wpmath0000014_7

Light_front_holography.html

  1. z z
  2. τ = x 0 + x 3 \tau=x^{0}+x^{3}
  3. z z
  4. P μ P μ | ϕ = 2 | ϕ P_{\mu}P^{\mu}|\phi\rangle=\mathcal{M}^{2}|\phi\rangle
  5. ( - d 2 d ζ 2 - 1 - 4 L 2 4 ζ 2 + U ( ζ ) ) ϕ ( ζ ) = M 2 ϕ ( ζ ) , \left(-\frac{d^{2}}{d\zeta^{2}}-\frac{1-4L^{2}}{4\zeta^{2}}+U(\zeta)\right)% \phi(\zeta)=M^{2}\phi(\zeta),
  6. L L
  7. ζ \zeta
  8. ζ \zeta
  9. z z
  10. U ( ζ ) U(\zeta)

Lill's_method.html

  1. θ θ
  2. x n - 1 , x^{n-1},
  3. x n - 2 , x^{n-2},
  4. θ θ
  5. θ θ
  6. a n x n + a n - 1 x n - 1 + a n - 2 x n - 2 + a_{n}x^{n}+a_{n-1}x^{n-1}+a_{n-2}x^{n-2}+\cdots
  7. a n x a_{n}x
  8. ( a n x + a n - 1 ) x (a_{n}x+a_{n-1})x
  9. ( ( a n x + a n - 1 ) x + a n - 2 ) x , ((a_{n}x+a_{n-1})x+a_{n-2})x,\ \dots

Limiting_parallel.html

  1. l l
  2. P P
  3. R R
  4. l l
  5. R R
  6. A a | | | B b Aa|||Bb
  7. A a Aa
  8. B b Bb
  9. A B AB
  10. B A a BAa
  11. B b Bb
  12. A B AB
  13. a a
  14. A B AB
  15. a a
  16. C C
  17. C A B + C B A < 2 right angles a A B + b B A > 2 right angles \angle CAB+\angle CBA<2\,\text{ right angles}\Rightarrow\angle aAB+\angle bBA>% 2\,\text{ right angles}

Lincoln_index.html

  1. L = E 1 E 2 S L={E_{1}E_{2}\over S}

Lindblad_superoperator.html

  1. C C
  2. ρ \rho
  3. L ( C ) ρ = C ρ C - 1 2 ( C C ρ + ρ C C ) L(C)\rho=C\rho C^{\dagger}-\frac{1}{2}\left(C^{\dagger}C\rho+\rho C^{\dagger}C\right)
  4. ρ ˙ = - i [ ω c a a , ρ ] + L ( 2 κ ( n ¯ + 1 ) a ) ρ + L ( 2 κ n ¯ a ) ρ \dot{\rho}=-i[\omega_{c}a^{\dagger}a,\rho]+L(\sqrt{2\kappa(\bar{n}+1)}a)\rho+L% (\sqrt{2\kappa\bar{n}}a^{\dagger})\rho
  5. ω c \omega_{c}
  6. a a
  7. κ \kappa
  8. n ¯ \bar{n}
  9. H = H S + H B + H B S H=H_{S}+H_{B}+H_{BS}\,
  10. χ ˙ = - i [ H , χ ] \dot{\chi}=-i[H,\chi]
  11. ρ = tr B χ \rho=\operatorname{tr}_{B}\chi
  12. M ~ = U M U \tilde{M}=UMU^{\dagger}
  13. M M
  14. U = e i ( H S + H B ) t U=e^{i(H_{S}+H_{B})t}
  15. χ ~ ˙ = - i [ H ~ B S , χ ~ ] \dot{\tilde{\chi}}=-i[\tilde{H}_{BS},\tilde{\chi}]\,
  16. H ~ B S = e i ( H S + H B ) t H B S e - i ( H S + H B ) t \tilde{H}_{BS}=e^{i(H_{S}+H_{B})t}H_{BS}e^{-i(H_{S}+H_{B})t}
  17. χ ~ ( t ) = χ ~ ( 0 ) - i 0 t d t [ H ~ B S ( t ) , χ ~ ( t ) ] \tilde{\chi}(t)=\tilde{\chi}(0)-i\int^{t}_{0}dt^{\prime}[\tilde{H}_{BS}(t^{% \prime}),\tilde{\chi}(t^{\prime})]
  18. χ ~ \tilde{\chi}
  19. χ ~ ˙ = - i [ H ~ B S , χ ~ ( 0 ) ] - 0 t d t [ H ~ B S ( t ) , [ H ~ B S ( t ) , χ ~ ( t ) ] ] \dot{\tilde{\chi}}=-i[\tilde{H}_{BS},\tilde{\chi}(0)]-\int^{t}_{0}dt^{\prime}[% \tilde{H}_{BS}(t),[\tilde{H}_{BS}(t^{\prime}),\tilde{\chi}(t^{\prime})]]
  20. t = 0 t=0
  21. χ ( 0 ) = ρ ( 0 ) R 0 \chi(0)=\rho(0)R_{0}
  22. R 0 R_{0}
  23. χ ~ = e i H S t ρ e - i H S t e i H B t R e - i H B t \tilde{\chi}=e^{iH_{S}t}\rho e^{-iH_{S}t}e^{iH_{B}t}Re^{-iH_{B}t}
  24. tr R χ ~ = ρ ~ \operatorname{tr}_{R}\tilde{\chi}=\tilde{\rho}
  25. ρ ~ ˙ = - 0 t d t tr R { [ H ~ B S ( t ) , [ H ~ B S ( t ) , χ ~ ( t ) ] ] } \dot{\tilde{\rho}}=-\int^{t}_{0}dt^{\prime}\operatorname{tr}_{R}\{[\tilde{H}_{% BS}(t),[\tilde{H}_{BS}(t^{\prime}),\tilde{\chi}(t^{\prime})]]\}
  26. χ ~ ( t ) = ρ ~ ( t ) R 0 \tilde{\chi}(t)=\tilde{\rho}(t)R_{0}
  27. ρ ~ ˙ = - 0 t d t tr R { [ H ~ B S ( t ) , [ H ~ B S ( t ) , ρ ~ ( t ) R 0 ] ] } \dot{\tilde{\rho}}=-\int^{t}_{0}dt^{\prime}\operatorname{tr}_{R}\{[\tilde{H}_{% BS}(t),[\tilde{H}_{BS}(t^{\prime}),\tilde{\rho}(t^{\prime})R_{0}]]\}
  28. ρ ( t ) ρ ( t ) \rho(t^{\prime})\rightarrow\rho(t)
  29. ρ ~ ˙ = - 0 t d t tr R { [ H ~ B S ( t ) , [ H ~ B S ( t ) , ρ ~ ( t ) R 0 ] ] } \dot{\tilde{\rho}}=-\int^{t}_{0}dt^{\prime}\operatorname{tr}_{R}\{[\tilde{H}_{% BS}(t),[\tilde{H}_{BS}(t^{\prime}),\tilde{\rho}(t)R_{0}]]\}
  30. H B S = α i Γ i H_{BS}=\sum\alpha_{i}\Gamma_{i}
  31. α i \alpha_{i}
  32. Γ i \Gamma_{i}
  33. ρ ~ ˙ = - 0 t d t tr R { [ α i ( t ) Γ i ( t ) , [ α j ( t ) Γ j ( t ) , ρ ~ ( t ) R 0 ] ] } \dot{\tilde{\rho}}=-\sum\int^{t}_{0}dt^{\prime}\operatorname{tr}_{R}\{[\alpha_% {i}(t)\Gamma_{i}(t),[\alpha_{j}(t^{\prime})\Gamma_{j}(t^{\prime}),\tilde{\rho}% (t)R_{0}]]\}
  34. ρ ~ ˙ = - 0 t d t ( α i ( t ) α j ( t ) ρ - α j ( t ) ρ ( t ) α i ( t ) ) Γ i ( t ) Γ j ( t ) + ( ρ ( t ) α j ( t ) α i ( t ) - α i ( t ) ρ ( t ) α j ( t ) ) Γ j ( t ) Γ i ( t ) \dot{\tilde{\rho}}=-\sum\int^{t}_{0}dt^{\prime}\left(\alpha_{i}(t)\alpha_{j}(t% ^{\prime})\rho-\alpha_{j}(t^{\prime})\rho(t)\alpha_{i}(t)\right)\langle\Gamma_% {i}(t)\Gamma_{j}(t^{\prime})\rangle+\left(\rho(t)\alpha_{j}(t^{\prime})\alpha_% {i}(t)-\alpha_{i}(t)\rho(t)\alpha_{j}(t^{\prime})\right)\langle\Gamma_{j}(t^{% \prime})\Gamma_{i}(t)\rangle
  35. Γ i Γ j = tr { Γ i Γ j R 0 } \langle\Gamma_{i}\Gamma_{j}\rangle=\operatorname{tr}\{\Gamma_{i}\Gamma_{j}R_{0}\}
  36. Γ i ( t ) Γ j ( t ) δ ( t , t ) \langle\Gamma_{i}(t)\Gamma_{j}(t^{\prime})\rangle\propto\delta(t,t^{\prime})

Lindström–Gessel–Viennot_lemma.html

  1. A = { a 1 , , a n } A=\{a_{1},\ldots,a_{n}\}
  2. B = { b 1 , , b n } B=\{b_{1},\ldots,b_{n}\}
  3. ω e \omega_{e}
  4. ω ( P ) \omega(P)
  5. e ( a , b ) = P : a b ω ( P ) e(a,b)=\sum_{P:a\to b}\omega(P)
  6. e ( a , b ) e(a,b)
  7. M = ( e ( a 1 , b 1 ) e ( a 1 , b 2 ) e ( a 1 , b n ) e ( a 2 , b 1 ) e ( a 2 , b 2 ) e ( a 2 , b n ) e ( a n , b 1 ) e ( a n , b 2 ) e ( a n , b n ) ) M=\begin{pmatrix}e(a_{1},b_{1})&e(a_{1},b_{2})&\cdots&e(a_{1},b_{n})\\ e(a_{2},b_{1})&e(a_{2},b_{2})&\cdots&e(a_{2},b_{n})\\ \vdots&\vdots&\ddots&\vdots\\ e(a_{n},b_{1})&e(a_{n},b_{2})&\cdots&e(a_{n},b_{n})\end{pmatrix}
  8. σ \sigma
  9. { 1 , 2 , , n } \left\{1,2,...,n\right\}
  10. a i a_{i}
  11. b σ ( i ) b_{\sigma(i)}
  12. i j i\neq j
  13. σ ( P ) \sigma(P)
  14. σ \sigma
  15. det ( M ) = ( P 1 , , P n ) : A B sign ( σ ( P ) ) i = 1 n ω ( P i ) . \det(M)=\sum_{(P_{1},\ldots,P_{n})\colon A\to B}\mathrm{sign}(\sigma(P))\prod_% {i=1}^{n}\omega(P_{i}).
  16. ( 1 , 2 , , n ) (1,2,\ldots,n)
  17. P i P_{i}
  18. a i a_{i}
  19. b σ ( i ) b_{\sigma(i)}
  20. ( a 1 , a 2 , , a n ) (a_{1},a_{2},...,a_{n})
  21. ( b 1 , b 2 , , b n ) (b_{1},b_{2},...,b_{n})
  22. ( P 1 , P 2 , , P n ) (P_{1},P_{2},...,P_{n})
  23. P i P_{i}
  24. a i a_{i}
  25. b i b_{i}
  26. i j i\neq j
  27. P = ( P 1 , P 2 , , P n ) P=(P_{1},P_{2},...,P_{n})
  28. ω ( P ) \omega(P)
  29. ω ( P 1 ) ω ( P 2 ) ω ( P n ) \omega(P_{1})\omega(P_{2})\cdots\omega(P_{n})
  30. ( a 1 , a 2 , , a n ) (a_{1},a_{2},...,a_{n})
  31. ( b 1 , b 2 , , b n ) (b_{1},b_{2},...,b_{n})
  32. ( a 1 , a 2 , , a n ) (a_{1},a_{2},...,a_{n})
  33. ( b σ ( 1 ) , b σ ( 2 ) , , b σ ( n ) ) \left(b_{\sigma(1)},b_{\sigma(2)},...,b_{\sigma(n)}\right)
  34. σ \sigma
  35. S n S_{n}
  36. σ \sigma
  37. σ ( P ) \sigma(P)
  38. σ ( P ) \sigma(P)
  39. det M = σ S n sign ( σ ) e ( a 1 , b σ ( 1 ) ) e ( a 2 , b σ ( 2 ) ) e ( a n , b σ ( n ) ) \det M=\sum_{\sigma\in S_{n}}\mathrm{sign}(\sigma)\cdot e\left(a_{1},b_{\sigma% (1)}\right)e\left(a_{2},b_{\sigma(2)}\right)\cdots e\left(a_{n},b_{\sigma(n)}\right)
  40. = σ S n sign ( σ ) P is an n -path from ( a 1 , a 2 , , a n ) to ( b σ ( 1 ) , b σ ( 2 ) , , b σ ( n ) ) ω ( P ) =\sum_{\sigma\in S_{n}}\mathrm{sign}(\sigma)\cdot\sum_{P\,\text{ is an }n\,% \text{-path from }\left(a_{1},a_{2},...,a_{n}\right)\,\text{ to }\left(b_{% \sigma(1)},b_{\sigma(2)},...,b_{\sigma(n)}\right)}\omega(P)
  41. e ( a i , b σ ( i ) ) e\left(a_{i},b_{\sigma(i)}\right)
  42. a i b σ ( i ) a_{i}\to b_{\sigma(i)}
  43. e ( a 1 , b σ ( 1 ) ) e ( a 2 , b σ ( 2 ) ) e ( a n , b σ ( n ) ) e\left(a_{1},b_{\sigma(1)}\right)e\left(a_{2},b_{\sigma(2)}\right)\cdots e% \left(a_{n},b_{\sigma(n)}\right)
  44. ( a 1 , a 2 , , a n ) (a_{1},a_{2},...,a_{n})
  45. ( b σ ( 1 ) , b σ ( 2 ) , , b σ ( n ) ) \left(b_{\sigma(1)},b_{\sigma(2)},...,b_{\sigma(n)}\right)
  46. P is a twisted n -path from ( a 1 , a 2 , , a n ) to ( b 1 , b 2 , , b n ) sign ( σ ( P ) ) ω ( P ) . \sum_{P\,\text{ is a twisted }n\,\text{-path from }\left(a_{1},a_{2},...,a_{n}% \right)\,\text{ to }\left(b_{1},b_{2},...,b_{n}\right)}\mathrm{sign}(\sigma(P)% )\omega(P).
  47. ( P 1 , , P n ) : A B sign ( σ ( P ) ) i = 1 n ω ( P i ) \sum_{(P_{1},\ldots,P_{n})\colon A\to B}\mathrm{sign}(\sigma(P))\prod_{i=1}^{n% }\omega(P_{i})
  48. = P is a nonintersecting twisted n -path from ( a 1 , a 2 , , a n ) to ( b 1 , b 2 , , b n ) sign ( σ ( P ) ) ω ( P ) . =\sum_{P\,\text{ is a nonintersecting twisted }n\,\text{-path from }\left(a_{1% },a_{2},...,a_{n}\right)\,\text{ to }\left(b_{1},b_{2},...,b_{n}\right)}% \mathrm{sign}(\sigma(P))\omega(P).
  49. sign ( σ ( P ) ) ω ( P ) \mathrm{sign}(\sigma(P))\omega(P)
  50. ( a 1 , a 2 , , a n ) \left(a_{1},a_{2},...,a_{n}\right)
  51. ( b 1 , b 2 , , b n ) \left(b_{1},b_{2},...,b_{n}\right)
  52. sign ( σ ( P ) ) ω ( P ) \mathrm{sign}(\sigma(P))\omega(P)
  53. ( a 1 , a 2 , , a n ) \left(a_{1},a_{2},...,a_{n}\right)
  54. ( b 1 , b 2 , , b n ) \left(b_{1},b_{2},...,b_{n}\right)
  55. ( a 1 , a 2 , , a n ) \left(a_{1},a_{2},...,a_{n}\right)
  56. ( b 1 , b 2 , , b n ) \left(b_{1},b_{2},...,b_{n}\right)
  57. sign ( σ ( P ) ) \mathrm{sign}(\sigma(P))
  58. ω ( P ) \omega(P)
  59. sign ( σ ( P ) ) ω ( P ) \mathrm{sign}(\sigma(P))\omega(P)
  60. ( a 1 , a 2 , , a n ) \left(a_{1},a_{2},...,a_{n}\right)
  61. ( b 1 , b 2 , , b n ) \left(b_{1},b_{2},...,b_{n}\right)
  62. P = ( P 1 , P 2 , , P n ) P=\left(P_{1},P_{2},...,P_{n}\right)
  63. ( a 1 , a 2 , , a n ) \left(a_{1},a_{2},...,a_{n}\right)
  64. ( b 1 , b 2 , , b n ) \left(b_{1},b_{2},...,b_{n}\right)
  65. P i P_{i}
  66. P j P_{j}
  67. σ ( f ( P ) ) \sigma(f(P))
  68. σ ( P ) \sigma(P)
  69. σ ( i ) \sigma(i)
  70. σ ( j ) \sigma(j)
  71. sign ( σ ( f ( P ) ) ) = - sign ( σ ( P ) ) \mathrm{sign}(\sigma(f(P)))=-\mathrm{sign}(\sigma(P))
  72. ω ( f ( P ) ) = ω ( P ) \omega(f(P))=\omega(P)
  73. λ = λ 1 + + λ r \lambda=\lambda_{1}+\cdots+\lambda_{r}
  74. s λ ( x 1 , , x n ) s_{\lambda}(x_{1},\ldots,x_{n})
  75. s λ ( x 1 , , x n ) = T w ( T ) , s_{\lambda}(x_{1},\ldots,x_{n})=\sum_{T}w(T),
  76. x 1 x 3 x 4 3 x 5 x 6 x 7 x_{1}x_{3}x_{4}^{3}x_{5}x_{6}x_{7}
  77. s λ ( x 1 , , x n ) = det ( ( h λ i + j - i ) i , j r × r ) , s_{\lambda}(x_{1},\ldots,x_{n})=\det\left((h_{\lambda_{i}+j-i})_{i,j}^{r\times r% }\right),
  78. s ( 3 , 2 , 2 , 1 ) = | h 3 h 4 h 5 h 6 h 1 h 2 h 3 h 4 1 h 1 h 2 h 3 0 0 1 h 1 | . s_{(3,2,2,1)}=\begin{vmatrix}h_{3}&h_{4}&h_{5}&h_{6}\\ h_{1}&h_{2}&h_{3}&h_{4}\\ 1&h_{1}&h_{2}&h_{3}\\ 0&0&1&h_{1}\end{vmatrix}.
  79. a i = ( r + 1 - i , 1 ) a_{i}=(r+1-i,1)
  80. b i = ( λ i + r + 1 - i , n ) b_{i}=(\lambda_{i}+r+1-i,n)
  81. 2 \mathbb{Z}^{2}
  82. e ( a , b ) e(a,b)
  83. e ( a , b ) e(a,b)

Line_integral_convolution.html

  1. 𝐮 \mathbf{u}
  2. s y m b o l σ ( s ) d s = 𝐮 ( s y m b o l σ ( s ) ) | 𝐮 ( s y m b o l σ ( s ) ) | \frac{symbol\sigma(s)}{ds}=\frac{\mathbf{u}(symbol{\sigma}(s))}{|\mathbf{u}(% symbol{\sigma}(s))|}
  3. s y m b o l σ 𝐫 ( s ) symbol{\sigma}_{\mathbf{r}}(s)
  4. 𝐫 \mathbf{r}
  5. s = 0 s=0
  6. 𝐫 \mathbf{r}
  7. D ( 𝐫 ) = - L / 2 L / 2 k ( s ) N ( s y m b o l σ 𝐫 ( s ) ) d s D(\mathbf{r})=\int_{-L/2}^{L/2}k(s)N(symbol{\sigma}_{\mathbf{r}}(s))ds
  8. k ( s ) k(s)
  9. N ( 𝐫 ) N(\mathbf{r})
  10. L L
  11. t t
  12. t + δ t t+\delta t

Linear_temporal_logic_to_Büchi_automaton.html

  1. \vDash
  2. \vDash
  3. \vDash

Linguistic_sequence_complexity.html

  1. C = U 1 U 2 U i . U w C=U_{1}U_{2}...U_{i}....U_{w}
  2. < v a r > i <var>i

LINPACK_benchmarks.html

  1. A x - b A x n ϵ O ( 1 ) {\lVert Ax-b\rVert\over\lVert A\rVert\lVert x\rVert n\epsilon}\leq O(1)
  2. ϵ \epsilon
  3. \lVert\cdot\rVert
  4. O ( 1 ) O(1)

Liouville_field_theory.html

  1. S = 1 4 π d 2 x g ( g μ ν μ ϕ ν ϕ + ( b + b - 1 ) R ϕ + 4 π e 2 b ϕ ) , S=\frac{1}{4\pi}\int d^{2}x\sqrt{g}(g^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}% \phi+(b+b^{-1})R\phi+4\pi e^{2b\phi}),
  2. μ = / x μ , g μ ν \partial_{\mu}=\partial/\partial x^{\mu},\ g_{\mu\nu}
  3. R R
  4. b b
  5. ϕ \phi
  6. Δ ϕ ( x ) = 1 2 ( b + b - 1 ) R ( x ) + 4 π b e 2 b ϕ ( x ) \Delta\phi(x)=\frac{1}{2}(b+b^{-1})R(x)+4\pi be^{2b\phi(x)}
  7. Δ = g - 1 / 2 μ ( g 1 / 2 g μ ν ν ) \Delta=g^{-1/2}\partial_{\mu}(g^{1/2}g^{\mu\nu}\partial_{\nu})
  8. ( 2 x 2 + 2 y 2 ) ϕ ( x , y ) = 4 π b e 2 b ϕ ( x , y ) \left(\frac{\partial^{2}}{\partial x^{2}}+\frac{\partial^{2}}{\partial y^{2}}% \right)\phi(x,y)=4\pi be^{2b\phi(x,y)}
  9. c c
  10. c = 1 + 6 ( b + 1 / b ) 2 c=1+6(b+1/b)^{2}
  11. S L ( 2 , R ) SL(2,R)
  12. N = 1 N=1
  13. A N A_{N}

Liouville_gravity.html

  1. S = 1 4 π d 2 x - g [ ( b + b - 1 ) Φ R + ( Φ ) 2 + 4 π μ e 2 b Φ ] , S=\frac{1}{4\pi}\int d^{2}x\sqrt{-g}\left[\left(b+b^{-1}\right)\Phi R+\left(% \nabla\Phi\right)^{2}+4\pi\mu e^{2b\Phi}\right],

Lipophilic_efficiency.html

  1. L i P E = p I C 50 - L o g P \ LiPE=pIC_{50}-LogP
  2. L E l i p o = l o g ( - Δ G P ) \ LElipo=log\bigg(\frac{\ -\ \Delta G}{P}\bigg)

List_of_films_with_associated_hip_hop_songs.html

  1. t = 1 t = N 11 - r t \sum_{t=1}^{t=N}11-r_{t}

List_of_formulas_in_elementary_geometry.html

  1. r 2 r^{2}
  2. V = s 3 V=s^{3}
  3. 6 s 2 6s^{2}
  4. 4 / 3 4/3
  5. r 3 r^{3}
  6. r 2 r^{2}
  7. r 2 r^{2}
  8. r 2 r^{2}
  9. C = 2 π r C=\,2\pi r
  10. A = π r 2 A=\,\pi r^{2}
  11. d A = d r r d θ dA=\,\mathrm{d}r\,r\,\mathrm{d}\theta
  12. A = 4 π r 2 A=4\pi r^{2}
  13. V = 4 3 π r 3 V=\frac{4}{3}\pi r^{3}
  14. d A = r 2 sin θ d θ d ϕ dA=r^{2}\sin\theta\,d\theta\,d\phi
  15. d V = r 2 sin θ d r d θ d φ \mathrm{d}V=r^{2}\sin\theta\,\mathrm{d}r\,\mathrm{d}\theta\,\mathrm{d}\varphi

List_of_integrals_of_Gaussian_functions.html

  1. ϕ ( x ) = e - x 2 2 2 π \phi(x)=\frac{e^{-\frac{x^{2}}{2}}}{\sqrt{2\pi}}
  2. Φ ( x ) = - x ϕ ( t ) d t = 1 2 ( 1 + erf ( x 2 ) ) \Phi(x)=\int_{-\infty}^{x}\phi(t)dt=\frac{1}{2}\left(1+\operatorname{erf}\left% (\frac{x}{\sqrt{2}}\right)\right)
  3. T ( h , a ) = ϕ ( h ) 0 a ϕ ( h x ) 1 + x 2 d x T(h,a)=\phi(h)\int_{0}^{a}\frac{\phi(hx)}{1+x^{2}}\,dx
  4. ϕ ( x ) d x = Φ ( x ) + C \int\phi(x)\,dx=\Phi(x)+C
  5. x ϕ ( x ) d x = - ϕ ( x ) + C \int x\phi(x)\,dx=-\phi(x)+C
  6. x 2 ϕ ( x ) d x = Φ ( x ) - x ϕ ( x ) + C \int x^{2}\phi(x)\,dx=\Phi(x)-x\phi(x)+C
  7. x 2 k + 1 ϕ ( x ) d x = - ϕ ( x ) j = 0 k ( 2 k ) ! ! ( 2 j ) ! ! x 2 j + C \int x^{2k+1}\phi(x)\,dx=-\phi(x)\sum_{j=0}^{k}\frac{(2k)!!}{(2j)!!}x^{2j}+C
  8. x 2 k + 2 ϕ ( x ) d x = - ϕ ( x ) j = 0 k ( 2 k + 1 ) ! ! ( 2 j + 1 ) ! ! x 2 j + 1 + ( 2 k + 1 ) ! ! Φ ( x ) + C \int x^{2k+2}\phi(x)\,dx=-\phi(x)\sum_{j=0}^{k}\frac{(2k+1)!!}{(2j+1)!!}x^{2j+% 1}+(2k+1)!!\,\Phi(x)+C
  9. ϕ ( x ) 2 d x = 1 2 π Φ ( x 2 ) + C \int\phi(x)^{2}\,dx=\tfrac{1}{2\sqrt{\pi}}\Phi(x\sqrt{2})+C
  10. ϕ ( x ) ϕ ( a + b x ) d x = 1 t ϕ ( a t ) Φ ( t x + a b t ) + C , t = 1 + b 2 \int\phi(x)\phi(a+bx)\,dx=\tfrac{1}{t}\phi(\tfrac{a}{t})\Phi(tx+\tfrac{ab}{t})% +C,\qquad t=\sqrt{1+b^{2}}
  11. x ϕ ( a + b x ) d x = - b - 2 ( ϕ ( a + b x ) + a Φ ( a + b x ) ) + C \int x\phi(a+bx)\,dx=-b^{-2}\left(\phi(a+bx)+a\Phi(a+bx)\right)+C
  12. x 2 ϕ ( a + b x ) d x = b - 3 ( ( a 2 + 1 ) Φ ( a + b x ) + ( a - b x ) ϕ ( a + b x ) ) + C \int x^{2}\phi(a+bx)\,dx=b^{-3}\left((a^{2}+1)\Phi(a+bx)+(a-bx)\phi(a+bx)% \right)+C
  13. ϕ ( a + b x ) n d x = 1 b n ( 2 π ) n - 1 Φ ( n ( a + b x ) ) + C \int\phi(a+bx)^{n}\,dx=\frac{1}{b\sqrt{n(2\pi)^{n-1}}}\Phi\left(\sqrt{n}(a+bx)% \right)+C
  14. Φ ( a + b x ) d x = b - 1 ( ( a + b x ) Φ ( a + b x ) + ϕ ( a + b x ) ) + C \int\Phi(a+bx)\,dx=b^{-1}\left((a+bx)\Phi(a+bx)+\phi(a+bx)\right)+C
  15. x Φ ( a + b x ) d x = 1 2 b 2 ( ( b 2 x 2 - a 2 - 1 ) Φ ( a + b x ) + ( b x - a ) ϕ ( a + b x ) ) + C \int x\Phi(a+bx)\,dx=\tfrac{1}{2b^{2}}\left((b^{2}x^{2}-a^{2}-1)\Phi(a+bx)+(bx% -a)\phi(a+bx)\right)+C
  16. x 2 Φ ( a + b x ) d x = 1 3 b 3 ( ( b 3 x 3 + a 3 + 3 a ) Φ ( a + b x ) + ( b 2 x 2 - a b x + a 2 + 2 ) ϕ ( a + b x ) ) + C \int x^{2}\Phi(a+bx)\,dx=\tfrac{1}{3b^{3}}\left((b^{3}x^{3}+a^{3}+3a)\Phi(a+bx% )+(b^{2}x^{2}-abx+a^{2}+2)\phi(a+bx)\right)+C
  17. x n Φ ( x ) d x = 1 n + 1 ( ( x n + 1 - n x n - 1 ) Φ ( x ) + x n ϕ ( x ) + n ( n - 1 ) x n - 2 Φ ( x ) d x ) + C \int x^{n}\Phi(x)\,dx=\frac{1}{n+1}\left(\left(x^{n+1}-nx^{n-1}\right)\Phi(x)+% x^{n}\phi(x)+n(n-1)\int x^{n-2}\Phi(x)\,dx\right)+C
  18. x ϕ ( x ) Φ ( a + b x ) d x = b t ϕ ( a t ) Φ ( x t + a b t ) - ϕ ( x ) Φ ( a + b x ) + C , t = 1 + b 2 \int x\phi(x)\Phi(a+bx)\,dx=\tfrac{b}{t}\phi(\tfrac{a}{t})\Phi(xt+\tfrac{ab}{t% })-\phi(x)\Phi(a+bx)+C,\qquad t=\sqrt{1+b^{2}}
  19. Φ ( x ) 2 d x = x Φ ( x ) 2 + 2 Φ ( x ) ϕ ( x ) - 1 π Φ ( x 2 ) + C \int\Phi(x)^{2}\,dx=x\Phi(x)^{2}+2\Phi(x)\phi(x)-\tfrac{1}{\sqrt{\pi}}\Phi(x% \sqrt{2})+C
  20. e c x ϕ ( b x ) n d x = e c 2 2 n b 2 b n ( 2 π ) n - 1 Φ ( b 2 x n - c b n ) + C , b 0 , n > 0 \int e^{cx}\phi(bx)^{n}\,dx=\frac{e^{\frac{c^{2}}{2nb^{2}}}}{b\sqrt{n(2\pi)^{n% -1}}}\Phi\left(\frac{b^{2}xn-c}{b\sqrt{n}}\right)+C,\qquad b\neq 0,n>0
  21. - x 2 ϕ ( x ) n d x = 1 n 3 ( 2 π ) n - 1 \int_{-\infty}^{\infty}x^{2}\phi(x)^{n}\,dx=\frac{1}{\sqrt{n^{3}(2\pi)^{n-1}}}
  22. - 0 ϕ ( a x ) Φ ( b x ) d x = ( 2 π | a | ) - 1 ( π 2 - arctan ( b | a | ) ) \int_{-\infty}^{0}\phi(ax)\Phi(bx)dx=(2\pi|a|)^{-1}\left(\tfrac{\pi}{2}-% \arctan(\tfrac{b}{|a|})\right)
  23. 0 ϕ ( a x ) Φ ( b x ) d x = ( 2 π | a | ) - 1 ( π 2 + arctan ( b | a | ) ) \int_{0}^{\infty}\phi(ax)\Phi(bx)\,dx=(2\pi|a|)^{-1}\left(\tfrac{\pi}{2}+% \arctan(\tfrac{b}{|a|})\right)
  24. 0 x ϕ ( x ) Φ ( b x ) d x = 1 2 2 π ( 1 + b 1 + b 2 ) \int_{0}^{\infty}x\phi(x)\Phi(bx)\,dx=\frac{1}{2\sqrt{2\pi}}\left(1+\frac{b}{% \sqrt{1+b^{2}}}\right)
  25. 0 x 2 ϕ ( x ) Φ ( b x ) d x = 1 4 + 1 2 π ( b 1 + b 2 + arctan ( b ) ) \int_{0}^{\infty}x^{2}\phi(x)\Phi(bx)\,dx=\frac{1}{4}+\frac{1}{2\pi}\left(% \frac{b}{1+b^{2}}+\arctan(b)\right)
  26. x ϕ ( x ) 2 Φ ( x ) d x = 1 4 π 3 \int x\phi(x)^{2}\Phi(x)\,dx=\frac{1}{4\pi\sqrt{3}}
  27. 0 Φ ( b x ) 2 ϕ ( x ) d x = ( 2 π ) - 1 ( arctan ( b ) + arctan 1 + 2 b 2 ) \int_{0}^{\infty}\Phi(bx)^{2}\phi(x)\,dx=(2\pi)^{-1}\left(\arctan(b)+\arctan% \sqrt{1+2b^{2}}\right)
  28. - Φ ( a + b x ) 2 ϕ ( x ) d x = Φ ( a 1 + b 2 ) - 2 T ( a 1 + b 2 , 1 1 + 2 b 2 ) \int_{-\infty}^{\infty}\Phi(a+bx)^{2}\phi(x)\,dx=\Phi\left(\frac{a}{\sqrt{1+b^% {2}}}\right)-2T\left(\frac{a}{\sqrt{1+b^{2}}},\frac{1}{\sqrt{1+2b^{2}}}\right)
  29. - x Φ ( a + b x ) 2 ϕ ( x ) d x = 2 b 1 + b 2 ϕ ( a t ) Φ ( a 1 + b 2 1 + 2 b 2 ) \int_{-\infty}^{\infty}x\Phi(a+bx)^{2}\phi(x)\,dx=\frac{2b}{\sqrt{1+b^{2}}}% \phi(\tfrac{a}{t})\Phi\left(\frac{a}{\sqrt{1+b^{2}}\sqrt{1+2b^{2}}}\right)
  30. - Φ ( b x ) 2 ϕ ( x ) d x = π - 1 arctan 1 + 2 b 2 \int_{-\infty}^{\infty}\Phi(bx)^{2}\phi(x)\,dx=\pi^{-1}\arctan\sqrt{1+2b^{2}}
  31. - x ϕ ( x ) Φ ( b x ) d x = - x ϕ ( x ) Φ ( b x ) 2 d x = b 2 π ( 1 + b 2 ) \int_{-\infty}^{\infty}x\phi(x)\Phi(bx)\,dx=\int_{-\infty}^{\infty}x\phi(x)% \Phi(bx)^{2}\,dx=\frac{b}{\sqrt{2\pi(1+b^{2})}}
  32. - Φ ( a + b x ) ϕ ( x ) d x = Φ ( a 1 + b 2 ) \int_{-\infty}^{\infty}\Phi(a+bx)\phi(x)\,dx=\Phi\left(\frac{a}{\sqrt{1+b^{2}}% }\right)
  33. - x Φ ( a + b x ) ϕ ( x ) d x = b t ϕ ( a t ) , t = 1 + b 2 \int_{-\infty}^{\infty}x\Phi(a+bx)\phi(x)\,dx=\tfrac{b}{t}\phi(\tfrac{a}{t}),% \qquad t=\sqrt{1+b^{2}}
  34. 0 x Φ ( a + b x ) ϕ ( x ) d x = b t ϕ ( a t ) Φ ( - a b t ) + ( 2 π ) - 1 / 2 Φ ( a ) , t = 1 + b 2 \int_{0}^{\infty}x\Phi(a+bx)\phi(x)\,dx=\tfrac{b}{t}\phi(\tfrac{a}{t})\Phi(-% \tfrac{ab}{t})+(2\pi)^{-1/2}\Phi(a),\qquad t=\sqrt{1+b^{2}}
  35. - ln ( x 2 ) 1 σ ϕ ( x σ ) d x = ln ( σ 2 ) - γ - ln 2 ln ( σ 2 ) - 1.27036 \int_{-\infty}^{\infty}\ln(x^{2})\tfrac{1}{\sigma}\phi\left(\tfrac{x}{\sigma}% \right)\,dx=\ln(\sigma^{2})-\gamma-\ln 2\approx\ln(\sigma^{2})-1.27036

