wpmath0000003_1

Big_push_model.html

  1. l l
  2. n n
  3. l / n {l}/{n}
  4. l / n {l}/{n}
  5. h h
  6. h h
  7. l / n {l/n}
  8. 1 / m {1}/{m}
  9. m m
  10. m m
  11. D 1 = 1 / n D_{1}={1}/{n}
  12. w 1 w_{1}
  13. D 1 D_{1}
  14. l * l^{*}
  15. w 1 l * w_{1}l^{*}
  16. w 1 l * < D 1 w_{1}l^{*}<D_{1}
  17. w 1 l * w_{1}l^{*}
  18. D 1 D_{1}
  19. w 2 w_{2}
  20. D 1 D_{1}
  21. w 2 l * > D 1 w_{2}l^{*}>D_{1}
  22. w 2 l * w_{2}l^{*}
  23. D 1 D_{1}
  24. D 2 D_{2}
  25. w 2 l * < D 2 w_{2}l^{*}<D_{2}

Big_Two.html

  1. ( 4 1 ) 13 ( 52 13 ) = 0.010568 % = 1 : 9462 \frac{{\textstyle\left({{4}\atop{1}}\right)}^{13}}{{\textstyle\left({{52}\atop% {13}}\right)}}=0.010568\%=1:9462
  2. 4 ( 52 13 ) = 0.000000000629908 % = 1 : 158753389900 \frac{4}{{\textstyle\left({{52}\atop{13}}\right)}}=0.000000000629908\%=1:15875% 3389900
  3. ( 48 9 ) ( 52 13 ) = 0.264106 % = 1 : 379 \frac{{\textstyle\left({{48}\atop{9}}\right)}}{{\textstyle\left({{52}\atop{13}% }\right)}}=0.264106\%=1:379

Bilinear_form.html

  1. B ( 𝐯 , 𝐰 ) = x T A y = i , j = 1 n a i j x i y j . B(\mathbf{v},\mathbf{w})=x^{\mathrm{T}}Ay=\sum_{i,j=1}^{n}a_{ij}x_{i}y_{j}.
  2. B ( x , y ) = 0 B(x,y)=0\,
  3. y V y\in V
  4. B ( x , y ) = 0 B(x,y)=0\,
  5. x V x\in V
  6. B ± = 1 2 ( B ± B * ) B^{\pm}=\frac{1}{2}(B\pm B^{*})
  7. W = { 𝐯 B ( 𝐯 , 𝐰 ) = 0 𝐰 W } . W^{\perp}=\{\mathbf{v}\mid B(\mathbf{v},\mathbf{w})=0\ \forall\mathbf{w}\in W% \}\ .
  8. k = 1 p x k y k - k = p + 1 n x k y k \sum_{k=1}^{p}x_{k}y_{k}-\sum_{k=p+1}^{n}x_{k}y_{k}
  9. B ( 𝐮 , 𝐯 ) C 𝐮 𝐯 . B(\mathbf{u},\mathbf{v})\leq C\|\mathbf{u}\|\|\mathbf{v}\|.
  10. B ( 𝐮 , 𝐮 ) c 𝐮 2 . B(\mathbf{u},\mathbf{u})\geq c\|\mathbf{u}\|^{2}.

Bimetallic_strip.html

  1. κ = 6 E 1 E 2 ( h 1 + h 2 ) h 1 h 2 ϵ E 1 2 h 1 4 + 4 E 1 E 2 h 1 3 h 2 + 6 E 1 E 2 h 1 2 h 2 2 + 4 E 1 E 2 h 2 3 h 1 + E 2 2 h 2 4 \kappa=\frac{6E_{1}E_{2}(h_{1}+h_{2})h_{1}h_{2}\epsilon}{E_{1}^{2}h_{1}^{4}+4E% _{1}E_{2}h_{1}^{3}h_{2}+6E_{1}E_{2}h_{1}^{2}h_{2}^{2}+4E_{1}E_{2}h_{2}^{3}h_{1% }+E_{2}^{2}h_{2}^{4}}
  2. E 1 E_{1}
  3. h 1 h_{1}
  4. E 2 E_{2}
  5. h 2 h_{2}
  6. ϵ \epsilon
  7. ϵ = ( α 1 - α 2 ) Δ T \epsilon=(\alpha_{1}-\alpha_{2})\Delta T\,

Bimodule.html

  1. ( r m ) s = r ( m s ) . (rm)s=r(ms).
  2. R S o p R\otimes_{\mathbb{Z}}S^{op}
  3. R S o p R\otimes_{\mathbb{Z}}S^{op}
  4. f : M N f:M\rightarrow N
  5. f ( m + m ) = f ( m ) + f ( m ) f(m+m^{\prime})=f(m)+f(m^{\prime})
  6. f ( r m s ) = r f ( m ) s f(rms)=rf(m)s
  7. ( f g ) ( f g ) = ( f f ) ( g g ) (f^{\prime}\otimes g^{\prime})\circ(f\otimes g)=(f^{\prime}\circ f)\otimes(g^{% \prime}\circ g)
  8. = K \otimes=\otimes_{K}

Bimonster_group.html

  1. B i = M 2 . Bi=M\wr\mathbb{Z}_{2}.\,

Binary_decision_diagram.html

  1. N N
  2. V N V_{N}
  3. V N V_{N}
  4. V N V_{N}
  5. f ( x 1 , x 2 , x 3 ) = x ¯ 1 x ¯ 2 x ¯ 3 + x 1 x 2 + x 2 x 3 f(x_{1},x_{2},x_{3})=\bar{x}_{1}\bar{x}_{2}\bar{x}_{3}+x_{1}x_{2}+x_{2}x_{3}
  6. f ( x 1 , , x n ) f(x_{1},\ldots,x_{n})
  7. f ( x 1 , , x 2 n ) = x 1 x 2 + x 3 x 4 + + x 2 n - 1 x 2 n . f(x_{1},\ldots,x_{2n})=x_{1}x_{2}+x_{3}x_{4}+\cdots+x_{2n-1}x_{2n}.
  8. x 1 < x 3 < < x 2 n - 1 < x 2 < x 4 < < x 2 n x_{1}<x_{3}<\cdots<x_{2n-1}<x_{2}<x_{4}<\cdots<x_{2n}
  9. x 1 < x 2 < x 3 < x 4 < < x 2 n - 1 < x 2 n x_{1}<x_{2}<x_{3}<x_{4}<\cdots<x_{2n-1}<x_{2n}

Binary_Golay_code.html

  1. M 23 M_{23}
  2. M 24 M_{24}
  3. x 23 - 1 x^{23}-1
  4. x 23 - 1 = ( x + 1 ) ( x 11 + x 9 + x 7 + x 6 + x 5 + x + 1 ) ( x 11 + x 10 + x 6 + x 5 + x 4 + x 2 + 1 ) . x^{23}-1=(x+1)(x^{11}+x^{9}+x^{7}+x^{6}+x^{5}+x+1)(x^{11}+x^{10}+x^{6}+x^{5}+x% ^{4}+x^{2}+1).
  5. ( x 11 + x 10 + x 6 + x 5 + x 4 + x 2 + 1 ) \left(x^{11}+x^{10}+x^{6}+x^{5}+x^{4}+x^{2}+1\right)
  6. 𝔽 2 4 \mathbb{F}_{2}^{4}

Binary_option.html

  1. σ \sigma
  2. Φ \Phi
  3. Φ ( x ) = 1 2 π - x e - 1 2 z 2 d z . \Phi(x)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{x}e^{-\frac{1}{2}z^{2}}dz.
  4. d 1 = ln S K + ( r - q + σ 2 / 2 ) T σ T , d 2 = d 1 - σ T . d_{1}=\frac{\ln\frac{S}{K}+(r-q+\sigma^{2}/2)T}{\sigma\sqrt{T}},\,d_{2}=d_{1}-% \sigma\sqrt{T}.\,
  5. C = e - r T Φ ( d 2 ) . C=e^{-rT}\Phi(d_{2}).\,
  6. P = e - r T Φ ( - d 2 ) . P=e^{-rT}\Phi(-d_{2}).\,
  7. C = S e - q T Φ ( d 1 ) . C=Se^{-qT}\Phi(d_{1}).\,
  8. P = S e - q T Φ ( - d 1 ) . P=Se^{-qT}\Phi(-d_{1}).\,
  9. r F O R r_{FOR}
  10. r D O M r_{DOM}
  11. C = e - r D O M T Φ ( d 2 ) C=e^{-r_{DOM}T}\Phi(d_{2})\,
  12. P = e - r D O M T Φ ( - d 2 ) P=e^{-r_{DOM}T}\Phi(-d_{2})\,
  13. C = S e - r F O R T Φ ( d 1 ) C=Se^{-r_{FOR}T}\Phi(d_{1})\,
  14. P = S e - r F O R T Φ ( - d 1 ) P=Se^{-r_{FOR}T}\Phi(-d_{1})\,
  15. σ \sigma
  16. σ ( K ) \sigma(K)
  17. C v C_{v}
  18. C = lim ϵ 0 C v ( K - ϵ ) - C v ( K ) ϵ C=\lim_{\epsilon\to 0}\frac{C_{v}(K-\epsilon)-C_{v}(K)}{\epsilon}
  19. C = - d C v d K C=-\frac{dC_{v}}{dK}
  20. σ \sigma
  21. K K
  22. C = - d C v ( K , σ ( K ) ) d K = - C v K - C v σ σ K C=-\frac{dC_{v}(K,\sigma(K))}{dK}=-\frac{\partial C_{v}}{\partial K}-\frac{% \partial C_{v}}{\partial\sigma}\frac{\partial\sigma}{\partial K}
  23. - C v K = - ( S Φ ( d 1 ) - K e - r T Φ ( d 2 ) ) K = e - r T Φ ( d 2 ) = C n o s k e w -\frac{\partial C_{v}}{\partial K}=-\frac{\partial(S\Phi(d_{1})-Ke^{-rT}\Phi(d% _{2}))}{\partial K}=e^{-rT}\Phi(d_{2})=C_{noskew}
  24. C v σ \frac{\partial C_{v}}{\partial\sigma}
  25. σ K \frac{\partial\sigma}{\partial K}
  26. C = C n o s k e w - V e g a v * S k e w C=C_{noskew}-Vega_{v}*Skew

Bingham_plastic.html

  1. τ 0 \tau_{0}
  2. u y = { 0 , τ < τ 0 ( τ - τ 0 ) / μ , τ τ 0 \frac{\partial u}{\partial y}=\left\{\begin{matrix}0&,\tau<\tau_{0}\\ (\tau-\tau_{0})/{\mu_{\infty}}&,\tau\geq\tau_{0}\end{matrix}\right.
  3. f = 2 h f g D L V 2 \ f=\ {2h_{f}gD\over LV^{2}}
  4. h f {\ h_{f}}
  5. f {\ f}
  6. L {\ L}
  7. g {\ g}
  8. D {\ D}
  9. V {\ V}
  10. f L = 64 R e [ 1 + H e 6 R e - 64 3 ( H e 4 f 3 R e 7 ) ] \ f_{L}=\ {64\over Re}\left[1+{He\over 6Re}-{64\over 3}\left({He^{4}\over{f}^{% 3}Re^{7}}\right)\right]
  11. f {\ f}
  12. R e {\ Re}
  13. H e {\ He}
  14. Re = ρ V D μ \mathrm{Re}={\rho{\ V}D\over{\mu}}
  15. He = ρ D 2 τ o μ 2 \mathrm{He}={\rho{\ D^{2}}{\ \tau_{o}}\over{{\mu}^{2}}}
  16. ρ {\rho}
  17. μ {\ \mu}
  18. τ o {\ \tau_{o}}
  19. f T = 10 a R e - 0.193 \ f_{T}=\ {10^{a}}\ {Re^{-0.193}}
  20. f T {\ f_{T}}
  21. a = - 1.47 [ 1 + 0.146 e - 2.9 × 10 - 5 H e ] \ a=-1.47\left[1+0.146{\ e^{-2.9\times{10^{-5}}He}}\right]
  22. f L = 64 R e + 64 R e ( H e 6.2218 R e ) 0.958 \ f_{L}=\ {64\over Re}+{64\over Re}\left({He\over{6.2218Re}}\right)^{0.958}
  23. f L = K 1 + 4 K 2 ( K 1 + K 1 K 2 K 1 4 + 3 K 2 ) 3 1 + 3 K 2 ( K 1 + K 1 K 2 K 1 4 + 3 K 2 ) 4 f_{L}=\frac{K_{1}+\dfrac{4K_{2}}{\left(K_{1}+\frac{K_{1}K_{2}}{K_{1}^{4}+3K_{2% }}\right)^{3}}}{1+\dfrac{3K_{2}}{\left(K_{1}+\frac{K_{1}K_{2}}{K_{1}^{4}+3K_{2% }}\right)^{4}}}
  24. K 1 = 16 R e + 16 H e 6 R e 2 \ K_{1}=\ {16\over Re}+{16He\over 6{Re^{2}}}
  25. K 2 = - 16 H e 4 3 R e 8 \ K_{2}=\ -{16{He^{4}}\over 3{Re^{8}}}
  26. f = [ f L m + f T m ] 1 m \ f=\ {\left[{f_{L}}^{m}+{f_{T}}^{m}\right]}^{1\over m}
  27. m = 1.7 + 40000 R e \ m=\ 1.7+{40000\over Re}

Binomial.html

  1. a x n - b x m , ax^{n}-bx^{m}\,,
  2. a a
  3. b b
  4. n n
  5. m m
  6. x x
  7. m m
  8. n n
  9. a x 1 n 1 x i n i - b x 1 m 1 x i m i ax_{1}^{n_{1}}\cdots x_{i}^{n_{i}}-bx_{1}^{m_{1}}\cdots x_{i}^{m_{i}}
  10. 3 x - 2 x 2 3x-2x^{2}
  11. x y + y x 2 xy+yx^{2}
  12. x 2 + y 2 x^{2}+y^{2}
  13. x 2 - y 2 x^{2}-y^{2}
  14. x 2 - y 2 = ( x + y ) ( x - y ) . x^{2}-y^{2}=(x+y)(x-y).
  15. x n + 1 - y n + 1 = ( x - y ) k = 0 n x k y n - k x^{n+1}-y^{n+1}=(x-y)\sum_{k=0}^{n}x^{k}\,y^{n-k}
  16. x 2 + y 2 = x 2 - ( i y ) 2 = ( x - i y ) ( x + i y ) x^{2}+y^{2}=x^{2}-(iy)^{2}=(x-iy)(x+iy)
  17. ( a x + b ) (ax+b)
  18. ( c x + d ) (cx+d)
  19. ( a x + b ) ( c x + d ) = a c x 2 + ( a d + b c ) x + b d . (ax+b)(cx+d)=acx^{2}+(ad+bc)x+bd.
  20. ( x + y ) n (x+y)^{n}
  21. ( x + y ) 2 (x+y)^{2}
  22. x + y x+y
  23. x 2 + 2 x y + y 2 x^{2}+2xy+y^{2}
  24. b = 2 m n b=2mn
  25. c = n 2 + m 2 c=n^{2}+m^{2}
  26. a 2 + b 2 = c 2 a^{2}+b^{2}=c^{2}
  27. x 3 + y 3 = ( x + y ) ( x 2 - x y + y 2 ) x^{3}+y^{3}=(x+y)(x^{2}-xy+y^{2})
  28. x 3 - y 3 = ( x - y ) ( x 2 + x y + y 2 ) x^{3}-y^{3}=(x-y)(x^{2}+xy+y^{2})

Biochemical_oxygen_demand.html

  1. U n s e e d e d : B O D 5 = ( D 0 - D 5 ) P Unseeded:BOD_{5}=\frac{(D_{0}-D_{5})}{P}
  2. S e e d e d : B O D 5 = ( D 0 - D 5 ) - ( B 0 - B 5 ) f P Seeded:BOD_{5}=\frac{(D_{0}-D_{5})-(B_{0}-B_{5})f}{P}
  3. D 0 D_{0}
  4. D 5 D_{5}
  5. P P
  6. B 0 B_{0}
  7. B 5 B_{5}
  8. f f
  9. I 0 / I = 1 + K S V [ O 2 ] I_{0}/I~{}=~{}1~{}+~{}K_{SV}~{}[O_{2}]
  10. I = L u m i n e s c e n c e i n t h e p r e s e n c e o f o x y g e n I~{}=~{}Luminescence~{}in~{}the~{}presence~{}of~{}oxygen
  11. I 0 = L u m i n e s c e n c e i n t h e a b s e n c e o f o x y g e n I_{0}~{}=~{}Luminescence~{}in~{}the~{}absence~{}of~{}oxygen
  12. K S V = S t e r n - V o l m e r c o n s t a n t f o r o x y g e n q u e n c h i n g K_{SV}~{}=~{}Stern-Volmer~{}constant~{}for~{}oxygen~{}quenching
  13. [ O 2 ] = D i s s o l v e d o x y g e n c o n c e n t r a t i o n [O_{2}]~{}=~{}Dissolved~{}oxygen~{}concentration

Birational_geometry.html

  1. x = 2 t 1 + t 2 x=\frac{2\,t}{1+t^{2}}
  2. y = 1 - t 2 1 + t 2 , y=\frac{1-t^{2}}{1+t^{2}}\,,

Birch_and_Swinnerton-Dyer_conjecture.html

  1. p X N p p \prod_{p\leq X}\frac{N_{p}}{p}
  2. p x N p p C log ( x ) r as x \prod_{p\leq x}\frac{N_{p}}{p}\approx C\log(x)^{r}\mbox{ as }~{}x\rightarrow\infty
  3. L ( r ) ( E , 1 ) r ! = # Sha ( E ) Ω E R E p | N c p ( # E Tor ) 2 \frac{L^{(r)}(E,1)}{r!}=\frac{\#\mathrm{Sha}(E)\Omega_{E}R_{E}\prod_{p|N}c_{p}% }{(\#E_{\mathrm{Tor}})^{2}}
  4. 2 x 2 + y 2 + 8 z 2 = n 2x^{2}+y^{2}+8z^{2}=n
  5. 2 x 2 + y 2 + 32 z 2 = n 2x^{2}+y^{2}+32z^{2}=n
  6. y 2 = x 3 - n 2 x y^{2}=x^{3}-n^{2}x
  7. y 2 = x 3 + a x + b y^{2}=x^{3}+ax+b

Bisimulation.html

  1. S S
  2. R R
  3. S S
  4. R R
  5. S S
  6. S S
  7. R R
  8. R R
  9. R R
  10. p , q p,q
  11. S S
  12. ( p , q ) (p,q)
  13. R R
  14. p p^{\prime}
  15. S S
  16. p 𝛼 p p\overset{\alpha}{\rightarrow}p^{\prime}
  17. q q^{\prime}
  18. S S
  19. q 𝛼 q q\overset{\alpha}{\rightarrow}q^{\prime}
  20. ( p , q ) R (p^{\prime},q^{\prime})\in R
  21. q q^{\prime}
  22. S S
  23. q 𝛼 q q\overset{\alpha}{\rightarrow}q^{\prime}
  24. p p^{\prime}
  25. S S
  26. p 𝛼 p p\overset{\alpha}{\rightarrow}p^{\prime}
  27. ( p , q ) R (p^{\prime},q^{\prime})\in R
  28. p p
  29. q q
  30. S S
  31. p p
  32. q q
  33. p q p\,\sim\,q
  34. R R
  35. ( p , q ) (p,q)
  36. R R
  37. \,\sim\,
  38. p p
  39. q q
  40. q q
  41. p p
  42. p p
  43. q q
  44. p p
  45. q q
  46. q q
  47. p p
  48. M = p . c ¯ . M + p . t ¯ . M + p . ( c ¯ . M + t ¯ . M ) M=p.\overline{c}.M+p.\overline{t}.M+p.(\overline{c}.M+\overline{t}.M)
  49. M = p . ( c ¯ . M + t ¯ . M ) M^{\prime}=p.(\overline{c}.M^{\prime}+\overline{t}.M^{\prime})
  50. ( S , Λ , ) (S,\Lambda,\rightarrow)
  51. R R
  52. S S
  53. R R
  54. S S
  55. S S
  56. α Λ \forall\alpha\in\Lambda
  57. R ; 𝛼 𝛼 ; R R\ ;\ \overset{\alpha}{\rightarrow}\quad{\subseteq}\quad\overset{\alpha}{% \rightarrow}\ ;\ R
  58. R - 1 ; 𝛼 𝛼 ; R - 1 R^{-1}\ ;\ \overset{\alpha}{\rightarrow}\quad{\subseteq}\quad\overset{\alpha}{% \rightarrow}\ ;\ R^{-1}
  59. S S
  60. F : 𝒫 ( S × S ) 𝒫 ( S × S ) F:\mathcal{P}(S\times S)\to\mathcal{P}(S\times S)
  61. S S
  62. S S
  63. R R
  64. S S
  65. F ( R ) F(R)
  66. ( p , q ) (p,q)
  67. S S
  68. S S
  69. α Λ . p S . p 𝛼 p q S . q 𝛼 q and ( p , q ) R \forall\alpha\in\Lambda.\,\forall p^{\prime}\in S.\,p\overset{\alpha}{% \rightarrow}p^{\prime}\,\Rightarrow\,\exists q^{\prime}\in S.\,q\overset{% \alpha}{\rightarrow}q^{\prime}\,\textrm{ and }\,(p^{\prime},q^{\prime})\in R
  70. α Λ . q S . q 𝛼 q p S . p 𝛼 p and ( p , q ) R \forall\alpha\in\Lambda.\,\forall q^{\prime}\in S.\,q\overset{\alpha}{% \rightarrow}q^{\prime}\,\Rightarrow\,\exists p^{\prime}\in S.\,p\overset{% \alpha}{\rightarrow}p^{\prime}\,\textrm{ and }\,(p^{\prime},q^{\prime})\in R
  71. F F
  72. α \alpha
  73. ( p , q ) (p,q)
  74. ( p , q ) 𝛼 ( p , q ) (p,q)\overset{\alpha}{\rightarrow}(p^{\prime},q)
  75. ( p , q ) 𝛼 ( p , q ) (p,q)\overset{\alpha}{\rightarrow}(p,q^{\prime})
  76. α \alpha
  77. ( p , q ) (p^{\prime},q)
  78. ( p , q ) (p,q^{\prime})
  79. α \alpha
  80. ( p , q ) 𝛼 ( p , q ) (p^{\prime},q)\overset{\alpha}{\rightarrow}(p^{\prime},q^{\prime})
  81. ( p , q ) 𝛼 ( p , q ) (p,q^{\prime})\overset{\alpha}{\rightarrow}(p^{\prime},q^{\prime})
  82. ( p , q ) (p,q)
  83. ( p , q ) (p,q)
  84. ( S , Λ , ) (S,\Lambda,\rightarrow)
  85. ξ \xi_{\rightarrow}
  86. S S
  87. S S
  88. Λ \Lambda
  89. 𝒫 ( Λ × S ) \mathcal{P}(\Lambda\times S)
  90. p { ( α , q ) S : p 𝛼 q } . p\mapsto\{(\alpha,q)\in S:p\overset{\alpha}{\rightarrow}q\}.
  91. π i : S × S S \pi_{i}\colon S\times S\to S
  92. i i
  93. ( p , q ) (p,q)
  94. p p
  95. q q
  96. i = 1 , 2 i=1,2
  97. 𝒫 ( Λ × π 1 ) \mathcal{P}(\Lambda\times\pi_{1})
  98. π 1 \pi_{1}
  99. P { ( α , p ) Λ × S : q . ( α , p , q ) P } P\mapsto\{(\alpha,p)\in\Lambda\times S:\exists q.(\alpha,p,q)\in P\}
  100. P P
  101. Λ × S × S \Lambda\times S\times S
  102. 𝒫 ( Λ × π 2 ) \mathcal{P}(\Lambda\times\pi_{2})
  103. R S × S R\subseteq S\times S
  104. ( S , Λ , ) (S,\Lambda,\rightarrow)
  105. γ : R 𝒫 ( Λ × R ) \gamma\colon R\to\mathcal{P}(\Lambda\times R)
  106. R R
  107. i = 1 , 2 i=1,2
  108. ξ π i = 𝒫 ( Λ × π i ) γ \xi_{\rightarrow}\circ\pi_{i}=\mathcal{P}(\Lambda\times\pi_{i})\circ\gamma
  109. ξ \xi_{\rightarrow}
  110. ( S , Λ , ) (S,\Lambda,\rightarrow)
  111. τ \tau
  112. p p
  113. q q
  114. p p
  115. p p^{\prime}
  116. q q^{\prime}
  117. q q
  118. q q^{\prime}

Black_hole_thermodynamics.html

  1. ( 1 / 2 ln 2 ) / 4 π (1/2\cdot\ln{2})/4\pi
  2. 1 / 4 1/4
  3. S BH = k A 4 P 2 S_{\,\text{BH}}=\frac{kA}{4\ell_{\mathrm{P}}^{2}}
  4. A A
  5. 4 π R 2 4\pi R^{2}
  6. k k
  7. P = G / c 3 \ell_{\mathrm{P}}=\sqrt{G\hbar/c^{3}}
  8. A A
  9. d E = κ 8 π d A + Ω d J + Φ d Q , dE=\frac{\kappa}{8\pi}\,dA+\Omega\,dJ+\Phi\,dQ,
  10. E E
  11. κ \displaystyle\kappa
  12. A A
  13. Ω \Omega
  14. J J
  15. Φ \Phi
  16. Q Q
  17. d A d t 0. \frac{dA}{dt}\geq 0.
  18. κ \displaystyle\kappa
  19. κ \displaystyle\kappa
  20. κ \displaystyle\kappa
  21. T H = κ 2 π . T_{\,\text{H}}=\frac{\kappa}{2\pi}.
  22. S BH = A 4 . S_{\,\text{BH}}=\frac{A}{4}.

Blackletter.html

  1. 𝔄 𝔅 𝔇 𝔈 𝔉 𝔊 𝔍 𝔎 𝔏 𝔐 𝔑 𝔒 𝔓 𝔔 𝔖 𝔗 𝔘 𝔙 𝔚 𝔛 𝔜 \mathfrak{A}\mathfrak{B}\mathfrak{C}\mathfrak{D}\mathfrak{E}\mathfrak{F}% \mathfrak{G}\mathfrak{H}\mathfrak{I}\mathfrak{J}\mathfrak{K}\mathfrak{L}% \mathfrak{M}\mathfrak{N}\mathfrak{O}\mathfrak{P}\mathfrak{Q}\mathfrak{R}% \mathfrak{S}\mathfrak{T}\mathfrak{U}\mathfrak{V}\mathfrak{W}\mathfrak{X}% \mathfrak{Y}\mathfrak{Z}
  2. 𝔞 𝔟 𝔠 𝔡 𝔢 𝔣 𝔤 𝔥 𝔦 𝔧 𝔨 𝔩 𝔪 𝔫 𝔬 𝔭 𝔮 𝔯 𝔰 𝔱 𝔲 𝔳 𝔴 𝔵 𝔶 𝔷 \mathfrak{a}\mathfrak{b}\mathfrak{c}\mathfrak{d}\mathfrak{e}\mathfrak{f}% \mathfrak{g}\mathfrak{h}\mathfrak{i}\mathfrak{j}\mathfrak{k}\mathfrak{l}% \mathfrak{m}\mathfrak{n}\mathfrak{o}\mathfrak{p}\mathfrak{q}\mathfrak{r}% \mathfrak{s}\mathfrak{t}\mathfrak{u}\mathfrak{v}\mathfrak{w}\mathfrak{x}% \mathfrak{y}\mathfrak{z}

Blade_element_theory.html

  1. v i = T A 1 2 ρ . v_{i}=\sqrt{\frac{T}{A}\cdot\frac{1}{2\rho}}.

BLAST.html

  1. p ( S x ) = 1 - exp ( - e - λ ( x - μ ) ) p\left(S\geq x\right)=1-\exp\left(-e^{-\lambda\left(x-\mu\right)}\right)
  2. μ = [ log ( K m n ) ] λ \mu={}^{\left[\log\left(Km^{\prime}n^{\prime}\right)\right]}\!\!\diagup\!\!{}_% {\lambda}\;
  3. λ \lambda
  4. K \mathrm{K}
  5. λ \lambda
  6. K \mathrm{K}
  7. m m^{\prime}
  8. n n^{\prime}
  9. m m - ( ln K m n ) H m^{\prime}\approx m-{}^{\left(\ln Kmn\right)}\!\!\diagup\!\!{}_{H}\;
  10. n n - ( ln K m n ) H n^{\prime}\approx n-{}^{\left(\ln Kmn\right)}\!\!\diagup\!\!{}_{H}\;
  11. H \mathrm{H}
  12. λ = 0.318 \lambda=0.318
  13. K = 0.13 \mathrm{K}=0.13
  14. H = 0.40 \mathrm{H}=0.40
  15. E 1 - e - p ( s > x ) D E\approx 1-e^{-p\left(s>x\right)D}
  16. p < 0.1 p<0.1
  17. E p D E\approx pD

Blind_signature.html

  1. r e mod N r^{e}\bmod N
  2. m m r e ( mod N ) m^{\prime}\equiv mr^{e}\ (\mathrm{mod}\ N)
  3. m m^{\prime}
  4. r r e mod N r\mapsto r^{e}\bmod N
  5. r e mod N r^{e}\bmod N
  6. m m^{\prime}
  7. s ( m ) d ( mod N ) . s^{\prime}\equiv(m^{\prime})^{d}\ (\mathrm{mod}\ N).
  8. s s r - 1 ( mod N ) s\equiv s^{\prime}\cdot r^{-1}\ (\mathrm{mod}\ N)
  9. r e d r ( mod N ) r^{ed}\equiv r\;\;(\mathop{{\rm mod}}N)
  10. s s r - 1 ( m ) d r - 1 m d r e d r - 1 m d r r - 1 m d ( mod N ) , s\equiv s^{\prime}\cdot r^{-1}\equiv(m^{\prime})^{d}r^{-1}\equiv m^{d}r^{ed}r^% {-1}\equiv m^{d}rr^{-1}\equiv m^{d}\;\;(\mathop{{\rm mod}}N),
  11. m m
  12. m m^{\prime}
  13. m ′′ \displaystyle m^{\prime\prime}
  14. m m^{\prime}
  15. m m
  16. s \displaystyle s^{\prime}
  17. ϕ ( n ) \phi(n)
  18. m = s r - 1 ( mod n ) \displaystyle m=s^{\prime}\cdot r^{-1}\;\;(\mathop{{\rm mod}}n)

Bloch_wave.html

  1. ψ ( 𝐫 ) = e i 𝐤 𝐫 u ( 𝐫 ) \psi(\mathbf{r})=\mathrm{e}^{\mathrm{i}\mathbf{k}\cdot\mathbf{r}}u(\mathbf{r})
  2. ψ ( 𝐫 ) = e i 𝐤 𝐫 u ( 𝐫 ) , \psi(\mathbf{r})=\mathrm{e}^{\mathrm{i}\mathbf{k}\cdot\mathbf{r}}u(\mathbf{r}),
  3. ψ \psi
  4. ψ ( 𝐫 ) = e i 𝐤 𝐫 u ( 𝐫 ) \psi(\mathbf{r})=\mathrm{e}^{\mathrm{i}\mathbf{k}\cdot\mathbf{r}}u(\mathbf{r})
  5. n 1 𝐚 1 + n 2 𝐚 2 + n 3 𝐚 3 n_{1}\mathbf{a}_{1}+n_{2}\mathbf{a}_{2}+n_{3}\mathbf{a}_{3}
  6. T ^ n 1 , n 2 , n 3 \hat{T}_{n_{1},n_{2},n_{3}}\!
  7. ψ ( 𝐫 + 𝐚 i ) = C i ψ ( 𝐫 ) \psi(\mathbf{r}+\mathbf{a}_{i})=C_{i}\psi(\mathbf{r})
  8. ψ ( 𝐫 + 𝐚 i ) = e 2 π i θ i ψ ( 𝐫 ) \psi(\mathbf{r}+\mathbf{a}_{i})=\mathrm{e}^{2\pi\mathrm{i}\theta_{i}}\psi(% \mathbf{r})
  9. u ( 𝐫 ) = e - i 𝐤 𝐫 ψ ( 𝐫 ) . u(\mathbf{r})=\mathrm{e}^{-\mathrm{i}\mathbf{k}\cdot\mathbf{r}}\psi(\mathbf{r}% )\,.
  10. u ( 𝐫 + 𝐚 i ) = e - i 𝐤 ( 𝐫 + 𝐚 i ) ψ ( 𝐫 + 𝐚 i ) = ( e - i 𝐤 𝐫 e - i 𝐤 𝐚 i ) ( e 2 π i θ i ψ ( 𝐫 ) ) = e - i 𝐤 𝐫 e - 2 π i θ i e 2 π i θ i ψ ( 𝐫 ) = u ( 𝐫 ) u(\mathbf{r}+\mathbf{a}_{i})=\mathrm{e}^{-\mathrm{i}\mathbf{k}\cdot(\mathbf{r}% +\mathbf{a}_{i})}\psi(\mathbf{r}+\mathbf{a}_{i})=\big(\mathrm{e}^{-\mathrm{i}% \mathbf{k}\cdot\mathbf{r}}\mathrm{e}^{-\mathrm{i}\mathbf{k}\cdot\mathbf{a}_{i}% }\big)\big(\mathrm{e}^{2\pi\mathrm{i}\theta_{i}}\psi(\mathbf{r})\big)=\mathrm{% e}^{-\mathrm{i}\mathbf{k}\cdot\mathbf{r}}\mathrm{e}^{-2\pi\mathrm{i}\theta_{i}% }\mathrm{e}^{2\pi\mathrm{i}\theta_{i}}\psi(\mathbf{r})=u(\mathbf{r})
  11. T ^ n 1 , n 2 , n 3 \hat{T}_{n_{1},n_{2},n_{3}}\!
  12. T ^ n 1 , n 2 , n 3 \hat{T}_{n_{1},n_{2},n_{3}}\!
  13. d 2 y d x 2 + ( θ 0 + 2 n = 1 θ n cos ( 2 n x ) ) y = 0 , \frac{d^{2}y}{dx^{2}}+\left(\theta_{0}+2\sum_{n=1}^{\infty}\theta_{n}\cos(2nx)% \right)y=0,

