wpmath0000008_0

10-Deacetylbaccatin.html

  1. \rightleftharpoons

12_Ophiuchi.html

  1. log ( g ) = 4.6 \log(g)=4.6

15_and_290_theorems.html

  1. Q i j Q_{ij}
  2. x x
  3. Q ( x ) = i , j Q i j x i x j Q(x)=\sum_{i,j}Q_{ij}x_{i}x_{j}
  4. Q Q
  5. Q ( x ) > 0 Q(x)>0
  6. x 0 x\neq 0
  7. Q ( x ) Q(x)
  8. Q Q
  9. Q i j Q_{ij}
  10. Q Q
  11. Q Q
  12. Q i j Q_{ij}
  13. w 2 + x 2 + y 2 + z 2 w^{2}+x^{2}+y^{2}+z^{2}
  14. w 2 + 2 x 2 + 5 y 2 + 5 z 2 , w^{2}+2x^{2}+5y^{2}+5z^{2},
  15. \mathbb{Z}

1951_USAF_resolution_test_chart.html

  1. Resolution (lp/mm) = 2 Group + ( element - 1 ) / 6 \,\text{Resolution (lp/mm)}=2^{\,\text{Group}+(\,\text{element}-1)/6}

215_(number).html

  1. 215 = ( 3 ! ) 3 - 1 215=(3!)^{3}-1
  2. n 2 - 17 n^{2}-17
  3. 215 2 - 17 = 2 × 152 2 215^{2}-17=2\times 152^{2}\,

217_(number).html

  1. 217 = 6 3 + 1 3 = 9 3 - 8 3 217=6^{3}+1^{3}=9^{3}-8^{3}

24_Game.html

  1. ( 4 + 13 - 1 4 ) = 1820 {\textstyle\left({{4+13-1}\atop{4}}\right)}=1820

252_(number).html

  1. ( 10 5 ) {\textstyle\left({{10}\atop{5}}\right)}
  2. τ ( 3 ) \tau(3)
  3. τ \tau
  4. σ 3 ( 6 ) \sigma_{3}(6)
  5. σ 3 \sigma_{3}
  6. 1 3 + 2 3 + 3 3 + 6 3 = ( 1 3 + 2 3 ) ( 1 3 + 3 3 ) = 252. 1^{3}+2^{3}+3^{3}+6^{3}=(1^{3}+2^{3})(1^{3}+3^{3})=252.

277_(number).html

  1. sec x = 1 + 1 2 x 2 + 5 24 x 4 + 61 720 x 6 + 277 8064 x 8 + \sec x=1+\frac{1}{2}x^{2}+\frac{5}{24}x^{4}+\frac{61}{720}x^{6}+\frac{277}{806% 4}x^{8}+\cdots

3-beta-HSD.html

  1. \rightleftharpoons

310_helix.html

  1. 3 cos Ω = 1 - 4 cos 2 ( φ + ψ 2 ) . 3\cos\Omega=1-4\cos^{2}\left(\frac{\varphi+\psi}{2}\right).

353_(number).html

  1. 353 4 = 30 4 + 120 4 + 272 4 + 315 4 . 353^{4}=30^{4}+120^{4}+272^{4}+315^{4}.

3671_Dionysus.html

  1. tan ( θ 2 ) = radius of moon distance from surface of asteroid to center of moon \scriptstyle{\mathrm{tan}\left(\frac{\theta}{2}\right)=\frac{\mathrm{radius~{}% of~{}moon}}{\mathrm{distance~{}from~{}surface~{}of~{}asteroid~{}to~{}center~{}% of~{}moon}}}

5'-nucleotidase.html

  1. \rightleftharpoons

5_(number).html

  1. 3 n - 1 3n-1
  2. 5 × x 5\times x
  3. 5 ÷ x 5\div x
  4. 1. 6 ¯ 1.\overline{6}
  5. 0.8 3 ¯ 0.8\overline{3}
  6. 0. 714285 ¯ 0.\overline{714285}
  7. 0. 5 ¯ 0.\overline{5}
  8. 0. 45 ¯ 0.\overline{45}
  9. 0.41 6 ¯ 0.41\overline{6}
  10. 0. 384615 ¯ 0.\overline{384615}
  11. 0. 357142 ¯ 0.\overline{357142}
  12. 0. 3 ¯ 0.\overline{3}
  13. x ÷ 5 x\div 5
  14. 5 x 5^{x}\,
  15. x 5 x^{5}\,

65537_(number).html

  1. 2 2 n + 1 2^{2^{n}}+1
  2. n = 4 n=4
  3. 2 2 0 + 1 = 2 1 + 1 = 3 , 2^{2^{0}}+1=2^{1}+1=3,
  4. 2 2 1 + 1 = 2 2 + 1 = 5 , 2^{2^{1}}+1=2^{2}+1=5,
  5. 2 2 2 + 1 = 2 4 + 1 = 17 , 2^{2^{2}}+1=2^{4}+1=17,
  6. 2 2 3 + 1 = 2 8 + 1 = 257 , 2^{2^{3}}+1=2^{8}+1=257,
  7. 2 2 4 + 1 = 2 16 + 1 = 65537. 2^{2^{4}}+1=2^{16}+1=65537.
  8. 2 2 5 + 1 = 2 32 + 1 = 4294967297 = 641 × 6700417 2^{2^{5}}+1=2^{32}+1=4294967297=641\times 6700417
  9. 2 2 6 + 1 = 2 64 + 1 = 274177 × 67280421310721 2^{2^{6}}+1=2^{64}+1=274177\times 67280421310721
  10. 10 n + 27 10^{n}+27
  11. 4 {}_{4}

A_(disambiguation).html

  1. × 10 10 \times 10^{−}10
  2. 𝔸 \mathbb{A}
  3. \forall

Abel's_test.html

  1. a n \sum a_{n}
  2. a n b n \sum a_{n}b_{n}
  3. a n \sum a_{n}
  4. lim n a n = 0 \lim_{n\rightarrow\infty}a_{n}=0\,
  5. f ( z ) = n = 0 a n z n f(z)=\sum_{n=0}^{\infty}a_{n}z^{n}\,
  6. z = e i θ z 1 2 - z - 1 2 = 2 i sin θ 2 0 z=e^{i\theta}\quad\Rightarrow\quad z^{\frac{1}{2}}-z^{-\frac{1}{2}}=2i\sin{% \textstyle\frac{\theta}{2}}\neq 0
  7. 2 i sin θ 2 ( S p - S q ) = n = q + 1 p a n ( z n + 1 2 - z n - 1 2 ) = [ n = q + 2 p ( a n - 1 - a n ) z n - 1 2 ] - a q + 1 z q + 1 2 + a p z p + 1 2 \begin{aligned}\displaystyle 2i\sin{\textstyle\frac{\theta}{2}}\left(S_{p}-S_{% q}\right)&\displaystyle=\sum_{n=q+1}^{p}a_{n}\left(z^{n+\frac{1}{2}}-z^{n-% \frac{1}{2}}\right)\\ &\displaystyle=\left[\sum_{n=q+2}^{p}\left(a_{n-1}-a_{n}\right)z^{n-\frac{1}{2% }}\right]-a_{q+1}z^{q+\frac{1}{2}}+a_{p}z^{p+\frac{1}{2}}\end{aligned}
  8. S p = n = 0 p a n z n . S_{p}=\sum_{n=0}^{p}a_{n}z^{n}.\,
  9. | 2 i sin θ 2 ( S p - S q ) | = | n = q + 1 p a n ( z n + 1 2 - z n - 1 2 ) | [ n = q + 2 p | ( a n - 1 - a n ) z n - 1 2 | ] + | a q + 1 z q + 1 2 | + | a p z p + 1 2 | = [ n = q + 2 p ( a n - 1 - a n ) ] + a q + 1 + a p = a q + 1 - a p + a q + 1 + a p = 2 a q + 1 . \begin{aligned}\displaystyle\left|2i\sin{\textstyle\frac{\theta}{2}}\left(S_{p% }-S_{q}\right)\right|&\displaystyle=\left|\sum_{n=q+1}^{p}a_{n}\left(z^{n+% \frac{1}{2}}-z^{n-\frac{1}{2}}\right)\right|\\ &\displaystyle\leq\left[\sum_{n=q+2}^{p}\left|\left(a_{n-1}-a_{n}\right)z^{n-% \frac{1}{2}}\right|\right]+\left|a_{q+1}z^{q+\frac{1}{2}}\right|+\left|a_{p}z^% {p+\frac{1}{2}}\right|\\ &\displaystyle=\left[\sum_{n=q+2}^{p}\left(a_{n-1}-a_{n}\right)\right]+a_{q+1}% +a_{p}\\ &\displaystyle=a_{q+1}-a_{p}+a_{q+1}+a_{p}=2a_{q+1}.\end{aligned}

Abel_equation.html

  1. f ( h ( x ) ) = h ( x + 1 ) f(h(x))=h(x+1)
  2. α ( f ( x ) ) = α ( x ) + 1 \alpha(f(x))=\alpha(x)+1
  3. f f
  4. α α
  5. α - 1 ( α ( f ( x ) ) ) = α - 1 ( α ( x ) + 1 ) . \alpha^{-1}(\alpha(f(x)))=\alpha^{-1}(\alpha(x)+1)\,.
  6. f ( α - 1 ( y ) ) = α - 1 ( y + 1 ) . f(\alpha^{-1}(y))=\alpha^{-1}(y+1)\,.
  7. f ( x ) f(x)
  8. s s
  9. Ψ ( f ( x ) ) = s Ψ ( x ) Ψ(f(x))=sΨ(x)
  10. ω ( ω ( x , u ) , v ) = ω ( x , u + v ) , \omega(\omega(x,u),v)=\omega(x,u+v)~{},
  11. ω ( x , 1 ) = f ( x ) \omega(x,1)=f(x)
  12. ω ( x , u ) = α - 1 ( α ( x ) + u ) \omega(x,u)=\alpha^{-1}(\alpha(x)+u)
  13. ω ( x , 0 ) = x ω(x,0)=x
  14. f = e x p f=exp
  15. α ( f ( f ( x ) ) ) = α ( x ) + 2 , \alpha(f(f(x)))=\alpha(x)+2~{},
  16. α ( f n ( x ) ) = α ( x ) + n . \alpha(f_{n}(x))=\alpha(x)+n~{}.

Aberth_method.html

  1. p ( x ) = p n x n + p n - 1 x n - 1 + + p 1 x + p 0 p(x)=p_{n}x^{n}+p_{n-1}x^{n-1}+\cdots+p_{1}x+p_{0}
  2. z 1 * , z 2 * , , z n * z^{*}_{1},\,z^{*}_{2},\dots,z^{*}_{n}
  3. p ( x ) = p n ( x - z 1 * ) ( x - z 2 * ) ( x - z n * ) . p(x)=p_{n}\cdot(x-z^{*}_{1})\cdot(x-z^{*}_{2})\cdots(x-z^{*}_{n}).
  4. z 1 , , z n z_{1},\dots,z_{n}\in\mathbb{C}
  5. w 1 , , w n w_{1},\dots,w_{n}\in\mathbb{C}
  6. w k = - p ( z k ) p ( z k ) 1 - p ( z k ) p ( z k ) j k 1 z k - z j , w_{k}=-\frac{\frac{p(z_{k})}{p^{\prime}(z_{k})}}{1-\frac{p(z_{k})}{p^{\prime}(% z_{k})}\cdot\sum_{j\neq k}\frac{1}{z_{k}-z_{j}}},
  7. z 1 + w 1 , , z n + w n z_{1}+w_{1},\dots,z_{n}+w_{n}
  8. F ( x ) = p ( x ) j = 0 ; j k n ( x - z j ) F(x)=\frac{p(x)}{\prod_{j=0;\,j\neq k}^{n}(x-z_{j})}
  9. z 1 , , z n z_{1},\dots,z_{n}
  10. z k z_{k}
  11. z 1 , , z k - 1 , z k + 1 , , z n z_{1},\dots,z_{k-1},z_{k+1},\dots,z_{n}
  12. z k z_{k}
  13. F ( x ) F ( x ) \tfrac{F(x)}{F^{\prime}(x)}
  14. F ( x ) F ( x ) = d d x ln | F ( x ) | = d d x ( ln | p ( x ) | - j = 0 ; j k n ln | x - z j | ) = p ( x ) p ( x ) - j = 0 ; j k n 1 x - z j \begin{aligned}\displaystyle\frac{F^{\prime}(x)}{F(x)}&\displaystyle=\frac{d}{% dx}\ln|F(x)|\\ &\displaystyle=\frac{d}{dx}\big(\ln|p(x)|-\sum_{j=0;\,j\neq k}^{n}\ln|x-z_{j}|% \big)\\ &\displaystyle=\frac{p^{\prime}(x)}{p(x)}-\sum_{j=0;\,j\neq k}^{n}\frac{1}{x-z% _{j}}\end{aligned}
  15. z k = z k - F ( z k ) F ( z k ) = z k - 1 p ( z k ) p ( z k ) - j = 0 ; j k n 1 z k - z j , z_{k}^{\prime}=z_{k}-\frac{F(z_{k})}{F^{\prime}(z_{k})}=z_{k}-\frac{1}{\frac{p% ^{\prime}(z_{k})}{p(z_{k})}-\sum_{j=0;\,j\neq k}^{n}\frac{1}{z_{k}-z_{j}}}\,,

Absolute_presentation_of_a_group.html

  1. G G
  2. S S
  3. R R
  4. G S R . G\simeq\langle S\mid R\rangle.
  5. G G
  6. S S
  7. r = 1 r=1
  8. r R r\in R
  9. G G
  10. G G
  11. S S
  12. 1. 1.
  13. I I
  14. i 1 i\neq 1
  15. i I i\in I
  16. G G
  17. S S
  18. R R
  19. I I
  20. G G
  21. S R , I . \langle S\mid R,I\rangle.
  22. G G
  23. S R . \langle S\mid R\rangle.
  24. h : G H h:G\rightarrow H
  25. I I
  26. h ( G ) h(G)
  27. G G
  28. h ( G ) h(G)
  29. N G N\triangleleft G
  30. G G
  31. I N { 1 } . I\cap N\neq\left\{1\right\}.
  32. a a 8 = 1 . \langle a\mid a^{8}=1\rangle.
  33. a 8 = 1 a^{8}=1\,
  34. a a 4 = 1 \langle a\mid a^{4}=1\rangle
  35. a a 2 = 1 \langle a\mid a^{2}=1\rangle
  36. a a = 1 . \langle a\mid a=1\rangle.
  37. a 4 1 a^{4}\neq 1
  38. a a 8 = 1 , a 4 1 . \langle a\mid a^{8}=1,a^{4}\neq 1\rangle.
  39. a a 8 = 1 , a 2 1 \langle a\mid a^{8}=1,a^{2}\neq 1\rangle
  40. a 2 1 a^{2}\neq 1
  41. G G\,
  42. H H\,
  43. G G\,
  44. G = x 1 , x 2 R G=\langle x_{1},x_{2}\mid R\rangle
  45. G * G^{*}\,
  46. H * H^{*}\,
  47. G G\,
  48. H * H^{*}\,
  49. h : G H * h:G\rightarrow H^{*}
  50. G * G^{*}\,

Absolutely_irreducible.html

  1. x 2 + y 2 - 1 x^{2}+y^{2}-1
  2. x 2 + y 2 x^{2}+y^{2}
  3. x 2 + y 2 = ( x + i y ) ( x - i y ) , x^{2}+y^{2}=(x+iy)(x-iy),
  4. x 2 + y 2 = 1 x^{2}+y^{2}=1
  5. x 2 + y 2 = 0 x^{2}+y^{2}=0
  6. x 2 + y 2 = ( x + y i ) ( x - y i ) , x^{2}+y^{2}=(x+yi)(x-yi),
  7. i i

Absorbing_set_(random_dynamical_systems).html

  1. t τ B ( ω ) φ ( t , ϑ - t ω ) B K ( ω ) . t\geq\tau_{B}(\omega)\implies\varphi(t,\vartheta_{-t}\omega)B\subseteq K(% \omega).

Absorption_(pharmacokinetics).html

  1. d W d t = D A ( C s - C ) L \frac{dW}{dt}=\frac{DA(C_{s}-C)}{L}
  2. d W d t \frac{dW}{dt}
  3. C s C_{s}

Abstract_rewriting_machine.html

  1. f ( x , y , z ) g ( x , h ( y ) , z ) f(\vec{x},\vec{y},\vec{z})\rightarrow g(\vec{x},h(\vec{y}),\vec{z})
  2. f ( x ) x f(x)\rightarrow x
  3. f ( x , g ( y ) , z ) h ( x , y , z ) f(\vec{x},g(\vec{y}),\vec{z})\rightarrow h(\vec{x},\vec{y},\vec{z})
  4. f ( x , z ) g ( x , y , z ) - for y x z f(\vec{x},\vec{z})\rightarrow g(\vec{x},y,\vec{z})-{\rm for}~{}y\in\vec{x}\cup% \vec{z}
  5. f ( x , y , z ) g ( x , z ) f(\vec{x},\vec{y},\vec{z})\rightarrow g(\vec{x},\vec{z})
  6. f ( x ) g ( x ) f(\vec{x})\rightarrow g(\vec{x})

Abstract_Wiener_space.html

  1. ( i 1 × i 2 ) * ( γ H 1 × H 2 ) = ( i 1 ) * ( γ H 1 ) ( i 2 ) * ( γ H 2 ) , (i_{1}\times i_{2})_{*}(\gamma^{H_{1}\times H_{2}})=(i_{1})_{*}\left(\gamma^{H% _{1}}\right)\otimes(i_{2})_{*}\left(\gamma^{H_{2}}\right),
  2. H := L 0 2 , 1 ( [ 0 , T ] ; n ) := { paths starting at 0 with first derivative L 2 } H:=L_{0}^{2,1}([0,T];\mathbb{R}^{n}):=\{\,\text{paths starting at 0 with first% derivative}\in L^{2}\}
  3. σ 1 , σ 2 L 0 2 , 1 := 0 T σ ˙ 1 ( t ) , σ ˙ 2 ( t ) n d t , \langle\sigma_{1},\sigma_{2}\rangle_{L_{0}^{2,1}}:=\int_{0}^{T}\langle\dot{% \sigma}_{1}(t),\dot{\sigma}_{2}(t)\rangle_{\mathbb{R}^{n}}\,\mathrm{d}t,
  4. σ C 0 := sup t [ 0 , T ] σ ( t ) n , \|\sigma\|_{C_{0}}:=\sup_{t\in[0,T]}\|\sigma(t)\|_{\mathbb{R}^{n}},

Abū_al-Ḥasan_ibn_ʿAlī_al-Qalaṣādī.html

  1. 2 x 3 + 3 x 2 - 4 x + 5 = 0 2x^{3}+3x^{2}-4x+5=0

ABX_test.html

  1. N / 2 + N N/2+\sqrt{N}

AC0.html

  1. × \times

AC_(complexity).html

  1. O ( log i n ) O(\log^{i}n)
  2. AC = i 0 AC i \mbox{AC}~{}=\bigcup_{i\geq 0}\mbox{AC}~{}^{i}
  3. NC i AC i NC . i + 1 \mbox{NC}~{}^{i}\subseteq\mbox{AC}~{}^{i}\subseteq\mbox{NC}~{}^{i+1}.

AC_motor.html

  1. N s = 120 F / p N_{s}=120F/p
  2. S = ( N s - N r ) / N s S=(N_{s}-N_{r})/N_{s}

Achilles_number.html

  1. 784 = 2 4 7 2 = ( 2 2 ) 2 7 2 = ( 2 2 7 ) 2 = 28 2 . 784=2^{4}\cdot 7^{2}=(2^{2})^{2}\cdot 7^{2}=(2^{2}\cdot 7)^{2}=28^{2}.\,

Acoustic_contrast_factor.html

  1. β \beta
  2. β \beta
  3. ρ \rho
  4. ρ \rho
  5. ϕ \phi
  6. ϕ = 5 ρ p - 2 ρ 2 ρ p + ρ - β p β \phi={\frac{5\rho_{p}-2\rho}{2\rho_{p}+\rho}}-{\frac{\beta_{p}}{\beta}}
  7. ϕ \phi

Acoustic_guitar.html

  1. f f

Acoustic_wave.html

  1. 2 p x 2 - 1 c 2 2 p t 2 = 0 {\partial^{2}p\over\partial x^{2}}-{1\over c^{2}}{\partial^{2}p\over\partial t% ^{2}}=0
  2. p p
  3. x x
  4. c c
  5. t t
  6. 2 u x 2 - 1 c 2 2 u t 2 = 0 {\partial^{2}u\over\partial x^{2}}-{1\over c^{2}}{\partial^{2}u\over\partial t% ^{2}}=0
  7. u u
  8. p = R cos ( ω t - k x ) + ( 1 - R ) cos ( ω t + k x ) p=R\cos(\omega t-kx)+(1-R)\cos(\omega t+kx)
  9. ω \omega
  10. t t
  11. k k
  12. R R
  13. R = 1 R=1
  14. R = 0 R=0
  15. R = 0.5 R=0.5
  16. p V = n R T \ pV=nRT
  17. p p
  18. V V
  19. n n
  20. R R
  21. 8.314 472 ( 15 ) J mol K 8.314\,472(15)~{}\frac{\mathrm{J}}{\mathrm{mol~{}K}}
  22. V V
  23. V V
  24. p p
  25. V V m = - 1 γ p p m {\partial V\over V_{m}}={-1\over\ \gamma}{\partial p\over p_{m}}
  26. γ \gamma
  27. x x
  28. x x m A = V V m = - 1 γ p p m {\partial x\over x_{m}}A={\partial V\over V_{m}}={-1\over\ \gamma}{\partial p% \over p_{m}}
  29. A A
  30. p = p 0 c o s ( ω t - k x ) p=p_{0}cos(\omega t-kx)
  31. x = x 0 s i n ( ω t - k x ) x=x_{0}sin(\omega t-kx)
  32. u = u 0 c o s ( ω t - k x ) u=u_{0}cos(\omega t-kx)
  33. T T m = γ - 1 γ p p m {\partial T\over T_{m}}={\gamma-1\over\ \gamma}{\partial p\over p_{m}}
  34. c = C ρ c=\sqrt{\frac{C}{\rho}}\,
  35. ρ \rho
  36. c c
  37. c 2 = p ρ c^{2}=\frac{\partial p}{\partial\rho}
  38. p p
  39. ρ \rho
  40. f = N c 2 d N { 1 , 2 , 3 , } f=\frac{Nc}{2d}\qquad\qquad N\in\{1,2,3,\dots\}
  41. c c
  42. d d
  43. R = I reflected I incident R=\frac{I_{\mathrm{reflected}}}{I_{\mathrm{incident}}}
  44. α = 1 - R 2 \alpha=1-R^{2}
  45. α \alpha
  46. R R
  47. L p = 10 log 10 ( p < m t p l > r m s 2 p ref 2 ) L_{p}=10\log_{10}\left(\frac{{p_{\mathrm{<}mtpl>{{rms}}}}^{2}}{{p_{\mathrm{ref% }}}^{2}}\right)
  48. p rms p_{\mathrm{rms}}
  49. p ref p_{\mathrm{ref}}
  50. L u = 10 log 10 ( u < m t p l > r m s 2 u ref 2 ) L_{u}=10\log_{10}\left(\frac{{u_{\mathrm{<}mtpl>{{rms}}}}^{2}}{{u_{\mathrm{ref% }}}^{2}}\right)
  51. u rms u_{\mathrm{rms}}
  52. u ref u_{\mathrm{ref}}
  53. L I = 10 log 10 I rms I ref L_{I}=10\log_{10}\frac{I_{\mathrm{rms}}}{I_{\mathrm{ref}}}
  54. I rms I_{\mathrm{rms}}
  55. I ref I_{\mathrm{ref}}

Action-angle_coordinates.html

  1. W ( 𝐪 ) W(\mathbf{q})
  2. S S
  3. K ( 𝐰 , 𝐉 ) K(\mathbf{w},\mathbf{J})
  4. H ( 𝐪 , 𝐩 ) H(\mathbf{q},\mathbf{p})
  5. 𝐰 \mathbf{w}
  6. 𝐉 \mathbf{J}
  7. W W
  8. 𝐰 \mathbf{w}
  9. J k p k d q k J_{k}\equiv\oint p_{k}\,dq_{k}
  10. E = E ( q k , p k ) E=E(q_{k},p_{k})
  11. J k J_{k}
  12. K K
  13. w k w_{k}
  14. d d t J k = 0 = K w k \frac{d}{dt}J_{k}=0=\frac{\partial K}{\partial w_{k}}
  15. w k w_{k}
  16. w k W J k w_{k}\equiv\frac{\partial W}{\partial J_{k}}
  17. K = K ( 𝐉 ) K=K(\mathbf{J})
  18. 𝐉 \mathbf{J}
  19. d d t w k = K J k ν k ( 𝐉 ) \frac{d}{dt}w_{k}=\frac{\partial K}{\partial J_{k}}\equiv\nu_{k}(\mathbf{J})
  20. J J
  21. w k = ν k ( 𝐉 ) t + β k w_{k}=\nu_{k}(\mathbf{J})t+\beta_{k}
  22. β k \beta_{k}
  23. T T
  24. w k w_{k}
  25. Δ w k = ν k ( 𝐉 ) T \Delta w_{k}=\nu_{k}(\mathbf{J})T
  26. ν k ( 𝐉 ) \nu_{k}(\mathbf{J})
  27. q k q_{k}
  28. w k w_{k}
  29. q k q_{k}
  30. Δ w k w k q k d q k = 2 W J k q k d q k = d d J k W q k d q k = d d J k p k d q k = d J k d J k = 1 \Delta w_{k}\equiv\oint\frac{\partial w_{k}}{\partial q_{k}}\,dq_{k}=\oint% \frac{\partial^{2}W}{\partial J_{k}\,\partial q_{k}}\,dq_{k}=\frac{d}{dJ_{k}}% \oint\frac{\partial W}{\partial q_{k}}\,dq_{k}=\frac{d}{dJ_{k}}\oint p_{k}\,dq% _{k}=\frac{dJ_{k}}{dJ_{k}}=1
  31. Δ w k \Delta w_{k}
  32. ν k ( 𝐉 ) = 1 T \nu_{k}(\mathbf{J})=\frac{1}{T}
  33. 𝐰 \mathbf{w}
  34. q k q_{k}
  35. q k = s 1 = - s 2 = - s N = - A s 1 , s 2 , , s N k e i 2 π s 1 w 1 e i 2 π s 2 w 2 e i 2 π s N w N q_{k}=\sum_{s_{1}=-\infty}^{\infty}\sum_{s_{2}=-\infty}^{\infty}\cdots\sum_{s_% {N}=-\infty}^{\infty}A^{k}_{s_{1},s_{2},\ldots,s_{N}}e^{i2\pi s_{1}w_{1}}e^{i2% \pi s_{2}w_{2}}\cdots e^{i2\pi s_{N}w_{N}}
  36. A s 1 , s 2 , , s N k A^{k}_{s_{1},s_{2},\ldots,s_{N}}
  37. q k q_{k}
  38. w k w_{k}
  39. q k = s k = - e i 2 π s k w k q_{k}=\sum_{s_{k}=-\infty}^{\infty}e^{i2\pi s_{k}w_{k}}
  40. J k J_{k}
  41. ν k \nu_{k}
  42. ν k = ν l \nu_{k}=\nu_{l}
  43. k l k\neq l

Active_load.html

  1. I C = V C C - V o u t R C I_{C}=\frac{V_{CC}-V_{out}}{R_{C}}

Active_pixel_sensor.html

  1. V n 2 = k T / 2 C V_{n}^{2}=kT/2C
  2. N e = k T C / 2 q N_{e}=\frac{\sqrt{kTC/2}}{q}
  3. V n 2 = k T / C V_{n}^{2}=kT/C
  4. N e = k T C q N_{e}=\frac{\sqrt{kTC}}{q}

Acyclic_object.html

  1. 𝒞 \mathcal{C}
  2. F : 𝒞 𝒟 F:\mathcal{C}\to\mathcal{D}
  3. F F
  4. F F
  5. A A
  6. 𝒞 \mathcal{C}
  7. R i F ( A ) = 0 {\rm R}^{i}F(A)=0\,\!
  8. i > 0 i>0\,\!
  9. R i F {\rm R}^{i}F
  10. F F

ADALINE.html

  1. θ \theta
  2. y = j = 1 n x j w j + θ y=\sum_{j=1}^{n}x_{j}w_{j}+\theta
  3. x n + 1 = 1 x_{n+1}=1
  4. w n + 1 = θ w_{n+1}=\theta
  5. y = x w y=x\cdot w
  6. η \eta
  7. y y
  8. o o
  9. w w + η ( o - y ) x w\leftarrow w+\eta(o-y)x
  10. E = ( o - y ) 2 E=(o-y)^{2}

Adams_chromatic_valence_color_space.html

  1. V Y , V_{Y},
  2. V Y 2 = 1.4742 Y - 0.004743 Y 2 V_{Y}^{2}=1.4742Y-0.004743Y^{2}
  3. V X - V Y V_{X}-V_{Y}
  4. V X V_{X}
  5. ( y n / x n ) X (y_{n}/x_{n})X
  6. V Z - V Y V_{Z}-V_{Y}
  7. V Z V_{Z}
  8. ( y n / z n ) Z (y_{n}/z_{n})Z
  9. V X - V Y V_{X}-V_{Y}
  10. 0.4 ( V Z - V Y ) 0.4(V_{Z}-V_{Y})
  11. ( y n / x n ) X = Y = ( y n / z n ) Z (y_{n}/x_{n})X=Y=(y_{n}/z_{n})Z
  12. V X - V Y = 0 V_{X}-V_{Y}=0
  13. V Z - V Y = 0 V_{Z}-V_{Y}=0
  14. y n x n X - Y \frac{y_{n}}{x_{n}}X-Y
  15. y n z n Z - Y \frac{y_{n}}{z_{n}}Z-Y
  16. x n , y n , z n x_{n},y_{n},z_{n}
  17. y n x n X Y = x / x n y / y n \frac{y_{n}}{x_{n}}\frac{X}{Y}=\frac{x/x_{n}}{y/y_{n}}
  18. y n z n Z Y = z / z n y / y n \frac{y_{n}}{z_{n}}\frac{Z}{Y}=\frac{z/z_{n}}{y/y_{n}}
  19. Y = 1.2219 V J - 0.23111 V J 2 + 0.23951 V J 3 - 0.021009 V J 4 + 0.0008404 V J 5 Y=1.2219V_{J}-0.23111V_{J}^{2}+0.23951V_{J}^{3}-0.021009V_{J}^{4}+0.0008404V_{% J}^{5}
  20. X / x n Y / y n - 1 = X / x n - Y / y n Y / y n \frac{X/x_{n}}{Y/y_{n}}-1=\frac{X/x_{n}-Y/y_{n}}{Y/y_{n}}
  21. Z / z n Y / y n - 1 = Z / z n - Y / y n Y / y n \frac{Z/z_{n}}{Y/y_{n}}-1=\frac{Z/z_{n}-Y/y_{n}}{Y/y_{n}}
  22. W X = ( x / x n y / y n - 1 ) V J W_{X}=\left(\frac{x/x_{n}}{y/y_{n}}-1\right)V_{J}
  23. W Z = ( z / z n y / y n - 1 ) V J W_{Z}=\left(\frac{z/z_{n}}{y/y_{n}}-1\right)V_{J}
  24. Δ E = ( 0.5 Δ V J ) 2 + ( Δ W X ) 2 + ( 0.4 Δ W Z ) 2 \Delta E=\sqrt{(0.5\Delta V_{J})^{2}+(\Delta W_{X})^{2}+(0.4\Delta W_{Z})^{2}}
  25. W X = V X - V Y W_{X}=V_{X}-V_{Y}
  26. W Z = V Z - V Y W_{Z}=V_{Z}-V_{Y}
  27. Δ E = ( 0.23 Δ V Y ) 2 + ( Δ W X ) 2 + ( 0.4 Δ W Z ) 2 \Delta E=\sqrt{(0.23\Delta V_{Y})^{2}+(\Delta W_{X})^{2}+(0.4\Delta W_{Z})^{2}}

Adapted_process.html

  1. ( Ω , , ) (\Omega,\mathcal{F},\mathbb{P})
  2. I I
  3. \leq
  4. I I
  5. \mathbb{N}
  6. 0 \mathbb{N}_{0}
  7. [ 0 , T ] [0,T]
  8. [ 0 , + ) [0,+\infty)
  9. = ( i ) i I \mathcal{F}_{\cdot}=\left(\mathcal{F}_{i}\right)_{i\in I}
  10. \mathcal{F}
  11. ( S , Σ ) (S,\Sigma)
  12. X : I × Ω S X:I\times\Omega\to S
  13. X X
  14. ( i ) i I \left(\mathcal{F}_{i}\right)_{i\in I}
  15. X i : Ω S X_{i}:\Omega\to S
  16. ( i , Σ ) (\mathcal{F}_{i},\Sigma)
  17. i I i\in I

Adaptive_expertise.html

  1. A E = 0.14 F - 0.36 C + 1.27 T AE=0.14F-0.36C+1.27T

Adaptive_grammar.html

  1. L = { w w | w is a letter } L=\{ww|w\mbox{ is a letter}~{}\}
  2. L = { w w | w { a , b } + } L=\{ww|w\in\{a,b\}^{+}\}

Adaptive_quadrature.html

  1. f ( x ) f(x)
  2. Q a b f ( x ) d x Q\approx\int_{a}^{b}f(x)\,\mbox{d}~{}x
  3. ε | Q - a b f ( x ) d x | \varepsilon\approx\left|Q-\int_{a}^{b}f(x)\,\mbox{d}~{}x\right|
  4. ε > τ \varepsilon>\tau
  5. Q Q
  6. f ( x ) f(x)
  7. [ a , b ] [a,b]
  8. ε \varepsilon
  9. τ \tau
  10. Q a b f ( x ) d x , Q\approx\int_{a}^{b}f(x)\,\mbox{d}~{}x,
  11. ε | Q - a b f ( x ) d x | , \varepsilon\approx\left|Q-\int_{a}^{b}f(x)\,\mbox{d}~{}x\right|,
  12. Q n = i = 0 n w i f ( x i ) a b f ( x ) d x Q_{n}\quad=\quad\sum_{i=0}^{n}w_{i}f(x_{i})\quad\approx\quad\int_{a}^{b}f(x)\,% \mbox{d}~{}x
  13. x i x_{i}
  14. w i w_{i}
  15. x i x_{i}
  16. x i = a + i n ( b - a ) x_{i}=a+\frac{i}{n}(b-a)
  17. f ( x ) f(x)
  18. x i = cos ( 2 i n π ) x_{i}=\cos\left(\frac{2i}{n}\pi\right)
  19. x i = cos ( 2 ( i + 0.5 ) n + 1 π ) x_{i}=\cos\left(\frac{2(i+0.5)}{n+1}\pi\right)

Adaptive_Simpson's_method.html

  1. | S ( a , c ) + S ( c , b ) - S ( a , b ) | / 15 < ϵ |S(a,c)+S(c,b)-S(a,b)|/15<\epsilon\,
  2. [ a , b ] [a,b]\,\!
  3. c c\,\!
  4. S ( a , b ) S(a,b)\,\!
  5. S ( a , c ) S(a,c)\,\!
  6. S ( c , b ) S(c,b)\,\!
  7. ϵ \epsilon\,\!
  8. S ( a , c ) + S ( c , b ) S(a,c)+S(c,b)\,
  9. S ( a , b ) S(a,b)\,
  10. [ S ( a , c ) + S ( c , b ) - S ( a , b ) ] / 15 [S(a,c)+S(c,b)-S(a,b)]/15\,

Adherent_point.html

  1. A A
  2. x x

ADHM_construction.html

  1. μ r = [ B 1 , B 1 ] + [ B 2 , B 2 ] + I I - J J , \mu_{r}=[B_{1},B_{1}^{\dagger}]+[B_{2},B_{2}^{\dagger}]+II^{\dagger}-J^{% \dagger}J,
  2. μ c = [ B 1 , B 2 ] + I J . \displaystyle\mu_{c}=[B_{1},B_{2}]+IJ.
  3. μ r = μ c = 0 \mu_{r}=\mu_{c}=0
  4. μ \vec{\mu}
  5. x i j = ( z 2 z 1 - z 1 ¯ z 2 ¯ ) . x_{ij}=\begin{pmatrix}z_{2}&z_{1}\\ -\bar{z_{1}}&\bar{z_{2}}\end{pmatrix}.
  6. Δ = ( I B 2 + z 2 B 1 + z 1 J - B 1 - z 1 ¯ B 2 + z 2 ¯ ) . \Delta=\begin{pmatrix}I&B_{2}+z_{2}&B_{1}+z_{1}\\ J^{\dagger}&-B_{1}^{\dagger}-\bar{z_{1}}&B_{2}^{\dagger}+\bar{z_{2}}\end{% pmatrix}.
  7. μ r = μ c = 0 \displaystyle\mu_{r}=\mu_{c}=0
  8. Δ Δ = ( f - 1 0 0 f - 1 ) \Delta\Delta^{\dagger}=\begin{pmatrix}f^{-1}&0\\ 0&f^{-1}\end{pmatrix}
  9. P = Δ ( f 0 0 f ) Δ . P=\Delta^{\dagger}\begin{pmatrix}f&0\\ 0&f\end{pmatrix}\Delta.
  10. P + U U = 1. P+UU^{\dagger}=1.\,
  11. A m = U m U . A_{m}=U^{\dagger}\partial_{m}U.

