_{0}(\varepsilon)
  22. B n < ( e - 0.6 + ε n ln ( n + 1 ) ) n B_{n}<\left(\frac{e^{-0.6+\varepsilon}n}{\ln(n+1)}\right)^{n}
  23. n 0 ( ε ) = max { e 4 , d - 1 ( ε ) } ~{}n_{0}(\varepsilon)=\max\left\{e^{4},d^{-1}(\varepsilon)\right\}~{}
  24. d ( x ) := ln ln ( x + 1 ) - ln ln x + 1 + e - 1 ln x . ~{}d(x):=\ln\ln(x+1)-\ln\ln x+\frac{1+e^{-1}}{\ln x}\,.
  25. B n 1 n ( n W ( n ) ) n + 1 2 exp ( n W ( n ) - n - 1 ) . B_{n}\sim\frac{1}{\sqrt{n}}\left(\frac{n}{W(n)}\right)^{n+\frac{1}{2}}\exp% \left(\frac{n}{W(n)}-n-1\right).
  26. B n + h = ( n + h ) ! W ( n ) n + h × exp ( e W ( n ) - 1 ) ( 2 π B ) 1 / 2 × ( 1 + P 0 + h P 1 + h 2 P 2 e W ( n ) + Q 0 + h Q 1 + h 2 Q 2 + h 3 Q 3 + h 4 Q 4 e 2 W ( n ) + O ( e - 3 W ( n ) ) ) B_{n+h}=\frac{(n+h)!}{W(n)^{n+h}}\times\frac{\exp(e^{W(n)}-1)}{(2\pi B)^{1/2}}% \times\left(1+\frac{P_{0}+hP_{1}+h^{2}P_{2}}{e^{W(n)}}+\frac{Q_{0}+hQ_{1}+h^{2% }Q_{2}+h^{3}Q_{3}+h^{4}Q_{4}}{e^{2W(n)}}+O(e^{-3W(n)})\right)
  27. h = O ( ln ( n ) ) h=O(\ln(n))
  28. n n\rightarrow\infty
  29. B B
  30. P i P_{i}
  31. Q i Q_{i}
  32. W ( n ) W(n)
  33. ln B n n = ln n - ln ln n - 1 + ln ln n ln n + 1 ln n + 1 2 ( ln ln n ln n ) 2 + O ( ln ln n ( ln n ) 2 ) as n \begin{aligned}\displaystyle\frac{\ln B_{n}}{n}&\displaystyle=\ln n-\ln\ln n-1% +\frac{\ln\ln n}{\ln n}+\frac{1}{\ln n}+\frac{1}{2}\left(\frac{\ln\ln n}{\ln n% }\right)^{2}+O\left(\frac{\ln\ln n}{(\ln n)^{2}}\right)\\ &\displaystyle{}\qquad\,\text{as }n\to\infty\end{aligned}

Bellman–Ford_algorithm.html

  1. | V | - 1 |V|-1
  2. | V | |V|
  3. O ( | V | | E | ) O(|V|\cdot|E|)
  4. | V | |V|
  5. | E | |E|
  6. i i
  7. | V | - 1 |V|-1
  8. | V | - 1 |V|-1
  9. | V | |V|

Bernoulli_distribution.html

  1. 1 - 6 p q p q \frac{1-6pq}{pq}
  2. - q ln ( q ) - p ln ( p ) -q\ln(q)-p\ln(p)\,
  3. q + p e t q+pe^{t}\,
  4. q + p e i t q+pe^{it}\,
  5. q + p z q+pz\,
  6. 1 p ( 1 - p ) \frac{1}{p(1-p)}
  7. p p
  8. q = 1 - p q=1-p
  9. X X
  10. P r ( X = 1 ) = 1 - P r ( X = 0 ) = 1 - q = p . Pr(X=1)=1-Pr(X=0)=1-q=p.\!
  11. p p
  12. 1 - p 1-p
  13. p = 0.5 p=0.5
  14. f f
  15. f ( k ; p ) = { p if k = 1 , 1 - p if k = 0. f(k;p)=\begin{cases}p&\,\text{if }k=1,\\ 1-p&\text{if }k=0.\end{cases}
  16. f ( k ; p ) = p k ( 1 - p ) 1 - k for k { 0 , 1 } . f(k;p)=p^{k}(1-p)^{1-k}\!\quad\,\text{for }k\in\{0,1\}.
  17. X X
  18. E ( X ) = p E\left(X\right)=p
  19. Var ( X ) = p ( 1 - p ) . \textrm{Var}\left(X\right)=p\left(1-p\right).
  20. n = 1 n=1
  21. p p
  22. p = 1 / 2 p=1/2
  23. 0 p 1 0\leq p\leq 1
  24. p p
  25. X 1 , , X n X_{1},\dots,X_{n}
  26. Y = k = 1 n X k B ( n , p ) Y=\sum_{k=1}^{n}X_{k}\sim\mathrm{B}(n,p)
  27. B ( 1 , p ) \mathrm{B}(1,p)

Bernoulli_polynomials.html

  1. B n ( x ) = k = 0 n ( n k ) b n - k x k , B_{n}(x)=\sum_{k=0}^{n}{n\choose k}b_{n-k}x^{k},
  2. t e x t e t - 1 = n = 0 B n ( x ) t n n ! . \frac{te^{xt}}{e^{t}-1}=\sum_{n=0}^{\infty}B_{n}(x)\frac{t^{n}}{n!}.
  3. 2 e x t e t + 1 = n = 0 E n ( x ) t n n ! . \frac{2e^{xt}}{e^{t}+1}=\sum_{n=0}^{\infty}E_{n}(x)\frac{t^{n}}{n!}.
  4. B n ( x ) = D e D - 1 x n B_{n}(x)={D\over e^{D}-1}x^{n}
  5. a x B n ( u ) d u = B n + 1 ( x ) - B n + 1 ( a ) n + 1 . \int_{a}^{x}B_{n}(u)~{}du=\frac{B_{n+1}(x)-B_{n+1}(a)}{n+1}~{}.
  6. x x + 1 B n ( u ) d u = x n . \int_{x}^{x+1}B_{n}(u)\,du=x^{n}.
  7. ( T f ) ( x ) = x x + 1 f ( u ) d u (Tf)(x)=\int_{x}^{x+1}f(u)\,du
  8. ( T f ) ( x ) = e D - 1 D f ( x ) = n = 0 D n ( n + 1 ) ! f ( x ) = f ( x ) + f ( x ) 2 + f ′′ ( x ) 6 + f ′′′ ( x ) 24 + . \begin{aligned}\displaystyle(Tf)(x)={e^{D}-1\over D}f(x)&\displaystyle{}=\sum_% {n=0}^{\infty}{D^{n}\over(n+1)!}f(x)\\ &\displaystyle{}=f(x)+{f^{\prime}(x)\over 2}+{f^{\prime\prime}(x)\over 6}+{f^{% \prime\prime\prime}(x)\over 24}+\cdots~{}.\end{aligned}
  9. B m ( x ) = n = 0 m 1 n + 1 k = 0 n ( - 1 ) k ( n k ) ( x + k ) m . B_{m}(x)=\sum_{n=0}^{m}\frac{1}{n+1}\sum_{k=0}^{n}(-1)^{k}{n\choose k}(x+k)^{m}.
  10. B n ( x ) = - n ζ ( 1 - n , x ) B_{n}(x)=-n\zeta(1-n,x)
  11. Δ n x m = k = 0 n ( - 1 ) n - k ( n k ) ( x + k ) m \Delta^{n}x^{m}=\sum_{k=0}^{n}(-1)^{n-k}{n\choose k}(x+k)^{m}
  12. B m ( x ) = n = 0 m ( - 1 ) n n + 1 Δ n x m . B_{m}(x)=\sum_{n=0}^{m}\frac{(-1)^{n}}{n+1}\Delta^{n}x^{m}.
  13. Δ = e D - 1 \Delta=e^{D}-1\,
  14. D e D - 1 = log ( Δ + 1 ) Δ = n = 0 ( - Δ ) n n + 1 . {D\over e^{D}-1}={\log(\Delta+1)\over\Delta}=\sum_{n=0}^{\infty}{(-\Delta)^{n}% \over n+1}.
  15. E m ( x ) = n = 0 m 1 2 n k = 0 n ( - 1 ) k ( n k ) ( x + k ) m . E_{m}(x)=\sum_{n=0}^{m}\frac{1}{2^{n}}\sum_{k=0}^{n}(-1)^{k}{n\choose k}(x+k)^% {m}\,.
  16. E m ( x ) = k = 0 m ( m k ) E k 2 k ( x - 1 2 ) m - k . E_{m}(x)=\sum_{k=0}^{m}{m\choose k}\frac{E_{k}}{2^{k}}\left(x-\frac{1}{2}% \right)^{m-k}\,.
  17. k = 0 x k p = B p + 1 ( x + 1 ) - B p + 1 ( 0 ) p + 1 \sum_{k=0}^{x}k^{p}=\frac{B_{p+1}(x+1)-B_{p+1}(0)}{p+1}
  18. B n = B n ( 0 ) . \textstyle B_{n}=B_{n}(0).
  19. ζ ( - n ) = - 1 n + 1 B n + 1 \textstyle\zeta(-n)=-\frac{1}{n+1}B_{n+1}
  20. n = 0 , 1 , 2 \textstyle n=0,1,2\cdots
  21. B n = B n ( 1 ) . \textstyle B_{n}=B_{n}(1).
  22. n = 0 n=0
  23. B 1 ( 1 ) = 1 / 2 = - B 1 ( 0 ) B_{1}(1)=1/2=-B_{1}(0)
  24. E n = 2 n E n ( 1 / 2 ) . E_{n}=2^{n}E_{n}(1/2).
  25. B 0 ( x ) = 1 B_{0}(x)=1\,
  26. B 1 ( x ) = x - 1 / 2 B_{1}(x)=x-1/2\,
  27. B 2 ( x ) = x 2 - x + 1 / 6 B_{2}(x)=x^{2}-x+1/6\,
  28. B 3 ( x ) = x 3 - 3 2 x 2 + 1 2 x B_{3}(x)=x^{3}-\frac{3}{2}x^{2}+\frac{1}{2}x\,
  29. B 4 ( x ) = x 4 - 2 x 3 + x 2 - 1 30 B_{4}(x)=x^{4}-2x^{3}+x^{2}-\frac{1}{30}\,
  30. B 5 ( x ) = x 5 - 5 2 x 4 + 5 3 x 3 - 1 6 x B_{5}(x)=x^{5}-\frac{5}{2}x^{4}+\frac{5}{3}x^{3}-\frac{1}{6}x\,
  31. B 6 ( x ) = x 6 - 3 x 5 + 5 2 x 4 - 1 2 x 2 + 1 42 . B_{6}(x)=x^{6}-3x^{5}+\frac{5}{2}x^{4}-\frac{1}{2}x^{2}+\frac{1}{42}.\,
  32. E 0 ( x ) = 1 E_{0}(x)=1\,
  33. E 1 ( x ) = x - 1 / 2 E_{1}(x)=x-1/2\,
  34. E 2 ( x ) = x 2 - x E_{2}(x)=x^{2}-x\,
  35. E 3 ( x ) = x 3 - 3 2 x 2 + 1 4 E_{3}(x)=x^{3}-\frac{3}{2}x^{2}+\frac{1}{4}\,
  36. E 4 ( x ) = x 4 - 2 x 3 + x E_{4}(x)=x^{4}-2x^{3}+x\,
  37. E 5 ( x ) = x 5 - 5 2 x 4 + 5 2 x 2 - 1 2 E_{5}(x)=x^{5}-\frac{5}{2}x^{4}+\frac{5}{2}x^{2}-\frac{1}{2}\,
  38. E 6 ( x ) = x 6 - 3 x 5 + 5 x 3 - 3 x . E_{6}(x)=x^{6}-3x^{5}+5x^{3}-3x.\,
  39. B 16 ( x ) = x 16 - 8 x 15 + 20 x 14 - 182 3 x 12 + 572 3 x 10 - 429 x 8 + 1820 3 x 6 - 1382 3 x 4 + 140 x 2 - 3617 510 B_{16}(x)=x^{16}-8x^{15}+20x^{14}-\frac{182}{3}x^{12}+\frac{572}{3}x^{10}-429x% ^{8}+\frac{1820}{3}x^{6}-\frac{1382}{3}x^{4}+140x^{2}-\frac{3617}{510}
  40. M n < 2 n ! ( 2 π ) n M_{n}<\frac{2n!}{(2\pi)^{n}}
  41. M n = 2 ζ ( n ) n ! ( 2 π ) n M_{n}=\frac{2\zeta(n)n!}{(2\pi)^{n}}
  42. ζ ( x ) \zeta(x)
  43. m n > - 2 n ! ( 2 π ) n m_{n}>\frac{-2n!}{(2\pi)^{n}}
  44. m n = - 2 ζ ( n ) n ! ( 2 π ) n . m_{n}=\frac{-2\zeta(n)n!}{(2\pi)^{n}}.
  45. Δ B n ( x ) = B n ( x + 1 ) - B n ( x ) = n x n - 1 , \Delta B_{n}(x)=B_{n}(x+1)-B_{n}(x)=nx^{n-1},\,
  46. Δ E n ( x ) = E n ( x + 1 ) - E n ( x ) = 2 ( x n - E n ( x ) ) . \Delta E_{n}(x)=E_{n}(x+1)-E_{n}(x)=2(x^{n}-E_{n}(x)).\,
  47. B n ( x ) = n B n - 1 ( x ) , B_{n}^{\prime}(x)=nB_{n-1}(x),\,
  48. E n ( x ) = n E n - 1 ( x ) . E_{n}^{\prime}(x)=nE_{n-1}(x).\,
  49. B n ( x + y ) = k = 0 n ( n k ) B k ( x ) y n - k B_{n}(x+y)=\sum_{k=0}^{n}{n\choose k}B_{k}(x)y^{n-k}
  50. E n ( x + y ) = k = 0 n ( n k ) E k ( x ) y n - k E_{n}(x+y)=\sum_{k=0}^{n}{n\choose k}E_{k}(x)y^{n-k}
  51. B n ( 1 - x ) = ( - 1 ) n B n ( x ) , n 0 , B_{n}(1-x)=(-1)^{n}B_{n}(x),\quad n\geq 0,
  52. E n ( 1 - x ) = ( - 1 ) n E n ( x ) E_{n}(1-x)=(-1)^{n}E_{n}(x)\,
  53. ( - 1 ) n B n ( - x ) = B n ( x ) + n x n - 1 (-1)^{n}B_{n}(-x)=B_{n}(x)+nx^{n-1}\,
  54. ( - 1 ) n E n ( - x ) = - E n ( x ) + 2 x n (-1)^{n}E_{n}(-x)=-E_{n}(x)+2x^{n}\,
  55. B n ( 1 2 ) = ( 1 2 n - 1 - 1 ) B n , n 0 from the multiplication theorems below. B_{n}\left(\frac{1}{2}\right)=\left(\frac{1}{2^{n-1}}-1\right)B_{n},\quad n% \geq 0\,\text{ from the multiplication theorems below.}
  56. r + s + t = n r+s+t=n
  57. x + y + z = 1 x+y+z=1
  58. r [ s , t ; x , y ] n + s [ t , r ; y , z ] n + t [ r , s ; z , x ] n = 0 , r[s,t;x,y]_{n}+s[t,r;y,z]_{n}+t[r,s;z,x]_{n}=0,
  59. [ s , t ; x , y ] n = k = 0 n ( - 1 ) k ( s k ) ( t n - k ) B n - k ( x ) B k ( y ) . [s,t;x,y]_{n}=\sum_{k=0}^{n}(-1)^{k}{s\choose k}{t\choose{n-k}}B_{n-k}(x)B_{k}% (y).
  60. B n ( x ) = - n ! ( 2 π i ) n k 0 e 2 π i k x k n = - 2 n ! k = 1 cos ( 2 k π x - n π 2 ) ( 2 k π ) n . B_{n}(x)=-\frac{n!}{(2\pi i)^{n}}\sum_{k\not=0}\frac{e^{2\pi ikx}}{k^{n}}=-2n!% \sum_{k=1}^{\infty}\frac{\cos\left(2k\pi x-\frac{n\pi}{2}\right)}{(2k\pi)^{n}}.
  61. B n ( x ) = - Γ ( n + 1 ) k = 1 exp ( 2 π i k x ) + e i π n exp ( 2 π i k ( 1 - x ) ) ( 2 π i k ) n . B_{n}(x)=-\Gamma(n+1)\sum_{k=1}^{\infty}\frac{\exp(2\pi ikx)+e^{i\pi n}\exp(2% \pi ik(1-x))}{(2\pi ik)^{n}}.
  62. S ν ( x ) = k = 0 sin ( ( 2 k + 1 ) π x ) ( 2 k + 1 ) ν S_{\nu}(x)=\sum_{k=0}^{\infty}\frac{\sin((2k+1)\pi x)}{(2k+1)^{\nu}}
  63. ν > 1 \nu>1
  64. C 2 n ( x ) = ( - 1 ) n 4 ( 2 n - 1 ) ! π 2 n E 2 n - 1 ( x ) C_{2n}(x)=\frac{(-1)^{n}}{4(2n-1)!}\pi^{2n}E_{2n-1}(x)
  65. S 2 n + 1 ( x ) = ( - 1 ) n 4 ( 2 n ) ! π 2 n + 1 E 2 n ( x ) . S_{2n+1}(x)=\frac{(-1)^{n}}{4(2n)!}\pi^{2n+1}E_{2n}(x).
  66. C ν C_{\nu}
  67. S ν S_{\nu}
  68. C ν ( x ) = - C ν ( 1 - x ) C_{\nu}(x)=-C_{\nu}(1-x)
  69. S ν ( x ) = S ν ( 1 - x ) . S_{\nu}(x)=S_{\nu}(1-x).
  70. χ ν \chi_{\nu}
  71. C ν ( x ) = Re χ ν ( e i x ) C_{\nu}(x)=\mbox{Re}~{}\chi_{\nu}(e^{ix})
  72. S ν ( x ) = Im χ ν ( e i x ) . S_{\nu}(x)=\mbox{Im}~{}\chi_{\nu}(e^{ix}).
  73. x n = 1 n + 1 k = 0 n ( n + 1 k ) B k ( x ) x^{n}=\frac{1}{n+1}\sum_{k=0}^{n}{n+1\choose k}B_{k}(x)
  74. x n = E n ( x ) + 1 2 k = 0 n - 1 ( n k ) E k ( x ) . x^{n}=E_{n}(x)+\frac{1}{2}\sum_{k=0}^{n-1}{n\choose k}E_{k}(x).
  75. ( x ) k (x)_{k}
  76. B n + 1 ( x ) = B n + 1 + k = 0 n n + 1 k + 1 { n k } ( x ) k + 1 B_{n+1}(x)=B_{n+1}+\sum_{k=0}^{n}\frac{n+1}{k+1}\left\{\begin{matrix}n\\ k\end{matrix}\right\}(x)_{k+1}
  77. B n = B n ( 0 ) B_{n}=B_{n}(0)
  78. { n k } = S ( n , k ) \left\{\begin{matrix}n\\ k\end{matrix}\right\}=S(n,k)
  79. ( x ) n + 1 = k = 0 n n + 1 k + 1 [ n k ] ( B k + 1 ( x ) - B k + 1 ) (x)_{n+1}=\sum_{k=0}^{n}\frac{n+1}{k+1}\left[\begin{matrix}n\\ k\end{matrix}\right]\left(B_{k+1}(x)-B_{k+1}\right)
  80. [ n k ] = s ( n , k ) \left[\begin{matrix}n\\ k\end{matrix}\right]=s(n,k)
  81. m 1 m≥1
  82. B n ( m x ) = m n - 1 k = 0 m - 1 B n ( x + k m ) B_{n}(mx)=m^{n-1}\sum_{k=0}^{m-1}B_{n}\left(x+\frac{k}{m}\right)
  83. E n ( m x ) = m n k = 0 m - 1 ( - 1 ) k E n ( x + k m ) for m = 1 , 3 , E_{n}(mx)=m^{n}\sum_{k=0}^{m-1}(-1)^{k}E_{n}\left(x+\frac{k}{m}\right)\quad% \mbox{ for }~{}m=1,3,\dots
  84. E n ( m x ) = - 2 n + 1 m n k = 0 m - 1 ( - 1 ) k B n + 1 ( x + k m ) for m = 2 , 4 , E_{n}(mx)=\frac{-2}{n+1}m^{n}\sum_{k=0}^{m-1}(-1)^{k}B_{n+1}\left(x+\frac{k}{m% }\right)\quad\mbox{ for }~{}m=2,4,\dots
  85. a x B n ( t ) d t = B n + 1 ( x ) - B n + 1 ( a ) n + 1 \int_{a}^{x}B_{n}(t)\,dt=\frac{B_{n+1}(x)-B_{n+1}(a)}{n+1}
  86. a x E n ( t ) d t = E n + 1 ( x ) - E n + 1 ( a ) n + 1 \int_{a}^{x}E_{n}(t)\,dt=\frac{E_{n+1}(x)-E_{n+1}(a)}{n+1}
  87. 0 1 B n ( t ) B m ( t ) d t = ( - 1 ) n - 1 m ! n ! ( m + n ) ! B n + m for m , n 1 \int_{0}^{1}B_{n}(t)B_{m}(t)\,dt=(-1)^{n-1}\frac{m!n!}{(m+n)!}B_{n+m}\quad% \mbox{ for }m,n\geq 1
  88. 0 1 E n ( t ) E m ( t ) d t = ( - 1 ) n 4 ( 2 m + n + 2 - 1 ) m ! n ! ( m + n + 2 ) ! B n + m + 2 \int_{0}^{1}E_{n}(t)E_{m}(t)\,dt=(-1)^{n}4(2^{m+n+2}-1)\frac{m!n!}{(m+n+2)!}B_% {n+m+2}
  89. x x
  90. x x
  91. P k ( x ) is continuous for all k 1 \displaystyle P_{k}(x)\,\text{ is continuous for all }k\neq 1

Bernoulli_process.html

  1. 2 = { H , T } . 2=\{H,T\}.
  2. 2 = { H , T } 2=\{H,T\}
  3. Ω = 2 = { H , T } \Omega=2^{\mathbb{N}}=\{H,T\}^{\mathbb{N}}
  4. Ω = 2 \Omega=2^{\mathbb{Z}}
  5. ( Ω , ) (\Omega,\mathcal{F})
  6. \mathcal{F}
  7. { p , 1 - p } \{p,1-p\}
  8. P = { p , 1 - p } P=\{p,1-p\}^{\mathbb{N}}
  9. P = { p , 1 - p } P=\{p,1-p\}^{\mathbb{Z}}
  10. [ ω 1 , ω 2 , ω n ] [\omega_{1},\omega_{2},\cdots\omega_{n}]
  11. 1 , 2 , , n 1,2,\cdots,n
  12. P ( [ ω 1 , ω 2 , , ω n ] ) = p k ( 1 - p ) n - k P([\omega_{1},\omega_{2},\cdots,\omega_{n}])=p^{k}(1-p)^{n-k}
  13. P ( X 1 = ω 1 , X 2 = ω 2 , , X n = ω n ) = p k ( 1 - p ) n - k P(X_{1}=\omega_{1},X_{2}=\omega_{2},\cdots,X_{n}=\omega_{n})=p^{k}(1-p)^{n-k}
  14. X i X_{i}
  15. x i x_{i}
  16. ω i \omega_{i}
  17. lim n p n = 0 \lim_{n\to\infty}p^{n}=0
  18. 0 p < 1 0\leq p<1
  19. ( Ω , , P ) (\Omega,\mathcal{F},P)
  20. E [ X i = H ] = p E[X_{i}=H]=p
  21. X i X_{i}
  22. N ( k , n ) = ( n k ) = n ! k ! ( n - k ) ! N(k,n)={n\choose k}=\frac{n!}{k!(n-k)!}
  23. E [ X i = H k out of n times ] = P ( k , n ) = ( n k ) p k ( 1 - p ) n - k E[X_{i}=H\mbox{ k out of n times}~{}]=P(k,n)={n\choose k}p^{k}(1-p)^{n-k}
  24. n n\to\infty
  25. n ! = 2 π n n n e - n ( 1 + 𝒪 ( 1 n ) ) n!=\sqrt{2\pi n}\;n^{n}e^{-n}\left(1+\mathcal{O}\left(\frac{1}{n}\right)\right)
  26. 2 n 2^{n}
  27. 2 n H 2^{nH}
  28. H 1 H\leq 1
  29. n n\to\infty
  30. H = - p log 2 p - ( 1 - p ) log 2 ( 1 - p ) H=-p\log_{2}p-(1-p)\log_{2}(1-p)
  31. Ω = 2 \Omega=2^{\mathbb{Z}}
  32. T X i = X i + 1 TX_{i}=X_{i+1}
  33. ω Ω \omega\in\Omega
  34. P ( T ω ) = P ( ω ) P(T\omega)=P(\omega)
  35. ( Ω , ) (\Omega,\mathcal{F})
  36. T : T:\mathcal{F}\to\mathcal{F}
  37. ( Ω , ) (\Omega,\mathcal{F})
  38. x = { n : X n ( x ) = 1 } \mathbb{Z}^{x}=\{n\in\mathbb{Z}:X_{n}(x)=1\}\,
  39. x \mathbb{Z}^{x}
  40. \mathbb{N}
  41. x \mathbb{Z}^{x}

Bernoulli_trial.html

  1. p p
  2. q q
  3. p \displaystyle p
  4. p : q p:q
  5. q : p . q:p.
  6. o f o_{f}
  7. o a : o_{a}:
  8. o f = p / q = p / ( 1 - p ) = ( 1 - q ) / q o a = q / p = ( 1 - p ) / p = q / ( 1 - q ) \begin{aligned}\displaystyle o_{f}&\displaystyle=p/q=p/(1-p)=(1-q)/q\\ \displaystyle o_{a}&\displaystyle=q/p=(1-p)/p=q/(1-q)\end{aligned}
  9. o f \displaystyle o_{f}
  10. S : F S:F
  11. F : S . F:S.
  12. p \displaystyle p
  13. n n
  14. p p
  15. B ( n , p ) B(n,p)
  16. k k
  17. B ( n , p ) B(n,p)
  18. P ( k ) = ( n k ) p k q n - k P(k)={n\choose k}p^{k}q^{n-k}
  19. ( n k ) {n\choose k}
  20. p = 1 2 p=\tfrac{1}{2}
  21. q q
  22. q = 1 - p = 1 - 1 2 = 1 2 q=1-p=1-\tfrac{1}{2}=\tfrac{1}{2}
  23. P ( 2 ) \displaystyle P(2)

Bertrand's_postulate.html

  1. n > 3 n>3
  2. p p
  3. n < p < 2 n - 2 n<p<2n-2
  4. n > 1 n>1
  5. p p
  6. n < p < 2 n n<p<2n
  7. p n p_{n}
  8. n n
  9. n 1 n\geq 1
  10. p n + 1 < 2 p n p_{n+1}<2p_{n}
  11. π ( x ) \scriptstyle\pi(x)\,
  12. π ( x ) \scriptstyle\pi(x)\,
  13. x \scriptstyle x\,
  14. π ( x ) - π ( x 2 ) 1 , \pi(x)-\pi(\tfrac{x}{2})\geq 1,\,
  15. x 2. \,x\geq 2.\,
  16. ϑ ( x ) = p = 2 x ln ( p ) \vartheta(x)=\sum_{p=2}^{x}\ln(p)
  17. ϵ > 0 \epsilon>0
  18. n 0 > 0 n_{0}>0
  19. n > n 0 n>n_{0}
  20. p p
  21. n < p < ( 1 + ϵ ) n n<p<(1+\epsilon)n
  22. lim n π ( ( 1 + ϵ ) n ) - π ( n ) n / log n = ϵ , \lim_{n\to\infty}\frac{\pi((1+\epsilon)n)-\pi(n)}{n/\log n}=\epsilon,
  23. π ( ( 1 + ϵ ) n ) - π ( n ) \pi((1+\epsilon)n)-\pi(n)
  24. n n
  25. [ x , x + O ( x 21 / 40 ) ] [x,\,x+O(x^{21/40})]
  26. x x

Best-first_search.html

  1. f ( n ) f(n)

