wpmath0000004_3

Commensurability_(mathematics).html

  1. [ H j : H 1 H 2 ] < for j = 1 , 2. [H_{j}:H_{1}\cap H_{2}]<\infty\,\text{ for }j=1,\,2.
  2. A \mathrm{A}
  3. B \mathrm{B}
  4. 𝒪 , \mathcal{O},
  5. 𝒪 \mathcal{O}
  6. \mathfrak{H}
  7. 𝔊 \mathfrak{G}
  8. 𝔊 \mathfrak{G}
  9. \mathfrak{H}

Commercial_paper.html

  1. d y C P dy_{CP}
  2. P f P_{f}
  3. P 0 P_{0}
  4. t t
  5. d y C P = ( P f - P 0 P f ) . 360 t dy_{CP}=(\frac{P_{f}-P_{0}}{P_{f}}).\frac{360}{t}
  6. b e y C P bey_{CP}
  7. b e y C P = ( P f - P 0 P 0 ) . 365 t bey_{CP}=(\frac{P_{f}-P_{0}}{P_{0}}).\frac{365}{t}

Compact-open_topology.html

  1. X X
  2. Y Y
  3. C ( X , Y ) C(X,Y)
  4. X X
  5. Y Y
  6. K K
  7. X X
  8. U U
  9. Y Y
  10. V ( K , U ) V(K,U)
  11. f C ( X , Y ) f∈C(X,Y)
  12. f ( K ) U . f(K)⊂U.
  13. V ( K , U ) V(K,U)
  14. C ( X , Y ) C(X,Y)
  15. C ( X , Y ) C(X,Y)
  16. K K
  17. X X
  18. * *
  19. C ( * , X ) C(*,X)
  20. X X
  21. X X
  22. Y Y
  23. X X
  24. S S
  25. Y Y
  26. C ( X , Y ) C(X,Y)
  27. Y Y
  28. Y Y
  29. f f
  30. K K
  31. X X
  32. f f
  33. K K
  34. X X
  35. Y Y
  36. X , Y X,Y
  37. Z Z
  38. Y Y
  39. C ( Y , Z ) × C ( X , Y ) C ( X , Z ) , C(Y,Z)×C(X,Y)→C(X,Z),
  40. ( f , g ) f g , (f,g)↦f∘g,
  41. C ( Y , Z ) × C ( X , Y ) C(Y,Z)×C(X,Y)
  42. Y Y
  43. e : C ( Y , Z ) × Y Z e:C(Y,Z)×Y→Z
  44. e ( f , x ) = f ( x ) e(f,x)=f(x)
  45. X X
  46. X X
  47. Y Y
  48. d d
  49. C ( X , Y ) C(X,Y)
  50. f , g C ( X , Y ) f,g∈C(X,Y)
  51. X X
  52. Y Y
  53. m m
  54. U X U⊆X
  55. Y Y
  56. p K ( f ) = sup { D j f ( x ) : x K , 0 j m } p_{K}(f)=\sup\left\{\left\|D^{j}f(x)\right\|\ :\ x\in K,0\leq j\leq m\right\}
  57. D < s u p > 0 f ( x ) = f ( x ) D<sup>0f(x)=f(x)

Compact_group.html

  1. 1 G 0 G π 0 ( G ) 1. 1\to G_{0}\to G\to\pi_{0}(G)\to 1.\,
  2. 1 A G ~ 0 G 0 1 1\to A\to\tilde{G}_{0}\to G_{0}\to 1\,
  3. A \sub Z ( G ~ 0 ) A\sub Z(\tilde{G}_{0})
  4. G 0 ~ \tilde{G_{0}}
  5. G ~ 0 𝕋 m × K . \tilde{G}_{0}\cong\mathbb{T}^{m}\times K.

Comparative_statics.html

  1. x x
  2. f ( x , a ) = 0 f(x,a)=0\,
  3. a a
  4. x x
  5. a a
  6. B d x + C d a = 0. B\,\text{d}x+C\,\text{d}a=0.
  7. d x \,\text{d}x
  8. d a \,\text{d}a
  9. x x
  10. a a
  11. B B
  12. C C
  13. f f
  14. x x
  15. a a
  16. x x
  17. a a
  18. x x
  19. d x = - B - 1 C d a . \,\text{d}x=-B^{-1}C\,\text{d}a.
  20. d x d a = - B - 1 C . \frac{{\,\text{d}x}}{{\,\text{d}a}}=-B^{-1}C.
  21. n n
  22. n n
  23. f ( x , a ) = 0 f(x,a)=0
  24. n n
  25. n n
  26. x x
  27. m m
  28. a a
  29. d a \,\text{d}a
  30. d x = - B - 1 C d a \,\text{d}x=-B^{-1}C\,\text{d}a
  31. B B
  32. n n
  33. n n
  34. f f
  35. x x
  36. C C
  37. n n
  38. m m
  39. f f
  40. a a
  41. B B
  42. C C
  43. x x
  44. a a
  45. B d x + C d a = 0 B\,\text{d}x+C\,\text{d}a\,=0
  46. x x
  47. B - 1 C B^{-1}C
  48. B - 1 B^{-1}
  49. B - 1 C d a B^{-1}C\,\text{d}a
  50. Q d ( P ) = a + b P Q^{d}(P)=a+bP
  51. Q s ( P ) = c + g P Q^{s}(P)=c+gP
  52. Q d Q^{d}
  53. Q s Q^{s}
  54. P e q b = a - c g - b . P^{eqb}=\frac{a-c}{g-b}.
  55. λ \lambda
  56. d P d t \frac{dP}{dt}
  57. d ( d P / d t ) d P \frac{d(dP/dt)}{dP}
  58. d ( d P / d t ) d P = - λ ( - b + g ) . \frac{d(dP/dt)}{dP}=-\lambda(-b+g).
  59. p ( x ; q ) p(x;q)
  60. x * ( q ) = arg max p ( x ; q ) x^{*}(q)=\arg\max p(x;q)
  61. f ( x ; q ) = D x p ( x ; q ) f(x;q)=D_{x}p(x;q)
  62. p ( x ; q ) p(x;q)
  63. x * ( q ) x^{*}(q)
  64. f ( x * ( q ) ; q ) = 0 f(x^{*}(q);q)=0
  65. x i * / q j , i = 1 , , n , j = 1 , , m \partial x^{*}_{i}/\partial q_{j},i=1,...,n,j=1,...,m
  66. x * ( q ) x^{*}(q)
  67. x * ( q ) x^{*}(q)
  68. D q x * ( q ) = - [ D x f ( x * ( q ) ; q ) ] - 1 D q f ( x * ( q ) ; q ) . D_{q}x^{*}(q)=-[D_{x}f(x^{*}(q);q)]^{-1}D_{q}f(x^{*}(q);q).
  69. D q p ( x * ( q ) , q ) = D q p ( x ; q ) | x = x * ( q ) . D_{q}p(x^{*}(q),q)=D_{q}p(x;q)|_{x=x^{*}(q)}.
  70. x 1 , , x n x_{1},...,x_{n}
  71. x 1 , , x n x_{1},...,x_{n}
  72. q 1 , , q m q_{1},...,q_{m}

Compartmental_models_in_epidemiology.html

  1. d S d t = - β I S N \frac{dS}{dt}=-\frac{\beta IS}{N}
  2. d I d t = β I S N - γ I \frac{dI}{dt}=\frac{\beta IS}{N}-\gamma I
  3. d R d t = γ I \frac{dR}{dt}=\gamma I
  4. d S d t + d I d t + d R d t = 0 \frac{dS}{dt}+\frac{dI}{dt}+\frac{dR}{dt}=0
  5. S ( t ) + I ( t ) + R ( t ) = Constant = N S(t)+I(t)+R(t)=\textrm{Constant}=N
  6. N N
  7. R 0 = β γ R_{0}=\frac{\beta}{\gamma}
  8. T c = β - 1 T_{c}=\beta^{-1}
  9. T r = γ - 1 T_{r}=\gamma^{-1}
  10. T r / T c . T_{r}/T_{c}.
  11. S ( t ) = S ( 0 ) e - R 0 ( R ( t ) - R ( 0 ) ) S(t)=S(0)e^{-R_{0}(R(t)-R(0))}
  12. t + t\rightarrow+\infty
  13. R = 1 - S ( 0 ) e - R 0 ( R - R ( 0 ) ) R_{\infty}=1-S(0)e^{-R_{0}(R_{\infty}-R(0))}
  14. d I d t = ( R 0 S / N - 1 ) γ I \frac{dI}{dt}=(R_{0}S/N-1)\gamma I
  15. R 0 > N S ( 0 ) , R_{0}>\frac{N}{S(0)},
  16. d I d t ( 0 ) > 0 , \frac{dI}{dt}(0)>0,
  17. R 0 < N S ( 0 ) , R_{0}<\frac{N}{S(0)},
  18. d I d t ( 0 ) < 0 , \frac{dI}{dt}(0)<0,
  19. F = β I , F=\beta I,
  20. N N
  21. F = β I N . F=\beta\frac{I}{N}.
  22. μ \mu
  23. Λ \Lambda
  24. d S d t = Λ - μ S - β I S \frac{dS}{dt}=\Lambda-\mu S-\beta IS
  25. d I d t = β I S - ( γ + μ ) I \frac{dI}{dt}=\beta IS-(\gamma+\mu)I
  26. d R d t = γ I - μ R \frac{dR}{dt}=\gamma I-\mu R
  27. ( S ( t ) , I ( t ) , R ( t ) ) = ( Λ μ , 0 , 0 ) . \left(S(t),I(t),R(t)\right)=\left(\frac{\Lambda}{\mu},0,0\right).
  28. R 0 = β Λ μ ( μ + γ ) , R_{0}=\frac{\beta\Lambda}{\mu(\mu+\gamma)},
  29. R 0 1 lim t ( S ( t ) , I ( t ) , R ( t ) ) = DFE = ( Λ μ , 0 , 0 ) R_{0}\leq 1\Rightarrow\lim_{t\rightarrow\infty}\left(S(t),I(t),R(t)\right)=% \textrm{DFE}=\left(\frac{\Lambda}{\mu},0,0\right)
  30. R 0 > 1 , I ( 0 ) > 0 lim t ( S ( t ) , I ( t ) , R ( t ) ) = EE = ( γ + μ β , μ β ( R 0 - 1 ) , γ β ( R 0 - 1 ) ) . R_{0}>1,I(0)>0\Rightarrow\lim_{t\rightarrow\infty}\left(S(t),I(t),R(t)\right)=% \textrm{EE}=\left(\frac{\gamma+\mu}{\beta},\frac{\mu}{\beta}\left(R_{0}-1% \right),\frac{\gamma}{\beta}\left(R_{0}-1\right)\right).
  31. R 0 R_{0}
  32. F = β ( t ) I N , β ( t + T ) = β ( t ) F=\beta(t)\frac{I}{N},\beta(t+T)=\beta(t)
  33. d S d t = μ N - μ S - β ( t ) I N S \frac{dS}{dt}=\mu N-\mu S-\beta(t)\frac{I}{N}S
  34. d I d t = β ( t ) I N S - ( γ + μ ) I \frac{dI}{dt}=\beta(t)\frac{I}{N}S-(\gamma+\mu)I
  35. R = N - S - I R=N-S-I
  36. 1 T 0 T β ( t ) μ + γ d t < 1 lim t + ( S ( t ) , I ( t ) ) = D F E = ( N , 0 ) , \frac{1}{T}\int_{0}^{T}\frac{\beta(t)}{\mu+\gamma}dt<1\Rightarrow\lim_{t% \rightarrow+\infty}\left(S(t),I(t)\right)=DFE=\left(N,0\right),
  37. d S d t = - β S I N + γ I \frac{dS}{dt}=-\frac{\beta SI}{N}+\gamma I
  38. d I d t = β S I N - γ I \frac{dI}{dt}=\frac{\beta SI}{N}-\gamma I
  39. d S d t + d I d t = 0 S ( t ) + I ( t ) = N \frac{dS}{dt}+\frac{dI}{dt}=0\Rightarrow S(t)+I(t)=N
  40. d I d t = ( β N - γ ) I - β I 2 \frac{dI}{dt}=\left(\beta N-\gamma\right)I-\beta I^{2}
  41. I ( 0 ) > 0 \forall I(0)>0
  42. β N γ 1 lim t + I ( t ) = 0 \frac{\beta N}{\gamma}\leq 1\Rightarrow\lim_{t\rightarrow+\infty}I(t)=0
  43. β N γ > 1 lim t + I ( t ) = β N - γ β \frac{\beta N}{\gamma}>1\Rightarrow\lim_{t\rightarrow+\infty}I(t)=\frac{\beta N% -\gamma}{\beta}
  44. I = y - 1 I=y^{-1}
  45. I ( t ) = I 1 + V e - χ ( t - t 0 ) I(t)=\frac{I_{\infty}}{1+Ve^{-\chi(t-t_{0})}}
  46. I = χ N / β I_{\infty}=\chi N/\beta
  47. χ = β - γ \chi=\beta-\gamma
  48. V = I / I 0 - 1 V=I_{\infty}/I_{0}-1
  49. S ( t ) = N - I ( t ) S(t)=N-I(t)
  50. a - 1 a^{-1}
  51. d S d t = μ N - μ S - β I N S \frac{dS}{dt}=\mu N-\mu S-\beta\frac{I}{N}S
  52. d E d t = β I N S - ( μ + a ) E \frac{dE}{dt}=\beta\frac{I}{N}S-(\mu+a)E
  53. d I d t = a E - ( γ + μ ) I \frac{dI}{dt}=aE-(\gamma+\mu)I
  54. d R d t = γ I - μ R . \frac{dR}{dt}=\gamma I-\mu R.
  55. S + E + I + R = N S+E+I+R=N
  56. N N
  57. R 0 = a μ + a β μ + γ . R_{0}=\frac{a}{\mu+a}\frac{\beta}{\mu+\gamma}.
  58. ( S ( 0 ) , E ( 0 ) , I ( 0 ) , R ( 0 ) ) { ( S , E , I , R ) [ 0 , N ] 4 : S 0 , E 0 , I 0 , R 0 , S + E + I + R = N } \left(S(0),E(0),I(0),R(0)\right)\in\left\{(S,E,I,R)\in[0,N]^{4}:S\geq 0,E\geq 0% ,I\geq 0,R\geq 0,S+E+I+R=N\right\}
  59. R 0 1 lim t + ( S ( t ) , E ( t ) , I ( t ) , R ( t ) ) = D F E = ( N , 0 , 0 , 0 ) R_{0}\leq 1\Rightarrow\lim_{t\rightarrow+\infty}\left(S(t),E(t),I(t),R(t)% \right)=DFE=\left(N,0,0,0\right)
  60. R 0 > 1 , I ( 0 ) > 0 lim t + ( S ( t ) , E ( t ) , I ( t ) , R ( t ) ) = E E . R_{0}>1,I(0)>0\Rightarrow\lim_{t\rightarrow+\infty}\left(S(t),E(t),I(t),R(t)% \right)=EE.
  61. β ( t ) \beta(t)
  62. d E 1 d t = β ( t ) I 1 - ( γ + a ) E 1 \frac{dE_{1}}{dt}=\beta(t)I_{1}-(\gamma+a)E_{1}
  63. d I 1 d t = a E 1 - ( γ + μ ) I 1 \frac{dI_{1}}{dt}=aE_{1}-(\gamma+\mu)I_{1}
  64. P ( 0 , 1 ) P\in(0,1)
  65. d S d t = μ N ( 1 - P ) - μ S - β I N S \frac{dS}{dt}=\mu N(1-P)-\mu S-\beta\frac{I}{N}S
  66. d I d t = β I N S - ( μ + γ ) I \frac{dI}{dt}=\beta\frac{I}{N}S-(\mu+\gamma)I
  67. d V d t = μ N P - μ V \frac{dV}{dt}=\mu NP-\mu V
  68. lim t + V ( t ) = N P , \lim_{t\rightarrow+\infty}V(t)=NP,
  69. R 0 ( 1 - P ) 1 lim t + ( S ( t ) , I ( t ) ) = D F E = ( N ( 1 - P ) , 0 ) R_{0}(1-P)\leq 1\Rightarrow\lim_{t\rightarrow+\infty}\left(S(t),I(t)\right)=% DFE=\left(N\left(1-P\right),0\right)
  70. R 0 ( 1 - P ) > 1 , I ( 0 ) > 0 lim t + ( S ( t ) , I ( t ) ) = E E = ( N R 0 ( 1 - P ) , N ( R 0 ( 1 - P ) - 1 ) ) . R_{0}(1-P)>1,I(0)>0\Rightarrow\lim_{t\rightarrow+\infty}\left(S(t),I(t)\right)% =EE=\left(\frac{N}{R_{0}(1-P)},N\left(R_{0}(1-P)-1\right)\right).
  71. P P * = 1 - 1 R 0 P\geq P^{*}=1-\frac{1}{R_{0}}
  72. P = P ( I ) , P ( I ) > 0. P=P(I),P^{\prime}(I)>0.
  73. P ( 0 ) P * , P(0)\geq P^{*},
  74. d S d t = μ N ( 1 - P ) - μ S - ρ S - β I N S \frac{dS}{dt}=\mu N(1-P)-\mu S-\rho S-\beta\frac{I}{N}S
  75. d V d t = μ N P + ρ S - μ V \frac{dV}{dt}=\mu NP+\rho S-\mu V
  76. P 1 - ( 1 + ρ μ ) 1 R 0 P\geq 1-\left(1+\frac{\rho}{\mu}\right)\frac{1}{R_{0}}
  77. d S d t = μ N - μ S - β I N S , S ( n T + ) = ( 1 - p ) S ( n T - ) n = 0 , 1 , 2 , \frac{dS}{dt}=\mu N-\mu S-\beta\frac{I}{N}S,S(nT^{+})=(1-p)S(nT^{-})n=0,1,2,\dots
  78. d V d t = - μ V , V ( n T + ) = V ( n T - ) + p S ( n T - ) n = 0 , 1 , 2 , \frac{dV}{dt}=-\mu V,V(nT^{+})=V(nT^{-})+pS(nT^{-})n=0,1,2,\dots
  79. I = 0 I=0
  80. S * ( t ) = 1 - p 1 - ( 1 - p ) E - μ T E - μ M O D ( t , T ) S^{*}(t)=1-\frac{p}{1-(1-p)E^{-\mu T}}E^{-\mu MOD(t,T)}
  81. R 0 0 T S * ( t ) d t < 1 R_{0}\int_{0}^{T}{S^{*}(t)dt}<1
  82. s ( t , a ) , i ( t , a ) , r ( t , a ) s(t,a),i(t,a),r(t,a)
  83. S ( t ) = 0 a M s ( t , a ) d a S(t)=\int_{0}^{a_{M}}{s(t,a)da}
  84. I ( t ) = 0 a M i ( t , a ) d a I(t)=\int_{0}^{a_{M}}{i(t,a)da}
  85. R ( t ) = 0 a M r ( t , a ) d a R(t)=\int_{0}^{a_{M}}{r(t,a)da}
  86. a M + a_{M}\leq+\infty
  87. t s ( t , a ) + a s ( t , a ) = - μ ( a ) s ( a , t ) - s ( a , t ) 0 a M k ( a , a 1 ; t ) i ( a 1 , t ) d a 1 \partial_{t}s(t,a)+\partial_{a}s(t,a)=-\mu(a)s(a,t)-s(a,t)\int_{0}^{a_{M}}{k(a% ,a_{1};t)i(a_{1},t)da_{1}}
  88. t i ( t , a ) + a i ( t , a ) = s ( a , t ) 0 a M k ( a , a 1 ; t ) i ( a 1 , t ) d a 1 - μ ( a ) i ( a , t ) - γ ( a ) i ( a , t ) \partial_{t}i(t,a)+\partial_{a}i(t,a)=s(a,t)\int_{0}^{a_{M}}{k(a,a_{1};t)i(a_{% 1},t)da_{1}}-\mu(a)i(a,t)-\gamma(a)i(a,t)
  89. t r ( t , a ) + a r ( t , a ) = - μ ( a ) r ( a , t ) + γ ( a ) i ( a , t ) \partial_{t}r(t,a)+\partial_{a}r(t,a)=-\mu(a)r(a,t)+\gamma(a)i(a,t)
  90. F ( a , t , i ( , ) ) = 0 a M k ( a , a 1 ; t ) i ( a 1 , t ) d a 1 F(a,t,i(\cdot,\cdot))=\int_{0}^{a_{M}}{k(a,a_{1};t)i(a_{1},t)da_{1}}
  91. k ( a , a 1 ; t ) k(a,a_{1};t)
  92. i ( t , 0 ) = r ( t , 0 ) = 0 i(t,0)=r(t,0)=0
  93. s ( t , 0 ) = 0 a M ( φ s ( a ) s ( a , t ) + φ i ( a ) i ( a , t ) + φ r ( a ) r ( a , t ) ) d a s(t,0)=\int_{0}^{a_{M}}{\left(\varphi_{s}(a)s(a,t)+\varphi_{i}(a)i(a,t)+% \varphi_{r}(a)r(a,t)\right)da}
  94. φ j ( a ) , j = s , i , r \varphi_{j}(a),j=s,i,r
  95. n ( t , a ) = s ( t , a ) + i ( t , a ) + r ( t , a ) n(t,a)=s(t,a)+i(t,a)+r(t,a)
  96. t n ( t , a ) + a n ( t , a ) = - μ ( a ) n ( a , t ) \partial_{t}n(t,a)+\partial_{a}n(t,a)=-\mu(a)n(a,t)
  97. φ ( . ) \varphi(.)
  98. μ ( a ) \mu(a)
  99. 1 = 0 a M φ ( a ) exp ( - 0 a μ ( q ) d q ) d a 1=\int_{0}^{a_{M}}{\varphi(a)\exp\left(-\int_{0}^{a}{\mu(q)dq}\right)da}
  100. n * ( a ) = C exp ( - 0 a μ ( q ) d q ) , n^{*}(a)=C\exp\left(-\int_{0}^{a}{\mu(q)dq}\right),
  101. D F S ( a ) = ( n * ( a ) , 0 , 0 ) . DFS(a)=\left(n^{*}(a),0,0\right).

Competitiveness.html

  1. T C I = FX Earnings - FX Expenses FX Earnings + FX Expenses TCI=\frac{\textrm{FX Earnings}-\textrm{FX Expenses}}{\textrm{FX Earnings}+% \textrm{FX Expenses}}

Complete_coloring.html

  1. G = ( V , E ) G=(V,E)
  2. k k
  3. V V
  4. k k
  5. V 1 , V 2 , , V k V_{1},V_{2},\ldots,V_{k}
  6. V i V_{i}
  7. G G
  8. V i , V j , V i V j V_{i},V_{j},V_{i}\cup V_{j}
  9. O ( | V | / log | V | ) O\left(|V|/\sqrt{\log|V|}\right)
  10. n 2 n \sqrt{n2^{n}}

Complete_group.html

  1. 1 N G G 1 1\rightarrow N\rightarrow G\rightarrow G^{\prime}\rightarrow 1

Complete_Heyting_algebra.html

  1. x s S s = s S ( x s ) . x\wedge\bigvee_{s\in S}s=\bigvee_{s\in S}(x\wedge s).
  2. x ( y z ) = ( x y ) ( x z ) x\wedge(y\vee z)=(x\wedge y)\vee(x\wedge z)
  3. f : X Y f\colon X\to Y
  4. f - 1 : P ( Y ) P ( X ) f^{-1}\colon P(Y)\to P(X)
  5. f - 1 : O ( Y ) O ( X ) f^{-1}\colon O(Y)\to O(X)
  6. f : X Y f\colon X\to Y
  7. O ( f ) : O ( X ) O ( Y ) O(f)\colon O(X)\to O(Y)
  8. f - 1 : O ( Y ) O ( X ) . f^{-1}\colon O(Y)\to O(X).
  9. f : A B f\colon A\to B
  10. f * : B A f^{*}\colon B\to A
  11. f : A B f\colon A\to B
  12. S ( A ) S ( B ) , S(A)\to S(B),
  13. p : P ( 1 ) A p\colon P(1)\to A
  14. p * : A P ( 1 ) . p^{*}\colon A\to P(1).
  15. f : A B f\colon A\to B
  16. S ( f ) : S ( A ) S ( B ) . S(f)\colon S(A)\to S(B).
  17. S S

Completely_multiplicative_function.html

  1. ( + , ) (\mathbb{Z}^{+},\cdot)
  2. μ f \mu f
  3. μ \mu
  4. f ( g * h ) = ( f g ) * ( f h ) f\cdot(g*h)=(f\cdot g)*(f\cdot h)
  5. \cdot
  6. f * f = τ f f*f=\tau\cdot f
  7. g = h = 1 g=h=1
  8. 1 ( n ) = 1 1(n)=1
  9. τ \tau
  10. f ( g * h ) ( n ) = f ( n ) d | n g ( d ) h ( n d ) = = d | n f ( n ) ( g ( d ) h ( n d ) ) = = d | n ( f ( d ) f ( n d ) ) ( g ( d ) h ( n d ) ) (since f is completely multiplicative) = = d | n ( f ( d ) g ( d ) ) ( f ( n d ) h ( n d ) ) = ( f g ) * ( f h ) . \begin{aligned}\displaystyle f\cdot\left(g*h\right)(n)&\displaystyle=f(n)\cdot% \sum_{d|n}g(d)h\left(\frac{n}{d}\right)=\\ &\displaystyle=\sum_{d|n}f(n)\cdot(g(d)h\left(\frac{n}{d}\right))=\\ &\displaystyle=\sum_{d|n}(f(d)f\left(\frac{n}{d}\right))\cdot(g(d)h\left(\frac% {n}{d}\right))\,\text{ (since }f\,\text{ is completely multiplicative) }=\\ &\displaystyle=\sum_{d|n}(f(d)g(d))\cdot(f\left(\frac{n}{d}\right)h\left(\frac% {n}{d}\right))\\ &\displaystyle=(f\cdot g)*(f\cdot h).\end{aligned}
  11. L ( s , a ) = n = 1 a ( n ) n s = p ( 1 - a ( p ) p s ) - 1 , L(s,a)=\sum^{\infty}_{n=1}\frac{a(n)}{n^{s}}=\prod_{p}\biggl(1-\frac{a(p)}{p^{% s}}\biggr)^{-1},

Complex_cobordism.html

  1. Σ - 2 𝐏 \Sigma^{\infty-2}\mathbb{C}\mathbf{P}^{\infty}
  2. 𝐏 \mathbb{C}\mathbf{P}^{\infty}
  3. π * R MU * ( R ) \pi_{*}R\to\operatorname{MU}_{*}(R)
  4. π n S \pi_{n}S
  5. π n S \pi_{n}S
  6. MU * ( S ) L \operatorname{MU}_{*}(S)\simeq L
  7. E * ( 𝐂𝐏 ) = E * ( point ) [ [ x ] ] E^{*}(\mathbf{CP}^{\infty})=E^{*}(\,\text{point})[[x]]
  8. E * ( 𝐂𝐏 ) × E * ( 𝐂𝐏 ) = E * ( point ) [ [ x 1 , 1 x ] ] E^{*}(\mathbf{CP}^{\infty})\times E^{*}(\mathbf{CP}^{\infty})=E^{*}(\,\text{% point})[[x\otimes 1,1\otimes x]]
  9. ψ ( b k ) = i + j = k ( b ) 2 i j + 1 b j \psi(b_{k})=\sum_{i+j=k}(b)_{2i}^{j+1}\otimes b_{j}
  10. x x + b 1 x 2 + b 2 x 3 + x\rightarrow x+b_{1}x^{2}+b_{2}x^{3}+\cdots

Complex_Mexican_hat_wavelet.html

  1. Ψ ^ ( ω ) = { 2 2 3 π - 1 / 4 ω 2 e - 1 2 ω 2 ω 0 0 ω 0. \hat{\Psi}(\omega)=\begin{cases}2\sqrt{\frac{2}{3}}\pi^{-1/4}\omega^{2}e^{-% \frac{1}{2}\omega^{2}}&\omega\geq 0\\ 0&\omega\leq 0.\end{cases}
  2. Ψ ( t ) = 2 3 π - 1 4 ( π ( 1 - t 2 ) e - 1 2 t 2 - ( 2 i t + π erf [ i 2 t ] ( 1 - t 2 ) e - 1 2 t 2 ) ) . \Psi(t)=\frac{2}{\sqrt{3}}\pi^{-\frac{1}{4}}\left(\sqrt{\pi}(1-t^{2})e^{-\frac% {1}{2}t^{2}}-\left(\sqrt{2}it+\sqrt{\pi}\operatorname{erf}\left[\frac{i}{\sqrt% {2}}t\right]\left(1-t^{2}\right)e^{-\frac{1}{2}t^{2}}\right)\right).
  3. O ( | t | - 3 ) O(|t|^{-3})
  4. | Ψ ( t ) | |\Psi(t)|
  5. Ψ ^ ( ω ) \hat{\Psi}(\omega)
  6. ω = 0 \omega=0

Complex_polygon.html

  1. ( a + i b ) (a+ib)
  2. a a
  3. b b
  4. i i
  5. - 1. -1.
  6. x x
  7. y , y,
  8. i x ix
  9. i y iy

Complex_projective_plane.html

  1. ( z 1 , z 2 , z 3 ) 𝐂 3 , ( z 1 , z 2 , z 3 ) ( 0 , 0 , 0 ) (z_{1},z_{2},z_{3})\in\mathbf{C}^{3},\qquad(z_{1},z_{2},z_{3})\neq(0,0,0)
  2. ( z 1 , z 2 , z 3 ) ( λ z 1 , λ z 2 , λ z 3 ) ; λ 𝐂 , λ 0. (z_{1},z_{2},z_{3})\equiv(\lambda z_{1},\lambda z_{2},\lambda z_{3});\quad% \lambda\in\mathbf{C},\qquad\lambda\neq 0.
  3. π 2 = π 5 = \pi_{2}=\pi_{5}=\mathbb{Z}

Complex_projective_space.html

  1. Z = ( Z 1 , Z 2 , , Z n + 1 ) n + 1 , ( Z 1 , Z 2 , , Z n + 1 ) ( 0 , 0 , , 0 ) Z=(Z_{1},Z_{2},\ldots,Z_{n+1})\in\mathbb{C}^{n+1},\qquad(Z_{1},Z_{2},\ldots,Z_% {n+1})\neq(0,0,\ldots,0)
  2. ( Z 1 , Z 2 , , Z n + 1 ) ( λ Z 1 , λ Z 2 , , λ Z n + 1 ) ; λ , λ 0. (Z_{1},Z_{2},\ldots,Z_{n+1})\equiv(\lambda Z_{1},\lambda Z_{2},\ldots,\lambda Z% _{n+1});\quad\lambda\in\mathbb{C},\qquad\lambda\neq 0.
  3. U i = { Z Z i 0 } U_{i}=\{Z\mid Z_{i}\neq 0\}
  4. z 1 = Z 1 / Z i , z 2 = Z 2 / Z i , , z i - 1 = Z i - 1 / Z i , z i = Z i + 1 / Z i , , z n = Z n + 1 / Z i . z_{1}=Z_{1}/Z_{i},z_{2}=Z_{2}/Z_{i},\dots,z_{i-1}=Z_{i-1}/Z_{i},z_{i}=Z_{i+1}/% Z_{i},\dots,z_{n}=Z_{n+1}/Z_{i}.
  5. S 3 S 2 S^{3}\to S^{2}
  6. 𝐂𝐏 n = 𝐂𝐏 n - 1 𝐂 n . \mathbf{CP}^{n}=\mathbf{CP}^{n-1}\cup\mathbf{C}^{n}.
  7. S 1 S 2 n + 1 𝐂𝐏 n S^{1}\hookrightarrow S^{2n+1}\twoheadrightarrow\mathbf{CP}^{n}
  8. U ( 1 ) S 2 n + 1 𝐂𝐏 n U(1)\hookrightarrow S^{2n+1}\twoheadrightarrow\mathbf{CP}^{n}
  9. K 𝐂 * ( 𝐂𝐏 n ) = K 𝐂 0 ( 𝐂𝐏 n ) = 𝐙 [ H ] / ( H - 1 ) n + 1 . K_{\mathbf{C}}^{*}(\mathbf{CP}^{n})=K_{\mathbf{C}}^{0}(\mathbf{CP}^{n})=% \mathbf{Z}[H]/(H-1)^{n+1}.
  10. T 𝐂𝐏 n ϑ 1 = H n + 1 , T\mathbf{CP}^{n}\oplus\vartheta^{1}=H^{\oplus n+1},
  11. ϑ 1 \vartheta^{1}
  12. P ( 1 × U ( n ) ) PU ( n ) . \mathrm{P}(1\times\mathrm{U}(n))\cong\mathrm{PU}(n).
  13. U ( n + 1 ) / ( U ( 1 ) × U ( n ) ) S U ( n + 1 ) / S ( U ( 1 ) × U ( n ) ) . U(n+1)/(U(1)\times U(n))\cong SU(n+1)/S(U(1)\times U(n)).
  14. S = n = 0 S n . S=\bigoplus_{n=0}^{\infty}S_{n}.
  15. n > 0 S n . \bigoplus_{n>0}S_{n}.
  16. V ( I ) = { p Proj S p I } V(I)=\{p\in\operatorname{Proj}S\mid p\supseteq I\}
  17. f ( λ z ) = λ k f ( z ) f(\lambda z)=\lambda^{k}f(z)
  18. 𝐂 n + 1 × 𝐂𝐏 n 𝐂𝐏 n \mathbf{C}^{n+1}\times\mathbf{CP}^{n}\to\mathbf{CP}^{n}
  19. { ( x , L ) x L } . \{(x,L)\mid x\in L\}.
  20. O ( k H ) O ( k ) . O(kH)\cong O(k).