Block_matrix.html

  1. n n
  2. m m
  3. M M
  4. n n
  5. r o w g r o u p s rowgroups
  6. m m
  7. c o l g r o u p s colgroups
  8. ( i , j ) (i,j)
  9. ( s , t ) (s,t)
  10. ( x , y ) (x,y)
  11. x r o w g r o u p s x\in rowgroups
  12. y c o l g r o u p s y\in colgroups
  13. 𝐏 = [ 1 1 2 2 1 1 2 2 3 3 4 4 3 3 4 4 ] \mathbf{P}=\begin{bmatrix}1&1&2&2\\ 1&1&2&2\\ 3&3&4&4\\ 3&3&4&4\end{bmatrix}
  14. 𝐏 11 = [ 1 1 1 1 ] , 𝐏 12 = [ 2 2 2 2 ] , 𝐏 21 = [ 3 3 3 3 ] , 𝐏 22 = [ 4 4 4 4 ] . \mathbf{P}_{11}=\begin{bmatrix}1&1\\ 1&1\end{bmatrix},\mathbf{P}_{12}=\begin{bmatrix}2&2\\ 2&2\end{bmatrix},\mathbf{P}_{21}=\begin{bmatrix}3&3\\ 3&3\end{bmatrix},\mathbf{P}_{22}=\begin{bmatrix}4&4\\ 4&4\end{bmatrix}.
  15. 𝐏 = [ 𝐏 11 𝐏 12 𝐏 21 𝐏 22 ] . \mathbf{P}=\begin{bmatrix}\mathbf{P}_{11}&\mathbf{P}_{12}\\ \mathbf{P}_{21}&\mathbf{P}_{22}\end{bmatrix}.
  16. A A
  17. B B
  18. ( m × p ) (m\times p)
  19. 𝐀 \mathbf{A}
  20. q q
  21. s s
  22. 𝐀 = [ 𝐀 11 𝐀 12 𝐀 1 s 𝐀 21 𝐀 22 𝐀 2 s 𝐀 q 1 𝐀 q 2 𝐀 q s ] \mathbf{A}=\begin{bmatrix}\mathbf{A}_{11}&\mathbf{A}_{12}&\cdots&\mathbf{A}_{1% s}\\ \mathbf{A}_{21}&\mathbf{A}_{22}&\cdots&\mathbf{A}_{2s}\\ \vdots&\vdots&\ddots&\vdots\\ \mathbf{A}_{q1}&\mathbf{A}_{q2}&\cdots&\mathbf{A}_{qs}\end{bmatrix}
  23. ( p × n ) (p\times n)
  24. 𝐁 \mathbf{B}
  25. s s
  26. r r
  27. 𝐁 = [ 𝐁 11 𝐁 12 𝐁 1 r 𝐁 21 𝐁 22 𝐁 2 r 𝐁 s 1 𝐁 s 2 𝐁 s r ] , \mathbf{B}=\begin{bmatrix}\mathbf{B}_{11}&\mathbf{B}_{12}&\cdots&\mathbf{B}_{1% r}\\ \mathbf{B}_{21}&\mathbf{B}_{22}&\cdots&\mathbf{B}_{2r}\\ \vdots&\vdots&\ddots&\vdots\\ \mathbf{B}_{s1}&\mathbf{B}_{s2}&\cdots&\mathbf{B}_{sr}\end{bmatrix},
  28. A A
  29. 𝐂 = 𝐀𝐁 \mathbf{C}=\mathbf{A}\mathbf{B}
  30. 𝐂 \mathbf{C}
  31. ( m × n ) (m\times n)
  32. q q
  33. r r
  34. 𝐂 \mathbf{C}
  35. 𝐂 α β = γ = 1 s 𝐀 α γ 𝐁 γ β . \mathbf{C}_{\alpha\beta}=\sum^{s}_{\gamma=1}\mathbf{A}_{\alpha\gamma}\mathbf{B% }_{\gamma\beta}.
  36. 𝐂 α β = 𝐀 α γ 𝐁 γ β . \mathbf{C}_{\alpha\beta}=\mathbf{A}_{\alpha\gamma}\mathbf{B}_{\gamma\beta}.
  37. [ 𝐀 𝐁 𝐂 𝐃 ] - 1 = [ 𝐀 - 1 + 𝐀 - 1 𝐁 ( 𝐃 - 𝐂𝐀 - 1 𝐁 ) - 1 𝐂𝐀 - 1 - 𝐀 - 1 𝐁 ( 𝐃 - 𝐂𝐀 - 1 𝐁 ) - 1 - ( 𝐃 - 𝐂𝐀 - 1 𝐁 ) - 1 𝐂𝐀 - 1 ( 𝐃 - 𝐂𝐀 - 1 𝐁 ) - 1 ] , \begin{bmatrix}\mathbf{A}&\mathbf{B}\\ \mathbf{C}&\mathbf{D}\end{bmatrix}^{-1}=\begin{bmatrix}\mathbf{A}^{-1}+\mathbf% {A}^{-1}\mathbf{B}(\mathbf{D}-\mathbf{CA}^{-1}\mathbf{B})^{-1}\mathbf{CA}^{-1}% &-\mathbf{A}^{-1}\mathbf{B}(\mathbf{D}-\mathbf{CA}^{-1}\mathbf{B})^{-1}\\ -(\mathbf{D}-\mathbf{CA}^{-1}\mathbf{B})^{-1}\mathbf{CA}^{-1}&(\mathbf{D}-% \mathbf{CA}^{-1}\mathbf{B})^{-1}\end{bmatrix},
  38. \,
  39. [ 𝐀 𝐁 𝐂 𝐃 ] - 1 = [ ( 𝐀 - 𝐁𝐃 - 1 𝐂 ) - 1 - ( 𝐀 - 𝐁𝐃 - 1 𝐂 ) - 1 𝐁𝐃 - 1 - 𝐃 - 1 𝐂 ( 𝐀 - 𝐁𝐃 - 1 𝐂 ) - 1 𝐃 - 1 + 𝐃 - 1 𝐂 ( 𝐀 - 𝐁𝐃 - 1 𝐂 ) - 1 𝐁𝐃 - 1 ] . \begin{bmatrix}\mathbf{A}&\mathbf{B}\\ \mathbf{C}&\mathbf{D}\end{bmatrix}^{-1}=\begin{bmatrix}(\mathbf{A}-\mathbf{BD}% ^{-1}\mathbf{C})^{-1}&-(\mathbf{A}-\mathbf{BD}^{-1}\mathbf{C})^{-1}\mathbf{BD}% ^{-1}\\ -\mathbf{D}^{-1}\mathbf{C}(\mathbf{A}-\mathbf{BD}^{-1}\mathbf{C})^{-1}&\quad% \mathbf{D}^{-1}+\mathbf{D}^{-1}\mathbf{C}(\mathbf{A}-\mathbf{BD}^{-1}\mathbf{C% })^{-1}\mathbf{BD}^{-1}\end{bmatrix}.
  40. \,
  41. 𝐀 = [ 𝐀 1 0 0 0 𝐀 2 0 0 0 𝐀 n ] \mathbf{A}=\begin{bmatrix}\mathbf{A}_{1}&0&\cdots&0\\ 0&\mathbf{A}_{2}&\cdots&0\\ \vdots&\vdots&\ddots&\vdots\\ 0&0&\cdots&\mathbf{A}_{n}\end{bmatrix}
  42. \oplus
  43. \oplus\,\ldots\,\oplus
  44. \ldots
  45. det 𝐀 = det 𝐀 1 × × det 𝐀 n \operatorname{det}\mathbf{A}=\operatorname{det}\mathbf{A}_{1}\times\ldots% \times\operatorname{det}\mathbf{A}_{n}
  46. tr 𝐀 = tr 𝐀 1 + + tr 𝐀 n . \operatorname{tr}\mathbf{A}=\operatorname{tr}\mathbf{A}_{1}+\cdots+% \operatorname{tr}\mathbf{A}_{n}.
  47. ( 𝐀 1 0 0 0 𝐀 2 0 0 0 𝐀 n ) - 1 = ( 𝐀 1 - 1 0 0 0 𝐀 2 - 1 0 0 0 𝐀 n - 1 ) . \begin{pmatrix}\mathbf{A}_{1}&0&\cdots&0\\ 0&\mathbf{A}_{2}&\cdots&0\\ \vdots&\vdots&\ddots&\vdots\\ 0&0&\cdots&\mathbf{A}_{n}\end{pmatrix}^{-1}=\begin{pmatrix}\mathbf{A}_{1}^{-1}% &0&\cdots&0\\ 0&\mathbf{A}_{2}^{-1}&\cdots&0\\ \vdots&\vdots&\ddots&\vdots\\ 0&0&\cdots&\mathbf{A}_{n}^{-1}\end{pmatrix}.
  48. A A
  49. A 1 A_{1}
  50. A 2 A_{2}
  51. A n A_{n}
  52. 𝐀 = [ 𝐁 1 𝐂 1 0 𝐀 2 𝐁 2 𝐂 2 𝐀 k 𝐁 k 𝐂 k 𝐀 n - 1 𝐁 n - 1 𝐂 n - 1 0 𝐀 n 𝐁 n ] \mathbf{A}=\begin{bmatrix}\mathbf{B}_{1}&\mathbf{C}_{1}&&&\cdots&&0\\ \mathbf{A}_{2}&\mathbf{B}_{2}&\mathbf{C}_{2}&&&&\\ &\ddots&\ddots&\ddots&&&\vdots\\ &&\mathbf{A}_{k}&\mathbf{B}_{k}&\mathbf{C}_{k}&&\\ \vdots&&&\ddots&\ddots&\ddots&\\ &&&&\mathbf{A}_{n-1}&\mathbf{B}_{n-1}&\mathbf{C}_{n-1}\\ 0&&\cdots&&&\mathbf{A}_{n}&\mathbf{B}_{n}\end{bmatrix}
  53. 𝐀 = [ 𝐀 ( 1 , 1 ) 𝐀 ( 1 , 2 ) 𝐀 ( 1 , n - 1 ) 𝐀 ( 1 , n ) 𝐀 ( 2 , 1 ) 𝐀 ( 1 , 1 ) 𝐀 ( 1 , 2 ) 𝐀 ( 1 , n - 1 ) 𝐀 ( 2 , 1 ) 𝐀 ( 1 , 1 ) 𝐀 ( 1 , 2 ) 𝐀 ( n - 1 , 1 ) 𝐀 ( 2 , 1 ) 𝐀 ( 1 , 1 ) 𝐀 ( 1 , 2 ) 𝐀 ( n , 1 ) 𝐀 ( n - 1 , 1 ) 𝐀 ( 2 , 1 ) 𝐀 ( 1 , 1 ) ] . \mathbf{A}=\begin{bmatrix}\mathbf{A}_{(1,1)}&\mathbf{A}_{(1,2)}&&&\cdots&% \mathbf{A}_{(1,n-1)}&\mathbf{A}_{(1,n)}\\ \mathbf{A}_{(2,1)}&\mathbf{A}_{(1,1)}&\mathbf{A}_{(1,2)}&&&&\mathbf{A}_{(1,n-1% )}\\ &\ddots&\ddots&\ddots&&&\vdots\\ &&\mathbf{A}_{(2,1)}&\mathbf{A}_{(1,1)}&\mathbf{A}_{(1,2)}&&\\ \vdots&&&\ddots&\ddots&\ddots&\\ \mathbf{A}_{(n-1,1)}&&&&\mathbf{A}_{(2,1)}&\mathbf{A}_{(1,1)}&\mathbf{A}_{(1,2% )}\\ \mathbf{A}_{(n,1)}&\mathbf{A}_{(n-1,1)}&\cdots&&&\mathbf{A}_{(2,1)}&\mathbf{A}% _{(1,1)}\end{bmatrix}.
  54. \oplus
  55. 𝐀 𝐁 = [ a 11 a 1 n 0 0 a m 1 a m n 0 0 0 0 b 11 b 1 q 0 0 b p 1 b p q ] . \mathbf{A}\oplus\mathbf{B}=\begin{bmatrix}a_{11}&\cdots&a_{1n}&0&\cdots&0\\ \vdots&\cdots&\vdots&\vdots&\cdots&\vdots\\ a_{m1}&\cdots&a_{mn}&0&\cdots&0\\ 0&\cdots&0&b_{11}&\cdots&b_{1q}\\ \vdots&\cdots&\vdots&\vdots&\cdots&\vdots\\ 0&\cdots&0&b_{p1}&\cdots&b_{pq}\end{bmatrix}.
  56. [ 1 3 2 2 3 1 ] [ 1 6 0 1 ] = [ 1 3 2 0 0 2 3 1 0 0 0 0 0 1 6 0 0 0 0 1 ] . \begin{bmatrix}1&3&2\\ 2&3&1\end{bmatrix}\oplus\begin{bmatrix}1&6\\ 0&1\end{bmatrix}=\begin{bmatrix}1&3&2&0&0\\ 2&3&1&0&0\\ 0&0&0&1&6\\ 0&0&0&0&1\end{bmatrix}.

Blood_flow.html

  1. P I = v s y s t o l e - v d i a s t o l e v m e a n PI=\frac{v_{systole}-v_{diastole}}{v_{mean}}
  2. σ = F A \sigma=\frac{F}{A}
  3. U s = 2 9 ( ρ p - ρ f ) μ g a 2 U_{s}=\frac{2}{9}\frac{\left(\rho_{p}-\rho_{f}\right)}{\mu}g\,a^{2}
  4. Q = K ( [ P c - P i ] S - [ P c - P i ] ) \ Q=K([P_{c}-P_{i}]S-[P_{c}-P_{i}])
  5. B L s = E B V ln H i H m \ BL_{s}=EBV\ln\frac{H_{i}}{H_{m}}
  6. A N H = B L s 450 ANH=\frac{BL_{s}}{450}
  7. R C M = E V B × ( H i - H m ) RCM=EVB\times(H_{i}-H_{m})
  8. B L H = R C M H H m BL_{H}=\frac{RCM_{H}}{H_{m}}
  9. B L i = B L H - B L s \ {BL_{i}}={BL_{H}}-{BL_{s}}
  10. R C M i = B L i × H m {RCM_{i}}={BL_{i}}\times{H_{m}}
  11. T = 70 × patient’s weight in kg 4900 T=\frac{70\times\,\text{patient's weight in kg}}{4900}

Bloom_filter.html

  1. x x
  2. y y
  3. z z
  4. w w
  5. x x
  6. y y
  7. z z
  8. m m
  9. k k
  10. m m
  11. k k
  12. m m
  13. k k
  14. k k
  15. k k
  16. k k
  17. k k
  18. k k
  19. k k
  20. k k
  21. m m
  22. k k
  23. k k
  24. k k
  25. k k
  26. p p
  27. n n
  28. m m
  29. k = ( m / n ) ln 2 k=(m/n)\ln 2
  30. 1 - 1 m . 1-\frac{1}{m}.
  31. ( 1 - 1 m ) k . \left(1-\frac{1}{m}\right)^{k}.
  32. ( 1 - 1 m ) k n ; \left(1-\frac{1}{m}\right)^{kn};
  33. 1 - ( 1 - 1 m ) k n . 1-\left(1-\frac{1}{m}\right)^{kn}.
  34. ( 1 - [ 1 - 1 m ] k n ) k ( 1 - e - k n / m ) k . \left(1-\left[1-\frac{1}{m}\right]^{kn}\right)^{k}\approx\left(1-e^{-kn/m}% \right)^{k}.
  35. 1 - q 1-q
  36. ( 1 - q ) k (1-q)^{k}
  37. E [ q ] = ( 1 - 1 m ) k n E[q]=\left(1-\frac{1}{m}\right)^{kn}
  38. Pr ( | q - E [ q ] | λ m ) 2 exp ( - 2 λ 2 / m ) \Pr(\left|q-E[q]\right|\geq\frac{\lambda}{m})\leq 2\exp(-2\lambda^{2}/m)
  39. t Pr ( q = t ) ( 1 - t ) k ( 1 - E [ q ] ) k = ( 1 - [ 1 - 1 m ] k n ) k ( 1 - e - k n / m ) k \sum_{t}\Pr(q=t)(1-t)^{k}\approx(1-E[q])^{k}=\left(1-\left[1-\frac{1}{m}\right% ]^{kn}\right)^{k}\approx\left(1-e^{-kn/m}\right)^{k}
  40. k = m n ln 2 , k=\frac{m}{n}\ln 2,
  41. 2 - k 0.6185 m / n . 2^{-k}\approx{0.6185}^{m/n}.
  42. p = ( 1 - e - ( m / n ln 2 ) n / m ) ( m / n ln 2 ) p=\left(1-e^{-(m/n\ln 2)n/m}\right)^{(m/n\ln 2)}
  43. ln p = - m n ( ln 2 ) 2 . \ln p=-\frac{m}{n}\left(\ln 2\right)^{2}.
  44. m = - n ln p ( ln 2 ) 2 . m=-\frac{n\ln p}{(\ln 2)^{2}}.
  45. ( 1 - e - k ( n + 0.5 ) / ( m - 1 ) ) k . \left(1-e^{-k(n+0.5)/(m-1)}\right)^{k}.
  46. n * = - m ln [ 1 - X m ] k n^{*}=-\tfrac{m\ln\left[1-\tfrac{X}{m}\right]}{k}
  47. n * n^{*}
  48. m m
  49. n ( A * ) = - m ln [ 1 - n ( A ) / m ] / k n(A^{*})=-m\ln\left[1-n(A)/m\right]/k
  50. n ( B * ) = - m ln [ 1 - n ( B ) / m ] / k n(B^{*})=-m\ln\left[1-n(B)/m\right]/k
  51. n ( A * B * ) = - m ln [ 1 - n ( A B ) / m ] / k n(A^{*}\cup B^{*})=-m\ln\left[1-n(A\cup B)/m\right]/k
  52. n ( A B ) n(A\cup B)
  53. n ( A * B * ) = n ( A * ) + n ( B * ) - n ( A * B * ) n(A^{*}\cap B^{*})=n(A^{*})+n(B^{*})-n(A^{*}\cup B^{*})
  54. 1.44 log 2 ( 1 / ϵ ) 1.44\log_{2}(1/\epsilon)
  55. ϵ \epsilon
  56. log 2 ( 1 / ϵ ) \log_{2}(1/\epsilon)
  57. ϵ \epsilon
  58. log ( 1 / ϵ ) \log(1/\epsilon)
  59. [ 0 , n / ε ] \left[0,n/\varepsilon\right]
  60. ϵ \epsilon
  61. n log 2 ( 1 / ϵ ) n\log_{2}(1/\epsilon)

Blower_door.html

  1. A C H n a t u r a l = A C H a t 50 p a s c a l 20 ACH_{natural}={ACH_{at50pascal}\over 20}\,\!
  2. A C H n a t u r a l ACH_{natural}\,\!
  3. A C H a t 50 p a s c a l ACH_{at50pascal}\,\!
  4. Q = C Δ P n Q=C{\Delta}P^{n}\,\!
  5. Q Q\,\!
  6. C C\,\!
  7. Δ P {\Delta}P\,\!
  8. n n\,\!
  9. Q F a n = C F a n Δ P F a n n F a n Q_{Fan}=C_{Fan}{{\Delta}P_{Fan}}^{n_{Fan}}\,\!
  10. Q B u i l d i n g = C B u i l d i n g Δ P B u i l d i n g n B u i l d i n g Q_{Building}=C_{Building}{{\Delta}P_{Building}}^{n_{Building}}\,\!
  11. Q F a n = Q B u i l d i n g Q_{Fan}=Q_{Building}\,\!
  12. C F a n Δ P F a n n F a n = C B u i l d i n g Δ P B u i l d i n g n B u i l d i n g C_{Fan}{{\Delta}P_{Fan}}^{n_{Fan}}=C_{Building}{{\Delta}P_{Building}}^{n_{% Building}}\,\!
  13. C B u i l d i n g C_{Building}\,\!
  14. n B u i l d i n g n_{Building}\,\!
  15. Q C o r r e c t e d = Q M e a s u r e d * ρ I n ρ O u t Q_{Corrected}=Q_{Measured}*{\rho_{In}\over\rho_{Out}}\,\!
  16. Q C o r r e c t e d Q_{Corrected}\,\!
  17. Q M e a s u r e d Q_{Measured}\,\!
  18. C F a n C_{Fan}\,\!
  19. n F a n n_{Fan}\,\!
  20. ρ I n \rho_{In}\,\!
  21. ρ O u t \rho_{Out}\,\!
  22. Q C o r r e c t e d = Q M e a s u r e d * ρ O u t ρ I n Q_{Corrected}=Q_{Measured}*{\rho_{Out}\over\rho_{In}}\,\!
  23. ρ O u t ρ I n {\rho_{Out}\over\rho_{In}}\,\!
  24. ρ I n ρ O u t {\rho_{In}\over\rho_{Out}}\,\!
  25. ρ I n \rho_{In}\,\!
  26. ρ I n = 0.07517 * ( 1 - 0.0035666 * E 528 ) 5.2553 * ( 528 T I n + 460 ) \rho_{In}=0.07517*(1-{0.0035666*E\over 528})^{5.2553}*({528\over T_{In}+460})\,\!
  27. ρ I n \rho_{In}\,\!
  28. E E\,\!
  29. T I n T_{In}\,\!
  30. ρ O u t \rho_{Out}\,\!
  31. ρ O u t = 0.07517 * ( 1 - 0.0035666 * E 528 ) 5.2553 * ( 528 T O u t + 460 ) \rho_{Out}=0.07517*(1-{0.0035666*E\over 528})^{5.2553}*({528\over T_{Out}+460}% )\,\!
  32. ρ O u t \rho_{Out}\,\!
  33. E E\,\!
  34. T O u t T_{Out}\,\!
  35. C F a n C_{Fan}\,\!
  36. n F a n n_{Fan}\,\!
  37. Q A c t u a l = Q F a n * ρ R e f ρ A c t u a l Q_{Actual}=Q_{Fan}*\sqrt{\rho_{Ref}\over\rho_{Actual}}\,\!
  38. Q A c t u a l Q_{Actual}\,\!
  39. Q F a n Q_{Fan}\,\!
  40. ρ R e f \rho_{Ref}\,\!
  41. ρ A c t u a l \rho_{Actual}\,\!
  42. ρ I n \rho_{In}\,\!
  43. ρ O u t \rho_{Out}\,\!
  44. A C H 50 = Q 50 * 60 V B u i l d i n g ACH_{50}={Q_{50}*60\over V_{Building}}\,\!
  45. A C H 50 ACH_{50}\,\!
  46. Q 50 Q_{50}\,\!
  47. V B u i l d i n g V_{Building}\,\!
  48. E L A = C B u i l d i n g * ρ 2 * Δ P R e f n B u i l d i n g - 0.5 ELA=C_{Building}*\sqrt{\rho\over 2}*{\Delta}P_{Ref}^{n_{Building}-0.5}\,\!
  49. E L A ELA\,\!
  50. C B u i l d i n g C_{Building}\,\!
  51. ρ \rho\,\!
  52. Δ P R e f {\Delta}P_{Ref}\,\!
  53. n B u i l d i n g n_{Building}\,\!
  54. N L = 1000 * ( E L A A F l o o r ) * ( H H R e f ) 0.3 NL=1000*({ELA\over A_{Floor}})*({H\over H_{Ref}})^{0.3}\,\!
  55. N L NL\,\!
  56. E L A ELA\,\!
  57. A F l o o r A_{Floor}\,\!
  58. H H\,\!
  59. H R e f H_{Ref}\,\!

Bohlen–Pierce_scale.html

  1. 3 13 = 3 1 / 13 = 1.08818... \sqrt[13]{3}=3^{1/13}=1.08818...
  2. 1200 log 2 ( 3 1 / 13 ) = 146.3... 1200\log_{2}(3^{1/13})=146.3...
  3. 3 0 13 3^{\frac{0}{13}}
  4. 1 1 \begin{matrix}\frac{1}{1}\end{matrix}
  5. 3 1 13 3^{\frac{1}{13}}
  6. 27 25 \begin{matrix}\frac{27}{25}\end{matrix}
  7. 3 2 13 3^{\frac{2}{13}}
  8. 25 21 \begin{matrix}\frac{25}{21}\end{matrix}
  9. 3 3 13 3^{\frac{3}{13}}
  10. 9 7 \begin{matrix}\frac{9}{7}\end{matrix}
  11. 3 4 13 3^{\frac{4}{13}}
  12. 7 5 \begin{matrix}\frac{7}{5}\end{matrix}
  13. 3 5 13 3^{\frac{5}{13}}
  14. 75 49 \begin{matrix}\frac{75}{49}\end{matrix}
  15. 3 6 13 3^{\frac{6}{13}}
  16. 5 3 \begin{matrix}\frac{5}{3}\end{matrix}
  17. 3 7 13 3^{\frac{7}{13}}
  18. 9 5 \begin{matrix}\frac{9}{5}\end{matrix}
  19. 3 8 13 3^{\frac{8}{13}}
  20. 49 25 \begin{matrix}\frac{49}{25}\end{matrix}
  21. 3 9 13 3^{\frac{9}{13}}
  22. 15 7 \begin{matrix}\frac{15}{7}\end{matrix}
  23. 3 10 13 3^{\frac{10}{13}}
  24. 7 3 \begin{matrix}\frac{7}{3}\end{matrix}
  25. 3 11 13 3^{\frac{11}{13}}
  26. 63 25 \begin{matrix}\frac{63}{25}\end{matrix}
  27. 3 12 13 3^{\frac{12}{13}}
  28. 25 9 \begin{matrix}\frac{25}{9}\end{matrix}
  29. 3 13 13 3^{\frac{13}{13}}
  30. 3 1 \begin{matrix}\frac{3}{1}\end{matrix}

Bohr_effect.html

  1. \rightleftharpoons
  2. \rightleftharpoons
  3. \rightleftharpoons

Bonaventura_Cavalieri.html

  1. y = x n y=x^{n}

Boole's_inequality.html

  1. ( i A i ) i ( A i ) . {\mathbb{P}}\biggl(\bigcup_{i}A_{i}\biggr)\leq\sum_{i}{\mathbb{P}}(A_{i}).
  2. n = 1 n=1
  3. ( A 1 ) ( A 1 ) . \mathbb{P}(A_{1})\leq\mathbb{P}(A_{1}).
  4. n n
  5. ( i = 1 n A i ) i = 1 n ( A i ) . {\mathbb{P}}\biggl(\bigcup_{i=_{1}}^{n}A_{i}\biggr)\leq\sum_{i=_{1}}^{n}{% \mathbb{P}}(A_{i}).
  6. ( A B ) = ( A ) + ( B ) - ( A B ) , \mathbb{P}(A\cup B)=\mathbb{P}(A)+\mathbb{P}(B)-\mathbb{P}(A\cap B),
  7. ( i = 1 n + 1 A i ) = ( i = 1 n A i ) + ( A n + 1 ) - ( i = 1 n A i A n + 1 ) . {\mathbb{P}}\biggl(\bigcup_{i=_{1}}^{n+1}A_{i}\biggr)={\mathbb{P}}\biggl(% \bigcup_{i=_{1}}^{n}A_{i}\biggr)+\mathbb{P}(A_{n+1})-{\mathbb{P}}\biggl(% \bigcup_{i=_{1}}^{n}A_{i}\cap A_{n+1}\biggr).
  8. ( i = 1 n A i A n + 1 ) 0 , {\mathbb{P}}\biggl(\bigcup_{i=_{1}}^{n}A_{i}\cap A_{n+1}\biggr)\geq 0,
  9. ( i = 1 n + 1 A i ) ( i = 1 n A i ) + ( A n + 1 ) {\mathbb{P}}\biggl(\bigcup_{i=_{1}}^{n+1}A_{i}\biggr)\leq{\mathbb{P}}\biggl(% \bigcup_{i=_{1}}^{n}A_{i}\biggr)+\mathbb{P}(A_{n+1})
  10. ( i = 1 n + 1 A i ) i = 1 n ( A i ) + ( A n + 1 ) = i = 1 n + 1 ( A i ) {\mathbb{P}}\biggl(\bigcup_{i=_{1}}^{n+1}A_{i}\biggr)\leq\sum_{i=_{1}}^{n}{% \mathbb{P}}(A_{i})+\mathbb{P}(A_{n+1})=\sum_{i=_{1}}^{n+1}{\mathbb{P}}(A_{i})
  11. S 1 := i = 1 n ( A i ) , S_{1}:=\sum_{i=1}^{n}{\mathbb{P}}(A_{i}),
  12. S 2 := 1 i < j n ( A i A j ) , S_{2}:=\sum_{1\leq i<j\leq n}{\mathbb{P}}(A_{i}\cap A_{j}),
  13. S k := 1 i 1 < < i k n ( A i 1 A i k ) S_{k}:=\sum_{1\leq i_{1}<\cdots<i_{k}\leq n}{\mathbb{P}}(A_{i_{1}}\cap\cdots% \cap A_{i_{k}})
  14. ( i = 1 n A i ) j = 1 k ( - 1 ) j - 1 S j , {\mathbb{P}}\biggl(\bigcup_{i=1}^{n}A_{i}\biggr)\leq\sum_{j=1}^{k}(-1)^{j-1}S_% {j},
  15. ( i = 1 n A i ) j = 1 k ( - 1 ) j - 1 S j . {\mathbb{P}}\biggl(\bigcup_{i=1}^{n}A_{i}\biggr)\geq\sum_{j=1}^{k}(-1)^{j-1}S_% {j}.

Borel_regular_measure.html

  1. μ ( A ) = μ ( A B ) + μ ( A B ) . \mu(A)=\mu(A\cap B)+\mu(A\setminus B).