Adjoint_filter.html

  1. h * h^{*}
  2. h h
  3. ( h * ) k = h - k ¯ (h^{*})_{k}=\overline{h_{-k}}
  4. 2 \ell_{2}
  5. h * x , y = x , h * * y \langle h*x,y\rangle=\langle x,h^{*}*y\rangle
  6. x x
  7. x * * x x^{*}*x
  8. h * * = h {h^{*}}^{*}=h
  9. ( h * g ) * = h * * g * (h*g)^{*}=h^{*}*g^{*}
  10. ( h k ) * = h * k (h\leftarrow k)^{*}=h^{*}\rightarrow k

Adleman–Pomerance–Rumely_primality_test.html

  1. O ( ( ln n ) c ln ln ln n ) . O((\ln n)^{c\,\ln\,\ln\,\ln n}).

Admissible_set.html

  1. A A\,
  2. A , \langle A,\in\rangle

Aerostatics.html

  1. ρ [ U j t + U i U j t ] = - P x j - τ i j x i + ρ g j \rho[{\partial U_{j}\over\partial t}+U_{i}{\partial U_{j}\over\partial t}]=-{% \partial P\over\partial x_{j}}-{\partial\tau_{ij}\over\partial x_{i}}+\rho g_{j}
  2. ρ \rho
  3. U j U_{j}
  4. P P
  5. g g
  6. τ i j \tau_{ij}
  7. U j = 0 U_{j}=0
  8. τ i j = 0 \tau_{ij}=0
  9. P x j = ρ g j {\partial P\over\partial x_{j}}=\rho g_{j}
  10. P ρ = R T {P\over\rho}=RT
  11. R R
  12. T T
  13. P x j = ρ g j ^ = P R T g j ^ {\partial P\over\partial x_{j}}=\rho\hat{g_{j}}={P\over\ RT}\hat{g_{j}}

Affiliated_operator.html

  1. G ( A ) = { ( x , A x ) : x D ( A ) } H H G(A)=\{(x,Ax):x\in D(A)\}\subseteq H\oplus H
  2. A = V | A | , A=V|A|,\,
  3. E ( [ 0 , N ] ) E([0,N])

Affine_action.html

  1. W W
  2. 𝔤 \mathfrak{g}
  3. 𝔥 \mathfrak{h}
  4. 𝔥 * \mathfrak{h}^{*}
  5. 𝔥 * \mathfrak{h}^{*}
  6. w λ := w ( λ + δ ) - δ w\cdot\lambda:=w(\lambda+\delta)-\delta
  7. δ \delta

Affine_shape_adaptation.html

  1. μ \mu
  2. I I
  3. x ¯ = ( x , y ) T \bar{x}=(x,y)^{T}
  4. Σ t \Sigma_{t}
  5. g ( x ¯ ; Σ ) = 1 2 π det Σ t e - x ¯ Σ t - 1 x ¯ / 2 g(\bar{x};\Sigma)=\frac{1}{2\pi\sqrt{\operatorname{det}\Sigma_{t}}}e^{-\bar{x}% \Sigma_{t}^{-1}\bar{x}/2}
  6. I L I_{L}
  7. L ( x ¯ ; Σ t ) = x i ¯ I L ( x - ξ ) g ( ξ ¯ ; Σ t ) d ξ ¯ . L(\bar{x};\Sigma_{t})=\int_{\bar{xi}}I_{L}(x-\xi)\,g(\bar{\xi};\Sigma_{t})\,d% \bar{\xi}.
  8. η = B ξ \eta=B\xi
  9. B B
  10. I R I_{R}
  11. I L ( ξ ¯ ) = I R ( η ¯ ) I_{L}(\bar{\xi})=I_{R}(\bar{\eta})
  12. L L
  13. R R
  14. I L I_{L}
  15. I R I_{R}
  16. L ( ξ ¯ , Σ L ) = R ( η ¯ , Σ R ) L(\bar{\xi},\Sigma_{L})=R(\bar{\eta},\Sigma_{R})
  17. Σ L \Sigma_{L}
  18. Σ R \Sigma_{R}
  19. Σ R = B Σ L B T \Sigma_{R}=B\Sigma_{L}B^{T}
  20. L = ( L x , L y ) T \nabla L=(L_{x},L_{y})^{T}
  21. Σ t \Sigma_{t}
  22. Σ s \Sigma_{s}
  23. μ L ( x ¯ ; Σ t , Σ s ) = g ( x ¯ - ξ ¯ ; Σ s ) ( L ( ξ ¯ ; Σ t ) L T ( ξ ¯ ; Σ t ) ) \mu_{L}(\bar{x};\Sigma_{t},\Sigma_{s})=g(\bar{x}-\bar{\xi};\Sigma_{s})\,\left(% \nabla_{L}(\bar{\xi};\Sigma_{t})\nabla_{L}^{T}(\bar{\xi};\Sigma_{t})\right)
  24. q ¯ = B p ¯ \bar{q}=B\bar{p}
  25. μ L ( p ¯ ; Σ t , Σ s ) = B T μ R ( q ; B Σ t B T , B Σ s B T ) B \mu_{L}(\bar{p};\Sigma_{t},\Sigma_{s})=B^{T}\mu_{R}(q;B\Sigma_{t}B^{T},B\Sigma% _{s}B^{T})B
  26. p ¯ \bar{p}
  27. q ¯ \bar{q}
  28. B B
  29. μ L \mu_{L}
  30. μ R \mu_{R}
  31. B B
  32. μ R \mu_{R}
  33. μ \mu
  34. μ - 1 \mu^{-1}
  35. μ \mu
  36. B ^ = μ 1 / 2 \hat{B}=\mu^{1/2}
  37. μ 1 / 2 \mu^{1/2}
  38. μ \mu
  39. B ^ - 1 \hat{B}^{-1}
  40. μ \mu

Airsoft_pellets.html

  1. A d r a g = 1 2 m ρ a i r v 2 0.47 π ( D 2 ) 2 A_{drag}=\frac{1}{2m}\rho_{air}v^{2}\cdot 0.47\cdot\pi\left(\frac{D}{2}\right)% ^{2}
  2. 576 m / s 2 576m/s^{2}
  3. v c r = 162.1 e - 0.38 m v_{cr}=162.1e^{-0.38\sqrt{m}}

Airy_points.html

  1. 1 / 3 = 0.57735... 1/\sqrt{3}=0.57735...
  2. 571428 ¯ \overline{571428}

Aitken's_delta-squared_process.html

  1. x = ( x n ) n 𝒩 x={(x_{n})}_{n\in\mathcal{N}}
  2. A x = ( x n + 2 x n - ( x n + 1 ) 2 x n + 2 - 2 x n + 1 + x n ) n \Z * , Ax={\left(\frac{x_{n+2}\,x_{n}-(x_{n+1})^{2}}{x_{n+2}-2\,x_{n+1}+x_{n}}\right)% }_{n\in\Z^{*}},
  3. ( A x ) n = x n - ( Δ x n ) 2 Δ 2 x n , (Ax)_{n}=x_{n}-\frac{(\Delta x_{n})^{2}}{\Delta^{2}x_{n}},
  4. = x n + 2 - ( Δ x n + 1 ) 2 Δ 2 x n =x_{n+2}-\frac{(\Delta x_{n+1})^{2}}{\Delta^{2}x_{n}}
  5. Δ x n = ( x n + 1 - x n ) , Δ x n + 1 = ( x n + 2 - x n + 1 ) , \Delta x_{n}={(x_{n+1}-x_{n})},\ \Delta x_{n+1}={(x_{n+2}-x_{n+1})},
  6. Δ 2 x n = x n - 2 x n + 1 + x n + 2 = Δ x n + 1 - Δ x n , \Delta^{2}x_{n}=x_{n}-2x_{n+1}+x_{n+2}=\Delta x_{n+1}-\Delta x_{n},
  7. n = 0 , 1 , 2 , 3 , n=0,1,2,3,\dots\,
  8. Δ x n + 1 - Δ x n = ( x n + 2 - x n + 1 ) - ( x n + 1 - x n ) \Delta x_{n+1}-\Delta x_{n}\ =(x_{n+2}-x_{n+1})-(x_{n+1}-x_{n})
  9. x n - 2 x n + 1 + x n + 2 x_{n}-2x_{n+1}+x_{n+2}
  10. x x
  11. \ell
  12. lim n | x n + 1 - | | x n - | = μ . \lim_{n\to\infty}\frac{|x_{n+1}-\ell|}{|x_{n}-\ell|}=\mu.
  13. x n x_{n}
  14. lim n ( A x ) n - x n - = 0. \lim_{n\to\infty}\frac{(Ax)_{n}-\ell}{x_{n}-\ell}=0.
  15. A A
  16. A [ x - ] = A x - A[x-\ell]=Ax-\ell
  17. \ell
  18. A x Ax
  19. Δ \Delta
  20. x n + 1 = f ( x n ) x_{n+1}=f(x_{n})
  21. f f
  22. x n = + a n + b n x_{n}=\ell+a^{n}+b^{n}
  23. 0 < b < a < 1 0<b<a<1
  24. A x Ax
  25. b n b^{n}
  26. x x
  27. \ell
  28. A x Ax
  29. A x Ax
  30. x x
  31. A x Ax
  32. x x
  33. 2 1.4142136 \sqrt{2}\approx 1.4142136
  34. a n a_{n}
  35. a n + 1 = a n + 2 a n 2 . a_{n+1}=\frac{a_{n}+\frac{2}{a_{n}}}{2}.
  36. a 0 = 1 : a_{0}=1:
  37. π 4 \frac{\pi}{4}
  38. π 4 = n = 0 ( - 1 ) n 2 n + 1 0.785398 \frac{\pi}{4}=\sum_{n=0}^{\infty}\frac{(-1)^{n}}{2n+1}\approx 0.785398
  39. x n + 1 = f ( x n ) x_{n+1}=f(x_{n})
  40. x 0 x_{0}
  41. α = f ( α ) \alpha=f(\alpha)
  42. x n + 1 = 1 2 ( x n + 2 x n ) x_{n+1}=\frac{1}{2}(x_{n}+\frac{2}{x_{n}})
  43. x 0 = 1 x_{0}=1
  44. 2 \sqrt{2}
  45. f ( x ) = 1 2 ( x + 2 x ) f(x)=\frac{1}{2}(x+\frac{2}{x})
  46. f ( α ) f^{\prime}(\alpha)

Aitoff_projection.html

  1. x = 2 a z e q x ( λ 2 , ϕ ) x=2\mathrm{azeq}_{x}\left(\frac{\lambda}{2},\phi\right)\,
  2. y = azeq y ( λ 2 , ϕ ) y=\mathrm{azeq}_{y}\left(\frac{\lambda}{2},\phi\right)
  3. azeq x \mathrm{azeq}_{x}
  4. azeq y \mathrm{azeq}_{y}
  5. x = 2 cos ( ϕ ) sin ( λ 2 ) sinc ( α ) x=\frac{2\cos(\phi)\sin\left(\frac{\lambda}{2}\right)}{\mathrm{sinc}(\alpha)}\,
  6. y = sin ( ϕ ) sinc ( α ) y=\frac{\sin(\phi)}{\mathrm{sinc}(\alpha)}\,
  7. α = arccos ( cos ( ϕ ) cos ( λ 2 ) ) \alpha=\arccos\left(\cos(\phi)\cos\left(\frac{\lambda}{2}\right)\right)\,
  8. sinc ( α ) \mathrm{sinc}(\alpha)
  9. λ \lambda
  10. ϕ \phi

Albert_Ingham.html

  1. ζ ( 1 / 2 + i t ) = O ( t c ) \zeta\left(1/2+it\right)=O\left(t^{c}\right)
  2. π ( x + x θ ) - π ( x ) x θ log x , \pi\left(x+x^{\theta}\right)-\pi(x)\sim\frac{x^{\theta}}{\log x},

Alcohol_sulfotransferase.html

  1. \rightleftharpoons

Aldose_reductase.html

  1. \rightleftharpoons
  2. \rightleftharpoons

Aleph_One_(disambiguation).html

  1. 1 \aleph_{1}

Algebraic_operation.html

  1. 3 x + 1 3\frac{x}{+1}
  2. ( 5 × 5 ) + 5 + 5 + 3 (5\times 5)+5+5+3
  3. 5 2 + ( 2 × 5 ) + 3 5^{2}+(2\times 5)+3
  4. ( b × b ) + b + b + a (b\times b)+b+b+a
  5. b 2 + 2 b + a b^{2}+2b+a
  6. 2 × b 2 b b + b + b 3 b b × b b 2 \begin{aligned}\displaystyle 2\times b&\displaystyle\equiv 2b\\ \displaystyle b+b+b&\displaystyle\equiv 3b\\ \displaystyle b\times b&\displaystyle\equiv b^{2}\end{aligned}
  7. ( 7 × 7 ) - 7 - 5 (7\times 7)-7-5
  8. 7 2 - 7 - 5 7^{2}-7-5
  9. ( b × b ) - b - a (b\times b)-b-a
  10. b 2 - b - a b^{2}-b-a
  11. b 2 - b b 3 b - b 2 b b 2 - b b ( b - 1 ) \begin{aligned}\displaystyle b^{2}-b&\displaystyle\not\equiv b\\ \displaystyle 3b-b&\displaystyle\equiv 2b\\ \displaystyle b^{2}-b&\displaystyle\equiv b(b-1)\end{aligned}
  12. 3 × 5 3\times 5
  13. 3 . 5 3\ .\ 5
  14. 3 5 3\cdot 5
  15. ( 3 ) ( 5 ) (3)(5)
  16. a × b a\times b
  17. a . b a.b
  18. a b a\cdot b
  19. a b ab
  20. a × a × a a\times a\times a
  21. a 3 a^{3}
  22. 12 ÷ 4 12\div 4
  23. 12 / 4 12/4
  24. 12 4 \frac{12}{4}
  25. b ÷ a b\div a
  26. b / a b/a
  27. b a \frac{b}{a}
  28. ( a + b ) 3 1 3 × ( a + b ) \frac{(a+b)}{3}\equiv\tfrac{1}{3}\times(a+b)
  29. 3 1 2 3^{\frac{1}{2}}
  30. 2 3 2^{3}
  31. a 1 2 a^{\frac{1}{2}}
  32. a 3 a^{3}
  33. a 1 2 a^{\frac{1}{2}}
  34. a \sqrt{a}
  35. a 3 a^{3}
  36. a × a × a a\times a\times a
  37. a a
  38. b b
  39. x x
  40. y y
  41. ( 3 + 5 ) = ( 5 + 3 ) (3+5)=(5+3)
  42. ( 3 × 5 ) = ( 5 × 3 ) (3\times 5)=(5\times 3)
  43. ( a + b ) = ( b + a ) (a+b)=(b+a)
  44. ( a × b ) = ( b × a ) (a\times b)=(b\times a)
  45. ( a - b ) ( b - a ) (a-b)\not\equiv(b-a)
  46. a = b a=b
  47. ( 3 + 5 ) + 7 = 3 + ( 5 + 7 ) (3+5)+7=3+(5+7)
  48. ( 3 × 5 ) × 7 = 3 × ( 5 × 7 ) (3\times 5)\times 7=3\times(5\times 7)
  49. ( a + b ) + c = a + ( b + c ) (a+b)+c=a+(b+c)
  50. ( a × b ) × c = a × ( b × c ) (a\times b)\times c=a\times(b\times c)

Algebraically_closed_group.html

  1. A A
  2. A A
  3. A A
  4. x x
  5. G G
  6. x 2 = 1 x^{2}=1
  7. x 3 = 1 x^{3}=1
  8. x 1 x\neq 1
  9. x = 1 x=1
  10. G G
  11. G G
  12. . .
  13. 1 ¯ \underline{1}
  14. a ¯ \underline{a}
  15. 1 ¯ \underline{1}
  16. 1 1
  17. a a
  18. a ¯ \underline{a}
  19. a a
  20. 1 1
  21. G G
  22. x 2 = 1 x^{2}=1
  23. x 1 x\neq 1
  24. G G
  25. x = a x=a
  26. x 4 = 1 x^{4}=1
  27. x 2 a - 1 = 1 x^{2}a^{-1}=1
  28. G G
  29. H H
  30. . .
  31. 1 ¯ \underline{1}
  32. a ¯ \underline{a}
  33. b ¯ \underline{b}
  34. c ¯ \underline{c}
  35. 1 ¯ \underline{1}
  36. 1 1
  37. a a
  38. b b
  39. c c
  40. a ¯ \underline{a}
  41. a a
  42. 1 1
  43. c c
  44. b b
  45. b ¯ \underline{b}
  46. b b
  47. c c
  48. a a
  49. 1 1
  50. c ¯ \underline{c}
  51. c c
  52. b b
  53. 1 1
  54. a a
  55. G G
  56. H H
  57. x = b x=b
  58. x = c x=c
  59. G G
  60. G G
  61. G G
  62. G G
  63. H H
  64. G G
  65. A A
  66. A A
  67. G G
  68. F F
  69. G G
  70. E E
  71. I I
  72. F G F\star G
  73. F F
  74. G G
  75. x i x_{i}
  76. g j g_{j}
  77. G G
  78. E E
  79. x 1 2 g 1 4 x 3 = 1 x_{1}^{2}g_{1}^{4}x_{3}=1
  80. x 3 2 g 2 x 4 g 1 = 1 x_{3}^{2}g_{2}x_{4}g_{1}=1
  81. \dots
  82. I I
  83. g 5 - 1 x 3 1 g_{5}^{-1}x_{3}\neq 1
  84. \dots
  85. G G
  86. f : F G f:F\rightarrow G
  87. f ~ ( e ) = 1 \tilde{f}(e)=1
  88. e E e\in E
  89. f ~ ( i ) 1 \tilde{f}(i)\neq 1
  90. i I i\in I
  91. f ~ \tilde{f}
  92. f ~ : F G G \tilde{f}:F\star G\rightarrow G
  93. f f
  94. F F
  95. G G
  96. G G
  97. x 1 g 6 , x 3 g 7 x_{1}\mapsto g_{6},x_{3}\mapsto g_{7}
  98. x 4 g 8 x_{4}\mapsto g_{8}
  99. g 6 2 g 1 4 g 7 = 1 g_{6}^{2}g_{1}^{4}g_{7}=1
  100. g 7 2 g 2 g 8 g 1 = 1 g_{7}^{2}g_{2}g_{8}g_{1}=1
  101. \dots
  102. g 5 - 1 g 7 1 g_{5}^{-1}g_{7}\neq 1
  103. \dots
  104. G G
  105. H H
  106. G G
  107. H H
  108. h : G H h:G\rightarrow H
  109. h ~ ( E ) \tilde{h}(E)
  110. h ~ ( I ) \tilde{h}(I)
  111. H H
  112. h ~ \tilde{h}
  113. h ~ : F G F H \tilde{h}:F\star G\rightarrow F\star H
  114. h h
  115. G G
  116. F F
  117. A A
  118. A A
  119. A A
  120. A A
  121. C C
  122. C C
  123. C 1 C_{1}
  124. C C
  125. C 1 C_{1}
  126. C 1 C_{1}
  127. C C
  128. S 0 , S 1 , S 2 , S_{0},S_{1},S_{2},\dots
  129. D 0 , D 1 , D 2 , D_{0},D_{1},D_{2},\dots
  130. D 0 = C D_{0}=C
  131. D i + 1 = { D i if S i is not consistent with D i D i , h 1 , h 2 , , h n if S i has a solution in H D i with x j h j 1 j n D_{i+1}=\left\{\begin{matrix}D_{i}&\mbox{if}~{}\ S_{i}\ \mbox{is not % consistent with}~{}\ D_{i}\\ \langle D_{i},h_{1},h_{2},\dots,h_{n}\rangle&\mbox{if}~{}\ S_{i}\ \mbox{has a % solution in}~{}\ H\supseteq D_{i}\ \mbox{with}~{}\ x_{j}\mapsto h_{j}\ 1\leq j% \leq n\end{matrix}\right.
  132. C 1 = i = 0 D i C_{1}=\cup_{i=0}^{\infty}D_{i}
  133. C = C 0 , C 1 , C 2 , C=C_{0},C_{1},C_{2},\dots
  134. A = i = 0 C i A=\cup_{i=0}^{\infty}C_{i}
  135. A A
  136. C C
  137. A A
  138. C i C_{i}
  139. C i + 1 C_{i+1}

Algebraically_compact_group.html

  1. \mathbb{Z}

Alkylglycerone_phosphate_synthase.html

  1. \rightleftharpoons

All-pass_filter.html

  1. H ( s ) = s R C - 1 s R C + 1 , H(s)=\frac{sRC-1}{sRC+1},\,
  2. | H ( i ω ) | = 1 and H ( i ω ) = 180 - 2 arctan ( ω R C ) . |H(i\omega)|=1\quad\,\text{and}\quad\angle H(i\omega)=180^{\circ}-2\arctan(% \omega RC).\,
  3. exp { - s T } , \exp\{-sT\},
  4. T T
  5. s s\in\mathbb{C}
  6. exp { - s T } = exp { - s T / 2 } exp { s T / 2 } 1 - s T / 2 1 + s T / 2 , \exp\{-sT\}=\frac{\exp\{-sT/2\}}{\exp\{sT/2\}}\approx\frac{1-sT/2}{1+sT/2},
  7. R C = T / 2 RC=T/2
  8. H ( s ) H(s)
  9. z 0 z_{0}
  10. H ( z ) = z - 1 - z 0 ¯ 1 - z 0 z - 1 H(z)=\frac{z^{-1}-\overline{z_{0}}}{1-z_{0}z^{-1}}
  11. 1 / z 0 ¯ 1/\overline{z_{0}}
  12. z ¯ \overline{z}
  13. z 0 z_{0}
  14. z 0 ¯ \overline{z_{0}}
  15. z 0 z_{0}
  16. H ( z ) = z - 1 - z 0 ¯ 1 - z 0 z - 1 × z - 1 - z 0 1 - z 0 ¯ z - 1 = z - 2 - 2 ( z 0 ) z - 1 + | z 0 | 2 1 - 2 ( z 0 ) z - 1 + | z 0 | 2 z - 2 , H(z)=\frac{z^{-1}-\overline{z_{0}}}{1-z_{0}z^{-1}}\times\frac{z^{-1}-z_{0}}{1-% \overline{z_{0}}z^{-1}}=\frac{z^{-2}-2\Re(z_{0})z^{-1}+\left|{z_{0}}\right|^{2% }}{1-2\Re(z_{0})z^{-1}+\left|z_{0}\right|^{2}z^{-2}},
  17. y [ k ] - 2 ( z 0 ) y [ k - 1 ] + | z 0 | 2 y [ k - 2 ] = x [ k - 2 ] - 2 ( z 0 ) x [ k - 1 ] + | z 0 | 2 x [ k ] , y[k]-2\Re(z_{0})y[k-1]+\left|z_{0}\right|^{2}y[k-2]=x[k-2]-2\Re(z_{0})x[k-1]+% \left|z_{0}\right|^{2}x[k],\,
  18. y [ k ] y[k]
  19. x [ k ] x[k]
  20. k k
  21. z 0 z_{0}

Alpha_max_plus_beta_min_algorithm.html

  1. | z | = a 2 + b 2 |z|=\sqrt{a^{2}+b^{2}}
  2. | z | = α 𝐌𝐚𝐱 + β 𝐌𝐢𝐧 |z|=\alpha\,\!\mathbf{Max}+\beta\,\!\mathbf{Min}
  3. 𝐌𝐚𝐱 \mathbf{Max}
  4. 𝐌𝐢𝐧 \mathbf{Min}
  5. α \alpha\,\!
  6. β \beta\,\!
  7. α 0 = 2 cos π 8 1 + cos π 8 = 0.96043387... \alpha_{0}=\frac{2\cos\frac{\pi}{8}}{1+\cos\frac{\pi}{8}}=0.96043387...
  8. β 0 = 2 sin π 8 1 + cos π 8 = 0.39782473... \beta_{0}=\frac{2\sin\frac{\pi}{8}}{1+\cos\frac{\pi}{8}}=0.39782473...
  9. α \alpha\,\!
  10. β \beta\,\!
  11. α 0 \alpha_{0}
  12. β 0 \beta_{0}