List_of_logic_systems.html

  1. { , ¬ } \{\to,\neg\}
  2. A , A B B . A,A\to B\vdash B.
  3. A ( B A ) A\to(B\to A)
  4. ( A ( B C ) ) ( ( A B ) ( A C ) ) (A\to(B\to C))\to((A\to B)\to(A\to C))
  5. ( A B ) ( ¬ B ¬ A ) (A\to B)\to(\neg B\to\neg A)
  6. ¬ ¬ A A \neg\neg A\to A
  7. A ¬ ¬ A A\to\neg\neg A
  8. A ( B A ) A\to(B\to A)
  9. ( A ( B C ) ) ( B ( A C ) ) (A\to(B\to C))\to(B\to(A\to C))
  10. ( B C ) ( ( A B ) ( A C ) ) (B\to C)\to((A\to B)\to(A\to C))
  11. A ( ¬ A B ) A\to(\neg A\to B)
  12. ( A B ) ( ( ¬ A B ) B ) (A\to B)\to((\neg A\to B)\to B)
  13. ( A B ) ( ( B C ) ( A C ) ) (A\to B)\to((B\to C)\to(A\to C))
  14. ( ¬ A A ) A (\neg A\to A)\to A
  15. A ( ¬ A B ) A\to(\neg A\to B)
  16. ( ( A B ) C ) ( ¬ A C ) ((A\to B)\to C)\to(\neg A\to C)
  17. ( ( A B ) C ) ( B C ) ((A\to B)\to C)\to(B\to C)
  18. ( ¬ A C ) ( ( B C ) ( ( A B ) C ) ) (\neg A\to C)\to((B\to C)\to((A\to B)\to C))
  19. A ( B A ) A\to(B\to A)
  20. ( A ( B C ) ) ( ( A B ) ( A C ) ) (A\to(B\to C))\to((A\to B)\to(A\to C))
  21. ( ¬ A ¬ B ) ( B A ) (\neg A\to\neg B)\to(B\to A)
  22. ( A B ) ( ( B C ) ( A C ) ) (A\to B)\to((B\to C)\to(A\to C))
  23. A ( ¬ A B ) A\to(\neg A\to B)
  24. ( ¬ A B ) ( ( B A ) A ) (\neg A\to B)\to((B\to A)\to A)
  25. [ ( A ( B A ) ) ( [ ( ¬ C ( D ¬ E ) ) [ ( C ( D F ) ) ( ( E D ) ( E F ) ) ] ] G ) ] ( H G ) [(A\to(B\to A))\to([(\neg C\to(D\to\neg E))\to[(C\to(D\to F))\to((E\to D)\to(E% \to F))]]\to G)]\to(H\to G)
  26. ( ( ( ( A B ) ( ¬ C ¬ D ) ) C ) E ) ( ( E A ) ( D A ) ) ((((A\to B)\to(\neg C\to\neg D))\to C)\to E)\to((E\to A)\to(D\to A))
  27. A ( B A ) A\to(B\to A)
  28. ( A ( B C ) ) ( ( A B ) ( A C ) ) (A\to(B\to C))\to((A\to B)\to(A\to C))
  29. ( ¬ A ¬ B ) ( ( ¬ A B ) A ) (\neg A\to\neg B)\to((\neg A\to B)\to A)
  30. A ( B A ) A\to(B\to A)
  31. ( A B ) ( ( B C ) ( A C ) ) (A\to B)\to((B\to C)\to(A\to C))
  32. ( A ( B C ) ) ( B ( A C ) ) (A\to(B\to C))\to(B\to(A\to C))
  33. ¬ ¬ A A \neg\neg A\to A
  34. ( A ¬ A ) ¬ A (A\to\neg A)\to\neg A
  35. ( A ¬ B ) ( B ¬ A ) (A\to\neg B)\to(B\to\neg A)
  36. ( A B ) ( ¬ B ( A C ) ) (A\to B)\to(\neg B\to(A\to C))
  37. A ( B ( C A ) ) A\to(B\to(C\to A))
  38. ( ¬ A B ) ( ( A B ) B ) (\neg A\to B)\to((A\to B)\to B)
  39. ¬ A ( A B ) \neg A\to(A\to B)
  40. A ( B ( C A ) ) A\to(B\to(C\to A))
  41. ( ¬ A C ) ( ( B C ) ( ( A B ) C ) ) (\neg A\to C)\to((B\to C)\to((A\to B)\to C))
  42. { , } \{\to,\bot\}
  43. ( A B ) ( ( B C ) ( A C ) ) (A\to B)\to((B\to C)\to(A\to C))
  44. A ( B A ) A\to(B\to A)
  45. ( ( A B ) A ) A ((A\to B)\to A)\to A
  46. A \bot\to A
  47. A ( B A ) A\to(B\to A)
  48. ( A ( B C ) ) ( ( A B ) ( A C ) ) (A\to(B\to C))\to((A\to B)\to(A\to C))
  49. ( ( A ) ) A ((A\to\bot)\to\bot)\to A
  50. ( ( ( ( A B ) ( C ) ) D ) E ) ( ( E A ) ( C A ) ) ((((A\to B)\to(C\to\bot))\to D)\to E)\to((E\to A)\to(C\to A))
  51. ( ( A B ) ( ( C ) D ) ) ( ( D A ) ( E ( F A ) ) ) ((A\to B)\to((\bot\to C)\to D))\to((D\to A)\to(E\to(F\to A)))
  52. { ¬ , } \{\neg,\lor\}
  53. A , ¬ A B B . A,\neg A\lor B\vdash B.
  54. ¬ ( ¬ B C ) ( ¬ ( A B ) ( A C ) ) \neg(\neg B\lor C)\lor(\neg(A\lor B)\lor(A\lor C))
  55. ¬ ( A B ) ( B A ) \neg(A\lor B)\lor(B\lor A)
  56. ¬ A ( B A ) \neg A\lor(B\lor A)
  57. ¬ ( A A ) A \neg(A\lor A)\lor A
  58. ¬ ( ¬ ( ¬ A B ) ( C ( D E ) ) ) ( ¬ ( ¬ D A ) ( C ( E A ) ) ) \neg(\neg(\neg A\lor B)\lor(C\lor(D\lor E)))\lor(\neg(\neg D\lor A)\lor(C\lor(% E\lor A)))
  59. ¬ ( ¬ ( ¬ A B ) ( C ( D E ) ) ) ( ¬ ( ¬ E D ) ( C ( A D ) ) ) \neg(\neg(\neg A\lor B)\lor(C\lor(D\lor E)))\lor(\neg(\neg E\lor D)\lor(C\lor(% A\lor D)))
  60. ¬ ( ¬ ( ¬ A B ) ( C ( D E ) ) ) ( ¬ ( ¬ C A ) ( E ( D A ) ) ) \neg(\neg(\neg A\lor B)\lor(C\lor(D\lor E)))\lor(\neg(\neg C\lor A)\lor(E\lor(% D\lor A)))
  61. A , A ( B C ) C . A,A\mid(B\mid C)\vdash C.
  62. ( A ( B C ) ) [ ( E ( E E ) ) ( ( D B ) [ ( A D ) ( A D ) ] ) ] (A\mid(B\mid C))\mid[(E\mid(E\mid E))\mid((D\mid B)\mid[(A\mid D)\mid(A\mid D)% ])]
  63. ( A ( B C ) ) [ ( D ( D D ) ) ( ( D B ) [ ( A D ) ( A D ) ] ) ] (A\mid(B\mid C))\mid[(D\mid(D\mid D))\mid((D\mid B)\mid[(A\mid D)\mid(A\mid D)% ])]
  64. ( A ( B C ) ) [ ( A ( C A ) ) ( ( D B ) [ ( A D ) ( A D ) ] ) ] (A\mid(B\mid C))\mid[(A\mid(C\mid A))\mid((D\mid B)\mid[(A\mid D)\mid(A\mid D)% ])]
  65. ( A ( B C ) ) [ ( ( D C ) [ ( A D ) ( A D ) ] ) ( A ( A B ) ) ] (A\mid(B\mid C))\mid[((D\mid C)\mid[(A\mid D)\mid(A\mid D)])\mid(A\mid(A\mid B% ))]
  66. ( A ( B C ) ) [ ( A ( B C ) ) ( ( D C ) [ ( C D ) ( A D ) ] ) ] (A\mid(B\mid C))\mid[(A\mid(B\mid C))\mid((D\mid C)\mid[(C\mid D)\mid(A\mid D)% ])]
  67. ( A ( B C ) ) [ ( [ ( B D ) ( A D ) ] ( D B ) ) ( ( C B ) A ) ] (A\mid(B\mid C))\mid[([(B\mid D)\mid(A\mid D)]\mid(D\mid B))\mid((C\mid B)\mid A)]
  68. A ( B A ) A\to(B\to A)
  69. ( A B ) ( ( B C ) ( A C ) ) (A\to B)\to((B\to C)\to(A\to C))
  70. ( ( A B ) A ) A ((A\to B)\to A)\to A
  71. [ ( A ( B A ) ) [ ( [ ( ( C D ) E ) F ] [ ( D F ) ( C F ) ] ) G ] ] G [(A\to(B\to A))\to[([((C\to D)\to E)\to F]\to[(D\to F)\to(C\to F)])\to G]]\to G
  72. [ ( A B ) ( ( C D ) E ) ] ( [ F ( ( C D ) E ) ] [ ( A F ) ( D E ) ] ) [(A\to B)\to((C\to D)\to E)]\to([F\to((C\to D)\to E)]\to[(A\to F)\to(D\to E)])
  73. ( ( A B ) ( C D ) ) ( E ( ( D A ) ( C A ) ) ) ((A\to B)\to(C\to D))\to(E\to((D\to A)\to(C\to A)))
  74. ( ( A B ) ( C D ) ) ( ( D A ) ( E ( C A ) ) ) ((A\to B)\to(C\to D))\to((D\to A)\to(E\to(C\to A)))
  75. ( ( A B ) C ) ( ( C A ) ( D A ) ) ((A\to B)\to C)\to((C\to A)\to(D\to A))
  76. { , , , } \{\to,\land,\lor,\bot\}
  77. A ( B A ) A\to(B\to A)
  78. ( A ( B C ) ) ( ( A B ) ( A C ) ) (A\to(B\to C))\to((A\to B)\to(A\to C))
  79. ( A B ) A (A\land B)\to A
  80. ( A B ) B (A\land B)\to B
  81. A ( B ( A B ) ) A\to(B\to(A\land B))
  82. A ( A B ) A\to(A\lor B)
  83. B ( A B ) B\to(A\lor B)
  84. ( A C ) ( ( B C ) ( ( A B ) C ) ) (A\to C)\to((B\to C)\to((A\lor B)\to C))
  85. A \bot\to A
  86. { , , , ¬ } \{\to,\land,\lor,\neg\}
  87. ( A ¬ A ) ¬ A (A\to\neg A)\to\neg A
  88. ¬ A ( A B ) \neg A\to(A\to B)
  89. ¬ A ¬ ¬ A . \neg A\lor\neg\neg A.
  90. ( A B ) ( B A ) . (A\to B)\lor(B\to A).
  91. A ( B A ) A\to(B\to A)
  92. ( A ( B C ) ) ( ( A B ) ( A C ) ) (A\to(B\to C))\to((A\to B)\to(A\to C))
  93. E ( ( A B ) ( ( ( D A ) ( B C ) ) ( A C ) ) ) E\to((A\to B)\to(((D\to A)\to(B\to C))\to(A\to C)))
  94. A ( B A ) A\to(B\to A)
  95. ( A B ) ( ( A ( B C ) ) ( A C ) ) (A\to B)\to((A\to(B\to C))\to(A\to C))
  96. ( ( A B ) C ) ( D ( ( B ( C E ) ) ( B E ) ) ) ((A\to B)\to C)\to(D\to((B\to(C\to E))\to(B\to E)))
  97. ( A ( A B ) ) ( A B ) (A\to(A\to B))\to(A\to B)
  98. ( B C ) ( ( A B ) ( A C ) ) (B\to C)\to((A\to B)\to(A\to C))
  99. ( A ( B C ) ) ( B ( A C ) ) (A\to(B\to C))\to(B\to(A\to C))
  100. A ( B A ) A\to(B\to A)
  101. ( A ( A B ) ) ( A B ) (A\to(A\to B))\to(A\to B)
  102. ( A B ) ( ( B C ) ( A C ) ) (A\to B)\to((B\to C)\to(A\to C))
  103. A ( B A ) A\to(B\to A)
  104. A A A\to A
  105. ( A B ) ( ( B C ) ( A C ) ) (A\to B)\to((B\to C)\to(A\to C))
  106. ( B C ) ( ( A B ) ( A C ) ) (B\to C)\to((A\to B)\to(A\to C))
  107. ( A ( A B ) ) ( A B ) (A\to(A\to B))\to(A\to B)
  108. { , , } \{\to,\land,\lor\}
  109. ( A B ) A (A\land B)\to A
  110. ( A B ) B (A\land B)\to B
  111. A ( B ( A B ) ) A\to(B\to(A\land B))
  112. A ( A B ) A\to(A\lor B)
  113. B ( A B ) B\to(A\lor B)
  114. ( A C ) ( ( B C ) ( ( A B ) C ) ) (A\to C)\to((B\to C)\to((A\lor B)\to C))
  115. \leftrightarrow
  116. ( A B ) ( A B ) (A\leftrightarrow B)\to(A\to B)
  117. ( A B ) ( B A ) (A\leftrightarrow B)\to(B\to A)
  118. ( A B ) ( ( B A ) ( A B ) ) (A\to B)\to((B\to A)\to(A\leftrightarrow B))
  119. \bot
  120. { , , , ¬ } \{\to,\land,\lor,\neg\}
  121. ( A ¬ B ) ( B ¬ A ) (A\to\neg B)\to(B\to\neg A)
  122. ( A B ) ( ¬ B ¬ A ) (A\to B)\to(\neg B\to\neg A)
  123. A ¬ ¬ A A\to\neg\neg A
  124. ( A ¬ B ) ( B ¬ A ) (A\to\neg B)\to(B\to\neg A)
  125. ¬ A ( A B ) \neg A\to(A\to B)
  126. ( A ¬ A ) ¬ A (A\to\neg A)\to\neg A
  127. ¬ A ( A B ) \neg A\to(A\to B)
  128. { , , , ¬ } \{\to,\land,\lor,\neg\}
  129. ( ¬ A ¬ B ) ( B A ) (\neg A\to\neg B)\to(B\to A)
  130. ( A ¬ B ) ( B ¬ A ) (A\to\neg B)\to(B\to\neg A)
  131. ¬ ¬ A A \neg\neg A\to A
  132. ¬ A ( A B ) \neg A\to(A\to B)
  133. A ¬ ¬ A A\leftrightarrow\neg\neg A
  134. ¬ ( A B ) ( ¬ A ¬ B ) \neg(A\lor B)\leftrightarrow(\neg A\land\neg B)
  135. ¬ ( A B ) ( ¬ A ¬ B ) \neg(A\land B)\leftrightarrow(\neg A\lor\neg B)
  136. \equiv
  137. A , A B B A,A\equiv B\vdash B
  138. ( ( A C ) ( B A ) ) ( C B ) ((A\equiv C)\equiv(B\equiv A))\equiv(C\equiv B)
  139. ( A ( B C ) ) ( ( A B ) C ) (A\equiv(B\equiv C))\equiv((A\equiv B)\equiv C)
  140. A A A\equiv A
  141. ( A B ) ( B A ) (A\equiv B)\equiv(B\equiv A)
  142. ( A B ) ( ( B C ) ( A C ) ) (A\equiv B)\equiv((B\equiv C)\equiv(A\equiv C))
  143. ( A ( B C ) ) ( ( A B ) C ) (A\equiv(B\equiv C))\equiv((A\equiv B)\equiv C)
  144. ( ( A C ) ( B A ) ) ( C B ) ((A\equiv C)\equiv(B\equiv A))\equiv(C\equiv B)
  145. ( A B ) ( B A ) (A\equiv B)\equiv(B\equiv A)
  146. ( ( A C ) ( B A ) ) ( C B ) ((A\equiv C)\equiv(B\equiv A))\equiv(C\equiv B)
  147. ( A B ) ( ( C B ) ( A C ) ) (A\equiv B)\equiv((C\equiv B)\equiv(A\equiv C))
  148. ( A B ) ( ( A C ) ( C B ) ) (A\equiv B)\equiv((A\equiv C)\equiv(C\equiv B))
  149. ( A B ) ( ( C A ) ( B C ) ) (A\equiv B)\equiv((C\equiv A)\equiv(B\equiv C))
  150. ( ( A B ) C ) ( B ( C A ) ) ((A\equiv B)\equiv C)\equiv(B\equiv(C\equiv A))
  151. A ( ( B ( A C ) ) ( C B ) ) A\equiv((B\equiv(A\equiv C))\equiv(C\equiv B))
  152. ( A ( B C ) ) ( C ( A B ) ) (A\equiv(B\equiv C))\equiv(C\equiv(A\equiv B))
  153. ( A B ) ( C ( ( B C ) A ) ) (A\equiv B)\equiv(C\equiv((B\equiv C)\equiv A))
  154. ( A B ) ( C ( ( C B ) A ) ) (A\equiv B)\equiv(C\equiv((C\equiv B)\equiv A))
  155. ( ( A ( B C ) ) C ) ( B A ) ((A\equiv(B\equiv C))\equiv C)\equiv(B\equiv A)
  156. ( ( A ( B C ) ) B ) ( C A ) ((A\equiv(B\equiv C))\equiv B)\equiv(C\equiv A)
  157. A ( ( B ( C A ) ) ( C B ) ) A\equiv((B\equiv(C\equiv A))\equiv(C\equiv B))
  158. A ( ( B C ) ( ( A C ) B ) ) A\equiv((B\equiv C)\equiv((A\equiv C)\equiv B))
  159. A ( ( B C ) ( ( C A ) B ) ) A\equiv((B\equiv C)\equiv((C\equiv A)\equiv B))
  160. A ( ( ( A B ) ( C B ) ) C ) A\equiv(((A\equiv B)\equiv(C\equiv B))\equiv C)

List_of_numeral_systems.html

  1. e e
  2. π \pi
  3. 2 \sqrt{2}
  4. 2 12 \sqrt[12]{2}

List_of_Russian_physicists.html

  1. E = k m c 2 E=kmc^{2}

List_of_Russian_scientists.html

  1. E = k m c 2 E=kmc^{2}

List_of_types_of_numbers.html

  1. \scriptstyle\mathbb{N}
  2. \scriptstyle\mathbb{Z}
  3. \scriptstyle\mathbb{Q}
  4. \scriptstyle\mathbb{R}
  5. 𝕀 \scriptstyle\mathbb{I}
  6. \scriptstyle\mathbb{C}
  7. \scriptstyle\mathbb{H}
  8. 𝕆 \scriptstyle\mathbb{O}
  9. 𝕊 \scriptstyle\mathbb{S}

List_update_problem.html

  1. σ \sigma
  2. σ \sigma
  3. σ \sigma
  4. O P T ( σ ) OPT(\sigma)
  5. A L G ( σ ) ALG(\sigma)
  6. α 0 \alpha\geq 0
  7. σ \sigma
  8. A L G ( σ ) - c . O P T ( σ ) α ALG(\sigma)-c.OPT(\sigma)\leq\alpha
  9. 2 - 2 l + 1 2-\frac{2}{l+1}
  10. 2 - 2 l + 1 2-\frac{2}{l+1}

Little_q-Jacobi_polynomials.html

  1. p n ( x ; a , b ; q ) = ϕ 1 2 ( q - n , a b q n + 1 ; a q ; q , x q ) \displaystyle p_{n}(x;a,b;q)={}_{2}\phi_{1}(q^{-n},abq^{n+1};aq;q,xq)

Little_q-Laguerre_polynomials.html

  1. p n ( x ; a | q ) = ϕ 1 2 ( q - n , 0 ; a q ; q , q x ) = 1 ( a - 1 q - n ; q ) n ϕ 0 2 ( q - n , x - 1 ; ; q , x / a ) \displaystyle p_{n}(x;a|q)={}_{2}\phi_{1}(q^{-n},0;aq;q,qx)=\frac{1}{(a^{-1}q^% {-n};q)_{n}}{}_{2}\phi_{0}(q^{-n},x^{-1};;q,x/a)

Littlewood_subordination_theorem.html

  1. C h ( f ) = f h C_{h}(f)=f\circ h
  2. H p ( D ) H^{p}(D)
  3. A p ( D ) A^{p}(D)
  4. f H p p = sup r 1 2 π 0 2 π | f ( r e i θ ) | p d θ \|f\|_{H^{p}}^{p}=\sup_{r}{1\over 2\pi}\int_{0}^{2\pi}|f(re^{i\theta})|^{p}\,d\theta
  5. f A p p = 1 π D | f ( z ) | p d x d y \|f\|_{A^{p}}^{p}={1\over\pi}\iint_{D}|f(z)|^{p}\,dx\,dy
  6. f 𝒟 2 = 1 π D | f ( z ) | 2 d x d y = 1 4 π D | x f | 2 + | y f | 2 d x d y \|f\|_{\mathcal{D}}^{2}={1\over\pi}\iint_{D}|f^{\prime}(z)|^{2}\,dx\,dy={1% \over 4\pi}\iint_{D}|\partial_{x}f|^{2}+|\partial_{y}f|^{2}\,dx\,dy
  7. C h f 2 f 2 , \displaystyle{\|C_{h}f\|^{2}\leq\|f\|^{2},}
  8. U f ( z ) = z f ( z ) . \displaystyle{Uf(z)=zf(z)}.
  9. U * f ( z ) = f ( z ) - f ( 0 ) z . U^{*}f(z)={f(z)-f(0)\over z}.
  10. f = a 0 + z U * f f=a_{0}+zU^{*}f
  11. C h f = a 0 + h C h U * f . C_{h}f=a_{0}+hC_{h}U^{*}f.
  12. C h f 2 = | a 0 | 2 + h C h U * f 2 | a 0 2 | + C h U * f 2 . \|C_{h}f\|^{2}=|a_{0}|^{2}+\|hC_{h}U^{*}f\|^{2}\leq|a_{0}^{2}|+\|C_{h}U^{*}f\|% ^{2}.
  13. C h U * f 2 U * f 2 = f 2 - | a 0 | 2 , \|C_{h}U^{*}f\|^{2}\leq\|U^{*}f\|^{2}=\|f\|^{2}-|a_{0}|^{2},
  14. C h f 2 f 2 . \|C_{h}f\|^{2}\leq\|f\|^{2}.
  15. 𝒟 . \mathcal{D}.
  16. f ( z ) = f i ( z ) f o ( z ) f(z)=f_{i}(z)f_{o}(z)
  17. C h f H p ( C h f i ) ( C h f o ) H p C h f o H p C h f o p / 2 H 2 2 / p f H p . \|C_{h}f\|_{H^{p}}\leq\|(C_{h}f_{i})(C_{h}f_{o})\|_{H^{p}}\leq\|C_{h}f_{o}\|_{% H^{p}}\leq\|C_{h}f_{o}^{p/2}\|_{H^{2}}^{2/p}\leq\|f\|_{H^{p}}.

LL_grammar.html

  1. Z a Y 1 Y n , n 0 Z\rightarrow aY_{1}\ldots Y_{n},n\geq 0
  2. Z Z
  3. a a

Local_Elevation.html

  1. 𝐐 ( 𝐫 ) \mathbf{Q}(\mathbf{r})
  2. 𝐫 \mathbf{r}
  3. U t o t ( 𝐫 ) = U p h y s ( 𝐫 ) + U b i a s L E ( 𝐐 ; t ) U_{tot}(\mathbf{r})=U_{phys}(\mathbf{r})+U_{bias}^{LE}(\mathbf{Q};t)
  4. U b i a s L E ( 𝐐 ; t ) U_{bias}^{LE}(\mathbf{Q};t)
  5. t t
  6. U b i a s L E ( 𝐐 ; t = 0 ) = 0 U_{bias}^{LE}(\mathbf{Q};t=0)=0
  7. U b i a s L E ( 𝐐 ; ( n + 1 ) Δ t ) = U b i a s L E ( 𝐐 ; n Δ t ) + k L E F ( 𝐐 - 𝐐 n + 1 ) U_{bias}^{LE}(\mathbf{Q};(n+1)\Delta t)=U_{bias}^{LE}(\mathbf{Q};n\Delta t)+k_% {LE}F(\mathbf{Q}-\mathbf{Q}_{n+1})
  8. k L E k_{LE}
  9. F ( 𝐐 - 𝐐 n + 1 ) F(\mathbf{Q}-\mathbf{Q}_{n+1})
  10. F ( 0 ) = 1 F(0)=1
  11. U b i a s L E ( 𝐐 ; n Δ t ) = i = 1 n k L E F ( 𝐐 - 𝐐 i ) U_{bias}^{LE}(\mathbf{Q};n\Delta t)=\sum_{i=1}^{n}k_{LE}F(\mathbf{Q}-\mathbf{Q% }_{i})
  12. F ( 𝐐 - 𝐐 i ) F(\mathbf{Q}-\mathbf{Q}_{i})

Local_feature_size.html

  1. M M
  2. x M x\in M
  3. x x
  4. M M
  5. x x
  6. x x

Localization_(algebra).html

  1. m s \frac{m}{s}
  2. 0 S 0\not\in S
  3. 1 S 1\in S
  4. m s \frac{m}{s}
  5. m s + n t := t m + s n s t \frac{m}{s}+\frac{n}{t}:=\frac{tm+sn}{st}
  6. m s n t := m n s t \frac{m}{s}\frac{n}{t}:=\frac{mn}{st}
  7. 0 / 1 0/1
  8. 1 / 1 1/1
  9. j : R S - 1 R , m m / 1 j:R\to S^{-1}R,m\mapsto m/1
  10. m s = n t \frac{m}{s}=\frac{n}{t}
  11. S - 1 R S^{-1}R
  12. r / s , r R , s R × r/s,r\in R,s\in R^{\times}
  13. S - 1 R S^{-1}R
  14. m m / 1 m\to m/1
  15. S - 1 R S^{-1}R
  16. R f R_{f}
  17. R [ f - 1 ] . R[f^{-1}].
  18. M M
  19. R R
  20. a m s := a m s a\cdot\frac{m}{s}:=\frac{am}{s}
  21. S - 1 M S^{-1}M
  22. R R
  23. m / s m/s
  24. M 𝔭 M_{\mathfrak{p}}
  25. 𝔭 \mathfrak{p}
  26. M f M_{f}
  27. M S - 1 M M\to S^{-1}M
  28. S - 1 R S^{-1}R
  29. S - 1 ( M R N ) S - 1 M S - 1 R S - 1 N S^{-1}(M\otimes_{R}N)\to S^{-1}M\otimes_{S^{-1}R}S^{-1}N
  30. M M
  31. S - 1 Hom R ( M , N ) Hom S - 1 R ( S - 1 M , S - 1 N ) S^{-1}\operatorname{Hom}_{R}(M,N)\to\operatorname{Hom}_{S^{-1}R}(S^{-1}M,S^{-1% }N)
  32. S - 1 M = 0 S^{-1}M=0
  33. t M = 0 tM=0
  34. t S t\in S
  35. S S
  36. R 𝔭 R_{\mathfrak{p}}
  37. 𝔭 \mathfrak{p}
  38. R = 𝔭 R 𝔭 = 𝔪 R 𝔪 R=\cap_{\mathfrak{p}}R_{\mathfrak{p}}=\cap_{\mathfrak{m}}R_{\mathfrak{m}}
  39. I \sqrt{I}
  40. I S - 1 R = I S - 1 R \sqrt{I}\cdot S^{-1}R=\sqrt{I\cdot S^{-1}R}
  41. 𝔭 \mathfrak{p}
  42. M 𝔭 M_{\mathfrak{p}}
  43. 𝔭 \mathfrak{p}
  44. M 𝔭 M_{\mathfrak{p}}
  45. 𝔭 \mathfrak{p}
  46. M 𝔪 M_{\mathfrak{m}}
  47. 𝔪 \mathfrak{m}
  48. f : M N f:M\to N
  49. p M p p\mapsto M_{p}

Locally_profinite_group.html

  1. F × F^{\times}
  2. M n ( F ) \operatorname{M}_{n}(F)
  3. GL n ( F ) \operatorname{GL}_{n}(F)
  4. ψ : G × \psi:G\to\mathbb{C}^{\times}
  5. ( ρ , V ) (\rho,V)
  6. ρ \rho
  7. V K V^{K}
  8. ρ \rho
  9. V K V^{K}
  10. G / K G/K
  11. V * V^{*}
  12. ρ * \rho^{*}
  13. ρ * ( g ) α , v = α , ρ * ( g - 1 ) v \langle\rho^{*}(g)\alpha,v\rangle=\langle\alpha,\rho^{*}(g^{-1})v\rangle
  14. ρ * \rho^{*}
  15. V ~ = K ( V * ) K \widetilde{V}=\bigcup_{K}(V^{*})^{K}
  16. K K
  17. ρ * \rho^{*}
  18. ρ ~ = ρ * \widetilde{\rho}=\rho^{*}
  19. ( ρ ~ , V ~ ) (\widetilde{\rho},\widetilde{V})
  20. ( ρ , V ) (\rho,V)
  21. ( ρ , V ) ( ρ ~ , V ~ ) (\rho,V)\mapsto(\widetilde{\rho},\widetilde{V})
  22. ρ \rho
  23. ρ ~ \widetilde{\rho}
  24. ρ ρ ~ ~ \rho\to\widetilde{\widetilde{\rho}}
  25. ρ \rho
  26. ρ \rho
  27. ρ ~ \widetilde{\rho}
  28. ρ \rho
  29. ρ ~ \widetilde{\rho}
  30. G G
  31. G / K G/K
  32. ρ \rho
  33. G G
  34. C c ( G ) C^{\infty}_{c}(G)
  35. G G
  36. ( f * h ) ( x ) = G f ( g ) h ( g - 1 x ) d μ ( g ) (f*h)(x)=\int_{G}f(g)h(g^{-1}x)d\mu(g)
  37. C c ( G ) C^{\infty}_{c}(G)
  38. \mathbb{C}
  39. ( G ) \mathfrak{H}(G)
  40. ( ρ , V ) (\rho,V)
  41. ρ ( f ) = G f ( g ) ρ ( g ) d μ ( g ) . \rho(f)=\int_{G}f(g)\rho(g)d\mu(g).
  42. ρ ρ \rho\mapsto\rho
  43. G G
  44. ( G ) \mathfrak{H}(G)
  45. ρ ( ( G ) ) V = V \rho(\mathfrak{H}(G))V=V

Locating_the_center_of_mass.html

  1. 3 × ( - 2.5 2 π ) + 5 × 10 2 + 13.33 × 10 2 2 - 2.5 2 π + 10 2 + 10 2 2 8.5 \frac{3\times(-2.5^{2}\pi)+5\times 10^{2}+13.33\times\frac{10^{2}}{2}}{-2.5^{2% }\pi+10^{2}+\frac{10^{2}}{2}}\approx 8.5

Lockman_Hole.html

  1. \sim

Loewner_differential_equation.html

  1. φ ( z ) = g - 1 ( f ( z ) ) . \displaystyle{\varphi(z)=g^{-1}(f(z)).}
  2. f ( D r ) g ( D r ) . \displaystyle{f(D_{r})\subseteq g(D_{r}).}
  3. U ( s ) U ( t ) U(s)\subsetneq U(t)
  4. U ( ) = . U(\infty)={\mathbb{C}}.
  5. s n t s_{n}\uparrow t
  6. U ( s n ) U ( t ) U(s_{n})\rightarrow U(t)
  7. f t ( D ) = U ( t ) , f t ( 0 ) = 0 , z f t ( 0 ) = 1 f_{t}(D)=U(t),\,\,\,f_{t}(0)=0,\,\,\,\partial_{z}f_{t}(0)=1
  8. a ( t ) = f t ( 0 ) a(t)=f^{\prime}_{t}(0)
  9. f t ( 0 ) = e t . f^{\prime}_{t}(0)=e^{t}.
  10. f t ( z ) = e t z + a 2 ( t ) z 2 + f_{t}(z)=e^{t}z+a_{2}(t)z^{2}+\cdots
  11. f s ( D ) f t ( D ) \displaystyle{f_{s}(D)\subsetneq f_{t}(D)}
  12. f s ( z ) = f t ( φ s , t ( z ) ) . \displaystyle{f_{s}(z)=f_{t}(\varphi_{s,t}(z)).}
  13. φ s , t φ t , r = φ s , r \displaystyle{\varphi_{s,t}\circ\varphi_{t,r}=\varphi_{s,r}}
  14. φ t , t ( z ) = z . \displaystyle{\varphi_{t,t}(z)=z.}
  15. w s ( z ) = t φ s , t ( z ) | t = s \displaystyle{w_{s}(z)=\partial_{t}\varphi_{s,t}(z)|_{t=s}}
  16. w s ( z ) = - z p s ( z ) \displaystyle{w_{s}(z)=-zp_{s}(z)}
  17. p s ( z ) > 0 \displaystyle{\Re\,p_{s}(z)>0}
  18. d w d t = - w p t ( w ) \displaystyle{{dw\over dt}=-wp_{t}(w)}
  19. f t ( z ) = f s ( φ s , t ( z ) ) \displaystyle{f_{t}(z)=f_{s}(\varphi_{s,t}(z))}
  20. t f t ( z ) = z p t ( z ) z f t ( z ) \displaystyle{\partial_{t}f_{t}(z)=zp_{t}(z)\partial_{z}f_{t}(z)}
  21. f t ( z ) | t = 0 = f 0 ( z ) . \displaystyle{f_{t}(z)|_{t=0}=f_{0}(z).}
  22. f s ( z ) = lim t e t ϕ s , t ( z ) . \displaystyle{f_{s}(z)=\lim_{t\rightarrow\infty}e^{t}\phi_{s,t}(z).}
  23. φ 0 , 1 ( z ) = ψ ( z ) . \displaystyle{\varphi_{0,1}(z)=\psi(z).}
  24. f 0 ( z ) = z , f 1 ( z ) = g ( z ) . \displaystyle{f_{0}(z)=z,\,\,\,f_{1}(z)=g(z).}
  25. p ( z ) = 0 2 π 1 + e - i θ z 1 - e - i θ z d μ ( θ ) , \displaystyle{p(z)=\int_{0}^{2\pi}{1+e^{-i\theta}z\over 1-e^{-i\theta}z}\,d\mu% (\theta),}
  26. p t ( z ) = 1 + κ ( t ) z 1 - κ ( t ) z \displaystyle{p_{t}(z)={1+\kappa(t)z\over 1-\kappa(t)z}}
  27. g ( z ) = f ( r z ) / r \displaystyle{g(z)=f(rz)/r}
  28. f t ( z ) = e t ( z + b 2 ( t ) z 2 + b 3 ( t ) z 3 + ) \displaystyle{f_{t}(z)=e^{t}(z+b_{2}(t)z^{2}+b_{3}(t)z^{3}+\cdots)}
  29. f 0 ( z ) = f ( z ) . \displaystyle{f_{0}(z)=f(z).}
  30. φ s , t ( z ) = f t - 1 f s ( z ) = e s - t ( z + a 2 ( s , t ) z 2 + a 3 ( s , t ) z 3 + ) \displaystyle{\varphi_{s,t}(z)=f_{t}^{-1}\circ f_{s}(z)=e^{s-t}(z+a_{2}(s,t)z^% {2}+a_{3}(s,t)z^{3}+\cdots)}
  31. p t ( z ) = 1 + κ ( t ) z 1 - κ ( t ) z \displaystyle{p_{t}(z)={1+\kappa(t)z\over 1-\kappa(t)z}}
  32. κ ( t ) = λ ( t ) - 1 . \displaystyle{\kappa(t)=\lambda(t)^{-1}.}
  33. f t ( λ ( t ) ) = c ( t ) . \displaystyle{f_{t}(\lambda(t))=c(t).}
  34. | a 3 | 3 \displaystyle{|a_{3}|\leq 3}
  35. f ( z ) = z + a 2 z 2 + a 3 z 3 + \displaystyle{f(z)=z+a_{2}z^{2}+a_{3}z^{3}+\cdots}
  36. φ 0 , t ( z ) = e - t ( z + a 2 ( t ) z 2 + a 3 ( t ) z 3 + ) \displaystyle{\varphi_{0,t}(z)=e^{-t}(z+a_{2}(t)z^{2}+a_{3}(t)z^{3}+\cdots)}
  37. a n ( 0 ) = 0 , a n ( ) = a n . \displaystyle{a_{n}(0)=0,\,\,a_{n}(\infty)=a_{n}.}
  38. α ( t ) = e - t κ ( t ) , \displaystyle{\alpha(t)=e^{-t}\kappa(t),}
  39. a 2 ˙ = - 2 α \displaystyle{\dot{a_{2}}=-2\alpha}
  40. a 3 ˙ = - 2 α 2 - 4 α a 2 . \displaystyle{\dot{a_{3}}=-2\alpha^{2}-4\alpha\,a_{2}.}
  41. a 2 = - 2 0 α ( t ) d t \displaystyle{a_{2}=-2\int_{0}^{\infty}\alpha(t)\,dt}
  42. | a 2 | 2. \displaystyle{|a_{2}|\leq 2.}
  43. a 3 = - 2 0 α 2 d t + 4 ( 0 α d t ) 2 \displaystyle{a_{3}=-2\int_{0}^{\infty}\alpha^{2}\,dt+4\left(\int_{0}^{\infty}% \alpha\,dt\right)^{2}}
  44. | a 3 | = 2 0 | α 2 | d t + 4 ( 0 α d t ) 2 2 0 | α 2 | d t + 4 ( 0 e - t d t ) ( 0 e t ( α ) 2 d t ) = 1 + 4 0 ( e - t - e - 2 t ) ( κ ) 2 d t 3 , \displaystyle{|a_{3}|=2\int_{0}^{\infty}|\Re\alpha^{2}|\,dt+4\left(\int_{0}^{% \infty}\Re\alpha\,dt\right)^{2}}\leq 2\int_{0}^{\infty}|\Re\alpha^{2}|\,dt+4% \left(\int_{0}^{\infty}e^{-t}\,dt\right)\left(\int_{0}^{\infty}e^{t}(\Re\alpha% )^{2}\,dt\right)=1+4\int_{0}^{\infty}(e^{-t}-e^{-2t})(\Re\kappa)^{2}\,dt\leq 3,

Log-Cauchy_distribution.html

  1. f ( x ; μ , σ ) \displaystyle f(x;\mu,\sigma)
  2. μ \mu
  3. σ > 0 \sigma>0
  4. σ \sigma
  5. e μ e^{\mu}
  6. μ \mu
  7. σ \sigma
  8. μ \mu
  9. σ \sigma
  10. μ = 0 \mu=0
  11. σ = 1 \sigma=1
  12. f ( x ; 0 , 1 ) = 1 x π ( 1 + ( ln x ) 2 ) , x > 0 f(x;0,1)=\frac{1}{x\pi(1+(\ln x)^{2})},\ \ x>0
  13. μ = 0 \mu=0
  14. σ = 1 \sigma=1
  15. F ( x ; 0 , 1 ) = 1 2 + 1 π arctan ( ln x ) , x > 0 F(x;0,1)=\frac{1}{2}+\frac{1}{\pi}\arctan(\ln x),\ \ x>0
  16. μ = 0 \mu=0
  17. σ = 1 \sigma=1
  18. S ( x ; 0 , 1 ) = 1 2 - 1 π arctan ( ln x ) , x > 0 S(x;0,1)=\frac{1}{2}-\frac{1}{\pi}\arctan(\ln x),\ \ x>0
  19. μ = 0 \mu=0
  20. σ = 1 \sigma=1
  21. λ ( x ; 0 , 1 ) = ( 1 x π ( 1 + ( ln x ) 2 ) ( 1 2 - 1 π arctan ( ln x ) ) ) - 1 , x > 0 \lambda(x;0,1)=\left(\frac{1}{x\pi\left(1+\left(\ln x\right)^{2}\right)}\left(% \frac{1}{2}-\frac{1}{\pi}\arctan(\ln x)\right)\right)^{-1},\ \ x>0
  22. μ \mu
  23. σ \sigma

Log-polar_coordinates.html

  1. r = e ρ . r=e^{\rho}.\,
  2. r r
  3. { ρ = log x 2 + y 2 , θ = arctan y / x if x > 0. \begin{cases}\rho=\log\sqrt{x^{2}+y^{2}},\\ \theta=\arctan y/x\hbox{ if }x>0.\end{cases}
  4. { x = e ρ cos θ , y = e ρ sin θ . \begin{cases}x=e^{\rho}\cos\theta,\\ y=e^{\rho}\sin\theta.\end{cases}
  5. x + i y = e ρ + i θ x+iy=e^{\rho+i\theta}\,
  6. 2 u x 2 + 2 u y 2 = 0 \frac{\partial^{2}u}{\partial x^{2}}+\frac{\partial^{2}u}{\partial y^{2}}=0
  7. r r ( r u r ) + 2 u θ 2 = 0 r\frac{\partial}{\partial r}\left(r\frac{\partial u}{\partial r}\right)+\frac{% \partial^{2}u}{\partial\theta^{2}}=0
  8. ( r r ) 2 u + 2 u θ 2 = 0 \left(r\frac{\partial}{\partial r}\right)^{2}u+\frac{\partial^{2}u}{\partial% \theta^{2}}=0
  9. r = e ρ r=e^{\rho}
  10. r r = ρ r\frac{\partial}{\partial r}=\frac{\partial}{\partial\rho}
  11. 2 u ρ 2 + 2 u θ 2 = 0 \frac{\partial^{2}u}{\partial\rho^{2}}+\frac{\partial^{2}u}{\partial\theta^{2}% }=0
  12. f ( x , y ) = u ( x , y ) + i v ( x , y ) f(x,y)=u(x,y)+iv(x,y)
  13. u x = v y , u y = - v x \frac{\partial u}{\partial x}=\frac{\partial v}{\partial y},\ \ \ \ \ \ \frac{% \partial u}{\partial y}=-\frac{\partial v}{\partial x}
  14. f ( r e i θ ) = R e i Φ f(re^{i\theta})=Re^{i\Phi}
  15. r log R r = Φ θ , log R θ = - r Φ r , r\frac{\partial\log R}{\partial r}=\frac{\partial\Phi}{\partial\theta},\ \ \ % \ \ \ \frac{\partial\log R}{\partial\theta}=-r\frac{\partial\Phi}{\partial r},
  16. P = log R P=\log R
  17. P ρ = Φ θ , P θ = - Φ ρ \frac{\partial P}{\partial\rho}=\frac{\partial\Phi}{\partial\theta},\ \ \ \ \ % \ \frac{\partial P}{\partial\theta}=-\frac{\partial\Phi}{\partial\rho}
  18. ( x + i y ) f ( x + i y ) = 0 \left(\frac{\partial}{\partial x}+i\frac{\partial}{\partial y}\right)f(x+iy)=0
  19. x \frac{\partial}{\partial x}
  20. y \frac{\partial}{\partial y}
  21. ρ \frac{\partial}{\partial\rho}
  22. θ \frac{\partial}{\partial\theta}
  23. ( ρ + i θ ) f ( e ρ + i θ ) = 0 \left(\frac{\partial}{\partial\rho}+i\frac{\partial}{\partial\theta}\right)f(e% ^{\rho+i\theta})=0
  24. u ( r , θ ) = R ( r ) Θ ( θ ) u(r,\theta)=R(r)\Theta(\theta)
  25. { Θ ′′ ( θ ) + ν 2 Θ ( θ ) = 0 r 2 R ′′ ( r ) + r R ( r ) - ν 2 R ( r ) = 0 \begin{cases}\Theta^{\prime\prime}(\theta)+\nu^{2}\Theta(\theta)=0\\ r^{2}R^{\prime\prime}(r)+rR^{\prime}(r)-\nu^{2}R(r)=0\end{cases}
  26. ν \nu
  27. r 2 R ′′ ( r ) + c r R ( r ) + d R ( r ) = 0 r^{2}R^{\prime\prime}(r)+crR^{\prime}(r)+dR(r)=0
  28. c , d c,d
  29. R ( r ) = r λ R(r)=r^{\lambda}
  30. P ′′ ( ρ ) + ( c - 1 ) P ( ρ ) + d P ( ρ ) = 0 P^{\prime\prime}(\rho)+(c-1)P^{\prime}(\rho)+dP(\rho)=0
  31. c = 1 c=1
  32. d = - ν 2 d=-\nu^{2}
  33. r r
  34. P ′′ ( ρ ) - ν 2 P ( ρ ) = 0 P^{\prime\prime}(\rho)-\nu^{2}P(\rho)=0
  35. x x
  36. y y
  37. π \pi
  38. γ \gamma
  39. Λ γ \Lambda_{\gamma}
  40. γ = 1 \gamma=1
  41. Λ γ 2 + 2 θ 2 = 0 \Lambda_{\gamma}^{2}+\frac{\partial^{2}\ }{\partial\theta^{2}}=0