Borel_subgroup.html

  1. 𝔤 \mathfrak{g}
  2. 𝔥 \mathfrak{h}
  3. 𝔥 \mathfrak{h}
  4. 𝔥 \mathfrak{h}
  5. 𝔤 \mathfrak{g}
  6. 𝔤 \mathfrak{g}

Borel–Kolmogorov_paradox.html

  1. 1 2 cos ϕ \frac{1}{2}\cos\phi
  2. f ( λ | ϕ = 0 ) = 1 2 π . f(\lambda|\phi=0)=\frac{1}{2\pi}.
  3. f ( ϕ | λ = 0 ) = 1 2 cos ϕ . f(\phi|\lambda=0)=\frac{1}{2}\cos\phi.
  4. x = r cos ϕ cos λ x=r\cos\phi\cos\lambda
  5. y = r cos ϕ sin λ y=r\cos\phi\sin\lambda
  6. z = r sin ϕ z=r\sin\phi
  7. ω r ( ϕ , λ ) = || ( x , y , z ) ϕ × ( x , y , z ) λ || = r 2 cos ϕ . \omega_{r}(\phi,\lambda)=\left|\left|{\partial(x,y,z)\over\partial\phi}\times{% \partial(x,y,z)\over\partial\lambda}\right|\right|=r^{2}\cos\phi\ .
  8. ω r ( λ ) = || ( x , y , z ) ϕ || = r , respectively \omega_{r}(\lambda)=\left|\left|{\partial(x,y,z)\over\partial\phi}\right|% \right|=r\ ,\quad\mathrm{respectively}
  9. ω r ( ϕ ) = || ( x , y , z ) λ || = r cos ϕ . \omega_{r}(\phi)=\left|\left|{\partial(x,y,z)\over\partial\lambda}\right|% \right|=r\cos\phi\ .
  10. μ Φ , Λ ( d ϕ , d λ ) = f Φ , Λ ( ϕ , λ ) ω r ( ϕ , λ ) d ϕ d λ \mu_{\Phi,\Lambda}(d\phi,d\lambda)=f_{\Phi,\Lambda}(\phi,\lambda)\omega_{r}(% \phi,\lambda)d\phi d\lambda
  11. ( Ω 1 × Ω 2 ) \mathcal{B}(\Omega_{1}\times\Omega_{2})
  12. f Φ , Λ f_{\Phi,\Lambda}
  13. ω r ( ϕ , λ ) d ϕ d λ \omega_{r}(\phi,\lambda)d\phi d\lambda
  14. μ Φ ( d ϕ ) = λ Ω 2 μ Φ , Λ ( d ϕ , d λ ) , \mu_{\Phi}(d\phi)=\int_{\lambda\in\Omega_{2}}\mu_{\Phi,\Lambda}(d\phi,d\lambda% )\ ,
  15. μ Λ ( d λ ) = ϕ Ω 1 μ Φ , Λ ( d ϕ , d λ ) . \mu_{\Lambda}(d\lambda)=\int_{\phi\in\Omega_{1}}\mu_{\Phi,\Lambda}(d\phi,d% \lambda)\ .
  16. f Φ , Λ f_{\Phi,\Lambda}
  17. μ Φ | Λ ( d ϕ | λ ) = μ Φ , Λ ( d ϕ , d λ ) μ Λ ( d λ ) = 1 2 r ω r ( ϕ ) d ϕ , and \mu_{\Phi|\Lambda}(d\phi|\lambda)={\mu_{\Phi,\Lambda}(d\phi,d\lambda)\over\mu_% {\Lambda}(d\lambda)}=\frac{1}{2r}\omega_{r}(\phi)d\phi\ ,\quad\mathrm{and}
  18. μ Λ | Φ ( d λ | ϕ ) = μ Φ , Λ ( d ϕ , d λ ) μ Φ ( d ϕ ) = 1 2 r π ω r ( λ ) d λ . \mu_{\Lambda|\Phi}(d\lambda|\phi)={\mu_{\Phi,\Lambda}(d\phi,d\lambda)\over\mu_% {\Phi}(d\phi)}=\frac{1}{2r\pi}\omega_{r}(\lambda)d\lambda\ .
  19. μ Φ | Λ \mu_{\Phi|\Lambda}
  20. ω r ( ϕ ) d ϕ \omega_{r}(\phi)d\phi
  21. μ Λ | Φ \mu_{\Lambda|\Phi}
  22. ω r ( λ ) d λ \omega_{r}(\lambda)d\lambda

Boundary_value_problem.html

  1. y ( t ) y(t)
  2. t = 0 t=0
  3. t = 1 t=1
  4. y ( t ) y(t)
  5. y ( t ) y^{\prime}(t)
  6. t = 0 t=0
  7. y ′′ ( x ) + y ( x ) = 0 y^{\prime\prime}(x)+y(x)=0\,
  8. y ( x ) y(x)
  9. y ( 0 ) = 0 , y ( π / 2 ) = 2. y(0)=0,\ y(\pi/2)=2.
  10. y ( x ) = A sin ( x ) + B cos ( x ) . y(x)=A\sin(x)+B\cos(x).\,
  11. y ( 0 ) = 0 y(0)=0
  12. 0 = A 0 + B 1 0=A\cdot 0+B\cdot 1
  13. B = 0. B=0.
  14. y ( π / 2 ) = 2 y(\pi/2)=2
  15. 2 = A 1 2=A\cdot 1
  16. A = 2. A=2.
  17. y ( x ) = 2 sin ( x ) . y(x)=2\sin(x).\,

Bounded_operator.html

  1. L v Y M v X . \|Lv\|_{Y}\leq M\|v\|_{X}.\,\,
  2. L op \|L\|_{\mathrm{op}}\,
  3. K : [ a , b ] × [ c , d ] K:[a,b]\times[c,d]\to{\mathbb{R}}\,
  4. L , L,\,
  5. C [ a , b ] C[a,b]\,
  6. [ a , b ] [a,b]\,
  7. C [ c , d ] , C[c,d],\,
  8. L L\,
  9. ( L f ) ( y ) = a b K ( x , y ) f ( x ) d x , (Lf)(y)=\int_{a}^{b}\!K(x,y)f(x)\,dx,\,
  10. Δ : H 2 ( n ) L 2 ( n ) \Delta:H^{2}({\mathbb{R}}^{n})\to L^{2}({\mathbb{R}}^{n})\,
  11. x 0 2 + x 1 2 + x 2 2 + < , x_{0}^{2}+x_{1}^{2}+x_{2}^{2}+\cdots<\infty,\,
  12. L ( x 0 , x 1 , x 2 , ) = ( 0 , x 0 , x 1 , x 2 , ) L(x_{0},x_{1},x_{2},\dots)=(0,x_{0},x_{1},x_{2},\dots)\,
  13. L ( v + h ) - L v = L h M h . \|L(v+h)-Lv\|=\|Lh\|\leq M\|h\|.\,
  14. h \mathit{h}\,
  15. δ > 0 \delta>0
  16. L ( h ) = L ( h ) - L ( 0 ) 1 \|L(h)\|=\|L(h)-L(0)\|\leq 1
  17. h δ \|h\|\leq\delta
  18. v v
  19. L v = v δ L ( δ v v ) = v δ L ( δ v v ) v δ 1 = 1 δ v . \|Lv\|=\left\|{\|v\|\over\delta}L\left(\delta{v\over\|v\|}\right)\right\|={\|v% \|\over\delta}\left\|L\left(\delta{v\over\|v\|}\right)\right\|\leq{\|v\|\over% \delta}\cdot 1={1\over\delta}\|v\|.
  20. P = - π π | P ( x ) | d x . \|P\|=\int_{-\pi}^{\pi}\!|P(x)|\,dx.
  21. v = e i n x v=e^{inx}
  22. v = 2 π , \|v\|=2\pi,
  23. L ( v ) = 2 π n \|L(v)\|=2\pi n\to\infty
  24. L v M v , \|Lv\|\leq M\|v\|,\,

Bounded_variation.html

  1. B V BV
  2. y y
  3. x x
  4. x x
  5. y y
  6. f f
  7. g h g−h
  8. g g
  9. h h
  10. f f
  11. Ω Ω
  12. V a b ( f ) = sup P 𝒫 i = 0 n P - 1 | f ( x i + 1 ) - f ( x i ) | . V_{a}^{b}(f)=\sup_{P\in\mathcal{P}}\sum_{i=0}^{n_{P}-1}|f(x_{i+1})-f(x_{i})|.\,
  13. 𝒫 = { P = { x 0 , , x n P } | P is a partition of [ a , b ] } \scriptstyle\mathcal{P}=\left\{P=\{x_{0},\dots,x_{n_{P}}\}|P\,\text{ is a % partition of }[a,b]\right\}
  14. V a b ( f ) = a b | f ( x ) | d x . V_{a}^{b}(f)=\int_{a}^{b}|f^{\prime}(x)|\,\mathrm{d}x.
  15. f f
  16. f B V ( [ a , b ] ) V a b ( f ) < + f\in BV([a,b])\iff V_{a}^{b}(f)<+\infty
  17. Ω \Omega
  18. u u
  19. L 1 ( Ω ) L^{1}(\Omega)
  20. u B V ( Ω ) u\in BV(\Omega)
  21. D u ( Ω , n ) \scriptstyle Du\in\mathcal{M}(\Omega,\mathbb{R}^{n})
  22. Ω u ( x ) div s y m b o l ϕ ( x ) d x = - Ω \langlesymbol ϕ , D u ( x ) \forallsymbol ϕ C c 1 ( Ω , n ) \int_{\Omega}u(x)\,\mathrm{div}symbol{\phi}(x)\mathrm{d}x=-\int_{\Omega}% \langlesymbol{\phi},Du(x)\rangle\qquad\forallsymbol{\phi}\in C_{c}^{1}(\Omega,% \mathbb{R}^{n})
  23. u u
  24. C c 1 ( Ω , n ) \scriptstyle C_{c}^{1}(\Omega,\mathbb{R}^{n})
  25. \scriptstylesymbol ϕ \scriptstylesymbol{\phi}
  26. Ω \Omega
  27. D u Du
  28. u u
  29. u u
  30. L 1 ( Ω ) L^{1}(\Omega)
  31. u u
  32. Ω \Omega
  33. V ( u , Ω ) := sup { Ω u ( x ) div s y m b o l ϕ ( x ) d x \colonsymbol ϕ C c 1 ( Ω , n ) , \Vertsymbol ϕ L ( Ω ) 1 } V(u,\Omega):=\sup\left\{\int_{\Omega}u(x)\mathrm{div}symbol{\phi}(x)\mathrm{d}% x\colonsymbol{\phi}\in C_{c}^{1}(\Omega,\mathbb{R}^{n}),\ \Vertsymbol{\phi}\|_% {L^{\infty}(\Omega)}\leq 1\right\}
  34. L ( Ω ) \scriptstyle\|\;\|_{L^{\infty}(\Omega)}
  35. Ω | D u | = V ( u , Ω ) \int_{\Omega}|Du|=V(u,\Omega)
  36. V ( u , Ω ) V(u,\Omega)
  37. u u
  38. u u
  39. C 1 C^{1}
  40. B V ( Ω ) = { u L 1 ( Ω ) : V ( u , Ω ) < + } BV(\Omega)=\{u\in L^{1}(\Omega)\colon V(u,\Omega)<+\infty\}
  41. V ( u , Ω ) < + \scriptstyle V(u,\Omega)<+\infty
  42. | Ω u ( x ) div s y m b o l ϕ ( x ) d x | V ( u , Ω ) \Vertsymbol ϕ | L ( Ω ) s y m b o l ϕ C c 1 ( Ω , n ) \left|\int_{\Omega}u(x)\,\mathrm{div}symbol{\phi}(x)\mathrm{d}x\right|\leq V(u% ,\Omega)\Vertsymbol{\phi}\|_{L^{\infty}(\Omega)}\qquad\forall symbol{\phi}\in C% _{c}^{1}(\Omega,\mathbb{R}^{n})
  43. s y m b o l ϕ Ω u ( x ) div s y m b o l ϕ ( x ) d x \scriptstyle symbol{\phi}\mapsto\,\int_{\Omega}u(x)\,\mathrm{div}symbol{\phi}(% x)dx
  44. C c 1 ( Ω , n ) \scriptstyle C_{c}^{1}(\Omega,\mathbb{R}^{n})
  45. C c 1 ( Ω , n ) C 0 ( Ω , n ) \scriptstyle C_{c}^{1}(\Omega,\mathbb{R}^{n})\subset C^{0}(\Omega,\mathbb{R}^{% n})
  46. C 0 ( Ω , n ) \scriptstyle C^{0}(\Omega,\mathbb{R}^{n})
  47. L l o c 1 ( Ω ) \scriptstyle L^{1}_{loc}(\Omega)
  48. V ( u , U ) := sup { Ω u ( x ) div s y m b o l ϕ ( x ) d x \colonsymbol ϕ C c 1 ( U , n ) , \Vertsymbol ϕ L ( Ω ) 1 } V(u,U):=\sup\left\{\int_{\Omega}u(x)\mathrm{div}symbol{\phi}(x)\mathrm{d}x% \colonsymbol{\phi}\in C_{c}^{1}(U,\mathbb{R}^{n}),\ \Vertsymbol{\phi}\|_{L^{% \infty}(\Omega)}\leq 1\right\}
  49. U 𝒪 c ( Ω ) \scriptstyle U\in\mathcal{O}_{c}(\Omega)
  50. 𝒪 c ( Ω ) \scriptstyle\mathcal{O}_{c}(\Omega)
  51. Ω \Omega
  52. B V l o c ( Ω ) = { u L l o c 1 ( Ω ) : V ( u , U ) < + U 𝒪 c ( Ω ) } BV_{loc}(\Omega)=\{u\in L^{1}_{loc}(\Omega)\colon V(u,U)<+\infty\;\forall U\in% \mathcal{O}_{c}(\Omega)\}
  53. B V ( Ω ) \scriptstyle BV(\Omega)
  54. B V l o c ( Ω ) \scriptstyle BV_{loc}(\Omega)
  55. B V ¯ ( Ω ) \scriptstyle\overline{BV}(\Omega)
  56. B V ( Ω ) \scriptstyle BV(\Omega)
  57. x 0 x_{0}
  58. [ a , b ] [a,b]
  59. u u
  60. lim x x 0 - u ( x ) = lim x x 0 + u ( x ) \lim_{x\rightarrow x_{0^{-}}}\!\!\!u(x)=\!\!\!\lim_{x\rightarrow x_{0^{+}}}\!% \!\!u(x)
  61. lim x x 0 - u ( x ) lim x x 0 + u ( x ) \lim_{x\rightarrow x_{0^{-}}}\!\!\!u(x)\neq\!\!\!\lim_{x\rightarrow x_{0^{+}}}% \!\!\!u(x)
  62. x 0 x_{0}
  63. Ω \Omega
  64. s y m b o l a ^ n \scriptstyle{symbol{\hat{a}}}\in\mathbb{R}^{n}
  65. Ω \Omega
  66. Ω ( s y m b o l a ^ , s y m b o l x 0 ) = Ω { s y m b o l x n | \langlesymbol x - s y m b o l x 0 , s y m b o l a ^ > 0 } Ω ( - s y m b o l a ^ , s y m b o l x 0 ) = Ω { s y m b o l x n | \langlesymbol x - s y m b o l x 0 , - s y m b o l a ^ > 0 } \Omega_{({symbol{\hat{a}}},symbol{x}_{0})}=\Omega\cap\{symbol{x}\in\mathbb{R}^% {n}|\langlesymbol{x}-symbol{x}_{0},{symbol{\hat{a}}}\rangle>0\}\qquad\Omega_{(% -{symbol{\hat{a}}},symbol{x}_{0})}=\Omega\cap\{symbol{x}\in\mathbb{R}^{n}|% \langlesymbol{x}-symbol{x}_{0},-{symbol{\hat{a}}}\rangle>0\}
  67. x 0 x_{0}
  68. Ω n \scriptstyle\Omega\in\mathbb{R}^{n}
  69. u u
  70. lim s y m b o l x Ω ( s y m b o l a ^ , s y m b o l x 0 ) s y m b o l x s y m b o l x 0 u ( s y m b o l x ) = lim s y m b o l x Ω ( - s y m b o l a ^ , s y m b o l x 0 ) s y m b o l x s y m b o l x 0 u ( s y m b o l x ) \lim_{\overset{symbol{x}\rightarrow symbol{x}_{0}}{symbol{x}\in\Omega_{({% symbol{\hat{a}}},symbol{x}_{0})}}}\!\!\!\!\!\!u(symbol{x})=\!\!\!\!\!\!\!\lim_% {\overset{symbol{x}\rightarrow symbol{x}_{0}}{symbol{x}\in\Omega_{(-{symbol{% \hat{a}}},symbol{x}_{0})}}}\!\!\!\!\!\!\!u(symbol{x})
  71. lim s y m b o l x Ω ( s y m b o l a ^ , s y m b o l x 0 ) s y m b o l x s y m b o l x 0 u ( s y m b o l x ) lim s y m b o l x Ω ( - s y m b o l a ^ , s y m b o l x 0 ) s y m b o l x s y m b o l x 0 u ( s y m b o l x ) \lim_{\overset{symbol{x}\rightarrow symbol{x}_{0}}{symbol{x}\in\Omega_{({% symbol{\hat{a}}},symbol{x}_{0})}}}\!\!\!\!\!\!u(symbol{x})\neq\!\!\!\!\!\!\!% \lim_{\overset{symbol{x}\rightarrow symbol{x}_{0}}{symbol{x}\in\Omega_{(-{% symbol{\hat{a}}},symbol{x}_{0})}}}\!\!\!\!\!\!\!u(symbol{x})
  72. x 0 x_{0}
  73. Ω \Omega
  74. n - 1 n-1
  75. lim s y m b o l x Ω ( s y m b o l a ^ , s y m b o l x 0 ) s y m b o l x s y m b o l x 0 u ( s y m b o l x ) = u s y m b o l a ^ ( s y m b o l x 0 ) lim s y m b o l x Ω ( - s y m b o l a ^ , s y m b o l x 0 ) s y m b o l x s y m b o l x 0 u ( s y m b o l x ) = u - s y m b o l a ^ ( s y m b o l x 0 ) \lim_{\overset{symbol{x}\rightarrow symbol{x}_{0}}{symbol{x}\in\Omega_{({% symbol{\hat{a}}},symbol{x}_{0})}}}\!\!\!\!\!\!u(symbol{x})=u_{symbol{\hat{a}}}% (symbol{x}_{0})\qquad\lim_{\overset{symbol{x}\rightarrow symbol{x}_{0}}{symbol% {x}\in\Omega_{(-{symbol{\hat{a}}},symbol{x}_{0})}}}\!\!\!\!\!\!\!u(symbol{x})=% u_{-symbol{\hat{a}}}(symbol{x}_{0})
  76. u u
  77. x 0 x_{0}
  78. V ( , Ω ) : B V ( Ω ) + \scriptstyle V(\cdot,\Omega):BV(\Omega)\rightarrow\mathbb{R}^{+}
  79. { u n } n \scriptstyle\{u_{n}\}_{n\in\mathbb{N}}
  80. u L 1 loc ( Ω ) \scriptstyle u\in L^{1}\text{loc}(\Omega)
  81. lim inf n V ( u n , Ω ) lim inf n Ω u n ( x ) div s y m b o l ϕ d x Ω lim n u n ( x ) div s y m b o l ϕ d x = Ω u ( x ) div s y m b o l ϕ d x \forallsymbol ϕ C c 1 ( Ω , n ) , \Vertsymbol ϕ | L ( Ω ) 1 \liminf_{n\rightarrow\infty}V(u_{n},\Omega)\geq\liminf_{n\rightarrow\infty}% \int_{\Omega}u_{n}(x)\,\mathrm{div}\,symbol{\phi}\,\mathrm{d}x\geq\int_{\Omega% }\lim_{n\rightarrow\infty}u_{n}(x)\,\mathrm{div}\,symbol{\phi}\,\mathrm{d}x=% \int_{\Omega}u(x)\,\mathrm{div}symbol{\phi}\,\mathrm{d}x\qquad\forallsymbol{% \phi}\in C_{c}^{1}(\Omega,\mathbb{R}^{n}),\quad\Vertsymbol{\phi}\|_{L^{\infty}% (\Omega)}\leq 1
  82. \scriptstylesymbol ϕ C c 1 ( Ω , n ) \scriptstylesymbol{\phi}\in C_{c}^{1}(\Omega,\mathbb{R}^{n})
  83. \Vertsymbol ϕ | L ( Ω ) 1 \scriptstyle\Vertsymbol{\phi}\|_{L^{\infty}(\Omega)}\leq 1
  84. lim inf n V ( u n , Ω ) V ( u , Ω ) \liminf_{n\rightarrow\infty}V(u_{n},\Omega)\geq V(u,\Omega)
  85. B V ( Ω ) BV(\Omega)
  86. L 1 ( Ω ) L^{1}(\Omega)
  87. Ω [ u ( x ) + v ( x ) ] div s y m b o l ϕ ( x ) d x = Ω u ( x ) div s y m b o l ϕ ( x ) d x + Ω v ( x ) div s y m b o l ϕ ( x ) d x = = - Ω \langlesymbol ϕ ( x ) , D u ( x ) - Ω s y m b o l ϕ ( x ) , D v ( x ) = - Ω s y m b o l ϕ ( x ) , [ D u ( x ) + D v ( x ) ] \begin{aligned}\displaystyle\int_{\Omega}[u(x)+v(x)]\,\mathrm{div}symbol{\phi}% (x)\mathrm{d}x&\displaystyle=\int_{\Omega}u(x)\,\mathrm{div}symbol{\phi}(x)% \mathrm{d}x+\int_{\Omega}v(x)\,\mathrm{div}symbol{\phi}(x)\mathrm{d}x=\\ &\displaystyle=-\int_{\Omega}\langlesymbol{\phi}(x),Du(x)\rangle-\int_{\Omega}% \langle symbol{\phi}(x),Dv(x)\rangle=-\int_{\Omega}\langle symbol{\phi}(x),[Du% (x)+Dv(x)]\rangle\end{aligned}
  88. ϕ C c 1 ( Ω , n ) \scriptstyle\phi\in C_{c}^{1}(\Omega,\mathbb{R}^{n})
  89. u + v B V ( Ω ) \scriptstyle u+v\in BV(\Omega)
  90. u , v B V ( Ω ) \scriptstyle u,v\in BV(\Omega)
  91. Ω c u ( x ) div s y m b o l ϕ ( x ) d x = c Ω u ( x ) div s y m b o l ϕ ( x ) d x = - c Ω s y m b o l ϕ ( x ) , D u ( x ) \int_{\Omega}c\cdot u(x)\,\mathrm{div}symbol{\phi}(x)\mathrm{d}x=c\!\int_{% \Omega}u(x)\,\mathrm{div}symbol{\phi}(x)\mathrm{d}x=-c\!\int_{\Omega}\langle symbol% {\phi}(x),Du(x)\rangle
  92. c \scriptstyle c\in\mathbb{R}
  93. c u B V ( Ω ) \scriptstyle cu\in BV(\Omega)
  94. u B V ( Ω ) \scriptstyle u\in BV(\Omega)
  95. c \scriptstyle c\in\mathbb{R}
  96. B V ( Ω ) BV(\Omega)
  97. L 1 ( Ω ) L^{1}(\Omega)
  98. B V : B V ( Ω ) + \scriptstyle\|\;\|_{BV}:BV(\Omega)\rightarrow\mathbb{R}^{+}
  99. u B V := u L 1 + V ( u , Ω ) \|u\|_{BV}:=\|u\|_{L^{1}}+V(u,\Omega)
  100. L 1 \scriptstyle\|\;\|_{L^{1}}
  101. L 1 ( Ω ) L^{1}(\Omega)
  102. B V ( Ω ) BV(\Omega)
  103. B V ( Ω ) BV(\Omega)
  104. { u n } n \scriptstyle\{u_{n}\}_{n\in\mathbb{N}}
  105. B V ( Ω ) BV(\Omega)
  106. L 1 ( Ω ) L^{1}(\Omega)
  107. u u
  108. L 1 ( Ω ) L^{1}(\Omega)
  109. u n u_{n}
  110. B V ( Ω ) BV(\Omega)
  111. n n
  112. u B V < + \scriptstyle\|u\|_{BV}<+\infty
  113. V ( , Ω ) \scriptstyle V(\cdot,\Omega)
  114. u u
  115. ε \scriptstyle\varepsilon
  116. u j - u k B V < ε j , k N V ( u k - u , Ω ) lim inf j + V ( u k - u j , Ω ) ε \|u_{j}-u_{k}\|_{BV}<\varepsilon\quad\forall j,k\geq N\in\mathbb{N}\quad% \Rightarrow\quad V(u_{k}-u,\Omega)\leq\liminf_{j\rightarrow+\infty}V(u_{k}-u_{% j},\Omega)\leq\varepsilon
  117. B V ( [ 0 , 1 ] ) BV([0,1])
  118. [ α , 1 ] [\alpha,1]
  119. [ 0 , 1 ] [0,1]
  120. χ α - χ β B V = 2 + | α - β | \|\chi_{\alpha}-\chi_{\beta}\|_{BV}=2+|\alpha-\beta|
  121. B V ( ] 0 , 1 [ ) BV(]0,1[)
  122. [ 0 , 1 ] [0,1]
  123. B α = { ψ B V ( [ 0 , 1 ] ) ; χ α - ψ B V 1 } B_{\alpha}=\left\{\psi\in BV([0,1]);\|\chi_{\alpha}-\psi\|_{BV}\leq 1\right\}
  124. [ 0 , 1 ] [0,1]
  125. B V ( [ 0 , 1 ] ) BV([0,1])
  126. B V l o c BV_{loc}
  127. f : p \scriptstyle f:\mathbb{R}^{p}\rightarrow\mathbb{R}
  128. C 1 C^{1}
  129. \scriptstylesymbol u ( s y m b o l x ) = ( u 1 ( s y m b o l x ) , , u p ( s y m b o l x ) ) \scriptstylesymbol{u}(symbol{x})=(u_{1}(symbol{x}),\ldots,u_{p}(symbol{x}))
  130. B V ( Ω ) BV(\Omega)
  131. Ω \Omega
  132. n \scriptstyle\mathbb{R}^{n}
  133. f \circsymbol u ( s y m b o l x ) = f ( s y m b o l u ( s y m b o l x ) ) B V ( Ω ) \scriptstyle f\circsymbol{u}(symbol{x})=f(symbol{u}(symbol{x}))\in BV(\Omega)
  134. f ( s y m b o l u ( s y m b o l x ) ) x i = k = 1 p f ¯ ( s y m b o l u ( s y m b o l x ) ) u k u k ( s y m b o l x ) x i i = 1 , , n \frac{\partial f(symbol{u}(symbol{x}))}{\partial x_{i}}=\sum_{k=1}^{p}\frac{% \partial\bar{f}(symbol{u}(symbol{x}))}{\partial u_{k}}\frac{\partial{u_{k}(% symbol{x})}}{\partial x_{i}}\qquad\forall i=1,\ldots,n
  135. f ¯ ( s y m b o l u ( s y m b o l x ) ) \scriptstyle\bar{f}(symbol{u}(symbol{x}))
  136. x Ω \scriptstyle x\in\Omega
  137. f ¯ ( s y m b o l u ( s y m b o l x ) ) = 0 1 f ( s y m b o l u s y m b o l a ^ ( s y m b o l x ) t + s y m b o l u - s y m b o l a ^ ( s y m b o l x ) ( 1 - t ) ) d t \bar{f}(symbol{u}(symbol{x}))=\int_{0}^{1}f\left(symbol{u}_{symbol{\hat{a}}}(% symbol{x})t+symbol{u}_{-symbol{\hat{a}}}(symbol{x})(1-t)\right)dt
  138. f : p s \scriptstyle f:\mathbb{R}^{p}\rightarrow\mathbb{R}^{s}
  139. f ( u ) = v ( s y m b o l x ) u ( s y m b o l x ) \scriptstyle f(u)=v(symbol{x})u(symbol{x})
  140. v ( s y m b o l x ) \scriptstyle v(symbol{x})
  141. B V BV
  142. B V BV
  143. v ( s y m b o l x ) u ( s y m b o l x ) x i = u ¯ ( s y m b o l x ) v ( s y m b o l x ) x i + v ¯ ( s y m b o l x ) u ( s y m b o l x ) x i \frac{\partial v(symbol{x})u(symbol{x})}{\partial x_{i}}={\bar{u}(symbol{x})}% \frac{\partial v(symbol{x})}{\partial x_{i}}+{\bar{v}(symbol{x})}\frac{% \partial u(symbol{x})}{\partial x_{i}}
  144. B V ( Ω ) BV(\Omega)
  145. B V ( Ω ) BV(\Omega)
  146. { v n } \{v_{n}\}
  147. { u n } \{u_{n}\}
  148. B V BV
  149. v v
  150. u u
  151. B V ( Ω ) BV(\Omega)
  152. v u n n v u v n u n v u v u B V ( Ω ) \begin{matrix}vu_{n}\xrightarrow[n\to\infty]{}vu\\ v_{n}u\xrightarrow[n\to\infty]{}vu\end{matrix}\quad\Longleftrightarrow\quad vu% \in BV(\Omega)
  153. B V ( Ω ) BV(\Omega)
  154. φ : [ 0 , + ) [ 0 , + ) \scriptstyle\varphi:[0,+\infty)\longrightarrow[0,+\infty)
  155. φ ( 0 ) = φ ( 0 + ) = lim x 0 + φ ( x ) = 0 \scriptstyle\varphi(0)=\varphi(0+)=\lim_{x\rightarrow 0_{+}}\varphi(x)=0
  156. f : [ 0 , T ] X \scriptstyle f:[0,T]\longrightarrow X
  157. [ 0 , T ] [0,T]
  158. X X
  159. s y m b o l φ \scriptstyle symbol\varphi
  160. f f
  161. [ 0 , T ] [0,T]
  162. φ -Var [ 0 , T ] ( f ) := sup j = 0 k φ ( | f ( t j + 1 ) - f ( t j ) | X ) , \mathop{\varphi\mbox{-Var}~{}}_{[0,T]}(f):=\sup\sum_{j=0}^{k}\varphi\left(|f(t% _{j+1})-f(t_{j})|_{X}\right),
  163. [ 0 , T ] [0,T]
  164. t i t_{i}
  165. 0 = t 0 < t 1 < < t k = T . 0=t_{0}<t_{1}<\ldots<t_{k}=T.
  166. φ \scriptstyle\varphi
  167. f f
  168. φ \scriptstyle\varphi
  169. φ \scriptstyle\varphi
  170. f B V φ ( [ 0 , T ] ; X ) φ -Var [ 0 , T ] ( f ) < + f\in BV_{\varphi}([0,T];X)\iff\mathop{\varphi\mbox{-Var}~{}}_{[0,T]}(f)<+\infty
  171. B V φ ( [ 0 , T ] ; X ) \scriptstyle BV_{\varphi}([0,T];X)
  172. f B V φ := f + φ -Var [ 0 , T ] ( f ) , \|f\|_{BV_{\varphi}}:=\|f\|_{\infty}+\mathop{\varphi\mbox{-Var}~{}}_{[0,T]}(f),
  173. f \scriptstyle\|f\|_{\infty}
  174. f f
  175. φ ( x ) = x p \scriptstyle\varphi(x)=x^{p}
  176. p p
  177. Ω \Omega
  178. S B V ( Ω ) SBV(\Omega)
  179. B V ( Ω ) BV(\Omega)
  180. n n
  181. n - 1 n-1
  182. u u
  183. u S B V ( Ω ) \scriptstyle u\in{S\!BV}(\Omega)
  184. f f
  185. g g
  186. Ω \Omega
  187. Ω | f | d H n + Ω | g | d H n - 1 < + . \int_{\Omega}|f|dH^{n}+\int_{\Omega}|g|dH^{n-1}<+\infty.
  188. ϕ \scriptstyle\phi
  189. Ω \Omega
  190. ϕ C c 1 ( Ω , n ) \scriptstyle\phi\in C_{c}^{1}(\Omega,\mathbb{R}^{n})
  191. Ω u div ϕ d H n = Ω ϕ , f d H n + Ω ϕ , g d H n - 1 . \int_{\Omega}u\mbox{div}~{}\phi dH^{n}=\int_{\Omega}\langle\phi,f\rangle dH^{n% }+\int_{\Omega}\langle\phi,g\rangle dH^{n-1}.
  192. H α H^{\alpha}
  193. α \alpha
  194. T V ( x ) = i = 1 | x i + 1 - x i | . TV(x)=\sum_{i=1}^{\infty}|x_{i+1}-x_{i}|.
  195. x b v = | x 1 | + T V ( x ) = | x 1 | + i = 1 | x i + 1 - x i | . \|x\|_{bv}=|x_{1}|+TV(x)=|x_{1}|+\sum_{i=1}^{\infty}|x_{i+1}-x_{i}|.
  196. lim n x n = 0. \lim_{n\to\infty}x_{n}=0.
  197. x b v 0 = T V ( x ) = i = 1 | x i + 1 - x i | . \|x\|_{bv_{0}}=TV(x)=\sum_{i=1}^{\infty}|x_{i+1}-x_{i}|.
  198. μ \mu
  199. ( X , Σ ) (X,\Sigma)
  200. μ = | μ | ( X ) \scriptstyle\|\mu\|=|\mu|(X)
  201. f ( x ) = { 0 , if x = 0 sin ( 1 / x ) , if x 0 f(x)=\begin{cases}0,&\mbox{if }~{}x=0\\ \sin(1/x),&\mbox{if }~{}x\neq 0\end{cases}
  202. [ 0 , 2 / π ] [0,2/\pi]
  203. f ( x ) = { 0 , if x = 0 x sin ( 1 / x ) , if x 0 f(x)=\begin{cases}0,&\mbox{if }~{}x=0\\ x\sin(1/x),&\mbox{if }~{}x\neq 0\end{cases}
  204. [ 0 , 2 / π ] [0,2/\pi]
  205. f ( x ) = { 0 , if x = 0 x 2 sin ( 1 / x ) , if x 0 f(x)=\begin{cases}0,&\mbox{if }~{}x=0\\ x^{2}\sin(1/x),&\mbox{if }~{}x\neq 0\end{cases}
  206. [ 0 , 2 / π ] [0,2/\pi]
  207. [ a , b ] [a,b]
  208. a > 0 a>0
  209. W 1 , 1 ( Ω ) W^{1,1}(\Omega)
  210. B V ( Ω ) BV(\Omega)
  211. u u
  212. W 1 , 1 ( Ω ) W^{1,1}(\Omega)
  213. μ := u \scriptstyle\mu:=\nabla u\mathcal{L}
  214. \scriptstyle\mathcal{L}
  215. Ω \Omega
  216. u div ϕ = - ϕ d μ = - ϕ u ϕ C c 1 \int u\mathrm{div}\phi=-\int\phi\,d\mu=-\int\phi\nabla u\qquad\forall\phi\in C% _{c}^{1}
  217. W 1 , 1 W^{1,1}
  218. f f
  219. [ a , b ] [a,b]
  220. f f
  221. f f
  222. ( a , b ] (a,b]
  223. [ a , b ) [a,b)
  224. f ( x ) f^{\prime}(x)
  225. B V BV
  226. B V BV
  227. B V ( Ω ) BV(Ω)
  228. B V BV
  229. B V BV
  230. S B V SBV
  231. B V BV

Boussinesq_approximation_(buoyancy).html

  1. ρ 1 \rho_{1}
  2. ρ 2 \rho_{2}
  3. ρ \rho
  4. Δ ρ = ρ 1 - ρ 2 \Delta\rho=\rho_{1}-\rho_{2}
  5. g g^{\prime}
  6. g = g ρ 1 - ρ 2 ρ . g^{\prime}=g{\rho_{1}-\rho_{2}\over\rho}.
  7. g ( Δ ρ / ρ ) 2 g(\Delta\rho/\rho)^{2}
  8. ρ 1 / ρ 2 \rho_{1}/\rho_{2}
  9. Δ ρ / ρ \Delta\rho/\rho
  10. g g^{\prime}

Bragg's_law.html

  1. π π
  2. 2 d sin θ = n λ , 2d\sin\theta=n\lambda\,,
  3. d d
  4. θ \theta
  5. ( A B + B C ) - ( A C ) . (AB+BC)-(AC^{\prime})\,.
  6. ( A B + B C ) - ( A C ) = n λ , (AB+BC)-(AC^{\prime})=n\lambda\,,
  7. n n
  8. λ \lambda
  9. A B = B C = d sin θ and A C = 2 d tan θ , AB=BC=\frac{d}{\sin\theta}\,\text{ and }AC=\frac{2d}{\tan\theta}\,,
  10. A C = A C cos θ = 2 d tan θ cos θ = ( 2 d sin θ cos θ ) cos θ = 2 d sin θ cos 2 θ . AC^{\prime}=AC\cdot\cos\theta=\frac{2d}{\tan\theta}\cos\theta=\left(\frac{2d}{% \sin\theta}\cos\theta\right)\cos\theta=\frac{2d}{\sin\theta}\cos^{2}\theta\,.
  11. n λ = 2 d sin θ ( 1 - cos 2 θ ) = 2 d sin θ sin 2 θ , n\lambda=\frac{2d}{\sin\theta}(1-\cos^{2}\theta)=\frac{2d}{\sin\theta}\sin^{2}% \theta\,,
  12. n λ = 2 d sin θ , n\lambda=2d\sin\theta\,,
  13. 2 n Λ sin ( θ + ϕ ) = λ B , 2n\Lambda\sin(\theta+\phi)=\lambda_{B}\,,
  14. d = a h 2 + k 2 + l 2 , d=\frac{a}{\sqrt{h^{2}+k^{2}+l^{2}}}\,,
  15. a a
  16. h h
  17. k k
  18. l l
  19. ( λ 2 a ) 2 = sin 2 θ h 2 + k 2 + l 2 . \left(\frac{\lambda\ }{2a}\right)^{2}=\frac{\sin^{2}\theta\ }{h^{2}+k^{2}+l^{2% }}\,.

