wpmath0000013_0

(5407)_1992_AX.html

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

(g,K)-module.html

  1. ( 𝔤 , K ) (\mathfrak{g},K)
  2. ( 𝔤 , K ) (\mathfrak{g},K)
  3. 𝔤 \mathfrak{g}
  4. 𝔤 \mathfrak{g}
  5. 𝔨 \mathfrak{k}
  6. ( 𝔤 , K ) (\mathfrak{g},K)
  7. 𝔤 \mathfrak{g}
  8. 𝔤 \mathfrak{g}
  9. k ( X v ) = ( Ad ( k ) X ) ( k v ) k\cdot(X\cdot v)=(\operatorname{Ad}(k)X)\cdot(k\cdot v)
  10. 𝔨 \mathfrak{k}
  11. ( d d t exp ( t Y ) v ) | t = 0 = Y v . \left.\left(\frac{d}{dt}\exp(tY)\cdot v\right)\right|_{t=0}=Y\cdot v.
  12. \cdot
  13. 𝔤 \mathfrak{g}
  14. 𝔤 \mathfrak{g}
  15. k v k\cdot v
  16. 𝔤 \mathfrak{g}
  17. k X v = k X k - 1 k v = ( k X k - 1 ) k v . kXv=kXk^{-1}kv=\left(kXk^{-1}\right)kv.
  18. 𝔤 \mathfrak{g}
  19. 𝔤 \mathfrak{g}
  20. 𝔨 \mathfrak{k}
  21. 𝔤 \mathfrak{g}

10_Metre.html

  1. 10.000 metres = L + B + 1 / 3 G + 3 d + 1 / 3 S - F 2 10.000\mbox{ metres}~{}=\frac{L+B+1/3G+3d+1/3\sqrt{S}-F}{2}
  2. L L
  3. B B
  4. G G
  5. d d
  6. S S
  7. F F
  8. 10.000 metres = L + 0.25 G + 2 d + S - F 2.5 10.000\mbox{ metres}~{}=\frac{L+0.25G+2d+\sqrt{S}-F}{2.5}
  9. L L
  10. G G
  11. d d
  12. S S
  13. F F

15-metre_class.html

  1. 15 m = L + B + 1 2 G + 3 d + 1 3 S - F 2 15~{}\mbox{m}~{}=\frac{\textrm{L}+B+\frac{1}{2}G+3d+\frac{1}{3}\sqrt{S}-F}{2}

1922–23_Beşiktaş_J.K._season.html

  1. \leftarrow
  2. \rightarrow

1923–24_Beşiktaş_J.K._season.html

  1. \leftarrow
  2. \rightarrow

1924–25_Beşiktaş_J.K._season.html

  1. \leftarrow
  2. \rightarrow

1925–26_Beşiktaş_J.K._season.html

  1. \leftarrow
  2. \rightarrow

1926–27_Beşiktaş_J.K._season.html

  1. \leftarrow
  2. \rightarrow

1928–29_Beşiktaş_J.K._season.html

  1. \leftarrow
  2. \rightarrow

1929–30_Beşiktaş_J.K._season.html

  1. \leftarrow
  2. \rightarrow

1956–57_Beşiktaş_J.K._season.html

  1. \leftarrow
  2. \rightarrow

1957–58_Beşiktaş_J.K._season.html

  1. \leftarrow
  2. \rightarrow

1958–59_Beşiktaş_J.K._season.html

  1. \leftarrow
  2. \rightarrow

1959–60_Beşiktaş_J.K._season.html

  1. \leftarrow
  2. \rightarrow

1963–64_Beşiktaş_J.K._season.html

  1. \leftarrow
  2. \rightarrow

1964_PRL_symmetry_breaking_papers.html

  1. μ 2 < 0 \mu^{2}<0
  2. ϕ 0 \phi_{0}
  3. ϕ ~ = ϕ - ϕ 0 \tilde{\phi}=\phi-\phi_{0}
  4. ϕ 0 \phi_{0}
  5. g ϕ 0 ψ ¯ ψ g\phi_{0}\bar{\psi}\psi
  6. ϕ 0 \phi_{0}
  7. g ϕ 0 g\phi_{0}
  8. ϕ ~ \tilde{\phi}

2002–03_Beşiktaş_J.K._season.html

  1. \leftarrow
  2. \rightarrow

2008_Puerto_Rico_Islanders_season.html

  1. \leftarrow
  2. \rightarrow

2009_Jupiter_impact_event.html

  1. ( 100 5 ) 25 - ( - 2.8 ) = 131 (\sqrt[5]{100})^{25-(-2.8)}=131
  2. × 10 1 1 \times 10^{1}1

2009_Puerto_Rico_Islanders_season.html

  1. \leftarrow
  2. \rightarrow

2009–10_Olympique_Lyonnais_season.html

  1. \leftarrow
  2. \rightarrow

2009–10_Paris_Saint-Germain_F.C._(Ladies)_season.html

  1. \leftarrow
  2. \rightarrow

21:9_aspect_ratio.html

  1. 3 ¯ \overline{3}
  2. 370 ¯ \overline{370}
  3. 7 ¯ \overline{7}
  4. 3 ¯ \overline{3}
  5. 4 3 4 3 = 16 9 \tfrac{4}{3}\cdot\tfrac{4}{3}=\tfrac{16}{9}
  6. 16 9 4 3 = 64 27 \tfrac{16}{9}\cdot\tfrac{4}{3}=\tfrac{64}{27}
  7. 1 3 \tfrac{1}{3}
  8. 370 ¯ \overline{370}

4-Nitrochlorobenzene.html

  1. ClC 6 H 5 + HNO 3 ClC 6 H 4 NO 2 + H 2 O \mathrm{ClC_{6}H_{5}+HNO_{3}\rightarrow ClC_{6}H_{4}NO_{2}+H_{2}O}

5_21_honeycomb.html

  1. E ~ 8 {\tilde{E}}_{8}
  2. E ~ 8 {\tilde{E}}_{8}
  3. E ~ 8 {\tilde{E}}_{8}
  4. A ~ 8 {\tilde{A}}_{8}
  5. E ~ 8 {\tilde{E}}_{8}
  6. A ~ 8 {\tilde{A}}_{8}
  7. A 8 A_{8}
  8. E ~ 8 {\tilde{E}}_{8}
  9. D ~ 8 {\tilde{D}}_{8}
  10. E ~ 8 {\tilde{E}}_{8}
  11. D ~ 8 {\tilde{D}}_{8}
  12. D 8 D_{8}

6.5_Metre.html

  1. L f S D 3 2 , 8 \frac{Lf\cdot\sqrt{S}}{\sqrt[3]{D}}\leq 2,8

6₂_knot.html

  1. Δ ( t ) = - t 2 + 3 t - 3 - 1 + 3 t - 1 - t - 2 , \Delta(t)=-t^{2}+3t-3-1+3t^{-1}-t^{-2},\,
  2. ( z ) = - z 4 - z 2 + 1 , \nabla(z)=-z^{4}-z^{2}+1,\,
  3. V ( q ) = q - 1 + 2 q - 1 - 2 q - 2 + 2 q - 3 - 2 q - 4 + q - 5 . V(q)=q-1+2q^{-1}-2q^{-2}+2q^{-3}-2q^{-4}+q^{-5}.\,

6₃_knot.html

  1. σ 1 - 1 σ 2 2 σ 1 - 2 σ 2 . \sigma_{1}^{-1}\sigma_{2}^{2}\sigma_{1}^{-2}\sigma_{2}.\,
  2. Δ ( t ) = t 2 - 3 t + 5 - 3 t - 1 + t - 2 , \Delta(t)=t^{2}-3t+5-3t^{-1}+t^{-2},\,
  3. ( z ) = z 4 + z 2 + 1 , \nabla(z)=z^{4}+z^{2}+1,\,
  4. V ( q ) = - q 3 + 2 q 2 - 2 q + 3 - 2 q - 1 + 2 q - 2 - q - 3 , V(q)=-q^{3}+2q^{2}-2q+3-2q^{-1}+2q^{-2}-q^{-3},\,
  5. L ( a , z ) = a z 5 + z 5 a - 1 + 2 a 2 z 4 + 2 z 4 a - 2 + 4 z 4 + a 3 z 3 + a z 3 + z 3 a - 1 + z 3 a - 3 - 3 a 2 z 2 - 3 z 2 a - 2 - 6 z 2 - a 3 z - 2 a z - 2 z a - 1 - z a - 3 + a 2 + a - 2 + 3. L(a,z)=az^{5}+z^{5}a^{-1}+2a^{2}z^{4}+2z^{4}a^{-2}+4z^{4}+a^{3}z^{3}+az^{3}+z^% {3}a^{-1}+z^{3}a^{-3}-3a^{2}z^{2}-3z^{2}a^{-2}-6z^{2}-a^{3}z-2az-2za^{-1}-za-3% +a^{2}+a^{-2}+3.\,

7_Metre.html

  1. 7.000 metres = L + B + 1 / 3 G + 3 d + 1 / 3 S - F 2 7.000\mbox{ metres}~{}=\frac{L+B+1/3G+3d+1/3\sqrt{S}-F}{2}
  2. L L
  3. B B
  4. G G
  5. d d
  6. S S
  7. F F
  8. 7.000 metres = L + 0.25 G + 2 d + S - F 2.5 7.000\mbox{ metres}~{}=\frac{L+0.25G+2d+\sqrt{S}-F}{2.5}
  9. L L
  10. G G
  11. d d
  12. S S
  13. F F

7₁_knot.html

  1. Δ ( t ) = t 3 - t 2 + t - 1 + t - 1 - t - 2 + t - 3 , \Delta(t)=t^{3}-t^{2}+t-1+t^{-1}-t^{-2}+t^{-3},\,
  2. ( z ) = z 6 + 5 z 4 + 6 z 2 + 1 , \nabla(z)=z^{6}+5z^{4}+6z^{2}+1,\,
  3. V ( q ) = q - 3 + q - 5 - q - 6 + q - 7 - q - 8 + q - 9 - q - 10 . V(q)=q^{-3}+q^{-5}-q^{-6}+q^{-7}-q^{-8}+q^{-9}-q^{-10}.\,

8.5_Metre.html

  1. J = L f S D 3 3 , 2 J=\frac{Lf\cdot\sqrt{S}}{\sqrt[3]{D}}\leq 3,2

9_Metre.html

  1. 9.000 metres = L + B + 1 / 3 G + 3 d + 1 / 3 S - F 2 9.000\mbox{ metres}~{}=\frac{L+B+1/3G+3d+1/3\sqrt{S}-F}{2}
  2. L L
  3. B B
  4. G G
  5. d d
  6. S S
  7. F F

Aanderaa–Karp–Rosenberg_conjecture.html

  1. Ω ( min { n min ( p , 1 - p ) , n 2 log n } ) \Omega\left(\min\left\{\frac{n}{\min(p,1-p)},\frac{n^{2}}{\log n}\right\}\right)

Absorptance.html

  1. A = Φ e a Φ e i , A=\frac{\Phi_{\mathrm{e}}^{\mathrm{a}}}{\Phi_{\mathrm{e}}^{\mathrm{i}}},
  2. A ν = Φ e , ν a Φ e , ν i , A_{\nu}=\frac{\Phi_{\mathrm{e},\nu}^{\mathrm{a}}}{\Phi_{\mathrm{e},\nu}^{% \mathrm{i}}},
  3. A λ = Φ e , λ a Φ e , λ i , A_{\lambda}=\frac{\Phi_{\mathrm{e},\lambda}^{\mathrm{a}}}{\Phi_{\mathrm{e},% \lambda}^{\mathrm{i}}},
  4. A Ω = L e , Ω a L e , Ω i , A_{\Omega}=\frac{L_{\mathrm{e},\Omega}^{\mathrm{a}}}{L_{\mathrm{e},\Omega}^{% \mathrm{i}}},
  5. A ν , Ω = L e , Ω , ν a L e , Ω , ν i , A_{\nu,\Omega}=\frac{L_{\mathrm{e},\Omega,\nu}^{\mathrm{a}}}{L_{\mathrm{e},% \Omega,\nu}^{\mathrm{i}}},
  6. A λ , Ω = L e , Ω , λ a L e , Ω , λ i , A_{\lambda,\Omega}=\frac{L_{\mathrm{e},\Omega,\lambda}^{\mathrm{a}}}{L_{% \mathrm{e},\Omega,\lambda}^{\mathrm{i}}},

Abstract_elementary_class.html

  1. K , K \langle K,\prec_{K}\rangle
  2. K K
  3. L = L ( K ) L=L(K)
  4. K \prec_{K}
  5. K K
  6. M K N M\prec_{K}N
  7. M M
  8. N N
  9. K K
  10. M , N , M , N K , M,N,M^{\prime},N^{\prime}\in K,
  11. f : M M , f\colon M\simeq M^{\prime},
  12. g : N N , g\colon N\simeq N^{\prime},
  13. f g , f\subseteq g,
  14. M K N , M\prec_{K}N,
  15. M K N . M^{\prime}\prec_{K}N^{\prime}.
  16. M 1 K M 3 , M_{1}\prec_{K}M_{3},
  17. M 2 K M 3 , M_{2}\prec_{K}M_{3},
  18. M 1 M 2 , M_{1}\subseteq M_{2},
  19. M 1 K M 2 . M_{1}\prec_{K}M_{2}.
  20. γ \gamma
  21. { M α α < γ } K \{\,M_{\alpha}\mid\alpha<\gamma\,\}\subseteq K
  22. α < β < γ M α K M β \alpha<\beta<\gamma\implies M_{\alpha}\prec_{K}M_{\beta}
  23. α < γ M α K \bigcup_{\alpha<\gamma}M_{\alpha}\in K
  24. M α K N M_{\alpha}\prec_{K}N
  25. α < γ \alpha<\gamma
  26. α < γ M α K N \bigcup_{\alpha<\gamma}M_{\alpha}\prec_{K}N
  27. μ | L ( K ) | + 0 \mu\geq|L(K)|+\aleph_{0}
  28. A A
  29. M M
  30. N N
  31. K K
  32. A A
  33. N | A | + μ \|N\|\leq|A|+\mu
  34. N K M N\prec_{K}M
  35. LS ( K ) \operatorname{LS}(K)
  36. μ \mu
  37. K K
  38. K K
  39. f : M N f:M\rightarrow N
  40. M , N K M,N\in K
  41. f [ M ] K N f[M]\prec_{K}N
  42. f f
  43. M M
  44. f [ M ] f[M]
  45. K K
  46. Mod ( T ) \operatorname{Mod}(T)
  47. ϕ \phi
  48. L ω 1 , ω L_{\omega_{1},\omega}
  49. \mathcal{F}
  50. ϕ \phi
  51. Mod ( T ) , \langle\operatorname{Mod}(T),\prec_{\mathcal{F}}\rangle
  52. 0 \aleph_{0}
  53. L κ , ω L_{\kappa,\omega}
  54. L ω 1 , ω ( Q ) L_{\omega_{1},\omega}(Q)
  55. Q Q
  56. 1 \aleph_{1}
  57. 2 0 2^{\aleph_{0}}
  58. K K
  59. M 0 , M 1 , M 2 K M_{0},M_{1},M_{2}\in K
  60. M 0 K M 1 M_{0}\prec_{K}M_{1}
  61. M 0 K M 2 M_{0}\prec_{K}M_{2}
  62. N K N\in K
  63. M 1 M_{1}
  64. M 2 M_{2}
  65. N N
  66. M 0 M_{0}
  67. \mathfrak{C}
  68. μ \mu
  69. LS ( K ) \operatorname{LS}(K)
  70. λ μ \lambda\geq\mu
  71. λ \lambda
  72. θ \theta
  73. θ μ \theta\geq\mu
  74. K K
  75. PC 2 LS ( K ) \operatorname{PC}_{2^{\operatorname{LS}(K)}}
  76. 2 LS ( K ) 2^{\operatorname{LS}(K)}
  77. K K
  78. ( 2 LS ( K ) ) + \beth_{(2^{\operatorname{LS}(K)})^{+}}
  79. λ \lambda
  80. λ + \lambda^{+}
  81. 2 λ < 2 λ + 2^{\lambda}<2^{\lambda^{+}}
  82. λ \lambda
  83. PC 0 \operatorname{PC}_{\aleph_{0}}
  84. 0 \aleph_{0}
  85. 0 \aleph_{0}
  86. 1 \aleph_{1}
  87. 2 \aleph_{2}
  88. L ω 1 , ω ( Q ) L_{\omega_{1},\omega}(Q)
  89. λ \lambda
  90. μ λ \mu\leq\lambda
  91. 2 2 LS ( K ) 2^{2^{\operatorname{LS}(K)}}
  92. 2 LS ( K ) 2^{\operatorname{LS}(K)}

Abstract_rewriting_system.html

  1. \rightarrow
  2. 1 2 = \rightarrow_{1}\cup\rightarrow_{2}=\rightarrow
  3. * \stackrel{*}{\rightarrow}
  4. = \rightarrow\cup=
  5. * \stackrel{*}{\rightarrow}
  6. \rightarrow
  7. \rightarrow
  8. \leftrightarrow
  9. - 1 \rightarrow\cup\rightarrow^{-1}
  10. \rightarrow
  11. * \stackrel{*}{\leftrightarrow}
  12. = \leftrightarrow\cup=
  13. * \stackrel{*}{\leftrightarrow}
  14. \rightarrow
  15. \rightarrow
  16. x y x\rightarrow y
  17. x * y x\stackrel{*}{\rightarrow}y
  18. x x\downarrow
  19. c = a = b c=a\downarrow=b\downarrow
  20. * \stackrel{*}{\leftrightarrow}
  21. * \stackrel{*}{\leftrightarrow}
  22. x * z * y x\stackrel{*}{\rightarrow}z\stackrel{*}{\leftarrow}y
  23. * * \stackrel{*}{\rightarrow}\circ\stackrel{*}{\leftarrow}
  24. \circ
  25. \downarrow
  26. x , y x\mathbin{\downarrow}y
  27. x * y x\stackrel{*}{\leftrightarrow}y
  28. x , y x\mathbin{\downarrow}y
  29. ( A , ) (A,\rightarrow)
  30. x * w * y x\stackrel{*}{\leftarrow}w\stackrel{*}{\rightarrow}y
  31. x , y x\mathbin{\downarrow}y
  32. x w * y x\leftarrow w\stackrel{*}{\rightarrow}y
  33. x , y x\mathbin{\downarrow}y
  34. x w y x\leftarrow w\rightarrow y
  35. x , y x\mathbin{\downarrow}y
  36. x * y x\stackrel{*}{\leftrightarrow}y
  37. x * y x\stackrel{*}{\rightarrow}y
  38. x * y x\stackrel{*}{\leftrightarrow}y
  39. { a b , b a , a x , b y } \{a\rightarrow b,b\rightarrow a,a\rightarrow x,b\rightarrow y\}
  40. x 0 x 1 x 2 x_{0}\rightarrow x_{1}\rightarrow x_{2}\rightarrow\cdots
  41. a b a b a\rightarrow b\rightarrow a\rightarrow b\rightarrow\cdots
  42. \rightarrow

Acceptance_sampling.html

  1. M M
  2. N < M N<M
  3. B B
  4. B B
  5. B B
  6. N N
  7. B B

Accounting_rate_of_return.html

  1. ARR = Average return during period Average investment \,\text{ARR}=\frac{\,\text{Average return during period }}{\,\text{Average % investment}}
  2. Average investment = Book value at beginning of year 1 + Book value at end of useful life 2 \,\text{Average investment}=\frac{\,\text{Book value at beginning of year 1 + % Book value at end of useful life}}{\,\text{2}}

Acoustic_metamaterials.html

  1. n 2 = ρ β n^{2}=\frac{\rho}{\beta}
  2. k = | n | ω c . \vec{k}=\frac{\ |n|\omega}{c}.\,
  3. s \scriptstyle\overleftarrow{s}
  4. k \scriptstyle\overrightarrow{k}

Acoustic_radiation_force.html

  1. | F | = 2 α I c |F|=\frac{2\alpha I}{c}

Activity_selection_problem.html

  1. S = { 1 , 2 , , n } S=\{1,2,\ldots,n\}
  2. B = ( A { k } ) { 1 } B=(A\setminus\{k\})\cup\{1\}
  3. f 1 f k f_{1}\leq f_{k}
  4. A = A { 1 } A^{\prime}=A\setminus\{1\}
  5. S = { i S : s i f 1 } S^{\prime}=\{i\in S:s_{i}\geq f_{1}\}
  6. k k
  7. k k
  8. k k
  9. O ( n 3 ) O(n^{3})
  10. ( i , j ) (i,j)
  11. ( i , t ) (i,t)
  12. t t
  13. j j
  14. ( i , j ) (i,j)
  15. O ( n 2 ) O(n^{2})
  16. ( i , j ) (i,j)
  17. ( 1 , j ) (1,j)
  18. O ( n l o g ( n ) ) O(nlog(n))

Adaptive_estimator.html

  1. ν ^ n \scriptstyle\hat{\nu}_{n}
  2. 𝒫 ν ( η 0 ) = { P θ : ν N , η = η 0 } . \mathcal{P}_{\nu}(\eta_{0})=\big\{P_{\theta}:\nu\in N,\,\eta=\eta_{0}\big\}.
  3. I ν η ( θ ) = E [ z ν z η ] = 0 for all θ , I_{\nu\eta}(\theta)=\operatorname{E}[\,z_{\nu}z_{\eta}^{\prime}\,]=0\quad\,% \text{for all }\theta,
  4. 𝒫 \scriptstyle\mathcal{P}
  5. 𝒫 = { f θ ( x ) = 1 2 π σ e - 1 2 σ 2 ( x - μ ) 2 | μ , σ > 0 } . \mathcal{P}=\Big\{\ f_{\theta}(x)=\tfrac{1}{\sqrt{2\pi}\sigma}e^{-\frac{1}{2% \sigma^{2}}(x-\mu)^{2}}\ \Big|\ \mu\in\mathbb{R},\sigma>0\ \Big\}.
  6. μ ^ = x ¯ \hat{\mu}\,=\,\bar{x}

Adaptive_learning.html

  1. 2 x 2 + x 3 2x^{2}+x^{3}
  2. 3 x 5 3x^{5}

Additive_K-theory.html

  1. A A
  2. k k
  3. 𝔤 l ( A ) {\mathfrak{g}l}(A)
  4. A A
  5. H ( 𝔤 l ( A ) , k ) H_{\cdot}({\mathfrak{g}l}(A),k)\,
  6. i i
  7. K i + ( A ) K^{+}_{i}(A)
  8. i i
  9. H C i ( A ) K i + 1 + ( A ) . HC_{i}(A)\cong K^{+}_{i+1}(A).\,

Additive_map.html

  1. f ( x + y ) = f ( x ) + f ( y ) . f(x+y)=f(x)+f(y).
  2. R 1 R_{1}
  3. R 2 R_{2}
  4. f : R 1 R 2 , f:R_{1}\to R_{2}\,,
  5. R 1 R_{1}
  6. R 2 R_{2}
  7. f f
  8. g g
  9. f + g f+g
  10. D D
  11. 0
  12. f : D D f:D\to D
  13. D D
  14. f ( x ) = f ( s ) 0 x f ( s ) 1 . f(x)={}_{(s)0}f\ x\ {}_{(s)1}f\,.
  15. s s
  16. f f
  17. f ( s ) 0 , f ( s ) 1 D {}_{(s)0}f,{}_{(s)1}f\in D

Additive_Markov_chain.html

  1. Pr ( X n = x n | X n - 1 = x n - 1 , X n - 2 = x n - 2 , , X n - m = x n - m ) = r = 1 m f ( x n , x n - r , r ) \Pr(X_{n}=x_{n}|X_{n-1}=x_{n-1},X_{n-2}=x_{n-2},\dots,X_{n-m}=x_{n-m})=\sum_{r% =1}^{m}f(x_{n},x_{n-r},r)
  2. Pr ( X n = 1 | X n - 1 = x n - 1 , X n - 2 = x n - 2 , ) = X ¯ + r = 1 m F ( r ) ( x n - r - X ¯ ) , \Pr(X_{n}=1|X_{n-1}=x_{n-1},X_{n-2}=x_{n-2},\dots)=\bar{X}+\sum_{r=1}^{m}F(r)(% x_{n-r}-\bar{X}),
  3. Pr ( X n = 0 | X n - 1 = x n - 1 , X n - 2 = x n - 2 , ) = 1 - Pr ( X n = 1 | X n - 1 = x n - 1 , X n - 2 = x n - 2 , ) . \Pr(X_{n}=0|X_{n-1}=x_{n-1},X_{n-2}=x_{n-2},\dots)=1-\Pr(X_{n}=1|X_{n-1}=x_{n-% 1},X_{n-2}=x_{n-2},\dots).
  4. X ¯ \bar{X}
  5. X ¯ \bar{X}
  6. X n X_{n}
  7. X k X_{k}
  8. n - k n-k
  9. K ( r ) = ( X n - X ¯ ) ( X n + r - X ¯ ) = X n X n + r - X ¯ 2 , K(r)=\langle(X_{n}-\bar{X})(X_{n+r}-\bar{X})\rangle=\langle X_{n}X_{n+r}% \rangle-{\bar{X}}^{2},
  10. \langle\cdots\rangle
  11. K ( - r ) = K ( r ) , K ( 0 ) = X ¯ ( 1 - X ¯ ) . K(-r)=K(r),K(0)=\bar{X}(1-\bar{X}).
  12. K ( r ) = s = 1 m K ( r - s ) F ( s ) , r = 1 , 2 , . K(r)=\sum_{s=1}^{m}K(r-s)F(s),\,\,\,\,r=1,2,\dots\,.

Adequate_equivalence_relation.html

  1. α , β Z * ( X ) \alpha,\beta\in Z^{*}(X)
  2. α Z * ( X ) \alpha^{\prime}\in Z^{*}(X)
  3. α \alpha
  4. α \alpha^{\prime}
  5. α \alpha^{\prime}
  6. β \beta
  7. α Z * ( X ) \alpha\in Z^{*}(X)
  8. β Z * ( X × Y ) \beta\in Z^{*}(X\times Y)
  9. β \beta
  10. α × Y \alpha\times Y
  11. α \alpha
  12. ( π Y ) * ( β ( α × Y ) ) (\pi_{Y})_{*}(\beta\cdot(\alpha\times Y))
  13. π Y : X × Y Y \pi_{Y}:X\times Y\to Y
  14. β ( α ) := ( π Y ) * ( β ( α × Y ) ) \beta(\alpha):=(\pi_{Y})_{*}(\beta\cdot(\alpha\times Y))
  15. β \beta
  16. ( Z - Z ) n (Z-Z^{\prime})^{\otimes n}

Adjoint_equation.html

  1. d X t = a ( X t ) d t + b ( X t ) d W dX_{t}=a(X_{t})dt+b(X_{t})dW
  2. X n + 1 = X n + a Δ t + ζ b Δ t X_{n+1}=X_{n}+a\Delta t+\zeta b\sqrt{\Delta t}
  3. ζ \zeta

Adjoint_state_method.html

  1. A * A^{*}
  2. A * = = A T A^{*}==A^{T}
  3. s t x s^{t}x
  4. A x = b Ax=b
  5. r t b r^{t}b
  6. A * r = s A^{*}r=s
  7. A * A^{*}
  8. A A
  9. r r

Adjusted_current_yield.html

  1. ( 100 - Clean price ) / Years to maturity Clean price * 100. \frac{(100-\,\text{Clean price})/\,\text{Years to maturity}}{\,\text{Clean % price}}*100.
  2. Annual coupon payments Clean price * 100 + ( 100 - Clean price ) / Years to maturity Clean price * 100. \frac{\,\text{Annual coupon payments}}{\,\text{Clean price}}*100+\frac{(100-\,% \text{Clean price})/\,\text{Years to maturity}}{\,\text{Clean price}}*100.

Adjusted_mutual_information.html

  1. S = { s 1 , s 2 , s N } S=\{s_{1},s_{2},\ldots s_{N}\}
  2. U = { U 1 , U 2 , , U R } U=\{U_{1},U_{2},\ldots,U_{R}\}
  3. V = { V 1 , V 2 , , V C } V=\{V_{1},V_{2},\ldots,V_{C}\}
  4. U i U j = V i V j = U_{i}\cap U_{j}=V_{i}\cap V_{j}=\varnothing
  5. i j i\neq j
  6. i = 1 R U i = j = 1 C V j = S \cup_{i=1}^{R}U_{i}=\cup_{j=1}^{C}V_{j}=S
  7. M = [ n i j ] j = 1 C i = 1 R M=[n_{ij}]^{i=1\ldots R}_{j=1\ldots C}
  8. n i j n_{ij}
  9. U i U_{i}
  10. V j V_{j}
  11. n i j = | U i V j | n_{ij}=\left|U_{i}\cap V_{j}\right|
  12. U i U_{i}
  13. P ( i ) = | U i | N P(i)=\frac{|U_{i}|}{N}
  14. H ( U ) = - i = 1 R P ( i ) log P ( i ) H(U)=-\sum_{i=1}^{R}P(i)\log P(i)
  15. H ( V ) = - j = 1 C P ( j ) log P ( j ) H(V)=-\sum_{j=1}^{C}P^{\prime}(j)\log P^{\prime}(j)
  16. P ( j ) = | V j | / N P^{\prime}(j)={|V_{j}|}/{N}
  17. M I ( U , V ) = i = 1 R j = 1 C P ( i , j ) log P ( i , j ) P ( i ) P ( j ) MI(U,V)=\sum_{i=1}^{R}\sum_{j=1}^{C}P(i,j)\log\frac{P(i,j)}{P(i)P^{\prime}(j)}
  18. U i U_{i}
  19. V j V_{j}
  20. P ( i , j ) = | U i V j | N P(i,j)=\frac{|U_{i}\cap V_{j}|}{N}
  21. E { M I ( U , V ) } = \displaystyle E\{MI(U,V)\}=
  22. ( a i + b j - N ) + (a_{i}+b_{j}-N)^{+}
  23. max ( 1 , a i + b j - N ) \max(1,a_{i}+b_{j}-N)
  24. a i a_{i}
  25. b j b_{j}
  26. a i = j = 1 C n i j a_{i}=\sum_{j=1}^{C}n_{ij}
  27. b j = i = 1 R n i j b_{j}=\sum_{i=1}^{R}n_{ij}
  28. A M I ( U , V ) = M I ( U , V ) - E { M I ( U , V ) } max { H ( U ) , H ( V ) } - E { M I ( U , V ) } AMI(U,V)=\frac{MI(U,V)-E\{MI(U,V)\}}{\max{\{H(U),H(V)\}}-E\{MI(U,V)\}}

Adolfas_Jucys.html

  1. [ S n ] \mathbb{C}[S_{n}]

Advance_ratio.html

  1. J = V a n D J=\frac{V_{a}}{nD}
  2. V r = ω × r \overrightarrow{V_{r}}=\overrightarrow{\omega}\times\overrightarrow{r}
  3. n = ω 2 π n=\frac{\omega}{2\pi}
  4. V = V a + V r \overrightarrow{V}=\overrightarrow{V_{a}}+\overrightarrow{V_{r}}

Advertising_elasticity_of_demand.html

  1. A E D = % change in quantity demanded % change in spending on advertising = Δ Q d / Q d Δ A / A AED=\frac{\%\ \mbox{change in quantity demanded}~{}}{\%\ \mbox{change in % spending on advertising}~{}}=\frac{\Delta Q_{d}/Q_{d}}{\Delta A/A}
  2. Advertising expenditure Sales revenue = - A E D P E D or, symbolically, A P . Q = - E A E P \frac{\mbox{Advertising expenditure}~{}}{\mbox{Sales revenue}~{}}=-\frac{AED}{% PED}\mbox{ or, symbolically, }~{}\frac{A}{P.Q}=-\frac{E_{A}}{E_{P}}

Afocal_photography.html

  1. α = d f × 180 π 180 d π f \alpha={d\over f}\times{180\over\pi}\equiv{180d\over\pi f}
  2. α = 2 arctan d 2 f \alpha=2\arctan{d\over 2f}

Afocal_system.html

  1. M = - f 2 f 1 , M=\frac{-f_{2}}{f_{1}},

AGM_method.html

  1. π \pi
  2. a 0 \displaystyle a_{0}
  3. N + , N\to+\infty,\,
  4. lim N a N = lim N b N = M ( a , b ) , \lim_{N\to\infty}a_{N}=\lim_{N\to\infty}b_{N}=M(a,b),\,
  5. π \pi
  6. π = 4 ( M ( 1 ; 1 2 ) ) 2 1 - j = 1 2 j + 1 c j 2 , \pi=\frac{4\left(M(1;\frac{1}{\sqrt{2}})\right)^{2}}{\displaystyle 1-\sum_{j=1% }^{\infty}2^{j+1}c_{j}^{2}},
  7. c j = 1 2 ( a j - 1 - b j - 1 ) c_{j}=\frac{1}{2}\left(a_{j-1}-b_{j-1}\right)
  8. c j = c j - 1 2 4 a j . c_{j}=\frac{c_{j-1}^{2}}{4a_{j}}.
  9. a 0 = 1 , b 0 = cos α , a_{0}=1,\quad b_{0}=\cos\alpha,
  10. lim N a N = π 2 K ( α ) , \lim_{N\to\infty}a_{N}=\frac{\pi}{2K(\alpha)},
  11. K ( α ) = 0 π / 2 ( 1 - α sin 2 θ ) - 1 / 2 d θ . K(\alpha)=\int_{0}^{\pi/2}(1-\alpha\sin^{2}\theta)^{-1/2}\,d\theta.