Logan_plot.html

  1. C p ( t ) C_{p}(t)
  2. d 𝐀 d t = 𝐊𝐀 + 𝐐 C p ( t ) \frac{d\mathbf{A}}{dt}=\mathbf{KA}+\mathbf{Q}C_{p}(t)
  3. 𝐀 \mathbf{A}
  4. t t
  5. 𝐊 \mathbf{K}
  6. 𝐐 \mathbf{Q}
  7. 0 t A ( τ ) d τ = - 𝐔 n T 𝐊 - 1 𝐐 0 t C p ( τ ) d τ + 𝐔 n T 𝐊 - 1 𝐀 \int_{0}^{t}A(\tau)\,d\tau=-\mathbf{U}_{n}^{T}\mathbf{K}^{-1}\mathbf{Q}\int_{0% }^{t}C_{p}(\tau)\,d\tau+\mathbf{U}_{n}^{T}\mathbf{K}^{-1}\mathbf{A}
  8. 𝐔 n T \mathbf{U}_{n}^{T}
  9. A ( t ) = 𝐔 n T 𝐀 A(t)=\mathbf{U}_{n}^{T}\mathbf{A}
  10. ROI ( t ) \mathrm{ROI}(t)
  11. V p V_{p}
  12. 0 t ROI ( τ ) d τ = ( - 𝐔 n T 𝐊 - 1 𝐐 + V p ) 0 t C p ( τ ) d τ + 𝐔 n T 𝐊 - 1 𝐀 \int_{0}^{t}\mathrm{ROI}(\tau)\,d\tau=(-\mathbf{U}_{n}^{T}\mathbf{K}^{-1}% \mathbf{Q}+V_{p})\int_{0}^{t}C_{p}(\tau)\,d\tau+\mathbf{U}_{n}^{T}\mathbf{K}^{% -1}\mathbf{A}
  13. ROI ( t ) \mathrm{ROI}(t)
  14. 0 t ROI ( τ ) d τ ROI ( t ) = ( - 𝐔 n T 𝐊 - 1 𝐐 + V p ) 0 t C p ( τ ) d τ ROI ( t ) + 𝐔 n T 𝐊 - 1 𝐀 𝐔 n T 𝐀 + V p C p {{\int_{0}^{t}\mathrm{ROI}(\tau)\,d\tau}\over\mathrm{ROI}(t)}=(-\mathbf{U}_{n}% ^{T}\mathbf{K}^{-1}\mathbf{Q}+V_{p}){{\int_{0}^{t}C_{p}(\tau)\,d\tau}\over% \mathrm{ROI}(t)}+{{\mathbf{U}_{n}^{T}\mathbf{K}^{-1}\mathbf{A}}\over{\mathbf{U% }_{n}^{T}\mathbf{A}+V_{p}C_{p}}}
  15. t > t t>t^{\prime}
  16. 𝐀 = - 𝐊 - 1 𝐐 C p ( t ) \mathbf{A}=-\mathbf{K}^{-1}\mathbf{Q}C_{p}(t)
  17. 0 t C p ( τ ) d τ ROI ( t ) {{\int_{0}^{t}C_{p}(\tau)d\tau}\over\mathrm{ROI}(t)}
  18. 0 t ROI ( τ ) d τ ROI ( t ) {{\int_{0}^{t}\mathrm{ROI}(\tau)d\tau}\over\mathrm{ROI}(t)}
  19. ( - 𝐔 n T 𝐊 - 1 𝐐 + V p ) (-\mathbf{U}_{n}^{T}\mathbf{K}^{-1}\mathbf{Q}+V_{p})
  20. 𝐔 n T 𝐊 - 1 𝐀 𝐔 n T 𝐀 + V p C p {{\mathbf{U}_{n}^{T}\mathbf{K}^{-1}\mathbf{A}}\over{\mathbf{U}_{n}^{T}\mathbf{% A}+V_{p}C_{p}}}
  21. K 1 K_{1}
  22. k 2 k_{2}
  23. k 3 k_{3}
  24. B max k on B_{\max}k_{\mathrm{on}}
  25. k 4 k_{4}
  26. V d V_{d}
  27. K 1 k 2 ( 1 + k 3 k 4 ) + V p \frac{K_{1}}{k_{2}}(1+\frac{k_{3}}{k_{4}})+V_{p}
  28. k 3 = B max k on k_{3}=B_{\max}k_{\mathrm{on}}
  29. k 4 = k off k_{4}=k_{\mathrm{off}}
  30. k 3 k 4 = B max K d \frac{k_{3}}{k_{4}}=\frac{B_{\max}}{K_{d}}
  31. K d = k off / k on K_{d}=k_{\mathrm{off}}/k_{\mathrm{on}}
  32. B max B_{\max}
  33. K d K_{d}
  34. k on k_{\mathrm{on}}
  35. k off k_{\mathrm{off}}
  36. K 1 K_{1}
  37. k 2 k_{2}
  38. λ + V p \lambda+V_{p}
  39. λ \lambda
  40. K 1 / k 2 {K_{1}}/{k_{2}}
  41. - 1 k 2 ( 1 + V p / λ ) \frac{-1}{k_{2}(1+V_{p}/\lambda)}

Logarithmic_Schrödinger_equation.html

  1. i ψ t + Δ ψ + ψ ln | ψ | 2 = 0. i\frac{\partial\psi}{\partial t}+\Delta\psi+\psi\ln|\psi|^{2}=0.
  2. ψ = ψ ( 𝐱 , t ) \psi=\psi(\mathrm{\mathbf{x}},t)
  3. Δ \Delta\,
  4. 𝐱 \mathrm{\mathbf{x}}

Logit-normal_distribution.html

  1. 1 2 [ 1 + erf ( logit ( x ) - μ 2 σ 2 ) ] \frac{1}{2}\Big[1+\operatorname{erf}\Big(\frac{\operatorname{logit}(x)-\mu}{% \sqrt{2\sigma^{2}}}\Big)\Big]
  2. P ( μ ) P(\mu)\,
  3. f X ( x ; μ , σ ) = 1 σ 2 π 1 x ( 1 - x ) e - ( logit ( x ) - μ ) 2 2 σ 2 f_{X}(x;\mu,\sigma)=\frac{1}{\sigma\sqrt{2\pi}}\,\frac{1}{x(1-x)}\,e^{-\frac{(% \operatorname{logit}(x)-\mu)^{2}}{2\sigma^{2}}}
  4. logit ( x ) = σ 2 ( 2 x - 1 ) + μ . \operatorname{logit}(x)=\sigma^{2}(2x-1)+\mu.
  5. f X ( 𝐱 ; s y m b o l μ , s y m b o l Σ ) = 1 | 2 π s y m b o l Σ | 1 2 1 i = 1 D x i e - 1 2 { log ( 𝐱 - D x D ) - s y m b o l μ } s y m b o l Σ - 1 { log ( 𝐱 - D x D ) - s y m b o l μ } , 𝐱 𝒮 D , f_{X}(\mathbf{x};symbol{\mu},symbol{\Sigma})=\frac{1}{|2\pi symbol{\Sigma}|^{% \frac{1}{2}}}\,\frac{1}{\prod\limits_{i=1}^{D}x_{i}}\,e^{-\frac{1}{2}\left\{% \log\left(\frac{\mathbf{x}_{-D}}{x_{D}}\right)-symbol{\mu}\right\}^{\top}% symbol{\Sigma}^{-1}\left\{\log\left(\frac{\mathbf{x}_{-D}}{x_{D}}\right)-% symbol{\mu}\right\}}\quad,\quad\mathbf{x}\in\mathcal{S}^{D}\;\;,
  6. 𝐱 - D \mathbf{x}_{-D}
  7. 𝐱 \mathbf{x}
  8. 𝒮 D \mathcal{S}^{D}
  9. 𝐲 𝒩 ( s y m b o l μ , s y m b o l Σ ) , 𝐲 D - 1 \mathbf{y}\sim\mathcal{N}\left(symbol{\mu},symbol{\Sigma}\right)\;,\;\mathbf{y% }\in\mathbb{R}^{D-1}
  10. 𝐱 = [ e y 1 1 + i = 1 D - 1 e y i , , e y D - 1 1 + i = 1 D - 1 e y i , 1 1 + i = 1 D - 1 e y i ] \mathbf{x}=\left[\frac{e^{y_{1}}}{1+\sum_{i=1}^{D-1}e^{y_{i}}},\dots,\frac{e^{% y_{D-1}}}{1+\sum_{i=1}^{D-1}e^{y_{i}}},\frac{1}{1+\sum_{i=1}^{D-1}e^{y_{i}}}\right]
  11. 𝐲 = [ log ( x 1 x D ) , , log ( x D - 1 x D ) ] \mathbf{y}=\left[\log\left(\frac{x_{1}}{x_{D}}\right),\dots,\log\left(\frac{x_% {D-1}}{x_{D}}\right)\right]
  12. n \mathbb{R}^{n}
  13. 𝒮 D \mathcal{S}^{D}
  14. D - 1 \mathbb{R}^{D-1}
  15. K ( p , q ) = 𝒮 D p ( 𝐱 | s y m b o l α ) log ( p ( 𝐱 | s y m b o l α ) q ( 𝐱 | s y m b o l μ , s y m b o l Σ ) ) d 𝐱 K(p,q)=\int_{\mathcal{S}^{D}}p\left(\mathbf{x}|symbol{\alpha}\right)\log\left(% \frac{p\left(\mathbf{x}|symbol{\alpha}\right)}{q\left(\mathbf{x}|symbol{\mu},% symbol{\Sigma}\right)}\right)\,d\mathbf{x}
  16. s y m b o l μ * = 𝐄 p [ log ( 𝐱 - D x D ) ] , s y m b o l Σ * = 𝐕𝐚𝐫 p [ log ( 𝐱 - D x D ) ] symbol{\mu}^{*}=\mathbf{E}_{p}\left[\log\left(\frac{\mathbf{x}_{-D}}{x_{D}}% \right)\right]\quad,\quad symbol{\Sigma}^{*}=\,\textbf{Var}_{p}\left[\log\left% (\frac{\mathbf{x}_{-D}}{x_{D}}\right)\right]
  17. ψ \psi
  18. ψ \psi^{\prime}
  19. μ i * = ψ ( α i ) - ψ ( α D ) , i = 1 , , D - 1 \mu_{i}^{*}=\psi\left(\alpha_{i}\right)-\psi\left(\alpha_{D}\right)\quad,\quad i% =1,\cdots,D-1
  20. Σ i i * = ψ ( α i ) + ψ ( α D ) , i = 1 , , D - 1 \Sigma_{ii}^{*}=\psi^{\prime}\left(\alpha_{i}\right)+\psi^{\prime}\left(\alpha% _{D}\right)\quad,\quad i=1,\cdots,D-1
  21. Σ i j * = ψ ( α D ) , i j \Sigma_{ij}^{*}=\psi^{\prime}\left(\alpha_{D}\right)\quad,\quad i\neq j
  22. s y m b o l α symbol{\alpha}
  23. α i , i = 1 , , D \alpha_{i}\rightarrow\infty,i=1,\cdots,D
  24. p ( 𝐱 | s y m b o l α ) q ( 𝐱 | s y m b o l μ * , s y m b o l Σ * ) p\left(\mathbf{x}|symbol{\alpha}\right)\rightarrow q\left(\mathbf{x}|symbol{% \mu}^{*},symbol{\Sigma}^{*}\right)

Lomax_distribution.html

  1. 1 - [ 1 + x λ ] - α 1-\left[{1+{x\over\lambda}}\right]^{-\alpha}
  2. λ α - 1 for α > 1 {\lambda\over{\alpha-1}}\,\text{ for }\alpha>1
  3. λ ( 2 α - 1 ) \lambda(\sqrt[\alpha]{2}-1)
  4. λ 2 α ( α - 1 ) 2 ( α - 2 ) for α > 2 {{\lambda^{2}\alpha}\over{(\alpha-1)^{2}(\alpha-2)}}\,\text{ for }\alpha>2
  5. for 1 < α 2 \infty\,\text{ for }1<\alpha\leq 2
  6. 2 ( 1 + α ) α - 3 α - 2 α for α > 3 \frac{2(1+\alpha)}{\alpha-3}\,\sqrt{\frac{\alpha-2}{\alpha}}\,\text{ for }% \alpha>3\,
  7. 6 ( α 3 + α 2 - 6 α - 2 ) α ( α - 3 ) ( α - 4 ) for α > 4 \frac{6(\alpha^{3}+\alpha^{2}-6\alpha-2)}{\alpha(\alpha-3)(\alpha-4)}\,\text{ % for }\alpha>4\,
  8. p ( x ) = α λ [ 1 + x λ ] - ( α + 1 ) , x 0 , p(x)={\alpha\over\lambda}\left[{1+{x\over\lambda}}\right]^{-(\alpha+1)},\qquad x% \geq 0,
  9. α > 0 \alpha>0
  10. λ > 0 \lambda>0
  11. p ( x ) = α λ α ( x + λ ) α + 1 p(x)={{\alpha\lambda^{\alpha}}\over{(x+\lambda)^{\alpha+1}}}
  12. { ( λ + x ) p ( x ) + ( α + 1 ) p ( x ) = 0 , p ( 0 ) = α λ } \left\{\begin{array}[]{l}(\lambda+x)p^{\prime}(x)+(\alpha+1)p(x)=0,\\ p(0)=\frac{\alpha}{\lambda}\end{array}\right\}
  13. If Y Pareto ( x m = λ , α ) , then Y - x m Lomax ( λ , α ) . \,\text{If }Y\sim\mbox{Pareto}~{}(x_{m}=\lambda,\alpha),\,\text{ then }Y-x_{m}% \sim\mbox{Lomax}~{}(\lambda,\alpha).
  14. If X Lomax ( λ , α ) then X P(II) ( x m = λ , α , μ = 0 ) . \,\text{If }X\sim\mbox{Lomax}~{}(\lambda,\alpha)\,\text{ then }X\sim\,\text{P(% II)}(x_{m}=\lambda,\alpha,\mu=0).
  15. μ = 0 , ξ = 1 α , σ = λ α . \mu=0,~{}\xi={1\over\alpha},~{}\sigma={\lambda\over\alpha}.
  16. α = 2 - q q - 1 , λ = 1 λ q ( q - 1 ) . \alpha={{2-q}\over{q-1}},~{}\lambda={1\over\lambda_{q}(q-1)}.
  17. ν \nu
  18. E [ X ν ] E[X^{\nu}]
  19. α \alpha
  20. ν \nu
  21. E ( X ν ) = λ ν Γ ( α - ν ) Γ ( 1 + ν ) Γ ( α ) E(X^{\nu})=\frac{\lambda^{\nu}\Gamma(\alpha-\nu)\Gamma(1+\nu)}{\Gamma(\alpha)}

Lorentz-violating_neutrino_oscillations.html

  1. Δ m 2 8 × 10 - 5 eV 2 \Delta m^{2}_{\odot}\simeq 8\times 10^{-5}\,\mbox{eV}~{}^{2}
  2. Δ m 2 atm 2.5 × 10 - 3 eV 2 \Delta m^{2}\text{atm}\simeq 2.5\times 10^{-3}\,\mbox{eV}~{}^{2}
  3. Δ m 2 LSND 1 eV 2 \Delta m^{2}\text{LSND}\simeq 1\,\mbox{eV}~{}^{2}
  4. ω 2 π / 23 h 56 min \omega_{\oplus}\sim 2\pi/23\,\mbox{h}~{}\,56\,\mbox{min}~{}
  5. P ν a ν b P ν ¯ b ν ¯ a P_{\nu_{a}\rightarrow\nu_{b}}\neq P_{\bar{\nu}_{b}\rightarrow\bar{\nu}_{a}}
  6. P ν a ν b = P ν ¯ b ν ¯ a P_{\nu_{a}\rightarrow\nu_{b}}=P_{\bar{\nu}_{b}\rightarrow\bar{\nu}_{a}}
  7. 0.1 eV \lesssim 0.1\,\,\text{eV}
  8. 10 - 17 \lesssim 10^{-17}
  9. ν μ ν τ \nu_{\mu}\leftrightarrow\nu_{\tau}
  10. ν μ ν e \nu_{\mu}\rightarrow\nu_{e}
  11. ν μ ν e \nu_{\mu}\rightarrow\nu_{e}
  12. θ 13 \theta_{13}
  13. U a a U_{a^{\prime}a}
  14. E a b = diag ( λ 1 , λ 2 , λ 3 ) E_{a^{\prime}b^{\prime}}=\mathrm{diag}(\lambda_{1},\lambda_{2},\lambda_{3})
  15. | ν b |\nu_{b}\rangle
  16. | ν a |\nu_{a}\rangle
  17. P ν b ν a = | ν a | ν b ( L ) | 2 = | a U a a * U a b e - i λ a L | 2 , P_{\nu_{b}\rightarrow\nu_{a}}=\left|\left\langle\nu_{a}|\nu_{b}(L)\right% \rangle\right|^{2}=\left|\sum_{a^{\prime}}U_{a^{\prime}a}^{*}U_{a^{\prime}b}\,% e^{-i\lambda_{a^{\prime}}L}\right|^{2},
  18. λ a \lambda_{a^{\prime}}\frac{}{}
  19. λ a = m a 2 / 2 E \lambda_{a^{\prime}}=m^{2}_{a^{\prime}}/2E
  20. h eff = ( | p | 0 0 | p | ) + 1 2 | p | ( ( m ~ 2 ) 0 0 ( m ~ 2 ) * ) + 1 | p | ( a ^ eff - c ^ eff - g ^ eff + H ^ eff - g ^ eff + H ^ eff - a ^ eff T - c ^ eff T ) , h\text{eff}=\begin{pmatrix}|\vec{p}|&0\\ \\ 0&|\vec{p}|\end{pmatrix}+\frac{1}{2|\vec{p}|}\begin{pmatrix}(\tilde{m}^{2})&0% \\ \\ 0&(\tilde{m}^{2})^{*}\end{pmatrix}+\frac{1}{|\vec{p}|}\begin{pmatrix}\widehat{% a}\text{eff}-\widehat{c}\text{eff}&-\widehat{g}\text{eff}+\widehat{H}\text{eff% }\\ \\ -\widehat{g}\text{eff}^{\dagger}+\widehat{H}\text{eff}^{\dagger}&-\widehat{a}% \text{eff}^{T}-\widehat{c}\text{eff}^{T}\end{pmatrix},
  21. v of = 1 - | m l | 2 2 | p | 2 + d j m ( d - 3 ) | p | d - 4 Y j m ( p ^ ) [ ( a of ( d ) ) j m - ( c of ( d ) ) j m ] , v\text{of}=1-\frac{|m_{l}|^{2}}{2|\vec{p}|^{2}}+\sum_{djm}(d-3)|\vec{p}|^{d-4}% \,Y_{jm}(\hat{p})\big[(a\text{of}^{(d)})_{jm}-(c\text{of}^{(d)})_{jm}\big],
  22. ( h eff ) A B = E ( δ a b 0 0 δ a ¯ b ¯ ) + 1 2 E ( ( m ~ 2 ) a b 0 0 ( m ~ 2 ) a ¯ b ¯ * ) + 1 E ( [ ( a L ) α p α - ( c L ) α β p α p β ] a b - i 2 p α ( ϵ + ) β [ ( g α β γ p γ - H α β ) ] a b ¯ i 2 p α ( ϵ + ) β * [ ( g α β γ p γ - H α β ) ] a ¯ b * [ ( a R ) α p α - ( c R ) α β p α p β ] a ¯ b ¯ ) . \begin{aligned}\displaystyle(h\text{eff})_{AB}&\displaystyle=E\begin{pmatrix}% \delta_{ab}&0\\ \\ 0&\delta_{\bar{a}\bar{b}}\end{pmatrix}+\frac{1}{2E}\begin{pmatrix}(\tilde{m}^{% 2})_{ab}&0\\ \\ 0&(\tilde{m}^{2})_{\bar{a}\bar{b}}^{*}\end{pmatrix}\\ \\ &\displaystyle\quad+\frac{1}{E}\begin{pmatrix}[(a_{L})^{\alpha}p_{\alpha}-(c_{% L})^{\alpha\beta}p_{\alpha}p_{\beta}]_{ab}&-i\sqrt{2}p_{\alpha}(\epsilon_{+})_% {\beta}[(g^{\alpha\beta\gamma}p_{\gamma}-H^{\alpha\beta})]_{a\bar{b}}\\ \\ i\sqrt{2}p_{\alpha}(\epsilon_{+})_{\beta}^{*}[(g^{\alpha\beta\gamma}p_{\gamma}% -H^{\alpha\beta})]_{\bar{a}b}^{*}&[(a_{R})^{\alpha}p_{\alpha}-(c_{R})^{\alpha% \beta}p_{\alpha}p_{\beta}]_{\bar{a}\bar{b}}\end{pmatrix}.\end{aligned}
  23. e ¯ \overline{e}
  24. μ ¯ \overline{μ}
  25. τ ¯ \overline{τ}
  26. a ¯ \overline{a}
  27. b ¯ \overline{b}
  28. e ¯ \overline{e}
  29. μ ¯ \overline{μ}
  30. τ ¯ \overline{τ}
  31. E | p | E\simeq|\vec{p}|
  32. ( a L ) a b α (a_{L})^{\alpha}_{ab}
  33. ( c L ) a b α β (c_{L})^{\alpha\beta}_{ab}
  34. ( a R ) a ¯ b ¯ α (a_{R})^{\alpha}_{\bar{a}\bar{b}}
  35. ( c R ) a ¯ b ¯ α β (c_{R})^{\alpha\beta}_{\bar{a}\bar{b}}
  36. g a b ¯ α β γ g^{\alpha\beta\gamma}_{a\bar{b}}
  37. H a b ¯ α β H^{\alpha\beta}_{a\bar{b}}
  38. ( h eff ) a b = 1 E [ ( a L ) α p α - ( c L ) α β p α p β ] a b . (h\text{eff})_{ab}=\frac{1}{E}[(a_{L})^{\alpha}p_{\alpha}-(c_{L})^{\alpha\beta% }p_{\alpha}p_{\beta}]_{ab}.
  39. S ( L ) = e - i h eff L 1 - i h eff L - 1 2 h 2 eff L 2 + . S(L)=e^{-ih\text{eff}L}\simeq 1-ih\text{eff}L-\frac{1}{2}h^{2}\text{eff}L^{2}+\cdots.
  40. P ν b ν a L 2 | ( h eff ) a b | 2 , a b . P_{\nu_{b}\rightarrow\nu_{a}}\simeq L^{2}|(h\text{eff})_{ab}|^{2},\quad a\neq b.
  41. 10 - 19 GeV 10^{-19}\,\,\text{GeV}
  42. ( a L ) a b α (a_{L})^{\alpha}_{ab}
  43. 10 - 17 10^{-17}
  44. ( c L ) a b α β (c_{L})^{\alpha\beta}_{ab}
  45. ( a L ) a b α (a_{L})^{\alpha}_{ab}
  46. ( c L ) a b α β (c_{L})^{\alpha\beta}_{ab}
  47. h = h 0 + δ h , h=h_{0}+\delta h,
  48. P ν b ν a = P ν b ν a ( 0 ) + P ν b ν a ( 1 ) + P ν b ν a ( 2 ) + , P_{\nu_{b}\rightarrow\nu_{a}}=P_{\nu_{b}\rightarrow\nu_{a}}^{(0)}+P_{\nu_{b}% \rightarrow\nu_{a}}^{(1)}+P_{\nu_{b}\rightarrow\nu_{a}}^{(2)}+\cdots,
  49. P ν b ν a ( 0 ) P_{\nu_{b}\rightarrow\nu_{a}}^{(0)}
  50. P ν μ ν τ ( 1 ) = - R e ( δ h μ τ ) L sin ( Δ m 32 2 L / 2 E ) . P_{\nu_{\mu}\rightarrow\nu_{\tau}}^{(1)}=-Re(\delta h_{\mu\tau})L\,\sin{(% \Delta m^{2}_{32}L/2E)}.
  51. ( a L ) a b α (a_{L})^{\alpha}_{ab}
  52. ( c L ) a b α β (c_{L})^{\alpha\beta}_{ab}
  53. Δ m 2 1 eV 2 \Delta m^{2}\sim 1\,\,\text{eV}^{2}

Lorenz_96_model.html

  1. d x i d t = - x i - 2 x i - 1 + x i - 1 x i + 1 - x i + F \frac{dx_{i}}{dt}=-x_{i-2}x_{i-1}+x_{i-1}x_{i+1}-x_{i}+F
  2. x i x_{i}
  3. F F
  4. F = 8 F=8

Lorenz_asymmetry_coefficient.html

  1. S = F ( μ ) + L ( μ ) S=F(\mu)+L(\mu)\,
  2. ( x 1 , , x m , x m + 1 , , x n ) (x_{1},...,x_{m},x_{m+1},...,x_{n})
  3. δ = μ - x m x m + 1 - x m \delta=\frac{\mu-x_{m}}{x_{m+1}-x_{m}}
  4. F ( μ ) = m + δ n F(\mu)=\frac{m+\delta}{n}
  5. L ( μ ) = L m + δ x m + 1 L n L(\mu)=\frac{L_{m}+\delta x_{m+1}}{L_{n}}
  6. L i = j = 1 i x j L_{i}=\sum_{j=1}^{i}x_{j}

Lossless-Join_Decomposition.html

  1. R R
  2. R 1 R_{1}
  3. R 2 R_{2}
  4. R 1 R_{1}
  5. R 2 R_{2}
  6. R R
  7. R R
  8. F F
  9. R R
  10. R 1 R_{1}
  11. R 2 R_{2}
  12. R R
  13. F F
  14. F F
  15. F F
  16. R 1 R_{1}
  17. R 2 R_{2}
  18. R 1 R_{1}
  19. R 1 R_{1}
  20. R 2 R_{2}
  21. R 2 R_{2}
  22. R = ( A , B , C , D ) R=(A,B,C,D)
  23. A A
  24. B B
  25. C C
  26. D D
  27. F = { A B C } F=\{A\rightarrow BC\}
  28. R 1 = ( A , B , C ) R_{1}=(A,B,C)
  29. R 2 = ( A , D ) R_{2}=(A,D)
  30. F F
  31. R 1 R 2 = ( A ) R_{1}\cap R_{2}=(A)
  32. A A
  33. R 1 R_{1}
  34. A B C A\rightarrow BC
  35. R 1 R 2 R 1 R_{1}\cap R_{2}\rightarrow R_{1}

Lottery_(probability).html

  1. \succsim\!
  2. α \alpha
  3. p q p\succsim\!q
  4. α p + ( 1 - α ) r α q + ( 1 - α ) r \alpha p+(1-\alpha)r\succsim\!\alpha q+(1-\alpha)r

Lovász_number.html

  1. u i T u j = { 1 , if i = j , 0 , if i j E . u_{i}^{\mathrm{T}}u_{j}=\begin{cases}1,&\mbox{if }~{}i=j,\\ 0,&\mbox{if }~{}ij\notin E.\end{cases}
  2. ϑ ( G ) = min c , U max i V 1 ( c T u i ) 2 , \vartheta(G)=\min_{c,U}\max_{i\in V}\frac{1}{(c^{\mathrm{T}}u_{i})^{2}},
  3. ϑ ( G ) = min A λ max ( A ) . \vartheta(G)=\min_{A}\lambda\text{max}(A).
  4. ϑ ( G ) = max B Tr ( B J ) . \vartheta(G)=\max_{B}\operatorname{Tr}(BJ).
  5. G ¯ \overline{G}
  6. G ¯ \overline{G}
  7. ϑ ( G ) = max d , V i V ( d T v i ) 2 . \vartheta(G)=\max_{d,V}\sum_{i\in V}(d^{\mathrm{T}}v_{i})^{2}.
  8. ϑ ( K n ) = 1 \vartheta(K_{n})=1
  9. ϑ ( K ¯ n ) = n \vartheta(\bar{K}_{n})=n
  10. ϑ ( C 5 ) = 5 \vartheta(C_{5})=\sqrt{5}
  11. ϑ ( C n ) = { n cos ( π / n ) 1 + cos ( π / n ) for odd n , n 2 for even n \vartheta(C_{n})=\begin{cases}\frac{n\cos(\pi/n)}{1+\cos(\pi/n)}&\,\text{for % odd }n,\\ \frac{n}{2}&\,\text{for even }n\end{cases}
  12. ϑ ( K G 5 , 2 ) = 4 \vartheta(KG_{5,2})=4
  13. ϑ ( K G n , k ) = ( n - 1 k - 1 ) \vartheta(KG_{n,k})={\left({{n-1}\atop{k-1}}\right)}
  14. ϑ ( K n 1 , , n k ) = max 1 i k n i \vartheta(K_{n_{1},\dots,n_{k}})=\max_{1\leq i\leq k}n_{i}
  15. ϑ ( G H ) = ϑ ( G ) ϑ ( H ) . \vartheta(G\boxtimes H)=\vartheta(G)\vartheta(H).
  16. G ¯ \overline{G}
  17. ϑ ( G ) ϑ ( G ¯ ) n , \vartheta(G)\vartheta(\bar{G})\geq n,
  18. ω ( G ) ϑ ( G ¯ ) χ ( G ) , \omega(G)\leq\vartheta(\bar{G})\leq\chi(G),
  19. Θ ( G ) = sup k α ( G k ) k = lim k α ( G k ) k , \Theta(G)=\sup_{k}\sqrt[k]{\alpha(G^{k})}=\lim_{k\rightarrow\infty}\sqrt[k]{% \alpha(G^{k})},
  20. α ( G ) Θ ( G ) ϑ ( G ) . \alpha(G)\leq\Theta(G)\leq\vartheta(G).
  21. [ u m r a d , u 5 ] [u^{\prime}mrad^{\prime},u^{\prime}5^{\prime}]
  22. [ u m r a d , u 5 ] [u^{\prime}mrad^{\prime},u^{\prime}5^{\prime}]
  23. u k = ( cos θ sin θ cos φ k sin θ sin φ k ) , cos θ = 1 5 4 , φ k = 2 π k 5 u_{k}=\begin{pmatrix}\cos{\theta}\\ \sin{\theta}\cos{\varphi_{k}}\\ \sin{\theta}\sin{\varphi_{k}}\end{pmatrix},\quad\cos{\theta}=\frac{1}{\sqrt[4]% {5}},\quad\varphi_{k}=\frac{2\pi k}{5}
  24. [ u m r a d , u 5 ] [u^{\prime}mrad^{\prime},u^{\prime}5^{\prime}]
  25. [ u m r a d , u 5 ] [u^{\prime}mrad^{\prime},u^{\prime}5^{\prime}]

Lovelock_theory_of_gravity.html

  1. D D
  2. D = 3 , 4 D=3,4
  3. D > 4 D>4
  4. = - g n = 0 t α n n , n = 1 2 n δ α 1 β 1 α n β n μ 1 ν 1 μ n ν n r = 1 n R μ r ν r α r β r \mathcal{L}=\sqrt{-g}\ \sum\limits_{n=0}^{t}\alpha_{n}\ \mathcal{R}^{n},\qquad% \mathcal{R}^{n}=\frac{1}{2^{n}}\delta_{\alpha_{1}\beta_{1}...\alpha_{n}\beta_{% n}}^{\mu_{1}\nu_{1}...\mu_{n}\nu_{n}}\prod\limits_{r=1}^{n}R_{\quad\mu_{r}\nu_% {r}}^{\alpha_{r}\beta_{r}}
  5. R μ ν α β R_{\quad\mu\nu}^{\alpha\beta}
  6. δ \delta
  7. δ α 1 β 1 α n β n μ 1 ν 1 μ n ν n = 1 n ! δ [ α 1 μ 1 δ β 1 ν 1 δ α n μ n δ β n ] ν n . \delta_{\alpha_{1}\beta_{1}\cdots\alpha_{n}\beta_{n}}^{\mu_{1}\nu_{1}...\mu_{n% }\nu_{n}}=\frac{1}{n!}\delta_{[\alpha_{1}}^{\mu_{1}}\delta_{\beta_{1}}^{\nu_{1% }}\cdots\delta_{\alpha_{n}}^{\mu_{n}}\delta_{\beta_{n}]}^{\nu_{n}}.
  8. n \mathcal{R}^{n}
  9. \mathcal{L}
  10. 2 n 2n
  11. n < D / 2 n<D/2
  12. t t
  13. D = 2 t + 2 D=2t+2
  14. D = 2 t + 1 D=2t+1
  15. α n \alpha_{n}
  16. \mathcal{L}
  17. 2 n - D {}^{2n-D}
  18. α 1 = ( 16 π G ) - 1 = l P 2 - D \alpha_{1}=(16\pi G)^{-1}=l_{P}^{2-D}
  19. \mathcal{L}
  20. = - g ( α 0 + α 1 R + α 2 ( R 2 + R α β μ ν R α β μ ν - 4 R μ ν R μ ν ) + α 3 𝒪 ( R 3 ) ) , \mathcal{L}=\sqrt{-g}\ (\alpha_{0}+\alpha_{1}R+\alpha_{2}\left(R^{2}+R_{\alpha% \beta\mu\nu}R^{\alpha\beta\mu\nu}-4R_{\mu\nu}R^{\mu\nu}\right)+\alpha_{3}% \mathcal{O}(R^{3})),
  21. α 0 \alpha_{0}
  22. Λ \Lambda
  23. α n \alpha_{n}
  24. n 2 n\geq 2
  25. R μ ν α β R_{\quad\mu\nu}^{\alpha\beta}
  26. 2 = R 2 + R α β μ ν R α β μ ν - 4 R μ ν R μ ν \mathcal{R}^{2}=R^{2}+R_{\alpha\beta\mu\nu}R^{\alpha\beta\mu\nu}-4R_{\mu\nu}R^% {\mu\nu}
  27. D D

Low-rank_approximation.html

  1. 𝒮 : n p m × n \mathcal{S}:\mathbb{R}^{n_{p}}\to\mathbb{R}^{m\times n}
  2. p n p p\in\mathbb{R}^{n_{p}}
  3. r r
  4. minimize over p ^ p - p ^ subject to rank ( 𝒮 ( p ^ ) ) r . \,\text{minimize}\quad\,\text{over }\widehat{p}\quad\|p-\widehat{p}\|\quad\,% \text{subject to}\quad\operatorname{rank}\big(\mathcal{S}(\widehat{p})\big)% \leq r.
  5. minimize over D ^ D - D ^ F subject to rank ( D ^ ) r \,\text{minimize}\quad\,\text{over }\widehat{D}\quad\|D-\widehat{D}\|_{\,\text% {F}}\quad\,\text{subject to}\quad\operatorname{rank}\big(\widehat{D}\big)\leq r
  6. D = U Σ V m × n , m n D=U\Sigma V^{\top}\in\mathbb{R}^{m\times n},\quad m\leq n
  7. D D
  8. U U
  9. Σ = : diag ( σ 1 , , σ m ) \Sigma=:\operatorname{diag}(\sigma_{1},\ldots,\sigma_{m})
  10. V V
  11. U = : [ U 1 U 2 ] , Σ = : [ Σ 1 0 0 Σ 2 ] , and V = : [ V 1 V 2 ] , U=:\begin{bmatrix}U_{1}&U_{2}\end{bmatrix},\quad\Sigma=:\begin{bmatrix}\Sigma_% {1}&0\\ 0&\Sigma_{2}\end{bmatrix},\quad\,\text{and}\quad V=:\begin{bmatrix}V_{1}&V_{2}% \end{bmatrix},
  12. Σ 1 \Sigma_{1}
  13. r × r r\times r
  14. U 1 U_{1}
  15. m × r m\times r
  16. V 1 V_{1}
  17. n × r n\times r
  18. r r
  19. D ^ * = U 1 Σ 1 V 1 , \widehat{D}^{*}=U_{1}\Sigma_{1}V_{1}^{\top},
  20. D - D ^ * F = min rank ( D ^ ) r D - D ^ F = σ r + 1 2 + + σ m 2 . \|D-\widehat{D}^{*}\|_{\,\text{F}}=\min_{\operatorname{rank}(\widehat{D})\leq r% }\|D-\widehat{D}\|_{\,\text{F}}=\sqrt{\sigma^{2}_{r+1}+\cdots+\sigma^{2}_{m}}.
  21. D ^ * \widehat{D}^{*}
  22. σ r + 1 σ r \sigma_{r+1}\neq\sigma_{r}
  23. A = U n Σ n V n A=U_{n}\Sigma_{n}V^{\top}_{n}
  24. U n U_{n}\quad
  25. V n \quad V^{\top}_{n}
  26. Σ n \Sigma_{n}
  27. ( σ 1 σ 2 σ n ) (\sigma_{1}\sigma_{2}\cdots\sigma_{n})
  28. ( σ n σ n - 1 σ 1 ) (\sigma_{n}\leq\sigma_{n-1}\leq\cdots\leq\sigma_{1})
  29. A A
  30. A k = Σ i = 1 k u i σ i v i A^{k}=\Sigma^{k}_{i=1}u_{i}\sigma_{i}v_{i}
  31. A k A^{k}\quad
  32. A - A k is minimum \|A-A^{k}\|\quad\,\text{is minimum}
  33. B s.t. A - B 2 2 < A - A k 2 2 = σ k + 1 2 \quad\exists\quad B\quad\,\text{s.t.}\|A-B\|^{2}_{2}<\|A-A^{k}\|^{2}_{2}\quad=% \quad\sigma^{2}_{k+1}
  34. rank ( B ) k (Assuming in Low Rank Approximation, we are approximating via a matrix whose rank k \operatorname{rank}(B)\leq k\quad\,\text{(Assuming in Low Rank Approximation, % we are approximating via a matrix whose rank}\leq k
  35. dim ( null ( B ) ) + rank ( B ) = n dim ( null ( B ) ) n - k \operatorname{dim}(\operatorname{null}(B))+\operatorname{rank}(B)=n\rightarrow% \operatorname{dim(\operatorname{null}(B))}\geq n-k
  36. w null ( B ) w\in\operatorname{null}(B)
  37. ( A - B ) w 2 = A w 2 < σ k + 1 \|(A-B)w\|_{2}=\|Aw\|_{2}<\sigma_{k+1}
  38. ( k + 1 ) \exists(k+1)\quad
  39. ( v 1 , v 2 , , v n ) (v_{1},v_{2},\cdots,v_{n})\quad
  40. V span ( v 1 , v 2 , , v n ) and A V 2 σ k + 1 V\in\operatorname{span}(v_{1},v_{2},\cdots,v_{n})\quad\,\text{and}\|AV\|_{2}% \geq\sigma_{k+1}
  41. n - k + k + 1 > n n-k+k+1>n\quad
  42. A k A^{k}
  43. D - D ^ D-\widehat{D}
  44. minimize over D ^ vec ( D - D ^ ) W vec ( D - D ^ ) subject to rank ( D ^ ) r , \,\text{minimize}\quad\,\text{over }\widehat{D}\quad\operatorname{vec}^{\top}(% D-\widehat{D})W\operatorname{vec}(D-\widehat{D})\quad\,\text{subject to}\quad% \operatorname{rank}(\widehat{D})\leq r,
  45. v e c ( A ) vec(A)
  46. A A
  47. W W
  48. rank ( D ^ ) r there are P \R m × r and L \R r × n such that D ^ = P L \operatorname{rank}(\widehat{D})\leq r\quad\iff\quad\,\text{there are }P\in\R^% {m\times r}\,\text{ and }L\in\R^{r\times n}\,\text{ such that }\widehat{D}=PL
  49. rank ( D ^ ) r there is full row rank R \R m - r × m such that R D ^ = 0 \operatorname{rank}(\widehat{D})\leq r\quad\iff\quad\,\text{there is full row % rank }R\in\R^{m-r\times m}\,\text{ such that }R\widehat{D}=0
  50. minimize over D ^ , P and L vec ( D - D ^ ) W vec ( D - D ^ ) subject to D ^ = P L \,\text{minimize}\quad\,\text{over }\widehat{D},P\,\text{ and }L\quad% \operatorname{vec}^{\top}(D-\widehat{D})W\operatorname{vec}(D-\widehat{D})% \quad\,\text{subject to}\quad\widehat{D}=PL
  51. minimize over D ^ and R vec ( D - D ^ ) W vec ( D - D ^ ) subject to R D ^ = 0 and R R = I r , \,\text{minimize}\quad\,\text{over }\widehat{D}\,\text{ and }R\quad% \operatorname{vec}^{\top}(D-\widehat{D})W\operatorname{vec}(D-\widehat{D})% \quad\,\text{subject to}\quad R\widehat{D}=0\quad\,\text{and}\quad RR^{\top}=I% _{r},
  52. I r I_{r}
  53. r r
  54. P P
  55. L L
  56. P P
  57. L L
  58. P P
  59. L L

Low_level_injection.html

  1. n n
  2. Δ n \Delta n
  3. n 0 n_{0}
  4. n = Δ n + n 0 n=\Delta n+n_{0}
  5. Δ n = Δ p \Delta n=\Delta p
  6. Δ n N D \Delta n<<N_{D}
  7. n = N D n=N_{D}
  8. p = Δ p + p 0 p=\Delta p+p_{0}

Lubin–Tate_formal_group_law.html

  1. e ( F ( x , y ) ) = F ( e ( x ) , e ( y ) ) . e(F(x,y))=F(e(x),e(y)).
  2. f ( F ( X , Y ) ) = G ( f ( X ) , f ( Y ) ) f(F(X,Y))=G(f(X),f(Y))

Lucas_aggregate_supply_function.html

  1. Y s = f ( P - P e x p e c t e d ) Y_{s}=f(P-P_{expected})
  2. Y N t Y_{N_{t}}
  3. P t P_{t}
  4. Ω t - 1 \Omega_{t-1}
  5. α \alpha
  6. Y s = Y N t + α [ P t - E ( P t | Ω t - 1 ) ] Y_{s}=Y_{N_{t}}+\alpha[P_{t}-E\left(P_{t}|\Omega_{t-1}\right)]

Lyapunov_vector.html

  1. x d x\in\mathbb{R}^{d}
  2. v ( k ) ( x ) v^{(k)}(x)
  3. ( k = 1 d ) (k=1\dots d)
  4. λ k \lambda_{k}
  5. x n + 1 = M t n t n + 1 ( x n ) x_{n+1}=M_{t_{n}\to t_{n+1}}(x_{n})
  6. x n x_{n}
  7. t n t_{n}
  8. x n + 1 x_{n+1}
  9. t n + 1 t_{n+1}
  10. J n ~{}J_{n}
  11. h n h_{n}
  12. M t n t n + 1 ( x n + h n ) M t n t n + 1 ( x n ) + J n h n = x n + 1 + h n + 1 M_{t_{n}\to t_{n+1}}(x_{n}+h_{n})\approx M_{t_{n}\to t_{n+1}}(x_{n})+J_{n}h_{n% }=x_{n+1}+h_{n+1}
  13. Q 0 = 𝕀 Q_{0}=\mathbb{I}~{}
  14. Q n + 1 R n + 1 = J n Q n Q_{n+1}R_{n+1}=J_{n}Q_{n}
  15. Q n + 1 R n + 1 Q_{n+1}R_{n+1}
  16. J n Q n J_{n}Q_{n}
  17. x n x_{n}
  18. Q 0 Q_{0}
  19. Q n Q_{n}
  20. k k
  21. k k
  22. R n R_{n}
  23. r k k ( n ) r^{(n)}_{kk}
  24. R n R_{n}
  25. λ k = lim m 1 t n + m - t n l = 1 m log r k k ( n + l ) \lambda_{k}=\lim_{m\to\infty}\frac{1}{t_{n+m}-t_{n}}\sum_{l=1}^{m}\log r^{(n+l% )}_{kk}
  26. λ 1 λ 2 λ d \lambda_{1}\geq\lambda_{2}\geq\dots\geq\lambda_{d}
  27. k k
  28. Q n Q_{n}
  29. Q n Q_{n}
  30. n n
  31. k k
  32. Q n + m Q_{n+m}
  33. k k
  34. n n
  35. m m
  36. c n = Q n T h n c_{n}=Q_{n}^{T}h_{n}
  37. c n - 1 = R n - 1 c n c_{n-1}=R_{n}^{-1}c_{n}
  38. k k
  39. c n + m c_{n+m}
  40. Q n c n Q_{n}c_{n}
  41. v ( k ) ( x n ) v^{(k)}(x_{n})
  42. k k
  43. m m
  44. n n