Brahmagupta–Fibonacci_identity.html

  1. ( a 2 + b 2 ) ( c 2 + d 2 ) \displaystyle\left(a^{2}+b^{2}\right)\left(c^{2}+d^{2}\right)
  2. ( 1 2 + 4 2 ) ( 2 2 + 7 2 ) = 26 2 + 15 2 = 30 2 + 1 2 . (1^{2}+4^{2})(2^{2}+7^{2})=26^{2}+15^{2}=30^{2}+1^{2}.\,
  3. ( a 2 + n b 2 ) ( c 2 + n d 2 ) = ( a c - n b d ) 2 + n ( a d + b c ) 2 ( 3 ) = ( a c + n b d ) 2 + n ( a d - b c ) 2 , ( 4 ) \begin{aligned}\displaystyle\left(a^{2}+nb^{2}\right)\left(c^{2}+nd^{2}\right)% &\displaystyle{}=\left(ac-nbd\right)^{2}+n\left(ad+bc\right)^{2}&&&% \displaystyle(3)\\ &\displaystyle{}=\left(ac+nbd\right)^{2}+n\left(ad-bc\right)^{2},&&&% \displaystyle(4)\end{aligned}
  4. | a + b i | | c + d i | = | ( a + b i ) ( c + d i ) | |a+bi||c+di|=|(a+bi)(c+di)|\,
  5. | a + b i | | c + d i | = | ( a c - b d ) + i ( a d + b c ) | , |a+bi||c+di|=|(ac-bd)+i(ad+bc)|,\,
  6. | a + b i | 2 | c + d i | 2 = | ( a c - b d ) + i ( a d + b c ) | 2 , |a+bi|^{2}|c+di|^{2}=|(ac-bd)+i(ad+bc)|^{2},\,
  7. ( a 2 + b 2 ) ( c 2 + d 2 ) = ( a c - b d ) 2 + ( a d + b c ) 2 . (a^{2}+b^{2})(c^{2}+d^{2})=(ac-bd)^{2}+(ad+bc)^{2}.\,
  8. N ( a + b i ) = a 2 + b 2 and N ( c + d i ) = c 2 + d 2 , N(a+bi)=a^{2}+b^{2}\,\text{ and }N(c+di)=c^{2}+d^{2},\,
  9. N ( ( a + b i ) ( c + d i ) ) = N ( ( a c - b d ) + i ( a d + b c ) ) = ( a c - b d ) 2 + ( a d + b c ) 2 . N((a+bi)(c+di))=N((ac-bd)+i(ad+bc))=(ac-bd)^{2}+(ad+bc)^{2}.\,
  10. N ( ( a + b i ) ( c + d i ) ) = N ( a + b i ) N ( c + d i ) . N((a+bi)(c+di))=N(a+bi)\cdot N(c+di).\,
  11. ( x 1 2 - N y 1 2 ) ( x 2 2 - N y 2 2 ) = ( x 1 x 2 + N y 1 y 2 ) 2 - N ( x 1 y 2 + x 2 y 1 ) 2 , (x_{1}^{2}-Ny_{1}^{2})(x_{2}^{2}-Ny_{2}^{2})=(x_{1}x_{2}+Ny_{1}y_{2})^{2}-N(x_% {1}y_{2}+x_{2}y_{1})^{2},\,
  12. ( x 1 x 2 + N y 1 y 2 , x 1 y 2 + x 2 y 1 , k 1 k 2 ) . (x_{1}x_{2}+Ny_{1}y_{2}\,,\,x_{1}y_{2}+x_{2}y_{1}\,,\,k_{1}k_{2}).

Braid_group.html

  1. n n
  2. n n
  3. n > 1 n>1
  4. n = 4 n=4
  5. n n
  6. σ σ
  7. τ τ
  8. σ τ στ
  9. i i
  10. i + 1 i+ 1
  11. i i
  12. i + 1 i+ 1
  13. n n
  14. B n = σ 1 , , σ n - 1 | σ i σ i + 1 σ i = σ i + 1 σ i σ i + 1 , σ i σ j = σ j σ i , B_{n}=\left\langle\sigma_{1},\ldots,\sigma_{n-1}|\sigma_{i}\sigma_{i+1}\sigma_% {i}=\sigma_{i+1}\sigma_{i}\sigma_{i+1},\sigma_{i}\sigma_{j}=\sigma_{j}\sigma_{% i}\right\rangle,
  15. 1 i n 2 1 ≤i≤n−2
  16. | i j | 2 |i−j|≥2
  17. 𝐙 \mathbf{Z}
  18. n n
  19. ( n + 1 ) (n+1)
  20. n n
  21. n 1 n≥1
  22. n 3 n≥3
  23. 1 + 1 1 + 1 + 1 = 3 1 + 1 − 1 + 1 + 1=3
  24. n n
  25. n n
  26. S n = s 1 , , s n - 1 | s i s i + 1 s i = s i + 1 s i s i + 1 , s i s j = s j s i for | i - j | 2 , s i 2 = 1 . S_{n}=\left\langle s_{1},\ldots,s_{n-1}|s_{i}s_{i+1}s_{i}=s_{i+1}s_{i}s_{i+1},% s_{i}s_{j}=s_{j}s_{i}\,\text{ for }|i-j|\geq 2,s_{i}^{2}=1\right\rangle.
  27. n n
  28. 1 F n - 1 P n P n - 1 1. 1\to F_{n-1}\to P_{n}\to P_{n-1}\to 1.
  29. P S L ( 2 , 𝐙 ) PSL(2,\mathbf{Z})
  30. SL ( 2 , 𝐑 ) ¯ PSL ( 2 , 𝐑 ) . \overline{\mathrm{SL}(2,\mathbf{R})}\to\mathrm{PSL}(2,\mathbf{R}).
  31. a = σ 1 σ 2 σ 1 , b = σ 1 σ 2 . a=\sigma_{1}\sigma_{2}\sigma_{1},\quad b=\sigma_{1}\sigma_{2}.
  32. c c
  33. σ 1 c σ 1 - 1 = σ 2 c σ 2 - 1 = c \sigma_{1}c\sigma_{1}^{-1}=\sigma_{2}c\sigma_{2}^{-1}=c
  34. c c
  35. C C
  36. c c
  37. σ 1 C R = [ 1 1 0 1 ] σ 2 C L - 1 = [ 1 0 - 1 1 ] \sigma_{1}C\mapsto R=\begin{bmatrix}1&1\\ 0&1\end{bmatrix}\qquad\sigma_{2}C\mapsto L^{-1}=\begin{bmatrix}1&0\\ -1&1\end{bmatrix}
  38. L L
  39. R R
  40. v , p | v 2 = p 3 = 1 \langle v,p\,|\,v^{2}=p^{3}=1\rangle
  41. v = [ 0 1 - 1 0 ] , p = [ 0 1 - 1 1 ] . v=\begin{bmatrix}0&1\\ -1&0\end{bmatrix},\qquad p=\begin{bmatrix}0&1\\ -1&1\end{bmatrix}.
  42. a a
  43. v v
  44. b b
  45. p p
  46. C C
  47. c c
  48. C C
  49. n n
  50. n n
  51. n n
  52. G G
  53. X X
  54. n n
  55. G G
  56. G G
  57. X X
  58. σ i ( x 1 , , x i - 1 , x i , x i + 1 , , x n ) = ( x 1 , , x i - 1 , x i + 1 , x i + 1 - 1 x i x i + 1 , x i + 2 , , x n ) . \sigma_{i}\left(x_{1},\ldots,x_{i-1},x_{i},x_{i+1},\ldots,x_{n}\right)=\left(x% _{1},\ldots,x_{i-1},x_{i+1},x_{i+1}^{-1}x_{i}x_{i+1},x_{i+2},\ldots,x_{n}% \right).
  59. x x
  60. X X
  61. n 5 n≥ 5
  62. S O ( 3 ) SO(3)
  63. n ( n 1 ) / 2 n(n−1)/2
  64. q q
  65. t t
  66. n 1 n−1
  67. n 1 n−1
  68. ( 0 , 1 / n ) (0,1/n)
  69. ( 0 , 1 / n , 0 ) (0,1/n,0)
  70. ( 0 , 1 / n , 1 ) (0,1/n,1)
  71. { ( x i ) i x i = x j for some i j } . \{(x_{i})_{i\in\mathbb{N}}\mid x_{i}=x_{j}\,\text{ for some }i\neq j\}.

Branch_and_bound.html

  1. n n
  2. f ( x ) f(x)
  3. x x
  4. S S
  5. f ( x ) f(x)
  6. g ( x ) = - f ( x ) g(x)=-f(x)
  7. S S
  8. f ( x ) f(x)
  9. x x
  10. S S
  11. S 1 , S 2 , S_{1},S_{2},\ldots
  12. S S
  13. f ( x ) f(x)
  14. S S
  15. min { v 1 , v 2 , } \min\{v_{1},v_{2},\ldots\}
  16. v i v_{i}
  17. f ( x ) f(x)
  18. S i S_{i}
  19. S S
  20. f ( x ) f(x)
  21. S S
  22. A A
  23. B B
  24. A A
  25. m m
  26. m m
  27. S S
  28. S S
  29. S S
  30. S S
  31. 𝐱 \mathbf{x}
  32. n \mathbb{R}^{n}
  33. f f
  34. g g
  35. f f
  36. x < s u b > h x<sub>h

Branch_point.html

  1. g ( z ) = exp ( z - 1 / k ) g(z)=\exp\left(z^{-1/k}\right)\,
  2. \mathbb{Z}
  3. F ( z ) = z 1 - z F(z)=\sqrt{z}\sqrt{1-z}\,
  4. ln z = ln r + i θ . \ln z=\ln r+i\theta.\,
  5. f a ( z ) = 1 z - a f_{a}(z)={1\over z-a}
  6. u ( z ) = a = - 1 a = 1 f a ( z ) d a = a = - 1 a = 1 1 z - a d a = log ( z + 1 z - 1 ) u(z)=\int_{a=-1}^{a=1}f_{a}(z)\,da=\int_{a=-1}^{a=1}{1\over z-a}\,da=\log\left% ({z+1\over z-1}\right)
  7. w = z k w=z^{k}
  8. e P = 1 2 π i γ f ( z ) f ( z ) - f ( P ) d z . e_{P}=\frac{1}{2\pi i}\int_{\gamma}\frac{f^{\prime}(z)}{f(z)-f(P)}\,dz.
  9. e P = v P ( t f ) e_{P}=v_{P}(t\circ f)
  10. t f t\circ f

Branching_process.html

  1. Z n + 1 = i = 1 Z n X n , i Z_{n+1}=\sum_{i=1}^{Z_{n}}X_{n,i}
  2. S i + 1 = S i + X i + 1 - 1 = j = 1 i + 1 X j - i S_{i+1}=S_{i}+X_{i+1}-1=\sum_{j=1}^{i+1}X_{j}-i
  3. lim n Pr ( Z n = 0 ) . \lim_{n\to\infty}\Pr(Z_{n}=0).
  4. 0 = d 0 d 1 d 2 1. 0=d_{0}\leq d_{1}\leq d_{2}\leq\cdots\leq 1.
  5. d m = p 0 + p 1 d m - 1 + p 2 ( d m - 1 ) 2 + p 3 ( d m - 1 ) 3 + . d_{m}=p_{0}+p_{1}d_{m-1}+p_{2}(d_{m-1})^{2}+p_{3}(d_{m-1})^{3}+\cdots.\,
  6. h ( z ) = p 0 + p 1 z + p 2 z 2 + . h(z)=p_{0}+p_{1}z+p_{2}z^{2}+\cdots.\,
  7. d m = h ( d m - 1 ) . d_{m}=h(d_{m-1}).\,
  8. d = h ( d ) . d=h(d).\,
  9. h ( z ) = p 1 + 2 p 2 z + 3 p 3 z 2 + 0 h^{\prime}(z)=p_{1}+2p_{2}z+3p_{3}z^{2}+\cdots\geq 0
  10. h ′′ ( z ) = 2 p 2 + 6 p 3 z + 12 p 4 z 2 + 0 h^{\prime\prime}(z)=2p_{2}+6p_{3}z+12p_{4}z^{2}+\cdots\geq 0
  11. d m = p 0 + p 1 d m - 1 + p 2 ( d m - 1 ) 2 . d_{m}=p_{0}+p_{1}d_{m-1}+p_{2}(d_{m-1})^{2}.\,

Brane_cosmology.html

  1. ( 3 + N ) + 1 (3+N)+1
  2. N N

Brauer_group.html

  1. n > 1 n>1
  2. 0 Br ( K ) v S Br ( K v ) 𝐐 / 𝐙 0 , 0\rightarrow\textrm{Br}(K)\rightarrow\bigoplus_{v\in S}\textrm{Br}(K_{v})% \rightarrow\mathbf{Q}/\mathbf{Z}\rightarrow 0,
  3. Br ( K ) H 2 ( Gal ( K s / K ) , K s * ) . \textrm{Br}(K)\cong H^{2}(\textrm{Gal}(K^{s}/K),{K^{s}}^{*}).

Brayton_cycle.html

  1. η = 1 - T 1 T 2 = 1 - ( P 1 P 2 ) ( γ - 1 ) / γ \eta=1-\frac{T_{1}}{T_{2}}=1-\left(\frac{P_{1}}{P_{2}}\right)^{(\gamma-1)/\gamma}
  2. γ \gamma

Brinell_scale.html

  1. BHN = 2 P π D ( D - D 2 - d 2 ) \operatorname{BHN}=\frac{2P}{\pi D\left(D-\sqrt{D^{2}-d^{2}}\right)}
  2. HBW = 0.102 2 F π D ( D - D 2 - d 2 ) \operatorname{HBW}=0.102\frac{2F}{\pi D\left(D-\sqrt{D^{2}-d^{2}}\right)}

Brownian_ratchet.html

  1. T 1 T_{1}
  2. T 2 T_{2}
  3. T 1 T_{1}
  4. T 2 T_{2}
  5. T 1 T_{1}
  6. T 2 T_{2}
  7. T 1 T_{1}

Buffer_amplifier.html

  1. R i n = v x i x = r π + ( β + 1 ) ( r O | | R L ) R_{in}=\frac{v_{x}}{i_{x}}=r_{\pi}+(\beta+1)({r_{O}}||{R_{L}})

Bumper_(automobile).html

  1. E k = 1 2 m v 2 E\text{k}=\tfrac{1}{2}mv^{2}

Butterworth_filter.html

  1. G ( ω ) = 1 1 + ω 2 n , G(\omega)=\sqrt{\frac{1}{1+{\omega}^{2n}}},
  2. 2 \sqrt{2}
  3. H ( s ) = V o ( s ) V i ( s ) = 1 1 + 2 s + 2 s 2 + s 3 . H(s)=\frac{V_{o}(s)}{V_{i}(s)}=\frac{1}{1+2s+2s^{2}+s^{3}}.
  4. G ( ω ) = | H ( j ω ) | = 1 1 + ω 6 , G(\omega)=|H(j\omega)|=\frac{1}{\sqrt{1+\omega^{6}}},
  5. G 2 ( ω ) = | H ( j ω ) | 2 = H ( j ω ) H * ( j ω ) = 1 1 + ω 6 , G^{2}(\omega)=|H(j\omega)|^{2}=H(j\omega)H^{*}(j\omega)=\frac{1}{1+\omega^{6}},
  6. Φ ( ω ) = arg ( H ( j ω ) ) . \Phi(\omega)=\arg(H(j\omega)).\!
  7. ω 0 = 1 \omega_{0}=1
  8. G ( ω ) G(\omega)
  9. G 2 ( ω ) = | H ( j ω ) | 2 = G 0 2 1 + ( ω ω c ) 2 n G^{2}(\omega)=\left|H(j\omega)\right|^{2}=\frac{{G_{0}}^{2}}{1+\left(\frac{% \omega}{\omega_{c}}\right)^{2n}}
  10. G 0 G_{0}
  11. G 0 G_{0}
  12. s = σ + j ω s=\sigma+j\omega
  13. | H ( s ) | 2 = H ( s ) H ( s ) ¯ \left|H(s)\right|^{2}=H(s)\overline{H(s)}
  14. s = j ω s=j\omega
  15. H ( - j ω ) = H ( j ω ) ¯ H(-j\omega)=\overline{H(j\omega)}
  16. H ( s ) H ( - s ) = G 0 2 1 + ( - s 2 ω c 2 ) n , H(s)H(-s)=\frac{{G_{0}}^{2}}{1+\left(\frac{-s^{2}}{\omega_{c}^{2}}\right)^{n}},
  17. s = j ω s=j\omega
  18. n n
  19. - s k 2 ω c 2 = ( - 1 ) 1 n = e j ( 2 k - 1 ) π n k = 1 , 2 , 3 , , n -\frac{s_{k}^{2}}{\omega_{c}^{2}}=(-1)^{\frac{1}{n}}=e^{\frac{j(2k-1)\pi}{n}}% \qquad\mathrm{k=1,2,3,\ldots,n}
  20. s k = ω c e j ( 2 k + n - 1 ) π 2 n k = 1 , 2 , 3 , , n . s_{k}=\omega_{c}e^{\frac{j(2k+n-1)\pi}{2n}}\qquad\mathrm{k=1,2,3,\ldots,n}.
  21. H ( s ) = G 0 k = 1 n ( s - s k ) / ω c . H(s)=\frac{G_{0}}{\prod_{k=1}^{n}(s-s_{k})/\omega_{c}}.
  22. s 1 s_{1}
  23. s n s_{n}
  24. ω c = 1 \omega_{c}=1
  25. B n ( s ) = k = 1 n 2 [ s 2 - 2 s cos ( 2 k + n - 1 2 n π ) + 1 ] n = even B_{n}(s)=\prod_{k=1}^{\frac{n}{2}}\left[s^{2}-2s\cos\left(\frac{2k+n-1}{2n}\,% \pi\right)+1\right]\qquad\mathrm{n=even}
  26. B n ( s ) = ( s + 1 ) k = 1 n - 1 2 [ s 2 - 2 s cos ( 2 k + n - 1 2 n π ) + 1 ] n = odd . B_{n}(s)=(s+1)\prod_{k=1}^{\frac{n-1}{2}}\left[s^{2}-2s\cos\left(\frac{2k+n-1}% {2n}\,\pi\right)+1\right]\qquad\mathrm{n=odd}.
  27. B n ( s ) B_{n}(s)
  28. ( s + 1 ) (s+1)
  29. s 2 + 1.4142 s + 1 s^{2}+1.4142s+1
  30. ( s + 1 ) ( s 2 + s + 1 ) (s+1)(s^{2}+s+1)
  31. ( s 2 + 0.7654 s + 1 ) ( s 2 + 1.8478 s + 1 ) (s^{2}+0.7654s+1)(s^{2}+1.8478s+1)
  32. ( s + 1 ) ( s 2 + 0.6180 s + 1 ) ( s 2 + 1.6180 s + 1 ) (s+1)(s^{2}+0.6180s+1)(s^{2}+1.6180s+1)
  33. ( s 2 + 0.5176 s + 1 ) ( s 2 + 1.4142 s + 1 ) ( s 2 + 1.9319 s + 1 ) (s^{2}+0.5176s+1)(s^{2}+1.4142s+1)(s^{2}+1.9319s+1)
  34. ( s + 1 ) ( s 2 + 0.4450 s + 1 ) ( s 2 + 1.2470 s + 1 ) ( s 2 + 1.8019 s + 1 ) (s+1)(s^{2}+0.4450s+1)(s^{2}+1.2470s+1)(s^{2}+1.8019s+1)
  35. ( s 2 + 0.3902 s + 1 ) ( s 2 + 1.1111 s + 1 ) ( s 2 + 1.6629 s + 1 ) ( s 2 + 1.9616 s + 1 ) (s^{2}+0.3902s+1)(s^{2}+1.1111s+1)(s^{2}+1.6629s+1)(s^{2}+1.9616s+1)
  36. ω c \omega_{c}
  37. H ( s ) = G 0 B n ( a ) H(s)=\frac{G_{0}}{B_{n}(a)}
  38. a = s ω c . a=\frac{s}{\omega_{c}}.
  39. ω c = 1 \omega_{c}=1
  40. G 0 = 1 G_{0}=1
  41. d G d ω = - n G 3 ω 2 n - 1 \frac{dG}{d\omega}=-nG^{3}\omega^{2n-1}
  42. ω \omega
  43. G ( ω ) = 1 - 1 2 ω 2 n + 3 8 ω 4 n + G(\omega)=1-\frac{1}{2}\omega^{2n}+\frac{3}{8}\omega^{4n}+\ldots
  44. ω = 0 \omega=0
  45. ω c = 1 \omega_{c}=1
  46. lim ω d log ( G ) d log ( ω ) = - n . \lim_{\omega\rightarrow\infty}\frac{d\log(G)}{d\log(\omega)}=-n.
  47. C k = 2 sin [ ( 2 k - 1 ) 2 n π ] k = odd C_{k}=2\sin\left[\frac{(2k-1)}{2n}\pi\right]\qquad\mathrm{k=odd}
  48. L k = 2 sin [ ( 2 k - 1 ) 2 n π ] k = even . L_{k}=2\sin\left[\frac{(2k-1)}{2n}\pi\right]\qquad\mathrm{k=even}.
  49. L k \scriptstyle L_{k}
  50. C k \scriptstyle C_{k}
  51. L k \scriptstyle L_{k}
  52. C k \scriptstyle C_{k}
  53. g k \scriptstyle g_{k}
  54. g k \scriptstyle g_{k}
  55. g k = 2 sin [ ( 2 k - 1 ) 2 n π ] k = 1 , 2 , 3 , , n . g_{k}=2\sin\left[\frac{(2k-1)}{2n}\pi\right]\qquad\mathrm{k=1,2,3,\ldots,n}.
  56. ω c = 1 \scriptstyle\omega_{c}=1
  57. g j = a j a j - 1 c j - 1 g j - 1 j = 2 , 3 , , n g_{j}=\frac{a_{j}a_{j-1}}{c_{j-1}g_{j-1}}\qquad\mathrm{j=2,3,\ldots,n}
  58. g 1 = a 1 g_{1}=a_{1}
  59. a j = sin π 2 [ ( 2 j - 1 ) n ] j = 1 , 2 , 3 , , n a_{j}=\sin\frac{\pi}{2}\left[\frac{(2j-1)}{n}\right]\qquad\mathrm{j=1,2,3,% \ldots,n}
  60. c j = cos 2 [ π j 2 n ] j = 1 , 2 , 3 , , n . c_{j}=\cos^{2}\left[\frac{\pi j}{2n}\right]\qquad\mathrm{j=1,2,3,\ldots,n}.
  61. n n
  62. H ( s ) = V o u t ( s ) V i n ( s ) = 1 1 + C 2 ( R 1 + R 2 ) s + C 1 C 2 R 1 R 2 s 2 . H(s)=\frac{V_{out}(s)}{V_{in}(s)}=\frac{1}{1+C_{2}(R_{1}+R_{2})s+C_{1}C_{2}R_{% 1}R_{2}s^{2}}.
  63. ω c = 1 \omega_{c}=1
  64. C 1 C 2 R 1 R 2 = 1 C_{1}C_{2}R_{1}R_{2}=1\,
  65. C 2 ( R 1 + R 2 ) = - 2 cos ( 2 k + n - 1 2 n π ) . C_{2}(R_{1}+R_{2})=-2\cos\left(\frac{2k+n-1}{2n}\pi\right).

Büchi_automaton.html

  1. Q final = Q { init } Q\text{final}=Q\cup\{\,\text{init}\}
  2. Σ = 2 A P \Sigma=2^{AP}
  3. I = { init } I=\{\,\text{init}\}
  4. F = Q { init } F=Q\cup\{\,\text{init}\}
  5. δ = q a p \delta=q\overrightarrow{a}p
  6. a \overrightarrow{a}

Cadence_(music).html

  1. 4 3 ( 9 8 ) 2 = 256 243 \textstyle{{{4\over 3}\over\left({9\over 8}\right)^{2}}={256\over 243}}\,\!

Calculus_of_communicating_systems.html

  1. P : := | a . P 1 | A | P 1 + P 2 | P 1 | P 2 | P 1 [ b / a ] | P 1 \ a P::=\emptyset\,\,\,|\,\,\,a.P_{1}\,\,\,|\,\,\,A\,\,\,|\,\,\,P_{1}+P_{2}\,\,\,|% \,\,\,P_{1}|P_{2}\,\,\,|\,\,\,P_{1}[b/a]\,\,\,|\,\,\,P_{1}{\backslash}a\,\,\,
  2. \emptyset
  3. a . P 1 a.P_{1}
  4. a a
  5. P 1 P_{1}
  6. A = def P 1 A\overset{\underset{\mathrm{def}}{}}{=}P_{1}
  7. A A
  8. P 1 P_{1}
  9. A A
  10. P 1 + P 2 P_{1}+P_{2}
  11. P 1 P_{1}
  12. P 2 P_{2}
  13. P 1 | P 2 P_{1}|P_{2}
  14. P 1 P_{1}
  15. P 2 P_{2}
  16. P 1 [ b / a ] P_{1}[b/a]
  17. P 1 P_{1}
  18. a a
  19. b b
  20. P 1 \ a P_{1}{\backslash}a
  21. P 1 P_{1}
  22. a a

Calculus_of_constructions.html

  1. A A
  2. B B
  3. ( A B ) (A~{}B)
  4. ( λ x : A . B ) (\mathbf{\lambda}x:A.B)
  5. ( x : A . B ) (\forall x:A.B)
  6. x 1 : A 1 , x 2 : A 2 , t : B x_{1}:A_{1},x_{2}:A_{2},\ldots\vdash t:B
  7. x 1 , x 2 , x_{1},x_{2},\ldots
  8. A 1 , A 2 , A_{1},A_{2},\ldots
  9. t t
  10. B B
  11. Γ \Gamma
  12. x 1 : A 1 , x 2 : A 2 , x_{1}:A_{1},x_{2}:A_{2},\ldots
  13. A : B : C A:B:C
  14. A A
  15. B B
  16. B B
  17. C C
  18. B ( x := N ) B(x:=N)
  19. N N
  20. x x
  21. B B
  22. Γ A : B Γ C : D {\Gamma\vdash A:B}\over{\Gamma^{\prime}\vdash C:D}
  23. Γ A : B \Gamma\vdash A:B
  24. Γ C : D \Gamma^{\prime}\vdash C:D
  25. Γ P : T {{}\over{}\Gamma\vdash P:T}
  26. Γ A : K Γ , x : A x : A {\Gamma\vdash A:K\over{\Gamma,x:A\vdash x:A}}
  27. Γ , x : A t : B : K Γ ( λ x : A . t ) : ( x : A . B ) : K {\Gamma,x:A\vdash t:B:K\over{\Gamma\vdash(\lambda x:A.t):(\forall x:A.B):K}}
  28. Γ M : ( x : A . B ) Γ N : A Γ M N : B ( x := N ) {\Gamma\vdash M:(\forall x:A.B)\qquad\qquad\Gamma\vdash N:A\over{\Gamma\vdash MN% :B(x:=N)}}
  29. Γ M : A A = β B B : K Γ M : B {\Gamma\vdash M:A\qquad\qquad A=_{\beta}B\qquad\qquad B:K\over{\Gamma\vdash M:% B}}
  30. \forall
  31. A B x : A . B ( x B ) A B C : P . ( A B C ) C A B C : P . ( A C ) ( B C ) C ¬ A C : P . ( A C ) x : A . B C : P . ( x : A . ( B C ) ) C \begin{matrix}A\Rightarrow B&\equiv&\forall x:A.B&(x\notin B)\\ A\wedge B&\equiv&\forall C:P.(A\Rightarrow B\Rightarrow C)\Rightarrow C&\\ A\vee B&\equiv&\forall C:P.(A\Rightarrow C)\Rightarrow(B\Rightarrow C)% \Rightarrow C&\\ \neg A&\equiv&\forall C:P.(A\Rightarrow C)&\\ \exists x:A.B&\equiv&\forall C:P.(\forall x:A.(B\Rightarrow C))\Rightarrow C&% \end{matrix}
  32. A : P . A A A \forall A:P.A\Rightarrow A\Rightarrow A
  33. A : P . ( A A ) ( A A ) \forall A:P.(A\Rightarrow A)\Rightarrow(A\Rightarrow A)
  34. A × B A\times B
  35. A B A\wedge B
  36. A + B A+B
  37. A B A\vee B
  38. x : A . B \forall x:A.B

Calibration_curve.html

  1. y u n k - y ¯ y_{unk}-\bar{y}
  2. y u n k = y ¯ y_{unk}=\bar{y}
  3. s x = s y | m | 1 n + 1 k + ( y u n k - y ¯ ) 2 m 2 ( x i - x ¯ ) 2 s_{x}=\frac{s_{y}}{|m|}\sqrt{\frac{1}{n}+\frac{1}{k}+\frac{(y_{unk}-\bar{y})^{% 2}}{m^{2}\sum{(x_{i}-\bar{x})^{2}}}}
  4. s y s_{y}
  5. = ( y i - m x i - b ) 2 n - 2 =\sqrt{\frac{\sum{(y_{i}-mx_{i}-b)}^{2}}{n-2}}
  6. m m
  7. b b
  8. n n
  9. k k
  10. y u n k n o w n y_{unknown}
  11. y ¯ \bar{y}
  12. x i x_{i}
  13. x ¯ \bar{x}

Candidate_key.html

  1. α + \alpha^{+}
  2. α \alpha
  3. α + \alpha^{+}
  4. α \alpha
  5. α \alpha
  6. minimize ( α ) \,\text{minimize}(\alpha)
  7. α \alpha
  8. minimize ( α ) \,\text{minimize}(\alpha)
  9. n ! n!
  10. 2 n 2^{n}
  11. 2 n 2\cdot n
  12. { A i B i : i { 1 , , n } } { B i A i : i { 1 , , n } } \{A_{i}\rightarrow B_{i}:i\in\{1,\dots,n\}\}\cup\{B_{i}\rightarrow A_{i}:i\in% \{1,\dots,n\}\}
  13. 2 n 2^{n}
  14. { A 1 , B 1 } × × { A n , B n } \{A_{1},B_{1}\}\times\dots\times\{A_{n},B_{n}\}
  15. α β \alpha\rightarrow\beta
  16. α ( K i β ) \alpha\cup(K_{i}\setminus\beta)

Canny_edge_detector.html

  1. H i j = 1 2 π σ 2 exp ( - ( i - k - 1 ) 2 + ( j - k - 1 ) 2 2 σ 2 ) H_{ij}=\frac{1}{2\pi\sigma^{2}}\exp(-\frac{(i-k-1)^{2}+(j-k-1)^{2}}{2\sigma^{2% }})
  2. σ \sigma
  3. 𝐁 = 1 159 [ 2 4 5 4 2 4 9 12 9 4 5 12 15 12 5 4 9 12 9 4 2 4 5 4 2 ] * 𝐀 . \mathbf{B}=\frac{1}{159}\begin{bmatrix}2&4&5&4&2\\ 4&9&12&9&4\\ 5&12&15&12&5\\ 4&9&12&9&4\\ 2&4&5&4&2\end{bmatrix}*\mathbf{A}.
  4. 𝐆 = 𝐆 x 2 + 𝐆 y 2 \mathbf{G}=\sqrt{{\mathbf{G}_{x}}^{2}+{\mathbf{G}_{y}}^{2}}
  5. 𝚯 = atan2 ( 𝐆 y , 𝐆 x ) \mathbf{\Theta}=\operatorname{atan2}\left(\mathbf{G}_{y},\mathbf{G}_{x}\right)
  6. d ( x , y ) = G x ( x , y ) 2 + G y ( x , y ) 2 d(x,y)=\sqrt{G_{x}(x,y)^{2}+G_{y}(x,y)^{2}}
  7. w ( x , y ) = exp ( - d ( x , y ) 2 h 2 ) w(x,y)=\exp\left(-\frac{\sqrt{d(x,y)}}{2h^{2}}\right)
  8. f ( x , y ) = 1 N i = - 1 1 j = - 1 1 f ( x + i , y + j ) w ( x + i , y + j ) f(x,y)=\frac{1}{N}\sum\limits_{i=-1}^{1}\sum\limits_{j=-1}^{1}f(x+i,y+j)w(x+i,% y+j)
  9. N = i = - 1 1 j = - 1 1 w ( x + i , y + j ) N=\sum\limits_{i=-1}^{1}\sum\limits_{j=-1}^{1}w(x+i,y+j)
  10. G x ( x , y ) = [ I ( i , j + 1 ) - I ( i , j - 1 ) + I ( i - 1 , j + 1 ) - I ( i - 1 , j - 1 ) + I ( i + 1 , j + 1 ) - I ( i + 1 , j - 1 ) ] / 2 G_{x}(x,y)=[I(i,j+1)-I(i,j-1)+I(i-1,j+1)-I(i-1,j-1)+I(i+1,j+1)-I(i+1,j-1)]/2
  11. G y ( x , y ) = [ I ( i + 1 , j ) - I ( i - 1 , j ) + I ( i + 1 , j - 1 ) - I ( i - 1 , j - 1 ) + I ( i + 1 , j + 1 ) - I ( i - 1 , j + 1 ) ] / 2 G_{y}(x,y)=[I(i+1,j)-I(i-1,j)+I(i+1,j-1)-I(i-1,j-1)+I(i+1,j+1)-I(i-1,j+1)]/2
  12. p i = n i n pi=\frac{n_{i}}{n}
  13. u T = i = 0 L - 1 i p i w 0 u_{T}=\sum\limits_{i=0}^{L-1}\frac{ip_{i}}{w_{0}}
  14. w 0 = i = 0 L - 1 p i w_{0}=\sum\limits_{i=0}^{L-1}p_{i}
  15. u 1 = i = T + 1 L - 1 i p i w 1 u_{1}=\sum\limits_{i=T+1}^{L-1}\frac{ip_{i}}{w_{1}}
  16. w 1 = 1 - w 0 w_{1}=1-w_{0}
  17. σ b 2 \sigma_{b}^{2}