Complexification.html

  1. V = V . V^{\mathbb{C}}=V\otimes_{\mathbb{R}}\mathbb{C}.
  2. α ( v β ) = v ( α β ) for all v V and α , β . \alpha(v\otimes\beta)=v\otimes(\alpha\beta)\qquad\mbox{for all }~{}v\in V\mbox% { and }~{}\alpha,\beta\in\mathbb{C}.
  3. v = v 1 1 + v 2 i v=v_{1}\otimes 1+v_{2}\otimes i
  4. v = v 1 + i v 2 . v=v_{1}+iv_{2}.\,
  5. ( a + i b ) ( v 1 + i v 2 ) = ( a v 1 - b v 2 ) + i ( b v 1 + a v 2 ) . (a+ib)(v_{1}+iv_{2})=(av_{1}-bv_{2})+i(bv_{1}+av_{2}).\,
  6. V V i V V^{\mathbb{C}}\cong V\oplus iV
  7. v v 1. v\mapsto v\otimes 1.
  8. dim V = dim V . \dim_{\mathbb{C}}V^{\mathbb{C}}=\dim_{\mathbb{R}}V.
  9. V := V V , V^{\mathbb{C}}:=V\oplus V,
  10. V V^{\mathbb{C}}
  11. J ( v , w ) := ( - w , v ) , J(v,w):=(-w,v),
  12. J = [ 0 - I V I V 0 ] . J=\begin{bmatrix}0&-I_{V}\\ I_{V}&0\end{bmatrix}.
  13. V V^{\mathbb{C}}
  14. V J V V\oplus JV
  15. V i V , V\oplus iV,
  16. χ : V V ¯ \chi:V^{\mathbb{C}}\to\overline{V^{\mathbb{C}}}
  17. χ ( v z ) = v z ¯ . \chi(v\otimes z)=v\otimes\bar{z}.
  18. V ¯ \overline{V^{\mathbb{C}}}
  19. V = { w W : χ ( w ) = w } . V=\{w\in W:\chi(w)=w\}.
  20. χ ( z 1 , , z n ) = ( z ¯ 1 , , z ¯ n ) \chi(z_{1},\ldots,z_{n})=(\bar{z}_{1},\ldots,\bar{z}_{n})
  21. f : V W f^{\mathbb{C}}:V^{\mathbb{C}}\to W^{\mathbb{C}}
  22. f ( v z ) = f ( v ) z . f^{\mathbb{C}}(v\otimes z)=f(v)\otimes z.
  23. ( id V ) = id V (\mathrm{id}_{V})^{\mathbb{C}}=\mathrm{id}_{V^{\mathbb{C}}}
  24. ( f g ) = f g (f\circ g)^{\mathbb{C}}=f^{\mathbb{C}}\circ g^{\mathbb{C}}
  25. ( f + g ) = f + g (f+g)^{\mathbb{C}}=f^{\mathbb{C}}+g^{\mathbb{C}}
  26. ( a f ) = a f a (af)^{\mathbb{C}}=af^{\mathbb{C}}\quad\forall a\in\mathbb{R}
  27. ( V * ) = V * Hom ( V , ) . (V^{*})^{\mathbb{C}}=V^{*}\otimes\mathbb{C}\cong\mathrm{Hom}_{\mathbb{R}}(V,% \mathbb{C}).
  28. ( φ 1 1 + φ 2 i ) φ 1 + i φ 2 (\varphi_{1}\otimes 1+\varphi_{2}\otimes i)\leftrightarrow\varphi_{1}+i\varphi% _{2}
  29. φ 1 + i φ 2 ¯ = φ 1 - i φ 2 \overline{\varphi_{1}+i\varphi_{2}}=\varphi_{1}-i\varphi_{2}
  30. φ ( v z ) = z φ ( v ) . \varphi(v\otimes z)=z\varphi(v).
  31. ( V * ) ( V ) * . (V^{*})^{\mathbb{C}}\cong(V^{\mathbb{C}})^{*}.
  32. Hom ( V , W ) Hom ( V , W ) . \mathrm{Hom}_{\mathbb{R}}(V,W)^{\mathbb{C}}\cong\mathrm{Hom}_{\mathbb{C}}(V^{% \mathbb{C}},W^{\mathbb{C}}).
  33. ( V W ) V W . (V\otimes_{\mathbb{R}}W)^{\mathbb{C}}\cong V^{\mathbb{C}}\otimes_{\mathbb{C}}W% ^{\mathbb{C}}.
  34. ( Λ k V ) Λ k ( V ) . (\Lambda_{\mathbb{R}}^{k}V)^{\mathbb{C}}\cong\Lambda_{\mathbb{C}}^{k}(V^{% \mathbb{C}}).

Compliance_(physiology).html

  1. C = Δ V Δ P C=\frac{\Delta V}{\Delta P}

Compound_Poisson_distribution.html

  1. N Poisson ( λ ) , N\sim\operatorname{Poisson}(\lambda),
  2. X 1 , X 2 , X 3 , X_{1},X_{2},X_{3},\dots
  3. N N
  4. N N
  5. Y N = n = 1 N X n Y\mid N=\sum_{n=1}^{N}X_{n}
  6. E Y ( Y ) = E N [ E Y N ( Y ) ] = E N [ N E X ( X ) ] = E N ( N ) E X ( X ) , \operatorname{E}_{Y}(Y)=\operatorname{E}_{N}\left[\operatorname{E}_{Y\mid N}(Y% )\right]=\operatorname{E}_{N}\left[N\operatorname{E}_{X}(X)\right]=% \operatorname{E}_{N}(N)\operatorname{E}_{X}(X),
  7. Var Y ( Y ) = E N [ Var Y N ( Y ) ] + Var N [ E Y N ( Y ) ] = E N [ N Var X ( X ) ] + Var N [ N E X ( X ) ] , \operatorname{Var}_{Y}(Y)=E_{N}\left[\operatorname{Var}_{Y\mid N}(Y)\right]+% \operatorname{Var}_{N}\left[E_{Y\mid N}(Y)\right]=\operatorname{E}_{N}\left[N% \operatorname{Var}_{X}(X)\right]+\operatorname{Var}_{N}\left[N\operatorname{E}% _{X}(X)\right],
  8. Var Y ( Y ) = E N ( N ) Var X ( X ) + ( E X ( X ) ) 2 Var N ( N ) . \operatorname{Var}_{Y}(Y)=\operatorname{E}_{N}(N)\operatorname{Var}_{X}(X)+% \left(\operatorname{E}_{X}(X)\right)^{2}\operatorname{Var}_{N}(N).
  9. E ( Y ) = E ( N ) E ( X ) , \operatorname{E}(Y)=\operatorname{E}(N)\operatorname{E}(X),
  10. Var ( Y ) = E ( N ) ( Var ( X ) + E ( X ) 2 ) = E ( N ) E ( X 2 ) . \operatorname{Var}(Y)=E(N)(\operatorname{Var}(X)+{E(X)}^{2})=E(N){E(X^{2})}.
  11. φ Y ( t ) = E ( e i t Y ) = E N ( ( E ( e i t X ) ) N ) = E N ( ( φ X ( t ) ) N ) , \varphi_{Y}(t)=\operatorname{E}(e^{itY})=\operatorname{E}_{N}(\left(% \operatorname{E}(e^{itX}))^{N}\right)=\operatorname{E}_{N}((\varphi_{X}(t))^{N% }),\,
  12. φ Y ( t ) = e λ ( φ X ( t ) - 1 ) . \varphi_{Y}(t)=\textrm{e}^{\lambda(\varphi_{X}(t)-1)}.\,
  13. K Y ( t ) = ln E [ e t Y ] = ln E [ E [ e t Y N ] ] = ln E [ e N K X ( t ) ] = K N ( K X ( t ) ) . K_{Y}(t)=\ln E[e^{tY}]=\ln E[E[e^{tY}\mid N]]=\ln E[e^{NK_{X}(t)}]=K_{N}(K_{X}% (t)).\,
  14. X 1 , X 2 , X 3 , X_{1},X_{2},X_{3},\dots
  15. P ( X 1 = k ) = α k , ( k = 0 , 1 , ) P(X_{1}=k)=\alpha_{k},\ (k=0,1,\ldots)
  16. Y Y
  17. P Y ( z ) = i = 0 P ( Y = i ) z i = exp ( k = 1 α k λ ( z k - 1 ) ) , ( | z | 1 ) P_{Y}(z)=\sum\limits_{i=0}^{\infty}P(Y=i)z^{i}=\exp\left(\sum\limits_{k=1}^{% \infty}\alpha_{k}\lambda(z^{k}-1)\right),\quad(|z|\leq 1)
  18. ( α 1 λ , α 2 λ , ) ( i = 1 α i = 1 , α i 0 , λ > 0 ) (\alpha_{1}\lambda,\alpha_{2}\lambda,\ldots)\in\mathbb{R}^{\infty}\left(\sum_{% i=1}^{\infty}\alpha_{i}=1,\alpha_{i}\geq 0,\lambda>0\right)
  19. X C P ( α 1 λ , α 2 λ , ) . X\sim CP(\alpha_{1}\lambda,\alpha_{2}\lambda,\ldots).
  20. X C P ( α 1 λ , α 2 λ , , α r λ ) X\sim CP(\alpha_{1}\lambda,\alpha_{2}\lambda,\ldots,\alpha_{r}\lambda)
  21. X X
  22. r r
  23. r = 1 , 2 r=1,2
  24. r = 3 , 4 r=3,4
  25. X X
  26. α k \alpha_{k}
  27. λ > 0 \lambda>0
  28. { Y ( t ) : t 0 } \{\,Y(t):t\geq 0\,\}
  29. Y ( t ) = i = 1 N ( t ) D i , Y(t)=\sum_{i=1}^{N(t)}D_{i},
  30. { N ( t ) : t 0 } \{\,N(t):t\geq 0\,\}
  31. λ \lambda
  32. { D i : i 1 } \{\,D_{i}:i\geq 1\,\}
  33. { N ( t ) : t 0 } . \{\,N(t):t\geq 0\,\}.\,

Compound_Poisson_process.html

  1. λ > 0 \lambda>0
  2. { Y ( t ) : t 0 } \{\,Y(t):t\geq 0\,\}
  3. Y ( t ) = i = 1 N ( t ) D i Y(t)=\sum_{i=1}^{N(t)}D_{i}
  4. { N ( t ) : t 0 } \{\,N(t):t\geq 0\,\}
  5. λ \lambda
  6. { D i : i 1 } \{\,D_{i}:i\geq 1\,\}
  7. { N ( t ) : t 0 } . \{\,N(t):t\geq 0\,\}.\,
  8. D i D_{i}
  9. E ( Y ( t ) ) = E ( E ( Y ( t ) | N ( t ) ) ) = E ( N ( t ) E ( D ) ) = E ( N ( t ) ) E ( D ) = λ t E ( D ) . \,E(Y(t))=E(E(Y(t)|N(t)))=E(N(t)E(D))=E(N(t))E(D)=\lambda tE(D).
  10. var ( Y ( t ) ) \displaystyle\operatorname{var}(Y(t))
  11. Pr ( Y ( t ) = i ) = n Pr ( Y ( t ) = i | N ( t ) = n ) Pr ( N ( t ) = n ) \,\Pr(Y(t)=i)=\sum_{n}\Pr(Y(t)=i|N(t)=n)\Pr(N(t)=n)
  12. E ( e s Y ) = i e s i Pr ( Y ( t ) = i ) = i e s i n Pr ( Y ( t ) = i | N ( t ) = n ) Pr ( N ( t ) = n ) = n Pr ( N ( t ) = n ) i e s i Pr ( Y ( t ) = i | N ( t ) = n ) = n Pr ( N ( t ) = n ) i e s i Pr ( D 1 + D 2 + + D n = i ) = n Pr ( N ( t ) = n ) M D ( s ) n = n Pr ( N ( t ) = n ) e n ln ( M D ( s ) ) = M N ( t ) ( ln ( M D ( s ) ) ) = e λ t ( M D ( s ) - 1 ) . \begin{aligned}\displaystyle E(e^{sY})&\displaystyle=\sum_{i}e^{si}\Pr(Y(t)=i)% \\ &\displaystyle=\sum_{i}e^{si}\sum_{n}\Pr(Y(t)=i|N(t)=n)\Pr(N(t)=n)\\ &\displaystyle=\sum_{n}\Pr(N(t)=n)\sum_{i}e^{si}\Pr(Y(t)=i|N(t)=n)\\ &\displaystyle=\sum_{n}\Pr(N(t)=n)\sum_{i}e^{si}\Pr(D_{1}+D_{2}+\cdots+D_{n}=i% )\\ &\displaystyle=\sum_{n}\Pr(N(t)=n)M_{D}(s)^{n}\\ &\displaystyle=\sum_{n}\Pr(N(t)=n)e^{n\ln(M_{D}(s))}\\ &\displaystyle=M_{N(t)}(\ln(M_{D}(s)))\\ &\displaystyle=e^{\lambda t\left(M_{D}(s)-1\right)}.\end{aligned}
  13. μ ( A ) = Pr ( D A ) . \mu(A)=\Pr(D\in A).\,
  14. exp ( λ t ( μ - δ 0 ) ) \exp(\lambda t(\mu-\delta_{0}))\,
  15. exp ( ν ) = n = 0 ν * n n ! \exp(\nu)=\sum_{n=0}^{\infty}{\nu^{*n}\over n!}
  16. ν * n = ν * * ν n factors \nu^{*n}=\underbrace{\nu*\cdots*\nu}_{n\,\text{ factors}}

Compression_(functional_analysis).html

  1. P K T | K : K K P_{K}T|_{K}:K\rightarrow K
  2. P K : H K P_{K}:H\rightarrow K
  3. H H
  4. W W
  5. H H
  6. W W
  7. T W = V * T V : W W T_{W}=V^{*}TV:W\rightarrow W
  8. V * V^{*}
  9. T W T_{W}
  10. I : W - > H I:W->H
  11. V * = I * = P K : H - > W V^{*}=I^{*}=P_{K}:H->W

Computable_function.html

  1. f : k f:\mathbb{N}^{k}\rightarrow\mathbb{N}
  2. k k
  3. 𝐱 \mathbf{x}
  4. f ( 𝐱 ) f(\mathbf{x})
  5. f : k f:\mathbb{N}^{k}\rightarrow\mathbb{N}
  6. f ( 𝐱 ) f(\mathbf{x})
  7. 𝐱 \mathbf{x}
  8. f ( 𝐱 ) f(\mathbf{x})
  9. f ( 𝐱 ) f(\mathbf{x})
  10. 𝐱 \mathbf{x}
  11. f f
  12. f f
  13. f f
  14. f ( 0 ) , f ( 1 ) , f(0),f(1),...
  15. f f
  16. f f
  17. f g f\circ g

Computation_tree_logic.html

  1. ϕ : := | | p | ( ¬ ϕ ) | ( ϕ and ϕ ) | ( ϕ ϕ ) | ( ϕ ϕ ) | ( ϕ ϕ ) | AX ϕ | EX ϕ | AF ϕ | EF ϕ | AG ϕ | EG ϕ | A [ ϕ U ϕ ] | E [ ϕ U ϕ ] \phi::=\bot|\top|p|(\neg\phi)|(\phi\and\phi)|(\phi\phi)|(\phi\Rightarrow\phi)|% (\phi\Leftrightarrow\phi)|\mbox{AX }~{}\phi|\mbox{EX }~{}\phi|\mbox{AF }~{}% \phi|\mbox{EF }~{}\phi|\mbox{AG }~{}\phi|\mbox{EG }~{}\phi|\mbox{A }~{}[\phi% \mbox{ U }~{}\phi]|\mbox{E }~{}[\phi\mbox{ U }~{}\phi]
  2. p p
  3. { ¬ , and , AX , AU , EU } \{\neg,\and,\mbox{AX}~{},\mbox{AU}~{},\mbox{EU}~{}\}
  4. A \mbox{A}~{}
  5. E \mbox{E}~{}
  6. EF ( EG p AF r \mbox{EF }~{}(\mbox{EG }~{}p\Rightarrow\mbox{AF }~{}r
  7. EF ( r U q ) \mbox{EF }~{}\big(r\mbox{ U }~{}q\big)
  8. U U
  9. A A
  10. E E
  11. ¬ , , and , \neg,,\and,\Rightarrow
  12. \Leftrightarrow
  13. ϕ \phi
  14. ϕ \phi
  15. ϕ \phi
  16. ϕ \phi
  17. ϕ \phi
  18. ϕ \phi
  19. ϕ \phi
  20. ϕ \phi
  21. ϕ \phi
  22. ϕ \phi
  23. ϕ \phi
  24. ψ \psi
  25. ϕ \phi
  26. ψ \psi
  27. ψ \psi
  28. ϕ \phi
  29. ψ \psi
  30. ϕ \phi
  31. ψ \psi
  32. ψ \psi
  33. , ¬ ,\neg
  34. ϕ \phi
  35. ϕ \phi
  36. ϕ \phi
  37. ϕ \phi
  38. ϕ \phi
  39. ¬ \neg
  40. ¬ \neg
  41. ϕ \phi
  42. ϕ \phi
  43. ¬ \neg
  44. ¬ \neg
  45. ϕ \phi
  46. ¬ \neg
  47. ¬ \neg
  48. ϕ \phi
  49. ϕ \phi
  50. ϕ \phi
  51. ¬ \neg
  52. ¬ \neg
  53. ϕ \phi
  54. ϕ \phi
  55. ψ \psi
  56. ¬ \neg
  57. ¬ \neg
  58. ψ \psi
  59. ¬ \neg
  60. ϕ \phi
  61. $\or$
  62. ψ \psi
  63. $\or$
  64. ¬ \neg
  65. ψ \psi
  66. = ( S , , L ) \mathcal{M}=(S,\rightarrow,L)
  67. S S
  68. S × S \rightarrow\subseteq S\times S
  69. L L
  70. = ( S , , L ) \mathcal{M}=(S,\rightarrow,L)
  71. s S , ϕ F s\in S,\phi\in F
  72. \mathcal{M}
  73. ( , s ϕ ) (\mathcal{M},s\models\phi)
  74. ϕ \phi
  75. ( ( , s ) ) ( ( , s ) ⊧̸ ) \Big((\mathcal{M},s)\models\top\Big)\Leftrightarrow\Big((\mathcal{M},s)\not% \models\bot\Big)
  76. ( ( , s ) p ) ( p L ( s ) ) \Big((\mathcal{M},s)\models p\Big)\Leftrightarrow\Big(p\in L(s)\Big)
  77. ( ( , s ) ¬ ϕ ) ( ( , s ) ⊧̸ ϕ ) \Big((\mathcal{M},s)\models\neg\phi\Big)\Leftrightarrow\Big((\mathcal{M},s)% \not\models\phi\Big)
  78. ( ( , s ) ϕ 1 ϕ 2 ) ( ( ( , s ) ϕ 1 ) ( ( , s ) ϕ 2 ) ) \Big((\mathcal{M},s)\models\phi_{1}\land\phi_{2}\Big)\Leftrightarrow\Big(\big(% (\mathcal{M},s)\models\phi_{1}\big)\land\big((\mathcal{M},s)\models\phi_{2}% \big)\Big)
  79. ( ( , s ) ϕ 1 ϕ 2 ) ( ( ( , s ) ϕ 1 ) ( ( , s ) ϕ 2 ) ) \Big((\mathcal{M},s)\models\phi_{1}\lor\phi_{2}\Big)\Leftrightarrow\Big(\big((% \mathcal{M},s)\models\phi_{1}\big)\lor\big((\mathcal{M},s)\models\phi_{2}\big)\Big)
  80. ( ( , s ) ϕ 1 ϕ 2 ) ( ( ( , s ) ⊧̸ ϕ 1 ) ( ( , s ) ϕ 2 ) ) \Big((\mathcal{M},s)\models\phi_{1}\Rightarrow\phi_{2}\Big)\Leftrightarrow\Big% (\big((\mathcal{M},s)\not\models\phi_{1}\big)\lor\big((\mathcal{M},s)\models% \phi_{2}\big)\Big)
  81. ( ( , s ) ϕ 1 ϕ 2 ) ( ( ( ( , s ) ϕ 1 ) ( ( , s ) ϕ 2 ) ) ( ¬ ( ( , s ) ϕ 1 ) ¬ ( ( , s ) ϕ 2 ) ) ) \bigg((\mathcal{M},s)\models\phi_{1}\Leftrightarrow\phi_{2}\bigg)% \Leftrightarrow\bigg(\Big(\big((\mathcal{M},s)\models\phi_{1}\big)\land\big((% \mathcal{M},s)\models\phi_{2}\big)\Big)\lor\Big(\neg\big((\mathcal{M},s)% \models\phi_{1}\big)\land\neg\big((\mathcal{M},s)\models\phi_{2}\big)\Big)\bigg)
  82. ( ( , s ) A X ϕ ) ( s s 1 ( ( , s 1 ) ϕ ) ) \Big((\mathcal{M},s)\models AX\phi\Big)\Leftrightarrow\Big(\forall\langle s% \rightarrow s_{1}\rangle\big((\mathcal{M},s_{1})\models\phi\big)\Big)
  83. ( ( , s ) E X ϕ ) ( s s 1 ( ( , s 1 ) ϕ ) ) \Big((\mathcal{M},s)\models EX\phi\Big)\Leftrightarrow\Big(\exists\langle s% \rightarrow s_{1}\rangle\big((\mathcal{M},s_{1})\models\phi\big)\Big)
  84. ( ( , s ) A G ϕ ) ( s 1 s 2 ( s = s 1 ) i ( ( , s i ) ϕ ) ) \Big((\mathcal{M},s)\models AG\phi\Big)\Leftrightarrow\Big(\forall\langle s_{1% }\rightarrow s_{2}\rightarrow\ldots\rangle(s=s_{1})\forall i\big((\mathcal{M},% s_{i})\models\phi\big)\Big)
  85. ( ( , s ) E G ϕ ) ( s 1 s 2 ( s = s 1 ) i ( ( , s i ) ϕ ) ) \Big((\mathcal{M},s)\models EG\phi\Big)\Leftrightarrow\Big(\exists\langle s_{1% }\rightarrow s_{2}\rightarrow\ldots\rangle(s=s_{1})\forall i\big((\mathcal{M},% s_{i})\models\phi\big)\Big)
  86. ( ( , s ) A F ϕ ) ( s 1 s 2 ( s = s 1 ) i ( ( , s i ) ϕ ) ) \Big((\mathcal{M},s)\models AF\phi\Big)\Leftrightarrow\Big(\forall\langle s_{1% }\rightarrow s_{2}\rightarrow\ldots\rangle(s=s_{1})\exists i\big((\mathcal{M},% s_{i})\models\phi\big)\Big)
  87. ( ( , s ) E F ϕ ) ( s 1 s 2 ( s = s 1 ) i ( ( , s i ) ϕ ) ) \Big((\mathcal{M},s)\models EF\phi\Big)\Leftrightarrow\Big(\exists\langle s_{1% }\rightarrow s_{2}\rightarrow\ldots\rangle(s=s_{1})\exists i\big((\mathcal{M},% s_{i})\models\phi\big)\Big)
  88. ( ( , s ) A [ ϕ 1 U ϕ 2 ] ) ( s 1 s 2 ( s = s 1 ) i ( ( ( , s i ) ϕ 2 ) ( ( j < i ) ( , s j ) ϕ 1 ) ) ) \bigg((\mathcal{M},s)\models A[\phi_{1}U\phi_{2}]\bigg)\Leftrightarrow\bigg(% \forall\langle s_{1}\rightarrow s_{2}\rightarrow\ldots\rangle(s=s_{1})\exists i% \Big(\big((\mathcal{M},s_{i})\models\phi_{2}\big)\land\big(\forall(j<i)(% \mathcal{M},s_{j})\models\phi_{1}\big)\Big)\bigg)
  89. ( ( , s ) E [ ϕ 1 U ϕ 2 ] ) ( s 1 s 2 ( s = s 1 ) i ( ( ( , s i ) ϕ 2 ) ( ( j < i ) ( , s j ) ϕ 1 ) ) ) \bigg((\mathcal{M},s)\models E[\phi_{1}U\phi_{2}]\bigg)\Leftrightarrow\bigg(% \exists\langle s_{1}\rightarrow s_{2}\rightarrow\ldots\rangle(s=s_{1})\exists i% \Big(\big((\mathcal{M},s_{i})\models\phi_{2}\big)\land\big(\forall(j<i)(% \mathcal{M},s_{j})\models\phi_{1}\big)\Big)\bigg)
  90. s s
  91. ϕ \phi
  92. ψ \psi
  93. ϕ ψ \phi\equiv\psi
  94. ¬ A ϕ E ¬ ϕ \neg A\phi\equiv E\neg\phi
  95. ¬ A F ϕ E G ¬ ϕ \neg AF\phi\equiv EG\neg\phi
  96. ¬ E F ϕ A G ¬ ϕ \neg EF\phi\equiv AG\neg\phi
  97. ¬ A X ϕ E X ¬ ϕ \neg AX\phi\equiv EX\neg\phi
  98. E U EU
  99. { A X , E X } \{AX,EX\}
  100. { E G , A F , A U } \{EG,AF,AU\}
  101. A G ϕ ϕ A X A G ϕ AG\phi\equiv\phi\land AXAG\phi
  102. E G ϕ ϕ E X E G ϕ EG\phi\equiv\phi\land EXEG\phi
  103. A F ϕ ϕ A X A F ϕ AF\phi\equiv\phi\lor AXAF\phi
  104. E F ϕ ϕ E X E F ϕ EF\phi\equiv\phi\lor EXEF\phi
  105. A [ ϕ U ψ ] ψ ( ϕ A X A [ ϕ U ψ ] ) A[\phi U\psi]\equiv\psi\lor(\phi\land AXA[\phi U\psi])
  106. E [ ϕ U ψ ] ψ ( ϕ E X E [ ϕ U ψ ] ) E[\phi U\psi]\equiv\psi\lor(\phi\land EXE[\phi U\psi])
  107. \Rightarrow
  108. \land

Computer_cooling.html

  1. C F M = Q C p × r × D T CFM=\frac{Q}{Cp\times r\times DT}
  2. C F M = 3.16 × W allowed temperature rise in F CFM=\frac{3.16\times W}{\,\text{allowed temperature rise in}^{\circ}F}
  3. C F M = 1.76 × W allowed temperature rise in C CFM=\frac{1.76\times W}{\,\text{allowed temperature rise in}^{\circ}C}
  4. C F M = 3.16 × 500 W ( 130 - 100 ) = 53 CFM=\frac{3.16\times 500W}{(130-100)}=53

Computer_experiment.html

  1. x x
  2. f f
  3. f ( x ) f(x)
  4. x x
  5. f ( x ) f(x)
  6. f ( ) f(\cdot)
  7. f ( x ) f(x)
  8. f f
  9. f GP ( m ( ) , C ( , ) ) , f\sim\operatorname{GP}(m(\cdot),C(\cdot,\cdot)),
  10. m m
  11. C C
  12. ν = 1 / 2 \nu=1/2
  13. ν \nu\rightarrow\infty
  14. n n
  15. 𝒪 ( n 3 ) \mathcal{O}(n^{3})

Concentration_of_measure.html

  1. ( X , d , μ ) (X,d,\mu)
  2. μ ( X ) = 1 \mu(X)=1
  3. α ( ϵ ) = sup { μ ( X A ϵ ) | μ ( A ) 1 / 2 } , \alpha(\epsilon)=\sup\left\{\mu(X\setminus A_{\epsilon})\,|\,\mu(A)\geq 1/2% \right\},
  4. A ϵ = { x | d ( x , A ) < ϵ } A_{\epsilon}=\left\{x\,|\,d(x,A)<\epsilon\right\}
  5. ϵ \epsilon
  6. A A
  7. α ( ) \alpha(\cdot)
  8. X X
  9. α ( ϵ ) = sup { μ ( { F 𝑀 + ϵ } ) } , \alpha(\epsilon)=\sup\left\{\mu(\{F\geq\mathop{M}+\epsilon\})\right\},
  10. F : X F:X\to\mathbb{R}
  11. M = M e d F M=\mathop{Med}F
  12. μ { F M } 1 / 2 , μ { F M } 1 / 2. \mu\{F\geq M\}\geq 1/2,\,\mu\{F\leq M\}\geq 1/2.
  13. X X
  14. α ( ϵ ) \alpha(\epsilon)
  15. ϵ \epsilon
  16. ( X n , d n , μ n ) (X_{n},d_{n},\mu_{n})
  17. α n \alpha_{n}
  18. ϵ > 0 α n ( ϵ ) 0 as n , \forall\epsilon>0\,\,\alpha_{n}(\epsilon)\to 0{\rm\;as\;}n\to\infty,
  19. ϵ > 0 α n ( ϵ ) C exp ( - c n ϵ 2 ) \forall\epsilon>0\,\,\alpha_{n}(\epsilon)\leq C\exp(-cn\epsilon^{2})
  20. c , C > 0 c,C>0
  21. A A
  22. S n S^{n}
  23. σ n ( A ) \sigma_{n}(A)
  24. { x S n | dist ( x , x 0 ) R } , \left\{x\in S^{n}|\mathrm{dist}(x,x_{0})\leq R\right\},
  25. R R
  26. ϵ \epsilon
  27. A ϵ A_{\epsilon}
  28. ϵ > 0 \epsilon>0
  29. σ n ( A ) = 1 / 2 \sigma_{n}(A)=1/2
  30. σ n ( S n ) = 1 \sigma_{n}(S^{n})=1
  31. σ n ( A ϵ ) 1 - C exp ( - c n ϵ 2 ) \sigma_{n}(A_{\epsilon})\geq 1-C\exp(-cn\epsilon^{2})
  32. C , c C,c
  33. ( S n ) n (S^{n})_{n}
  34. f ( μ ) f_{\ast}(\mu)

Conchoid_(mathematics).html

  1. r = α ( θ ) r=\alpha(\theta)
  2. r = α ( θ ) ± d r=\alpha(\theta)\pm d
  3. x = a x=a
  4. r = a cos θ r=\frac{a}{\cos\theta}
  5. x = a ± d cos θ , y = a tan θ ± d sin θ . x=a\pm d\cos\theta,\,y=a\tan\theta\pm d\sin\theta.

Conchoid_of_de_Sluze.html

  1. r = sec θ + a cos θ r=\sec\theta+a\cos\theta\,
  2. ( x - 1 ) ( x 2 + y 2 ) = a x 2 (x-1)(x^{2}+y^{2})=ax^{2}\,
  3. | a | ( 1 + a / 4 ) π |a|(1+a/4)\pi\,
  4. a < - 1 a<-1
  5. ( 1 - a 2 ) - ( a + 1 ) - a ( 2 + a 2 ) arcsin 1 - a . \left(1-\frac{a}{2}\right)\sqrt{-(a+1)}-a\left(2+\frac{a}{2}\right)\arcsin% \frac{1}{\sqrt{-a}}.
  6. a < - 1 a<-1
  7. ( 2 + a 2 ) a arccos 1 - a + ( 1 - a 2 ) - ( a + 1 ) . \left(2+\frac{a}{2}\right)a\arccos\frac{1}{\sqrt{-a}}+\left(1-\frac{a}{2}% \right)\sqrt{-(a+1)}.