Beta_distribution.html

  1. I x ( α , β ) I_{x}(\alpha,\beta)\!
  2. E [ X ] = α α + β \operatorname{E}[X]=\frac{\alpha}{\alpha+\beta}\!
  3. E [ ln X ] = ψ ( α ) - ψ ( α + β ) \operatorname{E}[\ln X]=\psi(\alpha)-\psi(\alpha+\beta)\!
  4. I 1 2 [ - 1 ] ( α , β ) (in general) α - 1 3 α + β - 2 3 for α , β > 1 \begin{matrix}I_{\frac{1}{2}}^{[-1]}(\alpha,\beta)\,\text{ (in general) }\\ \approx\frac{\alpha-\tfrac{1}{3}}{\alpha+\beta-\tfrac{2}{3}}\,\text{ for }% \alpha,\beta>1\end{matrix}
  5. α - 1 α + β - 2 \frac{\alpha-1}{\alpha+\beta-2}\!
  6. var [ X ] = α β ( α + β ) 2 ( α + β + 1 ) \operatorname{var}[X]=\frac{\alpha\beta}{(\alpha+\beta)^{2}(\alpha+\beta+1)}\!
  7. var [ ln X ] = ψ 1 ( α ) - ψ 1 ( α + β ) \operatorname{var}[\ln X]=\psi_{1}(\alpha)-\psi_{1}(\alpha+\beta)\!
  8. 2 ( β - α ) α + β + 1 ( α + β + 2 ) α β \frac{2\,(\beta-\alpha)\sqrt{\alpha+\beta+1}}{(\alpha+\beta+2)\sqrt{\alpha% \beta}}
  9. 6 [ ( α - β ) 2 ( α + β + 1 ) - α β ( α + β + 2 ) ] α β ( α + β + 2 ) ( α + β + 3 ) \frac{6[(\alpha-\beta)^{2}(\alpha+\beta+1)-\alpha\beta(\alpha+\beta+2)]}{% \alpha\beta(\alpha+\beta+2)(\alpha+\beta+3)}
  10. ln B ( α , β ) - ( α - 1 ) ψ ( α ) - ( β - 1 ) ψ ( β ) + ( α + β - 2 ) ψ ( α + β ) \begin{matrix}\ln B(\alpha,\beta)-(\alpha-1)\psi(\alpha)-(\beta-1)\psi(\beta)% \\ +(\alpha+\beta-2)\psi(\alpha+\beta)\end{matrix}
  11. 1 + k = 1 ( r = 0 k - 1 α + r α + β + r ) t k k ! 1+\sum_{k=1}^{\infty}\left(\prod_{r=0}^{k-1}\frac{\alpha+r}{\alpha+\beta+r}% \right)\frac{t^{k}}{k!}
  12. F 1 1 ( α ; α + β ; i t ) {}_{1}F_{1}(\alpha;\alpha+\beta;i\,t)\!
  13. var [ ln X ] cov [ ln X , ln ( 1 - X ) ] cov [ ln X , ln ( 1 - X ) ] var [ ln ( 1 - X ) ] \begin{matrix}\\ \operatorname{var}[\ln X]&\operatorname{cov}[\ln X,\ln(1-X)]\\ \operatorname{cov}[\ln X,\ln(1-X)]&\operatorname{var}[\ln(1-X)]\end{matrix}
  14. f ( x ; α , β ) \displaystyle f(x;\alpha,\beta)
  15. B B
  16. X Beta ( α , β ) X\sim{\rm Beta}(\alpha,\beta)
  17. X e ( α , β ) X\sim\mathcal{B}e(\alpha,\beta)
  18. X β α , β X\sim{\beta}_{\alpha,\beta}
  19. ( x - 1 ) x f ( x ) + ( α - 1 - ( α + β - 2 ) x ) f ( x ) = 0 (x-1)xf^{\prime}(x)+(\alpha-1-(\alpha+\beta-2)x)f(x)=0
  20. f ( x ) = f ( x ) ( α + β - 2 ) x - ( α - 1 ) ( x - 1 ) x = - x α - 2 ( 1 - x ) β - 2 B ( α , β ) ( ( α + β - 2 ) x - ( α - 1 ) ) \begin{aligned}\displaystyle f^{\prime}(x)&\displaystyle=f(x)\frac{(\alpha+% \beta-2)x-(\alpha-1)}{(x-1)x}\\ &\displaystyle=-\frac{x^{\alpha-2}(1-x)^{\beta-2}}{B(\alpha,\beta)}{((\alpha+% \beta-2)x-(\alpha-1))}\end{aligned}
  21. F ( x ; α , β ) = B ( x ; α , β ) B ( α , β ) = I x ( α , β ) F(x;\alpha,\beta)=\dfrac{B{}(x;\alpha,\beta)}{B{}(\alpha,\beta)}=I_{x}(\alpha,\beta)
  22. B ( x ; α , β ) B(x;\alpha,\beta)
  23. I x ( α , β ) I_{x}(\alpha,\beta)
  24. α - 1 α + β - 2 . \frac{\alpha-1}{\alpha+\beta-2}.
  25. x = I 1 2 [ - 1 ] ( α , β ) x=I_{\frac{1}{2}}^{[-1]}(\alpha,\beta)
  26. I x ( α , β ) = 1 2 I_{x}(\alpha,\beta)=\tfrac{1}{2}
  27. 1 - 2 - 1 β 1-2^{-\frac{1}{\beta}}
  28. 2 - 1 α 2^{-\frac{1}{\alpha}}
  29. lim β 0 median = lim α median = 1 , lim α 0 median = lim β median = 0. \begin{aligned}\displaystyle\lim_{\beta\to 0}\,\text{median}=\lim_{\alpha\to% \infty}\,\text{median}=1,\\ \displaystyle\lim_{\alpha\to 0}\,\text{median}=\lim_{\beta\to\infty}\,\text{% median}=0.\end{aligned}
  30. median α - 1 3 α + β - 2 3 for α , β 1. \,\text{median}\approx\frac{\alpha-\tfrac{1}{3}}{\alpha+\beta-\tfrac{2}{3}}\,% \text{ for }\alpha,\beta\geq 1.
  31. μ = E [ X ] = 0 1 x f ( x ; α , β ) d x = 0 1 x x α - 1 ( 1 - x ) β - 1 B ( α , β ) d x = α α + β = 1 1 + β α \begin{aligned}\displaystyle\mu=\operatorname{E}[X]&\displaystyle=\int_{0}^{1}% xf(x;\alpha,\beta)\,dx\\ &\displaystyle=\int_{0}^{1}x\,\frac{x^{\alpha-1}(1-x)^{\beta-1}}{B(\alpha,% \beta)}\,dx\\ &\displaystyle=\frac{\alpha}{\alpha+\beta}\\ &\displaystyle=\frac{1}{1+\frac{\beta}{\alpha}}\end{aligned}
  32. lim β α 0 μ = 1 \displaystyle\lim_{\frac{\beta}{\alpha}\to 0}\mu=1
  33. lim β 0 μ = lim α μ = 1 \displaystyle\lim_{\beta\to 0}\mu=\lim_{\alpha\to\infty}\mu=1
  34. ln G X = E [ ln X ] \ln G_{X}=\operatorname{E}[\ln X]
  35. E [ ln X ] \displaystyle\operatorname{E}[\ln X]
  36. G X = e E [ ln X ] = e ψ ( α ) - ψ ( α + β ) G_{X}=e^{\operatorname{E}[\ln X]}=e^{\psi(\alpha)-\psi(\alpha+\beta)}
  37. lim β 0 G X = lim α G X = 1 \displaystyle\lim_{\beta\to 0}G_{X}=\lim_{\alpha\to\infty}G_{X}=1
  38. G X α - 1 2 α + β - 1 2 if α , β > 1. G_{X}\approx\frac{\alpha\,-\frac{1}{2}}{\alpha+\beta-\frac{1}{2}}\,\text{ if }% \alpha,\beta>1.
  39. G ( X i Y i ) = G ( X i ) G ( Y i ) G\left(\frac{X_{i}}{Y_{i}}\right)=\frac{G(X_{i})}{G(Y_{i})}
  40. G ( 1 - X ) = e E [ ln ( 1 - X ) ] = e ψ ( β ) - ψ ( α + β ) G_{(1-X)}=e^{\operatorname{E}[\ln(1-X)]}=e^{\psi(\beta)-\psi(\alpha+\beta)}
  41. lim α = β 0 G ( 1 - X ) = 0 \displaystyle\lim_{\alpha=\beta\to 0}G_{(1-X)}=0
  42. lim β 0 G ( 1 - X ) = lim α G ( 1 - X ) = 0 \displaystyle\lim_{\beta\to 0}G_{(1-X)}=\lim_{\alpha\to\infty}G_{(1-X)}=0
  43. G ( 1 - X ) β - 1 2 α + β - 1 2 if α , β > 1. G_{(1-X)}\approx\frac{\beta-\frac{1}{2}}{\alpha+\beta-\frac{1}{2}}\,\text{ if % }\alpha,\beta>1.
  44. G X ( B ( α , β ) ) = G ( 1 - X ) ( B ( β , α ) ) . G_{X}(B(\alpha,\beta))=G_{(1-X)}(B(\beta,\alpha)).
  45. H X \displaystyle H_{X}
  46. lim α 0 H X = undefined \displaystyle\lim_{\alpha\to 0}H_{X}=\,\text{undefined}
  47. H ( 1 - X ) = 1 E [ 1 ( 1 - X ) ] = β - 1 α + β - 1 if β > 1 , & α > 0. H_{(1-X)}=\frac{1}{\operatorname{E}[\frac{1}{(1-X)}]}=\frac{\beta-1}{\alpha+% \beta-1}\,\text{ if }\beta>1,\&\alpha>0.
  48. lim β 0 H ( 1 - X ) = undefined \displaystyle\lim_{\beta\to 0}H_{(1-X)}=\,\text{undefined}
  49. H X ( B ( α , β ) ) = H ( 1 - X ) ( B ( β , α ) ) if α , β > 1. H_{X}(B(\alpha,\beta))=H_{(1-X)}(B(\beta,\alpha))\,\text{ if }\alpha,\beta>1.
  50. var ( X ) = E [ ( X - μ ) 2 ] = α β ( α + β ) 2 ( α + β + 1 ) \operatorname{var}(X)=\operatorname{E}[(X-\mu)^{2}]=\frac{\alpha\beta}{(\alpha% +\beta)^{2}(\alpha+\beta+1)}
  51. var ( X ) = 1 4 ( 2 β + 1 ) , \operatorname{var}(X)=\frac{1}{4(2\beta+1)},
  52. α \displaystyle\alpha
  53. var ( X ) = μ ( 1 - μ ) 1 + ν \operatorname{var}(X)=\frac{\mu(1-\mu)}{1+\nu}
  54. lim β 0 var ( X ) = lim α 0 var ( X ) = lim β var ( X ) = lim α var ( X ) = lim ν var ( X ) = lim μ 0 var ( X ) = lim μ 1 var ( X ) = 0 \displaystyle\lim_{\beta\to 0}\operatorname{var}(X)=\lim_{\alpha\to 0}% \operatorname{var}(X)=\lim_{\beta\to\infty}\operatorname{var}(X)=\lim_{\alpha% \to\infty}\operatorname{var}(X)=\lim_{\nu\to\infty}\operatorname{var}(X)=\lim_% {\mu\to 0}\operatorname{var}(X)=\lim_{\mu\to 1}\operatorname{var}(X)=0
  55. ln var GX \displaystyle\ln\,\operatorname{var_{GX}}
  56. var GX = e var [ ln X ] \operatorname{var_{GX}}=e^{\operatorname{var}[\ln X]}
  57. ln var G ( 1 - X ) \displaystyle\ln\,\operatorname{var_{G(1-X)}}
  58. var [ ln X ] = ψ 1 ( α ) - ψ 1 ( α + β ) \operatorname{var}[\ln X]=\psi_{1}(\alpha)-\psi_{1}(\alpha+\beta)
  59. var [ ln ( 1 - X ) ] = ψ 1 ( β ) - ψ 1 ( α + β ) \operatorname{var}[\ln(1-X)]=\psi_{1}(\beta)-\psi_{1}(\alpha+\beta)
  60. cov [ ln X , ln ( 1 - X ) ] = - ψ 1 ( α + β ) \operatorname{cov}[\ln X,\ln(1-X)]=-\psi_{1}(\alpha+\beta)
  61. ψ 1 ( α ) = d 2 ln Γ ( α ) d α 2 = d ψ ( α ) d α . \psi_{1}(\alpha)=\frac{d^{2}\ln\Gamma(\alpha)}{d\alpha^{2}}=\frac{d\,\psi(% \alpha)}{d\alpha}.
  62. ln var GX = var [ ln X ] = ψ 1 ( α ) - ψ 1 ( α + β ) \ln\,\operatorname{var_{GX}}=\operatorname{var}[\ln X]=\psi_{1}(\alpha)-\psi_{% 1}(\alpha+\beta)
  63. ln var G ( 1 - X ) = var [ ln ( 1 - X ) ] = ψ 1 ( β ) - ψ 1 ( α + β ) \ln\,\operatorname{var_{G(1-X)}}=\operatorname{var}[\ln(1-X)]=\psi_{1}(\beta)-% \psi_{1}(\alpha+\beta)
  64. ln cov GX , ( 1 - X ) = cov [ ln X , ln ( 1 - X ) ] = - ψ 1 ( α + β ) \ln\,\operatorname{cov_{G{X,(1-X)}}}=\operatorname{cov}[\ln X,\ln(1-X)]=-\psi_% {1}(\alpha+\beta)
  65. lim α 0 ln var GX = lim β 0 ln var G ( 1 - X ) = \displaystyle\lim_{\alpha\to 0}\ln\,\operatorname{var_{GX}}=\lim_{\beta\to 0}% \ln\,\operatorname{var_{G(1-X)}}=\infty
  66. lim α ( lim β ln var GX ) = lim β ( lim α ln var G ( 1 - X ) ) = lim α ( lim β 0 ln cov GX , ( 1 - X ) ) = lim β ( lim α 0 ln cov GX , ( 1 - X ) ) = 0 \displaystyle\lim_{\alpha\to\infty}(\lim_{\beta\to\infty}\ln\,\operatorname{% var_{GX}})=\lim_{\beta\to\infty}(\lim_{\alpha\to\infty}\ln\,\operatorname{var_% {G(1-X)}})=\lim_{\alpha\to\infty}(\lim_{\beta\to 0}\ln\,\operatorname{cov_{G{X% ,(1-X)}}})=\lim_{\beta\to\infty}(\lim_{\alpha\to 0}\ln\,\operatorname{cov_{G{X% ,(1-X)}}})=0
  67. ln var GX ( B ( α , β ) ) = ln var G ( 1 - X ) ( B ( β , α ) ) . \ln\,\operatorname{var_{GX}}(B(\alpha,\beta))=\ln\,\operatorname{var_{G(1-X)}}% (B(\beta,\alpha)).
  68. ln cov GX , ( 1 - X ) ( B ( α , β ) ) = ln cov GX , ( 1 - X ) ( B ( β , α ) ) \ln\,\operatorname{cov_{G{X,(1-X)}}}(B(\alpha,\beta))=\ln\,\operatorname{cov_{% G{X,(1-X)}}}(B(\beta,\alpha))
  69. E [ | X - E [ X ] | ] = 2 α α β β B ( α , β ) ( α + β ) α + β + 1 \operatorname{E}[|X-E[X]|]=\frac{2\alpha^{\alpha}\beta^{\beta}}{B(\alpha,\beta% )(\alpha+\beta)^{\alpha+\beta+1}}
  70. mean abs. dev. from mean standard deviation \displaystyle\frac{\,\text{mean abs. dev. from mean}}{\,\text{standard % deviation}}
  71. 2 π \sqrt{\frac{2}{\pi}}
  72. 3 2 \frac{\sqrt{3}}{2}
  73. E [ | X - E [ X ] | ] = 2 μ μ ν ( 1 - μ ) ( 1 - μ ) ν ν B ( μ ν , ( 1 - μ ) ν ) \operatorname{E}[|X-E[X]|]=\frac{2\mu^{\mu\nu}(1-\mu)^{(1-\mu)\nu}}{\nu B(\mu% \nu,(1-\mu)\nu)}
  74. E [ | X - E [ X ] | ] = 2 1 - ν ν B ( ν 2 , ν 2 ) \displaystyle\operatorname{E}[|X-E[X]|]=\frac{2^{1-\nu}}{\nu B(\tfrac{\nu}{2},% \tfrac{\nu}{2})}
  75. lim β 0 E [ | X - E [ X ] | ] \displaystyle\lim_{\beta\to 0}\operatorname{E}[|X-E[X]|]
  76. γ 1 = E [ ( X - μ ) 3 ] ( var ( X ) ) 3 / 2 = 2 ( β - α ) α + β + 1 ( α + β + 2 ) α β . \gamma_{1}=\frac{\operatorname{E}[(X-\mu)^{3}]}{(\operatorname{var}(X))^{3/2}}% =\frac{2(\beta-\alpha)\sqrt{\alpha+\beta+1}}{(\alpha+\beta+2)\sqrt{\alpha\beta% }}.
  77. α \displaystyle\alpha
  78. γ 1 = E [ ( X - μ ) 3 ] ( var ( X ) ) 3 / 2 = 2 ( 1 - 2 μ ) 1 + ν ( 2 + ν ) μ ( 1 - μ ) . \gamma_{1}=\frac{\operatorname{E}[(X-\mu)^{3}]}{(\operatorname{var}(X))^{3/2}}% =\frac{2(1-2\mu)\sqrt{1+\nu}}{(2+\nu)\sqrt{\mu(1-\mu)}}.
  79. γ 1 = E [ ( X - μ ) 3 ] ( var ( X ) ) 3 / 2 = 2 ( 1 - 2 μ ) var μ ( 1 - μ ) + var if var < μ ( 1 - μ ) \gamma_{1}=\frac{\operatorname{E}[(X-\mu)^{3}]}{(\operatorname{var}(X))^{3/2}}% =\frac{2(1-2\mu)\sqrt{\,\text{ var }}}{\mu(1-\mu)+\operatorname{var}}\,\text{ % if }\operatorname{var}<\mu(1-\mu)
  80. ( γ 1 ) 2 = ( E [ ( X - μ ) 3 ] ) 2 ( var ( X ) ) 3 = 4 ( 2 + ν ) 2 ( 1 var - 4 ( 1 + ν ) ) (\gamma_{1})^{2}=\frac{(\operatorname{E}[(X-\mu)^{3}])^{2}}{(\operatorname{var% }(X))^{3}}=\frac{4}{(2+\nu)^{2}}\bigg(\frac{1}{\,\text{var}}-4(1+\nu)\bigg)
  81. var = 1 4 ( 1 + ν ) \operatorname{var}=\frac{1}{4(1+\nu)}
  82. lim α = β 0 γ 1 = lim α = β γ 1 = lim ν 0 γ 1 = lim ν γ 1 = lim μ 1 2 γ 1 = 0 \lim_{\alpha=\beta\to 0}\gamma_{1}=\lim_{\alpha=\beta\to\infty}\gamma_{1}=\lim% _{\nu\to 0}\gamma_{1}=\lim_{\nu\to\infty}\gamma_{1}=\lim_{\mu\to\frac{1}{2}}% \gamma_{1}=0
  83. lim α 0 γ 1 = lim μ 0 γ 1 = \displaystyle\lim_{\alpha\to 0}\gamma_{1}=\lim_{\mu\to 0}\gamma_{1}=\infty
  84. excess kurtosis = kurtosis - 3 = E [ ( X - μ ) 4 ] ( var ( X ) ) 2 - 3 = 6 [ α 3 - α 2 ( 2 β - 1 ) + β 2 ( β + 1 ) - 2 α β ( β + 2 ) ] α β ( α + β + 2 ) ( α + β + 3 ) = 6 [ ( α - β ) 2 ( α + β + 1 ) - α β ( α + β + 2 ) ] α β ( α + β + 2 ) ( α + β + 3 ) . \begin{aligned}\displaystyle\,\text{excess kurtosis}&\displaystyle=\,\text{% kurtosis}-3\\ &\displaystyle=\frac{\operatorname{E}[(X-\mu)^{4}]}{{(\operatorname{var}(X))^{% 2}}}-3\\ &\displaystyle=\frac{6[\alpha^{3}-\alpha^{2}(2\beta-1)+\beta^{2}(\beta+1)-2% \alpha\beta(\beta+2)]}{\alpha\beta(\alpha+\beta+2)(\alpha+\beta+3)}\\ &\displaystyle=\frac{6[(\alpha-\beta)^{2}(\alpha+\beta+1)-\alpha\beta(\alpha+% \beta+2)]}{\alpha\beta(\alpha+\beta+2)(\alpha+\beta+3)}.\end{aligned}
  85. excess kurtosis = - 6 3 + 2 α if α = β \,\text{excess kurtosis}=-\frac{6}{3+2\alpha}\,\text{ if }\alpha=\beta
  86. α \displaystyle\alpha
  87. excess kurtosis = 6 3 + ν ( ( 1 - 2 μ ) 2 ( 1 + ν ) μ ( 1 - μ ) ( 2 + ν ) - 1 ) \,\text{excess kurtosis}=\frac{6}{3+\nu}\bigg(\frac{(1-2\mu)^{2}(1+\nu)}{\mu(1% -\mu)(2+\nu)}-1\bigg)
  88. excess kurtosis = 6 ( 3 + ν ) ( 2 + ν ) ( 1 var - 6 - 5 ν ) if var < μ ( 1 - μ ) \,\text{excess kurtosis}=\frac{6}{(3+\nu)(2+\nu)}\left(\frac{1}{\,\text{ var }% }-6-5\nu\right)\,\text{ if }\,\text{ var }<\mu(1-\mu)
  89. excess kurtosis = 6 var ( 1 - var - 5 μ ( 1 - μ ) ) ( var + μ ( 1 - μ ) ) ( 2 var + μ ( 1 - μ ) ) if var < μ ( 1 - μ ) \,\text{excess kurtosis}=\frac{6\,\text{ var }(1-\,\text{ var }-5\mu(1-\mu))}{% (\,\text{var }+\mu(1-\mu))(2\,\text{ var }+\mu(1-\mu))}\,\text{ if }\,\text{ % var }<\mu(1-\mu)
  90. excess kurtosis = 6 3 + ν ( ( 2 + ν ) 4 ( skewness ) 2 - 1 ) if (skewness) 2 - 2 < excess kurtosis < 3 2 ( skewness ) 2 \,\text{excess kurtosis}=\frac{6}{3+\nu}\bigg(\frac{(2+\nu)}{4}(\,\text{% skewness})^{2}-1\bigg)\,\text{ if (skewness)}^{2}-2<\,\text{excess kurtosis}<% \frac{3}{2}(\,\text{skewness})^{2}
  91. lim ν 0 excess kurtosis = ( skewness ) 2 - 2 lim ν excess kurtosis = 3 2 ( skewness ) 2 \begin{aligned}&\displaystyle\lim_{\nu\to 0}\,\text{excess kurtosis}=(\,\text{% skewness})^{2}-2\\ &\displaystyle\lim_{\nu\to\infty}\,\text{excess kurtosis}=\tfrac{3}{2}(\,\text% {skewness})^{2}\end{aligned}
  92. ( skewness ) 2 - 2 < excess kurtosis < 3 2 ( skewness ) 2 (\,\text{skewness})^{2}-2<\,\text{excess kurtosis}<\tfrac{3}{2}(\,\text{% skewness})^{2}
  93. lim α 0 excess kurtosis = lim β 0 excess kurtosis = lim μ 0 excess kurtosis = lim μ 1 excess kurtosis = \displaystyle\lim_{\alpha\to 0}\,\text{excess kurtosis}=\lim_{\beta\to 0}\,% \text{excess kurtosis}=\lim_{\mu\to 0}\,\text{excess kurtosis}=\lim_{\mu\to 1}% \,\text{excess kurtosis}=\infty
  94. φ X ( α ; β ; t ) = E [ e i t X ] = 0 1 e i t x f ( x ; α , β ) d x = F 1 1 ( α ; α + β ; i t ) = n = 0 α ( n ) ( i t ) n ( α + β ) ( n ) n ! = 1 + k = 1 ( r = 0 k - 1 α + r α + β + r ) ( i t ) k k ! \begin{aligned}\displaystyle\varphi_{X}(\alpha;\beta;t)&\displaystyle=% \operatorname{E}\left[e^{itX}\right]\\ &\displaystyle=\int_{0}^{1}e^{itx}f(x;\alpha,\beta)dx\\ &\displaystyle={}_{1}F_{1}(\alpha;\alpha+\beta;it)\\ &\displaystyle=\sum_{n=0}^{\infty}\frac{\alpha^{(n)}(it)^{n}}{(\alpha+\beta)^{% (n)}n!}\\ &\displaystyle=1+\sum_{k=1}^{\infty}\left(\prod_{r=0}^{k-1}\frac{\alpha+r}{% \alpha+\beta+r}\right)\frac{(it)^{k}}{k!}\end{aligned}
  95. x ( n ) = x ( x + 1 ) ( x + 2 ) ( x + n - 1 ) x^{(n)}=x(x+1)(x+2)\cdots(x+n-1)
  96. φ X ( α ; β ; 0 ) = F 1 1 ( α ; α + β ; 0 ) = 1 \varphi_{X}(\alpha;\beta;0)={}_{1}F_{1}(\alpha;\alpha+\beta;0)=1
  97. Re [ F 1 1 ( α ; α + β ; i t ) ] = Re [ F 1 1 ( α ; α + β ; - i t ) ] \textrm{Re}\left[{}_{1}F_{1}(\alpha;\alpha+\beta;it)\right]=\textrm{Re}\left[{% }_{1}F_{1}(\alpha;\alpha+\beta;-it)\right]
  98. Im [ F 1 1 ( α ; α + β ; i t ) ] = - Im [ F 1 1 ( α ; α + β ; - i t ) ] \textrm{Im}\left[{}_{1}F_{1}(\alpha;\alpha+\beta;it)\right]=-\textrm{Im}\left[% {}_{1}F_{1}(\alpha;\alpha+\beta;-it)\right]
  99. I α - 1 2 I_{\alpha-\frac{1}{2}}
  100. F 1 1 ( α ; 2 α ; i t ) \displaystyle{}_{1}F_{1}(\alpha;2\alpha;it)
  101. M X ( α ; β ; t ) = E [ e t X ] = 0 1 e t x f ( x ; α , β ) d x = F 1 1 ( α ; α + β ; t ) = n = 0 α ( n ) ( α + β ) ( n ) t n n ! = 1 + k = 1 ( r = 0 k - 1 α + r α + β + r ) t k k ! \begin{aligned}\displaystyle M_{X}(\alpha;\beta;t)&\displaystyle=\operatorname% {E}\left[e^{tX}\right]\\ &\displaystyle=\int_{0}^{1}e^{tx}f(x;\alpha,\beta)\,dx\\ &\displaystyle={}_{1}F_{1}(\alpha;\alpha+\beta;t)\\ &\displaystyle=\sum_{n=0}^{\infty}\frac{\alpha^{(n)}}{(\alpha+\beta)^{(n)}}% \frac{t^{n}}{n!}\\ &\displaystyle=1+\sum_{k=1}^{\infty}\left(\prod_{r=0}^{k-1}\frac{\alpha+r}{% \alpha+\beta+r}\right)\frac{t^{k}}{k!}\end{aligned}
  102. r = 0 k - 1 α + r α + β + r \prod_{r=0}^{k-1}\frac{\alpha+r}{\alpha+\beta+r}
  103. ( t k k ! ) \left(\frac{t^{k}}{k!}\right)
  104. E [ X k ] = α ( k ) ( α + β ) ( k ) = r = 0 k - 1 α + r α + β + r \operatorname{E}[X^{k}]=\frac{\alpha^{(k)}}{(\alpha+\beta)^{(k)}}=\prod_{r=0}^% {k-1}\frac{\alpha+r}{\alpha+\beta+r}
  105. E [ X k ] = α + k - 1 α + β + k - 1 E [ X k - 1 ] . \operatorname{E}[X^{k}]=\frac{\alpha+k-1}{\alpha+\beta+k-1}\operatorname{E}[X^% {k-1}].
  106. E [ 1 - X ] = β α + β \displaystyle\operatorname{E}[1-X]=\frac{\beta}{\alpha+\beta}
  107. var [ ( 1 - X ) ] = var [ X ] = - cov [ X , ( 1 - X ) ] = α β ( α + β ) 2 ( α + β + 1 ) \operatorname{var}[(1-X)]=\operatorname{var}[X]=-\operatorname{cov}[X,(1-X)]=% \frac{\alpha\beta}{(\alpha+\beta)^{2}(\alpha+\beta+1)}
  108. E [ 1 X ] = α + β - 1 α - 1 if α > 1 \displaystyle\operatorname{E}\left[\frac{1}{X}\right]=\frac{\alpha+\beta-1}{% \alpha-1}\,\text{ if }\alpha>1
  109. E [ X 1 - X ] = α β - 1 if β > 1 E [ 1 - X X ] = β α - 1 if α > 1 \begin{aligned}&\displaystyle\operatorname{E}\left[\frac{X}{1-X}\right]=\frac{% \alpha}{\beta-1}\,\text{ if }\beta>1\\ &\displaystyle\operatorname{E}\left[\frac{1-X}{X}\right]=\frac{\beta}{\alpha-1% }\,\text{ if }\alpha>1\end{aligned}
  110. var [ 1 X ] = E [ ( 1 X - E [ 1 X ] ) 2 ] = var [ 1 - X X ] = E [ ( 1 - X X - E [ 1 - X X ] ) 2 ] = β ( α + β - 1 ) ( α - 2 ) ( α - 1 ) 2 if α > 2 \operatorname{var}\left[\frac{1}{X}\right]=\operatorname{E}\left[\left(\frac{1% }{X}-\operatorname{E}\left[\frac{1}{X}\right]\right)^{2}\right]=\operatorname{% var}\left[\frac{1-X}{X}\right]=\operatorname{E}\left[\left(\frac{1-X}{X}-% \operatorname{E}\left[\frac{1-X}{X}\right]\right)^{2}\right]=\frac{\beta(% \alpha+\beta-1)}{(\alpha-2)(\alpha-1)^{2}}\,\text{ if }\alpha>2
  111. var [ 1 1 - X ] = E [ ( 1 1 - X - E [ 1 1 - X ] ) 2 ] = var [ X 1 - X ] = E [ ( X 1 - X - E [ X 1 - X ] ) 2 ] = α ( α + β - 1 ) ( β - 2 ) ( β - 1 ) 2 if β > 2 \operatorname{var}\left[\frac{1}{1-X}\right]=\operatorname{E}\left[\left(\frac% {1}{1-X}-\operatorname{E}\left[\frac{1}{1-X}\right]\right)^{2}\right]=% \operatorname{var}\left[\frac{X}{1-X}\right]=\operatorname{E}\left[\left(\frac% {X}{1-X}-\operatorname{E}\left[\frac{X}{1-X}\right]\right)^{2}\right]=\frac{% \alpha(\alpha+\beta-1)}{(\beta-2)(\beta-1)^{2}}\,\text{ if }\beta>2
  112. cov [ 1 X , 1 1 - X ] = cov [ 1 - X X , X 1 - X ] = cov [ 1 X , X 1 - X ] = cov [ 1 - X X , 1 1 - X ] = α + β - 1 ( α - 1 ) ( β - 1 ) if α , β > 1 \operatorname{cov}\left[\frac{1}{X},\frac{1}{1-X}\right]=\operatorname{cov}% \left[\frac{1-X}{X},\frac{X}{1-X}\right]=\operatorname{cov}\left[\frac{1}{X},% \frac{X}{1-X}\right]=\operatorname{cov}\left[\frac{1-X}{X},\frac{1}{1-X}\right% ]=\frac{\alpha+\beta-1}{(\alpha-1)(\beta-1)}\,\text{ if }\alpha,\beta>1
  113. E [ ln ( X ) ] \displaystyle\operatorname{E}[\ln(X)]
  114. ψ ( α ) = d ln Γ ( α ) d α \psi(\alpha)=\frac{d\ln\Gamma(\alpha)}{d\alpha}
  115. E [ ln ( X 1 - X ) ] \displaystyle\operatorname{E}\left[\ln\left(\frac{X}{1-X}\right)\right]
  116. E [ ln 2 ( X ) ] \displaystyle\operatorname{E}\left[\ln^{2}(X)\right]
  117. cov [ ln ( X ) , ln ( 1 - X ) ] \displaystyle\operatorname{cov}[\ln(X),\ln(1-X)]
  118. ψ 1 ( α ) = d 2 ln Γ ( α ) d α 2 = d ψ ( α ) d α \psi_{1}(\alpha)=\frac{d^{2}\ln\Gamma(\alpha)}{d\alpha^{2}}=\frac{d\psi(\alpha% )}{d\alpha}
  119. var [ ln ( 1 X ) ] \displaystyle\operatorname{var}\left[\ln\left(\frac{1}{X}\right)\right]
  120. var [ ln ( X 1 - X ) ] = var [ ln ( 1 - X X ) ] = - cov [ ln ( X 1 - X ) , ln ( 1 - X X ) ] = ψ 1 ( α ) + ψ 1 ( β ) \operatorname{var}\left[\ln\left(\frac{X}{1-X}\right)\right]=\operatorname{var% }\left[\ln\left(\frac{1-X}{X}\right)\right]=-\operatorname{cov}\left[\ln\left(% \frac{X}{1-X}\right),\ln\left(\frac{1-X}{X}\right)\right]=\psi_{1}(\alpha)+% \psi_{1}(\beta)
  121. h ( X ) \displaystyle h(X)
  122. f ( x ; α , β ) = 1 B ( α , β ) x α - 1 ( 1 - x ) β - 1 f(x;\alpha,\beta)=\frac{1}{B(\alpha,\beta)}x^{\alpha-1}(1-x)^{\beta-1}
  123. 0 1 1 - x α - 1 1 - x d x = ψ ( α ) - ψ ( 1 ) \int_{0}^{1}\frac{1-x^{\alpha-1}}{1-x}dx=\psi(\alpha)-\psi(1)