Air_changes_per_hour.html

  1. N = 60 Q V o l \quad N=\frac{60Q}{Vol}
  2. R p = A C P H * D * h 60 \quad Rp=\frac{ACPH*D*h}{60}

Air_pollutant_concentrations.html

  1. ppmv = mg / m 3 ( 0.08205 T ) M \mathrm{ppmv}=\mathrm{mg}/\mathrm{m}^{3}\cdot\frac{(0.08205\cdot T)}{M}
  2. mg / m 3 = ppmv M ( 0.08205 T ) \mathrm{mg}/\mathrm{m}^{3}=\mathrm{ppmv}\cdot\frac{M}{(0.08205\cdot T)}
  3. P h = P ( 288 - 6.5 h 288 ) 5.2558 P_{\mathrm{h}}=P\,\cdot\bigg(\frac{288-6.5h}{288}\bigg)^{5.2558}
  4. C h = C ( 288 - 6.5 h 288 ) 5.2558 C_{\mathrm{h}}=C\,\cdot\bigg(\frac{288-6.5h}{288}\bigg)^{5.2558}
  5. C dry basis = C wet basis 1 - w C_{\mathrm{dry\,basis}}=\frac{C_{\mathrm{wet\,basis}}}{1-w}
  6. C r = C m ( 20.9 - reference volume % O 2 ) ( 20.9 - measured volume % O 2 ) C_{\mathrm{r}}=C_{\mathrm{m}}\cdot\frac{(20.9-\mathrm{reference\,volume\,\%\,O% _{2}})}{(20.9-\mathrm{measured\,volume\,\%\,O_{2}})}
  7. C r = C m ( reference volume % CO 2 ) ( measured volume % CO 2 ) C_{\mathrm{r}}=C_{\mathrm{m}}\cdot\frac{(\mathrm{reference\,volume\,\%\,CO_{2}% })}{(\mathrm{measured\,volume\,\%\,CO_{2}})}

Airway_pressure_release_ventilation.html

  1. ( P h i g h * T h i g h ) + ( P l o w * T l o w ) T h i g h + T l o w \frac{(P_{high}*T_{high})\,+(P_{low}*T_{low})}{T_{high}+T_{low}}

Airy_beam.html

  1. i Φ ξ + 1 2 2 Φ s 2 = 0 i\frac{\partial\Phi}{\partial\xi}+\frac{1}{2}\frac{\partial^{2}\Phi}{\partial% \,s^{2}}=0
  2. Φ ( ξ , s ) = Ai ( s - ( ξ / 2 ) 2 ) exp ( i ( s ξ / 2 ) - i ( ξ 3 / 12 ) ) \Phi(\xi,\,s)=\mathrm{Ai}(\,s-(\xi/2)^{2})\exp(i(\,s\xi/2)-i(\xi^{3}/12))
  3. Ai \mathrm{Ai}
  4. Φ \Phi
  5. s = x / x 0 s=x/x_{0}
  6. x 0 x_{0}
  7. ξ = z / k x 0 2 \xi=z/kx_{0}^{2}
  8. k = 2 π n / λ 0 k=2\pi\,n/\lambda_{0}

Airy_zeta_function.html

  1. Ai ( x ) = 1 π 0 cos ( 1 3 t 3 + x t ) d t , \mathrm{Ai}(x)=\frac{1}{\pi}\int_{0}^{\infty}\cos\left(\tfrac{1}{3}t^{3}+xt% \right)\,dt,
  2. ζ Ai ( s ) = i = 1 1 | a i | s . \zeta_{\mathrm{Ai}}(s)=\sum_{i=1}^{\infty}\frac{1}{|a_{i}|^{s}}.
  3. ζ ( 2 ) = π 2 / 6 \zeta(2)=\pi^{2}/6
  4. ζ Ai ( 2 ) = i = 1 1 a i 2 = 3 5 / 3 Γ 4 ( 2 3 ) 4 π 2 , \zeta_{\mathrm{Ai}}(2)=\sum_{i=1}^{\infty}\frac{1}{a_{i}^{2}}=\frac{3^{5/3}% \Gamma^{4}(\frac{2}{3})}{4\pi^{2}},
  5. ζ Ai ( 1 ) = - 3 - 2 / 3 Γ ( 2 3 ) Γ ( 4 3 ) . \zeta_{\mathrm{Ai}}(1)=\frac{-3^{-2/3}\Gamma(\frac{2}{3})}{\Gamma(\frac{4}{3})}.

Aizik_Volpert.html

  1. n 1 n≥1
  2. f ( s y m b o l u ( s y m b o l x ) ) x i = k = 1 p f ¯ ( s y m b o l u ( s y m b o l x ) ) u k u k ( s y m b o l x ) x i i = 1 , , n \frac{\partial f(symbol{u}(symbol{x}))}{\partial x_{i}}=\sum_{k=1}^{p}\frac{% \partial\bar{f}(symbol{u}(symbol{x}))}{\partial u_{k}}\frac{\partial{u_{k}(% symbol{x})}}{\partial x_{i}}\qquad\forall i=1,\ldots,n
  3. [ u o v e r s e t , u m a t h b f 013 , u f ] ( 𝐮 ( 𝐱 ) ) [u^{\prime}overset^{\prime},u^{\prime}\\ mathbf{\u{2}013}^{\prime},u^{\prime}f^{\prime}](\mathbf{u}(\mathbf{x}))
  4. f f
  5. 𝐮 \mathbf{u}
  6. H δ H⋅δ
  7. H ( x ) H(x)
  8. δ ( x ) δ(x)
  9. 𝐟 : < s u p > p s \mathbf{f}:ℝ<sup>p→ℝ^{s}

AK_model.html

  1. Y = A K a L 1 - a Y=AK^{a}L^{1-a}\,
  2. a a
  3. a = 1 a=1
  4. n n
  5. δ \delta
  6. k k
  7. y y
  8. L L
  9. s s
  10. Y = A K Y=AK\,
  11. Y L = A . K L \frac{Y}{L}=A.\frac{K}{L}
  12. y = A k y=Ak
  13. A > 0 A>0
  14. k ( t ) = s . f ( k ) - n k k(t)=s.f(k)-nk
  15. k ( t ) k = s . f ( k ) k - n \frac{k(t)}{k}=s.\frac{f(k)}{k}-n
  16. f ( k ) k = A \frac{f(k)}{k}=A
  17. k ( t ) k = s . A - n \frac{k(t)}{k}=s.A-n

Ak_singularity.html

  1. ( φ , ψ ) f := ψ f φ - 1 (\varphi,\psi)\cdot f:=\psi\circ f\circ\varphi^{-1}
  2. orb ( f ) = { ψ f φ - 1 : φ diff ( R n ) , ψ diff ( R ) } . \mbox{orb}~{}(f)=\{\psi\circ f\circ\varphi^{-1}:\varphi\in\mbox{diff}~{}({R}^{% n}),\psi\in\mbox{diff}~{}({R})\}\ .
  3. f ( x 1 , , x n ) = 1 + ε 1 x 1 2 + + ε n - 1 x n - 1 2 ± x n k + 1 f(x_{1},\ldots,x_{n})=1+\varepsilon_{1}x_{1}^{2}+\cdots+\varepsilon_{n-1}x^{2}% _{n-1}\pm x_{n}^{k+1}
  4. ε i = ± 1 \varepsilon_{i}=\pm 1

Alan_M._Frieze.html

  1. ϵ \epsilon
  2. K K
  3. n n
  4. n n
  5. K K
  6. 1 / ϵ 1/\epsilon
  7. K K
  8. ϵ \epsilon
  9. γ \gamma
  10. G = ( V , E ) G=(V,E)
  11. n n
  12. P P
  13. V V
  14. V 0 , V 1 , , V k V_{0},V_{1},\ldots,V_{k}
  15. | V 1 | > 4 2 k |V_{1}|>4^{2k}
  16. 4 k > 600 γ 2 4^{k}>600\gamma^{2}
  17. γ k 2 \gamma k^{2}
  18. ( V r , V s ) (V_{r},V_{s})
  19. γ \gamma
  20. P P^{\prime}
  21. P P
  22. 1 + k 4 k 1+k4^{k}
  23. | V 0 | + n / 4 k |V_{0}|+n/4^{k}
  24. ind ( P ) ind ( P ) + γ 5 / 20 \operatorname{ind}(P^{\prime})\geq\operatorname{ind}(P)+\gamma^{5}/20
  25. W W
  26. R × C R\times C
  27. | R | = p |R|=p
  28. | C | = q |C|=q
  29. W inf 1 \|W\|_{\inf}\leq 1
  30. γ \gamma
  31. S R S\subseteq R
  32. T C T\subseteq C
  33. | S | γ p |S|\geq\gamma p
  34. | T | γ q |T|\geq\gamma q
  35. | W ( S , T ) | γ | S | | T | |W(S,T)|\geq\gamma|S||T|
  36. σ 1 ( W ) γ 3 p q \sigma_{1}(W)\geq\gamma^{3}\sqrt{pq}
  37. σ 1 ( W ) γ p q \sigma_{1}(W)\geq\gamma\sqrt{pq}
  38. S R S\subseteq R
  39. T C T\subseteq C
  40. | S | γ p |S|\geq\gamma^{\prime}p
  41. | T | γ q |T|\geq\gamma^{\prime}q
  42. W ( S , T ) γ | S | | T | W(S,T)\geq\gamma^{\prime}|S||T|
  43. γ = γ 3 / 108 \gamma^{\prime}=\gamma^{3}/108
  44. S S
  45. T T
  46. G G
  47. P 1 P_{1}
  48. V 0 , V 1 , , V b V_{0},V_{1},\ldots,V_{b}
  49. | V i | n / b |V_{i}|\lfloor n/b\rfloor
  50. | V 0 | < b |V_{0}|<b
  51. k 1 = b k_{1}=b
  52. ( V r , V s ) (V_{r},V_{s})
  53. P i P_{i}
  54. σ 1 ( W r , s ) \sigma_{1}(W_{r,s})
  55. ( V r , V s ) (V_{r},V_{s})
  56. ϵ - \epsilon-
  57. γ = ϵ 9 / 108 - \gamma=\epsilon^{9}/108-
  58. ϵ ( k 1 2 ) \epsilon\left(\begin{array}[]{c}k_{1}\\ 2\\ \end{array}\right)
  59. γ - \gamma-
  60. P i P_{i}
  61. ϵ - \epsilon-
  62. P = P i P=P_{i}
  63. k = k i k=k_{i}
  64. γ = ϵ 9 / 108 \gamma=\epsilon^{9}/108
  65. P P^{\prime}
  66. 1 + k i 4 k i 1+k_{i}4^{k_{i}}
  67. k i + 1 = k i 4 k i k_{i}+1=k_{i}4^{k_{i}}
  68. P i + 1 = P P_{i}+1=P^{\prime}
  69. i = i + 1 i=i+1

Albertson_conjecture.html

  1. cr ( K n ) = 1 4 n 2 n - 1 2 n - 2 2 n - 3 2 . \textrm{cr}(K_{n})=\frac{1}{4}\left\lfloor\frac{n}{2}\right\rfloor\left\lfloor% \frac{n-1}{2}\right\rfloor\left\lfloor\frac{n-2}{2}\right\rfloor\left\lfloor% \frac{n-3}{2}\right\rfloor.

Albert–Brauer–Hasse–Noether_theorem.html

  1. A K K v M d ( K v ) . A\otimes_{K}K_{v}\simeq M_{d}(K_{v}).

Albion_process.html

  1. M S 2 + 1 2 O 2 + H 2 S O 4 M S O 4 + S 0 + H 2 O MS_{2}+\frac{1}{2}O_{2}+H_{2}SO_{4}\rightarrow MSO_{4}+S^{0}+H_{2}O

Alexandru_Froda.html

  1. d y = f ( x , y ) d x dy=f(x,y)dx
  2. n = 1 n=1

Alexandru_Proca.html

  1. μ A μ = 0 \partial_{\mu}A^{\mu}=0\!
  2. A ν - ν ( μ A μ ) + m 2 A ν = j ν \Box A^{\nu}-\partial^{\nu}(\partial_{\mu}A^{\mu})+m^{2}A^{\nu}=j^{\nu}
  3. = ( 2 t 2 ) - 2 \Box=\left(\frac{\partial^{2}}{\partial t^{2}}\right)-\nabla^{2}
  4. A μ A^{\mu}
  5. \Box
  6. j ν j^{\nu}
  7. A ν A^{\nu}
  8. A ν = ( ϕ , 𝐀 ) A^{\nu}=(\phi,\mathbf{A})
  9. 𝐄 = - ϕ - 𝐀 t \mathbf{E}=-\mathbf{\nabla}\phi-\frac{\partial\mathbf{A}}{\partial t}
  10. 𝐁 = × 𝐀 . \mathbf{B}=\mathbf{\nabla}\times\mathbf{A}.
  11. μ ( μ A ν - ν A μ ) + ( m c ) 2 A ν = 0 \partial_{\mu}(\partial^{\mu}A^{\nu}-\partial^{\nu}A^{\mu})+\left(\frac{mc}{% \hbar}\right)^{2}A^{\nu}=0
  12. 1 c 2 2 t 2 ψ - 2 ψ + m 2 c 2 2 ψ = 0 \frac{1}{c^{2}}\frac{\partial^{2}}{\partial t^{2}}\psi-\nabla^{2}\psi+\frac{m^% {2}c^{2}}{\hbar^{2}}\psi=0
  13. - t 2 ψ + 2 ψ = m 2 ψ -\partial_{t}^{2}\psi+\nabla^{2}\psi=m^{2}\psi

Algae_bioreactor.html

  1. 6 CO 2 + 6 H 2 O C 6 H 12 O 6 + 6 O 2 Δ H 0 = + 2870 kJ mol \begin{matrix}\mathrm{6\;CO_{2}+6\;H_{2}O\quad\longrightarrow\;C_{6}H_{12}O_{6% }+6\;O_{2}}\qquad\Delta H^{0}=+2870\ \frac{\mathrm{kJ}}{\mathrm{mol}}\end{matrix}

Algorithmic_Lovász_local_lemma.html

  1. 𝒜 = { A 1 , , A n } \mathcal{A}=\{A_{1},\ldots,A_{n}\}
  2. A 𝒜 A\in\mathcal{A}
  3. Γ ( A ) \Gamma(A)
  4. 𝒜 \mathcal{A}
  5. A A
  6. 𝒜 ( { A } Γ ( A ) ) \mathcal{A}\setminus(\{A\}\cup\Gamma(A))
  7. x : 𝒜 ( 0 , 1 ) x:\mathcal{A}\rightarrow(0,1)
  8. A 𝒜 : Pr [ A ] x ( A ) B Γ ( A ) ( 1 - x ( B ) ) \forall A\in\mathcal{A}:\Pr[A]\leq x(A)\prod_{B\in\Gamma(A)}(1-x(B))
  9. 𝒜 \mathcal{A}
  10. Pr [ A 1 ¯ A n ¯ ] A 𝒜 ( 1 - x ( A ) ) . \Pr\left[\,\overline{A_{1}}\wedge\cdots\wedge\overline{A_{n}}\,\right]\geq% \prod_{A\in\mathcal{A}}(1-x(A)).
  11. Pr [ A 1 ¯ A n ¯ ] \Pr\left[\overline{A_{1}}\wedge\cdots\wedge\overline{A_{n}}\right]
  12. A 𝒜 ( 1 - x ( A ) ) \prod_{A\in\mathcal{A}}(1-x(A))
  13. 𝒜 \mathcal{A}
  14. 𝒫 \mathcal{P}
  15. 𝒫 \mathcal{P}
  16. 𝒜 \mathcal{A}
  17. A 𝒜 A\in\mathcal{A}
  18. | Γ ( A ) | < 2 k 48 |\Gamma(A)|<2^{\frac{k}{48}}
  19. | Γ ( A ) | < 2 k e , |\Gamma(A)|<\frac{2^{k}}{e},
  20. P 𝒫 , v P P\in\mathcal{P},v_{P}
  21. ( v P ) 𝒫 (v_{P})_{\mathcal{P}}
  22. 𝒫 \mathcal{P}
  23. ( v P ) 𝒫 (v_{P})_{\mathcal{P}}
  24. ( v P ) 𝒫 (v_{P})_{\mathcal{P}}
  25. 𝒜 \mathcal{A}
  26. 𝒫 \mathcal{P}
  27. P 𝒫 \forall P\in\mathcal{P}
  28. v P v_{P}\leftarrow
  29. A 𝒜 \exists A\in\mathcal{A}
  30. ( v P ) 𝒫 (v_{P})_{\mathcal{P}}
  31. A 𝒜 A\in\mathcal{A}
  32. P vbl ( A ) \forall P\in\,\text{vbl}(A)
  33. v P v_{P}\leftarrow
  34. ( v P ) 𝒫 (v_{P})_{\mathcal{P}}
  35. P 𝒫 P\in\mathcal{P}
  36. 𝒜 \mathcal{A}
  37. 𝒫 \mathcal{P}
  38. 𝒜 \mathcal{A}
  39. x : 𝒜 ( 0 , 1 ) x:\mathcal{A}\to(0,1)
  40. A 𝒜 : Pr [ A ] x ( A ) B Γ ( A ) ( 1 - x ( B ) ) \forall A\in\mathcal{A}:\Pr[A]\leq x(A)\prod_{B\in\Gamma(A)}(1-x(B))
  41. 𝒫 \mathcal{P}
  42. 𝒜 \mathcal{A}
  43. A 𝒜 A\in\mathcal{A}
  44. x ( A ) 1 - x ( A ) \frac{x(A)}{1-x(A)}
  45. A 𝒜 x ( A ) 1 - x ( A ) . \sum_{A\in\mathcal{A}}\frac{x(A)}{1-x(A)}.
  46. A 𝒜 : | Γ ( A ) | D \forall A\in\mathcal{A}:|\Gamma(A)|\leq D
  47. A 𝒜 : Pr [ A ] p \forall A\in\mathcal{A}:\Pr[A]\leq p
  48. e p ( D + 1 ) 1 ep(D+1)\leq 1
  49. 𝒫 \mathcal{P}
  50. 𝒜 \mathcal{A}
  51. 𝒫 \mathcal{P}
  52. 𝒜 \mathcal{A}
  53. A 𝒜 A\in\mathcal{A}
  54. 1 D \frac{1}{D}
  55. n D \frac{n}{D}
  56. 2 k k e \frac{2^{k}}{ke}
  57. 𝒫 \mathcal{P}
  58. Pr [ A j ] = p = 2 - k . \Pr[A_{j}]=p=2^{-k}.
  59. 2 k k e \frac{2^{k}}{ke}
  60. D = k ( 2 k k e - 1 ) 2 k e - 1 D=k\left(\frac{2^{k}}{ke}-1\right)\leq\frac{2^{k}}{e}-1
  61. D + 1 2 k e , D+1\leq\frac{2^{k}}{e},
  62. e p ( D + 1 ) e 2 - k 2 k e = 1 ep(D+1)\leq e2^{-k}\frac{2^{k}}{e}=1
  63. n 2 k e - k \frac{n}{\frac{2^{k}}{e}-k}
  64. k k
  65. 2 k 2^{k}
  66. A , B 𝒜 A,B\in\mathcal{A}
  67. vbl ( A ) vbl ( B ) = \,\text{vbl}(A)\cap\,\text{vbl}(B)=\emptyset
  68. A 𝒜 : Pr [ A ] ( 1 - ε ) x ( A ) B Γ ( A ) ( 1 - x ( B ) ) \forall A\in\mathcal{A}:\Pr[A]\leq(1-\varepsilon)x(A)\prod_{B\in\Gamma(A)}(1-x% (B))
  69. O ( 1 ε log A 𝒜 x ( A ) 1 - x ( A ) ) O\left(\frac{1}{\varepsilon}\log\sum_{A\in\mathcal{A}}\frac{x(A)}{1-x(A)}\right)

Algorithmic_version_for_Szemerédi_regularity_partition.html

  1. i , j i,j
  2. ϵ \epsilon
  3. ϵ k 2 \epsilon{k^{2}}
  4. ( V i , V j , ) (V_{i},V_{j},)
  5. ϵ \epsilon
  6. ϵ > 0 \epsilon>0
  7. m m
  8. N N
  9. M M
  10. G G
  11. N N
  12. k k
  13. m m
  14. k k
  15. M M
  16. ϵ \epsilon
  17. G G
  18. k k
  19. ϵ \epsilon
  20. V 0 V_{0}
  21. ϵ \epsilon
  22. G G
  23. γ \gamma
  24. G = ( V , E ) G=(V,E)
  25. n n
  26. P P
  27. V V
  28. V 0 , V 1 , , V k V_{0},V_{1},...,V_{k}
  29. | V 1 | > 4 2 k |V_{1}|>4^{2k}
  30. 4 k > 600 γ 2 4^{k}>600\gamma^{2}
  31. γ k 2 \gamma k^{2}
  32. ( V r , V s ) (V_{r},V_{s})
  33. γ \gamma
  34. P P^{\prime}
  35. P P
  36. 1 + k 4 k 1+k4^{k}
  37. | V 0 | + n / 4 k |V_{0}|+n/4^{k}
  38. i n d ( P ) ind(P^{\prime})
  39. i n d ( P ) + γ 5 / 20 ind(P)+\gamma^{5}/20
  40. W W
  41. R R
  42. C C
  43. | R | = p |R|=p
  44. | C | = q |C|=q
  45. W inf 1 \|W\|_{\inf}\leq 1
  46. γ \gamma
  47. S S
  48. R R
  49. T T
  50. C C
  51. | S | |S|
  52. γ p \gamma p
  53. | T | |T|
  54. γ q \gamma q
  55. | W ( S , T ) | |W(S,T)|
  56. γ | S | | T | \gamma|S||T|
  57. σ 1 ( W ) γ 3 p q \sigma_{1}(W)\geq\gamma^{3}\sqrt{pq}
  58. σ 1 ( W ) γ p q \sigma_{1}(W)\geq\gamma\sqrt{pq}
  59. S S
  60. R R
  61. T T
  62. C C
  63. | S | |S|
  64. γ p \gamma^{\prime}p
  65. | T | |T|
  66. γ q \gamma^{\prime}q
  67. W ( S , T ) W(S,T)
  68. γ | S | | T | \gamma^{\prime}|S||T|
  69. γ = γ 3 / 108 \gamma^{\prime}=\gamma^{3}/108
  70. S S
  71. T T
  72. G G
  73. P 1 P_{1}
  74. V 0 , V 1 , , V b V_{0},V_{1},...,V_{b}
  75. | V i | = n / b |V_{i}|=\lfloor n/b\rfloor
  76. | V 0 | < b |V_{0}|<b
  77. k 1 = b k_{1}=b
  78. ( V r , V s ) (V_{r},V_{s})
  79. P i P_{i}
  80. σ 1 ( W r , s ) \sigma_{1}(W_{r,s})
  81. ( V r , V s ) (V_{r},V_{s})
  82. ϵ - \epsilon-
  83. γ = ϵ 9 / 108 - \gamma=\epsilon^{9}/108-
  84. ϵ ( k 1 2 ) \epsilon{k_{1}\choose 2}
  85. γ - \gamma-
  86. P i P_{i}
  87. ϵ - \epsilon-
  88. P = P i P=P_{i}
  89. k = k i k=k_{i}
  90. γ = ϵ 9 / 108 \gamma=\epsilon^{9}/108
  91. P P^{\prime}
  92. 1 + k i 4 k i 1+k_{i}4^{k_{i}}
  93. k i + 1 = k i 4 k i k_{i}+1=k_{i}4^{k_{i}}
  94. P i + 1 = P P_{i}+1=P^{\prime}
  95. i = i + 1 i=i+1
  96. ϵ \epsilon
  97. O ( ϵ - 45 ) O(\epsilon^{-45})
  98. γ 5 / 20 = O ( ϵ 45 ) \gamma^{5}/20=O(\epsilon^{45})

Almost_simple_group.html

  1. S A Aut ( S ) . S\leq A\leq\operatorname{Aut}(S).
  2. n = 5 n=5
  3. n 7 , n\geq 7,
  4. S n S_{n}
  5. A n , A_{n},
  6. S n S_{n}
  7. n = 6 n=6
  8. S 6 S_{6}
  9. A 6 A_{6}
  10. Aut ( A 6 ) , \operatorname{Aut}(A_{6}),
  11. A 6 . A_{6}.
  12. M 10 M_{10}
  13. PGL 2 ( 9 ) \operatorname{PGL}_{2}(9)
  14. A 6 A_{6}
  15. Aut ( A 6 ) . \operatorname{Aut}(A_{6}).

Alpha_algorithm.html

  1. W T * W\subseteq T^{*}
  2. T T
  3. T W T_{W}
  4. T I T_{I}
  5. T O T_{O}
  6. W \succ_{W}
  7. a W b a\succ_{W}b
  8. a a
  9. b b
  10. a W b a\rightarrow_{W}b
  11. a W b b W a a\succ_{W}b\wedge b\not\succ_{W}a
  12. a # b W a\#{}_{W}b
  13. a W b b W a a\not\succ_{W}b\wedge b\not\succ_{W}a
  14. a | W b a\|_{W}b
  15. a W b b W a a\succ_{W}b\wedge b\succ_{W}a
  16. Y W Y_{W}
  17. ( A , B ) (A,B)
  18. A × A A\times A
  19. B × B B\times B
  20. W \succ_{W}
  21. A × B A\times B
  22. W \rightarrow_{W}
  23. P W P_{W}
  24. p ( A , B ) p_{(A,B)}
  25. Y W Y_{W}
  26. i W i_{W}
  27. o W o_{W}
  28. F W F_{W}
  29. { ( a , p ( A , B ) ) | ( A , B ) Y W a A } \{(a,p_{(A,B)})|(A,B)\in Y_{W}\wedge a\in A\}
  30. { ( p ( A , B ) , b ) | ( A , B ) Y W b B } \{(p_{(A,B)},b)|(A,B)\in Y_{W}\wedge b\in B\}
  31. { ( i W , t ) | t T I } \{(i_{W},t)|t\in T_{I}\}
  32. { ( t , i O ) | t T O } \{(t,i_{O})|t\in T_{O}\}
  33. α ( W ) = ( P W , T W , F W ) \alpha(W)=(P_{W},T_{W},F_{W})
  34. i W i_{W}
  35. o W o_{W}
  36. T W T_{W}
  37. F W F_{W}
  38. i W i_{W}
  39. o W o_{W}
  40. W \succ_{W}
  41. Y W Y_{W}
  42. W \succ_{W}
  43. T W T_{W}

Amoroso–Robinson_relation.html

  1. R x = p ( 1 + 1 ϵ x , p ) \frac{\partial R}{\partial x}=p\left(1+\frac{1}{\epsilon_{x,p}}\right)
  2. R x \scriptstyle\frac{\partial R}{\partial x}
  3. x x
  4. p p
  5. ϵ x , p < 0 \epsilon_{x,p}<0

Analogue_filter.html

  1. Z i = Z o Z s \scriptstyle Z_{i}=\sqrt{Z_{o}Z_{s}}
  2. [ 𝐀 ] = s 2 [ 𝐋 ] + s [ 𝐑 ] + [ 𝐃 ] = s [ 𝐙 ] \mathbf{[A]}=s^{2}\mathbf{[L]}+s\mathbf{[R]}+\mathbf{[D]}=s\mathbf{[Z]}
  3. s = σ + i ω \scriptstyle s=\sigma+i\omega
  4. Z p ( s ) = det [ 𝐀 ] s a 11 Z_{\mathrm{p}}(s)=\frac{\det\mathbf{[A]}}{s\,a_{11}}
  5. [ 𝐓 ] T [ 𝐀 ] [ 𝐓 ] \mathbf{[T]}^{T}\mathbf{[A]}\mathbf{[T]}
  6. [ 𝐓 ] = [ 1 0 0 T 21 T 22 T 2 n T n 1 T n 2 T n n ] \mathbf{[T]}=\begin{bmatrix}1&0\cdots 0\\ T_{21}&T_{22}\cdots T_{2n}\\ \cdot&\cdots\\ T_{n1}&T_{n2}\cdots T_{nn}\end{bmatrix}
  7. 1 1 + J F 2 \frac{1}{1+JF^{2}}

Analytic_polyhedron.html

  1. { z D : | f j ( z ) | < 1 , 1 j N } \{z\in D:|f_{j}(z)|<1,1\leq j\leq N\}\,
  2. D D
  3. f j f_{j}
  4. D D
  5. f j f_{j}
  6. σ j = { z D : | f j ( z ) | = 1 } , 1 j N . \sigma_{j}=\{z\in D:|f_{j}(z)|=1\},1\leq j\leq N.
  7. k k
  8. 2 n - k 2n-k

Analytical_regularization.html

  1. G X = Y GX=Y
  2. Y Y
  3. X X
  4. G G
  5. G G
  6. G 1 + G 2 G_{1}+G_{2}
  7. G 1 G_{1}
  8. G G
  9. G 2 G_{2}
  10. G 1 G_{1}
  11. X + G 1 - 1 G 2 X = G 1 - 1 Y X+G_{1}^{-1}G_{2}X=G_{1}^{-1}Y
  12. X + A X = B X+AX=B
  13. A A
  14. B B
  15. 𝐆 1 \mathbf{G}_{1}

András_Hajnal.html

  1. ω 1 \aleph_{\omega_{1}}
  2. 2 ω 1 < ( 2 1 ) + . 2^{\aleph_{\omega_{1}}}<\aleph_{(2^{\aleph_{1}})^{+}}.
  3. ω 1 ( α ) n 2 . \omega_{1}\to(\alpha)^{2}_{n}.
  4. κ + ( α ) 2 \kappa^{+}\to(\alpha)^{2}

André–Quillen_cohomology.html

  1. 0 Der B ( C , M ) Der A ( C , M ) Der A ( B , M ) . 0\to\operatorname{Der}_{B}(C,M)\to\operatorname{Der}_{A}(C,M)\to\operatorname{% Der}_{A}(B,M).
  2. D q ( B / A , M ) = H q ( A , B , M ) = def H q ( Der A ( P , M ) ) . D^{q}(B/A,M)=H^{q}(A,B,M)\stackrel{\,\text{def}}{=}H^{q}(\operatorname{Der}_{A% }(P,M)).
  3. D q ( B / A , M ) = H q ( A , B , M ) = def H q ( Ω P / A B M ) . D_{q}(B/A,M)=H_{q}(A,B,M)\stackrel{\,\text{def}}{=}H_{q}(\Omega_{P/A}\otimes_{% B}M).
  4. D q ( B / A , M ) = H q ( Hom B ( L B / A , M ) ) , D^{q}(B/A,M)=H^{q}(\operatorname{Hom}_{B}(L_{B/A},M)),
  5. D q ( B / A , M ) = H q ( L B / A B M ) . D_{q}(B/A,M)=H_{q}(L_{B/A}\otimes_{B}M).

Anisotropic_diffusion.html

  1. Ω 2 \Omega\subset\mathbb{R}^{2}
  2. I ( , t ) : Ω I(\cdot,t):\Omega\rightarrow\mathbb{R}
  3. I t = div ( c ( x , y , t ) I ) = c I + c ( x , y , t ) Δ I \frac{\partial I}{\partial t}=\mathrm{div}\left(c(x,y,t)\nabla I\right)=\nabla c% \cdot\nabla I+c(x,y,t)\Delta I
  4. Δ \Delta
  5. \nabla
  6. div ( ) \mathrm{div}(\dots)
  7. c ( x , y , t ) c(x,y,t)
  8. c ( x , y , t ) c(x,y,t)
  9. c ( I ) = e - ( I / K ) 2 c\left(\|\nabla I\|\right)=e^{-\left(\|\nabla I\|/K\right)^{2}}
  10. c ( I ) = 1 1 + ( I K ) 2 c\left(\|\nabla I\|\right)=\frac{1}{1+\left(\frac{\|\nabla I\|}{K}\right)^{2}}
  11. M M
  12. E : M E:M\rightarrow\mathbb{R}
  13. E [ I ] = 1 2 Ω g ( I ( x ) 2 ) d x E[I]=\frac{1}{2}\int_{\Omega}g\left(\|\nabla I(x)\|^{2}\right)\,dx
  14. g : g:\mathbb{R}\rightarrow\mathbb{R}
  15. h h
  16. d d t | t = 0 E [ I + t h ] \displaystyle\left.\frac{d}{dt}\right|_{t=0}E[I+th]
  17. E I \nabla E_{I}
  18. L 2 ( Ω , ) L^{2}(\Omega,\mathbb{R})
  19. E I = - div ( g ( I ( x ) 2 ) I ) \nabla E_{I}=-\mathrm{div}(g^{\prime}\left(\|\nabla I(x)\|^{2}\right)\nabla I)
  20. I t = - E I = div ( g ( I ( x ) 2 ) I ) \frac{\partial I}{\partial t}=-\nabla E_{I}=\mathrm{div}(g^{\prime}\left(\|% \nabla I(x)\|^{2}\right)\nabla I)
  21. c = g c=g^{\prime}
  22. I t = div ( c ( | D G σ * I | ) I ) \frac{\partial I}{\partial t}=\mathrm{div}\left(c(|DG_{\sigma}*I|)\nabla I\right)
  23. G σ = C σ - ( 1 / 2 ) e x p ( - | x | 2 / 4 σ ) G_{\sigma}=C{\sigma}^{-\left(1/2\right)}exp\left(-|x|^{2}/4{\sigma}\right)

Anisotropic_Network_Model.html

  1. V i j = γ 2 ( s i j - s i j o ) 2 V_{ij}={\gamma\over 2}{(s_{ij}-{s_{ij}}^{o})}^{2}
  2. 2 V i j x i 2 = 2 V i j x j 2 = γ s i j 2 ( x j - x i ) 2 {\partial^{2}V_{ij}\over\partial{x_{i}}^{2}}={\partial^{2}V_{ij}\over\partial{% x_{j}}^{2}}={\gamma\over{s_{ij}}^{2}}{(x_{j}-x_{i})}^{2}
  3. 2 V i j x i y j = - γ s i j 2 ( x j - x i ) ( y j - y i ) {\partial^{2}V_{ij}\over\partial x_{i}\partial y_{j}}={-\gamma\over{s_{ij}}^{2% }}{(x_{j}-x_{i})}{(y_{j}-y_{i})}
  4. \Eta = [ H i i H i j H j i H j j ] \Eta=\begin{bmatrix}{H_{ii}}&{H_{ij}}\\ {H_{ji}}&{H_{jj}}\end{bmatrix}
  5. H i j = [ 2 V i j x i x j 2 V i j x i y j 2 V i j x i z j 2 V i j y i x j 2 V i j y i y j 2 V i j y i z j 2 V i j z i x j 2 V i j z i y j 2 V i j z i z j ] H_{ij}=\begin{bmatrix}{\partial^{2}V_{ij}\over\partial x_{i}\partial x_{j}}&{% \partial^{2}V_{ij}\over\partial x_{i}\partial y_{j}}&{\partial^{2}V_{ij}\over% \partial x_{i}\partial z_{j}}\\ {\partial^{2}V_{ij}\over\partial y_{i}\partial x_{j}}&{\partial^{2}V_{ij}\over% \partial y_{i}\partial y_{j}}&{\partial^{2}V_{ij}\over\partial y_{i}\partial z% _{j}}\\ {\partial^{2}V_{ij}\over\partial z_{i}\partial x_{j}}&{\partial^{2}V_{ij}\over% \partial z_{i}\partial y_{j}}&{\partial^{2}V_{ij}\over\partial z_{i}\partial z% _{j}}\end{bmatrix}
  6. H i j = - γ s i j 2 [ ( x j - x i ) ( x j - x i ) ( x j - x i ) ( y j - y i ) ( x j - x i ) ( z j - z i ) ( y j - y i ) ( x j - x i ) ( y j - y i ) ( y j - y i ) ( y j - y i ) ( z j - z i ) ( z j - z i ) ( x j - x i ) ( z j - z i ) ( y j - y i ) ( z j - z i ) ( z j - z i ) ] H_{ij}={-\gamma\over{s_{ij}}^{2}}\begin{bmatrix}{(x_{j}-x_{i})(x_{j}-x_{i})}&{% (x_{j}-x_{i})(y_{j}-y_{i})}&{(x_{j}-x_{i})(z_{j}-z_{i})}\\ {(y_{j}-y_{i})(x_{j}-x_{i})}&{(y_{j}-y_{i})(y_{j}-y_{i})}&{(y_{j}-y_{i})(z_{j}% -z_{i})}\\ {(z_{j}-z_{i})(x_{j}-x_{i})}&{(z_{j}-z_{i})(y_{j}-y_{i})}&{(z_{j}-z_{i})(z_{j}% -z_{i})}\end{bmatrix}
  7. H i j = - γ s i j 2 [ x j - x i y j - y i z j - z i ] [ x j - x i y j - y i z j - z i ] H_{ij}={-\gamma\over{s_{ij}}^{2}}\begin{bmatrix}x_{j}-x_{i}\\ y_{j}-y_{i}\\ z_{j}-z_{i}\end{bmatrix}\begin{bmatrix}x_{j}-x_{i}&y_{j}-y_{i}&z_{j}-z_{i}\end% {bmatrix}
  8. H i j = - 1 s i j p + 2 [ ( X j - X i ) ( X j - X i ) ( X j - X i ) ( Y j - Y i ) ( X j - X i ) ( Z j - Z i ) ( Y j - Y i ) ( X j - X i ) ( Y j - Y i ) ( Y j - Y i ) ( Y j - Y i ) ( Z j - Z i ) ( Z j - Z i ) ( X j - X i ) ( Z j - Z i ) ( Y j - Y i ) ( Z j - Z i ) ( Z j - Z i ) ] H_{ij}={-1\over{s_{ij}}^{p+2}}\begin{bmatrix}{(X_{j}-X_{i})(X_{j}-X_{i})}&{(X_% {j}-X_{i})(Y_{j}-Y_{i})}&{(X_{j}-X_{i})(Z_{j}-Z_{i})}\\ {(Y_{j}-Y_{i})(X_{j}-X_{i})}&{(Y_{j}-Y_{i})(Y_{j}-Y_{i})}&{(Y_{j}-Y_{i})(Z_{j}% -Z_{i})}\\ {(Z_{j}-Z_{i})(X_{j}-X_{i})}&{(Z_{j}-Z_{i})(Y_{j}-Y_{i})}&{(Z_{j}-Z_{i})(Z_{j}% -Z_{i})}\end{bmatrix}
  9. H = U Λ U T H=U\Lambda{U^{T}}