Conditional_entropy.html

  1. Y Y
  2. X X
  3. Y Y
  4. X X
  5. H ( Y | X ) H(Y|X)
  6. H ( Y | X = x ) H(Y|X=x)
  7. Y Y
  8. X X
  9. x x
  10. H ( Y | X ) H(Y|X)
  11. H ( Y | X = x ) H(Y|X=x)
  12. x x
  13. X X
  14. X X
  15. 𝒳 \mathcal{X}
  16. Y Y
  17. 𝒴 \mathcal{Y}
  18. Y Y
  19. X X
  20. H ( Y | X ) x 𝒳 p ( x ) H ( Y | X = x ) = - x 𝒳 p ( x ) y 𝒴 p ( y | x ) log p ( y | x ) = - x 𝒳 y 𝒴 p ( x , y ) log p ( y | x ) = - x 𝒳 , y 𝒴 p ( x , y ) log p ( y | x ) = - x 𝒳 , y 𝒴 p ( x , y ) log p ( x , y ) p ( x ) . = x 𝒳 , y 𝒴 p ( x , y ) log p ( x ) p ( x , y ) . \begin{aligned}\displaystyle H(Y|X)&\displaystyle\equiv\sum_{x\in\mathcal{X}}% \,p(x)\,H(Y|X=x)\\ &\displaystyle=-\sum_{x\in\mathcal{X}}p(x)\sum_{y\in\mathcal{Y}}\,p(y|x)\,\log% \,p(y|x)\\ &\displaystyle=-\sum_{x\in\mathcal{X}}\sum_{y\in\mathcal{Y}}\,p(x,y)\,\log\,p(% y|x)\\ &\displaystyle=-\sum_{x\in\mathcal{X},y\in\mathcal{Y}}p(x,y)\log\,p(y|x)\\ &\displaystyle=-\sum_{x\in\mathcal{X},y\in\mathcal{Y}}p(x,y)\log\frac{p(x,y)}{% p(x)}.\\ &\displaystyle=\sum_{x\in\mathcal{X},y\in\mathcal{Y}}p(x,y)\log\frac{p(x)}{p(x% ,y)}.\\ \end{aligned}
  21. H ( Y | X ) = 0 H(Y|X)=0
  22. Y Y
  23. X X
  24. H ( Y | X ) = H ( Y ) H(Y|X)=H(Y)
  25. Y Y
  26. X X
  27. H ( X , Y ) H(X,Y)
  28. H ( X , Y ) H(X,Y)
  29. X X
  30. H ( X ) H(X)
  31. X X
  32. H ( X , Y ) - H ( X ) H(X,Y)-H(X)
  33. H ( Y | X ) H(Y|X)
  34. H ( Y | X ) = H ( X , Y ) - H ( X ) . H(Y|X)\,=\,H(X,Y)-H(X)\,.
  35. H ( Y | X ) = x 𝒳 , y 𝒴 p ( x , y ) log p ( x ) p ( x , y ) H(Y|X)=\sum_{x\in\mathcal{X},y\in\mathcal{Y}}p(x,y)\log\frac{p(x)}{p(x,y)}
  36. = - x 𝒳 , y 𝒴 p ( x , y ) log p ( x , y ) + x 𝒳 , y 𝒴 p ( x , y ) log p ( x ) =-\sum_{x\in\mathcal{X},y\in\mathcal{Y}}p(x,y)\log\,p(x,y)+\sum_{x\in\mathcal{% X},y\in\mathcal{Y}}p(x,y)\log\,p(x)
  37. = H ( X , Y ) + x 𝒳 p ( x ) log p ( x ) =H(X,Y)+\sum_{x\in\mathcal{X}}p(x)\log\,p(x)
  38. = H ( X , Y ) - H ( X ) . =H(X,Y)-H(X).
  39. H ( Y | X ) = H ( X | Y ) - H ( X ) + H ( Y ) . H(Y|X)\,=\,H(X|Y)-H(X)+H(Y)\,.
  40. H ( Y | X ) = H ( X , Y ) - H ( X ) H(Y|X)=H(X,Y)-H(X)
  41. H ( X | Y ) = H ( Y , X ) - H ( Y ) H(X|Y)=H(Y,X)-H(Y)
  42. H ( X , Y ) = H ( Y , X ) H(X,Y)=H(Y,X)
  43. H ( X , Y ) H ( Y , X ) H(X,Y)\neq H(Y,X)
  44. X X
  45. Y Y
  46. H ( Y | X ) H ( Y ) H(Y|X)\leq H(Y)\,
  47. H ( X , Y ) = H ( X | Y ) + H ( Y | X ) + I ( X ; Y ) , H(X,Y)=H(X|Y)+H(Y|X)+I(X;Y),\qquad
  48. H ( X , Y ) = H ( X ) + H ( Y ) - I ( X ; Y ) , H(X,Y)=H(X)+H(Y)-I(X;Y),\,
  49. I ( X ; Y ) H ( X ) , I(X;Y)\leq H(X),\,
  50. I ( X ; Y ) I(X;Y)
  51. X X
  52. Y Y
  53. X X
  54. Y Y
  55. H ( Y | X ) = H ( Y ) and H ( X | Y ) = H ( X ) H(Y|X)=H(Y)\,\text{ and }H(X|Y)=H(X)\,
  56. H ( X | Y = y ) H(X|Y=y)
  57. H ( X ) H(X)
  58. H ( X | Y ) H(X|Y)
  59. H ( X ) H(X)

Conditional_independence.html

  1. Pr ( R B Y ) = Pr ( R Y ) Pr ( B Y ) \Pr(R\cap B\mid Y)=\Pr(R\mid Y)\Pr(B\mid Y)\,
  2. Pr ( R B not Y ) Pr ( R not Y ) Pr ( B not Y ) . \Pr(R\cap B\mid\,\text{not }Y)\not=\Pr(R\mid\,\text{not }Y)\Pr(B\mid\,\text{% not }Y).\,
  3. Pr ( R B Y ) = Pr ( R Y ) Pr ( B Y ) , \Pr(R\cap B\mid Y)=\Pr(R\mid Y)\Pr(B\mid Y),\,
  4. Pr ( R B Y ) = Pr ( R Y ) . \Pr(R\mid B\cap Y)=\Pr(R\mid Y).\,
  5. Pr ( R B Σ ) = Pr ( R Σ ) Pr ( B Σ ) a . s . \Pr(R\cap B\mid\Sigma)=\Pr(R\mid\Sigma)\Pr(B\mid\Sigma)\ a.s.
  6. Pr ( A Σ ) \Pr(A\mid\Sigma)
  7. A A
  8. χ A \chi_{A}
  9. Σ \Sigma
  10. Pr ( A Σ ) := E [ χ A Σ ] . \Pr(A\mid\Sigma):=\operatorname{E}[\chi_{A}\mid\Sigma].
  11. X Y W X\perp\!\!\!\perp Y\mid W
  12. X Y W X\perp Y\mid W
  13. ( X Y ) W (X\perp\!\!\!\perp Y)\mid W
  14. X Y Y X X\perp\!\!\!\perp Y\Rightarrow Y\perp\!\!\!\perp X
  15. X Y K Y X K X\perp\!\!\!\perp Y\mid K\Rightarrow Y\perp\!\!\!\perp X\mid K
  16. X Y Y X X\perp\!\!\!\perp Y\quad\Rightarrow\quad Y\perp\!\!\!\perp X
  17. X A , B and { X A X B X\perp\!\!\!\perp A,B\quad\Rightarrow\quad\,\text{ and }\begin{cases}X\perp\!% \!\!\perp A\\ X\perp\!\!\!\perp B\end{cases}
  18. p X , A , B ( x , a , b ) = p X ( x ) p A , B ( a , b ) p_{X,A,B}(x,a,b)=p_{X}(x)p_{A,B}(a,b)
  19. X A , B X\perp A,B
  20. B p X , A , B ( x , a , b ) = B p X ( x ) p A , B ( a , b ) \int_{B}\!p_{X,A,B}(x,a,b)=\int_{B}\!p_{X}(x)p_{A,B}(a,b)
  21. p X , A ( x , a ) = p X ( x ) p A ( a ) p_{X,A}(x,a)=p_{X}(x)p_{A}(a)
  22. X A , B X A B X\perp\!\!\!\perp A,B\quad\Rightarrow\quad X\perp\!\!\!\perp A\mid B
  23. X A B X B } and X A , B \left.\begin{aligned}\displaystyle X\perp\!\!\!\perp A\mid B\\ \displaystyle X\perp\!\!\!\perp B\end{aligned}\right\}\,\text{ and }\quad% \Rightarrow\quad X\perp\!\!\!\perp A,B
  24. X A B X B } and X A , B and { X A B X B X B A X A \left.\begin{aligned}\displaystyle X\perp\!\!\!\perp A\mid B\\ \displaystyle X\perp\!\!\!\perp B\end{aligned}\right\}\,\text{ and }\quad\iff% \quad X\perp\!\!\!\perp A,B\quad\Rightarrow\quad\,\text{ and }\begin{cases}X% \perp\!\!\!\perp A\mid B\\ X\perp\!\!\!\perp B\\ X\perp\!\!\!\perp B\mid A\\ X\perp\!\!\!\perp A\\ \end{cases}
  25. X A B X B A } and X A , B \left.\begin{aligned}\displaystyle X\perp\!\!\!\perp A\mid B\\ \displaystyle X\perp\!\!\!\perp B\mid A\end{aligned}\right\}\,\text{ and }% \quad\Rightarrow\quad X\perp\!\!\!\perp A,B
  26. X A B X\perp\!\!\!\perp A\mid B
  27. 2 12 \frac{2}{12}
  28. 1 6 \frac{1}{6}
  29. 4 12 \frac{4}{12}
  30. 1 3 \frac{1}{3}
  31. 6 12 \frac{6}{12}
  32. 1 2 \frac{1}{2}

Conditional_quantum_entropy.html

  1. ρ A B \rho^{AB}
  2. S ( A | B ) ρ S(A|B)_{\rho}
  3. H ( A | B ) ρ H(A|B)_{\rho}
  4. ρ A | B \rho_{A|B}
  5. S ( ) S(\cdot)
  6. ρ A B \rho^{AB}
  7. S ( A B ) ρ = def S ( ρ A B ) S(AB)_{\rho}\ \stackrel{\mathrm{def}}{=}\ S(\rho^{AB})
  8. S ( A ) ρ = def S ( ρ A ) = S ( tr B ρ A B ) S(A)_{\rho}\ \stackrel{\mathrm{def}}{=}\ S(\rho^{A})=S(\mathrm{tr}_{B}\rho^{AB})
  9. S ( B ) ρ S(B)_{\rho}
  10. S ( A | B ) ρ = def S ( A B ) ρ - S ( B ) ρ S(A|B)_{\rho}\ \stackrel{\mathrm{def}}{=}\ S(AB)_{\rho}-S(B)_{\rho}

Cone.html

  1. L S A = π r l LSA=\pi rl
  2. r r
  3. l l
  4. l = r 2 + h 2 l=\sqrt{r^{2}+h^{2}}
  5. h h
  6. π r 2 \pi r^{2}
  7. S A = π r 2 + π r l SA=\pi r^{2}+\pi rl
  8. S A = π r ( r + l ) SA=\pi r(r+l)
  9. V V
  10. A B A_{B}
  11. H H
  12. V = 1 3 A B H V=\frac{1}{3}A_{B}H
  13. x 2 d x = 1 3 x 3 . \int x^{2}dx=\tfrac{1}{3}x^{3}.
  14. V = 0 H r 2 π d h V=\int_{0}^{H}r^{2}\pi\,dh
  15. r = R h H r=R\frac{h}{H}
  16. V = 0 H [ R h H ] 2 π d h V=\int_{0}^{H}\left[R\frac{h}{H}\right]^{2}\pi\,dh
  17. V = 1 3 π R 2 H . V=\frac{1}{3}\pi R^{2}H.
  18. A A
  19. A = π R 2 + π R S A=\pi R^{2}+\pi RS\,
  20. S = R 2 + H 2 S=\sqrt{R^{2}+H^{2}}
  21. π R 2 \pi R^{2}
  22. π R S \pi RS
  23. h h
  24. 2 θ 2\theta
  25. z z
  26. F ( s , t , u ) = ( u tan s cos t , u tan s sin t , u ) F(s,t,u)=\left(u\tan s\cos t,u\tan s\sin t,u\right)
  27. s , t , u s,t,u
  28. [ 0 , θ ) [0,\theta)
  29. [ 0 , 2 π ) [0,2\pi)
  30. [ 0 , h ] [0,h]
  31. { F ( x , y , z ) 0 , z 0 , z h } , \{F(x,y,z)\leq 0,z\geq 0,z\leq h\},
  32. F ( x , y , z ) = ( x 2 + y 2 ) ( cos θ ) 2 - z 2 ( sin θ ) 2 . F(x,y,z)=(x^{2}+y^{2})(\cos\theta)^{2}-z^{2}(\sin\theta)^{2}.\,
  33. d d
  34. 2 θ 2\theta
  35. F ( u ) = 0 F(u)=0
  36. F ( u ) = ( u d ) 2 - ( d d ) ( u u ) ( cos θ ) 2 F(u)=(u\cdot d)^{2}-(d\cdot d)(u\cdot u)(\cos\theta)^{2}
  37. F ( u ) = u d - | d | | u | cos θ F(u)=u\cdot d-|d||u|\cos\theta
  38. u = ( x , y , z ) u=(x,y,z)
  39. u d u\cdot d

Cone_(topology).html

  1. C X = ( X × I ) / ( X × { 0 } ) CX=(X\times I)/(X\times\{0\})\,
  2. { ( x , y , z ) 3 x 2 + y 2 = z 2 and 0 z 1 } . \{(x,y,z)\in\mathbb{R}^{3}\mid x^{2}+y^{2}=z^{2}\mbox{ and }~{}0\leq z\leq 1\}.
  3. ( X , x 0 ) (X,x_{0})
  4. X × [ 0 , 1 ] / ( X × { 0 } { x 0 } × [ 0 , 1 ] ) X\times[0,1]/(X\times\left\{0\right\}\cup\left\{x_{0}\right\}\times[0,1])
  5. x ( x , 1 ) x\mapsto(x,1)
  6. ( x 0 , 0 ) (x_{0},0)
  7. X C X X\mapsto CX
  8. C : T o p T o p C:{Top}\to{Top}

Confidence_region.html

  1. s y m b o l β symbol{\beta}
  2. 𝐘 = 𝐗 s y m b o l β + s y m b o l ε \mathbf{Y}=\mathbf{X}symbol{\beta}+symbol{\varepsilon}
  3. s y m b o l β symbol{\beta}
  4. s y m b o l ε symbol{\varepsilon}
  5. σ 2 \sigma^{2}
  6. s y m b o l β symbol{\beta}
  7. ( s y m b o l β - 𝐛 ) 𝐗 𝐗 ( s y m b o l β - 𝐛 ) p s 2 F 1 - α ( p , ν ) , (symbol{\beta}-\mathbf{b})^{\prime}\mathbf{X}^{\prime}\mathbf{X}(symbol{\beta}% -\mathbf{b})\leq ps^{2}F_{1-\alpha}(p,\nu),
  8. s y m b o l β , symbol{\beta},
  9. σ 2 \sigma^{2}
  10. s 2 = ε ε n - p . s^{2}=\frac{\varepsilon^{\prime}\varepsilon}{n-p}.
  11. ν = n - p \nu=n-p
  12. α \alpha
  13. X X^{\prime}
  14. X X
  15. s y m b o l β symbol\beta
  16. s y m b o l ε symbol{\varepsilon}
  17. s y m b o l ε symbol{\varepsilon}
  18. 𝐕 σ 2 \mathbf{V}\sigma^{2}
  19. 𝐈 σ 2 \mathbf{I}\sigma^{2}
  20. 𝐏 𝐏 = 𝐏𝐏 = 𝐕 \mathbf{P}^{\prime}\mathbf{P}=\mathbf{P}\mathbf{P}=\mathbf{V}
  21. 𝐘 = 𝐗 s y m b o l β + s y m b o l ε \mathbf{Y}=\mathbf{X}symbol{\beta}+symbol{\varepsilon}
  22. 𝐙 = 𝐐 s y m b o l β + 𝐟 , \mathbf{Z}=\mathbf{Q}symbol{\beta}+\mathbf{f},
  23. 𝐙 = 𝐏 - 1 𝐘 \mathbf{Z}=\mathbf{P}^{-1}\mathbf{Y}
  24. 𝐐 = 𝐏 - 1 𝐗 \mathbf{Q}=\mathbf{P}^{-1}\mathbf{X}
  25. 𝐟 = 𝐏 - 1 s y m b o l ε \mathbf{f}=\mathbf{P}^{-1}symbol{\varepsilon}
  26. s y m b o l β symbol{\beta}
  27. ( 𝐛 - s y m b o l β ) 𝐐 𝐐 ( 𝐛 - s y m b o l β ) = p n - p ( 𝐙 𝐙 - 𝐛 𝐐 𝐙 ) F 1 - α ( p , n - p ) . (\mathbf{b}-symbol{\beta})^{\prime}\mathbf{Q}^{\prime}\mathbf{Q}(\mathbf{b}-% symbol{\beta})={\frac{p}{n-p}}(\mathbf{Z}^{\prime}\mathbf{Z}-\mathbf{b}^{% \prime}\mathbf{Q}^{\prime}\mathbf{Z})F_{1-\alpha}(p,n-p).
  28. χ 2 \chi^{2}

Conformal_symmetry.html

  1. M μ ν i ( x μ ν - x ν μ ) , \displaystyle M_{\mu\nu}\equiv i(x_{\mu}\partial_{\nu}-x_{\nu}\partial_{\mu})\,,
  2. M μ ν M_{\mu\nu}
  3. P μ P_{\mu}
  4. D D
  5. K μ K_{\mu}
  6. [ D , K μ ] = - i K μ , \displaystyle[D,K_{\mu}]=-iK_{\mu}\,,
  7. η μ ν \eta_{\mu\nu}
  8. D D
  9. K μ K_{\mu}
  10. x μ x μ - a μ x 2 1 - 2 a x + a 2 x 2 x^{\mu}\to\frac{x^{\mu}-a^{\mu}x^{2}}{1-2a\cdot x+a^{2}x^{2}}
  11. a μ a^{\mu}
  12. x μ x μ x^{\mu}\to x^{\prime\mu}
  13. x μ x 2 = x μ x 2 - a μ , \frac{{x}^{\prime\mu}}{{x^{\prime}}^{2}}=\frac{x^{\mu}}{x^{2}}-a^{\mu},

Conical_surface.html

  1. C C
  2. C C
  3. p p
  4. θ \theta
  5. 2 θ 2\theta
  6. C C
  7. C C
  8. S S
  9. S ( t , u ) = v + u q ( t ) S(t,u)=v+uq(t)
  10. v v
  11. q q
  12. 2 θ 2\theta
  13. z z
  14. S ( t , u ) = ( u cos θ cos t , u cos θ sin t , u sin θ ) S(t,u)=(u\cos\theta\cos t,u\cos\theta\sin t,u\sin\theta)
  15. t t
  16. u u
  17. [ 0 , 2 π ) [0,2\pi)
  18. ( - , + ) (-\infty,+\infty)
  19. S ( x , y , z ) = 0 S(x,y,z)=0
  20. S ( x , y , z ) = ( x 2 + y 2 ) ( cos θ ) 2 - z 2 ( sin θ ) 2 . S(x,y,z)=(x^{2}+y^{2})(\cos\theta)^{2}-z^{2}(\sin\theta)^{2}.
  21. 𝐝 \mathbf{d}
  22. 2 θ 2\theta
  23. S ( 𝐱 ) = 0 S(\mathbf{x})=0
  24. S ( 𝐱 ) = ( 𝐱 𝐝 ) 2 - ( 𝐝 𝐝 ) ( 𝐱 𝐱 ) ( cos θ ) 2 S(\mathbf{x})=(\mathbf{x}\cdot\mathbf{d})^{2}-(\mathbf{d}\cdot\mathbf{d})(% \mathbf{x}\cdot\mathbf{x})(\cos\theta)^{2}
  25. S ( 𝐱 ) = 𝐱 𝐝 - | 𝐝 | | 𝐱 | cos θ S(\mathbf{x})=\mathbf{x}\cdot\mathbf{d}-|\mathbf{d}||\mathbf{x}|\cos\theta
  26. 𝐱 = ( x , y , z ) \mathbf{x}=(x,y,z)
  27. 𝐱 𝐝 \mathbf{x}\cdot\mathbf{d}
  28. S ( x , y , z ) = a x 2 + b y 2 + c z 2 + 2 u x y + 2 v y z + 2 w z x = 0 S(x,y,z)=ax^{2}+by^{2}+cz^{2}+2uxy+2vyz+2wzx=0

Conifold.html

  1. 4 \mathbb{CP}^{4}
  2. 4 \mathbb{CP}^{4}
  3. z 1 5 + z 2 5 + z 3 5 + z 4 5 + z 5 5 - 5 ψ z 1 z 2 z 3 z 4 z 5 = 0 z_{1}^{5}+z_{2}^{5}+z_{3}^{5}+z_{4}^{5}+z_{5}^{5}-5\psi z_{1}z_{2}z_{3}z_{4}z_% {5}=0
  4. z i z_{i}
  5. 4 \mathbb{CP}^{4}
  6. ψ \psi
  7. ψ \psi
  8. z i z_{i}
  9. S 2 × S 3 S^{2}\times S^{3}

Conjugate_prior.html

  1. θ p ( x θ ) \theta\mapsto p(x\mid\theta)\!
  2. p ( θ ) p(\theta)\!
  3. p ( x ) p(x)\!
  4. p ( θ | x ) = p ( x | θ ) p ( θ ) p ( x | θ ) p ( θ ) d θ . p(\theta|x)=\frac{p(x|\theta)\,p(\theta)}{\int p(x|\theta^{\prime})\,p(\theta^% {\prime})\,d\theta^{\prime}}.\!
  5. p ( x ) = ( n x ) q x ( 1 - q ) n - x p(x)={n\choose x}q^{x}(1-q)^{n-x}
  6. q q
  7. f ( q ) q a ( 1 - q ) b f(q)\propto q^{a}(1-q)^{b}
  8. a a
  9. b b
  10. a a
  11. b b
  12. q q
  13. α \alpha
  14. β \beta
  15. p ( q ) = q α - 1 ( 1 - q ) β - 1 B ( α , β ) p(q)={q^{\alpha-1}(1-q)^{\beta-1}\over B(\alpha,\beta)}
  16. α \alpha
  17. β \beta
  18. α \alpha
  19. β \beta
  20. α \alpha
  21. β \beta
  22. α \alpha
  23. β \beta
  24. P ( s , f | q = x ) = ( s + f s ) x s ( 1 - x ) f , P(s,f|q=x)={s+f\choose s}x^{s}(1-x)^{f},
  25. P ( q = x | s , f ) = P ( s , f | x ) P ( x ) P ( s , f | x ) P ( x ) d x = ( s + f s ) x s + α - 1 ( 1 - x ) f + β - 1 / B ( α , β ) y = 0 1 ( ( s + f s ) y s + α - 1 ( 1 - y ) f + β - 1 / B ( α , β ) ) d y = x s + α - 1 ( 1 - x ) f + β - 1 B ( s + α , f + β ) , \begin{aligned}\displaystyle P(q=x|s,f)&\displaystyle=\frac{P(s,f|x)P(x)}{\int P% (s,f|x)P(x)dx}\\ &\displaystyle={{{s+f\choose s}x^{s+\alpha-1}(1-x)^{f+\beta-1}/B(\alpha,\beta)% }\over\int_{y=0}^{1}\left({s+f\choose s}y^{s+\alpha-1}(1-y)^{f+\beta-1}/B(% \alpha,\beta)\right)dy}\\ &\displaystyle={x^{s+\alpha-1}(1-x)^{f+\beta-1}\over B(s+\alpha,f+\beta)},\\ \end{aligned}
  26. α \alpha
  27. β \beta
  28. α \alpha
  29. β \beta
  30. α - 1 \alpha-1
  31. β - 1 \beta-1
  32. α \alpha
  33. β \beta
  34. x 1 , , x n x_{1},\ldots,x_{n}
  35. α , β \alpha,\,\beta\!
  36. α + i = 1 n x i , β + n - i = 1 n x i \alpha+\sum_{i=1}^{n}x_{i},\,\beta+n-\sum_{i=1}^{n}x_{i}\!
  37. α - 1 \alpha-1
  38. β - 1 \beta-1
  39. p ( x ~ = 1 ) = α α + β p(\tilde{x}=1)=\frac{\alpha^{\prime}}{\alpha^{\prime}+\beta^{\prime}}
  40. α , β \alpha,\,\beta\!
  41. α + i = 1 n x i , β + i = 1 n N i - i = 1 n x i \alpha+\sum_{i=1}^{n}x_{i},\,\beta+\sum_{i=1}^{n}N_{i}-\sum_{i=1}^{n}x_{i}\!
  42. α - 1 \alpha-1
  43. β - 1 \beta-1
  44. BetaBin ( x ~ | α , β ) \operatorname{BetaBin}(\tilde{x}|\alpha^{\prime},\beta^{\prime})
  45. α , β \alpha,\,\beta\!
  46. α + i = 1 n x i , β + r n \alpha+\sum_{i=1}^{n}x_{i},\,\beta+rn\!
  47. α - 1 \alpha-1
  48. β - 1 \beta-1
  49. β - 1 r \frac{\beta-1}{r}
  50. r r
  51. k , θ k,\,\theta\!
  52. k + i = 1 n x i , θ n θ + 1 k+\sum_{i=1}^{n}x_{i},\ \frac{\theta}{n\theta+1}\!
  53. k k
  54. 1 / θ 1/\theta
  55. NB ( x ~ | k , θ 1 + θ ) \operatorname{NB}(\tilde{x}|k^{\prime},\frac{\theta^{\prime}}{1+\theta^{\prime% }})
  56. α , β \alpha,\,\beta\!
  57. α + i = 1 n x i , β + n \alpha+\sum_{i=1}^{n}x_{i},\ \beta+n\!
  58. α \alpha
  59. β \beta
  60. NB ( x ~ | α , 1 1 + β ) \operatorname{NB}(\tilde{x}|\alpha^{\prime},\frac{1}{1+\beta^{\prime}})
  61. s y m b o l α symbol\alpha\!
  62. s y m b o l α + ( c 1 , , c k ) , symbol\alpha+(c_{1},\ldots,c_{k}),
  63. c i c_{i}
  64. α i - 1 \alpha_{i}-1
  65. i i
  66. p ( x ~ = i ) = α i i α i p(\tilde{x}=i)=\frac{{\alpha_{i}}^{\prime}}{\sum_{i}{\alpha_{i}}^{\prime}}
  67. = α i + c i i α i + n =\frac{\alpha_{i}+c_{i}}{\sum_{i}\alpha_{i}+n}
  68. s y m b o l α symbol\alpha\!
  69. s y m b o l α + i = 1 n 𝐱 i symbol\alpha+\sum_{i=1}^{n}\mathbf{x}_{i}\!
  70. α i - 1 \alpha_{i}-1
  71. i i
  72. DirMult ( 𝐱 ~ | s y m b o l α ) \operatorname{DirMult}(\tilde{\mathbf{x}}|symbol\alpha^{\prime})
  73. n = N , α , β n=N,\alpha,\,\beta\!
  74. α + i = 1 n x i , β + i = 1 n N i - i = 1 n x i \alpha+\sum_{i=1}^{n}x_{i},\,\beta+\sum_{i=1}^{n}N_{i}-\sum_{i=1}^{n}x_{i}\!
  75. α - 1 \alpha-1
  76. β - 1 \beta-1
  77. α , β \alpha,\,\beta\!
  78. α + n , β + i = 1 n x i \alpha+n,\,\beta+\sum_{i=1}^{n}x_{i}\!
  79. α - 1 \alpha-1
  80. β - 1 \beta-1
  81. μ 0 , σ 0 2 \mu_{0},\,\sigma_{0}^{2}\!
  82. ( μ 0 σ 0 2 + i = 1 n x i σ 2 ) / ( 1 σ 0 2 + n σ 2 ) , \left.\left(\frac{\mu_{0}}{\sigma_{0}^{2}}+\frac{\sum_{i=1}^{n}x_{i}}{\sigma^{% 2}}\right)\right/\left(\frac{1}{\sigma_{0}^{2}}+\frac{n}{\sigma^{2}}\right),
  83. ( 1 σ 0 2 + n σ 2 ) - 1 \left(\frac{1}{\sigma_{0}^{2}}+\frac{n}{\sigma^{2}}\right)^{-1}
  84. 1 / σ 0 2 1/\sigma_{0}^{2}
  85. μ 0 \mu_{0}
  86. 𝒩 ( x ~ | μ 0 , σ 0 2 + σ 2 ) \mathcal{N}(\tilde{x}|\mu_{0}^{\prime},{\sigma_{0}^{2}}^{\prime}+\sigma^{2})
  87. μ 0 , τ 0 \mu_{0},\,\tau_{0}\!
  88. ( τ 0 μ 0 + τ i = 1 n x i ) / ( τ 0 + n τ ) , τ 0 + n τ \left.\left(\tau_{0}\mu_{0}+\tau\sum_{i=1}^{n}x_{i}\right)\right/(\tau_{0}+n% \tau),\,\tau_{0}+n\tau
  89. τ 0 \tau_{0}
  90. μ 0 \mu_{0}
  91. 𝒩 ( x ~ | μ 0 , 1 τ 0 + 1 τ ) \mathcal{N}\left(\tilde{x}|\mu_{0}^{\prime},\frac{1}{\tau_{0}^{\prime}}+\frac{% 1}{\tau}\right)
  92. α , β \mathbf{\alpha,\,\beta}
  93. α + n 2 , β + i = 1 n ( x i - μ ) 2 2 \mathbf{\alpha}+\frac{n}{2},\,\mathbf{\beta}+\frac{\sum_{i=1}^{n}{(x_{i}-\mu)^% {2}}}{2}
  94. 2 α 2\alpha
  95. β / α \beta/\alpha
  96. 2 β 2\beta
  97. μ \mu
  98. t 2 α ( x ~ | μ , σ 2 = β / α ) t_{2\alpha^{\prime}}(\tilde{x}|\mu,\sigma^{2}=\beta^{\prime}/\alpha^{\prime})
  99. ν , σ 0 2 \nu,\,\sigma_{0}^{2}\!
  100. ν + n , ν σ 0 2 + i = 1 n ( x i - μ ) 2 ν + n \nu+n,\,\frac{\nu\sigma_{0}^{2}+\sum_{i=1}^{n}(x_{i}-\mu)^{2}}{\nu+n}\!
  101. ν \nu
  102. σ 0 2 \sigma_{0}^{2}
  103. t ν ( x ~ | μ , σ 0 2 ) t_{\nu^{\prime}}(\tilde{x}|\mu,{\sigma_{0}^{2}}^{\prime})
  104. α , β \alpha,\,\beta\!
  105. α + n 2 , β + i = 1 n ( x i - μ ) 2 2 \alpha+\frac{n}{2},\,\beta+\frac{\sum_{i=1}^{n}(x_{i}-\mu)^{2}}{2}\!
  106. 2 α 2\alpha
  107. β / α \beta/\alpha
  108. 2 β 2\beta
  109. μ \mu
  110. t 2 α ( x ~ | μ , σ 2 = β / α ) t_{2\alpha^{\prime}}(\tilde{x}|\mu,\sigma^{2}=\beta^{\prime}/\alpha^{\prime})
  111. μ 0 , ν , α , β \mu_{0},\,\nu,\,\alpha,\,\beta
  112. ν μ 0 + n x ¯ ν + n , ν + n , α + n 2 , \frac{\nu\mu_{0}+n\bar{x}}{\nu+n},\,\nu+n,\,\alpha+\frac{n}{2},\,
  113. β + 1 2 i = 1 n ( x i - x ¯ ) 2 + n ν ν + n ( x ¯ - μ 0 ) 2 2 \beta+\tfrac{1}{2}\sum_{i=1}^{n}(x_{i}-\bar{x})^{2}+\frac{n\nu}{\nu+n}\frac{(% \bar{x}-\mu_{0})^{2}}{2}
  114. x ¯ \bar{x}
  115. ν \nu
  116. μ 0 \mu_{0}
  117. 2 α 2\alpha
  118. μ 0 \mu_{0}
  119. 2 β 2\beta
  120. t 2 α ( x ~ | μ , β ( ν + 1 ) ν α ) t_{2\alpha^{\prime}}\left(\tilde{x}|\mu^{\prime},\frac{\beta^{\prime}(\nu^{% \prime}+1)}{\nu^{\prime}\alpha^{\prime}}\right)
  121. μ 0 , ν , α , β \mu_{0},\,\nu,\,\alpha,\,\beta
  122. ν μ 0 + n x ¯ ν + n , ν + n , α + n 2 , \frac{\nu\mu_{0}+n\bar{x}}{\nu+n},\,\nu+n,\,\alpha+\frac{n}{2},\,
  123. β + 1 2 i = 1 n ( x i - x ¯ ) 2 + n ν ν + n ( x ¯ - μ 0 ) 2 2 \beta+\tfrac{1}{2}\sum_{i=1}^{n}(x_{i}-\bar{x})^{2}+\frac{n\nu}{\nu+n}\frac{(% \bar{x}-\mu_{0})^{2}}{2}
  124. x ¯ \bar{x}
  125. ν \nu
  126. μ 0 \mu_{0}
  127. 2 α 2\alpha
  128. μ 0 \mu_{0}
  129. 2 β 2\beta
  130. t 2 α ( x ~ | μ , β ( ν + 1 ) α ν ) t_{2\alpha^{\prime}}\left(\tilde{x}|\mu^{\prime},\frac{\beta^{\prime}(\nu^{% \prime}+1)}{\alpha^{\prime}\nu^{\prime}}\right)
  131. s y m b o l s y m b o l μ 0 , s y m b o l Σ 0 symbol{symbol\mu}_{0},\,symbol\Sigma_{0}
  132. ( s y m b o l Σ 0 - 1 + n s y m b o l Σ - 1 ) - 1 ( s y m b o l Σ 0 - 1 s y m b o l μ 0 + n s y m b o l Σ - 1 𝐱 ¯ ) , \left(symbol\Sigma_{0}^{-1}+nsymbol\Sigma^{-1}\right)^{-1}\left(symbol\Sigma_{% 0}^{-1}symbol\mu_{0}+nsymbol\Sigma^{-1}\mathbf{\bar{x}}\right),
  133. ( s y m b o l Σ 0 - 1 + n s y m b o l Σ - 1 ) - 1 \left(symbol\Sigma_{0}^{-1}+nsymbol\Sigma^{-1}\right)^{-1}
  134. 𝐱 ¯ \mathbf{\bar{x}}
  135. s y m b o l Σ 0 - 1 symbol\Sigma_{0}^{-1}
  136. s y m b o l μ 0 symbol\mu_{0}
  137. 𝒩 ( 𝐱 ~ | s y m b o l μ 0 , s y m b o l Σ 0 + s y m b o l Σ ) \mathcal{N}(\tilde{\mathbf{x}}|{symbol\mu_{0}}^{\prime},{symbol\Sigma_{0}}^{% \prime}+symbol\Sigma)
  138. 𝐬𝐲𝐦𝐛𝐨𝐥 μ 0 , s y m b o l Λ 0 \mathbf{symbol\mu}_{0},\,symbol\Lambda_{0}
  139. ( s y m b o l Λ 0 + n s y m b o l Λ ) - 1 ( s y m b o l Λ 0 s y m b o l μ 0 + n s y m b o l Λ 𝐱 ¯ ) , ( s y m b o l Λ 0 + n s y m b o l Λ ) \left(symbol\Lambda_{0}+nsymbol\Lambda\right)^{-1}\left(symbol\Lambda_{0}% symbol\mu_{0}+nsymbol\Lambda\mathbf{\bar{x}}\right),\,\left(symbol\Lambda_{0}+% nsymbol\Lambda\right)
  140. 𝐱 ¯ \mathbf{\bar{x}}
  141. s y m b o l Λ symbol\Lambda
  142. s y m b o l μ 0 symbol\mu_{0}
  143. 𝒩 ( 𝐱 ~ | s y m b o l μ 0 , ( s y m b o l Λ 0 - 1 + s y m b o l Λ - 1 ) - 1 ) \mathcal{N}\left(\tilde{\mathbf{x}}|{symbol\mu_{0}}^{\prime},({{symbol\Lambda_% {0}}^{\prime}}^{-1}+symbol\Lambda^{-1})^{-1}\right)
  144. ν , s y m b o l Ψ \nu,\,symbol\Psi
  145. n + ν , s y m b o l Ψ + i = 1 n ( 𝐱 𝐢 - s y m b o l μ ) ( 𝐱 𝐢 - s y m b o l μ ) T n+\nu,\,symbol\Psi+\sum_{i=1}^{n}(\mathbf{x_{i}}-symbol\mu)(\mathbf{x_{i}}-% symbol\mu)^{T}
  146. ν \nu
  147. s y m b o l Ψ symbol\Psi
  148. t ν - p + 1 ( 𝐱 ~ | s y m b o l μ , 1 ν - p + 1 s y m b o l Ψ ) t_{\nu^{\prime}-p+1}\left(\tilde{\mathbf{x}}|symbol\mu,\frac{1}{\nu^{\prime}-p% +1}symbol\Psi^{\prime}\right)
  149. ν , 𝐕 \nu,\,\mathbf{V}
  150. n + ν , ( 𝐕 - 1 + i = 1 n ( 𝐱 𝐢 - s y m b o l μ ) ( 𝐱 𝐢 - s y m b o l μ ) T ) - 1 n+\nu,\,\left(\mathbf{V}^{-1}+\sum_{i=1}^{n}(\mathbf{x_{i}}-symbol\mu)(\mathbf% {x_{i}}-symbol\mu)^{T}\right)^{-1}
  151. ν \nu
  152. 𝐕 - 1 \mathbf{V}^{-1}
  153. t ν - p + 1 ( 𝐱 ~ | s y m b o l μ , 1 ν - p + 1 𝐕 - 1 ) t_{\nu^{\prime}-p+1}\left(\tilde{\mathbf{x}}|symbol\mu,\frac{1}{\nu^{\prime}-p% +1}{\mathbf{V}^{\prime}}^{-1}\right)
  154. s y m b o l μ 0 , κ 0 , ν 0 , s y m b o l Ψ symbol\mu_{0},\,\kappa_{0},\,\nu_{0},\,symbol\Psi
  155. κ 0 s y m b o l μ 0 + n 𝐱 ¯ κ 0 + n , κ 0 + n , ν 0 + n , \frac{\kappa_{0}symbol\mu_{0}+n\mathbf{\bar{x}}}{\kappa_{0}+n},\,\kappa_{0}+n,% \,\nu_{0}+n,\,
  156. s y m b o l Ψ + 𝐂 + κ 0 n κ 0 + n ( 𝐱 ¯ - s y m b o l μ 0 ) ( 𝐱 ¯ - s y m b o l μ 0 ) T symbol\Psi+\mathbf{C}+\frac{\kappa_{0}n}{\kappa_{0}+n}(\mathbf{\bar{x}}-symbol% \mu_{0})(\mathbf{\bar{x}}-symbol\mu_{0})^{T}
  157. 𝐱 ¯ \mathbf{\bar{x}}
  158. 𝐂 = i = 1 n ( 𝐱 𝐢 - 𝐱 ¯ ) ( 𝐱 𝐢 - 𝐱 ¯ ) T \mathbf{C}=\sum_{i=1}^{n}(\mathbf{x_{i}}-\mathbf{\bar{x}})(\mathbf{x_{i}}-% \mathbf{\bar{x}})^{T}
  159. κ 0 \kappa_{0}
  160. s y m b o l μ 0 symbol\mu_{0}
  161. ν 0 \nu_{0}
  162. s y m b o l μ 0 symbol\mu_{0}
  163. s y m b o l Ψ = ν 0 s y m b o l Σ 0 symbol\Psi=\nu_{0}symbol\Sigma_{0}
  164. t ν 0 - p + 1 ( 𝐱 ~ | s y m b o l μ 0 , κ 0 + 1 κ 0 ( ν 0 - p + 1 ) s y m b o l Ψ ) t_{{\nu_{0}}^{\prime}-p+1}\left(\tilde{\mathbf{x}}|{symbol\mu_{0}}^{\prime},% \frac{{\kappa_{0}}^{\prime}+1}{{\kappa_{0}}^{\prime}({\nu_{0}}^{\prime}-p+1)}% symbol\Psi^{\prime}\right)
  165. s y m b o l μ 0 , κ 0 , ν 0 , 𝐕 symbol\mu_{0},\,\kappa_{0},\,\nu_{0},\,\mathbf{V}
  166. κ 0 s y m b o l μ 0 + n 𝐱 ¯ κ 0 + n , κ 0 + n , ν 0 + n , \frac{\kappa_{0}symbol\mu_{0}+n\mathbf{\bar{x}}}{\kappa_{0}+n},\,\kappa_{0}+n,% \,\nu_{0}+n,\,
  167. ( 𝐕 - 1 + 𝐂 + κ 0 n κ 0 + n ( 𝐱 ¯ - s y m b o l μ 0 ) ( 𝐱 ¯ - s y m b o l μ 0 ) T ) - 1 \left(\mathbf{V}^{-1}+\mathbf{C}+\frac{\kappa_{0}n}{\kappa_{0}+n}(\mathbf{\bar% {x}}-symbol\mu_{0})(\mathbf{\bar{x}}-symbol\mu_{0})^{T}\right)^{-1}
  168. 𝐱 ¯ \mathbf{\bar{x}}
  169. 𝐂 = i = 1 n ( 𝐱 𝐢 - 𝐱 ¯ ) ( 𝐱 𝐢 - 𝐱 ¯ ) T \mathbf{C}=\sum_{i=1}^{n}(\mathbf{x_{i}}-\mathbf{\bar{x}})(\mathbf{x_{i}}-% \mathbf{\bar{x}})^{T}
  170. κ 0 \kappa_{0}
  171. s y m b o l μ 0 symbol\mu_{0}
  172. ν 0 \nu_{0}
  173. s y m b o l μ 0 symbol\mu_{0}
  174. 𝐕 - 1 \mathbf{V}^{-1}
  175. t ν 0 - p + 1 ( 𝐱 ~ | s y m b o l μ 0 , κ 0 + 1 κ 0 ( ν 0 - p + 1 ) 𝐕 - 1 ) t_{{\nu_{0}}^{\prime}-p+1}\left(\tilde{\mathbf{x}}|{symbol\mu_{0}}^{\prime},% \frac{{\kappa_{0}}^{\prime}+1}{{\kappa_{0}}^{\prime}({\nu_{0}}^{\prime}-p+1)}{% \mathbf{V}^{\prime}}^{-1}\right)
  176. U ( 0 , θ ) U(0,\theta)\!
  177. x m , k x_{m},\,k\!
  178. max { x 1 , , x n , x m } , k + n \max\{\,x_{1},\ldots,x_{n},x_{\mathrm{m}}\},\,k+n\!
  179. k k
  180. x m x_{m}
  181. α , β \alpha,\,\beta\!
  182. α + n , β + i = 1 n ln x i x m \alpha+n,\,\beta+\sum_{i=1}^{n}\ln\frac{x_{i}}{x_{\mathrm{m}}}\!
  183. α \alpha
  184. β \beta
  185. x m x_{m}
  186. a , b a,b\!
  187. a + n , b + i = 1 n x i β a+n,\,b+\sum_{i=1}^{n}x_{i}^{\beta}\!
  188. a a
  189. b b
  190. μ 0 , τ 0 \mu_{0},\,\tau_{0}\!
  191. ( τ 0 μ 0 + τ i = 1 n ln x i ) / ( τ 0 + n τ ) , τ 0 + n τ \left.\left(\tau_{0}\mu_{0}+\tau\sum_{i=1}^{n}\ln x_{i}\right)\right/(\tau_{0}% +n\tau),\,\tau_{0}+n\tau
  192. τ 0 \tau_{0}
  193. μ 0 \mu_{0}
  194. α , β \alpha,\,\beta\!
  195. α + n 2 , β + i = 1 n ( ln x i - μ ) 2 2 \alpha+\frac{n}{2},\,\beta+\frac{\sum_{i=1}^{n}(\ln x_{i}-\mu)^{2}}{2}\!
  196. 2 α 2\alpha
  197. β α \frac{\beta}{\alpha}
  198. 2 β 2\beta
  199. α , β \alpha,\,\beta\!
  200. α + n , β + i = 1 n x i \alpha+n,\,\beta+\sum_{i=1}^{n}x_{i}\!
  201. α \alpha
  202. β \beta
  203. Lomax ( x ~ | β , α ) \operatorname{Lomax}(\tilde{x}|\beta^{\prime},\alpha^{\prime})
  204. α 0 , β 0 \alpha_{0},\,\beta_{0}\!
  205. α 0 + n α , β 0 + i = 1 n x i \alpha_{0}+n\alpha,\,\beta_{0}+\sum_{i=1}^{n}x_{i}\!
  206. α 0 \alpha_{0}
  207. β 0 \beta_{0}
  208. CG ( 𝐱 ~ | α , α 0 , β 0 ) = β ( 𝐱 ~ | α , α 0 , 1 , β 0 ) \operatorname{CG}(\tilde{\mathbf{x}}|\alpha,{\alpha_{0}}^{\prime},{\beta_{0}}^% {\prime})=\operatorname{\beta^{\prime}}(\tilde{\mathbf{x}}|\alpha,{\alpha_{0}}% ^{\prime},1,{\beta_{0}}^{\prime})
  209. α 0 , β 0 \alpha_{0},\,\beta_{0}\!
  210. α 0 + n α , β 0 + i = 1 n 1 x i \alpha_{0}+n\alpha,\,\beta_{0}+\sum_{i=1}^{n}\frac{1}{x_{i}}\!
  211. α 0 \alpha_{0}
  212. β 0 \beta_{0}
  213. a α - 1 β α c Γ ( α ) b \propto\frac{a^{\alpha-1}\beta^{\alpha c}}{\Gamma(\alpha)^{b}}
  214. a , b , c a,\,b,\,c\!
  215. a i = 1 n x i , b + n , c + n a\prod_{i=1}^{n}x_{i},\,b+n,\,c+n\!
  216. b b
  217. c c
  218. b b
  219. α \alpha
  220. c c
  221. β \beta
  222. a a
  223. p α - 1 e - β q Γ ( α ) r β - α s \propto\frac{p^{\alpha-1}e^{-\beta q}}{\Gamma(\alpha)^{r}\beta^{-\alpha s}}
  224. p , q , r , s p,\,q,\,r,\,s\!
  225. p i = 1 n x i , q + i = 1 n x i , r + n , s + n p\prod_{i=1}^{n}x_{i},\,q+\sum_{i=1}^{n}x_{i},\,r+n,\,s+n\!
  226. α \alpha
  227. r r
  228. p p
  229. β \beta
  230. s s
  231. q q
  232. 𝒩 ( ) \mathcal{N}()
  233. t n ( ) t_{n}()