  124. H ( X 1 , X 2 ) = 0 1 - f ( x ; α , β ) ln ( f ( x ; α , β ) ) d x = ln ( B ( α , β ) ) - ( α - 1 ) ψ ( α ) - ( β - 1 ) ψ ( β ) + ( α + β - 2 ) ψ ( α + β ) . \begin{aligned}\displaystyle H(X_{1},X_{2})&\displaystyle=\int_{0}^{1}-f(x;% \alpha,\beta)\ln(f(x;\alpha^{\prime},\beta^{\prime}))dx\\ &\displaystyle=\ln\left(B(\alpha^{\prime},\beta^{\prime})\right)-(\alpha^{% \prime}-1)\psi(\alpha)-(\beta^{\prime}-1)\psi(\beta)+(\alpha^{\prime}+\beta^{% \prime}-2)\psi(\alpha+\beta).\end{aligned}
  125. D KL ( X 1 , X 2 ) = 0 1 f ( x ; α , β ) ln ( f ( x ; α , β ) f ( x ; α , β ) ) d x = ( 0 1 f ( x ; α , β ) ln ( f ( x ; α , β ) ) d x ) - ( 0 1 f ( x ; α , β ) ln ( f ( x ; α , β ) ) d x ) = - h ( X 1 ) + H ( X 1 , X 2 ) = ln ( B ( α , β ) B ( α , β ) ) + ( α - α ) ψ ( α ) + ( β - β ) ψ ( β ) + ( α - α + β - β ) ψ ( α + β ) . \begin{aligned}\displaystyle D_{\mathrm{KL}}(X_{1},X_{2})&\displaystyle=\int_{% 0}^{1}f(x;\alpha,\beta)\ln\left(\frac{f(x;\alpha,\beta)}{f(x;\alpha^{\prime},% \beta^{\prime})}\right)dx\\ &\displaystyle=\left(\int_{0}^{1}f(x;\alpha,\beta)\ln(f(x;\alpha,\beta))dx% \right)-\left(\int_{0}^{1}f(x;\alpha,\beta)\ln(f(x;\alpha^{\prime},\beta^{% \prime}))dx\right)\\ &\displaystyle=-h(X_{1})+H(X_{1},X_{2})\\ &\displaystyle=\ln\left(\frac{B(\alpha^{\prime},\beta^{\prime})}{B(\alpha,% \beta)}\right)+(\alpha-\alpha^{\prime})\psi(\alpha)+(\beta-\beta^{\prime})\psi% (\beta)+(\alpha^{\prime}-\alpha+\beta^{\prime}-\beta)\psi(\alpha+\beta).\end{aligned}
  126. D KL ( X 1 , X 2 ) = D KL ( X 2 , X 1 ) , if h ( X 1 ) = h ( X 2 ) , for (skewed) α β D_{\mathrm{KL}}(X_{1},X_{2})=D_{\mathrm{KL}}(X_{2},X_{1}),\,\text{ if }h(X_{1}% )=h(X_{2}),\,\text{ for (skewed) }\alpha\neq\beta
  127. α - 1 α + β - 2 median α α + β , \frac{\alpha-1}{\alpha+\beta-2}\leq\,\text{median}\leq\frac{\alpha}{\alpha+% \beta},
  128. ( skewness ) 2 + 1 < kurtosis < 3 2 ( skewness ) 2 + 3 (\,\text{skewness})^{2}+1<\,\text{kurtosis}<\frac{3}{2}(\,\text{skewness})^{2}+3
  129. ( skewness ) 2 - 2 < excess kurtosis < 3 2 ( skewness ) 2 (\,\text{skewness})^{2}-2<\,\text{excess kurtosis}<\frac{3}{2}(\,\text{% skewness})^{2}
  130. p = β α + β p=\tfrac{\beta}{\alpha+\beta}
  131. q = 1 - p = α α + β q=1-p=\tfrac{\alpha}{\alpha+\beta}
  132. f ( x ; α , β ) = f ( 1 - x ; β , α ) f(x;\alpha,\beta)=f(1-x;\beta,\alpha)
  133. F ( x ; α , β ) = I x ( α , β ) = 1 - F ( 1 - x ; β , α ) = 1 - I 1 - x ( β , α ) F(x;\alpha,\beta)=I_{x}(\alpha,\beta)=1-F(1-x;\beta,\alpha)=1-I_{1-x}(\beta,\alpha)
  134. mode ( B ( α , β ) ) = 1 - mode ( B ( β , α ) ) , if B ( β , α ) B ( 1 , 1 ) \operatorname{mode}(B(\alpha,\beta))=1-\operatorname{mode}(B(\beta,\alpha)),\,% \text{ if }B(\beta,\alpha)\neq B(1,1)
  135. median ( B ( α , β ) ) = 1 - median ( B ( β , α ) ) \operatorname{median}(B(\alpha,\beta))=1-\operatorname{median}(B(\beta,\alpha))
  136. μ ( B ( α , β ) ) = 1 - μ ( B ( β , α ) ) \mu(B(\alpha,\beta))=1-\mu(B(\beta,\alpha))
  137. G X ( B ( α , β ) ) = G ( 1 - X ) ( B ( β , α ) ) G_{X}(B(\alpha,\beta))=G_{(1-X)}(B(\beta,\alpha))
  138. H X ( B ( α , β ) ) = H ( 1 - X ) ( B ( β , α ) ) if α , β > 1 H_{X}(B(\alpha,\beta))=H_{(1-X)}(B(\beta,\alpha))\,\text{ if }\alpha,\beta>1
  139. var ( B ( α , β ) ) = var ( B ( β , α ) ) \operatorname{var}(B(\alpha,\beta))=\operatorname{var}(B(\beta,\alpha))
  140. ln ( var GX ( B ( α , β ) ) ) = ln ( var G ( 1 - X ) ( B ( β , α ) ) ) \ln(\operatorname{var_{GX}}(B(\alpha,\beta)))=\ln(\operatorname{var_{G(1-X)}}(% B(\beta,\alpha)))
  141. ln cov GX , ( 1 - X ) ( B ( α , β ) ) = ln cov GX , ( 1 - X ) ( B ( β , α ) ) \ln\,\operatorname{cov_{G{X,(1-X)}}}(B(\alpha,\beta))=\ln\operatorname{cov_{G{% X,(1-X)}}}(B(\beta,\alpha))
  142. E [ | X - E [ X ] | ] ( B ( α , β ) ) = E [ | X - E [ X ] | ] ( B ( β , α ) ) \operatorname{E}[|X-E[X]|](B(\alpha,\beta))=\operatorname{E}[|X-E[X]|](B(\beta% ,\alpha))
  143. skewness ( B ( α , β ) ) = - skewness ( B ( β , α ) ) \operatorname{skewness}(B(\alpha,\beta))=-\operatorname{skewness}(B(\beta,% \alpha))
  144. excess kurtosis ( B ( α , β ) ) = excess kurtosis ( B ( β , α ) ) \,\text{excess kurtosis}(B(\alpha,\beta))=\,\text{excess kurtosis}(B(\beta,% \alpha))
  145. Re [ F 1 1 ( α ; α + β ; i t ) ] = Re [ F 1 1 ( α ; α + β ; - i t ) ] \,\text{Re}[{}_{1}F_{1}(\alpha;\alpha+\beta;it)]=\,\text{Re}[{}_{1}F_{1}(% \alpha;\alpha+\beta;-it)]
  146. Im [ F 1 1 ( α ; α + β ; i t ) ] = - Im [ F 1 1 ( α ; α + β ; - i t ) ] \,\text{Im}[{}_{1}F_{1}(\alpha;\alpha+\beta;it)]=-\,\text{Im}[{}_{1}F_{1}(% \alpha;\alpha+\beta;-it)]
  147. Abs [ F 1 1 ( α ; α + β ; i t ) ] = Abs [ F 1 1 ( α ; α + β ; - i t ) ] \,\text{Abs}[{}_{1}F_{1}(\alpha;\alpha+\beta;it)]=\,\text{Abs}[{}_{1}F_{1}(% \alpha;\alpha+\beta;-it)]
  148. h ( B ( α , β ) ) = h ( B ( β , α ) ) h(B(\alpha,\beta))=h(B(\beta,\alpha))
  149. D KL ( X 1 , X 2 ) = D KL ( X 2 , X 1 ) , if h ( X 1 ) = h ( X 2 ) , for (skewed) α β D_{\mathrm{KL}}(X_{1},X_{2})=D_{\mathrm{KL}}(X_{2},X_{1}),\,\text{ if }h(X_{1}% )=h(X_{2})\,\text{, for (skewed) }\alpha\neq\beta
  150. i , j = j , i {\mathcal{I}}_{i,j}={\mathcal{I}}_{j,i}
  151. κ = ( α - 1 ) ( β - 1 ) α + β - 3 α + β - 2 \kappa=\frac{\sqrt{\frac{(\alpha-1)(\beta-1)}{\alpha+\beta-3}}}{\alpha+\beta-2}
  152. x = mode ± κ = α - 1 ± ( α - 1 ) ( β - 1 ) α + β - 3 α + β - 2 x=\,\text{mode}\pm\kappa=\frac{\alpha-1\pm\sqrt{\frac{(\alpha-1)(\beta-1)}{% \alpha+\beta-3}}}{\alpha+\beta-2}
  153. x = mode + κ = 2 β x=\,\text{mode}+\kappa=\frac{2}{\beta}
  154. x = mode - κ = 1 - 2 α x=\,\text{mode}-\kappa=1-\frac{2}{\alpha}
  155. x = mode + κ = α - 1 + ( α - 1 ) ( β - 1 ) α + β - 3 α + β - 2 x=\,\text{mode}+\kappa=\frac{\alpha-1+\sqrt{\frac{(\alpha-1)(\beta-1)}{\alpha+% \beta-3}}}{\alpha+\beta-2}
  156. x = α - 1 + ( α - 1 ) ( β - 1 ) α + β - 3 α + β - 2 x=\frac{\alpha-1+\sqrt{\frac{(\alpha-1)(\beta-1)}{\alpha+\beta-3}}}{\alpha+% \beta-2}
  157. x = mode - κ = α - 1 - ( α - 1 ) ( β - 1 ) α + β - 3 α + β - 2 x=\,\text{mode}-\kappa=\frac{\alpha-1-\sqrt{\frac{(\alpha-1)(\beta-1)}{\alpha+% \beta-3}}}{\alpha+\beta-2}
  158. x = α - 1 - ( α - 1 ) ( β - 1 ) α + β - 3 α + β - 2 x=\frac{\alpha-1-\sqrt{\frac{(\alpha-1)(\beta-1)}{\alpha+\beta-3}}}{\alpha+% \beta-2}
  159. lim α = β 0 var ( X ) = 1 4 \lim_{\alpha=\beta\to 0}\operatorname{var}(X)=\tfrac{1}{4}
  160. lim α = β 0 excess kurtosis ( X ) = - 2 \lim_{\alpha=\beta\to 0}\operatorname{excess\ kurtosis}(X)=-2
  161. lim α = β var ( X ) = 0 \lim_{\alpha=\beta\to\infty}\operatorname{var}(X)=0
  162. lim α = β excess kurtosis ( X ) = 0 \lim_{\alpha=\beta\to\infty}\operatorname{excess\ kurtosis}(X)=0
  163. α - 1 α + β - 2 \tfrac{\alpha-1}{\alpha+\beta-2}
  164. mode = α - 1 α + β - 2 \,\text{mode }=\tfrac{\alpha-1}{\alpha+\beta-2}
  165. α = - 1 + 5 2 , β = 1 \alpha=\tfrac{-1+\sqrt{5}}{2},\beta=1
  166. α = 1 , β = - 1 + 5 2 \alpha=1,\beta=\tfrac{-1+\sqrt{5}}{2}
  167. median = 1 - 1 2 \,\text{median}=1-\tfrac{1}{\sqrt{2}}
  168. 0 < median < 1 - 1 2 0<\,\text{median}<1-\tfrac{1}{\sqrt{2}}
  169. 1 2 < median < 1 2 \tfrac{1}{2}<\,\text{median}<\tfrac{1}{\sqrt{2}}
  170. median = 1 2 \,\text{median}=\tfrac{1}{\sqrt{2}}
  171. 1 2 < median < 1 \tfrac{1}{\sqrt{2}}<\,\text{median}<1
  172. ( α ^ , β ^ ) (\hat{\alpha},\hat{\beta})
  173. sample mean(X) = x ¯ = 1 N i = 1 N X i \,\text{sample mean(X)}=\bar{x}=\frac{1}{N}\sum_{i=1}^{N}X_{i}
  174. sample variance(X) = v ¯ = 1 N - 1 i = 1 N ( X i - x ¯ ) 2 \,\text{sample variance(X)}=\bar{v}=\frac{1}{N-1}\sum_{i=1}^{N}(X_{i}-\bar{x})% ^{2}
  175. α ^ = x ¯ ( x ¯ ( 1 - x ¯ ) v ¯ - 1 ) , \hat{\alpha}=\bar{x}\left(\frac{\bar{x}(1-\bar{x})}{\bar{v}}-1\right),
  176. v ¯ < x ¯ ( 1 - x ¯ ) , \bar{v}<\bar{x}(1-\bar{x}),
  177. β ^ = ( 1 - x ¯ ) ( x ¯ ( 1 - x ¯ ) v ¯ - 1 ) , \hat{\beta}=(1-\bar{x})\left(\frac{\bar{x}(1-\bar{x})}{\bar{v}}-1\right),
  178. v ¯ < x ¯ ( 1 - x ¯ ) . \bar{v}<\bar{x}(1-\bar{x}).
  179. x ¯ \bar{x}
  180. y ¯ - a c - a , \frac{\bar{y}-a}{c-a},
  181. v ¯ \bar{v}
  182. v Y ¯ ( c - a ) 2 \frac{\bar{v_{Y}}}{(c-a)^{2}}
  183. sample mean(Y) = y ¯ = 1 N i = 1 N Y i \,\text{sample mean(Y)}=\bar{y}=\frac{1}{N}\sum_{i=1}^{N}Y_{i}
  184. sample variance(Y) = v Y ¯ = 1 N - 1 i = 1 N ( Y i - y ¯ ) 2 \,\text{sample variance(Y)}=\bar{v_{Y}}=\frac{1}{N-1}\sum_{i=1}^{N}(Y_{i}-\bar% {y})^{2}
  185. α ^ , β ^ , a ^ , c ^ \hat{\alpha},\hat{\beta},\hat{a},\hat{c}
  186. excess kurtosis = 6 3 + ν ( ( 2 + ν ) 4 ( skewness ) 2 - 1 ) if (skewness) 2 - 2 < excess kurtosis < 3 2 ( skewness ) 2 \,\text{excess kurtosis}=\frac{6}{3+\nu}\left(\frac{(2+\nu)}{4}(\,\text{% skewness})^{2}-1\right)\,\text{ if (skewness)}^{2}-2<\,\text{excess kurtosis}<% \tfrac{3}{2}(\,\text{skewness})^{2}
  187. ν ^ = α ^ + β ^ = 3 ( sample excess kurtosis ) - ( sample skewness ) 2 + 2 3 2 ( sample skewness ) 2 - (sample excess kurtosis) if (sample skewness) 2 - 2 < sample excess kurtosis < 3 2 ( sample skewness ) 2 \hat{\nu}=\hat{\alpha}+\hat{\beta}=3\frac{(\,\text{sample excess kurtosis})-(% \,\text{sample skewness})^{2}+2}{\frac{3}{2}(\,\text{sample skewness})^{2}-\,% \text{(sample excess kurtosis)}}\,\text{ if (sample skewness)}^{2}-2<\,\text{% sample excess kurtosis}<\tfrac{3}{2}(\,\text{sample skewness})^{2}
  188. α ^ = β ^ = ν ^ 2 = 3 2 ( sample excess kurtosis ) + 3 - (sample excess kurtosis) if sample skewness = 0 and - 2 < sample excess kurtosis < 0 \hat{\alpha}=\hat{\beta}=\frac{\hat{\nu}}{2}=\frac{\frac{3}{2}(\,\text{sample % excess kurtosis})+3}{-\,\text{(sample excess kurtosis)}}\,\text{ if sample % skewness}=0\,\text{ and }-2<\,\text{sample excess kurtosis}<0
  189. ν ^ \hat{\nu}
  190. a ^ , c ^ \hat{a},\hat{c}
  191. α ^ , β ^ \hat{\alpha},\hat{\beta}
  192. ( sample skewness ) 2 = 4 ( β ^ - α ^ ) 2 ( 1 + α ^ + β ^ ) α ^ β ^ ( 2 + α ^ + β ^ ) 2 (\,\text{sample skewness})^{2}=\frac{4(\hat{\beta}-\hat{\alpha})^{2}(1+\hat{% \alpha}+\hat{\beta})}{\hat{\alpha}\hat{\beta}(2+\hat{\alpha}+\hat{\beta})^{2}}
  193. sample excess kurtosis = 6 3 + α ^ + β ^ ( ( 2 + α ^ + β ^ ) 4 ( sample skewness ) 2 - 1 ) if (sample skewness) 2 - 2 < sample excess kurtosis < 3 2 ( sample skewness ) 2 \,\text{sample excess kurtosis}=\frac{6}{3+\hat{\alpha}+\hat{\beta}}\left(% \frac{(2+\hat{\alpha}+\hat{\beta})}{4}(\,\text{sample skewness})^{2}-1\right)% \,\text{ if (sample skewness)}^{2}-2<\,\text{sample excess kurtosis}<\tfrac{3}% {2}(\,\text{sample skewness})^{2}
  194. α ^ , β ^ = ν ^ 2 ( 1 ± 1 1 + 16 ( ν ^ + 1 ) ( ν ^ + 2 ) 2 ( sample skewness ) 2 ) if sample skewness 0 and ( sample skewness ) 2 - 2 < sample excess kurtosis < 3 2 ( sample skewness ) 2 \hat{\alpha},\hat{\beta}=\frac{\hat{\nu}}{2}\left(1\pm\frac{1}{\sqrt{1+\frac{1% 6(\hat{\nu}+1)}{(\hat{\nu}+2)^{2}(\,\text{sample skewness})^{2}}}}\right)\,% \text{ if sample skewness}\neq 0\,\text{ and }(\,\text{sample skewness})^{2}-2% <\,\text{sample excess kurtosis}<\tfrac{3}{2}(\,\text{sample skewness})^{2}
  195. α ^ > β ^ \hat{\alpha}>\hat{\beta}
  196. p = β α + β p=\tfrac{\beta}{\alpha+\beta}
  197. q = 1 - p = α α + β q=1-p=\tfrac{\alpha}{\alpha+\beta}
  198. a ^ , c ^ \hat{a},\hat{c}
  199. ( c ^ - a ^ ) (\hat{c}-\hat{a})
  200. ( c ^ - a ^ ) (\hat{c}-\hat{a})
  201. sample excess kurtosis = 6 ( 3 + ν ^ ) ( 2 + ν ^ ) ( ( c ^ - a ^ ) 2 (sample variance) - 6 - 5 ν ^ ) \,\text{sample excess kurtosis}=\frac{6}{(3+\hat{\nu})(2+\hat{\nu})}\bigg(% \frac{(\hat{c}-\hat{a})^{2}}{\,\text{(sample variance)}}-6-5\hat{\nu}\bigg)
  202. ( c ^ - a ^ ) = (sample variance) 6 + 5 ν ^ + ( 2 + ν ^ ) ( 3 + ν ^ ) 6 (sample excess kurtosis) (\hat{c}-\hat{a})=\sqrt{\,\text{(sample variance)}}\sqrt{6+5\hat{\nu}+\frac{(2% +\hat{\nu})(3+\hat{\nu})}{6}\,\text{(sample excess kurtosis)}}
  203. ( c ^ - a ^ ) (\hat{c}-\hat{a})
  204. ( c ^ - a ^ ) (\hat{c}-\hat{a})
  205. ( sample skewness ) 2 = 4 ( 2 + ν ^ ) 2 ( ( c ^ - a ^ ) 2 (sample variance) - 4 ( 1 + ν ^ ) ) (\,\text{sample skewness})^{2}=\frac{4}{(2+\hat{\nu})^{2}}\bigg(\frac{(\hat{c}% -\hat{a})^{2}}{\,\text{(sample variance)}}-4(1+\hat{\nu})\bigg)
  206. ( c ^ - a ^ ) = (sample variance) 2 ( 2 + ν ^ ) 2 ( sample skewness ) 2 + 16 ( 1 + ν ^ ) (\hat{c}-\hat{a})=\frac{\sqrt{\,\text{(sample variance)}}}{2}\sqrt{(2+\hat{\nu% })^{2}(\,\text{sample skewness})^{2}+16(1+\hat{\nu})}
  207. ( c ^ - a ^ ) , α ^ , ν ^ = α ^ + β ^ (\hat{c}-\hat{a}),\hat{\alpha},\hat{\nu}=\hat{\alpha}+\hat{\beta}
  208. a ^ = ( sample mean ) - ( α ^ ν ^ ) ( c ^ - a ^ ) \hat{a}=(\,\text{sample mean})-\left(\frac{\hat{\alpha}}{\hat{\nu}}\right)(% \hat{c}-\hat{a})
  209. c ^ = ( c ^ - a ^ ) + a ^ \hat{c}=(\hat{c}-\hat{a})+\hat{a}
  210. sample mean \displaystyle\,\text{sample mean}
  211. ln ( α , β | X ) \displaystyle\ln\,\mathcal{L}(\alpha,\beta|X)
  212. ln ( α , β | X ) α = i = 1 N ln X i - N ln B ( α , β ) α = 0 \frac{\partial\ln\mathcal{L}(\alpha,\beta|X)}{\partial\alpha}=\sum_{i=1}^{N}% \ln X_{i}-N\frac{\partial\ln B(\alpha,\beta)}{\partial\alpha}=0
  213. ln ( α , β | X ) β = i = 1 N ln ( 1 - X i ) - N ln B ( α , β ) β = 0 \frac{\partial\ln\mathcal{L}(\alpha,\beta|X)}{\partial\beta}=\sum_{i=1}^{N}\ln% (1-X_{i})-N\frac{\partial\ln\mathrm{B}(\alpha,\beta)}{\partial\beta}=0
  214. ln B ( α , β ) α = - ln Γ ( α + β ) α + ln Γ ( α ) α + ln Γ ( β ) α = - ψ ( α + β ) + ψ ( α ) + 0 \frac{\partial\ln B(\alpha,\beta)}{\partial\alpha}=-\frac{\partial\ln\Gamma(% \alpha+\beta)}{\partial\alpha}+\frac{\partial\ln\Gamma(\alpha)}{\partial\alpha% }+\frac{\partial\ln\Gamma(\beta)}{\partial\alpha}=-\psi(\alpha+\beta)+\psi(% \alpha)+0
  215. ln B ( α , β ) β = - ln Γ ( α + β ) β + ln Γ ( α ) β + ln Γ ( β ) β = - ψ ( α + β ) + 0 + ψ ( β ) \frac{\partial\ln B(\alpha,\beta)}{\partial\beta}=-\frac{\partial\ln\Gamma(% \alpha+\beta)}{\partial\beta}+\frac{\partial\ln\Gamma(\alpha)}{\partial\beta}+% \frac{\partial\ln\Gamma(\beta)}{\partial\beta}=-\psi(\alpha+\beta)+0+\psi(\beta)
  216. ψ ( α ) = ln Γ ( α ) α \psi(\alpha)=\frac{\partial\ln\Gamma(\alpha)}{\partial\alpha}
  217. 2 ln ( α , β | X ) α 2 = - N 2 ln B ( α , β ) α 2 < 0 \frac{\partial^{2}\ln\mathcal{L}(\alpha,\beta|X)}{\partial\alpha^{2}}=-N\frac{% \partial^{2}\ln B(\alpha,\beta)}{\partial\alpha^{2}}<0
  218. 2 ln ( α , β | X ) β 2 = - N 2 ln B ( α , β ) β 2 < 0 \frac{\partial^{2}\ln\mathcal{L}(\alpha,\beta|X)}{\partial\beta^{2}}=-N\frac{% \partial^{2}\ln B(\alpha,\beta)}{\partial\beta^{2}}<0
  219. 2 ln B ( α , β ) α 2 = ψ 1 ( α ) - ψ 1 ( α + β ) > 0 \frac{\partial^{2}\ln B(\alpha,\beta)}{\partial\alpha^{2}}=\psi_{1}(\alpha)-% \psi_{1}(\alpha+\beta)>0
  220. 2 ln B ( α , β ) β 2 = ψ 1 ( β ) - ψ 1 ( α + β ) > 0 \frac{\partial^{2}\ln B(\alpha,\beta)}{\partial\beta^{2}}=\psi_{1}(\beta)-\psi% _{1}(\alpha+\beta)>0
  221. ψ 1 ( α ) = 2 ln Γ ( α ) α 2 = ψ ( α ) α \psi_{1}(\alpha)=\frac{\partial^{2}\ln\Gamma(\alpha)}{\partial\alpha^{2}}=\,% \frac{\partial\,\psi(\alpha)}{\partial\alpha}
  222. var [ ln ( X ) ] = E [ ln 2 ( X ) ] - ( E [ ln ( X ) ] ) 2 = ψ 1 ( α ) - ψ 1 ( α + β ) \operatorname{var}[\ln(X)]=\operatorname{E}[\ln^{2}(X)]-(\operatorname{E}[\ln(% X)])^{2}=\psi_{1}(\alpha)-\psi_{1}(\alpha+\beta)
  223. var [ ln ( 1 - X ) ] = E [ ln 2 ( 1 - X ) ] - ( E [ ln ( 1 - X ) ] ) 2 = ψ 1 ( β ) - ψ 1 ( α + β ) \operatorname{var}[\ln(1-X)]=\operatorname{E}[\ln^{2}(1-X)]-(\operatorname{E}[% \ln(1-X)])^{2}=\psi_{1}(\beta)-\psi_{1}(\alpha+\beta)
  224. var [ ln ( X ) ] > 0 \operatorname{var}[\ln(X)]>0
  225. var [ ln ( 1 - X ) ] > 0 \operatorname{var}[\ln(1-X)]>0
  226. ψ 1 ( α ) - ψ 1 ( α + β ) = ln G X α > 0 \psi_{1}(\alpha)-\psi_{1}(\alpha+\beta)=\frac{\partial\,\ln G_{X}}{\partial% \alpha}>0
  227. ψ 1 ( β ) - ψ 1 ( α + β ) = ln G ( 1 - X ) β > 0 \psi_{1}(\beta)-\psi_{1}(\alpha+\beta)=\frac{\partial\,\ln G_{(1-X)}}{\partial% \beta}>0
  228. ln G X β , ln G ( 1 - X ) α < 0. \frac{\partial\,\ln G_{X}}{\partial\beta},\frac{\partial\,\ln G_{(1-X)}}{% \partial\alpha}<0.
  229. α ^ , β ^ \hat{\alpha},\hat{\beta}
  230. E ^ [ ln ( X ) ] = ψ ( α ^ ) - ψ ( α ^ + β ^ ) = 1 N i = 1 N ln X i = ln G ^ X E ^ [ ln ( 1 - X ) ] = ψ ( β ^ ) - ψ ( α ^ + β ^ ) = 1 N i = 1 N ln ( 1 - X i ) = ln G ^ ( 1 - X ) \begin{aligned}\displaystyle\hat{\operatorname{E}}[\ln(X)]&\displaystyle=\psi(% \hat{\alpha})-\psi(\hat{\alpha}+\hat{\beta})=\frac{1}{N}\sum_{i=1}^{N}\ln X_{i% }=\ln\hat{G}_{X}\\ \displaystyle\hat{\operatorname{E}}[\ln(1-X)]&\displaystyle=\psi(\hat{\beta})-% \psi(\hat{\alpha}+\hat{\beta})=\frac{1}{N}\sum_{i=1}^{N}\ln(1-X_{i})=\ln\hat{G% }_{(1-X)}\end{aligned}
  231. log G ^ X \log\hat{G}_{X}
  232. log G ^ ( 1 - X ) \log\hat{G}_{(1-X)}
  233. α ^ = β ^ \hat{\alpha}=\hat{\beta}
  234. G ^ X = G ^ ( 1 - X ) \hat{G}_{X}=\hat{G}_{(1-X)}
  235. G ^ X = i = 1 N ( X i ) 1 N G ^ ( 1 - X ) = i = 1 N ( 1 - X i ) 1 N \begin{aligned}\displaystyle\hat{G}_{X}&\displaystyle=\prod_{i=1}^{N}(X_{i})^{% \frac{1}{N}}\\ \displaystyle\hat{G}_{(1-X)}&\displaystyle=\prod_{i=1}^{N}(1-X_{i})^{\frac{1}{% N}}\end{aligned}
  236. α ^ , β ^ \hat{\alpha},\hat{\beta}
  237. α ^ , β ^ \hat{\alpha},\hat{\beta}
  238. ψ ( α ^ ) ln ( α ^ - 1 2 ) \psi(\hat{\alpha})\approx\ln(\hat{\alpha}-\tfrac{1}{2})
  239. ln α ^ - 1 2 α ^ + β ^ - 1 2 ln G ^ X \ln\frac{\hat{\alpha}-\frac{1}{2}}{\hat{\alpha}+\hat{\beta}-\frac{1}{2}}% \approx\ln\hat{G}_{X}
  240. ln β ^ - 1 2 α ^ + β ^ - 1 2 ln G ^ ( 1 - X ) \ln\frac{\hat{\beta}-\frac{1}{2}}{\hat{\alpha}+\hat{\beta}-\frac{1}{2}}\approx% \ln\hat{G}_{(1-X)}
  241. α ^ 1 2 + G ^ X 2 ( 1 - G ^ X - G ^ ( 1 - X ) ) if α ^ > 1 \hat{\alpha}\approx\tfrac{1}{2}+\frac{\hat{G}_{X}}{2(1-\hat{G}_{X}-\hat{G}_{(1% -X)})}\,\text{ if }\hat{\alpha}>1
  242. β ^ 1 2 + G ^ ( 1 - X ) 2 ( 1 - G ^ X - G ^ ( 1 - X ) ) if β ^ > 1 \hat{\beta}\approx\tfrac{1}{2}+\frac{\hat{G}_{(1-X)}}{2(1-\hat{G}_{X}-\hat{G}_% {(1-X)})}\,\text{ if }\hat{\beta}>1
  243. ln Y i - a c - a , \ln\frac{Y_{i}-a}{c-a},
  244. ln c - Y i c - a \ln\frac{c-Y_{i}}{c-a}
  245. α ^ β ^ \hat{\alpha}\neq\hat{\beta}
  246. E ^ [ ln ( X 1 - X ) ] = ψ ( α ^ ) - ψ ( β ^ ) = 1 N i = 1 N ln X i 1 - X i = ln G ^ X - ln ( G ^ ( 1 - X ) ) \hat{\operatorname{E}}\left[\ln\left(\frac{X}{1-X}\right)\right]=\psi(\hat{% \alpha})-\psi(\hat{\beta})=\frac{1}{N}\sum_{i=1}^{N}\ln\frac{X_{i}}{1-X_{i}}=% \ln\hat{G}_{X}-\ln\left(\hat{G}_{(1-X)}\right)
  247. ln X 1 - X \ln\frac{X}{1-X}
  248. β ^ \hat{\beta}
  249. α ^ \hat{\alpha}
  250. ψ ( α ^ ) = 1 N i = 1 N ln X i 1 - X i + ψ ( β ^ ) \psi(\hat{\alpha})=\frac{1}{N}\sum_{i=1}^{N}\ln\frac{X_{i}}{1-X_{i}}+\psi(\hat% {\beta})
  251. α ^ = ( Inverse digamma ) [ ln G ^ X - ln G ^ ( 1 - X ) + ψ ( β ^ ) ] \hat{\alpha}=(\,\text{Inverse digamma })[\ln\hat{G}_{X}-\ln\hat{G}_{(1-X)}+% \psi(\hat{\beta})]
  252. β ^ = 1 \hat{\beta}=1
  253. ψ ( α ^ ) - ψ ( α ^ + β ^ ) = ln G ^ X \psi(\hat{\alpha})-\psi(\hat{\alpha}+\hat{\beta})=\ln\hat{G}_{X}
  254. α ^ \hat{\alpha}
  255. α ^ = - 1 1 N i = 1 N ln X i = - 1 ln G ^ X \hat{\alpha}=-\frac{1}{\frac{1}{N}\sum_{i=1}^{N}\ln X_{i}}=-\frac{1}{\ln\hat{G% }_{X}}
  256. G ^ X < 1 \hat{G}_{X}<1
  257. ( - ln G ^ X ) > 0 (-\ln\hat{G}_{X})>0
  258. α ^ > 0 \hat{\alpha}>0
  259. ln ( α , β | X ) N = ( α - 1 ) ln G ^ X + ( β - 1 ) ln G ^ ( 1 - X ) - ln B ( α , β ) \frac{\ln\,\mathcal{L}(\alpha,\beta|X)}{N}=(\alpha-1)\ln\hat{G}_{X}+(\beta-1)% \ln\hat{G}_{(1-X)}-\ln B(\alpha,\beta)
  260. α ^ , β ^ \hat{\alpha},\hat{\beta}
  261. 2 ln ( α , β | X ) α 2 = - var [ ln X ] \frac{\partial^{2}\ln\mathcal{L}(\alpha,\beta|X)}{\partial\alpha^{2}}=-% \operatorname{var}[\ln X]
  262. 2 ln ( α , β | X ) β 2 = - var [ ln ( 1 - X ) ] \frac{\partial^{2}\ln\mathcal{L}(\alpha,\beta|X)}{\partial\beta^{2}}=-% \operatorname{var}[\ln(1-X)]
  263. α ^ \hat{\alpha}
  264. var ( α ^ ) 1 var [ ln X ] 1 ψ 1 ( α ^ ) - ψ 1 ( α ^ + β ^ ) \mathrm{var}(\hat{\alpha})\geq\frac{1}{\operatorname{var}[\ln X]}\geq\frac{1}{% \psi_{1}(\hat{\alpha})-\psi_{1}(\hat{\alpha}+\hat{\beta})}
  265. var ( β ^ ) 1 var [ ln ( 1 - X ) ] 1 ψ 1 ( β ^ ) - ψ 1 ( α ^ + β ^ ) \mathrm{var}(\hat{\beta})\geq\frac{1}{\operatorname{var}[\ln(1-X)]}\geq\frac{1% }{\psi_{1}(\hat{\beta})-\psi_{1}(\hat{\alpha}+\hat{\beta})}
  266. ln ( α , β | X ) N = ( α - 1 ) ( ψ ( α ^ ) - ψ ( α ^ + β ^ ) ) + ( β - 1 ) ( ψ ( β ^ ) - ψ ( α ^ + β ^ ) ) - ln B ( α , β ) \frac{\ln\,\mathcal{L}(\alpha,\beta|X)}{N}=(\alpha-1)(\psi(\hat{\alpha})-\psi(% \hat{\alpha}+\hat{\beta}))+(\beta-1)(\psi(\hat{\beta})-\psi(\hat{\alpha}+\hat{% \beta}))-\ln B(\alpha,\beta)