Lyapunov–Schmidt_reduction.html

  1. f ( x , λ ) = 0 f(x,\lambda)=0\,
  2. X , Λ , X,\Lambda,
  3. Y Y
  4. Λ \Lambda
  5. f ( x , λ ) f(x,\lambda)
  6. C p C^{p}
  7. ( x 0 , λ 0 ) X × Λ (x_{0},\lambda_{0})\in X\times\Lambda
  8. Y Y
  9. f ( x 0 , λ 0 ) = 0. f(x_{0},\lambda_{0})=0.
  10. f x ( x , λ ) f_{x}(x,\lambda)
  11. x ( λ ) x(\lambda)
  12. f ( x ( λ ) , λ ) = 0 f(x(\lambda),\lambda)=0
  13. λ 0 \lambda_{0}
  14. f x ( x , λ ) f_{x}(x,\lambda)
  15. f x ( x , λ ) f_{x}(x,\lambda)
  16. ker f x ( x 0 , λ 0 ) = X 1 \ker f_{x}(x_{0},\lambda_{0})=X_{1}
  17. X 1 X_{1}
  18. ran f x ( x 0 , λ 0 ) = Y 1 \mathrm{ran}f_{x}(x_{0},\lambda_{0})=Y_{1}
  19. Y Y
  20. ( x 0 , λ 0 ) = ( 0 , 0 ) . (x_{0},\lambda_{0})=(0,0).
  21. Y Y
  22. Y = Y 1 Y 2 Y=Y_{1}\oplus Y_{2}
  23. dim Y 2 < \dim Y_{2}<\infty
  24. Q Q
  25. Y 1 Y_{1}
  26. X = X 1 X 2 X=X_{1}\oplus X_{2}
  27. Q Q
  28. I - Q I-Q
  29. Q f ( x , λ ) = 0 Qf(x,\lambda)=0\,
  30. ( I - Q ) f ( x , λ ) = 0 (I-Q)f(x,\lambda)=0\,
  31. x 1 X 1 x_{1}\in X_{1}
  32. x 2 X 2 x_{2}\in X_{2}
  33. Q f ( x 1 + x 2 , λ ) = 0 Qf(x_{1}+x_{2},\lambda)=0\,
  34. x 2 x_{2}
  35. Q f ( x 1 + x 2 , λ ) : X 2 × ( X 1 × Λ ) Y 1 Qf(x_{1}+x_{2},\lambda):\quad X_{2}\times(X_{1}\times\Lambda)\to Y_{1}\,
  36. x 2 ( x 1 , λ ) x_{2}(x_{1},\lambda)
  37. Q f ( x 1 + x 2 ( x 1 , λ ) , λ ) = 0 Qf(x_{1}+x_{2}(x_{1},\lambda),\lambda)=0\,
  38. x 2 ( x 1 , λ ) x_{2}(x_{1},\lambda)
  39. ( I - Q ) f ( x 1 + x 2 ( x 1 , λ ) , λ ) = 0 (I-Q)f(x_{1}+x_{2}(x_{1},\lambda),\lambda)=0\,
  40. ( I - Q ) (I-Q)
  41. x 1 x_{1}
  42. λ \lambda

M-tree.html

  1. r r
  2. n n
  3. l l
  4. N N
  5. r r
  6. N N
  7. n n
  8. l l
  9. N N
  10. N N
  11. O O
  12. N N
  13. N N
  14. N N
  15. N N
  16. N N
  17. M T MT
  18. O n O_{n}
  19. M T MT
  20. M T MT
  21. O n O_{n}
  22. N e N_{e}
  23. N N
  24. N N
  25. N i n N_{in}
  26. N e N_{e}
  27. N R O N_{RO}
  28. d ( O r , O n ) r ( O r ) d(O_{r},O_{n})<=r(O_{r})
  29. N i n N_{in}
  30. O r * = min O r N i n d ( O r , O n ) O_{r}^{*}=\min_{O_{r}\in N_{in}}d(O_{r},O_{n})
  31. O r * = min O r N i n d ( O r , O n ) - r ( O r ) O_{r}^{*}=\min_{O_{r}\in N_{in}}d(O_{r},O_{n})-r(O_{r})
  32. r ( O r * ) r(O_{r}^{*})
  33. d ( O r * , O n ) d(O_{r}^{*},O_{n})
  34. T ( O r * ) T(O_{r}^{*})
  35. O n O_{n}
  36. N N
  37. N N
  38. O n O_{n}
  39. N N
  40. O n O_{n}
  41. N N
  42. N N
  43. N N
  44. N N
  45. N N
  46. N 1 N_{1}
  47. N 2 N_{2}
  48. N p N_{p}
  49. N N
  50. N p N_{p}
  51. N 2 N_{2}
  52. N N
  53. M T MT
  54. O n O_{n}
  55. M T MT
  56. N N NN
  57. N O N\cup O
  58. N N
  59. O p O_{p}
  60. N N
  61. N p N_{p}
  62. N N
  63. N N^{\prime}
  64. O p 1 O_{p1}
  65. O p 2 O_{p2}
  66. N N
  67. O p 1 O_{p1}
  68. O p 2 O_{p2}
  69. N N
  70. O p 1 O_{p1}
  71. O p 2 O_{p2}
  72. N 1 N_{1}
  73. N 2 N_{2}
  74. N 1 N_{1}
  75. N N
  76. N 2 N_{2}
  77. N N^{\prime}
  78. N N
  79. N p N_{p}
  80. O p 1 O_{p1}
  81. O p 2 O_{p2}
  82. N p N_{p}
  83. O p O_{p}
  84. O p 1 O_{p1}
  85. N p N_{p}
  86. N p N_{p}
  87. O p 2 O_{p2}
  88. N p N_{p}
  89. N p N_{p}
  90. O p 2 O_{p2}
  91. N N
  92. Q Q
  93. r ( Q ) r(Q)
  94. d ( O j , Q ) r ( Q ) d(Oj,Q)\leq r(Q)
  95. O p O_{p}
  96. N N
  97. N N
  98. O r O_{r}
  99. | d ( O p , Q ) - d ( O r , O p ) | r ( Q ) + r ( O r ) |d(O_{p},Q)-d(O_{r},O_{p})|\leq r(Q)+r(O_{r})
  100. d ( O r , Q ) d(O_{r},Q)
  101. d ( O r , Q ) r ( Q ) + r ( O r ) d(O_{r},Q)\leq r(Q)+r(O_{r})
  102. T ( O r T(O_{r}
  103. Q Q
  104. r ( Q ) r(Q)
  105. O j O_{j}
  106. N N
  107. | d ( O p , Q ) - d ( O j , O p ) | r ( Q ) |d(O_{p},Q)-d(O_{j},O_{p})|\leq r(Q)
  108. d ( O j , Q ) d(O_{j},Q)
  109. d ( O j , Q ) d(O_{j},Q)
  110. r ( Q ) r(Q)
  111. o i d ( O j ) oid(O_{j})
  112. o i d ( O j ) oid(O_{j})
  113. T ( O r ) T(O_{r})
  114. O r O_{r}

M._Stanley_Livingston.html

  1. < m t p l > ω = e B M c <mtpl>{{\omega}}={eB\over Mc}

M::G::1_queue.html

  1. π ( z ) = ( 1 - z ) ( 1 - ρ ) g ( λ ( 1 - z ) ) g ( λ ( 1 - z ) ) - z \pi(z)=\frac{(1-z)(1-\rho)g(\lambda(1-z))}{g(\lambda(1-z))-z}
  2. P = ( a 0 a 1 a 2 a 3 a 4 a 0 a 1 a 2 a 3 a 4 0 a 0 a 1 a 2 a 3 0 0 a 0 a 1 a 2 0 0 0 a 0 a 1 ) P=\begin{pmatrix}a_{0}&a_{1}&a_{2}&a_{3}&a_{4}&\cdots\\ a_{0}&a_{1}&a_{2}&a_{3}&a_{4}&\cdots\\ 0&a_{0}&a_{1}&a_{2}&a_{3}&\cdots\\ 0&0&a_{0}&a_{1}&a_{2}&\cdots\\ 0&0&0&a_{0}&a_{1}&\cdots\\ \vdots&\vdots&\vdots&\vdots&\vdots&\ddots\end{pmatrix}
  3. a v = 0 e - λ u ( λ u ) v v ! d F ( u ) for v 0 a_{v}=\int_{0}^{\infty}e^{-\lambda u}\frac{(\lambda u)^{v}}{v!}\,\text{d}F(u)~% {}\,\text{ for }v\geq 0
  4. ϕ ( s ) \phi(s)
  5. ϕ ( s ) = g [ s + λ - λ ϕ ( s ) ] \phi(s)=g[s+\lambda-\lambda\phi(s)]
  6. W ( s ) = ( 1 - ρ ) s g ( s ) s - λ ( 1 - g ( s ) ) W^{\ast}(s)=\frac{(1-\rho)sg(s)}{s-\lambda(1-g(s))}

M::G::k_queue.html

  1. E [ W M/G/ k ] = C 2 + 1 2 𝔼 [ W M/M/ c ] E[W^{\,\text{M/G/}k}]=\frac{C^{2}+1}{2}\mathbb{E}[W^{\,\text{M/M/}c}]

Machining_Time.html

  1. Drilling Time(T) = L f × N \,\text{Drilling Time(T)}=\frac{L}{f\times N}
  2. L L
  3. f f
  4. N N

Magnetic_field-assisted_finishing.html

  1. F = ( m B ) \vec{F}=\nabla(\vec{m}\cdot\vec{B})
  2. F = m B \vec{F}=\vec{m}\cdot\nabla\vec{B}
  3. m = M V \vec{m}=\vec{M}V
  4. M = H k χ \vec{M}=\vec{H_{k}}\chi
  5. B = μ 0 H a \vec{B}=\mu_{0}\vec{H_{a}}
  6. F = μ 0 χ V H k H a \vec{F}=\mu_{0}\chi V\vec{H_{k}}\nabla\vec{H_{a}}
  7. H k \vec{H_{k}}
  8. H a \vec{H_{a}}
  9. B \vec{B}
  10. M \vec{M}
  11. m \vec{m}
  12. H \nabla\vec{H}
  13. V V
  14. χ \chi
  15. μ 0 \mu_{0}
  16. W = W m + W f + W t W=W_{m}+W_{f}+W_{t}
  17. F n = m f n = B 2 2 μ 0 ( 1 - 1 μ m ) S F_{n}=mf_{n}=\frac{B^{2}}{2\mu_{0}}\left(1-\frac{1}{\mu_{m}}\right)S
  18. μ m = 2 + μ F - 2 ( 1 - μ F ) V i 2 + μ F - ( 1 - μ F ) V i \mu_{m}=\frac{2+\mu_{F}-2(1-\mu_{F})Vi}{2+\mu_{F}-(1-\mu_{F})Vi}
  19. f n = f_{n}=
  20. m = m=
  21. B = B=
  22. S = S=
  23. μ 0 = \mu_{0}=
  24. μ m = \mu_{m}=
  25. μ f = \mu_{f}=
  26. F H = d W d x F_{H}=\frac{dW}{dx}

Magnetic_levitation.html

  1. P m a g = B 2 2 μ 0 P_{mag}=\frac{B^{2}}{2\mu_{0}}
  2. P m a g P_{mag}
  3. B B
  4. μ 0 \mu_{0}
  5. B d B d z = μ 0 ρ g χ B\frac{dB}{dz}=\mu_{0}\,\rho\,\frac{g}{\chi}
  6. χ \chi
  7. ρ \rho
  8. g g
  9. μ 0 \mu_{0}
  10. B B
  11. d B d z \frac{dB}{dz}
  12. B d B d z 1400 T 2 / m B\frac{dB}{dz}\approx 1400\ \mathrm{T^{2}/m}
  13. B d B d z 375 T 2 / m . B\frac{dB}{dz}\approx 375\ \mathrm{T^{2}/m}.

Magnetic_mineralogy.html

  1. × 10 - 6 \times 10^{-6}
  2. × 10 - 6 \times 10^{-6}
  3. x x
  4. x = 0 x=0
  5. x = 1 x=1

Magnetic_radiation_reaction_force.html

  1. 𝐅 rad = - μ 0 q 2 R 24 π c 3 d 3 a d t 3 \mathbf{F}_{\mathrm{rad}}=-\frac{\mu_{0}q^{2}R}{24\pi c^{3}}\frac{\mathrm{d}^{% 3}\vec{a}}{\mathrm{d}t^{3}}
  2. d 3 a d t 3 \frac{\mathrm{d}^{3}\vec{a}}{\mathrm{d}t^{3}}
  3. P = μ 0 m ¨ 2 6 π c 3 P=\frac{\mu_{0}\ddot{m}^{2}}{6\pi c^{3}}
  4. 𝐦 = 1 2 q 𝐫 × 𝐯 \mathbf{m}=\frac{1}{2}\,q\,\mathbf{r}\times\mathbf{v}
  5. 𝐫 \mathbf{r}
  6. q q
  7. 𝐯 \mathbf{v}
  8. P = μ 0 q 2 r 2 a ˙ 2 24 π c 3 P=\frac{\mu_{0}q^{2}r^{2}\dot{a}^{2}}{24\pi c^{3}}
  9. τ 1 \tau_{1}
  10. τ 2 \tau_{2}
  11. τ 1 τ 2 𝐅 rad 𝐯 d t = τ 1 τ 2 - P d t = - τ 1 τ 2 μ 0 q 2 r 2 a ˙ 2 24 π c 3 d t = - τ 1 τ 2 μ 0 q 2 r 2 24 π c 3 d 𝐚 d t d 𝐚 d t d t \int_{\tau_{1}}^{\tau_{2}}\mathbf{F}_{\mathrm{rad}}\cdot\mathbf{v}dt=\int_{% \tau_{1}}^{\tau_{2}}-Pdt=-\int_{\tau_{1}}^{\tau_{2}}\frac{\mu_{0}q^{2}r^{2}% \dot{a}^{2}}{24\pi c^{3}}dt=-\int_{\tau_{1}}^{\tau_{2}}\frac{\mu_{0}q^{2}r^{2}% }{24\pi c^{3}}\frac{d\mathbf{a}}{dt}\cdot\frac{d\mathbf{a}}{dt}dt
  12. τ 1 τ 2 𝐅 rad 𝐯 d t = - μ 0 q 2 r 2 24 π c 3 d 𝐚 d t 𝐚 | τ 1 τ 2 + τ 1 τ 2 μ 0 q 2 r 2 24 π c 3 d 2 𝐚 d t 2 𝐚 d t = - 0 + τ 1 τ 2 μ 0 q 2 r 2 24 π c 3 𝐚 ¨ 𝐚 d t \int_{\tau_{1}}^{\tau_{2}}\mathbf{F}_{\mathrm{rad}}\cdot\mathbf{v}dt=-\frac{% \mu_{0}q^{2}r^{2}}{24\pi c^{3}}\frac{d\mathbf{a}}{dt}\cdot\mathbf{a}\bigg|_{% \tau_{1}}^{\tau_{2}}+\int_{\tau_{1}}^{\tau_{2}}\frac{\mu_{0}q^{2}r^{2}}{24\pi c% ^{3}}\frac{d^{2}\mathbf{a}}{dt^{2}}\cdot\mathbf{a}dt=-0+\int_{\tau_{1}}^{\tau_% {2}}\frac{\mu_{0}q^{2}r^{2}}{24\pi c^{3}}\mathbf{\ddot{a}}\cdot\mathbf{a}dt
  13. τ 1 τ 2 𝐅 rad 𝐯 d t = - μ 0 q 2 r 2 24 π c 3 d 𝐚 d t 𝐚 | τ 1 τ 2 + μ 0 q 2 r 2 24 π c 3 d 3 𝐯 d t 3 𝐯 | τ 1 τ 2 - τ 1 τ 2 μ 0 q 2 r 2 24 π c 3 d 3 𝐚 d t 3 𝐯 d t = - 0 + 0 - τ 1 τ 2 μ 0 q 2 r 2 24 π c 3 d 3 𝐚 d t 3 𝐯 d t \int_{\tau_{1}}^{\tau_{2}}\mathbf{F}_{\mathrm{rad}}\cdot\mathbf{v}dt=-\frac{% \mu_{0}q^{2}r^{2}}{24\pi c^{3}}\frac{d\mathbf{a}}{dt}\cdot\mathbf{a}\bigg|_{% \tau_{1}}^{\tau_{2}}+\frac{\mu_{0}q^{2}r^{2}}{24\pi c^{3}}\frac{d^{3}\mathbf{v% }}{dt^{3}}\cdot\mathbf{v}\bigg|_{\tau_{1}}^{\tau_{2}}-\int_{\tau_{1}}^{\tau_{2% }}\frac{\mu_{0}q^{2}r^{2}}{24\pi c^{3}}\frac{d^{3}\mathbf{a}}{dt^{3}}\cdot% \mathbf{v}dt=-0+0-\int_{\tau_{1}}^{\tau_{2}}\frac{\mu_{0}q^{2}r^{2}}{24\pi c^{% 3}}\frac{d^{3}\mathbf{a}}{dt^{3}}\cdot\mathbf{v}dt
  14. 𝐅 rad = - μ 0 q 2 r 2 24 π c 3 d 3 𝐚 d t 3 \mathbf{F}_{\mathrm{rad}}=-\frac{\mu_{0}q^{2}r^{2}}{24\pi c^{3}}\frac{d^{3}% \mathbf{a}}{dt^{3}}
  15. 𝐅 ext \mathbf{F}_{\mathrm{ext}}
  16. m 𝐯 ˙ = 𝐅 rad + 𝐅 ext = m t 0 < m t p l > v ¨ + 𝐅 ext . m\dot{\mathbf{v}}=\mathbf{F}_{\mathrm{rad}}+\mathbf{F}_{\mathrm{ext}}=mt_{0}% \ddot{\mathbf{<}mtpl>{{v}}}+\mathbf{F}_{\mathrm{ext}}.
  17. t 0 = μ 0 q 2 6 π m c . t_{0}=\frac{\mu_{0}q^{2}}{6\pi mc}.
  18. m 𝐯 ˙ = 1 t 0 t exp ( - t - t t 0 ) 𝐅 ext ( t ) d t . m\dot{\mathbf{v}}={1\over t_{0}}\int_{t}^{\infty}\exp\left(-{t^{\prime}-t\over t% _{0}}\right)\,\mathbf{F}_{\mathrm{ext}}(t^{\prime})\,dt^{\prime}.
  19. exp ( - t - t t 0 ) \exp\left(-{t^{\prime}-t\over t_{0}}\right)
  20. t 0 t_{0}
  21. t 0 t_{0}
  22. 10 - 24 10^{-24}

Magnetic_resonance_therapy.html

  1. ω \omega
  2. ( d B 0 d t 1 B 1 γ B 0 ) ( d B 0 d t 1 B 1 1 T 1 ) (\frac{dB_{0}}{dt}\cdot\frac{1}{B_{1}}\geq\gamma B_{0})\wedge(\frac{dB_{0}}{dt% }\cdot\frac{1}{B_{1}}\gg\frac{1}{T_{1}})
  3. B 1 B_{1}
  4. d B 0 / d t dB_{0}/dt
  5. γ \gamma

Magnetochemistry.html

  1. χ v \chi_{v}
  2. M = χ v H \vec{M}=\chi_{v}\vec{H}
  3. M \vec{M}
  4. H \vec{H}
  5. χ mol = M χ v / ρ \chi\text{mol}=M\chi_{v}/\rho
  6. e - Δ E / k T e^{-\Delta E/kT}
  7. χ = C T \chi={C\over T}
  8. C = N g 2 S ( S + 1 ) μ B 2 3 k C=\frac{Ng^{2}S(S+1)\mu_{B}^{2}}{3k}
  9. χ = C T - T c \chi=\frac{C}{T-T_{c}}
  10. μ eff = constant T χ \mu_{\,\text{eff}}=\mathrm{constant}\sqrt{T\chi}
  11. μ eff = 3 k N μ B 2 T χ 2.82787 T χ \mu_{\,\text{eff}}=\sqrt{3k\over N\mu_{B}^{2}}\sqrt{T\chi}\approx 2.82787\sqrt% {T\chi}
  12. μ eff = 3 k N μ 0 μ B 2 T χ 797.727 T χ \mu_{\,\text{eff}}=\sqrt{3k\over N\mu_{0}\mu_{B}^{2}}\sqrt{T\chi}\approx 797.7% 27\sqrt{T\chi}
  13. L \vec{L}
  14. S \vec{S}
  15. μ eff = L ( L + 1 ) + 4 S ( S + 1 ) μ B \mu_{\,\text{eff}}=\sqrt{\vec{L}(\vec{L}+1)+4\vec{S}(\vec{S}+1)}\mu_{B}
  16. L \vec{L}
  17. L = 0 \vec{L}=0
  18. μ eff = n ( n + 2 ) μ B \mu_{\,\text{eff}}=\sqrt{n(n+2)}\mu_{B}
  19. L = i l i \vec{L}=\sum_{i}\vec{l}_{i}
  20. S = i s i \vec{S}=\sum_{i}\vec{s}_{i}
  21. J = L + S \vec{J}=\vec{L}+\vec{S}
  22. μ eff = g J ( J + 1 ) ; g = 3 2 + S ( S + 1 ) - L ( L + 1 ) 2 J ( J + 1 ) \mu_{\,\text{eff}}=g\sqrt{\vec{J}(\vec{J}+1)};g={3\over 2}+\frac{\vec{S}(\vec{% S}+1)-\vec{L}(\vec{L}+1)}{2\vec{J}(\vec{J}+1)}
  23. J = i j i = i ( l i + s i ) \vec{J}=\sum_{i}\vec{j}_{i}=\sum_{i}(\vec{l}_{i}+\vec{s}_{i})

Magnetoelectric_effect.html

  1. P i = α i j H j + β i j k H j H k + P_{i}=\sum\alpha_{ij}H_{j}+\sum\beta_{ijk}H_{j}H_{k}+\ldots
  2. M i = α i j E j + β i j k E j E k + M_{i}=\sum\alpha_{ij}E_{j}+\sum\beta_{ijk}E_{j}E_{k}+\ldots

Magnonics.html

  1. 𝐌 \mathbf{M}
  2. 𝐇 eff \mathbf{H}_{\mathrm{eff}}
  3. 𝐦 t = - γ 𝐦 × 𝐇 eff + α 𝐦 × 𝐦 t . \frac{\partial\textbf{m}}{\partial t}\,=\,-\gamma\,\textbf{m}\times\,\textbf{H% }_{\mathrm{eff}}\,+\,\alpha\,\textbf{m}\times\frac{\partial\textbf{m}}{% \partial t}\,.\qquad
  4. α \alpha
  5. γ \gamma
  6. 𝐦 = 𝐌 / M S . \textbf{m}={\textbf{M}}/{\mathrm{M}_{S}}\,.

Mahler_polynomial.html

  1. g n ( x ) t n / n ! = exp ( x ( 1 + t - e t ) ) \displaystyle\sum g_{n}(x)t^{n}/n!=\exp(x(1+t-e^{t}))
  2. g 0 = 1 ; g_{0}=1;
  3. g 1 = 0 ; g_{1}=0;
  4. g 2 = - x ; g_{2}=-x;
  5. g 3 = - x ; g_{3}=-x;
  6. g 4 = - x + 3 x 2 ; g_{4}=-x+3x^{2};
  7. g 5 = - x + 10 x 2 ; g_{5}=-x+10x^{2};
  8. g 6 = - x + 25 x - 15 x 3 ; g_{6}=-x+25x^{-}15x^{3};
  9. g 7 = - x + 56 x 2 - 105 x 3 ; g_{7}=-x+56x^{2}-105x^{3};
  10. g 8 = - x + 119 x 2 - 490 x 3 + 105 x 4 ; g_{8}=-x+119x^{2}-490x^{3}+105x^{4};

Maintenance_philosophy.html

  1. R e l i a b i l i t y = 0.5 × ( 1 + e ( - λ × T i m e B e t w e e n M a i n t e n a n c e A c t i o n s ) ) Reliability=0.5\times\left(1+e^{\left(-\lambda\times Time\ Between\ % Maintenance\ Actions\right)}\right)
  2. 0.9 0.5 × ( 1 + e - 0.2 ) 0.9\approx 0.5\times\left(1+e^{-0.2}\right)
  3. λ \lambda
  4. λ = 1 M e a n T i m e B e t w e e n F a i l u r e \lambda=\frac{1}{Mean\ Time\ Between\ Failure}
  5. R e l i a b i l i t y = e ( - λ × T i m e ) Reliability=e^{\left(}-\lambda\times Time\right)
  6. P r o b a b i l i t y o f F a i l u r e = 1 - R e l i a b i l i t y Probability\ of\ Failure=1-Reliability
  7. λ \lambda
  8. λ = λ 1 + λ 2 + + λ n = k = 1 N λ k \lambda=\lambda_{1}+\lambda_{2}+...+\lambda_{n}=\sum_{k=1}^{N}\lambda_{k}
  9. λ = ( 1 1 λ 1 + 1 λ 2 + + 1 λ n ) = ( 1 k = 1 N 1 λ k ) \lambda=\left(\dfrac{1}{\dfrac{1}{\lambda_{1}}+\dfrac{1}{\lambda_{2}}+...+% \dfrac{1}{\lambda_{n}}}\right)=\left(\dfrac{1}{\sum_{k=1}^{N}\dfrac{1}{\lambda% _{k}}}\right)
  10. 5 o 5^{o}
  11. 60 o 60^{o}
  12. 30 o 30^{o}
  13. 25 o 25^{o}
  14. A v a i l a b i l i t y = A v a i l a b l e T i m e T o t a l T i m e Availability=\frac{Available\ Time}{Total\ Time}
  15. T o t a l T i m e = A v a i l a b l e T i m e + D o w n T i m e Total\ Time=Available\ Time+Down\ Time
  16. D o w n T i m e = M a i n t e n a n c e T i m e + F a u l t e d T i m e Down\ Time=Maintenance\ Time+Faulted\ Time
  17. C o v e r a g e > A v a i l a b i l i t y Coverage>Availability
  18. R e a d i n e s s = 1 - λ × M e a n T i m e T o R e c o v e r Readiness=1-\lambda\times Mean\ Time\ To\ Recover
  19. C o v e r a g e > R e a d i n e s s Coverage>Readiness
  20. C o v e r a g e = F a u l t s D e t e c t e d B y C B M + F a u l t s D e t e c t e d B y P M S T o t a l P o s s i b l e F a u l t s Coverage=\frac{Faults\ Detected\ By\ CBM+Faults\ Detected\ By\ PMS}{Total\ % Possible\ Faults}
  21. C o v e r a g e T o t a l F a u l t s E x c l u d i n g O p e r a t i o n a l F a i l u r e T o t a l F a u l t s I n c l u d i n g O p e r a t i o n a l F a i l u r e Coverage\approx\frac{Total\ Faults\ Excluding\ Operational\ Failure}{Total\ % Faults\ Including\ Operational\ Failure}

Major_index.html

  1. maj ( w ) = w ( i ) > w ( i + 1 ) i . \operatorname{maj}(w)=\sum_{w(i)>w(i+1)}i.
  2. 0 1 2 3 4 5 6 0 1 0 0 0 0 0 0 1 0 1 1 1 0 0 0 2 0 1 2 1 1 0 0 3 0 1 1 2 1 1 0 4 0 0 1 1 2 1 0 5 0 0 0 1 1 1 0 6 0 0 0 0 0 0 1 \begin{array}[]{c|ccccccc}&0&1&2&3&4&5&6\\ \hline 0&1&0&0&0&0&0&0\\ 1&0&1&1&1&0&0&0\\ 2&0&1&2&1&1&0&0\\ 3&0&1&1&2&1&1&0\\ 4&0&0&1&1&2&1&0\\ 5&0&0&0&1&1&1&0\\ 6&0&0&0&0&0&0&1\end{array}

Manifest_covariance.html

  1. F a b = a A b - b A a F_{ab}\,=\,\partial_{a}A_{b}\,-\,\partial_{b}A_{a}\,
  2. A a A_{a}
  3. a A b = a A b + Γ a b c A c \partial_{a}A_{b}=\nabla_{a}A_{b}+\Gamma^{c}_{ab}A_{c}
  4. b A a = b A a + Γ b a c A c \partial_{b}A_{a}=\nabla_{b}A_{a}+\Gamma^{c}_{ba}A_{c}
  5. Γ a b c - Γ b a c = 0 , \Gamma^{c}_{ab}-\Gamma^{c}_{ba}=0,
  6. F a b = a A b - b A a . F_{ab}\,=\,\nabla_{a}A_{b}\,-\,\nabla_{b}A_{a}.

Manifold_alignment.html

  1. X X
  2. Y Y
  3. X i m X_{i}\in\mathbb{R}^{m}
  4. Y i n Y_{i}\in\mathbb{R}^{n}
  5. X X
  6. Y Y
  7. ϕ X : m d \phi_{X}:\,\mathbb{R}^{m}\rightarrow\mathbb{R}^{d}
  8. ϕ Y : n d \phi_{Y}:\,\mathbb{R}^{n}\rightarrow\mathbb{R}^{d}
  9. W W
  10. X X
  11. Y Y
  12. W i , j = { 1 i f X i Y j 0 o t h e r w i s e W_{i,j}=\begin{cases}1&if\,X_{i}\leftrightarrow Y_{j}\\ 0&otherwise\end{cases}
  13. S X S_{X}
  14. S Y S_{Y}
  15. 0 μ 1 0\leq\mu\leq 1
  16. arg min ϕ X , ϕ Y μ i , j ϕ X ( X i ) - ϕ X ( X j ) 2 S X , i , j + μ i , j ϕ Y ( Y i ) - ϕ Y ( Y j ) 2 S Y , i , j + ( 1 - μ ) i , j ϕ X ( X i ) - ϕ Y ( Y j ) 2 W i , j \arg\min_{\phi_{X},\phi_{Y}}\mu\sum_{i,j}\left\|\phi_{X}\left(X_{i}\right)-% \phi_{X}\left(X_{j}\right)\right\|^{2}S_{X,i,j}+\mu\sum_{i,j}\left\|\phi_{Y}% \left(Y_{i}\right)-\phi_{Y}\left(Y_{j}\right)\right\|^{2}S_{Y,i,j}+\left(1-\mu% \right)\sum_{i,j}\|\phi_{X}\left(X_{i}\right)-\phi_{Y}\left(Y_{j}\right)\|^{2}% W_{i,j}
  17. G = [ μ S X ( 1 - μ ) W ( 1 - μ ) W T μ S Y ] G=\left[\begin{array}[]{cc}\mu S_{X}&\left(1-\mu\right)W\\ \left(1-\mu\right)W^{T}&\mu S_{Y}\end{array}\right]

Marconi's_law.html

  1. H = c D H=c\sqrt{D}

Marcum_Q-function.html

  1. Q M Q_{M}
  2. Q M ( a , b ) = b x ( x a ) M - 1 exp ( - x 2 + a 2 2 ) I M - 1 ( a x ) d x Q_{M}(a,b)=\int_{b}^{\infty}x\left(\frac{x}{a}\right)^{M-1}\exp\left(-\frac{x^% {2}+a^{2}}{2}\right)I_{M-1}\left(ax\right)dx
  3. Q M Q_{M}
  4. Q M ( a , b ) = exp ( - a 2 + b 2 2 ) k = 1 - M ( a b ) k I k ( a b ) Q_{M}(a,b)=\exp\left(-\frac{a^{2}+b^{2}}{2}\right)\sum_{k=1-M}^{\infty}\left(% \frac{a}{b}\right)^{k}I_{k}\left(ab\right)
  5. I M - 1 I_{M-1}

Margrabe's_formula.html

  1. σ = σ 1 2 + σ 2 2 - 2 σ 1 σ 2 ρ \textstyle\sigma=\sqrt{\sigma_{1}^{2}+\sigma_{2}^{2}-2\sigma_{1}\sigma_{2}\rho}
  2. e - q 1 T S 1 ( 0 ) N ( d 1 ) - e - q 2 T S 2 ( 0 ) N ( d 2 ) e^{-q_{1}T}S_{1}(0)N(d_{1})-e^{-q_{2}T}S_{2}(0)N(d_{2})
  3. N N
  4. d 1 = ( l n ( S 1 ( 0 ) / S 2 ( 0 ) ) + ( q 2 - q 1 + σ 2 / 2 ) T ) / σ T d_{1}=(ln(S_{1}(0)/S_{2}(0))+(q_{2}-q_{1}+\sigma^{2}/2)T)/\sigma\sqrt{T}
  5. d 2 = d 1 - σ T d_{2}=d_{1}-\sigma\sqrt{T}

Mark_Mahowald.html

  1. h 1 h j h_{1}h_{j}
  2. j 3 j\geq 3
  3. π * S 2 {}_{2}\pi_{*}^{S}

Mark_Miodownik.html

  1. 2 {}^{2}

Maschke's_theorem.html

  1. φ : K [ G ] V \varphi:K[G]\to V
  2. φ ( x ) = 1 # G s G s π ( s - 1 x ) . \varphi(x)=\frac{1}{\#G}\sum_{s\in G}s\cdot\pi(s^{-1}\cdot x).
  3. φ ( t x ) = 1 # G s G s π ( s - 1 t x ) = 1 # G u G t u π ( u - 1 x ) = t φ ( x ) , \begin{aligned}\displaystyle\varphi(t\cdot x)&\displaystyle=\frac{1}{\#G}\sum_% {s\in G}s\cdot\pi(s^{-1}\cdot t\cdot x)\\ &\displaystyle=\frac{1}{\#G}\sum_{u\in G}t\cdot u\cdot\pi(u^{-1}\cdot x)\\ &\displaystyle=t\cdot\varphi(x),\end{aligned}
  4. K [ G ] = V ker φ K[G]=V\oplus\ker\varphi

Mason–Stothers_theorem.html

  1. a ( t ) a(t)
  2. b ( t ) b(t)
  3. c ( t ) c(t)
  4. a + b = c a+b=c
  5. max { deg ( a ) , deg ( b ) , deg ( c ) } deg ( rad ( a b c ) ) - 1 , \max\{\deg(a),\deg(b),\deg(c)\}\leq\deg(\operatorname{rad}(abc))-1,
  6. r a d ( f ) rad(f)
  7. f f
  8. d e g ( r a d ( f ) ) deg(rad(f))
  9. f f
  10. p > 0 p>0
  11. t < s u p > p + 1 = ( t + 1 ) p t<sup>p+1=(t+1)^{p}

Mathematical_finance.html

  1. \mathbb{Q}
  2. \mathbb{Q}
  3. \mathbb{P}
  4. \mathbb{P}
  5. \mathbb{Q}

Mathematics_of_radio_engineering.html

  1. cos ( z ) = e - i z + e i z 2 \,\cos(z)=\frac{e^{-iz}+e^{iz}}{2}
  2. cos ( ω z ) cos ( ϕ z ) = [ e - i ω z + e i ω z 2 ] [ e - i ϕ z + e i ϕ z 2 ] \,\cos(\omega z)\cos(\phi z)=[\frac{e^{-i\omega z}+e^{i\omega z}}{2}][\frac{e^% {-i\phi z}+e^{i\phi z}}{2}]
  3. = e - i [ ω - ϕ ] z + e - i [ ω + ϕ ] z n e g a t i v e . f r e q u e n c i e s + e i [ ω - ϕ ] z + e i [ ω + ϕ ] z p o s i t i v e . f r e q u e n c i e s 4 \,=\frac{\overbrace{e^{-i[\omega-\phi]z}+e^{-i[\omega+\phi]z}}^{negative.% frequencies}+\overbrace{e^{i[\omega-\phi]z}+e^{i[\omega+\phi]z}}^{positive.% frequencies}}{4}
  4. ω \scriptstyle\omega
  5. ϕ \scriptstyle\phi
  6. x a - n x a n x\to a^{-n}xa^{n}
  7. det ( a - n x a n ) = det ( a - n ) det ( x ) det ( a n ) \,\det(a^{-n}xa^{n})=\det(a^{-n})\det(x)\det(a^{n})
  8. = det ( a - n ) det ( a n ) det ( x ) \,=\det(a^{-n})\det(a^{n})\det(x)
  9. = det ( x ) \,=\det(x)
  10. V = S 1 + I R x * 3 \,V=\overbrace{S}^{1}+\overbrace{IR}^{x*3}
  11. 1 + 3 x + 2 x 2 - 4 x 3 + \,1+3x+2x^{2}-4x^{3}+\cdots
  12. ( E + i B ) = 0 \nabla\cdot(E+iB)=0
  13. × ( E + i B ) = i t ( E + i B ) \nabla\times(E+iB)=i\frac{\partial}{\partial t}(E+iB)
  14. E + i B e i θ ( E + i B ) E+iB\rightarrow e^{i\theta}(E+iB)
  15. q = i a + b 𝐢 + c 𝐣 + d 𝐤 \,q=ia+b\mathbf{i}+c\mathbf{j}+d\mathbf{k}
  16. q q ¯ = - a 2 + b 2 + c 2 + d 2 \,q\overline{q}=-a^{2}+b^{2}+c^{2}+d^{2}
  17. e + 1 2 n θ q e - 1 2 n θ * \,e^{+{1\over 2}n\theta}q{e^{-{1\over 2}n\theta}}^{*}
  18. e + 1 2 i n θ q e - 1 2 i n θ * \,e^{+{1\over 2}in\theta}q{e^{-{1\over 2}in\theta}}^{*}
  19. θ \theta
  20. θ \theta
  21. D D ¯ A = - J D\overline{D}A=-J
  22. A = i ϕ + A 1 𝐢 + A 2 𝐣 + A 3 𝐤 \,A=i\phi+A_{1}\mathbf{i}+A_{2}\mathbf{j}+A_{3}\mathbf{k}\quad
  23. J = i ρ + J 1 𝐢 + J 2 𝐣 + J 3 𝐤 \,J=i\rho+J_{1}\mathbf{i}+J_{2}\mathbf{j}+J_{3}\mathbf{k}\quad
  24. D ¯ = i / t + / x 𝐢 + / y 𝐣 + / z 𝐤 \,\overline{D}=i{\partial/\partial t}+{\partial/\partial x}\mathbf{i}+{% \partial/\partial y}\mathbf{j}+{\partial/\partial z}\mathbf{k}\quad
  25. D = i / t - / x 𝐢 - / y 𝐣 - / z 𝐤 \,D=i{\partial/\partial t}-{\partial/\partial x}\mathbf{i}-{\partial/\partial y% }\mathbf{j}-{\partial/\partial z}\mathbf{k}\quad
  26. D D ¯ D\overline{D}
  27. D D ¯ = - 2 / t 2 + 2 / x 2 + 2 / y 2 + 2 / z 2 \,D\overline{D}=-{\partial^{2}/\partial t^{2}}+{\partial^{2}/\partial x^{2}}+{% \partial^{2}/\partial y^{2}}+{\partial^{2}/\partial z^{2}}\quad
  28. e k i + j \,e^{ki+j}
  29. e k + i j \,e^{k+ij}
  30. e + i g \,e+ig
  31. B = i ϕ 2 + B 1 𝐢 + B 2 𝐣 + B 3 𝐤 \,B=i\phi_{2}+B_{1}\mathbf{i}+B_{2}\mathbf{j}+B_{3}\mathbf{k}\qquad
  32. K = i ρ 2 + K 1 𝐢 + K 2 𝐣 + K 3 𝐤 \,K=i\rho_{2}+K_{1}\mathbf{i}+K_{2}\mathbf{j}+K_{3}\mathbf{k}\qquad
  33. A + i B = [ i ϕ 1 - ϕ 2 ] + [ A 1 + i B 1 ] 𝐢 + [ A 2 + i B 2 ] 𝐣 + [ A 3 + i B 3 ] 𝐤 \,A+iB=[i\phi_{1}-\phi_{2}]+[A_{1}+iB_{1}]\mathbf{i}+[A_{2}+iB_{2}]\mathbf{j}+% [A_{3}+iB_{3}]\mathbf{k}\quad
  34. J + i K = [ i ρ 1 - ρ 2 ] + [ J 1 + i K 1 ] 𝐢 + [ J 2 + i K 2 ] 𝐣 + [ J 3 + i K 3 ] 𝐤 \,J+iK=[i\rho_{1}-\rho_{2}]+[J_{1}+iK_{1}]\mathbf{i}+[J_{2}+iK_{2}]\mathbf{j}+% [J_{3}+iK_{3}]\mathbf{k}\quad
  35. { e + i g } = e i θ { e + i g } \,\{e+ig\}=e^{i\theta}\{e+ig\}
  36. { A + i B } = e i θ { A + i B } \,\{A+iB\}=e^{i\theta}\{A+iB\}
  37. { J + i K } = e i θ { J + i K } \,\{J+iK\}=e^{i\theta}\{J+iK\}
  38. { F u v + i F u v } = e i θ { F u v + i F u v } \,\{F^{uv}+iF^{uv}\}=e^{i\theta}\{F^{uv}+iF^{uv}\}
  39. 2 π i 2\pi i
  40. 2 π i 2\pi i
  41. 1 + 3 x \,1+3x
  42. e i ω t \,e^{i\omega t}
  43. e i Ω [ t 2 - x 2 - y 2 - z 2 ] \,e^{i\Omega[\sqrt{t^{2}-x^{2}-y^{2}-z^{2}}]}
  44. t 2 - x 2 - y 2 - z 2 t^{2}-x^{2}-y^{2}-z^{2}
  45. Ω \Omega
  46. e i S [ ( 1 - r s r ) c 2 d t 2 - ( 1 - r s r ) - 1 d r 2 - r 2 ( d θ 2 + sin 2 θ d φ 2 ) ] \,e^{iS[\sqrt{\left(1-\frac{r_{s}}{r}\right)c^{2}dt^{2}-\left(1-\frac{r_{s}}{r% }\right)^{-1}dr^{2}-r^{2}\left(d\theta^{2}+\sin^{2}\theta\,d\varphi^{2}\right)% }]}
  47. e i [ t E - x p ] / o n e \,e^{i[t\cdot E-x\cdot p]/one}
  48. { 2 , 2 * 3 , 2 * 3 * 5 , 2 * 3 * 5 * 7 } \,\{2,2*3,2*3*5,2*3*5*7\cdots\}
  49. 25 = [ 2 * 3 * 5 ] 2 [ 2 * 3 ] 2 \,25=\frac{[2*3*5]^{2}}{[2*3]^{2}}
  50. 14 = [ 2 ] * [ 2 * 3 * 5 * 7 ] [ 2 * 3 * 5 ] \,14=\frac{[2]*[2*3*5*7]}{[2*3*5]}
  51. { 1 , x , x 2 , x 3 , x 4 } \{1,x,x^{2},x^{3},x^{4}\cdots\}
  52. { 1 , 1 + x , 1 + x + x 2 , 1 + x + x 2 + x 3 } \{1,1+x,1+x+x^{2},1+x+x^{2}+x^{3}\cdots\}
  53. { e i n x | n } \{e^{inx}\quad|\quad n\in\mathbb{R}\}
  54. { 0 n = θ e i n x d n | θ } \Big\{\int_{0}^{n=\theta}e^{inx}dn\quad|\quad\theta\in\mathbb{R}\Big\}
  55. 𝔉 { f * g } = 𝔉 { f } 𝔉 { g } \mathfrak{F}\{f*g\}=\mathfrak{F}\{f\}\cdot\mathfrak{F}\{g\}
  56. 𝔉 { f g } = 𝔉 { f } * 𝔉 { g } \mathfrak{F}\{f\cdot g\}=\mathfrak{F}\{f\}*\mathfrak{F}\{g\}
  57. not ( a and b ) = not a or not b \,\,\text{ not}(a\,\text{ and }b)=\,\text{ not }a\,\text{ or}\,\text{ not }b
  58. not ( a or b ) = not a and not b \,\,\text{ not}(a\,\text{ or }b)=\,\text{ not }a\,\text{ and}\,\text{ not }b
  59. ( a b ) ¯ = a ¯ b ¯ \,\overline{(a\cap b)}=\overline{a}\cup\overline{b}
  60. ( a b ) ¯ = a ¯ b ¯ \,\overline{(a\cup b)}=\overline{a}\cap\overline{b}
  61. gcd ( a , b ) - 1 = lcm ( a - 1 , b - 1 ) \,\gcd(a,b)^{-1}=\operatorname{lcm}(a^{-1},b^{-1})
  62. lcm ( a , b ) - 1 = gcd ( a - 1 , b - 1 ) \,\operatorname{lcm}(a,b)^{-1}=\gcd(a^{-1},b^{-1})
  63. q q ¯ q\overline{q}
  64. lim n z ± n \lim_{n\to\infty}z^{\pm n}

MathMagic.html

  1. y = 3 x + 2 a - 2.5 b . y=3x+2a-2.5\sqrt{b}.