Conjunction_elimination.html

  1. P Q P \frac{P\land Q}{\therefore P}
  2. P Q Q \frac{P\land Q}{\therefore Q}
  3. P Q P\land Q
  4. P P
  5. Q Q
  6. ( P Q ) P (P\land Q)\vdash P
  7. ( P Q ) Q (P\land Q)\vdash Q
  8. \vdash
  9. P P
  10. P Q P\land Q
  11. Q Q
  12. P Q P\land Q
  13. ( P Q ) P (P\land Q)\to P
  14. ( P Q ) Q (P\land Q)\to Q
  15. P P
  16. Q Q

Connection_form.html

  1. ξ = α = 1 k e α ξ α ( 𝐞 ) \xi=\sum_{\alpha=1}^{k}e_{\alpha}\xi^{\alpha}(\mathbf{e})
  2. ξ = 𝐞 [ ξ 1 ( 𝐞 ) ξ 2 ( 𝐞 ) ξ k ( 𝐞 ) ] = 𝐞 ξ ( 𝐞 ) \xi={\mathbf{e}}\begin{bmatrix}\xi^{1}(\mathbf{e})\\ \xi^{2}(\mathbf{e})\\ \vdots\\ \xi^{k}(\mathbf{e})\end{bmatrix}={\mathbf{e}}\,\xi(\mathbf{e})
  3. D : Γ ( E ) Γ ( E Ω 1 M ) D:\Gamma(E)\rightarrow\Gamma(E\otimes\Omega^{1}M)
  4. D ( f v ) = v ( d f ) + f D v D(fv)=v\otimes(df)+fDv
  5. D : Γ ( E Ω * M ) Γ ( E Ω * M ) D:\Gamma(E\otimes\Omega^{*}M)\rightarrow\Gamma(E\otimes\Omega^{*}M)
  6. D ( v α ) = ( D v ) α + ( - 1 ) deg v v d α D(v\wedge\alpha)=(Dv)\wedge\alpha+(-1)^{\,\text{deg}\,v}v\wedge d\alpha
  7. D e α = β = 1 k e β ω α β . De_{\alpha}=\sum_{\beta=1}^{k}e_{\beta}\otimes\omega^{\beta}_{\alpha}.
  8. D ξ = α = 1 k D ( e α ξ α ( 𝐞 ) ) = α = 1 k e α d ξ α ( 𝐞 ) + α = 1 k β = 1 k e β ω α β ξ α ( 𝐞 ) . D\xi=\sum_{\alpha=1}^{k}D(e_{\alpha}\xi^{\alpha}(\mathbf{e}))=\sum_{\alpha=1}^% {k}e_{\alpha}\otimes d\xi^{\alpha}(\mathbf{e})+\sum_{\alpha=1}^{k}\sum_{\beta=% 1}^{k}e_{\beta}\otimes\omega^{\beta}_{\alpha}\xi^{\alpha}(\mathbf{e}).
  9. D ξ ( 𝐞 ) = d ξ ( 𝐞 ) + ω ξ ( 𝐞 ) = ( d + ω ) ξ ( 𝐞 ) D\xi(\mathbf{e})=d\xi(\mathbf{e})+\omega\xi(\mathbf{e})=(d+\omega)\xi(\mathbf{% e})
  10. 𝐞 = 𝐞 g , i.e., e α = β e β g α β . {\mathbf{e}}^{\prime}={\mathbf{e}}\,g,\quad\,\text{i.e., }\,e^{\prime}_{\alpha% }=\sum_{\beta}e_{\beta}g^{\beta}_{\alpha}.
  11. ω ( 𝐞 g ) = g - 1 d g + g - 1 ω ( 𝐞 ) g . \omega(\mathbf{e}\,g)=g^{-1}dg+g^{-1}\omega(\mathbf{e})g.
  12. ω ( 𝐞 q ) = ( 𝐞 p - 1 𝐞 q ) - 1 d ( 𝐞 p - 1 𝐞 q ) + ( 𝐞 p - 1 𝐞 q ) - 1 ω ( 𝐞 p ) ( 𝐞 p - 1 𝐞 q ) . \omega(\mathbf{e}_{q})=(\mathbf{e}_{p}^{-1}\mathbf{e}_{q})^{-1}d(\mathbf{e}_{p% }^{-1}\mathbf{e}_{q})+(\mathbf{e}_{p}^{-1}\mathbf{e}_{q})^{-1}\omega(\mathbf{e% }_{p})(\mathbf{e}_{p}^{-1}\mathbf{e}_{q}).
  13. Ω ( 𝐞 ) = d ω ( 𝐞 ) + ω ( 𝐞 ) ω ( 𝐞 ) . \Omega(\mathbf{e})=d\omega(\mathbf{e})+\omega(\mathbf{e})\wedge\omega(\mathbf{% e}).
  14. Ω ( 𝐞 g ) = g - 1 Ω ( 𝐞 ) g . \Omega(\mathbf{e}\,g)=g^{-1}\Omega(\mathbf{e})g.
  15. Ω = 𝐞 Ω ( 𝐞 ) 𝐞 * \Omega={\mathbf{e}}\Omega(\mathbf{e}){\mathbf{e}}^{*}
  16. Ω Γ ( Ω 2 M Hom ( E , E ) ) . \Omega\in\Gamma(\Omega^{2}M\otimes\,\text{Hom}(E,E)).
  17. Ω ( v ) = D ( D v ) = D 2 v \Omega(v)=D(Dv)=D^{2}v\,
  18. Γ ( E ) D Γ ( E Ω 1 M ) D Γ ( E Ω 2 M ) D D Γ ( E Ω n ( M ) ) \Gamma(E)\ \stackrel{D}{\to}\ \Gamma(E\otimes\Omega^{1}M)\ \stackrel{D}{\to}\ % \Gamma(E\otimes\Omega^{2}M)\ \stackrel{D}{\to}\ \dots\ \stackrel{D}{\to}\ % \Gamma(E\otimes\Omega^{n}(M))
  19. θ x : T x M E x \theta_{x}:T_{x}M\rightarrow E_{x}
  20. Θ = D θ . \Theta=D\theta.\,
  21. θ = i θ i ( 𝐞 ) e i . \theta=\sum_{i}\theta^{i}(\mathbf{e})e_{i}.
  22. Θ i ( 𝐞 ) = d θ i ( 𝐞 ) + j ω j i ( 𝐞 ) θ j ( 𝐞 ) . \Theta^{i}(\mathbf{e})=d\theta^{i}(\mathbf{e})+\sum_{j}\omega_{j}^{i}(\mathbf{% e})\wedge\theta^{j}(\mathbf{e}).
  23. Θ i ( 𝐞 g ) = j g j i Θ j ( 𝐞 ) . \Theta^{i}(\mathbf{e}\,g)=\sum_{j}g_{j}^{i}\Theta^{j}(\mathbf{e}).
  24. Θ = i e i Θ i ( 𝐞 ) . \Theta=\sum_{i}e_{i}\Theta^{i}(\mathbf{e}).
  25. e i e j = k = 1 n Γ i j k ( 𝐞 ) e k . \nabla_{e_{i}}e_{j}=\sum_{k=1}^{n}\Gamma_{ij}^{k}(\mathbf{e})e_{k}.
  26. ω i j ( 𝐞 ) = k Γ k i j ( 𝐞 ) θ k . \omega_{i}^{j}(\mathbf{e})=\sum_{k}\Gamma_{ki}^{j}(\mathbf{e})\theta^{k}.
  27. D v = k e k ( d v k ) + j , k e k ω j k ( 𝐞 ) v j . Dv=\sum_{k}e_{k}\otimes(dv^{k})+\sum_{j,k}e_{k}\otimes\omega^{k}_{j}(\mathbf{e% })v^{j}.
  28. e i v = D v , e i = k e k ( e i v k + Σ j Γ i j k ( 𝐞 ) v j ) \nabla_{e_{i}}v=\langle Dv,e_{i}\rangle=\sum_{k}e_{k}\left(\nabla_{e_{i}}v^{k}% +\Sigma_{j}\Gamma^{k}_{ij}(\mathbf{e})v^{j}\right)
  29. Ω i j ( 𝐞 ) = d ω i j ( 𝐞 ) + k ω k j ( 𝐞 ) ω i k ( 𝐞 ) . \Omega_{i}^{j}(\mathbf{e})=d\omega_{i}^{j}(\mathbf{e})+\sum_{k}\omega_{k}^{j}(% \mathbf{e})\wedge\omega_{i}^{k}(\mathbf{e}).
  30. Ω i j = d ( Γ q i j θ q ) + ( Γ p k j θ p ) ( Γ q i k θ q ) = θ p θ q ( p Γ q i j + Γ p k j Γ q i k ) ) = 1 2 θ p θ q R p q i j \begin{array}[]{ll}\Omega_{i}^{j}&=d(\Gamma^{j}_{qi}\theta^{q})+(\Gamma^{j}_{% pk}\theta^{p})\wedge(\Gamma^{k}_{qi}\theta^{q})\\ &\\ &=\theta^{p}\wedge\theta^{q}\left(\partial_{p}\Gamma^{j}_{qi}+\Gamma^{j}_{pk}% \Gamma^{k}_{qi})\right)\\ &\\ &=\tfrac{1}{2}\theta^{p}\wedge\theta^{q}R_{pqi}{}^{j}\end{array}
  31. Θ i ( 𝐞 ) = d θ i + j ω j i ( 𝐞 ) θ j . \Theta^{i}(\mathbf{e})=d\theta^{i}+\sum_{j}\omega^{i}_{j}(\mathbf{e})\wedge% \theta^{j}.
  32. Θ i = Γ k j i θ k θ j \Theta^{i}=\Gamma^{i}_{kj}\theta^{k}\wedge\theta^{j}
  33. e α = β e β g α β . e_{\alpha}^{\prime}=\sum_{\beta}e_{\beta}g_{\alpha}^{\beta}.
  34. Γ ( γ ) 0 t e α ( γ ( 0 ) ) = β e β ( γ ( t ) ) g α β ( t ) \Gamma(\gamma)_{0}^{t}e_{\alpha}(\gamma(0))=\sum_{\beta}e_{\beta}(\gamma(t))g_% {\alpha}^{\beta}(t)
  35. γ ˙ ( 0 ) e α = β e β ω α β ( γ ˙ ( 0 ) ) \nabla_{\dot{\gamma}(0)}e_{\alpha}=\sum_{\beta}e_{\beta}\omega_{\alpha}^{\beta% }(\dot{\gamma}(0))
  36. D e α = β e β ω α β ( 𝐞 ) De_{\alpha}=\sum_{\beta}e_{\beta}\otimes\omega_{\alpha}^{\beta}(\mathbf{e})
  37. e α = β e β g α β e_{\alpha}^{\prime}=\sum_{\beta}e_{\beta}g_{\alpha}^{\beta}
  38. ω α β ( 𝐞 g ) = ( g - 1 ) γ β d g α γ + ( g - 1 ) γ β ω δ γ ( 𝐞 ) g α δ . \omega_{\alpha}^{\beta}(\mathbf{e}\cdot g)=(g^{-1})_{\gamma}^{\beta}dg_{\alpha% }^{\gamma}+(g^{-1})_{\gamma}^{\beta}\omega_{\delta}^{\gamma}(\mathbf{e})g_{% \alpha}^{\delta}.
  39. ω ( 𝐞 g ) = g - 1 d g + g - 1 ω g . \omega({\mathbf{e}}\cdot g)=g^{-1}dg+g^{-1}\omega g.
  40. ω ( 𝐞 g ) = g * ω 𝔤 + Ad g - 1 ω ( 𝐞 ) \omega({\mathbf{e}}\cdot g)=g^{*}\omega_{\mathfrak{g}}+\,\text{Ad}_{g^{-1}}% \omega(\mathbf{e})
  41. 𝐞 V = 𝐞 U h U V {\mathbf{e}}_{V}={\mathbf{e}}_{U}\cdot h_{UV}
  42. F G E = U U × G / F_{G}E=\left.\coprod_{U}U\times G\right/\sim
  43. \sim
  44. ( ( x , g U ) U × G ) ( ( x , g V ) V × G ) 𝐞 V = 𝐞 U h U V and g U = h U V - 1 ( x ) g V . ((x,g_{U})\in U\times G)\sim((x,g_{V})\in V\times G)\iff{\mathbf{e}}_{V}={% \mathbf{e}}_{U}\cdot h_{UV}\,\text{ and }g_{U}=h_{UV}^{-1}(x)g_{V}.
  45. π 1 : U × G U , π 2 : U × G G \pi_{1}:U\times G\to U,\quad\pi_{2}:U\times G\to G
  46. ω ( x , g ) = A d g - 1 π 1 * ω ( 𝐞 U ) + π 2 * ω 𝐠 . \omega_{(x,g)}=Ad_{g^{-1}}\pi_{1}^{*}\omega(\mathbf{e}_{U})+\pi_{2}^{*}\omega_% {\mathbf{g}}.
  47. ω ( 𝐞 ) = 𝐞 * ω . \omega({\mathbf{e}})={\mathbf{e}}^{*}\omega.
  48. X , ( 𝐞 g ) * ω = [ d ( 𝐞 g ) ] ( X ) , ω \langle X,({\mathbf{e}}\cdot g)^{*}\omega\rangle=\langle[d(\mathbf{e}\cdot g)]% (X),\omega\rangle

Consequent.html

  1. ϕ \phi
  2. ψ \psi
  3. ϕ \phi
  4. ψ \psi

Conserved_current.html

  1. j μ j^{\mu}
  2. μ j μ = 0 \partial_{\mu}j^{\mu}=0
  3. V V
  4. t Q = 0 , {\partial\over\partial t}Q=0\;,
  5. Q = V j 0 d V Q=\int_{V}j^{0}dV
  6. ρ t + 𝐉 = 0 \frac{\partial\rho}{\partial t}+\nabla\cdot\mathbf{J}=0
  7. ρ \rho

Consistent_histories.html

  1. H i H_{i}
  2. i i
  3. P i , j P_{i,j}
  4. t i , j t_{i,j}
  5. j j
  6. H i = ( P i , 1 , P i , 2 , , P i , n i ) H_{i}=(P_{i,1},P_{i,2},\ldots,P_{i,n_{i}})
  7. P i , 1 P_{i,1}
  8. t i , 1 t_{i,1}
  9. P i , 2 P_{i,2}
  10. t i , 2 t_{i,2}
  11. \ldots
  12. t i , 1 < t i , 2 < < t i , n i t_{i,1}<t_{i,2}<\ldots<t_{i,n_{i}}
  13. H i H j H_{i}H_{j}
  14. P i , j P_{i,j}
  15. P ^ i , j \hat{P}_{i,j}
  16. H i H_{i}
  17. H ^ i = P ^ i , 1 P ^ i , 2 P ^ i , n i \hat{H}_{i}=\hat{P}_{i,1}\otimes\hat{P}_{i,2}\otimes\cdots\otimes\hat{P}_{i,n_% {i}}
  18. C ^ H i := T j = 1 n i P ^ i , j ( t i , j ) = P ^ i , 1 P ^ i , 2 P ^ i , n i \hat{C}_{H_{i}}:=T\prod_{j=1}^{n_{i}}\hat{P}_{i,j}(t_{i,j})=\hat{P}_{i,1}\hat{% P}_{i,2}\cdots\hat{P}_{i,n_{i}}
  19. T T
  20. t i , j t_{i,j}
  21. t t
  22. t t
  23. { H i } \{H_{i}\}
  24. Tr ( C ^ H i ρ C ^ H j ) = 0 \operatorname{Tr}(\hat{C}_{H_{i}}\rho\hat{C}^{\dagger}_{H_{j}})=0
  25. i j i\neq j
  26. ρ \rho
  27. Tr ( C ^ H i ρ C ^ H j ) 0 \operatorname{Tr}(\hat{C}_{H_{i}}\rho\hat{C}^{\dagger}_{H_{j}})\approx 0
  28. i j i\neq j
  29. H i H_{i}
  30. Pr ( H i ) = Tr ( C ^ H i ρ C ^ H i ) \operatorname{Pr}(H_{i})=\operatorname{Tr}(\hat{C}_{H_{i}}\rho\hat{C}^{\dagger% }_{H_{i}})
  31. H i H_{i}
  32. H i H_{i}
  33. H j H_{j}
  34. H i H_{i}
  35. H j H_{j}
  36. H i H_{i}
  37. H j H_{j}

Constitutive_equation.html

  1. σ = F / A \sigma=F/A\,\!
  2. ϵ = Δ D / D \epsilon=\Delta D/D\,\!
  3. E mod = σ / ϵ E_{\mathrm{mod}}=\sigma/\epsilon\,\!
  4. Y = σ / ( Δ L / L ) Y=\sigma/\left(\Delta L/L\right)\,\!
  5. G = Δ x / L G=\Delta x/L\,\!
  6. B = P / ( Δ V / V ) B=P/\left(\Delta V/V\right)\,\!
  7. C = 1 / B C=1/B\,\!
  8. F = μ f R F=\mu_{f}R\,
  9. F i = - k x i F_{i}=-kx_{i}\,
  10. σ = E ϵ \sigma=E\,\epsilon\,
  11. σ i j = C i j k l ϵ k l ϵ i j = S i j k l σ k l \sigma_{ij}=C_{ijkl}\,\epsilon_{kl}\,\rightleftharpoons\,\epsilon_{ij}=S_{ijkl% }\,\sigma_{kl}\,
  12. e = | 𝐯 | separation | 𝐯 | approach e=\frac{\left|\mathbf{v}\right|_{\mathrm{separation}}}{\left|\mathbf{v}\right|% _{\mathrm{approach}}}\,\!
  13. D = 1 2 c d ρ A v 2 D=\frac{1}{2}c_{d}\rho Av^{2}\,
  14. τ = μ u y , \tau=\mu\frac{\partial u}{\partial y},
  15. τ i j = 2 μ ( e i j - 1 3 Δ δ i j ) \tau_{ij}=2\mu\left(e_{ij}-\frac{1}{3}\Delta\delta_{ij}\right)
  16. e i j = 1 2 ( v i x j + v j x i ) e_{ij}=\frac{1}{2}\left(\frac{\partial v_{i}}{\partial x_{j}}+\frac{\partial v% _{j}}{\partial x_{i}}\right)
  17. Δ = k e k k = div 𝐯 , \Delta=\sum_{k}e_{kk}=\,\text{div}\;\mathbf{v},
  18. p V = n R T pV=nRT\,
  19. 𝐃 ( 𝐫 , t ) = ε 0 𝐄 ( 𝐫 , t ) + 𝐏 ( 𝐫 , t ) \mathbf{D}(\mathbf{r},t)=\varepsilon_{0}\mathbf{E}(\mathbf{r},t)+\mathbf{P}(% \mathbf{r},t)
  20. 𝐇 ( 𝐫 , t ) = 1 μ 0 𝐁 ( 𝐫 , t ) - 𝐌 ( 𝐫 , t ) , \mathbf{H}(\mathbf{r},t)=\frac{1}{\mu_{0}}\mathbf{B}(\mathbf{r},t)-\mathbf{M}(% \mathbf{r},t),
  21. 𝐃 = ε 0 𝐄 , 𝐇 = 𝐁 / μ 0 \mathbf{D}=\varepsilon_{0}\mathbf{E},\;\;\;\mathbf{H}=\mathbf{B}/\mu_{0}
  22. 𝐏 = ε 0 χ e 𝐄 , 𝐌 = χ m 𝐇 , \mathbf{P}=\varepsilon_{0}\chi_{e}\mathbf{E},\;\;\;\mathbf{M}=\chi_{m}\mathbf{% H},
  23. 𝐃 = ε 𝐄 , 𝐇 = 𝐁 / μ , \mathbf{D}=\varepsilon\mathbf{E},\;\;\;\mathbf{H}=\mathbf{B}/\mu,
  24. ϵ / ϵ 0 = ϵ r = ( χ E + 1 ) , μ / μ 0 = μ r = ( χ M + 1 ) \epsilon/\epsilon_{0}=\epsilon_{r}=\left(\chi_{E}+1\right)\,,\quad\mu/\mu_{0}=% \mu_{r}=\left(\chi_{M}+1\right)\,\!
  25. 𝐃 = ε 𝐄 , 𝐇 = μ - 1 𝐁 \mathbf{D}=\varepsilon\mathbf{E},\;\;\;\mathbf{H}=\mu^{-1}\mathbf{B}
  26. D i = j ε i j E j B i = j μ i j H j . D_{i}=\sum_{j}\varepsilon_{ij}E_{j}\;\;\;B_{i}=\sum_{j}\mu_{ij}H_{j}.
  27. 𝐏 ( 𝐫 , t ) = ε 0 d 3 𝐫 d t χ ^ e ( 𝐫 , 𝐫 , t , t ; 𝐄 ) 𝐄 ( 𝐫 , t ) \mathbf{P}(\mathbf{r},t)=\varepsilon_{0}\int{\rm d}^{3}\mathbf{r}^{\prime}{\rm d% }t^{\prime}\;\hat{\chi}_{e}(\mathbf{r},\mathbf{r}^{\prime},t,t^{\prime};% \mathbf{E})\,\mathbf{E}(\mathbf{r}^{\prime},t^{\prime})
  28. 𝐌 ( 𝐫 , t ) = 1 μ 0 d 3 𝐫 d t χ ^ m ( 𝐫 , 𝐫 , t , t ; 𝐁 ) 𝐁 ( 𝐫 , t ) , \mathbf{M}(\mathbf{r},t)=\frac{1}{\mu_{0}}\int{\rm d}^{3}\mathbf{r}^{\prime}{% \rm d}t^{\prime}\;\hat{\chi}_{m}(\mathbf{r},\mathbf{r}^{\prime},t,t^{\prime};% \mathbf{B})\,\mathbf{B}(\mathbf{r}^{\prime},t^{\prime}),
  29. 𝐃 = ε 𝐄 + ξ 𝐇 , 𝐁 = μ 𝐇 + ζ 𝐄 . \mathbf{D}=\varepsilon\mathbf{E}+\xi\mathbf{H}\,,\quad\mathbf{B}=\mu\mathbf{H}% +\zeta\mathbf{E}.
  30. E k = ρ k i j J i H j E_{k}=\rho_{kij}J_{i}H_{j}\,
  31. P i = d i j k σ j k P_{i}=d_{ijk}\sigma_{jk}\,
  32. ϵ i j = d i j k E k \epsilon_{ij}=d_{ijk}E_{k}\,
  33. M i = q i j k σ j k M_{i}=q_{ijk}\sigma_{jk}\,
  34. Δ P j = p j Δ T \Delta P_{j}=p_{j}\Delta T\,
  35. Δ S = p i Δ E i \Delta S=p_{i}\Delta E_{i}\,
  36. E i = - β i j T x j E_{i}=-\beta_{ij}\frac{\partial T}{\partial x_{j}}\,
  37. q j = Π j i J i q_{j}=\Pi_{ji}J_{i}\,
  38. n = c 0 c = ε μ ε 0 μ 0 = ε r μ r n=\frac{c_{0}}{c}=\sqrt{\frac{\varepsilon\mu}{\varepsilon_{0}\mu_{0}}}=\sqrt{% \varepsilon_{r}\mu_{r}}\,
  39. n A B = n A n B n_{AB}=\frac{n_{A}}{n_{B}}\,
  40. c = 1 / ε μ c=1/\sqrt{\varepsilon\mu}\,
  41. c 0 = 1 / ε 0 μ 0 c_{0}=1/\sqrt{\varepsilon_{0}\mu_{0}}\,
  42. a i j = Π i j p q σ p q a_{ij}=\Pi_{ijpq}\sigma_{pq}\,
  43. q = C T q=CT\,
  44. L / T = α L \partial L/\partial T=\alpha L\,\!
  45. ϵ i j = α i j Δ T \epsilon_{ij}=\alpha_{ij}\Delta T\,
  46. ( V / T ) p = γ V (\partial V/\partial T)_{p}=\gamma V\,\!
  47. λ = - P / ( 𝐀 T ) \lambda=-P/\left(\mathbf{A}\cdot\nabla T\right)\,\!
  48. U = λ / δ x U=\lambda/\delta x\,\!
  49. R = 1 / U = Δ x / λ R=1/U=\Delta x/\lambda\,\!
  50. R = V / I R=V/I\,\!
  51. ρ = R A / l \rho=RA/l\,\!
  52. ρ - ρ 0 = ρ 0 α ( T - T 0 ) \rho-\rho_{0}=\rho_{0}\alpha(T-T_{0})\,\!
  53. G = 1 / R G=1/R\,\!
  54. σ = 1 / ρ \sigma=1/\rho\,\!
  55. \mathcal{R}
  56. R m = / Φ B R_{\mathrm{m}}=\mathcal{M}/\Phi_{B}
  57. 𝒫 \mathcal{P}
  58. Λ = 1 / R m \Lambda=1/R_{\mathrm{m}}
  59. J j = - D i j C x i J_{j}=-D_{ij}\frac{\partial C}{\partial x_{i}}
  60. q j = - κ μ P x j q_{j}=-\frac{\kappa}{\mu}\frac{\partial P}{\partial x_{j}}
  61. V = I R V=IR\,
  62. V x i = ρ j i J i J j = σ j i V x i \frac{\partial V}{\partial x_{i}}=\rho_{ji}J_{i}\,\rightleftharpoons\,J_{j}=% \sigma_{ji}\frac{\partial V}{\partial x_{i}}\,
  63. q j = - λ i j T x i q_{j}=-\lambda_{ij}\frac{\partial T}{\partial x_{i}}\,
  64. I = ϵ σ T 4 I=\epsilon\sigma T^{4}\,
  65. I = ϵ σ ( T ext 4 - T sys 4 ) I=\epsilon\sigma\left(T_{\mathrm{ext}}^{4}-T_{\mathrm{sys}}^{4}\right)\,