  267. ln ( α , β | X ) N = - H = - h - D KL = - ln B ( α , β ) + ( α - 1 ) ψ ( α ^ ) + ( β - 1 ) ψ ( β ^ ) - ( α + β - 2 ) ψ ( α ^ + β ^ ) \frac{\ln\,\mathcal{L}(\alpha,\beta|X)}{N}=-H=-h-D_{\mathrm{KL}}=-\ln B(\alpha% ,\beta)+(\alpha-1)\psi(\hat{\alpha})+(\beta-1)\psi(\hat{\beta})-(\alpha+\beta-% 2)\psi(\hat{\alpha}+\hat{\beta})
  268. H = 0 1 - f ( X ; α ^ , β ^ ) ln ( f ( X ; α , β ) ) d X H=\int_{0}^{1}-f(X;\hat{\alpha},\hat{\beta})\ln(f(X;\alpha,\beta))\,{\rm d}X
  269. ln ( α , β , a , c | Y ) \displaystyle\ln\,\mathcal{L}(\alpha,\beta,a,c|Y)
  270. ln ( α , β , a , c | Y ) α = i = 1 N ln ( Y i - a ) - N ( - ψ ( α + β ) + ψ ( α ) ) - N ln ( c - a ) = 0 \frac{\partial\ln\mathcal{L}(\alpha,\beta,a,c|Y)}{\partial\alpha}=\sum_{i=1}^{% N}\ln(Y_{i}-a)-N(-\psi(\alpha+\beta)+\psi(\alpha))-N\ln(c-a)=0
  271. ln ( α , β , a , c | Y ) β = i = 1 N ln ( c - Y i ) - N ( - ψ ( α + β ) + ψ ( β ) ) - N ln ( c - a ) = 0 \frac{\partial\ln\mathcal{L}(\alpha,\beta,a,c|Y)}{\partial\beta}=\sum_{i=1}^{N% }\ln(c-Y_{i})-N(-\psi(\alpha+\beta)+\psi(\beta))-N\ln(c-a)=0
  272. ln ( α , β , a , c | Y ) a = - ( α - 1 ) i = 1 N 1 Y i - a + N ( α + β - 1 ) 1 c - a = 0 \frac{\partial\ln\mathcal{L}(\alpha,\beta,a,c|Y)}{\partial a}=-(\alpha-1)\sum_% {i=1}^{N}\frac{1}{Y_{i}-a}\,+N(\alpha+\beta-1)\frac{1}{c-a}=0
  273. ln ( α , β , a , c | Y ) c = ( β - 1 ) i = 1 N 1 c - Y i - N ( α + β - 1 ) 1 c - a = 0 \frac{\partial\ln\mathcal{L}(\alpha,\beta,a,c|Y)}{\partial c}=(\beta-1)\sum_{i% =1}^{N}\frac{1}{c-Y_{i}}\,-N(\alpha+\beta-1)\frac{1}{c-a}=0
  274. α ^ , β ^ , a ^ , c ^ \hat{\alpha},\hat{\beta},\hat{a},\hat{c}
  275. 1 N i = 1 N ln Y i - a ^ c ^ - a ^ = ψ ( α ^ ) - ψ ( α ^ + β ^ ) = ln G ^ X \frac{1}{N}\sum_{i=1}^{N}\ln\frac{Y_{i}-\hat{a}}{\hat{c}-\hat{a}}=\psi(\hat{% \alpha})-\psi(\hat{\alpha}+\hat{\beta})=\ln\hat{G}_{X}
  276. 1 N i = 1 N ln c ^ - Y i c ^ - a ^ = ψ ( β ^ ) - ψ ( α ^ + β ^ ) = ln G ^ 1 - X \frac{1}{N}\sum_{i=1}^{N}\ln\frac{\hat{c}-Y_{i}}{\hat{c}-\hat{a}}=\psi(\hat{% \beta})-\psi(\hat{\alpha}+\hat{\beta})=\ln\hat{G}_{1-X}
  277. 1 1 N i = 1 N c ^ - a ^ Y i - a ^ = α ^ - 1 α ^ + β ^ - 1 = H ^ X \frac{1}{\frac{1}{N}\sum_{i=1}^{N}\frac{\hat{c}-\hat{a}}{Y_{i}-\hat{a}}}=\frac% {\hat{\alpha}-1}{\hat{\alpha}+\hat{\beta}-1}=\hat{H}_{X}
  278. 1 1 N i = 1 N c ^ - a ^ c ^ - Y i = β ^ - 1 α ^ + β ^ - 1 = H ^ 1 - X \frac{1}{\frac{1}{N}\sum_{i=1}^{N}\frac{\hat{c}-\hat{a}}{\hat{c}-Y_{i}}}=\frac% {\hat{\beta}-1}{\hat{\alpha}+\hat{\beta}-1}=\hat{H}_{1-X}
  279. G ^ X = i = 1 N ( Y i - a ^ c ^ - a ^ ) 1 N \hat{G}_{X}=\prod_{i=1}^{N}\left(\frac{Y_{i}-\hat{a}}{\hat{c}-\hat{a}}\right)^% {\frac{1}{N}}
  280. G ^ ( 1 - X ) = i = 1 N ( c ^ - Y i c ^ - a ^ ) 1 N \hat{G}_{(1-X)}=\prod_{i=1}^{N}\left(\frac{\hat{c}-Y_{i}}{\hat{c}-\hat{a}}% \right)^{\frac{1}{N}}
  281. a ^ , c ^ \hat{a},\hat{c}
  282. α ^ , β ^ > 1 \hat{\alpha},\hat{\beta}>1
  283. α = 2 : E [ - 1 N 2 ln ( α , β , a , c | Y ) a 2 ] = a , a \alpha=2:\quad\operatorname{E}\left[-\frac{1}{N}\frac{\partial^{2}\ln\mathcal{% L}(\alpha,\beta,a,c|Y)}{\partial a^{2}}\right]={\mathcal{I}}_{a,a}
  284. β = 2 : E [ - 1 N 2 ln ( α , β , a , c | Y ) c 2 ] = c , c \beta=2:\quad\operatorname{E}\left[-\frac{1}{N}\frac{\partial^{2}\ln\mathcal{L% }(\alpha,\beta,a,c|Y)}{\partial c^{2}}\right]={\mathcal{I}}_{c,c}
  285. α = 2 : E [ - 1 N 2 ln ( α , β , a , c | Y ) α a ] = α , a \alpha=2:\quad\operatorname{E}\left[-\frac{1}{N}\frac{\partial^{2}\ln\mathcal{% L}(\alpha,\beta,a,c|Y)}{\partial\alpha\partial a}\right]={\mathcal{I}}_{\alpha% ,a}
  286. β = 1 : E [ - 1 N 2 ln ( α , β , a , c | Y ) β c ] = β , c \beta=1:\quad\operatorname{E}\left[-\frac{1}{N}\frac{\partial^{2}\ln\mathcal{L% }(\alpha,\beta,a,c|Y)}{\partial\beta\partial c}\right]={\mathcal{I}}_{\beta,c}
  287. ( α ) = E [ ( α ln ( α | X ) ) 2 ] , \mathcal{I}(\alpha)=\operatorname{E}\left[\left(\frac{\partial}{\partial\alpha% }\ln\mathcal{L}(\alpha|X)\right)^{2}\right],
  288. ( α ) = - E [ 2 α 2 ln ( ( α | X ) ) ] . \mathcal{I}(\alpha)=-\operatorname{E}\left[\frac{\partial^{2}}{\partial\alpha^% {2}}\ln(\mathcal{L}(\alpha|X))\right].
  289. var [ α ^ ] 1 ( α ) . \operatorname{var}[\hat{\alpha}]\geq\frac{1}{\mathcal{I}(\alpha)}.
  290. [ θ 1 θ 2 θ N ] , \begin{bmatrix}\theta_{1}\\ \theta_{2}\\ \dots\\ \theta_{N}\end{bmatrix},
  291. ( ( θ ) ) i , j = E [ ( θ i ln ) ( θ j ln ) ] . {(\mathcal{I}(\theta))}_{i,j}=\operatorname{E}\left[\left(\frac{\partial}{% \partial\theta_{i}}\ln\mathcal{L}\right)\left(\frac{\partial}{\partial\theta_{% j}}\ln\mathcal{L}\right)\right].
  292. ( ( θ ) ) i , j = - E [ 2 θ i θ j ln ( ) ] . {(\mathcal{I}(\theta))}_{i,j}=-\operatorname{E}\left[\frac{\partial^{2}}{% \partial\theta_{i}\,\partial\theta_{j}}\ln(\mathcal{L})\right]\,.
  293. ln ( ( α , β | X ) ) = ( α - 1 ) i = 1 N ln X i + ( β - 1 ) i = 1 N ln ( 1 - X i ) - N ln B ( α , β ) \ln(\mathcal{L}(\alpha,\beta|X))=(\alpha-1)\sum_{i=1}^{N}\ln X_{i}+(\beta-1)% \sum_{i=1}^{N}\ln(1-X_{i})-N\ln B(\alpha,\beta)
  294. 1 N ln ( ( α , β | X ) ) = ( α - 1 ) 1 N i = 1 N ln X i + ( β - 1 ) 1 N i = 1 N ln ( 1 - X i ) - ln B ( α , β ) \frac{1}{N}\ln(\mathcal{L}(\alpha,\beta|X))=(\alpha-1)\frac{1}{N}\sum_{i=1}^{N% }\ln X_{i}+(\beta-1)\frac{1}{N}\sum_{i=1}^{N}\ln(1-X_{i})-\,\ln B(\alpha,\beta)
  295. - 2 ln ( α , β | X ) N α 2 = var [ ln ( X ) ] = ψ 1 ( α ) - ψ 1 ( α + β ) = α , α = E [ - 2 ln ( α , β | X ) N α 2 ] = ln var GX -\frac{\partial^{2}\ln\mathcal{L}(\alpha,\beta|X)}{N\partial\alpha^{2}}=% \operatorname{var}[\ln(X)]=\psi_{1}(\alpha)-\psi_{1}(\alpha+\beta)={\mathcal{I% }}_{\alpha,\alpha}=\operatorname{E}\left[-\frac{\partial^{2}\ln\mathcal{L}(% \alpha,\beta|X)}{N\partial\alpha^{2}}\right]=\ln\,\operatorname{var_{GX}}
  296. - 2 ln ( α , β | X ) N β 2 = var [ ln ( 1 - X ) ] = ψ 1 ( β ) - ψ 1 ( α + β ) = β , β = E [ - 2 ln ( α , β | X ) N β 2 ] = ln var G ( 1 - X ) -\frac{\partial^{2}\ln\mathcal{L}(\alpha,\beta|X)}{N\partial\beta^{2}}=% \operatorname{var}[\ln(1-X)]=\psi_{1}(\beta)-\psi_{1}(\alpha+\beta)={\mathcal{% I}}_{\beta,\beta}=\operatorname{E}\left[-\frac{\partial^{2}\ln\mathcal{L}(% \alpha,\beta|X)}{N\partial\beta^{2}}\right]=\ln\,\operatorname{var_{G(1-X)}}
  297. - 2 ln ( α , β | X ) N α β = cov [ ln X , ln ( 1 - X ) ] = - ψ 1 ( α + β ) = α , β = E [ - 2 ln ( α , β | X ) N α β ] = ln cov GX , ( 1 - X ) -\frac{\partial^{2}\ln\mathcal{L}(\alpha,\beta|X)}{N{\partial\alpha}{\partial% \beta}}=\operatorname{cov}[\ln X,\ln(1-X)]=-\psi_{1}(\alpha+\beta)={\mathcal{I% }}_{\alpha,\beta}=\operatorname{E}\left[-\frac{\partial^{2}\ln\mathcal{L}(% \alpha,\beta|X)}{N{\partial\alpha}{\partial\beta}}\right]=\ln\,\operatorname{% cov_{G{X,(1-X)}}}
  298. α , β = β , α = ln cov GX , ( 1 - X ) {\mathcal{I}}_{\alpha,\beta}={\mathcal{I}}_{\beta,\alpha}=\ln\,\operatorname{% cov_{G{X,(1-X)}}}
  299. ψ 1 ( α ) = d 2 ln Γ ( α ) α 2 = ψ ( α ) α \psi_{1}(\alpha)=\frac{d^{2}\ln\Gamma(\alpha)}{\partial\alpha^{2}}=\,\frac{% \partial\psi(\alpha)}{\partial\alpha}
  300. α , α , β , β \mathcal{I}_{\alpha,\alpha},\mathcal{I}_{\beta,\beta}
  301. α , β {\mathcal{I}}_{\alpha,\beta}
  302. det ( ( α , β ) ) \displaystyle\det(\mathcal{I}(\alpha,\beta))
  303. f ( y ; α , β , a , c ) = f ( x ; α , β ) c - a = ( y - a c - a ) α - 1 ( c - y c - a ) β - 1 ( c - a ) B ( α , β ) = ( y - a ) α - 1 ( c - y ) β - 1 ( c - a ) α + β - 1 B ( α , β ) . f(y;\alpha,\beta,a,c)=\frac{f(x;\alpha,\beta)}{c-a}=\frac{\left(\frac{y-a}{c-a% }\right)^{\alpha-1}\left(\frac{c-y}{c-a}\right)^{\beta-1}}{(c-a)B(\alpha,\beta% )}=\frac{(y-a)^{\alpha-1}(c-y)^{\beta-1}}{(c-a)^{\alpha+\beta-1}B(\alpha,\beta% )}.
  304. 1 N ln ( ( α , β , a , c | Y ) ) = α - 1 N i = 1 N ln ( Y i - a ) + β - 1 N i = 1 N ln ( c - Y i ) - ln B ( α , β ) - ( α + β - 1 ) ln ( c - a ) \frac{1}{N}\ln(\mathcal{L}(\alpha,\beta,a,c|Y))=\frac{\alpha-1}{N}\sum_{i=1}^{% N}\ln(Y_{i}-a)+\frac{\beta-1}{N}\sum_{i=1}^{N}\ln(c-Y_{i})-\ln B(\alpha,\beta)% -(\alpha+\beta-1)\ln(c-a)
  305. - 1 N 2 ln ( α , β , a , c | Y ) α 2 = var [ ln ( X ) ] = ψ 1 ( α ) - ψ 1 ( α + β ) = α , α = E [ - 1 N 2 ln ( α , β , a , c | Y ) α 2 ] = ln ( var GX ) -\frac{1}{N}\frac{\partial^{2}\ln\mathcal{L}(\alpha,\beta,a,c|Y)}{\partial% \alpha^{2}}=\operatorname{var}[\ln(X)]=\psi_{1}(\alpha)-\psi_{1}(\alpha+\beta)% =\mathcal{I}_{\alpha,\alpha}=\operatorname{E}\left[-\frac{1}{N}\frac{\partial^% {2}\ln\mathcal{L}(\alpha,\beta,a,c|Y)}{\partial\alpha^{2}}\right]=\ln(% \operatorname{var_{GX}})
  306. - 1 N 2 ln ( α , β , a , c | Y ) β 2 = var [ ln ( 1 - X ) ] = ψ 1 ( β ) - ψ 1 ( α + β ) = β , β = E [ - 1 N 2 ln ( α , β , a , c | Y ) β 2 ] = ln ( var G ( 1 - X ) ) -\frac{1}{N}\frac{\partial^{2}\ln\mathcal{L}(\alpha,\beta,a,c|Y)}{\partial% \beta^{2}}=\operatorname{var}[\ln(1-X)]=\psi_{1}(\beta)-\psi_{1}(\alpha+\beta)% ={\mathcal{I}}_{\beta,\beta}=\operatorname{E}\left[-\frac{1}{N}\frac{\partial^% {2}\ln\mathcal{L}(\alpha,\beta,a,c|Y)}{\partial\beta^{2}}\right]=\ln(% \operatorname{var_{G(1-X)}})
  307. - 1 N 2 ln ( α , β , a , c | Y ) α β = cov [ ln X , ( 1 - X ) ] = - ψ 1 ( α + β ) = α , β = E [ - 1 N 2 ln ( α , β , a , c | Y ) α β ] = ln ( cov GX , ( 1 - X ) ) -\frac{1}{N}\frac{\partial^{2}\ln\mathcal{L}(\alpha,\beta,a,c|Y)}{{\partial% \alpha}{\partial\beta}}=\operatorname{cov}[\ln{X,(1-X)}]=-\psi_{1}(\alpha+% \beta)={\mathcal{I}}_{\alpha,\beta}=\operatorname{E}\left[-\frac{1}{N}\frac{% \partial^{2}\ln\mathcal{L}(\alpha,\beta,a,c|Y)}{\partial\alpha\partial\beta}% \right]=\ln(\operatorname{cov_{G{X,(1-X)}}})
  308. a , a {\mathcal{I}}_{a,a}
  309. α > 2 : E [ - 1 N 2 ln ( α , β , a , c | Y ) a 2 ] = a , a = β ( α + β - 1 ) ( α - 2 ) ( c - a ) 2 β > 2 : E [ - 1 N 2 ln ( α , β , a , c | Y ) c 2 ] = c , c = α ( α + β - 1 ) ( β - 2 ) ( c - a ) 2 E [ - 1 N 2 ln ( α , β , a , c | Y ) a c ] = a , c = ( α + β - 1 ) ( c - a ) 2 α > 1 : E [ - 1 N 2 ln ( α , β , a , c | Y ) α a ] = α , a = β ( α - 1 ) ( c - a ) E [ - 1 N 2 ln ( α , β , a , c | Y ) α c ] = α , c = 1 ( c - a ) E [ - 1 N 2 ln ( α , β , a , c | Y ) β a ] = β , a = - 1 ( c - a ) β > 1 : E [ - 1 N 2 ln ( α , β , a , c | Y ) β c ] = β , c = - α ( β - 1 ) ( c - a ) \begin{aligned}\displaystyle\alpha>2:\quad\operatorname{E}\left[-\frac{1}{N}% \frac{\partial^{2}\ln\mathcal{L}(\alpha,\beta,a,c|Y)}{\partial a^{2}}\right]&% \displaystyle={\mathcal{I}}_{a,a}=\frac{\beta(\alpha+\beta-1)}{(\alpha-2)(c-a)% ^{2}}\\ \displaystyle\beta>2:\quad\operatorname{E}\left[-\frac{1}{N}\frac{\partial^{2}% \ln\mathcal{L}(\alpha,\beta,a,c|Y)}{\partial c^{2}}\right]&\displaystyle=% \mathcal{I}_{c,c}=\frac{\alpha(\alpha+\beta-1)}{(\beta-2)(c-a)^{2}}\\ \displaystyle\operatorname{E}\left[-\frac{1}{N}\frac{\partial^{2}\ln\mathcal{L% }(\alpha,\beta,a,c|Y)}{\partial a\partial c}\right]&\displaystyle={\mathcal{I}% }_{a,c}=\frac{(\alpha+\beta-1)}{(c-a)^{2}}\\ \displaystyle\alpha>1:\quad\operatorname{E}\left[-\frac{1}{N}\frac{\partial^{2% }\ln\mathcal{L}(\alpha,\beta,a,c|Y)}{\partial\alpha\partial a}\right]&% \displaystyle=\mathcal{I}_{\alpha,a}=\frac{\beta}{(\alpha-1)(c-a)}\\ \displaystyle\operatorname{E}\left[-\frac{1}{N}\frac{\partial^{2}\ln\mathcal{L% }(\alpha,\beta,a,c|Y)}{\partial\alpha\partial c}\right]&\displaystyle={% \mathcal{I}}_{\alpha,c}=\frac{1}{(c-a)}\\ \displaystyle\operatorname{E}\left[-\frac{1}{N}\frac{\partial^{2}\ln\mathcal{L% }(\alpha,\beta,a,c|Y)}{\partial\beta\partial a}\right]&\displaystyle={\mathcal% {I}}_{\beta,a}=-\frac{1}{(c-a)}\\ \displaystyle\beta>1:\quad\operatorname{E}\left[-\frac{1}{N}\frac{\partial^{2}% \ln\mathcal{L}(\alpha,\beta,a,c|Y)}{\partial\beta\partial c}\right]&% \displaystyle=\mathcal{I}_{\beta,c}=-\frac{\alpha}{(\beta-1)(c-a)}\end{aligned}
  310. a , a \mathcal{I}_{a,a}
  311. c , c \mathcal{I}_{c,c}
  312. a , a \mathcal{I}_{a,a}
  313. c , c \mathcal{I}_{c,c}
  314. a , a \mathcal{I}_{a,a}
  315. α , a \mathcal{I}_{\alpha,a}
  316. α , α \mathcal{I}_{\alpha,\alpha}
  317. β , β \mathcal{I}_{\beta,\beta}
  318. α , a = E [ 1 - X X ] c - a = β ( α - 1 ) ( c - a ) if α > 1 \mathcal{I}_{\alpha,a}=\frac{\operatorname{E}\left[\frac{1-X}{X}\right]}{c-a}=% \frac{\beta}{(\alpha-1)(c-a)}\,\text{ if }\alpha>1
  319. β , c = - E [ X 1 - X ] c - a = - α ( β - 1 ) ( c - a ) if β > 1 \mathcal{I}_{\beta,c}=-\frac{\operatorname{E}\left[\frac{X}{1-X}\right]}{c-a}=% -\frac{\alpha}{(\beta-1)(c-a)}\,\text{ if }\beta>1
  320. α > 2 : a , a \displaystyle\alpha>2:\quad\mathcal{I}_{a,a}
  321. det ( ( α , β , a , c ) ) \displaystyle\det(\mathcal{I}(\alpha,\beta,a,c))
  322. a , a {\mathcal{I}}_{a,a}
  323. c , c {\mathcal{I}}_{c,c}
  324. a , a , c , c , α , a , β , c {\mathcal{I}}_{a,a},{\mathcal{I}}_{c,c},{\mathcal{I}}_{\alpha,a},{\mathcal{I}}% _{\beta,c}
  325. X Γ ( α , θ ) X\sim\Gamma(\alpha,\theta)
  326. Y Γ ( β , θ ) Y\sim\Gamma(\beta,\theta)
  327. X X + Y B ( α , β ) . \tfrac{X}{X+Y}\sim B(\alpha,\beta).
  328. B ( k , n + 1 - k ) B(k,n+1-k)
  329. X 1 - X β ( α , β ) \tfrac{X}{1-X}\sim{\beta^{{}^{\prime}}}(\alpha,\beta)
  330. m X n ( 1 - X ) F ( n , m ) \tfrac{mX}{n(1-X)}\sim F(n,m)
  331. X B ( 1 + λ m - min max - min , 1 + λ max - m max - min ) X\sim B\left(1+\lambda\tfrac{m-\min}{\max-\min},1+\lambda\tfrac{\max-m}{\max-% \min}\right)
  332. lim n n B ( 1 , n ) = Exponential ( 1 ) \lim_{n\to\infty}nB(1,n)={\rm\textrm{Exponential}}(1)
  333. lim n n B ( k , n ) = Gamma ( k , 1 ) \lim_{n\to\infty}nB(k,n)=\textrm{Gamma}(k,1)
  334. X X + Y B ( α , β ) \tfrac{X}{X+Y}\sim B(\alpha,\beta)\,
  335. X χ 2 ( α ) X\sim\chi^{2}(\alpha)\,
  336. Y χ 2 ( β ) Y\sim\chi^{2}(\beta)\,
  337. X X + Y B ( α 2 , β 2 ) \tfrac{X}{X+Y}\sim B(\tfrac{\alpha}{2},\tfrac{\beta}{2})
  338. Pr ( X α α + β x ) = Pr ( Y x ) \Pr(X\leq\tfrac{\alpha}{\alpha+\beta x})=\Pr(Y\geq x)\,
  339. B ( α , β ) = lim δ 0 NonCentralBeta ( α , β , δ ) B(\alpha,\beta)=\lim_{\delta\to 0}{\rm NonCentralBeta}(\alpha,\beta,\delta)
  340. U ( k ) B ( k , n + 1 - k ) . U_{(k)}\sim B(k,n+1-k).
  341. s + 1 n + 2 \frac{s+1}{n+2}
  342. B ( 1 , 1 ) B(1,1)
  343. P ( p ; α , β ) = p α - 1 ( 1 - p ) β - 1 B ( α , β ) . P(p;\alpha,\beta)=\frac{p^{\alpha-1}(1-p)^{\beta-1}}{B(\alpha,\beta)}.
  344. det ( ( α , β ) ) = ψ 1 ( α ) ψ 1 ( β ) - ( ψ 1 ( α ) + ψ 1 ( β ) ) ψ 1 ( α + β ) \scriptstyle\sqrt{\det(\mathcal{I}(\alpha,\beta))}=\sqrt{\psi_{1}(\alpha)\psi_% {1}(\beta)-(\psi_{1}(\alpha)+\psi_{1}(\beta))\psi_{1}(\alpha+\beta)}
  345. ln ( p | H ) = H ln ( p ) + ( 1 - H ) ln ( 1 - p ) . \ln\mathcal{L}(p|H)=H\ln(p)+(1-H)\ln(1-p).
  346. ( p ) \displaystyle\sqrt{\mathcal{I}(p)}
  347. ( p ) = n p ( 1 - p ) . \sqrt{\mathcal{I}(p)}=\frac{\sqrt{n}}{\sqrt{p(1-p)}}.
  348. 1 p ( 1 - p ) \frac{1}{\sqrt{p(1-p)}}
  349. B ( 1 2 , 1 2 ) = 1 π p ( 1 - p ) . B(\tfrac{1}{2},\tfrac{1}{2})=\frac{1}{\pi\sqrt{p(1-p)}}.
  350. 1 p ( 1 - p ) \scriptstyle\frac{1}{\sqrt{p(1-p)}}
  351. det ( ( α , β ) ) \displaystyle\sqrt{\det(\mathcal{I}(\alpha,\beta))}
  352. B ( 1 2 , 1 2 ) 1 θ ( 1 - θ ) B(\tfrac{1}{2},\tfrac{1}{2})\sim\frac{1}{\sqrt{\theta(1-\theta)}}
  353. ( s , f | x = p ) = ( s + f s ) x s ( 1 - x ) f = ( n s ) x s ( 1 - x ) n - s . \mathcal{L}(s,f|x=p)={s+f\choose s}x^{s}(1-x)^{f}={n\choose s}x^{s}(1-x)^{n-s}.
  354. PriorProbability ( x = p ; α Prior , β Prior ) = x α Prior - 1 ( 1 - x ) β Prior - 1 B ( α Prior , β Prior ) {\,\text{PriorProbability}}(x=p;\alpha\,\text{Prior},\beta\,\text{Prior})=% \frac{x^{\alpha\,\text{Prior}-1}(1-x)^{\beta\,\text{Prior}-1}}{B(\alpha\,\text% {Prior},\beta\,\text{Prior})}
  355. Posterior Probability ( x = p | s , n - s ) \displaystyle\,\text{Posterior Probability}(x=p|s,n-s)
  356. ( s + f s ) = ( n s ) = ( s + f ) ! s ! f ! = n ! s ! ( n - s ) ! {s+f\choose s}={n\choose s}=\frac{(s+f)!}{s!f!}=\frac{n!}{s!(n-s)!}
  357. x α Prior - 1 ( 1 - x ) β Prior - 1 x^{\alpha\,\text{Prior}-1}(1-x)^{\beta\,\text{Prior}-1}
  358. Posterior Probability ( p = x | s , f ) = x s ( 1 - x ) n - s B ( s + 1 , n - s + 1 ) , with mean = s + 1 n + 2 , (and mode= s n if 0 < s < n ) . \,\text{Posterior Probability}(p=x|s,f)=\frac{x^{s}(1-x)^{n-s}}{B(s+1,n-s+1)},% \,\text{ with mean = }\frac{s+1}{n+2},\,\text{ (and mode= }\frac{s}{n}\,\text{% if }0<s<n).
  359. Posterior Probability ( p = x | s , f ) = x s - 1 2 ( 1 - x ) n - s - 1 2 B ( s + 1 2 , n - s + 1 2 ) , with mean = s + 1 2 n + 1 , (and mode= s - 1 2 n - 1 if 1 2 < s < n - 1 2 ) . \,\text{Posterior Probability}(p=x|s,f)={x^{s-\tfrac{1}{2}}(1-x)^{n-s-\frac{1}% {2}}\over B(s+\tfrac{1}{2},n-s+\tfrac{1}{2})},\,\text{ with mean = }\frac{s+% \tfrac{1}{2}}{n+1},\,\text{ (and mode= }\frac{s-\tfrac{1}{2}}{n-1}\,\text{ if % }\tfrac{1}{2}<s<n-\tfrac{1}{2}).
  360. Posterior Probability ( p = x | s , f ) = x s - 1 ( 1 - x ) n - s - 1 B ( s , n - s ) , with mean = s n , (and mode= s - 1 n - 2 if 1 < s < n - 1 ) . \,\text{Posterior Probability}(p=x|s,f)=\frac{x^{s-1}(1-x)^{n-s-1}}{B(s,n-s)},% \,\text{ with mean = }\frac{s}{n},\,\text{ (and mode= }\frac{s-1}{n-2}\,\text{% if }1<s<n-1).
  361. 1 2 = 0.70710678... \scriptstyle\frac{1}{\sqrt{2}}=0.70710678...
  362. var = ( n - s + 1 ) ( s + 1 ) ( 3 + n ) ( 2 + n ) 2 , which for s = n 2 results in var = 1 12 + 4 n \,\text{var}=\frac{(n-s+1)(s+1)}{(3+n)(2+n)^{2}},\,\text{ which for }s=\frac{n% }{2}\,\text{ results in var}=\frac{1}{12+4n}
  363. var = ( n - s + 1 2 ) ( s + 1 2 ) ( 2 + n ) ( 1 + n ) 2 , which for s = n 2 results in var = 1 8 + 4 n \,\text{var}=\frac{(n-s+\frac{1}{2})(s+\frac{1}{2})}{(2+n)(1+n)^{2}},\,\text{ % which for }s=\frac{n}{2}\,\text{ results in var}=\frac{1}{8+4n}
  364. var = ( n - s ) s ( 1 + n ) n 2 , which for s = n 2 results in var = 1 4 + 4 n \,\text{var}=\frac{(n-s)s}{(1+n)n^{2}},\,\text{ which for }s=\frac{n}{2}\,% \text{ results in var}=\frac{1}{4+4n}
  365. var = μ ( 1 - μ ) 1 + ν = ( n - s ) s ( 1 + n ) n 2 \operatorname{var}=\frac{\mu(1-\mu)}{1+\nu}=\frac{(n-s)s}{(1+n)n^{2}}
  366. μ ( X ) = a + 4 b + c 6 σ ( X ) = c - a 6 \begin{aligned}\displaystyle\mu(X)&\displaystyle=\frac{a+4b+c}{6}\\ \displaystyle\sigma(X)&\displaystyle=\frac{c-a}{6}\end{aligned}
  367. μ ( X ) = a + 4 b + c 6 \mu(X)=\frac{a+4b+c}{6}
  368. σ ( X ) = ( c - a ) 2 1 + 2 α \sigma(X)=\frac{(c-a)}{2\sqrt{1+2\alpha}}
  369. - 6 3 + 2 α \frac{-6}{3+2\alpha}
  370. σ ( X ) = ( c - a ) α ( 6 - α ) 6 7 , \sigma(X)=\frac{(c-a)\sqrt{\alpha(6-\alpha)}}{6\sqrt{7}},
  371. ( 3 - α ) 7 2 α ( 6 - α ) \frac{(3-\alpha)\sqrt{7}}{2\sqrt{\alpha(6-\alpha)}}
  372. 21 α ( 6 - α ) - 3 \frac{21}{\alpha(6-\alpha)}-3
  373. α = 3 - 2 \alpha=3-\sqrt{2}
  374. 1 2 \frac{1}{\sqrt{2}}
  375. α = 3 + 2 \alpha=3+\sqrt{2}
  376. - 1 2 \frac{-1}{\sqrt{2}}
  377. α \displaystyle\alpha
  378. ν \displaystyle\nu
  379. y = x ( c - a ) + a , therefore x = y - a c - a . y=x(c-a)+a,\,\text{ therefore }x=\frac{y-a}{c-a}.
  380. f ( y ; α , β , a , c ) = f ( x ; α , β ) c - a = ( y - a c - a ) α - 1 ( c - y c - a ) β - 1 ( c - a ) B ( α , β ) = ( y - a ) α - 1 ( c - y ) β - 1 ( c - a ) α + β - 1 B ( α , β ) . f(y;\alpha,\beta,a,c)=\frac{f(x;\alpha,\beta)}{c-a}=\frac{\left(\frac{y-a}{c-a% }\right)^{\alpha-1}\left(\frac{c-y}{c-a}\right)^{\beta-1}}{(c-a)B(\alpha,\beta% )}=\frac{(y-a)^{\alpha-1}(c-y)^{\beta-1}}{(c-a)^{\alpha+\beta-1}B(\alpha,\beta% )}.
  381. Y B ( α , β , a , c ) . Y\sim B(\alpha,\beta,a,c).
  382. mean ( Y ) \displaystyle\,\text{mean}(Y)
  383. (mean deviaton around mean) ( Y ) = ( (mean deviaton around mean) ( X ) ) ( c - a ) = 2 α α β β B ( α , β ) ( α + β ) α + β + 1 ( c - a ) \,\text{(mean deviaton around mean)}(Y)=(\,\text{(mean deviaton around mean)}(% X))(c-a)=\frac{2\alpha^{\alpha}\beta^{\beta}}{B(\alpha,\beta)(\alpha+\beta)^{% \alpha+\beta+1}}(c-a)
  384. var ( Y ) = var ( X ) ( c - a ) 2 = α β ( c - a ) 2 ( α + β ) 2 ( α + β + 1 ) . \,\text{var}(Y)=\,\text{var}(X)(c-a)^{2}=\frac{\alpha\beta(c-a)^{2}}{(\alpha+% \beta)^{2}(\alpha+\beta+1)}.
  385. skewness ( Y ) = skewness ( X ) = 2 ( β - α ) α + β + 1 ( α + β + 2 ) α β . \,\text{skewness}(Y)=\,\text{skewness}(X)=\frac{2(\beta-\alpha)\sqrt{\alpha+% \beta+1}}{(\alpha+\beta+2)\sqrt{\alpha\beta}}.
  386. kurtosis excess ( Y ) = kurtosis excess ( X ) = 6 [ ( α - β ) 2 ( α + β + 1 ) - α β ( α + β + 2 ) ] α β ( α + β + 2 ) ( α + β + 3 ) \,\text{kurtosis excess}(Y)=\,\text{kurtosis excess}(X)=\frac{6[(\alpha-\beta)% ^{2}(\alpha+\beta+1)-\alpha\beta(\alpha+\beta+2)]}{\alpha\beta(\alpha+\beta+2)% (\alpha+\beta+3)}