Canonical_correlation.html

  1. X = ( x 1 , , x n ) X=(x_{1},\dots,x_{n})^{\prime}
  2. Y = ( y 1 , , y m ) Y=(y_{1},\dots,y_{m})^{\prime}
  3. Σ X Y = cov ( X , Y ) \Sigma_{XY}=\operatorname{cov}(X,Y)
  4. n × m n\times m
  5. ( i , j ) (i,j)
  6. cov ( x i , y j ) \operatorname{cov}(x_{i},y_{j})
  7. X X
  8. Y Y
  9. a a
  10. b b
  11. a X a^{\prime}X
  12. b Y b^{\prime}Y
  13. ρ = corr ( a X , b Y ) \rho=\operatorname{corr}(a^{\prime}X,b^{\prime}Y)
  14. U = a X U=a^{\prime}X
  15. V = b Y V=b^{\prime}Y
  16. min { m , n } \min\{m,n\}
  17. Σ X X = cov ( X , X ) \Sigma_{XX}=\operatorname{cov}(X,X)
  18. Σ Y Y = cov ( Y , Y ) \Sigma_{YY}=\operatorname{cov}(Y,Y)
  19. ρ = a Σ X Y b a Σ X X a b Σ Y Y b . \rho=\frac{a^{\prime}\Sigma_{XY}b}{\sqrt{a^{\prime}\Sigma_{XX}a}\sqrt{b^{% \prime}\Sigma_{YY}b}}.
  20. c = Σ X X 1 / 2 a , c=\Sigma_{XX}^{1/2}a,
  21. d = Σ Y Y 1 / 2 b . d=\Sigma_{YY}^{1/2}b.
  22. ρ = c Σ X X - 1 / 2 Σ X Y Σ Y Y - 1 / 2 d c c d d . \rho=\frac{c^{\prime}\Sigma_{XX}^{-1/2}\Sigma_{XY}\Sigma_{YY}^{-1/2}d}{\sqrt{c% ^{\prime}c}\sqrt{d^{\prime}d}}.
  23. ( c Σ X X - 1 / 2 Σ X Y Σ Y Y - 1 / 2 ) d ( c Σ X X - 1 / 2 Σ X Y Σ Y Y - 1 / 2 Σ Y Y - 1 / 2 Σ Y X Σ X X - 1 / 2 c ) 1 / 2 ( d d ) 1 / 2 , \left(c^{\prime}\Sigma_{XX}^{-1/2}\Sigma_{XY}\Sigma_{YY}^{-1/2}\right)d\leq% \left(c^{\prime}\Sigma_{XX}^{-1/2}\Sigma_{XY}\Sigma_{YY}^{-1/2}\Sigma_{YY}^{-1% /2}\Sigma_{YX}\Sigma_{XX}^{-1/2}c\right)^{1/2}\left(d^{\prime}d\right)^{1/2},
  24. ρ ( c Σ X X - 1 / 2 Σ X Y Σ Y Y - 1 Σ Y X Σ X X - 1 / 2 c ) 1 / 2 ( c c ) 1 / 2 . \rho\leq\frac{\left(c^{\prime}\Sigma_{XX}^{-1/2}\Sigma_{XY}\Sigma_{YY}^{-1}% \Sigma_{YX}\Sigma_{XX}^{-1/2}c\right)^{1/2}}{\left(c^{\prime}c\right)^{1/2}}.
  25. d d
  26. Σ Y Y - 1 / 2 Σ Y X Σ X X - 1 / 2 c \Sigma_{YY}^{-1/2}\Sigma_{YX}\Sigma_{XX}^{-1/2}c
  27. c c
  28. Σ X X - 1 / 2 Σ X Y Σ Y Y - 1 Σ Y X Σ X X - 1 / 2 \Sigma_{XX}^{-1/2}\Sigma_{XY}\Sigma_{YY}^{-1}\Sigma_{YX}\Sigma_{XX}^{-1/2}
  29. c c
  30. Σ X X - 1 / 2 Σ X Y Σ Y Y - 1 Σ Y X Σ X X - 1 / 2 \Sigma_{XX}^{-1/2}\Sigma_{XY}\Sigma_{YY}^{-1}\Sigma_{YX}\Sigma_{XX}^{-1/2}
  31. d d
  32. Σ Y Y - 1 / 2 Σ Y X Σ X X - 1 / 2 c \Sigma_{YY}^{-1/2}\Sigma_{YX}\Sigma_{XX}^{-1/2}c
  33. d d
  34. Σ Y Y - 1 / 2 Σ Y X Σ X X - 1 Σ X Y Σ Y Y - 1 / 2 \Sigma_{YY}^{-1/2}\Sigma_{YX}\Sigma_{XX}^{-1}\Sigma_{XY}\Sigma_{YY}^{-1/2}
  35. c c
  36. Σ X X - 1 / 2 Σ X Y Σ Y Y - 1 / 2 d \Sigma_{XX}^{-1/2}\Sigma_{XY}\Sigma_{YY}^{-1/2}d
  37. a a
  38. Σ X X - 1 Σ X Y Σ Y Y - 1 Σ Y X \Sigma_{XX}^{-1}\Sigma_{XY}\Sigma_{YY}^{-1}\Sigma_{YX}
  39. b b
  40. Σ Y Y - 1 Σ Y X Σ X X - 1 Σ X Y \Sigma_{YY}^{-1}\Sigma_{YX}\Sigma_{XX}^{-1}\Sigma_{XY}
  41. a a
  42. Σ X X - 1 Σ X Y b \Sigma_{XX}^{-1}\Sigma_{XY}b
  43. b b
  44. Σ Y Y - 1 Σ Y X a \Sigma_{YY}^{-1}\Sigma_{YX}a
  45. U = c Σ X X - 1 / 2 X = a X U=c^{\prime}\Sigma_{XX}^{-1/2}X=a^{\prime}X
  46. V = d Σ Y Y - 1 / 2 Y = b Y V=d^{\prime}\Sigma_{YY}^{-1/2}Y=b^{\prime}Y
  47. i i
  48. p p
  49. ρ ^ i \widehat{\rho}_{i}
  50. i = 1 , , min { m , n } i=1,\dots,\min\{m,n\}
  51. i i
  52. χ 2 = - ( p - 1 - 1 2 ( m + n + 1 ) ) ln j = i min { m , n } ( 1 - ρ ^ j 2 ) , \chi^{2}=-\left(p-1-\frac{1}{2}(m+n+1)\right)\ln\prod_{j=i}^{\min\{m,n\}}(1-% \widehat{\rho}_{j}^{2}),
  53. ( m - i + 1 ) ( n - i + 1 ) (m-i+1)(n-i+1)
  54. p p
  55. min { m , n } \min\{m,n\}
  56. p p
  57. X = x 1 X=x_{1}
  58. E ( X ) = 0 \operatorname{E}(X)=0
  59. Y = X Y=X
  60. X X
  61. Y Y
  62. a = 1 a=1
  63. b = 1 b=1
  64. U = X U=X
  65. V = Y = X V=Y=X
  66. Y = - X Y=-X
  67. X X
  68. Y Y
  69. a = 1 a=1
  70. b = - 1 b=-1
  71. U = X U=X
  72. V = - Y = X V=-Y=X
  73. U = V U=V
  74. X = ( x 1 , , x n ) X=(x_{1},\dots,x_{n})^{\prime}
  75. Y = ( y 1 , , y m ) Y=(y_{1},\dots,y_{m})^{\prime}
  76. E ( X ) = E ( Y ) = 0 \operatorname{E}(X)=\operatorname{E}(Y)=0
  77. Σ X X = Cov ( X , X ) = E [ X X ] \Sigma_{XX}=\operatorname{Cov}(X,X)=\operatorname{E}[XX^{\prime}]
  78. Σ Y Y = Cov ( Y , Y ) = E [ Y Y ] \Sigma_{YY}=\operatorname{Cov}(Y,Y)=\operatorname{E}[YY^{\prime}]
  79. X X
  80. Y Y
  81. x i x_{i}
  82. X X
  83. y j y_{j}
  84. Y Y
  85. cov ( x i , y j ) \operatorname{cov}(x_{i},y_{j})
  86. U U
  87. V V
  88. X X
  89. Y Y
  90. corr ( U , V ) \operatorname{corr}(U,V)

Canonical_form.html

  1. A = U * B U A=U^{*}BU
  2. A = U B V * A=UBV^{*}
  3. A = P - 1 B P A=P^{-1}BP
  4. A = P - 1 B P A=P^{-1}BP
  5. A = P - 1 B P A=P^{-1}BP
  6. A = P - 1 B Q A=P^{-1}BQ
  7. K n K^{n}
  8. 2 ( I ) \ell^{2}(I)
  9. C * C^{*}
  10. C * C^{*}
  11. C ( X ) C(X)
  12. ( x - h ) 2 + ( y - k ) 2 = r 2 (x-h)^{2}+(y-k)^{2}=r^{2}\,

Canonical_transformation.html

  1. ( 𝐪 , 𝐩 , t ) ( 𝐐 , 𝐏 , t ) (\mathbf{q},\mathbf{p},t)→(\mathbf{Q},\mathbf{P},t)
  2. 𝐪 𝐐 \mathbf{q}→\mathbf{Q}
  3. P i = L Q ˙ i . P_{i}=\frac{\partial L}{\partial\dot{Q}_{i}}.
  4. 𝐪 \mathbf{q}
  5. N N
  6. 𝐪 ( q 1 , q 2 , , q N - 1 , q N ) . \mathbf{q}\equiv\left(q_{1},q_{2},\ldots,q_{N-1},q_{N}\right).
  7. 𝐪 ˙ d 𝐪 d t . \dot{\mathbf{q}}\equiv\frac{d\mathbf{q}}{dt}.
  8. 𝐩 𝐪 k = 1 N p k q k . \mathbf{p}\cdot\mathbf{q}\equiv\sum_{k=1}^{N}p_{k}q_{k}.
  9. 𝐩 ˙ = - H 𝐪 𝐪 ˙ = H 𝐩 \begin{aligned}\displaystyle\dot{\mathbf{p}}&\displaystyle=-\frac{\partial H}{% \partial\mathbf{q}}\\ \displaystyle\dot{\mathbf{q}}&\displaystyle=\frac{\partial H}{\partial\mathbf{% p}}\end{aligned}
  10. 𝐏 ˙ = - K 𝐐 𝐐 ˙ = K 𝐏 \begin{aligned}\displaystyle\dot{\mathbf{P}}&\displaystyle=-\frac{\partial K}{% \partial\mathbf{Q}}\\ \displaystyle\dot{\mathbf{Q}}&\displaystyle=\frac{\partial K}{\partial\mathbf{% P}}\end{aligned}
  11. K ( 𝐐 , 𝐏 ) K(\mathbf{Q},\mathbf{P})
  12. ( 𝐪 , 𝐩 , t ) ( 𝐐 , 𝐏 , t ) (\mathbf{q},\mathbf{p},t)→(\mathbf{Q},\mathbf{P},t)
  13. ( 𝐪 , 𝐩 ) (\mathbf{q},\mathbf{p})
  14. ( 𝐐 , 𝐏 ) (\mathbf{Q},\mathbf{P})
  15. Q ˙ m = Q m 𝐪 𝐪 ˙ + Q m 𝐩 𝐩 ˙ = Q m 𝐪 H 𝐩 - Q m 𝐩 H 𝐪 = { Q m , H } \begin{aligned}\displaystyle\dot{Q}_{m}&\displaystyle=\frac{\partial Q_{m}}{% \partial\mathbf{q}}\cdot\dot{\mathbf{q}}+\frac{\partial Q_{m}}{\partial\mathbf% {p}}\cdot\dot{\mathbf{p}}\\ &\displaystyle=\frac{\partial Q_{m}}{\partial\mathbf{q}}\cdot\frac{\partial H}% {\partial\mathbf{p}}-\frac{\partial Q_{m}}{\partial\mathbf{p}}\cdot\frac{% \partial H}{\partial\mathbf{q}}\\ &\displaystyle=\{Q_{m},H\}\end{aligned}
  16. H P m = H 𝐪 𝐪 P m + H 𝐩 𝐩 P m \frac{\partial H}{\partial P_{m}}=\frac{\partial H}{\partial\mathbf{q}}\cdot% \frac{\partial\mathbf{q}}{\partial P_{m}}+\frac{\partial H}{\partial\mathbf{p}% }\cdot\frac{\partial\mathbf{p}}{\partial P_{m}}
  17. ( Q m p n ) 𝐪 , 𝐩 = - ( q n P m ) 𝐐 , 𝐏 ( Q m q n ) 𝐪 , 𝐩 = ( p n P m ) 𝐐 , 𝐏 \begin{aligned}\displaystyle\left(\frac{\partial Q_{m}}{\partial p_{n}}\right)% _{\mathbf{q},\mathbf{p}}&\displaystyle=-\left(\frac{\partial q_{n}}{\partial P% _{m}}\right)_{\mathbf{Q},\mathbf{P}}\\ \displaystyle\left(\frac{\partial Q_{m}}{\partial q_{n}}\right)_{\mathbf{q},% \mathbf{p}}&\displaystyle=\left(\frac{\partial p_{n}}{\partial P_{m}}\right)_{% \mathbf{Q},\mathbf{P}}\end{aligned}
  18. ( P m p n ) 𝐪 , 𝐩 = ( q n Q m ) 𝐐 , 𝐏 ( P m q n ) 𝐪 , 𝐩 = - ( p n Q m ) 𝐐 , 𝐏 \begin{aligned}\displaystyle\left(\frac{\partial P_{m}}{\partial p_{n}}\right)% _{\mathbf{q},\mathbf{p}}&\displaystyle=\left(\frac{\partial q_{n}}{\partial Q_% {m}}\right)_{\mathbf{Q},\mathbf{P}}\\ \displaystyle\left(\frac{\partial P_{m}}{\partial q_{n}}\right)_{\mathbf{q},% \mathbf{p}}&\displaystyle=-\left(\frac{\partial p_{n}}{\partial Q_{m}}\right)_% {\mathbf{Q},\mathbf{P}}\end{aligned}
  19. d 𝐪 d 𝐩 = d 𝐐 d 𝐏 \int d\mathbf{q}d\mathbf{p}=\int d\mathbf{Q}d\mathbf{P}
  20. J J
  21. d 𝐐 d 𝐏 = J d 𝐪 d 𝐩 \int d\mathbf{Q}d\mathbf{P}=\int Jd\mathbf{q}d\mathbf{p}
  22. J ( 𝐐 , 𝐏 ) ( 𝐪 , 𝐩 ) J\equiv\frac{\partial(\mathbf{Q},\mathbf{P})}{\partial(\mathbf{q},\mathbf{p})}
  23. J ( 𝐐 , 𝐏 ) ( 𝐪 , 𝐏 ) / ( 𝐪 , 𝐩 ) ( 𝐪 , 𝐏 ) J\equiv\frac{\partial(\mathbf{Q},\mathbf{P})}{\partial(\mathbf{q},\mathbf{P})}% \left/\frac{\partial(\mathbf{q},\mathbf{p})}{\partial(\mathbf{q},\mathbf{P})}\right.
  24. J ( 𝐐 ) ( 𝐪 ) / ( 𝐩 ) ( 𝐏 ) J\equiv\frac{\partial(\mathbf{Q})}{\partial(\mathbf{q})}\left/\frac{\partial(% \mathbf{p})}{\partial(\mathbf{P})}\right.
  25. J = 1 J=1
  26. ( 𝐪 , 𝐩 , H ) (\mathbf{q},\mathbf{p},H)
  27. ( 𝐐 , 𝐏 , K ) (\mathbf{Q},\mathbf{P},K)
  28. q p = 𝐩 𝐪 ˙ - H ( 𝐪 , 𝐩 , t ) \mathcal{L}_{qp}=\mathbf{p}\cdot\dot{\mathbf{q}}-H(\mathbf{q},\mathbf{p},t)
  29. Q P = 𝐏 𝐐 ˙ - K ( 𝐐 , 𝐏 , t ) \mathcal{L}_{QP}=\mathbf{P}\cdot\dot{\mathbf{Q}}-K(\mathbf{Q},\mathbf{P},t)
  30. δ t 1 t 2 [ 𝐩 𝐪 ˙ - H ( 𝐪 , 𝐩 , t ) ] d t \displaystyle\delta\int_{t_{1}}^{t_{2}}\left[\mathbf{p}\cdot\dot{\mathbf{q}}-H% (\mathbf{q},\mathbf{p},t)\right]dt
  31. λ [ 𝐩 𝐪 ˙ - H ( 𝐪 , 𝐩 , t ) ] = 𝐏 𝐐 ˙ - K ( 𝐐 , 𝐏 , t ) + d G d t \lambda\left[\mathbf{p}\cdot\dot{\mathbf{q}}-H(\mathbf{q},\mathbf{p},t)\right]% =\mathbf{P}\cdot\dot{\mathbf{Q}}-K(\mathbf{Q},\mathbf{P},t)+\frac{dG}{dt}
  32. λ λ
  33. d G d t \frac{dG}{dt}
  34. λ λ
  35. λ 1 λ≠1
  36. d G d t \frac{dG}{dt}
  37. G G
  38. 𝐪 \mathbf{q}
  39. 𝐩 \mathbf{p}
  40. 𝐐 \mathbf{Q}
  41. 𝐏 \mathbf{P}
  42. t t
  43. ( 𝐪 , 𝐩 ) ( 𝐐 , 𝐏 ) (\mathbf{q},\mathbf{p})→(\mathbf{Q},\mathbf{P})
  44. G G 1 ( 𝐪 , 𝐐 , t ) G\equiv G_{1}(\mathbf{q},\mathbf{Q},t)
  45. 𝐩 𝐪 ˙ - H ( 𝐪 , 𝐩 , t ) = 𝐏 𝐐 ˙ - K ( 𝐐 , 𝐏 , t ) + G 1 t + G 1 𝐪 𝐪 ˙ + G 1 𝐐 𝐐 ˙ \mathbf{p}\cdot\dot{\mathbf{q}}-H(\mathbf{q},\mathbf{p},t)=\mathbf{P}\cdot\dot% {\mathbf{Q}}-K(\mathbf{Q},\mathbf{P},t)+\frac{\partial G_{1}}{\partial t}+% \frac{\partial G_{1}}{\partial\mathbf{q}}\cdot\dot{\mathbf{q}}+\frac{\partial G% _{1}}{\partial\mathbf{Q}}\cdot\dot{\mathbf{Q}}
  46. 2 N + 1 2N+1
  47. 𝐩 = G 1 𝐪 𝐏 = - G 1 𝐐 K = H + G 1 t \begin{aligned}\displaystyle\mathbf{p}&\displaystyle=\frac{\partial G_{1}}{% \partial\mathbf{q}}\\ \displaystyle\mathbf{P}&\displaystyle=-\frac{\partial G_{1}}{\partial\mathbf{Q% }}\\ \displaystyle K&\displaystyle=H+\frac{\partial G_{1}}{\partial t}\end{aligned}
  48. ( 𝐪 , 𝐩 ) ( 𝐐 , 𝐏 ) (\mathbf{q},\mathbf{p})→(\mathbf{Q},\mathbf{P})
  49. N N
  50. 𝐩 = G 1 𝐪 \mathbf{p}=\frac{\partial G_{1}}{\partial\mathbf{q}}
  51. 𝐐 \mathbf{Q}
  52. ( 𝐪 , 𝐩 ) (\mathbf{q},\mathbf{p})
  53. 𝐐 \mathbf{Q}
  54. N N
  55. 𝐏 = - G 1 𝐐 \mathbf{P}=-\frac{\partial G_{1}}{\partial\mathbf{Q}}
  56. 𝐏 \mathbf{P}
  57. ( 𝐪 , 𝐩 ) (\mathbf{q},\mathbf{p})
  58. ( 𝐪 , 𝐩 ) (\mathbf{q},\mathbf{p})
  59. ( 𝐐 , 𝐏 ) (\mathbf{Q},\mathbf{P})
  60. K = H + G 1 t K=H+\frac{\partial G_{1}}{\partial t}
  61. K K
  62. ( 𝐐 , 𝐏 ) (\mathbf{Q},\mathbf{P})
  63. G 1 𝐪 𝐐 G_{1}\equiv\mathbf{q}\cdot\mathbf{Q}
  64. 𝐩 = G 1 𝐪 = 𝐐 𝐏 = - G 1 𝐐 = - 𝐪 \begin{aligned}\displaystyle\mathbf{p}&\displaystyle=\frac{\partial G_{1}}{% \partial\mathbf{q}}=\mathbf{Q}\\ \displaystyle\mathbf{P}&\displaystyle=-\frac{\partial G_{1}}{\partial\mathbf{Q% }}=-\mathbf{q}\end{aligned}
  65. K = H K=H
  66. G - 𝐐 𝐏 + G 2 ( 𝐪 , 𝐏 , t ) G\equiv-\mathbf{Q}\cdot\mathbf{P}+G_{2}(\mathbf{q},\mathbf{P},t)
  67. - 𝐐 𝐏 -\mathbf{Q}\cdot\mathbf{P}
  68. 𝐩 𝐪 ˙ - H ( 𝐪 , 𝐩 , t ) = - 𝐐 𝐏 ˙ - K ( 𝐐 , 𝐏 , t ) + G 2 t + G 2 𝐪 𝐪 ˙ + G 2 𝐏 𝐏 ˙ \mathbf{p}\cdot\dot{\mathbf{q}}-H(\mathbf{q},\mathbf{p},t)=-\mathbf{Q}\cdot% \dot{\mathbf{P}}-K(\mathbf{Q},\mathbf{P},t)+\frac{\partial G_{2}}{\partial t}+% \frac{\partial G_{2}}{\partial\mathbf{q}}\cdot\dot{\mathbf{q}}+\frac{\partial G% _{2}}{\partial\mathbf{P}}\cdot\dot{\mathbf{P}}
  69. 2 N + 1 2N+1
  70. 𝐩 = G 2 𝐪 𝐐 = G 2 𝐏 K = H + G 2 t \begin{aligned}\displaystyle\mathbf{p}&\displaystyle=\frac{\partial G_{2}}{% \partial\mathbf{q}}\\ \displaystyle\mathbf{Q}&\displaystyle=\frac{\partial G_{2}}{\partial\mathbf{P}% }\\ \displaystyle K&\displaystyle=H+\frac{\partial G_{2}}{\partial t}\end{aligned}
  71. ( 𝐪 , 𝐩 ) ( 𝐐 , 𝐏 ) (\mathbf{q},\mathbf{p})→(\mathbf{Q},\mathbf{P})
  72. N N
  73. 𝐩 = G 2 𝐪 \mathbf{p}=\frac{\partial G_{2}}{\partial\mathbf{q}}
  74. 𝐏 \mathbf{P}
  75. ( 𝐪 , 𝐩 ) (\mathbf{q},\mathbf{p})
  76. 𝐏 \mathbf{P}
  77. N N
  78. 𝐐 = G 2 𝐏 \mathbf{Q}=\frac{\partial G_{2}}{\partial\mathbf{P}}
  79. 𝐐 \mathbf{Q}
  80. ( 𝐪 , 𝐩 ) (\mathbf{q},\mathbf{p})
  81. ( 𝐪 , 𝐩 ) (\mathbf{q},\mathbf{p})
  82. ( 𝐐 , 𝐏 ) (\mathbf{Q},\mathbf{P})
  83. K = H + G 2 t K=H+\frac{\partial G_{2}}{\partial t}
  84. K K
  85. ( 𝐐 , 𝐏 ) (\mathbf{Q},\mathbf{P})
  86. G 2 𝐠 ( 𝐪 ; t ) 𝐏 G_{2}\equiv\mathbf{g}(\mathbf{q};t)\cdot\mathbf{P}
  87. 𝐠 \mathbf{g}
  88. N N
  89. 𝐐 = G 2 𝐏 = 𝐠 ( 𝐪 ; t ) \mathbf{Q}=\frac{\partial G_{2}}{\partial\mathbf{P}}=\mathbf{g}(\mathbf{q};t)
  90. G 𝐪 𝐩 + G 3 ( 𝐩 , 𝐐 , t ) G\equiv\mathbf{q}\cdot\mathbf{p}+G_{3}(\mathbf{p},\mathbf{Q},t)
  91. 𝐪 𝐩 \mathbf{q}\cdot\mathbf{p}
  92. - 𝐪 𝐩 ˙ - H ( 𝐪 , 𝐩 , t ) = 𝐏 𝐐 ˙ - K ( 𝐐 , 𝐏 , t ) + G 3 t + G 3 𝐩 𝐩 ˙ + G 3 𝐐 𝐐 ˙ -\mathbf{q}\cdot\dot{\mathbf{p}}-H(\mathbf{q},\mathbf{p},t)=\mathbf{P}\cdot% \dot{\mathbf{Q}}-K(\mathbf{Q},\mathbf{P},t)+\frac{\partial G_{3}}{\partial t}+% \frac{\partial G_{3}}{\partial\mathbf{p}}\cdot\dot{\mathbf{p}}+\frac{\partial G% _{3}}{\partial\mathbf{Q}}\cdot\dot{\mathbf{Q}}
  93. 2 N + 1 2N+1
  94. 𝐪 = - G 3 𝐩 𝐏 = - G 3 𝐐 K = H + G 3 t \begin{aligned}\displaystyle\mathbf{q}&\displaystyle=-\frac{\partial G_{3}}{% \partial\mathbf{p}}\\ \displaystyle\mathbf{P}&\displaystyle=-\frac{\partial G_{3}}{\partial\mathbf{Q% }}\\ \displaystyle K&\displaystyle=H+\frac{\partial G_{3}}{\partial t}\end{aligned}
  95. ( 𝐪 , 𝐩 ) ( 𝐐 , 𝐏 ) (\mathbf{q},\mathbf{p})→(\mathbf{Q},\mathbf{P})
  96. N N
  97. 𝐪 = - G 3 𝐩 \mathbf{q}=-\frac{\partial G_{3}}{\partial\mathbf{p}}
  98. 𝐐 \mathbf{Q}
  99. ( 𝐪 , 𝐩 ) (\mathbf{q},\mathbf{p})
  100. 𝐐 \mathbf{Q}
  101. N N
  102. 𝐏 = - G 3 𝐐 \mathbf{P}=-\frac{\partial G_{3}}{\partial\mathbf{Q}}
  103. 𝐏 \mathbf{P}
  104. ( 𝐪 , 𝐩 ) (\mathbf{q},\mathbf{p})
  105. ( 𝐪 , 𝐩 ) (\mathbf{q},\mathbf{p})
  106. ( 𝐐 , 𝐏 ) (\mathbf{Q},\mathbf{P})
  107. K = H + G 3 t K=H+\frac{\partial G_{3}}{\partial t}
  108. K K
  109. ( 𝐐 , 𝐏 ) (\mathbf{Q},\mathbf{P})
  110. G 4 ( 𝐩 , 𝐏 , t ) G_{4}(\mathbf{p},\mathbf{P},t)
  111. G 𝐪 𝐩 - 𝐐 𝐏 + G 4 ( 𝐩 , 𝐏 , t ) G\equiv\mathbf{q}\cdot\mathbf{p}-\mathbf{Q}\cdot\mathbf{P}+G_{4}(\mathbf{p},% \mathbf{P},t)
  112. 𝐪 𝐩 - 𝐐 𝐏 \mathbf{q}\cdot\mathbf{p}-\mathbf{Q}\cdot\mathbf{P}
  113. - 𝐪 𝐩 ˙ - H ( 𝐪 , 𝐩 , t ) = - 𝐐 𝐏 ˙ - K ( 𝐐 , 𝐏 , t ) + G 4 t + G 4 𝐩 𝐩 ˙ + G 4 𝐏 𝐏 ˙ -\mathbf{q}\cdot\dot{\mathbf{p}}-H(\mathbf{q},\mathbf{p},t)=-\mathbf{Q}\cdot% \dot{\mathbf{P}}-K(\mathbf{Q},\mathbf{P},t)+\frac{\partial G_{4}}{\partial t}+% \frac{\partial G_{4}}{\partial\mathbf{p}}\cdot\dot{\mathbf{p}}+\frac{\partial G% _{4}}{\partial\mathbf{P}}\cdot\dot{\mathbf{P}}
  114. 2 N + 1 2N+1
  115. 𝐪 = - G 4 𝐩 𝐐 = G 4 𝐏 K = H + G 4 t \begin{aligned}\displaystyle\mathbf{q}&\displaystyle=-\frac{\partial G_{4}}{% \partial\mathbf{p}}\\ \displaystyle\mathbf{Q}&\displaystyle=\frac{\partial G_{4}}{\partial\mathbf{P}% }\\ \displaystyle K&\displaystyle=H+\frac{\partial G_{4}}{\partial t}\end{aligned}
  116. ( 𝐪 , 𝐩 ) ( 𝐐 , 𝐏 ) (\mathbf{q},\mathbf{p})→(\mathbf{Q},\mathbf{P})
  117. N N
  118. 𝐪 = - G 4 𝐩 \mathbf{q}=-\frac{\partial G_{4}}{\partial\mathbf{p}}
  119. 𝐏 \mathbf{P}
  120. ( 𝐪 , 𝐩 ) (\mathbf{q},\mathbf{p})
  121. 𝐏 \mathbf{P}
  122. N N
  123. 𝐐 = G 4 𝐏 \mathbf{Q}=\frac{\partial G_{4}}{\partial\mathbf{P}}
  124. 𝐐 \mathbf{Q}
  125. ( 𝐪 , 𝐩 ) (\mathbf{q},\mathbf{p})
  126. ( 𝐪 , 𝐩 ) (\mathbf{q},\mathbf{p})
  127. ( 𝐐 , 𝐏 ) (\mathbf{Q},\mathbf{P})
  128. K = H + G 4 t K=H+\frac{\partial G_{4}}{\partial t}
  129. K K
  130. ( 𝐐 , 𝐏 ) (\mathbf{Q},\mathbf{P})
  131. 𝐐 ( t ) 𝐪 ( t + τ ) \mathbf{Q}(t)\equiv\mathbf{q}(t+\tau)
  132. 𝐏 ( t ) 𝐩 ( t + τ ) \mathbf{P}(t)\equiv\mathbf{p}(t+\tau)
  133. δ t 1 t 2 [ 𝐏 𝐐 ˙ - K ( 𝐐 , 𝐏 , t ) ] d t = δ t 1 + τ t 2 + τ [ 𝐩 𝐪 ˙ - H ( 𝐪 , 𝐩 , t + τ ) ] d t = 0 \delta\int_{t_{1}}^{t_{2}}\left[\mathbf{P}\cdot\dot{\mathbf{Q}}-K(\mathbf{Q},% \mathbf{P},t)\right]dt=\delta\int_{t_{1}+\tau}^{t_{2}+\tau}\left[\mathbf{p}% \cdot\dot{\mathbf{q}}-H(\mathbf{q},\mathbf{p},t+\tau)\right]dt=0
  134. ( 𝐪 ( t ) , 𝐩 ( t ) ) (\mathbf{q}(t),\mathbf{p}(t))
  135. i p i d q i \sum_{i}p_{i}\,dq^{i}
  136. 𝐪 \mathbf{q}
  137. q i q^{i}
  138. q i q_{i}

Cantor's_theorem.html

  1. n n
  2. B = { x A : x f ( x ) } . B=\left\{\,x\in A:x\not\in f(x)\,\right\}.
  3. x f ( x ) x B x f ( x ) x B . x\in f(x)\iff x\in B\iff x\notin f(x)\iff x\notin B.
  4. P ( ) = { , { 1 , 2 } , { 1 , 2 , 3 } , { 4 } , { 1 , 5 } , { 3 , 4 , 6 } , { 2 , 4 , 6 , } , } . P(\mathbb{N})=\{\varnothing,\{1,2\},\{1,2,3\},\{4\},\{1,5\},\{3,4,6\},\{2,4,6,% \dots\},\dots\}.
  5. { 1 { 4 , 5 } 2 { 1 , 2 , 3 } 3 { 4 , 5 , 6 } 4 { 1 , 3 , 5 } } P ( ) . \mathbb{N}\begin{Bmatrix}1&\longleftrightarrow&\{4,5\}\\ 2&\longleftrightarrow&\{1,2,3\}\\ 3&\longleftrightarrow&\{4,5,6\}\\ 4&\longleftrightarrow&\{1,3,5\}\\ \vdots&\vdots&\vdots\end{Bmatrix}P(\mathbb{N}).
  6. R A = { x A : x x } . R_{A}=\left\{\,x\in A:x\not\in x\,\right\}.
  7. R U R U R U R U . R_{U}\in R_{U}\iff R_{U}\notin R_{U}.
  8. R U R U ( R U U R U R U ) . R_{U}\in R_{U}\iff(R_{U}\in U\wedge R_{U}\notin R_{U}).
  9. R = { x : x x } R=\left\{\,x:x\not\in x\,\right\}

Cantor_distribution.html

  1. e i t / 2 i = 1 cos ( t 3 i ) e^{\mathrm{i}\,t/2}\prod_{i=1}^{\infty}\cos{\left(\frac{t}{3^{i}}\right)}
  2. C 0 = [ 0 , 1 ] C 1 = [ 0 , 1 / 3 ] [ 2 / 3 , 1 ] C 2 = [ 0 , 1 / 9 ] [ 2 / 9 , 1 / 3 ] [ 2 / 3 , 7 / 9 ] [ 8 / 9 , 1 ] C 3 = [ 0 , 1 / 27 ] [ 2 / 27 , 1 / 9 ] [ 2 / 9 , 7 / 27 ] [ 8 / 27 , 1 / 3 ] [ 2 / 3 , 19 / 27 ] [ 20 / 27 , 7 / 9 ] [ 8 / 9 , 25 / 27 ] [ 26 / 27 , 1 ] C 4 = . \begin{aligned}\displaystyle C_{0}=&\displaystyle[0,1]\\ \displaystyle C_{1}=&\displaystyle[0,1/3]\cup[2/3,1]\\ \displaystyle C_{2}=&\displaystyle[0,1/9]\cup[2/9,1/3]\cup[2/3,7/9]\cup[8/9,1]% \\ \displaystyle C_{3}=&\displaystyle[0,1/27]\cup[2/27,1/9]\cup[2/9,7/27]\cup[8/2% 7,1/3]\cup\\ &\displaystyle[2/3,19/27]\cup[20/27,7/9]\cup[8/9,25/27]\cup[26/27,1]\\ \displaystyle C_{4}=&\displaystyle\cdots.\end{aligned}
  3. var ( X ) \displaystyle\operatorname{var}(X)
  4. var ( X ) = 1 8 . \operatorname{var}(X)=\frac{1}{8}.
  5. κ 2 n = 2 2 n - 1 ( 2 2 n - 1 ) B 2 n n ( 3 2 n - 1 ) , \kappa_{2n}=\frac{2^{2n-1}(2^{2n}-1)B_{2n}}{n\,(3^{2n}-1)},\,\!