Constraint_satisfaction.html

  1. X = Y + 1 X=Y+1

Consumption_function.html

  1. C = a + b ( Y - T ) C=a+b(Y-T)
  2. C = c 0 + c 1 Y d C=c_{0}+c_{1}Y^{d}
  3. C = ( 1 / 2 ) ( B - ( y / ( U L C - C=(1/2)(B-(y/(ULC-
  4. ) ) + ( x / ))+(x/
  5. ) ) ))

Contingency_table.html

  1. ϕ = χ 2 N , \phi=\sqrt{\frac{\chi^{2}}{N}},
  2. C = χ 2 N + χ 2 C=\sqrt{\frac{\chi^{2}}{N+\chi^{2}}}
  3. V = χ 2 N ( k - 1 ) , V=\sqrt{\frac{\chi^{2}}{N(k-1)}},
  4. k - 1 k \sqrt{\frac{k-1}{k}}

Continuous-time_Markov_chain.html

  1. Pr ( X ( t + h ) = j | X ( t ) = i ) = δ i j + q i j h + o ( h ) \Pr(X(t+h)=j|X(t)=i)=\delta_{ij}+q_{ij}h+o(h)
  2. Pr ( X t n + 1 = i n + 1 | X t 0 = i 0 , X t 1 = i 1 , , X t n = i n ) = p i n i n + 1 ( t n + 1 - t n ) \Pr(X_{t_{n+1}}=i_{n+1}|X_{t_{0}}=i_{0},X_{t_{1}}=i_{1},\ldots,X_{t_{n}}=i_{n}% )=p_{i_{n}i_{n+1}}(t_{n+1}-t_{n})
  3. P ( t ) = P ( t ) Q P^{\prime}(t)=P(t)Q
  4. Pr i ( X t = j for some t 0 ) > 0. \operatorname{Pr}_{i}(X_{t}=j\,\text{ for some }t\geq 0)>0.
  5. Pr i ( { t 0 : X t = i } is unbounded ) = 1 \operatorname{Pr}_{i}(\{t\geq 0:X_{t}=i\}\,\text{ is unbounded})=1
  6. Pr i ( { t 0 : X t = i } is unbounded ) = 0. \operatorname{Pr}_{i}(\{t\geq 0:X_{t}=i\}\,\text{ is unbounded})=0.
  7. P ( t ) = P ( t ) Q P^{\prime}(t)=P(t)Q
  8. P ( t ) = e t Q P(t)=e^{tQ}
  9. Q = ( - α α β - β ) . Q=\begin{pmatrix}-\alpha&\alpha\\ \beta&-\beta\end{pmatrix}.
  10. P ( t ) = ( β α + β + α α + β e - ( α + β ) t α α + β - α α + β e - ( α + β ) t β α + β - β α + β e - ( α + β ) t α α + β + β α + β e - ( α + β ) t ) P(t)=\begin{pmatrix}\frac{\beta}{\alpha+\beta}+\frac{\alpha}{\alpha+\beta}e^{-% (\alpha+\beta)t}&\frac{\alpha}{\alpha+\beta}-\frac{\alpha}{\alpha+\beta}e^{-(% \alpha+\beta)t}\\ \frac{\beta}{\alpha+\beta}-\frac{\beta}{\alpha+\beta}e^{-(\alpha+\beta)t}&% \frac{\alpha}{\alpha+\beta}+\frac{\beta}{\alpha+\beta}e^{-(\alpha+\beta)t}\end% {pmatrix}
  11. P ( t + s ) = e ( t + s ) Q = e t Q e s Q = P ( t ) P ( s ) P(t+s)=e^{(t+s)Q}=e^{tQ}e^{sQ}=P(t)P(s)
  12. P ( t ) = ( β α + β + α α + β e - ( α + β ) t α α + β - α α + β e - ( α + β ) t β α + β - β α + β e - ( α + β ) t α α + β + β α + β e - ( α + β ) t ) P(t)=\begin{pmatrix}\frac{\beta}{\alpha+\beta}+\frac{\alpha}{\alpha+\beta}e^{-% (\alpha+\beta)t}&\frac{\alpha}{\alpha+\beta}-\frac{\alpha}{\alpha+\beta}e^{-(% \alpha+\beta)t}\\ \frac{\beta}{\alpha+\beta}-\frac{\beta}{\alpha+\beta}e^{-(\alpha+\beta)t}&% \frac{\alpha}{\alpha+\beta}+\frac{\beta}{\alpha+\beta}e^{-(\alpha+\beta)t}\end% {pmatrix}
  13. P π = ( β α + β α α + β β α + β α α + β ) P_{\pi}=\begin{pmatrix}\frac{\beta}{\alpha+\beta}&\frac{\alpha}{\alpha+\beta}% \\ \frac{\beta}{\alpha+\beta}&\frac{\alpha}{\alpha+\beta}\end{pmatrix}
  14. π Q = 0. \pi Q=0.
  15. i S π i = 1. \sum_{i\in S}\pi_{i}=1.
  16. Q = ( - 0.025 0.02 0.005 0.3 - 0.5 0.2 0.02 0.4 - 0.42 ) . Q=\begin{pmatrix}-0.025&0.02&0.005\\ 0.3&-0.5&0.2\\ 0.02&0.4&-0.42\end{pmatrix}.
  17. π = ( 0.885 0.071 0.044 ) . \pi=\begin{pmatrix}0.885&0.071&0.044\end{pmatrix}.
  18. k i A = 0 for i A - j S q i j k j A = 1 for i A . \begin{aligned}\displaystyle k_{i}^{A}=0&\displaystyle\,\text{ for }i\in A\\ \displaystyle-\sum_{j\in S}q_{ij}k_{j}^{A}=1&\displaystyle\,\text{ for }i% \notin A.\end{aligned}
  19. X ^ t = X T - t \scriptstyle\hat{X}_{t}=X_{T-t}
  20. s i j = { q i j k i q i k if i j 0 otherwise . s_{ij}=\begin{cases}\frac{q_{ij}}{\sum_{k\neq i}q_{ik}}&\,\text{if }i\neq j\\ 0&\,\text{otherwise}.\end{cases}
  21. S = I - ( diag ( Q ) ) - 1 Q S=I-\left(\operatorname{diag}(Q)\right)^{-1}Q
  22. ϕ \phi
  23. ϕ S = ϕ , \phi S=\phi,\,
  24. ϕ \phi
  25. ϕ \phi
  26. || ϕ || 1 ||\phi||_{1}
  27. π = - ϕ ( diag ( Q ) ) - 1 ϕ ( diag ( Q ) ) - 1 1 . \pi={-\phi(\operatorname{diag}(Q))^{-1}\over\left\|\phi(\operatorname{diag}(Q)% )^{-1}\right\|_{1}}.

Continuous_functional_calculus.html

  1. π ( f ) = f x . \pi(f)=f\circ x.

Continuous_phase_modulation.html

  1. s ( t ) = A c cos ( 2 π f c t + D f - t m ( α ) d α ) s(t)=A_{c}\cos\left(2\pi f_{c}t+D_{f}\int_{-\infty}^{t}m(\alpha)d\alpha\right)\,

Continuous_wavelet_transform.html

  1. x ( t ) x(t)
  2. a + * a\in\mathbb{R^{+*}}
  3. b b\in\mathbb{R}
  4. X w ( a , b ) = 1 | a | 1 / 2 - x ( t ) ψ ¯ ( t - b a ) d t X_{w}(a,b)=\frac{1}{|a|^{1/2}}\int_{-\infty}^{\infty}x(t)\overline{\psi}\left(% \frac{t-b}{a}\right)\,dt
  5. ψ ( t ) \psi(t)
  6. x ( t ) x(t)
  7. x ( t ) = C ψ - 1 - - X w ( a , b ) 1 | a | 1 / 2 ψ ~ ( t - b a ) d b d a a 2 x(t)=C_{\psi}^{-1}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}X_{w}(a,b)% \frac{1}{|a|^{1/2}}\tilde{\psi}\left(\frac{t-b}{a}\right)\,db\ \frac{da}{a^{2}}
  8. ψ ~ ( t ) \tilde{\psi}(t)
  9. ψ ( t ) \psi(t)
  10. C ψ = - ψ ^ ¯ ( ω ) ψ ~ ^ ( ω ) | ω | d ω C_{\psi}=\int_{-\infty}^{\infty}\frac{\overline{\hat{\psi}}(\omega)\hat{\tilde% {\psi}}(\omega)}{|\omega|}\,d\omega
  11. ψ ~ ( t ) = ψ ( t ) \tilde{\psi}(t)=\psi(t)
  12. C ψ = - + | ψ ^ ( ω ) | 2 | ω | d ω C_{\psi}=\int_{-\infty}^{+\infty}\frac{\left|\hat{\psi}(\omega)\right|^{2}}{% \left|\omega\right|}d\omega
  13. 0 < C ψ < 0<C_{\psi}<\infty
  14. ψ ^ ( 0 ) = 0 \hat{\psi}(0)=0
  15. x ( t ) x(t)
  16. x ( t ) = 1 2 π ψ ^ ¯ ( 1 ) - - 1 a 2 X w ( a , b ) exp ( i t - b a ) d b d a x(t)=\frac{1}{2\pi\overline{\hat{\psi}}(1)}\int_{-\infty}^{\infty}\int_{-% \infty}^{\infty}\frac{1}{a^{2}}X_{w}(a,b)\exp\left(i\frac{t-b}{a}\right)\,db\ da
  17. ψ ( t ) = w ( t ) exp ( i t ) \psi(t)=w(t)\exp(it)
  18. w ( t ) w(t)
  19. a a
  20. X w ( a , b ) X_{w}(a,b)
  21. | X w ( a , b ) | 2 |X_{w}(a,b)|^{2}

Continuously_variable_slope_delta_modulation.html

  1. τ \tau
  2. τ \tau

Convergence_of_Fourier_series.html

  1. f ^ ( n ) \widehat{f}(n)
  2. f ^ ( n ) = 1 2 π 0 2 π f ( t ) e - i n t d t , n 𝐙 . \widehat{f}(n)=\frac{1}{2\pi}\int_{0}^{2\pi}f(t)e^{-int}\,dt,\quad n\in\mathbf% {Z}.
  3. f n f ^ ( n ) e i n t . f\sim\sum_{n}\widehat{f}(n)e^{int}.
  4. S N ( f ; t ) = n = - N N f ^ ( n ) e i n t . S_{N}(f;t)=\sum_{n=-N}^{N}\widehat{f}(n)e^{int}.
  5. S N ( f ) S_{N}(f)
  6. f ^ ( n ) \widehat{f}(n)
  7. S N S_{N}
  8. S N ( f ) = f * D N S_{N}(f)=f*D_{N}\,
  9. D N D_{N}
  10. D n ( t ) = sin ( ( n + 1 2 ) t ) sin ( t / 2 ) . D_{n}(t)=\frac{\sin((n+\frac{1}{2})t)}{\sin(t/2)}.
  11. | D n ( t ) | d t \int|D_{n}(t)|\,dt\to\infty
  12. f f
  13. | f ^ ( n ) | K | n | \left|\widehat{f}(n)\right|\leq{K\over|n|}
  14. K K
  15. f f
  16. f f
  17. | f ^ ( n ) | var ( f ) 2 π | n | . \left|\widehat{f}(n)\right|\leq{{\rm var}(f)\over 2\pi|n|}.
  18. f C p f\in C^{p}
  19. | f ^ ( n ) | f ( p ) L 1 | n | p . \left|\widehat{f}(n)\right|\leq{\|f^{(p)}\|_{L_{1}}\over|n|^{p}}.
  20. f C p f\in C^{p}
  21. f ( p ) f^{(p)}
  22. ω p \omega_{p}
  23. | f ^ ( n ) | ω ( 2 π / n ) | n | p \left|\widehat{f}(n)\right|\leq{\omega(2\pi/n)\over|n|^{p}}
  24. f f
  25. | f ^ ( n ) | K | n | α . \left|\widehat{f}(n)\right|\leq{K\over|n|^{\alpha}}.
  26. 0 π | f ( x 0 + t ) + f ( x 0 - t ) 2 - | d t t < , \int_{0}^{\pi}\Bigl|\frac{f(x_{0}+t)+f(x_{0}-t)}{2}-\ell\Bigr|\,\frac{\mathrm{% d}t}{t}<\infty,
  27. f C p f\in C^{p}
  28. f ( p ) f^{(p)}
  29. ω \omega
  30. ω \omega
  31. | f ( x ) - ( S N f ) ( x ) | K ln N N p ω ( 2 π / N ) |f(x)-(S_{N}f)(x)|\leq K{\ln N\over N^{p}}\omega(2\pi/N)
  32. K K
  33. f f
  34. p p
  35. N N
  36. f f
  37. α \alpha
  38. | f ( x ) - ( S N f ) ( x ) | K ln N N α . |f(x)-(S_{N}f)(x)|\leq K{\ln N\over N^{\alpha}}.
  39. f f
  40. 2 π 2\pi
  41. [ 0 , 2 π ] [0,2\pi]
  42. f f
  43. f f
  44. f A := n = - | f ^ ( n ) | < . \|f\|_{A}:=\sum_{n=-\infty}^{\infty}|\widehat{f}(n)|<\infty.
  45. ( S N f ) ( t ) (S_{N}f)(t)
  46. ( S N f ) ( t ) (S_{N}f)(t)
  47. f f
  48. f A c α f Lip α , f K := n = - + | n | | f ^ ( n ) | 2 c α f Lip α 2 \|f\|_{A}\leq c_{\alpha}\|f\|_{{\rm Lip}_{\alpha}},\qquad\|f\|_{K}:=\sum_{n=-% \infty}^{+\infty}|n||\widehat{f}(n)|^{2}\leq c_{\alpha}\|f\|^{2}_{{\rm Lip}_{% \alpha}}
  49. f Lip α \|f\|_{{\rm Lip}_{\alpha}}
  50. c α c_{\alpha}
  51. α \alpha
  52. f K \|f\|_{K}
  53. O ( 1 / n α ) O(1/n^{\alpha})
  54. lim N 0 2 π | f ( x ) - S N ( f ) | 2 d x = 0 \lim_{N\rightarrow\infty}\int_{0}^{2\pi}\left|f(x)-S_{N}(f)\right|^{2}\,dx=0
  55. S N f S_{N}f
  56. S N ( f ) S_{N}(f)
  57. a n a_{n}
  58. lim n 1 n k = 1 n a k = a . \lim_{n\to\infty}\frac{1}{n}\sum_{k=1}^{n}a_{k}=a.
  59. S N S_{N}
  60. K N ( f ; t ) = 1 N n = 0 N - 1 S n ( f ; t ) , N 1 , K_{N}(f;t)=\frac{1}{N}\sum_{n=0}^{N-1}S_{n}(f;t),\quad N\geq 1,
  61. K N ( f ) K_{N}(f)
  62. K N K_{N}
  63. K N ( f ) = f * F N K_{N}(f)=f*F_{N}\,
  64. F N F_{N}
  65. F N = 1 N n = 0 N - 1 D n . F_{N}=\frac{1}{N}\sum_{n=0}^{N-1}D_{n}.
  66. K N f K_{N}f
  67. K N f K_{N}f
  68. L 1 L^{1}
  69. S N ( f ; t ) S_{N}(f;t)
  70. | D N ( t ) | d t = 4 π 2 log N + O ( 1 ) . \int|D_{N}(t)|\,dt=\frac{4}{\pi^{2}}\log N+O(1).
  71. 4 / π 2 4/\pi^{2}
  72. | D N ( t ) | d t > c log N + O ( 1 ) \int|D_{N}(t)|\,dt>c\log N+O(1)
  73. lim N S N ( f ; t ) log N = 0. \lim_{N\to\infty}\frac{S_{N}(f;t)}{\log N}=0.
  74. lim ¯ N S N ( f ; t ) ω ( N ) = . \overline{\lim}_{N\to\infty}\frac{S_{N}(f;t)}{\omega(N)}=\infty.
  75. lim ¯ N S N ( f ; t ) log N = . \overline{\lim}_{N\to\infty}\frac{S_{N}(f;t)}{\sqrt{\log N}}=\infty.
  76. S N ( f ; t 1 , t 2 ) = | n 1 | N , | n 2 | N f ^ ( n 1 , n 2 ) e i ( n 1 t 1 + n 2 t 2 ) S_{N}(f;t_{1},t_{2})=\sum_{|n_{1}|\leq N,|n_{2}|\leq N}\widehat{f}(n_{1},n_{2}% )e^{i(n_{1}t_{1}+n_{2}t_{2})}
  77. n 1 2 + n 2 2 N 2 \sum_{n_{1}^{2}+n_{2}^{2}\leq N^{2}}
  78. log 2 N \log^{2}N
  79. N \sqrt{N}
  80. L p L^{p}
  81. log n \sqrt{\log n}

Convex_combination.html

  1. x 1 , x 2 , , x n x_{1},x_{2},\dots,x_{n}\,
  2. α 1 x 1 + α 2 x 2 + + α n x n \alpha_{1}x_{1}+\alpha_{2}x_{2}+\cdots+\alpha_{n}x_{n}
  3. α i \alpha_{i}\,
  4. α i 0 \alpha_{i}\geq 0
  5. α 1 + α 2 + + α n = 1. \alpha_{1}+\alpha_{2}+\cdots+\alpha_{n}=1.
  6. [ 0 , 1 ] [0,1]
  7. X X
  8. Y i Y_{i}
  9. α i \alpha_{i}
  10. f X ( x ) = i = 1 n α i f Y i ( x ) f_{X}(x)=\sum_{i=1}^{n}\alpha_{i}f_{Y_{i}}(x)

Convex_conjugate.html

  1. X X
  2. X * X^{*}
  3. X X
  4. , : X * × X . \langle\cdot,\cdot\rangle:X^{*}\times X\to\mathbb{R}.
  5. f : X { + } f:X\to\mathbb{R}\cup\{+\infty\}
  6. f : X * { + } f^{\star}:X^{*}\to\mathbb{R}\cup\{+\infty\}
  7. f ( x * ) := sup { x * , x - f ( x ) | x X } , f^{\star}\left(x^{*}\right):=\sup\left\{\left.\left\langle x^{*},x\right% \rangle-f\left(x\right)\right|x\in X\right\},
  8. f ( x * ) := - inf { f ( x ) - x * , x | x X } . f^{\star}\left(x^{*}\right):=-\inf\left\{\left.f\left(x\right)-\left\langle x^% {*},x\right\rangle\right|x\in X\right\}.
  9. f ( x ) = a , x - b , a n , b f(x)=\left\langle a,x\right\rangle-b,\,a\in\mathbb{R}^{n},b\in\mathbb{R}
  10. f ( x * ) = { b , x * = a + , x * a . f^{\star}\left(x^{*}\right)=\begin{cases}b,&x^{*}=a\\ +\infty,&x^{*}\neq a.\end{cases}
  11. f ( x ) = 1 p | x | p , 1 < p < f(x)=\frac{1}{p}|x|^{p},\,1<p<\infty
  12. f ( x * ) = 1 q | x * | q , 1 < q < f^{\star}\left(x^{*}\right)=\frac{1}{q}|x^{*}|^{q},\,1<q<\infty
  13. 1 p + 1 q = 1. \tfrac{1}{p}+\tfrac{1}{q}=1.
  14. f ( x ) = | x | f(x)=\left|x\right|
  15. f ( x * ) = { 0 , | x * | 1 , | x * | > 1. f^{\star}\left(x^{*}\right)=\begin{cases}0,&\left|x^{*}\right|\leq 1\\ \infty,&\left|x^{*}\right|>1.\end{cases}
  16. f ( x ) = e x f(x)=\,\!e^{x}
  17. f ( x * ) = { x * ln x * - x * , x * > 0 0 , x * = 0 , x * < 0. f^{\star}\left(x^{*}\right)=\begin{cases}x^{*}\ln x^{*}-x^{*},&x^{*}>0\\ 0,&x^{*}=0\\ \infty,&x^{*}<0.\end{cases}
  18. f ( x ) := - x F ( u ) d u = E [ max ( 0 , x - X ) ] = x - E [ min ( x , X ) ] f(x):=\int_{-\infty}^{x}F(u)\,du=\operatorname{E}\left[\max(0,x-X)\right]=x-% \operatorname{E}\left[\min(x,X)\right]
  19. f ( p ) = 0 p F - 1 ( q ) d q = ( p - 1 ) F - 1 ( p ) + E [ min ( F - 1 ( p ) , X ) ] = p F - 1 ( p ) - E [ max ( 0 , F - 1 ( p ) - X ) ] . f^{\star}(p)=\int_{0}^{p}F^{-1}(q)\,dq=(p-1)F^{-1}(p)+\operatorname{E}\left[% \min(F^{-1}(p),X)\right]=pF^{-1}(p)-\operatorname{E}\left[\max(0,F^{-1}(p)-X)% \right].
  20. f inc ( x ) := arg sup t t x - 0 1 max { t - f ( u ) , 0 } d u , f\text{inc}(x):=\arg\sup_{t}\,t\cdot x-\int_{0}^{1}\max\{t-f(u),0\}\,\mathrm{d% }u,
  21. f inc = f f\text{inc}=f
  22. f g f\leq g
  23. f * g * f^{*}\geq g^{*}
  24. ( f g ) : ( x , f ( x ) g ( x ) ) . (f\leq g):\iff(\forall x,f(x)\leq g(x)).
  25. ( f α ) α \left(f_{\alpha}\right)_{\alpha}
  26. ( inf α f α ) * ( x ) = sup α f α * ( x ) , \left(\inf_{\alpha}f_{\alpha}\right)^{*}(x)=\sup_{\alpha}f_{\alpha}^{*}(x),
  27. ( sup α f α ) * ( x ) inf α f α * ( x ) . \left(\sup_{\alpha}f_{\alpha}\right)^{*}(x)\leq\inf_{\alpha}f_{\alpha}^{*}(x).
  28. f * * f^{**}
  29. f * * f f^{**}\leq f
  30. f = f * * f=f^{**}
  31. f f
  32. f * f*
  33. x X x∈X
  34. p X * p∈X*
  35. p , x f ( x ) + f * ( p ) . \left\langle p,x\right\rangle\leq f(x)+f^{*}(p).
  36. f 0 f_{0}
  37. f 1 f_{1}
  38. 0 λ 1 0\leq\lambda\leq 1
  39. ( ( 1 - λ ) f 0 + λ f 1 ) ( 1 - λ ) f 0 + λ f 1 \left((1-\lambda)f_{0}+\lambda f_{1}\right)^{\star}\leq(1-\lambda)f_{0}^{\star% }+\lambda f_{1}^{\star}
  40. \star
  41. ( f g ) ( x ) = inf { f ( x - y ) + g ( y ) | y n } . \left(f\Box g\right)(x)=\inf\left\{f(x-y)+g(y)\,|\,y\in\mathbb{R}^{n}\right\}.
  42. ( f 1 f m ) = f 1 + + f m . \left(f_{1}\Box\cdots\Box f_{m}\right)^{\star}=f_{1}^{\star}+\cdots+f_{m}^{% \star}.
  43. f f
  44. f ( x ) = x * ( x ) := arg sup x x , x - f ( x ) f^{\prime}(x)=x^{*}(x):=\arg\sup_{x^{\star}}{\langle x,x^{\star}\rangle}-f^{% \star}(x^{\star})
  45. f ( x ) = x ( x ) := arg sup x x , x - f ( x ) ; f^{\star\prime}(x^{\star})=x(x^{\star}):=\arg\sup_{x}{\langle x,x^{\star}% \rangle}-f(x);
  46. x = f ( f ( x ) ) , x=\nabla f^{\star}(\nabla f(x)),
  47. x = f ( f ( x ) ) , x^{\star}=\nabla f(\nabla f^{\star}(x^{\star})),
  48. f ′′ ( x ) f ′′ ( x ( x ) ) = 1 , f^{\prime\prime}(x)\cdot f^{\star\prime\prime}(x^{\star}(x))=1,
  49. f ′′ ( x ) f ′′ ( x ( x ) ) = 1. f^{\star\prime\prime}(x^{\star})\cdot f^{\prime\prime}(x(x^{\star}))=1.
  50. γ > 0 \gamma>0
  51. g ( x ) = α + β x + γ f ( λ x + δ ) \,g(x)=\alpha+\beta x+\gamma\cdot f(\lambda x+\delta)
  52. g ( x ) = - α - δ x - β λ + γ f ( x - β λ γ ) . g^{\star}(x^{\star})=-\alpha-\delta\frac{x^{\star}-\beta}{\lambda}+\gamma\cdot f% ^{\star}\left(\frac{x^{\star}-\beta}{\lambda\gamma}\right).
  53. f α ( x ) = - f α ( x ~ ) , f_{\alpha}(x)=-f_{\alpha}(\tilde{x}),
  54. x ~ \tilde{x}
  55. ( A f ) = f A \left(Af\right)^{\star}=f^{\star}A^{\star}
  56. ( A f ) ( y ) = inf { f ( x ) : x X , A x = y } (Af)(y)=\inf\{f(x):x\in X,Ax=y\}
  57. f ( A x ) = f ( x ) , x , A G f\left(Ax\right)=f(x),\;\forall x,\;\forall A\in G
  58. g ( x ) g(x)
  59. dom ( g ) \operatorname{dom}(g)
  60. g * ( x * ) g^{*}(x^{*})
  61. dom ( g * ) \operatorname{dom}(g^{*})
  62. f ( a x ) f(ax)
  63. a 0 a\neq 0
  64. X X
  65. f * ( x * a ) f^{*}\left(\frac{x^{*}}{a}\right)
  66. X * X^{*}
  67. f ( x + b ) f(x+b)
  68. X X
  69. f * ( x * ) - b , x * f^{*}(x^{*})-\langle b,x^{*}\rangle
  70. X * X^{*}
  71. a f ( x ) af(x)
  72. a > 0 a>0
  73. X X
  74. a f * ( x * a ) af^{*}\left(\frac{x^{*}}{a}\right)
  75. X * X^{*}
  76. α + β x + γ f ( λ x + δ ) \alpha+\beta x+\gamma\cdot f(\lambda x+\delta)
  77. X X
  78. - α - δ x * - β λ + γ f * ( x * - β γ λ ) ( γ > 0 ) -\alpha-\delta\frac{x^{*}-\beta}{\lambda}+\gamma\cdot f^{*}\left(\frac{x^{*}-% \beta}{\gamma\lambda}\right)\quad(\gamma>0)
  79. X * X^{*}
  80. | x | p p \frac{|x|^{p}}{p}
  81. p > 1 p>1
  82. \mathbb{R}
  83. | x * | q q \frac{|x^{*}|^{q}}{q}
  84. 1 p + 1 q = 1 \frac{1}{p}+\frac{1}{q}=1
  85. \mathbb{R}
  86. - x p p \frac{-x^{p}}{p}
  87. 0 < p < 1 0<p<1
  88. + \mathbb{R}_{+}
  89. - ( - x * ) q q \frac{-(-x^{*})^{q}}{q}
  90. 1 p + 1 q = 1 \frac{1}{p}+\frac{1}{q}=1
  91. - - \mathbb{R}_{--}
  92. 1 + x 2 \sqrt{1+x^{2}}
  93. \mathbb{R}
  94. - 1 - ( x * ) 2 -\sqrt{1-(x^{*})^{2}}
  95. [ - 1 , 1 ] [-1,1]
  96. - log ( x ) -\log(x)
  97. + + \mathbb{R}_{++}
  98. - ( 1 + log ( - x * ) ) -(1+\log(-x^{*}))
  99. - - \mathbb{R}_{--}
  100. e x e^{x}
  101. \mathbb{R}
  102. { x * log ( x * ) - x * if x * > 0 0 if x * = 0 \begin{cases}x^{*}\log(x^{*})-x^{*}&\,\text{if }x^{*}>0\\ 0&\,\text{if }x^{*}=0\end{cases}
  103. + \mathbb{R}_{+}
  104. log ( 1 + e x ) \log\left(1+e^{x}\right)
  105. \mathbb{R}
  106. { x * log ( x * ) + ( 1 - x * ) log ( 1 - x * ) if 0 < x * < 1 0 if x * = 0 , 1 \begin{cases}x^{*}\log(x^{*})+(1-x^{*})\log(1-x^{*})&\,\text{if }0<x^{*}<1\\ 0&\,\text{if }x^{*}=0,1\end{cases}
  107. [ 0 , 1 ] [0,1]
  108. - log ( 1 - e x ) -\log\left(1-e^{x}\right)
  109. \mathbb{R}
  110. { x * log ( x * ) - ( 1 + x * ) log ( 1 + x * ) if x * > 0 0 if x * = 0 \begin{cases}x^{*}\log(x^{*})-(1+x^{*})\log(1+x^{*})&\,\text{if }x^{*}>0\\ 0&\,\text{if }x^{*}=0\end{cases}
  111. + \mathbb{R}_{+}

Convex_polygon.html

  1. 0.5 × Area ( R ) Area ( C ) 2 × Area ( r ) 0.5\,\text{ × Area}(R)\leq\,\text{Area}(C)\leq 2\,\text{ × Area}(r)

Conway's_LUX_method_for_magic_squares.html

  1. L : 4 1 2 3 U : 1 4 2 3 X : 1 4 3 2 \mathrm{L}:\quad\begin{smallmatrix}4&&1\\ &\swarrow&\\ 2&\rightarrow&3\end{smallmatrix}\qquad\mathrm{U}:\quad\begin{smallmatrix}1&&4% \\ \downarrow&&\uparrow\\ 2&\rightarrow&3\end{smallmatrix}\qquad\mathrm{X}:\quad\begin{smallmatrix}1&&4% \\ &\searrow\!\!\!\!\!\!\nearrow&\\ 3&&2\end{smallmatrix}

Cook–Levin_theorem.html

  1. \bigvee
  2. \bigvee
  3. \bigvee
  4. \bigvee

Cooling_tower.html

  1. M X M = D X C + W X C = X C ( D + W ) MX_{M}=DX_{C}+WX_{C}=X_{C}(D+W)
  2. X C X M = Cycles of concentration = M ( D + W ) = M ( M - E ) = 1 + E ( D + W ) {X_{C}\over X_{M}}=\,\text{Cycles of concentration}={M\over(D+W)}={M\over(M-E)% }=1+{E\over(D+W)}
  3. E = C Δ T c p H V E={C\Delta Tc_{p}\over H_{V}}
  4. \cdot

Cooper_test.html

  1. VO 2 max = d 12 - 504.9 44.73 \mathrm{VO_{2}\;max}={d_{12}-504.9\over 44.73}
  2. VO 2 max = ( 35.97 * d m i l e s 12 ) - 11.29 \mathrm{VO_{2}\;max}={(35.97*dmiles_{12})-11.29}