Matrix_Chernoff_bound.html

  1. { 𝐗 k } \{\mathbf{X}_{k}\}
  2. Pr { λ max ( k 𝐗 k ) t } \Pr\left\{\lambda_{\max}\left(\sum_{k}\mathbf{X}_{k}\right)\geq t\right\}
  3. { 𝐀 k } \{\mathbf{A}_{k}\}
  4. d d
  5. { ξ k } \{\xi_{k}\}
  6. t 0 t\geq 0
  7. Pr { λ max ( k ξ k 𝐀 k ) t } d e - t 2 / 2 σ 2 \Pr\left\{\lambda_{\,\text{max}}\left(\sum_{k}\xi_{k}\mathbf{A}_{k}\right)\geq t% \right\}\leq d\cdot e^{-t^{2}/2\sigma^{2}}
  8. σ 2 = k 𝐀 k 2 . \sigma^{2}=\bigg\|\sum_{k}\mathbf{A}^{2}_{k}\bigg\|.
  9. { 𝐁 k } \{\mathbf{B}_{k}\}
  10. d 1 × d 2 d_{1}\times d_{2}
  11. { ξ k } \{\xi_{k}\}
  12. σ 2 = max { k 𝐁 k 𝐁 k * , k 𝐁 k * 𝐁 k } . \sigma^{2}=\max\left\{\bigg\|\sum_{k}\mathbf{B}_{k}\mathbf{B}_{k}^{*}\bigg\|,% \bigg\|\sum_{k}\mathbf{B}_{k}^{*}\mathbf{B}_{k}\bigg\|\right\}.
  13. t 0 t\geq 0
  14. Pr { k ξ k 𝐁 k t } ( d 1 + d 2 ) e - t 2 / 2 σ 2 . \Pr\left\{\bigg\|\sum_{k}\xi_{k}\mathbf{B}_{k}\bigg\|\geq t\right\}\leq(d_{1}+% d_{2})\cdot e^{-t^{2}/2\sigma^{2}}.
  15. { 𝐗 k } \{\mathbf{X}_{k}\}
  16. d d
  17. 𝐗 k 𝟎 and λ max ( 𝐗 k ) R \mathbf{X}_{k}\succeq\mathbf{0}\quad\,\text{and}\quad\lambda_{\,\text{max}}(% \mathbf{X}_{k})\leq R
  18. μ min = λ min ( k 𝔼 𝐗 k ) and μ max = λ max ( k 𝔼 𝐗 k ) . \mu_{\,\text{min}}=\lambda_{\,\text{min}}\left(\sum_{k}\mathbb{E}\,\mathbf{X}_% {k}\right)\quad\,\text{and}\quad\mu_{\,\text{max}}=\lambda_{\,\text{max}}\left% (\sum_{k}\mathbb{E}\,\mathbf{X}_{k}\right).
  19. Pr { λ min ( k 𝐗 k ) ( 1 - δ ) μ min } d [ e - δ ( 1 - δ ) 1 - δ ] μ min / R for δ [ 0 , 1 ] , and \Pr\left\{\lambda_{\,\text{min}}\left(\sum_{k}\mathbf{X}_{k}\right)\leq(1-% \delta)\mu_{\,\text{min}}\right\}\leq d\cdot\left[\frac{e^{-\delta}}{(1-\delta% )^{1-\delta}}\right]^{\mu_{\,\text{min}}/R}\quad\,\text{for }\delta\in[0,1]\,% \text{, and}
  20. Pr { λ max ( k 𝐗 k ) ( 1 + δ ) μ max } d [ e δ ( 1 + δ ) 1 + δ ] μ max / R for δ 0. \Pr\left\{\lambda_{\,\text{max}}\left(\sum_{k}\mathbf{X}_{k}\right)\geq(1+% \delta)\mu_{\,\text{max}}\right\}\leq d\cdot\left[\frac{e^{\delta}}{(1+\delta)% ^{1+\delta}}\right]^{\mu_{\,\text{max}}/R}\quad\,\text{for }\delta\geq 0.
  21. { 𝐗 k : k = 1 , 2 , , n } \{\mathbf{X}_{k}:k=1,2,\ldots,n\}
  22. 𝐗 k 𝟎 and λ max ( 𝐗 k ) 1 \mathbf{X}_{k}\succeq\mathbf{0}\quad\,\text{and}\quad\lambda_{\,\text{max}}(% \mathbf{X}_{k})\leq 1
  23. μ ¯ min = λ min ( 1 n k = 1 n 𝔼 𝐗 k ) and μ ¯ max = λ max ( 1 n k = 1 n 𝔼 𝐗 k ) . \bar{\mu}_{\,\text{min}}=\lambda_{\,\text{min}}\left(\frac{1}{n}\sum_{k=1}^{n}% \mathbb{E}\,\mathbf{X}_{k}\right)\quad\,\text{and}\quad\bar{\mu}_{\,\text{max}% }=\lambda_{\,\text{max}}\left(\frac{1}{n}\sum_{k=1}^{n}\mathbb{E}\,\mathbf{X}_% {k}\right).
  24. Pr { λ min ( 1 n k = 1 n 𝐗 k ) α } d e - n D ( α μ ¯ min ) for 0 α μ ¯ min , and \Pr\left\{\lambda_{\,\text{min}}\left(\frac{1}{n}\sum_{k=1}^{n}\mathbf{X}_{k}% \right)\leq\alpha\right\}\leq d\cdot e^{-nD(\alpha\|\bar{\mu}_{\,\text{min}})}% \quad\,\text{for }0\leq\alpha\leq\bar{\mu}_{\,\text{min}}\,\text{, and}
  25. Pr { λ max ( 1 n k = 1 n 𝐗 k ) α } d e - n D ( α μ ¯ max ) for μ ¯ max α 1. \Pr\left\{\lambda_{\,\text{max}}\left(\frac{1}{n}\sum_{k=1}^{n}\mathbf{X}_{k}% \right)\geq\alpha\right\}\leq d\cdot e^{-nD(\alpha\|\bar{\mu}_{\,\text{max}})}% \quad\,\text{for }\bar{\mu}_{\,\text{max}}\leq\alpha\leq 1.
  26. D ( a u ) = a ( log a - log u ) + ( 1 - a ) ( log ( 1 - a ) - log ( 1 - u ) ) D(a\|u)=a\left(\log a-\log u\right)+(1-a)\left(\log(1-a)-\log(1-u)\right)
  27. a , u [ 0 , 1 ] a,u\in[0,1]
  28. { 𝐗 k } \{\mathbf{X}_{k}\}
  29. d d
  30. 𝐗 k 𝟎 and λ max ( 𝐗 k ) R \mathbf{X}_{k}\succeq\mathbf{0}\quad\,\text{and}\quad\lambda_{\,\text{max}}(% \mathbf{X}_{k})\leq R
  31. σ 2 = k 𝔼 ( 𝐗 k 2 ) . \sigma^{2}=\bigg\|\sum_{k}\mathbb{E}\,(\mathbf{X}^{2}_{k})\bigg\|.
  32. t 0 t\geq 0
  33. Pr { λ max ( k 𝐗 k ) t } \displaystyle\Pr\left\{\lambda_{\,\text{max}}\left(\sum_{k}\mathbf{X}_{k}% \right)\geq t\right\}
  34. h ( u ) h(u)
  35. h ( u ) = ( 1 + u ) log ( 1 + u ) - u h(u)=(1+u)\log(1+u)-u
  36. u 0 u\geq 0
  37. { 𝐗 k } \{\mathbf{X}_{k}\}
  38. d d
  39. 𝔼 𝐗 k = 𝟎 and 𝔼 ( 𝐗 k p ) p ! 2 R p - 2 𝐀 k 2 \mathbb{E}\,\mathbf{X}_{k}=\mathbf{0}\quad\,\text{and}\quad\mathbb{E}\,(% \mathbf{X}_{k}^{p})\preceq\frac{p!}{2}\cdot R^{p-2}\mathbf{A}_{k}^{2}
  40. p = 2 , 3 , 4 , p=2,3,4,\ldots
  41. σ 2 = k 𝐀 k 2 . \sigma^{2}=\bigg\|\sum_{k}\mathbf{A}^{2}_{k}\bigg\|.
  42. t 0 t\geq 0
  43. Pr { λ max ( k 𝐗 k ) t } \displaystyle\Pr\left\{\lambda_{\,\text{max}}\left(\sum_{k}\mathbf{X}_{k}% \right)\geq t\right\}
  44. { 𝐙 k } \{\mathbf{Z}_{k}\}
  45. d 1 × d 2 d_{1}\times d_{2}
  46. 𝔼 𝐙 k = 𝟎 and 𝐙 k R \mathbb{E}\,\mathbf{Z}_{k}=\mathbf{0}\quad\,\text{and}\quad\|\mathbf{Z}_{k}\|\leq R
  47. σ 2 = max { k 𝔼 ( 𝐙 k 𝐙 k * ) , k 𝔼 ( 𝐙 k * 𝐙 k ) } . \sigma^{2}=\max\left\{\bigg\|\sum_{k}\mathbb{E}\,(\mathbf{Z}_{k}\mathbf{Z}_{k}% ^{*})\bigg\|,\bigg\|\sum_{k}\mathbb{E}\,(\mathbf{Z}_{k}^{*}\mathbf{Z}_{k})% \bigg\|\right\}.
  48. t 0 t\geq 0
  49. Pr { k 𝐙 k t } ( d 1 + d 2 ) exp ( - t 2 σ 2 + R t / 3 ) \Pr\left\{\bigg\|\sum_{k}\mathbf{Z}_{k}\bigg\|\geq t\right\}\leq(d_{1}+d_{2})% \cdot\exp\left(\frac{-t^{2}}{\sigma^{2}+Rt/3}\right)
  50. { 𝐗 k } \{\mathbf{X}_{k}\}
  51. d d
  52. { 𝐀 k } \{\mathbf{A}_{k}\}
  53. 𝔼 k - 1 𝐗 k = 𝟎 and 𝐗 k 2 𝐀 k 2 \mathbb{E}_{k-1}\,\mathbf{X}_{k}=\mathbf{0}\quad\,\text{and}\quad\mathbf{X}_{k% }^{2}\preceq\mathbf{A}_{k}^{2}
  54. σ 2 = k 𝐀 k 2 . \sigma^{2}=\bigg\|\sum_{k}\mathbf{A}^{2}_{k}\bigg\|.
  55. t 0 t\geq 0
  56. Pr { λ max ( k 𝐗 k ) t } d e - t 2 / 8 σ 2 \Pr\left\{\lambda_{\,\text{max}}\left(\sum_{k}\mathbf{X}_{k}\right)\geq t% \right\}\leq d\cdot e^{-t^{2}/8\sigma^{2}}
  57. 𝐗 k \mathbf{X}_{k}
  58. 𝐗 k \mathbf{X}_{k}
  59. 𝐀 k \mathbf{A}_{k}
  60. { 𝐗 k } \{\mathbf{X}_{k}\}
  61. d d
  62. { 𝐀 k } \{\mathbf{A}_{k}\}
  63. 𝔼 𝐗 k = 𝟎 and 𝐗 k 2 𝐀 k 2 \mathbb{E}\,\mathbf{X}_{k}=\mathbf{0}\quad\,\text{and}\quad\mathbf{X}_{k}^{2}% \preceq\mathbf{A}_{k}^{2}
  64. t 0 t\geq 0
  65. Pr { λ max ( k 𝐗 k ) t } d e - t 2 / 8 σ 2 \Pr\left\{\lambda_{\,\text{max}}\left(\sum_{k}\mathbf{X}_{k}\right)\geq t% \right\}\leq d\cdot e^{-t^{2}/8\sigma^{2}}
  66. σ 2 = k 𝐀 k 2 . \sigma^{2}=\bigg\|\sum_{k}\mathbf{A}^{2}_{k}\bigg\|.
  67. t 0 t\geq 0
  68. Pr { λ max ( k 𝐗 k ) t } d e - t 2 / 2 σ 2 \Pr\left\{\lambda_{\,\text{max}}\left(\sum_{k}\mathbf{X}_{k}\right)\geq t% \right\}\leq d\cdot e^{-t^{2}/2\sigma^{2}}
  69. σ 2 = 1 2 k 𝐀 k 2 + 𝔼 𝐗 k 2 k 𝐀 k 2 . \sigma^{2}=\frac{1}{2}\bigg\|\sum_{k}\mathbf{A}^{2}_{k}+\mathbb{E}\,\mathbf{X}% ^{2}_{k}\bigg\|\leq\bigg\|\sum_{k}\mathbf{A}^{2}_{k}\bigg\|.
  70. { 𝐙 k : k = 1 , 2 , , n } \{\mathbf{Z}_{k}:k=1,2,\ldots,n\}
  71. 𝐇 \mathbf{H}
  72. n n
  73. d d
  74. { 𝐀 k } \{\mathbf{A}_{k}\}
  75. ( 𝐇 ( z 1 , , z k , , z n ) - 𝐇 ( z 1 , , z k , , z n ) ) 2 𝐀 k 2 , \left(\mathbf{H}(z_{1},\ldots,z_{k},\ldots,z_{n})-\mathbf{H}(z_{1},\ldots,z^{% \prime}_{k},\ldots,z_{n})\right)^{2}\preceq\mathbf{A}_{k}^{2},
  76. z i z_{i}
  77. z i z^{\prime}_{i}
  78. Z i Z_{i}
  79. i i
  80. σ 2 = k 𝐀 k 2 . \sigma^{2}=\bigg\|\sum_{k}\mathbf{A}^{2}_{k}\bigg\|.
  81. t 0 t\geq 0
  82. Pr { λ max ( 𝐇 ( 𝐳 ) - 𝔼 𝐇 ( 𝐳 ) ) t } d e - t 2 / 8 σ 2 , \Pr\left\{\lambda_{\,\text{max}}\left(\mathbf{H}(\mathbf{z})-\mathbb{E}\,% \mathbf{H}(\mathbf{z})\right)\geq t\right\}\leq d\cdot e^{-t^{2}/8\sigma^{2}},
  83. 𝐳 = ( Z 1 , , Z n ) \mathbf{z}=(Z_{1},\ldots,Z_{n})
  84. { 𝐀 k } \{\mathbf{A}_{k}\}
  85. d d
  86. { ξ k } \{\xi_{k}\}
  87. Pr { λ max ( k ξ k 𝐀 k ) t } d e - t 2 / 2 σ 2 \Pr\left\{\lambda_{\,\text{max}}\left(\sum_{k}\xi_{k}\mathbf{A}_{k}\right)\geq t% \right\}\leq d\cdot e^{-t^{2}/2\sigma^{2}}
  88. σ 2 = k 𝐀 k 2 . \sigma^{2}=\bigg\|\sum_{k}\mathbf{A}^{2}_{k}\bigg\|.
  89. σ A W 2 = k λ max ( 𝐀 k 2 ) \sigma_{AW}^{2}=\sum_{k}\lambda_{\max}\left(\mathbf{A}_{k}^{2}\right)
  90. σ 2 \sigma^{2}
  91. Σ \Sigma
  92. λ max \lambda_{\max}
  93. 𝐘 \mathbf{Y}
  94. Pr { λ max ( Y ) t } inf θ > 0 { e - θ t E [ tr e θ 𝐘 ] } . \Pr\left\{\lambda_{\max}(Y)\geq t\right\}\leq\inf_{\theta>0}\left\{e^{-\theta t% }\cdot\operatorname{E}\left[\operatorname{tr}e^{\theta\mathbf{Y}}\right]\right\}.
  95. θ > 0 \theta>0
  96. Pr { λ max ( 𝐘 ) t } = Pr { λ max ( θ 𝐘 ) θ t } = Pr { e λ max ( θ 𝐘 ) e θ t } e - θ t E e λ max ( θ 𝐘 ) e - θ t E tr e ( θ 𝐘 ) \begin{aligned}\displaystyle\Pr\left\{\lambda_{\max}(\mathbf{Y})\geq t\right\}% &\displaystyle=\Pr\left\{\lambda_{\max}(\mathbf{\theta Y})\geq\theta t\right\}% \\ &\displaystyle=\Pr\left\{e^{\lambda_{\max}(\theta\mathbf{Y})}\geq e^{\theta t}% \right\}\\ &\displaystyle\leq e^{-\theta t}\operatorname{E}e^{\lambda_{\max}(\theta% \mathbf{Y})}\\ &\displaystyle\leq e^{-\theta t}\operatorname{E}\operatorname{tr}e^{(\theta% \mathbf{Y})}\end{aligned}
  97. e λ max θ 𝐘 = λ max e θ 𝐘 tr e θ 𝐘 e^{\lambda_{\max}\theta\mathbf{Y}}=\lambda_{\max}e^{\theta\mathbf{Y}}\leq% \operatorname{tr}e^{\theta\mathbf{Y}}
  98. θ \theta
  99. θ > 0 \theta>0
  100. E tr e θ 𝐘 \operatorname{E}\operatorname{tr}e^{\theta\mathbf{Y}}
  101. E e θ 𝐘 := 𝐌 𝐘 ( θ ) \operatorname{E}e^{\theta\mathbf{Y}}:=\mathbf{M}_{\mathbf{Y}}(\theta)
  102. tr 𝐌 𝐗 1 + 𝐗 2 ( θ ) tr [ ( E e θ 𝐗 1 ) ( E e θ 𝐗 2 ) ] = tr 𝐌 𝐗 1 ( θ ) 𝐌 𝐗 2 ( θ ) \operatorname{tr}\mathbf{M}_{\mathbf{X}_{1}+\mathbf{X}_{2}}(\theta)\leq% \operatorname{tr}\left[\left(\operatorname{E}e^{\theta\mathbf{X}_{1}}\right)% \left(\operatorname{E}e^{\theta\mathbf{X}_{2}}\right)\right]=\operatorname{tr}% \mathbf{M}_{\mathbf{X}_{1}}(\theta)\mathbf{M}_{\mathbf{X}_{2}}(\theta)
  103. 𝐘 = k 𝐗 k \mathbf{Y}=\sum_{k}\mathbf{X}_{k}
  104. tr 𝐌 𝐘 ( θ ) \operatorname{tr}\mathbf{M}_{\mathbf{Y}}(\theta)
  105. tr ( 𝐀𝐁 ) tr ( 𝐀 ) λ max ( 𝐁 ) \operatorname{tr}(\mathbf{AB})\leq\operatorname{tr}(\mathbf{A})\lambda_{\max}(% \mathbf{B})
  106. tr 𝐌 𝐘 ( θ ) tr [ ( E e k = 1 n - 1 θ 𝐗 k ) ( E e θ 𝐗 n ) ] tr ( E e k = 1 n - 1 θ 𝐗 k ) λ max ( E e θ 𝐗 n ) . \operatorname{tr}\mathbf{M}_{\mathbf{Y}}(\theta)\leq\operatorname{tr}\left[% \left(\operatorname{E}e^{\sum_{k=1}^{n-1}\theta\mathbf{X}_{k}}\right)\left(% \operatorname{E}e^{\theta\mathbf{X}_{n}}\right)\right]\leq\operatorname{tr}% \left(\operatorname{E}e^{\sum_{k=1}^{n-1}\theta\mathbf{X}_{k}}\right)\lambda_{% \max}(\operatorname{E}e^{\theta\mathbf{X}_{n}}).
  107. tr 𝐌 𝐘 ( θ ) ( tr 𝐈 ) [ Π k λ max ( E e θ 𝐗 k ) ] = d e k λ max ( log E e θ 𝐗 k ) \operatorname{tr}\mathbf{M}_{\mathbf{Y}}(\theta)\leq(\operatorname{tr}\mathbf{% I})\left[\Pi_{k}\lambda_{\max}(\operatorname{E}e^{\theta\mathbf{X}_{k}})\right% ]=de^{\sum_{k}\lambda_{\max}\left(\log\operatorname{E}e^{\theta\mathbf{X}_{k}}% \right)}
  108. θ \theta
  109. H H
  110. X X
  111. E tr e 𝐇 + 𝐗 tr e 𝐇 + log ( E e 𝐗 ) \operatorname{E}\operatorname{tr}e^{\mathbf{H}+\mathbf{X}}\leq\operatorname{tr% }e^{\mathbf{H}+\log(\operatorname{E}e^{\mathbf{X}})}
  112. 𝐘 = e 𝐗 \mathbf{Y}=e^{\mathbf{X}}
  113. f ( 𝐘 ) = tr e 𝐇 + log ( 𝐘 ) f(\mathbf{Y})=\operatorname{tr}e^{\mathbf{H}+\log(\mathbf{Y})}
  114. E tr e 𝐇 + log ( 𝐘 ) tr e 𝐇 + log ( E 𝐘 ) . \operatorname{E}\operatorname{tr}e^{\mathbf{H}+\log(\mathbf{Y})}\leq% \operatorname{tr}e^{\mathbf{H}+\log(\operatorname{E}\mathbf{Y})}.
  115. 𝐗 k \mathbf{X}_{k}
  116. θ \theta\in\mathbb{R}
  117. tr 𝐌 k 𝐗 k ( θ ) tr e k log 𝐌 𝐗 k ( θ ) \operatorname{tr}\mathbf{M}_{\sum_{k}\mathbf{X}_{k}}(\theta)\leq\operatorname{% tr}e^{\sum_{k}\log\mathbf{M}_{\mathbf{X}_{k}}(\theta)}
  118. θ = 1 \theta=1
  119. E tr e k θ 𝐗 k tr e k log E e θ 𝐗 k . \operatorname{E}\operatorname{tr}e^{\sum_{k}\theta\mathbf{X}_{k}}\leq% \operatorname{tr}e^{\sum_{k}\log\operatorname{E}e^{\theta\mathbf{X}_{k}}}.
  120. E k \operatorname{E}_{k}
  121. 𝐗 1 , , 𝐗 k \mathbf{X}_{1},\ldots,\mathbf{X}_{k}
  122. 𝐗 i \mathbf{X}_{i}
  123. E k - 1 e 𝐗 k = E e 𝐗 k . \operatorname{E}_{k-1}e^{\mathbf{X}_{k}}=\operatorname{E}e^{\mathbf{X}_{k}}.
  124. 𝚵 k = log E k - 1 e 𝐗 k = log 𝐌 𝐗 k ( θ ) \mathbf{\Xi}_{k}=\log\operatorname{E}_{k-1}e^{\mathbf{X}_{k}}=\log\mathbf{M}_{% \mathbf{X}_{k}}(\theta)
  125. E tr e k = 1 n 𝐗 k = E 0 E n - 1 tr e k = 1 n - 1 𝐗 k + 𝐗 n E 0 E n - 2 tr e k = 1 n - 1 𝐗 k + log ( E n - 1 e 𝐗 n ) = E 0 E n - 2 tr e k = 1 n - 2 𝐗 k + 𝐗 n - 1 + 𝚵 n = tr e k = 1 n 𝚵 k \begin{aligned}\displaystyle\operatorname{E}\operatorname{tr}e^{\sum_{k=1}^{n}% \mathbf{X}_{k}}&\displaystyle=\operatorname{E}_{0}\cdots\operatorname{E}_{n-1}% \operatorname{tr}e^{\sum_{k=1}^{n-1}\mathbf{X}_{k}+\mathbf{X}_{n}}\\ &\displaystyle\leq\operatorname{E}_{0}\cdots\operatorname{E}_{n-2}% \operatorname{tr}e^{\sum_{k=1}^{n-1}\mathbf{X}_{k}+\log(\operatorname{E}_{n-1}% e^{\mathbf{X}_{n}})}\\ &\displaystyle=\operatorname{E}_{0}\cdots\operatorname{E}_{n-2}\operatorname{% tr}e^{\sum_{k=1}^{n-2}\mathbf{X}_{k}+\mathbf{X}_{n-1}+\mathbf{\Xi}_{n}}\\ &\displaystyle\vdots\\ &\displaystyle=\operatorname{tr}e^{\sum_{k=1}^{n}\mathbf{\Xi}_{k}}\end{aligned}
  126. 𝐇 m = k = 1 m - 1 𝐗 k + k = m + 1 n 𝚵 k \mathbf{H}_{m}=\sum_{k=1}^{m-1}\mathbf{X}_{k}+\sum_{k=m+1}^{n}\mathbf{\Xi}_{k}
  127. Pr { λ max ( k 𝐗 k ) t } inf θ > 0 { e - θ t tr e k log 𝐌 𝐗 k ( θ ) } \Pr\left\{\lambda_{\max}\left(\sum_{k}\mathbf{X}_{k}\right)\geq t\right\}\leq% \inf_{\theta>0}\left\{e^{-\theta t}\operatorname{tr}e^{\sum_{k}\log\mathbf{M}_% {\mathbf{X}_{k}}(\theta)}\right\}

Matrix_difference_equation.html

  1. x t = A x t - 1 + B x t - 2 x_{t}=Ax_{t-1}+Bx_{t-2}
  2. x t + 2 = A x t + 1 + B x t x_{t+2}=Ax_{t+1}+Bx_{t}
  3. x n = A x n - 1 + B x n - 2 x_{n}=Ax_{n-1}+Bx_{n-2}
  4. x t = A x t - 1 + b x_{t}=Ax_{t-1}+b\,
  5. x t = x t - 1 = x * x_{t}=x_{t-1}=x^{*}
  6. x * = [ I - A ] - 1 b x^{*}=[I-A]^{-1}b\,
  7. I I
  8. [ I - A ] [I-A]
  9. [ x t - x * ] = A [ x t - 1 - x * ] . [x_{t}-x^{*}]=A[x_{t-1}-x^{*}].\,
  10. x t x_{t}
  11. y t = A y t - 1 y_{t}=Ay_{t-1}
  12. y 0 y_{0}
  13. y 1 = A y 0 , y_{1}=Ay_{0},
  14. y 2 = A y 1 = A A y 0 = A 2 y 0 , y_{2}=Ay_{1}=AAy_{0}=A^{2}y_{0},
  15. y 3 = A y 2 = A A 2 y 0 = A 3 y 0 , y_{3}=Ay_{2}=AA^{2}y_{0}=A^{3}y_{0},
  16. y t = A t y 0 = P D t P - 1 y 0 , y_{t}=A^{t}y_{0}=PD^{t}P^{-1}y_{0},
  17. A t A^{t}
  18. y t = A y t - 1 , y_{t}=Ay_{t-1},
  19. y 1 . y_{1}.
  20. y t y_{t}
  21. y 1 , t y_{1,t}
  22. y 1 y_{1}
  23. y 1 , y_{1},
  24. y 1 , t = a 1 y 1 , t - 1 + a 2 y 1 , t - 2 + + a n y 1 , t - n y_{1,t}=a_{1}y_{1,t-1}+a_{2}y_{1,t-2}+\dots+a_{n}y_{1,t-n}
  25. a i a_{i}
  26. λ n - a 1 λ n - 1 - a 2 λ n - 2 - - a n λ 0 = 0. \lambda^{n}-a_{1}\lambda^{n-1}-a_{2}\lambda^{n-2}-\dots-a_{n}\lambda^{0}=0.
  27. x t = A x t - 1 + B x t - 2 x_{t}=Ax_{t-1}+Bx_{t-2}
  28. ( x t x t - 1 ) = ( A B I 0 ) ( x t - 1 x t - 2 ) , \begin{pmatrix}x_{t}\\ x_{t-1}\\ \end{pmatrix}=\begin{pmatrix}\,\text{A}&\,\text{B}\\ \,\text{I}&0\\ \end{pmatrix}\begin{pmatrix}x_{t-1}\\ x_{t-2}\end{pmatrix},
  29. I I
  30. z t z_{t}
  31. z t = L t z 0 . z_{t}=L^{t}z_{0}.
  32. H t - 1 = K + A H t A - A H t C ( C H t C + R ) - 1 C H t A , H_{t-1}=K+A^{\prime}H_{t}A-A^{\prime}H_{t}C(C^{\prime}H_{t}C+R)^{-1}C^{\prime}% H_{t}A,\,
  33. H t H_{t}
  34. H t H_{t}
  35. X t + 1 = - ( E + B X t ) ( C + A X t ) - 1 X_{t+1}=-(E+BX_{t})(C+AX_{t})^{-1}
  36. X t = N t D t - 1 X_{t}=N_{t}D_{t}^{-1}
  37. X t + 1 = - ( E + B N t D t - 1 ) D t D t - 1 ( C + A N t D t - 1 ) - 1 X_{t+1}=-(E+BN_{t}D_{t}^{-1})D_{t}D_{t}^{-1}(C+AN_{t}D_{t}^{-1})^{-1}
  38. = - ( E D t + B N t ) [ ( C + A N t D t - 1 ) D t ] - 1 =-(ED_{t}+BN_{t})[(C+AN_{t}D_{t}^{-1})D_{t}]^{-1}
  39. = - ( E D t + B N t ) [ C D t + A N t ] - 1 =-(ED_{t}+BN_{t})[CD_{t}+AN_{t}]^{-1}
  40. = N t + 1 D t + 1 - 1 , =N_{t+1}D_{t+1}^{-1},
  41. X t = N t D t - 1 X_{t}=N_{t}D_{t}^{-1}
  42. ( N t + 1 D t + 1 ) = ( - B - E A C ) ( N t D t ) J ( N t D t ) . \begin{pmatrix}N_{t+1}\\ D_{t+1}\end{pmatrix}=\begin{pmatrix}-B&-E\\ A&C\end{pmatrix}\begin{pmatrix}N_{t}\\ D_{t}\end{pmatrix}\equiv J\begin{pmatrix}N_{t}\\ D_{t}\end{pmatrix}.
  43. ( N t D t ) = J t ( N 0 D 0 ) . \begin{pmatrix}N_{t}\\ D_{t}\end{pmatrix}=J^{t}\begin{pmatrix}N_{0}\\ D_{0}\end{pmatrix}.