Cantor_function.html

  1. max x [ 0 , 1 ] | f n + 1 ( x ) - f n ( x ) | 1 2 max x [ 0 , 1 ] | f n ( x ) - f n - 1 ( x ) | , n 1. \max_{x\in[0,1]}|f_{n+1}(x)-f_{n}(x)|\leq\frac{1}{2}\,\max_{x\in[0,1]}|f_{n}(x% )-f_{n-1}(x)|,\quad n\geq 1.
  2. max x [ 0 , 1 ] | f ( x ) - f n ( x ) | 2 - n + 1 max x [ 0 , 1 ] | f 1 ( x ) - f 0 ( x ) | . \max_{x\in[0,1]}|f(x)-f_{n}(x)|\leq 2^{-n+1}\,\max_{x\in[0,1]}|f_{1}(x)-f_{0}(% x)|.
  3. \ldots
  4. \ldots
  5. H D H_{D}
  6. D = log ( 2 ) / log ( 3 ) D=\log(2)/\log(3)
  7. f ( x ) = H D ( C ( 0 , x ) ) . f(x)=H_{D}(C\cap(0,x)).
  8. y = k = 1 b k 2 - k y=\sum_{k=1}^{\infty}b_{k}2^{-k}
  9. C z ( y ) = k = 1 b k z k . C_{z}(y)=\sum_{k=1}^{\infty}b_{k}z^{k}.

Cantor_space.html

  1. 2 2^{\mathbb{N}}
  2. n = 0 2 a n 3 n + 1 . \sum_{n=0}^{\infty}\frac{2a_{n}}{3^{n+1}}.
  3. 2 0 2^{\aleph_{0}}

Capillary_number.html

  1. Ca = μ V γ \mathrm{Ca}=\frac{\mu V}{\gamma}
  2. γ \gamma

Cardinal_assignment.html

  1. A c B ( f ) ( f : A B is injective ) A\leq_{c}B\quad\iff\quad(\exists f)(f:A\to B\ \mathrm{is\ injective})
  2. A c B A\leq_{c}B
  3. B c A B\leq_{c}A
  4. A = c B A=_{c}B
  5. c \leq_{c}

Carlson_symmetric_form.html

  1. R F ( x , y , z ) = 1 2 0 d t ( t + x ) ( t + y ) ( t + z ) R_{F}(x,y,z)=\tfrac{1}{2}\int_{0}^{\infty}\frac{dt}{\sqrt{(t+x)(t+y)(t+z)}}
  2. R J ( x , y , z , p ) = 3 2 0 d t ( t + p ) ( t + x ) ( t + y ) ( t + z ) R_{J}(x,y,z,p)=\tfrac{3}{2}\int_{0}^{\infty}\frac{dt}{(t+p)\sqrt{(t+x)(t+y)(t+% z)}}
  3. R C ( x , y ) = R F ( x , y , y ) = 1 2 0 d t ( t + y ) ( t + x ) R_{C}(x,y)=R_{F}(x,y,y)=\tfrac{1}{2}\int_{0}^{\infty}\frac{dt}{(t+y)\sqrt{(t+x% )}}
  4. R D ( x , y , z ) = R J ( x , y , z , z ) = 3 2 0 d t ( t + z ) ( t + x ) ( t + y ) ( t + z ) R_{D}(x,y,z)=R_{J}(x,y,z,z)=\tfrac{3}{2}\int_{0}^{\infty}\frac{dt}{(t+z)\,% \sqrt{(t+x)(t+y)(t+z)}}
  5. R C \scriptstyle{R_{C}}
  6. R D \scriptstyle{R_{D}}
  7. R F \scriptstyle{R_{F}}
  8. R J \scriptstyle{R_{J}}
  9. R F \scriptstyle{R_{F}}
  10. R J \scriptstyle{R_{J}}
  11. R F ( x , y , z ) \scriptstyle{R_{F}(x,y,z)}
  12. R J ( x , y , z , p ) \scriptstyle{R_{J}(x,y,z,p)}
  13. F ( ϕ , k ) = sin ϕ R F ( cos 2 ϕ , 1 - k 2 sin 2 ϕ , 1 ) F(\phi,k)=\sin\phi R_{F}\left(\cos^{2}\phi,1-k^{2}\sin^{2}\phi,1\right)
  14. E ( ϕ , k ) = sin ϕ R F ( cos 2 ϕ , 1 - k 2 sin 2 ϕ , 1 ) - 1 3 k 2 sin 3 ϕ R D ( cos 2 ϕ , 1 - k 2 sin 2 ϕ , 1 ) E(\phi,k)=\sin\phi R_{F}\left(\cos^{2}\phi,1-k^{2}\sin^{2}\phi,1\right)-\tfrac% {1}{3}k^{2}\sin^{3}\phi R_{D}\left(\cos^{2}\phi,1-k^{2}\sin^{2}\phi,1\right)
  15. Π ( ϕ , n , k ) = sin ϕ R F ( cos 2 ϕ , 1 - k 2 sin 2 ϕ , 1 ) + 1 3 n sin 3 ϕ R J ( cos 2 ϕ , 1 - k 2 sin 2 ϕ , 1 , 1 - n sin 2 ϕ ) \Pi(\phi,n,k)=\sin\phi R_{F}\left(\cos^{2}\phi,1-k^{2}\sin^{2}\phi,1\right)+% \tfrac{1}{3}n\sin^{3}\phi R_{J}\left(\cos^{2}\phi,1-k^{2}\sin^{2}\phi,1,1-n% \sin^{2}\phi\right)
  16. 0 ϕ 2 π 0\leq\phi\leq 2\pi
  17. 0 k 2 sin 2 ϕ 1 0\leq k^{2}\sin^{2}\phi\leq 1
  18. 1 / 2 {1}/{2}
  19. K ( k ) = R F ( 0 , 1 - k 2 , 1 ) K(k)=R_{F}\left(0,1-k^{2},1\right)
  20. E ( k ) = R F ( 0 , 1 - k 2 , 1 ) - 1 3 k 2 R D ( 0 , 1 - k 2 , 1 ) E(k)=R_{F}\left(0,1-k^{2},1\right)-\tfrac{1}{3}k^{2}R_{D}\left(0,1-k^{2},1\right)
  21. Π ( n , k ) = R F ( 0 , 1 - k 2 , 1 ) + 1 3 n R J ( 0 , 1 - k 2 , 1 , 1 - n ) \Pi(n,k)=R_{F}\left(0,1-k^{2},1\right)+\tfrac{1}{3}nR_{J}\left(0,1-k^{2},1,1-n\right)
  22. R F R_{F}
  23. t + x = u \sqrt{t+x}=u
  24. R C ( x , y ) = R F ( x , y , y ) = 1 2 0 1 t + x ( t + y ) d t = x 1 u 2 - x + y d u = { arccos x y y - x , x < y 1 y , x = y arccosh x y x - y , x > y R_{C}(x,y)=R_{F}(x,y,y)=\frac{1}{2}\int_{0}^{\infty}\frac{1}{\sqrt{t+x}(t+y)}% dt=\int_{\sqrt{x}}^{\infty}\frac{1}{u^{2}-x+y}du=\begin{cases}\frac{\arccos% \sqrt{\frac{x}{y}}}{\sqrt{y-x}},&x<y\\ \frac{1}{\sqrt{y}},&x=y\\ \frac{\mathrm{arccosh}\sqrt{\frac{x}{y}}}{\sqrt{x-y}},&x>y\\ \end{cases}
  25. R J R_{J}
  26. R J ( x , y , y , p ) = 3 x 1 ( u 2 - x + y ) ( u 2 - x + p ) d u = { 3 p - y ( R C ( x , y ) - R C ( x , p ) ) , y p 3 2 ( y - x ) ( R C ( x , y ) - 1 y x ) , y = p x 1 y 3 2 , y = p = x R_{J}(x,y,y,p)=3\int_{\sqrt{x}}^{\infty}\frac{1}{(u^{2}-x+y)(u^{2}-x+p)}du=% \begin{cases}\frac{3}{p-y}(R_{C}(x,y)-R_{C}(x,p)),&y\neq p\\ \frac{3}{2(y-x)}\left(R_{C}(x,y)-\frac{1}{y}\sqrt{x}\right),&y=p\neq x\\ \frac{1}{y^{\frac{3}{2}}},&y=p=x\\ \end{cases}
  27. t = κ u t=\kappa u
  28. κ \kappa
  29. R F ( κ x , κ y , κ z ) = κ - 1 / 2 R F ( x , y , z ) R_{F}\left(\kappa x,\kappa y,\kappa z\right)=\kappa^{-1/2}R_{F}(x,y,z)
  30. R J ( κ x , κ y , κ z , κ p ) = κ - 3 / 2 R J ( x , y , z , p ) R_{J}\left(\kappa x,\kappa y,\kappa z,\kappa p\right)=\kappa^{-3/2}R_{J}(x,y,z% ,p)
  31. R F ( x , y , z ) = 2 R F ( x + λ , y + λ , z + λ ) = R F ( x + λ 4 , y + λ 4 , z + λ 4 ) , R_{F}(x,y,z)=2R_{F}(x+\lambda,y+\lambda,z+\lambda)=R_{F}\left(\frac{x+\lambda}% {4},\frac{y+\lambda}{4},\frac{z+\lambda}{4}\right),
  32. λ = x y + y z + z x \lambda=\sqrt{xy}+\sqrt{yz}+\sqrt{zx}
  33. R J ( x , y , z , p ) = 2 R J ( x + λ , y + λ , z + λ , p + λ ) + 6 R C ( d 2 , d 2 + ( p - x ) ( p - y ) ( p - z ) ) = 1 4 R J ( x + λ 4 , y + λ 4 , z + λ 4 , p + λ 4 ) + 6 R C ( d 2 , d 2 + ( p - x ) ( p - y ) ( p - z ) ) \begin{aligned}\displaystyle R_{J}(x,y,z,p)&\displaystyle=2R_{J}(x+\lambda,y+% \lambda,z+\lambda,p+\lambda)+6R_{C}(d^{2},d^{2}+(p-x)(p-y)(p-z))\\ &\displaystyle=\frac{1}{4}R_{J}\left(\frac{x+\lambda}{4},\frac{y+\lambda}{4},% \frac{z+\lambda}{4},\frac{p+\lambda}{4}\right)+6R_{C}(d^{2},d^{2}+(p-x)(p-y)(p% -z))\end{aligned}
  34. d = ( p + x ) ( p + y ) ( p + z ) d=(\sqrt{p}+\sqrt{x})(\sqrt{p}+\sqrt{y})(\sqrt{p}+\sqrt{z})
  35. λ = x y + y z + z x \lambda=\sqrt{xy}+\sqrt{yz}+\sqrt{zx}
  36. R F \scriptstyle{R_{F}}
  37. R J \scriptstyle{R_{J}}
  38. R F \scriptstyle{R_{F}}
  39. A = ( x + y + z ) / 3 \scriptstyle{A=(x+y+z)/3}
  40. Δ x \scriptstyle{\Delta x}
  41. Δ y \scriptstyle{\Delta y}
  42. Δ z \scriptstyle{\Delta z}
  43. R F ( x , y , z ) = R F ( A ( 1 - Δ x ) , A ( 1 - Δ y ) , A ( 1 - Δ z ) ) = 1 A R F ( 1 - Δ x , 1 - Δ y , 1 - Δ z ) \begin{aligned}\displaystyle R_{F}(x,y,z)&\displaystyle=R_{F}(A(1-\Delta x),A(% 1-\Delta y),A(1-\Delta z))\\ &\displaystyle=\frac{1}{\sqrt{A}}R_{F}(1-\Delta x,1-\Delta y,1-\Delta z)\end{aligned}
  44. Δ x = 1 - x / A \scriptstyle{\Delta x=1-x/A}
  45. Δ x \scriptstyle{\Delta x}
  46. Δ y \scriptstyle{\Delta y}
  47. Δ z \scriptstyle{\Delta z}
  48. R F ( x , y , z ) \scriptstyle{R_{F}(x,y,z)}
  49. x \scriptstyle{x}
  50. y \scriptstyle{y}
  51. z \scriptstyle{z}
  52. Δ x \scriptstyle{\Delta x}
  53. Δ y \scriptstyle{\Delta y}
  54. Δ z \scriptstyle{\Delta z}
  55. R F \scriptstyle{R_{F}}
  56. Δ x \scriptstyle{\Delta x}
  57. Δ y \scriptstyle{\Delta y}
  58. Δ z \scriptstyle{\Delta z}
  59. E 1 = Δ x + Δ y + Δ z = 0 E_{1}=\Delta x+\Delta y+\Delta z=0
  60. E 2 = Δ x Δ y + Δ y Δ z + Δ z Δ x E_{2}=\Delta x\Delta y+\Delta y\Delta z+\Delta z\Delta x
  61. E 3 = Δ x Δ y Δ z E_{3}=\Delta x\Delta y\Delta z
  62. R F ( x , y , z ) = 1 2 A 0 1 ( t + 1 ) 3 - ( t + 1 ) 2 E 1 + ( t + 1 ) E 2 - E 3 d t = 1 2 A 0 ( 1 ( t + 1 ) 3 2 - E 2 2 ( t + 1 ) 7 2 + E 3 2 ( t + 1 ) 9 2 + 3 E 2 2 8 ( t + 1 ) 11 2 - 3 E 2 E 3 4 ( t + 1 ) 13 2 + O ( E 1 ) + O ( Δ 6 ) ) d t = 1 A ( 1 - 1 10 E 2 + 1 14 E 3 + 1 24 E 2 2 - 3 44 E 2 E 3 + O ( E 1 ) + O ( Δ 6 ) ) \begin{aligned}\displaystyle R_{F}(x,y,z)&\displaystyle=\frac{1}{2\sqrt{A}}% \int_{0}^{\infty}\frac{1}{\sqrt{(t+1)^{3}-(t+1)^{2}E_{1}+(t+1)E_{2}-E_{3}}}dt% \\ &\displaystyle=\frac{1}{2\sqrt{A}}\int_{0}^{\infty}\left(\frac{1}{(t+1)^{\frac% {3}{2}}}-\frac{E_{2}}{2(t+1)^{\frac{7}{2}}}+\frac{E_{3}}{2(t+1)^{\frac{9}{2}}}% +\frac{3E_{2}^{2}}{8(t+1)^{\frac{11}{2}}}-\frac{3E_{2}E_{3}}{4(t+1)^{\frac{13}% {2}}}+O(E_{1})+O(\Delta^{6})\right)dt\\ &\displaystyle=\frac{1}{\sqrt{A}}\left(1-\frac{1}{10}E_{2}+\frac{1}{14}E_{3}+% \frac{1}{24}E_{2}^{2}-\frac{3}{44}E_{2}E_{3}+O(E_{1})+O(\Delta^{6})\right)\end% {aligned}
  63. E 1 \scriptstyle{E_{1}}
  64. E 1 \scriptstyle{E_{1}}
  65. R J \scriptstyle{R_{J}}
  66. R J \scriptstyle{R_{J}}
  67. p \scriptstyle{p}
  68. x \scriptstyle{x}
  69. y \scriptstyle{y}
  70. z \scriptstyle{z}
  71. R J \scriptstyle{R_{J}}
  72. p \scriptstyle{p}
  73. A = x + y + z + 2 p 5 A=\frac{x+y+z+2p}{5}
  74. Δ x \scriptstyle{\Delta x}
  75. Δ y \scriptstyle{\Delta y}
  76. Δ z \scriptstyle{\Delta z}
  77. Δ p \scriptstyle{\Delta p}
  78. R J ( x , y , z , p ) = R J ( A ( 1 - Δ x ) , A ( 1 - Δ y ) , A ( 1 - Δ z ) , A ( 1 - Δ p ) ) = 1 A 3 2 R J ( 1 - Δ x , 1 - Δ y , 1 - Δ z , 1 - Δ p ) \begin{aligned}\displaystyle R_{J}(x,y,z,p)&\displaystyle=R_{J}(A(1-\Delta x),% A(1-\Delta y),A(1-\Delta z),A(1-\Delta p))\\ &\displaystyle=\frac{1}{A^{\frac{3}{2}}}R_{J}(1-\Delta x,1-\Delta y,1-\Delta z% ,1-\Delta p)\end{aligned}
  79. Δ x \scriptstyle{\Delta x}
  80. Δ y \scriptstyle{\Delta y}
  81. Δ z \scriptstyle{\Delta z}
  82. Δ p \scriptstyle{\Delta p}
  83. Δ p \scriptstyle{\Delta p}
  84. E 1 = Δ x + Δ y + Δ z + 2 Δ p = 0 E_{1}=\Delta x+\Delta y+\Delta z+2\Delta p=0
  85. E 2 = Δ x Δ y + Δ y Δ z + 2 Δ z Δ p + Δ p 2 + 2 Δ p Δ x + Δ x Δ z + 2 Δ y Δ p E_{2}=\Delta x\Delta y+\Delta y\Delta z+2\Delta z\Delta p+\Delta p^{2}+2\Delta p% \Delta x+\Delta x\Delta z+2\Delta y\Delta p
  86. E 3 = Δ z Δ p 2 + Δ x Δ p 2 + 2 Δ x Δ y Δ p + Δ x Δ y Δ z + 2 Δ y Δ z Δ p + Δ y Δ p 2 + 2 Δ x Δ z Δ p E_{3}=\Delta z\Delta p^{2}+\Delta x\Delta p^{2}+2\Delta x\Delta y\Delta p+% \Delta x\Delta y\Delta z+2\Delta y\Delta z\Delta p+\Delta y\Delta p^{2}+2% \Delta x\Delta z\Delta p
  87. E 4 = Δ y Δ z Δ p 2 + Δ x Δ z Δ p 2 + Δ x Δ y Δ p 2 + 2 Δ x Δ y Δ z Δ p E_{4}=\Delta y\Delta z\Delta p^{2}+\Delta x\Delta z\Delta p^{2}+\Delta x\Delta y% \Delta p^{2}+2\Delta x\Delta y\Delta z\Delta p
  88. E 5 = Δ x Δ y Δ z Δ p 2 E_{5}=\Delta x\Delta y\Delta z\Delta p^{2}
  89. E 2 \scriptstyle{E_{2}}
  90. E 3 \scriptstyle{E_{3}}
  91. E 4 \scriptstyle{E_{4}}
  92. E 1 = 0 \scriptstyle{E_{1}=0}
  93. R J ( x , y , z , p ) = 3 2 A 3 2 0 1 ( t + 1 ) 5 - ( t + 1 ) 4 E 1 + ( t + 1 ) 3 E 2 - ( t + 1 ) 2 E 3 + ( t + 1 ) E 4 - E 5 d t = 3 2 A 3 2 0 ( 1 ( t + 1 ) 5 2 - E 2 2 ( t + 1 ) 9 2 + E 3 2 ( t + 1 ) 11 2 + 3 E 2 2 - 4 E 4 8 ( t + 1 ) 13 2 + 2 E 5 - 3 E 2 E 3 4 ( t + 1 ) 15 2 + O ( E 1 ) + O ( Δ 6 ) ) d t = 1 A 3 2 ( 1 - 3 14 E 2 + 1 6 E 3 + 9 88 E 2 2 - 3 22 E 4 - 9 52 E 2 E 3 + 3 26 E 5 + O ( E 1 ) + O ( Δ 6 ) ) \begin{aligned}\displaystyle R_{J}(x,y,z,p)&\displaystyle=\frac{3}{2A^{\frac{3% }{2}}}\int_{0}^{\infty}\frac{1}{\sqrt{(t+1)^{5}-(t+1)^{4}E_{1}+(t+1)^{3}E_{2}-% (t+1)^{2}E_{3}+(t+1)E_{4}-E_{5}}}dt\\ &\displaystyle=\frac{3}{2A^{\frac{3}{2}}}\int_{0}^{\infty}\left(\frac{1}{(t+1)% ^{\frac{5}{2}}}-\frac{E_{2}}{2(t+1)^{\frac{9}{2}}}+\frac{E_{3}}{2(t+1)^{\frac{% 11}{2}}}+\frac{3E_{2}^{2}-4E_{4}}{8(t+1)^{\frac{13}{2}}}+\frac{2E_{5}-3E_{2}E_% {3}}{4(t+1)^{\frac{15}{2}}}+O(E_{1})+O(\Delta^{6})\right)dt\\ &\displaystyle=\frac{1}{A^{\frac{3}{2}}}\left(1-\frac{3}{14}E_{2}+\frac{1}{6}E% _{3}+\frac{9}{88}E_{2}^{2}-\frac{3}{22}E_{4}-\frac{9}{52}E_{2}E_{3}+\frac{3}{2% 6}E_{5}+O(E_{1})+O(\Delta^{6})\right)\end{aligned}
  94. R J \scriptstyle{R_{J}}
  95. E 1 \scriptstyle{E_{1}}
  96. R C \scriptstyle{R_{C}}
  97. R J \scriptstyle{R_{J}}
  98. p . v . R C ( x , - y ) = x x + y R C ( x + y , y ) , \mathrm{p.v.}\;R_{C}(x,-y)=\sqrt{\frac{x}{x+y}}\,R_{C}(x+y,y),
  99. p . v . R J ( x , y , z , - p ) = ( q - y ) R J ( x , y , z , q ) - 3 R F ( x , y , z ) + 3 y R C ( x z , - p q ) y + p = ( q - y ) R J ( x , y , z , q ) - 3 R F ( x , y , z ) + 3 x y z x z + p q R C ( x z + p q , p q ) y + p \begin{aligned}\displaystyle\mathrm{p.v.}\;R_{J}(x,y,z,-p)&\displaystyle=\frac% {(q-y)R_{J}(x,y,z,q)-3R_{F}(x,y,z)+3\sqrt{y}R_{C}(xz,-pq)}{y+p}\\ &\displaystyle=\frac{(q-y)R_{J}(x,y,z,q)-3R_{F}(x,y,z)+3\sqrt{\frac{xyz}{xz+pq% }}R_{C}(xz+pq,pq)}{y+p}\end{aligned}
  100. q = y + ( z - y ) ( y - x ) y + p . q=y+\frac{(z-y)(y-x)}{y+p}.
  101. R J ( x , y , z , q ) \scriptstyle{R_{J}(x,y,z,q)}
  102. R F ( x , y , z ) R_{F}(x,y,z)
  103. x 0 = x x_{0}=x
  104. y 0 = y y_{0}=y
  105. z 0 = z z_{0}=z
  106. λ n = x n y n + y n z n + z n x n , \lambda_{n}=\sqrt{x_{n}y_{n}}+\sqrt{y_{n}z_{n}}+\sqrt{z_{n}x_{n}},
  107. x n + 1 = x n + λ n 4 , y n + 1 = y n + λ n 4 , z n + 1 = z n + λ n 4 x_{n+1}=\frac{x_{n}+\lambda_{n}}{4},y_{n+1}=\frac{y_{n}+\lambda_{n}}{4},z_{n+1% }=\frac{z_{n}+\lambda_{n}}{4}
  108. x x
  109. y y
  110. z z
  111. μ \mu
  112. R F ( x , y , z ) = R F ( μ , μ , μ ) = μ - 1 / 2 . R_{F}\left(x,y,z\right)=R_{F}\left(\mu,\mu,\mu\right)=\mu^{-1/2}.
  113. x n y n \sqrt{x_{n}}\sqrt{y_{n}}
  114. x n y n \sqrt{x_{n}y_{n}}
  115. R C ( x , y ) R_{C}(x,y)
  116. R C ( x , y ) = R F ( x , y , y ) . R_{C}\left(x,y\right)=R_{F}\left(x,y,y\right).

Cartan_connection.html

  1. 𝔥 \mathfrak{h}
  2. X ξ = d d t R h ( t ) | t = 0 . X_{\xi}=\frac{\mathrm{d}}{\mathrm{d}t}R_{h(t)}\biggr|_{t=0}.\,
  3. ω : T P 𝔥 \omega\colon TP\to\mathfrak{h}
  4. 𝔥 \mathfrak{h}
  5. Ad ( h ) ( R h * ω ) = ω \hbox{Ad}(h)(R_{h}^{*}\omega)=\omega
  6. ξ 𝔥 \xi\in\mathfrak{h}
  7. 𝔥 \mathfrak{h}
  8. 𝔤 \mathfrak{g}
  9. 𝔤 \mathfrak{g}
  10. 𝔤 \mathfrak{g}
  11. 𝔥 \mathfrak{h}
  12. 𝔥 \mathfrak{h}
  13. 𝔤 \mathfrak{g}
  14. 𝔥 \mathfrak{h}
  15. 𝔤 \mathfrak{g}
  16. d η + 1 2 [ η , η ] = 0. d\eta+\tfrac{1}{2}[\eta,\eta]=0.
  17. X d d t τ t γ | t = 0 = θ ( X ) 𝔤 . X\mapsto\left.\frac{d}{dt}\tau_{t}^{\gamma}\right|_{t=0}=\theta(X)\in\mathfrak% {g}.
  18. θ p = A d ( h p - 1 ) θ p + h p * ω H , \theta^{\prime}_{p}=Ad(h^{-1}_{p})\theta_{p}+h^{*}_{p}\omega_{H},
  19. 𝔥 \mathfrak{h}
  20. 𝔤 \mathfrak{g}
  21. θ V = A d ( h - 1 ) θ U + h * ω H , \theta_{V}=Ad(h^{-1})\theta_{U}+h^{*}\omega_{H},\,
  22. Ω U = d θ U + 1 2 [ θ U , θ U ] . \Omega_{U}=d\theta_{U}+\tfrac{1}{2}[\theta_{U},\theta_{U}].
  23. P = ( U U × H ) / P=(\coprod_{U}U\times H)/\sim
  24. 𝔤 \mathfrak{g}
  25. 𝔥 \mathfrak{h}
  26. 𝔤 \mathfrak{g}
  27. 𝔥 \mathfrak{h}
  28. 𝔤 / 𝔥 \mathfrak{g}/\mathfrak{h}
  29. 𝔤 / 𝔥 \mathfrak{g}/\mathfrak{h}
  30. 𝔤 \mathfrak{g}
  31. 𝔥 \mathfrak{h}
  32. P × H 𝔤 / 𝔥 P\times_{H}\mathfrak{g}/\mathfrak{h}
  33. 𝔥 \mathfrak{h}
  34. 𝔤 \mathfrak{g}
  35. Ω = d η + 1 2 [ η η ] . \Omega=d\eta+\tfrac{1}{2}[\eta\wedge\eta].
  36. P × H 𝔤 / 𝔥 P\times_{H}\mathfrak{g}/\mathfrak{h}
  37. : Ω M 0 ( 𝐕 ) Ω M 1 ( 𝐕 ) , \nabla\colon\Omega^{0}_{M}(\mathbf{V})\to\Omega^{1}_{M}(\mathbf{V}),
  38. Ω M k ( 𝐕 ) \Omega^{k}_{M}(\mathbf{V})
  39. Ω M 0 ( 𝐕 ) \Omega^{0}_{M}(\mathbf{V})
  40. Ω M 1 ( 𝐕 ) \Omega^{1}_{M}(\mathbf{V})
  41. X ( f v ) = d f ( X ) v + f X v \nabla_{X}(fv)=df(X)v+f\nabla_{X}v
  42. 𝔤 \mathfrak{g}
  43. 𝔤 \mathfrak{g}
  44. X ¯ \bar{X}
  45. X v = d v ( X ¯ ) + η ( X ¯ ) v \nabla_{X}v=dv(\bar{X})+\eta(\bar{X})\cdot v
  46. X ¯ \bar{X}
  47. X X + X ξ X\mapsto X+X_{\xi}
  48. X ξ X_{\xi}
  49. ξ 𝔥 \xi\in\mathfrak{h}
  50. X ¯ + X ξ \bar{X}+X_{\xi}
  51. X v = d v ( X ¯ + X ξ ) + η ( X ¯ + X ξ ) ) v \nabla_{X}v=dv(\bar{X}+X_{\xi})+\eta(\bar{X}+X_{\xi}))\cdot v
  52. = d v ( X ¯ ) + d v ( X ξ ) + η ( X ¯ ) v + ξ v =dv(\bar{X})+dv(X_{\xi})+\eta(\bar{X})\cdot v+\xi\cdot v
  53. = d v ( X ¯ ) + η ( X ¯ ) v =dv(\bar{X})+\eta(\bar{X})\cdot v
  54. ξ v + d v ( X ξ ) = 0 \xi\cdot v+dv(X_{\xi})=0
  55. h R h * v = v h\cdot R_{h}^{*}v=v
  56. X ¯ \bar{X}
  57. d v ( X ¯ ) dv(\bar{X})
  58. η ( X ¯ ) v \eta(\bar{X})\cdot v
  59. Ω k ( P , V ) \Omega^{k}(P,V)
  60. φ : Ω k ( P , V ) Ω 0 ( P , V k 𝔤 * ) \varphi\colon\Omega^{k}(P,V)\cong\Omega^{0}(P,V\otimes\bigwedge\nolimits^{k}% \mathfrak{g}^{*})
  61. φ ( β ) ( ξ 1 , ξ 2 , , ξ k ) = β ( η - 1 ( ξ 1 ) , , η - 1 ( ξ k ) ) \varphi(\beta)(\xi_{1},\xi_{2},\dots,\xi_{k})=\beta(\eta^{-1}(\xi_{1}),\dots,% \eta^{-1}(\xi_{k}))
  62. β Ω k ( P , V ) \beta\in\Omega^{k}(P,V)
  63. ξ j 𝔤 \xi_{j}\in\mathfrak{g}
  64. d : Ω k ( P , V ) Ω k + 1 ( P , V ) d\colon\Omega^{k}(P,V)\rightarrow\Omega^{k+1}(P,V)\,
  65. φ d : Ω 0 ( P , V ) Ω 0 ( P , V 𝔤 * ) . \varphi\circ d\colon\Omega^{0}(P,V)\rightarrow\Omega^{0}(P,V\otimes\mathfrak{g% }^{*}).\,
  66. P × H ( 𝐕 𝔤 * ) P\times_{H}(\mathbf{V}\otimes\mathfrak{g}^{*})

Cash_flow_statement.html

  1. Net Cash Flows from Operating Activities = Net Income + Rule Items \,\text{Net Cash Flows from Operating Activities}=\,\text{Net Income}+\,\text{% Rule Items}
  2. Net Cash Flows from Financing Activities = [ Dividends received from 3 < m t p l > rd parties ] - [ Dividends paid to 3 rd parties ] - [ Dividends paid to NCI but not intracompany dividend payments ] \begin{aligned}\displaystyle\,\text{Net Cash Flows from Financing Activities}=% &\displaystyle\left[\,\text{Dividends received from }3^{<}mtpl>{{\rm rd}}\,% \text{ parties}\right]\\ &\displaystyle-\left[\,\text{Dividends paid to }3^{{\rm rd}}\,\text{ parties}% \right]\\ &\displaystyle-[\,\text{Dividends paid to NCI but not}\\ &\displaystyle\,\text{intracompany dividend payments}]\end{aligned}

Casimir_element.html

  1. 𝔤 \mathfrak{g}
  2. n n
  3. { X i } i = 1 n \{X_{i}\}_{i=1}^{n}
  4. 𝔤 \mathfrak{g}
  5. { X i } i = 1 n \{X^{i}\}_{i=1}^{n}
  6. 𝔤 \mathfrak{g}
  7. 𝔤 \mathfrak{g}
  8. Ω \Omega
  9. U ( 𝔤 ) U(\mathfrak{g})
  10. Ω = i = 1 n X i X i . \Omega=\sum_{i=1}^{n}X_{i}X^{i}.
  11. 𝔤 \mathfrak{g}
  12. U ( 𝔤 ) U(\mathfrak{g})
  13. 𝔤 \mathfrak{g}
  14. ρ ( Ω ) = i = 1 n ρ ( X i ) ρ ( X i ) . \rho(\Omega)=\sum_{i=1}^{n}\rho(X_{i})\rho(X^{i}).
  15. 𝔤 \mathfrak{g}
  16. 𝔤 \mathfrak{g}
  17. G G
  18. 𝔤 \mathfrak{g}
  19. 𝔤 \mathfrak{g}
  20. G G
  21. 𝔤 \mathfrak{g}
  22. G G
  23. 𝔤 \mathfrak{g}
  24. G G
  25. 𝔰 𝔬 ( 3 ) \mathfrak{so}(3)
  26. L x , L y , L z L_{x},\,L_{y},\,L_{z}
  27. L 2 = L x 2 + L y 2 + L z 2 . L^{2}=L_{x}^{2}+L_{y}^{2}+L_{z}^{2}.
  28. L 2 = L x 2 + L y 2 + L z 2 = ( + 1 ) e . L^{2}=L_{x}^{2}+L_{y}^{2}+L_{z}^{2}=\ell(\ell+1)e.
  29. \ell
  30. \ell
  31. \ell
  32. ( 2 + 1 ) (2\ell+1)
  33. = 1 \ell\,=\,1
  34. L x = ( 0 0 0 0 0 - 1 0 1 0 ) , L y = ( 0 0 1 0 0 0 - 1 0 0 ) , L z = ( 0 - 1 0 1 0 0 0 0 0 ) . L_{x}=\begin{pmatrix}0&0&0\\ 0&0&-1\\ 0&1&0\end{pmatrix},L_{y}=\begin{pmatrix}0&0&1\\ 0&0&0\\ -1&0&0\end{pmatrix},L_{z}=\begin{pmatrix}0&-1&0\\ 1&0&0\\ 0&0&0\end{pmatrix}.
  35. L 2 = L x 2 + L y 2 + L z 2 = 2 ( 1 0 0 0 1 0 0 0 1 ) L^{2}=L_{x}^{2}+L_{y}^{2}+L_{z}^{2}=2\begin{pmatrix}1&0&0\\ 0&1&0\\ 0&0&1\end{pmatrix}
  36. ( + 1 ) = 2 \ell(\ell+1)\,=\,2
  37. = 1 \ell\,=\,1
  38. Ω \Omega
  39. , \langle,\rangle
  40. Ω \Omega
  41. L ( λ ) L(\lambda)
  42. λ \lambda
  43. Ω \Omega
  44. L ( λ ) L(\lambda)
  45. λ , λ + 2 ρ , \langle\lambda,\lambda+2\rho\rangle,
  46. ρ \rho