Cooperative_game.html

  1. N N
  2. v : 2 N v:2^{N}\to\mathbb{R}
  3. v ( ) = 0 v(\emptyset)=0
  4. c : 2 N c:2^{N}\to\mathbb{R}
  5. c ( ) = 0 c(\emptyset)=0
  6. c c
  7. v v
  8. v v
  9. v * v^{*}
  10. v * ( S ) = v ( N ) - v ( N S ) , S N . v^{*}(S)=v(N)-v(N\setminus S),\forall~{}S\subseteq N.\,
  11. S S
  12. N N
  13. c * c^{*}
  14. c c
  15. S N S\subsetneq N
  16. v S : 2 S v_{S}:2^{S}\to\mathbb{R}
  17. S S
  18. v S ( T ) = v ( T ) , T S . v_{S}(T)=v(T),\forall~{}T\subseteq S.\,
  19. S S
  20. v ( S T ) v ( S ) + v ( T ) v(S\cup T)\geq v(S)+v(T)
  21. S , T N S,T\subseteq N
  22. S T = S\cap T=\emptyset
  23. S T v ( S ) v ( T ) S\subseteq T\Rightarrow v(S)\leq v(T)
  24. v v
  25. W W
  26. W W
  27. W W
  28. S W S\in W
  29. S T S\subseteq T
  30. T W T\in W
  31. W W
  32. S W S\in W
  33. N S W N\setminus S\notin W
  34. W W
  35. S W S\notin W
  36. N S W N\setminus S\in W
  37. W W
  38. S W S\in W
  39. N S W N\setminus S\notin W
  40. v v
  41. v ( S ) = 1 - v ( N S ) v(S)=1-v(N\setminus S)
  42. S S
  43. W W
  44. W := S W S \bigcap W:=\bigcap_{S\in W}S
  45. W W
  46. T N T\subseteq N
  47. S S
  48. S W S\in W
  49. S T W S\cap T\in W
  50. N N
  51. N N
  52. v ( N ) v(N)
  53. x N x\in\mathbb{R}^{N}
  54. i N x i = v ( N ) \sum_{i\in N}x_{i}=v(N)
  55. x i v ( { i } ) , i N x_{i}\geq v(\{i\}),\forall~{}i\in N
  56. v v
  57. v v
  58. | N | |N|
  59. x x
  60. x i = x j x_{i}=x_{j}
  61. i i
  62. j j
  63. i i
  64. j j
  65. v ( S { i } ) = v ( S { j } ) , S N { i , j } v(S\cup\{i\})=v(S\cup\{j\}),\forall~{}S\subseteq N\setminus\{i,j\}
  66. v v
  67. ω \omega
  68. ( v + ω ) (v+\omega)
  69. ( v + ω ) (v+\omega)
  70. v v
  71. ω \omega
  72. i i
  73. v ( S { i } ) = v ( S ) , S N { i } v(S\cup\{i\})=v(S),\forall~{}S\subseteq N\setminus\{i\}
  74. v v
  75. x x
  76. y y
  77. v v
  78. x x
  79. y y
  80. S S\neq\emptyset
  81. x i > y i , i S x_{i}>y_{i},\forall~{}i\in S
  82. i S x i v ( S ) \sum_{i\in S}x_{i}\leq v(S)
  83. S S
  84. x x
  85. y y
  86. y y
  87. x x
  88. n - 2 n-2
  89. n - 3 n-3
  90. v v
  91. v v
  92. C ( v ) = { x N : i N x i = v ( N ) ; i S x i v ( S ) , S N } . C(v)=\left\{x\in\mathbb{R}^{N}:\sum_{i\in N}x_{i}=v(N);\quad\sum_{i\in S}x_{i}% \geq v(S),\forall~{}S\subseteq N\right\}.\,
  93. X X
  94. p = ( i p ) i N p=(\succ_{i}^{p})_{i\in N}
  95. i p \succ_{i}^{p}
  96. X X
  97. x i p y x\succ_{i}^{p}y
  98. i i
  99. x x
  100. y y
  101. p p
  102. v v
  103. p p
  104. v p \succ^{p}_{v}
  105. X X
  106. x v p y x\succ^{p}_{v}y
  107. S S
  108. v ( S ) = 1 v(S)=1
  109. x i p y x\succ_{i}^{p}y
  110. i S i\in S
  111. C ( v , p ) C(v,p)
  112. v v
  113. p p
  114. v p \succ^{p}_{v}
  115. X X
  116. v p \succ^{p}_{v}
  117. x C ( v , p ) x\in C(v,p)
  118. y X y\in X
  119. y v p x y\succ^{p}_{v}x
  120. C ( v , p ) C(v,p)
  121. p p
  122. X X
  123. X X
  124. v v
  125. C ( v , p ) C(v,p)
  126. p p
  127. X X
  128. v v
  129. ε \varepsilon
  130. ε \varepsilon\in\mathbb{R}
  131. C ε ( v ) = { x N : i N x i = v ( N ) ; i S x i v ( S ) - ε , S N } . C_{\varepsilon}(v)=\left\{x\in\mathbb{R}^{N}:\sum_{i\in N}x_{i}=v(N);\quad\sum% _{i\in S}x_{i}\geq v(S)-\varepsilon,\forall~{}S\subseteq N\right\}.
  132. ε \varepsilon
  133. ε \varepsilon
  134. ε \varepsilon
  135. ε \varepsilon
  136. ε \varepsilon
  137. ε \varepsilon
  138. ε \varepsilon
  139. ε \varepsilon
  140. ε \varepsilon
  141. v : 2 N v:2^{N}\to\mathbb{R}
  142. x N x\in\mathbb{R}^{N}
  143. s i j v ( x ) = max { v ( S ) - k S x k : S N { j } , i S } , s_{ij}^{v}(x)=\max\left\{v(S)-\sum_{k\in S}x_{k}:S\subseteq N\setminus\{j\},i% \in S\right\},
  144. v v
  145. ( s i j v ( x ) - s j i v ( x ) ) × ( x j - v ( j ) ) 0 (s_{ij}^{v}(x)-s_{ji}^{v}(x))\times(x_{j}-v(j))\leq 0
  146. ( s j i v ( x ) - s i j v ( x ) ) × ( x i - v ( i ) ) 0 (s_{ji}^{v}(x)-s_{ij}^{v}(x))\times(x_{i}-v(i))\leq 0
  147. s i j v ( x ) > s j i v ( x ) s_{ij}^{v}(x)>s_{ji}^{v}(x)
  148. x j = v ( j ) x_{j}=v(j)
  149. v : 2 N v:2^{N}\to\mathbb{R}
  150. x N x\in\mathbb{R}^{N}
  151. x x
  152. S N S\subseteq N
  153. v ( S ) - i S x i v(S)-\sum_{i\in S}x_{i}
  154. S S
  155. N N
  156. x x
  157. v ( S ) v(S)
  158. θ ( x ) 2 N \theta(x)\in\mathbb{R}^{2^{N}}
  159. x x
  160. θ i ( x ) θ j ( x ) , i < j \theta_{i}(x)\geq\theta_{j}(x),\forall~{}i<j
  161. x x
  162. v v
  163. θ 1 ( x ) 0 \theta_{1}(x)\leq 0
  164. 2 N \mathbb{R}^{2^{N}}
  165. x , y x,y
  166. θ ( x ) \theta(x)
  167. θ ( y ) \theta(y)
  168. k k
  169. θ i ( x ) = θ i ( y ) , i < k \theta_{i}(x)=\theta_{i}(y),\forall~{}i<k
  170. θ k ( x ) < θ k ( y ) \theta_{k}(x)<\theta_{k}(y)
  171. v v
  172. C ε ( v ) C_{\varepsilon}(v)
  173. v v
  174. v ( S T ) + v ( S T ) v ( S ) + v ( T ) , S , T N . v(S\cup T)+v(S\cap T)\geq v(S)+v(T),\forall~{}S,T\subseteq N.\,
  175. v v
  176. v ( S { i } ) - v ( S ) v ( T { i } ) - v ( T ) , S T N { i } , i N ; v(S\cup\{i\})-v(S)\leq v(T\cup\{i\})-v(T),\forall~{}S\subseteq T\subseteq N% \setminus\{i\},\forall~{}i\in N;\,
  177. π : N N \pi:N\to N
  178. S i = { j N : π ( j ) i } S_{i}=\{j\in N:\pi(j)\leq i\}
  179. 1 1
  180. i i
  181. π \pi
  182. i = 0 , , n i=0,\ldots,n
  183. S 0 = S_{0}=\emptyset
  184. x N x\in\mathbb{R}^{N}
  185. x i = v ( S π ( i ) ) - v ( S π ( i ) - 1 ) , i N x_{i}=v(S_{\pi(i)})-v(S_{\pi(i)-1}),\forall~{}i\in N
  186. v v
  187. π \pi
  188. 2 N 2^{N}
  189. N N

Coordinate_vector.html

  1. B = { b 1 , b 2 , , b n } B=\{b_{1},b_{2},\ldots,b_{n}\}
  2. v V v\in V
  3. v = α 1 b 1 + α 2 b 2 + + α n b n . v=\alpha_{1}b_{1}+\alpha_{2}b_{2}+\cdots+\alpha_{n}b_{n}.
  4. [ v ] B = ( α 1 , α 2 , , α n ) . [v]_{B}=(\alpha_{1},\alpha_{2},\cdots,\alpha_{n}).
  5. [ v ] B = [ α 1 α n ] [v]_{B}=\begin{bmatrix}\alpha_{1}\\ \vdots\\ \alpha_{n}\end{bmatrix}
  6. [ v ] B = [ α 1 α 2 α n ] . [v]_{B}=\begin{bmatrix}\alpha_{1}&\alpha_{2}&\dots&\alpha_{n}\end{bmatrix}.
  7. ϕ B \phi_{B}
  8. ϕ B ( v ) = [ v ] B \phi_{B}(v)=[v]_{B}
  9. ϕ B \phi_{B}
  10. ϕ B - 1 : F n V \phi_{B}^{-1}:F^{n}\to V
  11. ϕ B - 1 ( α 1 , , α n ) = α 1 b 1 + + α n b n . \phi_{B}^{-1}(\alpha_{1},\ldots,\alpha_{n})=\alpha_{1}b_{1}+\cdots+\alpha_{n}b% _{n}.
  12. ϕ B - 1 \phi_{B}^{-1}
  13. ϕ B - 1 \phi_{B}^{-1}
  14. ϕ B \phi_{B}
  15. B P = { 1 , x , x 2 , x 3 } B_{P}=\{1,x,x^{2},x^{3}\}
  16. 1 := [ 1 0 0 0 ] ; x := [ 0 1 0 0 ] ; x 2 := [ 0 0 1 0 ] ; x 3 := [ 0 0 0 1 ] 1:=\begin{bmatrix}1\\ 0\\ 0\\ 0\end{bmatrix}\quad;\quad x:=\begin{bmatrix}0\\ 1\\ 0\\ 0\end{bmatrix}\quad;\quad x^{2}:=\begin{bmatrix}0\\ 0\\ 1\\ 0\end{bmatrix}\quad;\quad x^{3}:=\begin{bmatrix}0\\ 0\\ 0\\ 1\end{bmatrix}\quad
  17. p ( x ) = a 0 + a 1 x + a 2 x 2 + a 3 x 3 p\left(x\right)=a_{0}+a_{1}x+a_{2}x^{2}+a_{3}x^{3}
  18. [ a 0 a 1 a 2 a 3 ] \begin{bmatrix}a_{0}\\ a_{1}\\ a_{2}\\ a_{3}\end{bmatrix}
  19. D p ( x ) = P ( x ) ; [ D ] = [ 0 1 0 0 0 0 2 0 0 0 0 3 0 0 0 0 ] Dp(x)=P^{\prime}(x)\quad;\quad[D]=\begin{bmatrix}0&1&0&0\\ 0&0&2&0\\ 0&0&0&3\\ 0&0&0&0\\ \end{bmatrix}
  20. [ M ] C B [M]_{C}^{B}
  21. [ M ] C B = [ [ b 1 ] C [ b n ] C ] [M]_{C}^{B}=\begin{bmatrix}\ [b_{1}]_{C}&\cdots&[b_{n}]_{C}\end{bmatrix}
  22. [ v ] C = [ M ] C B [ v ] B . [v]_{C}=[M]_{C}^{B}[v]_{B}.
  23. v = [ M ] B [ v ] B . v=[M]^{B}[v]_{B}.\,
  24. v = [ v ] E , v=[v]_{E},\,
  25. [ M ] B = [ M ] E B . [M]^{B}=[M]_{E}^{B}.
  26. [ M ] C B [ M ] B C = [ M ] C C = Id [M]_{C}^{B}[M]_{B}^{C}=[M]_{C}^{C}=\mathrm{Id}
  27. [ M ] B C [ M ] C B = [ M ] B B = Id [M]_{B}^{C}[M]_{C}^{B}=[M]_{B}^{B}=\mathrm{Id}
  28. [ v ] C = [ M ] C B [ v ] B , [v]_{C}=[M]_{C}^{B}[v]_{B},

Copolymer.html

  1. χ \chi
  2. χ N \chi N
  3. χ N \chi N
  4. d [ M 1 ] d [ M 2 ] = [ M 1 ] ( r 1 [ M 1 ] + [ M 2 ] ) [ M 2 ] ( [ M 1 ] + r 2 [ M 2 ] ) \frac{d\left[M_{1}\right]}{d\left[M_{2}\right]}=\frac{\left[M_{1}\right]\left(% r_{1}\left[M_{1}\right]+\left[M_{2}\right]\right)}{\left[M_{2}\right]\left(% \left[M_{1}\right]+r_{2}\left[M_{2}\right]\right)}

CORDIC.html

  1. β \beta
  2. v 0 v_{0}
  3. v 0 = [ 1 0 ] v_{0}=\begin{bmatrix}1\\ 0\end{bmatrix}
  4. v 1 v_{1}
  5. v i - 1 v_{i-1}
  6. R i R_{i}
  7. v i = R i v i - 1 v_{i}=R_{i}v_{i-1}
  8. R i = [ cos γ i - sin γ i sin γ i cos γ i ] R_{i}=\begin{bmatrix}\cos\gamma_{i}&-\sin\gamma_{i}\\ \sin\gamma_{i}&\cos\gamma_{i}\end{bmatrix}
  9. cos γ i \displaystyle\cos\gamma_{i}
  10. R i = 1 1 + tan 2 γ i [ 1 - tan γ i tan γ i 1 ] R_{i}={1\over\sqrt{1+\tan^{2}\gamma_{i}}}\begin{bmatrix}1&-\tan\gamma_{i}\\ \tan\gamma_{i}&1\end{bmatrix}
  11. v i = R i v i - 1 v_{i}=R_{i}v_{i-1}
  12. v i = 1 1 + tan 2 γ i [ 1 - tan γ i tan γ i 1 ] [ x i - 1 y i - 1 ] v_{i}={1\over\sqrt{1+\tan^{2}\gamma_{i}}}\begin{bmatrix}1&-\tan\gamma_{i}\\ \tan\gamma_{i}&1\end{bmatrix}\begin{bmatrix}x_{i-1}\\ y_{i-1}\end{bmatrix}
  13. x i - 1 x_{i-1}
  14. y i - 1 y_{i-1}
  15. v i - 1 v_{i-1}
  16. γ i \gamma_{i}
  17. tan γ i \tan\gamma_{i}
  18. ± 2 - i \pm 2^{-i}
  19. v i = K i [ 1 - σ i 2 - i σ i 2 - i 1 ] [ x i - 1 y i - 1 ] v_{i}=K_{i}\begin{bmatrix}1&-\sigma_{i}2^{-i}\\ \sigma_{i}2^{-i}&1\end{bmatrix}\begin{bmatrix}x_{i-1}\\ y_{i-1}\end{bmatrix}
  20. K i = 1 1 + 2 - 2 i K_{i}={1\over\sqrt{1+2^{-2i}}}
  21. σ i \sigma_{i}
  22. β i \beta_{i}
  23. σ i \sigma_{i}
  24. K i K_{i}
  25. K ( n ) = i = 0 n - 1 K i = i = 0 n - 1 1 / 1 + 2 - 2 i K(n)=\prod_{i=0}^{n-1}K_{i}=\prod_{i=0}^{n-1}1/\sqrt{1+2^{-2i}}
  26. v 0 v_{0}
  27. K = lim n K ( n ) 0.6072529350088812561694 K=\lim_{n\to\infty}K(n)\approx 0.6072529350088812561694
  28. β \beta
  29. σ \sigma
  30. β \beta
  31. β n + 1 \beta_{n+1}
  32. β i = β i - 1 - σ i γ i . γ i = arctan 2 - i , \beta_{i}=\beta_{i-1}-\sigma_{i}\gamma_{i}.\quad\gamma_{i}=\arctan 2^{-i},
  33. γ n \gamma_{n}
  34. arctan ( γ n ) = γ n \arctan(\gamma_{n})=\gamma_{n}
  35. β \beta
  36. v n v_{n}
  37. β i \beta_{i}
  38. β i \beta_{i}
  39. x x
  40. 0 + i x , 0+ix,
  41. cos x + i sin x . \cos x+i\sin x.
  42. K n R sin ( θ ± ϕ ) \displaystyle K_{n}R\sin(\theta\pm\phi)
  43. K n = 1 + 2 - 2 n , tan ( ϕ ) = 2 - n . K_{n}=\sqrt{1+2^{-2n}},\tan(\phi)=2^{-n}.

Core_(group).html

  1. Core S ( H ) := s S s - 1 H s . \mathrm{Core}_{S}(H):=\bigcap_{s\in S}{s^{-1}Hs}.
  2. O p ( G ) O_{p}(G)
  3. O p ( G ) O_{p^{\prime}}(G)
  4. O ( G ) O(G)
  5. O p , p ( G ) O_{p^{\prime},p}(G)
  6. O p , p ( G ) / O p ( G ) = O p ( G / O p ( G ) ) O_{p^{\prime},p}(G)/O_{p^{\prime}}(G)=O_{p}(G/O_{p^{\prime}}(G))
  7. O π 1 , π 2 , , π n + 1 ( G ) / O π 1 , π 2 , , π n ( G ) = O π n + 1 ( G / O π 1 , π 2 , , π n ( G ) ) O_{\pi_{1},\pi_{2},\dots,\pi_{n+1}}(G)/O_{\pi_{1},\pi_{2},\dots,\pi_{n}}(G)=O_% {\pi_{n+1}}(G/O_{\pi_{1},\pi_{2},\dots,\pi_{n}}(G))
  8. C G ( O p , p ( G ) / O p ( G ) ) O p , p ( G ) C_{G}(O_{p^{\prime},p}(G)/O_{p^{\prime}}(G))\subseteq O_{p^{\prime},p}(G)
  9. O ( G ) O_{\infty}(G)

Correlation_function_(quantum_field_theory).html

  1. n n
  2. C n ( x 1 , x 2 , , x n ) := ϕ ( x 1 ) ϕ ( x 2 ) ϕ ( x n ) = D ϕ e - S [ ϕ ] ϕ ( x 1 ) ϕ ( x n ) D ϕ e - S [ ϕ ] C_{n}(x_{1},x_{2},\ldots,x_{n}):=\left\langle\phi(x_{1})\phi(x_{2})\ldots\phi(% x_{n})\right\rangle=\frac{\int D\phi\;e^{-S[\phi]}\phi(x_{1})\ldots\phi(x_{n})% }{\int D\phi\;e^{-S[\phi]}}
  3. T T

Correlation_ratio.html

  1. y ¯ x = i y x i n x \overline{y}_{x}=\frac{\sum_{i}y_{xi}}{n_{x}}
  2. y ¯ = x n x y ¯ x x n x , \overline{y}=\frac{\sum_{x}n_{x}\overline{y}_{x}}{\sum_{x}n_{x}},
  3. y ¯ x \overline{y}_{x}
  4. y ¯ \overline{y}
  5. η 2 = x n x ( y ¯ x - y ¯ ) 2 x , i ( y x i - y ¯ ) 2 \eta^{2}=\frac{\sum_{x}n_{x}(\overline{y}_{x}-\overline{y})^{2}}{\sum_{x,i}(y_% {xi}-\overline{y})^{2}}
  6. η 2 = σ y ¯ 2 σ y 2 , where σ y ¯ 2 = x n x ( y ¯ x - y ¯ ) 2 x n x and σ y 2 = x , i ( y x i - y ¯ ) 2 n , \eta^{2}=\frac{{\sigma_{\overline{y}}}^{2}}{{\sigma_{y}}^{2}},\,\text{ where }% {\sigma_{\overline{y}}}^{2}=\frac{\sum_{x}n_{x}(\overline{y}_{x}-\overline{y})% ^{2}}{\sum_{x}n_{x}}\,\text{ and }{\sigma_{y}}^{2}=\frac{\sum_{x,i}(y_{xi}-% \overline{y})^{2}}{n},
  7. x x\;
  8. y ¯ x \overline{y}_{x}
  9. η \eta
  10. η = 0 \eta=0
  11. η = 1 \eta=1
  12. η \eta
  13. 5 ( 36 - 52 ) 2 + 4 ( 33 - 52 ) 2 + 6 ( 78 - 52 ) 2 = 6780 5(36-52)^{2}+4(33-52)^{2}+6(78-52)^{2}=6780
  14. η 2 = 6780 9640 = 0.7033 \eta^{2}=\frac{6780}{9640}=0.7033\ldots
  15. η = 6780 9640 = 0.8386 \eta=\sqrt{\frac{6780}{9640}}=0.8386\ldots
  16. η = 1 \eta=1
  17. η = 0 \eta=0
  18. η 2 \eta^{2}

Cost_of_capital.html

  1. E s = R f + β s ( R m - R f ) E_{s}=R_{f}+\beta_{s}(R_{m}-R_{f})
  2. K c s = D i v i d e n d P a y m e n t / S h a r e ( 1 + G r o w t h ) P r i c e M a r k e t + G r o w t h r a t e . K_{cs}=\frac{Dividend_{Payment/Share}(1+Growth)}{Price_{Market}}+Growth_{rate}.\,

Cotton_tensor.html

  1. C i j k = k R i j - j R i k + 1 2 ( n - 1 ) ( j R g i k - k R g i j ) . C_{ijk}=\nabla_{k}R_{ij}-\nabla_{j}R_{ik}+\frac{1}{2(n-1)}\left(\nabla_{j}Rg_{% ik}-\nabla_{k}Rg_{ij}\right).
  2. C i j = k ( R l i - 1 4 R g l i ) ϵ k l j , C_{i}^{j}=\nabla_{k}\left(R_{li}-\frac{1}{4}Rg_{li}\right)\epsilon^{klj},
  3. g ~ = e 2 ω g \tilde{g}=e^{2\omega}g
  4. ω \omega
  5. Γ ~ β γ α = Γ β γ α + S β γ α \widetilde{\Gamma}^{\alpha}_{\beta\gamma}=\Gamma^{\alpha}_{\beta\gamma}+S^{% \alpha}_{\beta\gamma}
  6. S β γ α S^{\alpha}_{\beta\gamma}
  7. S β γ α = δ γ α β ω + δ β α γ ω - g β γ α ω S^{\alpha}_{\beta\gamma}=\delta^{\alpha}_{\gamma}\partial_{\beta}\omega+\delta% ^{\alpha}_{\beta}\partial_{\gamma}\omega-g_{\beta\gamma}\partial^{\alpha}\omega
  8. R ~ λ = μ α β R λ μ α β + α S β μ λ - β S α μ λ + S α ρ λ S β μ ρ - S β ρ λ S α μ ρ {\widetilde{R}^{\lambda}}{}_{\mu\alpha\beta}={R^{\lambda}}_{\mu\alpha\beta}+% \nabla_{\alpha}S^{\lambda}_{\beta\mu}-\nabla_{\beta}S^{\lambda}_{\alpha\mu}+S^% {\lambda}_{\alpha\rho}S^{\rho}_{\beta\mu}-S^{\lambda}_{\beta\rho}S^{\rho}_{% \alpha\mu}
  9. n n
  10. R ~ β μ = R β μ - g β μ α α ω - ( n - 2 ) μ β ω + ( n - 2 ) ( μ ω β ω - g β μ λ ω λ ω ) \widetilde{R}_{\beta\mu}=R_{\beta\mu}-g_{\beta\mu}\nabla^{\alpha}\partial_{% \alpha}\omega-(n-2)\nabla_{\mu}\partial_{\beta}\omega+(n-2)(\partial_{\mu}% \omega\partial_{\beta}\omega-g_{\beta\mu}\partial^{\lambda}\omega\partial_{% \lambda}\omega)
  11. R ~ = e - 2 ω R - 2 e - 2 ω ( n - 1 ) α α ω - ( n - 2 ) ( n - 1 ) e - 2 ω λ ω λ ω \widetilde{R}=e^{-2\omega}R-2e^{-2\omega}(n-1)\nabla^{\alpha}\partial_{\alpha}% \omega-(n-2)(n-1)e^{-2\omega}\partial^{\lambda}\omega\partial_{\lambda}\omega
  12. C ~ α β γ = C α β γ + ( n - 2 ) λ ω W β γ α λ \widetilde{C}_{\alpha\beta\gamma}=C_{\alpha\beta\gamma}+(n-2)\partial_{\lambda% }\omega{W_{\beta\gamma\alpha}}^{\lambda}
  13. C ~ = C + grad ω W , \tilde{C}=C\;+\;\operatorname{grad}\,\omega\;\lrcorner\;W,
  14. C i j k = - C i k j C_{ijk}=-C_{ikj}\,
  15. C [ i j k ] = 0. C_{[ijk]}=0.\,
  16. δ W = ( 3 - n ) C , \delta W=(3-n)C,\,
  17. δ \delta

Counts_per_minute.html

  1. 2 π 2\pi

Coupling_(probability).html

  1. X 1 X_{1}
  2. X 2 X_{2}
  3. ( Ω 1 , F 1 , P 1 ) (\Omega_{1},F_{1},P_{1})
  4. ( Ω 2 , F 2 , P 2 ) (\Omega_{2},F_{2},P_{2})
  5. X 1 X_{1}
  6. X 2 X_{2}
  7. ( Ω , F , P ) (\Omega,F,P)
  8. Y 1 Y_{1}
  9. Y 2 Y_{2}
  10. Y 1 Y_{1}
  11. X 1 X_{1}
  12. Y 2 Y_{2}
  13. X 2 X_{2}
  14. Y 1 Y_{1}
  15. Y 2 Y_{2}
  16. Pr ( X 1 + + X n > k ) Pr ( Y 1 + + Y n > k ) . \Pr(X_{1}+\cdots+X_{n}>k)\leq\Pr(Y_{1}+\cdots+Y_{n}>k).

Coupling_constant.html

  1. α = e 2 4 π ε 0 c \alpha=\frac{e^{2}}{4\pi\varepsilon_{0}\hbar c}
  2. e e
  3. ε 0 \varepsilon_{0}
  4. \hbar
  5. c c
  6. g g
  7. 1 4 g 2 Tr G μ ν G μ ν \frac{1}{4g^{2}}{\rm Tr}\,G_{\mu\nu}G^{\mu\nu}
  8. G G
  9. G G
  10. g g
  11. 4 π ε 0 α . \sqrt{4\pi\varepsilon_{0}\alpha}.
  12. Δ E Δ t , \Delta E\Delta t\geq\hbar,
  13. μ \mu
  14. β ( g ) = μ g μ = g ln μ , \beta(g)=\mu\frac{\partial g}{\partial\mu}=\frac{\partial g}{\partial\ln\mu},
  15. α \alpha
  16. α s ( k 2 ) = def g s 2 ( k 2 ) 4 π 1 β 0 ln ( k 2 / Λ 2 ) , \alpha_{s}(k^{2})\ \stackrel{\mathrm{def}}{=}\ \frac{g_{s}^{2}(k^{2})}{4\pi}% \approx\frac{1}{\beta_{0}\ln(k^{2}/\Lambda^{2})},
  17. Λ M S = 217 - 23 + 25 MeV . \Lambda_{MS}=217^{+25}_{-23}{\rm\ MeV}.

Covariance_and_contravariance_(computer_science).html

  1. C C
  2. C 2 C^{2}
  3. C C

Covering_problems.html

  1. i = 1 n c i x i \sum_{i=1}^{n}c_{i}x_{i}
  2. i = 1 n a i j x i b j for j = 1 , , m \sum_{i=1}^{n}a_{ij}x_{i}\geq b_{j}\,\text{ for }j=1,\dots,m
  3. x i 0 for i = 1 , , n x_{i}\geq 0\,\text{ for }i=1,\dots,n
  4. a i j , b j , c i 0 a_{ij},b_{j},c_{i}\geq 0
  5. i = 1 , , n i=1,\dots,n
  6. j = 1 , , m j=1,\dots,m
  7. n n
  8. i i
  9. c i c_{i}
  10. x i x_{i}
  11. i i
  12. A 𝐱 𝐛 A\mathbf{x}\geq\mathbf{b}
  13. 𝐱 \mathbf{x}

CPU_cache.html

  1. log 2 ( r ) \lceil\log_{2}(r)\rceil
  2. r r
  3. log 2 ( b ) \lceil\log_{2}(b)\rceil
  4. b b
  5. x x
  6. y y
  7. n n
  8. x = y m o d n x=ymodn

Craig_retroazimuthal_projection.html

  1. x = λ x=\lambda\,
  2. y = λ sin ( λ ) ( sin ( ϕ ) cos ( λ ) - tan ( ϕ 0 ) cos ( ϕ ) ) y=\frac{\lambda}{\sin(\lambda)}(\sin(\phi)\cos(\lambda)-\tan(\phi_{0})\cos(% \phi))\,
  3. ϕ \phi
  4. λ \lambda
  5. ϕ 0 \phi_{0}
  6. λ = 0 \lambda=0
  7. λ / sin ( λ ) = 1 \lambda/\sin(\lambda)=1