Alternatives_to_general_relativity.html

  1. c c\;
  2. G G\;
  3. η μ ν \eta_{\mu\nu}\;
  4. g μ ν g_{\mu\nu}\;
  5. μ ϕ \partial_{\mu}\phi\;
  6. ϕ , μ \phi_{,\mu}\;
  7. μ ϕ \nabla_{\mu}\phi\;
  8. ϕ ; μ \phi_{;\mu}\;
  9. L L\,
  10. S S\,
  11. S = L - g d 4 x S=\int L\sqrt{-g}\,\mathrm{d}^{4}x
  12. g = - 1 g=-1\,
  13. L R L\,\propto\,R
  14. g μ ν g_{\mu\nu}\,
  15. d τ 2 = - g μ ν d x μ d x ν {d\tau}^{2}=-g_{\mu\nu}\,dx^{\mu}\,dx^{\nu}\,
  16. μ \mu
  17. ν \nu
  18. 0 = ν T μ ν = T μ ν , ν + Γ σ ν μ T σ ν + Γ σ ν ν T μ σ 0=\nabla_{\nu}T^{\mu\nu}={T^{\mu\nu}}_{,\nu}+\Gamma^{\mu}_{\sigma\nu}T^{\sigma% \nu}+\Gamma^{\nu}_{\sigma\nu}T^{\mu\sigma}\,
  19. T μ ν T^{\mu\nu}\,
  20. ν \nabla_{\nu}
  21. Γ σ ν α \Gamma^{\alpha}_{\sigma\nu}\,
  22. ρ \rho\,
  23. ϕ \phi\,
  24. 2 ϕ x j x j = 4 π G ρ . {\partial^{2}\phi\over\partial x^{j}\partial x^{j}}=4\pi G\rho\,.
  25. \nabla
  26. 2 ϕ = 4 π G ρ . \nabla^{2}\phi=4\pi G\rho\,.
  27. d 2 x j d t 2 = - ϕ x j . {d^{2}x^{j}\over dt^{2}}=-{\partial\phi\over\partial x^{j}\,}.
  28. ϕ = - G M / r . \phi=-GM/r\,.
  29. δ d τ = 0 \delta\int d\tau=0\,
  30. d τ 2 = - η μ ν d x μ d x ν {d\tau}^{2}=-\eta_{\mu\nu}dx^{\mu}dx^{\nu}\,
  31. η μ ν \eta_{\mu\nu}\,
  32. μ \mu\,
  33. ν \nu\,
  34. δ d τ = 0 \delta\int d\tau=0\,
  35. d τ 2 = - g μ ν d x μ d x ν {d\tau}^{2}=-g_{\mu\nu}dx^{\mu}dx^{\nu}\,
  36. T μ ν = ρ d x μ d τ d x ν d τ T^{\mu\nu}=\rho{dx^{\mu}\over d\tau}{dx^{\nu}\over d\tau}\,
  37. c c\,
  38. ϕ \phi\,
  39. ϕ = ρ \Box\phi=\rho\,
  40. ρ \rho\,
  41. \Box\,
  42. m = m 0 exp ( ϕ / c 2 ) m=m_{0}\exp(\phi/c^{2})\,
  43. - ϕ x μ = u ˙ μ + u μ c 2 ϕ ˙ -{\partial\phi\over\partial x^{\mu}}=\dot{u}_{\mu}+{u_{\mu}\over c^{2}\dot{% \phi}}\,
  44. u u\,
  45. δ ψ d τ = 0 \delta\int\psi d\tau=0\,
  46. d τ 2 = - η μ ν d x μ d x ν {d\tau}^{2}=-\eta_{\mu\nu}dx^{\mu}dx^{\nu}\,
  47. ψ \psi\,
  48. - T μ ν x ν = T 1 ψ ψ x μ -{\partial T^{\mu\nu}\over\partial x^{\nu}}=T{1\over\psi}{\partial\psi\over% \partial x_{\mu}}\,
  49. δ d s = 0 \delta\int ds=0\,
  50. d s 2 = g μ ν d x μ d x ν {ds}^{2}=g_{\mu\nu}dx^{\mu}dx^{\nu}\,
  51. g μ ν = ψ 2 η μ ν g_{\mu\nu}=\psi^{2}\eta_{\mu\nu}\,
  52. T R . T\,\propto\,R\,.
  53. δ d s = 0 \delta\int ds=0\,
  54. d s 2 = g μ ν d x μ d x ν {ds}^{2}=g_{\mu\nu}dx^{\mu}dx^{\nu}\,
  55. R μ ν = 8 π G c 4 ( T μ ν - 1 2 g μ ν T ) R_{\mu\nu}=\frac{8\pi G}{c^{4}}\left(T_{\mu\nu}-\frac{1}{2}g_{\mu\nu}T\right)\,
  56. T μ ν = c 4 8 π G ( R μ ν - 1 2 g μ ν R ) . T^{\mu\nu}={c^{4}\over 8\pi G}\left(R^{\mu\nu}-\frac{1}{2}g^{\mu\nu}R\right)\,.
  57. S = c 4 16 π G R - g d 4 x + S m S={c^{4}\over 16\pi G}\int R\sqrt{-g}d^{4}x+S_{m}\,
  58. G G\,
  59. R = R μ μ R=R_{\mu}^{~{}\mu}\,
  60. g = det ( g μ ν ) g=\det(g_{\mu\nu})\,
  61. S m S_{m}\,
  62. δ f ( ϕ / c 2 ) d s = 0 \delta\int f(\phi/c^{2})ds=0\,
  63. ϕ = G M / r \phi=GM/r\,
  64. c c\,
  65. ϕ \phi\,
  66. f ( ϕ / c 2 ) = exp ( - ϕ / c 2 ) f(\phi/c^{2})=\exp(-\phi/c^{2})\,
  67. c = c c=c_{\infty}\,
  68. f ( ϕ / c 2 ) = exp ( - ϕ / c 2 - ( ϕ / c 2 ) 2 / 2 ) f(\phi/c^{2})=\exp(-\phi/c^{2}-(\phi/c^{2})^{2}/2)\,
  69. c = c c=c_{\infty}\,
  70. f ( ϕ / c 2 ) = 1 f(\phi/c^{2})=1\,
  71. c 2 = c 2 - 2 ϕ c^{2}=c_{\infty}^{2}-2\phi\,
  72. f ( ϕ / c 2 ) = exp ( - ϕ / c 2 ) f(\phi/c^{2})=\exp(-\phi/c^{2})\,
  73. c 2 = c 2 - 2 ϕ c^{2}=c_{\infty}^{2}-2\phi\,
  74. f ( ϕ / c 2 ) = ϕ / c 2 + α ( ϕ / c 2 ) 2 f(\phi/c^{2})=\phi/c^{2}+\alpha(\phi/c^{2})^{2}\,
  75. c 2 / c 2 = 1 + 4 ( ϕ / c 2 ) + ( 15 + 2 α ) ( ϕ / c 2 ) 2 c_{\infty}^{2}/c^{2}=1+4(\phi/c_{\infty}^{2})+(15+2\alpha)(\phi/c_{\infty}^{2}% )^{2}\,
  76. α = - 7 / 2 \alpha=-7/2\,
  77. S = 1 16 π G d 4 x - g L ϕ + S m S={1\over 16\pi G}\int d^{4}x\sqrt{-g}L_{\phi}+S_{m}\,
  78. L ϕ = ϕ R - 2 g μ ν μ ϕ ν ϕ L_{\phi}=\phi R-2g^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi\,
  79. ϕ R \phi R\,
  80. S m S_{m}\,
  81. ϕ = 4 π T μ ν [ η μ ν e - 2 ϕ + ( e 2 ϕ + e - 2 ϕ ) μ t ν t ] \Box\phi=4\pi T^{\mu\nu}[\eta_{\mu\nu}e^{-2\phi}+(e^{2\phi}+e^{-2\phi})% \partial_{\mu}t\partial_{\nu}t]\,
  82. t t\,
  83. f ( ϕ ) f(\phi)\,
  84. k ( ϕ ) k(\phi)\,
  85. d s 2 = e - 2 f ( ϕ ) d t 2 - e 2 f ( ϕ ) [ d x 2 + d y 2 + d z 2 ] ds^{2}=e^{-2f(\phi)}dt^{2}-e^{2f(\phi)}[dx^{2}+dy^{2}+dz^{2}]\,
  86. η μ ν μ ν ϕ = 4 π ρ * k ( ϕ ) \eta^{\mu\nu}\partial_{\mu}\partial_{\nu}\phi=4\pi\rho^{*}k(\phi)\,
  87. ϕ \phi\,
  88. ψ \psi\,
  89. d s 2 = ϕ 2 d t 2 - ψ 2 [ d x 2 + d y 2 + d z 2 ] ds^{2}=\phi^{2}dt^{2}-\psi^{2}[dx^{2}+dy^{2}+dz^{2}]\,
  90. L ϕ = e ϕ ( 1 2 e - ϕ α ϕ α ϕ + 3 2 e ϕ 0 ϕ 0 ϕ ) L_{\phi}=e^{\phi}(\textstyle\frac{1}{2}e^{-\phi}\partial_{\alpha}\phi\partial_% {\alpha}\phi+\textstyle\frac{3}{2}e^{\phi}\partial_{0}\phi\partial_{0}\phi)\,
  91. χ \chi\,
  92. L ϕ = e ( 3 ϕ + χ ) / 2 ( - 1 2 e - ϕ α ϕ α ϕ - e - ϕ α ϕ χ ϕ + 3 2 e - χ 0 ϕ 0 ϕ ) L_{\phi}=e^{(3\phi+\chi)/2}(-\textstyle\frac{1}{2}e^{-\phi}\partial_{\alpha}% \phi\partial_{\alpha}\phi-e^{-\phi}\partial_{\alpha}\phi\partial_{\chi}\phi+% \textstyle\frac{3}{2}e^{-\chi}\partial_{0}\phi\partial_{0}\phi)\,
  93. S = 1 64 π G d 4 x - η η μ ν g α β g γ δ ( g α γ | μ g α δ | ν - 1 2 g α β | μ g γ δ | ν ) + S m S={1\over 64\pi G}\int d^{4}x\sqrt{-\eta}\eta^{\mu\nu}g^{\alpha\beta}g^{\gamma% \delta}(g_{\alpha\gamma|\mu}g_{\alpha\delta|\nu}-\textstyle\frac{1}{2}g_{% \alpha\beta|\mu}g_{\gamma\delta|\nu})+S_{m}
  94. η \eta\,
  95. η g μ ν - g α β η γ δ g μ α | γ g ν β | δ = - 16 π G g / η ( T μ ν - 1 2 g μ ν T ) \Box_{\eta}g_{\mu\nu}-g^{\alpha\beta}\eta^{\gamma\delta}g_{\mu\alpha|\gamma}g_% {\nu\beta|\delta}=-16\pi G\sqrt{g/\eta}(T_{\mu\nu}-\textstyle\frac{1}{2}g_{\mu% \nu}T)\,
  96. B μ ν B_{\mu\nu}\,
  97. B = B μ ν η μ ν B=B_{\mu\nu}\eta^{\mu\nu}\,
  98. a a\,
  99. f f\,
  100. S = 1 16 π G d 4 x - η ( a B μ ν | α B μ ν | α + f B , α B , α ) + S m S={1\over 16\pi G}\int d^{4}x\sqrt{-\eta}(aB^{\mu\nu|\alpha}B_{\mu\nu|\alpha}+% fB_{,\alpha}B^{,\alpha})+S_{m}
  101. a η B μ ν + f η μ ν η B = - 4 π G g / η T α β ( g α β / B μ ν ) a\Box_{\eta}B^{\mu\nu}+f\eta^{\mu\nu}\Box_{\eta}B=-4\pi G\sqrt{g/\eta}T^{% \alpha\beta}(\partial g_{\alpha\beta}/\partial B_{\mu}\nu)
  102. S = 1 16 π G d 4 x - g F ( N ) K μ ; ν K μ ; ν + S m S={1\over 16\pi G}\int d^{4}x\sqrt{-g}F(N)K^{\mu;\nu}K_{\mu;\nu}+S_{m}
  103. F ( N ) = - N / ( 2 + N ) F(N)=-N/(2+N)\;
  104. N = g μ ν K μ K ν N=g^{\mu\nu}K_{\mu}K_{\nu}\;
  105. T μ ν T^{\mu\nu}\;
  106. K μ K_{\mu}\;
  107. g g\;
  108. η \eta\;
  109. g μ ν ( x α ) = η μ ν - 2 Σ - y μ - y ν - ( w - ) 3 [ - g ρ u α d Σ α ] - g_{\mu\nu}(x^{\alpha})=\eta_{\mu\nu}-2\int_{\Sigma^{-}}{y_{\mu}^{-}y_{\nu}^{-}% \over(w^{-})^{3}}[\sqrt{-g}\rho u^{\alpha}d\Sigma_{\alpha}]^{-}
  110. η \eta\;
  111. x α x^{\alpha}\;
  112. ( y μ ) - = x μ - ( x μ ) - (y^{\mu})^{-}=x^{\mu}-(x^{\mu})^{-}\;
  113. ( y μ ) - ( y μ ) - = 0 , (y^{\mu})^{-}(y_{\mu})^{-}=0,\;
  114. w - = ( y μ ) - ( u μ ) - w^{-}=(y^{\mu})^{-}(u_{\mu})^{-}\;
  115. ( u μ ) = d x μ / d σ , (u_{\mu})=dx^{\mu}/d\sigma,\;
  116. d σ 2 = η μ ν d x μ d x ν d\sigma^{2}=\eta_{\mu\nu}dx^{\mu}dx^{\nu}\;
  117. h μ ν h_{\mu\nu}\;
  118. g μ ν = η μ ν + h μ ν g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}\;
  119. S = 1 16 π G d 4 x - η [ 2 h | ν μ ν h μ λ | λ - 2 h | ν μ ν h λ | μ λ + h ν | μ ν h λ λ | μ - h μ ν | λ h μ ν | λ ] + S m S={1\over 16\pi G}\int d^{4}x\sqrt{-\eta}[2h_{|\nu}^{\mu\nu}h_{\mu\lambda}^{|% \lambda}-2h_{|\nu}^{\mu\nu}h_{\lambda|\mu}^{\lambda}+h_{\nu|\mu}^{\nu}h_{% \lambda}^{\lambda|\mu}-h^{\mu\nu|\lambda}h_{\mu\nu|\lambda}]+S_{m}\;
  120. h μ ν h_{\mu\nu}\;
  121. S S\;
  122. L ϕ L_{\phi}\;
  123. S = 1 16 π G d 4 x - g L ϕ + S m S={1\over 16\pi G}\int d^{4}x\sqrt{-g}L_{\phi}+S_{m}\;
  124. L ϕ = ϕ R - ω ( ϕ ) ϕ g μ ν μ ϕ ν ϕ + 2 ϕ λ ( ϕ ) L_{\phi}=\phi R-{\omega(\phi)\over\phi}g^{\mu\nu}\partial_{\mu}\phi\partial_{% \nu}\phi+2\phi\lambda(\phi)\;
  125. S m = d 4 x g G N L m S_{m}=\int d^{4}x\sqrt{g}G_{N}L_{m}\;
  126. T μ ν = def 2 g δ S m δ g μ ν T^{\mu\nu}\ \stackrel{\mathrm{def}}{=}\ {2\over\sqrt{g}}{\delta S_{m}\over% \delta g_{\mu\nu}}
  127. ω ( ϕ ) \omega(\phi)\;
  128. λ ( ϕ ) \lambda(\phi)\;
  129. G N G_{N}\;
  130. G G\;
  131. λ = 0 \lambda=0\;
  132. λ \lambda
  133. ω \omega\;
  134. r r\;
  135. q q\;
  136. ϕ = [ 1 - q f ( ϕ ) ] f ( ϕ ) - r \phi=[1-qf(\phi)]f(\phi)^{-r}\;
  137. f f\;
  138. ω ( ϕ ) = - 3 2 - 1 4 f ( ϕ ) [ ( 1 - 6 q ) q f ( ϕ ) - 1 ] [ r + ( 1 - r ) q f ( ϕ ) ] - 2 \omega(\phi)=-\textstyle\frac{3}{2}-\textstyle\frac{1}{4}f(\phi)[(1-6q)qf(\phi% )-1][r+(1-r)qf(\phi)]^{-2}\;
  139. ω ( ϕ ) = ( 4 - 3 ϕ ) / ( 2 ϕ - 2 ) \omega(\phi)=(4-3\phi)/(2\phi-2)\;
  140. ω ( ϕ ) \omega(\phi)\;
  141. ω \omega\rightarrow\infty\;
  142. K μ K_{\mu}\;
  143. S = 1 16 π G d 4 x - g [ R + ω K μ K μ R + η K μ K ν R μ ν - ϵ F μ ν F μ ν + τ K μ ; ν K μ ; ν ] + S m S={1\over 16\pi G}\int d^{4}x\sqrt{-g}[R+\omega K_{\mu}K^{\mu}R+\eta K^{\mu}K^% {\nu}R_{\mu\nu}-\epsilon F_{\mu\nu}F^{\mu\nu}+\tau K_{\mu;\nu}K^{\mu;\nu}]+S_{% m}\;
  144. ω \omega\;
  145. η \eta\;
  146. ϵ \epsilon\;
  147. τ \tau\;
  148. F μ ν = K ν ; μ - K μ ; ν F_{\mu\nu}=K_{\nu;\mu}-K_{\mu;\nu}\;
  149. T μ ν T^{\mu\nu}\;
  150. K μ K_{\mu}\;
  151. ω = η = ϵ = 0 \omega=\eta=\epsilon=0\;
  152. τ = 1 \tau=1\;
  153. τ = 0 \tau=0\;
  154. ϵ = 1 \epsilon=1\;
  155. η = - 2 ω \eta=-2\omega\;
  156. ω = η = ϵ = τ = 0 \omega=\eta=\epsilon=\tau=0\;
  157. L = 1 32 π G Ω ν μ g ν ξ x η x ζ ε ξ μ η ζ L={1\over 32\pi G}\Omega_{\nu}^{\mu}g^{\nu\xi}x^{\eta}x^{\zeta}\varepsilon_{% \xi\mu\eta\zeta}\;
  158. Ω ν μ = d ω ν μ + ω ξ η \Omega_{\nu}^{\mu}=d\omega^{\mu}_{\nu}+\omega^{\eta}_{\xi}\;
  159. x μ = - ω ν μ x ν \nabla x^{\mu}=-\omega^{\mu}_{\nu}x^{\nu}\;
  160. ω ν μ \omega^{\mu}_{\nu}\;
  161. ε ξ μ η ζ \varepsilon_{\xi\mu\eta\zeta}\;
  162. ε 0123 = - g \varepsilon_{0123}=\sqrt{-g}\;
  163. g ν ξ g^{\nu\xi}\,
  164. T μ ν = 1 16 π G ( g μ ν η η ξ - g ξ μ η η ν - g ξ ν η η μ ) Ω ξ η T^{\mu\nu}={1\over 16\pi G}(g^{\mu\nu}\eta^{\xi}_{\eta}-g^{\xi\mu}\eta^{\nu}_{% \eta}-g^{\xi\nu}\eta^{\mu}_{\eta})\Omega^{\eta}_{\xi}\;
  165. ρ \rho
  166. T T
  167. ρ = T μ ν u μ u ν \rho=T_{\mu\nu}u^{\mu}u^{\nu}
  168. ρ = T μ ν δ μ ν \rho=T_{\mu\nu}\delta^{\mu\nu}
  169. u u
  170. δ \delta
  171. η \eta\;
  172. × 10 - 9 \times 10^{-}9
  173. × 10 - 13 \times 10^{-}13
  174. × 10 - 3 \times 10^{-}3
  175. × 10 - 21 \times 10^{-}21
  176. × 10 - 4 \times 10^{-}4
  177. γ \gamma\;
  178. β \beta\;
  179. η \eta\;
  180. α 1 \alpha_{1}\;
  181. α 2 \alpha_{2}\;
  182. α 3 \alpha_{3}\;
  183. ζ 1 \zeta_{1}\;
  184. ζ 2 \zeta_{2}\;
  185. ζ 3 \zeta_{3}\;
  186. ζ 4 \zeta_{4}\;
  187. γ \gamma\;
  188. β \beta\;
  189. η \eta\;
  190. α 1 \alpha_{1}\;
  191. α 2 \alpha_{2}\;
  192. α 3 \alpha_{3}\;
  193. α \alpha
  194. ζ 1 \zeta_{1}\;
  195. ζ 2 \zeta_{2}\;
  196. ζ 3 \zeta_{3}\;
  197. ζ 4 \zeta_{4}\;
  198. α 3 \alpha_{3}\;
  199. γ \gamma
  200. β \beta
  201. ξ \xi
  202. α 1 \alpha_{1}
  203. α 2 \alpha_{2}
  204. α 3 \alpha_{3}
  205. ζ 1 \zeta_{1}
  206. ζ 2 \zeta_{2}
  207. ζ 3 \zeta_{3}
  208. ζ 4 \zeta_{4}
  209. 1 + ω 2 + ω \textstyle\frac{1+\omega}{2+\omega}
  210. β \beta
  211. 1 + ω 2 + ω \textstyle\frac{1+\omega}{2+\omega}
  212. β \beta
  213. 1 + ω 2 + ω \textstyle\frac{1+\omega}{2+\omega}
  214. γ \gamma
  215. β \beta
  216. α 1 \alpha_{1}
  217. α 2 \alpha_{2}
  218. α 2 \alpha_{2}
  219. c 0 / c 1 - 1 c_{0}/c_{1}-1
  220. α 2 \alpha_{2}
  221. γ \gamma
  222. β \beta
  223. α 1 \alpha_{1}
  224. α 2 \alpha_{2}
  225. a c 0 / c 1 ac_{0}/c_{1}
  226. β \beta
  227. ξ \xi
  228. α 1 \alpha_{1}
  229. α 2 \alpha_{2}
  230. a c 0 / c 1 ac_{0}/c_{1}
  231. b c 0 bc_{0}
  232. α 1 \alpha_{1}
  233. α 2 \alpha_{2}
  234. λ \lambda
  235. 3 4 + λ 4 \textstyle\frac{3}{4}+\textstyle\frac{\lambda}{4}
  236. - 4 - 4 λ -4-4\lambda
  237. γ \gamma
  238. β \beta
  239. - 4 - 4 γ -4-4\gamma
  240. - 2 - 2 γ -2-2\gamma
  241. ζ 2 \zeta_{2}
  242. ζ 4 \zeta_{4}
  243. - 1 -1
  244. 1 2 \textstyle\frac{1}{2}
  245. - 1 -1
  246. 1 2 \textstyle\frac{1}{2}
  247. - 1 -1
  248. 1 - q 1-q
  249. ζ 2 \zeta_{2}
  250. - 1 -1
  251. 1 - q 1-q
  252. - 1 -1
  253. 1 2 \textstyle\frac{1}{2}
  254. ζ 4 \zeta_{4}
  255. β = ξ \beta=\xi
  256. α 2 \alpha_{2}
  257. Λ \Lambda\;
  258. Λ = 0 \Lambda=0\;
  259. 2 ϕ = 4 π ρ \nabla^{2}\phi=4\pi\rho\;
  260. 2 ϕ - Λ ϕ = 4 π ρ \nabla^{2}\phi-\Lambda\phi=4\pi\rho\;
  261. S = 1 16 π G R - g d 4 x + S m S={1\over 16\pi G}\int R\sqrt{-g}\,d^{4}x\,+S_{m}\;
  262. S = 1 16 π G ( R - 2 Λ ) - g d 4 x + S m S={1\over 16\pi G}\int(R-2\Lambda)\sqrt{-g}\,d^{4}x\,+S_{m}\;
  263. T μ ν = 1 8 π G ( R μ ν - 1 2 g μ ν R ) T^{\mu\nu}={1\over 8\pi G}\left(R^{\mu\nu}-\frac{1}{2}g^{\mu\nu}R\right)\;
  264. T μ ν = 1 8 π G ( R μ ν - 1 2 g μ ν R + g μ ν Λ ) T^{\mu\nu}={1\over 8\pi G}\left(R^{\mu\nu}-\frac{1}{2}g^{\mu\nu}R+g^{\mu\nu}% \Lambda\right)\;
  265. λ ( ϕ ) \lambda(\phi)\;
  266. ϕ \phi\;
  267. λ ( ϕ ) \lambda(\phi)\;
  268. L ϕ L_{\phi}\;
  269. S = 1 16 π G d 4 x - g L ϕ + S m S={1\over 16\pi G}\int d^{4}x\,\sqrt{-g}L_{\phi}+S_{m}\;
  270. λ ( ϕ ) \lambda(\phi)\;
  271. K μ K ν g μ ν K^{\mu}K^{\nu}g_{\mu\nu}\;
  272. L = - a 0 2 8 π G f [ | ϕ | 2 a 0 2 ] - ρ ϕ L=-{a_{0}^{2}\over 8\pi G}f\left[{|\nabla\phi|^{2}\over a_{0}^{2}}\right]-\rho\phi\;
  273. L = - a 0 2 8 π G f ~ ( l 0 2 g μ ν μ ϕ ν ϕ ) L=-{a_{0}^{2}\over 8\pi G}\tilde{f}(l_{0}^{2}g^{\mu\nu}\,\partial_{\mu}\phi\,% \partial_{\nu}\phi)\;
  274. f f\;
  275. f ~ \tilde{f}
  276. l 0 = c 2 / a 0 l_{0}=c^{2}/a_{0}\;
  277. ϕ \phi\;
  278. σ \sigma\;
  279. U α U_{\alpha}\;
  280. S = S g + S s + S v + S m S=S_{g}+S_{s}+S_{v}+S_{m}\;
  281. S s = - 1 2 [ σ 2 h α β ϕ , α ϕ , β + 1 2 G l 0 - 2 σ 4 F ( k G σ 2 ) ] - g d 4 x S_{s}=-\textstyle\frac{1}{2}\int[\sigma^{2}h^{\alpha\beta}\phi_{,\alpha}\phi_{% ,\beta}+\textstyle\frac{1}{2}Gl_{0}^{-2}\sigma^{4}F(kG\sigma^{2})]\sqrt{-g}\,d% ^{4}x\;
  282. S v = - K 32 π G [ g α β g μ ν U [ α , μ ] U [ β , ν ] - 2 ( λ / K ) ( g μ ν U μ U ν + 1 ) ] - g d 4 x S_{v}=-{K\over 32\pi G}\int[g^{\alpha\beta}g^{\mu\nu}U_{[\alpha,\mu]}U_{[\beta% ,\nu]}-2(\lambda/K)(g^{\mu\nu}U_{\mu}U_{\nu}+1)]\sqrt{-g}\,d^{4}x\;
  283. S m = L ( g ~ μ ν , f α , f | μ α , ) - g d 4 x S_{m}=\int L(\tilde{g}_{\mu\nu},f^{\alpha},f^{\alpha}_{|\mu},\ldots)\sqrt{-g}% \,d^{4}x\;
  284. h α β = def g α β - U α U β h^{\alpha\beta}\ \stackrel{\mathrm{def}}{=}\ g^{\alpha\beta}-U^{\alpha}U^{% \beta}\;
  285. k k\;
  286. K K\;
  287. U [ α , μ ] U_{[\alpha,\mu]}\;
  288. λ \lambda\;
  289. g ~ α β = e 2 ϕ g α β + 2 U α U β sinh ( 2 ϕ ) \tilde{g}^{\alpha\beta}=e^{2\phi}g^{\alpha\beta}+2U^{\alpha}U^{\beta}\sinh(2% \phi)\;
  290. L L\;
  291. g ~ α β \tilde{g}^{\alpha\beta}\;
  292. G G
  293. G N e w t o n G_{Newton}
  294. F F\;
  295. F ( μ ) = 3 4 μ 2 ( μ - 2 ) 2 1 - μ F(\mu)=\textstyle\frac{3}{4}{\mu^{2}(\mu-2)^{2}\over 1-\mu}\;
  296. μ = 1 \mu=1\;
  297. α 1 = 4 G K ( ( 2 K - 1 ) e - 4 ϕ 0 - e 4 ϕ 0 + 8 ) - 8 \alpha_{1}=4\frac{G}{K}((2K-1)e^{-4\phi_{0}}-e^{4\phi_{0}}+8)-8
  298. α 2 = 6 G 2 - K - 2 G ( K + 4 ) e 4 ϕ 0 ( 2 - K ) 2 - 1 \alpha_{2}=\frac{6G}{2-K}-\frac{2G(K+4)e^{4\phi_{0}}}{(2-K)^{2}}-1
  299. c = G N e w t o n i a n = 1 c=G_{Newtonian}=1
  300. G - 1 = 2 2 - K + k 4 π G^{-1}=\frac{2}{2-K}+\frac{k}{4\pi}
  301. ϕ 0 \phi_{0}
  302. ϕ \phi
  303. K 2 - K = e - 4 ϕ 0 - 1 \frac{K}{2-K}=e^{-4\phi_{0}}-1
  304. S - S M - S ^ M = - c 4 16 π G [ β g 1 / 2 R + α g ^ 1 / 2 R ^ - 2 ( g g ^ ) 1 / 4 f ( κ ) l 0 - 2 ( l 0 m Υ ( m ) ) ] d 4 x S-S_{M}-\hat{S}_{M}=-{c^{4}\over 16\pi G}\int[\beta g^{1/2}R+\alpha\hat{g}^{1/% 2}\hat{R}-2(g\hat{g})^{1/4}f(\kappa)l_{0}^{-2}\mathcal{M}(l_{0}^{m}\Upsilon^{(% m)})]d^{4}x
  305. S M S_{M}\;
  306. S ^ M \hat{S}_{M}
  307. Υ \Upsilon
  308. κ = ( g / g ^ ) 1 4 \kappa=(g/\hat{g})^{\frac{1}{4}}
  309. α , β \alpha,\beta
  310. \mathcal{M}
  311. Υ \Upsilon
  312. \mathcal{M}
  313. Υ \Upsilon
  314. S - S M = - 1 8 π G [ β ( ϕ ) 2 + α ( ϕ ^ ) 2 - a 0 2 ( ( ϕ - ϕ ^ ) 2 / a 0 2 ) ] d 4 x S-S_{M}=-{1\over 8\pi G}\int[\beta(\nabla\phi)^{2}+\alpha(\nabla\hat{\phi})^{2% }-a_{0}^{2}\mathcal{M}((\nabla\phi-\nabla\hat{\phi})^{2}/a_{0}^{2})]d^{4}x
  315. S M = ρ ( v 2 / 2 - ϕ ) S_{M}=\rho(v^{2}/2-\phi)
  316. ( z ) \mathcal{M}(z)
  317. z - 1 / 4 z^{-1/4}
  318. z z
  319. g μ ν g_{\mu\nu}\;
  320. L = L R + L M L=L_{R}+L_{M}\;
  321. L M L_{M}\;
  322. L R = - g [ R ( W ) - 2 λ - 1 4 μ 2 g μ ν g [ μ ν ] ] - 1 6 g μ ν W μ W ν L_{R}=\sqrt{-g}\left[R(W)-2\lambda-\frac{1}{4}\mu^{2}g^{\mu\nu}g_{[\mu\nu]}% \right]-\frac{1}{6}g^{\mu\nu}W_{\mu}W_{\nu}\;
  323. R ( W ) R(W)\;
  324. λ \lambda\;
  325. μ 2 \mu^{2}\;
  326. g [ ν μ ] g_{[\nu\mu]}\;
  327. g ν μ g_{\nu\mu}\;
  328. W μ W_{\mu}\;
  329. W μ - 2 g [ μ ν ] , ν W_{\mu}\approx-2g^{,\nu}_{[\mu\nu]}\;
  330. G G\;
  331. G G\;
  332. A μ ν A_{\mu\nu}\;
  333. J μ J_{\mu}\;
  334. S = S G + S F + S F M + S M S=S_{G}+S_{F}+S_{FM}+S_{M}\;
  335. S F = d 4 x - g ( 1 12 F μ ν ρ F μ ν ρ - 1 4 μ 2 A μ ν A μ ν ) S_{F}=\int d^{4}x\,\sqrt{-g}\left(\frac{1}{12}F_{\mu\nu\rho}F^{\mu\nu\rho}-% \frac{1}{4}\mu^{2}A_{\mu\nu}A^{\mu\nu}\right)\;
  336. S F M = d 4 x ϵ α β μ ν A α β μ J ν S_{FM}=\int d^{4}x\,\epsilon^{\alpha\beta\mu\nu}A_{\alpha\beta}\partial_{\mu}J% _{\nu}\;
  337. F μ ν ρ = μ A ν ρ + ρ A μ ν F_{\mu\nu\rho}=\partial_{\mu}A_{\nu\rho}+\partial_{\rho}A_{\mu\nu}
  338. ϵ α β μ ν \epsilon^{\alpha\beta\mu\nu}\;
  339. S = S G + S K + S S + S M S=S_{G}+S_{K}+S_{S}+S_{M}\;
  340. K μ K_{\mu}
  341. G G\;
  342. ω \omega\;
  343. μ \mu\;
  344. S G S_{G}\;
  345. G G\;
  346. S K = - d 4 x - g ω ( 1 4 B μ ν B μ ν + V ( K ) ) S_{K}=-\int d^{4}x\,\sqrt{-g}\omega\left(\frac{1}{4}B_{\mu\nu}B^{\mu\nu}+V(K)% \right)\;
  347. B μ ν = μ K ν - ν K μ B_{\mu\nu}=\partial_{\mu}K_{\nu}-\partial_{\nu}K_{\mu}\;
  348. S S \displaystyle S_{S}
  349. V ( K ) = - 1 2 μ 2 ϕ μ ϕ μ - 1 4 g ( ϕ μ ϕ μ ) 2 V(K)=-\frac{1}{2}\mu^{2}\phi^{\mu}\phi_{\mu}-\frac{1}{4}g(\phi^{\mu}\phi_{\mu}% )^{2}\;
  350. g g\;

Alternatization.html

  1. α : S × S A \alpha:S\times S\to A
  2. α \alpha
  3. x S , α ( x , x ) = 0. \forall x\in S,\quad\alpha(x,x)=0.
  4. α : V × V K \alpha:V\times V\to K
  5. α \alpha
  6. x V , α ( x , x ) = 0. \forall x\in V,\quad\alpha(x,x)=0.
  7. α : M × M R \alpha:M\times M\to R
  8. α \alpha
  9. x M , α ( x , x ) = 0. \forall x\in M,\quad\alpha(x,x)=0.
  10. α : V × V × × V K \alpha:V\times V\times...\times V\to K
  11. α \alpha
  12. x 1 , x 2 , , x n V , i { 1 , 2 , , n - 1 } , x i = x i + 1 α ( x 1 , x 2 , , x n ) = 0. \forall x_{1},x_{2},...,x_{n}\in V,\quad\forall i\in\{1,2,...,n-1\},\quad x_{i% }=x_{i+1}\,\implies\,\alpha(x_{1},x_{2},...,x_{n})=0.
  13. α : M × M × × M R \alpha:M\times M\times...\times M\to R
  14. α \alpha
  15. x 1 , x 2 , , x n M , i { 1 , 2 , , n - 1 } , x i = x i + 1 α ( x 1 , x 2 , , x n ) = 0. \forall x_{1},x_{2},...,x_{n}\in M,\quad\forall i\in\{1,2,...,n-1\},\quad x_{i% }=x_{i+1}\,\implies\,\alpha(x_{1},x_{2},...,x_{n})=0.
  16. α : S × S A \alpha:S\times S\to A
  17. x , y S , \forall x,y\in S,
  18. α \alpha
  19. β : S × S A \beta:S\times S\to A
  20. ( x , y ) α ( x , y ) - α ( y , x ) . (x,y)\mapsto\alpha(x,y)-\alpha(y,x).
  21. x , y S , α ( x , y ) + α ( y , x ) = 0 \forall x,y\in S,\quad\alpha(x,y)+\alpha(y,x)=0
  22. x , y V , \forall x,y\in V,
  23. α ( x + y , x + y ) = 0 \alpha(x+y,x+y)=0
  24. = α ( x , x + y ) + α ( y , x + y ) =\alpha(x,x+y)+\alpha(y,x+y)
  25. = α ( x , x ) + α ( x , y ) + α ( y , x ) + α ( y , y ) =\alpha(x,x)+\alpha(x,y)+\alpha(y,x)+\alpha(y,y)
  26. = α ( x , y ) + α ( y , x ) =\alpha(x,y)+\alpha(y,x)

Ambient_construction.html

  1. δ ω g = ω 2 g \delta_{\omega}g=\omega^{2}g
  2. h p ( X p , Y p ) = g p ( π * X , π * Y ) . h_{p}(X_{p},Y_{p})=g_{p}(\pi_{*}X,\pi_{*}Y).
  3. δ ω * h = ω 2 h \delta^{*}_{\omega}h=\omega^{2}h
  4. h = t 2 g i j ( x , ρ ) d x i d x j + 2 ρ d t 2 + 2 t d t d ρ , h^{\sim}=t^{2}g_{ij}(x,\rho)dx^{i}dx^{j}+2\rho dt^{2}+2tdtd\rho,\,
  5. ρ g i j ′′ - ρ g k l g i k g j l + 1 2 ρ g k l g k l g i j + 2 - n 2 g i j - 1 2 g k l g k l g i j + Ric ( g ) i j = 0. \rho g_{ij}^{\prime\prime}-\rho g^{kl}g_{ik}^{\prime}g_{jl}+\tfrac{1}{2}\rho g% ^{kl}g_{kl}^{\prime}g_{ij}^{\prime}+\frac{2-n}{2}g_{ij}^{\prime}-\tfrac{1}{2}g% ^{kl}g_{kl}^{\prime}g_{ij}+\mathrm{Ric}(g)_{ij}=0.