Beta_function.html

  1. B ( x , y ) = 0 1 t x - 1 ( 1 - t ) y - 1 d t \mathrm{B}(x,y)=\int_{0}^{1}t^{x-1}(1-t)^{y-1}\,\mathrm{d}t\!
  2. Re ( x ) , Re ( y ) > 0. \textrm{Re}(x),\textrm{Re}(y)>0.\,
  3. B ( x , y ) = B ( y , x ) . B(x,y)=B(y,x).\!
  4. Γ \Gamma
  5. B ( x , y ) = ( x - 1 ) ! ( y - 1 ) ! ( x + y - 1 ) ! B(x,y)=\dfrac{(x-1)!\,(y-1)!}{(x+y-1)!}\!
  6. B ( x , y ) = Γ ( x ) Γ ( y ) Γ ( x + y ) B(x,y)=\dfrac{\Gamma(x)\,\Gamma(y)}{\Gamma(x+y)}\!
  7. B ( x , y ) = 2 0 π / 2 ( sin θ ) 2 x - 1 ( cos θ ) 2 y - 1 d θ , Re ( x ) > 0 , Re ( y ) > 0 B(x,y)=2\int_{0}^{\pi/2}(\sin\theta)^{2x-1}(\cos\theta)^{2y-1}\,\mathrm{d}% \theta,\qquad\mathrm{Re}(x)>0,\ \mathrm{Re}(y)>0\!
  8. B ( x , y ) = 0 t x - 1 ( 1 + t ) x + y d t , Re ( x ) > 0 , Re ( y ) > 0 B(x,y)=\int_{0}^{\infty}\dfrac{t^{x-1}}{(1+t)^{x+y}}\,\mathrm{d}t,\qquad% \mathrm{Re}(x)>0,\ \mathrm{Re}(y)>0\!
  9. B ( x , y ) = n = 0 < m t p l > ( n - y n ) x + n , B(x,y)=\sum_{n=0}^{\infty}\dfrac{<}{m}tpl>{{n-y\choose n}}{x+n},\!
  10. B ( x , y ) = x + y x y n = 1 ( 1 + x y n ( x + y + n ) ) - 1 , B(x,y)=\frac{x+y}{xy}\prod_{n=1}^{\infty}\left(1+\dfrac{xy}{n(x+y+n)}\right)^{% -1},\!
  11. B ( x , y ) = B ( x , y + 1 ) + B ( x + 1 , y ) B(x,y)=B(x,y+1)+B(x+1,y)\!
  12. B ( x + 1 , y ) = B ( x , y ) x x + y B(x+1,y)=B(x,y)\cdot\dfrac{x}{x+y}\!
  13. B ( x , y + 1 ) = B ( x , y ) y x + y B(x,y+1)=B(x,y)\cdot\dfrac{y}{x+y}\!
  14. B ( x , y ) ( t t + x + y - 1 ) = ( t t + x - 1 ) * ( t t + y - 1 ) x 1 , y 1 , B(x,y)\cdot(t\mapsto t_{+}^{x+y-1})=(t\to t_{+}^{x-1})*(t\to t_{+}^{y-1})% \qquad x\geq 1,y\geq 1,\!
  15. B ( x , y ) B ( x + y , 1 - y ) = π x sin ( π y ) , B(x,y)\cdot B(x+y,1-y)=\dfrac{\pi}{x\sin(\pi y)},\!
  16. t t + x t\mapsto t_{+}^{x}
  17. Γ ( 1 2 ) = π \Gamma(\tfrac{1}{2})=\sqrt{\pi}
  18. ( 1 - e 2 π i α ) ( 1 - e 2 π i β ) B ( α , β ) = C t α - 1 ( 1 - t ) β - 1 d t . \displaystyle(1-e^{2\pi i\alpha})(1-e^{2\pi i\beta})B(\alpha,\beta)=\int_{C}t^% {\alpha-1}(1-t)^{\beta-1}\,\mathrm{d}t.
  19. ( n k ) = 1 ( n + 1 ) B ( n - k + 1 , k + 1 ) . {n\choose k}=\frac{1}{(n+1)B(n-k+1,k+1)}.
  20. B B\,
  21. ( n k ) = ( - 1 ) n n ! sin ( π k ) π i = 0 n ( k - i ) . {n\choose k}=(-1)^{n}n!\cfrac{\sin(\pi k)}{\pi\prod_{i=0}^{n}(k-i)}.
  22. Γ ( x ) Γ ( y ) = 0 e - u u x - 1 d u 0 e - v v y - 1 d v = 0 0 e - u - v u x - 1 v y - 1 d u d v . \begin{aligned}\displaystyle\Gamma(x)\Gamma(y)&\displaystyle=\int_{0}^{\infty}% \ e^{-u}u^{x-1}\,\mathrm{d}u\int_{0}^{\infty}\ e^{-v}v^{y-1}\,\mathrm{d}v\\ &\displaystyle=\int_{0}^{\infty}\int_{0}^{\infty}\ e^{-u-v}u^{x-1}v^{y-1}\,% \mathrm{d}u\,\mathrm{d}v.\end{aligned}
  23. u = f ( z , t ) = z t u=f(z,t)=zt
  24. v = g ( z , t ) = z ( 1 - t ) v=g(z,t)=z(1-t)
  25. Γ ( x ) Γ ( y ) = z = 0 t = 0 1 e - z ( z t ) x - 1 ( z ( 1 - t ) ) y - 1 | J ( z , t ) | d t d z = z = 0 t = 0 1 e - z ( z t ) x - 1 ( z ( 1 - t ) ) y - 1 z d t d z = z = 0 e - z z x + y - 1 d z t = 0 1 t x - 1 ( 1 - t ) y - 1 d t , \begin{aligned}\displaystyle\Gamma(x)\Gamma(y)&\displaystyle=\int_{z=0}^{% \infty}\int_{t=0}^{1}e^{-z}(zt)^{x-1}(z(1-t))^{y-1}|J(z,t)|\,\mathrm{d}t\,% \mathrm{d}z\\ &\displaystyle=\int_{z=0}^{\infty}\int_{t=0}^{1}e^{-z}(zt)^{x-1}(z(1-t))^{y-1}% z\,\mathrm{d}t\,\mathrm{d}z\\ &\displaystyle=\int_{z=0}^{\infty}e^{-z}z^{x+y-1}\,\mathrm{d}z\int_{t=0}^{1}t^% {x-1}(1-t)^{y-1}\,\mathrm{d}t,\end{aligned}
  26. | J ( z , t ) | |J(z,t)|
  27. u = f ( z , t ) u=f(z,t)
  28. v = g ( z , t ) v=g(z,t)
  29. Γ ( x ) Γ ( y ) = Γ ( x + y ) B ( x , y ) . \Gamma(x)\,\Gamma(y)=\Gamma(x+y)B(x,y).
  30. f ( u ) := e - u u x - 1 1 \R + f(u):=e^{-u}u^{x-1}1_{\R_{+}}
  31. g ( u ) := e - u u y - 1 1 \R + g(u):=e^{-u}u^{y-1}1_{\R_{+}}
  32. Γ ( x ) Γ ( y ) = ( \R f ( u ) d u ) ( \R g ( u ) d u ) = \R ( f * g ) ( u ) d u = B ( x , y ) Γ ( x + y ) . \Gamma(x)\Gamma(y)=\left(\int_{\R}f(u)\mathrm{d}u\right)\left(\int_{\R}g(u)% \mathrm{d}u\right)=\int_{\R}(f*g)(u)\mathrm{d}u=B(x,y)\,\Gamma(x+y).
  33. x B ( x , y ) = B ( x , y ) ( Γ ( x ) Γ ( x ) - Γ ( x + y ) Γ ( x + y ) ) = B ( x , y ) ( ψ ( x ) - ψ ( x + y ) ) , {\partial\over\partial x}\mathrm{B}(x,y)=\mathrm{B}(x,y)\left({\Gamma^{\prime}% (x)\over\Gamma(x)}-{\Gamma^{\prime}(x+y)\over\Gamma(x+y)}\right)=\mathrm{B}(x,% y)(\psi(x)-\psi(x+y)),
  34. ψ ( x ) \ \psi(x)
  35. B ( x , y ) 2 π x x - 1 2 y y - 1 2 ( x + y ) x + y - 1 2 B(x,y)\sim\sqrt{2\pi}\frac{{x^{x-\frac{1}{2}}y^{y-\frac{1}{2}}}}{{\left({x+y}% \right)^{x+y-\frac{1}{2}}}}
  36. B ( x , y ) Γ ( y ) x - y . B(x,y)\sim\Gamma(y)\,x^{-y}.
  37. B ( x ; a , b ) = 0 x t a - 1 ( 1 - t ) b - 1 d t . B(x;\,a,b)=\int_{0}^{x}t^{a-1}\,(1-t)^{b-1}\,\mathrm{d}t.\!
  38. I x ( a , b ) = B ( x ; a , b ) B ( a , b ) . I_{x}(a,b)=\dfrac{B(x;\,a,b)}{B(a,b)}.\!
  39. F ( k ; n , p ) = Pr ( X k ) = I 1 - p ( n - k , k + 1 ) = 1 - I p ( k + 1 , n - k ) . F(k;n,p)=\Pr(X\leq k)=I_{1-p}(n-k,k+1)=1-I_{p}(k+1,n-k).
  40. I 0 ( a , b ) = 0 I_{0}(a,b)=0\,
  41. I 1 ( a , b ) = 1 I_{1}(a,b)=1\,
  42. I x ( a , 1 ) = x a I_{x}(a,1)=x^{a}\,
  43. I x ( 1 , b ) = 1 - ( 1 - x ) b I_{x}(1,b)=1-(1-x)^{b}\,
  44. I x ( a , b ) = 1 - I 1 - x ( b , a ) I_{x}(a,b)=1-I_{1-x}(b,a)\,
  45. I x ( a + 1 , b ) = I x ( a , b ) - x a ( 1 - x ) b a B ( a , b ) I_{x}(a+1,b)=I_{x}(a,b)-\frac{x^{a}(1-x)^{b}}{aB(a,b)}\,
  46. I x ( a , b + 1 ) = I x ( a , b ) + x a ( 1 - x ) b b B ( a , b ) I_{x}(a,b+1)=I_{x}(a,b)+\frac{x^{a}(1-x)^{b}}{bB(a,b)}\,
  47. B ( s y m b o l α ) = i = 1 K Γ ( α i ) Γ ( i = 1 K α i ) , \qquadsymbol α = ( α 1 , , α K ) . \mathrm{B}(symbol{\alpha})=\frac{\prod_{i=1}^{K}\Gamma(\alpha_{i})}{\Gamma% \left(\sum_{i=1}^{K}\alpha_{i}\right)},\qquadsymbol{\alpha}=(\alpha_{1},\ldots% ,\alpha_{K}).

Bézier_surface.html

  1. 𝐩 ( u , v ) = i = 0 n j = 0 m B i n ( u ) B j m ( v ) 𝐤 i , j \mathbf{p}(u,v)=\sum_{i=0}^{n}\sum_{j=0}^{m}B_{i}^{n}(u)\;B_{j}^{m}(v)\;% \mathbf{k}_{i,j}
  2. B i n ( u ) = ( n i ) u i ( 1 - u ) n - i B_{i}^{n}(u)={n\choose i}\;u^{i}(1-u)^{n-i}
  3. ( n i ) = n ! i ! ( n - i ) ! {n\choose i}=\frac{n!}{i!(n-i)!}

Bézier_triangle.html

  1. p ( s , t , u ) = ( α s + β t + γ u ) 3 = β 3 t 3 + 3 α β 2 s t 2 + 3 β 2 γ t 2 u + 3 α 2 β s 2 t + 6 α β γ s t u + 3 β γ 2 t u 2 + α 3 s 3 + 3 α 2 γ s 2 u + 3 α γ 2 s u 2 + γ 3 u 3 \begin{aligned}\displaystyle p(s,t,u)=(\alpha s+\beta t+\gamma u)^{3}=&% \displaystyle\beta^{3}\ t^{3}+3\ \alpha\beta^{2}\ st^{2}+3\ \beta^{2}\gamma\ t% ^{2}u+\\ &\displaystyle 3\ \alpha^{2}\beta\ s^{2}t+6\ \alpha\beta\gamma\ stu+3\ \beta% \gamma^{2}\ tu^{2}+\\ &\displaystyle\alpha^{3}\ s^{3}+3\ \alpha^{2}\gamma\ s^{2}u+3\ \alpha\gamma^{2% }\ su^{2}+\gamma^{3}\ u^{3}\end{aligned}
  2. ( s y m b o l α 3 s y m b o l α 2 β s y m b o l α β 2 s y m b o l β 3 s y m b o l α 2 γ s y m b o l α β γ s y m b o l β 2 γ s y m b o l α γ 2 s y m b o l β γ 2 s y m b o l γ 3 ) = ( 1 0 0 0 0 0 0 0 0 0 1 2 1 2 0 0 0 0 0 0 0 0 1 4 2 4 1 4 0 0 0 0 0 0 0 1 8 3 8 3 8 1 8 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 2 1 2 0 0 0 0 0 0 0 0 1 4 2 4 1 4 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 2 1 2 0 0 0 0 0 0 0 0 0 0 1 ) ( s y m b o l α 3 s y m b o l α 2 β s y m b o l α β 2 s y m b o l β 3 s y m b o l α 2 γ s y m b o l α β γ s y m b o l β 2 γ s y m b o l α γ 2 s y m b o l β γ 2 s y m b o l γ 3 ) \begin{pmatrix}symbol{\alpha^{3}}{{}^{\prime}}\\ symbol{\alpha^{2}\beta}{{}^{\prime}}\\ symbol{\alpha\beta^{2}}{{}^{\prime}}\\ symbol{\beta^{3}}{{}^{\prime}}\\ symbol{\alpha^{2}\gamma}{{}^{\prime}}\\ symbol{\alpha\beta\gamma}{{}^{\prime}}\\ symbol{\beta^{2}\gamma}{{}^{\prime}}\\ symbol{\alpha\gamma^{2}}{{}^{\prime}}\\ symbol{\beta\gamma^{2}}{{}^{\prime}}\\ symbol{\gamma^{3}}{{}^{\prime}}\end{pmatrix}=\begin{pmatrix}1&0&0&0&0&0&0&0&0&% 0\\ {1\over 2}&{1\over 2}&0&0&0&0&0&0&0&0\\ {1\over 4}&{2\over 4}&{1\over 4}&0&0&0&0&0&0&0\\ {1\over 8}&{3\over 8}&{3\over 8}&{1\over 8}&0&0&0&0&0&0\\ 0&0&0&0&1&0&0&0&0&0\\ 0&0&0&0&{1\over 2}&{1\over 2}&0&0&0&0\\ 0&0&0&0&{1\over 4}&{2\over 4}&{1\over 4}&0&0&0\\ 0&0&0&0&0&0&0&1&0&0\\ 0&0&0&0&0&0&0&{1\over 2}&{1\over 2}&0\\ 0&0&0&0&0&0&0&0&0&1\end{pmatrix}\cdot\begin{pmatrix}symbol{\alpha^{3}}\\ symbol{\alpha^{2}\beta}\\ symbol{\alpha\beta^{2}}\\ symbol{\beta^{3}}\\ symbol{\alpha^{2}\gamma}\\ symbol{\alpha\beta\gamma}\\ symbol{\beta^{2}\gamma}\\ symbol{\alpha\gamma^{2}}\\ symbol{\beta\gamma^{2}}\\ symbol{\gamma^{3}}\end{pmatrix}
  3. ( α s + β t + γ u ) n = i + j + k = n i , j , k 0 ( n i j k ) s i t j u k α i β j γ k = i + j + k = n i , j , k 0 n ! i ! j ! k ! s i t j u k α i β j γ k (\alpha s+\beta t+\gamma u)^{n}=\sum_{\begin{smallmatrix}i+j+k=n\\ i,j,k\geq 0\end{smallmatrix}}{n\choose i\ j\ k}s^{i}t^{j}u^{k}\alpha^{i}\beta^% {j}\gamma^{k}=\sum_{\begin{smallmatrix}i+j+k=n\\ i,j,k\geq 0\end{smallmatrix}}\frac{n!}{i!j!k!}s^{i}t^{j}u^{k}\alpha^{i}\beta^{% j}\gamma^{k}

Bézout's_theorem.html

  1. n + 1 n+1
  2. d 1 , , d n d_{1},\ldots,d_{n}
  3. d 1 d n , d_{1}\cdots d_{n},
  4. d 1 , , d n . d_{1},\ldots,d_{n}.
  5. d 1 d n . d_{1}\cdots d_{n}.
  6. d 1 d n d_{1}\cdots d_{n}
  7. δ \delta
  8. d 1 d_{1}
  9. d 2 d_{2}
  10. δ - 1 \delta-1
  11. d 1 d 2 . d_{1}d_{2}.
  12. ( x - a ) 2 + ( y - b ) 2 = r 2 (x-a)^{2}+(y-b)^{2}=r^{2}
  13. ( x - a z ) 2 + ( y - b z ) 2 - r 2 z 2 = 0 , (x-az)^{2}+(y-bz)^{2}-r^{2}z^{2}=0,
  14. x 2 + 4 y 2 - 1 = 0 : two intersections of multiplicity 2 x^{2}+4y^{2}-1=0:\ \hbox{two intersections of multiplicity 2}
  15. 5 x 2 + 6 x y + 5 y 2 + 6 y - 5 = 0 : an intersection of multiplicity 3 5x^{2}+6xy+5y^{2}+6y-5=0:\ \hbox{an intersection of multiplicity 3}
  16. 4 x 2 + y 2 + 6 x + 2 = 0 : an intersection of multiplicity 4 4x^{2}+y^{2}+6x+2=0:\ \hbox{an intersection of multiplicity 4}
  17. a 0 z m + a 1 z m - 1 + + a m - 1 z + a m = 0 a_{0}z^{m}+a_{1}z^{m-1}+\dots+a_{m-1}z+a_{m}=0
  18. b 0 z n + b 1 z n - 1 + + b n - 1 z + b n = 0 b_{0}z^{n}+b_{1}z^{n-1}+\dots+b_{n-1}z+b_{n}=0
  19. S = ( a 0 a 1 a 2 a 3 a 4 0 0 0 a 0 a 1 a 2 a 3 a 4 0 0 0 a 0 a 1 a 2 a 3 a 4 b 0 b 1 b 2 b 3 0 0 0 0 b 0 b 1 b 2 b 3 0 0 0 0 b 0 b 1 b 2 b 3 0 0 0 0 b 0 b 1 b 2 b 3 ) . S=\begin{pmatrix}a_{0}&a_{1}&a_{2}&a_{3}&a_{4}&0&0\\ 0&a_{0}&a_{1}&a_{2}&a_{3}&a_{4}&0\\ 0&0&a_{0}&a_{1}&a_{2}&a_{3}&a_{4}\\ b_{0}&b_{1}&b_{2}&b_{3}&0&0&0\\ 0&b_{0}&b_{1}&b_{2}&b_{3}&0&0\\ 0&0&b_{0}&b_{1}&b_{2}&b_{3}&0\\ 0&0&0&b_{0}&b_{1}&b_{2}&b_{3}\\ \end{pmatrix}.

Bialgebra.html

  1. ( id B Δ ) Δ = ( Δ id B ) Δ (\mathrm{id}_{B}\otimes\Delta)\circ\Delta=(\Delta\otimes\mathrm{id}_{B})\circ\Delta
  2. ( id B ϵ ) Δ = id B = ( ϵ id B ) Δ (\mathrm{id}_{B}\otimes\epsilon)\circ\Delta=\mathrm{id}_{B}=(\epsilon\otimes% \mathrm{id}_{B})\circ\Delta
  3. η 2 := ( η η ) : K K K ( B B ) \eta_{2}:=(\eta\otimes\eta):K\otimes K\equiv K\to(B\otimes B)
  4. 2 := ( ) ( i d τ i d ) : ( B B ) ( B B ) ( B B ) \nabla_{2}:=(\nabla\otimes\nabla)\circ(id\otimes\tau\otimes id):(B\otimes B)% \otimes(B\otimes B)\to(B\otimes B)
  5. 2 ( ( x 1 x 2 ) ( y 1 y 2 ) ) = ( x 1 y 1 ) ( x 2 y 2 ) \nabla_{2}((x_{1}\otimes x_{2})\otimes(y_{1}\otimes y_{2}))=\nabla(x_{1}% \otimes y_{1})\otimes\nabla(x_{2}\otimes y_{2})
  6. ( x 1 x 2 ) ( y 1 y 2 ) = x 1 y 1 x 2 y 2 (x_{1}\otimes x_{2})(y_{1}\otimes y_{2})=x_{1}y_{1}\otimes x_{2}y_{2}
  7. ϵ 2 := ( ϵ ϵ ) : ( B B ) K K K \epsilon_{2}:=(\epsilon\otimes\epsilon):(B\otimes B)\to K\otimes K\equiv K
  8. Δ 2 := ( i d τ i d ) ( Δ Δ ) : ( B B ) ( B B ) ( B B ) \Delta_{2}:=(id\otimes\tau\otimes id)\circ(\Delta\otimes\Delta):(B\otimes B)% \to(B\otimes B)\otimes(B\otimes B)
  9. Δ = 2 ( Δ Δ ) : ( B B ) ( B B ) \Delta\circ\nabla=\nabla_{2}\circ(\Delta\otimes\Delta):(B\otimes B)\to(B% \otimes B)
  10. Δ η = η 2 : K ( B B ) \Delta\circ\eta=\eta_{2}:K\to(B\otimes B)
  11. ϵ = 0 ( ϵ ϵ ) : ( B B ) K \epsilon\circ\nabla=\nabla_{0}\circ(\epsilon\otimes\epsilon):(B\otimes B)\to K
  12. ϵ η = η 0 : K K \epsilon\circ\eta=\eta_{0}:K\to K
  13. Δ 2 = Δ : ( B B ) ( B B ) , \nabla\otimes\nabla\circ\Delta_{2}=\Delta\circ\nabla:(B\otimes B)\to(B\otimes B),
  14. ϵ = 0 ϵ 2 : ( B B ) K \epsilon\circ\nabla=\nabla_{0}\circ\epsilon_{2}:(B\otimes B)\to K
  15. Δ η = η 2 Δ 0 : K ( B B ) , \Delta\circ\eta=\eta_{2}\circ\Delta_{0}:K\to(B\otimes B),
  16. ϵ η = η 0 ϵ 0 : K K \epsilon\circ\eta=\eta_{0}\circ\epsilon_{0}:K\to K
  17. \mathbb{R}
  18. G \mathbb{R}^{G}
  19. Δ ( 𝐞 g ) = 𝐞 g 𝐞 g , \Delta(\mathbf{e}_{g})=\mathbf{e}_{g}\otimes\mathbf{e}_{g}\,,
  20. G \mathbb{R}^{G}
  21. ε ( 𝐞 g ) = 1 , \varepsilon(\mathbf{e}_{g})=1\,,
  22. G \mathbb{R}^{G}
  23. ( 𝐞 g 𝐞 h ) = 𝐞 g h , \nabla\bigl(\mathbf{e}_{g}\otimes\mathbf{e}_{h}\bigr)=\mathbf{e}_{gh}\,,
  24. G G \mathbb{R}^{G}\otimes\mathbb{R}^{G}
  25. η = 𝐞 i , \eta=\mathbf{e}_{i}\;,

BibTeX.html

  1. BIB T E X {\mathrm{B{\scriptstyle{IB}}\!T\!_{\displaystyle E}\!X}}

Bifurcation_diagram.html

  1. x n + 1 = r x n ( 1 - x n ) . x_{n+1}=rx_{n}(1-x_{n}).\,
  2. x n + 1 = x n 2 - c x_{n+1}=x_{n}^{2}-c
  3. x ¨ + f ( x ; μ ) + ϵ g ( x ) = 0 \ddot{x}+f(x;\mu)+\epsilon g(x)=0
  4. μ 0 \mu\neq 0
  5. μ \mu
  6. ϵ \epsilon
  7. ϵ = 0 \epsilon=0
  8. ϵ 0 \epsilon\neq 0

Big_Rip.html

  1. w w
  2. w w
  3. w w
  4. w w
  5. w w
  6. w w
  7. w w
  8. w w
  9. w w
  10. w w
  11. w w

Bin_packing_problem.html

  1. S S
  2. V V
  3. n n
  4. a 1 , , a n a_{1},\dots,a_{n}
  5. B B
  6. B B
  7. S 1 S B S_{1}\cup\dots\cup S_{B}
  8. { 1 , , n } \{1,\dots,n\}
  9. i S k a i V \sum_{i\in S_{k}}a_{i}\leq V
  10. k = 1 , , B k=1,\dots,B
  11. B B
  12. B B
  13. B = i = 1 n y i B=\sum_{i=1}^{n}y_{i}
  14. B 1 , B\geq 1,
  15. j = 1 n a j x i j V y i , \sum_{j=1}^{n}a_{j}x_{ij}\leq Vy_{i},
  16. i { 1 , , n } \forall i\in\{1,\ldots,n\}
  17. i = 1 n x i j = 1 , \sum_{i=1}^{n}x_{ij}=1,
  18. j { 1 , , n } \forall j\in\{1,\ldots,n\}
  19. y i { 0 , 1 } , y_{i}\in\{0,1\},
  20. i { 1 , , n } \forall i\in\{1,\ldots,n\}
  21. x i j { 0 , 1 } , x_{ij}\in\{0,1\},
  22. i { 1 , , n } j { 1 , , n } \forall i\in\{1,\ldots,n\}\,\forall j\in\{1,\ldots,n\}
  23. y i = 1 y_{i}=1
  24. i i
  25. x i j = 1 x_{ij}=1
  26. j j
  27. i i
  28. i = 1 n a i > B - 1 2 V \sum_{i=1}^{n}a_{i}>\tfrac{B-1}{2}V
  29. i = 1 n a i V \tfrac{\sum_{i=1}^{n}a_{i}}{V}

Binary_logarithm.html

  1. 2 2
  2. n n
  3. 2 2
  4. n n
  5. x = log 2 n 2 x = n . x=\log_{2}n\quad\Longleftrightarrow\quad 2^{x}=n.
  6. 1 1
  7. 0
  8. 2 2
  9. 1 1
  10. 4 4
  11. 2 2
  12. 8 8
  13. 3 3
  14. 16 16
  15. 4 4
  16. 32 32
  17. 5 5
  18. n n
  19. l g n lgn
  20. l o g n logn
  21. 2 2
  22. l d n ldn
  23. l b n lbn
  24. l g n lgn
  25. n n
  26. log 2 n + 1. \lfloor\log_{2}n\rfloor+1.\,
  27. n n
  28. n n
  29. n n
  30. n n
  31. n n
  32. n n
  33. n n
  34. n n
  35. 1 1
  36. n n
  37. k k
  38. 1 1
  39. O ( l o g n ) O(logn)
  40. O ( n l o g n ) O(nlogn)
  41. O ( n l o g n ) O(nlogn)
  42. O ( l o g n ) O(logn)
  43. O ( n l o g n ) O(nlogn)
  44. n n
  45. 2 2
  46. 1 1
  47. 1 −1
  48. 2 : 1 2:1
  49. x x
  50. y y
  51. z z
  52. x x
  53. y y
  54. y y
  55. z z
  56. x x
  57. z z
  58. 1200 1200
  59. 12 12
  60. 100 100
  61. | 1200 log 2 f 1 f 2 | . \left|1200\log_{2}\frac{f_{1}}{f_{2}}\right|.
  62. 1000 1000
  63. 1200 1200
  64. 4 4
  65. 32 32
  66. n n
  67. n n
  68. 2.585 2.585
  69. 3 3
  70. 6 6
  71. 3 3
  72. 2 2
  73. log 2 N 2 t \log_{2}\frac{N^{2}}{t}
  74. N N
  75. t t
  76. l n ln
  77. l o g log
  78. log 2 n = ln n ln 2 = log 10 n log 10 2 , \log_{2}n=\frac{\ln n}{\ln 2}=\frac{\log_{10}n}{\log_{10}2},
  79. log 2 n 1.442695 ln n 3.321928 log 10 n . \log_{2}n\approx 1.442695\ln n\approx 3.321928\log_{10}n.
  80. log 2 ( n ) = log 2 ( n + 1 ) - 1 , if n 1. \lfloor\log_{2}(n)\rfloor=\lceil\log_{2}(n+1)\rceil-1,\,\text{ if }n\geq 1.
  81. log 2 ( 0 ) = - 1 \lfloor\log_{2}(0)\rfloor=-1
  82. x x
  83. n l z ( x ) nlz(x)
  84. log 2 ( n ) = 31 - nlz ( n ) . \lfloor\log_{2}(n)\rfloor=31-\operatorname{nlz}(n).
  85. 1 1
  86. 1 1
  87. log 2 x \lfloor\log_{2}x\rfloor
  88. x > 0 x>0
  89. n n
  90. n n
  91. log 2 x = n + log 2 y where y = 2 - n x and y [ 1 , 2 ) \log_{2}x=n+\log_{2}y\quad\,\text{where }y=2^{-n}x\,\text{ and }y\in[1,2)
  92. y y
  93. 1 , 2 ) ) 1,2))
  94. y = 1 y=1
  95. y y
  96. z z
  97. 2 , 4 ) ) 2,4))
  98. m m
  99. z = y < s u p > 2 < s u p > m z=y<sup>2<sup>m
  100. log 2 z = 2 m log 2 y log 2 y = log 2 z 2 m = 1 + log 2 ( z / 2 ) 2 m = 2 - m + 2 - m log 2 ( z / 2 ) \begin{aligned}\displaystyle\log_{2}z&\displaystyle=2^{m}\log_{2}y\\ \displaystyle\log_{2}y&\displaystyle=\frac{\log_{2}z}{2^{m}}\\ &\displaystyle=\frac{1+\log_{2}(z/2)}{2^{m}}\\ &\displaystyle=2^{-m}+2^{-m}\log_{2}(z/2)\end{aligned}
  101. z / 2 z/2
  102. 1 , 2 ) ) 1,2))
  103. z / 2 z/2
  104. m i m_{i}
  105. log 2 x \displaystyle\log_{2}x
  106. m < s u b > i > 0 m<sub>i>0
  107. k k
  108. 2 < s u p > k 2<sup>k

Binary_number.html

  1. n n
  2. n n
  3. $\or$
  4. 1 1
  5. $\or$
  6. 1 = 1 1=1
  7. 1 + 1 = 10 1+1=10
  8. and \and
  9. 1 2 \begin{matrix}\frac{1}{2}\end{matrix}
  10. ( 1 2 ) 2 = 1 4 \begin{matrix}(\frac{1}{2})^{2}=\frac{1}{4}\end{matrix}
  11. 1 2 \begin{matrix}\frac{1}{2}\end{matrix}
  12. ( 1 3 ) \begin{matrix}(\frac{1}{3})\end{matrix}
  13. 1 3 \begin{matrix}\frac{1}{3}\end{matrix}
  14. 1 3 × 2 = 2 3 < 1 \begin{matrix}\frac{1}{3}\times 2=\frac{2}{3}<1\end{matrix}
  15. 2 3 × 2 = 1 1 3 1 \begin{matrix}\frac{2}{3}\times 2=1\frac{1}{3}\geq 1\end{matrix}
  16. 1 3 × 2 = 2 3 < 1 \begin{matrix}\frac{1}{3}\times 2=\frac{2}{3}<1\end{matrix}
  17. 2 3 × 2 = 1 1 3 1 \begin{matrix}\frac{2}{3}\times 2=1\frac{1}{3}\geq 1\end{matrix}
  18. 3 ¯ \overline{3}
  19. 01 ¯ \overline{01}
  20. 0011 ¯ \overline{0011}
  21. x \displaystyle x
  22. x x
  23. x x
  24. x x
  25. x x
  26. p 2 a \frac{p}{2^{a}}
  27. 1 10 3 10 \frac{1_{10}}{3_{10}}
  28. 1 2 11 2 \frac{1_{2}}{11_{2}}
  29. 01 ¯ \overline{01}
  30. 12 10 17 10 \frac{12_{10}}{17_{10}}
  31. 1100 2 10001 2 \frac{1100_{2}}{10001_{2}}
  32. 10110100 ¯ \overline{10110100}
  33. 2 \sqrt{2}

Binary_symmetric_channel.html

  1. B S C p BSC_{p}
  2. e e
  3. B S C p BSC_{p}
  4. 1 1
  5. p p
  6. 0
  7. 1 - p 1-p
  8. e B S C p e\in BSC_{p}
  9. p < 1 2 p<\frac{1}{2}
  10. ϵ \epsilon
  11. 0 < ϵ < 1 2 - p 0<\epsilon<\frac{1}{2}-p
  12. n n
  13. p p
  14. ϵ \epsilon
  15. k ( 1 - H ( p + ϵ ) ) n k\leq\lfloor(1-H(p+\epsilon))n\rfloor
  16. E E
  17. { 0 , 1 } k { 0 , 1 } n \{0,1\}^{k}\rightarrow\{0,1\}^{n}
  18. D D
  19. { 0 , 1 } n \{0,1\}^{n}
  20. \rightarrow
  21. { 0 , 1 } k \{0,1\}^{k}
  22. m { 0 , 1 } k m\in\{0,1\}^{k}
  23. Pr e B S C p [ D ( E ( m ) + e ) m ] 2 - δ n \Pr_{e\in BSC_{p}}[D(E(m)+e)\neq m]\leq 2^{-{\delta}n}
  24. { 0 , 1 } k \{0,1\}^{k}
  25. E E
  26. B S C p BSC_{p}
  27. k k
  28. E E
  29. D D
  30. E E
  31. E E
  32. { 0 , 1 } k { 0 , 1 } n \{0,1\}^{k}\rightarrow\{0,1\}^{n}
  33. m { 0 , 1 } k m\in\{0,1\}^{k}
  34. E ( m ) { 0 , 1 } n E(m)\in\{0,1\}^{n}
  35. D D
  36. E E
  37. D D
  38. { 0 , 1 } n { 0 , 1 } k \{0,1\}^{n}\rightarrow\{0,1\}^{k}
  39. y y
  40. \in
  41. { 0 , 1 } n \{0,1\}^{n}
  42. m { 0 , 1 } k m\in\{0,1\}^{k}
  43. Δ ( y , E ( m ) ) \Delta(y,E(m))
  44. ( E , D ) (E,D)
  45. p p
  46. ϵ \epsilon
  47. m { 0 , 1 } k m\in\{0,1\}^{k}
  48. E E
  49. B S C p BSC_{p}
  50. m m
  51. m m
  52. p p
  53. ϵ \epsilon
  54. m { 0 , 1 } k m\in\{0,1\}^{k}
  55. m m
  56. 𝔼 E [ Pr e B S C p [ D ( E ( m ) + e ) m ] ] \mathbb{E}_{E}[\Pr_{e\in BSC_{p}}[D(E(m)+e)\neq m]]
  57. y y
  58. D ( y ) D(y)
  59. m m
  60. y y
  61. ( p + ϵ ) n (p+\epsilon)n
  62. E ( m ) E(m)
  63. m { 0 , 1 } k m^{\prime}\in\{0,1\}^{k}
  64. Δ ( y , E ( m ) ) Δ ( y , E ( m ) ) \Delta(y,E(m^{\prime}))\leq\Delta(y,E(m))
  65. P r e B S C p Pr_{e\in BSC_{p}}
  66. [ Δ ( y , E ( m ) ) > ( p + ϵ ) n ] 2 - ϵ 2 n [\Delta(y,E(m))>(p+\epsilon)n]\leq 2^{-{\epsilon^{2}}n}
  67. n n
  68. ϵ \epsilon
  69. E ( m ) B ( y , ( p + ϵ ) n ) E(m^{\prime})\in B(y,(p+\epsilon)n)
  70. V o l ( B ( y , ( p + ϵ ) n ) / 2 n Vol(B(y,(p+\epsilon)n)/2^{n}
  71. B ( x , r ) B(x,r)
  72. r r
  73. x x
  74. V o l ( B ( x , r ) ) Vol(B(x,r))
  75. V o l ( B ( y , ( p + ϵ ) n ) ) 2 H ( p ) n Vol(B(y,(p+\epsilon)n))\approx 2^{H(p)n}
  76. 2 H ( p ) n / 2 n = 2 H ( p ) n - n 2^{H(p)n}/2^{n}=2^{H(p)n-n}
  77. m { 0 , 1 } k m^{\prime}\in\{0,1\}^{k}
  78. 2 k + H ( p ) n - n \leq 2^{k+H(p)n-n}
  79. 2 - Ω ( n ) 2^{-\Omega(n)}
  80. k k
  81. p ( y | E ( m ) ) p(y|E(m))
  82. y y
  83. E ( m ) E(m)
  84. B ( E ( m ) , ( p + ϵ ) n ) B(E(m),(p+\epsilon)n)
  85. Ball \,\text{Ball}
  86. Pr e B S C p [ D ( E ( m ) + e ) m ] = y { 0 , 1 } n p ( y | E ( m ) ) 1 D ( y ) m y Ball p ( y | E ( m ) ) 1 D ( y ) m + y Ball p ( y | E ( m ) ) 1 D ( y ) m 2 - ϵ 2 n + y Ball p ( y | E ( m ) ) 1 D ( y ) m . \Pr_{e\in BSC_{p}}[D(E(m)+e)\neq m]=\sum_{y\in\{0,1\}^{n}}p(y|E(m))\cdot 1_{D(% y)\neq m}\leq\sum_{y\notin\,\text{Ball}}p(y|E(m))\cdot 1_{D(y)\neq m}+\sum_{y% \in\,\text{Ball}}p(y|E(m))\cdot 1_{D(y)\neq m}\leq 2^{-{\epsilon^{2}}n}+\sum_{% y\in\,\text{Ball}}p(y|E(m))\cdot 1_{D(y)\neq m}.
  87. 𝔼 E \mathbb{E}_{E}
  88. P r e B S C p Pr_{e\in BSC_{p}}
  89. D ( E ( m ) + e ) D(E(m)+e)
  90. \neq
  91. m ] m]
  92. \leq
  93. 2 - ϵ 2 n 2^{-{\epsilon^{2}}n}
  94. + +
  95. y Ball \sum_{y\in\,\text{Ball}}
  96. p ( y | E ( m ) ) p(y|E(m))
  97. 𝔼 [ 1 D ( y ) m ] \mathbb{E}[1_{D(y)\neq m}]
  98. y Ball p ( y | E ( m ) ) 1 \sum_{y\in\,\text{Ball}}p(y|E(m))\leq 1
  99. 𝔼 [ 1 D ( y ) m ] 2 k + H ( p + ϵ ) n - n \mathbb{E}[1_{D(y)\neq m}]\leq 2^{k+H(p+\epsilon)n-n}
  100. 𝔼 E [ Pr e B S C p [ D ( E ( m ) + e ) m ] ] 2 - ϵ 2 n + 2 k + H ( p + ϵ ) n - n 2 - δ n \mathbb{E}_{E}[\Pr_{e\in BSC_{p}}[D(E(m)+e)\neq m]]\leq 2^{-{\epsilon^{2}}n}+2% ^{k+H(p+\epsilon)n-n}\leq 2^{-\delta n}
  101. δ \delta
  102. 𝔼 m [ 𝔼 E [ Pr e B S C p [ D ( E ( m ) + e ) ] m ] ] 2 - δ n \mathbb{E}_{m}[\mathbb{E}_{E}[\Pr_{e\in BSC_{p}}[D(E(m)+e)]\neq m]]\leq 2^{-% \delta n}
  103. E E
  104. 𝔼 E [ 𝔼 m [ Pr e B S C p [ D ( E ( m ) + e ) ] m ] ] 2 - δ n \mathbb{E}_{E}[\mathbb{E}_{m}[\Pr_{e\in BSC_{p}}[D(E(m)+e)]\neq m]]\leq 2^{-% \delta n}
  105. E * E^{*}
  106. D * D^{*}
  107. 𝔼 m [ Pr e B S C p [ D * ( E * ( m ) + e ) m ] ] 2 - δ n \mathbb{E}_{m}[\Pr_{e\in BSC_{p}}[D^{*}(E^{*}(m)+e)\neq m]]\leq 2^{-\delta n}
  108. m m
  109. m m
  110. 2 k 2^{k}
  111. 2 k - 1 2^{k-1}
  112. 2.2 - δ n 2.2^{-\delta n}
  113. m m
  114. 2 k - 1 2^{k-1}
  115. E E^{\prime}
  116. D D^{\prime}
  117. 2 - δ n + 1 2^{-\delta n+1}
  118. δ \delta^{\prime}
  119. δ - 1 n \delta-\frac{1}{n}
  120. 2 - δ n 2^{-\delta^{\prime}n}
  121. 1 - H ( p ) 1-H(p)
  122. k k
  123. \geq
  124. \lceil
  125. ( 1 - H ( p + ϵ ) n ) (1-H(p+\epsilon)n)
  126. \rceil
  127. E E
  128. { 0 , 1 } k \{0,1\}^{k}
  129. \rightarrow
  130. { 0 , 1 } n \{0,1\}^{n}
  131. D D
  132. { 0 , 1 } n \{0,1\}^{n}
  133. \rightarrow
  134. { 0 , 1 } k \{0,1\}^{k}
  135. P r e B S C p Pr_{e\in BSC_{p}}
  136. D ( E ( m ) + e ) D(E(m)+e)
  137. \neq
  138. m ] m]
  139. \geq
  140. 1 2 \frac{1}{2}
  141. k k
  142. B S C p BSC_{p}
  143. H ( p ) H(p)
  144. 2 H ( p + ϵ ) n 2^{H(p+\epsilon)n}
  145. n n
  146. 2 k 2^{k}
  147. 2 n 2^{n}
  148. 2 k 2 H ( p + ϵ ) n 2 n 2^{k}2^{H(p+\epsilon)n}\geq 2^{n}
  149. k ( 1 - H ( p + ϵ ) n ) k\geq\lceil(1-H(p+\epsilon)n)\rceil
  150. B S C BSC
  151. B E C BEC
  152. B S C p BSC_{p}
  153. C * C^{*}
  154. = =
  155. C out C in C\text{out}\circ C\text{in}
  156. B S C p BSC_{p}
  157. C out C\text{out}
  158. N N
  159. 1 - ϵ 2 1-\frac{\epsilon}{2}
  160. F 2 k F_{2^{k}}
  161. k = O ( l o g N ) k=O(logN)
  162. D out D\text{out}
  163. C out C\text{out}
  164. γ \gamma
  165. t out ( N ) t\text{out}(N)
  166. C in C\text{in}
  167. n n
  168. k k
  169. 1 - H ( p ) - ϵ 2 1-H(p)-\frac{\epsilon}{2}
  170. D in D\text{in}
  171. C in C\text{in}
  172. γ 2 \frac{\gamma}{2}
  173. B S C p BSC_{p}
  174. t in ( N ) t\text{in}(N)
  175. C out C\text{out}
  176. C out C\text{out}
  177. C in C\text{in}
  178. n n
  179. k k
  180. B S C p BSC_{p}
  181. R ( C * ) = R ( C in ) × R ( C out ) = ( 1 - ϵ 2 ) ( 1 - H ( p ) - ϵ 2 ) 1 R(C^{*})=R(C\text{in})\times R(C\text{out})=(1-\frac{\epsilon}{2})(1-H(p)-% \frac{\epsilon}{2})\geq 1
  182. - -
  183. H ( p ) H(p)
  184. - -
  185. ϵ \epsilon
  186. B S C p BSC_{p}
  187. C * C^{*}
  188. N N
  189. C * C^{*}
  190. O ( N 2 ) + O ( N k 2 ) = O ( N 2 ) O(N^{2})+O(Nk^{2})=O(N^{2})
  191. N t in ( k ) + t out ( N ) = N O ( 1 ) Nt\text{in}(k)+t\text{out}(N)=N^{O(1)}
  192. t out ( N ) = N O ( 1 ) t\text{out}(N)=N^{O(1)}
  193. t in ( k ) = 2 O ( k ) t\text{in}(k)=2^{O(k)}
  194. C * C^{*}
  195. y i = D in ( y i ) , i ( 0 , N ) y_{i}^{\prime}=D\text{in}(y_{i}),\quad i\in(0,N)
  196. D out D\text{out}
  197. y = ( y 1 y N ) y^{\prime}=(y_{1}^{\prime}\ldots y_{N}^{\prime})
  198. C in C\text{in}
  199. C out C\text{out}
  200. i i
  201. D in D\text{in}
  202. γ 2 \frac{\gamma}{2}
  203. B S C p BSC_{p}
  204. D in D\text{in}
  205. γ N 2 \frac{\gamma N}{2}
  206. γ N \gamma N
  207. e - γ N 6 e^{\frac{-\gamma N}{6}}
  208. C out C\text{out}
  209. γ N \gamma N
  210. C * C^{*}
  211. 2 - Ω ( γ N ) 2^{-\Omega(\gamma N)}
  212. C * C^{*}
  213. C * C^{*}
  214. C in C\text{in}
  215. C o u t C{out}