Annualized_loss_expectancy.html

  1. A L E = A R O × S L E {ALE}={ARO}\times{SLE}

Anshel–Anshel–Goldfeld_key_exchange.html

  1. a ¯ = ( a 1 , , a n ) {\overline{a}}=(a_{1},\ldots,a_{n})
  2. a ¯ {\overline{a}}
  3. a i 1 ε 1 , , a i L ε L a_{i_{1}}^{\varepsilon_{1}},\ldots,a_{i_{L}}^{\varepsilon_{L}}
  4. a i k a ¯ a_{i_{k}}\in{\overline{a}}
  5. ε k = ± 1 \varepsilon_{k}=\pm 1
  6. A = a i 1 ε 1 a i L ε L A=a_{i_{1}}^{\varepsilon_{1}}\ldots a_{i_{L}}^{\varepsilon_{L}}
  7. b ¯ = ( b 1 , , b n ) {\overline{b}}=(b_{1},\ldots,b_{n})
  8. G G
  9. b ¯ {\overline{b}}
  10. b j 1 δ 1 , , b j L δ L b_{j_{1}}^{\delta_{1}},\ldots,b_{j_{L}}^{\delta_{L}}
  11. b j k b ¯ b_{j_{k}}\in{\overline{b}}
  12. δ k = ± 1 \delta_{k}=\pm 1
  13. B = b j 1 δ 1 b j L δ L B=b_{j_{1}}^{\delta_{1}}\ldots b_{j_{L}}^{\delta_{L}}
  14. a ¯ = ( A - 1 b 1 A , , A - 1 b n A ) {\overline{a}}=(A^{-1}b_{1}A,\ldots,A^{-1}b_{n}A)
  15. b ¯ = ( B - 1 a 1 B , , B - 1 a n B ) {\overline{b}}=(B^{-1}a_{1}B,\ldots,B^{-1}a_{n}B)
  16. K = A - 1 B - 1 A B G K=A^{-1}B^{-1}AB\in G
  17. A A
  18. B B
  19. K K
  20. A - 1 B - 1 a i 1 ε 1 B B - 1 a i L ε L B A^{-1}\cdot B^{-1}a_{i_{1}}^{\varepsilon_{1}}B\cdots B^{-1}a_{i_{L}}^{% \varepsilon_{L}}B
  21. K K
  22. ( A - 1 b i 1 ε 1 A A - 1 b i L ε L A ) - 1 B = A - 1 B - 1 A B (A^{-1}b_{i_{1}}^{\varepsilon_{1}}A\cdots A^{-1}b_{i_{L}}^{\varepsilon_{L}}A)^% {-1}\cdot B=A^{-1}B^{-1}AB

Antenna_equivalent_radius.html

  1. r e = exp { 1 L 2 ln | s y m b o l x - s y m b o l y | d x d y } r_{e}=\exp\left\{{1\over L^{2}}\oint_{\ell}\oint_{\ell}\ln|symbol{x}-symbol{y}% |\;dx\;dy\right\}
  2. \scriptstyle\ell
  3. L \scriptstyle{L}
  4. s y m b o l x \scriptstyle{symbol{x}}
  5. s y m b o l y \scriptstyle{symbol{y}}
  6. d x \scriptstyle{dx}
  7. d y \scriptstyle{dy}
  8. λ \scriptstyle{\lambda}
  9. λ \scriptstyle{\lambda}
  10. r s \sqrt{rs}
  11. exp { r 1 2 ln r 1 + r 2 2 ln r 2 + 2 r 1 r 2 ln S ( r 1 + r 2 ) 2 } \exp\left\{{r_{1}^{2}\ln r_{1}+r_{2}^{2}\ln r_{2}+2r_{1}r_{2}\ln S\over(r_{1}+% r_{2})^{2}}\right\}
  12. ( r s 2 ) 1 / 3 \left(rs^{2}\right)^{1/3}
  13. ( 2 r s 3 ) 1 / 4 \left(\sqrt{2}\,rs^{3}\right)^{1/4}
  14. ( 2.62 r s 4 ) 1 / 5 \left(2.62\,rs^{4}\right)^{1/5}
  15. ( 6 r s 5 ) 1 / 6 \left(6\,rs^{5}\right)^{1/6}
  16. N 6 N\gg 6
  17. r 1 / N R 1 - 1 / N \displaystyle{r^{1/N}R^{1-1/N}}
  18. W e - 3 / 2 0.22 W \displaystyle{We^{-3/2}\approx 0.22\,W}
  19. 0.58 W \displaystyle{0.58\,W}
  20. 0.41 W \displaystyle{0.41\,W}
  21. Q \scriptstyle{Q}
  22. s y m b o l y \scriptstyle{symbol{y}}
  23. d x \scriptstyle{dx}
  24. Q d x \scriptstyle{Q\,dx}
  25. - 2 k e Q d x ln | s y m b o l x - s y m b o l y | \scriptstyle{-2k_{e}\,Q\,dx\ln|symbol{x}-symbol{y}|}
  26. k e \scriptstyle{k_{e}}
  27. s y m b o l y \scriptstyle{symbol{y}}
  28. V ( s y m b o l y ) = - 2 k e Q ln | s y m b o l x - s y m b o l y | d x . V(symbol{y})=-2k_{e}Q\oint_{\ell}\ln|symbol{x}-symbol{y}|\;dx.
  29. V ¯ = 1 L V ( s y m b o l y ) d y = - 2 k e Q 1 L ln | s y m b o l x - s y m b o l y | d x d y . \bar{V}={1\over L}\oint_{\ell}V(symbol{y})\;dy\,=-2k_{e}Q{1\over L}\oint_{\ell% }\oint_{\ell}\ln|symbol{x}-symbol{y}|\;dx\;dy.
  30. V c = - 2 k e Q L ln ( r e ) . V_{c}=-2k_{e}QL\ln\left(r_{e}\right).
  31. V ¯ \scriptstyle{\bar{V}}
  32. V c \scriptstyle{V_{c}}
  33. ln ( r e ) = 1 L 2 ln | s y m b o l x - s y m b o l y | d x d y . \ln\left(r_{e}\right)={1\over L^{2}}\oint_{\ell}\oint_{\ell}\ln|symbol{x}-% symbol{y}|\;dx\;dy.
  34. α \scriptstyle{\alpha}
  35. | α | \scriptstyle{|\alpha|}

Antimetric_electrical_network.html

  1. Z i 1 R 0 = R 0 Z i 2 \frac{Z_{i1}}{R_{0}}=\frac{R_{0}}{Z_{i2}}
  2. Z i 1 Z i 2 = R 0 2 . Z_{i1}Z_{i2}={R_{0}}^{2}.
  3. z 11 = z 22 z_{11}=z_{22}
  4. z 11 = - z 22 z_{11}=-z_{22}
  5. z 12 = z 21 z_{12}=z_{21}
  6. [ 𝐳 ] = [ z 11 z 12 z 12 z 11 ] \left[\mathbf{z}\right]=\begin{bmatrix}z_{11}&z_{12}\\ z_{12}&z_{11}\end{bmatrix}
  7. [ 𝐳 ] = [ z 11 z 12 z 12 - z 11 ] \left[\mathbf{z}\right]=\begin{bmatrix}z_{11}&z_{12}\\ z_{12}&-z_{11}\end{bmatrix}
  8. S 11 = S 22 S_{11}=S_{22}
  9. S 11 = - S 22 S_{11}=-S_{22}
  10. z 12 = z 21 z_{12}=z_{21}
  11. [ 𝐒 ] = [ S 11 S 12 S 12 S 11 ] \left[\mathbf{S}\right]=\begin{bmatrix}S_{11}&S_{12}\\ S_{12}&S_{11}\end{bmatrix}
  12. [ 𝐒 ] = [ S 11 S 12 S 12 - S 11 ] \left[\mathbf{S}\right]=\begin{bmatrix}S_{11}&S_{12}\\ S_{12}&-S_{11}\end{bmatrix}

Antithetic_variates.html

  1. { ε 1 , , ε M } \{\varepsilon_{1},\dots,\varepsilon_{M}\}
  2. { - ε 1 , , - ε M } \{-\varepsilon_{1},\dots,-\varepsilon_{M}\}
  3. θ = E ( h ( X ) ) = E ( Y ) \theta=\mathrm{E}(h(X))=\mathrm{E}(Y)\,
  4. Y 1 and Y 2 Y_{1}\,\text{ and }Y_{2}\,
  5. θ {\theta}
  6. θ ^ = θ ^ 1 + θ ^ 2 2 . \hat{\theta}=\frac{\hat{\theta}_{1}+\hat{\theta}_{2}}{2}.
  7. Var ( θ ^ ) = Var ( Y 1 ) + Var ( Y 2 ) + 2 Cov ( Y 1 , Y 2 ) 4 \,\text{Var}(\hat{\theta})=\frac{\,\text{Var}(Y_{1})+\,\text{Var}(Y_{2})+2\,% \text{Cov}(Y_{1},Y_{2})}{4}
  8. Var ( Y 1 ) = Var ( Y 2 ) \,\text{Var}(Y_{1})=\,\text{Var}(Y_{2})
  9. Var ( θ ^ ) = Var ( Y 1 ) 2 = Var ( Y 2 ) 2 . \,\text{Var}(\hat{\theta})=\frac{\,\text{Var}(Y_{1})}{2}=\frac{\,\text{Var}(Y_% {2})}{2}.
  10. Y 1 Y_{1}
  11. Y 2 Y_{2}
  12. C o v ( Y 1 , Y 2 ) Cov(Y_{1},Y_{2})
  13. Var ( θ ^ ) \,\text{Var}(\hat{\theta})
  14. Var ( Y 1 ) 2 = Var ( Y 2 ) 2 \frac{\,\text{Var}(Y_{1})}{2}=\frac{\,\text{Var}(Y_{2})}{2}
  15. u 1 , , u n u_{1},\ldots,u_{n}
  16. u i u_{i}
  17. u 1 , , u n u^{\prime}_{1},\ldots,u^{\prime}_{n}
  18. u i = 1 - u i u^{\prime}_{i}=1-u_{i}
  19. u 1 u_{1}
  20. u i u^{\prime}_{i}
  21. I = 0 1 1 1 + x d x . I=\int_{0}^{1}\frac{1}{1+x}\,\mathrm{d}x.
  22. I = ln 2 0.69314718 I=\ln 2\approx 0.69314718
  23. f ( U ) f(U)
  24. f ( x ) = 1 1 + x f(x)=\frac{1}{1+x}

Apparent_infection_rate.html

  1. r = 1 t 2 - t 1 log e [ x 2 ( 1 - x 1 ) x 1 ( 1 - x 2 ) ] r=\frac{1}{t_{2}-t_{1}}\log_{e}\left[\frac{x_{2}(1-x_{1})}{x_{1}(1-x_{2})}\right]

Apparent_molar_property.html

  1. V = V 0 + V 1 ϕ = V ~ 0 n 0 + V ~ 1 ϕ n 1 V=V_{0}+{}^{\phi}{V}_{1}\ =\tilde{V}_{0}n_{0}+{}^{\phi}\tilde{V}_{1}n_{1}\,
  2. V ~ 0 \tilde{V}_{0}
  3. V ~ 1 ϕ {}^{\phi}\tilde{V}_{1}\,
  4. V ~ 1 ϕ {}^{\phi}\tilde{V}_{1}\,
  5. V ~ 1 ϕ {}^{\phi}\tilde{V}_{1}\,
  6. V ~ 1 ϕ {}^{\phi}\tilde{V}_{1}\,
  7. V ~ 1 \tilde{V}_{1}
  8. V = V ~ 0 ϕ n 0 + V ~ 1 n 1 V={}^{\phi}\tilde{V}_{0}n_{0}+\tilde{V}_{1}n_{1}\,
  9. V = V 0 + V 1 ϕ = v 0 m 0 + v 1 ϕ m 1 V=V_{0}+{}^{\phi}{V}_{1}\ =v_{0}m_{0}+{}^{\phi}{v}_{1}m_{1}\,
  10. V ~ 1 ϕ = 1 b ( 1 ρ - 1 ρ 0 0 ) + M 1 ρ {}^{\phi}\tilde{V}_{1}=\frac{1}{b}(\frac{1}{\rho}-\frac{1}{\rho_{0}^{0}})+% \frac{M_{1}}{\rho}
  11. V ~ i ϕ = 1 b j ( 1 ρ - 1 ρ 0 0 ) + b j M j b j ρ {}^{\phi}\tilde{V}_{i}=\frac{1}{\sum b_{j}}(\frac{1}{\rho}-\frac{1}{\rho_{0}^{% 0}})+\frac{\sum b_{j}M_{j}}{\sum b_{j}\rho}
  12. V 1 ¯ = V ~ 1 ϕ + b V ~ 1 ϕ b . \bar{V_{1}}={}^{\phi}\tilde{V}_{1}+b\frac{\partial{}^{\phi}\tilde{V}_{1}}{% \partial b}.
  13. V = V ~ 0 n 0 + V ~ 1 ϕ n 1 + V ~ 2 ϕ n 2 V=\tilde{V}_{0}n_{0}+{}^{\phi}\tilde{V}_{1}n_{1}+{}^{\phi}\tilde{V}_{2}n_{2}\,
  14. V = V ~ 0 n 0 + V ~ 1 n 1 + V ~ 2 ϕ n 2 V=\tilde{V}_{0}n_{0}+\tilde{V}_{1}n_{1}+{}^{\phi}\tilde{V}_{2}n_{2}\,
  15. V = V ~ 0 n 0 + V ~ 1 ϕ n 1 + V ~ 2 n 2 V=\tilde{V}_{0}n_{0}+{}^{\phi}\tilde{V}_{1}n_{1}+\tilde{V}_{2}n_{2}\,
  16. V = V ~ 0 ϕ n 0 + V ~ 1 n 1 + V ~ 2 n 2 V={}^{\phi}\tilde{V}_{0}n_{0}+\tilde{V}_{1}n_{1}+\tilde{V}_{2}n_{2}\,
  17. i V i ϕ = q V - ( q - 1 ) i V i \sum_{i}{}^{\phi}{V}_{i}=qV-(q-1)\sum_{i}V_{i}\,
  18. α V = i α i V i + α j ϕ V j ϕ \alpha V=\sum_{i}\alpha_{i}V_{i}+\alpha_{j}^{\phi}{}^{\phi}{V}_{j}
  19. V T = i V i T + V j ϕ T \frac{\partial V}{\partial T}=\sum_{i}\frac{\partial V_{i}}{\partial T}+\frac{% \partial{}^{\phi}{V}_{j}}{\partial T}
  20. : α V V = i α V , i V i + V j ϕ T :\alpha_{V}V=\sum_{i}\alpha_{V,i}V_{i}+\frac{\partial{}^{\phi}{V}_{j}}{% \partial T}

Appell–Humbert_theorem.html

  1. α ( u + v ) = e i π E ( u , v ) α ( u ) α ( v ) \alpha(u+v)=e^{i\pi E(u,v)}\alpha(u)\alpha(v)
  2. α ( u ) e π H ( z , u ) + H ( u , u ) π / 2 \alpha(u)e^{\pi H(z,u)+H(u,u)\pi/2}

Applied_element_method.html

  1. K n = E T d a K_{n}=\frac{E\cdot T\cdot d}{a}
  2. K s = G T d a K_{s}=\frac{G\cdot T\cdot d}{a}
  3. [ sin 2 ( θ + α ) K n - K n sin ( θ + α ) cos ( θ + α ) cos ( θ + α ) K s L sin ( α ) + cos 2 ( θ + α ) K s + K s sin ( θ + α ) cos ( θ + α ) - sin ( θ + α ) K n L cos ( α ) - K n sin ( θ + α ) cos ( θ + α ) sin 2 ( θ + α ) K s cos ( θ + α ) K n L cos ( α ) + K s sin ( θ + α ) cos ( θ + α ) + cos 2 ( θ + α ) K n + sin ( θ + α ) K s L sin ( α ) cos ( θ + α ) K s L sin ( α ) cos ( θ + α ) K n L cos ( α ) L 2 cos 2 ( α ) K n - sin ( θ + α ) K n L cos ( α ) + sin ( θ + α ) K s L sin ( α ) + L 2 sin 2 ( α ) K s ] \begin{bmatrix}\sin^{2}(\theta+\alpha)K_{n}&-K_{n}\sin(\theta+\alpha)\cos(% \theta+\alpha)&\cos(\theta+\alpha)K_{s}L\sin(\alpha)\\ +\cos^{2}(\theta+\alpha)K_{s}&+K_{s}\sin(\theta+\alpha)\cos(\theta+\alpha)&-% \sin(\theta+\alpha)K_{n}L\cos(\alpha)\\ \\ -K_{n}\sin(\theta+\alpha)\cos(\theta+\alpha)&\sin^{2}(\theta+\alpha)K_{s}&\cos% (\theta+\alpha)K_{n}L\cos(\alpha)\\ +K_{s}\sin(\theta+\alpha)\cos(\theta+\alpha)&+\cos^{2}(\theta+\alpha)K_{n}&+% \sin(\theta+\alpha)K_{s}L\sin(\alpha)\\ \\ \cos(\theta+\alpha)K_{s}L\sin(\alpha)&\cos(\theta+\alpha)K_{n}L\cos(\alpha)&L^% {2}\cos^{2}(\alpha)K_{n}\\ -\sin(\theta+\alpha)K_{n}L\cos(\alpha)&+\sin(\theta+\alpha)K_{s}L\sin(\alpha)&% +L^{2}\sin^{2}(\alpha)K_{s}\end{bmatrix}

Approximate_limit.html

  1. k \mathbb{R}^{k}
  2. lim x x 0 ap f ( x ) . \lim_{x\rightarrow x_{0}}\operatorname{ap}\ f(x).
  3. lim x x 0 ap a f ( x ) = a lim x x 0 ap f ( x ) lim x x 0 ap ( f ( x ) + g ( x ) ) = lim x x 0 ap f ( x ) + lim x x 0 ap g ( x ) lim x x 0 ap ( f ( x ) - g ( x ) ) = lim x x 0 ap f ( x ) - lim x x 0 ap g ( x ) lim x x 0 ap ( f ( x ) g ( x ) ) = lim x x 0 ap f ( x ) lim x x 0 ap g ( x ) lim x x 0 ap ( f ( x ) / g ( x ) ) = lim x x 0 ap f ( x ) / lim x x 0 ap g ( x ) \begin{aligned}\displaystyle\lim_{x\rightarrow x_{0}}\operatorname{ap}\ a\cdot f% (x)&\displaystyle=a\cdot\lim_{x\rightarrow x_{0}}\operatorname{ap}\ f(x)\\ \displaystyle\lim_{x\rightarrow x_{0}}\operatorname{ap}\ (f(x)+g(x))&% \displaystyle=\lim_{x\rightarrow x_{0}}\operatorname{ap}\ f(x)+\lim_{x% \rightarrow x_{0}}\operatorname{ap}\ g(x)\\ \displaystyle\lim_{x\rightarrow x_{0}}\operatorname{ap}\ (f(x)-g(x))&% \displaystyle=\lim_{x\rightarrow x_{0}}\operatorname{ap}\ f(x)-\lim_{x% \rightarrow x_{0}}\operatorname{ap}\ g(x)\\ \displaystyle\lim_{x\rightarrow x_{0}}\operatorname{ap}\ (f(x)\cdot g(x))&% \displaystyle=\lim_{x\rightarrow x_{0}}\operatorname{ap}\ f(x)\cdot\lim_{x% \rightarrow x_{0}}\operatorname{ap}\ g(x)\\ \displaystyle\lim_{x\rightarrow x_{0}}\operatorname{ap}\ (f(x)/g(x))&% \displaystyle=\lim_{x\rightarrow x_{0}}\operatorname{ap}\ f(x)/\lim_{x% \rightarrow x_{0}}\operatorname{ap}\ g(x)\end{aligned}
  4. lim x x 0 ap f ( x ) = f ( x 0 ) \lim_{x\rightarrow x_{0}}\operatorname{ap}\ f(x)=f(x_{0})
  5. f ( x 0 + h ) - f ( x 0 ) h \frac{f(x_{0}+h)-f(x_{0})}{h}

Arbitrarily_varying_channel.html

  1. n \textstyle n
  2. W n : X n × \textstyle W^{n}:X^{n}\times
  3. S n Y n \textstyle S^{n}\rightarrow Y^{n}
  4. X \textstyle X
  5. Y \textstyle Y
  6. W n ( y | x , s ) \textstyle W^{n}(y|x,s)
  7. S \textstyle S
  8. x = ( x 1 , , x n ) \textstyle x=(x_{1},\ldots,x_{n})
  9. y = ( y 1 , , y n ) \textstyle y=(y_{1},\ldots,y_{n})
  10. s i \textstyle s_{i}
  11. S \textstyle S
  12. i \textstyle i
  13. R \textstyle R
  14. 0 \textstyle 0
  15. ε \textstyle\varepsilon
  16. δ \textstyle\delta
  17. n \textstyle n
  18. n \textstyle n
  19. 1 n log N > R - δ \textstyle\frac{1}{n}\log N>R-\delta
  20. max s S n e ¯ ( s ) ε \displaystyle\max_{s\in S^{n}}\bar{e}(s)\leq\varepsilon
  21. N \textstyle N
  22. Y \textstyle Y
  23. e ¯ ( s ) \textstyle\bar{e}(s)
  24. s \textstyle s
  25. R \textstyle R
  26. c \textstyle c
  27. 0 \textstyle 0
  28. c \textstyle\leq c
  29. c \textstyle c
  30. c \textstyle c
  31. s S W ( y | x , s ) U ( s | x ) = s S W ( y | x , s ) U ( s | x ) \displaystyle\sum_{s\in S}W(y|x,s)U(s|x^{\prime})=\sum_{s\in S}W(y|x^{\prime},% s)U(s|x)
  32. ( x , x , y , s ) \textstyle(x,x^{\prime},y,s)
  33. x , x X \textstyle x,x^{\prime}\in X
  34. y Y \textstyle y\in Y
  35. U ( s | x ) \textstyle U(s|x)
  36. U : X S \textstyle U:X\rightarrow S
  37. X r \textstyle X_{r}
  38. S r \textstyle S_{r}
  39. Y r \textstyle Y_{r}
  40. X \textstyle X
  41. S \textstyle S
  42. Y \textstyle Y
  43. P X r ( x ) \textstyle P_{X_{r}}(x)
  44. X r \textstyle X_{r}
  45. x \textstyle x
  46. P S r ( s ) \textstyle P_{S_{r}}(s)
  47. S r \textstyle S_{r}
  48. s \textstyle s
  49. P X r S r Y r \textstyle P_{X_{r}S_{r}Y_{r}}
  50. P X r ( x ) \textstyle P_{X_{r}}(x)
  51. P S r ( s ) \textstyle P_{S_{r}}(s)
  52. W ( y | x , s ) \textstyle W(y|x,s)
  53. P X r S r Y r \textstyle P_{X_{r}S_{r}Y_{r}}
  54. P X r S r Y r ( x , s , y ) = P X r ( x ) P S r ( s ) W ( y | x , s ) \textstyle P_{X_{r}S_{r}Y_{r}}(x,s,y)=P_{X_{r}}(x)P_{S_{r}}(s)W(y|x,s)
  55. H ( X r ) \textstyle H(X_{r})
  56. X r \textstyle X_{r}
  57. H ( X r | Y r ) \textstyle H(X_{r}|Y_{r})
  58. X r \textstyle X_{r}
  59. Y r \textstyle Y_{r}
  60. I ( X r Y r ) \textstyle I(X_{r}\land Y_{r})
  61. X r \textstyle X_{r}
  62. Y r \textstyle Y_{r}
  63. H ( X r ) - H ( X r | Y r ) \textstyle H(X_{r})-H(X_{r}|Y_{r})
  64. I ( P ) = min Y r I ( X r Y r ) \displaystyle I(P)=\min_{Y_{r}}I(X_{r}\land Y_{r})
  65. Y r \textstyle Y_{r}
  66. X r \textstyle X_{r}
  67. S r \textstyle S_{r}
  68. Y r \textstyle Y_{r}
  69. P X r S r Y r \textstyle P_{X_{r}S_{r}Y_{r}}
  70. c > 0 \textstyle c>0
  71. c > 0 \textstyle c>0
  72. c = max P I ( P ) \displaystyle c=\max_{P}I(P)
  73. I ( P ) \textstyle I(P)
  74. c = max P I ( P ) \textstyle c=\max_{P}I(P)
  75. I ( P ) \textstyle I(P)
  76. 0 \textstyle 0
  77. I ( P ) \textstyle I(P)
  78. X r \textstyle X_{r}
  79. Y r \textstyle Y_{r}
  80. S r \textstyle S_{r}
  81. P X r S r Y r \textstyle P_{X_{r}S_{r}Y_{r}}
  82. P X r = P \textstyle P_{X_{r}}=P
  83. P Y r ( y ) = x X s S P ( x ) P S r ( s ) W ( y | x , s ) \displaystyle P_{Y_{r}}(y)=\sum_{x\in X}\sum_{s\in S}P(x)P_{S_{r}}(s)W(y|x,s)
  84. ( \textstyle\equiv(
  85. X r \textstyle X_{r}
  86. Y r \textstyle Y_{r}
  87. W ( y | x , s ) = W ( y | s ) \textstyle W(y|x,s)=W^{\prime}(y|s)
  88. W ) \textstyle W^{\prime})
  89. P Y r ( y ) = x X s S P ( x ) P S r ( s ) W ( y | s ) \displaystyle P_{Y_{r}}(y)=\sum_{x\in X}\sum_{s\in S}P(x)P_{S_{r}}(s)W^{\prime% }(y|s)
  90. ( \textstyle\equiv(
  91. P ( x ) \textstyle P(x)
  92. x \textstyle x
  93. ) \textstyle)
  94. P Y r ( y ) = s S P S r ( s ) W ( y | s ) [ x X P ( x ) ] \displaystyle P_{Y_{r}}(y)=\sum_{s\in S}P_{S_{r}}(s)W^{\prime}(y|s)\left[\sum_% {x\in X}P(x)\right]
  95. ( \textstyle\equiv(
  96. x X P ( x ) = 1 ) \displaystyle\sum_{x\in X}P(x)=1)
  97. P Y r ( y ) = s S P S r ( s ) W ( y | s ) \displaystyle P_{Y_{r}}(y)=\sum_{s\in S}P_{S_{r}}(s)W^{\prime}(y|s)
  98. Y r \textstyle Y_{r}
  99. X r \textstyle X_{r}
  100. s S W ( y | s ) P S r ( s ) = s S W ( y | s ) P S r ( s ) \displaystyle\sum_{s\in S}W^{\prime}(y|s)P_{S_{r}}(s)=\sum_{s\in S}W^{\prime}(% y|s)P_{S_{r}}(s)
  101. U ( s | x ) \textstyle U(s|x)
  102. W ( y | x , s ) \textstyle W(y|x,s)
  103. x \textstyle x
  104. s \textstyle s
  105. y \textstyle y
  106. P Y r ( y ) \textstyle P_{Y_{r}}(y)
  107. I ( P ) \textstyle I(P)
  108. 0 \textstyle 0
  109. I ( P ) \textstyle I(P)
  110. Γ \textstyle\Gamma
  111. g ( x ) = 1 n i = 1 n g ( x i ) \displaystyle g(x)=\frac{1}{n}\sum_{i=1}^{n}g(x_{i})
  112. x X \textstyle x\in X
  113. x = ( x 1 , , x n ) \textstyle x=(x_{1},\dots,x_{n})
  114. Λ \textstyle\Lambda
  115. l ( s ) = 1 n i = 1 n l ( s i ) \displaystyle l(s)=\frac{1}{n}\sum_{i=1}^{n}l(s_{i})
  116. s X \textstyle s\in X
  117. s = ( s 1 , , s n ) \textstyle s=(s_{1},\dots,s_{n})
  118. Λ 0 ( P ) = min x X , s S P ( x ) l ( s ) \displaystyle\Lambda_{0}(P)=\min\sum_{x\in X,s\in S}P(x)l(s)
  119. I ( P , Λ ) \textstyle I(P,\Lambda)
  120. I ( P ) \textstyle I(P)
  121. I ( P , Λ ) = min Y r I ( X r Y r ) \displaystyle I(P,\Lambda)=\min_{Y_{r}}I(X_{r}\land Y_{r})
  122. s \textstyle s
  123. S r \textstyle S_{r}
  124. l ( s ) Λ \textstyle l(s)\leq\Lambda
  125. g ( x ) \textstyle g(x)
  126. X \textstyle X
  127. l ( s ) \textstyle l(s)
  128. S \textstyle S
  129. 0 \textstyle 0
  130. g ( x ) \textstyle g(x)
  131. l ( s ) \textstyle l(s)
  132. x i \textstyle x_{i}
  133. Γ \textstyle\Gamma
  134. Λ \textstyle\Lambda
  135. R \textstyle R
  136. x 1 , , x N \textstyle x_{1},\dots,x_{N}
  137. g ( x i ) Γ \textstyle g(x_{i})\leq\Gamma
  138. s \textstyle s
  139. l ( s ) Λ \textstyle l(s)\leq\Lambda
  140. c ( Γ , Λ ) \textstyle c(\Gamma,\Lambda)
  141. Λ \textstyle\Lambda
  142. Λ 0 ( P ) \textstyle\Lambda_{0}(P)
  143. N - 1 2 N - 1 n l m a x 2 n ( Λ - Λ 0 ( P ) ) 2 \textstyle\frac{N-1}{2N}-\frac{1}{n}\frac{l_{max}^{2}}{n(\Lambda-\Lambda_{0}(P% ))^{2}}
  144. l m a x \textstyle l_{max}
  145. l ( s ) \textstyle l(s)
  146. N - 1 2 N \textstyle\frac{N-1}{2N}
  147. 1 2 \textstyle\frac{1}{2}
  148. ( Λ - Λ 0 ( P ) ) \textstyle(\Lambda-\Lambda_{0}(P))
  149. Λ \textstyle\Lambda
  150. Λ 0 ( P ) \textstyle\Lambda_{0}(P)
  151. Λ 0 ( P ) \textstyle\Lambda_{0}(P)
  152. Λ \textstyle\Lambda
  153. α > 0 \textstyle\alpha>0
  154. β > 0 \textstyle\beta>0
  155. δ > 0 \textstyle\delta>0
  156. n n 0 \textstyle n\geq n_{0}
  157. P \textstyle P
  158. Λ 0 ( P ) Λ + α \textstyle\Lambda_{0}(P)\geq\Lambda+\alpha
  159. min x X P ( x ) β \displaystyle\min_{x\in X}P(x)\geq\beta
  160. P X r = P \textstyle P_{X_{r}}=P
  161. x 1 , , x N \textstyle x_{1},\dots,x_{N}
  162. P \textstyle P
  163. 1 n log N > I ( P , Λ ) - δ \textstyle\frac{1}{n}\log N>I(P,\Lambda)-\delta
  164. max l ( s ) Λ e ¯ ( s ) exp ( - n γ ) \displaystyle\max_{l(s)\leq\Lambda}\bar{e}(s)\leq\exp(-n\gamma)
  165. n 0 \textstyle n_{0}
  166. γ \textstyle\gamma
  167. α \textstyle\alpha
  168. β \textstyle\beta
  169. δ \textstyle\delta
  170. W ζ ( y | x ) = s S W ( y | x , s ) P S r ( s ) \displaystyle W_{\zeta}(y|x)=\sum_{s\in S}W(y|x,s)P_{S_{r}}(s)
  171. I ( P , ζ ) \textstyle I(P,\zeta)
  172. I ( P ) \textstyle I(P)
  173. I ( P , ζ ) = min Y r I ( X r Y r ) \displaystyle I(P,\zeta)=\min_{Y_{r}}I(X_{r}\land Y_{r})
  174. P S r ( s ) \textstyle P_{S_{r}}(s)
  175. I ( P , ζ ) \textstyle I(P,\zeta)
  176. P X r S r Y r \textstyle P_{X_{r}S_{r}Y_{r}}
  177. W ζ ( y | x ) \textstyle W_{\zeta}(y|x)
  178. W ( y | x , s ) \textstyle W(y|x,s)
  179. c = m a x P I ( P , ζ ) \displaystyle c=max_{P}I(P,\zeta)