Ambient_isotopy.html

  1. F : M × [ 0 , 1 ] M F:M\times[0,1]\rightarrow M

Ambulatory_blood_pressure.html

  1. D i p = ( 1 - S B P S l e e p i n g S B P W a k i n g ) × 100 % Dip=(1-\frac{SBP_{Sleeping}}{SBP_{Waking}})\times 100\%

AMS-LaTeX.html

  1. y \displaystyle y
  2. a b c a\leq b\leq c
  3. a 2 + b 2 = c 2 a^{2}+b^{2}=c^{2}

Analogical_modeling.html

  1. n n
  2. n 2 n^{2}
  3. n i 2 n_{i}^{2}
  4. n i ( n - n i ) n_{i}(n-n_{i})
  5. n i 2 \sum{n_{i}^{2}}
  6. n i ( n - n i ) = n 2 - n i 2 \sum{n_{i}(n-n_{i})}=n^{2}-\sum{n_{i}^{2}}
  7. n = 5 n=5
  8. n r = 4 n_{r}=4
  9. n e = 1 n_{e}=1
  10. n 2 = 25 n^{2}=25
  11. n r 2 = 16 n_{r}^{2}=16
  12. n e 2 = 1 n_{e}^{2}=1
  13. n r ( n - n r ) = 4 n_{r}(n-n_{r})=4
  14. n e ( n - n e ) = 4 n_{e}(n-n_{e})=4
  15. n r 2 + n e 2 = 17 n_{r}^{2}+n_{e}^{2}=17
  16. n r ( n - n r ) + n e ( n - n e ) = n 2 - ( n r 2 + n e 2 ) = 8 n_{r}(n-n_{r})+n_{e}(n-n_{e})=n^{2}-(n_{r}^{2}+n_{e}^{2})=8
  17. 8 / 25 = .32 8/25=.32
  18. 2 m 2^{m}
  19. x ¯ \overline{x}
  20. 2 ¯ \overline{2}
  21. 1 ¯ \overline{1}
  22. 3 ¯ \overline{3}
  23. 1 ¯ \overline{1}
  24. 2 ¯ \overline{2}
  25. 1 ¯ \overline{1}
  26. 2 ¯ \overline{2}
  27. 3 ¯ \overline{3}
  28. 2 ¯ \overline{2}
  29. 3 ¯ \overline{3}
  30. 2 ¯ \overline{2}
  31. 3 ¯ \overline{3}
  32. 1 ¯ \overline{1}
  33. 3 ¯ \overline{3}
  34. 1 ¯ \overline{1}
  35. 3 ¯ \overline{3}
  36. 1 ¯ \overline{1}
  37. 2 ¯ \overline{2}
  38. 3 ¯ \overline{3}
  39. 1 ¯ \overline{1}
  40. 3 ¯ \overline{3}
  41. 2 ¯ \overline{2}
  42. 1 ¯ \overline{1}
  43. 2 ¯ \overline{2}
  44. 3 ¯ \overline{3}
  45. 1 ¯ \overline{1}
  46. 2 ¯ \overline{2}
  47. 2 ¯ \overline{2}
  48. 1 ¯ \overline{1}
  49. 3 ¯ \overline{3}
  50. 1 ¯ \overline{1}
  51. 2 ¯ \overline{2}
  52. 3 ¯ \overline{3}
  53. 2 ¯ \overline{2}
  54. 3 ¯ \overline{3}
  55. 1 ¯ \overline{1}
  56. 3 ¯ \overline{3}
  57. 1 ¯ \overline{1}
  58. 2 ¯ \overline{2}
  59. 2 ¯ \overline{2}
  60. 2 ¯ \overline{2}
  61. 2 ¯ \overline{2}
  62. χ 2 \chi^{2}

Analytic_space.html

  1. 𝒪 U \mathcal{O}_{U}
  2. 𝒪 Z = 𝒪 U / Z \mathcal{O}_{Z}=\mathcal{O}_{U}/\mathcal{I}_{Z}
  3. Z \mathcal{I}_{Z}
  4. ( X , 𝒪 X ) (X,\mathcal{O}_{X})
  5. ( U , 𝒪 U ) (U,\mathcal{O}_{U})
  6. 𝒪 \mathcal{O}
  7. 𝒪 \mathcal{O}

Anamorphism.html

  1. h = ana f h=\mathrm{ana}\ f
  2. fin h = F h f \mathrm{fin}\circ h=Fh\circ f
  3. [ ( f ) ] [\!(f)\!]

Andreas_Wirth.html

  1. - -

Andrica's_conjecture.html

  1. p n + 1 - p n < 1 \sqrt{p_{n+1}}-\sqrt{p_{n}}<1
  2. n n
  3. p n p_{n}
  4. g n = p n + 1 - p n g_{n}=p_{n+1}-p_{n}
  5. g n < 2 p n + 1. g_{n}<2\sqrt{p_{n}}+1.
  6. n n
  7. A n = p n + 1 - p n A_{n}=\sqrt{p_{n+1}}-\sqrt{p_{n}}
  8. A n A_{n}
  9. p n + 1 x - p n x = 1 , p_{n+1}^{x}-p_{n}^{x}=1,
  10. p n p_{n}
  11. n = 1 n=1
  12. p n + 1 x - p n x < 1 p_{n+1}^{x}-p_{n}^{x}<1
  13. x < x min . x<x_{\min}.

Angle-resolved_photoemission_spectroscopy.html

  1. E = ω - E k f - ϕ . E=\hbar\omega-E_{k_{f}}-\phi.
  2. k i = k f = 2 m E f sin θ \hbar k_{i\parallel}=\hbar k_{f\parallel}=\sqrt{2mE_{f}}\sin\theta
  3. E k f = E_{k_{f}}=
  4. ω = \hbar\omega=
  5. ϕ = \phi=
  6. k f = \hbar k_{f}=
  7. k i = \hbar k_{i}=
  8. k i k_{i\perp}
  9. k i = 1 2 m ( E f cos 2 θ + V 0 ) k_{i\perp}=\frac{1}{\hbar}\sqrt{2m(E_{f}\cos^{2}\theta+V_{0})}
  10. V 0 V_{0}
  11. ϕ \phi
  12. V 0 V_{0}
  13. E E
  14. 𝐤 i = 𝐤 i + 𝐤 i \mathbf{k}_{i}=\mathbf{k}_{i\parallel}+\mathbf{k}_{i\perp}

Angle_condition.html

  1. 1 + 𝐆 ( s ) = 0 1+\,\textbf{G}(s)=0
  2. 𝐆 ( s ) = 𝐏 ( s ) 𝐐 ( s ) \,\textbf{G}(s)=\frac{\,\textbf{P}(s)}{\,\textbf{Q}(s)}
  3. e j 2 π + 𝐆 ( s ) = 0 e^{j2\pi}+\,\textbf{G}(s)=0
  4. 𝐆 ( s ) = - 1 = e j ( π + 2 k π ) \,\textbf{G}(s)=-1=e^{j(\pi+2k\pi)}
  5. ( k = 0 , 1 , 2 , ) (k=0,1,2,...)
  6. 𝐆 ( s ) \,\textbf{G}(s)
  7. 𝐆 ( s ) = 𝐏 ( s ) 𝐐 ( s ) = K ( s - a 1 ) ( s - a 2 ) ( s - a n ) ( s - b 1 ) ( s - b 2 ) ( s - b m ) \,\textbf{G}(s)=\frac{\,\textbf{P}(s)}{\,\textbf{Q}(s)}=K\frac{(s-a_{1})(s-a_{% 2})\cdots(s-a_{n})}{(s-b_{1})(s-b_{2})\cdots(s-b_{m})}
  8. ( s - a p ) (s-a_{p})
  9. ( s - b q ) (s-b_{q})
  10. A p e j θ p A_{p}e^{j\theta_{p}}
  11. B q e j ϕ q B_{q}e^{j\phi_{q}}
  12. 𝐆 ( s ) \,\textbf{G}(s)
  13. 𝐆 ( s ) = K A 1 A 2 A n e j ( θ 1 + θ 2 + + θ n ) B 1 B 2 B m e j ( ϕ 1 + ϕ 2 + + ϕ m ) \,\textbf{G}(s)=K\frac{A_{1}A_{2}\cdots A_{n}e^{j(\theta_{1}+\theta_{2}+\cdots% +\theta_{n})}}{B_{1}B_{2}\cdots B_{m}e^{j(\phi_{1}+\phi_{2}+\cdots+\phi_{m})}}
  14. e j ( π + 2 k π ) = K A 1 A 2 A n e j ( θ 1 + θ 2 + + θ n ) B 1 B 2 B m e j ( ϕ 1 + ϕ 2 + + ϕ m ) = K A 1 A 2 A n B 1 B 2 B m e j ( θ 1 + θ 2 + + θ n - ( ϕ 1 + ϕ 2 + + ϕ m ) ) e^{j(\pi+2k\pi)}=K\frac{A_{1}A_{2}\cdots A_{n}e^{j(\theta_{1}+\theta_{2}+% \cdots+\theta_{n})}}{B_{1}B_{2}\cdots B_{m}e^{j(\phi_{1}+\phi_{2}+\cdots+\phi_% {m})}}=K\frac{A_{1}A_{2}\cdots A_{n}}{B_{1}B_{2}\cdots B_{m}}e^{j(\theta_{1}+% \theta_{2}+\cdots+\theta_{n}-(\phi_{1}+\phi_{2}+\cdots+\phi_{m}))}
  15. π + 2 k π = θ 1 + θ 2 + + θ n - ( ϕ 1 + ϕ 2 + + ϕ m ) \pi+2k\pi=\theta_{1}+\theta_{2}+\cdots+\theta_{n}-(\phi_{1}+\phi_{2}+\cdots+% \phi_{m})
  16. ( k = 0 , 1 , 2 , ) (k=0,1,2,...)
  17. θ 1 , θ 2 , θ n \theta_{1},\theta_{2},\cdots\theta_{n}
  18. ϕ 1 , ϕ 2 , ϕ m \phi_{1},\phi_{2},\cdots\phi_{m}

Angstrom_exponent.html

  1. τ λ τ λ 0 = ( λ λ 0 ) - α \frac{\tau_{\lambda}}{\tau_{\lambda_{0}}}=\left(\frac{\lambda}{\lambda_{0}}% \right)^{-\alpha}
  2. τ λ \tau_{\lambda}
  3. λ \lambda
  4. τ λ 0 \tau_{\lambda_{0}}
  5. λ 0 \lambda_{0}
  6. τ λ 1 \tau_{\lambda_{1}}\,
  7. τ λ 2 \tau_{\lambda_{2}}\,
  8. λ 1 \lambda_{1}\,
  9. λ 2 \lambda_{2}\,
  10. α = - log τ λ 1 τ λ 2 log λ 1 λ 2 \alpha=-\frac{\log\frac{\tau_{\lambda_{1}}}{\tau_{\lambda_{2}}}}{\log\frac{% \lambda_{1}}{\lambda_{2}}}\,

Angular_eccentricity.html

  1. α = sin - 1 e = cos - 1 ( b a ) . \alpha=\sin^{-1}e=\cos^{-1}\left(\frac{b}{a}\right).\,\!
  2. e e
  3. a 2 - b 2 a \frac{\sqrt{a^{2}-b^{2}}}{a}
  4. sin α \sin\alpha
  5. e e^{\prime}
  6. a 2 - b 2 b \frac{\sqrt{a^{2}-b^{2}}}{b}
  7. tan α \tan\alpha
  8. e ′′ e^{\prime\prime}
  9. a 2 - b 2 a 2 + b 2 \sqrt{\frac{a^{2}-b^{2}}{a^{2}+b^{2}}}
  10. sin α 2 - sin 2 α \frac{\sin\alpha}{\sqrt{2-\sin^{2}\alpha}}
  11. f f
  12. a - b a \frac{a-b}{a}
  13. 1 - cos α 1-\cos\alpha
  14. = 2 sin 2 ( α 2 ) =2\sin^{2}\left(\frac{\alpha}{2}\right)
  15. f f^{\prime}
  16. a - b b \frac{a-b}{b}
  17. sec α - 1 \sec\alpha-1
  18. = 2 sin 2 ( α 2 ) 1 - 2 sin 2 ( α 2 ) =\frac{2\sin^{2}(\frac{\alpha}{2})}{1-2\sin^{2}(\frac{\alpha}{2})}
  19. n n
  20. a - b a + b \frac{a-b}{a+b}
  21. 1 - cos α 1 + cos α \frac{1-\cos\alpha}{1+\cos\alpha}
  22. = tan 2 ( α 2 ) =\tan^{2}\left(\frac{\alpha}{2}\right)

Anonymous_function.html

  1. ( ( ( 1 × 2 ) × 3 ) × 4 ) × 5 = 120. \left(\left(\left(1\times 2\right)\times 3\right)\times 4\right)\times 5=120.

Answer-seizure_ratio.html

  1. A S R = 100 a n s w e r e d c a l l s t o t a l c a l l s ASR=100\ \frac{answered\ calls}{totalcalls}

Antiderivative_(complex_analysis).html

  1. U U
  2. g : U , g:U\to\mathbb{C},
  3. g g
  4. f : U f:U\to\mathbb{C}
  5. d f d z = g \frac{df}{dz}=g
  6. U U
  7. U U
  8. g : U g:U\to\mathbb{C}
  9. U U
  10. γ g ( ζ ) d ζ = f ( b ) - f ( a ) . \int_{\gamma}g(\zeta)d\zeta=f(b)-f(a).
  11. γ g ( ζ ) d ζ = 0 , \oint_{\gamma}g(\zeta)d\zeta=0,
  12. γ g ( ζ ) d ζ \oint_{\gamma}g(\zeta)d\zeta
  13. γ g ( z ) d z = a b g ( γ ( t ) ) γ ( t ) d t = a b f ( γ ( t ) ) γ ( t ) d t . \int_{\gamma}g(z)\,dz=\int_{a}^{b}g(\gamma(t))\gamma^{\prime}(t)\,dt=\int_{a}^% {b}f^{\prime}(\gamma(t))\gamma^{\prime}(t)\,dt.
  14. γ g ( z ) d z = a b d d t f ( γ ( t ) ) d t = f ( γ ( b ) ) - f ( γ ( a ) ) . \int_{\gamma}g(z)\,dz=\int_{a}^{b}\frac{d}{dt}f\left(\gamma(t)\right)\,dt=f% \left(\gamma(b)\right)-f\left(\gamma(a)\right).
  15. f ( z ) = γ g ( ζ ) d ζ f(z)=\int_{\gamma}\!g(\zeta)\,d\zeta
  16. | f ( w ) - f ( z ) w - z - g ( z ) | = | z w g ( ζ ) d ζ w - z - z w g ( z ) d ζ w - z | z w | g ( ζ ) - g ( z ) | | w - z | d ζ max ζ [ w , z ] | g ( ζ ) - g ( z ) | , \begin{aligned}\displaystyle\left|\frac{f(w)-f(z)}{w-z}-g(z)\right|&% \displaystyle=\left|\int_{z}^{w}\frac{g(\zeta)d\zeta}{w-z}-\int_{z}^{w}\frac{g% (z)d\zeta}{w-z}\right|\\ &\displaystyle\leq\int_{z}^{w}\frac{|g(\zeta)-g(z)|}{|w-z|}d\zeta\\ &\displaystyle\leq\max_{\zeta\in[w,z]}|g(\zeta)-g(z)|,\end{aligned}

Apartness_relation.html

  1. ¬ ( x # x ) \neg\;(x\#x)
  2. x # y y # x x\#y\;\to\;y\#x
  3. x # y ( x # z y # z ) x\#y\;\to\;(x\#z\;\vee\;y\#z)
  4. ¬ ( x # y ) x = y \neg\;(x\#y)\;\to\;x=y
  5. x , y : A . f ( x ) # B f ( y ) \Rarr x # A y \forall x,\,y:A.\,f(x)\;\#_{B}\;f(y)\Rarr x\;\#_{A}\;y

Arc_(projective_geometry).html

  1. π \pi

Archie's_law.html

  1. C t = 1 a C w ϕ m S w n C_{t}=\frac{1}{a}C_{w}\phi^{m}S_{w}^{n}
  2. ϕ \phi\,\!
  3. C t C_{t}
  4. C w C_{w}
  5. S w S_{w}
  6. m m
  7. n n
  8. a a
  9. R t = a ϕ - m S w - n R w R_{t}=a\phi^{-m}S_{w}^{-n}R_{w}
  10. R t R_{t}
  11. R w R_{w}
  12. F = a ϕ m = R o R w F=\frac{a}{\phi^{m}}=\frac{R_{o}}{R_{w}}
  13. R o R_{o}
  14. S w = 1 S_{w}=1
  15. I = R t R o = S w - n I=\frac{R_{t}}{R_{o}}=S_{w}^{-n}
  16. 1 - S w 1-S_{w}
  17. m m
  18. m m
  19. n n
  20. n n
  21. a a
  22. a a
  23. S w = 1 S_{w}=1
  24. log C t = log C w + m log ϕ \log{C_{t}}=\log{C_{w}}+m\log{\phi}\,\!
  25. m m

Area-to-area_Lee_model.html

  1. L = L 0 + γ log d - 10 log F A L\;=\;L_{0}\;+\;\gamma\log d\;-10\log{F_{A}}
  2. γ \gamma\;
  3. L 0 = G B + G M + 20 ( log λ - log d ) - 22 L_{0}\;=\;G_{B}\;+\;G_{M}\;+\;20\;(\log\lambda\;-\;\log d)\;-\;22
  4. λ \lambda
  5. F A = F B H F B G F M H F M G F F F_{A}\;=\;F_{BH}\;F_{BG}\;F_{MH}\;F_{MG}\;F_{F}
  6. F B H = ( h B 30.48 ) 2 F_{BH}\;=\;\left(\;\frac{h_{B}}{30.48}\;\right)^{2}
  7. F B H = ( h B 100 ) 2 F_{BH}\;=\;\left(\;\frac{h_{B}}{100}\;\right)^{2}
  8. F B G = G B 4 F_{BG}\;=\;\frac{G_{B}}{4}
  9. F M H = { h M 3 if, h M > 3 ( h M 3 ) 2 if, h M 3 F_{MH}\;=\;\begin{cases}\;\;\frac{h_{M}}{3}\;\;\;\;\mbox{ if, }~{}h_{M}>3\\ \Big(\frac{h_{M}}{3}\Big)^{2}\mbox{ if, }~{}h_{M}\leq 3\end{cases}
  10. F M G = G M F_{MG}\;=\;G_{M}
  11. F F = ( f 900 ) - n for 2 < n < 3 F_{F}\;=\;\big(\frac{f}{900}\big)^{-n}\mbox{ for }~{}2<n<3

Arf_invariant.html

  1. q ( x , y ) = a x 2 + x y + b y 2 q(x,y)=ax^{2}+xy+by^{2}
  2. a , b a,b
  3. a b ab
  4. q ( x , y ) = a x 2 + x y + b y 2 q^{\prime}(x,y)=a^{\prime}x^{2}+xy+b^{\prime}y^{2}
  5. q ( x , y ) q(x,y)
  6. a b ab
  7. a b a^{\prime}b^{\prime}
  8. u 2 + u u^{2}+u
  9. u u
  10. a b ab
  11. q q
  12. q q
  13. q q
  14. q = q 1 + + q r q=q_{1}+...+q_{r}
  15. q q
  16. q i q_{i}
  17. q q
  18. q q
  19. q q
  20. x y xy
  21. x 2 + x y + y 2 x^{2}+xy+y^{2}
  22. x y xy
  23. M \partial M
  24. 2 \mathbb{Z}_{2}
  25. H k ( M , M ; 2 ) H k - 1 ( M ; 2 ) H_{k}(M,\partial M;\mathbb{Z}_{2})\to H_{k-1}(\partial M;\mathbb{Z}_{2})
  26. H k ( M ; 2 ) H k ( M ; 2 ) H_{k}(\partial M;\mathbb{Z}_{2})\to H_{k}(M;\mathbb{Z}_{2})
  27. M M
  28. λ : H k ( M ; 2 ) × H k ( M ; 2 ) 2 \lambda\colon H_{k}(M;\mathbb{Z}_{2})\times H_{k}(M;\mathbb{Z}_{2})\to\mathbb{% Z}_{2}
  29. 2 \mathbb{Z}_{2}
  30. λ \lambda
  31. μ : H k ( M ; 2 ) 2 \mu\colon H_{k}(M;\mathbb{Z}_{2})\to\mathbb{Z}_{2}
  32. μ ( x + y ) + μ ( x ) + μ ( y ) λ ( x , y ) ( mod 2 ) x , y H k ( M ; 2 ) \mu(x+y)+\mu(x)+\mu(y)\equiv\lambda(x,y)\;\;(\mathop{{\rm mod}}2)\;\forall\,x,% y\in H_{k}(M;\mathbb{Z}_{2})
  33. { x , y } \{x,y\}
  34. H k ( M ; 2 ) H_{k}(M;\mathbb{Z}_{2})
  35. λ ( x , y ) = 1 \lambda(x,y)=1
  36. μ ( x + y ) , μ ( x ) , μ ( y ) \mu(x+y),\mu(x),\mu(y)
  37. H 1 , 1 H^{1,1}
  38. H 0 , 0 H^{0,0}
  39. { x , y } \{x,y\}
  40. λ \lambda
  41. H k ( M ; 2 ) H_{k}(M;\mathbb{Z}_{2})
  42. H 0 , 0 H^{0,0}
  43. H 1 , 1 H^{1,1}
  44. H 0 , 0 H 0 , 0 H 1 , 1 H 1 , 1 H^{0,0}\oplus H^{0,0}\cong H^{1,1}\oplus H^{1,1}
  45. A r f ( H k ( M ; 2 ) ; μ ) Arf(H_{k}(M;\mathbb{Z}_{2});\mu)
  46. H 1 , 1 H^{1,1}
  47. 2 \in\mathbb{Z}_{2}
  48. M M
  49. g g
  50. M \partial M
  51. M M
  52. S m S^{m}
  53. m 4 m\geq 4
  54. m = 3 m=3
  55. m 4 m\geq 4
  56. x 1 , x 2 , , x 2 g - 1 , x 2 g x_{1},x_{2},\dots,x_{2g-1},x_{2g}
  57. H 1 ( M ) = 2 g H_{1}(M)=\mathbb{Z}^{2g}
  58. x i : S 1 M x_{i}:S^{1}\subset M
  59. S 1 M S m S^{1}\subset M\subset S^{m}
  60. S 1 S m S^{1}\subset S^{m}
  61. S 1 S O ( m - 1 ) S^{1}\to SO(m-1)
  62. π 1 ( S O ( m - 1 ) ) 2 \pi_{1}(SO(m-1))\cong\mathbb{Z}_{2}
  63. m 4 m\geq 4
  64. S 1 S^{1}
  65. Ω 1 f r a m e d π m ( S m - 1 ) ( m 4 ) 2 \Omega^{framed}_{1}\cong\pi_{m}(S^{m-1})\,(m\geq 4)\cong\mathbb{Z}_{2}
  66. S 1 S^{1}
  67. μ ( x ) 2 \mu(x)\in\mathbb{Z}_{2}
  68. Φ ( M ) = A r f ( H 1 ( M , M ; 2 ) ; μ ) 2 \Phi(M)=Arf(H_{1}(M,\partial M;\mathbb{Z}_{2});\mu)\in\mathbb{Z}_{2}
  69. π 1 ( S O ( 2 ) ) \pi_{1}(SO(2))\cong\mathbb{Z}
  70. m m
  71. 2 \mathbb{Z}_{2}
  72. m = 3 m=3
  73. Φ ( M ) \Phi(M)
  74. H 1 , 1 H^{1,1}
  75. H 1 , 1 H^{1,1}
  76. T 2 T^{2}
  77. H 1 ( T 2 ; 2 ) H_{1}(T^{2};\mathbb{Z}_{2})
  78. π 1 ( S O ( 3 ) ) \pi_{1}(SO(3))
  79. Ω 2 f r a m e d π m ( S m - 2 ) ( m 4 ) 2 \Omega^{framed}_{2}\cong\pi_{m}(S^{m-2})\,(m\geq 4)\cong\mathbb{Z}_{2}
  80. T 2 T^{2}
  81. ( M 2 , M ) S 3 (M^{2},\partial M)\subset S^{3}
  82. M = K : S 1 S 3 \partial M=K\colon S^{1}\hookrightarrow S^{3}
  83. D 2 D^{2}
  84. x H 1 ( M ; 2 ) x\in H_{1}(M;\mathbb{Z}_{2})
  85. x x
  86. μ ( x ) \mu(x)
  87. S 3 S^{3}
  88. D 4 D^{4}
  89. M M
  90. D 4 D^{4}
  91. S 3 S^{3}
  92. x H 1 ( M , M ) x\in H_{1}(M,\partial M)
  93. M D 4 M\hookrightarrow D^{4}
  94. x x
  95. M M
  96. π 1 ( S O ( 3 ) ) \pi_{1}(SO(3))
  97. μ ( x ) \mu(x)
  98. μ \mu
  99. H 0 , 0 H^{0,0}
  100. 2 \mathbb{Z}_{2}
  101. H 2 k + 1 ( M ; 2 ) H_{2k+1}(M;\mathbb{Z}_{2})
  102. μ \mu
  103. k 0 , 1 , 3 k\neq 0,1,3
  104. x H 2 k + 1 ( M ; 2 ) x\in H_{2k+1}(M;\mathbb{Z}_{2})
  105. x : S 2 k + 1 M x:S^{2k+1}\subset M
  106. μ ( x ) 2 \mu(x)\in\mathbb{Z}_{2}
  107. x x
  108. μ \mu
  109. H 2 k + 1 ( M ; 2 ) H_{2k+1}(M;\mathbb{Z}_{2})
  110. π 4 k + 2 S 2 \pi_{4k+2}^{S}\to\mathbb{Z}_{2}
  111. 4 k + 2 4k+2
  112. ( f , b ) : M X (f,b):M\to X
  113. ( K 2 k + 1 ( M ; 2 ) , μ ) (K_{2k+1}(M;\mathbb{Z}_{2}),\mu)
  114. 2 \mathbb{Z}_{2}
  115. K 2 k + 1 ( M ; 2 ) = k e r ( f * : H 2 k + 1 ( M ; 2 ) H 2 k + 1 ( X ; 2 ) ) K_{2k+1}(M;\mathbb{Z}_{2})=ker(f_{*}:H_{2k+1}(M;\mathbb{Z}_{2})\to H_{2k+1}(X;% \mathbb{Z}_{2}))
  116. λ \lambda
  117. X = S 4 k + 2 X=S^{4k+2}
  118. μ \mu
  119. μ \mu
  120. μ ( x ) \mu(x)
  121. λ ( x , x ) \lambda(x,x)
  122. μ ( x ) = 0 \mu(x)=0
  123. ( f , b ) (f,b)
  124. L 4 k + 2 ( ) = 2 L_{4k+2}(\mathbb{Z})=\mathbb{Z}_{2}

Arf_invariant_of_a_knot.html

  1. V = v i , j V=v_{i,j}
  2. i = 1 g v 2 i - 1 , 2 i - 1 v 2 i , 2 i ( mod 2 ) . \sum\limits^{g}_{i=1}v_{2i-1,2i-1}v_{2i,2i}\;\;(\mathop{{\rm mod}}2).
  3. { a i , b i } , i = 1... g \{a_{i},b_{i}\},i=1...g
  4. A r f ( K ) = i = 1 g l k ( a i , a i + ) l k ( b i , b i + ) ( mod 2 ) . Arf(K)=\sum\limits^{g}_{i=1}lk(a_{i},a_{i}^{+})lk(b_{i},b_{i}^{+})\;\;(\mathop% {{\rm mod}}2).
  5. a + a^{+}
  6. Δ ( t ) = c 0 + c 1 t + + c n t n + + c 0 t 2 n \Delta(t)=c_{0}+c_{1}t+\cdots+c_{n}t^{n}+\cdots+c_{0}t^{2n}
  7. c n - 1 + c n - 3 + + c r c_{n-1}+c_{n-3}+\cdots+c_{r}
  8. \equiv

Arnold's_cat_map.html

  1. 𝕋 2 \mathbb{T}^{2}
  2. 2 / 2 \mathbb{R}^{2}/\mathbb{Z}^{2}
  3. Γ : 𝕋 2 𝕋 2 \Gamma:\mathbb{T}^{2}\to\mathbb{T}^{2}
  4. Γ : ( x , y ) ( 2 x + y , x + y ) mod 1. \Gamma\,:\,(x,y)\to(2x+y,x+y)\bmod 1.
  5. Γ ( [ x y ] ) = [ 2 1 1 1 ] [ x y ] mod 1 = [ 1 1 0 1 ] [ 1 0 1 1 ] [ x y ] mod 1. \Gamma\left(\begin{bmatrix}x\\ y\end{bmatrix}\right)=\begin{bmatrix}2&1\\ 1&1\end{bmatrix}\begin{bmatrix}x\\ y\end{bmatrix}\bmod 1=\begin{bmatrix}1&1\\ 0&1\end{bmatrix}\begin{bmatrix}1&0\\ 1&1\end{bmatrix}\begin{bmatrix}x\\ y\end{bmatrix}\bmod 1.
  6. q t + 1 - 3 q t + q t - 1 = 0 mod N q_{t+1}-3q_{t}+q_{t-1}=0\mod N
  7. p t + 1 = q t + p t mod N p_{t+1}=q_{t}+p_{t}\mod N
  8. q t - 1 = q t - p t mod N q_{t-1}=q_{t}-p_{t}\mod N
  9. p t - 1 = - q t + 2 p t mod N p_{t-1}=-q_{t}+2p_{t}\mod N
  10. n = 0 : T 0 ( x , y ) = Input Image ( x , y ) n = 1 : T 1 ( x , y ) = T 0 ( mod ( 2 x + y , N ) , mod ( x + y , N ) ) n = k : T k ( x , y ) = T k - 1 ( mod ( 2 x + y , N ) , mod ( x + y , N ) ) n = m : Output Image ( x , y ) = T m ( x , y ) \begin{array}[]{rrcl}n=0:&T^{0}(x,y)&=&\mbox{Input Image}~{}(x,y)\\ n=1:&T^{1}(x,y)&=&T^{0}\left(\bmod(2x+y,N),\bmod(x+y,N)\right)\\ &&\vdots\\ n=k:&T^{k}(x,y)&=&T^{k-1}\left(\bmod(2x+y,N),\bmod(x+y,N)\right)\\ &&\vdots\\ n=m:&\mbox{Output Image}~{}(x,y)&=&T^{m}(x,y)\end{array}

Ars_Magna_(Gerolamo_Cardano).html

  1. - q 2 + q 2 4 + p 3 27 3 + - q 2 - q 2 4 + p 3 27 3 , \sqrt[3]{-\frac{q}{2}+\sqrt{\frac{q^{2}}{4}+\frac{p^{3}}{27}}}+\sqrt[3]{-\frac% {q}{2}-\sqrt{\frac{q^{2}}{4}+\frac{p^{3}}{27}}},

Artin_billiard.html

  1. / Γ , \mathbb{H}/\Gamma,
  2. \mathbb{H}
  3. Γ = P S L ( 2 , ) \Gamma=PSL(2,\mathbb{Z})
  4. E = 1 / 4 E=1/4
  5. H ( p , q ) = 1 2 m p i p j g i j ( q ) H(p,q)=\frac{1}{2m}p_{i}p_{j}g^{ij}(q)
  6. q i , i = 1 , 2 q^{i},i=1,2
  7. p i p_{i}
  8. p i = m g i j d q j d t p_{i}=mg_{ij}\frac{dq^{j}}{dt}
  9. d s 2 = g i j ( q ) d q i d q j ds^{2}=g_{ij}(q)\,dq^{i}\,dq^{j}
  10. d s 2 = d y 2 y 2 ds^{2}=\frac{dy^{2}}{y^{2}}
  11. / Γ \mathcal{H}/\Gamma
  12. P S L ( 2 , ) PSL(2,\mathbb{Z})
  13. U = { z H : | z | > 1 , | Re ( z ) | < 1 2 } U=\left\{z\in H:\left|z\right|>1,\,\left|\,\mbox{Re}~{}(z)\,\right|<\frac{1}{2% }\right\}

Artin–Hasse_exponential.html

  1. E p ( x ) = exp ( x + x p p + x p 2 p 2 + x p 3 p 3 + ) . E_{p}(x)=\exp\left(x+\frac{x^{p}}{p}+\frac{x^{p^{2}}}{p^{2}}+\frac{x^{p^{3}}}{% p^{3}}+\cdots\right).
  2. e x = n 1 ( 1 - x n ) - μ ( n ) / n . e^{x}=\prod_{n\geq 1}(1-x^{n})^{-\mu(n)/n}.\,
  3. E p ( x ) = ( p , n ) = 1 ( 1 - x n ) - μ ( n ) / n . E_{p}(x)=\prod_{(p,n)=1}(1-x^{n})^{-\mu(n)/n}.
  4. E p ( x ) = n 0 t p , n n ! x n . E_{p}(x)=\sum_{n\geq 0}\frac{t_{p,n}}{n!}x^{n}.
  5. exp ( H G x [ G : H ] / [ G : H ] ) = n 0 a G , n n ! x n , \exp(\sum_{H\subset G}x^{[G:H]}/[G:H])=\sum_{n\geq 0}\frac{a_{G,n}}{n!}x^{n},
  6. exp ( d | m x d / d ) = n 0 a m , n n ! x n , \exp(\sum_{d|m}x^{d}/d)=\sum_{n\geq 0}\frac{a_{m,n}}{n!}x^{n},
  7. E p ( x ) E_{p}(x)
  8. 𝔽 p ( x ) \mathbb{F}_{p}(x)