Binoculars.html

  1. D = O H Mil × 1000 D=\frac{OH}{\,\text{Mil}}\times 1000
  2. D D
  3. O H OH
  4. Mil \,\text{Mil}
  5. 8000 m = 120 m 15 mil × 1000 8000\,\text{m}=\frac{120\,\text{m}}{15\,\text{mil}}\times 1000

Binomial_heap.html

  1. n n
  2. ( n d ) {\textstyle\left({{n}\atop{d}}\right)}
  3. d d
  4. 2 3 + 2 2 + 2 0 2^{3}+2^{2}+2^{0}

Binomial_options_pricing_model.html

  1. u u
  2. d d
  3. u 1 u\geq 1
  4. 0 < d 1 0<d\leq 1
  5. S S
  6. S u p = S u S_{up}=S\cdot u
  7. S d o w n = S d S_{down}=S\cdot d
  8. σ \sigma
  9. t t
  10. σ 2 t \sigma^{2}t
  11. u = e σ t u=e^{\sigma\sqrt{t}}
  12. d = e - σ t = 1 u . d=e^{-\sigma\sqrt{t}}=\frac{1}{u}.
  13. S n = S 0 × u N u - N d S_{n}=S_{0}\times u^{N_{u}-N_{d}}
  14. N u N_{u}
  15. N d N_{d}
  16. S n - K S_{n}-K
  17. K K
  18. S n S_{n}
  19. K K
  20. S n S_{n}
  21. n t h n^{th}
  22. C t - Δ t , i = e - r Δ t ( p C t , i + 1 + ( 1 - p ) C t , i - 1 ) C_{t-\Delta t,i}=e^{-r\Delta t}(pC_{t,i+1}+(1-p)C_{t,i-1})\,
  23. C t , i C_{t,i}\,
  24. i t h i^{th}\,
  25. t {t}\,
  26. p = e ( r - q ) Δ t - d u - d p=\frac{e^{(r-q)\Delta t}-d}{u-d}
  27. q q
  28. q = r q=r
  29. p p
  30. ( 0 , 1 ) (0,1)
  31. Δ t \Delta t
  32. Δ t < σ 2 ( r - q ) 2 \Delta t<\frac{\sigma^{2}}{(r-q)^{2}}
  33. q q
  34. i i
  35. P V ( D k ) \sum{PV(D_{k})}
  36. k < i k<i
  37. P V ( D k ) PV(D_{k})
  38. k k
  39. S S
  40. i i
  41. j j

Binomial_type.html

  1. p n ( x + y ) = k = 0 n ( n k ) p k ( x ) p n - k ( y ) . p_{n}(x+y)=\sum_{k=0}^{n}{n\choose k}\,p_{k}(x)\,p_{n-k}(y).
  2. ( x ) n = x ( x - 1 ) ( x - 2 ) ( x - n + 1 ) . (x)_{n}=x(x-1)(x-2)\cdot\cdots\cdot(x-n+1).
  3. x ( n ) = x ( x + 1 ) ( x + 2 ) ( x + n - 1 ) x^{(n)}=x(x+1)(x+2)\cdot\cdots\cdot(x+n-1)
  4. p n ( x ) = x ( x - a n ) n - 1 p_{n}(x)=x(x-an)^{n-1}\,
  5. p n ( x ) = k = 1 n S ( n , k ) x k p_{n}(x)=\sum_{k=1}^{n}S(n,k)x^{k}
  6. p n ( x ) n p n - 1 ( x ) p_{n}(x)\mapsto np_{n-1}(x)
  7. Q = n = 1 c n D n Q=\sum_{n=1}^{\infty}c_{n}D^{n}
  8. p 0 ( x ) = 1 , p_{0}(x)=1,\,
  9. p n ( 0 ) = 0 for n 1 , and p_{n}(0)=0\quad{\rm for\ }n\geq 1,{\rm\ and}
  10. Q p n ( x ) = n p n - 1 ( x ) . Qp_{n}(x)=np_{n-1}(x).\,
  11. p n ( x ) = k = 1 n B n , k ( a 1 , , a n - k + 1 ) x k . p_{n}(x)=\sum_{k=1}^{n}B_{n,k}(a_{1},\dots,a_{n-k+1})x^{k}.
  12. p n ( 0 ) = a n . p_{n}^{\prime}(0)=a_{n}.\,
  13. P ( t ) = n = 1 a n n ! t n . P(t)=\sum_{n=1}^{\infty}{a_{n}\over n!}t^{n}.
  14. P - 1 ( d d x ) P^{-1}\left({d\over dx}\right)\,
  15. ( a b ) n = j = 0 n ( n j ) a j b n - j . (a\diamondsuit b)_{n}=\sum_{j=0}^{n}{n\choose j}a_{j}b_{n-j}.
  16. a n k a_{n}^{k\diamondsuit}\,
  17. a a k factors . \underbrace{a\diamondsuit\cdots\diamondsuit a}_{k\,\text{ factors}}.\,
  18. p n ( x ) = k = 1 n a n k x k k ! p_{n}(x)=\sum_{k=1}^{n}{a_{n}^{k\diamondsuit}x^{k}\over k!}\,
  19. n = 0 p n ( x ) n ! t n = e x f ( t ) \sum_{n=0}^{\infty}{p_{n}(x)\over n!}t^{n}=e^{xf(t)}
  20. f ( t ) = n = 1 p n ( 0 ) n ! t n . f(t)=\sum_{n=1}^{\infty}{p_{n}\,^{\prime}(0)\over n!}t^{n}.
  21. f - 1 ( D ) p n ( x ) = n p n - 1 ( x ) . f^{-1}(D)p_{n}(x)=np_{n-1}(x).
  22. n = 0 a n n ! t n \sum_{n=0}^{\infty}{a_{n}\over n!}t^{n}
  23. n = 0 b n n ! t n \sum_{n=0}^{\infty}{b_{n}\over n!}t^{n}
  24. c n = k = 0 n ( n k ) a k b n - k c_{n}=\sum_{k=0}^{n}{n\choose k}a_{k}b_{n-k}
  25. g ( t ) x = e x f ( t ) g(t)^{x}=e^{xf(t)}
  26. p n ( x ) = k = 0 n a n , k x k . p_{n}(x)=\sum_{k=0}^{n}a_{n,k}\,x^{k}.
  27. ( p n q ) ( x ) = k = 0 n a n , k q k ( x ) (p_{n}\circ q)(x)=\sum_{k=0}^{n}a_{n,k}\,q_{k}(x)
  28. p n ( 0 ) = κ n = p_{n}^{\prime}(0)=\kappa_{n}=\,
  29. p n ( 1 ) = μ n = p_{n}(1)=\mu_{n}^{\prime}=\,
  30. f ( t ) = n = 1 κ n n ! t n f(t)=\sum_{n=1}^{\infty}\frac{\kappa_{n}}{n!}t^{n}
  31. f - 1 ( D ) f^{-1}(D)\,
  32. f - 1 ( D ) p n ( x ) = n p n - 1 ( x ) . f^{-1}(D)p_{n}(x)=np_{n-1}(x).\,

Bioluminescence.html

  1. L + O 2 + ( ATP ) ( Mg 2 + ) Luciferase oxy - L + CO 2 + AMP + PP + light \mathrm{L+O_{2}+(ATP)\xrightarrow[(Mg^{2+})]{Luciferase}\ oxy\,\text{-}L+CO_{2% }+AMP+PP+light}

Biot_number.html

  1. Bi = h L C k b \mathrm{Bi}=\frac{hL_{C}}{\ k_{b}}
  2. L C = V body A surface \mathit{L_{C}}=\frac{V_{\rm body}}{A_{\rm surface}}
  3. T - T T 0 - T = e - BiFo {T-T_{\infty}\over T_{0}-T_{\infty}}=e^{\mathrm{-BiFo}}
  4. Bi m \mathrm{Bi}_{m}
  5. Bi m = h m L C D A B \mathrm{Bi}_{m}=\frac{h_{m}L_{C}}{D_{AB}}

Biot–Savart_law.html

  1. d 𝐥 d\mathbf{l}
  2. 𝐫 = 𝐫 - 𝐥 \mathbf{r^{\prime}}=\mathbf{r}-\mathbf{l}
  3. 𝐥 \mathbf{l}
  4. 𝐫 \mathbf{r}
  5. 𝐁 ( 𝐫 ) = μ 0 4 π C I d 𝐥 × 𝐫 ^ | 𝐫 | 2 \mathbf{B}(\mathbf{r})=\frac{\mu_{0}}{4\pi}\int_{C}\frac{Id\mathbf{l}\times% \mathbf{\hat{r}^{\prime}}}{|\mathbf{r^{\prime}}|^{2}}
  6. 𝐫 ^ \mathbf{\hat{r}^{\prime}}
  7. 𝐫 \mathbf{r^{\prime}}
  8. 𝐫 \mathbf{r}
  9. 𝐉 \mathbf{J}
  10. 𝐁 ( 𝐫 ) = μ 0 2 π C ( 𝐉 d l ) × 𝐫 | 𝐫 | 2 = μ 0 2 π C ( 𝐉 d l ) × 𝐫 ^ | 𝐫 | \mathbf{B}(\mathbf{r})=\frac{\mu_{0}}{2\pi}\int_{C}\ \frac{(\mathbf{J}\,dl)% \times\mathbf{r}^{\prime}}{|\mathbf{r}^{\prime}|^{2}}=\frac{\mu_{0}}{2\pi}\int% _{C}\ \frac{(\mathbf{J}\,dl)\times\mathbf{\hat{r}^{\prime}}}{|\mathbf{r}^{% \prime}|}
  11. 𝐁 ( 𝐫 ) = μ 0 4 π V ( 𝐉 d V ) × 𝐫 ^ | 𝐫 | 2 \mathbf{B}(\mathbf{r})=\frac{\mu_{0}}{4\pi}\iiint_{V}\ \frac{(\mathbf{J}\,dV)% \times\mathbf{\hat{r}^{\prime}}}{|\mathbf{r}^{\prime}|^{2}}
  12. d V dV
  13. 𝐉 \mathbf{J}
  14. 𝐁 \mathbf{B}
  15. 𝐁 ( 𝐫 ) = μ 0 4 π I C d 𝐥 × 𝐫 ^ | 𝐫 | 2 \mathbf{B}(\mathbf{r})=\frac{\mu_{0}}{4\pi}I\int_{C}\frac{d\mathbf{l}\times% \mathbf{\hat{r}^{\prime}}}{|\mathbf{r^{\prime}}|^{2}}
  16. 𝐄 = q 4 π ϵ 0 1 - v 2 / c 2 ( 1 - v 2 sin 2 θ / c 2 ) 3 / 2 𝐫 ^ | 𝐫 | 2 \mathbf{E}=\frac{q}{4\pi\epsilon_{0}}\frac{1-v^{2}/c^{2}}{(1-v^{2}\sin^{2}% \theta/c^{2})^{3/2}}\frac{\mathbf{\hat{r}^{\prime}}}{|\mathbf{r}^{\prime}|^{2}}
  17. 𝐁 = 1 c 2 𝐯 × 𝐄 \mathbf{B}=\frac{1}{c^{2}}\mathbf{v}\times\mathbf{E}
  18. ^ r \mathbf{\hat{}}r^{\prime}
  19. 𝐯 \mathbf{v}
  20. 𝐫 \mathbf{r}^{\prime}
  21. 𝐄 = q 4 π ϵ 0 𝐫 ^ | 𝐫 | 2 \mathbf{E}=\frac{q}{4\pi\epsilon_{0}}\ \frac{\mathbf{\hat{r}^{\prime}}}{|% \mathbf{r}^{\prime}|^{2}}
  22. 𝐁 = μ 0 q 4 π 𝐯 × 𝐫 ^ | 𝐫 | 2 \mathbf{B}=\frac{\mu_{0}q}{4\pi}\mathbf{v}\times\frac{\mathbf{\hat{r}^{\prime}% }}{|\mathbf{r}^{\prime}|^{2}}
  23. d V dV
  24. d L dL
  25. Γ \Gamma
  26. v = Γ 2 π r v=\frac{\Gamma}{2\pi r}
  27. v = Γ 4 π r [ cos A - cos B ] v=\frac{\Gamma}{4\pi r}\left[\cos A-\cos B\right]
  28. 𝐁 ( 𝐫 ) = μ 0 4 π V d 3 l 𝐉 ( 𝐥 ) × 𝐫 - 𝐥 | 𝐫 - 𝐥 | 3 \mathbf{B}(\mathbf{r})=\frac{\mu_{0}}{4\pi}\iiint_{V}d^{3}l\mathbf{J}(\mathbf{% l})\times\frac{\mathbf{r}-\mathbf{l}}{|\mathbf{r}-\mathbf{l}|^{3}}
  29. 𝐫 - 𝐥 | 𝐫 - 𝐥 | 3 = - ( 1 | 𝐫 - 𝐥 | ) \frac{\mathbf{r}-\mathbf{l}}{|\mathbf{r}-\mathbf{l}|^{3}}=-\nabla\left(\frac{1% }{|\mathbf{r}-\mathbf{l}|}\right)
  30. 𝐫 \mathbf{r}
  31. 𝐁 ( 𝐫 ) = μ 0 4 π × V d 3 l 𝐉 ( 𝐥 ) | 𝐫 - 𝐥 | \mathbf{B}(\mathbf{r})=\frac{\mu_{0}}{4\pi}\nabla\times\iiint_{V}d^{3}l\frac{% \mathbf{J}(\mathbf{l})}{|\mathbf{r}-\mathbf{l}|}
  32. 𝐫 \mathbf{r}
  33. × 𝐁 = μ 0 4 π V d 3 l 𝐉 ( 𝐥 ) ( 1 | 𝐫 - 𝐥 | ) - μ 0 4 π V d 3 l 𝐉 ( 𝐥 ) 2 ( 1 | 𝐫 - 𝐥 | ) \nabla\times\mathbf{B}=\frac{\mu_{0}}{4\pi}\nabla\iiint_{V}d^{3}l\mathbf{J}(% \mathbf{l})\cdot\nabla\left(\frac{1}{|\mathbf{r}-\mathbf{l}|}\right)-\frac{\mu% _{0}}{4\pi}\iiint_{V}d^{3}l\mathbf{J}(\mathbf{l})\nabla^{2}\left(\frac{1}{|% \mathbf{r}-\mathbf{l}|}\right)
  34. ( 1 | 𝐫 - 𝐥 | ) = - l ( 1 | 𝐫 - 𝐥 | ) , \nabla\left(\frac{1}{|\mathbf{r}-\mathbf{l}|}\right)=-\nabla_{l}\left(\frac{1}% {|\mathbf{r}-\mathbf{l}|}\right),
  35. 2 ( 1 | 𝐫 - 𝐥 | ) = - 4 π δ ( 𝐫 - 𝐥 ) \nabla^{2}\left(\frac{1}{|\mathbf{r}-\mathbf{l}|}\right)=-4\pi\delta(\mathbf{r% }-\mathbf{l})
  36. × 𝐁 = μ 0 𝐉 \nabla\times\mathbf{B}=\mu_{0}\mathbf{J}
  37. 𝐄 / t = 0 \partial\mathbf{E}/\partial t=0

Bipartite_graph.html

  1. U U
  2. V V
  3. U U
  4. V V
  5. U U
  6. V V
  7. U U
  8. V V
  9. U U
  10. V V
  11. U U
  12. V V
  13. G = ( U , V , E ) G=(U,V,E)
  14. U U
  15. V V
  16. E E
  17. ( U , V , E ) (U,V,E)
  18. | U | = | V | |U|=|V|
  19. G G
  20. G G
  21. deg ( v ) \deg(v)
  22. v V deg ( v ) = u U deg ( u ) = | E | . \sum_{v\in V}\deg(v)=\sum_{u\in U}\deg(u)=|E|\,.
  23. U U
  24. V V
  25. ( 5 , 5 , 5 ) , ( 3 , 3 , 3 , 3 , 3 ) (5,5,5),(3,3,3,3,3)
  26. ( U , V , E ) (U,V,E)
  27. ( 0 , 1 ) (0,1)
  28. | U | × | V | |U|\times|V|
  29. ( U , V , E ) (U,V,E)
  30. U U
  31. V V
  32. E E
  33. v v
  34. e e
  35. v v
  36. v v
  37. n n
  38. ( 0 , 1 ) (0,1)
  39. n × n n\times n
  40. n n
  41. n n
  42. O ( n log n ) O(n\log n)
  43. Ω ( n 2 ) \Omega(n^{2})
  44. P P
  45. J J
  46. ( P , J , E ) (P,J,E)

Birefringence.html

  1. Δ n = n e - n o \Delta n=n_{e}-n_{o}\,
  2. μ = μ 0 \mu=\mu_{0}
  3. 𝐃 \mathbf{D}
  4. 𝐄 \mathbf{E}
  5. 𝐃 = ϵ 𝐄 . \mathbf{D}=\epsilon\,\mathbf{E}\,.
  6. 𝐃 \mathbf{D}
  7. 𝐃 \mathbf{D}
  8. 𝐄 \mathbf{E}
  9. c = 1 / μ 0 ϵ 0 c=1/\sqrt{\mu_{0}\epsilon_{0}}
  10. ω 4 c 4 - ω 2 c 2 ( k x 2 + k y 2 n z 2 + k x 2 + k z 2 n y 2 + k y 2 + k z 2 n x 2 ) + \frac{\omega^{4}}{c^{4}}-\frac{\omega^{2}}{c^{2}}\left(\frac{k_{x}^{2}+k_{y}^{% 2}}{n_{z}^{2}}+\frac{k_{x}^{2}+k_{z}^{2}}{n_{y}^{2}}+\frac{k_{y}^{2}+k_{z}^{2}% }{n_{x}^{2}}\right)+
  11. ( k x 2 n y 2 n z 2 + k y 2 n x 2 n z 2 + k z 2 n x 2 n y 2 ) ( k x 2 + k y 2 + k z 2 ) = 0 \left(\frac{k_{x}^{2}}{n_{y}^{2}n_{z}^{2}}+\frac{k_{y}^{2}}{n_{x}^{2}n_{z}^{2}% }+\frac{k_{z}^{2}}{n_{x}^{2}n_{y}^{2}}\right)(k_{x}^{2}+k_{y}^{2}+k_{z}^{2})=0

Birthday_attack.html

  1. 1 - ( 364 / 365 ) 30 1-(364/365)^{30}
  2. 1 - 365 ! / ( ( 365 - n ) ! 365 n ) 1-365!/((365-n)!\cdot 365^{n})
  3. f f
  4. x 1 , x 2 x_{1},x_{2}
  5. f ( x 1 ) = f ( x 2 ) f(x_{1})=f(x_{2})
  6. x 1 , x 2 x_{1},x_{2}
  7. f f
  8. f ( x ) f(x)
  9. H H
  10. H H
  11. x 1 x_{1}
  12. x 2 x_{2}
  13. f ( x 1 ) = f ( x 2 ) f(x_{1})=f(x_{2})
  14. 1.25 H 1.25\sqrt{H}
  15. p ( n ; H ) 1 - e - n ( n - 1 ) / ( 2 H ) 1 - e - n 2 / ( 2 H ) , p(n;H)\approx 1-e^{-n(n-1)/(2H)}\approx 1-e^{-n^{2}/(2H)},\,
  16. n ( p ; H ) 2 H ln 1 1 - p , n(p;H)\approx\sqrt{2H\ln\frac{1}{1-p}},
  17. n ( 0.5 ; H ) 1.1774 H . n(0.5;H)\approx 1.1774\sqrt{H}.\,
  18. Q ( H ) π 2 H . Q(H)\approx\sqrt{\frac{\pi}{2}H}.
  19. ln 1 1 - p \ln\frac{1}{1-p}
  20. n ( p ; H ) n(p;H)
  21. p p
  22. p ( n ) n 2 2 m p(n)\approx{n^{2}\over 2m}
  23. n 2 m × p ( n ) n\approx\sqrt{2m\times p(n)}
  24. m = 2 32 m=2^{32}
  25. p 2 - 20 p\approx 2^{-20}
  26. n 2 × 2 32 × 2 - 20 = 2 1 + 32 - 20 = 2 13 = 2 6.5 90.5 n\approx\sqrt{2\times 2^{32}\times 2^{-20}}=\sqrt{2^{1+32-20}}=\sqrt{2^{13}}=2% ^{6.5}\approx 90.5
  27. m m
  28. f ( m ) f(m)
  29. f f
  30. f ( m ) f(m)
  31. m m
  32. m m^{\prime}
  33. m m
  34. m m
  35. m m^{\prime}
  36. f ( m ) = f ( m ) f(m)=f(m^{\prime})
  37. n n
  38. 2 n 2n