Arbitration_inter-frame_spacing.html

  1. σ \sigma

Arcsine_distribution.html

  1. F ( x ) = 2 π arcsin ( x ) F(x)=\frac{2}{\pi}\arcsin\left(\sqrt{x}\right)
  2. 1 2 \frac{1}{2}
  3. 1 2 \frac{1}{2}
  4. x 0 , 1 x\in{0,1}
  5. 1 8 \tfrac{1}{8}
  6. 0
  7. - 3 2 -\tfrac{3}{2}
  8. ln π 4 \ln\tfrac{\pi}{4}
  9. 1 + k = 1 ( r = 0 k - 1 2 r + 1 2 r + 2 ) t k k ! 1+\sum_{k=1}^{\infty}\left(\prod_{r=0}^{k-1}\frac{2r+1}{2r+2}\right)\frac{t^{k% }}{k!}
  10. F 1 1 ( 1 2 ; 1 ; i t ) {}_{1}F_{1}(\tfrac{1}{2};1;i\,t)
  11. F ( x ) = 2 π arcsin ( x ) = arcsin ( 2 x - 1 ) π + 1 2 F(x)=\frac{2}{\pi}\arcsin\left(\sqrt{x}\right)=\frac{\arcsin(2x-1)}{\pi}+\frac% {1}{2}
  12. f ( x ) = 1 π x ( 1 - x ) f(x)=\frac{1}{\pi\sqrt{x(1-x)}}
  13. X X
  14. X Beta ( 1 2 , 1 2 ) X\sim{\rm Beta}(\tfrac{1}{2},\tfrac{1}{2})
  15. F ( x ) = 2 π arcsin ( x - a b - a ) F(x)=\frac{2}{\pi}\arcsin\left(\sqrt{\frac{x-a}{b-a}}\right)
  16. a + b 2 \frac{a+b}{2}
  17. a + b 2 \frac{a+b}{2}
  18. x a , b x\in{a,b}
  19. 1 8 ( b - a ) 2 \tfrac{1}{8}(b-a)^{2}
  20. 0
  21. - 3 2 -\tfrac{3}{2}
  22. F ( x ) = 2 π arcsin ( x - a b - a ) F(x)=\frac{2}{\pi}\arcsin\left(\sqrt{\frac{x-a}{b-a}}\right)
  23. f ( x ) = 1 π ( x - a ) ( b - x ) f(x)=\frac{1}{\pi\sqrt{(x-a)(b-x)}}
  24. f ( x ; α ) = sin π α π x - α ( 1 - x ) α - 1 f(x;\alpha)=\frac{\sin\pi\alpha}{\pi}x^{-\alpha}(1-x)^{\alpha-1}
  25. Beta ( 1 - α , α ) {\rm Beta}(1-\alpha,\alpha)
  26. α = 1 2 \alpha=\tfrac{1}{2}
  27. X Arcsine ( a , b ) then k X + c Arcsine ( a k + c , b k + c ) X\sim{\rm Arcsine}(a,b)\ \,\text{then }kX+c\sim{\rm Arcsine}(ak+c,bk+c)
  28. X Arcsine ( - 1 , 1 ) then X 2 Arcsine ( 0 , 1 ) X\sim{\rm Arcsine}(-1,1)\ \,\text{then }X^{2}\sim{\rm Arcsine}(0,1)
  29. { 2 ( x - 1 ) x f ( x ) + ( 2 x - 1 ) f ( x ) = 0 } \left\{2(x-1)xf^{\prime}(x)+(2x-1)f(x)=0\right\}
  30. sin ( U ) \sin(U)
  31. sin ( 2 U ) \sin(2U)
  32. - cos ( 2 U ) -\cos(2U)
  33. sin ( U + V ) \sin(U+V)
  34. sin ( U - V ) \sin(U-V)
  35. Arcsine ( - 1 , 1 ) {\rm Arcsine}(-1,1)
  36. X X
  37. α \alpha
  38. X - a b - a Beta ( 1 - α , α ) \frac{X-a}{b-a}\sim{\rm Beta}(1-\alpha,\alpha)

Arcsine_laws_(Wiener_process).html

  1. Pr [ X x ] = 2 π arcsin ( x ) . \Pr\left[X\leq x\right]=\frac{2}{\pi}\arcsin\left(\sqrt{x}\right).
  2. T + = | { t [ 0 , 1 ] : W t > 0 } | T_{+}=\left|\{\,t\in[0,1]\,\colon\,W_{t}>0\,\}\right|
  3. T + T_{+}
  4. Pr [ T + x ] = 2 π arcsin ( x ) , x [ 0 , 1 ] . \Pr\left[T_{+}\leq x\right]=\frac{2}{\pi}\arcsin\left(\sqrt{x}\right),\qquad% \forall x\in[0,1].
  5. L = sup { t [ 0 , 1 ] : W t = 0 } L=\sup\left\{t\in[0,1]\,\colon\,W_{t}=0\right\}
  6. Pr [ L x ] = 2 π arcsin ( x ) , x [ 0 , 1 ] . \Pr\left[L\leq x\right]=\frac{2}{\pi}\arcsin\left(\sqrt{x}\right),\qquad% \forall x\in[0,1].
  7. W M = sup { W s : s [ 0 , 1 ] } . W_{M}=\sup\{W_{s}\,\colon\,s\in[0,1]\}.
  8. Pr [ M x ] = 2 π arcsin ( x ) , x [ 0 , 1 ] . \Pr\left[M\leq x\right]=\frac{2}{\pi}\arcsin\left(\sqrt{x}\right),\qquad% \forall x\in[0,1].
  9. M t = sup { W s : s [ 0 , t ] } , M_{t}=\sup\{W_{s}\,\colon\,s\in[0,t]\},

Argument_of_latitude.html

  1. u u
  2. u = ν + ω u=\nu+\omega
  3. u u
  4. ν \nu
  5. ω \omega

Arias_Intensity.html

  1. I A = π 2 g 0 T d a ( t ) 2 d t I_{A}=\frac{\pi}{2g}\int_{0}^{T_{d}}a(t)^{2}dt

Arithmetic_circuit_complexity.html

  1. × \times
  2. perm ( X ) = ( - 1 ) n S { 1 , , n } ( - 1 ) | S | i = 1 n j S x i , j \operatorname{perm}(X)=(-1)^{n}\sum_{S\subseteq\{1,\ldots,n\}}(-1)^{|S|}\prod_% {i=1}^{n}\sum_{j\in S}x_{i,j}
  3. Ω \Omega
  4. Ω \Omega

Arithmetical_ring.html

  1. R 𝔪 R_{\mathfrak{m}}
  2. 𝔪 \mathfrak{m}
  3. 𝔪 \mathfrak{m}
  4. 𝔞 , 𝔟 \mathfrak{a},\mathfrak{b}
  5. 𝔠 \mathfrak{c}
  6. 𝔞 ( 𝔟 + 𝔠 ) = ( 𝔞 𝔟 ) + ( 𝔞 𝔠 ) \mathfrak{a}\cap(\mathfrak{b}+\mathfrak{c})=(\mathfrak{a}\cap\mathfrak{b})+(% \mathfrak{a}\cap\mathfrak{c})
  7. 𝔞 , 𝔟 \mathfrak{a},\mathfrak{b}
  8. 𝔠 \mathfrak{c}
  9. 𝔞 + ( 𝔟 𝔠 ) = ( 𝔞 + 𝔟 ) ( 𝔞 + 𝔠 ) \mathfrak{a}+(\mathfrak{b}\cap\mathfrak{c})=(\mathfrak{a}+\mathfrak{b})\cap(% \mathfrak{a}+\mathfrak{c})

Arnold_diffusion.html

  1. H ( I , p , q , ϕ , t ) = 1 2 I 2 + p 2 + ϵ cos ( q - 1 ) + μ cos ( q - 1 ) ( sin ϕ + cos t ) H(I,p,q,\phi,t)={1\over 2}I^{2}+p^{2}+\epsilon\cos{(q-1)}+\mu\cos{(q-1)}(\sin{% \phi+\cos t)}
  2. ϵ , δ , K > 0 , \epsilon,\delta,K>0,
  3. K δ K\gg\delta
  4. μ 0 > 0 \mu_{0}>0
  5. μ < μ 0 \mu<\mu_{0}
  6. I ( 0 ) < δ and I ( T ) > K I(0)<\delta\,\text{ and }I(T)>K\,
  7. T 0. T\gg 0.

Arruda–Boyce_model.html

  1. W = N k B θ n [ β λ chain - n ln ( sinh β β ) ] W=Nk_{B}\theta\sqrt{n}\left[\beta\lambda_{\mathrm{chain}}-\sqrt{n}\ln\left(% \cfrac{\sinh\beta}{\beta}\right)\right]
  2. n n
  3. k B k_{B}
  4. θ \theta
  5. N N
  6. λ chain = I 1 3 ; β = - 1 ( λ chain n ) \lambda_{\mathrm{chain}}=\sqrt{\tfrac{I_{1}}{3}}~{};~{}~{}\beta=\mathcal{L}^{-% 1}\left(\cfrac{\lambda_{\mathrm{chain}}}{\sqrt{n}}\right)
  7. I 1 I_{1}
  8. - 1 ( x ) \mathcal{L}^{-1}(x)
  9. - 1 ( x ) = { 1.31 tan ( 1.59 x ) + 0.91 x for | x | < 0.841 1 sgn ( x ) - x for 0.841 | x | < 1 \mathcal{L}^{-1}(x)=\begin{cases}1.31\tan(1.59x)+0.91x&\quad\mathrm{for}~{}|x|% <0.841\\ \tfrac{1}{\operatorname{sgn}(x)-x}&\quad\mathrm{for}~{}0.841\leq|x|<1\end{cases}
  10. W = C 1 [ 1 2 ( I 1 - 3 ) + 1 20 N ( I 1 2 - 9 ) + 11 1050 N 2 ( I 1 3 - 27 ) + 19 7000 N 3 ( I 1 4 - 81 ) + 519 673750 N 4 ( I 1 5 - 243 ) ] W=C_{1}\left[\tfrac{1}{2}(I_{1}-3)+\tfrac{1}{20N}(I_{1}^{2}-9)+\tfrac{11}{1050% N^{2}}(I_{1}^{3}-27)+\tfrac{19}{7000N^{3}}(I_{1}^{4}-81)+\tfrac{519}{673750N^{% 4}}(I_{1}^{5}-243)\right]
  11. C 1 C_{1}
  12. N N
  13. λ m \lambda_{m}
  14. W = C 1 [ 1 2 ( I 1 - 3 ) + 1 20 λ m 2 ( I 1 2 - 9 ) + 11 1050 λ m 4 ( I 1 3 - 27 ) + 19 7000 λ m 6 ( I 1 4 - 81 ) + 519 673750 λ m 8 ( I 1 5 - 243 ) ] W=C_{1}\left[\tfrac{1}{2}(I_{1}-3)+\tfrac{1}{20\lambda_{m}^{2}}(I_{1}^{2}-9)+% \tfrac{11}{1050\lambda_{m}^{4}}(I_{1}^{3}-27)+\tfrac{19}{7000\lambda_{m}^{6}}(% I_{1}^{4}-81)+\tfrac{519}{673750\lambda_{m}^{8}}(I_{1}^{5}-243)\right]
  15. W = C 1 i = 1 5 α i β i - 1 ( I 1 i - 3 i ) W=C_{1}~{}\sum_{i=1}^{5}\alpha_{i}~{}\beta^{i-1}~{}(I_{1}^{i}-3^{i})
  16. β := 1 N = 1 λ m 2 \beta:=\tfrac{1}{N}=\tfrac{1}{\lambda_{m}^{2}}
  17. α 1 := 1 2 ; α 2 := 1 20 ; α 3 := 11 1050 ; α 4 := 19 7000 ; α 5 := 519 673750 . \alpha_{1}:=\tfrac{1}{2}~{};~{}~{}\alpha_{2}:=\tfrac{1}{20}~{};~{}~{}\alpha_{3% }:=\tfrac{11}{1050}~{};~{}~{}\alpha_{4}:=\tfrac{19}{7000}~{};~{}~{}\alpha_{5}:% =\tfrac{519}{673750}.
  18. J = det ( s y m b o l F ) J=\det(symbol{F})
  19. s y m b o l F symbol{F}
  20. W = D 1 ( J 2 - 1 2 - ln J ) + C 1 i = 1 5 α i β i - 1 ( I ¯ 1 i - 3 i ) W=D_{1}\left(\tfrac{J^{2}-1}{2}-\ln J\right)+C_{1}~{}\sum_{i=1}^{5}\alpha_{i}~% {}\beta^{i-1}~{}(\overline{I}_{1}^{i}-3^{i})
  21. D 1 D_{1}
  22. I ¯ 1 = I 1 J - 2 / 3 \overline{I}_{1}={I}_{1}J^{-2/3}
  23. D 1 = κ 2 D_{1}=\tfrac{\kappa}{2}
  24. κ \kappa
  25. μ \mu
  26. W I 1 | I 1 = 3 = μ 2 . \cfrac{\partial W}{\partial I_{1}}\biggr|_{I_{1}=3}=\frac{\mu}{2}\,.
  27. W I 1 = C 1 i = 1 5 i α i β i - 1 I 1 i - 1 . \cfrac{\partial W}{\partial I_{1}}=C_{1}~{}\sum_{i=1}^{5}i~{}\alpha_{i}~{}% \beta^{i-1}~{}I_{1}^{i-1}\,.
  28. I 1 = 3 I_{1}=3
  29. μ = 2 C 1 i = 1 5 i α i β i - 1 I 1 i - 1 . \mu=2C_{1}~{}\sum_{i=1}^{5}i\,\alpha_{i}~{}\beta^{i-1}~{}I_{1}^{i-1}\,.
  30. α i \alpha_{i}
  31. μ = C 1 ( 1 + 3 5 λ m 2 + 99 175 λ m 4 + 513 875 λ m 6 + 42039 67375 λ m 8 ) . \mu=C_{1}\left(1+\tfrac{3}{5\lambda_{m}^{2}}+\tfrac{99}{175\lambda_{m}^{4}}+% \tfrac{513}{875\lambda_{m}^{6}}+\tfrac{42039}{67375\lambda_{m}^{8}}\right)\,.
  32. s y m b o l σ = - p s y m b o l 1 + 2 W I 1 s y m b o l B = - p s y m b o l 1 + 2 C 1 [ i = 1 5 i α i β i - 1 I 1 i - 1 ] s y m b o l B symbol{\sigma}=-p~{}symbol{\mathit{1}}+2~{}\cfrac{\partial W}{\partial I_{1}}~% {}symbol{B}=-p~{}symbol{\mathit{1}}+2C_{1}~{}\left[\sum_{i=1}^{5}i~{}\alpha_{i% }~{}\beta^{i-1}~{}I_{1}^{i-1}\right]symbol{B}
  33. 𝐧 1 \mathbf{n}_{1}
  34. λ 1 = λ , λ 2 = λ 3 \lambda_{1}=\lambda,~{}\lambda_{2}=\lambda_{3}
  35. λ 1 λ 2 λ 3 = 1 \lambda_{1}~{}\lambda_{2}~{}\lambda_{3}=1
  36. λ 2 2 = λ 3 2 = 1 / λ \lambda_{2}^{2}=\lambda_{3}^{2}=1/\lambda
  37. I 1 = λ 1 2 + λ 2 2 + λ 3 2 = λ 2 + 2 λ . I_{1}=\lambda_{1}^{2}+\lambda_{2}^{2}+\lambda_{3}^{2}=\lambda^{2}+\cfrac{2}{% \lambda}~{}.
  38. s y m b o l B = λ 2 𝐧 1 𝐧 1 + 1 λ ( 𝐧 2 𝐧 2 + 𝐧 3 𝐧 3 ) . symbol{B}=\lambda^{2}~{}\mathbf{n}_{1}\otimes\mathbf{n}_{1}+\cfrac{1}{\lambda}% ~{}(\mathbf{n}_{2}\otimes\mathbf{n}_{2}+\mathbf{n}_{3}\otimes\mathbf{n}_{3})~{}.
  39. σ 11 = - p + 2 C 1 λ 2 [ i = 1 5 i α i β i - 1 I 1 i - 1 ] σ 22 = - p + 2 C 1 λ [ i = 1 5 i α i β i - 1 I 1 i - 1 ] = σ 33 . \begin{aligned}\displaystyle\sigma_{11}&\displaystyle=-p+2C_{1}\lambda^{2}% \left[\sum_{i=1}^{5}i~{}\alpha_{i}~{}\beta^{i-1}~{}I_{1}^{i-1}\right]\\ \displaystyle\sigma_{22}&\displaystyle=-p+\cfrac{2C_{1}}{\lambda}\left[\sum_{i% =1}^{5}i~{}\alpha_{i}~{}\beta^{i-1}~{}I_{1}^{i-1}\right]=\sigma_{33}~{}.\end{aligned}
  40. σ 22 = σ 33 = 0 \sigma_{22}=\sigma_{33}=0
  41. p = 2 C 1 λ [ i = 1 5 i α i β i - 1 I 1 i - 1 ] . p=\cfrac{2C_{1}}{\lambda}\left[\sum_{i=1}^{5}i~{}\alpha_{i}~{}\beta^{i-1}~{}I_% {1}^{i-1}\right]~{}.
  42. σ 11 = 2 C 1 ( λ 2 - 1 λ ) [ i = 1 5 i α i β i - 1 I 1 i - 1 ] . \sigma_{11}=2C_{1}\left(\lambda^{2}-\cfrac{1}{\lambda}\right)\left[\sum_{i=1}^% {5}i~{}\alpha_{i}~{}\beta^{i-1}~{}I_{1}^{i-1}\right]~{}.
  43. λ - 1 \lambda-1\,
  44. T 11 = σ 11 / λ = 2 C 1 ( λ - 1 λ 2 ) [ i = 1 5 i α i β i - 1 I 1 i - 1 ] . T_{11}=\sigma_{11}/\lambda=2C_{1}\left(\lambda-\cfrac{1}{\lambda^{2}}\right)% \left[\sum_{i=1}^{5}i~{}\alpha_{i}~{}\beta^{i-1}~{}I_{1}^{i-1}\right]~{}.
  45. 𝐧 1 \mathbf{n}_{1}
  46. 𝐧 2 \mathbf{n}_{2}
  47. λ 1 = λ 2 = λ \lambda_{1}=\lambda_{2}=\lambda\,
  48. λ 1 λ 2 λ 3 = 1 \lambda_{1}~{}\lambda_{2}~{}\lambda_{3}=1
  49. λ 3 = 1 / λ 2 \lambda_{3}=1/\lambda^{2}\,
  50. I 1 = λ 1 2 + λ 2 2 + λ 3 2 = 2 λ 2 + 1 λ 4 . I_{1}=\lambda_{1}^{2}+\lambda_{2}^{2}+\lambda_{3}^{2}=2~{}\lambda^{2}+\cfrac{1% }{\lambda^{4}}~{}.
  51. s y m b o l B = λ 2 𝐧 1 𝐧 1 + λ 2 𝐧 2 𝐧 2 + 1 λ 4 𝐧 3 𝐧 3 . symbol{B}=\lambda^{2}~{}\mathbf{n}_{1}\otimes\mathbf{n}_{1}+\lambda^{2}~{}% \mathbf{n}_{2}\otimes\mathbf{n}_{2}+\cfrac{1}{\lambda^{4}}~{}\mathbf{n}_{3}% \otimes\mathbf{n}_{3}~{}.
  52. σ 11 = 2 C 1 ( λ 2 - 1 λ 4 ) [ i = 1 5 i α i β i - 1 I 1 i - 1 ] = σ 22 . \sigma_{11}=2C_{1}\left(\lambda^{2}-\cfrac{1}{\lambda^{4}}\right)\left[\sum_{i% =1}^{5}i~{}\alpha_{i}~{}\beta^{i-1}~{}I_{1}^{i-1}\right]=\sigma_{22}~{}.
  53. λ - 1 \lambda-1\,
  54. T 11 = σ 11 λ = 2 C 1 ( λ - 1 λ 5 ) [ i = 1 5 i α i β i - 1 I 1 i - 1 ] = T 22 . T_{11}=\cfrac{\sigma_{11}}{\lambda}=2C_{1}\left(\lambda-\cfrac{1}{\lambda^{5}}% \right)\left[\sum_{i=1}^{5}i~{}\alpha_{i}~{}\beta^{i-1}~{}I_{1}^{i-1}\right]=T% _{22}~{}.
  55. 𝐧 1 \mathbf{n}_{1}
  56. 𝐧 3 \mathbf{n}_{3}
  57. λ 1 = λ , λ 3 = 1 \lambda_{1}=\lambda,~{}\lambda_{3}=1
  58. λ 1 λ 2 λ 3 = 1 \lambda_{1}~{}\lambda_{2}~{}\lambda_{3}=1
  59. λ 2 = 1 / λ \lambda_{2}=1/\lambda\,
  60. I 1 = λ 1 2 + λ 2 2 + λ 3 2 = λ 2 + 1 λ 2 + 1 . I_{1}=\lambda_{1}^{2}+\lambda_{2}^{2}+\lambda_{3}^{2}=\lambda^{2}+\cfrac{1}{% \lambda^{2}}+1~{}.
  61. s y m b o l B = λ 2 𝐧 1 𝐧 1 + 1 λ 2 𝐧 2 𝐧 2 + 𝐧 3 𝐧 3 . symbol{B}=\lambda^{2}~{}\mathbf{n}_{1}\otimes\mathbf{n}_{1}+\cfrac{1}{\lambda^% {2}}~{}\mathbf{n}_{2}\otimes\mathbf{n}_{2}+\mathbf{n}_{3}\otimes\mathbf{n}_{3}% ~{}.
  62. σ 11 = 2 C 1 ( λ 2 - 1 λ 2 ) [ i = 1 5 i α i β i - 1 I 1 i - 1 ] ; σ 22 = 0 ; σ 33 = 2 C 1 ( 1 - 1 λ 2 ) [ i = 1 5 i α i β i - 1 I 1 i - 1 ] . \sigma_{11}=2C_{1}\left(\lambda^{2}-\cfrac{1}{\lambda^{2}}\right)\left[\sum_{i% =1}^{5}i~{}\alpha_{i}~{}\beta^{i-1}~{}I_{1}^{i-1}\right]~{};~{}~{}\sigma_{22}=% 0~{};~{}~{}\sigma_{33}=2C_{1}\left(1-\cfrac{1}{\lambda^{2}}\right)\left[\sum_{% i=1}^{5}i~{}\alpha_{i}~{}\beta^{i-1}~{}I_{1}^{i-1}\right]~{}.
  63. λ - 1 \lambda-1\,
  64. T 11 = σ 11 λ = 2 C 1 ( λ - 1 λ 3 ) [ i = 1 5 i α i β i - 1 I 1 i - 1 ] . T_{11}=\cfrac{\sigma_{11}}{\lambda}=2C_{1}\left(\lambda-\cfrac{1}{\lambda^{3}}% \right)\left[\sum_{i=1}^{5}i~{}\alpha_{i}~{}\beta^{i-1}~{}I_{1}^{i-1}\right]~{}.
  65. s y m b o l F = s y m b o l 1 + γ 𝐞 1 𝐞 2 symbol{F}=symbol{1}+\gamma~{}\mathbf{e}_{1}\otimes\mathbf{e}_{2}
  66. 𝐞 1 , 𝐞 2 \mathbf{e}_{1},\mathbf{e}_{2}
  67. γ = λ - 1 λ ; λ 1 = λ ; λ 2 = 1 λ ; λ 3 = 1 \gamma=\lambda-\cfrac{1}{\lambda}~{};~{}~{}\lambda_{1}=\lambda~{};~{}~{}% \lambda_{2}=\cfrac{1}{\lambda}~{};~{}~{}\lambda_{3}=1
  68. s y m b o l F = [ 1 γ 0 0 1 0 0 0 1 ] ; s y m b o l B = s y m b o l F \cdotsymbol F T = [ 1 + γ 2 γ 0 γ 1 0 0 0 1 ] symbol{F}=\begin{bmatrix}1&\gamma&0\\ 0&1&0\\ 0&0&1\end{bmatrix}~{};~{}~{}symbol{B}=symbol{F}\cdotsymbol{F}^{T}=\begin{% bmatrix}1+\gamma^{2}&\gamma&0\\ \gamma&1&0\\ 0&0&1\end{bmatrix}
  69. I 1 = tr ( s y m b o l B ) = 3 + γ 2 I_{1}=\mathrm{tr}(symbol{B})=3+\gamma^{2}
  70. s y m b o l σ = - p s y m b o l 1 + 2 C 1 [ i = 1 5 i α i β i - 1 ( 3 + γ 2 ) i - 1 ] s y m b o l B symbol{\sigma}=-p~{}symbol{\mathit{1}}+2C_{1}\left[\sum_{i=1}^{5}i~{}\alpha_{i% }~{}\beta^{i-1}~{}(3+\gamma^{2})^{i-1}\right]~{}symbol{B}
  71. N N
  72. l l
  73. r 0 = l N r_{0}=l\sqrt{N}
  74. d x 1 d x 2 d x 3 dx_{1}dx_{2}dx_{3}
  75. ( x 1 , x 2 , x 3 ) (x_{1},x_{2},x_{3})
  76. p ( x 1 , x 2 , x 3 ) = b 3 π 3 / 2 exp [ - b 2 ( x 1 2 + x 2 2 + x 3 2 ) ] ; b := 3 2 N l 2 p(x_{1},x_{2},x_{3})=\cfrac{b^{3}}{\pi^{3/2}}~{}\exp[-b^{2}(x_{1}^{2}+x_{2}^{2% }+x_{3}^{2})]~{};~{}~{}b:=\sqrt{\cfrac{3}{2Nl^{2}}}
  77. s = c - k B b 2 r 2 s=c-k_{B}b^{2}r^{2}
  78. c c
  79. n n
  80. Δ S = - 1 2 n k B ( λ 1 2 + λ 2 2 + λ 3 2 - 3 ) = - 1 2 n k B ( I 1 - 3 ) \Delta S=-\tfrac{1}{2}nk_{B}(\lambda_{1}^{2}+\lambda_{2}^{2}+\lambda_{3}^{2}-3% )=-\tfrac{1}{2}nk_{B}(I_{1}-3)
  81. W = - θ d S = 1 2 n k B θ ( I 1 - 3 ) W=-\theta\,dS=\tfrac{1}{2}nk_{B}\theta(I_{1}-3)
  82. θ \theta

Arturo_Rodas.html

  1. \infty

Ascending_chain_condition_on_principal_ideals.html

  1. ( X ) ( X / 2 ) ( X / 4 ) ( X / 8 ) , (X)\subset(X/2)\subset(X/4)\subset(X/8),...

Assumed_mean.html

  1. d i = x i - x 0 d_{i}=x_{i}-x_{0}\,
  2. A = i = 1 N d i A=\sum_{i=1}^{N}d_{i}\,
  3. B = i = 1 N d i 2 B=\sum_{i=1}^{N}d_{i}^{2}\,
  4. D = A N D=\frac{A}{N}\,
  5. x ¯ = x 0 + D \overline{x}=x_{0}+D\,
  6. σ = B - N D 2 N \sigma=\sqrt{\frac{B-ND^{2}}{N}}\,
  7. σ = B - N D 2 N - 1 \sigma=\sqrt{\frac{B-ND^{2}}{N-1}}\,
  8. x 0 + C S × A N = 175.5 + 3 × - 55 / 100 = 173.85 x_{0}+CS\times\frac{A}{N}=175.5+3\times-55/100=173.85
  9. C S B - A 2 N N - 1 = 5.57 CS\sqrt{\frac{B-\frac{A^{2}}{N}}{N-1}}=5.57

Asteroid_capture.html

  1. T T
  2. | V | |V|
  3. E = T + V E=T+V
  4. V V

Asymptotic_homogenization.html

  1. ( A ( x ϵ ) u ϵ ) = f \nabla\cdot\left(A\left(\frac{\vec{x}}{\epsilon}\right)\nabla u_{\epsilon}% \right)=f
  2. ϵ \epsilon
  3. A ( y ) A\left(\vec{y}\right)
  4. A ( y + e i ) = A ( y ) A\left(\vec{y}+\vec{e}_{i}\right)=A\left(\vec{y}\right)
  5. i = 1 , , n i=1,\dots,n
  6. ( A * u ) = f \nabla\cdot\left(A^{*}\nabla u\right)=f
  7. A * A^{*}
  8. A i j * = ( 0 , 1 ) n A ( y ) ( w j ( y ) + e j ) e i d y 1 d y n , i , j = 1 , , n A^{*}_{ij}=\int_{(0,1)^{n}}A(\vec{y})\left(\nabla w_{j}(\vec{y})+\vec{e}_{j}% \right)\cdot\vec{e}_{i}\,dy_{1}\dots dy_{n},\qquad i,j=1,\dots,n
  9. w j w_{j}
  10. y ( A ( y ) w j ) = - y ( A ( y ) e j ) . \nabla_{y}\cdot\left(A(\vec{y})\nabla w_{j}\right)=-\nabla_{y}\cdot\left(A(% \vec{y})\vec{e}_{j}\right).
  11. u ϵ u u_{\epsilon}\approx u
  12. ϵ \epsilon
  13. u ϵ u u_{\epsilon}\to u
  14. ϵ 0 \epsilon\to 0
  15. A * A^{*}
  16. y = x / ϵ \vec{y}=\vec{x}/\epsilon
  17. ϵ \epsilon
  18. u ϵ ( x ) = u ( x , y ) = u 0 ( x , y ) + ϵ u 1 ( x , y ) + ϵ 2 u 2 ( x , y ) + O ( ϵ 3 ) u_{\epsilon}(\vec{x})=u(\vec{x},\vec{y})=u_{0}(\vec{x},\vec{y})+\epsilon u_{1}% (\vec{x},\vec{y})+\epsilon^{2}u_{2}(\vec{x},\vec{y})+O(\epsilon^{3})\,
  19. u 1 ( x , x / ϵ ) u_{1}(\vec{x},\vec{x}/\epsilon)
  20. Γ \Gamma

Asymptotic_theory_(statistics).html

  1. n n\to\infty\,
  2. X ¯ n \scriptstyle\overline{X}_{n}
  3. X ¯ n \scriptstyle\overline{X}_{n}
  4. θ n = θ + h / n \scriptstyle\theta_{n}\,=\,\theta+h/\sqrt{n}
  5. θ = θ 0 + h / n \scriptstyle\theta\,=\,\theta_{0}+h/\sqrt{n}
  6. θ ^ n 𝑝 θ 0 \hat{\theta}_{n}\ \xrightarrow{p}\ \theta_{0}
  7. b n ( θ ^ n - a n ) 𝑑 G , b_{n}(\hat{\theta}_{n}-a_{n})\ \xrightarrow{d}\ G,
  8. θ ^ n \scriptstyle\hat{\theta}_{n}
  9. n ( θ ^ n - θ 0 ) 𝑑 𝒩 ( 0 , V ) . \sqrt{n}(\hat{\theta}_{n}-\theta_{0})\ \xrightarrow{d}\ \mathcal{N}(0,V).

Atiyah_conjecture.html

  1. l 2 l^{2}
  2. l 2 l^{2}
  3. l 2 l^{2}
  4. l 2 l^{2}
  5. l 2 l^{2}
  6. l 2 l^{2}
  7. l 2 l^{2}
  8. l 2 l^{2}
  9. l 2 l^{2}

ATLAS_of_Finite_Groups.html

  1. E n ( 2 ) E_{n}(2)

ATM_Adaptation_Layer_2.html

  1. G ( x ) = x 5 + x 2 + 1 G(x)=x^{5}+x^{2}+1

Aurifeuillean_factorization.html

  1. 2 4 k + 2 + 1 2^{4k+2}+1
  2. 2 4 k + 2 + 1 = ( 2 2 k + 1 - 2 k + 1 + 1 ) ( 2 2 k + 1 + 2 k + 1 + 1 ) . 2^{4k+2}+1=(2^{2k+1}-2^{k+1}+1)\cdot(2^{2k+1}+2^{k+1}+1).
  3. b n - 1 b^{n}-1
  4. Φ n ( b ) \Phi_{n}(b)
  5. b = s 2 t b=s^{2}\cdot t
  6. t t
  7. t 1 mod 4 t\equiv 1\mod 4
  8. n t ( mod 2 t ) ; n\equiv t\;\;(\mathop{{\rm mod}}2t);
  9. t 2 , 3 ( mod 4 ) t\equiv 2,3\;\;(\mathop{{\rm mod}}4)
  10. n 2 t ( mod 4 t ) . n\equiv 2t\;\;(\mathop{{\rm mod}}4t).
  11. Φ n ( b 2 k + 1 ) \Phi_{n}(b^{2k+1})
  12. b = s 2 t b=s^{2}\cdot t
  13. t t
  14. a 4 + 4 b 4 a^{4}+4b^{4}
  15. a 4 + 4 b 4 = ( a 2 - 2 a b + 2 b 2 ) ( a 2 + 2 a b + 2 b 2 ) . a^{4}+4b^{4}=(a^{2}-2ab+2b^{2})\cdot(a^{2}+2ab+2b^{2}).
  16. L 10 k + 5 L_{10k+5}
  17. L 10 k + 5 = L 2 k + 1 ( 5 F 2 k + 1 2 - 5 F 2 k + 1 + 1 ) ( 5 F 2 k + 1 2 + 5 F 2 k + 1 + 1 ) L_{10k+5}=L_{2k+1}\cdot(5{F_{2k+1}}^{2}-5F_{2k+1}+1)\cdot(5{F_{2k+1}}^{2}+5F_{% 2k+1}+1)
  18. 2 4 k + 2 + 1 2^{4k+2}+1
  19. 2 58 + 1 = 536838145 536903681. 2^{58}+1=536838145\cdot 536903681.\,\!
  20. 5 107367629. 5\cdot 107367629.

Authorship_of_the_Epistle_to_the_Hebrews.html

  1. 𝔓 \mathfrak{P}

Autoclave_(industrial).html

  1. d e l t a delta

Average_rectified_value.html

  1. k f = RMS ARV . k_{\mathrm{f}}=\frac{\mathrm{RMS}}{\mathrm{ARV}}.