Arylsulfatase.html

  1. \rightleftharpoons

Aspherical_space.html

  1. Γ \ G / K \Gamma\backslash G/K
  2. S 2 f * ω = c 1 ( T M ) , f * [ S 2 ] = 0 \int_{S^{2}}f^{*}\omega=\langle c_{1}(TM),f_{*}[S^{2}]\rangle=0
  3. f : S 2 M , f\colon S^{2}\to M,
  4. c 1 ( T M ) c_{1}(TM)

Asset_turnover.html

  1. Asset Turnover = Net Sales Revenue Average Total Assets \mbox{Asset Turnover}~{}=\frac{\mbox{Net Sales Revenue}~{}}{\mbox{Average % Total Assets}~{}}

Asymptotic_theory.html

  1. lim x π ( x ) ln ( x ) x \lim_{x\rightarrow\infty}\frac{\pi(x)\ln(x)}{x}
  2. Z i Z_{i}
  3. i = 1 i=1
  4. n n
  5. n n
  6. i i
  7. n n
  8. y = 1 x y=\frac{1}{x}
  9. y y
  10. x x
  11. φ n + 1 ( x ) = o ( φ n ( x ) ) ( x L ) \varphi_{n+1}(x)=o(\varphi_{n}(x))\ (x\rightarrow L)
  12. n = 0 a n φ n ( x ) \sum_{n=0}^{\infty}a_{n}\varphi_{n}(x)
  13. f ( x ) = n = 0 N a n φ n ( x ) + O ( φ N + 1 ( x ) ) ( x L ) . f(x)=\sum_{n=0}^{N}a_{n}\varphi_{n}(x)+O(\varphi_{N+1}(x))\ (x\rightarrow L).
  14. f ( x ) n = 0 a n φ n ( x ) ( x L ) f(x)\sim\sum_{n=0}^{\infty}a_{n}\varphi_{n}(x)\ (x\rightarrow L)
  15. e x x x 2 π x Γ ( x + 1 ) 1 + 1 12 x + 1 288 x 2 - 139 51840 x 3 - ( x ) \frac{e^{x}}{x^{x}\sqrt{2\pi x}}\Gamma(x+1)\sim 1+\frac{1}{12x}+\frac{1}{288x^% {2}}-\frac{139}{51840x^{3}}-\cdots\ (x\rightarrow\infty)
  16. x e x E 1 ( x ) n = 0 ( - 1 ) n n ! x n ( x ) xe^{x}E_{1}(x)\sim\sum_{n=0}^{\infty}\frac{(-1)^{n}n!}{x^{n}}\ (x\rightarrow\infty)
  17. ζ ( s ) n = 1 N - 1 n - s + N 1 - s s - 1 + N - s m = 1 B 2 m s 2 m - 1 ¯ ( 2 m ) ! N 2 m - 1 \zeta(s)\sim\sum_{n=1}^{N-1}n^{-s}+\frac{N^{1-s}}{s-1}+N^{-s}\sum_{m=1}^{% \infty}\frac{B_{2m}s^{\overline{2m-1}}}{(2m)!N^{2m-1}}
  18. B 2 m B_{2m}
  19. s 2 m - 1 ¯ s^{\overline{2m-1}}
  20. N > | s | N>|s|
  21. π x e x 2 erfc ( x ) = 1 + n = 1 ( - 1 ) n ( 2 n ) ! n ! ( 2 x ) 2 n . \sqrt{\pi}xe^{x^{2}}{\rm erfc}(x)=1+\sum_{n=1}^{\infty}(-1)^{n}\frac{(2n)!}{n!% (2x)^{2n}}.
  22. 1 1 - w = n = 0 w n \frac{1}{1-w}=\sum_{n=0}^{\infty}w^{n}
  23. w 1 w\neq 1
  24. | w | < 1 |w|<1
  25. e - w / t e^{-w/t}
  26. 0 e - w / t 1 - w d w = n = 0 t n + 1 0 e - u u n d u \int_{0}^{\infty}\frac{e^{-w/t}}{1-w}dw=\sum_{n=0}^{\infty}t^{n+1}\int_{0}^{% \infty}e^{-u}u^{n}du
  27. u = w / t u=w/t
  28. e - 1 / t Ei ( 1 t ) = n = 0 n ! t n + 1 e^{-1/t}\;\operatorname{Ei}\left(\frac{1}{t}\right)=\sum_{n=0}^{\infty}n!\;t^{% n+1}
  29. Ei ( 1 / t ) \operatorname{Ei}(1/t)
  30. x = - 1 / t x=-1/t
  31. Ei ( x ) = - E 1 ( - x ) \operatorname{Ei}(x)=-E_{1}(-x)

Atmospheric_model.html

  1. z z
  2. z z

Atmospheric_models.html

  1. ρ = M P R T \rho=\frac{MP}{RT}
  2. P A - ( P + d P ) A - ( ρ A d h ) g 0 = 0 PA-(P+\,\text{d}P)A-(\rho A\,\text{d}h)g_{0}=0\,
  3. d P = - g 0 ρ d h \,\text{d}P=-g_{0}\rho\,\text{d}h\,
  4. H = R T M g 0 H=\frac{RT}{Mg_{0}}
  5. ( 1 - 1 e ) (1-\frac{1}{e})
  6. g ( z ) = G m e ( r e + z ) 2 g(z)=\frac{Gm_{e}}{(r_{e}+z)^{2}}
  7. h = r e z r e + z h=\frac{r_{e}z}{r_{e}+z}
  8. g ( z ) d z = g 0 d h \frac{}{}g(z)dz=g_{0}dh
  9. g 0 = g ( 0 ) = G m e r e 2 g_{0}=g(0)=\frac{Gm_{e}}{{r_{e}}^{2}}

Atmospheric_sounding.html

  1. x \vec{x}
  2. y \vec{y}
  3. y = f ( x ) \vec{y}=\vec{f}(\vec{x})
  4. f \vec{f}
  5. f \vec{f}
  6. f - 1 \vec{f}^{-1}
  7. { x : y } \{\vec{x}:\vec{y}\}

Atmospheric_thermodynamics.html

  1. e s ( T ) = 6.1094 exp ( 17.625 T T + 243.04 ) e_{s}(T)=6.1094\exp\left(\frac{17.625T}{T+243.04}\right)
  2. e s ( T ) e_{s}(T)
  3. T T

Attack_rate.html

  1. number of new cases in the population at risk number of persons at risk in the population = Rate \frac{\mbox{number of new cases in the population at risk}~{}}{\mbox{number of% persons at risk in the population}~{}}=\mbox{Rate}~{}

Attributable_risk.html

  1. E F = N e - N n N e EF=\frac{N_{e}-N_{n}}{N_{e}}
  2. Combined PAR = 1 - ( 1 - PAR 1 ) ( 1 - PAR 2 ) ( 1 - PAR 3 ) . \,\text{Combined PAR}=1-(1-\,\text{PAR}_{1})(1-\,\text{PAR}_{2})(1-\,\text{PAR% }_{3})\cdots.\,

Attribute-value_system.html

  1. P 1 P_{1}
  2. P 1 P_{1}
  3. P 2 P_{2}
  4. P 3 P_{3}
  5. P 4 P_{4}
  6. P 5 P_{5}
  7. O 1 O_{1}
  8. O 2 O_{2}
  9. O 3 O_{3}
  10. O 4 O_{4}
  11. O 5 O_{5}
  12. O 6 O_{6}
  13. O 7 O_{7}
  14. O 8 O_{8}
  15. O 9 O_{9}
  16. O 10 O_{10}

Auction_theory.html

  1. B ( v ) = ( n - 1 n ) v B(v)=\left(\frac{n-1}{n}\right)v
  2. B ( x ) = ( n - 1 n ) x < b B(x)=\left(\frac{n-1}{n}\right)x<b
  3. x < ( n n - 1 ) b x<\left(\frac{n}{n-1}\right)b
  4. w ( b ) = Pr { b 2 < b } n - 1 = ( n n - 1 ) n - 1 b n - 1 w(b)=\Pr{{\{{{b}_{2}}<b\}}^{n-1}}={{\left(\frac{n}{n-1}\right)}^{n-1}}{{b}^{n-% 1}}
  5. U ( b ) = w ( b ) ( v - b ) = ( n n - 1 ) n - 1 b n - 1 ( v - b ) = ( n n - 1 ) n - 1 ( b n - 1 v - b n ) U(b)=w(b)(v-b)={{\left(\frac{n}{n-1}\right)}^{n-1}}{{b}^{n-1}}(v-b)={{\left(% \frac{n}{n-1}\right)}^{n-1}}({{b}^{n-1}}v-{{b}^{n}})
  6. B ( v ) = ( n - 1 n ) v B(v)=\left(\frac{n-1}{n}\right)v
  7. w = Pr { x < v } F ( v ) w=\Pr\{x<v\}\equiv F(v)
  8. C ( v ) = 0 v < m t p l > x f ( x ) d x C(v)=\int\limits_{0}^{v}<mtpl>{{}}xf(x)dx
  9. e ( v ) = C ( v ) F ( v ) = 0 v < m t p l > x f ( x ) d x F ( v ) e(v)=\frac{C(v)}{F(v)}=\frac{\int\limits_{0}^{v}<mtpl>{{}}xf(x)dx}{F(v)}
  10. e ( v ) F ( v ) + e ( v ) f ( v ) = v f ( v ) {e}^{\prime}(v)F(v)+e(v)f(v)=vf(v)
  11. e ( v ) = f ( v ) F ( v ) ( v - e ( v ) ) {e}^{\prime}(v)=\frac{f(v)}{F(v)}(v-e(v))
  12. w = Pr { b < B ( y ) } = Pr { B ( x ) < B ( y ) } = Pr { x < y } = F ( v ) w=\Pr\{b<B(y)\}=\Pr\{B(x)<B(y)\}=\Pr\{x<y\}=F(v)
  13. U = w ( v - B ( x ) ) = F ( x ) ( v - B ( x ) ) U=w(v-B(x))=F(x)(v-B(x))
  14. U ( x ) = f ( x ) ( v - B ( x ) ) - F ( x ) B ( x ) = F ( x ) ( f ( x ) F ( x ) ( v - B ( x ) ) - B ( x ) ) = 0 {U}^{\prime}(x)=f(x)(v-B(x))-F(x){B}^{\prime}(x)=F(x)\left(\frac{f(x)}{F(x)}(v% -B(x))-{B}^{\prime}(x)\right)=0
  15. f ( v ) F ( v ) ( v - B ( v ) ) - B ( v ) = 0 \frac{f(v)}{F(v)}(v-B(v))-{B}^{\prime}(v)=0

Audio_bit_depth.html

  1. ± 1 2 \scriptstyle{\pm\frac{1}{2}}
  2. SQNR = 20 log 10 ( 2 Q ) 6.02 Q dB \mathrm{SQNR}=20\log_{10}(2^{Q})\approx 6.02\cdot Q\ \mathrm{dB}\,\!
  3. number of samples = ( 2 n ) 2 = 2 2 n . \mathrm{number\ of\ samples}=(2^{n})^{2}=2^{2n}.

Autler–Townes_effect.html

  1. H = i 1 2 m i [ 𝐩 i - q i c 𝐀 ( 𝐫 𝐢 , 𝐭 ) ] 2 + V ( 𝐫 i ) , H=\sum_{i}\frac{1}{2m_{i}}\left[\mathbf{p}_{i}-\frac{q_{i}}{c}\mathbf{A(% \mathbf{r}_{i},t)}\right]^{2}+V(\mathbf{r}_{i}),\ \ \ \ \ \ \ \ \ \
  2. 𝐫 i \mathbf{r}_{i}\,
  3. 𝐩 i \mathbf{p}_{i}\,
  4. m i m_{i}\,
  5. q i q_{i}\,
  6. i i\,
  7. c c\,
  8. 𝐀 \mathbf{A}
  9. 𝐀 ( t + τ ) = 𝐀 ( t ) \mathbf{A}(t+\tau)=\mathbf{A}(t)
  10. H ( t + τ ) = H ( t ) . H(t+\tau)=H(t).
  11. i t ψ ( ξ , t ) = H ( t ) ψ ( ξ , t ) , i\hbar\frac{\partial}{\partial t}\psi(\mathbf{\xi},t)=H(t)\psi(\mathbf{\xi},t),
  12. ξ \xi
  13. ψ ( ξ , t ) = exp [ - i E b t / ] ϕ ( ξ , t ) . \psi(\mathbf{\xi},t)=\exp[-iE_{b}t/\hbar]\phi(\mathbf{\xi},t).
  14. E b E_{b}
  15. ϕ \phi\,
  16. ϕ ( ξ , t + τ ) = ϕ ( ξ , t ) \phi(\mathbf{\xi},t+\tau)=\phi(\mathbf{\xi},t)
  17. ϕ ( ξ , t + 2 π / ω ) = ϕ ( ξ , t ) \phi(\mathbf{\xi},t+2\pi/\omega)=\phi(\mathbf{\xi},t)
  18. ω = 2 π / τ \omega=2\pi/\tau
  19. ϕ ( ξ , t ) \phi(\mathbf{\xi},t)
  20. ψ ( ξ , t ) = exp [ - i E b t / ] k = - C k ( ξ ) exp [ - i k ω t ] \psi(\mathbf{\xi},t)=\exp[-iE_{b}t/\hbar]\sum_{k=-\infty}^{\infty}C_{k}(% \mathbf{\xi})\exp[-ik\omega t]
  21. ψ ( ξ , t ) = k = - C k ( ξ ) exp [ - i ( E b + k ω ) t / ] \psi(\mathbf{\xi},t)=\sum_{k=-\infty}^{\infty}C_{k}(\mathbf{\xi})\exp[-i(E_{b}% +k\omega)t/\hbar]\ \ \ \ \ \ \ \ \ \
  22. ω = 2 π / T \omega=2\pi/T\,
  23. E b + k ω E_{b}+k\omega
  24. ω 0 \omega_{0}
  25. Δ ω 0 \Delta<<\omega_{0}
  26. H ^ = H ^ A + H ^ i n t . \hat{H}=\hat{H}_{A}+\hat{H}_{int}.
  27. H ^ \hat{H}
  28. H ^ = - Δ | e e | + Ω 2 ( | e g | + | g e | ) . \hat{H}=-\hbar\Delta|e\rangle\langle e|+\frac{\hbar\Omega}{2}(|e\rangle\langle g% |+|g\rangle\langle e|).
  29. Ω \Omega
  30. | g , | e |g\rangle,|e\rangle
  31. E ± = - Δ 2 ± Ω 2 + Δ 2 2 E_{\pm}=\frac{-\hbar\Delta}{2}\pm\frac{\hbar\sqrt{\Omega^{2}+\Delta^{2}}}{2}
  32. E ± ± Ω 2 . E_{\pm}\approx\pm\frac{\hbar\Omega}{2}.
  33. | + |+\rangle
  34. | - |-\rangle
  35. | + |+\rangle
  36. | - |-\rangle
  37. Ω \hbar\Omega
  38. Ω \Omega
  39. | g |g\rangle
  40. | e |e\rangle
  41. | e |e\rangle
  42. | g |g\rangle
  43. | g |g\rangle

Autocorrelator.html

  1. E ( t ) E\left(t\right)
  2. D F D_{F}
  3. D V ( t ) D_{V}\left(t\right)
  4. E ( t - D F / c ) + E ( t - D V ( t ) / c ) E\left(t-D_{F}/c\right)+E\left(t-D_{V}\left(t\right)/c\right)
  5. c c
  6. D / c D/c
  7. E 2 ( t - D F / c ) + E 2 ( t - D V ( t ) / c ) + 2 E ( t - D F / c ) E ( t - D V ( t ) / c ) E^{2}\left(t-D_{F}/c\right)+E^{2}\left(t-D_{V}\left(t\right)/c\right)+2E\left(% t-D_{F}/c\right)E\left(t-D_{V}\left(t\right)/c\right)
  8. E 2 ( t ) + E 2 ( t - A ( t ) ) + 2 E ( t ) E ( t - A ( t ) ) where A ( t ) = ( D V ( t ) - D F ) / c and t = t - D F / c E^{2}\left(t^{{}^{\prime}}\right)+E^{2}\left(t^{{}^{\prime}}-A\left(t\right)% \right)+2E\left(t^{{}^{\prime}}\right)E\left(t^{{}^{\prime}}-A\left(t\right)% \right)\,\mbox{ where }~{}\,A\left(t\right)=\left(D_{V}\left(t\right)-D_{F}% \right)/c\,\mbox{ and }~{}\,t^{{}^{\prime}}=t-D_{F}/c
  9. E ( t ) E\left(t\right)
  10. S ( t ) = E 2 ( t ) + E 2 ( t - A ( t ) ) + 2 E ( t ) E ( t - A ( t ) ) d t S\left(t\right)=\int E^{2}\left(t^{{}^{\prime}}\right)+E^{2}\left(t^{{}^{% \prime}}-A\left(t\right)\right)+2E\left(t^{{}^{\prime}}\right)E\left(t^{{}^{% \prime}}-A\left(t\right)\right)\,dt^{{}^{\prime}}
  11. k F + k V = k S H G k_{F}+k_{V}=k_{SHG}
  12. S ( t ) S\left(t\right)
  13. S ( t ) = E ( t ) E ( t - A ( t ) ) d t S\left(t\right)=\int E\left(t^{{}^{\prime}}\right)E\left(t^{{}^{\prime}}-A% \left(t\right)\right)\,dt^{{}^{\prime}}
  14. A ( t ) A\left(t\right)

Automatic_sequence.html

  1. e E e\in E
  2. ϕ ( e , s ) = ϕ ( ϕ ( e , s 1 s 2 s t - 1 ) , s t ) . \phi(e,s)=\phi(\phi(e,s_{1}s_{2}...s_{t-1}),s_{t})\,.
  3. m ( n ) = π ( ϕ ( e , s ( n ) ) ) , m(n)=\pi(\phi(e,s(n)))\,,
  4. σ ( E ) E k \sigma(E)\subseteq E^{k}
  5. e E e\in E
  6. f u ( z ) = n j ( u ( n ) ) z n . f_{u}(z)=\sum_{n}j(u(n))z^{n}\ .
  7. ( 1 + z ) 3 T 2 + ( 1 + z ) 2 T + z = 0 (1+z)^{3}T^{2}+(1+z)^{2}T+z=0

Automatically_Tuned_Linear_Algebra_Software.html

  1. 𝐲 α 𝐱 + 𝐲 \mathbf{y}\leftarrow\alpha\mathbf{x}+\mathbf{y}\!
  2. 𝐲 α A 𝐱 + β 𝐲 \mathbf{y}\leftarrow\alpha A\mathbf{x}+\beta\mathbf{y}\!
  3. T 𝐱 = 𝐲 T\mathbf{x}=\mathbf{y}
  4. 𝐱 \mathbf{x}
  5. T T
  6. C α A B + β C C\leftarrow\alpha AB+\beta C\!
  7. B α T - 1 B B\leftarrow\alpha T^{-1}B
  8. T T
  9. N 2 N^{2}
  10. N 2 N^{2}
  11. 𝐲 α A 𝐱 + β 𝐲 \mathbf{y}\leftarrow\alpha A\mathbf{x}+\beta\mathbf{y}\!
  12. A α 𝐱𝐲 T + A A\leftarrow\alpha\mathbf{x}\mathbf{y}^{T}+A\!
  13. N 3 N^{3}
  14. N 2 N^{2}
  15. O ( n 3 ) O(n^{3})
  16. O ( n 2 ) O(n^{2})
  17. n 3 n^{3}
  18. n 2 n^{2}
  19. M p + N B K p + N B N B M\cdot p+NB\cdot Kp+NB\cdot NB
  20. 2 K p N B + N B N B 2\cdot Kp\cdot NB+NB\cdot NB

Available_energy_(particle_collision).html

  1. E a = 2 E t E k + ( m t c 2 ) 2 + ( m k c 2 ) 2 E_{a}=\sqrt{2E_{t}E_{k}+(m_{t}c^{2})^{2}+(m_{k}c^{2})^{2}}
  2. E t E_{t}
  3. E k E_{k}
  4. m t m_{t}
  5. m k m_{k}
  6. c c
  7. ( m c 2 ) 2 = E 2 - P 2 c 2 (mc^{2})^{2}=E^{2}-P^{2}c^{2}
  8. P a = P k P_{a}=P_{k}
  9. P a P_{a}
  10. P k P_{k}
  11. E T = E t + E k E_{T}=E_{t}+E_{k}
  12. E T E_{T}
  13. ( E a ) 2 = ( E T ) 2 - ( P a ) 2 c 2 (E_{a})^{2}=(E_{T})^{2}-(P_{a})^{2}c^{2}
  14. ( E a ) 2 = ( E t + E k ) 2 - ( P k ) 2 c 2 (E_{a})^{2}=(E_{t}+E_{k})^{2}-(P_{k})^{2}c^{2}
  15. ( E a ) 2 = ( E t ) 2 + ( E k ) 2 + 2 E t E k - ( P k ) 2 c 2 (E_{a})^{2}=(E_{t})^{2}+(E_{k})^{2}+2E_{t}E_{k}-(P_{k})^{2}c^{2}
  16. ( m k ) 2 c 4 = ( E k ) 2 - ( P k ) 2 c 2 (m_{k})^{2}c^{4}=(E_{k})^{2}-(P_{k})^{2}c^{2}
  17. ( m t ) 2 c 4 = ( E t ) 2 (m_{t})^{2}c^{4}=(E_{t})^{2}
  18. ( E a ) 2 = ( m k ) 2 c 4 + ( m t ) 2 c 4 + 2 E t E k (E_{a})^{2}=(m_{k})^{2}c^{4}+(m_{t})^{2}c^{4}+2E_{t}E_{k}
  19. E a = ( m t c 2 ) 2 + ( m k c 2 ) 2 + 2 E t E k E_{a}=\sqrt{(m_{t}c^{2})^{2}+(m_{k}c^{2})^{2}+2E_{t}E_{k}}

Avidity.html

  1. [ A b ] + [ A g ] [ A b A g ] [Ab]+[Ag]\leftrightharpoons[AbAg]
  2. K a = k o n k o f f = [ A b A g ] [ A b ] [ A g ] K_{a}=\frac{k_{on}}{k_{off}}=\frac{[AbAg]}{[Ab][Ag]}
  3. K d = k o f f k o n = [ A b ] [ A g ] [ A b A g ] K_{d}=\frac{k_{off}}{k_{on}}=\frac{[Ab][Ag]}{[AbAg]}

Avrami_equation.html

  1. α \alpha\,\!
  2. β \beta\,\!
  3. N ˙ \dot{N}\,\!
  4. G ˙ \dot{G}\,\!
  5. 0 < τ < t 0<\tau<t\,\!
  6. N = V N ˙ d τ N=V\dot{N}d\tau\,\!
  7. G ˙ ( t - τ ) \dot{G}(t-\tau)
  8. β \beta
  9. d V β e = 4 π 3 G ˙ 3 ( t - τ ) 3 V N ˙ d τ dV_{\beta}^{e}=\frac{4\pi}{3}\dot{G}^{3}(t-\tau)^{3}V\dot{N}d\tau\,\!
  10. τ = 0 \tau=0
  11. τ = t \tau=t
  12. V β e = π 3 V N ˙ G ˙ 3 t 4 V_{\beta}^{e}=\frac{\pi}{3}V\dot{N}\dot{G}^{3}t^{4}\,\!
  13. α \alpha
  14. d V β = d V β e ( 1 - V β / V ) dV_{\beta}=dV_{\beta}^{e}(1-V_{\beta}/V)\,\!
  15. 1 1 - V β / V d V β = d V β e \frac{1}{1-V_{\beta}/V}dV_{\beta}=dV_{\beta}^{e}\,\!
  16. ln ( 1 - Y ) = - V β e / V \ln(1-Y)=-V_{\beta}^{e}/V\,\!
  17. β \beta
  18. V β / V V_{\beta}/V
  19. Y = 1 - exp ( - K t n ) Y=1-\exp(-Kt^{n})\,\!
  20. K = π N ˙ G ˙ 3 / 3 K=\pi\dot{N}\dot{G}^{3}/3\,\!
  21. n = 4 n=4\,\!
  22. ln ( - ln [ 1 - Y ( t ) ] ) = ln K + n ln t \ln\,{(-\ln{[1-Y(t)]})}=\ln K+n\ln t\,\!

Axilrod–Teller_potential.html

  1. V i j k = E 0 [ 1 + 3 cos γ i cos γ j cos γ k ( r i j r j k r i k ) 3 ] V_{ijk}=E_{0}\left[\frac{1+3\cos\gamma_{i}\cos\gamma_{j}\cos\gamma_{k}}{\left(% r_{ij}r_{jk}r_{ik}\right)^{3}}\right]
  2. r i j r_{ij}
  3. i i
  4. j j
  5. γ i \gamma_{i}
  6. 𝐫 i j \mathbf{r}_{ij}
  7. 𝐫 i k \mathbf{r}_{ik}
  8. E 0 E_{0}
  9. V α 3 V\alpha^{3}
  10. V V
  11. α \alpha
  12. E 0 E_{0}
  13. p p

Axiom_A.html

  1. n f n ( U ) = Ω ( f ) . \cap_{n\in\mathbb{Z}}f^{n}(U)=\Omega(f).
  2. f ϵ h ( x ) = h f ( x ) , x Ω ( f ) . f_{\epsilon}\circ h(x)=h\circ f(x),\quad\forall x\in\Omega(f).

Axiom_of_limitation_of_size.html

  1. C ( ¬ \exist W ( C W ) \exist F ( x ( \exist W ( x W ) \exist s ( s C and s , x F ) ) and x y s ( ( s , x F and s , y F ) x = y ) ) ) . \forall C\biggl(\lnot\exist W\left(C\in W\right)\iff\exist F\Bigl(\forall x% \bigl(\exist W\left(x\in W\right)\Rightarrow\exist s\left(s\in C\and\langle s,% x\rangle\in F\right)\bigr)\and\forall x\forall y\forall s\bigl(\left(\langle s% ,x\rangle\in F\and\langle s,y\rangle\in F\right)\Rightarrow x=y\bigr)\Bigr)% \biggr).
  2. 0 \aleph_{0}

Ax–Kochen_theorem.html

  1. Y d = Y_{d}=\varnothing

Bach_tensor.html

  1. B a b = P c d W a c b d + c a P b c - c c P a b B_{ab}=P_{cd}{{{W_{a}}^{c}}_{b}}^{d}+\nabla^{c}\nabla_{a}P_{bc}-\nabla^{c}% \nabla_{c}P_{ab}
  2. W W
  3. P P
  4. R a b R_{ab}
  5. R R
  6. P a b = 1 n - 2 ( R a b - R 2 ( n - 1 ) g a b ) . P_{ab}=\frac{1}{n-2}\left(R_{ab}-\frac{R}{2(n-1)}g_{ab}\right).

Backspread.html

  1. K l \displaystyle K_{l}
  2. Maximum Profit = [ K u - 2 × ( K u - K l ) + C n ] × N \,\text{Maximum Profit}=\left[K_{u}-2\times\left(K_{u}-K_{l}\right)+C_{n}% \right]\times N
  3. Maximum Profit (upside) = C n \,\text{Maximum Profit (upside)}=C_{n}\,
  4. Maximum Loss = ( C n + K l - K u ) × N \,\text{Maximum Loss}=\left(C_{n}+K_{l}-K_{u}\right)\times N

Backward_Euler_method.html

  1. d y d t = f ( t , y ) \frac{\mathrm{d}y}{\mathrm{d}t}=f(t,y)
  2. y ( t 0 ) = y 0 . y(t_{0})=y_{0}.
  3. f f
  4. t 0 t_{0}
  5. y 0 y_{0}
  6. y y
  7. t t
  8. y 0 , y 1 , y 2 , y_{0},y_{1},y_{2},\ldots
  9. y k y_{k}
  10. y ( t 0 + k h ) y(t_{0}+kh)
  11. h h
  12. y k + 1 = y k + h f ( t k + 1 , y k + 1 ) . y_{k+1}=y_{k}+hf(t_{k+1},y_{k+1}).
  13. f ( t k , y k ) f(t_{k},y_{k})
  14. f ( t k + 1 , y k + 1 ) f(t_{k+1},y_{k+1})
  15. y k + 1 y_{k+1}
  16. y k + 1 y_{k+1}
  17. y k + 1 [ 0 ] = y k , y k + 1 [ i + 1 ] = y k + h f ( t k + 1 , y k + 1 [ i ] ) . y_{k+1}^{[0]}=y_{k},\quad y_{k+1}^{[i+1]}=y_{k}+hf(t_{k+1},y_{k+1}^{[i]}).
  18. y k + 1 y_{k+1}
  19. d y d t = f ( t , y ) \frac{\mathrm{d}y}{\mathrm{d}t}=f(t,y)
  20. t k t_{k}
  21. t k + 1 = t k + h t_{k+1}=t_{k}+h
  22. y ( t k + 1 ) - y ( t k ) = t k t k + 1 f ( t , y ( t ) ) d t . y(t_{k+1})-y(t_{k})=\int_{t_{k}}^{t_{k+1}}f(t,y(t))\,\mathrm{d}t.
  23. y ( t k + 1 ) - y ( t k ) h f ( t k + 1 , y ( t k + 1 ) ) . y(t_{k+1})-y(t_{k})\approx hf(t_{k+1},y(t_{k+1})).
  24. y k y_{k}
  25. y ( t k ) y(t_{k})
  26. O ( h 2 ) O(h^{2})
  27. t t
  28. O ( h ) O(h)
  29. 1 1 1 \begin{array}[]{c|c}1&1\\ \hline&1\\ \end{array}

Baer_ring.html

  1. X R X\subseteq R
  2. { r R r X = { 0 } } \{r\in R\mid rX=\{0\}\}
  3. * : R R *:R\rightarrow R
  4. { 0 } \{0\}
  5. { 0 } \{0\}

Bagnold_formula.html

  1. q = C ρ g d D u * 3 q=C\ \frac{\rho}{g}\ \sqrt{\frac{d}{D}}u_{*}^{3}
  2. ρ \rho
  3. u * u_{*}

Baire_one_star_function.html

  1. f : f:\mathbb{R}\to\mathbb{R}
  2. f 𝐁 1 * f\in\mathbf{B}^{*}_{1}
  3. P P\in\mathbb{R}
  4. I I\in\mathbb{R}
  5. P I P\cap I
  6. f | P I f|_{P\cap I}

Banach–Mazur_theorem.html

  1. ( X , [ u ! ! ] [ u ! ! ] ) (X,[u^{\prime}!!^{\prime}]⋅[u^{\prime}!!^{\prime}])
  2. X X
  3. K K
  4. j j
  5. X X
  6. C ( K ) C(K)
  7. K K
  8. K K
  9. X X′
  10. K K
  11. j j
  12. x X x∈X
  13. j ( x ) j(x)
  14. K K
  15. x K : j ( x ) ( x ) = x ( x ) . \forall x^{\prime}\in K:\qquad j(x)(x^{\prime})=x^{\prime}(x).
  16. j j
  17. α α
  18. C < s u p > 0 ( 0 , 11 α , 𝐑 ) C<sup>0(0,11^{α},\mathbf{R})

Band_(mathematics).html

  1. ( i , j ) ( k , ) = ( i , ) (i,j)\cdot(k,\ell)=(i,\ell)\,
  2. ( i x , j x ) ( i y , j y ) ( i x , j x ) = ( i x , j x ) (i_{x},j_{x})\cdot(i_{y},j_{y})\cdot(i_{x},j_{x})=(i_{x},j_{x})
  3. Set × Set \mathrm{Set}_{\neq\emptyset}\times\mathrm{Set}_{\neq\emptyset}
  4. Set \mathrm{Set}_{\neq\emptyset}

Band_brake.html

  1. e μ θ e^{\mu\theta}
  2. μ \mu
  3. θ \theta
  4. μ θ \mu\theta
  5. μ \mu
  6. μ \mu
  7. μ \mu
  8. μ \mu

Band_diagram.html

  1. - e ϕ -e\phi
  2. ϕ \phi

Band_emission.html

  1. F 0 , λ = 0 λ E λ , b d λ 0 E λ , b d λ = 0 λ E λ , b d λ σ T 4 = 0 λ T E λ , b σ T 5 d ( λ T ) = f ( λ T ) F_{0,\lambda}=\frac{\int_{0}^{\lambda}E_{\lambda,b}d\lambda}{\int_{0}^{\infty}% E_{\lambda,b}d\lambda}=\frac{\int_{0}^{\lambda}E_{\lambda,b}d\lambda}{\sigma T% ^{4}}=\int_{0}^{\lambda T}\frac{E_{\lambda,b}}{\sigma T^{5}}d(\lambda T)=f(% \lambda T)

Barium_chromate.html

  1. 𝖡𝖺 ( 𝖮𝖧 ) 𝟤 + 𝖪 𝟤 𝖢𝗋𝖮 𝟦 𝖡𝖺𝖢𝗋𝖮 𝟦 + 𝟤 𝖪𝖮𝖧 \mathsf{Ba(OH)_{2}+\ K_{2}CrO_{4}\longrightarrow\ BaCrO_{4}\downarrow+2\ KOH}
  2. 𝟦 𝖡 𝖺 𝖢 𝗋 𝖮 𝟦 + 𝟤 𝖡 𝖺 ( 𝖮𝖧 ) 𝟤 𝖭𝖺𝖭 𝟥 2 𝖡𝖺 𝟥 ( 𝖢𝗋𝖮 𝟦 ) 𝟤 + 𝖮 𝟤 + 𝟤 𝖧 𝟤 𝖮 ~{}\mathsf{4BaCrO_{4}+2Ba(OH)_{2}\xrightarrow{NaN_{3}}\ 2Ba_{3}(CrO_{4})_{2}+O% _{2}\uparrow\ +2H_{2}O\uparrow}