Birthday_problem.html

  1. n n
  2. N - 1 N-1
  3. 22 + 21 + 20 + + 1 = 253 22+21+20+\cdots+1=253
  4. ( 23 2 ) = 23 22 2 = 253 \textstyle{23\choose 2}=\frac{23\cdot 22}{2}=253
  5. n n
  6. P ( A ) P(A)
  7. A A
  8. P ( A ) P(A)
  9. P ( A ) P(A)
  10. P ( 1 ) × P ( 2 ) × P ( 3 ) × × P ( 23 ) P(1) ×P(2) ×P(3) × ... ×P(23)
  11. P ( 1 ) P(1)
  12. 1 1
  13. 365 365 \frac{365}{365}
  14. P ( 2 ) P(2)
  15. 364 365 \frac{364}{365}
  16. P ( 3 ) = 363 365 P(3)=\frac{363}{365}
  17. P ( 23 ) P(23)
  18. 343 365 \frac{343}{365}
  19. P ( A ) = 365 365 × 364 365 × 363 365 × 362 365 × × 343 365 P(A^{\prime})=\frac{365}{365}\times\frac{364}{365}\times\frac{363}{365}\times% \frac{362}{365}\times\cdots\times\frac{343}{365}
  20. P ( A ) = ( 1 365 ) 23 × ( 365 × 364 × 363 × × 343 ) P(A^{\prime})=\left(\frac{1}{365}\right)^{23}\times(365\times 364\times 363% \times\cdots\times 343)
  21. P ( A ) 1 0.492703 = 0.507297 P(A)≈ 1 − 0.492703=0.507297
  22. n n
  23. p ( n ) p(n)
  24. n n
  25. p ¯ ( n ) \overline{p}(n)
  26. n n
  27. p ¯ ( n ) \overline{p}(n)
  28. n > 365 n> 365
  29. n 365 n≤ 365
  30. p ¯ ( n ) \displaystyle\bar{p}(n)
  31. ! !
  32. ( 365 n ) \textstyle{365\choose n}
  33. P r k {{}_{k}P_{r}}
  34. 364 365 \frac{364}{365}
  35. 363 365 \frac{363}{365}
  36. n 1 n− 1
  37. n n
  38. n n
  39. p ( n ) p(n)
  40. p ( n ) = 1 - p ¯ ( n ) . p(n)=1-\bar{p}(n).\,
  41. 1 2 \frac{1}{2}
  42. n = 23 n=23
  43. n n
  44. 𝒮 \mathcal{S}
  45. N N
  46. \mathcal{B}
  47. b : 𝒮 b:\mathcal{S}\mapsto\mathcal{B}
  48. 𝒮 \mathcal{S}
  49. 𝒮 \mathcal{S}
  50. \mathcal{B}
  51. | 𝒮 | = N |\mathcal{S}|=N
  52. | | = 366 |\mathcal{B}|=366
  53. 366 N 366^{N}
  54. 366 ! ( 366 - N ) ! \dfrac{366!}{(366-N)!}
  55. 𝒮 \mathcal{S}
  56. P ( A ) = 366 ! 366 N ( 366 - N ) ! P(A)=\dfrac{366!}{366^{N}(366-N)!}
  57. P ( A ) = 1 - 366 ! 366 N ( 366 - ( N ) ) ! P(A^{\prime})=1-\dfrac{366!}{366^{N}(366-(N))!}
  58. e x = 1 + x + x 2 2 ! + e^{x}=1+x+\frac{x^{2}}{2!}+\cdots
  59. e x 1 + x . e^{x}\approx 1+x.
  60. x = - a / 365 x=-a/365
  61. e - a / 365 1 - a 365 . e^{-a/365}\approx 1-\frac{a}{365}.
  62. e - 1 / 365 1 - 1 365 . e^{-1/365}\approx 1-\frac{1}{365}.
  63. p ¯ ( n ) 1 × e - 1 / 365 × e - 2 / 365 e - ( n - 1 ) / 365 = 1 × e - ( 1 + 2 + + ( n - 1 ) ) / 365 = e - ( n ( n - 1 ) / 2 ) / 365 . \begin{aligned}\displaystyle\bar{p}(n)&\displaystyle\approx 1\times e^{-1/365}% \times e^{-2/365}\cdots e^{-(n-1)/365}\\ &\displaystyle=1\times e^{-(1+2+\cdots+(n-1))/365}\\ &\displaystyle=e^{-(n(n-1)/2)/365}.\end{aligned}
  64. p ( n ) = 1 - p ¯ ( n ) 1 - e - n ( n - 1 ) / ( 2 × 365 ) . p(n)=1-\bar{p}(n)\approx 1-e^{-n(n-1)/(2\times 365)}.
  65. p ( n ) 1 - e - n 2 / ( 2 × 365 ) , p(n)\approx 1-e^{-n^{2}/(2\times 365)},\,
  66. p ( n , d ) \displaystyle p(n,d)
  67. p ¯ ( n ) ( 364 365 ) C ( n , 2 ) . \bar{p}(n)\approx\left(\frac{364}{365}\right)^{C(n,2)}.
  68. p ( n ) 1 - ( 364 365 ) C ( n , 2 ) . p(n)\approx 1-\left(\frac{364}{365}\right)^{C(n,2)}.
  69. Poi ( C ( 23 , 2 ) 365 ) = Poi ( 253 365 ) Poi ( 0.6932 ) \mathrm{Poi}\left(\frac{C(23,2)}{365}\right)=\mathrm{Poi}\left(\frac{253}{365}% \right)\approx\mathrm{Poi}(0.6932)
  70. Pr ( X > 0 ) = 1 - Pr ( X = 0 ) 1 - e - 0.6932 1 - 0.499998 = 0.500002. \Pr(X>0)=1-\Pr(X=0)\approx 1-e^{-0.6932}\approx 1-0.499998=0.500002.
  71. p ( n ) n 2 2 m p(n)\approx{n^{2}\over 2m}
  72. n 2 m × p ( n ) n\approx\sqrt{2m\times p(n)}
  73. n 2 × 365 × 0.5 = 365 19 n\approx\sqrt{2\times 365\times 0.5}=\sqrt{365}\approx 19
  74. n 1 2 + 1 4 + 2 × ln ( 2 ) × 365 = 22.999943. n\approx\frac{1}{2}+\sqrt{\frac{1}{4}+2\times\ln(2)\times 365}=22.999943.
  75. 10 18 10^{−}18
  76. 10 15 10^{−}15
  77. 10 12 10^{−}12
  78. 10 9 10^{−}9
  79. 10 6 10^{−}6
  80. × 10 9 \times 10^{9}
  81. × 10 3 \times 10^{3}
  82. × 10 3 \times 10^{3}
  83. × 10 4 \times 10^{4}
  84. × 10 4 \times 10^{4}
  85. × 10 5 \times 10^{5}
  86. × 10 1 2 \times 10^{1}2
  87. × 10 3 \times 10^{3}
  88. × 10 4 \times 10^{4}
  89. × 10 5 \times 10^{5}
  90. × 10 5 \times 10^{5}
  91. × 10 6 \times 10^{6}
  92. × 10 6 \times 10^{6}
  93. × 10 1 4 \times 10^{1}4
  94. × 10 2 \times 10^{2}
  95. × 10 4 \times 10^{4}
  96. × 10 5 \times 10^{5}
  97. × 10 6 \times 10^{6}
  98. × 10 7 \times 10^{7}
  99. × 10 7 \times 10^{7}
  100. × 10 7 \times 10^{7}
  101. × 10 1 9 \times 10^{1}9
  102. × 10 2 \times 10^{2}
  103. × 10 3 \times 10^{3}
  104. × 10 5 \times 10^{5}
  105. × 10 6 \times 10^{6}
  106. × 10 8 \times 10^{8}
  107. × 10 8 \times 10^{8}
  108. × 10 9 \times 10^{9}
  109. × 10 9 \times 10^{9}
  110. × 10 9 \times 10^{9}
  111. × 10 2 8 \times 10^{2}8
  112. × 10 5 \times 10^{5}
  113. × 10 7 \times 10^{7}
  114. × 10 8 \times 10^{8}
  115. × 10 1 0 \times 10^{1}0
  116. × 10 1 1 \times 10^{1}1
  117. × 10 1 3 \times 10^{1}3
  118. × 10 1 3 \times 10^{1}3
  119. × 10 1 4 \times 10^{1}4
  120. × 10 1 4 \times 10^{1}4
  121. × 10 1 4 \times 10^{1}4
  122. × 10 3 8 \times 10^{3}8
  123. × 10 1 0 \times 10^{1}0
  124. × 10 1 1 \times 10^{1}1
  125. × 10 1 3 \times 10^{1}3
  126. × 10 1 4 \times 10^{1}4
  127. × 10 1 6 \times 10^{1}6
  128. × 10 1 7 \times 10^{1}7
  129. × 10 1 8 \times 10^{1}8
  130. × 10 1 9 \times 10^{1}9
  131. × 10 1 9 \times 10^{1}9
  132. × 10 1 9 \times 10^{1}9
  133. × 10 5 7 \times 10^{5}7
  134. × 10 2 0 \times 10^{2}0
  135. × 10 2 1 \times 10^{2}1
  136. × 10 2 3 \times 10^{2}3
  137. × 10 2 4 \times 10^{2}4
  138. × 10 2 6 \times 10^{2}6
  139. × 10 2 7 \times 10^{2}7
  140. × 10 2 8 \times 10^{2}8
  141. × 10 2 8 \times 10^{2}8
  142. × 10 2 8 \times 10^{2}8
  143. × 10 2 9 \times 10^{2}9
  144. × 10 7 7 \times 10^{7}7
  145. × 10 2 9 \times 10^{2}9
  146. × 10 3 1 \times 10^{3}1
  147. × 10 3 2 \times 10^{3}2
  148. × 10 3 4 \times 10^{3}4
  149. × 10 3 5 \times 10^{3}5
  150. × 10 3 7 \times 10^{3}7
  151. × 10 3 7 \times 10^{3}7
  152. × 10 3 8 \times 10^{3}8
  153. × 10 3 8 \times 10^{3}8
  154. × 10 3 8 \times 10^{3}8
  155. × 10 1 15 \times 10^{1}15
  156. × 10 4 8 \times 10^{4}8
  157. × 10 5 0 \times 10^{5}0
  158. × 10 5 1 \times 10^{5}1
  159. × 10 5 3 \times 10^{5}3
  160. × 10 5 4 \times 10^{5}4
  161. × 10 5 6 \times 10^{5}6
  162. × 10 5 6 \times 10^{5}6
  163. × 10 5 7 \times 10^{5}7
  164. × 10 5 7 \times 10^{5}7
  165. × 10 5 8 \times 10^{5}8
  166. × 10 1 54 \times 10^{1}54
  167. × 10 6 8 \times 10^{6}8
  168. × 10 6 9 \times 10^{6}9
  169. × 10 7 1 \times 10^{7}1
  170. × 10 7 2 \times 10^{7}2
  171. × 10 7 4 \times 10^{7}4
  172. × 10 7 5 \times 10^{7}5
  173. × 10 7 6 \times 10^{7}6
  174. × 10 7 6 \times 10^{7}6
  175. × 10 7 7 \times 10^{7}7
  176. × 10 7 7 \times 10^{7}7
  177. 1 - p ( n ) = p ¯ ( n ) = k = 1 n - 1 ( 1 - k 365 ) . 1-p(n)=\bar{p}(n)=\prod_{k=1}^{n-1}\left(1-{k\over 365}\right).
  178. p ¯ ( n ) = k = 1 n - 1 ( 1 - k 365 ) < k = 1 n - 1 ( e - k / 365 ) = e - ( n ( n - 1 ) ) / ( 2 × 365 ) . \bar{p}(n)=\prod_{k=1}^{n-1}\left(1-{k\over 365}\right)<\prod_{k=1}^{n-1}\left% (e^{-k/365}\right)=e^{-(n(n-1))/(2\times 365)}.
  179. e - ( n ( n - 1 ) ) / ( 2 365 ) < 1 2 e^{-(n(n-1))/(2\cdot 365)}<\frac{1}{2}
  180. 1 - ( 1 - 1 d ) ( 1 - 2 d ) ( 1 - n - 1 d ) 1 2 . 1-\left(1-\frac{1}{d}\right)\left(1-\frac{2}{d}\right)\cdots\left(1-\frac{n-1}% {d}\right)\geq\frac{1}{2}.
  181. 3 - 2 ln 2 6 < n ( d ) - 2 d ln 2 9 - 86 ln 2 . \frac{3-2\ln 2}{6}<n(d)-\sqrt{2d\ln 2}\leq 9-\sqrt{86\ln 2}.
  182. n ( d ) - 2 d ln 2 n(d)-\sqrt{2d\ln 2}
  183. ( 3 - 2 ln 2 ) / 6 0.27 (3-2\ln 2)/6\approx 0.27
  184. 9 - 86 ln 2 1.28 9-\sqrt{86\ln 2}\approx 1.28
  185. 2 d ln 2 \left\lceil\sqrt{2d\ln 2}\right\rceil
  186. 2 d ln 2 + 1 \left\lceil\sqrt{2d\ln 2}\right\rceil+1
  187. x \lceil x\rceil
  188. n ( d ) = 2 d ln 2 n(d)=\left\lceil\sqrt{2d\ln 2}\right\rceil
  189. n ( d ) = 2 d ln 2 + 3 - 2 ln 2 6 n(d)=\left\lceil\sqrt{2d\ln 2}+\frac{3-2\ln 2}{6}\right\rceil
  190. n ( d ) = 2 d ln 2 + 3 - 2 ln 2 6 + 9 - 4 ( ln 2 ) 2 72 2 d ln 2 n(d)=\left\lceil\sqrt{2d\ln 2}+\frac{3-2\ln 2}{6}+\frac{9-4(\ln 2)^{2}}{72% \sqrt{2d\ln 2}}\right\rceil
  191. n ( d ) = 2 d ln 2 + 3 - 2 ln 2 6 + 9 - 4 ( ln 2 ) 2 72 2 d ln 2 - 2 ( ln 2 ) 2 135 d n(d)=\left\lceil\sqrt{2d\ln 2}+\frac{3-2\ln 2}{6}+\frac{9-4(\ln 2)^{2}}{72% \sqrt{2d\ln 2}}-\frac{2(\ln 2)^{2}}{135d}\right\rceil
  192. p ( n ; d ) = { 1 - k = 1 n - 1 ( 1 - k d ) n d 1 n > d p(n;d)=\begin{cases}1-\prod_{k=1}^{n-1}\left(1-{k\over d}\right)&n\leq d\\ 1&n>d\end{cases}
  193. p ( n ; d ) 1 - e - n ( n - 1 ) / ( 2 d ) p(n;d)\approx 1-e^{-n(n-1)/(2d)}
  194. p ( n ; d ) 1 - ( d - 1 d ) n ( n - 1 ) / 2 p(n;d)\approx 1-\left(\frac{d-1}{d}\right)^{n(n-1)/2}
  195. n ( p ; d ) 2 d ln ( 1 1 - p ) . n(p;d)\approx\sqrt{2d\cdot\ln\left({1\over 1-p}\right)}.
  196. p 0 = 1 d m + n i = 1 m j = 1 n S 2 ( m , i ) S 2 ( n , j ) k = 0 i + j - 1 d - k p_{0}=\frac{1}{d^{m+n}}\sum_{i=1}^{m}\sum_{j=1}^{n}S_{2}(m,i)S_{2}(n,j)\prod_{% k=0}^{i+j-1}d-k
  197. n ( p ; 365 ) 2 × 365 ln ( 1 1 - p ) . n(p;365)\approx\sqrt{2\times 365\ln\left({1\over 1-p}\right)}.
  198. q ( n ) = 1 - ( 365 - 1 365 ) n q(n)=1-\left(\frac{365-1}{365}\right)^{n}
  199. q ( n ; d ) = 1 - ( d - 1 d ) n . q(n;d)=1-\left(\frac{d-1}{d}\right)^{n}.
  200. 253 = 23 × ( 23 - 1 ) 2 253=\frac{23\times(23-1)}{2}
  201. p ( n , k , m ) = 1 - ( m - n k - 1 ) ! m n - 1 ( m - n ( k + 1 ) ) ! \begin{aligned}\displaystyle p(n,k,m)&\displaystyle=1-{(m-nk-1)!\over m^{n-1}(% m-n(k+1))!}\end{aligned}
  202. k = 1 n q ( k - 1 ; d ) = n - d + d ( d - 1 d ) n . \sum_{k=1}^{n}q(k-1;d)=n-d+d\left(\frac{d-1}{d}\right)^{n}.
  203. n ¯ = 1 + Q ( M ) \overline{n}\,=\,1+Q(M)
  204. Q ( M ) = k = 1 M M ! ( M - k ) ! M k . Q(M)=\sum_{k=1}^{M}\frac{M!}{(M-k)!M^{k}}.
  205. Q ( M ) = 1 + M - 1 M + ( M - 1 ) ( M - 2 ) M 2 + + ( M - 1 ) ( M - 2 ) 1 M M - 1 Q(M)=1+\frac{M-1}{M}+\frac{(M-1)(M-2)}{M^{2}}+\cdots+\frac{(M-1)(M-2)\cdots 1}% {M^{M-1}}
  206. Q ( M ) π M 2 - 1 3 + 1 12 π 2 M - 4 135 M + . Q(M)\sim\sqrt{\frac{\pi M}{2}}-\frac{1}{3}+\frac{1}{12}\sqrt{\frac{\pi}{2M}}-% \frac{4}{135M}+\cdots.
  207. n ¯ = 1 + Q ( M ) 24.61659 \scriptstyle\overline{n}\,=\,1+Q(M)\approx 24.61659
  208. 1 , 000 , 000 N \scriptstyle 1,000,000\sqrt{N}
  209. 1 , 000 , 000 N \scriptstyle 1,000,000\sqrt{N}

Bisection.html

  1. a , b , c a,b,c
  2. s = ( a + b + c ) / 2 , s=(a+b+c)/2,
  3. a a
  4. 2 b c s ( s - a ) b + c . \frac{2\sqrt{bcs(s-a)}}{b+c}.
  5. t a t_{a}
  6. t a 2 + m n = b c t_{a}^{2}+mn=bc
  7. t a , t b , t_{a},t_{b},
  8. t c t_{c}
  9. ( b + c ) 2 b c t a 2 + ( c + a ) 2 c a t b 2 + ( a + b ) 2 a b t c 2 = ( a + b + c ) 2 . \frac{(b+c)^{2}}{bc}t_{a}^{2}+\frac{(c+a)^{2}}{ca}t_{b}^{2}+\frac{(a+b)^{2}}{% ab}t_{c}^{2}=(a+b+c)^{2}.
  10. p a = 2 a T a 2 + b 2 - c 2 , p_{a}=\tfrac{2aT}{a^{2}+b^{2}-c^{2}},
  11. p b = 2 b T a 2 + b 2 - c 2 , p_{b}=\tfrac{2bT}{a^{2}+b^{2}-c^{2}},
  12. p c = 2 c T a 2 - b 2 + c 2 , p_{c}=\tfrac{2cT}{a^{2}-b^{2}+c^{2}},
  13. a b c a\geq b\geq c
  14. T . T.
  15. 2 + 1 : 1 \sqrt{2}+1:1

Bit_rate.html

  1. T b T_{b}
  2. R b = 1 T b , R_{b}={1\over T_{b}},
  3. i = 1 n log 2 M i T i \sum_{i=1}^{n}\frac{\log_{2}{M_{i}}}{T_{i}}
  4. b i t r a t e = s a m p l e r a t e × b i t d e p t h × c h a n n e l s bit\ rate=sample\ rate\times bit\ depth\times channels
  5. 44 , 100 × 16 × 2 = 1 , 411 , 200 bit / s = 1 , 411.2 kbit / s 44,100\times 16\times 2=1,411,200\ \mathrm{bit/s}=1,411.2\ \mathrm{kbit/s}
  6. s i z e i n b i t s = s a m p l e r a t e × b i t d e p t h × c h a n n e l s × l e n g t h o f t i m e size\ in\ bits=sample\ rate\times bit\ depth\times channels\times length\ of\ time
  7. s i z e i n b y t e s = s i z e i n b i t s 8 size\ in\ bytes=\frac{size\ in\ bits}{8}
  8. 44 , 100 × 16 × 2 × 4 , 800 8 = 846 , 720 , 000 bytes 847 MB \frac{44,100\times 16\times 2\times 4,800}{8}=846,720,000\ \mathrm{bytes}% \approx 847\ \mathrm{MB}

Bitwise_operation.html

  1. x > y x>y
  2. NOT x = n = 0 b 2 n [ ( x 2 n mod 2 + 1 ) mod 2 ] \,\text{NOT }x=\sum_{n=0}^{b}2^{n}\left[\left(\left\lfloor\frac{x}{2^{n}}% \right\rfloor\bmod 2+1\right)\bmod 2\right]
  3. x AND y = n = 0 b 2 n ( x 2 n mod 2 ) ( y 2 n mod 2 ) x\,\text{ AND }y=\sum_{n=0}^{b}2^{n}\left(\left\lfloor\frac{x}{2^{n}}\right% \rfloor\bmod 2\right)\left(\left\lfloor\frac{y}{2^{n}}\right\rfloor\bmod 2\right)
  4. x OR y = n = 0 b 2 n [ [ ( x 2 n mod 2 ) + ( y 2 n mod 2 ) + ( x 2 n mod 2 ) ( y 2 n mod 2 ) mod 2 ] mod 2 ] x\,\text{ OR }y=\sum_{n=0}^{b}2^{n}\left[\left[\left(\left\lfloor\frac{x}{2^{n% }}\right\rfloor\bmod 2\right)+\left(\left\lfloor\frac{y}{2^{n}}\right\rfloor% \bmod 2\right)+\left(\left\lfloor\frac{x}{2^{n}}\right\rfloor\bmod 2\right)% \left(\left\lfloor\frac{y}{2^{n}}\right\rfloor\bmod 2\right)\bmod 2\right]% \bmod 2\right]
  5. x XOR y = n = 0 b 2 n [ [ ( x 2 n mod 2 ) + ( y 2 n mod 2 ) ] mod 2 ] x\,\text{ XOR }y=\sum_{n=0}^{b}2^{n}\left[\left[\left(\left\lfloor\frac{x}{2^{% n}}\right\rfloor\bmod 2\right)+\left(\left\lfloor\frac{y}{2^{n}}\right\rfloor% \bmod 2\right)\right]\bmod 2\right]
  6. b b
  7. x = log 2 x + 1 x=\lfloor\log_{2}x\rfloor+1
  8. x 0 x\neq 0

Black_model.html

  1. T T T^{\prime}\geq T
  2. c = e - r T [ F N ( d 1 ) - K N ( d 2 ) ] c=e^{-rT}[FN(d_{1})-KN(d_{2})]
  3. p = e - r T [ K N ( - d 2 ) - F N ( - d 1 ) ] p=e^{-rT}[KN(-d_{2})-FN(-d_{1})]
  4. d 1 = ln ( F / K ) + ( σ 2 / 2 ) T σ T d_{1}=\frac{\ln(F/K)+(\sigma^{2}/2)T}{\sigma\sqrt{T}}
  5. d 2 = ln ( F / K ) - ( σ 2 / 2 ) T σ T = d 1 - σ T , d_{2}=\frac{\ln(F/K)-(\sigma^{2}/2)T}{\sigma\sqrt{T}}=d_{1}-\sigma\sqrt{T},
  6. e - r T e^{-rT}
  7. e - r T e^{-rT^{\prime}}
  8. e - r ( T - t ) F ( t ) e^{-r(T-t)}F(t)
  9. e - r ( T - t ) [ F ( t ) - F ( 0 ) ] e^{-r(T-t)}[F(t)-F(0)]
  10. e - r ( T - t ) e^{-r(T-t)}
  11. e - r ( T - t ) F ( t ) e^{-r(T-t)}F(t)

Black–Scholes_model.html

  1. S S
  2. V ( S , t ) V(S,t)
  3. C ( S , t ) C(S,t)
  4. P ( S , t ) P(S,t)
  5. K K
  6. r r
  7. μ \mu
  8. S S
  9. σ \sigma
  10. t t
  11. Π \Pi
  12. N ( x ) N(x)
  13. N ( x ) = 1 2 π - x e - z 2 2 d z N(x)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{x}e^{-\frac{z^{2}}{2}}\,dz
  14. N ( x ) N^{\prime}(x)
  15. N ( x ) = 1 2 π e - x 2 2 N^{\prime}(x)=\frac{1}{\sqrt{2\pi}}e^{-\frac{x^{2}}{2}}
  16. V t + 1 2 σ 2 S 2 2 V S 2 + r S V S - r V = 0 \frac{\partial V}{\partial t}+\frac{1}{2}\sigma^{2}S^{2}\frac{\partial^{2}V}{% \partial S^{2}}+rS\frac{\partial V}{\partial S}-rV=0
  17. C ( S , t ) \displaystyle C(S,t)
  18. P ( S , t ) = K e - r ( T - t ) - S + C ( S , t ) = N ( - d 2 ) K e - r ( T - t ) - N ( - d 1 ) S \begin{aligned}\displaystyle P(S,t)&\displaystyle=Ke^{-r(T-t)}-S+C(S,t)\\ &\displaystyle=N(-d_{2})Ke^{-r(T-t)}-N(-d_{1})S\end{aligned}\,
  19. N ( ) N(\cdot)
  20. T - t T-t
  21. S S
  22. K K
  23. r r
  24. σ \sigma
  25. C ( F , τ ) \displaystyle C(F,\tau)
  26. τ = T - t \tau=T-t
  27. D = e - r τ D=e^{-r\tau}
  28. F = e r τ S = S D F=e^{r\tau}S=\frac{S}{D}
  29. S = D F S=DF
  30. C - P = D ( F - K ) = S - D K C-P=D(F-K)=S-DK
  31. P ( F , τ ) = D [ N ( - d - ) K - N ( - d + ) F ] P(F,\tau)=D\left[N(-d_{-})K-N(-d_{+})F\right]
  32. N ( d ± ) N(d_{\pm})
  33. d ± d_{\pm}
  34. d + d_{+}
  35. C = D [ N ( d + ) F - N ( d - ) K ] C=D\left[N(d_{+})F-N(d_{-})K\right]
  36. C = D N ( d + ) F - D N ( d - ) K C=DN(d_{+})F-DN(d_{-})K
  37. D N ( d + ) F DN(d_{+})F
  38. D N ( d - ) K DN(d_{-})K
  39. N ( d + ) F N(d_{+})~{}F
  40. N ( d - ) K N(d_{-})~{}K
  41. N ( d + ) F N(d_{+})F
  42. N ( d + ) N(d_{+})
  43. N ( d - ) K N(d_{-})K
  44. N ( d - ) , N(d_{-}),
  45. N ( d + ) N(d_{+})
  46. N ( d - ) N(d_{-})
  47. d ± d_{\pm}
  48. N ( d ± ) N(d_{\pm})
  49. N ( d - ) K N(d_{-})K
  50. N ( d + ) F N(d_{+})F
  51. d ± d_{\pm}
  52. 1 2 σ 2 \frac{1}{2}\sigma^{2}
  53. ( r ± 1 2 σ 2 ) τ , \left(r\pm\frac{1}{2}\sigma^{2}\right)\tau,
  54. m = 1 σ τ ln ( F K ) m=\frac{1}{\sigma\sqrt{\tau}}\ln\left(\frac{F}{K}\right)
  55. 1 2 σ 2 \frac{1}{2}\sigma^{2}
  56. N ( d 1 ) , N ( d 2 ) N(d_{1}),N(d_{2})
  57. S T ( 0 , ) S_{T}\in(0,\infty)
  58. p ( S , T ) = N [ d 2 ( S T ) ] S T σ T p(S,T)=\frac{N^{\prime}[d_{2}(S_{T})]}{S_{T}\sigma\sqrt{T}}
  59. d 2 = d 2 ( K ) d_{2}=d_{2}(K)
  60. N ( d 2 ) N(d_{2})
  61. N ( d 1 ) N(d_{1})
  62. S N ( d 1 ) SN(d_{1})
  63. C S \frac{\partial C}{\partial S}
  64. N ( d 1 ) N(d_{1})\,
  65. - N ( - d 1 ) = N ( d 1 ) - 1 -N(-d_{1})=N(d_{1})-1\,
  66. 2 C S 2 \frac{\partial^{2}C}{\partial S^{2}}
  67. N ( d 1 ) S σ T - t \frac{N^{\prime}(d_{1})}{S\sigma\sqrt{T-t}}\,
  68. C σ \frac{\partial C}{\partial\sigma}
  69. S N ( d 1 ) T - t SN^{\prime}(d_{1})\sqrt{T-t}\,
  70. C t \frac{\partial C}{\partial t}
  71. - S N ( d 1 ) σ 2 T - t - r K e - r ( T - t ) N ( d 2 ) -\frac{SN^{\prime}(d_{1})\sigma}{2\sqrt{T-t}}-rKe^{-r(T-t)}N(d_{2})\,
  72. - S N ( d 1 ) σ 2 T - t + r K e - r ( T - t ) N ( - d 2 ) -\frac{SN^{\prime}(d_{1})\sigma}{2\sqrt{T-t}}+rKe^{-r(T-t)}N(-d_{2})\,
  73. C r \frac{\partial C}{\partial r}
  74. K ( T - t ) e - r ( T - t ) N ( d 2 ) K(T-t)e^{-r(T-t)}N(d_{2})\,
  75. - K ( T - t ) e - r ( T - t ) N ( - d 2 ) -K(T-t)e^{-r(T-t)}N(-d_{2})\,
  76. [ t , t + d t ] [t,t+dt]
  77. q S t d t qS_{t}\,dt
  78. q q
  79. C ( S 0 , t ) = e - r ( T - t ) [ F N ( d 1 ) - K N ( d 2 ) ] C(S_{0},t)=e^{-r(T-t)}[FN(d_{1})-KN(d_{2})]\,
  80. P ( S 0 , t ) = e - r ( T - t ) [ K N ( - d 2 ) - F N ( - d 1 ) ] P(S_{0},t)=e^{-r(T-t)}[KN(-d_{2})-FN(-d_{1})]\,
  81. F = S 0 e ( r - q ) ( T - t ) F=S_{0}e^{(r-q)(T-t)}\,
  82. d 1 , d 2 d_{1},d_{2}
  83. d 1 = 1 σ T - t [ ln ( S 0 K ) + ( r - q + 1 2 σ 2 ) ( T - t ) ] d_{1}=\frac{1}{\sigma\sqrt{T-t}}\left[\ln\left(\frac{S_{0}}{K}\right)+(r-q+% \frac{1}{2}\sigma^{2})(T-t)\right]
  84. d 2 = d 1 - σ T - t d_{2}=d_{1}-\sigma\sqrt{T-t}
  85. δ \delta
  86. t 1 , t 2 , t_{1},t_{2},\ldots
  87. S t = S 0 ( 1 - δ ) n ( t ) e u t + σ W t S_{t}=S_{0}(1-\delta)^{n(t)}e^{ut+\sigma W_{t}}
  88. n ( t ) n(t)
  89. t t
  90. C ( S 0 , T ) = e - r T [ F N ( d 1 ) - K N ( d 2 ) ] C(S_{0},T)=e^{-rT}[FN(d_{1})-KN(d_{2})]\,
  91. F = S 0 ( 1 - δ ) n ( T ) e r T F=S_{0}(1-\delta)^{n(T)}e^{rT}\,
  92. V t + 1 2 σ 2 S 2 2 V S 2 + r S V S - r V 0 \frac{\partial V}{\partial t}+\frac{1}{2}\sigma^{2}S^{2}\frac{\partial^{2}V}{% \partial S^{2}}+rS\frac{\partial V}{\partial S}-rV\leq 0
  93. V ( S , T ) = H ( S ) V(S,T)=H(S)
  94. V ( S , t ) H ( S ) V(S,t)\geq H(S)
  95. H ( S ) H(S)
  96. S S
  97. s * s*
  98. S - X S-X

Blast_furnace.html

  1. \rightleftharpoons

Block_and_tackle.html

  1. M A = F B F A = n , MA=\frac{F_{B}}{F_{A}}=n,\!
  2. F a = L N 1 𝑒𝑓𝑓 F_{a}=\frac{L}{N}\frac{1}{\,\textit{eff}}
  3. F a F_{a}
  4. L L
  5. N N
  6. e f f eff
  7. S S
  8. x x
  9. 1 𝑒𝑓𝑓 1 + S x 100 . \frac{1}{\,\textit{eff}}\approx 1+S\frac{x}{100}.
  10. S S
  11. x x
  12. K K
  13. 𝑒𝑓𝑓 = K N - 1 K S N ( K - 1 ) . \,\textit{eff}=\frac{K^{N}-1}{K^{S}N(K-1)}.
  14. K K

Block_cipher_mode_of_operation.html

  1. C i = E K ( P i C i - 1 ) , C 0 = I V C_{i}=E_{K}(P_{i}\oplus C_{i-1}),C_{0}=IV
  2. P i = D K ( C i ) C i - 1 , C 0 = I V . P_{i}=D_{K}(C_{i})\oplus C_{i-1},C_{0}=IV.
  3. C i = E K ( P i P i - 1 C i - 1 ) , P 0 C 0 = I V C_{i}=E_{K}(P_{i}\oplus P_{i-1}\oplus C_{i-1}),P_{0}\oplus C_{0}=IV
  4. P i = D K ( C i ) P i - 1 C i - 1 , P 0 C 0 = I V P_{i}=D_{K}(C_{i})\oplus P_{i-1}\oplus C_{i-1},P_{0}\oplus C_{0}=IV
  5. C i = E K ( C i - 1 ) P i C_{i}=E_{K}(C_{i-1})\oplus P_{i}
  6. P i = E K ( C i - 1 ) C i P_{i}=E_{K}(C_{i-1})\oplus C_{i}
  7. C 0 = IV C_{0}=\ \mbox{IV}~{}
  8. P i = head ( E K ( S i - 1 ) , x ) C i P_{i}=\mbox{head}~{}(E_{K}(S_{i-1}),x)\oplus C_{i}
  9. S i = ( ( S i - 1 x ) + C i ) mod 2 n S_{i}=\ ((S_{i-1}<<x)+C_{i})\mbox{ mod }~{}2^{n}
  10. S 0 = IV S_{0}=\ \mbox{IV}~{}
  11. C j = P j O j C_{j}=P_{j}\oplus O_{j}
  12. P j = C j O j P_{j}=C_{j}\oplus O_{j}
  13. O j = E K ( I j ) O_{j}=\ E_{K}(I_{j})
  14. I j = O j - 1 I_{j}=\ O_{j-1}
  15. I 0 = IV I_{0}=\ \mbox{IV}~{}
  16. 2 32 2^{32}

Blood_sugar.html

  1. Glucose + Alkaline copper tartarate Reduction Cuprous oxide \mathrm{Glucose}+\mathrm{Alkaline\ copper\ tartarate}\xrightarrow{\mathrm{% Reduction}}\mathrm{Cuprous\ oxide}
  2. Cu + + + Phosphomolybdic acid Oxidation Phosphomolybdenum oxide \mathrm{Cu}^{++}+\mathrm{Phosphomolybdic\ acid}\xrightarrow{\mathrm{Oxidation}% }\mathrm{Phosphomolybdenum\ oxide}
  3. Cu + + + Arsenomolybdic acid Oxidation Arsenomolybdenum oxide \mathrm{Cu}^{++}+\mathrm{Arsenomolybdic\ acid}\xrightarrow{\mathrm{Oxidation}}% \mathrm{Arsenomolybdenum\ oxide}
  4. Cu + + + Neocuproine Oxidation Cu + + neocuproine complex \mathrm{Cu}^{++}+\mathrm{Neocuproine}\xrightarrow{\mathrm{Oxidation}}\mathrm{% Cu}^{++}\mathrm{neocuproine\ complex}
  5. Glucose + Alkaline ferricyanide Ferrocyanide \mathrm{Glucose}+\mathrm{Alkaline\ ferricyanide}\longrightarrow\mathrm{Ferrocyanide}
  6. Glucose + O 2 glucose oxidase Oxidation D-glucono-1,5-lactone + H 2 O 2 \mathrm{Glucose}+\mathrm{O}_{2}\xrightarrow[\mathrm{Oxidation}]{\mathrm{% glucose\ oxidase}}\textrm{D-glucono-1,5-lactone}+\mathrm{H_{2}O_{2}}
  7. H 2 O 2 + 𝑂 -dianisidine Oxidation peroxidase H 2 O + oxidized chromogen \mathrm{H_{2}O_{2}}+\,\textit{O}\,\text{-dianisidine}\xrightarrow[\mathrm{% Oxidation}]{\mathrm{peroxidase}}\mathrm{H_{2}O}+\mathrm{oxidized\ chromogen}
  8. Glucose + ATP Hexokinase + Mg + + Phosphorylation G-6PO 4 + ADP G-6PO 4 + NADP G-6PD Oxidation 6-Phosphogluconate + NADPH + H + \begin{aligned}&\displaystyle\mathrm{Glucose}+\mathrm{ATP}\xrightarrow[\mathrm% {Phosphorylation}]{\mathrm{Hexokinase}+\mathrm{Mg}^{++}}\textrm{G-6PO}_{4}+% \mathrm{ADP}\\ &\displaystyle\textrm{G-6PO}_{4}+\mathrm{NADP}\xrightarrow[\mathrm{Oxidation}]% {\textrm{G-6PD}}\textrm{6-Phosphogluconate}+\mathrm{NADPH}+\mathrm{H}^{+}\\ \end{aligned}

BMP_file_format.html

  1. RowSize = BitsPerPixel ImageWidth + 31 32 4 , \mbox{RowSize}~{}=\left\lfloor\frac{\mbox{BitsPerPixel}~{}\cdot\mbox{% ImageWidth}~{}+31}{32}\right\rfloor\cdot 4,
  2. PixelArraySize = RowSize | ImageHeight | \mbox{PixelArraySize}~{}=\mbox{RowSize}~{}\cdot\left|\mbox{ImageHeight}~{}\right|