Matsubara_frequency.html

  1. S η = 1 β i ω n g ( i ω n ) S_{\eta}=\frac{1}{\beta}\sum_{i\omega_{n}}g(i\omega_{n})
  2. ω n \omega_{n}
  3. n n\in\mathbb{Z}
  4. ω n = 2 n π β \omega_{n}=\frac{2n\pi}{\beta}
  5. ω n = ( 2 n + 1 ) π β \omega_{n}=\frac{(2n+1)\pi}{\beta}
  6. z - 1 z^{-1}
  7. T = 0 d ω g ( ω ) \int_{T=0}\mathrm{d}\omega\ g(\omega)
  8. S η . S_{\eta}.
  9. z = i ω z=i\omega
  10. S η = 1 β i ω g ( i ω ) = 1 2 π i β g ( z ) h η ( z ) d z S_{\eta}=\frac{1}{\beta}\sum_{i\omega}g(i\omega)=\frac{1}{2\pi i\beta}\oint g(% z)h_{\eta}(z)dz
  11. S η = - 1 β z 0 g ( z ) poles Res g ( z 0 ) h η ( z 0 ) S_{\eta}=-\frac{1}{\beta}\sum_{z_{0}\in g(z)\,\text{ poles}}\,\text{Res}\,g(z_% {0})h_{\eta}(z_{0})
  12. z = i ω n z=i\omega_{n}
  13. h B ( 1 ) ( z ) = β 1 - e - β z = - β n B ( - z ) = β ( 1 + n B ( z ) ) h_{B}^{(1)}(z)=\frac{\beta}{1-e^{-\beta z}}=-\beta n_{B}(-z)=\beta(1+n_{B}(z))
  14. h B ( 2 ) ( z ) = - β 1 - e β z = β n B ( z ) h_{B}^{(2)}(z)=\frac{-\beta}{1-e^{\beta z}}=\beta n_{B}(z)
  15. h B ( 1 ) ( z ) h_{B}^{(1)}(z)
  16. n B ( z ) = ( e β z - 1 ) - 1 n_{B}(z)=(e^{\beta z}-1)^{-1}
  17. z = i ω m z=i\omega_{m}
  18. h F ( 1 ) ( z ) = β 1 + e - β z = β n F ( - z ) = β ( 1 - n F ( z ) ) h_{F}^{(1)}(z)=\frac{\beta}{1+e^{-\beta z}}=\beta n_{F}(-z)=\beta(1-n_{F}(z))
  19. h F ( 2 ) ( z ) = - β 1 + e β z = - β n F ( z ) h_{F}^{(2)}(z)=\frac{-\beta}{1+e^{\beta z}}=-\beta n_{F}(z)
  20. h F ( 1 ) ( z ) h_{F}^{(1)}(z)
  21. n F ( z ) = ( e β z + 1 ) - 1 n_{F}(z)=(e^{\beta z}+1)^{-1}
  22. g ( z ) = G ( z ) e - z τ g(z)=G(z)e^{-z\tau}
  23. h η ( z ) = h η ( 1 ) ( z ) h_{\eta}(z)=h_{\eta}^{(1)}(z)
  24. S η = 1 β i ω g ( i ω ) S_{\eta}=\frac{1}{\beta}\sum_{i\omega}g(i\omega)
  25. g ( i ω ) g(i\omega)
  26. S η S_{\eta}
  27. ( i ω - ξ ) - 1 (i\omega-\xi)^{-1}
  28. - η n η ( ξ ) -\eta n_{\eta}(\xi)
  29. ( i ω - ξ ) - 2 (i\omega-\xi)^{-2}
  30. - η n η ( ξ ) = β n η ( ξ ) ( η + n η ( ξ ) ) -\eta n_{\eta}^{\prime}(\xi)=\beta n_{\eta}(\xi)(\eta+n_{\eta}(\xi))
  31. ( i ω - ξ ) - n (i\omega-\xi)^{-n}
  32. - η ( n - 1 ) ! ξ n - 1 n η ( ξ ) -\frac{\eta}{(n-1)!}\partial_{\xi}^{n-1}n_{\eta}(\xi)
  33. 1 ( i ω - ξ 1 ) ( i ω - ξ 2 ) \frac{1}{(i\omega-\xi_{1})(i\omega-\xi_{2})}
  34. - η ( n η ( ξ 1 ) - n η ( ξ 2 ) ) ξ 1 - ξ 2 -\frac{\eta(n_{\eta}(\xi_{1})-n_{\eta}(\xi_{2}))}{\xi_{1}-\xi_{2}}
  35. 1 ( i ω - ξ 1 ) 2 ( i ω - ξ 2 ) 2 \frac{1}{(i\omega-\xi_{1})^{2}(i\omega-\xi_{2})^{2}}
  36. η ( ξ 1 - ξ 2 ) 2 ( 2 ( n η ( ξ 1 ) - n η ( ξ 2 ) ) ξ 1 - ξ 2 - ( n η ( ξ 1 ) + n η ( ξ 2 ) ) ) \frac{\eta}{(\xi_{1}-\xi_{2})^{2}}\left(\frac{2(n_{\eta}(\xi_{1})-n_{\eta}(\xi% _{2}))}{\xi_{1}-\xi_{2}}-(n_{\eta}^{\prime}(\xi_{1})+n_{\eta}^{\prime}(\xi_{2}% ))\right)
  37. 1 ( i ω - ξ 1 ) 2 - ξ 2 2 \frac{1}{(i\omega-\xi_{1})^{2}-\xi_{2}^{2}}
  38. η c η ( ξ 1 , ξ 2 ) \eta c_{\eta}(\xi_{1},\xi_{2})
  39. 1 ( i ω ) 2 - ξ 2 \frac{1}{(i\omega)^{2}-\xi^{2}}
  40. η c η ( 0 , ξ ) = - 1 2 ξ ( 1 + 2 η n η ( ξ ) ) \eta c_{\eta}(0,\xi)=-\frac{1}{2\xi}(1+2\eta n_{\eta}(\xi))
  41. ( i ω ) 2 ( i ω ) 2 - ξ 2 \frac{(i\omega)^{2}}{(i\omega)^{2}-\xi^{2}}
  42. - ξ 2 ( 1 + 2 η n η ( ξ ) ) -\frac{\xi}{2}(1+2\eta n_{\eta}(\xi))
  43. 1 ( ( i ω ) 2 - ξ 2 ) 2 \frac{1}{((i\omega)^{2}-\xi^{2})^{2}}
  44. - η 2 ξ 2 ( c η ( 0 , ξ ) + n η ( ξ ) ) -\frac{\eta}{2\xi^{2}}(c_{\eta}(0,\xi)+n_{\eta}^{\prime}(\xi))
  45. ( i ω ) 2 ( ( i ω ) 2 - ξ 2 ) 2 \frac{(i\omega)^{2}}{((i\omega)^{2}-\xi^{2})^{2}}
  46. η 2 ( c η ( 0 , ξ ) - n η ( ξ ) ) \frac{\eta}{2}(c_{\eta}(0,\xi)-n_{\eta}^{\prime}(\xi))
  47. ( i ω ) 2 + ξ 2 ( ( i ω ) 2 - ξ 2 ) 2 \frac{(i\omega)^{2}+\xi^{2}}{((i\omega)^{2}-\xi^{2})^{2}}
  48. - η n η ( ξ ) = β n η ( ξ ) ( η + n η ( ξ ) ) -\eta n_{\eta}^{\prime}(\xi)=\beta n_{\eta}(\xi)(\eta+n_{\eta}(\xi))
  49. 1 ( ( i ω ) 2 - ξ 1 2 ) ( ( i ω ) 2 - ξ 2 2 ) \frac{1}{((i\omega)^{2}-\xi_{1}^{2})((i\omega)^{2}-\xi_{2}^{2})}
  50. η ( c η ( 0 , ξ 1 ) - c η ( 0 , ξ 2 ) ) ξ 1 2 - ξ 2 2 \frac{\eta(c_{\eta}(0,\xi_{1})-c_{\eta}(0,\xi_{2}))}{\xi_{1}^{2}-\xi_{2}^{2}}
  51. ( 1 ( i ω ) 2 - ξ 1 2 + 1 ( i ω ) 2 - ξ 2 2 ) 2 \left(\frac{1}{(i\omega)^{2}-\xi_{1}^{2}}+\frac{1}{(i\omega)^{2}-\xi_{2}^{2}}% \right)^{2}
  52. η ( 3 ξ 1 2 + ξ 2 2 2 ξ 1 2 ( ξ 1 2 - ξ 2 2 ) c η ( 0 , ξ 1 ) - n η ( ξ 1 ) 2 ξ 1 2 ) + ( 1 2 ) \eta\left(\frac{3\xi_{1}^{2}+\xi_{2}^{2}}{2\xi_{1}^{2}(\xi_{1}^{2}-\xi_{2}^{2}% )}c_{\eta}(0,\xi_{1})-\frac{n_{\eta}^{\prime}(\xi_{1})}{2\xi_{1}^{2}}\right)+(% 1\leftrightarrow 2)
  53. ( 1 ( i ω ) 2 - ξ 1 2 - 1 ( i ω ) 2 - ξ 2 2 ) 2 \left(\frac{1}{(i\omega)^{2}-\xi_{1}^{2}}-\frac{1}{(i\omega)^{2}-\xi_{2}^{2}}% \right)^{2}
  54. η ( - 5 ξ 1 2 - ξ 2 2 2 ξ 1 2 ( ξ 1 2 - ξ 2 2 ) c η ( 0 , ξ 1 ) - n η ( ξ 1 ) 2 ξ 1 2 ) + ( 1 2 ) \eta\left(-\frac{5\xi_{1}^{2}-\xi_{2}^{2}}{2\xi_{1}^{2}(\xi_{1}^{2}-\xi_{2}^{2% })}c_{\eta}(0,\xi_{1})-\frac{n_{\eta}^{\prime}(\xi_{1})}{2\xi_{1}^{2}}\right)+% (1\leftrightarrow 2)
  55. β \beta\rightarrow\infty
  56. 1 β i ω = - i i d ( i ω ) 2 π \frac{1}{\beta}\sum_{i\omega}=\int_{-i\infty}^{i\infty}\frac{\mathrm{d}(i% \omega)}{2\pi}
  57. Ω \Omega
  58. Ω \Omega
  59. Ω \Omega\rightarrow\infty
  60. η lim Ω [ - i Ω i Ω d ( i ω ) 2 π ( ln ( - i ω + ξ ) - π ξ 2 Ω ) - Ω π ( ln Ω - 1 ) ] = { 0 ξ 0 - η ξ ξ < 0 , \eta\lim_{\Omega\rightarrow\infty}\left[\int_{-i\Omega}^{i\Omega}\frac{\mathrm% {d}(i\omega)}{2\pi}\left(\ln(-i\omega+\xi)-\frac{\pi\xi}{2\Omega}\right)-\frac% {\Omega}{\pi}(\ln\Omega-1)\right]=\left\{\begin{array}[]{cc}0&\xi\geq 0\\ -\eta\xi&\xi<0\end{array}\right.,
  61. η lim Ω - i Ω i Ω d ( i ω ) 2 π ( 1 - i ω + ξ - π 2 Ω ) = { 0 ξ 0 - η ξ < 0 , \eta\lim_{\Omega\rightarrow\infty}\int_{-i\Omega}^{i\Omega}\frac{\mathrm{d}(i% \omega)}{2\pi}\left(\frac{1}{-i\omega+\xi}-\frac{\pi}{2\Omega}\right)=\left\{% \begin{array}[]{cc}0&\xi\geq 0\\ -\eta&\xi<0\end{array}\right.,
  62. G ( τ ) = 1 β i ω G ( i ω ) e - i ω τ G(\tau)=\frac{1}{\beta}\sum_{i\omega}G(i\omega)e^{-i\omega\tau}
  63. G ( τ ) = - 𝒯 τ ψ ( τ ) ψ * ( 0 ) G(\tau)=-\langle\mathcal{T}_{\tau}\psi(\tau)\psi^{*}(0)\rangle
  64. G η ( τ ) = { G B ( τ ) , if η = + 1 G F ( τ ) , if η = - 1 G_{\eta}(\tau)=\begin{cases}G_{B}(\tau),&\mbox{if }~{}\eta=+1\\ G_{F}(\tau),&\mbox{if }~{}\eta=-1\end{cases}
  65. G B ( τ ) = 1 β i ω n G ( i ω n ) e - i ω n τ G_{B}(\tau)=\frac{1}{\beta}\sum_{i\omega_{n}}G(i\omega_{n})e^{-i\omega_{n}\tau}
  66. G F ( τ ) = 1 β i ω m G ( i ω m ) e - i ω m τ G_{F}(\tau)=\frac{1}{\beta}\sum_{i\omega_{m}}G(i\omega_{m})e^{-i\omega_{m}\tau}
  67. G ( i ω ) G(i\omega)
  68. G η ( τ ) G_{\eta}(\tau)
  69. ( i ω - ξ ) - 1 (i\omega-\xi)^{-1}
  70. - e ξ ( β - τ ) n η ( ξ ) -e^{\xi(\beta-\tau)}n_{\eta}(\xi)
  71. ( i ω - ξ ) - 2 (i\omega-\xi)^{-2}
  72. e ξ ( β - τ ) n η ( ξ ) ( τ + η β n η ( ξ ) ) e^{\xi(\beta-\tau)}n_{\eta}(\xi)\left(\tau+\eta\beta n_{\eta}(\xi)\right)
  73. ( i ω - ξ ) - 3 (i\omega-\xi)^{-3}
  74. - 1 2 e ξ ( β - τ ) n η ( ξ ) ( τ 2 + η β ( β + 2 τ ) n η ( ξ ) + 2 β 2 n η 2 ( ξ ) ) -\frac{1}{2}e^{\xi(\beta-\tau)}n_{\eta}(\xi)\left(\tau^{2}+\eta\beta(\beta+2% \tau)n_{\eta}(\xi)+2\beta^{2}n^{2}_{\eta}(\xi)\right)
  75. ( i ω - ξ 1 ) - 1 ( i ω - ξ 2 ) - 1 (i\omega-\xi_{1})^{-1}(i\omega-\xi_{2})^{-1}
  76. - e ξ 1 ( β - τ ) n η ( ξ 1 ) - e ξ 2 ( β - τ ) n η ( ξ 2 ) ξ 1 - ξ 2 -\frac{e^{\xi_{1}(\beta-\tau)}n_{\eta}(\xi_{1})-e^{\xi_{2}(\beta-\tau)}n_{\eta% }(\xi_{2})}{\xi_{1}-\xi_{2}}
  77. ( ω 2 + m 2 ) - 1 (\omega^{2}+m^{2})^{-1}
  78. e - m τ 2 m + η m cosh m τ n η ( m ) \frac{e^{-m\tau}}{2m}+\frac{\eta}{m}\cosh{m\tau}\;n_{\eta}(m)
  79. i ω ( ω 2 + m 2 ) - 1 i\omega(\omega^{2}+m^{2})^{-1}
  80. e - m τ 2 - η sinh m τ n η ( m ) \frac{e^{-m\tau}}{2}-\eta\,\sinh{m\tau}\;n_{\eta}(m)
  81. ψ ψ * = 𝒯 τ ψ ( τ = 0 + ) ψ * ( 0 ) = - G η ( τ = 0 + ) = - 1 β i ω G ( i ω ) e - i ω 0 + \langle\psi\psi^{*}\rangle=\langle\mathcal{T}_{\tau}\psi(\tau=0^{+})\psi^{*}(0% )\rangle=-G_{\eta}(\tau=0^{+})=-\frac{1}{\beta}\sum_{i\omega}G(i\omega)e^{-i% \omega 0^{+}}
  82. ψ * ψ = η 𝒯 τ ψ ( τ = 0 - ) ψ * ( 0 ) = - η G η ( τ = 0 - ) = - η β i ω G ( i ω ) e i ω 0 + \langle\psi^{*}\psi\rangle=\eta\langle\mathcal{T}_{\tau}\psi(\tau=0^{-})\psi^{% *}(0)\rangle=-\eta G_{\eta}(\tau=0^{-})=-\frac{\eta}{\beta}\sum_{i\omega}G(i% \omega)e^{i\omega 0^{+}}
  83. G ( 0 ) = i ω ( i ω - ξ ) - 1 G(0)=\sum_{i\omega}(i\omega-\xi)^{-1}
  84. h η ( 1 ) ( z ) h_{\eta}^{(1)}(z)
  85. G ( τ = 0 + ) G(\tau=0^{+})
  86. h η ( 2 ) ( z ) h_{\eta}^{(2)}(z)
  87. G ( τ = 0 - ) G(\tau=0^{-})
  88. G B ( τ = 0 - ) = 1 β i ω n e i ω n 0 + i ω n - ξ = - n B ( ξ ) G_{B}(\tau=0^{-})=\frac{1}{\beta}\sum_{i\omega_{n}}\frac{e^{i\omega_{n}0^{+}}}% {i\omega_{n}-\xi}=-n_{B}(\xi)
  89. G B ( τ = 0 + ) = 1 β i ω n e - i ω n 0 + i ω n - ξ = - ( n B ( ξ ) + 1 ) G_{B}(\tau=0^{+})=\frac{1}{\beta}\sum_{i\omega_{n}}\frac{e^{-i\omega_{n}0^{+}}% }{i\omega_{n}-\xi}=-(n_{B}(\xi)+1)
  90. G F ( τ = 0 - ) = 1 β i ω m e i ω m 0 + i ω m - ξ = n F ( ξ ) G_{F}(\tau=0^{-})=\frac{1}{\beta}\sum_{i\omega_{m}}\frac{e^{i\omega_{m}0^{+}}}% {i\omega_{m}-\xi}=n_{F}(\xi)
  91. G F ( τ = 0 + ) = 1 β i ω m e - i ω m 0 + i ω m - ξ = - ( 1 - n F ( ξ ) ) G_{F}(\tau=0^{+})=\frac{1}{\beta}\sum_{i\omega_{m}}\frac{e^{-i\omega_{m}0^{+}}% }{i\omega_{m}-\xi}=-(1-n_{F}(\xi))
  92. 1 β i ω n ln ( β ( - i ω n + ξ ) ) = 1 β ln ( 1 - e - β ξ ) \frac{1}{\beta}\sum_{i\omega_{n}}\ln(\beta(-i\omega_{n}+\xi))=\frac{1}{\beta}% \ln(1-e^{-\beta\xi})
  93. - 1 β i ω m ln ( β ( - i ω m + ξ ) ) = - 1 β ln ( 1 + e - β ξ ) -\frac{1}{\beta}\sum_{i\omega_{m}}\ln(\beta(-i\omega_{m}+\xi))=-\frac{1}{\beta% }\ln(1+e^{-\beta\xi})
  94. Σ ( i ω m ) = - 1 β i ω n 1 i ω m + i ω n - ϵ 1 i ω n - Ω = n F ( ϵ ) + n B ( Ω ) i ω m - ϵ + Ω \Sigma(i\omega_{m})=-\frac{1}{\beta}\sum_{i\omega_{n}}\frac{1}{i\omega_{m}+i% \omega_{n}-\epsilon}\frac{1}{i\omega_{n}-\Omega}=\frac{n_{F}(\epsilon)+n_{B}(% \Omega)}{i\omega_{m}-\epsilon+\Omega}
  95. Π ( i ω n ) = 1 β i ω m 1 i ω m + i ω n - ϵ 1 i ω m - ϵ = - n F ( ϵ ) - n F ( ϵ ) i ω n - ϵ + ϵ \Pi(i\omega_{n})=\frac{1}{\beta}\sum_{i\omega_{m}}\frac{1}{i\omega_{m}+i\omega% _{n}-\epsilon}\frac{1}{i\omega_{m}-\epsilon^{\prime}}=-\frac{n_{F}(\epsilon)-n% _{F}\left(\epsilon^{\prime}\right)}{i\omega_{n}-\epsilon+\epsilon^{\prime}}
  96. Π ( i ω n ) = - 1 β i ω m 1 i ω m + i ω n - ϵ 1 - i ω m - ϵ = 1 - n F ( ϵ ) - n F ( ϵ ) i ω n - ϵ - ϵ . \Pi(i\omega_{n})=-\frac{1}{\beta}\sum_{i\omega_{m}}\frac{1}{i\omega_{m}+i% \omega_{n}-\epsilon}\frac{1}{-i\omega_{m}-\epsilon^{\prime}}=\frac{1-n_{F}(% \epsilon)-n_{F}\left(\epsilon^{\prime}\right)}{i\omega_{n}-\epsilon-\epsilon^{% \prime}}.
  97. n η n_{\eta}
  98. n η ( ξ ) = 1 e β ξ - η n_{\eta}(\xi)=\frac{1}{e^{\beta\xi}-\eta}
  99. n η ( ξ ) = { n B ( ξ ) , if η = + 1 n F ( ξ ) , if η = - 1 n_{\eta}(\xi)=\begin{cases}n_{B}(\xi),&\mbox{if }~{}\eta=+1\\ n_{F}(\xi),&\mbox{if }~{}\eta=-1\end{cases}
  100. n B ( ξ ) = 1 2 ( coth β ξ 2 - 1 ) n_{B}(\xi)=\frac{1}{2}\left(\mathrm{coth}\frac{\beta\xi}{2}-1\right)
  101. n F ( ξ ) = 1 2 ( 1 - tanh β ξ 2 ) n_{F}(\xi)=\frac{1}{2}\left(1-\mathrm{tanh}\frac{\beta\xi}{2}\right)
  102. n η ( - ξ ) = - η - n η ( ξ ) n_{\eta}(-\xi)=-\eta-n_{\eta}(\xi)
  103. c η c_{\eta}
  104. n η ( - ξ ) = n η ( ξ ) + 2 ξ c η ( 0 , ξ ) n_{\eta}(-\xi)=n_{\eta}(\xi)+2\xi c_{\eta}(0,\xi)
  105. n η ( i ω m + ξ ) = - n - η ( ξ ) n_{\eta}(i\omega_{m}+\xi)=-n_{-\eta}(\xi)
  106. n B ( ξ ) = - β 4 csch 2 β ξ 2 n_{B}^{\prime}(\xi)=-\frac{\beta}{4}\mathrm{csch}^{2}\frac{\beta\xi}{2}
  107. n F ( ξ ) = - β 4 sech 2 β ξ 2 n_{F}^{\prime}(\xi)=-\frac{\beta}{4}\mathrm{sech}^{2}\frac{\beta\xi}{2}
  108. n η ( ξ ) = - β n η ( ξ ) ( 1 + η n η ( ξ ) ) n_{\eta}^{\prime}(\xi)=-\beta n_{\eta}(\xi)(1+\eta n_{\eta}(\xi))
  109. n η ( ξ ) = η δ ( ξ ) n_{\eta}^{\prime}(\xi)=\eta\delta(\xi)
  110. β \beta\rightarrow\infty
  111. n B ′′ ( ξ ) = β 2 4 csch 2 β ξ 2 coth β ξ 2 n_{B}^{\prime\prime}(\xi)=\frac{\beta^{2}}{4}\mathrm{csch}^{2}\frac{\beta\xi}{% 2}\mathrm{coth}\frac{\beta\xi}{2}
  112. n F ′′ ( ξ ) = β 2 4 sech 2 β ξ 2 tanh β ξ 2 n_{F}^{\prime\prime}(\xi)=\frac{\beta^{2}}{4}\mathrm{sech}^{2}\frac{\beta\xi}{% 2}\mathrm{tanh}\frac{\beta\xi}{2}
  113. n η ( a + b ) - n η ( a - b ) = - sinh β b cosh β a - η cosh β b n_{\eta}(a+b)-n_{\eta}(a-b)=-\frac{\mathrm{sinh}\beta b}{\mathrm{cosh}\beta a-% \eta\,\mathrm{cosh}\beta b}
  114. n B ( b ) - n B ( - b ) = coth β b 2 n_{B}(b)-n_{B}(-b)=\mathrm{coth}\frac{\beta b}{2}
  115. n F ( b ) - n F ( - b ) = - tanh β b 2 n_{F}(b)-n_{F}(-b)=-\mathrm{tanh}\frac{\beta b}{2}
  116. n B ( a + b ) - n B ( a - b ) = coth β b 2 + n B ′′ ( b ) a 2 + n_{B}(a+b)-n_{B}(a-b)=\mathrm{coth}\frac{\beta b}{2}+n_{B}^{\prime\prime}(b)a^% {2}+\cdots
  117. n F ( a + b ) - n F ( a - b ) = - tanh β b 2 + n F ′′ ( b ) a 2 + n_{F}(a+b)-n_{F}(a-b)=-\mathrm{tanh}\frac{\beta b}{2}+n_{F}^{\prime\prime}(b)a% ^{2}+\cdots
  118. n B ( a + b ) - n B ( a - b ) = 2 n B ( a ) b + n_{B}(a+b)-n_{B}(a-b)=2n_{B}^{\prime}(a)b+\cdots
  119. n F ( a + b ) - n F ( a - b ) = 2 n F ( a ) b + n_{F}(a+b)-n_{F}(a-b)=2n_{F}^{\prime}(a)b+\cdots
  120. c η ( a , b ) - n η ( a + b ) - n η ( a - b ) 2 b c_{\eta}(a,b)\equiv-\frac{n_{\eta}(a+b)-n_{\eta}(a-b)}{2b}
  121. c B ( a , b ) c + ( a , b ) c_{B}(a,b)\equiv c_{+}(a,b)
  122. c F ( a , b ) c - ( a , b ) c_{F}(a,b)\equiv c_{-}(a,b)
  123. c η ( a , b ) = sinh β b 2 b ( cosh β a - η cosh β b ) c_{\eta}(a,b)=\frac{\mathrm{sinh}\beta b}{2b(\mathrm{cosh}\beta a-\eta\,% \mathrm{cosh}\beta b)}
  124. c F ( a , b ) c_{F}(a,b)
  125. c B ( a , b ) = 1 4 b ( coth β ( a - b ) 2 - coth β ( a + b ) 2 ) c_{B}(a,b)=\frac{1}{4b}\left(\mathrm{coth}\frac{\beta(a-b)}{2}-\mathrm{coth}% \frac{\beta(a+b)}{2}\right)
  126. c F ( a , b ) = 1 4 b ( tanh β ( a + b ) 2 - tanh β ( a - b ) 2 ) c_{F}(a,b)=\frac{1}{4b}\left(\mathrm{tanh}\frac{\beta(a+b)}{2}-\mathrm{tanh}% \frac{\beta(a-b)}{2}\right)
  127. c B ( 0 , b ) = - 1 2 b coth β b 2 c_{B}(0,b)=-\frac{1}{2b}\mathrm{coth}\frac{\beta b}{2}
  128. c F ( 0 , b ) = 1 2 b tanh β b 2 c_{F}(0,b)=\frac{1}{2b}\mathrm{tanh}\frac{\beta b}{2}
  129. c B ( a , 0 ) = β 4 csch 2 β a 2 c_{B}(a,0)=\frac{\beta}{4}\mathrm{csch}^{2}\frac{\beta a}{2}
  130. c F ( a , 0 ) = β 4 sech 2 β a 2 c_{F}(a,0)=\frac{\beta}{4}\mathrm{sech}^{2}\frac{\beta a}{2}
  131. c F ( 0 , b ) = 1 2 | b | c_{F}(0,b)=\frac{1}{2|b|}
  132. c F ( a , 0 ) = δ ( a ) c_{F}(a,0)=\delta(a)
  133. c F ( a , b ) = { 1 2 | b | , if | a | < | b | 0 , if | a | > | b | c_{F}(a,b)=\begin{cases}\frac{1}{2|b|},&\mbox{if }~{}|a|<|b|\\ 0,&\mbox{if }~{}|a|>|b|\end{cases}

Mattig_formula.html

  1. r 1 = c R 0 H 0 q 0 z + ( q 0 - 1 ) ( - 1 + 1 + 2 q 0 z ) q 0 2 ( 1 + z ) r_{1}=\frac{c}{R_{0}H_{0}}\frac{q_{0}z+(q_{0}-1)(-1+\sqrt{1+2q_{0}z})}{q_{0}^{% 2}(1+z)}
  2. r 1 = d p R = d c R 0 r_{1}=\frac{d_{p}}{R}=\frac{d_{c}}{R_{0}}
  3. d p d_{p}
  4. d c d_{c}
  5. q 0 = Ω 0 / 2 q_{0}=\Omega_{0}/2
  6. Ω 0 \Omega_{0}
  7. R 0 R_{0}
  8. R R
  9. H 0 H_{0}
  10. z z
  11. q 0 > 0 q_{0}>0
  12. q 0 0 q_{0}\leq 0
  13. r 1 r_{1}
  14. D L = R 0 r 1 ( 1 + z ) = c H 0 q 0 2 [ q 0 z + ( q 0 - 1 ) ( - 1 + 1 + 2 q 0 z ) ] D_{L}\ =\ R_{0}r_{1}(1+z)=\frac{c}{H_{0}q_{0}^{2}}\left[q_{0}z+(q_{0}-1)(-1+% \sqrt{1+2q_{0}z})\right]
  15. q 0 = 0 q_{0}=0
  16. D L = c H 0 ( z + z 2 2 ) D_{L}=\frac{c}{H_{0}}\left(z+\frac{z^{2}}{2}\right)
  17. q 0 0 q_{0}\geq 0
  18. D L = c H 0 z [ 1 + z ( 1 - q 0 ) 1 + q 0 z + 1 + 2 q 0 z ] D_{L}=\frac{c}{H_{0}}z\left[1+\frac{z(1-q_{0})}{1+q_{0}z+\sqrt{1+2q_{0}z}}\right]

Maurice_A._de_Gosson.html

  1. \sqrt{\hbar}

Max_Jakob.html

  1. J a = C p , f ( T s a t - T w ) h f . g Ja=\frac{C_{p,f}(T_{sat}-T_{w})}{h_{f.g}}

Maximal_information_coefficient.html

  1. H ( X b ) = H ( Y b ) = H ( X b , Y b ) \mathrm{H}\left(X_{b}\right)=\mathrm{H}\left(Y_{b}\right)=\mathrm{H}\left(X_{b% },Y_{b}\right)
  2. H ( X b ) \mathrm{H}(X_{b})
  3. H ( Y b ) \mathrm{H}(Y_{b})
  4. n x n_{x}
  5. n y n_{y}
  6. n x × n y N 0.6 n_{x}\times n_{y}\leq\mathrm{N}^{0.6}
  7. X b X_{b}
  8. Y b Y_{b}
  9. n x = 2 n_{x}=2
  10. n y = 2 n_{y}=2
  11. I ( X b ; Y b ) \mathrm{I}(X_{b};Y_{b})
  12. I ( X ; Y ) = H ( X ) + H ( Y ) - H ( X , Y ) I(X;Y)=H(X)+H(Y)-H(X,Y)
  13. n x , n y n_{x},n_{y}
  14. log min ( n x , n y ) \log\min\left(n_{x},n_{y}\right)
  15. n x n_{x}
  16. n y n_{y}
  17. n x × n y N 0.6 n_{x}\times n_{y}\leq\mathrm{N}^{0.6}

Maximum-entropy_Markov_model.html

  1. O 1 , , O n O_{1},\dots,O_{n}
  2. S 1 , , S n S_{1},\dots,S_{n}
  3. P ( S 1 , , S n | O 1 , , O n ) P(S_{1},\dots,S_{n}|O_{1},\dots,O_{n})
  4. P ( S 1 , , S n | O 1 , , O n ) = t = 1 n P ( S t | S t - 1 , O t ) . P(S_{1},\dots,S_{n}|O_{1},\dots,O_{n})=\prod_{t=1}^{n}P(S_{t}|S_{t-1},O_{t}).
  5. P ( s | s , o ) P(s|s^{\prime},o)
  6. s s^{\prime}
  7. s s
  8. P ( s | s , o ) = P s ( s | o ) = 1 Z ( o , s ) exp ( a λ a f a ( o , s ) ) . P(s|s^{\prime},o)=P_{s^{\prime}}(s|o)=\frac{1}{Z(o,s^{\prime})}\exp\left(\sum_% {a}\lambda_{a}f_{a}(o,s)\right).
  9. f a ( o , s ) f_{a}(o,s)
  10. Z ( o , s ) Z(o,s^{\prime})
  11. E e [ f a ( o , s ) ] = E p [ f a ( o , s ) ] for all a . \operatorname{E}_{e}\left[f_{a}(o,s)\right]=\operatorname{E}_{p}\left[f_{a}(o,% s)\right]\quad\,\text{ for all }a.
  12. λ a \lambda_{a}
  13. S 1 , , S n S_{1},\dots,S_{n}
  14. α t + 1 ( s ) = s S α t ( s ) P s ( s | o t + 1 ) . \alpha_{t+1}(s)=\sum_{s^{\prime}\in S}\alpha_{t}(s^{\prime})P_{s^{\prime}}(s|o% _{t+1}).

Maxwell-Bloch_equations.html

  1. ψ = c g ψ g + c e ψ e \psi=c_{g}\psi_{g}+c_{e}\psi_{e}
  2. | c g | 2 + | c e | 2 = 1 \left|c_{g}\right|^{2}+\left|c_{e}\right|^{2}=1
  3. ρ = [ ρ e e ρ e g ρ g e ρ g g ] = [ c e c e * c e c g * c g c e * c g c g * ] \rho=\begin{bmatrix}\rho_{ee}&\rho_{eg}\\ \rho_{ge}&\rho_{gg}\end{bmatrix}=\begin{bmatrix}c_{e}c_{e}^{*}&c_{e}c_{g}^{*}% \\ c_{g}c_{e}^{*}&c_{g}c_{g}^{*}\end{bmatrix}
  4. d ρ g g d t = γ ρ e e + i 2 ( Ω * ρ ¯ e g - Ω ρ ¯ g e ) \frac{d\rho_{gg}}{dt}=\gamma\rho_{ee}+\frac{i}{2}(\Omega^{*}\bar{\rho}_{eg}-% \Omega\bar{\rho}_{ge})
  5. d ρ e e d t = - γ ρ e e + i 2 ( Ω ρ ¯ g e - Ω * ρ ¯ e g ) \frac{d\rho_{ee}}{dt}=-\gamma\rho_{ee}+\frac{i}{2}(\Omega\bar{\rho}_{ge}-% \Omega^{*}\bar{\rho}_{eg})
  6. d ρ ¯ g e d t = - ( γ 2 + i δ ) ρ ¯ g e + i 2 Ω * ( ρ e e - ρ g g ) \frac{d\bar{\rho}_{ge}}{dt}=-\left(\frac{\gamma}{2}+i\delta\right)\bar{\rho}_{% ge}+\frac{i}{2}\Omega^{*}(\rho_{ee}-\rho_{gg})
  7. d ρ ¯ e g d t = - ( γ 2 - i δ ) ρ ¯ e g + i 2 Ω ( ρ g g - ρ e e ) \frac{d\bar{\rho}_{eg}}{dt}=-\left(\frac{\gamma}{2}-i\delta\right)\bar{\rho}_{% eg}+\frac{i}{2}\Omega(\rho_{gg}-\rho_{ee})
  8. ρ e g ( t ) \rho_{eg}(t)
  9. γ 2 \frac{\gamma}{2}
  10. Ω \Omega
  11. Ω = | χ g , e | 2 + δ 2 \Omega=\sqrt{|\chi_{g,e}|^{2}+\delta^{2}}
  12. δ = ω - ω 0 \delta=\omega-\omega_{0}
  13. ω \omega
  14. ω 0 \omega_{0}
  15. χ g , e = d g , e E 0 \chi_{g,e}={\vec{d}_{g,e}\cdot\vec{E}_{0}\over\hbar}
  16. d g , e \scriptstyle{\vec{d}_{g,e}}
  17. g e \scriptstyle{g\rightarrow e}
  18. E 0 = ϵ ^ E 0 \scriptstyle{\vec{E}_{0}=\hat{\epsilon}E_{0}}
  19. H = ω c a a + ω a σ σ + i g ( a σ - a σ ) + i J ( a e - i ω l t - a e i ω l t ) H=\omega_{c}a^{\dagger}a+\omega_{a}\sigma^{\dagger}\sigma+ig(a^{\dagger}\sigma% -a\sigma^{\dagger})+iJ(a^{\dagger}e^{-i\omega_{l}t}-ae^{i\omega_{l}t})
  20. a a
  21. σ = 1 2 ( σ x - i σ y ) \sigma=\frac{1}{2}\left(\sigma_{x}-i\sigma_{y}\right)
  22. | ψ e - i ω l t ( a a + σ σ ) | ψ |\psi\rangle\rightarrow\operatorname{e}^{-i\omega_{l}t\left(a^{\dagger}a+% \sigma^{\dagger}\sigma\right)}|\psi\rangle
  23. H = Δ c a a + Δ a σ σ + i g ( a σ - a σ ) + i J ( a - a ) H=\Delta_{c}a^{\dagger}a+\Delta_{a}\sigma^{\dagger}\sigma+ig(a^{\dagger}\sigma% -a\sigma^{\dagger})+iJ(a^{\dagger}-a)
  24. Δ i = ω i - ω l \Delta_{i}=\omega_{i}-\omega_{l}
  25. J J
  26. J = 2 P ( Δ c 2 + κ 2 ) / ( ω c κ ) J=\sqrt{2P(\Delta_{c}^{2}+\kappa^{2})/(\omega_{c}\kappa)}
  27. 2 κ 2\kappa
  28. ρ ˙ = - i [ H , ρ ] + 2 κ ( a ρ a - 1 2 ( a a ρ + ρ a a ) ) + 2 γ ( σ ρ σ - 1 2 ( σ σ ρ + ρ σ σ ) ) \dot{\rho}=-i[H,\rho]+2\kappa\left(a\rho a^{\dagger}-\frac{1}{2}\left(a^{% \dagger}a\rho+\rho a^{\dagger}a\right)\right)+2\gamma\left(\sigma\rho\sigma^{% \dagger}-\frac{1}{2}\left(\sigma^{\dagger}\sigma\rho+\rho\sigma^{\dagger}% \sigma\right)\right)
  29. O = tr ( O ρ ) \langle O\rangle=\operatorname{tr}\left(O\rho\right)
  30. O ˙ = tr ( O ρ ˙ ) \langle\dot{O}\rangle=\operatorname{tr}\left(O\dot{\rho}\right)
  31. a \langle a\rangle
  32. σ \langle\sigma\rangle
  33. σ z \langle\sigma_{z}\rangle
  34. d d t a = i ( - Δ c a - i g σ - i J ) - κ a \frac{d}{dt}\langle a\rangle=i\left(-\Delta_{c}\langle a\rangle-ig\langle% \sigma\rangle-iJ\right)-\kappa\langle a\rangle
  35. d d t σ = i ( - Δ a σ - i g a σ z ) - γ σ \frac{d}{dt}\langle\sigma\rangle=i\left(-\Delta_{a}\langle\sigma\rangle-ig% \langle a\sigma_{z}\rangle\right)-\gamma\langle\sigma\rangle
  36. d d t σ z = - 2 g ( a σ + a σ ) - 2 γ σ z - 2 γ \frac{d}{dt}\langle\sigma_{z}\rangle=-2g\left(\langle a^{\dagger}\sigma\rangle% +\langle a\sigma^{\dagger}\rangle\right)-2\gamma\langle\sigma_{z}\rangle-2\gamma
  37. a σ \langle a^{\dagger}\sigma\rangle
  38. a = ( γ / 2 g ) x \langle a\rangle=(\gamma/\sqrt{2}g)x
  39. σ = - p / 2 \langle\sigma\rangle=-p/\sqrt{2}
  40. σ z = - D \langle\sigma_{z}\rangle=-D
  41. Θ = Δ c / κ \Theta=\Delta_{c}/\kappa
  42. C = g 2 / 2 κ γ C=g^{2}/2\kappa\gamma
  43. y = 2 g J / κ γ y=\sqrt{2}gJ/\kappa\gamma
  44. Δ = Δ a / γ \Delta=\Delta_{a}/\gamma
  45. x ˙ = κ ( - 2 C p + y - ( i Θ + 1 ) x ) \dot{x}=\kappa\left(-2Cp+y-(i\Theta+1)x\right)
  46. p ˙ = γ ( - ( 1 + i Δ ) p + x D ) \dot{p}=\gamma\left(-(1+i\Delta)p+xD\right)
  47. D ˙ = γ ( 2 ( 1 - D ) - ( x * p + x p * ) ) \dot{D}=\gamma\left(2(1-D)-(x^{*}p+xp^{*})\right)

Max–min_inequality.html

  1. f : Z × W f:Z\times W\mapsto\mathbb{R}
  2. sup z Z inf w W f ( z , w ) inf w W sup z Z f ( z , w ) . \sup_{z\in Z}\inf_{w\in W}f(z,w)\leq\inf_{w\in W}\sup_{z\in Z}f(z,w).\,
  3. f , W , Z f,W,Z
  4. g ( z ) inf w W f ( z , w ) g(z)\triangleq\inf_{w\in W}f(z,w)
  5. g ( z ) f ( z , w ) , z , w \implies g(z)\leq f(z,w),\forall z,w
  6. sup z g ( z ) sup z f ( z , w ) , w \implies\sup_{z}g(z)\leq\sup_{z}f(z,w),\forall w
  7. sup z inf w f ( z , w ) sup z f ( z , w ) w \implies\sup_{z}\inf_{w}f(z,w)\leq\sup_{z}f(z,w)\forall w
  8. sup z inf w f ( z , w ) inf w sup z f ( z , w ) \implies\sup_{z}\inf_{w}f(z,w)\leq\inf_{w}\sup_{z}f(z,w)\qquad\square

McConnell_equation.html

  1. a a
  2. ρ \rho
  3. C 6 H 6 - C_{6}H_{6}^{-}
  4. a = Q ρ a=Q\rho
  5. Q Q

McDonald–Kreitman_test.html

  1. α = 1 - D s P n D n P s \alpha=1-\frac{D_{s}P_{n}}{D_{n}P_{s}}
  2. N I = P n / P s D n / D s NI=\frac{P_{n}/P_{s}}{D_{n}/D_{s}}

McKay_graph.html

  1. χ i , χ j \chi_{i},\chi_{j}
  2. χ i \chi_{i}
  3. χ j \chi_{j}
  4. χ j \chi_{j}
  5. V χ i V\otimes\chi_{i}
  6. V χ i V\otimes\chi_{i}
  7. c V = ( d δ i j - n i j ) i j c_{V}=(d\delta_{ij}-n_{ij})_{ij}
  8. ( ( χ i ( g ) ) i ((\chi_{i}(g))_{i}
  9. d - χ V ( g ) d-\chi_{V}(g)
  10. χ V \chi_{V}
  11. χ \chi
  12. { χ 1 , , χ d } \{\chi_{1},\ldots,\chi_{d}\}
  13. V χ i = j n i j χ j , V\otimes\chi_{i}=\sum_{j}n_{ij}\chi_{j},
  14. Γ G \Gamma_{G}
  15. Γ G \Gamma_{G}
  16. χ i \chi_{i}
  17. χ j \chi_{j}
  18. χ i n i j χ j \chi_{i}\xrightarrow{n_{ij}}\chi_{j}
  19. χ i \chi_{i}
  20. χ j \chi_{j}
  21. n i j = V χ i , χ j = 1 | G | g G V ( g ) χ i ( g ) χ j ( g ) ¯ , n_{ij}=\langle V\otimes\chi_{i},\chi_{j}\rangle=\frac{1}{|G|}\sum_{g\in G}V(g)% \chi_{i}(g)\overline{\chi_{j}(g)},
  22. , \langle\cdot,\cdot\rangle
  23. c V = ( d δ i j - n i j ) i j , c_{V}=(d\delta_{ij}-n_{ij})_{ij},
  24. δ i j \delta_{ij}
  25. χ i \chi_{i}
  26. ψ j \psi_{j}
  27. χ i × ψ j 1 i k , 1 j l \chi_{i}\times\psi_{j}\quad 1\leq i\leq k,\,\,1\leq j\leq l
  28. A × B A\times B
  29. χ i × ψ j ( a , b ) = χ i ( a ) ψ j ( b ) , ( a , b ) A × B \chi_{i}\times\psi_{j}(a,b)=\chi_{i}(a)\psi_{j}(b),(a,b)\in A\times B
  30. ( c A × c B ) ( χ i × ψ l ) , χ n × ψ p = c A χ k , χ n c B ψ l , ψ p . \langle(c_{A}\times c_{B})\otimes(\chi_{i}\times\psi_{l}),\chi_{n}\times\psi_{% p}\rangle=\langle c_{A}\otimes\chi_{k},\chi_{n}\rangle\cdot\langle c_{B}% \otimes\psi_{l},\psi_{p}\rangle.
  31. χ i × ψ j \chi_{i}\times\psi_{j}
  32. χ k × ψ l \chi_{k}\times\psi_{l}
  33. χ i \chi_{i}
  34. χ k \chi_{k}
  35. ψ j \psi_{j}
  36. ψ l \psi_{l}
  37. T ¯ \overline{T}
  38. S = ( i 0 0 - i ) , V = ( 0 i i 0 ) , U = 1 2 ( ϵ ϵ 3 ϵ ϵ 7 ) , S=\left(\begin{array}[]{cc}i&0\\ 0&-i\end{array}\right),V=\left(\begin{array}[]{cc}0&i\\ i&0\end{array}\right),U=\frac{1}{\sqrt{2}}\left(\begin{array}[]{cc}\epsilon&% \epsilon^{3}\\ \epsilon&\epsilon^{7}\end{array}\right),
  39. T ¯ \overline{T}
  40. T ¯ = { U k , S U k , V U k , S V U k | k = 0 , , 5 } . \overline{T}=\{U^{k},SU^{k},VU^{k},SVU^{k}|k=0,\ldots,5\}.
  41. T ¯ \overline{T}
  42. C 1 = { U 0 = I } , C_{1}=\{U^{0}=I\},
  43. C 2 = { U 3 = - I } , C_{2}=\{U^{3}=-I\},
  44. C 3 = { ± S , ± V , ± S V } , C_{3}=\{\pm S,\pm V,\pm SV\},
  45. C 4 = { U 2 , S U 2 , V U 2 , S V U 2 } , C_{4}=\{U^{2},SU^{2},VU^{2},SVU^{2}\},
  46. C 5 = { - U , S U , V U , S V U } , C_{5}=\{-U,SU,VU,SVU\},
  47. C 6 = { - U 2 , - S U 2 , - V U 2 , - S V U 2 } , C_{6}=\{-U^{2},-SU^{2},-VU^{2},-SVU^{2}\},
  48. C 7 = { U , - S U , - V U , - S V U } . C_{7}=\{U,-SU,-VU,-SVU\}.
  49. T ¯ \overline{T}
  50. C 1 C_{1}
  51. C 2 C_{2}
  52. C 3 C_{3}
  53. C 4 C_{4}
  54. C 5 C_{5}
  55. C 6 C_{6}
  56. C 7 C_{7}
  57. χ 1 \chi_{1}
  58. 1 1
  59. 1 1
  60. 1 1
  61. 1 1
  62. 1 1
  63. 1 1
  64. 1 1
  65. χ 2 \chi_{2}
  66. 1 1
  67. 1 1
  68. 1 1
  69. ω \omega
  70. ω 2 \omega^{2}
  71. ω \omega
  72. ω 2 \omega^{2}
  73. χ 3 \chi_{3}
  74. 1 1
  75. 1 1
  76. 1 1
  77. ω 2 \omega^{2}
  78. ω \omega
  79. ω 2 \omega^{2}
  80. ω \omega
  81. χ 4 \chi_{4}
  82. 3 3
  83. 3 3
  84. - 1 -1
  85. 0
  86. 0
  87. 0
  88. 0
  89. c c
  90. 2 2
  91. - 2 -2
  92. 0
  93. - 1 -1
  94. - 1 -1
  95. 1 1
  96. 1 1
  97. χ 5 \chi_{5}
  98. 2 2
  99. - 2 -2
  100. 0
  101. - ω -\omega
  102. - ω 2 -\omega^{2}
  103. ω \omega
  104. ω 2 \omega^{2}
  105. χ 6 \chi_{6}
  106. 2 2
  107. - 2 -2
  108. 0
  109. - ω 2 -\omega^{2}
  110. - ω -\omega
  111. ω 2 \omega^{2}
  112. ω \omega
  113. ω = e 2 π i / 3 \omega=e^{2\pi i/3}
  114. T ¯ \overline{T}
  115. E ~ 6 \tilde{E}_{6}

Mean_absolute_scaled_error.html

  1. MASE = 1 n t = 1 n ( | e t | 1 n - 1 i = 2 n | Y i - Y i - 1 | ) = t = 1 n | e t | n n - 1 i = 2 n | Y i - Y i - 1 | \mathrm{MASE}=\frac{1}{n}\sum_{t=1}^{n}\left(\frac{\left|e_{t}\right|}{\frac{1% }{n-1}\sum_{i=2}^{n}\left|Y_{i}-Y_{i-1}\right|}\right)=\frac{\sum_{t=1}^{n}% \left|e_{t}\right|}{\frac{n}{n-1}\sum_{i=2}^{n}\left|Y_{i}-Y_{i-1}\right|}

Mean_airway_pressure.html

  1. M P A W = f * T i 60 * ( P I P - P E E P ) + P E E P M_{PAW}=\frac{f*T_{i}}{60}*(P_{IP}-PEEP)+PEEP
  2. M P A W = F 1 F 1 + F E * P I P + ( 1 - F 1 F 1 + F E ) * P E E P M_{PAW}=\frac{F_{1}}{F_{1}+F_{E}}*P_{IP}+(1-\frac{F_{1}}{F_{1}+F_{E}})*PEEP
  3. M P A W = ( R ) ( T i ) ( P I ) + [ 60 - ( R ) ( T i ) ] ( P E E P ) 60 M_{PAW}=\frac{(R)(T_{i})(P_{I})+[60-(R)(T_{i})](PEEP)}{60}
  4. M P A W = f * T i 60 * ( P I P - P E E P ) + P E E P M_{PAW}=\frac{f*T_{i}}{60}*(P_{IP}-PEEP)+PEEP
  5. M P A W = ( T i * P I P ) + ( T e * P E E P ) T i + T e M_{PAW}=\frac{(T_{i}*P_{IP})+(T_{e}*PEEP)}{T_{i}+T_{e}}
  6. M P A W = ( 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 M_{PAW}=\frac{(P_{high}*T_{high})\,+(P_{low}*T_{low})}{T_{high}+T_{low}}

Mean_inter-particle_distance.html

  1. 1 / n 1/n
  2. r 1 / n 1 / 3 , \langle r\rangle\sim 1/n^{1/3},
  3. n = N / V n=N/V
  4. ( 3 4 π n ) 1 / 3 , \left(\frac{3}{4\pi n}\right)^{1/3},
  5. 1 / n 1/n
  6. 1 / n 1 / 3 1/n^{1/3}
  7. 1 / n 1/n
  8. 1.61 1.61
  9. N N
  10. V V
  11. n = N / V n=N/V
  12. r r
  13. N N
  14. N - 1 N-1
  15. r r
  16. r r
  17. r + d r r+dr
  18. ( 4 π r 2 N / V ) d r (4\pi r^{2}N/V)dr
  19. 1 - 4 π r 3 / 3 V 1-4\pi r^{3}/3V
  20. P N ( r ) d r = 4 π r 2 d r N V ( 1 - 4 π 3 r 3 / V ) N - 1 = 3 a ( r a ) 2 d r ( 1 - ( r a ) 3 1 N ) N - 1 P_{N}(r)dr=4\pi r^{2}dr\frac{N}{V}\left(1-\frac{4\pi}{3}r^{3}/V\right)^{N-1}=% \frac{3}{a}\left(\frac{r}{a}\right)^{2}dr\left(1-\left(\frac{r}{a}\right)^{3}% \frac{1}{N}\right)^{N-1}\,
  21. a = ( 3 4 π n ) 1 / 3 . a=\left(\frac{3}{4\pi n}\right)^{1/3}.
  22. N N\rightarrow\infty
  23. lim x ( 1 + 1 x ) x = e \lim_{x\rightarrow\infty}\left(1+\frac{1}{x}\right)^{x}=e
  24. P ( r ) = 3 a ( r a ) 2 e - ( r / a ) 3 . P(r)=\frac{3}{a}\left(\frac{r}{a}\right)^{2}e^{-(r/a)^{3}}\,.
  25. 0 P ( r ) d r = 1 . \int_{0}^{\infty}P(r)dr=1\,.
  26. r peak = ( 2 / 3 ) 1 / 3 a 0.874 a . r_{\,\text{peak}}=\left(2/3\right)^{1/3}a\approx 0.874a\,.
  27. r k = 0 P ( r ) r k d r = 3 a k 0 x k + 2 e - x 3 d x , \langle r^{k}\rangle=\int_{0}^{\infty}P(r)r^{k}dr=3a^{k}\int_{0}^{\infty}x^{k+% 2}e^{-x^{3}}dx\,,
  28. t = x 3 t=x^{3}
  29. r k = a k 0 t k / 3 e - t d t = a k Γ ( 1 + k 3 ) , \langle r^{k}\rangle=a^{k}\int_{0}^{\infty}t^{k/3}e^{-t}dt=a^{k}\Gamma(1+\frac% {k}{3})\,,
  30. Γ \Gamma
  31. r k = a k Γ ( 1 + k 3 ) . \langle r^{k}\rangle=a^{k}\Gamma(1+\frac{k}{3})\,.
  32. r = a Γ ( 4 3 ) = a 3 Γ ( 1 3 ) 0.893 a . \langle r\rangle=a\Gamma(\frac{4}{3})=\frac{a}{3}\Gamma(\frac{1}{3})\approx 0.% 893a\,.