Barrow's_inequality.html

  1. P A + P B + P C 2 ( P U + P V + P W ) , PA+PB+PC\geq 2(PU+PV+PW),\,

Base_flow_(random_dynamical_systems).html

  1. ϑ s : Ω Ω \vartheta_{s}:\Omega\to\Omega
  2. ( Ω , , ) (\Omega,\mathcal{F},\mathbb{P})
  3. ( Ω , , , ϑ ) (\Omega,\mathcal{F},\mathbb{P},\vartheta)
  4. ϑ s \vartheta_{s}
  5. s s
  6. \mathbb{R}
  7. [ 0 , + ) [0,+\infty)\subsetneq\mathbb{R}
  8. \mathbb{Z}
  9. { 0 } \mathbb{N}\cup\{0\}
  10. ϑ s \vartheta_{s}
  11. ( , ) (\mathcal{F},\mathcal{F})
  12. E E\in\mathcal{F}
  13. ϑ s - 1 ( E ) \vartheta_{s}^{-1}(E)\in\mathcal{F}
  14. \mathbb{P}
  15. E E\in\mathcal{F}
  16. ( ϑ s - 1 ( E ) ) = ( E ) \mathbb{P}(\vartheta_{s}^{-1}(E))=\mathbb{P}(E)
  17. ϑ s \vartheta_{s}
  18. ϑ 0 = id Ω : Ω Ω \vartheta_{0}=\mathrm{id}_{\Omega}:\Omega\to\Omega
  19. Ω \Omega
  20. ϑ s ϑ t = ϑ s + t \vartheta_{s}\circ\vartheta_{t}=\vartheta_{s+t}
  21. s s
  22. t t
  23. ϑ s - 1 = ϑ - s \vartheta_{s}^{-1}=\vartheta_{-s}
  24. - s -s
  25. ϑ s \vartheta_{s}
  26. s { 0 } s\in\mathbb{N}\cup\{0\}
  27. s [ 0 , + ) s\in[0,+\infty)
  28. s s\in\mathbb{Z}
  29. s s\in\mathbb{R}
  30. W : × Ω X W:\mathbb{R}\times\Omega\to X
  31. ( Ω , , ) (\Omega,\mathcal{F},\mathbb{P})
  32. ϑ s : Ω Ω \vartheta_{s}:\Omega\to\Omega
  33. W ( t , ϑ s ( ω ) ) = W ( t + s , ω ) - W ( s , ω ) W(t,\vartheta_{s}(\omega))=W(t+s,\omega)-W(s,\omega)
  34. ϑ s \vartheta_{s}
  35. s s

Basic_pitch_count_estimator.html

  1. 3.3 P A + 1.5 S O + 2.2 B B 3.3PA+1.5SO+2.2BB

Bass_diffusion_model.html

  1. f ( t ) 1 - F ( t ) = p + q F ( t ) \frac{f(t)}{1-F(t)}=p+qF(t)
  2. f ( t ) \ f(t)
  3. F ( t ) \ F(t)
  4. p \ p
  5. q \ q
  6. S ( t ) \ S(t)
  7. f ( t ) \ f(t)
  8. m \ m
  9. S ( t ) = m f ( t ) \ S(t)=mf(t)
  10. S ( t ) = m ( p + q ) 2 p e - ( p + q ) t ( 1 + q p e - ( p + q ) t ) 2 \ S(t)=m{\frac{(p+q)^{2}}{p}}\frac{e^{-(p+q)t}}{(1+\frac{q}{p}e^{-(p+q)t})^{2}}
  11. t * \ t^{*}
  12. t * = ln q - ln p p + q \ t^{*}=\frac{\ln q-\ln p}{p+q}
  13. f ( t ) 1 - F ( t ) = ( p + q F ( t ) ) x ( t ) \frac{f(t)}{1-F(t)}=(p+{q}F(t))x(t)
  14. x ( t ) \ x(t)
  15. S 1 , t = F ( t 1 ) m 1 ( 1 - F ( t 2 ) ) \ S_{1,t}=F(t_{1})m_{1}(1-F(t_{2}))
  16. S 2 , t = F ( t 2 ) ( m 2 + F ( t 1 ) m 1 ) ( 1 - F ( t 3 ) ) \ S_{2,t}=F(t_{2})(m_{2}+F(t_{1})m_{1})(1-F(t_{3}))
  17. S 3 , t = F ( t 3 ) ( m 3 + F ( t 2 ) ( m 2 + F ( t 1 ) m 1 ) ) \ S_{3,t}=F(t_{3})(m_{3}+F(t_{2})(m_{2}+F(t_{1})m_{1}))
  18. m i = a i M i \ m_{i}=a_{i}M_{i}
  19. M i \ M_{i}
  20. a i \ a_{i}
  21. t i \ t_{i}
  22. F ( t i ) = 1 - e - ( p + q ) t i 1 + q p e - ( p + q ) t i \ F(t_{i})=\frac{1-e^{-(p+q)t_{i}}}{1+\frac{q}{p}e^{-(p+q)t_{i}}}

Basu's_theorem.html

  1. P θ A ( B ) = P θ ( A - 1 B ) = T ( X ) P θ ( A - 1 B | T = t ) P θ T ( d t ) P_{\theta}^{A}(B)=P_{\theta}(A^{-1}B)=\int_{T(X)}P_{\theta}(A^{-1}B|T=t)\ P_{% \theta}^{T}(dt)\,
  2. T ( X ) [ P ( A - 1 B | T = t ) - P A ( B ) ] P θ T ( d t ) = 0 \int_{T(X)}\big[P(A^{-1}B|T=t)-P^{A}(B)\big]\ P_{\theta}^{T}(dt)=0\,
  3. P ( A - 1 B | T = t ) = P A ( B ) for all t P(A^{-1}B|T=t)=P^{A}(B)\quad\,\text{for all }t\,
  4. μ ^ = X i n , \widehat{\mu}=\frac{\sum X_{i}}{n},\,
  5. σ ^ 2 = ( X i - X ¯ ) 2 n - 1 , \widehat{\sigma}^{2}=\frac{\sum\left(X_{i}-\bar{X}\right)^{2}}{n-1},\,

Bathythermograph.html

  1. z ( t ) = a t 2 + b t z(t)=at^{2}+bt

Batting_average_on_balls_in_play.html

  1. B A B I P = H - H R A B - K - H R + S F BABIP=\frac{H-HR}{AB-K-HR+SF}

Bayes_estimator.html

  1. π \pi
  2. θ ^ = θ ^ ( x ) \widehat{\theta}=\widehat{\theta}(x)
  3. L ( θ , θ ^ ) L(\theta,\widehat{\theta})
  4. θ ^ \widehat{\theta}
  5. E π ( L ( θ , θ ^ ) ) E_{\pi}(L(\theta,\widehat{\theta}))
  6. θ \theta
  7. θ ^ \widehat{\theta}
  8. θ ^ \widehat{\theta}
  9. E ( L ( θ , θ ^ ) | x ) E(L(\theta,\widehat{\theta})|x)
  10. MSE = E [ ( θ ^ ( x ) - θ ) 2 ] , \mathrm{MSE}=E\left[(\widehat{\theta}(x)-\theta)^{2}\right],
  11. θ \theta
  12. x x
  13. θ ^ ( x ) = E [ θ | x ] = θ p ( θ | x ) d θ . \widehat{\theta}(x)=E[\theta|x]=\int\theta p(\theta|x)\,d\theta.
  14. θ ^ ( x ) = σ 2 σ 2 + τ 2 μ + τ 2 σ 2 + τ 2 x . \widehat{\theta}(x)=\frac{\sigma^{2}}{\sigma^{2}+\tau^{2}}\mu+\frac{\tau^{2}}{% \sigma^{2}+\tau^{2}}x.
  15. θ ^ ( X ) = n X ¯ + a n + 1 b . \widehat{\theta}(X)=\frac{n\overline{X}+a}{n+\frac{1}{b}}.
  16. F F
  17. a > 0 a>0
  18. L ( θ , θ ^ ) = a | θ - θ ^ | L(\theta,\widehat{\theta})=a|\theta-\widehat{\theta}|
  19. F ( θ ^ ( x ) | X ) = 1 2 . F(\widehat{\theta}(x)|X)=\tfrac{1}{2}.
  20. a , b > 0 a,b>0
  21. L ( θ , θ ^ ) = { a | θ - θ ^ | , for θ - θ ^ 0 b | θ - θ ^ | , for θ - θ ^ < 0 L(\theta,\widehat{\theta})=\begin{cases}a|\theta-\widehat{\theta}|,&\mbox{for % }~{}\theta-\widehat{\theta}\geq 0\\ b|\theta-\widehat{\theta}|,&\mbox{for }~{}\theta-\widehat{\theta}<0\end{cases}
  22. F ( θ ^ ( x ) | X ) = a a + b . F(\widehat{\theta}(x)|X)=\frac{a}{a+b}.
  23. K > 0 K>0
  24. L > 0 L>0
  25. L ( θ , θ ^ ) = { 0 , for | θ - θ ^ | < K L , for | θ - θ ^ | K . L(\theta,\widehat{\theta})=\begin{cases}0,&\mbox{for }~{}|\theta-\widehat{% \theta}|<K\\ L,&\mbox{for }~{}|\theta-\widehat{\theta}|\geq K.\end{cases}
  26. p p
  27. p ( θ ) d θ = 1. \int p(\theta)d\theta=1.
  28. p ( θ ) = 1 p(\theta)=1
  29. p ( θ ) d θ = . \int{p(\theta)d\theta}=\infty.
  30. p ( θ ) p(\theta)
  31. p ( θ | x ) = p ( x | θ ) p ( θ ) p ( x | θ ) p ( θ ) d θ . p(\theta|x)=\frac{p(x|\theta)p(\theta)}{\int p(x|\theta)p(\theta)d\theta}.
  32. L ( θ , a ) p ( θ | x ) d θ \int{L(\theta,a)p(\theta|x)d\theta}
  33. L ( a - θ ) L(a-\theta)
  34. θ \theta
  35. p ( x | θ ) = f ( x - θ ) p(x|\theta)=f(x-\theta)
  36. p ( θ ) = 1 p(\theta)=1
  37. p ( θ | x ) = p ( x | θ ) p ( θ ) p ( x ) = f ( x - θ ) p ( x ) p(\theta|x)=\frac{p(x|\theta)p(\theta)}{p(x)}=\frac{f(x-\theta)}{p(x)}
  38. E [ L ( a - θ ) | x ] = L ( a - θ ) p ( θ | x ) d θ = 1 p ( x ) L ( a - θ ) f ( x - θ ) d θ . E[L(a-\theta)|x]=\int{L(a-\theta)p(\theta|x)d\theta}=\frac{1}{p(x)}\int L(a-% \theta)f(x-\theta)d\theta.
  39. a ( x ) a(x)
  40. x x
  41. L ( a - θ ) f ( x - θ ) d θ \int L(a-\theta)f(x-\theta)d\theta
  42. x . x.
  43. x + a 0 x+a_{0}
  44. a 0 a_{0}
  45. a 0 a_{0}
  46. x = 0 x=0
  47. x 1 x_{1}
  48. L ( a - θ ) f ( x 1 - θ ) d θ = L ( a - x 1 - θ ) f ( - θ ) d θ . \int L(a-\theta)f(x_{1}-\theta)d\theta=\int L(a-x_{1}-\theta^{\prime})f(-% \theta^{\prime})d\theta^{\prime}.
  49. a a
  50. a - x 1 a-x_{1}
  51. a - x 1 = a 0 a-x_{1}=a_{0}
  52. a ( x ) = a 0 + x . a(x)=a_{0}+x.\,\!
  53. x 1 , , x n x_{1},\ldots,x_{n}
  54. f ( x i | θ i ) f(x_{i}|\theta_{i})
  55. θ n + 1 \theta_{n+1}
  56. x n + 1 x_{n+1}
  57. θ i \theta_{i}
  58. π \pi
  59. π \pi
  60. μ π \mu_{\pi}\,\!
  61. σ π . \sigma_{\pi}\,\!.
  62. π \pi
  63. μ m \mu_{m}\,\!
  64. σ m \sigma_{m}\,\!
  65. x 1 , , x n x_{1},\ldots,x_{n}
  66. μ ^ m = 1 n x i , \widehat{\mu}_{m}=\frac{1}{n}\sum{x_{i}},
  67. σ ^ m 2 = 1 n ( x i - μ ^ m ) 2 . \widehat{\sigma}_{m}^{2}=\frac{1}{n}\sum{(x_{i}-\widehat{\mu}_{m})^{2}}.
  68. μ m = E π [ μ f ( θ ) ] , \mu_{m}=E_{\pi}[\mu_{f}(\theta)]\,\!,
  69. σ m 2 = E π [ σ f 2 ( θ ) ] + E π [ μ f ( θ ) - μ m ] , \sigma_{m}^{2}=E_{\pi}[\sigma_{f}^{2}(\theta)]+E_{\pi}[\mu_{f}(\theta)-\mu_{m}],
  70. μ f ( θ ) \mu_{f}(\theta)
  71. σ f ( θ ) \sigma_{f}(\theta)
  72. f ( x i | θ i ) f(x_{i}|\theta_{i})
  73. μ f ( θ ) = θ \mu_{f}(\theta)=\theta
  74. σ f 2 ( θ ) = K \sigma_{f}^{2}(\theta)=K
  75. μ π = μ m , \mu_{\pi}=\mu_{m}\,\!,
  76. σ π 2 = σ m 2 - σ f 2 = σ m 2 - K . \sigma_{\pi}^{2}=\sigma_{m}^{2}-\sigma_{f}^{2}=\sigma_{m}^{2}-K.
  77. μ ^ π = μ ^ m , \widehat{\mu}_{\pi}=\widehat{\mu}_{m},
  78. σ ^ π 2 = σ ^ m 2 - K . \widehat{\sigma}_{\pi}^{2}=\widehat{\sigma}_{m}^{2}-K.
  79. x i | θ i N ( θ i , 1 ) x_{i}|\theta_{i}\sim N(\theta_{i},1)
  80. θ n + 1 N ( μ ^ π , σ ^ π 2 ) \theta_{n+1}\sim N(\widehat{\mu}_{\pi},\widehat{\sigma}_{\pi}^{2})
  81. θ n + 1 \theta_{n+1}
  82. x n + 1 x_{n+1}
  83. p > 2 p>2
  84. x 1 , x 2 , x_{1},x_{2},\ldots
  85. f ( x i | θ ) f(x_{i}|\theta)
  86. δ n = δ n ( x 1 , , x n ) \delta_{n}=\delta_{n}(x_{1},\ldots,x_{n})
  87. δ n \delta_{n}
  88. θ 0 \theta_{0}
  89. n ( δ n - θ 0 ) N ( 0 , 1 I ( θ 0 ) ) , \sqrt{n}(\delta_{n}-\theta_{0})\to N\left(0,\frac{1}{I(\theta_{0})}\right),
  90. δ n ( x ) = E [ θ | x ] = a + x a + b + n . \delta_{n}(x)=E[\theta|x]=\frac{a+x}{a+b+n}.
  91. δ n ( x ) = a + b a + b + n E [ θ ] + n a + b + n δ M L E . \delta_{n}(x)=\frac{a+b}{a+b+n}E[\theta]+\frac{n}{a+b+n}\delta_{MLE}.
  92. α α + β B + β α + β b \frac{\alpha}{\alpha+\beta}B+\frac{\beta}{\alpha+\beta}b
  93. 4 4 + n V + n 4 + n v \frac{4}{4+n}V+\frac{n}{4+n}v
  94. W = R v + C m v + m W={Rv+Cm\over v+m}
  95. W W
  96. R R
  97. v v
  98. m m
  99. C C

Bayesian_inference_in_phylogeny.html

  1. π j ( . ) \pi_{j}(.)
  2. j = 1 , 2 , , m j=1,2,\ldots,m
  3. π 1 = π \pi_{1}=\pi
  4. π j \pi_{j}
  5. j = 2 , 3 , , m j=2,3,\ldots,m
  6. π j ( θ ) = π ( θ ) 1 / [ 1 + λ ( j - 1 ) ] , λ > 0 , \pi_{j}(\theta)=\pi(\theta)^{1/[1+\lambda(j-1)]},\ \ \lambda>0,
  7. 2 , 3 , , m 2,3,\ldots,m
  8. π ( . ) \pi(.)
  9. 1 / T 1/T
  10. T > 1 T>1
  11. θ ( j ) \theta^{(j)}
  12. j j
  13. j = 1 , 2 , , m j=1,2,\ldots,m
  14. i i
  15. j j
  16. α = π i ( θ ( j ) ) π j ( θ ( i ) ) π i ( θ ( i ) ) π j ( θ ( j ) ) \alpha=\frac{\pi_{i}(\theta^{(j)})\pi_{j}(\theta^{(i)})}{\pi_{i}(\theta^{(i)})% \pi_{j}(\theta^{(j)})}
  17. π i ( θ ) / π j ( θ ) \pi_{i}(\theta)/\pi_{j}(\theta)
  18. m m
  19. MC 3 \mathrm{MC}^{3}
  20. t 8 t_{8}
  21. A A
  22. B B
  23. t 1 t_{1}
  24. t 9 t_{9}
  25. m = t 1 + t 8 + t 9 m=t_{1}+t_{8}+t_{9}
  26. m = m exp ( λ ( U 1 - 0.5 ) ) m^{\star}=m\exp(\lambda(U_{1}-0.5))
  27. U 1 U_{1}
  28. ( 0 , 1 ) (0,1)
  29. h ( y ) h ( x ) × m 3 m 3 \frac{h(y)}{h(x)}\times\frac{{m^{\star}}^{3}}{m^{3}}
  30. n 1 n_{1}
  31. n 2 n_{2}
  32. λ \lambda
  33. p ( t ) = λ e - λ t p(t)=\lambda e^{-\lambda t}
  34. 1 / 4 ( 1 / 4 + 3 / 4 e - 4 / 3 t ) 1/4\left(1/4+3/4e^{-4/3t}\right)
  35. 1 / 4 ( 1 / 4 - 1 / 4 e - 4 / 3 t ) 1/4\left(1/4-1/4e^{-4/3t}\right)
  36. h ( t ) = ( 1 / 4 ) n 1 + n 2 ( 1 / 4 + 3 / 4 e - 4 / 3 t n 1 ) h(t)=\left(1/4\right)^{n_{1}+n_{2}}\left(1/4+3/4{e^{-4/3t}}^{n_{1}}\right)
  37. h ( t ) = ( 1 / 4 - 1 / 4 e - 4 / 3 t n 2 ) ( λ e - λ t ) h(t)=\left(1/4-1/4{e^{-4/3t}}^{n_{2}}\right)(\lambda e^{-\lambda t})
  38. w w
  39. t = | t + U | t^{\star}=|t+U|
  40. U U
  41. - w -w
  42. w w
  43. h ( t ) / h ( t ) h(t^{\star})/h(t)
  44. n 1 = 70 n_{1}=70
  45. n 2 = 30 n_{2}=30
  46. w w
  47. w = 0.1 w=0.1
  48. w = 0.5 w=0.5
  49. 5 5
  50. 2000 2000

Bayesian_linear_regression.html

  1. i = 1 , , n i=1,...,n
  2. y i y_{i}
  3. k × 1 k\times 1
  4. 𝐱 i \mathbf{x}_{i}
  5. y i = 𝐱 i T s y m b o l β + ϵ i , y_{i}=\mathbf{x}_{i}^{\rm T}symbol\beta+\epsilon_{i},
  6. s y m b o l β symbol\beta
  7. k × 1 k\times 1
  8. ϵ i \epsilon_{i}
  9. ϵ i N ( 0 , σ 2 ) . \epsilon_{i}\sim N(0,\sigma^{2}).
  10. ρ ( 𝐲 | 𝐗 , s y m b o l β , σ 2 ) ( σ 2 ) - n / 2 exp ( - 1 2 σ 2 ( 𝐲 - 𝐗 s y m b o l β ) T ( 𝐲 - 𝐗 s y m b o l β ) ) . \rho(\mathbf{y}|\mathbf{X},symbol\beta,\sigma^{2})\propto(\sigma^{2})^{-n/2}% \exp\left(-\frac{1}{2{\sigma}^{2}}(\mathbf{y}-\mathbf{X}symbol\beta)^{\rm T}(% \mathbf{y}-\mathbf{X}symbol\beta)\right).
  11. s y m b o l β ^ = ( 𝐗 T 𝐗 ) - 1 𝐗 T 𝐲 \hat{symbol\beta}=(\mathbf{X}^{\rm T}\mathbf{X})^{-1}\mathbf{X}^{\rm T}\mathbf% {y}
  12. 𝐗 \mathbf{X}
  13. n × k n\times k
  14. 𝐱 i T \mathbf{x}_{i}^{\rm T}
  15. 𝐲 \mathbf{y}
  16. n n
  17. [ y 1 y n ] T [y_{1}\;\cdots\;y_{n}]^{\rm T}
  18. s y m b o l β symbol\beta
  19. s y m b o l β symbol\beta
  20. σ \sigma
  21. ρ ( s y m b o l β , σ 2 ) \rho(symbol\beta,\sigma^{2})
  22. s y m b o l β symbol\beta
  23. σ \sigma
  24. s y m b o l β symbol\beta
  25. ( s y m b o l β - s y m b o l β ^ ) (symbol\beta-\hat{symbol\beta})
  26. ( 𝐲 - 𝐗 s y m b o l β ) T ( 𝐲 - 𝐗 s y m b o l β ) = ( 𝐲 - 𝐗 s y m b o l β ^ ) T ( 𝐲 - 𝐗 s y m b o l β ^ ) + ( s y m b o l β - s y m b o l β ^ ) T ( 𝐗 T 𝐗 ) ( s y m b o l β - s y m b o l β ^ ) . \begin{aligned}\displaystyle(\mathbf{y}-\mathbf{X}symbol\beta)^{\rm T}(\mathbf% {y}-\mathbf{X}symbol\beta)&\displaystyle=(\mathbf{y}-\mathbf{X}\hat{symbol% \beta})^{\rm T}(\mathbf{y}-\mathbf{X}\hat{symbol\beta})\\ &\displaystyle+(symbol\beta-\hat{symbol\beta})^{\rm T}(\mathbf{X}^{\rm T}% \mathbf{X})(symbol\beta-\hat{symbol\beta}).\end{aligned}
  27. ρ ( 𝐲 | 𝐗 , s y m b o l β , σ 2 ) ( σ 2 ) - v / 2 exp ( - v s 2 2 σ 2 ) ( σ 2 ) - ( n - v ) / 2 × exp ( - 1 2 σ 2 ( s y m b o l β - s y m b o l β ^ ) T ( 𝐗 T 𝐗 ) ( s y m b o l β - s y m b o l β ^ ) ) , \begin{aligned}\displaystyle\rho(\mathbf{y}|\mathbf{X},symbol\beta,\sigma^{2})% &\displaystyle\propto(\sigma^{2})^{-v/2}\exp\left(-\frac{vs^{2}}{2{\sigma}^{2}% }\right)(\sigma^{2})^{-(n-v)/2}\\ &\displaystyle\times\exp\left(-\frac{1}{2{\sigma}^{2}}(symbol\beta-\hat{symbol% \beta})^{\rm T}(\mathbf{X}^{\rm T}\mathbf{X})(symbol\beta-\hat{symbol\beta})% \right),\end{aligned}
  28. v s 2 = ( 𝐲 - 𝐗 s y m b o l β ^ ) T ( 𝐲 - 𝐗 s y m b o l β ^ ) , and v = n - k , vs^{2}=(\mathbf{y}-\mathbf{X}\hat{symbol\beta})^{\rm T}(\mathbf{y}-\mathbf{X}% \hat{symbol\beta}),\,\text{and}\;v=n-k,
  29. k k
  30. ρ ( s y m b o l β , σ 2 ) = ρ ( σ 2 ) ρ ( s y m b o l β | σ 2 ) , \rho(symbol\beta,\sigma^{2})=\rho(\sigma^{2})\rho(symbol\beta|\sigma^{2}),
  31. ρ ( σ 2 ) \rho(\sigma^{2})
  32. ρ ( σ 2 ) ( σ 2 ) - ( v 0 / 2 + 1 ) exp ( - v 0 s 0 2 2 σ 2 ) . \rho(\sigma^{2})\propto(\sigma^{2})^{-(v_{0}/2+1)}\exp\left(-\frac{v_{0}s_{0}^% {2}}{2{\sigma}^{2}}\right).
  33. Inv-Gamma ( a 0 , b 0 ) \,\text{Inv-Gamma}(a_{0},b_{0})
  34. a 0 = v 0 / 2 a_{0}=v_{0}/2
  35. b 0 = 1 2 v 0 s 0 2 b_{0}=\frac{1}{2}v_{0}s_{0}^{2}
  36. v 0 v_{0}
  37. s 0 2 s_{0}^{2}
  38. v v
  39. s 2 s^{2}
  40. Scale-inv- χ 2 ( v 0 , s 0 2 ) . \mbox{Scale-inv-}~{}\chi^{2}(v_{0},s_{0}^{2}).
  41. ρ ( s y m b o l β | σ 2 ) \rho(symbol\beta|\sigma^{2})
  42. ρ ( s y m b o l β | σ 2 ) ( σ 2 ) - k / 2 exp ( - 1 2 σ 2 ( s y m b o l β - s y m b o l μ 0 ) T 𝚲 0 ( s y m b o l β - s y m b o l μ 0 ) ) . \rho(symbol\beta|\sigma^{2})\propto(\sigma^{2})^{-k/2}\exp\left(-\frac{1}{2{% \sigma}^{2}}(symbol\beta-symbol\mu_{0})^{\rm T}\mathbf{\Lambda}_{0}(symbol% \beta-symbol\mu_{0})\right).
  43. 𝒩 ( s y m b o l μ 0 , σ 2 𝚲 0 - 1 ) . \mathcal{N}\left(symbol\mu_{0},\sigma^{2}\mathbf{\Lambda}_{0}^{-1}\right).
  44. ρ ( s y m b o l β , σ 2 | 𝐲 , 𝐗 ) ρ ( 𝐲 | 𝐗 , s y m b o l β , σ 2 ) ρ ( s y m b o l β | σ 2 ) ρ ( σ 2 ) \rho(symbol\beta,\sigma^{2}|\mathbf{y},\mathbf{X})\propto\rho(\mathbf{y}|% \mathbf{X},symbol\beta,\sigma^{2})\rho(symbol\beta|\sigma^{2})\rho(\sigma^{2})
  45. ( σ 2 ) - n / 2 exp ( - 1 2 σ 2 ( 𝐲 - 𝐗 s y m b o l β ) T ( 𝐲 - 𝐗 s y m b o l β ) ) \propto(\sigma^{2})^{-n/2}\exp\left(-\frac{1}{2{\sigma}^{2}}(\mathbf{y}-% \mathbf{X}symbol\beta)^{\rm T}(\mathbf{y}-\mathbf{X}symbol\beta)\right)
  46. × ( σ 2 ) - k / 2 exp ( - 1 2 σ 2 ( s y m b o l β - s y m b o l μ 0 ) T s y m b o l Λ 0 ( s y m b o l β - s y m b o l μ 0 ) ) \times(\sigma^{2})^{-k/2}\exp\left(-\frac{1}{2{\sigma}^{2}}(symbol\beta-symbol% \mu_{0})^{\rm T}symbol\Lambda_{0}(symbol\beta-symbol\mu_{0})\right)
  47. × ( σ 2 ) - ( a 0 + 1 ) exp ( - b 0 σ 2 ) . \times(\sigma^{2})^{-(a_{0}+1)}\exp\left(-\frac{b_{0}}{{\sigma}^{2}}\right).
  48. s y m b o l μ n symbol\mu_{n}
  49. s y m b o l β symbol\beta
  50. s y m b o l β ^ \hat{symbol\beta}
  51. s y m b o l μ 0 symbol\mu_{0}
  52. s y m b o l Λ 0 symbol\Lambda_{0}
  53. s y m b o l μ n = ( 𝐗 T 𝐗 + s y m b o l Λ 0 ) - 1 ( 𝐗 T 𝐗 s y m b o l β ^ + s y m b o l Λ 0 s y m b o l μ 0 ) . symbol\mu_{n}=(\mathbf{X}^{\rm T}\mathbf{X}+symbol\Lambda_{0})^{-1}(\mathbf{X}% ^{\rm T}\mathbf{X}\hat{symbol\beta}+symbol\Lambda_{0}symbol\mu_{0}).
  54. s y m b o l μ n symbol\mu_{n}
  55. s y m b o l β - s y m b o l μ n symbol\beta-symbol\mu_{n}
  56. ( 𝐲 - 𝐗 s y m b o l β ) T ( 𝐲 - 𝐗 s y m b o l β ) + ( s y m b o l β - s y m b o l μ 0 ) T s y m b o l Λ 0 ( s y m b o l β - s y m b o l μ 0 ) = (\mathbf{y}-\mathbf{X}symbol\beta)^{\rm T}(\mathbf{y}-\mathbf{X}symbol\beta)+(% symbol\beta-symbol\mu_{0})^{\rm T}symbol\Lambda_{0}(symbol\beta-symbol\mu_{0})=
  57. ( s y m b o l β - s y m b o l μ n ) T ( 𝐗 T 𝐗 + s y m b o l Λ 0 ) ( s y m b o l β - s y m b o l μ n ) + 𝐲 T 𝐲 - s y m b o l μ n T ( 𝐗 T 𝐗 + s y m b o l Λ 0 ) s y m b o l μ n + s y m b o l μ 0 T s y m b o l Λ 0 s y m b o l μ 0 . (symbol\beta-symbol\mu_{n})^{\rm T}(\mathbf{X}^{\rm T}\mathbf{X}+symbol\Lambda% _{0})(symbol\beta-symbol\mu_{n})+\mathbf{y}^{\rm T}\mathbf{y}-symbol\mu_{n}^{% \rm T}(\mathbf{X}^{\rm T}\mathbf{X}+symbol\Lambda_{0})symbol\mu_{n}+symbol\mu_% {0}^{\rm T}symbol\Lambda_{0}symbol\mu_{0}.
  58. ρ ( s y m b o l β , σ 2 | 𝐲 , 𝐗 ) ( σ 2 ) - k / 2 exp ( - 1 2 σ 2 ( s y m b o l β - s y m b o l μ n ) T ( 𝐗 T 𝐗 + 𝚲 0 ) ( s y m b o l β - s y m b o l μ n ) ) \rho(symbol\beta,\sigma^{2}|\mathbf{y},\mathbf{X})\propto(\sigma^{2})^{-k/2}% \exp\left(-\frac{1}{2{\sigma}^{2}}(symbol\beta-symbol\mu_{n})^{\rm T}(\mathbf{% X}^{\rm T}\mathbf{X}+\mathbf{\Lambda}_{0})(symbol\beta-symbol\mu_{n})\right)
  59. × ( σ 2 ) - ( n + 2 a 0 ) / 2 - 1 exp ( - 2 b 0 + 𝐲 T 𝐲 - s y m b o l μ n T ( 𝐗 T 𝐗 + s y m b o l Λ 0 ) s y m b o l μ n + s y m b o l μ 0 T s y m b o l Λ 0 s y m b o l μ 0 2 σ 2 ) . \times(\sigma^{2})^{-(n+2a_{0})/2-1}\exp\left(-\frac{2b_{0}+\mathbf{y}^{\rm T}% \mathbf{y}-symbol\mu_{n}^{\rm T}(\mathbf{X}^{\rm T}\mathbf{X}+symbol\Lambda_{0% })symbol\mu_{n}+symbol\mu_{0}^{\rm T}symbol\Lambda_{0}symbol\mu_{0}}{2{\sigma}% ^{2}}\right).
  60. ρ ( s y m b o l β , σ 2 | 𝐲 , 𝐗 ) ρ ( s y m b o l β | σ 2 , 𝐲 , 𝐗 ) ρ ( σ 2 | 𝐲 , 𝐗 ) , \rho(symbol\beta,\sigma^{2}|\mathbf{y},\mathbf{X})\propto\rho(symbol\beta|% \sigma^{2},\mathbf{y},\mathbf{X})\rho(\sigma^{2}|\mathbf{y},\mathbf{X}),
  61. 𝒩 ( s y m b o l μ n , σ 2 s y m b o l Λ n - 1 ) \mathcal{N}\left(symbol\mu_{n},\sigma^{2}symbol\Lambda_{n}^{-1}\right)\,
  62. Inv-Gamma ( a n , b n ) \,\text{Inv-Gamma}\left(a_{n},b_{n}\right)
  63. s y m b o l Λ n = ( 𝐗 T 𝐗 + 𝚲 0 ) , s y m b o l μ n = ( s y m b o l Λ n ) - 1 ( 𝐗 T 𝐗 s y m b o l β ^ + s y m b o l Λ 0 s y m b o l μ 0 ) , symbol\Lambda_{n}=(\mathbf{X}^{\rm T}\mathbf{X}+\mathbf{\Lambda}_{0}),\quad symbol% \mu_{n}=(symbol\Lambda_{n})^{-1}(\mathbf{X}^{\rm T}\mathbf{X}\hat{symbol\beta}% +symbol\Lambda_{0}symbol\mu_{0}),
  64. a n = a 0 + n 2 , b n = b 0 + 1 2 ( 𝐲 T 𝐲 + s y m b o l μ 0 T s y m b o l Λ 0 s y m b o l μ 0 - s y m b o l μ n T s y m b o l Λ n s y m b o l μ n ) . a_{n}=a_{0}+\frac{n}{2},\qquad b_{n}=b_{0}+\frac{1}{2}(\mathbf{y}^{\rm T}% \mathbf{y}+symbol\mu_{0}^{\rm T}symbol\Lambda_{0}symbol\mu_{0}-symbol\mu_{n}^{% \rm T}symbol\Lambda_{n}symbol\mu_{n}).
  65. s y m b o l μ n = ( 𝐗 T 𝐗 + s y m b o l Λ 0 ) - 1 ( s y m b o l Λ 0 s y m b o l μ 0 + 𝐗 T 𝐗 s y m b o l β ^ ) = ( 𝐗 T 𝐗 + s y m b o l Λ 0 ) - 1 ( s y m b o l Λ 0 s y m b o l μ 0 + 𝐗 T 𝐲 ) , symbol\mu_{n}=(\mathbf{X}^{\rm T}\mathbf{X}+symbol\Lambda_{0})^{-1}(symbol% \Lambda_{0}symbol\mu_{0}+\mathbf{X}^{\rm T}\mathbf{X}\hat{symbol\beta})=(% \mathbf{X}^{\rm T}\mathbf{X}+symbol\Lambda_{0})^{-1}(symbol\Lambda_{0}symbol% \mu_{0}+\mathbf{X}^{\rm T}\mathbf{y}),
  66. s y m b o l Λ n = ( 𝐗 T 𝐗 + s y m b o l Λ 0 ) , symbol\Lambda_{n}=(\mathbf{X}^{\rm T}\mathbf{X}+symbol\Lambda_{0}),
  67. a n = a 0 + n 2 , a_{n}=a_{0}+\frac{n}{2},
  68. b n = b 0 + 1 2 ( 𝐲 T 𝐲 + s y m b o l μ 0 T s y m b o l Λ 0 s y m b o l μ 0 - s y m b o l μ n T s y m b o l Λ n s y m b o l μ n ) . b_{n}=b_{0}+\frac{1}{2}(\mathbf{y}^{\rm T}\mathbf{y}+symbol\mu_{0}^{\rm T}% symbol\Lambda_{0}symbol\mu_{0}-symbol\mu_{n}^{\rm T}symbol\Lambda_{n}symbol\mu% _{n}).
  69. p ( 𝐲 | m ) p(\mathbf{y}|m)
  70. m m
  71. p ( 𝐲 | 𝐗 , s y m b o l β , σ ) p(\mathbf{y}|\mathbf{X},symbol\beta,\sigma)
  72. p ( s y m b o l β , σ ) p(symbol\beta,\sigma)
  73. p ( 𝐲 , s y m b o l β , σ | 𝐗 ) p(\mathbf{y},symbol\beta,\sigma|\mathbf{X})
  74. s y m b o l β symbol\beta
  75. σ \sigma
  76. p ( 𝐲 | m ) = p ( 𝐲 | 𝐗 , s y m b o l β , σ ) p ( s y m b o l β , σ ) d s y m b o l β d σ p(\mathbf{y}|m)=\int p(\mathbf{y}|\mathbf{X},symbol\beta,\sigma)\,p(symbol% \beta,\sigma)\,dsymbol\beta\,d\sigma
  77. p ( 𝐲 | m ) = 1 ( 2 π ) n / 2 det ( s y m b o l Λ 0 ) det ( s y m b o l Λ n ) b 0 a 0 b n a n Γ ( a n ) Γ ( a 0 ) p(\mathbf{y}|m)=\frac{1}{(2\pi)^{n/2}}\sqrt{\frac{\det(symbol\Lambda_{0})}{% \det(symbol\Lambda_{n})}}\cdot\frac{b_{0}^{a_{0}}}{b_{n}^{a_{n}}}\cdot\frac{% \Gamma(a_{n})}{\Gamma(a_{0})}
  78. Γ \Gamma
  79. s y m b o l β symbol\beta
  80. σ \sigma
  81. p ( 𝐲 | m ) = p ( s y m b o l β , σ | m ) p ( 𝐲 | 𝐗 , s y m b o l β , σ , m ) p ( s y m b o l β , σ | 𝐲 , 𝐗 , m ) p(\mathbf{y}|m)=\frac{p(symbol\beta,\sigma|m)\,p(\mathbf{y}|\mathbf{X},symbol% \beta,\sigma,m)}{p(symbol\beta,\sigma|\mathbf{y},\mathbf{X},m)}
  82. s y m b o l μ 0 = 0 , 𝚲 0 = c 𝐈 symbol\mu_{0}=0,\mathbf{\Lambda}_{0}=c\mathbf{I}