Bode_plot.html

  1. log ( a b ) = log ( a ) + log ( b ) . \log(a\cdot b)=\log(a)+\log(b)\;.
  2. A sin ( ω t ) A\sin(\omega t)
  3. x x
  4. - Φ -\Phi
  5. ( A x ) sin ( ω t - Φ ) (\frac{A}{x})\sin(\omega t-\Phi)
  6. Φ \Phi
  7. T High ( f ) = j f f 1 1 + j f f 1 , \mathrm{T_{High}}(f)=\frac{\mathrm{j}\frac{f}{f_{1}}}{1+\mathrm{j}\frac{f}{f_{% 1}}}\;,
  8. | T High ( f ) | = f f 1 1 + ( f f 1 ) 2 , \left|\mathrm{T_{High}}(f)\right|=\frac{\frac{f}{f_{1}}}{\sqrt{1+\left(\frac{f% }{f_{1}}\right)^{2}}}\;,
  9. φ T High = 90 - tan - 1 ( f f 1 ) . \varphi_{\mathrm{T_{High}}}=90^{\circ}-\tan^{-1}\left(\frac{f}{f_{1}}\right)\;.
  10. 20 log 10 | T High ( f ) | = 20 log 10 ( f f 1 ) 20\log_{10}\left|\mathrm{T_{High}}(f)\right|=20\log_{10}\left(\frac{f}{f_{1}}\right)
  11. - 20 log 10 ( 1 + ( f f 1 ) 2 ) . \ -20\log_{10}\left(\sqrt{1+\left(\frac{f}{f_{1}}\right)^{2}}\right)\;.
  12. T Low ( f ) = 1 1 + j f f 1 . \mathrm{T_{Low}}(f)=\frac{1}{1+\mathrm{j}\frac{f}{f_{1}}}\;.
  13. f ( x ) = A ( x - c n ) a n f(x)=A\prod(x-c_{n})^{a_{n}}
  14. log ( f ( x ) ) = log ( A ) + a n log ( x - c n ) . \log(f(x))=\log(A)+\sum a_{n}\log(x-c_{n})\;.
  15. 𝐝𝐁 = 20 log 10 ( X ) \mathbf{dB}=20\log_{10}(X)
  16. H ( s ) = A ( s - x n ) a n ( s - y n ) b n H(s)=A\prod\frac{(s-x_{n})^{a_{n}}}{(s-y_{n})^{b_{n}}}
  17. x n x_{n}
  18. y n y_{n}
  19. s = j ω s=\mathrm{j}\omega
  20. a n , b n > 0 a_{n},b_{n}>0
  21. H H
  22. ω = x n \omega=x_{n}
  23. 20 a n dB 20\cdot a_{n}\,\mathrm{dB}
  24. ω = y n \omega=y_{n}
  25. 20 b n dB 20\cdot b_{n}\,\mathrm{dB}
  26. ω \omega
  27. | H ( j ω ) | |H(\mathrm{j}\omega)|
  28. a x 2 + b x + c ax^{2}+bx+c
  29. ( a x + c ) 2 (\sqrt{a}x+\sqrt{c})^{2}
  30. ω \omega
  31. x n x_{n}
  32. y n y_{n}
  33. H ( j ω ) H(\mathrm{j}\omega)
  34. | H ( j ω ) | = H H * |H(\mathrm{j}\omega)|=\sqrt{H\cdot H^{*}}
  35. ( s + x n ) (s+x_{n})
  36. ( x n + j ω ) ( x n - j ω ) = x n 2 + ω 2 \sqrt{(x_{n}+\mathrm{j}\omega)\cdot(x_{n}-\mathrm{j}\omega)}=\sqrt{x_{n}^{2}+% \omega^{2}}
  37. 3 a n dB 3\cdot a_{n}\ \mathrm{dB}
  38. 3 b n dB 3\cdot b_{n}\ \mathrm{dB}
  39. x n x_{n}
  40. y n y_{n}
  41. H ( s ) = A ( s - x n ) a n ( s - y n ) b n H(s)=A\prod\frac{(s-x_{n})^{a_{n}}}{(s-y_{n})^{b_{n}}}
  42. - arctan ( Im [ H ( s ) ] Re [ H ( s ) ] ) -\arctan\left(\tfrac{\mathrm{Im}[H(s)]}{\mathrm{Re}[H(s)]}\right)
  43. A A
  44. 0 deg 0\deg
  45. A A
  46. 180 deg 180\deg
  47. ω = | x n | \omega=|x_{n}|
  48. Re ( z ) < 0 \operatorname{Re}(z)<0
  49. 45 a n 45\cdot a_{n}
  50. ω = | x n | \omega=|x_{n}|
  51. | x n | 10 \frac{|x_{n}|}{10}
  52. ω = | y n | \omega=|y_{n}|
  53. Re ( p ) < 0 \operatorname{Re}(p)<0
  54. 45 b n 45\cdot b_{n}
  55. ω = | y n | \omega=|y_{n}|
  56. | y n | 10 \frac{|y_{n}|}{10}
  57. Re ( s ) > 0 \operatorname{Re}(s)>0
  58. 90 a n 90\cdot a_{n}
  59. 90 b n 90\cdot b_{n}
  60. H ( j f ) = 1 1 + j2 π f R C . H(\mathrm{j}f)=\frac{1}{1+\mathrm{j}2\pi fRC}\;.
  61. f c f_{\mathrm{c}}
  62. f c = 1 2 π R C f_{\mathrm{c}}={1\over{2\mathrm{\pi}RC}}
  63. ω c = 1 R C \omega_{\mathrm{c}}={1\over{RC}}
  64. ω c = 2 π f c \omega_{\mathrm{c}}=2\mathrm{\pi}f_{\mathrm{c}}
  65. H ( j ω ) = 1 1 + j ω ω c . H(\mathrm{j}\omega)={1\over 1+\mathrm{j}{\omega\over{{\omega_{\mathrm{c}}}}}}\;.
  66. A vdB A_{\mathrm{vdB}}
  67. A vdB \displaystyle A_{\mathrm{vdB}}
  68. ω \omega
  69. ω c \omega_{\mathrm{c}}
  70. ω ω c {\omega\over{\omega_{\mathrm{c}}}}
  71. ω c \omega_{\mathrm{c}}
  72. ω ω c {\omega\over{\omega_{\mathrm{c}}}}
  73. - 20 log ω ω c -20\log{\omega\over{\omega_{\mathrm{c}}}}
  74. - 20 dB -20\,\mathrm{dB}
  75. φ = - tan - 1 ω ω c \varphi=-\tan^{-1}{\omega\over{\omega_{\mathrm{c}}}}
  76. ω \omega
  77. ω \omega
  78. ω c \omega_{\mathrm{c}}
  79. ω ω c {\omega\over{\omega_{\mathrm{c}}}}
  80. ω = ω c \omega=\omega_{\mathrm{c}}
  81. ω ω c {\omega\over{\omega_{\mathrm{c}}}}
  82. ω c \omega_{\mathrm{c}}
  83. A FB = A OL 1 + β A OL , A_{\mathrm{FB}}=\frac{A_{\mathrm{OL}}}{1+\beta A_{\mathrm{OL}}}\;,
  84. β A OL ( f 180 ) = - | β A OL ( f 180 ) | = - | β A OL | 180 , \beta A_{\mathrm{OL}}\left(f_{180}\right)=-|\beta A_{\mathrm{OL}}\left(f_{180}% \right)|=-|\beta A_{\mathrm{OL}}|_{180}\;,
  85. | a + j b | = [ a 2 + b 2 ] 1 2 |a+\mathrm{j}b|=\left[a^{2}+b^{2}\right]^{\frac{1}{2}}
  86. | β A OL ( f 0 d B ) | = 1 . |\beta A_{\mathrm{OL}}\left(f_{\mathrm{0dB}}\right)|=1\;.
  87. 20 log 10 ( | β A OL | 180 ) = 20 log 10 ( | A OL | 180 ) - 20 log 10 ( β - 1 ) 20\log_{10}(|\beta A_{\mathrm{OL}}|_{180})=20\log_{10}(|A_{\mathrm{OL}}|_{180}% )-20\log_{10}(\beta^{-1})

Bogomol'nyi–Prasad–Sommerfield_bound.html

  1. E = d 3 x [ 1 2 D φ T D φ + 1 2 π T π + V ( φ ) + 1 2 g 2 Tr [ E E + B B ] ] E=\int d^{3}x\,\left[\frac{1}{2}\overrightarrow{D\varphi}^{T}\cdot% \overrightarrow{D\varphi}+\frac{1}{2}\pi^{T}\pi+V(\varphi)+\frac{1}{2g^{2}}% \operatorname{Tr}\left[\vec{E}\cdot\vec{E}+\vec{B}\cdot\vec{B}\right]\right]
  2. E d 3 x [ 1 2 Tr [ D φ D φ ] + 1 2 g 2 Tr [ B B ] ] d 3 x Tr [ 1 2 ( D φ 1 g B ) 2 ± 1 g D φ B ] ± 1 g d 3 x Tr [ D φ B ] = ± 1 g S 2 boundary Tr [ φ B d S ] . \begin{aligned}\displaystyle E&\displaystyle\geq\int d^{3}x\,\left[\frac{1}{2}% \operatorname{Tr}\left[\overrightarrow{D\varphi}\cdot\overrightarrow{D\varphi}% \right]+\frac{1}{2g^{2}}\operatorname{Tr}\left[\vec{B}\cdot\vec{B}\right]% \right]\\ &\displaystyle\geq\int d^{3}x\,\operatorname{Tr}\left[\frac{1}{2}\left(% \overrightarrow{D\varphi}\mp\frac{1}{g}\vec{B}\right)^{2}\pm\frac{1}{g}% \overrightarrow{D\varphi}\cdot\vec{B}\right]\\ &\displaystyle\geq\pm\frac{1}{g}\int d^{3}x\,\operatorname{Tr}\left[% \overrightarrow{D\varphi}\cdot\vec{B}\right]\\ &\displaystyle=\pm\frac{1}{g}\int_{S^{2}\ \mathrm{boundary}}\operatorname{Tr}% \left[\varphi\vec{B}\cdot d\vec{S}\right].\end{aligned}
  3. E S 2 Tr [ φ B d S ] . E\geq\left\|\int_{S^{2}}\operatorname{Tr}\left[\varphi\vec{B}\cdot d\vec{S}% \right]\right\|.
  4. π = 0 \pi=0
  5. D φ 1 g B = 0 \overrightarrow{D\varphi}\mp\frac{1}{g}\vec{B}=0

Bogosort.html

  1. ( e - 1 ) n ! (e-1)n!
  2. ( n - 1 ) n ! (n-1)n!
  3. n - 1 n-1

Bohr_magneton.html

  1. 1 / 2 {1}/{2}
  2. e m e \frac{e\hbar}{m_{\mathrm{e}}}
  3. μ B = e 2 m e \mu_{\mathrm{B}}=\frac{e\hbar}{2m_{\mathrm{e}}}
  4. μ B = e 2 m e c \mu_{\mathrm{B}}=\frac{e\hbar}{2m_{\mathrm{e}}c}

Bohr_radius.html

  1. a 0 a_{0}
  2. × 10 11 \times 10^{−}11
  3. a 0 = 4 π ε 0 2 m e e 2 = m e c α a_{0}=\frac{4\pi\varepsilon_{0}\hbar^{2}}{m_{\mathrm{e}}e^{2}}=\frac{\hbar}{m_% {\mathrm{e}}\,c\,\alpha}
  4. a 0 a_{0}
  5. ε 0 \varepsilon_{0}
  6. \hbar
  7. m e m_{\mathrm{e}}
  8. e e
  9. c c
  10. α \alpha
  11. a 0 = 2 m e e 2 a_{0}=\frac{\hbar^{2}}{m_{e}e^{2}}
  12. × 10 11 \times 10^{−}11
  13. λ e \lambda_{\mathrm{e}}
  14. r e r_{\mathrm{e}}
  15. m e m_{\mathrm{e}}
  16. \hbar
  17. e e
  18. m e m_{\mathrm{e}}
  19. \hbar
  20. c c
  21. m e m_{\mathrm{e}}
  22. c c
  23. e e
  24. α \alpha
  25. r e = α λ e 2 π = α 2 a 0 . r_{\mathrm{e}}=\frac{\alpha\lambda_{\mathrm{e}}}{2\pi}=\alpha^{2}a_{0}.
  26. a 0 * = λ p + λ e 2 π α , \ a_{0}^{*}\ =\frac{\lambda_{\mathrm{p}}+\lambda_{\mathrm{e}}}{2\pi\alpha},
  27. λ p \lambda_{\mathrm{p}}
  28. λ e \lambda_{\mathrm{e}}
  29. α \alpha

Bombardier_beetle.html

  1. H 2 O 2 ( a q ) H 2 O ( l ) + 1 2 O 2 ( g ) H_{2}O_{2(aq)}\longrightarrow H_{2}O_{(l)}+\tfrac{1}{2}O_{2(g)}
  2. C 6 H 4 ( O H ) 2 ( a q ) C 6 H 4 O 2 ( a q ) + H 2 ( g ) C_{6}H_{4}(OH)_{2(aq)}\longrightarrow C_{6}H_{4}O_{2(aq)}+H_{2(g)}
  3. C 6 H 4 ( O H ) 2 ( a q ) + H 2 O 2 ( a q ) C 6 H 4 O 2 ( a q ) + 2 H 2 O ( l ) C_{6}H_{4}(OH)_{2(aq)}+H_{2}O_{2(aq)}\longrightarrow C_{6}H_{4}O_{2(aq)}+2H_{2% }O_{(l)}

Boolean_prime_ideal_theorem.html

  1. \vee

Borwein's_algorithm.html

  1. A \displaystyle A
  2. - C 3 π = n = 0 ( 6 n ) ! ( 3 n ) ! ( n ! ) 3 A + n B C 3 n \frac{\sqrt{-C^{3}}}{\pi}=\sum_{n=0}^{\infty}{\frac{(6n)!}{(3n)!(n!)^{3}}\frac% {A+nB}{C^{3n}}}
  3. a 0 = 1 3 s 0 = 3 - 1 2 \begin{aligned}\displaystyle a_{0}&\displaystyle=\frac{1}{3}\\ \displaystyle s_{0}&\displaystyle=\frac{\sqrt{3}-1}{2}\end{aligned}
  4. r k + 1 = 3 1 + 2 ( 1 - s k 3 ) 1 / 3 s k + 1 = r k + 1 - 1 2 a k + 1 = r k + 1 2 a k - 3 k ( r k + 1 2 - 1 ) \begin{aligned}\displaystyle r_{k+1}&\displaystyle=\frac{3}{1+2(1-s_{k}^{3})^{% 1/3}}\\ \displaystyle s_{k+1}&\displaystyle=\frac{r_{k+1}-1}{2}\\ \displaystyle a_{k+1}&\displaystyle=r_{k+1}^{2}a_{k}-3^{k}(r_{k+1}^{2}-1)\end{aligned}
  5. A = 212175710912 61 + 1657145277365 B = 13773980892672 61 + 107578229802750 C = ( 5280 ( 236674 + 30303 61 ) ) 3 \begin{aligned}\displaystyle A&\displaystyle=212175710912\sqrt{61}+16571452773% 65\\ \displaystyle B&\displaystyle=13773980892672\sqrt{61}+107578229802750\\ \displaystyle C&\displaystyle=(5280(236674+30303\sqrt{61}))^{3}\end{aligned}
  6. 1 / π = 12 n = 0 ( - 1 ) n ( 6 n ) ! ( A + n B ) ( n ! ) 3 ( 3 n ) ! C n + 1 / 2 1/\pi=12\sum_{n=0}^{\infty}\frac{(-1)^{n}(6n)!\,(A+nB)}{(n!)^{3}(3n)!\,C^{n+1/% 2}}\,\!
  7. a 0 = 6 - 4 2 y 0 = 2 - 1 \begin{aligned}\displaystyle a_{0}&\displaystyle=6-4\sqrt{2}\\ \displaystyle y_{0}&\displaystyle=\sqrt{2}-1\end{aligned}
  8. y k + 1 \displaystyle y_{k+1}
  9. a 0 = 2 b 0 = 0 p 0 = 2 + 2 \begin{aligned}\displaystyle a_{0}&\displaystyle=\sqrt{2}\\ \displaystyle b_{0}&\displaystyle=0\\ \displaystyle p_{0}&\displaystyle=2+\sqrt{2}\end{aligned}
  10. a n + 1 = a n + 1 / a n 2 b n + 1 = ( 1 + b n ) a n a n + b n p n + 1 = ( 1 + a n + 1 ) p n b n + 1 1 + b n + 1 \begin{aligned}\displaystyle a_{n+1}&\displaystyle=\frac{\sqrt{a_{n}}+1/\sqrt{% a_{n}}}{2}\\ \displaystyle b_{n+1}&\displaystyle=\frac{(1+b_{n})\sqrt{a_{n}}}{a_{n}+b_{n}}% \\ \displaystyle p_{n+1}&\displaystyle=\frac{(1+a_{n+1})\,p_{n}b_{n+1}}{1+b_{n+1}% }\end{aligned}
  11. a 0 = 1 2 s 0 = 5 ( 5 - 2 ) \begin{aligned}\displaystyle a_{0}&\displaystyle=\frac{1}{2}\\ \displaystyle s_{0}&\displaystyle=5(\sqrt{5}-2)\end{aligned}
  12. x n + 1 = 5 s n - 1 y n + 1 = ( x n + 1 - 1 ) 2 + 7 z n + 1 = ( 1 2 x n + 1 ( y n + 1 + y n + 1 2 - 4 x n + 1 3 ) ) 1 / 5 a n + 1 = s n 2 a n - 5 n ( s n 2 - 5 2 + s n ( s n 2 - 2 s n + 5 ) ) s n + 1 = 25 ( z n + 1 + x n + 1 / z n + 1 + 1 ) 2 s n \begin{aligned}\displaystyle x_{n+1}&\displaystyle=\frac{5}{s_{n}}-1\\ \displaystyle y_{n+1}&\displaystyle=(x_{n+1}-1)^{2}+7\\ \displaystyle z_{n+1}&\displaystyle=\left(\frac{1}{2}x_{n+1}\left(y_{n+1}+% \sqrt{y_{n+1}^{2}-4x_{n+1}^{3}}\right)\right)^{1/5}\\ \displaystyle a_{n+1}&\displaystyle=s_{n}^{2}a_{n}-5^{n}\left(\frac{s_{n}^{2}-% 5}{2}+\sqrt{s_{n}(s_{n}^{2}-2s_{n}+5)}\right)\\ \displaystyle s_{n+1}&\displaystyle=\frac{25}{(z_{n+1}+x_{n+1}/z_{n+1}+1)^{2}s% _{n}}\end{aligned}
  13. 0 < a n - 1 π < 16 5 n e - 5 n π 0<a_{n}-\frac{1}{\pi}<16\cdot 5^{n}\cdot e^{-5^{n}}\pi\,\!
  14. a 0 = 1 3 r 0 = 3 - 1 2 s 0 = ( 1 - r 0 3 ) 1 / 3 \begin{aligned}\displaystyle a_{0}&\displaystyle=\frac{1}{3}\\ \displaystyle r_{0}&\displaystyle=\frac{\sqrt{3}-1}{2}\\ \displaystyle s_{0}&\displaystyle=(1-r_{0}^{3})^{1/3}\end{aligned}
  15. t n + 1 = 1 + 2 r n u n + 1 = ( 9 r n ( 1 + r n + r n 2 ) ) 1 / 3 v n + 1 = t n + 1 2 + t n + 1 u n + 1 + u n + 1 2 w n + 1 = 27 ( 1 + s n + s n 2 ) v n + 1 a n + 1 = w n + 1 a n + 3 2 n - 1 ( 1 - w n + 1 ) s n + 1 = ( 1 - r n ) 3 ( t n + 1 + 2 u n + 1 ) v n + 1 r n + 1 = ( 1 - s n + 1 3 ) 1 / 3 \begin{aligned}\displaystyle t_{n+1}&\displaystyle=1+2r_{n}\\ \displaystyle u_{n+1}&\displaystyle=(9r_{n}(1+r_{n}+r_{n}^{2}))^{1/3}\\ \displaystyle v_{n+1}&\displaystyle=t_{n+1}^{2}+t_{n+1}u_{n+1}+u_{n+1}^{2}\\ \displaystyle w_{n+1}&\displaystyle=\frac{27(1+s_{n}+s_{n}^{2})}{v_{n+1}}\\ \displaystyle a_{n+1}&\displaystyle=w_{n+1}a_{n}+3^{2n-1}(1-w_{n+1})\\ \displaystyle s_{n+1}&\displaystyle=\frac{(1-r_{n})^{3}}{(t_{n+1}+2u_{n+1})v_{% n+1}}\\ \displaystyle r_{n+1}&\displaystyle=(1-s_{n+1}^{3})^{1/3}\end{aligned}

Bose–Einstein_statistics.html

  1. N V n q \frac{N}{V}\geq n_{q}
  2. n i ( ε i ) = g i e ( ε i - μ ) / k T - 1 n_{i}(\varepsilon_{i})=\frac{g_{i}}{e^{(\varepsilon_{i}-\mu)/kT}-1}
  3. ϵ i \epsilon_{i}
  4. n ¯ i ( ϵ i ) = g i e ( ϵ i - μ ) / k T + 1 \bar{n}_{i}(\epsilon_{i})=\frac{g_{i}}{e^{(\epsilon_{i}-\mu)/kT}+1}
  5. k T ε i - μ kT\gg\varepsilon_{i}-\mu
  6. n i = g i k T ε i - μ n_{i}=\frac{g_{i}kT}{\varepsilon_{i}-\mu}
  7. 𝒵 \displaystyle\mathcal{Z}
  8. N = k B T 1 𝒵 ( 𝒵 μ ) V , T = 1 exp ( ( ϵ - μ ) / k B T ) - 1 \langle N\rangle=k_{B}T\frac{1}{\mathcal{Z}}\left(\frac{\partial\mathcal{Z}}{% \partial\mu}\right)_{V,T}=\frac{1}{\exp((\epsilon-\mu)/k_{B}T)-1}
  9. ( Δ N ) 2 = k B T ( d N d μ ) V , T = N 2 - N 2 \langle(\Delta N)^{2}\rangle=k_{B}T\left(\frac{d\langle N\rangle}{d\mu}\right)% _{V,T}=\langle N^{2}\rangle-\langle N\rangle^{2}
  10. ( Δ N ) 2 = N \langle(\Delta N)^{2}\rangle=\langle N\rangle
  11. i \displaystyle i
  12. ε i \displaystyle\varepsilon_{i}
  13. n i \displaystyle n_{i}
  14. g i \displaystyle g_{i}
  15. g i \displaystyle g_{i}
  16. i \displaystyle i
  17. w ( n , g ) \displaystyle w(n,g)
  18. n \displaystyle n
  19. g \displaystyle g
  20. n \displaystyle n
  21. w ( n , 1 ) = 1 \displaystyle w(n,1)=1
  22. ( n + 1 ) \displaystyle(n+1)
  23. n \displaystyle n
  24. w ( n , 2 ) = ( n + 1 ) ! n ! 1 ! . w(n,2)=\frac{(n+1)!}{n!1!}.
  25. n \displaystyle n
  26. w ( n , 3 ) = w ( n , 2 ) + w ( n - 1 , 2 ) + + w ( 1 , 2 ) + w ( 0 , 2 ) w(n,3)=w(n,2)+w(n-1,2)+\cdots+w(1,2)+w(0,2)
  27. w ( n , 3 ) = k = 0 n w ( n - k , 2 ) = k = 0 n ( n - k + 1 ) ! ( n - k ) ! 1 ! = ( n + 2 ) ! n ! 2 ! w(n,3)=\sum_{k=0}^{n}w(n-k,2)=\sum_{k=0}^{n}\frac{(n-k+1)!}{(n-k)!1!}=\frac{(n% +2)!}{n!2!}
  28. k = 0 n ( k + a ) ! k ! a ! = ( n + a + 1 ) ! n ! ( a + 1 ) ! . \sum_{k=0}^{n}\frac{(k+a)!}{k!a!}=\frac{(n+a+1)!}{n!(a+1)!}.
  29. w ( n , g ) \displaystyle w(n,g)
  30. w ( n , g ) = ( n + g - 1 ) ! n ! ( g - 1 ) ! . w(n,g)=\frac{(n+g-1)!}{n!(g-1)!}.
  31. n i \displaystyle n_{i}
  32. W = i w ( n i , g i ) = i ( n i + g i - 1 ) ! n i ! ( g i - 1 ) ! i ( n i + g i ) ! n i ! ( g i - 1 ) ! W=\prod_{i}w(n_{i},g_{i})=\prod_{i}\frac{(n_{i}+g_{i}-1)!}{n_{i}!(g_{i}-1)!}% \approx\prod_{i}\frac{(n_{i}+g_{i})!}{n_{i}!(g_{i}-1)!}
  33. n i 1 n_{i}\gg 1
  34. n i \displaystyle n_{i}
  35. W \displaystyle W
  36. ln ( W ) \displaystyle\ln(W)
  37. n i \displaystyle n_{i}
  38. f ( n i ) = ln ( W ) + α ( N - n i ) + β ( E - n i ε i ) f(n_{i})=\ln(W)+\alpha(N-\sum n_{i})+\beta(E-\sum n_{i}\varepsilon_{i})
  39. n i 1 n_{i}\gg 1
  40. ( x ! x x e - x 2 π x ) \left(x!\approx x^{x}\,e^{-x}\,\sqrt{2\pi x}\right)
  41. f ( n i ) = i ( n i + g i ) ln ( n i + g i ) - n i ln ( n i ) + α ( N - n i ) + β ( E - n i ε i ) + K . f(n_{i})=\sum_{i}(n_{i}+g_{i})\ln(n_{i}+g_{i})-n_{i}\ln(n_{i})+\alpha\left(N-% \sum n_{i}\right)+\beta\left(E-\sum n_{i}\varepsilon_{i}\right)+K.
  42. n i n_{i}
  43. n i \displaystyle n_{i}
  44. n i \displaystyle n_{i}
  45. n i = g i e α + β ε i - 1 . n_{i}=\frac{g_{i}}{e^{\alpha+\beta\varepsilon_{i}}-1}.
  46. d ln W = α d N + β d E d\ln W=\alpha\,dN+\beta\,dE
  47. S = k ln W S=k\,\ln W
  48. β = 1 k T \beta=\frac{1}{kT}
  49. α = - μ k T \alpha=-\frac{\mu}{kT}
  50. μ \mu
  51. n i = g i e ( ε i - μ ) / k T - 1 . n_{i}=\frac{g_{i}}{e^{(\varepsilon_{i}-\mu)/kT}-1}.
  52. n i = g i e ε i / k T / z - 1 , n_{i}=\frac{g_{i}}{e^{\varepsilon_{i}/kT}/z-1},
  53. z = exp ( μ / k T ) \displaystyle z=\exp(\mu/kT)
  54. α \alpha
  55. μ \mu
  56. ( g - 1 + n ) ! ( g - 1 ) ! n ! \frac{(g-1+n)!}{(g-1)!n!}
  57. n \displaystyle n
  58. { 1 , , g } \displaystyle\left\{1,\dots,g\right\}
  59. g 1 g\geq 1
  60. i \displaystyle i
  61. m i \displaystyle m_{i}
  62. ( i - 1 ) \displaystyle(i-1)
  63. m i - 1 \displaystyle m_{i-1}
  64. m i m i - 1 m_{i}\geq m_{i-1}
  65. ( m 1 , m 2 , , m n ) \displaystyle\left(m_{1},m_{2},\dots,m_{n}\right)
  66. m i m i - 1 m_{i}\geq m_{i-1}
  67. S ( n , g ) \displaystyle S(n,g)
  68. S ( n , g ) = { ( m 1 , m 2 , , m n ) | . m i m i - 1 , m i { 1 , , g } , i = 1 , , n } . S(n,g)=\Big\{\left(m_{1},m_{2},\dots,m_{n}\right)\Big|\Big.m_{i}\geq m_{i-1},m% _{i}\in\left\{1,\dots,g\right\},\forall i=1,\dots,n\Big\}.
  69. w ( n , g ) \displaystyle w(n,g)
  70. n \displaystyle n
  71. g \displaystyle g
  72. S ( n , g ) \displaystyle S(n,g)
  73. S ( n , g ) \displaystyle S(n,g)
  74. w ( n , g ) \displaystyle w(n,g)
  75. S ( n , g ) \displaystyle S(n,g)
  76. S ( 4 , 3 ) = { ( 1111 ) , ( 1112 ) , ( 1113 ) ( a ) , ( 1122 ) , ( 1123 ) , ( 1133 ) ( b ) , ( 1222 ) , ( 1223 ) , ( 1233 ) , ( 1333 ) ( c ) , S(4,3)=\left\{\underbrace{(1111),(1112),(1113)}_{(a)},\underbrace{(1122),(1123% ),(1133)}_{(b)},\underbrace{(1222),(1223),(1233),(1333)}_{(c)},\right.
  77. ( 2222 ) , ( 2223 ) , ( 2233 ) , ( 2333 ) , ( 3333 ) ( d ) } \left.\underbrace{(2222),(2223),(2233),(2333),(3333)}_{(d)}\right\}
  78. w ( 4 , 3 ) = 15 \displaystyle w(4,3)=15
  79. 15 \displaystyle 15
  80. S ( 4 , 3 ) \displaystyle S(4,3)
  81. ( a ) \displaystyle(a)
  82. m i \displaystyle m_{i}
  83. 1 \displaystyle 1
  84. m n \displaystyle m_{n}
  85. 1 \displaystyle 1
  86. g = 3 \displaystyle g=3
  87. ( b ) \displaystyle(b)
  88. m 1 = m 2 = 1 \displaystyle m_{1}=m_{2}=1
  89. m 3 \displaystyle m_{3}
  90. 2 \displaystyle 2
  91. g = 3 \displaystyle g=3
  92. m i m i - 1 \displaystyle m_{i}\geq m_{i-1}
  93. S ( n , g ) \displaystyle S(n,g)
  94. m 4 \displaystyle m_{4}
  95. { 2 , 3 } \displaystyle\left\{2,3\right\}
  96. ( c ) \displaystyle(c)
  97. ( d ) \displaystyle(d)
  98. S ( 4 , 3 ) \displaystyle S(4,3)
  99. n = 4 \displaystyle n=4
  100. { 1 , 2 , 3 } \displaystyle\left\{1,2,3\right\}
  101. g = 3 \displaystyle g=3
  102. 3 4 = ( 3 + 4 - 1 3 - 1 ) = ( 3 + 4 - 1 4 ) = 6 ! 4 ! 2 ! = 15 \displaystyle\left\langle\begin{matrix}3\\ 4\end{matrix}\right\rangle={3+4-1\choose 3-1}={3+4-1\choose 4}=\frac{6!}{4!2!}% =15
  103. S ( n , g ) \displaystyle S(n,g)
  104. n \displaystyle n
  105. { 1 , , g } \displaystyle\left\{1,\dots,g\right\}
  106. g \displaystyle g
  107. w ( n , g ) \displaystyle w(n,g)
  108. w ( n , g ) = g n = ( g + n - 1 g - 1 ) = ( g + n - 1 n ) = ( g + n - 1 ) ! n ! ( g - 1 ) ! \displaystyle w(n,g)=\left\langle\begin{matrix}g\\ n\end{matrix}\right\rangle={g+n-1\choose g-1}={g+n-1\choose n}=\frac{(g+n-1)!}% {n!(g-1)!}
  109. w ( n , g ) \displaystyle w(n,g)
  110. k = 0 n ( k + a ) ! k ! a ! = ( n + a + 1 ) ! n ! ( a + 1 ) ! . \sum_{k=0}^{n}\frac{(k+a)!}{k!a!}=\frac{(n+a+1)!}{n!(a+1)!}.
  111. w ( n , g ) = k = 0 n w ( n - k , g - 1 ) = w ( n , g - 1 ) + w ( n - 1 , g - 1 ) + + w ( 1 , g - 1 ) + w ( 0 , g - 1 ) \displaystyle w(n,g)=\sum_{k=0}^{n}w(n-k,g-1)=w(n,g-1)+w(n-1,g-1)+\cdots+w(1,g% -1)+w(0,g-1)
  112. n = 4 \displaystyle n=4
  113. g = 3 \displaystyle g=3
  114. w ( 4 , 3 ) = w ( 4 , 2 ) + w ( 3 , 2 ) + w ( 2 , 2 ) + w ( 1 , 2 ) + w ( 0 , 2 ) , \displaystyle w(4,3)=w(4,2)+w(3,2)+w(2,2)+w(1,2)+w(0,2),
  115. S ( 4 , 3 ) \displaystyle S(4,3)
  116. S ( 4 , 3 ) = { ( 1111 ) , ( 1112 ) , ( 1122 ) , ( 1222 ) , ( 2222 ) ( α ) , ( 111 \color R e d 3 = ) , ( 112 \color R e d 3 = ) , ( 122 \color R e d 3 = ) , ( 222 \color R e d 3 = ) ( β ) , S(4,3)=\left\{\underbrace{(1111),(1112),(1122),(1222),(2222)}_{(\alpha)},% \underbrace{(111{\color{Red}{\underset{=}{3}}}),(112{\color{Red}{\underset{=}{% 3}}}),(122{\color{Red}{\underset{=}{3}}}),(222{\color{Red}{\underset{=}{3}}})}% _{(\beta)},\right.
  117. ( 11 \color R e d 33 = = ) , ( 12 \color R e d 33 = = ) , ( 22 \color R e d 33 = = ) ( γ ) , ( 1 \color R e d 333 = = = ) , ( 2 \color R e d 333 = = = ) ( δ ) ( \color R e d 3333 = = = = ) ( ω ) } . \left.\underbrace{(11{\color{Red}{\underset{==}{33}}}),(12{\color{Red}{% \underset{==}{33}}}),(22{\color{Red}{\underset{==}{33}}})}_{(\gamma)},% \underbrace{(1{\color{Red}{\underset{===}{333}}}),(2{\color{Red}{\underset{===% }{333}}})}_{(\delta)}\underbrace{({\color{Red}{\underset{====}{3333}}})}_{(% \omega)}\right\}.
  118. ( α ) \displaystyle(\alpha)
  119. S ( 4 , 3 ) \displaystyle S(4,3)
  120. S ( 4 , 2 ) = { ( 1111 ) , ( 1112 ) , ( 1122 ) , ( 1222 ) , ( 2222 ) } \displaystyle S(4,2)=\left\{(1111),(1112),(1122),(1222),(2222)\right\}
  121. m 4 = 3 \displaystyle m_{4}=3
  122. ( β ) \displaystyle(\beta)
  123. S ( 4 , 3 ) \displaystyle S(4,3)
  124. S ( 3 , 2 ) = { ( 111 ) , ( 112 ) , ( 122 ) , ( 222 ) } \displaystyle S(3,2)=\left\{(111),(112),(122),(222)\right\}
  125. ( β ) \displaystyle(\beta)
  126. S ( 4 , 3 ) \displaystyle S(4,3)
  127. S ( 3 , 2 ) \displaystyle S(3,2)
  128. ( β ) S ( 3 , 2 ) \displaystyle(\beta)\longleftrightarrow S(3,2)
  129. ( γ ) S ( 2 , 2 ) = { ( 11 ) , ( 12 ) , ( 22 ) } \displaystyle(\gamma)\longleftrightarrow S(2,2)=\left\{(11),(12),(22)\right\}
  130. ( δ ) S ( 1 , 2 ) = { ( 1 ) , ( 2 ) } \displaystyle(\delta)\longleftrightarrow S(1,2)=\left\{(1),(2)\right\}
  131. ( ω ) S ( 0 , 2 ) = \displaystyle(\omega)\longleftrightarrow S(0,2)=\varnothing
  132. S ( 4 , 3 ) = k = 0 4 S ( 4 - k , 2 ) \displaystyle S(4,3)=\bigcup_{k=0}^{4}S(4-k,2)
  133. S ( n , g ) = k = 0 n S ( n - k , g - 1 ) \displaystyle S(n,g)=\bigcup_{k=0}^{n}S(n-k,g-1)
  134. S ( i , g - 1 ) , for i = 0 , , n \displaystyle S(i,g-1)\ ,\ {\rm for}\ i=0,\dots,n
  135. w ( n , g ) = k = 0 n w ( n - k , g - 1 ) \displaystyle w(n,g)=\sum_{k=0}^{n}w(n-k,g-1)
  136. w ( 0 , g ) = 1 , g , and w ( n , 0 ) = 1 , n \displaystyle w(0,g)=1\ ,\forall g\ ,{\rm and}\ w(n,0)=1\ ,\forall n
  137. w ( n , g ) = k 1 = 0 n k 2 = 0 n - k 1 w ( n - k 1 - k 2 , g - 2 ) = k 1 = 0 n k 2 = 0 n - k 1 k g = 0 n - j = 1 g - 1 k j w ( n - i = 1 g k i , 0 ) . \displaystyle w(n,g)=\sum_{k_{1}=0}^{n}\sum_{k_{2}=0}^{n-k_{1}}w(n-k_{1}-k_{2}% ,g-2)=\sum_{k_{1}=0}^{n}\sum_{k_{2}=0}^{n-k_{1}}\cdots\sum_{k_{g}=0}^{n-\sum_{% j=1}^{g-1}k_{j}}w(n-\sum_{i=1}^{g}k_{i},0).
  138. w ( n , g ) = k 1 = 0 n k 2 = 0 n - k 1 k g = 0 n - j = 1 g - 1 k j 1 , \displaystyle w(n,g)=\sum_{k_{1}=0}^{n}\sum_{k_{2}=0}^{n-k_{1}}\cdots\sum_{k_{% g}=0}^{n-\sum_{j=1}^{g-1}k_{j}}1,
  139. q \displaystyle q
  140. p \displaystyle p
  141. k = 0 q p = q p \displaystyle\sum_{k=0}^{q}p=qp
  142. w ( 4 , 3 ) \displaystyle w(4,3)
  143. w ( 3 , 3 ) \displaystyle w(3,3)
  144. w ( 3 , 2 ) \displaystyle w(3,2)