Casting_out_nines.html

  1. \bcancel 32 \bcancel 64 \bcancel{3}2\bcancel{6}4\,
  2. \Rightarrow
  3. \bcancel 6 \bcancel{6}\,
  4. \bcancel 8 \bcancel 4 \bcancel 1 \bcancel 5 \bcancel{8}\bcancel{4}\bcancel{1}\bcancel{5}\,
  5. \Rightarrow
  6. 0 0\,
  7. 2 \bcancel 946 2\bcancel{9}46\,
  8. \Rightarrow
  9. \bcancel 3 \bcancel{3}\,
  10. + \bcancel 320 \bcancel 6 ¯ \underline{+\bcancel{3}20\bcancel{6}}
  11. \Rightarrow
  12. 2 2\,
  13. \bcancel 17 \bcancel 831 \bcancel{1}7\bcancel{8}31\,
  14. \bigg\Downarrow
  15. \Downarrow
  16. 2 {2}\,
  17. \Leftrightarrow
  18. 2 2\,
  19. \bcancel 5 \bcancel 6 \bcancel 4 \bcancel 3 \bcancel{5}\bcancel{6}\bcancel{4}\bcancel{3}\,
  20. \Rightarrow
  21. 0 ( 9 ) 0(9)\,
  22. - 2 \bcancel 8 \bcancel 9 \bcancel 1 ¯ \underline{-2\bcancel{8}\bcancel{9}\bcancel{1}}\,
  23. \Rightarrow
  24. - 2 -2\,
  25. \bcancel 27 \bcancel 5 \bcancel 2 \bcancel{2}7\bcancel{5}\bcancel{2}\,
  26. \bigg\Downarrow
  27. \Downarrow
  28. 7 {7}\,
  29. \Leftrightarrow
  30. 7 7\,
  31. \bcancel 5 \bcancel 48 \bcancel{5}\bcancel{4}8\,
  32. \Rightarrow
  33. 8 8\,
  34. × 62 \bcancel 9 ¯ \underline{\times 62\bcancel{9}}\,
  35. \Rightarrow
  36. 8 8\,
  37. \bcancel 344 \bcancel 6 \bcancel 92 {\bcancel{3}44\bcancel{6}\bcancel{9}2}\,
  38. \bigg\Downarrow
  39. \Downarrow
  40. 1 {1}\,
  41. \Leftrightarrow
  42. 1 1\,
  43. \bcancel 27 \bcancel 5462 \bcancel{27}\bcancel{54}62\,
  44. ÷ \div
  45. 877 877\,
  46. = =
  47. 314 314\,
  48. r . r.
  49. 84 84\,
  50. \Downarrow
  51. \Downarrow
  52. \Downarrow
  53. \Downarrow
  54. 8 8\,
  55. \Leftrightarrow
  56. ( 4 (4\,
  57. × \times
  58. 8 ) 8)\,
  59. + +
  60. 3 3\,

Catastrophe_theory.html

  1. V = x 3 + a x V=x^{3}+ax\,
  2. V = x 4 + a x 2 + b x V=x^{4}+ax^{2}+bx\,
  3. V = x 5 + a x 3 + b x 2 + c x V=x^{5}+ax^{3}+bx^{2}+cx\,
  4. V = x 3 + y 3 + a x y + b x + c y V=x^{3}+y^{3}+axy+bx+cy\,
  5. V = x 3 3 - x y 2 + a ( x 2 + y 2 ) + b x + c y V=\frac{x^{3}}{3}-xy^{2}+a(x^{2}+y^{2})+bx+cy\,
  6. V = x 2 y + y 4 + a x 2 + b y 2 + c x + d y V=x^{2}y+y^{4}+ax^{2}+by^{2}+cx+dy\,
  7. V = x V=x
  8. V = ± x 2 + a x V=\pm x^{2}+ax
  9. V = x k + 1 + V=x^{k+1}+\cdots
  10. V = x 3 + y 4 + a x y 2 + b x y + c x + d y + e y 2 V=x^{3}+y^{4}+axy^{2}+bxy+cx+dy+ey^{2}

Category_of_topological_spaces.html

  1. ( X U A i ) I (X\to UA_{i})_{I}
  2. ( A A i ) I (A\to A_{i})_{I}

Cauchy_principal_value.html

  1. lim ε 0 + [ a b - ε f ( x ) d x + b + ε c f ( x ) d x ] \lim_{\varepsilon\rightarrow 0+}\left[\int_{a}^{b-\varepsilon}f(x)\,\mathrm{d}% x+\int_{b+\varepsilon}^{c}f(x)\,\mathrm{d}x\right]
  2. a b f ( x ) d x = ± \int_{a}^{b}f(x)\,\mathrm{d}x=\pm\infty
  3. b c f ( x ) d x = \int_{b}^{c}f(x)\,\mathrm{d}x=\mp\infty
  4. lim a - a a f ( x ) d x \lim_{a\rightarrow\infty}\int_{-a}^{a}f(x)\,\mathrm{d}x
  5. - 0 f ( x ) d x = ± \int_{-\infty}^{0}f(x)\,\mathrm{d}x=\pm\infty
  6. 0 f ( x ) d x = \int_{0}^{\infty}f(x)\,\mathrm{d}x=\mp\infty
  7. lim ε 0 + [ b - 1 ε b - ε f ( x ) d x + b + ε b + 1 ε f ( x ) d x ] . \lim_{\varepsilon\rightarrow 0+}\left[\int_{b-\frac{1}{\varepsilon}}^{b-% \varepsilon}f(x)\,\mathrm{d}x+\int_{b+\varepsilon}^{b+\frac{1}{\varepsilon}}f(% x)\,\mathrm{d}x\right].
  8. P L f ( z ) d z = L * f ( z ) d z = lim ε 0 L ( ε ) f ( z ) d z , \mathrm{P}\int_{L}f(z)\ \mathrm{d}z=\int_{L}^{*}f(z)\ \mathrm{d}z=\lim_{% \varepsilon\to 0}\int_{L(\varepsilon)}f(z)\ \mathrm{d}z,
  9. C c ( ) {C_{c}^{\infty}}(\mathbb{R})
  10. \mathbb{R}
  11. p . v . ( 1 x ) : C c ( ) \operatorname{p.\!v.}\left(\frac{1}{x}\right)\,:\,{C_{c}^{\infty}}(\mathbb{R})% \to\mathbb{C}
  12. [ p . v . ( 1 x ) ] ( u ) = lim ε 0 + [ - ε ; ε ] u ( x ) x d x = 0 + u ( x ) - u ( - x ) x d x for u C c ( ) \left[\operatorname{p.\!v.}\left(\frac{1}{x}\right)\right](u)=\lim_{% \varepsilon\to 0^{+}}\int_{\mathbb{R}\setminus[-\varepsilon;\varepsilon]}\frac% {u(x)}{x}\,\mathrm{d}x=\int_{0}^{+\infty}\frac{u(x)-u(-x)}{x}\,\mathrm{d}x% \quad\,\text{for }u\in{C_{c}^{\infty}}(\mathbb{R})
  13. 0 + u ( x ) - u ( - x ) x d x \int_{0}^{+\infty}\frac{u(x)-u(-x)}{x}\,\mathrm{d}x
  14. u ( x ) u(x)
  15. u ( x ) - u ( - x ) x \frac{u(x)-u(-x)}{x}
  16. [ 0 , ) [0,\infty)
  17. lim x 0 u ( x ) - u ( - x ) = 0 \lim\limits_{x\searrow 0}u(x)-u(-x)=0
  18. lim x 0 u ( x ) - u ( - x ) x = lim x 0 u ( x ) + u ( - x ) 1 = 2 u ( 0 ) , \lim\limits_{x\searrow 0}\frac{u(x)-u(-x)}{x}=\lim\limits_{x\searrow 0}\frac{u% ^{\prime}(x)+u^{\prime}(-x)}{1}=2u^{\prime}(0),
  19. u ( x ) u^{\prime}(x)
  20. 0 1 u ( x ) - u ( - x ) x d x \int\limits_{0}^{1}\frac{u(x)-u(-x)}{x}\,\mathrm{d}x
  21. u ( x ) - u ( - x ) u(x)-u(-x)
  22. | 0 1 u ( x ) - u ( - x ) x d x | 0 1 | u ( x ) - u ( - x ) | x d x 0 1 2 x x sup x | u ( x ) | d x 2 sup x | u ( x ) | \left|\int\limits_{0}^{1}\frac{u(x)-u(-x)}{x}\,\mathrm{d}x\right|\leq\int% \limits_{0}^{1}\frac{|u(x)-u(-x)|}{x}\,\mathrm{d}x\leq\int\limits_{0}^{1}\frac% {2x}{x}\sup\limits_{x\in\mathbb{R}}|u^{\prime}(x)|\,\mathrm{d}x\leq 2\sup% \limits_{x\in\mathbb{R}}|u^{\prime}(x)|
  23. | 1 u ( x ) - u ( - x ) x d x | 2 sup x | x u ( x ) | 1 1 x 2 d x = 2 sup x | x u ( x ) | , \left|\int\limits_{1}^{\infty}\frac{u(x)-u(-x)}{x}\,\mathrm{d}x\right|\leq 2% \sup\limits_{x\in\mathbb{R}}|x\cdot u(x)|\int\limits_{1}^{\infty}\frac{1}{x^{2% }}\,\mathrm{d}x=2\sup\limits_{x\in\mathbb{R}}|x\cdot u(x)|,
  24. p . v . ( 1 x ) : C c ( ) \operatorname{p.\!v.}\left(\frac{1}{x}\right)\,:\,{C_{c}^{\infty}}(\mathbb{R})% \to\mathbb{C}
  25. u u
  26. u u
  27. 0
  28. x u xu
  29. u u
  30. x x
  31. x f = 1 f = p . v . ( 1 x ) + K δ , xf=1\quad\Rightarrow\quad f=\operatorname{p.\!v.}\left(\frac{1}{x}\right)+K\delta,
  32. K K
  33. δ \delta
  34. n \mathbb{R}^{n}
  35. K K
  36. [ p . v . ( K ) ] ( f ) = lim ε 0 n B ε ( 0 ) f ( x ) K ( x ) d x . [\operatorname{p.\!v.}(K)](f)=\lim_{\varepsilon\to 0}\int_{\mathbb{R}^{n}% \setminus B_{\varepsilon(0)}}f(x)K(x)\,\mathrm{d}x.
  37. K K
  38. - n -n
  39. lim a 0 + ( - 1 - a d x x + a 1 d x x ) = 0 , \lim_{a\rightarrow 0+}\left(\int_{-1}^{-a}\frac{\mathrm{d}x}{x}+\int_{a}^{1}% \frac{\mathrm{d}x}{x}\right)=0,
  40. lim a 0 + ( - 1 - 2 a d x x + a 1 d x x ) = ln 2. \lim_{a\rightarrow 0+}\left(\int_{-1}^{-2a}\frac{\mathrm{d}x}{x}+\int_{a}^{1}% \frac{\mathrm{d}x}{x}\right)=\ln 2.
  41. - 1 1 d x x ( which gives - + ) . \int_{-1}^{1}\frac{\mathrm{d}x}{x}{\ }\left(\mbox{which}~{}\ \mbox{gives}~{}\ % -\infty+\infty\right).
  42. lim a - a a 2 x d x x 2 + 1 = 0 , \lim_{a\rightarrow\infty}\int_{-a}^{a}\frac{2x\,\mathrm{d}x}{x^{2}+1}=0,
  43. lim a - 2 a a 2 x d x x 2 + 1 = - ln 4. \lim_{a\rightarrow\infty}\int_{-2a}^{a}\frac{2x\,\mathrm{d}x}{x^{2}+1}=-\ln 4.
  44. - 2 x d x x 2 + 1 ( which gives - + ) . \int_{-\infty}^{\infty}\frac{2x\,\mathrm{d}x}{x^{2}+1}{\ }\left(\mbox{which}~{% }\ \mbox{gives}~{}\ -\infty+\infty\right).
  45. f f
  46. P V f ( x ) d x , PV\int f(x)\,\mathrm{d}x,
  47. L * f ( z ) d z , \int_{L}^{*}f(z)\,\mathrm{d}z,
  48. - f ( x ) d x , -\!\!\!\!\!\!\int f(x)\,\mathrm{d}x,
  49. P , P,
  50. 𝒫 , \mathcal{P},
  51. P v , P_{v},
  52. ( C P V ) , (CPV),

Cauchy–Binet_formula.html

  1. ( [ n ] m ) {\textstyle\left({{[n]}\atop{m}}\right)}
  2. ( n m ) {\textstyle\left({{n}\atop{m}}\right)}
  3. S ( [ n ] m ) S\in{\textstyle\left({{[n]}\atop{m}}\right)}
  4. det ( A B ) = S ( [ n ] m ) det ( A [ m ] , S ) det ( B S , [ m ] ) . \det(AB)=\sum_{S\in{\textstyle\left({{[n]}\atop{m}}\right)}}\det(A_{[m],S})% \det(B_{S,[m]}).
  5. A = ( 1 1 2 3 1 - 1 ) A=\begin{pmatrix}1&1&2\\ 3&1&-1\\ \end{pmatrix}
  6. B = ( 1 1 3 1 0 2 ) B=\begin{pmatrix}1&1\\ 3&1\\ 0&2\end{pmatrix}
  7. det ( A B ) = | 1 1 3 1 | | 1 1 3 1 | + | 1 2 1 - 1 | | 3 1 0 2 | + | 1 2 3 - 1 | | 1 1 0 2 | . \det(AB)=\left|\begin{matrix}1&1\\ 3&1\end{matrix}\right|\cdot\left|\begin{matrix}1&1\\ 3&1\end{matrix}\right|+\left|\begin{matrix}1&2\\ 1&-1\end{matrix}\right|\cdot\left|\begin{matrix}3&1\\ 0&2\end{matrix}\right|+\left|\begin{matrix}1&2\\ 3&-1\end{matrix}\right|\cdot\left|\begin{matrix}1&1\\ 0&2\end{matrix}\right|.
  8. A B = ( 4 6 6 2 ) AB=\begin{pmatrix}4&6\\ 6&2\end{pmatrix}
  9. - 2 × - 2 + - 3 × 6 + - 7 × 2 -2\times-2+-3\times 6+-7\times 2
  10. ( [ n ] m ) {\textstyle\left({{[n]}\atop{m}}\right)}
  11. ( [ n ] m ) = { [ n ] } {\textstyle\left({{[n]}\atop{m}}\right)}=\{[n]\}
  12. ( [ n ] 1 ) {\textstyle\left({{[n]}\atop{1}}\right)}
  13. j = 1 n A 1 , j B j , 1 \textstyle\sum_{j=1}^{n}A_{1,j}B_{j,1}
  14. L f = ( ( δ f ( i ) , j ) i [ m ] , j [ n ] ) and R g = ( ( δ j , g ( k ) ) j [ n ] , k [ m ] ) L_{f}=\bigl((\delta_{f(i),j})_{i\in[m],j\in[n]}\bigr)\quad\,\text{and}\quad R_% {g}=\bigl((\delta_{j,g(k)})_{j\in[n],k\in[m]}\bigr)
  15. δ \delta
  16. f : [ m ] [ n ] g : [ m ] [ n ] p ( f , g ) det ( L f R g ) = f : [ m ] [ n ] g : [ m ] [ n ] p ( f , g ) S ( [ n ] m ) det ( ( L f ) [ m ] , S ) det ( R g ) S , [ m ] ) , \sum_{f:[m]\to[n]}\sum_{g:[m]\to[n]}p(f,g)\det(L_{f}R_{g})=\sum_{f:[m]\to[n]}% \sum_{g:[m]\to[n]}p(f,g)\sum_{S\in{\textstyle\left({{[n]}\atop{m}}\right)}}% \det((L_{f})_{[m],S})\det(R_{g})_{S,[m]}),
  17. ( i = 1 m A i , f ( i ) ) ( k = 1 m B g ( k ) , k ) \textstyle(\prod_{i=1}^{m}A_{i,f(i)})(\prod_{k=1}^{m}B_{g(k),k})
  18. det ( ( L f ) [ m ] , S ) \det((L_{f})_{[m],S})
  19. det ( ( R g ) S , [ m ] ) \det((R_{g})_{S,[m]})
  20. f ( i ) g ( [ m ] ) f(i)\notin g([m])
  21. det ( L f R g ) = det ( ( L f ) [ m ] , S ) det ( R g ) S , [ m ] ) . \det(L_{f}R_{g})=\det((L_{f})_{[m],S})\det(R_{g})_{S,[m]}).\,
  22. f = h π - 1 f=h\circ\pi^{-1}
  23. g = h σ g=h\circ\sigma
  24. ( L f ) [ m ] , S (L_{f})_{[m],S}
  25. ( R g ) S , [ m ] (R_{g})_{S,[m]}
  26. π σ \pi\circ\sigma
  27. det ( L f R g ) = S ( [ n ] m ) det ( ( L f ) [ m ] , S ) det ( ( R g ) S , [ m ] ) , \det(L_{f}R_{g})=\sum_{S\in{\textstyle\left({{[n]}\atop{m}}\right)}}\det((L_{f% })_{[m],S})\det((R_{g})_{S,[m]}),
  28. L f = ( ( δ f ( i ) , j ) i [ m ] , j [ n ] ) and R g = ( ( δ j , g ( k ) ) j [ n ] , k [ m ] ) . L_{f}=\bigl((\delta_{f(i),j})_{i\in[m],j\in[n]}\bigr)\quad\,\text{and}\quad R_% {g}=\bigl((\delta_{j,g(k)})_{j\in[n],k\in[m]}\bigr).
  29. δ g ( 1 ) g ( m ) f ( 1 ) f ( m ) = k : [ m ] [ n ] k ( 1 ) < < k ( m ) δ k ( 1 ) k ( m ) f ( 1 ) f ( m ) δ g ( 1 ) g ( m ) k ( 1 ) k ( m ) . \delta^{f(1)\dots f(m)}_{g(1)\dots g(m)}=\sum_{k:[m]\to[n]\atop k(1)<\dots<k(m% )}\delta^{f(1)\dots f(m)}_{k(1)\dots k(m)}\delta^{k(1)\dots k(m)}_{g(1)\dots g% (m)}.
  30. ( n m ) {\textstyle\left({{n}\atop{m}}\right)}

Causal_system.html

  1. y ( t 0 ) y(t_{0})
  2. x ( t ) x(t)
  3. t t 0 t\leq t_{0}
  4. t < 0 t<0
  5. x x
  6. y y
  7. x 1 ( t ) x_{1}(t)
  8. x 2 ( t ) x_{2}(t)
  9. x 1 ( t ) = x 2 ( t ) , t t 0 , x_{1}(t)=x_{2}(t),\quad\forall\ t\leq t_{0},
  10. y 1 ( t ) = y 2 ( t ) , t t 0 . y_{1}(t)=y_{2}(t),\quad\forall\ t\leq t_{0}.
  11. h ( t ) h(t)
  12. H H
  13. H H
  14. h ( t ) = 0 , t < 0 h(t)=0,\quad\forall\ t<0
  15. x x
  16. y y
  17. y ( t ) = 1 - x ( t ) cos ( ω t ) y\left(t\right)=1-x\left(t\right)\cos\left(\omega t\right)
  18. y ( t ) = 0 x ( t - τ ) e - β τ d τ y\left(t\right)=\int_{0}^{\infty}x(t-\tau)e^{-\beta\tau}\,d\tau
  19. y ( t ) = - sin ( t + τ ) x ( τ ) d τ y(t)=\int_{-\infty}^{\infty}\sin(t+\tau)x(\tau)\,d\tau
  20. y n = 1 2 x n - 1 + 1 2 x n + 1 y_{n}=\frac{1}{2}\,x_{n-1}+\frac{1}{2}\,x_{n+1}
  21. y ( t ) = 0 sin ( t + τ ) x ( τ ) d τ y(t)=\int_{0}^{\infty}\sin(t+\tau)x(\tau)\,d\tau
  22. y n = x n + 1 y_{n}=x_{n+1}

Cavendish_experiment.html

  1. G = g R earth 2 M earth = 3 g 4 π R earth ρ earth G=g\frac{R\text{earth}^{2}}{M\text{earth}}=\frac{3g}{4\pi R\text{earth}\rho% \text{earth}}\,
  2. g g
  3. R earth R\text{earth}
  4. ρ earth \rho\text{earth}
  5. θ \theta
  6. κ θ \kappa\theta
  7. κ \kappa
  8. κ θ = L F \kappa\theta\ =LF\,
  9. F = G m M r 2 F=\frac{GmM}{r^{2}}\,
  10. κ θ = L G m M r 2 ( 1 ) \kappa\theta\ =L\frac{GmM}{r^{2}}\qquad\qquad\qquad(1)\,
  11. κ \kappa\,
  12. T = 2 π I / κ T=2\pi\sqrt{I/\kappa}
  13. I = m ( L / 2 ) 2 + m ( L / 2 ) 2 = 2 m ( L / 2 ) 2 = m L 2 / 2 I=m(L/2)^{2}+m(L/2)^{2}=2m(L/2)^{2}=mL^{2}/2\,
  14. T = 2 π m L 2 2 κ T=2\pi\sqrt{\frac{mL^{2}}{2\kappa}}\,
  15. κ \kappa
  16. G = 2 π 2 L r 2 M T 2 θ G=\frac{2\pi^{2}Lr^{2}}{MT^{2}}\theta\,
  17. m g = G m M e a r t h R e a r t h 2 mg=\frac{GmM_{earth}}{R_{earth}^{2}}\,
  18. M e a r t h = g R e a r t h 2 G M_{earth}=\frac{gR_{earth}^{2}}{G}\,
  19. ρ e a r t h = M e a r t h 4 π R e a r t h 3 / 3 = 3 g 4 π R e a r t h G \rho_{earth}=\frac{M_{earth}}{4\pi R_{earth}^{3}/3}=\frac{3g}{4\pi R_{earth}G}\,
  20. θ \theta\,
  21. radians \mbox{radians}~{}\,
  22. F F\,
  23. N \mbox{N}~{}\,
  24. G G\,
  25. m kg - 1 3 s - 2 \mbox{m}~{}^{3}{\mbox{kg}~{}}^{-1}\mbox{s}~{}^{-2}\,
  26. m m\,
  27. kg \mbox{kg}~{}\,
  28. M M\,
  29. kg \mbox{kg}~{}\,
  30. r r\,
  31. m \mbox{m}~{}\,
  32. L L\,
  33. m \mbox{m}~{}\,
  34. κ \kappa\,
  35. N m radian - 1 \mbox{N}~{}\,\mbox{m}~{}\,\mbox{radian}~{}^{-1}\,
  36. I I\,
  37. kg m 2 \mbox{kg}~{}\,\mbox{m}~{}^{2}\,
  38. T T\,
  39. s \mbox{s}~{}\,
  40. g g\,
  41. m s - 2 \mbox{m}~{}\,\mbox{s}~{}^{-2}\,
  42. M e a r t h M_{earth}\,
  43. kg \mbox{kg}~{}\,
  44. R e a r t h R_{earth}\,
  45. m \mbox{m}~{}\,
  46. ρ e a r t h \rho_{earth}\,
  47. kg m - 3 \mbox{kg}~{}\,\mbox{m}~{}^{-3}\,

Cayley_graph.html

  1. G G
  2. S S
  3. Γ = Γ ( G , S ) \Gamma=\Gamma(G,S)
  4. g g
  5. G G
  6. V ( Γ ) V(\Gamma)
  7. Γ \Gamma
  8. G . G.
  9. s s
  10. S S
  11. c s c_{s}
  12. g G , s S , g\in G,s\in S,
  13. g g
  14. g s gs
  15. c s . c_{s}.
  16. E ( Γ ) E(\Gamma)
  17. ( g , g s ) , (g,gs),
  18. s S s\in S
  19. S S
  20. S = S - 1 S=S^{-1}
  21. G = G=\mathbb{Z}
  22. G = n G=\mathbb{Z}_{n}
  23. C n C_{n}
  24. 2 \mathbb{Z}^{2}
  25. ( ± 1 , 0 ) , ( 0 , ± 1 ) (\pm 1,0),(0,\pm 1)
  26. 2 \mathbb{R}^{2}
  27. n × m \mathbb{Z}_{n}\times\mathbb{Z}_{m}
  28. n × m n\times m
  29. a , b | a 4 = b 2 = e , a b = b a 3 . \langle a,b|a^{4}=b^{2}=e,ab=ba^{3}\rangle.\,
  30. b , c | b 2 = c 2 = e , b c b c = c b c b . \langle b,c|b^{2}=c^{2}=e,bcbc=cbcb\rangle.\,
  31. { ( 1 x z 0 1 y 0 0 1 ) , x , y , z } \left\{\begin{pmatrix}1&x&z\\ 0&1&y\\ 0&0&1\\ \end{pmatrix},\ x,y,z\in\mathbb{Z}\right\}
  32. Z = X Y X - 1 Y - 1 , X Z = Z X , Y Z = Z Y Z=XYX^{-1}Y^{-1},\ XZ=ZX,\ YZ=ZY
  33. G G
  34. G G
  35. h G h\in G
  36. g V ( Γ ) g\in V(\Gamma)
  37. h g V ( Γ ) hg\in V(\Gamma)
  38. ( g , g s ) (g,gs)
  39. ( h g , h g s ) (hg,hgs)
  40. Γ \Gamma
  41. G G
  42. G G
  43. G G
  44. S S
  45. Γ = Γ ( G , S ) \Gamma=\Gamma(G,S)
  46. v 1 V ( Γ ) v_{1}\in V(\Gamma)
  47. v v
  48. Γ \Gamma
  49. G G
  50. v 1 v_{1}
  51. v . v.
  52. S S
  53. G G
  54. Γ \Gamma
  55. s s
  56. s = s - 1 s=s^{-1}
  57. Γ ( G , S ) \Gamma(G,S)
  58. S S
  59. S S
  60. k k
  61. k k
  62. k k
  63. S S
  64. r r
  65. r . r.
  66. S . S.
  67. 𝒫 \mathcal{P}
  68. 𝒫 \mathcal{P}
  69. f : G G f:G^{\prime}\to G
  70. S S^{\prime}
  71. G G^{\prime}
  72. f ¯ : Γ ( G , S ) Γ ( G , S ) , \bar{f}:\Gamma(G^{\prime},S^{\prime})\to\Gamma(G,S),\quad
  73. S = f ( S ) . S=f(S^{\prime}).
  74. G G
  75. k k
  76. S S
  77. Γ ( G , S ) \Gamma(G,S)
  78. 2 k 2k
  79. Γ ( G , S ) \Gamma(G,S)
  80. S S
  81. G . G.
  82. S S
  83. H H

Center_(algebra).html

  1. u : A - - A u:A\otimes-\rightarrow-\otimes A

Centered_hexagonal_number.html

  1. n n
  2. n 3 - ( n - 1 ) 3 = 3 n ( n - 1 ) + 1. n^{3}-(n-1)^{3}=3n(n-1)+1.\,
  3. 1 + 6 ( 1 2 n ( n - 1 ) ) 1+6\left({1\over 2}n(n-1)\right)
  4. n n
  5. ( n 1 ) (n−1)
  6. n n
  7. n < s u p > 3 n<sup>3

Central_angle.html

  1. 0 < Θ < 180 , Θ = ( 180 L π R ) = L R 0^{\circ}<\Theta<180^{\circ}\,,\,\,\Theta=\left({\frac{180L}{\pi R}}\right)^{% \circ}=\frac{L}{R}
  2. L = Θ 360 2 π R Θ = ( 180 L π R ) L=\frac{\Theta}{360^{\circ}}\cdot 2\pi R\,\Rightarrow\,\Theta=\left({\frac{180% L}{\pi R}}\right)^{\circ}
  3. L = Θ 2 π 2 π R Θ = L R L=\frac{\Theta}{2\pi}\cdot 2\pi R\,\Rightarrow\,\Theta=\frac{L}{R}
  4. 180 < Θ < 360 , Θ = ( 360 - 180 L π R ) = 2 π - L R 180^{\circ}<\Theta<360^{\circ}\,,\,\,\Theta=\left(360-\frac{180L}{\pi R}\right% )^{\circ}=2\pi-\frac{L}{R}

Central_simple_algebra.html

  1. ind ( D ) = i = 1 r p i m i \mathrm{ind}(D)=\prod_{i=1}^{r}p_{i}^{m_{i}}
  2. D = i = 1 r D i D=\otimes_{i=1}^{r}D_{i}
  3. p i m i p_{i}^{m_{i}}
  4. t + x 𝐢 + y 𝐣 + z 𝐤 ( t + x i y + z i - y + z i t - x i ) . t+x\mathbf{i}+y\mathbf{j}+z\mathbf{k}\leftrightarrow\left({\begin{array}[]{*{2% 0}c}t+xi&y+zi\\ -y+zi&t-xi\end{array}}\right).

Centrifugal_compressor.html

  1. M M
  2. L L
  3. T T
  4. Q = Q=
  5. ( L 3 T ) \left(\frac{L^{3}}{T}\right)
  6. H = H=
  7. ( M L T 2 ) \left(\frac{ML}{T^{2}}\right)
  8. U = U=
  9. ( L T ) \left(\frac{L}{T}\right)
  10. P = P=
  11. ( M L 2 T 3 ) \left(\frac{ML^{2}}{T^{3}}\right)
  12. ρ = \rho=
  13. ( M L 3 ) \left(\frac{M}{L^{3}}\right)
  14. μ = \mu=
  15. ( M L T ) \left(\frac{M}{LT}\right)
  16. D = D=
  17. ( L 1 ) \left(\frac{L}{1}\right)
  18. a = a=
  19. ( L T ) \left(\frac{L}{T}\right)
  20. Π 1 = \Pi_{1}=
  21. ( Q N D 3 ) \left(\frac{Q}{ND^{3}}\right)
  22. Π 2 = \Pi_{2}=
  23. ( g H N 2 D 2 ) \left(\frac{gH}{N^{2}D^{2}}\right)
  24. Π 4 = \Pi_{4}=
  25. ( N D a ) \left(\frac{ND}{a}\right)
  26. Π 3 = \Pi_{3}=
  27. ( P ρ N 3 D 5 ) \left(\frac{P}{\rho N^{3}D^{5}}\right)
  28. Π 5 = \Pi_{5}=
  29. ( ρ N D 2 μ ) \left(\frac{\rho ND^{2}}{\mu}\right)
  30. 1 1
  31. ( Π 1 * Π 4 ) = (\Pi_{1}*\Pi_{4})=
  32. ( m ( R t / k ) 0.5 p D 2 ) \left(\frac{m(Rt/k)^{0.5}}{pD^{2}}\right)
  33. 2 2
  34. ( m 1 ρ 1 a 1 D 1 2 C o s ( α 1 ) ) = \left(\frac{m_{1}}{\rho_{1}a_{1}D_{1}^{2}Cos(\alpha_{1})}\right)=
  35. ( m 2 ρ 2 a 2 D 2 2 C o s ( α 2 ) ) \left(\frac{m_{2}}{\rho_{2}a_{2}D_{2}^{2}Cos(\alpha_{2})}\right)
  36. 3 3
  37. ( m 1 ( t 1 ) 0.5 p 1 ) = \left(\frac{m_{1}(t_{1})^{0.5}}{p_{1}}\right)=
  38. ( m 2 ( t 2 ) 0.5 p 2 ) \left(\frac{m_{2}(t_{2})^{0.5}}{p_{2}}\right)
  39. 4 4
  40. ( Π 1 ) 0.5 ( Π 2 ) 0.75 = \frac{(\Pi_{1})^{0.5}}{(\Pi_{2})^{0.75}}=
  41. ( N Q 0.5 ( g H ) 0.75 ) \left(\frac{NQ^{0.5}}{(gH)^{0.75}}\right)
  42. 5 5
  43. ( Π 2 ) 0.25 ( Π 1 ) 0.5 = \frac{(\Pi_{2})^{0.25}}{(\Pi_{1})^{0.5}}=
  44. ( N D 0.5 ( g H ) 0.75 ) \left(\frac{ND^{0.5}}{(gH)^{0.75}}\right)
  45. ( Q 1 Q 2 ) = \left(\frac{Q_{1}}{Q_{2}}\right)=
  46. ( N 1 N 2 ) \left(\frac{N_{1}}{N_{2}}\right)
  47. ( H 1 H 2 ) = \left(\frac{H_{1}}{H_{2}}\right)=
  48. ( N 1 N 2 ) 2 \left(\frac{N_{1}}{N_{2}}\right)^{2}
  49. ( P 1 P 2 ) = \left(\frac{P_{1}}{P_{2}}\right)=
  50. ( N 1 N 2 ) 3 \left(\frac{N_{1}}{N_{2}}\right)^{3}
  51. ( Q 1 Q 2 ) = \left(\frac{Q_{1}}{Q_{2}}\right)=
  52. ( D 1 D 2 ) 3 \left(\frac{D_{1}}{D_{2}}\right)^{3}
  53. ( H 1 H 2 ) = \left(\frac{H_{1}}{H_{2}}\right)=
  54. ( D 1 D 2 ) 2 \left(\frac{D_{1}}{D_{2}}\right)^{2}
  55. ( P 1 P 2 ) = \left(\frac{P_{1}}{P_{2}}\right)=
  56. ( D 1 D 2 ) 5 \left(\frac{D_{1}}{D_{2}}\right)^{5}
  57. ρ t + ( ρ 𝐯 ) = 0 \frac{\partial\rho}{\partial t}+\nabla\cdot(\rho\mathbf{v})=0
  58. ρ ( 𝐯 t + 𝐯 𝐯 ) = - p + μ 2 𝐯 + ( 1 3 μ + μ v ) ( 𝐯 ) + 𝐟 \rho\left(\frac{\partial\mathbf{v}}{\partial t}+\mathbf{v}\cdot\nabla\mathbf{v% }\right)=-\nabla p+\mu\nabla^{2}\mathbf{v}+\left(\tfrac{1}{3}\mu+\mu^{v})% \nabla(\nabla\cdot\mathbf{v}\right)+\mathbf{f}
  59. δ Q = T d S \delta Q=TdS
  60. T T
  61. S S
  62. d U = T d S - p d V . dU=TdS-pdV.\,
  63. p V = n R T . {\ pV=nRT}.
  64. p = ρ ( γ - 1 ) U {\ p=\rho(\gamma-1)U}
  65. ρ \rho
  66. γ = C p / C v \gamma=C_{p}/C_{v}
  67. U = C v T U=C_{v}T
  68. C v C_{v}
  69. C p C_{p}

Cesàro_mean.html

  1. c n = 1 n i = 1 n a i c_{n}=\frac{1}{n}\sum_{i=1}^{n}a_{i}
  2. lim n a n = A \lim_{n\to\infty}a_{n}=A
  3. lim n c n = A . \lim_{n\to\infty}c_{n}=A.
  4. a n = { 1 if n = 2 k - 1 , 0 if n = 2 k a_{n}=\begin{cases}1&\mbox{if }~{}n=2k-1,\\ 0&\mbox{if }~{}n=2k\end{cases}
  5. 1 2 \frac{1}{2}
  6. a n = ( - 1 ) n a_{n}=(-1)^{n}
  7. 1 / 2 1/2
  8. 0

Champernowne_constant.html

  1. C 10 = n = 1 k = 10 n - 1 10 n - 1 k 10 n ( k - 10 n - 1 + 1 ) + 9 l = 1 n - 1 10 l - 1 l C_{10}=\sum_{n=1}^{\infty}\sum_{k=10^{n-1}}^{10^{n}-1}\frac{k}{10^{n(k-10^{n-1% }+1)+9\sum_{l=1}^{n-1}10^{l-1}l}}
  2. b b
  3. b b
  4. b 1 b−1
  5. C 10 C_{10}
  6. 10 / 81 = 0. 123456790 ¯ 10/81=0.\overline{123456790}
  7. 60499999499 490050000000 = 0.123456789 101112 96979900010203040506070809 ¯ , \frac{60499999499}{490050000000}=0.123456789\overline{101112\ldots 96979900010% 203040506070809},
  8. C 10 C_{10}
  9. μ ( C 10 ) = 10 \mu(C_{10})=10
  10. μ ( C b ) = b \mu(C_{b})=b
  11. b 2 b\geq 2

Characteristic_class.html

  1. [ M ] H n ( M ) [M]\in H_{n}(M)
  2. c 1 , , c k c_{1},\dots,c_{k}
  3. deg c i \mbox{deg}~{}\,c_{i}
  4. i 1 , , i l i_{1},\dots,i_{l}
  5. deg c i j = n \sum\mbox{deg}~{}\,c_{i_{j}}=n
  6. c i 1 c i 2 c i m ( [ M ] ) c_{i_{1}}\smile c_{i_{2}}\smile\dots\smile c_{i_{m}}([M])
  7. \smile
  8. c 1 2 c_{1}^{2}
  9. P 1 , 1 P_{1,1}
  10. p 1 2 p_{1}^{2}
  11. χ \chi
  12. 𝐙 / 2 𝐙 \mathbf{Z}/2\mathbf{Z}
  13. 𝐙 / 2 𝐙 \mathbf{Z}/2\mathbf{Z}
  14. c ( V 1 ) = c ( V ) c(V\oplus 1)=c(V)
  15. B G ( n ) BG(n)
  16. B G ( n + 1 ) BG(n+1)
  17. B G ( n ) B G ( n + 1 ) BG(n)\to BG(n+1)
  18. 𝐑 n 𝐑 n + 1 \mathbf{R}^{n}\to\mathbf{R}^{n+1}
  19. B G BG
  20. H k ( X ) H^{k}(X)
  21. H k ( B O ( k ) ) H^{k}(BO(k))
  22. H k + 1 H^{k+1}

Charles_Fefferman.html

  1. n \mathbb{C}^{n}

Charts_on_SO(3).html

  1. α = 2 cos - 1 ( w ) = 2 sin - 1 ( x 2 + y 2 + z 2 ) . \alpha=2\cos^{-1}(w)=2\sin^{-1}\left(\sqrt{x^{2}+y^{2}+z^{2}}\right).
  2. w = a z + b c z + d , w=\frac{az+b}{cz+d},

Chebotarev's_density_theorem.html

  1. 5 = ( 1 + 2 i ) ( 1 - 2 i ) 5=(1+2i)(1-2i)
  2. 3 3
  3. 2 = - i ( 1 + i ) 2 2=-i(1+i)^{2}
  4. [ i ] \mathbb{Z}\subset\mathbb{Z}[i]
  5. # X # G . \frac{\#X}{\#G}.
  6. | C | | G | ( li ( x ) + O ( x ( n log x + log | Δ | ) ) ) , \frac{|C|}{|G|}\Bigl(\mathrm{li}(x)+O\bigl(\sqrt{x}(n\log x+\log|\Delta|)\bigr% )\Bigr),
  7. μ ( X ) μ ( G ) . \frac{\mu(X)}{\mu(G)}.