Bayesian_multivariate_linear_regression.html

  1. 𝐱 i \mathbf{x}_{i}
  2. y i , 1 = 𝐱 i T s y m b o l β 1 + ϵ i , 1 y_{i,1}=\mathbf{x}_{i}^{\rm T}symbol\beta_{1}+\epsilon_{i,1}
  3. \cdots
  4. y i , m = 𝐱 i T s y m b o l β m + ϵ i , m y_{i,m}=\mathbf{x}_{i}^{\rm T}symbol\beta_{m}+\epsilon_{i,m}
  5. { ϵ i , 1 , , ϵ i , m } \{\epsilon_{i,1},\ldots,\epsilon_{i,m}\}
  6. 𝐲 i T \mathbf{y}_{i}^{\rm T}
  7. 𝐲 i T = 𝐱 i T 𝐁 + s y m b o l ϵ i T . \mathbf{y}_{i}^{\rm T}=\mathbf{x}_{i}^{\rm T}\mathbf{B}+symbol\epsilon_{i}^{% \rm T}.
  8. k × m k\times m
  9. s y m b o l β 1 , , s y m b o l β m symbol\beta_{1},\ldots,symbol\beta_{m}
  10. 𝐁 = [ ( s y m b o l β 1 ) ( s y m b o l β m ) ] = [ ( β 1 , 1 β k , 1 ) ( β 1 , m β k , m ) ] . \mathbf{B}=\begin{bmatrix}\begin{pmatrix}\\ symbol\beta_{1}\\ \\ \end{pmatrix}\cdots\begin{pmatrix}\\ symbol\beta_{m}\\ \\ \end{pmatrix}\end{bmatrix}=\begin{bmatrix}\begin{pmatrix}\beta_{1,1}\\ \vdots\\ \beta_{k,1}\\ \end{pmatrix}\cdots\begin{pmatrix}\beta_{1,m}\\ \vdots\\ \beta_{k,m}\\ \end{pmatrix}\end{bmatrix}.
  11. s y m b o l ϵ i symbol\epsilon_{i}
  12. s y m b o l ϵ i N ( 0 , s y m b o l Σ ϵ 2 ) . symbol\epsilon_{i}\sim N(0,symbol\Sigma_{\epsilon}^{2}).
  13. 𝐘 = 𝐗𝐁 + 𝐄 , \mathbf{Y}=\mathbf{X}\mathbf{B}+\mathbf{E},
  14. n × m n\times m
  15. n × k n\times k
  16. 𝐗 = [ 𝐱 1 T 𝐱 2 T 𝐱 n T ] = [ x 1 , 1 x 1 , k x 2 , 1 x 2 , k x n , 1 x n , k ] . \mathbf{X}=\begin{bmatrix}\mathbf{x}^{\rm T}_{1}\\ \mathbf{x}^{\rm T}_{2}\\ \vdots\\ \mathbf{x}^{\rm T}_{n}\end{bmatrix}=\begin{bmatrix}x_{1,1}&\cdots&x_{1,k}\\ x_{2,1}&\cdots&x_{2,k}\\ \vdots&\ddots&\vdots\\ x_{n,1}&\cdots&x_{n,k}\end{bmatrix}.
  17. 𝐁 ^ \hat{\mathbf{B}}
  18. 𝐁 ^ = ( 𝐗 T 𝐗 ) - 1 𝐗 T 𝐘 \hat{\mathbf{B}}=(\mathbf{X}^{\rm T}\mathbf{X})^{-1}\mathbf{X}^{\rm T}\mathbf{Y}
  19. ρ ( 𝐄 | s y m b o l Σ ϵ ) ( s y m b o l Σ ϵ 2 ) - n / 2 exp ( - 1 2 tr ( 𝐄 T 𝐄 s y m b o l Σ ϵ - 1 ) ) , \rho(\mathbf{E}|symbol\Sigma_{\epsilon})\propto(symbol\Sigma_{\epsilon}^{2})^{% -n/2}\exp(-\frac{1}{2}{\rm tr}(\mathbf{E}^{\rm T}\mathbf{E}symbol\Sigma_{% \epsilon}^{-1})),
  20. 𝐄 \mathbf{E}
  21. 𝐘 , 𝐗 , \mathbf{Y},\mathbf{X},
  22. 𝐁 \mathbf{B}
  23. ρ ( 𝐘 | 𝐗 , 𝐁 , s y m b o l Σ ϵ ) ( s y m b o l Σ ϵ 2 ) - n / 2 exp ( - 1 2 tr ( ( 𝐘 - 𝐗𝐁 ) T ( 𝐘 - 𝐗𝐁 ) s y m b o l Σ ϵ - 1 ) ) , \rho(\mathbf{Y}|\mathbf{X},\mathbf{B},symbol\Sigma_{\epsilon})\propto(symbol% \Sigma_{\epsilon}^{2})^{-n/2}\exp(-\frac{1}{2}{\rm tr}((\mathbf{Y}-\mathbf{X}% \mathbf{\mathbf{B}})^{\rm T}(\mathbf{Y}-\mathbf{X}\mathbf{\mathbf{B}})symbol% \Sigma_{\epsilon}^{-1})),
  24. ρ ( 𝐁 , Σ ϵ ) \rho(\mathbf{B},\Sigma_{\epsilon})
  25. 𝐁 \mathbf{B}
  26. ( 𝐁 - 𝐁 ^ ) (\mathbf{B}-\hat{\mathbf{B}})
  27. ρ ( 𝐘 | 𝐗 , 𝐁 , s y m b o l Σ ϵ ) s y m b o l Σ ϵ - ( n - k ) / 2 exp ( - tr ( 1 2 𝐒 T 𝐒 s y m b o l Σ ϵ - 1 ) ) ( s y m b o l Σ ϵ 2 ) - k / 2 exp ( - 1 2 tr ( ( 𝐁 - 𝐁 ^ ) T 𝐗 T 𝐗 ( 𝐁 - 𝐁 ^ ) s y m b o l Σ ϵ - 1 ) ) , \rho(\mathbf{Y}|\mathbf{X},\mathbf{B},symbol\Sigma_{\epsilon})\propto symbol% \Sigma_{\epsilon}^{-(n-k)/2}\exp(-{\rm tr}(\frac{1}{2}\mathbf{S}^{\rm T}% \mathbf{S}symbol\Sigma_{\epsilon}^{-1}))(symbol\Sigma_{\epsilon}^{2})^{-k/2}% \exp(-\frac{1}{2}{\rm tr}((\mathbf{B}-\hat{\mathbf{B}})^{\rm T}\mathbf{X}^{\rm T% }\mathbf{X}(\mathbf{B}-\hat{\mathbf{B}})symbol\Sigma_{\epsilon}^{-1})),
  28. 𝐒 = 𝐘 - 𝐁 ^ 𝐗 \mathbf{S}=\mathbf{Y}-\hat{\mathbf{B}}\mathbf{X}
  29. ρ ( 𝐁 , s y m b o l Σ ϵ ) = ρ ( s y m b o l Σ ϵ ) ρ ( 𝐁 | s y m b o l Σ ϵ ) , \rho(\mathbf{B},symbol\Sigma_{\epsilon})=\rho(symbol\Sigma_{\epsilon})\rho(% \mathbf{B}|symbol\Sigma_{\epsilon}),
  30. ρ ( s y m b o l Σ ϵ ) \rho(symbol\Sigma_{\epsilon})
  31. ρ ( 𝐁 | s y m b o l Σ ϵ ) \rho(\mathbf{B}|symbol\Sigma_{\epsilon})
  32. 𝐁 \mathbf{B}
  33. 𝐁 , 𝐁 ^ \mathbf{B},\hat{\mathbf{B}}
  34. s y m b o l β = vec ( 𝐁 ) , s y m b o l β ^ = vec ( 𝐁 ^ ) symbol\beta={\rm vec}(\mathbf{B}),\hat{symbol\beta}={\rm vec}(\hat{\mathbf{B}})
  35. tr ( ( 𝐁 - 𝐁 ^ ) T 𝐗 T 𝐗 ( 𝐁 - 𝐁 ^ ) s y m b o l Σ ϵ - 1 ) = vec ( 𝐁 - 𝐁 ^ ) T vec ( 𝐗 T 𝐗 ( 𝐁 - 𝐁 ^ ) s y m b o l Σ ϵ - 1 ) {\rm tr}((\mathbf{B}-\hat{\mathbf{B}})^{\rm T}\mathbf{X}^{\rm T}\mathbf{X}(% \mathbf{B}-\hat{\mathbf{B}})symbol\Sigma_{\epsilon}^{-1})={\rm vec}(\mathbf{B}% -\hat{\mathbf{B}})^{\rm T}{\rm vec}(\mathbf{X}^{\rm T}\mathbf{X}(\mathbf{B}-% \hat{\mathbf{B}})symbol\Sigma_{\epsilon}^{-1})
  36. vec ( 𝐗 T 𝐗 ( 𝐁 - 𝐁 ^ ) s y m b o l Σ ϵ - 1 ) = ( s y m b o l Σ ϵ - 1 𝐗 T 𝐗 ) vec ( 𝐁 - 𝐁 ^ ) , {\rm vec}(\mathbf{X}^{\rm T}\mathbf{X}(\mathbf{B}-\hat{\mathbf{B}})symbol% \Sigma_{\epsilon}^{-1})=(symbol\Sigma_{\epsilon}^{-1}\otimes\mathbf{X}^{\rm T}% \mathbf{X}){\rm vec}(\mathbf{B}-\hat{\mathbf{B}}),
  37. 𝐀 𝐁 \mathbf{A}\otimes\mathbf{B}
  38. m × n m\times n
  39. p × q p\times q
  40. m p × n q mp\times nq
  41. vec ( 𝐁 - 𝐁 ^ ) T ( s y m b o l Σ ϵ - 1 𝐗 T 𝐗 ) vec ( 𝐁 - 𝐁 ^ ) {\rm vec}(\mathbf{B}-\hat{\mathbf{B}})^{\rm T}(symbol\Sigma_{\epsilon}^{-1}% \otimes\mathbf{X}^{\rm T}\mathbf{X}){\rm vec}(\mathbf{B}-\hat{\mathbf{B}})
  42. = ( s y m b o l β - s y m b o l β ^ ) T ( s y m b o l Σ ϵ - 1 𝐗 T 𝐗 ) ( s y m b o l β - s y m b o l β ^ ) =(symbol\beta-\hat{symbol\beta})^{\rm T}(symbol\Sigma_{\epsilon}^{-1}\otimes% \mathbf{X}^{\rm T}\mathbf{X})(symbol\beta-\hat{symbol\beta})
  43. ( s y m b o l β - s y m b o l β ^ ) (symbol\beta-\hat{symbol\beta})

BB84.html

  1. a a
  2. b b
  3. n n
  4. n n
  5. | ψ = i = 1 n | ψ a i b i . |\psi\rangle=\bigotimes_{i=1}^{n}|\psi_{a_{i}b_{i}}\rangle.
  6. a i a_{i}
  7. b i b_{i}
  8. i th i^{\mathrm{th}}
  9. a a
  10. b b
  11. a i b i a_{i}b_{i}
  12. | ψ 00 = | 0 |\psi_{00}\rangle=|0\rangle
  13. | ψ 10 = | 1 |\psi_{10}\rangle=|1\rangle
  14. | ψ 01 = | + = 1 2 | 0 + 1 2 | 1 |\psi_{01}\rangle=|+\rangle=\frac{1}{\sqrt{2}}|0\rangle+\frac{1}{\sqrt{2}}|1\rangle
  15. | ψ 11 = | - = 1 2 | 0 - 1 2 | 1 . |\psi_{11}\rangle=|-\rangle=\frac{1}{\sqrt{2}}|0\rangle-\frac{1}{\sqrt{2}}|1\rangle.
  16. b i b_{i}
  17. a i a_{i}
  18. b b
  19. | ψ |\psi\rangle
  20. ε ρ = ε | ψ ψ | \varepsilon\rho=\varepsilon|\psi\rangle\langle\psi|
  21. ε \varepsilon
  22. b b
  23. b b^{\prime}
  24. b b
  25. a a^{\prime}
  26. b b
  27. b i b_{i}
  28. b i b^{\prime}_{i}
  29. a a
  30. a a^{\prime}
  31. b b
  32. b b^{\prime}
  33. k k
  34. k / 2 k/2

BBGKY_hierarchy.html

  1. f N = f N ( 𝐪 1 𝐪 N , 𝐩 1 𝐩 N , t ) f_{N}=f_{N}(\mathbf{q}_{1}\dots\mathbf{q}_{N},\mathbf{p}_{1}\dots\mathbf{p}_{N% },t)
  2. f N t + i = 1 N 𝐪 ˙ i f N 𝐪 i + i = 1 N ( - Φ i e x t 𝐪 i - j = 1 N Φ i j 𝐪 i ) f N 𝐩 i = 0. \frac{\partial f_{N}}{\partial t}+\sum_{i=1}^{N}\dot{\mathbf{q}}_{i}\frac{% \partial f_{N}}{\partial\mathbf{q}_{i}}+\sum_{i=1}^{N}\left(-\frac{\partial% \Phi_{i}^{ext}}{\partial\mathbf{q}_{i}}-\sum_{j=1}^{N}\frac{\partial\Phi_{ij}}% {\partial\mathbf{q}_{i}}\right)\frac{\partial f_{N}}{\partial\mathbf{p}_{i}}=0.
  3. f s = f s ( 𝐪 1 𝐪 s , 𝐩 1 𝐩 s , t ) f_{s}=f_{s}(\mathbf{q}_{1}\dots\mathbf{q}_{s},\mathbf{p}_{1}\dots\mathbf{p}_{s% },t)
  4. f s t + i = 1 s 𝐪 ˙ i f s 𝐪 i + i = 1 s ( - Φ i e x t 𝐪 i - j = 1 s Φ i j 𝐪 i ) f s 𝐩 i = ( N - s ) i = 1 s 𝐩 i Φ i s + 1 𝐪 i f s + 1 d 𝐪 s + 1 d 𝐩 s + 1 . \frac{\partial f_{s}}{\partial t}+\sum_{i=1}^{s}\dot{\mathbf{q}}_{i}\frac{% \partial f_{s}}{\partial\mathbf{q}_{i}}+\sum_{i=1}^{s}\left(-\frac{\partial% \Phi_{i}^{ext}}{\partial\mathbf{q}_{i}}-\sum_{j=1}^{s}\frac{\partial\Phi_{ij}}% {\partial\mathbf{q}_{i}}\right)\frac{\partial f_{s}}{\partial\mathbf{p}_{i}}=(% N-s)\sum_{i=1}^{s}\frac{\partial}{\partial\mathbf{p}_{i}}\int\frac{\partial% \Phi_{is+1}}{\partial\mathbf{q}_{i}}\cdot f_{s+1}\,d\mathbf{q}_{s+1}d\mathbf{p% }_{s+1}.
  5. 𝐪 i , 𝐩 i \mathbf{q}_{i},\mathbf{p}_{i}
  6. Φ e x t ( 𝐪 i ) \Phi^{ext}(\mathbf{q}_{i})
  7. Φ i j ( 𝐪 i , 𝐪 j ) \Phi_{ij}(\mathbf{q}_{i},\mathbf{q}_{j})
  8. 𝐪 s + 1 𝐪 N , 𝐩 s + 1 𝐩 N \mathbf{q}_{s+1}\dots\mathbf{q}_{N},\mathbf{p}_{s+1}\dots\mathbf{p}_{N}
  9. N N
  10. D f N = 0 Df_{N}=0
  11. f s f s + 1 f_{s}\sim\int f_{s+1}
  12. f s f_{s}
  13. D f s div 𝐩 grad 𝐪 Φ i , s + 1 f s + 1 . Df_{s}\propto\,\text{div}_{\mathbf{p}}\langle\,\text{grad}_{\mathbf{q}}\Phi_{i% ,s+1}\rangle_{f_{s+1}}.
  14. f s + 2 , f s + 3 , f_{s+2},f_{s+3},\dots
  15. f s f_{s}
  16. f s + 1 . f_{s+1}.

Beale_number.html

  1. B n = W o P V F B_{n}=\frac{Wo}{PVF}
  2. W o = B n P V F Wo=\frac{}{}B_{n}PVF

Beam_propagation_method.html

  1. ( 2 + k 0 2 n 2 ) ψ = 0 (\nabla^{2}+k_{0}^{2}n^{2})\psi=0
  2. E ( x , y , z , t ) = ψ ( x , y ) exp ( - j ω t ) E(x,y,z,t)=\psi(x,y)\exp(-j\omega t)
  3. ψ ( x , y ) = A ( x , y ) exp ( + j k o ν y ) \psi(x,y)=A(x,y)\exp(+jk_{o}\nu y)
  4. A ( x , y ) A(x,y)
  5. 2 ( A ( x , y ) ) y 2 = 0 \frac{\partial^{2}(A(x,y))}{\partial y^{2}}=0
  6. [ 2 x 2 + k 0 2 ( n 2 - ν 2 ) ] A ( x , y ) = ± 2 j k 0 ν A k ( x , y ) y \left[\frac{\partial^{2}}{\partial x^{2}}+k_{0}^{2}(n^{2}-\nu^{2})\right]A(x,y% )=\pm 2jk_{0}\nu\frac{\partial A_{k}(x,y)}{\partial y}
  7. A ( x , y ) A(x,y)
  8. ψ ( x , y ) \psi(x,y)

Bearing_capacity.html

  1. q u l t = 1.3 c N c + σ z D N q + 0.4 γ B N γ q_{ult}=1.3c^{\prime}N_{c}+\sigma^{\prime}_{zD}N_{q}+0.4\gamma^{\prime}BN_{\gamma}
  2. q u l t = c N c + σ z D N q + 0.5 γ B N γ q_{ult}=c^{\prime}N_{c}+\sigma^{\prime}_{zD}N_{q}+0.5\gamma^{\prime}BN_{\gamma}
  3. q u l t = 1.3 c N c + σ z D N q + 0.3 γ B N γ q_{ult}=1.3c^{\prime}N_{c}+\sigma^{\prime}_{zD}N_{q}+0.3\gamma^{\prime}BN_{\gamma}
  4. N q = e 2 π ( 0.75 - ϕ / 360 ) tan ϕ 2 cos 2 ( 45 + ϕ / 2 ) N_{q}=\frac{e^{2\pi\left(0.75-\phi^{\prime}/360\right)\tan\phi^{\prime}}}{2% \cos^{2}\left(45+\phi^{\prime}/2\right)}
  5. N c = 5.7 N_{c}=5.7
  6. N c = N q - 1 tan ϕ N_{c}=\frac{N_{q}-1}{\tan\phi^{\prime}}
  7. N γ = tan ϕ 2 ( K p γ cos 2 ϕ - 1 ) N_{\gamma}=\frac{\tan\phi^{\prime}}{2}\left(\frac{K_{p\gamma}}{\cos^{2}\phi^{% \prime}}-1\right)
  8. N γ = 2 ( N q + 1 ) tan ϕ 1 + 0.4 sin 4 ϕ N_{\gamma}=\frac{2\left(N_{q}+1\right)\tan\phi^{\prime}}{1+0.4\sin 4\phi^{% \prime}}
  9. q u l t = 0.867 c N c + σ z D N q + 0.4 γ B N γ q_{ult}=0.867c^{\prime}N^{\prime}_{c}+\sigma^{\prime}_{zD}N^{\prime}_{q}+0.4% \gamma^{\prime}BN^{\prime}_{\gamma}
  10. q u l t = 2 3 c N c + σ z D N q + 0.5 γ B N γ q_{ult}=\frac{2}{3}c^{\prime}N^{\prime}_{c}+\sigma^{\prime}_{zD}N^{\prime}_{q}% +0.5\gamma^{\prime}BN^{\prime}_{\gamma}
  11. q u l t = 0.867 c N c + σ z D N q + 0.3 γ B N γ q_{ult}=0.867c^{\prime}N^{\prime}_{c}+\sigma^{\prime}_{zD}N^{\prime}_{q}+0.3% \gamma^{\prime}BN^{\prime}_{\gamma}
  12. N c , N q a n d N y N^{\prime}_{c},N^{\prime}_{q}andN^{\prime}_{y}
  13. N c , N q , a n d N y N_{c},N_{q},andN_{y}
  14. ( ϕ ) (\phi^{\prime})
  15. : t a n - 1 ( 2 3 t a n ϕ ) :tan^{-1}\,(\frac{2}{3}tan\phi^{\prime})
  16. q a l l = q u l t F S q_{all}=\frac{q_{ult}}{FS}

Bellard's_formula.html

  1. π = 1 2 6 n = 0 ( - 1 ) n 2 10 n ( - 2 5 4 n + 1 \displaystyle\pi=\frac{1}{2^{6}}\sum_{n=0}^{\infty}\frac{(-1)^{n}}{2^{10n}}\,% \left(-\frac{2^{5}}{4n+1}\right.

Benaloh_cryptosystem.html

  1. ( / n ) * (\mathbb{Z}/n\mathbb{Z})^{*}
  2. r | ( p - 1 ) , gcd ( r , ( p - 1 ) / r ) = 1 , r|(p-1),\operatorname{gcd}(r,(p-1)/r)=1,
  3. gcd ( r , ( q - 1 ) ) = 1 \operatorname{gcd}(r,(q-1))=1
  4. n = p q , ϕ = ( p - 1 ) ( q - 1 ) n=pq,\phi=(p-1)(q-1)
  5. y n * y\in\mathbb{Z}^{*}_{n}
  6. y ϕ / r 1 mod n y^{\phi/r}\not\equiv 1\mod n
  7. D ( E ( m ) ) = m D(E(m))=m
  8. r = p 1 p 2 p k r=p_{1}p_{2}\dots p_{k}
  9. y n * y\in\mathbb{Z}^{*}_{n}
  10. p i p_{i}
  11. y ϕ / p i 1 mod n y^{\phi/p_{i}}\neq 1\mod n
  12. x = y ϕ / r mod n x=y^{\phi/r}\mod n
  13. y , n y,n
  14. ϕ , x \phi,x
  15. m r m\in\mathbb{Z}_{r}
  16. u n * u\in\mathbb{Z}^{*}_{n}
  17. E r ( m ) = y m u r mod n E_{r}(m)=y^{m}u^{r}\mod n
  18. c n * c\in\mathbb{Z}^{*}_{n}
  19. a = c ϕ / r mod n a=c^{\phi/r}\mod n
  20. m = log x ( a ) m=\log_{x}(a)
  21. x m a mod n x^{m}\equiv a\mod n
  22. m r m\in\mathbb{Z}_{r}
  23. u n * u\in\mathbb{Z}^{*}_{n}
  24. a = ( c ) ϕ / r ( y m u r ) ϕ / r ( y m ) ϕ / r ( u r ) ϕ / r ( y ϕ / r ) m ( u ) ϕ ( x ) m ( u ) 0 x m mod n a=(c)^{\phi/r}\equiv(y^{m}u^{r})^{\phi/r}\equiv(y^{m})^{\phi/r}(u^{r})^{\phi/r% }\equiv(y^{\phi/r})^{m}(u)^{\phi}\equiv(x)^{m}(u)^{0}\equiv x^{m}\mod n
  25. x i a mod n x^{i}\equiv a\mod n
  26. 0 ( r - 1 ) 0\dots(r-1)
  27. O ( r ) O(\sqrt{r})
  28. z x r mod n z\equiv x^{r}\mod n

Bending_(metalworking).html

  1. B A = A ( π 180 ) ( R + K × T ) BA=A\left(\frac{\pi}{180}\right)\left(R+K\times T\right)
  2. B D = 2 ( R + T ) tan A 2 - B A BD=2\left(R+T\right)\tan{\frac{A}{2}}-BA
  3. B D = R ( A - 2 ) + T ( k A - 2 ) BD=R\left(A-2\right)+T\left(kA-2\right)
  4. K = - R + B A π A / 180 T K=\frac{-R+\frac{BA}{\pi A/180}}{T}
  5. K = l o g ( m i n ( 100 , m a x ( 20 R , T ) T ) ) 2 l o g ( 100 ) K=\frac{log(min(100,\frac{max(20R,T)}{T}))}{2log(100)}