Measurable_Riemann_mapping_theorem.html

  1. μ < 1 \|\mu\|_{\infty}<1
  2. z ¯ f ( z ) = μ ( z ) z f ( z ) \partial_{\overline{z}}f(z)=\mu(z)\partial_{z}f(z)

Measurement_microphone_calibration.html

  1. i i
  2. j j
  3. k k
  4. i i
  5. j j
  6. Z a c Z_{ac}
  7. I i I_{i}
  8. U j U_{j}
  9. Z i j Z_{ij}
  10. M i M_{i}
  11. M j M_{j}
  12. Z i j = U j I i = M i Z a c M j Z_{ij}=\frac{U_{j}}{I_{i}}=M_{i}\;Z_{ac}\;M_{j}
  13. i k ik
  14. j k jk
  15. M i = 1 Z a c Z i j Z i k Z j k M_{i}=\sqrt{\frac{1}{Z_{ac}}\frac{Z_{ij}Z_{ik}}{Z_{jk}}}
  16. Z a c = p Q = F v S 2 Z_{ac}=\frac{p}{Q}=\frac{F}{vS^{2}}
  17. Z a c , f r e e = ρ 0 ω 4 π r e - m 2 r e - j ( k r - π 2 ) Z_{ac,free}=\frac{\rho_{0}\omega}{4\pi r}e^{-\frac{m}{2}r}e^{-j(kr-\frac{\pi}{% 2})}
  18. r r
  19. Z a c , d i f f = ρ 0 ω π A = ρ 0 ω 4 π d c Z_{ac,diff}=\frac{\rho_{0}\omega}{\sqrt{\pi A}}=\frac{\rho_{0}\omega}{4\pi d_{% c}}
  20. A A
  21. d c d_{c}
  22. Z a c , c o m p = ρ 0 c 2 j ω V 0 Z_{ac,comp}=\frac{\rho_{0}c^{2}}{j\omega V_{0}}
  23. V 0 V_{0}

Mechanical_advantage_device.html

  1. M A = π d m l MA=\frac{\pi d_{m}}{l}

Mechanical_index.html

  1. MI = PNP F c , \,\text{MI}=\frac{\,\text{PNP}}{\sqrt{F_{c}}},

Mechanism_of_sonoluminescence.html

  1. ( 1 - R ˙ c ) R R ¨ + 3 2 R 2 ˙ ( 1 - R ˙ 3 c ) = ( 1 + R ˙ c ) 1 ρ l [ p B ( R , t ) - p A ( t + R c ) - P ] + R d p B ( R , t ) ρ l c d t \left(1-\frac{\dot{R}}{c}\right)R\ddot{R}+\frac{3}{2}\dot{R^{2}}\left(1-\frac{% \dot{R}}{3c}\right)=\left(1+\frac{\dot{R}}{c}\right)\frac{1}{\rho_{l}}\left[p_% {B}(R,t)-p_{A}\left(t+\frac{R}{c}\right)-P_{\infty}\right]+\frac{Rdp_{B}(R,t)}% {\rho_{l}cdt}
  2. R \scriptstyle R
  3. ρ l \scriptstyle\rho_{l}
  4. c \scriptstyle c
  5. p B ( R , t ) \scriptstyle p_{B}(R,t)
  6. t \scriptstyle t
  7. p A ( t + R c ) \scriptstyle p_{A}(t+\frac{R}{c})
  8. p ˙ = 3 r ( ( γ - 1 ) K T r | R - γ p R ˙ ) \dot{p}=\frac{3}{r}\left((\gamma-1)K\frac{\partial T}{\partial r}\Bigg|_{R}-% \gamma p\dot{R}\right)
  9. T T
  10. K K
  11. r r
  12. ( 1 - R ˙ c ) R R ¨ + 3 2 R 2 ˙ ( 1 - R ˙ 3 c ) = ( 1 + R ˙ c ) 1 ρ l [ p B ( R , t ) - p A ( t ) - P ] + R ρ l c ( 1 - R ˙ c ) d p B ( R , t ) d t \left(1-\frac{\dot{R}}{c}\right)R\ddot{R}+\frac{3}{2}\dot{R^{2}}\left(1-\frac{% \dot{R}}{3c}\right)=\left(1+\frac{\dot{R}}{c}\right)\frac{1}{\rho_{l}}\left[p_% {B}(R,t)-p_{A}(t)-P_{\infty}\right]+\frac{R}{\rho_{l}c}\left(1-\frac{\dot{R}}{% c}\right)\frac{dp_{B}(R,t)}{dt}
  13. R \scriptstyle R
  14. ρ l \scriptstyle\rho_{l}
  15. c \scriptstyle c
  16. d p B ( R , t ) \scriptstyle dp_{B}(R,t)
  17. t \scriptstyle t
  18. p A ( t ) \scriptstyle p_{A}(t)
  19. R R ¨ + 3 2 R 2 ˙ = 1 ρ l ( p g - P 0 - P ( t ) - 4 μ R ˙ R - 2 γ R ) R\ddot{R}+\frac{3}{2}\dot{R^{2}}=\frac{1}{\rho_{l}}\left(p_{g}-P_{0}-P\left(t% \right)-4\mu\frac{\dot{R}}{R}-\frac{2\gamma}{R}\right)
  20. R \scriptstyle R
  21. R ¨ \scriptstyle\ddot{R}
  22. R ˙ \scriptstyle\dot{R}
  23. ρ l \scriptstyle\rho_{l}
  24. p g \scriptstyle p_{g}
  25. P 0 \scriptstyle P_{0}
  26. P ( t ) \scriptstyle P(t)
  27. μ \scriptstyle\mu
  28. γ \scriptstyle\gamma
  29. E = c v R g P 0 n b E=\frac{c_{v}}{R_{g}}P_{0}nb
  30. c v c_{v}
  31. R g R_{g}
  32. P 0 P_{0}
  33. n n
  34. b b
  35. μ m \mu m

Mehler_kernel.html

  1. H H
  2. x x
  3. E ( x , y ) = n = 0 ( ρ / 2 ) n n ! H n ( x ) H n ( y ) . E(x,y)=\sum_{n=0}^{\infty}\frac{(\rho/2)^{n}}{n!}~{}\mathit{H}_{n}(x)\mathit{H% }_{n}(y)~{}.
  4. φ ( x , t ) φ(x,t)
  5. φ t = 2 φ x 2 - x 2 φ D x φ . \frac{\partial\varphi}{\partial t}=\frac{\partial^{2}\varphi}{\partial x^{2}}-% x^{2}\varphi\equiv D_{x}\varphi~{}.
  6. D D
  7. n n
  8. φ n ( x , t ) = e - ( 2 n + 1 ) t H n ( x ) exp ( - x 2 / 2 ) . \varphi_{n}(x,t)=e^{-(2n+1)t}~{}H_{n}(x)\exp(-x^{2}/2)~{}.
  9. φ ( x , 0 ) φ(x,0)
  10. φ ( x , t ) = K ( x , y ; t ) φ ( y , 0 ) d y , \varphi(x,t)=\int K(x,y;t)\varphi(y,0)dy~{},
  11. K K
  12. K ( x , y ; t ) n 0 e - ( 2 n + 1 ) t π 2 n n ! H n ( x ) H n ( y ) exp ( - ( x 2 + y 2 ) / 2 ) . K(x,y;t)\equiv\sum_{n\geq 0}\frac{e^{-(2n+1)t}}{\sqrt{\pi}2^{n}n!}~{}H_{n}(x)H% _{n}(y)\exp(-(x^{2}+y^{2})/2)~{}.
  13. n 0 ( ρ / 2 ) n n ! H n ( x ) H n ( y ) exp ( - ( x 2 + y 2 ) / 2 ) = 1 ( 1 - ρ 2 ) exp 4 x y ρ - ( 1 + ρ 2 ) ( x 2 + y 2 ) 2 ( 1 - ρ 2 ) . \displaystyle{\sum_{n\geq 0}\frac{(\rho/2)^{n}}{n!}H_{n}(x)H_{n}(y)\exp(-(x^{2% }+y^{2})/2)={1\over\sqrt{(1-\rho^{2})}}\exp{4xy\rho-(1+\rho^{2})(x^{2}+y^{2})% \over 2(1-\rho^{2})}}~{}.
  14. K K
  15. t t
  16. ρ ρ
  17. t t
  18. x x
  19. y y
  20. K ( x , y ; 0 ) = δ ( x - y ) . K(x,y;0)=\delta(x-y)~{}.
  21. d y K ( x , y ; t ) K ( y , z ; t ) = K ( x , z ; t + t ) . \int dyK(x,y;t)K(y,z;t^{\prime})=K(x,z;t+t^{\prime})~{}.
  22. K K
  23. ( x , y ) 𝐌 ( x y ) , (x,y){\mathbf{M}}\begin{pmatrix}{x}\\ {y}\end{pmatrix}~{},~{}
  24. 𝐌 cosech ( 2 t ) ( cosh ( 2 t ) - 1 - 1 cosh ( 2 t ) ) , {\mathbf{M}}\equiv\,\text{cosech}(2t)\begin{pmatrix}\cosh(2t)&-1\\ -1&\cosh(2t)\end{pmatrix}~{},
  25. 𝐌 T ( 0 1 - 1 0 ) 𝐌 = ( 0 1 - 1 0 ) . {\mathbf{M}}\text{T}~{}\begin{pmatrix}0&1\\ -1&0\end{pmatrix}~{}{\mathbf{M}}=\begin{pmatrix}0&1\\ -1&0\end{pmatrix}~{}.
  26. x x / 2 x→x/√2
  27. y y / 2 y→y/√2
  28. H H
  29. x x
  30. H e He
  31. x x
  32. E E
  33. 1 1 - ρ 2 exp ( - ρ 2 ( x 2 + y 2 ) - 2 ρ x y 2 ( 1 - ρ 2 ) ) = n = 0 ρ n n ! 𝐻𝑒 n ( x ) 𝐻𝑒 n ( y ) . \frac{1}{\sqrt{1-\rho^{2}}}\exp\left(-\frac{\rho^{2}(x^{2}+y^{2})-2\rho xy}{2(% 1-\rho^{2})}\right)=\sum_{n=0}^{\infty}\frac{\rho^{n}}{n!}~{}\mathit{He}_{n}(x% )\mathit{He}_{n}(y)~{}.
  34. x , y x,y
  35. p ( x , y ) = 1 2 π 1 - ρ 2 exp ( - ( x 2 + y 2 ) - 2 ρ x y 2 ( 1 - ρ 2 ) ) , p(x,y)=\frac{1}{2\pi\sqrt{1-\rho^{2}}}\exp\left(-\frac{(x^{2}+y^{2})-2\rho xy}% {2(1-\rho^{2})}\right)~{},
  36. p ( x ) , p ( y ) p(x),p(y)
  37. x x
  38. y y
  39. p ( x , y ) = p ( x ) p ( y ) n = 0 ρ n n ! 𝐻𝑒 n ( x ) 𝐻𝑒 n ( y ) . p(x,y)=p(x)p(y)\sum_{n=0}^{\infty}\frac{\rho^{n}}{n!}~{}\mathit{He}_{n}(x)% \mathit{He}_{n}(y)~{}.
  40. p ( x , y ) p(x,y)
  41. c ( i u 1 , i u 2 ) = exp ( - ( u 1 2 + u 2 2 - 2 ρ u 1 u 2 ) / 2 ) . c(iu_{1},iu_{2})=\exp(-(u_{1}^{2}+u_{2}^{2}-2\rho u_{1}u_{2})/2)~{}.
  42. exp ( - ( u 1 2 + u 2 2 ) / 2 ) n = 0 ρ n n ! ( u 1 u 2 ) n . \exp(-(u_{1}^{2}+u_{2}^{2})/2)\sum_{n=0}^{\infty}\frac{\rho^{n}}{n!}(u_{1}u_{2% })^{n}~{}.
  43. [ ψ n ] ( y ) = ( - i ) n ψ n ( y ) , \mathcal{F}[\psi_{n}](y)=(-i)^{n}\psi_{n}(y)~{},
  44. [ f ] ( y ) = d x f ( x ) n 0 ( - i ) n ψ n ( x ) ψ n ( y ) . \mathcal{F}[f](y)=\int dxf(x)\sum_{n\geq 0}(-i)^{n}\psi_{n}(x)\psi_{n}(y)~{}.
  45. α α
  46. α = n 0 ( - i ) 2 α n / π ψ n ( x ) ψ n ( y ) . \mathcal{F}_{\alpha}=\sum_{n\geq 0}(-i)^{2\alpha n/\pi}\psi_{n}(x)\psi_{n}(y)~% {}.
  47. α = π / 2 α=π/2
  48. α = π / 2 α=−π/2
  49. ρ ρ
  50. α α
  51. α [ f ] ( y ) = 1 - i cot ( α ) 2 π e i cot ( α ) 2 y 2 - e - i ( csc ( α ) y x - cot ( α ) 2 x 2 ) f ( x ) d x . \mathcal{F}_{\alpha}[f](y)=\sqrt{\frac{1-i\cot(\alpha)}{2\pi}}~{}e^{i\frac{% \cot(\alpha)}{2}y^{2}}\int_{-\infty}^{\infty}e^{-i\left(\csc(\alpha)~{}yx-% \frac{\cot(\alpha)}{2}x^{2}\right)}f(x)\,\mathrm{d}x~{}.
  52. α α
  53. π π
  54. δ ( x y ) δ(x−y)
  55. δ ( x + y ) δ(x+y)
  56. α α
  57. π π
  58. 2 \mathcal{F}^{2}
  59. f f
  60. f f
  61. x x
  62. α \mathcal{F}_{\alpha}
  63. f f
  64. f ( x ) f(x)
  65. f ( x ) f(−x)
  66. α α
  67. π π

Mehler–Fock_transform.html

  1. F ( x ) = 0 P i t - 1 / 2 ( x ) f ( t ) d t , F(x)=\int_{0}^{\infty}P_{it-1/2}(x)f(t)dt,

Mehler–Heine_formula.html

  1. lim n P n ( cos z n ) = J 0 ( z ) \lim_{n\to\infty}P_{n}\Bigl(\cos{z\over n}\Bigr)=J_{0}(z)
  2. lim n n - α P n α , β ( cos z n ) = ( z 2 ) - α J α ( z ) . \lim_{n\to\infty}n^{-\alpha}P_{n}^{\alpha,\beta}\left(\cos\frac{z}{n}\right)=% \left(\frac{z}{2}\right)^{-\alpha}J_{\alpha}(z)~{}.

Mehmet_Nadir.html

  1. x 2 + y 2 - z 2 = u 5 x^{2}+y^{2}-z^{2}=u^{5}
  2. x = b ( a 2 + b 2 ) ( a 2 - b 2 ) x=b\cdot(a^{2}+b^{2})\cdot(a^{2}-b^{2})
  3. y = 1 2 ( ( a 2 - 1 ) ( a 2 + b 2 ) 2 - 4 b 2 ) y=\frac{1}{2}\cdot((a^{2}-1)\cdot(a^{2}+b^{2})^{2}-4\cdot b^{2})
  4. z = 1 2 ( ( a 2 + 1 ) ( a 2 + b 2 ) 2 + 4 b 2 ) z=\frac{1}{2}\cdot((a^{2}+1)\cdot(a^{2}+b^{2})^{2}+4\cdot b^{2})
  5. u = a 2 + b 2 u=a^{2}+b^{2}

Meixner–Pollaczek_polynomials.html

  1. P n ( λ ) ( x ; ϕ ) = ( 2 λ ) n n ! e i n ϕ F 1 2 ( - n , λ + i x ; 2 λ ; 1 - e - 2 i ϕ ) P_{n}^{(\lambda)}(x;\phi)=\frac{(2\lambda)_{n}}{n!}e^{in\phi}{}_{2}F_{1}(-n,% \lambda+ix;2\lambda;1-e^{-2i\phi})
  2. P n λ ( cos ϕ ; a , b ) = ( 2 λ ) n n ! e i n ϕ F 1 2 ( - n , λ + i ( a cos ϕ + b ) / sin ϕ ; 2 λ ; 1 - e - 2 i ϕ ) P_{n}^{\lambda}(\cos\phi;a,b)=\frac{(2\lambda)_{n}}{n!}e^{in\phi}{}_{2}F_{1}(-% n,\lambda+i(a\cos\phi+b)/\sin\phi;2\lambda;1-e^{-2i\phi})
  3. w ( x ; λ , ϕ ) = | Γ ( λ + i x ) | 2 e ( 2 ϕ - π ) x w(x;\lambda,\phi)=|\Gamma(\lambda+ix)|^{2}e^{(2\phi-\pi)x}
  4. - P n ( λ ) ( x ; ϕ ) P m ( λ ) ( x ; ϕ ) w ( x ; λ , ϕ ) d x = 2 π Γ ( n + 2 λ ) ( 2 sin ϕ ) 2 λ n ! δ m n \int_{-\infty}^{\infty}P_{n}^{(\lambda)}(x;\phi)P_{m}^{(\lambda)}(x;\phi)w(x;% \lambda,\phi)dx=\frac{2\pi\Gamma(n+2\lambda)}{(2\sin\phi)^{2\lambda}n!}\delta_% {mn}

Membrane_technology.html

  1. d V p d t = Q = Δ p μ A ( 1 R m + R ) \frac{dV_{p}}{dt}=Q=\frac{\Delta p}{\mu}\ A\left(\frac{1}{R_{m}+R}\right)
  2. S = C p C f S=\frac{C_{p}}{C_{f}}
  3. L p = J Δ p L_{p}=\frac{J}{\Delta p}

Membraneless_Fuel_Cells.html

  1. . J = - D ϕ x . \bigg.J=-D\frac{\partial\phi}{\partial x}\bigg.
  2. J J
  3. ( mol m 2 s ) \left(\tfrac{\mathrm{mol}}{m^{2}\cdot s}\right)
  4. J J
  5. D \,D
  6. ( m 2 s ) \left(\tfrac{m^{2}}{s}\right)
  7. ϕ \,\phi
  8. ( mol m 3 ) \left(\tfrac{\mathrm{mol}}{m^{3}}\right)
  9. x \,x

MEMS_magnetic_actuator.html

  1. F m a g = q v × B \vec{F}_{mag}=q\vec{v}\times B
  2. H ( z ) = I r 2 2 ( r 2 + z 2 ) 3 / 2 H(z)=\frac{Ir^{2}}{2(r^{2}+z^{2})^{3/2}}
  3. F z = B I A m a g z z + h m a g d H z d z d z F_{z}=B_{I}A_{mag}\int_{z}^{z+h_{mag}}\frac{dHz}{dz}dz

MEMS_magnetic_field_sensor.html

  1. μ \mu

Mertens-stable_equilibrium.html

  1. G G
  2. Σ \Sigma
  3. 0 < δ 1 0<\delta\leq 1
  4. P δ = { ϵ τ 0 ϵ δ , τ Σ } P_{\delta}=\{\,\epsilon\tau\mid 0\leq\epsilon\leq\delta,\tau\in\Sigma\,\}
  5. P δ \partial P_{\delta}
  6. η P 1 \eta\in P_{1}
  7. η ¯ \bar{\eta}
  8. η P 1 \eta\in P_{1}
  9. G ( η ) G(\eta)
  10. n n
  11. G G
  12. τ \tau
  13. G G
  14. σ = ( 1 - η ¯ ) τ + η \sigma=(1-\bar{\eta})\tau+\eta
  15. σ \sigma
  16. G ( η ) G(\eta)
  17. τ \tau
  18. G ( η ) G(\eta)
  19. \mathcal{E}
  20. P 1 P_{1}
  21. \mathcal{E}
  22. ( η , σ ) P 1 × Σ (\eta,\sigma)\in P_{1}\times\Sigma
  23. σ \sigma
  24. G ( η ) G(\eta)
  25. ( η , σ ) (\eta,\sigma)\in\mathcal{E}
  26. τ ( η , σ ) ( σ - η ) / ( 1 - η ¯ ) \tau(\eta,\sigma)\equiv(\sigma-\eta)/(1-\bar{\eta})
  27. G ( η ) G(\eta)
  28. p p
  29. \mathcal{E}
  30. P 1 P_{1}
  31. E E\subseteq\mathcal{E}
  32. E 0 = { ( 0 , σ ) E } E_{0}=\{\,(0,\sigma)\in E\,\}
  33. 0 < δ 1 0<\delta\leq 1
  34. ( E δ , E δ ) = p - 1 ( P δ , P δ ) E (E_{\delta},\partial E_{\delta})=p^{-1}(P_{\delta},\partial P_{\delta})\cap E
  35. H ˇ \check{H}
  36. S Σ S\subseteq\Sigma
  37. E E
  38. \mathcal{E}
  39. E 0 = { 0 } × S E_{0}=\{\,0\,\}\times S
  40. V V
  41. E 0 E_{0}
  42. E E
  43. V E 1 V\setminus\partial E_{1}
  44. E 0 E_{0}
  45. E E
  46. p * : H ˇ * ( P δ , P δ ) H ˇ * ( E δ , E δ ) p^{*}:\check{H}^{*}(P_{\delta},\partial P_{\delta})\to\check{H}^{*}(E_{\delta}% ,\partial E_{\delta})
  47. δ > 0 \delta>0

Meta-regression.html

  1. y j = β 0 + β 1 x 1 j + β 2 x 2 j + + ε y_{j}=\beta_{0}+\beta_{1}x_{1j}+\beta_{2}x_{2j}+\cdots+\varepsilon
  2. y j y_{j}
  3. j j
  4. β 0 \beta_{0}
  5. x i . ( i = 1 k ) x_{i.}(i=1\ldots k)
  6. ε \varepsilon
  7. θ \theta
  8. 𝒩 ( θ , σ θ ) \mathcal{N}(\theta,\sigma_{\theta})
  9. σ θ 2 \sigma_{\theta}^{2}
  10. θ \theta
  11. ε = 0 \varepsilon=0
  12. y j = β 0 + β 1 x 1 j + β 2 x 2 j + + η j y_{j}=\beta_{0}+\beta_{1}x_{1j}+\beta_{2}x_{2j}+\cdots+\eta_{j}\,
  13. σ η j 2 \sigma^{2}_{\eta_{j}}
  14. j j
  15. θ \theta
  16. 𝒩 ( θ , σ i ) \mathcal{N}(\theta,\sigma_{i})
  17. 𝒩 ( θ , σ θ ) . \mathcal{N}(\theta,\sigma_{\theta}).
  18. y j = β 0 + β 1 x 1 j + β 2 x 2 j + + η + ε j y_{j}=\beta_{0}+\beta_{1}x_{1j}+\beta_{2}x_{2j}+\cdots+\eta+\varepsilon_{j}\,
  19. σ ε j 2 \sigma^{2}_{\varepsilon_{j}}
  20. j j
  21. σ η 2 \sigma^{2}_{\eta}

Metal_ions_in_aqueous_solution.html

  1. { [ M ( O H ) ] ( z - 1 ) + } = K 1 , - 1 { M z + } { H + } - 1 \{[M(OH)]^{(z-1)+}\}=K_{1,-1}\{M^{z+}\}\{H^{+}\}^{-1}
  2. { [ M ( O H ) ] ( z - 1 ) + } = K 1 , 1 { M z + } { O H - } \{[M(OH)]^{(z-1)+}\}=K_{1,1}\{M^{z+}\}\{OH^{-}\}
  3. K 1 , - 1 = K 1 , 1 × K w K_{1,-1}=K_{1,1}\times K_{w}
  4. [ M x ( O H ) y ] = β x , - y * [ M ] x [ H ] - y [M_{x}(OH)_{y}]=\beta_{x,-y}*[M]^{x}[H]^{-y}
  5. β * \beta*
  6. rate = - ( d [ A ] d t ) T = k [ A ] \mathrm{rate}=-\left(\frac{d[A]}{dt}\right)_{T}=k[A]
  7. ( l n k P ) T = Δ V R T \left(\frac{\partial lnk}{\partial P}\right)_{T}=\frac{\Delta V^{\ddagger}}{RT}

Metal_oxide_adhesion.html

  1. W a d = γ m + γ o - γ m o W_{ad}=\gamma_{m}+\gamma_{o}-\gamma_{mo}
  2. J / m - 2 J/{m^{-2}}
  3. γ = E - T S \gamma=E-TS
  4. γ \scriptstyle\gamma
  5. γ \scriptstyle\gamma
  6. γ \scriptstyle\gamma
  7. D = N s N t D={N_{s}\over N_{t}}
  8. J = - D δ C δ x J=-D{\delta C\over\delta x}

Metaplectic_structure.html

  1. ( M , ω ) (M,\omega)
  2. π 𝐑 : 𝐑 M \pi_{\mathbf{R}}\colon{\mathbf{R}}\to M\,
  3. ρ : Mp ( n , ) Sp ( n , ) . \rho\colon{\mathrm{Mp}}(n,{\mathbb{R}})\to{\mathrm{Sp}}(n,{\mathbb{R}}).\,
  4. ( 𝐏 , F 𝐏 ) ({\mathbf{P}},F_{\mathbf{P}})
  5. π 𝐑 : 𝐑 M \pi_{\mathbf{R}}\colon{\mathbf{R}}\to M\,
  6. π 𝐏 : 𝐏 M \pi_{\mathbf{P}}\colon{\mathbf{P}}\to M\,
  7. Mp ( n , ) {\mathrm{Mp}}(n,{\mathbb{R}})
  8. M M
  9. F 𝐏 : 𝐏 𝐑 F_{\mathbf{P}}\colon{\mathbf{P}}\to{\mathbf{R}}\,
  10. 2 2
  11. π 𝐑 F 𝐏 = π 𝐏 \pi_{\mathbf{R}}\circ F_{\mathbf{P}}=\pi_{\mathbf{P}}
  12. F 𝐏 ( 𝐩 q ) = F 𝐏 ( 𝐩 ) ρ ( q ) F_{\mathbf{P}}({\mathbf{p}}q)=F_{\mathbf{P}}({\mathbf{p}})\rho(q)
  13. 𝐩 𝐏 {\mathbf{p}}\in{\mathbf{P}}
  14. q Mp ( n , ) . q\in{\mathrm{Mp}}(n,{\mathbb{R}}).
  15. π 𝐏 : 𝐏 M \pi_{\mathbf{P}}\colon{\mathbf{P}}\to M\,
  16. M M
  17. ( 𝐏 1 , F 𝐏 1 ) ({\mathbf{P}_{1}},F_{\mathbf{P}_{1}})
  18. ( 𝐏 2 , F 𝐏 2 ) ({\mathbf{P}_{2}},F_{\mathbf{P}_{2}})
  19. ( M , ω ) (M,\omega)
  20. Mp ( n , ) {\mathrm{Mp}}(n,{\mathbb{R}})
  21. f : 𝐏 1 𝐏 2 f\colon{\mathbf{P}_{1}}\to{\mathbf{P}_{2}}
  22. F 𝐏 2 f = F 𝐏 1 F_{\mathbf{P}_{2}}\circ f=F_{\mathbf{P}_{1}}
  23. f ( 𝐩 q ) = f ( 𝐩 ) q f({\mathbf{p}}q)=f({\mathbf{p}})q
  24. 𝐩 𝐏 1 {\mathbf{p}}\in{\mathbf{P}_{1}}
  25. q Mp ( n , ) . q\in{\mathrm{Mp}}(n,{\mathbb{R}}).
  26. F 𝐏 1 F_{\mathbf{P}_{1}}
  27. F 𝐏 2 F_{\mathbf{P}_{2}}
  28. Sp ( n , ) {\mathrm{Sp}}(n,{\mathbb{R}})
  29. π 𝐑 : 𝐑 M \pi_{\mathbf{R}}\colon{\mathbf{R}}\to M\,
  30. ( M , ω ) (M,\omega)
  31. M M
  32. ( M , ω ) (M,\omega)
  33. w 2 ( M ) H 2 ( M , 2 ) w_{2}(M)\in H^{2}(M,{\mathbb{Z}_{2}})
  34. M M
  35. 2 {}_{2}
  36. c 1 ( M ) H 2 ( M , ) c_{1}(M)\in H^{2}(M,{\mathbb{Z}})
  37. w 2 ( M ) w_{2}(M)
  38. ( M , ω ) (M,\omega)
  39. c 1 ( M ) c_{1}(M)
  40. w 2 ( M ) w_{2}(M)
  41. ( M , ω ) (M,\omega)
  42. H 1 ( M , 2 ) H^{1}(M,{\mathbb{Z}_{2}})
  43. M M
  44. 2 {\mathbb{Z}_{2}}
  45. M M
  46. w 1 ( M ) H 1 ( M , 2 ) w_{1}(M)\in H^{1}(M,{\mathbb{Z}_{2}})
  47. M M
  48. ( T N , θ ) , (T^{\ast}N,\theta)\,,
  49. N N
  50. 2 k + 1 , {\mathbb{P}}^{2k+1}{\mathbb{C}}\,,
  51. k 0 . \,k\in{\mathbb{N}}_{0}\,.
  52. 2 k + 1 {\mathbb{P}}^{2k+1}{\mathbb{C}}\,
  53. G r ( 2 , 4 ) , Gr(2,4)\,,

Method_of_mean_weighted_residuals.html

  1. ϕ i \phi_{i}
  2. ϕ i \phi_{i}
  3. u ( x ) u(x)
  4. R ( x , u , u x , , d n u d x n ) R\left(x,u,u_{x},\ldots,\frac{d^{n}u}{dx^{n}}\right)
  5. w i w_{i}
  6. a i a_{i}
  7. R ( x , u , u x , , d n u d x n ) = 0 R\left(x,u,u_{x},\ldots,\frac{d^{n}u}{dx^{n}}\right)=0
  8. ϕ i \phi_{i}
  9. R ( x , u , u x , , d n u d x n ) = 0 R\left(x,u,u_{x},\ldots,\frac{d^{n}u}{dx^{n}}\right)=0
  10. a i a_{i}
  11. ( R ( x , u , u x , , d n u d x n ) , w i ) = 0 \left(R\left(x,u,u_{x},\ldots,\frac{d^{n}u}{dx^{n}}\right),w_{i}\right)=0
  12. ( f , g ) (f,g)
  13. r ( x ) r(x)
  14. r ( x ) = 1 1 - x 2 r(x)=\frac{1}{\sqrt{1-x^{2}}}
  15. u ( x ) = v ( x ) + L ( x ) u(x)=v(x)+L(x)
  16. w i = u a i w_{i}=\frac{\partial u}{\partial a_{i}}
  17. x i x_{i}
  18. w i = R a i w_{i}=\frac{\partial R}{\partial a_{i}}
  19. R 2 \|{R}\|^{2}
  20. a i a_{i}
  21. x i x^{i}

Methodology_of_econometrics.html

  1. ( v - b ) Pr ( b wins ) (v-b)\Pr(b\ \textrm{wins})
  2. b * b^{*}
  3. ( v - b * ) Pr ( b * wins ) b - Pr ( b * wins ) = 0 (v-b^{*})\frac{\partial\Pr(b^{*}\ \textrm{wins})}{\partial b}-\Pr(b^{*}\ % \textrm{wins})=0
  4. v v
  5. v = b * + Pr ( b * wins ) Pr ( b * wins ) / b v=b^{*}+\frac{\Pr(b^{*}\ \textrm{wins})}{\partial\Pr(b^{*}\ \textrm{wins})/% \partial b}

Metric_differential.html

  1. | f ( x ) - f ( y ) | = 0 1 | χ [ 0 , x ] ( t ) - χ [ 0 , y ] ( t ) | d t = x y d t = | x - y | , |f(x)-f(y)|=\int_{0}^{1}|\chi_{[0,x]}(t)-\chi_{[0,y]}(t)|\,dt=\int_{x}^{y}\,dt% =|x-y|,
  2. M D ( f , z ) ( x ) = lim r 0 d X ( f ( z + r x ) , f ( z ) ) r MD(f,z)(x)=\lim_{r\rightarrow 0}\frac{d_{X}(f(z+rx),f(z))}{r}
  3. d X ( f ( x ) , f ( y ) ) - M D ( f , z ) ( x - y ) = o ( | x - z | + | y - z | ) . d_{X}(f(x),f(y))-MD(f,z)(x-y)=o(|x-z|+|y-z|).\,

Meyerhoff_manifold.html

  1. 12 ( 283 ) 3 / 2 ζ k ( 2 ) ( 2 π ) - 6 = 0.9812 12\cdot(283)^{3/2}\zeta_{k}(2)(2\pi)^{-6}=0.9812\ldots

Michelangelo_Ricci.html

  1. x m ( a - x ) n x^{m}(a-x)^{n}
  2. y m = k x n y^{m}=kx^{n}

Micro-mechanics_of_failure.html

  1. σ \sigma
  2. t t
  3. σ ¯ \bar{\sigma}
  4. Δ T \Delta T
  5. σ f = M f σ ¯ + A f Δ T σ m = M m σ ¯ + A m Δ T t i = M i σ ¯ + A i Δ T \begin{array}[]{lcl}\sigma_{\mathrm{f}}&=&M_{\mathrm{f}}\bar{\sigma}+A_{% \mathrm{f}}\Delta T\\ \sigma_{\mathrm{m}}&=&M_{\mathrm{m}}\bar{\sigma}+A_{\mathrm{m}}\Delta T\\ t_{\mathrm{i}}&=&M_{\mathrm{i}}\bar{\sigma}+A_{\mathrm{i}}\Delta T\end{array}
  6. σ \sigma
  7. σ ¯ \bar{\sigma}
  8. t t
  9. f {\mathrm{f}}
  10. m {\mathrm{m}}
  11. i {\mathrm{i}}
  12. M M
  13. A A
  14. M M
  15. A A
  16. M M
  17. A A
  18. Maximum stress failure criterion: - X f < σ 1 < X f Quadratic failure criterion: j = 1 6 i = 1 6 F i j σ i σ j + i = 1 6 F i σ i = 1 \begin{array}[]{lcl}\,\text{Maximum stress failure criterion:}-X^{\prime}_{% \mathrm{f}}<\sigma_{1}<X_{\mathrm{f}}\\ \,\text{Quadratic failure criterion: }\displaystyle\sum_{j=1}^{6}\displaystyle% \sum_{i=1}^{6}F_{ij}\sigma_{i}\sigma_{j}+\displaystyle\sum_{i=1}^{6}F_{i}% \sigma_{i}=1\end{array}
  19. F 11 = 1 X f X f , F 22 = F 33 = 1 Y f Y f F_{11}=\cfrac{1}{X_{\mathrm{f}}X^{\prime}_{\mathrm{f}}}\ ,\ F_{22}=F_{33}=% \cfrac{1}{Y_{\mathrm{f}}Y^{\prime}_{\mathrm{f}}}
  20. F 44 = 1 S f4 2 , F 55 = F 66 = 1 S f6 2 F_{44}=\cfrac{1}{S_{\mathrm{f}4}^{2}}\ ,\ F_{55}=F_{66}=\cfrac{1}{S_{\mathrm{f% }6}^{2}}
  21. F 1 = 1 X f - 1 X f , F 2 = F 3 = 1 Y f - 1 Y f F_{1}=\cfrac{1}{X_{\mathrm{f}}}-\cfrac{1}{X_{\mathrm{f}}^{\prime}}\ ,\ F_{2}=F% _{3}=\cfrac{1}{Y_{\mathrm{f}}}-\cfrac{1}{Y_{\mathrm{f}}^{\prime}}
  22. F 12 = F 21 = F 13 = F 31 = - 1 2 X f X f Y f Y f , F 23 = F 32 = - 1 2 Y f Y f F_{12}=F_{21}=F_{13}=F_{31}=-\cfrac{1}{2\sqrt{X_{\mathrm{f}}{X}_{\mathrm{f}}^{% \prime}Y_{\mathrm{f}}Y_{\mathrm{f}}^{\prime}}}\ ,\ F_{23}=F_{32}=-\cfrac{1}{2Y% _{\mathrm{f}}Y_{\mathrm{f}}^{\prime}}
  23. X f X_{\mathrm{f}}
  24. X f X_{\mathrm{f}}^{\prime}
  25. Y f Y_{\mathrm{f}}
  26. Y f Y_{\mathrm{f}}^{\prime}
  27. S f4 S_{\mathrm{f}4}
  28. S f6 S_{\mathrm{f}6}
  29. σ M i s e s 2 C m T m + ( 1 T m - 1 C m ) I 1 = 1 \begin{array}[]{lcl}\cfrac{\sigma_{Mises}^{2}}{C_{\mathrm{m}}T_{\mathrm{m}}}+% \left(\cfrac{1}{T_{\mathrm{m}}}-\cfrac{1}{C_{\mathrm{m}}}\right)I_{1}=1\end{array}
  30. T m {T}_{\mathrm{m}}
  31. C m {C}_{\mathrm{m}}
  32. σ M i s e s \sigma_{Mises}
  33. I 1 {\mathrm{I}}_{1}
  34. ( t n Y n ) 2 + ( t s Y s ) 2 = 1 \begin{array}[]{lcl}\left(\cfrac{\left\langle{t}_{n}\right\rangle}{{Y}_{n}}% \right)^{2}+\left(\cfrac{{t}_{s}}{{Y}_{s}}\right)^{2}=1\end{array}
  35. t n {t}_{n}
  36. t s {t}_{s}
  37. Y n {Y}_{n}
  38. Y s {Y}_{s}