Averaged_one-dependence_estimators.html

  1. P ^ ( y x 1 , x n ) = i : 1 i n F ( x i ) m P ^ ( y , x i ) j = 1 n P ^ ( x j y , x i ) y Y i : 1 i n F ( x i ) m P ^ ( y , x i ) j = 1 n P ^ ( x j y , x i ) \hat{P}(y\mid x_{1},\ldots x_{n})=\frac{\sum_{i:1\leq i\leq n\wedge F(x_{i})% \geq m}\hat{P}(y,x_{i})\prod_{j=1}^{n}\hat{P}(x_{j}\mid y,x_{i})}{\sum_{y^{% \prime}\in Y}\sum_{i:1\leq i\leq n\wedge F(x_{i})\geq m}\hat{P}(y^{\prime},x_{% i})\prod_{j=1}^{n}\hat{P}(x_{j}\mid y^{\prime},x_{i})}
  2. P ^ ( ) \hat{P}(\cdot)
  3. P ( ) P(\cdot)
  4. F ( ) F(\cdot)
  5. P ( y x 1 , x n ) = P ( y , x 1 , x n ) P ( x 1 , x n ) . P(y\mid x_{1},\ldots x_{n})=\frac{P(y,x_{1},\ldots x_{n})}{P(x_{1},\ldots x_{n% })}.
  6. 1 i n 1\leq i\leq n
  7. P ( y , x 1 , x n ) = P ( y , x i ) P ( x 1 , x n y , x i ) . P(y,x_{1},\ldots x_{n})=P(y,x_{i})P(x_{1},\ldots x_{n}\mid y,x_{i}).
  8. P ( y , x 1 , x n ) = P ( y , x i ) j = 1 n P ( x j y , x i ) . P(y,x_{1},\ldots x_{n})=P(y,x_{i})\prod_{j=1}^{n}P(x_{j}\mid y,x_{i}).
  9. O ( l n 2 ) O(ln^{2})
  10. O ( k n 2 ) O(kn^{2})

Averaging_argument.html

  1. N N
  2. B / 3 B/3
  3. M = ( N B ) / 3 M=(NB)/3
  4. N / 3 N/3
  5. N / 3 N/3
  6. B B
  7. ( N B ) / 3 (NB)/3
  8. M M
  9. \scriptstyle\blacksquare
  10. p : X × Y TRUE/FALSE p\colon X\times Y\to\,\text{TRUE/FALSE}
  11. f f
  12. 0 f 1 0\leq f\leq 1
  13. x X x\in X
  14. f f
  15. y Y y\in Y
  16. p ( x , y ) p(x,y)
  17. y Y y\in Y
  18. f f
  19. x X x\in X
  20. p ( x , y ) p(x,y)
  21. f f
  22. C C
  23. C ( x , y ) = f ( x ) C(x,y)=f(x)
  24. ρ \rho
  25. x x
  26. y y
  27. Y Y
  28. { 0 , 1 } m \{0,1\}^{m}
  29. y 0 { 0 , 1 } m y_{0}\in\{0,1\}^{m}
  30. Pr x [ C ( x , y 0 ) = f ( x ) ] ρ \Pr_{x}[C(x,y_{0})=f(x)]\geq\rho
  31. y y
  32. p y p_{y}
  33. Pr x [ C ( x , y ) = f ( x ) ] \Pr_{x}[C(x,y)=f(x)]
  34. Pr x , y [ C ( x , y ) = f ( x ) ] = E y [ p y ] \Pr_{x,y}[C(x,y)=f(x)]=E_{y}[p_{y}]\,
  35. Z Z
  36. E [ Z ] ρ E[Z]\geq\rho
  37. Pr [ Z ρ ] > 0 \Pr[Z\geq\rho]>0
  38. E [ Z ] E[Z]
  39. Z Z
  40. ρ \rho
  41. ρ \rho
  42. 𝒫 𝒫 𝒫 / poly \mathcal{BPP}\subsetneq\mathcal{P}/\,\text{poly}

B-admissible_representation.html

  1. D B ( V ) := ( B E V ) G D_{B}(V):=(B\otimes_{E}V)^{G}
  2. α B , V : B F D B ( V ) B E V \alpha_{B,V}:B\otimes_{F}D_{B}(V)\longrightarrow B\otimes_{E}V
  3. dim F D B ( V ) dim E V \dim_{F}D_{B}(V)\leq\dim_{E}V
  4. D = D K s ( V ) D=D_{K_{s}}(V)
  5. D K s D_{K_{s}}
  6. D B ( V ) := Hom G ( V , B ) D_{B}^{\ast}(V):=\mathrm{Hom}_{G}(V,B)

B-convex_space.html

  1. i = 1 n α i x i ( 1 - ε ) n . \left\|\sum_{i=1}^{n}\alpha_{i}x_{i}\right\|\leq(1-\varepsilon)n.
  2. p > 1 p>1

Backfitting_algorithm.html

  1. Y i = α + j = 1 p f j ( X i j ) + ϵ i Y_{i}=\alpha+\sum_{j=1}^{p}f_{j}(X_{ij})+\epsilon_{i}
  2. X 1 , X 2 , , X p X_{1},X_{2},\ldots,X_{p}
  3. p p
  4. X X
  5. Y Y
  6. ϵ \epsilon
  7. f j f_{j}
  8. X j X_{j}
  9. f j f_{j}
  10. α \alpha
  11. f j f_{j}
  12. α \alpha
  13. i = 1 N f j ( X i j ) = 0 \sum_{i=1}^{N}f_{j}(X_{ij})=0
  14. j j
  15. α = 1 / N i = 1 N y i \alpha=1/N\sum_{i=1}^{N}y_{i}
  16. α ^ = 1 / N i = 1 N y i , f j ^ 0 \hat{\alpha}=1/N\sum_{i=1}^{N}y_{i},\hat{f_{j}}\equiv 0
  17. j \forall j
  18. f j ^ \hat{f_{j}}
  19. f j ^ Smooth [ { y i - α ^ - k j f k ^ ( x i k ) } 1 N ] \hat{f_{j}}\leftarrow\,\text{Smooth}[\{y_{i}-\hat{\alpha}-\sum_{k\neq j}\hat{f% _{k}}(x_{ik})\}_{1}^{N}]
  20. f j ^ f j ^ - 1 / N i = 1 N f j ^ ( x i j ) \hat{f_{j}}\leftarrow\hat{f_{j}}-1/N\sum_{i=1}^{N}\hat{f_{j}}(x_{ij})
  21. Smooth \,\text{Smooth}
  22. min E [ Y - ( α + j = 1 p f j ( X j ) ) ] 2 \min E[Y-(\alpha+\sum_{j=1}^{p}f_{j}(X_{j}))]^{2}
  23. f i ( X i ) = E [ Y - ( α + j i p f j ( X j ) ) | X i ] f_{i}(X_{i})=E[Y-(\alpha+\sum_{j\neq i}^{p}f_{j}(X_{j}))|X_{i}]
  24. ( I P 1 P 1 P 2 I P 2 P p P p I ) ( f 1 ( X 1 ) f 2 ( X 2 ) f p ( X p ) ) = ( P 1 Y P 2 Y P p Y ) \begin{pmatrix}I&P_{1}&\cdots&P_{1}\\ P_{2}&I&\cdots&P_{2}\\ \vdots&&\ddots&\vdots\\ P_{p}&\cdots&P_{p}&I\end{pmatrix}\begin{pmatrix}f_{1}(X_{1})\\ f_{2}(X_{2})\\ \vdots\\ f_{p}(X_{p})\end{pmatrix}=\begin{pmatrix}P_{1}Y\\ P_{2}Y\\ \vdots\\ P_{p}Y\end{pmatrix}
  25. P i ( ) = E ( | X i ) P_{i}(\cdot)=E(\cdot|X_{i})
  26. S i S_{i}
  27. P i P_{i}
  28. S i Y S_{i}Y
  29. E ( Y | X ) E(Y|X)
  30. ( I S 1 S 1 S 2 I S 2 S p S p I ) ( f 1 f 2 f p ) = ( S 1 Y S 2 Y S p Y ) \begin{pmatrix}I&S_{1}&\cdots&S_{1}\\ S_{2}&I&\cdots&S_{2}\\ \vdots&&\ddots&\vdots\\ S_{p}&\cdots&S_{p}&I\end{pmatrix}\begin{pmatrix}f_{1}\\ f_{2}\\ \vdots\\ f_{p}\end{pmatrix}=\begin{pmatrix}S_{1}Y\\ S_{2}Y\\ \vdots\\ S_{p}Y\end{pmatrix}
  31. S ^ f = Q Y \hat{S}f=QY\,
  32. f i ( 0 ) f_{i}^{(0)}
  33. f i ( j ) f_{i}^{(j)}
  34. f i ^ ( j ) Smooth [ { y i - α ^ - k j f k ^ ( x i k ) } 1 N ] \hat{f_{i}}^{(j)}\leftarrow\,\text{Smooth}[\{y_{i}-\hat{\alpha}-\sum_{k\neq j}% \hat{f_{k}}(x_{ik})\}_{1}^{N}]
  35. f 1 = S 1 ( Y - f 2 ) , f 2 = S 2 ( Y - f 1 ) f_{1}=S_{1}(Y-f_{2}),f_{2}=S_{2}(Y-f_{1})
  36. f ^ 1 ( i ) \hat{f}_{1}^{(i)}
  37. f 1 f_{1}
  38. f ^ 1 ( i ) = S 1 [ Y - f ^ 2 ( i - 1 ) ] , f ^ 2 ( i ) = S 2 [ Y - f ^ 1 ( i - 1 ) ] \hat{f}_{1}^{(i)}=S_{1}[Y-\hat{f}_{2}^{(i-1)}],\hat{f}_{2}^{(i)}=S_{2}[Y-\hat{% f}_{1}^{(i-1)}]
  39. f ^ 1 ( i ) = Y - α = 0 i - 1 ( S 1 S 2 ) α ( I - S 1 ) Y - ( S 1 S 2 ) i - 1 S 1 f ^ 2 ( 0 ) \hat{f}_{1}^{(i)}=Y-\sum_{\alpha=0}^{i-1}(S_{1}S_{2})^{\alpha}(I-S_{1})Y-(S_{1% }S_{2})^{i-1}S_{1}\hat{f}_{2}^{(0)}
  40. f ^ 2 ( i ) = S 2 α = 0 i - 1 ( S 1 S 2 ) α ( I - S 1 ) Y + S 2 ( S 1 S 2 ) i - 1 S 1 f ^ 2 ( 0 ) \hat{f}_{2}^{(i)}=S_{2}\sum_{\alpha=0}^{i-1}(S_{1}S_{2})^{\alpha}(I-S_{1})Y+S_% {2}(S_{1}S_{2})^{i-1}S_{1}\hat{f}_{2}^{(0)}
  41. α \alpha
  42. f ^ 2 ( 0 ) = 0 \hat{f}_{2}^{(0)}=0
  43. f ^ 1 ( i ) = [ I - α = 0 i - 1 ( S 1 S 2 ) α ( I - S 1 ) ] Y \hat{f}_{1}^{(i)}=[I-\sum_{\alpha=0}^{i-1}(S_{1}S_{2})^{\alpha}(I-S_{1})]Y
  44. f ^ 2 ( i ) = [ S 2 α = 0 i - 1 ( S 1 S 2 ) α ( I - S 1 ) ] Y \hat{f}_{2}^{(i)}=[S_{2}\sum_{\alpha=0}^{i-1}(S_{1}S_{2})^{\alpha}(I-S_{1})]Y
  45. S 1 S 2 < 1 \|S_{1}S_{2}\|<1
  46. X i X_{i}
  47. b b
  48. S ^ b = 0 \hat{S}b=0
  49. f ^ \hat{f}
  50. f ^ + α b \hat{f}+\alpha b
  51. α \alpha\in\mathbb{R}
  52. 𝒱 1 ( S i ) \mathcal{V}_{1}(S_{i})
  53. S ^ b = 0 \hat{S}b=0
  54. b i 𝒱 1 ( S i ) i = 1 , , p b_{i}\in\mathcal{V}_{1}(S_{i})\forall i=1,\dots,p
  55. i = 1 p b i = 0. \sum_{i=1}^{p}b_{i}=0.
  56. A A
  57. 𝒱 1 ( S 1 ) + + 𝒱 1 ( S p ) \mathcal{V}_{1}(S_{1})+\dots+\mathcal{V}_{1}(S_{p})
  58. α ^ = 1 / N 1 N y i , f j ^ 0 \hat{\alpha}=1/N\sum_{1}^{N}y_{i},\hat{f_{j}}\equiv 0
  59. i , j \forall i,j
  60. f + ^ = α + f 1 ^ + + f p ^ \hat{f_{+}}=\alpha+\hat{f_{1}}+\dots+\hat{f_{p}}
  61. f j ^ \hat{f_{j}}
  62. y - f + ^ y-\hat{f_{+}}
  63. 𝒱 1 ( S i ) + + 𝒱 1 ( S p ) \mathcal{V}_{1}(S_{i})+\dots+\mathcal{V}_{1}(S_{p})
  64. a = A ( Y - f + ^ ) a=A(Y-\hat{f_{+}})
  65. ( Y - a ) (Y-a)
  66. ( I - A i ) S i (I-A_{i})S_{i}
  67. f j ^ \hat{f_{j}}

Bagger–Lambert–Gustavsson_action.html

  1. S = ( - 1 2 D μ X I D μ X I + i 2 Ψ ¯ Γ μ D μ Ψ + i 4 Ψ ¯ Γ I J [ X I , X J , Ψ ] - 1 12 [ X I , X J , X K ] [ X I , X J , X K ] + 1 2 ε a b c T r ( A a b A c + 2 3 A a A b A c ) ) d σ 3 S=\int{\left(-\frac{1}{2}D^{\mu}X_{I}D_{\mu}X_{I}+\frac{i}{2}\overline{\Psi}% \Gamma^{\mu}D_{\mu}\Psi+\frac{i}{4}\overline{\Psi}\Gamma_{IJ}\left[X^{I},X^{J}% ,\Psi\right]-\frac{1}{12}\left[X^{I},X^{J},X^{K}\right]\left[X^{I},X^{J},X^{K}% \right]+\frac{1}{2}\varepsilon^{abc}Tr(A_{a}\partial_{b}A_{c}+\frac{2}{3}A_{a}% A_{b}A_{c})\right)}d\sigma^{3}
  2. [ A , B , C ] η ε μ ν τ η A μ B ν C τ \left[A,B,C\right]_{\eta}\equiv\varepsilon^{\mu\nu\tau\eta}A_{\mu}B_{\nu}C_{\tau}

Bailey–Borwein–Plouffe_formula.html

  1. π \pi
  2. π \pi
  3. π = k = 0 [ 1 16 k ( 4 8 k + 1 - 2 8 k + 4 - 1 8 k + 5 - 1 8 k + 6 ) ] \pi=\sum_{k=0}^{\infty}\left[\frac{1}{16^{k}}\left(\frac{4}{8k+1}-\frac{2}{8k+% 4}-\frac{1}{8k+5}-\frac{1}{8k+6}\right)\right]
  4. π \pi
  5. α = k = 0 [ 1 b k p ( k ) q ( k ) ] \alpha=\sum_{k=0}^{\infty}\left[\frac{1}{b^{k}}\frac{p(k)}{q(k)}\right]
  6. P ( s , b , m , A ) = k = 0 [ 1 b k j = 1 m a j ( m k + j ) s ] P(s,b,m,A)=\sum_{k=0}^{\infty}\left[\frac{1}{b^{k}}\sum_{j=1}^{m}\frac{a_{j}}{% (mk+j)^{s}}\right]
  7. A = ( a 1 , a 2 , , a m ) A=(a_{1},a_{2},\dots,a_{m})
  8. ln 10 9 \displaystyle\ln\frac{10}{9}
  9. ln 2 \displaystyle\ln 2
  10. arctan 1 b \displaystyle\arctan\frac{1}{b}
  11. π \pi
  12. π \pi
  13. π \pi
  14. π \displaystyle\pi
  15. π = k = 0 [ 1 16 k ( 120 k 2 + 151 k + 47 512 k 4 + 1024 k 3 + 712 k 2 + 194 k + 15 ) ] . \pi=\sum_{k=0}^{\infty}\left[\frac{1}{16^{k}}\left(\frac{120k^{2}+151k+47}{512% k^{4}+1024k^{3}+712k^{2}+194k+15}\right)\right].
  16. π \pi
  17. π \pi
  18. π \pi
  19. π = 4 k = 0 1 ( 16 k ) ( 8 k + 1 ) - 2 k = 0 1 ( 16 k ) ( 8 k + 4 ) - k = 0 1 ( 16 k ) ( 8 k + 5 ) - k = 0 1 ( 16 k ) ( 8 k + 6 ) . \pi=4\sum_{k=0}^{\infty}\frac{1}{(16^{k})(8k+1)}-2\sum_{k=0}^{\infty}\frac{1}{% (16^{k})(8k+4)}-\sum_{k=0}^{\infty}\frac{1}{(16^{k})(8k+5)}-\sum_{k=0}^{\infty% }\frac{1}{(16^{k})(8k+6)}.\!
  20. k = 0 1 ( 16 k ) ( 8 k + 1 ) = k = 0 n 1 ( 16 k ) ( 8 k + 1 ) + k = n + 1 1 ( 16 k ) ( 8 k + 1 ) . \sum_{k=0}^{\infty}\frac{1}{(16^{k})(8k+1)}=\sum_{k=0}^{n}\frac{1}{(16^{k})(8k% +1)}+\sum_{k=n+1}^{\infty}\frac{1}{(16^{k})(8k+1)}.\!
  21. k = 0 16 n - k 8 k + 1 = k = 0 n 16 n - k 8 k + 1 + k = n + 1 16 n - k 8 k + 1 . \sum_{k=0}^{\infty}\frac{16^{n-k}}{8k+1}=\sum_{k=0}^{n}\frac{16^{n-k}}{8k+1}+% \sum_{k=n+1}^{\infty}\frac{16^{n-k}}{8k+1}.\!
  22. k = 0 n 16 n - k mod ( 8 k + 1 ) 8 k + 1 + k = n + 1 16 n - k 8 k + 1 . \sum_{k=0}^{n}\frac{16^{n-k}\bmod(8k+1)}{8k+1}+\sum_{k=n+1}^{\infty}\frac{16^{% n-k}}{8k+1}.\!
  23. π \pi
  24. 4 Σ 1 - 2 Σ 2 - Σ 3 - Σ 4 . 4\Sigma_{1}-2\Sigma_{2}-\Sigma_{3}-\Sigma_{4}.\,\!
  25. π \pi
  26. π \pi
  27. π \pi
  28. π \pi
  29. π 3 \pi^{3}
  30. log 3 2 \log^{3}2
  31. ζ ( 3 ) \zeta(3)
  32. ζ ( x ) \zeta(x)
  33. π 4 \pi^{4}
  34. log 4 2 \log^{4}2
  35. log 5 2 \log^{5}2
  36. ζ ( 5 ) \zeta(5)
  37. π \pi
  38. log 2 \log 2
  39. π \pi

Balaban_10-cage.html

  1. ( x - 3 ) ( x - 2 ) ( x - 1 ) 8 x 2 ( x + 1 ) 8 ( x + 2 ) ( x + 3 ) ( x 2 - 6 ) 2 ( x 2 - 5 ) 4 ( x 2 - 2 ) 2 ( x 4 - 6 x 2 + 3 ) 8 (x-3)(x-2)(x-1)^{8}x^{2}(x+1)^{8}(x+2)(x+3)(x^{2}-6)^{2}(x^{2}-5)^{4}(x^{2}-2)% ^{2}(x^{4}-6x^{2}+3)^{8}

Balaban_11-cage.html

  1. ( x - 3 ) x 12 ( x 2 - 6 ) 5 ( x 2 - 2 ) 12 ( x 3 - x 2 - 4 x + 2 ) 2 (x-3)x^{12}(x^{2}-6)^{5}(x^{2}-2)^{12}(x^{3}-x^{2}-4x+2)^{2}\cdot
  2. ( x 3 + x 2 - 6 x - 2 ) ( x 4 - x 3 - 6 x 2 + 4 x + 4 ) 4 ( x 5 + x 4 - 8 x 3 - 6 x 2 + 12 x + 4 ) 8 \cdot(x^{3}+x^{2}-6x-2)(x^{4}-x^{3}-6x^{2}+4x+4)^{4}(x^{5}+x^{4}-8x^{3}-6x^{2}% +12x+4)^{8}

Balance_equation.html

  1. π \scriptstyle{\pi}
  2. j S { i } π i q i j = j S { i } π j q j i . \sum_{j\in S\setminus\{i\}}\pi_{i}q_{ij}=\sum_{j\in S\setminus\{i\}}\pi_{j}q_{% ji}.
  3. π i q i j \pi_{i}q_{ij}
  4. π \pi
  5. j S { i } π i p i j = j S { i } π j p j i . \sum_{j\in S\setminus\{i\}}\pi_{i}p_{ij}=\sum_{j\in S\setminus\{i\}}\pi_{j}p_{% ji}.
  6. π i \pi_{i}
  7. π i q i j = π j q j i \pi_{i}q_{ij}=\pi_{j}q_{ji}
  8. π \pi
  9. π \pi
  10. π i p i j = π j p j i . \pi_{i}p_{ij}=\pi_{j}p_{ji}.
  11. π \pi

Balanced_polygamma_function.html

  1. ψ ( z , q ) = ζ ( z + 1 , q ) + ( ψ ( - z ) + γ ) ζ ( z + 1 , q ) Γ ( - z ) \psi(z,q)=\frac{\zeta^{\prime}(z+1,q)+(\psi(-z)+\gamma)\zeta(z+1,q)}{\Gamma(-z% )}\,
  2. ψ ( z , q ) = e - γ z z ( e γ z ζ ( z + 1 , q ) Γ ( - z ) ) , \psi(z,q)=e^{-\gamma z}\frac{\partial}{\partial z}\left(e^{\gamma z}\frac{% \zeta(z+1,q)}{\Gamma(-z)}\right),
  3. ψ ( z ) \psi(z)
  4. ζ ( z , q ) , \zeta(z,q),
  5. f ( 0 ) = f ( 1 ) f(0)=f(1)
  6. 0 1 f ( x ) d x = 0 \int_{0}^{1}f(x)dx=0
  7. ψ ( x ) = ψ ( 0 , x ) \psi(x)=\psi(0,x)\,
  8. ψ ( n ) ( x ) = ψ ( n , x ) ( n ) \psi^{(n)}(x)=\psi(n,x)\,\,\,(n\in\mathbb{N})
  9. Γ ( x ) = e ψ ( - 1 , x ) + 1 2 ln ( 2 π ) \Gamma(x)=e^{\psi(-1,x)+\frac{1}{2}\ln(2\pi)}\,\,\,
  10. ζ ( z , q ) = Γ ( 1 - z ) ( 2 - z ( ψ ( z - 1 , q 2 + 1 2 ) + ψ ( z - 1 , q 2 ) ) - ψ ( z - 1 , q ) ) ln ( 2 ) \zeta(z,q)=\frac{\Gamma(1-z)\left(2^{-z}\left(\psi\left(z-1,\frac{q}{2}+\frac{% 1}{2}\right)+\psi\left(z-1,\frac{q}{2}\right)\right)-\psi(z-1,q)\right)}{\ln(2)}
  11. ζ ( - 1 , x ) = ψ ( - 2 , x ) + x 2 2 - x 2 + 1 12 \zeta^{\prime}(-1,x)=\psi(-2,x)+\frac{x^{2}}{2}-\frac{x}{2}+\frac{1}{12}
  12. B n ( q ) = - Γ ( n + 1 ) ( 2 n - 1 ( ψ ( - n , q 2 + 1 2 ) + ψ ( - n , q 2 ) ) - ψ ( - n , q ) ) ln ( 2 ) B_{n}(q)=-\frac{\Gamma(n+1)\left(2^{n-1}\left(\psi\left(-n,\frac{q}{2}+\frac{1% }{2}\right)+\psi\left(-n,\frac{q}{2}\right)\right)-\psi(-n,q)\right)}{\ln(2)}
  13. B n ( q ) B_{n}(q)
  14. K ( z ) = A e ψ ( - 2 , z ) + z 2 - z 2 K(z)=Ae^{\psi(-2,z)+\frac{z^{2}-z}{2}}
  15. ψ ( - 2 , 1 4 ) = 1 8 ln ( 2 π ) + 9 8 ln A + G 4 π , \psi\left(-2,\frac{1}{4}\right)=\frac{1}{8}\ln(2\pi)+\frac{9}{8}\ln A+\frac{G}% {4\pi},
  16. A A
  17. G G
  18. ψ ( - 2 , 1 2 ) = 1 4 ln π + 3 2 ln A + 5 24 ln 2 \psi\left(-2,\frac{1}{2}\right)=\frac{1}{4}\ln\pi+\frac{3}{2}\ln A+\frac{5}{24% }\ln 2
  19. ψ ( - 2 , 1 ) = 1 2 ln ( 2 π ) \psi(-2,1)=\frac{1}{2}\ln(2\pi)
  20. ψ ( - 2 , 2 ) = ln ( 2 π ) - 1 \psi(-2,2)=\ln(2\pi)-1
  21. ψ ( - 3 , 1 2 ) = 1 16 ln ( 2 π ) + 1 2 ln A + 7 ζ ( 3 ) 32 π 2 \psi\left(-3,\frac{1}{2}\right)=\frac{1}{16}\ln(2\pi)+\frac{1}{2}\ln A+\frac{7% \,\zeta(3)}{32\,\pi^{2}}
  22. ψ ( - 3 , 1 ) = 1 4 ln ( 2 π ) + ln A \psi(-3,1)=\frac{1}{4}\ln(2\pi)+\ln A
  23. ψ ( - 3 , 2 ) = ln ( 2 π ) + 2 ln A - 3 4 \psi(-3,2)=\ln(2\pi)+2\ln A-\frac{3}{4}

Baranyai's_theorem.html

  1. 2 r < k 2\leq r<k
  2. K r k K^{k}_{r}
  3. K r k K^{k}_{r}
  4. ( k r ) r k {\left({{k}\atop{r}}\right)}\frac{r}{k}

Barlow's_formula.html

  1. P = 2 S t D P=\frac{2St}{D}

Barlow's_law.html

  1. I A L I\propto\sqrt{\frac{A}{L}}
  2. I d L I\propto\frac{d}{\sqrt{L}}
  3. I A L V I\propto\frac{A}{L}\cdot V

Barnes_interpolation.html

  1. w i j = exp ( - r m 2 κ ) , w_{ij}=\exp\left(-\frac{r_{m}^{2}}{\kappa}\right),\,
  2. κ \kappa
  3. κ = 5.052 ( 2 Δ n π ) 2 . \kappa=5.052\,\left(\frac{2\,\Delta n}{\pi}\right)^{2}.\,
  4. f k ( x , y ) f_{k}(x,y)
  5. g 0 ( x i , y j ) = k w i j f k ( x , y ) k w i j . g_{0}(x_{i},y_{j})=\frac{\displaystyle\sum_{k}w_{ij}f_{k}(x,y)}{\displaystyle% \sum_{k}w_{ij}}.
  6. g 1 ( x i , y j ) = g 0 ( x i , y j ) + ( f ( x , y ) - g 0 ( x , y ) ) exp ( - r m 2 γ κ ) . g_{1}(x_{i},y_{j})=g_{0}(x_{i},y_{j})+\sum(f(x,y)-g_{0}(x,y))\exp\left(-\frac{% r_{m}^{2}}{\gamma\kappa}\right).\,
  7. γ \gamma
  8. γ \gamma

Barnes_zeta_function.html

  1. ζ N ( s , w | a 1 , , a N ) = n 1 , , n N 0 1 ( w + n 1 a 1 + + n N a N ) s \zeta_{N}(s,w|a_{1},...,a_{N})=\sum_{n_{1},\dots,n_{N}\geq 0}\frac{1}{(w+n_{1}% a_{1}+\cdots+n_{N}a_{N})^{s}}

Barnsley_fern.html

  1. f ( x , y ) = [ a b c d ] [ x y ] + [ e f ] f(x,y)=\begin{bmatrix}a&b\\ c&d\end{bmatrix}\begin{bmatrix}x\\ y\end{bmatrix}+\begin{bmatrix}e\\ f\end{bmatrix}
  2. f ( x , y ) = [ 0.00 0.00 0.00 0.16 ] [ x y ] f(x,y)=\begin{bmatrix}\ 0.00&\ 0.00\\ 0.00&\ 0.16\end{bmatrix}\begin{bmatrix}\ x\\ y\end{bmatrix}
  3. f ( x , y ) = [ 0.85 0.04 - 0.04 0.85 ] [ x y ] + [ 0.00 1.60 ] f(x,y)=\begin{bmatrix}\ 0.85&\ 0.04\\ -0.04&\ 0.85\end{bmatrix}\begin{bmatrix}\ x\\ y\end{bmatrix}+\begin{bmatrix}\ 0.00\\ 1.60\end{bmatrix}
  4. f ( x , y ) = [ 0.20 - 0.26 0.23 0.22 ] [ x y ] + [ 0.00 1.60 ] f(x,y)=\begin{bmatrix}\ 0.20&\ -0.26\\ 0.23&\ 0.22\end{bmatrix}\begin{bmatrix}\ x\\ y\end{bmatrix}+\begin{bmatrix}\ 0.00\\ 1.60\end{bmatrix}
  5. f ( x , y ) = [ - 0.15 0.28 0.26 0.24 ] [ x y ] + [ 0.00 0.44 ] f(x,y)=\begin{bmatrix}\ -0.15&\ 0.28\\ 0.26&\ 0.24\end{bmatrix}\begin{bmatrix}\ x\\ y\end{bmatrix}+\begin{bmatrix}\ 0.00\\ 0.44\end{bmatrix}

Barrett_reduction.html

  1. c = a mod n . c=a\,\bmod\,n.\,
  2. n n
  3. a < n 2 a<n^{2}
  4. m = 1 / n m=1/n
  5. n n
  6. a mod n = a - a m n a\,\bmod\,n=a-\lfloor am\rfloor n
  7. x \lfloor x\rfloor
  8. 2 k > n 2^{k}>n
  9. n 2 - k n2^{-k}
  10. m = 4 k / n m=\lfloor 4^{k}/n\rfloor
  11. m 2 - k ( n 2 - k ) - 1 m2^{-k}\approx\left(n2^{-k}\right)^{-1}
  12. q = m a 4 k q=\left\lfloor\frac{ma}{4^{k}}\right\rfloor
  13. r = a - q n r=a-qn
  14. q q
  15. r a ( mod n ) r\equiv a\;\;(\mathop{{\rm mod}}n)
  16. a < n 2 a<n^{2}
  17. r < 2 n r<2n
  18. a mod n = { r if r < n r - n otherwise a\,\bmod\,n=\begin{cases}r&\,\text{if }r<n\\ r-n&\,\text{otherwise}\end{cases}

Barth–Nieto_quintic.html

  1. x 0 + x 1 + x 2 + x 3 + x 4 + x 5 = 0 \displaystyle x_{0}+x_{1}+x_{2}+x_{3}+x_{4}+x_{5}=0
  2. x 0 - 1 + x 1 - 1 + x 2 - 1 + x 3 - 1 + x 4 - 1 + x 5 - 1 = 0. \displaystyle x_{0}^{-1}+x_{1}^{-1}+x_{2}^{-1}+x_{3}^{-1}+x_{4}^{-1}+x_{5}^{-1% }=0.

Baryon_acoustic_oscillations.html

  1. a ( t ) a(t)
  2. a ( t ) ( 1 + z ( t ) ) - 1 a(t)\equiv(1+z(t))^{-1}\!
  3. H ( z ) H(z)
  4. H ( t ) a ˙ a H(t)\equiv\frac{\dot{a}}{a}\!
  5. a ˙ \dot{a}
  6. G N G_{N}
  7. P P
  8. ρ \rho\!
  9. k k
  10. Λ \Lambda\!
  11. H 2 = ( a ˙ a ) 2 = 8 π G 3 ρ - k c 2 a 2 + Λ c 2 3 H^{2}=\left(\frac{\dot{a}}{a}\right)^{2}=\frac{8\pi G}{3}\rho-\frac{kc^{2}}{a^% {2}}+\frac{\Lambda c^{2}}{3}
  12. H ˙ + H 2 = a ¨ a = - 4 π G 3 ( ρ + 3 p c 2 ) + Λ c 2 3 \dot{H}+H^{2}=\frac{\ddot{a}}{a}=-\frac{4\pi G}{3}\left(\rho+\frac{3p}{c^{2}}% \right)+\frac{\Lambda c^{2}}{3}
  13. a ¨ > 0 \ddot{a}>0
  14. w = P ρ < - 1 / 3 w=\frac{P}{\rho}<-1/3\!
  15. Λ \Lambda\!
  16. Ω \Omega\!
  17. x x
  18. x x
  19. ρ c \rho_{c}\!
  20. ρ c = 3 H 2 8 π G \rho_{c}=\frac{3H^{2}}{8\pi G}
  21. Ω x ρ x ρ c = 8 π G ρ x 3 H 2 \Omega_{x}\equiv\frac{\rho_{x}}{\rho_{c}}=\frac{8\pi G\rho_{x}}{3H^{2}}
  22. H 2 ( a ) = ( a ˙ a ) 2 = H 0 2 [ Ω m a - 3 + Ω r a - 4 + Ω k a - 2 + Ω Λ a - 3 ( 1 + w ) ] H^{2}(a)=\left(\frac{\dot{a}}{a}\right)^{2}=H_{0}^{2}\left[\Omega_{m}a^{-3}+% \Omega_{r}a^{-4}+\Omega_{k}a^{-2}+\Omega_{\Lambda}a^{-3(1+w)}\right]
  23. H ( z ) H(z)
  24. d A ( z ) d_{A}(z)
  25. ( z ) (z)
  26. Δ θ \Delta\theta
  27. Δ χ \Delta\chi
  28. Δ θ = Δ χ d A ( z ) \Delta\theta=\frac{\Delta\chi}{d_{A}(z)}\!
  29. d A ( z ) 0 z d z H ( z ) d_{A}(z)\propto\int_{0}^{z}\frac{dz^{\prime}}{H(z^{\prime})}\!
  30. Δ z \Delta z
  31. c Δ z = H ( z ) Δ χ c\Delta z=H(z)\Delta\chi\!