Bending_moment.html

  1. 𝐅 \mathbf{F}
  2. 𝐌 = 𝐫 × 𝐅 \mathbf{M}=\mathbf{r}\times\mathbf{F}
  3. 𝐌 \mathbf{M}
  4. 𝐫 \mathbf{r}
  5. × \times
  6. 𝐞 \mathbf{e}
  7. M = 𝐞 𝐌 = 𝐞 ( 𝐫 × 𝐅 ) M=\mathbf{e}\cdot\mathbf{M}=\mathbf{e}\cdot(\mathbf{r}\times\mathbf{F})
  8. \cdot
  9. F F
  10. 𝐞 x , 𝐞 y , 𝐞 z \mathbf{e}_{x},\mathbf{e}_{y},\mathbf{e}_{z}
  11. 𝐅 = 0 𝐞 x - F 𝐞 y + 0 𝐞 z and 𝐫 = x 𝐞 x + 0 𝐞 y + 0 𝐞 z . \mathbf{F}=0\,\mathbf{e}_{x}-F\,\mathbf{e}_{y}+0\,\mathbf{e}_{z}\quad\,\text{% and}\quad\mathbf{r}=x\,\mathbf{e}_{x}+0\,\mathbf{e}_{y}+0\,\mathbf{e}_{z}\,.
  12. 𝐌 = 𝐫 × 𝐅 = | 𝐞 x 𝐞 y 𝐞 z x 0 0 0 - F 0 | = - F x 𝐞 z . \mathbf{M}=\mathbf{r}\times\mathbf{F}=\left|\begin{matrix}\mathbf{e}_{x}&% \mathbf{e}_{y}&\mathbf{e}_{z}\\ x&0&0\\ 0&-F&0\end{matrix}\right|=-Fx\,\mathbf{e}_{z}\,.
  13. 𝐞 z \mathbf{e}_{z}
  14. M z = 𝐞 z 𝐌 = - F x . M_{z}=\mathbf{e}_{z}\cdot\mathbf{M}=-Fx\,.
  15. 𝐞 x , 𝐞 y , 𝐞 z \mathbf{e}_{x},\mathbf{e}_{y},\mathbf{e}_{z}
  16. 𝐄 x = 𝐞 x , 𝐄 y = - 𝐞 z , 𝐄 z = 𝐞 y \mathbf{E}_{x}=\mathbf{e}_{x},\mathbf{E}_{y}=-\mathbf{e}_{z},\mathbf{E}_{z}=% \mathbf{e}_{y}
  17. 𝐅 = 0 𝐄 x + 0 𝐄 y - F 𝐄 z and 𝐫 = x 𝐄 x + 0 𝐄 y + 0 𝐄 z . \mathbf{F}=0\,\mathbf{E}_{x}+0\,\mathbf{E}_{y}-F\,\mathbf{E}_{z}\quad\,\text{% and}\quad\mathbf{r}=x\,\mathbf{E}_{x}+0\,\mathbf{E}_{y}+0\,\mathbf{E}_{z}\,.
  18. 𝐌 = 𝐫 × 𝐅 = | 𝐄 x 𝐄 y 𝐄 z x 0 0 0 0 - F | = F x 𝐄 y and M y = 𝐄 y 𝐌 = F x . \mathbf{M}=\mathbf{r}\times\mathbf{F}=\left|\begin{matrix}\mathbf{E}_{x}&% \mathbf{E}_{y}&\mathbf{E}_{z}\\ x&0&0\\ 0&0&-F\end{matrix}\right|=Fx\,\mathbf{E}_{y}\quad\,\text{and}\quad M_{y}=% \mathbf{E}_{y}\cdot\mathbf{M}=Fx\,.
  19. 𝐌 x = A 𝐫 × ( σ x x 𝐞 x + σ x y 𝐞 y + σ x z 𝐞 z ) d A where 𝐫 = y 𝐞 y + z 𝐞 z . \mathbf{M}_{x}=\int_{A}\mathbf{r}\times(\sigma_{xx}\mathbf{e}_{x}+\sigma_{xy}% \mathbf{e}_{y}+\sigma_{xz}\mathbf{e}_{z})\,dA\quad\,\text{where}\quad\mathbf{r% }=y\,\mathbf{e}_{y}+z\,\mathbf{e}_{z}\,.
  20. 𝐌 x = A ( - y σ x x 𝐞 z + y σ x z 𝐞 x + z σ x x 𝐞 y - z σ x y 𝐞 x ) d A = : M x x 𝐞 x + M x y 𝐞 y + M x z 𝐞 z . \mathbf{M}_{x}=\int_{A}\left(-y\sigma_{xx}\mathbf{e}_{z}+y\sigma_{xz}\mathbf{e% }_{x}+z\sigma_{xx}\mathbf{e}_{y}-z\sigma_{xy}\mathbf{e}_{x}\right)dA=:M_{xx}\,% \mathbf{e}_{x}+M_{xy}\,\mathbf{e}_{y}+M_{xz}\,\mathbf{e}_{z}\,.
  21. [ M x x M x y M x z ] := A [ y σ x z - z σ x y z σ x x - y σ x x ] d A . \begin{bmatrix}M_{xx}\\ M_{xy}\\ M_{xz}\end{bmatrix}:=\int_{A}\begin{bmatrix}y\sigma_{xz}-z\sigma_{xy}\\ z\sigma_{xx}\\ -y\sigma_{xx}\end{bmatrix}\,dA\,.
  22. h h
  23. - F 𝐞 y -F\mathbf{e}_{y}
  24. 𝐑 O = R O 𝐞 y \mathbf{R}_{O}=R_{O}\mathbf{e}_{y}
  25. 𝐑 B = R B 𝐞 y \mathbf{R}_{B}=R_{B}\mathbf{e}_{y}
  26. R O + R B - F = 0 and - 𝐫 A × 𝐑 O + 𝐫 B × 𝐑 B = 0 . R_{O}+R_{B}-F=0\quad\,\text{and}\quad-\mathbf{r}_{A}\times\mathbf{R}_{O}+% \mathbf{r}_{B}\times\mathbf{R}_{B}=\mathbf{0}\,.
  27. L L
  28. 𝐫 A = x A 𝐞 x and 𝐫 B = ( L - x A ) 𝐞 x . \mathbf{r}_{A}=x_{A}\mathbf{e}_{x}\quad\,\text{and}\quad\mathbf{r}_{B}=(L-x_{A% })\mathbf{e}_{x}\,.
  29. R O = ( 1 - x A L ) F and R B = x A L F . R_{O}=\left(1-\frac{x_{A}}{L}\right)F\quad\,\text{and}\quad R_{B}=\frac{x_{A}}% {L}\,F\,.
  30. 𝐌 = - ( 𝐫 X - 𝐫 A ) × 𝐅 - 𝐫 X × 𝐑 O = [ ( x A - x ) 𝐞 x ] × ( - F 𝐞 y ) - ( x 𝐞 x ) × ( R O 𝐞 y ) . \mathbf{M}=-(\mathbf{r}_{X}-\mathbf{r}_{A})\times\mathbf{F}-\mathbf{r}_{X}% \times\mathbf{R}_{O}=\left[(x_{A}-x)\mathbf{e}_{x}\right]\times\left(-F\mathbf% {e}_{y}\right)-\left(x\mathbf{e}_{x}\right)\times\left(R_{O}\mathbf{e}_{y}% \right)\,.
  31. 𝐌 = | 𝐞 x 𝐞 y 𝐞 z x A - x 0 0 0 - F 0 | - | 𝐞 x 𝐞 y 𝐞 z x 0 0 0 R 0 0 | = F ( x - x A ) 𝐞 z - R 0 x 𝐞 z = - F x A L ( L - x ) 𝐞 z . \mathbf{M}=\left|\begin{matrix}\mathbf{e}_{x}&\mathbf{e}_{y}&\mathbf{e}_{z}\\ x_{A}-x&0&0\\ 0&-F&0\end{matrix}\right|-\left|\begin{matrix}\mathbf{e}_{x}&\mathbf{e}_{y}&% \mathbf{e}_{z}\\ x&0&0\\ 0&R_{0}&0\end{matrix}\right|=F(x-x_{A})\,\mathbf{e}_{z}-R_{0}x\,\mathbf{e}_{z}% =-\frac{Fx_{A}}{L}(L-x)\,\mathbf{e}_{z}\,.
  32. 𝐌 x z = - [ z - h / 2 h / 2 y σ x x d y d z ] 𝐞 z . \mathbf{M}_{xz}=-\left[\int_{z}\int_{-h/2}^{h/2}y\,\sigma_{xx}\,dy\,dz\right]% \mathbf{e}_{z}\,.
  33. 𝐞 z \mathbf{e}_{z}
  34. 𝐌 + 𝐌 x z = 𝟎 or - F x A L ( L - x ) + M x z = 0 or M x z = F x A L ( L - x ) . \mathbf{M}+\mathbf{M}_{xz}=\mathbf{0}\quad\,\text{or}\quad-\frac{Fx_{A}}{L}(L-% x)+M_{xz}=0\quad\,\text{or}\quad M_{xz}=\frac{Fx_{A}}{L}(L-x)\,.
  35. x = x A x=x_{A}
  36. M x z = F x A ( L - x A ) / L M_{xz}=Fx_{A}(L-x_{A})/L
  37. - σ 0 -\sigma_{0}
  38. σ 0 \sigma_{0}
  39. σ x x ( y ) = - y σ 0 \sigma_{xx}(y)=-y\sigma_{0}
  40. M x z = - [ z - h / 2 h / 2 y ( - y σ 0 ) d y d z ] = σ 0 I M_{xz}=-\left[\int_{z}\int_{-h/2}^{h/2}y\,(-y\sigma_{0})\,dy\,dz\right]=\sigma% _{0}\,I
  41. I I
  42. M x z M_{xz}
  43. 𝐌 x z = [ z - h / 2 h / 2 y σ x x d y d z ] 𝐞 z . \mathbf{M}_{xz}=\left[\int_{z}\int_{-h/2}^{h/2}y\,\sigma_{xx}\,dy\,dz\right]% \mathbf{e}_{z}\,.
  44. y y

Benjamin_Graham_formula.html

  1. V * = E P S × ( 8.5 + 2 g ) V*=EPS\times(8.5+2g)
  2. V * = E P S × ( 8.5 + 2 g ) × 4.4 Y V*=\cfrac{EPS\times(8.5+2g)\times 4.4}{Y}
  3. R G V = V * P RGV=\cfrac{V*}{P}

Berlekamp's_algorithm.html

  1. f ( x ) f(x)
  2. n n
  3. 𝔽 q \mathbb{F}_{q}
  4. g ( x ) g(x)
  5. g ( x ) g(x)
  6. f ( x ) f(x)
  7. f ( x ) f(x)
  8. f ( x ) f(x)
  9. R = 𝔽 q [ x ] f ( x ) . R=\frac{\mathbb{F}_{q}[x]}{\langle f(x)\rangle}.
  10. g ( x ) R g(x)\in R
  11. g ( x ) q g ( x ) ( mod f ( x ) ) . g(x)^{q}\equiv g(x)\;\;(\mathop{{\rm mod}}f(x)).\,
  12. n n
  13. 𝔽 q \mathbb{F}_{q}
  14. g ( x ) g(x)
  15. f ( x ) = s 𝔽 q gcd ( f ( x ) , g ( x ) - s ) . f(x)=\prod_{s\in\mathbb{F}_{q}}\gcd(f(x),g(x)-s).
  16. f ( x ) f(x)
  17. g ( x ) g(x)
  18. ( n + 1 ) × ( n + 1 ) (n+1)\times(n+1)
  19. 𝔽 q \mathbb{F}_{q}
  20. 𝒬 \mathcal{Q}
  21. 𝒬 = [ q i , j ] \mathcal{Q}=[q_{i,j}]
  22. q i , j q_{i,j}
  23. j j
  24. x i q x^{iq}
  25. f ( x ) f(x)
  26. x i q q i , n x n + q i , n - 1 x n - 1 + + q i , 0 ( mod f ( x ) ) . x^{iq}\equiv q_{i,n}x^{n}+q_{i,n-1}x^{n-1}+\ldots+q_{i,0}\;\;(\mathop{{\rm mod% }}f(x)).\,
  27. g ( x ) R g(x)\in R
  28. g ( x ) = g n x n + g n - 1 x n - 1 + + g 0 , g(x)=g_{n}x^{n}+g_{n-1}x^{n-1}+\ldots+g_{0},\,
  29. g = ( g 0 , g 1 , , g n ) . g=(g_{0},g_{1},\ldots,g_{n}).\,
  30. g 𝒬 g\mathcal{Q}
  31. g ( x ) q g(x)^{q}
  32. f ( x ) f(x)
  33. g ( x ) R g(x)\in R
  34. g ( 𝒬 - I ) = 0 g(\mathcal{Q}-I)=0
  35. I I
  36. ( n + 1 ) × ( n + 1 ) (n+1)\times(n+1)
  37. 𝒬 - I \mathcal{Q}-I
  38. 𝒬 - I \mathcal{Q}-I
  39. g ( x ) g(x)
  40. 𝔽 p n \mathbb{F}_{p^{n}}
  41. p p
  42. n 2 n\geq 2
  43. 𝔽 p n \mathbb{F}_{p^{n}}
  44. 𝔽 p \mathbb{F}_{p}
  45. n n

Bernays–Schönfinkel_class.html

  1. * * \exists^{*}\forall^{*}

Bertha_Swirles.html

  1. γ \gamma

Bertrand's_theorem.html

  1. V ( 𝐫 ) = - k r , V(\mathbf{r})=\frac{-k}{r},
  2. V ( 𝐫 ) = 1 2 k r 2 . V(\mathbf{r})=\frac{1}{2}kr^{2}.
  3. m d 2 r d t 2 - m r ω 2 = m d 2 r d t 2 - L 2 m r 3 = - d V d r m\frac{d^{2}r}{dt^{2}}-mr\omega^{2}=m\frac{d^{2}r}{dt^{2}}-\frac{L^{2}}{mr^{3}% }=-\frac{dV}{dr}
  4. ω d θ d t \omega\equiv\frac{d\theta}{dt}
  5. d V d r \frac{dV}{dr}
  6. d d t = L m r 2 d d θ \frac{d}{dt}=\frac{L}{mr^{2}}\frac{d}{d\theta}
  7. L r 2 d d θ ( L m r 2 d r d θ ) - L 2 m r 3 = - d V d r \frac{L}{r^{2}}\frac{d}{d\theta}\left(\frac{L}{mr^{2}}\frac{dr}{d\theta}\right% )-\frac{L^{2}}{mr^{3}}=-\frac{dV}{dr}
  8. u 1 r u\equiv\frac{1}{r}
  9. m r 2 L 2 \frac{mr^{2}}{L^{2}}
  10. d 2 u d θ 2 + u = - m L 2 d d u V ( 1 / u ) \frac{d^{2}u}{d\theta^{2}}+u=-\frac{m}{L^{2}}\frac{d}{du}V(1/u)
  11. d 2 u d θ 2 + u = J ( u ) - m L 2 d d u V ( 1 / u ) = - m L 2 u 2 f ( 1 / u ) \frac{d^{2}u}{d\theta^{2}}+u=J(u)\equiv-\frac{m}{L^{2}}\frac{d}{du}V(1/u)=-% \frac{m}{L^{2}u^{2}}f(1/u)
  12. u 0 = J ( u 0 ) = - m L 2 u 0 2 f ( 1 / u 0 ) u_{0}=J(u_{0})=-\frac{m}{L^{2}u_{0}^{2}}f(1/u_{0})
  13. u 0 1 / r 0 u_{0}\equiv 1/r_{0}
  14. η u - u 0 \eta\equiv u-u_{0}
  15. J ( u ) J ( u 0 ) + η J ( u 0 ) + 1 2 η 2 J ′′ ( u 0 ) + 1 6 η 3 J ′′′ ( u 0 ) + J(u)\approx J(u_{0})+\eta J^{\prime}(u_{0})+\frac{1}{2}\eta^{2}J^{\prime\prime% }(u_{0})+\frac{1}{6}\eta^{3}J^{\prime\prime\prime}(u_{0})+\cdots
  16. d 2 η d θ 2 + η = η J ( u 0 ) + 1 2 η 2 J ′′ ( u 0 ) + 1 6 η 3 J ′′′ ( u 0 ) + \frac{d^{2}\eta}{d\theta^{2}}+\eta=\eta J^{\prime}(u_{0})+\frac{1}{2}\eta^{2}J% ^{\prime\prime}(u_{0})+\frac{1}{6}\eta^{3}J^{\prime\prime\prime}(u_{0})+\cdots
  17. d 2 η d θ 2 + β 2 η = 1 2 η 2 J ′′ ( u 0 ) + 1 6 η 3 J ′′′ ( u 0 ) + \frac{d^{2}\eta}{d\theta^{2}}+\beta^{2}\eta=\frac{1}{2}\eta^{2}J^{\prime\prime% }(u_{0})+\frac{1}{6}\eta^{3}J^{\prime\prime\prime}(u_{0})+\cdots
  18. β 2 1 - J ( u 0 ) \beta^{2}\equiv 1-J^{\prime}(u_{0})
  19. η ( θ ) = h 1 cos ( β θ ) \eta(\theta)=h_{1}\cos\left(\beta\theta\right)
  20. J ( u 0 ) = 2 u 0 [ m L 2 u 0 2 f ( 1 / u 0 ) ] - [ m L 2 u 0 2 f ( 1 / u 0 ) ] 1 f ( 1 / u 0 ) d f d u = - 2 + u 0 f ( 1 / u 0 ) d f d u = 1 - β 2 J^{\prime}(u_{0})=\frac{2}{u_{0}}\left[\frac{m}{L^{2}u_{0}^{2}}f(1/u_{0})% \right]-\left[\frac{m}{L^{2}u_{0}^{2}}f(1/u_{0})\right]\frac{1}{f(1/u_{0})}% \frac{df}{du}=-2+\frac{u_{0}}{f(1/u_{0})}\frac{df}{du}=1-\beta^{2}
  21. d f d u \frac{df}{du}
  22. ( 1 / u 0 ) (1/u_{0})
  23. d f d r = ( β 2 - 3 ) f r \frac{df}{dr}=\left(\beta^{2}-3\right)\frac{f}{r}
  24. f ( r ) = - k r 3 - β 2 f(r)=-\frac{k}{r^{3-\beta^{2}}}
  25. J ( u ) = m k L 2 u 1 - β 2 J(u)=\frac{mk}{L^{2}}u^{1-\beta^{2}}
  26. η ( θ ) = h 0 + h 1 cos β θ + h 2 cos 2 β θ + h 3 cos 3 β θ + \eta(\theta)=h_{0}+h_{1}\cos\beta\theta+h_{2}\cos 2\beta\theta+h_{3}\cos 3% \beta\theta+\cdots
  27. h 1 2 h_{1}^{2}
  28. h 1 3 h_{1}^{3}
  29. h 0 , h 2 , h 3 , h_{0},h_{2},h_{3},\ldots
  30. h 0 = h 1 2 J ′′ ( u 0 ) 4 β 2 h_{0}=h_{1}^{2}\frac{J^{\prime\prime}(u_{0})}{4\beta^{2}}
  31. h 2 = - h 1 2 J ′′ ( u 0 ) 12 β 2 h_{2}=-h_{1}^{2}\frac{J^{\prime\prime}(u_{0})}{12\beta^{2}}
  32. h 3 = - 1 8 β 2 [ h 1 h 2 J ′′ ( u 0 ) 2 + h 1 3 J ′′′ ( u 0 ) 24 ] h_{3}=-\frac{1}{8\beta^{2}}\left[h_{1}h_{2}\frac{J^{\prime\prime}(u_{0})}{2}+h% _{1}^{3}\frac{J^{\prime\prime\prime}(u_{0})}{24}\right]
  33. 0 = ( 2 h 1 h 0 + h 1 h 2 ) J ′′ ( u 0 ) 2 + h 1 3 J ′′′ ( u 0 ) 8 = h 1 3 24 β 2 ( 3 β 2 J ′′′ ( u 0 ) + 5 J ′′ ( u 0 ) 2 ) 0=\left(2h_{1}h_{0}+h_{1}h_{2}\right)\frac{J^{\prime\prime}(u_{0})}{2}+h_{1}^{% 3}\frac{J^{\prime\prime\prime}(u_{0})}{8}=\frac{h_{1}^{3}}{24\beta^{2}}(3\beta% ^{2}J^{\prime\prime\prime}(u_{0})+5J^{\prime\prime}(u_{0})^{2})
  34. J ′′ ( u 0 ) = - β 2 ( 1 - β 2 ) u 0 J^{\prime\prime}(u_{0})=-\frac{\beta^{2}(1-\beta^{2})}{u_{0}}
  35. J ′′′ ( u 0 ) = β 2 ( 1 - β 2 ) ( 1 + β 2 ) u 0 2 J^{\prime\prime\prime}(u_{0})=\frac{\beta^{2}(1-\beta^{2})(1+\beta^{2})}{u_{0}% ^{2}}
  36. β 2 ( 1 - β 2 ) ( 4 - β 2 ) = 0 \beta^{2}\left(1-\beta^{2}\right)\left(4-\beta^{2}\right)=0
  37. V ( 𝐫 ) = - k r = - k u V(\mathbf{r})=\frac{-k}{r}=-ku
  38. d 2 u d θ 2 + u = - m L 2 d d u V ( 1 / u ) = k m L 2 \frac{d^{2}u}{d\theta^{2}}+u=-\frac{m}{L^{2}}\frac{d}{du}V(1/u)=\frac{km}{L^{2}}
  39. k m L 2 \frac{km}{L^{2}}
  40. u 1 r = k m L 2 [ 1 + e cos ( θ - θ 0 ) ] u\equiv\frac{1}{r}=\frac{km}{L^{2}}\left[1+e\cos\left(\theta-\theta_{0}\right)\right]
  41. e = 1 + 2 E L 2 k 2 m e=\sqrt{1+\frac{2EL^{2}}{k^{2}m}}
  42. E = - k 2 m 2 L 2 E=-\frac{k^{2}m}{2L^{2}}
  43. V ( 𝐫 ) = 1 2 k r 2 = 1 2 k ( x 2 + y 2 + z 2 ) V(\mathbf{r})=\frac{1}{2}kr^{2}=\frac{1}{2}k\left(x^{2}+y^{2}+z^{2}\right)
  44. d 2 x d t 2 + ω 0 2 x = 0 \frac{d^{2}x}{dt^{2}}+\omega_{0}^{2}x=0
  45. d 2 y d t 2 + ω 0 2 y = 0 \frac{d^{2}y}{dt^{2}}+\omega_{0}^{2}y=0
  46. d 2 z d t 2 + ω 0 2 z = 0 \frac{d^{2}z}{dt^{2}}+\omega_{0}^{2}z=0
  47. ω 0 2 k m \omega_{0}^{2}\equiv\frac{k}{m}
  48. x = A x cos ( ω 0 t + ϕ x ) x=A_{x}\cos\left(\omega_{0}t+\phi_{x}\right)
  49. y = A y cos ( ω 0 t + ϕ y ) y=A_{y}\cos\left(\omega_{0}t+\phi_{y}\right)
  50. z = A z cos ( ω 0 t + ϕ z ) z=A_{z}\cos\left(\omega_{0}t+\phi_{z}\right)
  51. T 2 π ω 0 T\equiv\frac{2\pi}{\omega_{0}}

Bessel_polynomials.html

  1. y n ( x ) = k = 0 n ( n + k ) ! ( n - k ) ! k ! ( x 2 ) k y_{n}(x)=\sum_{k=0}^{n}\frac{(n+k)!}{(n-k)!k!}\,\left(\frac{x}{2}\right)^{k}
  2. θ n ( x ) = x n y n ( 1 / x ) = k = 0 n ( n + k ) ! ( n - k ) ! k ! x n - k 2 k \theta_{n}(x)=x^{n}\,y_{n}(1/x)=\sum_{k=0}^{n}\frac{(n+k)!}{(n-k)!k!}\,\frac{x% ^{n-k}}{2^{k}}
  3. y 3 ( x ) = 15 x 3 + 15 x 2 + 6 x + 1 y_{3}(x)=15x^{3}+15x^{2}+6x+1\,
  4. θ 3 ( x ) = x 3 + 6 x 2 + 15 x + 15 \theta_{3}(x)=x^{3}+6x^{2}+15x+15\,
  5. y n ( x ) = x n θ n ( 1 / x ) y_{n}(x)=\,x^{n}\theta_{n}(1/x)\,
  6. θ n ( x ) = 2 π x n + 1 / 2 e x K n + 1 2 ( x ) \theta_{n}(x)=\sqrt{\frac{2}{\pi}}\,x^{n+1/2}e^{x}K_{n+\frac{1}{2}}(x)
  7. y n ( x ) = 2 π x e 1 / x K n + 1 2 ( 1 / x ) y_{n}(x)=\sqrt{\frac{2}{\pi x}}\,e^{1/x}K_{n+\frac{1}{2}}(1/x)
  8. y n ( x ) = 2 F 0 ( - n , n + 1 ; ; - x / 2 ) = ( 2 x ) - n U ( - n , - 2 n , 2 x ) = ( 2 x ) n + 1 U ( n + 1 , 2 n + 2 , 2 x ) . y_{n}(x)=\,_{2}F_{0}(-n,n+1;;-x/2)=\left(\frac{2}{x}\right)^{-n}U\left(-n,-2n,% \frac{2}{x}\right)=\left(\frac{2}{x}\right)^{n+1}U\left(n+1,2n+2,\frac{2}{x}% \right).
  9. θ n ( x ) = n ! ( - 2 ) n L n - 2 n - 1 ( 2 x ) \theta_{n}(x)=\frac{n!}{(-2)^{n}}\,L_{n}^{-2n-1}(2x)
  10. θ n ( x ) = ( - 2 n ) n ( - 2 ) n 1 F 1 ( - n ; - 2 n ; - 2 x ) \theta_{n}(x)=\frac{(-2n)_{n}}{(-2)^{n}}\,\,_{1}F_{1}(-n;-2n;-2x)
  11. ( 2 x ) n n ! = ( - 1 ) n j = 0 n n + 1 j + 1 ( j + 1 n - j ) L j - 2 j - 1 ( 2 x ) = 2 n n ! i = 0 n i ! ( 2 i + 1 ) ( 2 n + 1 n - i ) x i L i ( - 2 i - 1 ) ( 1 x ) . \frac{(2x)^{n}}{n!}=(-1)^{n}\sum_{j=0}^{n}\frac{n+1}{j+1}{j+1\choose n-j}L_{j}% ^{-2j-1}(2x)=\frac{2^{n}}{n!}\sum_{i=0}^{n}i!(2i+1){2n+1\choose n-i}x^{i}L_{i}% ^{(-2i-1)}\left(\frac{1}{x}\right).
  12. n = 0 2 π x n + 1 2 e x K n - 1 2 ( x ) t n n ! = e x ( 1 - 1 - 2 t ) . \sum_{n=0}\sqrt{\frac{2}{\pi}}x^{n+\frac{1}{2}}e^{x}K_{n-\frac{1}{2}}(x)\frac{% t^{n}}{n!}=e^{x(1-\sqrt{1-2t})}.
  13. y 0 ( x ) = 1 y_{0}(x)=1\,
  14. y 1 ( x ) = x + 1 y_{1}(x)=x+1\,
  15. y n ( x ) = ( 2 n - 1 ) x y n - 1 ( x ) + y n - 2 ( x ) y_{n}(x)=(2n\!-\!1)x\,y_{n-1}(x)+y_{n-2}(x)\,
  16. θ 0 ( x ) = 1 \theta_{0}(x)=1\,
  17. θ 1 ( x ) = x + 1 \theta_{1}(x)=x+1\,
  18. θ n ( x ) = ( 2 n - 1 ) θ n - 1 ( x ) + x 2 θ n - 2 ( x ) \theta_{n}(x)=(2n\!-\!1)\theta_{n-1}(x)+x^{2}\theta_{n-2}(x)\,
  19. x 2 d 2 y n ( x ) d x 2 + 2 ( x + 1 ) d y n ( x ) d x - n ( n + 1 ) y n ( x ) = 0 x^{2}\frac{d^{2}y_{n}(x)}{dx^{2}}+2(x\!+\!1)\frac{dy_{n}(x)}{dx}-n(n+1)y_{n}(x% )=0
  20. x d 2 θ n ( x ) d x 2 - 2 ( x + n ) d θ n ( x ) d x + 2 n θ n ( x ) = 0 x\frac{d^{2}\theta_{n}(x)}{dx^{2}}-2(x\!+\!n)\frac{d\theta_{n}(x)}{dx}+2n\,% \theta_{n}(x)=0
  21. y n ( x ; α , β ) := ( - 1 ) n n ! ( x β ) n L n ( 1 - 2 n - α ) ( β x ) , y_{n}(x;\alpha,\beta):=(-1)^{n}n!\left(\frac{x}{\beta}\right)^{n}L_{n}^{(1-2n-% \alpha)}\left(\frac{\beta}{x}\right),
  22. θ n ( x ; α , β ) := n ! ( - β ) n L n ( 1 - 2 n - α ) ( β x ) = x n y n ( 1 x ; α , β ) . \theta_{n}(x;\alpha,\beta):=\frac{n!}{(-\beta)^{n}}L_{n}^{(1-2n-\alpha)}(\beta x% )=x^{n}y_{n}\left(\frac{1}{x};\alpha,\beta\right).
  23. ρ ( x ; α , β ) := 1 F 1 ( 1 , α - 1 , - β x ) \rho(x;\alpha,\beta):=\,_{1}F_{1}\left(1,\alpha-1,-\frac{\beta}{x}\right)
  24. 0 = c ρ ( x ; α , β ) y n ( x ; α , β ) y m ( x ; α , β ) d x 0=\oint_{c}\rho(x;\alpha,\beta)y_{n}(x;\alpha,\beta)y_{m}(x;\alpha,\beta)% \mathrm{d}x
  25. B n ( α , β ) ( x ) = a n ( α , β ) x α e - β x ( d d x ) n ( x α + 2 n e - β x ) B_{n}^{(\alpha,\beta)}(x)=\frac{a_{n}^{(\alpha,\beta)}}{x^{\alpha}e^{-\frac{% \beta}{x}}}\left(\frac{d}{dx}\right)^{n}(x^{\alpha+2n}e^{-\frac{\beta}{x}})
  26. x 2 d 2 B n , m ( α , β ) ( x ) d x 2 + [ ( α + 2 ) x + β ] d B n , m ( α , β ) ( x ) d x - [ n ( α + n + 1 ) + m β x ] B n , m ( α , β ) ( x ) = 0 x^{2}\frac{d^{2}B_{n,m}^{(\alpha,\beta)}(x)}{dx^{2}}+[(\alpha+2)x+\beta]\frac{% dB_{n,m}^{(\alpha,\beta)}(x)}{dx}-\left[n(\alpha+n+1)+\frac{m\beta}{x}\right]B% _{n,m}^{(\alpha,\beta)}(x)=0
  27. 0 m n 0\leq m\leq n
  28. B n , m ( α , β ) ( x ) = a n , m ( α , β ) x α + m e - β x ( d d x ) n - m ( x α + 2 n e - β x ) B_{n,m}^{(\alpha,\beta)}(x)=\frac{a_{n,m}^{(\alpha,\beta)}}{x^{\alpha+m}e^{-% \frac{\beta}{x}}}\left(\frac{d}{dx}\right)^{n-m}(x^{\alpha+2n}e^{-\frac{\beta}% {x}})
  29. y 0 ( x ) = 1 y 1 ( x ) = x + 1 y 2 ( x ) = 3 x 2 + 3 x + 1 y 3 ( x ) = 15 x 3 + 15 x 2 + 6 x + 1 y 4 ( x ) = 105 x 4 + 105 x 3 + 45 x 2 + 10 x + 1 y 5 ( x ) = 945 x 5 + 945 x 4 + 420 x 3 + 105 x 2 + 15 x + 1 \begin{aligned}\displaystyle y_{0}(x)&\displaystyle=1\\ \displaystyle y_{1}(x)&\displaystyle=x+1\\ \displaystyle y_{2}(x)&\displaystyle=3x^{2}+3x+1\\ \displaystyle y_{3}(x)&\displaystyle=15x^{3}+15x^{2}+6x+1\\ \displaystyle y_{4}(x)&\displaystyle=105x^{4}+105x^{3}+45x^{2}+10x+1\\ \displaystyle y_{5}(x)&\displaystyle=945x^{5}+945x^{4}+420x^{3}+105x^{2}+15x+1% \end{aligned}

Bessel_process.html

  1. X t = W t , X_{t}=\|W_{t}\|,
  2. d X t = d Z t + n - 1 2 d t X t dX_{t}=dZ_{t}+\frac{n-1}{2}\frac{dt}{X_{t}}
  3. n n

Bessel–Clifford_function.html

  1. π ( x ) = 1 Π ( x ) = 1 Γ ( x + 1 ) \pi(x)=\frac{1}{\Pi(x)}=\frac{1}{\Gamma(x+1)}
  2. 𝒞 n ( z ) = k = 0 π ( k + n ) z k k ! {\mathcal{C}}_{n}(z)=\sum_{k=0}^{\infty}\pi(k+n)\frac{z^{k}}{k!}
  3. 𝒞 n ( x ) {\mathcal{C}}_{n}(x)
  4. x y ′′ + ( n + 1 ) y = y . xy^{\prime\prime}+(n+1)y^{\prime}=y.\qquad
  5. 𝒞 n ( z ) = π ( n ) 0 F 1 ( ; n + 1 ; z ) . {\mathcal{C}}_{n}(z)=\pi(n)\ _{0}F_{1}(;n+1;z).
  6. J n ( z ) = ( z 2 ) n 𝒞 n ( - z 2 4 ) ; J_{n}(z)=\left(\frac{z}{2}\right)^{n}{\mathcal{C}}_{n}\left(-\frac{z^{2}}{4}% \right);
  7. I n ( z ) = ( z 2 ) n 𝒞 n ( z 2 4 ) . I_{n}(z)=\left(\frac{z}{2}\right)^{n}{\mathcal{C}}_{n}\left(\frac{z^{2}}{4}% \right).
  8. 𝒞 n ( z ) = z - n / 2 I n ( 2 z ) ; {\mathcal{C}}_{n}(z)=z^{-n/2}I_{n}(2\sqrt{z});
  9. 𝒞 {\mathcal{C}}
  10. d d x 𝒞 n ( x ) = 𝒞 n + 1 ( x ) . \frac{d}{dx}{\mathcal{C}}_{n}(x)={\mathcal{C}}_{n+1}(x).
  11. 𝒞 {\mathcal{C}}
  12. x 𝒞 n + 2 ( x ) + ( n + 1 ) 𝒞 n + 1 ( x ) = 𝒞 n ( x ) , x{\mathcal{C}}_{n+2}(x)+(n+1){\mathcal{C}}_{n+1}(x)={\mathcal{C}}_{n}(x),
  13. 𝒞 n + 1 ( x ) 𝒞 n ( x ) = 1 n + 1 + x n + 2 + x n + 3 + x . \frac{{\mathcal{C}}_{n+1}(x)}{{\mathcal{C}}_{n}(x)}=\cfrac{1}{n+1+\cfrac{x}{n+% 2+\cfrac{x}{n+3+\cfrac{x}{\ddots}}}}.
  14. x y ′′ + ( n + 1 ) y = y xy^{\prime\prime}+(n+1)y^{\prime}=y\qquad
  15. 𝒞 {\mathcal{C}}
  16. 𝒦 n ( x ) = 1 2 0 exp ( - t - x t ) d t t n + 1 {\mathcal{K}}_{n}(x)=\frac{1}{2}\int_{0}^{\infty}\exp\left(-t-\frac{x}{t}% \right)\frac{dt}{t^{n+1}}
  17. ( x ) > 0 \Re(x)>0
  18. 𝒦 {\mathcal{K}}
  19. K n ( x ) = ( x 2 ) n 𝒦 n ( x 2 4 ) . K_{n}(x)=\left(\frac{x}{2}\right)^{n}{\mathcal{K}}_{n}\left(\frac{x^{2}}{4}% \right).
  20. Y n ( x ) = ( x 2 ) n 𝒦 n ( - x 2 4 ) . Y_{n}(x)=\left(\frac{x}{2}\right)^{n}{\mathcal{K}}_{n}\left(-\frac{x^{2}}{4}% \right).
  21. 𝒦 n ( x ) = x - n / 2 K n ( 2 x ) . {\mathcal{K}}_{n}(x)=x^{-n/2}K_{n}(2\sqrt{x}).
  22. 𝒞 {\mathcal{C}}
  23. 𝒦 {\mathcal{K}}
  24. 𝒞 n {\mathcal{C}}_{n}
  25. exp ( t + z t ) = n = - t n 𝒞 n ( z ) . \exp\left(t+\frac{z}{t}\right)=\sum_{n=-\infty}^{\infty}t^{n}{\mathcal{C}}_{n}% (z).
  26. 𝒞 n {\mathcal{C}}_{n}
  27. 𝒞 n ( z ) = 1 2 π i C exp ( z + z / t ) t n + 1 d t = 1 2 π 0 2 π exp ( z ( 1 + exp ( - i θ ) ) - n i θ ) ) d θ . {\mathcal{C}}_{n}(z)=\frac{1}{2\pi i}\oint_{C}\frac{\exp(z+z/t)}{t^{n+1}}\,dt=% \frac{1}{2\pi}\int_{0}^{2\pi}\exp(z(1+\exp(-i\theta))-ni\theta))\,d\theta.