Creation_and_annihilation_operators.html

  1. ( - 2 2 m d 2 d x 2 + 1 2 m ω 2 x 2 ) ψ ( x ) = E ψ ( x ) \left(-\frac{\hbar^{2}}{2m}\frac{d^{2}}{dx^{2}}+\frac{1}{2}m\omega^{2}x^{2}% \right)\psi(x)=E\psi(x)
  2. x = m ω q x\ =\ \sqrt{\frac{\hbar}{m\omega}}q
  3. ω 2 ( - d 2 d q 2 + q 2 ) ψ ( q ) = E ψ ( q ) \frac{\hbar\omega}{2}\left(-\frac{d^{2}}{dq^{2}}+q^{2}\right)\psi(q)=E\psi(q)
  4. ω = h ν \hbar\omega=h\nu
  5. - d 2 d q 2 + q 2 = ( - d d q + q ) ( d d q + q ) + d d q q - q d d q -\frac{d^{2}}{dq^{2}}+q^{2}=\left(-\frac{d}{dq}+q\right)\left(\frac{d}{dq}+q% \right)+\frac{d}{dq}q-q\frac{d}{dq}
  6. ( d d q q - q d d q ) f ( q ) = d d q ( q f ( q ) ) - q d f ( q ) d q = f ( q ) \left(\frac{d}{dq}q-q\frac{d}{dq}\right)f(q)=\frac{d}{dq}(qf(q))-q\frac{df(q)}% {dq}=f(q)
  7. d d q q - q d d q = 1 \frac{d}{dq}q-q\frac{d}{dq}=1
  8. - d 2 d q 2 + q 2 = ( - d d q + q ) ( d d q + q ) + 1 -\frac{d^{2}}{dq^{2}}+q^{2}=\left(-\frac{d}{dq}+q\right)\left(\frac{d}{dq}+q% \right)+1
  9. ω [ 1 2 ( - d d q + q ) 1 2 ( d d q + q ) + 1 2 ] ψ ( q ) = E ψ ( q ) \hbar\omega\left[\frac{1}{\sqrt{2}}\left(-\frac{d}{dq}+q\right)\frac{1}{\sqrt{% 2}}\left(\frac{d}{dq}+q\right)+\frac{1}{2}\right]\psi(q)=E\psi(q)
  10. a = 1 2 ( - d d q + q ) a^{\dagger}\ =\ \frac{1}{\sqrt{2}}\left(-\frac{d}{dq}+q\right)
  11. a = 1 2 ( d d q + q ) a\ \ =\ \frac{1}{\sqrt{2}}\left(\ \ \ \!\frac{d}{dq}+q\right)
  12. ω ( a a + 1 2 ) ψ ( q ) = E ψ ( q ) \hbar\omega\left(a^{\dagger}a+\frac{1}{2}\right)\psi(q)=E\psi(q)
  13. p = - i d d q p=-i\frac{d}{dq}
  14. [ q , p ] = i [q,p]=i\,
  15. a = 1 2 ( q + i p ) = 1 2 ( q + d d q ) a=\frac{1}{\sqrt{2}}(q+ip)=\frac{1}{\sqrt{2}}\left(q+\frac{d}{dq}\right)
  16. a = 1 2 ( q - i p ) = 1 2 ( q - d d q ) a^{\dagger}=\frac{1}{\sqrt{2}}(q-ip)=\frac{1}{\sqrt{2}}\left(q-\frac{d}{dq}\right)
  17. [ a , a ] = 1 2 [ q + i p , q - i p ] = 1 2 ( [ q , - i p ] + [ i p , q ] ) = - i 2 ( [ q , p ] + [ q , p ] ) = 1 [a,a^{\dagger}]=\frac{1}{2}[q+ip,q-ip]=\frac{1}{2}([q,-ip]+[ip,q])=\frac{-i}{2% }([q,p]+[q,p])=1
  18. a a
  19. a a^{\dagger}
  20. A = B + i C A=B+iC
  21. B , C B,C
  22. B C = C B BC=CB
  23. a a
  24. a = q + i p a=q+ip
  25. p , q p,q
  26. [ p , q ] = 1 [p,q]=1
  27. B B
  28. C C
  29. H ^ = ω ( a a - 1 2 ) = ω ( a a + 1 2 ) . \hat{H}=\hbar\omega\left(a\,a^{\dagger}-\frac{1}{2}\right)=\hbar\omega\left(a^% {\dagger}\,a+\frac{1}{2}\right).
  30. a a
  31. a a^{\dagger}
  32. [ H ^ , a ] = - ω a . [\hat{H},a]=-\hbar\omega\,a.
  33. [ H ^ , a ] = ω a . [\hat{H},a^{\dagger}]=\hbar\omega\,a^{\dagger}.
  34. ψ n \psi_{n}
  35. H ^ ψ n = E n ψ n \hat{H}\psi_{n}=E_{n}\,\psi_{n}
  36. H ^ a ψ n = ( E n - ω ) a ψ n . \hat{H}\,a\psi_{n}=(E_{n}-\hbar\omega)\,a\psi_{n}.
  37. H ^ a ψ n = ( E n + ω ) a ψ n . \hat{H}\,a^{\dagger}\psi_{n}=(E_{n}+\hbar\omega)\,a^{\dagger}\psi_{n}.
  38. a ψ n a\psi_{n}
  39. a ψ n a^{\dagger}\psi_{n}
  40. E n - ω E_{n}-\hbar\omega
  41. E n + ω E_{n}+\hbar\omega
  42. a a
  43. a a^{\dagger}
  44. Δ E = ω \Delta E=\hbar\omega
  45. a ψ 0 = 0 a\,\psi_{0}=0
  46. ψ 0 0 \psi_{0}\neq 0
  47. 0 = ω a a ψ 0 = ( H ^ - ω 2 ) ψ 0 . 0=\hbar\omega\,a^{\dagger}a\psi_{0}=\left(\hat{H}-\frac{\hbar\omega}{2}\right)% \,\psi_{0}.
  48. ψ 0 \psi_{0}
  49. E 0 = ω / 2 E_{0}=\hbar\omega/2
  50. ψ n \psi_{n}
  51. E n = ( n + 1 2 ) ω . E_{n}=\left(n+\frac{1}{2}\right)\hbar\omega.
  52. N = a a , N=a^{\dagger}a\,,
  53. a a , a\,a^{\dagger}\,,
  54. N + 1 . N+1\,.
  55. ω ( N + 1 2 ) ψ ( q ) = E ψ ( q ) \hbar\omega\,\left(N+\frac{1}{2}\right)\,\psi(q)=E\,\psi(q)
  56. ψ 0 ( q ) \ \psi_{0}(q)
  57. a ψ 0 ( q ) = 0 a\ \psi_{0}(q)=0
  58. q ψ 0 + d ψ 0 d q = 0 q\psi_{0}+\frac{d\psi_{0}}{dq}=0
  59. ψ 0 ( q ) = C exp ( - q 2 2 ) . \psi_{0}(q)=C\exp\left(-{q^{2}\over 2}\right).
  60. 1 π 4 1\over\sqrt[4]{\pi}
  61. - ψ 0 * ψ 0 d q = 1 \int_{-\infty}^{\infty}\psi_{0}^{*}\psi_{0}\,dq=1
  62. a = ( 0 0 0 0 1 0 0 0 0 2 0 0 0 0 3 0 0 0 0 0 n ) {a}^{\dagger}=\begin{pmatrix}0&0&0&\dots&0&\dots\\ \sqrt{1}&0&0&\dots&0&\dots\\ 0&\sqrt{2}&0&\dots&0&\dots\\ 0&0&\sqrt{3}&\dots&0&\dots\\ \vdots&\vdots&\vdots&\ddots&\vdots&\dots\\ 0&0&0&0&\sqrt{n}&\dots&\\ \vdots&\vdots&\vdots&\vdots&\vdots&\ddots\end{pmatrix}
  63. a = ( 0 1 0 0 0 0 0 2 0 0 0 0 0 3 0 0 0 0 0 n 0 0 0 0 0 ) {a}=\begin{pmatrix}0&\sqrt{1}&0&0&\dots&0&\dots\\ 0&0&\sqrt{2}&0&\dots&0&\dots\\ 0&0&0&\sqrt{3}&\dots&0&\dots\\ 0&0&0&0&\ddots&\vdots&\dots\\ \vdots&\vdots&\vdots&\vdots&\ddots&\sqrt{n}&\dots\\ 0&0&0&0&\dots&0&\ddots\\ \vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\ddots\end{pmatrix}
  64. a i j = ψ i | a | ψ j a^{\dagger}_{ij}=\langle\psi_{i}|{a}^{\dagger}|\psi_{j}\rangle
  65. a i j = ψ i | a | ψ j a_{ij}=\langle\psi_{i}|{a}|\psi_{j}\rangle
  66. [ a ( f ) , a ( g ) ] = [ a ( f ) , a ( g ) ] = 0 [a(f),a(g)]=[a^{\dagger}(f),a^{\dagger}(g)]=0
  67. [ a ( f ) , a ( g ) ] = f | g [a(f),a^{\dagger}(g)]=\langle f|g\rangle
  68. { a ( f ) , a ( g ) } = { a ( f ) , a ( g ) } = 0 \{a(f),a(g)\}=\{a^{\dagger}(f),a^{\dagger}(g)\}=0
  69. { a ( f ) , a ( g ) } = f | g \{a(f),a^{\dagger}(g)\}=\langle f|g\rangle
  70. | f ) |f)
  71. | f ) |f)
  72. a ( f ) | 0 = 0 a(f)|0\rangle=0
  73. | 0 ) |0)
  74. | f ) |f)
  75. ( f | f ) (f|f)
  76. | f ) |f)
  77. n i n_{i}
  78. i i
  79. d t dt
  80. n i d t n_{i}dt
  81. α n i d t \alpha n_{i}dt
  82. α n i d t \alpha n_{i}dt
  83. n i n_{i}
  84. 1 - 2 α n i d t 1-2\alpha n_{i}dt
  85. d t dt
  86. d t dt
  87. a | n = n | n - 1 a|n\rangle=\sqrt{n}\ |n-1\rangle
  88. a | n = n + 1 | n + 1 a^{\dagger}|n\rangle=\sqrt{n+1}\ |n+1\rangle
  89. [ a , a ] = 1 [a,a^{\dagger}]=1
  90. t | ψ = - α ( 2 a i a i - a i - 1 a i - a i + 1 a i ) | ψ = - α ( a i - a i - 1 ) ( a i - a i - 1 ) | ψ , \partial_{t}|\psi\rangle=-\alpha\sum(2a_{i}^{\dagger}a_{i}-a_{i-1}^{\dagger}a_% {i}-a_{i+1}^{\dagger}a_{i})|\psi\rangle=-\alpha\sum(a_{i}^{\dagger}-a_{i-1}^{% \dagger})(a_{i}-a_{i-1})|\psi\rangle,
  91. n n
  92. n ( n - 1 ) n(n-1)
  93. λ n ( n - 1 ) d t \lambda n(n-1)dt
  94. 1 - λ n ( n - 1 ) d t 1-\lambda n(n-1)dt
  95. λ ( a i a i - a i a i a i a i ) \lambda\sum(a_{i}a_{i}-a_{i}^{\dagger}a_{i}^{\dagger}a_{i}a_{i})
  96. t | ψ = - α ( a i - a i - 1 ) ( a i - a i - 1 ) | ψ + λ ( a i 2 - a i 2 a i 2 ) | ψ \partial_{t}|\psi\rangle=-\alpha\sum(a_{i}^{\dagger}-a_{i-1}^{\dagger})(a_{i}-% a_{i-1})|\psi\rangle+\lambda\sum(a_{i}^{2}-a_{i}^{\dagger 2}a_{i}^{2})|\psi\rangle
  97. a i a^{\dagger}_{i}
  98. a i a_{i}
  99. N = i n i = i a i a i N=\sum_{i}n_{i}=\sum_{i}a^{\dagger}_{i}a_{i}
  100. i i
  101. ( n , l , m , s ) (n,l,m,s)
  102. [ a i , a j ] a i a j - a j a i = δ i j , [a_{i},a^{\dagger}_{j}]\equiv a_{i}a^{\dagger}_{j}-a^{\dagger}_{j}a_{i}=\delta% _{ij},
  103. [ a i , a j ] = [ a i , a j ] = 0 , [a^{\dagger}_{i},a^{\dagger}_{j}]=[a_{i},a_{j}]=0,
  104. [ , ] [\ \ ,\ \ ]
  105. δ i j \delta_{ij}
  106. { , } \{\ \ ,\ \ \}
  107. { a i , a j } a i a j + a j a i = δ i j , \{a_{i},a^{\dagger}_{j}\}\equiv a_{i}a^{\dagger}_{j}+a^{\dagger}_{j}a_{i}=% \delta_{ij},
  108. { a i , a j } = { a i , a j } = 0. \{a^{\dagger}_{i},a^{\dagger}_{j}\}=\{a_{i},a_{j}\}=0.
  109. i j i\neq j

Creep_(deformation).html

  1. d ε d t = C σ m d b e - Q k T \frac{\mathrm{d}\varepsilon}{\mathrm{d}t}=\frac{C\sigma^{m}}{d^{b}}e^{\frac{-Q% }{kT}}
  2. ε {\varepsilon}
  3. d ε d t = A ( σ - σ t h ) n e - Q R ¯ T \frac{\mathrm{d}\varepsilon}{\mathrm{d}t}=A\left(\sigma-\sigma_{th}\right)^{n}% e^{\frac{-Q}{\bar{R}T}}
  4. τ \tau
  5. f ( τ ) f(\tau)

Critical_focus.html

  1. 1 / I + 1 / O = 1 / F 1/I+1/O=1/F
  2. I I
  3. O O
  4. F F

Cross-correlation.html

  1. ( f g ) ( τ ) = def - f * ( t ) g ( t + τ ) d t , (f\star g)(\tau)\ \stackrel{\mathrm{def}}{=}\int_{-\infty}^{\infty}f^{*}(t)\ g% (t+\tau)\,dt,
  2. f * f^{*}
  3. f f
  4. τ \tau
  5. ( f g ) [ n ] = def m = - f * [ m ] g [ m + n ] . (f\star g)[n]\ \stackrel{\mathrm{def}}{=}\sum_{m=-\infty}^{\infty}f^{*}[m]\ g[% m+n].
  6. X X
  7. Y Y
  8. Y - X Y-X
  9. f g f\star g
  10. f * g f*g
  11. X + Y X+Y
  12. f f
  13. g g
  14. g g
  15. f f
  16. g g
  17. ( f g ) (f\star g)
  18. f f
  19. g g
  20. f f
  21. f g = f * ( - t ) * g . f\star g=f^{*}(-t)*g.
  22. f g = f * g . f\star g=f*g.
  23. ( f g ) ( f g ) = ( f f ) ( g g ) . (f\star g)\star(f\star g)=(f\star f)\star(g\star g).
  24. { f g } = ( { f } ) * { g } , \mathcal{F}\{f\star g\}=(\mathcal{F}\{f\})^{*}\cdot\mathcal{F}\{g\},
  25. \mathcal{F}
  26. ( f * h ) g = h ( - ) * ( f g ) . (f*h)\star g=h(-)*(f\star g).
  27. ( X t , Y t ) (X_{t},Y_{t})
  28. γ X Y ( τ ) = E [ ( X t - μ X ) ( Y t + τ - μ Y ) ] , \gamma_{XY}(\tau)=\operatorname{E}[(X_{t}-\mu_{X})(Y_{t+\tau}-\mu_{Y})],
  29. ρ X Y ( τ ) = E [ ( X t - μ X ) ( Y t + τ - μ Y ) ] / ( σ X σ Y ) , \rho_{XY}(\tau)=\operatorname{E}[(X_{t}-\mu_{X})\,(Y_{t+\tau}-\mu_{Y})]/(% \sigma_{X}\sigma_{Y}),
  30. μ X \mu_{X}
  31. σ X \sigma_{X}
  32. ( X t ) (X_{t})
  33. ( Y t ) (Y_{t})
  34. ( X t , Y t ) (X_{t},Y_{t})
  35. τ delay = arg max 𝑡 ( ( f g ) ( t ) ) \tau_{\mathrm{delay}}=\underset{t}{\operatorname{arg\,max}}((f\star g)(t))
  36. t ( x , y ) t(x,y)
  37. f ( x , y ) f(x,y)
  38. 1 n x , y ( f ( x , y ) - f ¯ ) ( t ( x , y ) - t ¯ ) σ f σ t \frac{1}{n}\sum_{x,y}\frac{(f(x,y)-\overline{f})(t(x,y)-\overline{t})}{\sigma_% {f}\sigma_{t}}
  39. n n
  40. t ( x , y ) t(x,y)
  41. f ( x , y ) f(x,y)
  42. f ¯ \overline{f}
  43. σ f \sigma_{f}
  44. F ( x , y ) = f ( x , y ) - f ¯ F(x,y)=f(x,y)-\overline{f}
  45. T ( x , y ) = t ( x , y ) - t ¯ T(x,y)=t(x,y)-\overline{t}
  46. F F , T T \left\langle\frac{F}{\|F\|},\frac{T}{\|T\|}\right\rangle
  47. , \langle\cdot,\cdot\rangle
  48. \|\cdot\|

Cross-polytope.html

  1. { x n : x 1 1 } . \{x\in\mathbb{R}^{n}:\|x\|_{1}\leq 1\}.
  2. arccos ( 2 - n n ) \arccos\left(\frac{2-n}{n}\right)
  3. 2 k + 1 ( n k + 1 ) 2^{k+1}{n\choose{k+1}}
  4. 2 n n ! . \frac{2^{n}}{n!}.
  5. 2 k + 1 ( n k + 1 ) 2^{k+1}{n\choose k+1}

Crystallographic_point_group.html

  1. 3 ¯ \overline{3}
  2. 4 ¯ \overline{4}
  3. 3 ¯ \overline{3}
  4. 6 ¯ \overline{6}
  5. 6 ¯ \overline{6}
  6. 3 ¯ \overline{3}
  7. 3 ¯ \overline{3}
  8. 4 ¯ \overline{4}
  9. 4 ¯ \overline{4}
  10. 1 ¯ \overline{1}
  11. 1 1
  12. 1 ¯ \overline{1}
  13. 1 ¯ \overline{1}
  14. 2 ~ \tilde{2}
  15. 2 2
  16. m m
  17. 2 m \tfrac{2}{m}
  18. 2 : m 2:m
  19. 2 : 2 2:2
  20. 2 m 2\cdot m
  21. 2 m 2 m 2 m \tfrac{2}{m}\tfrac{2}{m}\tfrac{2}{m}
  22. m 2 : m m\cdot 2:m
  23. 4 4
  24. 4 ¯ \overline{4}
  25. 4 ¯ \overline{4}
  26. 4 ~ \tilde{4}
  27. 4 m \tfrac{4}{m}
  28. 4 : m 4:m
  29. 4 : 2 4:2
  30. 4 m 4\cdot m
  31. 4 ¯ \overline{4}
  32. 4 ¯ \overline{4}
  33. 4 ~ m \tilde{4}\cdot m
  34. 4 m 2 m 2 m \tfrac{4}{m}\tfrac{2}{m}\tfrac{2}{m}
  35. m 4 : m m\cdot 4:m
  36. 3 3
  37. 3 ¯ \overline{3}
  38. 3 ¯ \overline{3}
  39. 6 ~ \tilde{6}
  40. 3 : 2 3:2
  41. 3 m 3\cdot m
  42. 3 ¯ \overline{3}
  43. 2 m \tfrac{2}{m}
  44. 3 ¯ \overline{3}
  45. 6 ~ m \tilde{6}\cdot m
  46. 6 6
  47. 6 ¯ \overline{6}
  48. 6 ¯ \overline{6}
  49. 3 : m 3:m
  50. 6 m \tfrac{6}{m}
  51. 6 : m 6:m
  52. 6 : 2 6:2
  53. 6 m 6\cdot m
  54. 6 ¯ \overline{6}
  55. 6 ¯ \overline{6}
  56. m 3 : m m\cdot 3:m
  57. 6 m 2 m 2 m \tfrac{6}{m}\tfrac{2}{m}\tfrac{2}{m}
  58. m 6 : m m\cdot 6:m
  59. 3 / 2 3/2
  60. 2 m \tfrac{2}{m}
  61. 3 ¯ \overline{3}
  62. 3 ¯ \overline{3}
  63. 6 ~ / 2 \tilde{6}/2
  64. 3 / 4 3/4
  65. 4 ¯ \overline{4}
  66. 4 ¯ \overline{4}
  67. 3 / 4 ~ 3/\tilde{4}
  68. 4 m \tfrac{4}{m}
  69. 3 ¯ \overline{3}
  70. 2 m \tfrac{2}{m}
  71. 3 ¯ \overline{3}
  72. 6 ~ / 4 \tilde{6}/4

CTN.html

  1. cot θ . \cot\theta\,\!.

Cubic_graph.html

  1. K 3 , 3 K_{3,3}
  2. O ( 1.276 n ) O({1.276}^{n})

Cubic_Hermite_spline.html

  1. x 1 , x 2 , , x n x_{1},x_{2},\ldots,x_{n}
  2. x k x_{k}
  3. ( x k , x k + 1 ) (x_{k},x_{k+1})
  4. ( 0 , 1 ) (0,1)
  5. s y m b o l p 0 symbol{p}_{0}
  6. t = 0 t=0
  7. s y m b o l p 1 symbol{p}_{1}
  8. t = 1 t=1
  9. s y m b o l m 0 symbol{m}_{0}
  10. t = 0 t=0
  11. s y m b o l m 1 symbol{m}_{1}
  12. t = 1 t=1
  13. s y m b o l p ( t ) = ( 2 t 3 - 3 t 2 + 1 ) s y m b o l p 0 + ( t 3 - 2 t 2 + t ) s y m b o l m 0 + ( - 2 t 3 + 3 t 2 ) s y m b o l p 1 + ( t 3 - t 2 ) s y m b o l m 1 symbol{p}(t)=(2t^{3}-3t^{2}+1)symbol{p}_{0}+(t^{3}-2t^{2}+t)symbol{m}_{0}+(-2t% ^{3}+3t^{2})symbol{p}_{1}+(t^{3}-t^{2})symbol{m}_{1}
  14. x x
  15. ( x k , x k + 1 ) (x_{k},x_{k+1})
  16. [ 0 , 1 ] [0,1]
  17. s y m b o l p ( x ) = h 00 ( t ) s y m b o l p k + h 10 ( t ) ( x k + 1 - x k ) s y m b o l m k + h 01 ( t ) s y m b o l p k + 1 + h 11 ( t ) ( x k + 1 - x k ) s y m b o l m k + 1 . symbol{p}(x)=h_{00}(t)symbol{p}_{k}+h_{10}(t)(x_{k+1}-x_{k})symbol{m}_{k}+h_{0% 1}(t)symbol{p}_{k+1}+h_{11}(t)(x_{k+1}-x_{k})symbol{m}_{k+1}.
  18. t = ( x - x k ) / ( x k + 1 - x k ) t=(x-x_{k})/(x_{k+1}-x_{k})
  19. h h
  20. x k + 1 - x k x_{k+1}-x_{k}
  21. Q ( x ) Q(x)
  22. R ( x ) = Q ( x ) - P ( x ) R(x)=Q(x)-P(x)
  23. Q Q
  24. P P
  25. R R
  26. R ( 0 ) = Q ( 0 ) - P ( 0 ) = 0 R(0)=Q(0)-P(0)=0
  27. P P
  28. Q Q
  29. R ( 1 ) = 0 R(1)=0
  30. R R
  31. R ( x ) = a x ( x - 1 ) ( x - r ) R(x)=ax(x-1)(x-r)
  32. R ( x ) = a x ( x - 1 ) + a x ( x - r ) + a ( x - 1 ) ( x - r ) R^{\prime}(x)=ax(x-1)+ax(x-r)+a(x-1)(x-r)
  33. R ( 0 ) = Q ( 0 ) - P ( 0 ) = 0 R^{\prime}(0)=Q^{\prime}(0)-P^{\prime}(0)=0
  34. R ( 1 ) = Q ( 1 ) - P ( 1 ) = 0 R^{\prime}(1)=Q^{\prime}(1)-P^{\prime}(1)=0
  35. a = 0 a=0
  36. R = 0 R=0
  37. P ( x ) = Q ( x ) P(x)=Q(x)
  38. s y m b o l p ( t ) = h 00 ( t ) s y m b o l p 0 + h 10 ( t ) s y m b o l m 0 + h 01 ( t ) s y m b o l p 1 + h 11 ( t ) s y m b o l m 1 symbol{p}(t)=h_{00}(t)symbol{p}_{0}+h_{10}(t)symbol{m}_{0}+h_{01}(t)symbol{p}_% {1}+h_{11}(t)symbol{m}_{1}
  39. h 00 h_{00}
  40. h 10 h_{10}
  41. h 01 h_{01}
  42. h 11 h_{11}
  43. h 00 ( t ) h_{00}(t)
  44. 2 t 3 - 3 t 2 + 1 2t^{3}-3t^{2}+1
  45. ( 1 + 2 t ) ( 1 - t ) 2 (1+2t)(1-t)^{2}
  46. B 0 ( t ) + B 1 ( t ) B_{0}(t)+B_{1}(t)
  47. h 10 ( t ) h_{10}(t)
  48. t 3 - 2 t 2 + t t^{3}-2t^{2}+t
  49. t ( 1 - t ) 2 t(1-t)^{2}
  50. 1 3 B 1 ( t ) \frac{1}{3}\cdot B_{1}(t)
  51. h 01 ( t ) h_{01}(t)
  52. - 2 t 3 + 3 t 2 -2t^{3}+3t^{2}
  53. t 2 ( 3 - 2 t ) t^{2}(3-2t)
  54. B 3 ( t ) + B 2 ( t ) B_{3}(t)+B_{2}(t)
  55. h 11 ( t ) h_{11}(t)
  56. t 3 - t 2 t^{3}-t^{2}
  57. t 2 ( t - 1 ) t^{2}(t-1)
  58. - 1 3 B 2 ( t ) -\frac{1}{3}\cdot B_{2}(t)
  59. h 10 h_{10}
  60. h 11 h_{11}
  61. h 01 h_{01}
  62. h 11 h_{11}
  63. h 00 h_{00}
  64. h 10 h_{10}
  65. B k ( t ) = ( 3 k ) t k ( 1 - t ) 3 - k B_{k}(t)={3\choose k}\cdot t^{k}\cdot(1-t)^{3-k}
  66. s y m b o l p 0 , s y m b o l p 0 + s y m b o l m 0 3 , s y m b o l p 1 - s y m b o l m 1 3 , s y m b o l p 1 symbol{p}_{0},symbol{p}_{0}+\frac{symbol{m}_{0}}{3},symbol{p}_{1}-\frac{symbol% {m}_{1}}{3},symbol{p}_{1}
  67. ( t k , s y m b o l p k ) (t_{k},symbol{p}_{k})
  68. k = 1 , , n k=1,\ldots,n
  69. ( t 1 , t n ) (t_{1},t_{n})
  70. s y m b o l m k = s y m b o l p k + 1 - s y m b o l p k 2 ( t k + 1 - t k ) + s y m b o l p k - s y m b o l p k - 1 2 ( t k - t k - 1 ) symbol{m}_{k}=\frac{symbol{p}_{k+1}-symbol{p}_{k}}{2(t_{k+1}-t_{k})}+\frac{% symbol{p}_{k}-symbol{p}_{k-1}}{2(t_{k}-t_{k-1})}
  71. k = 2 , , n - 1 k=2,\ldots,n-1
  72. s y m b o l m k = ( 1 - c ) s y m b o l p k + 1 - s y m b o l p k - 1 t k + 1 - t k - 1 symbol{m}_{k}=(1-c)\frac{symbol{p}_{k+1}-symbol{p}_{k-1}}{t_{k+1}-t_{k-1}}
  73. c c
  74. ( 0 , 1 ) (0,1)
  75. c = 1 c=1
  76. c = 0 c=0
  77. s y m b o l m k = s y m b o l p k + 1 - s y m b o l p k - 1 t k + 1 - t k - 1 symbol{m}_{k}=\frac{symbol{p}_{k+1}-symbol{p}_{k-1}}{t_{k+1}-t_{k-1}}
  78. s y m b o l p k - 1 symbol{p}_{k-1}
  79. s y m b o l p k symbol{p}_{k}
  80. s y m b o l p k + 1 symbol{p}_{k+1}
  81. CINT x ( p - 1 , p 0 , p 1 , p 2 ) = 1 2 ( - x 3 + 2 x 2 - x 3 x 3 - 5 x 2 + 2 - 3 x 3 + 4 x 2 + x x 3 - x 2 ) ( p - 1 p 0 p 1 p 2 ) = 1 2 ( x ( ( 2 - x ) x - 1 ) x 2 ( 3 x - 5 ) + 2 x ( ( 4 - 3 x ) x + 1 ) ( x - 1 ) x 2 ) ( p - 1 p 0 p 1 p 2 ) \mathrm{CINT}_{x}(p_{-1},p_{0},p_{1},p_{2})=\frac{1}{2}\begin{pmatrix}-x^{3}+2% x^{2}-x\\ 3x^{3}-5x^{2}+2\\ -3x^{3}+4x^{2}+x\\ x^{3}-x^{2}\end{pmatrix}\cdot\begin{pmatrix}p_{-1}\\ p_{0}\\ p_{1}\\ p_{2}\end{pmatrix}=\frac{1}{2}\begin{pmatrix}x((2-x)x-1)\\ x^{2}(3x-5)+2\\ x((4-3x)x+1)\\ (x-1)x^{2}\end{pmatrix}\cdot\begin{pmatrix}p_{-1}\\ p_{0}\\ p_{1}\\ p_{2}\end{pmatrix}
  82. x [ 0 , 1 ] x\in[0,1]

Cup_product.html

  1. ( c p d q ) ( σ ) = c p ( σ ι 0 , 1 , p ) d q ( σ ι p , p + 1 , , p + q ) (c^{p}\smile d^{q})(\sigma)=c^{p}(\sigma\circ\iota_{0,1,...p})\cdot d^{q}(% \sigma\circ\iota_{p,p+1,...,p+q})
  2. ι S , S { 0 , 1 , , p + q } \iota_{S},S\subset\{0,1,...,p+q\}
  3. ( p + q ) (p+q)
  4. { 0 , , p + q } \{0,...,p+q\}
  5. σ ι 0 , 1 , , p \sigma\circ\iota_{0,1,...,p}
  6. σ ι p , p + 1 , , p + q \sigma\circ\iota_{p,p+1,...,p+q}
  7. δ ( c p d q ) = δ c p d q + ( - 1 ) p ( c p δ d q ) . \delta(c^{p}\smile d^{q})=\delta{c^{p}}\smile d^{q}+(-1)^{p}(c^{p}\smile\delta% {d^{q}}).
  8. H p ( X ) × H q ( X ) H p + q ( X ) . H^{p}(X)\times H^{q}(X)\to H^{p+q}(X).
  9. α p β q = ( - 1 ) p q ( β q α p ) \alpha^{p}\smile\beta^{q}=(-1)^{pq}(\beta^{q}\smile\alpha^{p})
  10. f : X Y f\colon X\to Y
  11. f * : H * ( Y ) H * ( X ) f^{*}\colon H^{*}(Y)\to H^{*}(X)
  12. f * ( α β ) = f * ( α ) f * ( β ) , f^{*}(\alpha\smile\beta)=f^{*}(\alpha)\smile f^{*}(\beta),
  13. : H p ( X ) × H q ( X ) H p + q ( X ) \smile\colon H^{p}(X)\times H^{q}(X)\to H^{p+q}(X)
  14. C ( X ) × C ( X ) C ( X × X ) Δ * C ( X ) \displaystyle C^{\bullet}(X)\times C^{\bullet}(X)\to C^{\bullet}(X\times X)% \overset{\Delta^{*}}{\to}C^{\bullet}(X)
  15. X X
  16. X × X X\times X
  17. Δ : X X × X \Delta\colon X\to X\times X
  18. Δ : X X × X \Delta\colon X\to X\times X
  19. Δ * : H ( X × X ) H ( X ) \Delta^{*}\colon H^{\bullet}(X\times X)\to H^{\bullet}(X)
  20. Δ * : H ( X ) H ( X × X ) \Delta_{*}\colon H_{\bullet}(X)\to H_{\bullet}(X\times X)
  21. ( u 1 + u 2 ) v = u 1 v + u 2 v (u_{1}+u_{2})\smile v=u_{1}\smile v+u_{2}\smile v
  22. u ( v 1 + v 2 ) = u v 1 + u v 2 . u\smile(v_{1}+v_{2})=u\smile v_{1}+u\smile v_{2}.
  23. X := S 2 S 1 S 1 X:=S^{2}\vee S^{1}\vee S^{1}
  24. S 1 S^{1}

Cupola_(geometry).html

  1. 1 / 2 {1}/{2}
  2. 1 / 2 {1}/{2}
  3. 1 / 2 {1}/{2}
  4. 1 / 2 {1}/{2}
  5. 1 / 2 {1}/{2}
  6. 1 / 2 {1}/{2}
  7. 1 / 2 {1}/{2}
  8. 1 / 2 {1}/{2}
  9. 1 / 2 {1}/{2}
  10. h = 1 - 1 4 sin 2 ( π d n ) h=\sqrt{1-\frac{1}{4\sin^{2}(\frac{\pi d}{n})}}

Current_algebra.html

  1. [ ρ a ( x ) , ρ b ( y ) ] = i f c a b δ ( x - y ) ρ c ( x ) [\rho^{a}(\vec{x}),\rho^{b}(\vec{y})]=if^{ab}_{c}\delta(\vec{x}-\vec{y})\rho^{% c}(\vec{x})

Current_source.html

  1. I R 2 ( = I E ) = V R 2 R 2 = V Z - V B E R 2 . I_{R2}(=I_{E})=\frac{V_{R2}}{R2}=\frac{V_{Z}-V_{BE}}{R2}.
  2. R 2 = V Z - V B E I R 2 R2=\frac{V_{Z}-V_{BE}}{I_{R2}}
  3. R 2 = V Z - 0.65 I R 2 R2=\frac{V_{Z}-0.65}{I_{R2}}
  4. R 1 = V S - V Z I Z + K I B R_{1}=\frac{V_{S}-V_{Z}}{I_{Z}+K\cdot I_{B}}
  5. I B = I C ( = I E = I R 2 ) h F E ( m i n ) I_{B}=\frac{I_{C}(=I_{E}=I_{R2})}{h_{FE(min)}}
  6. R 2 = V D - V B E I R 2 R_{2}=\frac{V_{D}-V_{BE}}{I_{R2}}
  7. R 1 = V S - V D I D + K I B R_{1}=\frac{V_{S}-V_{D}}{I_{D}+K\cdot I_{B}}
  8. R 2 = V Z + V D - V B E I R 2 R_{2}=\frac{V_{Z}+V_{D}-V_{BE}}{I_{R2}}
  9. R 2 = V Z I R 2 R_{2}=\frac{V_{Z}}{I_{R2}}
  10. R 1 = V S - V Z - V D I Z + K I B R_{1}=\frac{V_{S}-V_{Z}-V_{D}}{I_{Z}+K\cdot I_{B}}

Current_yield.html

  1. Current yield = Annual interest payment Clean price . \,\text{Current yield}=\frac{\,\text{Annual interest payment}}{\,\text{Clean % price}}.
  2. Current Yield = F × r P = $ 100 × 5.00 % $ 95.00 = $ 5.00 $ 95.00 = 5.2631 % \,\text{Current Yield}=\frac{F\times r}{P}=\frac{\$100\times 5.00\%}{\$95.00}=% \frac{\$5.00}{\$95.00}=5.2631\%

Curse_of_dimensionality.html

  1. O ( 2 d ) O(2^{d})
  2. r r
  3. d d
  4. 2 r 2r
  5. 2 r d π d / 2 d Γ ( d / 2 ) \frac{2r^{d}\pi^{d/2}}{d\;\Gamma(d/2)}
  6. ( 2 r ) d (2r)^{d}
  7. d d
  8. d d
  9. π d / 2 d 2 d - 1 Γ ( d / 2 ) 0 \frac{\pi^{d/2}}{d2^{d-1}\Gamma(d/2)}\rightarrow 0
  10. d d\rightarrow\infty
  11. r d r\sqrt{d}
  12. d \sqrt{d}
  13. lim d E ( dist max ( d ) - dist min ( d ) dist min ( d ) ) 0 \lim_{d\to\infty}E\left(\frac{\operatorname{dist}_{\max}(d)-\operatorname{dist% }_{\min}(d)}{\operatorname{dist}_{\min}(d)}\right)\to 0

Curvature_of_Riemannian_manifolds.html

  1. \nabla
  2. [ , ] [\cdot,\cdot]
  3. R ( u , v ) w = u v w - v u w - [ u , v ] w . R(u,v)w=\nabla_{u}\nabla_{v}w-\nabla_{v}\nabla_{u}w-\nabla_{[u,v]}w.
  4. R ( u , v ) R(u,v)
  5. u = / x i u=\partial/\partial x_{i}
  6. v = / x j v=\partial/\partial x_{j}
  7. [ u , v ] = 0 [u,v]=0
  8. R ( u , v ) w = u v w - v u w R(u,v)w=\nabla_{u}\nabla_{v}w-\nabla_{v}\nabla_{u}w
  9. w R ( u , v ) w w\mapsto R(u,v)w
  10. R ( u , v ) = - R ( v , u ) R(u,v)=-R(v,u)
  11. R ( u , v ) w , z = - R ( u , v ) z , w \langle R(u,v)w,z\rangle=-\langle R(u,v)z,w\rangle
  12. R ( u , v ) w + R ( v , w ) u + R ( w , u ) v = 0 R(u,v)w+R(v,w)u+R(w,u)v=0
  13. n 2 ( n 2 - 1 ) / 12 n^{2}(n^{2}-1)/12
  14. R ( u , v ) w , z = R ( w , z ) u , v \langle R(u,v)w,z\rangle=\langle R(w,z)u,v\rangle
  15. u R ( v , w ) + v R ( w , u ) + w R ( u , v ) = 0 \nabla_{u}R(v,w)+\nabla_{v}R(w,u)+\nabla_{w}R(u,v)=0
  16. K ( σ ) K(\sigma)
  17. σ \sigma
  18. σ \sigma
  19. σ \sigma
  20. σ \sigma
  21. σ \sigma
  22. v , u v,u
  23. σ \sigma
  24. K ( σ ) = K ( u , v ) / | u v | 2 where K ( u , v ) = R ( u , v ) v , u K(\sigma)=K(u,v)/|u\wedge v|^{2}\,\text{ where }K(u,v)=\langle R(u,v)v,u\rangle
  25. 6 R ( u , v ) w , z = 6\langle R(u,v)w,z\rangle=
  26. [ K ( u + z , v + w ) - K ( u + z , v ) - K ( u + z , w ) - K ( u , v + w ) - K ( z , v + w ) + K ( u , w ) + K ( v , z ) ] - [K(u+z,v+w)-K(u+z,v)-K(u+z,w)-K(u,v+w)-K(z,v+w)+K(u,w)+K(v,z)]-
  27. [ K ( u + w , v + z ) - K ( u + w , v ) - K ( u + w , z ) - K ( u , v + z ) - K ( w , v + z ) + K ( v , w ) + K ( u , z ) ] . [K(u+w,v+z)-K(u+w,v)-K(u+w,z)-K(u,v+z)-K(w,v+z)+K(v,w)+K(u,z)].
  28. R ( u , v ) w , z = 1 6 2 s t ( K ( u + s z , v + t w ) - K ( u + s w , v + t z ) ) | ( s , t ) = ( 0 , 0 ) \langle R(u,v)w,z\rangle=\frac{1}{6}\left.\frac{\partial^{2}}{\partial s% \partial t}\left(K(u+sz,v+tw)-K(u+sw,v+tz)\right)\right|_{(s,t)=(0,0)}
  29. Ω = Ω j i \Omega=\Omega^{i}_{\ j}
  30. so ( n ) \operatorname{so}(n)
  31. O ( n ) \operatorname{O}(n)
  32. e i e_{i}
  33. ω = ω j i \omega=\omega^{i}_{\ j}
  34. ω j k ( e i ) = e i e j , e k \omega^{k}_{\ j}(e_{i})=\langle\nabla_{e_{i}}e_{j},e_{k}\rangle
  35. Ω = Ω j i \Omega=\Omega^{i}_{\ j}
  36. Ω = d ω + ω ω \Omega=d\omega+\omega\wedge\omega
  37. R ( u , v ) w = Ω ( u v ) w . R(u,v)w=\Omega(u\wedge v)w.
  38. Ω θ = 0 \Omega\wedge\theta=0
  39. θ = θ i \theta=\theta^{i}
  40. θ i ( v ) = e i , v \theta^{i}(v)=\langle e_{i},v\rangle
  41. D Ω = 0 D\Omega=0
  42. Q Q
  43. Λ 2 ( T ) \Lambda^{2}(T)
  44. Q ( u v ) , w z = R ( u , v ) z , w . \langle Q(u\wedge v),w\wedge z\rangle=\langle R(u,v)z,w\rangle.
  45. { e i } \{e_{i}\}
  46. S c = i , j R ( e i , e j ) e j , e i = i Ric ( e i ) , e i , S\!c=\sum_{i,j}\langle R(e_{i},e_{j})e_{j},e_{i}\rangle=\sum_{i}\langle\,\text% {Ric}(e_{i}),e_{i}\rangle,
  47. { e i } \{e_{i}\}
  48. R i c ( u ) = i R ( u , e i ) e i . Ric(u)=\sum_{i}R(u,e_{i})e_{i}.
  49. e 2 f e^{2f}
  50. e 2 f ( R + ( Hess ( f ) - d f d f + 1 2 grad ( f ) 2 g ) g ) e^{2f}\left(R+\left(\,\text{Hess}(f)-df\otimes df+\frac{1}{2}\|\,\text{grad}(f% )\|^{2}g\right){~{}\wedge\!\!\!\!\!\!\bigcirc~{}}g\right)
  51. {~{}\wedge\!\!\!\!\!\!\bigcirc~{}}