Basel_III.html

  1. CET1 RWAs 4.5 % \frac{\mbox{CET1}~{}}{\mbox{RWAs}~{}}\geq 4.5\%
  2. Tier 1 Capital Total exposure 3 % \frac{\mbox{Tier 1 Capital}~{}}{\mbox{Total exposure}~{}}\geq 3\%
  3. L C R = High quality liquid assets Total net liquidity outflows over 30 days 100 % LCR=\frac{\mbox{High quality liquid assets}~{}}{\mbox{Total net liquidity % outflows over 30 days}~{}}\geq 100\%

Batchelor_scale.html

  1. λ B = ( η S c 1 / 2 ) = ( ν D 2 ϵ ) 1 4 \lambda_{B}=\left(\frac{\eta}{Sc^{1/2}}\right)=\left(\frac{\nu{}D^{2}}{% \epsilon}\right)^{\frac{1}{4}}

Bateman_function.html

  1. k n ( x ) = 2 π 0 π / 2 cos ( x tan θ - n θ ) d θ \displaystyle k_{n}(x)=\frac{2}{\pi}\int_{0}^{\pi/2}\cos(x\tan\theta-n\theta)% \,d\theta

Battle_of_Camp_Hill.html

  1. } \Big\}
  2. } \Big\}
  3. } \Big\}

Bcrypt.html

  1. \leftarrow
  2. \leftarrow
  3. \leftarrow
  4. \leftarrow
  5. π \pi
  6. \leftarrow
  7. \oplus
  8. \leftarrow
  9. \leftarrow
  10. \leftarrow
  11. \leftarrow
  12. \oplus
  13. \leftarrow
  14. \leftarrow
  15. \leftarrow
  16. \oplus
  17. \leftarrow
  18. \leftarrow
  19. \leftarrow
  20. \leftarrow
  21. \leftarrow

Beckstrom's_law.html

  1. i = 1 n V i , j = i = 1 n k = 1 m B i , j , k - C i , j , k ( 1 + r k ) t k \sum_{i=1}^{n}V_{i,j}=\sum_{i=1}^{n}\sum_{k=1}^{m}\frac{B_{i,j,k}-C_{i,j,k}}{(% 1+r_{k})^{t_{k}}}
  2. V i , j \sum{V_{i,j}}

Behavior_of_coupled_DEVS.html

  1. N N
  2. M M
  3. N N
  4. M M
  5. M M
  6. N = < X , Y , D , { M i } , C x x , C y x , C y y , S e l e c t > N=<X,Y,D,\{M_{i}\},C_{xx},C_{yx},C_{yy},Select>
  7. M = < X , Y , S , s 0 , t a , δ e x t , δ i n t , λ Align g t ; M=<X,Y,S,s_{0},ta,\delta_{ext},\delta_{int},\lambda&gt;
  8. X X
  9. Y Y
  10. S = × i D Q i S=\underset{i\in D}{\times}Q_{i}
  11. Q i = { ( s i , t e i ) | s i S i , t e i ( 𝕋 [ 0 , t a i ( s i ) ] ) } Q_{i}=\{(s_{i},t_{ei})|s_{i}\in S_{i},t_{ei}\in(\mathbb{T}\cap[0,ta_{i}(s_{i})% ])\}
  12. i D i\in D
  13. 𝕋 = [ 0 , ) \mathbb{T}=[0,\infty)
  14. s 0 = × i D q 0 i s_{0}=\underset{i\in D}{\times}q_{0i}
  15. q 0 i = ( s 0 i , 0 ) q_{0i}=(s_{0i},0)
  16. i D i\in D
  17. t a : S 𝕋 ta:S\rightarrow\mathbb{T}^{\infty}
  18. 𝕋 = [ 0 , ] \mathbb{T}^{\infty}=[0,\infty]
  19. s = ( , ( s i , t e i ) , ) s=(\ldots,(s_{i},t_{ei}),\ldots)
  20. t a ( s ) = min { t a i ( s i ) - t e i | i D } . ta(s)=\min\{ta_{i}(si)-t_{ei}|i\in D\}.
  21. δ e x t : Q × X S \delta_{ext}:Q\times X\rightarrow S
  22. q = ( s , t e ) q=(s,t_{e})
  23. s = ( , ( s i , t e i ) , ) , t e ( 𝕋 [ 0 , t a ( s ) ] ) s=(\ldots,(s_{i},t_{ei}),\ldots),t_{e}\in(\mathbb{T}\cap[0,ta(s)])
  24. x X x\in X
  25. δ e x t ( q , x ) = s = ( , ( s i , t e i ) , ) \delta_{ext}(q,x)=s^{\prime}=(\ldots,(s_{i}^{\prime},t_{ei}^{\prime}),\ldots)
  26. ( s i , t e i ) = { ( δ e x t ( s i , t e i , x i ) , 0 ) if ( x , x i ) C x x ( s i , t e i ) otherwise . (s_{i}^{\prime},t_{ei}^{\prime})=\begin{cases}(\delta_{ext}(s_{i},t_{ei},x_{i}% ),0)&\,\text{if }(x,x_{i})\in C_{xx}\\ (s_{i},t_{ei})&\,\text{otherwise}.\end{cases}
  27. s = ( , ( s i , t e i ) , ) S s=(\ldots,(s_{i},t_{ei}),\ldots)\in S
  28. I M M ( s ) = { i D | t a i ( s i ) = t a ( s ) } IMM(s)=\{i\in D|ta_{i}(s_{i})=ta(s)\}
  29. i * D i^{*}\in D
  30. i * = S e l e c t ( I M M ( s ) ) . i^{*}=Select(IMM(s)).
  31. δ i n t : S S \delta_{int}:S\rightarrow S
  32. s = ( , ( s i , t e i ) , ) s=(\ldots,(s_{i},t_{ei}),\ldots)
  33. δ i n t ( s ) = s = ( , ( s i , t e i ) , ) \delta_{int}(s)=s^{\prime}=(\ldots,(s_{i}^{\prime},t_{ei}^{\prime}),\ldots)
  34. ( s i , t e i ) = { ( δ i n t ( s i ) , 0 ) if i = i * ( δ e x t ( s i , t e i , x i ) , 0 ) if ( λ i * ( s i * ) , x i ) C y x ( s i , t e i ) otherwise . (s_{i}^{\prime},t_{ei}^{\prime})=\begin{cases}(\delta_{int}(s_{i}),0)&\,\text{% if }i=i^{*}\\ (\delta_{ext}(s_{i},t_{ei},x_{i}),0)&\,\text{if }(\lambda_{i^{*}}(s_{i^{*}}),x% _{i})\in C_{yx}\\ (s_{i},t_{ei})&\,\text{otherwise}.\end{cases}
  35. λ : S Y ϕ \lambda:S\rightarrow Y^{\phi}
  36. s = ( , ( s i , t e i ) , ) s=(\ldots,(s_{i},t_{ei}),\ldots)
  37. λ ( s ) = \lambda(s)=
  38. N = < X , Y , D , { M i } , C x x , C y x , C y y , S e l e c t > N=<X,Y,D,\{M_{i}\},C_{xx},C_{yx},C_{yy},Select>
  39. M = < X , Y , S , s 0 , t a , δ e x t , δ i n t , λ Align g t ; M=<X,Y,S,s_{0},ta,\delta_{ext},\delta_{int},\lambda&gt;
  40. X X
  41. Y Y
  42. S = × i D Q i S=\underset{i\in D}{\times}Q_{i}
  43. Q i = { ( s i , t s i , t e i ) | s i S i , t s i 𝕋 , t e i ( 𝕋 [ 0 , t s i ] ) } Q_{i}=\{(s_{i},t_{si},t_{ei})|s_{i}\in S_{i},t_{si}\in\mathbb{T}^{\infty},t_{% ei}\in(\mathbb{T}\cap[0,t_{si}])\}
  44. i D i\in D
  45. s 0 = × i D q 0 i s_{0}=\underset{i\in D}{\times}q_{0i}
  46. q 0 i = ( s 0 i , t a i ( s 0 i ) , 0 ) q_{0i}=(s_{0i},ta_{i}(s_{0i}),0)
  47. i D i\in D
  48. t a : S 𝕋 ta:S\rightarrow\mathbb{T}^{\infty}
  49. s = ( , ( s i , t s i , t e i ) , ) s=(\ldots,(s_{i},t_{si},t_{ei}),\ldots)
  50. t a ( s ) = min { t s i - t e i | i D } . ta(s)=\min\{t_{si}-t_{ei}|i\in D\}.
  51. δ e x t : Q × X S × { 0 , 1 } \delta_{ext}:Q\times X\rightarrow S\times\{0,1\}
  52. q = ( s , t s , t e ) q=(s,t_{s},t_{e})
  53. s = ( , ( s i , t s i , t e i ) , ) , t s 𝕋 , t e ( 𝕋 [ 0 , t s ] ) s=(\ldots,(s_{i},t_{si},t_{ei}),\ldots),t_{s}\in\mathbb{T}^{\infty},t_{e}\in(% \mathbb{T}\cap[0,t_{s}])
  54. x X x\in X
  55. δ e x t ( q , x ) = ( ( , ( s i , t s i , t e i ) , ) , b ) \delta_{ext}(q,x)=((\ldots,(s_{i}^{\prime},t_{si}^{\prime},t_{ei}^{\prime}),% \ldots),b)
  56. ( s i , t s i , t e i ) = { ( s i , t a i ( s i ) , 0 ) if ( x , x i ) C x x , δ e x t ( s i , t s i , t e i , x i ) = ( s i , 1 ) ( s i , t s i , t e i ) if ( x , x i ) C x x , δ e x t ( s i , t s i , t e i , x i ) = ( s i , 0 ) ( s i , t e i ) otherwise (s_{i}^{\prime},t_{si}^{\prime},t_{ei}^{\prime})=\begin{cases}(s_{i}^{\prime},% ta_{i}(s_{i}^{\prime}),0)&\,\text{if }(x,x_{i})\in C_{xx},\delta_{ext}(s_{i},t% _{si},t_{ei},x_{i})=(s_{i}^{\prime},1)\\ (s_{i}^{\prime},t_{si},t_{ei})&\,\text{if }(x,x_{i})\in C_{xx},\delta_{ext}(s_% {i},t_{si},t_{ei},x_{i})=(s_{i}^{\prime},0)\\ (s_{i},t_{ei})&\,\text{otherwise}\end{cases}
  57. b = { 1 if i D : ( x , x i ) C x x , δ e x t ( s i , t s i , t e i , x i ) = ( s i , 1 ) 0 otherwise . b=\begin{cases}1&\,\text{if }\exists i\in D:(x,x_{i})\in C_{xx},\delta_{ext}(s% _{i},t_{si},t_{ei},x_{i})=(s_{i}^{\prime},1)\\ 0&\,\text{otherwise}.\end{cases}
  58. s = ( , ( s i , t s i , t e i ) , ) S s=(\ldots,(s_{i},t_{si},t_{ei}),\ldots)\in S
  59. I M M ( s ) = { i D | t s i - t e i = t a ( s ) } IMM(s)=\{i\in D|t_{si}-t_{ei}=ta(s)\}
  60. i * D i^{*}\in D
  61. i * = S e l e c t ( I M M ( s ) ) . i^{*}=Select(IMM(s)).
  62. δ i n t : S S \delta_{int}:S\rightarrow S
  63. s = ( , ( s i , t s i , t e i ) , ) s=(\ldots,(s_{i},t_{si},t_{ei}),\ldots)
  64. δ i n t ( s ) = s = ( , ( s i , t s i , t e i ) , ) \delta_{int}(s)=s^{\prime}=(\ldots,(s_{i}^{\prime},t_{si}^{\prime},t_{ei}^{% \prime}),\ldots)
  65. ( s i , t s i , t e i ) = { ( s i , t a i ( s i ) , 0 ) if i = i * , δ i n t ( s i ) = s i , ( s i , t a i ( s i ) , 0 ) if ( λ i * ( s i * ) , x i ) C y x , δ e x t ( s i , t s i , t e i , x i ) = ( s , 1 ) ( s i , t s i , t e i ) if ( λ i * ( s i * ) , x i ) C y x , δ e x t ( s i , t s i , t e i , x i ) = ( s , 0 ) ( s i , t s i , t e i ) otherwise . (s_{i}^{\prime},t_{si}^{\prime},t_{ei}^{\prime})=\begin{cases}(s_{i}^{\prime},% ta_{i}(s_{i}^{\prime}),0)&\,\text{if }i=i^{*},\delta_{int}(s_{i})=s_{i}^{% \prime},\\ (s_{i}^{\prime},ta_{i}(s_{i}^{\prime}),0)&\,\text{if }(\lambda_{i^{*}}(s_{i^{*% }}),x_{i})\in C_{yx},\delta_{ext}(s_{i},t_{si},t_{ei},x_{i})=(s^{\prime},1)\\ (s_{i}^{\prime},t_{si},t_{ei})&\,\text{if }(\lambda_{i^{*}}(s_{i^{*}}),x_{i})% \in C_{yx},\delta_{ext}(s_{i},t_{si},t_{ei},x_{i})=(s^{\prime},0)\\ (s_{i},t_{si},t_{ei})&\,\text{otherwise}.\end{cases}
  66. λ : S Y ϕ \lambda:S\rightarrow Y^{\phi}
  67. s = ( , ( s i , t s i , t e i ) , ) s=(\ldots,(s_{i},t_{si},t_{ei}),\ldots)
  68. λ ( s ) = \lambda(s)=
  69. | D | > 0 |D|>0
  70. q = ( s , t e ) Q q=(s,t_{e})\in Q
  71. s = ( , ( s i , t e i ) , ) s=(\ldots,(s_{i},t_{ei}),\ldots)
  72. ω \omega
  73. ω = ϵ [ t , t + d t ] \omega=\epsilon_{[t,t+dt]}
  74. Δ ( q , ω ) = ( ( , ( s i , t e i + d t ) , ) , t e + d t ) . \Delta(q,\omega)=((\ldots,(s_{i},t_{ei}+dt),\ldots),t_{e}+dt).
  75. q = ( s , t s , t e ) Q q=(s,t_{s},t_{e})\in Q
  76. s = ( , ( s i , t s i , t e i ) , ) s=(\ldots,(s_{i},t_{si},t_{ei}),\ldots)
  77. ω \omega
  78. ω = ϵ [ t , t + d t ] \omega=\epsilon_{[t,t+dt]}
  79. Δ ( q , ω ) = ( ( , ( s i , t s i , t e i + d t ) , ) , t s , t e + d t ) . \Delta(q,\omega)=((\ldots,(s_{i},t_{si},t_{ei}+dt),\ldots),t_{s},t_{e}+dt).
  80. S e l e c t ( I M M ( s ) ) Select(IMM(s))

Behnke–Stein_theorem.html

  1. G k G_{k}
  2. G k G k + 1 G_{k}\subset G_{k+1}

Belt_filter.html

  1. Q 0 = m 0 s b L s l u d g e 0 Q_{0}=m_{0}s_{b}L_{sludge0}

Belt_friction.html

  1. T 2 = T 1 e μ s β T_{2}=T_{1}e^{\mu_{s}\beta}\,
  2. T 2 T_{2}
  3. T 1 T_{1}
  4. μ s \mu_{s}
  5. β \beta

Beltrami_vector_field.html

  1. 𝐅 × ( × 𝐅 ) = 0. \mathbf{F}\times(\nabla\times\mathbf{F})=0.
  2. 𝐅 \mathbf{F}
  3. 𝐅 = 0 \nabla\cdot\mathbf{F}=0
  4. × ( × 𝐅 ) - 2 𝐅 + ( 𝐅 ) \nabla\times(\nabla\times\mathbf{F})\equiv-\nabla^{2}\mathbf{F}+\nabla(\nabla% \cdot\mathbf{F})
  5. - 2 𝐅 = × ( λ 𝐅 ) -\nabla^{2}\mathbf{F}=\nabla\times(\lambda\mathbf{F})
  6. λ \lambda
  7. 2 𝐅 = - λ 2 𝐅 . \nabla^{2}\mathbf{F}=-\lambda^{2}\mathbf{F}.
  8. 𝐅 = - z 1 + z 2 𝐢 + 1 1 + z 2 𝐣 \mathbf{F}=-\frac{z}{\sqrt{1+z^{2}}}\mathbf{i}+\frac{1}{\sqrt{1+z^{2}}}\mathbf% {j}

Bennett_acceptance_ratio.html

  1. p ( State x State y ) = min ( e - β Δ U , 1 ) = M ( β Δ U ) p(\,\text{State}_{x}\rightarrow\,\text{State}_{y})=\min\left(e^{-\beta\,\Delta U% },1\right)=M(\beta\,\Delta U)
  2. M ( x ) min ( e - x , 1 ) M(x)\equiv\min(e^{-x},1)
  3. \left\langle\cdots\right\rangle
  4. f ( x ) / f ( - x ) e - x f(x)/f(-x)\equiv e^{-x}
  5. e - β ( Δ F - C ) = f ( β ( U B - U A - C ) ) A f ( β ( U A - U B + C ) ) B e^{-\beta(\Delta F-C)}=\frac{\left\langle f\left(\beta(U\text{B}-U\text{A}-C)% \right)\right\rangle\text{A}}{\left\langle f\left(\beta(U\text{A}-U\text{B}+C)% \right)\right\rangle\text{B}}
  6. e - β Δ F = M ( β ( U B - U A ) ) A M ( β ( U A - U B ) ) B e^{-\beta\Delta F}=\frac{\left\langle M\left(\beta(U\text{B}-U\text{A})\right)% \right\rangle\text{A}}{\left\langle M\left(\beta(U\text{A}-U\text{B})\right)% \right\rangle\text{B}}
  7. f ( x ) 1 1 + e x f(x)\equiv\frac{1}{1+e^{x}}
  8. C Δ F C\approx\Delta F
  9. e - β Δ F = e - β ( U B - U A ) A e^{-\beta\Delta F}=\left\langle e^{-\beta(U\text{B}-U\text{A})}\right\rangle% \text{A}
  10. Δ F = - k T log e β ( U A - U B ) A \Delta F=-kT\cdot\log\left\langle e^{\beta(U\text{A}-U\text{B})}\right\rangle% \text{A}
  11. C - C\rightarrow-\infty
  12. e β C e - β ( U B - U A ) A e^{\beta C}\left\langle e^{-\beta(U\text{B}-U\text{A})}\right\rangle\text{A}
  13. U B - U A k T U\text{B}-U\text{A}\ll kT
  14. Δ F U B - U A A - β 2 ( ( U B - U A ) 2 A - ( ( U B - U A ) A ) 2 ) \Delta F\approx\left\langle U\text{B}-U\text{A}\right\rangle\text{A}-\frac{% \beta}{2}\left(\left\langle(U\text{B}-U\text{A})^{2}\right\rangle\text{A}-% \left(\left\langle(U\text{B}-U\text{A})\right\rangle\text{A}\right)^{2}\right)
  15. U B - U A B Δ F U B - U A A \langle U\text{B}-U\text{A}\rangle\text{B}\leq\Delta F\leq\langle U\text{B}-U% \text{A}\rangle\text{A}
  16. U A = U ( λ = 0 ) , U B = U ( λ = 1 ) , U\text{A}=U(\lambda=0),U\text{B}=U(\lambda=1),
  17. F ( λ ) λ = U ( λ ) λ λ \frac{\partial F(\lambda)}{\partial\lambda}=\left\langle\frac{\partial U(% \lambda)}{\partial\lambda}\right\rangle_{\lambda}
  18. A = λ + , B = λ - \,\text{A}=\lambda^{+},\,\text{B}=\lambda^{-}
  19. Δ F = 0 1 U ( λ ) λ d λ \Delta F=\int_{0}^{1}\left\langle\frac{\partial U(\lambda)}{\partial\lambda}% \right\rangle\,d\lambda

Bent_function.html

  1. f ^ : \Z 2 n \Z \hat{f}:\Z_{2}^{n}\to\Z
  2. f ^ ( a ) = x \Z 2 n ( - 1 ) f ( x ) + a x \hat{f}(a)=\sum_{\scriptstyle{x\in\Z_{2}^{n}}}(-1)^{f(x)+a\cdot x}
  3. 2 n {}^{n}_{2}
  4. f ^ ( a ) = | S 0 ( a ) | - | S 1 ( a ) | = 2 | S 0 ( a ) | - 2 n . \hat{f}(a)=|S_{0}(a)|-|S_{1}(a)|=2|S_{0}(a)|-2^{n}.
  5. - 2 n f ^ ( a ) 2 n . -2^{n}\leq\hat{f}(a)\leq 2^{n}.
  6. f ^ 0 ( a ) = 2 n , f ^ 1 ( a ) = - 2 n . \hat{f}_{0}(a)=2^{n},~{}\hat{f}_{1}(a)=-2^{n}.
  7. f ^ ( a ) \hat{f}(a)
  8. f ^ ( a ) = W ( 2 n ) ( - 1 ) f ( a ) , \hat{f}(a)=W(2^{n})(-1)^{f(a)},
  9. | f ^ ( a ) | = 2 n / 2 |\hat{f}(a)|=2^{n/2}
  10. f ^ ( a ) = 2 n / 2 ( - 1 ) g ( a ) \hat{f}(a)=2^{n/2}(-1)^{g(a)}
  11. g ^ ( a ) = 2 n / 2 ( - 1 ) f ( a ) \hat{g}(a)=2^{n/2}(-1)^{f(a)}
  12. f : m n m f:\mathbb{Z}_{m}^{n}\to\mathbb{Z}_{m}
  13. f ^ ( a ) = x m n e 2 π i m ( f ( x ) - a x ) \hat{f}(a)=\sum_{x\in\mathbb{Z}_{m}^{n}}e^{\frac{2\pi i}{m}(f(x)-a\cdot x)}
  14. f : m n m f:\mathbb{Z}_{m}^{n}\to\mathbb{Z}_{m}
  15. f : m n m f:\mathbb{Z}_{m}^{n}\to\mathbb{Z}_{m}
  16. | f ^ | |\hat{f}|

Beppo-Levi_space.html

  1. D D′
  2. S S′
  3. α α
  4. v ^ \widehat{v}
  5. v v
  6. W ˙ r , p = { v D : | v | r , p , Ω < } , \dot{W}^{r,p}=\left\{v\in D^{\prime}\ :\ |v|_{r,p,\Omega}<\infty\right\},
  7. m 𝐍 , s 𝐑 m∈\mathbf{N},s∈\mathbf{R}
  8. - m + n 2 < s < n 2 -m+\tfrac{n}{2}<s<\tfrac{n}{2}
  9. H s \displaystyle H^{s}
  10. X < s u p > m , s X<sup>m,s

Bessel_potential.html

  1. ( I - Δ ) - s / 2 (I-\Delta)^{-s/2}

Beta_angle.html

  1. s y m b o l β symbol{\beta}
  2. β \beta
  3. β \beta

Beta_skeleton.html

  1. θ = { sin - 1 1 β , if β 1 π - sin - 1 β , if β 1 \theta=\begin{cases}\sin^{-1}\frac{1}{\beta},&\,\text{if }\beta\geq 1\\ \pi-\sin^{-1}{\beta},&\,\text{if }\beta\leq 1\end{cases}

Bethe–Slater_curve.html

  1. w i j = - 2 J S i S j w_{ij}=-2\cdot J\cdot S_{i}\cdot S_{j}

Bhaskara_I's_sine_approximation_formula.html

  1. sin x = 4 x ( 180 - x ) 40500 - x ( 180 - x ) \sin x^{\circ}=\frac{4x(180-x)}{40500-x(180-x)}
  2. sin x = 16 x ( π - x ) 5 π 2 - 4 x ( π - x ) \sin x=\frac{16x(\pi-x)}{5\pi^{2}-4x(\pi-x)}
  3. sin π n = 16 ( n - 1 ) 5 n 2 - 4 n + 4 . \sin\frac{\pi}{n}=\frac{16(n-1)}{5n^{2}-4n+4}.
  4. R sin x = R x ( 180 - x ) 10125 - 1 4 x ( 180 - x ) R\sin x^{\circ}=\frac{Rx(180-x)}{10125-\frac{1}{4}x(180-x)}
  5. 2 R sin x = 4 × 2 R × 2 R x × ( 360 R - 2 R x ) 1 4 × 5 × ( 360 R ) 2 - 2 R x × ( 360 R - 2 R x ) 2R\sin x^{\circ}=\frac{4\times 2R\times 2Rx\times(360R-2Rx)}{\frac{1}{4}\times 5% \times(360R)^{2}-2Rx\times(360R-2Rx)}
  6. sin x - 4 x ( 180 - x ) 40500 - x ( 180 - x ) \sin x^{\circ}-\frac{4x(180-x)}{40500-x(180-x)}
  7. 1 P M = A B A P × B P \frac{1}{PM}=\frac{AB}{AP\times BP}
  8. 1 P M < 2 R x ( 180 - x ) \frac{1}{PM}<\frac{2R}{x(180-x)}
  9. 1 P M = α 2 R x ( 180 - x ) + β \frac{1}{PM}=\alpha\frac{2R}{x(180-x)}+\beta
  10. sin x = a + b x + c x 2 p + q x + r x 2 \sin x=\frac{a+bx+cx^{2}}{p+qx+rx^{2}}
  11. k x ( 180 - x ) . kx(180-x).\,
  12. x ( 180 - x ) 90 × 90 = x ( 180 - x ) 8100 . \frac{x(180-x)}{90\times 90}=\frac{x(180-x)}{8100}.
  13. x ( 180 - x ) 2 × 30 × 150 = x ( 180 - x ) 9000 . \frac{x(180-x)}{2\times 30\times 150}=\frac{x(180-x)}{9000}.
  14. 8100 a + b x ( 180 - x ) . 8100a+bx(180-x).\,

Bhatia–Davis_inequality.html

  1. variance ( M - μ ) ( μ - m ) . \,\text{variance}\leq(M-\mu)(\mu-m).\,

Bicentric_quadrilateral.html

  1. { a + c = b + d A + C = B + D = π . \begin{cases}a+c=b+d\\ A+C=B+D=\pi.\end{cases}
  2. A W W B = D Y Y C \frac{AW}{WB}=\frac{DY}{YC}
  3. A C B D = A W + C Y B X + D Z \frac{AC}{BD}=\frac{AW+CY}{BX+DZ}
  4. K = a b c d . \displaystyle K=\sqrt{abcd}.
  5. K = a b c d . \displaystyle K=\sqrt{abcd}.
  6. K = e f g h 4 ( e + f + g + h ) . K=\sqrt[4]{efgh}(e+f+g+h).
  7. K = A I C I + B I D I . K=AI\cdot CI+BI\cdot DI.
  8. K = k l p q k 2 + l 2 . K=\frac{klpq}{k^{2}+l^{2}}.
  9. K = | m 2 - n 2 k 2 - l 2 | k l K=\left|\frac{m^{2}-n^{2}}{k^{2}-l^{2}}\right|kl
  10. K = 2 M N E I F I E F K=\frac{2MN\cdot EI\cdot FI}{EF}
  11. K = a c tan θ 2 = b d cot θ 2 . K=ac\tan{\frac{\theta}{2}}=bd\cot{\frac{\theta}{2}}.
  12. K = 2 r 2 ( 1 sin A + 1 sin B ) . K=2r^{2}\left(\frac{1}{\sin{A}}+\frac{1}{\sin{B}}\right).
  13. K = r ( r + 4 R 2 + r 2 ) sin θ K=r(r+\sqrt{4R^{2}+r^{2}})\sin\theta
  14. K = 2 M N E Q F Q K=2MN\sqrt{EQ\cdot FQ}
  15. 4 r 2 K 2 R 2 . \displaystyle 4r^{2}\leq K\leq 2R^{2}.
  16. K 4 3 r 4 R 2 + r 2 K\leq\tfrac{4}{3}r\sqrt{4R^{2}+r^{2}}
  17. K r ( r + 4 R 2 + r 2 ) K\leq r(r+\sqrt{4R^{2}+r^{2}})
  18. 2 K s r + r 2 + 4 R 2 ; 2\sqrt{K}\leq s\leq r+\sqrt{r^{2}+4R^{2}};
  19. 6 K a b + a c + a d + b c + b d + c d 4 r 2 + 4 R 2 + 4 r r 2 + 4 R 2 ; 6K\leq ab+ac+ad+bc+bd+cd\leq 4r^{2}+4R^{2}+4r\sqrt{r^{2}+4R^{2}};
  20. 4 K r 2 a b c d 16 9 r 2 ( r 2 + 4 R 2 ) . 4Kr^{2}\leq abcd\leq\frac{16}{9}r^{2}(r^{2}+4R^{2}).
  21. tan A 2 = b c a d = cot C 2 , \tan{\frac{A}{2}}=\sqrt{\frac{bc}{ad}}=\cot{\frac{C}{2}},
  22. tan B 2 = c d a b = cot D 2 . \tan{\frac{B}{2}}=\sqrt{\frac{cd}{ab}}=\cot{\frac{D}{2}}.
  23. sin A 2 = b c a d + b c = cos C 2 , \sin{\frac{A}{2}}=\sqrt{\frac{bc}{ad+bc}}=\cos{\frac{C}{2}},
  24. cos A 2 = a d a d + b c = sin C 2 , \cos{\frac{A}{2}}=\sqrt{\frac{ad}{ad+bc}}=\sin{\frac{C}{2}},
  25. sin B 2 = c d a b + c d = cos D 2 , \sin{\frac{B}{2}}=\sqrt{\frac{cd}{ab+cd}}=\cos{\frac{D}{2}},
  26. cos B 2 = a b a b + c d = sin D 2 . \cos{\frac{B}{2}}=\sqrt{\frac{ab}{ab+cd}}=\sin{\frac{D}{2}}.
  27. tan θ 2 = b d a c . \displaystyle\tan{\frac{\theta}{2}}=\sqrt{\frac{bd}{ac}}.
  28. r = a b c d a + c = a b c d b + d . \displaystyle r=\frac{\sqrt{abcd}}{a+c}=\frac{\sqrt{abcd}}{b+d}.
  29. R = 1 4 ( a b + c d ) ( a c + b d ) ( a d + b c ) a b c d . \displaystyle R=\frac{1}{4}\sqrt{\frac{(ab+cd)(ac+bd)(ad+bc)}{abcd}}.
  30. r = e g = f h . \displaystyle r=\sqrt{eg}=\sqrt{fh}.
  31. y 4 - 2 s y 3 + ( s 2 + 2 r 2 + 2 r 4 R 2 + r 2 ) y 2 - 2 r s ( 4 R 2 + r 2 + r ) y + r 2 s 2 = 0 y^{4}-2sy^{3}+(s^{2}+2r^{2}+2r\sqrt{4R^{2}+r^{2}})y^{2}-2rs(\sqrt{4R^{2}+r^{2}% }+r)y+r^{2}s^{2}=0
  32. R 2 r R\geq\sqrt{2}r
  33. r 2 R 1 2 ( sin A 2 cos B 2 + sin B 2 cos C 2 + sin C 2 cos D 2 + sin D 2 cos A 2 ) 1 \frac{r\sqrt{2}}{R}\leq\frac{1}{2}\left(\sin{\frac{A}{2}}\cos{\frac{B}{2}}+% \sin{\frac{B}{2}}\cos{\frac{C}{2}}+\sin{\frac{C}{2}}\cos{\frac{D}{2}}+\sin{% \frac{D}{2}}\cos{\frac{A}{2}}\right)\leq 1
  34. 8 r ( 4 R 2 + r 2 - r ) s 4 R 2 + r 2 + r \sqrt{8r\left(\sqrt{4R^{2}+r^{2}}-r\right)}\leq s\leq\sqrt{4R^{2}+r^{2}}+r
  35. 2 s r 2 a b c + a b d + a c d + b c d 2 r ( r + r 2 + 4 R 2 ) 2 2sr^{2}\leq abc+abd+acd+bcd\leq 2r(r+\sqrt{r^{2}+4R^{2}})^{2}
  36. a b c + a b d + a c d + b c d 2 K ( K + 2 R 2 ) . abc+abd+acd+bcd\leq 2\sqrt{K}(K+2R^{2}).
  37. 1 ( R - x ) 2 + 1 ( R + x ) 2 = 1 r 2 , \frac{1}{(R-x)^{2}}+\frac{1}{(R+x)^{2}}=\frac{1}{r^{2}},
  38. 2 r 2 ( R 2 + x 2 ) = ( R 2 - x 2 ) 2 . \displaystyle 2r^{2}(R^{2}+x^{2})=(R^{2}-x^{2})^{2}.
  39. x = R 2 + r 2 - r 4 R 2 + r 2 . x=\sqrt{R^{2}+r^{2}-r\sqrt{4R^{2}+r^{2}}}.
  40. x 2 0 x^{2}\geq 0
  41. R 2 r . R\geq\sqrt{2}r.
  42. 2 r 2 + x 2 R 2 2 r 2 + x 2 + 2 r x . 2r^{2}+x^{2}\leq R^{2}\leq 2r^{2}+x^{2}+2rx.
  43. x 2 = R 2 - 2 R r μ \displaystyle x^{2}=R^{2}-2Rr\cdot\mu
  44. μ = ( a b + c d ) ( a d + b c ) ( a + c ) 2 ( a c + b d ) = ( a b + c d ) ( a d + b c ) ( b + d ) 2 ( a c + b d ) \displaystyle\mu=\sqrt{\frac{(ab+cd)(ad+bc)}{(a+c)^{2}(ac+bd)}}=\sqrt{\frac{(% ab+cd)(ad+bc)}{(b+d)^{2}(ac+bd)}}
  45. 4 r e + f + g + h 4 r R 2 + x 2 R 2 - x 2 4r\leq e+f+g+h\leq 4r\cdot\frac{R^{2}+x^{2}}{R^{2}-x^{2}}
  46. 4 r 2 e 2 + f 2 + g 2 + h 2 4 ( R 2 + x 2 - r 2 ) 4r^{2}\leq e^{2}+f^{2}+g^{2}+h^{2}\leq 4(R^{2}+x^{2}-r^{2})
  47. 8 r a + b + c + d 8 r R 2 + x 2 R 2 - x 2 8r\leq a+b+c+d\leq 8r\cdot\frac{R^{2}+x^{2}}{R^{2}-x^{2}}
  48. 4 ( R 2 - x 2 + 2 r 2 ) a 2 + b 2 + c 2 + d 2 4 ( 3 R 2 - 2 r 2 ) . 4(R^{2}-x^{2}+2r^{2})\leq a^{2}+b^{2}+c^{2}+d^{2}\leq 4(3R^{2}-2r^{2}).
  49. 1 I A 2 + 1 I C 2 = 1 I B 2 + 1 I D 2 = 1 r 2 \frac{1}{IA^{2}}+\frac{1}{IC^{2}}=\frac{1}{IB^{2}}+\frac{1}{ID^{2}}=\frac{1}{r% ^{2}}
  50. 4 r 2 I A I C + I B I D 2 R 2 4r^{2}\leq IA\cdot IC+IB\cdot ID\leq 2R^{2}
  51. p q 4 r 2 - 4 R 2 p q = 1 \displaystyle\frac{pq}{4r^{2}}-\frac{4R^{2}}{pq}=1
  52. r = p q 2 p q + 4 R 2 r=\frac{pq}{2\sqrt{pq+4R^{2}}}
  53. p q = 2 r ( r + 4 R 2 + r 2 ) . pq=2r\left(r+\sqrt{4R^{2}+r^{2}}\right).
  54. 8 p q ( a + b + c + d ) 2 \displaystyle 8pq\leq(a+b+c+d)^{2}
  55. K = a b c d . \displaystyle K=\sqrt{abcd}.