Microbial_cooperation.html

  1. μ m \mu m

Microcontinuity.html

  1. x a x\approx a
  2. f ( x ) f ( a ) f(x)\approx f(a)
  3. \mathbb{R}
  4. c c\in\mathbb{R}
  5. f ( h a l ( c ) ) h a l ( f ( c ) ) f(hal(c))\subset hal(f(c))
  6. st f \,\text{st}\circ f
  7. f ( x ) = 1 x f(x)=\frac{1}{x}
  8. a > 0 a>0
  9. 1 a \frac{1}{a}
  10. 1 2 a \frac{1}{2a}
  11. f ( x ) = x 2 f(x)=x^{2}
  12. \mathbb{R}
  13. H * H\in\mathbb{R}^{*}
  14. e = 1 H e=\frac{1}{H}
  15. f n f_{n}
  16. f n * ( x ) f_{n}^{*}(x)
  17. f * ( x ) f^{*}(x)

Miller_theorem.html

  1. V 2 = K V 1 V_{2}=K{V_{1}}
  2. I i n 0 = V 1 Z I_{in0}=\frac{V_{1}}{Z}
  3. Z i n 0 = V 1 I i n 0 = Z . Z_{in0}=\frac{V_{1}}{I_{in0}}=Z.
  4. I i n = V 1 - V 2 Z = ( 1 - K ) Z V 1 = ( 1 - K ) I i n 0 I_{in}=\frac{V_{1}-V_{2}}{Z}=\frac{(1-K)}{Z}{V_{1}}={(1-K)}{I_{in0}}
  5. Z i n = V 1 I i n = Z 1 - K . Z_{in}=\frac{V_{1}}{I_{in}}=\frac{Z}{1-K}.
  6. V 2 = K V 1 V_{2}=K{V_{1}}
  7. Z i n 2 = K Z K - 1 . Z_{in2}=\frac{{K}{Z}}{K-1}.
  8. A V = K A_{V}=K
  9. Z i n t Z_{int}
  10. Z i n t Z_{int}
  11. Z i n t Z_{int}
  12. I 2 = K I 1 I_{2}=K{I_{1}}

Miller_twist_rule.html

  1. t 2 = 30 m s d 3 l ( 1 + l 2 ) {t}^{2}=\frac{30m}{sd^{3}l(1+l^{2})}
  2. t = T d {t}=\frac{T}{d}
  3. T T
  4. l = L d {l}=\frac{L}{d}
  5. L L
  6. s s
  7. s = 30 m t 2 d 3 l ( 1 + l 2 ) {s}=\frac{30m}{t^{2}d^{3}l(1+l^{2})}
  8. T T
  9. T = 30 m s d l ( 1 + l 2 ) {T}=\sqrt{\frac{30m}{sdl(1+l^{2})}}
  10. m m
  11. l ( 1 + l 2 ) l(1+l^{2})
  12. M M
  13. s s
  14. t = 30 m s d 3 l ( 1 + l 2 ) t=\sqrt{\frac{30m}{sd^{3}l(1+l^{2})}}
  15. t = 30 * 180 2.0 * .308 3 * 3.83 ( 1 + 3.83 2 ) = 39.2511937 t=\sqrt{\frac{30*180}{2.0*.308^{3}*3.83(1+3.83^{2})}}=39.2511937
  16. T T
  17. t t
  18. T = 39.2511937 * .308 = 12.0893677 T=39.2511937*.308=12.0893677
  19. T w i s t = C D 2 L × S G 10.9 Twist=\frac{CD^{2}}{L}\times\sqrt{\frac{SG}{10.9}}
  20. S = s 2 * m 2 C M α ÷ sin ( a ) * t * d * v 2 S=\frac{s^{2}*m^{2}}{C_{M}\alpha\div\sin(a)*t*d*v^{2}}
  21. v v
  22. T T
  23. f v = 1 / 2 [ v 2800 ] 1 / 6 f_{v}{{}^{1/2}}=[\frac{v}{2800}]^{1/6}
  24. v v
  25. s s
  26. f v = [ v 2800 ] 1 / 3 f_{v}=[\frac{v}{2800}]^{1/3}
  27. a a
  28. f a = e 3.158 x 10 - 5 * h f_{a}=e^{3.158x10^{-5}*h}
  29. h h

Milne-Thomson_circle_theorem.html

  1. w = f ( z ) w=f(z)
  2. | z | = a |z|=a
  3. | z | = a |z|=a
  4. w = f ( z ) + f ( a 2 z ¯ ) ¯ = f ( z ) + f ¯ ( a 2 < m t p l > z ) w=f(z)+\overline{f\left(\frac{a^{2}}{\bar{z}}\right)}=f(z)+\bar{f}\left(\frac{% a^{2}}{<}mtpl>{{z}}\right)

Min-plus_matrix_multiplication.html

  1. n × n n\times n
  2. A = ( a i j ) A=(a_{ij})
  3. B = ( b i j ) B=(b_{ij})
  4. C = ( c i j ) = A B C=(c_{ij})=A\star B
  5. n × n n\times n
  6. c i j = min k = 1 n { a i k + b k j } c_{ij}=\min_{k=1}^{n}\{a_{ik}+b_{kj}\}
  7. W W
  8. n × n n\times n
  9. W k W^{k}
  10. k k
  11. W n W^{n}

Mindlin–Reissner_plate_theory.html

  1. u α ( 𝐱 ) = u α 0 ( x 1 , x 2 ) - x 3 φ α ; α = 1 , 2 u 3 ( 𝐱 ) = w 0 ( x 1 , x 2 ) \begin{aligned}\displaystyle u_{\alpha}(\mathbf{x})&\displaystyle=u^{0}_{% \alpha}(x_{1},x_{2})-x_{3}~{}\varphi_{\alpha}~{};~{}~{}\alpha=1,2\\ \displaystyle u_{3}(\mathbf{x})&\displaystyle=w^{0}(x_{1},x_{2})\end{aligned}
  2. x 1 x_{1}
  3. x 2 x_{2}
  4. x 3 x_{3}
  5. u α 0 , α = 1 , 2 u^{0}_{\alpha},~{}\alpha=1,2
  6. w 0 w^{0}
  7. x 3 x_{3}
  8. φ 1 \varphi_{1}
  9. φ 2 \varphi_{2}
  10. x 3 x_{3}
  11. φ α \varphi_{\alpha}
  12. w 0 w^{0}
  13. φ 1 w , 1 0 \varphi_{1}\neq w^{0}_{,1}
  14. φ 2 w , 2 0 \varphi_{2}\neq w^{0}_{,2}
  15. ε α β = 1 2 ( u α , β 0 + u β , α 0 ) - x 3 2 ( φ α , β + φ β , α ) ε α 3 = 1 2 ( w , α 0 - φ α ) ε 33 = 0 \begin{aligned}\displaystyle\varepsilon_{\alpha\beta}&\displaystyle=\frac{1}{2% }(u^{0}_{\alpha,\beta}+u^{0}_{\beta,\alpha})-\frac{x_{3}}{2}~{}(\varphi_{% \alpha,\beta}+\varphi_{\beta,\alpha})\\ \displaystyle\varepsilon_{\alpha 3}&\displaystyle=\cfrac{1}{2}\left(w^{0}_{,% \alpha}-\varphi_{\alpha}\right)\\ \displaystyle\varepsilon_{33}&\displaystyle=0\end{aligned}
  16. κ \kappa
  17. ε α 3 = 1 2 κ ( w , α 0 - φ α ) \varepsilon_{\alpha 3}=\cfrac{1}{2}~{}\kappa~{}\left(w^{0}_{,\alpha}-\varphi_{% \alpha}\right)
  18. N α β , α = 0 M α β , β - Q α = 0 Q α , α + q = 0 \begin{aligned}&\displaystyle N_{\alpha\beta,\alpha}=0\\ &\displaystyle M_{\alpha\beta,\beta}-Q_{\alpha}=0\\ &\displaystyle Q_{\alpha,\alpha}+q=0\end{aligned}
  19. q q
  20. N α β := - h h σ α β d x 3 , N_{\alpha\beta}:=\int_{-h}^{h}\sigma_{\alpha\beta}~{}dx_{3}\,,
  21. M α β := - h h x 3 σ α β d x 3 , M_{\alpha\beta}:=\int_{-h}^{h}x_{3}~{}\sigma_{\alpha\beta}~{}dx_{3}\,,
  22. Q α := κ - h h σ α 3 d x 3 . Q_{\alpha}:=\kappa~{}\int_{-h}^{h}\sigma_{\alpha 3}~{}dx_{3}\,.
  23. δ U = Ω 0 - h h s y m b o l σ : \deltasymbol ϵ d x 3 d Ω = Ω 0 - h h [ σ α β δ ε α β + 2 σ α 3 δ ε α 3 ] d x 3 d Ω = Ω 0 - h h [ 1 2 σ α β ( δ u α , β 0 + δ u β , α 0 ) - x 3 2 σ α β ( δ φ α , β + δ φ β , α ) + κ σ α 3 ( δ w , α 0 - δ φ α ) ] d x 3 d Ω = Ω 0 [ 1 2 N α β ( δ u α , β 0 + δ u β , α 0 ) - 1 2 M α β ( δ φ α , β + δ φ β , α ) + Q α ( δ w , α 0 - δ φ α ) ] d Ω \begin{aligned}\displaystyle\delta U&\displaystyle=\int_{\Omega^{0}}\int_{-h}^% {h}symbol{\sigma}:\deltasymbol{\epsilon}~{}dx_{3}~{}d\Omega=\int_{\Omega^{0}}% \int_{-h}^{h}\left[\sigma_{\alpha\beta}~{}\delta\varepsilon_{\alpha\beta}+2~{}% \sigma_{\alpha 3}~{}\delta\varepsilon_{\alpha 3}\right]~{}dx_{3}~{}d\Omega\\ &\displaystyle=\int_{\Omega^{0}}\int_{-h}^{h}\left[\frac{1}{2}~{}\sigma_{% \alpha\beta}~{}(\delta u^{0}_{\alpha,\beta}+\delta u^{0}_{\beta,\alpha})-\frac% {x_{3}}{2}~{}\sigma_{\alpha\beta}~{}(\delta\varphi_{\alpha,\beta}+\delta% \varphi_{\beta,\alpha})+\kappa~{}\sigma_{\alpha 3}\left(\delta w^{0}_{,\alpha}% -\delta\varphi_{\alpha}\right)\right]~{}dx_{3}~{}d\Omega\\ &\displaystyle=\int_{\Omega^{0}}\left[\frac{1}{2}~{}N_{\alpha\beta}~{}(\delta u% ^{0}_{\alpha,\beta}+\delta u^{0}_{\beta,\alpha})-\frac{1}{2}M_{\alpha\beta}~{}% (\delta\varphi_{\alpha,\beta}+\delta\varphi_{\beta,\alpha})+Q_{\alpha}\left(% \delta w^{0}_{,\alpha}-\delta\varphi_{\alpha}\right)\right]~{}d\Omega\end{aligned}
  24. Q α := κ - h h σ α 3 d x 3 Q_{\alpha}:=\kappa~{}\int_{-h}^{h}\sigma_{\alpha 3}~{}dx_{3}
  25. δ U = Ω 0 [ - 1 2 ( N α β , β δ u α 0 + N α β , α δ u β 0 ) + 1 2 ( M α β , β δ φ α + M α β , α δ φ β ) - Q α , α δ w 0 - Q α δ φ α ] d Ω + Γ 0 [ 1 2 ( n β N α β δ u α 0 + n α N α β δ u β 0 ) - 1 2 ( n β M α β δ φ α + n α M α β δ φ β ) + n α Q α δ w 0 ] d Γ \begin{aligned}\displaystyle\delta U&\displaystyle=\int_{\Omega^{0}}\left[-% \frac{1}{2}~{}(N_{\alpha\beta,\beta}~{}\delta u^{0}_{\alpha}+N_{\alpha\beta,% \alpha}~{}\delta u^{0}_{\beta})+\frac{1}{2}(M_{\alpha\beta,\beta}~{}\delta% \varphi_{\alpha}+M_{\alpha\beta,\alpha}\delta\varphi_{\beta})-Q_{\alpha,\alpha% }~{}\delta w^{0}-Q_{\alpha}~{}\delta\varphi_{\alpha}\right]~{}d\Omega\\ &\displaystyle+\int_{\Gamma^{0}}\left[\frac{1}{2}~{}(n_{\beta}~{}N_{\alpha% \beta}~{}\delta u^{0}_{\alpha}+n_{\alpha}~{}N_{\alpha\beta}~{}\delta u^{0}_{% \beta})-\frac{1}{2}(n_{\beta}~{}M_{\alpha\beta}~{}\delta\varphi_{\alpha}+n_{% \alpha}M_{\alpha\beta}\delta\varphi_{\beta})+n_{\alpha}~{}Q_{\alpha}~{}\delta w% ^{0}\right]~{}d\Gamma\end{aligned}
  26. N α β = N β α N_{\alpha\beta}=N_{\beta\alpha}
  27. M α β = M β α M_{\alpha\beta}=M_{\beta\alpha}
  28. δ U = Ω 0 [ - N α β , α δ u β 0 + ( M α β , β - Q α ) δ φ α - Q α , α δ w 0 ] d Ω + Γ 0 [ n α N α β δ u β 0 - n β M α β δ φ α + n α Q α δ w 0 ] d Γ \begin{aligned}\displaystyle\delta U&\displaystyle=\int_{\Omega^{0}}\left[-N_{% \alpha\beta,\alpha}~{}\delta u^{0}_{\beta}+\left(M_{\alpha\beta,\beta}-Q_{% \alpha}\right)~{}\delta\varphi_{\alpha}-Q_{\alpha,\alpha}~{}\delta w^{0}\right% ]~{}d\Omega\\ &\displaystyle+\int_{\Gamma^{0}}\left[n_{\alpha}~{}N_{\alpha\beta}~{}\delta u^% {0}_{\beta}-n_{\beta}~{}M_{\alpha\beta}~{}\delta\varphi_{\alpha}+n_{\alpha}~{}% Q_{\alpha}~{}\delta w^{0}\right]~{}d\Gamma\end{aligned}
  29. q ( 𝐱 0 ) q(\mathbf{x}^{0})
  30. δ V ext = Ω 0 q δ w 0 d Ω \delta V_{\mathrm{ext}}=\int_{\Omega^{0}}q~{}\delta w^{0}~{}\mathrm{d}\Omega
  31. Ω 0 [ N α β , α δ u β 0 - ( M α β , β - Q α ) δ φ α + ( Q α , α + q ) δ w 0 ] d Ω = Γ 0 [ n α N α β δ u β 0 - n β M α β δ φ α + n α Q α δ w 0 ] d Γ \begin{aligned}&\displaystyle\int_{\Omega^{0}}\left[N_{\alpha\beta,\alpha}~{}% \delta u^{0}_{\beta}-\left(M_{\alpha\beta,\beta}-Q_{\alpha}\right)~{}\delta% \varphi_{\alpha}+\left(Q_{\alpha,\alpha}+q\right)~{}\delta w^{0}\right]~{}d% \Omega\\ &\displaystyle\qquad\qquad=\int_{\Gamma^{0}}\left[n_{\alpha}~{}N_{\alpha\beta}% ~{}\delta u^{0}_{\beta}-n_{\beta}~{}M_{\alpha\beta}~{}\delta\varphi_{\alpha}+n% _{\alpha}~{}Q_{\alpha}~{}\delta w^{0}\right]~{}d\Gamma\end{aligned}
  32. N α β , α = 0 M α β , β - Q α = 0 Q α , α + q = 0 \begin{aligned}&\displaystyle N_{\alpha\beta,\alpha}=0\\ &\displaystyle M_{\alpha\beta,\beta}-Q_{\alpha}=0\\ &\displaystyle Q_{\alpha,\alpha}+q=0\end{aligned}
  33. n α N α β or u β 0 n α M α β or φ α n α Q α or w 0 \begin{aligned}\displaystyle n_{\alpha}~{}N_{\alpha\beta}&\displaystyle\quad% \mathrm{or}\quad u^{0}_{\beta}\\ \displaystyle n_{\alpha}~{}M_{\alpha\beta}&\displaystyle\quad\mathrm{or}\quad% \varphi_{\alpha}\\ \displaystyle n_{\alpha}~{}Q_{\alpha}&\displaystyle\quad\mathrm{or}\quad w^{0}% \end{aligned}
  34. σ α β = C α β γ θ ε γ θ σ α 3 = C α 3 γ θ ε γ θ σ 33 = C 33 γ θ ε γ θ \begin{aligned}\displaystyle\sigma_{\alpha\beta}&\displaystyle=C_{\alpha\beta% \gamma\theta}~{}\varepsilon_{\gamma\theta}\\ \displaystyle\sigma_{\alpha 3}&\displaystyle=C_{\alpha 3\gamma\theta}~{}% \varepsilon_{\gamma\theta}\\ \displaystyle\sigma_{33}&\displaystyle=C_{33\gamma\theta}~{}\varepsilon_{% \gamma\theta}\end{aligned}
  35. σ 33 \sigma_{33}
  36. [ σ 11 σ 22 σ 23 σ 31 σ 12 ] = [ C 11 C 12 0 0 0 C 12 C 22 0 0 0 0 0 C 44 0 0 0 0 0 C 55 0 0 0 0 0 C 66 ] [ ε 11 ε 22 ε 23 ε 31 ε 12 ] \begin{bmatrix}\sigma_{11}\\ \sigma_{22}\\ \sigma_{23}\\ \sigma_{31}\\ \sigma_{12}\end{bmatrix}=\begin{bmatrix}C_{11}&C_{12}&0&0&0\\ C_{12}&C_{22}&0&0&0\\ 0&0&C_{44}&0&0\\ 0&0&0&C_{55}&0\\ 0&0&0&0&C_{66}\end{bmatrix}\begin{bmatrix}\varepsilon_{11}\\ \varepsilon_{22}\\ \varepsilon_{23}\\ \varepsilon_{31}\\ \varepsilon_{12}\end{bmatrix}
  37. [ N 11 N 22 N 12 ] = - h h [ C 11 C 12 0 C 12 C 22 0 0 0 C 66 ] [ ε 11 ε 22 ε 12 ] d x 3 = { - h h [ C 11 C 12 0 C 12 C 22 0 0 0 C 66 ] d x 3 } [ u 1 , 1 0 u 2 , 2 0 1 2 ( u 1 , 2 0 + u 2 , 1 0 ) ] \begin{bmatrix}N_{11}\\ N_{22}\\ N_{12}\end{bmatrix}=\int_{-h}^{h}\begin{bmatrix}C_{11}&C_{12}&0\\ C_{12}&C_{22}&0\\ 0&0&C_{66}\end{bmatrix}\begin{bmatrix}\varepsilon_{11}\\ \varepsilon_{22}\\ \varepsilon_{12}\end{bmatrix}dx_{3}=\left\{\int_{-h}^{h}\begin{bmatrix}C_{11}&% C_{12}&0\\ C_{12}&C_{22}&0\\ 0&0&C_{66}\end{bmatrix}~{}dx_{3}\right\}\begin{bmatrix}u^{0}_{1,1}\\ u^{0}_{2,2}\\ \frac{1}{2}~{}(u^{0}_{1,2}+u^{0}_{2,1})\end{bmatrix}
  38. [ M 11 M 22 M 12 ] = - h h x 3 [ C 11 C 12 0 C 12 C 22 0 0 0 C 66 ] [ ε 11 ε 22 ε 12 ] d x 3 = - { - h h x 3 2 [ C 11 C 12 0 C 12 C 22 0 0 0 C 66 ] d x 3 } [ φ 1 , 1 φ 2 , 2 1 2 ( φ 1 , 2 + φ 2 , 1 ) ] \begin{bmatrix}M_{11}\\ M_{22}\\ M_{12}\end{bmatrix}=\int_{-h}^{h}x_{3}~{}\begin{bmatrix}C_{11}&C_{12}&0\\ C_{12}&C_{22}&0\\ 0&0&C_{66}\end{bmatrix}\begin{bmatrix}\varepsilon_{11}\\ \varepsilon_{22}\\ \varepsilon_{12}\end{bmatrix}dx_{3}=-\left\{\int_{-h}^{h}x_{3}^{2}~{}\begin{% bmatrix}C_{11}&C_{12}&0\\ C_{12}&C_{22}&0\\ 0&0&C_{66}\end{bmatrix}~{}dx_{3}\right\}\begin{bmatrix}\varphi_{1,1}\\ \varphi_{2,2}\\ \frac{1}{2}(\varphi_{1,2}+\varphi_{2,1})\end{bmatrix}
  39. [ Q 1 Q 2 ] = κ - h h [ C 55 0 0 C 44 ] [ ε 31 ε 32 ] d x 3 = κ 2 { - h h [ C 55 0 0 C 44 ] d x 3 } [ w , 1 0 - φ 1 w , 2 0 - φ 2 ] \begin{bmatrix}Q_{1}\\ Q_{2}\end{bmatrix}=\kappa~{}\int_{-h}^{h}\begin{bmatrix}C_{55}&0\\ 0&C_{44}\end{bmatrix}\begin{bmatrix}\varepsilon_{31}\\ \varepsilon_{32}\end{bmatrix}dx_{3}=\cfrac{\kappa}{2}\left\{\int_{-h}^{h}% \begin{bmatrix}C_{55}&0\\ 0&C_{44}\end{bmatrix}~{}dx_{3}\right\}\begin{bmatrix}w^{0}_{,1}-\varphi_{1}\\ w^{0}_{,2}-\varphi_{2}\end{bmatrix}
  40. A α β := - h h C α β d x 3 A_{\alpha\beta}:=\int_{-h}^{h}C_{\alpha\beta}~{}dx_{3}
  41. D α β := - h h x 3 2 C α β d x 3 . D_{\alpha\beta}:=\int_{-h}^{h}x_{3}^{2}~{}C_{\alpha\beta}~{}dx_{3}\,.
  42. [ σ 11 σ 22 σ 12 ] = E 1 - ν 2 [ 1 ν 0 ν 1 0 0 0 1 - ν ] [ ε 11 ε 22 ε 12 ] . \begin{bmatrix}\sigma_{11}\\ \sigma_{22}\\ \sigma_{12}\end{bmatrix}=\cfrac{E}{1-\nu^{2}}\begin{bmatrix}1&\nu&0\\ \nu&1&0\\ 0&0&1-\nu\end{bmatrix}\begin{bmatrix}\varepsilon_{11}\\ \varepsilon_{22}\\ \varepsilon_{12}\end{bmatrix}\,.
  43. E E
  44. ν \nu
  45. ε α β \varepsilon_{\alpha\beta}
  46. σ 31 = 2 G ε 31 and σ 32 = 2 G ε 32 \sigma_{31}=2G\varepsilon_{31}\quad\,\text{and}\quad\sigma_{32}=2G\varepsilon_% {32}
  47. G = E / ( 2 ( 1 + ν ) ) G=E/(2(1+\nu))
  48. [ N 11 N 22 N 12 ] = 2 E h 1 - ν 2 [ 1 ν 0 ν 1 0 0 0 1 - ν ] [ u 1 , 1 0 u 2 , 2 0 1 2 ( u 1 , 2 0 + u 2 , 1 0 ) ] , \begin{bmatrix}N_{11}\\ N_{22}\\ N_{12}\end{bmatrix}=\cfrac{2Eh}{1-\nu^{2}}\begin{bmatrix}1&\nu&0\\ \nu&1&0\\ 0&0&1-\nu\end{bmatrix}\begin{bmatrix}u^{0}_{1,1}\\ u^{0}_{2,2}\\ \frac{1}{2}~{}(u^{0}_{1,2}+u^{0}_{2,1})\end{bmatrix}\,,
  49. [ M 11 M 22 M 12 ] = - 2 E h 3 3 ( 1 - ν 2 ) [ 1 ν 0 ν 1 0 0 0 1 - ν ] [ φ 1 , 1 φ 2 , 2 1 2 ( φ 1 , 2 + φ 2 , 1 ) ] , \begin{bmatrix}M_{11}\\ M_{22}\\ M_{12}\end{bmatrix}=-\cfrac{2Eh^{3}}{3(1-\nu^{2})}\begin{bmatrix}1&\nu&0\\ \nu&1&0\\ 0&0&1-\nu\end{bmatrix}\begin{bmatrix}\varphi_{1,1}\\ \varphi_{2,2}\\ \frac{1}{2}(\varphi_{1,2}+\varphi_{2,1})\end{bmatrix}\,,
  50. [ Q 1 Q 2 ] = κ G h [ w , 1 0 - φ 1 w , 2 0 - φ 2 ] . \begin{bmatrix}Q_{1}\\ Q_{2}\end{bmatrix}=\kappa Gh\begin{bmatrix}w^{0}_{,1}-\varphi_{1}\\ w^{0}_{,2}-\varphi_{2}\end{bmatrix}\,.
  51. D = 2 E h 3 3 ( 1 - ν 2 ) . D=\cfrac{2Eh^{3}}{3(1-\nu^{2})}\,.
  52. h h
  53. D = E h 3 12 ( 1 - ν 2 ) . D=\cfrac{Eh^{3}}{12(1-\nu^{2})}\,.
  54. M α β , β - Q α = 0 Q α , α + q = 0 . \begin{aligned}\displaystyle M_{\alpha\beta,\beta}-Q_{\alpha}&\displaystyle=0% \\ \displaystyle Q_{\alpha,\alpha}+q&\displaystyle=0\,.\end{aligned}
  55. 2 ( φ 1 x 1 + φ 2 x 2 ) = - q D 2 w 0 - φ 1 x 1 - φ 2 x 2 = - q κ G h 2 ( φ 1 x 2 - φ 2 x 1 ) = - 2 κ G h D ( 1 - ν ) ( φ 1 x 2 - φ 2 x 1 ) . \begin{aligned}&\displaystyle\nabla^{2}\left(\frac{\partial\varphi_{1}}{% \partial x_{1}}+\frac{\partial\varphi_{2}}{\partial x_{2}}\right)=-\frac{q}{D}% \\ &\displaystyle\nabla^{2}w^{0}-\frac{\partial\varphi_{1}}{\partial x_{1}}-\frac% {\partial\varphi_{2}}{\partial x_{2}}=-\frac{q}{\kappa Gh}\\ &\displaystyle\nabla^{2}\left(\frac{\partial\varphi_{1}}{\partial x_{2}}-\frac% {\partial\varphi_{2}}{\partial x_{1}}\right)=-\frac{2\kappa Gh}{D(1-\nu)}\left% (\frac{\partial\varphi_{1}}{\partial x_{2}}-\frac{\partial\varphi_{2}}{% \partial x_{1}}\right)\,.\end{aligned}
  56. M 11 x 1 + M 12 x 2 = Q 1 , M 21 x 1 + M 22 x 2 = Q 2 Q 1 x 1 + Q 2 x 2 = - q . \begin{aligned}\displaystyle\frac{\partial M_{11}}{\partial x_{1}}+\frac{% \partial M_{12}}{\partial x_{2}}&\displaystyle=Q_{1}\quad\,,\quad\frac{% \partial M_{21}}{\partial x_{1}}+\frac{\partial M_{22}}{\partial x_{2}}=Q_{2}% \\ \displaystyle\frac{\partial Q_{1}}{\partial x_{1}}+\frac{\partial Q_{2}}{% \partial x_{2}}&\displaystyle=-q\,.\end{aligned}
  57. M 11 = - D ( φ 1 x 1 + ν φ 2 x 2 ) , M 22 = - D ( φ 2 x 2 + ν φ 1 x 1 ) , M 12 = - D ( 1 - ν ) 2 ( φ 1 x 2 + φ 2 x 1 ) M_{11}=-D\left(\frac{\partial\varphi_{1}}{\partial x_{1}}+\nu\frac{\partial% \varphi_{2}}{\partial x_{2}}\right)~{},~{}~{}M_{22}=-D\left(\frac{\partial% \varphi_{2}}{\partial x_{2}}+\nu\frac{\partial\varphi_{1}}{\partial x_{1}}% \right)~{},~{}~{}M_{12}=-\frac{D(1-\nu)}{2}\left(\frac{\partial\varphi_{1}}{% \partial x_{2}}+\frac{\partial\varphi_{2}}{\partial x_{1}}\right)
  58. 3 φ 1 x 1 3 + 3 φ 1 x 1 x 2 2 + 3 φ 2 x 1 2 x 2 + 3 φ 2 x 2 3 = - q D . \frac{\partial^{3}\varphi_{1}}{\partial x_{1}^{3}}+\frac{\partial^{3}\varphi_{% 1}}{\partial x_{1}\partial x_{2}^{2}}+\frac{\partial^{3}\varphi_{2}}{\partial x% _{1}^{2}\partial x_{2}}+\frac{\partial^{3}\varphi_{2}}{\partial x_{2}^{3}}=-% \frac{q}{D}\,.
  59. := D ( φ 1 x 1 + φ 2 x 2 ) \mathcal{M}:=D\left(\frac{\partial\varphi_{1}}{\partial x_{1}}+\frac{\partial% \varphi_{2}}{\partial x_{2}}\right)
  60. 2 = - q . \nabla^{2}\mathcal{M}=-q\,.
  61. κ G h ( 2 w 0 - D ) = - q . \kappa Gh\left(\nabla^{2}w^{0}-\frac{\mathcal{M}}{D}\right)=-q\,.
  62. φ 1 \varphi_{1}
  63. φ 2 \varphi_{2}
  64. w 0 w^{0}
  65. 2 ( φ 1 x 2 - φ 2 x 1 ) = - 2 κ G h D ( 1 - ν ) ( φ 1 x 2 - φ 2 x 1 ) . \nabla^{2}\left(\frac{\partial\varphi_{1}}{\partial x_{2}}-\frac{\partial% \varphi_{2}}{\partial x_{1}}\right)=-\frac{2\kappa Gh}{D(1-\nu)}\left(\frac{% \partial\varphi_{1}}{\partial x_{2}}-\frac{\partial\varphi_{2}}{\partial x_{1}% }\right)\,.
  66. 2 ( φ 1 x 1 + φ 2 x 2 ) = - q D 2 w 0 - φ 1 x 1 - φ 2 x 2 = - q κ G h 2 ( φ 1 x 2 - φ 2 x 1 ) = - 2 κ G h D ( 1 - ν ) ( φ 1 x 2 - φ 2 x 1 ) . \begin{aligned}&\displaystyle\nabla^{2}\left(\frac{\partial\varphi_{1}}{% \partial x_{1}}+\frac{\partial\varphi_{2}}{\partial x_{2}}\right)=-\frac{q}{D}% \\ &\displaystyle\nabla^{2}w^{0}-\frac{\partial\varphi_{1}}{\partial x_{1}}-\frac% {\partial\varphi_{2}}{\partial x_{2}}=-\frac{q}{\kappa Gh}\\ &\displaystyle\nabla^{2}\left(\frac{\partial\varphi_{1}}{\partial x_{2}}-\frac% {\partial\varphi_{2}}{\partial x_{1}}\right)=-\frac{2\kappa Gh}{D(1-\nu)}\left% (\frac{\partial\varphi_{1}}{\partial x_{2}}-\frac{\partial\varphi_{2}}{% \partial x_{1}}\right)\,.\end{aligned}
  67. simply supported w 0 = 0 , M 11 = 0 ( or M 22 = 0 ) , φ 1 = 0 ( or φ 2 = 0 ) clamped w 0 = 0 , φ 1 = 0 , φ 2 = 0 . \begin{aligned}\displaystyle\,\text{simply supported}&\displaystyle\quad w^{0}% =0,M_{11}=0~{}(\,\text{or}~{}M_{22}=0),\varphi_{1}=0~{}(\,\text{or}~{}\varphi_% {2}=0)\\ \displaystyle\,\text{clamped}&\displaystyle\quad w^{0}=0,\varphi_{1}=0,\varphi% _{2}=0\,.\end{aligned}
  68. M 11 = D [ 𝒜 ( φ 1 x 1 + ν φ 2 x 2 ) - ( 1 - 𝒜 ) ( 2 w 0 x 1 2 + ν 2 w 0 x 2 2 ) ] + q 1 - ν M 22 = D [ 𝒜 ( φ 2 x 2 + ν φ 1 x 1 ) - ( 1 - 𝒜 ) ( 2 w 0 x 2 2 + ν 2 w 0 x 1 2 ) ] + q 1 - ν M 12 = D ( 1 - ν ) 2 [ 𝒜 ( φ 1 x 2 + φ 2 x 1 ) - 2 ( 1 - 𝒜 ) 2 w 0 x 1 x 2 ] Q 1 = 𝒜 κ G h ( φ 1 + w 0 x 1 ) Q 2 = 𝒜 κ G h ( φ 2 + w 0 x 2 ) . \begin{aligned}\displaystyle M_{11}&\displaystyle=D\left[\mathcal{A}\left(% \frac{\partial\varphi_{1}}{\partial x_{1}}+\nu\frac{\partial\varphi_{2}}{% \partial x_{2}}\right)-(1-\mathcal{A})\left(\frac{\partial^{2}w^{0}}{\partial x% _{1}^{2}}+\nu\frac{\partial^{2}w^{0}}{\partial x_{2}^{2}}\right)\right]+\frac{% q}{1-\nu}\,\mathcal{B}\\ \displaystyle M_{22}&\displaystyle=D\left[\mathcal{A}\left(\frac{\partial% \varphi_{2}}{\partial x_{2}}+\nu\frac{\partial\varphi_{1}}{\partial x_{1}}% \right)-(1-\mathcal{A})\left(\frac{\partial^{2}w^{0}}{\partial x_{2}^{2}}+\nu% \frac{\partial^{2}w^{0}}{\partial x_{1}^{2}}\right)\right]+\frac{q}{1-\nu}\,% \mathcal{B}\\ \displaystyle M_{12}&\displaystyle=\frac{D(1-\nu)}{2}\left[\mathcal{A}\left(% \frac{\partial\varphi_{1}}{\partial x_{2}}+\frac{\partial\varphi_{2}}{\partial x% _{1}}\right)-2(1-\mathcal{A})\,\frac{\partial^{2}w^{0}}{\partial x_{1}\partial x% _{2}}\right]\\ \displaystyle Q_{1}&\displaystyle=\mathcal{A}\kappa Gh\left(\varphi_{1}+\frac{% \partial w^{0}}{\partial x_{1}}\right)\\ \displaystyle Q_{2}&\displaystyle=\mathcal{A}\kappa Gh\left(\varphi_{2}+\frac{% \partial w^{0}}{\partial x_{2}}\right)\,.\end{aligned}
  69. h h
  70. 2 h 2h
  71. D = E h 3 / [ 12 ( 1 - ν 2 ) ] D=Eh^{3}/[12(1-\nu^{2})]
  72. = D [ 𝒜 ( φ 1 x 1 + φ 2 x 2 ) - ( 1 - 𝒜 ) 2 w 0 ] + 2 q 1 - ν 2 \mathcal{M}=D\left[\mathcal{A}\left(\frac{\partial\varphi_{1}}{\partial x_{1}}% +\frac{\partial\varphi_{2}}{\partial x_{2}}\right)-(1-\mathcal{A})\nabla^{2}w^% {0}\right]+\frac{2q}{1-\nu^{2}}\mathcal{B}
  73. Q 1 = x 1 + D ( 1 - ν ) 2 [ 𝒜 x 2 ( φ 1 x 2 - φ 2 x 1 ) ] - 1 + ν q x 1 Q 2 = x 2 - D ( 1 - ν ) 2 [ 𝒜 x 1 ( φ 1 x 2 - φ 2 x 1 ) ] - 1 + ν q x 2 . \begin{aligned}\displaystyle Q_{1}&\displaystyle=\frac{\partial\mathcal{M}}{% \partial x_{1}}+\frac{D(1-\nu)}{2}\left[\mathcal{A}\frac{\partial}{\partial x_% {2}}\left(\frac{\partial\varphi_{1}}{\partial x_{2}}-\frac{\partial\varphi_{2}% }{\partial x_{1}}\right)\right]-\frac{\mathcal{B}}{1+\nu}\frac{\partial q}{% \partial x_{1}}\\ \displaystyle Q_{2}&\displaystyle=\frac{\partial\mathcal{M}}{\partial x_{2}}-% \frac{D(1-\nu)}{2}\left[\mathcal{A}\frac{\partial}{\partial x_{1}}\left(\frac{% \partial\varphi_{1}}{\partial x_{2}}-\frac{\partial\varphi_{2}}{\partial x_{1}% }\right)\right]-\frac{\mathcal{B}}{1+\nu}\frac{\partial q}{\partial x_{2}}\,.% \end{aligned}
  74. 2 ( - 1 + ν q ) = - q κ G h ( 2 w 0 + D ) = - ( 1 - c 2 1 + ν ) q 2 ( φ 1 x 2 - φ 2 x 1 ) = c 2 ( φ 1 x 2 - φ 2 x 1 ) \begin{aligned}&\displaystyle\nabla^{2}\left(\mathcal{M}-\frac{\mathcal{B}}{1+% \nu}\,q\right)=-q\\ &\displaystyle\kappa Gh\left(\nabla^{2}w^{0}+\frac{\mathcal{M}}{D}\right)=-% \left(1-\cfrac{\mathcal{B}c^{2}}{1+\nu}\right)q\\ &\displaystyle\nabla^{2}\left(\frac{\partial\varphi_{1}}{\partial x_{2}}-\frac% {\partial\varphi_{2}}{\partial x_{1}}\right)=c^{2}\left(\frac{\partial\varphi_% {1}}{\partial x_{2}}-\frac{\partial\varphi_{2}}{\partial x_{1}}\right)\end{aligned}
  75. c 2 = 2 κ G h D ( 1 - ν ) . c^{2}=\frac{2\kappa Gh}{D(1-\nu)}\,.
  76. w 0 w^{0}
  77. φ 1 \varphi_{1}
  78. φ 2 \varphi_{2}
  79. x 2 x_{2}
  80. x 1 x_{1}
  81. 𝒜 = 1 \mathcal{A}=1
  82. = 0 \mathcal{B}=0
  83. κ \kappa
  84. 5 / 6 5/6
  85. w 0 w^{0}
  86. φ 1 \varphi_{1}
  87. φ 2 \varphi_{2}
  88. K := - D 2 w K \mathcal{M}^{K}:=-D\nabla^{2}w^{K}
  89. = K + 1 + ν q + D 2 Φ \mathcal{M}=\mathcal{M}^{K}+\frac{\mathcal{B}}{1+\nu}\,q+D\nabla^{2}\Phi
  90. Φ \Phi
  91. 2 2 Φ = 0 \nabla^{2}\nabla^{2}\Phi=0
  92. w K w^{K}
  93. w 0 = w K + K κ G h ( 1 - c 2 2 ) - Φ + Ψ w^{0}=w^{K}+\frac{\mathcal{M}^{K}}{\kappa Gh}\left(1-\frac{\mathcal{B}c^{2}}{2% }\right)-\Phi+\Psi
  94. Ψ \Psi
  95. 2 Ψ = 0 \nabla^{2}\Psi=0
  96. φ 1 = - w K x 1 - 1 κ G h ( 1 - 1 𝒜 - c 2 2 ) Q 1 K + x 1 ( D κ G h 𝒜 2 Φ + Φ - Ψ ) + 1 c 2 Ω x 2 φ 2 = - w K x 2 - 1 κ G h ( 1 - 1 𝒜 - c 2 2 ) Q 2 K + x 2 ( D κ G h 𝒜 2 Φ + Φ - Ψ ) + 1 c 2 Ω x 1 \begin{aligned}\displaystyle\varphi_{1}=-\frac{\partial w^{K}}{\partial x_{1}}% -\frac{1}{\kappa Gh}\left(1-\frac{1}{\mathcal{A}}-\frac{\mathcal{B}c^{2}}{2}% \right)Q_{1}^{K}+\frac{\partial}{\partial x_{1}}\left(\frac{D}{\kappa Gh% \mathcal{A}}\nabla^{2}\Phi+\Phi-\Psi\right)+\frac{1}{c^{2}}\frac{\partial% \Omega}{\partial x_{2}}\\ \displaystyle\varphi_{2}=-\frac{\partial w^{K}}{\partial x_{2}}-\frac{1}{% \kappa Gh}\left(1-\frac{1}{\mathcal{A}}-\frac{\mathcal{B}c^{2}}{2}\right)Q_{2}% ^{K}+\frac{\partial}{\partial x_{2}}\left(\frac{D}{\kappa Gh\mathcal{A}}\nabla% ^{2}\Phi+\Phi-\Psi\right)+\frac{1}{c^{2}}\frac{\partial\Omega}{\partial x_{1}}% \end{aligned}
  97. Q 1 K = - D x 1 ( 2 w K ) , Q 2 K = - D x 2 ( 2 w K ) , Ω := φ 1 x 2 - φ 2 x 1 . Q_{1}^{K}=-D\frac{\partial}{\partial x_{1}}\left(\nabla^{2}w^{K}\right)~{},~{}% ~{}Q_{2}^{K}=-D\frac{\partial}{\partial x_{2}}\left(\nabla^{2}w^{K}\right)~{},% ~{}~{}\Omega:=\frac{\partial\varphi_{1}}{\partial x_{2}}-\frac{\partial\varphi% _{2}}{\partial x_{1}}\,.