Chebyshev_filter.html

  1. ε = 1 \varepsilon=1
  2. ω \omega
  3. H n ( j ω ) H_{n}(j\omega)
  4. G n ( ω ) = | H n ( j ω ) | = 1 1 + ε 2 T n 2 ( ω ω 0 ) G_{n}(\omega)=\left|H_{n}(j\omega)\right|=\frac{1}{\sqrt{1+\varepsilon^{2}T_{n% }^{2}\left(\frac{\omega}{\omega_{0}}\right)}}
  5. ε \varepsilon
  6. ω 0 \omega_{0}
  7. T n T_{n}
  8. n n
  9. ε \varepsilon
  10. G = 1 / 1 + ε 2 G=1/\sqrt{1+\varepsilon^{2}}
  11. ω 0 \omega_{0}
  12. 1 / 1 + ε 2 1/\sqrt{1+\varepsilon^{2}}
  13. 20 log 10 1 + ε 2 20\log_{10}\sqrt{1+\varepsilon^{2}}
  14. ε = 1. \varepsilon=1.
  15. j ω j\omega
  16. ω 0 = 1 \omega_{0}=1
  17. ( ω p m ) (\omega_{pm})
  18. 1 + ε 2 T n 2 ( - j s ) = 0. 1+\varepsilon^{2}T_{n}^{2}(-js)=0.\,
  19. - j s = cos ( θ ) -js=\cos(\theta)
  20. 1 + ε 2 T n 2 ( cos ( θ ) ) = 1 + ε 2 cos 2 ( n θ ) = 0. 1+\varepsilon^{2}T_{n}^{2}(\cos(\theta))=1+\varepsilon^{2}\cos^{2}(n\theta)=0.\,
  21. θ \theta
  22. θ = 1 n arccos ( ± j ε ) + m π n \theta=\frac{1}{n}\arccos\left(\frac{\pm j}{\varepsilon}\right)+\frac{m\pi}{n}
  23. s p m = j cos ( θ ) s_{pm}=j\cos(\theta)\,
  24. = j cos ( 1 n arccos ( ± j ε ) + m π n ) . =j\cos\left(\frac{1}{n}\arccos\left(\frac{\pm j}{\varepsilon}\right)+\frac{m% \pi}{n}\right).
  25. s p m ± = ± sinh ( 1 n arsinh ( 1 ε ) ) sin ( θ m ) s_{pm}^{\pm}=\pm\sinh\left(\frac{1}{n}\mathrm{arsinh}\left(\frac{1}{% \varepsilon}\right)\right)\sin(\theta_{m})
  26. + j cosh ( 1 n arsinh ( 1 ε ) ) cos ( θ m ) +j\cosh\left(\frac{1}{n}\mathrm{arsinh}\left(\frac{1}{\varepsilon}\right)% \right)\cos(\theta_{m})
  27. θ m = π 2 2 m - 1 n . \theta_{m}=\frac{\pi}{2}\,\frac{2m-1}{n}.
  28. θ n \theta_{n}
  29. sinh ( arsinh ( 1 / ε ) / n ) \sinh(\mathrm{arsinh}(1/\varepsilon)/n)
  30. cosh ( arsinh ( 1 / ε ) / n ) . \cosh(\mathrm{arsinh}(1/\varepsilon)/n).
  31. H ( s ) = 1 2 n - 1 ε m = 1 n 1 ( s - s p m - ) H(s)=\frac{1}{2^{n-1}\varepsilon}\ \prod_{m=1}^{n}\frac{1}{(s-s_{pm}^{-})}
  32. s p m - s_{pm}^{-}
  33. τ g = - d d ω arg ( H ( j ω ) ) \tau_{g}=-\frac{d}{d\omega}\arg(H(j\omega))
  34. ε = 0.01 \varepsilon=0.01
  35. G n ( ω , ω 0 ) = 1 1 + 1 ε 2 T n 2 ( ω 0 / ω ) . G_{n}(\omega,\omega_{0})=\frac{1}{\sqrt{1+\frac{1}{\varepsilon^{2}T_{n}^{2}% \left(\omega_{0}/\omega\right)}}}.
  36. 1 1 + 1 ε 2 \frac{1}{\sqrt{1+\frac{1}{\varepsilon^{2}}}}
  37. ω o \omega_{o}
  38. ε = 1 10 0.1 γ - 1 . \varepsilon=\frac{1}{\sqrt{10^{0.1\gamma}-1}}.
  39. f H = f 0 cosh ( 1 n cosh - 1 1 ε ) . f_{H}=\frac{f_{0}}{\cosh\left(\frac{1}{n}\cosh^{-1}\frac{1}{\varepsilon}\right% )}.
  40. ω 0 = 1 \omega_{0}=1
  41. ( ω p m ) (\omega_{pm})
  42. 1 + ε 2 T n 2 ( - 1 / j s p m ) = 0. 1+\varepsilon^{2}T_{n}^{2}(-1/js_{pm})=0.
  43. 1 s p m ± = ± sinh ( 1 n arsinh ( 1 ε ) ) sin ( θ m ) \frac{1}{s_{pm}^{\pm}}=\pm\sinh\left(\frac{1}{n}\mathrm{arsinh}\left(\frac{1}{% \varepsilon}\right)\right)\sin(\theta_{m})
  44. + j cosh ( 1 n arsinh ( 1 ε ) ) cos ( θ m ) \qquad+j\cosh\left(\frac{1}{n}\mathrm{arsinh}\left(\frac{1}{\varepsilon}\right% )\right)\cos(\theta_{m})
  45. ( ω z m ) (\omega_{zm})
  46. ε 2 T n 2 ( - 1 / j s z m ) = 0. \varepsilon^{2}T_{n}^{2}(-1/js_{zm})=0.\,
  47. 1 / s z m = - j cos ( π 2 2 m - 1 n ) 1/s_{zm}=-j\cos\left(\frac{\pi}{2}\,\frac{2m-1}{n}\right)
  48. G 0 = 1 G_{0}=1
  49. G 1 = 2 A 1 γ G_{1}=\frac{2A_{1}}{\gamma}
  50. G k = 4 A k - 1 A k B k - 1 G k - 1 , k = 2 , 3 , 4 , , n G_{k}=\frac{4A_{k-1}A_{k}}{B_{k-1}G_{k-1}},\qquad k=2,3,4,\dots,n
  51. G n + 1 = { 1 if n odd coth 2 ( β 4 ) if n even G_{n+1}=\begin{cases}1&\,\text{if }n\,\text{ odd}\\ \coth^{2}\left(\frac{\beta}{4}\right)&\,\text{if }n\,\text{ even}\end{cases}
  52. f H = f 0 cosh ( 1 n cosh - 1 1 ε ) f_{H}=f_{0}\cosh\left(\frac{1}{n}\cosh^{-1}\frac{1}{\varepsilon}\right)
  53. γ = sinh ( β 2 n ) \gamma=\sinh\left(\frac{\beta}{2n}\right)
  54. β = ln [ coth ( R d b 17.37 ) ] \beta=\ln\left[\coth\left(\frac{R_{db}}{17.37}\right)\right]
  55. A k = sin ( 2 k - 1 ) π 2 n , k = 1 , 2 , 3 , , n A_{k}=\sin\frac{(2k-1)\pi}{2n},\qquad k=1,2,3,\dots,n
  56. B k = γ 2 + sin 2 ( k π n ) , k = 1 , 2 , 3 , , n B_{k}=\gamma^{2}+\sin^{2}\left(\frac{k\pi}{n}\right),\qquad k=1,2,3,\dots,n

Check_digit.html

  1. 5 0 + 5 2 + 5 4 + 5 6 + 5 8 + 0 mod 10 5\cdot 0+5\cdot 2+5\cdot 4+5\cdot 6+5\cdot 8+0\mod 10
  2. 1 , 2 , , 10 1,2,\dots,10

Chemical_energy.html

  1. Δ U f reactants \Delta{U_{f}^{\circ}}_{\mathrm{reactants}}
  2. Δ U f products \Delta{U_{f}^{\circ}}_{\mathrm{products}}

Chemical_oxygen_demand.html

  1. C H n O a N b + c ( n + a 4 - b 2 - 3 4 c ) O 2 n CO + 2 ( a 2 - 3 2 c ) H O 2 + c NH 3 \mbox{C}~{}_{n}\mbox{H}~{}_{a}\mbox{O}~{}_{b}\mbox{N}~{}_{c}+\left(n+\frac{a}{% 4}-\frac{b}{2}-\frac{3}{4}c\right)\mbox{O}~{}_{2}\rightarrow n\mbox{CO}~{}_{2}% +\left(\frac{a}{2}-\frac{3}{2}c\right)\mbox{H}~{}_{2}\mbox{O}~{}+c\mbox{NH}~{}% _{3}
  2. N H + 3 2 O 2 N O + 3 - H O 3 + \mbox{N}~{}\mbox{H}~{}_{3}+2\mbox{O}~{}_{2}\rightarrow\mbox{N}~{}\mbox{O}~{}_{% 3}^{-}+\mbox{H}~{}_{3}\mbox{O}~{}^{+}
  3. C n H a O b N c + dCr 2 O 7 2 - + ( 8 d + c ) H + nCO 2 + a + 8 d - 3 c 2 H 2 O + cNH 4 + + 2 dCr 3 + \mathrm{C_{n}H_{a}O_{b}N_{c}\ +\ dCr_{2}O_{7}^{2-}\ +\ (8d\ +\ c)H^{+}% \rightarrow nCO_{2}\ +\ \frac{a+8d-3c}{2}H_{2}O\ +\ cNH_{4}^{+}\ +\ 2dCr^{3+}}
  4. C O D = 8000 ( b - s ) n s a m p l e v o l u m e COD=\frac{8000(b-s)n}{sample\ volume}
  5. 6 C l - + Cr 2 O 7 2 - + 14 H + 3 C l 2 + 2 C r 3 + + 7 H 2 O \mathrm{6Cl^{-}+Cr_{2}O_{7}^{2-}+14H^{+}\rightarrow 3Cl_{2}+2Cr^{3+}+7H_{2}O}

Chen_Jingrun.html

  1. P x ( 1 , 2 ) 0.67 x C x ( log x ) 2 . P_{x}(1,2)\geq\frac{0.67xC_{x}}{(\log x)^{2}}.

Chern–Simons_theory.html

  1. H * ( M , ) H^{*}(M,\mathbb{R})
  2. d T f ( ω ) = f ( Ω k ) dTf(\omega)=f(\Omega^{k})
  3. T f ( ω ) = C f ( Ω k ) Tf(\omega)=\int_{C}f(\Omega^{k})
  4. C S ( M ) = s ( M ) 1 2 T p 1 / CS(M)=\int_{s(M)}\tfrac{1}{2}Tp_{1}\in\mathbb{R}/\mathbb{Z}
  5. p 1 p_{1}
  6. S = k 4 π M tr ( A d A + 2 3 A A A ) . S=\frac{k}{4\pi}\int_{M}\,\text{tr}\,(A\wedge dA+\tfrac{2}{3}A\wedge A\wedge A).
  7. F = d A + A A F=dA+A\wedge A\,
  8. 0 = δ S δ A = k 2 π F . 0=\frac{\delta S}{\delta A}=\frac{k}{2\pi}F.
  9. W R ( K ) W_{R}(K)
  10. W R ( K ) = Tr R 𝒫 exp i K A W_{R}(K)=\,\text{Tr}_{R}\,\mathcal{P}\,\exp{i\oint_{K}A}
  11. 𝒫 exp \mathcal{P}\,\exp
  12. sin ( π / ( k + N ) ) sin ( π N / ( k + N ) ) \frac{\sin(\pi/(k+N))}{\sin(\pi N/(k+N))}
  13. C S ( Γ ) = 1 2 π 2 d 3 x ϵ i j k ( Γ i q p j Γ k p q + 2 3 Γ i q p Γ j r q Γ k p r ) . CS(\Gamma)=\frac{1}{2\pi^{2}}\int d^{3}x\epsilon^{ijk}\biggl(\Gamma^{p}_{iq}% \partial_{j}\Gamma^{q}_{kp}+\frac{2}{3}\Gamma^{p}_{iq}\Gamma^{q}_{jr}\Gamma^{r% }_{kp}\biggr).
  14. = - 1 2 g ( ϵ m i j D i R j n + ϵ n i j D i R j m ) . =-\frac{1}{2\sqrt{g}}\bigl(\epsilon^{mij}D_{i}R^{n}_{j}+\epsilon^{nij}D_{i}R^{% m}_{j}).

Chiliagon.html

  1. A = 250 a 2 cot π 1000 79577.2 a 2 A=250a^{2}\cot\frac{\pi}{1000}\simeq 79577.2\,a^{2}

Chord_(peer-to-peer).html

  1. m m
  2. 2 m 2^{m}
  3. 0
  4. 2 m - 1 2^{m}-1
  5. m m
  6. 2 m - 1 2^{m}-1
  7. k k
  8. k k
  9. k k
  10. s u c c e s s o r ( k ) successor(k)
  11. k k
  12. s u c c e s s o r ( k ) successor(k)
  13. r r
  14. r r
  15. s u c c e s s o r ( k ) successor(k)
  16. O ( N ) O(N)
  17. m m
  18. i t h i^{th}
  19. n n
  20. s u c c e s s o r ( ( n + 2 i - 1 ) mod 2 m ) successor((n+2^{i-1})\,\bmod\,2^{m})
  21. k k
  22. k k
  23. k k
  24. O ( log N ) O(\log N)
  25. s u c c e s s o r ( k ) successor(k)
  26. n n
  27. n n
  28. n n
  29. s u c c e s s o r ( n ) successor(n)
  30. m m
  31. O ( M log N ) O(M\log N)
  32. i t h i^{th}
  33. ( i + 1 ) t h (i+1)^{th}
  34. O ( log 2 N ) O(\log^{2}N)
  35. O ( log N ) O(\log N)
  36. O ( log N ) O(\log N)
  37. N N
  38. n n
  39. k k
  40. p p
  41. k k
  42. n n
  43. p p
  44. n n
  45. k k
  46. f f
  47. f f
  48. i t h i^{th}
  49. n n
  50. f f
  51. p p
  52. 2 i - 1 2^{i-1}
  53. 2 i 2^{i}
  54. n n
  55. f f
  56. p p
  57. 2 i - 1 2^{i-1}
  58. f f
  59. p p
  60. n n
  61. f f
  62. p p
  63. t t
  64. p p
  65. 2 m / 2 t 2^{m}/2^{t}
  66. log N \log N
  67. 2 m / N 2^{m}/N
  68. log N \log N
  69. log N \log N
  70. O ( log N ) O(\log N)
  71. r = O ( log N ) r=O(\log N)
  72. r r
  73. ( 1 4 ) r = O ( 1 N ) \left({{1}\over{4}}\right)^{r}=O\left({{1}\over{N}}\right)
  74. ( n + 2 k - 1 ) mod 2 m , 1 k m (n+2^{k-1})\mbox{ mod }~{}2^{m},1\leq k\leq m
  75. \in
  76. \in
  77. \in
  78. \in
  79. 2 n e x t - 1 2^{next-1}

Chowla–Mordell_theorem.html

  1. p p
  2. χ \chi
  3. p p
  4. G ( χ ) = χ ( a ) ζ a G(\chi)=\sum\chi(a)\zeta^{a}
  5. ζ \zeta
  6. p p
  7. G ( χ ) | G ( χ ) | \frac{G(\chi)}{|G(\chi)|}
  8. χ \chi
  9. p p

Chowla–Selberg_formula.html

  1. w 4 r χ ( r ) log Γ ( r D ) = h 2 log ( 4 π | D | ) + τ log ( ( τ ) | η ( τ ) | 2 ) \frac{w}{4}\sum_{r}\chi(r)\log\Gamma\left(\frac{r}{D}\right)=\frac{h}{2}\log(4% \pi\sqrt{|D|})+\sum_{\tau}\log\left(\sqrt{\Im(\tau)}|\eta(\tau)|^{2}\right)
  2. η ( i ) = 2 - 1 π - 3 / 4 Γ ( 1 4 ) \eta(i)=2^{-1}\pi^{-3/4}\Gamma(\tfrac{1}{4})

Christen_Sørensen_Longomontanus.html

  1. 22 7 \frac{22}{7}

Chromatic_adaptation.html

  1. c c^{\prime}
  2. c = D 1 S T f 1 = D 2 S T f 2 c^{\prime}=D_{1}\,S^{T}\,f_{1}=D_{2}\,S^{T}\,f_{2}
  3. S S
  4. f f
  5. D = D 1 - 1 D 2 = [ L 2 / L 1 0 0 0 M 2 / M 1 0 0 0 S 2 / S 1 ] D=D_{1}^{-1}D_{2}=\begin{bmatrix}L_{2}/L_{1}&0&0\\ 0&M_{2}/M_{1}&0\\ 0&0&S_{2}/S_{1}\end{bmatrix}

Chromosome_(genetic_algorithm).html

  1. x x
  2. f ( x ) = x 2 f(x)=x^{2}

Chrysotile.html

  1. 3 {}_{3}
  2. 2 {}_{2}
  3. 5 {}_{5}
  4. 4 {}_{4}
  5. 3 {}_{3}
  6. 2 {}_{2}
  7. 5 {}_{5}
  8. 4 {}_{4}
  9. 2 M g 3 Si 2 O 5 ( OH ) 4 \mathrm{2Mg_{3}Si_{2}O_{5}(OH)_{4}}
  10. 3 M g 2 SiO 4 + SiO 2 + 4 H 2 O \mathrm{3Mg_{2}SiO_{4}+SiO_{2}+4H_{2}O}

Circle_group.html

  1. 𝕋 = { z : | z | = 1 } . \mathbb{T}=\{z\in\mathbb{C}:|z|=1\}.
  2. θ z = e i θ = cos θ + i sin θ . \theta\mapsto z=e^{i\theta}=\cos\theta+i\sin\theta.
  3. 𝕋 U ( 1 ) / SO ( 2 ) . \mathbb{T}\cong\mbox{U}~{}(1)\cong\mathbb{R}/\mathbb{Z}\cong\mbox{SO}~{}(2).
  4. θ e i θ = cos θ + i sin θ . \theta\mapsto e^{i\theta}=\cos\theta+i\sin\theta.
  5. e i θ 1 e i θ 2 = e i ( θ 1 + θ 2 ) . e^{i\theta_{1}}e^{i\theta_{2}}=e^{i(\theta_{1}+\theta_{2})}.\,
  6. 𝕋 / 2 π . \mathbb{T}\cong\mathbb{R}/2\pi\mathbb{Z}.\,
  7. e i θ [ cos θ - sin θ sin θ cos θ ] . e^{i\theta}\leftrightarrow\begin{bmatrix}\cos\theta&-\sin\theta\\ \sin\theta&\cos\theta\\ \end{bmatrix}.
  8. ϕ - n = ϕ n ¯ . \phi_{-n}=\overline{\phi_{n}}.\,
  9. Hom ( 𝕋 , 𝕋 ) . \mathrm{Hom}(\mathbb{T},\mathbb{T})\cong\mathbb{Z}.\,
  10. ρ n ( e i θ ) = [ cos n θ - sin n θ sin n θ cos n θ ] , n + , \rho_{n}(e^{i\theta})=\begin{bmatrix}\cos n\theta&-\sin n\theta\\ \sin n\theta&\cos n\theta\end{bmatrix},\quad n\in\mathbb{Z}^{+},
  11. ρ - n \rho_{-n}
  12. ρ n \rho_{n}
  13. 𝕋 ( / ) . \mathbb{T}\cong\mathbb{R}\oplus(\mathbb{Q}/\mathbb{Z}).\,
  14. × ( / ) \mathbb{C}^{\times}\cong\mathbb{R}\oplus(\mathbb{Q}/\mathbb{Z})

Circle_of_a_sphere.html

  1. O E ¯ \overline{OE}
  2. R R
  3. x 2 + y 2 + z 2 = R 2 . x^{2}+y^{2}+z^{2}=R^{2}.
  4. r r
  5. a a
  6. ( x - a ) 2 + y 2 + z 2 = r 2 . (x-a)^{2}+y^{2}+z^{2}=r^{2}.
  7. ( x - a ) 2 - x 2 = r 2 - R 2 a 2 - 2 a x = r 2 - R 2 x = a 2 + R 2 - r 2 2 a . \begin{aligned}\displaystyle(x-a)^{2}-x^{2}&\displaystyle=r^{2}-R^{2}\\ \displaystyle a^{2}-2ax&\displaystyle=r^{2}-R^{2}\\ \displaystyle x&\displaystyle=\frac{a^{2}+R^{2}-r^{2}}{2a}.\end{aligned}
  8. a = 0 a=0
  9. R = r R=r
  10. R r R\not=r

Circular_motion.html

  1. ω = 2 π T = d θ d t \omega=\frac{2\pi}{T}\ =\frac{d\theta}{dt}
  2. v = 2 π r T = ω r v\,=\frac{2\pi r}{T}=\omega r
  3. θ = 2 π t T = ω t \theta=2\pi\frac{t}{T}=\omega t\,
  4. α = d ω d t \alpha\ =\frac{d\omega}{dt}
  5. a = v 2 r = ω 2 r a\,=\frac{v^{2}}{r}\,={\omega^{2}}{r}
  6. F c = m a = m v 2 r F_{c}=ma=\frac{mv^{2}}{r}
  7. 𝐯 = s y m b o l ω × 𝐫 , \mathbf{v}=symbol\omega\times\mathbf{r}\ ,
  8. 𝐚 = s y m b o l ω × 𝐯 = s y m b o l ω × ( s y m b o l ω × 𝐫 ) , \mathbf{a}=symbol\omega\times\mathbf{v}=symbol\omega\times\left(symbol\omega% \times\mathbf{r}\right)\ ,
  9. r \stackrel{\vec{r}}{}
  10. r = R u ^ R ( t ) , \vec{r}=R\hat{u}_{R}(t)\ ,
  11. u ^ R ( t ) \hat{u}_{R}(t)
  12. u ^ R \hat{u}_{R}
  13. u ^ θ \hat{u}_{\theta}
  14. u ^ θ \hat{u}_{\theta}
  15. v = d d t r ( t ) = d R d t u ^ R + R d u ^ R d t . \vec{v}=\frac{d}{dt}\vec{r}(t)=\frac{dR}{dt}\hat{u}_{R}+R\frac{d\hat{u}_{R}}{% dt}\ .
  16. u ^ R \hat{u}_{R}
  17. r ( t ) \vec{r}(t)
  18. u ^ R \hat{u}_{R}
  19. d u ^ R d t = d θ d t u ^ θ , \frac{d\hat{u}_{R}}{dt}=\frac{d\theta}{dt}\hat{u}_{\theta}\ ,
  20. u ^ R \hat{u}_{R}
  21. u ^ θ \hat{u}_{\theta}
  22. u ^ R \hat{u}_{R}
  23. u ^ R \hat{u}_{R}
  24. u ^ R \hat{u}_{R}
  25. u ^ R \hat{u}_{R}
  26. u ^ θ \hat{u}_{\theta}
  27. v = d d t r ( t ) = R d u ^ R d t = R d θ d t u ^ θ = R ω u ^ θ . \vec{v}=\frac{d}{dt}\vec{r}(t)=R\frac{d\hat{u}_{R}}{dt}=R\frac{d\theta}{dt}% \hat{u}_{\theta}\ =R\omega\hat{u}_{\theta}\ .
  28. a = d d t v = d d t ( R ω u ^ θ ) . \vec{a}=\frac{d}{dt}\vec{v}=\frac{d}{dt}\left(R\ \omega\ \hat{u}_{\theta}\ % \right)\ .
  29. = R ( d ω d t u ^ θ + ω d u ^ θ d t ) . =R\left(\frac{d\omega}{dt}\ \hat{u}_{\theta}+\omega\ \frac{d\hat{u}_{\theta}}{% dt}\right)\ .
  30. u ^ θ \hat{u}_{\theta}
  31. u ^ R \hat{u}_{R}
  32. u ^ θ \hat{u}_{\theta}
  33. r ( t ) \vec{r}(t)
  34. u ^ θ \hat{u}_{\theta}
  35. u ^ θ \hat{u}_{\theta}
  36. u ^ R \hat{u}_{R}
  37. d u ^ θ d t = - d θ d t u ^ R = - ω u ^ R , \frac{d\hat{u}_{\theta}}{dt}=-\frac{d\theta}{dt}\hat{u}_{R}=-\omega\hat{u}_{R}\ ,
  38. u ^ θ \hat{u}_{\theta}
  39. u ^ R \hat{u}_{R}
  40. u ^ θ \hat{u}_{\theta}
  41. u ^ R \hat{u}_{R}
  42. a = R ( d ω d t u ^ θ + ω d u ^ θ d t ) \vec{a}=R\left(\frac{d\omega}{dt}\ \hat{u}_{\theta}+\omega\ \frac{d\hat{u}_{% \theta}}{dt}\right)
  43. = R d ω d t u ^ θ - ω 2 R u ^ R . =R\frac{d\omega}{dt}\ \hat{u}_{\theta}-\omega^{2}R\ \hat{u}_{R}\ .
  44. a R = - ω 2 R u ^ R , \vec{a}_{R}=-\omega^{2}R\hat{u}_{R}\ ,
  45. a θ = R d ω d t u ^ θ = d R ω d t u ^ θ = d | v | d t u ^ θ . \vec{a}_{\theta}=R\frac{d\omega}{dt}\ \hat{u}_{\theta}=\frac{dR\omega}{dt}\ % \hat{u}_{\theta}=\frac{d|\vec{v}|}{dt}\ \hat{u}_{\theta}\ .
  46. x x
  47. y y
  48. z z
  49. z = x + i y = R ( cos θ + i sin θ ) = R e i θ , z=x+iy=R(\cos\theta+i\sin\theta)=Re^{i\theta}\ ,
  50. i i
  51. θ = θ ( t ) , \theta=\theta(t)\ ,
  52. R ˙ = R ¨ = 0 , \dot{R}=\ddot{R}=0\ ,
  53. v = z ˙ = d ( R e i θ ) d t = R d ( e i θ ) d t = R ( e i θ ) d ( i θ ) d t = i R θ ˙ e i θ = i ω R e i θ = i ω z v=\dot{z}=\frac{d(Re^{i\theta})}{dt}=R\frac{d(e^{i\theta})}{dt}=R(e^{i\theta})% \frac{d(i\theta)}{dt}=iR\dot{\theta}e^{i\theta}=i\omega\cdot Re^{i\theta}=i\omega z
  54. a = v ˙ = i ω ˙ z + i ω z ˙ = ( i ω ˙ - ω 2 ) z a=\dot{v}=i\dot{\omega}z+i\omega\dot{z}=(i\dot{\omega}-\omega^{2})z
  55. = ( i ω ˙ - ω 2 ) R e i θ =\left(i\dot{\omega}-\omega^{2}\right)Re^{i\theta}
  56. = - ω 2 R e i θ + ω ˙ e i π 2 R e i θ . =-\omega^{2}Re^{i\theta}+\dot{\omega}e^{i\frac{\pi}{2}}Re^{i\theta}\ .
  57. v = r d θ d t = r ω v=r\frac{d\theta}{dt}=r\omega
  58. u a = 0. \vec{u}\cdot\vec{a}=0.
  59. α 2 = γ 4 a 2 + γ 6 ( u a ) 2 , \alpha^{2}=\gamma^{4}a^{2}+\gamma^{6}(\vec{u}\cdot\vec{a})^{2},
  60. α 2 = γ 4 a 2 . \alpha^{2}=\gamma^{4}a^{2}.
  61. α = γ 2 v 2 r . \alpha=\gamma^{2}\frac{v^{2}}{r}.
  62. a = v d θ d t = v ω = v 2 r a=v\frac{d\theta}{dt}=v\omega=\frac{v^{2}}{r}
  63. F n e t = m a F_{net}=ma\,
  64. F n e t = m a r F_{net}=ma_{r}\,
  65. F n e t = m v 2 / r F_{net}=mv^{2}/r\,
  66. F n e t = F c F_{net}=F_{c}\,
  67. F n e t = F c F_{net}=F_{c}\,
  68. F c F_{c}\,
  69. F c = ( n + m g ) F_{c}=(n+mg)\,
  70. a r 2 + a t 2 = a \sqrt{a_{r}^{2}+a_{t}^{2}}=a
  71. v 2 / r v^{2}/r
  72. a t = d v / d t a_{t}=dv/dt\,
  73. ( r , θ ) (r,\theta)
  74. a c = 2 ( d r / d t ) ( d θ / d t ) a_{c}=2(dr/dt)(d\theta/dt)
  75. a t a_{t}
  76. a r = - v 2 / r + d 2 r / d t 2 a_{r}=-v^{2}/r+d^{2}r/dt^{2}

Circular_segment.html

  1. R = h + d = h / 2 + c 2 / 8 h R=h+d=h/2+c^{2}/8h\frac{}{}
  2. s = α 180 π R = θ R = arcsin ( c h + c 2 4 h ) ( h + c 2 4 h ) s=\frac{\alpha}{180}\pi R={\theta}R=\arcsin\left(\frac{c}{h+\frac{c^{2}}{4h}}% \right)\left(h+\frac{c^{2}}{4h}\right)
  3. c = 2 R sin θ 2 = R 2 - 2 cos θ c=2R\sin\frac{\theta}{2}=R\sqrt{2-2\cos\theta}
  4. h = R ( 1 - cos θ 2 ) = R - R 2 - c 2 4 h=R\left(1-\cos\frac{\theta}{2}\right)=R-\sqrt{R^{2}-\frac{c^{2}}{4}}
  5. θ = 2 arctan c 2 d = 2 arccos d R = 2 arcsin c 2 R \theta=2\arctan\frac{c}{2d}=2\arccos\frac{d}{R}=2\arcsin\frac{c}{2R}
  6. A = R 2 2 ( θ - sin θ ) . A=\frac{R^{2}}{2}\left(\theta-\sin\theta\right).
  7. A = R 2 2 ( α π 180 - sin α ) . A=\frac{R^{2}}{2}\left(\frac{\alpha\pi}{180}-\sin\alpha\right).