Bidiakis_cube.html

  1. ( x - 3 ) ( x - 2 ) ( x 4 ) ( x + 1 ) ( x + 2 ) ( x 2 + x - 4 ) 2 (x-3)(x-2)(x^{4})(x+1)(x+2)(x^{2}+x-4)^{2}

Bidomain_model.html

  1. ( 𝚺 i v ) + ( 𝚺 i v e ) \displaystyle\nabla\cdot\left(\mathbf{\Sigma}_{i}\nabla v\right)+\nabla\cdot% \left(\mathbf{\Sigma}_{i}\nabla v_{e}\right)
  2. χ \chi
  3. C m C_{m}
  4. v = v i - v e v=v_{i}-v_{e}
  5. I ion I_{\mathrm{ion}}
  6. 𝕋 \mathbb{T}
  7. ( 𝚺 i v ) + ( 𝚺 i v e ) \displaystyle\nabla\cdot\left(\mathbf{\Sigma}_{i}\nabla v\right)+\nabla\cdot% \left(\mathbf{\Sigma}_{i}\nabla v_{e}\right)
  8. \mathbb{H}
  9. \partial\mathbb{H}
  10. 𝐱 \mathbf{x}
  11. \mathbb{H}
  12. v i v_{i}
  13. v e v_{e}
  14. J i J_{i}
  15. J e J_{e}
  16. 𝚺 i \mathbf{\Sigma}_{i}
  17. 𝚺 e \mathbf{\Sigma}_{e}
  18. J i \displaystyle J_{i}
  19. \mathbb{H}
  20. ( J i + J e ) \displaystyle\nabla\cdot\left(J_{i}+J_{e}\right)
  21. J t = χ ( C m v t + I ion ) , J_{t}=\chi\left(C_{m}\frac{\partial v}{\partial t}+I_{\mathrm{ion}}\right),
  22. χ \chi
  23. C m C_{m}
  24. v = v i - v e v=v_{i}-v_{e}
  25. I ion I_{\mathrm{ion}}
  26. ( 𝚺 i v i ) = χ ( C m v t + I ion ) , \nabla\cdot\left(\mathbf{\Sigma}_{i}\nabla v_{i}\right)=\chi\left(C_{m}\frac{% \partial v}{\partial t}+I_{\mathrm{ion}}\right),
  27. v = v i - v e v=v_{i}-v_{e}
  28. ( 𝚺 i v ) + ( 𝚺 i v e ) = χ ( C m v t + I ion ) . \nabla\cdot\left(\mathbf{\Sigma}_{i}\nabla v\right)+\nabla\cdot\left(\mathbf{% \Sigma}_{i}\nabla v_{e}\right)=\chi\left(C_{m}\frac{\partial v}{\partial t}+I_% {\mathrm{ion}}\right).
  29. 𝕋 \mathbb{T}
  30. 𝕋 \partial\mathbb{T}
  31. \mathbb{H}
  32. 𝕋 \mathbb{T}
  33. 𝚺 0 \mathbf{\Sigma}_{0}
  34. v 0 v_{0}
  35. 𝕋 \mathbb{T}
  36. 𝕋 \partial\mathbb{T}
  37. v e v_{e}
  38. v 0 v_{0}
  39. 𝕋 \mathbb{T}
  40. n ( 𝚺 i v i ) = 0 𝐱 \vec{n}\cdot\left(\mathbf{\Sigma}_{i}\nabla v_{i}\right)=0\,\,\,\,\,\,\,% \mathbf{x}\in\partial\mathbb{H}
  41. v = v i - v e v=v_{i}-v_{e}
  42. 𝚺 i = α 𝚺 e \mathbf{\Sigma}_{i}=\alpha\mathbf{\Sigma}_{e}
  43. α \alpha

Biexciton.html

  1. | 01 |01\rangle
  2. | 10 |10\rangle
  3. | 11 |11\rangle
  4. | 00 |00\rangle
  5. | 01 |01\rangle
  6. | 10 |10\rangle
  7. E b E_{b}
  8. E b = 2 E X - E X X E_{b}=2E_{X}-E_{XX}
  9. E X X E_{XX}
  10. E X E_{X}
  11. E b E_{b}
  12. B X X = c 1 a 2 + c 2 a + B b u l k B_{XX}=\frac{c_{1}}{a^{2}}+\frac{c_{2}}{a}+B_{bulk}
  13. B X X B_{XX}
  14. a a
  15. B b u l k B_{bulk}
  16. c 1 c_{1}
  17. c 2 c_{2}
  18. H X X = - 2 2 m e * ( 1 2 + 2 2 ) - 2 2 m h * ( a 2 + b 2 ) + V H_{XX}=-\frac{\hbar^{2}}{2m_{e}^{*}}({\nabla_{1}}^{2}+{\nabla_{2}}^{2})-\frac{% \hbar^{2}}{2m_{h}^{*}}({\nabla_{a}}^{2}+{\nabla_{b}}^{2})+V
  19. m e * m_{e}^{*}
  20. m h * m_{h}^{*}
  21. V = V 12 - V 1 a - V 1 b - V 2 a - V 2 b + V a b V=V_{12}-V_{1a}-V_{1b}-V_{2a}-V_{2b}+V_{ab}
  22. V i j V_{ij}
  23. i i
  24. j j
  25. i , j = 1 , 2 , a , b i,j=1,2,a,b
  26. V i j = e 2 ϵ | 𝐫 i - 𝐫 j | V_{ij}=\frac{e^{2}}{\epsilon|\mathbf{r}_{i}-\mathbf{r}_{j}|}
  27. ϵ \epsilon
  28. 𝐑 \mathbf{R}
  29. 𝐫 \mathbf{r}
  30. M = m e * + m h * M=m_{e}^{*}+m_{h}^{*}
  31. H X X = - 2 4 M R 2 - 2 M r 2 - 2 2 μ ( 1 a 2 + 2 b 2 ) + V H_{XX}=-\frac{\hbar^{2}}{4M}{\nabla_{R}}^{2}-\frac{\hbar^{2}}{M}{\nabla_{r}}^{% 2}-\frac{\hbar^{2}}{2\mu}({\nabla_{1a}}^{2}+{\nabla_{2b}}^{2})+V
  32. 1 / μ = 1 / m e * + 1 / m h * 1/\mu=1/{m_{e}^{*}}+1/{m_{h}^{*}}
  33. 1 a 2 {\nabla_{1a}}^{2}
  34. 2 b 2 {\nabla_{2b}}^{2}
  35. r 2 {\nabla_{r}}^{2}
  36. R 2 {\nabla_{R}}^{2}
  37. 𝐑 \mathbf{R}
  38. H X X = - ( 1 a 2 + 2 b 2 ) - 2 σ ( 1 + σ ) 2 r 2 + V H_{XX}=-({\nabla_{1a}}^{2}+{\nabla_{2b}}^{2})-{2\sigma}{(1+\sigma)^{2}}{\nabla% _{r}}^{2}+V
  39. σ = m e * / m h * \sigma={m_{e}^{*}}/{m_{h}^{*}}
  40. V V
  41. V = 2 ( 1 r 12 - 1 r 1 a - 1 r 1 b - 1 r 2 a - 1 r 2 b + 1 r a b ) V=2(\frac{1}{r_{12}}-\frac{1}{r_{1a}}-\frac{1}{r_{1b}}-\frac{1}{r_{2a}}-\frac{% 1}{r_{2b}}+\frac{1}{r_{ab}})
  42. ψ \psi
  43. H X X ψ = E X X ψ H_{XX}\psi=E_{XX}\psi
  44. E X X E_{XX}
  45. E b = 2 E X - E X X E_{b}=2E_{X}-E_{XX}
  46. E b E_{b}
  47. E X E_{X}
  48. r r
  49. r r
  50. E X X 0.195 e V r E_{XX}\approx\frac{0.195eV}{r}
  51. 78 a * 2 + 52 < m t p l > a * + 33 \frac{78}{{a^{*}}^{2}}+\frac{52}{<}mtpl>{{a^{*}}}+33
  52. a * a^{*}

Big_O_in_probability_notation.html

  1. X n = o p ( a n ) X_{n}=o_{p}(a_{n})\,
  2. lim n P ( | X n | ε ) = 0 , \lim_{n\to\infty}P(|X_{n}|\geq\varepsilon)=0,
  3. X n = O p ( a n ) , X_{n}=O_{p}(a_{n}),\,
  4. P ( | X n / a n | > M ) < ε , n . P(|X_{n}/a_{n}|>M)<\varepsilon,\ \forall n.
  5. ε N ε , δ ε such that P ( | X n | δ ε ) ε n > N ε \forall\varepsilon\quad\exists N_{\varepsilon},\delta_{\varepsilon}\quad\,% \text{ such that }P(|X_{n}|\geq\delta_{\varepsilon})\leq\varepsilon\quad% \forall n>N_{\varepsilon}
  6. ε , δ N ε such that P ( | X n | δ ) ε n > N ε \forall\varepsilon,\delta\quad\exists N_{\varepsilon}\quad\,\text{ such that }% P(|X_{n}|\geq\delta)\leq\varepsilon\quad\forall n>N_{\varepsilon}
  7. ( X n ) (X_{n})
  8. X n - E ( X n ) = O p ( var ( X n ) ) X_{n}-E(X_{n})=O_{p}(\sqrt{\operatorname{var}(X_{n})})\,
  9. a n - 2 var ( X n ) = var ( a n - 1 X n ) a_{n}^{-2}\operatorname{var}(X_{n})=\operatorname{var}(a_{n}^{-1}X_{n})
  10. ( a n ) (a_{n})
  11. a n - 1 ( X n - E ( X n ) ) a_{n}^{-1}(X_{n}-E(X_{n}))
  12. X n - E ( X n ) = o p ( a n ) X_{n}-E(X_{n})=o_{p}(a_{n})

Biggs–Smith_graph.html

  1. ( x - 3 ) ( x - 2 ) 18 x 17 ( x 2 - x - 4 ) 9 ( x 3 + 3 x 2 - 3 ) 16 (x-3)(x-2)^{18}x^{17}(x^{2}-x-4)^{9}(x^{3}+3x^{2}-3)^{16}

Bihari's_inequality.html

  1. u ( t ) α + 0 t f ( s ) w ( u ( s ) ) d s , t [ 0 , ) , u(t)\leq\alpha+\int_{0}^{t}f(s)\,w(u(s))\,ds,\qquad t\in[0,\infty),
  2. u ( t ) G - 1 ( G ( α ) + 0 t f ( s ) d s ) , t [ 0 , T ] , u(t)\leq G^{-1}\left(G(\alpha)+\int_{0}^{t}\,f(s)\,ds\right),\qquad t\in[0,T],
  3. G ( x ) = x 0 x d y w ( y ) , x 0 , x 0 > 0 , G(x)=\int_{x_{0}}^{x}\frac{dy}{w(y)},\qquad x\geq 0,\,x_{0}>0,
  4. G ( α ) + 0 t f ( s ) d s Dom ( G - 1 ) , t [ 0 , T ] . G(\alpha)+\int_{0}^{t}\,f(s)\,ds\in\,\text{Dom}(G^{-1}),\qquad\forall\,t\in[0,% T].

Binary_Independence_Model.html

  1. P ( R | x , q ) = P ( x | R , q ) * P ( R | q ) P ( x | q ) P(R|x,q)=\frac{P(x|R,q)*P(R|q)}{P(x|q)}
  2. P ( R = 1 | x , q ) + P ( R = 0 | x , q ) = 1 P(R=1|x,q)+P(R=0|x,q)=1
  3. p i p_{i}
  4. q i q_{i}
  5. i t h i^{th}
  6. i t h i^{th}
  7. Y i = p i * ( 1 - q i ) ( 1 - p i ) * q i Y_{i}=\frac{p_{i}*(1-q_{i})}{(1-p_{i})*q_{i}}
  8. Y i Y_{i}
  9. Y j Y_{j}
  10. i i
  11. j j
  12. i t h i^{th}
  13. l o g Y i logY_{i}

Binary_lambda_calculus.html

  1. λ x λ y . x \lambda x\,\lambda y.\,x
  2. λ x λ y . y \lambda x\,\lambda y.\,y
  3. , = λ x λ y λ z . z x y \langle,\rangle=\lambda x\,\lambda y\,\lambda z.\,zxy
  4. M , N = λ z . z M N \langle M,N\rangle=\lambda z.\,zMN
  5. B b 0 , B b 1 B b n - 1 , z \langle B_{b_{0}},\langle B_{b_{1}}\ldots\langle B_{b_{n-1}},z\rangle\ldots\rangle\rangle
  6. s : z s:z
  7. x , y M N = M x y N \langle x,y\rangle\ M\ N=M\ x\ y\ N
  8. N i l M N = N Nil\ M\ N=N
  9. λ M ^ = 00 M ^ \widehat{\lambda M}=00\widehat{M}
  10. M N ^ = 01 M ^ N ^ \widehat{M\ N}=01\widehat{M}\widehat{N}
  11. i ^ = 1 i 0 \widehat{i}=1^{i}0
  12. λ x λ y λ z . z x y \lambda x\lambda y\lambda z.zxy
  13. λ λ λ .132 \lambda\lambda\lambda.132
  14. 00 00 00 01 01 10 1110 110 00\ 00\ 00\ 01\ 01\ 10\ 1110\ 110
  15. λ 1 ^ = 0010 \widehat{\lambda 1}=0010
  16. ( λ 11 ) ( λ λ λ 1 ( λ λ λ λ 3 ( λ 5 ( 3 ( λ 2 ( 3 ( λ λ 3 ( λ 123 ) ) ) ( 4 ( λ 4 ( λ 31 ( 21 ) ) ) ) ) ) (\lambda 11)(\lambda\lambda\lambda 1(\lambda\lambda\lambda\lambda 3(\lambda 5(% 3(\lambda 2(3(\lambda\lambda 3(\lambda 123)))(4(\lambda 4(\lambda 31(21))))))
  17. ( 1 ( 2 ( λ 12 ) ) ( λ 4 ( λ 4 ( λ 2 ( 14 ) ) ) 5 ) ) ) ) ( 33 ) 2 ) ( λ 1 ( ( λ 11 ) ( λ 11 ) ) ) (1(2(\lambda 12))(\lambda 4(\lambda 4(\lambda 2(14)))5))))(33)2)(\lambda 1((% \lambda 11)(\lambda 11)))
  18. K S ( x | y 1 , , y k ) = min { ( p ) | U ( p : N i l ) y 1 y k = x } KS(x|y_{1},\ldots,y_{k})=\min\{\ell(p)\ |\ U\ (p:Nil)\ y_{1}\ \ldots\ y_{k}=\ % \,x\ \ \ \ \}
  19. K P ( x | y 1 , , y k ) = min { ( p ) | U ( p : z ) y 1 y k = x , z } KP(x|y_{1},\ldots,y_{k})=\min\{\ell(p)\ |\ U\ (p:\ z\ \ )\ y_{1}\ \ldots\ y_{k% }=\langle x,z\rangle\}
  20. λ 1 \lambda 1
  21. K S ( x ) ( x ) + 4 KS(x)\leq\ell(x)+4
  22. λ λ 1 ( ( λ 11 ) ( λ λ λ λ 2 ( 44 ) ( λ λ 32 ( 32 ( 2 ( 51 ( 21 ) ) ) ) ) ) ) ( λ λ 1 ) ( λ λ λ 1 ( λ 4 ( λ 4 ( λ 132 ) ) ) ) ( λ λ λ 1 ( 3 ( λ λ 1 ) ) 2 ) ( λ 1 ) 2 \lambda\lambda 1((\lambda 11)(\lambda\lambda\lambda\lambda 2(44)(\lambda% \lambda 32(32(2(51(21)))))))(\lambda\lambda 1)(\lambda\lambda\lambda 1(\lambda 4% (\lambda 4(\lambda 132))))(\lambda\lambda\lambda 1(3(\lambda\lambda 1))2)(% \lambda 1)2
  23. K P ( x | ( x ) ) ( x ) + 188 KP(x|\ell(x))\leq\ell(x)+188
  24. ( λ 11 ) ( λ λ λ 1 ( λ 1 ( 3 ( λ λ 1 ) ) ( 44 ( λ 1 ( λ λ λ 1 ( λ 4 ( λ λ 52 ( 52 ( 31 ( 21 ) ) ) ) ) ) 4 ( λ 1 ) ) ) ) ) ( λ λ λ 1 ( 3 ( ( λ 11 ) (\lambda 11)(\lambda\lambda\lambda 1(\lambda 1(3(\lambda\lambda 1))(44(\lambda 1% (\lambda\lambda\lambda 1(\lambda 4(\lambda\lambda 52(52(31(21))))))4(\lambda 1% )))))(\lambda\lambda\lambda 1(3((\lambda 11)
  25. ( λ λ λ λ 1 ( λ 55 ( λ λ 356 ( λ 1 ( λ λ 612 ) 3 ) ) ( λ λ 5 ( λ 143 ) ) ) ( 31 ) ) ( λ λ 1 ( λ λ 2 ) 2 ) ( λ 1 ) ) ( λ λ 1 ) ) 2 ) (\lambda\lambda\lambda\lambda 1(\lambda 55(\lambda\lambda 356(\lambda 1(% \lambda\lambda 612)3))(\lambda\lambda 5(\lambda 143)))(31))(\lambda\lambda 1(% \lambda\lambda 2)2)(\lambda 1))(\lambda\lambda 1))2)
  26. K P ( x ) ( x ¯ ) + 338 KP(x)\leq\ell(\overline{x})+338
  27. x ¯ \overline{x}
  28. 0 ¯ = 0 n + 1 ¯ = 1 ( n ) ¯ n \begin{array}[]{ll}\overline{0}&=0\\ \overline{n+1}&=1\ \overline{\ell(n)}\ n\\ \end{array}
  29. ( n ¯ ) ( n ) + ( ( n ) ) + + k - 1 ( n ) + O ( k ( n ) ) \ell(\overline{n})\leq\ell(n)+\ell(\ell(n))+\cdots+\ell^{k-1}(n)+O(\ell^{k}(n))
  30. λ ( λ 1 ( 1 ( ( λ 11 ) ( λ λ λ 1 ( λ λ 1 ) ( ( λ 441 ( ( λ 11 ) ( λ 2 ( 11 ) ) ) ) ( λ λ λ λ 13 ( 2 ( 64 ) ) ) ) ) ( λ λ λ 4 ( 13 ) ) ) ) ) ( λ λ 1 ( λ λ 2 ) 2 ) \lambda(\lambda 1(1((\lambda 11)(\lambda\lambda\lambda 1(\lambda\lambda 1)((% \lambda 441((\lambda 11)(\lambda 2(11))))(\lambda\lambda\lambda\lambda 13(2(64% )))))(\lambda\lambda\lambda 4(13)))))(\lambda\lambda 1(\lambda\lambda 2)2)
  31. 0011010100010100010100010000010100000100010100010000010000010100000100010100000100010000010000000100 001101010001010001010001000001010000010001010001000001000001010000010001010000% 0100010000010000000100
  32. 𝐾𝑆 ( 𝑃𝑅𝐼𝑀𝐸𝑆 ) 167 \mathit{KS}(\mathit{PRIMES})\leq 167
  33. 𝐾𝑃 ( 𝑃𝑅𝐼𝑀𝐸𝑆 ) 176 \mathit{KP}(\mathit{PRIMES})\leq 176
  34. ( λ 1 ( λ λ 2 ) ( λ 1 ( λ λ 1 ) ) ( λ λ λ ( λ 1 ( 32 ) ( λ λ 31 ( 611 ( 4 ( ( λ 11 ) ( λ λ λ λ λ 12 ( λ 1 ( λ λ 1 ) ( λ λ λ 82 ) ( λ λ 1 ) ( 4 ( 66 ) ( λ 4 ( λ 172 ) ) ) ) ) ) 5 ) (\lambda 1(\lambda\lambda 2)(\lambda 1(\lambda\lambda 1))(\lambda\lambda% \lambda(\lambda 1(32)(\lambda\lambda 31(611(4((\lambda 11)(\lambda\lambda% \lambda\lambda\lambda 12(\lambda 1(\lambda\lambda 1)(\lambda\lambda\lambda 82)% (\lambda\lambda 1)(4(66)(\lambda 4(\lambda 172))))))5)
  35. ( λ λ 1 ) ( λ λ λ λ 1 ) ( λ λ 2 ) ( λ λ 1 ) ) ( λ λ λ 1 ( λ 164 ) 2 ) ) ) ( 4 ( λ 1 ) ( λ 1 ) ( λ 1 ) 1 ) ) ( λ 1 ) ) ( λ λ ( λ 11 ) ( λ λ λ λ 1 ( λ ( λ λ λ λ 3 ( λ 6 ( 3 ( λ 2 ( 3 (\lambda\lambda 1)(\lambda\lambda\lambda\lambda 1)(\lambda\lambda 2)(\lambda% \lambda 1))(\lambda\lambda\lambda 1(\lambda 164)2)))(4(\lambda 1)(\lambda 1)(% \lambda 1)1))(\lambda 1))(\lambda\lambda(\lambda 11)(\lambda\lambda\lambda% \lambda 1(\lambda(\lambda\lambda\lambda\lambda 3(\lambda 6(3(\lambda 2(3
  36. ( λ 14 ( λ 3 ( λ 123 ) ) ) ) ( 4 ( λ 4 ( λ 14 ( 31 ) ( 21 ) ) ) ) ) ) ( 1 ( 2 ( λ 12 ) ) ( λ λ 5 ( λ 5 ( λ 2 ( 15 ) ) ) 76 ) ) ( λ 6 ( λ 132 ) ) ) ) ( λ 4 ( λ 132 ) ) ) ( 44 ) 3 ) ) (\lambda 14(\lambda 3(\lambda 123))))(4(\lambda 4(\lambda 14(31)(21))))))(1(2(% \lambda 12))(\lambda\lambda 5(\lambda 5(\lambda 2(15)))76))(\lambda 6(\lambda 1% 32))))(\lambda 4(\lambda 132)))(44)3))
  37. K P ( x , y ) K P ( x ) + K P ( y | x ) + 657 KP(x,y)\leq KP(x)+KP(y|x^{\ast})+657
  38. x x^{\ast}
  39. K P ( y | x ) K P ( x , y ) - K P ( x ) + O ( 1 ) KP(y|x^{\ast})\leq KP(x,y)-KP(x)+O(1)
  40. ( p ) - K P ( x ) \ell(p)-KP(x)
  41. Q = λ 1 ( ( λ 11 ) ( λ λ λ λ λ 14 ( 3 ( 55 ) 2 ) ) ) 1 Q=\lambda 1((\lambda 11)(\lambda\lambda\lambda\lambda\lambda 14(3(55)2)))1
  42. K S ( x x ) ( x ) + 66 KS(xx)\leq\ell(x)+66
  43. U ( Q ^ Q ^ : N i l ) = Q ^ Q ^ U(\widehat{Q}\widehat{Q}:Nil)=\widehat{Q}\widehat{Q}
  44. ( x ) > 66 \ell(x)>66
  45. x x xx
  46. ( x ) - 66 \ell(x)-66
  47. ( λ 1111 ( λ λ 1 ( λ λ 1 ) 2 ) ) ( λ λ 2 ( 21 ) ) (\lambda 1111(\lambda\lambda 1(\lambda\lambda 1)2))(\lambda\lambda 2(21))
  48. 2 2 2 2 = 65536 2^{2^{2^{2}}}=65536
  49. Ω λ = U ( p : z ) = x , z , x N F 2 - ( p ) \Omega_{\lambda}=\sum_{U(p:z)=\langle x,z\rangle,\,x\in NF}2^{-\ell(p)}
  50. Ω λ = .0001 2 \Omega_{\lambda}=.0001\ldots_{2}
  51. U 8 = λ 1 ( ( λ 11 ) ( λ ( λ λ λ 1 ( λ λ λ 2 ( λ λ λ ( λ 7 ( 10 ( λ 5 ( 2 ( λ λ 3 ( λ 123 ) ) ) ( 11 ( λ 3 ( λ 31 ( 21 ) ) ) ) ) 3 ) U8=\lambda 1((\lambda 11)(\lambda(\lambda\lambda\lambda 1(\lambda\lambda% \lambda 2(\lambda\lambda\lambda(\lambda 7(10(\lambda 5(2(\lambda\lambda 3(% \lambda 123)))(11(\lambda 3(\lambda 31(21)))))3)
  52. ( 4 ( 1 ( λ 15 ) 3 ) ( 10 ( λ 2 ( λ 2 ( 16 ) ) ) 6 ) ) ) 8 ) ( λ 1 ( λ 87 ( λ 162 ) ) ) ) ( λ 1 ( 43 ) ) ) ( 11 ) ) ( λ λ 2 ( ( λ 11 ) ( λ 11 ) ) ) ) (4(1(\lambda 15)3)(10(\lambda 2(\lambda 2(16)))6)))8)(\lambda 1(\lambda 87(% \lambda 162))))(\lambda 1(43)))(11))(\lambda\lambda 2((\lambda 11)(\lambda 11)% )))
  53. ( λ 11 ) ( λ ( λ λ λ 1 ( λ ( λ 2111 ( λ λ 133 ( λ λ 1 ( λ λ ( λ 7 ( 1 ( 3 ( λ λ λ λ λ 10 ¯ ( 1 ( λ 6143 ) ) ( λ 15 ( 65432 ) ) ) ( λ λ 2 ( ( λ 11 ) ( λ λ λ 2 (\lambda 11)(\lambda(\lambda\lambda\lambda 1(\lambda(\lambda 2111(\lambda% \lambda 133(\lambda\lambda 1(\lambda\lambda(\lambda 7(1(3(\lambda\lambda% \lambda\lambda\lambda\underline{10}(1(\lambda 6143))(\lambda 15(65432)))(% \lambda\lambda 2((\lambda 11)(\lambda\lambda\lambda 2
  54. ( λ λ λ 662 ( λ λ 6 ( λ 1 ( 26 ) 3 ) ( 15 ¯ ( 51 ( λ 1 ) ) ( 5 ( λ 1 ) 1 ) ) ) ) ( 12 ( λ λ λ 312 ) ) ) 1 ( λ λ 2 ) ) ) ) ) ( 3 ( 1 ( λ λ λ λ 9 ( 1 ( λ 51 ( λ 154 ) ) ) (\lambda\lambda\lambda 662(\lambda\lambda 6(\lambda 1(26)3)(\underline{15}(51(% \lambda 1))(5(\lambda 1)1))))(12(\lambda\lambda\lambda 312)))1(\lambda\lambda 2% )))))(3(1(\lambda\lambda\lambda\lambda 9(1(\lambda 51(\lambda 154)))
  55. ( 24 ( λ 142 ) ) ) ) ( 5 ( 11 ¯ ( λ 1 ) ) ( 12 ¯ ( λ 2 ( ( λ 11 ) ( λ λ λ 1 ( ( λ 11 ) ( λ λ λ 2 ( 1 ( 33 ) ) ( λ 8 ( 771 ) ) ) ) 21 ) ) ) ) ) ) ) ( λ 12 ¯ ( λ 12 ¯ ( λ 3 (24(\lambda 142))))(5(\underline{11}(\lambda 1))(\underline{12}(\lambda 2((% \lambda 11)(\lambda\lambda\lambda 1((\lambda 11)(\lambda\lambda\lambda 2(1(33)% )(\lambda 8(771))))21)))))))(\lambda\underline{12}(\lambda\underline{12}(\lambda 3
  56. ( 21 ) ) ) ) ) ) ) ) ( λ λ 1 ) ) ) ( 11 ) ) ( λ ( λ 11 ) ( λ λ 1 ( ( λ 1 ( 1 ( 1 ( λ λ 1 ( λ λ 2 ) 2 ) ) ) ) ( λ λ 2 ( 21 ) ) ( λ λ 1 ) ) ( 22 ) ) ( 1 ( λ λ λ λ λ λ 1 ) ) 1 ) (21))))))))(\lambda\lambda 1)))(11))(\lambda(\lambda 11)(\lambda\lambda 1((% \lambda 1(1(1(\lambda\lambda 1(\lambda\lambda 2)2))))(\lambda\lambda 2(21))(% \lambda\lambda 1))(22))(1(\lambda\lambda\lambda\lambda\lambda\lambda 1))1)
  57. λ x . x \lambda x.x

Binary_offset_carrier_modulation.html

  1. f s c f_{sc}
  2. ( f s c , f c ) (f_{sc}\;,f_{c})
  3. f s c f_{sc}
  4. f c f_{c}
  5. m = f s c / f r e f m=f_{sc}/f_{ref}
  6. n = f c / f r e f n=f_{c}/f_{ref}
  7. f r e f = 1.023 f_{ref}=1.023
  8. N B O C = 2 m / n N_{BOC}=2m/n
  9. N B O C N_{BOC}
  10. N B O C N_{BOC}
  11. N B O C N_{BOC}
  12. N B O C = 2 f s c f c = 2 m n N_{BOC}=2\frac{f_{sc}}{f_{c}}=2\frac{m}{n}

Binding_selectivity.html

  1. A + B A B ; K A B = [ A B ] [ A ] [ B ] A+B\leftrightharpoons AB;K_{AB}=\frac{[AB]}{[A][B]}
  2. A + C A C ; K A C = [ A C ] [ A ] [ C ] A+C\leftrightharpoons AC;K_{AC}=\frac{[AC]}{[A][C]}
  3. K B , C = K A C K A B K_{B,C}=\frac{K_{AC}}{K_{AB}}
  4. A B + C A C + B ; K B , C = [ A C ] [ B ] [ A B ] [ C ] = K A C [ A ] [ B ] [ C ] K A B [ A ] [ B ] [ C ] = K A C K A B AB+C\leftrightharpoons AC+B;K_{B,C}=\frac{[AC][B]}{[AB][C]}=\frac{K_{AC}[A][B]% [C]}{K_{AB}[A][B][C]}=\frac{K_{AC}}{K_{AB}}
  5. R S = N 4 ( α - 1 α ) ( k B 1 + k B ) R_{S}=\frac{\sqrt{N}}{4}\left(\frac{\alpha-1}{\alpha}\right)\left(\frac{k_{B}}% {1+k_{B}}\right)

Binet_equation.html

  1. r r
  2. θ \theta
  3. u = 1 / r u=1/r
  4. θ \theta
  5. h = L / m h=L/m
  6. L L
  7. m m
  8. F ( u ) = - m h 2 u 2 ( d 2 u d θ 2 + u ) . F(u)=-mh^{2}u^{2}\left(\frac{\mathrm{d}^{2}u}{\mathrm{d}\theta^{2}}+u\right).
  9. F ( r ) = m ( r ¨ - r θ ˙ 2 ) . F(r)=m(\ddot{r}-r\dot{\theta}^{2}).
  10. r 2 θ ˙ = h = constant . r^{2}\dot{\theta}=h=\,\text{constant}.
  11. r r
  12. u u
  13. d u d θ = d d t ( 1 r ) d t d θ = - r ˙ r 2 θ ˙ = - r ˙ h \displaystyle\frac{\mathrm{d}u}{\mathrm{d}\theta}=\frac{\mathrm{d}}{\mathrm{d}% t}\left(\frac{1}{r}\right)\frac{\mathrm{d}t}{\mathrm{d}\theta}=-\frac{{\dot{r}% }}{r^{2}\dot{\theta}}=-\frac{{\dot{r}}}{h}
  14. F = m ( r ¨ - r θ ˙ 2 ) = - m ( h 2 u 2 d 2 u d θ 2 + h 2 u 3 ) = - m h 2 u 2 ( d 2 u d θ 2 + u ) F=m(\ddot{r}-r\dot{\theta}^{2})=-m\left(h^{2}u^{2}\frac{\mathrm{d}^{2}u}{% \mathrm{d}\theta^{2}}+h^{2}u^{3}\right)=-mh^{2}u^{2}\left(\frac{\mathrm{d}^{2}% u}{\mathrm{d}\theta^{2}}+u\right)
  15. d 2 u d θ 2 + u = constant > 0. \frac{\mathrm{d}^{2}u}{\mathrm{d}\theta^{2}}+u=\,\text{constant}>0.
  16. θ \theta
  17. l u = 1 + ε cos θ . lu=1+\varepsilon\cos\theta.
  18. l l
  19. ε \varepsilon
  20. d 2 u d θ 2 + u = r s c 2 2 h 2 + 3 r s 2 u 2 \frac{\mathrm{d}^{2}u}{\mathrm{d}\theta^{2}}+u=\frac{r_{s}c^{2}}{2h^{2}}+\frac% {3r_{s}}{2}u^{2}
  21. c c
  22. r s r_{s}
  23. d 2 u d θ 2 + u = r s c 2 2 h 2 + 3 r s 2 u 2 - G Q 2 4 π ε 0 c 4 ( c 2 h 2 u + 2 u 3 ) \frac{\mathrm{d}^{2}u}{\mathrm{d}\theta^{2}}+u=\frac{r_{s}c^{2}}{2h^{2}}+\frac% {3r_{s}}{2}u^{2}-\frac{GQ^{2}}{4\pi\varepsilon_{0}c^{4}}\left(\frac{c^{2}}{h^{% 2}}u+2u^{3}\right)
  24. Q Q
  25. ε 0 \varepsilon_{0}
  26. l d 2 u d θ 2 = - ε cos θ . l\,\frac{\mathrm{d}^{2}u}{\mathrm{d}\theta^{2}}=-\varepsilon\cos\theta.
  27. F = - m h 2 u 2 ( - ε cos θ l + 1 + ε cos θ l ) = - m h 2 u 2 l = - m h 2 l r 2 , F=-mh^{2}u^{2}\left(\frac{-\varepsilon\cos\theta}{l}+\frac{1+\varepsilon\cos% \theta}{l}\right)=-\frac{mh^{2}u^{2}}{l}=-\frac{mh^{2}}{lr^{2}},
  28. h 2 / l h^{2}/l
  29. G M GM
  30. k e q 1 q 2 / m k_{e}q_{1}q_{2}/m
  31. F = - G M m u 2 ( 1 + 3 ( h u c ) 2 ) = - G M m r 2 ( 1 + 3 ( h r c ) 2 ) F=-GMmu^{2}\left(1+3\left(\frac{hu}{c}\right)^{2}\right)=-\frac{GMm}{r^{2}}% \left(1+3\left(\frac{h}{rc}\right)^{2}\right)
  32. F = - G M m r 2 ( 1 + ( 2 + 2 γ - β ) ( h r c ) 2 ) F=-\frac{GMm}{r^{2}}\left(1+(2+2\gamma-\beta)\left(\frac{h}{rc}\right)^{2}\right)
  33. γ = β = 1 \gamma=\beta=1
  34. γ = β = 0 \gamma=\beta=0
  35. F ( r ) = - k r 3 . F(r)=-\frac{k}{r^{3}}.
  36. d 2 u d θ 2 + u = k u m h 2 = C u . \frac{\mathrm{d}^{2}u}{\mathrm{d}\theta^{2}}+u=\frac{ku}{mh^{2}}=Cu.
  37. C < 1 C<1
  38. C = 0 C=0
  39. C = 1 C=1
  40. C > 1 C>1
  41. D D
  42. D u ( θ ) = sec θ . D\,u(\theta)=\sec\theta.
  43. u u
  44. D d 2 u d θ 2 = sec θ tan 2 θ + sec 3 θ = sec θ ( sec 2 θ - 1 ) + sec 3 θ = 2 D 3 u 3 - D u . D\,\frac{\mathrm{d}^{2}u}{\mathrm{d}\theta^{2}}=\sec\theta\tan^{2}\theta+\sec^% {3}\theta=\sec\theta(\sec^{2}\theta-1)+\sec^{3}\theta=2D^{3}u^{3}-D\,u.
  45. F = - m h 2 u 2 ( 2 D 2 u 3 - u + u ) = - 2 m h 2 D 2 u 5 = - 2 m h 2 D 2 r 5 . F=-mh^{2}u^{2}\left(2D^{2}u^{3}-u+u\right)=-2mh^{2}D^{2}u^{5}=-\frac{2mh^{2}D^% {2}}{r^{5}}.
  46. 1 / r 5 1/r^{5}
  47. d 2 u d θ 2 + u = C u 3 \frac{\mathrm{d}^{2}u}{\mathrm{d}\theta^{2}}+u=Cu^{3}

Biordered_set.html

  1. ω r = { ( e , f ) : f e = e } \omega^{r}=\{(e,f)\,:\,fe=e\}
  2. ω l = { ( e , f ) : e f = e } \omega^{l}=\{(e,f)\,:\,ef=e\}
  3. R = ω r ( ω r ) - 1 R=\omega^{r}\,\cap\,(\omega^{r})^{-1}
  4. L = ω l ( ω l ) - 1 L=\omega^{l}\,\cap\,(\omega^{l})^{-1}
  5. ω = ω r ω l \omega=\omega^{r}\,\cap\,\omega^{l}
  6. \prec
  7. g h e g ω r e h , g f ω l h f g\prec h\quad\Longleftrightarrow\quad eg\,\,\omega^{r}\,\,eh\,,\,\,\,gf\,\,% \omega^{l}\,\,hf\,
  8. S ( e , f ) = { h M ( e , f ) : g h for all g M ( e , f ) } S(e,f)=\{h\in M(e,f):g\prec h\,\text{ for all }g\in M(e,f)\}

BIRCH.html

  1. N N
  2. K K
  3. C F CF
  4. C F = ( N , L S , S S ) CF=(N,LS,SS)
  5. L S LS
  6. S S SS
  7. B B
  8. T T
  9. B B
  10. [ C F i , c h i l d i ] [CF_{i},child_{i}]
  11. c h i l d i child_{i}
  12. i i
  13. C F i CF_{i}
  14. L L
  15. [ C F i ] [CF_{i}]