Curvilinear_coordinates.html

  1. x = f ( q 1 , q 2 , q 3 ) , y = f ( q 1 , q 2 , q 3 ) , z = f ( q 1 , q 2 , q 3 ) x=f(q_{1},q_{2},q_{3}),\,y=f(q_{1},q_{2},q_{3}),\,z=f(q_{1},q_{2},q_{3})
  2. q 1 = f ( x , y , z ) , q 2 = f ( x , y , z ) , q 3 = f ( x , y , z ) q_{1}=f(x,y,z),\,q_{2}=f(x,y,z),\,q_{3}=f(x,y,z)
  3. 𝐫 = x 𝐞 x + y 𝐞 y + z 𝐞 z \mathbf{r}=x\mathbf{e}_{x}+y\mathbf{e}_{y}+z\mathbf{e}_{z}
  4. 𝐞 x = 𝐫 x ; 𝐞 y = 𝐫 y ; 𝐞 z = 𝐫 z . \mathbf{e}_{x}=\dfrac{\partial\mathbf{r}}{\partial x};\;\mathbf{e}_{y}=\dfrac{% \partial\mathbf{r}}{\partial y};\;\mathbf{e}_{z}=\dfrac{\partial\mathbf{r}}{% \partial z}.
  5. 𝐡 1 = 𝐫 q 1 ; 𝐡 2 = 𝐫 q 2 ; 𝐡 3 = 𝐫 q 3 . \mathbf{h}_{1}=\dfrac{\partial\mathbf{r}}{\partial q_{1}};\;\mathbf{h}_{2}=% \dfrac{\partial\mathbf{r}}{\partial q_{2}};\;\mathbf{h}_{3}=\dfrac{\partial% \mathbf{r}}{\partial q_{3}}.
  6. h 1 = | 𝐡 1 | ; h 2 = | 𝐡 2 | ; h 3 = | 𝐡 3 | h_{1}=|\mathbf{h}_{1}|;\;h_{2}=|\mathbf{h}_{2}|;\;h_{3}=|\mathbf{h}_{3}|
  7. 𝐛 1 = 𝐡 1 h 1 ; 𝐛 2 = 𝐡 2 h 2 ; 𝐛 3 = 𝐡 3 h 3 . \mathbf{b}_{1}=\dfrac{\mathbf{h}_{1}}{h_{1}};\;\mathbf{b}_{2}=\dfrac{\mathbf{h% }_{2}}{h_{2}};\;\mathbf{b}_{3}=\dfrac{\mathbf{h}_{3}}{h_{3}}.
  8. 3 \mathbb{R}^{3}
  9. d 𝐫 = 𝐫 q 1 d q 1 + 𝐫 q 2 d q 2 + 𝐫 q 3 d q 3 = h 1 d q 1 𝐛 1 + h 2 d q 2 𝐛 2 + h 3 d q 3 𝐛 3 d\mathbf{r}=\dfrac{\partial\mathbf{r}}{\partial q_{1}}dq_{1}+\dfrac{\partial% \mathbf{r}}{\partial q_{2}}dq_{2}+\dfrac{\partial\mathbf{r}}{\partial q_{3}}dq% _{3}=h_{1}dq_{1}\mathbf{b}_{1}+h_{2}dq_{2}\mathbf{b}_{2}+h_{3}dq_{3}\mathbf{b}% _{3}
  10. h i = | 𝐫 q i | h_{i}=\left|\frac{\partial\mathbf{r}}{\partial q_{i}}\right|
  11. h i = k x i q k h^{i}{}_{k}=\cfrac{\partial x^{i}}{\partial q^{k}}
  12. 𝐛 i = q i | q i | = h i q i \mathbf{b}^{i}=\dfrac{\nabla q_{i}}{\left|\nabla q_{i}\right|}=h_{i}\nabla q_{i}
  13. 𝐯 = v 1 𝐛 1 + v 2 𝐛 2 + v 3 𝐛 3 = v 1 𝐛 1 + v 2 𝐛 2 + v 3 𝐛 3 \mathbf{v}=v^{1}\mathbf{b}_{1}+v^{2}\mathbf{b}_{2}+v^{3}\mathbf{b}_{3}=v_{1}% \mathbf{b}^{1}+v_{2}\mathbf{b}^{2}+v_{3}\mathbf{b}^{3}
  14. 𝐯 𝐛 i = v k 𝐛 k 𝐛 i = v k δ k i = v i \mathbf{v}\cdot\mathbf{b}^{i}=v^{k}\mathbf{b}_{k}\cdot\mathbf{b}^{i}=v^{k}% \delta^{i}_{k}=v^{i}
  15. 𝐯 𝐛 i = v k 𝐛 k 𝐛 i = v k δ i k = v i \mathbf{v}\cdot\mathbf{b}_{i}=v_{k}\mathbf{b}^{k}\cdot\mathbf{b}_{i}=v_{k}% \delta_{i}^{k}=v_{i}
  16. 𝐯 𝐛 i = v k 𝐛 k 𝐛 i = g k i v k \mathbf{v}\cdot\mathbf{b}_{i}=v^{k}\mathbf{b}_{k}\cdot\mathbf{b}_{i}=g_{ki}v^{k}
  17. 𝐯 𝐛 i = v k 𝐛 k 𝐛 i = g k i v k \mathbf{v}\cdot\mathbf{b}^{i}=v_{k}\mathbf{b}^{k}\cdot\mathbf{b}^{i}=g^{ki}v_{k}
  18. cos α = | 𝐞 1 | | 𝐛 1 | | 𝐞 1 | = | 𝐛 1 | cos α \cos\alpha=\cfrac{|\mathbf{e}_{1}|}{|\mathbf{b}_{1}|}\quad\Rightarrow\quad|% \mathbf{e}_{1}|=|\mathbf{b}_{1}|\cos\alpha
  19. cos α = d x d q 1 = | 𝐞 1 | | 𝐛 1 | \cos\alpha=\cfrac{dx}{dq^{1}}=\frac{|\mathbf{e}_{1}|}{|\mathbf{b}_{1}|}
  20. p 1 = 𝐛 1 𝐞 1 | 𝐞 1 | = | 𝐛 1 | | 𝐞 1 | | 𝐞 1 | cos α = | 𝐛 1 | d x d q 1 p 1 | 𝐛 1 | = d x d q 1 p^{1}=\mathbf{b}_{1}\cdot\cfrac{\mathbf{e}_{1}}{|\mathbf{e}_{1}|}=|\mathbf{b}_% {1}|\cfrac{|\mathbf{e}_{1}|}{|\mathbf{e}_{1}|}\cos\alpha=|\mathbf{b}_{1}|% \cfrac{dx}{dq^{1}}\quad\Rightarrow\quad\cfrac{p^{1}}{|\mathbf{b}_{1}|}=\cfrac{% dx}{dq^{1}}
  21. q i x j \cfrac{\partial q^{i}}{\partial x_{j}}
  22. x i q j \cfrac{\partial x_{i}}{\partial q^{j}}
  23. 𝐛 1 = p 1 𝐞 1 + p 2 𝐞 2 + p 3 𝐞 3 = x 1 q 1 𝐞 1 + x 2 q 1 𝐞 2 + x 3 q 1 𝐞 3 \mathbf{b}_{1}=p^{1}\mathbf{e}_{1}+p^{2}\mathbf{e}_{2}+p^{3}\mathbf{e}_{3}=% \cfrac{\partial x_{1}}{\partial q^{1}}\mathbf{e}_{1}+\cfrac{\partial x_{2}}{% \partial q^{1}}\mathbf{e}_{2}+\cfrac{\partial x_{3}}{\partial q^{1}}\mathbf{e}% _{3}
  24. 𝐛 1 \displaystyle\mathbf{b}_{1}
  25. 𝐞 1 \displaystyle\mathbf{e}_{1}
  26. x i q k 𝐞 i = 𝐛 k , q i x k 𝐛 i = 𝐞 k \cfrac{\partial x_{i}}{\partial q^{k}}\mathbf{e}_{i}=\mathbf{b}_{k},\quad% \cfrac{\partial q^{i}}{\partial x_{k}}\mathbf{b}_{i}=\mathbf{e}_{k}
  27. 𝐉 = [ x 1 q 1 x 1 q 2 x 1 q 3 x 2 q 1 x 2 q 2 x 2 q 3 x 3 q 1 x 3 q 2 x 3 q 3 ] , 𝐉 - 1 = [ q 1 x 1 q 1 x 2 q 1 x 3 q 2 x 1 q 2 x 2 q 2 x 3 q 3 x 1 q 3 x 2 q 3 x 3 ] \mathbf{J}=\begin{bmatrix}\cfrac{\partial x_{1}}{\partial q^{1}}&\cfrac{% \partial x_{1}}{\partial q^{2}}&\cfrac{\partial x_{1}}{\partial q^{3}}\\ \cfrac{\partial x_{2}}{\partial q^{1}}&\cfrac{\partial x_{2}}{\partial q^{2}}&% \cfrac{\partial x_{2}}{\partial q^{3}}\\ \cfrac{\partial x_{3}}{\partial q^{1}}&\cfrac{\partial x_{3}}{\partial q^{2}}&% \cfrac{\partial x_{3}}{\partial q^{3}}\\ \end{bmatrix},\quad\mathbf{J}^{-1}=\begin{bmatrix}\cfrac{\partial q^{1}}{% \partial x_{1}}&\cfrac{\partial q^{1}}{\partial x_{2}}&\cfrac{\partial q^{1}}{% \partial x_{3}}\\ \cfrac{\partial q^{2}}{\partial x_{1}}&\cfrac{\partial q^{2}}{\partial x_{2}}&% \cfrac{\partial q^{2}}{\partial x_{3}}\\ \cfrac{\partial q^{3}}{\partial x_{1}}&\cfrac{\partial q^{3}}{\partial x_{2}}&% \cfrac{\partial q^{3}}{\partial x_{3}}\\ \end{bmatrix}
  28. det ( 𝐉 - 1 ) 0 \det(\mathbf{J}^{-1})\neq 0
  29. 𝐱 = i = 1 n x i 𝐞 i \mathbf{x}=\sum_{i=1}^{n}x_{i}\mathbf{e}^{i}
  30. 𝐯 = j = 1 n v ¯ j 𝐛 j = j = 1 n v ¯ j ( 𝐪 ) 𝐛 j ( 𝐪 ) \mathbf{v}=\sum_{j=1}^{n}\bar{v}^{j}\mathbf{b}_{j}=\sum_{j=1}^{n}\bar{v}^{j}(% \mathbf{q})\mathbf{b}_{j}(\mathbf{q})
  31. s y m b o l S = S i j 𝐛 i 𝐛 j = S i 𝐛 i j 𝐛 j = S i 𝐛 i j 𝐛 j = S i j 𝐛 i 𝐛 j symbol{S}=S^{ij}\mathbf{b}_{i}\otimes\mathbf{b}_{j}=S^{i}{}_{j}\mathbf{b}_{i}% \otimes\mathbf{b}^{j}=S_{i}{}^{j}\mathbf{b}^{i}\otimes\mathbf{b}_{j}=S_{ij}% \mathbf{b}^{i}\otimes\mathbf{b}^{j}
  32. \scriptstyle\otimes
  33. S i j = g i k S k = j g j k S i = k g i k g j S k S^{ij}=g^{ik}S_{k}{}^{j}=g^{jk}S^{i}{}_{k}=g^{ik}g^{j\ell}S_{k\ell}
  34. d 𝐱 d\mathbf{x}
  35. d 𝐱 d 𝐱 = x i q j x i q k d q j d q k d\mathbf{x}\cdot d\mathbf{x}=\cfrac{\partial x_{i}}{\partial q^{j}}\cfrac{% \partial x_{i}}{\partial q^{k}}dq^{j}dq^{k}
  36. x k q i x k q j = g i j ( q i , q j ) = 𝐛 i 𝐛 j \cfrac{\partial x_{k}}{\partial q^{i}}\cfrac{\partial x_{k}}{\partial q^{j}}=g% _{ij}(q^{i},q^{j})=\mathbf{b}_{i}\cdot\mathbf{b}_{j}
  37. v i = g i k v k v^{i}=g^{ik}v_{k}
  38. h i h j = g i j = 𝐛 i 𝐛 j h i = g i i = | 𝐛 i | = | 𝐱 q i | h_{i}h_{j}=g_{ij}=\mathbf{b}_{i}\cdot\mathbf{b}_{j}\quad\Rightarrow\quad h_{i}% =\sqrt{g_{ii}}=\left|\mathbf{b}_{i}\right|=\left|\cfrac{\partial\mathbf{x}}{% \partial q^{i}}\right|
  39. g i j = 𝐱 q i 𝐱 q j = ( h k i 𝐞 k ) ( h m j 𝐞 m ) = h k i h k j g_{ij}=\cfrac{\partial\mathbf{x}}{\partial q^{i}}\cdot\cfrac{\partial\mathbf{x% }}{\partial q^{j}}=\left(h_{ki}\mathbf{e}_{k}\right)\cdot\left(h_{mj}\mathbf{e% }_{m}\right)=h_{ki}h_{kj}
  40. g = g 11 g 22 g 33 = h 1 2 h 2 2 h 3 2 g = h 1 h 2 h 3 = J g=g_{11}g_{22}g_{33}=h_{1}^{2}h_{2}^{2}h_{3}^{2}\quad\Rightarrow\quad\sqrt{g}=% h_{1}h_{2}h_{3}=J
  41. ( x , y ) = ( r cos θ , r sin θ ) (x,y)=(r\cos\theta,r\sin\theta)\,\!
  42. s y m b o l = ϵ i j k 𝐞 i 𝐞 j 𝐞 k symbol{\mathcal{E}}=\epsilon_{ijk}\mathbf{e}^{i}\otimes\mathbf{e}^{j}\otimes% \mathbf{e}^{k}
  43. s y m b o l = i j k 𝐛 i 𝐛 j 𝐛 k = i j k 𝐛 i 𝐛 j 𝐛 k symbol{\mathcal{E}}=\mathcal{E}_{ijk}\mathbf{b}^{i}\otimes\mathbf{b}^{j}% \otimes\mathbf{b}^{k}=\mathcal{E}^{ijk}\mathbf{b}_{i}\otimes\mathbf{b}_{j}% \otimes\mathbf{b}_{k}
  44. i j k = 1 J ϵ i j k = 1 + g ϵ i j k \mathcal{E}^{ijk}=\cfrac{1}{J}\epsilon_{ijk}=\cfrac{1}{+\sqrt{g}}\epsilon_{ijk}
  45. 𝐛 i , j = 𝐛 i q j = Γ i j k 𝐛 k 𝐛 i , j 𝐛 k = Γ i j k \mathbf{b}_{i,j}=\frac{\partial\mathbf{b}_{i}}{\partial q^{j}}=\Gamma_{ijk}% \mathbf{b}^{k}\quad\Rightarrow\quad\mathbf{b}_{i,j}\cdot\mathbf{b}_{k}=\Gamma_% {ijk}
  46. g i j , k = ( 𝐛 i 𝐛 j ) , k = 𝐛 i , k 𝐛 j + 𝐛 i 𝐛 j , k = Γ i k j + Γ j k i g i k , j = ( 𝐛 i 𝐛 k ) , j = 𝐛 i , j 𝐛 k + 𝐛 i 𝐛 k , j = Γ i j k + Γ k j i g j k , i = ( 𝐛 j 𝐛 k ) , i = 𝐛 j , i 𝐛 k + 𝐛 j 𝐛 k , i = Γ j i k + Γ k i j \begin{aligned}\displaystyle g_{ij,k}&\displaystyle=(\mathbf{b}_{i}\cdot% \mathbf{b}_{j})_{,k}=\mathbf{b}_{i,k}\cdot\mathbf{b}_{j}+\mathbf{b}_{i}\cdot% \mathbf{b}_{j,k}=\Gamma_{ikj}+\Gamma_{jki}\\ \displaystyle g_{ik,j}&\displaystyle=(\mathbf{b}_{i}\cdot\mathbf{b}_{k})_{,j}=% \mathbf{b}_{i,j}\cdot\mathbf{b}_{k}+\mathbf{b}_{i}\cdot\mathbf{b}_{k,j}=\Gamma% _{ijk}+\Gamma_{kji}\\ \displaystyle g_{jk,i}&\displaystyle=(\mathbf{b}_{j}\cdot\mathbf{b}_{k})_{,i}=% \mathbf{b}_{j,i}\cdot\mathbf{b}_{k}+\mathbf{b}_{j}\cdot\mathbf{b}_{k,i}=\Gamma% _{jik}+\Gamma_{kij}\end{aligned}
  47. 𝐛 i , j = 𝐛 j , i Γ i j k = Γ j i k \mathbf{b}_{i,j}=\mathbf{b}_{j,i}\quad\Rightarrow\quad\Gamma_{ijk}=\Gamma_{jik}
  48. Γ i j k = 1 2 ( g i k , j + g j k , i - g i j , k ) = 1 2 [ ( 𝐛 i 𝐛 k ) , j + ( 𝐛 j 𝐛 k ) , i - ( 𝐛 i 𝐛 j ) , k ] \Gamma_{ijk}=\frac{1}{2}(g_{ik,j}+g_{jk,i}-g_{ij,k})=\frac{1}{2}[(\mathbf{b}_{% i}\cdot\mathbf{b}_{k})_{,j}+(\mathbf{b}_{j}\cdot\mathbf{b}_{k})_{,i}-(\mathbf{% b}_{i}\cdot\mathbf{b}_{j})_{,k}]
  49. Γ i j = k Γ j i , k 𝐛 i q j = Γ i j 𝐛 k k \Gamma_{ij}{}^{k}=\Gamma_{ji}{}^{k},\quad\cfrac{\partial\mathbf{b}_{i}}{% \partial q^{j}}=\Gamma_{ij}{}^{k}\mathbf{b}_{k}
  50. Γ i j = k 𝐛 i q j 𝐛 k = - 𝐛 i 𝐛 k q j \Gamma_{ij}{}^{k}=\cfrac{\partial\mathbf{b}_{i}}{\partial q^{j}}\cdot\mathbf{b% }^{k}=-\mathbf{b}_{i}\cdot\cfrac{\partial\mathbf{b}^{k}}{\partial q^{j}}
  51. 𝐛 i q j = - Γ i 𝐛 k j k , s y m b o l 𝐛 i = Γ i j 𝐛 k k 𝐛 j , s y m b o l 𝐛 i = - Γ j k 𝐛 k i 𝐛 j \cfrac{\partial\mathbf{b}^{i}}{\partial q^{j}}=-\Gamma^{i}{}_{jk}\mathbf{b}^{k% },\quad symbol{\nabla}\mathbf{b}_{i}=\Gamma_{ij}{}^{k}\mathbf{b}_{k}\otimes% \mathbf{b}^{j},\quad symbol{\nabla}\mathbf{b}^{i}=-\Gamma_{jk}{}^{i}\mathbf{b}% ^{k}\otimes\mathbf{b}^{j}
  52. 𝐱 λ 1 × 𝐱 λ 2 = ( 𝐱 q i q i λ 1 ) × ( 𝐱 q j q j λ 2 ) = k m p ( h k i q i λ 1 ) ( h m j q j λ 2 ) 𝐛 p {\partial\mathbf{x}\over\partial\lambda_{1}}\times{\partial\mathbf{x}\over% \partial\lambda_{2}}=\left({\partial\mathbf{x}\over\partial q^{i}}{\partial q^% {i}\over\partial\lambda_{1}}\right)\times\left({\partial\mathbf{x}\over% \partial q^{j}}{\partial q^{j}\over\partial\lambda_{2}}\right)=\mathcal{E}_{% kmp}\left(h_{ki}{\partial q^{i}\over\partial\lambda_{1}}\right)\left(h_{mj}{% \partial q^{j}\over\partial\lambda_{2}}\right)\mathbf{b}_{p}
  53. \mathcal{E}
  54. 𝐱 λ 1 × 𝐱 λ 2 = | 𝐞 1 𝐞 2 𝐞 3 h 1 i q i λ 1 h 2 i q i λ 1 h 3 i q i λ 1 h 1 j q j λ 2 h 2 j q j λ 2 h 3 j q j λ 2 | {\partial\mathbf{x}\over\partial\lambda_{1}}\times{\partial\mathbf{x}\over% \partial\lambda_{2}}=\begin{vmatrix}\mathbf{e}_{1}&\mathbf{e}_{2}&\mathbf{e}_{% 3}\\ h_{1i}\dfrac{\partial q^{i}}{\partial\lambda_{1}}&h_{2i}\dfrac{\partial q^{i}}% {\partial\lambda_{1}}&h_{3i}\dfrac{\partial q^{i}}{\partial\lambda_{1}}\\ h_{1j}\dfrac{\partial q^{j}}{\partial\lambda_{2}}&h_{2j}\dfrac{\partial q^{j}}% {\partial\lambda_{2}}&h_{3j}\dfrac{\partial q^{j}}{\partial\lambda_{2}}\end{vmatrix}
  55. C φ ( 𝐱 ) d s = a b φ ( 𝐱 ( λ ) ) | 𝐱 λ | d λ \int_{C}\varphi(\mathbf{x})ds=\int_{a}^{b}\varphi(\mathbf{x}(\lambda))\left|{% \partial\mathbf{x}\over\partial\lambda}\right|d\lambda
  56. C 𝐯 ( 𝐱 ) d 𝐬 = a b 𝐯 ( 𝐱 ( λ ) ) ( 𝐱 λ ) d λ \int_{C}\mathbf{v}(\mathbf{x})\cdot d\mathbf{s}=\int_{a}^{b}\mathbf{v}(\mathbf% {x}(\lambda))\cdot\left({\partial\mathbf{x}\over\partial\lambda}\right)d\lambda
  57. S φ ( 𝐱 ) d S = T φ ( 𝐱 ( λ 1 , λ 2 ) ) | 𝐱 λ 1 × 𝐱 λ 2 | d λ 1 d λ 2 \int_{S}\varphi(\mathbf{x})dS=\iint_{T}\varphi(\mathbf{x}(\lambda_{1},\lambda_% {2}))\left|{\partial\mathbf{x}\over\partial\lambda_{1}}\times{\partial\mathbf{% x}\over\partial\lambda_{2}}\right|d\lambda_{1}d\lambda_{2}
  58. S 𝐯 ( 𝐱 ) d S = T 𝐯 ( 𝐱 ( λ 1 , λ 2 ) ) ( 𝐱 λ 1 × 𝐱 λ 2 ) d λ 1 d λ 2 \int_{S}\mathbf{v}(\mathbf{x})\cdot dS=\iint_{T}\mathbf{v}(\mathbf{x}(\lambda_% {1},\lambda_{2}))\cdot\left({\partial\mathbf{x}\over\partial\lambda_{1}}\times% {\partial\mathbf{x}\over\partial\lambda_{2}}\right)d\lambda_{1}d\lambda_{2}
  59. V φ ( x , y , z ) d V = V χ ( q 1 , q 2 , q 3 ) J d q 1 d q 2 d q 3 \iiint_{V}\varphi(x,y,z)dV=\iiint_{V}\chi(q_{1},q_{2},q_{3})Jdq_{1}dq_{2}dq_{3}
  60. V 𝐮 ( x , y , z ) d V = V 𝐯 ( q 1 , q 2 , q 3 ) J d q 1 d q 2 d q 3 \iiint_{V}\mathbf{u}(x,y,z)dV=\iiint_{V}\mathbf{v}(q_{1},q_{2},q_{3})Jdq_{1}dq% _{2}dq_{3}
  61. φ = 1 h i φ q i 𝐛 i \nabla\varphi=\cfrac{1}{h_{i}}{\partial\varphi\over\partial q^{i}}\mathbf{b}^{i}
  62. 𝐯 = 1 h i 2 𝐯 q i 𝐛 i \nabla\mathbf{v}=\cfrac{1}{h_{i}^{2}}{\partial\mathbf{v}\over\partial q^{i}}% \otimes\mathbf{b}_{i}
  63. s y m b o l s y m b o l S = s y m b o l S q i 𝐛 i symbol{\nabla}symbol{S}=\cfrac{\partial symbol{S}}{\partial q^{i}}\otimes% \mathbf{b}^{i}
  64. 𝐯 = 1 j h j q i ( v i j i h j ) \nabla\cdot\mathbf{v}=\cfrac{1}{\prod_{j}h_{j}}\frac{\partial}{\partial q^{i}}% (v^{i}\prod_{j\neq i}h_{j})
  65. ( s y m b o l \cdotsymbol S ) 𝐚 = s y m b o l ( s y m b o l S 𝐚 ) (symbol{\nabla}\cdotsymbol{S})\cdot\mathbf{a}=symbol{\nabla}\cdot(symbol{S}% \cdot\mathbf{a})
  66. s y m b o l \cdotsymbol S = [ S i j q k - Γ k i l S l j - Γ k j l S i l ] g i k 𝐛 j symbol{\nabla}\cdotsymbol{S}=\left[\cfrac{\partial S_{ij}}{\partial q^{k}}-% \Gamma^{l}_{ki}S_{lj}-\Gamma^{l}_{kj}S_{il}\right]g^{ik}\mathbf{b}^{j}
  67. 2 φ = 1 j h j q i ( j h j h i 2 φ q i ) \nabla^{2}\varphi=\cfrac{1}{\prod_{j}h_{j}}\frac{\partial}{\partial q^{i}}% \left(\cfrac{\prod_{j}h_{j}}{h_{i}^{2}}\frac{\partial\varphi}{\partial q^{i}}\right)
  68. × 𝐯 = 1 h 1 h 2 h 3 𝐞 i ϵ i j k h i ( h k v k ) q j \nabla\times\mathbf{v}=\frac{1}{h_{1}h_{2}h_{3}}\mathbf{e}_{i}\epsilon_{ijk}h_% {i}\frac{\partial(h_{k}v_{k})}{\partial q^{j}}
  69. ϵ i j k \epsilon_{ijk}

Cut-elimination_theorem.html

  1. A 1 A_{1}
  2. A 2 A_{2}
  3. A 3 A_{3}
  4. B 1 B_{1}
  5. B 2 B_{2}
  6. B 3 B_{3}
  7. Γ A , Δ \Gamma\vdash A,\Delta
  8. Π , A Λ \Pi,A\vdash\Lambda
  9. Γ , Π Δ , Λ \Gamma,\Pi\vdash\Delta,\Lambda
  10. A A
  11. Γ A \Gamma\vdash A
  12. Π , A B \Pi,A\vdash B
  13. Γ , Π B \Gamma,\Pi\vdash B
  14. B B
  15. A A
  16. A A
  17. A A

Cut_rule.html

  1. Γ A , Δ Γ , A Δ Γ , Γ Δ , Δ \cfrac{\Gamma\vdash A,\Delta\qquad\Gamma^{\prime},A\vdash\Delta^{\prime}}{% \Gamma,\Gamma^{\prime}\vdash\Delta,\Delta^{\prime}}

Cuthill–McKee_algorithm.html

  1. R i R_{i}
  2. i = 1 , 2 , . . i=1,2,..
  3. R i + 1 R_{i+1}
  4. R i R_{i}
  5. R i R_{i}
  6. n × n n\times n
  7. i = 1 , 2 , i=1,2,\dots
  8. R i R_{i}
  9. R i R_{i}
  10. A i := Adj ( R i ) R A_{i}:=\operatorname{Adj}(R_{i})\setminus R
  11. A i A_{i}
  12. A i A_{i}

Cutting_stock_problem.html

  1. q j , j = 1 , , m q_{j},j=1,\ldots,m
  2. x i x_{i}
  3. min i = 1 n c i x i \min\sum_{i=1}^{n}c_{i}x_{i}
  4. s.t. i = 1 n a i j x i q j , j = 1 , , m \,\text{s.t.}\sum_{i=1}^{n}a_{ij}x_{i}\geq q_{j},\quad\quad\forall j=1,\dots,m
  5. x i 0 x_{i}\geq 0
  6. a i j a_{ij}
  7. j j
  8. i i
  9. c i c_{i}
  10. i i
  11. c i = 1 c_{i}=1
  12. q j i = 1 n a i j x i Q j , j = 1 , , m q_{j}\leq\sum_{i=1}^{n}a_{ij}x_{i}\leq Q_{j},\quad\quad\forall j=1,\dots,m

Cycle_detection.html

  1. x 0 , x 1 = f ( x 0 ) , x 2 = f ( x 1 ) , , x i = f ( x i - 1 ) , x_{0},\ x_{1}=f(x_{0}),\ x_{2}=f(x_{1}),\ \dots,\ x_{i}=f(x_{i-1}),\ \dots
  2. ( λ + μ ) ( 1 + c M - 1 / 2 ) (\lambda+\mu)(1+cM^{-1/2})
  3. ( λ + μ ) ( 1 + 1 M - 1 ) \scriptstyle(\lambda+\mu)(1+\frac{1}{M-1})

Cyclic_voltammetry.html

  1. E p a - E p c = 56.5 mV n E_{pa}-E_{pc}=\frac{56.5\,\text{ mV}}{n}

Cyclomatic_complexity.html

  1. E - N + P E-N+P
  2. M := b 1 ( G , t ) := rank H 1 ( G , t ) , M:=b_{1}(G,t):=\operatorname{rank}H_{1}(G,t),
  3. G ~ \tilde{G}
  4. M = b 1 ( G ~ ) = rank H 1 ( G ~ ) . M=b_{1}(\tilde{G})=\operatorname{rank}H_{1}(\tilde{G}).
  5. X X
  6. X X
  7. π 1 ( X ) = \Z n \pi_{1}(X)=\Z^{n}
  8. n + 1 n+1
  9. \leq
  10. \leq

Cyclonic_separation.html

  1. V i n V_{in}
  2. r r
  3. V t V_{t}
  4. V r V_{r}
  5. F d = - 6 π r p μ V r . F_{d}=-6\pi r_{p}\mu V_{r}.
  6. ρ p \rho_{p}
  7. F c = m V t 2 r F_{c}=m\frac{V_{t}^{2}}{r}
  8. = 4 3 π ρ p r p 3 V t 2 r . =\frac{4}{3}\pi\rho_{p}r_{p}^{3}\frac{V_{t}^{2}}{r}.
  9. ρ f \rho_{f}
  10. F b = - V p ρ f V t 2 r F_{b}=-V_{p}\rho_{f}\frac{V_{t}^{2}}{r}
  11. = - 4 π r p 3 3 V t 2 r ρ f . =-\frac{4\pi r_{p}^{3}}{3}\frac{V_{t}^{2}}{r}\rho_{f}.
  12. V p V_{p}
  13. m d V r d t = F d + F c + F b m\frac{dV_{r}}{dt}=F_{d}+F_{c}+F_{b}
  14. d V r d t \frac{dV_{r}}{dt}
  15. F d + F c + F b = 0 F_{d}+F_{c}+F_{b}=0
  16. - 6 π r p μ V r + 4 3 π r p 3 V t 2 r ρ p - 4 3 π r p 3 V t 2 r ρ f = 0 -6\pi r_{p}\mu V_{r}+\frac{4}{3}\pi r_{p}^{3}\frac{V_{t}^{2}}{r}\rho_{p}-\frac% {4}{3}\pi r_{p}^{3}\frac{V_{t}^{2}}{r}\rho_{f}=0
  17. V r V_{r}
  18. V r = 2 9 r p 2 μ V t 2 r ( ρ p - ρ f ) V_{r}=\frac{2}{9}\frac{r_{p}^{2}}{\mu}\frac{V_{t}^{2}}{r}(\rho_{p}-\rho_{f})
  19. d V r d t + 9 2 μ ρ p r p 2 V r - ( 1 - ρ f ρ p ) V t 2 r = 0 \frac{dV_{r}}{dt}+\frac{9}{2}\frac{\mu}{\rho_{p}r_{p}^{2}}V_{r}-\left(1-\frac{% \rho_{f}}{\rho_{p}}\right)\frac{V_{t}^{2}}{r}=0
  20. V r V_{r}
  21. x ′′ + c 1 x + c 2 = 0 x^{\prime\prime}+c_{1}x^{\prime}+c_{2}=0
  22. r 2 r^{2}
  23. V t r 2 . V_{t}\propto r^{2}.
  24. U r = U i n r R i n . U_{r}=U_{in}\frac{r}{R_{in}}.
  25. V t V_{t}