Birman–Wenzl_algebra.html

  1. G i G j = G j G i , if | i - j | 2 , G_{i}G_{j}=G_{j}G_{i},\mathrm{if}\left|i-j\right|\geqslant 2,
  2. G i G i + 1 G i = G i + 1 G i G i + 1 , G_{i}G_{i+1}G_{i}=G_{i+1}G_{i}G_{i+1},
  3. E i E i ± 1 E i = E i , E_{i}E_{i\pm 1}E_{i}=E_{i},
  4. G i + G i - 1 = m ( 1 + E i ) , G_{i}+{G_{i}}^{-1}=m(1+E_{i}),
  5. G i ± 1 G i E i ± 1 = E i G i ± 1 G i = E i E i ± 1 , G_{i\pm 1}G_{i}E_{i\pm 1}=E_{i}G_{i\pm 1}G_{i}=E_{i}E_{i\pm 1},
  6. G i ± 1 E i G i ± 1 = G i - 1 E i ± 1 G i - 1 , G_{i\pm 1}E_{i}G_{i\pm 1}={G_{i}}^{-1}E_{i\pm 1}{G_{i}}^{-1},
  7. G i ± 1 E i E i ± 1 = G i - 1 E i ± 1 , G_{i\pm 1}E_{i}E_{i\pm 1}={G_{i}}^{-1}E_{i\pm 1},
  8. E i ± 1 E i G i ± 1 = E i ± 1 G i - 1 , E_{i\pm 1}E_{i}G_{i\pm 1}=E_{i\pm 1}{G_{i}}^{-1},
  9. G i E i = E i G i = l - 1 E i , G_{i}E_{i}=E_{i}G_{i}=l^{-1}E_{i},
  10. E i G i ± 1 E i = l E i . E_{i}G_{i\pm 1}E_{i}=lE_{i}.
  11. E i E j = E j E i , if | i - j | 2 , E_{i}E_{j}=E_{j}E_{i},\mathrm{if}\left|i-j\right|\geqslant 2,
  12. ( E i ) 2 = ( m - 1 ( l + l - 1 ) - 1 ) E i , (E_{i})^{2}=(m^{-1}(l+l^{-1})-1)E_{i},\,\!
  13. G i 2 = m ( G i + l - 1 E i ) - 1. {G_{i}}^{2}=m(G_{i}+l^{-1}E_{i})-1.
  14. G i - G i - 1 = m ( 1 - E i ) , G_{i}-{G_{i}}^{-1}=m(1-E_{i}),
  15. ( E i ) 2 = ( m - 1 ( l - l - 1 ) + 1 ) E i , (E_{i})^{2}=(m^{-1}(l-l^{-1})+1)E_{i},\,\!
  16. G i G j = G j G i , if | i - j | 2 , and G i G i + 1 G i = G i + 1 G i G i + 1 , G_{i}G_{j}=G_{j}G_{i},\,\text{if }\left|i-j\right|\geqslant 2,\,\text{ and }G_% {i}G_{i+1}G_{i}=G_{i+1}G_{i}G_{i+1},\,\!
  17. E i E i ± 1 E i = E i and G i G i ± 1 E i = E i ± 1 E i , E_{i}E_{i\pm 1}E_{i}=E_{i}\,\text{ and }G_{i}G_{i\pm 1}E_{i}=E_{i\pm 1}E_{i},
  18. G i E i = E i G i = l - 1 E i and E i G i ± 1 E i = l E i . G_{i}E_{i}=E_{i}G_{i}=l^{-1}E_{i}\,\text{ and }E_{i}G_{i\pm 1}E_{i}=lE_{i}.
  19. ( 2 n ) ! / ( 2 n n ! ) (2n)!/(2^{n}n!)
  20. 𝔖 n , \mathfrak{S}_{n},
  21. n 𝒞 n {\mathcal{B}}_{n}\hookrightarrow{\mathcal{C}}_{n}
  22. ϕ : C n K T n \phi:C_{n}\to KT_{n}
  23. U i ( u ) = 1 - i sin u sin λ sin μ ( e i ( u - λ ) G i - e - i ( u - λ ) G i - 1 ) U_{i}(u)=1-\frac{i\sin u}{\sin\lambda\sin\mu}(e^{i(u-\lambda)}G_{i}-e^{-i(u-% \lambda)}{G_{i}}^{-1})
  24. λ \lambda
  25. μ \mu
  26. 2 cos λ = 1 + ( l - l - 1 ) / m 2\cos\lambda=1+(l-l^{-1})/m
  27. 2 cos λ = 1 + ( l - l - 1 ) / ( λ sin μ ) 2\cos\lambda=1+(l-l^{-1})/(\lambda\sin\mu)
  28. U i + 1 ( v ) U i ( u + v ) U i + 1 ( u ) = U i ( u ) U i + 1 ( u + v ) U i ( v ) U_{i+1}(v)U_{i}(u+v)U_{i+1}(u)=U_{i}(u)U_{i+1}(u+v)U_{i}(v)\,\!
  29. E i = U i ( λ ) E_{i}=U_{i}(\lambda)
  30. ρ ( u ) = sin ( λ - u ) sin ( μ + u ) sin λ sin μ \rho(u)=\frac{\sin(\lambda-u)\sin(\mu+u)}{\sin\lambda\sin\mu}
  31. u ± i u\to\pm i\infty
  32. G j ± {G_{j}}^{\pm}
  33. F F

Bivariate_von_Mises_distribution.html

  1. κ 1 = κ 2 = 200 \kappa_{1}=\kappa_{2}=200
  2. κ 3 = 0 \kappa_{3}=0
  3. κ 1 = κ 2 = 200 \kappa_{1}=\kappa_{2}=200
  4. κ 3 = 100 \kappa_{3}=100
  5. κ 1 = κ 2 = 20 , κ 3 = 0 \kappa_{1}=\kappa_{2}=20,\kappa_{3}=0
  6. S 1 × S 1 S^{1}\times S^{1}
  7. 3 \mathbb{R}^{3}
  8. ϕ , ψ [ 0 , 2 π ] \phi,\psi\in[0,2\pi]
  9. f ( ϕ , ψ ) exp [ κ 1 cos ( ϕ - μ ) + κ 2 cos ( ψ - ν ) + ( cos ( ϕ - μ ) , sin ( ϕ - μ ) ) 𝐀 ( cos ( ψ - ν ) , sin ( ψ - ν ) ) T ] , f(\phi,\psi)\propto\exp[\kappa_{1}\cos(\phi-\mu)+\kappa_{2}\cos(\psi-\nu)+(% \cos(\phi-\mu),\sin(\phi-\mu))\mathbf{A}(\cos(\psi-\nu),\sin(\psi-\nu))^{T}],
  10. μ \mu
  11. ν \nu
  12. ϕ \phi
  13. ψ \psi
  14. κ 1 \kappa_{1}
  15. κ 2 \kappa_{2}
  16. 𝐀 𝕄 ( 2 , 2 ) \mathbf{A}\in\mathbb{M}(2,2)
  17. f ( ϕ , ψ ) = Z c ( κ 1 , κ 2 , κ 3 ) exp [ κ 1 cos ( ϕ - μ ) + κ 2 cos ( ψ - ν ) - κ 3 cos ( ϕ - μ - ψ + ν ) ] , f(\phi,\psi)=Z_{c}(\kappa_{1},\kappa_{2},\kappa_{3})\ \exp[\kappa_{1}\cos(\phi% -\mu)+\kappa_{2}\cos(\psi-\nu)-\kappa_{3}\cos(\phi-\mu-\psi+\nu)],
  18. μ \mu
  19. ν \nu
  20. ϕ \phi
  21. ψ \psi
  22. κ 1 \kappa_{1}
  23. κ 2 \kappa_{2}
  24. κ 3 \kappa_{3}
  25. Z c Z_{c}
  26. κ 3 \kappa_{3}
  27. ϕ \phi
  28. ψ \psi
  29. f ( ϕ , ψ ) = Z s ( κ 1 , κ 2 , κ 3 ) exp [ κ 1 cos ( ϕ - μ ) + κ 2 cos ( ψ - ν ) + κ 3 sin ( ϕ - μ ) sin ( ψ - ν ) ] , f(\phi,\psi)=Z_{s}(\kappa_{1},\kappa_{2},\kappa_{3})\ \exp[\kappa_{1}\cos(\phi% -\mu)+\kappa_{2}\cos(\psi-\nu)+\kappa_{3}\sin(\phi-\mu)\sin(\psi-\nu)],

Bjerrum_plot.html

  1. [ CO 2 ] e q = [ H + ] e q 2 [ H + ] e q 2 + K 1 [ H + ] e q + K 1 K 2 × DIC , [\textrm{CO}_{2}]_{eq}=\frac{[\textrm{H}^{+}]_{eq}^{2}}{[\textrm{H}^{+}]_{eq}^% {2}+K_{1}[\textrm{H}^{+}]_{eq}+K_{1}K_{2}}\times\textrm{DIC},
  2. [ HCO 3 - ] e q = K 1 [ H + ] e q [ H + ] e q 2 + K 1 [ H + ] e q + K 1 K 2 × DIC , [\textrm{HCO}_{3}^{-}]_{eq}=\frac{K_{1}[\textrm{H}^{+}]_{eq}}{[\textrm{H}^{+}]% _{eq}^{2}+K_{1}[\textrm{H}^{+}]_{eq}+K_{1}K_{2}}\times\textrm{DIC},
  3. [ CO 3 2 - ] e q = K 1 K 2 [ H + ] e q 2 + K 1 [ H + ] e q + K 1 K 2 × DIC , [\textrm{CO}_{3}^{2-}]_{eq}=\frac{K_{1}K_{2}}{[\textrm{H}^{+}]_{eq}^{2}+K_{1}[% \textrm{H}^{+}]_{eq}+K_{1}K_{2}}\times\textrm{DIC},
  4. d [ CO 2 ] d t = - k 1 [ CO 2 ] + k - 1 [ H + ] [ HCO 3 - ] , \frac{\textrm{d}[\textrm{CO}_{2}]}{\textrm{d}t}=-k_{1}[\textrm{CO}_{2}]+k_{-1}% [\textrm{H}^{+}][\textrm{HCO}_{3}^{-}],
  5. d [ H + ] d t = k 1 [ CO 2 ] - k - 1 [ H + ] [ HCO 3 - ] + k 2 [ HCO 3 - ] - k - 2 [ H + ] [ CO 3 2 - ] , \frac{\textrm{d}[\textrm{H}^{+}]}{\textrm{d}t}=k_{1}[\textrm{CO}_{2}]-k_{-1}[% \textrm{H}^{+}][\textrm{HCO}_{3}^{-}]+k_{2}[\textrm{HCO}_{3}^{-}]-k_{-2}[% \textrm{H}^{+}][\textrm{CO}_{3}^{2-}],
  6. d [ HCO 3 - ] d t = k 1 [ CO 2 ] - k - 1 [ H + ] [ HCO 3 - ] - k 2 [ HCO 3 - ] + k - 2 [ H + ] [ CO 3 2 - ] , \frac{\textrm{d}[\textrm{HCO}_{3}^{-}]}{\textrm{d}t}=k_{1}[\textrm{CO}_{2}]-k_% {-1}[\textrm{H}^{+}][\textrm{HCO}_{3}^{-}]-k_{2}[\textrm{HCO}_{3}^{-}]+k_{-2}[% \textrm{H}^{+}][\textrm{CO}_{3}^{2-}],
  7. d [ CO 3 2 - ] d t = k 2 [ HCO 3 - ] - k - 2 [ H + ] [ CO 3 2 - ] , \frac{\textrm{d}[\textrm{CO}_{3}^{2-}]}{\textrm{d}t}=k_{2}[\textrm{HCO}_{3}^{-% }]-k_{-2}[\textrm{H}^{+}][\textrm{CO}_{3}^{2-}],
  8. K 1 = k 1 k - 1 = [ H + ] e q [ HCO 3 - ] e q [ CO 2 ] e q , K_{1}=\frac{k_{1}}{k_{-1}}=\frac{[\textrm{H}^{+}]_{eq}[\textrm{HCO}_{3}^{-}]_{% eq}}{[\textrm{CO}_{2}]_{eq}},
  9. K 2 = k 2 k - 2 = [ H + ] e q [ CO 3 2 - ] e q [ HCO 3 - ] e q . K_{2}=\frac{k_{2}}{k_{-2}}=\frac{[\textrm{H}^{+}]_{eq}[\textrm{CO}_{3}^{2-}]_{% eq}}{[\textrm{HCO}_{3}^{-}]_{eq}}.
  10. [ HCO 3 - ] e q = K 1 [ CO 2 ] e q [ H + ] e q , [\textrm{HCO}_{3}^{-}]_{eq}=\frac{K_{1}[\textrm{CO}_{2}]_{eq}}{[\textrm{H}^{+}% ]_{eq}},
  11. [ CO 3 2 - ] e q = K 2 [ HCO 3 - ] e q [ H + ] e q = K 1 K 2 [ CO 2 ] e q [ H + ] e q 2 . [\textrm{CO}_{3}^{2-}]_{eq}=\frac{K_{2}[\textrm{HCO}_{3}^{-}]_{eq}}{[\textrm{H% }^{+}]_{eq}}=\frac{K_{1}K_{2}[\textrm{CO}_{2}]_{eq}}{[\textrm{H}^{+}]_{eq}^{2}}.
  12. DIC = [ CO 2 ] + [ HCO 3 - ] + [ CO 3 2 - ] \textrm{DIC}=[\textrm{CO}_{2}]+[\textrm{HCO}_{3}^{-}]+[\textrm{CO}_{3}^{2-}]
  13. = [ CO 2 ] e q ( 1 + K 1 [ H + ] e q + K 1 K 2 [ H + ] e q 2 ) =[\textrm{CO}_{2}]_{eq}\left(1+\frac{K_{1}}{[\textrm{H}^{+}]_{eq}}+\frac{K_{1}% K_{2}}{[\textrm{H}^{+}]_{eq}^{2}}\right)
  14. = [ CO 2 ] e q ( [ H + ] e q 2 + K 1 [ H + ] e q + K 1 K 2 [ H + ] e q 2 ) . =[\textrm{CO}_{2}]_{eq}\left(\frac{[\textrm{H}^{+}]_{eq}^{2}+K_{1}[\textrm{H}^% {+}]_{eq}+K_{1}K_{2}}{[\textrm{H}^{+}]_{eq}^{2}}\right).
  15. [ CO 2 ] e q = [ H + ] e q 2 [ H + ] e q 2 + K 1 [ H + ] e q + K 1 K 2 × DIC . [\textrm{CO}_{2}]_{eq}=\frac{[\textrm{H}^{+}]_{eq}^{2}}{[\textrm{H}^{+}]_{eq}^% {2}+K_{1}[\textrm{H}^{+}]_{eq}+K_{1}K_{2}}\times\textrm{DIC}.

Black–Karasinski_model.html

  1. d ln ( r ) = [ θ t - ϕ t ln ( r ) ] d t + σ t d W t d\ln(r)=[\theta_{t}-\phi_{t}\ln(r)]\,dt+\sigma_{t}\,dW_{t}

Blandford–Znajek_process.html

  1. P = B 2 ( r r c ) 4 r c c = B 2 r 4 ω 2 c P=B^{2}\left(\frac{r}{r_{c}}\right)^{4}r_{c}c=\frac{B^{2}r^{4}\omega^{2}}{c}
  2. r c r_{c}

Blanuša_snarks.html

  1. ( x - 3 ) ( x - 1 ) 3 ( x + 1 ) ( x + 2 ) ( x 4 + x 3 - 7 x 2 - 5 x + 6 ) ( x 4 + x 3 - 5 x 2 - 3 x + 4 ) 2 (x-3)(x-1)^{3}(x+1)(x+2)(x^{4}+x^{3}-7x^{2}-5x+6)(x^{4}+x^{3}-5x^{2}-3x+4)^{2}
  2. ( x - 3 ) ( x - 1 ) 3 ( x 3 + 2 x 2 - 3 x - 5 ) ( x 3 + 2 x 2 - x - 1 ) ( x 4 + x 3 - 7 x 2 - 6 x + 7 ) ( x 4 + x 3 - 5 x 2 - 4 x + 3 ) . (x-3)(x-1)^{3}(x^{3}+2x^{2}-3x-5)(x^{3}+2x^{2}-x-1)(x^{4}+x^{3}-7x^{2}-6x+7)(x% ^{4}+x^{3}-5x^{2}-4x+3).
  3. B n 1 B_{n}^{1}
  4. B n 2 B_{n}^{2}
  5. B n 1 B_{n}^{1}
  6. 3 + 2 n 3+{\frac{2}{n}}
  7. B n 2 B_{n}^{2}
  8. 3 + 1 1 + 3 n / 2 3+{\frac{1}{\lfloor 1+3n/2\rfloor}}

Bloch-Siegert_shift.html

  1. ω \omega
  2. ω 0 \omega_{0}
  3. H a b = V a b 2 cos ( ω t ) H_{ab}=\frac{V_{ab}}{2}\cos{(\omega t)}
  4. ω , - ω \omega,-\omega
  5. ω \omega
  6. Ω = 1 2 ( | V a b / | ) 2 + ( ω - ω 0 ) 2 . \Omega=\frac{1}{2}\sqrt{(|V_{ab}/\hbar|)^{2}+(\omega-\omega_{0})^{2}}.
  7. - ω -\omega
  8. Ω e f f = 1 2 ( | V a b / | ) 2 + ( ω + ω 0 ) 2 . \Omega_{eff}=\frac{1}{2}\sqrt{(|V_{ab}/\hbar|)^{2}+(\omega+\omega_{0})^{2}}.
  9. 2 ω = ( | V a b / | ) 2 + ( ω + ω 0 ) 2 2\omega=\sqrt{(|V_{ab}/\hbar|)^{2}+(\omega+\omega_{0})^{2}}
  10. ω = ω 0 [ 1 + 1 4 ( V a b ω 0 ) 2 ] \omega=\omega_{0}\left[1+\frac{1}{4}\left(\frac{V_{ab}}{\hbar\omega_{0}}\right% )^{2}\right]
  11. ω = - 1 3 ω 0 [ 1 + 3 4 ( V a b ω 0 ) 2 ] . \omega=-\frac{1}{3}\omega_{0}\left[1+\frac{3}{4}\left(\frac{V_{ab}}{\hbar% \omega_{0}}\right)^{2}\right].
  12. ω 0 \omega_{0}
  13. δ ω B - S = 1 4 ( V a b ) 2 2 ω 0 \delta\omega_{B-S}=\frac{1}{4}\frac{(V_{ab})^{2}}{\hbar^{2}\omega_{0}}

Blossom_algorithm.html

  1. M 1 = M P = ( M P ) ( P M ) M_{1}=M\oplus P=(M\setminus P)\cup(P\setminus M)
  2. M B M\cap B
  3. { w , w } E M \{w^{\prime},w\}\in E\setminus M
  4. { u , u } E M \{u^{\prime},u\}\in E\setminus M
  5. \cup
  6. O ( | V | 4 ) O(|V|^{4})
  7. O ( | E | | V | 1 / 2 ) O(|E||V|^{1/2})

Body_size_and_species_richness.html

  1. N = M - 3 / 4 N=M^{-3/4}
  2. S = L - 2 S=L^{-2}

Bogomol'nyi–Prasad–Sommerfield_state.html

  1. { Q α A , Q ¯ β ˙ B } = 2 σ α β ˙ m P m δ B A { Q α A , Q β B } = 2 ϵ α β ϵ A B Z ¯ { Q ¯ α ˙ A , Q ¯ β ˙ B } = - 2 ϵ α ˙ β ˙ ϵ A B Z \begin{aligned}\displaystyle\{Q_{\alpha}^{A},\bar{Q}_{\dot{\beta}B}\}&% \displaystyle=2\sigma_{\alpha\dot{\beta}}^{m}P_{m}\delta^{A}_{B}\\ \displaystyle\{Q_{\alpha}^{A},Q_{\beta}^{B}\}&\displaystyle=2\epsilon_{\alpha% \beta}\epsilon^{AB}\bar{Z}\\ \displaystyle\{\bar{Q}_{\dot{\alpha}A},\bar{Q}_{\dot{\beta}B}\}&\displaystyle=% -2\epsilon_{\dot{\alpha}\dot{\beta}}\epsilon_{AB}Z\\ \end{aligned}
  2. α β ˙ \alpha\dot{\beta}
  3. R α A \displaystyle R_{\alpha}^{A}
  4. ( M , 0 , 0 , 0 ) (M,0,0,0)
  5. ( R 1 1 + ( R 1 1 ) ) 2 ψ \displaystyle(R_{1}^{1}+(R_{1}^{1})^{\dagger})^{2}\psi
  6. ξ \xi
  7. M | Z | \begin{aligned}\displaystyle M\geq|Z|\\ \end{aligned}

Bogomolov–Miyaoka–Yau_inequality.html

  1. c 1 2 3 c 2 c_{1}^{2}\leq 3c_{2}
  2. c 1 2 3 c 2 . c_{1}^{2}\leq 3c_{2}.\,
  3. c 2 ( X ) = e ( X ) c_{2}(X)=e(X)
  4. c 1 2 ( X ) = 2 e ( X ) + 3 σ ( X ) c_{1}^{2}(X)=2e(X)+3\sigma(X)
  5. σ ( X ) \sigma(X)
  6. σ ( X ) 1 3 e ( X ) , \sigma(X)\leq\frac{1}{3}e(X),
  7. σ ( X ) = ( 1 / 3 ) e ( X ) \sigma(X)=(1/3)e(X)
  8. c 1 2 = 3 c 2 c_{1}^{2}=3c_{2}
  9. 2 {\mathbb{C}}^{2}
  10. 1 2 {}^{2}_{1}
  11. 1 2 {}^{2}_{1}
  12. 1 2 {}^{2}_{1}
  13. 1 2 {}^{2}_{1}
  14. 1 2 {}^{2}_{1}
  15. 1 2 {}^{2}_{1}
  16. 1 2 {}^{2}_{1}

Bolt_thrust.html

  1. F b o l t = P m a x A i n t e r n a l . \vec{F}_{bolt}=P_{max}\cdot A_{internal}.
  2. A r e a = π r 2 3.1416 r 2 . Area=\pi r^{2}\approx 3{.}1416\cdot r^{2}.
  3. A r e a = π d 2 4 0.7854 d 2 . Area=\frac{\pi d^{2}}{4}\approx 0{.}7854\cdot d^{2}.
  4. F b o l t = P m a x A e x t e r n a l . \vec{F_{bolt}}=P_{max}\cdot A_{external}.

Bond_fluctuation_model.html

  1. 𝐁 = 𝐏 ± ( 2 0 0 ) 𝐏 ± ( 2 1 0 ) 𝐏 ± ( 2 1 1 ) 𝐏 ± ( 2 2 1 ) 𝐏 ± ( 3 0 0 ) 𝐏 ± ( 3 1 0 ) \mathbf{B}=\mathbf{P_{\pm}}\left(\begin{matrix}2\\ 0\\ 0\end{matrix}\right)\cup\!\ \mathbf{P_{\pm}}\left(\begin{matrix}2\\ 1\\ 0\end{matrix}\right)\cup\!\ \mathbf{P_{\pm}}\left(\begin{matrix}2\\ 1\\ 1\end{matrix}\right)\cup\!\ \mathbf{P_{\pm}}\left(\begin{matrix}2\\ 2\\ 1\end{matrix}\right)\cup\!\ \mathbf{P_{\pm}}\left(\begin{matrix}3\\ 0\\ 0\end{matrix}\right)\cup\!\ \mathbf{P_{\pm}}\left(\begin{matrix}3\\ 1\\ 0\end{matrix}\right)
  2. 2 , 5 , 6 , 3 2,\sqrt{5},\sqrt{6},3
  3. 10 \sqrt{10}
  4. 𝚫 𝐁 = 𝐏 ± ( 1 , 0 , 0 ) \mathbf{\Delta B}=\mathbf{P_{\pm}}\left(1,0,0\right)
  5. 1 , 2 , 3 1,\sqrt{2},\sqrt{3}
  6. Δ 𝐁 𝐏 ± ( 1 , 0 , 0 ) \Delta\mathbf{B}\in\mathbf{P}_{\pm}(1,0,0)
  7. Δ U \Delta U
  8. p M p_{M}
  9. p M = e - Δ U / k B T p_{M}=e^{-\Delta U/k_{B}T}\,
  10. # M C S = # attempts # monomers \#MCS=\frac{\#\,\text{ attempts}}{\#\,\text{ monomers}}

Bonse's_inequality.html

  1. p 1 p n > p n + 1 2 . p_{1}\cdots p_{n}>p_{n+1}^{2}.\,

Boole's_rule.html

  1. x 1 x 5 f ( x ) d x \int_{x_{1}}^{x_{5}}f(x)\,dx
  2. x 1 , x 2 = x 1 + h , x 3 = x 1 + 2 h , x 4 = x 1 + 3 h , x 5 = x 1 + 4 h . x_{1},\quad x_{2}=x_{1}+h,\quad x_{3}=x_{1}+2h,\quad x_{4}=x_{1}+3h,\quad x_{5% }=x_{1}+4h.\,
  3. x 1 x 5 f ( x ) d x = 2 h 45 ( 7 f ( x 1 ) + 32 f ( x 2 ) + 12 f ( x 3 ) + 32 f ( x 4 ) + 7 f ( x 5 ) ) + error term , \int_{x_{1}}^{x_{5}}f(x)\,dx=\frac{2h}{45}\left(7f(x_{1})+32f(x_{2})+12f(x_{3}% )+32f(x_{4})+7f(x_{5})\right)+\,\text{error term},
  4. - 8 945 h 7 f ( 6 ) ( c ) -\,\frac{8}{945}h^{7}f^{(6)}(c)

Boolean_model_(probability_theory).html

  1. λ \lambda
  2. {\mathcal{B}}
  3. λ \lambda
  4. ξ \xi
  5. C ξ C_{\xi}
  6. {\mathcal{B}}
  7. ξ ( ξ + C ξ ) \cup_{\xi}(\xi+C_{\xi})
  8. {\mathcal{B}}
  9. 1 - exp ( - λ A ) 1-\exp(-\lambda A)
  10. Γ \Gamma
  11. C ξ C_{\xi}
  12. A = E ( Γ ) . A=\operatorname{E}(\Gamma).

Bose–Mesner_algebra.html

  1. x X x\in X
  2. y X y\in X
  3. { x , y } R i \{x,y\}\in R_{i}
  4. x , y X x,y\in X
  5. { x , y } R k \{x,y\}\in R_{k}
  6. z X z\in X
  7. { x , z } R i \{x,z\}\in R_{i}
  8. { z , y } R j \{z,y\}\in R_{j}
  9. p i j k p^{k}_{ij}
  10. ( D i ) x , y = { 1 , if ( x , y ) R i , 0 , otherwise. ( 1 ) (D_{i})_{x,y}=\begin{cases}1,&\,\text{if }\left(x,y\right)\in R_{i},\\ 0,&\,\text{otherwise.}\end{cases}\qquad(1)
  11. 𝒜 \mathcal{A}
  12. i = 0 n a i D i \sideset{}{{}_{i=0}^{n}}{\sum}a_{i}D_{i}
  13. a i a_{i}
  14. D i D_{i}
  15. D i D_{i}
  16. i = 0 n D i = J \sum_{i=0}^{n}D_{i}=J
  17. D 0 = I , D_{0}=I,
  18. D i D j = k = 0 n p i j k D k = D j D i , i , j = 0 , , n . D_{i}D_{j}=\sum_{k=0}^{n}p^{k}_{ij}D_{k}=D_{j}D_{i},\qquad i,j=0,\ldots,n.
  19. D i D_{i}
  20. v i v_{i}
  21. D i J = J D i = v i J . ( 2 ) D_{i}J=JD_{i}=v_{i}J.\qquad(2)
  22. D 0 , , D n D_{0},\ldots,D_{n}
  23. 𝒜 \mathcal{A}
  24. n + 1 n+1
  25. 𝒜 \mathcal{A}
  26. 𝒜 \mathcal{A}
  27. 𝒜 \mathcal{A}
  28. S S
  29. A 𝒜 A\in\mathcal{A}
  30. Λ A \Lambda_{A}
  31. S - 1 A S = Λ A S^{-1}AS=\Lambda_{A}
  32. 𝒜 \mathcal{A}
  33. J 0 , , J n J_{0},\ldots,J_{n}
  34. J i 2 = J i , i = 0 , , n , ( 3 ) J_{i}^{2}=J_{i},i=0,\ldots,n,\qquad(3)
  35. J i J k = 0 , i k , ( 4 ) J_{i}J_{k}=0,i\neq k,\qquad(4)
  36. i = 0 n J i = I . ( 5 ) \sum_{i=0}^{n}J_{i}=I.\qquad(5)
  37. D i D_{i}
  38. E k E_{k}
  39. D i = k = 0 n p i ( k ) E k , ( 6 ) D_{i}=\sum_{k=0}^{n}p_{i}(k)E_{k},\qquad(6)
  40. | X | E k = i = 0 n q k ( i ) D i . ( 7 ) |X|E_{k}=\sum_{i=0}^{n}q_{k}\left(i\right)D_{i}.\qquad(7)
  41. p i ( k ) p_{i}(k)
  42. q k ( i ) q_{k}(i)
  43. D i D_{i}
  44. p i ( k ) p_{i}\left(k\right)
  45. q k ( i ) q_{k}\left(i\right)
  46. k = 0 n μ i p i ( k ) p ( k ) = v v i δ i , ( 8 ) \sum_{k=0}^{n}\mu_{i}p_{i}(k)p_{\ell}(k)=vv_{i}\delta_{i\ell},\quad(8)
  47. k = 0 n μ i q k ( i ) q ( i ) = v μ k δ k . ( 9 ) \sum_{k=0}^{n}\mu_{i}q_{k}(i)q_{\ell}(i)=v\mu_{k}\delta_{k\ell}.\quad(9)
  48. μ j p i ( j ) = v i q j ( i ) , i , j = 0 , , n . ( 10 ) \mu_{j}p_{i}(j)=v_{i}q_{j}(i),\quad i,j=0,\ldots,n.\quad(10)
  49. P T Δ μ P = v Δ v , ( 11 ) P^{T}\Delta_{\mu}P=v\Delta_{v},\quad(11)
  50. Q T Δ v Q = v Δ μ , ( 12 ) Q^{T}\Delta_{v}Q=v\Delta_{\mu},\quad(12)
  51. Δ v = diag { v 0 , v 1 , , v n } , Δ μ = diag { μ 0 , μ 1 , , μ n } . \Delta_{v}=\operatorname{diag}\{v_{0},v_{1},\ldots,v_{n}\},\qquad\Delta_{\mu}=% \operatorname{diag}\{\mu_{0},\mu_{1},\ldots,\mu_{n}\}.
  52. D i D D_{i}D_{\ell}
  53. p i ( k ) p ( k ) p_{i}(k)p_{\ell}(k)
  54. μ k \mu_{k}
  55. v v i δ i = trace D i D = k = 0 n μ i p i ( k ) p ( k ) , ( 13 ) vv_{i}\delta_{i\ell}=\operatorname{trace}D_{i}D_{\ell}=\sum_{k=0}^{n}\mu_{i}p_% {i}(k)p_{\ell}(k),\quad(13)
  56. ( 8 ) \left(8\right)
  57. ( 11 ) \left(11\right)
  58. Q = v P - 1 = Δ v - 1 P T Δ μ , ( 14 ) Q=vP^{-1}=\Delta_{v}^{-1}P^{T}\Delta_{\mu},\quad(14)
  59. ( 9 ) (9)
  60. ( 10 ) (10)
  61. ( 12 ) (12)
  62. \Box
  63. n n
  64. X = n X=\mathcal{F}^{n}
  65. \mathcal{F}
  66. ( , K ) \left(\mathcal{F},K\right)
  67. X = n X=\mathcal{F}^{n}
  68. n n
  69. ( , K ) n \left(\mathcal{F},K\right)_{\otimes}^{n}
  70. ( , K ) \left(\mathcal{F},K\right)
  71. X = n X=\mathcal{F}^{n}
  72. ( , K ) n \left(\mathcal{F},K\right)_{\otimes}^{n}
  73. F [ X ] F\left[X\right]
  74. 𝔽 \mathbb{F}

Boundary-incompressible_surface.html

  1. D S = α D\cap S=\alpha
  2. D M = β D\cap\partial M=\beta
  3. D \partial D
  4. α β = D \alpha\cup\beta=\partial D
  5. α β = α = β \alpha\cap\beta=\partial\alpha=\partial\beta
  6. α \alpha
  7. α \alpha
  8. S \partial S
  9. M \partial M
  10. S \partial S
  11. M \partial M
  12. V \partial V
  13. V \partial V

Branches_of_physics.html

  1. 𝐃 = ρ f 𝐁 = 0 × 𝐄 = - 𝐁 t × 𝐇 = 𝐉 f + 𝐃 t \begin{aligned}&\displaystyle\nabla\cdot\mathbf{D}=\rho_{f}\\ &\displaystyle\nabla\cdot\mathbf{B}=0\\ &\displaystyle\nabla\times\mathbf{E}=-\frac{\partial\mathbf{B}}{\partial t}\\ &\displaystyle\nabla\times\mathbf{H}=\mathbf{J}_{f}+\frac{\partial\mathbf{D}}{% \partial t}\end{aligned}

Brander–Spencer_model.html

  1. p ( x + y ) p(x+y)
  2. π h = x p ( x + y ) - c ( x ) + s x \pi^{h}=xp(x+y)-c(x)+sx
  3. x p ( x + y ) xp(x+y)
  4. c ( x ) c(x)
  5. π f = y p ( x + y ) - c ( y ) \pi^{f}=yp(x+y)-c(y)
  6. x p x + p - c q + s = 0 xp_{x}+p-c_{q}+s=0
  7. y p y + p - c y = 0 yp_{y}+p-c_{y}=0
  8. y = R 1 ( x ; s ) y=R1(x;s)
  9. y = R 2 ( x ) y=R2(x)
  10. p = a - b ( x + y ) p=a-b(x+y)
  11. u ( x , m ) = m + U ( x ) u(x,m)=m+U(x)

Brane.html

  1. α \alpha
  2. β \beta
  3. α \alpha
  4. β \beta

Bratteli_diagram.html

  1. A 0 A 1 A 2 A_{0}\subseteq A_{1}\subseteq A_{2}\subseteq\cdots
  2. A ^ i \hat{A}_{i}
  3. A i A_{i}
  4. A i λ A_{i}^{\lambda}
  5. λ A ^ i \lambda\in\hat{A}_{i}
  6. A i A i + 1 A_{i}\subseteq A_{i+1}
  7. A i + 1 A_{i+1}
  8. M M
  9. A i A_{i}
  10. g λ , μ g_{\lambda,\mu}
  11. A i + 1 μ A i A i + 1 = λ A ^ i g λ , μ A i λ . A_{i+1}^{\mu}\downarrow^{A_{i+1}}_{A_{i}}=\bigoplus_{\lambda\in\hat{A}_{i}}g_{% \lambda,\mu}A_{i}^{\lambda}.
  12. A 0 A 1 A 2 A_{0}\subseteq A_{1}\subseteq A_{2}\subseteq\cdots
  13. A ^ i \hat{A}_{i}
  14. i i
  15. λ \lambda
  16. i i
  17. μ \mu
  18. i + 1 i+1
  19. g λ , μ g_{\lambda,\mu}
  20. A i = S i A_{i}=S_{i}
  21. A i A_{i}
  22. k = 0 , 1 , 2 , , i / 2 k=0,1,2,\ldots,\lfloor i/2\rfloor
  23. A i A_{i}
  24. k = 0 , 1 , 2 , , i / 2 k=0,1,2,\ldots,\lfloor i/2\rfloor

Bray–Curtis_dissimilarity.html

  1. B C i j = 1 - 2 C i j S i + S j BC_{ij}=1-\frac{2C_{ij}}{S_{i}+S_{j}}
  2. C i j C_{ij}
  3. S i S_{i}
  4. S j S_{j}
  5. Q S i j QS_{ij}
  6. B C ¯ i j = 1 - Q S i j \overline{BC}_{ij}=1-QS_{ij}